,,growth ,,and ,,form ,,cambridge ,,university ,,press ,c4 ,f4 ,,clay1 ,,manager ,london3 ,,fetter ,,lane1 ,e4,c4 ,edinburgh3 #ajj ,,princes ,,street ,illustration ,new ,york3 ,g4 ,p4 ,,putnam',s ,,sons ,bombay1 ,calcutta and ,madras3 ,,macmillan ,,and ,co41 ,,ltd4 ,toronto3 ,j4 ,m4 ,,dent ,,and ,,sons1 ,,ltd4 ,tokyo3 ,,the ,,maruzen-,,kabushiki-,,kaisha ,all rights reserved ,,on ,,growth ,,and ,,form ,,by ,d',,arcy ,,wentworth ,,thompson ,cambridge3 at the ,university ,press #aiag 8,the reasonings about the wonderful and intricate operations of nature are so full of uncertainty1 that1 as the ,wise-man truly observes1 hardly do we guess aright at the things that are upon earth1 and with labour do we find the things that are before us40 ,stephen ,hales1 ,vegetable ,staticks 7#agbg71 p4 #cah1 #agch4 ,,prefatory ,,note ,this book of mine has little need of preface1 for indeed it is 8all preface0 from beginning to end4 ,i have written it as an easy introduction to the study of organic ,form1 by methods which are the common-places of physical science1 which are by no means novel in their application to natural history1 but which nevertheless naturalists are little accustomed to employ4 ,it is not the biologist with an inkling of mathematics1 but the skilled and learned mathematician who must ultimately deal with such problems as are merely sketched and adumbrated here4 ,i pretend to no mathematical skill1 but ,i have made what use ,i could of what tools ,i had2 ,i have dealt with simple cases1 and the mathematical methods which ,i have introduced are of the easiest and simplest kind4 ,elementary as they are1 my book has not been written without the help--the indispensable help--of many friends4 ,like ,mr ,pope translating ,homer1 when ,i felt myself deficient ,i sought assistance6 ,and the experience which ,johnson attributed to ,pope has been mine also1 that men of learning did not refuse to help me4 ,my debts are many1 and ,i will not try to proclaim them all3 but ,i beg to record my particular obligations to ,professor ,claxton ,fidler1 ,sir ,george ,greenhill1 ,sir ,joseph ,larmor1 and ,professor ,a4 ,mc,kenzie2 to a much younger but very helpful friend1 ,mr ,john ,marshall1 ,scholar of ,trinity2 lastly1 and 7if ,i may say so7 most of all1 to my colleague ,professor ,william ,peddie1 whose advice has made many useful additions to my book and whose criticism has spared me many a fault and blunder4 ,i am under obligations also to the authors and publishers of many books from which illustrations have been borrowed1 and especially to the following3 ,to the ,controller of ,h4,m4 ,stationery ,office1 for leave to reproduce a number of figures1 chiefly of ,foraminifera and of ,radiolaria1 from the ,reports of the ,challenger ,expedition4 vi ,to the ,council of the ,royal ,society of ,edinburgh1 and to that of the ,zoological ,society of ,london3--the former for letting me reprint from their ,transactions the greater part of the text and illustrations of my concluding chapter1 the latter for the use of a number of figures for my chapter on ,horns4 ,to ,professor ,e4 ,b4 ,wilson1 for his well-known and all but indispensable figures of the cell 7figs4 #db-ea1 #ec72 to ,m4 ,a4 ,prenant1 for other figures 7#da1 #dh7 in the same chapter2 to ,sir ,donald ,mac,alister and ,mr ,edwin ,arnold for certain figures 7#cce-g71 and to ,sir ,edward ,schfer and ,messrs ,longmans for another 7#ccd71 illustrating the minute trabecular structure of bone4 ,to ,mr ,gerhard ,heilmann1 of ,copenhagen1 for his beautiful diagrams 7figs4 #chh-ic1 #dja1 #djb7 included in my last chapter4 ,to ,professor ,claxton ,fidler and to ,messrs ,griffin1 for letting me use1 with more or less modification or simplification1 a number of illustrations 7figs4 #cci-cdf7 from ,professor ,fidler's ,textbook of ,bridge ,construction4 ,to ,messrs ,blackwood and ,sons1 for several cuts 7figs4 #abg-i1 #aca1 #agc7 from ,professor ,alleyne ,nicholson's ,palaeontology2 to ,mr ,heinemann1 for certain figures 7#eg1 #abb1 #abc1 #bje7 from ,dr ,stphane ,leduc's ,mechanism of ,life2 to ,mr ,a4 ,m4 ,worthington and to ,messrs ,longmans1 for figures 7#ga1 #ge7 from ,a ,study of ,splashes1 and to ,mr ,c4 ,r4 ,darling and to ,messrs ,e4 and ,s4 ,spon for those 7fig4 #he7 from ,mr ,darling's ,liquid ,drops and ,globules4 ,to ,messrs ,macmillan and ,co4 for two figures 7#cjd1 #cje7 from ,zittel's ,palaeontology1 to the ,oxford ,university ,press for a diagram 7fig4 #bh7 from ,mr ,j4 ,w4 ,jenkinson's ,experimental ,embryology2 and to the ,cambridge ,university ,press for a number of figures from ,professor ,henry ,woods's ,invertebrate ,palaeontology1 for one 7fig4 #baj7 from ,dr ,willey's ,zoological ,results1 and for another 7fig4 #cba7 from 8,thomson and ,tait40 ,many more1 and by much the greater part of my diagrams1 ,i owe to the untiring help of ,dr ,doris ,l4 ,mackinnon1 ,d4,sc41 and of ,miss ,helen ,ogilvie1 ,m4,a41 ,b4,sc41 of this ,college4 ,d',,arcy ,,wentworth ,,thompson4 ,,university ,,college1 ,,dundee4 ,december1 #aiaf4 ,,contents ,,chap4 ,,page ,i4 ,,introductory #a ,,ii4 ,,on ,,magnitude #af ,,iii4 ,,the ,,rate ,,of ,,growth #ej ,,iv4 ,,on ,,the ,,internal ,,form ,,and ,,structure ,,of ,,the ,,cell #aef ,v4 ,,the ,,forms ,,of ,,cells #bja ,,vi4 ,a ,,note ,,on ,,adsorption #bgg ,,vii4 ,,the ,,forms ,,of ,,tissues1 ,,or ,,cell-,,aggregates #bic ,,viii4 ,,the ,,same 7continued7 #cdf ,,ix4 ,,on ,,concretions1 ,,spicules1 ,,and ,,spicular ,,skeletons #daa ,x4 ,a ,,parenthetic ,,note ,,on ,,geodetics #dhh ,,xi4 ,,the ,,logarithmic ,,spiral #dic ,,xii4 ,,the ,,spiral ,,shells ,,of ,,the ,,foraminifera #ehg ,,xiii4 ,,the ,,shapes ,,of ,,horns1 ,,and ,,of ,,teeth ,,or ,,tusks3 ,,with ,a ,,note ,,on ,,torsion #fab ,,xiv4 ,,on ,,leaf-,,arrangement1 ,,or ,,phyllotaxis #fce ,,xv4 ,,on ,,the ,,shapes ,,of ,,eggs1 ,,and ,,of ,,certain ,,other ,,hollow ,,structures #feb ,,xvi4 ,,on ,,form ,,and ,,mechanical ,,efficiency #fgj ,,xvii4 ,,on ,,the ,,theory ,,of ,,transformations1 ,,or ,,the ,,comparison ,,of ,,related ,,forms #gai ,,epilogue #ggh ,,index #ghj ,,list ,,of ,,illustrations #a4 ,nerve-cells1 from larger and smaller animals 7,minot1 after ,irving ,hardesty7 4 4 4 #cg #b4 ,relative magnitudes of some minute organisms 7,zsigmondy7 4 4 4 #ci #c4 ,curves of growth in man 7,quetelet and ,bowditch7 4 4 4 #fa #d1 #e4 ,mean annual increments of stature and weight in man 7do47 4 4 4 #ff1 #fi #f4 ,the ratio1 throughout life1 of female weight to male 7do47 4 4 4 #ga #g-i4 ,curves of growth of child1 before and after birth 7,his and ,rssow7 4 4 4 #gd-f #aj4 ,curve of growth of bamboo 7,ostwald1 after ,kraus7 4 4 4 #gg #aa4 ,coefficients of variability in human stature 7,boas and ,wissler7 4 4 4 #hj #ab4 ,growth in weight of mouse 7,wolfgang ,ostwald7 4 4 4 #hc #ac4 ,do4 of silkworm 7,luciani and ,lo ,monaco7 4 4 4 #hd #ad4 ,do4 of tadpole 7,ostwald1 after ,schaper7 4 4 4 #he #ae4 ,larval eels1 or ,leptocephali1 and young elver 7,joh4 ,schmidt7 4 4 4 #hf #af4 ,growth in length of ,spirogyra 7,hofmeister7 4 4 4 #hg #ag4 ,pulsations of growth in ,crocus 7,bose7 4 4 4 #hh #ah4 ,relative growth of brain1 heart and body of man 7,quetelet7 4 4 4 #ij #ai4 ,ratio of stature to span of arms 7do47 4 4 4 #id #bj4 ,rates of growth near the tip of a bean-root 7,sachs7 4 4 4 #if #ba1 #bb4 ,the weight-length ratio of the plaice1 and its annual periodic changes 4 4 4 #ii1 #ajj #bc4 ,variability of tail-forceps in earwigs 7,bateson7 4 4 4 #ajd #bd4 ,variability of body-length in plaice 4 4 4 #aje #be4 ,rate of growth in plants in relation to temperature 7,sachs7 4 4 4 #aji #bf4 ,do4 in maize1 observed 7,kppen71 and calculated curves 4 4 4 #aab #bg4 ,do4 in roots of peas 7,miss ,i4 ,leitch7 4 4 4 #aac #bh1 #bi4 ,rate of growth of frog in relation to temperature 7,jenkinson1 after ,o4 ,hertwig71 and calculated curves of do4 4 4 4 #aae1 #f #cj4 ,seasonal fluctuation of rate of growth in man 7,daffner7 4 4 4 #aai #ca4 ,do4 in the rate of growth of trees 7,c4 ,e4 ,hall7 4 4 4 #abj #cb4 ,long-period fluctuation in the rate of growth of ,arizona trees 7,a4 ,e4 ,douglass7 4 4 4 #abb #cc1 #cd4 ,the varying form of brine-shrimps 7,artemia71 in relation to salinity 7,abonyi7 4 4 4 #abh1 #i #ce-ci4 ,curves of regenerative growth in tadpoles' tails 7,m4 ,l4 ,durbin7 4 4 4 #adj-ade #dj4 ,relation between amount of tail removed1 amount restored1 and time required for restoration 7,m4 ,m4 ,ellis7 4 4 4 #adh #da4 ,caryokinesis in trout's egg 7,prenant1 after ,prof4 ,p4 ,bouin7 4 4 4 #afi #db-ea4 ,diagrams of mitotic cell-division 7,prof4 ,e4 ,b4 ,wilson7 4 4 4 #aga-e #eb4 ,chromosomes in course of splitting and separation 7,hatschek and ,flemming7 4 4 4 #ahj #ec4 ,annular chromosomes of mole-cricket 7,wilson1 after vom ,rath7 4 4 4 #aha #ed-ef4 ,diagrams illustrating a hypothetic field of force in caryokinesis 7,prof4 ,w4 ,peddie7 4 4 4 #ahb-d #eg4 ,an artificial figure of caryokinesis 7,leduc7 4 4 4 #ahf #eh4 ,a segmented egg of ,cerebratulus 7,prenant1 after ,coe7 4 4 4 #ahi #ei4 ,diagram of a field of force with two like poles 4 4 4 #ahi #fj4 ,a budding yeast-cell 4 4 4 #bac #fa4 ,the roulettes of the conic sections 4 4 4 #bah #fb4 ,mode of development of an unduloid from a cylindrical tube 4 4 4 #bbj #fc-fe4 ,cylindrical1 unduloid1 nodoid and catenoid oil-globules 7,plateau7 4 4 4 #bbb1 #c #ff4 ,diagram of the nodoid1 or elastic curve 4 4 4 #bbd #fg4 ,diagram of a cylinder capped by the corresponding portion of a sphere 4 4 4 #bbf #fh4 ,a liquid cylinder breaking up into spheres 4 4 4 #bbg #fi4 ,the same phenomenon in a protoplasmic cell of ,trianea 4 4 4 #bcd #gj4 ,some phases of a splash 7,a4 ,m4 ,worthington7 4 4 4 #bce #ga4 ,a breaking wave 7do47 4 4 4 #bcf #gb4 ,the calycles of some campanularian zoophytes 4 4 4 #bcg #gc4 ,a flagellate monad1 ,distigma proteus 7,saville ,kent7 4 4 4 #bdf #gd4 ,noctiluca miliaris1 diagrammatic 4 4 4 #bdf #ge4 ,various species of ,vorticella 7,saville ,kent and others7 4 4 4 #bdg #gf4 ,various species of ,salpingoeca 7do47 4 4 4 #bdh #gg4 ,species of ,tintinnus1 ,dinobryon and ,codonella 7do47 4 4 4 #bdh #gh4 ,the tube or cup of ,vaginicola 4 4 4 #bdh #gi4 ,the same of ,folliculina 4 4 4 #bdi #hj4 ,trachelophyllum 7,wreszniowski7 4 4 4 #bdi #ha4 ,trichodina pediculus 4 4 4 #beb #hb4 ,dinenymplia gracilis 7,leidy7 4 4 4 #bec #hc4 ,a 8collar-cell0 of ,codosiga 4 4 4 #bed #hd4 ,various species of ,lagena 7,brady7 4 4 4 #bef #he4 ,hanging drops1 to illustrate the unduloid form 7,c4 ,r4 ,darling7 4 4 4 #beg #hf4 ,diagram of a fluted cylinder 4 4 4 #bfj #hg4 ,nodosaria scalaris 7,brady7 4 4 4 #bfb #hh4 ,fluted and pleated gonangia of certain ,campanularians 7,allman7 4 4 4 #bfb #hi4 ,various species of ,nodosaria1 ,sagrina and ,rheophax 7,brady7 4 4 4 #bfc #ij4 ,trypanosoma tineae and ,spirochaeta anodontae1 to shew undulating membranes 7,minchin and ,fantham7 4 4 4 #bff #ia4 ,some species of ,trichomastix and ,trichomonas 7,kofoid7 4 4 4 #bfg #ib4 ,herpetomonas assuming the undulatory membrane of a ,trypanosome 7,d4 ,l4 ,mackinnon7 4 4 4 #bfh #ic4 ,diagram of a human blood-corpuscle 4 4 4 #bga #id4 ,sperm-cells of decapod crustacea1 ,inachus and ,galathea 7,koltzoff7 4 4 4 #bgc #ie4 ,the same1 in saline solutions of varying density 7do47 4 4 4 #bgd #if4 ,a sperm-cell of ,dromia 7do47 4 4 4 #bge #ig4 ,chondriosomes in cells of kidney and pancreas 7,barratt and ,mathews7 4 4 4 #bhe #ih4 ,adsorptive concentration of potassium salts in various plant-cells 7,macallum7 4 4 4 #bij #ii-aja4 ,equilibrium of surface-tension in a floating drop 4 4 4 #bid1 #e #ajb4 ,plateau's 8bourrelet0 in plant-cells2 diagrammatic 7,berthold7 4 4 4 #bih #ajc4 ,parenchyma of maize1 shewing the same phenomenon 4 4 4 #bih #ajd1 #e4 ,diagrams of the partition-wall between two soap-bubbles 4 4 4 #bii1 #cjj #ajf4 ,diagram of a partition in a conical cell 4 4 4 #cjj #ajg4 ,chains of cells in ,nostoc1 ,anabaena and other low algae 4 4 4 #cjj #ajh4 ,diagram of a symmetrically divided soap-bubble 4 4 4 #cja #aji4 ,arrangement of partitions in dividing spores of ,pellia 7,campbell7 4 4 4 #cjb #aaj4 ,cells of ,dictyota 7,reinke7 4 4 4 #cjc #aaa1 #b4 ,terminal and other cells of ,chara1 and young antheridium of do4 4 4 4 #cjc #aac4 ,diagram of cell-walls and partitions under various conditions of tension 4 4 4 #cjd #aad1 #e4 ,the partition-surfaces of three interconnected bubbles 4 4 4 #cjg1 #h #aaf4 ,diagram of four interconnected cells or bubbles 4 4 4 #cji #aag4 ,various configurations of four cells in a frog's egg 7,rauber7 4 4 4 #caa #aah4 ,another diagram of two conjoined soap-bubbles 4 4 4 #cac #aai4 ,a froth of bubbles1 shewing its outer or 8epidermal0 layer 4 4 4 #cad #abj4 ,a tetrahedron1 or tetrahedral system1 shewing its centre of symmetry 4 4 4 #cag #aba4 ,a group of hexagonal cells 7,bonanni7 4 4 4 #cai #abb1 #c4 ,artificial cellular tissues 7,leduc7 4 4 4 #cbj #abd4 ,epidermis of ,girardia 7,goebel7 4 4 4 #cba #abe4 ,soap-froth1 and the same under compression 7,rhumbler7 4 4 4 #cbb #abf4 ,epidermal cells of ,elodea canadensis 7,berthold7 4 4 4 #cbb #abg4 ,lithostrotion ,martini 7,nicholson7 4 4 4 #cbe #abh4 ,cyathophyllum hexagonum 7,nicholson1 after ,zittel7 4 4 4 #cbe #abi4 ,arachnophyllum pentagonum 7,nicholson7 4 4 4 #cbf #acj4 ,heliolites 7,woods7 4 4 4 #cbf #aca4 ,confluent septa in ,thamnastraea and ,comoseris 7,nicholson1 after ,zittel7 4 4 4 #cbg #acb4 ,geometrical construction of a bee's cell 4 4 4 #ccj #acc4 ,stellate cells in the pith of a rush2 diagrammatic 4 4 4 #cce #acd4 ,diagram of soap-films formed in a cubical wire skeleton 7,plateau7 4 4 4 #ccg #ace4 ,polar furrows in systems of four soap-bubbles 7,robert7 4 4 4 #cda #acf-h4 ,diagrams illustrating the division of a cube by partitions of minimal area 4 4 4 #cdg-ej #aci4 ,cells from hairs of ,sphacelaria 7,berthold7 4 4 4 #cea #adj4 ,the bisection of an isosceles triangle by minimal partitions 4 4 4 #cec #ada4 ,the similar partitioning of spheroidal and conical cells 4 4 4 #cec #adb4 ,s-shaped partitions from cells of algae and mosses 7,reinke and others7 4 4 4 #cee #adc4 ,diagrammatic explanation of the ,s-shaped partitions 4 4 4 #cef #add4 ,development of ,erythrotrichia 7,berthold7 4 4 4 #cei #ade4 ,periclinal1 anticlinal and radial partitioning of a quadrant 4 4 4 #cei #adf4 ,construction for the minimal partitioning of a quadrant 4 4 4 #cfa #adg4 ,another diagram of anticlinal and periclinal partitions 4 4 4 #cfb #adh4 ,mode of segmentation of an artificially flattened frog's egg 7,roux7 4 4 4 #cfc #adi4 ,the bisection1 by minimal partitions1 of a prism of small angle 4 4 4 #cfd #aej4 ,comparative diagram of the various modes of bisection of a prismatic sector 4 4 4 #cfe #aea4 ,diagram of the further growth of the two halves of a quadrantal cell 4 4 4 #cfg #aeb4 ,diagram of the origin of an epidermic layer of cells 4 4 4 #cgj #aec4 ,a discoidal cell dividing into octants 4 4 4 #cga #aed4 ,a germinating spore of ,riccia 7after ,campbell71 to shew the manner of space-partitioning in the cellular tissue 4 4 4 #cgb #aee1 #f4 ,theoretical arrangement of successive partitions in a discoidal cell 4 4 4 #cgc #aeg4 ,sections of a moss-embryo 7,kienitz-,gerloff7 4 4 4 #cgd #aeh4 ,various possible arrangements of partitions in groups of four to eight cells 4 4 4 #cge #aei4 ,three modes of partitioning in a system of six cells 4 4 4 #cgf #afj1 #a4 ,segmenting eggs of ,trochus 7,robert71 and of ,cynthia 7,conklin7 4 4 4 #cgg #afb4 ,section of the apical cone of ,salvinia 7,pringsheim7 4 4 4 #cgg #afc1 #d4 ,segmenting eggs of ,pyrosoma 7,korotneff71 and of ,echinus 7,driesch7 4 4 4 #cgg #afe4 ,segmenting egg of a cephalopod 7,watase7 4 4 4 #cgh #aff1 #g4 ,eggs segmenting under pressure3 of ,echinus and ,nereis 7,driesch71 and of a frog 7,roux7 4 4 4 #cgh #afh4 ,various arrangements of a group of eight cells on the surface of a frog's egg 7,rauber7 4 4 4 #cha #afi4 ,diagram of the partitions and interfacial contacts in a system of eight cells 4 4 4 #chc #agj4 ,various modes of aggregation of eight oil-drops 7,roux7 4 4 4 #chd #aga4 ,forms1 or species1 of ,asterolampra 7,greville7 4 4 4 #chf #agb4 ,diagrammatic section of an alcyonarian polype 4 4 4 #chg #agc1 #d4 ,sections of ,heterophyllia 7,nicholson and ,martin ,duncan7 4 4 4 #chh1 #i #age4 ,diagrammatic section of a ctenophore 7,eucharis7 4 4 4 #cia #agf1 #g4 ,diagrams of the construction of a ,pluteus larva 4 4 4 #cib1 #c #agh1 #i4 ,diagrams of the development of stomata1 in ,sedum and in the hyacinth 4 4 4 #cid #ahj4 ,various spores and pollen-grains 7,berthold and others7 4 4 4 #cif #aha4 ,spore of ,anthoceros 7,campbell7 4 4 4 #cig #ahb1 #d1 #i4 ,diagrammatic modes of division of a cell under certain conditions of asymmetry 4 4 4 #djj-e #ahc4 ,development of the embryo of ,sphagnum 7,campbell7 4 4 4 #djb #ahe4 ,the gemma of a moss 7do47 4 4 4 #djc #ahf4 ,the antheridium of ,riccia 7do47 4 4 4 #djd #ahg4 ,section of growing shoot of ,selaginella1 diagrammatic 4 4 4 #djd #ahh4 ,an embryo of ,jungermannia 7,kienitz-,gerloff7 4 4 4 #djd #aij4 ,development of the sporangium of ,osmunda 7,bower7 4 4 4 #djf #aia4 ,embryos of ,phascum and of ,adiantum 7,kienitz-,gerloff7 4 4 4 #djh #aib4 ,a section of ,girardia 7,goebel7 4 4 4 #djh #aic4 ,an antheridium of ,pteris 7,strasburger7 4 4 4 #dji #aid4 ,spicules of ,siphonogorgia and ,anthogorgia 7,studer7 4 4 4 #dac #aie-g4 ,calcospherites1 deposited in white of egg 7,harting7 4 4 4 #dba1 #b #aih4 ,sections of the shell of ,mya 7,carpenter7 4 4 4 #dbb #aii4 ,concretions1 or spicules1 artificially deposited in cartilage 7,harting7 4 4 4 #dbc #bjj4 ,further illustrations of alcyonarian spicules3 ,eunicea 7,studer7 4 4 4 #dbd #bja-c4 ,associated1 aggregated and composite calcospherites 7,harting7 4 4 4 #dbe1 #f #bjd4 ,harting's 8conostats0 4 4 4 #dbg #bje4 ,liesegang's rings 7,leduc7 4 4 4 #dbh #bjf4 ,relay-crystals of common salt 7,bowman7 4 4 4 #dbi #bjg4 ,wheel-like crystals in a colloid medium 7do47 4 4 4 #dbi #bjh4 ,a concentrically striated calcospherite or spherocrystal 7,harting7 4 4 4 #dcb #bji4 ,otoliths of plaice1 shewing 8age-rings0 7,wallace7 4 4 4 #dcb #baj4 ,spicules1 or calcospherites1 of ,astrosclera 7,lister7 4 4 4 #dcf #baa4 #b4 ,c- and ,s-shaped spicules of sponges and holothurians 7,sollas and ,thel7 4 4 4 #ddb #bac4 ,an amphidisc of ,hyalonema 4 4 4 #ddb #bad-g4 ,spicules of calcareous1 tetractinellid and hexactinellid sponges1 and of various holothurians 7,haeckel1 ,schultze1 ,sollas and ,thel7 4 4 4 #dde-deb #bah4 ,diagram of a solid body confined by surface-energy to a liquid boundary-film 4 4 4 #dfj #bai4 ,astrorhiza limicola and arenaria 7,brady7 4 4 4 #dfd #bbj4 ,a nuclear 8reticulum plasmatique0 7,carnoy7 4 4 4 #dfh #bba4 ,a spherical radiolarian1 ,aulonia hexagona 7,haeckel7 4 4 4 #dfi #bbb4 ,actinomma arcadophorum 7do47 4 4 4 #dfi #bbc4 ,ethmosphaera conosiphonia 7do47 4 4 4 #dgj #bbd4 ,portions of shells of ,cenosphaera favosa and vesparia 7do47 4 4 4 #dgj #bbe4 ,aulastrum triceros 7do47 4 4 4 #dga #bbf4 ,part of the skeleton of ,cannorhaphis 7do47 4 4 4 #dgb #bbg4 ,a ,nassellarian skeleton1 ,callimitra carolotae 7do47 4 4 4 #dgb #bbh1 #i4 ,portions of ,dictyocha stapedia 7do47 4 4 4 #dgd #bcj4 ,diagram to illustrate the conformation of ,callimitra 4 4 4 #dgf #bca4 ,skeletons of various radiolarians 7,haeckel7 4 4 4 #dgi #bcb4 ,diagrammatic structure of the skeleton of ,dorataspis 7do47 4 4 4 #dha #bcc1 #d4 ,phatnaspis cristata 7,haeckel71 and a diagram of the same 4 4 4 #dhc #bce4 ,phractaspis prototypus 7,haeckel7 4 4 4 #dhd #bcf4 ,annular and spiral thickenings in the walls of plant-cells 4 4 4 #dhh #bcg4 ,a radiograph of the shell of ,nautilus 7,green and ,gardiner7 4 4 4 #did #bch4 ,a spiral foraminifer1 ,globigerina 7,brady7 4 4 4 #die #bci-db4 ,diagrams to illustrate the development or growth of a logarithmic spiral 4 4 4 #djg-eja #bdc4 ,a helicoid and a scorpioid cyme 4 4 4 #ejb #bdd4 ,an ,archimedean spiral 4 4 4 #ejc #bde-g4 ,more diagrams of the development of a logarithmic spiral 4 4 4 #eje1 #f #bdh-eg4 ,various diagrams illustrating the mathematical theory of gnomons 4 4 4 #ejh-ac #beh4 ,a shell of ,haliotis1 to shew how each increment of the shell constitutes a gnomon to the preexisting structure 4 4 4 #ead #bei1 #fj4 ,spiral foraminifera1 ,pulvinulina and ,cristellaria1 to illustrate the same principle 4 4 4 #ead1 #e #bfa4 ,another diagram of a logarithmic spiral 4 4 4 #eag #bfb4 ,a diagram of the logarithmic spiral of ,nautilus 7,moseley7 4 4 4 #eai #bfc1 #d4 ,opercula of ,turbo and of ,nerita 7,moseley7 4 4 4 #eba1 #b #bfe4 ,a section of the shell of ,melo ethiopicus 4 4 4 #ebe #bff4 ,shells of ,harpa and ,dolium1 to illustrate generating curves and gene 4 4 4 #ebf #bfg4 ,d',orbigny's ,helicometer 4 4 4 #ebi #bfh4 ,section of a nautiloid shell1 to shew the 8protoconch0 4 4 4 #eca #bfi-gc4 ,diagrams of logarithmic spirals1 of various angles 4 4 4 #ecb-e #bgd1 #f1 #g4 ,constructions for determining the angle of a logarithmic spiral 4 4 4 #ecg1 #h #bge4 ,an ammonite1 to shew its corrugated surface pattern 4 4 4 #ecg #bgh-hj4 ,illustrations of the 8angle of retardation0 4 4 4 #edb-d #bha4 ,a shell of ,macroscaphites1 to shew change of curvature 4 4 4 #eej #bhb4 ,construction for determining the length of the coiled spire 4 4 4 #eea #bhc4 ,section of the shell of ,triton corrugatus 7,woodward7 4 4 4 #eed #bhd4 ,lamellaria perspicua and ,sigaretus haliotoides 7do47 4 4 4 #eee #bhe1 #f4 ,sections of the shells of ,terebra maculata and ,trochus niloticus 4 4 4 #eei1 #fj #bhg-i4 ,diagrams illustrating the lines of growth on a lamellibranch shell 4 4 4 #efc-e #bij4 ,caprinella adversa 7,woodward7 4 4 4 #efg #bia4 ,section of the shell of ,productus 7,woods7 4 4 4 #efg #bib4 ,the 8skeletal loop0 of ,terebratula 7do47 4 4 4 #efh #bic1 #d4 ,the spiral arms of ,spirifer and of ,atrypa 7do47 4 4 4 #efi #bie-g4 ,shells of ,cleodora1 ,hyalaea and other pteropods 7,boas7 4 4 4 #egj1 #a #bih1 #i4 ,coordinate diagrams of the shell-outline in certain pteropods 4 4 4 #egb1 #c #cjj4 ,development of the shell of ,hyalaea tridentata 7,tesch7 4 4 4 #egc #cja4 ,pteropod shells1 of ,cleodora and ,hyalaea1 viewed from the side 7,boas7 4 4 4 #ege #cjb1 #c4 ,diagrams of septa in a conical shell 4 4 4 #egi #cjd4 ,a section of ,nautilus1 shewing the logarithmic spirals of the septa to which the shell-spiral is the evolute 4 4 4 #eha #cje4 ,cast of the interior of the shell of ,nautilus1 to shew the contours of the septa at their junction with the shell-wall 4 4 4 #ehb #cjf4 ,ammonites ,sowerbyi1 to shew septal outlines 7,zittel1 after ,steinmann and ,dderlein7 4 4 4 #ehd #cjg4 ,suture-line of ,pinacoceras 7,zittel1 after ,hauer7 4 4 4 #ehd #cjh4 ,shells of ,hastigerina1 to shew the 8mouth0 7,brady7 4 4 4 #ehh #cji4 ,nummulina antiquior 7,v4 von ,mller7 4 4 4 #eia #caj4 ,cornuspira foliacea and ,operculina complanata 7,brady7 4 4 4 #eid #caa4 ,miliolina pulchella and linnaeana 7,brady7 4 4 4 #eif #cab1 #c4 ,cyclammina cancellata 7do471 and diagrammatic figure of the same 4 4 4 #eif1 #g #cad4 ,orbulina universa 7,brady7 4 4 4 #eih #cae4 ,cristellaria reniformis 7do47 4 4 4 #fjj #caf4 ,discorbina bertheloti 7do47 4 4 4 #fjc #cag4 ,textularia trochus and concava 7do47 4 4 4 #fjd #cah4 ,diagrammatic figure of a ram's horns 7,sir ,v4 ,brooke7 4 4 4 #fae #cai4 ,head of an ,arabian wild goat 7,sclater7 4 4 4 #faf #cbj4 ,head of ,ovis ,ammon1 shewing ,st ,venant's curves 4 4 4 #fba #cba4 ,st ,venant's diagram of a triangular prism under torsion 7,thomson and ,tait7 4 4 4 #fbc #cbb4 ,diagram of the same phenomenon in a ram's horn 4 4 4 #fbc #cbc4 ,antlers of a ,swedish elk 7,lnnberg7 4 4 4 #fbi #cbd4 ,head and antlers of ,cervus duvauceli 7,lydekker7 4 4 4 #fcj #cbe1 #f4 ,diagrams of spiral phyllotaxis 7,p4 ,g4 ,tait7 4 4 4 #fdd1 #e #cbg4 ,further diagrams of phyllotaxis1 to shew how various spiral appearances may arise out of one and the same angular leaf-divergence 4 4 4 #fdh #cbh4 ,diagrammatic outlines of various sea-urchins 4 4 4 #ffd #cbi1 #cj4 ,diagrams of the angle of branching in blood-vessels 7,hess7 4 4 4 #ffg1 #h #cca1 #b4 ,diagrams illustrating the flexure of a beam 4 4 4 #fgd1 #h #ccc4 ,an example of the mode of arrangement of bast-fibres in a plant-stem 7,schwendener7 4 4 4 #fhj #ccd4 ,section of the head of a femur1 to shew its trabecular structure 7,schfer1 after ,robinson7 4 4 4 #fha #cce4 ,comparative diagrams of a crane-head and the head of a femur 7,culmann and ,h4 ,meyer7 4 4 4 #fhb #ccf4 ,diagram of stress-lines in the human foot 7,sir ,d4 ,mac,alister1 after ,h4 ,meyer7 4 4 4 #fhd #ccg4 ,trabecular structure of the os calcis 7do47 4 4 4 #fhe #cch4 ,diagram of shearing-stress in a loaded pillar 4 4 4 #fhf #cci4 ,diagrams of tied arch1 and bowstring girder 7,fidler7 4 4 4 #fic #cdj1 #a4 ,diagrams of a bridge3 shewing proposed span1 the corresponding stress-diagram and reciprocal plan of construction 7do47 4 4 4 #fif #cdb4 ,a loaded bracket and its reciprocal construction-diagram 7,culmann7 4 4 4 #fig #cdc1 #d4 ,a cantilever bridge1 with its reciprocal diagrams 7,fidler7 4 4 4 #fih #cde4 ,a two-armed cantilever of the ,forth ,bridge 7do47 4 4 4 #gjj #cdf4 ,a two-armed cantilever with load distributed over two pier-heads1 as in the quadrupedal skeleton 4 4 4 #gjj #cdg-i4 ,stress-diagrams4 or diagrams of bending moments1 in the backbones of the horse1 of a ,dinosaur1 and of ,titanotherium 4 4 4 #gja-d #cej4 ,the skeleton of ,stegosaurus 4 4 4 #gjg #cea4 ,bending-moments in a beam with fixed ends1 to illustrate the mechanics of chevron-bones 4 4 4 #gji #ceb1 #c4 ,coordinate diagrams of a circle1 and its deformation into an ellipse 4 4 4 #gbi #ced4 ,comparison1 by means of ,cartesian coordinates1 of the cannon-bones of various ruminant animals 4 4 4 #gbi #cee1 #f4 ,logarithmic coordinates1 and the circle of ,fig4 #ceb inscribed therein 4 4 4 #gbi1 #ca #ceg1 #h4 ,diagrams of oblique and radial coordinates 4 4 4 #gca #cei4 ,lanceolate1 ovate and cordate leaves1 compared by the help of radial coordinates 4 4 4 #gcb #cfj4 ,a leaf of ,begonia daedalea 4 4 4 #gcc #cfa4 ,a network of logarithmic spiral coordinates 4 4 4 #gce #cfb1 #c4 ,feet of ox1 sheep and giraffe1 compared by means of ,cartesian coordinates 4 4 4 #gch1 #dj #cfd1 #f4 8,proportional diagrams0 of human physiognomy 7,albert ,drer7 4 4 4 #gdj1 #b #cfe4 ,median and lateral toes of a tapir1 compared by means of rectangular and oblique coordinates 4 4 4 #gda #cfg1 #h4 ,a comparison of the copepods ,oithona and ,sapphirina 4 4 4 #gdb #cfi4 ,the carapaces of certain crabs1 ,geryon1 ,corystes and others1 compared by means of rectilinear and curvilinear coordinates 4 4 4 #gdd #cgj4 ,a comparison of certain amphipods1 ,harpinia1 ,stegocephalus and ,hyperia 4 4 4 #gdf #cga4 ,the calycles of certain campanularian zoophytes1 inscribed in corresponding ,cartesian networks 4 4 4 #gdg #cgb4 ,the calycles of certain species of ,aglaophenia1 similarly compared by means of curvilinear coordinates 4 4 4 #gdh #cgc1 #d4 ,the fishes ,argyropelecus and ,sternoptyx1 compared by means of rectangular and oblique coordinate systems 4 4 4 #gdh #cge1 #f4 ,scarus and ,pomacanthus1 similarly compared by means of rectangular and coaxial systems 4 4 4 #gdi #cgg-hj4 ,a comparison of the fishes ,polyprion1 ,pseudopriacanthus1 ,scorpaena and ,antigonia 4 4 4 #gej #cha1 #b4 ,a similar comparison of ,diodon and ,orthagoriscus 4 4 4 #gea #chc4 ,the same of various crocodiles3 ,c4 porosus1 ,c4 americanus and ,notosuchus terrestris 4 4 4 #gec #chd4 ,the pelvic girdles of ,stegosaurus and ,camptosaurus 4 4 4 #ged #che1 #f4 ,the shoulder-girdles of ,cryptocleidus and of ,ichthyosaurus 4 4 4 #gee #chg4 ,the skulls of ,dimorphodon and of ,pteranodon 4 4 4 #gef #chh-ib4 ,the pelves of ,archaeopteryx and of ,apatornis compared1 and a method illustrated whereby intermediate configurations may be found by interpolation 7,g4 ,heilmann7 4 4 4 #geg-i #cic4 ,the same pelves1 together with three of the intermediate or interpolated forms 4 4 4 #gfj #cid1 #e4 ,comparison of the skulls of two extinct rhinoceroses1 ,hyrachyus and ,aceratherium 7,osborn7 4 4 4 #gfa #cif4 ,occipital views of various extinct rhinoceroses 7do47 4 4 4 #gfb #cig-djj4 ,comparison with each other1 and with the skull of ,hyrachyus1 of the skulls of ,titanotherium1 tapir1 horse and rabbit 4 4 4 #gfc1 #d #dja1 #b4 ,coordinate diagrams of the skulls of ,eohippus and of ,equus1 with various actual and hypothetical intermediate types 7,heilmann7 4 4 4 #gfe-g #djc4 ,a comparison of various human scapulae 7,dwight7 4 4 4 #gfi #djd4 ,a human skull1 inscribed in ,cartesian coordinates 4 4 4 #ggj #dje4 ,the same coordinates on a new projection1 adapted to the skull of the chimpanzee 4 4 4 #ggj #djf4 ,chimpanzee's skull1 inscribed in the network of ,fig4 #dje 4 4 4 #gga #djg1 #h4 ,corresponding diagrams of a baboon's skull1 and of a dog's 4 4 4 #gga1 #c 8,cum formarum naturalium et corporalium esse non consistat nisi in unione ad materiam1 ejusdem agentis esse videtur eas producere cujus est materiam transmutare4 ,secundo1 quia cum hujusmodi formae non excedant virtutem et ordinem et facultatem principiorum agentium in natura1 nulla videtur necessitas eorum originem in principia reducere altiora40 ,aquinas1 ,de ,pot4 ,q4 iii1 a1 #aa4 7,quoted in ,brit4 ,assoc4 ,address1 ,section ,d1 #aiaa47 8444,i would that all other natural phenomena might similarly be deduced from mechanical principles4 ,for many things move me to suspect that everything depends upon certain forces1 in virtue of which the particles of bodies1 through forces not yet understood1 are either impelled together so as to cohere in regular figures1 or are repelled and recede from one another40 ,newton1 in ,preface to the ,principia4 7,quoted by ,mr ,w4 ,spottiswoode1 ,brit4 ,assoc4 ,presidential ,address1 #ahgh47 8,when ,science shall have subjected all natural phenomena to the laws of ,theoretical ,mechanics1 when she shall be able to predict the result of every combination as unerringly as ,hamilton predicted conical refraction1 or ,adams revealed to us the existence of ,neptune1--that we cannot say4 ,that day may never come1 and it is certainly far in the dim future4 ,we may not anticipate it1 we may not even call it possible4 ,but none the less are we bound to look to that day1 and to labour for it as the crowning triumph of ,science3--when ,theoretical ,mechanics shall be recognised as the key to every physical enigma1 the chart for every traveller through the dark ,infinite of ,nature40 ,j4 ,h4 ,jellett1 in ,brit4 ,assoc4 ,address1 ,section ,a1 #ahgd4 #a ,,chapter ,i ,,introductory ,of the chemistry of his day and generation1 ,kant declared that it was 8a science1 but not science10--8eine ,wissenschaft1 aber nicht ,wissenschaft02 for that the criterion of physical science lay in its relation to mathematics4 ,and a hundred years later ,du ,bois ,reymond1 profound student of the many sciences on which physiology is based1 recalled and reiterated the old saying1 declaring that chemistry would only reach the rank of science1 in the high and strict sense1 when it should be found possible to explain chemical reactions in the light of their causal relation to the velocities1 tensions and conditions of equilibrium of the component molecules2 that1 in short1 the chemistry of the future must deal with molecular mechanics1 by the methods and in the strict language of mathematics1 as the astronomy of ,newton and ,laplace dealt with the stars in their courses4 ,we know how great a step has been made towards this distant and once hopeless goal1 as ,kant defined it1 since van't ,hoff laid the firm foundations of a mathematical chemistry1 and earned his proud epitaph1 ,physicam chemiae adiunxit#a4 ,we need not wait for the full realisation of ,kant's desire1 in order to apply to the natural sciences the principle which he urged4 ,though chemistry fall short of its ultimate goal in mathematical mechanics1 nevertheless physiology is vastly strengthened and enlarged by making use of the chemistry1 as of the physics1 of the age4 ,little by little it draws nearer to our conception of a true science1 with each branch of physical science which it #b brings into relation with itself3 with every physical law and every mathematical theorem which it learns to take into its employ4 ,between the physiology of ,haller1 fine as it was1 and that of ,helmholtz1 ,ludwig1 ,claude ,bernard1 there was all the difference in the world4 ,as soon as we adventure on the paths of the physicist1 we learn to weigh and to measure1 to deal with time and space and mass and their related concepts1 and to find more and more our knowledge expressed and our needs satisfied through the concept of number1 as in the dreams and visions of ,plato and ,pythagoras2 for modern chemistry would have gladdened the hearts of those great philosophic dreamers4 ,but the zoologist or morphologist has been slow1 where the physiologist has long been eager1 to invoke the aid of the physical or mathematical sciences2 and the reasons for this difference lie deep1 and in part are rooted in old traditions4 ,the zoologist has scarce begun to dream of defining1 in mathematical language1 even the simpler organic forms4 ,when he finds a simple geometrical construction1 for instance in the honey-comb1 he would fain refer it to psychical instinct or design rather than to the operation of physical forces2 when he sees in snail1 or nautilus1 or tiny foraminiferal or radiolarian shell1 a close approach to the perfect sphere or spiral1 he is prone1 of old habit1 to believe that it is after all something more than a spiral or a sphere1 and that in this 8something more0 there lies what neither physics nor mathematics can explain4 ,in short he is deeply reluctant to compare the living with the dead1 or to explain by geometry or by dynamics the things which have their part in the mystery of life4 ,moreover he is little inclined to feel the need of such explanations or of such extension of his field of thought4 ,he is not without some justification if he feels that in admiration of nature's handiwork he has an horizon open before his eyes as wide as any man requires4 ,he has the help of many fascinating theories within the bounds of his own science1 which1 though a little lacking in precision1 serve the purpose of ordering his thoughts and of suggesting new objects of enquiry4 ,his art of classification becomes a ceaseless and an endless search after the blood-relationships of things living1 and the pedigrees of things #c dead and gone4 ,the facts of embryology become for him1 as ,wolff1 von ,baer and ,fritz ,mller proclaimed1 a record not only of the life-history of the individual but of the annals of its race4 ,the facts of geographical distribution or even of the migration of birds lead on and on to speculations regarding lost continents1 sunken islands1 or bridges across ancient seas4 ,every nesting bird1 every ant-hill or spider's web displays its psychological problems of instinct or intelligence4 ,above all1 in things both great and small1 the naturalist is rightfully impressed1 and finally engrossed1 by the peculiar beauty which is manifested in apparent fitness or 8adaptation10--the flower for the bee1 the berry for the bird4 ,time out of mind1 it has been by way of the 8final cause10 by the teleological concept of 8end10 of 8purpose10 or of 8design10 in one or another of its many forms 7for its moods are many71 that men have been chiefly wont to explain the phenomena of the living world2 and it will be so while men have eyes to see and ears to hear withal4 ,with ,galen1 as with ,aristotle1 it was the physician's way2 with ,john ,ray1 as with ,aristotle1 it was the naturalist's way2 with ,kant1 as with ,aristotle1 it was the philosopher's way4 ,it was the old ,hebrew way1 and has its splendid setting in the story that ,god made 8every plant of the field before it was in the earth1 and every herb of the field before it grew40 ,it is a common way1 and a great way2 for it brings with it a glimpse of a great vision1 and it lies deep as the love of nature in the hearts of men4 ,half overshadowing the 8efficient0 or physical cause1 the argument of the final cause appears in eighteenth century physics1 in the hands of such men as ,euler#b and ,maupertuis1 to whom ,leibniz#c had passed it on4 ,half overshadowed by the mechanical concept1 it runs through ,claude ,bernard's ,leons sur les #d phnomnes de la ,vie#d1 and abides in much of modern physiology#e4 ,inherited from ,hegel1 it dominated ,oken's ,naturphilosophie and lingered among his later disciples1 who were wont to liken the course of organic evolution not to the straggling branches of a tree1 but to the building of a temple1 divinely planned1 and the crowning of it with its polished minarets#f4 ,it is retained1 somewhat crudely1 in modern embryology1 by those who see in the early processes of growth a significance 8rather prospective than retrospective10 such that the embryonic phenomena must be 8referred directly to their usefulness in building the body of the future animal#g03--which is no more1 and no less1 than to say1 with ,aristotle1 that the organism is the 1 or final cause1 of its own processes of generation and development4 ,it is writ large in that ,entelechy#h which ,driesch rediscovered1 and which he made known to many who had neither learned of it from ,aristotle1 nor studied it with ,leibniz1 nor laughed at it with ,voltaire4 ,and1 though it is in a very curious way1 we are told that teleology was 8refounded1 reformed or rehabilitated#i0 by ,darwin's theory of natural selection1 whereby 8every variety of form and colour was urgently and absolutely called upon to produce its title to existence either as an active useful agent1 or as a survival0 of such active usefulness in the past4 ,but in this last1 and very important case1 we have reached a 8teleology0 without a 1 #e as men like ,butler and ,janet have been prompt to shew3 a teleology in which the final cause becomes little more1 if anything1 than the mere expression or resultant of a process of sifting out of the good from the bad1 or of the better from the worse1 in short of a process of mechanism#aj4 ,the apparent manifestations of 8purpose0 or adaptation become part of a mechanical philosophy1 according to which 8chaque chose finit toujours par s'accommoder son milieu#aa40 ,in short1 by a road which resembles but is not the same as ,maupertuis's road1 we find our way to the very world in which we are living1 and find that if it be not1 it is ever tending to become1 8the best of all possible worlds#ab40 ,but the use of the teleological principle is but one way1 not the whole or the only way1 by which we may seek to learn how things came to be1 and to take their places in the harmonious complexity of the world4 ,to seek not for ends but for 8antecedents0 is the way of the physicist1 who finds 8causes0 in what he has learned to recognise as fundamental properties1 or inseparable concomitants1 or unchanging laws1 of matter and of energy4 ,in ,aristotle's parable1 the house is there that men may live in it2 but it is also there because the builders have laid one stone upon another3 and it is as a mechanism1 or a mechanical construction1 that the physicist looks upon the world4 ,like warp and woof1 mechanism and teleology are interwoven together1 and we must not cleave to the one and despise the other2 for their union is 8rooted in the very nature of totality#ac40 ,nevertheless1 when philosophy bids us hearken and obey the lessons both of mechanical and of teleological interpretation1 the precept is hard to follow3 so that oftentimes it has come to pass1 just as in ,bacon's day1 that a leaning to the side of the final cause 8hath intercepted the severe and diligent inquiry of all #f real and physical causes10 and has brought it about that 8the search of the physical cause hath been neglected and passed in silence40 ,so long and so far as 8fortuitous variation#ad0 and the 8survival of the fittest0 remain engrained as fundamental and satisfactory hypotheses in the philosophy of biology1 so long will these 8satisfactory and specious causes0 tend to stay 8severe and diligent inquiry10 8to the great arrest and prejudice of future discovery40 ,the difficulties which surround the concept of active or 8real0 causation1 in ,bacon's sense of the word1 difficulties of which ,hume and ,locke and ,aristotle were little aware1 need scarcely hinder us in our physical enquiry4 ,as students of mathematical and of empirical physics1 we are content to deal with those antecedents1 or concomitants1 of our phenomena1 without which the phenomenon does not occur1--with causes1 in short1 which1 aliae ex aliis aptae et necessitate nexae1 are no more1 and no less1 than conditions sine qu non4 ,our purpose is still adequately fulfilled3 inasmuch as we are still enabled to correlate1 and to equate1 our particular phenomena with more and ever more of the physical phenomena around1 and so to weave a web of connection and interdependence which shall serve our turn1 though the metaphysician withhold from that interdependence the title of causality4 ,we come in touch with what the schoolmen called a ratio cognoscendi1 though the true ratio efficiendi is still enwrapped in many mysteries4 ,and so handled1 the quest of physical causes merges with another great ,aristotelian theme1--the search for relations between things apparently disconnected1 and for 8similitude in things to common view unlike40 ,newton did not shew the cause of the apple falling1 but he shewed a similitude between the apple and the stars4 ,moreover1 the naturalist and the physicist will continue to speak of 8causes10 just as of old1 though it may be with some mental reservations3 for1 as a ,french philosopher said1 in a kindred difficulty3 8ce sont l des manires de s'exprimer1 #g et si elles sont interdites il faut renoncer parler de ces choses40 ,the search for differences or essential contrasts between the phenomena of organic and inorganic1 of animate and inanimate things has occupied many mens' minds1 while the search for community of principles1 or essential similitudes1 has been followed by few2 and the contrasts are apt to loom too large1 great as they may be4 ,m4 ,dunan1 discussing the 8,problme de la ,vie#ae0 in an essay which ,m4 ,bergson greatly commends1 declares3 8,les lois physico-chimiques sont aveugles et brutales2 l o elles rgnent seules1 au lieu d'un ordre et d'un concert1 il ne peut y avoir qu'incohrence et chaos40 ,but the physicist proclaims aloud that the physical phenomena which meet us by the way have their manifestations of form1 not less beautiful and scarce less varied than those which move us to admiration among living things4 ,the waves of the sea1 the little ripples on the shore1 the sweeping curve of the sandy bay between its headlands1 the outline of the hills1 the shape of the clouds1 all these are so many riddles of form1 so many problems of morphology1 and all of them the physicist can more or less easily read and adequately solve3 solving them by reference to their antecedent phenomena1 in the material system of mechanical forces to which they belong1 and to which we interpret them as being due4 ,they have also1 doubtless1 their immanent teleological significance2 but it is on another plane of thought from the physicist's that we contemplate their intrinsic harmony and perfection1 and 8see that they are good40 ,nor is it otherwise with the material forms of living things4 ,cell and tissue1 shell and bone1 leaf and flower1 are so many portions of matter1 and it is in obedience to the laws of physics that their particles have been moved1 moulded and conformed#af4 #h ,they are no exception to the rule that 4 ,their problems of form are in the first instance mathematical problems1 and their problems of growth are essentially physical problems2 and the morphologist is1 ipso facto1 a student of physical science4 ,apart from the physico-chemical problems of modern physiology1 the road of physico-mathematical or dynamical investigation in morphology has had few to follow it2 but the pathway is old4 ,the way of the old ,ionian physicians1 of ,anaxagoras#ag1 of ,empedocles and his disciples in the days before ,aristotle1 lay just by that highwayside4 ,it was ,galileo's and ,borelli's way4 ,it was little trodden for long afterwards1 but once in a while ,swammerdam and ,raumur looked that way4 ,and of later years1 ,moseley and ,meyer1 ,berthold1 ,errera and ,roux have been among the little band of travellers4 ,we need not wonder if the way be hard to follow1 and if these wayfarers have yet gathered little4 ,a harvest has been reaped by others1 and the gleaning of the grapes is slow4 ,it behoves us always to remember that in physics it has taken great men to discover simple things4 ,they are very great names indeed that we couple with the explanation of the path of a stone1 the droop of a chain1 the tints of a bubble1 the shadows in a cup4 ,it is but the slightest adumbration of a dynamical morphology that we can hope to have1 until the physicist and the mathematician shall have made these problems of ours their own1 or till a new ,boscovich shall have written for the naturalist the new ,theoria ,philosophiae ,naturalis4 ,how far1 even then1 mathematics will suffice to describe1 and physics to explain1 the fabric of the body no man can foresee4 ,it may be that all the laws of energy1 and all the properties of matter1 and all the chemistry of all the colloids are as powerless to explain the body as they are impotent to comprehend the soul4 ,for my part1 ,i think it is not so4 ,of how it is that the soul informs the body1 physical science teaches me nothing3 consciousness is not explained to my comprehension by all the nerve-paths and 8neurones0 of the physiologist2 nor do ,i ask of physics how goodness shines in one man's face1 and evil betrays itself in another4 ,but of the construction and growth and working #i of the body1 as of all that is of the earth earthy1 physical science is1 in my humble opinion1 our only teacher and guide#ah4 ,often and often it happens that our physical knowledge is inadequate to explain the mechanical working of the organism2 the phenomena are superlatively complex1 the procedure is involved and entangled1 and the investigation has occupied but a few short lives of men4 ,when physical science falls short of explaining the order which reigns throughout these manifold phenomena1--an order more characteristic in its totality than any of its phenomena in themselves1--men hasten to invoke a guiding principle1 an entelechy1 or call it what you will4 ,but all the while1 so far as ,i am aware1 no physical law1 any more than that of gravity itself1 not even among the puzzles of chemical 8stereometry10 or of physiological 8surface-action0 or 8osmosis10 is known to be transgressed by the bodily mechanism4 ,some physicists declare1 as ,maxwell did1 that atoms or molecules more complicated by far than the chemist's hypotheses demand are requisite to explain the phenomena of life4 ,if what is implied be an explanation of psychical phenomena1 let the point be granted at once2 we may go yet further1 and decline1 with ,maxwell1 to believe that anything of the nature of physical complexity1 however exalted1 could ever suffice4 ,other physicists1 like ,auerbach#ai1 or ,larmor#bj1 or ,joly#ba1 assure us that our laws of thermodynamics do not suffice1 or are 8inappropriate10 to explain the maintenance or 7in ,joly's phrase7 the 8accelerative absorption0 #aj of the bodily energies1 and the long battle against the cold and darkness which is death4 ,with these weighty problems ,i am not for the moment concerned4 ,my sole purpose is to correlate with mathematical statement and physical law certain of the simpler outward phenomena of organic growth and structure or form3 while all the while regarding1 ex hypothesi1 for the purposes of this correlation1 the fabric of the organism as a material and mechanical configuration4 ,physical science and philosophy stand side by side1 and one upholds the other4 ,without something of the strength of physics1 philosophy would be weak2 and without something of philosophy's wealth1 physical science would be poor4 8,rien ne retirera du tissu de la science les fils d'or que la main du philosophe y a introduits#bb40 ,but there are fields where each1 for a while at least1 must work alone2 and where physical science reaches its limitations1 physical science itself must help us to discover4 ,meanwhile the appropriate and legitimate postulate of the physicist1 in approaching the physical problems of the body1 is that with these physical phenomena no alien influence interferes4 ,but the postulate1 though it is certainly legitimate1 and though it is the proper and necessary prelude to scientific enquiry1 may some day be proven to be untrue2 and its disproof will not be to the physicist's confusion1 but will come as his reward4 ,in dealing with forms which are so concomitant with life that they are seemingly controlled by life1 it is in no spirit of arrogant assertiveness that the physicist begins his argument1 after the fashion of a most illustrious exemplar1 with the old formulary of scholastic challenge1--,an ,vita sit8 ,dico quod non4 ,the terms ,form and ,growth1 which make up the title of this little book1 are to be understood1 as ,i need hardly say1 in their relation to the science of organisms4 ,we want to see how1 in some cases at least1 the forms of living things1 and of the parts of living things1 can be explained by physical considerations1 and to realise that1 in general1 no organic forms exist save such as are in conformity with ordinary physical laws4 ,and while growth is a somewhat vague word for a complex matter1 which may #aa depend on various things1 from simple imbibition of water to the complicated results of the chemistry of nutrition1 it deserves to be studied in relation to form1 whether it proceed by simple increase of size without obvious alteration of form1 or whether it so proceed as to bring about a gradual change of form and the slow development of a more or less complicated structure4 ,in the ,newtonian language of elementary physics1 force is recognised by its action in producing or in changing motion1 or in preventing change of motion or in maintaining rest4 ,when we deal with matter in the concrete1 force does not1 strictly speaking1 enter into the question1 for force1 unlike matter1 has no independent objective existence4 ,it is energy in its various forms1 known or unknown1 that acts upon matter4 ,but when we abstract our thoughts from the material to its form1 or from the thing moved to its motions1 when we deal with the subjective conceptions of form1 or movement1 or the movements that change of form implies1 then force is the appropriate term for our conception of the causes by which these forms and changes of form are brought about4 ,when we use the term force1 we use it1 as the physicist always does1 for the sake of brevity1 using a symbol for the magnitude and direction of an action in reference to the symbol or diagram of a material thing4 ,it is a term as subjective and symbolic as form itself1 and so is appropriately to be used in connection therewith4 ,the form1 then1 of any portion of matter1 whether it be living or dead1 and the changes of form that are apparent in its movements and in its growth1 may in all cases alike be described as due to the action of force4 ,in short1 the form of an object is a 8diagram of forces10 in this sense1 at least1 that from it we can judge of or deduce the forces that are acting or have acted upon it3 in this strict and particular sense1 it is a diagram1--in the case of a solid1 of the forces that have been impressed upon it when its conformation was produced1 together with those that enable it to retain its conformation2 in the case of a liquid 7or of a gas7 of the forces that are for the moment acting on it to restrain or balance its own inherent mobility4 ,in an organism1 great or small1 it is not merely the nature of the motions of the living substance that we must interpret in terms of force 7according to kinetics71 but also #ab the conformation of the organism itself1 whose permanence or equilibrium is explained by the interaction or balance of forces1 as described in statics4 ,if we look at the living cell of an ,amoeba or a ,spirogyra1 we see a something which exhibits certain active movements1 and a certain fluctuating1 or more or less lasting1 form2 and its form at a given moment1 just like its motions1 is to be investigated by the help of physical methods1 and explained by the invocation of the mathematical conception of force4 ,now the state1 including the shape or form1 of a portion of matter1 is the resultant of a number of forces1 which represent or symbolise the manifestations of various kinds of energy2 and it is obvious1 accordingly1 that a great part of physical science must be understood or taken for granted as the necessary preliminary to the discussion on which we are engaged4 ,but we may at least try to indicate1 very briefly1 the nature of the principal forces and the principal properties of matter with which our subject obliges us to deal4 ,let us imagine1 for instance1 the case of a so-called 8simple0 organism1 such as ,amoeba2 and if our short list of its physical properties and conditions be helpful to our further discussion1 we need not consider how far it be complete or adequate from the wider physical point of view#bc4 ,this portion of matter1 then1 is kept together by the intermolecular force of cohesion2 in the movements of its particles relatively to one another1 and in its own movements relative to adjacent matter1 it meets with the opposing force of friction4 ,it is acted on by gravity1 and this force tends 7though slightly1 owing to the ,amoeba's small mass1 and to the small difference between its density and that of the surrounding fluid71 to flatten it down upon the solid substance on which it may be creeping4 ,our ,amoeba tends1 in the next place1 to be deformed by any pressure from outside1 even though slight1 which may be applied to it1 and this circumstance shews it to consist of matter in a fluid1 or at least semi-fluid1 state3 which state is further indicated when we observe streaming or current motions in its interior4 #ac ,like other fluid bodies1 its surface1 whatsoever other substance1 gas1 liquid or solid1 it be in contact with1 and in varying degree according to the nature of that adjacent substance1 is the seat of molecular force exhibiting itself as a surface-tension1 from the action of which many important consequences follow1 which greatly affect the form of the fluid surface4 ,while the protoplasm of the ,amoeba reacts to the slightest pressure1 and tends to 8flow10 and while we therefore speak of it as a fluid1 it is evidently far less mobile than such a fluid1 for instance1 as water1 but is rather like treacle in its slow creeping movements as it changes its shape in response to force4 ,such fluids are said to have a high viscosity1 and this viscosity obviously acts in the way of retarding change of form1 or in other words of retarding the effects of any disturbing action of force4 ,when the viscous fluid is capable of being drawn out into fine threads1 a property in which we know that the material of some ,amoebae differs greatly from that of others1 we say that the fluid is also viscid1 or exhibits viscidity4 ,again1 not by virtue of our ,amoeba being liquid1 but at the same time in vastly greater measure than if it were a solid 7though far less rapidly than if it were a gas71 a process of molecular diffusion is constantly going on within its substance1 by which its particles interchange their places within the mass1 while surrounding fluids1 gases and solids in solution diffuse into and out of it4 ,in so far as the outer wall of the cell is different in character from the interior1 whether it be a mere pellicle as in ,amoeba or a firm cell-wall as in ,protococcus1 the diffusion which takes place through this wall is sometimes distinguished under the term osmosis4 ,within the cell1 chemical forces are at work1 and so also in all probability 7to judge by analogy7 are electrical forces2 and the organism reacts also to forces from without1 that have their origin in chemical1 electrical and thermal influences4 ,the processes of diffusion and of chemical activity within the cell result1 by the drawing in of water1 salts1 and food-material with or without chemical transformation into protoplasm1 in growth1 and this complex phenomenon we shall usually1 without discussing its nature and origin1 describe and picture as a force4 ,indeed we shall manifestly be inclined to use the term growth in two senses1 #ad just indeed as we do in the case of attraction or gravitation1 on the one hand as a process1 and on the other hand as a force4 ,in the phenomena of cell-division1 in the attractions or repulsions of the parts of the dividing nucleus and in the 8caryokinetic0 figures that appear in connection with it1 we seem to see in operation forces and the effects of forces1 that have1 to say the least of it1 a close analogy with known physical phenomena2 and to this matter we shall afterwards recur4 ,but though they resemble known physical phenomena1 their nature is still the subject of much discussion1 and neither the forms produced nor the forces at work can yet be satisfactorily and simply explained4 ,we may readily admit1 then1 that besides phenomena which are obviously physical in their nature1 there are actions visible as well as invisible taking place within living cells which our knowledge does not permit us to ascribe with certainty to any known physical force2 and it may or may not be that these phenomena will yield in time to the methods of physical investigation4 ,whether or no1 it is plain that we have no clear rule or guide as to what is 8vital0 and what is not2 the whole assemblage of so-called vital phenomena1 or properties of the organism1 cannot be clearly classified into those that are physical in origin and those that are sui generis and peculiar to living things4 ,all we can do meanwhile is to analyse1 bit by bit1 those parts of the whole to which the ordinary laws of the physical forces more or less obviously and clearly and indubitably apply4 ,morphology then is not only a study of material things and of the forms of material things1 but has its dynamical aspect1 under which we deal with the interpretation1 in terms of force1 of the operations of ,energy4 ,and here it is well worth while to remark that1 in dealing with the facts of embryology or the phenomena of inheritance1 the common language of the books seems to deal too much with the material elements concerned1 as the causes of development1 of variation or of hereditary transmission4 ,matter as such produces nothing1 changes nothing1 does nothing2 and however convenient it may afterwards be to abbreviate our nomenclature and our descriptions1 we must most carefully realise in the outset that the spermatozoon1 the nucleus1 #ae the chromosomes or the germ-plasm can never act as matter alone1 but only as seats of energy and as centres of force4 ,and this is but an adaptation 7in the light1 or rather in the conventional symbolism1 of modern physical science7 of the old saying of the philosopher3 4 #af ,,chapter ,,ii ,,on ,,magnitude ,to terms of magnitude1 and of direction1 must we refer all our conceptions of form4 ,for the form of an object is defined when we know its magnitude1 actual or relative1 in various directions2 and growth involves the same conceptions of magnitude and direction1 with this addition1 that they are supposed to alter in time4 ,before we proceed to the consideration of specific form1 it will be worth our while to consider1 for a little while1 certain phenomena of spatial magnitude1 or of the extension of a body in the several dimensions of space#bd4 ,we are taught by elementary mathematics that1 in similar solid figures1 the surface increases as the square1 and the volume as the cube1 of the linear dimensions4 ,if we take the simple case of a sphere1 with radius r1 the area of its surface is equal to #dr#b1 and its volume to 7#d_/c7r#c2 from which it follows that the ratio of volume to surface1 or ,v_/,s1 is 7#a_/c7r4 ,in other words1 the greater the radius 7or the larger the sphere7 the greater will be its volume1 or its mass 7if it be uniformly dense throughout71 in comparison with its superficial area4 ,and1 taking ,l to represent any linear dimension1 we may write the general equations in the form ,s ,l#b1 ,v ,l#c1 or ,s k ,l#b1 and ,v k ,l#c2 and ,v_/,s ,l4 ,from these elementary principles a great number of consequences follow1 all more or less interesting1 and some of them of great importance4 ,in the first place1 though growth in length 7let #ag us say7 and growth in volume 7which is usually tantamount to mass or weight7 are parts of one and the same process or phenomenon1 the one attracts our attention by its increase1 very much more than the other4 ,for instance a fish1 in doubling its length1 multiplies its weight by no less than eight times2 and it all but doubles its weight in growing from four inches long to five4 ,in the second place we see that a knowledge of the correlation between length and weight in any particular species of animal1 in other words a determination of k in the formula ,w k ,l#c1 enables us at any time to translate the one magnitude into the other1 and 7so to speak7 to weigh the animal with a measuring-rod2 this however being always subject to the condition that the animal shall in no way have altered its form1 nor its specific gravity4 ,that its specific gravity or density should materially or rapidly alter is not very likely2 but as long as growth lasts1 changes of form1 even though inappreciable to the eye1 are likely to go on4 ,now weighing is a far easier and far more accurate operation than measuring2 and the measurements which would reveal slight and otherwise imperceptible changes in the form of a fish--slight relative differences between length1 breadth and depth1 for instance1--would need to be very delicate indeed4 ,but if we can make fairly accurate determinations of the length1 which is very much the easiest dimension to measure1 and then correlate it with the weight1 then the value of k1 according to whether it varies or remains constant1 will tell us at once whether there has or has not been a tendency to gradual alteration in the general form4 ,to this subject we shall return1 when we come to consider more particularly the rate of growth4 ,but a much deeper interest arises out of this changing ratio of dimensions when we come to consider the inevitable changes of physical relations with which it is bound up4 ,we are apt1 and even accustomed1 to think that magnitude is so purely relative that differences of magnitude make no other or more essential difference2 that ,lilliput and ,brobdingnag are all alike1 according as we look at them through one end of the glass or the other4 ,but this is by no means so2 for scale has a very marked effect upon physical phenomena1 and the effect of scale constitutes what is known as the principle of similitude1 or of dynamical similarity4 #ah ,this effect of scale is simply due to the fact that1 of the physical forces1 some act either directly at the surface of a body1 or otherwise in proportion to the area of surface2 and others1 such as gravity1 act on all particles1 internal and external alike1 and exert a force which is proportional to the mass1 and so usually to the volume1 of the body4 ,the strength of an iron girder obviously varies with the cross-section of its members1 and each cross-section varies as the square of a linear dimension2 but the weight of the whole structure varies as the cube of its linear dimensions4 ,and it follows at once that1 if we build two bridges geometrically similar1 the larger is the weaker of the two#be4 ,it was elementary engineering experience such as this that led ,herbert ,spencer#bf to apply the principle of similitude to biology4 ,the same principle had been admirably applied1 in a few clear instances1 by ,lesage#bg1 a celebrated eighteenth century physician of ,geneva1 in an unfinished and unpublished work#bh4 ,lesage argued1 for instance1 that the larger ratio of surface to mass would lead in a small animal to excessive transpiration1 were the skin as 8porous0 as our own2 and that we may hence account for the hardened or thickened skins of insects and other small terrestrial animals4 ,again1 since the weight of a fruit increases as the cube of its dimensions1 while the strength of the stalk increases as the square1 it follows that the stalk should grow out of apparent due proportion to the fruit2 or alternatively1 that tall trees should not bear large fruit on slender branches1 and that melons and pumpkins must lie upon the ground4 ,and again1 that in quadrupeds a large head must be supported on a neck which is either #ai excessively thick and strong1 like a bull's1 or very short like the neck of an elephant4 ,but it was ,galileo who1 wellnigh #cjj years ago1 had first laid down this general principle which we now know by the name of the principle of similitude2 and he did so with the utmost possible clearness1 and with a great wealth of illustration1 drawn from structures living and dead#bi4 ,he showed that neither can man build a house nor can nature construct an animal beyond a certain size1 while retaining the same proportions and employing the same materials as sufficed in the case of a smaller structure#cj4 ,the thing will fall to pieces of its own weight unless we either change its relative proportions1 which will at length cause it to become clumsy1 monstrous and inefficient1 or else we must find a new material1 harder and stronger than was used before4 ,both processes are familiar to us in nature and in art1 and practical applications1 undreamed of by ,galileo1 meet us at every turn in this modern age of steel4 ,again1 as ,galileo was also careful to explain1 besides the questions of pure stress and strain1 of the strength of muscles to lift an increasing weight or of bones to resist its crushing stress1 we have the very important question of bending moments4 ,this question enters1 more or less1 into our whole range of problems2 it affects1 as we shall afterwards see1 or even determines the whole form of the skeleton1 and is very important in such a case as that of a tall tree#ca4 ,here we have to determine the point at which the tree will curve under its own weight1 if it be ever so little displaced from the perpendicular#cb4 ,in such an investigation we have to make #bj some assumptions1--for instance1 with regard to the trunk1 that it tapers uniformly1 and with regard to the branches that their sectional area varies according to some definite law1 or 7as ,ruskin assumed#cc7 tends to be constant in any horizontal plane2 and the mathematical treatment is apt to be somewhat difficult4 ,but ,greenhill has shewn that 7on such assumptions as the above71 a certain ,british ,columbian pine-tree1 which yielded the ,kew flagstaff measuring #bba ft4 in height with a diameter at the base of #ba inches1 could not possibly1 by theory1 have grown to more than about #cjj ft4 ,it is very curious that ,galileo suggested precisely the same height 7dugento braccia alta7 as the utmost limit of the growth of a tree4 ,in general1 as ,greenhill shews1 the diameter of a homogeneous body must increase as the power #c_/b of the height1 which accounts for the slender proportions of young trees1 compared with the stunted appearance of old and large ones#cd4 ,in short1 as ,goethe says in ,wahrheit und ,dichtung1 8,es ist dafr gesorgt dass die ,bume nicht in den ,himmel wachsen40 ,but ,eiffel's great tree of steel 7#ajjj feet high7 is built to a very different plan2 for here the profile of the tower follows the logarithmic curve1 giving equal strength throughout1 according to a principle which we shall have occasion to discuss when we come to treat of 8form and mechanical efficiency0 in connection with the skeletons of animals4 ,among animals1 we may see in a general way1 without the help of mathematics or of physics1 that exaggerated bulk brings with it a certain clumsiness1 a certain inefficiency1 a new element of risk and hazard1 a vague preponderance of disadvantage4 ,the case was well put by ,owen1 in a passage which has an interest of its own as a premonition 7somewhat like ,de ,candolle's7 of the 8struggle for existence40 ,owen wrote as follows#ce3 8,in proportion to the bulk of a species is the difficulty of the contest which1 as a living organised whole1 the individual of such species #ba has to maintain against the surrounding agencies that are ever tending to dissolve the vital bond1 and subjugate the living matter to the ordinary chemical and physical forces4 ,any changes1 therefore1 in such external conditions as a species may have been originally adapted to exist in1 will militate against that existence in a degree proportionate1 perhaps in a geometrical ratio1 to the bulk of the species4 ,if a dry season be greatly prolonged1 the large mammal will suffer from the drought sooner than the small one2 if any alteration of climate affect the quantity of vegetable food1 the bulky ,herbivore will first feel the effects of stinted nourishment40 ,but the principle of ,galileo carries us much further and along more certain lines4 ,the tensile strength of a muscle1 like that of a rope or of our girder1 varies with its cross-section2 and the resistance of a bone to a crushing stress varies1 again like our girder1 with its cross-section4 ,but in a terrestrial animal the weight which tends to crush its limbs or which its muscles have to move1 varies as the cube of its linear dimensions2 and so1 to the possible magnitude of an animal1 living under the direct action of gravity1 there is a definite limit set4 ,the elephant1 in the dimensions of its limb-bones1 is already shewing signs of a tendency to disproportionate thickness as compared with the smaller mammals2 its movements are in many ways hampered and its agility diminished3 it is already tending towards the maximal limit of size which the physical forces permit4 ,but1 as ,galileo also saw1 if the animal be wholly immersed in water1 like the whale1 7or if it be partly so1 as was in all probability the case with the giant reptiles of our secondary rocks71 then the weight is counterpoised to the extent of an equivalent volume of water1 and is completely counterpoised if the density of the animal's body1 with the included air1 be identical 7as in a whale it very nearly is7 with the water around4 ,under these circumstances there is no longer a physical barrier to the indefinite growth in magnitude of the animal#cf4 ,indeed1 #bb in the case of the aquatic animal there is1 as ,spencer pointed out1 a distinct advantage1 in that the larger it grows the greater is its velocity4 ,for its available energy depends on the mass of its muscles2 while its motion through the water is opposed1 not by gravity1 but by 8skin-friction10 which increases only as the square of its dimensions2 all other things being equal1 the bigger the ship1 or the bigger the fish1 the faster it tends to go1 but only in the ratio of the square root of the increasing length4 ,for the mechanical work 7,w7 of which the fish is capable being proportional to the mass of its muscles1 or the cube of its linear dimensions3 and again this work being wholly done in producing a velocity 7,v7 against a resistance 7,r7 which increases as the square of the said linear dimensions2 we have at once ,w l#c1 and also ,w ,r ,v#b l#b ,v#b4 ,therefore l#c l#b ,v#b1 and ,v l4 ,this is what is known as ,froude's ,law of the correspondence of speeds4 ,but there is often another side to these questions1 which makes them too complicated to answer in a word4 ,for instance1 the work 7per stroke7 of which two similar engines are capable should obviously vary as the cubes of their linear dimensions1 for it varies on the one hand with the surface of the piston1 and on the other1 with the length of the stroke2 so is it likewise in the animal1 where the corresponding variation depends on the cross-section of the muscle1 and on the space through which it contracts4 ,but in two precisely similar engines1 the actual available horse-power varies as the square of the linear dimensions1 and not as the cube2 and this for the obvious reason that the actual energy developed depends upon the heating-surface of the boiler#cg4 ,so likewise must there be a similar tendency1 among animals1 for the rate of supply of kinetic energy to vary with the surface of the #bc lung1 that is to say 7other things being equal7 with the square of the linear dimensions of the animal4 ,we may of course 7departing from the condition of similarity7 increase the heating-surface of the boiler1 by means of an internal system of tubes1 without increasing its outward dimensions1 and in this very way nature increases the respiratory surface of a lung by a complex system of branching tubes and minute air-cells2 but nevertheless in two similar and closely related animals1 as also in two steam-engines of precisely the same make1 the law is bound to hold that the rate of working must tend to vary with the square of the linear dimensions1 according to ,froude's law of steamship comparison4 ,in the case of a very large ship1 built for speed1 the difficulty is got over by increasing the size and number of the boilers1 till the ratio between boiler-room and engine-room is far beyond what is required in an ordinary small vessel#ch2 but though we find lung-space increased among animals where greater rate of working is required1 as in general among birds1 ,i do not know that it can be shewn to increase1 as in the 8over-boilered0 ship1 with the size of the animal1 and in a ratio which outstrips that of the other bodily dimensions4 ,if it be the case then1 that the working mechanism of the muscles should be able to exert a force proportionate to the cube of the linear bodily dimensions1 while the respiratory mechanism can only supply a store of energy at a rate proportional to the square of the said dimensions1 the singular result ought to follow that1 in swimming for instance1 the larger fish ought to be able to put on a spurt of speed far in excess of the smaller one2 but the distance travelled by the year's end should be very much alike for both of them4 ,and it should also follow that the curve of fatigue #bd should be a steeper one1 and the staying power should be less1 in the smaller than in the larger individual4 ,this is the case of long-distance racing1 where the big winner puts on his big spurt at the end4 ,and for an analogous reason1 wise men know that in the ',varsity boat-race it is judicious and prudent to bet on the heavier crew4 ,leaving aside the question of the supply of energy1 and keeping to that of the mechanical efficiency of the machine1 we may find endless biological illustrations of the principle of similitude4 ,in the case of the flying bird 7apart from the initial difficulty of raising itself into the air1 which involves another problem7 it may be shewn that the bigger it gets 7all its proportions remaining the same7 the more difficult it is for it to maintain itself aloft in flight4 ,the argument is as follows3 ,in order to keep aloft1 the bird must communicate to the air a downward momentum equivalent to its own weight1 and therefore proportional to the cube of its own linear dimensions4 ,but the momentum so communicated is proportional to the mass of air driven downwards1 and to the rate at which it is driven3 the mass being proportional to the bird's wing-area1 and also 7with any given slope of wing7 to the speed of the bird1 and the rate being again proportional to the bird's speed2 accordingly the whole momentum varies as the wing-area1 i4e4 as the square of the linear dimensions1 and also as the square of the speed4 ,therefore1 in order that the bird may maintain level flight1 its speed must be proportional to the square root of its linear dimensions4 ,now the rate at which the bird1 in steady flight1 has to work in order to drive itself forward1 is the rate at which it communicates energy to the air2 and this is proportional to m ,v#b1 i4e4 to the mass and to the square of the velocity of the air displaced4 ,but the mass of air displaced per second is proportional to the wing-area and to the speed of the bird's motion1 and therefore to the power #b of the linear dimensions2 and the speed at which it is displaced is proportional to the bird's speed1 and therefore to the square root of the linear dimensions4 ,therefore the energy communicated per second 7being proportional to the mass and to the square of the speed7 is jointly proportional to the power #b of the linear dimensions1 as above1 and to the first power thereof3 #be that is to say1 it increases in proportion to the power #c of the linear dimensions1 and therefore faster than the weight of the bird increases4 ,put in mathematical form1 the equations are as follows3 7m the mass of air thrust downwards2 ,v its velocity1 proportional to that of the bird2 ,m its momentum2 l a linear dimension of the bird2 w its weight2 ,w the work done in moving itself forward47 ,m w l#c4 ,but ,m m ,v1 and m l#b ,v4 ,therefore ,m l#b ,v#b1 and l#b ,v#b l#c1 or ,v l4 ,but1 again1 ,w m ,v#b l#b ,v ,v#b l#b l l l#c4 ,the work requiring to be done1 then1 varies as the power #c of the bird's linear dimensions1 while the work of which the bird is capable depends on the mass of its muscles1 and therefore varies as the cube of its linear dimensions#ci4 ,the disproportion does not seem at first sight very great1 but it is quite enough to tell4 ,it is as much as to say that1 every time we double the linear dimensions of the bird1 the difficulty of flight is increased in the ratio of #bc 3 #bc1 or #h 3 #aac1 or1 say1 #a 3 #ad4 ,if we take the ostrich to exceed the sparrow in linear dimensions as #be 3 #a1 which seems well within the mark1 we have the ratio between #bec and #bec1 or between #eg 3 #ef2 in other words1 flight is just five times more difficult for the larger than for the smaller bird#dj4 ,the above investigation includes1 besides the final result1 a number of others1 explicit or implied1 which are of not less importance4 ,of these the simplest and also the most important is #bf contained in the equation ,v l1 a result which happens to be identical with one we had also arrived at in the case of the fish4 ,in the bird's case it has a deeper significance than in the other2 because it implies here not merely that the velocity will tend to increase in a certain ratio with the length1 but that it must do so as an essential and primary condition of the bird's remaining aloft4 ,it is accordingly of great practical importance in aeronautics1 for it shews how a provision of increasing speed must accompany every enlargement of our aeroplanes4 ,if a given machine weighing1 say1 #ejj lbs4 be stable at #dj miles an hour1 then one geometrically similar which weighs1 say1 a couple of tons must have its speed determined as follows3 ,w 3 w 33 ,l#c 3 l#c 33 #h 3 #a4 ,therefore ,l 3 l 33 #b 3 #a4 ,but ,v#b 3 v#b 33 ,l 3 l4 ,therefore ,v 3 v 33 #b 3 #a #adad 3 #a4 ,that is to say1 the larger machine must be capable of a speed equal to #adad #dj1 or about #ef miles per hour4 ,it is highly probable1 as ,lanchester#da remarks1 that ,lilienthal met his untimely death not so much from any intrinsic fault in the design or construction of his machine1 but simply because his engine fell somewhat short of the power required to give the speed which was necessary for stability4 ,an arrow is a very imperfectly designed aeroplane1 but nevertheless it is evidently capable1 to a certain extent and at a high velocity1 of acquiring 8stability0 and hence of actual 8flight03 the duration and consequent range of its trajectory1 as compared with a bullet of similar initial velocity1 being correspondingly benefited4 ,when we return to our birds1 and again compare the ostrich with the sparrow1 we know little or nothing about the speed in flight of the latter1 but that of the swift is estimated#db to vary from a minimum of #bj to #ej feet or more per second1--say from #ad to #ce miles per hour4 ,let us take the same lower limit as not far from the minimal velocity of the sparrow's flight also2 and it #bg would follow that the ostrich1 of #be times the sparrow's linear dimensions1 would be compelled to fly 7if it flew at all7 with a minimum velocity of #e #ad1 or #gj miles an hour4 ,the same principle of necessary speed1 or the indispensable relation between the dimensions of a flying object and the minimum velocity at which it is stable1 accounts for a great number of observed phenomena4 ,it tells us why the larger birds have a marked difficulty in rising from the ground1 that is to say1 in acquiring to begin with the horizontal velocity necessary for their support2 and why accordingly1 as ,mouillard#dc and others have observed1 the heavier birds1 even those weighing no more than a pound or two1 can be effectively 8caged0 in a small enclosure open to the sky4 ,it tells us why very small birds1 especially those as small as humming-birds1 and fortiori the still smaller insects1 are capable of 8stationary flight10 a very slight and scarcely perceptible velocity relatively to the air being sufficient for their support and stability4 ,and again1 since it is in all cases velocity relative to the air that we are speaking of1 we comprehend the reason why one may always tell which way the wind blows by watching the direction in which a bird starts to fly4 ,it is not improbable that the ostrich has already reached a magnitude1 and we may take it for certain that the moa did so1 at which flight by muscular action1 according to the normal anatomy of a bird1 has become physiologically impossible4 ,the same reasoning applies to the case of man4 ,it would be very difficult1 and probably absolutely impossible1 for a bird to fly were it the bigness of a man4 ,but ,borelli1 in discussing this question1 laid even greater stress on the obvious fact that a man's pectoral muscles are so immensely less in proportion than those of a bird1 that however we may fit ourselves with wings we can never expect to move them by any power of our own relatively weaker muscles2 so it is that artificial flight only became possible when an engine was devised whose efficiency was extraordinarily great in comparison with its weight and size4 ,had ,leonardo da ,vinci known what ,galileo knew1 he would not have spent a great part of his life on vain efforts to make to himself wings4 ,borelli had learned the lesson thoroughly1 and #bh in one of his chapters he deals with the proposition1 8,est impossible1 ut homines propriis viribus artificiose volare possint#dd40 ,but just as it is easier to swim than to fly1 so is it obvious that1 in a denser atmosphere1 the conditions of flight would be altered1 and flight facilitated4 ,we know that in the carboniferous epoch there lived giant dragon-flies1 with wings of a span far greater than nowadays they ever attain2 and the small bodies and huge extended wings of the fossil pterodactyles would seem in like manner to be quite abnormal according to our present standards1 and to be beyond the limits of mechanical efficiency under present conditions4 ,but as ,harl suggests#de1 following upon a suggestion of ,arrhenius1 we have only to suppose that in carboniferous and jurassic days the terrestrial atmosphere was notably denser than it is at present1 by reason1 for instance1 of its containing a much larger proportion of carbonic acid1 and we have at once a means of reconciling the apparent mechanical discrepancy4 ,very similar problems1 involving in various ways the principle of dynamical similitude1 occur all through the physiology of locomotion3 as1 for instance1 when we see that a cockchafer can carry a plate1 many times his own weight1 upon his back1 or that a flea can jump many inches high4 ,problems of this latter class have been admirably treated both by ,galileo and by ,borelli1 but many later writers have remained ignorant of their work4 ,linnaeus1 for instance1 remarked that1 if an elephant were as strong in proportion as a stag-beetle1 it would be able to pull up rocks by the root1 and to level mountains4 ,and ,kirby and ,spence have a well-known passage directed to shew that such powers as have been conferred upon the insect have been withheld from the higher animals1 for the reason that had these latter been endued therewith they would have 8caused the early desolation of the world#df40 #bi ,such problems as that which is presented by the flea's jumping powers1 though essentially physiological in their nature1 have their interest for us here3 because a steady1 progressive diminution of activity with increasing size would tend to set limits to the possible growth in magnitude of an animal just as surely as those factors which tend to break and crush the living fabric under its own weight4 ,in the case of a leap1 we have to do rather with a sudden impulse than with a continued strain1 and this impulse should be measured in terms of the velocity imparted4 ,the velocity is proportional to the impulse 7x71 and inversely proportional to the mass 7,m7 moved3 ,v x_/,m4 ,but1 according to what we still speak of as 8,borelli's law10 the impulse 7i4e4 the work of the impulse7 is proportional to the volume of the muscle by which it is produced#dg1 that is to say 7in similarly constructed animals7 to the mass of the whole body2 for the impulse is proportional on the one hand to the cross-section of the muscle1 and on the other to the distance through which it contracts4 ,it follows at once from this that the velocity is constant1 whatever be the size of the animals3 in other words1 that all animals1 provided always that they are similarly fashioned1 with their various levers etc41 in like proportion1 ought to jump1 not to the same relative1 but to the same actual height#dh4 ,according to this1 then1 the flea is not a better1 but rather a worse jumper than a horse or a man4 ,as a matter of fact1 ,borelli is careful to point out that in the act of leaping the impulse is not actually instantaneous1 as in the blow of a hammer1 but takes some little time1 during which the levers are being extended by which the centre of gravity of the animal is being propelled forwards2 and this interval of time will be longer in the case of the longer levers of the larger animal4 ,to some extent1 then1 this principle acts as a corrective to the more general one1 #cj and tends to leave a certain balance of advantage1 in regard to leaping power1 on the side of the larger animal#di4 ,but on the other hand1 the question of strength of materials comes in once more1 and the factors of stress and strain and bending moment make it1 so to speak1 more and more difficult for nature to endow the larger animal with the length of lever with which she has provided the flea or the grasshopper4 ,to ,kirby and ,spence it seemed that 8,this wonderful strength of insects is doubtless the result of something peculiar in the structure and arrangement of their muscles1 and principally their extraordinary power of contraction40 ,this hypothesis1 which is so easily seen1 on physical grounds1 to be unnecessary1 has been amply disproved in a series of excellent papers by ,f4 ,plateau#ej4 ,a somewhat simple problem is presented to us by the act of walking4 ,it is obvious that there will be a great economy of work1 if the leg swing at its normal pendulum-rate2 and1 though this rate is hard to calculate1 owing to the shape and the jointing of the limb1 we may easily convince ourselves1 by counting our steps1 that the leg does actually swing1 or tend to swing1 just as a pendulum does1 at a certain definite rate#ea4 ,when we walk quicker1 we cause the leg-pendulum to describe a greater arc1 but we do not appreciably cause it to swing1 or vibrate1 quicker1 until we shorten the pendulum and begin to run4 ,now let two individuals1 ,a and ,b1 walk in a similar fashion1 that is to say1 with a similar angle of swing4 ,the arc through which the leg swings1 or the amplitude of each step1 will therefore vary as the length of leg1 or say as a_/b2 but the time of swing will vary as the square #ca root of the pendulum-length1 or a_/b4 ,therefore the velocity1 which is measured by amplitude_/time1 will also vary as the square-roots of the length of leg3 that is to say1 the average velocities of ,a and ,b are in the ratio of a 3 b4 ,the smaller man1 or smaller animal1 is so far at a disadvantage compared with the larger in speed1 but only to the extent of the ratio between the square roots of their linear dimensions3 whereas1 if the rate of movement of the limb were identical1 irrespective of the size of the animal1--if the limbs of the mouse for instance swung at the same rate as those of the horse1--then1 as ,f4 ,plateau said1 the mouse would be as slow or slower in its gait than the tortoise4 ,m4 ,delisle#eb observed a 8minute fly0 walk three inches in half-a-second4 ,this was good steady walking4 ,when we walk five miles an hour we go about #hh inches in a second1 or #hh_/f #adg times the pace of ,m4 ,delisle's fly4 ,we should walk at just about the fly's pace if our stature were #a_/7adg7b1 or #a_/baf of our present height1--say #gb_/baf inches1 or one-third of an inch high4 ,but the leg comprises a complicated system of levers1 by whose various exercise we shall obtain very different results4 ,for instance1 by being careful to rise upon our instep1 we considerably increase the length or amplitude of our stride1 and very considerably increase our speed accordingly4 ,on the other hand1 in running1 we bend and so shorten the leg1 in order to accommodate it to a quicker rate of pendulum-swing#ec4 ,in short1 the jointed structure of the leg permits us to use it as the shortest possible pendulum when it is swinging1 and as the longest possible lever when it is exerting its propulsive force4 ,apart from such modifications as that described in the last paragraph1--apart1 that is to say1 from differences in mechanical construction or in the manner in which the mechanism is used1--we have now arrived at a curiously simple and uniform result4 ,for in all the three forms of locomotion which we have attempted #cb to study1 alike in swimming1 in flight and in walking1 the general result1 attained under very different conditions and arrived at by very different modes of reasoning1 is in every case that the velocity tends to vary as the square root of the linear dimensions of the organism4 ,from all the foregoing discussion we learn that1 as ,crookes once upon a time remarked#ed1 the form as well as the actions of our bodies are entirely conditioned 7save for certain exceptions in the case of aquatic animals1 nicely balanced with the density of the surrounding medium7 by the strength of gravity upon this globe4 ,were the force of gravity to be doubled1 our bipedal form would be a failure1 and the majority of terrestrial animals would resemble short-legged saurians1 or else serpents4 ,birds and insects would also suffer1 though there would be some compensation for them in the increased density of the air4 ,while on the other hand if gravity were halved1 we should get a lighter1 more graceful1 more active type1 requiring less energy and less heat1 less heart1 less lungs1 less blood4 ,throughout the whole field of morphology we may find examples of a tendency 7referable doubtless in each case to some definite physical cause7 for surface to keep pace with volume1 through some alteration of its form4 ,the development of 8villi0 on the inner surface of the stomach and intestine 7which enlarge its surface much as we enlarge the effective surface of a bath-towel71 the various valvular folds of the intestinal lining1 including the remarkable 8spiral fold0 of the shark's gut1 the convolutions of the brain1 whose complexity is evidently correlated 7in part at least7 with the magnitude of the animal1--all these and many more are cases in which a more or less constant ratio tends to be maintained between mass and surface1 which ratio would have been more and more departed from had it not been for the alterations of surface-form#ee4 #cc ,in the case of very small animals1 and of individual cells1 the principle becomes especially important1 in consequence of the molecular forces whose action is strictly limited to the superficial layer4 ,in the cases just mentioned1 action is facilitated by increase of surface3 diffusion1 for instance1 of nutrient liquids or respiratory gases is rendered more rapid by the greater area of surface2 but there are other cases in which the ratio of surface to mass may make an essential change in the whole condition of the system4 ,we know1 for instance1 that iron rusts when exposed to moist air1 but that it rusts ever so much faster1 and is soon eaten away1 if the iron be first reduced to a heap of small filings2 this is a mere difference of degree4 ,but the spherical surface of the raindrop and the spherical surface of the ocean 7though both happen to be alike in mathematical form7 are two totally different phenomena1 the one due to surface-energy1 and the other to that form of mass-energy which we ascribe to gravity4 ,the contrast is still more clearly seen in the case of waves3 for the little ripple1 whose form and manner of propagation are governed by surface-tension1 is found to travel with a velocity which is inversely as the square root of its length2 while the ordinary big waves1 controlled by gravitation1 have a velocity directly proportional to the square root of their wave-length4 ,in like manner we shall find that the form of all small organisms is largely independent of gravity1 and largely if not mainly due to the force of surface-tension3 either as the direct result of the continued action of surface tension on the semi-fluid body1 or else as the result of its action at a prior stage of development1 in bringing about a form which subsequent chemical changes have rendered rigid and lasting4 ,in either case1 we shall find a very great tendency in small organisms to assume either the spherical form or other simple forms related to ordinary inanimate surface-tension phenomena2 which forms do not recur in the external morphology of large animals1 or if they in part recur it is for other reasons4 #cd ,now this is a very important matter1 and is a notable illustration of that principle of similitude which we have already discussed in regard to several of its manifestations4 ,we are coming easily to a conclusion which will affect the whole course of our argument throughout this book1 namely that there is an essential difference in kind between the phenomena of form in the larger and the smaller organisms4 ,i have called this book a study of ,growth and ,form1 because in the most familiar illustrations of organic form1 as in our own bodies for example1 these two factors are inseparably associated1 and because we are here justified in thinking of form as the direct resultant and consequence of growth3 of growth1 whose varying rate in one direction or another has produced1 by its gradual and unequal increments1 the successive stages of development and the final configuration of the whole material structure4 ,but it is by no means true that form and growth are in this direct and simple fashion correlative or complementary in the case of minute portions of living matter4 ,for in the smaller organisms1 and in the individual cells of the larger1 we have reached an order of magnitude in which the intermolecular forces strive under favourable conditions with1 and at length altogether outweigh1 the force of gravity1 and also those other forces leading to movements of convection which are the prevailing factors in the larger material aggregate4 ,however we shall require to deal more fully with this matter in our discussion of the rate of growth1 and we may leave it meanwhile1 in order to deal with other matters more or less directly concerned with the magnitude of the cell4 ,the living cell is a very complex field of energy1 and of energy of many kinds1 surface-energy included4 ,now the whole surface-energy of the cell is by no means restricted to its outer surface2 for the cell is a very heterogeneous structure1 and all its protoplasmic alveoli and other visible 7as well as invisible7 heterogeneities make up a great system of internal surfaces1 at every part of which one 8phase0 comes in contact with another 8phase10 and surface-energy is accordingly manifested4 ,but still1 the external surface is a definite portion of the system1 with a definite 8phase0 of its own1 and however little we may know of the distribution of the total energy of the system1 it is at least plain that #ce the conditions which favour equilibrium will be greatly altered by the changed ratio of external surface to mass which a change of magnitude1 unaccompanied by change of form1 produces in the cell4 ,in short1 however it may be brought about1 the phenomenon of division of the cell will be precisely what is required to keep approximately constant the ratio between surface and mass1 and to restore the balance between the surface-energy and the other energies of the system4 ,when a germ-cell1 for instance1 divides or 8segments0 into two1 it does not increase in mass2 at least if there be some slight alleged tendency for the egg to increase in mass or volume during segmentation1 it is very slight indeed1 generally imperceptible1 and wholly denied by some#ef4 ,the development or growth of the egg from a one-celled stage to stages of two or many cells1 is thus a somewhat peculiar kind of growth2 it is growth which is limited to increase of surface1 unaccompanied by growth in volume or in mass4 ,in the case of a soap-bubble1 by the way1 if it divide into two bubbles1 the volume is actually diminished#eg while the surface-area is greatly increased4 ,this is due to a cause which we shall have to study later1 namely to the increased pressure due to the greater curvature of the smaller bubbles4 ,an immediate and remarkable result of the principles just described is a tendency on the part of all cells1 according to their kind1 to vary but little about a certain mean size1 and to have1 in fact1 certain absolute limitations of magnitude4 ,sachs#eh pointed out1 in #ahie1 that there is a tendency for each nucleus to be only able to gather around itself a certain definite amount of protoplasm4 ,driesch#ei1 a little later1 found that1 by artificial subdivision of the egg1 it was possible to rear dwarf sea-urchin larvae1 one-half1 one-quarter1 or even one-eighth of their #cf normal size2 and that these dwarf bodies were composed of only a half1 a quarter or an eighth of the normal number of cells4 ,similar observations have been often repeated and amply confirmed4 ,for instance1 in the development of ,crepidula 7a little ,american 8slipper-limpet10 now much at home on our own oyster-beds71 ,conklin#fj has succeeded in rearing dwarf and giant individuals1 of which the latter may be as much as twenty-five times as big as the former4 ,but nevertheless1 the individual cells1 of skin1 gut1 liver1 muscle1 and of all the other tissues1 are just the same size in one as in the other1--in dwarf and in giant#fa4 ,driesch has laid particular stress upon this principle of a 8fixed cell-size40 ,we get an excellent1 and more familiar illustration of the same principle in comparing the large brain-cells or ganglion-cells1 both of the lower and of the higher animals#fb4 ,illustration3 ,fig4 #a4 ,motor ganglion-cells1 from the cervical spinal cord4 7,from ,minot1 after ,irving ,hardesty47 ,in ,fig4 #a we have certain identical nerve-cells taken from various mammals1 from the mouse to the elephant1 all represented on the same scale of magnification2 and we see at once that they are all of much the same order of magnitude4 ,the nerve-cell of the elephant is about twice that of the mouse in linear dimensions1 and therefore about eight times greater in volume1 or mass4 ,but making some allowance for difference of shape1 the linear dimensions of the elephant are to those of the mouse in a ratio certainly not less than one to fifty2 from which it would follow that the bulk of the larger animal is something like #abe1jjj times that of the less4 ,and it also follows1 the size of the nerve-cells being #cg about as eight to one1 that1 in corresponding parts of the nervous system of the two animals1 there are more than #ae1jjj times as many individual cells in one as in the other4 ,in short we may 7with ,enriques7 lay it down as a general law that among animals1 whether large or small1 the ganglion-cells vary in size within narrow limits2 and that1 amidst all the great variety of structural type of ganglion observed in different classes of animals1 it is always found that the smaller species have simpler ganglia than the larger1 that is to say ganglia containing a smaller number of cellular elements#fc4 ,the bearing of such simple facts as this upon the cell-theory in general is not to be disregarded2 and the warning is especially clear against exaggerated attempts to correlate physiological processes with the visible mechanism of associated cells1 rather than with the system of energies1 or the field of force1 which is associated with them4 ,for the life of #ch the body is more than the sum of the properties of the cells of which it is composed3 as ,goethe said1 8,das ,lebendige ist zwar in ,elemente zerlegt1 aber man kann es aus diesen nicht wieder zusammenstellen und beleben40 ,among certain lower and microscopic organisms1 such for instance as the ,rotifera1 we are still more palpably struck by the small number of cells which go to constitute a usually complex organ1 such as kidney1 stomach1 ovary1 etc4 ,we can sometimes number them in a few units1 in place of the thousands that make up such an organ in larger1 if not always higher1 animals4 ,these facts constitute one among many arguments which combine to teach us that1 however important and advantageous the subdivision of organisms into cells may be from the constructional1 or from the dynamical point of view1 the phenomenon has less essential importance in theoretical biology than was once1 and is often still1 assigned to it4 ,again1 just as ,sachs shewed that there was a limit to the amount of cytoplasm which could gather round a single nucleus1 so ,boveri has demonstrated that the nucleus itself has definite limitations of size1 and that1 in cell-division after fertilisation1 each new nucleus has the same size as its parent-nucleus#fd4 ,in all these cases1 then1 there are reasons1 partly no doubt physiological1 but in very large part purely physical1 which set limits to the normal magnitude of the organism or of the cell4 ,but as we have already discussed the existence of absolute and definite limitations1 of a physical kind1 to the possible increase in magnitude of an organism1 let us now enquire whether there be not also a lower limit1 below which the very existence of an organism is impossible1 or at least where1 under changed conditions1 its very nature must be profoundly modified4 ,among the smallest of known organisms we have1 for instance1 ,micromonas mesnili1 ,bonel1 a flagellate infusorian1 which measures about #cd 1 or #jjjcd mm41 by #jjjbe mm42 smaller even than this we have a pathogenic micrococcus of the rabbit1 ,m4 progrediens1 ,schrter1 the diameter of which is said to be only #jjjae mm4 or #ae 1 or #ae #aje cm41--about equal to the thickness of #ci the thinnest gold-leaf2 and as small if not smaller still are a few bacteria and their spores4 ,but here we have reached1 or all but reached the utmost limits of ordinary microscopic vision2 and there remain still smaller organisms1 the so-called 8filter-passers10 which the ultra-microscope reveals1 but which are mainly brought within our ken only by the maladies1 such as hydrophobia1 foot-and-mouth disease1 or the 8mosaic0 disease of the tobacco-plant1 to which these invisible micro-organisms give rise#fe4 ,accordingly1 since it is only by the diseases which they occasion that these tiny bodies are made known to us1 we might be tempted to suppose that innumerable other invisible organisms1 smaller and yet smaller1 exist unseen and unrecognised by man4 ,illustration3 ,fig4 #b4 ,relative magnitudes of3 ,a1 human blood-corpuscle 7#ge in diameter72 ,b1 ,bacillus anthracis 7#d - #ae #a 72 ,c1 various ,micrococci 7diam4 #je - #a 1 rarely #b 72 ,d1 ,micromonas progrediens1 ,schrter 7diam4 #jae 74 ,to illustrate some of these small magnitudes ,i have adapted the preceding diagram from one given by ,zsigmondy#ff4 ,upon the #dj same scale the minute ultramicroscopic particles of colloid gold would be represented by the finest dots which we could make visible to the naked eye upon the paper4 ,a bacillus of ordinary1 typical size is1 say1 #a in length4 ,the length 7or height7 of a man is about a million and three-quarter times as great1 i4e4 #age metres1 or #age #ajf 2 and the mass of the man is in the neighbourhood of five million1 million1 million 7#e #ajah7 times greater than that of the bacillus4 ,if we ask whether there may not exist organisms as much less than the bacillus as the bacillus is less than the dimensions of a man1 it is very easy to see that this is quite impossible1 for we are rapidly approaching a point where the question of molecular dimensions1 and of the ultimate divisibility of matter1 begins to call for our attention1 and to obtrude itself as a crucial factor in the case4 ,clerk ,maxwell dealt with this matter in his article 8,atom#fg10 and1 in somewhat greater detail1 ,errera discusses the question on the following lines#fh4 ,the weight of a hydrogen molecule is1 according to the physical chemists1 somewhere about #hf #b #ajbb milligrammes2 and that of any other element1 whose molecular weight is ,m1 is given by the equation 7,m7 #hf ,m #ajbb4 ,accordingly1 the weight of the atom of sulphur may be taken as #hf #cb #ajbb mgm4 #bge #ajbb mgm4 ,the analysis of ordinary bacteria shews them to consist#fi of about #he of water1 and #ae of solids2 while the solid residue of vegetable protoplasm contains about one part in a thousand of sulphur4 ,we may assume1 therefore1 that the living protoplasm contains about #a_/ajjj #ae_/ajj #ae #aje parts of sulphur1 taking the total weight as #a4 ,but our little micrococcus1 of #jae in diameter1 would1 if it were spherical1 have a volume of _/#f #jaec #ah #ajd cubic microns2 #da and therefore 7taking its density as equal to that of water71 a weight of #ah #ajd #aji #ah #ajac mgm4 ,but of this total weight1 the sulphur represents only #ah #ajac #ae #aje #bg #ajag mgm4 ,and if we divide this by the weight of an atom of sulphur1 we have 7#bg #ajag7 7#bge #ajbb7 #aj1jjj1 or thereby4 ,according to this estimate1 then1 our little ,micrococcus progrediens should contain only about #aj1jjj atoms of sulphur1 an element indispensable to its protoplasmic constitution2 and it follows that an organism of one-tenth the diameter of our micrococcus would only contain #aj sulphur-atoms1 and therefore only ten chemical 8molecules0 or units of protoplasm6 ,it may be open to doubt whether the presence of sulphur be really essential to the constitution of the proteid or 8protoplasmic0 molecule2 but ,errera gives us yet another illustration of a similar kind1 which is free from this objection or dubiety4 ,the molecule of albumin1 as is generally agreed1 can scarcely be less than a thousand times the size of that of such an element as sulphur3 according to one particular determination#gj1 serum albumin has a constitution corresponding to a molecular weight of #aj1aff1 and even this may be far short of the true complexity of a typical albuminoid molecule4 ,the weight of such a molecule is #hf #ajaff #ajbb #hg #ajah mgm4 ,now the bacteria contain about #ad of albuminoids1 these constituting by far the greater part of the dry residue2 and therefore 7from equation 7#e771 the weight of albumin in our micrococcus is about #ad_/ajj #ah #ajac #be #ajac mgm4 ,if we divide this weight by that which we have arrived at as the weight of an albumin molecule1 we have 7#be #ajac7 7#hg #ajah7 #bi #ajd1 in other words1 our micrococcus apparently contains something less than #cj1jjj molecules of albumin4 #db ,according to the most recent estimates1 the weight of the hydrogen molecule is somewhat less than that on which ,errera based his calculations1 namely about #af #ajbb mgms4 and according to this value1 our micrococcus would contain just about #bg1jjj albumin molecules4 ,in other words1 whichever determination we accept1 we see that an organism one-tenth as large as our micrococcus1 in linear dimensions1 would only contain some thirty molecules of albumin2 or1 in other words1 our micrococcus is only about thirty times as large1 in linear dimensions1 as a single albumin molecule#ga4 ,we must doubtless make large allowances for uncertainty in the assumptions and estimates upon which these calculations are based2 and we must also remember that the data with which the physicist provides us in regard to molecular magnitudes are1 to a very great extent1 maximal values1 above which the molecular magnitude 7or rather the sphere of the molecule's range of motion7 is not likely to lie3 but below which there is a greater element of uncertainty as to its possibly greater minuteness4 ,but nevertheless1 when we shall have made all reasonable allowances for uncertainty upon the physical side1 it will still be clear that the smallest known bodies which are described as organisms draw nigh towards molecular magnitudes1 and we must recognise that the subdivision of the organism cannot proceed to an indefinite extent1 and in all probability cannot go very much further than it appears to have done in these already discovered forms4 ,for1 even1 after giving all due regard to the complexity of our unit 7that is to say the albumin-molecule71 with all the increased possibilities of interrelation with its neighbours which this complexity implies1 we cannot but see that physiologically1 and comparatively speaking1 we have come down to a very simple thing4 ,while such considerations as these1 based on the chemical composition of the organism1 teach us that there must be a definite lower limit to its magnitude1 other considerations of a purely physical kind lead us to the same conclusion4 ,for our discussion of the principle of similitude has already taught us that1 long before we reach these almost infinitesimal magnitudes1 the #dc diminishing organism will have greatly changed in all its physical relations1 and must at length arrive under conditions which must surely be incompatible with anything such as we understand by life1 at least in its full and ordinary development and manifestation4 ,we are told1 for instance1 that the powerful force of surface-tension1 or capillarity1 begins to act within a range of about #a_/ejj1jjj of an inch1 or say #jje 4 ,a soap-film1 or a film of oil upon water1 may be attenuated to far less magnitudes than this2 the black spots upon a soap-bubble are known1 by various concordant methods of measurement1 to be only about #f #ajg cm41 or about #jjf thick1 and ,lord ,rayleigh and ,m4 ,devaux#gb have obtained films of oil of #jjb 1 or even #jja in thickness4 ,but while it is possible for a fluid film to exist in these almost molecular dimensions1 it is certain that1 long before we reach them1 there must arise new conditions of which we have little knowledge and which it is not easy even to imagine4 ,it would seem that1 in an organism of #a in diameter1 or even rather more1 there can be no essential distinction between the interior and the surface layers4 ,no hollow vesicle1 ,i take it1 can exist of these dimensions1 or at least1 if it be possible for it to do so1 the contained gas or fluid must be under pressures of a formidable kind#gc1 and of which we have no knowledge or experience4 ,nor1 ,i imagine1 can there be any real complexity1 or heterogeneity1 of its fluid or semi-fluid contents2 there can be no vacuoles within such a cell1 nor any layers defined within its fluid substance1 for something of the nature of a boundary-film is the necessary condition of the existence of such layers4 ,moreover1 the whole organism1 provided that it be fluid or semi-fluid1 can only be spherical in form4 ,what1 then1 can we attribute1 in the way of properties1 to an organism of a size as small as1 or smaller than1 say #je 8 ,it must1 in all probability1 be a homogeneous1 structureless sphere1 composed of a very small number of albuminoid or other molecules4 ,its vital properties and functions must be extraordinarily limited2 its specific outward characters1 even if we could see it1 must be nil2 and its specific properties must be little more than those of an ion-laden corpuscle1 enabling it to perform #dd this or that chemical reaction1 or to produce this or that pathogenic effect4 ,even among inorganic1 non-living bodies1 there must be a certain grade of minuteness at which the ordinary properties become modified4 ,for instance1 while under ordinary circumstances crystallisation starts in a solution about a minute solid fragment or crystal of the salt1 ,ostwald has shewn that we may have particles so minute that they fail to serve as a nucleus for crystallisation1--which is as much as to say that they are too minute to have the form and properties of a 8crystal02 and again1 in his thin oil-films1 ,lord ,rayleigh has noted the striking change of physical properties which ensues when the film becomes attenuated to something less than one close-packed layer of molecules#gd4 ,thus1 as ,clerk ,maxwell put it1 8molecular science sets us face to face with physiological theories4 ,it forbids the physiologist from imagining that structural details of infinitely small dimensions such as ,leibniz assumed1 one within another1 ad infinitum can furnish an explanation of the infinite variety which exists in the properties and functions of the most minute organisms40 ,and for this reason he reprobates1 with not undue severity1 those advocates of pangenesis and similar theories of heredity1 who would place 8a whole world of wonders within a body so small and so devoid of visible structure as a germ40 ,but indeed it scarcely needed ,maxwell's criticism to shew forth the immense physical difficulties of ,darwin's theory of ,pangenesis3 which1 after all1 is as old as ,democritus1 and is no other than that ,promethean particulam undique desectam of which we have read1 and at which we have smiled1 in our ,horace4 ,there are many other ways in which1 when we 8make a long excursion into space10 we find our ordinary rules of physical behaviour entirely upset4 ,a very familiar case1 analysed by ,stokes1 is that the viscosity of the surrounding medium has a relatively powerful effect upon bodies below a certain size4 ,a droplet of water1 a thousandth of an inch 7#be 7 in diameter1 cannot fall in still air quicker than about an inch and a half per second2 and as its size decreases1 its resistance varies as the diameter1 and not 7as with larger bodies7 as the surface of the #de drop4 ,thus a drop one-tenth of that size 7#be 71 the size1 apparently1 of the drops of water in a light cloud1 will fall a hundred times slower1 or say an inch a minute2 and one again a tenth of this diameter 7say #be 1 or about twice as big1 in linear dimensions1 as our micrococcus71 will scarcely fall an inch in two hours4 ,by reason of this principle1 not only do the smaller bacteria fall very slowly through the air1 but all minute bodies meet with great proportionate resistance to their movements in a fluid4 ,even such comparatively large organisms as the diatoms and the foraminifera1 laden though they are with a heavy shell of flint or lime1 seem to be poised in the water of the ocean1 and fall in it with exceeding slowness4 ,the ,brownian movement has also to be reckoned with1--that remarkable phenomenon studied nearly a century ago 7#ahbg7 by ,robert ,brown1 facile princeps botanicorum4 ,it is one more of those fundamental physical phenomena which the biologists have contributed1 or helped to contribute1 to the science of physics4 ,the quivering motion1 accompanied by rotation1 and even by translation1 manifested by the fine granular particles issuing from a crushed pollen-grain1 and which ,robert ,brown proved to have no vital significance but to be manifested also by all minute particles whatsoever1 organic and inorganic1 was for many years unexplained4 ,nearly fifty years after ,brown wrote1 it was said to be 8due1 either directly to some calorical changes continually taking place in the fluid1 or to some obscure chemical action between the solid particles and the fluid which is indirectly promoted by heat#ge40 ,very shortly after these last words were written1 it was ascribed by ,wiener to molecular action1 and we now know that it is indeed due to the impact or bombardment of molecules upon a body so small that these impacts do not for the moment1 as it were1 8average out0 to approximate equality on all sides4 ,the movement becomes manifest with particles of somewhere about #bj in diameter1 it is admirably displayed by particles of about #ab in diameter1 and becomes more marked the smaller the particles are4 ,the bombardment causes our particles to behave just like molecules of uncommon size1 and this #df behaviour is manifested in several ways#gf4 ,firstly1 we have the quivering movement of the particles2 secondly1 their movement backwards and forwards1 in short1 straight1 disjointed paths2 thirdly1 the particles rotate1 and do so the more rapidly the smaller they are1 and by theory1 confirmed by observation1 it is found that particles of #a in diameter rotate on an average through #ajj per second1 while particles of #ac in diameter turn through only #ad per minute4 ,lastly1 the very curious result appears1 that in a layer of fluid the particles are not equally distributed1 nor do they all ever fall1 under the influence of gravity1 to the bottom4 ,but just as the molecules of the atmosphere are so distributed1 under the influence of gravity1 that the density 7and therefore the number of molecules per unit volume7 falls off in geometrical progression as we ascend to higher and higher layers1 so is it with our particles1 even within the narrow limits of the little portion of fluid under our microscope4 ,it is only in regard to particles of the simplest form that these phenomena have been theoretically investigated#gg1 and we may take it as certain that more complex particles1 such as the twisted body of a ,spirillum1 would show other and still more complicated manifestations4 ,it is at least clear that1 just as the early microscopists in the days before ,robert ,brown never doubted but that these phenomena were purely vital1 so we also may still be apt to confuse1 in certain cases1 the one phenomenon with the other4 ,we cannot1 indeed1 without the most careful scrutiny1 decide whether the movements of our minutest organisms are intrinsically 8vital0 7in the sense of being beyond a physical mechanism1 or working model7 or not4 ,for example1 ,schaudinn has suggested that the undulating movements of ,spirochaete pallida must be due to the presence of a minute1 unseen1 8undulating membrane02 and ,doflein says of the same species that 8sie verharrt oft mit eigenthmlich zitternden ,bewegungen zu einem ,orte40 ,both movements1 the trembling or quivering #dg movement described by ,doflein1 and the undulating or rotating movement described by ,schaudinn1 are just such as may be easily and naturally interpreted as part and parcel of the ,brownian phenomenon4 ,while the ,brownian movement may thus simulate in a deceptive way the active movements of an organism1 the reverse statement also to a certain extent holds good4 ,one sometimes lies awake of a summer's morning watching the flies as they dance under the ceiling4 ,it is a very remarkable dance4 ,the dancers do not whirl or gyrate1 either in company or alone2 but they advance and retire2 they seem to jostle and rebound2 between the rebounds they dart hither or thither in short straight snatches of hurried flight2 and turn again sharply in a new rebound at the end of each little rush4 ,their motions are wholly 8erratic10 independent of one another1 and devoid of common purpose4 ,this is nothing else than a vastly magnified picture1 or simulacrum1 of the ,brownian movement2 the parallel between the two cases lies in their complete irregularity1 but this in itself implies a close resemblance4 ,one might see the same thing in a crowded market-place1 always provided that the bustling crowd had no business whatsoever4 ,in like manner ,lucretius1 and ,epicurus before him1 watched the dust-motes quivering in the beam1 and saw in them a mimic representation1 rei simulacrum et imago1 of the eternal motions of the atoms4 ,again the same phenomenon may be witnessed under the microscope1 in a drop of water swarming with ,paramoecia or suchlike ,infusoria2 and here the analogy has been put to a numerical test4 ,following with a pencil the track of each little swimmer1 and dotting its place every few seconds 7to the beat of a metronome71 ,karl ,przibram found that the mean successive distances from a common base-line obeyed with great exactitude the 8,einstein formula10 that is to say the particular form of the 8law of chance0 which is applicable to the case of the ,brownian movement#gh4 ,the phenomenon is 7of course7 merely analogous1 and by no means identical with the ,brownian movement2 for the range of motion of the little active organisms1 whether they be gnats or infusoria1 is vastly greater than that of the minute particles which are #dh passive under bombardment2 but nevertheless ,przibram is inclined to think that even his comparatively large infusoria are small enough for the molecular bombardment to be a stimulus1 though not the actual cause1 of their irregular and interrupted movements4 ,there is yet another very remarkable phenomenon which may come into play in the case of the minutest of organisms2 and this is their relation to the rays of light1 as ,arrhenius has told us4 ,on the waves of a beam of light1 a very minute particle 7in vacuo7 should be actually caught up1 and carried along with an immense velocity2 and this 8radiant pressure0 exercises its most powerful influence on bodies which 7if they be of spherical form7 are just about #jjjaf mm41 or #af in diameter4 ,this is just about the size1 as we have seen1 of some of our smallest known protozoa and bacteria1 while we have some reason to believe that others yet unseen1 and perhaps the spores of many1 are smaller still4 ,now we have seen that such minute particles fall with extreme slowness in air1 even at ordinary atmospheric pressures3 our organism measuring #af would fall but #hc metres in a year1 which is as much as to say that its weight offers practically no impediment to its transference1 by the slightest current1 to the very highest regions of the atmosphere4 ,beyond the atmosphere1 however1 it cannot go1 until some new force enable it to resist the attraction of terrestrial gravity1 which the viscosity of an atmosphere is no longer at hand to oppose4 ,but it is conceivable that our particle may go yet farther1 and actually break loose from the bonds of earth4 ,for in the upper regions of the atmosphere1 say fifty miles high1 it will come in contact with the rays and flashes of the ,northern ,lights1 which consist 7as ,arrhenius maintains7 of a fine dust1 or cloud of vapour-drops1 laden with a charge of negative electricity1 and projected outwards from the sun4 ,as soon as our particle acquires a charge of negative electricity it will begin to be repelled by the similarly laden auroral particles1 and the amount of charge necessary to enable a particle of given size 7such as our little monad of #af 7 to resist the attraction of gravity may be calculated1 and is found to be such as the actual conditions can easily supply4 ,finally1 when once set free from the entanglement of the earth's #di atmosphere1 the particle may be propelled by the 8radiant pressure0 of light1 with a velocity which will carry it4--like ,uriel gliding on a sunbeam1--as far as the orbit of ,mars in twenty days1 of ,jupiter in eighty days1 and as far as the nearest fixed star in three thousand years6 ,this1 and much more1 is ,arrhenius's contribution towards the acceptance of ,lord ,kelvin's hypothesis that life may be1 and may have been1 disseminated across the bounds of space1 throughout the solar system and the whole universe6 ,it may well be that we need attach no great practical importance to this bold conception2 for even though stellar space be shewn to be mare liberum to minute material travellers1 we may be sure that those which reach a stellar or even a planetary bourne are infinitely1 or all but infinitely1 few4 ,but whether or no1 the remote possibilities of the case serve to illustrate in a very vivid way the profound differences of physical property and potentiality which are associated in the scale of magnitude with simple differences of degree4 #ej ,,chapter ,,iii ,,the ,,rate ,,of ,,growth ,when we study magnitude by itself1 apart1 that is to say1 from the gradual changes to which it may be subject1 we are dealing with a something which may be adequately represented by a number1 or by means of a line of definite length2 it is what mathematicians call a scalar phenomenon4 ,when we introduce the conception of change of magnitude1 of magnitude which varies as we pass from one direction to another in space1 or from one instant to another in time1 our phenomenon becomes capable of representation by means of a line of which we define both the length and the direction2 it is 7in this particular aspect7 what is called a vector phenomenon4 ,when we deal with magnitude in relation to the dimensions of space1 the vector diagram which we draw plots magnitude in one direction against magnitude in another1--length against height1 for instance1 or against breadth2 and the result is simply what we call a picture or drawing of an object1 or 7more correctly7 a 8plane projection0 of the object4 ,in other words1 what we call ,form is a ratio of magnitudes1 referred to direction in space4 ,when in dealing with magnitude we refer its variations to successive intervals of time 7or when1 as it is said1 we equate it with time71 we are then dealing with the phenomenon of growth2 and it is evident1 therefore1 that this term growth has wide meanings4 ,for growth may obviously be positive or negative2 that is to say1 a thing may grow larger or smaller1 greater or less2 and by extension of the primitive concrete signification of the word1 we easily and legitimately apply it to non-material things1 such as temperature1 and say1 for instance1 that a body 8grows0 hot or cold4 ,when in a two-dimensional diagram1 we represent a magnitude 7for instance length7 in relation to time 7or 8plot0 #ea length against time1 as the phrase is71 we get that kind of vector diagram which is commonly known as a 8curve of growth40 ,we perceive1 accordingly1 that the phenomenon which we are now studying is a velocity 7whose 8dimensions0 are ,space_/,time or ,l_/,t72 and this phenomenon we shall speak of1 simply1 as a rate of growth4 ,in various conventional ways we can convert a two-dimensional into a three-dimensional diagram4 ,we do so1 for example1 by means of the geometrical method of 8perspective0 when we represent upon a sheet of paper the length1 breadth and depth of an object in three-dimensional space2 but we do it more simply1 as a rule1 by means of 8contour-lines10 and always when time is one of the dimensions to be represented4 ,if we superimpose upon one another 7or even set side by side7 pictures1 or plane projections1 of an organism1 drawn at successive intervals of time1 we have such a three-dimensional diagram1 which is a partial representation 7limited to two dimensions of space7 of the organism's gradual change of form1 or course of development2 and in such a case our contour-lines may1 for the purposes of the embryologist1 be separated by intervals representing a few hours or days1 or1 for the purposes of the palaeontologist1 by interspaces of unnumbered and innumerable years#gi4 ,such a diagram represents in two of its three dimensions form1 and in two1 or three1 of its dimensions growth2 and so we see how intimately the two conceptions are correlated or inter-related to one another4 ,in short1 it is obvious that the form of an animal is determined by its specific rate of growth in various directions2 accordingly1 the phenomenon of rate of growth deserves to be studied as a necessary preliminary to the theoretical study of form1 and1 mathematically speaking1 organic form itself appears to us as a function of time#hj4 #eb ,at the same time1 we need only consider this part of our subject somewhat briefly4 ,though it has an essential bearing on the problems of morphology1 it is in greater degree involved with physiological problems2 and furthermore1 the statistical or numerical aspect of the question is peculiarly adapted for the mathematical study of variation and correlation4 ,on these important subjects we shall scarcely touch2 for our main purpose will be sufficiently served if we consider the characteristics of a rate of growth in a few illustrative cases1 and recognise that this rate of growth is a very important specific property1 with its own characteristic value in this organism or that1 in this or that part of each organism1 and in this or that phase of its existence4 ,the statement which we have just made that 8the form of an organism is determined by its rate of growth in various directions10 is one which calls 7as we have partly seen in the foregoing chapter7 for further explanation and for some measure of qualification4 ,among organic forms we shall have frequent occasion to see that form is in many cases due to the immediate or direct action of certain molecular forces1 of which surface-tension is that which plays the greatest part4 ,now when surface-tension 7for instance7 causes a minute semi-fluid organism to assume a spherical form1 or gives the form of a catenary or an elastic curve to a film of protoplasm in contact with some solid skeletal rod1 or when it acts in various other ways which are productive of definite contours1 this is a process of conformation that1 both in appearance and reality1 is very different from the process by which an ordinary plant or animal grows into its specific form4 ,in both cases1 change of form is brought about by the movement of portions of matter1 and in both cases it is ultimately due to the action of molecular forces2 but in the one case the movements of the particles of matter lie for the most part within molecular range1 while in the other we have to deal chiefly with the transference of portions of matter into the system from without1 and from one widely distant part of the organism to another4 ,it is to this latter class of phenomena that we usually restrict the term growth2 and it is in regard to them that we are in a position to study the rate of action in different directions1 and to see that it is merely on a difference of velocities that the modification of form essentially depends4 #ec ,the difference between the two classes of phenomena is somewhat akin to the difference between the forces which determine the form of a rain-drop and those which1 by the flowing of the waters and the sculpturing of the solid earth1 have brought about the complex configuration of a river2 molecular forces are paramount in the conformation of the one1 and molar forces are dominant in the other4 ,at the same time it is perfectly true that all changes of form1 inasmuch as they necessarily involve changes of actual and relative magnitude1 may1 in a sense1 be properly looked upon as phenomena of growth2 and it is also true1 since the movement of matter must always involve an element of time#ha1 that in all cases the rate of growth is a phenomenon to be considered4 ,even though the molecular forces which play their part in modifying the form of an organism exert an action which is1 theoretically1 all but instantaneous1 that action is apt to be dragged out to an appreciable interval of time by reason of viscosity or some other form of resistance in the material4 ,from the physical or physiological point of view the rate of action even in such cases may be well worth studying2 for example1 a study of the rate of cell-division in a segmenting egg may teach us something about the work done1 and about the various energies concerned4 ,but in such cases the action is1 as a rule1 so homogeneous1 and the form finally attained is so definite and so little dependent on the time taken to effect it1 that the specific rate of change1 or rate of growth1 does not enter into the morphological problem4 ,to sum up1 we may lay down the following general statements4 ,the form of organisms is a phenomenon to be referred in part to the direct action of molecular forces1 in part to a more complex and slower process1 indirectly resulting from chemical1 osmotic and other forces1 by which material is introduced into the organism and transferred from one part of it to another4 ,it is this latter complex phenomenon which we usually speak of as 8growth40 #ed ,every growing organism1 and every part of such a growing organism1 has its own specific rate of growth1 referred to a particular direction4 ,it is the ratio between the rates of growth in various directions by which we must account for the external forms of all1 save certain very minute1 organisms4 ,this ratio between rates of growth in various directions may sometimes be of a simple kind1 as when it results in the mathematically definable outline of a shell1 or in the smooth curve of the margin of a leaf4 ,it may sometimes be a very constant one1 in which case the organism1 while growing in bulk1 suffers little or no perceptible change in form2 but such equilibrium seldom endures for more than a season1 and when the ratio tends to alter1 then we have the phenomenon of morphological 8development10 or steady and persistent change of form4 ,this elementary concept of ,form1 as determined by varying rates of ,growth1 was clearly apprehended by the mathematical mind of ,haller1--who had learned his mathematics of the great ,john ,bernoulli1 as the latter in turn had learned his physiology from the writings of ,borelli4 ,indeed it was this very point1 the apparently unlimited extent to which1 in the development of the chick1 inequalities of growth could and did produce changes of form and changes of anatomical 8structure10 that led ,haller to surmise that the process was actually without limits1 and that all development was but an unfolding1 or 8evolutio10 in which no part came into being which had not essentially existed before#hb4 ,in short the celebrated doctrine of 8preformation0 implied on the one hand a clear recognition of what1 throughout the later stages of development1 growth can do1 by hastening the increase in size of one part1 hindering that of another1 changing their relative magnitudes and positions1 and altering their forms2 while on the other hand it betrayed a failure 7inevitable in those days7 to recognise the essential difference between these movements of masses and the molecular processes which precede and accompany #ee them1 and which are characteristic of another order of magnitude4 ,by other writers besides ,haller the very general1 though not strictly universal connection between form and rate of growth has been clearly recognised4 ,such a connection is implicit in those 8proportional diagrams0 by which ,drer and some of his brother artists were wont to illustrate the successive changes of form1 or of relative dimensions1 which attend the growth of the child1 to boyhood and to manhood4 ,the same connection was recognised1 more explicitly1 by some of the older embryologists1 for instance by ,pander#hc1 and appears1 as a survival of the doctrine of preformation1 in his study of the development of the chick4 ,and long afterwards1 the embryological aspect of the case was emphasised by ,his1 who pointed out1 for instance1 that the various foldings of the blastoderm1 by which the neural and amniotic folds were brought into being1 were essentially and obviously the resultant of unequal rates of growth1--of local accelerations or retardations of growth1--in what to begin with was an even and uniform layer of embryonic tissue4 ,if we imagine a flat sheet of paper1 parts of which are caused 7as by moisture or evaporation7 to expand or to contract1 the plane surface is at once dimpled1 or 8buckled10 or folded1 by the resultant forces of expansion or contraction3 and the various distortions to which the plane surface of the 8germinal disc0 is subject1 as ,his shewed once and for all1 are precisely analogous4 ,an experimental demonstration still more closely comparable to the actual case of the blastoderm1 is obtained by making an 8artificial blastoderm10 of little pills or pellets of dough1 which are caused to grow1 with varying velocities1 by the addition of varying quantities of yeast4 ,here1 as ,roux is careful to point out#hd1 we observe that it is not only the growth of the individual cells1 but the traction exercised through their mutual interconnections1 which brings about the foldings and other distortions of the entire structure4 #ef ,but this again was clearly present to ,haller's mind1 and formed an essential part of his embryological doctrine4 ,for he has no sooner treated of incrementum1 or celeritas incrementi1 than he proceeds to deal with the contributory and complementary phenomena of expansion1 traction 7adtractio7#he1 and pressure1 and the more subtle influences which he denominates vis derivationis et revulsionis#hf3 these latter being the secondary and correlated effects on growth in one part1 brought about1 through such changes as are produced 7for instance7 in the circulation1 by the growth of another4 ,let us admit that1 on the physiological side1 ,haller's or ,his's methods of explanation carry us back but a little way2 yet even this little way is something gained4 ,nevertheless1 ,i can well remember the harsh criticism1 and even contempt1 which ,his's doctrine met with1 not merely on the ground that it was inadequate1 but because such an explanation was deemed wholly inappropriate1 and was utterly disavowed#hg4 ,hertwig1 for instance1 asserted that1 in embryology1 when we found one embryonic stage preceding another1 the existence of the former was1 for the embryologist1 an all-sufficient 8causal explanation0 of the latter4 8,we consider 7he says71 that we are studying and explaining a causal relation when we have demonstrated that the gastrula arises by invagination of a blastosphere1 or the neural canal by the infolding of a cell plate so as to constitute a tube#hh40 ,for ,hertwig1 therefore1 as #eg ,roux remarks1 the task of investigating a physical mechanism in embryology1--8der ,ziel das ,wirken zu erforschen10--has no existence at all4 ,for ,balfour also1 as for ,hertwig1 the mechanical or physical aspect of organic development had little or no attraction4 ,in one notable instance1 ,balfour himself adduced a physical1 or quasi-physical1 explanation of an organic process1 when he referred the various modes of segmentation of an ovum1 complete or partial1 equal or unequal and so forth1 to the varying amount or the varying distribution of food yolk in association with the germinal protoplasm of the egg#hi4 ,but in the main1 ,balfour1 like all the other embryologists of his day1 was engrossed by the problems of phylogeny1 and he expressly defined the aims of comparative embryology 7as exemplified in his own textbook7 as being 8twofold3 7#a7 to form a basis for ,phylogeny4 and 7#b7 to form a basis for ,organogeny or the origin and evolution of organs#ij40 ,it has been the great service of ,roux and his fellow-workers of the school of 8,entwickelungsmechanik10 and of many other students to whose work we shall refer1 to try1 as ,his tried#ia to import into embryology1 wherever possible1 the simpler concepts of physics1 to introduce along with them the method of experiment1 and to refuse to be bound by the narrow limitations which such teaching as that of ,hertwig would of necessity impose on the work and the thought and on the whole philosophy of the biologist4 ,before we pass from this general discussion to study some of the particular phenomena of growth1 let me give a single illustration1 from ,darwin1 of a point of view which is in marked contrast to ,haller's simple but essentially mathematical conception of ,form4 ,there is a curious passage in the ,origin of ,species#ib1 where ,darwin is discussing the leading facts of embryology1 and in particular ,von ,baer's 8law of embryonic resemblance40 ,here ,darwin says 8,we are so much accustomed to see a difference in #eh structure between the embryo and the adult1 that we are tempted to look at this difference as in some necessary manner contingent on growth4 ,but there is no reason why1 for instance1 the wing of a bat1 or the fin of a porpoise1 should not have been sketched out with all their parts in proper proportion1 as soon as any part became visible40 ,after pointing out with his habitual care various exceptions1 ,darwin proceeds to lay down two general principles1 viz4 8that slight variations generally appear at a not very early period of life10 and secondly1 that 8at whatever age a variation first appears in the parent1 it tends to reappear at a corresponding age in the offspring40 ,he then argues that it is with nature as with the fancier1 who does not care what his pigeons look like in the embryo1 so long as the full-grown bird possesses the desired qualities2 and that the process of selection takes place when the birds or other animals are nearly grown up1--at least on the part of the breeder1 and presumably in nature as a general rule4 ,the illustration of these principles is set forth as follows2 8,let us take a group of birds1 descended from some ancient form and modified through natural selection for different habits4 ,then1 from the many successive variations having supervened in the several species at a not very early age1 and having been inherited at a corresponding age1 the young will still resemble each other much more closely than do the adults1--just as we have seen with the breeds of the pigeon 4444 ,whatever influence long-continued use or disuse may have had in modifying the limbs or other parts of any species1 this will chiefly or solely have affected it when nearly mature1 when it was compelled to use its full powers to gain its own living2 and the effects thus produced will have been transmitted to the offspring at a corresponding nearly mature age4 ,thus the young will not be modified1 or will be modified only in a slight degree1 through the effects of the increased use or disuse of parts40 ,this whole argument is remarkable1 in more ways than we need try to deal with here2 but it is especially remarkable that ,darwin should begin by casting doubt upon the broad fact that a 8difference in structure between the embryo and the adult0 is 8in some necessary manner contingent on growth02 and that he should see no reason why complicated structures of the adult 8should not have been sketched out #ei with all their parts in proper proportion1 as soon as any part became visible40 ,it would seem to me that even the most elementary attention to form in its relation to growth would have removed most of ,darwin's difficulties in regard to the particular phenomena which he is here considering4 ,for these phenomena are phenomena of form1 and therefore of relative magnitude2 and the magnitudes in question are attained by growth1 proceeding with certain specific velocities1 and lasting for certain long periods of time4 ,and it is accordingly obvious that in any two related individuals 7whether specifically identical or not7 the differences between them must manifest themselves gradually1 and be but little apparent in the young4 ,it is for the same simple reason that animals which are of very different sizes when adult1 differ less and less in size 7as well as in form7 as we trace them backwards through the foetal stages4 ,though we study the visible effects of varying rates of growth throughout wellnigh all the problems of morphology1 it is not very often that we can directly measure the velocities concerned4 ,but owing to the obvious underlying importance which the phenomenon has to the morphologist we must make shift to study it where we can1 even though our illustrative cases may seem to have little immediate bearing on the morphological problem#ic4 ,in a very simple organism1 of spherical symmetry1 such as the single spherical cell of ,protococcus or of ,orbulina1 growth is reduced to its simplest terms1 and indeed it becomes so simple in its outward manifestations that it is no longer of special interest to the morphologist4 ,the rate of growth is measured by the rate of change in length of a radius1 i4e4 ,v 7,r ,r7_/,t1 and from this we may calculate1 as already indicated1 the rate of growth in terms of surface and of volume4 ,the growing body remains of constant form1 owing to the symmetry of the system2 because1 that is to say1 on the one hand the pressure exerted by the growing protoplasm is exerted equally in all directions1 after the manner of a hydrostatic pressure1 which indeed it actually is3 while on the other hand1 the 8skin0 or surface layer of the cell is sufficiently #fj homogeneous to exert at every point an approximately uniform resistance4 ,under these conditions then1 the rate of growth is uniform in all directions1 and does not affect the form of the organism4 ,but in a larger or a more complex organism the study of growth1 and of the rate of growth1 presents us with a variety of problems1 and the whole phenomenon becomes a factor of great morphological importance4 ,we no longer find that it tends to be uniform in all directions1 nor have we any right to expect that it should4 ,the resistances which it meets with will no longer be uniform4 ,in one direction but not in others it will be opposed by the important resistance of gravity2 and within the growing system itself all manner of structural differences will come into play1 setting up unequal resistances to growth by the varying rigidity or viscosity of the material substance in one direction or another4 ,at the same time1 the actual sources of growth1 the chemical and osmotic forces which lead to the intussusception of new matter1 are not uniformly distributed2 one tissue or one organ may well manifest a tendency to increase while another does not2 a series of bones1 their intervening cartilages1 and their surrounding muscles1 may all be capable of very different rates of increment4 ,the differences of form which are the resultants of these differences in rate of growth are especially manifested during that part of life when growth itself is rapid3 when the organism1 as we say1 is undergoing its development4 ,when growth in general has become slow1 the relative differences in rate between different parts of the organism may still exist1 and may be made manifest by careful observation1 but in many1 or perhaps in most cases1 the resultant change of form does not strike the eye4 ,great as are the differences between the rates of growth in different parts of an organism1 the marvel is that the ratios between them are so nicely balanced as they actually are1 and so capable1 accordingly1 of keeping for long periods of time the form of the growing organism all but unchanged4 ,there is the nicest possible balance of forces and resistances in every part of the complex body2 and when this normal equilibrium is disturbed1 then we get abnormal growth1 in the shape of tumours1 exostoses1 and malformations of every kind4 #fa ,the rate of growth in ,man4 ,man will serve us as well as another organism for our first illustrations of rate of growth2 and we cannot do better than go for our first data concerning him to ,quetelet's ,anthropomtrie#id1 an epoch-making book for the biologist4 ,for not only is it packed with information1 some of it still unsurpassed1 in regard to human growth and form1 but it also merits our highest admiration as the first great essay in scientific statistics1 and the first work in which organic variation was discussed from the point of view of the mathematical theory of probabilities4 ,illustration3 ,fig4 #c4 ,curve of ,growth in ,man1 from birth to #bj yrs 772 from ,quetelet's ,belgian data4 ,the upper curve of stature from ,bowditch's ,boston data4 ,if the child be some #bj inches1 or say #ej cm4 tall at birth1 and the man some six feet high1 or say #ahj cm41 at twenty1 we may say that his average rate of growth has been 7#ahj #ej7_/bj cm41 or #fe centimetres per annum4 ,but we know very well that this is #fb but a very rough preliminary statement1 and that the boy grew quickly during some1 and slowly during other1 of his twenty years4 ,it becomes necessary therefore to study the phenomenon of growth in successive small portions2 to study1 that is to say1 the successive lengths1 or the successive small differences1 or increments1 of length 7or of weight1 etc471 attained in successive short increments of time4 ,this we do in the first instance in the usual way1 by the 8graphic method0 of plotting length against time1 and so constructing our 8curve of growth40 ,our curve of growth1 whether of weight or length 7,fig4 #c71 has always a certain characteristic form1 or characteristic curvature4 ,this is our immediate proof of the fact that the rate of growth changes as time goes on2 for had it not been so1 had an equal increment of length been added in each equal interval of time1 our 8curve0 would have appeared as a straight line4 ,such as it is1 it tells us not only that the rate of growth tends to alter1 but that it alters in a definite and orderly way2 for1 subject to various minor interruptions1 due to secondary causes1 our curves of growth are1 on the whole1 8smooth0 curves4 ,the curve of growth for length or stature in man indicates a rapid increase at the outset1 that is to say during the quick growth of babyhood2 a long period of slower1 but still rapid and almost steady growth in early boyhood2 as a rule a marked quickening soon after the boy is in his teens1 when he comes to 8the growing age02 and finally a gradual arrest of growth as the boy 8comes to his full height10 and reaches manhood4 ,if we carried the curve further1 we should see a very curious thing4 ,we should see that a man's full stature endures but for a spell2 long before fifty#ie it has begun to abate1 by sixty it is notably lessened1 in extreme old age the old man's frame is shrunken and it is but a memory that 8he once was tall40 ,we have already seen1 and here we see again1 that growth may have a 8negative value40 ,the phenomenon of negative growth in old age extends to weight also1 and is evidently largely chemical in origin3 the organism can no longer add new material to its fabric fast enough to keep pace with the wastage of time4 ,our curve #fc of growth is in fact a diagram of activity1 or 8time-energy0 diagram#if4 ,as the organism grows it is absorbing energy beyond its daily needs1 and accumulating it at a rate depicted in our ,stature1 weight1 and span of outstretched arms4 7,after ,quetelet1 pp4 #aic1 #cdf47 ,stature in metres ,weight in kgm4 ,span of ratio arms1 of stature ,age ,male ,female ,f_/,m ,male ,female ,f_/,m male to span #j #jejj #jdid #ihh #cb #bi #ijg #jdif #ajjh #a #jfih #jfij #ihh #id #hh #icf #jfie #ajjd #b #jgia #jgha #ihg #aac #ajg #idg #jghi #ajjc #c #jhfd #jhed #ihh #abd #aah #ieb #jhfc #ajja #d #jibg #jiae #ihg #adb #acj #iae #jibg #ajjj #e #jihg #jigd #ihg #aeh #add #iaa #jihh #iii #f #ajdf #ajca #ihe #agb #afj #icj #ajdh #iih #g #aajd #ajhg #ihd #aia #age #iaf #aajg #iig #h #aafb #aadb #ihb #bjh #aia #iah #aaff #iif #i #abah #aaif #ihb #bbf #bad #idg #abbd #iie #aj #abgc #abdi #iha #bde #bce #iei #abha #iid #aa #acbe #acja #ihb #bga #bef #ide #acce #iib #ab #acge #aceb #ihc #bih #bih #ajjj #achh #iia #ac #adbc #adjj #ihd #cdd #cbi #ief #adch #ihi #ad #adfi #addf #ihd #chh #cfg #idf #adhi #ihg #ae #aeac #adhh #ihc #dcf #djd #ibg #aech #iid #af #aeed #aeba #igh #dig #dcf #hgg #aehd #iha #ag #aeid #aedf #igj #ebh #dgc #hif #afcj #igi #ah #afcj #aefc #iei #egh #dij #hdh #afgj #igf #ai #afee #aegj #idi #ehj #eaf #hij #agje #iga #bj #affi #aegd #idc #fja #ebc #hgj #agbh #iff #be #afhb #aegh #ich #fbi #ecc #hdg #agca #igb #cj #afhf #aehj #icg #fcg #edc #hec #agff #iee #dj #afhf #aehj #icg #fcg #eeb #hfg #agff #iee #ej #afhf #aehj #icg #fce #efb #hhd -- -- #fj #afgf #aega #icg #fai #edc #hgg -- -- #gj #affj #aeef #icg #eie #eae #hfe -- -- #hj #afcf #aecd #ich #egh #did #hee -- -- #ij #afaj #aeaj #ich #egh #dic #hec -- -- curve2 but the time comes when it accumulates no longer1 and at last it is constrained to draw upon its dwindling store4 ,but in part1 the slow decline in stature is an expression of an unequal contest between our bodily powers and the unchanging force of gravity1 #fd which draws us down when we would fain rise up#ig4 ,for against gravity we fight all our days1 in every movement of our limbs1 in every beat of our hearts2 it is the indomitable force that defeats us in the end1 that lays us on our deathbed1 that lowers us to the grave#ih4 ,side by side with the curve which represents growth in length1 or stature1 our diagram shows the curve of weight#ii4 ,that this curve is of a very different shape from the former one1 is accounted for in the main 7though not wholly7 by the fact which we have already dealt with1 that1 whatever be the law of increment in a linear dimension1 the law of increase in volume1 and therefore in weight1 will be that these latter magnitudes tend to vary as the cubes of the linear dimensions4 ,this however does not account for the change of direction1 or 8point of inflection0 which we observe in the curve of weight at about one or two years old1 nor for certain other differences between our two curves which the scale of our diagram does not yet make clear4 ,these differences are due to the fact that the form of the child is altering with growth1 that other linear dimensions are varying somewhat differently from length or stature1 and that consequently the growth in bulk or weight is following a more complicated law4 ,our curve of growth1 whether for weight or length1 is a direct picture of velocity1 for it represents1 as a connected series1 the successive epochs of time at which successive weights or lengths are attained4 ,but1 as we have already in part seen1 a great part of the interest of our curve lies in the fact that we can see from it1 not only that length 7or some other magnitude7 is changing1 but that the rate of change of magnitude1 or rate of growth1 is itself changing4 ,we have1 in short1 to study the phenomenon of acceleration3 we have begun by studying a velocity1 or rate of #fe change of magnitude2 we must now study an acceleration1 or rate of change of velocity4 ,the rate1 or velocity1 of growth is measured by the slope of the curve2 where the curve is steep1 it means that growth is rapid1 and when growth ceases the curve appears as a horizontal line4 ,if we can find a means1 then1 of representing at successive epochs the corresponding slope1 or steepness1 of the curve1 we shall have obtained a picture of the rate of change of velocity1 or the acceleration of growth4 ,the measure of the steepness of a curve is given by the tangent to the curve1 or we may estimate it by taking for equal intervals of time 7strictly speaking1 for each infinitesimal interval of time7 the actual increment added during that interval of time3 and in practice this simply amounts to taking the successive differences between the values of length 7or of weight7 for the successive ages which we have begun by studying4 ,if we then plot these successive differences against time1 we obtain a curve each point upon which represents a velocity1 and the whole curve indicates the rate of change of velocity1 and we call it an acceleration-curve4 ,it contains1 in truth1 nothing whatsoever that was not implicit in our former curve2 but it makes clear to our eye1 and brings within the reach of further investigation1 phenomena that were hard to see in the other mode of representation4 ,the acceleration-curve of height1 which we here illustrate1 in ,fig4 #d1 is very different in form from the curve of growth which we have just been looking at2 and it happens that1 in this case1 there is a very marked difference between the curve which we obtain from ,quetelet's data of growth in height and that which we may draw from any other series of observations known to me from ,british1 ,french1 ,american or ,german writers4 ,it begins 7as will be seen from our next table7 at a very high level1 such as it never afterwards attains2 and still stands too high1 during the first three or four years of life1 to be represented on the scale of the accompanying diagram4 ,from these high velocities it falls away1 on the whole1 until the age when growth itself ceases1 and when the rate of growth1 accordingly1 has1 for some years together1 the constant value of nil2 but the rate of fall1 or rate of change of velocity1 is subject to several changes or interruptions4 ,during the first three or four years of life the fall is continuous and rapid1 #ff but it is somewhat arrested for a while in childhood1 from about five years old to eight4 ,according to ,quetelet's data1 there is another slight interruption in the falling rate between the ages of about fourteen and sixteen2 but in place of this almost insignificant interruption1 the ,english and other statistics indicate a sudden ,illustration3 ,fig4 #d4 ,mean annual increments of stature 771 ,belgian and ,american4 and very marked acceleration of growth beginning at about twelve years of age1 and lasting for three or four years2 when this period of acceleration is over1 the rate begins to fall again1 and does so with great rapidity4 ,we do not know how far the absence of this striking feature in the ,belgian curve is due to the imperfections of ,quetelet's data1 or whether it is a real and significant feature in the small-statured race which he investigated4 ,annual ,increment of ,stature 7in cm47 from ,belgian and ,american ,statistics4 ,d3 ,belgian 7,quetelet1 p4 #cdd7 ,e3 ,paris99 7,variot et ,chaumet1 p4 #ee7 ,f3 ,toronto 7,boas1 p4 #aedg7 ,g3 ,worcester1 ,mass4 7,boas1 p4 #aedh7 ,h3 ,ann4 increment ,i3 ,increment ,j3 ,boys ,k3 ,girls ,v3 ,variability of do4 ,d ,e ,f ,g ,height ,height ,i ,height ,v ,h ,h ,age 7,boys7 ,h ,j ,k ,j ,k 7,boys7 7#f7 ,h 7,j7 ,v 7,k7 ,v #j #ejj -- -- -- -- -- -- -- -- -- -- -- -- #a #fih #aih #gdb #gcf -- -- -- -- -- -- -- -- -- #b #gia #ic #hbg #hah #he #hb -- -- -- -- -- -- -- #c #hfd #gc #hia #hhd #fd #ff -- -- -- -- -- -- -- #d #ibg #fc #ifh #ieh #gg #gd -- -- -- -- -- -- -- #e #ihg #fj #ajcc #ajai #fe #fa #ajeij #ddj -- -- -- -- -- #f #ajdj #ei #ajii #ajhi #ff #gj #aaaeh #dfb #efh #fee #aeg #ege #jhh #g #aajd #eh #aadd #aach #de #di #aafhc #dic #ebe #egj #jfh #eij #jih #h #aafb #eh #aaig #aaie #ec #eg #abbjd #ecd #eba #ecg #jhf #egj #aaj #i #abah #ef #abej #abdg #ec #dh #abfia #edi #dhg #dhi #jif #eej #jig #aj #abgc #ee #acjc #abie #ec #eb #acagh #ege #dhg #eaj #ajc #eig #abc #aa #acbe #eb #accf #acdd #cc #di #acfbj #fai #ddb #ejb #jhh #fag #ahe #ab #acge #ej #acgf #adae #dj #ga #adjgd #fff #ded #dii #abf #fih #ahi #ac #adbc #dh #adea #adhf #ge #ga #adfjj #ged #ebf #eia #ahf #fga #bjf #ad #adfi #df #aech #aebi #hg #dc #aebci #hdi #fci #ghh #bci #edd #bhi #ae #aeac #dd #aeif #aedb #eh #ac #aeigb #hgh #gcc #fbc #bia #ecd #bga #af #aeed #da -- -- -- -- #afdij #ggc #eah #efd #cdf -- -- #ag #aeid #dj -- -- -- -- #afhia #gbb #dja -- -- -- -- #ah #afcj #cf -- -- -- -- #agajg #fgd #baf -- -- -- -- #ai #afee #be -- -- -- -- -- -- -- -- -- -- -- #bj #afgj #ae -- -- -- -- -- -- -- -- -- -- -- 99 ,ages from #a-b1 #b-c1 etc4 ,the epochs are1 in this table1 #ee1 #fe1 years1 etc4 ,direct observations on actual1 or individualised1 increase of stature from year to year3 between the ages of #e-f1 #f-g1 etc4 ,even apart from these data of ,quetelet's 7which seem to constitute an extreme case71 it is evident that there are very #fh marked differences between different races1 as we shall presently see there are between the two sexes1 in regard to the epochs of acceleration of growth1 in other words1 in the 8phase0 of the curve4 ,it is evident that1 if we pleased1 we might represent the rate of change of acceleration on yet another curve1 by constructing a table of 8second differences02 this would bring out certain very interesting phenomena1 which here however we must not stay to discuss4 ,annual ,increment of ,weight in ,man 7kgm474 7,after ,quetelet1 ,anthropomtrie1 p4 #cdf9947 ,increment ,age ,male ,female #j-a #ei #ef #a-b #bj #bd #b-c #ae #ad #c-d #ae #ae #d-e #ai #ad #e-f #ai #ad #f-g #ai #aa #g-h #ai #ab #h-i #ai #bj #i-aj #ag #ba #aj-aa #ah #bd #aa-ab #bj #ce #ab-ac #da #ce #ac-ad #dj #ch #ad-ae #da #cg #ae-af #db #ce #af-ag #dc #cc #ag-ah #db #cj #ah-ai #cg #bc #ai-bj #ai #aa #bj-ba #ag #aa #ba-bb #ag #je #bb-bc #af #jd #bc-bd #ji #jb #bd-be #jh #jb 99 ,the values given in this table are not in precise accord with those of the ,table on p4 #fc4 ,the latter represent ,quetelet's results arrived at in #ahce2 the former are the means of his determinations in #ahce-dj4 ,the acceleration-curve for man's weight 7,fig4 #e71 whether we draw it from ,quetelet's data1 or from the ,british1 ,american and other statistics of later writers1 is on the whole similar to that which we deduce from the statistics of these latter writers in regard to height or stature2 that is to say1 it is not a curve which continually descends1 but it indicates a rate of growth which is subject to important fluctuations at certain epochs of life4 ,we see that it begins at a high level1 and falls continuously and rapidly#ajj #fi during the first two or three years of life4 ,after a slight recovery1 it runs nearly level during boyhood from about five to twelve years old2 it then rapidly rises1 in the 8growing period0 of the early teens1 and slowly and steadily falls from about the age of sixteen onwards4 ,it does not reach the base-line till the man is about seven or eight and twenty1 for normal increase of weight continues during the years when the man is 8filling out10 long after growth in height has ceased2 but at last1 somewhere about thirty1 the velocity reaches zero1 and even falls below it1 for then the man usually begins to lose weight a little4 ,the subsequent slow changes in this acceleration-curve we need not stop to deal with4 ,illustration3 ,fig4 #e4 ,mean annual increments of weight1 in man and woman2 from ,quetelet's data4 ,in the same diagram 7,fig4 #e7 ,i have set forth the acceleration-curves in respect of increment of weight for both man and woman1 according to ,quetelet4 ,that growth in boyhood and growth in girlhood follow a very different course is a matter of common knowledge2 but if we simply plot the ordinary curve of growth1 or velocity-curve1 the difference1 on the small scale of our diagrams1 #gj is not very apparent4 ,it is admirably brought out1 however1 in the acceleration-curves4 ,here we see that1 after infancy1 say from three years old to eight1 the velocity in the girl is steady1 just as in the boy1 but it stands on a lower level in her case than in his3 the little maid at this age is growing slower than the boy4 ,but very soon1 and while his acceleration-curve is still represented by a straight line1 hers has begun to ascend1 and until the girl is about thirteen or fourteen it continues to ascend rapidly4 ,after that age1 as after sixteen or seventeen in the boy's case1 it begins to descend4 ,in short1 throughout all this period1 it is a very similar curve in the two sexes2 but it has its notable differences1 in amplitude and especially in phase4 ,last of all1 we may notice that while the acceleration-curve falls to a negative value in the male about or even a little before the age of thirty years1 this does not happen among women4 ,they continue to grow in weight1 though slowly1 till very much later in life2 until there comes a final period1 in both sexes alike1 during which weight1 and height and strength all alike diminish4 ,from certain corrected1 or 8typical0 values1 given for ,american children by ,boas and ,wissler 7l4c4 p4 #db71 we obtain the following still clearer comparison of the annual increments of stature in boys and girls3 the typical stature at the commencement of the period1 i4e4 at the age of eleven1 being #acea cm4 and #acfi cm4 for the boys and girls respectively1 and the annual increments being as follows3 ,age #ab #ac #ad #ae #af #ag #ah #ai #bj ,boys 7cm47 #da #fc #hg #gi #eb #cb #ai #ji #jc ,girls 7cm47 #ge #gj #df #ba #ji #jd #ja #jj #jj ,difference #cd #jg #da #eh #dc #bh #ah #ji #jc ,the result of these differences 7which are essentially phase-differences7 between the two sexes in regard to the velocity of growth and to the rate of change of that velocity1 is to cause the ratio between the weights of the two sexes to fluctuate in a somewhat complicated manner4 ,at birth the baby-girl weighs on the average nearly #aj per cent4 less than the boy4 ,till about two years old she tends to gain upon him1 but she then loses again until the age of about five2 from five she gains for a few years somewhat rapidly1 and the girl of ten to twelve is only some #c per cent4 less in weight than the boy4 ,the boy in his teens gains #ga steadily1 and the young woman of twenty is nearly #ae per cent4 lighter than the man4 ,this ratio of difference again slowly diminishes1 and between fifty and sixty stands at about #ab per cent41 or not far from the mean for all ages2 but once more as old age advances1 the difference tends1 though very slowly1 to increase 7,fig4 #f74 ,illustration3 ,fig4 #f4 ,percentage ratio1 throughout life1 of female weight to male2 from ,quetelet's data4 ,while careful observations on the rate of growth in other animals are somewhat scanty1 they tend to show so far as they go that the general features of the phenomenon are always much the same4 ,whether the animal be long-lived1 as man or the elephant1 or short-lived1 like horse or dog1 it passes through the same phases of growth#aja4 ,in all cases growth begins slowly2 it attains a maximum velocity early in its course1 and afterwards slows down 7subject to temporary accelerations7 towards a point where growth ceases altogether4 ,but especially in the cold-blooded animals1 such as fishes1 the slowing-down period is very greatly protracted1 and the size of the creature would seem never actually to reach1 but only to approach asymptotically1 to a maximal limit4 ,the size ultimately attained is a resultant of the rate1 and of #gb the duration1 of growth4 ,it is in the main true1 as ,minot has said1 that the rabbit is bigger than the guinea-pig because he grows the faster2 but that man is bigger than the rabbit because he goes on growing for a longer time4 ,in ordinary physical investigations dealing with velocities1 as for instance with the course of a projectile1 we pass at once from the study of acceleration to that of momentum and so to that of force2 for change of momentum1 which is proportional to force1 is the product of the mass of a body into its acceleration or change of velocity4 ,but we can take no such easy road of kinematical investigation in this case4 ,the 8velocity0 of growth is a very different thing from the 8velocity0 of the projectile4 ,the forces at work in our case are not susceptible of direct and easy treatment2 they are too varied in their nature and too indirect in their action for us to be justified in equating them directly with the mass of the growing structure4 ,it was apparently from a feeling that the velocity of growth ought in some way to be equated with the mass of the growing structure that ,minot#ajb introduced a curious1 and 7as it seems to me7 an unhappy method of representing growth1 in the form of what he called 8percentage-curves02 a method which has been followed by a number of other writers and experimenters4 ,minot's method was to deal1 not with the actual increments added in successive periods1 such as years or days1 but with these increments represented as percentages of the amount which had been reached at the end of the former period4 ,for instance1 taking ,quetelet's values for the height in centimetres of a male infant from birth to four years old1 as follows3 ,years #j #a #b #c #d cm4 #ejj #fih #gia #hfd #ibg ,minot would state the percentage growth in each of the four annual periods at #cif1 #acc1 #if and #gc per cent4 respectively4 ,now when we plot actual length against time1 we have a perfectly definite thing4 ,when we differentiate this ,l_/,t1 we have d,l_/d,t1 which is 7of course7 velocity2 and from this1 by a second differentiation1 we obtain d#b ,l_/d,t#b1 that is to say1 the acceleration4 #gc ,but when you take percentages of y1 you are determining dy_/y1 and when you plot this against dx1 you have 7dy_/y7_/dx1 or dy_/7y dx71 or 7#a_/y7 7dy_/dx71 that is to say1 you are multiplying the thing you wish to represent by another quantity which is itself continually varying2 and the result is that you are dealing with something very much less easily grasped by the mind than the original factors4 ,professor ,minot is1 of course1 dealing with a perfectly legitimate function of x and y2 and his method is practically tantamount to plotting log y against x1 that is to say1 the logarithm of the increment against the time4 ,this could only be defended and justified if it led to some simple result1 for instance if it gave us a straight line1 or some other simpler curve than our usual curves of growth4 ,as a matter of fact1 it is manifest that it does nothing of the kind4 ,pre-natal and post-natal growth4 ,in the acceleration-curves which we have shown above 7,figs4 #b1 #c71 it will be seen that the curve starts at a considerable interval from the actual date of birth2 for the first two increments which we can as yet compare with one another are those attained during the first and second complete years of life4 ,now we can in many cases 8interpolate0 with safety between known points upon a curve1 but it is very much less safe1 and is not very often justifiable 7at least until we understand the physical principle involved1 and its mathematical expression71 to 8extrapolate0 beyond the limits of our observations4 ,in short1 we do not yet know whether our curve continued to ascend as we go backwards to the date of birth1 or whether it may not have changed its direction1 and descended1 perhaps1 to zero-value4 ,in regard to length1 or stature1 however1 we can obtain the requisite information from certain tables of ,rssow's#ajc1 who gives the stature of the infant month by month during the first year of its life1 as follows3 ,age in months #j #a #b #c #d #e #f #g #h #i #aj #aa #ab ,length in cm4 7#ej7 #ed #eh #fj #fb #fd #fe #ff #fge #fh #fi #gje #gb ,differences 7in cm47 #d #d #b #b #b #a #a #ae #e #a #ae #ae ,if we multiply these monthly differences1 or mean monthly velocities1 by #ab1 to bring them into a form comparable with the #gd annual velocities already represented on our acceleration-curves1 we shall see that the one series of observations joins on very well with the other2 and in short we see at once that our acceleration-curve rises steadily and rapidly as we pass back towards the date of birth4 ,illustration3 ,fig4 #g4 ,curve of growth 7in length or stature7 of child1 before and after birth4 7,from ,his and ,rssow's data47 ,but birth itself1 in the case of a viviparous animal1 is but an unimportant epoch in the history of growth4 ,it is an epoch whose relative date varies according to the particular animal3 the foal and the lamb are born relatively later1 that is to say when development has advanced much farther1 than in the case of man2 the kitten and the puppy are born earlier and therefore more helpless than we are2 and the mouse comes into the world still earlier and more inchoate1 so much so that even the little marsupial is scarcely more unformed and embryonic4 ,in all these cases alike1 we must1 in order to study the curve of growth in its entirety1 take full account of prenatal or intra-uterine growth4 #ge ,according to ,his#ajd1 the following are the mean lengths of the unborn human embryo1 from month to month4 ,months #j #a #b #c #d #e #f #g #h #i #aj 7,birth7 ,length in mm4 #j #ge #dj #hd #afb #bge #ceb #djb #ddc #dgb #dij-ejj ,increment per month in mm4 -- #ge #cbe #dd #gh #aac #gg #ej #da #bi #ah-bh ,illustration3 ,fig4 #h4 ,mean monthly increments of length or stature of child 7in cms474 ,these data link on very well to those of ,rssow1 which we have just considered1 and 7though ,his's measurements for the pre-natal months are more detailed than are those of ,rssow for the first year of post-natal life7 we may draw a continuous curve of growth 7,fig4 #g7 and curve of acceleration of growth 7,fig4 #h7 for the combined periods4 ,it will at once be seen that there is a 8point of inflection0 somewhere about the fifth month of intra-uterine life#aje3 up to that date growth proceeds with a continually increasing #gf velocity2 but after that date1 though growth is still rapid1 its velocity tends to fall away4 ,there is a slight break between our two separate sets of statistics at the date of birth1 while this is the very epoch regarding which we should particularly like to have precise and continuous information4 ,undoubtedly there is a certain slight arrest of growth1 or diminution of the rate of growth1 about the epoch of birth3 the sudden change in the #gg method of nutrition has its inevitable effect2 but this slight temporary set-back is immediately followed by a secondary1 and temporary1 acceleration4 ,illustration3 ,fig4 #i4 ,curve of pre-natal growth 7length or stature7 of child2 and corresponding curve of mean monthly increments 7mm474 ,illustration3 ,fig4 #aj4 ,curve of growth of bamboo 7from ,ostwald1 after ,kraus74 ,it is worth our while to draw a separate curve to illustrate on a larger scale ,his's careful data for the ten months of pre-natal life 7,fig4 #i74 ,we see that this curve of growth is a beautifully regular one1 and is nearly symmetrical on either side of that point of inflection of which we have already spoken2 it is a curve for which we might well hope to find a simple mathematical expression4 ,the acceleration-curve shown in ,fig4 #i together with the pre-natal curve of growth1 is not taken directly from ,his's recorded data1 but is derived from the tangents drawn to a smoothed curve1 corresponding as nearly as possible to the actual curve of growth3 the rise to a maximal velocity about the fifth month and the subsequent gradual fall are now demonstrated even more clearly than before4 ,in ,fig4 #aj1 which is a curve of growth of the bamboo#ajf1 we see 7so far as it goes7 the same essential features1 #gh the slow beginning1 the rapid increase of velocity1 the point of inflection1 and the subsequent slow negative acceleration#ajg4 ,variability and ,correlation of ,growth4 ,the magnitudes and velocities which we are here dealing with are1 of course1 mean values derived from a certain number1 sometimes a large number1 of individual cases4 ,but no statistical account of mean values is complete unless we also take account of the amount of variability among the individual cases from which the mean value is drawn4 ,to do this throughout would lead us into detailed investigations which lie far beyond the scope of this elementary book2 but we may very briefly illustrate the nature of the process1 in connection with the phenomena of growth which we have just been studying4 ,it was in connection with these phenomena1 in the case of man1 that ,quetelet first conceived the statistical study of variation1 on lines which were afterwards expounded and developed by ,galton1 and which have grown1 in the hands of ,karl ,pearson and others1 into the modern science of ,biometrics4 ,when ,quetelet tells us1 for instance1 that the mean stature of the ten-year old boy is #abgc metres1 this implies1 according to the law of error1 or law of probabilities1 that all the individual measurements of ten-year-old boys group themselves in an orderly way1 that is to say according to a certain definite law1 about this mean value of #abgc4 ,when these individual measurements are grouped and plotted as a curve1 so as to show the number of individual cases at each individual length1 we obtain a characteristic curve of error or curve of frequency2 and the 8spread0 of this curve is a measure of the amount of variability in this particular case4 ,a certain mathematical measure of this 8spread10 as described in works upon statistics1 is called the ,index of ,variability1 or ,standard ,deviation1 and is usually denominated by the letter 4 ,it is practically equivalent to a determination of the point upon the frequency curve where it changes its curvature on either side of the mean1 and where1 from being concave towards the middle line1 it spreads out to be convex thereto4 ,when we divide this #gi value by the mean1 we get a figure which is independent of any particular units1 and which is called the ,coefficient of ,variability4 7,it is usually multiplied by #ajj1 to make it of a more convenient amount2 and we may then define this coefficient1 ,c1 as 7_/,m7 #ajj47 ,in regard to the growth of man1 ,pearson has determined this coefficient of variability as follows3 in male new-born infants1 the coefficient in regard to weight is #aeff1 and in regard to stature1 #fej2 in male adults1 for weight #ajhc1 and for stature1 #cff4 ,the amount of variability tends1 therefore1 to decrease with growth or age4 ,similar determinations have been elaborated by ,bowditch1 by ,boas and ,wissler1 and by other writers for intermediate ages1 especially from about five years old to eighteen1 so covering a great part of the whole period of growth in man#ajh4 ,coefficient of ,variability 7_/,m #ajj7 in ,man1 at various ages4 ,age #e #f #g #h #i ,stature 7,bowditch7 #dgf #dfj #ddb #ddi #ddj ,stature 7,boas and ,wissler7 #dae #dad #dbb #dcg #dcc ,weight 7,bowditch7 #aaef #ajbh #aajh #iib #aajd ,age #aj #aa #ab #ac #ad ,stature 7,bowditch7 #dee #dgj #dij #edg #egi ,stature 7,boas and ,wissler7 #dcf #ded #dgc #eaf #eeg ,weight 7,bowditch7 #aafj #aagf #acgb #acfj #afhj ,age #ae #af #ag #ah ,stature 7,bowditch7 #eeg #dej #dee #cfi ,stature 7,boas and ,wissler7 #eej #dfi #dbg #cid ,weight 7,bowditch7 #aecb #acbh #abif #ajdj ,the result is very curious indeed4 ,we see1 from ,fig4 #aa1 that the curve of variability is very similar to what we have called the acceleration-curve 7,fig4 #d73 that is to say1 it descends when the rate of growth diminishes1 and rises very markedly again when1 in late boyhood1 the rate of growth is temporarily accelerated4 ,we #hj see1 in short1 that the amount of variability in stature or in weight is a function of the rate of growth in these magnitudes1 though we are not yet in a position to equate the terms precisely1 one with another4 ,illustration3 ,fig4 #aa4 ,coefficients of variability of stature in ,man 774 from ,boas and ,wissler's data4 ,if we take not merely the variability of stature or weight at a given age1 but the variability of the actual successive increments in each yearly period1 we see that this latter coefficient of variability tends to increase steadily1 and more and more rapidly1 within the limits of age for which we have information2 and this phenomenon is1 in the main1 easy of explanation4 ,for a great part of the difference1 in regard to rate of growth1 between one individual and another is a difference of phase1--a difference in the epochs of acceleration and retardation1 and finally in the epoch when growth comes to an end4 ,and it follows that the variability of rate will be more and more marked1 as we approach and reach the period when some individuals still continue1 and others have already ceased1 to grow4 ,in the following epitomised table1 #ha ,i have taken ,boas's determinations of variability 77 7op4 cit4 p4 #aedh71 converted them into the corresponding coefficients of variability 77_/,m7 #ajj71 and then smoothed the resulting numbers4 ,coefficients of ,variability in ,annual ,increment of ,stature4 ,age #g #h #i #aj #aa #ab #ac #ad #ae ,boys #agc #aeh #ahf #aia #baj #bdg #bij #cfb #dfa ,girls #aga #agh #aib #bbg #bei #bic #cgj #ddh -- ,the greater variability of annual increment in the girls1 as compared with the boys1 is very marked1 and is easily explained by the more rapid rate at which the girls run through the several phases of the phenomenon4 ,just as there is a marked difference in 8phase0 between the growth-curves of the two sexes1 that is to say a difference in the periods when growth is rapid or the reverse1 so also1 within each sex1 will there be room for similar1 but individual phase-differences4 ,thus we may have children of accelerated development1 who at a given epoch after birth are both rapidly growing and already 8big for their age02 and others of retarded development who are comparatively small and have not reached the period of acceleration which1 in greater or less degree1 will come to them in turn4 ,in other words1 there must under such circumstances be a strong positive 8coefficient of correlation0 between stature and rate of growth1 and also between the rate of growth in one year and the next4 ,but it does not by any means follow that a child who is precociously big will continue to grow rapidly1 and become a man or woman of exceptional stature4 ,on the contrary1 when in the case of the precocious or 8accelerated0 children growth has begun to slow down1 the backward ones may still be growing rapidly1 and so making up 7more or less completely7 to the others4 ,in other words1 the period of high positive correlation between stature and increment will tend to be followed by one of negative correlation4 ,this interesting and important point1 due to ,boas and ,wissler#aji1 is confirmed by the following table3 ,correlation of ,stature and ,increment in ,boys and ,girls4 7,from ,boas and ,wissler47 ,age #f #g #h #i #aj #aa #ab #ac #ad #ae ,stature 7,b7 #aabg #aaee #abcb #abgd #accb #acfh #adbg #adgc #aeei #afbb 7,g7 #aaad #aagg #abad #abgi #acah #acfg #addf #adig #aech #aegb ,increment 7,b7 #eg #ec #di #ea #ej #dg #ei #ge #fb #eb 7,g7 #ei #ee #ee #ei #fb #gb #fe #ed #cc #ag ,correlation 7,b7 #be #aa #jh #be #ah #ah #dh #bi #db #dd 7,g7 #dd #ad #bd #dg #ah #ah #db #ci #fc #aa #hb ,a minor1 but very curious point brought out by the same investigators is that1 if instead of stature we deal with height in the sitting posture 7or1 practically speaking1 with length of trunk or back71 then the correlations between this height and its annual increment are throughout negative4 ,in other words1 there would seem to be a general tendency for the long trunks to grow slowly throughout the whole period under investigation4 ,it is a well-known anatomical fact that tallness is in the main due not to length of body but to length of limb4 ,the whole phenomenon of variability in regard to magnitude and to rate of increment is in the highest degree suggestive3 inasmuch as it helps further to remind and to impress upon us that specific rate of growth is the real physiological factor which we want to get at1 of which specific magnitude1 dimensions and form1 and all the variations of these1 are merely the concrete and visible resultant4 ,but the problems of variability1 though they are intimately related to the general problem of growth1 carry us very soon beyond our present limitations4 ,rate of growth in other organisms#aaj4 ,just as the human curve of growth has its slight but well-marked interruptions1 or variations in rate1 coinciding with such epochs as birth and puberty1 so is it with other animals1 and this phenomenon is particularly striking in the case of animals which undergo a regular metamorphosis4 ,in the accompanying curve of growth in weight of the mouse 7,fig4 #ab71 based on ,w4 ,ostwald's observations#aaa1 we see a distinct slackening of the rate when the mouse is about a fortnight old1 at which period it opens its eyes and very soon afterwards is weaned4 ,at about six weeks old there is another well-marked retardation of growth1 following on a very rapid period1 and coinciding with the epoch of puberty4 #hc ,fig4 #ac shews the curve of growth of the silkworm#aab1 during its whole larval life1 up to the time of its entering the chrysalis stage4 ,the silkworm moults four times1 at intervals of about a week1 the first moult being on the sixth or seventh day after hatching4 ,a distinct retardation of growth is exhibited on our curve in the case of the third and fourth moults2 while a similar retardation accompanies the first and second moults also1 but the scale of our diagram does not render it visible4 ,when the worm is about seven weeks old1 a remarkable process of 8purgation0 takes place1 as a preliminary to entering on the pupal1 or chrysalis1 stage2 and the great and sudden loss of weight which accompanies this process is the most marked feature of our curve4 ,illustration3 ,fig4 #ab4 ,growth in weight of ,mouse4 7,after ,w4 ,ostwald47 ,the rate of growth in the tadpole#aac 7,fig4 #ad7 is likewise marked by epochs of retardation1 and finally by a sudden and drastic change4 ,there is a slight diminution in weight immediately after #hd the little larva frees itself from the egg2 there is a retardation of growth about ten days later1 when the external gills disappear2 and finally1 the complete metamorphosis1 with the loss of the tail1 the growth of the legs and the cessation of branchial respiration1 is accompanied by a loss of weight amounting to wellnigh half the weight of the full-grown larva4 #he ,illustration3 ,fig4 #ac4 ,growth in weight of ,silkworm4 7,from ,ostwald1 after ,luciani and ,lo ,monaco47 ,while as a general rule1 the better the animals be fed the quicker they grow and the sooner they metamorphose1 ,barfrth has pointed out the curious fact that a short spell of starvation1 just before metamorphosis is due1 appears to hasten the change4 ,illustration3 ,fig4 #ad4 ,growth in weight of ,tadpole4 7,from ,ostwald1 after ,schaper47 ,the negative growth1 or actual loss of bulk and weight which often1 and perhaps always1 accompanies metamorphosis1 is well shewn in the case of the eel#aad4 ,the contrast of size is great between #hg the flattened1 lancet-shaped ,leptocephalus larva and the little black cylindrical1 almost thread-like elver1 whose magnitude is less than that of the ,leptocephalus in every dimension1 even1 at first1 in length 7,fig4 #ae74 ,illustration3 ,fig4 #ae4 ,development of ,eel2 from ,leptocephalus larvae to young ,elver4 7,from ,ostwald after ,joh4 ,schmidt47 ,illustration3 ,fig4 #af4 ,growth in length of ,spirogyra4 7,from ,ostwald1 after ,hofmeister47 ,from the higher study of the physiology of growth we learn that such fluctuations as we have described are but special interruptions in a process which is never actually continuous1 but is perpetually interrupted in a rhythmic manner#aae4 ,hofmeister shewed1 for instance1 that the growth of ,spirogyra proceeds by fits and starts1 by periods of activity and rest1 which alternate with one another at intervals of so many minutes 7,fig4 #af74 ,and ,bose1 by very refined methods of experiment1 has shewn that plant-growth really proceeds by tiny and perfectly rhythmical pulsations recurring at regular intervals of a few seconds of time4 ,fig4 #ag shews1 according to ,bose's observations#aaf1 the growth of a crocus1 under a very high magnification4 ,the stalk grows by little jerks1 each with an amplitude of about #jjb mm41 every #hh twenty seconds or so1 and after each little increment there is a partial recoil4 ,illustration3 ,fig4 #ag4 ,pulsations of growth in ,crocus1 in micro-millimetres4 7,after ,bose47 ,the rate of growth of various parts or organs#aag4 ,the differences in regard to rate of growth between various parts or organs of the body1 internal and external1 can be amply illustrated in the case of man1 and also1 but chiefly in regard to external form1 in some few other creatures#aah4 ,it is obvious that there lies herein an endless field for the mathematical study of correlation and of variability1 but with this aspect of the case we cannot deal4 ,in the accompanying table1 ,i shew1 from some of ,vierordt's data1 the relative weights1 at various ages1 compared with the weight at birth1 of the entire body1 of the brain1 heart and liver2 #hi and also the percentage relation which each of these organs bears1 at the several ages1 to the weight of the whole body4 ,weight of ,various ,organs1 compared with the ,total ,weight of the ,human ,body 7male74 7,after ,vierordt1 ,anatom4 ,tabellen1 pp4 #ch1 #ci47 ,percentage weights compared ,weight ,relative weights of with total body-weights of body ,age in kg4 ,body ,brain ,heart ,liver ,body ,brain ,heart ,liver #j #ca #a #a #a #a #ajj #abbi #jgf #deg #a #ij #bij #bdh #age #bce #ajj #ajej #jdf #cgj #b #aaj #cee #bfi #bbj #cjb #ajj #icb #jdg #chi #c #abe #djc #bia #bge #cdb #ajj #hhf #jeb #chh #d #adj #deb #cdi #cad #dae #ajj #iej #jec #dbj #e #aei #eac #ccb #cdc #chj #ajj #gid #jea #cci #f #agh #egd #ceg #cfj #dcd #ajj #gfc #jdh #cde #g #aig #fce #ced #cie #dhf #ajj #fhd #jdg #cdi #h #baf #fig #cfb #djb #dei #ajj #fch #jdd #cja #i #bce #geh #cgd #dei #die #ajj #fjf #jdf #bii #aj #beb #hac #cgj #eda #eij #ajj #eei #jea #ccb #aa #bgj #hga #ceg #eig #fad #ajj #ejd #jeb #cbb #ab #bij #ice #cgh 7#dac7 #fba #ajj #dhh 7#jcd7 #cjc #ac #cca #ajfh #cij #fie #gca #ajj #ddi #jej #cac #ad #cga #aaig #cch #iaf #hci #ajj #cdg #jeh #cbj #ae #dab #acbi #cia #hde #ibb #ajj #cfb #jdh #cag #af #dei #adha #cgg #igf #ide #ajj #caf #jea #bie #ag #dig #afjc #cgj #ajfc #ajdf #ajj #bhd #jea #bih #ah #eci #agci #cgc #ajcc #ajfe #ajj #bfd #jdf #bhj #ai #egf #aheh #cfg #aadb #aafa #ajj #bdc #jea #bhf #bj #eie #aiai #cgi #abid #aaja #ajj #bdc #jea #bfb #ba #fab #aigd #cga #abei #aadh #ajj #bca #jdi #bff #bb #fbi #bjbi #ced #acbd #aahb #ajj #bad #jej #bff #bc #fde #bjha #cff #abdb #ajgi #ajj #baf #jdf #bcg #bd -- -- #cgd #acji #acjd #ajj -- -- -- #be #ffb #bacf #cgf #abgd #abhd #ajj #baf #jdf #bge ,from ,quetelet4 ,from the first portion of the table1 it will be seen that none of these organs by any means keep pace with the body as a whole in regard to growth in weight2 in other words1 there must be some other part of the fabric1 doubtless the muscles and the bones1 which increase more rapidly than the average increase of the body4 ,heart and liver both grow nearly at the same rate1 and by the #ij age of twenty-five they have multiplied their weight at birth by about thirteen times1 while the weight of the entire body has been multiplied by about twenty-one2 but the weight of the brain has meanwhile been multiplied only about three and a quarter times4 ,in the next place1 we see the very remarkable phenomenon that the brain1 growing rapidly till the child is about four years old1 then grows more much slowly till about eight or nine years old1 and after that time there is scarcely any further perceptible increase4 ,these phenomena are diagrammatically illustrated in ,fig4 #ah4 ,illustration3 ,fig4 #ah4 ,relative growth in weight 7in ,man7 of ,brain1 ,heart1 and whole ,body4 ,many statistics indicate a decrease of brain-weight during adult life4 ,boas#aai was inclined to attribute this apparent phenomenon to our statistical methods1 and to hold that it could 8hardly be explained in any other way than by assuming an increased death-rate among men with very large brains1 at an age of about twenty years40 ,but ,raymond ,pearl has shewn that there is evidence of a steady and very gradual decline in the weight of the brain with advancing age1 beginning at or before the twentieth year1 and continuing throughout adult life#abj4 #ia ,the second part of the table shews the steadily decreasing weights of the organs in question as compared with the body2 the brain falling from over #ab per cent4 at birth to little over #b per cent4 at five and twenty2 the heart from #ge to #df per cent42 and the liver from #deg to #bge per cent4 of the whole bodily weight4 ,it is plain1 then1 that there is no simple and direct relation1 holding good throughout life1 between the size of the body as a whole and that of the organs we have just discussed2 and the changing ratio of magnitude is especially marked in the case of the brain1 which1 as we have just seen1 constitutes about one-eighth of the whole bodily weight at birth1 and but one-fiftieth at five and twenty4 ,the same change of ratio is observed in other animals1 in equal or even greater degree4 ,for instance1 ,max ,weber#aba tells us that in the lion1 at five weeks1 four months1 eleven months1 and lastly when full-grown1 the brain-weight represents the following fractions of the weight of the whole body1 viz4 #a_/ah1 #a_/hj1 #a_/ahd1 and #a_/edf4 ,and ,kellicott has1 in like manner1 shewn that in the dogfish1 while some organs 7e4g4 rectal gland1 pancreas1 etc47 increase steadily and very nearly proportionately to the body as a whole1 the brain1 and some other organs also1 grow in a diminishing ratio1 which is capable of representation1 approximately1 by a logarithmic curve#abb4 ,but if we confine ourselves to the adult1 then1 as ,raymond ,pearl has shewn in the case of man1 the relation of brain-weight to age1 to stature1 or to weight1 becomes a comparatively simple one1 and may be sensibly expressed by a straight line1 or simple equation4 ,thus1 if ,w be the brain-weight 7in grammes71 and ,a be the age1 or ,s the stature1 of the individual1 then 7in the case of ,swedish males7 the following simple equations suffice to give the required ratios3 ,w #adhgh #aid ,a #iaejf #bhf ,s4 #ib ,these equations are applicable to ages between fifteen and eighty2 if we take narrower limits1 say between fifteen and fifty1 we can get a closer agreement by using somewhat altered constants4 ,in the two sexes1 and in different races1 these empirical constants will be greatly changed#abc4 ,donaldson has further shewn that the correlation between brain-weight and body-weight is very much closer in the rat than in man#abd4 ,the falling ratio of weight of brain to body with increase of size or age finds its parallel in comparative anatomy1 in the general law that the larger the animal the less is the relative weight of the brain4 ,weight of ,weight of entire animal brain gms4 gms4 ,ratio ,marmoset #cce #abe #a 3 #bf ,spider monkey #ahde #abf #a 3 #ae ,felis minuta #abcd #bcf #a 3 #ef ,f4 domestica #ccjj #ca #a 3 #ajg ,leopard #bg1gjj #afd #a 3 #afh ,lion #aai1ejj #bai #a 3 #edf ,elephant #c1jdh1jjj #edcj #a 3 #efj ,whale 7,globiocephalus7 #a1jjj1jjj #beaa #a 3 #djj ,for much information on this subject1 see ,dubois1 8,abhngigkeit des ,hirngewichtes von der ,krpergrsse bei den ,sugethieren10 ,arch4 f4 ,anthropol4 ,,xxv1 #ahig4 ,dubois has attempted1 but ,i think with very doubtful success1 to equate the weight of the brain with that of the animal4 ,we may do this1 in a very simple way1 by representing the weight of the body as a power of that of the brain2 thus1 in the above table of the weights of brain and body in four species of cat1 if we call ,w the weight of the body 7in grammes71 and w the weight of the brain1 then if in all four cases we express the ratio by ,w wn1 we find that n is almost constant1 and differs little from #bbd in all four species3 the values being respectively1 in the order of the table #bcf1 #bbd1 #bah1 and #bag4 ,but this evidently amounts to no more than an empirical rule2 for we can easily see that it depends on the particular scale which we have used1 and that if the weights had been taken1 for instance1 in kilogrammes or in milligrammes1 the agreement or coincidence would not have occurred#abe4 #ic ,the ,length of the ,head in ,man at various ,ages4 7,after ,quetelet1 p4 #bjg47 ,men ,women ,age ,total height ,head ,ratio ,height ,head ,ratio m4 m4 m4 m4 ,birth #jejj #jaaa #dej #jdid #jaaa #dde #a year #jfih #jaed #dec #jfij #jaed #ddh #b years #jgia #jagc #deg #jgha #jagb #ded #c years #jhfd #jahb #dgd #jhed #jahj #dgd #e years #jihg #jaib #ead #jigd #jahh #eah #aj years #abgc #jbje #fba #abdi #jbja #fba #ae years #aeac #jbae #gjd #adhh #jbac #fii #bj years #affi #jbbg #gce #aegd #jbbj #gae #cj years #afhf #jbbh #gci #aehj #jbba #gae #dj years #afhf #jbbh #gci #aehj #jbba #gae ,a smooth curve1 very similar to this1 for the growth in 8auricular height0 of the girl's head1 is given by ,pearson1 in ,biometrika1 ,,iii1 p4 #ada4 #aijd4 ,as regards external form1 very similar differences exist1 which however we must express in terms not of weight but of length4 ,thus the annexed table shews the changing ratios of the vertical length of the head to the entire stature2 and while this ratio constantly diminishes1 it will be seen that the rate of change is greatest 7or the coefficient of acceleration highest7 between the ages of about two and five years4 ,in one of ,quetelet's tables 7supra1 p4 #fc71 he gives measurements of the total span of the outstretched arms in man1 from year to year1 compared with the vertical stature4 ,the two measurements are so nearly identical in actual magnitude that a direct comparison by means of curves becomes unsatisfactory2 but ,i have reduced ,quetelet's data to percentages1 and it will be seen from ,fig4 #ai that the percentage proportion of span to height undergoes a remarkable and steady change from birth to the age of twenty years2 the man grows more rapidly in stretch of arms than he does in height1 and the span which was less than #id the stature at birth by about #a per cent4 exceeds it at the age of twenty by about #d per cent4 ,after the age of twenty1 ,quetelet's data are few and irregular1 but it is clear that the span goes on for a long while increasing in proportion to the stature4 ,how far the phenomenon is due to actual growth of the arms and how far to the increasing breadth of the chest is not yet ascertained4 ,illustration3 ,fig4 #ai4 ,ratio of stature in ,man1 to span of outstretched arms4 7,from ,quetelet's data47 ,the differences of rate of growth in different parts of the body are very simply brought out by the following table1 which shews the relative growth of certain parts and organs of a young trout1 at intervals of a few days during the period of most rapid development4 ,it would not be difficult1 from a picture of the little trout at any one of these stages1 to draw its approximate form at any other1 by the help of the numerical data here set forth#abf4 #ie ,trout 7,salmo fario73 proportionate growth of various organs4 7,from ,jenkinson's data47 ,days ,total #ast ,ventral #bnd ,breadth old length ,eye ,head dorsal fin dorsal ,tail-fin of tail #di #ajj #ajj #ajj #ajj #ajj #ajj #ajj #ajj #fc #abii #abid #adhc #adhf #adhe #ajhd #agch #aeei #gg #aedi #adgc #ahib 7#bjcf7 7#aicf7 #acib #begi #bbjd #ib #agcd #agid #bbjj 7#aicb7 7#ahba7 #aede #cjgf #bgbb #ajf #aidf #aibe #bdbe #agcb #afec #agcd #ccgc #bhgg ,while it is inequality of growth in different directions that we can most easily comprehend as a phenomenon leading to gradual change of outward form1 we shall see in another chapter#abg that differences of rate at different parts of a longitudinal system1 though always in the same direction1 also lead to very notable and regular transformations4 ,of this phenomenon1 the difference in rate of longitudinal growth between head and body is a simple case1 and the difference which accompanies and results from it in the bodily form of the child and the man is easy to see4 ,a like phenomenon has been studied in much greater detail in the case of plants1 by ,sachs and certain other botanists1 after a method in use by ,stephen ,hales a hundred and fifty years before#abh4 ,on the growing root of a bean1 ten narrow zones were marked off1 starting from the apex1 each zone a millimetre in breadth4 ,after twenty-four hours' growth1 at a certain constant temperature1 the whole marked portion had grown from #aj mm4 to #cc mm4 in length2 but the individual zones had grown at very unequal rates1 as shewn in the annexed table#abi4 ,zone ,increment mm4 ,apex #ae #bnd #eh #crd #hb #dth #ce #eth #af #fth #ac #gth #je #hth #jc #ith #jb #ajth #ja #if ,illustration3 ,fig4 #bj4 ,rate of growth in successive zones near the tip of the bean-root4 ,the several values in this table lie very nearly 7as we see by ,fig4 #bj7 in a smooth curve2 in other words a definite law1 or principle of continuity1 connects the rates of growth at successive points along the growing axis of the root4 ,moreover this curve1 in its general features1 is singularly like those acceleration-curves which we have already studied1 in which we plotted the rate of growth against successive intervals of time1 as here we have plotted it against