,transcriber's notes3 ,the text of this e-book has mostly been preserved in its original form including some inconsistency of hyphenation and use of diacritics 7aeriform_/ariform74 ,three spelling typos have been corrected 7arrangment arrangement1 pully pulley1 dye die7 as have typos in equations on pages #dj and #dc4 ,some missing punctuation has been corrected silently 7periods1 commas1 incorrect quotes74 ,footnotes have been positioned below the relevant paragraphs4 ,in this plain text version1 italic text is denoted by underscores and bold text by 99asterisks994 ,a caret 77 indicates that the following character is superscripted1 e4g4 ,d#b 7,d squared74 ,a ,,treatise ,,on ,,mechanics1 ,,by ,,captain ,,henry ,,kater1 ,v4 ,,pres3 ,r4,s4 ----and---- ,,dionysius ,,lardner1 ,d4,c4,l4 ,f4,r4,s4 c4 c4 ,a ,,new ,,edition ,,revised ,,corrected4 #aheb4 ,illustration3 ,h4 ,corbould del4 ,e4 ,finder fc4 ,london3 ,,printed ,,for ,,longman1 ,,brown1 ,,green ,,longmans4 ,,paternoster ,,row3 ,,advertisement4 ,this ,treatise on ,mechanics1 which was originally published in #ahcj1 is the work of ,dr4 ,lardner1 with the exception of the twenty-first chapter1 which was written by the late ,captain ,kater4 ,the present edition has been revised and corrected by ,dr4 ,lardner4 ,london1 ,january1 #aheb4 ,,contents4 ,,chap4 ,i4 ,,properties ,,of ,,matter4 ,organs of ,sense4--,sensations4--,properties or ,qualities4--,observation4 --,comparison and ,generalisation4--,particular and general ,qualities4-- ,magnitude4--,size4--,volume4--,lines4-- ,surfaces4--,edges4--,area4--,length4 --,impenetrability4--,apparent ,penetration4--,figure4--,different from ,volume4--,atoms4--,molecules4--,matter separable4--,particles4--,force4-- ,cohesion of ,atoms4--,hypothetical ,phrases unnecessary4--,attraction4 #a ,,chap4 ,,ii4 ,,properties ,,of ,,matter1 ,,continued4 ,divisibility4--,unlimited ,divisibility4--,wollaston's micrometric ,wire4 --,method of making it4--,thickness of a ,soap ,bubble4--,wings of ,insects4 --,gilding of ,wire for ,embroidery4--,globules of the ,blood4-- ,animalcules4--,their minute ,organisation4--,ultimate ,atoms4--,crystals4-- ,porosity4--,volume4--,density4--,quicks ilver passing through ,pores of ,wood4--,filtration4--,porosity of ,hydrophane4--,compressibility4-- ,elasticity4--,dilatability4--,heat4--,c ontraction of ,metal used to restore the ,perpendicular to ,walls of a ,building4--,impenetrability of ,air4--,compressibility of it4--,elasticity of it4--,liquids not absolutely incompressible4--,experiments4--,elastic ity of ,fluids4-- ,aeriform ,fluids4--,domestic ,fire ,box4--,evolution of ,heat by compressed ,air4 #i ,,chap4 ,,iii4 ,,inertia4 ,inertia4--,matter ,incapable of spontaneous ,change4--,impediments to ,motion4--,motion of the ,solar ,system4--,law of ,nature4--,language used to express ,inertia sometimes faulty4--,familiar ,examples of ,inertia4 #bg ,,chap4 ,,iv4 ,,action ,,and ,,reaction4 ,inertia in a single ,body4--,consequences of ,inertia in two or more ,bodies4--,examples4--,effects of ,impact4--,motion not estimated by ,speed or ,velocity alone4--,examples4--,rule for estimating the ,quantity of ,motion4--,action and ,reaction4--,examples of4--,velocity of two ,bodies after ,impact4--,rule for finding the common ,velocity after ,impact4--,magnet and ,iron4--,feather and ,cannon ,ball impinging4 --,newton's ,laws of ,motion4--,inutility of4--,familiar ,effects resulting from ,consequences of ,inertia4 #cd ,,chap4 ,v4 ,,composition ,,and ,,resolution ,,of ,,force4 ,motion and ,pressure4--,force4--,attraction4--,para llelogram of ,forces4 --,resultant4--,components4--,compositio n of ,force4--,resolution of ,force4--,illustrative ,experiments4--,composition of ,pressures4-- ,theorems regulating ,pressures also regulate ,motion4--,examples4-- ,resolution of ,motion4--,forces in ,equilibrium4--,composition of ,motion and ,pressure4--,illustrations4--,boat in a ,current4--,motions of ,fishes4 --,flight of ,birds4--,sails of a ,vessel4--,tacking4--,equestrian ,feats4 --,absolute and relative ,motion4 #dh ,,chap4 ,,vi4 ,,attraction4 ,impulse4--,mechanical ,state of ,bodies4--,absolute ,rest4--,uniform and rectilinear ,motion4--,attractions4--,molecular or atomic4--,interstitial ,spaces in ,bodies4--,repulsion and ,attraction4--,cohesion4--,in ,solids and ,fluids4--,manufacture of ,shot4--,capillary ,attractions4--,shortening of ,rope by ,moisture4--,suspension of ,liquids in capillary ,tubes4-- ,capillary ,siphon4--,affinity between ,quicksilver and ,gold4--,examples of ,affinity4--,sulphuric ,acid and ,water4--,oxygen and ,hydrogen4--,oxygen and ,quicksilver4--,magnetism4--,electricity and ,electro-,magnetism4-- ,gravitation4--,its ,law4--,examples of4--,depends on the ,mass4-- ,attraction between the ,earth and detached ,bodies on its ,surface4-- ,weight4--,gravitation of the ,earth4--,illustrated by ,projectiles4-- ,plumb-,line4--,cavendish's ,experiments4 #fc ,,chap4 ,,vii4 ,,terrestrial ,,gravity4 ,phenomena of falling ,bodies4--,gravity greater at the ,poles than ,equator4--,heavy and light ,bodies fall with equal ,speed to the ,earth4 --,experiment4--,increased ,velocity of falling ,bodies4--,principles of uniformly accelerated ,motion4--,relations between the ,height1 ,time1 and ,velocity4--,attwood's ,machine4--,retarded ,motion4 #hd ,,chap4 ,,viii4 ,,of ,,the ,,motion ,,of ,,bodies ,,on ,,inclined ,,planes ,,and ,,curves4 ,force perpendicular to a ,plane4--,oblique ,force4--,inclined ,plane4-- ,weight produces ,pressure and ,motion4--,motion uniformly accelerated4-- ,space moved through in a given ,time4--,increased ,elevation produces increased ,force4--,perpendicular and horizontal ,plane4--,final ,velocity4--,motion down a ,curve4--,depends upon ,velocity and ,curvature4 --,centrifugal ,force4--,circle of ,curvature4--,radius of ,curvature4-- ,whirling ,table4--,experiments4--,solar ,system4--,examples of centrifugal ,force4 #he ,,chap4 ,,ix4 ,,the ,,centre ,,of ,,gravity4 ,terrestrial ,attraction the combined ,action of parallel ,forces4-- ,single equivalent ,force4--,examples4--,method of finding the ,centre of ,gravity4--,line of ,direction4--,globe4--,oblate ,spheroid4--,prolate ,spheroid4--,cube4--,straight ,wand4--,flat ,plate4--,triangular ,plate4-- ,centre of ,gravity not always within the ,body4--,a ,ring4--,experiments4 --,stable1 instable1 and neutral ,equilibrium4--,motion and ,position of the ,arms and ,feet4--,effect of the ,knee-,joint4--,positions of a ,dancer4 --,porter under a ,load4--,motion of a ,quadruped4--,rope ,dancing4-- ,centre of ,gravity of two ,bodies separated from each other4-- ,mathematical and experimental ,examples4--,the ,conservation of the ,motion of the ,centre of ,gravity4--,solar ,system4--,centre of ,gravity sometimes called ,centre of ,inertia4 #ajg ,,chap4 ,x4 ,,the ,,mechanical ,,properties ,,of ,,an ,,axis4 ,an ,axis4--,planets and common spinning ,top4--,oscillation or ,vibration4 --,instantaneous and continued ,forces4--,percussion4--,continued ,force4 --,rotation4--,impressed ,forces4--,properties of a fixed ,axis4--,movement of the ,force round the ,axis4--,leverage of the ,force4--,impulse perpendicular to1 but not crossing1 the ,axis4--,radius of ,gyration4-- ,centre of ,gyration4--,moment of ,inertia4--,principal ,axes4--,centre of ,percussion4 #abh ,,chap4 ,,xi4 ,,of ,,the ,,pendulum4 ,isochronism4--,experiments4--,simple ,pendulum4--,examples illustrative of4--,length of4--,experiments of ,kater1 ,biot1 ,sabine1 and others4-- ,huygens' ,cycloidal ,pendulum4 #ade ,,chap4 ,,xii4 ,,of ,,simple ,,machines4 ,statics4--,dynamics4--,force4--,power4- -,weight4--,lever4--,cord4-- ,inclined ,plane4 #afj ,,chap4 ,,xiii4 ,,of ,,the ,,lever4 ,arms4--,fulcrum4--,three ,kinds of ,levers4--,crow ,bar4--,handspike4--,oar4 --,nutcrackers4--,turning ,lathe4--,steelyard4--,rectangular ,lever4-- ,hammer4--,load between two ,bearers4--,combination of ,levers4-- ,equivalent ,lever4 #afg ,,chap4 ,,xiv4 ,,of ,,wheel-,,work4 ,wheel and ,axle4--,thickness of the ,rope4--,ways of applying the ,power4 --,projecting ,pins4--,windlass4--,winch4--,axle4--,ho rizontal ,wheel4-- ,tread-,mill4--,cranes4--,water-,wheels4 --,paddle-,wheel4--,rachet-,wheel4-- ,rack4--,spring of a ,watch4--,fusee4--,straps or ,cords4--,examples of4-- ,turning ,lathe4--,revolving ,shafts4--,spinning ,machinery4--,saw-,mill4-- ,pinion4--,leaves4--,crane4--,spur-,whee ls4--,crown-,wheels4--,bevelled ,wheels4--,hunting-,cog4--,chronometers4 --,hair-,spring4--,balance-,wheel4 #agh ,,chap4 ,,xv4 ,,of ,,the ,,pulley4 ,cord4--,sheave4--,fixed ,pulley4--,fire ,escapes4--,single moveable ,pulley4--,systems of ,pulleys4--,smeaton's ,tackle4--,white's ,pulley4-- ,advantage of4--,runner4--,spanish ,bartons4 #aii ,,chap4 ,,xvi4 ,,on ,,the ,,inclined ,,plane1 ,,wedge1 ,,and ,,screw4 ,inclined ,plane4--,effect of a ,weight on4--,power of4--,roads4--,power ,oblique to the ,plane4--,plane sometimes moves under the ,weight4-- ,wedge4--,sometimes formed of two inclined ,planes4--,more powerful as its ,angle is acute4--,where used4--,limits to the ,angle4--,screw4 --,hunter's ,screw4--,examples4--,micrometer ,screw4 #bji ,,chap4 ,,xvii4 ,,on ,,the ,,regulation ,,and ,,accumulation ,,of ,,force4 ,uniformity of ,operation4--,irregularity of prime ,mover4--,water-,mill4 --,wind-,mill4--,steam ,pressure4--,animal ,power4--,spring4--,regulators4 --,steam-,engine4--,governor4--,self-act ing ,damper4--,tachometer4-- ,accumulation of ,power4--,examples4--,hammer4--,flail4-- ,bow-string4-- ,fire ,arms4--,air-,gun4--,steam-,gun4--,inert ,matter a ,magazine for ,force4--,fly-,wheel4--,condensed ,air4--,rolling ,metal4--,coining-,press4 #bbd ,,chap4 ,,xviii4 ,,mechanical ,,contrivances ,,for ,,modifying ,,motion4 ,division of ,motion into rectilinear and rotatory4--,continued and reciprocating4--,examples4--,flowing ,water4--,wind4--,animal ,motion4-- ,falling of a ,body4--,syringe-,pump4--,hammer4--,stea m-,engine4--,fulling ,mill4--,rose-,engine4--,apparatus of ,zureda4--,leupold's ,application of it4--,hooke's universal ,joint4--,circular and alternate ,motion4-- ,examples4--,watt's ,methods of connecting the ,motion of the ,piston with that of the ,beam4--,parallel ,motion4 #bde ,,chap4 ,,xix4 ,,of ,,friction ,,and ,,the ,,rigidity ,,of ,,cordage4 ,friction and ,rigidity4--,laws of ,friction4--,rigidity of ,cordage4-- ,strength of ,materials4--,resistance from ,friction4--,independent of the ,magnitude of ,surfaces4--,examples4--,vince's ,experiments4--,effect of ,velocity4--,means for diminishing ,friction4--,friction ,wheels4-- ,angle of ,repose4--,best ,angle of ,draught4--,rail-,roads4--,stiffness of ,ropes4 #bfj ,,chap4 ,,xx4 ,,on ,,the ,,strength ,,of ,,materials4 ,difficulty of determining the ,laws which govern the ,strength of ,materials4--,forces tending to separate the ,parts of a ,solid4--,laws by which ,solids resist ,compression4--,euler's theory4--,transverse ,strength of ,solids4--,strength diminished by the ,increase of ,height4 --,lateral or ,transverse ,strain4--,limits of ,magnitude4--,relative ,strength of small ,animals greater than large ones4 #bgb ,,chap4 ,,xxi4 ,,on ,,balances ,,and ,,pendulums4 ,weight4--,time4--,the ,balance4--,fulcrum4--,centre of ,gravity of4-- ,sensibility of4--,positions of the ,fulcrum4--,beam variously constructed4--,troughton's ,balance4--,robinson's ,balance4--,kater's ,balance4--,method of adjusting a ,balance4--,use of it4--,precautions necessary4--,of ,weights4--,adjustment of4--,dr4 ,black's ,balance4-- ,steelyard4--,roman ,statera or ,steelyard4--,convenience of4--,c4 ,paul's ,steelyard4--,chinese ,steelyard4--,danish ,balance4--,bent ,lever ,balance4 --,brady's ,balance4--,weighing ,machine for ,turnpike ,roads4--,instruments for ,weighing by means of a ,spring4--,spring ,steelyard4--,salter's ,spring ,balance4--,marriott's ,dial ,weighing ,machine4--,dynamometer4-- ,compensation ,pendulums4--,barton's ,gridiron ,pendulum4--,table of linear ,expansion4--,second ,table4--,harrison's ,pendulum4--,troughton's ,pendulum4--,benzenberg's ,pendulum4--,ward's ,compensation ,pendulum4-- ,compensation ,tube of ,julien le ,roy4--,deparcieux's ,compensation4-- ,kater's ,pendulum4--,reed's ,pendulum4--,ellicott's ,pendulum4--,mercurial ,pendulum4--,graham's ,pendulum4--,compensation ,pendulum of ,wood and ,lead4--,smeaton's ,pendulum4--,brown's ,mode of ,adjustment4 #bgh ,,the ,,elements ,,of ,,mechanics4 ,,chap4 ,i4 ,,properties ,,of ,,matter--,,magnitude--,,impenetrability --,,figure--,,force4 7#a47 ,placed in the material world1 ,man is continually exposed to the action of an infinite variety of objects by which he is surrounded4 ,the body1 to which the thinking and living principles have been united1 is an apparatus exquisitely contrived to receive and to transmit impressions4 ,its various parts are organised with obvious reference to the several external agents by which it is to be effected4 ,each organ is designed to convey to the mind immediate notice of some peculiar action1 and is accordingly endued with a corresponding susceptibility4 ,this adaptation of such organs to the particular influences of material agents1 is rendered still more conspicuous when we consider that1 however delicate its structure1 each organ is wholly insensible to every influence except that to which it appears to be specially appropriated4 ,the eye1 so intensely susceptible of impressions from light1 is not at all affected by those of sound2 while the fine mechanism of the ear1 so sensitively alive to every effect of the latter class1 is altogether insensible to the former4 ,the splendour of excessive light may occasion blindness1 and deafness may result from the roar of a cannonade2 but neither the sight nor the hearing can be injured by the most extreme action of that principle which is designed to affect the other4 ,thus the organs of sense are instruments by which the mind is enabled to determine the existence and the qualities of external things4 ,the effects which these objects produce upon the mind through the organs1 are called sensations1 and these sensations are the immediate elements of all human knowledge4 ,,matter is the general name which has been given to that substance1 which1 under forms infinitely various1 affects the senses4 ,metaphysicians have differed in defining this principle4 ,some have even doubted of its existence4 ,but these discussions are beyond the sphere of mechanical philosophy1 the conclusions of which are in nowise affected by them4 ,our investigations here relate1 not to matter as an abstract existence1 but to those qualities which we discover in it by the senses1 and of the existence of which we are sure1 however the question as to matter itself may be decided4 ,when we speak of 8bodies10 we mean those things1 whatever they be1 which excite in our minds certain sensations2 and the powers to excite those sensations are called 8properties10 or 8qualities40 7#b47 ,to ascertain by observation the properties of bodies1 is the first step towards obtaining a knowledge of nature4 ,hence man becomes a natural philosopher the moment he begins to feel and to perceive4 ,the first stage of life is a state of constant and curious excitement4 ,observation and attention1 ever awake1 are engaged upon a succession of objects new and wonderful4 ,the large repository of the memory is opened1 and every hour pours into it unbounded stores of natural facts and appearances1 the rich materials of future knowledge4 ,the keen appetite for discovery implanted in the mind for the highest ends1 continually stimulated by the presence of what is novel1 renders torpid every other faculty1 and the powers of reflection and comparison are lost in the incessant activity and unexhausted vigour of observation4 ,after a season1 however1 the more ordinary classes of phenomena cease to excite by their novelty4 ,attention is drawn from the discovery of what is new1 to the examination of what is familiar4 ,from the external world the mind turns in upon itself1 and the feverish astonishment of childhood gives place to the more calm contemplation of incipient maturity4 ,the vast and heterogeneous mass of phenomena collected by past experience is brought under review4 ,the great work of comparison begins4 ,memory produces her stores1 and reason arranges them4 ,then succeed those first attempts at generalisation which mark the dawn of science in the mind4 ,to compare1 to classify1 to generalise1 seem to be instinctive propensities peculiar to man4 ,they separate him from inferior animals by a wide chasm4 ,it is to these powers that all the higher mental attributes may be traced1 and it is from their right application that all progress in science must arise4 ,without these powers1 the phenomena of nature would continue a confused heap of crude facts1 with which the memory might be loaded1 but from which the intellect would derive no advantage4 ,comparison and generalisation are the great digestive organs of the mind1 by which only nutrition can be extracted from this mass of intellectual food1 and without which1 observation the most extensive1 and attention the most unremitting1 can be productive of no real or useful advancement in knowledge4 7#c47 ,upon reviewing those properties of bodies which the senses most frequently present to us1 we observe that very few of them are essential to1 and inseparable from1 matter4 ,the greater number may be called particular or peculiar qualities1 being found in some bodies but not in others4 ,thus the property of attracting iron is peculiar to the loadstone1 and not observable in other substances4 ,one body excites the sensation of green1 another of red1 and a third is deprived of all colour4 ,a few characteristic and essential qualities are1 however1 inseparable from matter in whatever state1 or under whatever form it exist4 ,such properties alone can be considered as tests of materiality4 ,where their presence is neither manifest to sense1 nor demonstrable by reason1 there matter is not4 ,the principal of these qualities are magnitude and impenetrability4 7#d47 ,magnitude4--,every body occupies space1 that is1 it has magnitude4 ,this is a property observable by the senses in all bodies which are not so minute as to elude them1 and which the understanding can trace to the smallest particle of matter4 ,it is impossible1 by any stretch of imagination1 even to conceive a portion of matter so minute as to have no magnitude4 ,the quantity of space which a body occupies is sometimes called its magnitude4 ,in colloquial phraseology1 the word size is used to express this notion2 but the most correct term1 and that which we shall generally adopt is volume4 ,thus we say1 the volume of the earth is so many cubic miles1 the volume of this room is so many cubic feet4 ,the external limits of the magnitude of a body are lines and surfaces1 lines being the limits which separate the several surfaces of the same body4 ,the linear limits of a body are also called edges4 ,thus the line which separates the top of a chest from one of its sides is called an edge4 ,the quantity of a surface is called its area1 and the quantity of a line is called its length4 ,thus we say1 the area of a field is so many acres1 the length of a rope is so many yards4 ,the word 8magnitude0 is1 however1 often used indifferently for volume1 area1 and length4 ,if the objects of investigation were of a more complex and subtle character1 as in metaphysics1 this unsteady application of terms might be productive of confusion1 and even of error2 but in this science the meaning of the term is evident1 from the way in which it is applied1 and no inconvenience is found to arise4 7#e47 ,impenetrability4--,this property will be most clearly explained by defining the positive quality from which it takes its name1 and of which it merely signifies the absence4 ,a substance would be penetrable if it were such as to allow another to pass through the space which it occupies1 without disturbing its component parts4 ,thus1 if a comet striking the earth could enter it at one side1 and1 passing through it1 emerge from the other without separating or deranging any bodies on or within the earth1 then the earth would be penetrable by the comet4 ,when bodies are said to be impenetrable1 it is therefore meant that one cannot pass through another without displacing some or all of the component parts of that other4 ,there are many instances of apparent penetration2 but in all these1 the parts of the body which seem to be penetrated are displaced4 ,thus1 if the point of a needle be plunged in a vessel of water1 all the water which previously filled the space into which the needle enters will be displaced1 and the level of the water will rise in the vessel to the same height as it would by pouring in so much more water as would fill the space occupied by the needle4 7#f47 ,figure4--,if the hand be placed upon a solid body1 we become sensible of its impenetrability1 by the obstruction which it opposes to the entrance of the hand within its dimensions4 ,we are also sensible that this obstruction commences at certain places2 that it has certain determinate limits2 that these limitations are placed in certain directions relatively to each other4 ,the mutual relation which is found to subsist between these boundaries of a body1 gives us the notion of its figure4 ,the figure and volume of a body should be carefully distinguished4 ,each is entirely independent of the other4 ,bodies having very different volumes may have the same figure2 and in like manner bodies differing in figure may have the same volume4 ,the figure of a body is what in popular language is called its shape or form4 ,the volume of a body is that which is commonly called its size4 ,it will hence be easily understood1 that one body 7a globe1 for example7 may have ten times the volume of another 7globe71 and yet have the same figure2 and that two bodies 7as a die and a globe7 may have figures altogether different1 and yet have equal volumes4 ,what we have here observed of volumes will also be applicable to lengths and areas4 ,the arc of a circle and a straight line may have the same length1 although they have different figures2 and1 on the other hand1 two arcs of different circles may have the same figure1 but very unequal lengths4 ,the surface of a ball is curved1 that of the table plane2 and yet the area of the surface of the ball may be equal to that of the table4 7#g47 ,atoms--,molecules4--,impenetrability must not be confounded with inseparability4 ,every body which has been brought under human observation is separable into parts2 and these parts1 however small1 are separable into others1 still more minute4 ,to this process of division no practical limit has ever been found4 ,nevertheless1 many of the phenomena which the researches of those who have successfully examined the laws of nature have developed1 render it highly probable that all bodies are composed of elementary parts which are indivisible and unalterable4 ,the component parts1 which may be called atoms1 are so minute1 as altogether to elude the senses1 even when aided by the most powerful scientific instruments4 ,the word molecule is often used to signify component parts of a body so small as to escape sensible observation1 but not ultimate atoms1 each molecule being supposed to be formed of several atoms1 arranged according to some determinate figure4 ,particle is used also to express small component parts1 but more generally is applied to those which are not too minute to be discoverable by observation4 7#h47 ,force4--,if the particles of matter were endued with no property in relation to one another1 except their mutual impenetrability1 the universe would be like a mass of sand1 without variety of state or form4 ,atoms1 when placed in juxtaposition1 would neither cohere1 as in solid bodies1 nor repel each other1 as in aeriform substances4 ,on the contrary1 we find that in some cases the atoms which compose bodies are not simply placed together1 but a certain effect is manifested in their strong coherence4 ,if they were merely placed in juxtaposition1 their separation would be effected as easily as any one of them could be removed from one place to another4 ,take a piece of iron1 and attempt to separate its parts3 the effort will be strongly resisted1 and it will be a matter of much greater facility to move the whole mass4 ,it appears1 therefore1 that in such cases the parts which are in juxtaposition cohere and resist their mutual separation4 ,this effect is denominated force2 and the constituent atoms are said to cohere with a greater or less degree of force1 according as they oppose a greater or less resistance to their mutual separation4 ,the coherence of particles in juxtaposition is an effect of the same class as the mutual approach of particles placed at a distance from each other4 ,it is not difficult to perceive that the same influence which causes the bodies ,a and ,b to approach each other1 when placed at some distance asunder1 will1 when they unite1 retain them together1 and oppose a resistance to their separation4 ,hence this effect of the mutual approximation of bodies towards each other is also called force4 ,force is generally defined to be 8whatever produces or opposes the production of motion in matter40 ,in this sense1 it is a name for the unknown cause of a known effect4 ,it would1 however1 be more philosophical to give the name1 not to the cause1 of which we are ignorant1 but to the effect1 of which we have sensible evidence4 ,to observe and to classify is the whole business of the natural philosopher4 ,when causes are referred to1 it is implied1 that effects of the same class arise from the agency of the same cause4 ,however probable this assumption may be1 it is altogether unnecessary4 ,all the objects of science1 the enlargement of mind1 the extension and improvement of knowledge1 the facility of its acquisition1 are obtained by generalisation alone1 and no good can arise from tainting our conclusions with the possible errors of hypotheses4 ,it may be here1 once for all1 observed1 that the phraseology of causation and hypotheses has become so interwoven with the language of science1 that it is impossible to avoid the frequent use of it4 ,thus1 we say1 8the magnet attracts iron20 the expression attract intimating the cause of the observed effect4 ,in such cases1 however1 we must be understood to mean the effect itself1 finding it less inconvenient to continue the use of the received phrases1 modifying their signification1 than to introduce new ones4 ,force1 when manifested by the mutual approach or cohesion of bodies1 is also called attraction1 and it is variously denominated1 according to the circumstances under which it is observed to act4 ,thus1 the force which holds together the atoms of solid bodies is called cohesive attraction4 ,the force which draws bodies to the surface of the earth1 when placed above it1 is called the attraction of gravitation4 ,the force which is exhibited by the mutual approach1 or adhesion1 of the loadstone and iron1 is called magnetic attraction1 and so on4 ,when force is manifested by the motion of bodies from each other1 it is called repulsion4 ,thus1 if a piece of glass1 having been briskly rubbed with a silk handkerchief1 touch successively two feathers1 these feathers1 if brought near each other1 will move asunder4 ,this effect is called repulsion1 and the feathers are said to repel each other4 7#i47 ,the influence which forces have upon the form1 state1 arrangement1 and motions of material substances is the principal object of physical science4 ,in its strict sense1 ,,mechanics is a term of very extensive signification4 ,according to the more popular usage1 however1 it has been generally applied to that part of physical science which includes the investigation of the phenomena of motion and rest1 pressure and other effects developed by the mutual action of solid masses4 ,the consideration of similar phenomena1 exhibited in bodies of the liquid form1 is consigned to ,,hydrostatics1 and that of aeriform fluids to ,,pneumatics4 ,,chap4 ,,ii4 ,,divisibility--,,porosity--,,density--, ,compressibility--,,elasticity--,,dilata bility4 7#aj47 ,besides the qualities of magnitude and impenetrability1 there are several other general properties of bodies contemplated in mechanical philosophy1 and to which we shall have frequent occasion to refer4 ,those which we shall notice in the present chapter are1 #a4 ,divisibility4 #b4 ,porosity--,density4 #c4 ,compressibility--,elasticity4 #d4 ,dilatability4 7#aa47 ,divisibility4--,observation and experience prove that all bodies of sensible magnitude1 even the most solid1 consist of parts which are separable4 ,to the practical subdivision of matter there seems to be no assignable limit4 ,numerous examples of the division of matter1 to a degree almost exceeding belief1 may be found in experimental enquiries instituted in physical science2 the useful arts furnish many instances not less striking2 but1 perhaps1 the most conspicuous proofs which can be produced1 of the extreme minuteness of which the parts of matter are susceptible1 arise from the consideration of certain parts of the organised world4 7#ab47 ,the relative places of stars in the heavens1 as seen in the field of view of a telescope1 are marked by fine lines of wire placed before the eye-glass1 and which cross each other at right angles4 ,the stars appearing in the telescope as mere lucid points without sensible magnitude1 it is necessary that the wires which mark their places should have a corresponding tenuity4 ,but these wires being magnified by the eye-glass would have an apparent thickness1 which would render them inapplicable to this purpose1 unless their real dimensions were of a most uncommon degree of minuteness4 ,to obtain wire for this purpose1 ,dr4 ,wollaston invented the following process3--,a piece of fine platinum wire1 a b1 is extended along the axis of a cylindrical mould1 ,a ,b1 fig4 #a4 ,into this mould1 at ,a1 molten silver is poured4 ,since the heat necessary for the fusion of platinum is much greater than that which retains silver in the liquid form1 the wire a b remains solid1 while the mould ,a ,b is filled with the silver4 ,when the metal has become solid by being cooled1 and has been removed from the mould1 a cylindrical bar of silver is obtained1 having a platinum wire in its axis4 ,this bar is then wire-drawn1 by forcing it successively through holes ,c1 ,d1 ,e1 ,f1 ,g1 ,h1 diminishing in magnitude1 the first hole being a little less than the wire at the beginning of the process4 ,by these means the platinum a b is wire-drawn at the same time and in the same proportion with the silver1 so that whatever be the original proportion of the thickness of the wire a b to that of the mould ,a ,b1 the same will be the proportion of the platinum wire to the whole at the several thicknesses ,c1 ,d1 c4 ,if we suppose the mould ,a ,b to be ten times the thickness of the wire a b1 then the silver wire1 throughout the whole process1 will be ten times the thickness of the platinum wire which it includes within it4 ,the silver wire may be drawn to a thickness not exceeding the #cjjth of an inch4 ,the platinum will thus not exceed the #cjjjth of an inch4 ,the wire is then dipped in nitric acid1 which dissolves the silver1 but leaves the platinum solid4 ,by this method ,dr4 ,wollaston succeeded in obtaining wire1 the diameter of which did not exceed the #ahjjjth of an inch4 ,a quantity of this wire1 equal in bulk to a common die used in games of chance1 would extend from ,paris to ,rome4 7#ac47 ,newton succeeded in determining the thickness of very thin lamin of transparent substances by observing the colours which they reflect4 ,a soap bubble is a thin shell of water1 and is observed to reflect different colours from different parts of its surface4 ,immediately before the bubble bursts1 a black spot may be observed near the top4 ,at this part the thickness has been proved not to exceed the #b1ejj1jjjth of an inch4 ,the transparent wings of certain insects are so attenuated in their structure that #ej1jjj of them placed over each other would not form a pile a quarter of an inch in height4 7#ad47 ,in the manufacture of embroidery it is necessary to obtain very fine gilt silver threads4 ,to accomplish this1 a cylindrical bar of silver1 weighing #cfj ounces1 is covered with about two ounces of gold4 ,this gilt bar is then wire-drawn1 as in the first example1 until it is reduced to a thread so fine that #cdjj feet of it weigh less than an ounce4 ,the wire is then flattened by passing it between rollers under a severe pressure1 a process which increases its length1 so that about #djjj feet shall weigh one ounce4 ,hence1 one foot will weigh the #djjjth part of an ounce4 ,the proportion of the gold to the silver in the original bar was that of #b to #cfj1 or #a to #ahj4 ,since the same proportion is preserved after the bar has been wire-drawn1 it follows that the quantity of gold which covers one foot of the fine wire is the #ahjth part of the #djjjth of an ounce2 that is the #gbj1jjjth part of an ounce4 ,the quantity of gold which covers one inch of this wire will be twelve times less than that which covers one foot4 ,hence1 this quantity will be the #h1fdj1jjjth part of an ounce4 ,if this inch be again divided into #ajj equal parts1 every part will be distinctly visible without the aid of microscopes4 ,the gold which covers this small but visible portion is the #hfd1jjj1jjjth part of an ounce4 ,but we may proceed even further2 this portion of the wire may be viewed by a microscope which magnifies #ejj times1 so that the #ejjth part of it will thus become visible4 ,in this manner1 therefore1 an ounce of gold may be divided into #dcb1jjj1jjj1jjj visible parts1 each of which will possess all the characters and qualities found in the largest masses of the metal4 ,it will retain its solidity1 texture1 and colour2 it will resist the same agents1 and enter into combination with the same substances4 ,if the gilt wire be dipped in nitric acid1 the silver within the coating will be dissolved1 but the hollow tube of gold which surrounded it will still cohere and remain suspended4 7#ae47 ,the organised world offers still more remarkable examples of the inconceivable subtilty of matter4 ,the blood which flows in the veins of animals is not1 as it seems1 an uniformly red liquid4 ,it consists of flat discs of a red colour1 floating in a transparent fluid called serum4 ,in different species these discs differ both in figure and in magnitude4 ,in man and all animals which suckle their young1 they are perfectly circular or nearly so4 ,in birds1 reptiles1 and fishes1 they are of oval form4 ,in the human species1 the diameter of these discs is about the #cejjth of an inch4 ,hence it follows1 that in a drop of blood which would remain suspended from the point of a fine needle1 there must be about #c1jjj1jjj of such discs4 ,small as these discs are1 the animal kingdom presents beings whose whole bodies are still more minute4 ,animalcules have been discovered1 whose magnitude is such1 that a million of them do not exceed the bulk of a grain of sand2 and yet each of these creatures is composed of members as curiously organised as those of the largest species2 they have life and spontaneous motion1 and are endued with sense and instinct4 ,in the liquids in which they live1 they are observed to move with astonishing speed and activity2 nor are their motions blind and fortuitous1 but evidently governed by choice1 and directed to an end4 ,they use food and drink1 from which they derive nutrition1 and are therefore furnished with a digestive apparatus4 ,they have great muscular power1 and are furnished with limbs and muscles of strength and flexibility4 ,they are susceptible of the same appetites1 and obnoxious to the same passions1 the gratification of which is attended with the same results as in our own species4 ,spallanzani observes1 that certain animalcules devour others so voraciously1 that they fatten and become indolent and sluggish by over-feeding4 ,after a meal of this kind1 if they be confined in distilled water1 so as to be deprived of all food1 their condition becomes reduced2 they regain their spirit and activity1 and amuse themselves in the pursuit of the more minute animals1 which are supplied to them2 they swallow these without depriving them of life1 for1 by the aid of the microscope1 the one has been observed moving within the body of the other4 ,these singular appearances are not matters of idle and curious observation4 ,they lead us to enquire what parts are necessary to produce such results4 ,must we not conclude that these creatures have heart1 arteries1 veins1 muscles1 sinews1 tendons1 nerves1 circulating fluids1 and all the concomitant apparatus of a living organised body8 ,and if so1 how inconceivably minute must those parts be6 ,if a globule of their blood bears the same proportion to their whole bulk as a globule of our blood bears to our magnitude1 what powers of calculation can give an adequate notion of its minuteness8 7#af47 ,these and many other phenomena observed in the immediate productions of nature1 or developed by mechanical and chemical processes1 prove that the materials of which bodies are formed are susceptible of minuteness which infinitely exceeds the powers of sensible observation1 even when those powers have been extended by all the aids of science4 ,shall we then conclude that matter is infinitely divisible1 and that there are no original constituent atoms of determinate magnitude and figure at which all subdivision must cease8 ,such an inference would be unwarranted1 even had we no other means of judging the question1 except those of direct observation2 for it would be imposing that limit on the works of nature which she has placed upon our powers of observing them4 ,aided by reason1 however1 and a due consideration of certain phenomena which come within our immediate powers of observation1 we are frequently able to determine other phenomena which are beyond those powers4 ,the diurnal motion of the earth is not perceived by us1 because all things around us participate in it1 preserve their relative position1 and appear to be at rest4 ,but reason tells us that such a motion must produce the alternations of day and night1 and the rising and setting of all the heavenly bodies2 appearances which are plainly observable1 and which betray the cause from which they arise4 ,again1 we cannot place ourselves at a distance from the earth1 and behold the axis on which it revolves1 and observe its peculiar obliquity to the orbit in which the earth moves2 but we see and feel the vicissitudes of the seasons1 an effect which is the immediate consequence of that inclination1 and by which we are able to detect it4 7#ag47 ,so it is in the present case4 ,although we are unable by direct observation to prove the existence of constituent material atoms of determinate figure1 yet there are many observable phenomena which render their existence in the highest degree probable1 if not morally certain4 ,the most remarkable of this class of effects is observed in the crystallisation of salts4 ,when salt is dissolved in a sufficient quantity of pure water1 it mixes with the water in such a manner as wholly to disappear to the sight and touch1 the mixture being one uniform transparent liquid like the water itself1 before its union with the salt4 ,the presence of the salt in the water may1 however1 be ascertained by weighing the mixture1 which will be found to exceed the original weight of the water by the exact amount of the weight of the salt4 ,it is a well-known fact1 that a certain degree of heat will convert water into vapour1 and that the same degree of heat does not produce the same effect upon salt4 ,the mixture of salt and water being exposed to this temperature1 the water will gradually evaporate1 disengaging itself from the salt with which it has been combined4 ,when so much of the water has evaporated1 that what remains is insufficient to keep in solution the whole of the salt1 a part of the latter thus disengaged from the water will return to the solid state4 ,the saline constituent will not in this case collect in irregular solid molecules2 but will exhibit itself in particles of regular figure1 terminated by plane surfaces1 the figure being always the same for the same species of salt1 but different for different species4 ,these particles are called crystals4 ,there are several circumstances in the formation of these crystals which merit attention4 ,if one of them be detached from the others1 and the progress of its formation observed1 it will be found gradually to increase1 always preserving its original figure4 ,since its increase must be caused by the continued accession of saline molecules1 disengaged by the evaporation of the water1 it follows that these molecules must be so formed1 that by attaching themselves successively to the crystal1 they maintain the regularity of its bounding planes1 and preserve their mutual inclinations unvaried4 ,suppose a crystal to be taken from the liquid during the process of crystallisation1 and a piece broken from it so as to destroy the regularity of its form3 if the crystal thus broken be restored to the liquid1 it will be observed gradually to resume its regular form1 the atoms of salt successively dismissed by the vaporising water filling up the irregular cavities produced by the fracture4 ,hence it follows1 that the saline particles which compose the surface of the crystal1 and those which form the interior of its mass1 are similar1 and exert similar attractions on the atoms disengaged by the water4 ,all these details of the process of crystallisation are very evident indications of a determinate figure in the ultimate atoms of the substances which are crystallised4 ,but besides the substances which are thus reduced by art to the form of crystals1 there are larger classes which naturally exist in that state4 ,there are certain planes1 called planes of cleavage1 in the directions of which natural crystals are easily divided4 ,these planes1 in substances of the same kind1 always have the same relative position1 but differ in different substances4 ,the surfaces of the planes of cleavage are quite invisible before the crystal is divided2 but when the parts are separated1 these surfaces exhibit a most intense polish1 which no effort of art can equal4 ,we may conceive crystallised substances to be regular mechanical structures formed of atoms of a certain figure1 on which the figure of the whole structure must depend4 ,the planes of cleavage are parallel to the sides of the constituent atoms2 and their directions1 therefore1 form so many conditions for the determination of their figure4 ,the shape of the atoms being thus determined1 it is not difficult to assign all the various ways in which they may be arranged1 so as to produce figures which are accordingly found to correspond with the various forms of crystals of the same substance4 7#ah47 ,when these phenomena are duly considered and compared1 little doubt can remain that all substances susceptible of crystallisation1 consist of atoms of determinate figure4 ,this is the case with all solid bodies whatever1 which have come under scientific observation1 for they have been severally found in or reduced to a crystallised form4 ,liquids crystallise in freezing1 and if ariform fluids could by any means be reduced to the solid form1 they would probably also manifest the same effect4 ,hence it appears reasonable to presume1 that all bodies are composed of atoms2 that the different qualities with which we find different substances endued1 depend on the magnitude and figure of these atoms2 that these atoms are indestructible and immutable by any natural process1 for we find the qualities which depend on them unchangeably the same under all the influences to which they have been submitted since their creation2 that these atoms are so minute in their magnitude1 that they cannot be observed by any means which human art has yet contrived2 but still that magnitudes can be assigned which they do not exceed4 ,it is proper1 however1 to observe here1 that the various theorems of mechanical science do not rest upon any hypothesis concerning these atoms as a basis4 ,these theorems are not inferred from this or any other supposition1 and therefore their truth would not be in anywise disturbed1 even though it should be established that matter is physically divisible in infinitum4 ,the basis of mechanical science is observed facts1 and1 since the reasoning is demonstrative1 the conclusions have the same degree of certainty as the facts from which they are deduced4 7#ai47 ,porosity4--,the volume of a body is the quantity of space included within its external surfaces4 ,the mass of a body1 is the collection of atoms or material particles of which it consists4 ,two atoms or particles are said to be in contact1 when they have approached each other until arrested by their mutual impenetrability4 ,if the component particles of a body were in contact1 the volume would be completely occupied by the mass4 ,but this is not the case4 ,we shall presently prove1 that the component particles of no known substance are in absolute contact4 ,hence it follows that the volume consists partly of material particles1 and partly of interstitial spaces1 which spaces are either absolutely void and empty1 or filled by some substance of a different kind from the body in question4 ,these interstitial spaces are called pores4 ,in bodies which are constituted uniformly throughout their entire dimensions1 the component particles and the pores are uniformly distributed through the volume2 that is1 a given space in one part of the volume will contain the same quantity of matter and the same quantity of pores as an equal space in another part4 7#bj47 ,the proportion of the quantity of matter to the magnitude is called the density4 ,thus if of two substances1 one contain in a given space twice as much matter as the other1 it is said to be 8twice as dense40 ,the density of bodies is1 therefore1 proportionate to the closeness or proximity of their particles2 and it is evident1 that the greater the density1 the less will be the porosity4 ,the pores of a body are frequently filled with another body of a more subtle nature4 ,if the pores of a body on the surface of the earth1 and exposed to the atmosphere1 be greater than the atoms of air1 then the air may pervade the pores4 ,this is found to be the case with many sorts of wood which have an open grain4 ,if a piece of such wood1 or of chalk1 or of sugar1 be pressed to the bottom of a vessel of water1 the air which fills the pores will be observed to escape in bubbles and to rise to the surface1 the water entering the pores1 and taking its place4 ,if a tall vessel or tube1 having a wooden bottom1 be filled with quicksilver1 the liquid metal will be forced by its own weight through the pores of the wood1 and will be seen escaping in a silver shower from the bottom4 7#ba47 ,the process of filtration1 in the arts1 depends on the presence of pores of such a magnitude as to allow a passage to the liquid1 but to refuse it to those impurities from which it is to be disengaged4 ,various substances are used as filtres2 but1 whatever be used1 this circumstance should always be remembered1 that no substance can be separated from a liquid by filtration1 except one whose particles are larger than those of the liquid4 ,in general1 filtres are used to separate solid impurities from a liquid4 ,the most ordinary filtres are soft stone1 paper1 and charcoal4 7#bb47 ,all organised substances in the animal and vegetable kingdoms are1 from their very natures1 porous in a high degree4 ,minerals are porous in various degrees4 ,among the silicious stones is one called hydrophane1 which manifests its porosity in a very remarkable manner4 ,the stone1 in its ordinary state1 is semi-transparent4 ,if1 however1 it be plunged in water1 when it is withdrawn it is as translucent as glass4 ,the pores1 in this case1 previously filled with air1 are pervaded by the water1 between which and the stone there subsists a physical relation1 by which the one renders the other perfectly transparent4 ,larger mineral masses exhibit degrees of porosity not less striking4 ,water percolates through the sides and roofs of caverns and grottoes1 and being impregnated with calcareous and other earths1 forms stalactites1 or pendant protuberances1 which present a curious appearance4 7#bc47 ,compressibility4--,that quality1 in virtue of which a body allows its volume to be diminished without diminishing its mass1 is called compressibility4 ,this effect is produced by bringing the constituent particles more close together1 and thereby increasing the density and diminishing the pores4 ,this effect may be produced in several ways2 but the name 8compressibility0 is only applied to it when it is caused by the agency of mechanical force1 as by pressure or percussion4 ,all known bodies1 whatever be their nature1 are capable of having their dimensions reduced without diminishing their mass2 and this is one of the most conclusive proofs that all bodies are porous1 or that the constituent atoms are not in contact2 for the space by which the volume may be diminished must1 before the diminution1 consist of pores4 7#bd47 ,elasticity4--,some bodies1 when compressed by mechanical agency1 will resume their former dimensions with a certain energy when relieved from the operation of the force which has compressed them4 ,this property is called elasticity2 and it follows1 from this definition1 that all elastic bodies must be compressible1 although the converse is not true1 compressibility not necessarily implying elasticity4 7#be47 ,dilatability4--,this quality is the opposite of compressibility4 ,it is the capability observed in bodies to have their volume enlarged without increasing their mass4 ,this effect may be produced in several ways4 ,in ordinary circumstances1 a body may exist under the constant action of a pressure by which its volume and density are determined4 ,it may happen1 that on the occasional removal of that pressure1 the body will dilate by a quality inherent in its constitution4 ,this is the case with common air4 ,dilatation may also be the effect of heat1 as will presently appear4 ,the several qualities of bodies which we have noticed in this chapter1 when viewed in relation to each other1 present many circumstances worthy of attention4 7#bf47 ,it is a physical law1 of high generality1 that an increase in the temperature1 or degree of heat by which a body is affected1 is accompanied by an increase of volume2 and that a diminution of temperature is accompanied by a diminution of volume4 ,the exceptions to this law will be noticed and explained in our treatise on ,,heat4 ,hence it appears that the reduction of temperature is an effect which1 considered mechanically1 is equivalent to compression or condensation1 since it diminishes the volume without altering the mass2 and since this is an effect of which all bodies whatever are susceptible1 it follows that all bodies whatever have pores4 7#bc47 ,the fact1 that the elevation of temperature produces an increase of volume1 is manifested by numerous experiments4 7#bg47 ,if a flaccid bladder be tied at the mouth1 so as to stop the escape of air1 and be then held before a fire1 it will gradually swell1 and assume the appearance of being fully inflated4 ,the small quantity of air contained in the bladder is1 in this case1 so much dilated by the heat1 that it occupies a considerably increased space1 and fills the bladder1 of which it before only occupied a small part4 ,when the bladder is removed from the fire1 and allowed to resume its former temperature1 the air returns to its former dimensions1 and the bladder becomes again flaccid4 7#bh47 ,let ,a ,b1 fig4 #b4 be a glass tube1 with a bulb at the end ,a2 and let the bulb ,a1 and a part of the tube1 be filled with any liquid1 coloured so as to be visible4 ,let ,c be the level of the liquid in the tube4 ,if the bulb be now exposed to heat1 by being plunged in hot water1 the level of the liquid ,c will rapidly rise towards ,b4 ,this effect is produced by the dilatation of the liquid in the bulb1 which filling a greater space1 a part of it is forced into the tube4 ,this experiment may easily be made with a common glass tube and a little port wine4 ,thermometers are constructed on this principle1 the rise of the liquid in the tube being used as an indication of the degree of heat which causes it4 ,a particular account of these useful instruments will be found in our treatise on ,,heat4 7#bi47 ,the change of dimension of solids produced by changes of temperature being much less than that of bodies in the liquid or aeriform state1 is not so easily observable4 ,a remarkable instance occurs in the process of shoeing the wheels of carriages4 ,the rim of iron with which the wheel is to be bound1 is made in the first instance of a diameter somewhat less than that of the wheel2 but being raised by the application of fire to a very high temperature1 its volume receives such an increase1 that it will be sufficient to embrace and surround the wheel4 ,when placed upon the wheel it is cooled1 and suddenly contracting its dimensions1 binds the parts of the wheel firmly together1 and becomes securely seated in its place upon the fellies4 7#cj47 ,it frequently happens that the stopper of a glass bottle or decanter becomes fixed in its place so firmly1 that the exertion of force sufficient to withdraw it would endanger the vessel4 ,in this case1 if a cloth wetted with hot-water be applied to the neck of the bottle1 the glass will expand1 and the neck will be enlarged1 so as to allow the stopper to be easily withdrawn4 7#ca47 ,the contraction of metal consequent upon change of temperature was applied some time ago in ,paris to restore the walls of a tottering building to their proper position4 ,in the ,conservatoire des ,arts et ,mtiers1 the walls of a part of the building were forced out of the perpendicular by the weight of the roof1 so that each wall was leaning outwards4 ,m4 ,molard conceived the notion of applying the irresistible force with which metals contract in cooling1 to draw the walls together4 ,bars of iron were placed in parallel directions across the building1 and at right-angles to the direction of the walls4 ,being passed through the walls1 nuts were screwed on their ends1 outside the building4 ,every alternate bar was then heated by lamps1 and the nuts screwed close to the walls4 ,the bars were then cooled1 and the lengths being diminished by contraction1 the nuts on their extremities were drawn together1 and with them the walls were drawn through an equal space4 ,the same process was repeated with the intermediate bars1 and so on alternately until the walls were brought into a perpendicular position4 7#cb47 ,since there is a continual change of temperature in all bodies on the surface of the globe1 it follows1 that there is also a continual change of magnitude4 ,the substances which surround us are constantly swelling and contracting1 under the vicissitudes of heat and cold4 ,they grow smaller in winter1 and dilate in summer4 ,they swell their bulk on a warm day1 and contract it on a cold one4 ,these curious phenomena are not noticed1 only because our ordinary means of observation are not sufficiently accurate to appreciate them4 ,nevertheless1 in some familiar instances the effect is very obvious4 ,in warm weather the flesh swells1 the vessels appear filled1 the hand is plump1 and the skin distended4 ,in cold weather1 when the body has been exposed to the open air1 the flesh appears to contract1 the vessels shrink1 and the skin shrivels4 7#cc47 ,the phenomena attending change of temperature are conclusive proofs of the universal porosity of material substances1 but they are not the only proofs4 ,many substances admit of compression by the mere agency of mechanical force4 ,let a small piece of cork be placed floating on the surface of water in a basin or other vessel1 and an empty glass goblet be inverted over the cork1 so that its edge just meets the water4 ,a portion of air will then be confined in the goblet1 and detached from the remainder of the atmosphere4 ,if the goblet be now pressed downwards1 so as to be entirely immersed1 it will be observed1 that the water will not fill it1 being excluded by the impenetrability of the air inclosed in it4 ,this experiment1 therefore1 is decisive of the fact1 that air1 one of the most subtle and attenuated substances we know of1 possesses the quality of impenetrability4 ,it absolutely excludes any other body from the space which it occupies at any given moment4 ,but although the water does not fill the goblet1 yet if the position of the cork which floats upon its surface be noticed1 it will be found that the level of the water within has risen above its edge or rim4 ,in fact1 the water has partially filled the goblet1 and the air has been forced to contract its dimensions4 ,this effect is produced by the pressure of the incumbent water forcing the surface in the goblet against the air1 which yields until it is so far compressed that it acquires a force able to withstand this pressure4 ,thus it appears that air is capable of being reduced in its dimensions by mechanical pressure1 independently of the agency of heat4 ,it is compressible4 ,that this effect is the consequence of the pressure of the liquid will be easily made manifest by showing that1 as the pressure is increased1 the air is proportionally contracted in its dimensions2 and as it is diminished1 the dimensions are on the other hand enlarged4 ,if the depth of the goblet in the water be increased1 the cork will be seen to rise in it1 showing that the increased pressure1 at the greater depth1 causes the air in the goblet to be more condensed4 ,if1 on the other hand1 the goblet be raised toward the surface1 the cork will be observed to descend toward the edge1 showing that as it is relieved from the pressure of the liquid1 the air gradually approaches to its primitive dimensions4 7#cd47 ,these phenomena also prove1 that air has the property of elasticity4 ,if it were simply compressible1 and not elastic1 it would retain the dimensions to which it was reduced by the pressure of the liquid2 but this is not found to be the result4 ,as the compressing force is diminished1 so in the same proportion does the air1 by its elastic virtue1 exert a force by which it resumes its former dimensions4 ,that it is the air alone which excludes the water from the goblet1 in the preceding experiments1 can easily be proved4 ,when the goblet is sunk deep in the vessel of water1 let it be inclined a little to one side until its mouth is presented towards the side of the vessel2 let this inclination be so regulated1 that the surface of the water in the goblet shall just reach its edge4 ,upon a slight increase of inclination1 air will be observed to escape from the goblet1 and to rise in bubbles to the surface of the water4 ,if the goblet be then restored to its position1 it will be found that the cork will rise higher in it than before the escape of the air4 ,the water in this case rises and fills the space which the air allowed to escape has deserted4 ,the same process may be repeated until all the air has escaped1 and then the goblet will be completely filled by the water4 7#ce47 ,liquids are compressible by mechanical force in so slight a degree1 that they are considered in all hydrostatical treatises as incompressible fluids4 ,they are1 however1 not absolutely incompressible1 but yield slightly to very intense pressure4 ,the question of the compressibility of liquids was raised at a remote period in the history of science4 ,nearly two centuries ago1 an experiment was instituted at the ,academy del ,cimento in ,florence1 to ascertain whether water be compressible4 ,with this view1 a hollow ball of gold was filled with the liquid1 and the aperture exactly and firmly closed4 ,the globe was then submitted to a very severe pressure1 by which its figure was slightly changed4 ,now it is proved in geometry1 that a globe has this peculiar property1 that any change whatever in its figure must necessarily diminish its volume or contents4 ,hence it was inferred1 that if the water did not issue through the pores of the gold1 or burst the globe1 its compressibility would be established4 ,the result of the experiment was1 that the water did ooze through the pores1 and covered the surface of the globe1 presenting the appearance of dew1 or of steam cooled by the metal4 ,but this experiment was inconclusive4 ,it is quite true1 that if the water had not escaped upon the change of figure of the globe1 the compressibility of the liquid would have been established4 ,the escape of the water does not1 however1 prove its incompressibility4 ,to accomplish this1 it would be necessary first to measure accurately the volume of water which transuded by compression1 and next to measure the diminution of volume which the vessel suffered by its change of figure4 ,if this diminution were greater than the volume of water which escaped1 it would follow that the water remaining in the globe had been compressed1 notwithstanding the escape of the remainder4 ,but this could never be accomplished with the delicacy and exactitude necessary in such an experiment2 and1 consequently1 as far as the question of the compressibility of water was concerned1 nothing was proved4 ,it forms1 however1 a very striking illustration of the porosity of so dense a substance as gold1 and proves that its pores are larger than the elementary particles of water1 since these are capable of passing through them4 7#cf47 ,it has since been proved1 that water1 and other liquids1 are compressible4 ,in the year #agfa1 ,canton communicated to the ,royal ,society the results of some experiments which proved this fact4 ,he provided a glass tube with a bulb1 such as that described in 7#bh71 and filled the bulb and a part of the tube with water well purified from air4 ,he then placed this in an apparatus called a condenser1 by which he was enabled to submit the surface of the liquid in the tube to very intense pressure of condensed air4 ,he found that the level of the liquid in the tube fell in a perceptible degree upon the application of the pressure4 ,the same experiment established the fact1 that liquids are elastic2 for upon removing the pressure1 the liquid rose to its original level1 and therefore resumed its former dimensions4 7#cg47 ,elasticity does not always accompany compressibility4 ,if lead or iron be submitted to the hammer1 it may be hardened and diminished in its volume2 but it will not resume its former volume after each stroke of the hammer4 7#ch47 ,there are some bodies which maintain the state of density in which they are commonly found by the continual agency of mechanical pressure2 and such bodies are endued with a quality1 in virtue of which they would enlarge their dimensions without limit1 if the pressure which confines them were removed4 ,such bodies are called elastic fluids or gases1 and always exist in the form of common air1 in whose mechanical properties they participate4 ,they are hence often called aeriform fluids4 ,those who are provided with an air-pump can easily establish this property experimentally4 ,take a flaccid bladder1 such as that already described in 7#bg471 and place it under the glass receiver of an air-pump3 by this instrument we shall be able to remove the air which surrounds the bladder under the receiver1 so as to relieve the small quantity of air which is inclosed in the bladder from the pressure of the external air3 when this is accomplished1 the bladder will be observed to swell1 as if it were inflated1 and will be perfectly distended4 ,the air contained in it1 therefore1 has a tendency to dilate1 which takes effect when it ceases to be resisted by the pressure of surrounding air4 7#ci47 ,it has been stated that the increase or diminution of temperature is accompanied by an increase or diminution of volume4 ,related to this1 there is another phenomenon too remarkable to pass unnoticed1 although this is not the proper place to dwell upon it3 it is the converse of the former2 viz4 that an increase or diminution of bulk is accompanied by a diminution or increase of temperature4 ,as the application of heat from some foreign source produces an increase of dimensions1 so if the dimensions be increased from any other cause1 a corresponding portion of the heat which the body had before the enlargement1 will be absorbed in the process1 and the temperature will be thereby diminished4 ,in the same way1 since the abstraction of heat causes a diminution of volume1 so if that diminution be caused by any other means1 the body will give out the heat which in the other case was abstracted1 and will rise in its temperature4 ,numerous and well-known facts illustrate these observations4 ,a smith by hammering a piece of bar iron1 and thereby compressing it1 will render it red hot4 ,when air is violently compressed1 it becomes so hot as to ignite cotton and other substances4 ,an ingenious instrument for producing a light for domestic uses has been constructed1 consisting of a small cylinder1 in which a solid piston moves air-tight3 a little tinder1 or dry sponge1 is attached to the bottom of the piston1 which is then violently forced into the cylinder3 the air between the bottom of the cylinder and the piston becomes intensely compressed1 and evolves so much heat as to light the tinder4 ,in all the cases where friction or percussion produces heat or fire1 it is because they are means of compression4 ,the effects of flints1 of pieces of wood rubbed together1 the warmth produced by friction on the flesh1 are all to be attributed to the same cause4 ,,chap4 ,,iii4 ,,inertia4 7#dj47 ,the quality of matter which is of all others the most important in mechanical investigations1 is that which has been called ,inertia4 ,matter is incapable of spontaneous change4 ,this is one of the earliest and most universal results of human observation3 it is equivalent to stating that mere matter is deprived of life2 for spontaneous action is the only test of the presence of the living principle4 ,if we see a mass of matter undergo any change1 we never seek for the cause of that change in the body itself2 we look for some external cause producing it4 ,this inability for voluntary change of state or qualities is a more general principle than inertia4 ,at any given moment of time a body must be in one or other of two states1 rest or motion4 ,inertia1 or inactivity1 signifies the total absence of power to change this state4 ,a body endued with inertia cannot of itself1 and independent of all external influence1 commence to move from a state of rest2 neither can it when moving arrest its progress and become quiescent4 7#da47 ,the same property by which a body is unable by any power of its own to pass from a state of rest to one of motion1 or vice vers1 also renders it incapable of increasing or diminishing any motion which it may have received from an external cause4 ,if a body be moving in a certain direction at the rate of ten miles per hour1 it cannot by any energy of its own change its rate of motion to eleven or nine miles an hour4 ,this is a direct consequence of that manifestation of inertia which has just been explained4 ,for the same power which would cause a body moving at ten miles an hour to increase its rate to eleven miles1 would also cause the same body at rest to commence moving at the rate of one mile an hour2 and the same power which would cause a body moving at the rate of ten miles an hour to move at the rate of nine miles in the hour1 would cause the same body moving at the rate of one mile an hour to become quiescent4 ,it therefore appears1 that to increase or diminish the motion of a body is an effect of the same kind as to change the state of rest into that of motion1 or vice vers4 7#db47 ,the effects and phenomena which hourly fall under our observation afford unnumbered examples of the inability of lifeless matter to put itself into motion1 or to increase any motion which may have been communicated to it4 ,but it does not happen that we have the same direct and frequent evidence of its inability to destroy or diminish any motion which it may have received4 ,and hence it arises1 that while no one will deny to matter the former effect of inertia1 few will at first acknowledge the latter4 ,indeed1 even so late as the time of ,,kepler1 philosophers themselves held it as a maxim1 that 8matter is more inclined to rest than to motion20 we ought not1 therefore1 to be surprised if in the present day those who have not been conversant with physical science are slow to believe that a body once put in motion would continue for ever to move with the same velocity1 if it were not stopped by some external cause4 ,reason1 assisted by observation1 will1 however1 soon dispel this illusion4 ,experience shows us in various ways1 that the same causes which destroy motion in one direction are capable of producing as much motion in the opposite direction4 ,thus1 if a wheel1 spinning on its axis with a certain velocity1 be stopped by a hand seizing one of the spokes1 the effort which accomplishes this is exactly the same as1 had the wheel been previously at rest1 would have put it in motion in the opposite direction with the same velocity4 ,if a carriage drawn by horses be in motion1 the same exertion of power in the horses is necessary to stop it1 as would be necessary to back it1 if it were at rest4 ,now1 if this be admitted as a general principle1 it must be evident that a body which can destroy or diminish its own motion must also be capable of putting itself into motion from a state of rest1 or of increasing any motion which it has received4 ,but this latter is contrary to all experience1 and therefore we are compelled to admit that a body cannot diminish or destroy any motion which it has received4 ,let us enquire why we are more disposed to admit the inability of matter to produce than to destroy motion in itself4 ,we see most of those motions which take place around us on the surface of the earth subject to gradual decay1 and if not renewed from time to time1 at length cease4 ,a stone rolled along the ground1 a wheel revolving on its axis1 the heaving of the deep after a storm1 and all other motions produced in bodies by external causes1 decay1 when the exciting cause is suspended2 and if that cause do not renew its action1 they ultimately cease4 ,but is there no exciting cause1 on the other hand1 which thus gradually deprives those bodies of their motion8--and if that cause were removed1 or its intensity diminished1 would not the motion continue1 or be more slowly retarded8 ,when a stone is rolled along the ground1 the inequalities of its shape as well as those of the ground are impediments1 which retard and soon destroy its motion4 ,render the stone round1 and the ground level1 and the motion will be considerably prolonged4 ,but still small asperities will remain on the stone1 and on the surface over which it rolls3 substitute for the stone a ball of highly-polished steel1 moving on a highly-polished steel plane1 truly level1 and the motion will continue without sensible diminution for a very long period2 but even here1 and in every instance of motions produced by art1 minute asperities must exist on the surfaces which move in contact with each other1 which must resist1 gradually diminish1 and ultimately destroy the motion4 ,independently of the obstructions to the continuation of motion arising from friction1 there is another impediment to which all motions on the surface of the earth are liable--the resistance of the air4 ,how much this may affect the continuation of motion appears by many familiar effects4 ,on a calm day carry an open umbrella with its concave side presented in the direction in which you are moving1 and a powerful resistance will be opposed to your progress1 which will increase with every increase of the speed with which you move4 7#dc47 ,we are not1 however1 without direct experience to prove1 that motions when unresisted will for ever continue4 ,in the heavens we find an apparatus1 which furnishes a sublime verification of this principle4 ,there1 removed from all casual obstructions and resistances1 the vast bodies of the universe roll on in their appointed paths with unerring regularity1 preserving without diminution all that motion which they received at their creation from the hand which launched them into space4 ,this alone1 unsupported by other reasons1 would be sufficient to establish the quality of inertia2 but viewed in connection with the other circumstances previously mentioned1 no doubt can remain that this is an universal law of nature4 7#dd47 ,it has been proved1 that inability to change the quantity of motion is a consequence of inertia4 ,the inability to change the direction of motion is another consequence of this quality4 ,the same cause which increases or diminishes motion1 would also give motion to a body at rest2 and therefore we infer that the same inability which prevents a body from moving itself1 will also prevent it from increasing or diminishing any motion which it has received4 ,in the same manner we can show1 that any cause which changes the direction of motion would also give motion to a body at rest2 and therefore if a body change the direction of its own motion1 the same body might move itself from a state of rest2 and therefore the power of changing the direction of any motion which it may have received is inconsistent with the quality of inertia4 7#de47 ,if a body1 moving from ,a1 fig4 #c4 to ,b1 receive at ,b a blow in the direction ,c ,b ,e1 it will immediately change its direction to that of another line ,b ,d4 ,the cause which produces this change of direction would have put the body in motion in the direction ,b ,e1 had it been quiescent at ,b when it sustained the blow4 7#df47 ,again1 suppose ,g ,h to be a hard plane surface2 and let the body be supposed to be perfectly inelastic4 ,when it strikes the surface at ,b1 it will commence to move along it in the direction ,b ,h4 ,this change of direction is produced by the resistance of the surface4 ,if the body1 instead of meeting the surface in the direction ,a ,b1 had moved in the direction ,e ,b1 perpendicular to it1 all motion would have been destroyed1 and the body reduced to a state of rest4 7#dg47 ,by the former example it appears that the deflecting cause would have put a quiescent body in motion1 and by the latter it would have reduced a moving body to a state of rest4 ,hence the phenomenon of a change of direction is to be referred to the same class as the change from rest to motion1 or from motion to rest4 ,the quality of inertia is1 therefore1 inconsistent with any change in the direction of motion which does not arise from an external cause4 7#dh47 ,from all that has been here stated1 we may infer generally1 that an inanimate parcel of matter is incapable of changing its state of rest or motion2 that1 in whatever state it be1 in that state it must for ever continue1 unless disturbed by some external cause2 that if it be in motion1 that motion must always be uniform1 or must proceed at the same rate1 equal spaces being moved over in the same time3 any increase of its rate must betray some impelling cause2 any diminution must proceed from an impeding cause1 and neither of these causes can exist in the body itself2 that such motion must not only be constantly at the same uniform rate1 but also must be always in the same direction1 any deflection from one uniform direction necessarily arising from some external influence4 ,the language sometimes used to explain the property of inertia in popular works1 is eminently calculated to mislead the student4 ,the terms resistance and stubbornness to move are faulty in this respect4 ,inertia implies absolute passiveness1 a perfect indifference to rest or motion4 ,it implies as strongly the absence of all resistance to the reception of motion1 as it does the absence of all power to move itself4 ,the term vis inerti or force of inactivity1 so frequently used even by authors pretending to scientific accuracy1 is still more reprehensible4 ,it is a contradiction in terms2 the term inactivity implying the absence of all force4 99 99 99 99 99 7#di47 ,before we close this chapter1 it may be advantageous to point out some practical and familiar examples of the general law of inertia4 ,the student must1 however1 recollect1 that the great object of science is generalisation1 and that his mind is to be elevated to the contemplation of the laws of nature1 and to receive a habit the very reverse of that which disposes us to enjoy the descent from generals to particulars4 ,instances1 taken from the occurrences of ordinary life1 may1 however1 be useful in verifying the general law1 and in impressing it upon the memory2 and for this reason1 we shall occasionally in the present treatise refer to such examples2 always1 however1 keeping them in subservience to the general principles of which they are manifestations1 and on which the attention of the student should never cease to be fixed4 7#ej47 ,if a carriage1 a horse1 or a boat1 moving with speed1 be suddenly retarded or stopped1 by any cause which does not at the same time affect passengers1 riders1 or any loose bodies which are carried1 they will be precipitated in the direction of the motion2 because by reason of their inertia1 they persevere in the motion which they shared in common with that which transported them1 and are not deprived of that motion by the same cause4 7#ea47 ,if a passenger leap from a carriage in rapid motion1 he will fall in the direction in which the carriage is moving at the moment his feet meet the ground2 because his body1 on quitting the vehicle1 retains1 by its inertia1 the motion which it had in common with it4 ,when he reaches the ground1 this motion is destroyed by the resistance of the ground to the feet1 but is retained in the upper and heavier part of the body2 so that the same effect is produced as if the feet had been tripped4 7#eb47 ,when a carriage is once put in motion with a determinate speed on a level road1 the only force necessary to sustain the motion is that which is sufficient to overcome the friction of the road2 but at starting a greater expenditure of force is necessary1 inasmuch as not only the friction is to be overcome1 but the force with which the vehicle is intended to move must be communicated to it4 ,hence we see that horses make a much greater exertion at starting than subsequently1 when the carriage is in motion2 and we may also infer the inexpediency of attempting to start at full speed1 especially with heavy carriages4 7#ec47 ,coursing owes all its interest to the instinctive consciousness of the nature of inertia which seems to govern the measures of the hare4 ,the greyhound is a comparatively heavy body moving at the same or greater speed in pursuit4 ,the hare doubles1 that is1 suddenly changes the direction of her course1 and turns back at an oblique angle with the direction in which she had been running4 ,the greyhound1 unable to resist the tendency of its body to persevere in the rapid motion it had acquired1 is urged forward many yards before it is able to check its speed and return to the pursuit4 ,meanwhile the hare is gaining ground in the other direction1 so that the animals are at a very considerable distance asunder when the pursuit is recommenced4 ,in this way a hare1 though much less fleet than a greyhound1 will often escape it4 ,in racing1 the horses shoot far beyond the winning-post before their course can be arrested4 ,,chap4 ,,iv4 ,,action ,,and ,,reaction4 7#ed47 ,the effects of inertia or inactivity1 considered in the last chapter1 are such as may be manifested by a single insulated body1 without reference to1 or connection with1 any other body whatever4 ,they might all be recognised if there were but one body existing in the universe4 ,there are1 however1 other important results of this law1 to the development of which two bodies at least are necessary4 7#ee47 ,if a mass ,a1 fig4 #d41 moving towards ,c1 impinge upon an equal mass1 which is quiescent at ,b1 the two masses will move together towards ,c after the impact4 ,but it will be observed1 that their speed after the impact will be only half that of ,a before it4 ,thus1 after the impact1 ,a loses half its velocity2 and ,b1 which was before quiescent1 receives exactly this amount of motion4 ,it appears1 therefore1 in this case1 that ,b receives exactly as much motion as ,a loses3 so that the real quantity of motion from ,b to ,c is the same as the quantity of motion from ,a to ,b4 ,now1 suppose that ,b consisted of two masses1 each equal to ,a1 it would be found that in this case the velocity of the triple mass after impact would be one-third of the velocity from ,a to ,b4 ,thus1 after impact1 ,a loses two-thirds of its velocity and1 ,b consisting of two masses each equal to ,a1 each of these two receives one-third of ,a's motion2 so that the whole motion received by ,b is two-thirds of the motion of ,a before impact4 ,by the impact1 therefore1 exactly as much motion is received by ,b as is lost by ,a4 ,a similar result will be obtained1 whatever proportion may subsist between the masses ,a and ,b4 ,suppose ,b to be ten times ,a2 then the whole motion of ,a must1 after the impact1 be distributed among the parts of the united masses of ,a and ,b3 but these united masses are1 in this case1 eleven times the mass of ,a4 ,now1 as they all move with a common motion1 it follows that ,a's former motion must be equally distributed among them2 so that each part shall have an eleventh part of it4 ,therefore the velocity after impact will be the eleventh part of the velocity of ,a before it4 ,thus ,a loses by the impact ten-eleventh parts of its motion1 which are precisely what ,b receives4 ,again1 if the masses of ,a and ,b be #e and #g1 then the united mass after impact will be #ab4 ,the motion of ,a before impact will be equally distributed between these twelve parts1 so that each part will have a twelfth of it2 but five of these parts belong to the mass ,a1 and seven to ,b4 ,hence ,b will receive seven-twelfths1 while ,a retains five-twelfths4 7#ef47 ,in general1 therefore1 when a mass ,a in motion impinges on a mass ,b at rest1 to find the motion of the united mass after impact1 8divide the whole motion of ,a into as many equal parts as there are equal component masses in ,a and ,b together1 and then ,b will receive by the impact as many parts of this motion as it has equal component masses40 ,this is an immediate consequence of the property of inertia1 explained in the last chapter4 ,if we were to suppose that by their mutual impact ,a were to give to ,b either more or less motion than that which it 7,a7 loses1 it would necessarily follow1 that either ,a or ,b must have a power of producing or of resisting motion1 which would be inconsistent with the quality of inertia already defined4 ,for if ,a give to ,b more motion than it loses1 all the overplus or excess must be excited in ,b by the action of ,a2 and1 therefore1 ,a is not inactive1 but is capable of exciting motion which it does not possess4 ,on the other hand1 ,b cannot receive from ,a less motion than ,a loses1 because then ,b must be admitted to have the power by its resistance of destroying all the deficiency2 a power essentially active1 and inconsistent with the quality of inertia4 7#eg47 ,if we contemplate the effects of impact1 which we have now described1 as facts ascertained by experiment 7which they may be71 we may take them as further verification of the universality of the quality of inertia4 ,but1 on the other hand1 we may view them as phenomena which may certainly be predicted from the previous knowledge of that quality2 and this is one of many instances of the advantage which science possesses over knowledge merely practical4 ,having obtained by observation or experience a certain number of simple facts1 and thence deduced the general qualities of bodies1 we are enabled1 by demonstrative reasoning1 to discover other facts which have never fallen under our observation1 or1 if so1 may have never excited attention4 ,in this way philosophers have discovered certain small motions and slight changes which have taken place among the heavenly