successive spatial intervals of an actual growing structure4 ,if we suppose for a moment that the velocities of growth had been transverse to the axis1 instead of1 as in this case1 longitudinal and parallel with it1 it is obvious that these same velocities would have given us a leaf-shaped structure1 of which our curve in ,fig4 #bj 7if drawn to a suitable scale7 would represent the actual outline on either side of the median axis2 or1 again1 if growth had been not confined to one plane but symmetrical about the axis1 we should have had a sort of turnip-shaped root1 #ig having the form of a surface of revolution generated by the same curve4 ,this then is a simple and not unimportant illustration of the direct and easy passage from velocity to form4 ,a kindred problem occurs when1 instead of 8zones0 artificially marked out in a stem1 we deal with the rates of growth in successive actual 8internodes02 and an interesting variation of this problem occurs when we consider1 not the actual growth of the internodes1 but the varying number of leaves which they successively produce4 ,where we have whorls of leaves at each node1 as in ,equisetum and in many water-weeds1 then the problem presents itself in a simple form1 and in one such case1 namely in ,ceratophyllum1 it has been carefully investigated by ,mr ,raymond ,pearl#acj4 ,it is found that the mean number of leaves per whorl increases with each successive whorl2 but that the rate of increment diminishes from whorl to whorl1 as we ascend the axis4 ,in other words1 the increase in the number of leaves per whorl follows a logarithmic ratio2 and if y be the mean number of leaves per whorl1 and x the successional number of the whorl from the root or main stem upwards1 then y ,a ,c log7x a71 where ,a1 ,c1 and a are certain specific constants1 varying with the part of the plant which we happen to be considering4 ,on the main stem1 the rate of change in the number of leaves per whorl is very slow2 when we come to the small twigs1 or 8tertiary branches10 it has become rapid1 as we see from the following abbreviated table3 ,number of leaves per whorl on the tertiary branches of ,ceratophyllum4 ,position of whorl #a #b #c #d #e #f ,mean number of leaves #fee #hjg #ijj #ibj #ige #ajjj ,increment -- #aeb #ic #bj 7#ee7 7#be7 ,we have seen that a slow but definite change of form is a common accompaniment of increasing age1 and is brought about as the simple and natural result of an altered ratio between the rates of growth in different dimensions3 or rather by the progressive change necessarily brought about by the difference in their accelerations4 ,there are many cases however in which the change is all but imperceptible to ordinary measurement1 and many others in which some one dimension is easily measured1 but others are hard to measure with corresponding accuracy4 #ih ,for instance1 in any ordinary fish1 such as a plaice or a haddock1 the length is not difficult to measure1 but measurements of breadth or depth are very much more uncertain4 ,in cases such as these1 while it remains difficult to define the precise nature of the change of form1 it is easy to shew that such a change is taking place if we make use of that ratio of length to weight which we have spoken of in the preceding chapter4 ,assuming1 as we may fairly do1 that weight is directly proportional to bulk or volume1 we may express this relation in the form ,w_/,l#c k1 where k is a constant1 to be determined for each particular case4 7,w and ,l are expressed in grammes and centimetres1 and it is usual to multiply the result by some figure1 such as #ajjj1 so as to give the constant k a value near to unity47 ,plaice caught in a certain area1 ,march1 #aijg4 ,variation of k 7the weight-length coefficient7 with size4 7,data taken from the ,department of ,agriculture and ,fisheries' ,plaice-,report1 vol4 ,i1 p4 #ajg1 #aijh47 ,size in cm4 ,weight in gm4 ,w_/,l#c #aj1jjj ,w_/,l#c 7smoothed7 #bc #aac #ibh -- #bd #abh #ibf #idc #be #aeb #igc #ifa #bf #agc #ihd #igi #bg #aic #iha #iij #bh #bba #ajjf #ajjd #bi #bej #ajbe #ajab #cj #bga #ajjd #ajab #ca #cjj #ajjg #ajjd #cb #cbh #ajja #iih #cc #ced #ihe #ihh #cd #chd #igg #ihj #ce #dai #igg #igf #cf #ded #igc #ifg #cg #dib #ieb #ifc #ch #ebi #ifd #ief #ci #efd #iea #iej #dj #fad #iei #iej #da #fdg #ici #ich #db #fgi #iaf #ibe #dc #gcb #iba #ibe #dd #hjj #ici #idj #de #hge #ifj -- #ii ,now while this k may be spoken of as a 8constant10 having a certain mean value specific to each species of organism1 and depending on the form of the organism1 any change to which it may be subject will be a very delicate index of progressive changes of form2 for we know that our measurements of length are1 on the average1 very accurate1 and weighing is a still more delicate method of comparison than any linear measurement4 ,illustration3 ,fig4 #ba4 ,changes in the weight-length ratio of ,plaice1 with increasing size4 ,thus1 in the case of plaice1 when we deal with the mean values for a large number of specimens1 and when we are careful to deal only with such as are caught in a particular locality and at a particular time1 we see that k is by no means constant1 but steadily increases to a maximum1 and afterwards slowly declines with the increasing size of the fish 7,fig4 #ba74 ,to begin with1 therefore1 the weight is increasing more rapidly than the cube of the length1 and it follows that the length itself is increasing less rapidly than some other linear dimension2 while in later life this condition is reversed4 ,the maximum is reached when the length of the fish is somewhere near to #cj cm41 and it is tempting to suppose that with this 8point of inflection0 there is associated some well-marked epoch in the fish's life4 ,as a matter of fact1 the size of #cj cm4 is approximately that at which sexual maturity may be said to begin1 or is at least near enough to suggest a close connection between the two phenomena4 ,the first step towards further investigation of the #ajj apparent coincidence would be to determine the coefficient k of the two sexes separately1 and to discover whether or not the point of inflection is reached 7or sexual maturity is reached7 at a smaller size in the male than in the female plaice2 but the material for this investigation is at present scanty4 ,illustration3 ,fig4 #bb4 ,periodic annual change in the weight-length ratio of ,plaice4 ,a still more curious and more unexpected result appears when we compare the values of k for the same fish at different seasons of the year#aca4 ,when for simplicity's sake 7as in the accompanying table and ,fig4 #bb7 we restrict ourselves to fish of one particular size1 it is not necessary to determine the value of k1 because a change in the ratio of length to weight is obvious enough2 but when we have small numbers1 and various sizes1 to deal with1 the determination of k may help us very much4 ,it will be seen1 then1 that in the case of plaice the ratio of weight to length exhibits a regular periodic variation with the course of the seasons4 #aja ,relation of ,weight to ,length in ,plaice of #ee cm4 long1 from ,month to ,month4 7,data taken from the ,department of ,agriculture and ,fisheries ,plaice-,report1 vol4 ,,ii1 p4 #ib1 #aiji47 ,average weight in grammes ,w_/,l#c #ajj ,w_/,l#c 7smoothed7 ,jan4 #bjci #abbf #aaeg ,feb4 #agce #ajdc #ajhj ,march #afaf #jiga #jihi ,april #aehe #jiec #jifg ,may #afbd #jigf #jihe ,june #agjg #ajbf #ajje ,july #afhf #ajac #ajcg ,august #aghc #ajgb #ajdb ,sept4 #agcc #ajdb #aaaa ,oct4 #bjbi #abbj #aafj ,nov4 #bjbf #abah #abac ,dec4 #aiih #abja #abae ,with unchanging length1 the weight and therefore the bulk of the fish falls off from about ,november to ,march or ,april1 and again between ,may or ,june and ,november the bulk and weight are gradually restored4 ,the explanation is simple1 and depends wholly on the process of spawning1 and on the subsequent building up again of the tissues and the reproductive organs4 ,it follows that1 by this method1 without ever seeing a fish spawn1 and without ever dissecting one to see the state of its reproductive system1 we can ascertain its spawning season1 and determine the beginning and end thereof1 with great accuracy4 ,as a final illustration of the rate of growth1 and of unequal growth in various directions1 ,i give the following table of data regarding the ox1 extending over the first three years1 or nearly so1 of the animal's life4 ,the observed data are 7#a7 the weight of the animal1 month by month1 7#b7 the length of the back1 from the occiput to the root of the tail1 and 7#c7 the height to the withers4 ,to these data ,i have added 7#a7 the ratio of length to height1 7#b7 the coefficient 7k7 expressing the ratio of weight to the cube of the length1 and 7#c7 a similar coefficient 7k7 for the height of the animal4 ,it will be seen that1 while all these ratios tend to alter continuously1 shewing that the animal's form is steadily altering as it approaches maturity1 the ratio between length and weight #ajb changes comparatively little4 ,the simple ratio between length and height increases considerably1 as indeed we should expect2 for we know that in all ,ungulate animals the legs are remarkably ,relations between the ,weight and certain ,linear ,dimensions of the ,ox4 7,data from ,przibram1 after ,cornevin47 k k ,age in ,w1 wt4 ,l1 length ,h1 ,w_/,l#c ,w_/,h#c months in kg4 of back height ,l_/,h #aj #aj #j #cg #gh #gj #aaad #ggi #ajgi #a #eec #id #gg #abba #ffe #abaj #b #hfc #aji #he #abhb #fff #adjf #c #abac #abjg #id #abhd #fij #adfj #d #aejc #acad #ie #achc #ffb #aged #e #agic #adjd #ajdj #acej #fdi #afjj #f #bajc #adhd #ajhg #acfe #fdd #afch #g #bdgc #aebd #aabb #aceh #fii #agea #h #bfgc #aeha #aadg #acgh #fgg #agia #i #bhbh #afba #aafb #acie #ffd #ahjb #aj #cjcg #afea #aaib #ache #fge #agic #aa #cbgg #afid #abae #acid #fgd #agid #ab #cejg #agdj #abch #adje #fff #ahdi #ac #cgdg #agfe #abed #adjg #fhb #aijj #ad #ciac #aghe #abfd #adab #fhh #aich #ae #djei #ahjd #abgj #adbj #fib #aihb #af #dagi #ahad #abhj #adag #gjj #bjib #ag #dbci #ahcb #abij #adbj #fhi #aigd #ah #dbci #ahei #abig #adcc #ffj #aidc #ai #dbgi #ahge #acjg #adce #fdi #aiaf #bj #dcgi #ahhd #acaa #adcg #fee #aidd #ba #ddgi #ahic #acba #adcc #ffa #aidc #bb #dfdd #aija #accc #adbf #fgf #aifj #bc #dhji #aiji #acde #adai #fia #aigg #bd #ejji #aiad #aceb #adaf #gad #bjbg #be #ebji #aiai #acei #adab #gcg #bjge #bf #ecda #aibd #acfa #adad #gej #baai #bg #edgc #aibi #acfc #adae #gfb #bafb #bh #eede #aibi #acfc #adae #ggb #baij #bi #efag #aibi #acfc #adae #ghb #bbah #cj #ehfb #aidi #achc #adji #gib #bbaf #ca #fajg #aifi #adjc #adjc #hjj #bbaa #cb #fbeg #aihc #adbj #acif #hjc #bahf #cc #fdjg #aiig #adcg #acij #hje #baei #cd #feeg #bjaa #aded #achc #hjf #bacc ,cornevin1 ,ch41 tudes sur la croissance1 ,arch4 de ,physiol4 norm4 et pathol4 7#e71 ,,iv1 p4 #dgg1 #ahib4 #ajc long at birth in comparison with other dimensions of the body4 ,it is somewhat curious1 however1 that this ratio seems to fall off a little in the third year of growth1 the animal continuing to grow in height to a marked degree after growth in length has become very slow4 ,the ratio between height and weight is by much the most variable of our three ratios2 the coefficient ,w_/,h#c steadily increases1 and is more than twice as great at three years old as it was at birth4 ,this illustrates the important1 but obvious fact1 that the coefficient k is most variable in the case of that dimension which grows most uniformly1 that is to say most nearly in proportion to the general bulk of the animal4 ,in short1 the successive values of k1 as determined 7at successive epochs7 for one dimension1 are a measure of the variability of the others4 ,from the whole of the foregoing discussion we see that a certain definite rate of growth is a characteristic or specific phenomenon1 deep-seated in the physiology of the organism2 and that a very large part of the specific morphology of the organism depends upon the fact that there is not only an average1 or aggregate1 rate of growth common to the whole1 but also a variation of rate in different parts of the organism1 tending towards a specific rate characteristic of each different part or organ4 ,the smallest change in the relative magnitudes of these partial or localised velocities of growth will be soon manifested in more and more striking differences of form4 ,this is as much as to say that the time-element1 which is implicit in the idea of growth1 can never 7or very seldom7 be wholly neglected in our consideration of form#acb4 ,it is scarcely necessary to enlarge here upon our statement1 for not only is the truth of it self-evident1 but it will find illustration again and again throughout this book4 ,nevertheless1 let us go out of our way for a moment to consider it in reference to a particular case1 and to enquire whether it helps to remove any of the difficulties which that case appears to present4 #ajd ,illustration3 ,fig4 #bc4 ,variability of length of tail-forceps in a sample of ,earwigs4 7,after ,bateson1 ,p4 ,z4 ,s4 #ahib1 p4 #ehh47 ,in a very well-known paper1 ,bateson shewed that1 among a large number of earwigs1 collected in a particular locality1 the males fell into two groups1 characterised by large or by small tail-forceps1 with very few instances of intermediate magnitude4 ,this distribution into two groups1 according to magnitude1 is illustrated in the accompanying diagram 7,fig4 #bc72 and the phenomenon was described1 and has been often quoted1 as one of dimorphism1 or discontinuous variation4 ,in this diagram the time-element does not appear2 but it is certain1 and evident1 that it lies close behind4 ,suppose we take some organism which is born not at all times of the year 7as man is7 but at some one particular season 7for instance a fish71 then any random sample will consist of individuals whose ages1 and therefore whose magnitudes1 will form a discontinuous series2 and by plotting these magnitudes on a curve in relation to the number of individuals of each particular magnitude1 we obtain a curve such as that shewn in ,fig4 #bd1 the first practical use of which is to enable us to analyse our sample into its constituent 8age-groups10 or in other words to determine approximately the age1 or ages of the fish4 ,and if1 instead of measuring the whole length of our fish1 we had confined ourselves to particular parts1 such as head1 or #aje tail or fin1 we should have obtained discontinuous curves of distribution1 precisely analogous to those for the entire animal4 ,now we know that the differences with which ,bateson was dealing were entirely a question of magnitude1 and we cannot help seeing that the discontinuous distributions of magnitude represented by his earwigs' tails are just such as are illustrated by the magnitudes of the older and younger fish2 we may indeed go so far as to say that the curves are precisely comparable1 for in both cases we see a characteristic feature of detail1 namely that the 8spread0 of the curve is greater in the second wave than in the first1 that is to say 7in the case of the fish7 in the older as well as larger series4 ,over the reason for this phenomenon1 which is simple and all but obvious1 we need not pause4 ,illustration3 ,fig4 #bd4 ,variability of length of body in a sample of ,plaice4 ,it is evident1 then1 that in this case of 8dimorphism10 the tails of the one group of earwigs 7which ,bateson calls the 8high males07 have either grown faster1 or have been growing for a longer period of time1 than those of the 8low males40 ,if we could be certain that the whole random sample of earwigs were of one and the same age1 then we should have to refer the phenomenon of dimorphism to a physiological phenomenon1 simple in kind 7however remarkable and unexpected72 viz4 that there were two alternative #ajf values1 very different from one another1 for the mean velocity of growth1 and that the individual earwigs varied around one or other of these mean values1 in each case according to the law of probabilities4 ,but on the other hand1 if we could believe that the two groups of earwigs were of different ages1 then the phenomenon would be simplicity itself1 and there would be no more to be said about it#acc4 ,before we pass from the subject of the relative rate of growth of different parts or organs1 we may take brief note of the fact that various experiments have been made to determine whether the normal ratios are maintained under altered circumstances of nutrition1 and especially in the case of partial starvation4 ,for instance1 it has been found possible to keep young rats alive for many weeks on a diet such as is just sufficient to maintain life without permitting any increase of weight4 ,the rat of three weeks old weighs about #be gms41 and under a normal diet should weigh at ten weeks old about #aej gms41 in the male1 or #aae gms4 in the female2 but the underfed rat is still kept at ten weeks old to the weight of #be gms4 ,under normal diet the proportions of the body change very considerably between the ages of three and ten weeks4 ,for instance the tail gets relatively longer2 and even when the total growth of the rat is prevented by underfeeding1 the form continues to alter so that this increasing length of the tail is still manifest#acd4 #ajg ,full-fed ,rats4 ,age in ,length of ,length of ,total weeks body 7mm47 tail 7mm47 length of tail #j #dhg #afi #fef #beh #a #fde #bid #ici #cac #c #ijd #eia #adie #cie #f #abhj #aajj #bchj #dfb #aj #agcj #aejj #cbcj #dfd ,underfed ,rats4 #f #ihj #gbc #agjc #dbe #aj #iif #hci #ahce #deg ,again as physiologists have long been aware1 there is a marked difference in the variation of weight of the different organs1 according to whether the animal's total weight remain constant1 or be caused to diminish by actual starvation2 and further striking differences appear when the diet is not only scanty1 but ill-balanced4 ,but these phenomena of abnormal growth1 however interesting from the physiological view1 are of little practical importance to the morphologist4 ,the effect of temperature#ace4 ,the rates of growth which we have hitherto dealt with are based on special investigations1 conducted under particular local conditions4 ,for instance1 ,quetelet's data1 so far as we have used them to illustrate the rate of growth in man1 are drawn from his study of the population of ,belgium4 ,but apart from that 8fortuitous0 individual variation which we have already considered1 it is obvious that the normal rate of growth will be found to vary1 in man and in other animals1 just as the average stature varies1 in different localities1 and in different 8races40 ,this phenomenon is a very complex one1 and is doubtless a resultant of many undefined contributory causes2 but we at least gain something in regard to it1 when we discover that the rate of growth is directly affected by temperature1 and probably by other physical #ajh conditions4 ,raumur was the first to shew1 and the observation was repeated by ,bonnet#acf1 that the rate of growth or development of the chick was dependent on temperature1 being retarded at temperatures below and somewhat accelerated at temperatures above the normal temperature of incubation1 that is to say the temperature of the sitting hen4 ,in the case of plants the fact that growth is greatly affected by temperature is a matter of familiar knowledge2 the subject was first carefully studied by ,alphonse ,de ,candolle1 and his results and those of his followers are discussed in the textbooks of ,botany#acg4 ,that variation of temperature constitutes only one factor in determining the rate of growth is admirably illustrated in the case of the ,bamboo4 ,it has been stated 7by ,lock7 that in ,ceylon the rate of growth of the ,bamboo is directly proportional to the humidity of the atmosphere3 and again 7by ,shibata7 that in ,japan it is directly proportional to the temperature4 ,the two statements have been ingeniously and satisfactorily reconciled by ,blackman#ach1 who suggests that in ,ceylon the temperature-conditions are all that can be desired1 but moisture is apt to be deficient3 while in ,japan there is rain in abundance but the average temperature is somewhat too low4 ,so that in the one country it is the one factor1 and in the other country it is the other1 which is essentially variable4 ,the annexed diagram 7,fig4 #be71 shewing the growth in length of the roots of some common plants during an identical period of forty-eight hours1 at temperatures varying from about #ad to #cg ,c41 is a sufficient illustration of the phenomenon4 ,we see that in all cases there is a certain optimum temperature at which the rate of growth is a maximum1 and we can also see that on either side of this optimum temperature the acceleration of growth1 positive or negative1 with increase of temperature is rapid1 while at a distance from the optimum it is very slow4 ,from the data given by ,sachs and others1 we see further that this optimum temperature is very much the same for all the common plants of our own climate which have as yet been studied2 in them it is #aji somewhere about #bf ,c4 7or say #gg ,f471 or about the temperature of a warm summer's day2 while it is found1 very naturally1 to be considerably higher in the case of plants such as the melon or the maize1 which are at home in warmer regions that our own4 ,illustration3 ,fig4 #be4 ,relation of rate of growth to temperature in certain plants4 7,from ,sachs's data47 ,in a large number of physical phenomena1 and in a very marked degree in all chemical reactions1 it is found that rate of action is affected1 and for the most part accelerated1 by rise of temperature2 and this effect of temperature tends to follow a definite 8exponential0 law1 which holds good within a considerable range of temperature1 but is altered or departed from when we pass beyond certain normal limits4 ,the law1 as laid down by van't ,hoff for chemical reactions1 is1 that for an interval of n degrees the velocity varies as xn1 x being called the 8temperature coefficient0#aci for the reaction in question4 #aaj ,van't ,hoff's law1 which has become a fundamental principle of chemical mechanics1 is likewise applicable 7with certain qualifications7 to the phenomena of vital chemistry2 and it follows that1 on very much the same lines1 we may speak of the 8temperature coefficient0 of growth4 ,at the same time we must remember that there is a very important difference 7though we can scarcely call it a fundamental one7 between the purely physical and the physiological phenomenon1 in that in the former we study 7or seek and profess to study7 one thing at a time1 while in the latter we have always to do with various factors which intersect and interfere2 increase in the one case 7or change of any kind7 tends to be continuous1 in the other case it tends to be brought to arrest4 ,this is the simple meaning of that ,law of ,optimum1 laid down by ,errera and by ,sachs as a general principle of physiology3 namely that every physiological process which varies 7like growth itself7 with the amount or intensity of some external influence1 does so according to a law in which progressive increase is followed by progressive decrease2 in other words the function has its optimum condition1 and its curve shews a definite maximum4 ,in the case of temperature1 as ,jost puts it1 it has on the one hand its accelerating effect which tends to follow van't ,hoff's law4 ,but it has also another and a cumulative effect upon the organism3 8,sie schdigt oder sie ermdet ihn1 und je hher sie steigt1 desto rascher macht sie die ,schdigung geltend und desto schneller schreitet sie voran40 ,it would seem to be this double effect of temperature in the case of the organism which gives us our 8optimum0 curves1 which are the expression1 accordingly1 not of a primary phenomenon1 but of a more or less complex resultant4 ,moreover1 as ,blackman and others have pointed out1 our 8optimum0 temperature is very ill-defined until we take account also of the duration of our experiment2 for obviously1 a high temperature may lead to a short1 but exhausting1 spell of rapid growth1 while the slower rate manifested at a lower temperature may be the best in the end4 #aaa ,the mile and the hundred yards are won by different runners2 and maximum rate of working1 and maximum amount of work done1 are two very different things#adj4 ,in the case of maize1 a certain series of experiments shewed that the growth in length of the roots varied with the temperature as follows#ada3 ,temperature ,growth in #dh hours ,c4 mm4 #ahj #aa #bce #ajh #bff #bif #bhe #bfe #cjb #fdf #cce #fie #cfe #bjg ,let us write our formula in the form ,v7tn7_/,vt xn4 ,then choosing two values out of the above experimental series 7say the second and the second-last71 we have t #bce1 n #aj1 and ,v1 ,v #ajh and #fie respectively4 ,accordingly #fie_/ajh #fd x#aj4 ,therefore 7log #fd7_/aj1 or #jhjf log x4 ,and1 x #abjd 7for an interval of #a ,c474 ,this first approximation might be considerably improved by taking account of all the experimental values1 two only of which we have as yet made use of2 but even as it is1 we see by ,fig4 #bf that it is in very fair accordance with the actual results of observation1 within those particular limits of temperature to which the experiment is confined4 #aab ,for an experiment on ,lupinus albus1 quoted by ,asa ,gray#adb1 ,i have worked out the corresponding coefficient1 but a little more carefully4 ,its value ,i find to be #aaf1 or very nearly identical with that we have just found for the maize2 and the correspondence between the calculated curve and the actual observations is now a close one4 ,illustration3 ,fig4 #bf4 ,relation of rate of growth to temperature in ,maize4 ,observed values 7after ,kppen71 and calculated curve4 ,since the above paragraphs were written1 new data have come to hand4 ,miss ,i4 ,leitch has made careful observations of the rate of growth of rootlets of the ,pea2 and ,i have attempted a further analysis of her principal results#adc4 ,in ,fig4 #bg are shewn the mean rates of growth 7based on about a hundred experiments7 at some thirty-four different temperatures between #jh and #bic1 each experiment lasting rather less than twenty-four hours4 ,working out the mean temperature coefficient for a great many combinations of these values1 ,i obtain a value of #ajib per ,c41 or #bda for an interval of #aj1 and a mean value for the whole series showing a rate of growth of just about #a mm4 per hour at a temperature of #bj4 ,my curve in ,fig4 #bg is drawn from these determinations2 and it will be seen that1 while it is by no means exact at the lower temperatures1 and will of course fail us altogether at very high #aac temperatures1 yet it serves as a very satisfactory guide to the relations between rate and temperature within the ordinary limits of healthy growth4 ,miss ,leitch holds that the curve is not a van't ,hoff curve2 and this1 in strict accuracy1 we need not dispute4 ,but the phenomenon seems to me to be one into which the van't ,hoff ratio enters largely1 though doubtless combined with other factors which we cannot at present determine or eliminate4 ,illustration3 ,fig4 #bg4 ,relation of rate of growth to temperature in rootlets of ,pea4 7,from ,miss ,i4 ,leitch's data47 ,while the above results conform fairly well to the law of the temperature coefficient1 it is evident that the imbibition of water plays so large a part in the process of elongation of the root or stem that the phenomenon is rather a physical than a chemical one3 and on this account1 as ,blackman has remarked1 the data commonly given for the rate of growth in plants are apt to be #aad irregular1 and sometimes 7we might even say7 misleading#add4 ,the fact also1 which we have already learned1 that the elongation of a shoot tends to proceed by jerks1 rather than smoothly1 is another indication that the phenomenon is not purely and simply a chemical one4 ,we have abundant illustrations1 however1 among animals1 in which we may study the temperature coefficient under circumstances where1 though the phenomenon is always complicated by osmotic factors1 true metabolic growth or chemical combination plays a larger role4 ,thus ,mlle4 ,maltaux and ,professor ,massart#ade have studied the rate of division in a certain flagellate1 ,chilomonas paramoecium1 and found the process to take #bi minutes at #ae ,c41 #ab at #be1 and only #e minutes at #ce ,c4 ,these velocities are in the ratio of #a 3 #bd 3 #egf1 which ratio corresponds precisely to a temperature coefficient of #bd for each rise of #aj1 or about #ajib for each degree centigrade4 ,by means of this principle we may throw light on the apparently complicated results of many experiments4 ,for instance1 ,fig4 #bh is an illustration1 which has been often copied1 of ,o4 ,hertwig's work on the effect of temperature on the rate of development of the tadpole#adf4 ,from inspection of this diagram1 we see that the time taken to attain certain stages of development 7denoted by the numbers ,,iii-,,vii7 was as follows1 at #bj and at #aj ,c41 respectively4 ,at #bj ,at #aj ,stage ,,iii #bj #fe days ,stage ,,iv #bg #ha days ,stage ,v #cj #ajg days ,stage ,,vi #dj #ace days ,stage ,,vii #ej #afh days ,total #afg #eef days ,that is to say1 the time taken to produce a given result at #aae #aj was 7on the average7 somewhere about #eef_/afg1 or #ccc1 times as long as was required at #bj4 ,illustration3 ,fig4 #bh4 ,diagram shewing time taken 7in days71 at various temperatures 7,c471 to reach certain stages of development in the ,frog3 viz4 ,i1 gastrula2 ,,ii1 medullary plate2 ,,iii1 closure of medullary folds2 ,,iv1 tail-bud2 ,v1 tail and gills2 ,,vi1 tail-fin2 ,,vii1 operculum beginning2 ,,viii1 do4 closing2 ,,ix1 first appearance of hind-legs4 7,from ,jenkinson1 after ,o4 ,hertwig1 #ahih47 ,we may then put our equation again in the simple form1 #aaf x#aj #ccc4 ,or1 #aj log x log #ccc #ebbdd4 ,therefore log x #jebbd1 and x #aabh4 ,that is to say1 between the intervals of #aj and #bj ,c41 if it take m days1 at a certain given temperature1 for a certain stage of development to be attained1 it will take m #aabhn days1 when the temperature is n degrees less1 for the same stage to be arrived at4 ,illustration3 ,fig4 #bi4 ,calculated values1 corresponding to preceding figure4 ,fig4 #bi is calculated throughout from this value2 and it will be seen that it is extremely concordant with the original diagram1 as regards all the stages of development and the whole range of temperatures shewn3 in spite of the fact that the coefficient on which it is based was derived by an easy method from a very few points in the original curves4 #aag ,karl ,peter#adg1 experimenting chiefly on echinoderm eggs1 and also making use of ,hertwig's experiments on young tadpoles1 gives the normal temperature coefficients for intervals of #aj ,c4 7commonly written ,q#aj7 as follows4 ,sphaerechinus #bae1 ,echinus #bac1 ,rana #bhf4 ,these values are not only concordant1 but are evidently of the same order of magnitude as the temperature-coefficient in ordinary chemical reactions4 ,peter has also discovered the very interesting fact that the temperature-coefficient alters with age1 usually but not always becoming smaller as age increases4 ,sphaerechinus2 ,segmentation ,q#aj #bbi1 ,later stages ,q#aj #bjc4 ,echinus2 ,segmentation ,q#aj #bcj1 ,later stages ,q#aj #bjh4 ,rana2 ,segmentation ,q#aj #bbc1 ,later stages ,q#aj #ccd4 ,furthermore1 the temperature coefficient varies with the temperature1 diminishing as the temperature rises1--a rule which van't ,hoff has shewn to hold in ordinary chemical operations4 ,thus1 in ,rana the temperature coefficient at low temperatures may be as high as #ef3 which is just another way of saying that at low temperatures development is exceptionally retarded4 ,in certain fish1 such as plaice and haddock1 ,i and others have found clear evidence that the ascending curve of growth is subject to seasonal interruptions1 the rate during the winter months being always slower than in the months of summer3 it is as though we superimposed a periodic1 annual1 sine-curve upon the continuous curve of growth4 ,and further1 as growth itself grows less and less from year to year1 so will the difference between the winter and the summer rate also grow less and less4 ,the fluctuation in rate #aah will represent a vibration which is gradually dying out2 the amplitude of the sine-curve will gradually diminish till it disappears2 in short1 our phenomenon is simply expressed by what is known as a 8damped sine-curve40 ,exactly the same thing occurs in man1 though neither in his case nor in that of the fish have we sufficient data for its complete illustration4 ,we can demonstrate the fact1 however1 in the case of man by the help of certain very interesting measurements which have been recorded by ,daffner#adh1 of the height of ,german cadets1 measured at half-yearly intervals4 ,growth in height of ,german military ,cadets1 in half-yearly periods4 7,daffner47 ,height in cent4 ,increment in cm4 ,number ,winter ,summer observed ,age ,october ,april ,october -year -year ,year #ab #aa-ab #acid #adaj #adcc #af #bc #ci #hj #ab-ac #adcj #adde #adgd #ae #bi #dd #adf #ac-ad #adge #adie #aebe #bj #cj #ej #afb #ad-ae #aebb #aeej #aehe #be #ce #fj #afb #ae-af #aehe #afjh #afch #bc #cj #ec #aej #af-ag #afce #afed #afgg #ai #bc #db #hb #ag-ah #afgg #afhi #agjd #ab #ae #bg #bb #ah-ai #afih #agjf #agae #jh #ji #ag #f #ai-bj #agjg #agaa #agae #jd #jd #jh ,in the accompanying diagram 7,fig4 #cj7 the half-yearly increments are set forth1 from the above table1 and it will be seen that they form two even and entirely separate series4 ,the curve joining up each series of points is an acceleration-curve2 and the comparison of the two curves gives a clear view of the relative rates of growth during winter and summer1 and the fluctuation which these velocities undergo during the years in question4 ,the dotted line represents1 approximately1 the acceleration-curve in its continuous fluctuation of alternate seasonal decrease and increase4 ,in the case of trees1 the seasonal fluctuations of growth#adi admit #aai of easy determination1 and it is a point of considerable interest to compare the phenomenon in evergreen and in deciduous trees4 ,i happen to have no measurements at hand with which to make this comparison in the case of our native trees1 but from a paper by ,mr ,charles ,e4 ,hall#aej ,i have compiled certain mean values for growth in the climate of ,uruguay4 ,illustration3 ,fig4 #cj4 ,half-yearly increments of growth1 in cadets of various ages4 7,from ,daffner's data47 ,mean monthly increase in ,girth of ,evergreen and ,deciduous ,trees1 at ,san ,jorge1 ,uruguay4 7,after ,c4 ,e4 ,hall47 ,values expressed as percentages of total annual increase4 ,jan4 ,feb4 ,mar4 ,apr4 ,may ,june ,july ,aug4 ,sept4 ,oct4 ,nov4 ,dec4 ,evergreens #ia #hh #hf #hi #gg #ed #dc #fj #ia #aaa #ajh #ajb ,deciduous trees #bjc #adf #ij #bc #jh #jc #jg #ac #ce #ii #afg #baj ,the measurements taken were those of the girth of the tree1 in mm41 at three feet from the ground4 ,the evergreens included species of ,pinus1 ,eucalyptus and ,acacia2 the deciduous trees included ,quercus1 ,populus1 ,robinia and ,melia4 ,i have merely taken mean values for these two groups1 and expressed the monthly values as percentages of the mean annual increase4 ,the result 7as shewn by ,fig4 #ca7 is very much what we might have expected4 ,the growth of the deciduous trees is completely arrested in winter-time1 and the arrest is all but complete over #abj a considerable period of time2 moreover1 during the warm season1 the monthly values are regularly graded 7approximately in a sine-curve7 with a clear maximum 7in the southern hemisphere7 about the month of ,december4 ,in the evergreen trees1 on the other hand1 the amplitude of the periodic wave is very much less2 there is a notable amount of growth all the year round1 and1 while there is a marked diminution in rate during the coldest months1 there is a tendency towards equality over a considerable part of the warmer season4 ,it is probable that some of the species examined1 and especially the pines1 were definitely retarded in growth1 either by a temperature above their optimum1 or by deficiency of moisture1 during the hottest period of the year2 with the result that the seasonal curve in our diagram has 7as it were7 its region of maximum cut off4 ,illustration3 ,fig4 #ca4 ,periodic annual fluctuation in rate of growth of trees 7in the southern hemisphere74 ,in the case of trees1 the seasonal periodicity of growth is so well marked that we are entitled to make use of the phenomenon in a converse way1 and to draw deductions as to variations in #aba climate during past years from the record of varying rates of growth which the tree1 by the thickness of its annual rings1 has preserved for us4 ,mr4 ,a4 ,e4 ,douglass1 of the ,university of ,arizona1 has made a careful study of this question#aea1 and ,i have received 7through ,professor ,h4 ,h4 ,turner of ,oxford7 some measurements of the average width of the successive annual rings in 8yellow pine10 #ejj years old1 from ,arizona1 in which trees the annual rings are very clearly distinguished4 ,from the year #acia to #aeah1 the mean of two trees was used2 from #aeai to #aiab1 the mean of five2 and the means of these1 and sometimes of larger numbers1 were found to be very concordant4 ,a correction was applied by drawing a long1 nearly straight line through the curve for the whole period1 which line was assumed to represent the slowly diminishing mean width of ring accompanying the increase of size1 or age1 of the tree2 and the actual growth as measured was equated with this diminishing mean4 ,the figures used give1 accordingly1 the ratio of the actual growth in each year to the mean growth corresponding to the age or magnitude of the tree at that epoch4 ,it was at once manifest that the rate of growth so determined shewed a tendency to fluctuate in a long period of between #ajj and #bjj years4 ,i then smoothed in groups of #ajj 7according to ,gauss's method7 the yearly values1 so that each number thus found represented the mean annual increase during a century3 that is to say1 the value ascribed to the year #aejj represented the average annual growth during the whole period between #adej and #aeej1 and so on4 ,these values give us a curve of beautiful and surprising smoothness1 from which we seem compelled to draw the direct conclusion that the climate of ,arizona1 during the last #ejj years1 has fluctuated with a regular periodicity of almost precisely #aej years4 ,here again we should be left in doubt 7so far as these #abc observations go7 whether the essential factor be a fluctuation of temperature or an alternation of moisture and aridity2 but the character of the ,arizona climate1 and the known facts of recent years1 encourage the belief that the latter is the more direct and more important factor4 ,illustration3 ,fig4 #cb4 ,long-period fluctuation in rate of growth of ,arizona trees 7smoothed in #ajj-year periods71 from ,a4,d4 #acij-adij to ,a4,d4 #ahaj-aiaj4 ,it has been often remarked that our common ,european trees1 such for instance as the elm or the cherry1 tend to have larger leaves the further north we go2 but in this case the phenomenon is to be ascribed rather to the longer hours of daylight than to any difference of temperature#aeb4 ,the point is a physiological one1 and consequently of little importance to us here#aec2 the main point for the morphologist is the very simple one that physical or climatic conditions have greatly influenced the rate of growth4 ,the case is analogous to the direct influence of temperature in modifying the colouration of organisms1 such as certain butterflies4 ,now if temperature affects the rate of growth in strict uniformity1 alike in all directions and in all parts or organs1 its direct effect must be limited to the production of local races or varieties differing from one another in actual magnitude1 as the ,siberian goldfinch or bullfinch1 for instance1 differ from our own4 ,but if there be even ever so little of a discriminating action in the enhancement of growth by temperature1 such that it accelerates the growth of one tissue or one organ more than another1 then it is evident that it must at once lead to an actual difference of racial1 or even 8specific0 form4 ,it is not to be doubted that the various factors of climate have some such discriminating influence4 ,the leaves of our northern trees may themselves be an instance of it2 and we have1 #abd probably1 a still better instance of it in the case of ,alpine plants#aed1 whose general habit is dwarfed1 though their floral organs suffer little or no reduction4 ,the subject1 however1 has been little investigated1 and great as its theoretic importance would be to us1 we must meanwhile leave it alone4 ,osmotic factors in growth4 ,the curves of growth which we have now been studying represent phenomena which have at least a two-fold interest1 morphological and physiological4 ,to the morphologist1 who recognises that form is a 8function0 of growth1 the important facts are mainly these3 7#a7 that the rate of growth is an orderly phenomenon1 with general features common to very various organisms1 while each particular organism has its own characteristic phenomena1 or 8specific constants02 7#b7 that rate of growth varies with temperature1 that is to say with season and with climate1 and with various other physical factors1 external and internal2 7#c7 that it varies in different parts of the body1 and according to various directions or axes2 such variations being definitely correlated with one another1 and thus giving rise to the characteristic proportions1 or form1 of the organism1 and to the changes in form which it undergoes in the course of its development4 ,but to the physiologist1 the phenomenon suggests many other important considerations1 and throws much light on the very nature of growth itself1 as a manifestation of chemical and physical energies4 ,to be content to shew that a certain rate of growth occurs in a certain organism under certain conditions1 or to speak of the phenomenon as a 8reaction0 of the living organism to its environment or to certain stimuli1 would be but an example of that 8lack of particularity#aee0 in regard to the actual mechanism of physical cause and effect with which we are apt in biology to be too easily satisfied4 ,but in the case of rate of growth we pass somewhat #abe beyond these limitations2 for the affinity with certain types of chemical reaction is plain1 and has been recognised by a great number of physiologists4 ,a large part of the phenomenon of growth1 both in animals and still more conspicuously in plants1 is associated with 8turgor10 that is to say1 is dependent on osmotic conditions2 in other words1 the velocity of growth depends in great measure 7as we have already seen1 p4 #aac7 on the amount of water taken up into the living cells1 as well as on the actual amount of chemical metabolism performed by them#aef4 ,of the chemical phenomena which result in the actual increase of protoplasm we shall speak presently1 but the rle of water in growth deserves also a passing word1 even in our morphological enquiry4 ,it has been shewn by ,loeb that in ,cerianthus or ,tubularia1 for instance1 the cells in order to grow must be turgescent2 and this turgescence is only possible so long as the salt water in which the cells lie does not overstep a certain limit of concentration4 ,the limit1 in the case of ,tubularia1 is passed when the salt amounts to about #ed per cent4 ,sea-water contains some #cj to #ce p4c4 of salts2 but it is when the salinity falls much below this normal1 to about #bb p4c41 that ,tubularia exhibits its maximal turgescence1 and maximal growth4 ,a further dilution is said to act as a poison to the animal4 ,loeb has also shewn#aeg that in certain eggs 7e4g4 those of the little fish ,fundulus7 an increasing concentration of the sea-water 7leading to a diminishing 8water-content0 of the egg7 retards the rate of segmentation and at length renders segmentation impossible2 though nuclear division1 by the way1 goes on for some time longer4 ,among many other observations of the same kind1 those of ,bialaszewicz#aeh1 on the early growth of the frog1 are notable4 ,he shews that the growth of the embryo while still within the #abf vitelline membrane depends wholly on the absorption of water2 that whether rate of growth be fast or slow 7in accordance with temperature7 the quantity of water absorbed is constant2 and that successive changes of form correspond to definite quantities of water absorbed4 ,the solid residue1 as ,davenport has also shewn1 may actually and notably diminish1 while the embryo organism is increasing rapidly in bulk and weight4 ,on the other hand1 in later stages and especially in the higher animals1 the percentage of water tends to diminish4 ,this has been shewn by ,davenport in the frog1 by ,potts in the chick1 and particularly by ,fehling in the case of man#aei4 ,fehling's results are epitomised as follows3 ,age in weeks #f #ag #bb #bd #bf #cj #ce #ci ,percentage of water #ige #iah #ibj #hii #hfd #hcg #hbi #gdb ,and the following illustrate ,davenport's results for the frog3 ,age in weeks #a #b #e #g #i #ad #da #hd ,percentage of water #efc #ehe #gfg #hic #ica #iej #ijb #hge ,to such phenomena of osmotic balance as the above1 or in other words to the dependence of growth on the uptake of water1 ,hber#afj and also ,loeb are inclined to refer the modifications of form which certain phyllopod crustacea undergo1 when the highly saline waters which they inhabit are further concentrated1 or are abnormally diluted4 ,their growth1 according to ,schmankewitsch1 is retarded by increase of concentration1 so that the individuals from the more saline waters appear stunted and dwarfish2 and they become altered or transformed in other ways1 which for the most part suggest 8degeneration10 or a failure to attain full and perfect development#afa4 ,important physiological changes also ensue4 ,the rate of multiplication is increased1 and parthenogenetic reproduction is encouraged4 ,male individuals become plentiful in the less saline waters1 and here the females bring forth #abg their young alive2 males disappear altogether in the more concentrated brines1 and then the females lay eggs1 which1 however1 only begin to develop when the salinity is somewhat reduced4 ,the best-known case is the little 8brine-shrimp10 ,artemia salina1 found1 in one form or another1 all the world over1 and first discovered more than a century and a half ago in the salt-pans at ,lymington4 ,among many allied forms1 one1 ,a4 milhausenii1 inhabits the natron-lakes of ,egypt and ,arabia1 where1 under the name of 8loul10 or 8,fezzan-worm10 it is eaten by the ,arabs#afb4 ,this fact is interesting1 because it indicates 7and investigation has apparently confirmed7 that the tissues of the creature are not impregnated with salt1 as is the medium in which it lives4 ,the fluids of the body1 the milieu interne 7as ,claude ,bernard called them#afc71 are no more salt than are those of any ordinary crustacean or other animal1 but contain only some #jh per cent4 of ,na,cl#afd1 while the milieu externe may contain #aj1 #bj1 or more per cent4 of this and other salts2 which is as much as to say that the skin1 or body-wall1 of the creature acts as a 8semi-permeable membrane10 through which the dissolved salts are not permitted to diffuse1 though water passes through freely3 until a statical equilibrium 7doubtless of a complex kind7 is at length attained4 ,among the structural changes which result from increased concentration of the brine 7partly during the life-time of the individual1 but more markedly during the short season which suffices for the development of three or four1 or perhaps more1 successive generations71 it is found that the tail comes to bear fewer and fewer bristles1 and the tail-fins themselves tend at last to disappear2 these changes corresponding to what have been #abh described as the specific characters of ,a4 milhausenii1 and of a still more extreme form1 ,a4 kppeniana2 while on the other hand1 progressive dilution of the water tends to precisely opposite conditions1 resulting in forms which have also been described as separate species1 and even referred to a separate genus1 ,callaonella1 closely akin to ,branchipus 7,fig4 #cc74 ,pari passu with these changes1 there is a marked change in the relative lengths of the fore and hind portions of the body1 that is to say1 of the 8cephalothorax0 and abdomen3 the latter growing relatively longer1 the salter the water4 ,in other words1 not only is the rate of growth of the whole ,illustration3 ,fig4 #cc4 ,brine-shrimps 7,artemia71 from more or less saline water4 ,upper figures shew tail-segment and tail-fins2 lower figures1 relative length of cephalothorax and abdomen4 7,after ,abonyi47 animal lessened by the saline concentration1 but the specific rates of growth in the parts of its body are relatively changed4 ,this latter phenomenon lends itself to numerical statement1 and ,abonyi has lately shewn that we may construct a very regular curve1 by plotting the proportionate length of the creature's abdomen against the salinity1 or density1 of the water2 and the several species of ,artemia1 with all their other correlated specific characters1 are then found to occupy successive1 more or less well-defined1 and more or less extended1 regions of the curve 7,fig4 #cc74 ,in short1 the density of the water is so clearly a 8function0 of the specific #abi character1 that we may briefly define the species ,artemia 7,callaonella7 ,jelskii1 for instance1 as the ,artemia of density #ajjj-ajaj 7,na,cl71 or the typical ,a4 salina1 or principalis1 as the ,artemia of density #ajah-ajbe1 and so forth4 ,it is a most interesting fact that these ,artemiae1 under the protection of their semi-permeable skins1 are capable of living in waters not only of great density1 but of very varied chemical composition4 ,the natron-lakes1 for instance1 contain large quantities of magnesium ,illustration3 ,fig4 #cd4 ,percentage ratio of length of abdomen to cephalothorax in brine-shrimps1 at various salinities4 7,after ,abonyi47 sulphate2 and the ,artemiae continue to live equally well in artificial solutions where this salt1 or where calcium chloride1 has largely taken the place of sodium chloride in the more common habitat4 ,furthermore1 such waters as those of the natron-lakes are subject to very great changes of chemical composition as concentration proceeds1 owing to the different solubilities of the constituent salts4 ,it appears that the forms which the ,artemiae assume1 and the changes which they undergo1 are identical or #acj indistinguishable1 whichever of the above salts happen to exist1 or to predominate1 in their saline habitat4 ,at the same time we still lack 7so far as ,i know7 the simple1 but crucial experiments which shall tell us whether1 in solutions of different chemical composition1 it is at equal densities1 or at 8isotonic0 concentrations 7that is to say1 under conditions where the osmotic pressure1 and consequently the rate of diffusion1 is identical71 that the same structural changes are produced1 or corresponding phases of equilibrium attained4 ,while ,hber and others#afe have referred all these phenomena to osmosis1 ,abonyi is inclined to believe that the viscosity1 or mechanical resistance1 of the fluid also reacts upon the organism2 and other possible modes of operation have been suggested4 ,but we may take it for certain that the phenomenon as a whole is not a simple one2 and that it includes besides the passive phenomena of intermolecular diffusion1 some other form of activity which plays the part of a regulatory mechanism#aff4 ,growth and catalytic action4 ,in ordinary chemical reactions we have to deal 7#a7 with a specific velocity proper to the particular reaction1 7#b7 with variations due to temperature and other physical conditions1 7#c7 according to van't ,hoff's 8,law of ,mass10 with variations due to the actual quantities present of the reacting substances1 and 7#d7 in certain cases1 with variations due to the presence of 8catalysing agents40 ,in the simpler reactions1 the law of mass involves a steady1 gradual slowing-down of the process1 according to a logarithmic ratio1 as the reaction proceeds and as the initial amount of substance diminishes2 a phenomenon1 however1 which need not necessarily #aca occur in the organism1 part of whose energies are devoted to the continual bringing-up of fresh supplies4 ,catalytic action occurs when some substance1 often in very minute quantity1 is present1 and by its presence produces or accelerates an action1 by opening 8a way round10 without the catalytic agent itself being diminished or used up#afg4 ,here the velocity curve1 though quickened1 is not necessarily altered in form1 for gradually the law of mass exerts its effect and the rate of the reaction gradually diminishes4 ,but in certain cases we have the very remarkable phenomenon that a body acting as a catalyser is necessarily formed as a product1 or bye-product1 of the main reaction1 and in such a case as this the reaction-velocity will tend to be steadily accelerated4 ,instead of dwindling away1 the reaction will continue with an ever-increasing velocity3 always subject to the reservation that limiting conditions will in time make themselves felt1 such as a failure of some necessary ingredient1 or a development of some substance which shall antagonise or finally destroy the original reaction4 ,such an action as this we have learned1 from ,ostwald1 to describe as 8autocatalysis40 ,now we know that certain products of protoplasmic metabolism1 such as the enzymes1 are very powerful catalysers1 and we are entitled to speak of an autocatalytic action on the part of protoplasm itself4 ,this catalytic activity of protoplasm is a very important phenomenon4 ,as ,blackman says1 in the address already quoted1 the botanists 7or the zoologists7 8call it growth1 attribute it to a specific power of protoplasm for assimilation1 and leave it alone as a fundamental phenomenon2 but they are much concerned as to the distribution of new growth in innumerable specifically distinct forms40 ,while the chemist1 on the other hand1 recognises it as a familiar phenomenon1 and refers it to the same category as his other known examples of autocatalysis4 #acb ,this very important1 and perhaps even fundamental phenomenon of growth would seem to have been first recognised by ,professor ,chodat of ,geneva1 as we are told by his pupil ,monnier#afh4 8,on peut bien1 ainsi que ,m4 ,chodat l'a propos1 considrer l'accroissement comme une raction chimique complexe1 dans laquelle le catalysateur est la cellule vivante1 et les corps en prsence sont l'eau1 les sels1 et l'acide carbonique40 ,very soon afterwards a similar suggestion was made by ,loeb#afi1 in connection with the synthesis of nuclein or nuclear protoplasm2 for he remarked that1 as in an autocatalysed chemical reaction1 the velocity of the synthesis increases during the initial stage of cell-division in proportion to the amount of nuclear matter already synthesised4 ,in other words1 one of the products of the reaction1 i4e4 one of the constituents of the nucleus1 accelerates the production of nuclear from cytoplasmic material4 ,the phenomenon of autocatalysis is by no means confined to living or protoplasmic chemistry1 but at the same time it is characteristically1 and apparently constantly1 associated therewith4 ,and it would seem that to it we may ascribe a considerable part of the difference between the growth of the organism and the simpler growth of the crystal#agj3 the fact1 for instance1 that the cell can grow in a very low concentration of its nutritive solution1 while the crystal grows only in a supersaturated one2 and the fundamental fact that the nutritive solution need only contain the more or less raw materials of the complex constituents of the cell1 while the crystal grows only in a solution of its own actual substance#aga4 ,as ,f4 ,f4 ,blackman has pointed out1 the multiplication of an organism1 for instance the prodigiously rapid increase of a bacterium1 #acc which tends to double its numbers every few minutes1 till 7were it not for limiting factors7 its numbers would be all but incalculable in a day#agb1 is a simple but most striking illustration of the potentialities of protoplasmic catalysis2 and 7apart from the large share taken by mere 8turgescence0 or imbibition of water7 the same is true of the growth1 or cell-multiplication1 of a multicellular organism in its first stage of rapid acceleration4 ,it is not necessary for us to pursue this subject much further1 for it is sufficiently clear that the normal 8curve of growth0 of an organism1 in all its general features1 very closely resembles the velocity-curve of chemical autocatalysis4 ,we see in it the first and most typical phase of greater and greater acceleration2 this is followed by a phase in which limiting conditions 7whose details are practically unknown7 lead to a falling off of the former acceleration2 and in most cases we come at length to a third phase1 in which retardation of growth is succeeded by actual diminution of mass4 ,here we may recognise the influence of processes1 or of products1 which have become actually deleterious2 their deleterious influence is staved off for a while1 as the organism draws on its accumulated reserves1 but they lead ere long to the stoppage of all activity1 and to the physical phenomenon of death4 ,but when we have once admitted that the limiting conditions of growth1 which cause a phase of retardation to follow a phase of acceleration1 are very imperfectly known1 it is plain that1 ipso facto1 we must admit that a resemblance rather than an identity between this phenomenon and that of chemical autocatalysis is all that we can safely assert meanwhile4 ,indeed1 as ,enriques has shewn1 points of contrast between the two phenomena are not lacking2 for instance1 as the chemical reaction draws to a close1 it is by the gradual attainment of chemical equilibrium3 but when organic growth draws to a close1 it is by reason of a very different kind of equilibrium1 due in the main to the gradual differentiation of the organism into parts1 among whose peculiar #acd and specialised functions that of cell-multiplication tends to fall into abeyance#agc4 ,it would seem to follow1 as a natural consequence1 from what has been said1 that we could without much difficulty reduce our curves of growth to logarithmic formulae#agd akin to those which the physical chemist finds applicable to his autocatalytic reactions4 ,this has been diligently attempted by various writers#age2 but the results1 while not destructive of the hypothesis itself1 are only partially successful4 ,the difficulty arises mainly from the fact that1 in the life-history of an organism1 we have usually to deal 7as indeed we have seen7 with several recurrent periods of relative acceleration and retardation4 ,it is easy to find a formula which shall satisfy the conditions during any one of these periodic phases1 but it is very difficult to frame a comprehensive formula which shall apply to the entire period of growth1 or to the whole duration of life4 ,but if it be meanwhile impossible to formulate or to solve in precise mathematical terms the equation to the growth of an organism1 we have yet gone a very long way towards the solution of such problems when we have found a 8qualitative expression10 as ,poincar puts it2 that is to say1 when we have gained a fair approximate knowledge of the general curve which represents the unknown function4 ,as soon as we have touched on such matters as the chemical phenomenon of catalysis1 we are on the threshold of a subject which1 if we were able to pursue it1 would soon lead us far into the special domain of physiology2 and there it would be necessary to follow it if we were dealing with growth as a phenomenon in itself1 instead of merely as a help to our study and comprehension of form4 ,for instance the whole question of diet1 of overfeeding and underfeeding1 would present itself for discussion#agf4 ,but without attempting to open up this large subject1 we may say a #ace further passing word upon the essential fact that certain chemical substances have the power of accelerating or of retarding1 or in some way regulating1 growth1 and of so influencing directly the morphological features of the organism4 ,thus lecithin has been shewn by ,hatai#agg1 ,danilewsky#agh and others to have a remarkable power of stimulating growth in various animals2 and the so-called 8auximones10 which ,professor ,bottomley prepares by the action of bacteria upon peat appear to be1 after a somewhat similar fashion1 potent accelerators of the growth of plants4 ,but by much the most interesting cases1 from our point of view1 are those where a particular substance appears to exert a differential effect1 stimulating the growth of one part or organ of the body more than another4 ,it has been known for a number of years that a diseased condition of the pituitary body accompanies the phenomenon known as 8acromegaly10 in which the bones are variously enlarged or elongated1 and which is more or less exemplified in every skeleton of a 8giant02 while on the other hand1 disease or extirpation of the thyroid causes an arrest of skeletal development1 and1 if it take place early1 the subject remains a dwarf4 ,these1 then1 are well-known illustrations of the regulation of function by some internal glandular secretion1 some enzyme or 8hormone0 7as ,bayliss and ,starling call it71 or 8harmozone10 as ,gley calls it in the particular case where the function regulated is that of growth1 with its consequent influence on form4 ,among other illustrations 7which are plentiful7 we have1 for instance the growth of the placental decidua1 which ,loeb has shewn to be due to a substance given off by the corpus luteum of the ovary1 giving to the uterine tissues an abnormal capacity for growth1 which in turn is called into action by the contact of the ovum1 or even of any foreign body4 ,and various sexual characters1 such as the plumage1 comb and spurs of the cock1 are believed in like manner to arise in response to some particular internal secretion4 ,when the source of such a secretion is removed by castration1 well-known morphological changes take place in various animals2 and when a converse change takes place1 the female acquires1 in greater or less degree1 characters which are #acf proper to the male1 as in certain extreme cases1 known from time immemorial1 when late in life a hen assumes the plumage of the cock4 ,there are some very remarkable experiments by ,gudernatsch1 in which he has shewn that by feeding tadpoles 7whether of frogs or toads7 on thyroid gland substance1 their legs may be made to grow out at any time1 days or weeks before the normal date of their appearance#agi4 ,no other organic food was found to produce the same effect2 but since the thyroid gland is known to contain iodine#ahj1 ,morse experimented with this latter substance1 and found that if the tadpoles were fed with iodised amino-acids the legs developed precociously1 just as when the thyroid gland itself was used4 ,we may take it1 then1 as an established fact1 whose full extent and bearings are still awaiting investigation1 that there exist substances both within and without the organism which have a marvellous power of accelerating growth1 and of doing so in such a way as to affect not only the size but the form or proportions of the organism4 ,if we once admit1 as we are now bound to do1 the existence of such factors as these1 which1 by their physiological activity and apart from any direct action of the nervous system1 tend towards the acceleration of growth and consequent modification of form1 we are led into wide fields of speculation by an easy and a legitimate pathway4 ,professor ,gley carries such speculations a long1 long way3 for he says#aha that by these chemical influences 8,toute une partie de la construction des tres parait s'expliquer d'une faon toute mcanique4 ,la forteresse1 si longtemps inaccessible1 du vitalisme est entame4 ,car la notion morphognique tait1 suivant le mot de ,dastre#ahb1 comme #8le dernier rduit de la force vitale4' 0 ,the physiological speculations we need not discuss3 but1 to take a single example from morphology1 we begin to understand the possibility1 and to comprehend the probable meaning1 of the #acg all but sudden appearance on the earth of such exaggerated and almost monstrous forms as those of the great secondary reptiles and the great tertiary mammals#ahc4 ,we begin to see that it is in order to account1 not for the appearance1 but for the disappearance of such forms as these that natural selection must be invoked4 ,and we then1 ,i think1 draw near to the conclusion that what is true of these is universally true1 and that the great function of natural selection is not to originate1 but to remove3 donec ad interitum genus id natura redegit#ahd4 ,the world of things living1 like the world of things inanimate1 grows of itself1 and pursues its ceaseless course of creative evolution4 ,it has room1 wide but not unbounded1 for variety of living form and structure1 as these tend towards their seemingly endless1 but yet strictly limited1 possibilities of permutation and degree3 it has room for the great and for the small1 room for the weak and for the strong4 ,environment and circumstance do not always make a prison1 wherein perforce the organism must either live or die2 for the ways of life may be changed1 and many a refuge found1 before the sentence of unfitness is pronounced and the penalty of extermination paid4 ,but there comes a time when 8variation10 in form1 dimensions1 or other qualities of the organism1 goes farther than is compatible with all the means at hand of health and welfare for the individual and the stock2 when1 under the active and creative stimulus of forces from within and from without1 the active and creative energies of growth pass the bounds of physical and physiological equilibrium3 and so reach the limits which1 as again ,lucretius tells us1 natural law has set between what may and what may not be1 8et quid quaeque queant per foedera naturai quid porro nequeant40 ,then1 at last1 we are entitled to use the customary metaphor1 and to see in natural selection an inexorable force1 whose function #ach is not to create but to destroy1--to weed1 to prune1 to cut down and to cast into the fire#ahe4 ,regeneration1 or growth and repair4 ,the phenomenon of regeneration1 or the restoration of lost or amputated parts1 is a particular case of growth which deserves separate consideration4 ,as we are all aware1 this property is manifested in a high degree among invertebrates and many cold-blooded vertebrates1 diminishing as we ascend the scale1 until at length1 in the warm-blooded animals1 it lessens down to no more than that vis medicatrix which heals a wound4 ,ever since the days of ,aristotle1 and especially since the experiments of ,trembley1 ,raumur and ,spallanzani in the middle of the eighteenth century1 the physiologist and the psychologist have alike recognised that the phenomenon is both perplexing and important4 ,the general phenomenon is amply discussed elsewhere1 and we need only deal with it in its immediate relation to growth#ahf4 ,regeneration1 like growth in other cases1 proceeds with a velocity which varies according to a definite law2 the rate varies with the time1 and we may study it as velocity and as acceleration4 ,let us take1 as an instance1 ,miss ,m4 ,l4 ,durbin's measurements of the rate of regeneration of tadpoles' tails3 the rate being here measured in terms1 not of mass1 but of length1 or longitudinal increment#ahg4 ,from a number of tadpoles1 whose average length was #cdb mm41 their tails being on an average #bab mm4 long1 about half the tail #aci 7#aae mm47 was cut off1 and the amounts regenerated in successive periods are shewn as follows3 ,days after operation #c #g #aj #ad #ah #bd #cj 7#a7 ,amount regenerated in mm4 #ad #cd #dc #eb #ee #fb #fe 7#b7 ,increment during each period #ad #bj #ji #ji #jc #jg #jc 7#c7 787 ,rate per day during each period #jdf #jej #jcj #jbe #jjg #jab #jje ,the first line of numbers in this table1 if plotted as a curve against the number of days1 will give us a very satisfactory view of the 8curve of growth0 within the period of the observations3 that is to say1 of the successive relations of length to time1 or the velocity of the process4 ,but the third line is not so satisfactory1 and must not be plotted directly as an acceleration curve4 ,for it is evident that the 8rates0 here determined do not correspond to velocities at the dates to which they are referred1 but are the mean velocities over a preceding period2 and moreover the periods over which these means are taken are here of very unequal length4 ,but we may draw a good deal more information from this experiment1 if we begin by drawing a smooth curve1 as nearly as possible through the points corresponding to the amounts regenerated 7according to the first line of the table72 and if we then interpolate from this smooth curve the actual lengths attained1 day by day1 and derive from these1 by subtraction1 the successive daily increments1 which are the measure of the daily mean velocities 7,table1 p4 #ada74 7,the more accurate and strictly correct method would be to draw successive tangents to the curve47 ,in our curve of growth 7,fig4 #ce7 we cannot safely interpolate values for the first three days1 that is to say for the dates between amputation and the first actual measurement of the regenerated part4 ,what goes on in these three days is very important2 but we know nothing about it1 save that our curve descended to zero somewhere or other within that period4 ,as we have already learned1 we can more or less safely interpolate between known points1 or actual observations2 but here we have no known starting-point4 ,in short1 for all that the observations tell us1 and for all that the appearance of the curve can suggest1 the curve of growth may have descended evenly to the base-line1 which it would then have reached about the end of the second #adj day2 or it may have had within the first three days a change of direction1 or 8point of inflection10 and may then have sprung at once from the base-line at zero4 ,that is to say1 there may ,illustration3 ,fig4 #ce4 ,curve of regenerative growth in tadpoles' tails4 7,from ,m4 ,l4 ,durbin's data47 have been an intervening 8latent period10 during which no growth ,illustration3 ,fig4 #cf4 ,mean daily increments1 corresponding to ,fig4 #ce4 #ada occurred1 between the time of injury and the first measurement of regenerative growth2 or1 for all we yet know1 regeneration may have begun at once1 but with a velocity much less than that which it afterwards attained4 ,this apparently trifling difference would correspond to a very great difference in the nature of the phenomenon1 and would lead to a very striking difference in the curve which we have next to draw4 ,the curve already drawn 7,fig4 #ce7 illustrates1 as we have seen1 the relation of length to time1 i4e4 ,l_/,t ,v4 ,the second 7,fig4 #cf7 represents the rate of change of velocity2 it sets ,v against ,t2 ,the foregoing table1 extended by graphic interpolation4 ,total ,daily ,days increment increment ,logs of do4 #a -- -- -- #b -- -- -- #c #adj #fj #agh #d #bjj #eb #agb #e #beb #de #afe #f #big #dc #afc #g #cdj #cb #aea #h #cgb #cj #adh #i #djb #bh #ade #aj #dcj #bb #acd #aa #deb #ba #acb #ab #dgc #ai #abh #ac #dib #ah #abf #ad #eaj #ag #abc #ae #ebg #ac #aaa #af #edj #ad #aae #ag #eed #ac #aaa #ah #efg #aa #ajd #ai #egh #aj #ajj #bj #ehh #aj #ajj #ba #eih #ji #ie #bb #fjg #jg #he #bc #fad #jg #hd #bd #fba #jh #ij #be #fbi #jf #gh #bf #fce #jf #gh #bg #fda #je #gj #bh #fdf #jd #fj #bi #fej #jc #dh #cj #fec #adb and ,v_/,t or ,l_/,t#b1 represents 7as we have learned7 the acceleration of growth1 this being simply the 8differential coefficient10 the first derivative of the former curve4 ,illustration3 ,fig4 #cg4 ,logarithms of values shewn in ,fig4 #cf4 ,now1 plotting this acceleration curve from the date of the first measurement made three days after the amputation of the tail 7,fig4 #cf71 we see that it has no point of inflection1 but falls steadily1 only more and more slowly1 till at last it comes down nearly to the base-line4 ,the velocities of growth are continually diminishing4 ,as regards the missing portion at the beginning of the curve1 we cannot be sure whether it bent round and came down to zero1 or whether1 as in our ordinary acceleration curves of growth from birth onwards1 it started from a maximum4 ,the former is1 in this case1 obviously the more probable1 but we cannot be sure4 ,as regards that large portion of the curve which we are acquainted with1 we see that it resembles the curve known as a rectangular hyperbola1 which is the form assumed when two variables 7in this case ,v and ,t7 vary inversely as one another4 ,if we take the logarithms of the velocities 7as given in the table7 and plot them against time 7,fig4 #cg71 we see that they fall1 approximately1 into a straight line2 and if this curve be plotted on the #adc proper scale we shall find that the angle which it makes with the base is about #be1 of which the tangent is #df1 or in round numbers 4 ,had the angle been #de 7tan #de #a71 the curve would have been actually a rectangular hyperbola1 with ,v ,t constant4 ,as it is1 we may assume1 provisionally1 that it belongs to the same family of curves1 so that ,vm ,tn1 or ,vm_/n ,t1 or ,v ,tn_/m1 are all severally constant4 ,in other words1 the velocity varies inversely as some power of the time1 or vice versa4 ,and in this particular case1 the equation ,v ,t#b constant1 holds very nearly true2 that is to say the velocity varies1 or tends to vary1 inversely as the square of the time4 ,if some general law akin to this could be established as a general law1 or even as a common rule1 it would be of great importance4 ,illustration3 ,fig4 #ch4 ,rate of regenerative growth in larger tadpoles4 ,but though neither in this case nor in any other can the minute increments of growth during the first few hours1 or the first couple of days1 after injury1 be directly measured1 yet the most important point is quite capable of solution4 ,what the foregoing curve leaves us in ignorance of1 is simply whether growth starts at zero1 with zero velocity1 and works up quickly to a maximum velocity from which it afterwards gradually falls away2 or whether after a latent period1 it begins1 so to speak1 in full force4 ,the answer #add to this question-depends on whether1 in the days following the first actual measurement1 we can or cannot detect a daily increment in velocity1 before that velocity begins its normal course of diminution4 ,now this preliminary ascent to a maximum1 or point of inflection of the curve1 though not shewn in the above-quoted experiment1 has been often observed3 as for instance1 in another similar experiment by the author of the former1 the tadpoles being in this case of larger size 7average #dia mm47#ahh4 ,days #c #e #g #aj #ab #ad #ag #bd #bh #ca ,increment #jhf #bae #cff #ebj #eie #fch #gaj #gfj #hbj #hdj ,or1 by graphic interpolation1 ,total ,daily ,days increment do4 #a #bc #bc #b #ec #cj #c #hf #cc #d #acj #dd #e #bjj #gj #f #bgh #gh #g #ceh #hj #h #dcj #gb #i #dij #fj #aj #ebi #ci #aa #efb #cc #ab #eij #bh #ac #fac #bc #ad #fch #be #ae #ffa #bc #af #fha #bj #ag #gjj #ai etc4 ,the acceleration curve is drawn in ,fig4 #ci4 ,here we have just what we lacked in the former case1 namely a visible point of inflection in the curve about the seventh day 7,figs4 #ch1 #ci71 whose existence is confirmed by successive observations on the #crd1 #eth1 #gth and #ajth days1 and which justifies to some extent our extrapolation for the otherwise unknown period up to and ending with the third day2 but even here there is a short space near the very beginning during which we are not quite sure of the precise slope of the curve4 ,we have now learned that1 according to these experiments1 with which many others are in substantial agreement1 the rate of growth in the regenerative process is as follows4 ,after a very short latent period1 not yet actually proved but whose existence is highly probable1 growth commences with a velocity which very #ade rapidly increases to a maximum4 ,the curve quickly1--almost suddenly1--changes its direction1 as the velocity begins to fall2 and the rate of fall1 that is1 the negative acceleration1 proceeds at a slower and slower rate1 which rate varies inversely as some power of the time1 and is found in both of the above-quoted experiments to be very approximately as #a_/,t#b4 ,but it is obvious that the value which we have found for the latter portion of the curve 7however closely it be conformed to7 is only an empirical value2 it has only a temporary usefulness1 and must in time give place to a formula which shall represent the entire phenomenon1 from start to finish4 ,illustration3 ,fig4 #ci4 ,daily increment1 or amount regenerated1 corresponding to ,fig4 #ch4 ,while the curve of regenerative growth is apparently different from the curve of ordinary growth as usually drawn 7and while this apparent difference has been commented on and treated as valid by certain writers7 we are now in a position to see that it only looks different because we are able to study it1 if not from the beginning1 at least very nearly so3 while an ordinary curve of growth1 as it is usually presented to us1 is one which dates1 not #adf from the beginning of growth1 but from the comparatively late1 and unimportant1 and even fallacious epoch of birth4 ,a complete curve of growth1 starting from zero1 has the same essential characteristics as the regeneration curve4 ,indeed the more we consider the phenomenon of regeneration1 the more plainly does it shew itself to us as but a particular case of the general phenomenon of growth#ahi1 following the same lines1 obeying the same laws1 and merely started into activity by the special stimulus1 direct or indirect1 caused by the infliction of a wound4 ,neither more nor less than in other problems of physiology are we called upon1 in the case of regeneration1 to indulge in metaphysical speculation1 or to dwell upon the beneficent purpose which seemingly underlies this process of healing and restoration4 ,it is a very general rule1 though apparently not a universal one1 that regeneration tends to fall somewhat short of a complete restoration of the lost part2 a certain percentage only of the lost tissues is restored4 ,this fact was well known to some of those old investigators1 who1 like the ,abb ,trembley and like ,voltaire1 found a fascination in the study of artificial injury and the regeneration which followed it4 ,sir ,john ,graham ,dalyell1 for instance1 says1 in the course of an admirable paragraph on regeneration#aij3 8,the reproductive faculty 444 is not confined to one portion1 but may extend over many2 and it may ensue even in relation to the regenerated portion more than once4 ,nevertheless1 the faculty gradually weakens1 so that in general every successive regeneration is smaller and more imperfect than the organisation preceding it2 and at length it is exhausted40 ,in certain minute animals1 such as the ,infusoria1 in which the capacity for 8regeneration0 is so great that the entire animal may be restored from the merest fragment1 it becomes of great interest to discover whether there be some definite size at which the fragment ceases to display this power4 ,this question has #adg been studied by ,lillie#aia1 who found that in ,stentor1 while still smaller fragments were capable of surviving for days1 the smallest portions capable of regeneration were of a size equal to a sphere of about #hj in diameter1 that is to say of a volume equal to about one twenty-seventh of the average entire animal4 ,he arrives at the remarkable conclusion that for this1 and for all other species of animals1 there is a 8minimal organisation mass10 that is to say a 8minimal mass of definite size consisting of nucleus and cytoplasm within which the organisation of the species can just find its latent expression40 ,and in like manner1 ,boveri#aib has shewn that the fragment of a sea-urchin's egg capable of growing up into a new embryo1 and so discharging the complete functions of an entire and uninjured ovum1 reaches its limit at about one-twentieth of the original egg1--other writers having found a limit at about one-fourth4 ,these magnitudes1 small as they are1 represent objects easily visible under a low power of the microscope1 and so stand in a very different category to the minimal magnitudes in which life itself can be manifested1 and which we have discussed in chapter ,,ii4 ,a number of phenomena connected with the linear rate of regeneration are illustrated and epitomised in the accompanying diagram 7,fig4 #dj71 which ,i have constructed from certain data given by ,ellis in a paper on the relation of the amount of tail regenerated to the amount removed1 in ,tadpoles4 ,these data are summarised in the next ,table4 ,the tadpoles were all very much of a size1 about #dj mm42 the average length of tail was very near to #bf mm41 or #fe per cent4 of the whole body-length2 and in four series of experiments about #aj1 #bj1 #dj and #fj per cent4 of the tail were severally removed4 ,the amount regenerated in successive intervals of three days is shewn in our table4 ,by plotting the actual amounts regenerated against these three-day intervals of time1 we may interpolate values for the time taken to regenerate definite percentage amounts1 #e per cent41 #aj per cent41 etc4 of the #adh ,illustration3 ,fig4 #dj4 ,relation between the percentage amount of tail removed1 the percentage restored1 and the time required for its restoration4 7,from ,m4 ,m4 ,ellis's data47 ,the ,rate of ,regenerative ,growth in ,tadpoles' ,tails4 7,after ,m4 ,m4 ,ellis1 ,j4 ,exp4 ,zool4 ,,vii1 p4 #dba1 #aiji47 ,body ,tail ,amount ,per cent4 amount regenerated in days length length removed of tail ,series mm4 mm4 mm4 removed #c #f #i #ab #ae #ah #cb ,o #ciege #behie #cb #abcf #ac #ca #dd #dd #dd #dd #dd ,p #djba #bfac #ebh #bjbj #aj #bi #dj #dd #dd #dd #dd ,r #cihf #begj #ajd #djej #f #bj #ca #dj #dh #dh #dh ,s #djcd #bfaa #adh #efg #j #af #cc #ci #de #dh #dh ,each series gives the mean of #bj experiments4 amount removed2 and my diagram is constructed from the four sets of values thus obtained1 that is to say from the four sets of experiments which differed from one another in the amount of tail amputated4 ,to these we have to add the general result of a fifth series of experiments1 which shewed that when as much as #ge per cent4 of the tail was cut off1 no regeneration took place at all1 but the animal presently died4 ,in our diagram1 then1 each #adi curve indicates the time taken to regenerate n per cent4 of the amount removed4 ,all the curves converge towards infinity1 when the amount removed 7as shewn by the ordinate7 approaches #ge per cent42 and all of the curves start from zero1 for nothing is regenerated where nothing had been removed4 ,each curve approximates in form to a cubic parabola4 ,the amount regenerated varies also with the age of the tadpole and with other factors1 such as temperature2 in other words1 for any given age1 or size1 of tadpole and also for various specific temperatures1 a similar diagram might be constructed4 ,the power of reproducing1 or regenerating1 a lost limb is particularly well developed in arthropod animals1 and is sometimes accompanied by remarkable modification of the form of the regenerated limb4 ,a case in point1 which has attracted much attention1 occurs in connection with the claws of certain ,crustacea#aic4 ,in many ,crustacea we have an asymmetry of the great claws1 one being larger than the other and also more or less different in form4 ,for instance1 in the common lobster1 one claw1 the larger of the two1 is provided with a few great 8crushing0 teeth1 while the smaller claw has more numerous teeth1 small and serrated4 ,though ,aristotle thought otherwise1 it appears that the crushing-claw may be on the right or left side1 indifferently2 whether it be on one or the other is a problem of 8chance40 ,it is otherwise in many other ,crustacea1 where the larger and more powerful claw is always left or right1 as the case may be1 according to the species3 where1 in other words1 the 8probability0 of the large or the small claw being left or being right is tantamount to certainty#aid4 ,the one claw is the larger because it has grown the faster2 #aej it has a higher 8coefficient of growth10 and accordingly1 as age advances1 the disproportion between the two claws becomes more and more evident4 ,moreover1 we must assume that the characteristic form of the claw is a 8function0 of its magnitude2 the knobbiness is a phenomenon coincident with growth1 and we never1 under any circumstances1 find the smaller claw with big crushing teeth and the big claw with little serrated ones4 ,there are many other somewhat similar cases where size and form are manifestly correlated1 and we have already seen1 to some extent1 that the phenomenon of growth is accompanied by certain ratios of velocity that lead inevitably to changes of form4 ,meanwhile1 then1 we must simply assume that the essential difference between the two claws is one of magnitude1 with which a certain differentiation of form is inseparably associated4 ,if we amputate a claw1 or if1 as often happens1 the crab 8casts it off10 it undergoes a process of regeneration1--it grows anew1 and evidently does so with an accelerated velocity1 which acceleration will cease when equilibrium of the parts is once more attained3 the accelerated velocity being a case in point to illustrate that vis revulsionis of ,haller1 to which we have already referred4 ,with the help of this principle1 ,przibram accounts for certain curious phenomena which accompany the process of regeneration4 ,as his experiments and those of ,morgan shew1 if the large or knobby claw 7,a7 be removed1 there are certain cases1 e4g4 the common lobster1 where it is directly regenerated4 ,in other cases1 e4g4 ,alpheus#aie1 the other claw 7,b7 assumes the size and form of that which was amputated1 while the latter regenerates itself in the form of the other and weaker one2 ,a and ,b have apparently changed places4 ,in a third case1 as in the crabs1 the ,a-claw regenerates itself as a small or ,b-claw1 but the ,b-claw remains for a time unaltered1 though slowly and in the course of repeated moults it later on assumes the large and heavily toothed ,a-form4 ,much has been written on this phenomenon1 but in essence it is very simple4 ,it depends upon the respective rates of growth1 upon a ratio between the rate of regeneration and the rate of growth of the uninjured limb3 complicated a little1 however1 by #aea the possibility of the uninjured limb growing all the faster for a time after the animal has been relieved of the other4 ,from the time of amputation1 say of ,a1 ,a begins to grow from zero1 with a high 8regenerative0 velocity2 while ,b1 starting from a definite magnitude1 continues to increase1 with its normal or perhaps somewhat accelerated velocity4 ,the ratio between the two velocities of growth will determine whether1 by a given time1 ,a has equalled1 outstripped1 or still fallen short of the magnitude of ,b4 ,that this is the gist of the whole problem is confirmed 7if confirmation be necessary7 by certain experiments of ,wilson's4 ,it is known that by section of the nerve to a crab's claw1 its growth is retarded1 and as the general growth of the animal proceeds the claw comes to appear stunted or dwarfed4 ,now in such a case as that of ,alpheus1 we have seen that the rate of regenerative growth in an amputated large claw fails to let it reach or overtake the magnitude of the growing little claw3 which latter1 in short1 now appears as the big one4 ,but if at the same time as we amputate the big claw we also sever the nerve to the lesser one1 we so far slow down the latter's growth that the other is able to make up to it1 and in this case the two claws continue to grow at approximately equal rates1 or in other words continue of coequal size4 ,the phenomenon of regeneration goes some way towards helping us to comprehend the phenomenon of 8multiplication by fission10 as it is exemplified at least in its simpler cases in many worms and worm-like animals4 ,for physical reasons which we shall have to study in another chapter1 there is a natural tendency for any tube1 if it have the properties of a fluid or semi-fluid substance1 to break up into segments after it comes to a certain length2 and nothing can prevent its doing so1 except the presence of some controlling force1 such for instance as may be due to the pressure of some external support1 or some superficial thickening or other intrinsic rigidity of its own substance4 ,if we add to this natural tendency towards fission of a cylindrical or tubular worm1 the ordinary phenomenon of regeneration1 we have all that is essentially implied in 8reproduction by fission40 ,and in so far #aeb as the process rests upon a physical principle1 or natural tendency1 we may account for its occurrence in a great variety of animals1 zoologically dissimilar2 and also for its presence here and absence there1 in forms which1 though materially different in a physical sense1 are zoologically speaking very closely allied4 ,,conclusion ,,and ,,summary4 ,but the phenomena of regeneration1 like all the other phenomena of growth1 soon carry us far afield1 and we must draw this brief discussion to a close4 ,for the main features which appear to be common to all curves of growth we may hope to have1 some day1 a physical explanation4 ,in particular we should like to know the meaning of that point of inflection1 or abrupt change from an increasing to a decreasing velocity of growth which all our curves1 and especially our acceleration curves1 demonstrate the existence of1 provided only that they include the initial stages of the whole phenomenon3 just as we should also like to have a full physical or physiological explanation of the gradually diminishing velocity of growth which follows1 and which 7though subject to temporary interruption or abeyance7 is on the whole characteristic of growth in all cases whatsoever4 ,in short1 the characteristic form of the curve of growth in length 7or any other linear dimension7 is a phenomenon which we are at present unable to explain1 but which presents us with a definite and attractive problem for future solution4 ,it would seem evident that the abrupt change in velocity must be due1 either to a change in that pressure outwards from within1 by which the 8forces of growth0 make themselves manifest1 or to a change in the resistances against which they act1 that is to say the tension of the surface2 and this latter force we do not by any means limit to 8surface-tension0 proper1 but may extend to the development of a more or less resistant membrane or 8skin10 or even to the resistance of fibres or other histological elements1 binding the boundary layers to the parts within4 ,i take it that the sudden arrest of velocity is much more likely to be due to a sudden increase of resistance than to a sudden diminution of internal energies3 in other words1 ,i suspect that it is coincident with some notable event of histological differentiation1 such as #aec the rapid formation of a comparatively firm skin2 and that the dwindling of velocities1 or the negative acceleration1 which follows1 is the resultant or composite effect of waning forces of growth on the one hand1 and increasing superficial resistance on the other4 ,this is as much as to say that growth1 while its own energy tends to increase1 leads also1 after a while1 to the establishment of resistances which check its own further increase4 ,our knowledge of the whole complex phenomenon of growth is so scanty that it may seem rash to advance even this tentative suggestion4 ,but yet there are one or two known facts which seem to bear upon the question1 and to indicate at least the manner in which a varying resistance to expansion may affect the velocity of growth4 ,for instance1 it has been shewn by ,frazee#aif that electrical stimulation of tadpoles1 with small current density and low voltage1 increases the rate of regenerative growth4 ,as just such an electrification would tend to lower the surface-tension1 and accordingly decrease the external resistance1 the experiment would seem to support1 in some slight degree1 the suggestion which ,i have made4 ,delage#aig has lately made use of the principle of specific rate of growth1 in considering the question of heredity itself4 ,we know that the chromatin of the fertilised egg comes from the male and female parent alike1 in equal or nearly equal shares2 we know that the initial chromatin1 so contributed1 multiplies many thousand-fold1 to supply the chromatin for every cell of the offspring's body2 and it has1 therefore1 a high 8coefficient of growth40 ,if we admit1 with ,van ,beneden and others1 that the initial contributions of male and female chromatin continue to be transmitted to the succeeding generations of cells1 we may then conceive these chromatins to retain each its own coefficient of growth2 and if these differed ever so little1 a gradual preponderance of one or other would make itself felt in time1 and might conceivably explain the preponderating influence of one parent or the other upon the characters of the offspring4 ,indeed ,o4 ,hertwig is said 7according to ,delage's interpretation7 to have actually shewn that we can artificially modify the rate of growth of one or other chromatin1 and so increase or diminish the influence of the maternal or paternal heredity4 ,this theory of ,delage's has its fascination1 but it calls for somewhat large assumptions2 and in particular1 it seems 7like so many other theories relating to the chromosomes7 to rest far too much upon material elements1 rather than on the imponderable dynamic factors of the cell4 #aed ,we may summarise1 as follows1 the main results of the foregoing discussion3 7#a7 ,except in certain minute organisms and minute parts of organisms1 whose form is due to the direct action of molecular forces1 we may look upon the form of the organism as a 8function of growth10 or a direct expression of a rate of growth which varies according to its different directions4 7#b7 ,rate of growth is subject to definite laws1 and the velocities in different directions tend to maintain a ratio which is more or less constant for each specific organism2 and to this regularity is due the fact that the form of the organism is in general regular and constant4 7#c7 ,nevertheless1 the ratio of velocities in different directions is not absolutely constant1 but tends to alter or fluctuate in a regular way2 and to these progressive changes are due the changes of form which accompany 8development10 and the slower changes of form which continue perceptibly in after life4 7#d7 ,the rate of growth is a function of the age of the organism1 it has a maximum somewhat early in life1 after which epoch of maximum it slowly declines4 7#e7 ,the rate of growth is directly affected by temperature1 and by other physical conditions4 7#f7 ,it is markedly affected1 in the way of acceleration or retardation1 at certain physiological epochs of life1 such as birth1 puberty1 or metamorphosis4 7#g7 ,under certain circumstances1 growth may be negative1 the organism growing smaller3 and such negative growth is a common accompaniment of metamorphosis1 and a frequent accompaniment of old age4 7#h7 ,the phenomenon of regeneration is associated with a large temporary increase in the rate of growth 7or 8acceleration0 of growth7 of the injured surface2 in other respects1 regenerative growth is similar to ordinary growth in all its essential phenomena4 ,in this discussion of growth1 we have left out of account a vast number of processes1 or phenomena1 by which1 in the physiological mechanism of the body1 growth is effected and controlled4 ,we have dealt with growth in its relation to magnitude1 and to #aee that relativity of magnitudes which constitutes form2 and so we have studied it as a phenomenon which stands at the beginning of a morphological1 rather than at the end of a physiological enquiry4 ,under these restrictions1 we have treated it as far as possible1 or in such fashion as our present knowledge permits1 on strictly physical lines4 ,in all its aspects1 and not least in its relation to form1 the growth of organisms has many analogies1 some close and some perhaps more remote1 among inanimate things4 ,as the waves grow when the winds strive with the other forces which govern the movements of the surface of the sea1 as the heap grows when we pour corn out of a sack1 as the crystal grows when from the surrounding solution the proper molecules fall into their appropriate places3 so in all these cases1 very much as in the organism itself1 is growth accompanied by change of form1 and by a development of definite shapes and contours4 ,and in these cases 7as in all other mechanical phenomena71 we are led to equate our various magnitudes with time1 and so to recognise that growth is essentially a question of rate1 or of velocity4 ,the differences of form1 and changes of form1 which are brought about by varying rates 7or 8laws07 of growth1 are essentially the same phenomenon whether they be1 so to speak1 episodes in the life-history of the individual1 or manifest themselves as the normal and distinctive characteristics of what we call separate species of the race4 ,from one form1 or ratio of magnitude1 to another there is but one straight and direct road of transformation1 be the journey taken fast or slow2 and if the transformation take place at all1 it will in all likelihood proceed in the self-same way1 whether it occur within the life-time of an individual or during the long ancestral history of a race4 ,no small part of what is known as ,wolff's or von ,baer's law1 that the individual organism tends to pass through the phases characteristic of its ancestors1 or that the life-history of the individual tends to recapitulate the ancestral history of its race1 lies wrapped up in this simple account of the relation between rate of growth and form4 ,but enough of this discussion4 ,let us leave for a while the subject of the growth of the organism1 and attempt to study the conformation1 within and without1 of the individual cell4 #aef ,,chapter ,,iv ,,on ,,the ,,internal ,,form ,,and ,,structure ,,of ,,the ,,cell ,in the early days of the cell-theory1 more than seventy years ago1 ,goodsir was wont to speak of cells as 8centres of growth0 or 8centres of nutrition10 and to consider them as essentially 8centres of force40 ,he looked forward to a time when the forces connected with the cell should be particularly investigated3 when1 that is to say1 minute anatomy should be studied in its dynamical aspect4 8,when this branch of enquiry10 he says 8shall have been opened up1 we shall expect to have a science of organic forces1 having direct relation to anatomy1 the science of organic forms#aih40 ,and likewise1 long afterwards1 ,giard contemplated a science of morphodynamique1--but still looked upon it as forming so guarded and hidden a 8territoire scientifique1 que la plupart des naturalistes de nos jours ne le verront que comme ,mose vit la terre promise1 seulement de loin et sans pouvoir y entrer#aii40 ,to the external forms of cells1 and to the forces which produce and modify these forms1 we shall pay attention in a later chapter4 ,but there are forms and configurations of matter within the cell1 which also deserve to be studied with due regard to the forces1 known or unknown1 of whose resultant they are the visible expression4 ,in the long interval since ,goodsir's day1 the visible structure1 the conformation and configuration1 of the cell1 has been studied far more abundantly than the purely dynamic problems that are associated therewith4 ,the overwhelming progress of microscopic observation has multiplied our knowledge of cellular and intracellular structure2 and to the multitude of visible structures it #aeg has been often easier to attribute virtues than to ascribe intelligible functions or modes of action4 ,but here and there nevertheless1 throughout the whole literature of the subject1 we find recognition of the inevitable fact that dynamical problems lie behind the morphological problems of the cell4 ,btschli pointed out forty years ago1 with emphatic clearness1 the failure of morphological methods1 and the need for physical methods1 if we were to penetrate deeper into the essential nature of the cell#bjj4 ,and such men as ,loeb and ,whitman1 ,driesch and ,roux1 and not a few besides1 have pursued the same train of thought and similar methods of enquiry4 ,whitman#bja1 for instance1 puts the case in a nutshell when1 in speaking of the so-called 8caryokinetic0 phenomena of nuclear division1 he reminds us that the leading idea in the term 8caryokinesis0 is motion1--8motion viewed as an exponent of forces residing in1 or acting upon1 the nucleus4 ,it regards the nucleus as a seat of energy1 which displays itself in phenomena of motion#bjb40 ,in short it would seem evident that1 except in relation to a dynamical investigation1 the mere study of cell structure has but little value of its own4 ,that a given cell1 an ovum for instance1 contains this or that visible substance or structure1 germinal vesicle or germinal spot1 chromatin or achromatin1 chromosomes or centrosomes1 obviously gives no explanation of the activities of the cell4 ,and in all such hypotheses as that of 8pangenesis10 in all the theories which attribute specific properties to micellae1 #aeh idioplasts1 ids1 or other constituent particles of protoplasm or of the cell1 we are apt to fall into the error of attributing to matter what is due to energy and is manifested in force3 or1 more strictly speaking1 of attributing to material particles individually what is due to the energy of their collocation4 ,the tendency is a very natural one1 as knowledge of structure increases1 to ascribe particular virtues to the material structures themselves1 and the error is one into which the disciple is likely to fall1 but of which we need not suspect the master-mind4 ,the dynamical aspect of the case was in all probability kept well in view by those who1 like ,goodsir himself1 first attacked the problem of the cell and originated our conceptions of its nature and functions4 ,but if we speak1 as ,weismann and others speak1 of an 8hereditary substance10 a substance which is split off from the parent-body1 and which hands on to the new generation the characteristics of the old1 we can only justify our mode of speech by the assumption that that particular portion of matter is the essential vehicle of a particular charge or distribution of energy1 in which is involved the capability of producing motion1 or of doing 8work40 ,for1 as ,newton said1 to tell us that a thing 8is endowed with an occult specific quality1 by which it acts and produces manifest effects1 is to tell us nothing2 but to derive two or three general principles of motion#bjc from phenomena would be a very great step in philosophy1 though the causes of these principles were not yet discovered40 ,the things which we see in the cell are less important than the actions which we recognise in the cell2 and these latter we must especially scrutinize1 in the hope of discovering how far they may be attributed to the simple and well-known physical forces1 and how far they be relevant or irrelevant to the phenomena which we associate with1 and deem essential to1 the manifestation of life4 ,it may be that in this way we shall in time draw nigh to the recognition of a specific and ultimate residuum4 #aei ,and lacking1 as we still do lack1 direct knowledge of the actual forces inherent in the cell1 we may yet learn something of their distribution1 if not also of their nature1 from the outward and inward configuration of the cell1 and from the changes taking place in this configuration2 that is to say from the movements of matter1 the kinetic phenomena1 which the forces in action set up4 ,the fact that the germ-cell develops into a very complex structure1 is no absolute proof that the cell itself is structurally a very complicated mechanism3 nor yet1 though this is somewhat less obvious1 is it sufficient to prove that the forces at work1 or latent1 within it are especially numerous and complex4 ,if we blow into a bowl of soapsuds and raise a great mass of many-hued and variously shaped bubbles1 if we explode a rocket and watch the regular and beautiful configuration of its falling streamers1 if we consider the wonders of a limestone cavern which a filtering stream has filled with stalactites1 we soon perceive that in all these cases we have begun with an initial system of very slight complexity1 whose structure in no way foreshadowed the result1 and whose comparatively simple intrinsic forces only play their part by complex interaction with the equally simple forces of the surrounding medium4 ,in an earlier age1 men sought for the visible embryo1 even for the homunculus1 within the reproductive cells2 and to this day1 we scrutinize these cells for visible structure1 unable to free ourselves from that old doctrine of 8pre-formation#bjd40 ,moreover1 the microscope seemed to substantiate the idea 7which we may trace back to ,leibniz#bje and to ,hobbes#bjf71 that there is no limit to the mechanical complexity which we may postulate in an organism1 and no limit1 therefore1 to the hypotheses which we may rest thereon4 ,but no microscopical examination of a stick of sealing-wax1 no study of the material of which it is composed1 can enlighten #afj us as to its electrical manifestations or properties4 ,matter of itself has no power to do1 to make1 or to become3 it is in energy that all these potentialities reside1 energy invisibly associated with the material system1 and in interaction with the energies of the surrounding universe4 ,that 8function presupposes structure0 has been declared an accepted axiom of biology4 ,who it was that so formulated the aphorism ,i do not know2 but as regards the structure of the cell it harks back to ,brcke1 with whose demand for a mechanism1 or organisation1 within the cell histologists have ever since been attempting to comply#bjg4 ,but unless we mean to include thereby invisible1 and merely chemical or molecular1 structure1 we come at once on dangerous ground4 ,for we have seen1 in a former chapter1 that some minute 8organisms0 are already known of such all but infinitesimal magnitudes that everything which the morphologist is accustomed to conceive as 8structure0 has become physically impossible2 and moreover recent research tends generally to reduce1 rather than to extend1 our conceptions of the visible structure necessarily inherent in living protoplasm4 ,the microscopic structure which1 in the last resort or in the simplest cases1 it seems to shew1 is that of a more or less viscous colloid1 or rather mixture of colloids1 and nothing more4 ,now1 as ,clerk ,maxwell puts it1 in discussing this very problem1 8one material system can differ from another only in the configuration and motion which it has at a given instant#bjh40 ,if we cannot assume differences in structure1 we must assume differences in motion1 that is to say1 in energy4 ,and if we cannot do this1 then indeed we are thrown back upon modes of reasoning unauthorised in physical science1 and shall find ourselves constrained to assume1 or to 8admit1 that the properties of a germ are not those of a purely material system40 #afa ,but we are by no means necessarily in this dilemma4 ,for though we come perilously near to it when we contemplate the lowest orders of magnitude to which life has been attributed1 yet in the case of the ordinary cell1 or ordinary egg or germ which is going to develop into a complex organism1 if we have no reason to assume or to believe that it comprises an intricate 8mechanism10 we may be quite sure1 both on direct and indirect evidence1 that1 like the powder in our rocket1 it is very heterogeneous in its structure4 ,it is a mixture of substances of various kinds1 more or less fluid1 more or less mobile1 influenced in various ways by chemical1 electrical1 osmotic1 and other forces1 and in their admixture separated by a multitude of surfaces1 or boundaries1 at which these1 or certain of these forces are made manifest4 ,indeed1 such an arrangement as this is already enough to constitute a 8mechanism02 for we must be very careful not to let our physical or physiological concept of mechanism be narrowed to an interpretation of the term derived from the delicate and complicated contrivances of human skill4 ,from the physical point of view1 we understand by a 8mechanism0 whatsoever checks or controls1 and guides into determinate paths1 the workings of energy2 in other words1 whatsoever leads in the degradation of energy to its manifestation in some determinate form of work1 at a stage short of that ultimate degradation which lapses in uniformly diffused heat4 ,this1 as ,warburg has well explained1 is the general effect or function of the physiological machine1 and in particular of that part of it which we call 8cell-structure#bji40 ,the normal muscle-cell is something which turns energy1 derived from oxidation1 into work2 it is a mechanism which arrests and utilises the chemical energy of oxidation in its downward course2 but the same cell when injured or disintegrated1 loses its 8usefulness10 and sets free a greatly increased proportion of its energy in the form of heat4 ,but very great and wonderful things are done after this manner by means of a mechanism 7whether natural or artificial7 of extreme simplicity4 ,a pool of water1 by virtue of its surface1 #afb is an admirable mechanism for the making of waves2 with a lump of ice in it1 it becomes an efficient and self-contained mechanism for the making of currents4 ,the great cosmic mechanisms are stupendous in their simplicity2 and1 in point of fact1 every great or little aggregate of heterogeneous matter 7not identical in 8phase07 involves1 ipso facto1 the essentials of a mechanism4 ,even a non-living colloid1 from its intrinsic heterogeneity1 is in this sense a mechanism1 and one in which energy is manifested in the movement and ceaseless rearrangement of the constituent particles4 ,for this reason ,graham 7if ,i remember rightly7 speaks somewhere or other of the colloid state as 8the dynamic state of matter02 or in the same philosopher's phrase 7of which ,mr ,hardy#baj has lately reminded us71 it possesses 8energia#baa40 ,let us turn then to consider1 briefly and diagrammatically1 the structure of the cell1 a fertilised germ-cell or ovum for instance1 not in any vain attempt to correlate this structure with the structure or properties of the resulting and yet distant organism2 but merely to see how far1 by the study of its form and its changing internal configuration1 we may throw light on certain forces which are for the time being at work within it4 ,we may say at once that we can scarcely hope to learn more of these forces1 in the first instance1 than a few facts regarding their direction and magnitude2 the nature and specific identity of the force or forces is a very different matter4 ,this latter problem is likely to be very difficult of elucidation1 for the reason1 among others1 that very different forces are often very much alike in their outward and visible manifestations4 ,so it has come to pass that we have a multitude of discordant hypotheses as to the nature of the forces acting within the cell1 and producing1 in cell division1 the 8caryokinetic0 figures of which we are about to speak4 ,one student may1 like ,rhumbler1 choose to account for them by an hypothesis of mechanical traction1 acting on a reticular web of protoplasm#bab2 another1 like ,leduc1 may shew us how in #afc many of their most striking features they may be admirably simulated by the diffusion of salts in a colloid medium2 others again1 like ,gallardo#bac and ,hartog1 and ,rhumbler 7in his earlier papers7#bad1 insist on their resemblance to the phenomena of electricity and magnetism#bae2 while ,hartog believes that the force in question is only analogous to these1 and has a specific identity of its own#baf4 ,all these conflicting views are of secondary importance1 so long as we seek only to account for certain configurations which reveal the direction1 rather than the nature1 of a force4 ,one and the same system of lines of force may appear in a field of magnetic or of electrical energy1 of the osmotic energy of diffusion1 of the gravitational energy of a flowing stream4 ,in short1 we may expect to learn something of the pure or abstract dynamics1 long before we can deal with the special physics of the cell4 ,for indeed 7as ,maillard has suggested71 just as uniform expansion about a single centre1 to whatsoever physical cause it may be due will lead to the configuration of a sphere1 so will any two centres or foci of potential 7of whatsoever kind7 lead to the configurations with which ,faraday made us familiar under the name of 8lines of force#bag02 and this is as much as to say that the phenomenon1 #afd though physical in the concrete1 is in the abstract purely mathematical1 and in its very essence is neither more nor less than a property of three-dimensional space4 ,but as a matter of fact1 in this instance1 that is to say in trying to explain the leading phenomena of the caryokinetic division of the cell1 we shall soon perceive that any explanation which is based1 like ,rhumbler's1 on mere mechanical traction1 is obviously inadequate1 and we shall find ourselves limited to the hypothesis of some polarised and polarising force1 such as we deal with1 for instance1 in the phenomena of magnetism or electricity4 ,let us speak first of the cell itself1 as it appears in a state of rest1 and let us proceed afterwards to study the more active phenomena which accompany its division4 ,our typical cell is a spherical body2 that is to say1 the uniform surface-tension at its boundary is balanced by the outward resistance of uniform forces within4 ,but at times the surface-tension may be a fluctuating quantity1 as when it produces the rhythmical contractions or 8,ransom's waves0 on the surface of a trout's egg2 or again1 while the egg is in contact with other bodies1 the surface-tension may be locally unequal and variable1 giving rise to an amoeboid figure1 as in the egg of ,hydra#bah4 ,within the ovum is a nucleus or germinal vesicle1 also spherical1 and consisting as a rule of portions of 8chromatin10 aggregated together within a more fluid drop4 ,the fact has often been commented upon that1 in cells generally1 there is no correlation of form 7though there apparently is of size7 between the nucleus and the 8cytoplasm10 or main body of the cell4 ,so ,whitman#bai remarks that 8except during the process of division the nucleus seldom departs from its typical spherical form4 ,it divides and sub-divides1 ever returning to the same round or oval form 4444 ,how different with the cell4 ,it preserves the spherical form as rarely as the nucleus departs from it4 ,variation in form marks the beginning and the end of every important chapter in its #afe history40 ,on simple dynamical grounds1 the contrast is easily explained4 ,so long as the fluid substance of the nucleus is qualitatively different from1 and incapable of mixing with1 the fluid or semi-fluid protoplasm which surrounds it1 we shall expect it to be1 as it almost always is1 of spherical form4 ,for1 on the one hand1 it is bounded by a liquid film1 whose surface-tension is uniform2 and on the other1 it is immersed in a medium which transmits on all sides a uniform fluid pressure#bbj4 ,for a similar reason the contractile vacuole of a ,protozoon is spherical in form3 it is just a 8drop0 of fluid1 bounded by a uniform surface-tension and through whose boundary-film diffusion is taking place4 ,but here1 owning to the small difference between the fluid constituting1 and that surrounding1 the drop1 the surface-tension equilibrium is unstable2 it is apt to vanish1 and the rounded outline of the drop1 like a burst bubble1 disappears in a moment#bba4 ,the case of the spherical nucleus is closely akin to the spherical form of the yolk within the bird's egg#bbb4 ,but if the substance of the cell acquire a greater solidity1 as for instance in a muscle #aff cell1 or by reason of mucous accumulations in an epithelium cell1 then the laws of fluid pressure no longer apply1 the external pressure on the nucleus tends to become unsymmetrical1 and its shape is modified accordingly4 8,amoeboid0 movements may be set up in the nucleus by anything which disturbs the symmetry of its own surface-tension4 ,and the cases1 as in many ,rhizopods1 where 8nuclear material0 is scattered in small portions throughout the cell instead of being aggregated in a single nucleus1 are probably capable of very simple explanation by supposing that the 8phase difference0 7as the chemists say7 between the nuclear and the protoplasmic substance is comparatively slight1 and the surface-tension which tends to keep them separate is correspondingly small#bbc4 ,it has been shewn that ordinary nuclei1 isolated in a living or fresh state1 easily flow together2 and this fact is enough to suggest that they are aggregations of a particular substance rather than bodies deserving the name of particular organs4 ,it is by reason of the same tendency to confluence or aggregation of particles that the ordinary nucleus is itself formed1 until the imposition of a new force leads to its disruption4 ,apart from that invisible or ultra-microscopic heterogeneity which is inseparable from our notion of a 8colloid10 there is a visible heterogeneity of structure within both the nucleus and the outer protoplasm4 ,the former1 for instance1 contains a rounded nucleolus or 8germinal spot10 certain conspicuous granules or strands of the peculiar substance called chromatin1 and a coarse meshwork of a protoplasmic material known as 8linin0 or achromatin2 the outer protoplasm1 or cytoplasm1 is generally believed to consist throughout of a sponge-work1 or rather alveolar meshwork1 of more and less fluid substances2 and lastly1 there are generally to be detected one or more very minute bodies1 usually in the cytoplasm1 sometimes within the nucleus1 known as the centrosome or centrosomes4 ,the morphologist is accustomed to speak of a 8polarity0 of #afg the cell1 meaning thereby a symmetry of visible structure about a particular axis4 ,for instance1 whenever we can recognise in a cell both a nucleus and a centrosome1 we may consider a line drawn through the two as the morphological axis of polarity2 in an epithelium cell1 it is obvious that the cell is morphologically symmetrical about a median axis passing from its free surface to its attached base4 ,again1 by an extension of the term 8polarity10 as is customary in dynamics1 we may have a 8radial0 polarity1 between centre and periphery2 and lastly1 we may have several apparently independent centres of polarity within the single cell4 ,only in cells of quite irregular1 or amoeboid form1 do we fail to recognise a definite and symmetrical 8polarity40 ,the morphological 8polarity0 is accompanied by1 and is but the outward expression 7or part of it7 of a true dynamical polarity1 or distribution of forces2 and the 8lines of force0 are rendered visible by concatenation of particles of matter1 such as come under the influence of the forces in action4 ,when the lines of force stream inwards from the periphery towards a point in the interior of the cell1 the particles susceptible of attraction either crowd towards the surface of the cell1 or1 when retarded by friction1 are seen forming lines or 8fibrillae0 which radiate outwards from the centre and constitute a so-called 8aster40 ,in the cells of columnar or ciliated epithelium1 where the sides of the cell are symmetrically disposed to their neighbours but the free and attached surfaces are very diverse from one another in their external relations1 it is these latter surfaces which constitute the opposite poles2 and in accordance with the parallel lines of force so set up1 we very frequently see parallel lines of granules which have ranged themselves perpendicularly to the free surface of the cell 7cf4 fig4 #ig74 ,a simple manifestation of 8polarity0 may be well illustrated by the phenomenon of diffusion1 where we may conceive1 and may automatically reproduce1 a 8field of force10 with its poles and visible lines of equipotential1 very much as in ,faraday's conception of the field of force of a magnetic system4 ,thus1 in one of ,leduc's experiments#bbd1 if we spread a layer of salt solution over a level #afh plate of glass1 and let fall into the middle of it a drop of indian ink1 or of blood1 we shall find the coloured particles travelling outwards from the central 8pole of concentration0 along the lines of diffusive force1 and so mapping out for us a 8monopolar field0 of diffusion3 and if we set two such drops side by side1 their lines of diffusion will oppose1 and repel1 one another4 ,or1 instead of the uniform layer of salt solution1 we may place at a little distance from one another a grain of salt and a drop of blood1 representing two opposite poles3 and so obtain a picture of a 8bipolar field0 of diffusion4 ,in either case1 we obtain results closely analogous to the 8morphological10 but really dynamical1 polarity of the organic cell4 ,but in all probability1 the dynamical polarity1 or asymmetry of the cell is a very complicated phenomenon3 for the obvious reason that1 in any system1 one asymmetry will tend to beget another4 ,a chemical asymmetry will induce an inequality of surface-tension1 which will lead directly to a modification of form2 the chemical asymmetry may in turn be due to a process of electrolysis in a polarised electrical field2 and again the chemical heterogeneity may be intensified into a chemical 8polarity10 by the tendency of certain substances to seek a locus of greater or less surface-energy4 ,we need not attempt to grapple with a subject so complicated1 and leading to so many problems which lie beyond the sphere of interest of the morphologist4 ,but yet the morphologist1 in his study of the cell1 cannot quite evade these important issues2 and we shall return to them again when we have dealt somewhat with the form of the cell1 and have taken account of some of the simpler phenomena of surface-tension4 ,we are now ready1 and in some measure prepared1 to study the numerous and complex phenomena which usually accompany the division of the cell1 for instance of the fertilised egg4 ,division of the cell is essentially accompanied1 and preceded1 by a change from radial or monopolar to a definitely bipolar polarity4 ,in the hitherto quiescent1 or apparently quiescent cell1 we perceive certain movements1 which correspond precisely to what must accompany and result from a 8polarisation0 of forces within the #afi cell3 of forces which1 whatever may be their specific nature1 at least are capable of polarisation1 and of producing consequent attraction or repulsion between charged particles of matter4 ,the opposing forces which were distributed in equilibrium throughout the substance of the cell become focussed at two 8centrosomes10 which may or may not be already distinguished as visible portions of matter2 in the egg1 one of these is always near to1 and the other remote from1 the 8animal pole0 of the egg1 which pole is visibly as well as chemically different from the other1 and is the region in which the more rapid and conspicuous developmental changes will presently begin4 ,between the two centrosomes1 a spindle-shaped ,illustration3 ,fig4 #da4 ,caryokinetic figure in a dividing cell 7or blastomere7 of the ,trout's egg4 7,after ,prenant1 from a preparation by ,prof4 ,p4 ,bouin47 figure appears1 whose striking resemblance to the lines of force made visible by iron-filings between the poles of a magnet1 was at once recognised by ,hermann ,fol1 when in #ahgc he witnessed for the first time the phenomenon in question4 ,on the farther side of the centrosomes are seen star-like figures1 or 8asters10 in which we can without difficulty recognise the broken lines of force which run externally to those stronger lines which lie nearer to the polar axis and which constitute the 8spindle40 ,the lines of force are rendered visible or 8material10 just as in the experiment of the iron-filings1 by the fact that1 in the heterogeneous substance of the cell1 certain portions of matter are more 8permeable0 to the acting force than the rest1 become themselves polarised after the #agj fashion of a magnetic or 8paramagnetic0 body1 arrange themselves in an orderly way between the two poles of the field of force1 cling to one another as it were in threads#bbe1 and are only prevented by the friction of the surrounding medium from approaching and congregating around the adjacent poles4 ,as the field of force strengthens1 the more will the lines of force be drawn in towards the interpolar axis1 and the less evident will be those remoter lines which constitute the terminal1 or extrapolar1 asters3 a clear space1 free from materialised lines of force1 may thus tend to be set up on either side of the spindle1 the so-called 8,btschli space0 of the histologists#bbf4 ,on the other hand1 the lines of force constituting the spindle will be less concentrated if they find a path of less resistance at the periphery of the cell3 as happens1 in our experiment of the iron-filings1 when we encircle the field of force with an iron ring4 ,on this principle1 the differences observed between cells in which the spindle is well developed and the asters small1 and others in which the spindle is weak and the asters enormously developed1 can be easily explained by variations in the potential of the field1 the large1 conspicuous asters being probably correlated with a marked permeability of the surface of the cell4 ,the visible field of force1 though often called the 8nuclear spindle10 is formed outside of1 but usually near to1 the nucleus4 ,let us look a little more closely into the structure of this body1 and into the changes which it presently undergoes4 ,within its spherical outline 7,fig4 #db71 it contains an 8alveolar0 #aga meshwork 7often described1 from its appearance in optical section1 as a 8reticulum071 consisting of more solid substances1 with more fluid matter filling up the interalveolar meshes4 ,this phenomenon is nothing else than what we call in ordinary language1 a 8froth0 or a 8foam40 ,it is a surface-tension phenomenon1 due to the interacting surface-tensions of two intermixed fluids1 not very different in density1 as they strive to separate4 ,of precisely the same kind 7as ,btschli was the first to shew7 are the minute alveolar networks which are to be discerned in the cytoplasm of the cell#bbg1 and which we now know to be not inherent in the nature of protoplasm1 or of living matter in general1 but to be due to various causes1 natural as well as artificial4 ,the microscopic honeycomb structure of cast metal under various conditions of cooling1 even on a grand scale the columnar structure of basaltic rock1 is an example of the same surface-tension phenomenon4 #agb ,illustration3 ,fig4 #db4 ,illustration3 ,fig4 #dc4 ,but here we touch the brink of a subject so important that we must not pass it by without a word1 and yet so contentious that we must not enter into its details4 ,the question involved is simply whether the great mass of recorded observations and accepted beliefs with regard to the visible structure of protoplasm and of the cell constitute a fair picture of the actual living cell1 or be based on appearances which are incident to death itself and to the artificial treatment which the microscopist is accustomed to apply4 ,the great bulk of histological work is done by methods which involve the sudden killing of the cell or organism by strong reagents1 the assumption being that death is so rapid that the visible phenomena exhibited during life are retained or 8fixed0 in our preparations4 ,while this assumption is reasonable and justified as regards the general outward form of small organisms or of individual cells1 enough has been done of late years to shew that the case is totally different in the case of the minute internal networks1 granules1 etc41 which represent the alleged structure of protoplasm4 ,for1 as ,hardy puts it1 8,it is notorious that the various fixing reagents are coagulants of organic colloids1 and that they produce precipitates which have a certain figure or structure1 444 and that the figure varies1 other things being equal1 according to the reagent used40 ,so it comes to pass that some writers#bbh have altogether denied the existence in the living cell-protoplasm of a network or alveolar 8foam02 others#bbi have cast doubts on the main tenets of recent histology regarding nuclear structure2 and ,hardy1 discussing the structure of certain gland-cells1 declares that 8there is no evidence that the structure discoverable in the cell-substance of these cells after fixation has any counterpart in the cell when living40 8,a large part of it0 he goes on to say 8is an artefact4 ,the profound difference in the minute structure of a secretory cell of a mucous gland according to the reagent which is used to fix it would1 it seems to me1 almost suffice to establish this statement in the absence of other evidence40 ,nevertheless1 histological study proceeds1 especially on the part of the morphologists1 with but little change in theory or in method1 in spite of these and many other warnings4 ,that certain visible structures1 nucleus1 vacuoles1 8attraction-spheres0 or centrosomes1 etc41 are actually present in the living cell1 we know for certain2 and to this class belong the great majority of structures 7including the nuclear 8spindle0 itself7 with which we are at present concerned4 ,that many other alleged structures are artificial has also been placed beyond a doubt2 but where to draw the dividing line we often do not know#bcj4 #agc ,the following is a brief epitome of the visible changes undergone by a typical cell1 leading up to the act of segmentation1 and constituting the phenomenon of mitosis or caryokinetic division4 ,in the egg of a sea-urchin1 we see with almost diagrammatic completeness what is set forth here#bca4 ,illustration3 ,fig4 #dd4 ,illustration3 ,fig4 #de4 #a4 ,the chromatin1 which to begin with was distributed in granules on the otherwise achromatic reticulum 7,fig4 #db71 concentrates to form a skein or spireme1 which may be a continuous thread from the first 7,figs4 #dc1 #dd71 or from the first segmented4 ,in any case it divides transversely sooner or later into a number of chromosomes 7,fig4 #de71 which as a rule have the shape of little rods1 straight or curved1 often bent into a ,v1 but which may also be ovoid1 or round1 or even annular4 ,certain deeply staining masses1 the nucleoli1 which may be present in the resting nucleus1 do not take part in the process of chromosome formation2 they are either cast out of the nucleus and are dissolved in the cytoplasm1 or fade away in situ4 #b4 ,meanwhile1 the deeply staining granule 7here extra-nuclear71 known as the centrosome1 has divided in two4 ,the two resulting granules travel to opposite poles of the nucleus1 and #agd there each becomes surrounded by a system of radiating lines1 the asters2 immediately around the centrosome is a clear space1 the centrosphere 7,figs4 #dc-de74 ,between the two centrosomes with their asters stretches a bundle of achromatic fibres1 the spindle4 #c4 ,the surface-film bounding the nucleus has broken down1 the definite nuclear boundaries are lost1 and the spindle now stretches through the nuclear material1 in which lie the chromosomes 7,figs4 #de1 #df74 ,these chromosomes now arrange themselves midway between the poles of the spindle1 where they form what is called the equatorial plate 7,fig4 #dg74 ,illustration3 ,fig4 #df4 ,illustration3 ,fig4 #dg4 #d4 ,each chromosome splits longitudinally into two3 usually at this stage1--but it is to be noticed that the splitting may have taken place so early as the spireme stage 7,fig4 #dh74 #e4 ,the halves of the split chromosomes now separate from one another1 and travel in opposite directions towards the two poles 7,fig4 #di74 ,as they move1 it becomes apparent that the spindle consists of a median bundle of 8fibres10 the central spindle1 running from pole to pole1 and a more superficial sheath of 8mantle-fibres10 to which the chromosomes seem to be attached1 and by which they seem to be drawn towards the asters4 #f4 ,the daughter chromosomes1 arranged now in two groups1 become closely crowded in a mass near the centre of each aster #age 7,fig4 #ej74 ,they fuse together and form once more an alveolar reticulum and may occasionally at this stage form another spireme4 ,illustration3 ,fig4 #dh4 ,illustration3 ,fig4 #di4 ,a boundary or surface wall is now developed round each reconstructed nuclear mass1 and the spindle-fibres disappear 7,fig4 #ea74 ,the centrosome remains1 as a rule1 outside the nucleus4 ,illustration3 ,fig4 #ej4 ,illustration3 ,fig4 #ea4 #g4 ,on the central spindle1 in the position of the equatorial plate1 there has appeared during the migration of the chromosomes1 a 8cell-plate0 of deeply staining thickenings 7,figs4 #ej1 #ea74 ,this is more conspicuous in plant-cells4 #agf #h4 ,a constriction has meanwhile appeared in the cytoplasm1 and the cell divides through the equatorial plane4 ,in plant-cells the line of this division is foreshadowed by the 8cell-plate10 which extends from the spindle across the entire cell1 and splits into two layers1 between which appears the membrane by which the daughter cells are cleft asunder4 ,in animal cells the cell-plate does not attain such dimensions1 and no cell-wall is formed4 ,the whole1 or very nearly the whole of these nuclear phenomena may be brought into relation with that polarisation of forces1 in the cell as a whole1 whose field is made manifest by the 8spindle0 and 8asters0 of which we have already spoken3 certain particular phenomena1 directly attributable to surface-tension and diffusion1 taking place in more or less obvious and inevitable dependence upon the polar system4 ,the reference numbers in the following account refer to the paragraphs and figures of the preceding summary of visible nuclear phenomena4 ,at the same time1 in attempting to explain the phenomena1 we cannot say too clearly1 or too often1 that all that we are meanwhile justified in doing is to try to shew that such and such actions lie within the range of known physical actions and phenomena1 or that known physical phenomena produce effects similar to them4 ,we want to feel sure that the whole phenomenon is not sui generis1 but is somehow or other capable of being referred to dynamical laws1 and to the general principles of physical science4 ,but when we speak of some particular force or mode of action1 using it as an illustrative hypothesis1 we must stop far short of the implication that this or that force is necessarily the very one which is actually at work within the living cell2 and certainly we need not attempt the formidable task of trying to reconcile1 or to choose between1 the various hypotheses which have already been enunciated1 or the several assumptions on which they depend4 ,any region of space within which action is manifested is a field of force2 and a simple example is a bipolar field1 in which the action is symmetrical with reference to the line joining two points1 or poles1 and also with reference to the 8equatorial0 plane equidistant from both4 ,we have such a 8field of force0 in #agg the neighbourhood of the centrosome of the ripe cell or ovum1 when it is about to divide2 and by the time the centrosome has divided1 the field is definitely a bipolar one4 ,the quality of a medium filling the field of force may be uniform1 or it may vary from point to point4 ,in particular1 it may depend upon the magnitude of the field2 and the quality of one medium may differ from that of another4 ,such variation of quality1 within one medium1 or from one medium to another1 is capable of diagrammatic representation by a variation of the direction or the strength of the field 7other conditions being the same7 from the state manifested in some uniform medium taken as a standard4 ,the medium is said to be permeable to the force1 in greater or less degree than the standard medium1 according as the variation of the density of the lines of force from the standard case1 under otherwise identical conditions1 is in excess or defect4 ,a body placed in the medium will tend to move towards regions of greater or less force according as its permeability is greater or less than that of the surrounding medium#bcb4 ,in the common experiment of placing iron-filings between the two poles of a magnetic field1 the filings have a very high permeability2 and not only do they themselves become polarised so as to attract one another1 but they tend to be attracted from the weaker to the stronger parts of the field1 and as we have seen1 were it not for friction or some other resistance1 they would soon gather together around the nearest pole4 ,but if we repeat the same experiment with such a metal as bismuth1 which is very little permeable to the magnetic force1 then the conditions are reversed1 and the particles1 being repelled from the stronger to the weaker parts of the field1 tend to take up their position as far from the poles as possible4 ,the particles have become polarised1 but in a sense opposite to that of the surrounding1 or adjacent1 field4 ,now1 in the field of force whose opposite poles are marked by #agh the centrosomes the nucleus appears to act as a more or less permeable body1 as a body more permeable than the surrounding medium1 that is to say the 8cytoplasm0 of the cell4 ,it is accordingly attracted by1 and drawn into1 the field of force1 and tries1 as it were1 to set itself between the poles and as far as possible from both of them4 ,in other words1 the centrosome-foci will be apparently drawn over its surface1 until the nucleus as a whole is involved within the field of force1 which is visibly marked out by the 8spindle0 7par4 #c1 ,figs4 #dd1 #de74 ,if the field of force be electrical1 or act in a fashion analogous to an electrical field1 the charged nucleus will have its surface-tensions diminished#bcc3 with the double result that the inner alveolar meshwork will be broken up 7par4 #a71 and that the spherical boundary of the whole nucleus will disappear 7par4 #b74 ,the break-up of the alveoli 7by thinning and rupture of their partition walls7 leads to the formation of a net1 and the further break-up of the net may lead to the unravelling of a thread or 8spireme0 7,figs4 #dc1 #dd74 ,here there comes into play a fundamental principle which1 in so far as we require to understand it1 can be explained in simple words4 ,the effect 7and we might even say the object7 of drawing the more permeable body in between the poles1 is to obtain an 8easier path0 by which the lines of force may travel2 but it is obvious that a longer route through the more permeable body may at length be found less advantageous than a shorter route through the less permeable medium4 ,that is to say1 the more permeable body will only tend to be drawn in to the field of force until a point is reached where 7so to speak7 the way round and the way through are equally advantageous4 ,we should accordingly expect that 7on our hypothesis7 there would be found cases in which the nucleus was wholly1 and others in which it was only partially1 and in greater or less degree1 drawn in to the field between the centrosomes4 ,this is precisely what is found to occur in actual fact4 ,figs4 #dd and #de represent two so-called 8types10 of a phase which follows that represented in ,fig4 #dc4 ,according to the usual descriptions 7and in particular to ,professor #agi ,e4 ,b4 ,wilson's#bcd71 we are told that1 in such a case as ,fig4 #dd1 the 8primary spindle0 disappears and the centrosomes diverge to opposite poles of the nucleus2 such a condition being found in many plant-cells1 and in the cleavage-stages of many eggs4 ,in ,fig4 #de1 on the other hand1 the primary spindle persists1 and subsequently comes to form the main or 8central0 spindle2 while at the same time we see the fading away of the nuclear membrane1 the breaking up of the spireme into separate chromosomes1 and an ingrowth into the nuclear area of the 8astral rays10--all as in ,fig4 #df1 which represents the next succeeding phase of ,fig4 #de4 ,this condition1 of ,fig4 #df1 occurs in a variety of cases2 it is well seen in the epidermal cells of the salamander1 and is also on the whole characteristic of the mode of formation of the 8polar bodies40 ,it is clear and obvious that the two 8types0 correspond to mere differences of degree1 and are such as would naturally be brought about by differences in the relative permeabilities of the nuclear mass and of the surrounding cytoplasm1 or even by differences in the magnitude of the former body4 ,but now an important change takes place1 or rather an important difference appears2 for1 whereas the nucleus as a whole tended to be drawn in to the stronger parts of the field1 when it comes to break up we find1 on the contrary1 that its contained spireme-thread or separate chromosomes tend to be repelled to the weaker parts4 ,whatever this difference may be due to1--whether1 for instance1 to actual differences of permeability1 or possibly to differences in 8surface-charge10--the fact is that the chromatin substance now behaves after the fashion of a 8diamagnetic0 body1 and is repelled from the stronger to the weaker parts of the field4 ,in other words1 its particles1 lying in the inter-polar field1 tend to travel towards the equatorial plane thereof 7,figs4 #dg1 #dh71 and further tend to move outwards towards the periphery of that plane1 towards what the histologist calls the 8mantle-fibres10 or outermost of the lines of force of which the spindle is made up 7par4 #e1 ,fig4 #dg74 ,and if this comparatively non-permeable chromatin substance come to consist of separate portions1 more or less elongated in form1 these portions1 or separate 8chromosomes10 will adjust themselves longitudinally1 #ahj in a peripheral equatorial circle 7,figs4 #dh1 #di74 ,this is precisely what actually takes place4 ,moreover1 before the breaking up of the nucleus1 long before the chromatin material has broken up into separate chromosomes1 and at the very time when it is being fashioned into a 8spireme10 this body already lies in a polar field1 and must already have a tendency to set itself in the equatorial plane thereof4 ,but the long1 continuous spireme thread is unable1 so long as the nucleus retains its spherical boundary wall1 to adjust itself in a simple equatorial annulus2 in striving to do so1 it must tend to coil and 8kink0 itself1 and in so doing 7if all this be so71 it must tend to assume the characteristic convolutions of the 8spireme40 ,illustration3 ,fig4 #eb4 ,chromosomes1 undergoing splitting and separation4 7,after ,hatschek and ,flemming1 diagrammatised47 ,after the spireme has broken up into separate chromosomes1 these particles come into a position of temporary1 and unstable1 equilibrium near the periphery of the equatorial plane1 and here they tend to place themselves in a symmetrical arrangement 7,fig4 #eb74 ,the particles are rounded1 linear1 sometimes annular1 similar in form and size to one another2 and lying as they do in a fluid1 and subject to a symmetrical system of forces1 it is not surprising that they arrange themselves in a symmetrical manner1 the precise arrangement depending on the form of the particles themselves4 ,this symmetry may perhaps be due1 as has already been suggested1 to induced electrical charges4 ,in discussing ,brauer's observations on the splitting of the chromatic filament1 and the symmetrical arrangement of the separate granules1 in ,ascaris megalocephala1 ,lillie#bce #aha remarks3 8,this behaviour is strongly suggestive of the division of a colloidal particle under the influence of its surface electrical charge1 and of the effects of mutual repulsion in keeping the products of division apart40 ,it is also probable that surface-tensions between the particles and the surrounding protoplasm would bring about an identical result1 and would sufficiently account for the obvious1 and at first sight1 very curious1 symmetry4 ,we know that if we float a couple of matches in water they tend to approach one another1 till they lie close together1 side by side2 and1 if we lay upon a smooth wet plate four matches1 half broken across1 a precisely similar attraction brings the four matches together in the form of a symmetrical cross4 ,whether one of these1 or some other1 be the actual explanation of the phenomenon1 it is at least plain that by some physical cause1 some mutual and symmetrical attraction or repulsion of the particles1 we must seek to account for the curious symmetry of these so-called 8tetrads40 ,the remarkable annular chromosomes1 shewn in ,fig4 #ec1 can also be easily imitated by means of loops of thread upon a soapy film when the film within the annulus is broken or its tension reduced4 ,illustration3 ,fig4 #ec4 ,annular chromosomes1 formed in the spermatogenesis of the ,mole-cricket4 7,from ,wilson1 after ,vom ,rath47 ,so far as we have now gone1 there is no great difficulty in pointing to simple and familiar phenomena of a field of force which are similar1 or comparable1 to the phenomena which we witness within the cell4 ,but among these latter phenomena there are others for which it is not so easy to suggest1 in accordance with known laws1 a simple mode of physical causation4 ,it is not at once obvious how1 in any simple system of symmetrical forces1 #ahb the chromosomes1 which had at first been apparently repelled from the poles towards the equatorial plane1 should then be split asunder1 and should presently be attracted in opposite directions1 some to one pole and some to the other4 ,remembering that it is not our purpose to assert that some one particular mode of action is at work1 but merely to shew that there do exist physical forces1 or distributions of force1 which are capable of producing the required result1 ,i give the following suggestive hypothesis1 which ,i owe to my colleague ,professor ,w4 ,peddie4 ,as we have begun by supposing that the nuclear1 or chromosomal matter differs in permeability from the medium1 that is to say the cytoplasm1 in which it lies1 let us now make the further assumption that its permeability is variable1 and depends upon the strength of the field4 ,illustration3 ,fig4 #ed4 ,in ,fig4 #ed1 we have a field of force 7representing our cell71 consisting of a homogeneous medium1 and including two opposite poles3 lines of force are indicated by full lines1 and loci of constant magnitude of force are shewn by dotted lines4 ,let us now consider a body whose permeability 77 depends on the strength of the field ,f4 ,at two field-strengths1 such as ,fa1 ,fb1 let the permeability of the body be equal to that of the #ahc medium1 and let the curved line in ,fig4 #ee represent generally its permeability at other field-strengths2 and let the outer and inner dotted curves in ,fig4 #ed represent respectively the loci of the field-strengths ,fb and ,fa4 ,the body if it be placed in the medium within either branch of the inner curve1 or outside the outer curve1 will tend to move into the neighbourhood of the adjacent pole4 ,if it be placed in the region intermediate to the two dotted curves1 it will tend to move towards regions of weaker field-strength4 ,illustration3 ,fig4 #ee4 ,the locus ,fb is therefore a locus of stable position1 towards which the body tends to move2 the locus ,fa is a locus of unstable position1 from which it tends to move4 ,if the body were placed across ,fa1 it might be torn asunder into two portions1 the split coinciding with the locus ,fa4 ,suppose a number of such bodies to be scattered throughout the medium4 ,let at first the regions ,fa and ,fb be entirely outside the space where the bodies are situated3 and1 in making this supposition we may1 if we please1 suppose that the loci which we are calling ,fa and ,fb are meanwhile situated somewhat farther from the axis than in our figure1 that 7for instance7 ,fa is situated where we have drawn ,fb1 and that ,fb is still further out4 ,the bodies then tend towards the poles2 but the tendency may be very small if1 in ,fig4 #ee1 the curve and its intersecting straight line do not diverge very far from one another beyond ,fa2 in other #ahd words1 if1 when situated in this region1 the permeability of the bodies is not very much in excess of that of the medium4 ,let the poles now tend to separate farther and farther from one another1 the strength of each pole remaining unaltered2 in other words1 let the centrosome-foci recede from one another1 as they actually do1 drawing out the spindle-threads between them4 ,the loci ,fa1 ,fb1 will close in to nearer relative distances from the poles4 ,in doing so1 when the locus ,fa crosses one of the bodies1 the body may be torn asunder2 if the body be of elongated shape1 and be crossed at more points than one1 the forces at work will tend to exaggerate its foldings1 and the tendency to rupture is greatest when ,fa is in some median position 7,fig4 #ef74 ,illustration3 ,fig4 #ef4 ,when the locus ,fa has passed entirely over the body1 the body tends to move towards regions of weaker force2 but when1 in turn1 the locus ,fb has crossed it1 then the body again moves towards regions of stronger force1 that is to say1 towards the nearest pole4 ,and1 in thus moving towards the pole1 it will do so1 as appears actually to be the case in the dividing cell1 along the course of the outer lines of force1 the so-called 8mantle-fibres0 of the histologist#bcf4 ,such considerations as these give general results1 easily open to modification in detail by a change of any of the arbitrary postulates which have been made for the sake of simplicity4 ,doubtless there are many other assumptions which would more or less meet the case2 for instance1 that of ,ida ,h4 ,hyde that1 #ahe during the active phase of the chromatin molecule 7during which it decomposes and sets free nucleic acid7 it carries a charge opposite to that which it bears during its resting1 or alkaline phase2 and that it would accordingly move towards different poles under the influence of a current1 wandering with its negative charge in an alkaline fluid during its acid phase to the anode1 and to the kathode during its alkaline phase4 ,a whole field of speculation is opened up when we begin to consider the cell not merely as a polarised electrical field1 but also as an electrolytic field1 full of wandering ions4 ,indeed it is high time we reminded ourselves that we have perhaps been dealing too much with ordinary physical analogies3 and that our whole field of force within the cell is of an order of magnitude where these grosser analogies may fail to serve us1 and might even play us false1 or lead us astray4 ,but our sole object meanwhile1 as ,i have said more than once1 is to demonstrate1 by such illustrations as these1 that1 whatever be the actual and as yet unknown modus operandi1 there are physical conditions and distributions of force which could produce just such phenomena of movement as we see taking place within the living cell4 ,this1 and no more1 is precisely what ,descartes is said to have claimed for his description of the human body as a 8mechanism#bcg40 ,the foregoing account is based on the provisional assumption that the phenomena of caryokinesis are analogous to1 if not identical with those of a bipolar electrical field2 and this comparison1 in my opinion1 offers without doubt the best available series of analogies4 ,but we must on no account omit to mention the fact that some of ,leduc's diffusion-experiments offer very remarkable analogies to the diagrammatic phenomena of caryokinesis1 as shewn in the annexed figure#bch4 ,here we have two identical 7not opposite7 poles of osmotic concentration1 formed by placing a drop of indian ink in salt water1 and then on either side of this central drop1 a hypertonic drop of salt solution more lightly coloured4 ,on either side the pigment of the central drop has been drawn towards the focus nearest to it2 but in the middle line1 the pigment #ahf is drawn in opposite directions by equal forces1 and so tends to remain undisturbed1 in the form of an 8equatorial plate40 ,nor should we omit to take account 7however briefly and inadequately7 of a novel and elegant hypothesis put forward by ,a4 ,b4 ,lamb4 ,this hypothesis makes use of a theorem of ,bjerknes1 to the effect that synchronously vibrating or pulsating bodies in a liquid field attract or repel one another according as their oscillations are identical or opposite in phase4 ,under such circumstances1 true currents1 or hydrodynamic lines of force1 are produced1 identical in form with the lines of force of a magnetic field2 and other particles floating1 though not necessarily pulsating1 in the liquid field1 tend to be attracted or repelled by the pulsating bodies according as they are lighter or heavier than the surrounding fluid4 ,moreover 7and this is the most remarkable point of all71 the lines of force set up by the oppositely pulsating bodies are the same as those which are produced by opposite magnetic poles3 though in the former case repulsion1 and in the latter case attraction1 takes place between the two poles#bci4 ,illustration3 ,fig4 #eg4 ,artificial caryokinesis 7after ,leduc71 for comparison with ,fig4 #da1 p4 #afi4 ,but to return to our general discussion4 ,while it can scarcely be too often repeated that our enquiry is not directed towards the solution of physiological problems1 save #ahg only in so far as they are inseparable from the problems presented by the visible configurations of form and structure1 and while we try1 as far as possible1 to evade the difficult question of what particular forces are at work when the mere visible forms produced are such as to leave this an open question1 yet in this particular case we have been drawn into the use of electrical analogies1 and we are bound to justify1 if possible1 our resort to this particular mode of physical action4 ,there is an important paper by ,r4 ,s4 ,lillie1 on the 8,electrical ,convection of certain ,free ,cells and ,nuclei#bdj10 which1 while ,i cannot quote it in direct support of the suggestions which ,i have made1 yet gives just the evidence we need in order to shew that electrical forces act upon the constituents of the cell1 and that their action discriminates between the two species of colloids represented by the cytoplasm and the nuclear chromatin4 ,and the difference is such that1 in the presence of an electrical current1 the cell substance and the nuclei 7including sperm-cells7 tend to migrate1 the former on the whole with the positive1 the latter with the negative stream3 a difference of electrical potential being thus indicated between the particle and the surrounding medium1 just as in the case of minute suspended particles of various kinds in various feebly conducing media#bda4 ,and the electrical difference is doubtless greatest1 in the case of the cell constituents1 just at the period of mitosis3 when the chromatin is invariably in its most deeply staining1 most strongly acid1 and therefore1 presumably1 in its most electrically negative phase4 ,in short1 #ahh ,lillie comes easily to the conclusion that 8electrical theories of mitosis are entitled to more careful consideration than they have hitherto received40 ,among other investigations1 all leading towards the same general conclusion1 namely that differences of electric potential play a great part in the phenomenon of cell division1 ,i would mention a very noteworthy paper by ,ida ,h4 ,hyde#bdb1 in which the writer shews 7among other important observations7 that not only is there a measurable difference of potential between the animal and vegetative poles of a fertilised egg 7,fundulus1 toad1 turtle1 etc471 but that this difference is not constant1 but fluctuates1 or actually reverses its direction1 periodically1 at epochs coinciding with successive acts of segmentation or other important phases in the development of the egg#bdc2 just as other physical rhythms1 for instance in the production of ,,co#b1 had already been shewn to do4 ,hence we shall be by no means surprised to find that the 8materialised0 lines of force1 which in the earlier stages form the convergent curves of the spindle1 are replaced in the later phases of caryokinesis by divergent curves1 indicating that the two foci1 which are marked out within the field by the divided and reconstituted nuclei1 are now alike in their polarity 7,figs4 #eh1 #ei74 ,it is certain1 to my mind1 that these observations of ,miss ,hyde's1 and of ,lillie's1 taken together with those of many writers on the behaviour of colloid particles generally in their relation to an electrical field1 have a close bearing upon the physiological side of our problem1 the full discussion of which lies outside our present field4 ,the break-up of the nucleus1 already referred to and ascribed to a diminution of its surface-tension1 is accompanied by certain diffusion phenomena which are sometimes visible to the eye2 and we are reminded of ,lord ,kelvin's view that diffusion is implicitly #ahi associated with surface-tension changes1 of which the first step is a minute puckering of the surface-skin1 a sort of interdigitation with the surrounding medium4 ,for instance1 ,schewiakoff has observed in ,euglypha#bdd that1 just before the break-up of the nucleus1 a system of rays appears1 concentred about it1 but having nothing to do with the polar asters3 and during the existence of this striation1 the nucleus enlarges very considerably1 evidently by imbibition of fluid from the surrounding protoplasm4 ,in short1 diffusion is at work1 hand in hand with1 and as it were in opposition to1 the surface-tensions which define the nucleus4 ,by diffusion1 hand in hand with surface-tension1 the alveoli of the nuclear meshwork are formed1 enlarged1 and finally ruptured3 diffusion sets up the movements which give rise to the appearance of rays1 or striae1 around the nucleus3 and through increasing diffusion1 and weakening surface-tension1 the rounded outline of the nucleus finally disappears4 #aij ,illustration3 ,fig4 #eh4 ,final stage in the first segmentation of the egg of ,cerebratulus4 7,from ,prenant1 after ,coe47#bde ,illustration3 ,fig4 #ei4 ,diagram of field of force with two similar poles4 ,as we study these manifold phenomena1 in the individual cases of particular plants and animals1 we recognise a close identity of type1 coupled with almost endless variation of specific detail2 and in particular1 the order of succession in which certain of the phenomena occur is variable and irregular4 ,the precise order of the phenomena1 the time of longitudinal and of transverse fission of the chromatin thread1 of the break-up of the nuclear wall1 and so forth1 will depend upon various minor contingencies and 8interferences40 ,and it is worthy of particular note that these variations1 in the order of events and in other subordinate details1 while doubtless attributable to specific physical conditions1 would seem to be without any obvious classificatory value or other biological significance#bdf4 ,as regards the actual mechanical division of the cell into two halves1 we shall see presently that1 in certain cases1 such as that of a long cylindrical filament1 surface-tension1 and what is known as the principle of 8minimal area10 go a long way to explain the mechanical process of division2 and in all cells whatsoever1 the process of division must somehow be explained as the result of a conflict between surface-tension and its opposing forces4 ,but in such a case as our spherical cell1 it is not very easy to see what physical cause is at work to disturb its equilibrium and its integrity4 ,the fact that1 when actual division of the cell takes place1 it does so at right angles to the polar axis and precisely in the direction of the equatorial plane1 would lead us to suspect that the new surface formed in the equatorial plane sets up an annular tension1 directed inwards1 where it meets the outer surface layer of the cell itself4 ,but at this point1 the problem becomes more complicated4 ,before we could hope to comprehend it1 we should have not only to enquire into the potential distribution at the surface of the cell in relation to that which we have seen to exist in its interior1 but we should probably also have to take account of the differences of potential which the material arrangements along the lines of force must themselves tend to produce4 ,only #aia thus could we approach a comprehension of the balance of forces which cohesion1 friction1 capillarity and electrical distribution combine to set up4 ,the manner in which we regard the phenomenon would seem to turn1 in great measure1 upon whether or no we are justified in assuming that1 in the liquid surface-film of a minute spherical cell1 local1 and symmetrically localised1 differences of surface-tension are likely to occur4 ,if not1 then changes in the conformation of the cell such as lead immediately to its division must be ascribed not to local changes in its surface-tension1 but rather to direct changes in internal pressure1 or to mechanical forces due to an induced surface-distribution of electrical potential4 ,it has seemed otherwise to many writers1 and we have a number of theories of cell division which are all based directly on inequalities or asymmetry of surface-tension4 ,for instance1 ,btschli suggested1 some forty years ago#bdg1 that cell division is brought about by an increase of surface-tension in the equatorial region of the cell4 ,this explanation1 however1 can scarcely hold2 for it would seem that such an increase of surface-tension in the equatorial plane would lead to the cell becoming flattened out into a disc1 with a sharply curved equatorial edge1 and to a streaming of material towards the equator4 ,in #ahie1 ,loeb shewed that the streaming went on from the equator towards the divided nuclei1 and he supposed that the violence of these streaming movements brought about actual division of the cell3 a hypothesis which was adopted by many other physiologists#bdh4 ,this streaming movement would suggest1 as ,robertson has pointed out1 a diminution of surface-tension in the region of the equator4 ,now ,quincke has shewn that the formation of soaps at the surface of an oil-droplet results in a diminution of the surface-tension of the latter2 and that if the saponification be local1 that part of the surface tends to spread4 ,by laying a thread moistened with a dilute solution of caustic alkali1 or even merely smeared with soap1 across a drop of oil1 ,robertson has further shewn that the drop at once divides into two3 the edges of the drop1 that is to say the ends of the #aib diameter across which the thread lies1 recede from the thread1 so forming a notch at each end of the diameter1 while violent streaming motions are set up at the surface1 away from the thread in the direction of the two opposite poles4 ,robertson#bdi suggests1 accordingly1 that the division of the cell is actually brought about by a lowering of the equatorial surface-tension1 and that this in turn is due to a chemical action1 such as a liberation of cholin1 or of soaps of cholin1 through the splitting of lecithin in nuclear synthesis4 ,but purely chemical changes are not of necessity the fundamental cause of alteration in the surface-tension of the egg1 for the action of electrolytes on surface-tension is now well known and easily demonstrated4 ,so1 according to other views than those with which we have been dealing1 electrical charges are sufficient in themselves to account for alterations of surface-tension2 while these in turn account for that protoplasmic streaming which1 as so many investigators agree1 initiates the segmentation of the egg#bej4 ,a great part of our difficulty arises from the fact that in such a case as this the various phenomena are so entangled and apparently concurrent that it is hard to say which initiates another1 and to which this or that secondary phenomenon may be considered due4 ,of recent years the phenomenon of adsorption has been adduced 7as we have already briefly said7 in order to account for many of the events and appearances which are associated with the asymmetry1 and lead towards the division1 of the cell4 ,but our short discussion of this phenomenon may be reserved for another chapter4 ,however1 we are not directly concerned here with the phenomena of segmentation or cell division in themselves1 except only in so far as visible changes of form are capable of easy and obvious correlation with the play of force4 ,the very fact of 8development0 indicates that1 while it lasts1 the equilibrium of the egg is never complete#bea4 ,and we may simply conclude the #aic matter by saying that1 if you have caryokinetic figures developing inside the cell1 that of itself indicates that the dynamic system and the localised forces arising from it are in continual alteration2 and1 consequently1 changes in the outward configuration of the system are bound to take place4 ,as regards the phenomena of fertilisation1--of the union of the spermatozoon with the 8pronucleus0 of the egg1--we might study these also in illustration1 up to a certain point1 of the polarised forces which are manifestly at work4 ,but we shall merely take1 as a single illustration1 the paths of the male and female pronuclei1 as they travel to their ultimate meeting place4 ,the spermatozoon1 when within a very short distance of the egg-cell1 is attracted by it4 ,of the nature of this attractive force we have no certain knowledge1 though we would seem to have a pregnant hint in ,loeb's discovery that1 in the neighbourhood of other substances1 such even as a fragment1 or bead1 of glass1 the spermatozoon undergoes a similar attraction4 ,but1 whatever the force may be1 it is one acting normally to the surface of the ovum1 and accordingly1 after entry1 the sperm-nucleus points straight towards the centre of the egg2 from the fact that other spermatozoa1 subsequent to the first1 fail to effect an entry1 we may safely conclude that an immediate consequence of the entry of the spermatozoon is an increase in the surface-tension of the egg#beb4 ,somewhere or other1 near or far away1 within the egg1 lies its own nuclear body1 the so-called female pronucleus1 and we find after a while that this has fused with the head of the spermatozoon 7or male pronucleus71 and that the body resulting from their fusion has come to occupy the centre of the egg4 ,this must be due 7as ,whitman pointed out long ago7 to a force of attraction acting between the two bodies1 and another force acting upon one or other or both in the direction of the centre of the cell4 ,did we know the magnitude of these several forces1 it would be a very easy task to calculate the precise path which the two pronuclei would follow1 leading to conjugation and the central #aid position4 ,as we do not know the magnitude1 but only the direction1 of these forces we can only make a general statement3 7#a7 the paths of both moving bodies will lie wholly within a plane triangle drawn between the two bodies and the centre of the cell2 7#b7 unless the two bodies happen to lie1 to begin with1 precisely on a diameter of the cell1 their paths until they meet one another will be curved paths1 the convexity of the curve being towards the straight line joining the two bodies2 7#c7 the two bodies will meet a little before they reach the centre2 and1 having met and fused1 will travel on to reach the centre in a straight line4 ,the actual study and observation of the path followed is not very easy1 owing to the fact that what we usually see is not the path itself1 but only a projection of the path upon the plane of the microscope2 but the curved path is particularly well seen in the frog's egg1 where the path of the spermatozoon is marked by a little streak of brown pigment1 and the fact of the meeting of the pronuclei before reaching the centre has been repeatedly seen by many observers4 ,the problem is nothing else than a particular case of the famous problem of three bodies1 which has so occupied the astronomers2 and it is obvious that the foregoing brief description is very far from including all possible cases4 ,many of these are particularly described in the works of ,fol1 ,roux1 ,whitman and others#bec4 ,the intracellular phenomena of which we have now spoken have assumed immense importance in biological literature and discussion during the last forty years2 but it is open to us to doubt whether they will be found in the end to possess more than a remote and secondary biological significance4 ,most1 if not all of them1 would seem to follow immediately and inevitably from very simple assumptions as to the physical constitution of the cell1 and from an extremely simple distribution of polarised forces within it4 ,we have already seen that how a thing grows1 and what it grows into1 is a dynamic and not a merely material problem2 so far as the material substance is concerned1 it is so only by reason #aie of the chemical1 electrical or other forces which are associated with it4 ,but there is another consideration which would lead us to suspect that many features in the structure and configuration of the cell are of very secondary biological importance2 and that is1 the great variation to which these phenomena are subject in similar or closely related organisms1 and the apparent impossibility of correlating them with the peculiarities of the organism as a whole4 8,comparative study has shewn that almost every detail of the processes 7of mitosis7 described above is subject to variation in different forms of cells#bed40 ,a multitude of cells divide to the accompaniment of caryokinetic phenomena2 but others do so without any visible caryokinesis at all4 ,sometimes the polarised field of force is within1 sometimes it is adjacent to1 and at other times it lies remote from the nucleus4 ,the distribution of potential is very often symmetrical and bipolar1 as in the case described2 but a less symmetrical distribution often occurs1 with the result that we have1 for a time at least1 numerous centres of force1 instead of the two main correlated poles3 this is the simple explanation of the numerous stellate figures1 or 8,strahlungen10 which have been described in certain eggs1 such as those of ,chaetopterus4 ,in one and the same species of worm 7,ascaris megalocephala71 one group or two groups of chromosomes may be present4 ,and remarkably constant1 in general1 as the number of chromosomes in any one species undoubtedly is1 yet we must not forget that1 in plants and animals alike1 the whole range of observed numbers is but a small one2 for 7as regards the germ-nuclei7 few organisms have less than six chromosomes1 and fewer still have more than sixteen#bee4 ,in closely related animals1 such as various species of ,copepods1 and even in the same species of worm or insect1 the form of the chromosomes1 and their arrangement in relation to the nuclear spindle1 have been found to differ in the various ways alluded to above4 ,in short1 there seem to be strong grounds for believing that these and many similar phenomena are in no way specifically related to the particular organism in which they have #aif been observed1 and are not even specially and indisputably connected with the organism as such4 ,they include such manifestations of the physical forces1 in their various permutations and combinations1 as may also be witnessed1 under appropriate conditions1 in non-living things4 ,when we attempt to separate our purely morphological or 8purely embryological0 studies from physiological and physical investigations1 we tend ipso facto to regard each particular structure and configuration as an attribute1 or a particular 8character10 of this or that particular organism4 ,from this assumption we are apt to go on to the drawing of new conclusions or the framing of new theories as to the ancestral history1 the classificatory position1 the natural affinities of the several organisms3 in fact1 to apply our embryological knowledge mainly1 and at times exclusively1 to the study of phylogeny4 ,when we find1 as we are not long of finding1 that our phylogenetic hypotheses1 as drawn from embryology1 become complex and unwieldy1 we are nevertheless reluctant to admit that the whole method1 with its fundamental postulates1 is at fault4 ,and yet nothing short of this would seem to be the case1 in regard to the earlier phases at least of embryonic development4 ,all the evidence at hand goes1 as it seems to me1 to shew that embryological data1 prior to and even long after the epoch of segmentation1 are essentially a subject for physiological and physical investigation and have but the very slightest link with the problems of systematic or zoological classification4 ,comparative embryology has its own facts to classify1 and its own methods and principles of classification4 ,thus we may classify eggs according to the presence or absence1 the paucity or abundance1 of their associated food-yolk1 the chromosomes according to their form and their number1 the segmentation according to its various 8types10 radial1 bilateral1 spiral1 and so forth4 ,but we have little right to expect1 and in point of fact we shall very seldom and 7as it were7 only accidentally find1 that these embryological categories coincide with the lines of 8natural0 or 8phylogenetic0 classification which have been arrived at by the systematic zoologist4 ,the cell1 which ,goodsir spoke of as a 8centre of force10 is in #aig reality a 8sphere of action0 of certain more or less localised forces2 and of these1 surface-tension is the particular force which is especially responsible for giving to the cell its outline and its morphological individuality4 ,the partially segmented differs from the totally segmented egg1 the unicellular ,infusorian from the minute multicellular ,turbellarian1 in the intensity and the range of those surface-tensions which in the one case succeed and in the other fail to form a visible separation between the 8cells40 ,adam ,sedgwick used to call attention to the fact that very often1 even in eggs that appear to be totally segmented1 it is yet impossible to discover an actual separation or cleavage1 through and through between the cells which on the surface of the egg are so clearly delimited2 so far and no farther have the physical forces effectuated a visible 8cleavage40 ,the vacuolation of the protoplasm in ,actinophrys or ,actinosphaerium is due to localised surface-tensions1 quite irrespective of the multinuclear nature of the latter organism4 ,in short1 the boundary walls due to surface-tension may be present or may be absent with or without the delimination of the other specific fields of force which are usually correlated with these boundaries and with the independent individuality of the cells4 ,what we may safely admit1 however1 is that one effect of these circumscribed fields of force is usually such a separation or segregation of the protoplasmic constituents1 the more fluid from the less fluid and so forth1 as to give a field where surface-tension may do its work and bring a visible boundary into being4 ,when the formation of a 8surface0 is once effected1 its physical condition1 or phase1 will be bound to differ notably from that of the interior of the cell1 and under appropriate chemical conditions the formation of an actual cell-wall1 cellulose or other1 is easily intelligible4 ,to this subject we shall return again1 in another chapter4 ,from the moment that we enter on a dynamical conception of the cell1 we perceive that the old debates were in vain as to what visible portions of the cell were active or passive1 living or non-living4 ,for the manifestations of force can only be due to the interaction of the various parts1 to the transference of energy from one to another4 ,certain properties may be manifested1 certain functions may be carried on1 by the protoplasm apart #aih from the nucleus2 but the interaction of the two is necessary1 that other and more important properties or functions may be manifested4 ,we know1 for instance1 that portions of an ,infusorian are incapable of regenerating lost parts in the absence of a nucleus1 while nucleated pieces soon regain the specific form of the organism3 and we are told that reproduction by fission cannot be initiated1 though apparently all its later steps can be carried on1 independently of nuclear action4 ,nor1 as ,verworn pointed out1 can the nucleus possibly be regarded as the 8sole vehicle of inheritance10 since only in the conjunction of cell and nucleus do we find the essentials of cell-life4 8,kern und ,protoplasma sind nur vereint lebensfhig10 as ,nussbaum said4 ,indeed we may1 with ,e4 ,b4 ,wilson1 go further1 and say that 8the terms #8nucleus' and #8cell-body' should probably be regarded as only topographical expressions denoting two differentiated areas in a common structural basis40 ,endless discussion has taken place regarding the centrosome1 some holding that it is a specific and essential structure1 a permanent corpuscle derived from a similar pre-existing corpuscle1 a 8fertilising element0 in the spermatozoon1 a special 8organ of cell-division10 a material 8dynamic centre0 of the cell 7as ,van ,beneden and ,boveri call it72 while on the other hand1 it is pointed out that many cells live and multiply without any visible centrosomes1 that a centrosome may disappear and be created anew1 and even that under artificial conditions abnormal chemical stimuli may lead to the formation of new centrosomes4 ,we may safely take it that the centrosome1 or the 8attraction sphere10 is essentially a 8centre of force10 and that this dynamic centre may or may not be constituted by 7but will be very apt to produce7 a concrete and visible concentration of matter4 ,it is far from correct to say1 as is often done1 that the cell-wall1 or cell-membrane1 belongs 8to the passive products of protoplasm rather than to the living cell itself02 or to say that in the animal cell1 the cell-wall1 because it is 8slightly developed10 is relatively unimportant compared with the important role which it assumes in plants4 ,on the contrary1 it is quite certain that1 whether visibly differentiated into a semi-permeable membrane1 or merely constituted by a liquid film1 the surface of the cell is the seat of #aii important forces1 capillary and electrical1 which play an essential part in the dynamics of the cell4 ,even in the thickened1 largely solidified cellulose wall of the plant-cell1 apart from the mechanical resistances which it affords1 the osmotic forces developed in connection with it are of essential importance4 ,but if the cell acts1 after this fashion1 as a whole1 each part interacting of necessity with the rest1 the same is certainly true of the entire multicellular organism3 as ,schwann said of old1 in very precise and adequate words1 8the whole organism subsists only by means of the reciprocal action of the single elementary parts#bef40 ,as ,wilson says again1 8the physiological autonomy of the individual cell falls into the background 444 and the apparently composite character which the multicellular organism may exhibit is owing to a secondary distribution of its energies among local centres of action#beg40 ,it is here that the homology breaks down which is so often drawn1 and overdrawn1 between the unicellular organism and the individual cell of the metazoon#beh4 ,whitman1 ,adam ,sedgwick#bei1 and others have lost no opportunity of warning us against a too literal acceptation of the cell-theory1 against the view that the multicellular organism is a colony 7or1 as ,haeckel called it 7in the case of the plant71 a 8republic07 of independent units of life#bfj4 ,as ,goethe said long ago1 8,das lebendige ist zwar in ,elemente #bjj zerlegt1 aber man kann es aus diesen nicht wieder zusammenstellen und beleben20 the dictum of the ,cellularpathologie being just the opposite1 8,jedes ,thier erscheint als eine ,summe vitaler ,einheiten1 von denen jede den vollen ,charakter des ,lebens an sich trgt40 ,hofmeister and ,sachs have taught us that in the plant the growth of the mass1 the growth of the organ1 is the primary fact1 that 8cell formation is a phenomenon very general in organic life1 but still only of secondary significance40 8,comparative embryology0 says ,whitman1 8reminds us at every turn that the organism dominates cell-formation1 using for the same purpose one1 several1 or many cells1 massing its material and directing its movements and shaping its organs1 as if cells did not exist#bfa40 ,so ,rauber declared that1 in the whole world of organisms1 8das ,ganze liefert die ,theile1 nicht die ,theile das ,ganze3 letzteres setzt die ,theile zusammen1 nicht diese jenes#bfb40 ,and on the botanical side ,de ,bary has summed up the matter in an aphorism1 8,die ,pflanze bildet ,zellen1 nicht die ,zelle bildet ,pflanzen40 ,discussed almost wholly from the concrete1 or morphological point of view1 the question has for the most part been made to turn on whether actual protoplasmic continuity can be demonstrated between one cell and another1 whether the organism be an actual reticulum1 or syncytium4 ,but from the dynamical point of view the question is much simpler4 ,we then deal not with material continuity1 not with little bridges of connecting protoplasm1 but with a continuity of forces1 a comprehensive field of force1 which runs through and through the entire organism and is by no means restricted in its passage to a protoplasmic continuum4 ,and such a continuous field of force1 somehow shaping the whole organism1 independently of the number1 magnitude and form of the individual cells1 which enter1 like a froth1 into its fabric1 seems to me certainly and obviously to exist4 ,as ,whitman says1 8the fact that physiological unity is not broken by cell-boundaries is confirmed in so many ways that it must be accepted as one of the fundamental truths of biology#bfc40 #bja ,,chapter ,v ,,the ,,forms ,,of ,,cells ,protoplasm1 as we have already said1 is a fluid or rather a semifluid substance1 and we need not pause here to attempt to describe the particular properties of the semifluid1 colloid1 or jelly-like substances to which it is allied2 we should find it no easy matter4 ,nor need we appeal to precise theoretical definitions of fluidity1 lest we come into a debateable land4 ,it is in the most general sense that protoplasm is 8fluid40 ,as ,graham said 7of colloid matter in general71 8its softness partakes of fluidity1 and enables the colloid to become a vehicle for liquid diffusion1 like water itself#bfd40 ,when we can deal with protoplasm in sufficient quantity we see it flow2 particles move freely through it1 air-bubbles and liquid droplets shew round or spherical within it2 and we shall have much to say about other phenomena manifested by its own surface1 which are those especially characteristic of liquids4 ,it may encompass and contain solid bodies1 and it may 8secrete0 within or around itself solid substances2 and very often in the complex living organism these solid substances formed by the living protoplasm1 like shell or nail or horn or feather1 may remain when the protoplasm which formed them is dead and gone2 but the protoplasm itself is fluid or semifluid1 and accordingly permits of free 7though not necessarily rapid7 diffusion and easy convection of particles within itself4 ,this simple fact is of elementary importance in connection with form1 and with what appear at first sight to be common characteristics or peculiarities of the forms of living things4 ,the older naturalists1 in discussing the differences between inorganic and organic bodies1 laid stress upon the fact or statement that the former grow by 8agglutination10 and the latter by #bjb what they termed 8intussusception40 ,the contrast is true1 rather1 of solid as compared with jelly-like bodies of all kinds1 living or dead1 the great majority of which as it so happens1 but by no means all1 are of organic origin4 ,a crystal 8grows0 by deposition of new molecules1 one by one and layer by layer1 superimposed or aggregated upon the solid substratum already formed4 ,each particle would seem to be influenced1 practically speaking1 only by the particles in its immediate neighbourhood1 and to be in a state of freedom and independence from the influence1 either direct or indirect1 of its remoter neighbours4 ,as ,lord ,kelvin and others have explained the formation and the resulting forms of crystals1 so we believe that each added particle takes up its position in relation to its immediate neighbours already arranged1 generally in the holes and corners that their arrangement leaves1 and in closest contact with the greatest number#bfe4 ,and hence we may repeat or imitate this process of arrangement1 with great or apparently even with precise accuracy 7in the case of the simpler crystalline systems71 by piling up spherical pills or grains of shot4 ,in so doing1 we must have regard to the fact that each particle must drop into the place where it can go most easily1 or where no easier place offers4 ,in more technical language1 each particle is free to take up1 and does take up1 its position of least potential energy relative to those already deposited2 in other words1 for each particle motion is induced until the energy of the system is so distributed that no tendency or resultant force remains to move it more4 ,the application of this principle has been shewn to lead to the production of planes#bff 7in all cases where by the limitation of material1 surfaces must occur72 and where we have planes1 straight edges and solid angles must obviously also occur2 and1 if equilibrium is #bjc to follow1 must occur symmetrically4 ,our piling up of shot1 or manufacture of mimic crystals1 gives us visible demonstration that the result is actually to obtain1 as in the natural crystal1 plane surfaces and sharp angles1 symmetrically disposed4 ,but the living cell grows in a totally different way1 very much as a piece of glue swells up in water1 by 8imbibition10 or by interpenetration into and throughout its entire substance4 ,the semifluid colloid mass takes up water1 partly to combine chemically with its individual molecules#bfg1 partly by physical diffusion into the interstices between these molecules1 and partly1 as it would seem1 in other ways2 so that the entire phenomenon is a very complex and even an obscure one4 ,but1 so far as we are concerned1 the net result is a very simple one4 ,for the equilibrium or tendency to equilibrium of fluid pressure in all parts of its interior while the process of imbibition is going on1 the constant rearrangement of its fluid mass1 the contrast in short with the crystalline method of growth where each particle comes to rest to move 7relatively to the whole7 no more1 lead the mass of jelly to swell up1 very much as a bladder into which we blow air1 and so1 by a graded and harmonious distribution of forces1 to assume everywhere a rounded and more or less bubble-like external form#bfh4 ,so1 when the same school of older naturalists called attention to a new distinction or contrast of form between the organic and inorganic objects1 in that the contours of the former tended to roundness and curvature1 and those of the latter to be bounded by straight lines1 planes and sharp angles1 we see that this contrast was not a new and different one1 but only another aspect of their former statement1 and an immediate consequence of the difference between the processes of agglutination and intussusception4 ,this common and general contrast between the form of the crystal on the one hand1 and of the colloid or of the organism on the other1 must by no means be pressed too far4 ,for ,lehmann1 #bjd in his great work on so-called ,fluid ,crystals#bfi1 to which we shall afterwards return1 has shewn how1 under certain circumstances1 surface-tension phenomena may coexist with crystallisation1 and produce a form of minimal potential which is a resultant of both3 the fact being that the bonds maintaining the crystalline arrangement are now so much looser than in the solid condition that the tendency to least total surface-area is capable of being satisfied4 ,thus the phenomenon of 8liquid crystallisation0 does not destroy the distinction between crystalline and colloidal forms1 but gives added unity and continuity to the whole series of phenomena#bgj4 ,lehmann has also demonstrated phenomena within the crystal1 known for instance as transcrystallisation1 which shew us that we must not speak unguardedly of the growth of crystals as limited to deposition upon a surface1 and ,btschli has already pointed out the possible great importance to the biologist of the various phenomena which ,lehmann has described#bga4 ,so far then1 as growth goes on1 unaffected by pressure or other external force1 the fluidity of protoplasm1 its mobility internal and external1 and the manner in which particles move with comparative freedom from place to place within1 all manifestly tend to the production of swelling1 rounded surfaces1 and to their great predominance over plane surfaces in the contour of the organism4 ,these rounded contours will tend to be preserved1 for a while1 in the case of naked protoplasm by its viscosity1 and in the presence of a cell-wall by its very lack of fluidity4 ,in a general way1 the presence of curved boundary surfaces will be especially obvious in the unicellular organisms1 and still more generally in the external forms of all organisms2 and wherever mutual pressure between adjacent cells1 or other adjacent parts1 has not come into play to flatten the rounded surfaces into planes4 ,but the rounded contours that are assumed and exhibited by #bje a piece of hard glue1 when we throw it into water and see it expand as it sucks the water up1 are not nearly so regular or so beautiful as are those which appear when we blow a bubble1 or form a drop1 or pour water into a more or less elastic bag4 ,for these curving contours depend upon the properties of the bag itself1 of the film or membrane that contains the mobile gas1 or that contains or bounds the mobile liquid mass4 ,and hereby1 in the case of the fluid or semifluid mass1 we are introduced to the subject of surface tension3 of which indeed we have spoken in the preceding chapter1 but which we must now examine with greater care4 ,among the forces which determine the forms of cells1 whether they be solitary or arranged in contact with one another1 this force of surface-tension is certainly of great1 and is probably of paramount importance4 ,but while we shall try to separate out the phenomena which are directly due to it1 we must not forget that1 in each particular case1 the actual conformation which we study may be1 and usually is1 the more or less complex resultant of surface tension acting together with gravity1 mechanical pressure1 osmosis1 or other physical forces4 ,surface tension is that force by which we explain the form of a drop or of a bubble1 of the surfaces external and internal of a 8froth0 or collocation of bubbles1 and of many other things of like nature and in like circumstances#bgb4 ,it is a property of liquids 7in the sense at least with which our subject is concerned71 and it is manifested at or very near the surface1 where the liquid comes into contact with another liquid1 a solid or a gas4 ,we note here that the term surface is to be interpreted in a wide sense2 for wherever we have solid particles imbedded in a fluid1 wherever we have a non-homogeneous fluid or semi-fluid such as a particle #bjf of protoplasm1 wherever we have the presence of 8impurities10 as in a mass of molten metal1 there we have always to bear in mind the existence of 8surfaces0 and of surface tensions1 not only on the exterior of the mass but also throughout its interstices1 wherever like meets unlike4 ,surface tension is due to molecular force1 to force that is to say arising from the action of one molecule upon another1 and it is accordingly exerted throughout a small thickness of material1 comparable to the range of the molecular forces4 ,we imagine that within the interior of the liquid mass such molecular interactions negative one another3 but that at and near the free surface1 within a layer or film approximately equal to the range of the molecular force1 there must be a lack of such equilibrium and consequently a manifestation of force4 ,the action of the molecular forces has been variously explained4 ,but one simple explanation 7or mode of statement7 is that the molecules of the surface layer 7whose thickness is definite and constant7 are being constantly attracted into the interior by those which are more deeply situated1 and that consequently1 as molecules keep quitting the surface for the interior1 the bulk of the latter increases while the surface diminishes2 and the process continues till the surface itself has become a minimum1 the surface-shrinkage exhibiting itself as a surface-tension4 ,this is a sufficient description of the phenomenon in cases where a portion of liquid is subject to no other than its own molecular forces1 and 7since the sphere has1 of all solids1 the smallest surface for a given volume7 it accounts for the spherical form of the raindrop1 of the grain of shot1 or of the living cell in many simple organisms4 ,it accounts also1 as we shall presently see1 for a great number of much more complicated forms1 manifested under less simple conditions4 ,let us here briefly note that surface tension is1 in itself1 a comparatively small force1 and easily measurable3 for instance that of water is equivalent to but a few grains per linear inch1 or a few grammes per metre4 ,but this small tension1 when it exists in a curved surface of very great curvature1 gives rise to a very great pressure directed towards the centre of curvature4 ,we can easily calculate this pressure1 and so satisfy ourselves that1 when the radius of curvature is of molecular dimensions1 the #bjg pressure is of the magnitude of thousands of atmospheres1--a conclusion which is supported by other physical considerations4 ,the contraction of a liquid surface and other phenomena of surface tension involve the doing of work1 and the power to do work is what we call energy4 ,it is obvious1 in such a simple case as we have just considered1 that the whole energy of the system is diffused throughout its molecules2 but of this whole stock of energy it is only that part which comes into play at or very near to the surface which normally manifests itself in work1 and hence we may speak 7though the term is open to some objections7 of a specific surface energy4 ,the consideration of surface energy1 and of the manner in which its amount is increased and multiplied by the multiplication of surfaces due to the subdivision of the organism into cells1 is of the highest importance to the physiologist2 and even the morphologist cannot wholly pass it by1 if he desires to study the form of the cell in its relation to the phenomena of surface tension or 8capillarity40 ,the case has been set forth with the utmost possible lucidity by ,tait and by ,clerk ,maxwell1 on whose teaching the following paragraphs are based3 they having based their teaching upon that of ,gauss1--who rested on ,laplace4 ,let ,e be the whole potential energy of a mass ,m of liquid2 let e#j be the energy per unit mass of the interior liquid 7we may call it the internal energy72 and let e be the energy per unit mass for a layer of the skin1 of surface ,s1 of thickness t1 and density 7e being what we call the surface energy74 ,it is obvious that the total energy consists of the internal plus the surface energy1 and that the former is distributed through the whole mass1 minus its surface layers4 ,that is to say1 in mathematical language1 ,e 7,m ,s t 7 e#j ,s t e4 ,but this is equivalent to writing3 ,m e#j ,s t 7e e#j72 and this is as much as to say that the total energy of the system may be taken to consist of two portions1 one uniform throughout the whole mass1 and another1 which is proportional on the one hand to the amount of surface1 and on the other hand is proportional to the difference between e and e#j1 that is to say to the difference between the unit values of the internal and the surface energy4 #bjh ,it was ,gauss who first shewed after this fashion how1 from the mutual attractions between all the particles1 we are led to an expression which is what we now call the potential energy of the system2 and we know1 as a fundamental theorem of dynamics1 that the potential energy of the system tends to a minimum1 and in that minimum finds1 as a matter of course1 its stable equilibrium4 ,we see in our last equation that the term ,m e#j is irreducible1 save by a reduction of the mass itself4 ,but the other term may be diminished 7#a7 by a reduction in the area of surface1 ,s1 or 7#b7 by a tendency towards equality of e and e#j1 that is to say by a diminution of the specific surface energy1 e4 ,these then are the two methods by which the energy of the system will manifest itself in work4 ,the one1 which is much the more important for our purposes1 leads always to a diminution of surface1 to the so-called 8principle of minimal areas02 the other1 which leads to the lowering 7under certain circumstances7 of surface tension1 is the basis of the theory of ,adsorption1 to which we shall have some occasion to refer as the modus operandi in the development of a cell-wall1 and in a variety of other histological phenomena4 ,in the technical phraseology of the day1 the 8capacity factor0 is involved in the one case1 and the 8intensity factor0 in the other4 ,inasmuch as we are concerned with the form of the cell it is the former which becomes our main postulate3 telling us that the energy equations of the surface of a cell1 or of the free surfaces of cells partly in contact1 or of the partition-surfaces of cells in contact with one another or with an adjacent solid1 all indicate a minimum of potential energy in the system1 by which the system is brought1 ipso facto1 into equilibrium4 ,and we shall not fail to observe1 with something more than mere historical interest and curiosity1 how deeply and intrinsically there enter into this whole class of problems the 8principle of least action0 of ,maupertuis1 the 8lineae curvae maximi minimive proprietate gaudentes0 of ,euler1 by which principles these old natural philosophers explained correctly a multitude of phenomena1 and drew the lines whereon the foundations of great part of modern physics are well and truly laid4 #bji ,in all cases where the principle of maxima and minima comes into play1 as it conspicuously does in the systems of liquid films which are governed by the laws of surface-tension1 the figures and conformations produced are characterised by obvious and remarkable symmetry4 ,such symmetry is in a high degree characteristic of organic forms1 and is rarely absent in living things1--save in such cases as amoeba1 where the equilibrium on which symmetry depends is likewise lacking4 ,and if we ask what physical equilibrium has to do with formal symmetry and regularity1 the reason is not far to seek2 nor can it be put better than in the following words of ,mach's#bgc4 8,in every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it4 ,in each deformation positive and negative work is done4 ,one condition1 therefore1 though not an absolutely sufficient one1 that a maximum or minimum of work corresponds to the form of equilibrium1 is thus supplied by symmetry4 ,regularity is successive symmetry4 ,there is no reason1 therefore1 to be astonished that the forms of equilibrium are often symmetrical and regular40 ,as we proceed in our enquiry1 and especially when we approach the subject of tissues1 or agglomerations of cells1 we shall have from time to time to call in the help of elementary mathematics4 ,but already1 with very little mathematical help1 we find ourselves in a position to deal with some simple examples of organic forms4 ,when we melt a stick of sealing-wax in the flame1 surface tension 7which was ineffectively present in the solid but finds play in the now fluid mass71 rounds off its sharp edges into curves1 so striving towards a surface of minimal area2 and in like manner1 by melting the tip of a thin rod of glass1 ,leeuwenhoek made the little spherical beads which served him for a microscope#bgd4 ,when any drop of protoplasm1 either over all its surface or at some free end1 as at the extremity of the pseudopodium of an