bodies1 and have directed the attention of astronomical observers to them1 instructing them with the greatest precision as to the exact moment of time and the point of the firmament to which they should direct the telescope1 in order to witness the predicted event4 7#eh47 ,since by the quality of inertia a body can neither generate nor destroy motion1 it follows that when two bodies act upon each other in any way whatever1 the total quantity of motion in a given direction1 after the action takes place1 must be the same as before it1 for otherwise some motion would be produced by the action of the bodies1 which would contradict the principle that they are inert4 ,the word 8action0 is here applied1 perhaps improperly1 but according to the usage of mechanical writers1 to express a certain phenomenon or effect4 ,it is1 therefore1 not to be understood as implying any active principle in the bodies to which it is attributed4 7#ei47 ,in the cases of collision of which we have spoken1 one of the masses ,b was supposed to be quiescent before the impact4 ,we shall now suppose it to be moving in the same direction as ,a1 that is1 towards ,c1 but with a less velocity1 so that ,a shall overtake it1 and impinge upon it4 ,after the impact1 the two masses will move towards ,c with a common velocity1 the amount of which we now propose to determine4 ,if the masses ,a and ,b be equal1 then their motions or velocities added together must be the motion of the united mass after impact1 since no motion can either be created or destroyed by that event4 ,but as ,a and ,b move with a common motion1 this sum must be equally distributed between them1 and therefore each will move with a velocity equal to half the sum of their velocities before the impact4 ,thus1 if ,a have the velocity #g1 and ,b have #e1 the velocity of the united mass after impact is #f1 being the half of #ab1 the sum of #g and #e4 ,if ,a and ,b be not equal1 suppose them divided into equal component parts1 and let ,a consist of #h1 and ,b of #f1 equal masses3 let the velocity of ,a be #ag1 so that the motion of each of the #h parts being #ag1 the motion of the whole will be #acf4 ,in the same manner1 let the velocity of ,b be #aj1 the motion of each part being #aj1 the whole motion of the #f parts will be #fj4 ,the sum of the two motions1 therefore1 towards ,c is #aif2 and since none of this can be lost by the impact1 nor any motion added to it1 this must also be the whole motion of the united masses after impact4 ,being equally distributed among the #ad component parts of which these united masses consist1 each part will have a fourteenth of the whole motion4 ,hence1 #aif being divided by #ad1 we obtain the quotient #ad1 which is the velocity with which the whole moves4 7#fj47 ,in general1 therefore1 when two masses moving in the same direction impinge one upon the other1 and after impact move together1 their common velocity may be determined by the following rule3 8,express the masses and velocities by numbers in the usual way1 and multiply the numbers expressing the masses by the numbers which express the velocities2 the two products thus obtained being added together1 and their sum divided by the sum of the numbers expressing the masses1 the quotient will be the number expressing the required velocity40 7#fa47 ,from the preceding details1 it appears that motion is not adequately estimated by speed or velocity4 ,for example1 a certain mass ,a1 moving at a determinate rate1 has a certain quantity of motion4 ,if another equal mass ,b be added to ,a1 and a similar velocity be given to it1 as much more motion will evidently be called into existence4 ,in other words1 the two equal masses ,a and ,b united have twice as much motion as the single mass ,a had when moving alone1 and with the same speed4 ,the same reasoning will show that three equal masses will with the same speed have three times the motion of any one of them4 ,in general1 therefore1 the velocity being the same1 the quantity of motion will always be increased or diminished in the same proportion as the mass moved is increased or diminished4 7#fb47 ,on the other hand1 the quantity of motion does not depend on the mass only1 but also on the speed4 ,if a certain determinate mass move with a certain determinate speed1 another equal mass which moves with twice the speed1 that is1 which moves over twice the space in the same time1 will have twice the quantity of motion4 ,in this manner1 the mass being the same1 the quantity of motion will increase or diminish in the same proportion as the velocity4 7#fc47 ,the true estimate1 then1 of the quantity of motion is found by multiplying together the numbers which express the mass and the velocity4 ,thus1 in the example which has been last given of the impact of masses1 the quantities of motion before and after impact appear to be as follow3 ,before ,impact4 ,after ,impact4 ,mass of ,a #h ,mass of ,a #h ,velocity of ,a #ag ,common velocity #ad ----------------- -------------- ,quantity of #h #aga or #acf ,quantity of #h #ad or #aab motion of ,a motion of ,a ----------------- -------------- ,mass of ,b #f ,mass of ,b #f ,velocity of ,b #aj ,common velocity #ad ----------------- -------------- ,quantity of #f #aj or #fj ,quantity of #f #ad #hd motion of ,b motion of ,b ----------------- -------------- 99 ,the sign placed between two numbers meant that they are to be multiplied together4 ,by this calculation it appears that in the impact ,a has lost a quantity of motion expressed by #bd1 and that ,b has received exactly that amount4 ,the effect1 therefore1 of the impact is a transfer of motion from ,a to ,b2 but no new motion is produced in the direction ,a ,c which did not exist before4 ,this is obviously consistent with the property of inertia1 and indeed an inevitable result of it4 ,these results may be generalised and more clearly and concisely expressed by the aid of the symbols of arithmetic4 ,let a express the velocity of ,a4 ,let b express the velocity of ,b4 ,let x express the velocity of the united masses of ,a and ,b after impact1 each of these velocities being expressed in feet per second1 and the masses of ,a and ,b being expressed by the weight in pounds4 ,we shall then have the momenta or moving forces of ,a and ,b before impact1 expressed by ,a a and ,b b1 and the moving force of the united mass after impact will be expressed by 7,a ,b7 x4 ,the moving force of ,a after impact is ,a x1 and therefore the force it loses by the collision will be 7,a a - ,a x74 ,the force of ,b after impact will be ,b x1 and therefore the force it gains will be ,b x - ,b b4 ,but since the force lost by ,a must be equal to the force gained by ,b1 we shall have ,a a - ,a x ,b x - ,b b from which it is easy to infer 7,a ,b7 x ,a a ,b b and if it be required to express the velocity of the united masses after impact1 we have x 7,a a ,b b7_/7,a ,b7 ,when it is said that ,a a and ,b b express the moving forces of ,a and ,b1 it must be understood that the unit of momentum or moving force is in the case here supposed1 the force with which a mass of matter weighing #a lb4 would move if its velocity were #a foot per second1 and accordingly the forces with which ,a and ,b move before impact are as many times this as there are units respectively in the numbers signified by the general symbols ,a a and ,b b4 ,in like manner1 the force of the united masses after impact is as many times greater than that of #a lb4 moving through #a foot per second as there are units in the numbers expressed by 7,a ,b7 x4 7#fd47 ,these phenomena present an example of a law deduced from the property of inertia1 and generally expressed thus--8action and reaction are equal1 and in contrary directions40 ,the student must1 however1 be cautious not to receive these terms in their ordinary acceptation4 ,after the full explanation of inertia given in the last chapter1 it is1 perhaps1 scarcely necessary here to repeat1 that in the phenomena manifested by the motion of two bodies1 there can be neither 8action0 nor 8reaction10 properly so called4 ,the bodies are absolutely incapable either of action or resistance4 ,the sense in which these words must be received1 as used in the law1 is merely an expression of the transfer of a certain quantity of motion from one body to another1 which is called an action in the body which loses the motion1 and a reaction in the body which receives it4 ,the accession of motion to the latter is said to proceed from the action of the former2 and the loss of the same motion in the former is ascribed to the reaction of the latter4 ,the whole phraseology is1 however1 most objectionable and unphilosophical1 and is calculated to create wrong notions4 7#fe47 ,the bodies impinging were1 in the last case1 supposed to move in the same direction4 ,we shall now consider the case in which they move in opposite directions4 ,first1 let the masses ,a and ,b be supposed to be equal1 and moving in opposite directions1 with the same velocity4 ,let ,c1 fig4 #e41 be the point at which they meet4 ,the equal motions in opposite directions will1 in this case1 destroy each other1 and both masses will be reduced to a state of rest4 ,thus1 the mass ,a loses all its motion in the direction ,a ,c1 which it may be supposed to transfer to ,b at the moment of impact4 ,but ,b having previously had an equal quantity of motion in the direction ,b ,c1 will now have two equal motions impressed upon it1 in directions immediately opposite2 and these motions neutralising each other1 the mass becomes quiescent4 ,in this case1 therefore1 as in all the former examples1 each body transfers to the other all the motion which it loses1 consistently with the principle of 8action and reaction40 ,the masses ,a and ,b being still supposed equal1 let them move towards ,c with different velocities4 ,let ,a move with the velocity #aj1 and ,b with the velocity #f4 ,of the #aj parts of motion with which ,a is endued1 #f being transferred to ,b1 will destroy the equal velocity #f1 which ,b has in the direction ,b ,c4 ,the bodies will then move together in the direction ,c ,b1 the four remaining parts of ,a's motion being equally distributed between them4 ,each body will1 therefore1 have two parts of ,a's original motion1 and #b therefore will be their common velocity after impact4 ,in this case1 ,a loses #h of the #aj parts of its motion in the direction ,a ,c4 ,on the other hand1 ,b loses the entire of its #f parts of motion in the direction ,b ,c1 and receives #b parts in the direction ,a ,c4 ,this is equivalent to receiving #h parts of ,a's motion in the direction ,a ,c4 ,thus1 according to the law of 8action and reaction10 ,b receives exactly what ,a loses4 ,finally1 suppose that both the masses and velocities of ,a and ,b are unequal4 ,let the mass of ,a be #h1 and its velocity #i3 and let the mass of ,b be #f1 and its velocity #e4 ,the quantity of motion of ,a will be #gb1 and that of ,b1 in the opposite direction1 will be #cj4 ,of the #gb parts of motion1 which ,a has in the direction ,a ,c1 #cj being transferred to ,b1 will destroy all its #cj parts of motion in the direction ,b ,c1 and the two masses will move in the direction ,c ,b1 with the remaining #db parts of motion1 which will be equally distributed among their #ad component masses4 ,each component part will1 therefore1 receive #c parts of motion2 and accordingly #c will be the common velocity of the united mass after impact4 7#ff47 ,when two masses moving in opposite directions impinge and move together1 their common velocity after impact may be found by the following rule3--8,multiply the numbers expressing the masses by those which express the velocities respectively1 and subtract the lesser product from the greater2 divide the remainder by the sum of the numbers expressing the masses1 and the quotient will be the common velocity2 the direction will be that of the mass which has the greater quantity of motion40 ,it may be shown without difficulty1 that the example which we have just given obeys the law of 8action and reaction40 ,before impact4 ,after impact4 ,mass of ,a #h ,mass of ,a #h ,velocity of ,a #i ,common velocity #c ------------ ----------- ,quantity of motion #h #i or #gb ,quantity of motion #h #c or #bd in direction ,a ,c in direction ,a ,c ------------ ----------- ,mass of ,b #f ,mass of ,b #f ,velocity of ,b #e ,common velocity #c ------------ ----------- ,quantity of motion #f #e or #cj ,quantity of motion #f #c or #ah in direction ,b ,c in direction ,a ,c ------------ ----------- ,hence it appears that the quantity of motion in the direction ,a ,c of which ,a has been deprived by the impact is #dh1 the difference between #gb and #bd4 ,on the other hand1 ,b loses by the impact the quantity #cj in the direction ,b ,c1 which is equivalent to receiving #cj in the direction ,a ,c4 ,but it also acquires a quantity #ah in the direction ,a ,c1 which1 added to the former #cj1 gives a total of #dh received by ,b in the direction ,a ,c4 ,thus the same quantity of motion which ,a loses in the direction ,a ,c1 is received by ,b in the same direction4 ,the law of 8action and reaction0 is1 therefore1 fulfilled4 ,this result may in like manner be generalised4 ,retaining the former symbols1 the moving forces of ,a and ,b before impact will be ,a a and ,b b and their forces after impact will be ,a x and ,b x4 ,the force lost by ,a will therefore be ,a a - ,a x4 ,the mass ,b will have lost all the force ,b b which it had in its former direction1 and will have received the force ,b x in the opposite direction4 ,therefore the actual force imparted to ,b by the collision will be ,b b ,b x4 ,but since the force lost by ,a must be equal to that imparted to ,b1 we shall have ,a a - ,a x ,b b ,b x and therefore 7,a ,b7 x ,a a - ,b b and if the common velocity after impact be required1 we have x 7,a a - ,b b7_/7,a ,b7 ,as a general rule1 therefore1 to find the common velocity after impact4 ,multiply the weights by the previous velocities and take their sum if the bodies move in the same direction1 and their difference if they move in opposite directions1 and divide the one or the other by the sum of their weights4 ,the greatest will be the velocity after impact4 7#fg47 ,the examples of the equality of action and reaction in the collision of bodies may be exhibited experimentally by a very simple apparatus4 ,let ,a1 fig4 #f41 and ,b be two balls of soft clay1 or any other substance which is inelastic1 or nearly so1 and let these be suspended from ,c by equal strings1 so that they may be in contact2 and let a graduated arc1 of which the centre is ,c1 be placed so that the balls may oscillate over it4 ,one of the balls being moved from its place of rest along the arc1 and allowed to descend upon the other through a certain number of degrees1 will strike the other with a velocity corresponding to that number of degrees1 and both balls will then move together with a velocity which may be estimated by the number of degrees of the arc through which they rise4 7#fh47 ,in all these cases in which we have explained the law of 8action and reaction10 the transfer of motion from one body to the other has been made by impact or collision4 ,the phenomenon has been selected only because it is the most ordinary way in which bodies are seen to affect each other4 ,the law is1 however1 universal1 and will be fulfilled in whatever manner the bodies may affect each other4 ,thus ,a may be connected with ,b by a flexible string1 which1 at the commencement of ,a's motion1 is slack4 ,until the string becomes stretched1 that is1 until ,a's distance from ,b becomes equal to the length of the string1 ,a will continue to have all the motion first impressed upon it4 ,but when the string is stretched1 a part of that motion is transferred to ,b1 which is then drawn after ,a2 and whatever motion ,b in this way receives1 ,a must lose4 ,all that has been observed of the effect of motion transferred by impact will be equally applicable in this case4 ,again1 if ,b1 fig4 #d41 be a magnet moving in the direction ,b ,c with a certain quantity of motion1 and while it is so moving a mass of iron be placed at rest at ,a1 the attraction of the magnet will draw the iron after it towards ,c1 and will thus communicate to the iron a certain quantity of motion in the direction of ,c4 ,all the motion thus communicated to the iron ,a must be lost by the magnet ,b4 ,if the magnet and the iron were both placed quiescent at ,b and ,a1 the attraction of the magnet would cause the iron to move from ,a towards ,b2 but the magnet in this case not having any motion1 cannot be literally said to transfer a motion to the iron4 ,at the moment1 however1 when the iron begins to move from ,a towards ,b1 the magnet will be observed to begin also to move from ,b towards ,a2 and if the velocities of the two bodies be expressed by numbers1 and respectively multiplied by the numbers expressing their masses1 the quantities of motion thus obtained will be found to be exactly equal4 ,we have already explained why a quantity of motion received in the direction ,b ,a1 is equivalent to the same quantity lost in the direction ,a ,b4 ,hence it appears1 that the magnet in receiving as much motion in the direction ,b ,a1 as it gives in the direction ,a ,b1 suffers an effect which is equivalent to losing as much motion directed towards ,c as it has communicated to the iron in the same direction4 ,in the same manner1 if the body ,b had any property in virtue of which it might repel ,a1 it would itself be repelled with the same quantity of motion4 ,in a word1 whatever be the manner in which the bodies may affect each other1 whether by collision1 traction1 attraction1 or repulsion1 or by whatever other name the phenomenon may be designated1 still it is an inevitable consequence1 that any motion1 in a given direction1 which one of the bodies may receive1 must be accompanied by a loss of motion in the same direction1 and to the same amount1 by the other body1 or the acquisition of as much motion in the contrary direction2 or1 finally1 by a loss in the same direction1 and an acquisition of motion in the contrary direction1 the combined amount of which is equal to the motion received by the former4 7#fi47 ,from the principle1 that the force of a body in motion depends on the mass and the velocity1 it follows1 that any body1 however small1 may be made to move with the same force as any other body1 however great1 by giving to the smaller body a velocity which bears to that of the greater the same proportion as the mass of the greater bears to the mass of the smaller4 ,thus a feather1 ten thousand of which would have the same weight as a cannon-ball1 would move with the same force if it had ten thousand times the velocity2 and in such a case1 these two bodies encountering in opposite directions1 would mutually destroy each other's motion4 7#gj47 ,the consequences of the property of inertia1 which have been explained in the present and preceding chapters1 have been given by ,newton1 in his ,,principia1 and1 after him1 in most ,english treatises on mechanics1 under the form of three propositions1 which are called the 8laws of motion40 ,they are as follow3-- ,i4 8,every body must persevere in its state of rest1 or of uniform motion in a straight line1 unless it be compelled to change that state by forces impressed upon it40 ,,ii4 8,every change of motion must be proportional to the impressed force1 and must be in the direction of that straight line in which the force is impressed40 ,,iii4 8,action must always be equal and contrary to reaction2 or the actions of two bodies upon each other must be equal1 and directed towards contrary sides40 ,when inertia and force are defined1 the first law becomes an identical proposition4 ,the second law cannot be rendered perfectly intelligible until the student has read the chapter on the composition and resolution of forces1 for1 in fact1 it is intended as an expression of the whole body of results in that chapter4 ,the third law has been explained in the present chapter1 as far as it can be rendered intelligible in the present stage of our progress4 ,we have noticed these formularies more from a respect for the authorities by which they have been proposed and adopted1 than from any persuasion of their utility4 ,their full import cannot be comprehended until nearly the whole of elementary mechanics has been acquired1 and then all such summaries become useless4 99 99 99 99 99 7#ga47 ,the consequences deduced from the consideration of the quality of inertia in this chapter1 will account for many effects which fall under our notice daily1 and with which we have become so familiar1 that they have almost ceased to excite curiosity4 ,one of the facts of which we have most frequent practical illustration is1 that the quantity of motion or moving force1 as it is sometimes called1 is estimated by the velocity of the motion1 and the weight or mass of the thing moved conjointly4 ,if the same force impel two balls1 one of one pound weight1 and the other of two pounds1 it follows1 since the balls can neither give force to themselves1 nor resist that which is impressed upon them1 that they will move with the same force4 ,but the lighter ball will move with twice the speed of the heavier4 ,the impressed force which is manifested by giving velocity to a double mass in the one1 is engaged in giving a double velocity to the other4 ,if a cannon-ball were forty times the weight of a musket-ball1 but the musket-ball moved with forty times the velocity of the cannon-ball1 both would strike any obstacle with the same force1 and would overcome the same resistance2 for the one would acquire from its velocity as much force as the other derives from its weight4 ,a very small velocity may be accompanied by enormous force1 if the mass which is moved with that velocity be proportionally great4 ,a large ship1 floating near the pier wall1 may approach it with so small a velocity as to be scarcely perceptible1 and yet the force will be so great as to crush a small boat4 ,a grain of shot flung from the hand1 and striking the person1 will occasion no pain1 and indeed will scarcely be felt1 while a block of stone having the same velocity would occasion death4 ,if a body in motion strike a body at rest1 the striking body must sustain as great a shock from the collision as if it had been at rest1 and struck by the other body with the same force4 ,for the loss of force which it sustains in the one direction1 is an effect of the same kind as if1 being at rest1 it had received as much force in the opposite direction4 ,if a man1 walking rapidly or running1 encounters another standing still1 he suffers as much from the collision as the man against whom he strikes4 ,if a leaden bullet be discharged against a plank of hard wood1 it will be found that the round shape of the ball is destroyed1 and that it has itself suffered a force by the impact1 which is equivalent to the effect which it produces upon the plank4 ,when two bodies moving in opposite directions meet1 each body sustains as great a shock as if1 being at rest1 it had been struck by the other body with the united forces of the two4 ,thus1 if two equal balls1 moving at the rate of ten feet in a second1 meet1 each will be struck with the same force as if1 being at rest1 the other had moved against it at the rate of twenty feet in a second4 ,in this case one part of the shock sustained arises from the loss of force in one direction1 and another from the reception of force in the opposite direction4 ,for this reason1 two persons walking in opposite directions receive from their encounter a more violent shock than might be expected4 ,if they be of nearly equal weight1 and one be walking at the rate of three and the other four miles an hour1 each sustains the same shock as if he had been at rest1 and struck by the other running at the rate of seven miles an hour4 ,this principle accounts for the destructive effects arising from ships running foul of each other at sea4 ,if two ships of #ejj tons burden encounter each other1 sailing at ten knots an hour1 each sustains the shock which1 being at rest1 it would receive from a vessel of #ajjj tons burden sailing ten knots an hour4 ,it is a mistake to suppose1 that when a large and small body encounter1 the small body suffers a greater shock than the large one4 ,the shock which they sustain must be the same2 but the large body may be better able to bear it4 ,when the fist of a pugilist strikes the body of his antagonist1 it sustains as great a shock as it gives2 but the fist being more fitted to endure the blow1 the injury and pain are inflicted on his opponent4 ,this is not the case1 however1 when fist meets fist4 ,then the parts in collision are equally sensitive and vulnerable1 and the effect is aggravated by both having approached each other with great force4 ,the effect of the blow is the same as if one fist1 being held at rest1 were struck by the other with the combined force of both4 ,,chap4 ,v4 ,,the ,,composition ,,and ,,resolution ,,of ,,force4 7#gb47 ,motion and pressure are terms too familiar to need explanation4 ,it may be observed1 generally1 that definitions in the first rudiments of a science are seldom1 if ever1 comprehended4 ,the force of words is learned by their application2 and it is not until a definition becomes useless1 that we are taught the meaning of the terms in which it is expressed4 ,moreover1 we are perhaps justified in saying1 that in the mathematical sciences the fundamental notions are of so uncompounded a character1 that definitions1 when developed and enlarged upon1 often draw us into metaphysical subtleties and distinctions1 which1 whatever be their merit or importance1 would be here altogether misplaced4 ,we shall1 therefore1 at once take it for granted1 that the words motion and pressure express phenomena or effects which are the subjects of constant experience and hourly observation2 and if the scientific use of these words be more precise than their general and popular application1 that precision will soon be learned by their frequent use in the present treatise4 7#gc47 ,,force is the name given in mechanics to whatever produces motion or pressure4 ,this word is also often used to express the motion or pressure itself2 and when the cause of the motion or pressure is not known1 this is the only correct use of the word4 ,thus1 when a piece of iron moves toward a magnet1 it is usual to say that the cause of the motion is 8the attraction of the magnet20 but in effect we are ignorant of the cause of this phenomenon2 and the name attraction would be better applied to the effect of which we have experience4 ,in like manner the attraction and repulsion of electrified bodies should be understood1 not as names for unknown causes1 but as words expressing observed appearances or effects4 ,when a certain phraseology has1 however1 gotten into general use1 it is neither easy nor convenient to supersede it4 ,we shall1 therefore1 be compelled1 in speaking of motion and pressure1 to use the language of causation2 but must advise the student that it is effects and not causes which will be expressed4 7#gd47 ,if two forces act upon the same point of a body in different directions1 a single force may be assigned1 which1 acting on that point1 will produce the same result as the united effects of the other two4 ,let ,p1 fig4 #g41 be the point on which the two forces act1 and let their directions be ,p ,a and ,p ,b4 ,from the point ,p1 upon the line ,p ,a1 take a length ,p a1 consisting of as many inches as there are ounces in the force ,p ,a2 and1 in like manner1 take ,p b1 in the direction ,p ,b1 consisting of as many inches as there are ounces in the force ,p ,b4 ,through a draw a line parallel to ,p ,b1 and through b draw a line parallel to ,p ,a1 and suppose that these lines meet at c4 ,then draw ,p ,c4 ,a single force1 acting in the direction ,p ,c1 and consisting of as many ounces as the line ,p c consists of inches1 will produce upon the point ,p the same effect as the two forces ,p ,a and ,p ,b produce acting together4 7#ge47 ,the figure ,p a c b is called in ,,geometry a parallelogram2 the lines ,p a1 ,p b1 are called its sides1 and the line ,p c is called its diagonal4 ,thus the method of finding an equivalent for two forces1 which we have just explained1 is generally called 8the parallelogram of forces10 and is usually expressed thus3 8,if two forces be represented in quantity and direction by the sides of a parallelogram1 an equivalent force will be represented in quantity and direction by its diagonal40 7#gf47 ,a single force1 which is thus mechanically equivalent to two or more other forces1 is called their resultant1 and relatively to it they are called its components4 ,in any mechanical investigation1 when the resultant is used for the components1 which it always may be1 the process is called 8the composition of force40 ,it is1 however1 frequently expedient to substitute for a single force two or more forces1 to which it is mechanically equivalent1 or of which it is the resultant4 ,this process is called 8the resolution of force40 7#gg47 ,to verify experimentally the theorem of the parallelogram of forces is not difficult4 ,let two small wheels1 ,m ,n1 fig4 #h41 with grooves in their edges to receive a thread1 be attached to an upright board1 or to a wall4 ,let a thread be passed over them1 having weights ,a and ,b1 hooked upon loops at its extremities4 ,from any part ,p of the thread between the wheels let a weight ,c be suspended3 it will draw the thread downwards1 so as to form an angle ,m ,p ,n1 and the apparatus will settle itself at rest in some determinate position4 ,in this state it is evident that since the weight ,c1 acting in the direction ,p ,c1 balances the weights ,a and ,b1 acting in the directions ,p ,m and ,p ,n1 these two forces must be mechanically equivalent to a force equal to the weight ,c1 and acting directly upwards from ,p4 ,the weight ,c is therefore the quantity of the resultant of the forces ,p ,m and ,p ,n2 and the direction of the resultant is that of a line drawn directly upwards from ,p4 ,to ascertain how far this is consistent with the theorem of 8the parallelogram of forces10 let a line ,p ,o be drawn upon the upright board to which the wheels are attached1 from the point ,p upward1 in the direction of the thread ,c ,p4 ,also1 let lines be drawn upon the board immediately under the threads ,p ,m and ,p ,n4 ,from the point ,p1 on the line ,p ,o1 take as many inches as there are ounces in the weight ,c4 ,let the part of ,p ,o thus measured be ,p c1 and from c draw c a parallel to ,p ,n1 and c b parallel to ,p ,m4 ,if the sides ,p a and ,p b of the parallelogram thus formed be measured1 it will be found that ,p a will consist of as many inches as there are ounces in the weight ,a1 and ,p b of as many inches as there are ounces in the weight ,b4 ,in this illustration1 ounces and inches have been used as the subdivisions of weight and length4 ,it is scarcely necessary to state1 that any other measures of these quantities would serve as well1 only observing that the same denominations must be preserved in all parts of the same investigation4 7#gh47 ,among the philosophical apparatus of the ,university of ,london1 is a very simple and convenient instrument which ,i constructed for the experimental illustration of this important theorem4 ,the wheels ,m ,n are attached to the tops of two tall stands1 the heights of which may be varied at pleasure by an adjusting screw4 ,a jointed parallelogram1 ,a ,b ,c ,d1 fig4 #i41 is formed1 whose sides are divided into inches1 and the joints at ,a and ,b are moveable1 so as to vary the lengths of the sides at pleasure4 ,the joint ,c is fixed at the extremity of a ruler1 also divided into inches1 while the opposite joint ,a is attached to a brass loop1 which surrounds the diagonal ruler loosely1 so as to slide freely along it4 ,an adjusting screw is provided in this loop so as to clamp it in any required position4 ,in making the experiment1 the sides ,a ,b and ,a ,d1 ,c ,b and ,c ,d are adjusted by the joints ,b and ,a to the same number of inches respectively as there are ounces in the weights ,a and ,b1 fig4 #h4 ,then the diagonal ,a ,c is adjusted by the loop and screw at ,a1 to as many inches as there are ounces in the weight ,c4 ,this done1 the point ,a is placed behind ,p1 fig4 #h41 and the parallelogram is held upright1 so that the diagonal ,a ,c shall be in the direction of the vertical thread ,p ,c4 ,the sides ,a ,b and ,a ,d will then be found to take the direction of the threads ,p ,m and ,p ,n4 ,by changing the weights and the lengths of the diagonal and sides of the parallelogram1 the experiment may be easily varied at pleasure4 7#gi47 ,in the examples of the composition of forces which we have here given1 the effects of the forces are the production of pressures1 or1 to speak more correctly1 the theorem which we have illustrated1 is 8the composition of pressures40 ,for the point ,p is supposed to be at rest1 and to be drawn or pressed in the directions ,p ,m and ,p ,n4 ,in the definition which has been given of the word force1 it is declared to include motions as well as pressures4 ,in fact1 if motion be resisted1 the effect is converted into pressure4 ,the same cause acting upon a body1 will either produce motion or pressure1 according as the body is free or restrained4 ,if the body be free1 motion ensues2 if restrained1 pressure1 or both these effects together4 ,it is therefore consistent with analogy to expect that the same theorems which regulate pressures1 will also be applicable to motions2 and we find accordingly a most exact correspondence4 7#hj47 ,if a body have a motion in the direction ,a ,b1 and at the point ,p it receive another motion1 such as would carry it in the direction ,p ,c1 fig4 #aj41 were it previously quiescent at ,p1 it is required to determine the direction which the body will take1 and the speed with which it will move1 under these circumstances4 ,let the velocity with which the body is moving from ,a to ,b be such1 that it would move through a certain space1 suppose ,p ,n1 in one second of time1 and let the velocity of the motion impressed upon it at ,p be such1 that if it had no previous motion it would move from ,p to ,m in one second4 ,from the point ,m draw a line parallel to ,p ,b1 and from ,n draw a line parallel to ,p ,c1 and suppose these lines to meet at some point1 as ,o4 ,then draw the line ,p ,o4 ,in consequence of the two motions1 which are at the same time impressed upon the body at ,p1 it will move in the straight line from ,p to ,o4 ,thus the two motions1 which are expressed in quantity and direction by the sides of a parallelogram1 will1 when given to the same body1 produce a single motion1 expressed in quantity and direction by its diagonal2 a theorem which is to motions exactly what the former was to pressures4 ,there are various methods of illustrating experimentally the composition of motion4 ,an ivory ball1 being placed upon a perfectly level square table1 at one of the corners1 and receiving two equal impulses1 in the directions of the sides of the table1 will move along the diagonal4 ,apparatus for this experiment differ from each other only in the way of communicating the impulses to the ball4 7#ha47 ,as two motions simultaneously communicated to a body are equivalent to a single motion in an intermediate direction1 so also a single motion may be mechanically replaced1 by two motions in directions expressed by the sides of any parallelogram1 whose diagonal represents the single motion4 ,this process is 8the resolution of motion10 and gives considerable clearness and facility to many mechanical investigations4 7#hb47 ,it is frequently necessary to express the portion of a given force1 which acts in some given direction different from the immediate direction of the force itself4 ,thus1 if a force act from ,a1 fig4 #aa41 in the direction ,a ,c1 we may require to estimate what part of that force acts in the direction ,a ,b4 ,if the force be a pressure1 take as many inches ,a ,p from ,a1 on the line ,a ,c1 as there are ounces in the force1 and from ,p draw ,p ,m perpendicular to ,a ,b2 then the part of the force which acts along ,a ,b will be as many ounces as there are inches in ,a ,m4 ,the force ,a ,b is mechanically equivalent to two forces1 expressed by the sides ,a ,m and ,a ,n of the parallelogram2 but ,a ,n1 being perpendicular to ,a ,b1 can have no effect on a body at ,a1 in the direction of ,a ,b1 and therefore the effective part of the force ,a ,p in the direction ,a ,b is expressed by ,a ,m4 7#hc47 ,any number of forces acting on the same point of a body may be replaced by a single force1 which is mechanically equivalent to them1 and which is1 therefore1 their resultant4 ,this composition may be effected by the successive application of the parallelogram of forces4 ,let the several forces be called ,a1 ,b1 ,c1 ,d1 ,e1 c4 ,draw the parallelogram whose sides express the forces ,a and ,b1 and let its diagonal be ,a4 ,the force expressed by ,a will be equivalent to ,a and ,b4 ,then draw the parallelogram whose sides express the forces ,a and ,c1 and let its diagonal be ,b4 ,this diagonal will express a force mechanically equivalent to ,a and ,c4 ,but ,a is mechanically equivalent to ,a and ,b1 and therefore ,b is mechanically equivalent to ,a1 ,b1 and ,c4 ,next construct a parallelogram1 whose sides express the forces ,b and ,d1 and let its diagonal be ,c4 ,the force expressed by ,c will be mechanically equivalent to the forces ,b and ,d2 but the force ,b is equivalent to ,a1 ,b1 ,c1 and therefore ,c is equivalent to ,a1 ,b1 ,c1 and ,d4 ,by continuing this process it is evident1 that a single force may be found1 which will be equivalent to1 and may be always substituted for1 any number of forces which act upon the same point4 ,if the forces which act upon the point neutralise each other1 so that no motion can ensue1 they are said to be in equilibrium4 7#hd47 ,examples of the composition of motion and pressure are continually presenting themselves4 ,they occur in almost every instance of motion or force which falls under our observation4 ,the difficulty is to find an example which1 strictly speaking1 is a simple motion4 ,when a boat is rowed across a river1 in which there is a current1 it will not move in the direction in which it is impelled by the oars4 ,neither will it take the direction of the stream1 but will proceed exactly in that intermediate direction which is determined by the composition of force4 ,let ,a1 fig4 #ab41 be the place of the boat at starting2 and suppose that the oars are so worked as to impel the boat towards ,b with a force which would carry it to ,b in one hour1 if there were no current in the river4 ,but1 on the other hand1 suppose the rapidity of the current is such1 that without any exertion of the rowers the boat would float down the stream in one hour to ,c4 ,from ,c draw ,c ,d parallel to ,a ,b1 and draw the straight line ,a ,d diagonally4 ,the combined effect of the oars and the current will be1 that the boat will be carried along ,a ,d1 and will arrive at the opposite bank in one hour1 at the point ,d4 ,if the object be1 therefore1 to reach the point ,b1 starting from ,a1 the rowers must calculate1 as nearly as possible1 the velocity of the current4 ,they must imagine a certain point ,e at such a distance above ,b that the boat would be floated by the stream from ,e to ,b in the time taken in crossing the river in the direction ,a ,e1 if there were no current4 ,if they row towards the point ,e1 the boat will arrive at the point ,b1 moving in the line ,a ,b4 ,in this case the boat is impelled by two forces1 that of the oars in the direction ,a ,e1 and that of the current in the direction ,a ,c4 ,the result will be1 according to the parallelogram of forces1 a motion in the diagonal ,a ,b4 ,the wind and tide acting upon a vessel is a case of a similar kind4 ,suppose that the wind is made to impel the vessel in the direction of the keel2 while the tide may be acting in any direction oblique to that of the keel4 ,the course of the vessel is determined exactly in the same manner as that of the boat in the last example4 ,the action of the oars themselves1 in impelling the boat1 is an example of the composition of force4 ,let ,a1 fig4 #ac41 be the head1 and ,b the stern of the boat4 ,the boatman presents his face towards ,b1 and places the oars so that their blades press against the water in the directions ,c ,e1 ,d ,f4 ,the resistance of the water produces forces on the side of the boat1 in the directions ,g ,l and ,h ,l1 which1 by the composition of force1 are equivalent to die diagonal force ,k ,l1 in the direction of the keel4 ,similar observations will apply to almost every body impelled by instruments projecting from its sides1 and acting against a fluid4 ,the motions of fishes1 the act of swimming1 the flight of birds1 are all instances of the same kind4 7#he47 ,the action of wind upon the sails of a vessel1 and the force thereby transmitted to the keel1 modified by the rudder1 is a problem which is solved by the principles of the composition and resolution of force2 but it is of too complicated and difficult a nature to be introduced with all its necessary conditions and limitations in this place4 ,the question may1 however1 be simplified1 if we consider the canvass of the sails to be stretched so completely as to form a plane surface4 ,let ,a ,b1 fig4 #ad41 be the position of the sail1 and let the wind blow in the direction ,c ,d4 ,if the line ,c ,d be taken to express the force of the wind1 let ,d ,e ,c ,f be a parallelogram1 of which it is the diagonal4 ,the force ,c ,d is equivalent to two forces1 one in the direction ,f ,d of the plane of the canvass1 and the other ,e ,d perpendicular to the sail4 ,the effect1 therefore1 is the same as if there were two winds1 one blowing in the direction of ,f ,d or ,b ,a1 that is against the edge of the sail1 and the other1 ,e ,d1 blowing full against its face4 ,it is evident that the former will produce no effect whatever upon the sail1 and that the latter will urge the vessel in the direction ,d ,g4 ,let us now consider this force ,d ,g as acting in the diagonal of the parallelogram ,d ,h ,g ,i4 ,it will be equivalent to two forces1 ,d ,h and ,d ,i1 acting along the sides4 ,one of these forces1 ,d ,h1 is in the direction of the keel1 and the other1 ,d ,i1 at right angles to the length of the vessel1 so as to urge it sideways4 ,the form of the vessel is evidently such as to offer a great resistance to the latter force1 and very little to the former4 ,it consequently proceeds with considerable velocity in the direction ,d ,h of its keel1 and makes way very slowly in the sideward direction ,d ,i4 ,the latter effect is called lee-way4 ,from this explanation it will be easily understood1 how a wind which is nearly opposed to the course of a vessel may1 nevertheless1 be made to impel it by the effect of sails4 ,the angle ,b ,d ,v1 formed by the sail and the direction of the keel1 may be very oblique1 as may also be the angle ,c ,d ,b formed by the direction of the wind and that of the sail4 ,therefore the angle ,c ,d ,v1 made up of these two1 and which is that formed by the direction of the wind and that of the keel1 may be very oblique4 ,in fig4 #ae4 the wind is nearly contrary to the direction of the keel1 and yet there is an impelling force expressed by the line ,d ,h1 the line ,c ,d expressing1 as before1 the whole force of the wind4 ,in this example there are two successive decompositions of force4 ,first1 the original force of the wind ,c ,d is resolved into two1 ,e ,d and ,f ,d2 and next the element ,e ,d1 or its equal ,d ,g1 is resolved into ,d ,i and ,d ,h2 so that the original force is resolved into three1 viz4 ,f ,d1 ,d ,i1 ,d ,h1 which1 taken together1 are mechanically equivalent to it4 ,the part ,f ,d is entirely ineffectual2 it glides off on the surface of the canvass without producing any effect upon the vessel4 ,the part ,d ,i produces lee-way1 and the part ,d ,h impels4 ,illustration3 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 7#hf47 ,if the wind1 however1 be directly contrary to the course which it is required that the vessel should take1 there is no position which can be given to the sails which will impel the vessel4 ,in this case the required course itself is resolved into two1 in which the vessel sails alternately1 a process which is called tacking4 ,thus1 suppose the vessel is required to move from ,a to ,e1 fig4 #af41 the wind setting from ,e to ,a4 ,the motion ,a ,b being resolved into two1 by being assumed as the diagonal of a parallelogram1 the sides ,a a1 a ,b of the parallelogram are successively sailed over1 and the vessel by this means arrives at ,b1 instead of moving along the diagonal ,a ,b4 ,in the same manner she moves along ,b b1 b ,c1 ,c c1 c ,d1 ,d d1 d ,e1 and arrives at ,e4 ,she thus sails continually at a sufficient angle with the wind to obtain an impelling force1 yet at a sufficiently small angle to make way in her proposed course4 ,the consideration of the effect of the rudder1 which we have omitted in the preceding illustration1 affords another instance of the resolution of force4 ,we shall not1 however1 pursue this example further4 7#hg47 ,a body falling from the top of the mast when the vessel is in full sail1 is an example of the composition of motion4 ,it might be expected1 that during the descent of the body1 the vessel having sailed forward1 would leave it behind1 and that1 therefore1 it would fall in the water behind the stern1 or at least on the deck1 considerably behind the mast4 ,on the other hand1 it is found to fall at the foot of the mast1 exactly as it would if the vessel were not in motion4 ,to account for this1 let ,a ,b1 fig4 #ag41 be the position of the mast when the body at the top is disengaged4 ,the mast is moving onwards with the vessel in the direction ,a ,c1 so that in the time which the body would take to fall to the deck1 the top of the mast would move from ,a to ,c4 ,but the body being on the mast at the moment it is disengaged1 has this motion ,a ,c in common with the mast2 and therefore in its descent it is affected by two motions1 viz4 that of the vessel expressed by ,a ,c1 and its descending motion expressed by ,a ,b4 ,hence1 by the composition of motion1 it will be found at the opposite angle ,d of the parallelogram1 at the end of the fall4 ,during the fall1 however1 the mast has moved with the vessel1 and has advanced to ,c ,d1 so that the body falls at the foot of the mast4 7#hh47 ,an instance of the composition of motion1 which is worthy of some attention1 as it affords a proof of the diurnal motion of the earth1 is derived from observing the descent of a body from a very high tower4 ,to render the explanation of this more simple1 we shall suppose the tower to be on the equator of the earth4 ,let ,e ,p ,q1 fig4 #ah41 be a section of the earth through the equator1 and let ,p ,t be the tower4 ,let us suppose that the earth moves on its axis in the direction ,e ,p ,q4 ,the foot ,p of the tower will1 therefore1 in one day move over the circle ,e ,p ,q1 while the top ,t moves over the greater circle ,t ,t ,r4 ,hence it is evident1 that the top of the tower moves with greater speed than the foot1 and therefore in the same time moves through a greater space4 ,now suppose a body placed at the top2 it participates in the motion which the top of the tower has in common with the earth4 ,if it be disengaged1 it also receives the descending motion ,t ,p4 ,let us suppose that the body would take five seconds to fall from ,t to ,p1 and that in the same time the top ,t is moved by the rotation of the earth from ,t to ,t1 the foot being moved from ,p to ,p4 ,the falling body is therefore endued with two motions1 one expressed by ,t ,t1 and the other by ,t ,p4 ,the combined effect of these will be found in the usual way by the parallelogram4 ,take ,t p equal to ,t ,t2 the body will move from ,t to p in the time of the fall1 and will meet the ground at p4 ,but since ,t ,t is greater than ,p ,p1 it follows that the point p must be at a distance from ,p equal to the excess of ,t ,t above ,p ,p4 ,hence the body will not fall exactly at the foot of the tower1 but at a certain distance from it1 in the direction of the earth's motion1 that is1 eastward4 ,this is found1 by experiment1 to be actually the case2 and the distance from the foot of the tower1 at which the body is observed to fall1 agrees with that which is computed from the motion of the earth1 to as great a degree of exactness as could be expected from the nature of the experiment4 7#hi47 ,the properties of compounded motions cause some of the equestrian feats exhibited at public spectacles to be performed by a kind of exertion very different from that which the spectators generally attribute to the performer4 ,for example1 the horseman standing on the saddle leaps over a garter extended over the horse at right angles to his motion2 the horse passing under the garter1 the rider lights upon the saddle at the opposite side4 ,the exertion of the performer1 in this case1 is not that which he would use were he to leap from the ground over a garter at the same height4 ,in the latter case1 he would make an exertion to rise1 and1 at the same time1 to project his body forward4 ,in the case1 however1 of the horseman1 he merely makes that exertion which is necessary to rise directly upwards to a sufficient height to clear the garter4 ,the motion which he has in common with the horse1 compounded with the elevation acquired by his muscular power1 accomplishes the leap4 ,to explain this more fully1 let ,a ,b ,c1 fig4 #ai41 be the direction in which the horse moves1 ,a being the point at which the rider quits the saddle1 and ,c the point at which he returns to it4 ,let ,d be the highest point which is to be cleared in the leap4 ,at ,a the rider makes a leap towards the point ,e1 and this must be done at such a distance from ,b1 that he would rise from ,b to ,e in the time in which the horse moves from ,a to ,b4 ,on departing from ,a1 the rider has1 therefore1 two motions1 represented by the lines ,a ,e and ,a ,b1 by which he will move from the point ,a to the opposite angle ,d of the parallelogram4 ,at ,d1 the exertion of the leap being overcome by the weight of his body1 he begins to return downward1 and would fall from ,d to ,b in the time in which the horse moves from ,b to ,c4 ,but at ,d he still retains the motion which he had in common with the horse2 and therefore1 in leaving the point ,d1 he has two motions1 expressed by the lines ,d ,f and ,d ,b4 ,the compounded effects of these motions carry him from ,d to ,c4 ,strictly speaking1 his motion from ,a to ,d1 and from ,d to ,c1 is not in straight lines1 but in a curve4 ,it is not necessary here1 however1 to attend to this circumstance4 7#ij47 ,if a billiard-ball strike the cushion of the table obliquely1 it will be reflected from it in a certain direction1 forming an angle with the direction in which it struck it4 ,this affords an example of the resolution and composition of motion4 ,we shall first consider the effect which would ensue if the ball struck the cushion perpendicularly4 ,let ,a ,b1 fig4 #bj41 be the cushion1 and ,c ,d the direction in which the ball moves towards it4 ,if the ball and the cushion were perfectly inelastic1 the resistance of the cushion would destroy the motion of the ball1 and it would be reduced to a state of rest at ,d4 ,if1 on the other hand1 the ball were perfectly elastic1 it would be reflected from the cushion1 and would receive as much motion from ,d to ,c after the impact1 as it had from ,c to ,d before it4 ,perfect elasticity1 however1 is a quality which is never found in these bodies4 ,they are always elastic1 but imperfectly so4 ,consequently the ball after the impact will be reflected from ,d towards ,c1 but with a less motion than that with which it approached from ,c to ,d4 ,now let us suppose that the ball1 instead of moving from ,c to ,d1 moves from ,e to ,d4 ,the force with which it strikes ,d being expressed by ,d ,e1 equal to ,e ,d1 may be resolved into two1 ,d ,f and ,d ,c4 ,the resistance of the cushion destroys ,d ,c1 and the elasticity produces a contrary force in the direction ,d ,c1 but less than ,d ,c or ,d ,c1 because that elasticity is imperfect4 ,the line ,d ,c expressing the force in the direction ,c ,d1 let ,d ,g 7less than ,d ,c7 express the reflective force in the direction ,d ,c4 ,the other element ,d ,f1 into which the force ,d ,e is resolved by the impact1 is not destroyed or modified by the cushion1 and therefore1 on leaving the cushion at ,d1 the ball is influenced by two forces1 ,d ,f 7which is equal to ,c ,e7 and ,d ,g4 ,consequently it will move in the diagonal ,d ,h4 7#ia47 ,the angle ,e ,d ,c is in this case called the 8angle of incidence10 and ,c ,d ,h is called 8the angle of reflection40 ,it is evident1 from what has been just inferred1 that the ball1 being imperfectly elastic1 the angle of incidence must always be less than the angle of reflection1 and with the same obliquity of incidence1 the more imperfect the elasticity is1 the less will be the angle of reflection4 ,in the impact of a perfectly elastic body1 the angle of reflection would be equal to the angle of incidence4 ,for then the line ,d ,g1 expressing the reflective force1 would be taken equal to ,c ,d1 and the angle ,c ,d ,h would be equal to ,c ,d ,e4 ,this is found by experiment to be the case when light is reflected from a polished surface of glass or metal4 ,motion is sometimes distinguished into absolute and relative4 ,what 8relative motion0 means is easily explained4 ,if a man walk upon the deck of a ship from stem to stern1 he has a relative motion which is measured by the space upon the deck over which he walks in a given time4 ,but while he is thus walking from stem to stern1 the ship and its contents1 including himself1 are impelled through the deep in the opposite direction4 ,if it so happen that the motion of the man1 from stem to stern1 be exactly equal to the motion of the ship in the contrary way1 the man will be1 relatively to the surface of the sea and that of the earth1 at rest4 ,thus1 relatively to the ship1 he is in motion1 while1 relatively to the surface of the earth1 he is at rest4 ,but still this is not absolute rest4 ,the surface itself is moving by the diurnal rotation of the earth upon its axis1 as well as by the annual motion in its orbit round the sun4 ,these motions1 and others to which the earth is subject1 must be all compounded by the theorem of the parallelogram of forces before we can obtain the absolute state of the body with respect to motion or rest4 ,,chap4 ,,vi4 ,,attraction4 7#ib47 ,whatever produces1 or tends to produce1 a change in the state of a particle or mass of matter with respect to motion or rest1 is a force4 ,rest1 or uniform rectilinear motion1 are therefore the only states in which any body can exist which is not subject to the present action of some force4 ,we are not1 however1 entitled to conclude1 that because a body is observed in one or other of these states1 it is therefore uninfluenced by any forces4 ,it may be under the immediate action of forces which neutralise each other3 thus two forces may be acting upon it which are equal1 and in opposite directions4 ,in such a case1 its state of rest1 or of uniform rectilinear motion1 will be undisturbed4 ,the state of uniform rectilinear motion declares more with respect to the body than the state of rest2 for the former betrays the action of a force upon the body at some antecedent period2 this action having been suspended1 while its effect continues to be observed in the motion which it has produced4 7#ic47 ,when the state of a body is changed from rest to uniform rectilinear motion1 the action of the force is only momentary1 in which case it is called an impulse4 ,if a body in uniform rectilinear motion receive an impulse in the direction in which it is moving1 the effect will be1 that it will continue to move uniformly in the same direction1 but its velocity will be increased by the amount of speed which the impulse would have given it had it been previously quiescent4 ,thus1 if the previous motion be at the rate of ten feet in a second1 and the impulse be such as would move it from a state of rest at five feet in a second1 the velocity1 after the impulse1 will be fifteen feet in a second4 ,but if the impulse be received in a direction immediately opposed to the previous motion1 then it will diminish the speed by that amount of velocity which it would give to the body had it been previously at rest4 ,in the example already given1 if the impulse were opposed to the previous motion1 the velocity of the body after the impulse would be five feet in a second4 ,if the impulse received in the direction opposed to the motion be such as would give to the body at rest a velocity equal to that with which it is moving1 then the effect will be1 that after the impulse no motion will exist2 and if the impulse would give it a still greater velocity1 the body will be moved in the opposite direction with an uniform velocity equal to the excess of that due to the impulse over that which the body previously had4 ,when a body in a state of uniform motion receives an impulse in a direction not coinciding with that of its motion1 it will move uniformly after the impulse in an intermediate direction1 which may be determined by the principles established for the composition of motion in the last chapter4 ,thus it appears1 that whenever the state of a body is changed either from rest to uniform rectilinear motion or vice versa1 or from one state of uniform rectilinear motion to another1 differing from that either in velocity or direction1 or in both1 the phenomenon is produced by that peculiar modification of force whose action continues but for a single instant1 and which has been called an impulse4 7#id47 ,in most cases1 however1 the mechanical state of a body is observed to be subject to a continual change or tendency to change4 ,we are surrounded by innumerable examples of this4 ,a body is placed on the table4 ,a continual pressure is excited on the surface of the table4 ,this pressure is only the consequence of the continual tendency of the body to move downwards4 ,if the body were excited by a force of the nature of an impulse1 the effect upon the table would be instantaneous1 and would immediately cease4 ,it would1 in fact1 be a blow4 ,but the continuation of the pressure proves the continuation of the action of the force4 ,if the table be removed from beneath the body1 the force which excites it being no longer resisted1 will produce motion2 it is manifested1 not as before1 by a tendency to produce motion1 but by the actual exhibition of that phenomenon4 ,now if the exciting force were an impulse1 the body would descend to the ground with an uniform velocity4 ,on the other hand1 as will hereafter appear1 every moment of its fall increases its speed1 and that speed is greatest at the instant it meets the ground4 ,a piece of iron placed at a distance from a magnet approaches it1 but not with an uniform velocity4 ,the force of the magnet continues to act during the approach of the iron1 and each moment gives it increased motion4 7#ie47 ,the forces which are thus in constant operation1 proceed from secret agencies which the human mind has never been able to detect4 ,all the analogies of nature prove that they are not the immediate results of the divine will1 but are secondary causes1 that is1 effects of some more remote principles4 ,to ascend to these secondary causes1 and thus as it were approach one step nearer to the ,creator1 is the great business of philosophy2 and the most certain means for accomplishing this1 is diligently to observe1 to compare1 and to classify the phenomena1 and to avoid assuming the existence of any thing which has not either been directly observed1 or which cannot be inferred demonstratively from natural phenomena4 ,philosophy should follow nature1 and not lead her4 ,while the law of inertia1 established by observation and reason1 declares the inability of matter1 from any principle resident in it1 to change its state1 all the phenomena of the universe prove that state to be in constant but regular fluctuation4 ,there is not in existence a single instance of the phenomenon of absolute rest1 or of motion which is absolutely uniform and rectilinear4 ,in bodies1 or the parts of bodies1 there is no known instance of simple passive juxtaposition unaccompanied by pressure or tension1 or some other 8tendency to motion40 ,innumerable secret powers are ever at work1 compensating1 as it were1 for inertia1 and supplying the material world with a substitute for the principles of action and will1 which give such immeasurable superiority to the character of life4 7#if47 ,the forces which are thus in continual operation1 whose existence is demonstrated by their observed effects1 but whose nature1 seat1 and mode of operation are unknown to us1 are called by the general name attractions4 ,these forces are classified according to the analogies which prevail among their effects1 in the same manner1 and according to the same principles1 as organised beings are grouped in natural history4 ,in that department of natural science1 when individuals are distributed in classes1 the object is merely to generalise1 and thereby promote the enlargement of knowledge2 but nothing is or ought to be thus assumed respecting the essence1 or real internal constitution of the individuals4 ,according to their external and observable characters and qualities they are classed2 and this classification should never be adduced as an evidence of any thing except that similitude of qualities to which it owed its origin4 ,phenomena are to the natural philosopher what organised beings are to the naturalist4 ,he groups and classifies them on the same principles1 and with a like object4 ,and as the naturalist gives to each species a name applicable to the individual beings which exhibit corresponding qualities1 so the philosopher gives to each force or attraction a name corresponding to the phenomena of which it is the cause4 ,the naturalist is ignorant of the real essence or internal constitution of the thing which he nominates1 and of the manner in which it comes to possess or exhibit those qualities which form the basis of his classification2 and the natural philosopher is equally ignorant of the nature1 seat1 and mode of operation of the force which he assigns as the cause of an observed class of effects4 ,these observations respecting the true import of the term 8attraction0 seem the more necessary to be premised1 because the general phraseology of physical science1 taken as language is commonly received1 will seem to convey something more4 ,the names of the several attractions which we shall have to notice1 frequently refer the seat of the cause to specific objects1 and seem to imply something respecting its mode of operation4 ,thus1 when we say 8the magnet attracts a piece of iron10 the true philosophical import of the words is1 8that a piece of iron placed in the vicinity of the magnet1 will move towards it1 or placed in contact1 will adhere to it1 so that some force is necessary to separate them40 ,in the ordinary sense1 however1 something more than this simple fact is implied4 ,it is insinuated that the magnet is the seat of the force which gives motion to the iron2 that in the production of the phenomenon1 the magnet is an agent exerting a certain influence1 of which the iron is the subject4 ,of all this1 however1 there is no proof2 on the contrary1 since the magnet must move towards the iron with just as much force as the iron moves towards the magnet1 there is as much reason to place the seat of the force in the iron1 and consider it as an agent affecting the magnet4 ,but1 in fact1 the influence which produces this phenomenon may not be resident in either the one body or the other4 ,it may be imagined to be a property of a medium in which both are placed1 or to arise from some third body1 the presence of which is not immediately observed4 ,however attractive these and like speculations may be1 they cannot be allowed a place in physical investigations1 nor should consequences drawn from such hypotheses be allowed to taint our conclusions with their uncertainty4 ,the student ought1 therefore1 to be aware1 that whatever may seem to be implied by the language used in this science in relation to attractions1 nothing is permitted to form the basis of reasoning respecting them except their effects2 and whatever be the common signification of the terms used1 it is to these effects1 and to these alone1 they should be referred4 7#ig47 ,attractions may be primarily distributed into two classes2 one consisting of those which exist between the molecules or constituent parts of bodies1 and the other between bodies themselves4 ,the former are sometimes called1 for distinction1 molecular or atomic attractions4 ,without the agency of molecular forces1 the whole face of nature would be deprived of variety and beauty2 the universe would be a confused heap of material atoms dispersed through space1 without form1 shape1 coherence1 or motion4 ,bodies would neither have the forms of solid1 liquid1 or air2 heat and light would no longer produce their wonted effects2 organised beings could not exist2 life itself1 as connected with body1 would be extinct4 ,atoms of matter1 whether distant or in juxtaposition1 would have no tendency to change their places1 and all would be eternal stillness and rest4 ,if1 then1 we are asked for a proof of the existence of molecular forces1 we may point to the earth and to the heavens2 we may name every object which can be seen or felt4 ,the whole material world is one great result of the influence of these powerful agents4 7#ih47 ,it has been proved 7#aa4 et seq47 that the constituent particles of bodies are of inconceivable minuteness1 and that they are not in immediate contact 7#bc71 but separated from each other by interstitial spaces1 which1 like the atoms themselves1 although too small to be directly observed1 yet are incontestably proved to exist1 by observable phenomena1 from which their existence demonstratively follows4 ,the resistance which every body opposes to compression1 proves that a repulsive influence prevails between the particles1 and that this repulsion is the cause which keeps the atoms separate1 and maintains the interstitial spaces just mentioned4 ,although this repulsion is found to exist between the molecules of all substances whatever1 yet it has different degrees of energy in different bodies4 ,this is proved by the fact1 that some substances admit of easy compression1 while in others1 the exertion of considerable force is necessary to produce the smallest diminution in bulk4 ,the space around each atom of a body1 through which this repulsive influence extends1 is generally limited1 and immediately beyond it1 a force of the opposite kind is manifested1 viz4 attraction4 ,thus1 in solid bodies1 the particles resist separation as well as compression1 and the application of force is as necessary to break the body1 or divide it into separate parts1 as to force its particles into closer aggregation4 ,it is by virtue of this attraction that solid bodies maintain their figure1 and that their parts are not separated and scattered like those of fluids1 merely by their own weight4 ,this force is called the attraction of cohesion4 ,the cohesive force acts in different substances with different degrees of energy3 in some its intensity is very great2 but the sphere of its influence apparently very limited4 ,this is the case with all bodies which are hard1 strong1 and brittle1 which no force can extend or stretch in any perceptible degree1 and which require a great force to break or tear them asunder4 ,such1 for example1 is cast iron1 certain stones1 and various other substances4 ,in some bodies the cohesive force is weak1 but the sphere of its action considerable4 ,bodies which are easily extended1 without being broken or torn asunder1 furnish examples of this4 ,such are ,indian-rubber1 or caoutchouc1 several animal and vegetable products1 and1 in general1 all solids of a soft and viscid kind4 ,between these extremes1 the cohesive force may be observed in various degrees4 ,in lead and other soft metals1 its sphere of action is greater1 and its energy less1 than in the former examples2 but its sphere less1 and energy greater1 than in the latter ones4 ,it is from the influence of this force1 and that of the repulsion1 whose sphere of action is still closer to the component atoms1 that all the varieties of texture which we denominate hard1 soft1 tough1 brittle1 ductile1 pliant1 c4 arise4 ,after having been broken1 or otherwise separated1 the parts of a solid may be again united by their cohesion1 provided any considerable number of points be brought into sufficiently close contact4 ,when this is done by mechanical means1 however1 the cohesion is not so strong as before their separation1 and a comparatively small force will be sufficient again to disunite them4 ,two pieces of lead freshly cut1 with smooth surfaces1 will adhere when pressed together1 and will require a considerable force to separate them4 ,in the same manner if a piece of ,indian-rubber be torn1 the parts separated will again cohere1 by being brought together with a slight pressure4 ,the union of the parts in such instances is easy1 because the sphere through which the influence of cohesion extends is considerable2 but even in bodies in which this influence extends through a more limited space1 the cohesion of separate pieces will be manifested1 provided their surfaces be highly polished1 so as to insure the near approach of a great number of their particles4 ,thus1 two polished surfaces of glass1 metal1 or stone1 will adhere when brought into contact4 ,in all these cases1 if the bodies be disunited by mechanical force1 they will separate at exactly the parts at which they had been united1 so that after their separation no part of the one will adhere to the other2 proving that the force of cohesion of the surfaces brought into contact is less than that which naturally held the particles of each together4 7#ii47 ,when a body is in the liquid form1 the weight of its particles greatly predominates over their mutual cohesion1 and consequently if such a body be unconfined it will be scattered by its own weight2 if it be placed in any vessel1 it will settle itself1 by the force of its weight1 into the lowest parts1 so that no space in the vessel below the upper surface of the liquid will be unoccupied4 ,the particles of a solid body placed in the vessel have exactly the same tendency1 by reason of their weight2 but this tendency is resisted and prevented from taking effect by their strong cohesion4 ,although this cohesion in solids is much greater than in liquids1 and productive of more obvious effects1 yet the principle is not altogether unobserved in liquids4 ,water converted into vapour by heat1 is divided into inconceivably minute particles1 which ascend in the atmosphere4 ,when it is there deprived of a part of that heat which gave it the vaporous form1 the particles1 in virtue of their cohesive force1 collect into round drops1 in which form they descend to the earth4 ,in the same manner1 if a liquid be allowed to fall gradually from the lip of a vessel1 it will not be dismissed in particles indefinitely small1 as if its mass were incoherent1 like sand or powder1 but will fall in drops of considerable magnitude4 ,in proportion as the cohesive force is greater1 these drops affect a greater size4 ,thus1 oil and viscid liquids fall in large drops2 ether1 alcohol1 and others in small ones4 ,two drops of rain trickling down a window pane will coalesce when they approach each other2 and the same phenomenon is still more remarkable1 if a few drops of quicksilver be scattered on an horizontal plate of glass4 ,it is the cohesive principle which gives rotundity to grains of shot3 the liquid metal is allowed to fall like rain from a great elevation4 ,in its descent the drops become truly globular1 and before they reach the end of their fall they are hardened by cooling1 so that they retain their shape4 ,it is also1 probably1 to the cohesive attraction that we should assign the globular forms of all the great bodies of the universe2 the sun1 planets1 satellites1 c41 which originally may have been in the liquid state4 7#ajj47 ,molecular attraction is also exhibited between the particles of liquids and solids4 ,a drop of water will not descend freely when it is in contact with a perpendicular glass plane3 it will adhere to the glass2 its descent will be retarded2 and if its weight be insufficient to overcome the adhesive force1 it will remain suspended4 ,if a plate of glass be placed upon the surface of water without being permitted to sink1 it will require more force to raise it from the water than is sufficient merely to balance the weight of the glass4 ,this shows the adhesion of the water and glass1 and also the cohesive force with which the particles of the water resist separation4 ,if a needle be dipped in certain liquids1 a drop will remain suspended at its point when withdrawn from them3 and1 in general1 when a solid body has been immersed in a liquid and withdrawn1 it is wet2 that is1 some of the liquid has adhered to its surfaces4 ,if no attraction existed between the solid and liquid1 the solid would be in the same state after immersion as before4 ,this is proved by liquids and solids between which no attraction exists4 ,if a piece of glass be immersed in mercury1 it will be in the same state when withdrawn as before it was immersed4 ,no mercury will adhere to it2 it will not be wet4 ,when it rains1 the person and vesture are affected only because this attraction exists between them and water4 ,if it rained mercury1 none would adhere to them4 7#aja47 ,when molecular attraction is exhibited by liquids pervading the interstices of porous bodies1 ascending in crevices or in the bores of small tubes1 it is called capillary attraction4 ,instances of this are innumerable4 ,liquids are thus drawn into the pores of sponge1 sugar1 lamp-wick1 c4 ,the animal and vegetable kingdom furnish numerous examples of this class of effects4 ,a weight being suspended by a dry rope1 will be drawn upwards through a considerable height1 if the rope be moistened with a wet sponge4 ,the attraction of the particles composing the rope for the water is in this case so powerful1 that the tension produced by several hundred weight cannot expel them4 ,a glass tube1 of small bore1 being dipped in water tinged by mixture with a little ink1 will retain a quantity of the liquid suspended when withdrawn4 ,the height of the liquid in the tube will be seen by looking through it4 ,it is found that the less the bore of the tube is1 the greater will be the height of the column sustained4 ,a series of such tubes fixed in the same frame1 with their lower orifices at the same level1 and with bores gradually decreasing1 being dipped in the liquid1 will exhibit columns gradually increasing4 ,a capillary syphon is formed of a hank of cotton threads1 one end of which is immersed in the vessel containing the liquid1 and the other is carried into the vessel into which the liquid is to be transferred4 ,the liquid may be thus drawn from the one vessel into the other4 ,the same effect may be produced by a glass syphon with a small bore4 7#ajb47 ,it frequently happens that a molecular repulsion is exhibited between a solid and a liquid4 ,if a piece of wood be immersed in quicksilver1 the liquid will be depressed at that part of the surface which is near the wood2 and in like manner1 if it be contained in a glass vessel1 it will be depressed at the edges4 ,in a barometer tube1 the surface of the mercury is convex1 owing partly to the repulsion between the glass and mercury4 ,all solids1 however1 do not repel mercury4 ,if any golden trinket be dipped in that liquid1 or even be exposed for a moment to contact with it1 the gold will be instantly intermingled with particles of quicksilver1 the metal changes its colour1 and becomes white like silver1 and the mercury can only be extricated by a difficult process4 ,chains1 seals1 rings1 c4 should always be laid aside by those engaged in experiments or other processes in which mercury is used4 7#ajc47 ,of all the forms under which molecular force is exhibited1 that in which it takes the name of affinity is attended with the most conspicuous effects4 ,affinity is in chemistry what inertia is in mechanics1 the basis of the science4 ,the present treatise is not the proper place for any detailed account of this important class of natural phenomena4 ,those who seek such knowledge are referred to our treatise on ,,chemistry4 ,since1 however1 affinity sometimes influences the mechanical state of bodies1 and affects their mechanical properties1 it will be necessary here to state so much respecting it as to render intelligible those references which we may have occasion to make to such effects4 ,when the particles of different bodies are brought into close contact1 and more especially when1 being in a fluid state1 they are mixed together1 their union is frequently observed to produce a compound body1 differing in its qualities from either of the component bodies4 ,thus the bulk of the compound is often greater or less than the united volumes of the component bodies4 ,the component bodies may be of the ordinary temperature of the atmosphere1 and yet the compound may be of a much higher or lower temperature4 ,the components may be liquid1 and the compound solid4 ,the colour of the compound may bear no resemblance whatever to that of the components4 ,the species of molecular action between the components1 which produce these and similar1 effects1 is called affinity4 7#ajd47 ,we shall limit ourselves here to the statement of a few examples of these phenomena4 ,if a pint of water and a pint of sulphuric acid be mixed1 the compound will be considerably less than a quart4 ,the density of the mixture is1 therefore1 greater than that which would result from the mere diffusion of the particles of the one fluid through those of the other4 ,the particles have assumed a greater proximity1 and therefore exhibit a mutual attraction4 ,in this experiment1 although the liquids before being mixed be of the temperature of the surrounding air1 the mixture will be so intensely hot1 that the vessel which contains it cannot be touched without pain4 ,if the two aeriform fluids1 called oxygen and hydrogen1 be mixed together in a certain proportion1 the compound will be water4 ,in this case1 the components are different from the compound1 not merely in the one being air and the other liquid1 but in other respects not less striking4 ,the compound water extinguishes fire1 and yet of the components1 hydrogen is one of the most inflammable substances in nature1 and the presence of oxygen is indispensably necessary to sustain the phenomenon of combustion4 ,oxygen gas1 united with quicksilver1 produces a compound of a black colour1 the quicksilver being white and the gas colourless4 ,when these substances are combined in another proportion1 they give a red compound4 7#aje47 ,having noticed the principal molecular forces1 we shall now proceed to the consideration of those attractions which are exhibited between bodies existing in masses4 ,the influence of molecular attractions is limited to insensible distances4 ,on the contrary1 the forces which are now to be noticed act at considerable distances1 and to the influence of some there is no limit1 the effect1 however1 decreasing as the distance increases4 ,the effect of the loadstone on iron is well known1 and is one of this class of forces4 ,for a detailed account of this force1 and the various phenomena of which it is the cause1 the reader is referred to our treatise on ,,magnetism4 ,when glass1 wax1 amber1 and other substances are submitted to friction with silken or woollen cloth1 they are observed to attract feathers1 and other light bodies placed near them4 ,a like effect is produced in several other ways1 and is attended with other phenomena1 the discussion of which forms a principal part of physical science4 ,the force thus exhibited is called electricity4 ,for details respecting it1 and for its connection with magnetism1 the reader is referred to our treatises on ,,electricity and ,,electro-,,magnetism4 7#ajf47 ,these attractions exist either between bodies of particular kinds1 or are developed by reducing the bodies which manifest them to a certain state by friction1 or some other means4 ,there is1 however1 an attraction1 which is manifested between bodies of all species1 and under all circumstances whatever2 an attraction1 the intensity of which is wholly independent of the nature of the bodies1 and only depends on their masses and mutual distances4 ,thus1 if a mass of metal and a mass of clay be placed in the vast abyss of space1 at a mile asunder1 they will instantly commence to approach each other with certain velocities4 ,again1 if a mass of stone and of wood respectively equal to the former1 be placed at a like distance1 they will also commence to approach each other with the same velocities as the former4 ,this universal attraction1 which only depends on the quantity of the masses and their mutual distances1 is called the 8attraction of gravitation40 ,we shall first explain the 8law0 of this attraction1 and shall then point out some of the principal phenomena by which its existence and its laws are known4 7#ajg47 ,the 8law of gravitation0 sometimes from its universality called the 8law of nature10 may be explained as follows3 ,let us suppose two masses1 ,a and ,b1 placed beyond the influence or attraction of any other bodies1 in a state of rest1 and at any proposed distance from each other4 ,by their mutual attraction they will approach each other1 but not with the same velocity4 ,the velocity of ,a will be greater than that of ,b1 in the same proportion as its mass is less than that of ,b4 ,thus1 if the mass of ,b be twice that of ,a1 while ,a approaches ,b through a space of two feet1 ,b will approach ,a through a space of one foot4 ,hence it follows1 that the force with which ,a moves towards ,b is equal to the force with which ,b moves towards ,a 7#fh74 ,this is only a consequence of the property of inertia1 and is an example of the equality of action and reaction1 as explained in ,chapter ,,iv4 ,the velocity with which ,a and ,b approach each other is estimated by the diminution of their distance1 ,a ,b1 by their mutual approach in a given time4 ,thus1 if in one second ,a move towards ,b through a space of two feet1 and in the same time ,b moves towards ,a through the space of one foot1 they will approach each other through a space of three feet in a second1 which will be their relative velocity 7#ia74 ,if the mass of ,b be doubled1 it will attract ,a with double the former force1 or1 what is the same1 will cause ,a to approach ,b with double the former velocity4 ,if the mass of ,b be trebled1 it will attract ,a with treble the first force1 and1 in general1 while the distance ,a ,b remains the same1 the attractive force of ,b upon ,a will increase or diminish in exactly the same proportion as the mass of ,b is increased or diminished4 ,in the same manner1 if the mass ,a be doubled1 it will be attracted by ,b with a double force1 because ,b exerts the same degree of attraction on every part of the mass ,a1 and any addition which it may receive will not diminish or otherwise affect the influence of ,b on its former mass4 ,to express this in general arithmetical symbols let a and b express the space through which ,a and ,b respectively would be moved towards each other by their mutual attraction4 ,we would then have ,a a ,b b4 ,thus1 it is a general law of gravitation1 that so long as the distance between two bodies remains the same1 each will attract and be attracted by the other1 in proportion to its mass2 and any increase or decrease of the mass will cause a corresponding increase or decrease in the amount of the attraction4 7#ajh47 ,we shall now explain the law1 according to which the attraction is changed1 by changing the distance between the bodies4 ,at the distance of one mile the body ,b attracts ,a with a certain force4 ,at the distance of two miles1 the masses not being changed1 the