amoeba1 is #baj seen likewise to 8round itself off10 that is not an effect of 8vital contractility10 but 7as ,hofmeister shewed so long ago as #ahfg7 a simple consequence of surface tension2 and almost immediately afterwards ,engelmann#bge argued on the same lines1 that the forces which cause the contraction of protoplasm in general may 8be just the same as those which tend to make every non-spherical drop of fluid become spherical60 ,we are not concerned here with the many theories and speculations which would connect the phenomena of surface tension with contractility1 muscular movement or other special physiological functions1 but we find ample room to trace the operation of the same cause in producing1 under conditions of rest and equilibrium1 certain definite and inevitable forms of surface4 ,it is however of great importance to observe that the living cell is one of those cases where the phenomena of surface tension are by no means limited to the outer surface2 for within the heterogeneous substance of the cell1 between the protoplasm and its nuclear and other contents1 and in the alveolar network of the cytoplasm itself 7so far as that 8alveolar structure0 is actually present in life71 we have a multitude of interior surfaces2 and1 especially among plants1 we may have a large inner surface of 8interfacial0 contact1 where the protoplasm contains cavities or 8vacuoles0 filled with a different and more fluid material1 the 8cell-sap40 ,here we have a great field for the development of surface tension phenomena3 and so long ago as #ahfe1 ,ngeli and ,schwendener shewed that the streaming currents of plant cells might be very plausibly explained by this phenomenon4 ,even ten years earlier1 ,weber had remarked upon the resemblance between these protoplasmic streamings and the streamings to be observed in certain inanimate drops1 for which no cause but surface tension could be assigned#bgf4 ,the case of amoeba1 though it is an elementary case1 is at the same time a complicated one4 ,while it remains 8amoeboid10 it is never at rest or in equilibrium2 it is always moving1 from one to another of its protean changes of configuration2 its surface tension is constantly varying from point to point4 ,where the #baa surface tension is greater1 that portion of the surface will contract into spherical or spheroidal forms2 where it is less the surface will correspondingly extend4 ,while generally speaking the surface energy has a minimal value1 it is not necessarily constant4 ,it may be diminished by a rise of temperature2 it may be altered by contact with adjacent substances#bgg1 by the transport of constituent materials from the interior to the surface1 or again by actual chemical and fermentative change4 ,within the cell1 the surface energies developed about its heterogeneous contents will constantly vary as these contents are affected by chemical metabolism4 ,as the colloid materials are broken down and as the particles in suspension are diminished in size the 8free surface energy0 will be increased1 but the osmotic energy will be diminished#bgh4 ,thus arise the various fluctuations of surface tension and the various phenomena of amoeboid form and motion1 which ,btschli and others have reproduced or imitated by means of the fine emulsions which constitute their 8artificial amoebae40 ,a multitude of experiments shew how extraordinarily delicate is the adjustment of the surface tension forces1 and how sensitive they are to the least change of temperature or chemical state4 ,thus1 on a plate which we have warmed at one side1 a drop of alcohol runs towards the warm area1 a drop of oil away from it2 and a drop of water on the glass plate exhibits lively movements when #bab we bring into its neighbourhood a heated wire1 or a glass rod dipped in ether4 ,when we find that a plasmodium of ,aethalium1 for instance1 creeps towards a damp spot1 or towards a warm spot1 or towards substances that happen to be nutritious1 and again creeps away from solutions of sugar or of salt1 we seem to be dealing with phenomena every one of which can be paralleled by ordinary phenomena of surface tension#bgi4 ,even the soap-bubble itself is imperfectly in equilibrium1 for the reason that its film1 like the protoplasm of amoeba or ,aethalium1 is an excessively heterogeneous substance4 ,its surface tensions vary from point to point1 and chemical changes and changes of temperature increase and magnify the variation4 ,the whole surface of the bubble is in constant movement as the concentrated portions of the soapy fluid make their way outwards from the deeper layers2 it thins and it thickens1 its colours change1 currents are set up in it1 and little bubbles glide over it2 it continues in this state of constant movement1 as its parts strive one with another in all their interactions towards equilibrium#bhj4 ,in the case of the naked protoplasmic cell1 as the amoeboid phase is emphatically a phase of freedom and activity1 of chemical and physiological change1 so1 on the other hand1 is the spherical form indicative of a phase of rest or comparative inactivity4 ,in the one phase we see unequal surface tensions manifested in the creeping movements of the amoeboid body1 in the rounding off of the ends of the pseudopodia1 in the flowing out of its substance over a particle of 8food10 and in the current-motions in the interior of its mass2 till finally1 in the other phase1 when internal homogeneity and equilibrium have been attained and the potential #bac energy of the system is for the time being at a minimum1 the cell assumes a rounded or spherical form1 passing into a state of 8rest10 and 7for a reason which we shall presently see7 becoming at the same time 8encysted40 ,illustration3 ,fig4 #fj4 ,in a budding yeast-cell 7,fig4 #fj71 we see a more definite and restricted change of surface tension4 ,when a 8bud0 appears1 whether with or without actual growth by osmosis or otherwise of the mass1 it does so because at a certain part of the cell-surface the surface tension has more or less suddenly diminished1 and the area of that portion expands accordingly2 but in turn the surface tension of the expanded area will make itself felt1 and the bud will be rounded off into a more or less spherical form4 ,the yeast-cell with its bud is a simple example of a principle which we shall find to be very important4 ,our whole treatment of cell-form in relation to surface-tension depends on the fact 7which ,errera was the first to point out1 or to give clear expression to7 that the incipient cell-wall retains with but little impairment the properties of a liquid film#bha1 and that the growing cell1 in spite of the membrane by which it has already begun to be surrounded1 behaves very much like a fluid drop4 ,but even the ordinary yeast-cell shows1 by its ovoid and non-spherical form1 that it has acquired its shape under the influence of some force other than that uniform and symmetrical surface-tension which would be productive of a sphere2 and this or any other asymmetrical form1 once acquired1 may be retained by virtue of the solidification and consequent rigidity of the membranous wall of the cell4 ,unless such rigidity ensue1 it is plain that such a conformation as that of the cell with its attached bud could not be long retained1 amidst the constantly varying conditions1 as a figure of even partial equilibrium4 ,but as a matter of fact1 the cell in this case is not in equilibrium at all2 it is in process of budding1 and is slowly altering its shape by rounding off the bud4 ,it is plain that over its surface the surface-energies are unequally distributed1 owing to some heterogeneity of the substance2 and to this matter we shall afterwards return4 ,in like manner the developing egg #bad through all its successive phases of form is never in complete equilibrium2 but is merely responding to constantly changing conditions1 by phases of partial1 transitory1 unstable and conditional equilibrium4 ,it is obvious that there are innumerable solitary plant-cells1 and unicellular organisms in general1 which1 like the yeast-cell1 do not correspond to any of the simple forms that may be generated under the influence of simple and homogeneous surface-tension2 and in many cases these forms1 which we should expect to be unstable and transitory1 have become fixed and stable by reason of the comparatively sudden or rapid solidification of the envelope4 ,this is the case1 for instance1 in many of the more complicated forms of diatoms or of desmids1 where we are dealing1 in a less striking but even more curious way than in the budding yeast-cell1 not with one simple act of formation1 but with a complicated result of successive stages of localised growth1 interrupted by phases of partial consolidation4 ,the original cell has acquired or assumed a certain form1 and then1 under altering conditions and new distributions of energy1 has thickened here or weakened there1 and has grown out or tended 7as it were7 to branch1 at particular points4 ,we can often1 or indeed generally1 trace in each particular stage of growth or at each particular temporary growing point1 the laws of surface tension manifesting themselves in what is for the time being a fluid surface2 nay more1 even in the adult and completed structure1 we have little difficulty in tracing and recognising 7for instance in the outline of such a desmid as ,euastrum7 the rounded lobes that have successively grown or flowed out from the original rounded and flattened cell4 ,what we see in a many chambered foraminifer1 such as ,globigerina or ,rotalia1 is just the same thing1 save that it is carried out in greater completeness and perfection4 ,the little organism as a whole is not a figure of equilibrium or of minimal area2 but each new bud or separate chamber is such a figure1 conditioned by the forces of surface tension1 and superposed upon the complex aggregate of similar bubbles after these latter have become consolidated one by one into a rigid system4 ,let us now make some enquiry regarding the various forms #bae which1 under the influence of surface tension1 a surface can possibly assume4 ,in doing so1 we are obviously limited to conditions under which other forces are relatively unimportant1 that is to say where the 8surface energy0 is a considerable fraction of the whole energy of the system2 and this in general will be the case when we are dealing with portions of liquid so small that their dimensions come within what we have called the molecular range1 or1 more generally1 in which the 8specific surface0 is large#bhb3 in other words it will be small or minute organisms1 or the small cellular elements of larger organisms1 whose forms will be governed by surface-tension2 while the general forms of the larger organisms will be due to other and non-molecular forces4 ,for instance1 a large surface of water sets itself level because here gravity is predominant2 but the surface of water in a narrow tube is manifestly curved1 for the reason that we are here dealing with particles which are mutually within the range of each other's molecular forces4 ,the same is the case with the cell-surfaces and cell-partitions which we are presently to study1 and the effect of gravity will be especially counteracted and concealed when1 as in the case of protoplasm in a watery fluid1 the object is immersed in a liquid of nearly its own specific gravity4 ,we have already learned1 as a fundamental law of surface-tension phenomena1 that a liquid film in equilibrium assumes a form which gives it a minimal area under the conditions to which it is subject4 ,and these conditions include 7#a7 the form of the boundary1 if such exist1 and 7#b7 the pressure1 if any1 to which the film is subject2 which pressure is closely related to the volume1 of air or of liquid1 which the film 7if it be a closed one7 may have to contain4 ,in the simplest of cases1 when we take up a soap-film on a plane wire ring1 the film is exposed to equal atmospheric pressure on both sides1 and it obviously has its minimal area in the form of a plane4 ,so long as our wire ring lies in one plane 7however irregular in outline71 the film stretched across it will still be in a plane2 but if we bend the ring so that it lies no longer in a plane1 then our film will become curved into a surface which may be extremely complicated1 but is still the smallest possible #baf surface which can be drawn continuously across the uneven boundary4 ,the question of pressure involves not only external pressures acting on the film1 but also that which the film itself is capable of exerting4 ,for we have seen that the film is always contracting to its smallest limits2 and when the film is curved1 this obviously leads to a pressure directed inwards1--perpendicular1 that is to say1 to the surface of the film4 ,in the case of the soap-bubble1 the uniform contraction of whose surface has led to its spherical form1 this pressure is balanced by the pressure of the air within2 and if an outlet be given for this air1 then the bubble contracts with perceptible force until it stretches across the mouth of the tube1 for instance the mouth of the pipe through which we have blown the bubble4 ,a precisely similar pressure1 directed inwards1 is exercised by the surface layer of a drop of water or a globule of mercury1 or by the surface pellicle on a portion or 8drop0 of protoplasm4 ,only we must always remember that in the soap-bubble1 or the bubble which a glass-blower blows1 there is a twofold pressure as compared with that which the surface-film exercises on the drop of liquid of which it is a part2 for the bubble consists 7unless it be so thin as to consist of a mere layer of molecules#bhc7 of a liquid layer1 with a free surface within and another without1 and each of these two surfaces exercises its own independent and coequal tension1 and corresponding pressure#bhd4 ,if we stretch a tape upon a flat table1 whatever be the tension of the tape it obviously exercises no pressure upon the table below4 ,but if we stretch it over a curved surface1 a cylinder for instance1 it does exercise a downward pressure2 and the more curved the surface the greater is this pressure1 that is to say the greater is this share of the entire force of tension which is resolved in the downward direction4 ,in mathematical language1 the pressure 7p7 varies directly as the tension 7,t71 and inversely as the radius of curvature 7,r73 that is to say1 p ,t_/,r1 per unit of surface4 #bag ,if instead of a cylinder1 which is curved only in one direction1 we take a case where there are curvatures in two dimensions 7as for instance a sphere71 then the effects of these must be simply added to one another1 and the resulting pressure p is equal to ,t_/,r ,t_/,r or p ,t7#a_/,r #a_/,r7#bhe4 ,and if in addition to the pressure p1 which is due to surface tension1 we have to take into account other pressures1 p1 p1 etc41 which are due to gravity or other forces1 then we may say that the total pressure1 ,p p p ,t7#a_/,r #a_/,r74 ,while in some cases1 for instance in speaking of the shape of a bird's egg1 we shall have to take account of these extraneous pressures1 in the present part of our subject we shall for the most part be able to neglect them4 ,our equation is an equation of equilibrium4 ,the resistance to compression1--the pressure outwards1--of our fluid mass1 is a constant quantity 7,p72 the pressure inwards1 ,t7#a_/,r #a_/,r71 is also constant2 and if 7unlike the case of the mobile amoeba7 the surface be homogeneous1 so that ,t is everywhere equal1 it follows that throughout the whole surface #a_/,r #a_/,r ,c 7a constant74 ,now equilibrium is attained after the surface contraction has done its utmost1 that is to say when it has reduced the surface to the smallest possible area2 and so we arrive1 from the physical side1 at the conclusion that a surface such that #a_/,r #a_/,r ,c1 in other words a surface which has the same mean curvature at all points1 is equivalent to a surface of minimal area3 and to the same conclusion we may also arrive through purely analytical mathematics4 ,it is obvious that the plane and the sphere are two examples of such surfaces1 for in both cases the radius of curvature is everywhere constant1 being equal to infinity in the case of the plane1 and to some definite magnitude in the case of the sphere4 ,from the fact that we may extend a soap-film across a ring of wire however fantastically the latter may be bent1 we realise that there is no limit to the number of surfaces of minimal area which may be constructed or may be imagined2 and while some of these are very complicated indeed1 some1 for instance a spiral helicoid screw1 are relatively very simple4 ,but if we limit ourselves to #bah surfaces of revolution 7that is to say1 to surfaces symmetrical about an axis71 we find1 as ,plateau was the first to shew1 that those which meet the case are very few in number4 ,they are six in all1 namely the plane1 the sphere1 the cylinder1 the catenoid1 the unduloid1 and a curious surface which ,plateau called the nodoid4 ,these several surfaces are all closely related1 and the passage from one to another is generally easy4 ,their mathematical interrelation is expressed by the fact 7first shewn by ,delaunay#bhf1 in #ahda7 that the plane curves by whose rotation they are generated are themselves generated as 8roulettes0 of the conic sections4 ,let us imagine a straight line upon which a circle1 an ellipse or other conic section rolls2 the focus of the conic section will describe a line in some relation to the fixed axis1 and this line 7or roulette71 rotating around the axis1 will describe in space one or other of the six surfaces of revolution with which we are dealing4 ,illustration3 ,fig4 #fa4 ,if we imagine an ellipse so to roll over a line1 either of its foci will describe a sinuous or wavy line 7,fig4 #fa;,b7 at a distance alternately maximal and minimal from the axis2 and this wavy line1 by rotation about the axis1 becomes the meridional line of the surface which we call the unduloid4 ,the more unequal the two axes are of our ellipse1 the more pronounced will be the sinuosity of the described roulette4 ,if the two axes be equal1 then our ellipse becomes a circle1 and the path described by its rolling centre is a straight line parallel to the axis 7,a72 and obviously the solid of revolution generated therefrom will be a cylinder4 ,if one axis of our ellipse vanish1 while the other remain of finite length1 then the ellipse is reduced to a straight line1 and its roulette will appear as a succession of semicircles touching one another upon the axis 7,c72 the solid of revolution will be a series of equal spheres4 ,if as before one axis of the ellipse vanish1 but the other be infinitely long1 then the curve described by the rotation #bai of this latter will be a circle of infinite radius1 i4e4 a straight line infinitely distant from the axis2 and the surface of rotation is now a plane4 ,if we imagine one focus of our ellipse to remain at a given distance from the axis1 but the other to become infinitely remote1 that is tantamount to saying that the ellipse becomes transformed into a parabola2 and by the rolling of this curve along the axis there is described a catenary 7,d71 whose solid of revolution is the catenoid4 ,lastly1 but this is a little more difficult to imagine1 we have the case of the hyperbola4 ,we cannot well imagine the hyperbola rolling upon a fixed straight line so that its focus shall describe a continuous curve4 ,but let us suppose that the fixed line is1 to begin with1 asymptotic to one branch of the hyperbola1 and that the rolling proceed until the line is now asymptotic to the other branch1 that is to say touching it at an infinite distance2 there will then be mathematical continuity if we recommence rolling with this second branch1 and so in turn with the other1 when each has run its course4 ,we shall see1 on reflection1 that the line traced by one and the same focus will be an 8elastic curve0 describing a succession of kinks or knots 7,e71 and the solid of revolution described by this meridional line about the axis is the so-called nodoid4 ,the physical transition of one of these surfaces into another can be experimentally illustrated by means of soap-bubbles1 or better still1 after the method of ,plateau1 by means of a large globule of oil1 supported when necessary by wire rings1 within a fluid of specific gravity equal to its own4 ,to prepare a mixture of alcohol and water of a density precisely equal to that of the oil-globule is a troublesome matter1 and a method devised by ,mr ,c4 ,r4 ,darling is a great improvement on ,plateau's#bhg4 ,mr ,darling uses the oily liquid orthotoluidene1 which does not mix with water1 has a beautiful and conspicuous red colour1 and has precisely the same density as water when both are kept at a temperature of #bd ,c4 ,we have therefore only to run the liquid into water at this temperature in order to produce beautifully spherical drops of any required size3 and by adding #bbj a little salt to the lower layers of water1 the drop may be made to float or rest upon the denser liquid4 ,we have already seen that the soap-bubble1 spherical to begin with1 is transformed into a plane when we relieve its internal pressure and let the film shrink back upon the orifice of the pipe4 ,if we blow a small bubble and then catch it up on a second pipe1 so that it stretches between1 we may gradually draw the two pipes apart1 with the result that the spheroidal surface will be gradually flattened in a longitudinal direction1 and the bubble will be transformed into a cylinder4 ,but if we draw the pipes yet farther apart1 the cylinder will narrow in the middle into a sort of hourglass form1 the increasing curvature of its transverse section being balanced by a gradually increasing negative curvature in the longitudinal section4 ,the cylinder has1 in turn1 been converted into an unduloid4 ,when we hold a portion of a soft glass tube in the flame1 and 8draw it out10 we are in the same identical fashion converting a cylinder into an unduloid 7,fig4 #fb;,a72 when on the other hand we stop the end and blow1 we again convert the cylinder into an unduloid 7,b71 but into one which is now positively1 while the former was negatively curved4 ,the two figures are essentially the same1 save that the two halves of the one are reversed in the other4 ,illustration3 ,fig4 #fb4 ,that spheres1 cylinders and unduloids are of the commonest occurrence among the forms of small unicellular organism1 or of individual cells in the simpler aggregates1 and that in the processes of growth1 reproduction and development transitions are frequent from one of these forms to another1 is obvious to the naturalist1 and we shall deal presently with a few illustrations of these phenomena4 ,but before we go further in this enquiry1 it will be necessary to consider1 to some small extent at least1 the curvatures of the six different surfaces1 that is to say1 to determine what modification #bba is required1 in each case1 of the general equation which applies to them all4 ,we shall find that with this question is closely connected the question of the pressures exercised by1 or impinging on the film1 and also the very important question of the limitations which1 from the nature of the case1 exist to prevent the extension of certain of the figures beyond certain bounds4 ,the whole subject is mathematical1 and we shall only deal with it in the most elementary way4 ,we have seen that1 in our general formula1 the expression #a_/,r #a_/,r ,c1 a constant2 and that this is1 in all cases1 the condition of our surface being one of minimal area4 ,in other words1 it is always true for one and all of the six surfaces which we have to consider4 ,but the constant ,c may have any value1 positive1 negative1 or nil4 ,in the case of the plane1 where ,r and ,r are both infinite1 it is obvious that #a_/,r #a_/,r #j4 ,the expression therefore vanishes1 and our dynamical equation of equilibrium becomes ,p p4 ,in short1 we can only have a plane film1 or we shall only find a plane surface in our cell1 when on either side thereof we have equal pressures or no pressure at all4 ,a simple case is the plane partition between two equal and similar cells1 as in a filament of spirogyra4 ,in the case of the sphere1 the radii are all equal1 ,r ,r2 they are also positive1 and ,t 7#a_/,r #a_/,r71 or #b ,t_/,r1 is a positive quantity1 involving a positive pressure ,p1 on the other side of the equation4 ,in the cylinder1 one radius of curvature has the finite and positive value ,r2 but the other is infinite4 ,our formula becomes ,t_/,r1 to which corresponds a positive pressure ,p1 supplied by the surface-tension as in the case of the sphere1 but evidently of just half the magnitude developed in the latter case for a given value of the radius ,r4 ,the catenoid has the remarkable property that its curvature in one direction is precisely equal and opposite to its curvature in the other1 this property holding good for all points of the surface4 ,that is to say1 ,r ,r2 and the expression becomes 7#a_/,r #a_/,r7 7#a_/,r #a_/,r7 #j2 in other words1 the surface1 as in the case of the plane1 has no #bbb curvature1 and exercises no pressure4 ,there are no other surfaces1 save these two1 which share this remarkable property2 and it follows1 as a simple corollary1 that we may expect at times to have the catenoid and the plane coexisting1 as parts of one and the same boundary system2 just as1 in a cylindrical drop or cell1 the cylinder is capped by portions of spheres1 such that the cylindrical and spherical portions of the wall exert equal positive pressures4 ,in the unduloid1 unlike the four surfaces which we have just been considering1 it is obvious that the curvatures change from one point to another4 ,at the middle of one of the swollen portions1 or 8beads10 the two curvatures are both positive2 the expression 7#a_/,r #a_/,r7 is therefore positive1 and it is also finite4 ,the film1 accordingly1 exercises a positive tension inwards1 which must be compensated by a finite and positive outward pressure ,p4 ,at the middle of one of the narrow necks1 between two adjacent beads1 there is obviously1 in the transverse direction1 a much stronger curvature than in the former case1 and the curvature which balances it is now a negative one4 ,but the sum of the two must remain positive1 as well as constant2 and we therefore see that the convex or positive curvature must always be greater than the concave or negative curvature at the same point4 ,this is plainly the case in our figure of the unduloid4 ,the nodoid is1 like the unduloid1 a continuous curve which keeps altering its curvature as it alters its distance from the axis2 but in this case the resultant pressure inwards is negative instead of positive4 ,but this curve is a complicated one1 and a full discussion of it would carry us beyond our scope4 ,illustration3 ,fig4 #fc4 ,in one of ,plateau's experiments1 a bubble of oil 7protected from gravity by the specific gravity of the surrounding fluid being identical with its own7 is balanced between two annuli4 ,it may then be brought to assume the form of ,fig4 #fc1 that is to say the form of a cylinder with spherical ends2 and there is then everywhere1 owing to the convexity of the surface film1 a pressure inwards upon the fluid contents of the bubble4 ,if the surrounding liquid be ever so little heavier or lighter than that which constitutes the drop1 then the conditions of equilibrium will be accordingly #bbc modified1 and the cylindrical drop will assume the form of an unduloid 7,fig4 #fd ,a1 ,b71 with its dilated portion below or above1 ,illustration3 ,fig4 #fd4 as the case may be2 and our cylinder may also1 of course1 be converted into an unduloid either by elongating it further1 or by abstracting a portion of its oil1 until at length rupture ensues and the cylinder breaks up into two new spherical drops4 ,in all cases alike1 the unduloid1 like the original cylinder1 will be capped by spherical ends1 which are the sign1 and the consequence1 of the positive pressure produced by the curved walls of the unduloid4 ,but if our initial cylinder1 instead of being tall1 be a flat or dumpy one 7with certain definite relations of height to breadth71 then new phenomena may be exhibited4 ,for now1 if a little oil be cautiously withdrawn from the mass by help of a small syringe1 the cylinder may be made to flatten down so that its upper and lower surfaces become plane2 which is of itself an indication that the pressure inwards is now nil4 ,but at the very moment when the upper and lower surfaces become plane1 it will be found that the sides curve inwards1 in the fashion shewn in ,fig4 #fe;,b4 ,this figure is a catenoid1 which1 as ,illustration3 ,fig4 #fe4 we have already seen1 is1 like the plane itself1 a surface exercising no pressure1 and which therefore may coexist with the plane as part of one and the same system4 ,we may continue to withdraw more oil from our bubble1 drop by drop1 and now the upper and lower surfaces dimple down into concave portions of spheres1 as the result of the negative internal pressure2 and thereupon the peripheral catenoid surface alters its form 7perhaps1 on this small scale1 imperceptibly71 and becomes a portion of a nodoid 7,fig4 #fe;,a74 #bbd ,it represents1 in fact1 that portion of the nodoid1 which in ,fig4 #ff lies between such points as ,o1 ,p4 ,while it is easy to ,illustration3 ,fig4 #ff4 draw the outline1 or meridional section1 of the nodoid 7as in ,fig4 #ff71 it is obvious that the solid of revolution to be derived from it1 can never be realised in its entirety3 for one part of the solid figure would cut1 or entangle with1 another4 ,all that we can ever do1 accordingly1 is to realise isolated portions of the nodoid4 ,if1 in a sequel to the preceding experiment of ,plateau's1 we use solid discs instead of annuli1 so as to enable us to exert direct mechanical pressure upon our globule of oil1 we again begin by adjusting the pressure of these discs so that the oil assumes the form of a cylinder3 our discs1 that is to say1 are adjusted to exercise a mechanical pressure equal to what in the former case was supplied by the surface-tension of the spherical caps or ends of the bubble4 ,if we now increase the pressure slightly1 the peripheral walls will become convexly curved1 exercising a precisely corresponding pressure4 ,under these circumstances the form assumed by the sides of our figure will be that of a portion of an unduloid4 ,if we increase the pressure between the discs1 the peripheral surface of oil will bulge out more and more1 and will presently constitute a portion of a sphere4 ,but we may continue the process yet further1 and within certain limits we shall find that the system remains perfectly stable4 ,what is this new curved surface which has arisen out of the sphere1 as the latter was produced from the unduloid8 ,it is no other than a portion of a nodoid1 that part which in ,fig4 #ff lies between such limits as ,m and ,n4 ,but this surface1 which is concave in both directions towards the surface of the oil within1 is exerting a pressure upon the latter1 just as did the sphere out of which a moment ago it was transformed2 and we had just stated1 in considering the previous experiment1 that the pressure inwards exerted by the nodoid was a negative one4 ,the explanation of this seeming discrepancy lies in the simple fact that1 if we follow the outline #bbe of our nodoid curve in ,fig4 #ff from ,o1 ,p1 the surface concerned in the former case1 to ,m1 ,n1 that concerned in the present1 we shall see that in the two experiments the surface of the liquid is not homologous1 but lies on the positive side of the curve in the one case and on the negative side in the other4 ,of all the surfaces which we have been describing1 the sphere is the only one which can enclose space2 the others can only help to do so1 in combination with one another or with the sphere itself4 ,thus we have seen that1 in normal equilibrium1 the cylindrical vesicle is closed at either end by a portion of a sphere1 and so on4 ,moreover the sphere is not only the only one of our figures which can enclose a finite space2 it is also1 of all possible figures1 that which encloses the greatest volume with the least area of surface2 it is strictly and absolutely the surface of minimal area1 and it is therefore the form which will be naturally assumed by a unicellular organism 7just as by a raindrop71 when it is practically homogeneous and when1 like ,orbulina floating in the ocean1 its surroundings are likewise practically homogeneous and symmetrical4 ,it is only relatively speaking that all the rest are surfaces minimae areae2 they are so1 that is to say1 under the given conditions1 which involve various forms of pressure or restraint4 ,such restraints are imposed1 for instance1 by the pipes or annuli with the help of which we draw out our cylindrical or unduloid oil-globule or soap-bubble2 and in the case of the organic cell1 similar restraints are constantly supplied by solidification1 partial or complete1 local or general1 of the cell-wall4 ,before we pass to biological illustrations of our surface-tension figures1 we have still another preliminary matter to deal with4 ,we have seen from our description of two of ,plateau's classical experiments1 that at some particular point one type of surface gives place to another2 and again1 we know that1 when we draw out our soap-bubble into and then beyond a cylinder1 there comes a certain definite point at which our bubble breaks in two1 and leaves us with two bubbles of which each is a sphere1 or a portion of a sphere4 ,in short there are certain definite limits to the dimensions of our figures1 within which limits equilibrium is stable but at which it becomes unstable1 and above which it #bbf breaks down4 ,moreover in our composite surfaces1 when the cylinder for instance is capped by two spherical cups or lenticular discs1 there is a well-defined ratio which regulates their respective curvatures1 and therefore their respective dimensions4 ,these two matters we may deal with together4 ,let us imagine a liquid drop which by appropriate conditions has been made to assume the form of a cylinder2 we have already seen that its ends will be terminated by portions of spheres4 ,since one and the same liquid film covers the sides and ends of the drop 7or since one and the same delicate membrane encloses the sides and ends of the cell71 we assume the surface-tension 7,t7 to be everywhere identical2 and it follows1 since the internal fluid-pressure is also everywhere identical1 that the expression 7#a_/,r #a_/,r7 for the cylinder is equal to the corresponding expression1 which we may call 7#a_/r #a_/r71 in the case of the terminal spheres4 ,but in the cylinder #a_/,r #j1 and in the sphere #a_/r #a_/r4 ,therefore our relation of equality becomes #a_/,r #b_/r1 or r #b ,r2 that is to say1 the sphere in question has just twice the radius of the cylinder of which it forms a cap4 ,illustration3 ,fig4 #fg4 ,and if ,ob1 the radius of the sphere1 be equal to twice the radius 7,oa7 of the cylinder1 it follows that the angle a,ob is an angle of #fj1 and b,oc is also an angle of #fj2 that is to say1 the arc bc is equal to 7#a_/c7 4 ,in other words1 the spherical disc which 7under the given conditions7 caps our cylinder1 is not a portion taken at haphazard1 but is neither more nor less than that portion of a sphere which is subtended by a cone of #fj4 ,moreover1 it is plain that the height of the spherical cap1 de1 ,ob ab ,r 7#b #c7 #jbg ,r1 where ,r is the radius of our cylinder1 or one-half the radius of our spherical cap3 in other words the normal height of the spherical cap over the end of the cylindrical cell is just a very little more than one-eighth of the diameter of the cylinder1 or of the radius of the #bbg sphere4 ,and these are the proportions which we recognise1 under normal circumstances1 in such a case as the cylindrical cell of ,spirogyra where its free end is capped by a portion of a sphere4 ,among the many important theoretical discoveries which we owe to ,plateau1 one to which we have just referred is of peculiar importance3 namely that1 with the exception of the sphere and the plane1 the surfaces with which we have been dealing are only in complete equilibrium within certain dimensional limits1 or in other words1 have a certain definite limit of stability2 only the plane and the sphere1 or any portions of a sphere1 are perfectly stable1 because they are perfectly symmetrical1 figures4 ,for experimental demonstration1 the case of the cylinder is the simplest4 ,if we produce a liquid film having the form of a cylinder1 either by ,illustration3 ,fig4 #fh4 drawing out a bubble or by supporting between two rings a globule of oil1 the experiment proceeds easily until the length of the cylinder becomes just about three times as great as its diameter4 ,but somewhere about this limit the cylinder alters its form2 it begins to narrow at the waist1 so passing into an unduloid1 and the deformation progresses quickly until at last our cylinder breaks in two1 and its two halves assume a spherical form4 ,it is found1 by theoretical considerations1 that the precise limit of stability is at the point when the length of the cylinder is exactly equal to its circumference1 that is to say1 when ,l #b,r1 or when the ratio of length to diameter is represented by 4 ,in the case of the catenoid1 ,plateau's experimental procedure was as follows4 ,to support his globule of oil 7in1 as usual1 a mixture of alcohol and water of its own specific gravity71 he used #bbh a pair of metal rings1 which happened to have a diameter of #ga millimetres2 and1 in a series of experiments1 he set these rings apart at distances of #ee1 #di1 #dg1 #de1 and #dc mm4 successively4 ,in each case he began by bringing his oil-globule into a cylindrical form1 by sucking superfluous oil out of the drop until this result was attained2 and always1 for the reason with which we are now acquainted1 the cylindrical sides were associated with spherical ends to the cylinder4 ,on continuing to withdraw oil in the hope of converting these spherical ends into planes1 he found1 naturally1 that the sides of the cylinder drew in to form a concave surface2 but it was by no means easy to get the extremities actually plane3 and unless they were so1 thus indicating that the surface-pressure of the drop was nil1 the curvature of the sides could not be that of a catenoid4 ,for in the first experiment1 when the rings were #ee mm4 apart1 as soon as the convexity of the ends was to a certain extent diminished1 it spontaneously increased again2 and the transverse constriction of the globule correspondingly deepened1 until at a certain point equilibrium set in anew4 ,indeed1 the more oil he removed1 the more convex became the ends1 until at last the increasing transverse constriction led to the breaking of the oil-globule into two4 ,in the third experiment1 when the rings were #dg mm4 apart1 it was easy to obtain end-surfaces that were actually plane1 and they remained so even though more oil was withdrawn1 the transverse constriction deepening accordingly4 ,only after a considerable amount of oil had been sucked up did the plane terminal surface become gradually convex1 and presently the narrow waist1 narrowing more and more1 broke across in the usual way4 ,finally in the fifth experiment1 where the rings were still nearer together1 it was again possible to bring the ends of the oil-globule to a plane surface1 as in the third and fourth experiments1 and to keep this surface plane in spite of some continued withdrawal of oil4 ,but very soon the ends became gradually concave1 and the concavity deepened as more and more oil was withdrawn1 until at a certain limit1 the whole oil-globule broke up in general disruption4 ,we learn from this that the limiting size of the catenoid was reached when the distance of the supporting rings was to their diameter as #dg to #ga1 or1 as nearly as possible1 as two to three2 #bbi and as a matter of fact it can be shewn that #b_/c is the true theoretical value4 ,above this limit of #b_/c1 the inevitable convexity of the end-surfaces shows that a positive pressure inwards is being exerted by the surface film1 and this teaches us that the sides of the figure actually constitute not a catenoid but an unduloid1 whose spontaneous changes tend to a form of greater stability4 ,below the #b_/c limit the catenoid surface is essentially unstable1 and the form into which it passes under certain conditions of disturbance such as that of the excessive withdrawal of oil1 is that of a nodoid 7,fig4 #fe;,a74 ,the unduloid has certain peculiar properties as regards its limitations of stability4 ,but as to these we need mention two facts only3 7#a7 that when the unduloid1 which we produce with our soap-bubble or our oil-globule1 consists of the figure containing a complete constriction1 it has somewhat wide limits of stability2 but 7#b7 if it contain the swollen portion1 then equilibrium is limited to the condition that the figure consists simply of one complete unduloid1 that is to say that its ends are constituted by the narrowest portions1 and its middle by the widest portion of the entire curve4 ,the theoretical proof of this latter fact is difficult1 but if we take the proof for granted1 the fact will serve to throw light on what we have learned regarding the stability of the cylinder4 ,for1 when we remember that the meridional section of our unduloid is generated by the rolling of an ellipse upon a straight line in its own plane1 we shall easily see that the length of the entire unduloid is equal to the circumference of the generating ellipse4 ,as the unduloid becomes less and less sinuous in outline1 it gradually approaches1 and in time reaches1 the form of a cylinder2 and correspondingly1 the ellipse which generated it has its foci more and more approximated until it passes into a circle4 ,the cylinder of a length equal to the circumference of its generating circle is therefore precisely homologous to an unduloid whose length is equal to the circumference of its generating ellipse2 and this is just what we recognise as constituting one complete segment of the unduloid4 ,while the figures of equilibrium which are at the same time surfaces of revolution are only six in number1 there is an infinite #bcj number of figures of equilibrium1 that is to say of surfaces of constant mean curvature1 which are not surfaces of revolution2 and it can be shewn mathematically that any given contour can be occupied by a finite portion of some one such surface1 in stable equilibrium4 ,the experimental verification of this theorem lies in the simple fact 7already noted7 that however we may bend a wire into a closed curve1 plane or not plane1 we may always1 under appropriate precautions1 fill the entire area with an unbroken film4 ,of the regular figures of equilibrium1 that is to say surfaces of constant mean curvature1 apart from the surfaces of revolution which we have discussed1 the helicoid spiral is the most interesting to the biologist4 ,this is a helicoid generated by a straight line perpendicular to an axis1 about which it turns at a uniform rate while at the same time it slides1 also uniformly1 along this same axis4 ,at any point in this surface1 the curvatures are equal and of opposite sign1 and the sum of the curvatures is accordingly nil4 ,among what are called 8ruled surfaces0 7which we may describe as surfaces capable of being defined by a system of stretched strings71 the plane and the helicoid are the only two whose mean curvature is null1 while the cylinder is the only one whose curvature is finite and constant4 ,as this simplest of helicoids corresponds1 in three dimensions1 to what in two dimensions is merely a plane 7the latter being generated by the rotation of a straight line about an axis without the superadded gliding motion which generates the helicoid71 so there are other and much more complicated helicoids which correspond to the sphere1 the unduloid and the rest of our figures of revolution1 the generating planes of these latter being supposed to wind spirally about an axis4 ,in the case of the cylinder it is obvious that the resulting figure is indistinguishable from the cylinder itself4 ,in the case of the unduloid we obtain a grooved spiral1 such as we may meet with in nature 7for instance in ,spirochtes1 ,bodo gracilis1 etc471 and which accordingly it is of interest to us to be able to recognise as a surface of minimal area or constant curvature4 ,the foregoing considerations deal with a small part only of the theory of surface tension1 or of capillarity3 with that part1 namely1 which relates to the forms of surface which are #bca capable of subsisting in equilibrium under the action of that force1 either of itself or subject to certain simple constraints4 ,and as yet we have limited ourselves to the case of a single surface1 or of a single drop or bubble1 leaving to another occasion a discussion of the forms assumed when such drops or vesicles meet and combine together4 ,in short1 what we have said may help us to understand the form of a cell1--considered1 as with certain limitations we may legitimately consider it1 as a liquid drop or liquid vesicle2 the conformation of a tissue or cell-aggregate must be dealt with in the light of another series of theoretical considerations4 ,in both cases1 we can do no more than touch upon the fringe of a large and difficult subject4 ,there are many forms capable of realisation under surface tension1 and many of them doubtless to be recognised among organisms1 which we cannot touch upon in this elementary account4 ,the subject is a very general one2 it is1 in its essence1 more mathematical than physical2 it is part of the mathematics of surfaces1 and only comes into relation with surface tension1 because this physical phenomenon illustrates and exemplifies1 in a concrete way1 most of the simple and symmetrical conditions with which the general mathematical theory is capable of dealing4 ,and before we pass to illustrate by biological examples the physical phenomena which we have described1 we must be careful to remember that the physical conditions which we have hitherto presupposed will never be wholly realised in the organic cell4 ,its substance will never be a perfect fluid1 and hence equilibrium will be more or less slowly reached2 its surface will seldom be perfectly homogeneous1 and therefore equilibrium will 7in the fluid condition7 seldom be perfectly attained2 it will very often1 or generally1 be the seat of other forces1 symmetrical or unsymmetrical2 and all these causes will more or less perturb the effects of surface tension acting by itself4 ,but we shall find that1 on the whole1 these effects of surface tension though modified are not obliterated nor even masked2 and accordingly the phenomena to which ,i have devoted the foregoing pages will be found manifestly recurring and repeating themselves among the phenomena of the organic cell4 ,in a spider's web we find exemplified several of the principles #bcb of surface tension which we have now explained4 ,the thread is formed out of the fluid secretion of a gland1 and issues from the body as a semi-fluid cylinder1 that is to say in the form of a surface of equilibrium1 the force of expulsion giving it its elongation and that of surface tension giving it its circular section4 ,it is prevented1 by almost immediate solidification on exposure to the air1 from breaking up into separate drops or spherules1 as it would otherwise tend to do as soon as the length of the cylinder had passed its limit of stability4 ,but it is otherwise with the sticky secretion which1 coming from another gland1 is simultaneously poured over the issuing thread when it is to form the spiral portion of the web4 ,this latter secretion is more fluid than the first1 and retains its fluidity for a very much longer time1 finally drying up after several hours4 ,by capillarity it 8wets0 the thread1 spreading itself over it in an even film1 which film is now itself a cylinder4 ,but this liquid cylinder has its limit of stability when its length equals its own circumference1 and therefore just at the points so defined it tends to disrupt into separate segments3 or rather1 in the actual case1 at points somewhat more distant1 owing to the imperfect fluidity of the viscous film1 and still more to the frictional drag upon it of the inner solid cylinder1 or thread1 with which it is in contact4 ,the cylinder disrupts in the usual manner1 passing first into the wavy outline of an unduloid1 whose swollen portions swell more and more till the contracted parts break asunder1 and we arrive at a series of spherical drops or beads1 of equal size1 strung at equal intervals along the thread4 ,if we try to spread varnish over a thin stretched wire1 we produce automatically the same identical result#bhh2 unless our varnish be such as to dry almost instantaneously1 it gathers into beads1 and do what we can1 we fail to spread it smooth4 ,it follows that1 according to the viscidity and drying power of the varnish1 the process may stop or seem to stop at any point short of the formation of the perfect spherules2 it is quite possible1 therefore1 that as our final stage we may only obtain half-formed beads1 or the wavy outline of an unduloid4 ,the formation of the beads may be facilitated or hastened by jerking the stretched thread1 as the spider actually does3 the #bcc effect of the jerk being to disturb and destroy the unstable equilibrium of the viscid cylinder#bhi4 ,another very curious phenomenon here presents itself4 ,in ,plateau's experimental separation of a cylinder of oil into two spherical portions1 it was noticed that1 when contact was nearly broken1 that is to say when the narrow neck of the unduloid had become very thin1 the two spherical bullae1 instead of absorbing the fluid out of the narrow neck into themselves as they had done with the preceding portion1 drew out this small remaining part of the liquid into a thin thread as they completed their spherical form and consequently receded from one another3 the reason being that1 after the thread or 8neck0 has reached a certain tenuity1 the internal friction of the fluid prevents or retards its rapid exit from the little thread to the adjacent spherule4 ,it is for the same reason that we are able to draw a glass rod or tube1 which we have heated in the middle1 into a long and uniform cylinder or thread1 by quickly separating the two ends4 ,but in the case of the glass rod1 the long thin intermediate cylinder quickly cools and solidifies1 while in the ordinary separation of a liquid cylinder the corresponding intermediate cylinder remains liquid2 and therefore1 like any other liquid cylinder1 it is liable to break up1 provided that its dimensions exceed the normal limit of stability4 ,and its length is generally such that it breaks at two points1 thus leaving two terminal portions continuous with the spheres and becoming confluent with these1 and one median portion which resolves itself into a comparatively tiny spherical drop1 midway between the original and larger two4 ,occasionally1 the same process of formation of a connecting thread repeats itself a second time1 between the small intermediate spherule and the large spheres2 and in this case we obviously obtain two additional spherules1 still smaller in size1 and lying one on either side of our first little one4 ,this whole phenomenon1 of equal and regularly interspaced beads1 often with little beads regularly interspaced between the larger ones1 and possibly also even a third series of still smaller beads regularly intercalated1 may be easily observed in a spider's web1 such as that of ,epeira1 very often with beautiful regularity1--which #bcd naturally1 however1 is sometimes interrupted and disturbed owing to a slight want of homogeneity in the secreted fluid2 and the same phenomenon is repeated on a grosser scale when the web is bespangled with dew1 and every thread bestrung with pearls innumerable4 ,to the older naturalists1 these regularly arranged and beautifully formed globules on the spider's web were a cause of great wonder and admiration4 ,blackwall1 counting some twenty globules in a tenth of an inch1 calculated that a large garden-spider's web comprised about #abj1jjj globules2 the net was spun and finished in about forty minutes1 and ,blackwall was evidently filled with astonishment at the skill and quickness with which the spider manufactured these little beads4 ,and no wonder1 for according to the above estimate they had to be made at the rate of about #ej per second#bij4 ,illustration3 ,fig4 #fi4 ,hair of ,trianea1 in glycerine4 7,after ,berthold47 ,the little delicate beads which stud the long thin pseudopodia of a foraminifer1 such as ,gromia1 or which in like manner appear upon the cylindrical film of protoplasm which covers the long radiating spicules of ,globigerina1 represent an identical phenomenon4 ,indeed there are many cases1 in which we may study in a protoplasmic filament the whole process of formation of such beads4 ,if we squeeze out on to a slide the viscid contents of a mistletoe berry1 the long sticky threads into which the substance runs shew the whole phenomenon particularly well4 ,another way to demonstrate it was noticed many years ago by ,hofmeister and afterwards explained by ,berthold4 ,the hairs of certain water-plants1 such as ,hydrocharis or ,trianea1 constitute very long cylindrical cells1 the protoplasm being supported1 and maintained in equilibrium by its contact with the cell-wall4 ,but if we immerse the filament in some dense fluid1 a little sugar-solution for instance1 or dilute glycerine1 the cell-sap tends to diffuse outwards1 the protoplasm parts company with its surrounding and supporting wall1 #bce and lies free as a protoplasmic cylinder in the interior of the cell4 ,thereupon it immediately shews signs of instability1 and commences to disrupt4 ,it tends to gather into spheres1 which however1 as in our illustration1 may be prevented by their narrow quarters from assuming the complete spherical form2 and in between these spheres1 we have more or less regularly alternate ones1 of smaller size#bia4 ,similar1 but less regular1 beads or droplets may be caused to appear1 under stimulation by an alternating current1 in the protoplasmic threads within the living cells of the hairs of ,tradescantia4 ,the explanation usually given is1 that the viscosity of the protoplasm is reduced1 or its fluidity increased2 but an increase of the surface tension would seem a more likely reason#bib4 ,illustration3 ,fig4 #gj4 ,phases of a ,splash4 7,from ,worthington47 ,we may take note here of a remarkable series of phenomena1 which1 though they seem at first sight to be of a very different order1 are closely related to the phenomena which attend and which bring about the breaking-up of a liquid cylinder or thread4 ,in some of ,mr ,worthington's most beautiful experiments on #bcf splashes1 it was found that the fall of a round pebble into water from a considerable height1 caused the rise of a filmy sheet of water in the form of a cup or cylinder2 and the edge of this cylindrical film tended to be cut up into alternate lobes and notches1 and the prominent lobes or 8jets0 tended1 in more extreme cases1 to break off or to break up into spherical beads 7,fig4 #gj7bic4 ,a precisely similar appearance is seen1 on a great scale1 in the thin edge of a breaking wave3 when the smooth cylindrical edge1 at a given moment1 shoots out an array of tiny jets which break up into the droplets which constitute 8spray0 7,fig4 #ga1 a1 b74 ,we are at once reminded of the beautifully symmetrical notching on the calycles of many hydroids1 which little cups before they became stiff and rigid had begun their existence as liquid or semi-liquid films4 ,illustration3 ,fig4 #ga4 ,a breaking wave4 7,from ,worthington47 ,the phenomenon is two-fold4 ,in the first place1 the edge of our tubular or crater-like film forms a liquid ring or annulus1 which is closely comparable with the liquid thread or cylinder which we have just been considering1 if only we conceive the thread to be bent round into the ring4 ,and accordingly1 just as the thread spontaneously segments1 first into an unduloid1 and then into separate spherical drops1 so likewise will the edge of our annulus tend to do4 ,this phase of notching1 or beading1 of the edge of the film is beautifully seen in many of ,worthington's experiments#bid4 ,in the second place1 the very fact of the rising of the crater means that liquid is flowing up from below towards the rim2 and the segmentation of the rim means that channels of easier flow are #bcg created1 along which the liquid is led1 or is driven1 into the protuberances3 and these are thus exaggerated into the jets or arms which are sometimes so conspicuous at the edge of the crater4 ,in short1 any film or film-like cup1 fluid or semi-fluid in its consistency1 will1 like the straight liquid cylinder1 be unstable3 and its instability will manifest itself 7among other ways7 in a tendency to segmentation or notching of the edge2 and just such a peripheral notching is a conspicuous feature of many minute organic cup-like structures4 ,in the case of the hydroid calycle 7,fig4 #gb71 we are led to the conclusion that the two common and conspicuous features of notching or indentation of the cup1 and of constriction or annulation of the long cylindrical stem1 are phenomena of the same order and are due to surface-tension in both cases alike4 ,illustration3 ,fig4 #gb4 ,calycles of ,campanularian zoophytes4 7,a7 ,c4 integra2 7,b7 ,c4 groenlandica2 7,c7 ,c4 bispinosa2 7,d7 ,c4 raridentata4 ,another phenomenon displayed in the same experiments is the formation of a rope-like or cord-like thickening of the edge of the annulus4 ,this is due to the more or less sudden checking at the rim of the flow of liquid rising from below3 and a similar peripheral thickening is frequently seen1 not only in some of our hydroid cups1 but in many ,vorticellas 7cf4 ,fig4 #ge71 and other organic cup-like conformations4 ,a perusal of ,mr ,worthington's book will soon suggest that these are not the only manifestations of surface-tension in connection with splashes which present curious resemblances and analogies to phenomena of organic form4 ,the phenomena of an ordinary liquid splash are so swiftly #bch transitory that their study is only rendered possible by 8instantaneous0 photography3 but this excessive rapidity is not an essential part of the phenomenon4 ,for instance1 we can repeat and demonstrate many of the simpler phenomena1 in a permanent or quasi-permanent form1 by splashing water on to a surface of dry sand1 or by firing a bullet into a soft metal target4 ,there is nothing1 then1 to prevent a slow and lasting manifestation1 in a viscous medium such as a protoplasmic organism1 of phenomena which appear and disappear with prodigious rapidity in a more mobile liquid4 ,nor is there anything peculiar in the 8splash0 itself2 it is simply a convenient method of setting up certain motions or currents1 and producing certain surface-forms1 in a liquid medium1--or even in such an extremely imperfect fluid as is represented 7in another series of experiments7 by a bed of sand4 ,accordingly1 we have a large range of possible conditions under which the organism might conceivably display configurations analogous to1 or identical with1 those which ,mr ,worthington has shewn us how to exhibit by one particular experimental method4 ,to one who has watched the potter at his wheel1 it is plain that the potter's thumb1 like the glass-blower's blast of air1 depends for its efficacy upon the physical properties of the medium on which it operates1 which for the time being is essentially a fluid4 ,the cup and the saucer1 like the tube and the bulb1 display 7in their simple and primitive forms7 beautiful surfaces of equilibrium as manifested under certain limiting conditions4 ,they are neither more nor less than glorified 8splashes10 formed slowly1 under conditions of restraint which enhance or reveal their mathematical symmetry4 ,we have seen1 and we shall see again before we are done1 that the art of the glass-blower is full of lessons for the naturalist as also for the physicist3 illustrating as it does the development of a host of mathematical configurations and organic conformations which depend essentially on the establishment of a constant and uniform pressure within a closed elastic shell or fluid envelope4 ,in like manner the potter's art illustrates the somewhat obscurer and more complex problems 7scarcely less frequent in biology7 of a figure of equilibrium which is an open surface1 or solid1 of revolution4 ,it is clear1 at the same time1 that the two series of problems are closely akin2 for the #bci glass-blower can make most things that the potter makes1 by cutting off portions of his hollow ware4 ,and besides1 when this fails1 and the glass-blower1 ceasing to blow1 begins to use his rod to trim the sides or turn the edges of wineglass or of beaker1 he is merely borrowing a trick from the craft of the potter4 ,it would be venturesome indeed to extend our comparison with these liquid surface-tension phenomena from the cup or calycle of the hydrozoon to the little hydroid polype within3 and yet ,i feel convinced that there is something to be learned by such a comparison1 though not without much detailed consideration and mathematical study of the surfaces concerned4 ,the cylindrical body of the tiny polype1 the jet-like row of tentacles1 the beaded annulations which these tentacles exhibit1 the web-like film which sometimes 7when they stand a little way apart7 conjoins their bases1 the thin annular film of tissue which surrounds the little organism's mouth1 and the manner in which this annular 8peristome0 contracts#bie1 like a shrinking soap-bubble1 to close the aperture1 are every one of them features to which we may find a singular and striking parallel in the surface-tension phenomena which ,mr ,worthington has illustrated and demonstrated in the case of the splash4 ,here however1 we may freely confess that we are for the present on the uncertain ground of suggestion and conjecture2 and so must we remain1 in regard to many other simple and symmetrical organic forms1 until their form and dynamical stability shall have been investigated by the mathematician3 in other words1 until the mathematicians shall have become persuaded that there is an immense unworked field wherein they may labour1 in the detailed study of organic form4 ,according to ,plateau1 the viscidity of the liquid1 while it helps to retard the breaking up of the cylinder and so increases the length of the segments beyond that which theory demands1 has nevertheless less influence in this direction than we might have expected4 ,on the other hand1 any external support or adhesion1 such as contact with a solid body1 will be equivalent to a reduction of surface-tension and so will very greatly increase the #bdj stability of our cylinder4 ,it is for this reason that the mercury in our thermometer tubes does not as a rule separate into drops1 though it occasionally does so1 much to our inconvenience4 ,and again it is for this reason that the protoplasm in a long and growing tubular or cylindrical cell does not necessarily divide into separate cells and internodes1 until the length of these far exceeds the theoretic limits4 ,of course however and whenever it does so1 we must1 without ever excluding the agency of surface tension1 remember that there may be other forces affecting the latter1 and accelerating or retarding that manifestation of surface tension by which the cell is actually rounded off and divided4 ,in most liquids1 ,plateau asserts that1 on the average1 the influence of viscosity is such as to cause the cylinder to segment when its length is about four times1 or at most from four to six times that of its diameter3 instead of a fraction over three times as1 in a perfect fluid1 theory would demand4 ,if we take it at four times1 it may then be shewn that the resulting spheres would have a diameter of about #ah times1 and their distance apart would be equal to about #bb times the diameter of the original cylinder4 ,the calculation is not difficult which would shew how these numbers are altered in the case of a cylinder formed around a solid core1 as in the case of the spider's web4 ,plateau has also made the interesting observation that the time taken in the process of division of the cylinder is directly proportional to the diameter of the cylinder1 while varying considerably with the nature of the liquid4 ,this question1 of the time occupied in the division of a cell or filament1 in relation to the dimensions of the latter1 has not so far as ,i know been enquired into by biologists4 ,from the simple fact that the sphere is of all surfaces that whose surface-area for a given volume is an absolute minimum1 we have already seen it to be plain that it is the one and only figure of equilibrium which will be assumed under surface-tension by a drop or vesicle1 when no other disturbing factors are present4 ,one of the most important of these disturbing factors will be introduced1 in the form of complicated tensions and pressures1 when one drop is in contact with another drop and when a system of intermediate films or partition walls is developed between them4 #bda ,this subject we shall discuss later1 in connection with cell-aggregates or tissues1 and we shall find that further theoretical considerations are needed as a preliminary to any such enquiry4 ,meanwhile let us consider a few cases of the forms of cells1 either solitary1 or in such simple aggregates that their individual form is little disturbed thereby4 ,let us clearly understand that the cases we are about to consider are those cases where the perfect symmetry of the sphere is replaced by another symmetry1 less complete1 such as that of an ellipsoidal or cylindrical cell4 ,the cases of asymmetrical deformation or displacement1 such as is illustrated in the production of a bud or the development of a lateral branch1 are much simpler4 ,for here we need only assume a slight and localised variation of surface-tension1 such as may be brought about in various ways through the heterogeneous chemistry of the cell2 to this point we shall return in our chapter on ,adsorption4 ,but the diffused and graded asymmetry of the system1 which brings about for instance the ellipsoidal shape of a yeast-cell1 is another matter4 ,if the sphere be the one surface of complete symmetry and therefore of independent equilibrium1 it follows that in every cell which is otherwise conformed there must be some definite force to cause its departure from sphericity2 and if this cause be the very simple and obvious one of the resistance offered by a solidified envelope1 such as an egg-shell or firm cell-wall1 we must still seek for the deforming force which was in action to bring about the given shape1 prior to the assumption of rigidity4 ,such a cause may be either external to1 or may lie within1 the cell itself4 ,on the one hand it may be due to external pressure or to some form of mechanical restraint3 as it is in all our experiments in which we submit our bubble to the partial restraint of discs or rings or more complicated cages of wire2 and on the other hand it may be due to intrinsic causes1 which must come under the head either of differences of internal pressure1 or of lack of homogeneity or isotropy in the surface itself#bif4 #bdb ,our full formula of equilibrium1 or equation to an elastic surface1 is ,p pe 7,t_/,r ,t_/,r71 where ,p is the internal pressure1 pe any extraneous pressure normal to the surface1 ,r1 ,r the radii of curvature at a point1 and ,t1 ,t1 the corresponding tensions1 normal to one another1 of the envelope4 ,now in any given form which we are seeking to account for1 ,r1 ,r are known quantities2 but all the other factors of the equation are unknown and subject to enquiry4 ,and somehow or other1 by this formula1 we must account for the form of any solitary cell whatsoever 7provided always that it be not formed by successive stages of solidification71 the cylindrical cell of ,spirogyra1 the ellipsoidal yeast-cell1 or 7as we shall see in another chapter7 the shape of the egg of any bird4 ,in using this formula hitherto1 we have taken it in a simplified form1 that is to say we have made several limiting assumptions4 ,we have assumed that ,p was simply the uniform hydrostatic pressure1 equal in all directions1 of a body of liquid2 we have assumed that the tension ,t was simply due to surface-tension in a homogeneous liquid film1 and was therefore equal in all directions1 so that ,t ,t2 and we have only dealt with surfaces1 or parts of a surface1 where extraneous pressure1 pn1 was non-existent4 ,now in the case of a bird's egg1 the external pressure pn1 that is to say the pressure exercised by the walls of the oviduct1 will be found to be a very important factor2 but in the case of the yeast-cell or the ,spirogyra1 wholly immersed in water1 no such external pressure comes into play4 ,we are accordingly left1 in such cases as these last1 with two hypotheses1 namely that the departure from a spherical form is due to inequalities in the internal pressure ,p1 or else to inequalities in the tension ,t1 that is to say to a difference between ,t and ,t4 ,in other words1 it is theoretically possible that the oval form of a yeast-cell is due to a greater internal pressure1 a greater 8tendency to grow10 in the direction of the longer axis of the ellipse1 or alternatively1 that with equal and symmetrical tendencies to growth there is associated a difference of external resistance in #bdc respect of the tension of the cell-wall4 ,now the former hypothesis is not impossible2 the protoplasm is far from being a perfect fluid2 it is the seat of various internal forces1 sometimes manifestly polar2 and accordingly it is quite possible that the internal forces1 osmotic and other1 which lead to an increase of the content of the cell and are manifested in pressure outwardly directed upon its wall may be unsymmetrical1 and such as to lead to a deformation of what would otherwise be a simple sphere4 ,but while this hypothesis is not impossible1 it is not very easy of acceptance4 ,the protoplasm1 though not a perfect fluid1 has yet on the whole the properties of a fluid2 within the small compass of the cell there is little room for the development of unsymmetrical pressures2 and1 in such a case as ,spirogyra1 where a large part of the cavity is filled by a fluid and watery cell-sap1 the conditions are still more obviously those under which a uniform hydrostatic pressure is to be expected4 ,but in variations of ,t1 that is to say of the specific surface-tension per unit area1 we have an ample field for all the various deformations with which we shall have to deal4 ,our condition now is1 that 7,t_/,r ,t_/,r7 a constant2 but it no longer follows1 though it may still often be the case1 that this will represent a surface of absolute minimal area4 ,as soon as ,t and ,t become unequal1 it is obvious that we are no longer dealing with a perfectly liquid surface film2 but its departure from a perfect fluidity may be of all degrees1 from that of a slight non-isotropic viscosity to the state of a firm elastic membrane#big4 ,and it matters little whether this viscosity or semi-rigidity be manifested in the self-same layer which is still a part of the protoplasm of the cell1 or in a layer which is completely differentiated into a distinct and separate membrane4 ,as soon as1 by secretion or 8adsorption10 the molecular constitution of the surface layer is altered1 it is clearly conceivable that the alteration1 or the secondary chemical changes which follow it1 may be such as to produce an anisotropy1 and to render the molecular forces less capable in one direction than another of exerting that contractile force by which they are striving to reduce to an absolute minimum the #bdd surface area of the cell4 ,a slight inequality in two opposite directions will produce the ellipsoid cell1 and a very great inequality will give rise to the cylindrical cell#bih4 ,i take it therefore1 that the cylindrical cell of ,spirogyra1 or any other cylindrical cell which grows in freedom from any manifest external restraint1 has assumed that particular form simply by reason of the molecular constitution of its developing surface-membrane2 and that this molecular constitution was anisotropous1 in such a way as to render extension easier in one direction than another4 ,such a lack of homogeneity or of isotropy1 in the cell-wall is often rendered visible1 especially in plant-cells1 in various ways1 in the form of concentric lamellae1 annular and spiral striations1 and the like4 ,but this phenomenon1 while it brings about a certain departure from complete symmetry1 is still compatible with1 and coexistent with1 many of the phenomena which we have seen to be associated with surface-tension4 ,the symmetry of tensions still leaves the cell a solid of revolution1 and its surface is still a surface of equilibrium4 ,the fluid pressure within the cylinder still causes the film or membrane which caps its ends to be of a spherical form4 ,and in the young cell1 where the surface pellicle is absent or but little differentiated1 as for instance in the ogonium of ,achlya1 or in the young zygospore of ,spirogyra1 we always see the tendency of the entire structure towards a spherical form reasserting itself3 unless1 as in the latter case1 it be overcome by direct compression within the cylindrical mother-cell4 ,moreover1 in those cases where the adult filament consists of cylindrical cells1 we see that the young1 germinating spore1 at first spherical1 very soon assumes with growth an elliptical or ovoid form3 the direct result of an incipient anisotropy of its envelope1 which when more developed will convert the ovoid into a cylinder4 ,we may also notice that a truly cylindrical cell is comparatively rare2 for in most cases1 what we call a cylindrical cell shews a distinct bulging of its sides2 it is not truly a cylinder1 but a portion of a spheroid or ellipsoid4 #bde ,unicellular organisms in general1 including the protozoa1 the unicellular cryptogams1 the various bacteria1 and the free1 isolated cells1 spores1 ova1 etc4 of higher organisms1 are referable for the most part to a very small number of typical forms2 but besides a certain number of others which may be so referable1 though obscurely1 there are obviously many others in which either no symmetry is to be recognized1 or in which the form is clearly not one of equilibrium4 ,among these latter we have ,amoeba itself1 and all manner of amoeboid organisms1 and also many curiously shaped cells1 such as the ,trypanosomes and various other aberrant ,infusoria4 ,we shall return to the consideration of these2 but in the meanwhile it will suffice to say that1 as their surfaces are not equilibrium-surfaces1 so neither are the living cells themselves in any stable equilibrium4 ,on the contrary1 they are in continual flux and movement1 each portion of the surface constantly changing its form1 and passing from one phase to another of an equilibrium which is never stable for more than a moment4 ,the former class1 which rest in stable equilibrium1 must fall 7as we have seen7 into two classes1--those whose equilibrium arises from liquid surface-tension alone1 and those in whose conformation some other pressure or restraint has been superimposed upon ordinary surface-tension4 ,to the fact that these little organisms belong to an order of magnitude in which form is mainly1 if not wholly1 conditioned and controlled by molecular forces1 is due the limited range of forms which they actually exhibit4 ,these forms vary according to varying physical conditions4 ,sometimes they do so in so regular and orderly a way that we instinctively explain them merely as 8phases of a life-history10 and leave physical properties and physical causation alone3 but many of their variations of form we treat as exceptional1 abnormal1 decadent or morbid1 and are apt to pass these over in neglect1 while we give our attention to what we suppose to be the typical or 8characteristic0 form or attitude4 ,in the case of the smallest organisms1 the bacteria1 micrococci1 and so forth1 the range of form is especially limited1 owing to their minuteness1 the powerful pressure which their highly curved surfaces exert1 and the comparatively homogeneous nature of their substance4 ,but within their narrow range of possible diversity #bdf these minute organisms are protean in their changes of form4 ,a certain species will not only change its shape from stage to stage of its little 8cycle0 of life2 but it will be remarkably different in outward form according to the circumstances under which we find it1 or the histological treatment to which we submit it4 ,hence the pathological student1 commencing the study of bacteriology1 is early warned to pay little heed to differences of form1 for purposes of recognition or specific identification4 ,whatever grounds we may have for attributing to these organisms a permanent or stable specific identity 7after the fashion of the higher plants and animals71 we can seldom safely do so on the ground of definite and always recognisable form3 we may often be inclined1 in short1 to ascribe to them a physiological 7sometimes a 8pathogenic071 rather than a morphological specificity4 ,illustration3 ,fig4 #gc4 ,a flagellate 8monad10 ,distigma proteus1 ,ehr4 7,after ,saville ,kent47 ,illustration3 ,fig4 #gd4 ,noctiluca miliaris4 ,among the ,infusoria1 we have a small number of forms whose symmetry is distinctly spherical1 for instance among the small flagellate monads2 but even these are seldom actually spherical except when we see them in a non-flagellate and more or less encysted or 8resting0 stage4 ,in this condition1 it need hardly be remarked that the spherical form is common and general among a great variety of unicellular organisms4 ,when our little monad developes a flagellum1 that is in itself an indication of 8polarity0 or symmetrical non-homogeneity of the cell2 and accordingly1 we #bdg usually see signs of an unequal tension of the membrane in the neighbourhood of the base of the flagellum4 ,here the tension is usually less than elsewhere1 and the radius of curvature is accordingly less3 in other words that end of the cell is drawn out to a tapering point 7,fig4 #gc74 ,but sometimes it is the other way1 as in ,noctiluca1 where the large flagellum springs from a depression in the otherwise uniformly rounded cell4 ,in this case the explanation seems to lie in the many strands of radiating protoplasm which converge upon this point1 and may be supposed to keep it relatively fixed by their viscosity1 while the rest of the cell-surface is free to expand 7,fig4 #gd74 ,illustration3 ,fig4 #ge4 ,various species of ,vorticella4 7,mostly after ,saville ,kent47 ,a very large number of ,infusoria represent unduloids1 or portions of unduloids1 and this type of surface appears and reappears in a great variety of forms4 ,the cups of the various species of ,vorticella 7,fig4 #ge7 are nothing in the world but a beautiful series of unduloids1 or partial unduloids1 in every gradation from a form that is all but cylindrical to one that is all but a perfect sphere4 ,these unduloids are not completely symmetrical1 but they are such unduloids as develop themselves when we suspend an oil-globule between two unequal rings1 or blow a soap-bubble between two unequal pipes2 for1 just as in these cases1 the surface of our ,vorticella bell finds its terminal supports1 on the one hand in its attachment to its narrow stalk1 and on the other in the thickened ring from which spring its circumoral cilia4 ,and here let me say1 that a point or zone from which cilia arise would seem always to have a peculiar relation to the surrounding tensions4 ,it usually forms a sharp salient1 a prominent point or ridge1 as in our little monads of ,fig4 #gc2 shewing that1 in its formation1 the surface tension had here locally diminished4 ,but if such a ridge or fillet consolidate in the least degree1 it becomes a source of strength1 and a point d'appui for the adjacent film4 ,we shall deal with this point again in the next chapter4 #bdh ,illustration3 ,fig4 #gf4 ,various species of ,salpingoeca4 ,illustration3 ,fig4 #gg4 ,various species of ,tintinnus1 ,dinobryon and ,codonella4 7,after ,saville ,kent and others47 ,illustration3 ,fig4 #gh4 ,vaginicola4 ,illustration3 ,fig4 #gi4 ,folliculina4 ,illustration3 ,fig4 #hj4 ,trachelophyllum4 7,after ,wreszniowski47 ,precisely the same series of unduloid forms may be traced in even greater variety among various other families or genera of the ,infusoria4 ,sometimes1 as in ,vorticella itself1 the unduloid is seen merely in the contour of the soft semifluid body of the living animal4 ,at other times1 as in ,salpingoeca1 ,tintinnus1 and many other genera1 we have a distinct membranous cup1 separate from the animal1 but originally secreted by1 and moulded upon1 its semifluid living surface4 ,here we have an excellent illustration of the contrast between the different ways in which such a structure may be regarded and interpreted4 ,the teleological explanation is that it is developed for the sake of protection1 as a domicile and shelter for the little organism within4 ,the mechanical explanation of the physicist 7seeking only after the 8efficient10 and not the 8final0 cause71 is that it is #bdi present1 and has its actual conformation1 by reason of certain chemico-physical conditions3 that it was inevitable1 under the given conditions1 that certain constituent substances actually present in the protoplasm should be aggregated by molecular forces in its surface layer2 that under this adsorptive process1 the conditions continuing favourable1 the particles should accumulate and concentrate till they formed 7with the help of the surrounding medium7 a pellicle or membrane1 thicker or thinner as the case might be2 that this surface pellicle or membrane was inevitably bound1 by molecular forces1 to become a surface of the least possible area which the circumstances permitted2 that in the present case1 the symmetry and 8freedom0 of the system permitted1 and ipso facto caused1 this surface to be a surface of revolution2 and that of the few surfaces of revolution which1 as being also surfaces minimae areae1 were available1 the unduloid was manifestly the one permitted1 and ipso facto caused1 by the dimensions of the organisms and other circumstances of the case4 ,and just as the thickness or thinness of the pellicle was obviously a subordinate matter1 a mere matter of degree1 so we also see that the actual outline of this or that particular unduloid is also a very subordinate matter1 such as physico-chemical variants of a minute kind would suffice to bring about2 for between the various unduloids which the various species of ,vorticella represent1 there is no more real difference than that difference of ratio or degree which exists between two circles of different diameter1 or two lines of unequal length4 #bej ,in very many cases 7of which ,fig4 #hj is an example71 we have an unduloid form exhibited1 not by a surrounding pellicle or shell1 but by the soft1 protoplasmic body of a ciliated organism4 ,in such cases the form is mobile1 and continually changes from one to another unduloid contour1 according to the movements of the animal4 ,we have here1 apparently1 to deal with an unstable equilibrium1 and also sometimes with the more complicated problem of 8stream-lines10 as in the difficult problems suggested by the form of a fish4 ,but this whole class of cases1 and of problems1 we can merely take note of in passing1 for their treatment is too hard for us4 ,in considering such series of forms as the various unduloids which we have just been regarding1 we are brought sharply up 7as in the case of our ,bacteria or ,micrococci7 against the biological concept of organic species4 ,in the intense classificatory activity of the last hundred years1 it has come about that every form which is apparently characteristic1 that is to say which is capable of being described or portrayed1 and capable of being recognised when met with again1 has been recorded as a species1--for we need not concern ourselves with the occasional discussions1 or individual opinions1 as to whether such and such a form deserve 8specific rank10 or be 8only a variety40 ,and this secular labour is pursued in direct obedience to the precept of the ,systema ,naturae1--8ut sic in summa confusione rerum apparenti1 summus conspiciatur ,naturae ordo40 ,in like manner the physicist records1 and is entitled to record1 his many hundred 8species0 of snow-crystals#bii1 or of crystals of calcium carbonate4 ,but regarding these latter species1 the physicist makes no assumptions3 he records them simpliciter1 as specific 8forms02 he notes1 as best he can1 the circumstances 7such as temperature or humidity7 under which they occur1 in the hope of elucidating the conditions determining their formation2 but above all1 he does not introduce #bea the element of time1 and of succession1 or discuss their origin and affiliation as an historical sequence of events4 ,but in biology1 the term species carries with it many large1 though often vague assumptions4 ,though the doctrine or concept of the 8permanence of species0 is dead and gone1 yet a certain definite value1 or sort of quasi-permanency1 is still connoted by the term4 ,thus if a tiny foraminiferal shell1 a ,lagena for instance1 be found living to-day1 and a shell indistinguishable from it to the eye be found fossil in the ,chalk or some other remote geological formation1 the assumption is deemed legitimate that that species has 8survived10 and has handed down its minute specific character or characters1 from generation to generation1 unchanged for untold myriads of years#cjj4 ,or if the ancient forms be like to1 rather than identical with the recent1 we still assume an unbroken descent1 accompanied by the hereditary transmission of common characters and progressive variations4 ,and if two identical forms be discovered at the ends of the earth1 still 7with occasional slight reservations on the score of possible 8homoplasy071 we build hypotheses on this fact of identity1 taking it for granted that the two appertain to a common stock1 whose dispersal in space must somehow be accounted for1 its route traced1 its epoch determined1 and its causes discussed or discovered4 ,in short1 the naturalist admits no exception to the rule that a 8natural classification0 can only be a genealogical one1 nor ever doubts that 8,the fact that we are able to classify organisms at all in accordance with the structural characteristics which they present1 is due to the fact of their being related by descent#cja40 ,but this great generalisation is apt in my opinion1 to carry us too far4 ,it may be safe and sure and helpful and illuminating when we apply it to such complex entities1--such thousand-fold resultants of the combination and permutation of many variable characters1--as a horse1 a lion or an eagle2 but 7to my mind7 it has a very different look1 and a far less firm foundation1 when we attempt to extend it to minute organisms whose specific characters are few and simple1 whose simplicity #beb becomes much more manifest when we regard it from the point of view of physical and mathematical description and analysis1 and whose form is referable1 or 7to say the least of it7 is very largely referable1 to the direct and immediate action of a particular physical force4 ,when we come to deal with the minute skeletons of the ,radiolaria we shall again find ourselves dealing with endless modifications of form1 in which it becomes still more difficult to discern1 or to apply1 the guiding principle of affiliation or genealogy4 ,illustration3 ,fig4 #ha4 ,among the more aberrant forms of ,infusoria is a little species known as ,trichodina pedicidus1 a parasite on the ,hydra1 or fresh-water polype 7,fig4 #ha47 ,this ,trichodina has the form of a more or less flattened circular disc1 with a ring of cilia around both its upper and lower margins4 ,the salient ridge from which these cilia spring may be taken1 as we have already said1 to play the part of a strengthening 8fillet40 ,the circular base of the animal is flattened1 in contact with the flattened surface of the ,hydra over which it creeps1 and the opposite1 upper surface may be flattened nearly to a plane1 or may at other times appear slightly convex or slightly concave4 ,the sides of the little organism are contracted1 forming a symmetrical equatorial groove between the upper and lower discs2 and1 on account of the minute size of the animal and its constant movements1 we cannot submit the curvature of this concavity to measurement1 nor recognise by the eye its exact contour4 ,but it is evident that the conditions are precisely similar to those described on p4 #bbc1 where we were considering the conditions of stability of the catenoid4 ,and it is further evident that1 when the upper disc is actually plane1 the equatorial groove is strictly a catenoid surface of revolution2 and when on the other hand it is depressed1 then the equatorial groove will tend to assume the form of a nodoidal surface4 ,another curious type is the flattened spiral of ,dinenympha#cjb #bec which reminds us of the cylindrical spiral of a ,spirillum among the bacteria4 ,in ,dinenympha we have a symmetrical figure1 whose two opposite surfaces each constitute a surface of constant mean curvature2 it is evidently a figure of equilibrium under certain special conditions of restraint4 ,the cylindrical coil of the ,spirillum1 on the other hand1 is a surface of constant mean curvature1 and therefore of equilibrium1 as truly1 and in the same sense1 as the cylinder itself4 ,illustration3 ,fig4 #hb4 ,dinenympha gracilis1 ,leidy4 ,illustration3 ,fig4 #hc4 ,a very curious conformation is that of the vibratile 8collar10 found in ,codosiga and the other 8,choanoflagellates10 and which we also meet with in the 8collar-cells0 which line the interior cavities of a sponge4 ,such collar-cells are always very minute1 and the collar is constituted of a very delicate film1 which shews an undulatory or rippling motion4 ,it is a surface of revolution1 and as it maintains itself in equilibrium 7though a somewhat unstable and fluctuating one71 it must be1 under the restricted circumstances of its case1 a surface of minimal area4 ,but it is not so easy to see what these special circumstances are2 and it is obvious that the collar1 if left to itself1 must at once #bed contract downwards towards its base1 and become confluent with the general surface of the cell2 for it has no longitudinal supports and no strengthening ring at its periphery4 ,but in all these collar-cells1 there stands within the annulus of the collar a large and powerful cilium or flagellum1 in constant movement2 and by the action of this flagellum1 and doubtless in part also by the intrinsic vibrations of the collar itself1 there is set up a constant steady current in the surrounding water1 whose direction would seem to be such that it passes up the outside of the collar1 down its inner side1 and out in the middle in the direction of the flagellum2 and there is a distinct eddy1 in which foreign particles tend to be caught1 around the peripheral margin of the collar4 ,when the cell dies1 that is to say when motion ceases1 the collar immediately shrivels away and disappears4 ,it is notable1 by the way1 that the edge of this little mobile cup is always smooth1 never notched or lobed as in the cases we have discussed on p4 #bcf3 this latter condition being the outcome of a definite instability1 marking the close of a period of equilibrium2 while in the vibratile collar of ,codosiga the equilibrium1 such as it is1 is being constantly renewed and perpetuated like that of a juggler's pole1 by the motions of the system4 ,i take it that1 somehow1 its existence 7in a state of partial equilibrium7 is due to the current motions1 and to the traction exerted upon it through the friction of the stream which is constantly passing by4 ,i think1 in short1 that it is formed very much in the same way as the cup-like ring of streaming ribbons1 which we see fluttering and vibrating in the air-current of a ventilating fan4 ,it is likely enough1 however1 that a different and much better explanation may yet be found2 and if we turn once more to ,mr ,worthington's ,study of ,splashes1 we may find a curious suggestion of analogy in the beautiful craters encircling a central jet 7as the collar of ,codosiga encircles the flagellum71 which we see produced in the later stages of the splash of a pebble#cjc4 #bee ,among the ,foraminifera we have an immense variety of forms1 which1 in the light of surface tension and of the principle of minimal area1 are capable of explanation and of reduction to a small number of characteristic types4 ,many of the ,foraminifera are composite structures1 formed by the successive imposition of cell upon cell1 and these we shall deal with later on2 let us glance here at the simpler conformations exhibited by the single chambered or 8monothalamic0 genera1 and perhaps one or two of the simplest composites4 ,we begin with forms1 like ,astrorhiza 7,fig4 #bai1 p4 #dfd71 which are in a high degree irregular1 and end with others which manifest a perfect and mathematical regularity4 ,the broad difference between these two types is that the former are characterised1 like ,amoeba1 by a variable surface tension1 and consequently by unstable equilibrium2 but the strong contrast between these and the regular forms is bridged over by various transition-stages1 or differences of degree4 ,indeed1 as in all other ,rhizopods1 the very fact of the emission of pseudopodia1 which reach their highest development in this group of animals1 is a sign of unstable surface-equilibrium2 and we must therefore consider that those forms which indicate symmetry and equilibrium in their shells have secreted these during periods when rest and uniformity of surface conditions alternated with the phases of pseudopodial activity4 ,the irregular forms are in almost all cases arenaceous1 that is to say they have no solid shells formed by steady adsorptive secretion1 but only a looser covering of sand grains with which the protoplasmic body has come in contact and cohered4 ,sometimes1 as in ,ramulina1 we have a calcareous shell combined with irregularity of form2 but here we can easily see a partial and as it were a broken regularity1 the regular forms of sphere and cylinder being repeated in various parts of the ramified mass4 ,when we look more closely at the arenaceous forms1 we find that the same thing is true of them2 they represent1 either in whole or part1 approximations to the form of surfaces of equilibrium1 spheres1 cylinders and so forth4 ,in ,aschemonella we have a precise replica of the calcareous ,ramulina2 and in ,astrorhiza itself1 in the forms distinguished by naturalists as ,a4 crassatina1 what is described as the 8subsegmented interior#cjd0 #bef seems to shew the natural1 physical tendency of the long semifluid cylinder of protoplasm to contract1 at its limit of stability1 into unduloid constrictions1 as a step towards the breaking up into separate spheres3 the completion of which process is restrained or prevented by the rigidity and friction of the arenaceous covering4 ,illustration3 ,fig4 #hd4 ,various species of ,lagena4 7,after ,brady47 ,passing to the typical1 calcareous-shelled ,foraminifera1 we have the most symmetrical of all possible types in the perfect sphere of ,orbulina2 this is a pelagic organism1 whose floating habitat places it in a position of perfect symmetry towards all external forces4 ,save for one or two other forms which are also spherical1 or approximately so1 like ,thurammina1 the rest of the monothalamic calcareous ,foraminifera are all comprised by naturalists within the genus ,lagena4 ,this large and varied genus consists of 8flask-shaped0 shells1 whose surface is simply that of an unduloid1 or more frequently1 like that of a flask itself1 an unduloid combined with a portion of a sphere4 ,we do not know the circumstances #beg under which the shell of ,lagena is formed1 nor the nature of the force by which1 during its formation1 the surface is stretched out into the unduloid form2 but we may be pretty sure that it is suspended vertically in the sea1 that is to say in a position of symmetry as regards its vertical axis1 about which the unduloid surface of revolution is symmetrically formed4 ,at the same time we have other types of the same shell in which the form is more or less flattened2 and these are doubtless the cases in which such symmetry of position was not present1 or was replaced by a broader1 lateral contact with the surface pellicle#cje4 ,illustration3 ,fig4 #he4 7,after ,darling47 ,while ,orbulina is a simple spherical drop1 ,lagena suggests to our minds a 8hanging drop10 drawn out to a long and slender neck by its own weight1 aided by the viscosity of the material4 ,indeed the various hanging drops1 such as ,mr ,c4 ,r4 ,darling shews us1 are the most beautiful and perfect unduloids1 with spherical ends1 that it is possible to conceive4 ,a suitable liquid1 a little denser than water and incapable of mixing with it 7such as ethyl benzoate71 is poured on a surface of water4 ,it spreads #beh over the surface and gradually forms a hanging drop1 approximately hemispherical2 but as more liquid is added the drop sinks or rather grows downwards1 still adhering to the surface film2 and the balance of forces between gravity and surface tension results in the unduloid contour1 as the increasing weight of the drop tends to stretch it out and finally break it in two4 ,at the moment of rupture1 by the way1 a tiny droplet is formed in the attenuated neck1 such as we described in the normal division of a cylindrical thread 7p4 #bcc74 ,to pass to a much more highly organised class of animals1 we find the unduloid beautifully exemplified in the little flask-shaped shells of certain ,pteropod mollusca1 e4g4 ,cuvierina#cjf4 ,here again the symmetry of the figure would at once lead us to suspect that the creature lived in a position of symmetry to the surrounding forces1 as for instance if it floated in the ocean in an erect position1 that is to say with its long axis coincident with the direction of gravity2 and this we know to be actually the mode of life of the little ,pteropod4 ,many species of ,lagena are complicated and beautified by a pattern1 and some by the superaddition to the shell of plane extensions or 8wings40 ,these latter give a secondary1 bilateral symmetry to the little shell1 and are strongly suggestive of a phase or period of growth in which it lay horizontally on the surface1 instead of hanging vertically from the surface-film3 in which1 that is to say1 it was a floating and not a hanging drop4 ,the pattern is of two kinds4 ,sometimes it consists of a sort of fine reticulation1 with rounded or more or less hexagonal interspaces3 in other cases it is produced by a symmetrical series of ridges or folds1 usually longitudinal1 on the body of the flask-shaped cell1 but occasionally transversely arranged upon the narrow neck4 ,the reticulated and folded patterns we may consider separately4 ,the netted pattern is very similar to the wrinkled surface of a dried pea1 or to the more regular wrinkled patterns upon many other seeds and even pollen-grains4 ,if a spherical body after developing a 8skin0 begin to shrink a little1 and if the skin have so far lost its elasticity as to be unable to keep pace with the shrinkage of the inner mass1 it will tend to fold or wrinkle2 and if the shrinkage be uniform1 and the elasticity and flexibility of the skin be also uniform1 then the amount of #bei folding will be uniformly distributed over the surface4 ,little concave depressions will appear1 regularly interspaced1 and separated by convex folds4 ,the little concavities being of equal size 7unless the system be otherwise perturbed7 each one will tend to be surrounded by six others2 and when the process has reached its limit1 the intermediate boundary-walls1 or raised folds1 will be found converted into a regular pattern of hexagons4 ,but the analogy of the mechanical wrinkling of the coat of a seed is but a rough and distant one2 for we are evidently dealing with molecular rather than with mechanical forces4 ,in one of ,darling's experiments1 a little heavy tar-oil is dropped onto a saucer of water1 over which it spreads in a thin film showing beautiful interference colours after the fashion of those of a soap-bubble4 ,presently tiny holes appear in the film1 which gradually increase in size till they form a cellular pattern or honeycomb1 the oil gathering together in the meshes or walls of the cellular net4 ,some action of this sort is in all probability at work in a surface-film of protoplasm covering the shell4 ,as a physical phenomenon the actions involved are by no means fully understood1 but surface-tension1 diffusion and cohesion doubtless play their respective parts therein#cjg4 ,the very perfect cellular patterns obtained by ,leduc 7to which we shall have occasion to refer in a subsequent chapter7 are diffusion patterns on a larger scale1 but not essentially different4 ,illustration3 ,fig4 #hf4 ,the folded or pleated pattern is doubtless to be explained1 in a general way1 by the shrinkage of a surface-film under certain #bfj conditions of viscous or frictional restraint4 ,a case which 7as it seems to me7 is closely analogous to that of our foraminiferal shells is described by ,quincke#cjh1 who let a film of albumin or of resin set and harden upon a surface of quicksilver1 and found that the little solid pellicle had been thrown into a pattern of symmetrical folds4 ,if the surface thus thrown into folds be that of a cylinder1 or any other figure with one principal axis of symmetry1 such as an ellipsoid or unduloid1 the direction of the folds will tend to be related to the axis of symmetry1 and we might expect accordingly to find regular longitudinal1 or regular transverse wrinkling4 ,now as a matter of fact we almost invariably find in the ,lagena the former condition3 that is to say1 in our ellipsoid or unduloid cell1 the puckering takes the form of the vertical fluting on a column1 rather than that of the transverse pleating of an accordion4 ,and further1 there is often a tendency for such longitudinal flutings to be more or less localised at the end of the ellipsoid1 or in the region where the unduloid merges into its spherical base4 ,in this latter region we often meet with a regular series of short longitudinal folds1 as we do in the forms of ,lagena denominated ,l4 semistriata4 ,all these various forms of surface can be imitated1 or rather can be precisely reproduced1 by the art of the glass-blower#cji4 ,furthermore1 they remind one1 in a striking way1 of the regular ribs or flutings in the film or sheath which splashes up to envelop a smooth ball which has been dropped into a liquid1 as ,mr ,worthington has so beautifully shewn#caj4 #bfa ,in ,mr ,worthington's experiment1 there appears to be something of the nature of a viscous drag in the surface-pellicle2 but whatever be the actual cause of variation of tension1 it is not difficult to see that there must be in general a tendency towards longitudinal puckering or 8fluting0 in the case of a thin-walled cylindrical or other elongated body1 rather than a tendency towards transverse puckering1 or 8pleating40 ,for let us suppose that some change takes place involving an increase of surface-tension in some small area of the curved wall1 and leading therefore to an increase of pressure3 that is to say let ,t become ,t t1 and ,p become ,p p4 ,our new equation of equilibrium1 then1 in place of ,p ,t_/r ,t_/r becomes ,p p 7,t t7_/r 7,t t7_/r1 and by subtraction1 p t_/r t_/r4 ,now if r r1 t_/r t_/r4 ,therefore1 in order to produce the small increment of pressure p1 it is easier to do so by increasing t_/r than t_/r2 that is to say1 the easier way is to alter1 or diminish r4 ,and the same will hold good if the tension and pressure be diminished instead of increased4 ,this is as much as to say that1 when corrugation or 8rippling0 of the walls takes place owing to small changes of surface-tension1 and consequently of pressure1 such corrugation is more likely to take place in the plane of r1--that is to say1 in the plane of greatest curvature4 ,and it follows that in such a figure as an ellipsoid1 wrinkling will be most likely to take place not only in a longitudinal direction but near the extremities of the figure1 that is to say again in the region of greatest curvature4 ,illustration3 ,fig4 #hg4 ,nodosaria scalaris1 ,batsch4 ,illustration3 ,fig4 #hh4 ,gonangia of ,campanularians4 7a7 ,c4 gracilis2 7b7 ,c4 grandis4 7,after ,allman47 ,the longitudinal wrinkling of the flask-shaped bodies of our ,lagenae1 and of the more or less cylindrical cells of many other ,foraminifera 7,fig4 #hg71 is in complete accord with the above theoretical considerations2 but nevertheless1 we soon find that our result is not a general one1 but is defined by certain limiting conditions1 and is accordingly subject to what are1 at first sight1 important exceptions4 ,for instance1 when we turn to the narrow neck of the ,lagena we see at once that our theory no longer holds2 for #bfb the wrinkling which was invariably longitudinal in the body of the cell is as invariably transverse in the narrow neck4 ,the reason for the difference is not far to seek4 ,the conditions in the neck are very different from those in the expanded portion of the cell3 the main difference being that the thickness of the wall is no longer insignificant1 but is of considerable magnitude as compared with the diameter1 or circumference1 of the neck4 ,we must accordingly take it into account in considering the bending moments at any point in this region of the shell-wall4 ,and it is at once obvious that1 in any portion of the narrow neck1 flexure of a wall in a transverse direction will be very difficult1 while flexure in a longitudinal direction will be comparatively easy2 just as1 in the case of a long narrow strip of iron1 we may easily bend it into folds running transversely to its long axis1 but not the other way4 ,the manner in which our little ,lagena-shell tends to fold or wrinkle1 longitudinally in its wider part1 and transversely or annularly in its narrow neck1 is thus completely and easily explained4 ,an identical phenomenon is apt to occur in the little flask-shaped gonangia1 or reproductive capsules1 of some of the hydroid zoophytes4 ,in the annexed drawings of these gonangia in two species of ,campanularia1 we see that in one case the little vesicle #bfc has the flask-shaped or unduloid configuration of a ,lagena2 and here the walls of the flask are longitudinally fluted1 just after the manner we have witnessed in the latter genus4 ,but in the other ,campanularian the vesicles are long1 narrow and tubular1 and here a transverse folding or pleating takes the place of the longitudinally fluted pattern4 ,and the very form of the folds or pleats is enough to suggest that we are not dealing here with a simple phenomenon of surface-tension1 but with a condition in which surface-tension and stiffness are both present1 and play their parts in the resultant form4 ,illustration3 ,fig4 #hi4 ,various ,foraminifera 7after ,brady71 a1 ,nodosaria simplex2 b1 ,n4 pygmaea2 c1 ,n4 costulata2 e1 ,n4 hispida2 f1 ,n4 elata2 d1 ,rheophax 7,lituola7 distans2 g1 ,sagrina virgata4 ,passing from the solitary flask-shaped cell of ,lagena1 we have1 in another series of forms1 a constricted cylinder1 or succession of unduloids2 such as are represented in ,fig4 #hi1 illustrating certain species of ,nodosaria1 ,rheophax and ,sagrina4 ,in some of these cases1 and certainly in that of the arenaceous genus ,rheophax1 we have to do with the ordinary phenomenon of a segmenting or partially segmenting cylinder4 ,but in others1 the structure is not developed out of a continuous protoplasmic cylinder1 but as we can see by examining the interior of the shell1 it has been formed in successive stages1 beginning with a simple unduloid 8,lagena10 about whose neck1 after its solidification1 another drop of protoplasm accumulated1 and in turn assumed the unduloid1 or lagenoid1 form4 ,the chains of interconnected bubbles which #bfd ,morey and ,draper made many years ago of melted resin are a very similar if not identical phenomenon#caa4 ,there now remain for our consideration1 among the ,protozoa1 the great oceanic group of the ,radiolaria1 and the little group of their freshwater allies1 the ,heliozoa4 ,in nearly all these forms we have this specific chemical difference from the ,foraminifera1 that when they secrete1 as they generally do secrete1 a hard skeleton1 it is composed of silica instead of lime4 ,these organisms and the various beautiful and highly complicated skeletal fabrics which they develop give us many interesting illustrations of physical phenomena1 among which the manifestations of surface-tension are very prominent4 ,but the chief phenomena connected with their skeletons we shall deal with in another place1 under the head of spicular concretions4 ,in a simple and typical ,heliozoan1 such as the ,sun-animalcule1 ,actinophrys sol1 we have a 8drop0 of protoplasm1 contracted by its surface tension into a spherical form4 ,within the heterogeneous protoplasmic mass are more fluid portions1 and at the surface which separates these from the surrounding protoplasm a similar surface tension causes them also to assume the form of spherical 8vacuoles10 which in reality are little clear drops within the big one2 unless indeed they become numerous and closely packed1 in which case1 instead of isolated spheres or droplets they will constitute a 8froth10 their mutual pressures and tensions giving rise to regular configurations such as we shall study in the next chapter4 ,one or more of such clear spaces may be what is called a 8contractile vacuole03 that is to say1 a droplet whose surface tension is in unstable equilibrium and is apt to vanish altogether1 so that the definite outline of the vacuole suddenly disappears#cab4 ,again1 within the protoplasm are one or more nuclei1 whose own surface tension 7at the surface between the nucleus and the surrounding protoplasm71 has drawn them in turn into the shape #bfe of spheres4 ,outwards through the protoplasm1 and stretching far beyond the spherical surface of the cell1 there run stiff linear threads of modified or differentiated protoplasm1 replaced or reinforced in some cases by delicate siliceous needles4 ,in either case we know little or nothing about the forces which lead to their production1 and we do not hide our ignorance when we ascribe their development to a 8radial polarisation0 of the cell4 ,in the case of the protoplasmic filament1 we may 7if we seek for a hypothesis71 suppose that it is somehow comparable to a viscid stream1 or 8liquid vein10 thrust or squirted out from the body of the cell4 ,but when it is once formed1 this long and comparatively rigid filament is separated by a distinct surface from the neighbouring protoplasm1 that is to say from the more fluid surface-protoplasm of the cell2 and the latter begins to creep up the filament1 just as water would creep up the interior of a glass tube1 or the sides of a glass rod immersed in the liquid4 ,it is the simple case of a balance between three separate tensions3 7#a7 that between the filament and the adjacent protoplasm1 7#b7 that between the filament and the adjacent water1 and 7#c7 that between the water and the protoplasm4 ,calling these tensions respectively ,tfp1 ,tfw1 and ,twp1 equilibrium will be attained when the angle of contact between the fluid protoplasm and the filament is such that cos 7,tfw ,twp7_/,tfp4 ,it is evident in this case that the angle is a very small one4 ,the precise form of the curve is somewhat different from that which1 under ordinary circumstances1 is assumed by a liquid which creeps up a solid surface1 as water in contact with air creeps up a surface of glass2 the difference being due to the fact that here1 owing to the density of the protoplasm being practically identical with that of the surrounding medium1 the whole system is practically immune from gravity4 ,under normal circumstances the curve is part of the 8elastic curve0 by which that surface of revolution is generated which we have called1 after ,plateau1 the nodoid2 but in the present case it is apparently a catenary4 ,whatever curve it be1 it obviously forms a surface of revolution around the filament4 ,since the attraction exercised by this surface tension is symmetrical around the filament1 the latter will be pulled equally #bff in all directions2 in other words it will tend to be set normally to the surface of the sphere1 that is to say radiating directly outwards from the centre4 ,if the distance between two adjacent filaments be considerable1 the curve will simply meet the filament at the angle already referred to2 but if they be sufficiently near together1 we shall have a continuous catenary curve forming a hanging loop between one filament and the other4 ,and when this is so1 and the radial filaments are more or less symmetrically interspaced1 we may have a beautiful system of honeycomb-like depressions over the surface of the organism1 each cell of the honeycomb having a strictly defined geometric configuration4 ,illustration3 ,fig4 #ij4 ,a1 ,trypanosoma tineae 7after ,minchin72 ,b1 ,spirochaeta anodontae 7after ,fantham74 ,in the simpler ,radiolaria1 the spherical form of the entire organism is equally well-marked2 and here1 as also in the more complicated ,heliozoa 7such as ,actinosphaerium71 the organism is differentiated into several distinct layers1 each boundary surface tending to be spherical1 and so constituting sphere within sphere4 ,one of these layers at least is close packed with vacuoles1 forming an 8alveolar meshwork10 with the configurations of which we shall attempt in another chapter to correlate the characteristic structure of certain complex types of skeleton4 ,an exceptional form of cell1 but a beautiful manifestation of surface-tension 7or so ,i take it to be71 occurs in ,trypanosomes1 those tiny parasites of the blood that are associated with sleeping-sickness and many other grave or dire maladies4 ,these tiny organisms consist of elongated solitary cells down one side of which runs a very delicate frill1 or 8undulating membrane10 the free edge of which is seen to be slightly thickened1 and the whole of #bfg which undergoes rhythmical and beautiful wavy movements4 ,when certain ,trypanosomes are artificially cultivated 7for instance ,t4 rotatorium1 from the blood of the frog71 phases of growth are witnessed in which the organism has no undulating membrane1 but possesses a long cilium or 8flagellum10 springing from near the front end1 and exceeding the whole body in length#cac4 ,again1 in ,t4 lewisii1 when it reproduces by 8multiple fission10 the products of this division are likewise devoid of an undulating membrane1 but are provided with a long free flagellum#cad4 ,it is a plausible assumption to suppose that1 as the flagellum waves about1 it comes to lie near and parallel to the body of the cell1 and that the frill or undulating membrane is formed by the clear1 fluid protoplasm of the surface layer springing up in a film to run up and along the flagellum1 just as a soap-film would be formed in similar circumstances4 ,illustration3 ,fig4 #ia4 ,a1 ,trichomonas muris1 ,hartmann2 ,b1 ,trichomastix serpentis1 ,dobell2 ,c1 ,trichomonas angusta1 ,alexeieff4 7,after ,kofoid47 ,this mode of formation of the undulating membrane or frill appears to be confirmed by the appearances shewn in ,fig4 #ia4 #bfh ,here we have three little organisms closely allied to the ordinary ,trypanosomes1 of which one1 ,trichomastix 7,b71 possesses four flagella1 and the other two1 ,trichomonas1 apparently three only3 the two latter possess the frill1 which is lacking in the first#cae4 ,but it is impossible to doubt that when the frill is present 7as in ,a and ,c71 its outer edge is constituted by the apparently missing flagellum 7a71 which has become attached to the body of the creature at the point c1 near its posterior end2 and all along its course1 the superficial protoplasm has been drawn out into a film1 between the flagellum 7a7 and the adjacent surface or edge of the body 7b74 ,illustration3 ,fig4 #ib4 ,herpetomonas assuming the undulatory membrane of a ,trypanosome4 7,after ,d4 ,l4 ,mackinnon47 ,moreover1 this mode of formation has been actually witnessed and described1 though in a somewhat exceptional case4 ,the little flagellate monad ,herpetomonas is normally destitute of an undulating membrane1 but possesses a single long terminal flagellum4 ,according to ,dr ,d4 ,l4 ,mackinnon1 the cytoplasm in a certain stage of growth becomes somewhat 8sticky10 a phrase which we may in all probability interpret to mean that its surface tension is being reduced4 ,for this stickiness is shewn in two ways4 ,in the first place1 the long body1 in the course of its various bending movements1 is apt to adhere head to tail 7so to speak71 giving a rounded or sometimes annular form to the organism1 such as has also been described in certain species or stages of ,trypanosomes4 ,but again1 the long flagellum1 if it get bent backwards upon the body1 tends to adhere to its surface4 8,where the flagellum was pretty long and active1 its efforts to continue movement under these abnormal conditions resulted in the gradual lifting up from the cytoplasm of the body of a sort of pseudo-undulating membrane 7,fig4 #ib74 ,the movements of this structure were so exactly those of a true undulating membrane that it was #bfi difficult to believe one was not dealing with a small1 blunt trypanosome#caf40 ,this in short is a precise description of the mode of development which1 from theoretical considerations alone1 we should conceive to be the natural if not the only possible way in which the undulating membrane could come into existence4 ,there is a genus closely allied to ,trypanosoma1 viz4 ,trypanoplasma1 which possesses one free flagellum1 together with an undulating membrane2 and it resembles the neighbouring genus ,bodo1 save that the latter has two flagella and no undulating membrane4 ,in like manner1 ,trypanosoma so closely resembles ,herpetomonas that1 when individuals ascribed to the former genus exhibit a free flagellum only1 they are said to be in the 8,herpetomonas stage40 ,in short all through the order1 we have pairs of genera1 which are presumed to be separate and distinct1 viz4 ,trypanosoma-,herpetomonas1 ,trypanoplasma-,bodo1 ,trichomastix-,trichomonas1 in which one differs from the other mainly if not solely in the fact that a free flagellum in the one is replaced by an undulating membrane in the other4 ,we can scarcely doubt that the two structures are essentially one and the same4 ,the undulating membrane of a ,trypanosome1 then1 according to our interpretation of it1 is a liquid film and must obey the law of constant mean curvature4 ,it is under curious limitations of freedom3 for by one border it is attached to the comparatively motionless body1 while its free border is constituted by a flagellum which retains its activity and is being constantly thrown1 like the lash of a whip1 into wavy curves4 ,it follows that the membrane1 for every alteration of its longitudinal curvature1 must at the same instant become curved in a direction perpendicular thereto2 it bends1 not as a tape bends1 but with the accompaniment of beautiful but tiny waves of double curvature1 all tending towards the establishment of an 8equipotential surface02 and its characteristic undulations are not originated by an active mobility of the membrane but are due to the molecular tensions which produce the very same result in a soap-film under similar circumstances4 ,in certain ,spirochaetes1 ,s4 anodontae 7,fig4 #ij7 and ,s4 balbiani #bgj 7which we find in oysters71 a very similar undulating membrane exists1 but it is coiled in a regular spiral round the body of the cell4 ,it forms a 8screw-surface10 or helicoid1 and1 though we might think that nothing could well be more curved1 yet its mathematical properties are such that it constitutes a 8ruled surface0 whose 8mean curvature0 is everywhere nil2 and this property 7as we have seen7 it shares with the plane1 and with the plane alone4 ,precisely such a surface1 and of exquisite beauty1 may be produced by bending a wire upon itself so that part forms an axial rod and part a spiral wrapping round the axis1 and then dipping the whole into a soapy solution4 ,these undulating and helicoid surfaces are exactly reproduced among certain forms of spermatozoa4 ,the tail of a spermatozoon consists normally of an axis surrounded by clearer and more fluid protoplasm1 and the axis sometimes splits up into two or more slender filaments4 ,to surface tension operating between these and the surface of the fluid protoplasm 7just as in the case of the flagellum of the ,trypanosome71 ,i ascribe the formation of the undulating membrane which we find1 for instance1 in the spermatozoa of the newt or salamander2 and of the helicoid membrane1 wrapped in a far closer and more beautiful spiral than that which we saw in ,spirochaeta1 which is characteristic of the spermatozoa of many birds4 ,before we pass from the subject of the conformation of the solitary cell we must take some account of certain other exceptional forms1 less easy of explanation1 and still less perfectly understood4 ,such is the case1 for instance1 with the red blood-corpuscles of man and other vertebrates2 and among the sperm-cells of the decapod crustacea we find forms still more aberrant and not less perplexing4 ,these are among the comparatively few cells or cell-like structures whose form seems to be incapable of explanation by theories of surface-tension4 ,in all the mammalia 7save a very few7 the red blood-corpuscles are flattened circular discs1 dimpled in upon their two opposite sides4 ,this configuration closely resembles that of an india-rubber ball when we pinch it tightly between finger and thumb2 and we may also compare it with that experiment of ,plateau's #bga 7described on p4 #bbc71 where a flat cylindrical oil-drop1 of certain relative dimensions1 can1 by sucking away a little of the contained oil1 be made to assume the form of a biconcave disc1 whose periphery is part of a nodoidal surface4 ,from the relation of the nodoid to the 8elastic curve10 we perceive that these two examples are closely akin one to the other4 ,illustration3 ,fig4 #ic4 ,the form of the corpuscle is symmetrical1 and its surface is a surface of revolution2 but it is obviously not a surface of constant mean curvature1 nor of constant pressure4 ,for we see at once that1 in the sectional diagram 7,fig4 #ic71 the pressure inwards due to surface tension is positive at ,a1 and negative at ,c2 at ,b there is no curvature in the plane of the paper1 while perpendicular to it the curvature is negative1 and the pressure therefore is also negative4 ,accordingly1 from the point of view of surface tension alone1 the blood-corpuscle is not a surface of equilibrium2 or in other words1 it is not a fluid drop suspended in another liquid4 ,it is obvious therefore that some other force or forces must be at work1 and the simple effect of mechanical pressure is here excluded1 because the blood-corpuscle exhibits its characteristic shape while floating freely in the blood4 ,in the lower vertebrates the blood-corpuscles have the form of a flattened oval disc1 with rather sharp edges and ellipsoidal surfaces1 and this again is manifestly not a surface of equilibrium4 ,two facts are especially noteworthy in connection with the form of the blood-corpuscle4 ,in the first place1 its form is only maintained1 that is to say it is only in equilibrium1 in relation to certain properties of the medium in which it floats4 ,if we add a little water to the blood1 the corpuscle quickly loses its characteristic shape and becomes a spherical drop1 that is to say a true surface of minimal area and of stable equilibrium4 ,if on the other hand we add a strong solution of salt1 or a little glycerine1 the corpuscle contracts1 and its surface becomes puckered and uneven4 ,in these phenomena it is so far obeying the laws of diffusion and of surface tension4 #bgb ,in the second place1 it can be exactly imitated artificially by means of other colloid substances4 ,many years ago ,norris made the very interesting observation that in an emulsion of glue the drops assumed a biconcave form resembling that of the mammalian corpuscles#cag4 ,the glue was impure1 and doubtless contained lecithin2 and it is possible 7as ,professor ,waymouth ,reid tells me7 to make a similar emulsion with cerebrosides and cholesterin oleate1 in which the same conformation of the drops or particles is beautifully shewn4 ,now such cholesterin bodies have an important place among those in which ,lehmann and others have shewn and studied the formation of fluid crystals1 that is to say of bodies in which the forces of crystallisation and the forces of surface tension are battling with one another#cah2 and1 for want of a better explanation1 we may in the meanwhile suggest that some such cause is at the bottom of the conformation the explanation of which presents so many difficulties4 ,but we must not1 perhaps1 pass from this subject without adding that the case is a difficult and complex one from the physiological point of view4 ,for the surface of a blood-corpuscle consists of a 8semi-permeable membrane10 through which certain substances pass freely and not others 7for the most part anions and not cations71 and it may be1 accordingly1 that we have in life a continual state of osmotic inequilibrium1 of negative osmotic tension within1 to which comparatively simple cause the imperfect distension of the corpuscle may be also due#cai4 ,the whole phenomenon would be comparatively easy to understand if we might postulate a stiffer peripheral region to the corpuscle1 in the form for instance of a peripheral elastic ring4 ,such an annular thickening or stiffening1 like the 8collapse-rings0 which an engineer inserts in a boiler1 has been actually asserted to exist1 but its presence is not authenticated4 ,but it is not at all improbable that we have still much to learn about the phenomena of osmosis itself1 as manifested in the case of minute bodies such as a blood-corpuscle2 and 7as ,professor ,peddie suggests to me7 it is by no means impossible that curvature #bgc of the surface may itself modify the osmotic or perhaps the adsorptive action4 ,if it should be found that osmotic action tended to stop1 or to reverse1 on change of curvature1 it would follow that this phenomenon would give rise to internal currents2 and the change of pressure consequent on these would tend to intensify the change of curvature when once started#cbj4 ,illustration3 ,fig4 #id4 ,sperm-cells of ,decapod ,crustacea 7after ,koltzoff74 a1 ,inachus scorpio2 b1 ,galathea squamifera2 c1 do4 after maceration1 to shew spiral fibrillae4 ,the sperm-cells of the ,decapod crustacea exhibit various singular shapes4 ,in the ,crayfish they are flattened cells with stiff curved processes radiating outwards like a ,st ,catherine's wheel2 in ,inachus there are two such circles of stiff processes2 in ,galathea we have a still more complex form1 with long and slightly twisted processes4 ,in all these cases1 just as in the case of the blood-corpuscle1 the structure alters1 and finally loses1 its characteristic form when the nature or constitution 7or as we may assume in particular--the density7 of the surrounding medium is changed4 ,here again1 as in the blood-corpuscle1 we have to do with a very important force1 which we had not hitherto considered in this connection1--the force of osmosis1 manifested under conditions similar to those of ,pfeffer's classical experiments on the plant-cell4 ,the surface of the cell acts as a 8semi-permeable membrane10 #bgd permitting the passage of certain dissolved substances 7or their 8ions07 and including or excluding others2 and thus rendering manifest and measurable the existence of a definite 8osmotic pressure40 ,in the case of the sperm-cells of ,inachus1 certain quantitative experiments have been performed#cba4 ,the sperm-cell exhibits its characteristic conformation while lying in the serous fluid of the animal's body1 in ordinary sea-water1 or in a #e per cent4 solution of potassium nitrate2 these three fluids being all 8isotonic0 with one another4 ,as we alter the concentration of potassium nitrate1 the cell assumes certain definite forms corresponding to definite concentrations of the salt2 and1 as a further and final proof that the phenomenon is entirely physical1 it is found that other salts produce an identical effect when their concentration is proportionate to their molecular weight1 and whatever identical effect is produced by various salts in their respective concentrations1 a similarly identical effect is produced when these concentrations are doubled or otherwise proportionately changed#cbb4 ,illustration3 ,fig4 #ie4 ,sperm-cells of ,inachus1 as they appear in saline solutions of varying density4 7,after ,koltzoff47 ,thus the following table shews the percentage concentrations of certain salts necessary to bring the cell into the forms a and c of ,fig4 #ie2 in each case the quantities are proportional to the molecular weights1 and in each case twice the quantity is necessary to produce the effect of ,fig4 #ie;c compared with that which gives rise to the all but spherical form of ,fig4 #ie;a4 #bge concentration of salts in which the sperm-cell of ,inachus assumes the form of fig4 a fig4 c ,sodium chloride #jf #ab ,sodium nitrate #jhe #ag ,potassium nitrate #aj #bj ,acetic acid #bb #de ,cane sugar #ej #ajj ,illustration3 ,fig4 #if4 ,sperm-cell of ,dromia4 7,after ,koltzoff47 ,if we look then1 upon the spherical form of the cell as its true condition of symmetry and of equilibrium1 we see that what we call its normal appearance is just one of many intermediate phases of shrinkage1 brought about by the abstraction of fluid from its interior as the result of an osmotic pressure greater outside than inside the cell1 and where the shrinkage of volume is not kept pace with by a contraction of the surface-area4 ,in the case of the blood-corpuscle1 the shrinkage is of no great amount1 and the resulting deformation is symmetrical2 such structural inequality as may be necessary to account for it need be but small4 ,but in the case of the sperm-cells1 we must have1 and we actually do find1 a somewhat complicated arrangement of more or less rigid or elastic structures in the wall of the cell1 which like the wire framework in ,plateau's experiments1 restrain and modify the forces acting on the drop4 ,in one form of ,plateau's experiments1 instead of supporting his drop on rings or frames of wire1 he laid upon its surface one or more elastic coils2 and then1 on withdrawing oil from the centre of his globule1 he saw its uniform shrinkage counteracted by the spiral springs1 with the result that the centre of each elastic coil seemed to shoot out into a prominence4 ,just such spiral coils are figured 7after ,koltzoff7 in ,fig4 #if2 and they may be regarded as precisely akin to those local thickenings1 spiral and other1 to which we have already ascribed the cylindrical form of the ,spirogyra cell4 ,in all probability we must in like manner attribute the peculiar spiral and other forms1 for instance of many ,infusoria1 to the #bgf presence1 among the multitudinous other differentiations of their protoplasmic substance1 of such more or less elastic fibrillae1 which play as it were the part of a microscopic skeleton#cbc4 ,but these cases which we have just dealt with1 lead us to another consideration4 ,in a semi-permeable membrane1 through which water passes freely in and out1 the conditions of a liquid surface are greatly modified2 and1 in the ideal or ultimate case1 there is neither surface nor surface tension at all4 ,and this would lead us somewhat to reconsider our position1 and to enquire whether the true surface tension of a liquid film is actually responsible for all that we have ascribed to it1 or whether certain of the phenomena which we have assigned to that cause may not in part be due to the contractility of definite and elastic membranes4 ,but to investigate this question1 in particular cases1 is rather for the physiologist3 and the morphologist may go on his way1 paying little heed to what is no doubt a difficulty4 ,in surface tension we have the production of a film with the properties of an elastic membrane1 and with the special peculiarity that contraction continues with the same energy however far the process may have already gone2 while the ordinary elastic membrane contracts to a certain extent1 and contracts no more4 ,but within wide limits the essential phenomena are the same in both cases4 ,our fundamental equations apply to both cases alike4 ,and accordingly1 so long as our purpose is morphological1 so long as what we seek to explain is regularity and definiteness of form1 it matters little if we should happen1 here or there1 to confuse surface tension with elasticity1 the contractile forces manifested at a liquid surface with those which come into play at the complex internal surfaces of an elastic solid4 #bgg ,,chapter ,,vi ,a ,,note ,,on ,,adsorption ,a very important corollary to1 or amplification of the theory of surface tension is to be found in the modern chemico-physical doctrine of ,adsorption#cbd4 ,in its full statement this subject soon becomes complicated1 and involves physical conceptions and mathematical treatment which go beyond our range4 ,but it is necessary for us to take account of the phenomenon1 though it be in the most elementary way4 ,in the brief account of the theory of surface tension with which our last chapter began1 it was pointed out that1 in a drop of liquid1 the potential energy of the system could be diminished1 and work manifested accordingly1 in two ways4 ,in the first place we saw that1 at our liquid surface1 surface tension tends to set up an equilibrium of form1 in which the surface is reduced or contracted either to the absolute minimum of a sphere1 or at any rate to the least possible area which is permitted by the various circumstances and conditions2 and if the two bodies which comprise our system1 namely the drop of liquid and its surrounding medium1 be simple substances1 and the system be uncomplicated by other distributions of force1 then the energy of the system will have done its work when this equilibrium of form1 this minimal area of surface1 is once attained4 ,this phenomenon of the production of a minimal surface-area we have now seen to be of fundamental importance in the external morphology of the cell1 and especially 7so far as we have yet gone7 of the solitary cell or unicellular organism4 #bgh ,but we also saw1 according to ,gauss's equation1 that the potential energy of the system will be diminished 7and its diminution will accordingly be manifested in work7 if from any cause the specific surface energy be diminished1 that is to say if it be brought more nearly to an equality with the specific energy of the molecules in the interior of the liquid mass4 ,this latter is a phenomenon of great moment in modern physiology1 and1 while we need not attempt to deal with it in detail1 it has a bearing on cell-form and cell-structure which we cannot afford to overlook4 ,in various ways a diminution of the surface energy may be brought about4 ,for instance1 it is known that every isolated drop of fluid has1 under normal circumstances1 a surface-charge of electricity3 in such a way that a positive or negative charge 7as the case may be7 is inherent in the surface of the drop1 while a corresponding charge1 of contrary sign1 is inherent in the immediately adjacent molecular layer of the surrounding medium4 ,now the effect of this distribution1 by which all the surface molecules of our drop are similarly charged1 is that by virtue of this charge they tend to repel one another1 and possibly also to draw other molecules1 of opposite charge1 from the interior of the mass2 the result being in either case to antagonise or cancel1 more or less1 that normal tendency of the surface molecules to attract one another which is manifested in surface tension4 ,in other words1 an increased electrical charge concentrating at the surface of a drop tends1 whether it be positive or negative1 to lower the surface tension4 ,but a still more important case has next to be considered4 ,let us suppose that our drop consists no longer of a single chemical substance1 but contains other substances either in suspension or in solution4 ,suppose 7as a very simple case7 that it be a watery fluid1 exposed to air1 and containing droplets of oil3 we know that the specific surface tension of oil in contact with air is much less than that of water1 and it follows that1 if the watery surface of our drop be replaced by an oily surface the specific surface energy of the system will be notably diminished4 ,now under these circumstances it is found that 7quite apart from gravity1 by which the oil might float to the surface7 the oil has a tendency to be drawn to the surface2 and this phenomenon of molecular attraction #bgi or 8adsorption0 represents the work done1 equivalent to the diminished potential energy of the system#cbe4 ,in more general terms1 if a liquid 7or one or other of two adjacent liquids7 be a chemical mixture1 some one constituent in which1 if it entered into or increased in amount in the surface layer1 would have the effect of diminishing its surface tension1 then that constituent will have a tendency to accumulate or concentrate at the surface3 the surface tension may be said1 as it were1 to exercise an attraction on this constituent substance1 drawing it into the surface layer1 and this tendency will proceed until at a certain 8surface concentration0 equilibrium is reached1 its opponent being that osmotic force which tends to keep the substance in uniform solution or diffusion4 ,in the complex mixtures which constitute the protoplasm of the living cell1 this phenomenon of 8adsorption0 has abundant play3 for many of these constituents1 such as oils1 soaps1 albumens1 etc4 possess the required property of diminishing surface tension4 ,moreover1 the more a substance has the power of lowering the surface tension of the liquid in which it happens to be dissolved1 the more will it tend to displace another and less effective substance from the surface layer4 ,thus we know that protoplasm always contains fats or oils1 not only in visible drops1 but also in the finest suspension or 8colloidal solution40 ,if under any impulse1 such for instance as might arise from the ,brownian movement1 a droplet of oil be brought close to the surface1 it is at once drawn into that surface1 and tends to spread itself in a thin layer over the whole surface of the cell4 ,but a soapy surface 7for instance7 would have in contact with the surrounding water a surface tension even less than that of the film of oil3 and consequently1 if soap be present in the water it will in turn be adsorbed1 and will tend to displace the oil from the surface pellicle#cbf4 ,and this is all as #bhj much as to say that the molecules of the dissolved or suspended substance or substances will so distribute themselves throughout the drop as to lead towards an equilibrium1 for each small unit of volume1 between the superficial and internal energy2 or so1 in other words1 as to lead towards a reduction to a minimum of the potential energy of the system4 ,this tendency to concentration at the surface of any substance within the cell by which the surface tension tends to be diminished1 or vice versa1 constitutes1 then1 the phenomenon of ,adsorption2 and the general statement by which it is defined is known as the ,willard-,gibbs1 or ,gibbs-,thomson law#cbg4 ,among the many important physical features or concomitants of this phenomenon1 let us take note at present that we need not conceive of a strictly superficial distribution of the adsorbed substance1 that is to say of its direct association with the surface layer of molecules such as we imagined in the case of the electrical charge2 but rather of a progressive tendency to concentrate1 more and more1 as the surface is nearly approached4 ,indeed we may conceive the colloid or gelatinous precipitate in which1 in the case of our protoplasmic cell1 the dissolved substance tends often to be thrown down1 to constitute one boundary layer after another1 the general effect being intensified and multiplied by the repeated addition of these new surfaces4 ,moreover1 it is not less important to observe that the process of adsorption1 in the neighbourhood of the surface of a heterogeneous liquid mass1 is a process which takes time2 the tendency to surface concentration is a gradual and progressive one1 and will fluctuate with every minute change in the composition of our substance and with every change in the area of its surface4 ,in other words1 it involves 7in every heterogeneous substance7 a continual instability of equilibrium3 and a constant manifestation #bha of motion1 sometimes in the mere invisible transfer of molecules but often in the production of visible currents of fluid or manifest alterations in the form or outline of the system4 ,the physiologist1 as we have already remarked1 takes account of the general phenomenon of adsorption in many ways3 particularly in connection with various results and consequences of osmosis1 inasmuch as this process is dependent on the presence of a membrane1 or membranes1 such as the phenomenon of adsorption brings into existence4 ,for instance it plays a leading part in all modern theories of muscular contraction1 in which phenomenon a connection with surface tension was first indicated by ,fitz,gerald and d',arsonval nearly forty years ago#cbh4 ,and1 as ,w4 ,ostwald was the first to shew1 it gives us an entirely new conception of the relation of gases 7that is to say1 of oxygen and carbon dioxide7 to the red corpuscles of the blood#cbi4 ,but restricting ourselves1 as much as may be1 to our morphological aspect of the case1 there are several ways in which adsorption begins at once to throw light upon our subject4 ,in the first place1 our preliminary account1 such as it is1 is already tantamount to a description of the process of development of a cell-membrane1 or cell-wall4 ,the so-called 8secretion0 of this cell-wall is nothing more than a sort of exudation1 or striving towards the surface1 of certain constituent molecules or particles within the cell2 and the ,gibbs-,thomson law formulates1 in part at least1 the conditions under which they do so4 ,the adsorbed material may range from the almost unrecognisable pellicle of a blood-corpuscle to the distinctly differentiated 8ectosarc0 of a protozoan1 and again to the development of a fully formed cell-wall1 as in the cellulose partitions of a vegetable tissue4 ,in such cases1 the dissolved and adsorbable material has not only the property of lowering the surface tension1 and hence #bhb of itself accumulating at the surface1 but has also the property of increasing the viscosity and mechanical rigidity of the material in which it is dissolved or suspended1 and so of constituting a visible and tangible 8membrane#ccj40 ,the 8zoogloea0 around a group of bacteria is probably a phenomenon of the same order4 ,in the superficial deposition of inorganic materials we see the same process abundantly exemplified4 ,not only do we have the simple case of the building of a shell or 8test0 upon the outward surface of a living cell1 as for instance in a ,foraminifer1 but in a subsequent chapter1 when we come to deal with various spicules and spicular skeletons such as those of the sponges and of the ,radiolaria1 we shall see that it is highly characteristic of the whole process of spicule-formation for the deposits to be laid down just in the 8interfacial0 boundaries between cells or vacuoles1 and that the form of the spicular structures tends in many cases to be regulated and determined by the arrangement of these boundaries4 ,in physical chemistry1 an important distinction is drawn between adsorption and pseudo-adsorption#cca1 the former being a reversible1 the latter an irreversible or permanent phenomenon4 ,that is to say1 adsorption1 strictly speaking1 implies the surface-concentration of a dissolved substance1 under circumstances which1 if they be altered or reversed1 will cause the concentration to diminish or disappear4 ,but pseudo-adsorption includes cases1 doubtless originating in adsorption proper1 where subsequent changes leave the concentrated substance incapable of re-entering the liquid system4 ,it is obvious that many 7though not all7 of our biological illustrations1 for instance the formation of spicules or of permanent cell-membranes1 belong to the class of so-called pseudo-adsorption phenomena4 ,but the apparent contrast between the two is in the main a secondary one1 and however important to the chemist is of little consequence to us4 #bhc ,while this brief sketch of the theory of membrane-formation is cursory and inadequate1 it is enough to shew that the physical theory of adsorption tends in part to overturn1 in part to simplify enormously1 the older histological descriptions4 ,we can no longer be content with such statements as that of ,strasbrger1 that membrane-formation in general is associated with the 8activity of the kinoplasm10 or that of ,harper that a certain spore-membrane arises directly from the astral rays#ccb4 ,in short1 we have easily reached the general conclusion that1 the formation of a cell-wall or cell-membrane is a chemico-physical phenomenon1 which the purely objective methods of the biological microscopist do not suffice to interpret4 ,if the process of adsorption1 on which the formation of a membrane depends1 be itself dependent on the power of the adsorbed substance to lower the surface tension1 it is obvious that adsorption can only take place when the surface tension already present is greater than zero4 ,it is for this reason that films or threads of creeping protoplasm shew little tendency1 or none1 to cover themselves with an encysting membrane2 and that it is only when1 in an altered phase1 the protoplasm has developed a positive surface tension1 and has accordingly gathered itself up into a more or less spherical body1 that the tendency to form a membrane is manifested1 and the organism develops its 8cyst0 or cell-wall4 ,it is found that a rise of temperature greatly reduces the adsorbability of a substance1 and this doubtless comes1 either in part or whole1 from the fact that a rise of temperature is itself a cause of the lowering of surface tension4 ,we may in all probability ascribe to this fact and to its converse1 or at least associate with it1 such phenomena as the encystment of unicellular organisms at the approach of winter1 or the frequent formation of strong shells or membranous capsules in 8winter-eggs40 ,again1 since a film or a froth 7which is a system of films7 can only be maintained by virtue of a certain viscosity or rigidity of #bhd the liquid1 it may be quickly caused to disappear by the presence in its neighbourhood of some substance capable of reducing the surface tension2 for this substance1 being adsorbed1 may displace from the adsorptive layer a material to which was due the rigidity of the film4 ,in this way a 8bathytonic0 substance such as ether causes most foams to subside1 and the pouring oil on troubled waters not only stills the waves but still more quickly dissipates the foam of the breakers4 ,the process of breaking up an alveolar network1 such as occurs at a certain stage in the nuclear division of the cell1 may perhaps be ascribed in part to such a cause1 as well as to the direct lowering of surface tension by electrical agency4 ,our last illustration has led us back to the subject of a previous chapter1 namely to the visible configuration of the interior of the cell2 and in connection with this wide subject there are many phenomena on which light is apparently thrown by our knowledge of adsorption1 and of which we took little or no account in our former discussion4 ,one of these phenomena is that visible or concrete 8polarity10 which we have already seen to be in some way associated with a dynamical polarity of the cell4 ,this morphological polarity may be of a very simple kind1 as when1 in an epithelial cell1 it is manifested by the outward shape of the elongated or columnar cell itself1 by the essential difference between its free surface and its attached base1 or by the presence in the neighbourhood of the former of mucous or other products of the cell's activity4 ,but in a great many cases1 this 8polarised0 symmetry is supplemented by the presence of various fibrillae1 or of linear arrangements of particles1 which in the elongated or 8monopolar0 cell run parallel with its axis1 and which tend to a radial arrangement in the more or less rounded or spherical cell4 ,of late years especially1 an immense importance has been attached to these various linear or fibrillar arrangements1 as they occur 7after staining7 in the cell-substance of intestinal epithelium1 of spermatocytes1 of ganglion cells1 and most abundantly and most frequently of all in gland cells4 ,various functions1 which seem somewhat arbitrarily chosen1 have been assigned1 and many hard names given to them2 for these structures now include your mitochondria and your chondriokonts 7both of these being varieties #bhe of chondriosomes71 your ,altmann's granules1 your microsomes1 pseudo-chromosomes1 epidermal fibrils and basal filaments1 your archeoplasm and ergastoplasm1 and probably your idiozomes1 plasmosomes1 and many other histological minutiae#ccc4 ,illustration3 ,fig4 #ig4 ,a1 ,b1 ,chondriosomes in kidney-cells1 prior to and during secretory activity 7after ,barratt72 ,c1 do4 in pancreas of frog 7after ,mathews74 ,the position of these bodies with regard to the other cell-structures is carefully described4 ,sometimes they lie in the neighbourhood of the nucleus itself1 that is to say in proximity to the fluid boundary surface which separates the nucleus from the cytoplasm2 and in this position they often form a somewhat cloudy sphere which constitutes the ,nebenkern4 ,in the majority of cases1 as in the epithelial cells1 they form filamentous structures1 and rows of granules1 whose main direction is parallel to the axis of the cell1 and which may1 in some cases1 and in some forms1 be conspicuous at the one end1 and in some cases at the other end of the cell4 ,but ,i do not find that the histologists attempt to explain1 or to correlate with other phenomena1 the tendency of these bodies to lie parallel with the axis1 and perpendicular to the extremities of the cell2 it is merely noted as a peculiarity1 or a specific character1 of these particular structures4 ,extraordinarily complicated and diverse functions have been ascribed to them4 ,engelmann's 8,fibrillenkonus10 which was almost certainly another aspect of the same phenomenon1 was held by him and by cytologists like ,breda and ,heidenhain1 to be an apparatus connected in some #bhf unexplained way with the mechanism of ciliary movement4 ,meves looked upon the chondriosomes as the actual carriers or transmitters of heredity#ccd4 ,altmann invented a new aphorism1 ,omne granulum e granulo1 as a refinement of ,virchow's omnis cellula e cellula2 and many other histologists1 more or less in accord1 accepted the chondriosomes as important entities1 sui generis1 intermediate in grade between the cell itself and its ultimate molecular components4 ,the extreme cytologists of the ,munich school1 ,popoff1 ,goldschmidt and others1 following ,richard ,hertwig1 declaring these structures to be identical with 8chromidia0 7under which name ,hertwig ranked all extra-nuclear chromatin71 would assign them complex functions in maintaining the balance between nuclear and cytoplasmic material2 and the 8chromidial hypothesis10 as every reader of recent cytological literature knows1 has become a very abstruse and complicated thing#cce4 ,with the help of the 8binuclearity hypothesis0 of ,schaudinn and his school1 it has given us the chromidial net1 the chromidial apparatus1 the trophochromidia1 idiochromidia1 gametochromidia1 the protogonoplasm1 and many other novel and original conceptions4 ,the names are apt to vary somewhat in significance from one writer to another4 ,the outstanding fact1 as it seems to me1 is that physiological science has been heavily burdened in this matter1 with a jargon of names and a thick cloud of hypotheses2 while1 from the physical point of view we are tempted to see but little mystery in the whole phenomenon1 and to ascribe it1 in all probability and in general terms1 to the gathering or 8clumping0 together1 under surface tension1 of various constituents of the heterogeneous cell-content1 and to the drawing out of these little clumps along the axis of the cell towards one or other of its extremities1 in relation to osmotic currents1 as these in turn are set up in direct relation #bhg to the phenomena of surface energy and of adsorption#ccf4 ,and all this implies that the study of these minute structures1 if it teach us nothing else1 at least surely and certainly reveals to us the presence of a definite 8field of force10 and a dynamical polarity within the cell4 ,our next and last illustration of the effects of adsorption1 which we owe to the investigations of ,professor ,macallum1 is of great importance2 for it introduces us to a series of phenomena in regard to which we seem now to stand on firmer ground than in some of the foregoing cases1 though we cannot yet consider that the whole story has been told4 ,in our last chapter we were restricted mainly1 though not entirely1 to a consideration of figures of equilibrium1 such as the sphere1 the cylinder or the unduloid2 and we began at once to find ourselves in difficulties when we were confronted by departures from symmetry1 as for instance in the simple case of the ellipsoidal yeast-cell and the production of its bud4 ,we found the cylindrical cell of ,spirogyra1 with its plane or spherical ends1 a comparatively simple matter to understand2 but when this uniform cylinder puts out a lateral outgrowth1 in the act of conjugation1 we have a new and very different system of forces to explain4 ,the analogy of the soap-bubble1 or of the simple liquid drop1 was apt to lead us to suppose that the surface tension was1 on the whole1 uniform over the surface of our cell2 and that its departures from symmetry of form were therefore likely to be due to variations in external resistance4 ,but if we have been inclined to make such an assumption we must now #bhh reconsider it1 and be prepared to deal with important localised variations in the surface tension of the cell4 ,for1 as a matter of fact1 the simple case of a perfectly symmetrical drop1 with uniform surface1 at which adsorption takes place with similar uniformity1 is probably rare in physics1 and rarer still 7if it exist at all7 in the fluid or fluid-containing system which we call in biology a cell4 ,we have mostly to do with cells whose general heterogeneity of substance leads to qualitative differences of surface1 and hence to varying distributions of surface tension4 ,we must accordingly investigate the case of a cell which displays some definite and regular heterogeneity of its liquid surface1 just as ,amoeba displays a heterogeneity which is complex1 irregular and continually fluctuating in amount and distribution4 ,such heterogeneity as we are speaking of must be essentially chemical1 and the preliminary problem is to devise methods of 8microchemical0 analysis1 which shall reveal localised accumulations of particular substances within the narrow limits of a cell1 in the hope that1 their normal effect on surface tension being ascertained1 we may then correlate with their presence and distribution the actual indications of varying surface tension which the form or movement of the cell displays4 ,in theory the method is all that we could wish1 but in practice we must be content with a very limited application of it2 for the substances which may have such action as we are looking for1 and which are also actual or possible constituents of the cell1 are very numerous1 while the means are very seldom at hand to demonstrate their precise distribution and localisation4 ,but in one or two cases we have such means1 and the most notable is in connection with the element potassium4 ,as ,professor ,macallum has shewn1 this element can be revealed1 in very minute quantities1 by means of a certain salt1 a nitrite of cobalt and sodium#ccg4 ,this salt penetrates readily into the tissues and into the interior of the cell2 it combines with potassium to form a sparingly soluble nitrite of cobalt1 sodium and potassium2 and this1 on subsequent treatment with ammonium sulphide1 is converted into a characteristic black precipitate of cobaltic sulphide#cch4 #bhi ,by this means ,macallum demonstrated some years ago the unexpected presence of accumulations of potassium 7i4e4 of chloride or other salts of potassium7 localised in particular parts of various cells1 both solitary cells and tissue cells2 and he arrived at the conclusion that the localised accumulations in question were simply evidences of concentration of the dissolved potassium salts1 formed and localised in accordance with the ,gibbs-,thomson law4 ,in other words1 these accumulations1 occurring as they actually do in connection with various boundary surfaces1 are evidence1 when they appear irregularly distributed over such a surface1 of inequalities in its surface tension#cci2 and we may safely take it that our potassium salts1 like inorganic substances in general1 tend to raise the surface tension1 and will therefore be found concentrating at a portion of the surface whose tension is weak#cdj4 ,in ,professor ,macallum's figure 7,fig4 #ih1 #a7 of the little green alga ,pleurocarpus1 we see that one side of the cell is beginning to bulge out in a wide convexity4 ,this bulge is1 in the first place1 a sign of weakened surface tension on one side of the cell1 which as a whole had hitherto been a symmetrical cylinder2 in the second place1 we see that the bulging area corresponds to the position of a great concentration of the potassic salt2 while in the third place1 from the physiological point of view1 we call the phenomenon the first stage in the process of conjugation4 ,in ,fig4 #ih1 #b1 of ,mesocarpus 7a close ally of ,spirogyra71 we see the same phenomenon admirably exemplified in a later stage4 ,from the adjacent cells distinct outgrowths are being emitted1 where the surface tension has been weakened3 just as the glass-blower warms and softens a small part of his tube to blow out the softened area into a bubble or diverticulum2 and in our ,mesocarpus cells 7besides a certain amount of potassium rendered visible over the boundary which #bij separates the green protoplasm from the cell-sap71 there is a very large accumulation precisely at the point where the tension of the originally cylindrical cell is weakening to produce the bulge4 ,but in a still later stage1 when the boundary between the two conjugating cells is lost and the cytoplasm of the two cells becomes fused together1 then the signs of potassic concentration quickly disappear1 the salt becoming generally diffused through the now symmetrical and spherical 8zygospore40 ,illustration3 ,fig4 #ih4 ,adsorptive concentration of potassium salts in 7#a7 cell of ,pleurocarpus about to conjugate2 7#b7 conjugating cells of ,mesocarpus2 7#c7 sprouting spores of ,equisetum4 7,after ,macallum47 ,in a spore of ,equisetum 7,fig4 #ih1 #c71 while it is still a single cell1 no localised concentration of potassium is to be discerned2 but as soon as the spore has divided1 by an internal partition1 into two cells1 the potassium salt is found to be concentrated in the smaller one1 and especially towards its outer wall1 which is marked by a pronounced convexity4 ,and as this convexity 7which corresponds to one pole of the now asymmetrical1 or quasi-ellipsoidal spore7 grows out into the root-hair1 the potassium salt accompanies its growth1 and is concentrated under its wall4 ,the concentration is1 #bia accordingly1 a concomitant of the diminished surface tension which is manifested in the altered configuration of the system4 ,in the case of ciliate or flagellate cells1 there is to be found a characteristic accumulation of potassium at and near the base of the cilia4 ,the relation of ciliary movement to surface tension lies beyond our range1 but the fact which we have just mentioned throws light upon the frequent or general presence of a little protuberance of the cell-surface just where a flagellum is given off 7cf4 p4 #bdg71 and of a little projecting ridge or fillet at the base of an isolated row of cilia1 such as we find in ,vorticella4 ,yet another of ,professor ,macallum's demonstrations1 though its interest is mainly physiological1 will help us somewhat further to comprehend what is implied in our phenomenon4 ,in a normal cell of ,spirogyra1 a concentration of potassium is revealed along the whole surface of the spiral coil of chlorophyll-bearing1 or 8chromatophoral10 protoplasm1 the rest of the cell being wholly destitute of the former substance3 the indication being that1 at this particular boundary1 between chromatophore and cell-sap1 the surface tension is small in comparison with any other interfacial surface within the system4 ,now as ,macallum points out1 the presence of potassium is known to be a factor1 in connection with the chlorophyll-bearing protoplasm1 in the synthetic production of starch from ,,co#b under the influence of sunlight4 ,but we are left in some doubt as to the consecutive order of the phenomena4 ,for the lowered surface tension1 indicated by the presence of the potassium1 may be itself a cause of the carbohydrate synthesis2 while on the other hand1 this synthesis may be attended by the production of substances 7e4g4 formaldehyde7 which lower the surface tension1 and so conduce to the concentration of potassium4 ,all we know for certain is that the several phenomena are associated with one another1 as apparently inseparable parts or inevitable concomitants of a certain complex action4 ,and now to return1 for a moment1 to the question of cell-form4 ,when we assert that the form of a cell 7in the absence of mechanical pressure7 is essentially dependent on surface tension1 and even when we make the preliminary assumption that protoplasm is essentially #bib a fluid1 we are resting our belief on a general consensus of evidence1 rather than on compliance with any one crucial definition4 ,the simple fact is that the agreement of cell-forms with the forms which physical experiment and mathematical theory assign to liquids under the influence of surface tension1 is so frequently and often so typically manifested1 that we are led1 or driven1 to accept the surface tension hypothesis as generally applicable and as equivalent to a universal law4 ,the occasional difficulties or apparent exceptions are such as call for further enquiry1 but fall short of throwing doubt upon our hypothesis4 ,macallum's researches introduce a new element of certainty1 a 8nail in a sure place10 when they demonstrate that1 in certain movements or changes of form which we should naturally attribute to weakened surface tension1 a chemical concentration which would naturally accompany such weakening actually takes place4 ,they further teach us that in the cell a chemical heterogeneity may exist of a very marked kind1 certain substances being accumulated here and absent there1 within the narrow bounds of the system4 ,such localised accumulations can as yet only be demonstrated in the case of a very few substances1 and of a single one in particular2 and these are substances whose presence does not produce1 but whose concentration tends to follow1 a weakening of surface tension4 ,the physical cause of the localised inequalities of surface tension remains unknown4 ,we may assume1 if we please1 that it is due to the prior accumulation1 or local production1 of chemical bodies which would have this direct effect2 though we are by no means limited to this hypothesis4 ,but in spite of some remaining difficulties and uncertainties1 we have arrived at the conclusion1 as regards unicellular organisms1 that not only their general configuration but also their departures from symmetry may be correlated with the molecular forces manifested in their fluid or semi-fluid surfaces4 #bic ,,chapter ,,vii ,,the ,,forms ,,of ,,tissues ,,or ,,cell-,,aggregates ,we now pass from the consideration of the solitary cell to that of cells in contact with one another1--to what we may call in the first instance 8cell-aggregates10--through which we shall be led ultimately to the study of complex tissues4 ,in this part of our subject1 as in the preceding chapters1 we shall have to give some consideration to the effects of various forces2 but1 as in the case of the conformation of the solitary cell1 we shall probably find1 and we may at least begin by assuming1 that the agency of surface tension is especially manifest and important4 ,the effect of this surface tension will chiefly manifest itself in the production of surfaces minimae areae3 where1 as ,plateau was always careful to point out1 we must understand by this expression not an absolute1 but a relative minimum1 an area1 that is to say1 which approximates to an absolute minimum as nearly as circumstances and the conditions of the case permit4 ,there are certain fundamental principles1 or fundamental equations1 besides those which we have already considered1 which we shall need in our enquiry4 ,for instance the case which we briefly touched upon 7on p4 #bfe7 of the angle of contact between the protoplasm and the axial filament in a ,heliozoan we shall now find to be but a particular case of a general and elementary theorem4 ,let us re-state as follows1 in terms of ,energy1 the general principle which underlies the theory of surface tension or capillarity4 ,when a fluid is in contact with another fluid1 or with a solid or a gas1 a portion of the energy of the system 7that1 namely1 which we call surface energy71 is proportional to the area of the surface of contact3 it is also proportional to a coefficient which is specific for each particular pair of substances1 and which is constant for these1 save only in so far as it may be modified by #bid changes of temperature or of electric charge4 ,the condition of minimum potential energy in the system1 which is the condition of equilibrium1 will accordingly be obtained by the utmost possible diminution in the area of the surfaces in contact4 ,when we have three bodies in contact1 the case becomes a little more complex4 ,suppose for instance we have a drop of some fluid1 ,a1 floating on another fluid1 ,b1 and exposed to air1 ,c4 ,the whole surface energy of the system may now be considered as divided into two parts1 one at the surface of the drop1 and the other outside of the same2 the latter portion is inherent in the surface ,,bc1 between the mass of fluid ,b and the superincumbent air1 ,c2 but the former portion consists of two parts1 for it is divided between the two surfaces ,,ab and ,,ac1 that namely which separates the drop from the surrounding fluid and that which separates it from the atmosphere4 ,so far as ,illustration3 ,fig4 #ii4 the drop is concerned1 then1 equilibrium depends on a proper balance between the energy1 per unit area1 which is resident in its own two surfaces1 and that which is external thereto3 that is to say1 if we call ,ebc the energy at the surface between the two fluids1 and so on with the other two pairs of surface energies1 the condition of equilibrium1 or of maintenance of the drop1 is that ,ebc ,eab ,eac4 ,if1 on the other hand1 the fluid ,a happens to be oil and the fluid ,b1 water1 then the energy per unit area of the water-air surface is greater than that of the oil-air surface and that of the oil-water surface together2 i4e4 ,ewa ,eoa ,eow4 ,here there is no equilibrium1 and in order to obtain it the water-air surface must always tend to decrease and the other two interfacial surfaces to increase2 which is as much as to say that the water tends to become covered by a spreading film of oil1 and the water-air surface to be abolished4 #bie ,the surface energy of which we have here spoken is manifested in that contractile force1 or 8tension10 of which we have already had so much to say#cda4 ,in any part of the free water surface1 for instance1 one surface particle attracts another surface particle1 and the resultant of these multitudinous attractions is an equilibrium of tension throughout this particular surface4 ,in the case of our three bodies in contact with one another1 and within a small area very near to the point of contact1 a water particle 7for instance7 will be pulled outwards by another water particle2 but on the opposite side1 so to speak1 there will be no water surface1 and no water particle1 to furnish the counterbalancing pull2 this counterpull1 ,illustration3 ,fig4 #ajj4 ,illustration3 ,fig4 #aja4 which is necessary for equilibrium1 must therefore be provided by the tensions existing in the other two surfaces of contact4 ,in short1 if we could imagine a single particle placed at the very point of contact1 it would be drawn upon by three different forces1 whose directions would lie in the three surface planes1 and whose magnitude would be proportional to the specific tensions characteristic of the two bodies which in each case combine to form the 8interfacial0 surface4 ,now for three forces acting at a point to be in equilibrium1 they must be capable of representation1 in magnitude and direction1 by the three sides of a triangle1 taken in order1 in accordance with the elementary theorem of the ,triangle of ,forces4 ,so1 if we know the form of our floating drop 7,fig4 #ajj71 then by drawing tangents from ,o 7the point of mutual contact71 #bif we determine the three angles of our triangle 7,fig4 #aja71 and we therefore know the relative magnitudes of the three surface tensions1 which magnitudes are proportional to its sides2 and conversely1 if we know the magnitudes1 or relative magnitudes1 of the three sides of the triangle1 we also know its angles1 and these determine the form of the section of the drop4 ,it is scarcely necessary to mention that1 since all points on the edge of the drop are under similar conditions1 one with another1 the form of the drop1 as we look down upon it from above1 must be circular1 and the whole drop must be a solid of revolution4 ,the principle of the ,triangle of ,forces is expanded1 as follows1 by an old seventeenth-century theorem1 called ,lami's ,theorem3 8,if three forces acting at a point be in equilibrium1 each force is proportional to the sine of the angle contained between the directions of the other two40 ,that is to say ,p 3 ,q 3 ,r 3 sin ,,qor 3 sin ,,por 3 sin ,,poq4 or ,p_/sin ,,qor ,q_/sin ,,rop ,r_/sin ,,poq4 ,and from this1 in turn1 we derive the equivalent formulae1 by which each force is expressed in terms of the other two1 and of the angle between them3 ,p#b ,q#b ,r#b #b,q ,r cos7,,qor71 etc4 ,from this and the foregoing1 we learn the following important and useful deductions3 7#a7 ,the three forces can only be in equilibrium when any one of them is less than the sum of the other two3 for otherwise1 the triangle is impossible4 ,now in the case of a drop of olive-oil upon a clean water surface1 the relative magnitudes of the three tensions 7at #ae ,c47 have been determined as follows3 ,water-air surface #ge ,oil-air surface #cb ,oil-water surface #ba ,no triangle having sides of these relative magnitudes is possible2 and no such drop therefore can remain in equilibrium4 #big 7#b7 ,the three surfaces may be all alike3 as when a soap-bubble floats upon soapy water1 or when two soap-bubbles are joined together1 on either side of a partition-film4 ,in this case1 the three tensions are all equal1 and therefore the three angles are all equal2 that is to say1 when three similar liquid surfaces meet together1 they always do so at an angle of #abj4 ,whether our two conjoined soap-bubbles be equal or unequal1 this is still the invariable rule2 because the specific tension of a particular surface is unaffected by any changes of magnitude or form4 7#c7 ,if two only of the surfaces be alike1 then two of the angles will be alike1 and the other will be unlike2 and this last will be the difference between #cfj and the sum of the other two4 ,a particular case is when a film is stretched between solid and parallel walls1 like a soap-film within a cylindrical tube4 ,here1 so long as there is no external pressure applied to either side1 so long as both ends of the tube are open or closed1 the angles on either side of the film will be equal1 that is to say the film will set itself at right angles to the sides4 ,many years ago ,sachs laid it down as a principle1 which has become celebrated in botany under the name of ,sachs's ,rule1 that one cell-wall always tends to set itself at right angles to another cell-wall4 ,this rule applies to the case which we have just illustrated2 and such validity as the rule possesses is due to the fact that among plant-tissues it very frequently happens that one cell-wall has become solid and rigid before another and later partition-wall is developed in connection with it4 7#d7 ,there is another important principle which arises not out of our equations but out of the general considerations by which we were led to them4 ,we have seen that1 at and near the point of contact between our several surfaces1 there is a continued balance of forces1 carried1 so to speak1 across the interval2 in other words1 there is physical continuity between one surface and another4 ,it follows necessarily from this that the surfaces merge one into another by a continuous curve4 ,whatever be the form of our surfaces and whatever the angle between them1 this small intervening surface1 approximately spherical1 is always there to bridge over the line of contact#cdb2 and this little fillet1 or 8bourrelet10 #bih as ,plateau called it1 is large enough to be a common and conspicuous feature in the microscopy of tissues 7,fig4 #ajb74 ,for instance1 the so-called 8splitting0 of the cell-wall1 which is conspicuous at the angles of the large 8parenchymatous0 cells in the succulent tissues of all higher plants 7,fig4 #ajc71 is nothing more than a manifestation of ,plateau's 8bourrelet10 or surface of continuity#cdc4 ,we may now illustrate some of the foregoing principles1 before we proceed to the more complex cases in which more bodies than three are in mutual contact4 ,but in doing so1 we must constantly bear in mind the principles set forth in our chapter on the forms of cells1 and especially those relating to the pressure exercised by a curved film4 ,illustration3 ,fig4 #ajb4 7,after ,berthold47 ,illustration3 ,fig4 #ajc4 ,parenchyma of ,maize4 ,let us look for a moment at the case presented by the partition-wall in a double soap-bubble4 ,as we have just seen1 the three films in contact 7viz4 the outer walls of the two bubbles and the partition-wall between7 being all composed of the same substance #bii and all alike in contact with air1 the three surface tensions must be equal2 and the three films must therefore1 in all cases1 meet at an angle of #abj4 ,but1 unless the two bubbles be of precisely equal size 7and therefore of equal curvature7 it is obvious that the tangents to the spheres will not meet the plane of their circle of contact at equal angles1 and therefore that the partition-wall must be a curved surface3 it is only plane when it divides two equal and symmetrical cells4 ,it is also obvious1 from the symmetry of the figure1 that the centres of the spheres1 the centre of the partition1 and the centres of the two spherical surfaces are all on one and the same straight line4 ,illustration3 ,fig4 #ajd4 ,now the surfaces of the two bubbles exert a pressure inwards which is inversely proportional to their radii3 that is to say p 3 p 33 #a_/r 3 #a_/r2 and the partition wall must1 for equilibrium1 exert a pressure 7,p7 which is equal to the difference between these two pressures1 that is to say1 ,p #a_/,r #a_/r #a_/r 7r r7_/r r4 ,it follows that the curvature of the partition wall must be just such a curvature as is capable of exerting this pressure1 that is to say1 ,r r r_/7r r74 ,the partition wall1 then1 is always a portion of a spherical surface1 whose radius is equal to the product1 divided by the difference1 of the radii of the two vesicles4 ,it follows at once from this that if the two bubbles be equal1 the radius of curvature of the partition is infinitely great1 that is to say the partition is 7as we have already seen7 a plane surface4 ,the geometrical construction by which we obtain the position of the centres of the two spheres and also of the partition surface is very simple1 always provided that the surface tensions are uniform throughout the system4 ,if p be a point of contact between the two spheres1 and cp be a radius of one of them1 then make the angle cpm #fj1 and mark off on pm1 pc equal to the #cjj radius of the other sphere2 in like manner1 make the angle cpn #fj1 cutting the line cc in c2 then c will be the centre of the second sphere1 and c that of the spherical partition4 ,illustration3 ,fig4 #aje4 ,illustration3 ,fig4 #ajf4 ,whether the partition be or be not a plane surface1 it is obvious that its line of junction with the rest of the system lies in a plane1 and is at right angles to the axis of symmetry4 ,the actual curvature of the partition-wall is easily seen in optical section2 but in surface view1 the line of junction is projected as a plane 7,fig4 #ajf71 perpendicular to the axis1 and this appearance has also helped to lend support and authority to 8,sachs's ,rule40 ,illustration3 ,fig4 #ajg4 ,filaments1 or chains of cells1 in various lower ,algae4 7,a7 ,nostoc2 7,b7 ,anabaena2 7,c7 ,rivularia2 7,d7 ,oscillatoria4 ,many spherical cells1 such as ,protococcus1 divide into two equal halves1 which are therefore separated by a plane partition4 ,among the other lower ,algae1 akin to ,protococcus1 such as the ,nostocs and ,oscillatoriae1 in which the cells are imbedded in a gelatinous matrix1 we find a series of forms such as are represented in ,fig4 #ajg4 ,sometimes the cells are solitary or disunited2 sometimes they run in pairs or in rows1 separated one from another by flat partitions2 and sometimes the conjoined cells are approximately hemispherical1 but at other times each half is more than a hemisphere4 ,these various conditions depend1 #cja according to what we have already learned1 upon the relative magnitudes of the tensions at the surface of the cells and at the boundary between them#cdd4 ,in the typical case of an equally divided cell1 such as a double and co-equal soap-bubble1 where the partition-wall and the outer walls are similar to one another and in contact with similar substances1 we can easily determine the form of the system4 ,for1 at any point of the boundary of the partition-wall1 ,o1 the tensions being equal1 the angles ,,qop1 ,,rop1 ,,qor are all equal1 and each is1 therefore1 an angle of #abj4 ,but ,,oq1 ,,or being tangents1 the centres of the two spheres 7or circular arcs in the figure7 lie on perpendiculars to them2 therefore the radii ,,co1 ,c,o meet at an ,illustration3 ,fig4 #ajh4 angle of #fj1 and ,,coc is an equilateral triangle4 ,that is to say1 the centre of each circle lies on the circumference of the other2 the partition lies midway between the two centres2 and the length 7i4e4 the diameter7 of the partition-wall1 ,,po1 is #b sin #fj #agcb times the radius1 or #hff times the diameter1 of each of the cells4 ,this gives us1 then1 the form of an aggregate of two equal cells under uniform conditions4 ,as soon as the tensions become unequal1 whether from changes in their own substance or from differences in the substances with which they are in contact1 then the form alters4 ,if the tension #cjb along the partition1 ,p1 diminishes1 the partition itself enlarges1 and the angle ,,qor increases3 until1 when the tension ,p is very small compared to ,q or ,r1 the whole figure becomes a circle1 and the partition-wall1 dividing it into two hemispheres1 stands at right angles to the outer wall4 ,this is the case when the outer wall of the cell is practically solid4 ,on the other hand1 if ,p begins to increase relatively to ,q and ,r1 then the partition-wall contracts1 and the two adjacent cells become larger and larger segments of a sphere1 until at length the system becomes divided into two separate cells4 ,illustration3 ,fig4 #aji4 ,spore of ,pellia4 7,after ,campbell47 ,in the spores of ,liverworts 7such as ,pellia71 the first partition-wall 7the equatorial partition in ,fig4 #aji1 a7 divides the spore into two equal halves1 and is therefore a plane surface1 normal to the surface of the cell2 but the next partitions arise near to either end of the original spherical or elliptical cell4 ,each of these latter partitions will 7like the first7 tend to set itself normally to the cell-wall2 at least the angles on either side of the partition will be identical1 and their magnitude will depend upon the tension existing between the cell-wall and the surrounding medium4 ,they will only be right angles if the cell-wall is already practically solid1 and in all probability 7rigidity of the cell-wall not being quite attained7 they will be somewhat greater4 ,in either case the partition itself will be a portion of a sphere1 whose curvature will now denote a difference of pressures in the two chambers or cells1 which it serves to separate4 7,the later stages of cell-division1 represented in the figures b and c1 we are not yet in a position to deal with47 ,we have innumerable cases1 near the tip of a growing filament1 where in like manner the partition-wall which cuts off the terminal #cjc cell constitutes a spherical lens-shaped surface1 set normally to the adjacent walls4 ,at the tips of the branches of many ,florideae1 for instance1 we find such a lenticular partition4 ,in ,dictyota dichotoma1 as figured by ,reinke1 we have a succession of such partitions2 and1 by the way1 in such cases as these1 where the tissues are very transparent1 we have often in optical section a puzzling confusion of lines2 one being the optical section of the curved partition-wall1 the other being the straight linear projection of its outer edge to which we have already referred4 ,in the conical terminal cell of ,chara1 we have the same lens-shaped curve1 but a little lower down1 where the sides of the shoot are approximately parallel1 we have flat transverse partitions1 at the edges of which1 however1 we recognise a convexity of the outer cell-wall and a definite angle of contact1 equal on the two sides of the partition4 ,illustration3 ,fig4 #aaj4 ,cells of ,dictyota4 7,after ,reinke47 ,illustration3 ,fig4 #aaa4 ,terminal and other cells of ,chara4 ,illustration3 ,fig4 #aab4 ,young antheridium of ,chara4 ,in the young antheridia of ,chara 7,fig4 #aab71 and in the not dissimilar case of the sporangium 7or conidiophore7 of ,mucor1 we easily recognise the hemispherical form of the septum which shuts off the large spherical cell from the cylindrical filament4 ,here1 in the first phase of development1 we should have to take into consideration the different pressures exerted by the single curvature of the cylinder and the double curvature of its spherical cap 7p4 #bba72 and we should find that the partition would have a somewhat low curvature1 with a radius less than the diameter of the cylinder2 which it would have exactly equalled but for the additional pressure inwards which it receives #cjd from the curvature of the large surrounding sphere4 ,but as the latter continues to grow1 its curvature decreases1 and so likewise does the inward pressure of its surface2 and accordingly the little convex partition bulges out more and more4 ,in order to epitomise the foregoing facts let the annexed diagrams 7,fig4 #aac7 represent a system of three films1 of which one is a partition-wall between the other two2 and let the tensions at the three surfaces1 or the tractions exercised upon a point at their meeting-place1 be proportional to ,t1 ,t and t4 ,let 1 1 be1 as in the figure1 the opposite angles4 ,then3 7#a7 ,if ,t be equal to ,t1 and t be relatively insignificant1 the angles 1 will be of #ij4 ,illustration3 ,fig4 #aac4 7#b7 ,if ,t ,t1 but be a little greater than t1 then t will exert an appreciable traction1 and 1 will be more than #ij1 say1 for instance1 #ajj4 7#c7 ,if ,t ,t t1 then 1 1 will all equal #abj4 ,the more complicated cases1 when t1 ,t and ,t are all unequal1 are already sufficiently explained4 ,the biological facts which the foregoing considerations go a long way to explain and account for have been the subject of much argument and discussion1 especially on the part of the botanists4 ,let me recapitulate1 in a very few words1 the history of this long discussion4 ,some fifty years ago1 ,hofmeister laid it down as a general law that 8,the partition-wall stands always perpendicular to what was previously the principal direction of growth in the cell10--or1 in other words1 perpendicular to the long axis of the cell#cde4 ,ten #cje years later1 ,sachs formulated his rule1 or principle1 of 8rectangular section10 declaring that in all tissues1 however complex1 the cell-walls cut one another 7at the time of their formation7 at right angles#cdf4 ,years before1 ,schwendener had found1 in the final results of cell-division1 a universal system of 8orthogonal trajectories#cdg02 and this idea ,sachs further developed1 introducing complicated systems of confocal ellipses and hyperbol1 and distinguishing between periclinal walls1 whose curves approximate to the peripheral contours1 radial partitions1 which cut these at an angle of #ij1 and finally anticlines1 which stand at right angles to the other two4 ,reinke1 in #ahhj1 was the first to throw some doubt upon this explanation4 ,he pointed out various cases where the angle was not a right angle1 but was very definitely an acute one2 and he saw1 apparently1 in the more common rectangular symmetry merely what he calls a necessary1 but secondary1 result of growth#cdh4 ,within the next few years1 a number of botanical writers were content to point out further exceptions to ,sachs's ,rule#cdi2 and in some cases to show that the curvatures of the partition-walls1 especially such cases of lenticular curvature as we have described1 were by no means accounted for by either ,hofmeister or ,sachs2 while within the same period1 ,sachs himself1 and also ,rauber1 attempted to extend the main generalisation to animal tissues#cej4 ,while these writers regarded the form and arrangement of the cell-walls as a biological phenomenon1 with little if any direct relation to ordinary physical laws1 or with but a vague reference to 8mechanical conditions10 the physical side of the case was soon urged by others1 with more or less force and cogency4 ,indeed the general resemblance between a cellular tissue and a 8froth0 #cjf had been pointed out long before1 by ,melsens1 who had made an 8artificial tissue0 by blowing into a solution of white of egg#cea4 ,in #ahhf1 ,berthold published his ,protoplasmamechanik1 in which he definitely adopted the principle of 8minimal areas10 and1 following on the lines of ,plateau1 compared the forms of many cell-surfaces and the arrangement of their partitions with those assumed under surface tension by a system of 8weightless films40 ,but1 as ,klebs#ceb points out in reviewing ,berthold's book1 ,berthold was careful to stop short of attributing the biological phenomena to a definite mechanical cause4 ,they remained for him1 as they had done for ,sachs1 so many 8phenomena of growth10 or 8properties of protoplasm40 ,in the same year1 but while still apparently unacquainted with ,berthold's work1 ,errera#cec published a short but very lucid article1 in which he definitely ascribed to the cell-wall 7as ,hofmeister had already done7 the properties of a semi-liquid film and drew from this as a logical consequence the deduction that it must assume the various configurations which the law of minimal areas imposes on the soap-bubble4 ,so what we may call ,errera's ,law is formulated as follows3 ,a cellular membrane1 at the moment of its formation1 tends to assume the form which would be assumed1 under the same conditions1 by a liquid film destitute of weight4 ,soon afterwards ,chabry1 in discussing the embryology of the ,ascidians1 indicated many of the points in which the contacts between cells repeat the surface-tension phenomena of the soap-bubble1 and came to the conclusion that part1 at least1 of the embryological phenomena were purely physical#ced2 and the same line of investigation and thought were pursued and developed by ,robert1 in connection with the embryology of the ,mollusca#cee4 ,driesch again1 in a series of papers1 continued to draw attention to the presence of capillary phenomena in the segmenting cells #cjg of various embryos1 and came to the conclusion that the mode of segmentation was of little importance as regards the final result#cef4 ,lastly de ,wildeman#ceg1 in a somewhat wider1 but also vaguer generalisation than ,errera's1 declared that 8,the form of the cellular framework of vegetables1 and also of animals1 in its essential features1 depends upon the forces of molecular physics40 ,illustration3 ,fig4 #aad4 ,let us return to our problem of the arrangement of partition films4 ,when we have three bubbles in contact1 instead of two as in the case already considered1 the phenomenon is strictly analogous to our former case4 ,the three bubbles will be separated by three partition surfaces1 whose curvature will depend upon the relative size of the spheres1 and which will be plane if the latter are all of the same dimensions2 but whether plane or curved1 the three partitions will meet one another at an angle of #abj1 in an axial line4 ,various pretty geometrical corollaries accompany this arrangement4 ,for instance1 if ,fig4 #aad represent the three associated bubbles in a plane drawn through their centres1 c1 c1 c 7or what is the same thing1 if it represent the base of three bubbles resting on a plane71 then the lines uc1 uc1 or sc1 sc1 etc41 drawn to the #cjh centres from the points of intersection of the circular arcs1 will always enclose an angle of #fj4 ,again 7,fig4 #aae71 if we make the angle cuf equal to #fj1 and produce uf to meet cc in f1 f will be the centre of the circular arc which constitutes the partition ,ou2 and further1 the three points f1 g1 h1 successively determined in this ,illustration3 ,fig4 #aae4 manner1 will lie on one and the same straight line4 ,in the case of coequal bubbles or cells 7as in ,fig4 #aad1 ,b71 it is obvious that the lines joining their centres form an equilateral triangle2 and consequently1 that the centre of each circle 7or sphere7 lies on the circumference of the other two2 it is also obvious that uf is now #cji parallel to cc1 and accordingly that the centre of curvature of the partition is now infinitely distant1 or 7as we have already said71 that the partition itself is plane4 ,illustration3 ,fig4 #aaf4 ,when we have four bubbles in conjunction1 they would seem to be capable of arrangement in two symmetrical ways3 either1 as in ,fig4 #aaf 7,a71 with the four partition-walls meeting at right angles1 or1 as in 7,b71 with five partitions meeting1 three and three1 at angles of #abj4 ,this latter arrangement is strictly analogous to the arrangement of three bubbles in ,fig4 #aad4 ,now1 though both of these figures1 from their symmetry1 are apparently figures of equilibrium1 yet1 physically1 the former turns out to be of unstable and the latter of stable equilibrium4 ,if we try to bring our four bubbles into the form of ,fig4 #aaf1 ,a1 such an arrangement endures only for an instant2 the partitions glide upon each other1 a median wall springs into existence1 and the system at once assumes the form of our second figure 7,b74 ,this is a direct consequence of the law of minimal areas3 for it can be shewn1 by somewhat difficult mathematics 7as was first done by ,lamarle71 that1 in dividing a closed space into a given number of chambers by means of partition-walls1 the least possible area of these partition-walls1 taken together1 can only be attained when they meet together in groups of three1 at equal angles1 that is to say at angles of #abj4 #caj ,wherever we have a true cellular complex1 an arrangement of cells in actual physical contact by means of a boundary film1 we find this general principle in force2 we must only bear in mind that1 for its perfect recognition1 we must be able to view the object in a plane at right angles to the boundary walls4 ,for instance1 in any ordinary section of a vegetable parenchyma1 we recognise the appearance of a 8froth10 precisely resembling that which we can construct by imprisoning a mass of soap-bubbles in a narrow vessel with flat sides of glass2 in both cases we see the cell-walls everywhere meeting1 by threes1 at angles of #abj1 irrespective of the size of the individual cells3 whose relative size1 on the other hand1 determines the curvature of the partition-walls4 ,on the surface of a honey-comb we have precisely the same conjunction1 between cell and cell1 of three boundary walls1 meeting at #abj4 ,in embryology1 when we examine a segmenting egg1 of four 7or more7 segments1 we find in like manner1 in the great majority of cases1 if not in all1 that the same principle is still exemplified2 the four segments do not meet in a common centre1 but each cell is in contact with two others1 and the three1 and only three1 common boundary walls meet at the normal angle of #abj4 ,a so-called polar furrow#ceh1 the visible edge of a vertical partition-wall1 joins 7or separates7 the two triple contacts1 precisely as in ,fig4 #aaf1 ,b4 ,in the four-celled stage of the frog's egg1 ,rauber 7an exceptionally careful observer7 shews us three alternative modes in which the four cells may be found to be conjoined 7,fig4 #aag74 ,in 7,a7 we have the commonest arrangement1 which is that which we have just studied and found to be the simplest theoretical one2 that namely where a straight 8polar furrow0 intervenes1 and where1 at its extremities1 the partition-walls are conjoined three by three4 ,in 7,b71 we have again a polar furrow1 which is now seen to be a portion of the first 8segmentation-furrow0 7cf4 ,fig4 #aee etc47 by which the egg was originally divided into two2 the four-celled stage being reached by the appearance of the transverse furrows #caa and their corresponding partitions4 ,in this case1 the polar furrow is seen to be sinuously curved1 and ,rauber tells us that its curvature gradually alters3 as a matter of fact1 it 7or rather the partition-wall corresponding to it7 is gradually setting itself into a position of equilibrium1 that is to say of equiangular contact with its neighbours1 which position of equilibrium is already attained or nearly so in ,fig4 #aag1 ,a4 ,in ,fig4 #aag1 ,c1 we have a very different condition1 with which we shall deal in a moment4 ,illustration3 ,fig4 #aag4 ,various ways in which the four cells are co-arranged in the four-celled stage of the frog's egg4 7,after ,rauber47 ,according to the relative magnitude of the bodies in contact1 this 8polar furrow0 may be longer or shorter1 and it may be so minute as to be not easily discernible2 but it is quite certain that no simple and homogeneous system of fluid films such as we are dealing with is in equilibrium without its presence4 ,in the accounts given1 however1 by embryologists of the segmentation of the egg1 while the polar furrow is depicted in the great majority of cases1 there are others in which it has not been seen and some in which its absence is definitely asserted#cei4 ,the cases where four cells1 lying in one plane1 meet in a point1 such as were frequently figured by the older embryologists1 are very difficult to verify1 and ,i have not come across a single clear case in recent literature4 ,considering the physical stability of the other arrangement1 the great preponderance of cases in which it is known to occur1 the difficulty of recognising the polar furrow in cases where it is very small and unless it be specially looked for1 and the natural tendency of the draughtsman to make an all but symmetrical structure appear wholly so1 ,i am much inclined to attribute to #cab error or imperfect observation all those cases where the junction-lines of four cells are represented 7after the manner of ,fig4 #aaf1 ,a7 as a simple cross#cfj4 ,but while a true four-rayed intersection1 or simple cross1 is theoretically impossible 7save as a transitory and highly unstable condition71 there is another condition which may closely simulate it1 and which is common enough4 ,there are plenty of representations of segmenting eggs1 in which1 instead of the triple junction and polar furrow1 the four cells 7and in like manner their more numerous successors7 are represented as rounded off1 and separated from one another by an empty space1 or by a little drop of an extraneous fluid1 evidently not directly miscible with the fluid surfaces of the cells4 ,such is the case in the obviously accurate figure which ,rauber gives 7,fig4 #aag1 ,c7 of the third mode of conjunction in the four-celled stage of the frog's egg4 ,here ,rauber is most careful to point out that the furrows do not simply 8cross10 or meet in a point1 but are separated by a little space1 which he calls the ,polgrbchen1 and asserts to be constantly present whensoever the polar furrow1 or ,brechungslinie1 is not to be discerned4 ,this little interposed space1 with its contained drop of fluid1 materially alters the case1 and implies a new condition of theoretical and actual equilibrium4 ,for1 on the one hand1 we see that now the four intercellular partitions do not meet one another at all2 but really impinge upon four new and separate partitions1 which constitute interfacial contacts1 not between cell and cell1 but between the respective cells and the intercalated drop4 ,and secondly1 the angles at which these four little surfaces will meet the four cell-partitions1 will be determined1 in the usual way1 by the balance between the respective tensions of these several surfaces4 ,in an extreme case 7as in some pollen-grains7 it may be found that the cells under the observed circumstances are not truly in surface contact3 that they are so many drops which touch but do not 8wet0 one another1 and which are merely held together by the pressure of the surrounding envelope4 ,but even supposing1 #cac as is in all probability the actual case1 that they are in actual fluid contact1 the case from the point of view of surface tension presents no difficulty4 ,in the case of the conjoined soap-bubbles1 we were dealing with similar contacts and with equal surface tensions throughout the system2 but in the system of protoplasmic cells which constitute the segmenting egg we must make allowance for an inequality of tensions1 between the surfaces where cell meets cell1 and where on the other hand cell-surface is in contact with the surrounding medium1--in this case generally water or one of the fluids of the body4 ,remember that our general condition is that1 in our ,illustration3 ,fig4 #aah4 entire system1 the sum of the surface energies is a minimum2 and1 while this is attained by the sum of the surfaces being a minimum in the case where the energy is uniformly distributed1 it is not necessarily so under non-uniform conditions4 ,in the diagram 7,fig4 #aah7 if the energy per unit area be greater along the contact surface cc1 where cell meets cell1 than along ca or cb1 where cell-surface is in contact with the surrounding medium1 these latter surfaces will tend to increase and the surface of cell-contact to diminish4 ,in short there will be the usual balance of forces between the tension along the surface cc1 and the two opposing tensions along ca and cb4 ,if the former be greater than either of the other two1 the outside angle will be less than #abj2 and if the tension along the surface cc be as much or more than the sum of the other two1 then the drops will stand in contact only1 save for the possible effect of external pressure1 at a point4 ,this is the explanation1 in general terms1 of the peculiar conditions obtaining in ,nostoc and its allies 7p4 #cjj71 and it also leads us to a consideration of the general