attraction of ,b upon ,a will be one-fourth of its amount at the distance of one mile4 ,at the distance of three miles1 it will be one-ninth of its original amount2 at four miles1 it is reduced to a sixteenth1 and so on4 ,the following table exhibits the diminution of the attraction corresponding to the successive increase of distance3 ---------------------------------------- ------ ,distance #a #b #c #d #e #f #g #h c4 ---------------------------------------- ------ ,attraction #a #a_/d #a_/ia_/afa_/bea_/cfa_/dia_/fd c4 ---------------------------------------- ------ ,in ,,arithmetic1 that number which is found by multiplying any proposed number by itself1 is called its square4 ,thus #d1 that is1 #b multiplied by #b1 is the square of #b2 #i that is1 #c times #c1 is the square of #c1 and so on4 ,on inspecting the above table1 it will be apparent1 therefore1 that the attraction of gravitation decreases in the same proportion as the square of the distance from the attracting body increases1 the mass of both bodies in this case being supposed to remain the same2 but if the mass of either be increased or diminished1 the attraction will be increased or diminished in the same proportion4 7#aji47 ,hence the law of gravitation may be thus expressed3 8,the mutual attraction of two bodies increases in the same proportion as their masses are increased1 and as the square of their distance is decreased2 and it decreases in proportion as their masses are decreased1 and as the square of their distance is increased40 ,this law may be more clearly expressed by means of general symbols4 ,let f express the force with which a mass weighing #a lb4 will attract another mass weighing #a lb41 at the distance of #a foot4 ,the force with which they will mutually attract1 when removed to the distance expressed in feet by ,d1 will be f_/,d#b that is1 the force f divided by the square of the number ,d4 ,if one of the bodies1 instead of weighing #a lb41 weigh the number of pounds expressed by ,a1 their mutual attraction will be increased ,a times1 and will therefore be expressed by 7,a f7_/,d#b ,in fine1 if the other be also the number of pounds expressed by ,b1 their mutual attraction will be 7,a ,b f7_/,d#b 7#aaj47 ,having explained the law of gravitation1 we shall now proceed to show how the existence of this force is proved1 and its law discovered4 ,the earth is known to be a globular mass of matter1 incomparably greater than any of the detached bodies which are found upon its surface4 ,if one of these bodies suspended at any proposed height above the surface of the earth be disengaged1 it will be observed to descend perpendicularly to the earth1 that is1 in the direction of the earth's centre4 ,the force with which it descends will also be found to be in proportion to the mass1 without any regard to the species of the body4 ,these circumstances are consistent with the account which we have given of gravitation4 ,but by that account we should expect1 that as the falling body is attracted with a certain force towards the earth1 the earth itself should be attracted towards it by the same force2 and instead of the falling body moving towards the earth1 which is the phenomenon observed1 the earth and it should move towards each other1 and meet at some intermediate point4 ,this1 in fact1 is the case1 although it is impossible to render the motion of the earth observable1 for reasons which will easily be understood4 ,since all the bodies around us participate in this motion1 it would not be directly observable1 even though its quantity were sufficiently great to be perceived under other circumstances4 ,but setting aside this consideration1 the space through which the earth moves in such a case is too minute to be the subject of sensible observation4 ,it has been stated 7#ajg71 that when two bodies attract each other1 the space through which the greater approaches the lesser1 bears to that through which the lesser approaches the greater1 the same proportion as the mass of the lesser bears to the mass of the greater4 ,now the mass of the earth is more than #ajjj1jjj1jjj1jjj1jjj times the mass of any body which is observed to fall on its surface2 and therefore if even the largest body which can come under observation were to fall through an height of #ejj feet1 the corresponding motion of the earth would be through a space less than the #ajjj1jjj1jjj1jjj1jjjth part of #ejj feet1 which is less than the #ajj1jjj1jjj1jjjth part of an inch4 ,the attraction between the earth and detached bodies on its surface is not only exhibited by the descent of these bodies when unsupported1 but by their pressure when supported4 ,this pressure is what is called weight4 ,the phenomena of weight1 and the descent of heavy bodies1 will be fully investigated in the next chapter4 7#aaa47 ,it is not alone by the direct fall of bodies that the gravitation of the earth is manifested4 ,the curvilinear motion of bodies projected in directions different from the perpendicular1 is a combination of the effects of the uniform velocity which has been given to the projectile by the impulse which it has received1 and the accelerated velocity which it receives from the earth's attraction4 ,suppose a body placed at any point ,p1 fig4 #ba41 above the surface of the earth1 and let ,p ,c be the direction of the earth's centre4 ,if the body were allowed to move without receiving any impulse1 it would descend to the earth in the direction ,p ,a1 with an accelerated motion4 ,but suppose that at the moment of its departure from ,p1 it receives an impulse in the direction ,p ,b1 which would carry it to ,b in the time the body would fall from ,p to ,a1 then1 by the composition of motion1 the body must at the end of that time be found in the line ,b ,d1 parallel to ,p ,a4 ,if the motion in the direction of ,p ,a were uniform1 the body ,p would in this case move in the straight line from ,p to ,d4 ,but this is not the case4 ,the velocity of the body in the direction ,p ,a is at first so small as to produce very little deflection of its motion from the line ,p ,b4 ,as the velocity1 however1 increases1 this deflection increases1 so that it moves from ,p to ,d in a curve1 which is convex1 towards ,p ,b4 ,the greater the velocity of the projectile in the direction ,p ,a1 the greater sweep the curve will take4 ,thus it will successively take the forms ,p ,d1 ,p ,e1 ,p ,f1 c41 and that velocity can be computed1 which 7setting aside the resistance of the air7 would cause the projectile to go completely round the earth1 and return to the point ,p from which it departed4 ,in this case1 the body ,p would continue to revolve round the earth like the moon4 ,hence it is obvious1 that the phenomenon of the revolution of the moon round the earth1 is nothing more than the combined effects of the earth's attraction1 and the impulse which it received when launched into space by the hand of its ,creator4 7#aab47 ,this is a great step in the analysis of the phenomenon of gravitation4 ,we have thus reduced to the same class two effects apparently very dissimilar1 the rectilinear descent of a heavy body1 and the nearly circular revolution of the moon round the earth4 ,hence we are conducted to a generalisation still more extensive4 ,as the moon's revolution round the earth1 in an orbit nearly circular1 is caused by the combination of the earth's attraction1 and an original projectile impulse1 so also the singular phenomena of the planets' revolution round the sun in orbits nearly circular1 must be considered an effect of the same class1 as well as the revolution of the satellites of those planets which are attended by such bodies4 ,although the orbits in which the comets move deviate very much from circles1 yet this does not hinder the application of the same principle to them1 their deviation from circles not depending on the sun's attraction1 but only on the direction and force of the original impulse which put them in motion4 7#aac47 ,we therefore conclude that gravitation is the principle which1 as it were1 animates the universe4 ,all the great changes and revolutions of the bodies which compose our system1 can be traced to or derived from this principle4 ,it still remains to show how that remarkable law1 by which this force is declared to increase or decrease in the same proportion as the square of the distance from the attracting body is decreased or increased1 may be verified and established4 ,it has been shown1 that the curvilinear path of a projectile depends on1 and can be derived1 by mathematical reasoning1 from the consideration of the intensity of the earth's attraction1 and the force of the original impulse1 or the velocity of projection4 ,in the same manner1 by a reverse process1 when we know the curve in which a projectile moves1 we can infer the amount of the attracting force which gives the curvature to its path4 ,in this way1 from our knowledge of the curvature of the moon's orbit1 and the velocity with which she moves1 the intensity of the attraction which the earth exerts upon her can be exactly ascertained4 ,upon comparing this with the force of gravitation at the earth's surface1 it is found that the latter is as many times greater than the former1 as the square of the moon's distance is greater than the square of the distance of a body on the surface of the earth from its centre4 7#aad47 ,if this were the only fact which could be brought to establish the law of gravitation1 it might be thought to be an accidental relation1 not necessarily characterising the attraction of gravitation4 ,upon examining the orbits and velocities of the several planets1 the same result is1 however1 obtained4 ,it is found that the forces with which they are severally attracted by the sun are great1 in exactly the same proportion as the squares of the several numbers expressing their distances are small4 ,the mutual gravitation of bodies on the surface of the earth towards each other is lost in the predominating force exerted by the earth upon all of them4 ,nevertheless1 in some cases1 this effect has not only been observed1 but actually measured4 ,a plumb-line1 under ordinary circumstances1 hangs in a direction truly vertical2 but if it be near a large mass of matter1 as a mountain1 it has been observed to be deflected from the true vertical1 towards the mountain4 ,this effect was observed by ,dr4 ,maskeline near the mountain called ,skehallien1 in ,scotland1 and by ,french astronomers near ,chimboraco4 ,for particulars of these observations1 see our treatise on ,,geod,,sy4 ,cavendish succeeded in exhibiting the effects of the mutual gravitation of metallic spheres4 ,two globes of lead ,a1 ,b1 each about a foot in diameter1 were placed at a certain distance asunder4 ,a light rod1 to the ends of which were attached small metallic balls ,c1 ,d1 was suspended at its centre ,e from a fine wire1 and the rod was placed as in fig4 #bb41 so that the attractions of each of the leaden globes had a tendency to turn the rod round the centre ,e in the same direction4 ,a manifest effect was produced upon the balls ,c1 ,d1 by the gravitation of the spheres4 ,in this experiment1 care must be taken that no magnetic substance is intermixed with the materials of the balls4 ,having so far stated the principles on which the law of gravitation is established1 we shall dismiss this subject without further details1 since it more properly belongs to the subject of ,,physical ,,astronomy2 to which we refer the reader for a complete demonstration of the law1 and for the detailed development of its various and important consequences4 ,,chap4 ,,vii4 ,,terrestrial ,,gravity4 7#aae47 ,,gravitation is the general name given to this attraction1 by whatever masses of matter it may be manifested4 ,as exhibited in the effects produced by the earth upon surrounding bodies1 it is called 8terrestrial gravity40 ,as the attraction of the earth is directed towards its centre1 it might be expected that two plumb-lines should appear not to be parallel1 but so inclined to each other as to converge to a point under the surface of the earth4 ,thus1 if ,a ,b and ,c ,d1 fig4 #bc41 be two plumb-lines1 each will be directed to the centre ,o1 where1 if their directions were continued1 they would meet4 ,in like manner1 if two bodies were allowed to fall from ,a and ,c1 they would descend in the directions ,a ,b and ,c ,d1 which converge to ,o4 ,observation1 on the contrary1 shows1 that plumb-lines suspended in places not far distant from each other are truly parallel2 and that bodies allowed to fall descend in parallel lines4 ,this apparent parallelism of the direction of terrestrial gravity is accounted for by the enormous proportion which the magnitude of the earth bears to the distance between the two plumb-lines or the two falling bodies which are compared4 ,if the distance between the places ,b1 ,d1 were #abjj feet1 the inclination of the lines ,a ,b and ,c ,d would not amount to a quarter of a minute1 or the #bdjth part of a degree4 ,but the distance1 in cases where the parallelism is assumed1 is never greater than1 and seldom so great as1 a few yards2 and hence the inclination of the directions ,a ,b and ,c ,d is too small to be appreciated by any practical measure4 ,in the investigation of the phenomena of falling bodies1 we shall1 therefore1 assume1 that all the particles of the same body are attracted in parallel directions1 perpendicular to an horizontal plane4 7#aaf47 ,since the intensity of terrestrial gravity increases as the square of the distance decreases1 it might be expected that1 as a falling body approaches the earth1 the force which accelerates it should be continually increasing1 and1 strictly speaking1 it is so4 ,but any height through which we observe falling bodies to descend bears so very small a proportion to the whole distance from the centre1 that the change of intensity of the force of gravity is quite beyond any practical means of estimating it4 ,the radius1 or the distance from the surface of the earth to its centre1 is #djjj miles4 ,now1 suppose a body descended through the height of half a mile1 a distance very much beyond those used in experimental enquiries1 the distances from the centre1 at the beginning and end of the fall1 are then in the proportion of #hjjj to #hjja1 and therefore the proportion of the force of attraction at the commencement to the force at the end1 being that of the squares of these numbers1 is #fd1jjj1jjj to #fd1jaf1jja1 which1 in the whole descent1 is an increase of about one part in #djjj2 a quantity practically insignificant4 ,we shall1 therefore1 in explaining the laws of falling bodies1 assume that1 in the entire descent1 the body is urged by a force of uniform intensity4 ,although the force which attracts all parts of the same body during its descent in a given place is the same1 yet the force of gravity1 at different parts of the earth's surface1 has different intensities4 ,the intensity diminishes with the latitude1 so that it is greater towards the poles1 and lesser towards the equator4 ,the causes of this variation1 its law1 and the experimental proofs of it1 will be explained1 when we shall treat of centrifugal force1 and the motion of pendulums4 ,it is sufficient merely to advert to it in this place4 7#aag47 ,since the earth's attraction acts separately and equally on every particle of matter1 without regard to the nature or species of the body1 it follows that all bodies1 of whatever kind1 or whatever be their masses1 must be moved with the same velocity4 ,if two equal particles of matter be placed at a certain distance above the surface of the earth1 they will fall in parallel lines1 and with exactly the same speed1 because the earth attracts them equally4 ,in the same manner1 a thousand particles would fall with equal velocities4 ,now1 these circumstances will in no wise be changed if those #ajjj particles1 instead of existing separately1 be aggregated into two solid masses1 one consisting of #iij particles1 and the other of #aj4 ,we shall thus have a heavy body and a light one1 and1 according to our reasoning1 they must fall to the earth with the same speed4 ,common experience1 however1 is not always consistent with this doctrine4 ,what are called light substances1 as feathers1 gold-leaf1 paper1 c41 are observed to fall slowly and irregularly1 while heavier masses1 as solid pieces of metal1 stones1 c41 fall rapidly4 ,nay1 there are not a few instances in which the earth1 instead of attracting bodies1 seems to repel them1 as in the case of smoke1 vapours1 balloons1 and other substances which actually ascend4 ,we are to consider that the mass of the earth is not the only agent engaged in these phenomena4 ,the earth is surrounded by an atmosphere composed of an elastic or aeriform fluid4 ,this atmosphere has certain properties1 which will be explained in our treatise on ,,pneumatics1 and which are the causes of the anomalous circumstances alluded to4 ,light bodies rise in the atmosphere1 for the same reason that a piece of cork rises from the bottom of a vessel of water2 and other light bodies fall more slowly than heavy ones1 for the same reason that an egg in water falls to the bottom more slowly than a leaden bullet4 ,this treatise is not the place to give a direct explanation of these phenomena4 ,it will be sufficient for our present purpose to show1 that if there were no atmosphere1 all bodies1 heavy and light1 would fall at the same rate4 ,this may easily be accomplished by the aid of an air-pump4 ,having by that instrument abstracted the air from a tall glass vessel1 we are enabled1 by means of a wire passing air-tight through a hole in the top1 to let fall several bodies from the top of the vessel to the bottom4 ,these1 whether they be feathers1 paper1 gold-leaf1 pieces of money1 c4 all descend with the same speed1 and strike the bottom at the same moment4 7#aah47 ,every one who has seen a heavy body fall from a height1 has witnessed the fact1 that its velocity increases as it approaches the ground4 ,but if this were not observable by the eye1 it would be betrayed by the effects4 ,it is well known1 that the force with which a body strikes the ground increases with the height from whence it has fallen4 ,this force1 however1 is proportional to the velocity which it has at the moment it meets the ground1 and therefore this velocity increases with the height4 ,when the observations on attraction in the last chapter are well understood1 it will be evident that the velocity which a body has acquired in falling from any height1 is the accumulated effects of the attraction of terrestrial gravity during the whole time of the fall4 ,each instant of the fall a new impulse is given to the body1 from which it receives additional velocity2 and its final velocity is composed of the aggregation of all the small increments of velocity which are thus communicated4 ,as we are at present to suppose the intensity of the attraction invariable1 it will follow that the velocity communicated to the body in each instant of time will be the same1 and therefore that the whole quantity of velocity produced or accumulated at the end of any time is proportional to the length of that time4 ,thus1 if a certain velocity be produced in a body having fallen for one second1 twice that velocity will be produced when it has fallen for two seconds1 thrice that velocity in three seconds1 and so on4 ,such is the fundamental principle or characteristic of uniformly accelerated motion4 7#aai47 ,in examining the circumstances of the descent of a body1 the time of the fall and the velocity at each instant of that time are not the only things to be attended to4 ,the spaces through which it falls in given intervals of time1 counted either from the commencement of its fall1 or from any proposed epoch of the descent1 are equally important objects of enquiry4 ,to estimate the space in reference to the time and the final velocity1 we must consider that this space has been moved through with varying speed4 ,from a state of rest at the beginning of the fall1 the speed gradually increases with the time1 and the final velocity is greater still than that which the body had at any preceding instant during its descent4 ,we cannot1 therefore1 directly appreciate the space moved through in this case by the time and final velocity4 ,but as the velocity increases uniformly with the time1 we shall obtain the average speed1 by finding that which the body had in the middle of the interval which elapsed between the beginning and end of the fall1 and thus the space through which the body has actually fallen is that through which it would move in the same time with this average velocity uniformly continued4 ,but since the velocity which the body receives in any time1 counted from the beginning of its descent1 is in the proportion of that time1 it follows that the velocity of the body after half the whole time of descent is half the final velocity4 ,from whence it appears1 that the height from which a body falls in any proposed time is equal to the space through which a body would move in the same time with half the final velocity1 and it is therefore equal to half the space which would be moved through in the same time with the final velocity4 7#abj47 ,it follows from this reasoning1 that between the three quantities1 the height1 the time1 and the final velocity1 which enter into the investigation of the phenomena of falling bodies1 there are two fixed relations3 ,first1 the time1 counted from the beginning of the fall and the final velocity1 are proportional the one to the other2 so that as one increases1 the other increases in the same proportion4 ,secondly1 the height being equal to half the space which would be moved through in the time of the fall1 with the final velocity1 must have a fixed proportion to these two quantities1 viz4 the time and the final velocity1 or must be proportional to the product of the two numbers which express them4 ,but since the time is always proportional to the final velocity1 they may be expressed by equal numbers1 and the product of equal numbers is the square of either of them4 ,hence1 the product of the numbers expressing the time and final velocity is equivalent to the square of the number expressing the time1 or to the square of the number expressing the final velocity4 ,hence we infer1 that the height is always proportional to the square of the time of the fall1 or to the square of the final velocity4 7#aba47 ,the use of a few mathematical characters will render these results more distinct1 even to students not conversant with mathematical science4 ,let ,s the height from which the body falls1 expressed in feet4 ,v the velocity at the end of the fall in feet per second4 ,t the number of seconds in the time of the fall4 g the number of feet through which a body would fall in one second4 ,it will therefore follow that the velocity acquired in one second will be #b;g1 and the velocity acquired in ,t seconds will therefore be #b;g ,t2 so that ,v #b;g ,t #a ,since the space which a body falls through in ,t seconds is found by multiplying the space it falls through in one second by ,t#b1 we shall have ,s g ,t#b #b from which1 combined with #a we deduce ,s ,v#b_/7d;g7 #c ,s 7#a_/b7,v ,t #d ,by these formularies1 if the height through which a body falls freely in one second be known1 the height through which it will fall in any proposed time may be computed4 ,for since the height is proportional to the square of the time1 the height through which it will fall in two seconds will be four times that which it falls through in one second4 ,in three seconds it will fall through nine times that space2 in four seconds1 sixteen times2 in five seconds1 twenty-five times1 and so on4 ,the following1 therefore1 is a general rule to find the height through which a body will fall in any given time3 8,reduce the given time to seconds1 take the square of the number of seconds in it1 and multiply the height through which a body falls in one second by that number2 the result will be the height sought40 ,the following table exhibits the heights and corresponding times as far as #aj seconds3 ---------------------------------------- ----- ,time #a #b #c #d #e #f #g #h #i #aj ---------------------------------------- ----- ,height #a #d #i #af #be #cf #di #fd #ha #ajj ---------------------------------------- ----- ,each unit in the numbers of the first row expresses a second of time1 and each unit in those of the second row expresses the height through which a body falls freely in a second4 7#abb47 ,if a body fall continually for several successive seconds1 the spaces which it falls through in each succeeding second have a remarkable relation among each other1 which may be easily deduced from the preceding table4 ,taking the space moved through in the first second still as our unit1 four times that space will be moved through in the first two seconds4 ,subtract from this #a1 the space moved through in the first second1 and the remainder #c is the space through which the body falls in the second second4 ,in like manner if #d1 the height fallen through in the first two seconds1 be subtracted from #i1 the height fallen through in the first three seconds1 the remainder #e will be the space fallen through in the third second4 ,to find the space fallen through in the fourth second1 subtract #i1 the space fallen through in the first three seconds1 from #af1 the space fallen through in the first four seconds1 and the result is #g1 and so on4 ,it thus appears that if the space fallen through in the first second be called #a1 the spaces described in the second1 third1 fourth1 fifth1 c4 seconds1 will be expressed by the odd numbers respectively1 #c1 #e1 #g1 #i1 c4 ,this places in a striking point of view the accelerated motion of a falling body1 the spaces moved through in each succeeding second being continually increased4 7#abc47 ,if velocity be estimated by the space through which the body would move uniformly in one second1 then the final velocity of a body falling for one second will be #b2 for with that final velocity the body would in one second move through twice the height through which it has fallen4 7#abd47 ,since the final velocity increases in the same proportion as the time1 it follows that after two seconds it is twice its amount after one1 and after three seconds thrice that1 and so on4 ,thus1 the following table exhibits the final velocities corresponding to the times of descent3 ---------------------------------------- ----------- ,time #a #b #c #d #e #f #g #h #i #aj ---------------------------------------- ----------- ,final velocity #b #d #f #h #aj #ab #ad #af #ah #bj ---------------------------------------- ----------- ,the numbers in the second row express the spaces through which a body with the final velocity would move in one second1 the unit being1 as usual1 the space through which a body falls freely in one second4 7#abe47 ,having thus developed theoretically the laws which characterise the descent of bodies1 falling freely by the force of gravity1 or by any other uniform force of the same kind1 it is necessary that we should show how these laws can be exhibited by actual experiment4 ,there are some circumstances attending the fall of heavy bodies which would render it difficult1 if not impossible1 to illustrate1 by the direct observation of this phenomenon1 the properties which have been explained in this chapter4 ,a body falling freely by the force of gravity1 as we shall hereafter prove1 descends in one second of time through a height of about #af feet#a2 in two seconds1 it would1 therefore1 fall through four times that space1 or #fd feet2 in three seconds1 through #i times the height1 or #add feet2 and in four seconds1 through #bef feet4 ,in order1 therefore1 to be enabled to observe the phenomena for only four seconds1 we should command an height of at least #bef feet4 ,but further2 the velocity at the end of the first second would be at the rate of #cb feet per second2 at the end of the second second1 it would be #fd feet per second2 and towards the end of the fall it would be about #abj feet per second4 ,it is evident that this great degree of rapidity would be a serious impediment to accurate observation1 even though we should be able to command the requisite height4 ,it appears therefore that the number expressed by g in the preceding formul is #afjhc4 #a ,more exactly through #af-a_/ab feet1 or #aic inches4 ,it occurred to ,mr4 ,george ,attwood1 a mathematician and natural philosopher of the last century1 that all the phenomena of falling bodies might be experimentally exhibited and accurately observed1 if a force of the same kind as gravity1 viz4 an uniformly accelerating force1 be used1 but of a much less intensity2 so that while the motion continues to be governed by the same laws1 its quantity may be so much diminished1 that the final velocity1 even after a descent of many seconds1 shall be so moderated as to admit of most deliberate and exact observation4 ,this being once accomplished1 nothing more would remain but to find the height through which a body would fall in one second1 or1 what is the same1 the proportion of the force of gravity to the mitigated but uniform accelerating force thus substituted for it4 7#abf47 ,to realise this notion1 ,attwood constructed a wheel turning on its axle with very little friction1 and having a groove on its edge to receive a string4 ,over this wheel1 and in the groove1 he placed a fine silken cord1 to the ends of which were attached equal cylindrical weights4 ,thus placed1 the weights perfectly balance each other1 and no motion ensues4 ,to one of the weights he then added a small quantity1 so as to give it a slight preponderance4 ,the loaded weight now began to descend1 drawing up on the other side the unloaded weight4 ,the descent of the loaded weight1 under these circumstances1 is a motion exactly of the same kind as the descent of a heavy body falling freely by the force of gravity2 that is1 it increases according to the same laws1 though at a very diminished rate4 ,to explain this1 suppose that the loaded weight descends from a state of rest through one inch in a second1 it will descend through #d inches in two seconds1 through #i in three1 through #af in four1 and so on4 ,thus in #bj seconds1 it would descend through #djj inches1 or #cc feet #d inches1 a height which1 if it were necessary1 could easily be commanded4 ,it might1 perhaps1 be thought1 that since the weights suspended at the ends of the thread are in equilibrium1 and therefore have no tendency either to move or to resist motion1 the additional weight placed upon one of them ought to descend as rapidly as it would if it were allowed to fall freely and unconnected with them4 ,it is very true that this weight will receive from the attraction of the earth the same force when placed upon one of the suspended weights1 as it would if it were disengaged from them2 but in the consequences which ensue1 there is this difference4 ,if it were unconnected with the suspended weights1 the whole force impressed upon it would be expended in accelerating its descent2 but being connected with the equal weights which sustain each other in equilibrium1 by the silken cord passing over the wheel1 the force which is impressed upon the added weight is expended1 not as before1 in giving velocity to the added weight alone1 but to it together with the two equal weights appended to the string1 one of which descends with the added weight1 and the other rises on the opposite side of the wheel4 ,hence1 setting aside any effect which the wheel itself produces1 the velocity of the descent must be lessened just in proportion as the mass among which the impressed force is to be distributed is increased2 and therefore the rate of the fall bears to that of a body falling freely the same proportion as the added weight bears to the sum of the masses of the equal suspended weights and the added weight4 ,thus the smaller the added weight is1 and the greater the equal suspended weights are1 the slower will the rate of descent be4 ,to render the circumstances of the fall conveniently observable1 a vertical shaft 7see fig4 #bd47 is usually provided1 which is placed behind the descending weight4 ,this pillar is divided to inches and halves1 and of course may be still more minutely graduated1 if necessary4 ,a stage to receive the falling weight is moveable on this pillar1 and capable of being fixed in any proposed position by an adjusting screw4 ,a pendulum vibrating seconds1 the beat of which ought to be very audible1 is placed near the observer4 ,the loaded weight being thus allowed to descend for any proposed time1 or from any required height1 all the circumstances of the descent may be accurately observed1 and the several laws already explained in this chapter may be experimentally verified4 7#abg47 ,the laws which govern the descent of bodies by gravity1 being reversed1 will be applicable to the ascent of bodies projected upwards4 ,if a body be projected directly upwards with any given velocity1 it will rise to the height from which it should have fallen to acquire that velocity4 ,the earth's attraction will1 in this case1 gradually deprive the body of the velocity which is communicated to it at the moment at which it is projected4 ,consequently1 the phenomenon will be that of retarded motion4 ,at each part of its ascent it will have the same velocity which it would have if it descended to the same place from the highest point to which it rises4 ,hence it is clear1 that all the particulars relative to the ascent of bodies may be immediately inferred from those of their descent1 and therefore this subject demands no further notice4 ,to complete the investigation of the phenomena of falling bodies1 it would now only remain to explain the method of ascertaining the exact height through which a body would descend in one second1 if unresisted by the atmosphere1 or any other disturbing cause4 ,as the solution of this problem1 however1 requires the aid of principles not yet explained1 it must for the present be postponed4 ,,chap4 ,,viii4 ,,of ,,the ,,motion ,,of ,,bodies ,,on ,,inclined ,,planes ,,and ,,curves4 7#abh47 ,in the last chapter1 we investigated the phenomena of bodies descending freely in the vertical direction1 and determined the laws which govern1 not their motion alone1 but that of bodies urged by any uniformly accelerating force whatever4 ,we shall now consider some of the most ordinary cases in which the free descent of bodies is impeded1 and the effects of their gravitation modified4 7#abi47 ,if a body1 urged by any forces whatever1 be placed upon a hard unyielding surface1 it will evidently remain at rest1 if the resultant 7#gf7 of all the forces which are applied to it be directed perpendicularly against the surface4 ,in this case1 the effect produced is pressure1 but no motion ensues4 ,if only one force act upon the body1 it will remain at rest1 provided the direction of that force be perpendicular to the surface4 ,but the effect will be different1 if the resultant of the forces which are applied to the body be oblique to the surface4 ,in that case this resultant1 which1 for simplicity1 may be taken as a single force1 may be considered as mechanically equivalent to two forces 7#gf71 one in the direction of the surface1 and the other perpendicular to it4 ,the latter element will be resisted1 and will produce a pressure2 the former will cause the body to move4 ,this will perhaps be more clearly apprehended by the aid of a diagram4 ,let ,a ,b1 fig4 #be41 be the surface1 and let ,p be a particle of matter placed upon it1 and urged by a force in the direction ,p ,d1 perpendicular to ,a ,b4 ,it is manifest1 that this force can only press the particle ,p against ,a ,b1 but cannot give it any motion4 ,but let us suppose1 that the force which urges ,p is in a direction ,p ,f1 oblique to ,a ,b4 ,taking ,p ,f as the diagonal of a parallelogram1 whose sides are ,p ,d and ,p ,c 7#gd71 the force ,p ,f is mechanically equivalent to two forces1 expressed by the lines ,p ,d and ,p ,c4 ,but ,p ,d1 being perpendicular to ,a ,b1 produces pressure without motion1 and ,p ,c1 being in the direction of ,a ,b1 produces motion without pressure4 ,thus the effect of the force ,p ,f is distributed between motion and pressure in a certain proportion1 which depends on the obliquity of its direction to that of the surface4 ,the two extreme cases are1 #a4 ,when it is in the direction of the surface2 it then produces motion without pressure3 and1 #b4 ,when it is perpendicular to the surface2 it then produces pressure without motion4 ,in all intermediate directions1 however1 it will produce both these effects4 7#acj47 ,it will be very apparent1 that the more oblique the direction of the force ,p ,f is to ,a ,b1 the greater will be that part of it which produces motion1 and the less will be that which produces pressure4 ,this will be evident by inspecting fig4 #bf4 ,in this figure the line ,p ,f1 which represents the force1 is equal to ,p ,f in fig4 #be4 ,but ,p ,d1 which expresses the pressure1 is less in fig4 #bf4 than in fig4 #be41 while ,p ,c1 which expresses the motion1 is greater4 ,so long1 then1 as the obliquity of the directions of the surface and the force remain unchanged1 so long will the distribution of the force between motion and pressure remain the same2 and therefore1 if the force itself remain the same1 the parts of it which produce motion and pressure will be respectively equal4 7#aca47 ,these general principles being understood1 no difficulty can arise in applying them to the motion of bodies urged on inclined planes or curves by the force of gravity4 ,if a body be placed on an unyielding horizontal plane1 it will remain at rest1 producing a pressure on the plane equal to the total amount of its weight4 ,for in this case the force which urges the body1 being that of terrestrial gravity1 its direction is vertical1 and therefore perpendicular to the horizontal plane4 ,but if the body ,p1 fig4 #be41 be placed upon a plane ,a ,b1 oblique to the direction of the force of gravity1 then1 according to what has been proved 7#abi71 the weight of the body will be distributed into two parts1 ,p ,c and ,p ,d2 one1 ,p ,d1 producing a pressure on the plane ,a ,b1 and the other1 ,p ,c1 producing motion down the plane4 ,since the obliquity of the perpendicular direction ,p ,f of the weight to that of the plane ,a ,b must be the same on whatever part of the plane the weight may be placed1 it follows 7#acj71 that the proportion ,p ,c of the weight which urges the body down the plane must be the same throughout its whole descent4 7#acb47 ,hence it may easily be inferred1 that the force down the plane is uniform2 for since the weight of the body ,p is always the same1 and since its proportion to that part which urges it down the plane is the same1 it follows that the quantity of this part cannot vary4 ,the motion of a heavy body down an inclined plane is therefore an uniformly-accelerated motion1 and is characterised by all the properties of uniformly-accelerated motion1 explained in the last chapter4 ,since ,p ,f represents the force of gravity1 that is1 the force with which the body would descend freely in the vertical direction1 and ,p ,c the force with which it moves down the plane1 it follows that a body would fall freely in the vertical direction from ,p to ,f in the same time as on the plane it would move from ,p to ,c4 ,in this manner1 therefore1 when the height through which a body would fall vertically is known1 the space through which it would descend in the same time down any given inclined plane may be immediately determined4 ,for let ,a ,b1 fig4 #be41 be the given inclined plane1 and let ,p ,f be the space through which the body would fall in one second4 ,from ,f draw ,f ,c perpendicular to the plane1 and the space ,p ,c is that through which the body ,p will fall in one second on the plane4 7#acc47 ,as the angle ,b ,a ,h1 which measures the elevation of the plane1 is increased1 the obliquity of the vertical direction ,p ,f with the plane is also increased4 ,consequently1 according to what has been proved 7#acj71 it follows1 that as the elevation of the plane is increased1 the force which urges the body down the plane is also increased1 and as the elevation is diminished1 the force suffers a corresponding diminution4 ,the two extreme cases are1 #a4 ,when the plane is raised until it becomes perpendicular1 in which case the weight is permitted to fall freely1 without exerting any pressure upon the plane2 and1 #b4 ,when the plane is depressed until it becomes horizontal1 in which case the whole weight is supported1 and there is no motion4 ,from these circumstances it follows1 that by means of an inclined plane we can obtain an uniformly-accelerating force of any magnitude less than that of gravity4 ,we have here omitted1 and shall for the present in every instance omit1 the effects of friction1 by which the motion down the plane is retarded4 ,having first investigated the mechanical properties of bodies supposed to be free from friction1 we shall consider friction separately1 and show how the present results are modified by it4 7#acd47 ,the accelerating forces on different inclined planes may be compared by the principle explained in 7#aca74 ,let figs4 #be4 and #bf4 be two inclined planes1 and take the lines ,p ,f in each figure equal1 both expressing the force of gravity1 then ,p ,c will be the force which in each case urges the body down the plane4 ,as the force down an inclined plane is less than that which urges a body falling freely in the vertical direction1 the space through which the body must fall to attain a certain final velocity must be just so much greater as the accelerating force is less4 ,on this principle we shall be able to determine the final velocity in descending through any space on a plane1 compared with the final velocity attained in falling freely in the vertical direction4 ,suppose the body ,p1 fig4 #bg41 placed at the top of the plane1 and from ,h draw the perpendicular ,h ,c4 ,if ,b ,h represent the force of gravity1 ,b ,c will represent the force down the plane 7#aca74 ,in order that the body moving down the plane shall have a final velocity equal to that of one which has fallen freely from ,b to ,h1 it will be necessary that it should move from ,b down the plane1 through a space which bears the same proportion to ,b ,h as ,b ,h does to ,b ,c4 ,but since the triangle ,a ,b ,h is in all respects similar to ,h ,b ,c1 only made upon a larger scale1 the line ,a ,b bears the same proportion to ,b ,h as ,b ,h bears to ,b ,c4 ,hence1 in falling on the inclined plane from ,b to ,a1 the final velocity is the same as in falling freely from ,b to ,h4 ,it is evident that the same will be true at whatever level an horizontal line be drawn4 ,thus1 if ,i ,k be horizontal1 the final velocity in falling on the plane from ,b to ,i will be the same as the final velocity in falling freely from ,b to ,k4 7#ace47 ,the motion of a heavy body down a curve differs in an important respect from the motion down an inclined plane4 ,every part of the plane being equally inclined to the vertical direction1 the effect of gravity in the direction of the plane is uniform2 and1 consequently1 the phenomena obey all the established laws of uniformly-accelerated motion4 ,if1 however1 we suppose the line ,b ,a1 on which the body ,p descends1 to be curved as in fig4 #bh41 the obliquity of its direction at different parts1 to the direction ,p ,f of gravity1 will evidently vary4 ,in the present instance1 this obliquity is greater towards ,b and less towards ,a1 and hence the part of the force of gravity which gives motion to the body is greater towards ,b than towards ,a 7#acj74 ,the force1 therefore1 which urges the body1 instead of being uniform as in the inclined plane1 is here gradually diminished4 ,the rate of this diminution depends entirely on the nature of the curve1 and can be deduced from the properties of the curve by mathematical reasoning4 ,the details of such an investigation are not1 however1 of a sufficiently elementary character to allow of being introduced with advantage into this treatise4 ,we must therefore limit ourselves to explain such of the results as may be necessary for the development of the other parts of the science4 7#acf47 ,when a heavy body is moved down an inclined plane by the force of gravity1 the plane has been proved to sustain a pressure1 arising from a certain part of the weight ,p ,d1 fig4 #be41 which acts perpendicularly to the plane4 ,this is also the case in moving down a curve such as ,b ,a1 fig4 #bh4 ,in this case1 also1 the whole weight is distributed between that part which is directed down the curve1 and that which1 being perpendicular to the curve1 produces a pressure upon it4 ,there is1 however1 another cause which produces pressure upon the curve1 and which has no operation in the case of the inclined plane4 ,by the property of inertia1 when a body is put in motion in any direction1 it must persevere in that direction1 unless it be deflected from it by an efficient force4 ,in the motion down an inclined plane the direction is never changed1 and therefore by its inertia the falling body retains all the motion impressed upon it continually in the same direction2 but when it descends upon a curve1 its direction is constantly varying1 and the resistance of the curve being the deflecting cause1 the curve must sustain a pressure equal to that force1 which would thus be capable of continually deflecting the body from the rectilinear path in which it would move in virtue of its inertia4 ,this pressure entirely depends on the curvature of the path in which the body is constrained to move1 and on its inertia1 and is therefore altogether independent of the weight1 and would1 in fact1 exist if the weight were without effect4 7#acg47 ,this pressure has been denominated centrifugal force1 because it evinces a tendency of the moving body to fly from the centre of the curve in which it is moved4 ,its quantity depends conjointly on the velocity of the motion and the curvature of the path through which the body is moved4 ,as circles may be described with every degree of curvature1 according to the length of the radius1 or the distance from their circumference to their centre1 it follows that1 whatever be the curve in which the body moves1 a circle can always be assigned which has the same curvature as is found at any proposed point of the given curve4 ,such a circle is called 8the circle of curvature0 at that point of the curve2 and as all curves1 except the circle1 vary their degrees of curvature at different points1 it follows that different parts of the same curve will have different circles of curvature4 ,it is evident that the greater the radius of a circle is1 the less is its curvature3 thus the circle with the radius ,a ,b1 fig4 #bi41 is more curved than that whose radius is ,c ,d1 and that in the exact proportion of the radius ,c ,d to the radius ,a ,b4 ,the radius of the circle of curvature for any part of a curve is called 8the radius of curvature0 of that part4 7#ach47 ,the centrifugal pressure increases as the radius of curvature increases2 but it also has a dependence on the velocity with which the moving body swings round the centre of the circle of curvature4 ,this velocity is estimated either by the actual space through which the body moves1 or by the angular velocity of a line drawn from the centre of the circle to the moving body4 ,that body carries one end of this line with it1 while the other remains fixed at the centre4 ,as this angular swing round the centre increases1 the centrifugal pressure increases4 ,to estimate the rate at which this pressure in general varies1 it is necessary to multiply the square of the number expressing the angular velocity by that which expresses the radius of curvature1 and the force increases in the same proportion as the product thus obtained4 7#aci47 ,we have observed that the same causes which produce pressure on a body restrained1 will produce motion if the body be free4 ,accordingly1 if a body be moved by any efficient cause in a curve1 it will1 by reason of the centrifugal force1 fly off1 and the moving force with which it will thus retreat from the centre round which it is whirled will be a measure of the centrifugal force4 ,upon this principle an apparatus called a whirling table has been constructed1 for the purpose of exhibiting experimental illustrations of the laws of centrifugal force4 ,by this machine we are enabled to place any proposed weights at any given distances from centres round which they are whirled1 either with the same angular velocity1 or with velocities having a certain proportion4 ,threads attached to the whirling weights are carried to the centres round which they respectively revolve1 and there1 passing over pulleys1 are connected with weights which may be varied at pleasure4 ,when the whirling weights fly from their respective centres1 by reason of the centrifugal force1 they draw up the weights attached to the other ends of the threads1 and the amount of the centrifugal force is estimated by the weight which it is capable of raising4 ,with this instrument the following experiments may be exhibited3-- ,exp4 #a4 ,equal weights whirled with the same velocity at equal distances from the centre raise the same weight1 and therefore have the same centrifugal force4 ,exp4 #b4 ,equal weights whirled with the same angular velocity at distances from the centre in the proportion of one to two1 will raise weights in the same proportion4 ,therefore the centrifugal forces are in that proportion4 ,exp4 #c4 ,equal weights whirled at equal distances with angular velocities which are as one to two1 will raise weights as one to four1 that is1 as the squares of the angular velocities4 ,therefore the centrifugal forces are in that proportion4 ,exp4 #d4 ,equal weights whirled at distances which are as two to three1 with angular velocities which are as one to two1 will raise weights which are as two to twelve2 that is1 as the products of the distances two and three1 and the squares one and four1 of the angular velocities4 ,hence1 the centrifugal forces are in this proportion4 ,the centrifugal force must also increase as the mass of the body moved increases2 for1 like attraction1 each particle of the moving body is separately and equally affected by it4 ,hence a double mass1 moving at the same distance1 and with the same velocity1 will have a double force4 ,the following experiment verifies this3-- ,exp4 #e4 ,if weights1 which are as one to two1 be whirled at equal distances with the same velocity1 they will raise weights which are as one to two4 ,the law which governs centrifugal force may then be expressed in general symbols briefly thus3-- ,let c the centrifugal force with which a weight of one lb4 revolving in a circle in one second1 the radius of which is one foot1 would act on a string connecting it with the centre4 ,the force with which it would act on a string1 the length of which is ,r feet1 would be c ,r2 and if instead of revolving in one second it revolved in ,t seconds1 the force would be 7c ,r7_/,t#b2 and if the revolving mass were ,w lbs4 the force would be ,c 7c ,w ,r7_/,t#b4 ,this formula includes the entire theory of centrifugal force4 ,but it can be shown that the number expressed by c is #abbf1 and consequently ,c 7#abbf ,w ,r7_/,t#b4 ,it is often more convenient to use the number of revolutions made in a given time than the time of one revolution4 ,let ,n then express the number of revolutions1 or fraction of a revolution1 made in one second1 and we shall have ,t #a_/,n4 ,therefore ,c #abbf ,w ,r ,n#b4 7#adj47 ,the consideration of centrifugal force proves1 that if a body be observed to move in a curvilinear path1 some efficient cause must exist which prevents it from flying off1 and which compels it to revolve round the centre4 ,if the body be connected with the centre by a thread1 cord1 or rod1 then the effect of the centrifugal force is to give tension to the thread1 cord1 or rod4 ,if an unyielding curved surface be placed on the convex side of the path1 then the force will produce pressure on this surface4 ,but if a body is observed to move in a curve without any visible material connection with its centre1 and without any obstruction on the convex side of its path to resist its retreat1 as is the case with the motions of the planets round the sun1 and the satellites round the planets1 it is usual to assign the cause to the attraction of the body which occupies the centre3 in the present instance the sun is that body1 and it is customary to say that the attraction of the sun1 neutralising the effects of the centrifugal force of the planets1 retains them in their orbits4 ,we have elsewhere animadverted on the inaccurate and unphilosophical style of this phraseology1 in which terms are admitted which intimate not only an unknown cause1 but assign its seat1 and intimate something of its nature4 ,all that we are entitled to declare in this case is1 that a motion is continually impressed upon the planet2 that this motion is directed towards the sun2 that it counteracts the centrifugal force2 but from whence this motion proceeds1 whether it be a virtue resident in the sun1 or a property of the medium or space in which both sun and planets are placed1 or whatever other influence may be its proximate cause1 we are altogether ignorant4 99 99 99 99 99 7#ada47 ,numerous examples of the effects of centrifugal force may be produced4 ,if a stone or other weight be placed in a sling1 which is whirled round by the hand in a direction perpendicular to the ground1 the stone will not fall out of the sling1 even when it is at the top of its circuit1 and1 consequently1 has no support beneath it4 ,the centrifugal force1 in this case1 acting from the hand1 which is the centre of rotation1 is greater than the weight of the body1 and therefore prevents its fall4 ,in like manner1 a glass of water may be whirled so rapidly that even when the mouth of the glass is presented downwards1 the water will still be retained in it by the centrifugal force4 ,if a bucket of water be suspended by a number of threads1 and these threads be twisted by turning round the bucket many times in the same direction1 on allowing the cords to untwist1 the bucket will be whirled rapidly round1 and the water will be observed to rise on its sides and sink at its centre1 owing to the centrifugal force with which it is driven from the centre4 ,this effect might be carried so far1 that all the water would flow over and leave the bucket nearly empty4 7#adb47 ,a carriage1 or horseman1 or pedestrian1 passing a corner moves in a curve1 and suffers a centrifugal force1 which increases with the velocity1 and which impresses on the body a force directed from the corner4 ,an animal causes its weight to resist this force1 by voluntarily inclining its body towards the corner4 ,in this case1 let ,a ,b1 fig4 #cj41 be the body2 ,c ,d is the direction of the weight perpendicular to the ground1 and ,c ,f is the direction of the centrifugal force parallel to the ground and from the corner4 ,the body ,a ,b is inclined to the corner1 so that the diagonal force 7#gd71 which is mechanically equivalent to the weight and centrifugal force1 shall be in the direction ,c ,a1 and shall therefore produce the pressure of the feet upon the ground4 ,as the velocity is increased1 the centrifugal force is also increased1 and therefore a greater inclination of the body is necessary to resist it4 ,we accordingly find that the more rapidly a corner is turned1 the more the animal inclines his body towards it4 ,a carriage1 however1 not having voluntary motion1 cannot make this compensation for the disturbing force which is called into existence by the gradual change of direction of the motion2 consequently it will1 under certain circumstances1 be overturned1 falling of course outwards1 or from the corner4 ,if ,a ,b be the carriage1 and ,c1 fig4 #ca41 the place at which the weight is principally collected1 this point ,c will be under the influence of two forces3 the weight1 which may be represented by the perpendicular ,c ,d1 and the centrifugal force1 which will be represented by a line ,c ,f1 which shall have the same proportion to ,c ,d as the centrifugal force has to the weight4 ,now the combined effect of these two forces will be the same as the effect of a single force1 represented by ,c ,g4 ,thus1 the pressure of the carriage on the road is brought nearer to the outer wheel ,b4 ,if the centrifugal force bear the same proportion to the weight as ,c ,f 7or ,d ,b71 fig4 #cb41 bears to ,c ,d1 the whole pressure is thrown upon the wheel ,b4 ,if the centrifugal force bear to the weight a greater proportion than ,d ,b has to ,c ,d1 then the line ,c ,f1 which represents it1 fig4 #cc41 will be greater than ,d ,b4 ,the diagonal ,c ,g1 which represents the combined effects of the weight and centrifugal force1 will in this case pass outside the wheel ,b1 and therefore this resultant will be unresisted4 ,to perceive how far it will tend to overturn the carriage1 let the force ,c ,g be resolved into two1 one in the direction of ,c ,b1 and the other ,c ,k1 perpendicular to ,c ,b4 ,the former ,c ,b will be resisted by the road1 but the latter ,c ,k will tend to lift the carriage over the external wheel4 ,if the velocity and the curvature of the course be continued for a sufficient time to enable this force ,c ,k to elevate the weight1 so that the line of direction shall fall on ,b1 the carriage will be overthrown4 ,it is evident from what has been now stated1 that the chances of overthrow under these circumstances depend on the proportion of ,b ,d to ,c ,d1 or what is to the same purpose1 of the distance between the wheels to the height of the principal seat of the load4 ,it will be shown in the next chapter1 that there is a certain point1 called the centre of gravity1 at which the entire weight of the vehicle and its load may be conceived to be concentrated4 ,this is the point which in the present investigation we have marked ,c4 ,the security of the carriage1 therefore1 depends on the greatness of the distance between the wheels and the smallness of the elevation of the centre of gravity above the road2 for either or both of these circumstances will increase the proportion of ,b ,d to ,c ,d4 7#adc47 ,in the equestrian feat exhibited in the ring at the amphitheatre1 when the horse moves round with the performer standing on the saddle1 both the horse and rider incline continually towards the centre of the ring1 and the inclination increases with the velocity of the motion3 by this inclination their weights counteract the effect of the centrifugal force1 exactly as in the case already mentioned 7#adb47 ,illustration3 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 7#add47 ,if a body be allowed to fall by its weight down a convex surface1 such as ,a ,b1 fig4 #cd41 it would continue upon the surface until it arrive at ,b but for the effect of the centrifugal force3 this1 giving it a motion from the centre of the curve1 will cause it to quit the curve at a certain point ,c1 which can be easily found by mathematical computation4 7#ade47 ,the most remarkable and important manifestation of centrifugal force is observed in the effects produced by the rotation of the earth upon its axis4 ,let the circle in fig4 #ce4 represent a section of the earth1 ,a ,b being the axis on which it revolves4 ,this rotation causes the matter which composes the mass of the earth to revolve in circles round the different points of the axis as centres at the various distances at which the component parts of this mass are placed4 ,as they all revolve with the same angular velocity1 they will be affected by centrifugal forces1 which will be greater or less in proportion as their distances from the centre are greater or less4 ,consequently the parts of the earth which are situated about the equator1 ,d1 will be more strongly affected by centrifugal force than those about the poles1 ,a ,b4 ,the effect of this difference has been that the component matter about the equator has actually been driven farther from the centre than that about the poles1 so that the figure of the earth has swelled out at the sides1 and appears proportionally depressed at the top and bottom1 resembling the shape of an orange4 ,an exaggerated representation of this figure is given in fig4 #cf42 the real difference between the distances of the poles and equator from the centre being too small to be perceptible in a diagram4 ,the exact proportion of ,c ,a to ,c ,d has never yet been certainly ascertained4 ,some observations make ,c ,d exceed ,c ,a by #a_/bgg1 and others by only #a_/ccc4 ,the latter1 however1 seems the more probable4 ,it may be considered to be included between these limits4 ,the same cause operates more powerfully in other planets which revolve more rapidly on their axes4 ,jupiter and ,saturn have forms which are considerably more elliptical4 7#adf47 ,the centrifugal force of the earth's rotation also affects detached bodies on its surface4 ,if such bodies were not held upon the surface by the earth's attraction1 they would be immediately flung off by the whirling motion in which they participate4 ,the centrifugal force1 however1 really diminishes the effects of the earth's attraction on those bodies1 or1 what is the same1 diminishes their weights4 ,if the earth did not revolve on its axis1 the weight of bodies in all places equally distant from the centre would be the same2 but this is not so when the bodies1 as they do1 move round with the earth4 ,they acquire from the centrifugal force a tendency to fly from the axis1 which increases with their distance from that axis1 and is therefore greater the nearer they are to the equator1 and less as they approach the pole4 ,but there is another reason why the centrifugal force is more efficient1 in the opposition which it gives to gravity near the equator than near the poles4 ,this force does not act from the centre of the earth1 but is directed from the earth's axis4 ,it is1 therefore1 not directly opposed to gravity1 except on the equator itself4 ,on leaving the equator1 and proceeding towards the poles1 it is less and less opposed to gravity1 as will be plain on inspecting fig4 #ce41 where the lines ,p ,c all represent the direction of gravity1 and the lines ,p ,f represent the direction of the centrifugal force4 ,since1 then1 as we proceed from the equator towards the poles1 not only the amount of the centrifugal force is continually diminished1 but also it acts less and less in opposition to gravity1 it follows that the weights of bodies are most diminished by it at the equator1 and less so towards the poles4 ,since bodies are commonly weighed by balancing them against other bodies of known weight1 it may be asked1 how the phenomena we have been just describing can be ascertained as a matter of fact8 for whatever be the body against which it may be balanced1 that body must suffer just as much diminution of weight as every other1 and consequently1 all being diminished in the same proportion1 the balance will be preserved though the weights be changed4 ,to render this effect observable1 it will be necessary to compare the effects of gravity with some phenomenon which is not affected by the centrifugal force of the earth's rotation1 and which will be the same at every part of the earth4 ,the means of accomplishing this will be explained in a subsequent chapter4 ,,chap4 ,,ix4 ,,the ,,centre ,,of ,,gravity4 7#adg47 ,by the earth's attraction1 all the particles which compose the mass of a body are solicited by equal forces in parallel directions downwards4 ,if these component particles were placed in mere juxtaposition1 without any mechanical connection1 the force impressed on any one of them could in nowise affect the others1 and the mass would1 in such a case1 be contemplated as an aggregation of small particles of matter1 each urged by an independent force4 ,but the bodies which are the subjects of investigation in mechanical science are not found in this state4 ,solid bodies are coherent masses1 the particles of which are firmly bound together1 so that any force which affects one1 being modified according to circumstances1 will be transmitted through the whole body4 ,liquids accommodate themselves to the shape of the surfaces on which they rest1 and forces affecting any one part are transmitted to others1 in a manner depending on the peculiar properties of this class of bodies4 ,as all bodies1 which are subjects of mechanical enquiry1 on the surface of the earth1 must be continually influenced by terrestrial gravity1 it is desirable to obtain some easy and summary method of estimating the effect of this force4 ,to consider it1 as is unavoidable in the first instance1 the combined action of an infinite number of equal and parallel forces soliciting the elementary molecules downwards1 would be attended with manifest inconvenience4 ,an infinite number of forces1 and an infinite subdivision of the mass1 would form parts of every mechanical problem4 ,to overcome this difficulty1 and to obtain all the ease and simplicity which can be desired in elementary investigations1 it is only necessary to determine some force1 whose single effect shall be equivalent to the combined effects of the gravitation of all the molecules of the body4 ,if this can be accomplished1 that single force might be introduced into all problems to represent the whole effect of the earth's attraction1 and no regard need be had to any particles of the body1 except that on which this force acts4 7#adh47 ,to discover such a force1 if it exist1 we shall first enquire what properties must necessarily characterise it4 ,let ,a ,b1 fig4 #cg41 be a solid body placed near the surface of the earth4 ,its particles are all solicited downwards1 in the directions represented by the arrows4 ,now1 if there be any single force equivalent to these combined effects1 two properties may be at once assigned to it3 #a4 ,it must be presented downwards1 in the common direction of those forces to which it is mechanically equivalent2 and1 #b4 it must be equal in intensity to their sum1 or1 what is the same1 to the force with which the whole mass would descend4 ,we shall then suppose it to have this intensity1 and to have the direction of the arrow ,d ,e4 ,now1 if the single force1 in the direction ,d ,e1 be equivalent to all the separate attractions which affect the particles1 we may suppose all these attractions removed1 and the body ,a ,b influenced only by a single attraction1 acting in the direction ,d ,e4 ,this being admitted1 it follows that if the body be placed upon a prop1 immediately under the direction of the line ,d ,e1 or be suspended from a fixed point immediately above its direction1 it will remain motionless4 ,for the whole attracting force in the direction ,d ,e will1 in the one case1 press the body on the prop1 and1 in the other case1 will give tension to the cord1 rod1 or whatever other means of suspension be used4 7#adi47 ,but suppose the body were suspended from some point ,p1 not in the direction of the line ,d ,e4 ,let ,p ,c be the direction of the thread by which the body is suspended4 ,its whole weight1 according to the supposition which we have adopted1 must then act in the direction ,c ,e4 ,taking ,c ,f to represent the weight2 it may be considered as mechanically equivalent to two forces 7#gd71 ,c ,i and ,c ,h4 ,of these ,c ,h1 acting directly from the point ,p1 merely produces pressure upon it1 and gives tension to the cord ,p ,c2 but ,c ,i1 acting at right angles to ,c ,p1 produces motion round ,p as a centre1 and in the direction ,c ,i1 towards a vertical line ,p ,g1 drawn through the point ,p4 ,if the body ,a ,b had been on the other side of the line ,p ,g1 it would have moved in like manner towards it1 and therefore in the direction contrary to its present motion4 ,hence we must infer1 that when the body is suspended from a fixed point1 it cannot remain at rest1 if that fixed point be not placed in the direction of the line ,d ,e2 and1 on the other hand1 that if the fixed point be in the direction of that line1 it cannot move4 ,a practical test is thus suggested1 by which the line ,d ,e may be at once discovered4 ,let a thread be attached to any point of the body1 and let it be suspended by this thread from a hook or other fixed point4 ,the direction of the thread1 when the body becomes quiescent1 will be that of a single force equivalent to the gravitation of all the component parts of the mass4 7#aej47 ,an enquiry is here suggested3 does the direction of the equivalent force thus determined depend on the position of the body with respect to the surface of the earth1 and how is the direction of the equivalent force affected by a change in that position8 ,this question may be at once solved if the body be suspended by different points1 and the directions which the suspending thread takes in each case relatively to the figure and dimensions of the body examined4 ,the body being suspended in this manner from any point1 let a small hole be bored through it1 in the exact direction of the thread1 so that if the thread were continued below the point where it is attached to the body1 it would pass through this hole4 ,the body being successively suspended by several different points on its surface1 let as many small holes be bored through it in the same manner4 ,if the body be then cut through1 so as to discover the directions which the several holes have taken1 they will be all found to cross each other at one point within the body2 or the same fact may be discovered thus3 a thin wire1 which nearly fills the holes being passed through any one of them1 it will be found to intercept the passage of a similar wire through any other4 ,this singular fact teaches us1 what indeed can be proved by mathematical reasoning without experiment1 that there is one point in every body through which the single force1 which is equivalent to the gravitation of all its particles1 must pass1 in whatever position the body be placed4 ,this point is called the centre of gravity4 7#aea47 ,in whatever situation a body may be placed1 the centre of gravity will have a tendency to descend in the direction of a line perpendicular to the horizon1 and which is called the line of direction of the weight4 ,if the body be altogether free and unrestricted by any resistance or impediment1 the centre of gravity will actually descend in this direction1 and all the other points of the body will move with the same velocity in parallel directions1 so that during its fall the position of the parts of the body1 with respect to the ground1 will be unaltered4 ,but if the body1 as is most usual1 be subject to some resistance or restraint1 it will either remain unmoved1 its weight being expended in exciting pressure on the restraining points or surfaces1 or it will move in a direction and with a velocity depending on the circumstances which restrain it4 ,in order to determine these effects1 to predict the pressure produced by the weight if the body be quiescent1 or the mixed effects of motion and pressure1 if it be not so1 it is necessary in all cases to be able to assign the place of the centre of gravity4 ,when the magnitude and figure of the body1 and the density of the matter which occupies its dimensions1 are known1 the place of the centre of gravity can be determined with the greatest precision by mathematical calculation4 ,the process by which this is accomplished1 however1 is not of a sufficiently elementary nature to be properly introduced into this treatise4 ,to render it intelligible would require the aid of some of the most advanced analytical principles2 and even to express the position of the point in question1 except in very particular instances1 would be impossible1 without the aid of peculiar symbols4 7#aeb47 ,there are certain particular forms of body in which1 when they are uniformly dense1 the place of the centre of gravity can be easily assigned1 and proved by reasoning1 which is generally intelligible2 but in all cases whatever1 this point may be easily determined by experiment4 7#aec47 ,if a body uniformly dense have such a shape that a point may be found on either side of which in all directions around it the materials of the body are similarly distributed1 that point will obviously be the centre of gravity4 ,for if it be supported1 the gravitation of the particles on one side drawing them downwards1 is resisted by an effect of exactly the same kind and of equal amount on the opposite side1 and so the body remains balanced on the point4 ,the most remarkable body of this kind is a globe1 the centre of which is evidently its centre of gravity4 ,a figure1 such as fig4 #ch41 called an oblate spheroid1 has its centre of gravity at its centre1 ,c4 ,such is the figure of the earth4 ,the same may be observed of the elliptical solid1 fig4 #ci41 which is called a prolate spheroid4 ,a cube1 and some other regular solids1 bounded by plane surfaces1 have a point within them1 such as above described1 and which is1 therefore1 their centre of gravity4 ,such are fig4 #dj4 ,a straight wand of uniform thickness has its centre of gravity at the centre of its length2 and a cylindrical body has its centre of gravity in its centre1 at the middle of its length or axis4 ,such is the point ,c1 fig4 #da4 ,a flat plate of any uniform substance1 and which has in every part an equal thickness1 has its centre of gravity at the middle of its thickness1 and under a point of its surface1 which is to be determined by its shape4 ,if it be circular or elliptical1 this point is its centre4 ,if it have any regular form1 bounded by straight edges1 it is that point which is equally distant from its several angles1 as ,c in fig4 #db4 7#aed47 ,there are some cases in which1 although the place of the centre of gravity is not so obvious as in the examples just given1 still it may be discovered without any mathematical process1 which is not easily understood4 ,suppose ,a ,b ,c1 fig4 #dc41 to be a flat triangular plate of uniform thickness and density4 ,let it be imagined to be divided into narrow bars1 by lines parallel to the side ,a ,c1 as represented in the figure4 ,draw ,b ,d from the angle ,b to the middle point ,d of the side ,a ,c4 ,it is not difficult to perceive1 that ,b ,d will divide equally all the bars into which the triangle is conceived to be divided4 ,now if the flat triangular plate ,a ,b ,c be placed in a horizontal position on a straight edge coinciding with the line ,b ,d1 it will be balanced3 for the bars parallel to ,a ,c will be severally balanced by the edge immediately under their middle point2 since that middle point is the centre of gravity of each bar4 ,since1 then1 the triangle is balanced on the edge1 the centre of gravity must be somewhere immediately over it1 and must1 therefore1 be within the plate at some point under the line ,b ,d4 ,the same reasoning will prove that the centre of gravity of the plate is under the line ,a ,e1 drawn from the angle ,a to the middle point ,e of the side ,b ,c4 ,to perceive this1 it is only necessary to consider the triangle divided into bars parallel to ,b ,c1 and thence to show that it will be balanced on an edge placed under ,a ,e4 ,since then the centre of gravity of the plate is under the line ,b ,d1 and also under ,a ,e1 it must be under the point ,g1 at which these lines cross each other2 and it is accordingly at a depth beneath ,g1 equal to half the thickness of the plate4 ,this may be experimentally verified by taking a piece of tin or card1 and cutting it into a triangular form4 ,the point ,g being found by drawing ,b ,d and ,a ,e1 which divide two sides equally1 it will be balanced if placed upon the point of a pin at ,g4 ,the centre of gravity of a triangle being thus determined1 we shall be able to find the position of the centre of gravity of any plate of uniform thickness and density which is bounded by straight edges1 as will be shown hereafter4 7#agc47 7#aee47 ,the centre of gravity is not always included within the volume of the body1 that is1 it is not enclosed by its surfaces4 ,numerous examples of this can be produced4 ,if a piece of wire be bent into any form1 the centre of gravity will rarely be in the wire4 ,suppose it be brought to the form of a ring4 ,in that case1 the centre of gravity of the wire will be the centre of the circle1 a point not forming any part of the wire itself3 nevertheless this point may be proved to have the characteristic property of the centre of gravity2 for if the ring be suspended by any point1 the centre of the ring must always settle itself under the point of suspension4 ,if this centre could be supposed to be connected with the ring by very fine threads1 whose weight would be insignificant1 and which might be united by a knot or otherwise at the centre1 the ring would be balanced upon a point placed under the knot4 ,in like manner1 if the wire be formed into an ellipse1 or any other curve similarly arranged round a centre point1 that point will be its centre of gravity4 7#aef47 ,to find the centre of gravity experimentally1 the method described in 7#adi1 #aej7 may be used4 ,in this case two points of suspension will be sufficient to determine it2 for the directions of the suspending cord being continued through the body1 will cross each other at the centre of gravity4 ,these directions may also be found by placing the body on a sharp point1 and adjusting it so as to be balanced upon it4 ,in this case a line drawn through the body directly upwards from the point will pass through the centre of gravity1 and therefore two such lines must cross at that point4 7#aeg47 ,if the body have two flat parallel surfaces like sheet metal1 stiff paper1 card1 board1 c41 the centre of gravity may be found by balancing the body in two positions on an horizontal straight edge4 ,the point where the lines marked by the edge cross each other will be immediately under the centre of gravity4 ,this may be verified by showing that the body will be balanced on a point thus placed1 or that if it be suspended1 the point thus determined will always come under the point of suspension4 ,the position of the centre of gravity of such bodies may also be found by placing the body on an horizontal table having a straight edge4 ,the body being moved beyond the edge until it is in that position in which the slightest disturbance will cause it to fall1 the centre of gravity will then be immediately over the edge4 ,this being done in two positions1 the centre of gravity will be determined as before4 7#aeh47 ,it has been already stated1 that when the body is perfectly free1 the centre of gravity must necessarily move downwards1 in a direction perpendicular to an horizontal plane4 ,when the body is not free1 the circumstances which restrain it generally permit the centre of gravity to move in certain directions1 but obstruct its motion in others4 ,thus if a body be suspended from a fixed point by a flexible cord1 the centre of gravity is free to move in every direction except those which would carry it farther from the point of suspension than the length of the cord4 ,hence if we conceive a globe or sphere to surround the point of suspension on every side to a distance equal to that of the centre of gravity from the point of suspension1 when the cord is fully stretched1 the centre of gravity will be at liberty to move in every direction within this sphere4 ,there are an infinite variety of circumstances under which the motion of a body may be restrained1 and in which a most important and useful class of mechanical problems originate4 ,before we notice others1 we shall1 however1 examine that which has just been described more particularly4 ,let ,p1 fig4 #dd41 be the point of suspension1 and ,c the centre of gravity1 and suppose the body so placed that ,c shall be within the sphere already described4 ,the cord will therefore be slackened1 and in this state the body will be free4 ,the centre of gravity will therefore descend in the perpendicular direction until the cord becomes fully extended2 the tension will then prevent its further motion in the perpendicular direction4 ,the downward force must now be considered as the diagonal of a parallelogram1 and equivalent to two forces ,c ,d and ,c ,e1 in the directions of the sides1 as already explained in 7#adi74 ,the force ,c ,d will bring the centre of gravity into the direction ,p ,f1 perpendicularly under the point of suspension4 ,since the force of gravity acts continually on ,c in its approach to ,p ,f1 it will move towards that line with accelerated speed1 and when it has arrived there it will have acquired a force to which no obstruction is immediately opposed1 and consequently by its inertia it retains this force1 and moves beyond ,p ,f on the other side4 ,but when the point ,c gets into the line ,p ,f1 it is in the lowest possible position2 for it is at the lowest point of the sphere which limits its motion4 ,when it passes to the other side of ,p ,f1 it must therefore begin to ascend1 and the force of gravity1 which1 in the former case1 accelerated its descent1 will now for the same reason1 and with equal energy1 oppose its ascent4 ,this will be easily understood4 ,let ,c be any point which it may have attained in ascending2 ,c ,g1 the force of gravity1 is now equivalent to ,c ,d and ,c ,e4 ,the latter as before produces tension2 but the former ,c ,d is in a direction immediately opposed to the motion1 and therefore retards it4 ,this retardation will continue until all the motion acquired by the body in its descent from the first position has been destroyed1 and then it will begin to return to ,p ,f1 and so it will continue to vibrate from the one side to the other until the friction on the point ,p1 and the resistance of the air1 gradually deprive it of its motion1 and bring it to a state of rest in the direction ,p ,f4 ,but for the effects of friction and atmospheric resistance1 the body would continue for ever to oscillate equally from side to side of the line ,p ,f4 7#aei47 ,the phenomenon just developed is only an example of an extensive class4 ,whenever the circumstances which restrain the body are of such a nature that the centre of gravity is prevented from descending below a certain level1 but not1 on the other hand1 restrained from rising above it1 the body will remain at rest if the centre of gravity be placed at the lowest limit of its level2 any disturbance will cause it to oscillate around this state1 and it cannot return to a state of rest until friction or some other cause have deprived it of the motion communicated by the disturbing force4 7#afj47 ,under the circumstances which we have just described1 the body could not maintain itself in a state of rest in any position except that in which the centre of gravity is1 at the lowest point of the space in which it is free to move4 ,this1 however1 is not always the case4 ,suppose it were suspended by an inflexible rod instead of a flexible string2 the centre of gravity would then not only be prevented from receding from the point of suspension1 but also from approaching it2 in fact1 it would be always kept at the same distance from it4 ,thus1 instead of being capable of moving anywhere within the sphere1 it is now capable of moving on its surface only4 ,the reasoning used in the last case may also be applied here1 to prove that when the centre of gravity is on either side of the perpendicular ,p ,f1 it will fall towards ,p ,f and oscillate1 and that if it be placed in the line ,p ,f1 it will remain in equilibrium4 ,but in this case there is another position1 in which the centre of gravity may be placed so as to produce equilibrium4 ,if it be placed at the highest point of the sphere in which it moves1 the whole force acting on it will then be directed on the point of suspension1 perpendicularly downwards1 and will be entirely expended in producing pressure on that point2 consequently1 the body will in this case be in equilibrium4 ,but this state of equilibrium is of a character very different from that in which the centre of gravity was at the lowest part of the sphere4 ,in the present case any displacement1 however slight1 of the centre of gravity1 will carry it to a lower level1 and the force of gravity will then prevent its return to its former state1 and will impel it downwards until it attain the lowest point of the sphere1 and round that point it will oscillate4 