properties and characters of an 8epidermal0 layer4 ,while the inner cells of the honey-comb are symmetrically situated1 sharing with their neighbours in equally distributed pressures or tensions1 and therefore all tending with great accuracy #cad to identity of form1 the case is obviously different with the cells at the borders of the system4 ,so it is1 in like manner1 with our froth of soap-bubbles4 ,the bubbles1 or cells1 in the interior of the mass are all alike in general character1 and if they be equal in size are alike in every respect3 their sides are uniformly flattened#cfa1 and tend to meet at equal angles of #abj4 ,but the bubbles which constitute the outer layer retain their spherical surfaces1 which however still tend to meet the partition-walls connected with them at constant angles of #abj4 ,this outer layer of bubbles1 which forms the surface of our froth1 constitutes after a fashion what we should call in botany an 8epidermal0 layer4 ,but in our froth of soap-bubbles we have1 as a rule1 the same kind of contact 7that is to say1 contact with air7 both within and without the bubbles2 while in our living cell1 the outer wall of the epidermal cell is exposed to air on the one side1 but is in contact with the ,illustration3 ,fig4 #aai4 protoplasm of the cell on the other3 and this involves a difference of tensions1 so that the outer walls and their adjacent partitions are no longer likely to meet at equal angles of #abj4 ,moreover1 a chemical change1 due for instance to oxidation or possibly also to adsorption1 is very likely to affect the external wall1 and may tend to its consolidation2 and this process1 as we have seen1 is tantamount to a large increase1 and at the same time an equalisation1 of tension in that outer wall1 and will lead the adjacent partitions to impinge upon it at angles more and more nearly approximating to #ij3 the bubble-like1 or spherical1 surfaces of the individual cells being more and more flattened in consequence4 ,lastly1 the chemical changes which affect the outer walls of the superficial cells may extend1 in greater or less degree1 to their inner walls also3 with the result that these #cae cells will tend to become more or less rectangular throughout1 and will cease to dovetail into the interstices of the next subjacent layer4 ,these then are the general characters which we recognise in an epidermis2 and we perceive that the fundamental character of an epidermis simply is that it lies on the outside1 and that its main physical characteristics follow1 as a matter of course1 from the position which it occupies and from the various consequences which that situation entails4 ,we have however by no means exhausted the subject in this short account2 for the botanist is accustomed to draw a sharp distinction between a true epidermis and what is called epidermal tissue4 ,the latter1 which is found in such a sea-weed as ,laminaria and in very many other cryptogamic plants1 consists1 as in the hypothetical case we have described1 of a more or less simple and direct modification of the general or fundamental tissue4 ,but a 8true epidermis10 such as we have it in the higher plants1 is something with a long morphological history1 something which has been laid down or differentiated in an early stage of the plant's growth1 and which afterwards retains its separate and independent character4 ,we shall see presently that a physical reason is again at hand to account1 under certain circumstances1 for the early partitioning off1 from a mass of embryonic tissue1 of an outer layer of cells which from their first appearance are marked off from the rest by their rectangular and flattened form4 ,we have hitherto considered our cells1 or bubbles1 as lying in a plane of symmetry1 and further1 we have only considered the appearance which they present as projected on that plane3 in simpler words1 we have been considering their appearance in surface or in sectional view4 ,but we have further to consider them as solids1 whether they be still grouped in relation to a single plane 7like the four cells in ,fig4 #aaf7 or heaped upon one another1 as for instance in a tetrahedral form like four cannon-balls2 and in either case we have to pass from the problems of plane to those of solid geometry4 ,in short1 the further development of our theme must lead us along two paths of enquiry1 which continually intercross1 namely 7#a7 the study of more complex cases of partition and of contact in a plane1 and 7#b7 the whole question of the surfaces #caf and angles presented by solid figures in symmetrical juxtaposition4 ,let us take a simple case of the latter kind1 and again afterwards1 so far as possible1 let us try to keep the two themes separate4 ,where we have three spheres in contact1 as in ,fig4 #aad or in either half of ,fig4 #aaf1 ,b1 let us consider the point of contact 7,o1 ,fig4 #aad7 not as a point in the plane section of the diagram1 but as a point where three furrows meet on the surface of the system4 ,at this point1 three cells meet2 but it is also obvious that there meet here six surfaces1 namely the outer1 spherical walls of the three bubbles1 and the three partition-walls which divide them1 two and two4 ,also1 four lines or edges meet here2 viz4 the three external arcs which form the outer boundaries of the partition-walls 7and which correspond to what we commonly call the 8furrows0 in the segmenting egg72 and as a fourth edge1 the 8arris0 or junction of the three partitions 7perpendicular to the plane of the paper71 where they all three meet together1 as we have seen1 at equal angles of #abj4 ,lastly1 there meet at the point four solid angles1 each bounded by three surfaces3 to wit1 within each bubble a solid angle bounded by two partition-walls and by the surface wall2 and 7fourthly7 an external solid angle bounded by the outer surfaces of all three bubbles4 ,now in the case of the soap-bubbles 7whose surfaces are all in contact with air1 both outside and in71 the six films meeting at the point1 whether surface films or partition films1 are all similar1 with similar tensions4 ,in other words the tensions1 or forces1 acting at the point are all similar and symmetrically arranged1 and it at once follows from this that the angles1 solid as well as plane1 are all equal4 ,it is also obvious that1 as regards the point of contact1 the system will still be symmetrical1 and its symmetry will be quite unchanged1 if we add a fourth bubble in contact with the other three3 that is to say1 if where we had merely the outer air before1 we now replace it by the air in the interior of another bubble4 ,the only difference will be that the pressure exercised by the walls of this fourth bubble will alter the curvature of the surfaces of the others1 so far as it encloses them2 and1 if all four bubbles be identical in size1 these surfaces which formerly we called external and which have now come to be internal partitions1 will1 like the others1 be flattened by equal and opposite pressure1 into planes4 ,we are now dealing1 in short1 #cag with six planes1 meeting symmetrically in a point1 and constituting there four equal solid angles4 ,illustration3 ,fig4 #abj4 ,if we make a wire cage1 in the form of a regular tetrahedron1 and dip it into soap-solution1 then when we withdraw it we see that to each one of the six edges of the tetrahedron1 i4e4 to each one of the six wires which constitute the little cage1 a film has attached itself2 and these six films meet internally at a point1 and constitute in every respect the symmetrical figure which we have just been describing4 ,in short1 the system of films we have hereby automatically produced is precisely the system of partition-walls which exist in our tetrahedral aggregation of four spherical bubbles3--precisely the same1 that is to say1 in the neighbourhood of the meeting-point1 and only differing in that we have made the wires of our tetrahedron straight1 instead of imitating the circular arcs which actually form the intersections of our bubbles4 ,this detail we can easily introduce in our wire model if we please4 ,let us look for a moment at the geometry of our figure4 ,let o 7,fig4 #abj7 be the centre of the tetrahedron1 i4e4 the centre of symmetry where our films meet2 and let oa1 ob1 oc1 od1 be lines drawn to the four corners of the tetrahedron4 ,produce ao to meet the base in p2 then apd is a right-angled triangle4 ,it is not difficult to prove that in such a figure1 o 7the centre of gravity of the system7 #cah lies just three-quarters of the way between an apex1 a1 and a point1 p1 which is the centre of gravity of the opposite base4 ,therefore op oa_/#c od_/#c4 ,therefore cos dop #a_/c and cos aod #a_/c4 ,that is to say1 the angle aod is just1 as nearly as possible1 #aji #bh #af4 ,this angle1 then1 of #aji #bh #af1 or very nearly #aji degrees and a half1 is the angle at which1 in this and every other solid system of liquid films1 the edges of the partition-walls meet one another at a point4 ,it is the fundamental angle in the solid geometry of our systems1 just as #abj was the fundamental angle of symmetry so long as we considered only the plane projection1 or plane section1 of three films meeting in an edge4 ,out of these two angles1 we may construct a great variety of figures1 plane and solid1 which become all the more varied and complex when1 by considering the case of unequal as well as equal cells1 we admit curved 7e4g4 spherical7 as well as plane boundary surfaces4 ,let us consider some examples and illustrations of these1 beginning with those which we need only consider in reference to a plane4 ,let us imagine a system of equal cylinders1 or equal spheres1 in contact with one another in a plane1 and represented in section by the equal and contiguous circles of ,fig4 #aba4 ,i borrow my figure1 by the way1 from an old ,italian naturalist1 ,bonanni 7a contemporary of ,borelli1 of ,hay and ,willoughby and of ,martin ,lister71 who dealt with this matter in a book chiefly devoted to molluscan shells#cfb4 ,it is obvious1 as a simple geometrical fact1 that each of these equal circles is in contact with six surrounding circles4 ,imagine now that the whole system comes under some uniform stress4 ,it may be of uniform surface tension at the boundaries of all the cells2 it may be of pressure caused by uniform growth or expansion within the cells2 or it may be due to some uniformly applied constricting pressure from without4 ,in all of these cases the points of contact between the circles in the diagram will be extended into #cai lines of contact1 representing surfaces of contact in the actual spheres or cylinders2 and the equal circles of our diagram will be converted into regular and equal hexagons4 ,the angles of these hexagons1 at each of which three hexagons meet1 are of course angles of #abj4 ,so far as the form is concerned1 so long as we are concerned only with a morphological result and not with a physiological process1 the result is precisely the same whatever be the force which brings the bodies together in symmetrical apposition2 it is by no means necessary for us1 in the first instance1 even to enquire whether it be surface tension or mechanical pressure or some other physical force which is the cause1 or the main cause1 of the phenomenon4 ,illustration3 ,fig4 #aba4 ,diagram of hexagonal cells4 7,after ,bonanni47 ,the production by mutual interaction of polyhedral cells1 which1 under conditions of perfect symmetry1 become regular hexagons1 is very beautifully illustrated by ,prof4 ,bnard's 8tourbillons cellulaires0 7cf4 p4 #bei71 and also in some of ,leduc's diffusion experiments4 ,a weak 7#e per cent47 solution of gelatine is allowed to set on a plate of glass1 and little drops of a #e or #aj per cent4 solution of ferrocyanide of potassium are then placed at regular intervals upon the gelatine4 ,immediately each little drop becomes the centre1 or pole1 of a system of diffusion currents1 #cbj and the several systems conflict with and repel one another1 so that presently each little area becomes the seat of a double current system1 from its centre outwards and back again2 until at length the concentration of the field becomes equalised and the currents #cba ,illustration3 ,fig4 #abb4 ,an 8artificial tissue10 formed by coloured drops of sodium chloride solution diffusing in a less dense solution of the same salt4 7,after ,leduc47 ,illustration3 ,fig4 #abc4 ,an artificial cellular tissue1 formed by the diffusion in gelatine of drops of a solution of potassium ferrocyanide4 7,after ,leduc47 cease4 ,after equilibrium is attained1 and when the gelatinous mass is permitted to dry1 we have an artificial tissue of more or less regularly hexagonal 8cells10 which simulate in the closest way an organic parenchyma4 ,and by varying the experiment1 in ways which ,leduc describes1 we may simulate various forms of tissue1 and produce cells with thick walls or with thin1 cells in close contact or with wide intercellular spaces1 cells with plane or with curved partitions1 and so forth4 ,illustration3 ,fig4 #abd4 ,epidermis of ,girardia4 7,after ,goebel47 ,the hexagonal pattern is illustrated among organisms in countless cases1 but those in which the pattern is perfectly regular1 by reason of perfect uniformity of force and perfect equality of the individual cells1 are not so numerous4 ,the hexagonal epithelium-cells of the pigment layer of the eye1 external to the retina1 are a good example4 ,here we have a single layer of uniform cells1 reposing on the one hand upon a basement membrane1 supported behind by the solid wall of the sclerotic1 and exposed on the other hand to the uniform fluid pressure of the vitreous humour4 ,the conditions all point1 and lead1 to a perfectly symmetrical result3 that is to say1 the cells1 uniform in size1 are flattened out to a uniform thickness by the fluid pressure acting radially2 and their reaction on each other converts the flattened discs into regular hexagons4 ,in an ordinary columnar epithelium1 such as that of the intestine1 we see again that the columnar cells have been compressed into hexagonal prisms2 but here as a rule the cells are less uniform in size1 small cells are apt to be intercalated among the larger1 and the perfect symmetry is accordingly lost4 ,the same is true of ordinary vegetable parenchyma2 the originally spherical cells are approximately equal in size1 but only approximately2 and there are accordingly all degrees in the regularity and symmetry of the resulting tissue4 ,but obviously1 wherever we #cbb have1 in addition to the forces which tend to produce the regular hexagonal symmetry1 some other asymmetrical component arising from growth or traction1 then our regular hexagons will be distorted in various simple ways4 ,this condition is illustrated in the accompanying diagram of the epidermis of ,girardia2 it also accounts for the more or less pointed or fusiform cells1 each still in contact 7as a rule7 with six others1 which form the epithelial lining of the blood-vessels3 and other similar1 or analogous1 instances are very common4 ,illustration3 ,fig4 #abe4 ,soap-froth under pressure4 7,after ,rhumbler47 ,in a soap-froth imprisoned between two glass plates1 we have a symmetrical system of cells1 which appear in optical section 7as in ,fig4 #abe1 ,b7 as regular hexagons2 but if we press the plates a little closer together1 the hexagons become deformed or flattened 7,fig4 #abe1 ,a74 ,in this case1 however1 if we cease to apply further pressure1 the tension of the films throughout the system soon adjusts itself again1 and in a short time the system has regained the former symmetry of ,fig4 #abe1 ,b4 ,illustration3 ,fig4 #abf4 ,from leaf of ,elodea canadensis4 7,after ,berthold47 ,in the growth of an ordinary dicotyledonous leaf1 we once more see reflected in the form of its epidermal cells the tractions1 irregular but on the whole longitudinal1 which growth has superposed on the tensions of the partition-walls 7,fig4 #abf74 ,in the narrow elongated leaf of a ,monocotyledon1 such as a hyacinth1 the elongated1 apparently quadrangular #cbc cells of the epidermis appear as a necessary consequence of the simpler laws of growth which gave its simple form to the leaf as a whole4 ,in this last case1 however1 as in all the others1 the rule still holds that only three partitions 7in surface view7 meet in a point2 and at their point of meeting the walls are for a short distance manifestly curved1 so as to permit the junction to take place at or nearly at the normal angle of #abj4 ,briefly speaking1 wherever we have a system of cylinders or spheres1 associated together with sufficient mutual interaction to bring them into complete surface contact1 there1 in section or in surface view1 we tend to get a pattern of hexagons4 ,while the formation of an hexagonal pattern on the basis of ready-formed and symmetrically arranged material units is a very common1 and indeed the general way1 it does not follow that there are not others by which such a pattern can be obtained4 ,for instance1 if we take a little triangular dish of mercury and set it vibrating 7either by help of a tuning-fork1 or by simply tapping on the sides7 we shall have a series of little waves or ripples starting inwards from each of the three faces2 and the intercrossing1 or interference of these three sets of waves produces crests and hollows1 and intermediate points of no disturbance1 whose loci are seen as a beautiful pattern of minute hexagons4 ,it is possible that the very minute and astonishingly regular pattern of hexagons which we see1 for instance1 on the surface of many diatoms1 may be a phenomenon of this order#cfc4 ,the same may be the case also in ,arcella1 where an apparently hexagonal pattern is found not to consist of simple hexagons1 but of 8straight lines in three sets of parallels1 the lines of each set making an angle of sixty degrees with those of the other two sets#cfd40 ,we must also bear in mind1 in the case of the minuter forms1 the large possibilities of optical illusion4 ,for instance1 in one of ,abbe's 8diffraction-plates10 a pattern of dots1 set at equal interspaces1 is reproduced on a very minute scale by photography2 but under certain conditions of microscopic illumination and focussing1 these isolated dots appear as a pattern of hexagons4 ,a symmetrical arrangement of hexagons1 such as we have just been studying1 suggests various simple geometrical corollaries1 of which the following may perhaps be a useful one4 ,we may sometimes desire to estimate the number of hexagonal areas or facets in some structure where these are numerous1 such for instance as the #cbd cornea of an insect's eye1 or in the minute pattern of hexagons on many diatoms4 ,an approximate enumeration is easily made as follows4 ,for the area of a hexagon 7if we call the short diameter1 that namely which bisects two of the opposite sides7 is #b 7#c7_/b1 the area of a circle being d#b _/#d4 ,then1 if the diameter 7d7 of a circular area include n hexagons1 the area of that circle equals 7n 7#b _/#d4 ,and1 dividing this number by the area of a single hexagon1 we obtain for the number of areas in the circle1 each equal to a hexagonal facet1 the expression n#b _/#d #b_/7c7 #jijgn#b1 or 7#i_/aj7 n#b1 nearly4 ,this calculation deals1 not only with the complete facets1 but with the areas of the broken hexagons at the periphery of the circle4 ,if we neglect these latter1 and consider our whole field as consisting of successive rings of hexagons about a central one1 we may obtain a still simpler rule#cfe4 ,for obviously1 around our central hexagon there stands a zone of six1 and around these a zone of twelve1 and around these a zone of eighteen1 and so on4 ,and the total number1 excluding the central hexagon1 is accordingly3 ,for one zone #f #b #c #c #a #b1 ,for two zones #ah #c #f #c #b #c1 ,for three zones #cf #d #i #c #c #d1 ,for four zones #fj #e #ab #c #d #e1 ,for five zones #ij #f x #ae #c #e #f1 and so forth4 ,if ,n be the number of zones1 and if we add one to the above numbers for the odd central hexagon1 the rule evidently is1 that the total number1 ,h1 #c,n7,n #a7 #a4 ,thus1 if in a preparation of a fly's cornea1 ,i can count twenty-five facets in a line from a central one1 the total number in the entire circular field is 7#c #be #bf7 #a #aieacff4 ,the same principles which account for the development of hexagonal symmetry hold true1 as a matter of course1 not only of individual cells 7in the biological sense71 but of any close-packed bodies of uniform size and originally circular outline2 and the hexagonal pattern is therefore of very common occurrence1 under widely different circumstances4 ,the curious reader may consult ,sir ,thomas ,browne's quaint and beautiful account1 in the ,garden of ,cyrus1 of hexagonal 7and also of quincuncial7 symmetry in plants and animals1 which 8doth neatly declare how nature ,geometrizeth1 and observeth order in all things40 #cbe ,we have many varied examples of this principle among corals1 wherever the polypes are in close juxtaposition1 with neither empty space nor accumulations of matrix between their adjacent walls4 ,favosites gothlandica1 for instance1 furnishes us with an excellent example4 ,in the great genus ,lithostrotion we have some species that are 8massive0 and others that are 8fasciculate02 in other words in some the long cylindrical corallites are in close contact with one another1 and in others they are separate and loosely bundled 7,fig4 #abg74 ,accordingly in the former the corallites are ,illustration3 ,fig4 #abg4 ,lithostrotion ,martini4 7,after ,nicholson47 ,illustration3 ,fig4 #abh4 ,cyathophyllum hexagonum4 7,from ,nicholson1 after ,zittel47 squeezed into hexagonal prisms1 while in the latter they retain their cylindrical form4 ,where the polypes are comparatively few1 and so have room to spread1 the mutual pressure ceases to work or only tends to push them asunder1 letting them remain circular in outline 7e4g4 ,thecosmilia74 ,where they vary gradually in size1 as for instance in ,cyathophyllum hexagonum1 they are more or less hexagonal but are not regular hexagons2 and where there is greater and more irregular variation in size1 the cells will be on the average hexagonal1 but some will have fewer and some more sides than six1 as in the annexed figure of ,arachnophyllum 7,fig4 #abi74 #cbf ,where larger and smaller cells1 corresponding to two different kinds of zooids1 are mixed together1 we may get various results4 ,if the larger cells are numerous enough to be more or less in contact with one another 7e4g4 various ,monticuliporae7 they will be irregular hexagons1 while the smaller cells between them will be crushed into all manner of irregular angular forms4 ,if on the other hand the large cells are comparatively few and are large and strong-walled compared with their smaller neighbours1 then the latter alone will be squeezed into hexagons1 while the larger ones will tend to retain their circular outline undisturbed 7e4g4 ,heliopora1 ,heliolites1 etc474 ,illustration3 ,fig4 #abi4 ,arachnophyllum pentagonum4 7,after ,nicholson47 ,illustration3 ,fig4 #acj4 ,heliolites4 7,after ,woods47 ,when1 as happens in certain corals1 the peripheral walls or 8thecae0 of the individual polypes remain undeveloped but the radiating septa are formed and calcified1 then we obtain new and beautiful mathematical configurations 7,fig4 #aca74 ,for the radiating septa are no longer confined to the circular or hexagonal bounds of a polypite1 but tend to meet and become confluent with their neighbours on every side2 and1 tending to assume positions of equilibrium1 or of minimal area1 under the restraints to which they are subject1 they fall into congruent curves2 and these correspond1 in a striking manner1 to the lines of force running1 in a common field of force1 between a number of secondary centres4 ,similar patterns may be produced in various ways1 by the play of osmotic or magnetic forces2 and a particular and very curious case is to be found in those complicated forms of nuclear division #cbg known as triasters1 polyasters1 etc41 whose relation to a field of force ,hartog has explained#cfg4 ,it is obvious that1 in our corals1 these curving septa are all orthogonal to the non-existent hexagonal boundaries4 ,as the phenomenon is wholly due to the imperfect development or non-existence of a thecal wall1 it is not surprising that we find identical configurations among various corals1 or families of corals1 not otherwise related to one another2 we find the same or very similar patterns displayed1 for instance1 in ,synhelia 7,oculinidae71 in ,phillipsastraea 7,rugosa71 in ,thamnastraea 7,fungida71 and in many more4 ,illustration3 ,fig4 #aca4 ,surface-views of ,corals with undeveloped thecae and confluent septa4 ,a1 ,thamnastraea2 ,b1 ,comoseris4 7,from ,nicholson1 after ,zittel47 ,the most famous of all hexagonal conformations and perhaps the most beautiful is that of the bee's cell4 ,here we have1 as in our last examples1 a series of equal cylinders1 compressed by symmetrical forces into regular hexagonal prisms4 ,but in this case we have two rows of such cylinders1 set opposite to one another1 end to end2 and we have accordingly to consider also the conformation of their ends4 ,we may suppose our original cylindrical cells to have spherical ends1 which is their normal and symmetrical mode of termination2 and1 for closest packing1 it is obvious that the end of any one cylinder will touch1 and fit in between1 the ends of three cylinders in the opposite row4 ,it is just as when we pile round-shot in a heap2 each sphere that we #cbh set down fits into its nest between three others1 and the four form a regular tetrahedral arrangement4 ,just as it was obvious1 then1 that by mutual pressure from the six laterally adjacent cells1 any one cell would be squeezed into a hexagonal prism1 so is it also obvious that1 by mutual pressure against the three terminal neighbours1 the end of any one cell will be compressed into a solid trihedral angle whose edges will meet1 as in the analogous case already described of a system of soap-bubbles1 at a plane angle of #aji and so many minutes and seconds4 ,what we have to comprehend1 then1 is how the six sides of the cell are to be combined with its three terminal facets4 ,this is done by bevelling off three alternate angles of the prism1 in a uniform manner1 until we have tapered the prism to a point2 and by so doing1 we evidently produce three rhombic surfaces1 each of which is double of the triangle formed by joining the apex to the three untouched angles of the prism4 ,if we experiment1 not with cylinders1 but with spheres1 if for instance we pile together a mass of bread-pills 7or pills of plasticine71 and then submit the whole to a uniform pressure1 it is obvious that each ball 7like the seeds in a pomegranate1 as ,kepler said71 will be in contact with twelve others1--six in its own plane1 three below and three above1 and in compression it will therefore develop twelve plane surfaces4 ,it will in short repeat1 above and below1 the conditions to which the bee's cell is subject at one end only2 and1 since the sphere is symmetrically situated towards its neighbours on all sides1 it follows that the twelve plane sides to which its surface has been reduced will be all similar1 equal and similarly situated4 ,moreover1 since we have produced this result by squeezing our original spheres close together1 it is evident that the bodies so formed completely fill space4 ,the regular solid which fulfils all these conditions is the rhombic dodecahedron4 ,the bee's cell1 then1 is this figure incompletely formed3 it is a hexagonal prism with one open or unfinished end1 and one trihedral apex of a rhombic dodecahedron4 ,the geometrical form of the bee's cell must have attracted the attention and excited the admiration of mathematicians from time immemorial4 ,pappus the ,alexandrine has left us 7in the introduction to the ,fifth ,book of his ,collections7 an account of its hexagonal plan1 and he drew from its mathematical symmetry the #cbi conclusion that the bees were endowed with reason3 8,there being1 then1 three figures which of themselves can fill up the space round a point1 viz4 the triangle1 the square and the hexagon1 the bees have wisely selected for their structure that which contains most angles1 suspecting indeed that it could hold more honey than either of the other two40 ,erasmus ,bartholinus was apparently the first to suggest that this hypothesis was not warranted1 and that the hexagonal form was no more than the necessary result of equal pressures1 each bee striving to make its own little circle as large as possible4 ,the investigation of the ends of the cell was a more difficult matter1 and came later1 than that of its sides4 ,in general terms this arrangement was doubtless often studied and described3 as for instance1 in the ,garden of ,cyrus3 8,and the ,combes themselves so regularly contrived that their mutual intersections make three ,lozenges at the bottom of every ,cell2 which severally regarded make three ,rows of neat ,rhomboidall ,figures1 connected at the angles1 and so continue three several chains throughout the whole comb40 ,but ,maraldi#cfh 7,cassini's nephew7 was the first to measure the terminal solid angle or determine the form of the rhombs in the pyramidal ending of the cell4 ,he tells us that the angles of the rhomb are #aaj and #gj3 8,chaque base d'alvole est forme par trois rhombes presque toujours gaux et semblables1 qui1 suivant les mesures que nous avons prises1 ont les deux angles obtus chacun de #aaj degrs1 et par consquent les deux aigus chacun de #gj40 ,he also stated that the angles of the trapeziums which form the sides of the body of the cell were identical angles1 of #aaj and #gj2 but in the same paper he speaks of the angles as being1 respectively1 #aji #bh and #gj #cb4 ,here a singular confusion at once arose1 and has been perpetuated in the books#cfi4 8,unfortunately ,raumur chose to look upon this second determination of ,maraldi's as being1 as well as the first1 a direct result of measurement1 whereas it is in reality theoretical4 ,he speaks of it as ,maraldi's more precise measurement1 and this error has been repeated in spite of its absurdity to the present day2 nobody #ccj appears to have thought of the impossibility of measuring such a thing as the end of a bee's cell to the nearest minute40 ,at any rate1 it now occurred to ,raumur 7as curiously enough1 it had not done to ,maraldi7 that1 just as the closely packed hexagons gave the minimal extent of boundary in a plane1 so the actual solid figure1 as determined by ,maraldi1 might be that which1 for a given solid content1 gives the minimum of surface3 or which1 in other words1 would hold the most honey for the least wax4 ,he set this problem before ,koenig1 and the geometer confirmed his conjecture1 the result of his calculations agreeing within two minutes 7#aji #bf and #gj #cd7 with ,maraldi's determination4 ,but again1 ,maclaurin#cgj and ,lhuilier#cga1 by different methods1 obtained a result identical with ,maraldi's2 and were able to shew that the discrepancy of #b was due to an error in ,koenig's calculation 7of tan #b71--that is to say to the imperfection of his logarithmic tables1--not 7as the books say#cgb7 8to a mistake on the part of the ,bee40 8,not to a mistake on the part of ,maraldi0 is1 of course1 all that we are entitled to say4 ,illustration3 ,fig4 #acb4 ,the theorem may be proved as follows3 ,,abcdef1 abcdef1 is a right prism upon a regular hexagonal base4 ,the corners ,,bdf are cut off by planes through the lines ,,ac1 ,,ce1 ,,ea1 meeting in a point ,v on the axis ,,vn of the prism1 and intersecting ,bb1 ,dd1 ,ff1 at ,x1 ,y1 ,z4 ,it is evident that the volume of the figure thus formed is the same as that of the original prism with hexagonal ends4 ,for1 if the axis cut the hexagon ,,abcdef in ,n1 the volumes ,,acvn1 ,,acbx are equal4 #cca ,it is required to find the inclination of the faces forming the trihedral angle at ,v to the axis1 such that the surface of the figure may be a minimum4 ,let the angle ,,nvx1 which is half the solid angle of the prism1 2 the side of the hexagon1 as ,,ab1 a2 and the height1 as ,aa1 h4 ,then1 ,,ac #b;a cos #cj a#c4 ,and ,,vx a_/sin 7from inspection of the triangle ,,lxb7 ,therefore the area of the rhombus ,,vaxc 7a#b #c7_/7b sin 74 ,and the area of ,aab,x 7a_/#b77b;h ,,vx cos 7 7a_/#b77b;h a_/#b cot 74 ,therefore the total area of the figure hexagon abcdef #c;a7#b;h 7a_/#b7 cot 7 #c77;a#b #c7_/7b sin 774 ,therefore d7,area7_/d 7#c;a#b_/b777a_/sin#b 7 7#c cos 7_/7sin#b 774 ,but this expression vanishes1 that is to say1 d7,area7_/d #j1 when cos #a_/c1 that is when #ed #dd #h 7#aji #bh #af74 ,this then is the condition under which the total area of the figure has its minimal value4 ,that the beautiful regularity of the bee's architecture is due to some automatic play of the physical forces1 and that it were fantastic to assume 7with ,pappus and ,raumur7 that the bee intentionally seeks for a method of economising wax1 is certain1 but the precise manner of this automatic action is not so clear4 ,when the hive-bee builds a solitary cell1 or a small cluster of cells1 as it does for those eggs which are to develop into queens1 it makes but a rude production4 ,the queen-cells are lumps of coarse wax hollowed out and roughly bitten into shape1 bearing the marks of the bee's jaws1 like the marks of a blunt adze on a rough-hewn log4 ,omitting the simplest of all cases1 when 7as among some humble-bees7 the old cocoons are used to hold honey1 the cells built by the 8solitary0 wasps and bees are of various kinds4 ,they may be formed by partitioning off little chambers in a hollow stem2 #ccb they may be rounded or oval capsules1 often very neatly constructed1 out of mud1 or vegetable fibre or little stones1 agglutinated together with a salivary glue2 but they shew1 except for their rounded or tubular form1 no mathematical symmetry4 ,the social wasps and many bees build1 usually out of vegetable matter chewed into a paste with saliva1 very beautiful nests of 8combs02 and the close-set papery cells which constitute these combs are just as regularly hexagonal as are the waxen cells of the hive-bee4 ,but in these cases 7or nearly all of them7 the cells are in a single row2 their sides are regularly hexagonal1 but their ends1 from the want of opponent forces1 remain simply spherical4 ,in ,melipona domestica 7of which ,darwin epitomises ,pierre ,huber's description7 8the large waxen honey-cells are nearly spherical1 nearly equal in size1 and are aggregated into an irregular mass40 ,but the spherical form is only seen on the outside of the mass2 for inwardly each cell is flattened into 8two1 three or more flat surfaces1 according as the cell adjoins two1 three or more other cells4 ,when one cell rests on three other cells1 which from the spheres being nearly of the same size is very frequently and necessarily the case1 the three flat surfaces are united into a pyramid2 and this pyramid1 as ,huber has remarked1 is manifestly a gross imitation of the three-sided pyramidal base of the cell of the hive-bee#cgc40 ,the question is1 to what particular force are we to ascribe the plane surfaces and definite angles which define the sides of the cell in all these cases1 and the ends of the cell in cases where one row meets and opposes another4 ,we have seen that ,bartholin suggested1 and it is still commonly believed1 that this result is due to simple physical pressure1 each bee enlarging as much as it can the cell which it is a-building1 and nudging its wall outwards till it fills every intervening gap and presses hard against the similar efforts of its neighbour in the cell next door#cgd4 ,but it is very doubtful #ccc whether such physical or mechanical pressure1 more or less intermittently exercised1 could produce the all but perfectly smooth1 plane surfaces and the all but perfectly definite and constant angles which characterise the cell1 whether it be constructed of wax or papery pulp4 ,it seems more likely that we have to do with a true surface-tension effect2 in other words1 that the walls assume their configuration when in a semi-fluid state1 while the papery pulp is still liquid1 or while the wax is warm under the high temperature of the crowded hive#cge4 ,under these circumstances1 the direct efforts of the wasp or bee may be supposed to be limited to the making of a tubular cell1 as thin as the nature of the material permits1 and packing these little cells as close as possible together4 ,it is then easily conceivable that the symmetrical tensions of the adjacent films 7though somewhat retarded by viscosity7 should suffice to bring the whole system into equilibrium1 that is to say into the precise configuration which the comb actually presents4 ,in short1 the ,maraldi pyramids which terminate the bee's cell are precisely identical with the facets of a rhombic dodecahedron1 such as we have assumed to constitute 7and which doubtless under certain conditions do constitute7 the surfaces of contact in the interior of a mass of soap-bubbles or of uniform parenchymatous cells2 and there is every reason to believe that the physical explanation is identical1 and not merely mathematically analogous4 ,the remarkable passage in which ,buffon discusses the bee's cell and the hexagonal configuration in general is of such historical importance1 and tallies so closely with the whole trend of our enquiry1 that ,i will quote it in full3 8,dirai-je encore un mot2 ces cellules des abeilles1 tant vantes1 tant admires1 me fournissent une preuve de plus contre l'enthousiasme et l'admiration2 cette figure1 toute gomtrique et toute rgulire qu'elle nous parat1 et qu'elle est en effet dans la spculation1 n'est ici qu'un rsultat mcanique et assez imparfait qui se trouve souvent dans la nature1 #ccd et que l'on remarque mme dans les productions les plus brutes2 les cristaux et plusieurs autres pierres1 quelques sels1 etc41 prennent constamment cette figure dans leur formation4 ,qu'on observe les petites cailles de la peau d'une roussette1 on verra qu'elles sont hexagones1 parce que chaque caille croissant en mme temps se fait obstacle1 et tend occuper le plus d'espace qu'il est possible dans un espace donn3 on voit ces mmes hexagones dans le second estomac des animaux ruminans1 on les trouve dans les graines1 dans leurs capsules1 dans certaines fleurs1 etc4 ,qu'on remplisse un vaisseau de pois1 ou pltot de quelque autre graine cylindrique1 et qu'on le ferme exactement aprs y avoir vers autant d'eau que les intervalles qui restent entre ces graines peuvent en recevoir2 qu'on fasse bouillir cette eau1 tous ces cylindres deviendront de colonnes six pans#cgf4 ,on y voit clairement la raison1 qui est purement mcanique2 chaque graine1 dont la figure est cylindrique1 tend par son renflement occuper le plus d'espace possible dans un espace donn1 elles deviennent donc toutes ncessairement hexagones par la compression rciproque4 ,chaque abeille cherche occuper de mme le plus d'espace possible dans un espace donn1 il est donc ncessaire aussi1 puisque le corps des abeilles est cylindrique1 que leurs cellules sont hexagones1--par la mme raison des obstacles rciproques4 ,on donne plus d'esprit aux mouches dont les ouvrages sont les plus rguliers2 les abeilles sont1 dit-on1 plus ingnieuses que les gupes1 que les frlons1 etc41 qui savent aussi l'architecture1 mais dont les constructions sont plus grossires et plus irrgulires que celles des abeilles3 on ne veut pas voir1 ou l'on ne se doute pas que cette rgularit1 plus ou moins grande1 dpend uniquement du nombre et de la figure1 et nullement de l'intelligence de ces petites btes2 plus elles sont nombreuses1 plus il y a des forces qui agissent galement et s'opposent de mme1 plus il y a par consquent de contrainte mcanique1 de rgularit force1 et de perfection apparente dans leurs productions#cgg40 #cce ,a very beautiful hexagonal symmetry1 as seen in section1 or dodecahedral1 as viewed in the solid1 is presented by the cells which form the pith of certain rushes 7e4g4 ,juncus effusus71 and somewhat less diagrammatically by those which make the pith of the banana4 ,these cells are stellate in form1 and the tissue presents in section the appearance of a network of six-rayed stars 7,fig4 #acc1 c71 linked together by the tips of the rays1 and separated by symmetrical1 air-filled1 intercellular spaces4 ,in thick sections1 the solid twelve-rayed stars may be very beautifully seen under the binocular microscope4 ,illustration3 ,fig4 #acc4 ,diagram of development of 8stellate cells10 in pith of ,juncus4 7,the dark1 or shaded1 areas represent the cells2 the light areas being the gradually enlarging 8intercellular spaces407 ,what has happened here is not difficult to understand4 ,imagine1 as before1 a system of equal spheres all in contact1 each one therefore touching six others in an equatorial plane2 and let the cells be not only in contact1 but become attached at the points of contact4 ,then instead of each cell expanding1 so as to encroach on and fill up the intercellular spaces1 let each cell tend to contract or shrivel up1 by the withdrawal of fluid from its interior4 ,the #ccf result will obviously be that the intercellular spaces will increase2 the six equatorial attachments of each cell 7,fig4 #acc1 a7 7or its twelve attachments in all1 to adjacent cells7 will remain fixed1 and the portions of cell-wall between these points of attachment will be withdrawn in a symmetrical fashion 7b7 towards the centre4 ,as the final result 7c7 we shall have a 8dodecahedral star0 or star-polygon1 which appears in section as a six-rayed figure4 ,it is obviously necessary that the pith-cells should not only be attached to one another1 but that the outermost layer should be firmly attached to a boundary wall1 so as to preserve the symmetry of the system4 ,what actually occurs in the rush is tantamount to this1 but not absolutely identical4 ,here it is not so much the pith-cells which tend to shrivel within a boundary of constant size1 but rather the boundary wall 7that is1 the peripheral ring of woody and other tissues7 which continues to expand after the pith-cells which it encloses have ceased to grow or to multiply4 ,the twelve points of attachment on the spherical surface of each little pith-cell are uniformly drawn asunder2 but the content1 or volume1 of the cell does not increase correspondingly2 and the remaining portions of the surface1 accordingly1 shrink inwards and gradually constitute the complicated surface of a twelve-pointed star1 which is still a symmetrical figure and is still also a surface of minimal area under the new conditions4 ,a few years after the publication of ,plateau's book1 ,lord ,kelvin shewed1 in a short but very beautiful paper#cgh1 that we must not hastily assume from such arguments as the foregoing1 that a close-packed assemblage of rhombic dodecahedra will be the true and general solution of the problem of dividing space with a minimum partitional area1 or will be present in a cellular liquid 8foam10 in which it is manifest that the problem is actually and automatically solved4 ,the general mathematical solution of the problem 7as we have already indicated7 is1 that every interface or partition-wall must have constant curvature throughout2 that where such partitions meet in an edge1 they must intersect at angles such that equal forces1 in planes perpendicular to the line #ccg of intersection1 shall balance2 and finally1 that no more than three such interfaces may meet in a line or edge1 whence it follows that the angle of intersection of the film-surfaces must be exactly #abj4 ,an assemblage of equal and similar rhombic dodecahedra goes far to meet the case3 it completely fills up space2 all its surfaces or interfaces are planes1 that is to say1 surfaces of constant curvature throughout2 and these surfaces all meet together at angles of #abj4 ,nevertheless1 the proof that our rhombic dodecahedron 7such as we find exemplified in the bee's cell7 is a surface of minimal area1 is not a comprehensive proof2 it is limited to certain conditions1 and practically amounts to no more than this1 that of the regular solids1 with all sides plane and similar1 this one has the least surface for its solid content4 ,illustration3 ,fig4 #acd4 ,the rhombic dodecahedron has six tetrahedral angles1 and eight trihedral angles2 and it is obvious1 on consideration1 that at each of the former six dodecahedra meet in a point1 and that1 where the four tetrahedral facets of each coalesce with their neighbours1 we have twelve plane films1 or interfaces1 meeting in a point4 ,in a precisely similar fashion1 we may imagine twelve plane films1 drawn inwards from the twelve edges of a cube1 to meet at a point in the centre of the cube4 ,but1 as ,plateau discovered#cgi1 when we dip a cubical wire skeleton into soap-solution and take it out again1 the twelve films which are thus generated do not meet in a point1 but are grouped around a small central1 plane1 quadrilateral film 7,fig4 #acd74 ,in other words1 twelve plane films1 meeting in a point1 are essentially unstable4 ,if we blow upon our artificial film-system1 the little quadrilateral alters its place1 setting itself parallel now to one and now to another of the paired faces of the cube2 but we never get rid of it4 ,moreover1 the size and shape of the quadrilateral1 as of all the other films in the system1 are perfectly definite4 ,of the twelve films 7which we had #cch expected to find all plane and all similar7 four are plane isosceles triangles1 and eight are slightly curved quadrilateral figures4 ,the former have two curved sides1 meeting at an angle of #aji #bh1 and their apices coincide with the corners of the central quadrilateral1 whose sides are also curved1 and also meet at this identical angle2--which 7as we observe7 is likewise an angle which we have been dealing with in the simpler case of the bee's cell1 and indeed in all the regular solids of which we have yet treated4 ,by completing the assemblage of polyhedra of which ,plateau's skeleton-cube gives a part1 ,lord ,kelvin shewed that we should obtain a set of equal and similar fourteen-sided figures1 or 8tetrakaidecahedra02 and that by means of an assemblage of these figures space is homogeneously partitioned--that is to say1 into equal1 similar and similarly situated cells--with an economy of surface in relation to area even greater than in an assemblage of rhombic dodecahedra4 ,in the most generalised case1 the tetrakaidecahedron is bounded by three pairs of equal and parallel quadrilateral faces1 and four pairs of equal and parallel hexagonal faces1 neither the quadrilaterals nor the hexagons being necessarily plane4 ,in a certain particular case1 the quadrilaterals are plane surfaces1 but the hexagons slightly curved 8anticlastic0 surfaces2 and these latter have at every point equal and opposite curvatures1 and are surfaces of minimal curvature for a boundary of six curved edges4 ,the figure has the remarkable property that1 like the plane rhombic dodecahedron1 it so partitions space that three faces meeting in an edge do so everywhere at equal angles of #abj #chj4 ,we may take it as certain that1 in a system of perfectly fluid films1 like the interior of a mass of soap-bubbles1 where the films are perfectly free to glide or to rotate over one another1 the mass is actually divided into cells of this remarkable conformation4 #cci ,and it is quite possible1 also1 that in the cells of a vegetable parenchyma1 by carefully macerating them apart1 the same conformation may yet be demonstrated under suitable conditions2 that is to say when the whole tissue is highly symmetrical1 and the individual cells are as nearly as possible equal in size4 ,but in an ordinary microscopic section1 it would seem practically impossible to distinguish the fourteen-sided figure from the twelve-sided4 ,moreover1 if we have anything whatsoever interposed so as to prevent our twelve films meeting in a point1 and 7so to speak7 to take the place of our little central quadrilateral1--if we have1 for instance1 a tiny bead or droplet in the centre of our artificial system1 or even a little thickening1 or 8bourrelet0 as ,plateau called it1 of the cell-wall1 then it is no longer necessary that the tetrakaidecahedron should be formed4 ,accordingly1 it is very probably the case that1 in the parenchymatous tissue1 under the actual conditions of restraint and of very imperfect fluidity1 it is after all the rhombic dodecahedral configuration which1 even under perfectly symmetrical conditions1 is generally assumed4 ,it follows from all that we have said1 that the problems connected with the conformation of cells1 and with the manner in which a given space is partitioned by them1 soon become exceedingly complex4 ,and while this is so even when all our cells are equal and symmetrically placed1 it becomes vastly more so when cells varying even slightly in size1 in hardness1 rigidity or other qualities1 are packed together4 ,the mathematics of the case very soon become too hard for us2 but in its essence1 the phenomenon remains the same4 ,we have little reason to doubt1 and no just cause to disbelieve1 that the whole configuration1 for instance of an egg in the advanced stages of segmentation1 is accurately determined by simple physical laws1 just as much as in the early stages of two or four cells1 during which early stages we are able to recognise and demonstrate the forces and their resultant effects4 ,but when mathematical investigation has become too difficult1 it often happens that physical experiment can reproduce for us the phenomena which ,nature exhibits to us1 and which we are striving to comprehend4 ,for instance1 in an admirable research1 ,m4 ,robert shewed1 some years ago1 not only that the early segmentation of #cdj the egg of ,trochus 7a marine univalve mollusc7 proceeded in accordance with the laws of surface tension1 but he also succeeded in imitating by means of soap-bubbles1 several stages1 one after another1 of the developing egg4 ,illustration3 ,fig4 #ace4 ,aggregations of four soap-bubbles1 to shew various arrangements of the intermediate partition and polar furrows4 7,after ,robert47 ,m4 ,robert carried his experiments as far as the stage of sixteen cells1 or bubbles4 ,it is not easy to carry the artificial system quite so far1 but in the earlier stages the experiment is easy2 we have merely to blow our bubbles in a little dish1 adding one to another1 and adjusting their sizes to produce a symmetrical system4 ,one of the simplest and prettiest parts of his investigation concerned the 8polar furrow0 of which we have spoken on p4 #caj4 ,on blowing four little contiguous bubbles he found 7as we may all find with the greatest ease7 that they form a symmetrical system1 two in contact with one another by a laminar film1 and two1 which are elevated a little above the others1 and which are separated by the length of the aforesaid lamina4 ,the bubbles are thus in contact three by three1 their partition-walls making with one another equal angles of #abj4 ,the upper and lower edges of the intermediate lamina 7the lower one visible through the transparent system7 constitute the two polar furrows of the embryologist 7,fig4 #ace1 #a-c74 ,the lamina itself is plane when the system is symmetrical1 but it responds by a corresponding curvature to the least inequality of the bubbles on either side4 ,in the experiment1 the upper polar furrow is usually a little shorter than the lower1 but parallel to it2 that is to say1 the lamina is of trapezoidal form3 this lack of perfect symmetry being due 7in the experimental case7 to the lower portion of the bubbles being somewhat drawn asunder by the tension of their attachments to the sides of the dish 7,fig4 #ace1 #d74 ,a similar phenomenon is usually found in ,trochus1 according to ,robert1 and many other observers have likewise found the upper furrow to be shorter than the one below4 ,in the various species of the genus ,crepidula1 ,conklin asserts that the two furrows are equal in ,c4 convexa1 that the upper one is the shorter in ,c4 fornicata1 and that the upper one all but disappears in ,c4 plana2 but we may well be permitted to doubt1 without the evidence of very special investigations1 whether these slight physical differences are actually characteristic of1 and constant in1 particular allied species4 #cda ,returning to the experimental case1 ,robert found that by withdrawing a little air from1 and so diminishing the bulk of the two terminal bubbles 7i4e4 those at the ends of the intermediate lamina71 the upper polar furrow was caused to elongate1 till it became equal in length to the lower2 and by continuing the process it became the longer in its turn4 ,these two conditions have again been described by investigators as characteristic of this embryo or that2 for instance in ,unio1 ,lillie has described the two furrows as gradually altering their respective lengths#cha2 and ,wilson 7as ,lillie remarks7 had already pointed out that 8the reduction of the apical cross-furrow1 as compared with that at the vegetative pole #cdb in molluscs and annelids #8stands in obvious relation to the different size of the cells produced at the two poles#chb4' 0 ,when the two lateral bubbles are gradually reduced in size1 or the two terminal ones enlarged1 the upper furrow becomes shorter and shorter2 and at the moment when it is about to vanish1 a new furrow makes its instantaneous appearance in a direction perpendicular to the old one2 but the inferior furrow1 constrained by its attachment to the base1 remains unchanged1 and accordingly our two polar furrows1 which were formerly parallel1 are now at right angles to one another4 ,instead of a single plane quadrilateral partition1 we have now two triangular ones1 meeting in the middle of the system by their apices1 and lying in planes at right angles to one another 7,fig4 #ace1 #e-g7chc4 ,two such polar furrows1 equal in length and arranged in a cross1 have again been frequently described by the embryologists4 ,robert himself found this condition in ,trochus1 as an occasional or exceptional occurrence3 it has been described as normal in ,asterina by ,ludwig1 in ,branchipus by ,spangenberg1 and in ,podocoryne and ,hydractinia by ,bunting4 ,it is evident that it represents a state of unstable equilibrium1 only to be maintained under certain conditions of restraint within the system4 ,so1 by slight and delicate modifications in the relative size of the cells1 we may pass through all the possible arrangements of the median partition1 and of the 8furrows0 which correspond to its upper and lower edges2 and every one of these arrangements has been frequently observed in the four-celled stage of various embryos4 ,as the phases pass one into the other1 they are accompanied by changes in the curvature of the partition1 which in like manner correspond precisely to phenomena which the embryologists have witnessed and described4 ,and all these configurations belong to that large class of phenomena whose distribution among embryos1 or among organisms in general1 bears no relation to the boundaries of zoological classification2 through molluscs1 worms1 #cdc coelenterates1 vertebrates and what not1 we meet with now one and now another1 in a medley which defies classification4 ,they are not 8vital phenomena10 or 8functions0 of the organism1 or special characteristics of this or that organism1 but purely physical phenomena4 ,the kindred but more complicated phenomena which correspond to the polar furrow when a larger number of cells than four are associated together1 we shall deal with in the next chapter4 ,having shewn that the capillary phenomena are patent and unmistakable during the earlier stages of embryonic development1 but soon become more obscure and incapable of experimental reproduction in the later stages1 when the cells have increased in number1 various writers including ,robert himself have been inclined to argue that the physical phenomena die away1 and are overpowered and cancelled by agencies of a very different order4 ,here we pass into a region where direct observation and experiment are not at hand to guide us1 and where a man's trend of thought1 and way of judging the whole evidence in the case1 must shape his philosophy4 ,we must remember that1 even in a froth of soap-bubbles1 we can apply an exact analysis only to the simplest cases and conditions of the phenomenon2 we cannot describe1 but can only imagine1 the forces which in such a froth control the respective sizes1 positions and curvatures of the innumerable bubbles and films of which it consists2 but our knowledge is enough to leave us assured that what we have learned by investigation of the simplest cases includes the principles which determine the most complex4 ,in the case of the growing embryo we know from the beginning that surface tension is only one of the physical forces at work2 and that other forces1 including those displayed within the interior of each living cell1 play their part in the determination of the system4 ,but we have no evidence whatsoever that at this point1 or that point1 or at any1 the dominion of the physical forces over the material system gives place to a new condition where agencies at present unknown to the physicist impose themselves on the living matter1 and become responsible for the conformation of its material fabric4 ,before we leave for the present the subject of the segmenting #cdd egg1 we must take brief note of two associated problems3 viz4 7#a7 the formation and enlargement of the segmentation cavity1 or central interspace around which the cells tend to group themselves in a single layer1 and 7#b7 the formation of the gastrula1 that is to say 7in a typical case7 the conversion 8by invagination10 of the one-layered ball into a two-layered cup4 ,neither problem is free from difficulty1 and all we can do meanwhile is to state them in general terms1 introducing some more or less plausible assumptions4 ,the former problem is comparatively easy1 as regards the tendency of a segmentation cavity to enlarge1 when once it has been established4 ,we may then assume that subdivision of the cells is due to the appearance of a new-formed septum within each cell1 that this septum has a tendency to shrink under surface tension1 and that these changes will be accompanied on the whole by a diminution of surface energy in the system4 ,this being so1 it may be shewn that the volume of the divided cells must be less than it was prior to division1 or in other words that part of their contents must exude during the process of segmentation#chd4 ,accordingly1 the case where the segmentation cavity enlarges and the embryo developes into a hollow blastosphere may1 under the circumstances1 be simply described as the case where that outflow or exudation from the cells of the blastoderm is directed on the whole inwards4 ,the physical forces involved in the invagination of the cell-layer to form the gastrula have been repeatedly discussed#che1 but the true explanation seems as yet to be by no means clear4 ,the case1 however1 is probably not a very difficult one1 provided that we may assume a difference of osmotic pressure at the two poles of the blastosphere1 that is to say between the cells which are being differentiated into outer and inner1 into epiblast and hypoblast4 ,it is plain that a blastosphere1 or hollow vesicle bounded by a layer of vesicles1 is under very different physical conditions from a single1 simple vesicle or bubble4 ,the blastosphere has no effective surface tension of its own1 such as to exert pressure on #cde its contents or bring the whole into a spherical form2 nor will local variations of surface energy be directly capable of affecting the form of the system4 ,but if the substance of our blastosphere be sufficiently viscous1 then osmotic forces may set up currents which1 reacting on the external fluid pressure1 may easily cause modifications of shape2 and the particular case of invagination itself will not be difficult to account for on this assumption of non-uniform exudation and imbibition4 #cdf ,,chapter ,,viii ,,the ,,forms ,,of ,,tissues ,,or ,,cell-,,aggregates 7continued7 ,the problems which we have been considering1 and especially that of the bee's cell1 belong to a class of 8isoperimetrical0 problems1 which deal with figures whose surface is a minimum for a definite content or volume4 ,such problems soon become difficult1 but we may find many easy examples which lead us towards the explanation of biological phenomena2 and the particular subject which we shall find most easy of approach is that of the division1 in definite proportions1 of some definite portion of space1 by a partition-wall of minimal area4 ,the theoretical principles so arrived at we shall then attempt to apply1 after the manner of ,berthold and ,errera1 to the actual biological phenomena of cell-division4 ,this investigation we may approach in two ways3 by considering1 namely1 the partitioning off from some given space or area of one-half 7or some other fraction7 of its content2 or again1 by dealing simultaneously with the partitions necessary for the breaking up of a given space into a definite number of compartments4 ,if we take1 to begin with1 the simple case of a cubical cell1 it is obvious that1 to divide it into two halves1 the smallest possible partition-wall is one which runs parallel to1 and midway between1 two of its opposite sides4 ,if we call a the length of one of the edges of the cube1 then a#b is the area1 alike of one of its sides1 and of the partition which we have interposed parallel1 or normal1 thereto4 ,but if we now consider the bisected cube1 and wish to divide the one-half of it again1 it is obvious that another partition parallel to the first1 so far from being the smallest possible1 is precisely twice the size of a cross-partition perpendicular to it2 #cdg for the area of this new partition is a a_/#b4 ,and again1 for a third bisection1 our next partition must be perpendicular to the other two1 and it is obviously a little square1 with an area of 7a7#b 7a#b74 ,from this we may draw the simple rule that1 for a rectangular body or parallelopiped to be divided equally by means of a partition of minimal area1 7#a7 the partition must cut across the longest axis of the figure2 and 7#b7 in the event of successive bisections1 each partition must run at right angles to its immediate predecessor4 ,illustration3 ,fig4 #acf4 7,after ,berthold47 ,we have already spoken of 8,sachs's ,rules10 which are an empirical statement of the method of cell-division in plant-tissues2 and we may now set them forth in full4 7#a7 ,the cell typically tends to divide into two co-equal parts4 7#b7 ,each new plane of division tends to intersect at right angles the preceding plane of division4 ,the first of these rules is a statement of physiological fact1 not without its exceptions1 but so generally true that it will justify us in limiting our enquiry1 for the most part1 to cases of equal subdivision4 ,that it is by no means universally true for cells generally is shewn1 for instance1 by such well-known cases #cdh as the unequal segmentation of the frog's egg4 ,it is true when the dividing cell is homogeneous1 and under the influence of symmetrical forces2 but it ceases to be true when the field is no longer dynamically symmetrical1 for instance1 when the parts differ in surface tension or internal pressure4 ,this latter condition1 of asymmetry of field1 is frequent in segmenting eggs#chf1 and is then equivalent to the principle upon which ,balfour laid stress1 as leading to 8unequal0 or to 8partial0 segmentation of the egg1--viz4 the unequal or asymmetrical distribution of protoplasm and of food-yolk4 ,the second rule1 which also has its exceptions1 is true in a large number of cases2 and it owes its validity1 as we may judge from the illustration of the repeatedly bisected cube1 solely to the guiding principle of minimal areas4 ,it is in short subordinate to1 and covers certain cases included under1 a much more important and fundamental rule1 due not to ,sachs but to ,errera2 that 7#c7 the incipient partition-wall of a dividing cell tends to be such that its area is the least possible by which the given space-content can be enclosed4 ,let us return to the case of our cube1 and let us suppose that1 instead of bisecting it1 we desire to shut off some small portion only of its volume4 ,it is found in the course of experiments upon soap-films1 that if we try to bring a partition-film too near to one side of a cubical 7or rectangular7 space1 it becomes unstable2 and is easily shifted to a totally new position1 in which it constitutes a curved cylindrical wall1 cutting off one corner of the cube4 ,it meets the sides of the cube at right angles 7for reasons which we have already considered72 and1 as we may see from the symmetry #cdi of the case1 it constitutes precisely one-quarter of a cylinder4 ,our plane transverse partition1 wherever it was placed1 had always the same area1 viz4 a#b2 and it is obvious that a cylindrical wall1 if it cut off a small corner1 may be much less than this4 ,we want1 accordingly1 to determine what is the particular volume which might be partitioned off with equal economy of wall-space in one way as the other1 that is to say1 what area of cylindrical wall would be neither more nor less than the area a#b4 ,the calculation is very easy4 ,the surface-area of a cylinder of length a is #br a1 and that of our quarter-cylinder is1 therefore1 a r_/#b2 and this being1 by hypothesis1 a#b1 we have a r_/#b1 or r #b;a_/4 ,the volume of a cylinder1 of length a1 is ar#b1 and that of our quarter-cylinder is 7a r#b7_/d1 which 7by substituting the value of r7 is equal to 7a#c7_/4 ,now precisely this same volume is1 obviously1 shut off by a transverse partition of area a#b1 if the third side of the rectangular space be equal to a_/4 ,and this fraction1 if we take a #a1 is equal to #jcah4441 or rather less than one-third4 ,and1 as we have just seen1 the radius1 or side1 of the corresponding quarter-cylinder will be twice that fraction1 or equal to #fcf times the side of the cubical cell4 ,illustration3 ,fig4 #acg4 ,if then1 in the process of division of a cubical cell1 it so divide that the two portions be not equal in volume but that one portion by anything less than about three-tenths of the whole1 or three-sevenths of the other portion1 there will be a tendency for the cell to divide1 not by means of a plane transverse partition1 but by means of a curved1 cylindrical wall cutting off one corner of the original cell2 and the part so cut off will be one-quarter of a cylinder4 ,by a similar calculation we can shew that a spherical wall1 cutting off one solid angle of the cube1 and constituting an octant of a sphere1 would likewise be of less area than a plane partition as soon as the volume to be enclosed was not greater than about #cej one-quarter of the original cell#chg4 ,but while both the cylindrical wall and the spherical wall would be of less area than the plane transverse partition after that limit 7of one-quarter volume7 was passed1 the cylindrical would still be the better of the two up to a further limit4 ,it is only when the volume to be partitioned off #cea is no greater than about #jae1 or somewhere about one-seventh1 of the whole1 that the spherical cell-wall in an angle of the cubical cell1 that is to say the octant of a sphere1 is definitely of less area than the quarter-cylinder4 ,in the accompanying diagram 7,fig4 #ach7 the relative areas of the three partitions are shewn for all fractions1 less than one-half1 of the divided cell4 ,illustration3 ,fig4 #ach4 ,in this figure1 we see that the plane transverse partition1 whatever fraction of the cube it cut off1 is always of the same dimensions1 that is to say is always equal to a#b1 or #a4 ,if one-half of the cube have to be cut off1 this plane transverse partition is much the best1 for we see by the diagram that a cylindrical partition cutting off an equal volume would have an area about #be1 and a spherical partition would have an area about #ej greater4 ,the point ,a in the diagram corresponds to the point where the cylindrical partition would begin to have an advantage over the plane1 that is to say 7as we have seen7 when the fraction to be cut off is about one-third1 or #cah of the whole4 ,in like manner1 at ,b the spherical octant begins to have an advantage over the plane2 and it is not till we reach the point ,c that the spherical octant becomes of less area than the quarter-cylinder4 ,illustration3 ,fig4 #aci4 ,the case we have dealt with is of little practical importance to the biologist1 because the cases in which a cubical1 or rectangular1 cell divides unequally1 and unsymmetrically1 are apparently few2 but we can find1 as ,berthold pointed out1 a few examples1 for instance in the hairs within the reproductive 8conceptacles0 of certain ,fuci 7,sphacelaria1 etc41 ,fig4 #aci71 or in the 8paraphyses0 of mosses 7,fig4 #adb74 ,but it is of great theoretical importance3 as serving to introduce us to a large class of cases1 in which the shape and the relative dimensions of the original cavity lead1 according to the principle of minimal areas1 to cell-division in very definite and sometimes unexpected ways4 ,it is not easy1 nor indeed possible1 to give a generalised account of these cases1 for the limiting conditions are somewhat complex1 and the mathematical treatment soon becomes difficult4 ,but it is easy to comprehend a few simple cases1 which of themselves will carry us a good long way2 and which will go far to convince the student that1 in other cases #ceb which we cannot fully master1 the same guiding principle is at the root of the matter4 ,the bisection of a solid 7or the subdivision of its volume in other definite proportions7 soon leads us into a geometry which1 if not necessarily difficult1 is apt to be unfamiliar2 but in such problems we can go a long way1 and often far enough for our particular purpose1 if we merely consider the plane geometry of a side or section of our figure4 ,for instance1 in the case of the cube which we have been just considering1 and in the case of the plane and cylindrical partitions by which it has been divided1 it is obvious that1 since these two partitions extend symmetrically from top to bottom of our cube1 that we need only consider 7so far as they are concerned7 the manner in which they subdivide the base of the cube4 ,the whole problem of the solid1 up to a certain point1 is contained in our plane diagram of ,fig4 #ach4 ,and when our particular solid is a solid of revolution1 then it is obvious that a study of its plane of symmetry 7that is to say any plane passing through its axis of rotation7 gives us the solution of the whole problem4 ,the right cone is a case in point1 for here the investigation of its modes of symmetrical subdivision is completely met by an examination of the isosceles triangle which constitutes its plane of symmetry4 ,the bisection of an isosceles triangle by a line which shall be the shortest possible is a very easy problem4 ,let ,,abc be such a triangle of which ,a is the apex2 it may be shewn that1 for its shortest line of bisection1 we are limited to three cases3 viz4 to a vertical line ,,ad1 bisecting the angle at ,a and the side ,,bc2 to a transverse line parallel to the base ,,bc2 or to an oblique line parallel to ,,ab or to ,,ac4 ,the respective magnitudes1 or lengths1 of these partition lines follow at once from the magnitudes of the angles of our triangle4 ,for we know1 to begin with1 since the areas of similar figures vary as the squares of their linear dimensions1 that1 in order to bisect the area1 a line parallel to one side of our triangle must always have a length equal to #a_/b of that side4 ,if then1 we take our base1 ,,bc1 in all cases of a length #b1 the transverse partition drawn parallel to it will always have a length equal to #b_/b1 or #b4 ,the vertical #cec partition1 ,,ad1 since ,,bd #a1 will always equal tan 7 being the angle ,,abc74 ,and the oblique partition1 ,,gh1 being equal to ,,ab_/#b #a_/7b cos 74 ,if then we call our vertical1 transverse ,illustration3 ,fig4 #adj4 and oblique partitions1 ,v1 ,t1 and ,o1 we have ,v tan 2 ,t #b2 and ,o #a_/7b cos 71 or ,v 3 ,t 3 ,o tan _/#b 3 #a 3 #a_/7b cos 74 ,and1 working out these equations for various values of 1 we very soon see that the vertical partition 7,v7 is the least of the three until #de1 at which limit ,v and ,o are each equal to #a_/b #gjg2 and that again1 when #fj1 ,o and ,t are each #a1 after which ,t 7whose value always #a7 is the shortest of the three partitions4 ,and1 as we have seen1 these results are at once applicable1 not only to the case of the plane triangle1 but also to that of the conical cell4 ,illustration3 ,fig4 #ada4 ,in like manner1 if we have a spheroidal body1 less than a hemisphere1 such for instance as a low1 watch-glass shaped cell 7,fig4 #ada1 a71 it is obvious that the smallest possible partition by which we can divide it into two equal halves #ced is 7as in our flattened disc7 a median vertical one4 ,and likewise1 the hemisphere itself can be bisected by no smaller partition meeting the walls at right angles than that median one which divides it into two similar quadrants of a sphere4 ,but if we produce our hemisphere into a more elevated1 conical body1 or into a cylinder with spherical cap1 it is obvious that there comes a point where a transverse1 horizontal partition will bisect the figure with less area of partition-wall than a median vertical one 7c74 ,and furthermore1 there will be an intermediate region1 a region where height and base have their relative dimensions nearly equal 7as in b71 where an oblique partition will be better than either the vertical or the transverse1 though here the analogy of our triangle does not suffice to give us the precise limiting values4 ,we need not examine these limitations in detail1 but we must look at the curvatures which accompany the several conditions4 ,we have seen that a film tends to set itself at equal angles to the surface which it meets1 and therefore1 when that surface is a solid1 to meet it 7or its tangent if it be a curved surface7 at right angles4 ,our vertical partition is1 therefore1 everywhere normal to the original cell-walls1 and constitutes a plane surface4 ,but in the taller1 conical cell with transverse partition1 the latter still meets the opposite sides of the cell at right angles1 and it follows that it must itself be curved2 moreover1 since the tension1 and therefore the curvature1 of the partition is everywhere uniform1 it follows that its curved surface must be a portion of a sphere1 concave towards the apex of the original1 now divided1 cell4 ,in the intermediate case1 where we have an oblique partition1 meeting both the base and the curved sides of the mother-cell1 the contact must still be everywhere at right angles3 provided we continue to suppose that the walls of the mother-cell 7like those of our diagrammatic cube7 have become practically rigid before the partition appears1 and are therefore not affected and deformed by the tension of the latter4 ,in such a case1 and especially when the cell is elliptical in cross-section1 or is still more complicated in form1 it is evident that the partition1 in adapting itself to circumstances and in maintaining itself as a surface of minimal area subject to all the conditions of the case1 may have to assume a complex curvature4 #cee ,illustration3 ,fig4 #adb4 ,s-shaped partitions3 ,a1 from ,taonia atomaria 7after ,reinke72 ,b1 from paraphyses of ,fucus2 ,c1 from rhizoids of ,moss2 ,d1 from paraphyses of ,polytrichum4 ,while in very many cases the partitions 7like the walls of the original cell7 will be either plane or spherical1 a more complex curvature will be assumed under a variety of conditions4 ,it will be apt to occur1 for instance1 when the mother-cell is irregular in shape1 and one particular case of such asymmetry will be that in which 7as in ,fig4 #adc7 the cell has begun to branch1 or give off a diverticulum1 before division takes place4 ,a very complicated case of a different kind1 though not without its analogies to the cases we are considering1 will occur in the partitions of minimal area which subdivide the spiral tube of a nautilus1 as we shall presently see4 ,and again1 whenever we have a marked internal asymmetry of the cell1 leading to irregular and anomalous modes of division1 in which the cell is not necessarily divided into two equal halves and in which the partition-wall may assume an oblique position1 then apparently anomalous curvatures will tend to make their appearance#chh4 ,suppose that a more or less oblong cell have a tendency to divide by means of an oblique partition 7as may happen through various causes or conditions of asymmetry71 such a partition will still have a tendency to set itself at right angles to the rigid walls #cef of the mother-cell3 and it will at once follow that our oblique partition1 throughout its whole extent1 will assume the form of a complex1 saddle-shaped or anticlastic surface4 ,illustration3 ,fig4 #adc4 ,diagrammatic explanation of ,s-shaped partition4 ,many such cases of partitions with complex or double curvature exist1 but they are not always easy of recognition1 nor is the particular case where they appear in a terminal cell a common one4 ,we may see them1 for instance1 in the roots 7or rhizoids7 of ,mosses1 especially at the point of development of a new rootlet 7,fig4 #adb1 ,c72 and again among ,mosses1 in the 8paraphyses0 of the male prothalli 7e4g4 in ,polytrichum71 we find more or less similar partitions 7,d74 ,they are frequent also among many ,fuci1 as in the hairs or paraphyses of ,fucus itself 7,b74 ,in ,taonia atomaria1 as figured in ,reinke's memoir on the ,dictyotaceae of the ,gulf of ,naples#chi1 we see1 in like manner1 oblique partitions1 which on more careful examination are seen to be curves of double curvature 7,fig4 #adb1 ,a74 ,the physical cause and origin of these ,s-shaped partitions is somewhat obscure1 but we may attempt a tentative explanation4 ,when we assert a tendency for the cell to divide transversely to its long axis1 we are not only stating empirically that the partition tends to appear in a small1 rather than a large cross-section of the cell3 but we are also implicitly ascribing to the cell a longitudinal polarity 7,fig4 #adc1 ,a71 and implicitly asserting that it tends to #ceg divide 7just as the segmenting egg does71 by a partition transverse to its polar axis4 ,such a polarity may conceivably be due to a chemical asymmetry1 or anisotropy1 such as we have learned of 7from ,professor ,macallum's experiments7 in our chapter on ,adsorption4 ,now if the chemical concentration1 on which this anisotropy or polarity 7by hypothesis7 depends1 be unsymmetrical1 one of its poles being as it were deflected to one side1 where a little branch or bud is being 7or about to be7 given off1--all in precise accordance with the adsorption phenomena described on p4 #bhi1--then our 8polar axis0 would necessarily be a curved axis1 and the partition1 being constrained 7again ex hypothesi7 to arise transversely to the polar axis1 would lie obliquely to the apparent axis of the cell 7,fig4 #adc1 ,b1 ,c74 ,and if the oblique partition be so situated that it has to meet the opposite walls 7as in ,c71 then1 in order to do so symmetrically 7i4e4 either perpendicularly1 as when the cell-wall is already solidified1 or at least at equal angles on either side71 it is evident that the partition1 in its course from one side of the cell to the other1 must necessarily assume a more or less ,s-shaped curvature 7,fig4 #adc1 ,d74 ,as a matter of fact1 while we have abundant simple illustrations of the principles which we have now begun to study1 apparent exceptions to this simplicity1 due to an asymmetry of the cell itself1 or of the system of which the single cell is but a part1 are by no means rare4 ,for example1 we know that in cambium-cells1 division frequently takes place parallel to the long axis of the cell1 when a partition of much less area would suffice if it were set cross-ways3 and it is only when a considerable disproportion has been set up between the length and breadth of the cell1 that the balance is in part redressed by the appearance of a transverse partition4 ,it was owing to such exceptions that ,berthold was led to qualify and even to depreciate the importance of the law of minimal areas as a factor in cell-division1 after he himself had done so much to demonstrate and elucidate it#cij4 ,he was deeply and rightly impressed by the fact that other forces besides surface #ceh tension1 both external and internal to the cell1 play their part in the determination of its partitions1 and that the answer to our problem is not to be given in a word4 ,how fundamentally important it is1 however1 in spite of all conflicting tendencies and apparent exceptions1 we shall see better and better as we proceed4 ,but let us leave the exceptions and return to a consideration of the simpler and more general phenomena4 ,and in so doing1 let us leave the case of the cubical1 quadrangular or cylindrical cell1 and examine the case of a spherical cell and of its successive divisions1 or the still simpler case of a circular1 discoidal cell4 ,when we attempt to investigate mathematically the position and form of a partition of minimal area1 it is plain that we shall be dealing with comparatively simple cases wherever even one dimension of the cell is much less than the other two4 ,where two dimensions are small compared with the third1 as in a thin cylindrical filament like that of ,spirogyra1 we have the problem at its simplest2 for it is at once obvious1 then1 that the partition must lie transversely to the long axis of the thread4 ,but even where one dimension only is relatively small1 as for instance in a flattened plate1 our problem is so far simplified that we see at once that the partition cannot be parallel to the extended plane1 but must cut the cell1 somehow1 at right angles to that plane4 ,in short1 the problem of dividing a much flattened solid becomes identical with that of dividing a simple surface of the same form4 ,there are a number of small ,algae1 growing in the form of small flattened discs1 consisting 7for a time at any rate7 of but a single layer of cells1 which1 as ,berthold shewed1 exemplify this comparatively simple problem2 and we shall find presently that it is also admirably illustrated in the cell-divisions which occur in the egg of a frog or a sea-urchin1 when the egg for the sake of experiment is flattened out under artificial pressure4 ,illustration3 ,fig4 #add4 ,development of ,erythrotrichia4 7,after ,berthold47 ,illustration3 ,fig4 #ade4 ,fig4 #add 7taken from ,berthold's ,monograph of the ,naples ,bangiaciae7 represents younger and older discs of the little alga ,erythrotrichia discigera2 and it will be seen that1 in all stages save the first1 we have an arrangement of cell-partitions which looks somewhat complex1 but into which we must attempt to throw some light and order4 ,starting with the original single1 and flattened1 #cei cell1 we have no difficulty with the first two cell-divisions2 for we know that no bisecting partitions can possibly be shorter than the two diameters1 which divide the cell into halves and into quarters4 ,we have only to remember that1 for the sum total of partitions to be a minimum1 three only must meet in a point2 and therefore1 the four quadrantal walls must shift a little1 producing the usual little median partition1 or cross-furrow1 instead of one common1 central point of junction4 ,this little intermediate wall1 however1 will be very small1 and to all intents and purposes we may deal with the case as though we had now to do with four equal cells1 each one of them a perfect quadrant4 ,and so our problem is1 to find the shortest line which shall divide the quadrant of a circle into two halves of equal area4 ,a radial partition 7,fig4 #ade1 ,a71 starting from the apex of the quadrant1 is at once excluded1 for a reason similar to that just referred to2 our choice must lie therefore between two modes of division such as are illustrated in ,fig4 #ade1 where the partition is either 7as in ,b7 #cfj concentric with the outer border of the cell1 or else 7as in ,c7 cuts that outer border2 in other words1 our partition may 7,b7 cut both radial walls1 or 7,c7 may cut one radial wall and the periphery4 ,these are the two methods of division which ,sachs called1 respectively1 7,b7 periclinal1 and 7,c7 anticlinal#cia4 ,we may either treat the walls of the dividing quadrant as already solidified1 or at least as having a tension compared with which that of the incipient partition film is inconsiderable4 ,in either case the partition must meet the cell-wall1 on either side1 at right angles1 and 7its own tension and curvature being everywhere uniform7 it must take the form of a circular arc4 ,now we find that a flattened cell which is approximately a quadrant of a circle invariably divides after the manner of ,fig4 #ade1 ,c1 that is to say1 by an approximately circular1 anticlinal wall1 such as we now recognise in the eight-celled stage of ,erythrotrichia 7,fig4 #add72 let us then consider that ,nature has solved our problem for us1 and let us work out the actual geometric conditions4 ,let the quadrant ,,oab 7in ,fig4 #adf7 be divided into two parts of equal area1 by the circular arc ,,mp4 ,it is required to determine 7#a7 the position of ,p upon the arc of the quadrant1 that is to say the angle ,,bop2 7#b7 the position of the point ,m on the side ,,oa2 and 7#c7 the length of the arc ,,mp in terms of a radius of the quadrant4 7#a7 ,draw ,,op2 also ,,pc a tangent1 meeting ,,oa in ,c2 and ,,pn1 perpendicular to ,,oa4 ,let us call a a radius2 and the angle at ,c1 which is obviously equal to ,,opn1 or ,,pob4 ,then ,,cp a cot 2 ,,pn a cos 2 ,,nc ,,cp cos a 7cos#b 7_/7sin 74 ,the area of the portion ,,pmn ,c7,p#b7 ,,pn ,,nc a#b7;cot#b 7 a7cos 7 a7cos#b 7_/7sin 7 a#b7;cot#b 7cos#c 7_/7sin 774 #cfa ,and the area of the portion ,,pna a#b7_/b 7 ,,on ,,np a#b7_/b 7 a7sin 7 a7cos 7 a#b7_/b sin cos 74 ,therefore the area of the whole portion ,,pma 7a#b7_/b 7_/#b cot#b 7cos#c 7_/sin sin cos 7 7a#b7_/b 7_/#b cot#b cot 71 and also1 by hypothesis1 area of the quadrant1 7a#b7_/h4 ,illustration3 ,fig4 #adf4 ,hence is defined by the equation a#b_/b 7_/#b cot#b cot 7 7a#b7_/h1 or _/#d cot#b cot #j4 ,we may solve this equation by constructing a table 7of which the following is a small portion7 for various values of 4 _/#d cot cot#b x #cd #cd #ghed #fjcc #adead #abgji #jjaf #ce #ghed #fjcf #adeje #abgjj #jjac #cf #ghed #fjci #addif #abfij #jjji #cg #ghed #fjdb #addhg #abfhj #jjje #ch #ghed #fjde #addgh #abfga #jjjb #ci #ghed #fjdh #addfi #abffa #jjjb #dj #ghed #fjea #addfj #abfeb #jjje #cfb ,we see accordingly that the equation is solved 7as accurately as need be7 when is an angle somewhat over #cd #ch1 or say #cd #ch4 ,that is to say1 a quadrant of a circle is bisected by a circular arc cutting the side and the periphery of the quadrant at right angles1 when the arc is such as to include 7#ij #cd #ch71 i4e4 #ee #bb of the quadrantal arc4 ,this determination of ours is practically identical with that which ,berthold arrived at by a rough and ready method1 without the use of mathematics4 ,he simply tried various ways of dividing a quadrant of paper by means of a circular arc1 and went on doing so till he got the weights of his two pieces of paper approximately equal4 ,the angle1 as he thus determined it1 was #cdf1 or say #cd #cf4 7#b7 ,the position of ,m on the side of the quadrant ,,oa is given by the equation ,,om a cosec a cot 2 the value of which expression1 for the angle which we have just discovered1 is #cjbh4 ,that is to say1 the radius 7or side7 of the quadrant will be divided by the new partition into two parts1 in the proportions of nearly three to seven4 7#c7 ,the length of the arc ,,mp is equal to a cot 2 and the value of this for the given angle is #hgea4 ,this is as much as to say that the curved partition-wall which we are considering is shorter than a radial partition in the proportion of #h to #aj1 or seven-eights almost exactly4 ,but we must also compare the length of this curved 8anticlinal0 partition-wall 7,,mp7 with that of the concentric1 or periclinal1 one 7,,rs1 ,fig4 #adg7 by which the quadrant might also be bisected4 ,the length of this partition is obviously equal to the arc of the quadrant 7i4e4 the peripheral wall of the cell7 divided by #b2 or1 in terms of the radius1 _/#bb #aaaa4 ,so that1 not only is the anticlinal partition 7such as we actually find in nature7 notably the best1 but the periclinal one1 when it comes to dividing an entire quadrant1 is very considerably larger even than a radial partition4 ,illustration3 ,fig4 #adg4 ,the two cells into which our original quadrant is now divided1 while they are equal in volume1 are of very different shapes2 the #cfc one is a triangle 7,,map7 with two sides formed of circular arcs1 and the other is a four-sided figure 7,,mobp71 which we may call approximately oblong4 ,we cannot say as yet how the triangular portion ought to divide2 but it is obvious that the least possible partition-wall which shall bisect the other must run across the long axis of the oblong1 that is to say periclinally4 ,this1 also1 is precisely what tends actually to take place4 ,in the following diagrams 7,fig4 #adh7 of a frog's egg dividing under pressure1 that is to say when reduced to the form of a flattened plate1 we see1 firstly1 the division into four quadrants 7by the partitions #a1 #b72 secondly1 the division of each quadrant by means of an anticlinal circular arc 7#c1 #c71 cutting the peripheral wall of the quadrant approximately in the proportions of three to seven2 and thirdly1 we see that of the eight cells 7four triangular and four oblong7 into which the whole egg is now divided1 the four which we have called oblong now proceed to divide by partitions transverse to their long axes1 or roughly parallel to the periphery of the egg4 ,illustration3 ,fig4 #adh4 ,segmentation of frog's egg1 under artificial compression4 7,after ,roux47 ,the question how the other1 or triangular1 portion of the divided quadrant will next divide leads us to another well-defined problem1 which is only a slight extension1 making allowance for the circular arcs1 of that elementary problem of the triangle we have already considered4 ,we know now that an entire quadrant must divide 7so that its bisecting wall shall have the least possible area7 by means of an anticlinal partition1 but how about any smaller sectors of circles8 ,it is obvious in the case of a small prismatic #cfd sector1 such as that shewn in ,fig4 #adi1 that a periclinal partition is the smallest by which we can possibly bisect the cell2 we want1 accordingly1 to know the limits below which the periclinal partition is always the best1 and above which the anticlinal arc1 as in the case of the whole quadrant1 has the advantage in regard to smallness of surface area4 ,this may be easily determined2 for the preceding investigation is a perfectly general one1 and the results hold good for sectors of any other arc1 as well as for the quadrant1 or arc of #ij4 ,that is to say1 the length of the partition-wall ,,mp is always determined by the angle 1 according to our equation ,,mp a cot 2 and the angle has a definite relation to 1 the angle of arc4 ,illustration3 ,fig4 #adi4 ,moreover1 in the case of the periclinal boundary1 ,,rs 7,fig4 #adg7 7or ab1 ,fig4 #adi71 we know that1 if it bisect the cell1 ,,rs a _/#b4 ,accordingly1 the arc ,,rs will be just equal to the arc ,,mp when cot _/#b4 ,when cot _/#b or ,,mp ,,rs1 then division will take place as in ,,rs4 ,when cot _/#b1 or ,,mp ,,rs1 then division will take place as in ,,mp4 ,in the accompanying diagram 7,fig4 #aej71 ,i have plotted the various magnitudes with which we are concerned1 in order to exhibit the several limiting values4 ,here we see1 in the first place1 the curve marked 1 which shews on the 7left-hand7 vertical scale the various possible magnitudes of that angle 7viz4 the angle #cfe of arc of the whole sector which we wish to divide71 and on the horizontal scale the corresponding values of 1 or the angle which ,illustration3 ,fig4 #aej4 determines the point on the periphery where it is cut by the partition-wall1 ,,mp4 ,two limiting cases are to be noticed here3 7#a7 at #ij 7point ,a in diagram71 because we are at present only #cff dealing with arcs no greater than a quadrant2 and 7#b71 the point 7,b7 where the angle comes to equal the angle 1 for after that point the construction becomes impossible1 since an anticlinal bisecting partition-wall would be partly outside the cell4 ,the only partition which1 after the point1 can possibly exist1 is a periclinal one4 ,this point1 as our diagram shews us1 occurs when the angles 7 and 7 are each rather under #eb4 ,next ,i have plotted1 on the same diagram1 and in relation to the same scales of angles1 the corresponding lengths of the two partitions1 viz4 ,,rs and ,,mp1 their lengths being expressed 7on the right-hand side of the diagram7 in relation to the radius of the circle 7a71 that is to say the side wall1 ,,oa1 of our cell4 ,the limiting values here are 7#a71 ,c1 ,c1 where the angle of arc is #ij1 and where1 as we have already seen1 the two partition-walls have the relative magnitudes of ,,mp 3 ,,rs #jhge 3 #aaaa2 7#b7 the point ,d1 where ,,rs equals unity1 that is to say where the periclinal partition has the same length as a radial one2 this occurs when is rather under #hb 7cf4 the points ,d1 ,d72 7#c7 the point ,e1 where ,,rs and ,,mp intersect2 that is to say the point at which the two partitions1 periclinal and anticlinal1 are of the same magnitude2 this is the case1 according to our diagram1 when the angle of arc is just over #fb4 ,we see from this1 then1 that what we have called an anticlinal partition1 as ,,mp1 is only likely to occur in a triangular or prismatic cell whose angle of arc lies between #ij and #fb4 ,in all narrower or more tapering cells1 the periclinal partition will be of less area1 and will therefore be more and more likely to occur4 ,the case 7,f7 where the angle is just #fj is of some interest4 ,here1 owing to the curvature of the peripheral border1 and the consequent fact that the peripheral angles are somewhat greater than the apical angle 1 the periclinal partition has a very slight and almost imperceptible advantage over the anticlinal1 the relative proportions being about as ,,mp 3 ,,rs #jgc 3 #jgb4 ,but if the equilateral triangle be a plane spherical triangle1 i4e4 a plane triangle bounded by circular arcs1 then we see that there is no longer any distinction at all between our two partitions2 ,,mp and ,,rs are now identical4 ,on the same diagram1 ,i have inserted the curve for values of #cfg cosec cot ,,om1 that is to say the distances from the centre1 along the side of the cell1 of the starting-point 7,m7 of the anticlinal partition4 ,the point ,c represents its position in the case of a quadrant1 and shews it to be 7as we have already said7 about #c_/aj of the length of the radius from the centre4 ,if1 on the other hand1 our cell be an equilateral triangle1 then we have to read off the point on this curve corresponding to #fj1 and we find it at the point ,f 7vertically under ,f71 which tells us that the partition now starts #de_/aj1 or nearly halfway1 along the radial wall4 ,the foregoing considerations carry us a long way in our investigations of many of the simpler forms of cell-division4 ,strictly speaking they are limited to the case of flattened cells1 in which we can treat the problem as though we were simply partitioning a plane surface4 ,but it is obvious that1 though they do not teach us the whole conformation of the partition which divides a more complicated solid into two halves1 yet they do1 even in such a case1 enlighten us so far1 that they tell us the appearance presented in one plane of the actual solid4 ,and as this is all that we see in a microscopic section1 it follows that the results we have arrived at will greatly help us in the interpretation of microscopic appearances1 even in comparatively complex cases of cell-division4 ,illustration3 ,fig4 #aea4 ,let us now return to our quadrant cell 7,,oapb71 which we have found to be divided into a triangular and a quadrilateral portion1 as in ,fig4 #adg or ,fig4 #aea2 and let us now suppose the whole system to grow1 in a uniform fashion1 as a prelude to further subdivision4 ,the whole quadrant1 growing uniformly 7or with equal radial increments71 will still remain a quadrant1 and it is obvious1 therefore1 that for every new increment of size1 more will be added to the margin of its triangular portion than to the #cfh narrower margin of its quadrilateral portion2 and these increments will be in proportion to the angles of arc1 viz4 #ee #bb 3 #cd #ch1 or as #if 3 #fj1 i4e4 as #h 3 #e4 ,and accordingly1 if we may assume 7and the assumption is a very plausible one71 that1 just as the quadrant itself divided into two halves after it got to a certain size1 so each of its two halves will reach the same size before again dividing1 it is obvious that the triangular portion will be doubled in size1 and therefore ready to divide1 a considerable time before the quadrilateral part4 ,to work out the problem in detail would lead us into troublesome mathematics2 but if we simply assume that the increments are proportional to the increasing radii of the circle1 we have the following equations3 ,let us call the triangular cell ,t1 and the quadrilateral1 ,q 7,fig4 #aea72 let the radius1 ,,oa1 of the original quadrantal cell a #a2 and let the increment which is required to add on a portion equal to ,t 7such as ,,pp,a,a7 be called x1 and let that required1 similarly1 for the doubling of ,q be called x4 ,then we see that the area of the original quadrant ,t ,q a#b #ghed;a#b1 while the area of ,t ,q #cibg;a#b4 ,the area of the enlarged sector1 p,,oa1 7a x7#b 7#ee #bb7 #b #dhca7;a x7#b1 and the area ,,opa a#b 7#ee #bb7 #b #dhca;a#b4 ,therefore the area of the added portion1 ,t1 #dhca 77a x7#b a#b74 ,and this1 by hypothesis1 ,t #cibg;a#b4 ,we get1 accordingly1 since a #a1 x#b #bx #cibg_/dhca #haj1 and1 solving1 x #a #aha #acde1 or x #jcde4 ,working out x in the same way1 we arrive at the approximate value1 x #a #aeag4 #cfi ,this is as much as to say that1 supposing each cell tends to divide into two halves when 7and not before7 its original size is doubled1 then1 in our flattened disc1 the triangular cell ,t will tend to divide when the radius of the disc has increased by about a third 7from #a to #acde71 but the quadrilateral cell1 ,q1 will not tend to divide until the linear dimensions of the disc have increased by about a half 7from #a to #aeag74 ,the case here illustrated is of no small general importance4 ,for it shews us that a uniform and symmetrical growth of the organism 7symmetrical1 that is to say1 under the limitations of a plane surface1 or plane section7 by no means involves a uniform or symmetrical growth of the individual cells1 but may1 under certain conditions1 actually lead to inequality among these2 and this inequality may be further emphasised by differences which arise out of it1 in regard to the order of frequency of further subdivision4 ,this phenomenon 7or to be quite candid1 this hypothesis1 which is due to ,berthold7 is entirely independent of any change or variation in individual surface tensions2 and accordingly it is essentially different from the phenomenon of unequal segmentation 7as studied by ,balfour71 to which we have referred on p4 #cdh4 ,in this fashion1 we might go on to consider the manner1 and the order of succession1 in which the subsequent cell-divisions would tend to take place1 as governed by the principle of minimal areas4 ,but the calculations would grow more difficult1 or the results got by simple methods would grow less and less exact4 ,at the same time1 some of these results would be of great interest1 and well worth the trouble of obtaining4 ,for instance1 the precise manner in which our triangular cell1 ,t1 would next divide would be interesting to know1 and a general solution of this problem is certainly troublesome to calculate4 ,but in this particular case we can see that the width of the triangular cell near ,p is so obviously less than that near either of the other two angles1 that a circular arc cutting off that angle is bound to be the shortest possible bisecting line2 and that1 in short1 our triangular cell will tend to subdivide1 just like the original quadrant1 into a triangular and a quadrilateral portion4 ,but the case will be different next time1 because in this new #cgj triangle1 ,,prq1 the least width is near the innermost angle1 that at ,q2 and the bisecting circular arc will therefore be opposite to ,q1 or 7approximately7 parallel to ,,pr4 ,the importance of this fact is at once evident2 for it means to say that there soon comes a time when1 whether by the division of triangles or of quadrilaterals1 we find only quadrilateral cells adjoining the periphery of our circular disc4 ,in the subsequent division of these quadrilaterals1 the partitions will arise transversely to their long axes1 that is to say1 radially 7as ,u1 ,v72 and we shall consequently have a superficial or peripheral layer of quadrilateral cells1 with sides approximately parallel1 that is to say what we are accustomed to call an epidermis4 ,and this epidermis or superficial layer will be in clear contrast with the more irregularly shaped cells1 the products of triangles and quadrilaterals1 which make up the deeper1 underlying layers of tissue4 ,illustration3 ,fig4 #aeb4 ,in following out these theoretic principles and others like to them1 in the actual division of living cells1 we must always bear in mind certain conditions and qualifications4 ,in the first place1 the law of minimal area and the other rules which we have arrived at are not absolute but relative3 they are links1 and very important links1 in a chain of physical causation2 they are always at work1 but their effects may be overridden and concealed by the operation of other forces4 ,secondly1 we must remember that1 in the great majority of cases1 the cell-system which we have in view is constantly increasing in magnitude by active growth2 and by this means the form and also the proportions of the cells are continually liable to alteration1 of which phenomenon we have already had an example4 ,thirdly1 we must carefully remember that1 until our cell-walls become absolutely solid and rigid1 they are always apt to be modified in form owing to the tension of the adjacent #cga walls2 and again1 that so long as our partition films are fluid or semifluid1 their points and lines of contact with one another may shift1 like the shifting outlines of a system of soap-bubbles4 ,this is the physical cause of the movements frequently seen among segmenting cells1 like those to which ,rauber called attention in the segmenting ovum of the frog1 and like those more striking movements or accommodations which give rise to a so-called 8spiral0 type of segmentation4 ,illustration3 ,fig4 #aec4 ,diagram of flattened or discoid cell dividing into octants3 to shew gradual tendency towards a position of equilibrium4 ,bearing in mind1 then1 these considerations1 let us see what our flattened disc is likely to look like1 after a few successive divisions into component cells4 ,in ,fig4 #aec1 a1 we have a diagrammatic representation of our disc1 after it has divided into four quadrants1 and each of these in turn into a triangular and a quadrilateral portion2 but as yet1 this figure scarcely suggests to us anything like the normal look of an aggregate of living cells4 ,but let us go a little further1 still limiting ourselves1 however1 to the consideration of the eight-celled stage4 ,wherever one of our radiating partitions meets the peripheral wall1 there will 7as we know7 be a mutual tension between the three convergent films1 which will tend to set their edges at equal angles to one another1 angles that is to say of #abj4 ,in consequence of this1 the outer wall of each individual cell will 7in this surface view of our disc7 #cgb be an arc of a circle of which we can determine the centre by the method used on p4 #cjg2 and1 furthermore1 the narrower cells1 that is to say the quadrilaterals1 will have this outer border somewhat more curved than their broader neighbours4 ,we arrive1 then1 at the condition shewn in ,fig4 #aec1 b4 ,within the cell1 also1 wherever wall meets wall1 the angle of contact must tend1 in every case1 to be an angle of #abj2 and in no case may more than three films 7as seen in section7 meet in a point 7c72 and this condition1 of the partitions meeting three by three1 and at co-equal angles1 will obviously involve the curvature of some1 if not all1 of the partitions 7d7 which in our preliminary investigation we treated as plane4 ,to solve this problem in a general way is no easy matter2 but it is a problem which ,nature solves in every case where1 as in the case we are considering1 eight bubbles1 or eight cells1 meet together in a 7plane or curved7 surface4 ,an approximate solution has been given in ,fig4 #aec1 d2 and it will now at once be recognised that this figure has vastly more resemblance to an aggregate of living cells than had the diagram of ,fig4 #aec1 a with which we began4 ,illustration3 ,fig4 #aed4 ,just as we have constructed in this case a series of purely diagrammatic or schematic figures1 so it will be as a rule possible to diagrammatise1 with but little alteration1 the complicated appearances presented by any ordinary aggregate of cells4 ,the accompanying little figure 7,fig4 #aed71 of a germinating spore of a ,liverwort 7,riccia71 after a drawing of ,professor ,campbell's1 scarcely needs further explanation3 for it is well-nigh a typical diagram of the method of space-partitioning which we are now considering4 ,let us look again at our figures 7on p4 #cei7 of the disc of ,erythrotrichia1 from ,berthold's ,monograph of the ,bangiaceae and redraw the earlier stages in diagrammatic fashion4 ,in the following series of diagrams the new partitions1 or those just about to form1 are in each case outlined2 and in the next succeeding stage they are shewn after settling down into position1 and after exercising their respective tractions on the walls previously laid down4 ,it is clear1 ,i think1 that these four diagrammatic figures represent all that is shewn in the first five stages drawn by ,berthold from the plant itself2 but the correspondence cannot #cgc in this case be precisely accurate1 for the simple reason that ,berthold's figures are taken from different individuals1 and are therefore only approximately consecutive and not strictly continuous4 ,the last of the six drawings in ,fig4 #add is already too ,illustration3 ,fig4 #aee4 ,theoretical arrangement of successive partitions in a discoid cell2 for comparison with ,fig4 #add4 complicated for diagrammatisation1 that is to say it is too complicated for us to decipher with certainty the precise order of appearance of the numerous partitions which it contains4 ,but in ,fig4 #aef ,i shew one more diagrammatic figure1 of a disc which ,illustration3 ,fig4 #aef4 ,theoretical division of a discoid cell into sixty-four chambers3 no allowance being made for the mutual tractions of the cell-walls4 has divided1 according to the theoretical plan1 into about sixty-four cells2 and making due allowance for the successive changes which the mutual tensions and tractions of the partitions must #cgd bring about1 increasing in complexity with each succeeding stage1 we can see1 even at this advanced and complicated stage1 a very considerable resemblance between the actual picture 7,fig4 #add7 and the diagram which we have here constructed in obedience to a few simple rules4 ,in like manner1 in the annexed figures1 representing sections through a young embryo of a ,moss1 we have very little difficulty in discerning the successive stages that must have intervened between the two stages shewn3 so as to lead from the just divided quadrants 7one of which1 by the way1 has not yet divided in our figure 7a77 to the stage 7b7 in which a well-marked epidermal layer surrounds an at first sight irregular agglomeration of 8fundamental0 tissue4 ,illustration3 ,fig4 #aeg4 ,sections of embryo of a moss4 7,after ,kienitz-,gerloff47 ,in the last paragraph but one1 ,i have spoken of the difficulty of so arranging the meeting-places of a number of cells that at each junction only three cell-walls shall meet in a line1 and all three shall meet it at equal angles of #abj4 ,as a matter of fact1 the problem is soluble in a number of ways2 that is to say1 when we have a number of cells1 say eight as in the case considered1 enclosed in a common boundary1 there are various ways in which their walls can be made to meet internally1 three by three1 at equal angles2 and these differences will entail differences also in the curvature of the walls1 and consequently in the shape of the cells4 ,the question is somewhat complex2 it has been dealt with by ,plateau1 and treated mathematically by ,m4 ,van ,rees#cib4 ,illustration3 ,fig4 #aeh4 ,various possible arrangements of intermediate partitions1 in groups of #d1 #e1 #f1 #g or #h cells4 ,if within our boundary we have three cells all meeting #cge internally1 they must meet in a point2 furthermore1 they tend to do so at equal angles of #abj1 and there is an end of the matter4 ,if we have four cells1 then1 as we have already seen1 the conditions are satisfied by interposing a little intermediate wall1 the two extremities of which constitute the meeting-points of three cells each1 and the upper edge of which marks the 8polar furrow40 ,similarly1 in the case of five cells1 we require two little intermediate walls1 and two polar furrows2 and we soon arrive at the rule that1 for n cells1 we require n #c little longitudinal partitions 7and corresponding polar furrows71 connecting the triple junctions of the cells2 and these little walls1 like all the rest within the system1 must be inclined to one another at angles of #abj4 ,where we have only one such wall 7as in the case of four cells71 or only two 7as in the case of five cells71 there is no room for ambiguity4 ,but where we have three little connecting-walls1 as in the case of six cells1 it is obvious that we can arrange them in three different ways1 as in the annexed ,fig4 #aei4 ,in the system of seven cells1 the four partitions can be arranged in four ways2 and the five partitions required in the case of eight cells can be arranged in no less than thirteen different ways1 of which ,fig4 #aeh shews some half-dozen only4 ,it does not follow that1 so to speak1 these various #cgf arrangements are all equally good2 some are known to be much more stable than others1 and some have never yet been realised in actual experiment4 ,the conditions which lead to the presence of any one of them1 in preference to another1 are as yet1 so far as ,i am aware1 undetermined1 but to this point we shall return4 ,examples of these various arrangements meet us at every turn1 and not only in cell-aggregates1 but in various cases where non-rigid and semi-fluid partitions 7or partitions that were so to begin with7 meet together4 ,and it is a necessary consequence of this physical phenomenon1 and of the limited and very small number of possible arrangements1 that we get similar appearances1 capable of representation by the same diagram1 in the most diverse fields of biology#cic4 ,illustration3 ,fig4 #aei4 ,among the published figures of embryonic stages and other cell aggregates1 we only discern these little intermediate partitions in cases where the investigator has drawn carefully just what lay before him1 without any preconceived notions as to radial or other symmetry2 but even in other cases we can generally recognise1 without much difficulty1 what the actual arrangement was whereby the cell-walls met together in equilibrium4 ,i have a strong suspicion that a leaning towards ,sachs's ,rule1 that one cell-wall tends to set itself at right angles to another cell-wall 7a rule whose strict limitations1 and narrow range of application1 we have already #cgg considered7 is responsible for many inaccurate or incomplete representations of the mutual arrangement of aggregated cells4 ,illustration3 ,fig4 #afj4 ,segmenting egg of ,trochus4 7,after ,robert47 ,illustration3 ,fig4 #afa4 ,two views of segmenting egg of ,cynthia partita4 7,after ,conklin47 ,illustration3 ,fig4 #afb4 7a7 ,section of apical cone of ,salvinia4 7,after ,pringsheim#cid47 7b7 ,diagram of probable actual arrangement4 ,illustration3 ,fig4 #afc4 ,egg of ,pyrosoma4 7,after ,korotneff74 ,illustration3 ,fig4 #afd4 ,egg of ,echinus1 segmenting under pressure4 7,after ,driesch47 ,in the accompanying series of figures 7,figs4 #afj-afg7 ,i have #cgh set forth a few aggregates of eight cells1 mostly from drawings of segmenting eggs4 ,in some cases they shew clearly the manner in which the cells meet one another1 always at angles of #abj1 and always with the help of five intermediate boundary walls within the eight-celled system2 in other cases ,i have added a slightly altered drawing1 so as to shew1 with as little change as #cgi possible1 the arrangement of boundaries which probably actually existed1 and gave rise to the appearance which the observer drew4 ,these drawings may be compared with the various diagrams of ,fig4 #aeh1 in which some seven out of the possible thirteen arrangements of five intermediate partitions 7for a system of eight cells7 have been already set forth4 ,illustration3 ,fig4 #afe4 7a7 ,part of segmenting egg of ,cephalopod 7after ,watase72 7b7 probable actual arrangement4 ,illustration3 ,fig4 #aff4 7a7 ,egg of ,echinus2 7b7 do4 of ,nereis1 under pressure4 7,after ,driesch74 ,illustration3 ,fig4 #afg4 7a7 ,egg of frog1 under pressure 7after ,roux72 7b7 probable actual arrangement4 ,it will be seen that ,m4 ,robert-,tornow's figure of the segmenting egg of ,trochus 7,fig4 #afj7 clearly shews the cells grouped after the fashion of ,fig4 #aeh1 a4 ,in like manner1 ,mr ,conklin's figure of the ascidian egg 7,cynthia7 shews equally clearly the arrangement g4 ,a sea-urchin egg1 segmenting under pressure1 as figured by ,driesch1 scarcely requires any modification of the drawing to appear as a diagram of the type d4 ,turning for a moment to a botanical illustration1 we have a figure of ,pringsheim's shewing an eight-celled stage in the apex of the young cone of ,salvinia2 it is in all probability referable1 as in my modified diagram1 to type c4 ,beside it is figured a very different object1 a segmenting egg of the ,ascidian ,pyrosoma1 after ,korotneff2 it may be that this also is to be referred to type c1 but ,i think it is more easily referable to type b4 ,for there is a difference between this diagram and that of ,salvinia1 in that here apparently1 of the pairs of lateral cells1 the upper and the lower cell are alternately the larger1 while in the diagram of ,salvinia the lower lateral cells both appear much larger than the upper ones2 and this difference tallies with the appearance produced if we fill in the eight cells according to the type b or the type c4 ,in the segmenting cuttlefish egg1 there is again a slight dubiety as to which type it should be referred to1 but it is in all probability referable1 like ,driesch's ,echinus egg1 to d4 ,lastly1 ,i have copied from ,roux a curious figure of the egg of ,rana esculenta1 viewed from the animal pole1 which appears to me referable1 in all probability1 to type g4 ,of type f1 in which the five partitions form a figure with four re-entrant angles1 that is to say a figure representing the five sides of a hexagon1 ,i have found no examples among segmenting eggs1 and that arrangement in all probability is a very unstable one4 ,it is obvious enough1 without more ado1 that these phenomena are in the strictest and completest way common to both plants #chj and animals4 ,in other words they tally with1 and they further extend1 the general and fundamental conclusions laid down by ,schwann1 in his ,mikroskopische ,untersuchungen ber die ,uebereinstimmung in der ,struktur und dem ,wachsthum der ,thiere und ,pflanzen4 ,but now that we have seen how a certain limited number of types of eight-celled segmentation 7or of arrangements of eight cell-partitions7 appear and reappear1 here and there1 throughout the whole world of organisms1 there still remains the very important question1 whether in each particular organism the conditions are such as to lead to one particular arrangement being predominant1 characteristic1 or even invariable4 ,in short1 is a particular arrangement of cell-partitions to be looked upon 7as the published figures of the embryologist are apt to suggest7 as a specific character1 or at least a constant or normal character1 of the particular organism8 ,the answer to this question is a direct negative1 but it is only in the work of the most careful and accurate observers that we find it revealed4 ,rauber 7whom we have more than once had occasion to quote7 was one of those embryologists who recorded just what he saw1 without prejudice or preconception2 as ,boerhaave said of ,swammerdam1 quod vidit id asseruit4 ,now ,rauber has put on record a considerable number of variations in the arrangement of the first eight cells1 which form a discoid surface about the dorsal 7or 8animal07 pole of the frog's egg4 ,in a certain number of cases these figures are identical with one another in type1 identical 7that is to say7 save for slight differences in magnitude1 relative proportions1 or orientation4 ,but ,i have selected 7,fig4 #afh7 six diagrammatic figures1 which are all essentially different1 and these diagrams seem to me to bear intrinsic evidence of their accuracy3 the curvatures of the partition-walls1 and the angles at which they meet agree closely with the requirements of theory1 and when they depart from theoretical symmetry they do so only to the slight extent which we should naturally expect in a material and imperfectly homogeneous system#cie4 #cha ,illustration3 ,fig4 #afh4 ,various modes of grouping of eight cells1 at the dorsal or epiblastic pole of the frog's egg4 7,after ,rauber47 ,of these six illustrations1 two are exceptional4 ,in ,fig4 #afh1 #e1 we observe that one of the eight cells is surrounded on all sides by the other seven4 ,this is a perfectly natural condition1 and represents1 like the rest1 a phase of partial or conditional equilibrium4 ,but it is not included in the series we are now considering1 which is restricted to the case of eight cells extending outwards to a common boundary4 ,the condition shewn in ,fig4 #afh1 #f1 is again peculiar1 and is probably rare2 but it is included under the cases considered on p4 #cab1 in which the cells are not in complete fluid contact1 but are separated by little droplets of extraneous matter2 it needs no further comment4 ,but the other four cases are beautiful diagrams of space-partitioning1 similar to those we have just been considering1 but so exquisitely clear that they need no modification1 no 8touching-up10 to exhibit their mathematical regularity4 ,it will easily be recognised that in ,fig4 #afh1 #a and #b1 we have the arrangements corresponding to a and d of our diagram 7,fig4 #aeh73 but the other two 7i4e4 #c and #d7 represent other of the thirteen possible arrangements1 which are not included in that #chb diagram4 ,it would be a curious and interesting investigation to ascertain1 in a large number of frogs' eggs1 all at this stage of development1 the percentage of cases in which these various arrangements occur1 with a view of correlating their frequency with the theoretical conditions 7so far as they are known1 or can be ascertained7 of relative stability4 ,one thing stands out as very certain indeed3 that the elementary diagram of the frog's egg commonly given in text-books of embryology1--in which the cells are depicted as uniformly symmetrical quadrangular bodies1--is entirely inaccurate and grossly misleading#cif4 ,we now begin to realise the remarkable fact1 which may even appear a startling one to the biologist1 that all possible groupings or arrangements whatsoever of eight cells 7where all take part in the surface of the group1 none being submerged or wholly enveloped by the rest7 are referable to some one or other of thirteen types or forms4 ,and that all the thousands and thousands of drawings which diligent observers have made of such eight-celled structures1 animal or vegetable1 anatomical1 histological or embryological1 are one and all representations of some one or another of these thirteen types3--or rather indeed of somewhat less than the whole thirteen1 for there is reason to believe that1 out of the total number of possible groupings1 a certain small number are essentially unstable1 and have at best1 in the concrete1 but a transitory and evanescent existence4 ,before we leave this subject1 on which a vast deal more might be said1 there are one or two points which we must not omit to consider4 ,let us note1 in the first place1 that the appearance which our plane diagrams suggest of inequality of the several cells is apt to be deceptive2 for the differences of magnitude apparent in one plane may well be1 and probably generally are1 balanced by equal and opposite differences in another4 ,secondly1 let us remark that the rule which we are considering refers only #chc to angles1 and to the number1 not to the length of the intermediate partitions2 it is to a great extent by variations in the length of these that the magnitudes of the cells may be equalised1 or otherwise balanced1 and the whole system brought into equilibrium4 ,lastly1 there is a curious point to consider1 in regard to the number of actual contacts1 in the various cases1 between cell and cell4 ,if we inspect the diagrams in ,fig4 #afi 7which represent three out of our thirteen possible arrangements of eight cells7 we shall see that1 in the case of type b1 two cells are each in contact with two others1 two cells with three others1 and four cells each with four other cells4 ,in type a four cells are each in contact with two1 two with four1 and two with five4 ,in type f1 two are in contact with two1 four with three1 and one with no less than seven4 ,in all cases the ,illustration3 ,fig4 #afi4 number of contacts is twenty-six in all2 or1 in other words1 there are thirteen internal partitions1 besides the eight peripheral walls4 ,for it is easy to see that1 in all cases of n cells with a common external boundary1 the number of internal partitions is #bn #c2 or the number of what we call the internal or interfacial contacts is #b7bn #c74 ,but it would appear that the most stable arrangements are those in which the total number of contacts is most evenly divided1 and the least stable are those in which some one cell has1 as in type f1 a predominant number of contacts4 ,in a well-known series of experiments1 ,roux has shewn how1 by means of oil-drops1 various arrangements1 or aggregations1 of cells can be simulated2 and in ,fig4 #agj ,i shew a number of ,roux's figures1 and have ascribed them to what seem to be their appropriate 8types0 among those which we have just been considering2 but #chd it will be observed that in these figures of ,roux's the drops are not always in complete contact1 a little air-bubble often keeping them apart at their apical junctions1 so that we see the configuration towards which the system is tending rather than that which it has fully attained#cig4 ,the type which we have called f was found by ,roux to be unstable1 the large 7or apparently large7 drop a quickly passing into the centre of the system1 and here taking up a position of equilibrium in which1 as usual1 three cells meet throughout in a point1 at equal angles1 and in which1 in this case1 all the cells have an equal number of 8interfacial0 contacts4 ,illustration3 ,fig4 #agj4 ,aggregations of oil-drops4 7,after ,roux47 ,figs4 #d-f represent successive changes in a single system4 ,we need by no means be surprised to find that1 in such arrangements1 the commonest and most stable distributions are those in which the cell-contacts are distributed as uniformly as possible between the several cells4 ,we always expect to find some such tendency to equality in cases where we have to do with small oscillations on either side of a symmetrical condition4 #che ,the rules and principles which we have arrived at from the point of view of surface tension have a much wider bearing than is at once suggested by the problems to which we have applied them2 for in this elementary study of the cell-boundaries in a segmenting egg or tissue we are on the verge of a difficult and important subject in pure mathematics4 ,it is a subject adumbrated by ,leibniz1 studied somewhat more deeply by ,euler1 and greatly developed of recent years4 ,it is the ,geometria ,situs of ,gauss1 the ,analysis ,situs of ,riemann1 the ,theory of ,partitions of ,cayley1 and of ,spatial ,complexes of ,listing#cih4 ,the crucial point for the biologist to comprehend is1 that in a closed surface divided into a number of faces1 the arrangement of all the faces1 lines and points in the system is capable of analysis1 and that1 when the number of faces or areas is small1 the number of possible arrangements is small also4 ,this is the simple reason why we meet in such a case as we have been discussing 7viz4 the arrangement of a group or system of eight cells7 with the same few types recurring again and again in all sorts of organisms1 plants as well as animals1 and with no relation to the lines of biological classification3 and why1 further1 we find similar configurations occurring to mark the symmetry1 not of cells merely1 but of the parts and organs of entire animals4 ,the phenomena are not 8functions10 or specific characters1 of this or that tissue or organism1 but involve general principles which lie within the province of the mathematician4 ,the theory of space-partitioning1 to which the segmentation of the egg gives us an easy practical introduction1 is illustrated in much more complex ways in other fields of natural history4 ,a very beautiful but immensely complicated case is furnished by the 8venation0 of the wings of insects4 ,here we have sometimes 7as in the dragon-flies71 a general reticulum of small1 more or less hexagonal 8cells03 but in most other cases1 in flies1 bees1 butterflies1 etc41 we have a moderate number of cells1 whose partitions always impinge upon one another three by three1 and whose arrangement1 therefore1 includes of necessity a number of small intermediate partitions1 analogous to our polar furrows4 ,i think #chf that a mathematical study of these1 including an investigation of the 8deformation0 of the wing 7that is to say1 of the changes in shape and changes in the form of its 8cells0 which it undergoes during the life of the individual1 and from one species to another7 would be of great interest4 ,in very many cases1 the entomologist relies upon this venation1 and upon the occurrence of this or that intermediate vein1 for his classification1 and therefore for his hypothetical phylogeny of particular groups2 which latter procedure hardly commends itself to the physicist or the mathematician4 ,illustration3 ,fig4 #aga4 7,a7 ,asterolampra marylandica1 ,ehr42 7,b1 ,c7 ,a4 variabilis1 ,grev4 7,after ,greville47 ,another case1 geometrically akin but biologically very different1 is to be found in the little diatoms of the genus ,asterolampra1 and their immediate congeners#cii4 ,in ,asterolampra we have a little disc1 in which we see 7as it were7 radiating spokes of one material1 alternating with intervals occupied on the flattened wheel-like disc by another 7,fig4 #aga74 ,the spokes vary in number1 but the general appearance is in a high degree suggestive of the ,chladni figures produced by the vibration of a circular plate4 ,the spokes broaden out towards the centre1 and interlock by visible junctions1 which obey the rule of triple intersection1 and accordingly exemplify the partition-figures with which we are dealing4 ,but whereas we have found the particular arrangement in which one cell is in contact with all the rest to be unstable1 according to ,roux's oil-drop experiments1 and to be conspicuous #chg by its absence from our diagrams of segmenting eggs1 here in ,asterolampra1 on the other hand1 it occurs frequently1 and is indeed the commonest arrangement#djj 7,fig4 #aga1 ,b74 ,in all probability1 we are entitled to consider this marked difference natural enough4 ,for we may suppose that in ,asterolampra 7unlike the case of the segmenting egg7 the tendency is to perfect radial symmetry1 all the spokes emanating from a point in the centre3 such a condition would be eminently unstable1 and would break down under the least asymmetry4 ,a very simple1 perhaps the simplest case1 would be that one single spoke should differ slightly from the rest1 and should so tend to be drawn in amid the others1 these latter remaining similar and symmetrical among themselves4 ,such a configuration would be vastly less unstable than the original one in which all the boundaries meet in a point2 and the fact that further progress is not made towards other configurations of still greater stability may be sufficiently accounted for by viscosity1 rapid solidification1 or other conditions of restraint4 ,a perfectly stable condition would of course be obtained if1 as in the case of ,roux's oil-drop 7,fig4 #agj1 #f71 one of the cellular spaces passed into the centre of the system1 the other partitions radiating outwards from its circular wall to the periphery of the whole system4 ,precisely such a condition occurs among our diatoms2 but when it does so1 it is looked upon as the mark and characterisation of the allied genus ,arachnoidiscus4 ,illustration3 ,fig4 #agb4 ,section of ,alcyonarian polype4 ,in a diagrammatic section of an ,alcyonarian polype 7,fig4 #agb71 we have eight chambers set1 symmetrically1 about a ninth1 which constitutes the 8stomach40 ,in this arrangement there is no difficulty1 for it is obvious that1 throughout the system1 three boundaries meet 7in plane section7 in a point4 ,in many corals we have as #chh simple1 or even simpler conditions1 for the radiating calcified partitions either converge upon a central chamber1 or fail to meet it and end freely4 ,but in a few cases1 the partitions or 8septa0 converge to meet one another1 there being no central chamber on which they may impinge2 and here the manner in which contact is effected becomes complicated1 and involves problems identical with those which we are now studying4 ,illustration3 ,fig4 #agc4 ,heterophyllia angulata4 7,after ,nicholson47 ,in the great majority of corals we have as simple or even simpler conditions than those of ,alcyonium2 for as a rule the calcified partitions or septa of the coral either converge upon a central chamber 7or central 8columella071 or else fail to meet it and end freely4 ,in the latter case the problem of space-partitioning does not arise2 in the former1 however numerous the septa be1 their separate contacts with the wall of the central chamber comply with our fundamental rule according to which three lines and no more meet in a point1 and from this simple and symmetrical arrangement there is little tendency to variation4 ,but in a few cases1 the septal partitions converge to meet one another1 there being no central chamber on which they may impinge2 and here the manner in which contact is effected becomes complicated1 and involves problems of space-partitioning identical with those which we are now studying4 ,in the genus ,heterophyllia and in a few allied forms we have such conditions1 and students of the ,coelenterata have found them very puzzling4 ,mc,coy#dja1 their first discoverer1 pronounced these corals to be 8totally unlike0 any other group1 recent or fossil2 and ,professor ,martin ,duncan1 writing a memoir on ,heterophyllia and its allies#djb1 described them as 8paradoxical in their anatomy40 ,illustration3 ,fig4 #agd4 ,heterophyllia sp4 7,after ,martin ,duncan47 ,the simplest or youngest ,heterophylliae known have six septa 7as in ,fig4 #agd1 a72 in the case figured1 four of these septa are conjoined two and two1 thus forming the usual triple junctions together with their intermediate partition-walls3 and in the #chi case of the other two we may fairly assume that their proper and original arrangement was that of our type #f;b 7,fig4 #aeh71 though the central intermediate partition has been crowded out by partial coalescence4 ,when with increasing age the septa become more numerous1 their arrangement becomes exceedingly variable2 for the simple reason that1 from the mathematical point of view1 the number of possible arrangements1 of #aj1 #ab or more cellular partitions in triple contact1 tends to increase with great rapidity1 and there is little to choose between many of them in regard to symmetry and equilibrium4 ,but while1 mathematically speaking1 each particular case among the multitude of possible cases is an orderly and definite arrangement1 from the purely biological point of view on the other hand no law or order is recognisable2 and so ,mc,coy described the genus as being characterised by the possession of septa 8destitute of any order of arrangement1 but irregularly branching and coalescing in their passage from the solid external walls towards some indefinite point near the centre where the few main lamellae irregularly anastomose40 #cij ,in the two examples figured 7,fig4 #agd71 both comparatively simple ones1 it will be seen that1 of the main chambers1 one is in each case an unsymmetrical one2 that is to say1 there is one chamber which is in contact with a greater number of its neighbours than any other1 and which at an earlier stage must have had contact with them all2 this was the case of our type f1 in the eight-celled system 7,fig4 #aeh74 ,such an asymmetrical chamber 7which may occur in a system of any number of cells greater than six71 constitutes what is known to students of the ,coelenterata as a 8fossula02 and we may recognise it not only here1 but also in ,zaphrentis and its allies1 and in a good many other corals besides4 ,moreover certain corals are described as having more than one fossula3 this appearance being naturally produced under certain of the other asymmetrical variations of normal space-partitioning4 ,where a single fossula occurs1 we are usually told that it is a symptom of 8bilaterality02 and this is in turn interpreted as an indication of a higher grade of organisation than is implied in the purely 8radial symmetry0 of the commoner types of coral4 ,the mathematical aspect of the case gives no warrant for this interpretation4 ,let us carefully notice 7lest we run the risk of confusing two distinct problems7 that the space-partitioning of ,heterophyllia by no means agrees with the details of that which we have studied in 7for instance7 the case of the developing disc of ,erythrotrichia3 the difference simply being that ,heterophyllia illustrates the general case of cell-partitioning as ,plateau and ,van ,rees studied it1 while in ,erythrotrichia1 and in our other embryological and histological instances1 we have found ourselves justified in making the additional assumption that each new partition divided a cell into co-equal parts4 ,no such law holds in ,heterophyllia1 whose case is essentially different from the others3 inasmuch as the chambers whose partition we are discussing in the coral are mere empty spaces 7empty save for the mere access of sea-water72 while in our histological and embryological instances1 we were speaking of the division of a cellular unit of living protoplasm4 ,accordingly1 among other differences1 the 8transverse0 or 8periclinal0 partitions1 which were bound to appear at regular intervals and in definite positions1 when co-equal bisection was a feature of the #cia case1 are comparatively few and irregular in the earlier stages of ,heterophyllia1 though they begin to appear in numbers after the main1 more or less radial1 partitions have become numerous1 and when accordingly these radiating partitions come to bound narrow and almost parallel-sided interspaces2 then it is that the transverse or periclinal partitions begin to come in1 and form what the student of the ,coelenterata calls the 8dissepiments0 of the coral4 ,we need go no further into the configuration and anatomy of the corals2 but it seems to me beyond a doubt that the whole question of the complicated arrangement of septa and dissepiments throughout the group 7including the curious vesicular or bubble-like tissue of the ,cyathophyllidae and the general structural plan of the ,tetracoralla1 such as ,streptoplasma and its allies7 is well worth investigation from the physical and mathematical point of view1 after the fashion which is here slightly adumbrated4 ,illustration3 ,fig4 #age4 ,diagrammatic section of a ,ctenophore 7,eucharis74 ,the method of dividing a circular1 or spherical1 system into eight parts1 equal as to their areas but unequal in their peripheral boundaries1 is probably of wide biological application2 that is to say1 without necessarily supposing it to be rigorously followed1 the typical configuration which it yields seems to recur again and again1 with more or less approximation to precision1 and under widely different circumstances4 ,i am inclined to think1 for instance1 that the unequal division of the surface of a ,ctenophore by its #cib meridian-like ciliated bands is a case in point 7,fig4 #age74 ,here1 if we imagine each quadrant to be twice bisected by a curved anticline1 we shall get what is apparently a close approximation to the actual position of the ciliated bands4 ,the case however is complicated by the fact that the sectional plan of the organism is never quite circular1 but always more or less elliptical4 ,one point1 at least1 is clearly seen in the symmetry of the ,ctenophores2 and that is that the radiating canals which pass outwards to correspond in position with the ciliated bands1 have no common centre1 but diverge from one another by repeated bifurcations1 in a manner comparable to the conjunctions of our cell-walls4 ,in like manner ,i am inclined to suggest that the same principle may help us to understand the apparently complex arrangement of the skeletal rods of a larval ,echinoderm1 and the very complex conformation of the larva which is brought about by the presence of these long1 slender skeletal radii4 ,illustration3 ,fig4 #agf4 ,diagrammatic arrangement of partitions1 represented by skeletal rods1 in larval ,echinoderm 7,ophiura74 ,in ,fig4 #agf ,i have divided a circle into its four quadrants1 and have bisected each quadrant by a circular arc 7,,bc71 passing from radius to periphery1 as in the foregoing cases of cell-division2 and ,i have again bisected1 in a similar way1 the triangular halves of each quadrant 7,,dd74 ,i have also inserted a small circle in the middle of the figure1 concentric with the large one4 ,if now we imagine those lines in the figure which ,i have drawn black to be replaced by solid rods we shall have at once the frame-work of an ,ophiurid 7,pluteus7 larva4 ,let us imagine all these arms to be #cic bent symmetrically downwards1 so that the plane of the paper is transformed into a spheroidal surface1 such as that of a hemisphere1 or that of a tall conical figure with curved sides2 let a membrane be spread1 umbrella-like1 between the outstretched skeletal rods1 and let its margin loop from rod to rod in curves which are possibly catenaries1 but are more probably portions of an 8elastic curve10 and the outward resemblance to a ,pluteus larva is now complete4 ,by various slight modifications1 by altering the relative lengths of the rods1 by modifying their curvature or by replacing the curved rod by a tangent to itself1 we can ring the changes which lead us from one known type of ,pluteus to another4 ,the case of the ,bipinnaria larvae of ,echinids is certainly analogous1 but it becomes very much more complicated2 we have to do with a more complex partitioning of space1 and ,i confess that ,i am not yet able to represent the more complicated forms in so simple a way4 ,illustration3 ,fig4 #agg4 ,pluteus-larva of ,ophiurid4 ,illustration3 ,fig4 #agh4 ,diagrammatic development of ,stomata in ,sedum4 7,cf4 fig4 in ,sachs's ,botany1 #ahhb1 p4 #ajc47 ,there are a few notable exceptions 7besides the various unequally segmenting eggs7 to the general rule that in cell-division the mother-cell tends to divide into equal halves2 and one of these exceptional cases is to be found in connection with the development of 8stomata0 in the leaves of plants4 ,the epidermal cells by which the leaf is covered may be of various shapes2 sometimes1 as in a hyacinth1 they are oblong1 but more often they have an irregular shape in which we can recognise1 more or less clearly1 a distorted or imperfect hexagon4 ,in the case of the oblong cells1 a transverse partition will be the least possible1 whether the cell be equally or unequally divided1 unless 7as we have already seen7 #cid the space to be cut off be a very small one1 not more than about three-tenths the area of a square based on the short side of the original rectangular cell4 ,as the portion usually cut off is not nearly so small as this1 we get the form of partition shewn in ,fig4 #agi1 and the cell so cut off is next bisected by a partition at right angles to the first2 this latter partition splits1 and the two last-formed cells constitute the so-called 8guard-cells0 of the ,illustration3 ,fig4 #agi4 ,diagrammatic development of stomata in ,hyacinth4 stoma4 ,in other cases1 as in ,fig4 #agh1 there will come a point where the minimal partition necessary to cut off the required fraction of the cell-content is no longer a transverse one1 but is a portion of a cylindrical wall 7#b7 cutting off one corner of the mother-cell4 ,the cell so cut off is now a certain segment of a circle1 with an arc of approximately #abj2 and its next division will be by means of a curved wall cutting it into a triangular and a quadrangular portion 7#c74 ,the triangular portion will continue to divide in a similar way 7#d1 #e71 and at length 7for a reason which is not yet clear7 the partition wall #cie between the new-formed cells splits1 and again we have the phenomenon of a 8stoma0 with its attendant guard-cells4 ,in ,fig4 #agi are shewn the successive stages of division1 and the changing curvatures of the various walls which ensue as each subsequent partition appears1 introducing a new tension into the system4 ,it is obvious that in the case of the oblong cells of the epidermis in the hyacinth the stomata will be found arranged in regular rows1 while they will be irregularly distributed over the surface of the leaf in such a case as we have depicted in ,sedum4 ,while1 as ,i have said1 the mechanical cause of the split which constitutes the orifice of the stoma is not quite clear1 yet there can be little or no doubt that it1 like the rest of the phenomenon1 is related to surface tension4 ,it might well be that it is directly due to the presence underneath this portion of epidermis of the hollow air-space which the stoma is apparently developed 8for the purpose0 of communicating with2 this air-surface on both sides of the delicate epidermis might well cause such an alteration of tensions that the two halves of the dividing cell would tend to part company4 ,in short1 if the surface-energy in a cell-air contact were half or less than half that in a contact between cell and cell1 then it is obvious that our partition would tend to split1 and give us a two-fold surface in contact with air1 instead of the original boundary or interface between one cell and the other4 ,in ,professor ,macallum's experiments1 which we have briefly discussed in our short chapter on ,adsorption1 it was found that large quantities of potassium gathered together along the outer walls of the guard-cells of the stoma1 thereby indicating a low surface-tension along these outer walls4 ,the tendency of the guard-cells to bulge outwards is so far explained1 and it is possible that1 under the existing conditions of restraint1 we may have here a force tending1 or helping1 to split the two cells asunder4 ,it is clear enough1 however1 that the last stage in the development of a stoma1 is1 from the physical point of view1 not yet properly understood4 ,illustration3 ,fig4 #ahj4 ,various pollen-grains and spores 7after ,berthold1 ,campbell1 ,goebel and others74 7#a7 ,epilobium2 7#b7 ,passiflora2 7#c7 ,neottia2 7#d7 ,periploca graeca2 7#e7 ,apocynum2 7#f7 ,erica2 7#g7 ,spore of ,osmunda2 7#h7 ,tetraspore of ,callithamnion4 ,illustration3 ,fig4 #aha4 ,dividing spore of ,anthoceros4 7,after ,campbell47 ,in all our foregoing examples of the development of a 8tissue0 we have seen that the process consists in the successive division of cells1 each act of division being accompanied by the formation #cif of a boundary-surface1 which1 whether it become at once a solid or semi-solid partition or whether it remain semi-fluid1 exercises in all cases an effect on the position and the form of the boundary which comes into being with the next act of division4 ,in contrast to this general process stands the phenomenon known as 8free cell-formation10 in which1 out of a common mass of protoplasm1 a number of separate cells are simultaneously1 or all but simultaneously1 differentiated4 ,in a number of cases it happens that1 to begin with1 a number of 8mother-cells0 are formed simultaneously1 and each of these divides1 by two successive divisions1 into four 8daughter-cells40 ,these daughter-cells will tend to group themselves1 just as would four soap-bubbles1 into a 8tetrad10 the four cells corresponding to the angles of a regular tetrahedron4 ,for the system of four bodies is evidently here in perfect symmetry2 the partition-walls and their respective edges meet at equal angles3 three walls everywhere meeting in an edge1 and the four edges converging to a point in the geometrical centre of the system4 ,this is the typical mode of development of pollen-grains1 common among ,monocotyledons and all but universal among ,dicotyledonous plants4 ,by a loosening of the surrounding tissue and an expansion of the cavity1 or anther-cell1 in which #cig they lie1 the pollen-grains afterwards fall apart1 and their individual form will depend upon whether or no their walls have solidified before this liberation takes place4 ,for if not1 then the separate grains will be free to assume a spherical form as a consequence of their own individual and unrestricted growth2 but if they become solid or rigid prior to the separation of the tetrad1 then they will conserve more or less completely the plane interfaces and sharp angles of the elements of the tetrahedron4 ,the latter is the case1 for instance1 in the pollen-grains of ,epilobium 7,fig4 #ahj1 #a7 and in many others4 ,in the ,passion-flower 7#b7 we have an intermediate condition3 where we can still see an indication of the facets where the grains abutted on one another in the tetrad1 but the plane faces have been swollen by growth into spheroidal or spherical surfaces4 ,it is obvious that there may easily be cases where the tetrads of daughter-cells are prevented from assuming the tetrahedral form3 cases1 that is to say1 where the four cells are forced and crushed into one plane4 ,the figures given by ,goebel of the development of the pollen of ,neottia 7#c1 a-e3 all the figures referring to grains taken from a single anther71 illustrate this to perfection2 and it will be seen that1 when the four cells lie in a plane1 they conform exactly to our typical diagram of the first four cells in a segmenting ovum4 ,occasionally1 though the four cells lie in a plane1 the diagram seems to fail us1 for the cells appear to meet in a simple cross 7as in #e72 but here we soon perceive that the cells are not in complete interfacial contact1 but are kept apart by a little intervening drop of fluid or bubble of air4 ,the spores of ferns 7#g7 develop in very much the same way as pollen-grains2 and they also very often retain traces of the shape which they assumed as members of a tetrahedral figure4 ,among the 8tetraspores0 7#h7 of the ,florideae1 or ,red ,seaweeds1 we have a phenomenon which is in every respect analogous4 ,here again it is obvious that1 apart from differences in actual magnitude1 and apart from superficial or 8accidental0 differences 7referable to other physical phenomena7 in the way of colour1 #cih texture and minute sculpture or pattern1 it comes to pass1 through the laws of surface-tension and the principles of the geometry of position1 that a very small number of diagrammatic figures will sufficiently represent the outward forms of all the tetraspores1 four-celled pollen-grains1 and other four-celled aggregates which are known or are even capable of existence4 ,we have been dealing hitherto 7save for some slight exceptions7 with the partitioning of cells on the assumption that the system either remains unaltered in size or else that growth has proceeded uniformly in all directions4 ,but we extend the scope of our enquiry very greatly when we begin to deal with unequal growth1 with growth1 that is to say1 which produces a greater extension along some one axis than another4 ,and here we come close in touch with that great and still 7as ,i think7 insufficiently appreciated generalisation of ,sachs1 that the manner in which the cells divide is the result1 and not the cause1 of the form of the dividing structure3 that the form of the mass is caused by its growth as a whole1 and is not a resultant of the growth of the cells individually considered#djc4 ,such asymmetry of growth may be easily imagined1 and may conceivably arise from a variety of causes4 ,in any individual cell1 for instance1 it may arise from molecular asymmetry of the structure of the cell-wall1 giving it greater rigidity in one direction than another1 while all the while the hydrostatic pressure within the cell remains constant and uniform4 ,in an aggregate of cells1 it may very well arise from a greater chemical1 or osmotic1 activity in one than another1 leading to a localised increase in the fluid pressure1 and to a corresponding bulge over a certain area of the external surface4 ,it might conceivably occur as a direct result of the preceding cell-divisions1 when these are such as to produce many peripheral or concentric walls in one part and few or none in another1 with the obvious result of strengthening the common boundary wall and resisting the outward pressure of growth in parts where the former is the case2 that is to say1 in our dividing quadrant1 if #cii its quadrangular portion subdivide by periclines1 and the triangular portion by oblique anticlines 7as we have seen to be the natural tendency71 then we might expect that external growth would be more manifest over the latter than over the former areas4 ,as a direct and immediate consequence of this we might expect a tendency for special outgrowths1 or 8buds10 to arise from the triangular rather than from the quadrangular cells2 and this turns out to be not merely a tendency towards which theoretical considerations point1 but a widespread and important factor in the morphology of the cryptogams4 ,but meanwhile1 without enquiring further into this complicated question1 let us simply take it that1 if we start from such a simple case as a round cell which has divided into two halves1 or four quarters 7as the case may be71 we shall at once get bilateral symmetry about a main axis1 and other secondary results arising therefrom1 as soon as one of the halves1 or one of the quarters1 begins to shew a rate of growth in advance of the others2 for the more rapidly growing cell1 or the peripheral wall common to two or more such rapidly growing cells1 will bulge out into an ellipsoid form1 and may finally extend into a cylinder with rounded or ellipsoid end4 ,this latter very simple case is illustrated in the development of a pollen-tube1 where the rapidly growing cell develops into the elongated cylindrical tube1 and the slow-growing or quiescent part remains behind as the so-called 8vegetative0 cell or cells4 ,just as we have found it easier to study the segmentation of a circular disc than that of a spherical cell1 so let us begin in the same way1 by enquiring into the divisions which will ensue if the disc tend to grow1 or elongate1 in some one particular direction1 instead of in radial symmetry4 ,the figures which we shall then obtain will not only apply to the disc1 but will also represent1 in all essential features1 a projection or longitudinal section of a solid body1 spherical to begin with1 preserving its symmetry as a solid of revolution1 and subject to the same general laws as we have studied in the disc#djd4 #djj 7#a7 ,suppose1 in the first place1 that the axis of growth lies symmetrically in one of the original quadrantal cells of a segmenting disc2 and let this growing cell elongate with comparative rapidity before it subdivides4 ,when it does divide1 it will necessarily do so by a transverse partition1 concave towards the apex of the cell3 and1 as further elongation takes place1 the cylindrical structure which will be developed thereby will tend to be again and again subdivided by similar concave transverse partitions4 ,if at any time1 through this process of concurrent elongation and subdivision1 the apical cell become equivalent to1 or less than1 a hemisphere1 it will next divide by means of a longitudinal1 or vertical partition2 and similar longitudinal partitions will arise in the other segments of the cylinder1 as soon as it comes about that their length 7in the direction of the axis7 is less than their breadth4 ,illustration3 ,fig4 #ahb4 ,but when we think of this structure in the solid1 we at once perceive that each of these flattened segments of the cylinder1 into which our cylinder has divided1 is equivalent to a flattened circular disc2 and its further division will accordingly tend to proceed like any other flattened disc1 namely into four quadrants1 and afterwards by anticlines and periclines in the usual way4 #dja ,a section across the cylinder1 then1 will tend to shew us precisely the same arrangements as we have already so fully studied in connection with the typical division of a circular cell into quadrants1 and of these quadrants into triangular and quadrangular portions1 and so on4 ,but there are other possibilities to be considered1 in regard to the mode of division of the elongating quasi-cylindrical portion1 as it gradually develops out of the growing and bulging quadrantal cell2 for the manner in which this latter cell divides will simply depend upon the form it has assumed before each successive act of division takes place1 that is to say upon the ratio between its rate of growth and the frequency of its successive divisions4 ,for1 as we have already seen1 if the growing cell attain a markedly oblong or cylindrical form before division ensues1 then the partition will arise transversely to the long axis2 if it be but a little more than a hemisphere1 it will divide by an oblique partition2 and if it be less than a hemisphere 7as it may come to be after successive transverse divisions7 it will divide by a vertical partition1 that is to say by one coinciding with its axis of growth4 ,an immense number of permutations and combinations may arise in this way1 and we must confine our illustrations to a small number of cases4 ,the important thing is not so much to trace out the various conformations which may arise1 but to grasp the fundamental principle3 which is1 that the forces which dominate the form of each cell regulate the manner of its subdivision1 that is to say the form of the new cells into which it subdivides2 or in other words1 the form of the growing organism regulates the form and number of the cells which eventually constitute it4 ,the complex cell-network is not the cause but the result of the general configuration1 which latter has its essential cause in whatsoever physical and chemical processes have led to a varying velocity of growth in one direction as compared with another4 ,illustration3 ,fig4 #ahc4 ,development of ,sphagnum4 7,after ,campbell47 ,in the annexed figure of an embryo of ,sphagnum we see a mode of development almost precisely corresponding to the hypothetical case which we have just described1--the case1 that is to say1 where one of the four original quadrants of the mother-cell is the chief agent in future growth and development4 ,we see at the base of our first figure 7a71 the three stationary1 or #djb undivided quadrants1 one of which has further slowly divided in the stage b4 ,the active quadrant has grown quickly into a cylindrical structure1 which inevitably divides1 in the next place1 into a series of transverse partitions2 and accordingly1 this mode of development carries with it the presence of a single 8apical cell10 whose lower wall is a spherical surface with its convexity downwards4 ,each cell of the subdivided cylinder now appears as a more or less flattened disc1 whose mode of further sub-division we may prognosticate according to our former investigation1 to which subject we shall presently return4 ,illustration3 ,fig4 #ahd4 7#b7 ,in the next place1 still keeping to the case where only one of the original quadrant-cells continues to grow and develop1 let us suppose that this growing cell falls to be divided when by growth it has become just a little greater than a hemisphere2 it will then divide1 as in ,fig4 #ahd1 #b1 by an oblique partition1 in the usual way1 whose precise position and inclination to the base will depend entirely on the configuration of the cell itself1 save only1 of course1 that we may have also to take into account the possibility of the division being into two unequal halves4 ,by our hypothesis1 #djc the growth of the whole system is mainly in a vertical direction1 which is as much as to say that the more actively growing protoplasm1 or at least the strongest osmotic force1 will be found near the apex2 where indeed there is obviously more external surface for osmotic action4 ,it will therefore be that one of the two cells which contains1 or constitutes1 the apex which will grow more rapidly than the other1 and which therefore will be the first to divide1 and indeed in any case1 it will usually be this one of the two which will tend to divide first1 inasmuch as the triangular and not the quadrangular half is bound to constitute the apex#dje4 ,it is obvious that 7unless the act of division be so long postponed that the cell has become quasi-cylindrical7 it will divide by another oblique partition1 starting from1 and running at right angles to1 the first4 ,and so division will proceed1 ,illustration3 ,fig4 #ahe4 ,gemma of ,moss4 7,after ,campbell47 by oblique alternate partitions1 each one tending to be1 at first1 perpendicular to that on which it is based and also to the peripheral wall2 but all these points of contact soon tending1 by reason of the equal tensions of the three films or surfaces which meet there1 to form angles of #abj4 ,there will always be1 in such a case1 a single apical cell1 of a more or less distinctly triangular form4 ,the annexed figure of the developing antheridium of a ,liverwort 7,riccia7 is a typical example of such a case4 ,in ,fig4 #ahe which represents a 8gemma0 of a ,moss1 we see just the same thing2 with this addition1 that here the lower of the two original cells has grown even more quickly than the other1 constituting a long cylindrical stalk1 and dividing in accordance with its shape1 by means of transverse septa4 ,in all such cases as these1 the cells whose development we have studied will in turn tend to subdivide1 and the manner in which they will do so must depend upon their own proportions2 and in all cases1 as we have already seen1 there will sooner or later be a tendency to the formation of periclinal walls1 cutting off an 8epidermal layer of cells10 as ,fig4 #ahf illustrates very well4 ,illustration3 ,fig4 #ahf4 ,development of antheridium of ,riccia4 7,after ,campbell47 ,illustration3 ,fig4 #ahg4 ,section of growing shoot of ,selaginella1 diagrammatic4 ,illustration3 ,fig4 #ahh4 ,embryo of ,jungermannia4 7,after ,kienitz-,gerloff47 ,the method of division by means of oblique partitions is a common one in the case of #8growing points'2 for it evidently #djd includes all cases in which the act of cell-division does not lag far behind that elongation which is determined by the specific rate of growth4 ,and it is also obvious that1 under a common type1 there must here be included a variety of cases which will1 at first sight1 present a very different appearance one from another4 ,for instance1 in ,fig4 #ahg which represents a growing shoot of ,selaginella1 and somewhat less diagrammatically in the young embryo of ,jungermannia 7,fig4 #ahh71 we have the appearance of an almost straight vertical partition running up in the axis of the system1 and the primary cell-walls are set almost at right angles to it1--almost transversely1 that is to say to the outer walls and to the long axis of the structure4 ,we soon recognise1 however1 #dje that the difference is merely a difference of degree4 ,the more remote the partitions are1 that is to say the greater the velocity of growth relatively to division1 the less abrupt will be the alternate kinks or curvatures of the portions which lie in the neighbourhood of the axis1 and the more will these portions appear to constitute a single unbroken wall4 ,illustration3 ,fig4 #ahi4 7#c7 ,but an appearance nearly1 if not quite1 indistinguishable from this may be got in another way1 namely1 when the original growing cell is so nearly hemispherical that it is actually divided by a vertical partition1 into two quadrants2 and from this vertical partition1 as it elongates1 lateral partition-walls will arise on either side4 ,and by the tensions exercised by these1 the vertical partition will be bent into little portions set at #abj one to another1 and the whole will come to look just like that which1 in the former case1 was made up of portions of many successive oblique partitions4 ,let us now1 in one or two cases1 follow out a little further the stages of cell-division whose beginning we have studied in the last paragraphs4 ,in the antheridium of ,riccia1 after the successive oblique partitions have produced the longitudinal series of cells shewn in ,fig4 #ahf1 it is plain that the next partitions will arise periclinally1 that is to say parallel to the outer wall1 which in this particular case represents the short axis of the oblong cells4 ,the effect is at once to produce an epidermal layer1 whose cells will tend to subdivide further by means of partitions perpendicular to the free surface1 that is to say crossing the flattened cells by their shortest diameter4 ,the inner mass1 beneath the epidermis1 consists of cells which are still more or less oblong1 or which become #djf definitely so in process of growth2 and these again divide1 parallel to their short axes1 into squarish cells1 which as usual1 by the mutual tension of their walls1 become hexagonal1 as seen in a plane section4 ,there is a clear distinction1 then1 in form as well as in position1 between the outer covering-cells and those which lie within this envelope2 the latter are reduced to a condition which merely fulfils the mechanical function of a protective coat1 while the former undergo less modification1 and give rise to the actively living1 reproductive elements4 ,illustration3 ,fig4 #aij4 ,development of sporangium of ,osmunda4 7,after ,bower47 ,in ,fig4 #aij is shewn the development of the sporangium of a fern 7,osmunda74 ,we may trace here the common phenomenon of a series of oblique partitions1 built alternately on one another1 and cutting off a conspicuous triangular apical cell4 ,over the whole system an epidermal layer has been formed1 in the manner we have described2 and in this case it covers the apical cell also1 owing to the fact that it was of such dimensions that1 at one stage of growth1 a periclinal partition wall1 cutting off its outer end1 was indicated as of less area than an anticlinal one4 ,this periclinal wall cuts down the apical cell to the proportions1 very nearly1 of an equilateral triangle1 but the solid form of the cell is obviously that of a tetrahedron with curved faces2 and accordingly1 the least possible partitions by which further subdivision can be effected will run successively parallel to its four sides 7or its three sides when we confine ourselves to the appearances as seen in #djg section74 ,the effect1 as seen in section1 is to cut off on each side a characteristically flattened cell1 oblong as seen in section1 still leaving a triangular 7or strictly speaking1 a tetrahedral7 one in the centre4 ,the former cells1 which constitute no specific structure or perform no specific physiological function1 but which merely represent certain directions in space towards which the whole system of partitioning has gradually led1 are called by botanists the 8tapetum40 ,the active growing tetrahedral cell which lies between them1 and from which in a sense every other cell in the system has been either directly or indirectly segmented off1 still manifests1 as it were1 its vigour and activity1 and now1 by internal subdivision1 becomes the mother-cell of the spores4 ,in all these cases1 for simplicity's sake1 we have merely considered the appearances presented in a single1 longitudinal1 plane of optical section4 ,but it is not difficult to interpret from these appearances what would be seen in another plane1 for instance in a transverse section4 ,in our first example1 for instance1 that of the developing embryo of ,sphagnum 7,fig4 #ahc71 we can see that1 at appropriate levels1 the cells of the original cylindrical row have divided into transverse rows of four1 and then of eight cells4 ,we may be sure that the four cells represent1 approximately1 quadrants of a cylindrical disc1 the four cells1 as usual1 not meeting in a point1 but intercepted by a small intermediate partition4 ,again1 where we have a plate of eight cells1 we may well imagine that the eight octants are arranged in what we have found to be the way naturally resulting from the division of four quadrants1 that is to say into alternately triangular and quadrangular portions2 and this is found by means of sections to be the case4 ,the accompanying figure is precisely comparable to our previous diagrams of the arrangement of an aggregate of eight cells in a dividing disc1 save only that1 in two cases1 the cells have already undergone a further subdivision4 ,illustration3 ,fig4 #aia4 7,a1 ,b17 ,sections of younger and older embryos of ,phascum2 7,c7 do4 of ,adiantum4 7,after ,kienitz-,gerloff47 ,illustration3 ,fig4 #aib4 ,section through frond of ,girardia sphacelaria4 7,after ,goebel47 ,it follows in like manner1 that in a host of cases we meet with this characteristic figure1 in one or other of its possible1 and strictly limited1 variations1--in the cross sections of growing embryonic structures1 just as we have already seen that it appears in a host of cases where the entire system 7or a portion of its #djh surface7 consists of eight cells only4 ,for example1 in ,fig4 #aia1 we have it again1 in a section of a young embryo of a moss 7,phascum71 and in a section of an embryo of a fern 7,adiantum74 ,in ,fig4 #aib shewing a section through a growing frond of a sea-weed 7,girardia7 we have a case where the partitions forming the eight octants have conformed to the usual type2 but instead of the usual division by periclines of the four quadrangular spaces1 these latter are dividing by means of oblique septa1 apparently owing to the fact that the cell is not dividing into two equal1 but into two unequal portions4 ,in this last figure we have a peculiar look of stiffness or formality1 such that it appears at first to bear little resemblance to the rest4 ,the explanation is of the simplest4 ,the mode of partitioning differs little 7except to some slight extent in the way already mentioned7 from the normal type2 but in this case the partition walls are so thick and become so quickly comparatively solid and rigid1 that the secondary curvatures due to their successive mutual tractions are here imperceptible4 ,illustration3 ,fig4 #aic4 ,development of antheridium of ,pteris4 7,after ,strasbrger47 ,a curious and beautiful case1 apparently aberrant but which would doubtless be found conforming strictly to physical laws1 if #dji only we clearly understood the actual conditions1 is indicated in the development of the antheridium of a fern1 as described by ,strasbrger4 ,here the antheridium develops from a single cell1 whose form has grown to be something more than a hemisphere2 and the first partition1 instead of stretching transversely across the cell1 as we should expect it to do if the cell were actually spherical1 has as it were sagged down to come in contact with the base1 and so to develop into an annular partition1 running round the lower margin of the cell4 ,the phenomenon is akin to that cutting off of the corner of a cubical cell by a spherical partition1 of which we have spoken on p4 #cdi1 and the annular film is very easy to reproduce by means of a soap-bubble in the bottom of a cylindrical dish or beaker4 ,the next partition is a periclinal one1 concentric with the outer surface of the young antheridium2 and this in turn is followed by a concave partition which cuts off the apex of the original cell3 but which becomes connected with the second1 or periclinal partition in precisely the same annular fashion as the first partition did with the base of the little antheridium4 ,the result is that1 at this stage1 we have four cell-cavities in the little antheridium3 7#a7 a central cavity2 7#b7 an annular space around the lower margin2 7#c7 a narrow annular or cylindrical space around the sides of the antheridium2 and 7#d7 a small terminal or apical cell4 ,it is evident that the tendency1 in the next place1 will be to subdivide the flattened external cells by means of anticlinal partitions1 and so to convert the whole structure into a single layer of epidermal cells1 surrounding a central cell within which1 in course of time1 the antherozoids are developed4 ,the foregoing account deals only with a few elementary phenomena1 and may seem to fall far short of an attempt to deal in general with 8the forms of tissues40 ,but it is the principle involved1 and not its ultimate and very complex results1 that we can alone #daj attempt to grapple with4 ,the stock-in-trade of mathematical physics1 in all the subjects with which that science deals1 is for the most part made up of simple1 or simplified1 cases of phenomena which in their actual and concrete manifestations are usually too complex for mathematical analysis2 and when we attempt to apply its methods to our biological and histological phenomena1 in a preliminary and elementary way1 we need not wonder if we be limited to illustrations which are obviously of a simple kind1 and which cover but a small part of the phenomena with which the histologist has become familiar4 ,but it is only relatively that these phenomena to which we have found the method applicable are to be deemed simple and few4 ,they go already far beyond the simplest phenomena of all1 such as we see in the dividing ,protococcus1 and in the first stages1 two-celled or four-celled1 of the segmenting egg4 ,they carry us into stages where the cells are already numerous1 and where the whole conformation has become by no means easy to depict or visualise1 without the help and guidance which the phenomena of surface-tension1 the laws of equilibrium and the principle of minimal areas are at hand to supply4 ,and so far as we have gone1 and so far as we can discern1 we see no sign of the guiding principles failing us1 or of the simple laws ceasing to hold good4 #daa ,,chapter ,,ix ,,on ,,concretions1 ,,spicules1 ,,and ,,spicular ,,skeletons ,the deposition of inorganic material in the living body1 usually in the form of calcium salts or of silica1 is a very common and wide-spread phenomenon4 ,it begins in simple ways1 by the appearance of small isolated particles1 crystalline or non-crystalline1 whose form has little relation or sometimes none to the structure of the organism2 it culminates in the complex skeletons of the vertebrate animals1 in the massive skeletons of the corals1 or in the polished1 sculptured and mathematically regular molluscan shells4 ,even among many very simple organisms1 such as the ,diatoms1 the ,radiolarians1 the ,foraminifera1 or the ,sponges1 the skeleton displays extraordinary variety and beauty1 whether by reason of the intrinsic form of its elementary constituents or the geometric symmetry with which these are arranged and interconnected4 ,with regard to the form of these various structures 7and this is all that immediately concerns us here71 it is plain that we have to do with two distinct problems1 which however1 though theoretically distinct1 may merge with one another4 ,for the form of the spicule or other skeletal element may depend simply upon its chemical nature1 as for instance1 to take a simple but not the only case1 when the form is purely crystalline2 or the inorganic solid material may be laid down in conformity with the shapes assumed by the cells1 tissues or organs1 and so be1 as it were1 moulded to the shape of the living organism2 and again1 there may well be intermediate stages in which both phenomena may be simultaneously recognised1 the molecular forces playing their part in conjunction with1 and under the restraint of1 the other forces inherent in the system4 #dab ,so far as the problem is a purely chemical one1 we must deal with it very briefly indeed2 and all the more because special investigations regarding it have as yet been few1 and even the main facts of the case are very imperfectly known4 ,this at least is evident1 that the whole series of phenomena with which we are about to deal go deep into the subject of colloid chemistry1 and especially with that branch of the science which deals with the properties of colloids in connection with capillary or surface phenomena4 ,it is to the special student of colloid chemistry that we must ultimately and chiefly look for the elucidation of our problem#djf4 ,in the first and simplest part of our subject1 the essential problem is the problem of crystallisation in presence of colloids4 ,in the cells of plants1 true crystals are found in comparative abundance1 and they consist1 in the great majority of cases1 of calcium oxalate4 ,in the stem and root of the rhubarb1 for instance1 in the leaf-stalk of ,begonia1 and in countless other cases1 sometimes within the cell1 sometimes in the substance of the cell-wall1 we find large and well-formed crystals of this salt2 their varieties of form1 which are extremely numerous1 are simply the crystalline forms proper to the salt itself1 and belong to the two systems1 cubic and monoclinic1 in one or other of which1 according to the amount of water of crystallisation1 this salt is known to crystallise4 ,when calcium oxalate crystallises according to the latter system 7as it does when its molecule is combined with two molecules of water of crystallisation71 the microscopic crystals have the form of fine needles1 or 8raphides10 such as are very common in plants2 and it has been found that these are artificially produced when the salt is crystallised out in presence of glucose or of dextrin#djg4 ,illustration3 ,fig4 #aid4 ,alcyonarian spicules3 ,siphonogorgia and ,anthogorgia4 7,after ,studer47 ,calcium carbonate1 on the other hand1 when it occurs in plant-cells 7as it does abundantly1 for instance in the 8cystoliths0 of the ,urticaceae and ,acanthaceae1 and in great quantities in ,melobesia #dac and the other calcareous or 8stony0 algae71 appears in the form of fine rounded granules1 whose inherent crystalline structure is not outwardly visible1 but is only revealed 7like that of a molluscan shell7 under polarised light4 ,among animals1 a skeleton of carbonate of lime occurs under a multitude of forms1 of which we need only mention now a very few of the most conspicuous4 ,the spicules of the calcareous sponges are triradiate1 occasionally quadriradiate1 bodies1 with pointed rays1 not crystalline in outward form but with a definitely crystalline internal structure4 ,we shall return again to these1 and find for them what would seem to be a satisfactory explanation of their form4 ,among the ,alcyonarian zoophytes we have a great variety of spicules#djh1 which are sometimes straight and slender rods1 sometimes flattened and more or less striated plates1 and still more often rounded or branched concretions with rough or knobby surfaces 7,figs4 #aid1 #bjj74 ,a third type1 presented by several very different things1 such as a pearl1 or the ear-bone of a bony fish1 consists of a more or less #dad rounded body1 sometimes spherical1 sometimes flattened1 in which the calcareous matter is laid down in concentric zones1 denser and clearer layers alternating with one another4 ,in the development of the molluscan shell and in the calcification of a bird's egg or the shell of a crab1 for instance1 spheroidal bodies with similar concentric striation make their appearance2 but instead of remaining separate they become crowded together1 and as they coalesce they combine to form a pattern of hexagons4 ,in some cases1 the carbonate of lime on being dissolved away by acid leaves behind it a certain small amount of organic residue2 in most cases other salts1 such as phosphates of lime1 ammonia or magnesia are present in small quantities2 and in most cases if not all the developing spicule or concretion is somehow or other so associated with living cells that we are apt to take it for granted that it owes its peculiarities of form to the constructive or plastic agency of these4 ,the appearance of direct association with living cells1 however1 is apt to be fallacious2 for the actual precipitation takes place1 as a rule1 not in actively living1 but in dead or at least inactive tissue#dji3 that is to say in the 8formed material0 or matrix which 7as for instance in cartilage7 accumulates round the living cells1 in the interspaces between these latter1 or at least1 as often happens1 in connection with the cell-wall or cell-membrane rather than within the substance of the protoplasm itself4 ,we need not go the length of asserting that this is a rule without exception2 but1 so far as it goes1 it is of great importance and to its consideration we shall presently return#daj4 ,cognate with this is the fact that it is known1 at least in some cases1 that the organism can go on living and multiplying with apparently unimpaired health1 when stinted or even wholly deprived of the material of which it is wont to make its spicules #dae or its shell4 ,thus1 ,pouchet and ,chabry#daa have shown that the eggs of sea-urchins reared in lime-free water develop in apparent health1 into larvae entirely destitute of the usual skeleton of calcareous rods1 and in which1 accordingly1 the long arms of the ,pluteus larva1 which the rods support and distend1 are entirely suppressed4 ,and again1 when ,foraminifera are kept for generations in water from which they gradually exhaust the lime1 their shells grow hyaline and transparent1 and seem to consist only of chitinous material4 ,on the other hand1 in the presence of excess of lime1 the shells become much altered1 strengthened with various 8ornaments10 and assuming characters described as proper to other varieties and even species#dab4 ,the crucial experiment1 then1 is to attempt the formation of similar structures or forms1 apart from the living organism3 but1 however feasible the attempt may be in theory1 we shall be prepared from the first to encounter difficulties1 and to realise that1 though the actions involved may be wholly within the range of chemistry and physics1 yet the actual conditions of the case may be so complex1 subtle and delicate1 that only now and then1 and in the simplest of cases1 shall we find ourselves in a position to imitate them completely and successfully4 ,such an investigation is only part of that much wider field of enquiry through which ,stephane ,leduc and many other workers#dac have sought to produce1 by synthetic means1 forms similar to those of living things2 but it is a well-defined and circumscribed part of that wider investigation4 ,when by chemical or physical experiment we obtain configurations similar1 for instance1 to the phenomena of nuclear division1 or conformations similar to a pattern of hexagonal cells1 or a group of vesicles which resemble some particular tissue or cell-aggregate1 we indeed prove what it is the main object of this book to illustrate1 namely1 that the physical forces are capable of producing particular organic forms4 ,but it is by no means always that we can feel perfectly assured that the physical forces which we deal with in our experiment are identical with1 and not merely analogous to1 #daf the physical forces which1 at work in nature1 are bringing about the result which we have succeeded in imitating4 ,in the present case1 however1 our enquiry is restricted and apparently simplified2 we are seeking in the first instance to obtain by purely chemical means a purely chemical result1 and there is little room for ambiguity in our interpretation of the experiment4 ,when we find ourselves investigating the forms assumed by chemical compounds under the peculiar circumstances of association with a living body1 and when we find these forms to be characteristic or recognisable1 and somehow different from those which1 under other circumstances1 the same substance is wont to assume1 an analogy presents itself to our minds1 captivating though perhaps somewhat remote1 between this subject of ours and certain synthetic problems of the organic chemist4 ,there is doubtless an essential difference1 as well as a difference of scale1 between the visible form of a spicule or concretion and the hypothetical form of an individual molecule2 but molecular form is a very important concept2 and the chemist has not only succeeded1 since the days of ,whler1 in synthesising many substances which are characteristically associated with living matter1 but his task has included the attempt to account for the molecular forms of certain 8asymmetric0 substances1 glucose1 malic acid and many more1 as they occur in nature4 ,these are bodies which1 when artificially synthesised1 have no optical activity1 but which1 as we actually find them in organisms1 turn 7when in solution7 the plane of polarised light in one direction or the other2 thus dextro-glucose and laevomalic acid are common products of plant metabolism2 but dextromalic acid and laevo-glucose do not occur in nature at all4 ,the optical activity of these bodies depends1 as ,pasteur shewed more than fifty years ago#dad1 upon the form1 right-handed or left-handed1 of their molecules1 which molecular asymmetry further gives rise to a corresponding right or left-handedness 7or enantiomorphism7 in the crystalline aggregates4 ,it is a distinct problem in organic or physiological chemistry1 #dag and by no means without its interest for the morphologist1 to discover how it is that nature1 for each particular substance1 habitually builds up1 or at least selects1 its molecules in a one-sided fashion1 right-handed or left-handed as the case may be4 ,it will serve us no better to assert that this phenomenon has its origin in 8fortuity10 than to repeat the ,abb ,galiani's saying1 8les ds de la nature sont pips40 ,the problem is not so closely related to our immediate subject that we need discuss it at length2 but at the same time it has its clear relation to the general question of form in relation to vital phenomena1 and moreover it has acquired interest as a theme of long-continued discussion and new importance from some comparatively recent discoveries4 ,according to ,pasteur1 there lay in the molecular asymmetry of the natural bodies and the symmetry of the artificial products1 one of the most deep-seated differences between vital and non-vital phenomena3 he went further1 and declared that 8this was perhaps the only well-marked line of demarcation that can at present #ahfj be drawn between the chemistry of dead and of living matter40 ,nearly forty years afterwards the same theme was pursued and elaborated by ,japp in a celebrated lecture#dae1 and the distinction still has its weight1 ,i believe1 in the minds of many if not most chemists4 8,we arrive at the conclusion10 said ,professor ,japp1 8that the production of single asymmetric compounds1 or their isolation from the mixture of their enantiomorphs1 is1 as ,pasteur firmly held1 the prerogative of life4 ,only the living organism1 or the living intelligence with its conception of asymmetry1 can produce this result4 ,only asymmetry can beget asymmetry40 ,in these last words 7which1 so far as the chemist and the biologist are concerned1 we may acknowledge to be perfectly true#daf7 lies the #dah crux of the difficulty2 for they at once bid us enquire whether in nature1 external to and antecedent to life1 there be not some asymmetry to which we may refer the further propagation or 8begetting0 of the new asymmetries3 or whether in default thereof1 we be rigorously confined to the conclusion1 from which ,japp 8saw no escape10 that 8at the moment when life first arose1 a directive force came into play1--a force of precisely the same character as that which enables the intelligent operator1 by the exercise of his will1 to select one crystallised enantiomorph and reject its asymmetric opposite#dag40 ,observe that it is only the first beginnings of chemical asymmetry that we need to discover2 for when asymmetry is once manifested1 it is not disputed that it will continue 8to beget asymmetry40 ,a plausible suggestion is now at hand1 which if it be confirmed and extended will supply or at least sufficiently illustrate the kind of explanation which is required#dah4 ,we know in the first place that in cases where ordinary non-polarised light acts upon a chemical substance1 the amount of chemical action is proportionate to the amount of light absorbed4 ,we know in the second place#dai1 in certain cases1 that light circularly polarised is absorbed in different amounts by the right-handed or left-handed varieties1 as the case may be1 of an asymmetric substance4 ,and thirdly1 we know that a portion of the light which comes to us from the sun is already plane-polarised light1 which becomes in part circularly polarised1 by reflection 7according to ,jamin7 at the surface of the sea1 and then rotated in a particular direction under the influence of terrestrial magnetism4 ,we only require to be assured that the relation between absorption of light and chemical activity will continue to hold good in the case of circularly polarised light2 that is to say #dai that the formation of some new substance or other1 under the influence of light so polarised1 will proceed asymmetrically in consonance with the asymmetry of the light itself2 or conversely1 that the asymmetrically polarised light will tend to more rapid decomposition of those molecules by which it is chiefly absorbed4 ,this latter proof is now said to be furnished by ,byk#dbj1 who asserts that certain tartrates become unsymmetrical under the continued influence of the asymmetric rays4 ,here then we seem to have an example1 of a particular kind and in a particular instance1 an example limited but yet crucial 7if confirmed71 of an asymmetric force1 non-vital in its origin1 which might conceivably be the starting-point of that asymmetry which is characteristic of so many organic products4 ,the mysteries of organic chemistry are great1 and the differences between its processes or reactions as they are carried out in the organism and in the laboratory are many#dba4 ,the actions1 catalytic and other1 which go on in the living cell are of extraordinary complexity4 ,but the contention that they are different in kind from what we term ordinary chemical operations1 or that in the production of single asymmetric compounds there is actually to be witnessed1 as ,pasteur maintained1 a 8prerogative of life10 would seem to be no longer safely tenable4 ,and furthermore1 it behoves us to remember that1 even though failure continued to attend all artificial attempts to originate the asymmetric or optically active compounds which organic nature produces in abundance1 this would only prove that a certain physical force1 or mode of physical action1 is at work among living things though unknown elsewhere4 ,it is a mode of action which we can easily imagine1 though the actual mechanism we cannot set agoing when we please4 ,and it follows that such a difference between living matter and dead would carry us but a little way1 for it would still be confined strictly to the physical or mechanical plane4 ,our historic interest in the whole question is increased by the #dbj fact1 or the great probability1 that 8the tenacity with which ,pasteur fought against the doctrine of spontaneous generation was not unconnected with his belief that chemical compounds of one-sided symmetry could not arise save under the influence of life#dbb40 ,but the question whether spontaneous generation be a fact or not does not depend upon theoretical considerations2 our negative response is based1 and is so far soundly based1 on repeated failures to demonstrate its occurrence4 ,many a great law of physical science1 not excepting gravitation itself1 has no higher claim on our acceptance4 ,let us return then1 after this digression1 to the general subject of the forms assumed by certain chemical bodies when deposited or precipitated within the organism1 and to the question of how far these forms may be artificially imitated or theoretically explained4 ,mr ,george ,rainey1 of ,st ,bartholomew's ,hospital 7to whom we have already referred71 and ,professor ,p4 ,harting1 of ,utrecht1 were the first to deal with this specific problem4 ,mr ,rainey published1 between #aheg and #ahfa1 a series of valuable and thoughtful papers to shew that shell and bone and certain other organic structures were formed 8by a process of molecular coalescence1 demonstrable in certain artificially-formed products#dbc40 ,professor ,harting1 after thirty years of experimental work1 published in #ahgb a paper1 which has become classical1 entitled ,recherches de ,morphologie ,synthtique1 sur la production artificielle de quelques formations calcaires organiques2 his aim was to pave the way for a 8morphologie synthtique10 as ,whler had laid the foundations of a 8chimie synthtique10 by his classical discovery forty years before4 #dba ,illustration3 ,fig4 #aie4 ,calcospherites1 or concretions of calcium carbonate1 deposited in white of egg4 7,after ,harting47 ,illustration3 ,fig4 #aif4 ,a single calcospherite1 with central 8nucleus10 and striated1 iridescent border4 7,after ,harting47 ,illustration3 ,fig4 #aig4 ,later stages in the same experiment4 ,illustration3 ,fig4 #aih1 ,a4 ,section of shell of ,mya2 ,b4 ,section of hinge-tooth of do4 7,after ,carpenter47 ,rainey and ,harting used similar methods1 and these were such as many other workers have continued to employ1--partly with the direct object of explaining the genesis of organic forms and partly as an integral part of what is now known as ,colloid ,chemistry4 ,the whole gist of the method was to bring some soluble salt of lime1 such as the chloride or nitrate1 into solution within a colloid medium1 such as gum1 gelatine or albumin2 and then to precipitate it out in the form of some insoluble compound1 such as the carbonate or oxalate4 ,harting found that1 when he added a little sodium or potassium carbonate to a concentrated solution of calcium chloride in albumin1 he got at first a gelatinous mass1 or 8colloid precipitate03 which slowly transformed by the appearance of tiny microscopic particles1 at first motionless1 but afterwards as they grew larger shewing the typical ,brownian movement4 ,so far1 very much the same phenomena were witnessed whether the solution were albuminous or not1 and similar appearances indeed had been witnessed and recorded by ,gustav ,rose1 so far back as #ahcgdbd2 but in the later stages the presence of albuminoid matter made a great difference4 ,now1 after a few days1 the calcium carbonate was seen to be deposited in the form of large rounded concretions1 with a more or less distinct central nucleus1 and with a surrounding structure at once radiate and #dbb concentric2 the presence of concentric zones or lamellae1 alternately dark and clear1 was especially characteristic4 ,these round 8calcospherites0 shewed a tendency to aggregate together in layers1 and then to assume polyhedral1 or often regularly hexagonal1 outlines4 ,in this latter condition they closely resemble the early stages of calcification in a molluscan 7,fig4 #aih71 or still more in a crustacean shell#dbe2 while in their isolated condition #dbc they very closely resemble the little calcareous bodies in the tissues of a trematode or a cestode worm1 or in the oesophageal glands of an earthworm#dbf4 ,illustration3 ,fig4 #aii4 ,large irregular calcareous concretions1 or spicules1 deposited in a piece of dead cartilage1 in presence of calcium phosphate4 7,after ,harting47 ,when the albumin was somewhat scanty1 or when it was mixed with gelatine1 and especially when a little phosphate of lime was #dbd added to the mixture1 the spheroidal globules tended to become rough1 by an outgrowth of spinous or digitiform projections2 and in some cases1 but not without the presence of the phosphate1 the result was an irregularly shaped knobby spicule1 precisely similar to those which are characteristic of the ,alcyonaria#dbg4 ,illustration3 ,fig4 #bjj4 ,additional illustrations of ,alcyonarian spicules3 ,eunicea4 7,after ,studer47 ,the rough spicules of the ,alcyonaria are extraordinarily variable in shape and size1 as1 looking at them from the chemist's or the physicist's point of view1 we should expect them to be4 ,partly upon the form of these spicules1 and partly on the general form or mode of branching of the entire colony of polypes1 a vast number of separate 8species0 have been based by systematic zoologists4 ,but it is now admitted that even in specimens of a single species1 from one and the same locality1 the spicules may vary immensely in shape and size3 and ,professor ,hickson declares 7in a paper published while these sheets are passing through the press7 that after many years of laborious work in striving to determine species of these animal colonies1 he feels 8quite convinced that we have been engaged in a more or less fruitless task#dbh04 ,the formation of a tooth has very lately been shown to be a phenomenon of the same order4 ,that is to say1 8calcification in both dentine and enamel #dbe is in great part a physical phenomenon2 the actual deposit in both tissues occurs in the form of calcospherites1 and the process in mammalian tissue is identical in every point with the same process occurring in lower organisms#dbi40 ,the ossification of bone1 we may be sure1 is in the same sense and to the same extent a physical phenomenon4 ,the typical structure of a calcospherite is no other than that of a pearl1 nor does it differ essentially from that of the otolith of a mollusc or of a bony fish4 7,the otoliths1 by the way1 of the elasmobranch fishes1 like those of reptiles and birds1 are not developed after this fashion1 but are true crystals of calc-spar47 ,illustration3 ,fig4 #bja4 ,a 8crust0 of close-packed calcareous concretions1 precipitated at the surface of an albuminous solution4 7,after ,harting47 ,throughout these phenomena1 the effect of surface-tension is manifest4 ,it is by surface-tension that ultra-microscopic particles are brought together in the first floccular precipitate or coagulum2 by the same agency1 the coarser particles are in turn agglutinated into visible lumps2 and the form of the calcospherites1 whether it be that of the solitary spheres or that assumed in various stages of aggregation 7e4g4 ,fig4 #bjb7dcj1 is likewise due to the same agency4 ,illustration3 ,fig4 #bjb4 ,aggregated calcospherites4 7,after ,harting47 ,from the point of view of colloid chemistry the whole phenomenon is very important and significant2 and not the least significant part is this tendency of the solidified deposits to assume the form of 8spherulites10 and other rounded contours4 ,in the phraseology of that science1 we are dealing with a two-phase system1 which finally consists of solid particles in suspension in a liquid 7the former being styled the disperse phase1 the latter the #dbf dispersion medium74 ,in accordance with a rule first recognised by ,ostwald#dca1 when a substance begins to separate out from a solution1 so making its appearance as a new phase1 it always makes its appearance first as a liquid#dcb4 ,here is a case in point4 ,the minute quantities of material1 on their way from a state of solution to a state of 8suspension10 pass through a liquid to a solid form2 and their temporary sojourn in the former leaves its impress in the rounded contours which surface-tension brought about while the little aggregate was still labile or fluid3 while coincidently with this surface-tension effect upon the surface1 crystallisation tended to take place throughout the little liquid mass1 or in such portion of it as had not yet consolidated and crystallised4 ,illustration3 ,fig4 #bjc4 7,after ,harting47 ,where we have simple aggregates of two or three calcospherites1 the resulting figure is precisely that of so many contiguous soap-bubbles4 ,in other cases1 composite forms result which are not so easily explained1 but which1 if we could only account for them1 would be of very great interest to the biologist4 ,for instance1 when smaller calcospheres seem1 as it were1 to invade the substance of a larger one1 we get curious conformations which in the closest possible way resemble the outlines of certain of the ,diatoms 7,fig4 #bjc74 ,another very curious formation1 which ,harting calls a 8conostat10 is of frequent occurrence1 and in it we see at least a suggestion of analogy with the configuration which1 in a protoplasmic structure1 we have spoken of as a 8collar-cell40 ,the #dbg conostats1 which are formed in the surface layer of the solution1 consist of a portion of a spheroidal calcospherite1 whose upper part is continued into a thin spheroidal collar1 of somewhat larger radius than the solid sphere2 but the precise manner in which the collar is formed1 possibly around a bubble of gas1 possibly about a vortex-like diffusion-current#dcc is not obvious4 ,among these various phenomena1 the concentric striation observed in the calcospherite has acquired a special interest and importance#dcd4 ,it is part of a phenomenon now widely known1 and recognised as an important factor in colloid chemistry1 under the name of 8,liesegang's ,rings#dce40 ,illustration3 ,fig4 #bjd4 ,conostats4 7,after ,harting47 ,if we dissolve1 for instance1 a little bichromate of potash in gelatine1 pour it on to a glass plate1 and after it is set place upon it a drop of silver nitrate solution1 there appears in the course of a few hours the phenomenon of ,liesegang's rings4 ,at first the silver forms a central patch of abundant reddish brown chromate precipitate2 but around this1 as the silver nitrate diffuses slowly through the gelatine1 the precipitate no longer comes down in a continuous1 uniform layer1 but forms a series of zones1 beautifully regular1 which alternate with clear interspaces of jelly1 and which stand farther and farther apart1 in logarithmic ratio1 as they recede from the centre4 ,for a discussion of the raison d'tre of #dbh this phenomenon1 still somewhat problematic1 the student must consult the text-books of physical and colloid chemistry#dcf4 ,but1 speaking very generally1 we may say the appearance of ,liesegang's rings is but a particular and striking case of a more general phenomenon1 namely the influence on crystallisation of the presence of foreign bodies or 8impurities10 represented in this case by the 8gel0 or colloid matrix#dcg4 ,faraday shewed long ago that to the presence of slight impurities might be ascribed the banded structure of ice1 of banded quartz or agate1 onyx1 etc42 and ,quincke and ,tomlinson have added to our scanty knowledge of the same phenomenon#dch4 ,illustration3 ,fig4 #bje4 ,liesegang's ,rings4 7,after ,leduc47 ,besides the tendency to rhythmic action1 as manifested in ,liesegang's rings1 the association of colloid matter with a crystalloid in solution may lead to other well-marked effects4 ,these1 according to ,professor ,j4 ,h4 ,bowman#dci1 may be grouped somewhat as follows3 7#a7 total prevention of crystallisation2 7#b7 suppression of certain of the lines of crystalline growth2 7#c7 extension of the crystal to abnormal proportions1 with a tendency for it to become a compound crystal2 7#d7 a curving or gyrating of the crystal or its parts4 #dbi ,illustration3 ,fig4 #bjf4 ,relay-crystals of common salt4 7,after ,bowman47 ,illustration3 ,fig4 #bjg4 ,wheel-like crystals in a colloid4 7,after ,bowman47 ,for instance1 it would seem that1 if the supply of material to the growing crystal be not forthcoming in sufficient quantity 7as may well happen in a colloid medium1 for lack of convection-currents71 then growth will follow only the strongest lines of crystallising force1 and will be suppressed or partially suppressed along other axes4 ,the crystal will have a tendency to become filiform1 or 8fibrous02 and the raphides of our plant-cells are a case in point4 ,again1 the long slender crystal so formed1 pushing its way into new material1 may initiate a new centre of crystallisation3 we get the phenomenon known as a 8relay10 along the principal lines of force1 and sometimes along subordinate axes as well4 ,this phenomenon is illustrated in the accompanying figure of crystallisation in a colloid medium of common salt2 and it may possibly be that we have here an explanation1 or part of an explanation1 of the compound siliceous spicules of the ,hexactinellid sponges4 ,lastly1 when the crystallising force is nearly equalled by the resistance of the viscous medium1 the crystal takes the line of least resistance1 with very various results4 ,one of these results would seem to be a gyratory course1 giving to the crystal a curious wheel-like shape1 as in ,fig4 #bjg2 and other results are the feathery1 fern-like #dcj or arborescent shapes so frequently seen in microscopic crystallisation4 ,to return to ,liesegang's rings1 the typical appearance of concentric rings upon a gelatinous plate may be modified in various experimental ways4 ,for instance1 our gelatinous medium may be placed in a capillary tube immersed in a solution of the precipitating salt1 and in this case we shall obtain a vertical succession of bands or zones regularly interspaced3 the result being very closely comparable to the banded pigmentation which we see in the hair of a rabbit or a rat4 ,in the ordinary plate preparation1 the free surface of the gelatine is under different conditions to the lower layers and especially to the lowest layer in contact with the glass2 and therefore it often happens that we obtain a double series of rings1 one deep and the other superficial1 which by occasional blending or interlacing1 may produce a netted pattern4 ,in some cases1 as when only the inner surface of our capillary tube is covered with a layer of gelatine1 there is a tendency for the deposit to take place in a continuous spiral line1 rather than in concentric and separate zones4 ,by such means1 according to ,kster#ddj various forms of annular1 spiral and reticulated thickenings in the vascular tissue of plants may be closely imitated2 and he and certain other writers have of late been inclined to carry the same chemico-physical phenomenon a very long way1 in the explanation of various banded1 striped1 and other rhythmically successional types of structure or pigmentation4 ,for example1 the striped pigmentation of the leaves in many plants 7such as ,eulalia japonica71 the striped or clouded colouring of many feathers or of a cat's skin1 the patterns of many fishes1 such for instance as the brightly coloured tropical ,chaetodonts and the like1 are all regarded by him as so many instances of 8diffusion-figures0 closely related to the typical ,liesegang phenomenon4 ,gebhardt has made a particular study of the same subject in the case of insects#dda4 ,he declares1 for instance1 that the banded wings of ,papilio podalirius are precisely imitated in ,liesegang's experiments2 that the finer markings on the wings of the ,goatmoth 7,cossus ligniperda7 shew the double arrangement of larger and of #dca smaller intermediate rhythms1 likewise manifested in certain cases of the same kind2 that the alternate banding of the antennae 7for instance in ,sesia spheciformis71 a pigmentation not concurrent with the segmented structure of the antenna1 is explicable in the same way2 and that the 8ocelli10 for instance of the ,emperor moth1 are typical illustrations of the common concentric type4 ,darwin's well-known disquisition#ddb on the ocellar pattern of the feathers of the ,argus ,pheasant1 as a result of sexual selection1 will occur to the reader's mind1 in striking contrast to this or to any other direct physical explanation#ddc4 ,to turn from the distribution of pigment to more deeply seated structural characters1 ,leduc has shewn how1 for instance1 the laminar structure of the cornea or the lens is again1 apparently1 a similar phenomenon4 ,in the lens of the fish's eye1 we have a very curious appearance1 the consecutive lamellae being roughened or notched by close-set1 interlocking sinuosities2 and precisely the same appearance1 save that it is not quite so regular1 is presented in one of ,kster's figures as the effect of precipitating a little sodium phosphate in a gelatinous medium4 ,biedermann has studied1 from the same point of view1 the structure and development of the molluscan shell1 the problem which ,rainey had first attacked more than fifty years before#ddd2 and ,liesegang himself has applied his results to the formation of pearls1 and to the development of bone#dde4 #dcb ,among all the many cases where this phenomenon of ,liesegang's comes to the naturalist's aid in explanation of rhythmic or zonary configurations in organic forms1 it has a special interest where the presence of concentric zones or rings appears1 at first sight1 as a sure and certain sign of periodicity of growth1 depending on the seasons1 and capable therefore of serving as a mark and record of the creature's age4 ,this is the case1 for instance1 with the scales1 bones and otoliths of fishes2 and a kindred phenomena in starch-grains has given rise1 in like manner1 to the belief that they indicate a diurnal and nocturnal periodicity of activity and rest#ddf4 ,illustration3 ,fig4 #bjh4 ,that this is actually the case in growing starch-grains is generally believed1 on the authority of ,meyer#ddg2 but while under certain circumstances a marked alternation of growing and resting periods may occur1 and may leave its impress on the structure of the grain1 there is now great reason to believe that1 apart from such external influences1 the internal phenomena of diffusion may1 just as in the typical ,liesegang experiment1 produce the well-known concentric rings4 ,the spherocrystals of inulin1 in like manner1 shew1 like the 8calcospherites0 of ,harting 7,fig4 #bjh71 a concentric structure which in all likelihood has had no causative impulse save from within4 ,illustration3 ,fig4 #bji4 ,otoliths of ,plaice1 showing four zones or 8age-rings40 7,after ,wallace47 ,the striation1 or concentric lamellation1 of the scales and otoliths of fishes has been much employed of recent years as a trustworthy and unmistakeable mark of the fish's age4 ,there are difficulties in the way of accepting this hypothesis1 not the least of which is the fact that the otolith-zones1 for instance1 are extremely well marked even in the case of some fishes which spend their lives in deep water1 #dcc where the temperature and other physical conditions shew little or no appreciable fluctuation with the seasons of the year4 ,there are1 on the other hand1 phenomena which seem strongly confirmatory of the hypothesis3 for instance the fact 7if it be fully established7 that in such a fish as the cod1 zones of growth1 identical in number1 are found both on the scales and in the otoliths#ddh4 ,the subject has become a much debated one1 and this is not the place for its discussion2 but it is at least obvious1 with the ,liesegang phenomenon in view1 that we have no right to assume that an appearance of rhythm and periodicity in structure and growth is necessarily bound up with1 and indubitably brought about by1 a periodic recurrence of particular external conditions4 ,but while in the ,liesegang phenomenon we have rhythmic precipitation which depends only on forces intrinsic to the system1 and is independent of any corresponding rhythmic changes in temperature or other external conditions1 we have not far to seek for instances of chemico-physical phenomena where rhythmic alternations of appearance or structure are produced in close relation to periodic fluctuations of temperature4 ,a well-known instance is that of the ,stassfurt deposits1 where the rock-salt alternates regularly with thin layers of 8anhydrite10 or 7in another series of beds7 with 8polyhalite#ddi03 and where these zones are commonly regarded as marking years1 and their alternate bands as having been formed in connection with the seasons4 ,a discussion1 however1 of this remarkable and significant phenomenon1 and of how the chemist explains it1 by help of the 8phase-rule10 in connection with temperature conditions1 would lead us far beyond our scope#dej4 ,we now see that the methods by which we attempt to study the chemical or chemico-physical phenomena which accompany the development of an inorganic concretion or spicule within the #dcd body of an organism soon introduce us to a multitude of kindred phenomena1 of which our knowledge is still scanty1 and which we must not attempt to discuss at greater length4 ,as regards our main point1 namely the formation of spicules and other elementary skeletal forms1 we have seen that certain of them may be safely ascribed to simple precipitation or crystallisation of inorganic materials1 in ways more or less modified by the presence of albuminous or other colloid substances4 ,the effect of these latter is found to be much greater in the case of some crystallisable bodies than in others4 ,for instance1 ,harting1 and ,rainey also1 found as a rule that calcium oxalate was much less affected by a colloid medium than was calcium carbonate2 it shewed in their hands no tendency to form rounded concretions or 8calcospherites0 in presence of a colloid1 but continued to crystallise1 either normally1 or with a tendency to form needles or raphides4 ,it is doubtless for this reason that1 as we have seen1 crystals of calcium oxalate are so common in the tissues of plants1 while those of other calcium salts are rare4 ,but true calcospherites1 or spherocrystals1 of the oxalate are occasionally found1 for instance in certain ,cacti1 and ,btschli#dea has succeeded in making them artificially in ,harting's usual way1 that is to say by crystallisation in a colloid medium4 ,there link on to these latter observations1 and to the statement already quoted that calcareous deposits are associated with the dead products rather than with the living cells of the organism1 certain very interesting facts in regard to the solubility of salts in colloid media1 which have been made known to us of late1 and which go far to account for the presence 7apart from the form7 of calcareous precipitates within the organism#deb4 ,it has been shewn1 in the first place1 that the presence of albumin has a notable effect on the solubility in a watery solution of calcium salts1 increasing the solubility of the phosphate in a marked degree1 and that of the carbonate in still greater proportion2 but the #dce sulphate is only very little more soluble in presence of albumin than in pure water1 and the rarity of its occurrence within the organism is so far accounted for4 ,on the other hand1 the bodies derived from the breaking down of the albumins1 their 8catabolic0 products1 such as the peptones1 etc41 dissolve the calcium salts to a much less degree than albumin itself2 and in the case of the phosphate1 its solubility in them is scarcely greater than in water4 ,the probability is1 therefore1 that the actual precipitation of the calcium salts is not due to the direct action of carbonic acid1 etc4 on a more soluble salt 7as was at one time believed72 but to catabolic changes in the proteids of the organism1 which tend to throw down the salts already formed1 which had remained hitherto in albuminous solution4 ,the very slight solubility of calcium phosphate under such circumstances accounts for its predominance in1 for instance1 mammalian bone#dec2 and wherever1 in short1 the supply of this salt has been available to the organism4 ,to sum up1 we see that1 whether from food or from sea-water1 calcium sulphate will tend to pass but little into solution in the albuminoid substances of the body3 calcium carbonate will enter more freely1 but a considerable part of it will tend to remain in solution3 while calcium phosphate will pass into solution in considerable amount1 but will be almost wholly precipitated again1 as the albumin becomes broken down in the normal process of metabolism4 ,we have still to wait for a similar and equally illuminating study of the solution and precipitation of silica1 in presence of organic colloids4 ,from the comparatively small group of inorganic formations which1 arising within living organisms1 owe their form solely to precipitation or to crystallisation1 that is to say to chemical or other molecular forces1 we shall presently pass to that other and larger group which appear to be conformed in direct relation to the forms and the arrangement of the cells or other protoplasmic elements#ded4 #dcf ,the two principles of conformation are both illustrated in the spicular skeletons of the ,sponges4 ,illustration3 ,fig4 #baj4 ,close-packed calcospherites1 or so-called 8spicules10 of ,astrosclera4 7,after ,lister47 ,in a considerable number1 but withal a minority of cases1 the form of the sponge-spicule may be deemed sufficiently explained on the lines of ,harting's and ,rainey's experiments1 that is to say as the direct result of chemical or physical phenomena associated with the deposition of lime or of silica in presence of colloids#dee4 ,this is the case1 for instance1 with various small spicules of a globular or spheroidal form1 formed of amorphous silica1 concentrically striated within1 and often developing irregular knobs or tiny tubercles over their surfaces4 ,in the aberrant sponge ,astrosclera#def1 we have1 to begin with1 rounded1 striated discs or globules1 which in like manner are nothing more or less than the #dcg 8calcospherites0 of ,harting's experiments2 and as these grow they become closely aggregated together 7,fig4 #baj71 and assume an angular1 polyhedral form1 once more in complete accordance with the results of experiment#deg4 ,again1 in many ,monaxonid sponges1 we have irregularly shaped1 or branched spicules1 roughened or tuberculated by secondary superficial deposits1 and reminding one of the spicules of some ,alcyonaria4 ,these also must be looked upon as the simple result of chemical deposition1 the form of the deposit being somewhat modified in conformity with the surrounding tissues1 just as in the simple experiment the form of the concretionary precipitate is affected by the heterogeneity1 visible or invisible1 of the matrix4 ,lastly1 the simple needles of amorphous silica1 which constitute one of the commonest types of spicule1 call for little in the way of explanation2 they are accretions or deposits about a linear axis1 or fine thread of organic material1 just as the ordinary rounded calcospherite is deposited about some minute point or centre of crystallisation1 and as ordinary crystallisation is often started by a particle of atmospheric dust2 in some cases they also1 like the others1 are apt to be roughened by more irregular secondary deposits1 which probably1 as in ,harting's experiments1 appear in this irregular form when the supply of material has become relatively scanty4 ,our few foregoing examples1 diverse as they are in look and kind and ranging from the spicules of ,astrosclera or ,alcyonium to the otoliths of a fish1 seem all to have their free origin in some larger or smaller fluid-containing space1 or cavity of the body3 pretty much as ,harting's calcospheres made their appearance in the albuminous content of a dish4 ,but we now come at last to a much larger class of spicular and skeletal structures1 for whose regular and often complex forms some other explanation than the intrinsic forces of crystallisation or molecular adhesion is manifestly necessary4 ,as we enter on this subject1 which is certainly no small or easy one1 it may conduce to simplicity1 and to brevity1 #dch if we try to make a rough classification1 by way of forecast1 of the chief conditions which we are likely to meet with4 ,just as we look upon animals as constituted1 some of a vast number of cells1 and others of a single cell or of a very few1 and just as the shape of the former has no longer a visible relation to the individual shapes of its constituent cells1 while in the latter it is cell-form which dominates or is actually equivalent to the form of the organism1 so shall we find it to be1 with more or less exact analogy1 in the case of the skeleton4 ,for example1 our own skeleton consists of bones1 in the formation of each of which a vast number of minute living cellular elements are necessarily concerned2 but the form and even the arrangement of these bone-forming cells or corpuscles are monotonously simple1 and we cannot find in these a physical explanation of the outward and visible configuration of the bone4 ,it is as part of a far larger field of force1--in which we must consider gravity1 the action of various muscles1 the compressions1 tensions and bending moments due to variously distributed loads1 the whole interaction of a very complex mechanical system1--that we must explain 7if we are to explain at all7 the configuration of a bone4 ,in contrast to these massive skeletons1 or constituents of a skeleton1 we have other skeletal elements whose whole magnitude1 or whose magnitude in some dimension or another1 is commensurate with the magnitude of a single living cell1 or 7as comes to very much the same thing7 is comparable to the range of action of the molecular forces4 ,such is the case with the ordinary spicules of a sponge1 with the delicate skeleton of a ,radiolarian1 or with the denser and robuster shells of the ,foraminifera4 ,the effect of scale1 then1 of which we had so much to say in our introductory chapter on ,magnitude1 is bound to be apparent in the study of skeletal fabrics1 and to lead to essential differences between the big and the little1 the massive and the minute1 in regard to their controlling forces and their resultant forms4 ,and if all this be so1 and if the range of action of the molecular forces be in truth the important and fundamental thing1 then we may somewhat extend our statement of the case1 and include in it not only association with the living cellular elements of the body1 but also association with any bubbles1 drops1 vacuoles or vesicles which #dci may be comprised within the bounds of the organism1 and which are 7as their names and characters connote7 of the order of magnitude of which we are speaking4 ,proceeding a little farther in our classification1 we may conceive each little skeletal element to be associated1 in one case1 with a single cell or vesicle1 and in another with a cluster or 8system0 of consociated cells4 ,in either case there are various possibilities4 ,for instance1 the calcified or other skeletal material may tend to overspread the entire outer surface of the cell or cluster of cells1 and so tend accordingly to assume some configuration comparable to that of a fluid drop or of an aggregation of drops2 this1 in brief1 is the gist and essence of our story of the foraminiferal shell4 ,another common1 but very different condition will arise if1 in the case of the cell-aggregates1 the skeletal material tends to accumulate in the interstices between the cells1 in the partition-walls which separate them1 or in the still more restricted distribution indicated by the lines of junction between these partition-walls4 ,conditions such as these will go a very long way to help us in our understanding of many sponge-spicules and of an immense variety of radiolarian skeletons4 ,and lastly 7for the present71 there is a possible and very interesting case of a skeletal element associated with the surface of a cell1 not so as to cover it like a shell1 but only so as to pursue a course of its own within it1 and subject to the restraints imposed by such confinement to a curved and limited surface4 ,with this curious condition we shall deal immediately4 ,this preliminary and much simplified classification of skeletal forms 7as is evident enough7 does not pretend to completeness4 ,it leaves out of account some kinds of conformation and configuration with which we shall attempt to deal1 and others which we must perforce omit4 ,but nevertheless it may help to clear or to mark our way towards the subjects which this chapter has to consider1 and the conditions by which they are at least partially defined4 ,among the several possible1 or conceivable1 types of microscopic skeletons let us choose1 to begin with1 the case of a spicule1 more or less simply linear as far as its intrinsic powers of growth are #ddj concerned1 but which owes its now somewhat complicated form to a restraint imposed by the individual cell to which it is confined1 and within whose bounds it is generated4 ,the conception of a spicule developed under such conditions we owe to a distinguished physicist1 the late ,professor ,g4 ,f4 ,fitz,gerald4 ,many years ago1 ,sollas pointed out that if a spicule begin to grow in some particular way1 presumably under the control or constraint imposed by the organism1 it continues to grow by further chemical deposition in the same form or direction even after it has got beyond the boundaries of the organism or its cells4 ,this phenomenon is what we see in1 and this imperfect explanation goes so far to account for1 the continued growth in straight lines of the long calcareous spines of ,globigerina or ,hastigerina1 or the similarly radiating but siliceous spicules of many ,radiolaria4 ,in physical language1 if our crystalline structure has once begun to be laid down in a definite orientation1 further additions tend to accrue in a like regular fashion and in an identical direction2 and this corresponds to the phenomenon of so-called 8orientirte ,adsorption10 as described by ,lehmann4 ,in ,globigerina or in ,acanthocystis the long needles grow out freely into the surrounding medium1 with nothing to impede their rectilinear growth and their approximately radiate distribution4 ,but let us consider some simple cases to illustrate the forms which a spicule will tend to assume when1 striving 7as it were7 to grow straight1 it comes under the influence of some simple and constant restraint or compulsion4 ,if we take any two points on some curved surface1 such as that of a sphere or an ellipsoid1 and imagine a string stretched between them1 we obtain what is known in mathematics as a 8geodetic0 curve4 ,it is the shortest line which can be traced between the two points1 upon the surface itself2 and the most familiar of all cases1 from which the name is derived1 is that curve upon the earth's surface which the navigator learns to follow in the practice of 8great-circle sailing40 ,where the surface is spherical1 the geodetic is always literally a 8great circle10 a circle1 that is to say1 whose centre is the centre of the sphere4 ,if instead of a sphere we be dealing with an ellipsoid1 the geodetic becomes a variable figure1 according to the position of our two points4 #dda ,for obviously1 if they lie in a line perpendicular to the long axis of the ellipsoid1 the geodetic which connects them is a circle1 also perpendicular to that axis2 and if they lie in a line parallel to the axis1 their geodetic is a portion of that ellipse about which the whole figure is a solid of revolution4 ,but if our two points lie1 relatively to one another1 in any other direction1 then their geodetic is part of a spiral curve in space1 winding over the surface of the ellipsoid4 ,to say1 as we have done1 that the geodetic is the shortest line between two points upon the surface1 is as much as to say that it is a projection of some particular straight line upon the surface in question2 and it follows that1 if any linear body be confined to that surface1 while retaining a tendency to grow by successive increments always 7save only for its confinement to that surface7 in a straight line1 the resultant form which it will assume will be that of a geodetic4 ,in mathematical language1 it is a property of a geodetic that the plane of any two consecutive elements is a plane perpendicular to that in which the geodetic lies2 or1 in simpler words1 any two consecutive elements lie in a straight line in the plane of the surface1 and only diverge from a straight line in space by the actual curvature of the surface to which they are restrained4 ,illustration3 ,fig4 #baa4 ,sponge and ,holothurian spicules4 ,illustration3 ,fig4 #bab4 ,illustration3 ,fig4 #bac4 ,an 8amphidisc0 of ,hyalonema4 ,let us now imagine a spicule1 whose natural tendency is to grow into a straight linear element1 either by reason of its own molecular anisotropy1 or because it is deposited about a thread-like axis2 and let us suppose that it is confined either within a cell-wall or in adhesion thereto2 it at once follows that its line of growth will be simply a geodetic to the surface of the cell4 ,and if the cell be an imperfect sphere1 or a more or less regular ellipsoid1 the spicule will tend to grow into one or other of three forms3 either a plane curve of circular arc2 or1 more commonly1 a plane curve which is a portion of an ellipse2 or1 most commonly of all1 a curve which is a portion of a spiral in space4 ,in the latter case1 the number of turns of the spiral will depend1 not only on the length of the spicule1 but on the relative dimensions of the ellipsoidal cell1 as well as upon the angle by which the spicule is inclined to the ellipsoid axes2 but a very common case will probably be that in which the spicule looks at first sight to be #ddb a plane ,c-shaped figure1 but is discovered1 on more careful inspection1 to lie not in one plane but in a more complicated spiral twist4 ,this investigation includes a series of forms which are abundantly represented among actual sponge-spicules1 as illustrated in ,figs4 #baa and #bab4 ,if the spicule be not restricted to linear growth1 but have a tendency to expand1 or to branch out from a main axis1 we shall obtain a series of more complex figures1 all related to the geodetic system of curves4 ,a very simple case will arise where the spicule occupies1 in the first instance1 the axis of the containing cell1 and then1 on reaching its boundary1 tends to branch or spread outwards4 ,we shall now get various figures1 in some of which the spicule will appear as an axis expanding into a disc or wheel at either end2 and in other cases1 the terminal disc will be replaced1 or represented1 by a series of rays or spokes1 with a reflex curvature1 corresponding to the spherical or ellipsoid curvature of the surface of the cell4 ,such spicules as these are again exceedingly common among various sponges 7,fig4 #bac74 ,furthermore1 if these mechanical methods of conformation1 and others like to these1 be the true cause of the shapes which the spicules assume1 it is plain that the production of these spicular shapes is not a specific function of sponges or of any particular sponge1 but that we should expect #ddc the same or very similar phenomena to occur in other organisms1 wherever the conditions of inorganic secretion within closed cells was very much the same4 ,as a matter of fact1 in the group of ,holothuroidea1 where the formation of intracellular spicules is a characteristic feature of the group1 all the principal types of conformation which we have just described can be closely paralleled4 ,indeed in many cases1 the forms of the ,holothurian spicules are identical and indistinguishable from those of the sponges#deh4 ,but the ,holothurian spicules are composed of calcium carbonate while those which we have just described in the case of sponges are usually1 if not always1 siliceous3 this being just another proof of the fact that in such cases the form of the spicule is not due to its chemical nature or molecular structure1 but to the external forces to which1 during its growth1 the spicule is submitted4 ,so much for that comparatively limited class of sponge-spicules whose forms seem capable of explanation on the hypothesis that they are developed within1 or under the restraint imposed by1 the surface of a cell or vesicle4 ,such spicules are usually of small size1 as well as of comparatively simple form2 and they are greatly outstripped in number1 in size1 and in supposed importance as guides to zoological classification1 by another class of spicules4 ,this new class includes such as we have supposed to be capable of explanation on the assumption that they develop in association 7of some sort or another7 with the lines of junction of contiguous cells4 ,they include the triradiate spicules of the calcareous sponges1 the quadriradiate or 8tetractinellid0 spicules which occur in the same group1 but more characteristically in certain siliceous sponges known as the ,tetractinellidae1 and lastly perhaps 7though these last are admittedly somewhat harder to understand7 the six-rayed spicules of the ,hexactinellids4 ,the spicules of the calcareous sponges are commonly triradiate1 and the three radii are usually inclined to one another at equal1 or nearly equal angles2 in certain cases1 two of the three rays are nearly in a straight line1 and at right angles to the #ddd third#dei4 ,they are seldom in a plane1 but are usually inclined to one another in a solid1 trihedral angle1 not easy of precise measurement under the microscope4 ,the three rays are very often supplemented by a fourth1 which is set tetrahedrally1 making1 that is to say1 coequal angles with the other three4 ,the calcareous spicule consists mainly of carbonate of lime1 in the form of calcite1 with 7according to von ,ebner7 some admixture of soda and magnesia1 of sulphates and of water4 ,according to the same writer 7but the fact1 though it would seem easy to test1 is still disputed7 there is no organic matter in the spicule1 either in the form of an axial filament or otherwise1 and the appearance of stratification1 often simulating the presence of an axial fibre1 is due to 8mixed crystallisation0 of the various constituents4 ,the spicule is a true crystal1 and therefore its existence and its form are primarily due to the molecular forces of crystallisation2 moreover it is a single crystal and not a group of crystals1 as is at once seen by its behaviour in polarised light4 ,but its axes are not crystalline axes1 and its form neither agrees with1 nor in any way resembles1 any one of the many polymorphic forms in which calcite is capable of crystallising4 ,it is as though it were carved out of a solid crystal2 it is1 in fact1 a crystal under restraint1 a crystal growing1 as it were1 in an artificial mould2 and this mould is constituted by the surrounding cells1 or structural vesicles of the sponge4 ,illustration3 ,fig4 #bad4 ,spicules of ,grantia and other calcareous sponges4 7,after ,haeckel47 ,we have already studied in an elementary way1 but amply for our present purpose1 the manner in which three or more cells1 or bubbles1 tend to meet together under the influence of surface-tension1 and also the outwardly similar phenomena which may be brought about by a uniform distribution of mechanical pressure4 ,we have seen that when we confine ourselves to a plane assemblage of such bodies1 we find them meeting one another in threes2 that in a section or plane projection of such an assemblage we see the partition-walls meeting one another at equal angles of #abj2 that when the bodies are uniform in size1 the partitions are straight lines1 which combine to form regular hexagons2 and that when #dde the bodies are unequal in size1 the partitions are curved1 and combine to form other and less regular polygons4 ,it is plain1 accordingly1 that in any flattened or stratified assemblage of such cells1 a solidified skeletal deposit which originates or accumulates either between the cells or within the thickness of their mutual partitions1 will tend to take the form of triradiate bodies1 whose rays 7in a typical case7 will be set at equal angles of #abj 7,fig4 #bad1 ,f74 ,and this latter condition of equality will be open to modification in various ways4 ,it will be modified by any inequality in the specific tensions of adjacent cells2 as a special case1 it will be apt to be greatly modified at the surface of the system1 where a spicule happens to be formed in a plane perpendicular to the cell-layer1 so that one of its three rays lies between two adjacent cells and the other two are associated with the surface of contact between the cells and the surrounding medium2 in such a case 7as in the cases considered in connection with the forms of the cells themselves #ddf on p4 #cad71 we shall tend to obtain a spicule with two equal angles and one unequal 7,fig4 #bad1 ,a1 ,c74 ,in the last case1 the two outer1 or superficial rays1 will tend to be markedly curved4 ,again1 the equiangular condition will be departed from1 and more or less curvature will be imparted to the rays1 wherever the cells of the system cease to be uniform in size1 and when the hexagonal symmetry of the system is lost accordingly4 ,lastly1 although we speak of the rays as meeting at certain definite angles1 this statement applies to their axes1 rather than to the rays themselves4 ,for1 if the triradiate spicule be developed in the interspace between three juxtaposed cells1 it is obvious that its sides will tend to be concave1 for the interspace between our three contiguous equal circles is an equilateral1 curvilinear triangle2 and even if our spicule be deposited1 not in the space between our three cells1 but in the thickness of the intervening wall1 then we may recollect 7from p4 #big7 that the several partitions never actually meet at sharp angles1 but the angle of contact is always bridged over by a small accumulation of material 7varying in amount according to its fluidity7 whose boundary takes the form of a circular arc1 and which constitutes the 8bourrelet0 of ,plateau4 ,in any sample of the triradiate spicules of ,grantia1 or in any series of careful drawings1 such as those of ,haeckel among others1 we shall find that all these various configurations are precisely and completely illustrated4 ,the tetrahedral1 or rather tetractinellid1 spicule needs no explanation in detail 7,fig4 #bad1 ,d1 ,e74 ,for just as a triradiate spicule corresponds to the case of three cells in mutual contact1 so does the four-rayed spicule to that of a solid aggregate of four cells3 these latter tending to meet one another in a tetrahedral system1 shewing four edges1 at each of which four surfaces meet1 the edges being inclined to one another at equal angles of about #aji4 ,and even in the case of a single layer1 or superficial layer1 of cells1 if the skeleton originate in connection with all the edges of mutual contact1 we shall1 in complete and typical cases1 have a four-rayed spicule1 of which one straight limb will correspond to the line of junction between the three cells1 and the other three limbs 7which will then be curved limbs7 will correspond to the edges where two cells meet one another on the surface of the system4 #ddg ,but if such a physical explanation of the forms of our spicules is to be accepted1 we must seek at once for some physical agency by which we may explain the presence of the solid material just at the junctions or interfaces of the cells1 and for the forces by which it is confined to1 and moulded to the form of1 these intercellular or interfacial contacts4 ,it is to ,dreyer that we chiefly owe the physical or mechanical theory of spicular conformation which ,i have just described1--a theory which ultimately rests on the form assumed1 under surface-tension1 by an aggregation of cells or vesicles4 ,but this fundamental point being granted1 we have still several possible alternatives by which to explain the details of the phenomenon4 ,dreyer1 if ,i understand him aright1 was content to assume that the solid material1 secreted or excreted by the organism1 accumulated in the interstices between the cells1 and was there subjected to mechanical pressure or constraint as the cells got more and more crowded together by their own growth and that of the system generally4 ,as far as the general form of the spicules goes1 such explanation is not inadequate1 though under it we may have to renounce some of our assumptions as to what takes place at the outer surface of the system4 ,but in all 7or most7 cases where1 but a few years ago1 the concepts of secretion or excretion seemed precise enough1 we are now-a-days inclined to turn to the phenomenon of adsorption as a further stage towards the elucidation of our facts4 ,here we have a case in point4 ,in the tissues of our sponge1 wherever two cells meet1 there we have a definite surface of contact1 and there accordingly we have a manifestation of surface-energy2 and the concentration of surface-energy will tend to be a maximum at the lines or edges whereby the three1 or four1 such surfaces are conjoined4 ,of the micro-chemistry of the sponge-cells our ignorance is great2 but 7without venturing on any hypothesis involving the chemical details of the process7 we may safely assert that there is an inherent probability that certain substances will tend to be concentrated and ultimately deposited just in these lines of intercellular contact and conjunction4 ,in other words1 adsorptive concentration1 under osmotic pressure1 at and in the surface-film which constitutes the mutual boundary between contiguous #ddh cells1 emerges as an alternative 7and1 as it seems to me1 a highly preferable alternative7 to ,dreyer's conception of an accumulation under mechanical pressure in the vacant spaces left between one cell and another4 ,but a purely chemical1 or purely molecular adsorption1 is not the only form of the hypothesis on which we may rely4 ,for from the purely physical point of view1 angles and edges of contact between adjacent cells will be loci in the field of distribution of surface-energy1 and any material particles whatsoever will tend to undergo a diminution of freedom on entering one of those boundary regions4 ,in a very simple case1 let us imagine a couple of soap bubbles in contact with one another4 ,over the surface of each bubble there glide in every direction1 as usual1 a multitude of tiny bubbles and droplets2 but as soon as these find their way into the groove or re-entrant angle between the two bubbles1 there their freedom of movement is so far restrained1 and out of that groove they have little or no tendency to emerge4 ,a cognate phenomenon is to be witnessed in microscopic sections of steel or other metals4 ,here1 amid the 8crystalline0 structure of the metal 7where in cooling its imperfectly homogeneous material has developed a cellular structure1 shewing 7in section7 hexagonal or polygonal contours71 we can easily observe1 as ,professor ,peddie has shewn me1 that the little particles of graphite and other foreign bodies common in the matrix1 have tended to aggregate themselves in the walls and at the angles of the polygonal cells--this being a direct result of the diminished freedom which the particles undergo on entering one of these boundary regions#dfj4 ,it is by a combination of these two principles1 chemical adsorption on the one hand1 and physical quasi-adsorption or concentration of grosser particles on the other1 that ,i conceive the substance of the sponge-spicule to be concentrated and aggregated at the cell boundaries2 and the forms of the triradiate and tetractinellid spicules are in precise conformity with this hypothesis4 ,a few general matters1 and a few particular cases1 remain to be considered4 ,it matters little or not at all1 for the phenomenon in question1 #ddi what is the histological nature or 8grade0 of the vesicular structures on which it depends4 ,in some cases 7apart from sponges71 they may be no more than the little alveoli of the intracellular protoplasmic network1 and this would seem to be the case at least in one known case1 that of the protozoan ,entosolenia aspera1 in which1 within the vesicular protoplasm of the single cell1 ,mbius has described tiny spicules in the shape of little tetrahedra with concave sides4 ,it is probably also the case in the small beginnings of the ,echinoderm spicules1 which are likewise intracellular1 and are of similar shape4 ,in the case of our sponges we have many varying conditions1 which we need not attempt to examine in detail4 ,in some cases there is evidence for believing that the spicule is formed at the boundaries of true cells or histological units4 ,but in the case of the larger triradiate or tetractinellid spicules of the sponge-body1 they far surpass in size the actual 8cells02 we find them lying1 regularly and symmetrically arranged1 between the 8pore-canals0 or 8ciliated chambers10 and it is in conformity with the shape and arrangement of these rounded or spheroidal structures that their shape is assumed4 ,again1 it is not necessarily at variance with our hypothesis to find that1 in the adult sponge1 the larger spicules may greatly outgrow the bounds not only of actual cells but also of the ciliated chambers1 and may even appear to project freely from the surface of the sponge4 ,for we have already seen that the spicule is capable of growing1 without marked change of form1 by further deposition1 or crystallisation1 of layer upon layer of calcareous molecules1 even in an artificial solution2 and we are entitled to believe that the same process may be carried on in the tissues of the sponge1 without greatly altering the symmetry of the spicule1 long after it has established its characteristic form of a system of slender trihedral or tetrahedral rays4 ,neither is it of great importance to our hypothesis whether the rayed spicule necessarily arises as a single structure1 or does so from separate minute centres of aggregation4 ,minchin has shewn that1 in some cases at least1 the latter is the case2 the spicule begins1 he tells us1 as three tiny rods1 separate from one another1 each developed in the interspace between two sister-cells1 which are themselves the results of the division of one of a #dej little trio of cells2 and the little rods meet and fuse together while still very minute1 when the whole spicule is only about #a_/bjj of a millimetre long4 ,at this stage1 it is interesting to learn that the spicule is non-crystalline2 but the new accretions of calcareous matter are soon deposited in crystalline form4 ,this observation threw considerable difficulties in the way of former mechanical theories of the conformation of the spicule1 and was quite at variance with ,dreyer's theory1 according to which the spicule was bound to begin from a central nucleus coinciding with the meeting-place of the three contiguous cells1 or rather the interspace between them4 ,but the difficulty is removed when we import the concept of adsorption2 for by this agency it is natural enough1 or conceivable enough1 that the process of deposition should go on at separate parts of a common system of surfaces2 and if the cells tend to meet one another by their interfaces before these interfaces extend to the angles and so complete the polygonal cell1 it is again conceivable and natural that the spicule should first arise in the form of separate and detached limbs or rays4 ,illustration3 ,fig4 #bae4 ,spicules of tetractinellid sponges 7after ,sollas74 a-e1 anatriaenes2 d-f1 protriaenes4 ,among the tetractinellid sponges1 whose spicules are composed of amorphous silica or opal1 all or most of the above-described main types of spicule occur1 and1 as the name of the group implies1 the four-rayed1 tetrahedral spicules are especially represented4 ,a somewhat frequent type of spicule is one in which one of the four rays is greatly developed1 and the other three constitute small prongs diverging at equal angles from the main or axial ray4 ,in all probability1 as ,dreyer suggests1 we have here had to do with a group of four vesicles1 of which three were large and co-equal1 while a fourth and very much smaller one lay above and between the other three4 ,in certain cases where we have likewise one large and three much smaller #dea rays1 the latter are recurved1 as in ,fig4 #bae4 ,this type1 save for the constancy of the number of rays1 and the limitation of the terminal ones to three1 and save also for the more important difference that they occur only at one and not at both ends of the long axis1 is similar to the type of spicule illustrated in ,fig4 #bac1 which we have explained as being probably developed within an oval cell1 by whose walls its branches have been conformed to geodetic curves4 ,but it is much more probable that we have here to do with a spicule developed in the midst of a group of three coequal and more or less elongated or cylindrical cells or vesicles1 the long axial ray corresponding to their common line of contact1 and the three short rays having each lain in the surface furrow between two out of the three adjacent cells4 ,illustration3 ,fig4 #baf4 ,various holothurian spicules4 7,after ,thel47 ,just as in the case of the little curved or ,s-shaped spicules1 formed apparently within the bounds of a single cell1 so also in the case of the larger tetractinellid and analogous types do we find among the ,holothuroidea the same configurations reproduced as we have dealt with in the sponges4 ,the holothurian spicules are a little less neatly formed1 a little rougher1 than the sponge-spicules2 and certain forms occur among the former group which do not present themselves among the latter2 but for the most part a community of type is obvious and striking 7,fig4 #baf74 ,a curious and1 physically speaking1 strictly analogous formation to the tetrahedral spicules of the sponges is found in the #deb spores of a certain little group of parasitic protozoa1 the ,actinomyxidia4 ,these spores are formed from clusters of six cells1 of which three come to constitute the capsule of the spore2 and this capsule1 always triradiate in its symmetry1 is in some species drawn out into long rays1 of which one constitutes a straight central axis1 while the others1 coming off from it at equal angles1 are recurved in wide circular arcs4 ,the account given of the development of this structure by its discoverers#dfa is somewhat obscure to me1 but ,i think that1 on physical grounds1 there can be no doubt whatever that the quadriradiate capsule has been somehow modelled upon a group of three surrounding cells1 its axis lying between the three1 and its three radial arcs occupying the furrows between adjacent pairs4 ,illustration3 ,fig4 #bag4 ,spicules of hexactinellid sponges4 7,after ,f4 ,e4 ,schultze47 ,the typically six-rayed siliceous spicules of the hexactinellid sponges1 while they are perhaps the most regular and beautifully formed spicules to be found within the entire group1 have been found very difficult to explain1 and ,dreyer has confessed his complete inability to account for their conformation4 ,but1 though it is doubtless only throwing the difficulty a little further back1 we may so far account for them by considering that the cells or vesicles by which they are conformed are not arranged in #dec what is known as 8closest packing10 but in linear series2 so that in their arrangement1 and by their mutual compression1 we tend to get a pattern1 not of hexagons1 but of squares3 or1 looking to the solid1 not of dodecahedra but of cubes or parallelopipeda4 ,this indeed appears to be the case1 not with the individual cells 7in the histological sense71 but with the larger units or vesicles which make up the body of the hexactinellid4 ,and this being so1 the spicules formed between the linear1 or cubical series of vesicles1 will have the same tendency towards a 8hexactinellid0 shape1 corresponding to the angles and adjacent edges of a system of cubes1 as in our former case they had to a triradiate or a tetractinellid form1 when developed in connection with the angles and edges of a system of hexagons1 or a system of dodecahedra4 ,histologically1 the case is illustrated by a well-known phenomenon in embryology4 ,in the segmenting ovum1 there is a tendency for the cells to be budded off in linear series2 and so they often remain1 in rows side by side1 at least for a considerable time and during the course of several consecutive cell divisions4 ,such an arrangement constitutes what the embryologists call the 8radial type0 of segmentation#dfb4 ,but in what is described as the 8spiral type0 of segmentation1 it is stated that1 as soon as the first horizontal furrow has divided the cells into an upper and a lower layer1 those of 8the upper layer are shifted in respect to the lower layer1 by means of a rotation about the vertical axis#dfc40 ,it is1 of course1 evident that the whole process is merely that which is familiar to physicists as 8close packing40 ,it is a very simple case of what ,lord ,kelvin used to call 8a problem in tactics40 ,it is a mere question of the rigidity of the system1 of the freedom of movement on the part of its constituent cells1 whether or at what stage this tendency to slip into the closest propinquity1 or position of minimum potential1 will be found to manifest itself4 ,however the hexactinellid spicules be arranged 7and this is #ded not at all easy to determine7 in relation to the tissues and chambers of the sponge1 it is at least clear that1 whether they be separate or be fused together 7as often happens7 in a composite skeleton1 they effect a symmetrical partitioning of space according to the cubical system1 in contrast to that closer packing which is represented and effected by the tetrahedral system#dfd4 ,this question of the origin and causation of the forms of sponge-spicules1 with which we have now briefly dealt1 is all the more important and all the more interesting because it has been discussed time and again1 from points of view which are characteristic of very different schools of thought in biology4 ,haeckel found in the form of the sponge-spicule a typical illustration of his theory of 8bio-crystallisation02 he considered that these 8biocrystals0 represented 8something midway--ein ,mittelding--between an inorganic crystal and an organic secretion02 that there was a 8compromise between the crystallising efforts of the calcium carbonate and the formative activity of the fused cells of the syncytium02 and that the semi-crystalline secretions of calcium carbonate 8were utilised by natural selection as #8spicules' for building up a skeleton1 and afterwards1 by the interaction of adaptation and heredity1 became modified in form and differentiated in a vast variety of ways in the struggle for existence#dfe40 ,what ,haeckel precisely signified by these words is not clear to me4 ,f4 ,e4 ,schultze1 perceiving that identical forms of spicule were developed whether the material were crystalline or non-crystalline1 abandoned all theories based upon crystallisation2 he simply saw in the form and arrangement of the spicules something which was 8best fitted0 for its purpose1 that is to say for the support and strengthening of the porous walls of the sponge1 and found clear evidence of 8utility0 in the specific structure of these skeletal elements4 #dee ,sollas and ,dreyer1 as we have seen1 introduced in various ways the conception of physical causation1--as indeed ,haeckel himself had done in regard to one particular1 when he supposed the position of the spicules to be due to the constant passage of the water-currents4 ,though even here1 by the way1 if ,i understand ,haeckel aright1 he was thinking not merely of a direct or immediate physical causation1 but of one manifesting itself through the agency of natural selection#dff4 ,sollas laid stress upon the 8path of least resistance0 as determining the direction of growth2 while ,dreyer dealt in greater detail with the various tensions and pressures to which the growing spicule was exposed1 amid the alveolar or vesicular structure which was represented alike by the chambers of the sponge1 by the reticulum of constituent cells1 or by the minute structure of the intracellular protoplasm4 ,but neither of these writers1 so far as ,i can discover1 was inclined to doubt for a moment the received canon of biology1 which sees in such structures as these the characteristics of true organic species1 and the indications of an hereditary affinity by which blood-relationship and the succession of evolutionary descent throughout geologic time can be ultimately deduced4 ,lastly1 ,minchin1 in a well-known paper#dfg1 took sides with ,schultze1 and gave reasons for dissenting from such mechanical theories as those of ,sollas and of ,dreyer4 ,for example1 after pointing out that all protoplasm contains a number of 8granules0 or microsomes1 contained in the alveolar framework and lodged at the nodes of the reticulum1 he argued that these also ought to acquire a form such as the spicules possess1 if it were the case that these latter owed their form to their very similar or identical position4 8,if vesicular tension cannot in any other instance cause the granules at the nodes to assume a tetraxon form1 why should it do so for the sclerites80 ,in all probability the answer to this question is not far to seek4 ,if the force which the 8mechanical0 hypothesis has in view were simply that of mechanical pressure1 #def as between solid bodies1 then indeed we should expect that any substances whatsoever1 lying between the impinging spheres1 would tend 7unless they were infinitely hard7 to assume the quadriradiate or 8tetraxon0 form2 but this conclusion does not follow at all1 in so far as it is to surface-energy that we ascribe the phenomenon4 ,here the specific nature of the substances involved makes all the difference4 ,we cannot argue from one substance to another2 adsorptive attraction shews its effect on one and not on another2 and we have not the least reason to be surprised if we find that the little granules of protoplasmic material1 which as they lie bathed in the more fluid protoplasm have 7presumably1 and as their shape indicates7 a strong surface-tension of their own1 behave towards the adjacent vesicles in a very different fashion to the incipient aggregations of calcareous or siliceous matter in a colloid medium4 8,the ontogeny of the spicules10 says ,professor ,minchin1 8points clearly to their regular form being a phylogenetic adaptation1 which has become fixed and handed on by heredity1 appearing in the ontogeny as a prophetic adaptation40 ,and again1 8,the forms of the spicules are the result of adaptation to the requirements of the sponge as a whole1 produced by the action of natural selection upon variation in every direction40 ,it would scarcely be possible to illustrate more briefly and more cogently than by these few words 7or the similar words of ,haeckel quoted on p4 #ded71 the fundamental difference between the ,darwinian conception of the causation and determination of ,form1 and that which is characteristic of the physical sciences4 ,if ,i have dealt comparatively briefly with the inorganic skeleton of sponges1 in spite of the obvious importance of this part of our subject from the physical or mechanical point of view1 it has been owing to several reasons4 ,in the first place1 though the general trend of the phenomena is clear1 it must be at once admitted that many points are obscure1 and could only be discussed at the cost of a long argument4 ,in the second place1 the physical theory is 7as ,i have shewn7 in manifest conflict with the accounts given by various embryologists of the development of the spicules1 and of the current biological theories which their descriptions embody2 it is beyond our scope to deal with such descriptions #deg in detail4 ,lastly1 we find ourselves able to illustrate the same physical principles with greater clearness and greater certitude in another group of animals1 namely the ,radiolaria4 ,in our description of the skeletons occurring within this group we shall by no means abandon the preliminary classification of microscopic skeletons which we have laid down2 but we shall have occasion to blend with it the consideration of certain other more or less correlated phenomena4 ,the group of microscopic organisms known as the ,radiolaria is extraordinarily rich in diverse forms1 or 8species40 ,i do not know how many of such species have been described and defined by naturalists1 but some thirty years ago the number was said to be over four thousand1 arranged in more than seven hundred genera#dfh4 ,of late years there has been a tendency to reduce the number1 it being found that some of the earlier species and even genera are but growth-stages of one and the same form1 sometimes mere fragments or 8fission-products0 common to several species1 or sometimes forms so similar and so interconnected by intermediate forms that the naturalist denominates them not 8species0 but 8varieties40 ,it has to be admitted1 in short1 that the conception of species among the ,radiolaria has not hitherto been1 and is not yet1 on the same footing as that among most other groups of animals4 ,but apart from the extraordinary multiplicity of forms among the ,radiolaria1 there are certain other features in this multiplicity which arrest our attention4 ,for instance1 the distribution of species in space is curious and vague2 many species are found all over the world1 or at least every here and there1 with no evidence of specific limitations of geographical habitat2 others occur in the neighbourhood of the two poles2 some are confined to warm and others to cold currents of the ocean4 ,in time also their distribution is not less vague3 so much so that it has been asserted of them that 8from the ,cambrian age downwards1 the families and even genera appear identical with those now living40 ,lastly1 except perhaps in the case of a few large 8colonial forms10 we seldom if ever find1 as is usual #deh in most animals1 a local predominance of one particular species4 ,on the contrary1 in a little pinch of deep-sea mud or of some fossil 8,radiolarian earth10 we shall probably find scores1 and it may be even hundreds1 of different forms4 ,moreover1 the radiolarian skeletons are of quite extraordinary delicacy and complexity1 in spite of their minuteness and the comparative simplicity of the 8unicellular0 organisms within which they grow2 and these complex conformations have a wonderful and unusual appearance of geometric regularity4 ,all these general considerations seem such as to prepare us for the special need of some physical hypothesis of causation4 ,the little skeletal fabrics remind us of such objects as snow-crystals 7themselves almost endless in their diversity71 rather than of a collection of distinct animals1 constructed in apparent accordance with functional needs1 and distributed in accordance with their fitness for particular situations4 ,nevertheless great efforts have been made of recent years to attach 8a biological meaning0 to these elaborate structures2 and 8to justify the hope that in time the utilitarian character of the skeleton will be more completely recognised#dfi40 ,in the majority of cases1 the skeleton of the ,radiolaria is composed1 like that of so many sponges1 of silica2 in one large family1 the ,acantharia 7and perhaps in some others71 it is composed1 in great part at least1 of a very unusual constituent1 namely strontium sulphate#dgj4 ,there is no fundamental or important morphological character in which the shells formed of these two constituents differ from one another2 and in no case can the chemical properties of these inorganic materials be said to influence the form of the complex skeleton or shell1 save only in this general way that1 by their rigidity and toughness1 they may give rise to a fabric far more delicate and slender than we find developed among calcareous organisms4 ,a slight exception to this rule is found in the presence of true crystals1 which occur within the central capsules of certain #dei ,radiolaria1 for instance the genus ,collosphaera#dga4 ,johannes ,mller 7whose knowledge and insight never fail to astonish us7 remarked that these were identical in form with crystals of celestine1 a sulphate of strontium and barium2 and ,btschli's discovery of sulphates of strontium and of barium in kindred forms render it all but certain that they are actually true crystals of celestine#dgb4 ,in its typical form1 the ,radiolarian body consists of a spherical mass of protoplasm1 around which1 and separated from it by some sort of porous 8capsule10 lies a frothy mass1 composed of protoplasm honeycombed into a multitude of alveoli or vacuoles1 filled with a fluid which can scarcely differ much from sea-water#dgc4 ,according to their surface-tension conditions1 these vacuoles may appear more or less isolated and spherical1 or joining together in a 8froth0 of polygonal cells2 and in the latter1 which is the commoner condition1 the cells tend to be of equal size1 and the resulting polygonal meshwork beautifully regular4 ,in many cases1 a large number of such simple individual organisms are associated together1 forming a floating colony1 and it is highly probable that many other forms1 with whose scattered skeletons we are alone acquainted1 had in life formed part likewise of a colonial organism4 ,in contradistinction to the sponges1 in which the skeleton always begins as a loose mass of isolated spicules1 which only in a few exceptional cases 7such as ,euplectella and ,farrea7 fuse into a continuous network1 the characteristic feature of the ,radiolarians lies in the possession of a continuous skeleton1 in the form of a netted mesh or perforated lacework1 sometimes however replaced by and often associated with minute independent spicules4 ,before we proceed to treat of the more complex skeletons1 we may begin1 then1 by dealing with these comparatively simple cases where either the entire skeleton or a considerable part of it is represented1 not by a continuous fabric1 but by a quantity of loose1 separate spicules1 or aciculae1 which seem1 like the spicules of ,alcyonium1 #dfj to be developed as free and isolated formations or deposits1 precipitated in the colloid matrix1 with no relation of form to the cellular or vesicular boundaries4 ,these simple acicular spicules occupy a definite position in the organism4 ,sometimes1 as for instance among the fresh-water ,heliozoa 7e4g4 ,raphidiophrys71 they lie on the outer surface of the organism1 and not infrequently 7when the spicules are few in number7 they tend to collect round the bases of the pseudopodia1 or around the large radiating spicules1 or axial rays1 in the cases where these latter are present4 ,when the spicules are thus localised around some prominent centre1 they tend to take up a position of symmetry in regard to it2 instead of forming a tangled or felted layer1 they come to lie side by side1 in a radiating cluster round the focus4 ,in other cases 7as for instance in the well-known ,radiolarian ,aulacantha scolymantha7 the felted layer of aciculae lies at some depth below the surface1 forming a sphere concentric with the entire spherical organism4 ,in either case1 whether the layer of spicules be deep or be superficial1 it tends to mark a 8surface of discontinuity10 a meeting place between two distinct layers of protoplasm or between the protoplasm and the water around2 and it is obvious that1 in either case1 there are manifestations of surface-energy at the boundary1 which cause the spicules to be retained there1 and to take up their position in its plane4 ,the case is somewhat1 though not directly1 analogous to that of a cirrus cloud1 which marks the place of a surface of discontinuity in a stratified atmosphere4 ,illustration3 ,fig4 #bah4 ,we have1 then1 to enquire what are the conditions which shall1 apart from gravity1 confine an extraneous body to a surface-film2 and we may do this very simply1 by considering the surface-energy of the entire system4 ,in ,fig4 #bah we have two fluids in contact with one another 7let us call them water and protoplasm71 and a body 7b7 which may be immersed in either1 or may be restricted to the boundary #dfa between4 ,we have here three possible 8interfacial contacts0 each with its own specific surface-energy1 per unit of surface area3 namely1 that between our particle and the water 7let us call it 71 that between the particle and the protoplasm 771 and that between water and protoplasm 774 ,when the body lies in the boundary of the two fluids1 let us say half in one and half in the other1 the surface-energies concerned are equivalent to 7,s_/#b7 7,s_/#b72 but we must also remember that1 by the presence of the particle1 a small portion 7equal to its sectional area s7 of the original contact-surface between water and protoplasm has been obliterated1 and with it a proportionate quantity of energy1 equivalent to s1 has been set free4 ,when1 on the other hand1 the body lies entirely within one or other fluid1 the surface-energies of the system 7so far as we are concerned7 are equivalent to ,s s1 or ,s s1 as the case may be4 ,according as be less or greater than 1 the particle will have a tendency to remain immersed in the water or in the protoplasm2 but if 7,s_/#b77 7 s be less than either ,s or ,s1 then the condition of minimal potential will be found when the particle lies1 as we have said1 in the boundary zone1 half in one fluid and half in the other2 and1 if we were to attempt a more general solution of the problem1 we should evidently have to deal with possible conditions of equilibrium under which the necessary balance of energies would be attained by the particle rising or sinking in the boundary zone1 so as to adjust the relative magnitudes of the surface-areas concerned4 ,it is obvious that this principle may1 in certain cases1 help us to explain the position even of a radial spicule1 which is just a case where the surface of the solid spicule is distributed between the fluids with a minimal disturbance1 or minimal replacement1 of the original surface of contact between the one fluid and the other4 ,in like manner we may provide for the case 7a common and an important one7 where the protoplasm 8creeps up0 the spicule1 covering it with a delicate film4 ,in ,acanthocystis we have yet another special case1 where the radial spicules plunge only a certain distance into the protoplasm of the cell1 being arrested at a boundary-surface between an inner and an outer layer of cytoplasm2 here we have only to assume that there is a tension #dfb at this surface1 between the two layers of protoplasm1 sufficient to balance the tensions which act directly on the spicule#dgd4 ,in various ,acanthometridae1 besides such typical characters as the radial symmetry1 the concentric layers of protoplasm1 and the capillary surfaces in which the outer1 vacuolated protoplasm is festooned upon the projecting radii1 we have another curious feature4 ,on the surface of the protoplasm where it creeps up the sides of the long radial spicules1 we find a number of elongated bodies1 forming in each case one or several little groups1 and lying neatly arranged in parallel bundles4 ,a ,russian naturalist1 ,schewiakoff1 whose views have been accepted in the text-books1 tells us that these are muscular structures1 serving to raise or lower the conical masses of protoplasm about the radial spicules1 which latter serve as so many 8tent-poles0 or masts1 on which the protoplasmic membranes are hoisted up2 and the little elongated bodies are dignified with various names1 such as 8myonemes0 or 8myophriscs10 in allusion to their supposed muscular nature#dge4 ,this explanation is by no means convincing4 ,to begin with1 we have precisely similar festoons of protoplasm in a multitude of other cases where the 8myonemes0 are lacking2 from their minute size 7#jjf-jab mm47 and the amount of contraction they are said to be capable of1 the myonemes can hardly be very efficient instruments of traction2 and further1 for them to act 7as is alleged7 for a specific purpose1 namely the 8hydrostatic regulation0 of the organism giving it power to sink or to swim1 would seem to imply a mechanism of action and of coordination which is difficult to conceive in these minute and simple organisms4 ,the fact is 7as it seems to me71 that the whole method of explanation is unnecessary4 ,just as the supposed 8hauling up0 of the protoplasmic festoons is at once explained by capillary phenomena1 so also1 in all probability1 is the position and arrangement of the little elongated bodies4 ,whatever the actual nature of these bodies may be1 whether they are truly portions of differentiated protoplasm1 or whether they are foreign bodies or spicular structures 7as bodies occupying a similar position in other cases undoubtedly are71 we can explain their situation on the surface #dfc of the protoplasm1 and their arrangement around the radial spicules1 all on the principles of surface-tension#dgf4 ,this last case is not of the simplest2 and ,i do not forget that my explanation of it1 which is wholly theoretical1 implies a doubt of ,schewiakoff's statements1 which are founded on direct personal observation4 ,this ,i am none too willing to do2 but whether it be justly done in this case or not1 ,i hold that it is in principle justifiable to look with great suspicion upon a number of kindred statements where it is obvious that the observer has left out of account the purely physical aspect of the phenomenon1 and all the opportunities of simple explanation which the consideration of that aspect might afford4 ,whether it be wholly applicable to this particular and complex case or no1 our general theorem of the localisation and arrestment of solid particles in a surface-film is of very great biological importance2 for on it depends the power displayed by many little naked protoplasmic organisms of covering themselves with an 8agglutinated0 shell4 ,sometimes1 as in ,difflugia1 ,astrorhiza 7,fig4 #bai7 and others1 this covering consists of sand-grains picked up from the surrounding medium1 and sometimes1 on the other hand1 as in ,quadrula1 it consists of solid particles which are said to arise1 as inorganic deposits or concretions1 within the protoplasm itself1 and which find their way outwards to a position of equilibrium in the surface-layer2 and in both cases1 the mutual capillary attractions between the particles1 confined to the boundary-layer but enjoying a certain measure of freedom therein1 tends to the orderly arrangement of the particles one with another1 and even to the appearance of a regular 8pattern0 as the result of this arrangement4 ,illustration3 ,fig4 #bai4 ,arenaceous ,foraminifera2 ,astrorhiza limicola and arenaria4 7,from ,brady's ,challenger ,monograph47 ,the 8picking up0 by the protoplasmic organism of a solid particle with which 8to build its house0 7for it is hard to avoid this customary use of anthropomorphic figures of speech1 misleading though they be71 is a physical phenomenon kindred to that by which an ,amoeba 8swallows0 a particle of food4 ,this latter process has been reproduced or imitated in various pretty experimental #dfe ways4 ,for instance1 ,rhumbler has shewn that if a thread of glass be covered with shellac and brought near a drop of chloroform suspended in water1 the drop takes in the spicule1 robs it of its shellac covering1 and then passes it out again#dgg4 ,it is all a question of relative surface-energies1 leading to different degrees of 8adhesion0 between the chloroform and the glass or its covering4 ,thus it is that the ,amoeba takes in the diatom1 dissolves off its proteid covering1 and casts out the shell4 ,furthermore1 as the whole phenomenon depends on a distribution of surface-energy1 the amount of which is specific to certain particular substances in contact with one another1 we have no difficulty in understanding the selective action1 which is very often a conspicuous feature in the phenomenon#dgh4 ,just as some caddis-worms make their houses of twigs1 and others of shells and again others of stones1 so some ,rhizopods construct their agglutinated 8test0 out of stray sponge-spicules1 or frustules of diatoms1 or again of tiny mud particles or of larger grains of sand4 ,in all these cases1 we have apparently to deal with differences in specific #dff surface-energies1 and also doubtless with differences in the total available amount of surface-energy in relation to gravity or other extraneous forces4 ,in my early student days1 ,wyville ,thomson used to tell us that certain deep-sea 8,difflugias10 after constructing a shell out of particles of the black volcanic sand common in parts of the ,north ,atlantic1 finished it off with 8a clean white collar0 of little grains of quartz4 ,even this phenomenon may be accounted for on surface-tension principles1 if we assume that the surface-energy ratios have tended to change1 either with the growth of the protoplasm or by reason of external variation of temperature or the like2 and we are by no means obliged to attribute the phenomenon to a manifestation of volition1 or taste1 or aesthetic skill1 on the part of the microscopic organism4 ,nor1 when certain ,radiolaria tend more than others to attract into their own substance diatoms and such-like foreign bodies1 is it scientifically correct to speak1 as some text-books do1 of species 8in which diatom selection has become a regular habit40 ,to do so is an exaggerated misuse of anthropomorphic phraseology4 ,the formation of an 8agglutinated0 shell is thus seen to be a purely physical phenomenon1 and indeed a special case of a more general physical phenomenon which has many other important consequences in biology4 ,for the shell to assume the solid and permanent character which it acquires1 for instance1 in ,difflugia1 we have only to make the additional assumption that some small quantities of a cementing substance are secreted by the animal1 and that this substance flows or creeps by capillary attraction between all the interstices of the little quartz grains1 and ends by binding them all firmly together4 ,rhumbler#dgi has shewn us how these agglutinated tests1 of spicules or of sand-grains1 can be precisely imitated1 and how they are formed with greater or less ease1 and greater or less rapidity1 according to the nature of the materials employed1 that is to say1 according to the specific surface-tensions which are involved4 ,for instance if we mix up a little powdered glass with chloroform1 and set a drop of the mixture in water1 the glass particles gather neatly round the surface of the drop so quickly that the eye cannot follow the #dfg operation4 ,if we perform the same experiment with oil and fine sand1 dropped into #gj per cent4 alcohol1 a still more beautiful artificial ,rhizopod shell is formed1 but it takes some three hours to do4 ,it is curious that1 just at the very time when ,rhumbler was thus demonstrating the purely physical nature of the ,difflugian shell1 ,verworn was studying the same and kindred organisms from the older standpoint of an incipient psychology#dhj4 ,but1 as ,rhumbler himself admits1 ,verworn was very careful not to overestimate the apparent signs of volition1 or selective choice1 in the little organism's use of the material of its dwelling4 ,this long parenthesis has led us away1 for the time being1 from the subject of the ,radiolarian skeleton1 and to that subject we must now return4 ,leaving aside1 then1 the loose and scattered spicules1 which we have sufficiently discussed1 the more perfect ,radiolarian skeletons consist of a continuous and regular structure2 and the siliceous 7or other inorganic7 material of which this framework is composed tends to be deposited in one or other of two ways or in both combined3 7#a7 in the form of long spicular axes1 usually conjoined at1 or emanating from1 the centre of the protoplasmic body1 and forming a symmetric radial system2 7#b7 in the form of a crust1 developed in various ways1 either on the outer surface of the organism or in relation to the various internal surfaces which separate its concentric layers or its component vesicles4 ,not unfrequently1 this superficial skeleton comes to constitute a spherical shell1 or a system of concentric or otherwise associated spheres4 ,illustration3 ,fig4 #bbj4 8,reticulum plasmatique40 7,after ,carnoy47 ,we have already learned that a great part of the body of the ,radiolarian1 and especially that outer portion to which ,haeckel has given the name of the 8calymma10 is built up of a great mass of 8vesicles10 forming a sort of stiff froth1 and equivalent in the physical sense 7though not necessarily in the biological sense7 to 8cells10 inasmuch as the little vesicles have their own well-defined boundaries1 and their own surface phenomena4 ,in short1 all that we have said of cell-surfaces1 and cell conformations1 in our discussion of cells and of tissues1 will apply in like manner1 and under appropriate conditions1 to these4 ,in certain cases1 even in #dfh so common and simple a one as the vacuolated substance of an ,actinosphaerium1 we may see a very close resemblance1 or formal analogy1 to an ordinary cellular or 8parenchymatous0 tissue1 in the close-packed arrangement and consequent configuration of these vesicles1 and even at times in a slight membranous hardening of their walls4 ,leidy has figured#dha some curious little bodies1 like small masses of consolidated froth1 which seem to be nothing else than the dead and empty husks1 or filmy skeletons1 of ,actinosphaerium4 ,and ,carnoy#dhb has demonstrated in certain cell-nuclei an all but precisely similar framework1 of extreme delicacy and minuteness1 as the result of partial solidification of interstitial matter in a close-packed system of alveoli 7,fig4 #bbj74 ,illustration3 ,fig4 #bba4 ,aulonia hexagona1 ,hkl4 ,illustration3 ,fig4 #bbb4 ,actinomma arcadophorum1 ,hkl4 ,let us now suppose that1 in our ,radiolarian1 the outer surface of the animal is covered by a layer of froth-like vesicles1 uniform or nearly so in size4 ,we know that their tensions will tend to conform them into a 8honeycomb10 or regular meshwork of hexagons1 and that the free end of each hexagonal prism will be a little spherical cap4 ,suppose now that it be at the outer surface of the protoplasm 7that namely which is in contact with the surrounding sea-water71 that the siliceous particles have a tendency to be secreted or adsorbed2 it will at once follow that they will show a tendency to aggregate in the grooves which separate the vesicles1 and the result will be the development of a most delicate sphere composed of tiny rods arranged in a regular hexagonal network 7e4g4 ,aulonia74 ,such a conformation is #dfi extremely common1 and among its many variants may be found cases in which 7e4g4 ,actinomma71 the vesicles have been less regular in size1 and some in which the hexagonal meshwork has been developed not only on one outer surface1 but at successive #dgj surfaces1 producing a system of concentric spheres4 ,if the siliceous material be not limited to the linear junctions of the cells1 but spread over a portion of the outer spherical surfaces or caps1 then we shall have the condition represented in ,fig4 #bbc 7,ethmosphaera71 where the shell appears perforated by circular instead of hexagonal apertures1 and the circular pores are set on slight spheroidal eminences2 and1 interconnected with such types as this1 we have others in which the accumulating pellicles of skeletal matter have extended from the edges into the substance of the boundary walls and have so produced a system of films1 normal to the surface of the sphere1 constituting a very perfect honeycomb1 as in ,cenosphaera favosa and vesparia#dhc4 ,illustration3 ,fig4 #bbc4 ,ethmosphaera conosiphonia1 ,hkl4 ,illustration3 ,fig4 #bbd4 ,portions of shells of two 8species0 of ,cenosphaera3 upper figure1 ,c4 favosa1 lower1 ,c4 vesparia1 ,hkl4 ,in one or two very simple forms1 such as the fresh-water ,clathrulina1 just such a spherical perforated shell is produced out of some organic1 acanthin-like substance2 and in some examples of ,clathrulina the chitinous lattice-work of the shell is just as #dga regular and delicate1 with the meshes just as beautifully hexagonal1 as in the siliceous shells of the oceanic ,radiolaria4 ,this is only another proof 7if proof be needed7 that the peculiar conformation of these little skeletons is not due to the material of which they are composed1 but to the moulding of that material upon an underlying vesicular structure4 ,illustration3 ,fig4 #bbe4 ,aulastrum triceros1 ,hkl4 ,let us next suppose that1 upon some such lattice-work as has just been described1 another and external layer of cells or vesicles is developed1 and that instead of 7or perhaps only in addition to7 a second hexagonal lattice-work1 which might develop concentrically to the first in the boundary-furrows of this new layer of cells1 the siliceous matter now tends to be deposited radially1 or normally to the surface of the sphere1 just in the lines where the external layer of vesicles meet one another1 three by three4 ,the result will be that1 when the vesicles themselves are removed1 a series of radiating spicules will be revealed1 directed outwards from each of the angles of the original hexagon2 as is seen in ,fig4 #bbe4 ,and it may further happen that these radiating skeletal rods are continued at their distal ends into divergent rays1 forming a triple fork1 and corresponding 7after a fashion #dgb which we have already described as occurring in certain sponge-spicules7 to the three superficial furrows between the adjacent cells4 ,this last is1 as it were1 an intermediate stage between the simple rods and the complete formation of another concentric sphere of latticed hexagons4 ,another possible case is when the large and uniform vesicles of the outer protoplasm are mixed with1 or replaced by1 much smaller vesicles1 piled on one another in more or less concentric layers2 in this case the radiating rods will no longer be straight1 but will be bent into a zig-zag pattern1 with angles in three vertical planes1 corresponding to the successive contacts of the groups of cells around the axis 7,fig4 #bbf74 ,illustration3 ,fig4 #bbf4 ,illustration3 ,fig4 #bbg4 ,a ,nassellarian skeleton1 ,callimitra carolotae1 ,hkl4 ,among a certain group called the ,nassellaria1 we find geometrical forms of peculiar simplicity and beauty1--such for instance as that which ,i have represented in ,fig4 #bbg4 ,it is obvious at a glance that this is such a skeleton as may have been formed #dgc 7,i think we may go so far as to say must have been formed7 at the interfaces of a little tetrahedral group of cells1 the four equal cells of the tetrahedron being in this particular case supplemented by a little one in the centre of the system4 ,we see1 precisely as in the internal boundary-system of an artificial group of four soap-bubbles1 the plane surfaces of contact1 six in number2 the relation to one another of each triple set of interfacial planes1 meeting one another at equal angles of #abj2 and finally the relation of the four lines or edges of triple contact1 which tend 7but for the little central vesicle7 to meet at co-equal solid angles in the centre of the system1 all as we have described on p4 #cah4 ,in short1 each triple-walled re-entrant angle of the little shell has essentially the configuration 7or a part thereof7 of what we have called a 8,maraldi pyramid0 in our account of the architecture of the honeycomb1 on p4 #cbidhd4 ,there are still two or three remarkable or peculiar features in this all but mathematically perfect shell1 and they are in part easy and in part they seem more difficult of interpretation4 ,illustration3 ,fig4 #bbh4 ,an isolated portion of the skeleton of ,dictyocha4 ,illustration3 ,fig4 #bbi4 ,dictyocha stapedia1 ,hkl4 ,we notice that the amount of solid matter deposited in the plane interfacial boundaries is greatly increased at the outer margin of each boundary wall1 where it merges or coincides with the superficial furrow which separates the free1 spherical surfaces of the bubbles from one another2 and we may sometimes find that1 along these edges1 the skeleton remains complete and strong1 while it shows signs of imperfect development or of breaking away over great part of the rest of the interfacial surfaces4 ,in this there is nothing anomalous1 for we have already recognised that it is at the edges or margins of the interfacial partition-walls that the manifestation of surface-energy will tend to reach its maximum4 ,and just as we have seen that1 in certain of our 8multicellular0 spherical ,radiolarians1 it is at the superficial #dgd edges or borders of the partitions1 and here only1 that skeletal formation occurs 7giving rise to the netted shell with its hexagonal meshes of ,fig4 #bba71 so also at times1 in the case of such little aggregates of cells or vesicles as the four-celled system of ,callimitra1 it may happen that about the external boundary-lines1 and not in the interior boundary-planes1 the whole of the skeletal matter is aggregated4 ,in ,fig4 #bbh we see a curious little skeletal structure or complex spicule1 whose conformation is easily accounted for after this fashion4 ,little spicules such as this form isolated portions of the skeleton in the genus ,dictyocha1 and occur scattered over the spherical surface of the organism 7,fig4 #bbi74 ,the more or less basket-shaped spicule has evidently been developed about a little cluster of four cells or vesicles1 lying in or on the plane of the surface of the organism1 and therefore arranged1 not in the tetrahedral form of ,callimitra1 but in the manner in which four contiguous cells lying side by side normally set themselves1 like the four cells of a segmenting egg3 that is to say with an intervening 8polar furrow10 whose ends mark the meeting place1 at equal angles1 of the cells in groups of three4 ,the little projecting spokes1 or spikes1 which are set normally to the main basket-work1 seem to be incompleted portions of a larger basket1 or in other words imperfectly formed elements corresponding to the interfacial contacts in the surrounding parts #dge of the system4 ,similar but more complex formations1 all explicable as basket-like frameworks developed around a cluster of cells1 are known in great variety4 ,in our ,nassellarian itself1 and in many other cases where the plane interfacial boundary-walls are skeletonised1 we see that the siliceous matter is not deposited in an even and continuous layer1 like the waxen walls of a bee's cell1 but constitutes a meshwork of fine curvilinear threads2 and the curves seem to run1 on the whole1 isogonally1 and to form three main series1 one approximately parallel to1 or concentric with1 the outer or free edge of the partition1 and the other two related severally to its two edges of attachment4 ,sometimes 7as may also be seen in our figure71 the system is still further complicated by a fourth series of linear elements1 which tend to run radially from the centre of the system to the free edge of each partition4 ,as regards the former1 their arrangement is such as would result if deposition or solidification had proceeded in waves1 starting independently from each of the three boundaries of the little partition-wall2 and something of this kind is doubtless what has happened4 ,we are reminded at once of the wave-like periodicity of the ,liesegang phenomenon4 ,but apart from this we might conceive of other explanations4 ,for instance1 the liquid film which originally constitutes the partition must easily be thrown into vibrations1 and 7like the dust upon a ,chladni's plate7 minute particles of matter in contact with the film would tend to take up their position in a symmetrical arrangement1 in direct relation to the nodal points or lines of the vibrating surface#dhe4 ,some such explanation as this 7to my thinking7 must be invoked to account for the minute and varied and very beautiful patterns upon many diatoms1 the resemblance of which patterns 7in certain of their simpler cases7 to the ,chladni figures is sometimes striking and obvious4 ,but the many special problems which the diatom skeleton suggests ,i have not attempted to consider4 ,illustration3 ,fig4 #bcj4 ,the last peculiarity of our ,nassellarian lies in an apparent departure from what we should at first expect in the way of its #dgf external symmetry4 ,were the system actually composed of four spherical vesicles in mutual contact1 the outer margin of each of the six interfacial planes would obviously be a circular arc2 and accordingly1 at each angle of the tetrahedron1 we should expect to have a depressed1 or re-entrant angle1 instead of a prominent cusp4 ,this is all doubtless due to some simple balance of tensions1 whose precise nature and distribution is meanwhile a matter of conjecture4 ,but it seems as though an extremely simple explanation would go a long way1 and possibly the whole way1 to meet this particular case4 ,in our ordinary plane diagram of three cells1 or soap-bubbles1 in contact1 we know 7and we have just said7 that the tensions of the three partitions draw inwards the outer walls of the system1 till at each point of triple contact 7,p7 we tend to get a triradiate1 equiangular junction4 ,but if we introduce another bubble into the centre of the system 7,fig4 #bcj71 then1 as ,plateau shewed1 the tensions of its walls and those of the three partitions by which it is now suspended1 again balance one another1 and the central bubble appears 7in plane projection7 as a curvilinear1 equilateral triangle4 ,we have only got to convert this plane diagram into that of a tetrahedral solid to obtain almost precisely the configuration which we are seeking to explain4 ,now we observe that1 so far as our figure of ,callimitra informs us1 this is just the shape of the little bubble which occupies the centre of the tetrahedral system in that ,radiolarian skeleton4 ,and ,i conceive1 accordingly1 that the entire organism was not limited to the four cells or vesicles 7together with the little central #dgg fifth7 which we have hitherto been imagining1 but there must have been an outer tetrahedral system1 enclosing the cells which fabricated the skeleton1 just as these latter enclosed1 and deformed1 the little bubble in the centre of all4 ,we have only to suppose that this hypothetical tetrahedral series1 forming the outer layer or surface of the whole system1 was for some chemico-physical reason incapable of secreting at its interfacial contacts a skeletal fabric#dhf4 ,in this hypothetical case1 the edges of the skeletal system would be circular arcs1 meeting one another at an angle of #abj1 or1 in the solid pyramid1 of #aji3 and this latter is very nearly the condition which our little skeleton actually displays4 ,but we observe in ,fig4 #bbg that1 in the immediate neighbourhood of the tetrahedral angle1 the circular arcs are slightly drawn out into projecting cusps 7cf4 ,fig4 #bcj1 ,b74 ,there is no ,s-shaped curvature of the tetrahedral edges as a whole1 but a very slight one1 a very slight change of curvature2 close to the apex4 ,this1 ,i conceive1 is nothing more than what1 in a material system1 we are bound to have1 to represent a 8surface of continuity40 ,it is a phenomenon precisely analogous to ,plateau's 8bourrelet10 which we have already seen to be a constant feature of all cellular systems1 rounding off the sharp angular contacts by which 7in our more elementary treatment7 we expect one film to make its junction with another#dhg4 ,in the foregoing examples of ,radiolaria1 the symmetry which the organism displays would seem to be identical with that symmetry of forces which is due to the assemblage of surface-tensions in the whole system2 this symmetry being displayed1 in one class of cases1 in a complex spherical mass of froth1 and in #dgh another class in a simpler aggregate of a few1 otherwise isolated1 vesicles4 ,but among the vast number of other known ,radiolaria1 there are certain forms 7especially among the ,phaeodaria and ,acantharia7 which display a still more remarkable symmetry1 the origin of which is by no means clear1 though surface-tension doubtless plays a part in its causation4 ,these are cases in which 7as in some of those already described7 the skeleton consists 7#a7 of radiating spicular rods1 definite in number and position1 and 7#b7 of interconnecting rods or plates1 tangential to the more or less spherical body of the organism1 whose form becomes1 accordingly1 that of a geometric1 polyhedral solid4 ,it may be that there is no mathematical difference1 save one of degree1 between such a hexagonal polyhedron as we have seen in ,aulacantha1 and those which we are about to describe2 but the greater regularity1 the numerical symmetry1 and the apparent simplicity of these latter1 makes of them a class apart1 and suggests problems which have not been solved nor even investigated4 ,illustration3 ,fig4 #bca4 ,skeletons of various ,radiolarians1 after ,haeckel4 #a4 ,circoporus sexfurcus2 #b4 ,c4 octahedrus2 #c4 ,circogonia icosahedra2 #d4 ,circospathis novena2 #e4 ,circorrhegma dodecahedra4 ,the matter is sufficiently illustrated by the accompanying figures1 all drawn from ,haeckel's ,monograph of the ,challenger ,radiolaria#dhh4 ,in one of these we see a regular octahedron1 in another a regular1 or pentagonal dodecahedron1 in a third a regular icosahedron4 ,in all cases the figure appears to be perfectly symmetrical1 though neither the triangular facets of the octahedron and icosahedron1 nor the pentagonal facets of the dodecahedron1 are necessarily plane surfaces4 ,in all of these cases1 the radial spicules correspond to the solid angles of the figure2 and they are1 accordingly1 six in number in the octahedron1 twenty in the dodecahedron1 and twelve in the icosahedron4 ,if we add to these three figures the regular tetrahedron1 which we have had frequent occasion to study1 and the cube 7which is represented1 at least in outline1 in the skeleton of the hexactinellid sponges71 we have completed the series of the five regular polyhedra known to geometers1 the ,platonic bodies#dhi of the older mathematicians4 ,it is at first sight all the more remarkable that we should here meet #dhj with the whole five regular polyhedra1 when we remember that1 among the vast variety of crystalline forms known among minerals1 the regular dodecahedron and icosahedron1 simple as they are from the mathematical point of view1 never occur4 ,not only do these latter never occur in ,crystallography1 but 7as is explained in text-books of that science7 it has been shewn that they cannot occur1 owing to the fact that their indices 7or numbers expressing the relation of the faces to the three primary axes7 involve an irrational quantity3 whereas it is a fundamental law of crystallography1 involved in the whole theory of space-partitioning1 that 8the indices of any and every face of a crystal are small whole numbers#dij40 ,at the same time1 an imperfect pentagonal dodecahedron1 whose pentagonal sides are non-equilateral1 is common among crystals4 ,if we may safely judge from ,haeckel's figures1 the pentagonal dodecahedron of the ,radiolarian is perfectly regular1 and we must presume1 accordingly1 that it is not brought about by principles of space-partitioning similar to those which manifest themselves in the phenomenon of crystallisation4 ,it will be observed that in all these radiolarian polyhedral shells1 the surface of each external facet is formed of a minute hexagonal network1 whose probable origin1 in relation to a vesicular structure1 is such as we have already discussed4 ,illustration3 ,fig4 #bcb4 ,dorataspis sp42 diagrammatic4 ,in certain allied ,radiolaria 7,fig4 #bcb71 which1 like the dodecahedral form figured in ,fig4 #bca1 #e1 have twenty radial spines1 these latter are commonly described as being arranged in a certain very singular way4 ,it is stated that their arrangement may be referred #dha to a series of five parallel circles on the sphere1 corresponding to the equator 7c71 the tropics 7b1 d7 and the polar circles 7a1 e72 and that beginning with four equidistant spines in the equator1 we have alternating whorls of four1 radiating outwards from the sphere in each of the other parallel zones4 ,this rule was laid down by the celebrated ,johannes ,mller1 and has ever since been used and quoted as ,mller's law4 ,the chief point in this alleged arrangement which strikes us at first sight as very curious1 is that there is said to be no spine at either pole2 and when we come to examine carefully the figure of the organism1 we find that the received description does not do justice to the facts4 ,we see1 in the first place1 from such figures as ,figs4 #bcb1 #bcd1 that here1 unlike our former cases1 the radial spines issue through the facets 7and through all the facets7 of the polyhedron1 instead of through its solid angles2 and accordingly1 that our twenty spines correspond 7not1 as before1 to a dodecahedron7 but to some sort of an icosahedron4 ,we see in the next place1 that this icosahedron is composed of faces1 or plates1 of two different kinds1 some hexagonal and some pentagonal2 and when we look closer1 we discover that the whole figure is that of a hexagonal prism1 whose twelve solid angles are replaced by pentagonal facets4 ,both hexagons and pentagons #dhb appear to be perfectly equilateral1 but if we try to construct a plane-sided polyhedron of this kind1 we soon find that it is impossible2 for into the angles between the six equatorial hexagons those of the six united pentagons will not fit4 ,the figure however can be easily constructed if we replace the straight edges 7or some of them7 by curves1 and the plane facets by corresponding1 slightly curved1 surfaces4 ,the true symmetry of this figure1 then1 is hexagonal1 with a polar axis1 produced into two polar spicules2 with six equatorial spicules1 or rays2 and with two sets of six spicular rays1 interposed between the polar axis and the equatorial rays1 and alternating in position with the latter4 ,mller's description was emended by ,brandt1 and what is now known as 8,brandt's law10 viz4 that the symmetry consists of two polar rays1 and three whorls of six each1 coincides with the above description so far as the spicular axes go3 save only that ,brandt specifically states that the intermediate whorls stand equidistant between the equator and the poles1 i4e4 in latitude #de4 ,while not far from the truth1 this statement is not exact2 for according to the geometry of the figure1 the intermediate cycles obviously stand in a slightly higher latitude1 but this latitude ,i have not attempted to determine2 for the calculation seems to be a little troublesome owing to the curvature of the sides of the figure1 and the enquiring mathematician will perform it more easily than ,i4 ,brandt1 if ,i understand him rightly1 did not propose his 8law0 as a substitute for ,mller's law1 but as a second law applicable to a few particular cases4 ,i on the other hand can find no case to which ,mller's law properly applies4 ,if we construct such a polyhedron1 and set it in the position of ,fig4 #bcb1 ,b1 we shall easily see that it is capable of explanation 7though improperly7 in accordance with ,mller's law2 for the four equatorial rays of ,mller 7c7 now correspond to the two polar and to two opposite equatorial facets of our polyhedron3 the four 8polar0 rays of ,mller 7a or e7 correspond to two adjacent hexagons and two intermediate pentagons of the figure3 and ,mller's 8tropical0 rays 7b or d7 are those which emanate from the remaining four pentagonal facets1 in each half of the figure4 ,in some cases1 such as ,haeckel's ,phatnaspis cristata 7,fig4 #bcc71 we have an ellipsoidal body1 from which the spines emerge in the order described1 but which is not obviously divided by facets4 ,in ,fig4 #bcd ,i have indicated the facets corresponding to the rays1 and dividing the surface in the usual symmetrical way4 #dhc ,illustration3 ,fig4 #bcc4 ,phatnaspis cristata1 ,hkl4 ,illustration3 ,fig4 #bcd4 ,the same1 diagrammatic4 #dhd ,within any polyhedron we may always inscribe another polyhedron1 whose corners correspond in number to the sides or facets of the original figure1 or 7in alternative cases7 to a certain number of these sides2 and a similar result is obtained by bevelling off the corners of the original polyhedron4 ,we may obtain a precisely similar symmetrical result if 7in such a case as these ,radiolarians which we are describing71 we imagine the radial spines to be interconnected by tangential rods1 instead of by the complete facets which we have just been dealing with4 ,in our complicated polyhedron with its twenty radial spines arranged in the manner described there are various symmetrical ways in which we may imagine these interconnecting bars to be arranged4 ,the most symmetrical of these is one in which the whole surface is divided into eighteen rhomboidal areas1 obtained by systematically connecting each group of four adjacent radii4 ,this figure has eighteen faces 7,f71 twenty corners 7,c71 and therefore thirty-six edges 7,e71 in conformity with ,euler's theorem1 ,f ,c ,e #b4 ,illustration3 ,fig4 #bce4 ,phractaspis prototypus1 ,hkl4 ,another symmetrical arrangement will divide the surface into fourteen rhombs and eight triangles4 ,this latter arrangement is obtained by linking up the radial rods as follows3 aaaa1 aba1 abcb1 bcdc1 etc4 ,here we have again twenty corners1 but we have twenty-two faces2 the number of edges1 or tangential spicular bars1 will be found1 therefore1 by the above formula1 to be forty4 ,in ,haeckel's figure of ,phractaspis prototypus we have a spicular skeleton which appears to be constructed precisely upon this plan1 and to be derivable from the faceted polyhedron precisely after this manner4 ,in all these latter cases it is the arrangement of the axial rods1 or in other words the 8polar symmetry0 of the entire organism1 which lies at the root of the matter1 and which1 if only #dhe we could account for it1 would make it comparatively easy to explain the superficial configuration4 ,but there are no obvious mechanical forces by which we can so explain this peculiar polarity4 ,this at least is evident1 that it arises in the central mass of protoplasm1 which is the essential living portion of the organism as distinguished from that frothy peripheral mass whose structure has helped us to explain so many phenomena of the superficial or external skeleton4 ,to say that the arrangement depends upon a specific polarisation of the cell is merely to refer the problem to other terms1 and to set it aside for future solution4 ,but it is possible that we may learn something about the lines in which to seek for such a solution by considering the case of ,lehmann's 8fluid crystals10 and the light which they throw upon the phenomena of molecular aggregation4 ,the phenomenon of 8fluid crystallisation0 is found in a number of chemical bodies2 it is exhibited at a specific temperature for each substance2 and it would seem to be limited to bodies in which there is a more or less elongated1 or 8chain-like0 arrangement of the atoms in the molecule4 ,such bodies1 at the appropriate temperature1 tend to aggregate themselves into masses1 which are sometimes spherical drops or globules 7the so-called 8spherulites071 and sometimes have the definite form of needle-like or prismatic crystals4 ,in either case they remain liquid1 and are also doubly refractive1 polarising light in brilliant colours4 ,together with them are formed ordinary solid crystals1 also with characteristic polarisation1 and into such solid crystals all the fluid material ultimately turns4 ,it is evident that in these liquid crystals1 though the molecules are freely mobile1 just as are those of water1 they are yet subject to1 or endowed with1 a 8directive force10 a force which confers upon them a definite configuration or 8polarity10 the ,gestaltungskraft of ,lehmann4 ,such an hypothesis as this had been gradually extruded from the theories of mathematical crystallography#dia2 and it had come to be believed that the symmetrical conformation of a homogeneous crystalline structure was sufficiently explained by the mere mechanical fitting together of appropriate structural units along the easiest and simplest lines of 8close packing03 just as #dhf a pile of oranges becomes definite1 both in outward form and inward structural arrangement1 without the play of any specific directive force4 ,but while our conceptions of the tactical arrangement of crystalline molecules remain the same as before1 and our hypotheses of 8modes of packing0 or of 8space-lattices0 remain as useful as ever for the definition and explanation of the molecular arrangements1 an entirely new theoretical conception is introduced when we find such space-lattices maintained in what has hitherto been considered the molecular freedom of a liquid field2 and we are constrained1 accordingly1 to postulate a specific molecular force1 or 8,gestaltungskraft0 7not unlike ,kepler's 8facultas formatrix071 to account for the phenomenon4 ,now just as some sort of specific 8,gestaltungskraft0 had been of old the deus ex machina accounting for all crystalline phenomena 7gnara totius geometri1 et in ea exercita1 as ,kepler said71 and as such an hypothesis1 after being dethroned and repudiated1 has now fought its way back and has made good its right to be heard1 so it may be also in biology4 ,we begin by an easy and general assumption of specific properties1 by which each organism assumes its own specific form2 we learn later 7as it is the purpose of this book to shew7 that throughout the whole range of organic morphology there are innumerable phenomena of form which are not peculiar to living things1 but which are more or less simple manifestations of ordinary physical law4 ,but every now and then we come to certain deep-seated signs of protoplasmic symmetry or polarisation1 which seem to lie beyond the reach of the ordinary physical forces4 ,it by no means follows that the forces in question are not essentially physical forces1 more obscure and less familiar to us than the rest2 and this would seem to be the crucial lesson for us to draw from ,lehmann's surprising and most beautiful discovery4 ,for ,lehmann seems actually to have demonstrated1 in non-living1 chemical bodies1 the existence of just such a determinant1 just such a 8,gestaltungskraft10 as would be of infinite help to us if we might postulate it for the explanation 7for instance7 of our ,radiolarian's axial symmetry4 ,but further than this we cannot go2 for such analogy as we seem to see in the ,lehmann phenomenon soon evades us1 and refuses to be pressed home4 ,not only is it the case1 as we have already #dhg seen1 that certain of the geometric forms assumed by the symmetrical ,radiolarian shells are just such as the 8space-lattice0 theory would seem to be inapplicable to1 but it is in other ways obvious that symmetry of crystallisation1 whether liquid or solid1 has no close parallel1 but only a series of analogies1 in the protoplasmic symmetry of the living cell4 #dhh ,,chapter ,x ,a ,,parenthetic ,,note ,,on ,,geodetics ,we have made use in the last chapter of the mathematical principle of ,geodetics 7or ,geodesics7 in order to explain the conformation of a certain class of sponge-spicules2 but the principle is of much wider application in morphology1 and would seem to deserve attention which it has not yet received4 ,illustration3 ,fig4 #bcf4 ,annular and spiral thickenings in the walls of plant-cells4 ,defining1 meanwhile1 our geodetic line 7as we have already done7 as the shortest distance between two points on the surface of a solid of revolution1 we find that the geodetics of the cylinder give us one of the simplest of cases4 ,here it is plain that the geodetics are of three kinds3 7#a7 a series of annuli around the cylinder1 that is to say1 a system of circles1 in planes parallel to one another and at right angles to the axis of the cylinder 7,fig4 #bcf1 a72 7#b7 a series of straight lines parallel to the axis2 and 7#c7 a series of spiral curves winding round the wall of the cylinder 7b1 c74 ,these three systems are all of frequent occurrence1 and are all illustrated in the local thickenings of the wall of the cylindrical cells or vessels of plants4 ,the spiral1 or rather helicoid1 geodetic is particularly common in cylindrical structures1 and is beautifully shewn for instance in the spiral coil which stiffens the tracheal tubes of an insect1 or the so-called 8tracheides0 of a woody stem4 ,a similar #dhi phenomenon is often witnessed in the splitting of a glass tube4 ,if a crack appear in a thin tube1 such as a test-tube1 it has a tendency to be prolonged in its own direction1 and the more perfectly homogeneous and isotropic be the glass the more evenly will the split tend to follow the straight course in which it began4 ,as a result1 the crack in our test-tube is often seen to continue till the tube is split into a continuous spiral ribbon4 ,in a right cone1 the spiral geodetic falls into closer and closer coils as the diameter of the cone narrows2 and a very beautiful geodetic of this kind is exemplified in the sutural line of a spiral shell1 such as ,turritella1 or in the striations which run parallel with the spiral suture4 ,similarly1 in an ellipsoidal surface1 we have a spiral geodetic1 whose coils get closer together as we approach the ends of the long axis of the ellipse2 in the splitting of the integument of an ,equisetum-spore1 by which are formed the spiral 8elaters0 of the spore1 we have a case of this kind1 though the spiral is not sufficiently prolonged to shew all its features in detail4 ,we have seen in these various cases1 that our original definition of a geodetic requires to be modified2 for it is only subject to conditions that it is 8the shortest distance between two points on the surface of the solid10 and one of the commonest of these restricting conditions is that our geodetic may be constrained to go twice1 or many times1 round the surface on its way4 ,in short1 we must redefine our geodetic1 as a curve drawn upon a surface1 such that1 if we take any two adjacent points on the curve1 the curve gives the shortest distance between them4 ,again1 in the geodetic systems which we meet with in morphology1 it sometimes happens that we have two opposite systems of geodetic spirals separate and distinct from one another1 as in ,fig4 #bcf1 c2 and it is also common to find the two systems interfering with one another1 and forming a criss-cross1 or reticulated arrangement4 ,this is a very common source of reticulated patterns4 ,among the ciliated ,infusoria1 we have in the spiral lines along which their cilia are arranged a great variety of beautiful geodetic curves2 though it is probable enough that in some complicated cases these are not simple geodetics1 but projections of curves other than a straight line upon the surface of the solid4 #dij ,lastly1 a very instructive case is furnished by the arrangement of the muscular fibres on the surface of a hollow organ1 such as the heart or the stomach4 ,here we may consider the phenomenon from the point of view of mechanical efficiency1 as well as from that of purely descriptive or objective anatomy4 ,in fact we have an a priori right to expect that the muscular fibres covering such hollow or tubular organs will coincide with geodetic lines1 in the sense in which we are now using the term4 ,for if we imagine a contractile fibre1 or elastic band1 to be fixed by its two ends upon a curved surface1 it is obvious that its first effort of contraction will tend to expend itself in accommodating the band to the form of the surface1 in 8stretching it tight10 or in other words in causing it to assume a direction which is the shortest possible line upon the surface between the two extremes3 and it is only then that further contraction will have the effect of constricting the tube and so exercising pressure on its contents4 ,thus the muscular fibres1 as they wind over the curved surface of an organ1 arrange themselves automatically in geodesic curves3 in precisely the same manner as we also automatically construct complex systems of geodesics whenever we wind a ball of wool or a spindle of tow1 or when the skilful surgeon bandages a limb4 ,in these latter cases we see the production of those 8figures-of-eight10 to which1 in the case for instance of the heart-muscles1 ,pettigrew and other anatomists have ascribed peculiar importance4 ,in the case of both heart and stomach we must look upon these organs as developed from a simple cylindrical tube1 after the fashion of the glass-blower1 as is further discussed on p4 #gcg of this book1 the modification of the simple cylinder consisting of various degrees of dilatation and of twisting4 ,in the primitive undistorted cylinder1 as in an artery or in the intestine1 the muscular fibres run in geodetic lines1 which as a rule are not spiral1 but are merely either annular or longitudinal2 these are the ordinary 8circular and longitudinal coats10 which form the normal musculature of all tubular organs1 or of the body-wall of a cylindrical worm#dib4 ,if we consider each muscular fibre as an elastic strand1 imbedded in the elastic membrane which constitutes the wall of the organ1 it #dia is evident that1 whatever be the distortion suffered by the entire organ1 the individual fibre will follow the same course1 which will still1 in a sense1 be a geodetic4 ,but if the distortion be considerable1 as for instance if the tube become bent upon itself1 or if at some point its walls bulge outwards in a diverticulum or pouch1 it is obvious that the old system of geodetics will only mark the shortest distance between two points more or less approximate to one another1 and that new systems of geodetics will tend to appear1 peculiar to the new surface1 and linking up points more remote from one another4 ,this is evidently the case in the human stomach4 ,we still have the systems1 or their unobliterated remains1 of circular and longitudinal muscles2 but we also see two new systems of fibres1 both obviously geodetic 7or rather1 when we look more closely1 both parts of one and the same geodetic system71 in the form of annuli encircling the pouch or diverticulum at the cardiac end of the stomach1 and of oblique fibres taking a spiral course from the neighbourhood of the oesophagus over the sides of the organ4 ,in the heart we have a similar1 but more complicated phenomenon4 ,its musculature consists1 in great part1 of the original simple system of circular and longitudinal muscles which enveloped the original arterial tubes1 which tubes1 after a process of local thickening1 expansion1 and especially twisting1 came together to constitute the composite1 or double1 mammalian heart2 and these systems of muscular fibres1 geodetic to begin with1 remain geodetic 7in the sense in which we are using the word7 after all the twisting to which the primitive cylindrical tube or tubes have been subjected4 ,that is to say1 these fibres still run their shortest possible course1 from start to finish1 over the complicated curved surface of the organ2 and it is only because they do so that their contraction1 or longitudinal shortening1 is able to produce its direct effect1 as ,borelli well understood1 in the contraction or systole of the heart#dic4 #dib ,as a parenthetic corollary to the case of the spiral pattern upon the wall of a cylindrical cell1 we may consider for a moment the spiral line which many small organisms tend to follow in their path of locomotion#did4 ,the helicoid spiral1 traced around the wall of our cylinder1 may be explained as a composition of two velocities1 one a uniform velocity in the direction of the axis of the cylinder1 the other a uniform velocity in a circle perpendicular to the axis4 ,in a somewhat analogous fashion1 the smaller ciliated organisms1 such as the ciliate and flagellate ,infusoria1 the ,rotifers1 the swarm-spores of various ,protists1 and so forth1 have a tendency to combine a direct with a revolving path in their ordinary locomotion4 ,the means of locomotion which they possess in their cilia are at best somewhat primitive and inefficient2 they have no apparent means of steering1 or modifying their direction2 and1 if their course tended to swerve ever so little to one side1 the result would be to bring them round and round again in an approximately circular path 7such as a man astray on the prairie is said to follow71 with little or no progress in a definite longitudinal direction4 ,but as a matter of fact1 either through the direct action of their cilia or by reason of a more or less unsymmetrical form of the body1 all these creatures tend more or less to rotate about their long axis while they swim4 ,and this axial rotation1 just as in the case of a rifle-bullet1 causes their natural swerve1 which is always in the same direction as regards their own bodies1 to be in a continually changing direction as regards space3 in short1 to make a spiral course around1 and more or less approximate to1 a straight axial line4 #dic ,,chapter ,,xi ,,the ,,logarithmic ,,spiral ,the very numerous examples of spiral conformation which we meet with in our studies of organic form are peculiarly adapted to mathematical methods of investigation4 ,but ere we begin to study them1 we must take care to define our terms1 and we had better also attempt some rough preliminary classification of the objects with which we shall have to deal4 ,in general terms1 a ,spiral ,curve is a line which1 starting from a point of origin1 continually diminishes in curvature as it recedes from that point2 or1 in other words1 whose radius of curvature continually increases4 ,this definition is wide enough to include a number of different curves1 but on the other hand it excludes at least one which in popular speech we are apt to confuse with a true spiral4 ,this latter curve is the simple ,screw1 or cylindrical ,helix1 which curve1 as is very evident1 neither starts from a definite origin1 nor varies in its curvature as it proceeds4 ,the 8spiral0 thickening of a woody plant-cell1 the 8spiral0 thread within an insect's tracheal tube1 or the 8spiral0 twist and twine of a climbing stem are not1 mathematically speaking1 spirals at all1 but screws or helices4 ,they belong to a distinct1 though by no means very remote1 family of curves4 ,some of these helical forms we have just now treated of1 briefly and parenthetically1 under the subject of ,geodetics4 ,illustration3 ,fig4 #bcg4 ,the shell of ,nautilus pompilius1 from a radiograph3 to shew the logarithmic spiral of the shell1 together with the arrangement of the internal septa4 7,from ,messrs ,green and ,gardiner1 in ,proc4 ,malacol4 ,soc4 ,,ii1 #ahig47 ,of true organic spirals we have no lack#die4 ,we think at once of the beautiful spiral curves of the horns of ruminants1 and of the still more varied1 if not more beautiful1 spirals of molluscan shells4 ,closely related spirals may be traced in the arrangement #did of the florets in the sunflower2 a true spiral1 though not1 by the way1 so easy of investigation1 is presented to us by the outline of a cordate leaf2 and yet again1 we can recognise typical though transitory spirals in the coil of an elephant's trunk1 in the 8circling #die spires0 of a snake1 in the coils of a cuttle-fish's arm1 or of a monkey's or a chameleon's tail4 ,among such forms as these1 and the many others which we might easily add to them1 it is obvious that we have to do with things which1 though mathematically similar1 are biologically speaking fundamentally different4 ,and not only are they biologically remote1 but they are also physically different1 in regard to the nature of the forces to which they are severally due4 ,for in the first place1 the spiral coil of the elephant's trunk or of the chameleon's tail is1 as we have said1 but a transitory configuration1 and is plainly the result of certain muscular forces acting upon a structure of a definite1 and normally an essentially different1 form4 ,it is rather a position1 or an attitude1 than a form1 in the sense in which we have been using this latter term2 and1 unlike most of the forms which we have been studying1 it has little or no direct relation to the phenomenon of ,growth4 ,illustration3 ,fig4 #bch4 ,a ,foraminiferal shell 7,globigerina74 ,again1 there is a manifest and not unimportant difference between such a spiral conformation as is built up by the separate and successive florets in the sunflower1 and that which1 in the snail or ,nautilus shell1 is apparently a single and indivisible unit4 ,and a similar1 if not identical difference is apparent between the ,nautilus shell and the minute shells of the ,foraminifera1 which so closely simulate it2 inasmuch as the spiral shells of these latter are essentially composite structures1 combined out of successive and separate chambers1 while the molluscan shell1 though it may 7as in ,nautilus7 become secondarily subdivided1 has grown as one continuous tube4 ,it follows from all this that there cannot #dif possibly be a physical or dynamical1 though there may well be a mathematical ,law of ,growth1 which is common to1 and which defines1 the spiral form in the ,nautilus1 in the ,globigerina1 in the ram's horn1 and in the disc of the sunflower4 ,of the spiral forms which we have now mentioned1 every one 7with the single exception of the outline of the cordate leaf7 is an example of the remarkable curve known as the ,logarithmic ,spiral4 ,but before we enter upon the mathematics of the logarithmic spiral1 let us carefully observe that the whole of the organic forms in which it is clearly and permanently exhibited1 however different they may be from one another in outward appearance1 in nature and in origin1 nevertheless all belong1 in a certain sense1 to one particular class of conformations4 ,in the great majority of cases1 when we consider an organism in part or whole1 when we look 7for instance7 at our own hand or foot1 or contemplate an insect or a worm1 we have no reason 7or very little7 to consider one part of the existing structure as older than another2 through and through1 the newer particles have been merged and commingled1 by intussusception1 among the old2 the whole outline1 such as it is1 is due to forces which for the most part are still at work to shape it1 and which in shaping it have shaped it as a whole4 ,but the horn1 or the snail-shell1 is curiously different2 for in each of these1 the presently existing structure is1 so to speak1 partly old and partly new2 it has been conformed by successive and continuous increments2 and each successive stage of growth1 starting from the origin1 remains as an integral and unchanging portion of the still growing structure1 and so continues to represent what at some earlier epoch constituted for the time being the structure in its entirety4 ,in a slightly different1 but closely cognate way1 the same is true of the spirally arranged florets of the sunflower4 ,for here again we are regarding serially arranged portions of a composite structure1 which portions1 similar to one another in form1 differ in age2 and they differ also in magnitude in a strict ratio according to their age4 ,somehow or other1 in the logarithmic spiral the time-element always enters in2 and to this important fact1 full of curious biological as well as mathematical significance1 we shall afterwards return4 #dig ,it is1 as we have so often seen1 an essential part of our whole problem1 to try to understand what distribution of forces is capable of producing this or that organic form1--to give1 in short1 a dynamical expression to our descriptive morphology4 ,now the general distribution of forces which lead to the formation of a spiral 7whether logarithmic or other7 is very easily understood2 and need not carry us beyond the use of very elementary mathematics4 ,illustration3 ,fig4 #bci4 ,if we imagine growth to act in a perpendicular direction1 as for example the upward force of growth in a growing stem 7,,oa71 then1 in the absence of other forces1 elongation will as a matter of course proceed in an unchanging direction1 that is to say the stem will grow straight upwards4 ,suppose now that there be some constant external force1 such as the wind1 impinging on the growing stem2 and suppose 7for simplicity's sake7 that this external force be in a constant direction 7,,ab7 perpendicular to the intrinsic force of growth4 ,the direction of actual growth will be in the line of the resultant of the two forces3 and1 since the external force is 7by hypothesis7 constant in direction1 while the internal force tends always to act in the line of actual growth1 it is obvious that our growing organism will tend to be bent into a curve1 to which1 for the time being1 #dih the actual force of growth will be acting at a tangent4 ,so long as the two forces continue to act1 the curve will approach1 but will never attain1 the direction of ,,ab1 perpendicular to the original direction ,,oa4 ,if the external force be constant in amount the curve will approximate to the form of a hyperbola2 and1 at any rate1 it is obvious that it will never tend to assume a spiral form4 ,in like manner1 if we consider a horizontal beam1 fixed at one end1 the imposition of a weight at the other will bend the beam into a curve1 which1 as the beam elongates or the weight increases1 will bring the weighted end nearer and nearer to the vertical4 ,but such a force1 constant in direction1 will obviously never curve the beam into a spiral1--a fact so patent and obvious that it would be superfluous to state it1 were it not that some naturalists have been in the habit of invoking gravity as the force to which may be attributed the spiral flexure of the shell4 ,but if1 on the other hand1 the deflecting force be inherent in the growing body1 or so connected with it in a system that its direction 7instead of being constant1 as in the former case7 changes with the direction of growth1 and is perpendicular 7or inclined at some constant angle7 to this changing direction of the growing force1 then it is plain that there is no such limit to the deflection from the normal1 but the growing curve will tend to wind round and round its point of origin4 ,in the typical case of the snail-shell1 such an intrinsic force is manifestly present in the action of the columellar muscle4 ,many other simple illustrations can be given of a spiral course being impressed upon what is primarily rectilinear motion1 by any steady deflecting force which the moving body carries1 so to speak1 along with it1 and which continually gives a lop-sided tendency to its forward movement4 ,for instance1 we have been told that a man or a horse1 travelling over a great prairie1 is very apt to find himself1 after a long day's journey1 back again near to his starting point4 ,here some small and imperceptible bias1 such as might for instance be caused by one leg being in a minute degree longer or stronger than the other1 has steadily deflected the forward movement to one side2 and has gradually brought the traveller back1 perhaps in a circle to the very point from which he set out1 #dii or else by a spiral curve1 somewhere within reach and recognition of it4 ,illustration3 ,fig4 #bdj4 ,we come to a similar result when we consider1 for instance1 a cylindrical body in which forces of growth are at work tending to its elongation1 but these forces are unsymmetrically distributed4 ,let the tendency to elongation along ,,ab be of a magnitude proportional to ,,bb1 and that along ,,cd be of a magnitude proportional to ,,dd2 and in each element parallel to ,,ab and ,,cd1 let a parallel force of growth1 proportionately intermediate in magnitude1 be at work3 and let ,,eff be the middle line4 ,then at any cross-section ,,bfd1 if we deduct the mean force ,,ff1 we have a certain positive force at ,b1 equal to ,bb1 and an equal and opposite force at ,d1 equal to ,dd4 ,but ,,ab and ,,cd are not separate structures1 but are connected together1 either by a solid core1 or by the walls of a tubular shell2 and the forces which tend to separate ,b and ,d are opposed1 accordingly1 by a tension in ,,bd4 ,it follows therefore1 that there will be a resultant force ,,bg1 acting in a direction intermediate between ,bb and ,,bd1 and also a resultant1 ,,dh1 acting at ,d in an opposite direction2 and accordingly1 after a small increment of growth1 the growing end of the cylinder will come to lie1 not in the direction ,,bd1 but in the direction ,,gh4 ,the problem is therefore analogous to that of a beam to which we apply a bending moment2 and it is plain that the unequal force of growth is equivalent to a 8couple0 which will impart to our structure a curved form4 ,for1 if we regard the part ,,abdc as practically rigid1 and the part ,,bb,d,d as pliable1 this couple #ejj will tend to turn strips such as ,b,d about an axis perpendicular to the plane of the diagram1 and passing through an intermediate point ,f4 ,it is plain1 also1 since all the forces under consideration are intrinsic to the system1 that this tendency will be continuous1 and that as growth proceeds the curving body will assume either a circular or a spiral form4 ,but the tension which we have here assumed to exist in the direction ,,bd will obviously disappear if we suppose a sufficiently rapid rate of growth in that direction4 ,for if we may regard the mouth of our tubular shell as perfectly extensible in its own plane1 so that it exerts no traction whatsoever on the sides1 then it will be drawn out into more and more elongated ellipses1 forming the more and more oblique orifices of a straight tube4 ,in other words1 in such a structure as we have presupposed1 the existence or maintenance of a constant ratio between the rates of extension or growth in the vertical and transverse directions will lead1 in general1 to the development of a logarithmic spiral2 the magnitude of that ratio will determine the character 7that is to say1 the constant angle7 of the spiral2 and the spirals so produced will include1 as special or limiting cases1 the circle and the straight line4 ,illustration3 ,fig4 #bda4 ,illustration3 ,fig4 #bdb4 ,we may dispense with the hypothesis of bending moments1 if we simply presuppose that the increments of growth take place at a constant angle to the growing surface 7as ,,ab71 but more rapidly at ,a 7which we shall call the 8outer edge07 than at ,b1 and that this difference of velocity maintains a constant ratio4 ,let us also assume that the whole structure is rigid1 the new accretions solidifying as soon as they are laid on4 ,for example1 #eja let ,fig4 #bdb represent in section the early growth of a ,nautilus-shell1 and let the part ,,arb represent the earliest stage of all1 which in ,nautilus is nearly semicircular4 ,we have to find a law governing the growth of the shell1 such that each edge shall develop into an equiangular spiral2 and this law1 accordingly1 must be the same for each edge1 namely that at each instant the direction of growth makes a constant angle with a line drawn from a fixed point 7called the pole of the spiral7 to the point at which growth is taking place4 ,this growth1 we now find1 may be considered as effected by the continuous addition of similar quadrilaterals4 ,thus1 in ,fig4 #bda1 ,,aedb is a quadrilateral with ,,ae1 ,,db parallel1 and with the angle ,,eab of a certain definite magnitude1 4 ,let ,,ab and ,,ed meet1 when produced1 in ,c2 and call the angle ,,ace 7or x,cy7 4 ,make the angle y,cz angle x,cy1 4 ,draw ,,eg1 so that the angle y,e,g 1 meeting ,cz in ,g2 and draw ,,df parallel to ,,eg4 ,it is then easy to show that ,,aedb and ,,egfd are similar quadrilaterals4 ,and1 when we consider the quadrilateral ,,aedb as having infinitesimal sides1 ,,ae and ,,bd1 the angle tends to 1 the constant angle of an equiangular spiral which passes through the points ,,aeg1 and of a similar spiral which passes through the points ,,bdf2 and the point ,c is the pole of both of these spirals4 ,in a particular limiting case1 when our quadrilaterals are all equal as well as similar1--which will be the case when the angle 7or the angles ,,eac1 etc47 is a #ejb right angle1--the 8spiral0 curve will be a circular arc1 ,c being the centre of the circle4 ,another1 and a very simple illustration may be drawn from the 8cymose inflorescences0 of the botanists1 though the actual mode of development of some of these structures is open to dispute1 and their nomenclature is involved in extraordinary historical confusion#dif4 ,illustration3 ,fig4 #bdc4 ,a1 a helicoid1 ,b1 a scorpioid cyme4 ,in ,fig4 #bdc;,b 7which represents the ,cicinnus of ,schimper1 or cyme unipare scorpioide of ,bravais1 as seen in the ,borage71 we begin with a primary shoot from which is given off1 at a certain definite angle1 a secondary shoot3 and from that in turn1 on the same side and at the same angle1 another shoot1 and so on4 ,the deflection1 or curvature1 is continuous and progressive1 for it is caused by no external force but only by causes intrinsic in the system4 ,and the whole system is symmetrical3 the angles at which the successive shoots are given off being all equal1 and the lengths of the shoots diminishing in constant ratio4 ,the result is that the successive shoots1 or successive increments of growth1 are tangents to a curve1 and this curve is a true logarithmic spiral4 ,but while1 in this simple case1 the successive shoots are depicted as lying in a plane1 it may also happen that1 in addition to their successive angular divergence from one another within that plane1 they also tend to diverge by successive equal angles from that plane of reference2 and by this means1 there will be superposed upon the logarithmic spiral a helicoid twist or screw4 ,and1 in the particular case where this latter angle of divergence is just equal to #ahj1 or two right angles1 the successive shoots will once more come to lie in a plane1 but they will appear to come off from one another on alternate sides1 as in ,fig4 #bdc ,a4 ,this is the ,schraubel or ,bostryx of ,schimper1 the cyme unipare hlicoide of ,bravais4 ,the logarithmic spiral is still latent in it1 as in the other2 but is concealed from view by the deformation resulting from the helicoid4 ,the confusion of nomenclature would seem to have arisen from the fact that many botanists did not recognise 7as the brothers ,bravais did7 the mathematical significance of the latter case2 but were led1 by the snail-like spiral of the scorpioid cyme1 to transfer the name 8helicoid0 to it4 ,in the study of such curves as these1 then1 we speak of the point of origin as the pole 7,o72 a straight line having its extremity in the pole and revolving about it1 is called the radius vector2 #ejc and a point 7,p7 which is conceived as travelling along the radius vector under definite conditions of velocity1 will then describe our spiral curve4 ,of several mathematical curves whose form and development may be so conceived1 the two most important 7and the only two with which we need deal71 are those which are known as 7#a7 the equable spiral1 or spiral of ,archimedes1 and 7#b7 the logarithmic1 or equiangular spiral4 ,illustration3 ,fig4 #bdd4 ,the former may be illustrated by the spiral coil in which a sailor coils a rope upon the deck2 as the rope is of uniform thickness1 so in the whole spiral coil is each whorl of the same breadth as that which precedes and as that which follows it4 ,using its ancient definition1 we may define it by saying1 that 8,if a straight line revolve uniformly about its extremity1 a point which likewise travels uniformly along it will describe the equable spiral#dig40 ,or1 putting the same thing into our more modern words1 8,if1 while the radius vector revolve uniformly about the pole1 a point 7,p7 travel with uniform velocity along it1 the curve described will be that called the equable spiral1 or spiral of ,archimedes40 #ejd ,it is plain that the spiral of ,archimedes may be compared to a cylinder coiled up4 ,and it is plain also that a radius 7r ,,op71 made up of the successive and equal whorls1 will increase in arithmetical progression3 and will equal a certain constant quantity 7a7 multiplied by the whole number of whorls1 or 7more strictly speaking7 multiplied by the whole angle 77 through which it has revolved3 so that r a4 ,but1 in contrast to this1 in the logarithmic spiral of the ,nautilus or the snail-shell1 the whorls gradually increase in breadth1 and do so in a steady and unchanging ratio4 ,our definition is as follows3 8,if1 instead of travelling with a uniform velocity1 our point move along the radius vector with a velocity increasing as its distance from the pole1 then the path described is called a logarithmic spiral40 ,each whorl which the radius vector intersects will be broader than its predecessor in a definite ratio2 the radius vector will increase in length in geometrical progression1 as it sweeps through successive equal angles2 and the equation to the spiral will be r a4 ,as the spiral of ,archimedes1 in our example of the coiled rope1 might be looked upon as a coiled cylinder1 so may the logarithmic spiral1 in the case of the shell1 be pictured as a cone coiled upon itself4 ,now it is obvious that if the whorls increase very slowly indeed1 the logarithmic spiral will come to look like a spiral of ,archimedes1 with which however it never becomes identical2 for it is incorrect to say1 as is sometimes done1 that the ,archimedean spiral is a 8limiting case0 of the logarithmic spiral4 ,the ,nummulite is a case in point4 ,here we have a large number of whorls1 very narrow1 very close together1 and apparently of equal breadth1 which give rise to an appearance similar to that of our coiled rope4 ,and1 in a case of this kind1 we might actually find that the whorls were of equal breadth1 being produced 7as is apparently the case in the ,nummulite7 not by any very slow and gradual growth in thickness of a continuous tube1 but by a succession of similar cells or chambers laid on1 round and round1 determined as to their size by constant surface-tension conditions and therefore of unvarying dimensions4 ,but even in this case we should have no ,archimedean spiral1 but only a logarithmic spiral in which the constant angle approximated to #ij4 #eje ,for1 in the logarithmic spiral1 when tends to #ij1 the expression r a cot tends to r a7#a cot 72 while the equation to the ,archimedean spiral is r b4 ,the nummulite must always have a central core1 or initial cell1 around which the coil is not only wrapped1 but out of which it springs2 and this initial chamber corresponds to our a in the expression r a a cot 4 ,the outer whorls resemble those of an ,archimedean spiral1 because of the other term a cot in the same expression4 ,it follows from this that in all such cases the whorls must be of excessively small breadth4 ,there are many other specific properties of the logarithmic spiral1 so interrelated to one another that we may choose pretty well any one of them as the basis of our definition1 and deduce the others from it either by analytical methods or by the methods of elementary geometry4 ,for instance1 the equation r a may be written in the form log r log a1 or 7log r7_/7log a71 or 7since a is a constant71 k log r4 ,which is as much as to say that the vector angles about the pole are proportional to the logarithms of the successive radii2 from which circumstance the name of the 8logarithmic spiral0 is derived4 ,illustration3 ,fig4 #bde4 ,let us next regard our logarithmic spiral from the dynamical point of view1 as when we consider the forces concerned in the growth of a material1 concrete spiral4 ,in a growing structure1 let the forces of growth exerted at any point ,p be a force ,f acting along the line joining ,p to a pole ,o and a force ,t acting in a direction perpendicular to ,,op2 and let the magnitude of these forces be in the same constant ratio at all points4 ,it follows that the resultant of the forces ,f and ,t 7as ,,pq7 makes a constant angle with the radius vector4 ,but the constancy of the angle between tangent and radius vector at any point is a fundamental property of the logarithmic spiral1 and may be shewn to follow from our definition of the curve3 it gives to the curve its alternative name of equiangular spiral4 ,hence in a structure growing under the above conditions the form of the boundary will be a logarithmic spiral4 #ejf ,illustration3 ,fig4 #bdf4 ,in such a spiral1 radial growth and growth in the direction of the curve bear a constant ratio to one another4 ,for1 if we consider a consecutive radius vector1 ,,op1 whose increment as compared with ,,op is dr1 while ds is the small arc ,,pp1 then dr_/ds cos constant4 ,illustration3 ,fig4 #bdg4 ,in the concrete case of the shell1 the distribution of forces will be1 originally1 a little more complicated than this1 though by resolving the forces in question1 the system may be reduced to this simple form4 ,and furthermore1 the actual distribution of forces will not always be identical2 for example1 there is a distinct difference between the cases 7as in the snail7 where a columellar muscle exerts a definite traction in the direction of the pole1 and those 7such as ,nautilus7 where there is no columellar muscle1 and where some other force must be discovered1 or postulated1 to account for the flexure4 ,in the most frequent case1 we have1 as in ,fig4 #bdg1 three forces to deal with1 acting at a point1 p3 ,l1 acting in the direction of the tangent to the curve1 and representing the force of longitudinal growth2 ,t1 perpendicular to ,l1 and representing the organism's tendency to grow in breadth2 and ,p1 the traction exercised1 in the direction of the pole1 by the columellar muscle4 ,let us resolve ,l and ,t into components along ,p 7namely ,a1 ,b71 and perpendicular to ,p 7namely ,a1 ,b72 we have now only two forces to consider1 viz4 ,p ,a ,b1 and ,a ,b4 ,and these two latter we can again resolve1 if we please1 so as to deal only with forces in the direction of ,p and ,t4 ,now1 the ratio of these forces remaining constant1 the locus of the point p is an equiangular spiral4 #ejg ,furthermore we see how any slight change in any one of the forces ,p1 ,t1 ,l will tend to modify the angle 1 and produce a slight departure from the absolute regularity of the logarithmic spiral4 ,such slight departures from the absolute simplicity and uniformity of the theoretic law we shall not be surprised to find1 more or less frequently1 in ,nature1 in the complex system of forces presented by the living organism4 ,in the growth of a shell1 we can conceive no simpler law than this1 namely1 that it shall widen and lengthen in the same unvarying proportions3 and this simplest of laws is that which ,nature tends to follow4 ,the shell1 like the creature within it1 grows in size but does not change its shape2 and the existence of this constant relativity of growth1 or constant similarity of form1 is of the essence1 and may be made the basis of a definition1 of the logarithmic spiral4 ,such a definition1 though not commonly used by mathematicians1 has been occasionally employed2 and it is one from which the other properties of the curve can be deduced with great ease and simplicity4 ,in mathematical language it would run as follows3 8,any plane curve proceeding from a fixed point 7which is called the pole71 and such that the arc intercepted between this point and any other whatsoever on the curve is always similar to itself1 is called an equiangular1 or logarithmic1 spiral#dih40 ,in this definition1 we have what is probably the most fundamental and 8intrinsic0 property of the curve1 namely the property of continual similarity3 and this is indeed the very property by reason of which it is peculiarly associated with organic growth in such structures as the horn or the shell1 or the scorpioid cyme which is described on p4 #ejb4 ,for it is peculiarly characteristic of the spiral of a shell1 for instance1 that 7under all normal circumstances7 it does not alter its shape as it grows2 each increment is geometrically similar to its predecessor1 and the whole1 at any epoch1 is similar to what constituted the whole at another and an earlier epoch4 ,we feel no surprise when the animal which secretes the shell1 or any other animal whatsoever1 grows by such #ejh symmetrical expansion as to preserve its form unchanged2 though even there1 as we have already seen1 the unchanging form denotes a nice balance between the rates of growth in various directions1 which is but seldom accurately maintained for long4 ,but the shell retains its unchanging form in spite of its asymmetrical growth2 it grows at one end only1 and so does the horn4 ,and this remarkable property of increasing by terminal growth1 but nevertheless retaining unchanged the form of the entire figure1 is characteristic of the logarithmic spiral1 and of no other mathematical curve4 ,illustration3 ,fig4 #bdh4 ,we may at once illustrate this curious phenomenon by drawing the outline of a little ,nautilus shell within a big one4 ,we know1 or we may see at once1 that they are of precisely the same shape2 so that1 if we look at the little shell through a magnifying glass1 it becomes identical with the big one4 ,but we know1 on the other hand1 that the little ,nautilus shell grows into the big one1 not by uniform growth or magnification in all directions1 as is 7though only approximately7 the case when the boy grows into the man1 but by growing at one end only4 ,though of all curves1 this property of continued similarity is found only in the logarithmic spiral1 there are very many rectilinear figures in which it may be observed4 ,for instance1 as we may easily see1 it holds good of any right cone2 for evidently1 in ,fig4 #bdh1 the little inner cone 7represented in its triangular section7 may become identical with the larger one either by magnification all round 7as in a71 or simply by an increment at one end 7as in b72 indeed1 in the case of the cone1 we have yet a third possibility1 for the same result is attained when it increases all round1 save only at the base1 that is to say when the triangular section increases #eji on two of its sides1 as in c4 ,all this is closely associated with the fact1 which we have already noted1 that the ,nautilus shell is but a cone rolled up2 in other words1 the cone is but a particular variety1 or 8limiting case10 of the spiral shell4 ,this property1 which we so easily recognise in the cone1 would seem to have engaged the particular attention of the most ancient mathematicians even from the days of ,pythagoras1 and so1 with little doubt1 from the more ancient days of that ,egyptian school whence he derived the foundations of his learning#dii2 and its bearing on our biological problem of the shell1 though apparently indirect1 is yet so close that it deserves our further consideration4 ,illustration3 ,fig4 #bdi4 ,illustration3 ,fig4 #bej4 ,if1 as in ,fig4 #bdi1 we add to two sides of a square a symmetrical ,l-shaped portion1 similar in shape to what we call a 8carpenter's square10 the resulting figure is still a square2 and the portion which we have added is called1 by ,aristotle 7,phys4 ,,iii1 #d71 a 8gnomon40 ,euclid extends the term to include the case of any parallelogram#ejj1 whether rectangular or not 7,fig4 #bej72 and ,hero of ,alexandria specifically defines a 8gnomon0 7as indeed ,aristotle implicitly defines it71 as any figure which1 being added to any figure whatsoever1 leaves the resultant figure similar to the original4 ,included in this important definition is the case of numbers1 considered geometrically2 that is to say1 the 1 which can be translated into form1 by means of rows of dots or other signs 7cf4 ,arist4 ,metaph4 #ajib b #ab71 or in the pattern of a tiled floor3 all according to 8the mystical way of #eaj ,pythagoras1 and the secret magick of numbers40 ,thus for example1 the odd numbers are 8gnomonic numbers10 because #j #a #ab1 #ab #c #bb1 #bb #e #cb1 #cb #g #db etc41 which relation we may illustrate graphically by the successive numbers of dots which keep the annexed figure a perfect square#eja3 as follows3 ,illustration3 ,fig4 #bea4 ,illustration3 ,fig4 #beb4 ,there are other gnomonic figures more curious still4 ,for instance1 if we make a rectangle 7,fig4 #bea7 such that the two sides are in the ratio of #a 3 #b1 it is obvious that1 on doubling it1 we obtain a precisely similar figure2 for #a 3 #b 33 #b 3 #b2 and #eaa each half of the figure1 accordingly1 is now a gnomon to the other4 ,another elegant example is when we start with a rectangle 7,a7 whose sides are in the proportion of #a 3 7#e #a71 or1 approximately1 #a 3 #jfah4 ,the gnomon to this figure is a square 7,b7 erected on its longer side1 and so on successively 7,fig4 #beb74 ,illustration3 ,fig4 #bec4 ,illustration3 ,fig4 #bed4 ,in any triangle1 as ,aristotle tells us1 one part is always a gnomon to the other part4 ,for instance1 in the triangle ,,abc 7,fig4 #bec71 let us draw ,,cd1 so as to make the angle ,,bcd equal to the angle ,a4 ,then the part ,,bcd is a triangle similar to the whole triangle ,,abc1 and ,,adc is a gnomon to ,,bcd4 ,a very elegant case is when the original triangle ,,abc is an isosceles triangle having one angle of #cf1 and the other two angles1 therefore1 each equal to #gb 7,fig4 #bed74 ,then1 by bisecting one of the angles of the base1 we subdivide the large isosceles triangle into two isosceles triangles1 of which one is similar to the whole figure and the other is its gnomon#ejb4 ,there is good reason to believe that this triangle was especially studied by the ,pythagoreans2 for it lies at the root of many interesting geometrical constructions1 such as the regular pentagon1 and the mystical 8pentalpha10 and a whole range of other curious figures beloved of the ancient mathematicians#ejc4 #eab ,illustration3 ,fig4 #bee4 ,if we take any one of these figures1 for instance the isosceles triangle which we have just described1 and add to it 7or subtract from it7 in succession a series of gnomons1 so converting it into larger and larger 7or smaller and smaller7 triangles all similar to the first1 we find that the apices 7or other corresponding points7 of all these triangles have their locus upon a logarithmic spiral3 a result which follows directly from that alternative definition of the logarithmic spiral which ,i have quoted from ,whitworth 7p4 #ejg74 ,illustration3 ,fig4 #bef4 ,logarithmic spiral derived from corresponding points in a system of squares4 ,again1 we may build up a series of right-angled triangles1 each of which is a gnomon to the preceding figure2 and here again1 a logarithmic spiral is the locus of corresponding points in these successive triangles4 ,and lastly1 whensoever we fill up space with a #eac collection of either equal or similar figures1 similarly situated1 as in ,figs4 #bef1 #beg1 there we can always discover a series of inscribed or escribed logarithmic spirals4 ,illustration3 ,fig4 #beg4 ,the same in a system of hexagons4 ,once more1 then1 we may modify our definition1 and say that3 8,any plane curve proceeding from a fixed point 7or pole71 and such that the vectorial area of any sector is always a gnomon to the whole preceding figure1 is called an equiangular1 or logarithmic1 spiral40 ,and we may now introduce this new concept and nomenclature into our description of the ,nautilus shell and other related organic forms1 by saying that3 7#a7 if a growing structure be built up of successive parts1 similar and similarly situated1 we can always trace through corresponding points a series of logarithmic spirals 7,figs4 #beh1 #bei1 etc472 7#b7 it is characteristic of the growth of the horn1 of the shell1 and of all other organic forms in which a logarithmic spiral can be recognised1 that each successive increment of growth is a gnomon to the entire pre-existing structure4 ,and conversely 7#c7 it follows obviously1 that in the logarithmic spiral outline of the shell or of the horn we can always inscribe an endless variety of other gnomonic figures1 having no necessary relation1 save as a #ead mathematical accident1 to the nature or mode of development of the actual structure#ejd4 #eae ,illustration3 ,fig4 #beh4 ,a shell of ,haliotis1 with two of the many lines of growth1 or generating curves1 marked out in black3 the areas bounded by these lines of growth being in all cases 8gnomons0 to the pre-existing shell4 ,illustration3 ,fig4 #bei4 ,a spiral foraminifer 7,pulvinulina71 to show how each successive chamber continues the symmetry of1 or constitutes a gnomon to1 the rest of the structure4 ,illustration3 ,fig4 #bfj4 ,another spiral foraminifer1 ,cristellaria4 ,of these three propositions1 the second is of very great use and advantage for our easy understanding and simple description of the molluscan shell1 and of a great variety of other structures whose mode of growth is analogous1 and whose mathematical properties are therefore identical4 ,we see at once that the successive chambers of a spiral ,nautilus 7,fig4 #bcg7 or of a straight ,orthoceras 7,fig4 #cjj71 each whorl or part of a whorl of a periwinkle or other gastropod 7,fig4 #beh71 each new increment of the operculum of a gastropod 7,fig4 #bfc71 each additional increment of an elephant's tusk1 or each new chamber of a spiral foraminifer 7,figs4 #bei and #bfj71 has its leading characteristic at once described and its form so far explained by the simple statement that it constitutes a gnomon to the whole previously existing structure4 ,and herein lies the explanation of that 8time-element0 in the development of organic spirals of which we have spoken already1 in a preliminary and empirical way4 ,for it follows as a simple corollary to this theorem of gnomons that we must not expect to find the logarithmic spiral manifested in a structure whose parts are simultaneously produced1 as for instance in the margin of a leaf1 or among the many curves that make the contour of a fish4 ,but we must rather look for it wherever the organism retains for us1 and still presents to us at a single view1 the successive phases of preceding growth1 the successive magnitudes attained1 the successive outlines occupied1 as the organism or a part thereof pursued the even tenour of its growth1 year by year and day by day4 ,and it easily follows from this1 that it is in the hard parts of organisms1 and not the soft1 fleshy1 actively growing parts1 that this spiral is commonly and characteristically found2 not in the fresh mobile tissues whose form is constrained merely by the active forces of the moment2 but in things like shell and tusk1 and horn and claw1 where the object is visibly composed of parts #eaf successively1 and permanently1 laid down4 ,in the main1 the logarithmic spiral is characteristic1 not of the living tissues1 but of the dead4 ,and for the same reason1 it will always or nearly always be accompanied1 and adorned1 by a pattern formed of 8lines of growth10 the lasting record of earlier and successive stages of form and magnitude4 ,it is evident that the spiral curve of the shell is1 in a sense1 a vector diagram of its own growth2 for it shews at each instant of time1 the direction1 radial and tangential1 of growth1 and the unchanging ratio of velocities in these directions4 ,regarding the actual velocity of growth in the shell1 we know very little 7or practically nothing71 by way of experimental measurement2 but if we make a certain simple assumption1 then we may go a good deal further in our description of the logarithmic spiral as it appears in this concrete case4 ,let us make the assumption that similar increments are added to the shell in equal times2 that is to say1 that the amount of growth in unit time is measured by the areas subtended by equal angles4 ,thus1 in the outer whorl of a spiral shell a definite area marked out by ridges1 tubercles1 etc41 has very different linear dimensions to the corresponding areas of the inner whorl1 but the symmetry of the figure implies that it subtends an equal angle with these2 and it is reasonable to suppose that the successive regions1 marked out in this way by successive natural boundaries or patterns1 are produced in equal intervals of time4 ,if this be so1 the radii measured from the pole to the boundary of the shell will in each case be proportional to the velocity of growth at this point upon the circumference1 and at the time when it corresponded with the outer lip1 or region of active growth2 and while the direction of the radius vector corresponds with the direction of growth in thickness of the animal1 so does the tangent to the curve correspond with the direction1 for the time being1 of the animal's growth in length4 ,the successive radii are a measure of the acceleration of growth1 and the spiral curve of the shell itself is no other than the hodograph of the growth of the contained organism4 #eag ,so far as we have now gone1 we have studied the elementary properties of the logarithmic spiral1 including its fundamental property of continued similarity2 and we have accordingly learned that the shell or the horn tends necessarily to assume the form of this mathematical figure1 because in these structures growth proceeds by successive increments1 which are always similar in form1 similarly situated1 and of constant relative magnitude one to another4 ,our chief objects in enquiring further into the mathematical properties of the logarithmic spiral will be3 7#a7 to find means of confirming and verifying the fact that the shell 7or other organic curve7 is actually a logarithmic spiral2 7#b7 to learn how1 by the properties of the curve1 we may further extend our knowledge or simplify our descriptions of the shell2 and 7#c7 to understand the factors by which the characteristic form of any particular logarithmic spiral is determined1 and so to comprehend the nature of the specific or generic characters by which one spiral shell is found to differ from another4 ,of the elementary properties of the logarithmic spiral1 so far as we have now enumerated them1 the following are those which we may most easily investigate in the concrete case1 such as we have to do with in the molluscan shell3 7#a7 that the polar radii of points whose vectorial angles are in arithmetical progression1 are themselves in geometrical progression2 and 7#b7 that the tangent at any point of a logarithmic spiral makes a constant angle 7called the angle of the spiral7 with the polar radius vector4 ,illustration3 ,fig4 #bfa4 ,the former of these two propositions may be written in what is1 perhaps1 a simpler form1 as follows3 radii which form equal angles about the pole of the logarithmic spiral1 are themselves continued proportionals4 ,that is to say1 in ,fig4 #bfa1 when the angle ,,roq is equal to the angle ,,qop1 then ,,or 3 ,,oq 33 ,,oq 3 ,,op4 ,a particular case of this proposition is when the equal angles are each angles of #cfj3 that is to say when in each case the radius vector makes a complete revolution1 and when1 therefore ,p1 ,q and ,r all lie upon the same radius4 #eah ,it was by observing1 with the help of very careful measurement1 this continued proportionality1 that ,moseley was enabled to verify his first assumption1 based on the general appearance of the shell1 that the shell of ,nautilus was actually a logarithmic spiral1 and this demonstration he was immediately afterwards in a position to generalise by extending it to all the spiral ,ammonitoid and ,gastropod mollusca#eje4 ,for1 taking a median transverse section of a ,nautilus pompilius1 and carefully measuring the successive breadths of the whorls 7from the dark line which marks what was originally the outer surface1 before it was covered up by fresh deposits on the part of the growing and advancing shell71 ,moseley found that 8the distance of any two of its whorls measured upon a radius vector is one-third that of the two next whorls measured upon the same radius vector#ejf4 ,thus 7in ,fig4 #bfb71 ab is one-third of bc1 de of ef1 gh of hi1 and kl of lm4 ,the curve is therefore a logarithmic spiral40 ,the numerical ratio in the case of the ,nautilus happens to be one of unusual simplicity4 ,let us take1 with ,moseley1 a somewhat more complicated example4 ,from the apex of a large specimen of ,turbo duplicatus#ejg a #eai line was drawn across its whorls1 and their widths were measured upon it in succession1 beginning with the last but one4 ,the measurements were1 as before1 made with a fine pair of compasses and a diagonal scale4 ,the sight was assisted by a magnifying glass4 ,in a parallel column to the following admeasurements are the terms of a geometric progression1 whose first term is the width of the widest whorl measured1 and whose common ratio is #aahjd4 ,illustration3 ,fig4 #bfb4 ,widths of successive ,terms of a geometrical progression1 whorls measured in inches whose first term is the width of and parts of an inch the widest whorl1 and whose common ratio is #aahjd #aca #aca #aab #aajih #id #idjah #hj #gifea #fg #fgdgf #eg #egafd #dh #dhdbg #da #dajbf ,the close coincidence between the observed and the calculated figures is very remarkable1 and is amply sufficient to justify the conclusion that we are here dealing with a true logarithmic spiral4 ,nevertheless1 in order to verify his conclusion still further1 and to get partially rid of the inaccuracies due to successive small #ebj measurements1 ,moseley proceeded to investigate the same shell1 measuring not single whorls1 but groups of whorls1 taken several at a time3 making use of the following property of a geometrical progression1 that 8if represent the ratio of the sum of every even number 7m7 of its terms to the sum of half that number of terms1 then the common ratio 7r7 of the series is represented by the formula r 7 #a7b_/m 40 ,accordingly1 ,moseley made the following measurements1 beginning from the second and third whorls respectively3 ,width of ,six whorls ,three whorls ,ratio #ecg #bjc #bfde #dee #agb #bfde ,four whorls ,two whorls #dae #agd #bche #ceb #adg #bcid 8,by the ratios of the two first admeasurements1 the formula gives r 7#afde7a_/c #aahjd4 ,by the mean of the ratios deduced from the second two admeasurements1 it gives r 7#achi7 #aahjf4 8,it is scarcely possible to imagine a more accurate verification than is deduced from these larger admeasurements1 and we may with safety annex to the species ,turbo duplicatus the characteristic number #aah40 ,by similar and equally concordant observations1 ,moseley found for ,turbo phasianus the characteristic ratio1 #age2 and for ,buccinum subulatum that of #aac4 ,from the table referring to ,turbo duplicatus1 on page #eai1 it is perhaps worth while to illustrate the logarithmic statement of the same facts3 that is to say1 the elementary corollary to the fact that the successive radii are in geometric progression1 that their logarithms differ from one another by a constant amount4 #eba ,turbo duplicatus4 ,relative widths of ,logarithms of ,difference of successive whorls successive whorls successive logarithms #aca #baagbg -- #aab #bjdibb #jfhje #id #aigcac #jgfji #hj #aijcji #jgjjd #fg #ahbfjg #jggjb #eg #ageehg #jgjbj #dh #afhabd #jgdfc #da #aafabgh #jfhdf ,mean difference #jgbjg ,and #jgbjg is the logarithm of #aahje4 ,illustration3 ,fig4 #bfc4 ,operculum of ,turbo4 ,the logarithmic spiral is not only very beautifully manifested in the molluscan shell1 but also1 in certain cases1 in the little lid or 8operculum0 by which the entrance to the tubular shell is closed after the animal has withdrawn itself within4 ,in the spiral shell of ,turbo1 for instance1 the operculum is a thick calcareous structure1 with a beautifully curved outline1 which grows by successive increments applied to one portion of its edge1 and shews1 accordingly1 a spiral line of growth upon its surface4 ,the successive increments leave their traces on the surface of the operculum #ebb 7,fig4 #bfd1 #a71 which traces have the form of curved lines in ,turbo1 and of straight lines in 7e4g47 ,nerita 7,fig4 #bfd1 #b72 that is to say1 apart from the side constituting the outer edge of the operculum 7which side is always and of necessity curved7 the successive increments constitute curvilinear triangles in the one case1 and rectilinear triangles in the other4 ,the sides of these triangles are tangents to the spiral line of the operculum1 and may be supposed to generate it by their consecutive intersections4 ,illustration3 ,fig4 #bfd4 ,opercula of 7#a7 ,turbo1 7#b7 ,nerita4 7,after ,moseley47 ,in a number of such opercula1 ,moseley measured the breadths of the successive whorls along a radius vector#ejh1 just in the same way as he did with the entire shell in the foregoing cases2 and here is one example of his results4 ,operculum of ,turbo sp42 breadth 7in inches7 of successive whorls1 measured from the pole4 ,distance ,ratio ,distance ,ratio ,distance ,ratio ,distance ,ratio #bd #af #b #ah #bbh #bca #bcj #bcj #ee #cg #f #db #bcb #bcj #bcj #bbd #abh #he #ach #id #ebc ,the ratio is approximately constant1 and this spiral also is1 therefore1 a logarithmic spiral4 ,but here comes in a very beautiful illustration of that property of the logarithmic spiral which causes its whole shape to remain unchanged1 in spite of its apparently unsymmetrical1 or unilateral1 mode of growth4 ,for the mouth of the tubular shell1 into which the operculum has to fit1 is growing or widening on all sides3 while the operculum is increasing1 not by additions made at the same time all round its margin1 but by additions made only on one side of it at each successive stage4 ,one edge of the operculum thus remains unaltered as it is advanced into each new position1 and as it is placed in a newly formed section of the tube1 similar to but greater than the last4 ,nevertheless1 the two apposed structures1 the chamber and its plug1 at all times fit one another to perfection4 ,the mechanical problem 7by no means an easy one71 is thus solved3 8,how to shape a tube of a variable section1 so that a piston driven along it shall1 by one side of its margin1 coincide continually with its surface as it advances1 provided only that the piston be made at the same time continually to revolve in its own plane40 ,as ,moseley puts it3 8,that the same edge which fitted a portion of the first less section should be capable of adjustment1 so as to fit a portion of the next similar but greater section1 supposes a geometrical provision in the curved form of the chamber of great apparent complication and difficulty4 ,but ,god hath bestowed upon this humble architect the practical skill of a learned geometrician1 and he makes this provision with admirable precision in that curvature of the logarithmic spiral which he gives to the section of the shell4 ,this curvature obtaining1 he has only to turn his operculum slightly round in its own plane as he advances it into each newly formed portion of his chamber1 to adapt one margin of it to a new and larger surface and a different curvature1 leaving the space to be filled up by increasing the operculum wholly on the other margin40 ,but in many1 and indeed more numerous ,gastropod mollusca1 the operculum does not grow in this remarkable spiral fashion1 but by the apparently much simpler process of accretion by concentric rings4 ,this suggests to us another mathematical #ebd feature of the logarithmic spiral4 ,we have already seen that the logarithmic spiral has a number of 8limiting cases10 apparently very diverse from one another4 ,thus the right cone is a logarithmic spiral in which the revolution of the radius vector is infinitely slow2 and1 in the same sense1 the straight line itself is a limiting case of the logarithmic spiral4 ,the spiral of ,archimedes1 though not a limiting case of the logarithmic spiral1 closely resembles one in which the angle of the spiral is very near to #ij1 and the spiral is coiled around a central core4 ,but if the angle of the spiral were actually #ij1 the radius vector would describe a circle1 identical with the 8core0 of which we have just spoken2 and accordingly it may be said that the circle is1 in this sense1 a true limiting case of the logarithmic spiral4 ,in this sense1 then1 the circular concentric operculum1 for instance of ,turritella or ,littorina1 does not represent a breach of continuity1 but a 8limiting case0 of the spiral operculum of ,turbo2 the successive 8gnomons0 are now not lateral or terminal additions1 but complete concentric rings4 ,viewed in regard to its own fundamental properties and to those of its limiting cases1 the logarithmic spiral is the simplest of all known curves2 and the rigid uniformity of the simple laws1 or forces1 by which it is developed sufficiently account for its frequent manifestation in the structures built up by the slow and steady growth of organisms4 ,in order to translate into precise terms the whole form and growth of a spiral shell1 we should have to employ a mathematical notation1 considerably more complicated than any that ,i have attempted to make use of in this book4 ,but1 in the most elementary language1 we may now at least attempt to describe the general method1 and some of the variations1 of the mathematical development of the shell4 ,let us imagine a closed curve in space1 whether circular or elliptical or of some other and more complex specific form1 not necessarily in a plane3 such a curve as we see before us when we consider the mouth1 or terminal orifice1 of our tubular shell2 and let us imagine some one characteristic point within this closed curve1 such as its centre of gravity4 ,then1 starting from a fixed #ebe origin1 let this centre of gravity describe an equiangular spiral in space1 about a fixed axis 7namely the axis of the shell71 while at the same time the generating curve grows1 with each angular increment of rotation1 in such a way as to preserve the symmetry of the entire figure1 with or without a simultaneous movement of translation along the axis4 ,illustration3 ,fig4 #bfe4 ,melo ethiopicus1 ,l4 ,it is plain that the entire resulting shell may now be looked upon in either of two ways4 ,it is1 on the one hand1 an ensemble of similar closed curves spirally arranged in space1 gradually increasing in dimensions1 in proportion to the increase of their vectorial angle from the pole4 ,in other words1 we can imagine our shell cut up into a system of rings1 following one another in continuous spiral succession from that terminal and largest one1 which constitutes the lip of the orifice of the shell4 ,or1 on the other hand1 we may figure to ourselves the whole shell as made up of an ensemble of spiral lines in space1 each spiral having been #ebf traced out by the gradual growth and revolution of a radius vector from the pole to a given point of the generating curve4 ,both systems of lines1 the generating spirals 7as these latter may be called71 and the closed generating curves corresponding to successive margins or lips of the shell1 may be easily traced in a great variety of cases4 ,thus1 for example1 in ,dolium1 ,eburnea1 and a host of others1 the generating spirals are beautifully marked out by ridges1 tubercles or bands of colour4 ,in ,trophon1 ,scalaria1 and 7among countless others7 in the ,ammonites1 it is the successive generating curves which more conspicuously leave their impress on the shell4 ,and in not a few cases1 as in ,harpa1 ,dolium perdix1 etc41 both alike are conspicuous1 ridges and colour-bands intersecting one another in a beautiful isogonal system4 #ebg ,illustration3 ,fig4 #bff4 #a1 ,harpa2 #b1 ,dolium4 ,the ridges on the shell correspond in 7#a7 to generating curves1 in 7#b7 to generating spirals4 ,in the complete mathematical formula 7such as ,i have not ventured to set forth#eji7 for any given turbinate shell1 we should have1 accordingly1 to include factors for at least the following elements3 7#a7 for the specific form of the section of the tube1 which we have called the generating curve2 7#b7 for the specific rate of growth of this generating curve2 7#c7 for its specific rate of angular rotation about the pole1 perpendicular to the axis2 7#d7 in turbinate 7as opposed to nautiloid7 shells1 for its rate of shear1 or screw-translation parallel to the axis4 ,there are also other factors of which we should have to take account1 and which would help to make our whole expression a very complicated one4 ,we should find1 for instance1 7#e7 that in very many cases our generating curve was not a plane curve1 but a sinuous curve in three dimensions2 and we should also have to take account 7#f7 of the inclination of the plane of this generating curve to the axis1 a factor which will have a very important influence on the form and appearance of the shell4 ,for instance in ,haliotis it is obvious that the generating curve lies in a plane very oblique to the axis of the shell4 ,lastly1 we at once perceive that the ratios which happen to exist between these various factors1 the ratio for instance between the growth-factor and the rate of angular revolution1 will give us endless possibilities of permutation of form4 ,for instance 7#g7 with a given velocity of vectorial rotation1 a certain rate of growth in the generating curve will give us a spiral shell of which each successive whorl will just touch its predecessor and no more2 with a slower growth-factor1 the whorls will stand asunder1 as in a ram's horn2 with a quicker growth-factor1 each whorl will cut or intersect its predecessor1 as in an ,ammonite or the majority of gastropods1 and so on 7cf4 p4 #eda74 ,in like manner 7#h7 the ratio between the growth-factor and the rate of screw-translation parallel to the axis will determine the apical angle of the resulting conical structure3 will give us the difference1 for example1 between the sharp1 pointed cone of ,turritella1 the less acute one of ,fusus or ,buccinum1 and the #ebh obtuse one of ,harpa or ,dolium4 ,in short it is obvious that all the differences of form which we observe between one shell and another are referable to matters of degree1 depending1 one and all1 upon the relative magnitudes of the various factors in the complex equation to the curve4 ,the paper in which1 nearly eighty years ago1 ,canon ,moseley thus gave a simple mathematical expression to the spiral forms of univalve shells1 is one of the classics of ,natural ,history4 ,but other students before him had come very near to recognising this mathematical simplicity of form and structure4 ,about the year #ahah1 ,reinecke had suggested that the relative breadths of the adjacent whorls in an ,ammonite formed a constant and diagnostic character2 and ,leopold von ,buch accepted and developed the idea#eaj4 ,but long before1 ,swammerdam1 with a deeper insight1 had grasped the root of the whole matter3 for1 taking a few diverse examples1 such as ,helix and ,spirula1 he shewed that they and all other spiral shells whatsoever were referable to one common type1 namely to that of a simple tube1 variously curved according to definite mathematical laws2 that all manner of ornamentation1 in the way of spines1 tuberosities1 colour-bands and so forth1 might be superposed upon them1 but the type was one throughout1 and specific differences were of a geometrical kind4 8,omnis enim quae inter eas animadvertitur differentia ex sola nascitur diversitate gyrationum3 quibus si insuper externa quaedam adjunguntur ornamenta pinnarum1 sinuum1 anfractuum1 planitierum1 eminentiarum1 profunditatum1 extensionum1 impressionum1 circumvolutionum1 colorumque3 444 tunc deinceps facile est1 quarumcumque ,cochlearum figuras geometricas1 curvosque1 obliquos atque rectos angulos1 ad unicam omnes speciem redigere3 ad oblongum videlicet tubulum1 qui vario modo curvatus1 crispatus1 extrorsum et introrsum flexus1 ita concrevit#eaa40 #ebi ,for some years after the appearance of ,moseley's paper1 a number of writers followed in his footsteps1 and attempted1 in various ways1 to put his conclusions to practical use4 ,for instance1 ,d',orbigny devised a very simple protractor1 which he called a ,helicometer#eab1 and which is represented in ,fig4 #bfg4 ,by means of this little instrument1 the apical angle of the turbinate shell was immediately read off1 and could then be used as a specific and diagnostic character4 ,by keeping one limb of the protractor parallel to the side of the cone while the other was brought into line with the suture between two adjacent whorls1 another specific angle1 the 8sutural angle10 could in like manner be recorded4 ,and1 by the linear scale upon the instrument1 the relative breadths of the consecutive whorls1 and that of the terminal chamber to the rest of the shell1 might also1 though somewhat roughly1 be determined4 ,for instance1 in ,terebra dimidiata1 the apical angle was found to be #ac1 the sutural angle #aji1 and so forth4 ,illustration3 ,fig4 #bfg4 ,d',orbigny's ,helicometer4 ,it was at once obvious that1 in such a shell as is represented in ,fig4 #bfg the entire outline of the shell 7always excepting that of the immediate neighbourhood of #ecj the mouth7 could be restored from a broken fragment4 ,for if we draw our tangents to the cone1 it follows from the symmetry of the figure that we can continue the projection of the sutural line1 and so mark off the successive whorls1 by simply drawing a series of consecutive parallels1 and by then filling into the quadrilaterals so marked off a series of curves similar to one another1 and to the whorls which are still intact in the broken shell4 ,but the use of the helicometer soon shewed that it was by no means universally the case that one and the same right cone was tangent to all the turbinate whorls2 in other words1 there was not always one specific apical angle which held good for the entire system4 ,in the great majority of cases1 it is true1 the same tangent touches all the whorls1 and is a straight line4 ,but in others1 as in the large ,cerithium nodosum1 such a line is slightly convex to the axis of the shell2 and in the short spire of ,dolium1 for instance1 the convexity is marked1 and the apex of the spire is a distinct cusp4 ,on the other hand1 in ,pupa and ,clausilia1 the common tangent is concave to the axis of the shell4 ,so also is it1 as we shall presently see1 among the ,ammonites3 where there are some species in which the ratio of whorl to whorl remains1 to all appearance1 perfectly constant2 others in which it gradually1 though only slightly increases2 and others again in which it slightly and gradually falls away4 ,it is obvious that1 among the manifold possibilities of growth1 such conditions as these are very easily conceivable4 ,it is much more remarkable that1 among these shells1 the relative velocities of growth in various dimensions should be as constant as it is1 than that there should be an occasional departure from perfect regularity4 ,in such cases as these latter1 the logarithmic law of growth is only approximately true4 ,the shell is no longer to be represented as a right cone which has been rolled up1 but as a cone which had grown trumpet-shaped1 or conversely whose mouth had narrowed in1 and which in section is a curvilinear instead of a rectilinear triangle4 ,but all that has happened is that a new factor1 usually of small or all but imperceptible magnitude1 has been introduced into the case2 so that the ratio1 log r log 1 is no longer constant1 but varies slightly1 and in accordance with some simple law4 #eca ,some writers1 such as ,naumann and ,grabau1 maintained that the molluscan spiral was no true logarithmic spiral1 but differed from it specifically1 and they gave to it the name of ,conchospiral4 ,they pointed out that the logarithmic spiral originates in a mathematical point1 while the molluscan shell starts with a little embryonic shell1 or central chamber 7the 8protoconch0 of the conchologists71 around which the spiral is subsequently wrapped4 ,it is plain that this undoubted and obvious fact need not affect the logarithmic law of the shell as a whole2 we have only to add a small constant to our equation1 which becomes r m a4 ,there would seem1 by the way1 to be considerable confusion in the books with regard to the so-called 8protoconch40 ,in many cases it is a definite structure1 of simple form1 representing the more or less globular embryonic shell before it began to elongate into its conical or spiral form4 ,but in many cases what is described as the 8protoconch0 is merely an empty space in the middle of the spiral coil1 resulting from the fact that the actual spiral shell has a definite magnitude to begin with1 and that we cannot follow it down to its vanishing point in infinity4 ,for instance1 in the accompanying figure1 the large space a is styled the protoconch1 but it is the little bulbous or hemispherical chamber within it1 at the end of the spire1 which is the real beginning of the tubular shell4 ,the form and magnitude of the space a are determined by the 8angle of retardation10 or ratio of rate of growth between the inner and outer curves of the spiral shell4 ,they are independent of the shape and size of the embryo1 and depend only 7as we shall see better presently7 on the direction and relative rate of growth of the double contour of the shell4 ,illustration3 ,fig4 #bfh4 ,now that we have dealt1 in a very general way1 with some of the more obvious properties of the logarithmic spiral1 let us consider certain of them a little more particularly1 keeping in #ecb view as our chief object the investigation 7on elementary lines7 of the possible manner and range of variation of the molluscan shell4 ,illustration3 ,fig4 #bfi4 ,there is yet another equation to the logarithmic spiral1 very commonly employed1 and without the help of which we shall find that we cannot get far4 ,it is as follows3 r cot 4 ,this follows directly from the fact that the angle 7the angle between the radius vector and the tangent to the curve7 is constant4 ,for1 then1 tan 7 tan 7 r d_/dr1 therefore dr_/r d cot 1 and1 integrating1 log r cot 1 or r cot 4 ,as we have seen throughout our preliminary discussion1 the two most important constants 7or chief 8specific characters10 as the naturalist would say7 in any given logarithmic spiral1 are 7#a7 the magnitude of the angle of the spiral1 or 8constant angle10 1 and 7#b7 the rate of increase of the radius vector for any given angle of revolution1 4 ,of this latter1 the simplest case is when #b1 or #cfj2 that is to say when we compare the breadths1 along the same radius vector1 of two successive whorls4 ,as our two magnitudes1 that of the constant angle1 and that of the ratio of the radii or breadths of whorl1 are related to one another1 we may determine either of them by actual measurement and proceed to calculate the other4 ,in any complete spiral1 such as that of ,nautilus1 it is 7as we have seen7 easy to measure any two radii 7r71 or the breadths in #ecc a radial direction of any two whorls 7,w74 ,we have then merely to apply the formula rn #a_/rn e cot 1 or ,wn #a_/,wn e cot 1 which we may simply write r e cot 1 etc42 since our first radius or whorl is regarded1 for the purpose of comparison1 as being equal to unity4 ,thus1 in the diagram1 ,,oc_/,,oe1 or ,,ef_/,,bd1 or ,,dc_/,,ef1 being in each case radii1 or diameters1 at right angles to one another1 are all equal to e_/#b cot 4 ,while in like manner1 ,,eo_/,,of1 ,,eg_/,,fh1 or ,,go_/,,ho1 all equal e cot 2 and ,,bc_/,,ba1 or ,,co_/,,ob e#b cot 4 ,illustration3 ,fig4 #bgj4 ,as soon1 then1 as we have prepared tables for these values1 the determination of the constant angle in a particular shell becomes a very simple matter4 ,a complete table would be cumbrous1 and it will be sufficient to deal with the simple case of the ratio between the breadths of adjacent1 or immediately succeeding1 whorls4 ,here we have r e#b cot 1 or log r log e #b cot 1 from which we obtain the following figures#eac3 #ecd ,ratio of breadth of each whorl to the next preceding ,constant angle r_/#a #aa #hi #h #abe #hg #eh #ae #hf #ah #bj #hc #db #be #ha #db #cj #hj #e #ce #gh #dc #dj #gg #cd #de #gf #cb #ej #ge #ch #ajj #fi #ec #bjj #fd #ca #ejj #eh #e #ajjj #ec #df #a1jjjj #db #ag #aj1jjj #cd #ai #ajj1jjj #bh #cg #a1jjj1jjj #bd #bh #aj1jjj1jjj #ba #ah #ajj1jjj1jjj #ah #ej #a1jjj1jjj1jjj #af #eb ,illustration3 ,fig4 #bga4 ,we learn several interesting things from this short table4 ,we see1 in the first place1 that where each whorl is about three times the breadth of its neighbour and predecessor1 as is the case in ,nautilus1 the constant angle is in the neighbourhood of #hj2 and hence also that1 in all the ordinary ,ammonitoid shells1 and in all the typically spiral shells of the ,gastropods#ead1 the constant angle is also a large one1 being very seldom less than #hj1 and usually between #hj and #he4 ,in the next place1 we see that with smaller angles the apparent form of the spiral is greatly altered1 and the very fact of its being a spiral soon ceases to be apparent 7,figs4 #bga1 #bgb74 ,suppose one whorl to be an inch in breadth1 then1 if the angle of the spiral were #hj1 the #ece next whorl would 7as we have just seen7 be about three inches broad2 if it were #gj1 the next whorl would be nearly ten inches1 and if it were #fj1 the next whorl would be nearly four feet broad4 ,if the angle were #bh1 the next whorl would be a mile and a half in breadth2 and if it were #ag1 the next would be some #ae1jjj miles broad4 ,illustration3 ,fig4 #bgb4 ,in other words1 the spiral shells of gentle curvature1 or of small constant angle1 such as ,dentalium or ,nodosaria1 are true logarithmic spirals1 just as are those of ,nautilus or ,rotalia3 from which they differ only in degree1 in the magnitude of an angular constant4 ,but this diminished magnitude of the angle causes the spiral to dilate with such immense rapidity that1 so to speak1 8it never comes round02 and so1 in such a shell as ,dentalium1 we never see but a small portion of the initial whorl4 ,illustration3 ,fig4 #bgc4 ,we might perhaps be inclined to suppose that1 in such a shell as ,dentalium1 the lack of a visible spiral convolution was only due to our seeing but a small portion of the curve1 at a distance from the pole1 and when1 therefore1 its #ecf curvature had already greatly diminished4 ,that is to say we might suppose that1 however small the angle a1 and however rapidly the whorls accordingly increased1 there would nevertheless be a manifest spiral convolution in the immediate neighbourhood of the pole1 as the starting point of the curve4 ,but it may be shewn that this is not so4 ,for1 taking the formula r a cot 1 this1 for any given spiral1 is equivalent to ak4 ,therefore log7r_/a7 k1 or1 #a_/k _/log7r_/a74 ,then1 if increase by #b1 while r increases to r#a1 #a_/k 7 #b7_/log7r#a_/;a71 which leads1 by subtraction to #a_/k log7r#a_/r7 #b4 ,now1 as tends to #j1 k 7i4e4 cot 7 tends to 1 and therefore1 as k 1 log7r#a_/r7 and also r#a_/r 4 ,therefore if one whorl exists1 the radius vector of the other is infinite2 in other words1 there is nowhere1 even in the near neighbourhood of the pole1 a complete revolution of the spire4 ,our spiral shells of small constant angle1 such as ,dentalium1 may accordingly be considered to represent sufficiently well the true commencement of their respective spirals4 ,let us return to the problem of how to ascertain1 by direct measurement1 the spiral angle of any particular shell4 ,the method already employed is only applicable to complete spirals1 that is to say to those in which the angle of the spiral is large1 and furthermore it is inapplicable to portions1 or broken fragments1 of a shell4 ,in the case of the broken fragment1 it is plain that the determination of the angle is not merely of theoretic interest1 but may be of great practical use to the conchologist as being the one and only way by which he may restore the outline of the missing portions4 ,we have a considerable choice of methods1 which have been summarised by1 and are partly due to1 a very careful student of the ,cephalopoda1 the late ,rev4 ,j4 ,f4 ,blake#eae4 #ecg 7#a7 ,the following method is useful and easy when we have a portion of a single whorl1 such as to shew both its inner and its outer edge4 ,a broken whorl of an ,ammonite1 a curved shell such as ,dentalium1 or a horn of similar form to the latter1 will fall under this head4 ,we have merely to draw a tangent1 ,,geh1 to the outer whorl at any point ,e2 then draw to the inner whorl a tangent parallel to ,,geh1 touching the curve in some point ,f4 ,the straight line joining the points of contact1 ,,ef1 must evidently pass through the pole3 and1 accordingly1 the angle ,,gef is the angle required4 ,in shells which bear longitudinal striae or other ornaments1 any pair of these will suffice for our purpose1 instead of the actual boundaries of the whorl4 ,but it is obvious that this method will be apt to fail us when the angle is very small2 and when1 consequently1 the points ,e and ,f are very remote4 ,illustration3 ,fig4 #bgd4 7#b7 ,in shells 7or horns7 shewing rings1 or other transverse ornamentation1 we may take it that these ornaments are set at a constant angle to the spire1 and therefore to the radii4 ,the angle 77 between two of them1 as ,,ac1 ,,bd1 is therefore equal to the angle between the polar radii from ,a and ,b1 or from ,c and ,d2 and therefore ,,bd_/,,ac e cot 1 which gives us the angle in terms of known quantities4 #ech ,illustration3 ,fig4 #bge4 ,an ,ammonite1 to shew corrugated surface-pattern4 ,illustration3 ,fig4 #bgf4 7#c7 ,if only the outer edge be available1 we have the ordinary geometrical problem1--given an arc of an equiangular spiral1 to find its pole and spiral angle4 ,the methods we may employ depend 7#a7 on determining directly the position of the pole1 and 7#b7 on determining the radius of curvature4 ,illustration3 ,fig4 #bgg4 ,the first method is theoretically simple1 but difficult in practice2 for it requires great accuracy in determining the points4 ,let ,,ad1 ,,db1 be two tangents drawn to the curve4 ,then a circle drawn through the points ,,abd will pass through the pole ,o2 since the angles ,,oad1 ,,obe 7the supplement of ,,obd71 are equal4 ,the point ,o may be determined by the intersection of two such circles2 and the angle ,,dbo is then the angle1 1 required4 ,or we may determine1 graphically1 at two points1 the radii of curvature1 #a #b4 ,then1 if s be the length of the arc between them 7which may be determined with fair accuracy by rolling the margin of the shell along a ruler7 cot 7#a #b7_/s4 ,the following method#eaf1 given by ,blake1 will save actual determination of the radii of curvature4 ,measure along a tangent to the curve1 the distance1 ,,ac1 at which a certain small offset1 ,,cd1 is made by the curve2 and from another point ,b1 measure the distance at which the curve makes an equal offset4 ,then1 calling the offset 2 the arc ,,ab1 s2 and ,,ac1 ,,be1 respectively x#a1 x#b1 we have #a 7x#ab #b7_/b1 approximately1 and cot 7x#bb x#ab7_/bs4 ,of all these methods by which the mathematical constants1 or specific characters1 of a given spiral shell may be determined1 the only one of which much use has been made is that which ,moseley first employed1 namely1 the simple method of determining #eci the relative breadths of the whorl at distances separated by some convenient vectorial angle 7such as #ij1 #ahj1 or #cfj74 ,very elaborate measurements of a number of ,ammonites have been made by ,naumann#eag1 by ,sandberger#eah1 and by ,grabau#eai1 among which we may choose a couple of cases for consideration4 ,in the following table ,i have taken a portion of ,grabau's determinations of the breadth of the whorls in ,ammonites 7,arcestes7 ,ammonites intuslabiatus4 ,ratio of breadth of ,breadth of whorls successive whorls ,the angle 77 7#ahj apart7 7#cfj apart7 as calculated #jcj mm4 -- -- -- #jcj #accc #hg #bc #jdj #aejj #hf #ai #jde #aejj #hf #ai #jfj #addd #hf #ci #jfe #adag #hf #di #jhe #afib #he #ac #aaj #aehh #he #dg #ace #aede #hf #b #agj #afcj #he #cc #bbj #adda #hf #dj #bde #adcb #hf #dc #cae #agce #he #j #dbe #afhc #he #af #ecj #adhb #hf #be #fcj #aeai #hf #ab #hje #afce #he #cb #ajcj #adaf #hf #ej #aadj #abeb #hg #eg #abij -- -- -- ,mean #hf #ae #edj intuslabiatus2 these measurements ,grabau gives for every #de of arc1 but ,i have only set forth one quarter of these measurements1 that is to say1 the breadths of successive whorls measured along one diameter on both sides of the pole4 ,the ratio between alternate measurements is therefore the same ratio as ,moseley adopted1 namely the ratio of breadth between contiguous whorls along a radius vector4 ,i have then added to these observed values the corresponding calculated values of the angle 1 as obtained from our usual formula4 ,there is considerable irregularity in the ratios derived from these measurements1 but it will be seen that this irregularity only implies a variation of the angle of the spiral between about #he and #hg2 and the values fluctuate pretty regularly about the mean1 which is #hf #ae4 ,considering the difficulty of measuring the whorls1 especially towards the centre1 and in particular the difficulty of determining with precise accuracy the position of the pole1 it is clear that in such a case as this we are scarcely justified in asserting that the law of the logarithmic spiral is departed from4 ,in some cases1 however1 it is undoubtedly departed from4 ,here for instance is another table from ,grabau1 shewing the corresponding ratios in an ,ammonite of the group of ,arcestes tornatus4 ,in this case we see a distinct tendency of the ratios to ,ammonites tornatus4 ,ratio of breadth of ,breadth of whorls successive whorls ,the spiral angle 7#ahj apart7 7#cfj apart7 77 as calculated #jbe mm4 -- -- -- #jcj #adjj #hf #ef #jce #affg #he #ba #jej #bjjj #hc #db #jgj #bjjj #hc #db #ajj #bjjj #hc #db #adj #bajj #hc #af #baj #bagi #hb #ef #cje #bbch #hb #db #dgj #bdib #ha #dd #gfj #begd #ha #bg #abaj #bedf #ha #cc #aice -- -- -- ,mean #hc #bb #eda increase as we pass from the centre of the coil outwards1 and consequently for the values of the angle to diminish4 ,the case is precisely comparable to that of a cone with slightly curving sides3 in which1 that is to say1 there is a slight acceleration of growth in a transverse as compared with the longitudinal direction4 ,in a tubular spiral1 whether plane or helicoid1 the consecutive whorls may either be 7#a7 isolated and remote from one another2 or 7#b7 they may precisely meet1 so that the outer border of one and the inner border of the next just coincide2 or 7#c7 they may overlap1 the vector plane of each outer whorl cutting that of its immediate predecessor or predecessors4 ,looking1 as we have done1 upon the spiral shell as being essentially a cone rolled up1 it is plain that1 for a given spiral angle1 intersection or non-intersection of the successive whorls will depend upon the apical angle of the original cone4 ,for the wider the cone1 the more rapidly will its inner border tend to encroach on the outer border of the preceding whorl4 ,but it is also plain that the greater be the apical angle of the cone1 and the broader1 consequently1 the cone itself be1 the greater difference will there be between the total lengths of its inner and outer border1 under given conditions of flexure4 ,and1 since the inner and outer borders are describing precisely the same spiral about the pole1 it is plain that we may consider the inner border as being retarded in growth as compared with the outer1 and as being always identical with a smaller and earlier part of the latter4 ,if be the ratio of growth between the outer and the inner curve1 then1 the outer curve being represented by r a e cot 1 the equation to the inner one will be r ae cot 1 or r a e7 7cot 1 and may then be called the angle of retardation1 to which the inner curve is subject by virtue of its slower rate of growth4 #edb ,dispensing with mathematical formulae1 the several conditions may be illustrated as follows3 ,illustration3 ,fig4 #bgh4 ,in the diagrams 7,fig4 #bgh71 ,o ,p#a ,p#b ,p#c1 etc4 represents a radius1 on which ,p#a1 ,p#b1 ,p#c1 are the points attained by the outer border of the tubular shell after as many entire consecutive revolutions4 ,and ,p#a1 ,p#b1 ,p#c1 are the points similarly intersected by the inner border2 ,,op_/,,op being always 1 which is the ratio of growth1 or 8cutting-down factor40 ,then1 obviously1 when ,o ,p#a is less than ,o ,p#b the whorls will be separated by an interspace 7a72 7#b7 when ,o ,p#a ,o ,p#b they will be in contact 7b71 and 7#c7 when ,o ,p#a is greater than ,o ,p#b there will a greater or less extent of overlapping1 that is to say of concealment of the surfaces of the earlier by the later whorls 7c74 ,and as a further case 7#d71 it is plain that if be very large1 that is to say if ,o ,p#a be greater1 not only than ,o ,p#b but also than ,o ,p#c1 ,o ,p#d1 etc41 we shall have complete1 or all but complete concealment by the last formed whorl1 of the whole of its predecessors4 ,this latter condition is completely attained in ,nautilus pompilius1 and approached1 though not quite attained1 in ,n4 umbilicatus2 and the difference between these two forms1 or 8species10 is constituted accordingly by a difference in the value of 4 7#e7 ,there is also a final case1 not easily distinguishable externally from 7#d71 where ,p lies on #edc the opposite side of the radius vector to ,p1 and is therefore imaginary4 ,this final condition is exhibited in ,argonauta4 ,the limiting values of are easily ascertained4 ,illustration3 ,fig #bgi ,in ,fig4 #bgi we have portions of two successive whorls1 whose corresponding points on the same radius vector 7as ,r and ,r7 are1 therefore1 at a distance apart corresponding to #b4 ,let r and r refer to the inner1 and ,r1 ,r to the outer sides of the two whorls4 ,then1 if we consider ,r a e cot 1 it follows that ,r a e7 #b7;cot 1 r a e cot a e7 7cot 1 and r a e7 #b7;cot a e7 #b 7cot 4 ,now in the three cases 7a1 b1 c7 represented in ,fig4 #bgh1 it is plain that r ,r1 respectively4 ,that is to say1 a e7 #b7;cot a e cot 1 and e#b cot #a4 ,the case in which e#b cot #a1 or log #b cot log 1 is the case represented in ,fig4 #bgh1 b3 that is to say1 the particular case1 for each value of 1 where the consecutive whorls just touch1 without interspace or overlap4 ,for such cases1 then1 we may tabulate the values of 1 as follows3 ,constant angle ,ratio 77 of rate of growth of inner border of tube1 of spiral as compared with that of the outer border #hi #hif #hh #hjc #hg #gbj #hf #fde #he #egg #hj #ccj #ge #bcd #gj #ajaf #fe #jecd #edd ,we see1 accordingly1 that in plane spirals whose constant angle lies1 say1 between #fe and #gj1 we can only obtain contact between consecutive whorls if the rate of growth of the inner border of the tube be a small fraction1--a tenth or a twentieth--of that of the outer border4 ,in spirals whose constant angle is #hj1 contact is attained when the respective rates of growth are1 approximately1 as #c to #a2 while in spirals of constant angle from about #he to #hi1 contact is attained when the rates of growth are in the ratio of from about #c_/e to #i_/aj4 ,illustration3 ,fig4 #bhj4 ,if on the other hand we have1 for any given value of 1 a value of greater or less than the value given in the above table1 then we have1 respectively1 the conditions of separation or of overlap which are exemplified in ,fig4 #bgh1 a and c4 ,and1 just as we have constructed this table of values of for the particular case of simple contact between the whorls1 so we could construct similar tables for various degrees of separation1 or degrees of overlap4 ,for instance1 a case which admits of simple solution is that in which the interspace between the whorls is everywhere a mean proportional between the breadths of the whorls themselves 7,fig4 #bhj74 #ede ,in this case1 let us call ,,oa ,r1 ,,oc ,r#a and ,,ob r4 ,we then have ,r#a ,,oa a e cot 1 ,r#b ,,oc a e7 #b7 cot 1 ,r#a ,r#b a e#b7 7 cot r#b #ebj4 ,and r#b 7#a_/7b #b cot 1 whence1 equating1 #a_/ e cot 4 ,the corresponding values of are as follows3 ,ratio 77 of rates of growth of outer and inner border1 such as to produce a spiral with interspaces between the whorls1 the breadth of which interspaces is a mean proportional between the ,constant angle 77 breadths of the whorls themselves #ij #ajj 7imaginary7 #hi #ie #hh #hi #hg #he #hf #ha #he #gf #hj #eg #ge #dc #gj #cb #fe #bc #fj #ah #ee #ac #ej #jij #de #jfc #dj #jdb #ce #jbf #cj #jaf ,as regards the angle of retardation1 1 in the formula r e cot 1 or r e7 7cot 1 and in the case r e7#b 7cot 1 or log 7#b 7cot 1 #edf it is evident that when #b1 that will mean that #a4 ,in other words1 the outer and inner borders of the tube are identical1 and the tube is constituted by one continuous line4 ,when is a very small fraction1 that is to say when the rates of growth of the two borders of the tube are very diverse1 then will tend towards infinity--tend that is to say towards a condition in which the inner border of the tube never grows at all4 ,this condition is not infrequently approached