7#afa47 ,the two states of equilibrium which have been just noticed1 are called stable and instable equilibrium4 ,the character of the former is1 that any disturbance of the state produces oscillation about it2 but any disturbance of the latter state produces a total overthrow1 and finally causes oscillation around the state of stable equilibrium4 ,let ,a ,b1 fig4 #de41 be an elliptical board resting on its edge on an horizontal plane4 ,in the position here represented1 the extremity ,p of the lesser axis being the point of support1 the board is in stable equilibrium2 for any motion on either side must cause the centre of gravity ,c to ascend in the directions ,c ,o1 and oscillation will ensue4 ,if1 however1 it rest upon the smaller end1 as in fig4 #df41 the position would still be a state of equilibrium1 because the centre of gravity is directly above the point of support2 but it would be instable equilibrium1 because the slightest displacement of the centre of gravity would cause it to descend4 ,thus an egg or a lemon may be balanced on the end1 but the least disturbance will overthrow it4 ,on the contrary1 it will easily rest on the side1 and any disturbance will produce oscillation4 7#afb47 ,when the circumstances under which the body is placed allow the centre of gravity to move only in an horizontal line1 the body is in a state which may be called neutral equilibrium4 ,the slightest force will move the centre of gravity1 but will neither produce oscillation nor overthrow the body1 as in the last two cases4 ,an example of this state is furnished by a cylinder placed upon an horizontal plane4 ,as the cylinder is rolled upon the plane1 the centre of gravity ,c1 fig4 #dg41 moves in a line parallel to the plane ,a ,b1 and distant from it by the radius of the cylinder4 ,the body will thus rest indifferently in any position1 because the line of direction always falls upon a point ,p at which the body rests upon the plane4 ,if the plane were inclined1 as in fig4 #dh41 a body might be so shaped1 that while it would roll the centre of gravity would move horizontally4 ,in this case the body would rest indifferently on any part of the plane1 as if it were horizontal1 provided the friction be sufficient to prevent the body from sliding down the plane4 ,if the centre of gravity of a cylinder happen not to coincide with its centre by reason of the want of uniformity in the materials of which it is composed1 it will not be in a state of neutral equilibrium on an horizontal plane1 as in fig4 #dg4 ,in this case let ,g1 fig4 #di41 be the centre of gravity4 ,in the position here represented1 where the centre of gravity is immediately below the centre ,c1 the state will be stable equilibrium1 because a motion on either side would cause the centre of gravity to ascend2 but in fig4 #ej41 where ,g is immediately above ,c1 the state is instable equilibrium1 because a motion on either side would cause ,g to descend1 and the body would turn into the position fig4 #di4 7#afc47 ,a cylinder of this kind will1 under certain circumstances1 roll up an inclined plane4 ,let ,a ,b1 fig4 #ea41 be the inclined plane1 and let the cylinder be so placed that the line of direction from ,g shall be above the point ,p at which the cylinder rests upon the plane4 ,the whole weight of the body acting in the direction ,g ,d will obviously cause the cylinder to roll towards ,a1 provided the friction be sufficient to prevent sliding2 but although the cylinder in this case ascends1 the centre of gravity ,g really descends4 ,when ,g is so placed that the line of direction ,g ,d shall fall on the point ,p1 the cylinder will be in equilibrium1 because its weight acts upon the point on which it rests4 ,there are two cases represented in fig4 #eb4 and fig4 #ec41 in which ,g takes this position4 ,fig4 #eb4 represents the state of stable1 and fig4 #ec4 of instable equilibrium4 7#afd47 ,when a body is placed upon a base1 its stability depends upon the position of the line of direction and the height of the centre of gravity above the base4 ,if the line of direction fall within the base1 the body will stand firm2 if it fall on the edge of the base1 it will be in a state in which the slightest force will overthrow it on that side at which the line of direction falls2 and if the line of direction fall without the base1 the body must turn over that edge which is nearest to the line of direction4 ,in fig4 #ed4 and fig4 #ee41 the line of direction ,g ,p falls within the base1 and it is obvious that the body will stand firm2 for any attempt to turn it over either edge would cause the centre of gravity to ascend4 ,but in fig4 #ef4 the line of direction falls upon the edge1 and if the body be turned over1 the centre of gravity immediately commences to descend4 ,until it be turned over1 however1 the centre of gravity is supported by the edge4 ,in fig4 #eg4 the line of direction falls outside the base1 the centre of gravity has a tendency to descend from ,g towards ,a1 and the body will accordingly fall in that direction4 7#afe47 ,when the line of direction falls within the base1 bodies will always stand firm1 but not with the same degree of stability4 ,in general1 the stability depends on the height through which the centre of gravity must be elevated before the body can be overthrown4 ,the greater this height is1 the greater in the same proportion will be the stability4 ,let ,b ,a ,c1 fig4 #eh41 be a pyramid1 the centre of gravity being at ,g4 ,to turn this over the edge ,b1 the centre of gravity2 must be carried over the arch ,g ,e1 and must therefore be raised through the height ,h ,e4 ,if1 however1 the pyramid were taller relatively to its base1 as in fig4 #ei41 the height ,h ,e would be proportionally less2 and if the base were very small in reference to the height1 as in fig4 #fj41 the height ,h ,e would be very small1 and a slight force would throw it over the edge ,b4 ,it is obvious that the same observations may be applied to all figures whatever1 the conclusions just deduced depending only on the distance of the line of direction from the edge of the base1 and the height of the centre of gravity above it4 7#aff47 ,hence we may perceive the principle on which the stability of loaded carriages depends4 ,when the load is placed at a considerable elevation above the wheels1 the centre of gravity is elevated1 and the carriage becomes proportionally insecure4 ,in coaches for the conveyance of passengers1 the luggage is therefore sometimes placed below the body of the coach2 light parcels of large bulk may be placed on the top with impunity4 ,when the centre of gravity of a carriage is much elevated1 there is considerable danger of overthrow1 if a corner be turned sharply and with a rapid pace2 for the centrifugal force then acting on the centre of gravity will easily raise it through the small height which is necessary to turn the carriage over the external wheels 7#adb74 7#afg47 ,the same waggon will have greater stability when loaded with a heavy substance which occupies a small space1 such as metal1 than when it carries the same weight of a lighter substance1 such as hay2 because the centre of gravity in the latter case will be much more elevated4 ,illustration3 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,if a large table be placed upon a single leg in its centre1 it will be impracticable to make it stand firm2 but if the pillar on which it rests terminate in a tripod1 it will have the same stability as if it had three legs attached to the points directly over the places where the feet of the tripod rest4 7#afh47 ,when a solid body is supported by more points than one1 it is not necessary for its stability that the line of direction should fall on one of those points4 ,if there be only two points of support1 the line of direction must fall between them4 ,the body is in this case supported as effectually as if it rested on an edge coinciding with a straight line drawn from one point of support to the other4 ,if there be three points of support1 which are not ranged in the same straight line1 the body will be supported in the same manner as it would be by a base coinciding with the triangle formed by straight lines joining the three points of support4 ,in the same manner1 whatever be the number of points on which the body may rest1 its virtual base will be found by supposing straight lines drawn1 joining the several points successively4 ,when the line of direction falls within this base1 the body will always stand firm1 and otherwise not4 ,the degree of stability is determined in the same manner as if the base were a continued surface4 7#afi47 ,necessity and experience teach an animal to adapt its postures and motions to the position of the centre of gravity of his body4 ,when a man stands1 the line of direction of his weight must fall within the base formed by his feet4 ,if ,a ,b1 ,c ,d1 fig4 #fa41 be the feet1 this base is the space ,a ,b ,d ,c4 ,it is evident1 that the more his toes are turned outwards1 the more contracted the base will be in the direction ,e ,f1 and the more liable he will be to fall backwards or forwards4 ,also1 the closer his feet are together1 the more contracted the base will be in the direction ,g ,h1 and the more liable he will be to fall towards either side4 ,when a man walks1 the legs are alternately lifted from the ground1 and the centre of gravity is either unsupported or thrown from the one side to the other4 ,the body is also thrown a little forward1 in order that the tendency of the centre of gravity to fall in the direction of the toes may assist the muscular action in propelling the body4 ,this forward inclination of the body increases with the speed of the motion4 ,but for the flexibility of the knee-joint the labour of walking would be much greater than it is2 for the centre of gravity would be more elevated by each step4 ,the line of motion of the centre of gravity in walking is represented by fig4 #fb41 and deviates but little from a regular horizontal line1 so that the elevation of the centre of gravity is subject to very slight variation4 ,but if there were no knee-joint1 as when a man has wooden legs1 the centre of gravity would move as in fig4 #fc41 so that at each step the weight of the body would be lifted through a considerable height1 and therefore the labour of walking would be much increased4 ,if a man stand on one leg1 the line of direction of his weight must fall within the space on which his foot treads4 ,the smallness of this space1 compared with the height of the centre of gravity1 accounts for the difficulty of this feat4 ,the position of the centre of gravity of the body changes with the posture and position of the limbs4 ,if the arm be extended from one side1 the centre of gravity is brought nearer to that side than it was when the arm hung perpendicularly4 ,when dancers1 standing on one leg1 extend the other at right angles to it1 they must incline the body in the direction opposite to that in which the leg is extended1 in order to bring the centre of gravity over the foot which supports them4 ,when a porter carries a load1 his position must be regulated by the centre of gravity of his body and the load taken together4 ,if he bore the load on his back1 the line of direction would pass beyond his heels1 and he would fall backwards4 ,to bring the centre of gravity over his feet he accordingly leans forward1 fig4 #fd4 ,if a nurse carry a child in her arms1 she leans back for a like reason4 ,when a load is carried on the head1 the bearer stands upright1 that the centre of gravity may be over his feet4 ,in ascending a hill1 we appear to incline forward2 and in descending1 to lean backward1 but in truth1 we are standing upright with respect to a level plane4 ,this is necessary to keep the line of direction between the feet1 as is evident from fig4 #fe4 ,a person sitting on a chair which has no back cannot rise from it without either stooping forward to bring the centre of gravity over the feet1 or drawing back the feet to bring them under the centre of gravity4 ,a quadruped never raises both feet on the same side simultaneously1 for the centre of gravity would then be unsupported4 ,let ,a ,b ,c ,d1 fig4 #ff41 be the feet4 ,the base on which it stands is ,a ,b ,c ,d1 and the centre of gravity is nearly over the point ,o1 where the diagonals cross each other4 ,the legs ,a and ,c being raised together1 the centre of gravity is supported by the legs ,b and ,d1 since it falls between them2 and when ,b and ,d are raised it is1 in like manner1 supported by the feet ,a and ,c4 ,the centre of gravity1 however1 is often unsupported for a moment2 for the leg ,b is raised from the ground before ,a comes to it1 as is plain from observing the track of a horse's feet1 the mark of ,a being upon or before that of ,b4 ,in the more rapid paces of all animals the centre of gravity is at intervals unsupported4 ,the feats of rope-dancers are experiments on the management of the centre of gravity4 ,the evolutions of the performer are found to be facilitated by holding in his hand a heavy pole4 ,his security in this case depends1 not on the centre of gravity of his body1 but on that of his body and the pole taken together4 ,this point is near the centre of the pole1 so that1 in fact1 he may be said to hold in his hands the point on the position of which the facility of his feats depends4 ,without the aid of the pole the centre of gravity would be within the trunk of the body1 and its position could not be adapted to circumstances with the same ease and rapidity4 7#agj47 ,the centre of gravity of a mass of fluid is that point which would have the properties which have been proved to belong to the centre of gravity of a solid1 if the fluid were solidified without changing in any respect the quantity or arrangement of its parts4 ,this point also possesses other properties1 in reference to fluids1 which will be investigated in ,,hydrostatics and ,,pneumatics4 7#aga47 ,the centre of gravity of two bodies separated from one another1 is that point which would possess the properties ascribed to the centre of gravity1 if the two bodies were united by an inflexible line1 the weight of which might be neglected4 ,to find this point mathematically is a very simple problem4 ,let ,a and ,b1 fig4 #fg41 be the two bodies1 and a and b their centres of gravity4 ,draw the right line a b1 and divide it at ,c1 in such a manner that a ,c shall have the same proportion to b ,c as the mass of the body ,b has to the mass of the body ,a4 ,this may easily be verified experimentally4 ,let ,a and ,b be two bodies1 whose weight is considerable1 in comparison with that of the rod a b1 which joins them4 ,let a fine silken string1 with its ends attached to them1 be hung upon a pin2 and on the same pin let a plumb-line be suspended4 ,in whatever position the bodies may be hung1 it will be observed that the plumb-line will cross the rod a b at the same point1 and that point will divide the line a b into parts a ,c and b ,c1 which are in the proportion of the mass of ,b to the mass of ,a4 7#agb47 ,the centre of gravity of three separate bodies is defined in the same manner as that of two1 and may be found by first determining the centre of gravity of two2 and then supposing their masses concentrated at that point1 so as to form one body1 and finding the centre of gravity of that and the third4 ,in the same manner the centre of gravity of any number of bodies may be determined4 7#agc47 ,if a plate of uniform thickness be bounded by straight edges1 its centre of gravity may be found by dividing it into triangles by diagonal lines1 as in fig4 #fh41 and having determined by 7#aed7 the centres of gravity of the several triangles1 the centre of gravity of the whole plate will be their common centre of gravity1 found as above4 7#agd47 ,although the centre of gravity takes its name from the familiar properties which it has in reference to detached bodies of inconsiderable magnitude1 placed on or near the surface of the earth1 yet it possesses properties of a much more general and not less important nature4 ,one of the most remarkable of these is1 that the centre of gravity of any number of separate bodies is never affected by the mutual attraction1 impact1 or other influence which the bodies may transmit from one to another4 ,this is a necessary consequence of the equality of action and reaction explained in ,chapter ,,iv4 ,for if ,a and ,b1 fig4 #fg41 attract each other1 and change their places to ,a and ,b1 the space a a will have to b b the same proportion as ,b has to ,a1 and therefore by what has just been proved 7#aga7 the same proportion as a ,c has to b ,c4 ,it follows1 that the remainders a ,c and b ,c will be in the proportion of ,b to ,a1 and that ,c will continue to be the centre of gravity of the bodies after they have approached by their mutual attraction4 ,suppose1 for example1 that ,a and ,b were #ablbs4 and #hlbs4 respectively1 and that a b were #dj feet4 ,the point ,c must 7#aga7 divide a b into two parts1 in the proportion of #h to #ab1 or of #b to #c4 ,hence it is obvious that a ,c will be #af feet1 and b ,c #bd feet4 ,now1 suppose that ,a and ,b attract each other1 and that ,a approaches ,b through two feet4 ,then ,b must approach ,a through three feet4 ,their distances from ,c will now be #ad feet and #ba feet1 which1 being in the proportion of ,b to ,a1 the point ,c will still be their centre of gravity4 ,hence it follows1 that if a system of bodies1 placed at rest1 be permitted to obey their mutual attractions1 although the bodies will thereby be severally moved1 yet their common centre of gravity must remain quiescent4 7#age47 ,when one of two bodies is moving in a straight line1 the other being at rest1 their common centre of gravity must move in a parallel straight line4 ,let ,a and ,b1 fig4 #fi41 be the centres of gravity of the bodies1 and let ,a move from ,a to a1 ,b remaining at rest4 ,draw the lines ,a ,b and a ,b4 ,in every position which the body ,b assumes during its motion1 the centre of gravity ,c divides the line joining them into parts ,a ,c1 ,b ,c1 which are in the proportion of the mass ,b to the mass ,a4 ,now1 suppose any number of lines drawn from ,b to the line ,a a2 a parallel ,c c to ,a a through ,c divides all these lines in the same proportion2 and therefore1 while the body ,a moves from ,a to a1 the common centre of gravity moves from ,c to c4 ,if both the bodies ,a and ,b moved uniformly in straight lines1 the centre of gravity would have a motion compounded 7#gd7 of the two motions with which it would be affected1 if each moved while the other remained at rest4 ,in the same manner1 if there were three bodies1 each moving uniformly in a straight line1 their common centre of gravity would have a motion compounded of that motion which it would have if one remained at rest while the other two moved1 and that which the motion of the first would give it if the last two remained at rest2 and in the same manner it may be proved1 that when any number of bodies move each in a straight line1 their common centre of gravity will have a motion compounded of the motions which it receives from the bodies severally4 ,it may happen that the several motions which the centre of gravity receives from the bodies of the system will neutralise each other2 and this does1 in fact1 take place for such motions as are the consequences of the mutual action of the bodies upon one another4 7#agf47 ,if a system of bodies be not under the immediate influence of any forces1 and their mutual attraction be conceived to be suspended1 they must severally be either at rest or in uniform rectilinear motion in virtue of their inertia4 ,hence1 their common centre of gravity must also be either at rest or in uniform rectilinear motion4 ,now1 if we suppose their mutual attractions to take effect1 the state of their common centre of gravity will not be changed1 but the bodies will severally receive motions compounded of their previous uniform rectilinear motions and those which result from their mutual attractions4 ,the combined effects will cause each body to revolve in an orbit round the common centre of gravity1 or will precipitate it towards that point4 ,but still that point will maintain its former state undisturbed4 ,this constitutes one of the general laws of mechanical science1 and is of great importance in physical astronomy4 ,it is known by the title 8the conservation of the motion of the centre of gravity40 7#agg47 ,the solar system is an instance of the class of phenomena to which we have just referred4 ,all the motions of the bodies which compose it can be traced to certain uniform rectilinear motions1 received from some former impulse1 or from a force whose action has been suspended1 and those motions which necessarily result from the principle of gravitation4 ,but we shall not here insist further on this subject1 which more properly belongs to another department of the science4 7#agh47 ,if a solid body suffer an impact in the direction of a line passing through its centre of gravity1 all the particles of the body will be driven forward with the same velocity in lines parallel to the direction of the impact1 and the whole force of the motion will be equal to that of the impact4 ,the common velocity of the parts of the body will in this case be determined by the principles explained in ,chapter ,,iv4 ,the impelling force being equally distributed among all the parts1 the velocity will be found by dividing the numerical value of that force by the number expressing the mass4 ,if any number of impacts be given simultaneously to different points of a body1 a certain complex motion will generally ensue4 ,the mass will have a relative motion round the centre of gravity as if it were fixed1 while that point will move forward uniformly in a straight line1 carrying the body with it4 ,the relative motion of the mass round the centre of gravity may be found by considering the centre of gravity as a fixed point1 round which the mass is free to move1 and then determining the motion which the applied forces would produce4 ,this motion being supposed to continue uninterrupted1 let all the forces be imagined to be applied in their proper directions and quantities to the centre of gravity4 ,by the principles for the composition of force they will be mechanically equivalent to a single force through that point4 ,in the direction of this single force the centre of gravity will move and have the same velocity as if the whole mass were there concentrated and received the impelling forces4 7#agi47 ,these general properties1 which are entirely independent of gravity1 render the 8centre of gravity0 an inadequate title for this important point4 ,some physical writers have1 consequently1 called it the 8centre of inertia40 ,the 8centre of gravity10 however1 is the name by which it is still generally designated4 ,,chap4 ,x4 ,,the ,,mechanical ,,properties ,,of ,,an ,,axis4 7#ahj47 ,when a body has a motion of rotation1 the line round which it revolves is called an axis4 ,every point of the body must in this case move in a circle1 whose centre lies in the axis1 and whose radius is the distance of the point from the axis4 ,sometimes while the body revolves1 the axis itself is moveable1 and not unfrequently in a state of actual motion4 ,the motions of the earth and planets1 or that of a common spinning-top1 are examples of this4 ,the cases1 however1 which will be considered in the present chapter1 are chiefly those in which the axis is immovable1 or at least where its motion has no relation to the phenomena under investigation4 ,instances of this are so frequent and obvious1 that it seems scarcely necessary to particularise them4 ,wheel-work of every description1 the moving parts of watches and clocks1 turning lathes1 mill-work1 doors and lids on hinges1 are all obvious examples4 ,in tools or other instruments which work on joints or pivots1 such as scissors1 shears1 pincers1 although the joint or pivot be not absolutely fixed1 it is to be considered so in reference to the mechanical effect4 ,illustration3 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,in some cases1 as in most of the wheels of watches and clocks1 fly-wheels and chucks of the turning lathe1 and the arms of wind-mills1 the body turns continually in the same direction1 and each of its points traverses a complete circle during every revolution of the body round its axis4 ,in other instances the motion is alternate or reciprocating1 its direction being at intervals reversed4 ,such is the case in pendulums of clocks1 balance-wheels of chronometers1 the treddle of the lathe1 doors and lids on hinges1 scissors1 shears1 pincers1 c4 ,when the alternation is constant and regular1 it is called oscillation or vibration1 as in pendulums and balance-wheels4 7#aha47 ,to explain the properties of an axis of rotation it will be necessary to consider the different kinds of forces to the action of which a body moveable on such an axis may be submitted1 to show how this action depends on their several quantities and directions1 to distinguish the cases in which the forces neutralise each other and mutually equilibrate from those in which motion ensues1 to determine the effect which the axis suffers1 and1 in the cases where motion is produced1 to estimate the effects of those centrifugal forces 7#acg47 which are created by the mass of the body whirling round the axis4 ,forces in general have been distinguished by the duration of their action into instantaneous and continued forces4 ,the effect of an instantaneous force is produced in an infinitely short time4 ,if the body which sustains such an action be previously quiescent and free1 it will move with a uniform velocity in the direction of the impressed force4 7#ic47 ,if1 on the other hand1 the body be not free1 but so restrained that the impulse cannot put it in motion1 then the fixed points or lines which resist the motion sustain a corresponding shock at the moment of the impulse4 ,this effect1 which is called percussion1 is1 like the force which causes it1 instantaneous4 ,a continued force produces a continued effect4 ,if the body be free and previously quiescent1 this effect is a continual increase of velocity4 ,if the body be so restrained that the applied force cannot put it in motion1 the effect is a continued pressure on the points or lines which sustain it4 7#id47 ,it may happen1 however1 that although the body be not absolutely free to move in obedience to the force applied to it1 yet still it may not be altogether so restrained as to resist the effect of that force and remain at rest4 ,if the point at which a force is applied be free to move in a certain direction not coinciding with that of the applied force1 that force will be resolved into two elements2 one of which is in the direction in which the point is free to move1 and the other at right angles to that direction4 ,the point will move in obedience to the former element1 and the latter will produce percussion or pressure on the points or lines which restrain the body4 ,in fact1 in such cases the resistance offered by the circumstances which confine the motion of the body modifies the motion which it receives1 and as every change of motion must be the consequence of a force applied 7#dd471 the fixed points or lines which offer the resistance must suffer a corresponding effect4 ,it may happen that the forces impressed on the body1 whether they be continued or instantaneous1 are such as1 were it free1 would communicate to it a motion which the circumstances which restrain it do not forbid it to receive4 ,in such a case the fixed points or lines which restrain the body sustain no force1 and the phenomena will be the same in all respects as if these points or lines were not fixed4 ,it will be easy to apply these general reflections to the case in which a solid body is moveable on a fixed axis4 ,such a body is susceptible of no motion except one of rotation on that axis4 ,if it be submitted to the action of instantaneous forces1 one or other of the following effects must ensue4 #a4 ,the axis may resist the forces1 and prevent any motion4 #b4 ,the axis may modify the effect of the forces sustaining a corresponding percussion1 and the body receiving a motion of rotation4 #c4 ,the forces applied may be such as would cause the body to spin round the axis even were it not fixed1 in which case the body will receive a motion of rotation1 but the axis will suffer no percussion4 ,what has been just observed of the effect of instantaneous forces is likewise applicable to continued ones4 #a4 ,the axis may entirely resist the effect of such forces1 in which case it will suffer a pressure which may be estimated by the rules for the composition of force4 #b4 ,it may modify the effect of the applied forces1 in which case it must also sustain a pressure1 and the body must receive a motion of rotation which is subject to constant variation1 owing to the incessant action of the forces4 #c4 ,the forces may be such as would communicate to the body the same rotatory motion if the axis were not fixed4 ,in this case the forces will produce no pressure on the axis4 ,the impressed forces are not the only causes which affect the axis of a body during the phenomenon of rotation4 ,this species of motion calls into action other forces depending on the inertia of the mass1 which produce effects upon the axis1 and which play a prominent part in the theory of rotation4 ,while the body revolves on its axis1 the component particles of its mass move in circles1 the centres of which are placed in the axis4 ,the radius of the circle in which each particle moves is the line drawn from that particle perpendicular to the axis4 ,it has been already proved that a particle of matter1 moving round a centre1 is attended with a centrifugal force proportionate to the radius of the circle in which it moves and to the square of its angular velocity4 ,when a solid body revolves on its axis1 all its parts are whirled round together1 each performing a complete revolution in the same time4 ,the angular velocity is consequently the same for all1 and the difference of the centrifugal forces of different particles must entirely depend upon their distances from the axis4 ,the tendency of each particle to fly from the axis1 arising from the centrifugal force1 is resisted by the cohesion of the parts of the mass1 and in general this tendency is expended in exciting a pressure or strain upon the axis4 ,it ought to be recollected1 however1 that this pressure or strain is altogether different from that already mentioned1 and produced by the forces which give motion to the body4 ,the latter depends entirely upon the quantity and directions of the applied forces in relation to the axis3 the former depends on the figure and density of the body1 and the velocity of its motion4 ,these very complex effects render a simple and elementary exposition of the mechanical properties of a fixed axis a matter of considerable difficulty4 ,indeed1 the complete mathematical development of this theory long eluded the skill of the most acute geometers1 and it was only at a comparatively late period that it yielded to the searching analysis of modern science4 7#ahb47 ,to commence with the most simple case1 we shall consider the body as submitted to the action of a single force4 ,the effect of this force will vary according to the relation of its direction to that of the axis4 ,there are two ways in which a body may be conceived to be moveable around an axis4 #a4 ,by having pivots at two points which rest in sockets1 so that when the body is moved it must revolve round the right line joining the pivots as an axis4 #b4 ,a thin cylindrical rod may pass through the body1 on which it may turn in the same manner as a wheel upon its axle4 ,if the force be applied to the body in the direction of the axis1 it is evident that no motion can ensue1 and the effect produced will be a pressure on that pivot towards which the force is directed4 ,if in this case the body revolved on a cylindrical rod1 the tendency of the force would be to make it slide along the rod without revolving round it4 ,let us next suppose the force to be applied not in the direction of the axis itself1 but parallel to it4 ,let ,a ,b1 fig4 #gj41 be the axis1 and let ,c ,d be the direction of the force applied4 ,the pivots being supposed to be at ,a and ,b1 draw ,a ,g and ,b ,f perpendicular to ,a ,b4 ,the force ,c ,d will be equivalent to three forces1 one acting from ,b towards ,a1 equal in quantity to the force ,c ,d4 ,this force will evidently produce a corresponding pressure on the pivot ,a4 ,the other two forces will act in the directions ,a ,g and ,b ,f1 and will have respectively to the force ,c ,d the same proportion as ,a ,e has to ,a ,b4 ,such will be the mechanical effect of a force ,c ,d parallel to the axis4 ,and as these effects are all directed on the pivots1 no motion can ensue4 ,if the body revolve on a cylindrical rod1 the forces ,a ,g and ,b ,f would produce a strain upon the axis1 while the third force in the direction ,b ,a would have a tendency to make the body slide along it4 7#ahc47 ,if the force applied to the body be directed upon the axis1 and at right angles to it1 no motion can be produced4 ,in this case1 if the body be supported by pivots at ,a and ,b1 the force ,k ,l1 perpendicular to the line ,a ,b1 will be distributed between the pivots1 producing a pressure on each proportional to its distance from the other4 ,the pressure on ,a having to the pressure on ,b the same proportion as ,l ,b has to ,l ,a4 ,if the force ,k ,h be directed obliquely to the axis1 it will be equivalent to two forces 7#gf471 one ,k ,l perpendicular to the axis1 and the other ,k ,m parallel to it4 ,the effect of each of these may be investigated as in the preceding cases4 ,in all these observations the body has been supposed to be submitted to the action of one force only4 ,if several forces act upon it1 the direction of each of them crossing the axis either perpendicularly or obliquely1 or taking the direction of the axis or any parallel direction1 their effects may be similarly investigated4 ,in the same manner we may determine the effects of any number of forces whose combined results are mechanically equivalent to forces which either intersect the axis or are parallel to it4 7#ahd47 ,if any force be applied whose direction lies in a plane oblique to the axis1 it can always be resolved into two elements 7#gf471 one of which is parallel to the axis1 and the other in a plane perpendicular to it4 ,the effect of the former has been already determined1 and therefore we shall at present confine our attention to the latter4 ,suppose the axis to be perpendicular to the paper1 and to pass through the point ,g1 fig4 #ga4 and let ,a ,b ,c be a section of the body4 ,it will be convenient to consider the section vertical and the axis horizontal1 omitting1 however1 any notice of the effect of the weight of the body4 ,let a weight ,w be suspended by a cord ,q ,w from any point ,q4 ,this weight will evidently have a tendency to turn the body round in the direction ,a ,b ,c4 ,let another cord be attached to any other point ,p1 and1 being carried over a wheel ,r1 let a dish ,s be attached to it1 and let fine sand be poured into this dish until the tendency of ,s to turn the body round the axis in the direction of ,c ,b ,a balances the opposite tendency of ,w4 ,let the weights of ,w and ,s be then exactly ascertained1 and also let the distances ,g ,i and ,g ,h of the cords from the axis be exactly measured4 ,it will be found that1 if the number of ounces in the weight ,s be multiplied by the number of inches in ,g ,h1 and also the number of ounces in ,w by the number of inches in ,g ,i1 equal products will be obtained4 ,this experiment may be varied by varying the position of the wheel ,r1 and thereby changing the direction of the string ,p ,r1 in which cases it will be always found necessary to vary the weight of ,s in such a manner1 that when the number of ounces in it is multiplied by the number of inches in the distance of the string from the axis1 the product obtained shall be equal to that of the weight ,w by the distance ,g ,i4 ,we have here used ounces and inches as the measures of weight and distance2 but it is obvious that any other measures would be equally applicable4 ,from what has been just stated it follows1 that the energy of the weight of ,s to move the body on its axis1 does not depend alone upon the actual amount of that weight1 but also upon the distance of the string from the axis4 ,if1 while the position of the string remains unaltered1 the weight of ,s be increased or diminished1 the resisting weight ,w must be increased or diminished in the same proportion4 ,but if1 while the weight of ,s remains unaltered1 the distance of the string ,p ,r from the axis ,g be increased or diminished1 it will be found necessary to increase or diminish the resisting weight ,w in exactly the same proportion4 ,it therefore appears that the increase or diminution of the distance of the direction of a force from the axis has the same effect upon its power to give rotation as a similar increase or diminution of the force itself4 ,the power of a force to produce rotation is1 therefore1 accurately estimated1 not by the force alone1 but by the product found by multiplying the force by the distance of its direction from the axis4 ,it is frequently necessary in mechanical science to refer to this power of a force1 and1 accordingly1 the product just mentioned has received a particular denomination4 ,it is called the moment of the force round the axis4 7#ahe47 ,the distance of the direction of a force from the axis is sometimes called the leverage of the force4 ,the moment of a force is therefore found by multiplying the force by its leverage1 and the energy of a given force to turn a body round an axis is proportional to the leverage of that force4 ,from all that has been observed it may easily be inferred that1 if several forces affect a body moveable on an axis1 having tendencies to turn it in different directions1 they will mutually neutralise each other and produce equilibrium1 if the sum of the moments of those forces which tend to turn the body in one direction be equal to the sum of the moments of those which tend to turn it in the opposite direction4 ,thus1 if the forces ,a1 ,b1 ,c1 444 tend to turn the body from right to left1 and the distances of their directions from the axis be a1 b1 c1 444 and the forces ,a1 ,b1 ,c1 444 tend to move it from left to right1 and the distances of their directions from the axis be a1 b1 c1 4442 then these forces will produce equilibrium1 if the products found by multiplying the ounces in ,a1 ,b1 ,c1 444 respectively by the inches in a1 b1 c1 444 when added together be equal to the products found by multiplying the ounces in ,a1 ,b1 ,c1 444 by the inches in a1 b1 c1 444 respectively when added together4 ,but if either of these sets of products when added together exceed the other1 the corresponding set of forces will prevail1 and the body will revolve on its axis4 7#ahf47 ,when a body receives an impulse in a direction perpendicular to the axis1 but not crossing it1 a uniform rotatory motion is produced4 ,the velocity of this motion depends on the force of the impulse1 the distance of the direction of the impulse from the axis1 and the manner in which the mass of the body is distributed round the axis4 ,it is to be considered that the whole force of the impulse is shared amongst the various parts of the mass1 and is transmitted to them from the point where the impulse is applied by reason of the cohesion and tenacity of the parts1 and the impossibility of one part yielding to a force without carrying all the other parts with it4 ,the force applied acts upon those particles nearer to the axis than its own direction under advantageous circumstances2 for1 according to what has been already explained1 their power to resist the effect of the applied force is small in the same proportion with their distance4 ,on the other hand1 the applied force acts upon particles of the mass1 at a greater distance than its own direction1 under circumstances proportionably disadvantageous2 for their resistance to the applied force is great in proportion to their distances from the axis4 ,let ,c ,d1 fig4 #gb41 be a section of the body made by a plane passing through the axis ,a ,b4 ,suppose the impulse to be applied at ,p1 perpendicular to this plane1 and at the distance ,p ,o from the axis4 ,the effect of the impulse being distributed through the mass will cause the body to revolve on ,a ,b1 with a uniform velocity4 ,there is a certain point ,g1 at which1 if the whole mass were concentrated1 it would receive from the impulse the same velocity round the axis4 ,the distance ,o ,g is called the radius of gyration of the axis ,a ,b1 and the point ,g is called the centre of gyration relatively to that axis4 ,the effect of the impulse upon the mass concentrated at ,g is great in exactly the same proportion as ,o ,g is small4 ,this easily follows from the property of moments which has been already explained2 from whence it may be inferred1 that the greater the radius of gyration is1 the less will be the velocity which the body will receive from a given impulse4 7#ahg47 ,since the radius of gyration depends on the manner in which the mass is arranged round the axis1 it follows that for different axes in the same body there will be different radii of gyration4 ,of all axes taken in the same body parallel to each other1 that which passes through the centre of gravity has the least radius of gyration4 ,if the radius of gyration of any axis passing through the centre of gravity be given1 that of any parallel axis can be found2 for the square of the radius of gyration of any axis is equal to the square of the distance of that axis from the centre of gravity added to the square of the radius of gyration of the parallel axis through the centre of gravity4 7#ahh47 ,the product of the numerical expressions for the mass of the body and the square of the radius of gyration is a quantity much used in mechanical science1 and has been called the moment of inertia4 ,the moments of inertia1 therefore1 for