in nature4 ,the nearly parallel-sided cone of ,dentalium1 or the widely separated whorls of ,lituites1 are evidently cases where nearly approaches unity in the one case1 and is still large in the other1 being correspondingly small2 while we can easily find cases where is very large1 and is a small fraction1 for instance in ,haliotis1 or in ,gryphaea4 ,for the purposes of the morphologist1 then1 the main result of this last general investigation is to shew that all the various types of 8open0 and 8closed0 spirals1 all the various degrees of separation or overlap of the successive whorls1 are simply the outward expression of a varying ratio in the rate of growth of the outer as compared with the inner border of the tubular shell4 ,the foregoing problem of contact1 or intersection1 of the successive whorls1 is a very simple one in the case of the discoid shell but a more complex one in the turbinate4 ,for in the discoid shell contact will evidently take place when the retardation of the inner as compared with the outer whorl is just #cfj1 and the shape of the whorls need not be considered4 ,as the angle of retardation diminishes from #cfj1 the whorls will stand further and further apart in an open coil2 as it increases beyond #cfj1 they will more and more overlap2 and when the angle of retardation is infinite1 that is to say when the true inner edge of the whorl does not grow at all1 then the shell is said to be completely involute4 ,of this latter condition we have a striking example in ,argonauta1 and one a little more obscure in ,nautilus pompilius4 ,in the turbinate shell1 the problem of contact is twofold1 for we have to deal with the possibilities of contact on the same side of the axis 7which is what we have dealt with in the discoid7 and #edg also with the new possibility of contact or intersection on the opposite side2 it is this latter case which will determine the presence or absence of an umbilicus1 and whether1 if present1 it will be an open conical space or a twisted cone4 ,it is further obvious that1 in the case of the turbinate1 the question of contact or no contact will depend on the shape of the generating curve2 and if we take the simple case where this generating curve may be considered as an ellipse1 then contact will be found to depend on the angle which the major axis of this ellipse makes with the axis of the shell4 ,the question becomes a complicated one1 and the student will find it treated in ,blake's paper already referred to4 ,when one whorl overlaps another1 so that the generating curve cuts its predecessor 7at a distance of #b7 on the same radius vector1 the locus of intersection will follow a spiral line upon the shell1 which is called the 8suture0 by conchologists4 ,it is evidently one of that ensemble of spiral lines in space of which1 as we have seen1 the whole shell may be conceived to be constituted2 and we might call it a 8contact-spiral10 or 8spiral of intersection40 ,in discoid shells1 such as an ,ammonite or a ,planorbis1 or in ,nautilus umbilicatus1 there are obviously two such contact-spirals1 one on each side of the shell1 that is to say one on each side of a plane perpendicular to the axis4 ,in turbinate shells such a condition is also possible1 but is somewhat rare4 ,we have it for instance1 in ,solarium perspectivum1 where the one contact-spiral is visible on the exterior of the cone1 and the other lies internally1 winding round the open cone of the umbilicus#eba2 but this second contact-spiral is usually imaginary1 or concealed within the whorls of the turbinated shell4 ,again1 in ,haliotis1 one of the contact-spirals is non-existent1 because of the extreme obliquity of the plane of the generating curve4 ,in ,scalaria pretiosa and in ,spirula there is no contact-spiral1 because the growth of the generating curve has been too slow1 in comparison with the vector rotation of its plane4 ,in ,argonauta and in ,cypraea1 there is no contact-spiral1 because the growth of the generating curve has been too quick4 ,nor1 of course1 is there any contact-spiral in ,patella or in ,dentalium1 because the angle is too small ever to give us a complete revolution of the spire4 #edh ,the various forms of straight or spiral shells among the ,cephalopods1 which we have seen to be capable of complete definition by the help of elementary mathematics1 have received a very complicated descriptive nomenclature from the palaeontologists4 ,for instance1 the straight cones are spoken of as orthoceracones or bactriticones1 the loosely coiled forms as gyroceracones or mimoceracones1 the more closely coiled shells1 in which one whorl overlaps the other1 as nautilicones or ammoniticones1 and so forth4 ,in such a succession of forms the biologist sees undoubted and unquestioned evidence of ancestral descent4 ,for instance we read in ,zittel's ,palaeontology#ebb3 8,the bactriticone obviously represents the primitive or primary radical of the ,ammonoidea1 and the mimoceracone the next or secondary radical of this order02 while precisely the opposite conclusion was drawn by ,owen1 who supposed that the straight chambered shells of such fossil cephalopods as ,orthoceras had been produced by the gradual unwinding of a coiled nautiloid shell#ebc4 ,to such phylogenetic hypotheses the mathematical or dynamical study of the forms of shells lends no valid support4 ,if we have two shells in which the constant angle of the spire be respectively #hj and #fj1 that fact in itself does not at all justify an assertion that the one is more primitive1 more ancient1 or more 8ancestral0 than the other4 ,nor1 if we find a third in which the angle happens to be #gj1 does that fact entitle us to say that this shell is intermediate between the other two1 in time1 or in blood relationship1 or in any other sense whatsoever save only the strictly formal and mathematical one4 ,for it is evident that1 though these particular arithmetical constants manifest themselves in visible and recognisable differences of form1 yet they are not necessarily more deep-seated or significant than are those which manifest themselves only in difference of magnitude2 and the student of phylogeny scarcely ventures to draw conclusions as to the relative antiquity of two allied organisms on the ground that one happens to be bigger or less1 or longer or shorter1 than the other4 #edi ,at the same time1 while it is obviously unsafe to rest conclusions upon such features as these1 unless they be strongly supported and corroborated in other ways1--for the simple reason that there is unlimited room for coincidence1 or separate and independent attainment of this or that magnitude or numerical ratio1--yet on the other hand it is certain that1 in particular cases1 the evolution of a race has actually involved gradual increase or decrease in some one or more numerical factors1 magnitude itself included1--that is to say increase or decrease in some one or more of the actual and relative velocities of growth4 ,when we do meet with a clear and unmistakable series of such progressive magnitudes or ratios1 manifesting themselves in a progressive series of 8allied0 forms1 then we have the phenomenon of 8orthogenesis40 ,for orthogenesis is simply that phenomenon of continuous lines or series of form 7and also of functional or physiological capacity71 which was the foundation of the ,theory of ,evolution1 alike to ,lamarck and to ,darwin and ,wallace2 and which we see to exist whatever be our ideas of the 8origin of species10 or of the nature and origin of 8functional adaptations40 ,and to my mind1 the mathematical 7as distinguished from the purely physical7 study of morphology bids fair to help us to recognise this phenomenon of orthogenesis in many cases where it is not at once patent to the eye2 and also1 on the other hand1 to warn us1 in many other cases1 that even strong and apparently complex resemblances in form may be capable of arising independently1 and may sometimes signify no more than the equally accidental numerical coincidences which are manifested in identity of length or weight1 or any other simple magnitudes4 ,illustration3 ,fig4 #bha4 ,an ammonitoid shell 7,macroscaphites7 to shew change of curvature4 ,i have already referred to the fact that1 while in general a very great and remarkable regularity of form is characteristic of the molluscan shell1 that complete regularity is apt to be departed from4 ,we have clear cases of such a departure in ,pupa1 ,clausilia1 and various ,bulimi1 where the enveloping cone of the spire is not a right cone but a cone whose sides are curved4 ,it is plain that this condition may arise in two ways3 either by a gradual change in the ratio of growth of the whorls1 that is to say in the logarithmic spiral itself1 or by a change in the velocity of #eej translation along the axis1 that is to say in the helicoid which1 in all turbinate shells1 is superposed upon the spiral4 ,very careful measurements will be necessary to determine to which of these factors1 or in what proportions to each1 the phenomenon is due4 ,but in many ,ammonitoidea where the helicoid factor does not enter into the case1 we have a clear illustration of gradual and marked changes in the spiral angle itself1 that is to say of the ratio of growth corresponding to increase of vectorial angle4 ,we have seen from some of ,naumann's and ,grabau's measurements that such a tendency to vary1 such an acceleration or retardation1 may be detected even in ,ammonites which present nothing abnormal to the eye4 ,but let us suppose that the spiral angle increases somewhat rapidly2 we shall then get a spiral with gradually narrowing whorls1 and this condition is characteristic of ,oekotraustes1 a subgenus of ,ammonites4 ,if on the other hand1 the angle gradually diminishes1 and even falls away to zero1 we shall have the spiral curve opening out1 as it does in ,scaphites1 ,ancyloceras and ,lituites1 until the spiral coil is replaced by a spiral curve so gentle as to seem all but straight4 ,lastly1 there are a few cases1 such as ,bellerophon expansus and some ,goniatites1 where the outer spiral does not perceptibly change1 but the whorls become more 8embracing0 or the whole shell more involute4 ,here it is the angle of retardation1 the ratio of growth between the outer and inner parts of the whorl1 which undergoes a gradual change4 ,in order to understand the relation of a close-coiled shell to one of its straighter congeners1 to compare 7for example7 an #eea ,ammonite with an ,orthoceras1 it is necessary to estimate the length of the right cone which has1 so to speak1 been coiled up into the spiral shell4 ,our problem then is1 ,to find the length of a plane logarithmic spiral1 in terms of the radius and the constant angle 4 ,in the annexed diagram1 if ,,op be a radius vector1 ,,oq a line of reference perpendicular to ,,op1 and ,,pq a tangent to the curve1 ,,pq1 or sec 1 is equal in length to the spiral arc ,,op4 ,and this is practically obvious3 for ,,pp_/,,pr ds_/dr sec 1 and therefore sec s_/r1 or the ratio of arc to radius vector4 ,illustration3 ,fig4 #bhb4 ,accordingly1 the ratio of l1 the total length1 to r1 the radius vector up to which the total length is to be measured1 is expressed by a simple table of secants2 as follows3 l_/r #e #ajjd #aj #ajae #bj #ajfd #cj #aafe #dj #acje #ej #aef #fj #bj #gj #bi #ge #ci #hj #eh #he #aae #hf #adc #hg #aia #hh #bhg #hi #egc #hi #aj #fhh #bj #hei #cj #aadf #dj #agai #ej #cdch #ee #fhge #ei #cdcgg #ij ,infinite ,putting the same table inversely1 so as to shew the total #eeb length in whole numbers1 in terms of the radius1 we have as follows3 ,total length 7in terms of the radius7 ,constant angle #b #fj #c #gj #ca #d #ge #cb #e #gh #bh #aj #hd #af #bj #hg #h #cj #hh #f #dj #hh #cd #ej #hh #ea #ajj #hi #bf #ajjj #hi #ef #cf #aj1jjj #hi #ei #cj ,accordingly1 we see that 7#a71 when the constant angle of the spiral is small1 the spiral itself is scarcely distinguishable from a straight line1 and its length is but very little greater than that of its own radius vector4 ,this remains pretty much the case for a considerable increase of angle1 say from #j to #bj or more2 7#b7 for a very considerably greater increase of the constant angle1 say to #ej or more1 the shell would only have the appearance of a gentle curve2 7#c7 the characteristic close coils of the ,nautilus or ,ammonite would be typically represented only when the constant angle lies within a few degrees on either side of about #hj4 ,the coiled up spiral of a ,nautilus1 with a constant angle of about #hj1 is about six times the length of its radius vector1 or rather more than three times its own diameter2 while that of an ,ammonite1 with a constant angle of1 say1 from #he to #hh1 is from about six to fifteen times as long as its own diameter4 ,and 7#d7 as we approach an angle of #ij 7at which point the spiral vanishes in a circle71 the length of the coil increases with enormous rapidity4 ,our spiral would soon assume the appearance of the close coils of a ,nummulite1 and the successive increments of breadth in the successive whorls would become inappreciable to the eye4 ,the logarithmic spiral of high constant angle would1 as we have already seen1 tend to become indistinguishable1 without the most careful measurement1 from an ,archimedean spiral4 ,and it is obvious1 moreover1 that our ordinary methods of #eec determining the constant angle of the spiral would not in these cases be accurate enough to enable us to measure the length of the coil3 we should have to devise a new method1 based on the measurement of radii or diameters over a large number of whorls4 ,the geometrical form of the shell involves many other beautiful properties1 of great interest to the mathematician1 but which it is not possible to reduce to such simple expressions as we have been content to use4 ,for instance1 we may obtain an equation which shall express completely the surface of any shell1 in terms of polar or of rectangular coordinates 7as has been done by ,moseley and by ,blake71 or in ,hamiltonian vector notation4 ,it is likewise possible 7though of little interest to the naturalist7 to determine the area of a conchoidal surface1 or the volume of a conchoidal solid1 and to find the centre of gravity of either surface or solid#ebd4 ,and ,blake has further shewn1 with considerable elaboration1 how we may deal with the symmetrical distortion1 due to pressure1 which fossil shells are often found to have undergone1 and how we may reconstitute by calculation their original undistorted form1--a problem which1 were the available methods only a little easier1 would be very helpful to the palaeontologist2 for1 as ,blake himself has shewn1 it is easy to mistake a symmetrically distorted specimen of 7for instance7 an ,ammonite1 for a new and distinct species of the same genus4 ,but it is evident that to deal fully with the mathematical problems contained in1 or suggested by1 the spiral shell1 would require a whole treatise1 rather than a single chapter of this elementary book4 ,let us then1 leaving mathematics aside1 attempt to summarise1 and perhaps to extend1 what has been said about the general possibilities of form in this class of organisms4 ,the ,univalve ,shell3 a summary4 ,the surface of any shell1 whether discoid or turbinate1 may be imagined to be generated by the revolution about a fixed axis of a closed curve1 which1 remaining always geometrically similar to itself1 increases continually its dimensions3 and1 since the rate of growth of the generating curve and its velocity of rotation follow the same law1 the curve traced in space by corresponding points #eed in the generating curve is1 in all cases1 a logarithmic spiral4 ,in discoid shells1 the generating figure revolves in a plane perpendicular to the axis1 as in ,nautilus1 the ,argonaut and the ,ammonite4 ,in turbinate shells1 it slides continually along the axis of revolution1 and the curve in space generated by any given point partakes1 therefore1 of the character of a helix1 as well as of a logarithmic spiral2 it may be strictly entitled a helico-spiral4 ,such turbinate or helico-spiral shells include the snail1 the periwinkle and all the common typical ,gastropods4 ,illustration3 ,fig4 #bhc4 ,section of a spiral1 or turbinate1 univalve1 ,triton corrugatus1 ,lam4 7,from ,woodward47 ,the generating figure1 as represented by the mouth of the shell1 is sometimes a plane curve1 of simple form2 in other and more numerous cases1 it becomes more complicated in form and its boundaries do not lie in one plane3 but in such cases as these we may replace it by its 8trace10 on a plane at some definite angle to the direction of growth1 for instance by its form as it appears in a section through the axis of the helicoid shell4 ,the generating curve is of very various shapes4 ,it is circular in ,scalaria or ,cyclostoma1 and in ,spirula2 it may be considered as a segment of a circle in ,natica or in ,planorbis4 ,it is approximately triangular in ,conus1 and rhomboidal in ,solarium or ,potamides4 ,it is very commonly more or less elliptical3 the long axis of the ellipse being parallel to the axis of the shell in ,oliva and ,cypraea2 all but perpendicular to it in many ,trochi2 and oblique to it in many well-marked cases1 such as ,stomatella1 ,lamellaria1 ,sigaretus haliotoides 7,fig4 #bhd7 and ,haliotis4 ,in ,nautilus pompilius it is approximately a semi-ellipse1 and in ,n4 umbilicatus rather more than a semi-ellipse1 the long axis lying in both cases perpendicular to the axis of the shell#ebe4 ,its #eee form is seldom open to easy mathematical expression1 save when it is an actual circle or ellipse2 but an exception to this rule may be found in certain ,ammonites1 forming the group 8,cordati10 where 7as ,blake points out7 the curve is very nearly represented by a cardioid1 whose equation is r a7#a cos 74 ,the generating curve may grow slowly or quickly2 its growth-factor is very slow in ,dentalium or ,turritella1 very rapid in ,nerita1 or ,pileopsis1 or ,haliotis or the ,limpet4 ,it may contain the axis in its plane1 as in ,nautilus2 it may be parallel to the axis1 as in the majority of ,gastropods2 or it may be inclined to the axis1 as it is in a very marked degree in ,haliotis4 ,in fact1 in ,haliotis the generating curve is so oblique to the axis of the shell that the latter appears to grow by additions to one margin only 7cf4 ,fig4 #beh71 as in the case of the opercula of ,turbo and ,nerita referred to on p4 #ebb2 and this is what ,moseley supposed it to do4 ,illustration3 ,fig4 #bhd4 ,a1 ,lamellaria perspicua2 ,b1 ,sigaretus haliotoides4 7,after ,woodward47 ,the general appearance of the entire shell is determined 7apart from the form of its generating curve7 by the magnitude of three angles2 and these in turn are determined1 as has been sufficiently explained1 by the ratios of certain velocities of growth4 ,these angles are 7#a7 the constant angle of the logarithmic spiral 772 7#b7 in turbinate shells1 the enveloping angle of the cone1 or 7taking half that angle7 the angle 77 which a tangent to the whorls makes with the axis of the shell2 and 7#c7 an angle called the 8angle of retardation0 771 which expresses the retardation in growth of #eef the inner as compared with the outer part of each whorl1 and therefore measures the extent to which one whorl overlaps1 or the extent to which it is separated from1 another4 ,the spiral angle 77 is very small in a limpet1 where it is usually taken as #j2 but it is evidently of a significant amount1 though obscured by the shortness of the tubular shell4 ,in ,dentalium it is still small1 but sufficient to give the appearance of a regular curve2 it amounts here probably to about #cj to #dj4 ,in ,haliotis it is from about #gj to #ge2 in ,nautilus about #hj2 and it lies between #hj and #he1 or even more1 in the majority of ,gastropods4 ,the case of ,fissurella is curious4 ,here we have1 apparently1 a conical shell with no trace of spiral curvature1 or 7in other words7 with a spiral angle which approximates to #j2 but in the minute embryonic shell 7as in that of the limpet7 a spiral convolution is distinctly to be seen4 ,it would seem1 then1 that what we have to do with here is an unusually large growth-factor in the generating curve1 which causes the shell to dilate into a cone of very wide angle1 the apical portion of which has become lost or absorbed1 and the remaining part of which is too short to show clearly its intrinsic curvature4 ,in the closely allied ,emarginula1 there is likewise a well-marked spiral in the embryo1 which however is still manifested in the curvature of the adult1 nearly conical1 shell4 ,in both cases we have to do with a very wide-angled cone1 and with a high retardation-factor for its inner1 or posterior1 border4 ,the series is continued1 from the apparently simple cone to the complete spiral1 through such forms as ,calyptraea4 ,the angle 1 as we have seen1 is not always1 nor rigorously1 a constant angle4 ,in some ,ammonites it may increase with age1 the whorls becoming closer and closer2 in others it may decrease rapidly1 and even fall to zero1 the coiled shell then straightening out1 as in ,lituites and similar forms4 ,it diminishes somewhat1 also1 in many ,orthocerata1 which are slightly curved in youth1 but straight in age4 ,it tends to increase notably in some common land-shells1 the ,pupae and ,bulimi2 and it decreases in ,succinea4 ,directly related to the angle is the ratio which subsists between the breadths of successive whorls4 ,the following table gives a few illustrations of this ratio in particular cases1 in addition to those which we have already studied4 #eeg ,ratio of breadth of consecutive whorls4 ,pointed ,turbinates ,telescopium fuscum #aad ,acus subulatus #aaf 99,turritella terebellata #aah 99,turritella imbricata #abj ,cerithium palustre #abb ,turritella duplicata #abc ,melanopsis terebralis #abc ,cerithium nodulosum #abd 99,turritella carinata #abe ,acus crenulatus #abe ,terebra maculata 7,fig4 #bhe7 #abe 99,cerithium lignitarum #abf ,acus dimidiatus #abh ,cerithium sulcatum #acb ,fusus longissimus #acd 99,pleurotomaria conoidea #acd ,trochus niloticus 7,fig4 #bhf7 #ada ,mitra episcopalis #adc ,fusus antiquus #aej ,scalaria pretiosa #aef ,fusus colosseus #aga ,phasianella bulloides #ahj ,helicostyla polychroa #bjj ,obtuse ,turbinates and ,discoids ,conus virgo #abe ,conus litteratus #adj ,conus betulina #adc 99,helix nemoralis #aej 99,solarium perspectivum #aej ,solarium trochleare #afb ,solarium magnificum #age 99,natica aperta #bjj ,euomphalus pentangulatus #bjj ,planorbis corneas #bjj ,solaropsis pellis-serpentis #bjj ,dolium zonatum #baj 99,natica glaucina #cjj ,nautilus pompilius #cjj ,haliotis excavatus #dbj ,haliotis parvus #fjj ,delphinula atrata #fjj ,haliotis rugoso-plicata #icj ,haliotis viridis #ajjj ,those marked 99 from ,naumann2 the rest from ,macalister#ebf4 ,in the case of turbinate shells1 we require to take into account the angle 1 in order to determine the spiral angle from the ratio of the breadths of consecutive whorls2 for the short table given on p4 #ecd is only applicable to discoid shells1 in which the angle is an angle of #ij4 ,our formula1 as mentioned on p4 #eah now becomes ,r #b sin cot 4 ,for this formula ,i have worked out the following table4 #eeh ,table shewing values of the spiral angle corresponding to certain ratios of breadth of successive whorls of the shell1 for various values of the apical semi-angle 4 ,ratio ,r_/#a #e #aj #ae #bj #cj #dj #ej #fj #gj #hj #ij #aa #hj #h #he #j #hf #dd #hg #bh #hh #af #hh #ci #hh #eb #hi #j #hi #d #hi #g #hi #h #abe #fg #ea #gh #bg #hb #aa #hd #e #he #ef #hf #ej #hg #ba #hg #ci #hg #ej #hg #ef #hg #eh #ae #ec #cj #fi #cg #gf #j #gi #ba #hb #ci #hd #af #he #ac #he #dd #hf #d #hf #ae #hf #ah #bj #ch #bj #eg #ce #ff #ee #gc #aa #gg #cd #hj #af #ha #eb #hb #de #hc #ah #hc #cg #hc #db #be #cj #ec #ej #j #fj #ce #fg #j #gc #de #gg #ac #gi #ai #hj #bf #ha #aa #ha #ce #ha #db #cj #bf #cb #dd #ej #ef #j #fc #j #gj #de #gd #de #gg #ag #gh #ce #gi #bh #gi #ef #hj #e #ce #bc #cg #da #e #eb #be #ei #ej #fh #ae #gb #de #ge #ce #gg #b #gh #a #gh #cc #gh #dc #dj #ba #ce #ch #aj #di #ce #eg #ae #ff #aj #ga #c #gd #i #ge #db #gf #dg #gg #bb #gg #cd #de #bj #j #cf #j #dg #ae #ee #e #fd #be #fi #ce #gb #ed #gd #cc #ge #dc #gf #bj #gf #ce #ej #ah #de #cd #aj #de #bj #ec #ae #fb #ee #fh #ae #ga #dh #gc #ca #gd #de #ge #be #ge #ch #ajj #ac #be #be #bj #ce #ae #dc #e #ec #de #fj #bj #fd #eg #fg #d #fh #db #fi #ce #fi #ec #bjj #aj #be #bj #j #bh #cj #ce #de #df #be #ec #be #eh #eb #fa #aj #fc #f #fd #aj #fd #ca #ejj #h #j #ae #ce #bb #ce #bh #ej #ch #de #de #ee #eb #a #ed #ah #ef #bh #eg #db #eh #f #ajjj #f #ej #ac #bj #ai #cj #be #e #cd #bj #da #ae #dg #ce #di #de #eb #c #ec #bj #ec #df #eei ,from this table1 by interpolation1 we may easily fill in the approximate values of 1 as soon as we have determined the apical angle and measured the ratio ,r2 as follows3 ,r ,turritella sp4 #aab #g #ha ,cerithium nodulosum #abd #ae #hb ,conus virgo #abe #gj #hh ,mitra episcopalis #adc #af #gh ,scalaria pretiosa #aef #bf #ha ,phasianella bulloides #ahj #bf #hj ,solarium perspectivum #aej #ec #he ,natica aperta #bjj #gj #hc ,planorbis corneus #bjj #ij #hd ,euomphalus pentangulatus #bjj #ij #hd ,we see from this that shells so different in appearance as ,cerithium1 ,solarium1 ,natica and ,planorbis differ very little indeed in the magnitude of the spiral angle 1 that is to say in the relative velocities of radial and tangential growth4 ,it is upon the angle that the difference in their form mainly depends3 that is to say the amount of longitudinal shearing1 or displacement parallel to the axis of the shell4 ,illustration3 ,fig4 #bhe4 ,terebra maculata1 ,l4 ,the enveloping angle1 or rather semi-angle 771 of the cone may be taken as #ij in the discoid shells1 such as ,nautilus and ,planorbis4 ,it is still a large angle1 of #gj or #ge1 in ,conus or in ,cymba1 somewhat less in ,cassis1 ,harpa1 ,dolium or ,natica2 it is about #ej to #ee in the various species of ,solarium1 about #ce in the typical ,trochi1 such as ,t4 niloticus or ,t4 zizyphinus1 and about #be or #bf in ,scalaria pretiosa and ,phasianella bulloides2 it becomes a very acute angle1 of #ae1 #aj1 or even less1 in ,eulima1 ,turritella or ,cerithium4 ,the costly ,conus gloria-maris1 one of the #efj great treasures of the conchologist1 differs from its congeners in no important particular save in the somewhat 8produced0 spire1 that is to say in the comparatively low value of the angle 4 ,illustration3 ,fig4 #bhf4 ,trochus niloticus1 ,l4 ,a variation with advancing age of is common1 but 7as ,blake points out7 it is often not to be distinguished or disentangled from an alteration of 4 ,whether alone1 or combined with a change in 1 we find it in all those many ,gastropods whose whorls cannot all be touched by the same enveloping cone1 and whose spire is accordingly described as concave or convex4 ,the former condition1 as we have it in ,cerithium1 and in the cusp-like spire of ,cassis1 ,dolium and some ,cones1 is much the commoner of the two4 ,and such tendency to decrease may lead to becoming a negative angle2 in which case we have a depressed spire1 as in the ,cypraeae4 ,when we find a 8reversed shell10 a whelk or a snail for instance whose spire winds to the left instead of to the right1 we may describe it mathematically by the simple statement that the angle has changed sign4 ,in the genus ,ampullaria1 or ,apple-snails1 inhabiting tropical or sub-tropical rivers1 we have a remarkable condition2 for in certain 8species0 the spiral turns to the right1 in others to the left1 and in others again we have a flattened #efa 8discoid0 shell2 and furthermore we have numerous intermediate stages1 on either side1 shewing right and left-handed spirals of varying degrees of acuteness#ebg4 ,in this case1 the angle may be said to vary1 within the limits of a genus1 from somewhere about #ce to somewhere about #abe4 ,the angle of retardation 77 is very small in ,dentalium and ,patella2 it is very large in ,haliotis4 ,it becomes infinite in ,argonauta and in ,cypraea4 ,connected with the angle of retardation are the various possibilities of contact or separation1 in various degrees1 between adjacent whorls in the discoid1 and between both adjacent and opposite whorls in the turbinated shell4 ,but with these phenomena we have already dealt sufficiently4 ,of ,bivalve ,shells4 ,hitherto we have dealt only with univalve shells1 and it is in these that all the mathematical problems connected with the spiral1 or helico-spiral1 are best illustrated4 ,but the case of the bivalve shell1 of ,lamellibranchs or of ,brachiopods1 presents no essential difference1 save only that we have here to do with two conjugate spirals1 whose two axes have a definite relation to one another1 and some freedom of rotatory movement relatively to one another4 ,the generating curve is particularly well seen in the bivalve1 where it simply constitutes what we call 8the outline of the shell40 ,it is for the most part a plane curve1 but not always2 for there are forms1 such as ,hippopus1 ,tridacna and many ,cockles1 or ,rhynchonella and ,spirifer among the ,brachiopods1 in which the edges of the two valves interlock1 and others1 such as ,pholas1 ,mya1 etc41 where in part they fail to meet4 ,in such cases as these the generating curves are conjugate1 having a similar relation1 but of opposite sign1 to a median plane of reference4 ,a great variety of form is exhibited by these generating curves among the bivalves4 ,in a good many cases the curve is approximately circular1 as in ,anomia1 ,cyclas1 ,artemis1 ,isocardia2 it is nearly semi-circular in ,argiope4 ,it is approximately elliptical in ,orthis and in ,anodon2 it may be called semi-elliptical in ,spirifer4 ,it is a nearly rectilinear #efb triangle in ,lithocardium1 and a curvilinear triangle in ,mactra4 ,many apparently diverse but more or less related forms may be shewn to be deformations of a common type1 by a simple application of the mathematical theory of 8,transformations10 which we shall have to study in a later chapter4 ,in such a series as is furnished1 for instance1 by ,gervillea1 ,perna1 ,avicula1 ,modiola1 ,mytilus1 etc41 a 8simple shear0 accounts for most1 if not all1 of the apparent differences4 ,upon the surface of the bivalve shell we usually see with great clearness the 8lines of growth0 which represent the successive margins of the shell1 or in other words the successive positions assumed during growth by the growing generating curve2 and we have a good illustration1 accordingly1 of how it is characteristic of the generating curve that it should constantly increase1 while never altering its geometric similarity4 ,underlying these 8lines of growth10 which are so characteristic of a molluscan shell 7and of not a few other organic formations71 there is1 then1 a 8law of growth0 which we may attempt to enquire into and which may be illustrated in various ways4 ,the simplest cases are those in which we can study the lines of growth on a more or less flattened shell1 such as the one valve of an oyster1 a ,pecten or a ,tellina1 or some such bivalve mollusc4 ,here around an origin1 the so-called 8umbo0 of the shell1 we have a series of curves1 sometimes nearly circular1 sometimes elliptical1 and often asymmetrical2 and such curves are obviously not 8concentric10 though we are often apt to call them so1 but are always 8co-axial40 ,this manner of arrangement may be illustrated by various analogies4 ,we might for instance compare it to a series of waves1 radiating outwards from a point1 through a medium which offered a resistance increasing1 with the angle of divergence1 according to some simple law4 ,we may find another1 and perhaps a simpler illustration as follows3 ,in a very simple and beautiful theorem1 ,galileo shewed that1 if we imagine a number of inclined planes1 or gutters1 sloping downwards 7in a vertical plane7 at various angles from a common starting-point1 and if we imagine a number of balls rolling each down its own gutter under the influence of gravity 7and without hindrance from friction71 then1 at any given instant1 the locus of #efc all these moving bodies is a circle passing through the point of origin4 ,for the acceleration along any one of the sloping paths1 for instance ,,ab 7,fig4 #bhg71 is such that ,,ab g cos t#b g ,,ab_/,,ac t#b4 ,therefore t#b #b_/;g ,,ac4 ,illustration3 ,fig4 #bhg4 ,that is to say1 all the balls reach the circumference of the circle at the same moment as the ball which drops vertically from ,a to ,c4 ,where1 then1 as often happens1 the generating curve of the shell is approximately a circle passing through the point of origin1 we may consider the acceleration of growth along various radiants to be governed by a simple mathematical law1 closely akin to that simple law of acceleration which governs the movements of a falling body4 ,and1 mutatis mutandis1 a similar definite law underlies the cases where the generating curve is continually elliptical1 or where it assumes some more complex1 but still regular and constant form4 ,it is easy to extend the proposition to the particular case where the lines of growth may be considered elliptical4 ,in such a case we have x#b_/;a#b y#b_/;b#b #a1 where a and b are the major and minor axes of the ellipse4 ,or1 changing the origin to the vertex of the figure x#b_/;a#b #bx_/a y#b_/;b#b #j1 giving 7x a7#b_/;a#b y#b_/;b#b #a4 ,then1 transferring to polar coordinates1 where r cos x1 r sin y1 we have 7r cos#b 7_/a#b 7#b cos 7_/a 7r sin 7_/b#b #j1 #efd which is equivalent to r 7#b;a b#b cos 7_/77b#b cos#b 7 7a#b sin#b 771 or1 eliminating the sine-function1 r 7#b;a b#b cos 7_/77b#b a#b7 cos#b a#b74 ,obviously1 in the case when a b1 this gives us the circular system which we have already considered4 ,for other values1 or ratios1 of a and b1 and for all values of 1 we can easily construct a table1 of which the following is a sample3 ,chords of an ellipse1 whose major and minor axes 7a1 b7 are in certain given ratios4 a_/b #a_/c #a_/b #b_/c #a_/a #c_/b #b_/a #c_/a #j #aj #aj #aj #aj #aj #aj #aj #aj #aja #aja #ajjb #ihe #idh #ijb #gic #bj #aje #ajc #ajje #idj #hbj #fie #dhe #cj #aaae #ajfe #ajje #hff #fff #die #bhi #dj #aba #aaa #iie #gff #eje #cdb #agh #ej #acd #aade #ieb #fdc #cgb #bcb #aac #fj #aej #aadb #heg #ejj #beh #aeb #jga #gj #aei #ajae #fgj #cdb #afc #jib #jdb #hj #abce #fce #cge #agd #jgh #jde #jbj #ij #jj #jj #jj #jj #jj #jj #jj ,illustration3 ,fig4 #bhh4 ,the coaxial ellipses which we then draw1 from the values given in the table1 are such as are shewn in ,fig4 #bhh for the ratio a_/b #c_/a1 and in ,fig4 #bhi for the ratio a_/b 2 these are fair approximations to the actual outlines1 and to the actual arrangement of the lines of growth1 in such forms as ,solecurtus or ,cultellus1 and in ,tellina or ,psammobia4 ,it is not difficult to introduce a constant into our equation to meet the case of a shell which is somewhat unsymmetrical on either side of the median axis4 ,it is a somewhat more troublesome matter1 however1 to bring these configurations into relation with a 8law of growth10 as was so easily done in the case of the circular figure3 in other words1 to #efe formulate a law of acceleration according to which points starting from the origin ,o1 and moving along radial lines1 would all lie1 at any future epoch1 on an ellipse passing through ,o2 and this calculation we need not enter into4 ,illustration3 ,fig4 #bhi4 ,all that we are immediately concerned with is the simple fact that where a velocity1 such as our rate of growth1 varies with its direction1--varies that is to say as a function of the angular divergence from a certain axis1--then1 in a certain simple case1 we get lines of growth laid down as a system of coaxial circles1 and1 when the function is a more complex one1 as a system of ellipses or of other more complicated coaxial figures1 which figures may or may not be symmetrical on either side of the axis4 ,among our bivalve mollusca we shall find the lines of growth to be approximately circular in1 for instance1 ,anomia2 in ,lima 7e4g4 ,l4 subauriculata7 we have a system of nearly symmetrical ellipses with the vertical axis about twice the transverse2 in ,solen pellucidus1 we have again a system of lines of growth which are not far from being symmetrical ellipses1 in which however the transverse is between three and four times as great as the vertical axis4 ,in the great majority of cases1 we have a similar phenomenon with the further complication of slight1 but occasionally very considerable1 lateral asymmetry4 ,in certain little ,crustacea 7of the genus ,estheria7 the carapace takes the form of a bivalve shell1 closely simulating that of a #eff lamellibranchiate mollusc1 and bearing lines of growth in all respects analogous to or even identical with those of the latter4 ,the explanation is very curious and interesting4 ,in ordinary ,crustacea the carapace1 like the rest of the chitinised and calcified integument1 is shed off in successive moults1 and is restored again as a whole4 ,but in ,estheria 7and one or two other small crustacea7 the moult is incomplete3 the old carapace is retained1 and the new1 growing up underneath it1 adheres to it like a lining1 and projects beyond its edge3 so that in course of time the margins of successive old carapaces appear as 8lines of growth0 upon the surface of the shell4 ,in this mode of formation1 then 7but not in the usual one71 we obtain a structure which 8is partly old and partly new10 and whose successive increments are all similar1 similarly situated1 and enlarged in a continued progression4 ,we have1 in short1 all the conditions appropriate and necessary for the development of a logarithmic spiral2 and this logarithmic spiral 7though it is one of small angle7 gives its own character to the structure1 and causes the little carapace to partake of the characteristic conformation of the molluscan shell4 ,the essential simplicity1 as well as the great regularity of the 8curves of growth0 which result in the familiar configurations of our bivalve shells1 sufficiently explain1 in a general way1 the ease with which they may be imitated1 as for instance in the so-called 8artificial shells0 which ,kappers has produced from the conchoidal form and lamination of lumps of melted and quickly cooled paraffin#ebh4 ,in the above account of the mathematical form of the bivalve shell1 we have supposed1 for simplicity's sake1 that the pole or origin of the system is at a point where all the successive curves touch one another4 ,but such an arrangement is neither theoretically probable1 nor is it actually the case2 for it would mean that in a certain direction growth fell1 not merely to a minimum1 but to zero4 ,as a matter of fact1 the centre of the system 7the 8umbo0 of the conchologists7 lies not at the edge of the system1 but very near to it2 in other words1 there is a certain amount of growth all round4 ,but to take account of this condition would involve more troublesome mathematics1 and it is obvious that the foregoing illustrations are a sufficiently near approximation to the actual case4 #efg ,among the bivalves the spiral angle 77 is very small in the flattened shells1 such as ,orthis1 ,lingula or ,anomia4 ,it is larger1 as a rule1 in the ,lamellibranchs than in the ,brachiopods1 but in the latter it is of considerable magnitude among the ,pentameri4 ,among the ,lamellibranchs it is largest in such forms as ,isocardia and ,diceras1 and in the very curious genus ,caprinella2 in all of these last-named genera its magnitude leads to the production of a spiral shell of several whorls1 precisely as in the univalves4 ,the angle is usually equal1 but of opposite sign1 in the two valves of the ,lamellibranch1 and usually of opposite sign but unequal in the two valves of the ,brachiopod4 ,it is very unequal in many ,ostreidae1 and especially in such forms as ,gryphaea1 or in ,caprinella1 which is a kind of exaggerated ,gryphaea4 ,occasionally it is of the same sign in both valves 7that is to say1 both valves curve the same way7 as we see sometimes in ,anomia1 and much better in ,productus or ,strophomena4 ,illustration3 ,fig4 #bij4 ,caprinella adversa4 7,after ,woodward47 ,illustration3 ,fig4 #bia4 ,section of ,productus 7,strophomena7 sp4 7,from ,woods47 ,owing to the large growth-factor of the generating curve1 and the comparatively small angle of the spiral1 the whole shell seldom assumes a spiral form so conspicuous as to manifest in a typical way the helical twist or shear which is so conspicuous in the #efh majority of univalves1 or to let us measure or estimate the magnitude of the apical angle 77 of the enveloping cone4 ,this however we can do in forms like ,isocardia and ,diceras2 while in ,caprinella we see that the whorls lie in a plane perpendicular to the axis1 forming a discoidal spire4 ,as in the latter shell1 so also universally among the ,brachiopods1 there is no lateral asymmetry in the plane of the generating curve such as to lead to the development of a helix2 but in the majority of the ,lamellibranchiata it is obvious1 from the obliquity of the lines of growth1 that the angle is significant in amount4 ,illustration3 ,fig4 #bib4 ,skeletal loop of ,terebratula4 7,from ,woods47 ,the so-called 8spiral arms0 of ,spirifer and many other ,brachiopods are not difficult to explain4 ,they begin as a single structure1 in the form of a loop of shelly substance1 attached to the dorsal valve of the shell1 in the neighbourhood of the hinge4 ,this loop has a curvature of its own1 similar to but not necessarily identical with that of the valve to which it is attached2 and this curvature will tend to be developed1 by continuous and symmetrical growth1 into a fully formed logarithmic spiral1 so far as it is permitted to do so under the constraint of the shell in which it is contained4 ,in various ,terebratulae we see the spiral growth of the loop1 more or less flattened and distorted by the restraining pressure of the ventral valve4 ,in a number of cases the loop remains small1 but gives off two nearly parallel branches or offshoots1 which continue to grow4 ,and these1 starting with just such a slight curvature as the loop itself possessed1 grow on and on till they may form close-wound spirals1 always provided that the 8spiral angle0 of the curve is such that the resulting spire can be freely contained within the cavity of the shell4 ,owing to the bilateral symmetry of the whole system1 the case will be rare1 and unlikely to occur1 in which each separate arm will coil strictly in a plane1 so as to constitute a discoid spiral2 for the original #efi direction of each of the two branches1 parallel to the valve 7or nearly so7 and outwards from the middle line1 will tend to constitute a curve of double curvature1 and so1 on further growth1 to develop into a helicoid4 ,this is what actually occurs1 in the great majority of cases4 ,but the curvature may be such that the helicoid grows outwards from the middle line1 or inwards towards the middle line1 a very slight difference in the initial curvature being sufficient to direct the spire the one way or the other2 the middle course of an undeviating discoid spire will be rare1 from the usual lack of any obvious controlling force to prevent its deviation4 ,the cases in which the helicoid spires point towards1 or point away from1 the middle line are ascribed1 in zoological classification1 to particular 8families0 of ,brachiopods1 the former ,illustration3 ,fig4 #bic4 ,spiral arms of ,spirifer4 7,from ,woods47 ,illustration3 ,fig4 #bid4 ,inwardly directed spiral arms of ,atrypa4 condition defining 7or helping to define7 the ,atrypidae and the latter the ,spiriferidae and ,athyridae4 ,it is obvious that the incipient curvature of the arms1 and consequently the form and direction of the spirals1 will be influenced by the surrounding pressures1 and these in turn by the general shape of the shell4 ,we shall expect1 accordingly1 to find the long outwardly directed spirals associated with shells which are transversely elongated1 as ,spirifer is2 while the more rounded ,atrypas will tend to the opposite condition4 ,in a few cases1 as in ,cyrtina or ,reticularia1 where the shell is comparatively narrow but long1 and where the uncoiled basal support of the arms is long also1 the spiral coils into which the latter grow are turned backwards1 in the direction where there is room for them4 ,and in the few cases where the shell is very considerably flattened1 the spirals 7if they find room #egj to grow at all7 will be constrained to do so in a discoid or nearly discoid fashion1 and this is actually the case in such flattened forms as ,koninckina or ,thecidium4 ,the ,shells of ,pteropods4 ,while mathematically speaking we are entitled to look upon the bivalve shell of the ,lamellibranch as consisting of two distinct elements1 each comparable to the entire shell of the univalve1 we have no biological grounds for such a statement2 for the shell arises from a single embryonic origin1 and afterwards becomes split into portions which constitute the two separate valves4 ,we can perhaps throw some indirect light upon this phenomenon1 and upon several other phenomena connected with shell-growth1 by a consideration of the simple conical or tubular shells of the ,pteropods4 ,the shells of the latter are in few cases suitable for simple mathematical investigation1 but nevertheless they are of very considerable interest in connection with our general problem4 ,the morphology of the ,pteropods is by no means well understood1 and in speaking of them ,i will assume that there are still grounds for believing 7in spite of ,boas' and ,pelseneer's arguments7 that they are directly related to1 or may at least be directly compared with1 the ,cephalopoda#ebi4 ,illustration3 ,fig4 #bie4 ,pteropod shells3 7#a7 ,cuvierina columnella2 7#b7 ,cleodora chierchiae2 7#c7 ,c4 pygmaea4 7,after ,boas47 ,the simplest shells among the ,pteropods have the form of a tube1 more or less cylindrical 7,cuvierina71 more often conical 7,creseis1 ,clio72 and this tubular shell 7as we have already had occasion to remark1 on p4 #beh71 frequently tends1 when it is very small and delicate1 to assume the character of an unduloid4 7,in such a case it is more than likely that the tiny shell1 or that portion of it which constitutes the unduloid1 has not grown by successive #ega increments or 8rings of growth10 but has developed as a whole47 ,a thickened 8rib0 is often1 perhaps generally1 present on the dorsal side of the little conical shell4 ,in a few cases 7,limacina1 ,peraclis7 the tube becomes spirally coiled1 in a normal logarithmic spiral or helico-spiral4 ,illustration3 ,fig4 #bif4 ,diagrammatic transverse sections1 or outlines of the mouth1 in certain ,pteropod shells3 ,a1 ,b1 ,cleodora australis2 ,c1 ,c4 pyramidalis2 ,d1 ,c4 balantium2 ,e1 ,c4 cuspidata4 7,after ,boas47 ,illustration3 ,fig4 #big4 ,shells of thecosome ,pteropods 7after ,boas74 7#a7 ,cleodora cuspidata2 7#b7 ,hyalaea trispinosa2 7#c7 ,h4 globulosa2 7#d7 ,h4 uncinata2 7#e7 ,h4 inflexa4 ,in certain cases 7e4g4 ,cleodora1 ,hyalaea7 the tube or cone is curiously modified4 ,in the first place1 its cross-section1 originally #egb circular or nearly so1 becomes flattened or compressed dorso-ventrally2 and the angle1 or rather edge1 where dorsal and ventral walls meet1 becomes more and more drawn out into a ridge or keel4 ,along the free margin1 both of the dorsal and the ventral portion of the shell1 growth proceeds with a regularly varying velocity1 so that these margins1 or lips1 of the shell become regularly curved or markedly sinuous4 ,at the same time1 growth in a transverse direction proceeds with an acceleration which manifests itself in a curvature of the sides1 replacing the straight borders of the original cone4 ,in other words1 the cross-section of the cone1 or what we have been calling the generating curve1 increases its dimensions more rapidly than its distance from the pole4 ,illustration3 ,fig4 #bih4 ,cleodora cuspidata4 ,in the above figures1 for instance in that of ,cleodora cuspidata1 the markings of the shell which represent the successive edges of the lip at former stages of growth1 furnish us at once with a 8graph0 of the varying velocities of growth as measured1 radially1 from the apex4 ,we can reveal more clearly the nature of these variations in the following way which is simply tantamount to converting our radial into rectangular coordinates4 ,neglecting curvature 7if any7 of the sides and treating the shell 7for simplicity's sake7 as a right cone1 we lay off equal angles from the apex ,o1 along the radii ,oa1 ,ob1 etc4 ,if we then plot1 as vertical equidistant ordinates1 the magnitudes ,oa1 ,ob 444 ,,oy1 and again on to ,oa1 we obtain a diagram such as the following 7,fig4 #bii72 by #egc help of which we not only see more clearly the way in which the growth-rate varies from point to point1 but we also recognise much better than before1 the similar nature of the law which governs this variation in the different species4 ,illustration3 ,fig4 #bii4 ,curves obtained by transforming radial ordinates1 as in ,fig4 #bih1 into vertical equidistant ordinates4 #a1 ,hyalaea trispinosa2 #b1 ,cleodora cuspidata4 ,illustration3 ,fig4 #cjj4 ,development of the shell of ,hyalaea 7,cavolinia7 tridentata1 ,forskal3 the earlier stages being the 8,pleuropus longifilis0 of ,troschel4 7,after ,tesch47 ,furthermore1 the young shell having become differentiated into a dorsal and a ventral part1 marked off from one another by a lateral edge or keel1 and the inequality of growth being such as to cause each portion to increase most rapidly in the median line1 it follows that the entire shell will appear to have been split into a dorsal and a ventral plate1 both connected with1 and projecting from1 #egd what remains of the original undivided cone4 ,putting the same thing in other words1 we may say that the generating figure1 which lay at first in a plane perpendicular to the axis of the cone1 has now1 by unequal growth1 been sharply bent or folded1 so as to lie approximately in two planes1 parallel to the anterior and posterior faces of the cone4 ,we have only to imagine the apical connecting portion to be further reduced1 and finally to disappear or rupture1 and we should have a bivalve shell developed out of the original simple cone4 ,in its outer and growing portion1 the shell of our ,pteropod now consists of two parts which1 though still connected together at the apex1 may be treated as growing practically independently4 ,the shell is no longer a simple tube1 or simple cone1 in which regular inequalities of growth will lead to the development of a spiral2 and this for the simple reason that we have now two opposite maxima of growth1 instead of a maximum on the one side and a minimum on the other side of our tubular shell4 ,as a matter of fact1 the dorsal and the ventral plate tend to curve in opposite directions1 towards the middle line1 the dorsal curving ventrally and the ventral curving towards the dorsal side4 ,in the case of the ,lamellibranch or the ,brachiopod1 it is quite possible for both valves to grow into more or less pronounced spirals1 for the simple reason that they are hinged upon one another2 and each growing edge1 instead of being brought to a standstill by the growth of its opposite neighbour1 is free to move out of the way1 by the rotation about the hinge of the plane in which it lies4 ,illustration3 ,fig4 #cja4 ,pteropod shells1 from the side3 7#a7 ,cleodora cuspidata2 7#b7 ,hyalaea longirostris2 7#c7 ,h4 trispinosa4 7,after ,boas47 ,but where1 as in the ,pteropod1 there is no such hinge1 the dorsal and ventral halves of the shell 7or dorsal and ventral valves1 if we may call them so71 if they curved towards one another 7as they do in a cockle71 would soon interfere with one another's progress1 and the development of a pair of conjugate spirals would become impossible4 ,nevertheless1 there is obviously1 in both dorsal and ventral valve1 a tendency to the development of a spiral curve1 that of the ventral valve being more marked than that of the larger and overlapping dorsal one1 exactly as in the two unequal valves of ,terebratula4 ,in many cases 7e4g4 ,cleodora cuspidata71 the dorsal valve or plate1 #ege strengthened and stiffened by its midrib1 is nearly straight1 while the curvature of the other is well displayed4 ,but the case will be materially altered and simplified if growth be arrested or retarded in either half of the shell4 ,suppose for instance that the dorsal valve grew so slowly that after a while1 in comparison with the other1 we might speak of it as being absent altogether3 or suppose that it merely became so reduced in relative size as to form no impediment to the continued growth of the ventral one2 the latter would continue to grow in the direction of its natural curvature1 and would end by forming a complete and coiled logarithmic spiral4 ,it would be precisely analogous to the spiral shell of ,nautilus1 and1 in regard to its ventral position1 concave towards the dorsal side1 it would even deserve to be called directly homologous with it4 ,suppose1 on the other hand1 that the ventral valve were to be greatly reduced1 and even to disappear1 the dorsal valve would then pursue its unopposed growth2 and1 were it to be markedly curved1 it would come to form a logarithmic spiral1 concave towards the ventral side1 as is the case in the shell of ,spirula#ecj4 ,were the dorsal valve to be destitute of any marked curvature 7or in other words1 to have but a low spiral angle71 it would form a simple plate1 as in the shells of ,sepia or ,loligo4 ,indeed1 in the shells of these latter1 and especially in that of ,sepia1 we seem to recognise a manifest resemblance to the dorsal plate of the ,pteropod shell1 as we have it 7e4g47 in ,cleodora or ,hyalaea2 #egf the little 8rostrum0 of ,sepia is but the apex of the primitive cone1 and the rounded anterior extremity has grown according to a law precisely such as that which has produced the curved margin of the dorsal valve in the ,pteropod4 ,the ventral portion of the original cone is nearly1 but not wholly1 wanting4 ,it is represented by the so-called posterior wall of the 8siphuncular space40 ,in many decapod cuttle-fishes also 7e4g4 ,todarodes1 ,illex1 etc47 we still see at the posterior end of the 8pen10 a vestige of the primitive cone1 whose dorsal margin only has continued to grow2 and the same phenomenon1 on an exaggerated scale1 is represented in the ,belemnites4 ,it is not at all impossible that we may explain on the same lines the development of the curious 8operculum0 of the ,ammonites4 ,this consists of a single horny plate 7,anaptychus71 or of a thicker1 more calcified plate divided into two symmetrical halves 7,aptychi71 often found inside the terminal chamber of the ,ammonite1 and occasionally to be seen lying in situ1 as an operculum which partially closes the mouth of the shell2 this structure is known to exist even in connection with the early embryonic shell4 ,in form the ,anaptychus1 or the pair of conjoined ,aptychi1 shew an upper and a lower border1 the latter strongly convex1 the former sometimes slightly concave1 sometimes slightly convex1 and usually shewing a median projection or slightly developed rostrum4 ,from this 8rostral0 border the curves of growth start1 and course round parallel to1 finally constituting1 the convex border4 ,it is this convex border which fits into the free margin of the mouth of the ,ammonite's shell1 while the other is applied to and overlaps the preceding whorl of the spire4 ,now this relationship is precisely what we should expect1 were we to imagine as our starting-point a shell similar to that of ,hyalaea1 in which however the dorsal part of the split cone had become separate from the ventral half1 had remained flat1 and had grown comparatively slowly1 while at the same time it kept slipping forward over the growing and coiling spire into which the ventral half of the original shell develops#eca4 ,in short1 ,i think there is reason to believe1 or at least to suspect1 that we #egg have in the shell and ,aptychus of the ,ammonites1 two portions of a once united structure2 of which other ,cephalopods retain not both parts but only one or other1 one as the ventrally situated shell of ,nautilus1 the other as the dorsally placed shell for example of ,sepia or of ,spirula4 ,in the case of the bivalve shells of the ,lamellibranchs or of the ,brachiopods1 we have to deal with a phenomenon precisely analogous to the split and flattened cone of our ,pteropods1 save only that the primitive cone has been split into two portions1 not incompletely as in the ,pteropod 7,hyalaea71 but completely1 so as to form two separate valves4 ,though somewhat greater freedom is given to growth now that the two valves are separate and hinged1 yet still the two valves oppose and hamper one another1 so that in the longitudinal direction each is capable of only a moderate curvature4 ,this curvature1 as we have seen1 is recognisable as a logarithmic spiral1 but only now and then does the growth of the spiral continue so far as to develop successive coils3 as it does in a few symmetrical forms such as ,isocardia cor2 and as it does still more conspicuously in a few others1 such as ,gryphaea and ,caprinella1 where one of the two valves is stunted1 and the growth of the other is 7relatively speaking7 unopposed4 ,of ,septa4 ,before we leave the subject of the molluscan shell1 we have still another problem to deal with1 in regard to the form and arrangement of the septa which divide up the tubular shell into chambers1 in the ,nautilus1 the ,ammonite and their allies 7,fig4 #cjd1 etc474 ,the existence of septa in a ,nautiloid shell may probably be accounted for as follows4 ,we have seen that it is a property of a cone that1 while growing by increments at one end only1 it conserves its original shape3 therefore the animal within1 which 7though growing by a different law7 also conserves its shape1 will continue to fill the shell if it actually fills it to begin with3 as does a snail or other ,gastropod4 ,but suppose that our mollusc fills a part only of a conical shell 7as it does in the case of ,nautilus72 then1 unless it alter its shape1 it must move upward as it grows in the growing cone1 until it come to occupy a space similar in form #egh to that which it occupied before3 just1 indeed1 as a little ball drops far down into the cone1 but a big one must stay farther up4 ,then1 when the animal after a period of growth has moved farther up in the shell1 the mantle-surface continues its normal secretory activity1 and that portion which had been in contact with the former septum secretes a septum anew4 ,in short1 at any given epoch1 the creature is not secreting a tube and a septum by separate operations1 but is secreting a shelly case about its rounded body1 of which case one part appears to us as the continuation of the tube1 and the other part1 merging with it by indistinguishable boundaries1 appears to us as the septum#ecb4 ,the various forms assumed by the septa in spiral shells#ecc present us with a number of problems of great beauty1 simple in their essence1 but whose full investigation would soon lead us into mathematics of a very high order4 ,we do not know in great detail how these septa are laid down2 but the essential facts are clear#ecd4 ,the septum begins as a very thin cuticular membrane 7composed apparently of a substance called conchyolin71 which is secreted by the skin1 or mantle-surface1 of the animal2 and upon this membrane nacreous matter is gradually laid down on the mantle-side 7that is to say between the animal's body and the cuticular membrane which has been thrown off from it71 so that the membrane remains as a thin pellicle over the hinder surface of the septum1 and so that1 to begin with1 the membranous septum is moulded on the flexible and elastic surface of the animal1 within which the fluids of the body must exercise a uniform1 or nearly uniform pressure4 ,let us think1 then1 of the septa as they would appear in their uncalcified condition1 formed of1 or at least superposed upon1 an #egi elastic membrane4 ,they must then follow the general law1 applicable to all elastic membranes under uniform pressure1 that the tension varies inversely as the radius of curvature2 and we come back once more to our old equation of ,laplace1 that ,p ,t7#a_/r #a_/r74 ,moreover1 since the cavity below the septum is practically closed1 and is filled either with air or with water1 ,p will be constant over the whole area of the septum4 ,and further1 we must assume1 at least to begin with1 that the membrane constituting the incipient septum is homogeneous or isotropic4 ,let us take first the case of a straight cone1 of circular section1 more or less like an ,orthoceras2 and let us suppose that the septum is attached to the shell in a plane perpendicular to its axis4 ,the septum itself must then obviously be spherical4 ,moreover the extent of the spherical surface is constant1 and easily determined4 ,for obviously1 in ,fig4 #cjb1 the angle ,,lcl equals the supplement of the angle 7,,lol7 of the cone2 that is to say1 the circle of contact subtends an angle at the centre of the spherical surface1 which is constant1 and which is equal to #b4 ,the case is not excluded where1 owing to an asymmetry of tensions1 the septum meets the side walls of the cone at other than a right angle1 as in ,fig4 #cjc2 and here1 while the septa still remain portions of spheres1 the geometrical construction for the position of their centres is equally easy4 ,illustration3 ,fig4 #cjb4 ,illustration3 ,fig4 #cjc4 ,if1 on the other hand1 the attachment of the septum to the inner walls of the cone be in a plane oblique to the axis1 then it is evident that the outline of the septum will be an ellipse1 and its surface an #ehj ellipsoid4 ,if the attachment of the septum be not in one plane1 but form a sinuous line of contact with the cone1 then the septum will be a saddle-shaped surface1 of great complexity and beauty4 ,in all cases1 provided only that the membrane be isotropic1 the form assumed will be precisely that of a soap-bubble under similar conditions of attachment3 that is to say1 it will be 7with the usual limitations or conditions7 a surface of minimal area4 ,if our cone be no longer straight1 but curved1 then the septa will be symmetrically deformed in consequence4 ,a beautiful and interesting case is afforded us by ,nautilus itself4 ,here the outline of the septum1 referred to a plane1 is approximately bounded by two elliptic curves1 similar and similarly situated1 whose areas are to one another in a definite ratio1 namely as ,a#a_/;,a#b 7r#a r#a7_/7r#b r#b7 #d cot 1 and a similar ratio exists in ,ammonites and all other close-whorled spirals1 in which however we cannot always make the simple assumption of elliptical form4 ,in a median section of ,nautilus1 we see each septum forming a tangent to the inner and to the outer wall1 just as it did in a section of the straight ,orthoceras2 but the curvatures in the neighbourhood of these two points of contact are not identical1 for they now vary inversely as the radii1 drawn from the pole of the spiral shell4 ,the contour of the septum in this median plane is a spiral curve identical with the original logarithmic spiral4 ,of this it is the 8invert10 and the fact that the original curve and its invert are both identical is one of the most beautiful properties of the logarithmic spiral#ece4 ,illustration3 ,fig4 #cjd4 ,section of ,nautilus1 shewing the contour of the septa in the median plane3 the septa being 7in this plane7 logarithmic spirals1 of which the shell-spiral is the evolute4 ,but while the outline of the septum in median section is simple and easy to determine1 the curved surface of the septum in its entirety is a very complicated matter1 even in ,nautilus which is one of the simplest of actual cases4 ,for1 in the first place1 since the form of the septum1 as seen in median section1 is that of a logarithmic spiral1 and as therefore its curvature is constantly altering1 it follows that1 in successive transverse sections1 the #eha curvature is also constantly altering4 ,but in the case of ,nautilus1 there are other aspects of the phenomenon1 which we can illustrate1 but only in part1 in the following simple manner4 ,let us imagine a pack of cards1 in which we have cut out of each card a similar concave arc of a logarithmic spiral1 such as we actually see in the median section of the septum of a ,nautilus4 ,then1 while we hold the cards together1 foursquare1 in the ordinary position of the #ehb pack1 we have a simple 8ruled0 surface1 which in any longitudinal section has the form of a logarithmic spiral but in any transverse section is a straight horizontal line4 ,if we shear or slide the cards upon one another1 thrusting the middle cards of the pack forward in advance of the others1 till the one end of the pack is a convex1 and the other a concave1 ellipse1 the cut edges which combine to represent our septum will now form a curved surface ,illustration3 ,fig4 #cje4 ,cast of the interior of ,nautilus3 to shew the contours of the septa at their junction with the shell-wall4 of much greater complexity2 and this is part1 but not by any means all1 of the deformation produced as a direct consequence of the form in ,nautilus of the section of the tube within which the septum has to lie4 ,and the complex curvature of the surface will be manifested in a sinuous outline of the edge1 or line of attachment of the septum to the tube1 and will vary according to the configuration of the latter4 ,in the case of ,nautilus1 it is easy to shew empirically 7though not perhaps easy to demonstrate #ehc mathematically7 that the sinuous or saddle-shaped form of the 8suture0 7or line of attachment of the septum to the tube7 is such as can be precisely accounted for in this manner4 ,it is also easy to see that1 when the section of the tube 7or 8generating curve07 is more complicated in form1 when it is flattened1 grooved1 or otherwise ornamented1 the curvature of the septum and the outline of its sutural attachment will become very complicated indeed#ecf2 but it will be comparatively simple in the case of the first few sutures of the young shell1 laid down before any overlapping of whorls has taken place1 and this comparative simplicity of the first-formed sutures is a marked feature among ,ammonites#ecg4 ,we have other sources of complication1 besides those which are at once introduced by the sectional form of the tube4 ,for instance1 the siphuncle1 or little inner tube which perforates the septa1 exercises a certain amount of tension1 sometimes evidently considerable1 upon the latter2 so that we can no longer consider each septum as an isotropic surface1 under uniform pressure2 and there may be other structural modifications1 or inequalities1 in that portion of the animal's body with which the septum is in contact1 and by which it is conformed4 ,it is hardly likely1 for all these reasons1 that we shall ever attain to a full and particular explanation of the septal surfaces and their sutural outlines throughout the whole range of ,cephalopod shells2 but in general terms1 the problem is probably not beyond the reach of mathematical analysis4 ,the problem might be approached experimentally1 after the manner of ,plateau's experiments1 by bending #ehd a wire into the complicated form of the suture-line1 and studying the form of the liquid film which constitutes the corresponding surface minimae areae4 ,illustration3 ,fig4 #cjf4 ,ammonites 7,sonninia7 ,sowerbyi4 7,from ,zittel1 after ,steinmann and ,dderlein47 ,illustration3 ,fig4 #cjg4 ,suture-line of a ,triassic ,ammonite 7,pinacoceras74 7,from ,zittel1 after ,hauer47 ,in certain ,ammonites the septal outline is further complicated in another way4 ,superposed upon the usual sinuous outline1 with its 8lobes0 and 8saddles10 we have here a minutely ramified1 or arborescent outline1 in which all the branches terminate in wavy1 more or less circular arcs1--looking just like the #8landscape marble' from the ,bristol ,rhaetic4 ,we have no difficulty in recognising in this a surface-tension phenomenon4 ,the figures are precisely such as we can imitate 7for instance7 by pouring a #ehe few drops of milk upon a greasy plate1 or of oil upon an alkaline solution4 ,we have very far from exhausted1 we have perhaps little more than begun1 the study of the logarithmic spiral and the associated curves which find exemplification in the multitudinous diversities of molluscan shells4 ,but1 with a closing word or two1 we must now bring this chapter to an end4 ,in the spiral shell we have a problem1 or a phenomenon1 of growth1 immensely simplified by the fact that each successive increment is irrevocably fixed in regard to magnitude and position1 instead1 of remaining in a state of flux and sharing in the further changes which the organism undergoes4 ,in such a structure1 then1 we have certain primary phenomena of growth manifested in their original simplicity1 undisturbed by secondary and conflicting phenomena4 ,what actually grows is merely the lip of an orifice1 where there is produced a ring of solid material1 whose form we have treated of under the name of the generating curve2 and this generating curve grows in magnitude without alteration of its form4 ,besides its increase in areal magnitude1 the growing curve has certain strictly limited degrees of freedom1 which define its motions in space3 that is to say1 it has a vector motion at right angles to the axis of the shell2 and it has a sliding motion along that axis4 ,and1 though we may know nothing whatsoever about the actual velocities of any of these motions1 we do know that they are so correlated together that their relative velocities remain constant1 and accordingly the form and symmetry of the whole system remain in general unchanged4 ,but there is a vast range of possibilities in regard to every one of these factors3 the generating curve may be of various forms1 and even when of simple form1 such as an ellipse1 its axes may be set at various angles to the system2 the plane also in which it lies may vary1 almost indefinitely1 in its angle relatively to that of any plane of reference in the system2 and in the several velocities of growth1 of rotation and of translation1 and therefore in the ratios between all these1 we have again a vast range of possibilities4 ,we have then a certain definite type1 or group of forms1 mathematically isomorphous1 but presenting infinite diversities of outward appearance3 which diversities1 as ,swammerdam #ehf said1 ex sola nascuntur diversitate gyrationum2 and which accordingly are seen to have their origin in differences of rate1 or of magnitude1 and so to be1 essentially1 neither more nor less than differences of degree4 ,in nature1 we find these forms presenting themselves with but little relation to the character of the creature by which they are produced4 ,spiral forms of certain particular kinds are common to ,gastropods and to ,cephalopods1 and to diverse families of each2 while outside the class of molluscs altogether1 among the ,foraminifera and among the worms 7as in ,spirorbis1 ,spirographis1 and in the ,dentalium-like shell of ,ditrupa71 we again meet with similar and corresponding forms4 ,again1 we find the same forms1 or forms which 7save for external ornament7 are mathematically identical1 repeating themselves in all periods of the world's geological history2 and1 irrespective of climate or local conditions1 we see them mixed up1 one with another1 in the depths and on the shores of every sea4 ,it is hard indeed 7to my mind7 to see where ,natural ,selection necessarily enters in1 or to admit that it has had any share whatsoever in the production of these varied conformations4 ,unless indeed we use the term ,natural ,selection in a sense so wide as to deprive it of any purely biological significance2 and so recognise as a sort of natural selection whatsoever nexus of causes suffices to differentiate between the likely and the unlikely1 the scarce and the frequent1 the easy and the hard3 and leads accordingly1 under the peculiar conditions1 limitations and restraints which we call 8ordinary circumstances10 one type of crystal1 one form of cloud1 one chemical compound1 to be of frequent occurrence and another to be rare4 #ehg ,,chapter ,,xii ,,the ,,spiral ,,shells ,,of ,,the ,,foraminifera ,we have already dealt in a few simple cases with the shells of the ,foraminifera#ech2 and we have seen that wherever the shell is but a single unit or single chamber1 its form may be explained in general by the laws of surface tension3 the assumption being that the little mass of protoplasm which makes the simple shell behaves as a fluid drop1 the form of which is perpetuated when the protoplasm acquires its solid covering4 ,thus the spherical ,orbulinae and the flask-shaped ,lagenae represent drops in equilibrium1 under various conditions of freedom or constraint2 while the irregular1 amoeboid body of ,astrorhiza is a manifestation not of equilibrium1 but of a varying and fluctuating distribution of surface energy4 ,when the foraminiferal shell becomes multilocular1 the same general principles continue to hold2 the growing protoplasm increases drop by drop1 and each successive drop has its particular phenomena of surface energy1 manifested at its fluid surface1 and tending to confer upon it a certain place in the system and a certain shape of its own4 ,it is characteristic and even diagnostic of this particular group of ,protozoa 7#a7 that development proceeds by a well-marked alternation of rest and of activity--of activity during which the protoplasm increases1 and of rest during which the shell is formed2 7#b7 that the shell is formed at the outer surface of the protoplasmic organism1 and tends to constitute a continuous or all but continuous covering2 and it follows 7#c7 from these two factors taken together that each successive increment is added on outside of and distinct from its predecessors1 that the successive parts or chambers of #ehh the shell are of different and successive ages1 that one part of the shell is always relatively new1 and the rest old in various grades of seniority4 ,the forms which we set together in the sister-group of ,radiolaria are very differently characterised4 ,here the cells or vesicles of which each little composite organism is made up are but little separated1 and in no way walled off1 from one another2 the hard skeletal matter tends to be deposited in the form of isolated spicules or of little connected rods or plates1 at the angles1 the edges or the interfaces of the vesicles2 the cells or vesicles form a coordinated and cotemporaneous rather than a successive series4 ,in a word1 the whole quasi-fluid protoplasmic body may be likened to a little mass of froth or foam3 that is to say1 to an aggregation of simultaneously formed drops or bubbles1 whose physical properties and geometrical relations are very different from those of a system of drops or bubbles which are formed one after another1 each solidifying before the next is formed4 ,illustration3 ,fig4 #cjh4 ,hastigerina sp42 to shew the 8mouth40 ,with the actual origin or mode of development of the foraminiferal shell we are now but little concerned4 ,the main factor is the adsorption1 and subsequent precipitation at the surface of the organism1 of calcium carbonate1--the shell so formed being interrupted by pores or by some larger interspace or 8mouth0 7,fig4 #cjh71 which interruptions we may doubtless interpret as being due to unequal distributions of surface energy4 ,in many #ehi cases the fluid protoplasm 8picks up0 sand-grains and other foreign particles1 after a fashion which we have already described 7p4 #dfc72 and it cements these together with more or less of calcareous material4 ,the calcareous shell is a crystalline structure1 and the micro-crystals of calcium carbonate are so set that their little prisms radiate outwards in each chamber through the thickness of the wall3--which symmetry is subject to corresponding modification when the spherical chambers are more or less symmetrically deformed#eci4 ,in various ways the rounded1 drop-like shells of the ,foraminifera1 both simple and compound1 have been artificially imitated4 ,thus1 if small globules of mercury be immersed in water in which a little chromic acid is allowed to dissolve1 as the little beads of quicksilver become slowly covered with a crystalline coat of mercuric chromate they assume various forms reminiscent of the monothalamic ,foraminifera4 ,the mercuric chromate has a higher atomic volume than the mercury which it replaces1 and therefore the fluid contents of the drop are under pressure1 which increases with the thickness of the pellicle2 hence at some weak spot in the latter the contents will presently burst forth1 so forming a mouth to the little shell4 ,sometimes a long thread is formed1 just as in ,rhabdammina linearis2 and sometimes unduloid swellings make their appearance on such a thread1 just as in ,r4 discreta4 ,and again1 by appropriate modifications of the experimental conditions1 it is possible 7as ,rhumbler has shewn7 to build up a chambered shell#edj4 ,in a few forms1 such as ,globigerina and its close allies1 the shell is beset during life with excessively long and delicate calcareous spines or needles4 ,it is only in oceanic forms that these are present1 because only when poised in water can such #eij delicate structures endure2 in dead shells1 such as we are much more familiar with1 every trace of them is broken and rubbed away4 ,the growth of these long needles is explained 7as we have already briefly mentioned1 on p4 #ddj7 by the phenomenon which ,lehmann calls orientirte ,adsorption--the tendency for a crystalline structure to grow by accretion1 not necessarily in the outward form of a 8crystal10 but continuing in any direction or orientation which has once been impressed upon it3 in this case the spicular growth is simply in direct continuation of the radial symmetry of the micro-crystalline elements of the shell-wall4 ,over the surface of the shell the radiating spicules tend to occur in a hexagonal pattern1 symmetrically grouped around the pores which perforate the shell4 ,rhumbler has suggested that this arrangement is due to diffusion-currents1 forming little eddies about the base of the pseudopodia issuing from the pores3 the idea being borrowed from ,bnard1 to whom is due the discovery of this type or order of vortices#eda4 ,in one of ,bnard's experiments a thin layer of paraffin is strewn with particles of graphite1 then warmed to melting1 whereupon each little solid granule becomes the centre of a vortex2 by the interaction of these vortices the particles tend to be repelled to equal distances from one another1 and in the end they are found to be arranged in a hexagonal pattern#edb4 ,the analogy is plain between this experiment and those diffusion experiments by which ,leduc produces his beautiful hexagonal systems of artificial cells1 with which we have dealt in a previous chapter 7p4 #cbj74 ,but let us come back to the shell itself1 and consider particularly its spiral form4 ,that the shell in the ,foraminifera should tend towards a spiral form need not surprise us2 for we have learned that one of the fundamental conditions of the production of a concrete spiral is just precisely what we have here1 namely the gradual development of a structure by means of successive increments superadded to its exterior1 which then form part1 successively1 of a permanent and rigid structure4 ,this condition #eia is obviously forthcoming in the foraminiferal1 but not at all in the radiolarian1 shell4 ,our second fundamental condition of the production of a logarithmic spiral is that each successive increment shall be so posited and so conformed that its addition to the system leaves the form of the whole system unchanged4 ,we have now to enquire into this latter condition2 and to determine whether the successive increments1 or successive chambers1 of the foraminiferal shell actually constitute gnomons to the entire structure4 ,it is obvious enough that the spiral shells of the ,foraminifera closely resemble true logarithmic spirals4 ,indeed so precisely do the minute shells of many ,foraminifera repeat or simulate the spiral shells of ,nautilus and its allies that to the naturalists of the early nineteenth century they were known as the ,cphalopodes microscopiques#edc1 until ,dujardin shewed that their little bodies comprised no complex anatomy of organs1 but consisted merely of that slime-like organic matter which he taught us to call 8sarcode10 and which we learned afterwards from ,schwann to speak of as 8protoplasm40 ,illustration3 ,fig4 #cji4 ,nummulina antiquior1 ,r4 and ,v4 7,after ,v4 von ,mller47 ,one striking difference1 however1 is apparent between the shell of ,nautilus and the little nautiloid or rotaline shells of the ,foraminifera3 namely that the septa in these latter1 and in all other #eib chambered ,foraminifera1 are convex outwards 7,fig4 #cjh71 whereas they are concave outwards in ,nautilus 7,fig4 #cjd7 and in the rest of the chambered molluscan shells4 ,the reason is perfectly simple4 ,in both cases the curvature of the septum was determined before it became rigid1 and at a time when it had the properties either of a fluid film or an elastic membrane4 ,in both cases the actual curvature is determined by the tensions of the membrane and the pressures to which it was exposed4 ,now it is obvious that the extrinsic pressure which the tension of the membrane has to withstand is on opposite sides in the two cases4 ,in ,nautilus1 the pressure to be resisted is that produced by the growing body of the animal1 lying to the outer side of the septum1 in the outer1 wider portion of the tubular shell4 ,in the ,foraminifer the septum at the time of its formation was no septum at all2 it was but a portion of the convex surface of a drop-that portion namely which afterwards became overlapped and enclosed by the succeeding drop2 and the curvature of the septum is concave towards the pressure to be resisted1 which latter is inside the septum1 being simply the hydrostatic pressure of the fluid contents of the drop4 ,the one septum is1 speaking generally1 the reverse of the other2 the organism1 so to speak1 is outside the one and inside the other2 and in both cases alike1 the septum tends to assume the form of a surface of minimal area1 as permitted1 or as defined1 by all the circumstances of the case4 ,the logarithmic spiral is easily recognisable in typical cases#edd 7and especially where the spire makes more than one visible revolution about the pole71 by its fundamental property of continued similarity3 that is to say1 by reason of the fact that the big many-chambered shell is of just the same shape as the smaller and younger shell--which phenomenon is apparent and even obvious in the nautiloid ,foraminifera1 as in ,nautilus itself3 but nevertheless the nature of the curve must be verified by careful measurement1 just as ,moseley determined or verified it in his #eic original study of nautilus 7cf4 p4 #eah74 ,this has accordingly been done1 by various writers3 and in the first instance by ,valerian von ,mller1 in an elaborate study of ,fusulina--a palaeozoic genus whose little shells have built up vast tracts of carboniferous limestone over great part of ,european ,russia#ede4 ,in this genus a growing surface of protoplasm may be conceived as wrapping round and round a small initial chamber1 in such a way as to produce a fusiform or ellipsoidal shell--a transverse section of which reveals the close-wound spiral coil4 ,the following are examples of measurements of the successive whorls in a couple of species of this genus4 ,f4 cylindrica1 ,fischer ,f4 ,bcki1 v4 ,mller ,breadth 7in millimetres74 ,whorl ,observed ,calculated ,observed ,calculated ,i #acb -- #jgi -- ,,ii #aie #aih #abj #aai ,,iii #cjj #big #ahj #agi ,,iv #ddi #dde #bfd #bfg ,v -- -- #cif #dja ,in both cases the successive whorls are very nearly in the ratio of #a 3 #ae2 and on this ratio the calculated values are based4 ,here is another of von ,mller's series of measurements of ,f4 cylindrica1 the measurements being those of opposite whorls--that is to say of whorls #ahj apart3 ,breadth in mm4 #jif #aag #add #agf #baf #bfd #cbc #cie ,log4 of mm4 #ihb #jfh #aeh #bdf #ccd #dbb #eji #eig ,diff4 of logs4 -- #jhf #jij #jhh #jhh #jhh #jhg #jhh ,the mean logarithmic difference is here #jhh1 log #abbe2 or the mean difference of alternate logs 7corresponding to a vector angle of #b1 i4e4 to consecutive measurements along the same radius7 is #agf1 log #ae1 the same value as before4 ,and this ratio of #ae between the breadths of successive whorls corresponds 7as we see by our table on p4 #ecd7 to a constant angle of about #eid #hf1 or just such a spiral as we commonly meet with in the ,ammonites#edf 7cf4 p4 #eci74 ,in ,fusulina1 and in some few other ,foraminifera 7cf4 ,fig4 #caj1 ,a71 the spire seems to wind evenly on1 with little or no external sign of the successive periods of growth1 or successive chambers of the shell4 ,the septa which mark off the chambers1 and correspond to retardations or cessations in the periodicity of growth1 are still to be found in sections of the shell of ,fusulina2 but they are somewhat irregular and comparatively inconspicuous2 the measurements we have just spoken of are taken without reference to the segments or chambers1 but only with reference to the whorls1 or in other words with direct reference to the vectorial angle4 ,illustration3 ,fig4 #caj4 ,a1 ,cornuspira foliacea1 ,phil42 ,b1 ,operculina complanata1 ,defr4 ,the linear dimensions of successive chambers have been #eie measured in a number of cases4 ,van ,iterson#edg has done so in various ,miliolinidae1 with such results as the following3 ,triloculina rotunda1 d',orb4 ,no4 of chamber #a #b #c #d #e #f #g #h #i #aj ,breadth of chamber in -- #cd #de #fa #hd #aad #adb #ahb #bdf #cai ,breadth of chamber in 1 calculated -- #cd #de #fj #gi #aje #adj #ahg #bdc #cai ,here the mean ratio of breadth of consecutive chambers may be taken as #acbc 7that is to say1 the eighth root of #cai_/cd72 and the calculated values1 as given above1 are based on this determination4 ,again1 ,rhumbler has measured the linear dimensions of a number of rotaline forms1 for instance ,pulvinulina menardi 7,fig4 #bei73 in which common species he finds the mean linear ratio of consecutive chambers to be about #aahg4 ,in both cases1 and especially in the latter1 the ratio is not strictly constant from chamber to chamber1 but is subject to a small secondary fluctuation#edh4 ,illustration3 ,fig4 #caa4 #a1 #b1 ,miliolina pulchella1 d',orb42 #c-e1 ,m4 linnaeana1 d',orb4 7,after ,brady47 ,illustration3 ,fig4 #cab4 ,cyclammina cancellata1 ,brady4 ,when the linear dimensions of successive chambers are in continued proportion1 then1 in order that the whole shell may constitute a logarithmic spiral1 it is necessary that the several chambers should subtend equal angles of revolution at the pole4 ,in the case of the ,miliolidae this is obviously the case 7,fig4 #caa72 for in this family the chambers lie in two rows 7,biloculina71 or three rows 7,triloculina71 or in some other small number of series3 so that the angles subtended by them are large1 simple fractions of the circular arc1 such as #ahj or #abj4 ,in many of the nautiloid forms1 such as ,cyclammina 7,fig4 #cab71 the angles subtended1 though of less magnitude1 are still remarkably constant1 as we #eig may see by ,fig4 #cac2 where the angle subtended by each chamber is made equal to #bj1 and this diagrammatic figure is not perceptibly different from the other4 ,in some cases the subtended angle is less constant2 and in these it would be necessary to equate the several linear dimensions with the corresponding vector angles1 according to our equation r e cot 4 ,it is probable that1 by so taking account of variations of 1 such variations of r as 7according to ,rhumbler's measurements7 ,pulvinulina and other genera appear to shew1 would be found to diminish or even to disappear4 ,illustration3 ,fig4 #cac4 ,cyclammina sp4 7,diagrammatic47 ,the law of increase by which each chamber bears a constant ratio of magnitude to the next may be looked upon as a simple consequence of the structural uniformity or homogeneity of the organism2 we have merely to suppose 7as this uniformity would naturally lead us to do7 that the rate of increase is at each instant proportional to the whole existing mass4 ,for if ,v#j1 ,v#a etc41 be the volumes of the successive chambers1 let ,v#a bear a constant proportion to ,v#j1 so that ,v#a q ,v#j1 and let ,v#b bear the same proportion to the whole pre-existing volume3 then ,v#b q7,v#j ,v#a7 q7,v#j q ,v#j7 q ,v#j7a q7 and ,v#b_/,v#a #a q4 #eih ,this ratio of #a_/7a q7 is easily shewn to be the constant ratio running through the whole series1 from chamber to chamber2 and if this ratio of volumes be constant1 so also are the ratios of corresponding surfaces1 and of corresponding linear dimensions1 provided always that the successive increments1 or successive chambers1 are similar in form4 ,we have still to discuss the similarity of form and the symmetry of position which characterise the successive chambers1 and which1 together with the law of continued proportionality of size1 are the distinctive characters and the indispensable conditions of a series of 8gnomons40 ,the minute size of the foraminiferal shell or at least of each successive increment thereof1 taken in connection with the fluid or semi-fluid nature of the protoplasmic substance1 is enough to suggest that the molecular forces1 and especially the force of surface-tension1 must exercise a controlling influence over the form of the whole structure2 and this suggestion1 or belief1 is already implied in our statement that each successive increment of growing protoplasm constitutes a separate drop4 ,these 8drops10 partially concealed by their successors1 but still shewing in part their rounded outlines1 are easily recognisable in the various foraminiferal shells which are illustrated in this chapter4 ,illustration3 ,fig4 #cad4 ,orbulina universa1 d',orb4 ,the accompanying figure represents1 to begin with1 the spherical shell characteristic of the common1 floating1 oceanic ,orbulina4 ,in the specimen illustrated1 a second chamber1 superadded to the #eii first1 has arisen as a drop of protoplasm which exuded through the pores of the first chamber1 accumulated on its surface1 and spread over the latter till it came to rest in a position of equilibrium4 ,we may take it that this position of equilibrium is determined1 at least in the first instance1 by the 8law of the constant angle10 which holds1 or tends to hold1 in all cases where the free surface of a given liquid is in contact with a given solid1 in presence of another liquid or a gas4 ,the corresponding equations are precisely the same as those which we have used in discussing the form of a drop 7on p4 #bid72 though some slight modification must be made in our definitions1 inasmuch as the consideration of surface-tension is no longer appropriate at the solid surfaces1 and the concept of surface-energy must take its place4 ,be that as it may1 it is enough for us to observe that1 in such a case as ours1 when a given fluid 7namely protoplasm7 is in surface contact with a solid 7viz4 a calcareous shell71 in presence of another fluid 7sea-water71 then the angle of contact1 or angle by which the common surface 7or interface7 of the two liquids abuts against the solid wall1 tends to be constant3 and that being so1 the drop will have a certain definite form1 depending 7inter alia7 on the form of the surface with which it is in contact4 ,after a period of rest1 during which the surface of our second drop becomes rigid by calcification1 a new period of growth will recur and a new drop of protoplasm be accumulated4 ,circumstances remaining the same1 this new drop will meet the solid surface of the shell at the same angle as did the former one2 and1 the other forces at work on the system remaining the same1 the form of the whole drop1 or chamber1 will be the same as before4 ,according to ,rhumbler1 this 8law of the constant angle0 is the fundamental principle in the mechanical conformation of the foraminiferal shell1 and provides for the symmetry of form as well as of position in each succeeding drop of protoplasm3 which form and position1 once acquired1 become rigid and fixed with the onset of calcification4 ,but ,rhumbler's explanation brings with it its own difficulties4 ,it is by no means easy of verification1 for on the very complicated curved surfaces of the shell it seems to me extraordinarily difficult to measure1 or even to recognise1 the actual angle of contact3 of which angle of contact1 by the way1 #fjj but little is known1 save only in the particular case where one of the three bodies is air1 as when a surface of water is exposed to air and in contact with glass4 ,it is easy moreover to see that in many of our ,foraminifera the angle of contact1 though it may be constant in homologous positions from chamber to chamber1 is by no means constant at all points along the boundary of each chamber4 ,in ,cristellaria1 for instance 7,fig4 #cae71 it would seem to be 7and ,rhumbler asserts that it actually is7 about #ij on the outer side and only about #ej on the inner side of each septal partition2 in ,pulvinulina 7,fig4 #bei71 according to ,rhumbler1 the angles adjacent to the mouth are of #ij1 and the opposite angles are of #fj1 in each chamber4 ,for these and other similar discrepancies ,rhumbler would account by simply invoking the heterogeneity of the protoplasmic drop3 that is to say1 by assuming that the protoplasm has a different composition and different properties 7including a very different distribution of surface-energy71 at points near to and remote from the mouth of the shell4 ,whether the differences in angle of contact be as great as ,rhumbler takes them to be1 whether marked heterogeneities of the protoplasm occur1 and whether these be enough to account for the differences of angle1 ,i cannot tell4 ,but it seems to me that we had better rest content with a general statement1 and that ,rhumbler has taken too precise and narrow a view4 ,illustration3 ,fig4 #cae4 ,cristellaria reniformis1 d',orb4 #fja ,in the molecular growth of a crystal1 although we must of necessity assume that each molecule settles down in a position of minimum potential energy1 we find it very hard indeed to explain precisely1 even in simple cases and after all the labours of modern crystallographers1 why this or that position is actually a place of minimum potential4 ,in the case of our little ,foraminifer 7just as in the case of the crystal71 let us then be content to assert that each drop or bead of protoplasm takes up a position of minimum potential energy1 in relation to all the circumstances of the case2 and let us not attempt1 in the present state of our knowledge1 to define that position of minimum potential by reference to angle of contact or any other particular condition of equilibrium4 ,in most cases the whole exposed surface1 on some portion of which the drop must come to rest1 is an extremely complicated one1 and the forces involved constitute a system which1 in its entirety1 is more complicated still2 but from the symmetry of the case and the continuity of the whole phenomenon1 we are entitled to believe that the conditions are just the same1 or very nearly the same1 time after time1 from one chamber to another3 as the one chamber is conformed so will the next tend to be1 and as the one is situated relatively to the system so will its successor tend to be situated in turn4 ,the physical law of minimum potential 7including also the law of minimal area7 is all that we need in order to explain1 in general terms1 the continued similarity of one chamber to another2 and the physiological law of growth1 by which a continued proportionality of size tends to run through the series of successive chambers1 impresses upon this series of similar increments the form of a logarithmic spiral4 ,in each particular case the nature of the logarithmic spiral1 as defined by its constant angle1 will be chiefly determined by the rate of growth2 that is to say by the particular ratio in which each new chamber exceeds its predecessor in magnitude4 ,but shells having the same constant angle 77 may still differ from one another in many ways--in the general form and relative position of the chambers1 in their extent of overlap1 and hence in the actual contour and appearance of the shell2 and these variations must correspond to particular distributions of energy within the system1 which is governed as a whole by the law of minimum potential4 #fjb ,our problem1 then1 becomes reduced to that of investigating the possible configurations which may be derived from the successive symmetrical apposition of similar bodies whose magnitudes are in continued proportion2 and it is obvious1 mathematically speaking1 that the various possible arrangements all come under the head of the logarithmic spiral1 together with the limiting cases which it includes4 ,since the difference between one such form and another depends upon the numerical value of certain coefficients of magnitude1 it is plain that any one must tend to pass into any other by small and continuous gradations2 in other words1 that a classification of these forms must 7like any classification whatsoever of logarithmic spirals or of any other mathematical curves71 be theoretic or 8artificial40 ,but we may easily make such an artificial classification1 and shall probably find it to agree1 more or less1 with the usual methods of classification recognised by biological students of the ,foraminifera4 ,firstly we have the typically spiral shells1 which occur in great variety1 and which 7for our present purpose7 we need hardly describe further4 ,we may merely notice how in certain cases1 for instance ,globigerina1 the individual chambers are little removed from spheres2 in other words1 the area of contact between the adjacent chambers is small4 ,in such forms as ,cyclammina and ,pulvinulina1 on the other hand1 each chamber is greatly overlapped by its successor1 and the spherical form of each is lost in a marked asymmetry4 ,furthermore1 in ,globigerina and some others we have a tendency to the development of a helicoid spiral in space1 as in so many of our univalve molluscan shells4 ,the mathematical problem of how a shell should grow1 under the assumptions which we have made1 would probably find its most general statement in such a case as that of ,globigerina1 where the whole organism lives and grows freely poised in a medium whose density is little different from its own4 ,the majority of spiral forms1 on the other hand1 are plane or discoid spirals1 and we may take it that in these cases some force has exercised a controlling influence1 so as to keep all the chambers in a plane4 ,this is especially the case in forms like ,rotalia or ,discorbina 7,fig4 #caf71 where the organism lives attached to a rock or a frond of sea-weed2 for here 7just as in the case of #fjc the coiled tubes which little worms such as ,serpula and ,spirorbis make1 under similar conditions7 the spiral disc is itself asymmetrical1 its whorls being markedly flattened on their attached surfaces4 ,illustration3 ,fig4 #caf4 ,discorbina bertheloti1 d',orb4 ,we may also conceive1 among other conditions1 the very curious case in which the protoplasm may entirely overspread the surface of the shell without reaching a position of equilibrium2 in which case a new shell will be formed enclosing the old one1 #fjd whether the old one be in the form of a single1 solitary chamber1 or have already attained to the form of a chambered or spiral shell4 ,this is precisely what often happens in the case of ,orbulina1 when within the spherical shell we find a small1 but perfectly formed1 spiral 8,globigerina#edi40 ,the various ,miliolidae 7,fig4 #caa71 only differ from the typical spiral1 or rotaline forms1 in the large angle subtended by each chamber1 and the consequent abruptness of their inclination to each other4 ,in these cases the outward appearance of a spiral tends to be lost2 and it behoves us to recollect1 all the more1 that our spiral curve is not necessarily identical with the outline of the shell1 but is always a line drawn through corresponding points in the successive chambers of the latter4 ,illustration3 ,fig4 #cag4 ,a1 ,tertularia trochus1 d',orb4 ,b1 ,t4 concava1 ,karrer4 ,we reach a limiting case of the logarithmic spiral when the chambers are arranged in a straight line2 and the eye will tend to associate with this limiting case the much more numerous forms in which the spiral angle is small1 and the shell only exhibits a gentle curve with no succession of enveloping whorls4 ,this constitutes the ,nodosarian type 7,fig4 #hg1 p4 #bfb72 and here again1 we must postulate some force which has tended to keep the chambers in a rectilinear series3 such for instance as gravity1 acting on a system of 8hanging drops40 #fje ,in ,textularia and its allies 7,fig4 #cag71 we have a precise parallel to the helicoid cyme of the botanists 7cf4 p4 #ejb73 that is to say we have a screw translation1 perpendicular to the plane of the underlying logarithmic spiral4 ,in other words1 in tracing a genetic spiral through the whole succession of chambers1 we do so by a continuous vector rotation1 through successive angles of #ahj 7or #abj in some cases71 while the pole moves along an axis perpendicular to the original plane of the spiral4 ,another type is furnished by the 8cyclic0 shells of the ,orbitolitidae1 where small and numerous chambers tend to be added on round and round the system1 so building up a circular flattened disc4 ,this again we perceive to be1 mathematically1 a limiting case of the logarithmic spiral1 where the spiral has become a circle and the constant angle is now an angle of #ij4 ,lastly there are a certain number of ,foraminifera in which1 without more ado1 we may simply say that the arrangement of the chambers is irregular1 neither the law of constant ratio of magnitude nor that of constant form being obeyed4 ,the chambers are heaped pell-mell upon one another1 and such forms are known to naturalists as the ,acervularidae4 ,while in these last we have an extreme lack of regularity1 we must not exaggerate the regularity or constancy which the more ordinary forms display4 ,we may think it hard to believe that the simple causes1 or simple laws1 which we have described should operate1 and operate again and again1 in millions of individuals to produce the same delicate and complex conformations4 ,but we are taking a good deal for granted if we assert that they do so1 and in particular we are assuming1 with very little proof1 the 8constancy of species0 in this group of animals4 ,just as ,verworn has shewn that the typical ,amoeba proteus1 when a trace of alkali is added to the water in which it lives1 tends1 by alteration of surface tensions1 to protrude the more delicate pseudopodia characteristic of ,a4 radiosa1--and again when the water is rendered a little more alkaline1 to turn apparently into the so-called ,a4 limax1--so it is evident that a very slight modification in the surface-energies concerned1 might tend to turn one so-called species into another among the ,foraminifera4 ,to what extent this process actually occurs1 we do not know4 #fjf ,but that this1 or something of the kind1 does actually occur we can scarcely doubt4 ,for example in the genus ,peneroplis1 the first portion of the shell consists of a series of chambers arranged in a spiral or nautiloid series2 but as age advances the spiral is apt to be modified in various ways#eej4 ,sometimes the successive chambers grow rapidly broader1 the whole shell becoming fan-shaped4 ,sometimes the chambers become narrower1 till they no longer enfold the earlier chambers but only come in contact each with its immediate predecessor3 the result being that the shell straightens out1 and 7taking into account the earlier spiral portion7 may be described as crozier-shaped4 ,between these extremes of shape1 and in regard to other variations of thickness or thinness1 roughness or smoothness1 and so on1 there are innumerable gradations passing one into another and intermixed without regard to geographical distribution3--8wherever ,peneroplides abound this wide variation exists1 and nothing can be more easy than to pick out a number of striking specimens and give to each a distinctive name1 but in no other way can they be divided into #8species4'#eea0 ,some writers have wondered at the peculiar variability of this particular shell#eeb2 but for all we know of the life-history of the ,foraminifera1 it may well be that a great number of the other forms which we distinguish as separate species and even genera are no more than temporary manifestations of the same variability#eec4 #fjg ,conclusion4 ,if we can comprehend and interpret on some such lines as these the form and mode of growth of the foraminiferal shell1 we may also begin to understand two striking features of the group1 namely1 on the one hand the large number of diverse types or families which exist and the large number of species and varieties within each1 and on the other the persistence of forms which in many cases seem to have undergone little change or none at all from the ,cretaceous or even from earlier periods to the present day4 ,in few other groups1 perhaps only among the ,radiolaria1 do we seem to possess so nearly complete a picture of all possible transitions between form and form1 and of the whole branching system of the evolutionary tree3 as though little or nothing of it had ever perished1 and the whole web of life1 past and present1 were as complete as ever4 ,it leads one to imagine that these shells have grown according to laws so simple1 so much in harmony with their material1 with their environment1 and with all the forces internal and external to which they are exposed1 that none is better than another and none fitter or less fit to survive4 ,it invites one also to contemplate the possibility of the lines of possible variation being here so narrow and determinate that identical forms may have come independently into being again and again4 ,while we can trace in the most complete and beautiful manner the passage of one form into another among these little shells1 and ascribe them all at last 7if we please7 to a series which starts with the simple sphere of ,orbulina or with the amoeboid body of ,astrorhiza1 the question stares us in the face whether this be an 8evolution0 which we have any right to correlate with historic time4 ,the mathematician can trace one conic section into another1 and 8evolve0 for example1 through innumerable graded ellipses1 the circle from the straight line3 which tracing of continuous steps is a true 8evolution10 though time has no part therein4 ,it was after this fashion that ,hegel1 and for that matter ,aristotle himself1 was an evolutionist--to whom evolution was #fjh a mental concept1 involving order and continuity in thought1 but not an actual sequence of events in time4 ,such a conception of evolution is not easy for the modern biologist to grasp1 and harder still to appreciate4 ,and so it is that even those who1 like ,dreyer#eed and like ,rhumbler1 study the foraminiferal shell as a physical system1 who recognise that its whole plan and mode of growth is closely akin to the phenomena exhibited by fluid drops under particular conditions1 and who explain the conformation of the shell by help of the same physical principles and mathematical laws--yet all the while abate no jot or tittle of the ordinary postulates of modern biology1 nor doubt the validity and universal applicability of the concepts of ,darwinian evolution4 ,for these writers the biogenetisches ,grundgesetz remains impregnable4 ,the ,foraminifera remain for them a great family tree1 whose actual pedigree is traceable to the remotest ages2 in which historical evolution has coincided with progressive change2 and in which structural fitness for a particular function 7or functions7 has exercised its selective action and ensured 8the survival of the fittest40 ,by successive stages of historic evolution we are supposed to pass from the irregular ,astrorhiza to a ,rhabdammina with its more concentrated disc2 to the forms of the same genus which consist of but a single tube with central chamber2 to those where this chamber is more and more distinctly segmented2 so to the typical many-chambered ,nodosariae2 and from these1 by another definite advance and later evolution to the spiral ,trochamminae4 ,after this fashion1 throughout the whole varied series of the ,foraminifera1 ,dreyer and ,rhumbler 7following ,neumayr7 recognise so many successions of related forms1 one passing into another1 and standing towards it in a definite relationship of ancestry or descent4 ,each evolution of form1 from simpler to more complex1 is deemed to have been attended by an advantage to the organism1 an enhancement of its chances of survival or perpetuation2 hence the historically older forms are1 on the whole1 structurally the simpler2 or conversely the simpler forms1 such as the simple sphere1 were the first to come into being in primeval seas2 and finally1 the gradual development and increasing #fji complication of the individual within its own lifetime is held to be at least a partial recapitulation of the unknown history of its race and dynasty#eee4 ,we encounter many difficulties when we try to extend such concepts as these to the ,foraminifera4 ,we are led for instance to assert1 as ,rhumbler does1 that the increasing complexity of the shell1 and of the manner in which one chamber is fitted on another1 makes for advantage2 and the particular advantage on which ,rhumbler rests his argument is strength4 ,increase of strength1 die ,festigkeitssteigerung1 is according to him the guiding principle in foraminiferal evolution1 and marks the historic stages of their development in geologic time4 ,but in days gone by ,i used to see the beach of a little ,connemara bay bestrewn with millions upon millions of foraminiferal shells1 simple ,lagenae1 less simple ,nodosariae1 more complex ,rotaliae3 all drifted by wave and gentle current from their sea-cradle to their sandy grave3 all lying bleached and dead3 one more delicate than another1 but all 7or vast multitudes of them7 perfect and unbroken4 ,and so ,i am not inclined to believe that niceties of form affect the case very much3 nor in general that foraminiferal life involves a struggle for existence wherein breakage is a constant danger to be averted1 and increased strength an advantage to be ensured#eef4 ,in the course of the same argument ,rhumbler remarks that ,foraminifera are absent from the coarse sands and gravels#eeg1 as ,williamson indeed had observed many years ago3 so averting1 or #faj at least escaping1 the dangers of concussion4 ,but this is after all a very simple matter of mechanical analysis4 ,the coarseness or fineness of the sediment on the sea-bottom is a measure of the current3 where the current is strong the larger stones are washed clean1 where there is perfect stillness the finest mud settles down2 and the light1 fragile shells of the ,foraminifera find their appropriate place1 like every other graded sediment1 in this spontaneous order of lixiviation4 ,the theorem of ,organic ,evolution is one thing2 the problem of deciphering the lines of evolution1 the order of phylogeny1 the degrees of relationship and consanguinity1 is quite another4 ,among the higher organisms we arrive at conclusions regarding these things by weighing much circumstantial evidence1 by dealing with the resultant of many variations1 and by considering the probability or improbability of many coincidences of cause and effect2 but even then our conclusions are at best uncertain1 our judgments are continually open to revision and subject to appeal1 and all the proof and confirmation we can ever have is that which comes from the direct1 but fragmentary evidence of palaeontology#eeh4 ,but in so far as forms can be shewn to depend on the play of physical forces1 and the variations of form to be directly due to simple quantitative variations in these1 just so far are we thrown back on our guard before the biological conception of consanguinity1 and compelled to revise the vague canons which connect classification with phylogeny4 ,the physicist explains in terms of the properties of matter1 and classifies according to a mathematical analysis1 all the drops and forms of drops and associations of drops1 all the kinds of froth and foam1 which he may discover among inanimate things2 and his task ends there4 ,but when such forms1 such conformations and configurations1 occur among living things1 then at once the biologist introduces his concepts of heredity1 of historical evolution1 of succession in time1 of recapitulation of remote ancestry in individual growth1 of common origin 7unless contradicted by direct evidence7 of similar forms remotely separated by geographic space or geologic time1 of fitness for a function1 of #faa adaptation to an environment1 of higher and lower1 of 8better0 and 8worse40 ,this is the fundamental difference between the 8explanations0 of the physicist and those of the biologist4 ,in the order of physical and mathematical complexity there is no question of the sequence of historic time4 ,the forces that bring about the sphere1 the cylinder or the ellipsoid are the same yesterday and to-morrow4 ,a snow-crystal is the same to-day as when the first snows fell4 ,the physical forces which mould the forms of ,orbulina1 of ,astrorhiza1 of ,lagena or of ,nodosaria to-day were still the same1 and for aught we have reason to believe the physical conditions under which they worked were not appreciably different1 in that yesterday which we call the ,cretaceous epoch2 or1 for aught we know1 throughout all that duration of time which is marked1 but not measured1 by the geological record4 ,in a word1 the minuteness of our organism brings its conformation as a whole within the range of the molecular forces2 the laws of its growth and form appear to lie on simple lines2 what ,bergson calls#eei the 8ideal kinship0 is plain and certain1 but the 8material affiliation0 is problematic and obscure2 and1 in the end and upshot1 it seems to me by no means certain that the biologist's usual mode of reasoning is appropriate to the case1 or that the concept of continuous historical evolution must necessarily1 or may safely and legitimately1 be employed4 #fab ,,chapter ,,xiii ,,the ,,shapes ,,of ,,horns1 ,,and ,,of ,,teeth ,,or ,,tusks3 ,,with ,a ,,note ,,on ,,torsion ,we have had so much to say on the subject of shell-spirals that we must deal briefly with the analogous problems which are presented by the horns of sheep1 goats1 antelopes and other horned quadrupeds2 and all the more1 because these horn-spirals are on the whole less symmetrical1 less easy of measurement than those of the shell1 and in other ways also are less easy of investigation4 ,let us dispense altogether in this case with mathematics2 and be content with a very simple account of the configuration of a horn4 ,there are three types of horn which deserve separate consideration3 firstly1 the horn of the rhinoceros2 secondly the horns of the sheep1 the goat1 the ox or the antelope1 that is to say1 of the so-called hollow-horned ruminants2 and thirdly1 the solid bony horns1 or 8antlers10 which are characteristic of the deer4 ,the horn of the rhinoceros presents no difficulty4 ,it is physiologically equivalent to a mass of consolidated hairs1 and1 like ordinary hair1 it consists of non-living or 8formed0 material1 continually added to by the living tissues at its base4 ,in section1 that is to say in the form of its 8generating curve10 the horn is approximately elliptical1 with the long axis fore-and-aft1 or1 in some species1 nearly circular4 ,its longitudinal growth proceeds with a maximum velocity anteriorly1 and a minimum posteriorly2 and the ratio of these velocities being constant1 the horn curves into the form of a logarithmic spiral in the manner that we have already studied4 ,the spiral is of small angle1 but in the longer-horned species1 such as the great white rhinoceros 7,ceratorhinus71 the spiral form is distinctly to be recognised4 ,as the horn #fac occupies a median position on the head1--a position1 that is to say1 of symmetry in respect to the field of force on either side1--there is no tendency towards a lateral twist1 and the horn accordingly develops as a plane logarithmic spiral4 ,when two horns coexist1 the hinder one is much the smaller of the two3 which is as much as to say that the force1 or rate1 of growth diminishes as we pass backwards1 just as it does within the limits of the single horn4 ,and accordingly1 while both horns have essentially the same shape1 the spiral curvature is less manifest in the second one1 simply by reason of its comparative shortness4 ,the paired horns of the ordinary hollow-horned ruminants1 such as the sheep or the goat1 grow under conditions which are in some respects similar1 but which differ in other and important respects from the conditions under which the horn grows in the rhinoceros4 ,as regards its structure1 the entire horn now consists of a bony core with a covering of skin2 the inner1 or dermal1 layer of the latter is richly supplied with nutrient blood-vessels1 while the outer layer1 or epidermis1 develops the fibrous or chitinous material1 chemically and morphologically akin to a mass of cemented or consolidated hairs1 which constitutes the 8sheath0 of the horn4 ,a zone of active growth at the base of the horn keeps adding to this sheath1 ring by ring1 and the specific form of this annular zone is1 accordingly1 that of the 8generating curve0 of the horn4 ,each horn no longer lies1 as it does in the rhinoceros1 in the plane of symmetry of the animal of which it forms a part2 and the limited field of force concerned in the genesis and growth of the horn is bound1 accordingly1 to be more or less laterally asymmetrical4 ,but the two horns are in symmetry one with another2 they form 8conjugate0 spirals1 one being the 8mirror-image0 of the other4 ,just as in the hairy coat of the animal each hair1 on either side of the median 8parting10 tends to have a certain definite direction of its own axis1 inclined away from the median axial plane of the whole system1 so is it both with the bony core of the horn and with the consolidated mass of hairs or hair-like substance which constitutes its sheath2 the primary axis of the horn is more or less inclined to1 and may even be nearly perpendicular to1 the axial plane of the animal4 ,the growth of the horny sheath is not continuous1 but more or #fad less definitely periodic3 sometimes1 as in the sheep1 this periodicity is particularly well-marked1 and causes the horny sheath to be composed of a series of all but separate rings1 which are supposed to be formed year by year1 and so to record the age of the animal#efj4 ,just as we sought for the true generating curve in the orifice1 or 8lip10 of the molluscan shell1 so we might be apt to assume that in the spiral horn the generating curve corresponded to the lip or margin of one of the horny rings or annuli4 ,this annular margin1 or boundary of the ring1 is usually a sinuous curve1 not lying in a plane1 but such as would form the boundary of an anticlastic surface of great complexity3 to the meaning and origin of which phenomenon we shall return presently4 ,but1 as we have already seen in the case of the molluscan shell1 the complexities of the lip itself1 or of the corresponding lines of growth upon the shell1 need not concern us in our study of the development of the spiral3 inasmuch as we may substitute for these actual boundary lines1 their 8trace10 or projection on a plane perpendicular to the axis--in other words the simple outline of a transverse section of the whorl4 ,in the horn1 this transverse section is often circular or nearly so1 as in the oxen and many antelopes3 it now and then becomes of somewhat complicated polygonal outline1 as in a highland ram2 but in many antelopes1 and in most of the sheep1 the outline is that of an isosceles1 or sometimes nearly equilateral triangle1 a form which is typically displayed1 for instance1 in ,ovis ,ammon4 ,the horn in this latter case is a trihedral prism1 whose three faces are1 7#a7 an upper1 or frontal face1 in continuation of the plane of the frontal bone2 7#b7 an outer1 or orbital1 starting from the upper margin of the orbit2 and 7#c7 an inner1 or 8nuchal10 abutting on the parietal bone#efa4 ,along these three faces1 and their corresponding angles or edges1 we can trace in the fibrous substance of the horn a series of homologous spirals1 such as we #fae have called in a preceding chapter the 8ensemble of generating spirals0 which constitute the surface4 ,illustration3 ,fig4 #cah4 ,diagram of ,ram's horns4 7,after ,sir ,vincent ,brooke1 from ,p4,z4,s47 a1 frontal2 b1 orbital2 c1 nuchal surface4 ,in some few cases1 of which the male musk ox is one of the most notable1 the horn is not developed in a continuous spiral curve4 ,it changes its shape as growth proceeds2 and this1 as we have seen1 is enough to show that it does not constitute a logarithmic spiral4 ,the reason is that the bony exostoses1 or horn-cores1 about which the horny sheath is shaped and moulded1 neither grow continuously nor even remain of constant size after attaining their full growth4 ,but as the horns grow heavy the bony core is bent downwards by their weight1 and so guides the growth of the horn in a new direction4 ,moreover as age advances1 the horn-core is further weakened and to a great extent absorbed3 and the horny sheath or horn proper1 deprived of its support1 continues to grow1 but in a flattened curve very different from its original spiral#efb4 ,the chamois is a somewhat analogous case4 ,here the terminal1 or oldest1 part of the horn is curved2 it tends to assume a spiral form1 though from its comparative shortness it seems merely to be bent into a hook4 ,but later on1 the bony core within1 as it grows and strengthens1 stiffens the horn1 and guides it into a straighter course or form4 ,the same phenomenon #faf of change of curvature1 manifesting itself at the time when1 or the place where1 the horn is freed from the support of the internal core1 is seen in a good many other antelopes 7such as the hartebeest7 and in many buffaloes2 and the cases where it is most manifest appear to be those where the bony core is relatively short1 or relatively weak4 ,illustration3 ,fig4 #cai4 ,head of ,arabian ,wild ,goat1 ,capra sinaitica4 7,after ,sclater1 from ,p4,z4,s47 ,but in the great majority of horns1 we have no difficulty in recognising a continuous logarithmic spiral1 nor in referring it1 as before1 to an unequal rate of growth 7parallel to the axis7 on two opposite sides of the horn1 the inequality maintaining a constant ratio as long as growth proceeds4 ,in certain antelopes1 such as the gemsbok1 the spiral angle is very small1 or in other words the horn is very nearly straight2 in other species of the same genus ,oryx1 such as the ,beisa antelope and the ,leucoryx1 a gentle #fag curve 7not unlike though generally less than that of a ,dentalium shell7 is evident2 and the spiral angle1 according to the few measurements ,i have made1 is found to measure from about #bj to nearly #dj4 ,in some of the large wild goats1 such as the ,scinde wild goat1 we have a beautiful logarithmic spiral1 with a constant angle of rather less than #gj2 and we may easily arrange a series of forms1 such for example as the ,siberian ibex1 the moufflon1 ,ovis ,ammon1 etc41 and ending with the long-horned ,highland ram3 in which1 as we pass from one to another1 we recognise precisely homologous spirals1 with an increasing angular constant1 the spiral angle being1 for instance1 about #ge or rather less in ,ovis ,ammon1 and in the ,highland ram a very little more4 ,we have already seen that in the neighbourhood of #gj or #hj a small change of angle makes a marked difference in the appearance of the spire2 and we know also that the actual length of the horn makes a very striking difference1 for the spiral becomes especially conspicuous to the eye when a horn or shell is long enough to shew several whorls1 or at least a considerable part of one entire whorl4 ,even in the simplest cases1 such as the wild goats1 the spiral is never 7strictly speaking7 a plane or discoid spiral3 but in greater or less degree there is always superposed upon the plane logarithmic spiral a helical spiral in space4 ,sometimes the latter is scarcely apparent1 for the helical curvature is comparatively small1 and the horn 7though long1 as in the said wild goats7 is not nearly long enough to shew a complete convolution3 at other times1 as in the ram1 and still better in many antelopes1 such as the koodoo1 the helicoid or corkscrew curve of the horn is its most characteristic feature4 ,accordingly we may study1 as in the molluscan shell1 the helicoid component of the spire--in other words the variation in what we have called 7on p4 #eee7 the angle 4 ,this factor it is which1 more than the constant angle of the logarithmic spiral1 imparts a characteristic appearance to the various species of sheep1 for instance to the various closely allied species of ,asiatic wild sheep1 or ,argali4 ,in all of these the constant angle of the logarithmic spiral is very much the same1 but the shearing component differs greatly4 ,and thus the long drawn out horns of #fah ,ovis ,poli1 four feet or more from tip to tip1 differ conspicuously from those of ,ovis ,ammon or ,o4 hodgsoni1 in which a very similar logarithmic spiral is wound 7as it were7 round a much blunter cone4 ,the ram's horn then1 like the snail's shell1 is a curve of double curvature1 in which one component has imposed upon the structure a plane logarithmic spiral1 and the other has produced a continuous displacement1 or 8shear10 proportionate in magnitude to1 and perpendicular or otherwise inclined in direction to1 the axis of the former spiral curvature4 ,the result is precisely analogous to that which we have studied in the snail and other spiral univalves2 but while the form1 and therefore the resultant forces1 are similar1 the original distribution of force is not the same3 for we have not here1 as we had in the snail-shell1 a 8columellar0 muscle1 to introduce the component acting in the direction of the axis4 ,we have1 it is true1 the central bony core1 which in part performs an analogous function2 but the main phenomenon here is apparently a complex distribution of rates of growth1 perpendicular to the plane of the generating curve4 ,let us continue to dispense with mathematics1 for the mathematical treatment of a curve of double curvature is never very simple1 and let us deal with the matter by experiment4 ,we have seen that the generating curve1 or transverse section1 of a typical ram's horn is triangular in form4 ,measuring 7along the curve of the horn7 the length of the three edges of the trihedral structure in a specimen of ,ovis ,ammon1 and calling them respectively the outer1 inner1 and hinder edges 7from their position at the base of the horn1 relatively to the skull71 ,i find the outer edge to measure #hj cm41 the inner #gd cm41 and the posterior #de cm42 let us say that1 roughly1 they are in the ratio of #i 3 #h 3 #e4 ,then1 if we make a number of little cardboard triangles1 equip each with three little legs 7,i make them of cork71 whose relative lengths are as #i 3 #h 3 #e1 and pile them up and stick them all together1 we straightway build up a curve of double curvature precisely analogous to the ram's horn3 except only that1 in this first approximation1 we have not allowed for the gradual increment 7or decrement7 of the triangular surfaces1 that is to say1 for the tapering of the horn due to the growth in its own plane of the generating curve4 #fai ,in this case then1 and in most other trihedral or three-sided horns1 one of the three components1 or three unequal velocities of growth1 is of relatively small magnitude1 but the other two are nearly equal one to the other4 ,it would involve but little change for these latter to become precisely equal2 and again but little to turn the balance of inequality the other way4 ,but the immediate consequence of this altered ratio of growth would be that the horn would appear to wind the other way1 as it does in the antelopes1 and also in certain goats1 e4g4 the markhor1 ,capra falconeri4 ,for these two opposite directions of twist ,dr ,wherry has introduced a convenient nomenclature4 ,when the horn winds so that we follow it from base to apex in the direction of the hands of a watch1 it is customary to call it a 8left-handed0 spiral4 ,such a spiral we have in the horn on the left-hand side of a ram's head4 ,accordingly1 ,dr ,wherry calls the condition homonymous1 where1 as in the sheep1 a right-handed spiral is on the right side of the head1 and a left-handed spiral on the left side2 while he calls the opposite condition heteronymous1 as we have it in the antelopes1 where the right-handed twist is on the left side of the head1 and the left-handed twist on the right-hand side4 ,among the goats1 we may have either condition4 ,thus the domestic and most of the wild goats agree with the sheep2 but in the markhor the twisted horns are heteronymous1 as in the antelopes4 ,the difference1 as we have seen1 is easily explained2 and 7very much as in the case of our opposite spirals in the apple-snail1 referred to on p4 #efj71 it has no very deep importance4 ,summarised then1 in a very few words1 the argument by which we account for the spiral conformation of the horn is as follows3 ,the horn elongates by dint of continual growth within a narrow zone1 or annulus1 at its base4 ,if the rate of growth be identical on all sides of this zone1 the horn will grow straight2 if it be greater on one side than on the other1 the horn will become curved3 and it probably will be greater on one side than on the other1 because each single horn occupies an unsymmetrical field with reference to the plane of symmetry of the animal4 ,if the maximal and minimal velocities of growth be precisely at opposite sides of the zone of growth1 the resultant spiral will be a plane spiral2 but if they be not precisely or diametrically opposite1 then the spiral will be a spiral in space1 with a winding or helical component2 and it is by no means likely that the maximum and minimum will occur at precisely opposite ends of a diameter1 for #fbj no such plane of symmetry is manifested in the field of force to which the growing annulus corresponds or appertains4 ,now we must carefully remember that the rates of growth of which we are here speaking are the net rates of longitudinal increment1 in which increment the activity of the living cells in the zone of growth at the base of the horn is only one 7though it is the fundamental7 factor4 ,in other words1 if the horny sheath were continually being added to with equal rapidity all round its zone of active growth1 but at the same time had its elongation more retarded on one side than the other 7prior to its complete solidification7 by varying degrees of adhesion or membranous attachment to the bone core within1 then the net result would be a spiral curve precisely such as would have arisen from initial inequalities in the rate of growth itself4 ,it seems highly probable that this is a very important factor1 and sometimes even the chief factor in the case4 ,the same phenomenon of attachment to the bony core1 and the consequent friction or retardation with which the sheath slides over its surface1 will lead to various subsidiary phenomena3 among others to the presence of transverse folds or corrugations upon the horn1 and to their unequal distribution upon its several faces or edges4 ,and while it is perfectly true that nearly all the characters of the horn can be accounted for by unequal velocities of longitudinal growth upon its different sides1 it is also plain that the actual field of force is a very complicated one indeed4 ,for example1 we can easily see that 7at least in the great majority of cases7 the direction of growth of the horny fibres of the sheath is by no means parallel to the axis of the core within2 accordingly these fibres will tend to wind in a system of helicoid curves around the core1 and not only this helicoid twist but any other tendency to spiral curvature on the part of the sheath will tend to be opposed or modified by the resistance of the core within4 ,but on the other hand living bone is a very plastic structure1 and yields easily though slowly to any forces tending to its deformation2 and so1 to a considerable extent1 the bony core itself will tend to be modelled by the curvature which the growing sheath assumes1 and the final result will be determined by an equilibrium between these two systems of forces4 #fba ,while it is not very safe1 perhaps1 to lay down any general rule as to what horns are more1 and what are less spirally curved1 ,i think it may be said that1 on the whole1 the thicker the horn1 the greater is its spiral curvature4 ,it is the slender horns1 of such forms as the ,beisa antelope1 which are gently curved1 and it is the robust horns of goats or of sheep in which the curvature is more pronounced4 ,other things being the same1 this is what we should expect to find2 for it is where the transverse section of the horn is large that we may expect to find the more marked differences in the intensity of the field of force1 whether of active growth or of retardation1 on opposite sides or in different sectors thereof4 ,illustration3 ,fig4 #cbj4 ,head of ,ovis ,ammon1 shewing ,st ,venant's curves4 ,but there is yet another and a very remarkable phenomenon which we may discern in the growth of a horn1 when it takes the form of a curve of double curvature1 namely1 an effect of torsional strain2 and this it is which gives rise to the sinuous 8lines of growth10 or sinuous boundaries of the separate horny rings1 of which we have already spoken4 ,it is not at first sight obvious that a mechanical strain of torsion is necessarily involved in the growth of the horn4 ,in our experimental illustration 7p4 #fah71 we built up a twisted coil of separate elements1 and no torsional strain attended the development of the system4 ,so would it be if the horny sheath grew by successive annular increments1 free save for their relation to one another1 and having no attachment to the solid core within4 ,but as a matter of fact there is #fbb such an attachment1 by subcutaneous connective tissue1 to the bony core2 and accordingly a torsional strain will be set up in the growing horny sheath1 again provided that the forces of growth therein be directed more or less obliquely to the axis of the core2 for a 8couple0 is thus introduced1 giving rise to a strain which the sheath would not experience were it free 7so to speak7 to slip along1 impelled only by the pressure of its own growth from below4 ,and furthermore1 the successive small increments of the growing horn 7that is to say1 of the horny sheath7 are not instantaneously converted from living to solid and rigid substance2 but there is an intermediate stage1 probably long-continued1 during which the new-formed horny substance in the neighbourhood of the zone of active growth is still plastic and capable of deformation4 ,now we know1 from the celebrated experiments of ,st ,venant#efc1 that in the torsion of an elastic body1 other than a cylinder of circular section1 a very remarkable state of strain is introduced4 ,if the body be thus cylindrical 7whether solid or hollow71 then a twist leaves each circular section unchanged1 in dimensions and in figure4 ,but in all other cases1 such as an elliptic rod or a prism of any particular sectional form1 forces are introduced which act parallel to the axis of the structure1 and which warp each section into a complex anticlastic surface4 ,thus in the case of a triangular and equilateral prism1 such as is shewn in section in ,fig4 #cba1 if the part of the rod represented in the section be twisted by a force acting in the direction of the arrow1 then the originally plane section will be warped as indicated in the diagram3--where the full contour-lines represent elevation above1 and the dotted lines represent depression below1 the original level4 ,on the external surface of the prism1 then1 contour-lines which were originally parallel and horizontal1 will be found warped into sinuous curves1 such that1 on each of the three faces1 the curve will be convex upwards on one half1 and concave upwards on the other half of the face4 ,the ram's horn1 and still better that of ,ovis ,ammon1 is comparable to such a prism1 save that in section it is not quite equilateral1 and that its three faces are not plane4 ,the warping is therefore not precisely identical on the three faces #fbc of the horn2 but1 in the general distribution of the curves1 it is in complete accordance with theory4 ,similar anticlastic curves are well seen in many antelopes2 but they are conspicuous by their absence in the cylindrical horns of oxen4 ,illustration3 ,fig4 #cba4 ,illustration3 ,fig4 #cbb4 ,the better to illustrate this phenomenon1 the nature of which is indeed obvious enough from a superficial examination of the horn1 ,i made a plaster cast of one of the horny rings in a horn of ,ovis ,ammon1 so as to get an accurate pattern of its sinuous edge3 and then1 filling the mould up with wet clay1 ,i modelled an anticlastic surface1 such as to correspond as nearly as possible with the sinuous outline#efd4 ,finally1 after making a plaster cast of this sectional surface1 ,i drew its contour-lines 7as shewn in ,fig4 #cbb71 with the help of a simple form of spherometer4 ,it will be seen that in great part this diagram is precisely similar to ,st ,venant's diagram of the cross-section of a twisted triangular prism2 and this is especially the case in the neighbourhood of the sharp angle of our prismatic section4 ,that in parts the diagram is somewhat asymmetrical is not to be wondered at3 and 7apart from inaccuracies due to the somewhat rough means by which it was made7 this asymmetry can be sufficiently accounted for by anisotropy of the material1 by inequalities in thickness of different parts of the horny sheath1 and especially 7,i think7 by unequal distributions of rigidity due to the presence of the smaller corrugations of the #fbd horn4 ,it is apparently on account of these minor corrugations that1 in such horns as the ,highland ram's1 where they are strongly marked1 the main ,st ,venant effect is not nearly so well shewn as in the smoother horns such as those of ,o4 ,ammon and its immediate congeners#efe4 ,a further ,note upon ,torsion4 ,the phenomenon of torsion1 to which we have been thus introduced1 opens up many wide questions in connection with form4 ,some of the associated phenomena are admirably illustrated in the case of climbing plants2 but we can only deal with these still more briefly and parenthetically4 ,the subject of climbing plants has been elaborately dealt with not only in ,darwin's books#eff1 but also by a very large number of earlier and later writers4 ,in 8twining0 plants1 which constitute the greater number of 8climbers10 the essential phenomenon is a tendency of the growing shoot to revolve about a vertical axis--a tendency long ago discussed and investigated by such writers as ,palm1 ,h4 von ,mohl and ,dutrochet#efg4 ,this tendency to revolution--8circumvolution10 as ,darwin calls it1 8revolving nutation10 as ,sachs puts it--is very closely comparable to the process by which an antelope's horn 7such as the koodoo's7 grows into its spiral or rather helicoid form2 and it is simply due1 in like manner1 to inequalities in the rate of growth on different sides of the growing stem4 ,there is only this difference between the two cases1 that in the antelope's horn the zone of active growth is confined to the base of the horn1 while in the climbing stem the same phenomenon is at work throughout the whole length of the growing structure4 ,this growth is in the main due to 8turgescence10 that is to the extension1 or elongation1 of ready-formed cells through the imbibition of water2 it is a phenomenon due to osmotic pressure4 ,the particular stimuli to which these movements 7that is to say1 these inequalities of growth7 have been #fbe ascribed1 such as contact 7thigmotaxis71 exposure to light 7heliotropism71 and so forth1 need not be discussed here#efh4 ,a simple stem growing upright in the dark1 or in uniformly diffused light1 would be in a position of equilibrium to a field of force radially symmetrical about its vertical axis4 ,but this complete radial symmetry will not often occur2 and the radial anomalies may be such as arise intrinsically from structural peculiarities in the stem itself1 or externally to it by reason of unequal illumination or through various other localised forces4 ,the essential fact1 so far as we are concerned1 is that in twining plants we have a very marked tendency to inequalities in longitudinal growth on different aspects of the stem--a tendency which is but an exaggerated manifestation of one which is more or less present1 under certain conditions1 in all plants whatsoever4 ,just as in the case of the ruminants' horns so we find here1 that this inequality may be1 so to speak1 positive or negative1 the maximum lying to the one side or the other of the twining stem2 and so it comes to pass that some climbers twine to the one side and some to the other3 the hop and the honeysuckle following the sun1 and the field-convolvulus twining in the reverse direction2 there are also some1 like the woody nightshade 7,solanum ,dulcamara7 which twine indifferently either way4 ,together with this circumnutatory movement1 there is very generally to be seen an actual torsion of the twining stem--a twist1 that is to say1 about its own axis2 and ,mohl made the curious observation1 confirmed by ,darwin1 that when a stem twines around a smooth cylindrical stick the torsion does not take place1 save 8only in that degree which follows as a mechanical necessity from the spiral winding03 but that stems which had climbed around a rough stick were all more or less1 and generally much1 twisted4 ,here ,darwin did not refrain from introducing that teleological argument which pervades his whole train of reasoning3 8,the stem10 he says1 8probably gains rigidity by being twisted 7on the same principle that a much twisted rope #fbf is stiffer than a slackly twisted one71 and is thus indirectly benefited so as to be able to pass over inequalities in its spiral ascent1 and to carry its own weight when allowed to revolve freely40 ,the mechanical explanation would appear to be very simple1 and such as to render the teleological hypothesis unnecessary4 ,in the case of the roughened support1 there is a temporary adhesion or 8clinging0 between it and the growing stem which twines around it2 and a system of forces is thus set up1 producing a 8couple10 just as it was in the case of the ram's or antelope's horn through direct adhesion of the bony core to the surrounding sheath4 ,the twist is the direct result of this couple1 and it disappears when the support is so smooth that no such force comes to be exerted4 ,another important class of climbers includes the so-called 8leaf-climbers40 ,in these1 some portion of the leaf1 generally the petiole1 sometimes 7as in the fumitory7 the elongated midrib1 curls round a support2 and a phenomenon of like nature occurs in many1 though not all1 of the so-called 8tendril-bearers40 ,except that a different part of the plant1 leaf or tendril instead of stem1 is concerned in the twining process1 the phenomenon here is strictly analogous to our former case2 but in the resulting helix there is1 as a rule1 this obvious difference1 that1 while the twining stem1 for instance of the hop1 makes a slow revolution about its support1 the typical leaf-climber makes a close1 firm coil3 the axis of the latter is nearly perpendicular and parallel to the axis of its support1 while in the twining stem the angle between the two axes is comparatively small4 ,mathematically speaking1 the difference merely amounts to this1 that the component in the direction of the vertical axis is large in the one case1 and the corresponding component is small1 if not absent1 in the other2 in other words1 we have in the climbing stem a considerable vertical component1 due to its own tendency to grow in height1 while this longitudinal or vertical extension of the whole system is not apparent1 or little apparent1 in the other cases4 ,but from the fact that the twining stem tends to run obliquely to its support1 and the coiling petiole of the leaf-climber tends to run transversely to the axis of its support1 there immediately follows this marked difference1 that the phenomenon #fbg of torsion1 so manifest in the former case1 will be absent in the latter4 ,there is one other phenomenon which meets us in the twining and twisted stem1 and which is doubtless illustrated also1 though not so well1 in the antelope's horn2 it is a phenomenon which forms the subject of a second chapter of ,st ,venant's researches on the effects of torsional strain in elastic bodies4 ,we have already seen how one effect of torsion1 in for instance a prism1 is to produce strains parallel to the axis1 elevating parts and depressing other parts of each transverse section4 ,but in addition to this1 the same torsion has the effect of materially altering the form of the section itself1 as we may easily see by twisting a square or oblong piece of india-rubber4 ,if we start with a cylinder1 such as a round piece of catapult india-rubber1 and twist it on its own long axis1 we have already seen that it suffers no other distortion2 it still remains a cylinder1 that is to say1 it is still in section everywhere circular4 ,but if it be of any other shape than cylindrical the case is quite different1 for now the sectional shape tends to alter under the strain of torsion4 ,thus1 if our rod be elliptical in section to begin with1 it will1 under torsion1 become a more elongated ellipse2 if it be square1 its angles will become more prominent1 and its sides will curve inwards1 till at length the square assumes the appearance of a four-pointed star1 with rounded angles4 ,furthermore1 looking at the results of this process of modification1 we find experimentally that the resultant figures are more easily twisted1 less resistant to torsion1 than were those from which we evolved them2 and this is a very curious physical or mathematical fact4 ,so a cylinder1 which is especially resistant to torsion1 is very easily bent or flexed2 while projecting ribs or angles1 such as an engineer makes in a bar or pillar of iron for the purpose of greatly increasing its strength in the way of resistance to bending1 actually make it much weaker than before 7for the same amount of metal per unit length7 in the way of resistance to torsion4 ,in the hop itself1 and in a very considerable number of other twining and twisting stems1 the ribbed or channelled form of the stem is a conspicuous feature4 ,we may safely take it1 7#a7 that #fbh such stems are especially susceptible of torsion2 and 7#b7 that the effect of torsion will be to intensify any such peculiarities of sectional outline which they may possess1 though not to initiate them in an originally cylindrical structure4 ,in the leaf-climbers the case does not present itself1 for there1 as we have seen1 torsion itself is not1 or is very slightly1 manifested4 ,there are very distinct traces of the phenomenon in the horns of certain antelopes1 but the reason why it is not a more conspicuous feature of the antelope's horn or of the ram's is apparently a very simple one3 namely1 that the presence of the bony core within tends to check that deformation which is perpendicular1 while it permits that which is parallel1 to the axis of the horn4 ,of ,deer's ,antlers4 ,but let us return to our subject of the shapes of horns1 and consider briefly our last class of these structures1 namely the bony antlers of the various species of elk and deer#efi4 ,the problems which these present to us are very different from those which we have had to do with in the antelope or the sheep4 ,illustration3 ,fig4 #cbc4 ,antlers of ,swedish ,elk4 7,after ,lnnberg1 from ,p4,z4,s47 ,with regard to its structure1 it is plain that the bony antler corresponds1 upon the whole1 to the bony core of the antelope's horn2 while in place of the hard horny sheath of the latter1 we have the soft 8velvet10 which every season covers the new growing antler1 and protects the large nutrient blood-vessels by help of which the antler grows#egj4 ,the main difference lies in the fact that1 in the one case1 the bony core1 imprisoned within its sheath1 is rendered incapable of branching and incapable also of lateral expansion1 and the whole horn is only permitted to grow in length1 while retaining a sectional contour that is identical with 7or but little altered from7 that which it possesses at its growing base3 #fbi but in the antler1 on the other hand1 no such restraint is imposed1 and the living1 growing fabric of bone may expand into a broad flattened plate over which the blood-vessels run4 ,in the immediate neighbourhood of the main blood-vessels growth will be most active2 in the interspaces between1 it may wholly fail3 with the result that we may have great notches cut out of the flattened plate1 or may at length find it reduced to the form of a simple branching structure4 ,the main point1 as it seems to me1 is that the 8horn0 is essentially an axial rod1 while the 8antler0 is essentially an outspread surface#ega4 ,in other words1 ,i believe that the whole configuration of an antler is more easily understood by conceiving it as a plate or a surface1 more and more notched and scolloped till but a slender skeleton may remain1 than to look upon it the other way1 namely as an axial stem 7or beam7 giving #fcj off branches 7or tines71 the interspaces between which latter may sometimes be filled up to form a continuous plate4 ,illustration3 ,fig4 #cbd4 ,head and antlers of a ,stag 7,cervus ,duvauceli74 7,after ,lydekker1 from ,p4,z4,s47 ,in a sense it matters very little whether we regard the broad plate-like antlers of the elk or the slender branching antlers of the stag as the more primitive type2 for we are not concerned here with the question of hypothetical phylogeny4 ,and even from the mathematical point of view it makes little or no difference whether we describe the plate as constituted by the interconnection of the branches1 or the branches derived by a process of notching or incision from the plate4 ,the important point for us is to recognise that 7save for occasional slight irregularities7 the branching system in the one conforms essentially to the curved plate or surface which we see plainly in the other4 ,in short the arrangement of the branches is more or less comparable to that of the veins in a leaf1 or to that of the blood-vessels as they course over the curved surface of an organ4 ,it is a process of ramification1 not1 like that of a tree1 in various planes1 but strictly limited #fca to a single surface4 ,and just as the veins within a leaf are not necessarily confined 7as they happen to be in most ordinary leaves7 to a plane surface1 but1 as in the petal of a tulip or the capsule of a poppy1 may have to run their course within a curved surface1 so does the analogy of the leaf lead us directly to the mode of branching which is characteristic of the antler4 ,the surface to which the branches of the antler tend to be confined is a more or less spheroidal1 or occasionally an ellipsoidal one2 and furthermore1 when we inspect any well-developed pair of antlers1 such as those of a red deer1 a sambur or a wapiti1 we have no difficulty in seeing that the two antlers make up between them a single surface1 and constitute a symmetrical figure1 each half being the mirror-image of the other4 ,to put the case in another way1 a pair of antlers 7apart from occasional slight irregularities7 tends to constitute a figure such that we could conceive an elastic sheet stretched over or round the entire system1 so as to form one continuous and even surface2 and not only would the surface curvature be on the whole smooth and even1 but the boundary of the surface would also tend to be an even curve3 that is to say the tips of all the tines would approximately have their locus in a continuous curve4 ,it follows from this that if we want to make a simple model of a set of antlers1 we shall be very greatly helped by taking some appropriate spheroidal surface as our groundwork or scaffolding4 ,the best form of surface is a matter for trial and investigation in each particular case2 but even in a sphere1 by selecting appropriate areas thereof1 we can obtain sufficient varieties of surface to meet all ordinary cases4 ,with merely a bit of sculptor's clay or plasticine1 we should be put hard to it to model the horns of a wapiti or a reindeer3 but if we start with an orange 7or a round florence flask7 and lay our little tapered rolls of plasticine upon it1 in simple natural curves1 it is surprising to see how quickly and successfully we can imitate one type of antler after another4 ,in doing so1 we shall be struck by the fact that our model may vary in its mode of branching within very considerable limits1 and yet look perfectly natural4 ,for the same wide range of variation is characteristic of the natural antlers themselves4 ,as ,sir ,v4 ,brooke says 7op4 cit4 p4 #hib71 8,no two antlers are ever exactly alike2 and the #fcb variation to which the antlers are subject is so great that in the absence of a large series they would be held to be indicative of several distinct species#egb40 ,but all these many variations lie within a limited range1 for they are all subject to our general rule that the entire structure is essentially confined to a single curved surface4 ,it is plain that in the curvatures both of the beam and of its tines1 in the angles by which these latter meet the beam1 and in the contours of the entire system1 there are involved many elegant mathematical problems with which we cannot at present attempt to deal4 ,nor must we attempt meanwhile to enquire into the physical meaning or origin of these phenomena1 for as yet the clue seems to be lacking and we should only heap one hypothesis upon another4 ,that there is a complete contrast of mathematical properties between the horn and the antler is the main lesson with which1 in the meantime1 we must rest content4 ,of ,teeth1 and of ,beak and ,claw4 ,in a fashion similar to that manifested in the shell or the horn1 we find the logarithmic spiral to be implicit in a great many other organic structures where the phenomena of growth proceed in a similar way3 that is to say1 where about an axis there is some asymmetry leading to unequal rates of longitudinal growth1 and where the structure is of such a kind that each new increment is added on as a permanent and unchanging part of the entire conformation4 ,nail and claw1 beak and tooth1 all come under this category4 ,the logarithmic spiral always tends to manifest itself in such structures as these1 though it usually only attracts our attention in elongated structures1 where 7that is to say7 the radius vector has described a considerable angle4 ,when the canary-bird's claws grow long from lack of use1 or when the incisor tooth of a rabbit or a rat grows long by reason of an injury to the opponent tooth against which it was wont to bite1 we know that the tooth or claw tends to grow into a spiral curve1 and we speak of it as a malformation4 ,but there has been no fundamental change of form1 save only an abnormal increase in length2 #fcc the elongated tooth or claw has the selfsame curvature that it had when it was short1 but the spiral curvature becomes more and more manifest the longer it grows4 ,a curious analogous case is that of the ,new ,zealand huia bird1 in which the beak of the female is described as being comparatively short and straight1 while that of the male is long and curved2 it is easy to see that there is a slight curvature also in the beak of the female1 and that the beak of the male shows nothing but the same curve produced4 ,in the case of the more curved beaks1 such as those of an eagle or a parrot1 we may1 if we please1 determine the constant angle of the logarithmic spiral1 just as we have done in the case of the ,nautilus shell2 and here again1 as the bird grows older or the beak longer1 the spiral nature of the curve becomes more and more apparent1 as in the hooked beak of an old eagle1 or as in the great beak of some large parrot such as a hyacinthine macaw4 ,let us glance at one or two instances to illustrate the spiral curvature of teeth4 ,a dentist knows that every tooth has a curvature of its own1 and that in pulling the tooth he must follow the direction of the curve2 but in an ordinary tooth this curvature is scarcely visible1 and is least so when the diameter of the tooth is large compared with its length4 ,in the simply formed1 more or less conical teeth1 such as are those of the dolphin1 and in the more or less similarly shaped canines and incisors of mammals in general1 the curvature of the tooth is particularly well seen4 ,we see it in the little teeth of a hedgehog1 and in the canines of a dog or a cat it is very obvious indeed4 ,when the great canine of the carnivore becomes still further enlarged or elongated1 as in ,machairodus1 it grows into the strongly curved sabre-tooth of that great extinct tiger4 ,in rodents1 it is the incisors which undergo a great elongation2 their rate of growth differs1 though but slightly1 on the two sides1 anterior and posterior1 of the axis1 and by summation of these slight differences in the rapid growth of the tooth an unmistakeable logarithmic spiral is gradually built up4 ,we see it admirably in the beaver1 or in the great ground-rat1 ,geomys4 ,the elephant is a similar case1 save that the tooth1 or tusk1 remains1 owing to comparative lack of wear1 in a more perfect condition4 ,in the rodent 7save only in those abnormal cases mentioned on the last page7 the #fcd anterior1 first-formed1 part of the tooth wears away as fast as it is added to from behind2 and in the grown animal1 all those portions of the tooth near to the pole of the logarithmic spiral have long disappeared4 ,in the elephant1 on the other hand1 we see1 practically speaking1 the whole unworn tooth1 from point to root2 and its actual tip nearly coincides with the pole of the spiral4 ,if we assume 7as with no great inaccuracy we may do7 that the tip actually coincides with the pole1 then we may very easily construct the continuous spiral of which the existing tusk constitutes a part2 and by so doing1 we see the short1 gently curved tusk of our ordinary elephant growing gradually into the spiral tusk of the mammoth4 ,no doubt1 just as in the case of our molluscan shells1 we have a tendency to variation1 both individual and specific1 in the constant angle of the spiral2 some elephants1 and some species of elephant1 undoubtedly have a higher spiral angle than others4 ,but in most cases1 the angle would seem to be such that a spiral configuration would become very manifest indeed if only the tusk pursued its steady growth1 unchanged otherwise in form1 till it attained the dimensions which we meet with in the mammoth4 ,in a species such as ,mastodon angustidens1 or ,m4 arvernensis1 the specific angle is low and the tusk comparatively straight2 but the ,american mastodons and the existing species of elephant have tusks which do not differ appreciably1 except in size1 from the great spiral tusks of the mammoth1 though from their comparative shortness the spiral is little developed and only appears to the eye as a gentle curve4 ,wherever the tooth is very long indeed1 as in the mammoth or the beaver1 the effect of some slight and all but inevitable lateral asymmetry in the rate of growth begins to shew itself3 in other words1 the spiral is seen to lie not absolutely in a plane1 but to be a curve of double curvature1 like a twisted horn4 ,we see this condition very well in the huge canine tusks of the ,babirussa2 it is a conspicuous feature in the mammoth1 and it is more or less perceptible in any large tusk of the ordinary elephants4 ,the form of a molar tooth1 which is essentially a branching or budding system1 and in which such longitudinal growth as gives rise to a spiral curve is but little manifest1 constitutes an entirely different problem with which ,i shall not at present attempt to deal4 #fce ,,chapter ,,xiv ,,on ,,leaf-,,arrangement1 ,,or ,,phyllotaxis ,the beautiful configurations produced by the orderly arrangement of leaves or florets on a stem have long been an object of admiration and curiosity4 ,leonardo da ,vinci would seem1 as ,sir ,theodore ,cook tells us1 to have been the first to record his thoughts upon this subject2 but the old ,greek and ,egyptian geometers are not likely to have left unstudied or unobserved the spiral traces of the leaves upon a palm-stem1 or the spiral curves of the petals of a lotus or the florets in a sunflower4 ,the spiral leaf-order has been regarded by many learned botanists as involving a fundamental law of growth1 of the deepest and most far-reaching importance2 while others1 such as ,sachs1 have looked upon the whole doctrine of 8phyllotaxis0 as 8a sort of geometrical or arithmetical playing with ideas10 and 8the spiral theory as a mode of view gratuitously introduced into the plant40 ,sachs even goes so far as to declare this doctrine 8in direct opposition to scientific investigation1 and based upon the idealistic direction of the ,naturphilosophie10--the mystical biology of ,oken and his school4 ,the essential facts of the case are not difficult to understand2 but the theories built upon them are so varied1 so conflicting1 and sometimes so obscure1 that we must not attempt to submit them to detailed analysis and criticism4 ,there are two chief ways by which we may approach the question1 according to whether we regard1 as the more fundamental and typical1 one or other of the two chief modes in which the phenomenon presents itself4 ,that is to say1 we may hold that the phenomenon is displayed in its essential simplicity by the corkscrew spirals1 or helices1 which mark the position of the leaves upon a cylindrical stem or on an #fcf elongated fir-cone2 or1 on the other hand1 we may be more attracted by1 and regard as of greater importance1 the logarithmic spirals which we trace in the curving rows of florets in the discoidal inflorescence of a sunflower4 ,whether one way or the other be the better1 or even whether one be not positively correct and the other radically wrong1 has been vehemently debated4 ,in my judgment they are1 both mathematically and biologically1 to be regarded as inseparable and correlative phenomena4 ,the helical arrangement 7as in the fir-cone7 was carefully studied in the middle of the eighteenth century by the celebrated ,bonnet1 with the help of ,calandrini1 the mathematician4 ,memoirs published about #ahce1 by ,schimper and ,braun1 greatly amplified ,bonnet's investigations1 and introduced a nomenclature which still holds its own in botanical textbooks4 ,naumann and the brothers ,bravais are among those who continued the investigation in the years immediately following1 and ,hofmeister1 in #ahfh1 gave an admirable account and summary of the work of these and many other writers#egc4 ,starting from some given level and proceeding upwards1 let us mark the position of some one leaf 7,a7 upon a cylindrical stem4 ,another1 and a younger leaf 7,b7 will be found standing at a certain distance around the stem1 and a certain distance along the stem1 #fcg from the first4 ,the former distance may be expressed as a fractional 8divergence0 7such as two-fifths of the circumference of the stem7 as the botanists describe it1 or by an 8angle of azimuth0 7such as #add7 as the mathematician would be more likely to state it4 ,the position of ,b relatively to ,a must be determined1 not only by this angle 1 in the horizontal plane1 but also by an angle 77 in the vertical plane2 for the height of ,b above the level of ,a1 in comparison with the diameter of the cylinder1 will obviously make a great difference in the appearance of the whole system1 in short the position of each leaf must be expressed by ,f7 sin 74 ,but this matter botanical students have not concerned themselves with2 in other words1 their studies have been limited 7or mainly limited7 to the relation of the leaves to one another in azimuth4 ,whatever relation we have found between ,a and ,b1 let precisely the same relation subsist between ,b and ,c3 and so on4 ,let the growth of the system1 that is to say1 be continuous and uniform2 it is then evident that we have the elementary conditions for the development of a simple cylindrical helix2 and this 8primary helix0 or 8genetic spiral0 we can now trace1 winding round and round the stem1 through ,a1 ,b1 ,c1 etc4 ,but if we can trace such a helix through ,a1 ,b1 ,c1 it follows from the symmetry of the system1 that we have only to join ,a to some other leaf to trace another spiral helix1 such1 for instance1 as ,a1 ,c1 ,e1 etc42 parallel to which will run another and similar one1 namely in this case ,b1 ,d1 ,f1 etc4 ,and these spirals will run in the opposite direction to the spiral ,,abc4 ,in short1 the existence of one helical arrangement of points implies and involves the existence of another and then another helical pattern1 just as1 in the pattern of a wall-paper1 our eye travels from one linear series to another4 ,a modification of the helical system will be introduced when1 instead of the leaves appearing1 or standing1 in singular succession1 we get two or more appearing simultaneously upon the same level4 ,if there be two such1 then we shall have two generating spirals precisely equivalent to one another2 and we may call them ,a1 ,b1 ,c1 etc41 and ,a1 ,b1 ,c1 and so on4 ,these are the cases which we call 8whorled0 leaves1 or in the simplest case1 where #fch the whorl consists of two opposite leaves only1 we call them decussate4 ,among the phenomena of phyllotaxis1 two points in particular have been found difficult of explanation1 and have aroused discussion4 ,these are 7#a71 the presence of the logarithmic spirals such as we have already spoken of in the sunflower2 and 7#b7 the fact that1 as regards the number of the helical or spiral rows1 certain numerical coincidences are apt to recur again and again1 to the exclusion of others1 and so to become characteristic features of the phenomenon4 ,the first of these appears to me to present no difficulty4 ,it is a mere matter of strictly mathematical 8deformation40 ,the stem which we have begun to speak of as a cylinder is not strictly so1 inasmuch as it tapers off towards its summit4 ,the curve which winds evenly around this stem is1 accordingly1 not a true helix1 for that term is confined to the curve which winds evenly around the cylinder3 it is a curve in space which 7like the spiral curve we have studied in our turbinate shells7 partakes of the characters of a helix and of a logarithmic spiral1 and which is in fact a logarithmic spiral with its pole drawn out of its original plane by a force acting in the direction of the axis4 ,if we imagine a tapering cylinder1 or cone1 projected1 by vertical projection1 on a plane1 it becomes a circular disc2 and a helix described about the cone necessarily becomes in the disc a logarithmic spiral described about a focus which corresponds to the apex of our cone4 ,in like manner we may project an identical spiral in space upon such surfaces as 7for instance7 a portion of a sphere or of an ellipsoid2 and in all these cases we preserve the spiral configuration1 which is the more clearly brought into view the more we reduce the vertical component by which it was accompanied4 ,the converse is1 of course1 equally true1 and equally obvious1 namely that any logarithmic spiral traced upon a circular disc or spheroidal surface will be transformed into a corresponding spiral helix when the plane or spheroidal disc is extended into an elongated cone approximating to a cylinder4 ,this mathematical conception is translated1 in botany1 into actual fact4 ,the fir-cone may be looked upon as a cylindrical axis contracted at both ends1 until #fci it becomes approximately an ellipsoidal solid of revolution1 generated about the long axis of the ellipse2 and the semi-ellipsoidal capitulum of the teasel1 the more or less hemispherical one of the thistle1 and the flattened but still convex one of the sunflower1 are all beautiful and successive deformations of what is typically a long1 conical1 and all but cylindrical stem4 ,on the other hand1 every stem as it grows out into its long cylindrical shape is but a deformation of the little spheroidal or ellipsoidal surface1 or cone1 which was its forerunner in the bud4 ,this identity of the helical spirals around the stem with spirals projected on a plane was clearly recognised by ,hofmeister1 who was accustomed to represent his diagrams of leaf-arrangement either in one way or the other1 though not in a strictly geometrical projection#egd4 ,according to ,mr ,a4 ,h4 ,church#ege1 who has dealt very carefully and elaborately with the whole question of phyllotaxis1 the logarithmic spirals such as we see in the disc of the sunflower have a far greater importance and a far deeper meaning than this brief treatment of mine would accord to them3 and ,sir ,theodore ,cook1 in his book on the ,curves of ,life1 has adopted and has helped to expound and popularise ,mr ,church's investigations4 ,mr ,church1 regarding the problem as one of 8uniform growth10 easily arrives at the conclusion that1 if this growth can be conceived as taking place symmetrically about a central point or 8pole10 the uniform growth would then manifest itself in logarithmic spirals1 including of course the limiting cases of the circle and straight line4 ,with this statement ,i have little fault to find2 it is in essence identical with much that ,i have said in a previous chapter4 ,but other statements of ,mr ,church's1 and many theories woven about them by ,sir ,t4 ,cook and himself1 ,i am less able to follow4 ,mr ,church tells us that the essential phenomenon in the sunflower disc is a series of orthogonally intersecting logarithmic spirals4 ,unless ,i wholly misapprehend ,mr ,church's meaning1 ,i should say that this is very far from essential4 ,the spirals #fdj intersect isogonally1 but orthogonal intersection would be only one particular case1 and in all probability a very infrequent one1 in the intersection of logarithmic spirals developed about a common pole4 ,again on the analogy of the hydrodynamic lines of force in certain vortex movements1 and of similar lines of force in certain magnetic phenomena1 ,mr ,church proceeds to argue that the energies of life follow lines comparable to those of electric energy1 and that the logarithmic spirals of the sunflower are1 so to speak1 lines of equipotential#egf4 ,and ,sir ,t4 ,cook remarks that this 8theory1 if correct1 would be fundamental for all forms of growth1 though it would be more easily observed in plant construction than in animals40 ,the parallel ,i am not able to follow4 ,mr ,church sees in phyllotaxis an organic mystery1 a something for which we are unable to suggest any precise cause3 a phenomenon which is to be referred1 somehow1 to waves of growth emanating from a centre1 but on the other hand not to be explained by the division of an apical cell1 or any other histological factor4 ,as ,sir ,t4 ,cook puts it1 8at the growing point of a plant where the new members are being formed1 there is simply nothing to see40 ,but it is impossible to deal satisfactorily1 in brief space1 either with ,mr ,church's theories1 or my own objections to them#egg4 ,let it suffice to say that ,i1 for my part1 see no subtle mystery in the matter1 other than what lies in the steady production of similar growing parts1 similarly situated1 at similar successive intervals of time4 ,if such be the case1 then we are bound to have in #fda consequence a series of symmetrical patterns1 whose nature will depend upon the form of the entire surface4 ,if the surface be that of a cylinder we shall have a system1 or systems1 of spiral helices3 if it be a plane1 with an infinitely distant focus1 such as we obtain by 8unwrapping0 our cylindrical surface1 we shall have straight lines2 if it be a plane containing the focus within itself1 or if it be any other symmetrical surface containing the focus1 then we shall have a system of logarithmic spirals4 ,the appearance of these spirals is sometimes spoken of as a 8subjective0 phenomenon1 but the description is inaccurate3 it is a purely mathematical phenomenon1 an inseparable secondary result of other arrangements which we1 for the time being1 regard as primary4 ,when the bricklayer builds a factory chimney1 he lays his bricks in a certain steady1 orderly way1 with no thought of the spiral patterns to which this orderly sequence inevitably leads1 and which spiral patterns are by no means 8subjective40 ,the designer of a wall-paper not only has no intention of producing a pattern of criss-cross lines1 but on the contrary he does his best to avoid them2 nevertheless1 so long as his design is a symmetrical one1 the criss-cross intersections inevitably come4 ,let us1 however1 leave this discussion1 and return to the facts of the case4 ,our second question1 which relates to the numerical coincidences so familiar to all students of phyllotaxis1 is not to be set and answered in a word4 ,let us1 for simplicity's sake1 avoid consideration of simultaneous or whorled leaf origins1 and consider only the more frequent cases where a single 8genetic spiral0 can be traced throughout the entire system4 ,it is seldom that this primary1 genetic spiral catches the eye1 for the leaves which immediately succeed one another in this genetic order are usually far apart on the circumference of the stem1 and it is only in close-packed arrangements that the eye readily apprehends the continuous series4 ,accordingly in such a case as a fir-cone1 for instance1 it is certain of the secondary spirals or 8parastichies0 which catch the eye2 and among fir-cones1 we can easily count these1 and we find them to be #fdb on the whole very constant in number1 according to the species4 ,thus in many cones1 such as those of the ,norway spruce1 we can trace five rows of scales winding steeply up the cone in one direction1 and three rows winding less steeply the other way2 in certain other species1 such as the common larch1 the normal number is eight rows in the one direction and five in the other2 while in the ,american larch we have again three in the one direction and five in the other4 ,it not seldom happens that two arrangements grade into one another on different parts of one and the same cone4 ,among other cases in which such spiral series are readily visible we have1 for instance1 the crowded leaves of the stone-crops and mesembryanthemums1 and 7as we have said7 the crowded florets of the composites4 ,among these we may find plenty of examples in which the numbers of the serial rows are similar to those of the fir-cones2 but in some cases1 as in the daisy and others of the smaller composites1 we shall be able to trace thirteen rows in one direction and twenty-one in the other1 or perhaps twenty-one and thirty-four2 while in a great big sunflower we may find 7in one and the same species7 thirty-four and fifty-five1 fifty-five and eighty-nine1 or even as many as eighty-nine and one hundred and forty-four4 ,on the other hand1 in an ordinary 8pentamerous0 flower1 such as a ranunculus1 we may be able to trace1 in the arrangement of its sepals1 petals and stamens1 shorter spiral series1 three in one direction and two in the other4 ,it will be at once observed that these arrangements manifest themselves in connection with very different things1 in the orderly interspacing of single leaves and of entire florets1 and among all kinds of leaf-like structures1 foliage-leaves1 bracts1 cone-scales1 and the various parts or members of the flower4 ,again we must be careful to note that1 while the above numerical characters are by much the most common1 so much so as to be deemed 8normal10 many other combinations are known to occur4 ,the arrangement1 as we have seen1 is apt to vary when the entire structure varies greatly in size1 as in the disc of the sunflower4 ,it is also subject to less regular variation within one and the same species1 as can always be discovered when we examine a sufficiently large sample of fir-cones4 ,for instance1 out of #eje #fdc cones of the ,norway spruce1 ,beal#egh found #ib per cent4 in which the spirals were in five and eight rows2 in #f per cent4 the rows were four and seven1 and in #d per cent4 they were four and six4 ,in each case they were nearly equally divided as regards direction2 for instance of the #dfg cones shewing the five-eight arrangement1 the five-series ran in right-handed spirals in #bbd cases1 and in left-handed spirals in #bdc4 ,omitting the 8abnormal0 cases1 such as we have seen to occur in a small percentage of our cones of the spruce1 the arrangements which we have just mentioned may be set forth as follows1 7the fractional number used being simply an abbreviated symbol for the number of associated helices or parastichies which we can count running in the opposite directions73 #b_/c1 #c_/e1 #e_/h1 #h_/ac1 #ac_/ba1 #ba_/cd1 #cd_/ee1 #ee_/hi1 #hi_/add4 ,now these numbers form a very interesting series1 which happens to have a number of curious mathematical properties#egi4 ,we see1 for instance1 that the denominator of each fraction is the numerator of the next2 and further1 that each successive numerator1 or denominator1 is the sum of the preceding two4 ,our immediate problem1 then1 is to determine1 if possible1 how these numerical coincidences come about1 and why these particular numbers should be so commonly met with #fdd as to be considered 8normal0 and characteristic features of the general phenomenon of phyllotaxis4 ,the following account is based on a short paper by ,professor ,p4 ,g4 ,tait#ehj4 ,illustration3 ,fig4 #cbe4 ,of the two following diagrams1 ,fig4 #cbe represents the general case1 and ,fig4 #cbf a particular one1 for the sake of possibly greater simplicity4 ,both diagrams represent a portion of a branch1 or fir-cone1 regarded as cylindrical1 and unwrapped to form a plane surface4 ,a1 a1 at the two ends of the base-line1 represent the same initial leaf or scale3 ,o is a leaf which can be reached from ,a by m steps in a right-hand spiral 7developed into the straight line ,,ao71 and by n steps from a in a left-handed spiral a,o4 ,now it is obvious in our fir-cone1 that we can include all the scales upon the cone by taking so many spirals in the one direction1 and again include them all by so many in the other4 ,accordingly1 in our diagrammatic construction1 the spirals ,,ao and a,o must1 and always can1 be so taken that m spirals parallel to a,o1 and n spirals parallel to ,,ao1 shall separately include all the leaves upon the stem or cone4 ,if m and n have a common factor1 l1 it can easily be shewn that the arrangement is composite1 and that there are l fundamental1 or genetic spirals1 and l leaves 7including ,a7 which are situated exactly on the line ,aa4 ,that is to say1 we have here a whorled arrangement1 which we have agreed to leave unconsidered in favour of the simpler case4 ,we restrict ourselves1 accordingly1 to the cases where there is but one genetic spiral1 and when therefore m and n are prime to one another4 ,our fundamental1 or genetic1 spiral1 as we have seen1 is that which passes from ,a 7or a7 to the leaf which is situated nearest to the base-line ,aa4 ,the fundamental spiral will thus be right-handed 7,a1 ,p1 etc47 if ,p1 which is nearer to ,a than to a1 be this leaf--left-handed if it be p4 ,that is to say1 we make it a convention that we shall always1 for our fundamental spiral1 run #fde round the system1 from one leaf to the next1 by the shortest way4 ,illustration3 ,fig4 #cbf4 ,now it is obvious1 from the symmetry of the figure 7as further shewn in ,fig4 #cbf71 that1 besides the spirals running along ,,ao and a,o1 we have a series running from the steps on a,o to the steps on ,,ao4 ,in other words we can find a leaf 7,s7 upon ,,ao1 which1 like the leaf ,o1 is reached directly by a spiral series from ,a and from a1 such that a,s includes n steps1 and ,,as 7being part of the old spiral line ,,ao7 now includes m n steps4 ,and1 since m and n are prime to one another 7for otherwise the system would have been a composite or whorled one71 it is evident that we can continue this process of convergence until we come down to a #a1 #a arrangement1 that is to say to a leaf which is reached by a single step1 in opposite directions from ,a and from a1 which leaf is therefore the first leaf1 next to ,a1 of the fundamental or generating spiral4 #fdf ,if our original lines along ,,ao and a,o contain1 for instance1 #ac and #h steps respectively 7i4e4 m #ac1 n #h71 then our next series1 observable in the same cone1 will be #h and 7#ac #h7 or #e2 the next #e and 7#h #e7 or #c2 the next #c1 #b2 and the next #b1 #a2 leading to the ultimate condition of #a1 #a4 ,these are the very series which we have found to be common1 or normal2 and so far as our investigation has yet gone1 it has proved to us that1 if one of these exists1 it entails1 ipso facto1 the presence of the rest4 ,in following down our series1 according to the above construction1 we have seen that at every step we have changed direction1 the longer and the shorter sides of our triangle changing places every time4 ,let us stop for a moment1 when we come to the #a1 #b series1 or ,,at1 a,t of ,fig4 #cbf4 ,it is obvious that there is nothing to prevent us making a new #a1 #c series if we please1 by continuing the generating spiral through three leaves1 and connecting the leaf so reached directly with our initial one4 ,but in the case represented in ,fig4 #cbf1 it is obvious that these two series 7,a1 #a1 #b1 #c1 etc41 and a1 #c1 #f1 etc47 will be running in the same direction2 i4e4 they will both be right-handed1 or both left-handed spirals4 ,the simple meaning of this is that the third leaf of the generating spiral was distant from our initial leaf by more than the circumference of the cylindrical stem2 in other words1 that there were more than two1 but less than three leaves in a single turn of the fundamental spiral4 ,less than two there can obviously never be4 ,when there are exactly two1 we have the simplest of all possible arrangements1 namely that in which the leaves are placed alternately on opposite sides of the stem4 ,when there are more than two1 but less than three1 we have the elementary condition for the production of the series which we have been considering1 namely #a1 #b2 #b1 #c2 #c1 #e1 etc4 ,to put the latter part of this argument in more precise language1 let us say that3 ,if1 in our descending series1 we come to steps #a and t1 where t is determined by the condition that #a and t #a would give spirals both right-handed1 or both left-handed2 it follows that there are less than t #a leaves in a single turn of the fundamental spiral4 ,and1 determined in this manner1 it is found in the great majority of cases1 in fir-cones and a host of other examples of phyllotaxis1 that t #b4 ,in other words1 in the #fdg great majority of cases1 we have what corresponds to an arrangement next in order of simplicity to the simplest case of all3 next1 that is to say1 to the arrangement which consists of opposite and alternate leaves4 8,these simple considerations10 as ,tait says1 8explain completely the so-called mysterious appearance of terms of the recurring series #a1 #b1 #c1 #e1 #h1 #ac1 etc4#eha ,the other natural series1 usually but misleadingly represented by convergents to an infinitely extended continuous fraction1 are easily explained1 as above1 by taking t #c1 #d1 #e1 etc41 etc40 ,many examples of these latter series have been given by ,dickson#ehb and other writers4 ,we have now learned1 among other elementary facts1 that wherever any one system of helical spirals is present1 certain others invariably and of necessity accompany it1 and are definitely related to it4 ,in any diagram1 such as ,fig4 #cbf1 in which we represent our leaf-arrangement by means of uniform and regularly interspaced dots1 we can draw one series of spirals after another1 and one as easily as another4 ,but in our fir-cone1 for instance1 one particular series1 or rather two conjugate series1 are always conspicuous1 while the others are sought and found with comparative difficulty4 ,illustration3 ,fig4 #cbg4 ,the phenomenon is illustrated by ,fig4 #cbg1 a-d4 ,the ground-plan of all these diagrams is identically the same4 ,the generating spiral in each case represents a divergence of #c_/h1 or #ace of azimuth2 and the points succeed one another at the same successional distances parallel to the axis4 ,the rectangular outlines1 which correspond to the exposed surface of the leaves or cone-scales1 are of equal area1 and of equal number4 ,nevertheless the appearances presented by these diagrams are very different2 for in one the eye catches a #e_/h arrangement1 in another a #c_/e2 and so on1 down to an arrangement of #a_/a4 ,the mathematical side of this very curious phenomenon ,i have not attempted to investigate4 ,but it is quite obvious that1 in a system within #fdh which various spirals are implicitly contained1 the conspicuousness of one set or another does not depend upon angular divergence4 ,it depends on the relative proportions in length and breadth of the leaves themselves2 or1 more strictly speaking1 on the ratio of the diagonals of the rhomboidal figure by which each leaf-area is circumscribed4 ,when1 as in the fir-cone1 the scales by mutual compression conform to these rhomboidal outlines1 their inclined edges at once guide the eye in the direction of some one particular spiral2 and we shall not fail to notice that in such cases the usual #fdi result is to give us arrangements corresponding to the middle diagrams in ,fig4 #cbg1 which are the configurations in which the quadrilateral outlines approach most nearly to a rectangular form1 and give us accordingly the least possible ratio 7under the given conditions7 of sectional boundary-wall to surface area4 ,the manner in which one system of spirals may be caused to slide1 so to speak1 into another1 has been ingeniously demonstrated by ,schwendener on a mechanical model1 consisting essentially of a framework which can be opened or closed to correspond with one after another of the above series of diagrams#ehc4 ,the determination of the precise angle of divergence of two consecutive leaves of the generating spiral does not enter into the above general investigation 7though ,tait gives1 in the same paper1 a method by which it may be easily determined72 and the very fact that it does not so enter shews it to be essentially unimportant4 ,the determination of so-called 8orthostichies10 or precisely vertical successions of leaves1 is also unimportant4 ,we have no means1 other than observation1 of determining that one leaf is vertically above another1 and spiral series such as we have been dealing with will appear1 whether such orthostichies exist1 whether they be near or remote1 or whether the angle of divergence be such that no precise vertical superposition ever occurs4 ,and lastly1 the fact that the successional numbers1 expressed as fractions1 #a_/b1 #b_/c1 #c_/e1 represent a convergent series1 whose final term is equal to #jfahjc4441 the sectio aurea or 8golden mean0 of unity1 is seen to be a mathematical coincidence1 devoid of biological significance2 it is but a particular case of ,lagrange's theorem that the roots of every numerical equation of the second degree can be expressed by a periodic continued fraction4 ,the same number has a multitude of curious arithmetical properties4 ,it is the final term of all similar series to that with which we have been dealing1 such for instance as #a_/c1 #c_/d1 #d_/g1 etc41 or #a_/d1 #d_/e1 #e_/i1 etc4 ,it is a number beloved of the circle-squarer1 and of all those who seek to find1 and then to penetrate1 the secrets of the ,great ,pyramid4 ,it is deep-set in ,pythagorean as well as in ,euclidean geometry4 ,it enters 7as the chord of an angle of #cf71 #fej into the thrice-isosceles triangle of which we have spoken on p4 #eaa2 it is a number which becomes 7by the addition of unity7 its own reciprocal2 its properties never end4 ,to ,kepler 7as ,naber tells us7 it was a symbol of ,creation1 or ,generation4 ,its recent application to biology and art-criticism by ,sir ,theodore ,cook and others is not new4 ,naber's book1 already quoted1 is full of it4 ,zeising1 in #ahed1 found in it the key to all morphology1 and the same writer1 later on#ehd1 declared it to dominate both architecture and music4 ,but indeed1 to use ,sir ,thomas ,browne's words 7though it was of another number that he spoke73 8,to enlarge this contemplation into all the mysteries and secrets accommodable unto this number1 were inexcusable ,pythagorisme40 ,if this number has any serious claim at all to enter into the biological question of phyllotaxis1 this must depend on the fact1 first emphasized by ,chauncey ,wright#ehe1 that1 if the successive leaves of the fundamental spiral be placed at the particular azimuth which divides the circle in this 8sectio aurea10 then no two leaves will ever be superposed2 and thus we are said to have 8the most thorough and rapid distribution of the leaves round the stem1 each new or higher leaf falling over the angular space between the two older ones which are nearest in direction1 so as to divide it in the same ratio 7,k71 in which the first two or any two successive ones divide the circumference4 ,now #e_/h and all successive fractions differ inappreciably from ,k40 ,to this view there are many simple objections4 ,in the first place1 even #e_/h1 or #fbe1 is but a moderately close approximation to the 8golden mean02 in the second place the arrangements by which a better approximation is got1 such as #h_/ac1 #ac_/ba1 and the very close approximations such as #cd_/ee1 #ee_/hi1 #hi_/add1 etc41 are comparatively rare1 while the much less close approximations of #c_/e or #b_/c1 or even #a_/b1 are extremely common4 ,again1 the general type of argument such as that which asserts that the plant is 8aiming at0 something which we may call an 8ideal angle0 is one that cannot commend itself to a plain student of physical science3 nor is the hypothesis rendered more acceptably when ,sir ,t4 ,cook qualifies it by telling us that 8all that a plant can do #fea is to vary1 to make blind shots at constructions1 or to #8mutate' as it is now termed2 and the most suitable of these constructions will in the long run be isolated by the action of ,natural ,selection40 ,finally1 and this is the most concrete objection of all1 the supposed isolation of the leaves1 or their most complete 8distribution to the action of the surrounding atmosphere0 is manifestly very little affected by any conditions which are confined to the angle of azimuth4 ,if we could imagine a case in which all the leaves of the stem1 or all the scales of a fir-cone1 were crushed down to one and the same level1 into a simple ring or whorl of leaves1 then indeed they would have their most equable distribution under the condition of the 8ideal angle10 that is to say of the 8golden mean40 ,but if it be 7so to speak7 ,nature's object to set them further apart than they actually are1 to give them freer exposure to the air than they actually have1 then it is surely manifest that the simple way to do so is to elongate the axis1 and to set the leaves further apart1 lengthways on the stem4 ,this has at once a far more potent effect than any nice manipulation of the 8angle of divergence40 ,for it is obvious that in ,f7 sin 7 we have a greater range of variation by altering than by altering 4 ,we come then1 without more ado1 to the conclusion that the 8,fibonacci series10 and its supposed usefulness1 and the hypothesis of its introduction into plant-structure through natural selection1 are all matters which deserve no place in the plain study of botanical phenomena4 ,as ,sachs shrewdly recognised years ago1 all such speculations as these hark back to a school of mystical idealism4 #feb ,,chapter ,,xv ,,on ,,the ,,shapes ,,of ,,eggs1 ,,and ,,of ,,certain ,,other ,,hollow ,,structures ,the eggs of birds and all other hard-shelled eggs1 such as those of the tortoise and the crocodile1 are simple solids of revolution2 but they differ greatly in form1 according to the configuration of the plane curve by the revolution of which the egg is1 in a mathematical sense1 generated4 ,some few eggs1 such as those of the owl1 the penguin1 or the tortoise1 are spherical or very nearly so2 a few more1 such as the grebe's1 the cormorant's or the pelican's1 are approximately ellipsoidal1 with symmetrical or nearly symmetrical ends1 and somewhat similar are the so-called 8cylindrical0 eggs of the megapodes and the sand-grouse2 the great majority1 like the hen's egg1 are ovoid1 a little blunter at one end than the other2 and some1 by an exaggeration of this lack of antero-posterior symmetry1 are blunt at one end but characteristically pointed at the other1 as is the case with the eggs of the guillemot and puffin1 the sandpiper1 plover and curlew4 ,it is an obvious but by no means negligible fact that the egg1 while often pointed1 is never flattened or discoidal2 it is a prolate1 but never an oblate1 spheroid4 ,the careful study and collection of birds' eggs would seem to have begun with the ,count de ,marsigli#ehf1 the same celebrated naturalist who first studied the 8flowers0 of the coral1 and who wrote the ,histoire physique de la mer2 and the specific form1 as well as the colour and other attributes of the egg have been again and again discussed1 and not least by the many dilettanti naturalists of the eighteenth century who soon followed in ,marsigli's footsteps#ehg4 #fec ,we need do no more than mention ,aristotle's belief1 doubtless old in his time1 that the more pointed egg produces the male chicken1 and the blunter egg the hen2 though this theory survived into modern times#ehh and perhaps still lingers on4 ,several naturalists1 such as ,gnther 7#aggb7 and ,bhle 7#ahah71 have taken the trouble to disprove it by experiment4 ,a more modern and more generally accepted explanation has been that the form of the egg is in direct relation to that of the bird which has to be hatched within--a view that would seem to have been first set forth by ,naumann and ,bhle1 in their great treatise on eggs#ehi1 and adopted by ,des ,murs#eij and many other well-known writers4 ,in a treatise by de ,lafresnaye#eia1 an elaborate comparison is made between the skeleton and the egg of the various birds1 to shew1 for instance1 how those birds with a deep-keeled sternum laid rounded eggs1 which alone could accommodate the form of the young4 ,according to this view1 that 8,nature had foreseen#eib0 the form adapted to and necessary for the growing embryo1 it was easy to correlate the owl with its spherical egg1 the diver with its elliptical one1 and in like manner the round egg of the tortoise and the elongated one of the crocodile with the shape of the creatures which had afterwards to be hatched therein4 ,a few writers1 such as ,thienemann#eic1 looked at the same facts the other way1 and asserted that the form of the egg was determined by that of the bird by which it was laid1 and in whose body it had been conformed4 ,in more recent times1 other theories1 based upon the principles of ,natural ,selection1 have been current and very generally accepted1 to account for these diversities of form4 ,the pointed1 conical egg of the guillemot is generally supposed to be an adaptation1 #fed advantageous to the species in the circumstances under which the egg is laid2 the pointed egg is less apt than a spherical one to roll off the narrow ledge of rock on which this bird is said to lay its solitary egg1 and the more pointed the egg1 so much the fitter and likelier is it to survive4 ,the fact that the plover or the sandpiper1 breeding in very different situations1 lay eggs that are also conical1 elicits another explanation1 to the effect that here the conical form permits the many large eggs to be packed closely under the mother bird#eid4 ,whatever truth there be in these apparent adaptations to existing circumstances1 it is only by a very hasty logic that we can accept them as a vera causa1 or adequate explanation of the facts2 and it is obvious that1 in the bird's egg1 we have an admirable case for the direct investigation of the mechanical or physical significance of its form#eie4 ,of all the many naturalists of the eighteenth and nineteenth centuries who wrote on the subject of eggs1 one alone 7so far as ,i am aware7 ascribed the form of the egg to direct mechanical causes4 ,gnther#eif1 in #aggb1 declared that the more or less rounded or pointed form of the egg is a mechanical consequence of the pressure of the oviduct at a time when the shell is yet unformed or unsolidified2 and that accordingly1 to explain the round egg of the owl or the kingfisher1 we have only to admit that the oviduct of these birds is somewhat larger than that of most others1 or less subject to violent contractions4 ,this statement contains1 in essence1 the whole story of the mechanical conformation of the egg4 ,let us consider1 very briefly1 the conditions to which the egg is subject in its passage down the oviduct#eig4 7#a7 ,the 8egg10 as it enters the oviduct1 consists of the yolk only1 enclosed in its vitelline membrane4 ,as it passes down the first portion of the oviduct1 the white is gradually superadded1 #fee and becomes in turn surrounded by the 8shell-membrane40 ,about this latter the shell is secreted1 rapidly and at a late period2 the egg having meanwhile passed on into a wider portion of the oviducal tube1 called 7by loose analogy1 as ,owen says7 the 8uterus40 ,here the egg assumes its permanent form1 here it becomes rigid1 and it is to this portion of the 8oviduct0 that our argument principally refers4 7#b7 ,both the yolk and the entire egg tend to fill completely their respective membranes1 and1 whether this be due to growth or imbibition on the part of the contents or to contraction on the part of the surrounding membranes1 the resulting tendency is for both yolk and egg to be1 in the first instance1 spherical1 unless otherwise distorted by external pressure4 7#c7 ,the egg is subject to pressure within the oviduct1 which is an elastic1 muscular tube1 along the walls of which pass peristaltic waves of contraction4 ,these muscular contractions may be described as the contraction of successive annuli of muscle1 giving annular 7or radial7 pressure to successive portions of the egg2 they drive the egg forward against the frictional resistance of the tube1 while tending at the same time to distort its form4 ,while nothing is known1 so far as ,i am aware1 of the muscular physiology of the oviduct1 it is well known in the case of the intestine that the presence of an obstruction leads to the development of violent contractions in its rear1 which waves of contraction die away1 and are scarcely if at all propagated in advance of the obstruction4 7#d7 ,it is known by observation that a hen's egg is always laid blunt end foremost4 7#e7 ,it can be shown1 at least as a very common rule1 that those eggs which are most unsymmetrical1 or most tapered off posteriorly1 are also eggs of a large size relatively to the parent bird4 ,the guillemot is a notable case in point1 and so also are the curlews1 sandpipers1 phaleropes and terns4 ,we may accordingly presume that the more pointed eggs are those that are large relatively to the tube or oviduct through which they have to pass1 or1 in other words1 are those which are subject to the greatest pressure while being forced along4 ,so general is this relation that we may go still further1 and presume with great plausibility #fef in the few exceptional cases 7of which the apteryx is the most conspicuous7 where the egg is relatively large though not markedly unsymmetrical1 that in these cases the oviduct itself is in all probability large 7as ,gnther had suggested7 in proportion to the size of the bird4 ,in the case of the common fowl we can trace a direct relation between the size and shape of the egg1 for the first eggs laid by a young pullet are usually smaller1 and at the same time are much more nearly spherical than the later ones2 and1 moreover1 some breeds of fowls lay proportionately smaller eggs than others1 and on the whole the former eggs tend to be rounder than the latter#eih4 ,we may now proceed to inquire more particularly how the form of the egg is controlled by the pressures to which it is subjected4 ,the egg1 just prior to the formation of the shell1 is1 as we have seen1 a fluid body1 tending to a spherical shape and enclosed within a membrane4 ,our problem1 then1 is3 ,given a practically incompressible fluid1 contained in a deformable capsule1 which is either 7a7 entirely inextensible1 or 7b7 slightly extensible1 and which is placed in a long elastic tube the walls of which are radially contractile1 to determine the shape under pressure4 ,if the capsule be spherical1 inextensible1 and completely filled with the fluid1 absolutely no deformation can take place4 ,the few eggs that are actually or approximately spherical1 such as those of the tortoise or the owl1 may thus be alternatively explained as cases where little or no deforming pressure has been applied prior to the solidification of the shell1 or else as cases where the capsule was so little capable of extension and so completely filled as to preclude the possibility of deformation4 ,if the capsule be not spherical1 but be inextensible1 then deformation can take place under the external radial compression1 #feg only provided that the pressure tends to make the shape more nearly spherical1 and then only on the further supposition that the capsule is also not entirely filled as the deformation proceeds4 ,in other words1 an incompressible fluid contained in an inextensible envelope cannot be deformed without puckering of the envelope taking place4 ,let us next assume1 as the conditions by which this result may be avoided1 7a7 that the envelope is to some extent extensible1 or 7b7 that the whole structure grows under relatively fixed conditions4 ,the two suppositions are practically identical with one another in effect4 ,it is obvious that1 on the presumption that the envelope is only moderately extensible1 the whole structure can only be distorted to a moderate degree away from the spherical or spheroidal form4 ,at all points the shape is determined by the law of the distribution of radial pressure within the given region of the tube1 surface friction helping to maintain the egg in position4 ,if the egg be under pressure from the oviduct1 but without any marked component either in a forward or backward direction1 the egg will be compressed in the middle1 and will tend more or less to the form of a cylinder with spherical ends4 ,the eggs of the grebe1 cormorant1 or crocodile may be supposed to receive their shape in such circumstances4 ,when the egg is subject to the peristaltic contraction of the oviduct during its formation1 then from the nature and direction of motion of the peristaltic wave the pressure will be greatest somewhere behind the middle of the egg2 in other words1 the tube is converted for the time being into a more conical form1 and the simple result follows that the anterior end of the egg becomes the broader and the posterior end the narrower4 ,with a given shape and size of body1 equilibrium in the tube may be maintained under greater radial pressure towards one end than towards the other4 ,for example1 a cylinder having conical ends1 of semi-angles and respectively1 remains in equilibrium1 apart from friction1 if p cos#b p cos#b 1 so that at the more tapered end where is small p is small4 ,therefore the whole structure might assume such a configuration1 or grow under such conditions1 finally becoming rigid by solidification of the envelope4 #feh ,according to the preceding paragraph1 we must assume some initial distribution of pressure1 some squeeze applied to the posterior part of the egg1 in order to give it its tapering form4 ,but1 that form once acquired1 the egg may remain in equilibrium both as regards form and position within the tube1 even after that excess of pressure on the posterior part is relieved4 ,moreover1 the above equation shews that a normal pressure no greater and 7within certain limits7 actually less acting upon the posterior part than on the anterior part of the egg after the shell is formed will be sufficient to communicate to it a forward motion4 ,this is an important consideration1 for it shews that the ordinary form of an egg1 and even the conical form of an extreme case such as the guillemot's1 is directly favourable to the movement of the egg within the oviduct1 blunt end foremost4 ,the mathematical statement of the whole case is as follows3 ,in our egg1 consisting of an extensible membrane filled with an incompressible fluid and under external pressure1 the equation of the envelope is pn ,t7#a_/r #a_/r7 ,p1 where pn is the normal component of external pressure at a point where r and r are the radii of curvature1 ,t is the tension of the envelope1 and ,p the internal fluid pressure4 ,this is simply the equation of an elastic surface where ,t represents the coefficient of elasticity2 in other words1 a flexible elastic shell has the same mathematical properties as our fluid1 membrane-covered egg4 ,and this is the identical equation which we have already had so frequent occasion to employ in our discussion of the forms of cells2 save only that in these latter we had chiefly to study the tension ,t 7i4e4 the surface-tension of the semi-fluid cell7 and had little or nothing to do with the factor of external pressure 7pn71 which in the case of the egg becomes of chief importance4 ,the above equation is the equation of equilibrium1 so that it must be assumed either that the whole body is at rest or that its motion while under pressure is not such as to affect the result4 ,tangential forces1 which have been neglected1 could modify the form by alteration of ,t4 ,in our case we must1 and may very reasonably1 assume that any movement of the egg down the oviduct during the period when its form is being impressed upon it is very slow1 being possibly balanced by the advance of the #fei peristaltic wave which causes the movement1 as well as by friction4 ,the quantity ,t is the tension of the enclosing capsule--the surrounding membrane4 ,if ,t be constant or symmetrical about the axis of the body1 the body is symmetrical4 ,but the abnormal eggs that a hen sometimes lays1 cylindrical1 annulated1 or quite irregular1 are due to local weakening of the membrane1 in other words1 to asymmetry of ,t4 ,not only asymmetry of ,t1 but also asymmetry of pn1 will render the body subject to deformation1 and this factor1 the unknown but regularly varying1 largely radial1 pressure applied by successive annuli of the oviduct1 is the essential cause of the form1 and variations of form1 of the egg4 ,in fact1 in so far as the postulates correspond near enough to actualities1 the above equation is the equation of all eggs in the universe4 ,at least this is so if we generalise it in the form pn ,t_/r ,t_/r ,p in recognition of a possible difference between the principal tensions4 ,in the case of the spherical egg it is obvious that pn is everywhere equal4 ,the simplest case is where pn #j1 in other words1 where the egg is so small as practically to escape deforming pressure from the tube4 ,but we may also conceive the tube to be so thin-walled and extensible as to press with practically equal force upon all parts of the contained sphere4 ,if while our egg be in process of conformation the envelope be free at any part from external pressure 7that is to say1 if pn #j71 then it is obvious that that part 7if of circular section7 will be a portion of a sphere4 ,this is not unlikely to be the case actually or approximately at one or both poles of the egg1 and is evidently the case over a considerable portion of the anterior end of the plover's egg4 ,in the case of the conical egg with spherical ends1 as is more or less the case in the plover's and the guillemot's1 then at either end of the egg r and r are identical1 and they are greater at the blunt anterior end than at the other4 ,if we may assume that pn vanishes at the poles of the egg1 then it is plain that ,t varies in the neighbourhood of these poles1 and1 further1 that the tension ,t is greatest at and near the small end of the egg4 ,it is here1 in short1 that the egg is most likely to be irregularly distorted or #ffj even to burst1 and it is here that we most commonly find irregularities of shape in abnormal eggs4 ,if one portion of the envelope were to become practically stiff before p ceases to vary1 that would be tantamount to a sudden variation of ,t1 and would introduce asymmetry by the imposition of a boundary condition in addition to the above equation4 ,within the egg lies the yolk1 and the yolk is invariably spherical or very nearly so1 whatever be the form of the entire egg4 ,the reason is simple1 and lies in the fact that the yolk is itself enclosed in another membrane1 between which and the outer membrane lies a fluid the presence of which makes pn for the inner membrane practically constant4 ,the smallness of friction is indicated by the well-known fact that the 8germinal spot0 on the surface of the yolk is always found uppermost1 however we may place and wherever we may open the egg2 that is to say1 the yolk easily rotates within the egg1 bringing its lighter pole uppermost4 ,so1 owing to this lack of friction in the outer fluid1 or white1 whatever shear is produced within the egg will not be easily transmitted to the yolk1 and1 moreover1 owing to the same fluidity1 the yolk will easily recover its normal sphericity after the egg-shell is formed and the unequal pressure relieved4 ,these1 then1 are the general principles involved in1 and illustrated by1 the configuration of an egg2 and they take us as far as we can safely go without actual quantitative determinations1 in each particular case1 of the forces concerned4 ,in certain cases among the invertebrates1 we again find instances of hard-shelled eggs which have obviously been moulded by the oviduct1 or so-called 8ootype10 in which they have lain3 and not merely in such a way as to shew the effects of peristaltic pressure upon a uniform elastic envelope1 but so as to impress upon the egg the more or less irregular form of the cavity1 within which it had been for a time contained and compressed4 ,after this fashion ,dr ,looss#eii of ,cairo has #ffa explained the curious form of the egg in ,bilharzia 7,schistosoma7 haematobium1 a formidable parasitic worm to which is due a disease wide-spread in ,africa and ,arabia1 and an especial scourge of the ,mecca pilgrims4 ,the egg in this worm is provided at one end with a little spine1 which now and then is found to be placed not terminally but laterally or ventrally1 and which when so placed has been looked upon as the mark of a supposed new species1 ,s4 ,mansoni4 ,as ,looss has now shewn1 the little spine must be explained as having been moulded within a little funnel-shaped expansion of the uterus1 just where it communicates with the common duct leading from the ovary and yolk-gland2 by the accumulation of eggs in the ootype1 the one last formed is crowded into a sideways position1 and then1 where the side-wall of the egg bulges in the funnel-shaped orifice of the duct1 a little lateral 8spine0 is formed4 ,in another species1 ,s4 japonicum1 the egg is described as bulging into a so-called 8calotte10 or bubble-like convexity at the end opposite to the spine4 ,this1 ,i think1 may1 with very little doubt1 be ascribed to hardening of the egg-shell having taken place just at the period when partial relief from pressure was being experienced by the egg in the neighbourhood of the dilated orifice of the oviduct4 ,this case of ,bilharzia is not1 from our present point of view1 a very important one1 but nevertheless it is interesting4 ,it ascribes to a mechanical cause a curious peculiarity of form2 it shews1 by reference to this mechanical principle1 that two conditions which were very different to the systematic naturalist's eye1 were really only two simple mechanical modifications of the same thing2 and it destroys the chief evidence for the existence of a supposed new species of worm1 a continued belief in which1 among worms of such great pathogenic importance1 might lead to gravely erroneous pathological deductions4 ,on the ,form of ,sea-urchins ,as a corollary to the problem of the bird's egg1 we may consider for a moment the forms assumed by the shells of the sea-urchins4 ,these latter are commonly divided into two classes1 the ,regular and the ,irregular ,echinids4 ,the regular sea-urchins1 save in #ffb slight details which do not affect our problem1 have a complete radial symmetry4 ,the axis of the animal's body is vertical1 with mouth below and the intestinal outlet above2 and around this axis the shell is built as a symmetrical system4 ,it follows that in horizontal section the shell is everywhere circular1 and we shall have only to consider its form as seen in vertical section or projection4 ,the irregular urchins 7very inaccurately so-called7 have the anal extremity of the body removed from its central1 dorsal situation2 and it follows that they have now a single plane of symmetry1 about which the organism1 shell and all1 is bilaterally symmetrical4 ,we need not concern ourselves in detail with the shapes of their shells1 which may be very simply interpreted1 by the help of radial co-ordinates1 as deformations of the circular or 8regular0 type4 ,the sea-urchin shell consists of a membrane1 stiffened into rigidity by calcareous deposits1 which constitute a beautiful skeleton of separate1 neatly fitting 8ossicles40 ,the rigidity of the shell is more apparent than real1 for the entire structure is1 in a sluggish way1 plastic2 inasmuch as each little ossicle is capable of growth1 and the entire shell grows by increments to each and all of these multitudinous elements1 whose individual growth involves a certain amount of freedom to move relatively to one another2 in a few cases the ossicles are so little developed that the whole shell appears soft and flexible4 ,the viscera of the animal occupy but a small part of the space within the shell1 the cavity being mainly filled by a large quantity of watery fluid1 whose density must be very near to that of the external sea-water4 ,apart from the fact that the sea-urchin continues to grow1 it is plain that we have here the same general conditions as in the egg-shell1 and that the form of the sea-urchin is subject to a similar equilibrium of forces4 ,but there is this important difference1 that an external muscular pressure 7such as the oviduct administers during the consolidation of egg-shell71 is now lacking4 ,in its place we have the steady continuous influence of gravity1 and there is yet another force which in all probability we require to take into consideration4 ,while the sea-urchin is alive1 an immense number of delicate 8tube-feet10 with suckers at their tips1 pass through minute pores #ffc in the shell1 and1 like so many long cables1 moor the animal to the ground4 ,they constitute a symmetrical system of forces1 with one resultant downwards1 in the direction of gravity1 and another outwards in a radial direction2 and if we look upon the shell as originally spherical1 both will tend to depress the sphere into a flattened cake4 ,we need not consider the radial component1 but may treat the case as that of a spherical shell symmetrically depressed under the influence of gravity4 ,this is precisely the condition which we have to deal with in a drop of liquid lying on a plate2 the form of which is determined by its own uniform surface-tension1 plus gravity1 acting against the uniform internal hydrostatic pressure4 ,simple as this system is1 the full mathematical investigation of the form of a drop is not easy1 and we can scarcely hope that the systematic study of the ,echinodermata will ever be conducted by methods based on ,laplace's differential equation#fjj2 but we have no difficulty in seeing that the various forms represented in a series of sea-urchin shells are no other than those which we may easily and perfectly imitate in drops4 ,in the case of the drop of water 7or of any other particular liquid7 the specific surface-tension is always constant1 and the pressure varies inversely as the radius of curvature2 therefore the smaller the drop the more nearly is it able to conserve the spherical form1 and the larger the drop the more does it become flattened under gravity4 ,we can represent the phenomenon by using india-rubber balls filled with water1 of different sizes2 the little ones will remain very nearly spherical1 but the larger will fall down 8of their own weight10 into the form of more and more flattened cakes2 and we see the same thing when we let drops of heavy oil 7such as the orthotoluidene spoken of on p4 #bai71 fall through a tall column of water1 the little ones remaining round1 and the big ones getting more and more flattened as they sink4 ,in the case of the sea-urchin1 the same series of forms may be assumed to occur1 irrespective of size1 through variations in ,t1 the specific tension1 or 8strength10 of the enveloping shell4 ,accordingly we may study1 entirely from this point of view1 such a series as the following 7,fig4 #cbh74 ,in a very few cases1 such as the fossil ,palaeechinus1 we have an approximately spherical #ffd shell1 that is to say a shell so strong that the influence of gravity becomes negligible as a cause of deformation4 ,the ordinary species of ,echinus begin to display a pronounced depression1 and this reaches its maximum in such soft-shelled flexible forms as ,phormosoma4 ,on the general question ,i took the opportunity of consulting ,mr ,c4 ,r4 ,darling1 who is an acknowledged expert in drops1 and he at once agreed with me that such forms as are represented in ,fig4 #cbh are no other than diagrammatic illustrations ,illustration3 ,fig4 #cbh4 ,diagrammatic vertical outlines of various ,sea-urchins3 ,a1 ,palaeechinus2 ,b1 ,echinus acutus2 ,c1 ,cidaris2 ,d1 ,d ,coelopleurus2 ,e1 ,e ,genicopatagus2 ,f1 ,phormosoma luculenter2 ,g1 ,p4 tenuis2 ,h1 ,asthenosoma2 ,i1 ,urechinus4 of various kinds of drops1 8most of which can easily be reproduced in outline by the aid of liquids of approximately equal density to water1 although some of them are fugitive40 ,he found a difficulty in the case of the outline which represents ,asthenosoma1 but the reason for the anomaly is obvious2 the flexible shell has flattened down until it has come in contact with the hard skeleton of the jaws1 or 8,aristotle's lantern10 within1 and the curvature of the outline is accordingly disturbed4 ,the elevated1 conical shells such as those of ,urechinus and ,coelopleurus evidently call for some further explanation2 for there is here some cause at work #ffe to elevate1 rather than to depress the shell4 ,mr ,darling tells me that these forms 8are nearly identical in shape with globules ,i have frequently obtained1 in which1 on standing1 bubbles of gas rose to the summit and pressed the skin upwards1 without being able to escape40 ,the same condition may be at work in the sea-urchin2 but a similar tendency would also be manifested by the presence in the upper part of the shell of any accumulation of substance lighter than water1 such as is actually present in the masses of fatty1 oily eggs4 ,on the ,form and ,branching of ,blood-vessels ,passing to what may seem a very different subject1 we may investigate a number of interesting points in connection with the form and structure of the blood-vessels1 on the same principle and by help of the same equations as those we have used1 for instance1 in studying the egg-shell4 ,we know that the fluid pressure 7,p7 within the vessel is balanced by 7#a7 the tension 7,t7 of the wall1 divided by the radius of curvature1 and 7#b7 the external pressure 7pn71 normal to the wall3 according to our formula ,p pn ,t7#a_/r #a_/r74 ,if we neglect the external pressure1 that is to say any support which may be given to the vessel by the surrounding tissues1 and if we deal only with a cylindrical vein or artery1 this formula becomes simplified to the form ,p ,t_/,r4 ,that is to say1 under constant pressure1 the tension varies as the radius4 ,but the tension1 per unit area of the vessel1 depends upon the thickness of the wall1 that is to say on the amount of membranous and especially of muscular tissue of which it is composed4 ,therefore1 so long as the pressure is constant1 the thickness of the wall should vary as the radius1 or as the diameter1 of the blood-vessel4 ,but it is not the case that the pressure is constant1 for it gradually falls off1 by loss through friction1 as we pass from the large arteries to the small2 and accordingly we find that while1 for a time1 the cross-sections of the larger and smaller vessels are symmetrical figures1 with the wall-thickness proportional to the size of the tube1 this proportion is gradually lost1 and the walls #fff of the small arteries1 and still more of the capillaries1 become exceedingly thin1 and more so than in strict proportion to the narrowing of the tube4 ,in the case of the heart we have1 within each of its cavities1 a pressure which1 at any given moment1 is constant over the whole wall-area1 but the thickness of the wall varies very considerably4 ,for instance1 in the left ventricle1 the apex is by much the thinnest portion1 as it is also that with the greatest curvature4 ,we may assume1 therefore 7or at least suspect71 that the formula1 t7#a_/r #a_/r7 ,c1 holds good2 that is to say1 that the thickness 7t7 of the wall varies inversely as the mean curvature4 ,this may be tested experimentally1 by dilating a heart with alcohol under a known pressure1 and then measuring the thickness of the walls in various parts after the whole organ has become hardened4 ,by this means it is found that1 for each of the cavities1 the law holds good with great accuracy#fja4 ,moreover1 if we begin by dilating the right ventricle and then dilate the left in like manner1 until the whole heart is equally and symmetrically dilated1 we find 7#a7 that we have had to use a pressure in the left ventricle from six to seven times as great as in the right ventricle1 and 7#b7 that the thickness of the walls is just in the same proportion#fjb4 ,a great many other problems of a mechanical or hydrodynamical kind arise in connection with the blood-vessels#fjc1 and while these are chiefly interesting to the physiologist they have also their interest for the morphologist in so far as they bear upon structure and form4 ,as an example of such mechanical problems #ffg we may take the conditions which determine or help to determine the manner of branching of an artery1 or the angle at which its branches are given off2 for1 as ,john ,hunter said#fjd1 8,to keep up a circulation sufficient for the part1 and no more1 ,nature has varied the angle of the origin of the arteries accordingly40 ,the general principle is that the form and arrangement of the blood-vessels is such that the circulation proceeds with a minimum of effort1 and with a minimum of wall-surface1 the latter condition leading to a minimum of friction and being therefore included in the first4 ,what1 then1 should be the angle of branching1 such that there shall be the least possible loss of energy in the course of the circulation8 ,in order to solve this problem in any particular case we should obviously require to know 7#a7 how the loss of energy depends upon the distance travelled1 and 7#b7 how the loss of energy varies with the diameter of the vessel4 ,the loss of energy is evidently greater in a narrow tube than in a wide one1 and greater1 obviously1 in a long journey than a short4 ,if the ,illustration3 ,fig4 #cbi4 large artery1 ,,ab1 give off a comparatively narrow branch leading to ,p 7such as ,,cp1 or ,,dp71 the route ,,acp is evidently shorter than ,,adp1 but on the other hand1 by the latter path1 the blood has tarried longer in the wide vessel ,,ab1 and has had a shorter course in the narrow branch4 ,the relative advantage of the two paths will depend on the loss of energy in the portion ,,cd1 as compared with that in the alternative portion ,,cd1 the latter being short and narrow1 the former long and wide4 ,if we ask1 then1 which factor is the more important1 length or width1 we may safely take it that the question is one of degree3 and that the factor of width will become much the more important wherever the artery and its branch are markedly unequal in size4 ,in other words1 it would seem that for small branches a large angle of bifurcation1 and for large branches a small one1 is always the better4 ,roux has laid down certain rules in regard to the branching of arteries1 which correspond with the general #ffh conclusions which we have just arrived at4 ,the most important of these are as follows3 7#a7 ,if an artery bifurcate into two equal branches1 these branches come off at equal angles to the main stem4 7#b7 ,if one of the two branches be smaller than the other1 then the main branch1 or continuation of the original artery1 makes with the latter a smaller angle than does the smaller or 8lateral0 branch4 ,and 7#c7 all branches which are so small that they scarcely seem to weaken or diminish the main stem come off from it at a large angle1 from about #gj to #ij4 ,illustration3 ,fig4 #ccj4 ,we may follow ,hess in a further investigation of this phenomenon4 ,let ,,ab be an artery1 from which a branch has to be given off so as to reach ,p1 and let ,,acp1 ,,adp1 etc41 be alternative courses which the branch may follow3 ,,cd1 ,,de1 etc41 in the diagram1 being equal distances 7 l7 along ,,ab4 ,let us call the angles ,,pcd1 ,,pce1 x#a1 x#b1 etc43 and the distances ,,cd1 ,,de1 by which each branch exceeds the next in length1 we shall call l#a1 l#b1 etc4 ,now it is evident that1 of the courses shewn1 ,,acp is the shortest which the blood can take1 but it is also that by which its transit through the narrow branch is the longest4 ,we may reduce its transit through the narrow branch more and more1 till we come to ,,cgp1 or rather to a point where the branch comes off at right angles to the main stem2 but in so doing we very considerably increase the whole distance travelled4 ,we may take it that there will be some intermediate point which will strike the balance of advantage4 ,now it is easy to shew that if1 in ,fig4 #ccj1 the route ,,adp and ,,aep 7two contiguous routes7 be equally favourable1 then any other route on either side of these1 such as ,,acp or ,,afp1 must be less favourable than either4 ,let ,,adp and ,,aep1 then1 be equally favourable2 that is to say1 let the loss of energy which the blood suffers in its passage along these two routes be equal4 #ffi ,then1 if we make the distance ,,de very small1 the angles x#b and x#c are nearly equal1 and may be so treated4 ,and again1 if ,,de be very small1 then ,,de,e becomes a right angle1 and l#b 7or ,,de7 l cos x#b4 ,but if ,l be the loss of energy per unit distance in the wide tube ,,ab1 and ,l be the corresponding loss of energy in the narrow tube ,,dp1 etc41 then l,l l#b ,l1 because1 as we have assumed1 the loss of energy on the route ,,dp is equal to that on the whole route ,,dep4 ,therefore l,l l,l cos x#b1 and cos x#b ,l_/,l4 ,that is to say1 the most favourable angle of branching will be such that the cosine of the angle is equal to the ratio of the loss of energy which the blood undergoes1 per unit of length1 in the main vessel1 as compared with that which it undergoes in the branch4 ,while these statements are so far true1 and while they undoubtedly cover a great number of observed facts1 yet it is plain that1 as in all such cases1 we must regard them not as a complete explanation1 but as factors in a complicated phenomenon3 not forgetting that 7as the most learned of all students of the heart and arteries1 ,dr ,thomas ,young1 said in his ,croonian lecture#fje7 all such questions as these1 and all matters connected with the muscular and elastic powers of the blood-vessels1 8belong to the most refined departments of hydraulics40 ,some other explanation must be sought in order to account for a phenomenon which particularly impressed ,john ,hunter's mind1 namely the gradually altering angle at which the successive intercostal arteries are given off from the thoracic aorta3 the special interest of this case arising from the regularity and symmetry of the series1 for 8there is not another set of arteries in the body whose origins are so much the same1 whose offices are so much the same1 whose distances from their origin to the place of use1 and whose uses 8 sizes#fjf are so much the same40 #fgj ,,chapter ,,xvi ,,on ,,form ,,and ,,mechanical ,,efficiency ,there is a certain large class of morphological problems of which we have not yet spoken1 and of which we shall be able to say but little4 ,nevertheless they are so important1 so full of deep theoretical significance1 and are so bound up with the general question of form and of its determination as a result of growth1 that an essay on growth and form is bound to take account of them1 however imperfectly and briefly4 ,the phenomena which ,i have in mind are just those many cases where adaptation1 in the strictest sense1 is obviously present1 in the clearly demonstrable form of mechanical fitness for the exercise of some particular function or action which has become inseparable from the life and well-being of the organism4 ,when we discuss certain so-called 8adaptations0 to outward circumstance1 in the way of form1 colour and so forth1 we are often apt to use illustrations convincing enough to certain minds but unsatisfying to others--in other words1 incapable of demonstration4 ,with regard to colouration1 for instance1 it is by colours 8cryptic10 8warning10 8signalling10 8mimetic10 and so on#fjg1 that we prosaically expound1 and slavishly profess to justify1 the vast ,aristotelian synthesis that ,nature makes all things with a purpose and 8does nothing in vain40 ,only for a moment let us glance at some few instances by which the modern teleologist accounts for this or that manifestation of colour1 and is led on and on to beliefs and doctrines to which it becomes more and more difficult to subscribe4 #fga ,some dangerous and malignant animals are said 7in sober earnest7 to wear a perpetual war-paint1 in order to 8remind their enemies that they had better leave them alone#fjh40 ,the wasp and the hornet1 in gallant black and gold1 are terrible as an army with banners2 and the ,gila ,monster 7the poison-lizard of the ,arizona desert7 is splashed with scarlet--its dread and black complexion stained with heraldry more dismal4 ,but the wasp-like livery of the noisy1 idle hover-flies and drone-flies is but stage armour1 and in their tinsel suits the little counterfeit cowardly knaves mimic the fighting crew4 ,the jewelled splendour of the peacock and the humming-bird1 and the less effulgent glory of the lyre-bird and the ,argus pheasant1 are ascribed to the unquestioned prevalence of vanity in the one sex and wantonness in the other#fji4 ,the zebra is striped that it may graze unnoticed on the plain1 the tiger that it may lurk undiscovered in the jungle2 the banded ,chaetodont and ,pomacentrid fishes are further bedizened to the hues of the coral-reefs in which they dwell#faj4 ,the tawny lion is yellow as the desert sand2 but the leopard wears its dappled hide to blend1 as it crouches on the branch1 with the sun-flecks peeping through the leaves4 ,the ptarmigan and the snowy owl1 the arctic fox and the polar bear1 are white among the snows2 but go he north or go he south1 the raven 7like the jackdaw7 is boldly and impudently black4 ,the rabbit has his white scut1 and sundry antelopes their piebald flanks1 that one timorous fugitive may hie after another1 spying the warning signal4 ,the primeval terrier or collie-dog #fgb had brown spots over his eyes that he might seem awake when he was sleeping#faa3 so that an enemy might let the sleeping dog lie1 for the singular reason that he imagined him to be awake4 ,and a flock of flamingos1 wearing on rosy breast and crimson wings a garment of invisibility1 fades away into the sky at dawn or sunset like a cloud incarnadine#fab4 ,to buttress the theory of natural selection the same instances of 8adaptation0 7and many more7 are used1 which in an earlier but not distant age testified to the wisdom of the ,creator and revealed to simple piety the high purpose of ,god4 ,in the words of a certain learned theologian#fac1 8,the free use of final causes to explain what seems obscure was temptingly easy 4444 ,hence the finalist was often the man who made a liberal use of the ignava ratio1 or lazy argument3 when you failed to explain a thing by the ordinary process of causality1 you could 8explain0 it by reference to some purpose of nature or of its ,creator4 ,this method lent itself with dangerous facility to the well-meant endeavours of the older theologians to expound and emphasise the beneficence of the divine purpose40 ,mutatis mutandis1 the passage carries its plain message to the naturalist4 ,the fate of such arguments or illustrations is always the same4 ,they attract and captivate for awhile2 they go to the building of a creed1 which contemporary orthodoxy defends under its severest penalties3 but the time comes when they lose their fascination1 they somehow cease to satisfy and to convince1 their foundations are discovered to be insecure1 and in the end no man troubles to controvert them4 ,but of a very different order from all such 8adaptations0 as these1 are those very perfect adaptations of form which1 for instance1 fit a fish for swimming or a bird for flight4 ,here we are #fgc far above the region of mere hypothesis1 for we have to deal with questions of mechanical efficiency where statical and dynamical considerations can be applied and established in detail4 ,the naval architect learns a great part of his lesson from the investigation of the stream-lines of a fish2 and the mathematical study of the stream-lines of a bird1 and of the principles underlying the areas and curvatures of its wings and tail1 has helped to lay the very foundations of the modern science of aeronautics4 ,when1 after attempting to comprehend the exquisite adaptation of the swallow or the albatross to the navigation of the air1 we try to pass beyond the empirical study and contemplation of such perfection of mechanical fitness1 and to ask how such fitness came to be1 then indeed we may be excused if we stand wrapt in wonderment1 and if our minds be occupied and even satisfied with the conception of a final cause4 ,and yet all the while1 with no loss of wonderment nor lack of reverence1 do we find ourselves constrained to believe that somehow or other1 in dynamical principles and natural law1 there lie hidden the steps and stages of physical causation by which the material structure was so shapen to its ends#fad4 ,but the problems associated with these phenomena are difficult at every stage1 even long before we approach to the unsolved secrets of causation2 and for my part ,i readily confess that ,i lack the requisite knowledge for even an elementary discussion of the form of a fish or of a bird4 ,but in the form of a bone we have a problem of the same kind and order1 so far simplified and particularised that we may to some extent deal with it1 and may possibly even find1 in our partial comprehension of it1 a partial clue to the principles of causation underlying this whole class of problems4 ,before we speak of the form of a bone1 let us say a word about1 the mechanical properties of the material of which it is built#fae1 in #fgd relation to the strength it has to manifest or the forces it has to resist3 understanding always that we mean thereby the properties of fresh or living bone1 with all its organic as well as inorganic constituents1 for dead1 dry bone is a very different thing4 ,in all the structures raised by the engineer1 in beams1 pillars and girders of every kind1 provision has to be made1 somehow or other1 for strength of two kinds1 strength to resist compression or crushing1 and strength to resist tension or pulling asunder4 ,the evenly loaded column is designed with a view to supporting a downward pressure1 the wire-rope1 like the tendon of a muscle1 is adapted only to resist a tensile stress2 but in many or most cases the two functions are very closely inter-related and combined4 ,the case of a loaded beam is a familiar one2 though1 by the way1 we are now told that it is by no means so simple as it looks1 and indeed that 8the stresses and strains in this log of timber are so complex that the problem has not yet been solved in a manner that reasonably accords with the known strength of the beam as found by actual experiment#faf40 ,however1 be that as it may1 we know1 ,illustration3 ,fig4 #cca4 roughly1 that when the beam is loaded in the middle and supported at both ends1 it tends to be bent into an arc1 in which condition its lower fibres are being stretched1 or are undergoing a tensile stress1 while its upper fibres are undergoing compression4 ,it follows that in some intermediate layer there is a 8neutral zone10 where the fibres of the wood are subject to no stress of either kind4 ,in like manner1 a vertical pillar if unevenly loaded 7as1 for instance1 the shaft of our thigh-bone normally is7 will tend to bend1 and so to endure compression on its concave1 and tensile stress upon its convex side4 ,in many cases it is the business of the engineer to separate out1 as far as possible1 the pressure-lines from the tension-lines1 in order to use separate modes of construction1 or even different materials for each4 ,in a #fge suspension-bridge1 for instance1 a great part of the fabric is subject to tensile strain only1 and is built throughout of ropes or wires2 but the massive piers at either end of the bridge carry the weight of the whole structure and of its load1 and endure all the 8compression-strains0 which are inherent in the system4 ,very much the same is the case in that wonderful arrangement of struts and ties which constitute1 or complete1 the skeleton of an animal4 ,the 8skeleton10 as we see it in a ,museum1 is a poor and even a misleading picture of mechanical efficiency#fag4 ,from the engineer's point of view1 it is a diagram showing all the compression-lines1 but by no means all of the tension-lines of the construction2 it shews all the struts1 but few of the ties1 and perhaps we might even say none of the principal ones2 it falls all to pieces unless we clamp it together1 as best we can1 in a more or less clumsy and immobilised way4 ,but in life1 that fabric of struts is surrounded and interwoven with a complicated system of ties3 ligament and membrane1 muscle and tendon1 run between bone and bone2 and the beauty and strength of the mechanical construction lie not in one part or in another1 but in the complete fabric which all the parts1 soft and hard1 rigid and flexible1 tension-bearing and pressure-bearing1 make up together#fah4 ,however much we may find a tendency1 whether in nature or art1 to separate these two constituent factors of tension and compression1 we cannot do so completely2 and accordingly the engineer seeks for a material which shall1 as nearly as possible1 offer equal resistance to both kinds of strain4 ,in the following table--,i borrow it from ,sir ,donald ,mac,alister--we see approximately the relative breaking 7or tearing7 limit and crushing limit in a few substances4 #fgf ,average ,strength of ,materials 7in kg4 per sq4 mm474 ,tensile ,crushing strength strength ,steel #ajj #ade ,wrought ,iron #dj #bj ,cast ,iron #ab #gb ,wood #d #b ,bone #i-ab #ac-af ,at first sight1 bone seems weak indeed2 but it has the great and unusual advantage that it is very nearly as good for a tie as for a strut1 nearly as strong to withstand rupture1 or tearing asunder1 as to resist crushing4 ,we see that wrought-iron is only half as strong to withstand the former as the latter2 while in cast-iron there is a still greater discrepancy the other way1 for it makes a good strut but a very bad tie indeed4 ,cast-steel is not only actually stronger than any of these1 but it also possesses1 like bone1 the two kinds of strength in no very great relative disproportion4 ,when the engineer constructs an iron or steel girder1 to take the place of the primitive wooden beam1 we know that he takes advantage of the elementary principle we have spoken of1 and saves weight and economises material by leaving out as far as possible all the middle portion1 all the parts in the neighbourhood of the 8neutral zone02 and in so doing he reduces his girder to an upper and lower 8flange10 connected together by a 8web10 the whole resembling1 in cross-section1 an ,i or an 4 ,but it is obvious that1 if the strains in the two flanges are to be equal as well as opposite1 and if the material be such as cast-iron or wrought-iron1 one or other flange must be made much thicker than the other in order that it may be equally strong2 and if at times the two flanges have1 as it were1 to change places1 or play each other's parts1 then there must be introduced a margin of safety by making both flanges thick enough to meet that kind of stress in regard to which the material happens to be weakest4 ,there is great economy1 then1 in any material which is1 as nearly as possible1 equally strong in both ways2 and so we see that1 from the engineer's or contractor's point of view1 bone is a very good and suitable material for purposes of construction4 #fgg ,the ,i or the ,h-girder or rail is designed to resist bending in one particular direction1 but if1 as in a tall pillar1 it be necessary to resist bending in all directions alike1 it is obvious that the tubular or cylindrical construction best meets the case2 for it is plain that this hollow tubular pillar is but the ,i-girder turned round every way1 in a 8solid of revolution10 so that on any two opposite sides compression and tension are equally met and resisted1 and there is now no need for any substance at all in the way of web or 8filling0 within the hollow core of the tube4 ,and it is not only in the supporting pillar that such a construction is useful2 it is appropriate in every case where stiffness is required1 where bending has to be resisted4 ,the long bone of a bird's wing has little or no weight to carry1 but it has to withstand powerful bending moments2 and in the arm-bone of a long-winged bird1 such as an albatross1 we see the tubular construction manifested in its perfection1 the bony substance being reduced to a thin1 perfectly cylindrical1 and almost empty shell4 ,the quill of the bird's feather1 the hollow shaft of a reed1 the thin tube of the wheat-straw bearing its heavy burden in the ear1 are all illustrations which ,galileo used in his account of this mechanical principle#fai4 ,two points1 both of considerable importance1 present themselves here1 and we may deal with them before we go further4 ,in the first place1 it is not difficult to see that1 in our bending beam1 the strain is greatest at its middle2 if we press our walking-stick hard against the ground1 it will tend to snap midway4 ,hence1 if our cylindrical column be exposed to strong bending stresses1 it will be prudent and economical to make its walls thickest in the middle and thinning off gradually towards the ends2 and if we look at a longitudinal section of a thigh-bone1 we shall see that this is just what nature has done4 ,the thickness of the walls is nothing less than a diagram1 or 8graph10 of the 8bending-moments0 from one point to another along the length of the bone4 ,illustration3 ,fig4 #ccb4 ,the second point requires a little more explanation4 ,if we #fgh imagine our loaded beam to be supported at one end only 7for instance1 by being built into a wall71 so as to form what is called a 8bracket0 or 8cantilever10 then we can see1 without much difficulty1 that the lines of stress in the beam run somewhat as in the accompanying diagram4 ,immediately under the load1 the 8compression-lines0 tend to run vertically downward2 but where the bracket is fastened to the wall1 there is pressure directed horizontally against the wall in the lower part of the surface of attachment2 and the vertical beginning and the horizontal end of these pressure-lines must be continued into one another in the form of some even mathematical curve--which1 as it happens1 is part of a parabola4 ,the tension-lines are identical in form with the compression-lines1 of which they constitute the 8mirror-image02 and where the two systems intercross1 they do so at right angles1 or 8orthogonally0 to one another4 ,such systems of stress-lines as these we shall deal with again2 but let us take note here of the important1 though well-nigh obvious fact1 that while in the beam they both unite to carry the load1 yet it is always possible to weaken one set of lines at the expense of the other1 and in some cases to do altogether away with one set or the other4 ,for example1 when we replace our end-supported beam by a curved bracket1 bent upwards or downwards as the case may be1 we have evidently cut away in the one case the greater part of the tension-lines1 and in the other the greater part of the compression-lines4 ,and if instead of bridging a stream with our beam of wood we bridge it with a rope1 it is evident that this new construction contains all the tension-lines1 but none of the compression-lines of the old4 ,the biological interest connected with this principle lies chiefly in the mechanical construction of the rush or the straw1 or any other typically cylindrical stem4 ,the material of which the stalk is constructed is very weak to withstand compression1 but parts of it have a very great tensile strength4 ,schwendener1 who was both botanist and engineer1 has elaborately investigated the factor of strength in the cylindrical stem1 which ,galileo was the first to call attention to4 #fgi ,schwendener#fbj shewed that the strength was concentrated in the little bundles of 8bast-tissue0 but that these bast-fibres had a tensile strength per square mm4 of section1 up to the limit of elasticity1 not less than that of steel-wire of such quality as was in use in his day4 ,for instance1 we see in the following table the load which various fibres1 and various wires1 were found capable of sustaining1 not up to the breaking-point1 but up to the 8elastic limit10 or point beyond which complete recovery to the original length took place no longer after release of the load4 ,stress1 or load in gms4 ,strain1 or amount per sq4 mm41 at of stretching1 ,limit of ,elasticity per mille ,secale cereale #ae-bj #dd ,lilium auratum #ai #gf ,phormium tenax #bj #acj ,papyrus antiquorum #bj #aeb ,molinia coerulea #bb #aaj ,pincenectia recurvata #be #ade ,copper wire #aba #aj ,brass wire #acc #ace ,iron wire #bai #aj ,steel wire #bdf #ab ,in other respects1 it is true1 the plant-fibres were inferior to the wires2 for the former broke asunder very soon after the limit of elasticity was passed1 while the iron-wire could stand1 before snapping1 three times the load which was measured by its limit of elasticity3 in the language of a modern engineer1 the bast-fibres had a low 8yield-point10 little above the elastic limit4 ,but nevertheless1 within certain limits1 plant-fibre and wire were just as good and strong one as the other4 ,and then ,schwendener proceeds to shew1 in many beautiful diagrams1 the various ways in which these strands of strong tensile tissue are arranged in various cases3 sometimes1 in the simpler cases1 forming numerous small bundles arranged in a peripheral ring1 not quite at the periphery1 for a certain amount of space has to be left for living and active tissue2 sometimes in a sparser ring of larger and #fhj stronger bundles2 sometimes with these bundles further strengthened by radial balks or ridges2 sometimes with all the fibres set ,illustration3 ,fig4 #ccc4 close together in a continuous hollow cylinder4 ,in the case figured in ,fig4 #ccc ,schwendener calculated that the resistance to bending was at least twenty-five times as great as it would have been had the six main bundles been brought close together in a solid core4 ,in many cases the centre of the stem is altogether empty2 in all other cases it is filled with soft tissue1 suitable for the ascent of sap or other functions1 but never such as to confer mechanical rigidity4 ,in a tall conical stem1 such as that of a palm-tree1 we can see not only these principles in the construction of the cylindrical trunk1 but we can observe1 towards the apex1 the bundles of fibre curving over and intercrossing orthogonally with one another1 exactly after the fashion of our stress-lines in ,fig4 #ccb2 but of course1 in this case1 we are still dealing with tensile members1 the opposite bundles taking on in turn1 as the tree sways1 the alternate function of resisting tensile strain#fba4 ,let us now come1 at last1 to the mechanical structure of bone1 of which we find a well-known and classical illustration in the various bones of the human leg4 ,in the case of the tibia1 the bone is somewhat widened out above1 and its hollow shaft is capped by an almost flattened roof1 on which the weight of the body directly rest4 ,it is obvious that1 under these circumstances1 the engineer would find it necessary to devise means for supporting this flat roof1 and for distributing the vertical pressures which impinge upon it to the cylindrical walls of the shaft4 #fha ,illustration3 ,fig4 #ccd4 ,head of the human femur in section4 7,after ,schfer1 from a photo by ,prof4 ,a4 ,robinson47 ,in the case of the bird's wing-bone1 the hollow of the bone is practically empty1 as we have already said1 being filled only with air save for a thin layer of living tissue immediately within the cylinder of bone2 but in our own bones1 and all weight-carrying bones in general1 the hollow space is filled with marrow1 blood-vessels and other tissues2 and among these living tissues lies a fine lattice-work of little interlaced 8trabeculae0 of bone1 forming the so-called 8cancellous tissue40 ,the older anatomists were content to describe this cancellous tissue as a sort of 8spongy network10 or irregular honeycomb1 until1 some fifty years ago1 a remarkable discovery was made regarding it4 ,it was found by ,hermann ,meyer 7and afterwards shewn in greater detail by ,julius ,wolff and others7 that the trabeculae1 as seen in a longitudinal section of a long bone1 were arranged in a very definite and orderly way2 in the femur1 they spread in beautiful curving #fhb lines from the head to the tubular shaft of the bone1 and these bundles of lines were crossed by others1 with so nice a regularity of arrangement that each intercrossing was as nearly as possible an orthogonal one3 that is to say1 the one set of fibres crossed the other everywhere at right angles4 ,a great engineer1 ,professor ,culmann of ,zrich 7to whom1 by the way1 we owe the whole modern method of 8graphic statics071 happened to see some of ,meyer's drawings and preparations1 and he recognised in a moment that in the arrangement of the trabeculae we had ,illustration3 ,fig4 #cce4 ,crane-head and femur4 7,after ,culmann and ,h4 ,meyer47 nothing more nor less than a diagram of the lines of stress1 or directions of compression and tension1 in the loaded structure3 in short1 that nature was strengthening the bone in precisely the manner and direction in which strength was needed4 ,in the accompanying diagram of a crane-head1 by ,culmann1 we recognise a slight modification 7caused entirely by the curved shape of the structure7 of the still simpler lines of tension and compression which we have already seen in our end-supported beam as represented in ,fig4 #ccb4 ,in the shaft of the crane1 the concave #fhc or inner side1 overhung by the loaded head1 is the 8compression-member02 the outer side is the 8tension-member02 and the pressure-lines1 starting from the loaded surface1 gather themselves together1 always in the direction of the resultant pressure1 till they form a close bundle running down the compressed side of the shaft3 while the tension-lines1 running upwards along the opposite side of the shaft1 spread out through the head1 orthogonally to1 and linking together1 the system of compression-lines4 ,the head of the femur 7,fig4 #cce7 is a little more complicated in form and a little less symmetrical than ,culmann's diagrammatic crane1 from which it chiefly differs in the fact that the load is divided into two parts1 that namely which is borne by the head of the bone1 and that smaller portion which rests upon the great trochanter2 but this merely amounts to saying that a notch has been cut out of the curved upper surface of the structure1 and we have no difficulty in seeing that the anatomical arrangement of the trabeculae follows precisely the mechanical distribution of compressive and tensile stress or1 in other words1 accords perfectly with the theoretical stress-diagram of the crane4 ,the lines of stress are bundled close together along the sides of the shaft1 and lost or concealed there in the substance of the solid wall of bone2 but in and near the head of the bone1 a peripheral shell of bone does not suffice to contain them1 and they spread out through the central mass in the actual concrete form of bony trabeculae#fbb4 #fhd ,mutatis mutandis1 the same phenomenon may be traced in any other bone which carries weight and is liable to flexure2 and in the os calcis and the tibia1 and more or less in all the bones of the lower limb1 the arrangement is found to be very simple and clear4 ,illustration3 ,fig4 #ccf4 ,diagram of stress-lines in the human foot4 7,from ,sir ,d4 ,mac,alister1 after ,h4 ,meyer47 ,thus1 in the os calcis1 the weight resting on the head of the bone has to be transmitted partly through the backward-projecting heel to the ground1 and partly forwards through its articulation with the cuboid bone1 to the arch of the foot4 ,we thus have1 very much as in a triangular roof-tree1 two compression-members1 sloping apart from one another2 and these have to be bound together by a 8tie0 or tension-member1 corresponding to the third1 horizontal member of the truss4 ,illustration3 ,fig4 #ccg4 ,trabecular structure of the os calcis4 7,from ,mac,alister47 ,so far1 dealing wholly with the stresses and strains due to tension and compression1 we have altogether omitted to speak of a third very important factor in the engineer's calculations1 namely what is known as 8shearing stress40 ,a shearing force is one which produces 8angular distortion0 in a figure1 or 7what comes to the same thing7 which tends to cause its particles to #fhe slide over one another4 ,a shearing stress is a somewhat complicated thing1 and we must try to illustrate it 7however imperfectly7 in the simplest possible way4 ,if we build up a pillar1 for instance1 of a pile of flat horizontal slates1 or of a pack of cards1 a vertical load placed upon it will produce compression1 but will have no tendency to cause one card to slide1 or shear1 upon another2 and in like manner1 if we make up a cable of parallel wires and1 letting it hang vertically1 load it evenly with a weight1 again the tensile stress produced has no tendency to cause one wire to slip or shear upon another4 ,but the case would have been very different if we had built up our pillar of cards or slates lying obliquely to the lines of pressure1 for then at once there would have been a tendency for the elements of the pile to slip and slide asunder1 and to produce what the geologists call 8a fault0 in the structure4 ,somewhat more generally1 if ,,ab be a bar1 or pillar1 of cross-section a under a direct load ,p1 giving a stress per unit area p1 then the whole pressure ,p pa4 ,let ,,cd be an oblique section1 inclined at an angle to the cross-section2 the pressure on ,,cd will evidently be pa cos 4 ,but at any point ,o in ,,cd1 the pressure ,p may be resolved into the force ,q acting along ,,cd1 and ,n perpendicular to it3 where ,n ,p cos 1 and ,q ,p sin pa sin 4 ,the whole force ,q upon ,,cd q area of ,,cd1 which is q a_/7cos 74 #fhf ,therefore qa_/7cos 7 pa sin 1 therefore q p sin cos 1 p sin #b4 ,therefore when sin #b #a1 that is1 when #de1 q is a maximum1 and p_/#b2 and when sin #b #j1 that is when #j or #ij1 then q vanishes altogether4 ,illustration3 ,fig4 #cch4 ,this is as much as to say1 that a shearing stress vanishes altogether along the lines of maximum compression or tension2 it has a definite value in all other positions1 and a maximum value when it is inclined at #de to either1 or half-way between the two4 ,this may be further illustrated in various simple ways4 ,when we submit a cubical block of iron to compression in the testing machine1 it does not tend to give way by crumbling all to pieces2 but as a rule it disrupts by shearing1 and along some plane approximately at #de to the axis of compression4 ,again1 in the beam which we have already considered under a bending moment1 we know that if we substitute for it a pack of cards1 they will be strongly sheared on one another2 and the shearing stress is greatest in the 8neutral zone10 where neither tension nor compression is manifested3 that is to say in the line which cuts at equal angles of #de the orthogonally intersecting lines of pressure and tension4 ,in short we see that1 while shearing stresses can by no means be got rid of1 the danger of rupture or breaking-down under shearing stress is completely got rid of when we arrange the materials of our construction wholly along the pressure-lines and tension-lines of the system2 for along these lines there is no shear4 ,to apply these principles to the growth and development of our bone1 we have only to imagine a little trabecula 7or group of trabeculae7 being secreted and laid down fortuitously in any direction within the substance of the bone4 ,if it lie in the direction of one of the pressure-lines1 for instance1 it will be in a position of comparative equilibrium1 or minimal disturbance2 but if it be inclined obliquely to the pressure-lines1 the shearing force will at once tend to act upon it and move it away4 ,this is neither more nor less than what happens when we comb our #fhg hair1 or card a lock of wool3 filaments lying in the direction of the comb's path remain where they were2 but the others1 under the influence of an oblique component of pressure1 are sheared out of their places till they too come into coincidence with the lines of force4 ,so straws show how the wind blows--or rather how it has been blowing4 ,for every straw that lies askew to the wind's path tends to be sheared into it2 but as soon as it has come to lie the way of the wind it tends to be disturbed no more1 save 7of course7 by a violence such as to hurl it bodily away4 ,in the biological aspect of the case1 we must always remember that our bone is not only a living1 but a highly plastic structure2 the little trabeculae are constantly being formed and deformed1 demolished and formed anew4 ,here1 for once1 it is safe to say that 8heredity0 need not and cannot be invoked to account for the configuration and arrangement of the trabeculae3 for we can see them1 at any time of life1 in the making1 under the direct action and control of the forces to which the system is exposed4 ,if a bone be broken and so repaired that its parts lie somewhat out of their former place1 so that the pressure-and tension-lines have now a new distribution1 before many weeks are over the trabecular system will be found to have been entirely remodelled1 so as to fall into line with the new system of forces4 ,and as ,wolff pointed out1 this process of reconstruction extends a long way off from the seat of injury1 and so cannot be looked upon as a mere accident of the physiological process of healing and repair2 for instance1 it may happen that1 after a fracture of the shaft of a long bone1 the trabecular meshwork is wholly altered and reconstructed within the distant extremities of the bone4 ,moreover1 in cases of transplantation of bone1 for example when a diseased metacarpal is repaired by means of a portion taken from the lower end of the ulna1 with astonishing quickness the plastic capabilities of the bony tissue are so manifested that neither in outward form nor inward structure can the old portion be distinguished from the new4 ,herein then lies1 so far as we can discern it1 a great part at least of the physical causation of what at first sight strikes us as a purely functional adaptation3 as a phenomenon1 in other words1 #fhh whose physical cause is as obscure as its final cause or end is1 apparently1 manifest4 ,partly associated with the same phenomenon1 and partly to be looked upon 7meanwhile at least7 as a fact apart1 is the very important physiological truth that a condition of strain1 the result of a stress1 is a direct stimulus to growth itself4 ,this indeed is no less than one of the cardinal facts of theoretical biology4 ,the soles of our boots wear thin1 but the soles of our feet grow thick1 the more we walk upon them3 for it would seem that the living cells are 8stimulated0 by pressure1 or by what we call 8exercise10 to increase and multiply4 ,the surgeon knows1 when he bandages a broken limb1 that his bandage is doing something more than merely keeping the parts together3 and that the even1 constant pressure which he skilfully applies is a direct encouragement of growth and an active agent in the process of repair4 ,in the classical experiments of ,sdillot#fbc1 the greater part of the shaft of the tibia was excised in some young puppies1 leaving the whole weight of the body to rest upon the fibula4 ,the latter bone is normally about one-fifth or sixth of the diameter of the tibia2 but under the new conditions1 and under the 8stimulus0 of the increased load1 it grew till it was as thick or even thicker than the normal bulk of the larger bone4 ,among plant tissues this phenomenon is very apparent1 and in a somewhat remarkable way2 for a strain caused by a constant or increasing weight 7such as that in the stalk of a pear while the pear is growing and ripening7 produces a very marked increase of strength without any necessary increase of bulk1 but rather by some histological1 or molecular1 alteration of the tissues4 ,hegler1 and also ,pfeffer1 have investigated this subject1 by loading the young shoot of a plant nearly to its breaking point1 and then redetermining the breaking-strength after a few days4 ,some young shoots of the sunflower were found to break with a strain of #afj gms42 but when loaded with #aej gms41 and retested after two days1 they were able to support #bej gms42 and being again loaded with something short of this1 by next day they sustained #cjj gms41 and a few days later even #djj gms4 #fhi ,such experiments have been amply confirmed1 but so far as ,i am aware1 we do not know much more about the matter3 we do not know1 for instance1 how far the change is accompanied by increase in number of the bast-fibres1 through transformation of other tissues2 or how far it is due to increase in size of these fibres2 or whether it be not simply due to strengthening of the original fibres by some molecular change4 ,but ,i should be much inclined to suspect that the latter had a good deal to do with the phenomenon4 ,we know nowadays that a railway axle1 or any other piece of steel1 is weakened by a constant succession of frequently interrupted strains2 it is said to be 8fatigued10 and its strength is restored by a period of rest4 ,the converse effect of continued strain in a uniform direction may be illustrated by a homely example4 ,the confectioner takes a mass of boiled sugar or treacle 7in a particular molecular condition determined by the temperature to which it has been exposed71 and draws the soft sticky mass out into a rope2 and then1 folding it up lengthways1 he repeats the process again and again4 ,at first the rope is pulled out of the ductile mass without difficulty2 but as the work goes on it gets harder to do1 until all the man's force is used to stretch the rope4 ,here we have the phenomenon of increasing strength1 following mechanically on a rearrangement of molecules1 as the original isotropic condition is transmuted more and more into molecular asymmetry or anisotropy2 and the rope apparently 8adapts itself0 to the increased strain which it is called on to bear1 all after a fashion which at least suggests a parallel to the increasing strength of the stretched and weighted fibre in the plant4 ,for increase of strength by rearrangement of the particles we have already a rough illustration in our lock of wool or hank of tow4 ,the piece of tow will carry but little weight while its fibres are tangled and awry3 but as soon as we have carded it out1 and brought all its long fibres parallel and side by side1 we may at once make of it a strong and useful cord4 ,in some such ways as these1 then1 it would seem that we may co-ordinate1 or hope to co-ordinate1 the phenomenon of growth with certain of the beautiful structural phenomena which present themselves to our eyes as 8provisions10 or mechanical adaptations1 for the display of strength where strength is most required4 #fij ,that is to say1 the origin1 or causation1 of the phenomenon would seem to lie1 partly in the tendency of growth to be accelerated under strain3 and partly in the automatic effect of shearing strain1 by which it tends to displace parts which grow obliquely to the direct lines of tension and of pressure1 while leaving those in place which happen to lie parallel or perpendicular to those lines3 an automatic effect which we can probably trace as working on all scales of magnitude1 and as accounting therefore for the rearrangement of minute particles in the metal or the fibre1 as well as for the bringing into line of the fibres themselves within the plant1 or of the little trabeculae within the bone4 ,but we may now attempt to pass from the study of the individual bone to the much wider and not less beautiful problems of mechanical construction which are presented to us by the skeleton as a whole4 ,certain problems of this class are by no means neglected by writers on anatomy1 and many have been handed down from ,borelli1 and even from older writers4 ,for instance1 it is an old tradition of anatomical teaching to point out in the human body examples of the three orders of levers#fbd2 again1 the principle that the limb-bones tend to be shortened in order to support the weight of a very heavy animal is well understood by comparative anatomists1 in accordance with ,euler's law1 that the weight which a column liable to flexure is capable of supporting varies inversely as the square of its length2 and again1 the statical equilibrium of the body1 in relation for instance to the erect posture of man1 has long been a favourite theme of the philosophical anatomist4 ,but the general method1 based upon that of graphic statics1 to which we have been introduced in our study of a bone1 has not1 so far as ,i know1 been applied to the general fabric of the skeleton4 ,yet it is plain that each bone plays #fia a part in relation to the whole body1 analogous to that which a little trabecula1 or a little group of trabeculae1 plays within the bone itself3 that is to say1 in the normal distribution of forces in the body1 the bones tend to follow the lines of stress1 and especially the pressure-lines4 ,to demonstrate this in a comprehensive way would doubtless be difficult2 for we should be dealing with a framework of very great complexity1 and should have to take account of a great variety of conditions#fbe4 ,this framework is complicated as we see it in the skeleton1 where 7as we have said7 it is only1 or chiefly1 the struts of the whole fabric which are represented2 but to understand the mechanical structure in detail1 we should have to follow out the still more complex arrangement of the ties1 as represented by the muscles and ligaments1 and we should also require much detailed information as to the weights of the various parts and as to the other forces concerned4 ,without these latter data we can only treat the question in a preliminary and imperfect way4 ,but1 to take once again a small and simplified part of a big problem1 let us think of a quadruped 7for instance1 a horse7 in a standing posture1 and see whether the methods and terminology of the engineer may not help us1 as they did in regard to the minute structure of the single bone4 ,standing four-square upon its forelegs and hindlegs1 with the weight of the body suspended between1 the quadruped at once suggests to us the analogy of a bridge1 carried by its two piers4 ,and if it occurs to us1 as naturalists1 that we never look at a standing quadruped without contemplating a bridge1 so1 conversely1 a similar idea has occurred to the engineer2 for ,professor ,fidler1 in this ,treatise on ,bridge-,construction1 deals with the chief descriptive part of his subject under the heading of 8,the ,comparative ,anatomy of ,bridges40 ,the designation is most just1 for in studying the various types of bridges we are studying a series of well-planned skeletons#fbf2 and 7at the cost of a little pedantry7 #fib we might go even further1 and study 7after the fashion of the anatomist7 the 8osteology0 and 8desmology0 of the structure1 that is to say the bones which are represented by 8struts10 and the ligaments1 etc41 which are represented by 8ties40 ,furthermore after the methods of the comparative anatomist1 we may classify the families1 genera and species of bridges according to their distinctive mechanical features1 which correspond to certain definite conditions and functions4 ,in more ways than one1 the quadrupedal bridge is a remarkable one2 and perhaps its most remarkable peculiarity is that it is a jointed and flexible bridge1 remaining in equilibrium under considerable and sometimes great modifications of its curvature1 such as we see1 for instance1 when a cat humps or flattens her back4 ,the fact that flexibility is an essential feature in the quadrupedal bridge1 while it is the last thing which an engineer desires and the first which he seeks to provide against1 will impose certain important limiting conditions upon the design of the skeletal fabric2 but to this matter we shall afterwards return4 ,let us begin by considering the quadruped at rest1 when he stands upright and motionless upon his feet1 and when his legs exercise no function save only to carry the weight of the whole body4 ,so far as that function is concerned1 we might now perhaps compare the horse's legs with the tall and slender piers of some railway bridge2 but it is obvious that these jointed legs are ill-adapted to receive the horizontal thrust of any arch that may be placed atop of them4 ,hence it follows that the curved backbone of the horse1 which appears to cross like an arch the span between his shoulders and his flanks1 cannot be regarded as an arch1 in the #fic engineer's sense of the word4 ,it resembles an arch in form1 but not in function1 for it cannot act as an arch unless it be held back at each end 7as every arch is held back7 by abutments capable of resisting the horizontal thrust2 and these necessary abutments are not present in the structure4 ,but in various ways the engineer can modify his superstructure so as to supply the place of these external reactions1 which in the simple arch are obviously indispensable4 ,thus1 for example1 we may begin by inserting a straight steel tie1 ,,ab 7,fig4 #cci71 uniting the ends of the curved rib ,aa,b2 and this tie will supply the place of the external reactions1 converting the structure into a 8tied arch10 such as we may see in the roofs of many railway-stations4 ,or we may go on to fill in the space between arch and tie by a 8web-system10 converting it into what the engineer describes as a 8parabolic bowstring girder0 7,fig4 #cci;b74 ,in either case1 the structure becomes an ,illustration3 ,fig4 #cci4 independent 8detached girder10 supported at each end but not otherwise fixed1 and consisting essentially of an upper compression-member1 ,aa,b1 and a lower tension-member1 ,,ab4 ,but again1 in the skeleton of the quadruped1 the necessary tie1 ,,ab1 is not to be found2 and it follows that these comparatively simple types of bridge do not correspond to1 nor do they help us to understand1 the type of bridge which nature has designed in the skeleton of the quadruped4 ,nevertheless if we try to look1 as an engineer would look1 at the actual design of the animal skeleton and the actual distribution of its load1 we find that1 the one is most admirably adapted to the other1 according to the strict principles of engineering construction4 ,the structure is not an arch1 nor a tied arch1 nor a bowstring girder3 but it is strictly and beautifully #fid comparable to the main girder of a double-armed cantilever bridge4 ,obviously1 in our quadrupedal bridge1 the superstructure does not terminate 7as it did in our former diagram7 at the two points of support1 but it extends beyond them at each end1 carrying the head at one end and the tail at the other1 upon a pair of projecting arms or 8cantilevers0 7,fig4 #cdf74 ,in a typical cantilever bridge1 such as the ,forth ,bridge 7,fig4 #cde71 a certain simplification is introduced4 ,for each pier carries1 in this case1 its own double-armed cantilever1 linked by a short connecting girder to the next1 but so jointed to it that no weight is transmitted from one cantilever to another4 ,the bridge in short is cut into separate sections1 practically independent of one another2 at the joints a certain amount of bending is not precluded1 but shearing strain is evaded2 and each pier carries only its own load4 ,by this arrangement the engineer finds that design and construction are alike simplified and facilitated4 ,in the case of the horse1 it is obvious that the two piers of the bridge1 that is to say the fore-legs and the hind-legs1 do not bear 7as they do in the ,forth ,bridge7 separate and independent loads1 but the whole system forms a continuous structure4 ,in this case1 the calculation of the loads will be a little more difficult and the corresponding design of the structure a little more complicated4 ,we shall accordingly simplify our problem very considerably if1 to begin with1 we look upon the quadrupedal skeleton as constituted of two separate systems1 that is to say of two balanced cantilevers1 one supported on the fore-legs and the other on the hind2 and we may deal afterwards with the fact that these two cantilevers are not independent1 but are bound up in one common field of force and plan of construction4 ,in the horse it is plain that the two cantilever systems into which we may thus analyse the quadrupedal bridge are unequal in magnitude and importance4 ,the fore-part of the animal is much bulkier than its hind quarters1 and the fact that the fore-legs carry1 as they so evidently do1 a greater weight than the hind-legs has long been known and is easily proved2 we have only to walk a horse onto a weigh-bridge1 weigh first his fore-legs and then his hind-legs1 to discover that what we may call his front half weighs #fie a good deal more than what is carried on his hind feet1 say about three-fifths of the whole weight of the animal4 ,the great 7or anterior7 cantilever then1 in the horse1 is constituted by the heavy head and still heavier neck on one side of the pier which is represented by the fore-legs1 and by the dorsal vertebrae carrying a large part of the weight of the trunk upon the other side2 and this weight is so balanced over the fore-legs that the cantilever1 while 8anchored0 to the other parts of the structure1 transmits but little of its weight to the hind-legs1 and the amount so transmitted will vary with the position of the head and with the position of any artificial load#fbg4 ,under certain conditions1 as when the head is thrust well forward1 it is evident that the hind-legs will be actually relieved of a portion of the comparatively small load which is their normal share4 ,our problem now is to discover1 in a rough and approximate way1 some of the structural details which the balanced load upon the double cantilever will impress upon the fabric4 ,working by the methods of graphic statics1 the engineer's task is1 in theory1 one of great simplicity4 ,he begins by drawing in outline the structure which he desires to erect2 he calculates the stresses and bending-moments necessitated by the dimensions and load on the structure2 he draws a new diagram representing these forces1 and he designs and builds his fabric on the lines of this statical diagram4 ,he does1 in short1 precisely what we have seen nature doing in the case of the bone4 ,for if we had begun1 as it were1 by blocking out the femur roughly1 and considering its position and dimensions1 its means of support and the load which it has to bear1 we could have proceeded at once to draw the system of stress-lines which must occupy the field of force3 and to precisely these stress-lines has nature kept in the building of the bone1 down to the minute arrangement of its trabeculae4 ,the essential function of a bridge is to stretch across a certain span1 and carry a certain definite load2 and this being so1 the #fif chief problem in the designing of a bridge is to provide due resistance to the 8bending-moments0 which result from the load4 ,these bending-moments will vary from point to point along the girder1 and taking the simplest case of a uniform load supported at both ends1 they will be represented by points on a parabola4 ,if the girder be of uniform depth1 that is to say if its two flanges1 ,illustration3 ,fig4 #cdj4 ,a1 ,span of proposed bridge4 ,b1 ,stress diagram1 or diagram of bending-moments#fbh4 respectively under tension and compression1 be parallel to one another1 then the stress upon these flanges will vary as the bending-moments1 and will accordingly be very severe in the middle and will dwindle towards the ends4 ,but if we make the depth of the girder everywhere proportional to the bending-moments1 that is ,illustration3 ,fig4 #cda4 ,the bridge constructed1 as a parabolic girder4 to say if we copy in the girder the outlines of the bending-moment diagram1 then our design will automatically meet the circumstances of the case1 for the horizontal stress in each flange will now be uniform throughout the length of the girder4 ,in short1 in #fig ,professor ,fidler's words1 8,every diagram of moments represents the outline of a framed structure which will carry the given load with a uniform horizontal stress in the principal members40 ,illustration3 ,fig4 #cdb4 ,in the following diagrams 7,fig4 #cdb1 a1 b7 7which are taken from the original ones of ,culmann71 we see at once that the loaded beam or bracket 7a7 has a 8danger-point0 close to its fixed base1 that is to say at the point remotest from its load4 ,but in the parabolic bracket 7b7 there is no danger-point at all1 for the dimensions of the structure are made to increase pari passu with the bending-moments3 stress and resistance vary together4 ,again in ,fig4 #cdj1 we have a simple span 7,a71 with its stress diagram 7,b72 and in ,fig4 #cda we have the corresponding parabolic girder1 whose stresses are now uniform throughout4 ,in fact we see that1 by a process of conversion1 the stress diagram in each case becomes the structural diagram in the other#fbi4 ,now all this is but the modern rendering of one of ,galileo's most famous propositions4 ,in the ,dialogue which we have already quoted more than once#fcj1 ,sagredo says 8,it would be a fine thing if one could discover the proper shape to give a solid in order to make it equally resistant at every point1 in which case a load placed at the middle would not produce fracture more easily than if placed at any other point40 ,and ,galileo 7in the person of ,salviati7 first puts the problem into its more general form2 and then shews us how1 by giving a parabolic outline to our beam1 we have its simple and comprehensive solution4 ,in the case of our cantilever bridge1 we shew the primitive girder #fih in ,fig4 #cdc1 ,a1 with its bending-moment diagram 7,b72 and it is evident that1 if we turn this diagram upside down1 it will still be illustrative1 just as before1 of the bending-moments from point to point3 for as yet it is merely a diagram1 or graph1 of relative magnitudes4 ,to either of these two stress diagrams1 direct or inverted1 we may fit the design of the construction1 as in ,figs4 #cdc1 ,c and #cdd4 ,illustration3 ,fig4 #cdc4 ,illustration3 ,fig4 #cdd4 ,now in different animals the amount and distribution of the load differs so greatly that we can expect no single diagram1 drawn from the comparative anatomy of bridges1 to apply equally well to all the cases met with in the comparative anatomy of quadrupeds2 but nevertheless we have already gained an insight into the general principles of 8structural design0 in the quadrupedal bridge4 ,in our last diagram the upper member of the cantilever is under #fii tension2 it is represented in the quadruped by the ligamentum nuchae on the one side of the cantilever1 and by the supraspinous ligaments of the dorsal vertebrae on the other4 ,the compression member is similarly represented1 on both sides of the cantilever1 by the vertebral column1 or rather by the bodies of the vertebrae2 while the web1 or 8filling10 of the girders1 that is to say the upright or sloping members which extend from one flange to the other1 is represented on the one hand by the spines of the vertebrae1 and on the other hand1 by the oblique interspinous ligaments and muscles4 ,the high spines over the quadruped's withers are no other than the high struts which rise over the supporting piers in the parabolic girder1 and correspond to the position of the maximal bending-moments4 ,the fact that these tall vertebrae of the withers usually slope backwards1 sometimes steeply1 in a quadruped1 is easily and obviously explained#fca4 ,for each vertebra tends to act as a 8hinged lever10 and its spine1 acted on by the tensions transmitted by the ligaments on either side1 takes up its position as the diagonal of the parallelogram of forces to which it is exposed4 ,it happens that in these comparatively simple types of cantilever bridge the whole of the parabolic curvature is transferred to one or other of the principal members1 either the tension-member or the compression-member as the case may be4 ,but it is of course equally permissible to have both members curved1 in opposite directions4 ,this1 though not exactly the case in the ,forth ,bridge1 is approximately so2 for here the main compression-member is curved or arched1 and the main tension-member slopes downwards on either side from its maximal height above the piers4 ,in short1 the ,forth ,bridge is a nearer approach than either of the other cantilever bridges which we have #gjj illustrated to the plan of the quadrupedal skeleton2 for the main compression-member almost exactly recalls the form of the vertebral column1 while the main tension-member1 though not so closely similar to the supraspinous and nuchal ligaments1 corresponds to the plan of these in a somewhat simplified form4 ,illustration3 ,fig4 #cde4 ,a two-armed cantilever of the ,forth ,bridge4 ,thick lines1 compression-members 7bones72 thin lines1 tension-members 7ligaments74 ,we may now pass without difficulty from the two-armed cantilever supported on a single pier1 as it is in each separate section of the ,forth ,bridge1 or as we have imagined it to be in the forequarters of a horse1 to the condition which actually exists in that quadruped1 where a two-armed cantilever has its load distributed over two separate piers4 ,this is not precisely what an engineer calls a 8continuous0 girder1 for that term is applied to a girder which1 as a continuous structure1 crosses two or more spans1 while here there is only one4 ,but nevertheless1 this girder ,illustration3 ,fig4 #cdf4 is effectively continuous from the head to the tip of the tail2 and at each point of support 7,a and ,b7 it is subjected to the negative bending-moment due to the overhanging load on each of the projecting cantilever arms ,,ah and ,,bt4 ,the diagram of bending-moments will 7according to the ordinary conventions7 lie below #gja the base line 7because the moments are negative71 and must take some such form as that shown in the diagram3 for the girder must suffer its greatest bending stress not at the centre1 but at the two points of support ,a and ,b1 where the moments are measured by the vertical ordinates4 ,it is plain that this figure only differs from a representation of two independent two-armed cantilevers in the fact that there is no point midway in the span where the bending-moment vanishes1 but only a region between the two piers in which its magnitude tends to diminish4 ,illustration3 ,fig4 #cdg4 ,stress-diagram of horse's backbone4 ,the diagram effects a graphic summation of the positive and negative moments1 but its form may assume various modifications according to the method of graphic summation which we may choose to adopt2 and it is obvious also that the form of the diagram may assume many modifications of detail according to the actual distribution of the load4 ,in all cases the essential points to be observed are these3 firstly that the girder which is to resist the bending-moments induced by the load must possess its two principal members--an upper tension-member or tie1 represented by ligament1 and a lower compression-member represented by bone3 these members being united by a web represented by the vertebral spines with their interspinous ligaments1 and being placed one above the other in the order named because the moments are negative2 secondly we observe that the depth of the web1 or distance apart of the principal members1--that is to say the height of the vertebral spines1--must be proportional to the bending-moment at each point along the length of the girder4 ,in the case of an animal carrying two-thirds of his weight upon his fore-legs and only one-third upon his hind-legs1 the bending-moment diagram will be unsymmetrical1 after the fashion of ,fig4 #cdg1 the vertical ordinate at ,a being thrice the height of that at ,b4 #gjb ,on the other hand the ,dinosaur1 with his light head and enormous tail would give us a moment-diagram with the opposite kind of asymmetry1 the greatest bending stress being now found over the haunches1 at ,b 7,fig4 #cdh74 ,a glance at the skeleton of ,diplodocus ,carnegii will shew us the high vertebral spines over the loins1 in precise correspondence with the requirements of this diagram3 just as in the horse1 under the opposite conditions of load1 the highest vertebral spines are those of the withers1 that is to say those of the posterior cervical and anterior dorsal vertebrae4 ,illustration3 ,fig4 #cdh4 ,stress-diagram of backbone of ,dinosaur4 ,we have now not only dealt with the general resemblance1 both in structure and in function1 of the quadrupedal backbone with its associated ligaments to a double-armed cantilever girder1 but we have begun to see how the characters of the vertebral system must differ in different quadrupeds1 according to the conditions imposed by the varying distribution of the load3 and in particular how the height of the vertebral spines which constitute the web will be in a definite relation1 as regards magnitude and position1 to the bending-moments induced thereby4 ,we should require much detailed information as to the actual weights of the several parts of the body before we could follow out quantitatively the mechanical efficiency of each type of skeleton2 but in an approximate way what we have already learnt will enable us to trace many interesting correspondences between structure and function in this particular part of comparative anatomy4 ,we must1 however1 be careful to note that the great cantilever system is not of necessity constituted by the vertical column and its ligaments alone1 but that the pelvis1 firmly united as it is to the sacral vertebrae1 and stretching backwards far beyond the acetabulum1 becomes an intrinsic part of the system2 and helping 7as it does7 to carry the load of the abdominal viscera1 #gjc constitutes a great portion of the posterior cantilever arm1 or even its chief portion in cases where the size and weight of the tail are insignificant1 as is the case in the majority of terrestrial mammals4 ,we may also note here1 that just as a bridge is often a 8combined0 or composite structure1 exhibiting a combination of principles in its construction1 so in the quadruped we have1 as it were1 another girder supported by the same piers to carry the viscera2 and consisting of an inverted parabolic girder1 whose compression-member is again constituted by the backbone1 its tension-member by the line of the sternum and the abdominal muscles1 while the ribs and intercostal muscles play the part of the web or filling4 ,a very few instances must suffice to illustrate the chief variations in the load1 and therefore in the bending-moment diagram1 and therefore also in the plan of construction1 of various quadrupeds4 ,but let us begin by setting forth1 in a few cases1 the actual weights which are borne by the fore-limbs and the hind-limbs1 in our quadrupedal bridge#fcb4 ,on ,on ,fore- ,hind- on on ,gross weight4 feet feet ,fore- ,hind- ton cwts4 cwts4 cwts4 feet4 feet4 ,camel 7,bactrian7 -- #adbe #ibe #de #fgc #cbg ,llama -- #bge #age #hge #ffg #ccc ,elephant 7,indian7 #a #aege #bje #adge #ehb #dah ,horse -- #hbe #dge #ce #egf #dbd ,horse 7large ,clydesdale7 -- #aee #he #gj #edh #deb ,it will be observed that in all these animals the load upon the fore-feet preponderates considerably over that upon the hind1 the preponderance being rather greater in the elephant than in the horse1 and markedly greater in the camel and the llama than in the other two4 ,but while these weights are helpful and suggestive1 it is obvious that they do not go nearly far enough to give us a full insight into the constructional diagram to which the animals are conformed4 ,for such a purpose we should #gjd require to weigh the total load1 not in two portions1 but in many2 and we should also have to take close account of the general form of the animal1 of the relation between that form and the distribution of the load1 and of the actual directions of each bone and ligament by which the forces of compression and tension were transmitted4 ,all this lies beyond us for the present2 but nevertheless we may consider1 very briefly1 the principal cases involved in our enquiry1 of which the above animals form only a partial and preliminary illustration4 ,illustration3 ,fig4 #cdi4 ,stress-diagram of ,titanotherium4 7#a7 ,wherever we have a heavily loaded anterior cantilever arm1 that is to say whenever the head and neck represent a considerable fraction of the whole weight of the body1 we tend to have large bending-moments over the fore-legs1 and correspondingly high spines over the vertebrae of the withers4 ,this is the case in the great majority of four-footed1 terrestrial animals1 the chief exceptions being found in animals with comparatively small heads but large and heavy tails1 such as the anteaters or the ,dinosaurian reptiles1 and also 7very naturally7 in animals such as the crocodile1 where the 8bridge0 can scarcely be said to be developed1 for the long heavy body sags down to rest upon the ground4 ,the case is sufficiently exemplified by the horse1 and still more notably by the stag1 the ox1 or the pig4 ,it is illustrated in the accompanying diagram of the conditions in the great extinct ,titanotherium4 7#b7 ,in the elephant and the camel we have similar conditions1 but slightly modified4 ,in both cases1 and especially in the latter1 the weight on the fore-quarters is relatively large2 and in both cases the bending-moments are all the larger1 by reason of the length and forward extension of the camel's neck1 and the forward #gje position of the heavy tusks of the elephant4 ,in both cases the dorsal spines are large1 but they do not strike us as exceptionally so2 but in both cases1 and especially in the elephant1 they slope backwards in a marked degree4 ,each spine1 as already explained1 must in all cases assume the position of the diagonal in the parallelogram of forces defined by the tensions acting on it at its extremity2 for it constitutes a 8hinged lever10 by which the bending-moments on either side are automatically balanced2 and it is plain that the more the spine slopes backwards the more it indicates a relatively large strain thrown upon the great ligament of the neck1 and a relief of strain upon the more directly acting1 but weaker1 ligaments of the back and loins4 ,in both cases1 the bending-moments would seem to be more evenly distributed over the region of the back than1 for instance1 in the stag1 with its light hind-quarters and heavy load of antlers3 and in both cases the high 8girder0 is considerably prolonged1 by an extension of the tall spines backwards in the direction of the loins4 ,when we come to such a case as the mammoth1 with its immensely heavy and immensely elongated tusks1 we perceive at once that the bending-moments over the fore-legs are now very severe2 and we see also that the dorsal spines in this region are much more conspicuously elevated than in the ordinary elephant4 7#c7 ,in the case of the giraffe we have1 without doubt1 a very heavy load upon the fore-legs1 though no weighings are at hand to define the ratio2 but as far as possible this disproportionate load would seem to be relieved1 by help of a downward as well as backward thrust1 through the sloping back1 to the unusually low hind-quarters4 ,the dorsal spines of the vertebrae are very high and strong1 and the whole girder-system very perfectly formed4 ,the elevated1 rather than protruding position of the head lessens the anterior bending-moment as far as possible2 but it leads to a strong compressional stress transmitted almost directly downwards through the neck3 in correlation with which we observe that the bodies of the cervical vertebrae are exceptionally large and strong and steadily increase in size and strength from the head downwards4 7#d7 ,in the kangaroo1 the fore-limbs are entirely relieved of their load1 and accordingly the tall spines over the withers1 which #gjf were so conspicuous in all heavy-headed quadrupeds1 have now completely vanished4 ,the creature has become bipedal1 and body and tail form the extremities of a single balanced cantilever1 whose maximal bending-moments are marked by strong1 high lumbar and sacral vertebrae1 and by iliac bones of peculiar form and exceptional strength4 ,precisely the same condition is illustrated in the ,iguanodon1 and better still by reason of the great bulk of the creature1 and of the heavy load which falls to be supported by the great cantilever and by the hind-legs which form its piers4 ,the long and heavy body and neck require a balance-weight 7as in the kangaroo7 in the form of a long heavy tail4 ,and the double-armed cantilever1 so constituted1 shews a beautiful parabolic curvature in the graded heights of the whole series of vertebral spines1 which rise to a maximum over the haunches and die away slowly towards the neck and the tip of the tail4 7#e7 ,in the case of some of the great ,american fossil reptiles1 such as ,diplodocus1 it has always been a more or less disputed question whether or not they assumed1 like ,iguanodon1 an erect1 bipedal attitude4 ,in all of these we see an elongated pelvis1 and1 in still more marked degree1 we see elevated spinous processes of the vertebrae over the hind-limbs2 in all of them we have a long heavy tail1 and in most of them we have a marked reduction in size and weight both of the fore-limb and of the head itself4 ,the great size of these animals is not of itself a proof against the erect attitude2 because it might well have been accompanied by an aquatic or partially submerged habitat1 and the crushing stress of the creature's huge bulk proportionately relieved4 ,but we must consider each such case in the whole light of its own evidence2 and it is easy to see that1 just as the quadrupedal mammal may carry the greater part but not all of its weight upon its fore-limbs1 so a heavy-tailed reptile may carry the greater part upon its hind-limbs1 without this process going so far as to relieve its fore-limbs of all weight whatsoever4 ,this would seem to be the case in such a form as ,diplodocus1 and also in ,stegosaurus1 whose restoration by ,marsh is doubtless substantially correct#fcc4 ,the fore-limbs1 #gjg though comparatively small1 are obviously fashioned for support1 but the weight which they have to carry is far less than that which the hind-limbs bear4 ,the head is small and the neck short1 while on the other hand the hind-quarters and the tail are big and massive4 ,the backbone bends into a great1 double-armed cantilever1 culminating over the pelvis and the hind-limbs1 and here furnished with its highest and strongest spines to separate the tension-member from the compression-member of the girder4 ,the fore-legs form a secondary supporting pier to this great cantilever1 the greater part of whose weight is poised upon the hind-limbs alone4 ,illustration3 ,fig4 #cej4 ,diagram of ,stegosaurus4 7#f7 ,in a bird1 such as an ostrich or a common fowl1 the bipedal habit necessitates the balancing of the load upon a single double-armed cantilever-girder1 just as in the ,iguanodon and the kangaroo1 but the construction is effected in a somewhat different way4 ,the great heavy tail has entirely disappeared2 but1 though from the skeleton alone it would seem that nearly all the bulk of the animal lay in front of the hind-limbs1 yet in the living bird we can easily perceive that the great weight of the abdominal organs lies suspended behind the socket for the thigh-bone1 and so hangs from the posterior lever-arm of the cantilever1 balancing the head and neck and thorax whose combined weight hangs from #gjh the anterior arm4 ,the great cantilever girder appears1 accordingly1 balanced over the hind-legs4 ,it is now constituted in part by the posterior dorsal or lumbar vertebrae1 all traces of special elevation having disappeared from the anterior dorsals2 but the greater part of the girder is made up of the great iliac bones1 placed side by side1 and gripping firmly the sacral vertebrae1 often almost to the extinction of these latter4 ,in the form of these iliac bones1 the arched curvature of their upper border1 in their elongation fore-and-aft to overhang both ways their supporting pier1 and in the coincidence of their greatest height with the median line of support over the centre of gravity1 we recognise all the characteristic properties of the typical balanced cantilever#fcd4 7#g7 ,we find a highly important corollary in the case of aquatic animals4 ,for here the effect of gravity is neutralised2 we have neither piers nor cantilevers2 and we find accordingly in all aquatic mammals of whatsoever group--whales1 seals or sea-cows--that the high arched vertebral spines over the withers1 or corresponding structures over the hind-limbs1 have both entirely disappeared4 ,just as the cantilever girder tended to become obsolete in the aquatic mammal so does it tend to weaken and disappear in the aquatic bird4 ,there is a very marked contrast between the high-arched strongly-built pelvis in the ostrich or the hen1 and the long1 thin1 comparatively straight and weakly bone which represents it in a diver1 a grebe or a penguin4 ,but in the aquatic mammal1 such as a whale or a dolphin 7and not less so in the aquatic bird71 stiffness must be ensured in order to enable the muscles to act against the resistance of the water in the act of swimming2 and accordingly nature must provide against bending-moments irrespective of gravity4 ,in the dolphin1 at any rate as regards its tail end1 the conditions will be not very different from those of a column or beam with fixed ends1 in which1 under deflexion1 there will be two points of contrary flexure1 as at ,c1 ,d1 in ,fig4 #cea4 #gji ,illustration3 ,fig4 #cea4 ,here1 between ,c and ,d we have a varying bending-moment1 represented by a continuous curve with its maximal elevation midway between the points of inflexion4 ,and correspondingly1 in our dolphin1 we have a continuous series of high dorsal spines1 rising to a maximum about the middle of the animal's body1 and falling to nil at some distance from the end of the tail4 ,it is their business 7as usual7 to keep the tension-member1 represented by the strong supraspinous ligaments1 wide apart from the compression-member1 which is as usual represented by the backbone itself4 ,but in our diagram we see that on the further side of ,c and ,d we have a negative curve of bending-moments1 or bending-moments in a contrary direction4 ,without inquiring how these stresses are precisely met towards the dolphin's head 7where the coalesced cervical vertebrae suggest themselves as a partial explanation71 we see at once that towards the tail they are met by the strong series of chevron-bones1 which in the caudal region1 where tall dorsal spines are no longer needed1 take their place below the vertebrae1 in precise correspondence with the bending-moment diagram4 ,in many cases other than these aquatic ones1 when we have to deal with animals with long and heavy tails 7like the ,iguanodon and the kangaroo of which we have already spoken71 we are apt to meet with similar1 though usually shorter chevron-bones2 and in all these cases we may see without difficulty that a negative bending-moment is there to be resisted4 ,in the dolphin we may find a good illustration of the fact that not only is it necessary to provide for rigidity in the vertical direction1 but also in the horizontal1 where a tendency to bending must be resisted on either side4 ,this function is effected in part by the ribs with their associated muscles1 but they extend but a little way and their efficacy for this purpose can be but small4 ,we have1 however1 behind the region of the ribs and on either side of the backbone a strong series of elongated and flattened transverse processes1 forming a web for the support of a tension-member in the usual form of ligament1 and so playing a part precisely analogous to that performed by the dorsal spines in the same #gaj animal4 ,in an ordinary fish1 such as a cod or a haddock1 we see precisely the same thing3 the backbone is stiffened by the indispensable help of its three series of ligament-connected processes1 the dorsal and the two transverse series4 ,and here we see 7as we see partly also among the whales71 that these three series of processes1 or struts1 tend to be arranged well-nigh at equal angles1 of #abj1 with one another1 giving the greatest and most uniform strength of which such a system is capable4 ,on the other hand1 in a flat fish1 such as a plaice1 where from the natural mode of progression it is necessary that the backbone should be flexible in one direction while stiffened in another1 we find the whole outline of the fish comparable to that of a double bowstring girder1 the compression-member being 7as usual7 the backbone1 the tension-member on either side being constituted by the interspinous ligaments and muscles1 while the web or filling is very beautifully represented by the long and evenly graded spines1 which spring symmetrically from opposite sides of each individual vertebra4 ,the main result at which we have now arrived1 in regard to the construction of the vertebral column and its associated parts1 is that we may look upon it as a certain type of girder1 whose depth1 as we cannot help seeing1 is everywhere very nearly proportional to the height of the corresponding ordinate in the diagram of moments3 just as it is in the girder of a cantilever bridge as designed by a modern engineer4 ,in short1 after the nineteenth or twentieth century engineer has done his best in framing the design of a big cantilever1 he may find that some of his best ideas bad1 so to speak1 been anticipated ages ago in the fabric of the great saurians and the larger mammals4 ,but it is possible that the modern engineer might be disposed to criticise the skeleton girder at two or three points2 and in particular he might think the girder1 as we see it for instance in ,diplodocus or ,stegosaurus1 not deep enough for carrying the animal's enormous weight of some twenty tons4 ,if we adopt a much greater depth 7or ratio of depth to length7 as in the modern cantilever1 we shall greatly increase the strength of the structure2 but at the same time we should greatly increase its rigidity1 and #gaa this is precisely what1 in the circumstances of the case1 it would seem that nature is bound to avoid4 ,we need not suppose that the great saurian was by any means active and limber2 but a certain amount of activity and flexibility he was bound to have1 and in a thousand ways he would find the need of a backbone that should be flexible as well as strong4 ,now this opens up a new aspect of the matter and is the beginning of a long1 long story1 for in every direction this double requirement of strength and flexibility imposes new conditions upon our design4 ,to represent all the correlated quantities we should have to construct not only a diagram of moments but also a diagram of elastic deflexion and its so-called 8curvature02 and the engineer would want to know something more about the material of the ligamentous tension-member--its modulus of elasticity in direct tension1 its elastic limit1 and its safe working stress4 ,in various ways our structural problem is beset by 8limiting conditions40 ,not only must rigidity be associated with flexibility1 but also stability must be ensured in various positions and attitudes2 and the primary function of support or weight-carrying must be combined with the provision of points d'appui for the muscles concerned in locomotion4 ,we cannot hope to arrive at a numerical or quantitative solution of this complicate problem1 but we have found it possible to trace it out in part towards a qualitative solution4 ,and speaking broadly we may certainly say that in each case the problem has been solved by nature herself1 very much as she solves the difficult problems of minimal areas in a system of soap-bubbles2 so that each animal is fitted with a backbone adapted to his own individual needs1 or 7in other words7 corresponding exactly to the mean resultant of the stresses to which as a mechanical system it is exposed4 ,throughout this short discussion of the principles of construction1 limited to one part of the skeleton1 we see the same general principles at work which we recognise in the plan and construction of an individual bone4 ,that is to say1 we see a tendency for material to be laid down just in the lines of stress1 and so as to evade thereby the distortions and disruptions due to shear4 ,in these phenomena there lies a definite law of growth1 #gab whatever its ultimate expression or explanation may come to be4 ,let us not press either argument or hypothesis too far3 but be content to see that skeletal form1 as brought about by growth1 is to a very large extent determined by mechanical considerations1 and tends to manifest itself as a diagram1 or reflected image1 of mechanical stress4 ,if we fail1 owing to the immense complexity of the case1 to unravel all the mathematical principles involved in the construction of the skeleton1 we yet gain something1 and not a little1 by applying this method to the familiar objects of our anatomical study3 obvia conspicimus1 nubem pellente mathesi#fce4 ,before we leave this subject of mechanical adaptation1 let us dwell once more for a moment upon the considerations which arise from our conception of a field of force1 or field of stress1 in which tension and compression 7for instance7 are inevitably combined1 and are met by the materials naturally fitted to resist them4 ,it has been remarked over and over again how harmoniously the whole organism hangs together1 and how throughout its fabric one part is related and fitted to another in strictly functional correlation4 ,but this conception1 though never denied1 is sometimes apt to be forgotten in the course of that process of more and more minute analysis by which1 for simplicity's sake1 we seek to unravel the intricacies of a complex organism4 ,we tend1 as we analyse a thing into its parts or into its properties1 to magnify these1 to exaggerate their apparent independence1 and to hide from ourselves 7at least for a time7 the essential integrity and individuality of the composite whole4 ,we divide the body into its organs1 the skeleton into its bones1 as in very much the same fashion we make a subjective analysis of the mind1 according to the teachings of psychology1 into component factors3 but we know very well that judgment and knowledge1 courage or gentleness1 love or fear1 have no separate existence1 but are somehow mere manifestations1 or imaginary co-efficients1 of a most complex integral4 ,and likewise1 as biologists1 we may go so far as to say that even the bones themselves are only in a limited and even a deceptive sense1 separate and individual things4 ,the skeleton begins as a continuum1 and a continuum it remains all life long4 ,the things that link bone with bone1 #gac cartilage1 ligaments1 membranes1 are fashioned out of the same primordial tissue1 and come into being pari passu1 with the bones themselves4 ,the entire fabric has its soft parts and its hard1 its rigid and its flexible parts2 but until we disrupt and dismember its bony1 gristly and fibrous parts1 one from another1 it exists simply as a 8skeleton10 as one integral and individual whole4 ,a bridge was once upon a time a loose heap of pillars and rods and rivets of steel4 ,but the identity of these is lost1 just as if they were fused into a solid mass1 when once the bridge is built2 their separate functions are only to be recognised and analysed in so far as we can analyse the stresses1 the tensions and the pressures1 which affect this part of the structure or that2 and these forces are not themselves separate entities1 but are the resultants of an analysis of the whole field of force4 ,moreover when the bridge is broken it is no longer a bridge1 and all its strength is gone4 ,so is it precisely with the skeleton4 ,in it is reflected a field of force3 and keeping pace1 as it were1 in action and interaction with this field of force1 the whole skeleton and every part thereof1 down to the minute intrinsic structure of the bones themselves1 is related in form and in position to the lines of force1 to the resistances it has to encounter2 for by one of the mysteries of biology1 resistance begets resistance1 and where pressure falls there growth springs up in strength to meet it4 ,and1 pursuing the same train of thought1 we see that all this is true not of the skeleton alone but of the whole fabric of the body4 ,muscle and bone1 for instance1 are inseparably associated and connected2 they are moulded one with another2 they come into being together1 and act and react together#fcf4 ,we may study them apart1 but it is as a concession to our weakness and to the narrow outlook of our minds4 ,we see1 dimly perhaps1 but yet with all the assurance of conviction1 that between muscle and bone there can be no change in the one but it is correlated with changes in the other2 that through and through they are linked in indissoluble association2 that they are only separate entities #gad in this limited and subordinate sense1 that they are parts of a whole which1 when it loses its composite integrity1 ceases to exist4 ,the biologist1 as well as the philosopher1 learns to recognise that the whole is not merely the sum of its parts4 ,it is this1 and much more than this4 ,for it is not a bundle of parts but an organisation of parts1 of parts in their mutual arrangement1 fitting one with another1 in what ,aristotle calls 8a single and indivisible principle of unity02 and this is no merely metaphysical conception1 but is in biology the fundamental truth which lies at the basis of ,geoffroy's 7or ,goethe's7 law of 8compensation10 or 8balancement of growth40 ,nevertheless ,darwin found no difficulty in believing that 8natural selection will tend in the long run to reduce any part of the organisation1 as soon as1 through changed habits1 it becomes superfluous3 without by any means causing some other part to be largely developed in a corresponding degree4 ,and conversely1 that natural selection may perfectly well succeed in largely developing an organ without requiring as a necessary compensation the reduction of some adjoining part#fcg40 ,this view has been developed into a doctrine of the 8independence of single characters0 7not to be confused with the germinal 8unit characters0 of ,mendelism71 especially by the palaeontologists4 ,thus ,osborn asserts a 8principle of hereditary correlation10 combined with a 8principle of hereditary separability whereby the body is a colony1 a mosaic1 of single individual and separable characters#fch40 ,i cannot think that there is more than a small element of truth in this doctrine4 ,as ,kant said1 8die ,ursache der ,art der ,existenz bei jedem ,theile eines lebenden ,krpers ist im ,ganzen enthalten40 ,and1 according to the trend or aspect of our thought1 we may look upon the co-ordinated parts1 now as related and fitted to the end or function of the whole1 and now as related to or resulting from the physical causes inherent in the entire system of forces to which the whole has been exposed1 and under whose influence it has come into being#fci4 #gae ,it would seem to me that the mechanical principles and phenomena which we have dealt with in this chapter are of no small importance to the morphologist1 all the more when he is inclined to direct his study of the skeleton exclusively to the problem of phylogeny2 and especially when1 according to the methods of modern comparative morphology1 he is apt to take the skeleton to pieces1 and to draw from the comparison of a series of scapulae1 humeri1 or individual vertebrae1 conclusions as to the descent and relationship of the animals to which they belong4 ,it would1 ,i dare say1 be a gross exaggeration to see in every bone nothing more than a resultant of immediate and direct physical or mechanical conditions2 for to do so would be to deny the existence1 in this connection1 of a principle of heredity4 ,and though ,i have tried throughout this book to lay emphasis on the direct action of causes other than heredity1 in short to circumscribe the employment of the latter as a working hypothesis in morphology1 there can still be no question whatsoever but that heredity is a vastly important as well as a mysterious thing2 it is one of the great factors in biology1 however we may attempt to figure to ourselves1 or howsoever we may fail even to imagine1 its underlying physical explanation4 ,but ,i maintain that it is no less an exaggeration if we tend to neglect these direct physical and mechanical modes of causation altogether1 and to see in the characters of a bone merely the results of variation and of heredity1 and to trust1 in consequence1 to those characters as a sure and certain and unquestioned guide to affinity and phylogeny4 ,comparative anatomy has its physiological side1 which filled men's minds in ,john ,hunter's day1 and in ,owen's day2 it has its #gaf classificatory and phylogenetic aspect1 which has all but filled men's minds during the last couple of generations2 and we can lose sight of neither aspect without risk of error and misconception4 ,it is certain that the question of phylogeny1 always difficult1 becomes especially so in cases where a great change of physical or mechanical conditions has come about1 and where accordingly the physical and physiological factors in connection with change of form are bound to be large4 ,to discuss these questions at length would be to enter on a discussion of ,lamarck's philosophy of biology1 and of many other things besides4 ,but let us take one single illustration4 ,the affinities of the whales constitute1 as will be readily admitted1 a very hard problem in phylogenetic classification4 ,we know now that the extinct ,zeuglodons are related to the old ,creodont carnivores1 and thereby 7though distantly7 to the seals2 and it is supposed1 but it is by no means so certain1 that in turn they are to be considered as representing1 or as allied to1 the ancestors of the modern toothed whales#fdj4 ,the proof of any such a contention becomes1 to my mind1 extraordinarily difficult and complicated2 and the arguments commonly used in such cases may be said 7in ,bacon's phrase7 to allure1 rather than to extort assent4 ,though the ,zeuglodonts were aquatic animals1 we do not know1 and we have no right to suppose or to assume1 that they swam after the fashion of a whale 7any more than the seal does71 that they dived like a whale1 and leaped like a whale4 ,but the fact that the whale does these things1 and the way in which he does them1 is reflected in many parts of his skeleton--perhaps more or less in all3 so much so that the lines of stress which these actions impose are the very plan and working-diagram of great part of his structure4 ,that the ,zeuglodon has a scapula like that of a whale is to my mind no necessary argument that he is akin by blood-relationship to a whale3 that his dorsal vertebrae are very different from a whale's is no conclusive argument that #gag such blood-relationship is lacking4 ,the former fact goes a long way to prove that he used his flippers very much as a whale does2 the latter goes still farther to prove that his general movements and equilibrium in the water were totally different4 ,the whale may be descended from the ,carnivora1 or might for that matter1 as an older school of naturalists believed1 be descended from the ,ungulates2 but whether or no1 we need not expect to find in him the scapula1 the pelvis or the vertebral column of the lion or of the cow1 for it would be physically impossible that he could live the life he does with any one of them4 ,in short1 when we hope to find the missing links between a whale and his terrestrial ancestors1 it must be not by means of conclusions drawn from a scapula1 an axis1 or even from a tooth1 but by the discovery of forms so intermediate in their general structure as to indicate an organisation and1 ipso facto1 a mode of life1 intermediate between the terrestrial and the ,cetacean form4 ,there is no valid syllogism to the effect that ,a has a flat curved scapula like a seal's1 and ,b has a flat1 curved scapula like a seal's3 and therefore ,a and ,b are related to the seals and to each other2 it is merely a flagrant case of an 8undistributed middle40 ,but there is validity in an argument that ,b shews in its general structure1 extending over this bone and that bone1 resemblances both to ,a and to the seals3 and that therefore he may be presumed to be related to both1 in his hereditary habits of life and in actual kinship by blood4 ,it is cognate to this argument that 7as every palaeontologist knows7 we find clues to affinity more easily1 that is to say with less confusion and perplexity1 in certain structures than in others4 ,the deep-seated rhythms of growth which1 as ,i venture to think1 are the chief basis of morphological heredity1 bring about similarities of form1 which endure in the absence of conflicting forces2 but a new system of forces1 introduced by altered environment and habits1 impinging on those particular parts of the fabric which lie within this particular field of force1 will assuredly not be long of manifesting itself in notable and inevitable modifications of form4 ,and if this be really so1 it will further imply that modifications of form will tend to manifest themselves1 not so much in small and isolated phenomena1 in this part of the fabric or in that1 in a scapula for instance or a humerus3 but rather in #gah some slow1 general1 and more or less uniform or graded modification1 spread over a number of correlated parts1 and at times extending over the whole1 or over great portions1 of the body4 ,whether any such general tendency to widespread and correlated transformation exists1 we shall attempt to discuss in the following chapter4 #gai ,,chapter ,,xvii ,,on ,,the ,,theory ,,of ,,transformations1 ,,or ,,the ,,comparison ,,of ,,related ,,forms#fda ,in the foregoing chapters of this book we have attempted to study the inter-relations of growth and form1 and the part which certain of the physical forces play in this complex interaction2 and1 as part of the same enquiry1 we have tried in comparatively simple cases to use mathematical methods and mathematical terminology in order to describe and define the forms of organisms4 ,we have learned in so doing that our own study of organic form1 which we call by ,goethe's name of ,morphology1 is but a portion of that wider ,science of ,form which deals with the forms assumed by matter under all aspects and conditions1 and1 in a still wider sense1 with forms which are theoretically imaginable4 ,the study of form may be descriptive merely1 or it may become analytical4 ,we begin by describing the shape of an object in the simple words of common speech3 we end by defining it in the precise language of mathematics2 and the one method tends to follow the other in strict scientific order and historical continuity4 ,thus1 for instance1 the form of the earth1 of a raindrop or a rainbow1 the shape of the hanging chain1 or the path of a stone thrown up into the air1 may all be described1 however inadequately1 in common words2 but when we have learned to comprehend and to define the sphere1 the catenary1 or the parabola1 we have made a wonderful and perhaps a manifold advance4 ,the mathematical definition of a 8form0 has a quality of precision which was quite lacking in our earlier stage of mere description2 it is expressed in few words1 or in still briefer symbols1 and these #gbj words or symbols are so pregnant with meaning that thought itself is economised2 we are brought by means of it in touch with ,galileo's aphorism 7as old as ,plato1 as old as ,pythagoras1 as old perhaps as the wisdom of the ,egyptians71 that 8the ,book of ,nature is written in characters of ,geometry40 ,next1 we soon reach through mathematical analysis to mathematical synthesis2 we discover homologies or identities which were not obvious before1 and which our descriptions obscured rather than revealed3 as for instance1 when we learn that1 however we hold our chain1 or however we fire our bullet1 the contour of the one or the path of the other is always mathematically homologous4 ,lastly1 and this is the greatest gain of all1 we pass quickly and easily from the mathematical conception of form in its statical aspect to form in its dynamical relations3 we pass from the conception of form to an understanding of the forces which gave rise to it2 and in the representation of form and in the comparison of kindred forms1 we see in the one case a diagram of forces in equilibrium1 and in the other case we discern the magnitude and the direction of the forces which have sufficed to convert the one form into the other4 ,here1 since a change of material form is only effected by the movement of matter1 we have once again the support of the schoolman's and the philosopher's axiom1 8,ignorato motu1 ignoratur ,natura40 ,in the morphology of living things the use of mathematical methods and symbols has made slow progress2 and there are various reasons for this failure to employ a method whose advantages are so obvious in the investigation of other physical forms4 ,to begin with1 there would seem to be a psychological reason lying in the fact that the student of living things is by nature and training an observer of concrete objects and phenomena1 and the habit of mind which he possesses and cultivates is alien to that of the theoretical mathematician4 ,but this is by no means the only reason2 for in the kindred subject of mineralogy1 for instance1 crystals were still treated in the days of ,linnaeus as wholly within the province of the naturalist1 and were described by him after the simple methods in use for animals and plants3 but as soon as ,hay showed the application of mathematics to #gba the description and classification of crystals1 his methods were immediately adopted and a new science came into being4 ,a large part of the neglect and suspicion of mathematical methods in organic morphology is due 7as we have partly seen in our opening chapter7 to an ingrained and deep-seated belief that even when we seem to discern a regular mathematical figure in an organism1 the sphere1 the hexagon1 or the spiral which we so recognise merely resembles1 but is never entirely explained by1 its mathematical analogue2 in short1 that the details in which the figure differs from its mathematical prototype are more important and more interesting than the features in which it agrees1 and even that the peculiar aesthetic pleasure with which we regard a living thing is somehow bound up with the departure from mathematical regularity which it manifests as a peculiar attribute of life4 ,this view seems to me to involve a misapprehension4 ,there is no such essential difference between these phenomena of organic form and those which are manifested in portions of inanimate matter#fdb4 ,no chain hangs in a perfect catenary and no raindrop is a perfect sphere3 and this for the simple reason that forces and resistances other than the main one are inevitably at work4 ,the same is true of organic form1 but it is for the mathematician to unravel the conflicting forces which are at work together4 ,and this process of investigation may lead us on step by step to new phenomena1 as it has done in physics1 where sometimes a knowledge of form leads us to the interpretation of forces1 and at other times a knowledge of the forces at work guides us towards a better insight into form4 ,i would illustrate this by the case of the earth itself4 ,after the fundamental advance had been made which taught us that the world was round1 ,newton showed that the forces at work upon it must lead to its being imperfectly spherical1 and in the course of time its oblate spheroidal shape was actually verified4 ,but now1 in turn1 it has been shown that its form is still more complicated1 and the next step will be to seek for the forces that have deformed the oblate spheroid4 #gbb ,the organic forms which we can define1 more or less precisely1 in mathematical terms1 and afterwards proceed to explain and to account for in terms of force1 are of many kinds1 as we have seen2 but nevertheless they are few in number compared with ,nature's all but infinite variety4 ,the reason for this is not far to seek4 ,the living organism represents1 or occupies1 a field of force which is never simple1 and which as a rule is of immense complexity4 ,and just as in the very simplest of actual cases we meet with a departure from such symmetry as could only exist under conditions of ideal simplicity1 so do we pass quickly to cases where the interference of numerous1 though still perhaps very simple1 causes leads to a resultant which lies far beyond our powers of analysis4 ,nor must we forget that the biologist is much more exacting in his requirements1 as regards form1 than the physicist2 for the latter is usually content with either an ideal or a general description of form1 while the student of living things must needs be specific4 ,the physicist or mathematician can give us perfectly satisfying expressions for the form of a wave1 or even of a heap of sand2 but we never ask him to define the form of any particular wave of the sea1 nor the actual form of any mountain-peak or hill#fdc4 #gbc ,for various reasons1 then1 there are a vast multitude of organic forms which we are unable to account for1 or to define1 in mathematical terms2 and this is not seldom the case even in forms which are apparently of great simplicity and regularity4 ,the curved outline of a leaf1 for instance1 is such a case2 its ovate1 lanceolate1 or cordate shape is apparently very simple1 but the difficulty of finding for it a mathematical expression is very great indeed4 ,to define the complicated outline of a fish1 for instance1 or of a vertebrate skull1 we never even seek for a mathematical formula4 ,but in a very large part of morphology1 our essential task lies in the comparison of related forms rather than in the precise definition of each2 and the deformation of a complicated figure may be a phenomenon easy of comprehension1 though the figure itself have to be left unanalysed and undefined4 ,this process of comparison1 of recognising in one form a definite permutation or deformation of another1 apart altogether from a precise and adequate understanding of the original 8type0 or standard of comparison1 lies within the immediate province of mathematics1 and finds its solution in the elementary use of a certain method of the mathematician4 ,this method is the ,method of ,co-ordinates1 on which is based the ,theory of ,transformations4 ,i imagine that when ,descartes conceived the method of co-ordinates1 as a generalisation from the proportional diagrams of the artist and the architect1 and long before the immense possibilities of this analysis could be foreseen1 he had in mind a very simple purpose2 it was perhaps no more than to find a way of translating the form of a curve into numbers and into words4 ,this is precisely what we do1 by the method of co-ordinates1 every time we study a statistical curve2 and conversely1 we translate numbers into form whenever we 8plot a curve0 to illustrate a table of mortality1 a rate of growth1 or the daily variation of temperature or barometric pressure4 ,in precisely the same way it is possible to inscribe in a net of rectangular co-ordinates the outline1 for instance1 of a fish1 and so to translate #gbd it into a table of numbers1 from which again we may at pleasure reconstruct the curve4 ,but it is the next step in the employment of co-ordinates which is of special interest and use to the morphologist2 and this step consists in the alteration1 or 8transformation10 of our system of co-ordinates and in the study of the corresponding transformation of the curve or figure inscribed in the co-ordinate network4 ,let us inscribe in a system of ,cartesian co-ordinates the outline of an organism1 however complicated1 or a part thereof3 such as a fish1 a crab1 or a mammalian skull4 ,we may now treat this complicated figure1 in general terms1 as a function of x1 y4 ,if we submit our rectangular system to 8deformation10 on simple and recognised lines1 altering1 for instance1 the direction of the axes1 the ratio of x_/y1 or substituting for x and y some more complicated expressions1 then we shall obtain a new system of co-ordinates1 whose deformation from the original type the inscribed figure will precisely follow4 ,in other words1 we obtain a new figure1 which represents the old figure under strain1 and is a function of the new co-ordinates in precisely the same way as the old figure was of the original co-ordinates x and y4 ,the problem is closely akin to that of the cartographer who transfers identical data to one projection or another2 and whose object is to secure 7if it be possible7 a complete correspondence1 in each small unit of area1 between the one representation and the other4 ,the morphologist will not seek to draw his organic forms in a new and artificial projection2 but1 in the converse aspect of the problem1 he will inquire whether two different but more or less obviously related forms can be so analysed and interpreted that each may be shown to be a transformed representation of the other4 ,this once demonstrated1 it will be a comparatively easy task 7in all probability7 to postulate the direction and magnitude of the force capable of effecting the required transformation4 ,again1 if such a simple alteration of the system of forces can be proved adequate to meet the case1 we may find ourselves able to dispense with many widely current and more complicated hypotheses of biological causation4 ,for it is a maxim in physics that an effect ought not to be ascribed to #gbe the joint operation of many causes if few are adequate to the production of it3 ,frustra fit per plura1 quod fieri potest per pauciora4 ,it is evident that by the combined action of appropriate forces any material form can be transformed into any other3 just as out of a 8shapeless0 mass of clay the potter or the sculptor models his artistic product2 or just as we attribute to ,nature herself the power to effect the gradual and successive transformation of the simplest into the most complex organism4 ,in like manner it is possible1 at least theoretically1 to cause the outline of any closed curve to appear as a projection of any other whatsoever4 ,but we need not let these theoretical considerations deter us from our method of comparison of related forms4 ,we shall strictly limit ourselves to cases where the transformation necessary to effect a comparison shall be of a simple kind1 and where the transformed1 as well as the original1 co-ordinates shall constitute an harmonious and more or less symmetrical system4 ,we should fall into deserved and inevitable confusion if1 whether by the mathematical or any other method1 we attempted to compare organisms separated far apart in ,nature and in zoological classification4 ,we are limited1 not by the nature of our method1 but by the whole nature of the case1 to the comparison of organisms such as are manifestly related to one another and belong to the same zoological class4 ,our inquiry lies1 in short1 just within the limits which ,aristotle himself laid down when1 in defining a 8genus10 he showed that 7apart from those superficial characters1 such as colour1 which he called 8accidents07 the essential differences between one 8species0 and another are merely differences of proportion1 of relative magnitude1 or 7as he phrased it7 of 8excess and defect40 8,save only for a difference in the way of excess or defect1 the parts are identical in the case of such animals as are of one and the same genus2 and by #8genus' ,i mean1 for instance1 ,bird or ,fish40 ,and again3 8,within the limits of the same genus1 as a general rule1 most of the parts exhibit differences 444 in the way of multitude or fewness1 magnitude or parvitude1 in short1 in the way of excess or defect4 ,for #8the more' and #8the less' may be represented as #gbf #8excess' and #8defect#fdd4' 0 ,it is precisely this difference of relative magnitudes1 this ,aristotelian 8excess and defect0 in the case of form1 which our co-ordinate method is especially adapted to analyse1 and to reveal and demonstrate as the main cause of what 7again in the ,aristotelian sense7 we term 8specific0 differences4 ,the applicability of our method to particular cases will depend upon1 or be further limited by1 certain practical considerations or qualifications4 ,of these the chief1 and indeed the essential1 condition is1 that the form of the entire structure under investigation should be found to vary in a more or less uniform manner1 after the fashion of an approximately homogeneous and isotropic body4 ,but an imperfect isotropy1 provided always that some 8principle of continuity0 run through its variations1 will not seriously interfere with our method2 it will only cause our transformed co-ordinates to be somewhat less regular and harmonious than are those1 for instance1 by which the physicist depicts the motions of a perfect fluid or a theoretic field of force in a uniform medium4 ,again1 it is essential that our structure vary in its entirety1 or at least that 8independent variants0 should be relatively few4 ,that independent variations occur1 that localised centres of diminished or exaggerated growth will now and then be found1 is not only probable but manifest2 and they may even be so pronounced as to appear to constitute new formations altogether4 ,such independent variants as these ,aristotle himself clearly recognised3 8,it happens further that some have parts that others have not2 for instance1 some birds have spurs and others not1 some have crests1 or combs1 and others not2 but1 as a general rule1 most parts and those that go to make up the bulk of the body are either identical with one another1 or differ from one another in the way of contrast and of excess and defect4 ,for #8the more' and #8the less' may be represented as #8excess' or #8defect4' 0 ,if1 in the evolution of a fish1 for instance1 it be the case that its several and constituent parts--head1 body1 and tail1 or this fin and that fin--represent so many independent variants1 then our co-ordinate system will at once become too complex to be intelligible2 we shall be making not one comparison but several #gbg separate comparisons1 and our general method will be found inapplicable4 ,now precisely this independent variability of parts and organs--here1 there1 and everywhere within the organism--would appear to be implicit in our ordinary accepted notions regarding variation2 and1 unless ,i am greatly mistaken1 it is precisely on such a conception of the easy1 frequent1 and normal independent variability of parts that our conception of the process of natural selection is fundamentally based4 ,for the morphologist1 when comparing one organism with another1 describes the differences between them point by point1 and 8character0 by 8character#fde40 ,if he is from time to time constrained to admit the existence of 8correlation0 between characters 7as a hundred years ago ,cuvier first showed the way71 yet all the while he recognises this fact of correlation somewhat vaguely1 as a phenomenon due to causes which1 except in rare instances1 he can hardly hope to trace2 and he falls readily into the habit of thinking and talking of evolution as though it had proceeded on the lines of his own descriptions1 point by point1 and character by character#fdf4 ,but if1 on the other hand1 diverse and dissimilar fishes can be referred as a whole to identical functions of very different co-ordinate systems1 this fact will of itself constitute a proof that variation has proceeded on definite and orderly lines1 that a comprehensive 8law of growth0 has pervaded the whole structure in its integrity1 and that some more or less simple and recognisable system of forces has been at work4 ,it will not only show how real and deep-seated is the phenomenon of 8correlation10 in regard to form1 but it will also demonstrate the fact that a correlation which had seemed too complex for analysis or #gbh comprehension is1 in many cases1 capable of very simple graphic expression4 ,this1 after many trials1 ,i believe to be in general the case1 bearing always in mind that the occurrence of independent or localised variations must often be considered4 ,we are dealing in this chapter with the forms of related organisms1 in order to shew that the differences between them are as a general rule simple and symmetrical1 and just such as might have been brought about by a slight and simple change in the system of forces to which the living and growing organism was exposed4 ,mathematically speaking1 the phenomenon is identical with one met with by the geologist1 when he finds a bed of fossils squeezed flat or otherwise symmetrically deformed by the pressures to which they1 and the strata which contain them1 have been subjected4 ,in the first step towards fossilisation1 when the body of a fish or shellfish is silted over and buried1 we may take it that the wet sand or mud exercises1 approximately1 a hydrostatic pressure--that is to say a pressure which is uniform in all directions1 and by which the form of the buried object will not be appreciably changed4 ,as the strata consolidate and accumulate1 the fossil organisms which they contain will tend to be flattened by the vast superincumbent load1 just as the stratum which contains them will also be compressed and will have its molecular arrangement more or less modified#fdg4 ,but the deformation due to direct vertical pressure in a horizontal stratum is not nearly so striking as are the deformations produced by the oblique or shearing stresses to which inclined and folded strata have been exposed1 and by which their various 8dislocations0 have been brought about4 ,and especially in mountain regions1 where these dislocations are especially numerous and complicated1 the contained fossils are apt to be so curiously and yet so symmetrically deformed 7usually by a simple shear7 that they may easily be interpreted as so many distinct and separate 8species#fdh40 ,a great number of described species1 and here and there a new genus 7as the genus ,ellipsolithes for an obliquely deformed ,goniatite or ,nautilus7 are said to rest on no other foundation#fdi4 ,illustration3 ,fig4 #ceb4 ,illustration3 ,fig4 #cec4 ,illustration3 ,fig4 #ced4 ,illustration3 ,fig4 #cee4 ,if we begin by drawing a net of rectangular equidistant co-ordinates 7about the axes x and y71 we may alter or deform this #gbi network in various ways1 several of which are very simple indeed4 ,thus 7#a7 we may alter the dimensions of our system1 extending it along one or other axis1 and so converting each little square into a corresponding and directly proportionate oblong 7,fig4 #cec74 ,it follows that any figure which we may have inscribed in the original net1 and which we transfer to the new1 will thereby be deformed in strict proportion to the deformation of the entire configuration1 being still defined by corresponding points in the network and being throughout in conformity with the original figure4 ,for instance1 a circle inscribed in the original 8,cartesian0 net will now1 after extension in the y-direction1 be found elongated #gcj into an ellipse4 ,in elementary mathematical language1 for the original x and y we have substituted x#a and c y#a1 and the equation to our original circle1 x#b y#b a#b1 becomes that of the ellipse1 x#ab c#b y#ab a#b4 ,if ,i draw the cannon-bone of an ox 7,fig4 #ced1 ,a71 for instance1 within a system of rectangular co-ordinates1 and then transfer the same drawing1 point for point1 to a system in which for the x of the original diagram we substitute x #bx_/#c1 we obtain a drawing 7,b7 which is a very close approximation to the cannon-bone of the sheep4 ,in other words1 the main 7and perhaps the only7 difference between the two bones is simply that that of the sheep is elongated1 along the vertical axis1 as compared with that of the ox in the relation of #c_/b4 ,and similarly1 the long slender cannon-bone of the giraffe 7,c7 is referable to the same identical type1 subject to a reduction of breadth1 or increase of length1 corresponding to x x_/#c4 7#b7 ,the second type is that where extension is not equal or uniform at all distances from the origin3 but grows greater or less1 as1 for instance1 when we stretch a tapering elastic band4 ,in such cases1 as ,i have represented it in ,fig4 #cee1 the ordinate increases logarithmically1 and for y we substitute y4 ,it is obvious that this logarithmic extension may involve both abscissae and ordinates1 x becoming x1 while y becomes y4 ,the circle in our original figure is now deformed into some such shape as that of ,fig4 #cef4 ,this method of deformation is a common one1 and will often be of use to us in our comparison of organic forms4 7#c7 ,our third type is the 8simple shear10 where the rectangular co-ordinates become 8oblique10 their axes being inclined to one another at a certain angle 4 ,our original rectangle now becomes such a figure as that of ,fig4 #ceg4 ,the system may now be described in terms of the oblique axes ,x1 ,y2 or may be directly referred to new rectangular co-ordinates 1 by the simple transposition x cot 1 y cosec 4 ,illustration3 ,fig4 #cef4 ,illustration3 ,fig4 #ceg4 ,illustration3 ,fig4 #ceh4 7#d7 ,yet another important class of deformations may be represented by the use of radial co-ordinates1 in which one set of lines are represented as radiating from a point or 8focus10 while the other set are transformed into circular arcs cutting the radii orthogonally4 ,these radial co-ordinates are especially applicable #gca to cases where there exists 7either within or without the figure7 some part which is supposed to suffer no deformation2 a simple illustration is afforded by the diagrams which illustrate the flexure of a beam 7,fig4 #ceh74 ,in biology these co-ordinates will be especially applicable in cases where the growing structure includes a 8node10 or point where growth is absent or at a minimum2 and about which node the rate of growth may be assumed to increase symmetrically4 ,precisely such a case is furnished us in a leaf of an ordinary dicotyledon4 ,the leaf of a #gcb typical monocotyledon--such as a grass or a hyacinth1 for instance--grows continuously from its base1 and exhibits no node or 8point of arrest40 ,its sides taper off gradually from its broad base to its slender tip1 according to some law of decrement specific to the plant2 and any alteration in the relative velocities of longitudinal and transverse growth will merely make the leaf a little broader or narrower1 and will effect no other conspicuous alteration in its contour4 ,but if there once come into existence a node1 or 8locus of no growth10 about which we may assume the growth--which in the hyacinth leaf was longitudinal and transverse--to take place radially and transversely to the radii1 then we shall ,illustration3 ,fig4 #cei4 at once see1 in the first place1 that the sloping and slightly curved sides of the hyacinth leaf suffer a transformation into what we consider a more typical and 8leaf-like0 shape1 the sides of the figure broadening out to a zone of maximum breadth and then drawing inwards to the pointed apex4 ,if we now alter the ratio between the radial and tangential velocities of growth--in other words1 if we increase the angles between corresponding radii--we pass successively through the various configurations which the botanist describes as the lanceolate1 the ovate1 and finally the cordate leaf4 ,these successive changes may to some extent1 and in appropriate cases1 be traced as the individual leaf grows #gcc to maturity2 but as a much more general rule1 the balance of forces1 the ratio between radial and tangential velocities of growth1 remains so nicely and constantly balanced that the leaf increases in size without conspicuous modification of form4 ,it is rather what we may call a long-period variation1 a tendency for the relative velocities to alter from one generation to another1 whose result is brought into view by this method of illustration4 ,illustration3 ,fig4 #cfj4 ,begonia daedalea4 ,there are various corollaries to this method of describing the form of a leaf which may be here alluded to1 for we shall not return again to the subject of radial co-ordinates4 ,for instance1 the so-called unsymmetrical leaf#fej of a begonia1 in which one side of the leaf may be merely ovate while the other has a cordate outline1 is seen to be really a case of unequal1 and not truly asymmetrical1 growth on either side of the midrib4 ,there is nothing more mysterious in its conformation than1 for instance1 in that of a forked twig in which one limb of the fork has grown longer than the other4 ,the case of the begonia leaf is of sufficient interest to deserve illustration1 and in ,fig4 #cfj ,i have outlined a leaf of the large ,begonia daedalea4 ,on the smaller left-hand side of the leaf ,i have taken at random three points1 a1 b1 c1 and have measured the angles1 ,a,oa1 etc41 which the radii from the hilus of the leaf to these points make with the median axis4 ,on the other side of the leaf ,i have marked the points a1 b1 c1 such that the radii drawn to this margin of the leaf are equal to the former1 ,oa to ,oa1 etc4 ,now if the two sides of the leaf are #gcd mathematically similar to one another1 it is obvious that the respective angles should be in continued proportion1 i4e4 as ,a,oa is to ,a,oa1 so should ,a,ob be to ,a,ob4 ,this proves to be very nearly the case4 ,for ,i have measured the three angles on one side1 and one on the other1 and have then compared1 as follows1 the calculated with the observed values of the other two3 ,a,oa ,a,ob ,a,oc ,a,oa ,a,ob ,a,oc ,observed values #ab #bh4e #hh -- -- #aeg ,calculated values -- -- -- #ba4e #ea4a -- ,observed values -- -- -- #bj #eb -- ,the agreement is very close1 and what discrepancy there is may be amply accounted for1 firstly1 by the slight irregularity of the sinuous margin of the leaf2 and secondly1 by the fact that the true axis or midrib of the leaf is not straight but slightly curved1 and therefore that it is curvilinear and not rectilinear triangles which we ought to have measured4 ,when we understand these few points regarding the peripheral curvature of the leaf1 it is easy to see that its principal veins approximate closely to a beautiful system of isogonal co-ordinates4 ,it is also obvious that we can easily pass1 by a process of shearing1 from those cases where the principal veins start from the base of the leaf to those1 as in most dicotyledons1 where they arise successively from the midrib4 ,it may sometimes happen that the node1 or 8point of arrest10 is at the upper instead of the lower end of the leaf-blade2 and occasionally there may be a node at both ends4 ,in the former case1 as we have it in the daisy1 the form of the leaf will be1 as it were1 inverted1 the broad1 more or less heart-shaped1 outline appearing at the upper end1 while below the leaf tapers gradually downwards to an ill-defined base4 ,in the latter case1 as in ,dionaea1 we obtain a leaf equally expanded1 and similarly ovate or cordate1 at both ends4 ,we may notice1 lastly1 that the shape of a solid fruit1 such as an apple or a cherry1 is a solid of revolution1 developed from similar curves and to be explained on the same principle4 ,in the cherry we have a 8point of arrest0 at the base of the berry1 where it joins its peduncle1 and about this point the fruit 7in imaginary section7 swells out into a cordate outline2 while in the #gce apple we have two such well-marked points of arrest1 above and below1 and about both of them the same conformation tends to arise4 ,the bean and the human kidney owe their 8reniform0 shape to precisely the same phenomenon1 namely1 to the existence of a node or 8hilus10 about which the forces of growth are radially and symmetrically arranged4 ,most of the transformations which we have hitherto considered 7other than that of the simple shear7 are particular cases of a general transformation1 obtainable by the method of conjugate functions and equivalent to the projection of the original figure on a new plane4 ,appropriate transformations1 on these general lines1 provide for the cases of a coaxial system where the ,cartesian co-ordinates are replaced by coaxial circles1 or a confocal system in which they are replaced by confocal ellipses and hyperbolas4 ,illustration3 ,fig4 #cfa4 ,yet another curious and important transformation1 belonging to the same class1 is that by which a system of straight lines becomes transformed into a conformal system of logarithmic spirals3 the straight line ,y ,,ax c corresponding to the logarithmic spiral ,a log r c 7,fig4 #cfa74 ,this beautiful and simple transformation lets us at once convert1 for instance1 the straight conical shell of the ,pteropod or the ,orthoceras into the logarithmic spiral of the ,nautiloid2 it involves a mathematical symbolism which is but a slight extension of that which we have employed in our elementary treatment of the logarithmic spiral4 ,these various systems of coordinates1 which we have now briefly considered1 are sometimes called 8isothermal co-ordinates10 from the fact that1 when employed in this particular branch of physics1 they perfectly represent the phenomena of the conduction of heat1 the contour lines of equal temperature appearing1 under appropriate conditions1 as the orthogonal lines of the co-ordinate system4 ,and it follows that #gcf the 8law of growth0 which our biological analysis by means of orthogonal co-ordinate systems presupposes1 or at least foreshadows1 is one according to which the organism grows or develops along stream lines1 which may be defined by a suitable mathematical transformation4 ,when the system becomes no longer orthogonal1 as in many of the following illustrations--for instance1 that of ,orthagoriscus 7,fig4 #chb71--then the transformation is no longer within the reach of comparatively simple mathematical analysis4 ,such departure from the typical symmetry of a 8stream-line0 system is1 in the first instance1 sufficiently accounted for by the simple fact that the developing organism is very far from being homogeneous and isotropic1 or1 in other words1 does not behave like a perfect fluid4 ,but though under such circumstances our co-ordinate systems may be no longer capable of strict mathematical analysis1 they will still indicate graphically the relation of the new co-ordinate system to the old1 and conversely will furnish us with some guidance as to the 8law of growth10 or play of forces1 by which the transformation has been effected4 ,before we pass from this brief discussion of transformations in general1 let us glance at one or two cases in which the forces applied are more or less intelligible1 but the resulting transformations are1 from the mathematical point of view1 exceedingly complicated4 ,the 8marbled papers0 of the bookbinder are a beautiful illustration of visible 8stream lines40 ,on a dishful of a sort of semi-liquid gum the workman dusts a few simple lines or patches of colouring matter2 and then1 by passing a comb through the liquid1 he draws the colour-bands into the streaks1 waves1 and spirals which constitute the marbled pattern1 and which he then transfers to sheets of paper laid down upon the gum4 ,by some such system of shears1 by the effect of unequal traction or unequal growth in various directions and superposed on an originally simple pattern1 we may account for the not dissimilar marbled patterns which we recognise1 for instance1 on a large serpent's skin4 ,but it must be remarked1 in the case of the marbled paper1 that though the method of application of the forces is simple1 yet in the aggregate the system of forces set up by the many #gcg teeth of the comb is exceedingly complex1 and its complexity is revealed in the complicated 8diagram of forces0 which constitutes the pattern4 ,to take another and still more instructive illustration4 ,to turn one circle 7or sphere7 into two circles would be1 from the point of view of the mathematician1 an extraordinarily difficult transformation2 but1 physically speaking1 its achievement may be extremely simple4 ,the little round gourd grows naturally1 by its symmetrical forces of expansive growth1 into a big1 round1 or somewhat oval pumpkin or melon4 ,but the ,moorish husbandman ties a rag round its middle1 and the same forces of growth1 unaltered save for the presence of this trammel1 now expand the globular structure into two superposed and connected globes4 ,and again1 by varying the position of the encircling band1 or by applying several such ligatures instead of one1 a great variety of artificial forms of 8gourd0 may be1 and actually are1 produced4 ,it is clear1 ,i think1 that we may account for many ordinary biological processes of development or transformation of form by the existence of trammels or lines of constraint1 which limit and determine the action of the expansive forces of growth that would otherwise be uniform and symmetrical4 ,this case has a close parallel in the operations of the glassblower1 to which we have already1 more than once1 referred in passing#fea4 ,the glassblower starts his operations with a tube1 which he first closes at one end so as to form a hollow vesicle1 within which his blast of air exercises a uniform pressure on all sides2 but the spherical conformation which this uniform expansive force would naturally tend to produce is modified into all kinds of forms by the trammels or resistances set up as the workman lets one part or another of his bubble be unequally heated or cooled4 ,it was ,oliver ,wendell ,holmes who first shewed this curious parallel between the operations of the glassblower and those of ,nature1 when she starts1 as she so often does1 with a simple tube#feb4 ,the alimentary canal1 #gch the arterial system including the heart1 the central nervous system of the vertebrate1 including the brain itself1 all begin as simple tubular structures4 ,and with them ,nature does just what the glassblower does1 and1 we might even say1 no more than he4 ,for she can expand the tube here and narrow it there2 thicken its walls or thin them2 blow off a lateral offshoot or caecal diverticulum2 bend the tube1 or twist and coil it2 and infold or crimp its walls as1 so to speak1 she pleases4 ,such a form as that of the human stomach is easily explained when it is regarded from this point of view2 it is simply an ill-blown bubble1 a bubble that has been rendered lopsided by a trammel or restraint along one side1 such as to prevent its symmetrical expansion--such a trammel as is produced if the glassblower lets one side of his bubble get cold1 and such as is actually present in the stomach itself in the form of a muscular band4 ,we may now proceed to consider and illustrate a few permutations or transformations of organic form1 out of the vast multitude which are equally open to this method of inquiry4 ,illustration3 ,fig4 #cfb4 ,we have already compared in a preliminary fashion the metacarpal or cannon-bone of the ox1 the sheep1 and the giraffe 7,fig4 #ced72 and we have seen that the essential difference in form between these three bones is a matter of relative length and breadth1 such that1 if we reduce the figures to an identical standard of length 7or identical values of y71 the breadth 7or value of x7 will be approximately two-thirds that of the ox in the case of the sheep and one-third that of the ox in the case of the giraffe4 ,we may easily1 for the sake of closer comparison1 determine these ratios more accurately1 for instance1 if it be our purpose to compare the different racial varieties within the limits of a single species4 ,and in such cases1 by the way1 as when we compare with one another various breeds or races of cattle or of horses1 the ratios #gci of length and breadth in this particular bone are extremely significant#fec4 ,if1 instead of limiting ourselves to the cannon-bone1 we inscribe the entire foot of our several ,ungulates in a co-ordinate system1 the same ratios of x that served us for the cannon-bones still give us a first approximation to the required comparison2 but even in the case of such closely allied forms as the ox and the sheep there is evidently something wanting in the comparison4 ,the reason is that the relative elongation of the several parts1 or individual bones1 has not proceeded equally or proportionately in all cases2 in other words1 that the equations for x will not suffice without some simultaneous modification of the values of y 7,fig4 #cfb74 ,in such a case it may be found possible to satisfy the varying values of y by some logarithmic or other formula2 but1 even if that be possible1 it will probably be somewhat difficult of discovery or verification in such a case as the present1 owing to the fact that we have too few well-marked points of correspondence between the one object and the other1 and that especially along the shaft of such long bones as the cannon-bone of the ox1 the deer1 the llama1 or the giraffe there is a complete lack of easily recognisable corresponding points4 ,in such a case a brief tabular statement of apparently corresponding values of y1 or of those obviously corresponding values which coincide with the boundaries of the several bones of the foot1 will1 as in the following example1 enable us to dispense with a fresh equation4 a b c d y 7,ox7 #j #ah #bg #db #ajj y 7,sheep7 #j #aj #ai #cf #ajj y 7,giraffe7 #j #e #aj #bd #ajj ,this summary of values of y1 coupled with the equations for the #gdj value of x1 will enable us1 from any drawing of the ox's foot1 to construct a figure of that of the sheep or of the giraffe with remarkable accuracy4 ,illustration3 ,fig4 #cfc4 ,that underlying the varying amounts of extension to which the parts or segments of the limb have been subject there is a law1 or principle of continuity1 may be discerned from such a diagram as the above 7,fig4 #cfc71 where the values of y in the case of the ox are plotted as a straight line1 and the corresponding values for the sheep 7extracted from the above table7 are seen to form a more or less regular and even curve4 ,this simple graphic result implies the existence of a comparatively simple equation between y and y4 ,illustration3 ,fig4 #cfd4 7,after ,albert ,drer47 ,an elementary application of the principle of co-ordinates to the study of proportion1 as we have here used it to illustrate the varying proportions of a bone1 was in common use in the sixteenth and seventeenth centuries by artists in their study of the human form4 ,the method is probably much more ancient1 and may even be classical#fed2 it is fully described and put in practice by ,albert ,drer in his ,geometry1 and especially in his ,treatise on ,proportion#fee4 ,in this latter work1 the manner in which the #gda human figure1 features1 and facial expression are all transformed and modified by slight variations in the relative magnitude of the parts is admirably and copiously illustrated 7,fig4 #cfd74 ,illustration3 ,fig4 #cfe4 ,in a tapir's foot there is a striking difference1 and yet at the same time there is an obvious underlying resemblance1 between the middle toe and either of its unsymmetrical lateral neighbours4 ,let us take the median terminal phalanx and inscribe its outline in a net of rectangular equidistant co-ordinates 7,fig4 #cfe1 a74 ,let us then make a similar network about axes which are no longer at right angles1 but inclined to one another at an angle of about #ej 7b74 ,if into this new network we fill in1 point for point1 an outline precisely corresponding to our original drawing of the middle toe1 we shall find that we have already represented the main features of the adjacent lateral one4 ,we shall1 however1 perceive that our new diagram looks a little too bulky on one side1 the inner side1 of the lateral toe4 ,if now we substitute for our equidistant ordinates1 ordinates which get gradually closer and closer together as we pass towards the median side of the toe1 then we shall obtain a diagram which differs in no essential respect from an actual outline copy of the lateral toe 7c74 ,in short1 the difference between the outline of the middle toe of the tapir and the next lateral toe may be almost completely expressed by saying that if the one be represented by rectangular equidistant co-ordinates1 the other will be represented by oblique co-ordinates1 whose axes make an angle of #ej1 and in which the abscissal interspaces decrease in a certain logarithmic ratio4 ,we treated our original complex curve or projection of the tapir's toe as a function of the form ,f 7x1 y7 #j4 ,the figure of the tapir's lateral #gdb toe is a precisely identical function of the form ,f 7ex1 y#a7 #j1 where x#a1 y#a are oblique co-ordinate axes inclined to one another at an angle of #ej4 ,illustration3 ,fig4 #cff4 7,after ,albert ,drer47 ,drer was acquainted with these oblique co-ordinates also1 and ,i have copied two illustrative figures from his book#fef4 ,illustration3 ,fig4 #cfg4 ,oithona nana4 ,illustration3 ,fig4 #cfh4 ,sapphirina4 ,in ,fig4 #cfg ,i have sketched the common ,copepod ,oithona nana1 #gdc and have inscribed it in a rectangular net1 with abscissae three-fifths the length of the ordinates4 ,side by side 7,fig4 #cfh7 is drawn a very different ,copepod1 of the genus ,sapphirina2 and about it is drawn a network such that each co-ordinate passes 7as nearly as possible7 through points corresponding to those of the former figure4 ,it will be seen that two differences are apparent4 7#a7 ,the values of y in ,fig4 #cfh are large in the upper part of the figure1 and diminish rapidly towards its base4 7#b7 ,the values of x are very large in the neighbourhood of the origin1 but diminish rapidly as we pass towards either side1 away from the median vertical axis2 and it is probable that they do so according to a definite1 but somewhat complicated1 ratio4 ,if1 instead of seeking for an actual equation1 we simply tabulate our values of x and y in the second figure as compared with the first 7just as we did in comparing the feet of the ,ungulates71 we get the dimensions of a net in which1 by simply projecting the figure of ,oithona1 we obtain that of ,sapphirina without further trouble1 e4g43 x 7,oithona7 #j #c #f #i #ab #ae -- x 7,sapphirina7 #j #h #aj #ab #ac #ad -- y 7,oithona7 #j #e #aj #ae #bj #be #cj y 7,sapphirina7 #j #b #g #c #bc #cb #dj ,in this manner1 with a single model or type to copy from1 we may record in very brief space the data requisite for the production of approximate outlines of a great number of forms4 ,for instance the difference1 at first sight immense1 between the attenuated body of a ,caprella and the thick-set body of a ,cyamus is obviously little1 and is probably nothing1 more than a difference of relative magnitudes1 capable of tabulation by numbers and of complete expression by means of rectilinear co-ordinates4 ,the ,crustacea afford innumerable instances of more complex deformations4 ,thus we may compare various higher ,crustacea with one another1 even in the case of such dissimilar forms as a lobster and a crab4 ,it is obvious that the whole body of the former is elongated as compared with the latter1 and that the crab is relatively broad in the region of the carapace1 while it tapers off rapidly towards its attenuated and abbreviated tail4 ,in a general way1 the elongated rectangular system of co-ordinates #gdd in which we may inscribe the outline of the lobster becomes a shortened triangle in the case of the crab4 ,in a little more detail we may compare the outline of the carapace in various crabs one with another3 and the comparison will be found easy and significant1 even1 in many cases1 down to minute details1 such as the number and situation of the marginal spines1 though these are in other cases subject to independent variability4 ,illustration3 ,fig4 #cfi4 ,carapaces of various crabs4 #a1 ,geryon2 #b1 ,corystes2 #c1 ,scyramathia2 #d1 ,paralomis2 #e1 ,lupa2 #f1 ,chorinus4 ,if we choose1 to begin with1 such a crab as ,geryon 7,fig4 #cfi1 #a71 and inscribe it in our equidistant rectangular co-ordinates1 we shall see that we pass easily to forms more elongated in a transverse #gde direction1 such as ,matuta or ,lupa 7#e71 and conversely1 by transverse compression1 to such a form as ,corystes 7#b74 ,in certain other cases the carapace conforms to a triangular diagram1 more or less curvilinear1 as in ,fig4 #d1 which represents the genus ,paralomis4 ,here we can easily see that the posterior border is transversely elongated as compared with that of ,geryon1 while at the same time the anterior part is longitudinally extended as compared with the posterior4 ,a system of slightly curved and converging ordinates1 with orthogonal and logarithmically interspaced abscissal lines1 as shown in the figure1 appears to satisfy the conditions4 ,in an interesting series of cases1 such as the genus ,chorinus1 or ,scyramathia1 and in the spider-crabs generally1 we appear to have just the converse of this4 ,while the carapace of these crabs presents a somewhat triangular form1 which seems at first sight more or less similar to those just described1 we soon see that the actual posterior border is now narrow instead of broad1 the broadest part of the carapace corresponding precisely1 not to that which is broadest in ,paralomis1 but to that which was broadest in ,geryon2 while the most striking difference from the latter lies in an antero-posterior lengthening of the forepart of the carapace1 culminating in a great elongation of the frontal region1 with its two spines or 8horns40 ,the curved ordinates here converge posteriorly and diverge widely in front 7,figs4 #c and #f71 while the decremental interspacing of the abscissae is very marked indeed4 ,we put our method to a severer test when we attempt to sketch an entire and complicated animal than when we simply compare corresponding parts such as the carapaces of various ,malacostraca1 or related bones as in the case of the tapir's toes4 ,nevertheless1 up to a certain point1 the method stands the test very well4 ,in other words1 one particular mode and direction of variation is often 7or even usually7 so prominent and so paramount throughout the entire organism1 that one comprehensive system of co-ordinates suffices to give a fair picture of the actual phenomenon4 ,to take another illustration from the ,crustacea1 ,i have drawn roughly in ,fig4 #cgj1 #a a little amphipod of the family ,phoxocephalidae 7,harpinia sp474 ,deforming the co-ordinates of the figure into the #gdf curved orthogonal system in ,fig4 #b1 we at once obtain a very fair representation of an allied genus1 belonging to a different family of amphipods1 namely ,stegocephalus4 ,as we proceed further from our type our co-ordinates will require greater deformation1 and the resultant figure will usually be somewhat less accurate4 ,in ,fig4 #c ,i show a network1 to which1 if we transfer our diagram of ,harpinia or of ,stegocephalus1 we shall obtain a tolerable representation of the aberrant genus ,hyperia1 with its narrow abdomen1 its reduced pleural lappets1 its great eyes1 and its inflated head4 ,illustration3 ,fig #cgj4 #a4 ,harpinia plumosa ,kr4 #b4 ,stegocephalus inflatus ,kr4 #c4 ,hyperia galba4 ,the hydroid zoophytes constitute a 8polymorphic0 group1 within which a vast number of species have already been distinguished2 and the labours of the systematic naturalist are constantly adding to the number4 ,the specific distinctions are for the most part based1 not upon characters directly presented #gdg by the living animal1 but upon the form1 size and arrangement of the little cups1 or 8calycles10 secreted and inhabited by the little individual polypes which compose the compound organism4 ,the variations1 which are apparently infinite1 of these conformations are easily seen to be a question of relative magnitudes1 and are capable of complete expression1 sometimes by very simple1 sometimes by somewhat more complex1 co-ordinate networks4 ,illustration3 ,fig4 #cga4 a1 ,campanularia macroscyphus1 ,allm42 b1 ,gonothyraea hyalina1 ,hincks2 c1 ,clytia ,johnstoni1 ,alder4 ,for instance1 the varying shapes of the simple wineglass-shaped cups of the ,campanularidae are at once sufficiently represented and compared by means of simple ,cartesian co-ordinates 7,fig4 #cga74 ,in the two allied families of ,plumulariidae and ,aglaopheniidae the calycles are set unilaterally upon a jointed stem1 and small cup-like structures 7holding rudimentary polypes7 are associated with the large calycles in definite number and position4 ,these small calyculi are variable in number1 but in the great majority of cases they accompany the large calycle in groups of three--two standing by its upper border1 and one1 which is especially variable in form and magnitude1 lying at its base4 ,the stem is liable to flexure and1 in a high degree1 to extension or compression2 and these variations extend1 often on an exaggerated scale1 to the related calycles4 ,as a result we find that we can draw various systems of curved or sinuous co-ordinates1 which express1 all but completely1 the configuration of the various #gdh hydroids which we inscribe therein 7,fig4 #cgb74 ,the comparative smoothness or denticulation of the margin of the calycle1 and the number of its denticles1 constitutes an independent variation1 and requires separate description2 we have already seen 7p4 #bcf7 that this denticulation is in all probability due to a particular physical cause4 ,illustration3 ,fig4 #cgb4 a1 ,cladocarpus crenatus1 ,f42 b1 ,aglaophenia pluma1 ,l42 c1 ,a4 rhynchocarpa1 ,a42 d1 ,a cornuta1 ,k42 e1 ,a4 ramulosa1 ,k4 ,illustration3 ,fig4 #cgc4 ,argyropelecus ,olfersi4 ,illustration3 ,fig4 #cgd4 ,sternoptyx diaphana4 ,among the fishes we discover a great variety of deformations1 some of them of a very simple kind1 while others are more striking and more unexpected4 ,a comparatively simple case1 involving a simple shear1 is illustrated by ,figs4 #cgc and #cgd4 ,fig4 #cgc represents1 within ,cartesian co-ordinates1 a certain little oceanic fish known as ,argyropelecus ,olfersi4 ,fig4 #dgd represents precisely the same outline1 transferred to a system of oblique co-ordinates whose #gdi axes are inclined at an angle of #gj2 but this is now 7as far as can be seen on the scale of the drawing7 a very good figure of an allied fish1 assigned to a different genus1 under the name of ,sternoptyx diaphana4 ,the deformation illustrated by this case of ,argyropelecus is precisely analogous to the simplest and commonest kind of deformation to which fossils are subject 7as we have seen on p4 #eec7 as the result of shearing-stresses in the solid rock4 ,illustration3 ,fig4 #cge4 ,scarus sp4 ,illustration3 ,fig4 #cgf4 ,pomacanthus4 ,fig4 #cge is an outline diagram of a typical ,scaroid fish4 ,let us deform its rectilinear co-ordinates into a system of 7approximately7 coaxial circles1 as in ,fig4 #cgf1 and then filling into the new system1 space by space and point by point1 our former diagram of ,scarus1 we obtain a very good outline of an allied fish1 belonging to a neighbouring family1 of the genus ,pomacanthus4 ,this case is all the more interesting1 because upon the body of our ,pomacanthus there are striking colour bands1 which correspond in direction very closely to the lines of our new curved ordinates4 ,in like manner1 the still more bizarre outlines of other fishes of the same family of ,chaetodonts will be found to correspond to very slight modifications of similar co-ordinates2 in other words1 to small variations in the values of the constants of the coaxial curves4 ,in ,figs4 #cgg-chj ,i have represented another series of ,acanthopterygian fishes1 not very distantly related to the foregoing4 ,if we start this series with the figure of ,polyprion1 in ,fig4 #cgg1 we see that the outlines of ,pseudopriacanthus 7,fig4 #cgh7 and of ,sebastes or ,scorpaena 7,fig4 #cgi7 are easily derived by substituting a system of triangular1 or radial1 co-ordinates for the rectangular ones in #gej which we had inscribed ,polyprion4 ,the very curious fish ,antigonia capros1 an oceanic relative of our own 8boar-fish10 conforms closely to the peculiar deformation represented in ,fig4 #chj4 ,illustration3 ,fig4 #cgg4 ,polyprion4 ,illustration3 ,fig4 #cgh4 ,pseudopriacanthus altus4 ,illustration3 ,fig4 #cgi4 ,scorpaena sp4 ,illustration3 ,fig4 #chj4 ,antigonia capros4 ,illustration3 ,fig4 #cha4 ,diodon4 ,illustration3 ,fig4 #chb4 ,orthagoriscus4 ,fig4 #cha is a common1 typical ,diodon or porcupine-fish1 and in ,fig4 #chb ,i have deformed its vertical co-ordinates into a system of concentric circles1 and its horizontal co-ordinates into a system of curves which1 approximately and provisionally1 are made to resemble a system of hyperbolas#feg4 ,the old outline1 transferred #gea in its integrity to the new network1 appears as a manifest representation of the closely allied1 but very different looking1 sunfish1 ,orthagoriscus mola4 ,this is a particularly instructive case of deformation or transformation4 ,it is true that1 in a mathematical sense1 it is not a perfectly satisfactory or perfectly regular deformation1 for the system is no longer isogonal2 but nevertheless1 it is symmetrical to the eye1 and obviously approaches to an isogonal system under certain conditions of friction or constraint4 ,and as such it accounts1 by one single integral transformation1 for all the apparently separate and distinct external differences between the two fishes4 ,it leaves the parts near to the origin of the system1 the whole region of the head1 the opercular orifice and the pectoral fin1 practically unchanged #geb in form1 size and position2 and it shews a greater and greater apparent modification of size and form as we pass from the origin towards the periphery of the system4 ,in a word1 it is sufficient to account for the new and striking contour in all its essential details1 of rounded body1 exaggerated dorsal and ventral fins1 and truncated tail4 ,in like manner1 and using precisely the same co-ordinate networks1 it appears to me possible to shew the relations1 almost bone for bone1 of the skeletons of the two fishes2 in other words1 to reconstruct the skeleton of the one from our knowledge of the skeleton of the other1 under the guidance of the same correspondence as is indicated in their external configuration4 ,the family of the crocodiles has had a special interest for the evolutionist ever since ,huxley pointed out that1 in a degree only second to the horse and its ancestors1 it furnishes us with a close and almost unbroken series of transitional forms1 running down in continuous succession from one geological formation to another4 ,i should be inclined to transpose this general statement into other terms1 and to say that the ,crocodilia constitute a case in which1 with unusually little complication from the presence of independent variants1 the trend of one particular mode of transformation is visibly manifested4 ,if we exclude meanwhile from our comparison a few of the oldest of the crocodiles1 such as ,belodon1 which differ more fundamentally from the rest1 we shall find a long series of genera in which we can refer not only the changing contours of the skull1 but even the shape and size of the many constituent bones and their intervening spaces or 8vacuities10 to one and the same simple system of transformed co-ordinates4 ,the manner in which the skulls of various ,crocodilians differ from one another may be sufficiently illustrated by three or four examples4 ,illustration3 ,fig4 #chc4 ,a1 ,crocodilus porosus4 ,b1 ,c4 americanus4 ,c1 ,notosuchus terrestris4 ,let us take one of the typical modern crocodiles as our standard of form1 e4g4 ,c4 porosus1 and inscribe it1 as in ,fig4 #chc1 a1 in the usual ,cartesian co-ordinates4 ,by deforming the rectangular network into a triangular system1 with the apex of the triangle a little way in front of the snout1 as in b1 we pass to such a form as ,c4 americanus4 ,by an exaggeration of the same process we at once get an approximation to the form of one of the sharp-snouted1 #gec or longirostrine1 crocodiles1 such as the genus ,tomistoma2 and1 in the species figured1 the oblique position of the orbits1 the arched contour of the occipital border1 and certain other characters suggest a certain amount of curvature1 such as ,i have represented in the diagram 7,fig4 #chc1 b71 on the part of the horizontal co-ordinates4 ,in the still more elongated skull of such a form as the ,indian ,gavial1 the whole skull has undergone a great longitudinal extension1 or1 in other words1 the ratio of x_/y is greatly diminished2 and this extension is not uniform1 but is at a maximum in the region of the nasal and maxillary bones4 ,this especially elongated region is at the same time narrowed in an exceptional degree1 and its excessive narrowing is represented by a curvature1 convex towards the median axis1 on the part of the vertical ordinates4 ,let us take as a last illustration one of the ,mesozoic crocodiles1 the little ,notosuchus1 from the ,cretaceous formation4 ,this little crocodile is very different from our type in the proportions of its skull4 ,the region of the snout1 in front of and including the frontal bones1 is greatly shortened2 from constituting fully two-thirds of the whole length of the skull in ,crocodilus1 it now constitutes less than half1 or1 say1 three-sevenths of the whole2 and the whole skull1 and especially its posterior part1 is curiously compact1 broad1 and squat4 ,the orbit is unusually large4 ,if in the diagram of this skull we select a number of points obviously corresponding #ged to points where our rectangular co-ordinates intersect particular bones or other recognisable features in our typical crocodile1 we shall easily discover that the lines joining these points in ,notosuchus fall into such a co-ordinate network as that which is represented in ,fig4 #chc1 c4 ,to all intents and purposes1 then1 this not very complex system1 representing one harmonious 8deformation10 accounts for all the differences between the two figures1 and is sufficient to enable one at any time to reconstruct a detailed drawing1 bone for bone1 of the skull of ,notosuchus from the model furnished by the common crocodile4 ,illustration3 ,fig4 #chd4 ,pelvis of 7,a7 ,stegosaurus2 7,b7 ,camptosaurus4 ,the many diverse forms of ,dinosaurian reptiles1 all of which manifest a strong family likeness underlying much superficial diversity1 furnish us with plentiful material for comparison by the method of transformations4 ,as an instance1 ,i have figured the pelvic bones of ,stegosaurus and of ,camptosaurus 7,fig4 #chd1 a1 b7 to show that1 when the former is taken as our ,cartesian type1 a slight curvature and an approximately logarithmic extension of the x-axis brings us easily to the configuration of the other4 ,in the original specimen of ,camptosaurus described by ,marsh#feh1 the anterior portion of the iliac bone is missing2 and in ,marsh's restoration this part of the bone is drawn as though it came somewhat abruptly to a sharp point4 ,in my figure ,i #gee have completed this missing part of the bone in harmony with the general co-ordinate network which is suggested by our comparison of the two entire pelves2 and ,i venture to think that the result is more natural in appearance1 and more likely to be correct than was ,marsh's conjectural restoration4 ,it would seem1 in fact1 that there is an obvious field for the employment of the method of co-ordinates in this task of reproducing missing portions of a structure to the proper scale and in harmony with related types4 ,to this subject we shall presently return4 ,illustration3 ,fig4 #che4 ,shoulder-girdle of ,cryptocleidus4 a1 young2 b1 adult4 ,illustration3 ,fig4 #chf4 ,shoulder-girdle of ,ichthyosaurus4 ,in ,fig4 #che1 a1 b1 ,i have drawn the shoulder-girdle of ,cryptocleidus1 a ,plesiosaurian reptile1 half-grown in the one case and full-grown in the other4 ,the change of form during growth in this region of the body is very considerable1 and its nature is well brought out by the two co-ordinate systems4 ,in ,fig4 #chf ,i have drawn the shoulder-girdle of an ,ichthyosaur1 referring it to ,cryptocleidus as a standard of comparison4 ,the interclavicle1 which is present in ,ichthyosaurus1 is minute and hidden in ,cryptocleidus2 but the numerous other differences between the two #gef forms1 chief among which is the great elongation in ,ichthyosaurus of the two clavicles1 are all seen by our diagrams to be part and parcel of one general and systematic deformation4 ,illustration3 ,fig4 #chg4 a1 ,skull of ,dimorphodon4 b1 ,skull of ,pteranodon4 ,before we leave the group of reptiles we may glance at the very strangely modified skull of ,pteranodon1 one of the extinct flying reptiles1 or ,pterosauria4 ,in this very curious skull the region of the jaws1 or beak1 is greatly elongated and pointed2 the occipital bone is drawn out into an enormous backwardly-directed crest2 the posterior part of the lower jaw is similarly produced backwards2 the orbit is small2 and the quadrate bone is strongly inclined downwards and forwards4 ,the whole skull has a configuration which stands1 apparently1 in the strongest possible contrast to that of a more normal ,ornithosaurian such as ,dimorphodon4 ,but if we inscribe the latter in ,cartesian coordinates 7,fig4 #chg1 a71 and refer our ,pteranodon to a system of oblique co-ordinates 7b71 in which the two co-ordinate systems of parallel lines become each a pencil of diverging rays1 we make manifest a correspondence which extends uniformly throughout all parts of these very different-looking skulls4 ,illustration3 ,fig4 #chh4 ,pelvis of ,archaeopteryx4 ,illustration3 ,fig4 #chi4 ,pelvis of ,apatornis4 ,we have dealt so far1 and for the most part we shall continue to deal1 with our co-ordinate method as a means of comparing one known structure with another4 ,but it is obvious1 as ,i have said1 #geg that it may also be employed for drawing hypothetical structures1 on the assumption that they have varied from a known form in some definite way4 ,and this process may be especially useful1 and will be most obviously legitimate1 when we apply it to the particular case of representing intermediate stages between two forms which are actually known to exist1 in other words1 of reconstructing the transitional stages through which the course of evolution must have successively travelled if it has brought about the change from some ancestral type to its presumed descendant4 ,some little time ago ,i sent to my friend1 ,mr ,gerhard ,heilmann of ,copenhagen1 a few of my own rough co-ordinate diagrams1 including some in which the pelves of certain ancient and primitive birds were compared one with another4 ,mr ,heilmann1 who is both a skilled draughtsman and an able morphologist1 returned me a set of diagrams which are a vast improvement on my own1 #geh and which are reproduced in ,figs4 #chh-cic4 ,here we have1 as extreme cases1 the pelvis of ,archaeopteryx1 the most ancient of known birds1 and that of ,apatornis1 one of the fossil 8toothed0 ,illustration3 ,fig4 #cij4 ,the co-ordinate systems of ,figs4 #chh and #chi1 with three intermediate systems interpolated4 ,illustration3 ,fig4 #cia4 ,the first intermediate co-ordinate network1 with its corresponding inscribed pelvis4 birds from the ,north ,american ,cretaceous formations--a bird shewing some resemblance to the modern terns4 ,the pelvis of ,archaeopteryx is taken as our type1 and referred accordingly to #gei ,cartesian co-ordinates 7,fig4 #chh72 while the corresponding coordinates of the very different pelvis of ,apatornis are represented in ,fig4 #chi4 ,in ,fig4 #cij the outlines of these two co-ordinate systems are superposed upon one another1 and those of three intermediate and equidistant co-ordinate systems are interpolated between them4 ,from each of these latter systems1 so determined by direct interpolation1 a complete co-ordinate diagram is drawn1 and the corresponding outline of a pelvis is found from each of these systems of co-ordinates1 as in ,figs4 #cia1 #cib4 ,finally1 in ,fig4 #cic the complete series is represented1 beginning with the known pelvis of ,archaeopteryx1 and leading up by our three intermediate hypothetical types to the known pelvis of ,apatornis4 ,illustration3 ,fig4 #cib4 ,the second and third intermediate co-ordinate networks1 with their corresponding inscribed pelves4 ,among mammalian skulls ,i will take two illustrations only1 one drawn from a comparison of the human skull with that of the higher apes1 and another from the group of ,perissodactyle #gfj ,ungulates1 the group which includes the rhinoceros1 the tapir1 and the horse4 ,illustration3 ,fig4 #cic4 ,the pelves of ,archaeopteryx and of ,apatornis1 with three transitional types interpolated between them4 ,let us begin by choosing as our type the skull of ,hyrachyus agrarius1 ,cope1 from the ,middle ,eocene of ,north ,america1 as figured by ,osborn in his ,monograph of the ,extinct ,rhinoceroses#fei 7,fig4 #cid74 ,illustration3 ,fig4 #cid4 ,skull of ,hyrachyus agrarius4 7,after ,osborn47 ,illustration3 ,fig4 #cie4 ,skull of ,aceratherium tridactylum4 7,after ,osborn47 ,the many other forms of primitive rhinoceros described in the monograph differ from ,hyrachyus in various details--in the characters of the teeth1 sometimes in the number of the toes1 and so forth2 and they also differ very considerably in the general #gfa appearance of the skull4 ,but these differences in the conformation of the skull1 conspicuous as they are at first sight1 will be found easy to bring under the conception of a simple and homogeneous transformation1 such as would result from the application of some not very complicated stress4 ,for instance1 the corresponding co-ordinates of ,aceratherium tridactylum1 as shown in ,fig4 #cie1 indicate that the essential difference between this skull and the former one may be summed up by saying that the long axis of the skull of ,aceratherium has undergone a slight double curvature1 while the upper parts of the skull have at the same time been #gfb subject to a vertical expansion1 or to growth in somewhat greater proportion than the lower parts4 ,precisely the same changes1 on a somewhat greater scale1 give us the skull of an existing rhinoceros4 ,illustration3 ,fig4 #cif4 ,occipital view of the skulls of various extinct rhinoceroses 7,aceratherium spp474 7,after ,osborn47 ,among the species of ,aceratherium1 the posterior1 or occipital1 view of the skull presents specific differences which are perhaps more conspicuous than those furnished by the side view2 and these differences are very strikingly brought out by the series of conformal transformations which ,i have represented in ,fig4 #cif4 ,in this case it will perhaps be noticed that the correspondence is not always quite accurate in small details4 ,it could easily have been made much more accurate by giving a slightly sinuous curvature to certain of the co-ordinates4 ,but as they stand1 the correspondence indicated is very close1 and the simplicity of the figures illustrates all the better the general character of the transformation4 ,by similar and not more violent changes we pass easily to such allied forms as the ,titanotheres 7,fig4 #cig72 and the well-known series of species of ,titanotherium1 by which ,professor ,osborn has #gfc illustrated the evolution of this genus1 constitutes a simple and suitable case for the application of our method4 ,but our method enables us to pass over greater gaps than these1 and to discern the general1 and to a very large extent even the detailed1 resemblances between the skull of the rhinoceros and those of the tapir or the horse4 ,from the ,cartesian co-ordinates in which we have begun by inscribing the skull of a primitive rhinoceros1 we pass to the tapir's skull 7,fig4 #cih71 firstly1 by converting the rectangular into a triangular network1 by which we represent the depression of the anterior and the progressively increasing elevation of the posterior part of the skull2 and secondly1 by giving to the vertical ordinates a curvature such as to bring about a certain longitudinal compression1 or condensation1 in the forepart of the skull1 especially in the nasal and orbital regions4 ,illustration3 ,fig4 #cig4 ,titanotherium robustum4 ,illustration3 ,fig4 #cih4 ,tapir's skull4 ,the conformation of the horse's skull departs from that of our primitive ,perissodactyle 7that is to say our early type of rhinoceros1 ,hyrachyus7 in a direction that is nearly the opposite of that taken by ,titanotherium and by the recent species of rhinoceros4 ,for we perceive1 by ,fig4 #cii1 that the horizontal co-ordinates1 which in these latter cases became transformed into curves with the concavity upwards1 are curved1 in the case of the horse1 in the opposite direction4 ,and the vertical ordinates1 which are also curved1 somewhat in the same fashion as in the tapir1 are very nearly equidistant1 instead of being1 as in that animal1 crowded together anteriorly4 ,ordinates and abscissae form an oblique #gfd system1 as is shown in the figure4 ,in this case ,i have attempted to produce the network beyond the region which is actually required to include the diagram of the horse's skull1 in order to show better the form of the general transformation1 with a part only of which we have actually to deal4 ,illustration3 ,fig4 #cii4 ,horse's skull4 ,illustration3 ,fig4 #djj4 ,rabbit's skull4 ,it is at first sight not a little surprising to find that we can pass1 by a cognate and even simpler transformation1 from our ,perissodactyle skulls to that of the rabbit2 but the fact that we can easily do so is a simple illustration of the undoubted affinity which exists between the ,rodentia1 especially the family of the ,leporidae1 and the more primitive ,ungulates4 ,for my part1 ,i would go further2 for ,i think there is strong reason to believe that the ,perissodactyles are more closely related to the ,leporidae than the former are to the other ,ungulates1 or than the ,leporidae are to the rest of the ,rodentia4 ,be that as it may1 it is obvious from ,fig4 #djj that the rabbit's skull conforms to a system of #gfe co-ordinates corresponding to the ,cartesian co-ordinates in which we have inscribed the skull of ,hyrachyus1 with the difference1 firstly1 that the horizontal ordinates of the latter are transformed into equidistant curved lines1 approximately arcs of circles1 with their concavity directed downwards2 and secondly1 that the vertical ordinates are transformed into a pencil of rays approximately orthogonal to the circular arcs4 ,in short1 the configuration of the rabbit's skull is derived from that of our primitive rhinoceros by the unexpectedly simple process of submitting the latter to a ,illustration3 ,fig4 #dja4 ,a1 outline diagram of the ,cartesian co-ordinates of the skull of ,hyracotherium or ,eohippus1 as shewn in ,fig4 #djb1 ,a4 ,h1 outline of the corresponding projection of the horse's skull4 ,b-,g1 intermediate1 or interpolated1 outlines4 strong and uniform flexure in the downward direction 7cf4 ,fig4 #ceh1 p4 #gca74 ,in the case of the rabbit the configuration of the individual bones does not conform quite so well to the general transformation as it does when we are comparing the several ,perissodactyles one with another2 and the chief departures from conformity will be found in the size of the orbit and in the outline of the immediately surrounding bones4 ,the simple fact is that the relatively enormous eye of the rabbit constitutes an independent variation1 which cannot be brought into the general and fundamental transformation1 but must be dealt with #gfh separately4 ,the enlargement of the eye1 like the modification in form and number of the teeth1 is a separate phenomenon1 which supplements but in no way contradicts our general comparison of the skulls taken in their entirety4 ,illustration3 two printed pages ,fig4 #djb4 ,a1 skull of ,hyracotherium1 from the ,eocene1 after ,w4 ,b4 ,scott2 ,h1 skull of horse1 represented as a co-ordinate transformation of that of ,hyracotherium1 and to the same scale of magnitude2 ,b-,g1 various artificial or imaginary types1 reconstructed as intermediate stages between ,a and ,h2 ,m1 skull of ,mesohippus1 from the ,oligocene1 after ,scott1 for comparison with ,c2 ,p1 skull of ,protohippus1 from the ,miocene1 after ,cope1 for comparison with ,e2 ,pp1 lower jaw of ,protohippus placidus 7after ,matthew and ,gidley71 for comparison with ,f2 ,mi1 ,miohippus 7after ,osborn71 ,pa1 ,parahippus 7after ,peterson71 shewing resemblance1 but less perfect agreement1 with ,c and ,d4 ,before we leave the ,perissodactyla and their allies1 let us look a little more closely into the case of the horse and its immediate relations or ancestors1 doing so with the help of a set of diagrams which ,i again owe to ,mr ,gerard ,heilmann#ffj4 ,here we start afresh1 with the skull 7,fig4 #djb1 ,a7 of ,hyracotherium 7or ,eohippus71 inscribed in a simple ,cartesian network4 ,at the other end of the series 7,h7 is a skull of ,equus1 in its own corresponding network2 and the intermediate stages 7,b-,g7 are all drawn by direct and simple interpolation1 as in ,mr ,heilmann's former series of drawings of ,archaeopteryx and ,apatornis4 ,in this present case1 the relative magnitudes are shewn1 as well as the forms1 of the several skulls4 ,alongside of these reconstructed diagrams1 are set figures of certain extinct 8horses0 7,equidae or ,palaeotheriidae71 and in two cases1 viz4 ,mesohippus and ,protohippus 7,m1 ,p71 it will be seen that the actual fossil skull coincides in the most perfect fashion with one of the hypothetical forms or stages which our method shews to be implicitly involved in the transition from ,hyracotherium to ,equus4 ,in a third case1 that of ,parahippus 7,pa71 the correspondence 7as ,mr ,heilmann points out7 is by no means exact4 ,the outline of this skull comes nearest to that of the hypothetical transition stage ,d1 but the 8fit0 is now a bad one2 for the skull of ,parahippus is evidently a longer1 straighter and narrower skull1 and differs in other minor characters besides4 ,in short1 though some writers have placed ,parahippus in the direct line of descent between ,equus and ,eohippus1 we see at once that there is no place for it there1 and that it must1 accordingly1 represent a somewhat divergent branch or offshoot of the ,equidae#ffa4 ,it may be noticed1 especially in the case of ,protohippus #gfi 7,p71 that the configuration of the angle of the jaw does not tally quite so accurately with that of our hypothetical diagrams as do other parts of the skull4 ,as a matter of fact1 this region is somewhat variable1 in different species of a genus1 and even in different individuals of the same species2 in the small figure 7,pp7 of ,protohippus placidus the correspondence is more exact4 ,illustration3 ,fig4 #djc4 ,human scapulae 7after ,dwight74 ,a1 ,caucasian2 ,b1 ,negro2 ,c1 ,north ,american ,indian 7from ,kentucky ,mountains74 ,in considering this series of figures we cannot but be struck1 not only with the regularity of the succession of 8transformations10 but also with the slight and inconsiderable differences which separate the known and recorded stages1 and even the two extremes of the whole series4 ,these differences are no greater 7save in regard to actual magnitude7 than those between one human skull and another1 at least if we take into account the older or remoter races2 and they are again no greater1 but if anything less1 than the range of variation1 racial and individual1 in certain other human bones1 for instance the scapula#ffb4 ,the variability of this latter bone is great1 but it is neither #ggj surprising nor peculiar2 for it is linked with all the considerations of mechanical efficiency and functional modification which we dealt with in our last chapter4 ,the scapula occupies1 as it were1 a focus in a very important field of force2 and the lines of force converging on it will be very greatly modified by the varying development of the muscles over a large area of the body and of the uses to which they are habitually put4 ,illustration3 ,fig4 #djd4 ,human skull4 ,illustration3 ,fig4 #dje4 ,co-ordinates of chimpanzee's skull1 as a projection of the ,cartesian co-ordinates of ,fig4 #djd4 ,let us now inscribe in our ,cartesian co-ordinates the outline of a human skull 7,fig4 #djd71 for the purpose of comparing it with the skulls of some of the higher apes4 ,we know beforehand that the main differences between the human and the simian types depend upon the enlargement or expansion of the brain and braincase in man1 and the relative diminution or enfeeblement of his jaws4 ,together with these changes1 the 8facial angle0 increases from an oblique angle to nearly a right angle in man1 #gga and the configuration of every constituent bone of the face and skull undergoes an alteration4 ,we do not know to begin with1 and we are not shewn by the ordinary methods of comparison1 how far these various changes form part of one harmonious and congruent transformation1 or whether we are to look1 for instance1 upon the changes undergone by the frontal1 the occipital1 the maxillary1 and the mandibular regions as a congeries of separate modifications or independent variants4 ,but as soon as we have marked out a number of points in the gorilla's or chimpanzee's skull1 corresponding with those which our co-ordinate network intersected in the human skull1 we find that these corresponding points may be at once linked up by smoothly curved lines of intersection1 which form a new system of co-ordinates and constitute a simple 8projection0 of our human skull4 ,the network represented in ,fig4 #dje constitutes such a projection of the human skull on what we may call1 figuratively speaking1 the 8plane0 of the chimpanzee2 and the full diagram in ,fig4 #djf demonstrates the correspondence4 ,in ,fig4 #djg ,i have shewn the similar deformation in the case of a baboon1 and it is obvious that the transformation is of precisely the same order1 and differs only in an increased intensity or degree of deformation4 ,illustration3 ,fig4 #djf4 ,skull of chimpanzee4 ,illustration3 ,fig4 #djg4 ,skull of baboon4 ,in both dimensions1 as we pass from above downwards and from behind forwards1 the corresponding areas of the network are seen to increase in a gradual and approximately logarithmic order in the lower as compared with the higher type of skull2 and1 in short1 it becomes at once manifest that the modifications of jaws1 braincase1 and the regions between are all portions of one continuous and integral process4 ,it is of course easy to draw the #ggb inverse diagrams1 by which the ,cartesian co-ordinates of the ape are transformed into curvilinear and non-equidistant co-ordinates in man4 ,from this comparison of the gorilla's or chimpanzee's with the human skull we realise that an inherent weakness underlies the anthropologist's method of comparing skulls by reference to a small number of axes4 ,the most important of these are the 8facial0 and 8basicranial0 axes1 which include between them the 8facial angle40 ,but it is1 in the first place1 evident that these axes are merely the principal axes of a system of co-ordinates1 and that their restricted and isolated use neglects all that can be learned from the filling in of the rest of the co-ordinate network4 ,and1 in the second place1 the 8facial axis10 for instance1 as ordinarily used in the anthropological comparison of one human skull with another1 or of the human skull with the gorilla's1 is in all cases treated as a straight line2 but our investigation has shewn that rectilinear axes only meet the case in the simplest and most closely related transformations2 and that1 for instance1 in the anthropoid skull no rectilinear axis is homologous with a rectilinear axis in a man's skull1 but what is a straight line in the one has become a certain definite curve in the other4 ,mr ,heilmann tells me that he has tried1 but without success1 to obtain a transitional series between the human skull and some prehuman1 anthropoid type1 which series 7as in the case of the ,equidae7 should be found to contain other known types in direct linear sequence4 ,it appears impossible1 however1 to obtain such a series1 or to pass by successive and continuous gradations through such forms as ,mesopithecus1 ,pithecanthropus1 ,homo neanderthalensis1 and the lower or higher races of modern man4 ,the failure is not the fault of our method4 ,it merely indicates that no one straight line of descent1 or of consecutive transformation1 exists2 but on the contrary1 that among human and anthropoid types1 recent and extinct1 we have to do with a complex problem of divergent1 rather than of continuous1 variation4 ,and in like manner1 easy as it is to correlate the baboon's and chimpanzee's skulls severally with that of man1 and easy as it is to see that the chimpanzee's skull is much nearer to the human type than is the baboon's1 it is also not difficult to perceive that the series is not1 #ggc strictly speaking1 continuous1 and that neither of our two apes lies precisely on the same direct line or sequence of deformation by which we may hypothetically connect the other with man4 ,illustration3 ,fig4 #djh4 ,skull of dog1 compared with the human skull of ,fig4 #djd4 ,as a final illustration ,i have drawn the outline of a dog's skull 7,fig4 #djh71 and inscribed it in a network comparable with the ,cartesian network of the human skull in ,fig4 #djd4 ,here we attempt to bridge over a wider gulf than we have crossed in any of our former comparisons4 ,but1 nevertheless1 it is obvious that our method still holds good1 in spite of the fact that there are various specific differences1 such as the open or closed orbit1 etc41 which have to be separately described and accounted for4 ,we see that the chief essential differences in plan between the dog's skull and the man's lie in the fact that1 relatively speaking1 the former tapers away in front1 a triangular taking the place of a rectangular conformation2 secondly1 that1 coincident with the tapering off1 there is a progressive elongation1 or pulling out1 of the whole forepart of the skull2 and lastly1 as a minor difference1 that the straight vertical ordinates of the human skull become curved1 with their convexity directed forwards1 in the dog4 ,while the net result is that in the dog1 just as in the chimpanzee1 the brain-pan is smaller and the jaws are larger than in man1 it is now conspicuously evident that the co-ordinate network of the ape is by no means intermediate between those which fit the other two4 ,the mode of deformation is on different lines2 and1 while it may be correct to say that the chimpanzee and the baboon are more brute-like1 it would be by no means accurate to assert that they are more dog-like1 than man4 #ggd ,in this brief account of co-ordinate transformations and of their morphological utility ,i have dealt with plane co-ordinates only1 and have made no mention of the less elementary subject of co-ordinates in three-dimensional space4 ,in theory there is no difficulty whatsoever in such an extension of our method2 it is just as easy to refer the form of our fish or of our skull to the rectangular co-ordinates x1 y1 z1 or to the polar co-ordinates 1 1 1 as it is to refer their plane projections to the two axes to which our investigation has been confined4 ,and that it would be advantageous to do so goes without saying2 for it is the shape of the solid object1 not that of the mere drawing of the object1 that we want to understand2 and already we have found some of our easy problems in solid geometry leading us 7as in the case of the form of the bivalve and even of the univalve shell7 quickly in the direction of co-ordinate analysis and the theory of conformal transformations4 ,but this extended theme ,i have not attempted to pursue1 and it must be left to other times1 and to other hands4 ,nevertheless1 let us glance for a moment at the sort of simple cases1 the simplest possible cases1 with which such an investigation might begin2 and we have found our plane co-ordinate systems so easily and effectively applicable to certain fishes that we may seek among them for our first and tentative introduction to the three-dimensional field4 ,it is obvious enough that the same method of description and analysis which we have applied to one plane1 we may apply to another3 drawing by observation1 and by a process of trial and error1 our various cross-sections and the co-ordinate systems which seem best to correspond4 ,but the new and important problem which now emerges is to correlate the deformation or transformation which we discover in one plane with that which we have observed in another3 and at length1 perhaps1 after grasping the general principles of such correlation1 to forecast approximately what is likely to take place in the other two planes of reference when we are acquainted with one1 that is to say1 to determine the values along one axis in terms of the other two4 ,let us imagine a common 8round0 fish1 and a common 8flat0 fish1 such as a haddock and a plaice4 ,these two fishes are not as nicely adapted for comparison by means of plane co-ordinates as #gge some which we have studied1 owing to the presence of essentially unimportant1 but yet conspicuous differences in the position of the eyes1 or in the number of the fins1--that is to say in the manner in which the continuous dorsal fin of the plaice appears in the haddock to be cut or scolloped into a number of separate fins4 ,but speaking broadly1 and apart from such minor differences as these1 it is manifest that the chief factor in the case 7so far as we at present see7 is simply the broadening out of the plaice's body1 as compared with the haddock's1 in the dorso-ventral direction1 that is to say1 along the y axis2 in other words1 the ratio x_/y is much less1 7and indeed little more than half as great71 in the haddock than in the plaice4 ,but we also recognise at once that while the plaice 7as compared with the haddock7 is expanded in one direction1 it is also flattened1 or thinned out1 in the other3 y increases1 but z diminishes1 relatively to x4 ,and furthermore1 we soon see that this is a common or even a general phenomenon4 ,the high1 expanded body in our ,antigonia or in our sun-fish is at the same time flattened or compressed from side to side1 in comparison with the related fishes which we have chosen as standards of reference or comparison2 and conversely1 such a fish as the skate1 while it is expanded from side to side in comparison with a shark or dogfish1 is at the same time flattened or depressed in its vertical section4 ,we proceed then1 to enquire whether there be any simple relation of magnitude discernible between these twin factors of expansion and compression2 and the very fact that the two dimensions tend to vary inversely already assures us that1 in the general process of deformation1 the volume is less affected than are the linear dimensions4 ,some years ago1 when ,i was studying the length-weight co-efficient in fishes 7of which we have already spoken in ,chap4 ,,iii1 p4 #ih71 that is to say the coefficient k in the formula ,w k ,l#c1 or k ,w_/,l#c1 ,i was not a little surprised to find that k was all but identical in two such different looking fishes as our haddock and our plaice3 thus indicating that these two fishes1 little as they resemble one another externally 7though they belong to two closely related families71 have approximately the same volume when they are equal in length2 or1 in other words1 that the extent to which the plaice's body has become expanded or broadened is just about #ggf compensated for by the extent to which it has also got flattened or thinned4 ,in short1 if we could permit ourselves to conceive of a haddock being directly transformed into a plaice1 a very large part of the change would be simply accounted for by supposing the former fish to be 8rolled out10 as a baker rolls a piece of dough4 ,this is1 as it were1 an extreme case of the balancement des organes1 or 8compensation of parts40 ,simple ,cartesian co-ordinates will not suffice very well to compare the haddock with the plaice1 for the deformation undergone by the former in comparison with the latter is more on the lines of that by which we have compared our ,antigonia with our ,polyprion2 that is to say1 the expansion is greater towards the middle of the fish's length1 and dwindles away towards either end4 ,but again simplifying our illustration to the utmost1 and being content with a rough comparison1 we may assert that1 when haddock and plaice are brought to the same standard of length1 we can inscribe them both 7approximately7 in rectangular co-ordinate networks1 such that ,y in the plaice is about twice as great as y in the haddock4 ,but if the volumes of the two fishes be equal1 this is as much as to say that xyz in the one case 7or rather the summation of all these values7 is equal to ,,xyz in the other2 and therefore 7since ,x x1 and ,y #by71 it follows that ,z z_/#b4 ,when we have drawn our vertical transverse section of the haddock 7or projected that fish in the yz plane71 we have reason accordingly to anticipate that we can draw a similar projection 7or section7 of the plaice by simply doubling the y's and halving the z's3 and1 very approximately1 this turns out to be the case4 ,the plaice is 7in round numbers7 just about twice as broad and also just about half as thick as the haddock2 and therefore the ratio of breadth to thickness 7or y to z7 is just about four times as great in the one case as in the other4 ,it is true that this simple1 or simplified1 illustration carries us but a very little way1 and only half prepares us for much greater complications4 ,for instance1 we have no right or reason to presume that the equality of weights1 or volumes1 is a common1 much less a general rule4 ,and again1 in all cases of more complex deformation1 such as that by which we have compared ,diodon with the sunfish1 we must be prepared for very much more #ggg recondite methods of comparison and analysis1 leading doubtless to very much more complicated results4 ,in this last case1 of ,diodon and the sunfish1 we have seen that the vertical expansion of the latter as compared with the former fish1 increases rapidly as we go backwards towards the tail2 but we can by no means say that the lateral compression increases in like proportion4 ,if anything1 it would seem that the said expansion and compression tend to vary inversely2 for the ,diodon is very thick in front and greatly thinned away behind1 while the flattened sunfish is more nearly of the same thickness all the way along4 ,interesting as the whole subject is we must meanwhile leave it alone2 recognising1 however1 that if the difficulties of description and representation could be overcome1 it is by means of such co-ordinates in space that we should at last obtain an adequate and satisfying picture of the processes of deformation and of the directions of growth#ffc4 #ggh ,,epilogue4 ,in the beginning of this book ,i said that its scope and treatment were of so prefatory a kind that of other preface it had no need2 and now1 for the same reason1 with no formal and elaborate conclusion do ,i bring it to a close4 ,the fact that ,i set little store by certain postulates 7often deemed to be fundamental7 of our present-day biology the reader will have discovered and ,i have not endeavoured to conceal4 ,but it is not for the sake of polemical argument that ,i have written1 and the doctrines which ,i do not subscribe to ,i have only spoken of by the way4 ,my task is finished if ,i have been able to shew that a certain mathematical aspect of morphology1 to which as yet the morphologist gives little heed1 is interwoven with his problems1 complementary to his descriptive task1 and helpful1 nay essential1 to his proper study and comprehension of ,form4 ,hic artem remumque repono4 ,and while ,i have sought to shew the naturalist how a few mathematical concepts and dynamical principles may help and guide him1 ,i have tried to shew the mathematician a field for his labour1--a field which few have entered and no man has explored4 ,here may be found homely problems1 such as often tax the highest skill of the mathematician1 and reward his ingenuity all the more for their trivial associations and outward semblance of simplicity4 ,that ,i am no skilled mathematician ,i have had little need to confess1 but something of the use and beauty of mathematics ,i think ,i am able to understand4 ,i know that in the study of material things1 number1 order and position are the threefold clue to exact knowledge2 that these three1 in the mathematician's hands1 furnish the 8first outlines for a sketch of the ,universe02 that by square and circle we are helped1 like ,emile ,verhaeren's carpenter1 to conceive 8,les lois indubitables et fcondes ,qui sont la rgle et la clart du monde40 ,for the harmony of the world is made manifest in ,form and ,number1 and the heart and soul and all the poetry of ,natural #ggi ,philosophy are embodied in the concept of mathematical beauty4 ,a greater than ,verhaeren had this in mind when he told of 8the golden compasses1 prepared ,in ,god's eternal store40 ,a greater than ,milton had magnified the theme and glorified ,him 8who sitteth upon the circle of the earth10 saying3 ,he measureth the waters in the hollow of his hand1 he meteth out the heavens with his span1 he comprehendeth the dust of the earth in a measure4 ,moreover the perfection of mathematical beauty is such 7as ,maclaurin learned of the bee71 that whatsoever is most beautiful and regular is also found to be most useful and excellent4 ,the living and the dead1 things animate and inanimate1 we dwellers in the world and this world wherein we dwell1-- 1--are bound alike by physical and mathematical law4 8,conterminous with space and coeval with time is the kingdom of ,mathematics2 within this range her dominion is supreme2 otherwise than according to her order nothing can exist1 and nothing takes place in contradiction to her laws40 ,so said1 some forty years ago1 a certain mathematician2 and ,philolaus the ,pythagorean had said much the same4 ,but with no less love and insight has the science of ,form and ,number been appraised in our own day and generation by a very great ,naturalist indeed3--by that old man eloquent1 that wise student and pupil of the ant and the bee1 who died but yesterday1 and who in his all but saecular life tasted of the firstfruits of immortality2 who curiously conjoined the wisdom of antiquity with the learning of to-day2 whose ,provenal verse seems set to ,dorian music2 in whose plainest words is a sound as of bees' industrious murmur2 and who1 being of the same blood and marrow with ,plato and ,pythagoras1 saw in ,number 8la clef de la vote10 and found in it 8le comment et le pourquoi des choses40 ,,notes3 #a ,these sayings of ,kant and of ,du ,bois1 and others like to them1 have been the text of many discourses3 see1 for instance1 ,stallo's ,concepts1 p4 #ba1 #ahhb2 ,hber1 ,biol4 ,centralbl4 ,,xix1 p4 #bhd1 #ahij1 etc4 ,cf4 also ,jellett1 ,rep4 ,brit4 ,ass4 #ahgd1 p4 #a4 #b 8,quum enim mundi universi fabrica sit perfectissima1 atque a ,creatore sapientissimo absoluta1 nihil omnino in mundo contingit in quo non maximi minimive ratio quaepiam eluceat2 quamobrem dubium prorsus est nullum quin omnes mundi effectus ex causis finalibus1 ope methodi maximorum et minimorum1 aeque feliciter determinari queant atque ex ipsis causis efficientibus40 ,methodus inveniendi1 etc4 #agdd 7cit4 ,mach1 ,science of ,mechanics1 #aijb1 p4 #dee74 #c ,cf4 ,opp4 7ed4 ,erdmann71 p4 #ajf1 8,bien loin d'exclure les causes finales4441 c'est de l qu'il faut tout dduire en ,physique40 #d ,cf4 p4 #afb4 8,la force vitale dirige des phnomnes qu'elle ne produit pas3 les agents physiques produisent des phnomnes qu'ils ne dirigent pas40 #e ,it is now and then conceded with reluctance4 ,thus ,enriques1 a learned and philosophic naturalist1 writing 8della economia di sostanza nelle osse cave0 7,arch4 f4 ,entw4 ,mech4 ,,xx1 #aijf71 says 8una certa impronta di teleologismo qu e l rimasta1 mio malgrado1 in questo scritto40 #f ,cf4 ,cleland1 ,on ,terminal ,forms of ,life1 ,j4 ,anat4 and ,phys4 ,,xviii1 #ahhd4 #g ,conklin1 ,embryology of ,crepidula1 ,journ4 of ,morphol4 ,,xiii1 p4 #bjc1 #ahig2 ,lillie1 ,f4 ,r41 ,adaptation in ,cleavage1 ,woods ,holl ,biol4 ,lectures1 pp4 #dc-fg1 #ahii4 #h ,i am inclined to trace back ,driesch's teaching of ,entelechy to no less a person than ,melanchthon4 ,when ,bacon 7de ,augm4 ,,iv1 #c7 states with disapproval that the soul 8has been regarded rather as a function than as a substance10 ,r4 ,l4 ,ellis points out that he is referring to ,melanchthon's exposition of the ,aristotelian doctrine4 ,for ,melanchthon1 whose view of the peripatetic philosophy had long great influence in the ,protestant ,universities1 affirmed that1 according to the true view of ,aristotle's opinion1 the soul is not a substance1 but an 1 or function4 ,he defined it as quaedam ciens actiones--a description all but identical with that of ,claude ,bernard's 8force vitale40 #i ,ray ,lankester1 ,encycl4 ,brit4 7#ith ed471 art4 8,zoology10 p4 #hjf1 #ahhh4 #aj ,alfred ,russel ,wallace1 especially in his later years1 relied upon a direct but somewhat crude teleology4 ,cf4 his ,world of ,life1 a ,manifestation of ,creative ,power1 ,directive ,mind and ,ultimate ,purpose1 #aiaj4 #aa ,janet1 ,les ,causes ,finales1 #ahgf1 p4 #cej4 #ab ,the phrase is ,leibniz's1 in his ,thodice4 #ac ,cf4 7int4 al47 ,bosanquet1 ,the ,meaning of ,teleology1 ,proc4 ,brit4 ,acad4 #aije-f1 pp4 #bce-bde4 ,cf4 also ,leibniz 7,discours de ,mtaphysique2 ,lettres indites1 ed4 de ,careil1 #aheg1 p4 #ced2 cit4 ,janet1 p4 #fdc71 8,l'un et l'autre est bon1 l'un et l'autre peut tre utile 444 et les auteurs qui suivent ces routes diffrentes ne devraient point se maltraiter3 et seq40 #ad ,the reader will understand that ,i speak1 not of the 8severe and diligent inquiry0 of variation or of 8fortuity10 but merely of the easy assumption that these phenomena are a sufficient basis on which to rest1 with the all-powerful help of natural selection1 a theory of definite and progressive evolution4 #ae ,revue ,philosophique4 ,,xxxiii1 #ahib4 #af ,this general principle was clearly grasped by ,dr ,george ,rainey 7a learned physician of ,st ,bartholomew's7 many years ago1 and expressed in such words as the following3 8444444it is illogical to suppose that in the case of vital organisms a distinct force exists to produce results perfectly within the reach of physical agencies1 especially as in many instances no end could be attained were that the case1 but that of opposing one force by another capable of effecting exactly the same purpose40 7,on ,artificial ,calculi1 ,q4,j4,m4,s4 7,trans4 ,microsc4 ,soc471 ,,vi1 p4 #di1 #aheh47 ,cf4 also ,helmholtz1 infra cit41 p4 #i4 #ag ,whereby he incurred the reproach of ,socrates1 in the ,phaedo4 #ah ,in a famous lecture 7,conservation of ,forces applied to ,organic ,nature1 ,proc4 ,roy4 ,instit41 ,april #ab1 #ahfa71 ,helmholtz laid it down1 as 8the fundamental principle of physiology10 that 8,there may be other agents acting in the living body than those agents which act in the inorganic world2 but those forces1 as far as they cause chemical and mechanical influence in the body1 must be quite of the same character as inorganic forces3 in this at least1 that their effects must be ruled by necessity1 and must always be the same when acting in the same conditions2 and so there cannot exist any arbitrary choice in the direction of their actions40 ,it would follow from this1 that1 like the other 8physical0 forces1 they must be subject to mathematical analysis and deduction4 ,cf4 also ,dr ,t4 ,young's ,croonian ,lecture ,on the ,heart and ,arteries1 ,phil4 ,trans4 #ahji1 p4 #a2 ,coll4 ,works1 ,i1 #eaa4 #ai ,ektropismus1 oder die physikalische ,theorie des ,lebens1 ,leipzig1 #aiaj4 #bj ,wilde ,lecture1 ,nature1 ,march #ab1 #aijh2 ibid4 ,sept4 #f1 #aijj1 p4 #dhe2 ,aether and ,matter1 p4 #bhh4 ,cf4 also ,lord ,kelvin1 ,fortnightly ,review1 #ahib1 p4 #cac4 #ba ,joly1 ,the ,abundance of ,life1 ,proc4 ,roy4 ,dublin ,soc4 ,,vii1 #ahij2 and in ,scientific ,essays1 etc4 #aiae1 p4 #fj et seq4 #bb ,papillon1 ,histoire de la philosophie moderne1 ,i1 p4 #cjj4 #bc ,with the special and important properties of colloidal matter we are1 for the time being1 not concerned4 #bd ,cf4 ,hans ,przibram1 ,anwendung elementarer ,mathematik auf ,biologische ,probleme 7in ,roux's ,vortrge1 ,heft ,,iii71 ,leipzig1 #aijh1 p4 #aj4 #be ,the subject is treated from an engineering point of view by ,prof4 ,james ,thomson1 ,comparisons of ,similar ,structures as to ,elasticity1 ,strength1 and ,stability1 ,trans4 ,inst4 ,engineers1 ,scotland1 #ahgf 7,collected ,papers1 #aiab1 pp4 #cfa-cgb71 and by ,prof4 ,a4 ,barr1 ibid4 #ahii2 see also ,rayleigh1 ,nature1 ,april #bb1 #aiae4 #bf ,cf4 ,spencer1 ,the ,form of the ,earth1 etc41 ,phil4 ,mag4 ,,xxx1 pp4 #aid-f1 #ahdg2 also ,principles of ,biology1 pt4 ,,ii1 ch4 ,i1 #ahfd 7p4 #abc1 etc474 #bg ,george ,louis ,lesage 7#agbd-ahjc71 well known as the author of one of the few attempts to explain gravitation4 7,cf4 ,leray1 ,constitution de la ,matire1 #ahfi2 ,kelvin1 ,proc4 ,r4 ,s4 ,e4 ,,vii1 p4 #egg1 #ahgb1 etc42 ,clerk ,maxwell1 ,phil4 ,trans4 vol4 #aeg1 p4 #ej1 #ahfg2 art4 8,atom10 ,encycl4 ,brit4 #ahge1 p4 #df47 #bh ,cf4 ,pierre ,prvost1 ,notices de la vie et des crits de ,lesage1 #ahje2 quoted by ,janet1 ,causes ,finales1 app4 ,,iii4 #bi ,discorsi e ,dimostrazioni matematiche1 intorno due nuove scienze1 attenenti alla ,mecanica1 ed ai ,movimenti ,locali3 appresso gli ,elzevirii1 ,,mdcxxxviii4 ,opere1 ed4 ,favaro1 ,,viii1 p4 #afi seq4 ,transl4 by ,henry ,crew and ,a4 de ,salvio1 #aiad1 p4 #acj1 etc4 ,see ,nature1 ,june #ag1 #aiae4 #cj ,so ,werner remarked that ,michael ,angelo and ,bramanti could not have built of gypsum at ,paris on the scale they built of travertin in ,rome4 #ca ,sir ,g4 ,greenhill1 ,determination of the greatest height to which a ,tree of given proportions can grow1 ,cambr4 ,phil4 ,soc4 ,pr4 ,,iv1 p4 #fe1 #ahha1 and ,chree1 ibid4 ,,vii1 #ahib4 ,cf4 ,poynting and ,thomson's ,properties of ,matter1 #aijg1 p #ii4 #cb ,in like manner the wheat-straw bends over under the weight of the loaded ear1 and the tip of the cat's tail bends over when held upright1--not because they 8possess flexibility10 but because they outstrip the dimensions within which stable equilibrium is possible in a vertical position4 ,the kitten's tail1 on the other hand1 stands up spiky and straight4 #cc ,modern ,painters4 #cd ,the stem of the giant bamboo may attain a height of #fj metres1 while not more than about #dj cm4 in diameter near its base1 which dimensions are not very far short of the theoretical limits 7,a4 ,j4 ,ewart1 ,phil4 ,trans4 vol4 #aih1 p4 #ga1 #aijf74 #ce ,trans4 ,zool4 ,soc4 ,,iv1 #ahej1 p4 #bg4 #cf ,it would seem to be a common if not a general rule that marine organisms1 zoophytes1 molluscs1 etc41 tend to be larger than the corresponding and closely related forms living in fresh water4 ,while the phenomenon may have various causes1 it has been attributed 7among others7 to the simple fact that the forces of growth are less antagonised by gravity in the denser medium 7cf4 ,houssay1 ,la ,forme et la ,vie1 #aijj1 p4 #hae74 ,the effect of gravity on outward form is illustrated1 for instance1 by the contrast between the uniformly upward branching of a sea-weed and the drooping curves of a shrub or tree4 #cg ,the analogy is not a very strict one4 ,we are not taking account1 for instance1 of a proportionate increase in thickness of the boiler-plates4 #ch ,let ,l be the length1 ,s the 7wetted7 surface1 ,t the tonnage1 ,d the displacement 7or volume7 of a ship2 and let it cross the ,atlantic at a speed ,v4 ,then1 in comparing two ships1 similarly constructed but of different magnitudes1 we know that ,l ,v#b1 ,s ,l#b ,v#d1 ,d ,t ,l#c ,v#f2 also ,r 7resistance7 ,s ,v#b ,v#f2 ,h 7horse-power7 ,r ,v ,v#g2 and the coal 7,c7 necessary for the voyage ,h_/,v ,v#f4 ,that is to say1 in ordinary engineering language1 to increase the speed across the ,atlantic by #a per cent4 the ship's length must be increased #b per cent41 her tonnage or displacement #f per cent41 her coal-consumpt also #f per cent41 her horse-power1 and therefore her boiler-capacity1 #g per cent4 ,her bunkers1 accordingly1 keep pace with the enlargement of the ship1 but her boilers tend to increase out of proportion to the space available4 #ci ,this is the result arrived at by ,helmholtz1 ,ueber ein ,theorem geometrisch hnliche ,bewegungen flssiger ,krper betreffend1 nebst ,anwendung auf das ,problem ,luftballons zu lenken1 ,monatsber4 ,akad4 ,berlin1 #ahgc1 pp4 #eja-ad4 ,it was criticised and challenged 7somewhat rashly7 by ,k4 ,mllenhof1 ,die ,grsse der ,flugflchen1 etc41 ,pflger's ,archiv1 ,,xxxv1 p4 #djg1 ,,xxxvi1 p4 #edh1 #ahhe4 #dj ,cf4 also ,chabrier1 ,vol des ,insectes1 ,mm4 ,mus4 ,hist4 ,nat4 ,paris1 ,,vi-,,viii1 #ahbj-bb4 #da ,aerial ,flight1 vol4 ,,ii 7,aerodonetics71 #aijh1 p4 #aej4 #db ,by ,lanchester1 op4 cit4 p4 #aca4 #dc ,cf4 ,l'empire de l'air2 ornithologie applique l'aviation4 #ahha4 #dd ,de ,motu ,animalium1 ,i1 prop4 cciv1 ed4 #afhe1 p4 #bdc4 #de ,harl1 ,on ,atmospheric ,pressure in past ,geological ,ages1 ,bull4 ,geol4 ,soc4 ,fr4 ,,xi1 pp4 #aah-aba2 or ,cosmos1 p4 #cj1 ,july #h1 #aiaa4 #df ,introduction to ,entomology1 #ahbf1 ,,ii1 p4 #aij4 ,k4 and ,s41 like many less learned authors1 are fond of popular illustrations of the 8wonders of ,nature10 to the neglect of dynamical principles4 ,they suggest1 for instance1 that if the white ant were as big as a man1 its tunnels would be 8magnificent cylinders of more than three hundred feet in diameter02 and that if a certain noisy ,brazilian insect were as big as a man1 its voice would be heard all the world over3 8so that ,stentor becomes a mute when compared with these insects60 ,it is an easy consequence of anthropomorphism1 and hence a common characteristic of fairy-tales1 to neglect the principle of dynamical1 while dwelling on the aspect of geometrical1 similarity4 #dg ,i4e4 the available energy of muscle1 in ft4-lbs4 per lb4 of muscle1 is the same for all animals3 a postulate which requires considerable qualification when we are comparing very different kinds of muscle1 such as the insect's and the mammal's4 #dh ,prop4 clxxvii4 ,animalia minora et minus ponderosa majores saltus efficiunt respectu sui corporis1 si caetera fuerint paria4 #di ,see also 7int4 al471 ,john ,bernoulli1 de ,motu ,musculorum1 ,basil41 #afid2 ,chabry1 ,mcanisme du ,saut1 ,j4 de l',anat4 et de la ,physiol4 ,,xix1 #ahhc2 ,sur la longueur des membres des animaux sauteurs1 ibid4 ,,xxi1 p4 #cef1 #ahhe2 ,le ,hello1 ,de l'action des organes locomoteurs1 etc41 ibid4 ,,xxix1 p4 #fe-ic1 #ahic1 etc4 #ej ,recherches sur la force absolue des muscles des ,invertbrs1 ,bull4 ,acad4 ,e4 de ,belgique 7#c71 ,,vi1 ,,vii1 #ahhc-hd2 see also ibid4 7#b71 ,,xx1 #ahfe1 ,,xxii1 #ahff2 ,ann4 ,mag4 ,n4 ,h4 ,,xvii1 p4 #aci1 #ahff1 ,,xix1 p4 #ie1 #ahfg4 ,the subject was also well treated by ,straus-,drckheim1 in his ,considrations gnrales sur l'anatomie compare des animaux articuls1 #ahbh4 #ea ,the fact that the limb tends to swing in pendulum-time was first observed by the brothers ,weber 7,mechanik der menschl4 ,gehwerkzeuge1 ,gttingen1 #ahcf74 ,some later writers have criticised the statement 7e4g4 ,fischer1 ,die ,kinematik des ,beinschwingens etc41 ,abh4 math4 phys4 ,kl4 k4 ,schs4 ,ges4 ,,xxv-,,xxviii1 #ahii-aijc71 but for all that1 with proper qualifications1 it remains substantially true4 #eb ,quoted in ,mr ,john ,bishop's interesting article in ,todd's ,cyclopaedia1 ,,iii1 p4 #ddc4 #ec ,there is probably also another factor involved here3 for in bending1 and therefore shortening1 the leg we bring its centre of gravity nearer to the pivot1 that is to say1 to the joint1 and so the muscle tends to move it the more quickly4 #ed ,proc4 ,psychical ,soc4 ,,xii1 pp4 #cch-cee1 #ahig4 #ee ,for various calculations of the increase of surface due to histological and anatomical subdivision1 see ,e4 ,babak1 ,ueber die ,oberflchenentwickelung bei ,organismen1 ,biol4 ,centralbl4 ,,xxx1 pp4 #bbe-bci1 #beg-bfg1 #aiaj4 ,in connection with the physical theory of surface-energy1 ,wolfgang ,ostwald has introduced the conception of specific surface1 that is to say the ratio of surface to volume1 or ,s_/,v4 ,in a cube1 ,v l#c1 and ,s #fl#b2 therefore ,s_/,v #f_/l4 ,therefore if the side l measure #f cm41 the ratio ,s_/,v #a1 and such a cube may be taken as our standard1 or unit of specific surface4 ,a human blood-corpuscle has1 accordingly1 a specific surface of somewhere about #ad1jjj or #ae1jjj4 ,it is found in physical chemistry that surface energy becomes an important factor when the specific surface reaches a value of #aj1jjj or thereby4 #ef ,though the entire egg is not increasing in mass1 this is not to say that its living protoplasm is not increasing all the while at the expense of the reserve material4 #eg ,cf4 ,tait1 ,proc4 ,r4,s4,e4 ,v1 #ahff1 and ,,vi1 #ahfh4 #eh ,physiolog4 ,notizen 7#i71 p4 #dbe1 #ahie4 ,cf4 ,strasbrger1 ,ueber die ,wirkungssphre der ,kerne und die ,zellgrsse1 ,histolog4 ,beitr4 7#e71 pp4 #ie-abi1 #ahic2 ,j4 ,j4 ,gerassimow1 ,ueber die ,grsse des ,zellkernes1 ,beih4 ,bot4 ,centralbl4 ,,xviii1 #aije2 also ,g4 ,levi and ,t4 ,terni1 ,le variazioni dell' indice plasmatico-nucleare durante l'intercinesi1 ,arch4 ,ital4 di ,anat4 ,x1 p4 #ede1 #aiaa4 #ei ,arch4 f4 ,entw4 ,mech4 ,,iv1 #ahih1 pp4 #ge1 #bdg4 #fj ,conklin1 ,e4 ,g41 ,cell-size and nuclear-size1 ,j4 ,exp4 ,zool4 ,,xii4 pp4 #a-ih1 #aiab4 #fa ,thus the fibres of the crystalline lens are of the same size in large and small dogs2 ,rabl1 ,z4 f4 w4 ,z4 ,,lxvii1 #ahii4 ,cf4 7int4 al47 ,pearson1 ,on the ,size of the ,blood-corpuscles in ,rana1 ,biometrika1 ,,vi1 p4 #djc1 #aiji4 ,dr ,thomas ,young caught sight of the phenomenon1 early in last century3 8,the solid particles of the blood do not by any means vary in magnitude in the same ratio with the bulk of the animal10 ,natural ,philosophy1 ed4 #ahde1 p4 #dff2 and ,leeuwenhoek and ,stephen ,hales were aware of it a hundred years before4 ,but in this case1 though the blood-corpuscles show no relation of magnitude to the size of the animal1 they do seem to have some relation to its activity4 ,at least the corpuscles in the sluggish ,amphibia are much the largest known to us1 while the smallest are found among the deer and other agile and speedy mammals4 7,cf4 ,gulliver1 ,p4,z4,s4 #ahge1 p4 #dgd1 etc47 ,this apparent correlation may have its bearing on modern views of the surface-condensation or adsorption of oxygen in the blood-corpuscles1 a process which would be greatly facilitated and intensified by the increase of surface due to their minuteness4 #fb ,cf4 ,p4 ,enriques1 ,la forma come funzione della grandezza3 ,ricerche sui gangli nervosi degli ,invertebrati1 ,arch4 f4 ,entw4 ,mech4 ,,xxv1 p4 #fee1 #aijg-h4 #fc ,while the difference in cell-volume is vastly less than that between the volumes1 and very much less also than that between the surfaces1 of the respective animals1 yet there is a certain difference2 and this it has been attempted to correlate with the need for each cell in the many-celled ganglion of the larger animal to possess a more complex 8exchange-system0 of branches1 for intercommunication with its more numerous neighbours4 ,another explanation is based on the fact that1 while such cells as continue to divide throughout life tend to uniformity of size in all mammals1 those which do not do so1 and in particular the ganglion cells1 continue to grow1 and their size becomes1 therefore1 a function of the duration of life4 ,cf4 ,g4 ,levi1 ,studii sulla grandezza delle cellule1 ,arch4 ,ital4 di ,anat4 e di ,embryolog4 ,v1 p4 #bia1 #aijf4 #fd ,boveri4 ,zellen-studien1 ,v4 ,ueber die ,abhngigkeit der ,kerngrsse und ,zellenzahl der ,seeigellarven von der ,chromosomenzahl der ,ausgangszellen4 ,jena1 #aije4 #fe ,recent important researches suggest that such ultra-minute 8filter-passers0 are the true cause of certain acute maladies commonly ascribed to the presence of much larger organisms2 cf4 ,hort1 ,lakin and ,benians1 ,the true infective ,agent in ,cerebrospinal ,fever1 etc41 ,j4 ,roy4 ,army ,med4 ,corps1 ,feb4 #aiaj4 #ff ,zur ,erkenntniss der ,kolloide1 #aije1 p4 #abb2 where there will be found an interesting discussion of various molecular and other minute magnitudes4 #fg ,encyclopaedia ,britannica1 #ith edit41 vol4 ,,iii1 p4 #db1 #ahge4 #fh ,sur la limite de petitesse des organismes1 ,bull4 ,soc4 ,r4 des ,sc4 md4 et nat4 de ,bruxelles1 ,jan4 #aijc2 ,rec4 d'uvres 7,physiol4 gnrale71 p4 #cbe4 #fi ,cf4 ,a4 ,fischer1 ,vorlesungen ber ,bakterien1 #ahig1 p4 #ej4 #gj ,f4 ,hofmeister1 quoted in ,cohnheim's ,chemie der ,eiweisskrper1 #aijj1 p4 #ah4 #ga ,mc,kendrick arrived at a still lower estimate1 of about #abej proteid molecules in the minutest organisms4 ,brit4 ,ass4 ,rep4 #aija1 p4 #hjh4 #gb ,cf4 ,perrin1 ,les ,atomes1 #aiad1 p4 #gd4 #gc ,cf4 ,tait1 ,on ,compression of ,air in small ,bubbles1 ,proc4 ,r4 ,s4 ,e4 ,v1 #ahfe4 #gd ,phil4 ,mag4 ,,xlviii1 #ahii2 ,collected ,papers1 ,,iv1 p4 #dcj4 #ge ,carpenter1 ,the ,microscope1 edit4 #ahfb1 p4 #ahe4 #gf ,the modern literature on the ,brownian ,movement is very large1 owing to the value which the phenomenon is shewn to have in determining the size of the atom4 ,for a fuller1 but still elementary account1 see ,j4 ,cox1 ,beyond the ,atom1 #aiac1 pp4 #aah-abh2 and see1 further1 ,perrin1 ,les ,atomes1 pp4 #aai-ahi4 #gg ,cf4 ,r4 ,gans1 ,wie fallen ,stbe und ,scheiben in einer reibenden ,flssigkeit8 ,mnchener ,bericht1 #aiaa1 p4 #aia2 ,k4 ,przibram1 ,ueber die ,brown'sche ,bewegung nicht kugelfrmiger ,teilchen1 ,wiener ,ber4 #aiab1 p4 #bcci4 #gh ,ueber die ungeordnete ,bewegung niederer ,thiere1 ,pflger's ,archiv1 ,,cliii1 p4 #dja1 #aiac4 #gi ,sometimes we find one and the same diagram suffice1 whether the intervals of time be great or small2 and we then invoke 8,wolff's ,law10 and assert that the life-history of the individual repeats1 or recapitulates1 the history of the race4 #hj ,our subject is one of ,bacon's 8,instances of the ,course10 or studies wherein we 8measure ,nature by periods of ,time40 ,in ,bacon's ,catalogue of ,particular ,histories1 one of the odd hundred histories or investigations which he foreshadowed is precisely that which we are engaged on1 viz4 a 8,history of the ,growth and ,increase of the ,body1 in the whole and in its parts40 #ha ,cf4 ,aristotle1 ,phys4 vi1 #e1 #bce a #aa1 1 4 ,bacon emphasised1 in like manner1 the fact that 8all motion or natural action is performed in time3 some more quickly1 some more slowly1 but all in periods determined and fixed in the nature of things4 ,even those actions which seem to be performed suddenly1 and 7as we say7 in the twinkling of an eye1 are found to admit of degree in respect of duration40 ,nov4 ,org4 ,,xlvi4 #hb ,cf4 7e4g47 ,elem4 ,physiol4 ed4 #agff1 ,,viii1 p4 #aad1 8,ducimur autem ad evolutionem potissimum1 quando a perfecto animale retrorsum progredimur1 et incrementorum atque mutationum seriem relegimus4 ,ita inveniemus perfectum illud animal fuisse imperfectius1 alterius figurae et fabricae1 et denique rude et informe3 et tamen idem semper animal sub iis diversis phasibus fuisse1 quae absque ullo saltu perpetuos parvosque per gradus cohaereant40 #hc ,beitrge zur ,entwickelungsgeschichte des ,hhnchens im ,ei1 p4 #dj1 #ahag4 ,roux ascribes the same views also to ,von ,baer and to ,r4 ,h4 ,lotze 7,allg4 ,physiologie1 p4 #cec1 #ahea74 #hd ,roux1 ,die ,entwickelungsmechanik1 p4 #ii1 #aije4 #he ,op4 cit4 p4 #cjb1 8,magnum hoc naturae instrumentum1 etiam in corpore animato evolvendo potenter operatur2 etc40 #hf ,ibid4 p4 #cjf4 8,subtiliora ista1 et aliquantum hypothesi mista1 tamen magnum mihi videntur speciem veri habere40 #hg ,cf4 ,his1 ,on the ,principles of ,animal ,morphology1 ,proc4 ,r4 ,s4 ,e4 ,,xv1 #ahhh1 p4 #bid3 8,my own attempts to introduce some elementary mechanical or physiological conceptions into embryology have not generally been agreed to by morphologists4 ,to one it seemed ridiculous to speak of the elasticity of the germinal layers2 another thought that1 by such considerations1 we #8put the cart before the horse'3 and one more recent author states1 that we have better things to do in embryology than to discuss tensions of germinal layers and similar questions1 since all explanations must of necessity be of a phylogenetic nature4 ,this opposition to the application of the fundamental principles of science to embryological questions would scarcely be intelligible had it not a dogmatic background4 ,no other explanation of living forms is allowed than heredity1 and any which is founded on another basis must be rejected 4444444 ,to think that heredity will build organic beings without mechanical means is a piece of unscientific mysticism40 #hh ,hertwig1 ,o41 ,zeit und ,streitfragen der ,biologie1 ,,ii4 #ahig4 #hi ,cf4 ,roux1 ,gesammelte ,abhandlungen1 ,,ii1 p4 #ca1 #ahie4 #ij ,treatise on ,comparative ,embryology1 ,i1 p4 #d1 #ahha4 #ia ,cf4 ,fick1 ,anal4 ,anzeiger1 ,,xxv1 p4 #aij1 #aijd4 #ib #ast ed4 p4 #ddd2 #fth ed4 p4 #cij4 ,the student should not fail to consult the passage in question2 for there is always a risk of misunderstanding or misinterpretation when one attempts to epitomise ,darwin's carefully condensed arguments4 #ic 8,in omni rerum naturalium historia utile est mensuras definiri et numeros10 ,haller1 ,elem4 ,physiol4 ,,ii1 p4 #beh1 #agfj4 ,cf4 ,hales1 ,vegetable ,staticks1 ,introduction4 #id ,brussels1 #ahga4 ,cf4 the same author's ,physique sociale1 #ahce1 and ,lettres sur la thorie des probabilits1 #ahdf4 ,see also1 for the general subject1 ,boyd1 ,r41 ,tables of weights of the ,human ,body1 etc4 ,phil4 ,trans4 vol4 ,,cli1 #ahfa2 ,roberts1 ,c41 ,manual of ,anthropometry1 #ahgh2 ,daffner1 ,f41 ,das ,wachsthum des ,menschen 7#bnd ed471 #aijb1 etc4 #ie ,dr ,johnson was not far wrong in saying that 8life declines from thirty-five02 though the ,autocrat of the ,breakfast-table1 like ,cicero1 declares that 8the furnace is in full blast for ten years longer40 #if ,joly1 ,the ,abundance of ,life1 #aiae 7#ahij71 p4 #hf4 #ig 8,lou pes1 mstre de tout ,le poids1 matre de tout1 mstre snso vergougno1 ,que te tirasso en bas de sa brutalo pougno10 ,j4 ,h4 ,fabre1 ,oubreto prouvenalo1 p4 #fa4 #ih ,the continuity of the phenomenon of growth1 and the natural passage from the phase of increase to that of decrease or decay1 are admirably discussed by ,enriques1 in 8,la morte10 ,riv4 di ,scienza1 #aijg1 and in 8,wachsthum und seine analytische ,darstellung10 ,biol4 ,centralbl4 ,june1 #aiji4 ,haller 7,elem4 ,,vii1 p4 #fh7 recognised decrementum as a phase of growth1 not less important 7theoretically7 than incrementum3 8tristis1 sed copiosa1 haec est materies40 #ii ,cf4 7int4 al471 ,friedenthal1 ,h41 ,das ,wachstum des ,krpergewichtes 444 in verschiedenen ,lebensltern1 ,zeit4 f4 allg4 ,physiol4 ,,ix1 pp4 #dhg-ead1 #aiji4 #ajj ,as ,haller observed it to do in the chick 7,elem4 ,,viii1 p4 #bid73 8,hoc iterum incrementum miro ordine ita distribuitur1 ut in principio incubationis maximum est3 inde perpetuo minuatur40 #aja ,there is a famous passage in ,lucretius 7v4 #hhc7 where he compares the course of life1 or rate of growth1 in the horse and his boyish master3 ,principio circum tribus actis impiger annis ,floret equus1 puer hautquaquam1 etc4 #ajb ,minot1 ,c4 ,s41 ,senescence and ,rejuvenation1 ,journ4 of ,physiol4 ,,xii1 pp4 #ig-aec1 #ahia2 ,the ,problem of ,age1 ,growth and ,death1 ,pop4 ,science ,monthly 7,june-,dec471 #aijg4 #ajc ,quoted in ,vierordt's ,anatomische 444 ,daten und ,tabellen1 #aijf4 p4 #ac4 #ajd ,unsere ,krperform1 ,leipzig1 #ahgd4 #aje ,no such point of inflection appears in the curve of weight according to ,c4 ,m4 ,jackson's data 7,on the ,prenatal ,growth of the ,human ,body1 etc41 ,amer4 ,journ4 of ,anat4 ,,ix1 #ajji1 pp4 #abf1 #aef71 nor in those quoted by him from ,ahlfeld1 ,fehling and others4 ,but it is plain that the very rapid increase of the monthly weights1 approximately in the ratio of the cubes of the corresponding lengths1 would tend to conceal any such breach of continuity1 unless it happened to be very marked indeed4 ,moreover in the case of ,jackson's data 7and probably also in the others7 the actual age of the embryos was not determined1 but was estimated from their lengths4 ,the following is ,jackson's estimate of average weights at intervals of a lunar month3 ,months #j #a #b #c #d #e #f #g #h #i #aj ,wt in gms4 #j #jd #c #cf #abj #ccj #fjj #ajjj #aejj #bbjj #cbjj #ajf ,g4 ,kraus 7after ,wallich-,martius71 ,ann4 du ,jardin bot4 de ,buitenzorg1 ,,xii1 #a1 #ahid1 p4 #baj4 ,cf4 ,w4 ,ostwald1 ,zeitliche ,eigenschaften1 etc4 p4 #ef4 #ajg ,cf4 ,chodat1 ,r41 et ,monnier1 ,a41 ,sur la courbe de croissance des vgtaux1 ,bull4 ,herb4 ,boissier 7#b71 ,v1 pp4 #fae1 #faf1 #aije4 #ajh ,cf4 ,fr4 ,boas1 ,growth of ,toronto ,children1 ,rep4 of ,u4,s4 ,comm4 of ,education1 #ahif-g1 pp4 #aeda-aeii1 #ahih2 ,boas and ,clark ,wissler1 ,statistics of ,growth1 ,education ,rep4 #aijd1 pp4 #be-acb1 #aijf2 ,h4 ,p4 ,bowditch1 ,rep4 ,mass4 ,state ,board of ,health1 #ahgg2 ,k4 ,pearson1 ,on the ,magnitude of certain coefficients of ,correlation in ,man1 ,pr4 ,r4 ,s4 ,,lxvi1 #aijj4 #aji l4c4 p4 #db1 and other papers there quoted4 #aaj ,see1 for an admirable rsum of facts1 ,wolfgang ,ostwald1 ,ueber die ,zeitliche ,eigenschaften der ,entwickelungsvorgnge 7#ga pp471 ,leipzig1 #aijh 7,roux's ,vortrge1 ,heft ,v73 to which work ,i am much indebted4 ,a long list of observations on the growth-rate of various animals is also given by ,h4 ,przibram1 ,exp4 ,zoologie1 #aiac1 pt4 ,,iv 7,vitalitt71 pp4 #he-hg4 #aaa ,cf4 ,st ,loup1 ,vitesse de croissance chez les ,souris1 ,bull4 ,soc4 ,zool4 ,fr4 ,,xviii1 #bdb1 #ahic2 ,robertson1 ,arch4 f4 ,entwickelungsmech4 ,,xxv1 p4 #ehg1 #aijh2 ,donaldson4 ,boas ,memorial ,volume1 ,new ,york1 #aijf4 #aab ,luciani e ,lo ,monaco1 ,arch4 ,ital4 de ,biologie1 ,,xxvii1 p4 #cdj1 #ahig4 #aac ,schaper1 ,arch4 f4 ,entwickelungsmech4 ,,xiv1 p4 #cef1 #aijb4 ,cf4 ,barfurth1 ,versuche ber die ,verwandlung der ,froschlarven1 ,arch4 f4 mikr4 ,anat4 ,,xxix1 #ahhg4 #aad ,joh4 ,schmidt1 ,contributions to the ,life-history of the ,eel1 ,rapports du ,conseil ,intern4 pour l'exploration de la ,mer1 vol4 ,v1 pp4 #acg-bgd1 ,copenhague1 #aijf4 #aae ,that the metamorphoses of an insect are but phases in a process of growth1 was firstly clearly recognised by ,swammerdam1 ,biblia ,naturae1 #agcg1 pp4 #f1 #egi etc4 #aaf ,from ,bose1 ,j4 ,c41 ,plant ,response1 ,london1 #aijf1 p4 #dag4 #aag ,this phenomenon1 of incrementum inequale1 as opposed to incrementum in universum1 was most carefully studied by ,haller3 8,incrementum inequale multis modis fit1 ut aliae partes corporis aliis celerius increscant4 ,diximus hepar minus fieri1 majorem pulmonem1 minimum thymum1 etc40 7,elem4 ,,viii 7#b71 p4 #cd74 #aah ,see 7inter alia7 ,fischel1 ,a41 ,variabilitt und ,wachsthum des embryonalen ,krpers1 ,morphol4 ,jahrb4 ,,xxiv1 pp4 #cfi-djd1 #ahif4 ,oppel1 ,vergleichung des ,entwickelungsgrades der ,organe zu verschiedenen ,entwickelungszeiten bei ,wirbelthieren1 ,jena1 #ahia4 ,faucon1 ,a41 ,peses et ,mensurations ftales diffrents ges de la grossesse4 7,thse47 ,paris1 #ahig4 ,loisel1 ,g41 ,croissance compare en poids et en longueur des ftus mle et femelle dans l'espce humaine1 ,c4 ,r4 ,soc4 de ,biologie1 ,paris1 #aijc4 ,jackson1 ,c4 ,m41 ,pre-natal growth of the human body and the relative growth of the various organs and parts1 ,am4 ,j4 of ,anat4 ,,ix1 #aiji2 ,post-natal growth and variability of the body and of the various organs in the albino rat1 ibid4 ,,xv1 #aiac4 #aai l4c4 p4 #aedb4 #abj ,variation and ,correlation in ,brain-weight1 ,biometrika1 ,,iv1 pp4 #ac-ajd1 #aije4 #aba ,die ,sugethiere1 p4 #aag4 #abb ,amer4 ,j4 of ,anatomy1 ,,viii1 pp4 #cai-cec1 #aijh4 ,donaldson 7,journ4 ,comp4 ,neur4 and ,psychol4 ,,xviii1 pp4 #cde-cib1 #aijh7 also gives a logarithmic formula for brain-weight 7y7 as compared with body-weight 7x71 which in the case of the white rat is y #eed #efi log7x #hg71 and the agreement is very close4 ,but the formula is admittedly empirical and as ,raymond ,pearl says 7,amer4 ,nat4 #aiji1 p4 #cjc71 8no ulterior biological significance is to be attached to it40 #abc ,biometrika1 ,,iv1 pp4 #ac-ajd1 #aijd4 #abd ,donaldson1 ,h4 ,h41 ,a ,comparison of the ,white ,rat with ,man in respect to the ,growth of the entire ,body1 ,boas ,memorial ,vol41 ,new ,york1 #aijf1 pp4 #e-bf4 #abe ,besides many papers quoted by ,dubois on the growth and weight of the brain1 and numerous papers in ,biometrika1 see also the following3 ,ziehen1 ,th41 ,das ,gehirn3 ,massverhltnisse1 in ,bardeleben's ,handb4 der ,anat4 des ,menschen1 ,,iv1 pp4 #cec-chf1 #ahii4 ,spitzka1 ,e4 ,a41 ,brain-weight of ,animals with special reference to the ,weight of the ,brain in the ,macaque ,monkey1 ,j4 ,comp4 ,neurol4 ,,xiii1 pp4 #i-ag1 #aijc4 ,warneke1 ,p41 ,mitteilung neuer ,gehirn und ,krpergewichtsbestimmungen bei ,sugern1 nebst ,zusammenstellung der gesammten bisher beobachteten absoluten und relativen ,gehirngewichte bei den verschiedenen ,species1 ,j4 f4 ,psychol4 u4 ,neurol4 ,,xiii1 pp4 #cee-djc1 #aiji4 ,donaldson1 ,h4 ,h41 ,on the regular seasonal ,changes in the relative ,weight of the ,central ,nervous ,system of the ,leopard ,frog1 ,journ4 of ,morph4 ,,xxii1 pp4 #ffc-fid1 #aiaa4 #abf ,cf4 ,jenkinson1 ,growth1 ,variability and ,correlation in ,young ,trout1 ,biometrika1 ,,viii1 pp4 #ddd-dee1 #aiab4 #abg ,cf4 chap4 xvii1 p4 #gci4 #abh 8 444,i marked in the same manner as the ,vine1 young ,honeysuckle shoots1 etc44442 and ,i found in them all a gradual scale of unequal extensions1 those parts extending most which were tenderest10 ,vegetable ,staticks1 ,exp4 cxxiii4 #abi ,from ,sachs1 ,textbook of ,botany1 #ahhb1 p4 #hbj4 #acj ,variation and ,differentiation in ,ceratophyllum1 ,carnegie ,inst4 ,publications1 ,no4 #eh1 ,washington1 #aijg4 #aca ,cf4 ,lmmel1 ,ueber periodische ,variationen in ,organismen1 ,biol4 ,centralbl4 ,,xxii1 pp4 #cfh-cgf1 #aijc4 #acb ,herein lies the easy answer to a contention frequently raised by ,bergson1 and to which he ascribes great importance1 that 8a mere variation of size is one thing1 and a change of form is another40 ,thus he considers 8a change in the form of leaves0 to constitute 8a profound morphological difference40 ,creative ,evolution1 p4 #ga4 #acc ,i do not say that the assumption that these two groups of earwigs were of different ages is altogether an easy one2 for of course1 even in an insect whose metamorphosis is so simple as the earwig's1 consisting only in the acquisition of wings or wing-cases1 we usually take it for granted that growth proceeds no more after the final stage1 or 8adult form0 is attained1 and further that this adult form is attained at an approximately constant age1 and constant magnitude4 ,but even if we are not permitted to think that the earwig may have grown1 or moulted1 after once the elytra were produced1 it seems to me far from impossible1 and far from unlikely1 that prior to the appearance of the elytra one more stage of growth1 or one more moult took place in some cases than in others3 for the number of moults is known to be variable in many species of ,orthoptera4 ,unfortunately ,bateson tells us nothing about the sizes or total lengths of his earwigs2 but his figures suggest that it was bigger earwigs that had the longer tails2 and that the rate of growth of the tails had had a certain definite ratio to that of the bodies1 but not necessarily a simple ratio of equality4 #acd ,jackson1 ,c4 ,m41 ,j4 of ,exp4 ,zool4 ,,xix1 #aiae1 p4 #ii2 cf4 also ,hans ,aron1 ,unters4 ber die ,beeinflssung der ,wachstum durch die ,ernhrung1 ,berl4 klin4 ,wochenbl4 ,,li1 pp4 #igb-igg1 #aiac1 etc4 #ace ,the temperature limitations of life1 and to some extent of growth1 are summarised for a large number of species by ,davenport1 ,exper4 ,morphology1 cc4 viii1 xviii1 and by ,hans ,przibram1 ,exp4 ,zoologie1 ,,iv1 c4 v4 #acf ,raumur3 ,l'art de faire clore et lever en toute saison des oiseaux domestiques1 foit par le moyen de la chaleur du fumier1 ,paris1 #agdi4 #acg ,cf4 7int4 al47 de ,vries1 ,h41 ,matriaux pour la connaissance de l'influence de la temprature sur les plantes1 ,arch4 ,nerl4 ,v1 #che-dja1 #ahgj4 ,kppen1 ,wrme und ,pflanzenwachstum1 ,bull4 ,soc4 ,imp4 ,nat4 ,moscou4 ,,xliii1 pp4 #da-aaj1 #ahgj4 #ach ,blackman1 ,f4 ,f41 ,ann4 of ,botany1 ,,xix1 p4 #bha1 #aije4 #aci ,for various instances of a 8temperature coefficient0 in physiological processes1 see ,kanitz1 ,zeitschr4 f4 ,elektrochemie1 #aijg1 p4 #gjg2 ,biol4 ,centralbl4 ,,xxvii1 p4 #aa1 #aijg2 ,hertzog1 ,r4 ,o41 ,temperatureinfluss auf die ,entwicklungsgeschwindigkeit der ,organismen1 ,zeitschr4 f4 ,elektrochemie1 ,,xi1 p #hbj1 #aije2 ,krogh1 ,quantitative ,relation between ,temperature and ,standard ,metabolism1 ,int4 ,zeitschr4 f4 physik4-chem4 ,biologie1 ,i1 p4 #dia1 #aiad2 ,ptter1 ,a41 ,ueber ,temperaturkoefficienten1 ,zeitschr4 f4 allgem4 ,physiol4 ,,xvi1 p4 #egd1 #aiad4 ,also ,cohen1 ,physical ,chemistry for ,physicians and ,biologists 7,english edition71 #aijc2 ,pike1 ,f4 ,h41 and ,scott4 ,e4 ,l41 ,the ,regulation of the ,physico-chemical ,condition of the ,organism1 ,american ,naturalist1 ,jan4 #aiae1 and various papers quoted therein4 #adj ,cf4 ,errera1 ,l41 ,l',optimum1 #ahif 7,rec4 d',oeuvres1 ,physiol4 gnrale1 pp4 #cch-cfh1 #aiaj72 ,sachs1 ,physiologie d4 ,pflanzen1 #ahhb1 p4 #bcc2 ,pfeffer1 ,pflanzenphysiologie1 ii1 p4 #gh1 #aijd2 and cf4 ,jost1 ,ueber die ,reactionsgeschwindigkeit im ,organismus1 ,biol4 ,centralbl4 ,,xxvi1 pp4 #bbe-bdd1 #aijf4 #ada ,after ,kppen1 ,bull4 ,soc4 ,nat4 ,moscou1 ,,xliii1 pp4 #da-aaj1 #ahga4 #adb ,botany1 p4 #chg4 #adc ,leitch1 ,i41 ,some ,experiments on the ,influence of ,temperature on the ,rate of ,growth in ,pisum sativum1 ,ann4 of ,botany1 ,,xxx1 pp4 #be-df1 #aiaf4 7,cf4 especially ,table ,,iii1 p4 #de47 #add ,blackman1 ,f4 ,f41 ,presidential ,address in ,botany1 ,brit4 ,ass4 ,dublin1 #aijh4 #ade ,rec4 de l',inst4 ,bot4 de ,bruxelles1 ,,vi1 #aijf4 #adf ,hertwig1 ,o41 ,einfluss der ,temperatur auf die ,entwicklung von ,rana fusca und ,r4 esculenta1 ,arch4 f4 mikrosk4 ,anat4 ,,li1 p4 #cai1 #ahih4 ,cf4 also ,bialaszewicz1 ,k41 ,beitrge z4 ,kenntniss d4 ,wachsthumsvorgnge bei ,amphibienembryonen1 ,bull4 ,acad4 ,sci4 de ,cracovie1 p4 #ghc1 #aijh2 ,abstr4 in ,arch4 f4 ,entwicklungsmech4 ,,xxviii1 p4 #afj1 #aiji4 #adg ,der ,grad der ,beschleunigung tierischer ,entwickelung durch erhhte ,temperatur1 ,a4 f4 ,entw4 ,mech4 ,,xx4 p4 #acj1 #aije4 ,more recently1 ,bialaszewicz has determined the coefficient for the rate of segmentation in ,rana as being #bd per #aj ,c4 #adh ,das ,wachstum des ,menschen1 p4 #cbi1 #aijb4 #adi ,the diurnal periodicity is beautifully shewn in the case of the ,hop by ,joh4 ,schmidt 7,c4 ,r4 du ,laboratoire de ,carlsberg1 ,x1 pp4 #bce-bdh1 ,copenhague1 #aiac74 #aej ,trans4 ,botan4 ,soc4 ,edinburgh1 ,,xviii1 #ahia1 p4 #def4 #aea ,i had not received1 when this was written1 ,mr ,douglass's paper1 ,on a method of estimating ,rainfall by the ,growth of ,trees1 ,bull4 ,amer4 ,geograph4 ,soc4 ,,xlvi1 pp4 #cba-cce1 #aiad4 ,mr ,douglass does not fail to notice the long period here described2 but he lays more stress on the occurrence of shorter cycles 7of #aa1 #ba and #cc years71 well known to meteorologists4 ,mr ,douglass is inclined 7and ,i think rightly7 to correlate the variations in growth directly with fluctuations in rainfall1 that is to say with alternate periods of moisture and aridity2 but he points out that the temperature curves 7and also the sunspot curves7 are markedly similar4 #aeb ,it may well be that the effect is not due to light after all2 but to increased absorption of heat by the soil1 as a result of the long hours of exposure to the sun4 #aec ,on growth in relation to light1 see ,davenport1 ,exp4 ,morphology1 ,,ii1 ch4 xvii4 ,in some cases 7as in the roots of ,peas71 exposure to light seems to have no effect on growth2 in other cases1 as in diatoms 7according to ,whipple's experiments1 quoted by ,davenport1 ,,ii1 p4 #dbc71 the effect of light on growth or multiplication is well-marked1 measurable1 and apparently capable of expression by a logarithmic formula4 ,the discrepancy would seem to arise from the fact that1 while light-energy always tends to be absorbed by the chlorophyll of the plant1 converted into chemical energy1 and stored in the shape of starch or other reserve materials1 the actual rate of growth depends on the rate at which these reserves are drawn on3 and this is another matter1 in which light-energy is no longer directly concerned4 #aed ,cf4 for instance1 ,ngeli's classical account of the effect of change of habitat on ,alpine and other plants3 ,sitzungsber4 ,baier4 ,akad4 ,wiss4 #ahfe1 pp4 #bbh-bhd4 #aee ,cf4 ,blackman1 ,f4 ,f41 ,presidential ,address in ,botany1 ,brit4 ,ass4 ,dublin1 #aijh4 ,the fact was first enunciated by ,baudrimont and ,st ,ange1 ,recherches sur le dveloppement du ftus1 ,mm4 ,acad4 ,sci4 ,,xi1 p4 #dfi1 #ahea4 #aef ,cf4 ,loeb1 ,untersuchungen zur physiol4 ,morphologie der ,thiere1 #ahib2 also ,experiments on ,cleavage1 ,j4 of ,morph4 ,,vii1 p4 #bec1 #ahib2 ,zusammenstellung der ,ergebnisse einiger ,arbeiten ber die ,dynamik des thierischen ,wachsthum1 ,arch4 f4 ,entw4 ,mech4 ,,xv1 #aijb-c1 p4 #ffi2 ,davenport1 ,on the ,rle of ,water in ,growth1 ,boston ,soc4 ,n4 ,h4 #ahig2 ,ida ,h4 ,hyde1 ,am4 ,j4 of ,physiol4 ,,xii1 #aije1 p4 #bda1 etc4 #aeg ,pflger's ,archiv1 ,,lv1 #ahic4 #aeh ,beitrge zur ,kenntniss der ,wachstumsvorgnge bei ,amphibienembryonen1 ,bull4 ,acad4 ,sci4 de ,cracovie1 #aijh1 p4 #ghc2 cf4 ,arch4 f4 ,entw4 ,mech4 ,,xxviii1 p4 #afj1 #aiji2 ,,xxxiv1 p4 #dhi1 #aiab4 #aei ,fehling1 ,h41 ,arch4 fr ,gynaekologie1 ,,xi1 #ahgg2 cf4 ,morgan1 ,experimental ,zoology1 p4 #bdj1 #aijg4 #afj ,hber1 ,r41 ,bedeutung der ,theorie der ,lsungen fr ,physiologie und ,medizin1 ,biol4 ,centralbl4 ,,xix1 #ahii2 cf4 pp4 #bgb-bgd4 #afa ,schmankewitsch has made other interesting observations on change of size and form1 after some generations1 in relation to change of density2 e4g4 in the flagellate infusorian ,anisonema acinus1 ,btschli 7,z4 f4 w4 ,z4 ,,xxix1 p4 #dbi1 #ahgg74 #afb ,these 8,fezzan-worms10 when first described1 were supposed to be 8insects' eggs02 cf4 ,humboldt1 ,personal ,narrative1 ,,vi1 i1 #h1 note2 ,kirby and ,spence1 ,letter ,x4 #afc ,cf4 ,introd4 l'tude de la mdecine exprimentale1 #ahhe1 p4 #aaj4 #afd ,cf4 ,abonyi1 ,z4 f4 w4 ,z4 ,,cxiv1 p4 #acd1 #aiae4 ,but ,frdricq has shewn that the amount of ,na,cl in the blood of ,crustacea 7,carcinus moenas7 varies1 and all but corresponds1 with the density of the water in which the creature has been kept 7,arch4 de ,zool4 ,exp4 et ,gn4 7#b71 ,,iii1 p4 xxxv1 #ahhe72 and other results of ,frdricq's1 and various data given or quoted by ,bottazzi 7,osmotischer ,druck und elektrische ,leitungsfhigkeit der ,flssigkeiten der ,organismen1 in ,asher-,spiro's ,ergebn4 d4 ,physiologie1 ,,vii1 pp4 #afj-djb1 #aijh7 suggest that the case of the brine-shrimps must be looked upon as an extreme or exceptional one4 #afe ,cf4 ,schmankewitsch1 ,z4 f4 w4 ,zool4 ,,xxv1 #ahge1 ,,xxix1 #ahgg1 etc42 transl4 in appendix to ,packard's ,monogr4 of ,n4 ,american ,phyllopoda1 #ahhc1 pp4 #dff-ead2 ,daday de ,des1 ,ann4 ,sci4 ,nat4 7,zool471 7#i71 ,,xi1 #aiaj2 ,samter und ,heymons1 ,abh4 d4 ,k4 pr4 ,akad4 ,wiss4 #aijb2 ,bateson1 ,mat4 for the ,study of ,variation1 #ahid1 pp4 #if-aja2 ,anikin1 ,mitth4 ,kais4 ,univ4 ,tomsk1 ,,xiv3 ,zool4 ,centralbl4 ,,vi1 pp4 #gef-gfj1 #aijh2 ,abonyi1 ,z4 f4 w4 ,z4 ,,cxiv1 pp4 #if-afh1 #aiae 7with copious bibliography71 etc4 #aff ,according to the empirical canon of physiology1 that 7as ,frdricq expresses it7 8,l'tre vivant est agenc de telle manire que chaque influence perturbatrice provoque d'elle-mme la mise en activit de l'appareil compensateur qui doit neutraliser et rparer le dommage40 #afg ,such phenomena come precisely under the head of what ,bacon called ,instances of ,magic3 8,by which ,i mean those wherein the material or efficient cause is scanty and small as compared with the work or effect produced2 so that even when they are common1 they seem like miracles1 some at first sight1 others even after attentive consideration4 ,these magical effects are brought about in three ways 444 of which one is by excitation or invitation in another body1 as in the magnet which excites numberless needles without losing any of its virtue1 or in yeast and such-like40 ,nov4 ,org41 cap4 li4 #afh ,monnier1 ,a41 ,les matires minrales1 et la loi d'accroissement des ,vgtaux1 ,publ4 de l',inst4 de ,bot4 de l',univ4 de ,genve 7#g71 ,,iii1 #aije4 ,cf4 ,robertson1 ,on the ,normal ,rate of ,growth of an ,individual1 and its ,biochemical ,significance1 ,arch4 f4 ,entw4 ,mech4 ,,xxv1 pp4 #eha-fad1 ,,xxvi1 pp4 #ajh-aah1 #aijh2 ,wolfgang ,ostwald1 ,die zeitlichen ,eigenschaften der ,entwickelungsvorgnge1 #aijh2 ,hatai1 ,s41 ,interpretation of ,growth-curves from a ,dynamical ,standpoint1 ,anat4 ,record1 ,v1 p4 #cgc1 #aiaa4 #afi ,biochem4 ,zeitschr4 ,,ii1 #aijf1 p4 #cd4 #agj ,even a crystal may be said1 in a sense1 to display 8autocatalysis03 for the bigger its surface becomes1 the more rapidly does the mass go on increasing4 #aga ,cf4 ,loeb1 ,the ,stimulation of ,growth1 ,science1 ,may #ad1 #aiae4 #agb ,b4 coli-communis1 according to ,buchner1 tends to double in #bb minutes2 in #bd hours1 therefore1 a single individual would be multiplied by something like #ajbh2 ,sitzungsber4 ,mnchen4 ,ges4 ,morphol4 u4 ,physiol4 ,,iii1 pp4 #fe-ga1 #ahhh4 ,cf4 ,marshall ,ward1 ,biology of ,bacillus ramosus1 etc4 ,pr4 ,r4 ,s4 ,,lviii1 #bfe-dfh1 #ahie4 ,the comparatively large infusorian ,stylonichia1 according to ,maupas1 would multiply in a month by #ajdc4 #agc ,cf4 ,enriques1 ,wachsthum und seine analytisehe ,darstellung1 ,biol4 ,centralbl4 #aiji1 p4 #ccg4 #agd ,cf4 7int4 al47 ,mellor1 ,chemical ,statics and ,dynamics1 #aijd1 p4 #bia4 #age ,cf4 ,robertson1 l4c4 #agf ,see1 for a brief resum of this subject1 ,morgan's ,experimental ,zoology1 chap4 xvi4 #agg ,amer4 ,j4 of ,physiol41 ,x1 #aijd4 #agh ,c4,r4 ,,cxxi1 ,,cxxii1 #ahie-if4 #agi ,cf4 ,loeb1 ,science1 ,may #ad1 #aiae4 #ahj ,cf4 ,baumann u4 ,roos1 ,vorkommen von ,iod im ,thierkrper1 ,zeitschr4 fr ,physiol4 ,chem4 ,,xxi1 ,,xxii1 #ahie1 #f4 #aha ,le ,no-,vitalisme1 ,rev4 ,scientifique1 ,mars #aiaa1 p4 #bb 7of reprint74 #ahb ,la vie et la mort1 p4 #dc1 #aijb4 #ahc ,cf4 ,dendy1 ,evolutionary ,biology1 #aiab1 p4 #djh2 ,brit4 ,ass4 ,report 7,portsmouth71 #aiaa1 p4 #bgh4 #ahd ,lucret4 v1 #hgg4 8,lucretius nowhere seems to recognise the possibility of improvement or change of species by #8natural selection'2 the animals remain as they were at the first1 except that the weaker and more useless kinds have been crushed out4 ,hence he stands in marked contrast with modern evolutionists40 ,kelsey's note1 ad loc4 #ahe ,even after we have so narrowed the scope and sphere of natural selection1 it is still hard to understand2 for the causes of extinction are often wellnigh as hard to comprehend as are those of the origin of species4 ,if we assert 7as has been lightly done7 that ,smilodon perished owing to its gigantic tusks1 that ,teleosaurus was handicapped by its exaggerated snout1 or ,stegosaurus weighed down by its intolerable load of armour1 we may be reminded of other kindred forms to show that similar conditions did not necessarily lead to extermination1 or that rapid extinction ensued apart from any such visible or apparent disadvantages4 ,cf4 ,lucas1 ,f4 ,a41 ,on ,momentum in ,variation1 ,amer4 ,nat4 xli1 p4 #df1 #aijg4 #ahf ,see ,professor ,t4 ,h4 ,morgan's ,regeneration 7#caf pp471 #aija for a full account and copious bibliography4 ,the early experiments on regeneration1 by ,vallisneri1 ,raumur1 ,bonnet1 ,trembley1 ,baster1 and others1 are epitomised by ,haller1 ,elem4 ,physiologiae1 ,,viii1 p4 #aef seq4 #ahg ,journ4 ,experim4 ,zool4 ,,vii1 p4 #cig1 #aiji4 #ahh ,op4 cit4 p4 #djf1 ,exp4 ,,iv4 #ahi ,the experiments of ,loeb on the growth of ,tubularia in various saline solutions1 referred to on p4 #abe1 might as well or better have been referred to under the heading of regeneration1 as they were performed on cut pieces of the zoophyte4 7,cf4 ,morgan1 op4 cit4 p4 #ce47 #aij ,powers of the ,creator1 ,i1 p4 #g1 #ahea4 ,see also ,rare and ,remarkable ,animals1 ,,ii1 pp4 #ag-ai1 #ij1 #ahdg4 #aia ,lillie1 ,f4 ,r41 ,the smallest ,parts of ,stentor capable of ,regeneration1 ,journ4 of ,morphology1 ,,xii1 p4 #bci1 #ahig4 #aib ,boveri1 ,entwicklungsfhigkeit kernloser ,seeigeleier1 etc41 ,arch4 f4 ,entw4 ,mech4 ,,ii1 #ahie4 ,see also ,morgan1 ,studies of the partial larvae of ,sphaerechinus1 ibid4 #ahie2 ,j4 ,loeb1 ,on the ,limits of ,divisibility of ,living ,matter1 ,biol4 ,lectures1 #ahid1 etc4 #aic ,cf4 ,przibram1 ,h41 ,scheerenumkehr bei dekapoden ,crustaceen1 ,arch4 f4 ,entw4 ,mech4 ,,xix1 #aha-bdg1 #aije2 ,,xxv1 #bff-cdd1 #aijg4 ,emmel1 ibid4 ,,xxii1 #edb1 #aijf2 ,regeneration of lost parts in ,lobster1 ,rep4 ,comm4 ,inland ,fisheries1 ,rhode ,island1 ,,xxxv1 ,,xxxvi1 #aije-f2 ,science 7n4s471 ,,xxvi1 #hc-hg1 #aijg4 ,zeleny1 ,compensatory ,regulation1 ,j4 ,exp4 ,zool4 ,,ii1 #a-ajb1 #cdg-cfi1 #aije2 etc4 #aid ,lobsters are occasionally found with two symmetrical claws3 which are then usually serrated1 sometimes 7but very rarely7 both blunt-toothed4 ,cf4 ,calman1 ,p4,z4,s4 #aijf1 pp4 #fcc1 #fcd1 and reff4 #aie ,wilson1 ,e4 ,b41 ,reversal of ,symmetry in ,alpheus heterochelis1 ,biol4 ,bull4 ,,iv1 p4 #aig1 #aijc4 #aif ,j4 ,exp4 ,zool4 ,,vii1 p4 #deg1 #aiji4 #aig ,biologica1 ,,iii1 p4 #afa1 ,june4 #aiac4 #aih ,anatomical and ,pathological ,observations1 p4 #c1 #ahde2 ,anatomical ,memoirs1 ,,ii1 p4 #cib1 #ahfh4 #aii ,giard1 ,a41 ,l'uf et les dbuts de l'volution1 ,bull4 ,sci4 du ,nord de la ,fr4 ,,viii1 pp4 #beb-beh1 #ahgf4 #bjj ,entwickelungsvorgnge der ,eizelle1 #ahgf2 ,investigations on ,microscopic ,foams and ,protoplasm1 p4 #a1 #ahid4 #bja ,journ4 of ,morphology1 ,i1 p4 #bbi1 #ahhg4 #bjb ,while it has been very common to look upon the phenomena of mitosis as sufficiently explained by the results towards which they seem to lead1 we may find here and there a strong protest against this mode of interpretation4 ,the following is a case in point3 8,on a tent d'tablir dans la mitose dite primitive plusieurs catgories1 plusieurs types de mitose4 ,on a choisi le plus souvent comme base de ces systmes des concepts abstraits et tlologiques3 rpartition plus ou moins exacte de la chromatine entre les deux noyaux-fils suivant qu'il y a ou non des chromosomes 7,dangeard71 distribution particulire et signification dualiste des substances nuclaires 7substance kintique et substance gnrative ou hrditaire1 ,hartmann et ses lves71 etc4 ,pour moi tous ces essais sont rejeter catgoriquement cause de leur caractre finaliste2 de plus1 ils sont construits sur des concepts non dmontrs1 et qui parfois reprsentent des gnralisations absolument errones40 ,a4 ,alexeieff1 ,archiv fr ,protistenkunde1 ,,xix1 p4 #cdd1 #aiac4 #bjc ,this is the old philosophic axiom writ large3 ,ignorato motu1 ignoratur natura2 which again is but an adaptation of ,aristotle's phrase1 1 as equivalent to the 8,efficient ,cause40 ,fitz,gerald holds that 8all explanation consists in a description of underlying motions02 ,scientific ,writings1 #aijb1 p4 #che4 #bjd ,as when ,ngeli concluded that the organism is1 in a certain sense1 8vorgebildet02 ,beitr4 zur wiss4 ,botanik1 ,,ii1 #ahfj4 ,cf4 ,e4 ,b4 ,wilson1 ,the ,cell1 etc41 p4 #cjb4 #bje 8,la matire arrange par une sagesse divine doit tre essentiellement organise partout 444 il y a machine dans les parties de la machine ,naturelle l'infini40 ,sur le principe de la ,vie1 p4 #dca 7,erdmann74 ,this is the very converse of the doctrine of the ,atomists1 who could not conceive a condition 8ubi dimidiae partis pars semper habebit ,dimidiam partem1 nec res praefiniet ulla40 #bjf ,cf4 an interesting passage from the ,elements 7,i1 p4 #dde1 ,molesworth's edit471 quoted by ,owen1 ,hunterian ,lectures on the ,invertebrates1 #bnd ed4 pp4 #dj1 #da1 #ahee4 #bjg 8,wir mssen deshalb den lebenden ,zellen1 abgesehen von der ,molekularstructur der organischen ,verbindungen welche sie enthlt1 noch eine andere und in anderer ,weise complicirte ,structur zuschreiben1 und diese es ist welche wir mit dem ,namen ,organisation bezeichnen10 ,brcke1 ,die ,elementarorganismen1 ,wiener ,sitzungsber4 ,,xliv1 #ahfa1 p4 #chf2 quoted by ,wilson1 ,the ,cell1 etc4 p4 #bhi4 ,cf4 also ,hardy1 ,journ4 of ,physiol4 ,,xxiv1 #ahii1 p4 #aei4 #bjh ,precisely as in the ,lucretian concursus1 motus1 ordo1 positura1 figurae1 whereby bodies mutato ordine mutant naturam4 #bji ,otto ,warburg1 ,beitrge zur ,physiologie der ,zelle1 insbesondere ber die ,oxidationsgeschwindigkeit in ,zellen2 in ,asher-,spiro's ,ergebnisse der ,physiologie1 ,,xiv1 pp4 #bec-ccg1 #aiad 7see p4 #cae74 7,cf4 ,bayliss1 ,general ,physiology1 #aiae1 p4 #eij74 #baj ,hardy1 ,w4 ,b41 ,on some ,problems of ,living ,matter 7,guthrie ,lecture71 ,tr4 ,physical ,soc4 ,london1 xxviii1 p4 #ii-aah1 #aiaf4 #baa ,as a matter of fact both phrases occur1 side by side1 in ,graham's classical paper on 8,liquid ,diffusion applied to ,analysis10 ,phil4 ,trans4 ,,cli1 p4 #ahd1 #ahfa2 ,chem4 and ,phys4 ,researches 7ed4 ,angus ,smith71 #ahgf1 p4 #eed4 #bab ,l4 ,rhumbler1 ,mechanische ,erklrung der ,aehnlichkeit zwischen ,magnetischen ,kraftliniensystemen und ,zelltheilungsfiguren1 ,arch4 f4 ,entw4 ,mech4 ,,xv1 p4 #dhb1 #aijc4 #bac ,gallardo1 ,a41 ,essai d'interpretation des figures caryocintiques1 ,anales del ,museo de ,buenos-,aires 7#b71 ,,ii1 #ahif2 ,la division de la cellule1 phenomne bipolaire de caractre electro-colloidal1 ,arch4 f4 ,entw4 ,mech4 ,,xxviii1 #aiji1 etc4 #bad ,arch4 f4 ,entw4 ,mech4 ,,iii1 ,,iv1 #ahif-ig4 #bae ,on various theories of the mechanism of mitosis1 see 7e4g47 ,wilson1 ,the ,cell in ,development1 etc41 pp4 #ajj-aad2 ,meves1 ,zelltheilung1 in ,merkel u4 ,bonnet's ,ergebnisse der ,anatomie1 etc41 ,,vii1 ,,viii1 #ahig-h2 ,ida ,h4 ,hyde1 ,amer4 ,journ4 of ,physiol4 ,,xii1 pp4 #bda-bge1 #aije2 and especially ,prenant1 ,a41 ,theories et interprtations physiques de la mitose1 ,j4 de l',anat4 et ,physiol4 ,,xlvi1 pp4 #eaa-egh1 #aiaj4 #baf ,hartog1 ,m41 ,une force nouvelle3 le mitokintisme1 ,c4,r4 #aa ,juli1 #aiaj2 ,mitokinetism in the ,mitotic ,spindle and in the ,polyasters1 ,arch4 f4 ,entw4 ,mech4 ,,xxvii1 pp4 #ada-ade1 #aiji2 cf4 ibid4 ,,xl1 pp4 #cc-fd1 #aiad4 ,cf4 also ,hartog's papers in ,proc4 ,r4 ,s4 7,b71 ,,lxxvi1 #aije2 ,science ,progress 7n4 s471 ,i1 #aijg2 ,riv4 di ,scienza1 ,,ii1 #aijh2 ,c4 ,r4 ,assoc4 fr4 pour l',avancem4 des ,sc4 #aiad1 etc4 #bag ,the configurations1 as obtained by the usual experimental methods1 were of course known long before ,faraday's day1 and constituted the 8convergent and divergent magnetic curves0 of eighteenth century mathematicians4 ,as ,leslie said1 in #ahba1 they were 8regarded with wonder by a certain class of dreaming philosophers1 who did not hesitate to consider them as the actual traces of an invisible fluid1 perpetually circulating between the poles of the magnet40 ,faraday's great advance was to interpret them as indications of stress in a medium1--of tension or attraction along the lines1 and of repulsion transverse to the lines1 of the diagram4 #bah ,cf4 also the curious phenomenon in a dividing egg described as 8spinning0 by ,mrs ,g4 ,f4 ,andrews1 ,j4 of ,morph4 ,,xii1 pp4 #cfg-chi1 #ahig4 #bai ,whitman1 ,j4 of ,morph4 ,,ii1 p4 #dj1 #ahhi4 #bbj 8,souvent il n'y a qu'une sparation physique entre le cytoplasme et le suc nuclaire1 comme entre deux liquides immiscibles1 etc420 ,alexeieff1 ,sur la mitose dite 8primitive10 ,arch4 f4 ,protistenk4 ,,xxix1 p4 #ceg1 #aiac4 #bba ,the appearance of 8vacuolation0 is a result of endosmosis or the diffusion of a less dense fluid into the denser plasma of the cell4 ,caeteris paribus1 it is less apparent in marine organisms than in those of freshwater1 and in many or most marine ,ciliates and even ,rhizopods a contractile vacuole has not been observed 7,btschli1 in ,bronn's ,protozoa1 p4 #adad72 it is also absent1 and probably for the same reason1 in parasitic ,protozoa1 such as the ,gregarines and the ,entamoebae4 ,rossbach shewed that the contractile vacuole of ordinary freshwater ,ciliates was very greatly diminished in a #e per cent4 solution of ,na,cl1 and all but disappeared in a #a per cent4 solution of sugar 7,arb4 z4 z4 ,inst4 ,wrzburg1 #ahgb1 cf4 ,massart1 ,arch4 de ,biol4 ,,lx1 p4 #eae1 #ahhi74 ,actinophrys sol1 when gradually acclimatised to sea-water1 loses its vacuoles1 and vice versa 7,gruber1 ,biol4 ,centralbl4 ,,ix1 p4 #bb1 #ahhi72 and the same is true of ,amoeba 7,zuelzer1 ,arch4 f4 ,entw4 ,mech4 #aiaj1 p4 #fcb74 ,the gradual enlargement of the contractile vacuole is precisely analogous to the change of size of a bubble until the gases on either side of the film are equally diffused1 as described long ago by ,draper 7,phil4 ,mag4 7n4 s471 ,,xi1 p4 #eei1 #ahcg74 ,rhumbler has shewn that contractile or pulsating vacuoles may be well imitated in chloroform-drops1 suspended in water in which various substances are dissolved 7,arch4 f4 ,entw4 ,mech4 ,,vii1 #ahih1 p4 #ajc74 ,the pressure within the contractile vacuole1 always greater than without1 diminishes with its size1 being inversely proportional to its radius2 and when it lies near the surface of the cell1 as in a ,heliozoon1 it bursts as soon as it reaches a thinness which its viscosity or molecular cohesion no longer permits it to maintain4 #bbb ,cf4 p4 #ffj4 #bbc ,the elongated or curved 8macronucleus0 of an ,infusorian is to be looked upon as a single mass of chromatin1 rather than as an aggregation of particles in a fluid drop1 as in the case described4 ,it has a shape of its own1 in which ordinary surface-tension plays a very subordinate part4 #bbd ,thorie physico-chimique de la ,vie1 p4 #gc1 #aiaj2 ,mechanism of ,life1 p4 #ef1 #aiaa4 #bbe ,whence the name 8mitosis0 7,greek 1 a thread71 applied first by ,flemming to the whole phenomenon4 ,kollmann 7,biol4 ,centralbl4 ,,ii1 p4 #ajg1 #ahhb7 called it divisio per fila1 or divisio laqueis implicata4 ,many of the earlier students1 such as ,van ,beneden 7,rech4 sur la maturation de l'uf1 ,arch4 de ,biol4 ,,iv1 #ahhc71 and ,hermann 7,zur ,lehre v4 d4 ,entstehung d4 karyokinetischen ,spindel1 ,arch4 f4 mikrosk4 ,anat4 ,,xxxvii1 #ahia7 thought they recognised actual muscular threads1 drawing the nuclear material asunder towards the respective foci or poles2 and some such view was long maintained by other writers1 ,boveri1 ,heidenhain1 ,flemming1 ,r4 ,hertwig1 and many more4 ,in fact1 the existence of contractile threads1 or the ascription to the spindle rather than to the poles or centrosomes of the active forces concerned in nuclear division1 formed the main tenet of all those who declined to go beyond the 8contractile properties of protoplasm0 for an explanation of the phenomenon4 7,cf4 also ,j4 ,w4 ,jenkinson1 ,q4 ,j4 ,m4 ,s4 ,,xlviii1 p4 #dga1 #aijd47 #bbf ,cf4 ,btschli1 ,o41 ,ueber die knstliche ,nachahmung der karyokinetischen ,figur1 ,verh4 ,med4 ,nat4 ,ver4 ,heidelberg1 ,v1 pp4 #bh-da 7#ahib71 #ahig4 #bbg ,arrhenius1 in describing a typical colloid precipitate1 does so in terms that are very closely applicable to the ordinary microscopic appearance of the protoplasm of the cell4 ,the precipitate consists1 he says1 8en un rseau d'une substance solide contenant peu d'eau1 dans les mailles duquel est inclus un fluide contenant un peu de colloide dans beaucoup d'eau 444 ,evidemment cette structure se forme cause de la petite diffrence de poids spcifique des deux phases1 et de la consistance gluante des particules spares1 qui s'attachent en forme de rseau40 ,rev4 ,scientifique1 ,feb4 #aiaa4 #bbh ,f4 ,schwartz1 in ,cohn's ,beitr4 z4 ,biologie der ,pflanzen1 ,v1 p4 #a1 #ahhg4 #bbi ,fischer1 ,anat4 ,anzeiger1 ,,ix1 p4 #fgh1 #ahid1 ,x1 p4 #gfi1 #ahie4 #bcj ,see1 in particular1 ,w4 ,b4 ,hardy1 ,on the structure of ,cell ,protoplasm1 ,journ4 of ,physiol4 ,,xxiv1 pp4 #aeh-bjg1 #ahhi2 also ,hber1 ,physikalische ,chemie der ,zelle und der ,gewebe1 #aijb4 ,cf4 7int4 al47 ,flemming1 ,zellsubstanz1 ,kern und ,zelltheilung #ahhb1 p4 #ea1 etc4 #bca ,my description and diagrams 7,figs #db-ea7 are based on those of ,professor ,e4 ,b4 ,wilson4 #bcb ,if the word permeability be deemed too directly suggestive of the phenomena of magnetism we may replace it by the more general term of specific inductive capacity4 ,this would cover the particular case1 which is by no means an improbable one1 of our phenomena being due to a 8surface charge0 borne by the nucleus itself and also by the chromosomes3 this surface charge being in turn the result of a difference in inductive capacity between the body or particle and its surrounding medium4 7,cf4 footnote1 p4 #ahg47 #bcc ,on the effect of electrical influences in altering the surface-tensions of the colloid particles1 see ,bredig1 ,anorganische ,fermente1 pp4 #ae1 #af1 #aija4 #bcd ,the ,cell1 etc4 p4 #ff4 #bce ,lillie1 ,r4 ,s41 ,amer4 ,j4 of ,physiol4 ,,viii1 p4 #bhb1 #aijc4 #bcf ,we have not taken account in the above paragraphs of the obvious fact that the supposed symmetrical field of force is distorted by the presence in it of the more or less permeable bodies2 nor is it necessary for us to do so1 for to that distorted field the above argument continues to apply1 word for word4 #bcg ,m4 ,foster1 ,lectures on the ,history of ,physiology1 #aija1 p4 #fb4 #bch ,op4 cit4 pp4 #aaj and #ia4 #bci ,lamb1 ,a4 ,b41 ,a new ,explanation of the ,mechanism of ,mitosis1 ,journ4 ,exp4 ,zool4 ,v1 pp4 #bg-cc1 #aijh4 #bdj ,amer4 ,j4 of ,physiol4 ,,viii1 pp4 #bgc-bhc1 #aijc 7vide supra1 p4 #aha72 cf4 ibid4 ,,xv1 pp4 #df-hd1 #aije4 ,cf4 also ,biological ,bulletin1 ,,iv1 p4 #age4 #aijc4 #bda ,in like manner ,hardy has shewn that colloid particles migrate with the negative stream if the reaction of the surrounding fluid be alkaline1 and vice versa4 ,the whole subject is much wider than these brief allusions suggest1 and is essentially part of ,quincke's theory of ,electrical ,diffusion or ,endosmosis3 according to which the particles and the fluid in which they float 7or the fluid and the capillary walls through which it flows7 each carry a charge1 there being a discontinuity of potential at the surface of contact1 and hence a field of force leading to powerful tangential or shearing stresses1 communicating to the particles a velocity which varies with the density per unit area of the surface charge4 ,see ,w4 ,b4 ,hardy's paper on ,coagulation by ,electricity1 ,journ4 of ,physiol4 ,,xxiv1 p4 #bhh-cjd1 #ahii1 also ,hardy and ,h4 ,w4 ,harvey1 ,surface ,electric ,charges of ,living ,cells1 ,proc4 ,r4 ,s4 ,,lxxxiv 7,b71 pp4 #bag-bbf1 #aiaa1 and papers quoted therein4 ,cf4 also ,e4 ,n4 ,harvey's observations on the convection of unicellular organisms in an electric field 7,studies on the ,permeability of ,cells1 ,journ4 of ,exper4 ,zool4 ,x1 pp4 #ejh-eef1 #aiaa74 #bdb ,on ,differences in ,electrical ,potential in ,developing ,eggs1 ,amer4 ,journ4 of ,physiol4 ,,xii1 pp4 #bda-bge1 #aije4 ,this paper contains an excellent summary of various physical theories of the segmentation of the cell4 #bdc ,gray has recently demonstrated a temporary increase of electrical conductivity in sea-urchin eggs during the process of fertilisation 7,the ,electrical ,conductivity of fertilised and unfertilised ,eggs1 ,journ4 ,mar4 ,biol4 ,assoc4 ,x1 pp4 #ej-ei1 #aiac74 #bdd ,schewiakoff1 ,ueber die karyokinetische ,kerntheilung der ,euglypha alveolata1 ,morph4 ,jahrb4 ,,xiii1 pp4 #aic-beh1 #ahhh 7see p4 #baf74 #bde ,coe1 ,w4 ,r41 ,maturation and ,fertilization of the ,egg of ,cerebratulus1 ,zool4 ,jahrbcher 7,anat4 ,abth471 ,,xii1 pp4 #dbe-dgf1 #ahii4 #bdf ,thus1 for example1 ,farmer and ,digby 7,on ,dimensions of ,chromosomes considered in relation to ,phylogeny1 ,phil4 ,trans4 7,b71 ,,ccv1 pp4 #a-bc1 #aiad7 have been at pains to shew1 in confutation of ,meek 7ibid4 ,,cciii1 pp4 #a-gd1 #aiab71 that the width of the chromosomes cannot be correlated with the order of phylogeny4 #bdg ,cf4 also ,arch4 f4 ,entw4 ,mech4 ,x1 p4 #eb1 #aijj4 #bdh ,cf4 ,loeb1 ,am4 ,j4 of ,physiol4 ,,vi1 p4 #cb1 #aijb2 ,erlanger1 ,biol4 ,centralbl4 ,,xvii1 pp4 #aeb1 #cci1 #ahig2 ,conklin1 ,biol4 ,lectures1 ,woods ,holl1 p4 #fi1 etc4 #ahih-i4 #bdi ,robertson1 ,t4 ,b41 ,note on the ,chemical ,mechanics of ,cell ,division1 ,arch4 f4 ,entw4 ,mech4 ,,xxvii1 p4 #bi1 #aiji1 ,,xxxv1 p4 #fib4 #aiac4 ,cf4 ,r4 ,s4 ,lillie1 ,j4 ,exp4 ,zool4 ,,xxi1 pp4 #cfi-djb1 #aiaf4 #bej ,cf4 ,d',arsonval1 ,arch4 de ,physiol4 p4 #dfj1 #ahhi2 ,ida ,h4 ,hyde1 op4 cit4 p4 #bdb4 #bea ,cf4 ,plateau's remarks 7,statique des liquides1 ,,ii1 p4 #aed7 on the tendency towards equilibrium1 rather than actual equilibrium1 in many of his systems of soap-films4 #beb ,but under artificial conditions1 8polyspermy0 may take place1 e4g4 under the action of dilute poisons1 or of an abnormally high temperature1 these being all1 doubtless1 conditions under which the surface-tension is diminished4 #bec ,fol1 ,h41 ,recherches sur la fcondation1 #ahgi4 ,roux1 ,w41 ,beitrge zur ,entwickelungsmechanik des ,embryo1 ,arch4 f4 ,mikr4 ,anat4 ,,xix1 #ahhg4 ,whitman1 ,c4 ,o41 ,okinesis1 ,journ4 of ,morph4 ,i1 #ahhg4 #bed ,wilson4 ,the ,cell1 p4 #gg4 #bee ,eight and twelve are by much the commonest numbers1 six and sixteen coming next in order4 ,if we may judge by the list given by ,e4 ,b4 ,wilson 7,the ,cell1 p4 #bjf71 over #hj of the observed cases lie between #f and #af1 and nearly #fj between #h and #ab4 #bef ,theory of ,cells1 p4 #aia4 #beg ,the ,cell in ,development1 etc4 p4 #ei2 cf4 pp4 #chh1 #dac4 #beh ,e4g4 ,brcke1 ,elementarorganismen1 p4 #chg3 8,wir mssen in der ,zelle einen kleinen ,thierleib sehen1 und drfen die ,analogien1 welche zwischen ihr und den kleinsten ,thierformen existiren1 niemals aus den ,augen lassen40 #bei ,whitman1 ,c4 ,o41 ,the ,inadequacy of the ,cell-theory1 ,journ4 of ,morphol4 ,,viii1 pp4 #fci-feh1 #ahic2 ,sedgwick1 ,a41 ,on the ,inadequacy of the ,cellular ,theory of ,development1 ,q4,j4,m4,s4 ,,xxxvii1 pp4 #hg-aja1 #ahie1 ,,xxxviii1 pp4 #cca-ccg1 #ahif4 ,cf4 ,bourne1 ,g4 ,c41 ,a ,criticism of the ,cell-theory2 being an answer to ,mr ,sedgwick's article1 etc41 ibid4 ,,xxxviii1 pp4 #acg-agd1 #ahif4 #bfj ,cf4 ,hertwig1 ,o41 ,die ,zelle und die ,gewebe1 #ahic1 p4 #a2 8,die ,zellen1 in welche der ,anatom die pflanzlichen und thierischen ,organismen zerlegt1 sind die ,trger der ,lebensfunktionen2 sie sind1 wie ,virchow sich ausgedrckt hat1 die #8,lebenseinheiten4' ,von diesem ,gesichtspunkt aus betrachtet1 erscheint der ,gesammtlebensprocess eines zusammengesetzten ,organismus nichts ,anderes zu sein als das hchst verwickelte ,resultat der einzelnen ,lebensprocesse seiner zahlreichen1 verschieden functionirenden ,zellen40 #bfa ,journ4 of ,morph4 ,,viii1 p4 #fec1 #ahic4 #bfb ,neue ,grundlegungen zur ,kenntniss der ,zelle1 ,morph4 ,jahrb4 ,,viii1 pp4 #bgb1 #cac1 #ccc1 #ahhc4 #bfc ,journ4 of ,morph4 ,,ii1 p4 #di1 #ahhi4 #bfd ,phil4 ,trans4 ,,cli1 p4 #ahc1 #ahfa2 ,researches1 ed4 ,angus ,smith1 #ahgg1 p4 #eec4 #bfe ,cf4 ,kelvin1 ,on the ,molecular ,tactics of a ,crystal1 ,the ,boyle ,lecture1 ,oxford1 #ahic1 ,baltimore ,lectures1 #aijd1 pp4 #fab-fdb4 ,here ,kelvin was mainly following ,bravais's 7and ,frankenheim's7 theory of 8space-lattices10 but he had been largely anticipated by the crystallographers4 ,for an account of the development of the subject in modern crystallography1 by ,sohncke1 von ,fedorow1 ,schnfliess1 ,barlow and others1 see ,tutton's ,crystallography1 chap4 ix1 pp4 #aah-acd1 #aiaa4 #bff ,in a homogeneous crystalline arrangement1 symmetry compels a locus of one property to be a plane or set of planes2 the locus in this case being that of least surface potential energy4 #bfg ,this is what ,graham called the water of gelatination1 on the analogy of water of crystallisation2 ,chem4 and ,phys4 ,researches1 p4 #eig4 #bfh ,here1 in a non-crystalline or random arrangement of particles1 symmetry ensures that the potential energy shall be the same per unit area of all surfaces2 and it follows from geometrical considerations that the total surface energy will be least if the surface be spherical4 #bfi ,lehmann1 ,o41 ,flssige ,krystalle1 sowie ,plasticitt von ,krystallen im allgemeinen1 etc41 #bfd pp4 #ci pll41 ,leipsig1 #aijd4 ,for a semi-popular1 illustrated account1 see ,tutton's ,crystals 7,int4 ,sci4 ,series71 #aiaa4 #bgj ,as ,graham said of an allied phenomenon 7the so-called blood-crystals of ,funke71 it 8illustrates the maxim that in nature there are no abrupt transitions1 and that distinctions of class are never absolute40 #bga ,cf4 ,przibram1 ,h41 ,kristall-analogien zur ,entwickelungsmechanik der ,organismen1 ,arch4 f4 ,entw4 ,mech4 ,,xxii1 p4 #bjg1 #aijf 7with copious bibliography72 ,lehmann1 ,scheinbar lebende ,kristalle und ,myelinformen1 ibid4 ,,xxvi1 p4 #dhc1 #aijh4 #bgb ,the idea of a 8surface-tension0 in liquids was first enunciated by ,segner1 ,de figuris superficierum fluidarum1 in ,comment4 ,soc4 ,roy4 ,gttingen1 #agea1 p4 #cja4 ,hooke1 in the ,micrographia 7#affe1 ,obs4 ,,viii1 etc471 had called attention to the globular or spherical form of the little morsels of steel struck off by a flint1 and had shewn how to make a powder of such spherical grains1 by heating fine filings to melting point4 8,this ,phaenomenon0 he said 8proceeds from a propriety which belongs to all kinds of fluid ,bodies more or less1 and is caused by the ,incongruity of the ,ambient and included ,fluid1 which so acts and modulates each other1 that they acquire1 as neer as is possible1 a spherical or globular form44440 #bgc ,science of ,mechanics1 #aijb1 p4 #cie2 see also ,mach's article ,ueber die physikalische ,bedeutung der ,gesetze der ,symmetrie1 ,lotos1 ,,xxi1 pp4 #aci-adg1 #ahga4 #bgd ,similarly1 ,sir ,david ,brewster and others made powerful lenses by simply dropping small drops of ,canada balsam1 castor oil1 or other strongly refractive liquids1 on to a glass plate3 ,on ,new ,philosophical ,instruments 7,description of a new ,fluid ,microscope71 ,edinburgh1 #ahac1 p4 #dac4 #bge ,beitrge z4 ,physiologie d4 ,protoplasma1 ,pflger's ,archiv1 ,,ii1 p4 #cjg1 #ahfi4 #bgf ,poggend4 ,annalen1 ,,xciv1 pp4 #ddg-dei1 #ahee4 ,cf4 ,strethill ,wright1 ,phil4 ,mag4 ,feb4 #ahfj4 #bgg ,haycraft and ,carlier pointed out 7,proc4 ,r4,s4,e4 ,,xv1 pp4 #bbj-bbd1 #ahhh7 that the amoeboid movements of a white blood-corpuscle are only manifested when the corpuscle is in contact with some solid substance3 while floating freely in the plasma or serum of the blood1 these corpuscles are spherical1 that is to say they are at rest and in equilibrium4 ,the same fact has recently been recorded anew by ,ledingham 7,on ,phagocytosis from an adsorptive point of view1 ,journ4 of ,hygiene1 ,,xii1 p4 #cbd1 #aiab74 ,on the emission of pseudopodia as brought about by changes in surface tension1 see also 7int4 al47 ,jensen1 ,ueber den ,geotropismus niederer ,organismen1 ,pflger's ,archiv1 ,,liii1 #ahic4 ,jensen remarks that in ,orbitolites1 the pseudopodia issuing through the pores of the shell first float freely1 then as they grow longer bend over till they touch the ground1 whereupon they begin to display amoeboid and streaming motions4 ,verworn indicates 7,allg4 ,physiol4 #ahie1 p4 #dbi71 and ,davenport says 7,experim4 ,morphology1 ,,ii1 p4 #cgf7 that 8this persistent clinging to the substratum is a #8thigmotropic' reaction1 and one which belongs clearly to the category of #8response4' 0 7,cf4 ,ptter1 ,thigmotaxis bei ,protisten1 ,a4 f4 ,physiol4 #aijj1 ,suppl4 p4 #bdg47 ,but it is not clear to my mind that to account for this simple phenomenon we need invoke other factors than gravity and surface-action4 #bgh ,cf4 ,pauli1 ,allgemeine physikalische ,chemie d4 ,zellen u4 ,gewebe1 in ,asher-,spiro's ,ergebnisse der ,physiologie1 #aiab2 ,przibram1 ,vitalitt1 #aiac1 p4 #f4 #bgi ,the surface-tension theory of protoplasmic movement has been denied by many4 ,cf4 7e4g471 ,jennings1 ,h4 ,s41 ,contributions to the ,study of the ,behaviour of the ,lower ,organisms1 ,carnegie ,inst4 #aijd1 pp4 #acj-bcj2 ,dellinger1 ,o4 ,p41 ,locomotion of ,amoebae1 etc4 ,journ4 ,exp4 ,zool4 ,,iii1 pp4 #ccg-ceg1 #aijf2 also various papers by ,max ,heidenhain1 in ,anatom4 ,hefte 7,merkel und ,bonnet71 etc4 #bhj ,these various movements of a liquid surface1 and other still more striking movements such as those of a piece of camphor floating on water1 were at one time ascribed by certain physicists to a peculiar force1 sui generis1 the force pipolique of ,dutrochet3 until van der ,mensbrugghe shewed that differences of surface tension were enough to account for this whole series of phenomena 7,sur la tension superficielle des liquides considre au point de vue de certains mouvements observs leur surface1 ,mm4 ,cour4 ,acad4 de ,belgique1 ,,xxxiv1 #ahfi2 cf4 ,plateau1 p4 #bhc74 #bha ,cf4 infra1 p4 #cjf4 #bhb ,cf4 p4 #cb4 #bhc ,or1 more strictly speaking1 unless its thickness be less than twice the range of the molecular forces4 #bhd ,it follows that the tension1 depending only on the surface-conditions1 is independent of the thickness of the film4 #bhe ,this simple but immensely important formula is due to ,laplace 7,mcanique ,cleste1 ,bk4 x4 suppl4 ,thorie de l'action capillaire1 #ahjf74 #bhf ,sur la surface de rvolution dont la courbure moyenne est constante1 ,journ4 de ,m4 ,liouville1 ,,vi1 p4 #cji1 #ahda4 #bhg ,see ,liquid ,drops and ,globules1 #aiad1 p4 #aa4 ,robert ,boyle used turpentine in much the same way4 ,for other methods see ,plateau1 op4 cit4 p4 #aed4 #bhh ,felix ,plateau recommends the use of a weighted thread1 or plumb-line1 drawn up out of a jar of water or oil2 ,phil4 ,mag4 ,,xxxiv1 p4 #bdf1 #ahfg4 #bhi ,cf4 ,boys1 ,c4 ,v41 ,on ,quartz ,fibres1 ,nature1 ,july #aa1 #ahhi2 ,warburton1 ,c41 ,the ,spinning ,apparatus of ,geometric ,spiders1 ,q4,j4,m4,s4 ,,xxxi1 pp4 #bi-ci1 #ahij4 #bij ,j4 ,blackwall1 ,spiders of ,great ,britain 7,ray ,society71 #ahei1 p4 #aj2 ,trans4 ,linn4 ,soc4 ,,xvi1 p4 #dgg1 #ahcc4 #bia ,the intermediate spherules appear1 with great regularity and beauty1 whenever a liquid jet breaks up into drops2 see the instantaneous photographs in ,poynting and ,thomson's ,properties of ,matter1 pp4 #aea1 #aeb1 7ed4 #aijg74 #bib ,khne1 ,untersuchungen ber das ,protoplasma1 #ahfd1 p4 #ge1 etc4 #bic ,a ,study of ,splashes1 #aijh1 p4 #ch1 etc42 ,segmentation of a ,liquid ,annulus1 ,proc4 ,roy4 ,soc4 ,,xxx1 pp4 #di-fj1 #ahhj4 #bid ,cf4 ibid4 pp4 #ag1 #gg4 ,the same phenomenon is beautifully and continuously evident when a strong jet of water from a tap impinges on a curved surface and then shoots off it4 #bie ,see a ,study of ,splashes1 p4 #ed4 #bif ,a case which we have not specially considered1 but which may be found to deserve consideration in biology1 is that of a cell or drop suspended in a liquid of varying density1 for instance in the upper layers of a fluid 7e4g4 sea-water7 at whose surface condensation is going on1 so as to produce a steady density-gradient4 ,in this case the normally spherical drop will be flattened into an oval form1 with its maximum surface-curvature lying at the level where the densities of the drop and the surrounding liquid are just equal4 ,the sectional outline of the drop has been shewn to be not a true oval or ellipse1 but a somewhat complicated quartic curve4 7,rice1 ,phil4 ,mag4 ,jan4 #aiae47 #big ,indeed any non-isotropic stiffness1 even though ,t remained uniform1 would simulate1 and be indistinguishable from1 a condition of non-stiffness and non-isotropic ,t4 #bih ,a non-symmetry of ,t and ,t might also be capable of explanation as a result of 8liquid crystallisation40 ,this hypothesis is referred to1 in connection with the blood-corpuscles1 on p4 #bgb4 #bii ,the case of the snow-crystals is a particularly interesting one2 for their 8distribution0 is in some ways analogous to what we find1 for instance1 among our microscopic skeletons of ,radiolarians4 ,that is to say1 we may one day meet with myriads of some one particular form or species only1 and another day with myriads of another2 while at another time and place we may find species intermingled in inexhaustible variety4 7,cf4 e4g4 ,j4 ,glaisher1 ,ill4 ,london ,news1 ,feb4 #ag1 #ahee2 ,q4,j4,m4,s4 ,,iii1 pp4 #agi-ahe1 #ahee74 #cjj ,cf4 ,bergson1 ,creative ,evolution1 p4 #ajg3 8,certain ,foraminifera have not varied since the ,silurian epoch4 ,unmoved witnesses of the innumerable revolutions that have upheaved our planet1 the ,lingulae are today what they were at the remotest times of the palaeozoic era40 #cja ,ray ,lankester1 ,a4,m4,n4,h4 7#d71 ,,xi1 p4 #cba1 #ahgc4 #cjb ,leidy1 ,parasites of the ,termites1 ,j4 ,nat4 ,sci41 ,philadelphia1 ,,viii1 pp4 #dbe-ddg1 #ahgd-ha2 cf4 ,saville ,kent's ,infusoria1 ,,ii1 p4 #eea4 #cjc ,op4 cit4 p4 #gi4 #cjd ,brady1 ,challenger ,monograph1 pl4 ,,xx1 p4 #bcc4 #cje ,that the ,foraminifera not only can but do hang from the surface of the water is confirmed by the following apt quotation which ,i owe to ,mr ,e4 ,heron-,allen3 8,quand on place1 comme il a t dit1 le dpt provenant du lavage des fucus dans un flacon que l'on remplit de nouvelle eau1 on voit au bout d'une heure environ les animaux ,gromia dujardinii se mettre en mouvement et commencer grimper4 ,six heures aprs ils tapissent l'extrieur du flacon1 de sorte que les plus levs sont trente-six ou quarante-deux millimetres du fond2 le lendemain beaucoup d'entre eux1 aprs avoir atteint le niveau du liquide1 ont continu ramper sa surface1 en se laissant pendre au-dessous comme certains mollusques gastropodes40 7,dujardin1 ,f41 ,observations nouvelles sur les prtendus cphalopodes microscopiques1 ,ann4 des ,sci4 ,nat4 7#b71 ,,iii1 p4 #cab1 #ahce47 #cjf ,cf4 ,boas1 ,spolia ,atlantica1 #ahhf1 pl4 #f4 #cjg ,this cellular pattern would seem to be related to the 8cohesion figures0 described by ,tomlinson in various surface-films 7,phil4 ,mag4 #ahfa to #ahgj72 to the 8tesselated structure0 in liquids described by ,professor ,james ,thomson in #ahhb 7,collected ,papers1 p4 #acf72 and to the tourbillons cellulaires of ,prof4 ,h4 ,bnard 7,ann4 de ,chimie 7#g71 ,,xxiii1 pp4 #fb-add1 #aija1 7#h71 ,,xxiv1 pp4 #efc-eff1 #aiaa71 ,rev4 gnr4 des ,sci4 ,,xi1 p4 #abfh1 #aijj2 cf4 also ,e4 ,h4 ,weber4 7,poggend4 ,ann4 ,,xciv1 p4 #deb1 #ahee1 etc474 ,the phenomenon is of great interest and various appearances have been referred to it1 in biology1 geology1 metallurgy and even astronomy3 for the flocculent clouds in the solar photosphere shew an analogous configuration4 7,see letters by ,kerr ,grant1 ,larmor1 ,wager and others1 in ,nature1 ,april #af to ,june #aa1 #aiad47 ,in many instances1 marked by strict symmetry or regularity1 it is very possible that the interference of waves or ripples may play its part in the phenomenon4 ,but in the majority of cases1 it is fairly certain that localised centres of action1 or of diminished tension1 are present1 such as might be provided by dust-particles in the case of ,darling's experiment 7cf4 infra1 p4 #eij74 #cjh ,ueber physikalischen ,eigenschaften dnner1 fester ,lamellen1 ,s4,b4 ,berlin4 ,akad4 #ahhh1 pp4 #ghi1 #gij4 #cji ,certain palaeontologists 7e4g4 ,haeusler and ,spandel7 have maintained that in each family or genus the plain smooth-shelled forms are the primitive and ancient ones1 and that the ribbed and otherwise ornamented shells make their appearance at later dates in the course of a definite evolution 7cf4 ,rhumbler1 ,foraminiferen der ,plankton-,expedition1 #aiaa1 i1 p4 #ba74 ,if this were true it would be of fundamental importance3 but this book of mine would not deserve to be written4 #caj ,a ,study of ,splashes1 p4 #aaf4 #caa ,see ,silliman's ,journal1 ,,ii1 p4 #agi1 #ahbj2 and cf4 ,plateau1 op4 cit4 ,,ii1 pp4 #acd1 #dfa4 #cab ,the presence or absence of the contractile vacuole or vacuoles is one of the chief distinctions1 in systematic zoology1 between the ,heliozoa and the ,radiolaria4 ,as we have seen on p4 #afe 7footnote71 it is probably no more than a physical consequence of the different conditions of existence in fresh water and in salt4 #cac ,cf4 ,doflein1 ,lehrbuch der ,protozoenkunde1 #aiaa1 p4 #dbb4 #cad ,cf4 ,minchin1 ,introduction to the ,study of the ,protozoa1 #aiad p4 #bic1 ,fig4 #abg4 #cae ,cf4 ,c4 ,a4 ,kofoid and ,olive ,swezy1 ,on ,trichomonad ,flagellates1 etc4 ,pr4 ,amer4 ,acad4 of ,arts and ,sci4 ,,li1 pp4 #bhi-cgh1 #aiae4 #caf ,d4 ,l4 ,mackinnon1 ,herpetomonads from the ,alimentary ,tract of certain ,dungflies1 ,parasitology1 ,,iii1 p4 #bfh1 #aiaj4 #cag ,proc4 ,roy4 ,soc4 ,,xii1 pp4 #bea-beg1 #ahfb-c4 #cah ,cf4 7int4 al47 ,lehmann1 ,ueber scheinbar lebende ,kristalle und ,myelinformen1 ,arch4 f4 ,entw4 ,mech4 ,,xxvi1 p4 #dhc1 #aijh2 ,ann4 d4 ,physik1 ,,xliv1 p4 #ifi1 #aiad4 #cai ,cf4 ,b4 ,moore and ,h4 ,c4 ,roaf1 ,on the ,osmotic ,equilibrium of the ,red ,blood ,corpuscle1 ,biochem4 ,journal1 ,,iii1 p4 #ee1 #aijh4 #cbj ,for an attempt to explain the form of a blood-corpuscle by surface-tension alone1 see ,rice1 ,phil4 ,mag4 ,nov4 #aiad2 but cf4 ,shorter1 ibid4 ,jan4 #aiae4 #cba ,koltzoff1 ,n4 ,k41 ,studien ber die ,gestalt der ,zelle1 ,arch4 f4 mikrosk4 ,anat4 ,,lxvii1 pp4 #cfd-ega1 #aije2 ,biol4 ,centralbl4 ,,xxiii1 pp4 #fhj-fif1 #aijc1 ,,xxvi1 pp4 #hed-hfc1 #aijf2 ,arch4 f4 ,zellforschung1 ,,ii1 pp4 #a-fe1 #aijh1 ,,vii1 pp4 #cdd-dbc1 #aiaa2 ,anat4 ,anzeiger1 ,,xli1 pp4 #ahc-bjf1 #aiab4 #cbb ,cf4 supra1 p4 #abi4 #cbc ,as ,bethe points out 7,zellgestalt1 ,plateausche ,flssigkeitstigur und ,neurofibrille1 ,anat4 ,anz4 ,,xl4 p4 #bji1 #aiaa71 the spiral fibres of which ,koltzoff speaks must lie in the surface1 and not within the substance1 of the cell whose conformation is affected by them4 #cbd ,see for a further but still elementary account1 ,michaelis1 ,dynamics of ,surfaces1 #aiad1 p4 #bb seq42 ,macallum1 ,oberflchenspannung und ,lebenserscheinungen1 in ,asher-,spiro's ,ergebnisse der ,physiologie1 ,,xi1 pp4 #eih-feh1 #aiaa2 see also ,w4 ,w4 ,taylor's ,chemistry of ,colloids1 #aiae1 p4 #bba seq41 ,wolfgang ,ostwald1 ,grundriss der ,kolloidchemie1 #aiji1 and other text-books of physical chemistry2 and ,bayliss's ,principles of ,general ,physiology1 pp4 #ed-gc1 #aiae4 #cbe ,the first instance of what we now call an adsorptive phenomenon was observed in soap-bubbles4 ,leidenfrost1 in #agef1 was aware that the outer layer of the bubble was covered by an 8oily0 layer4 ,a hundred years later ,dupr shewed that in a soap-solution the soap tends to concentrate at the surface1 so that the surface-tension of a very weak solution is very little different from that of a strong one 7,thorie mcanique de la chaleur1 #ahfi1 p4 #cgf2 cf4 ,plateau1 ,,ii1 p4 #ajj74 #cbf ,this identical phenomenon was the basis of ,quincke's theory of amoeboid movement 7,ueber periodische ,ausbreitung von ,flssigkeitsoberflchen1 etc41 ,,sb4 ,berlin4 ,akad4 #ahhh1 pp4 #gia-hjf2 cf4 ,pflger's ,archiv1 #ahgi1 p4 #acf74 #cbg ,j4 ,willard ,gibbs1 ,equilibrium of ,heterogeneous ,substances1 ,tr4 ,conn4 ,acad4 ,,iii1 pp4 #chj-djj1 #ahgf1 also in ,collected ,papers1 ,i1 pp4 #ahe-bah1 ,london1 #aijf2 ,j4 ,j4 ,thomson1 ,applications of ,dynamics to ,physics and ,chemistry1 #ahhh 7,surface tension of solutions71 p4 #aij4 ,see also 7int4 al47 the various papers by ,c4 ,m4 ,lewis1 ,phil4 ,mag4 7#f71 ,,xv1 p4 #dii1 #aijh1 ,,xvii1 p4 #dff1 #aiji1 ,zeitschr4 f4 physik4 ,chemie1 ,,lxx1 p4 #abi1 #aiaj2 ,milner1 ,phil4 ,mag4 7#f71 ,,xiii1 p4 #if1 #aijg1 etc4 #cbh ,g4 ,f4 ,fitz,gerald1 ,on the ,theory of ,muscular ,contraction1 ,brit4 ,ass4 ,rep4 #ahgh2 also in ,scientific ,writings1 ed4 ,larmor1 #aijb1 pp4 #cd1 #ge4 ,a4 d',arsonval1 ,relations entre l'lectricit animale et la tension superficielle1 ,c4 ,r4 ,,cvi1 p4 #agdj4 #ahhh2 cf4 ,a4 ,imbert1 ,le mcanisme de la contraction musculaire1 dduit de la considration des forces de tension superficielle1 ,arch4 de ,phys4 7#e71 ,,ix1 pp4 #bhi-cja1 #ahig4 #cbi ,ueber die ,natur der ,bindung der ,gase im ,blut und in seinen ,bestandtheilen1 ,kolloid4 ,zeitschr4 ,,ii1 pp4 #bfd-bgb1 #bid-cja1 #aijh2 cf4 ,loewy1 ,dissociationsspannung des ,oxyhaemoglobin im ,blut1 ,arch4 f4 ,anat4 und ,physiol4 #aijd1 p4 #bca4 #ccj ,we may trace the first steps in the study of this phenomenon to ,melsens1 who found that thin films of white of egg become firm and insoluble 7,sur les modifications apportes l'albumine 444 par l'action purement mcanique1 ,c4 ,r4 ,acad4 ,sci4 ,,xxxiii1 p4 #bdg1 #ahea72 and ,harting made similar observations about the same time4 ,ramsden has investigated the same subject1 and also the more general phenomenon of the formation of albuminoid and fatty membranes by adsorption3 cf4 ,koagulierung der ,eiweisskrper auf mechanischer ,wege1 ,arch4 f4 ,anat4 u4 ,phys4 7,phys4 ,abth47 #ahid1 p4 #eag2 ,abscheidung fester ,krper in ,oberflchenschichten ,z4 f4 phys4 ,chem4 ,,xlvii1 p4 #cda1 #aijb2 ,proc4 ,r4 ,s4 ,,lxxii1 p4 #aef1 #aijd4 ,for a general review of the whole subject see ,h4 ,zangger1 ,ueber ,membranen und ,membranfunktionen1 in ,asher-,spiro's ,ergebnisse der ,physiologie1 ,,vii1 pp4 #ii-afj1 #aijh4 #cca ,cf4 ,taylor1 ,chemistry of ,colloids1 p4 #beb4 #ccb ,strasbrger1 ,ueber ,cytoplasmastrukturen1 etc4 ,jahrb4 f4 wiss4 ,bot4 ,,xxx1 #ahig2 ,r4 ,a4 ,harper1 ,kerntheilung und freie ,zellbildung im ,ascus1 ibid42 cf4 ,wilson1 ,the ,cell in ,development1 etc4 pp4 #ec-ee4 #ccc ,cf4 ,a4 ,gurwitsch1 ,morphologie und ,biologie der ,zelle1 #aijd1 pp4 #afi-ahe2 ,meves1 ,die ,chondriosomen als ,trger erblicher ,anlagen1 ,arch4 f4 mikrosk4 ,anat4 #aijh1 p4 #gb2 ,j4 ,o4 ,w4 ,barratt1 ,changes in ,chondriosomes1 etc4 ,q4,j4,m4,s4 ,,lviii1 pp4 #eec-eff1 #aiac1 etc42 ,a4 ,mathews1 ,changes in ,structure of the ,pancreas ,cell1 etc41 ,j4 of ,morph4 ,,xv 7,suppl471 pp4 #aga-bbb1 #ahii4 #ccd ,the question whether chromosomes1 chondriosomes or chromidia be the true vehicles or transmitters of 8heredity0 is not without its analogy to the older problem of whether the pineal gland or the pituitary body were the actual seat and domicile of the soul4 #cce ,cf4 ,c4 ,c4 ,dobell1 ,chromidia and the ,binuclearity ,hypotheses2 a review and a criticism1 ,q4,j4,m4,s4 ,,liii1 #bgi-cbf1 #aiji2 ,prenant1 ,a41 ,les ,mitochondries et l',ergastoplasme1 ,journ4 de l',anat4 et de la ,physiol4 ,,xlvi1 pp4 #bag-bhe1 #aiaj 7both with copious bibliography74 #ccf ,traube in particular has maintained that in differences of surface-tension we have the origin of the active force productive of osmotic currents1 and that herein we find an explanation1 or an approach to an explanation1 of many phenomena which were formerly deemed peculiarly 8vital0 in their character4 8,die ,differenz der ,oberflchenspannungen oder der ,oberflchendruck eine ,kraft darstellt1 welche als treibende ,kraft der ,osmose1 an die ,stelle des nicht mit dem ,oberflchendruck identischen osmotischen ,druckes1 zu setzen ist1 etc40 7,oberflchendruck und seine ,bedeutung im ,organismus1 ,pflger's ,archiv1 ,,cv1 p4 #eei1 #aijd47 ,cf4 also ,hardy 7,pr4 ,phys4 ,soc4 ,,xxviii1 p4 #aaf1 #aiaf71 8,if the surface film of a colloid membrane separating two masses of fluid were to change in such a way as to lower the potential of the water in it1 water would enter the region from both sides at once4 ,but if the change of state were to be propagated as a wave of change1 starting at one face and dying out at the other face1 water would be carried along from one side of the membrane to the other4 ,a succession of such waves would maintain a flow of fluid40 #ccg ,on the ,distribution of ,potassium in animal and vegetable ,cells2 ,journ4 of ,physiol4 ,,xxxii1 p4 #ie1 #aije4 #cch ,the reader will recognise that there is a fundamental difference1 and contrast1 between such experiments as these of ,professor ,macallum's and the ordinary staining processes of the histologist4 ,the latter are 7as a general rule7 purely empirical1 while the former endeavour to reveal the true microchemistry of the cell4 8,on peut dire que la microchimie n'est encore qu' la priode d'essai1 et que l'avenir de l'histologie et spcialement de la cytologie est tout entier dans la microchimie0 7,prenant1 ,a41 ,mthodes et rsultats de la ,microchimie1 ,journ4 de l',anat4 et de la ,physiol4 ,,xlvi1 pp4 #cdc-djd1 #aiaj74 #cci ,cf4 ,macallum1 ,presidential ,address1 ,section ,i1 ,brit4 ,ass4 ,rep4 7,sheffield71 #aiaj1 p4 #gdd4 #cdj ,in accordance with a simple corollary to the ,gibbs-,thomson law4 #cda ,it can easily be proved 7by equating the increase of energy stored in an increased surface to the work done in increasing that surface71 that the tension measured per unit breadth1 ,tab1 is equal to the energy per unit area1 ,eab4 #cdb ,the presence of this little liquid 8bourrelet10 drawn from the material of which the partition-walls themselves are composed1 is obviously tending to a reduction of the internal surface-area4 ,and it may be that it is as well1 or better1 accounted for on this ground than on ,plateau's assumption that it represents a 8surface of continuity40 #cdc ,a similar 8bourrelet0 is admirably seen at the line of junction between a floating bubble and the liquid on which it floats2 in which case it constitutes a 8masse annulaire10 whose mathematical properties and relation to the form of the nearly hemispherical bubble1 have been investigated by van der ,mensbrugghe 7cf4 ,plateau1 op4 cit41 p4 #chf74 ,the form of the superficial vacuoles in ,actinophrys or ,actinosphaerium involves an identical problem4 #cdd ,in an actual calculation we must of course always take account of the tensions on both sides of each film or membrane4 #cde ,hofmeister1 ,pringsheim's ,jahrb4 ,,iii1 p4 #bgb1 #ahfc2 ,hdb4 d4 physiol4 ,bot4 ,i1 #ahfg1 p4 #abi4 #cdf ,sachs1 ,ueber die ,anordnung der ,zellen in jngsten ,pflanzentheilen1 ,verh4 phys4 med4 ,ges4 ,wrzburg1 ,,xi1 pp4 #bai-bdb1 #ahgg2 ,ueber ,zellenanordnung und ,wachsthum1 ibid4 ,,xii1 #ahgh2 ,ueber die durch ,wachsthum bedingte ,verschiebung kleinster ,theilchen in trajectorischen ,curven1 ,monatsber4 k4 ,akad4 ,wiss4 ,berlin1 #ahhj2 ,physiology of ,plants1 chap4 xxvii1 pp4 #dca-dei1 ,oxford1 #ahhg4 #cdg ,schwendener1 ,ueber den ,bau und das ,wachsthum des ,flechtenthallus1 ,naturf4 ,ges4 ,zrich1 ,febr4 #ahfj1 pp4 #bgb-bif4 #cdh ,reinke1 ,lehrbuch der ,botanik1 #ahhj1 p4 #eai4 #cdi ,cf4 ,leitgeb1 ,unters4 ber die ,lebermoose1 ,,ii1 p4 #d1 ,graz1 #ahha4 #cej ,rauber1 ,neue ,grundlegungen zur ,kenntniss der ,zelle1 ,morph4 ,jahrb4 ,,viii1 pp4 #bgi1 #ccd1 #ahhb4 #cea ,c4 ,r4 ,acad4 ,sc4 ,,xxxiii1 p4 #bdg1 #ahea2 ,ann4 de chimie et de phys4 7#c71 ,,xxxiii1 p4 #agj1 #ahea2 ,bull4 ,r4 ,acad4 ,belg4 ,,xxiv1 p4 #eca1 #aheg4 #ceb ,klebs1 ,biolog4 ,centralbl4 ,,vii1 pp4 #aic-bja1 #ahhg4 #cec ,l4 ,errera1 ,sur une condition fondamentale d'quilibre des cellules vivantes1 ,c4 ,r41 ,,ciii1 p4 #hbb1 #ahhf2 ,bull4 ,soc4 ,belge de ,microscopie1 ,,xiii1 ,oct4 #ahhf2 ,recueil d'uvres 7,physiologie gnrale71 #aiaj1 pp4 #bja-bje4 #ced ,l4 ,chabry1 ,embryologie des ,ascidiens1 ,j4 ,anat4 et ,physiol4 ,,xxiii1 p4 #bff1 #ahhg4 #cee ,robert1 ,embryologie des ,troques1 ,arch4 de ,zool4 exp4 et gn4 7#c71 ,x1 #ahib4 #cef 8,dass der ,furchungsmodus etwas fr das ,zuknftige unwesentliches ist10 ,z4 f4 w4 ,z4 ,,lv1 #ahic1 p4 #cg4 ,with this statement compare1 or contrast1 that of ,conklin1 quoted on p4 #d2 cf4 also pp4 #aeg1 #cdh 7footnotes74 #ceg de ,wildeman1 ,etudes sur l'attache des cloisons cellulaires1 ,mm4 ,couronn4 de l',acad4 ,r4 de ,belgique1 ,,liii1 #hd pp41 #ahic-d4 #ceh ,it was so termed by ,conklin in #ahig1 in his paper on ,crepidula 7,j4 of ,morph4 ,,xiii1 #ahig74 ,it is the ,querfurche of ,rabl 7,morph4 ,jahrb4 ,v1 #ahgi72 the ,polarfurche of ,o4 ,hertwig 7,jen4 ,zeitschr4 ,,xiv1 #ahhj72 the ,brechungslinie of ,rauber 7,neue ,grundlage zur ,k4 der ,zelle1 ,m4 ,jb4 ,,viii1 #ahhb74 ,it is carefully discussed by ,robert1 ,dv4 des ,troques1 ,arch4 de ,zool4 ,exp4 et ,gn4 7#c71 ,x1 #ahib1 p4 #cjg seq4 #cei ,thus ,wilson 7,j4 of ,morph4 ,,viii1 #ahie7 declared that in ,amphioxus the polar furrow was occasionally absent1 and ,driesch took occasion to criticise and to throw doubt upon the statement 7,arch4 f4 ,entw4 ,mech4 ,i1 #ahie1 p4 #dah74 #cfj ,precisely the same remark was made long ago by ,driesch3 8,das so oft sehematisch gezeichnete ,vierzellenstadium mit zwei sich in zwei ,punkten scheidende ,medianen kann man wohl getrost aus der ,reihe des ,existierenden streichen10 ,entw4 mech4 ,studien1 ,z4 f4 w4 ,z4 ,,liii1 p4 #aff1 #ahib4 ,cf4 also his ,math4 mechanische ,bedeutung morphologischer ,probleme der ,biologie1 ,jena1 #ei pp4 #ahia4 #cfa ,compare1 however1 p4 #bii4 #cfb ,ricreatione dell' occhio e della mente1 nell' ,osservatione delle ,chiocciole1 ,roma1 #afha4 #cfc ,cf4 some of ,j4 ,h4 ,vincent's photographs of ripples1 in ,phil4 ,mag4 #ahig-ahii2 or those of ,f4 ,r4 ,watson1 in ,phys4 ,review1 #ahig1 #aija1 #aiaf4 ,the appearance will depend on the rate of the wave1 and in turn on the surface-tension2 with a low tension one would probably see only a moving 8jabble40 ,fitz,gerald thought diatom-patterns might be due to electromagnetic vibrations 7,works1 p4 #ejc1 #aijb74 #cfd ,cushman1 ,j4 ,a4 and ,henderson1 ,w4 ,p41 ,amer4 ,nat4 ,,xl1 pp4 #gig-hjb1 #aijf4 #cfe ,this does not merely neglect the broken ones but all whose centres lie between this circle and a hexagon inscribed in it4 #cff ,for more detailed calculations see a paper by 8,h4,m40 8 ,h4 ,munro1 in ,q4 ,j4 ,m4 ,s4 ,,vi1 p4 #hc1 #aheh4 #cfg ,cf4 ,hartog1 ,the ,dual ,force of the ,dividing ,cell1 ,science ,progress 7n4s471 ,i1 ,oct4 #aijg1 and other papers4 ,also ,baltzer1 ,ueber mehrpolige ,mitosen bei ,seeigeleiern1 ,inaug4 ,diss4 #aijh4 #cfh ,observations sur les ,abeilles1 ,mm4 ,acad4 ,sc4 ,paris1 #agab1 p4 #bii4 #cfi ,as explained by ,leslie ,ellis1 in his essay 8,on the ,form of ,bees' ,cells10 in ,mathematical and other ,writings1 #ahec1 p4 #cec2 cf4 ,o4 ,terquem1 ,nouv4 ,ann4 ,math4 #ahef1 p4 #agh4 #cgj ,phil4 ,trans4 ,,xlii1 #agdc1 pp4 #efe-ega4 #cga ,mm4 de l',acad4 de ,berlin1 #agha4 #cgb ,cf4 ,gregory1 ,examples1 p4 #ajf1 ,wood's ,homes without ,hands1 #ahfe1 p4 #dbh1 ,mach1 ,science of ,mechanics1 #aijb1 p4 #dec1 etc41 etc4 #cgc ,origin of ,species1 ch4 ,,viii 7#fth ed41 p4 #bba74 ,the cells of various bees1 humble-bees and social wasps have been described and mathematically investigated by ,k4 ,mllenhoff1 ,pflger's ,archiv ,,xxxii1 p4 #ehi1 #ahhc2 but his many interesting results are too complex to epitomise4 ,for figures of various nests and combs see 7e4g47 von ,bttel-,reepen1 ,biol4 ,centralbl4 ,,xxxiii1 pp4 #d1 #hi1 #abi1 #ahc1 #aijc4 #cgd ,darwin had a somewhat similar idea1 though he allowed more play to the bee's instinct or conscious intention4 ,thus1 when he noticed certain half-completed cell-walls to be concave on one side and convex on the other1 but to become perfectly flat when restored for a short time to the hive1 he says3 8,it was absolutely impossible1 from the extreme thinness of the little plate1 that they could have effected this by gnawing away the convex side2 and ,i suspect that the bees in such cases stand on opposite sides and push and bend the ductile and warm wax 7which as ,i have tried is easily done7 into its proper intermediate plane1 and thus flatten it40 #cge ,since writing the above1 ,i see that ,mllenhoff gives the same explanation1 and declares that the waxen wall is actually a ,flssigkeitshutchen1 or liquid film4 #cgf ,bonnet criticised ,buffon's explanation1 on the ground that his description was incomplete2 for ,buffon took no account of the ,maraldi pyramids4 #cgg ,buffon1 ,histoire ,naturelle1 ,,iv1 p4 #ii4 ,among many other papers on the ,bee's cell1 see ,barclay1 ,mem4 ,wernerian ,soc4 ,,ii1 p4 #bei 7#ahab71 #ahah2 ,sharpe1 ,phil4 ,mag4 ,,iv1 #ahbh1 pp4 #ai-ba2 ,l4 ,lalanne1 ,ann4 ,sci4 ,nat4 7#b7 ,zool4 ,,xiii1 pp4 #ceh-cgd1 #ahdj2 ,haughton1 ,ann4 ,mag4 ,nat4 ,hist4 7#c71 ,,xi1 pp4 #dae-dbi1 #ahfc2 ,a4 ,r4 ,wallace1 ibid4 ,,xii1 p4 #cjc1 #ahfc2 ,jeffries ,wyman4 ,pr4 ,amer4 ,acad4 of ,arts and ,sc4 ,,vii1 pp4 #fh-hc1 #ahfh2 ,chauncey ,wright1 ibid4 ,,iv1 p4 #dcb1 #ahfj4 #cgh ,sir ,w4 ,thomson1 ,on the ,division of ,space with ,minimum ,partitional ,area1 ,phil4 ,mag4 7#e71 ,,xxiv1 pp4 #ejc-ead1 ,dec4 #ahhg2 cf4 ,baltimore ,lectures1 #aijd1 p4 #fae4 #cgi ,also discovered independently by ,sir ,david ,brewster1 ,trans4 ,r4,s4,e4 ,,xxiv1 p4 #eje1 #ahfg1 ,,xxv1 p4 #aae1 #ahfi4 #chj ,von ,fedorow had already described 7in ,russian7 the same figure1 under the name of cubo-octahedron1 or hepta-parallelohedron1 limited however to the case where all the faces are plane4 ,this figure1 together with the cube1 the hexagonal prism1 the rhombic dodecahedron and the 8elongated dodecahedron10 constituted the five plane-faced1 parallel-sided figures by which space is capable of being completely filled and symmetrically partitioned2 the series so forming the foundation of ,von ,fedorow's theory of crystalline structure4 ,the elongated dodecahedron is1 essentially1 the figure of the bee's cell4 #cha ,f4 ,r4 ,lillie1 ,embryology of the ,unionidae1 ,journ4 of ,morphology1 ,x1 p4 #ab1 #ahie4 #chb ,e4 ,b4 ,wilson1 ,the ,cell-lineage of ,nereis1 ,journ4 of ,morphology1 ,,vi1 p4 #deb1 #ahib4 #chc ,it is highly probable1 and we may reasonably assume1 that the two little triangles do not actually meet at an apical point1 but merge into one another by a twist1 or minute surface of complex curvature1 so as not to contravene the normal conditions of equilibrium4 #chd ,professor ,peddie has given me this interesting and important result1 but the mathematical reasoning is too lengthy to be set forth here4 #che ,cf4 ,rhumbler1 ,arch4 f4 ,entw4 ,mech4 ,,xiv1 p4 #dja1 #aijb2 ,assheton1 ibid4 ,,xxxi1 pp4 #df-gh1 #aiaj4 #chf ,m4 ,robert 7l4 c4 p4 #cje7 has compiled a long list of cases among the molluscs and the worms1 where the initial segmentation of the egg proceeds by equal or unequal division4 ,the two cases are about equally numerous4 ,but like many other writers1 he would ascribe this equality or inequality rather to a provision for the future than to a direct effect of immediate physical causation3 8,il semble assez probable1 comme on l'a dit souvent1 que la plus grande taille d'un blastomre est lie l'importance et au dveloppement prcoce des parties du corps qui doivent en natre3 il y aurait l une sorte de reflet des stades postrieures du dveloppement sur les premires phnomnes1 ce que ,m4 ,ray ,lankester appelle precocious segregation4 ,il faut avouer pourtant qu'on est parfois assez embarrass pour assigner une cause pareilles diffrences40 #chg ,the principle is well illustrated in an experiment of ,sir ,david ,brewster's 7,trans4 ,r4,s4,e4 ,,xxv1 p4 #aaa1 #ahfi74 ,a soap-film is drawn over the rim of a wine-glass1 and then covered by a watch-glass4 ,the film is inclined or shaken till it becomes attached to the glass covering1 and it then immediately changes place1 leaving its transverse position to take up that of a spherical segment extending from one side of the wine-glass to its cover1 and so enclosing the same volume of air as formerly but with a great economy of surface1 precisely as in the case of our spherical partition cutting off one corner of a cube4 #chh ,cf4 ,wildeman1 ,attache des ,cloisons1 etc41 pls4 #a1 #b4 #chi ,nova ,acta ,k4 ,leop4 ,akad4 ,,xi1 #a1 pl4 ,,iv4 #cij ,cf4 ,protoplasmamechanik1 p4 #bbi3 8,insofern liegen also die ,verhltnisse hier wesentlich anders als bei der ,zertheilung hohler ,krperformen durch flssige ,lamellen4 ,wenn die ,membran bei der ,zelltheilung die von dem ,prinzip der kleinsten ,flchen geforderte ,lage und ,krmmung annimmt1 so werden wir den ,grund dafr in andrer ,weise abzuleiten haben40 #cia ,there is1 ,i think1 some ambiguity or disagreement among botanists as to the use of this latter term3 the sense in which ,i am using it1 viz4 for any partition which meets the outer or peripheral wall at right angles 7the strictly radial partition being for the present excluded71 is1 however1 clear4 #cib ,cit4 ,plateau1 ,statique des ,liquides1 i1 p4 #ceh4 #cic ,even in a ,protozoon 7,euglena viridis71 when kept alive under artificial compression1 ,ryder found a process of cell-division to occur which he compares to the segmenting blastoderm of a fish's egg1 and which corresponds in its essential features with that here described4 ,contrib4 ,zool4 ,lab4 ,univ4 ,pennsylvania1 ,i1 pp4 #cg-ej1 #ahic4 #cid ,this1 like many similar figures1 is manifestly drawn under the influence of ,sachs's theoretical views1 or assumptions1 regarding orthogonal trajectories1 coaxial circles1 confocal ellipses1 etc4 #cie ,such preconceptions as ,rauber entertained were all in a direction likely to lead him away from such phenomena as he has faithfully depicted4 ,rauber had no idea whatsoever of the principles by which we are guided in this discussion1 nor does he introduce at all the analogy of surface-tension1 or any other purely physical concept4 ,but he was deeply under the influence of ,sachs's rule of rectangular intersection2 and he was accordingly disposed to look upon the configuration represented above in ,fig4 #afh1 #f1 as the most typical or most primitive4 #cif ,cf4 ,rauber1 ,neue ,grundlage z4 ,k4 der ,zelle1 ,morph4 ,jahrb4 ,,viii1 #ahhc1 pp4 #bgc1 #bgd3 8,ich betone noch1 dass unter meinen ,figuren diejenige gar nicht enthalten ist1 welche zum ,typus der ,batrachierfurchung gehrig am meisten bekannt ist 4444 ,es haben so ausgezeichnete ,beobachter sie als vorhanden beschrieben1 dass es mir nicht einfallen kann1 sie berhaupt nicht anzuerkennen40 #cig ,roux's experiments were performed with drops of paraffin suspended in dilute alcohol1 to which a little calcium acetate was added to form a soapy pellicle over the drops and prevent them from reuniting with one another4 #cih ,cf4 7e4g47 ,clerk ,maxwell1 ,on ,reciprocal ,figures1 etc41 ,trans4 ,r4 ,s4 ,e4 ,,xxvi1 p4 #i1 #ahgj4 #cii ,see ,greville1 ,k4 ,r41 ,monograph of the ,genus ,asterolampra1 ,q4,j4,m4,s4 ,,viii1 7,trans471 pp4 #ajb-abd1 #ahfj2 cf4 ,,ibid4 7n4s471 ,,ii1 pp4 #da-ee1 #ahfb4 #djj ,the same is true of the insect's wing2 but in this case ,i do not hazard a conjectural explanation4 #dja ,ann4 ,mag4 ,n4 ,h4 7#b71 ,,iii1 p4 #abf1 #ahdi4 #djb ,phil4 ,trans4 ,,clvii1 pp4 #fdc-fef1 #ahfg4 #djc ,sachs1 ,pflanzenphysiologie 7,vorlesung ,,xxiv71 #ahhb2 cf4 ,rauber1 ,neue ,grundlage zur ,kenntniss der ,zelle1 ,morphol4 ,jahrb4 ,,viii1 p4 #cjc seq41 #ahhc2 ,e4 ,b4 ,wilson1 ,cell-lineage of ,nereis1 ,journ4 of ,morphology1 ,,vi1 p4 #ddh1 #ahib1 etc4 #djd ,in the following account ,i follow closely on the lines laid down by ,berthold2 ,protoplasmamechanik1 cap4 vii4 ,many botanical phenomena identical and similar to those here dealt with1 are elaborately discussed by ,sachs in his ,physiology of ,plants 7chap4 xxvii1 pp4 #dca-dei1 ,oxford1 #ahhg72 and in his earlier papers1 ,ueber die ,anordnung der ,zellen in jngsten ,pflanzentheilen1 and ,ueber ,zellenanordnung und ,wachsthum 7,arb4 d4 botan4 ,inst4 ,wrzburg1 #ahgh1 #ahgi74 ,but ,sachs's treatment differs entirely from that which ,i adopt and advocate here3 his explanations being based on his 8law0 of rectangular succession1 and involving complicated systems of confocal conics1 with their orthogonally intersecting ellipses and hyperbolas4 #dje ,cf4 p4 #cfi4 #djf ,there is much information regarding the chemical composition and mineralogical structure of shells and other organic products in ,h4 ,c4 ,sorby's ,presidential ,address to the ,geological ,society 7,proc4 ,geol4 ,soc4 #ahgi1 pp4 #ef-ic72 but ,sorby failed to recognise that association with 8organic0 matter1 or with colloid matter whether living or dead1 introduced a new series of purely physical phenomena4 #djg ,vesque1 ,ann4 des ,sc4 ,nat4 7,bot47 7#e71 ,,xix1 p4 #caj1 #ahgd4 #djh ,cf4 ,klliker1 ,icones ,histiologicae1 #ahfd1 pp4 #aai1 etc4 #dji ,in an interesting paper by ,irvine and ,sims ,woodhead on the 8,secretion of ,carbonate of ,lime by ,animals0 7,proc4 ,r4 ,s4 ,e4 ,,xvi1 #ahhi1 p4 #cea7 it is asserted that 8lime salts1 of whatever form1 are deposited only in vitally inactive tissue40 #daj ,the tube of ,teredo shews no trace of organic matter1 but consists of irregular prismatic crystals3 the whole structure 8being identical with that of small veins of calcite1 such as are seen in thin sections of rocks0 7,sorby1 ,proc4 ,geol4 ,soc4 #ahgi1 p4 #eh74 ,this1 then1 would seem to be a somewhat exceptional case of a shell laid down completely outside of the animal's external layer of organic or colloid substance4 #daa ,c4 ,r4 ,soc4 ,biol4 ,paris 7#i71 ,i1 pp4 #ag-bj1 #ahhi2 ,c4 ,r4 ,ac4 ,sc4 ,,cviii1 pp4 #aif-h1 #ahhi4 #dab ,cf4 ,heron-,allen1 ,phil4 ,trans4 7,b71 vol4 ,,ccvi1 p4 #bfb1 #aiae4 #dac ,see ,leduc1 ,mechanism of ,life 7#aiaa71 ch4 ,x1 for copious references to other works on the artificial production of 8organic0 forms4 #dad ,lectures on the ,molecular ,asymmetry of ,natural ,organic ,compounds1 ,chemical ,soc4 of ,paris1 #ahfj1 and also in ,ostwald's ,klassiker d4 ex4 ,wiss4 ,no4 #bh1 and in ,alembic ,club ,reprints1 ,no4 #ad1 ,edinburgh1 #ahig2 cf4 ,richardson1 ,g4 ,m41 ,foundations of ,stereochemistry1 ,n4 ,y4 #aija4 #dae ,japp1 ,stereometry and ,vitalism1 ,brit4 ,ass4 ,rep4 7,bristol71 p4 #hac1 #ahih2 cf4 also a voluminous discussion in ,nature1 #ahih-i4 #daf ,they represent the general theorem of which particular cases are found1 for instance1 in the asymmetry of the ferments 7or enzymes7 which act upon asymmetrical bodies1 the one fitting the other1 according to ,emil ,fischer's well-known phrase1 as lock and key4 ,cf4 his ,bedeutung der ,stereochemie fr die ,physiologie1 ,z4 f4 physiol4 ,chemie1 ,v1 p4 #fj1 #ahii1 and various papers in the ,ber4 d4 d4 chem4 ,ges4 from #ahid4 #dag ,in accordance with ,emil ,fischer's conception of 8asymmetric synthesis10 it is now held to be more likely that the process is synthetic than analytic3 more likely1 that is to say1 that the plant builds up from the first one asymmetric body to the exclusion of the other1 than that it 8selects0 or 8picks out0 7as ,japp supposed7 the right-handed or the left-handed molecules from an original1 optically inactive1 mixture of the two2 cf4 ,a4 ,mc,kenzie1 ,studies in ,asymmetric ,synthesis1 ,journ4 ,chem4 ,soc4 7,trans471 ,,lxxxv1 p4 #abdi1 #aijd4 #dah ,see for a fuller discussion1 ,hans ,przibram1 ,vitalitt1 #aiac1 ,kap4 iv1 ,stoffwechsel 7,assimilation und ,katalyse74 #dai ,cf4 ,cotton1 ,ann4 de ,chim4 et de ,phys4 7#g71 ,,viii1 pp4 #cdg-dcb 7cf4 p4 #cgc71 #ahif4 #dbj ,byk1 ,a41 ,zur ,frage der ,spaltbarkeit von ,razemverbindungen durch ,zirkularpolarisiertes ,licht1 ein ,beitrag zur primren ,entstehung optisch-activer ,substanzen1 ,zeitsch4 f4 physikal4 ,chemie1 ,,xlix1 p4 #fda1 #aijd4 ,it must be admitted that further positive evidence on these lines is still awanting4 #dba ,cf4 7int4 al47 ,emil ,fischer1 ,untersuchungen ber ,aminosuren1 ,proteine1 etc4 ,berlin1 #aijf4 #dbb ,japp1 l4 c4 p4 #hbh4 #dbc ,rainey1 ,g41 ,on the ,elementary ,formation of the ,skeletons of ,animals1 and other ,hard ,structures formed in connection with ,living ,tissue1 ,brit4 ,for4 ,med4 ,ch4 ,rev4 ,,xx1 pp4 #dea-dgf1 #aheg2 published separately with additions1 #hvo4 ,london1 #aheh4 ,for other papers by ,rainey on kindred subjects see ,q4 ,j4 ,m4 ,s4 ,,vi 7,tr4 ,microsc4 ,soc471 pp4 #da-ej1 #aheh1 ,,vii1 pp4 #bab-bbe1 #ahei1 ,,viii1 pp4 #a-aj1 #ahfj1 ,i 7n4 s471 pp4 #bc-cb1 #ahfa4 ,cf4 also ,ord1 ,w4 ,m41 ,on ,molecular ,coalescence1 and on the influence exercised by ,colloids upon the ,forms of ,inorganic ,matter1 ,q4 ,j4 ,m4 ,s4 ,,xii1 pp4 #bai-bci1 #ahgb2 and also the early but still interesting observations of ,mr ,charles ,hatchett1 ,chemical ,experiments on ,zoophytes2 with some observations on the component parts of ,membrane1 ,phil4 ,trans4 #ahjj4 pp4 #cbg-djb4 #dbd ,cf4 ,quincke1 ,ueber unsichtbare ,flssigkeitsschichten1 ,ann4 der ,physik1 #aijb4 #dbe ,see for instance other excellent illustrations in ,carpenter's article 8,shell10 in ,todd's ,cyclopdia1 vol4 ,,iv4 pp4 #eej-ega1 #ahdg-di4 ,according to ,carpenter1 the shells of the mollusca 7and also of the crustacea7 are 8essentially composed of cells1 consolidated by a deposit of carbonate of lime in their interior40 ,that is to say1 ,carpenter supposed that the spherulites1 or calcospherites of ,harting1 were1 to begin with1 just so many living protoplasmic cells4 ,soon afterwards however1 ,huxley pointed out that the mode of formation1 while at first sight 8irresistibly suggesting a cellular structure1 444 is in reality nothing of the kind10 but 8is simply the result of the concretionary manner in which the calcareous matter is deposited02 ibid4 art4 8,tegumentary ,organs10 vol4 ,v1 p4 #dhg1 #ahei4 ,quekett 7,lectures on ,histology1 vol4 ,,ii1 p4 #cic1 #ahed1 and ,q4 ,j4 ,m4 ,s4 ,,xi1 pp4 #ie-ajd1 #ahfc7 supported ,carpenter2 but ,williamson 7,histological ,features in the ,shells of the ,crustacea1 ,q4 ,j4 ,m4 ,s4 ,,viii1 pp4 #ce-dg1 #ahfj7 amply confirmed ,huxley's view1 which in the end ,carpenter himself adopted 7,the ,microscope1 #ahfb1 p4 #fjd74 ,a like controversy arose later in regard to corals4 ,mrs ,gordon 7,m4 ,m4 ,ogilvie7 asserted that the coral was built up 8of successive layers of calcified cells1 which hang together at first by their cell-walls1 and ultimately1 as crystalline changes continue1 form the individual laminae of the skeletal structures0 7,phil4 ,trans4 ,,clxxxvii1 p4 #ajb1 #ahif73 whereas v4 ,koch had figured the coral as formed out of a mass of 8,kalkconcremente0 or 8crystalline spheroids10 laid down outside the ectoderm1 and precisely similar both in their early rounded and later polygonal stages 7though von ,koch was not aware of the fact7 to the calcospherites of ,harting 7,entw4 d4 ,kalkskelettes von ,asteroides1 ,mitth4 ,zool4 ,st4 ,neapel1 ,,iii1 pp4 #bhd-bij1 pl4 ,,xx1 #ahhb74 ,lastly ,duerden shewed that external to1 and apparently secreted by the ectoderm lies a homogeneous organic matrix or membrane1 8in which the minute calcareous crystals forming the skeleton are laid down0 7,the ,coral ,siderastraea radians1 etc41 ,carnegie ,inst4 ,washington1 #aijd1 p4 #cd74 ,cf4 also ,m4 ,m4 ,ogilvie-,gordon1 ,q4 ,j4 ,m4 ,s4 ,,xlix1 p4 #bjc1 #aije1 etc4 #dbf ,cf4 ,claparde1 ,z4 f4 w4 ,z4 ,,xix1 p4 #fjd1 #ahfi4 #dbg ,spicules extremely like those of the ,alcyonaria occur also in a few sponges2 cf4 7e4g471 ,vaughan ,jennings1 ,journ4 ,linn4 ,soc4 ,,xxiii1 p4 #eca1 pl4 #ac1 fig4 #h1 #ahia4 #dbh ,mem4 ,manchester ,lit4 and ,phil4 ,soc4 ,,lx1 p4 #aa1 #aiaf4 #dbi ,mummery1 ,j4 ,h41 ,on ,calcification in ,enamel and ,dentine1 ,phil4 ,trans4 ,,ccv 7,b71 pp4 #ie-aaa1 #aiad4 #dcj ,the artificial concretion represented in ,fig4 #bjb is identical in appearance with the concretions found in the kidney of ,nautilus1 as figured by ,willey 7,zoological ,results1 p4 lxxvi1 ,fig4 #b1 #aijb74 #dca ,cf4 ,taylor's ,chemistry of ,colloids1 p4 #ah1 etc41 #aiae4 #dcb ,this rule1 undreamed of by ,errera1 supports and justifies the cardinal assumption 7of which we have had so much to say in discussing the forms of cells and tissues7 that the incipient cell-wall behaves as1 and indeed actually is1 a liquid film 7cf4 p4 #cjf74 #dcc ,cf4 p4 #bed4 #dcd ,cf4 ,harting1 op4 cit41 pp4 #bb1 #ej3 8,j'avais cru d'abord que ces couches concentriques taient produites par l'alternance de la chaleur ou de la lumire1 pendant le jour et la nuit4 ,mais l'exprience1 expressment institue pour examiner cette question1 y a rpondu ngativement40 #dce ,liesegang1 ,r4 ,e41 ,ueber die ,schichtungen bei ,diffusionen1 ,leipzig1 #aijg1 and other earlier papers4 #dcf ,cf4 ,taylor's ,chemistry of ,colloids1 pp4 #adf-adh1 #aiae4 #dcg ,cf4 ,s4 ,c4 ,bradford1 ,the ,liesegang ,phenomenon and ,concretionary ,structure in ,rocks1 ,nature1 ,,xcvii1 p4 #hj1 #aiaf2 cf4 ,sci4 ,progress1 ,x1 p4 #cfi1 #aiaf4 #dch ,cf4 ,faraday1 ,on ,ice of ,irregular ,fusibility1 ,phil4 ,trans41 #aheh1 p4 #bbh2 ,researches in ,chemistry1 etc41 #ahei1 p4 #cgd2 ,tyndall1 ,forms of ,water1 p4 #agh1 #ahgb2 ,tomlinson1 ,c41 ,on some effects of small ,quantities of ,foreign ,matter on ,crystallisation1 ,phil4 ,mag4 7#e7 ,,xxxi1 p4 #cic1 #ahia1 and other papers4 #dci ,a ,study in ,crystallisation1 ,j4 of ,soc4 of ,chem4 ,industry1 ,,xxv1 p4 #adc1 #aijf4 #ddj ,ueber ,zonenbildung in kolloidalen ,medien1 ,jena1 #aiac4 #dda ,verh4 d4 d4 ,zool4 ,gesellsch4 p4 #agi1 #aiab4 #ddb ,descent of ,man1 ,,ii1 pp4 #acb-aec1 #ahga4 #ddc ,as a matter of fact1 the phenomena associated with the development of an 8ocellus0 are or may be of great complexity1 inasmuch as they involve not only a graded distribution of pigment1 but also1 in 8optical0 coloration1 a symmetrical distribution of structure or form4 ,the subject therefore deserves very careful discussion1 such as ,bateson gives to it 7,variation1 chap4 xii74 ,this1 by the way1 is one of the very rare cases in which ,bateson appears inclined to suggest a purely physical explanation of an organic phenomenon3 8,the suggestion is strong that the whole series of rings 7in ,morpho7 may have been formed by some one central disturbance1 somewhat as a series of concentric waves may be formed by the splash of a stone thrown into a pool1 etc40 #ddd ,cf4 also ,sir ,d4 ,brewster1 ,on optical properties of ,mother of ,pearl1 ,phil4 ,trans4 #ahad1 p4 #cig4 #dde ,biedermann1 ,w41 ,ueber die ,bedeutung von ,kristallisationsprozessen der ,skelette wirbelloser ,thiere1 namentlich der ,molluskenschalen1 ,z4 f4 allg4 ,physiol4 ,i1 p4 #aed1 #aijb2 ,ueber ,bau und ,entstehung der ,molluskenschale1 ,jen4 ,zeitschr4 ,,xxxvi1 pp4 #a-afd1 #aijb4 ,cf4 also ,steinmann1 ,ueber ,schale und ,kalksteinbildungen1 ,ber4 ,naturf4 ,ges4 ,freiburg i4 ,br ,,iv1 #ahhi2 ,liesegang1 ,naturw4 ,wochenschr4 p4 #fda1 #aiaj4 #ddf ,cf4 ,btschli1 ,ueber die ,herstellung knstlicher ,strkekrner oder von ,sphrokrystallen der ,strke1 ,verh4 nat4 med4 ,ver4 ,heidelberg1 ,v1 pp4 #deg-dgb1 #ahif4 #ddg ,untersuchungen ber die ,strkekrner1 ,jena1 #aije4 #ddh ,cf4 ,winge1 ,meddel4 fra ,komm4 for ,havundersgelse 7,fiskeri71 ,,iv1 p4 #bj1 ,copenhagen1 #aiae4 #ddi ,the anhydrite is sulphate of lime 7,ca,s,o#d72 the polyhalite is a triple sulphate of lime1 magnesia and potash 7#b ,ca,s,o#d4 ,mg,s,o#d4 ,k#b,,so#d #b ,h#b,o74 #dej ,cf4 van't ,hoff1 ,physical ,chemistry in the ,service of the ,sciences1 p4 #ii seq4 ,chicago1 #aijc4 #dea ,sphrocrystalle von ,kalkoxalat bei ,kakteen1 ,ber4 d4 d4 ,bot4 ,gesellsch4 p4 #agh1 #ahhe4 #deb ,pauli1 ,w4 u4 ,samec1 ,m41 ,ueber ,lslichkeitsbeeinflssung von ,elektrolyten durch ,eiweisskrper1 ,biochem4 ,zeitschr4 ,,xvii1 p4 #bce1 #aiaj4 ,some of these results were known much earlier2 cf4 ,fokker in ,pflger's ,archiv1 ,,vii1 p4 #bgd1 #ahgc2 also ,irvine and ,sims ,woodhead1 op4 cit4 p4 #cdg4 #dec ,which1 in #ajjj parts of ash1 contains about #hdj parts of phosphate and #gf parts of calcium carbonate4 #ded ,cf4 ,dreyer1 ,fr41 ,die ,principien der ,gerstbildung bei ,rhizopoden1 ,spongien und ,echinodermen1 ,jen4 ,zeitschr4 ,,xxvi1 pp4 #bjd-dfh1 #ahib4 #dee ,in an anomalous and very remarkable ,australian sponge1 just described by ,professor ,dendy 7,nature1 ,may #ah1 #aiaf1 p4 #bec7 under the name of ,collosclerophora1 the spicules are 8gelatinous10 consisting of a gel of colloid silica with a high percentage of water4 ,it is not stated whether an organic colloid is present together with the silica4 ,these gelatinous spicules arise as exudations on the outer surface of cells1 and come to lie in intercellular spaces or vesicles4 #def ,lister1 in ,willey's ,zoological ,results1 pt ,,iv1 p4 #dei1 #aijj4 #deg ,the peculiar spicules of ,astrosclera are now said to consist of spherules1 or calcospherites1 of aragonite1 spores of a certain red seaweed forming the nuclei1 or starting-points1 of the concretions 7,r4 ,kirkpatrick1 ,proc4 ,r4 ,s4 ,,lxxxiv 7,b71 p4 #egi1 #aiaa74 #deh ,see for instance the plates in ,thel's ,monograph of the ,challenger ,holothuroidea2 also ,sollas's ,tetractinellida1 p4 lxi4 #dei ,for very numerous illustrations of the triradiate and quadriradiate spicules of the calcareous sponges1 see 7int4 al471 papers by ,dendy 7,q4 ,j4 ,m4 ,s4 ,,xxxv1 #ahic71 ,minchin 7,p4 ,z4 ,s4 #aijd71 ,jenkin 7,p4 ,z4 ,s4 #aijh71 etc4 #dfj ,cf4 again ,bnard's ,tourbillons cellulaires1 ,ann4 de ,chimie1 #aija1 p4 #hd4 #dfa ,lger1 ,stolc and others1 in ,doflein's ,lehrbuch d4 ,protozoenkunde1 #aiaa1 p4 #iab4 #dfb ,see1 for instance1 the figures of the segmenting egg of ,synapta 7after ,selenka71 in ,korschelt and ,heider's ,vergleichende ,entwicklungsgeschichte 7,allgem4 ,th41 #cte ,lief471 p4 #ai1 #aiji4 ,on the spiral type of segmentation as a secondary derivative1 due to mechanical causes1 of the 8radial0 type of segmentation1 see ,e4 ,b4 ,wilson1 ,cell-lineage of ,nereis1 ,journ4 of ,morphology1 ,,vi1 p4 #dej1 #ahib4 #dfc ,korschelt and ,heider1 p4 #af4 #dfd ,chall4 ,rep4 ,hexactinellida1 pls4 xvi1 liii1 lxxvi1 lxxxviii4 #dfe 8,hierbei nahm der kohlensaure ,kalk eine halb-krystallinische ,beschaffenheit an1 und gestaltete sich unter ,aufnahme von ,krystallwasser und in ,verbindung mit einer geringen ,quantitt von organischer ,substanz zu jenen individuellen1 festen ,krpern1 welche durch die natrliche ,zchtung als ,spicula zur ,skeletbildung bentzt1 und spterhin durch die ,wechselwirkung von ,anpassung und ,vererbung im ,kampfe ums ,dasein auf das ,vielfltigste umgebildet und differenziert wurden40 ,die ,kalkschwmme1 ,i1 p4 #cgg1 #ahgb2 cf4 also pp4 #dhb1 #dhc4 #dff ,op4 cit4 p4 #dhc4 8,die geordnete1 oft so sehr regelmssige und zierliche ,zusammensetzung des ,skeletsystems ist zum grssten ,theile unmittelbares ,product der ,wasserstrmung2 die characteristische ,lagerung der ,spicula ist von der constanten ,richtung des ,wasserstroms hervorgebracht2 zum kleinsten ,theile ist sie die ,folge von ,anpassungen an untergeordnete ussere ,existenzbedingungen40 #dfg ,materials for a ,monograph of the ,ascones1 ,q4 ,j4 ,m4 ,s4 ,,xl4 pp4 #dfi-ehg1 #ahih4 #dfh ,haeckel1 in his ,challenger ,monograph1 p4 clxxxviii 7#ahhg7 estimated the number of known forms at #dcad species1 included in #gci genera4 ,of these1 #cejh species were described for the first time in that work4 #dfi ,cf4 ,gamble1 ,radiolaria 7,lankester's ,treatise on ,zoology71 vol4 ,i1 p4 #aca1 #aiji4 ,cf4 also papers by ,v4 ,hcker1 in ,jen4 ,zeitschr4 ,,xxxix1 p4 #eha1 #aije1 ,z4 f4 wiss4 ,zool4 ,,lxxxiii1 p4 #ccf1 #aije1 ,arch4 f4 ,protistenkunde1 ,,ix1 p4 #aci1 #aijg1 etc4 #dgj ,btschli1 ,ueber die chemische ,natur der ,skeletsubstanz der ,acantharia1 ,zool4 ,anz4 ,,xxx1 p4 #ghd1 #aijf4 #dga ,for figures of these crystals see ,brandt1 ,f4 u4 ,fl4 d4 ,golfes von ,neapel1 ,,xiii1 ,radiolaria1 #ahhe1 pl4 v4 ,cf4 ,j4 ,mller1 ,ueber die ,thalassicollen1 etc4 ,abh4 ,k4 ,akad4 ,wiss4 ,berlin1 #aheh4 #dgb ,celestine1 or celestite1 is ,sr,s,o#d with some ,ba,o replacing ,sr,o4 #dgc ,with the colloid chemists1 we may adopt 7as ,rhumbler has done7 the terms spumoid or emulsoid to denote an agglomeration of fluid-filled vesicles1 restricting the name froth to such vesicles when filled with air or some other gas4 #dgd ,cf4 ,koltzoff1 ,zur ,frage der ,zellgestalt1 ,anat4 ,anzeiger1 ,,xli1 p4 #aij1 #aiab4 #dge ,mm4 de l',acad4 des ,sci41 ,st4 ,ptersbourg1 ,,xii1 ,nr4 #aj1 #aijb4 #dgf ,the manner in which the minute spicules of ,raphidiophrys arrange themselves round the bases of the pseudopodial rays is a similar phenomenon4 #dgg ,rhumbler1 ,physikalische ,analyse von ,lebenserscheinungen der ,zelle1 ,arch4 f4 ,entw4 ,mech4 ,,vii1 p4 #ajc1 #ahih4 #dgh ,the whole phenomenon is described by biologists as a 8surprising exhibition of constructive and selective activity10 and is ascribed1 in varying phraseology1 to intelligence1 skill1 purpose1 psychical activity1 or 8microscopic mentality03 that is to say1 to ,galen's 1 or 8artistic creativeness0 7cf4 ,brock's ,galen1 #aiaf1 p4 xxix74 ,cf4 ,carpenter1 ,mental ,physiology1 #ahgd1 p4 #da2 ,norman1 ,architectural achievements of ,little ,masons1 etc41 ,ann4 ,mag4 ,nat4 ,hist4 7#e71 ,i1 p4 #bhd1 #ahgh2 ,heron-,allen1 ,contributions 444 to the ,study of the ,foraminifera1 ,phil4 ,trans4 7,b71 ,,ccvi1 pp4 #bbg-bgi1 #aiae2 ,theory and ,phenomena of ,purpose and ,intelligence exhibited by the ,protozoa1 as illustrated by selection and behaviour in the ,foraminifera1 ,journ4 ,r4 ,microscop4 ,soc4 pp4 #edg-eeg1 #aiae2 ibid41 pp4 #acg-adj1 #aiaf4 ,prof4 ,j4 ,a4 ,thomson 7,new ,statesman1 ,oct4 #bc1 #aiae7 describes a certain little foraminifer1 whose protoplasmic body is overlaid by a crust of sponge-spicules1 as 8a psycho-physical individuality whose experiments in self-expression include a masterly treatment of sponge-spicules1 and illustrate that organic skill which came before the dawn of ,art40 ,sir ,ray ,lankester finds it 8not difficult to conceive of the existence of a mechanism in the protoplasm of the ,protozoa which selects and rejects building-material1 and determines the shapes of the structures built1 comparable to that mechanism which is assumed to exist in the nervous system of insects and other animals which #8automatically' go through wonderfully elaborate series of complicated actions40 ,and he agrees with 8,darwin and others who have attributed the building up of these inherited mechanisms to the age-long action of ,natural ,selection1 and the survival of those individuals possessing qualities or #8tricks' of life-saving value10 ,j4 ,r4 ,microsc4 ,soc4 ,april1 #aiaf1 p4 #acf4 #dgi ,rhumbler1 ,das ,protoplasma als physikalisches ,system1 ,jena1 p4 #eia1 #aiad2 also in ,arch4 f4 ,entwickelungsmech4 ,,vii1 pp4 #bgi-cce1 #ahih4 #dhj ,verworn1 ,psycho-physiologische ,protisten-,studien1 ,jena1 #ahhi 7#bai pp474 #dha ,leidy1 ,j41 ,fresh-water ,rhizopods of ,n4 ,america1 #ahgi1 p4 #bfb1 pl4 xli1 figs4 #aa1 #ab4 #dhb ,carnoy1 ,biologie ,cellulaire1 p4 #bdd1 fig4 #ajh2 cf4 ,dreyer1 op4 cit4 #ahib1 fig4 #ahe4 #dhc ,in all these latter cases we recognise a relation to1 or extension of1 the principle of ,plateau's bourrelet1 or van der ,mensbrugghe's masse annulaire1 of which we have already spoken 7p4 #big74 #dhd ,apart from the fact that the apex of each pyramid is interrupted1 or truncated1 by the presence of the little central cell1 it is also possible that the solid angles are not precisely equivalent to those of ,maraldi's pyramids1 owing to the fact that there is a certain amount of distortion1 or axial asymmetry1 in the ,nassellarian system4 ,in other words 7to judge from ,haeckel's figures71 the tetrahedral symmetry in ,nassellaria is not absolutely regular1 but has a main axis about which three of the trihedral pyramids are symmetrical1 the fourth having its solid angle somewhat diminished4 #dhe ,cf4 ,faraday's beautiful experiments1 ,on the ,moving ,groups of ,particles found on ,vibrating ,elastic ,surfaces1 etc41 ,phil4 ,trans4 #ahca1 p4 #bii2 ,researches in ,chem4 and ,phys4 #ahei1 pp4 #cad-ceh4 #dhf ,we need not go so far as to suppose that the external layer of cells wholly lacked the power of secreting a skeleton4 ,in many of the ,nassellariae figured by ,haeckel 7for there are many variant forms or species besides that represented here71 the skeleton of the partition-walls is very slightly and scantily developed4 ,in such a case1 if we imagine its few and scanty strands to be broken away1 the central tetrahedral figure would be set free1 and would have all the appearance of a complete and independent structure4 #dhg ,the 8bourrelet0 is not only1 as ,plateau expresses it1 a 8surface of continuity10 but we also recognise that it tends 7so far as material is available for its production7 to further lessen the free surface-area4 ,on its relation to vapour-pressure and to the stability of foam1 see ,fitz,gerald's interesting note in ,nature1 ,feb4 #a1 #ahid 7,works1 p4 #cji74 #dhh ,of the many thousand figures in the hundred and forty plates of this beautifully illustrated book1 there is scarcely one which does not depict1 now patently1 now in pregnant suggestion1 some subtle and elegant geometrical configuration4 #dhi ,they were known 7of course7 long before ,plato3 4 #dij ,if the equation of any plane face of a crystal be written in the form h x k y l z #a1 then h1 k1 l are the indices of which we are speaking4 ,they are the reciprocals of the parameters1 or reciprocals of the distances from the origin at which the plane meets the several axes4 ,in the case of the regular or pentagonal dodecahedron these indices are #b1 #a #e1 #j4 ,kepler described as follows1 briefly but adequately1 the common characteristics of the dodecahedron and icosahedron3 8,duo sunt corpora regularia1 dodecaedron et icosaedron1 quorum illud quinquangulis figuratur expresse1 hoc triangulis quidem sed in quinquanguli formam coaptatis4 ,utriusque horum corporum ipsiusque adeo quinquanguli structura perfici non potest sine proportione illa1 quam hodierni geometrae divinam appellant0 7,de nive sexangula 7#afaa71 ,opera1 ed4 ,frisch1 ,,vii1 p4 #gbc74 ,here ,kepler was dealing1 somewhat after the manner of ,sir ,thomas ,browne1 with the mysteries of the quincunx1 and also of the hexagon2 and was seeking for an explanation of the mysterious or even mystical beauty of the #e-petalled or #c-petalled flower1--pulchritudinis aut proprietatis figurae1 quae animam harum plantarum characterisavit4 #dia ,cf4 ,tutton1 ,crystallography1 p4 #icb1 #aiaa4 #dib ,however1 we can often recognise1 in a small artery for instance1 that the so-called 8circular0 fibres tend to take a slightly oblique1 or spiral1 course4 #dic ,the spiral fibres1 or a large portion of them1 constitute what ,searle called 8the rope of the heart0 7,todd's ,cyclopaedia1 ,,ii1 p4 #fba1 #ahcf74 ,the 8twisted sinews of the heart0 were known to early anatomists1 and have been frequently and elaborately studied3 for instance1 by ,gerdy 7,bull4 ,fac4 ,med4 ,paris1 #ahbj1 pp4 #dj-adh71 and by ,pettigrew 7,phil4 ,trans4 #ahfd71 and of late by ,j4 ,b4 ,macallum 7,johns ,hopkins ,hospital ,report1 ,,ix1 #aijj7 and by ,franklin ,p4 ,mall 7,amer4 ,j4 of ,anat4 ,,xi1 #aiaa74 #did ,cf4 ,btschli1 8,protozoa10 in ,bronn's ,thierreich1 ,,ii1 p4 #hdh1 ,,iii1 p4 #aghe1 etc41 #ahhc-hg2 ,jennings1 ,amer4 ,nat4 ,,xxxv1 p4 #cfi1 #aija2 ,ptter1 ,thigmotaxie bei ,protisten1 ,arch4 f4 ,anat4 u4 ,phys4 7,phys4 ,abth4 ,suppl471 pp4 #bdc-cjb1 #aijj4 #die ,a great number of spiral forms1 both organic and artificial1 are described and beautifully illustrated in ,sir ,t4 ,a4 ,cook's ,curves of ,life1 #aiad1 and ,spirals in ,nature and ,art1 #aijc4 #dif ,cf4 ,vines1 ,the ,history of the ,scorpioid ,cyme1 ,journ4 of ,botany 7n4s471 ,x1 pp4 #c-i1 #ahha4 #dig ,leslie's ,geometry of ,curved ,lines1 p4 #dag1 #ahba4 ,this is practically identical with ,archimedes' own definition 7ed4 ,torelli1 p4 #bai72 cf4 ,cantor1 ,geschichte der ,mathematik1 ,i1 p4 #bfb1 #ahhj4 #dih ,see an interesting paper by ,whitworth1 ,w4 ,a41 8,the ,equiangular ,spiral1 its chief properties proved geometrically10 in the ,messenger of ,mathematics 7#a71 ,i1 p4 #e1 #ahfb4 #dii ,i am well aware that the debt of ,greek science to ,egypt and the ,east is vigorously denied by many scholars1 some of whom go so far as to believe that the ,egyptians never had any science1 save only some 8rough rules of thumb for measuring fields and pyramids0 7,burnet's ,greek ,philosophy1 #aiad1 p4 #e74 #ejj ,euclid 7,,ii1 def4 #b74 #eja ,cf4 ,treutlein1 ,z4 f4 ,math4 u4 ,phys4 7,hist4 litt4 ,abth471 ,,xxviii1 p4 #bji1 #ahhc4 #ejb ,this is the so-called ,dreifachgleichschenkelige ,dreieck2 cf4 ,naber1 op4 infra cit4 ,the ratio #a 3 #jfah is again not hard to find in this construction4 #ejc ,see1 on the mathematical history of the ,gnomon1 ,heath's ,euclid1 ,i1 passim1 #aijh2 ,zeuthen1 ,theorme de ,pythagore1 ,genve1 #aijd2 also a curious and interesting book1 ,das ,theorem des ,pythagoras1 by ,dr4 ,h4 ,a4 ,naber1 ,haarlem1 #aijh4 #ejd ,for many beautiful geometrical constructions based on the molluscan shell1 see ,colman1 ,s4 and ,coan1 ,c4 ,a41 ,nature's ,harmonic ,unity 7ch4 ix1 ,conchology71 ,new ,york1 #aiab4 #eje ,the ,rev4 ,h4 ,moseley1 ,on the ,geometrical ,forms of ,turbinated and ,discoid ,shells1 ,phil4 ,trans4 pp4 #cea-cgj4 #ahch4 #ejf ,it will be observed that here ,moseley1 speaking as a mathematician and considering the linear spiral1 speaks of whorls when he means the linear boundaries1 or lines traced by the revolving radius vector2 while the conchologist usually applies the term whorl to the whole space between the two boundaries4 ,as conchologists1 therefore1 we call the breadth of a whorl what ,moseley looked upon as the distance between two consecutive whorls4 ,but this latter nomenclature ,moseley himself often uses4 #ejg ,in the case of ,turbo1 and all other 8turbinate0 shells1 we are dealing not with a plane logarithmic spiral1 as in ,nautilus1 but with a 8gauche0 spiral1 such that the radius vector no longer revolves in a plane perpendicular to the axis of the system1 but is inclined to that axis at some constant angle 774 ,the figure still preserves its continued similarity1 and may with strict accuracy be called a logarithmic spiral in space4 ,it is evident that its envelope will be a right circular cone2 and indeed it is commonly spoken of as a logarithmic spiral wrapped upon a cone1 its pole coinciding with the apex of the cone4 ,it follows that the distances of successive whorls of the spiral measured on the same straight line passing through the apex of the cone1 are in geometrical progression1 and conversely just as in the former case4 ,but the ratio between any two consecutive interspaces 7i4e4 ,r#c ,r#b_/,r#b ,r#a7 is now equal to #b sin cot 1 being the semi-angle of the enveloping cone4 7,cf4 ,moseley1 ,phil4 ,mag4 ,,xxi1 p4 #cjj1 #ahdb47 #ejh ,as the successive increments evidently constitute similar figures1 similarly related to the pole 7,p71 it follows that their linear dimensions are to one another as the radii vectores drawn to similar points in them3 for instance as ,p ,p#a1 ,p ,p#b1 which 7in ,fig4 #bfd1 #a7 are radii vectores drawn to the points where they meet the common boundary4 #eji ,the equation to the surface of a turbinate shell is discussed by ,moseley 7,phil4 ,trans4 tom4 cit4 p4 #cgj71 both in terms of polar coordinates and of the rectangular coordinates x1 y1 z4 ,a more elegant representation can be given in vector notation1 by the method of quaternions4 #eaj ,j4 ,c4 ,m4 ,reinecke1 ,maris protogaei ,nautilos1 etc41 ,coburg1 #ahah4 ,leopold von ,buch1 ,ueber die ,ammoniten in den lteren ,gebirgsschichten1 ,abh4 ,berlin4 ,akad41 ,phys4 ,kl4 pp4 #ace-aeh1 #ahcj2 ,ann4 ,sc4 ,nat4 ,,xxviii1 pp4 #e-dc1 #ahcc2 cf4 ,elie de ,beaumont1 ,sur l'enroulement des ,ammonites1 ,soc4 ,philom41 ,pr4 verb4 pp4 #de-dh1 #ahda4 #eaa ,biblia ,naturae sive ,historia ,insectorum1 ,leydae1 #agcg1 p4 #aeb4 #eab ,alcide ,d',orbigny1 ,bull4 de la soc4 gol4 ,fr4 ,,xiii1 p4 #bjj1 #ahdb2 ,cours lm4 de ,palontologie1 ,,ii1 p4 #e1 #ahea4 ,a somewhat similar instrument was described by ,boube4 in ,bull4 soc4 gol4 ,i1 p4 #bcb1 #ahca4 ,naumann's ,conchyliometer 7,poggend4 ,ann4 ,,liv1 p4 #edd1 #ahde7 was an application of the screw-micrometer2 it was provided also with a rotating stage1 for angular measurement4 ,it was adapted for the ,study of a discoid or ammonitoid shell1 while ,d',orbigny's instrument was meant for the study of a turbinate shell4 #eac ,it is obvious that the ratios of opposite whorls1 or of radii #ahj apart1 are represented by the square roots of these values2 and the ratios of whorls or radii #ij apart1 by the square roots of these again4 #ead ,for the correction to be applied in the case of the helicoid1 or 8turbinate0 shells1 see p4 #eeg4 #eae ,on the ,measurement of the ,curves formed by ,cephalopods and other ,mollusks4 ,phil4 ,mag4 7#e71 ,,vi1 pp4 #bda-bfc1 #ahgh4 #eaf ,for an example of this method1 see ,blake1 l4c4 p4 #bea4 #eag ,naumann1 ,c4 ,f41 ,ueber die ,spiralen von ,conchylien1 ,abh4 k4 schs4 ,ges4 pp4 #aec-aif1 #ahdf2 ,ueber die cyclocentrische ,conchospirale u4 ber das ,windungsgesetz von ,planorbis corneus1 ibid4 ,i1 pp4 #aga-aie1 #ahdi2 ,spirale von ,nautilus u4 ,ammonites galeatus1 ,ber4 k4 schs4 ,ges4 ,,ii1 p4 #bf1 #ahdh2 ,spirale von ,amm4 ,ramsaueri1 ibid4 ,,xvi1 p4 #ba1 #ahfd2 see also ,poggendorff's ,annalen1 ,l1 p4 #bbc1 #ahdj2 ,,li1 p4 #bde1 #ahda2 ,,liv1 p4 #eda1 #ahde1 etc4 #eah ,sandberger1 ,g41 ,spiralen des ,ammonites ,amaltheus1 ,a4 ,gaytani1 und ,goniatites intumescens1 ,zeitschr4 d4 d4 ,geol4 ,gesellsch4 ,x1 pp4 #ddf-ddi1 #aheh4 #eai ,grabau1 ,a4 ,h41 ,ueber die ,naumannsche ,conchospirale1 etc4 ,inauguraldiss4 ,leipzig1 #ahgb2 ,die ,spiralen von ,conchylien1 etc4 ,programm1 ,nr4 #ejb1 ,leipzig1 #ahhb4 #ebj ,it has been pointed out to me that it does not follow at once and obviously that1 because the interspace ,,ab is a mean proportional between the breadths of the adjacent whorls1 therefore the whole distance ,,ob is a mean proportional between ,,oa and ,,oc4 ,this is a corollary which requires to be proved2 but the proof is easy4 #eba ,a beautiful construction3 stupendum ,naturae artificium1 ,linnaeus4 #ebb ,english edition1 p4 #ecg1 #aijj4 ,the chapter is revised by ,prof4 ,alpheus ,hyatt1 to whom the nomenclature is largely due4 ,for a more copious terminology1 see ,hyatt1 ,phylogeny of an ,acquired ,characteristic1 p4 #dbb seq41 #ahid4 #ebc ,this latter conclusion is adopted by ,willey1 ,zoological ,results1 p4 #gdg1 #aijb4 #ebd ,see ,moseley1 op4 cit4 pp4 #cfa seq4 #ebe ,in ,nautilus1 the 8hood0 has somewhat different dimensions in the two sexes1 and these differences are impressed upon the shell1 that is to say upon its 8generating curve40 ,the latter constitutes a somewhat broader ellipse in the male than in the female4 ,but this difference is not to be detected in the young2 in other words1 the form of the generating curve perceptibly alters with advancing age4 ,somewhat similar differences in the shells of ,ammonites were long ago suspected1 by ,d',orbigny1 to be due to sexual differences4 7,cf4 ,willey1 ,natural ,science1 ,,vi1 p4 #daa1 #ahie2 ,zoological ,results1 p4 #gdb1 #aijb47 #ebf ,macalister1 ,alex41 ,observations on the ,mode of ,growth of ,discoid and ,turbinated ,shells1 ,p4 ,r4 ,s4 ,,xviii1 pp4 #ebi-ecb1 #ahgj4 #ebg ,see figures in ,arnold ,lang's ,comparative ,anatomy 7,english translation71 ,,ii1 p4 #afa1 #aijb4 #ebh ,kappers1 ,c4 ,u4 ,a41 ,die ,bildung knstlicher ,molluskenschalen1 ,zeitschr4 f4 allg4 ,physiol4 ,,vii1 p4 #aff1 #aijh4 #ebi ,we need not assume a close relationship1 nor indeed any more than such a one as permits us to compare the shell of a ,nautilus with that of a ,gastropod4 #ecj ,cf4 ,owen1 8,these shells ,nautilus and ,ammonites are revolutely spiral or coiled over the back of the animal1 not involute like ,spirula03 ,palaeontology1 #ahfa1 p4 #ig2 cf4 ,mem4 on the ,pearly ,nautilus1 #ahcb2 also ,p4,z4,s4 #ahgh1 p4 #iee4 #eca ,the case of ,terebratula or of ,gryphaea would be closely analogous1 if the smaller valve were less closely connected and co-articulated with the larger4 #ecb 8,it has been suggested1 and ,i think in some quarters adopted as a dogma1 that the formation of successive septa in ,nautilus is correlated with the recurrence of reproductive periods4 ,this is not the case1 since1 according to my observations1 propagation only takes place after the last septum is formed20 ,willey1 ,zoological ,results1 p4 #gdf1 #aijb4 #ecc ,cf4 ,woodward1 ,henry1 ,on the ,structure of ,camerated ,shells1 ,pop4 ,sci4 ,rev4 ,,xi1 pp4 #aac-abj1 #ahgb4 #ecd ,see ,willey1 ,contributions to the ,natural ,history of the ,pearly ,nautilus1 ,zoological ,results1 etc4 p4 #gdi1 #aijb4 ,cf4 also ,bather1 ,shell-growth in ,cephalopoda1 ,ann4 ,mag4 ,n4 ,h4 7#f71 ,i1 pp #bih-caj1 #ahhh2 ibid4 pp4 #dba-dbg1 and other papers by ,blake1 ,riefstahl1 etc4 quoted therein4 #ece ,it was this that led ,james ,bernoulli1 in imitation of ,archimedes1 to have the logarithmic spiral graven on his tomb1 with the pious motto1 ,eadem mutata resurgam4 ,on ,goodsir's grave the same symbol is reinscribed4 #ecf ,the 8lobes0 and 8saddles0 which arise in this manner1 and on whose arrangement the modern classification of the nautiloid and ammonitoid shells largely depends1 were first recognised and named by ,leopold von ,buch1 ,ann4 ,sci4 ,nat4 ,,xxvii1 ,,xxviii1 #ahbi4 #ecg ,blake has remarked upon the fact 7op4 cit4 p4 #bdh7 that in some ,cyrtocerata we may have a curved shell in which the ornaments approximately run at a constant angular distance from the pole1 while the septa approximate to a radial direction2 and that 8thus one law of growth is illustrated by the inside1 and another by the outside40 ,in this there is nothing at which we need wonder4 ,it is merely a case where the generating curve is set very obliquely to the axis of the shell2 but where the septa1 which have no necessary relation to the mouth of the shell1 take their places1 as usual1 at a certain definite angle to the walls of the tube4 ,this relation of the septa to the walls of the tube arises after the tube itself is fully formed1 and the obliquity of growth of the open end of the tube has no relation to the matter4 #ech ,cf4 pp4 #bee1 #dfc1 etc4 #eci ,in a few cases1 according to ,awerinzew and ,rhumbler1 where the chambers are added on in concentric series1 as in ,orbitolites1 we have the crystalline structure arranged radially in the radial walls but tangentially in the concentric ones3 whereby we tend to obtain1 on a minute scale1 a system of orthogonal trajectories1 comparable to that which we shall presently study in connection with the structure of bone4 ,cf4 ,s4 ,awerinzew1 ,kalkschale der ,rhizopoden1 ,z4 f4 w4 ,z4 ,,lxxiv1 pp4 #dgh-dij1 #aijc4 #edj ,rhumbler1 ,l41 ,die ,doppelschalen von ,orbitolites und anderer ,foraminiferen1 etc41 ,arch4 f4 ,protistenkunde1 ,i1 pp4 #aic-bif1 #aijb2 and other papers4 ,also ,die ,foraminiferen der ,planktonexpedition1 ,i1 #aiaa1 pp4 #ej-ef4 #eda ,bnard1 ,h1 ,les tourbillons cellulaires1 ,ann4 de ,chimie 7#h71 ,,xxiv1 #aija4 ,cf4 also the pattern of cilia on an ,infusorian1 as figured by ,btschli in ,bronn's ,protozoa1 ,,iii1 p4 #abha1 #ahhg4 #edb ,a similar hexagonal pattern is obtained by the mutual repulsion of floating magnets in ,mr ,r4 ,w4 ,wood's experiments1 ,phil4 ,mag4 ,,xlvi1 pp4 #afb-afd1 #ahih4 #edc ,cf4 ,d',orbigny1 ,alc41 ,tableau mthodique de la classe des ,cphalopodes1 ,ann4 des ,sci4 ,nat4 7#a71 ,,vii1 pp4 #bde-cae1 #ahbf2 ,dujardin4 ,flix1 ,observations nouvelles sur les prtendus ,cphalopodes microscopiques1 ibid4 7#b71 ,,iii1 pp4 #ajh1 #aji1 #cab-cae1 #ahce2 ,recherches sur les organismes infrieurs1 ibid4 ,,iv1 pp4 #cdc-cgg1 #ahce1 etc4 #edd ,it is obvious that the actual outline of a foraminiferal1 just as of a molluscan shell1 may depart widely from a logarithmic spiral4 ,when we say here1 for short1 that the shell is a logarithmic spiral1 we merely mean that it is essentially related to one3 that it can be inscribed in such a spiral1 or that corresponding points 7such1 for instance1 as the centres of gravity of successive chambers1 or the extremities of successive septa7 wall always be found to lie upon such a spiral4 #ede von ,mller1 ,v41 ,die spiral-gewundenen ,foraminifera des russischen ,kohlenkalks1 ,mm4 de l',acad4 ,imp4 ,sci41 ,st ,ptersbourg 7#g71 ,,xxv1 #ahgh4 #edf ,as von ,mller is careful to explain1 ,naumann's formula for the 8cyclocentric conchospiral0 is appropriate to this and other spiral ,foraminifera1 since we have in all these cases a central or initial chamber1 approximately spherical1 about which the logarithmic spiral is coiled 7cf4 ,fig4 #cji74 ,in species where the central chamber is especially large1 ,naumann's formula is all the more advantageous4 ,but it is plain that it is only required when we are dealing with diameters1 or with radii2 so long as we are merely comparing the breadths of successive whorls1 the two formulae come to the same thing4 #edg ,van ,iterson1 ,g41 ,mathem4 u4 mikrosk4-anat4 ,studien ber ,blattstellungen1 nebst ,betrachtungen ber den ,schalenbau der ,miliolinen1 #cca pp41 ,jena1 #aijg4 #edh ,hans ,przibram asserts that the linear ratio of successive chambers tends in many ,foraminifera to approximate to #abf1 which #b2 in other words1 that the volumes of successive chambers tend to double4 ,this ,przibram would bring into relation with another law1 viz4 that insects and other arthropods tend to moult1 or to metamorphose1 just when they double their weights1 or increase their linear dimensions in the ratio of #a 3 #b4 7,die ,kammerprogression der ,foraminiferen als ,parallele zur ,hutungsprogression der ,mantiden1 ,arch4 f4 ,entw4 ,mech4 ,,xxxiv p4 #fhj1 #ahac47 ,neither rule seems to me to be well grounded4 #edi ,cf4 ,schacko1 ,g41 ,ueber ,globigerina-,einschluss bei ,orbulina1 ,wiegmann's ,archiv1 ,,xlix1 p4 #dbh1 #ahhc2 ,brady1 ,chall4 ,rep41 p4 #fjg1 #ahhd4 #eej ,cf4 ,brady1 ,h4 ,b41 ,challenger ,rep41 ,foraminifera1 #ahhd1 p4 #bjc1 pl4 ,,xiii4 #eea ,brady1 op4 cit41 p4 #bjf2 ,batsch1 one of the earliest writers on ,foraminifera1 had already noticed that this whole series of ear-shaped and crozier-shaped shells was filled in by gradational forms2 ,conchylien des ,seesandes1 #agia1 p4 #d1 pl4 ,,vi1 fig4 #ae;a-f4 ,see also1 in particular1 ,dreyer1 ,peneroplis2 eine ,studie zur biologischen ,morphologie und zur ,speciesfrage1 ,leipzig1 #ahih2 also ,eimer und ,fickert1 ,artbildung und ,verwandschaft bei den ,foraminiferen1 ,tbinger zool4 ,arbeiten1 ,,iii1 p4 #ce1 #ahii4 #eeb ,doflein1 ,protozoenkunde1 #aiaa1 p4 #bfc2 8,was diese ,art veranlsst in dieser ,weise gelegentlich zu varren1 ist vorlufig noch ganz rthselhaft40 #eec ,in the case of ,globigerina1 some fourteen species 7out of a very much larger number of described forms7 were allowed by ,brady 7in #ahhd7 to be distinct2 and this list has been1 ,i believe1 rather added to than diminished4 ,but these so-called species depend for the most part on slight differences of degree1 differences in the angle of the spiral1 in the ratio of magnitude of the segments1 or in their area of contact one with another4 ,moreover with the exception of one or two 8dwarf0 forms1 said to be limited to ,arctic and ,antarctic waters1 there is no principle of geographical distribution to be discerned amongst them4 ,a species found fossil in ,new ,britain turns up in the ,north ,atlantic3 a species described from the ,west ,indies is rediscovered at the ice-barrier of the ,antarctic4 #eed ,dreyer1 ,f41 ,principien der ,gerstbildung bei ,rhizopoden1 etc41 ,jen4 ,zeitschr4 ,,xxvi1 pp4 #bjd-dfh1 #ahib4 #eee ,a difficulty arises in the case of forms 7like ,peneroplis7 where the young shell appears to be more complex than the old1 the first formed portion being closely coiled while the later additions become straight and simple3 8die biformen ,arten verhalten sich1 kurz gesagt4 gerade umgekehrt als man nach dem biogenetischen ,grundgesetz erwarten sollte10 ,rhumbler1 op4 cit41 p4 #cc etc4 #eef 8,das ,festigkeitsprinzip als ,movens der ,weiterentwicklung ist zu interessant und fr die ,aufstellung meines ,systems zu wichtig um die ,frage unerrtert zu lassen1 warum diese ,bevorzgung der ,festigkeit stattgefunden hat4 ,meiner ,ansicht nach lautet die ,antwort auf diese ,frage einfach1 weil die ,foraminiferen meistens unter ,verhltnissen leben1 die ihre ,schalen in hohem ,grade der ,gefahr des ,zerbrechens aussetzen2 es muss also eine fortwahrende ,auslese des ,festeren stattfinden10 ,rhumbler1 op4 cit41 p4 #bb4 #eeg 8,die ,foraminiferen kiesige oder grobsandige ,gebiete des ,meeresbodens nicht lieben1 u4s4w403 where the last two words have no particular meaning1 save only that 7as ,m4 ,aurelius says7 8of things that use to be1 we say commonly that they love to be40 #eeh ,in regard to the ,foraminifera1 8die ,palaeontologie lsst uns leider an ,anfang der ,stammesgeschichte fast gnzlich im ,stiche10 ,rhumbler1 op4 cit41 p4 #ad4 #eei ,the evolutionist theory1 as ,bergson puts it1 8consists above all in establishing relations of ideal kinship1 and in maintaining that wherever there is this relation of1 so to speak1 logical affiliation between forms1 there is also a relation of chronological succession between the species in which these forms are materialised03 ,creative ,evolution1 #aiaa1 p4 #bf4 ,cf4 supra1 p4 #bea4 #efj ,in the case of the ram's horn1 the assumption that the rings are annual is probably justified4 ,in cattle they are much less conspicuous1 but are sometimes well-marked in the cow2 and in ,sweden they are then called 8calf-rings10 from a belief that they record the number of offspring4 ,that is to say1 the growth of the horn is supposed to be retarded during gestation1 and to be accelerated after parturition1 when superfluous nourishment seeks a new outlet4 7,cf4 ,lnnberg1 ,p4,z4,s41 p4 #fhi1 #aijj47 #efa ,cf4 ,sir ,v4 ,brooke1 ,on the ,large ,sheep of the ,thian ,shan1 ,p4,z4,s41 p4 #eaa1 #ahge4 #efb ,cf4 ,lnnberg1 ,e41 ,on the ,structure of the ,musk ,ox1 ,p4,z4,s41 pp4 #fhf-gah1 #aijj4 #efc ,st ,venant1 ,de la torsion des prismes1 avec des considrations sur leur flexion1 etc41 ,mm4 des ,savants trangers1 ,paris1 ,,xiv1 pp4 #bcc-efj1 #ahef4 #efd ,this is not difficult to do1 with considerable accuracy1 if the clay be kept well wetted1 or semi-fluid1 and the smoothing be done with a large wet brush4 #efe ,the curves are well shewn in most of ,sir ,v4 ,brooke's figures of the various species of ,argali1 in the paper quoted on p4 #fad4 #eff ,climbing ,plants1 #ahfe 7#bnd edit4 #ahge72 ,power of ,movement in ,plants1 #ahhj4 #efg ,palm1 ,ueber das ,winden der ,pflanzen1 #ahbg2 von ,mohl1 ,bau und ,winden der ,ranken1 etc41 #ahbg2 ,dutrochet1 ,mouvements rvolutifs spontans1 ,c4,r4 #ahdc1 etc4 #efh ,cf4 7e4g47 ,lepeschkin1 ,zur ,kenntnis des ,mechanismus der ,variationsbewegungen1 ,ber4 d4 d4 ,bot4 ,gesellsch4 ,,xxvi ,a1 pp4 #gbd-gce1 #aijh2 also ,a4 ,trndle1 ,der ,einfluss des ,lichtes auf die ,permeabilitt des ,plasmahaut1 ,jahrb4 wiss4 ,bot4 ,,xlviii1 pp4 #aga-bhb1 #aiaj4 #efi ,for an elaborate study of antlers1 see ,rrig1 ,a41 ,arch4 f4 ,entw4 ,mech4 ,x1 pp4 #ebe-fdd1 #aijj1 ,,xi1 pp4 #fe-adh1 #bbe-cji1 #aija2 ,hoffmann1 ,c41 ,zur ,morphologie der rezenten ,hirschen1 #ge pp41 #bc pls41 #aija3 also ,sir ,victor ,brooke1 ,on the ,classification of the ,cervidae1 ,p4,z4,s41 pp4 #hhc-ibh1 #ahgh4 ,for a discussion of the development of horns and antlers1 see ,gadow1 ,h41 ,p4,z4,s41 pp4 #bjf-bbb1 #aijb1 and works quoted therein4 #egj ,cf4 ,rhumbler1 ,l41 ,ueber die ,abhngigkeit des ,geweihwachstums der ,hirsche1 speziell des ,edelhirsches1 vom ,verlauf der ,blutgefsse im ,kolbengeweih1 ,zeitschr4 f4 ,forst4 und ,jagdwesen1 #aiaa1 pp4 #bie-cad4 #ega ,the fact that in one very small deer1 the little ,south ,american ,coassus1 the antler is reduced to a simple short spike1 does not preclude the general distinction which ,i have drawn4 ,in ,coassus we have the beginnings of an antler1 which has not yet manifested its tendency to expand2 and in the many allied species of the ,american genus ,cariacus1 we find the expansion manifested in various simple modes of ramification or bifurcation4 7,cf4 ,sir ,v4 ,brooke1 ,classification of the ,cervidae1 p4 #hig47 #egb ,cf4 also the immense range of variation in elks' horns1 as described by ,lnnberg1 ,p4,z4,s4 ,,ii1 pp4 #ceb-cfj1 #aijb4 #egc ,besides papers referred to below1 and many others quoted in ,sach's ,botany and elsewhere1 the following are important3 ,braun1 ,alex41 ,vergl4 ,untersuchung ber die ,ordnung der ,schuppen an den ,tannenzapfen1 etc41 ,verh4 ,car4 ,leop4 ,akad4 ,,xv1 pp4 #aii-dja1 #ahca2 ,dr ,c4 ,schimper's ,vortrge ber die ,mglichkeit eines wissenschaftlichen ,verstndnisses der ,blattstellung1 etc41 ,flora1 ,,xviii1 pp4 #ade-aia1 #gcg-gef1 #ahce2 ,schimper1 ,c4 ,f41 ,geometrische ,anordnung der um eine ,axe peripherische ,blattgebilde1 ,verhandl4 ,schweiz4 ,ges41 pp4 #aac-aag1 #ahcf2 ,bravais1 ,l4 and ,a41 ,essai sur la disposition des feuilles curvisries1 ,ann4 ,sci4 ,nat4 7#b71 ,,vii1 pp4 #db-aaj1 #ahcg2 ,sur la disposition symmtrique des inflorescences1 ibid41 pp4 #aic-bba1 #bia-cdh1 ,,viii1 pp4 #aa-db1 #ahch2 ,sur la disposition gnrale des feuilles rectisries1 ibid4 ,,xii1 pp4 #e-da1 #fe-gg1 #ahci2 ,zeising1 ,normalverhltniss der chemischen und morphologischen ,proportionen1 ,leipzig1 #ahef2 ,naumann1 ,c4 ,f41 ,ueber den ,quincunx als ,gesetz der ,blattstellung bei ,sigillaria1 etc41 ,neues ,jahrb4 f4 ,miner4 #ahdb1 pp4 #daj-dag2 ,lestiboudois1 ,t41 ,phyllotaxie anatomique1 ,paris1 #ahdh2 ,henslow1 ,g41 ,phyllotaxis1 ,london1 #ahga2 ,wiesner1 ,bemerkungen ber rationale und irrationale ,divergenzen1 ,flora1 ,,lviii1 pp4 #aac-aae1 #aci-adc1 #ahge2 ,airy1 ,h41 ,on ,leaf ,arrangement1 ,proc4 ,r4 ,s4 ,,xxi1 p4 #agf1 #ahgc2 ,schwendener1 ,s41 ,mechanische ,theorie der ,blattstellungen1 ,leipzig1 #ahgh2 ,delpino1 ,f41 ,causa meccanica della filotassi quincunciale1 ,genova1 #ahhj2 de ,candolle1 ,c41 tude de ,phyllotaxie1 ,genve1 #ahha4 #egd ,allgemeine ,morphologie der ,gewchse1 p4 #ddb1 etc4 #ahfh4 #ege ,relation of ,phyllotaxis to ,mechanical ,laws1 ,oxford1 #aija-aijc2 cf4 ,ann4 of ,botany1 ,,xv1 p4 #dha1 #aija4 #egf 8,the proposition is that the genetic spiral is a logarithmic spiral1 homologous with the line of current-flow in a spiral vortex2 and that in such a system the action of orthogonal forces will be mapped out by other orthogonally intersecting logarithmic spirals--the #8parastichies' 02 ,church1 op4 cit4 ,i1 p4 #db4 #egg ,mr ,church's whole theory1 if it be not based upon1 is interwoven with1 ,sachs's theory of the orthogonal intersection of cell-walls1 and the elaborate theories of the symmetry of a growing point or apical cell which are connected therewith4 ,according to ,mr ,church1 8the law of the orthogonal intersection of cell-walls at a growing apex may be taken as generally accepted0 7p4 #cb72 but ,i have taken a very different view of ,sachs's law1 in the eighth chapter of the present book4 ,with regard to his own and ,sachs's hypotheses1 ,mr ,church makes the following curious remark 7p4 #db73 8,nor are the hypotheses here put forward more imaginative than that of the paraboloid apex of ,sachs which remains incapable of proof1 or his construction for the apical cell of ,pteris which does not satisfy the evidence of his own drawings40 #egh ,amer4 ,naturalist1 ,,vii1 p4 #ddi1 #ahgc4 #egi ,this celebrated series1 which appears in the continued fraction 7#a_/a7 7#a_/7a 77 etc4 and is closely connected with the ,sectio aurea or ,golden ,mean1 is commonly called the ,fibonacci series1 after a very learned twelfth century arithmetician 7known also as ,leonardo of ,pisa71 who has some claims to be considered the introducer of ,arabic numerals into christian ,europe4 ,it is called ,lami's series by some1 after ,father ,bernard ,lami1 a contemporary of ,newton's1 and one of the co-discoverers of the parallelogram of forces4 ,it was well-known to ,kepler1 who1 in his paper ,de nive sexangula 7cf4 supra1 p4 #dhj71 discussed it in connection with the form of the dodecahedron and icosahedron1 and with the ternary or quinary symmetry of the flower4 7,cf4 ,ludwig1 ,f41 ,kepler ber das ,vorkommen der ,fibonaccireihe im ,pflanzenreich1 ,bot4 ,centralbl4 ,,lxviii1 p4 #g1 #ahif74 ,professor ,william ,allman1 ,professor of ,botany in ,dublin 7father of the historian of ,greek geometry71 speculating on the same facts1 put forward the curious suggestion that the cellular tissue of the dicotyledons1 or exogens1 would be found to consist of dodecahedra4 and that of the monocotyledons or endogens of icosahedra 7,on the mathematical connexion between the parts of ,vegetables3 abstract of a ,memoir read before the ,royal ,society in the year #ahaa 7privately printed1 n4d474 ,cf4 ,de ,candolle1 ,organognie vgtale1 ,i1 p4 #ecd74 #ehj ,proc4 ,roy4 ,soc4 ,edin4 ,,vii1 p4 #cia1 #ahgb4 #eha ,the necessary existence of these recurring spirals is also proved1 in a somewhat different way1 by ,leslie ,ellis1 ,on the ,theory of ,vegetable ,spirals1 in ,mathematical and other ,writings1 #ahec1 pp4 #ceh-cgb4 #ehb ,proc4 ,roy4 ,soc4 ,edin4 ,,vii1 p4 #cig1 #ahgb2 ,trans4 ,roy4 ,soc4 ,edin4 ,,xxvi1 p4 #eje1 #ahgj-ga4 #ehc ,a common form of pail-shaped waste-paper basket1 with wide rhomboidal meshes of cane1 is well-nigh as good a model as is required4 #ehd ,deutsche ,vierteljahrsschrift1 p4 #bfa1 #ahfh4 #ehe ,memoirs of ,amer4 ,acad4 ,,ix1 p4 #chi4 #ehf ,de avibus circa aquas ,danubii vagantibus et de ipsarum ,nidis 7,vol4 ,v of the ,danubius ,pannonico-mysicus71 ,hagae ,com41 #agbf4 #ehg ,sir ,thomas ,browne had a collection of eggs at ,norwich1 according to ,evelyn1 in #afga4 #ehh ,cf4 ,lapierre1 in ,buffon's ,histoire ,naturelle1 ed4 ,sonnini1 #ahjj4 #ehi ,eier der ,vgel ,deutschlands1 #ahah-bh 7cit4 des ,murs1 p4 #cf74 #eij ,trait d',oologie1 #ahfj4 #eia ,lafresnaye1 ,f4 de1 ,comparaison des ufs des ,oiseaux avec leurs squelettes1 comme seul moven de reconnatre la cause de leurs diffrentes formes1 ,rev4 ,zool41 #ahde1 pp4 #ahj-ahg1 #bci-bdd4 #eib ,cf4 ,des ,murs1 p4 #fg3 8,elle devait encore penser au moment o ce germe aurait besoin de l'espace ncessaire son accroissement1 ce moment o 444 il devra remplir exactement l'intervalle circonscrit par sa fragile prison1 etc40 #eic ,thienemann1 ,f4 ,a4 ,l41 ,syst4 ,darstellung der ,fortpflanzung der ,vgel ,europas4 ,leipzig1 #ahbe-ch4 #eid ,cf4 ,newton's ,dictionary of ,birds1 #ahic1 p4 #aia2 ,szielasko1 ,gestalt der ,vogeleier1 ,j4 f4 ,ornith4 ,,liii1 pp4 #bgc-big1 #aije4 #eie ,jacob ,steiner suggested a ,cartesian oval1 r mr c1 as a general formula for all eggs 7cf4 ,fechner1 ,ber4 schs4 ,ges41 #ahdi1 p4 #eg72 but this formula 7which fails in such a case as the guillemot71 is purely empirical1 and has no mechanical foundation4 #eif ,gnther1 ,f4 ,c41 ,sammlung von ,nestern und ,eyern verschiedener ,vgel1 ,nrnb4 #aggb4 ,cf4 also ,raymond ,pearl1 ,morphogenetic ,activity of the ,oviduct1 ,j4 ,exp4 ,zool4 ,,vi1 pp4 #cci-cei1 #aiji4 #eig ,the following account is in part reprinted from ,nature1 ,june #d1 #aijh4 #eih ,in so far as our explanation involves a shaping or moulding of the egg by the uterus or 8oviduct0 7an agency supplemented by the proper tensions of the egg71 it is curious to note that this is very much the same as that old view of ,telesius regarding the formation of the embryo 7,de rerum natura1 ,,vi1 cc4 #d and #aj71 which he had inherited from ,galen1 and of which ,bacon speaks 7,nov4 ,org4 cap4 #ej2 cf4 ,ellis's note74 ,bacon expressly remarks that 8,telesius should have been able to shew the like formation in the shells of eggs40 ,this old theory of embryonic modelling survives only in our usage of the term 8matrix0 for a 8mould40 #eii ,journal of ,tropical ,medicine1 #aeth ,june1 #aiaa4 ,i leave this paragraph as it was written1 though it is now once more asserted that the terminal and lateral-spined eggs belong to separate and distinct species of ,bilharzia 7,leiper1 ,brit4 ,med4 ,journ41 #ahth ,march1 #aiaf1 p4 #daa74 #fjj ,cf4 ,bashforth and ,adams1 ,theoretical ,forms of ,drops1 etc41 ,cambridge1 #ahhc4 #fja ,woods1 ,r4 ,h41 ,on a ,physical ,theorem applied to tense ,membranes1 ,journ4 of ,anat4 and ,phys4 ,,xxvi1 pp4 #cfb-cga1 #ahib4 ,a similar investigation of the tensions in the uterine wall1 and of the varying thickness of its muscles1 was attempted by ,haughton in his ,animal ,mechanics1 pp4 #aea-aeh1 #ahgc4 #fjb ,this corresponds with a determination of the normal pressures 7in systole7 by ,krohl1 as being in the ratio of #a 3 #fh4 #fjc ,cf4 ,schwalbe1 ,g41 ,ueber ,wechselbeziehungen und ihr ,einfluss auf die ,gestaltung des ,arteriensystem1 ,jen4 ,zeitschr4 ,,xii1 p4 #bfg1 #ahgh1 ,roux1 ,ueber die ,verzweigungen der ,blutgefssen des ,menschen1 ibid4 ,,xii1 p4 #bje1 #ahgh2 ,ueber die ,bedeutung der ,ablenkung des ,arterienstmmen bei der ,astaufgabe1 ibid4 ,,xiii1 p4 #cja1 #ahgi2 ,hess1 ,walter1 ,eine mechanisch bedingte ,gesetzmssigkeit im ,bau des ,blutgefsssystems1 ,a4 f4 ,entw4 ,mech4 ,,xvi1 p4 #fcb1 #aijc2 ,thoma1 ,r41 ,ueber die ,histogenese und ,histomechanik des ,blutgefsssystems1 #ahic4 #fjd ,essays1 etc41 edited by ,owen1 ,i1 p4 #acd1 #ahfa4 #fje ,on the ,functions of the ,heart and ,arteries1 ,phil4 ,trans4 #ahji1 pp4 #a-ca1 cf4 #ahjh1 pp4 #afd-ahf2 ,collected ,works1 ,i1 pp4 #eaa-ecd1 #ahee4 ,the same lesson is conveyed by all such work as that of ,volkmann1 ,e4 ,h4 ,weber and ,poiseuille4 ,cf4 ,stephen ,hales' ,statical ,essays1 ,,ii1 ,introduction3 8,especially considering that they i4e4 animal ,bodies are in a manner framed of one continued ,maze of innumerable ,canals1 in which ,fluids are incessantly circulating1 some with great ,force and ,rapidity1 others with very different ,degrees of rebated ,velocity3 ,hence1 etc40 #fjf 8,sizes0 is ,owen's editorial emendation1 which seems amply justified4 #fjg ,for a more elaborate classification1 into colours cryptic1 procryptic1 anticryptic1 apatetic1 epigamic1 sematic1 episematic1 aposematic1 etc41 see ,poulton's ,colours of ,animals 7,int4 ,scientific ,series1 ,,lxviii71 #ahij2 cf4 also ,meldola1 ,r41 ,variable ,protective ,colouring in ,insects1 ,p4,z4,s4 #ahgc1 pp4 #aec-afb1 etc4 #fjh ,dendy1 ,evolutionary ,biology1 p4 #ccf1 #aiab4 #fji ,delight in beauty is one of the pleasures of the imagination2 there is no limit to its indulgence1 and no end to the results which we may ascribe to its exercise4 ,but as for the particular 8standard of beauty0 which the bird 7for instance7 admires and selects 7as ,darwin says in the ,origin1 p4 #gj1 edit4 #ahhd71 we are very much in the dark1 and we run the risk of arguing in a circle3 for wellnigh all we can safely say is what ,addison says 7in the #dabth ,spectator7--that each different species 8is most affected with the beauties of its own kind 4444 ,hinc merula in nigro se oblectat nigra marito2 444 hinc noctua tetram ,canitiem alarum et glaucos miratur ocellos40 #faj ,cf4 ,bridge1 ,t4 ,w41 ,cambridge ,natural ,history 7,fishes71 ,,vii1 p4 #agc1 #aijd2 also ,frisch1 ,k4 v41 ,ueber farbige ,anpassung bei ,fische1 ,zool4 ,jahrb4 7,abt4 ,allg4 ,zool471 ,,xxxii1 pp4 #aga-bcj1 #aiad4 #faa ,nature1 ,l1 p4 #egb2 ,,li1 pp4 #cc1 #eg1 #ecc1 #ahid-ie4 #fab ,they are 8wonderfully fitted for #8vanishment' against the flushed1 rich-coloured skies of early morning and evening 4444 their chief feeding-times02 and 8look like a real sunset or dawn1 repeated on the opposite side of the heavens1--either east or west as the case may be03 ,thayer1 ,concealing-coloration in the ,animal ,kingdom1 ,new ,york1 #aiji1 pp4 #aed-aee4 ,this hypothesis1 like the rest1 is not free from difficulty4 ,twilight is apt to be short in the homes of the flamingo3 and moreover1 ,mr ,abel ,chapman1 who watched them on the ,guadalquivir1 tells us that they feed by day4 #fac ,principal ,galloway1 ,philosophy of ,religion1 p4 #cdd1 #aiad4 #fad ,cf4 ,professor ,flint1 in his ,preface to ,affleck's translation of ,janet's ,causes finales3 8,we are1 no doubt1 still a long way from a mechanical theory of organic growth1 but it may be said to be the quaesitum of modern science1 and no one can say that it is a chimaera40 #fae ,cf4 ,sir ,donald ,mac,alister1 ,how a ,bone is ,built1 ,engl4 ,ill4 ,mag4 #ahhd4 #faf ,professor ,claxton ,fidler1 ,on ,bridge ,construction1 p4 #bb 7#dth ed471 #aiji2 cf4 7int4 al47 ,love's ,elasticity1 p4 #bj 7,historical ,introduction71 #bnd ed41 #aijf4 #fag ,in preparing or 8macerating0 a skeleton1 the naturalist nowadays carries on the process till nothing is left but the whitened bones4 ,but the old anatomists1 whose object was not the study of 8comparative0 morphology but the wider theme of comparative physiology1 were wont to macerate by easy stages2 and in many of their most instructive preparations1 the ligaments were intentionally left in connection with the bones1 and as part of the 8skeleton40 #fah ,in a few anatomical diagrams1 for instance in some of the drawings in ,schmaltz's ,atlas der ,anatomie des ,pferdes1 we may see the system of 8ties0 diagrammatically inserted in the figure of the skeleton4 ,cf4 ,gregory1 ,on the principles of ,quadrupedal ,locomotion1 ,ann4 ,n4 ,y4 ,acad4 of ,sciences1 ,,xxii1 p4 #bhi1 #aiab4 #fai ,galileo1 ,dialogues concerning ,two ,new ,sciences 7#afch71 ,crew and ,salvio's translation1 ,new ,york1 #aiad1 p4 #aej2 ,opere1 ed4 ,favaro1 ,,viii1 p4 #ahf4 ,cf4 ,borelli1 ,de ,motu ,animalium1 ,i1 prop4 ,,clxxx1 #afhe4 ,cf4 also ,camper1 ,p41 ,la structure des os dans les oiseaux1 ,opp4 ,,iii1 p4 #dei1 ed4 #ahjc2 ,rauber1 ,a41 ,galileo ber ,knochenformen1 ,morphol4 ,jahrb4 ,,vii1 pp4 #cbg1 #cbh1 #ahha2 ,paolo ,enriques1 ,della economia di sostanza nelle osse cave1 ,arch4 f4 ,ent4 ,mech4 ,,xx1 pp4 #dbg-dfe1 #aijf4 #fbj ,das mechanische ,prinzip4 im anatomischen ,bau der ,monocotylen1 ,leipzig1 #ahgd4 #fba ,for further botanical illustrations1 see 7int4 al47 ,hegler1 ,einfluss der ,zugkraften auf die ,festigkeit und die ,ausbildung mechanischer ,gewebe in ,pflanzen1 ,,sb4 schs4 ,ges4 d4 ,wiss4 p4 #fch1 #ahia2 ,kny1 ,l41 ,einfluss von ,zug und ,druck auf die ,richtung der ,scheidewande in sich teilenden ,pflanzenzellen1 ,ber4 d4 bot4 ,gesellsch4 ,,xiv1 #ahif2 ,sachs1 ,mechanomorphose und ,phylogenie1 ,flora1 ,,lxxviii1 #ahid2 cf4 also ,pflger1 ,einwirkung der ,schwerkraft1 etc41 ber die ,richtung der ,zelltheilung1 ,archiv1 ,,xxxiv1 #ahhd4 #fbb ,among other works on the mechanical construction of bone see3 ,bourgery1 ,trait de l'anatomie 7,i4 ,ostologie71 #ahcb 7with admirable illustrations of trabecular structure72 ,fick1 ,l41 ,die ,ursachen der ,knochenformen1 ,gttingen1 #aheg2 ,meyer1 ,h41 ,die ,architektur der ,spongiosa1 ,archiv f4 ,anat4 und ,physiol4 ,,xlvii1 pp4 #fae-fbh1 #ahfg2 ,statik u4 ,mechanik des menschlichen ,knochengerstes1 ,leipzig1 #ahgc2 ,wolff1 ,j41 ,die innere ,architektur der ,knochen1 ,arch4 f4 ,anat1 und ,phys4 ,l1 #ahgj2 ,das ,gesetz der ,transformation bei ,knochen1 #ahib2 von ,ebner1 ,v41 ,der feinere ,bau der ,knochensubstanz1 ,wiener ,bericht1 ,,lxxii1 #ahge2 ,rauber1 ,anton1 ,elastizitt und ,festigkeit der ,knochen1 ,leipzig1 #ahgf2 ,o4 ,meserer1 ,elast1 u4 ,festigk4 d4 menschlichen ,knochen1 ,stuttgart1 #ahhj2 ,mac,alister1 ,sir ,donald1 ,how a ,bone is ,built1 ,english ,illustr4 ,mag4 pp4 #fdj-fdi1 #ahhd2 ,rasumowsky1 ,architektonik des ,fussskelets1 ,int4 ,monatsschr4 f4 ,anat4 p4 #aig1 #ahhi2 ,zschokke1 ,weitere ,unters4 ber das ,verhltniss der ,knochenbildung zur ,statik und ,mechanik des ,vertebratenskelets1 ,zrich1 #ahib2 ,roux1 ,w41 ,ges4 ,abhandlungen ber ,entwicklungsmechanik der ,organismen1 ,bd4 ,i1 ,funktionelle ,anpassung1 ,leipzig1 #ahie2 ,triepel1 ,h41 ,die ,stossfestigkeit der ,knochen1 ,arch4 f4 ,anat4 u4 ,phys4 #aijj2 ,gebhardt1 ,funktionell wichtige ,anordnungsweisen der feineren und grberen ,bauelemente des ,wirbelthierknochens1 etc41 ,arch4 f4 ,entw4 ,mech4 #aijj-aiaj2 ,kirchner4 ,a41 ,architektur der ,metatarsalien1 ,a4 f4 ,e4 ,m4 ,,xxiv1 #aijg2 ,triepel1 ,herm41 ,die trajectorielle ,structuren 7in ,einf4 in die ,physikalische ,anatomie1 #aijh72 ,dixon1 ,a4 ,f41 ,architecture of the ,cancellous ,tissue forming the ,upper ,end of the ,femur1 ,journ4 of ,anat4 and ,phys4 7#c7 ,,xliv1 pp4 #bbc-bcj1 #aiaj4 #fbc ,sdillot1 ,de l'influence des fonctions sur la structure et la forme des organes2 ,c4 ,r4 ,,lix1 p4 #eci1 #ahfd2 cf4 ,,lx1 p4 #ig1 #ahfe1 ,,lxviii4 p4 #addd4 #ahfi4 #fbd ,e4g4 7#a7 the head1 nodding backwards and forwards on a fulcrum1 represented by the atlas vertebra1 lying between the weight and the power2 7#b7 the foot1 raising on tip-toe the weight of the body against the fulcrum of the ground1 where the weight is between the fulcrum and the power1 the latter being represented by the tendo ,achillis2 7#c7 the arm1 lifting a weight in the hand1 with the power 7i4e4 the biceps muscle7 between the fulcrum and the weight4 7,the second case1 by the way1 has been much disputed2 cf4 ,haycraft in ,schfer's ,textbook of ,physiology1 p4 #bea1 #aijj47 #fbe ,our problem is analogous to ,dr ,thomas ,young's problem of the best disposition of the timbers in a wooden ship 7,phil4 ,trans4 #ahad1 p4 #cjc74 ,he was not long of finding that the forces which may act upon the fabric are very numerous and very variable1 and that the best mode of resisting them1 or best structural arrangement for ultimate strength1 becomes an immensely complicated problem4 #fbf ,in like manner1 ,clerk ,maxwell could not help employing the term 8skeleton0 in defining the mathematical conception of a 8frame10 constituted by points and their interconnecting lines3 in studying the equilibrium of which1 we consider its different points as mutually acting on each other with forces whose directions are those of the lines joining each pair of points4 ,hence 7says ,maxwell71 8in order to exhibit the mechanical action of the frame in the most elementary manner1 we may draw it as a skeleton1 in which the different points are joined by straight lines1 and we may indicate by numbers attached to these lines the tensions or compressions in the corresponding pieces of the frame0 7,trans4 ,r4 ,s4 ,e4 ,,xxvi1 p4 #a1 #ahgj74 ,it follows that the diagram so constructed represents a 8diagram of forces10 in this limited sense that it is geometrical as regards the position and direction of the forces1 but arithmetical as regards their magnitude4 ,it is to just such a diagram that the animal's skeleton tends to approximate4 #fbg ,when the jockey crouches over the neck of his race-horse1 and when ,tod ,sloan introduced the 8,american seat10 the object in both cases is to relieve the hind-legs of weight1 and so leave them free for the work of propulsion4 ,nevertheless1 we must not exaggerate the share taken by the hind-limbs in this latter duty2 cf4 ,stillman1 ,the ,horse in ,motion1 p4 #fi1 #ahhb4 #fbh ,this and the following diagrams are borrowed and adapted from ,professor ,fidler's ,bridge ,construction4 #fbi ,the method of constructing reciprocal diagrams1 in which one should represent the outlines of a frame1 and the other the system of forces necessary to keep it in equilibrium1 was first indicated in ,culmann's ,graphische ,statik2 it was greatly developed soon afterwards by ,macquorn ,rankine 7,phil4 ,mag4 ,feb4 #ahfd1 and ,applied ,mechanics1 passim71 to whom is mainly due the general application of the principle to engineering practice4 #fcj ,dialogues concerning ,two ,new ,sciences 7#afch73 ,crew and ,salvio's translation1 p4 #adj seq4 #fca ,the form and direction of the vertebral spines have been frequently and elaborately described2 cf4 7e4g47 ,gottlieb1 ,h41 ,die ,anticlinie der ,wirbelsule der ,sugethiere1 ,morphol4 ,jahrb4 ,,lxix1 pp4 #agi-bbj1 #aiae1 and many works quoted therein4 ,according to ,morita1 ,ueber die ,ursachen der ,richtung und ,gestalt der thoracalen ,dornfortstze der ,sugethierwirbelsule 7ibi cit4 p4 #bja71 various changes take place in the direction or inclination of these processes in rabbits1 after section of the interspinous ligaments and muscles4 ,these changes seem to be very much what we should expect1 on simple mechanical grounds4 ,see also ,fischer1 ,o41 ,theoretische ,grundlagen fr eine ,mechanik der lebenden ,krper1 ,leipzig1 pp4 #c1 #cgb1 #aijf4 #fcb ,i owe the first four of these determinations to the kindness of ,dr ,chalmers ,mitchell1 who had them made for me at the ,zoological ,society's ,gardens2 while the great ,clydesdale carthorse was weighed for me by a friend in ,dundee4 #fcc ,this pose of ,diplodocus1 and of other ,sauropodous reptiles1 has been much discussed4 ,cf4 7int4 al47 ,abel1 ,o41 ,abh4 k4 k4 zool4 bot4 ,ges4 ,wien1 ,v4 #aiji-aj 7#fj pp472 ,tornier1 ,,sb4 ,ges4 ,naturf4 ,fr4 ,berlin1 pp4 #aic-bji1 #aiji2 ,hay1 ,o4 ,p41 ,amer4 ,nat4 ,oct4 #aijh2 ,tr4 ,wash4 ,acad4 ,sci4 ,,xlii1 pp4 #a-be1 #aiaj2 ,holland1 ,amer4 ,nat4 ,may1 #aiaj1 pp4 #bei-bhc2 ,matthew1 ibid4 pp4 #edg-efj2 ,gilmore1 ,c4 ,w4 7,restoration of ,stegosaurus74 ,pr4 ,u4,s4 ,nat4 ,museum1 #aiae4 #fcd ,the form of the cantilever is much less typical in the small flying birds1 where the strength of the pelvic region is insured in another way1 with which we need not here stop to deal4 #fce ,the motto was ,macquorn ,rankine's4 #fcf ,john ,hunter was seldom wrong2 but ,i cannot believe that he was right when he said 7,scientific ,works1 ed4 ,owen1 ,i1 p4 #cga71 8,the bones1 in a mechanical view1 appear to be the first that are to be considered4 ,we can study their shape1 connexions1 number1 uses1 etc41 without considering any other part of the body40 #fcg ,origin of ,species1 #fth ed4 p4 #aah4 #fch ,amer4 ,naturalist1 ,april1 #aiae1 p4 #aih1 etc4 ,cf4 infra1 p4 #gbg4 #fci ,driesch sees in 8,entelechy0 that something which differentiates the whole from the sum of its parts in the case of the organism3 8,the organism1 we know1 is a system the single constituents of which are inorganic in themselves2 only the whole constituted by them in their typical order or arrangement owes its specificity to #8,entelechy' 0 7,gifford ,lectures1 p4 #bbi1 #aijh73 and ,i think it could be shewn that many other philosophers have said precisely the same thing4 ,so far as the argument goes1 ,i fail to see how this ,entelechy is shewn to be peculiarly or specifically related to the living organism4 ,the conception that the whole is always something very different from its parts is a very ancient doctrine4 ,the reader will perhaps remember how1 in another vein1 the theme is treated by ,martinus ,scriblerus3 8,in every ,jack there is a meat-roasting ,quality1 which neither resides in the fly1 nor in the weight1 nor in any particular wheel of the ,jack1 but is the result of the whole composition2 etc41 etc40 #fdj 8,there can be no doubt that ,fraas is correct in regarding this type 7,procetus7 as an annectant form between the ,zeuglodonts and the ,creodonta1 but1 although the origin of the ,zeuglodonts is thus made clear1 it still seems to be by no means so certain as that author believes1 that they may not themselves be the ancestral forms of the ,odontoceti02 ,andrews1 ,tertiary ,vertebrata of the ,fayum1 #aijf1 p4 #bce4 #fda ,reprinted1 with some changes and additions1 from a paper in the ,trans4 ,roy4 ,soc4 ,edin4 ,l1 pp4 #heg-ie1 #aiae4 #fdb ,m4 ,bergson repudiates1 with peculiar confidence1 the application of mathematics to biology4 ,cf4 ,creative ,evolution1 p4 #ba1 8,calculation touches1 at most1 certain phenomena of organic destruction4 ,organic creation1 on the contrary1 the evolutionary phenomena which properly constitute life1 we cannot in any way subject to a mathematical treatment40 #fdc ,in this there lies a certain justification for a saying of ,minot's1 of the greater part of which1 nevertheless1 ,i am heartily inclined to disapprove4 8,we biologists10 he says1 8cannot deplore too frequently or too emphatically the great mathematical delusion by which men often of great if limited ability have been misled into becoming advocates of an erroneous conception of accuracy4 ,the delusion is that no science is accurate until its results can be expressed mathematically4 ,the error comes from the assumption that mathematics can express complex relations4 ,unfortunately mathematics have a very limited scope1 and are based upon a few extremely rudimentary experiences1 which we make as very little children and of which no adult has any recollection4 ,the fact that from this basis men of genius have evolved wonderful methods of dealing with numerical relations should not blind us to another fact1 namely1 that the observational basis of mathematics is1 psychologically speaking1 very minute compared with the observational basis of even a single minor branch of biology 4444 ,while therefore here and there the mathematical methods may aid us1 we need a kind and degree of accuracy of which mathematics is absolutely incapable 4444 ,with human minds constituted as they actually are1 we cannot anticipate that there will ever be a mathematical expression for any organ or even a single cell1 although formulae will continue to be useful for dealing now and then with isolated details4440 7op4 cit41 p4 #ai1 #aiaa74 ,it were easy to discuss and criticise these sweeping assertions1 which perhaps had their origin and parentage in an obiter dictum of ,huxley's1 to the effect that 8,mathematics is that study which knows nothing of observation1 nothing of experiment1 nothing of induction1 nothing of causation0 7cit4 ,cajori1 ,hist of ,elem4 ,mathematics1 p4 #bhc74 ,but ,gauss called mathematics 8a science of the eye02 and ,sylvester assures us that 8most1 if not all1 of the great ideas of modern mathematics have had their origin in observation0 7,brit4 ,ass4 ,address1 #ahfi1 and ,laws of ,verse1 p4 #abj1 #ahgj74 #fdd ,historia ,animalium ,i1 #a4 #fde ,cf4 supra1 p4 #gad4 #fdf ,cf4 ,osborn1 ,h4 ,f41 ,on the ,origin of ,single ,characters1 as observed in fossil and living ,animals and ,plants1 ,amer4 ,nat4 ,,xlix1 pp4 #aic-bci1 #aiae 7and other papers72 ibid4 p4 #aid1 8,each individual is composed of a vast number of somewhat similar new or old characters1 each character has its independent and separate history1 each character is in a certain stage of evolution1 each character is correlated with the other characters of the individual 4444 ,the real problem has always been that of the origin and development of characters4 ,since the ,origin of ,species appeared1 the terms variation and variability have always referred to single characters2 if a species is said to be variable1 we mean that a considerable number of the single characters or groups of characters of which it is composed are variable10 etc4 #fdg ,cf4 ,sorby1 ,quart4 ,journ4 ,geol4 ,soc4 7,proc471 #ahgi1 p4 #hh4 #fdh ,cf4 ,d',orbigny1 ,alc41 ,cours lm4 de ,palontologie1 etc41 ,i1 pp4 #add-adh1 #ahdi2 see also ,sharpe1 ,daniel1 ,on ,slaty ,cleavage1 ,q4,j4,g4,s4 ,,iii1 p4 #gd1 #ahdg4 #fdi ,thus ,ammonites erugatus1 when compressed1 has been described as ,a4 planorbis3 cf4 ,blake1 ,j4 ,f41 ,phil4 ,mag4 7#e71 ,,vi1 p4 #bfj1 #ahgh4 ,wettstein has shewn that several species of the fish-genus ,lepidopus have been based on specimens artificially deformed in various ways3 ,ueber die ,fischfauna des ,tertiren ,glarnerschiefers1 ,abh4 ,schw4 ,palaeont4 ,gesellsch4 ,,xiii1 #ahhf 7see especially pp4 #bc-ch1 pl4 ,i74 ,the whole subject1 interesting as it is1 has been little studied3 both ,blake and ,wettstein deal with it mathematically4 #fej ,cf4 ,sir ,thomas ,browne1 in ,the ,garden of ,cyrus3 8,but why ofttimes one side of the leaf is unequall unto the other1 as in ,hazell and ,oaks1 why on either side the master vein the lesser and derivative channels stand not directly opposite1 nor at equall angles1 respectively unto the adverse side1 but those of one side do often exceed the other1 as the ,wallnut and many more1 deserves another enquiry40 #fea ,where gourds are common1 the glass-blower is still apt to take them for a prototype1 as the prehistoric potter also did4 ,for instance1 a tall1 annulated ,florence oil-flask is an exact but no longer a conscious imitation of a gourd which has been converted into a bottle in the manner described4 #feb ,cf4 ,elsie ,venner1 chap4 ii4 #fec ,this significance is particularly remarkable in connection with the development of speed1 for the metacarpal region is the seat of very important leverage in the propulsion of the body4 ,in the ,museum of the ,royal ,college of ,surgeons in ,edinburgh1 there stand side by side the skeleton of an immense carthorse 7celebrated for having drawn all the stones of the ,bell ,rock ,lighthouse to the shore71 and a beautiful skeleton of a racehorse1 which 7though the fact is disputed7 there is good reason to believe is the actual skeleton of ,eclipse4 ,when ,i was a boy my grandfather used to point out to me that the cannon-bone of the little racer is not only relatively1 but actually1 longer than that of the great ,clydesdale4 #fed ,cf4 ,vitruvius1 ,,iii1 #a4 #fee ,les quatres livres d',albert ,drer de la proportion des parties et pourtraicts des corps humains1 ,arnheim1 #afac1 folio 7and earlier editions74 ,cf4 also ,lavater1 ,essays on ,physiognomy1 ,,iii1 p4 #bga1 #agii4 #fef ,it was these very drawings of ,drer's that gave to ,peter ,camper his notion of the 8facial angle40 ,camper's method of comparison was the very same as ours1 save that he only drew the axes1 without filling in the network1 of his coordinate system2 he saw clearly the essential fact1 that the skull varies as a whole1 and that the 8facial angle0 is the index to a general deformation4 8,the great object was to shew that natural differences might be reduced to rules1 of which the direction of the facial line forms the norma or canon2 and that these directions and inclinations are always accompanied by correspondent form1 size and position of the other parts of the cranium10 etc42 from ,dr ,t4 ,cogan's preface to ,camper's work ,on the ,connexion between the ,science of ,anatomy and the ,arts of ,drawing1 ,painting and ,sculpture 7#agfh871 quoted in ,dr ,r4 ,hamilton's ,memoir of ,camper1 in ,lives of ,eminent ,naturalists 7,nat4 ,libr471 ,edin4 #ahdj4 #feg ,the co-ordinate system of ,fig4 #chb is somewhat different from that which ,i drew and published in my former paper4 ,it is not unlikely that further investigation will further simplify the comparison1 and shew it to involve a still more symmetrical system4 #feh ,dinosaurs of ,north ,america1 pl4 ,,lxxxi1 etc4 #ahif4 #fei ,mem4 ,amer4 ,mus4 of ,nat4 ,hist4 ,i1 ,,iii1 #ahih4 #ffj ,these and also other coordinate diagrams will be found in ,mr ,g4 ,heilmann's book ,fuglenes ,afstamning1 #cih pp41 ,copenhagen1 #aiaf2 see especially pp4 #cfh-chj4 #ffa ,cf4 ,w4 ,b4 ,scott 7,amer4 ,journ4 of ,science1 ,,xlviii1 pp4 #cce-cgd1 #ahid71 8,we find that any mammalian series at all complete1 such as that of the horses1 is remarkably continuous1 and that the progress of discovery is steadily filling up what few gaps remain4 ,so closely do successive stages follow upon one another that it is sometimes extremely difficult to arrange them all in order1 and to distinguish clearly those members which belong in the main line of descent1 and those which represent incipient branches4 ,some phylogenies actually suffer from an embarrassment of riches40 #ffb ,cf4 ,dwight1 ,t41 ,the ,range of ,variation of the ,human ,scapula1 ,amer4 ,nat4 ,,xxi1 pp4 #fbg-fch1 #ahhg4 ,cf4 also ,turner1 ,challenger ,rep4 ,,xlvii1 on ,human ,skeletons1 p4 #hf1 #ahhf3 8,i gather both from my own measurements1 and those of other observers1 that the range of variation in the relative length and breadth of the scapula is very considerable in the same race1 so that it needs a large number of bones to enable one to obtain an accurate idea of the mean of the race40 #ffc ,there is a paper on the mathematical study of organic forms and organic processes by the learned and celebrated ,gustav ,theodor ,fechner1 which ,i have only lately read1 but which would have been of no little use and help to our argument had ,i known it before4 7,ueber die mathematische ,behandlung organischer ,gestalten und ,processe1 ,berichte d4 k4 schs4 ,gesellsch41 ,math4-phys4 ,cl41 ,leipzig1 #ahdi1 pp4 #ej-fd47 ,fechner's treatment is more purely mathematical and less physical in its scope and bearing than ours1 and his paper is but a short one2 but the conclusions to which he is led differ little from our own4 ,let me quote a single sentence which1 together with its context1 runs precisely on the lines of the discussion with which this chapter of ours began4 8,so ist also die mathematische ,bestimmbarkeit im ,gebiete des ,organischen ganz eben so gut vorhanden als in dem des ,unorganischen1 und in letzterem eben solchen oder quivalenten ,beschrnkungen unterworfen als in ersterem2 und nur sofern die unorganischen ,formen und das unorganische ,geschehen sich einer einfacheren ,gesetzlichkeit mehr nhern als die organischen1 kann die ,approximation im unorganischen ,gebiet leichter und weiter getrieben werden als im organischen4 ,dies wre der ganze1 sonach rein relative1 ,unterschied40 ,here in a nutshell1 in words written some seventy years ago1 is the gist of the whole matter4 ,an interesting little book of ,schiaparelli's 7which ,i ought to have known long ago7--,forme organiche naturali e forme geometriche pure1 ,milano1 ,hoepli1 #ahih--;has likewise come into my hands too late for discussion4 #ghj ,,index4 ,abbe's diffraction plates1 #cbc ,abel1 ,o41 #gjf ,abonyi1 ,a41 #abg ,acantharia1 spicules of1 #deh ,acanthometridae1 #dfb ,acceleration1 #fd ,aceratherium1 #gfa ,achlya1 #bdd ,acromegaly1 #ace ,actinomma1 #dfi ,actinomyxidia1 #deb ,actinophrys1 #afe1 #aig1 #bfd1 #bih ,actinosphaerium1 #aig1 #bff1 #bih1 #dfh ,adams1 ,j4 ,c41 #ffc ,adaptation1 #fgj ,addison1 ,joseph1 #fga ,adiantum1 #djh ,adsorption1 #aib1 #bjh1 #bda1 #bgg1 #ceg2 orientirte1 #ddj1 #eij2 pseudo1 #bhb ,agglutination1 #bja ,aglaophenia1 #gdh ,airy1 ,h41 #fcf ,albumin molecule1 #da ,alcyonaria1 #chg1 #dac1 #dbd1 #dei ,alexeieff1 ,a41 #aeg1 #afe ,allmann1 ,w41 #fdc ,alpheus1 claws of1 #aej ,alpine 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#ach1 #adi1 #aeh1 #eji1 #fec1 #gad1 #gbe1 #gbf ,arizona trees1 #aba ,arrhenius1 ,sv41 #bh1 #dh1 #aga ,artemia1 #abg ,artemis1 #efa ,ascaris megalocephala1 #ahj1 #aie ,aschemonella1 #bee ,assheton1 ,r41 #cdd ,asterina1 #cdb ,asteroides1 #dbc ,asterolampra1 #chf ,asters1 #afg1 #agd ,asthenosoma1 #ffd ,astrorhiza1 #bee1 #dfc1 #ehg1 #fjg ,astrosclera1 #dcf ,asymmetric substances1 #daf ,asymmetry1 #bda ,atrypa1 #efi ,auerbach1 ,f41 #i ,aulacantha1 #dfj ,aulastrum1 #dga ,aulonia1 #dfh ,auricular height1 #ic ,autocatalysis1 #aca ,auximones1 #ace ,awerinzew1 ,s41 #ehi ,babak1 ,e41 #cb ,babirussa1 teeth of1 #fcd ,baboon1 skull of1 #gga ,bacillus1 #ci2 ,b4 ramosus1 #acc ,bacon1 ,lord1 #d1 #e1 #ea1 #ec1 #aca1 #fef1 #gaf ,bacteria1 #bde1 #bej ,baer1 ,k4 ,e41 von1 #c1 #ee1 #eg1 #aee ,balancement1 #gad1 #ggf ,balfour1 ,f4 ,m41 #eg1 #cdh ,baltzer1 ,fr41 #cbg ,bamboo1 growth of1 #gg ,barclay1 ,j41 #ccd ,barfurth1 ,d41 #he ,barlow1 ,w41 #bjb ,barratt1 ,j4 ,o4 ,w41 #bhe ,bartholinus1 ,e41 #cbi 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,research4 ,by ,,james ,,johnstone1 ,fisheries ,laboratory1 ,university of ,liverpool4 ,with chart and #ca illustrations4 ,demy #hvo4 #is net4 ,,mendel',s ,,principles ,,of ,,heredity4 ,by ,w4 ,,bateson1 ,m4,a41 ,f4,r4,s41 ,v4,m4,h41 ,director of the ,john ,innes ,horticultural ,institution4 ,third impression with additions4 ,with #c portraits1 #f coloured plates1 and #ch figures4 ,royal #hvo4 #abs net4 ,,the ,,elements ,,of ,,botany4 ,by ,sir ,,francis ,,darwin1 ,sc4,d41 ,m4,b41 ,f4,r4,s41 ,fellow of ,christ's ,college4 ,second edition4 ,with #id illustrations4 ,crown #hvo4 #ds #f;d net4 ,cambridge ,biological ,series4 ,,practical ,,physiology ,,of ,,plants4 ,by ,sir ,,francis ,,darwin1 ,sc4,d41 ,f4,r4,s41 and ,e4 ,,hamilton ,,acton1 ,m4,a4 ,third edition4 ,with #de illustrations4 ,crown #hvo4 #ds #f;d net4 ,cambridge ,biological ,series4 ,,botany4 ,a ,text-,book for ,senior ,students4 ,by ,d4 ,,thoday1 ,m4,a41 ,lecturer in ,physiological ,botany and ,assistant ,director of the ,botanical ,laboratories in the ,university of ,manchester4 ,with #bje figures4 ,large crown #hvo4 #es #f;d net4 ,,alg4 ,volume ,i1 ,myxophyce1 ,peridinie1 ,bacillarie1 ,chlorophyce1 together with a brief summary of the ,occurrence and ,distribution of ,freshwater ,alg4 ,by ,g4 ,s4 ,,west1 ,m4,a41 ,d4,sc41 ,a4,r4,c4,s41 ,f4,l4,s41 ,mason ,professor of ,botany in the ,university of ,birmingham4 ,with #bga illustrations4 ,large royal #hvo4 #bes net4 ,cambridge ,botanical ,handbooks4 ,,the ,,philosophy ,,of ,,biology4 ,by ,,james ,,johnstone1 ,d4,sc4 ,demy #hvo4 #is net4 ,,cambridge ,,manuals ,,of ,,science ,,and ,,literature4 ,general ,editors3 ,p4 ,,giles1 ,litt4,d41 and ,a4 ,c4 ,,seward1 ,m4,a41 ,f4,r4,s4 ,royal #afmo4 ,cloth1 #as #c;d net each4 ,leather1 #bs #f;d net each4 ,,the ,,coming ,,of ,,evolution4 ,by ,,john ,w4 ,,judd1 ,c4,b41 ,,ll4,d41 ,f4,r4,s4 ,with #d plates4 ,,heredity ,,in ,,the ,,light ,,of ,,recent ,,research4 ,by ,l4 ,,doncaster1 ,sc4,d4 ,with #ab figures4 ,,prehistoric ,,man4 ,by ,w4 ,l4 ,h4 ,,duckworth1 ,m4,a41 ,m4,d41 ,sc4,d4 ,with #b tables and #bh figures4 ,,primitive ,,animals4 ,by ,g4 ,,smith1 ,m4,a4 ,with #bf figures4 ,,the ,,life-,,story ,,of ,,insects4 ,by ,prof4 ,g4 ,h4 ,,carpenter4 ,with #bd illustrations4 ,,earthworms ,,and ,,their ,,allies4 ,by ,,frank ,e4 ,,beddard1 ,m4,a4 7,oxon471 ,f4,r4,s41 ,f4,r4,s4,e4 ,with #ac figures4 ,,spiders4 ,by ,,cecil ,,warburton1 ,m4,a4 ,with #ac figures4 ,,bees ,,and ,,wasps4 ,by ,o4 ,h4 ,,latter1 ,m4,a41 ,f4,e4,s4 ,with #ba illustrations4 ,,the ,,flea4 ,by ,h4 ,,russell4 ,with #i illustrations4 ,,life ,,in ,,the ,,sea4 ,by ,,james ,,johnstone1 ,b4,sc4 ,with frontispiece1 #d figures and #e tailpieces4 ,,plant-,,animals4 ,by ,f4 ,w4 ,,keeble1 ,sc4,d4 ,with #bc figures4 ,,plant-,,life ,,on ,,land ,,considered ,,in ,,some ,,of ,,its ,,biological ,,aspects4 ,by ,f4 ,o4 ,,bower1 ,sc4,d41 ,f4,r4,s4 ,with #bg figures4 ,,links ,,with ,,the ,,past ,,in ,,the ,,plant ,,world4 ,by ,a4 ,c4 ,,seward1 ,m4,a41 ,f4,r4,s4 ,with frontispiece and #bj figures4 ,cambridge ,university ,press ,fetter ,lane1 ,london3 ,c4 ,f4 ,clay1 ,manager ,,transcriber',s ,,note ,original spelling and grammar have been generally retained1 with some exceptions noted below4 ,original printed page numbers are shown like this3 #eb4 ,original small caps are now uppercase4 ,italics look like this4 ,footnotes have been renumbered #a-ffc and relocated to the end of the book1 ahead of the ,index4 ,the transcriber produced the cover image and hereby assigns it to the public domain4 ,original page images are available from archive4org--search for 8ongrowthform#aiagthom04 ,some tables and illustrations have been moved from their original locations within paragraphs of text to nearby locations between paragraphs4 ,this includes1 for example1 the full-page table printed on page #fg1 which page number is removed from these ebook editions4 ,some other tables and illustrations have been left where they originally lay1 in the middle of paragraphs of text4 ,some1 but not all ditto marks 1 including 8do401 have been eliminated4 ,enlarged curly brackets turned horizontal1 used as graphic devices in the printed book to combine information in two or more columns of a table1 have been eliminated4 ,enlarged curly brackets 1 used as graphic devices to suggest combination of information over two or more lines of text1 have been eliminated4 ,for example1 on page #ge1 in the last column of the table1 two lengths1 #dij and #ejj1 were printed1 the latter under the former1 with a large right curly bracket combining them4 ,the transcriber has changed that construction to 8#dij-ejj01 taking the original to mean a range4 ,page #ajf4 ,changed 8it we could believe0 to 8if we could believe04 ,page #ajg4 ,changed 87m470 to 87mm4701 in column #c of the table4 ,page #aag4 ,both 8,q#aj0 and 8,q#aj0 appear on the page as originally printed4 ,page #bgbn4 ,changed 8,proc4 ,r y4 ,soc4 ,,xii0 to 8,proc4 ,roy4 ,soc4 ,,xii04 ,page #cfh4 ,perhaps1 the original 8,the area of the enlarged sector1 p,,oa0 should read 8,the area of the enlarged sector1 ,p,,oa04 ,page #dbhn4 ,changed 8,phenonemon0 to 8,phenomenon04 ,page #dfcn4 ,changed 8,raphidophrys0 to 8,raphidiophrys04 ,page #edc4 ,the ,unicode character u#b;a#h;c greater-than above double-line equal above less-than is pretty rare1 and may not display properly in most fonts4 ,an image is used instead of the ,unicode in all but the simple text edition4 ,page #fgf4 ,the ,unicode character u#bccf ,,apl functional symbol i-beam is also unusual4 ,an image is substituted in all but the simple text edition4 ,page #gdh4 ,changed 8,fig4 #dgd0 to 8,fig4 #cgd04 ,page #gfh4 ,changed 8in the case of ,pro ohippus0 to 8in the case of ,protohippus04 ,there were three footnotes on this page1 but only two footnote anchors4 ,the second footnote1 missing an anchor1 said 8 ,cf4 ,zittel1 ,grundzge d4 ,palaeontologie1 p4 #dfc1 #aiaa40