different axes in the same body are proportional to the squares of the corresponding radii of gyration2 and consequently increase as the distances of the axes from the centre of gravity increase4 7#ahg47 7#ahi47 ,from what has been explained in 7#ahg471 it follows1 that the moment of inertia of any axis may be computed by common arithmetic1 if the moment of inertia of a parallel axis through the centre of gravity be previously known4 ,to determine this last1 however1 would require analytical processes altogether unsuitable to the nature and objects of the present treatise4 ,the velocity of rotation which a body receives from a given impulse is great in exactly the same proportion as the moment of inertia is small4 ,thus the moment of inertia may be considered in rotatory motion analogous to the mass of the body in rectilinear motion4 ,from what has been explained in 7#ahg47 it follows that a given impulse at a given distance from the axis will communicate the greatest angular velocity when the axis passes through the centre of gravity1 and that the velocity which it will communicate round other axes will be diminished in the same proportion as the squares of their distances from the centre of gravity added to the square of the radius of gyration for a parallel axis through the centre of gravity are augmented4 7#aij47 ,if any point whatever be assumed in a body1 and right lines be conceived to diverge in all directions from that point1 there are generally two of these lines1 which being taken as axes of rotation1 one has a greater and the other a less moment of inertia than any of the others4 ,it is a remarkable circumstance1 that1 whatever be the nature of the body1 whatever be its shape1 and whatever be the position of the point assumed1 these two axes of greatest and least moment will always be at right angles to each other4 ,these axes and a third through the same point1 and at right angles to both of them1 are called the principal axes of that point from which they diverge4 ,to form a distinct notion of their relative position1 let the axis of greatest moment be imagined to lie horizontally from north to south1 and the axis of least moment from east to west2 then the third principal axis will be presented perpendicularly upwards and downwards4 ,the first two being called the principal axes of greatest and least moment1 the third may be called the intermediate principal axis4 7#aia47 ,although the moments of the three principal axes be in general unequal1 yet bodies may be found having certain axes for which these moments may be equal4 ,in some cases the moment of the intermediate axis is equal to that of the principal axis of greatest moment3 in others it is equal to that of the principal axis of least moment1 and in others the moments of all the three principal axes are equal to each other4 ,if the moments of any two of three principal axes be equal1 the moments of all axes through the same point and in their plane will also be equal2 and if the moments of the three principal axes through a point be equal1 the moments of all axes whatever1 through the same point1 will be equal4 7#aib47 ,if the moments of the principal axes through the centre of gravity be known1 the moments for all other axes through that point may be easily computed4 ,to effect this it is only necessary to multiply the moments of the principal axes by the squares of the co-sines of the angles formed by them respectively with the axis whose moment is sought4 ,the products being added together will give the required moment4 7#aic47 ,by combining this result with that of 7#ahi471 it will be evident that the moment of all axes whatever may be determined1 if those of the principal axes through the centre of gravity be known4 7#aid47 ,it is obvious that the principal axis of least moment through the centre of gravity has a less moment of inertia than any other axis whatever4 ,for it has1 by its definition 7#aij47 a less moment of inertia than any other axis through the centre of gravity1 and every other axis through the centre of gravity has a less moment of inertia than a parallel axis through any other point 7#ahg47 and 7#ahi47 7#aie47 ,if two of the principal axes through the centre of gravity have equal moments of inertia1 all axes in any plane parallel to the plane of these axes1 and passing through the point where a perpendicular from the centre of gravity meets that plane1 must have equal moments of inertia4 ,for by 7#aia47 all axes in the plane of those two have equal moments1 and by 7#ahi47 the axes in the parallel plane have moments which exceed these by the same quantity1 being equally distant from them4 7#ahg47 ,hence it is obvious that if the three principal axes through the centre of gravity have equal moments1 all axes situated in any given plane1 and passing through the point where the perpendicular from the centre of gravity meets that plane1 will have equal moments1 being equally distant from parallel axes through the centre of gravity4 7#aif47 ,if the three principal axes through the centre of gravity have unequal moments1 there is no point whatever for which all axes will have equal moments2 but if the principal axis of least moment and the intermediate principal axis through the centre of gravity have equal moments1 then there will be two points on the principal axis of greatest moment1 equally distant at opposite sides of the centre of gravity1 at which all axes will have equal moments4 ,if the three principal axes through the centre of gravity have equal moments1 no other point of the body can have principal axes of equal moment4 7#aig47 ,when a body revolves on a fixed axis1 the parts of its mass are whirled in circles round the axis2 and since they move with a common angular velocity1 they will have centrifugal forces proportional to their distances from the axis4 ,if the component parts of the mass were not united together by cohesive forces of energies greater than these centrifugal forces1 they would be separated1 and would fly off from the axis2 but their cohesion prevents this1 and causes the effects of the different centrifugal forces1 which affect the different parts of the mass1 to be transmitted so as to modify each other1 and finally to produce one or more forces mechanically equivalent to the whole1 and which are exerted upon the axis and resisted by it4 ,we propose now to explain these effects1 as far as it is possible to render them intelligible without the aid of mathematical language4 ,it is obvious that any number of equal parts of the mass1 which are uniformly arranged in a circle round the axis1 have equal centrifugal forces acting from the centre of the circle in every direction4 ,these mutually neutralise each other1 and therefore exert no force on the axis4 ,the same may be said of all parts of the mass which are regularly and equally distributed on every side of the axis4 ,also if equal masses be placed at equal distances on opposite sides of the axis1 their centrifugal forces will destroy each other4 ,hence it appears that the pressure which the axis of rotation sustains from the centrifugal forces of the revolving mass1 arises from the unequal distribution of the matter around it4 ,from this reasoning it will be easily perceived that in the following examples the axis of rotation will sustain no pressure4 ,a globe revolving on any of its diameters1 the density being the same at equal distances from the centre4 ,a spheroid or a cylinder revolving on its axis1 the density being equal at equal distances from the axis4 ,a cube revolving on an axis which passes through the centre of two opposite bases1 being of uniform density4 ,a circular plate of uniform thickness and density revolving on one of its diameters as an axis4 7#aih47 ,in all these examples it will be observed that the axis of rotation passes through the centre of gravity4 ,the general theorem1 of which they are only particular instances1 is1 8if a body revolve on a principal axis1 passing through the centre of gravity1 the axis will sustain no pressure from the centrifugal force of the revolving mass40 ,this is a property in which the principal axes through the centre of gravity are unique4 ,there is no other axis on which a body could revolve without pressure4 ,if two of the principal axes through the centre of gravity have equal moments1 every axis in their plane has the same moment1 and is to be considered equally as a principal axis4 ,in this case the body would revolve on any of these axes without pressure4 ,a homogeneous spheroid furnishes an example of this4 ,if any of the diameters of the earth's equator were a fixed axis1 the earth would revolve on it without producing pressure4 ,if the three principal axes through the centre of gravity have equal moments1 all axes through the centre of gravity are to be considered as principal axes4 ,in this case the body would revolve without pressure on any axis through the centre of gravity4 ,a globe1 in which the density of the mass at equal distances from the centre is the same1 is an example of this4 ,such a body would revolve without pressure on any axis through its centre4 7#aii47 ,since no pressure is excited on the axis in these cases1 the state of the body will not be changed1 if during its rotation the axis cease to be fixed4 ,the body will notwithstanding continue to revolve round the axis1 and the axis will maintain its position4 ,thus a spinning-top of homogeneous material and symmetrical form will revolve steadily in the same position1 until the friction of its point with the surface on which it rests deprives it of motion4 ,this is a phenomenon which can only be exhibited when the axis of rotation is a principal axis through the centre of gravity4 7#bjj47 ,if the body revolve round any axis through the centre of gravity1 which is not a principal axis1 the centrifugal pressure is represented by two forces1 which are equal and parallel1 but which act in opposite directions on different points of the axis4 ,the effect of these forces is to produce a strain upon the axis1 and give the body a tendency to move round another axis at right angles to the former4 7#bja47 ,if the fixed axis on which a body revolves be a principal axis through any point different from the centre of gravity1 then a pressure will be produced by the centrifugal force of the revolving mass1 and this pressure will act at right angles to the axis on the point to which it is a principal axis1 and in the plane through that axis and the centre of gravity4 ,the amount of the pressure will be proportional to the mass of the body1 the distance of the centre of gravity from the axis1 and the square of the velocity of rotation4 7#bjb47 ,since the whole pressure is in this case excited on a single point1 the stability of the axis will not be disturbed1 provided that point alone be fixed4 ,so that even though the axis should be free to turn on that point1 no motion will ensue as long as no external forces act upon the body4 7#bjc47 ,if the axis of rotation be not a principal axis1 the centrifugal forces will produce an effect which cannot be represented by a single force4 ,the effect may be understood by conceiving two forces to act on different points of the axis at right angles to it and to each other4 ,the quantities of these pressures and their directions depend on the figure and density of the mass and the position of the axis1 in a manner which cannot be explained without the aid of mathematical language and principles4 7#bjd47 ,the effects upon the axis which have been now explained are those which arise from the motion of rotation1 from whatever cause that motion may have arisen4 ,the forces which produce that motion1 however1 are attended with effects on the axis which still remain to be noticed4 ,when these forces1 whether they be of the nature of instantaneous actions or continued forces1 are entirely resisted by the axis1 their directions must severally be in a plane passing through the axis1 or they must1 by the principles of the composition of force 7#gd47 et seq41 be mechanically equivalent to forces in that plane4 ,in every other case the impressed forces must produce motion1 and1 except in certain cases1 must also produce effects upon the axis4 ,by the rules for the composition of force it is possible in all cases to resolve the impressed forces into others which are either in planes through the axis1 or in planes perpendicular to it1 or1 finally1 some in planes through it1 and others in planes perpendicular to it4 ,the effect of those which are in planes through the axis has been already explained2 and we shall now confine our attention to those impelling forces which act at right angles to the axis1 and which produce motion4 ,it will be sufficient to consider the effect of a single force at right angles to the axis2 for whatever be the number of forces which act either simultaneously or successively1 the effect of the whole will be decided by combining their separate effects4 ,the effect which a single force produces depends on two circumstances1 #a4 ,the position of the axis with respect to the figure and mass of the body1 and #b4 ,the quantity and direction of the force itself4 ,in general the shock which the axis sustains from the impact may be represented by two impacts applied to it at different points1 one parallel to the impressed force1 and the other perpendicular to it1 but both perpendicular to the axis4 ,there are certain circumstances1 however1 under which this effect will be modified4 ,if the impulse which the body receives be in a direction perpendicular to a plane through the axis and the centre of gravity1 and at a distance from the axis which bears to the radius of gyration 7#ahf47 the same proportion as that line bears to the distance of the centre of gravity from the axis1 there are certain cases in which the impulse will produce no percussion4 ,to characterise these cases generally would require analytical formul which cannot conveniently be translated into ordinary language4 ,that point of the plane1 however1 where the direction of the impressed force meets it1 when no percussion on the axis is produced1 is called the centre of percussion4 ,if the axis of rotation be a principal axis1 the centre of percussion must be in the right line drawn through the centre of gravity1 intersecting the axis at right angles1 and at the distance from the axis already explained4 ,if the axis of rotation be parallel to a principal axis through the centre of gravity1 the centre of percussion will be determined in the same manner4 7#bje47 ,there are many positions which the axis may have in which there will be no centre of percussion2 that is1 there will be no direction in which an impulse could be applied without producing a shock upon the axis4 ,one of these positions is when it is a principal axis through the centre of gravity4 ,this is the only case of rotation round an axis in which no effect arises from the centrifugal force2 and therefore it follows that the only case in which the axis sustains no effect from the motion produced1 is one in which it must necessarily suffer an effect from that which produces the motion4 ,if the body be acted upon by continued forces1 their effect is at each instant determined by the general principles for the composition of force4 ,,chap4 ,,xi4 ,,on ,,the ,,pendulum4 7#bjf47 ,when a body is placed on a horizontal axis which does not pass through its centre of gravity1 it will remain in permanent equilibrium only when the centre of gravity is immediately below the axis4 ,if this point be placed in any other situation1 the body will oscillate from side to side1 until the atmospherical resistance and the friction of the axis destroy its motion4 7#aei1 #afj47 ,such a body is called a pendulum4 ,the swinging motion which it receives is called oscillation or vibration4 7#bjg47 ,the use of the pendulum1 not only for philosophical purposes1 but in the ordinary economy of life1 renders it a subject of considerable importance4 ,it furnishes the most exact means of measuring time1 and of determining with precision various natural phenomena4 ,by its means the variation of the force of gravity in different latitudes is discovered1 and the law of that variation experimentally exhibited4 ,in the present chapter1 we propose to explain the general principles which regulate the oscillation of pendulums4 ,minute details concerning their construction will be given in the twenty-first chapter of this volume4 7#bjh47 ,a simple pendulum is composed of a heavy molecule attached to the end of a flexible thread1 and suspended by a fixed point ,o1 fig4 #gc4 ,when the pendulum is placed in the position ,o ,c1 the molecule being vertically below the point of suspension1 it will remain in equilibrium2 but if it be drawn into the position ,o ,a and there liberated1 it will descend towards ,c1 moving through the arc ,a ,c with accelerated motion4 ,having arrived at ,c and acquired a certain velocity1 it will1 by reason of its inertia1 continue to move in the same direction4 ,it will therefore commence to ascend the arc ,c ,a with the velocity so acquired4 ,during its ascent1 the weight of the molecule retards its motion in exactly the same manner as it had accelerated it in descending from ,a to ,c2 and when the molecule has ascended through the arc ,c ,a equal to ,c ,a1 its entire velocity will be destroyed1 and it will cease to move in that direction4 ,it will thus be placed at ,a in the same manner as in the first instance it had been placed at ,a1 and consequently it will descend from ,a to ,c with accelerated motion1 in the same manner as it first moved from ,a to ,c4 ,it will then ascend from ,c to ,a1 and so on1 continually4 ,in this case the thread1 by which the molecule is suspended1 is supposed to be perfectly flexible1 inextensible1 and of inconsiderable weight4 ,the point of suspension is supposed to be without friction1 and the atmosphere to offer no resistance to the motion4 ,it is evident from what has been stated1 that the times of moving from ,a to ,a and from ,a to ,a are equal1 and will continue to be equal so long as the pendulum continues to vibrate4 ,if the number of vibrations performed by the pendulum were registered1 and the time of each vibration known1 this instrument would become a chronometer4 ,the rate at which the motion of the pendulum is accelerated in its descent towards its lowest position is not uniform1 because the force which impels it is continually decreasing1 and altogether disappears at the point ,c4 ,the impelling force arises from the effect of gravity on the suspended molecule1 and this effect is always produced in the vertical direction ,a ,v4 ,the greater the angle ,o ,a ,v is1 the less efficient the force of gravity will be in accelerating the molecule3 this angle evidently increases as the molecule approaches ,c1 which will appear by inspecting fig4 #gc4 ,at ,c1 the force of gravity acting in the direction ,c ,b is totally expended in giving tension to the thread1 and is inefficient in moving the molecule4 ,it follows1 therefore1 that the impelling force is greatest at ,a1 and continually diminishes from ,a to ,c1 where it altogether vanishes4 ,the same observations will be applicable to the retarding force from ,c to ,a1 and to the accelerating force from ,a to ,c1 and so on4 ,when the length of the thread and the intensity of the force of gravity are given1 the time of vibration depends on the length of the arc ,a ,c1 or on the magnitude of the angle ,a ,o ,c4 ,if1 however1 this angle do not exceed a certain limit of magnitude1 the time of vibration will be subject to no sensible variation1 however that angle may vary4 ,thus the time of oscillation will be the same1 whether the angle ,a ,o ,c be #b1 or #a #cj1 or #a1 or any lesser magnitude4 ,this property of a pendulum is expressed by the word isochronism4 ,the strict demonstration of this property depends on mathematical principles1 the details of which would not be suitable to the present treatise4 ,it is not difficult1 however1 to explain generally how it happens that the same pendulum will swing through greater and smaller arcs of vibration in the same time4 ,if it swing from ,a1 the force of gravity at the commencement of its motion impels it with an effect depending on the obliquity of the lines ,o ,a and ,a ,v4 ,if it commence its motion from a1 the impelling effect from the force of gravity will be considerably less than at ,a2 consequently1 the pendulum begins to move at a slower rate1 when it swings from a than when it moves from ,a3 the greater magnitude of the swing is therefore compensated by the increased velocity1 so that the greater and the smaller arcs of vibration are moved through in the same time4 7#bji47 ,to establish this property experimentally1 it is only necessary to suspend a small ball of metal1 or other heavy substance1 by a flexible thread1 and to put it in a state of vibration1 the entire arc of vibration not exceeding #d or #e1 the friction on the point of suspension and other causes will gradually diminish the arc of vibration1 so that after the lapse of some hours it will be so small1 that the motion will scarcely be discerned without microscopic aid4 ,if the vibration of this pendulum be observed in reference to a correct timekeeper1 at the commencement1 at the middle1 and towards the end of its motion1 the rate will be found to suffer no sensible change4 ,this remarkable law of isochronism was one of the earliest discoveries of ,galileo4 ,it is said1 that when very young1 he observed a chandelier suspended from the roof of a church in ,pisa swinging with a pendulous motion1 and was struck with the uniformity of the rate even when the extent of the swing was subject to evident variation4 7#baj47 ,it has been stated in 7#aag47 that the attraction of gravity affects all bodies equally1 and moves them with the same velocity1 whatever be the nature or quantity of the materials of which they are composed4 ,since it is the force of gravity which moves the pendulum1 we should therefore expect that the circumstances of that motion should not be affected either by the quantity or quality of the pendulous body4 ,and we find this1 in fact1 to be the case2 for if small pieces of different heavy substances such as lead1 brass1 ivory1 c41 be suspended by fine threads of equal length1 they will vibrate in the same time1 provided their weights bear a considerable proportion to the atmospherical resistance1 or that they be suspended in vacuo4 7#baa47 ,since the time of vibration of a pendulum1 which oscillates in small arcs1 depends neither on the magnitude of the arc of vibration nor on the quality or weight of the pendulous body1 it will be necessary to explain the circumstances on which the variation of this time depends4 ,the first and most striking of these circumstances is the length of the suspending thread4 ,the rudest experiments will demonstrate the fact1 that every increase in the length of this thread will produce a corresponding increase in the time of vibration2 but according to what law does this increase proceed8 ,if the length of the thread be doubled or trebled1 will the time of vibration also be increased in a double or treble proportion8 ,this problem is capable of exact mathematical solution1 and the result shows that the time of vibration increases not in the proportion of the increased length of the thread1 but as the square root of that length2 that is to say1 if the length of the thread be increased in a four-fold proportion1 the time of vibration will be augmented in a two-fold proportion4 ,if the thread be increased to nine times its length1 the time of vibration will be trebled1 and so on4 ,this relation is exactly the same as that which was proved to subsist between the spaces through which a body falls freely1 and the times of fall4 ,in the table1 page #hi1 if the figures representing the height be understood to express the length of different pendulums1 the figures immediately above them will express the corresponding times of vibration4 ,this law of the proportion of the lengths of pendulums to the squares of the time of vibration may be experimentally established in the following manner3-- ,let ,a1 ,b1 ,c1 fig4 #gd41 be three small pieces of metal each attached by threads to two points of suspension1 and let them be placed in the same vertical line under the point ,o2 suppose them so adjusted that the distances ,o ,a1 ,o ,b1 and ,o ,c shall be in the proportion of the numbers #a1 #d1 and #i4 ,let them be removed from the vertical in a direction at right angles to the plane of the paper1 so that the threads shall be in the same plane1 and therefore the three pendulums will have the same angle of vibration4 ,being now liberated1 the pendulum ,a will immediately gain upon ,b1 and ,b upon ,c1 so that ,a will have completed one vibration before ,b or ,c4 ,at the end of the second vibration of ,a1 the pendulum ,b will have arrived at the end of its first vibration1 so that the suspending threads of ,a and ,b will then be separated by the whole angle of vibration2 at the end of the fourth vibration of ,a the suspending threads of ,a and ,b will return to their first position1 ,b having completed two vibrations2 thus the proportion of the times of vibration of ,b and ,a will be #b to #a1 the proportion of their lengths being #d to #a4 ,at the end of the third vibration of ,a1 ,c will have completed one vibration1 and the suspending strings will coincide in the position distant by the whole angle of vibration from their first position4 ,so that three vibrations of ,a are performed in the same time as one of ,c3 the proportion of the time of vibration of ,c and ,a are1 therefore1 #c to #a1 the proportion of their lengths being #i to #a1 conformably to the law already explained4 7#bab47 ,in all the preceding observations we have assumed that the material of the pendulous body is of inconsiderable magnitude1 its whole weight being conceived to be collected in a physical point4 ,this is generally called a simple pendulum2 but since the conditions of a suspending thread without weight1 and a heavy molecule without magnitude1 cannot have practical existence1 the simple pendulum must be considered as imaginary1 and merely used to establish hypothetical theorems1 which1 though inapplicable in practice1 are nevertheless the means of investigating the laws which govern the real phenomena of pendulous bodies4 ,a pendulous body being of determinate magnitude1 its several parts will be situated at different distances from the axis of suspension4 ,if each component part of such a body were separately connected with the axis of suspension by a fine thread1 it would1 being unconnected with the other particles1 be an independent simple pendulum1 and would oscillate according to the laws already explained4 ,it therefore follows that those particles of the body which are nearest to the axis of suspension would1 if liberated from their connection with the others1 vibrate more rapidly than those which are more remote4 ,the connection1 however1 which the particles of the body have1 by reason of their solidity1 compels them all to vibrate in the same time4 ,consequently1 those particles which are nearer the axis are retarded by the slower motion of those which are more remote2 while the more remote particles1 on the other hand1 are urged forward by the greater tendency of the nearer particles to rapid vibration4 ,this will be more readily comprehended1 if we conceive two particles of matter ,a and ,b1 fig4 #ge41 to be connected with the same axis ,o by an inflexible wire ,o ,c1 the weight of which may be neglected4 ,if ,b were removed1 ,a would vibrate in a certain time depending upon the distance ,o ,a4 ,if ,a were removed1 and ,b placed upon the wire at a distance ,b ,o equal to four times ,a ,o1 ,b would vibrate in twice the former time4 ,now if both be placed on the wire at the distances just mentioned1 the tendency of ,a to vibrate more rapidly will be transmitted to ,b by means of the wire1 and will urge ,b forward more quickly than if ,a were not present3 on the other hand1 the tendency of ,b to vibrate more slowly will be transmitted by the wire to ,a1 and will cause it to move more slowly than if ,b were not present4 ,the inflexible quality of the connecting wire will in this case compel ,a and ,b to vibrate simultaneously1 the time of vibration being greater than that of ,a1 and less than that of ,b1 if each vibrated unconnected with the other4 ,if1 instead of supposing two particles of matter placed on the wire1 a greater number were supposed to be placed at various distances from ,o1 it is evident the same reasoning would be applicable4 ,they would mutually affect each other's motion2 those placed nearest to point ,o accelerating the motion of those more remote1 and being themselves retarded by the latter4 ,among these particles one would be found in which all these effects would be mutually neutralised1 all the particles nearer ,o being retarded in reference to that motion which they would have if unconnected with the rest1 and those more remote being in the same respect accelerated4 ,the point at which such a particle is placed is called the centre of oscillation4 ,what has been here observed of the effects of particles of matter placed upon rigid wire will be equally applicable to the particles of a solid body4 ,those which are nearer to the axis are urged forward by those which are more remote1 and are in their turn retarded by them2 and as with the particles placed upon the wire1 there is a certain particle of the body at which the effects are mutually neutralised1 and which vibrates in the same time as it would if it were unconnected with the other parts of the body1 and simply connected by a fine thread to the axis4 ,by this centre of oscillation the calculations respecting the vibration of a solid body are rendered as simple as those of a molecule of inconsiderable magnitude4 ,all the properties which have been explained as belonging to a simple pendulum may thus be transferred to a vibrating body of any magnitude and figure1 by considering it as equivalent to a single particle of matter vibrating at its centre of oscillation4 7#bac47 ,it follows from this reasoning1 that the virtual length of a pendulum is to be estimated by the distance of its centre of oscillation from the axis of suspension1 and therefore that the times of vibration of different pendulums are in the same proportion as the square roots of the distances of their centres of oscillation from their axes4 ,the investigation of the position of the centre of oscillation is1 in most cases1 a subject of intricate mathematical calculation4 ,it depends on the magnitude and figure of the pendulous body1 the manner in which the mass is distributed through its volume1 or the density of its several parts1 and the position of the axis on which it swings4 ,the place of the centre of oscillation may be determined when the position of the centre of gravity and the centre of gyration are known2 for the distance of the centre of oscillation from the axis will always be obtained by dividing the square of the radius of gyration 7#ahf47 by the distance of the centre of gravity from the axis4 ,thus if #f be the radius of gyration1 and #i the distance of gravity from the axis1 #cf divided by #i1 which is #d1 will be the distance of the centre of oscillation from the axis4 ,hence it may be inferred generally1 that the greater the proportion which the radius of gyration bears to the distance of the centre of gravity from the axis1 the greater will be the distance of the centre of oscillation4 ,it follows from this reasoning1 that the length of a pendulum is not limited by the dimensions of its volume4 ,if the axis be so placed that the centre of gravity is near it1 and the centre of gyration comparatively removed from it1 the centre of oscillation may be placed far beyond the limits of the pendulous body4 ,suppose the centre of gravity is at a distance of one inch from the axis1 and the centre of gyration #ab inches1 the centre of oscillation will then be at the distance of #add inches1 or #ab feet4 ,such a pendulum may not in its greatest dimensions exceed one foot1 and yet its time of vibration would be equal to that of a simple pendulum whose length is #ab feet4 ,by these means pendulums of small dimensions may be made to vibrate as slowly as may be desired4 ,the instruments called metronomes1 used for marking the time of musical performances1 are constructed on this principle4 7#bad47 ,the centre of oscillation is distinguished by a very remarkable property in relation to the axis of suspension4 ,if ,a1 fig4 #gf41 be the point of suspension1 and ,o the corresponding centre of oscillation1 the time of vibration of the pendulum will not be changed if it be raised from its support1 inverted1 and suspended from the point ,o4 ,it follows1 therefore1 that if ,o be taken as the point of suspension1 ,a will be the corresponding centre of oscillation4 ,these two points are1 therefore1 convertible4 ,this property may be verified experimentally in the following manner4 ,a pendulum being put into a state of vibration1 let a small heavy body be suspended by a fine thread1 the length of which is so adjusted that it vibrates simultaneously with the pendulum4 ,let the distance from the point of suspension to the centre of the vibrating body be measured1 and take this distance on the pendulum from the axis of suspension downwards2 the place of the centre of oscillation will thus be obtained1 since the distance so measured from the axis is the length of the equivalent simple pendulum4 ,if the pendulum be now raised from its support1 inverted1 and suspended from the centre of oscillation thus obtained1 it will be found to vibrate simultaneously with the body suspended by the thread4 7#bae47 ,this property of the interchangeable nature of the centres of oscillation and suspension has been1 at a late period1 adopted by ,captain ,kater1 as an accurate means of determining the length of a pendulum4 ,having ascertained with great accuracy two points of suspension at which the same body will vibrate in the same time1 the distance between these points being accurately measured1 is the length of the equivalent simple pendulum4 ,see ,chapter ,,xxi4 7#baf47 ,the manner in which the time of vibration of a pendulum depends on its length being explained1 we are next to consider how this time is affected by the attraction of gravity4 ,it is obvious that1 since the pendulum is moved by this attraction1 the rapidity of its motion will be increased1 if the impelling force receive any augmentation2 but it still is to be decided1 in what exact proportion the time of oscillation will be diminished by any proposed increase in the intensity of the earth's attraction4 ,it can be demonstrated mathematically1 that the time of one vibration of a pendulum has the same proportion to the time of falling freely in the perpendicular direction1 through a height equal to half the length of the pendulum1 as the circumference of a circle has to its diameter4 ,since1 therefore1 the times of vibration of pendulums are in a fixed proportion to the times of falling freely through spaces equal to the halves of their lengths1 it follows that these times have the same relation to the force of attraction as the times of falling freely through their lengths have to that force4 ,if the intensity of the force of gravity were increased in a four-fold proportion1 the time of falling through a given height would be diminished in a two-fold proportion2 if the intensity were increased to a nine-fold proportion1 the time of falling through a given space would be diminished in a three-fold proportion1 and so on2 the rate of diminution of the time being always as the square root of the increased force4 ,by what has been just stated this law will also be applicable to the vibration of pendulums4 ,any increase in the intensity of the force of gravity would cause a given pendulum to vibrate more rapidly1 and the increased rapidity of the vibration would be in the same proportion as the square root of the increased intensity of the force of gravity4 7#bag47 ,the laws which regulate the times of vibration of pendulums in relation to one another being well understood1 the whole theory of these instruments will be completed1 when the method of ascertaining the actual time of vibration of any pendulum1 in reference to its length1 has been explained4 ,in such an investigation1 the two elements to be determined are1 #a4 the exact time of a single vibration1 and1 #b4 the exact distance of the centre of oscillation from the point of suspension4 ,the former is ascertained by putting a pendulum in motion in the presence of a good chronometer1 and observing precisely the number of oscillations which are made in any proposed number of hours4 ,the entire time during which the pendulum swings1 being divided by the number of oscillations made during that time1 the exact time of one oscillation will be obtained4 ,the distance of the centre of oscillation from the point of suspension may be rendered a matter of easy calculation1 by giving a certain uniform figure and material to the pendulous body4 7#bah47 ,the time of vibration of one pendulum of known length being thus obtained1 we shall be enabled immediately to solve either of the following problems4 8,to find the length of a pendulum which shall vibrate in a given time40 8,to find the time of vibration of a pendulum of a given length40 ,the former is solved as follows3 the time of vibration of the known pendulum is to the time of vibration of the required pendulum1 as the square root of the length of the known pendulum is to the square root of the length of the required pendulum4 ,this length is therefore found by the ordinary rules of arithmetic4 ,the latter may be solved as follows3 the length of the known pendulum is to the length of the proposed pendulum1 as the square of the time of vibration of the known pendulum is to the square of the time of vibration of the proposed pendulum4 ,the latter time may therefore be found by arithmetic4 7#bai47 ,since the rate of a pendulum has a known relation to the intensity of the earth's attraction1 we are enabled1 by this instrument1 not only to detect certain variations in that attraction in various parts of the earth1 but also to discover the actual amount of the attraction at any given place4 ,the actual amount of the earth's attraction at any given place is estimated by the height through which a body would fall freely at that place in any given time1 as in one second4 ,to determine this1 let the length of a pendulum which would vibrate in one second at that place be found4 ,as the circumference of a circle is to its diameter#b 7a known proportion71 so will one second be to the time of falling through a height equal to half the length of this pendulum4 ,this time is therefore a matter of arithmetical calculation4 ,it has been proved in 7#abj471 that the heights1 through which a body falls freely1 are in the same proportion as the squares of the times2 from whence it follows1 that the square of the time of falling through a height equal to half the length of the pendulum is to one second as half the length of that pendulum is to the height through which a body would fall in one second4 ,this height1 therefore1 may be immediately computed1 and thus the actual amount of the force of gravity at any given place may be ascertained4 #b ,this ratio is that of #ca1daf to #aj1jjj very nearly4 7#bbj47 ,to compare the force of gravity in different parts of the earth1 it is only necessary to swing the same pendulum in the places under consideration1 and to observe the rapidity of its vibrations4 ,the proportion of the force of gravity in the several places will be that of the squares of the velocity of the vibration4 ,observations to this effect have been made at several places1 by ,biot1 ,kater1 ,sabine1 and others4 ,the earth being a mass of matter of a form nearly spherical1 revolving with considerable velocity on an axis1 its component parts are affected by a centrifugal force2 in virtue of which1 they have a tendency to fly off in a direction perpendicular to the axis4 ,this tendency increases in the same proportion as the distance of any part from the axis increases1 and consequently those parts of the earth which are near the equator1 are more strongly affected by this influence than those near the pole4 ,it has been already explained 7#ade47 that the figure of the earth is affected by this cause1 and that it has acquired a spheroidal form4 ,the centrifugal force1 acting in opposition to the earth's attraction1 diminishes its effects2 and consequently1 where this force is more efficient1 a pendulum will vibrate more slowly4 ,by these means the rate of vibration of a pendulum becomes an indication of the amount of the centrifugal force4 ,but this latter varies in proportion to the distance of the place from the earth's axis2 and thus the rate of a pendulum indicates the relation of the distances of different parts of the earth's surface from its axis4 ,the figure of the earth may be thus ascertained1 and that which theory assigns to it1 it may be practically proved to have4 ,this1 however1 is not the only method by which the figure of the earth may be determined4 ,the meridians being sections of the earth through its axis1 if their figure were exactly determined1 that of the earth would be known4 ,measurements of arcs of meridians on a large scale have been executed1 and are still being made in various parts of the earth1 with a view to determine the curvature of a meridian at different latitudes4 ,this method is independent of every hypothesis concerning the density and internal structure of the earth1 and is considered by some to be susceptible of more accuracy than that which depends on the observations of pendulums4 7#bba47 ,it has been stated that1 when the arc of vibration of a pendulum is not very small1 a variation in its length will produce a sensible effect on the time of vibration4 ,to construct a pendulum such that the time of vibration may be independent of the extent of the swing1 was a favourite speculation of geometers4 ,this problem was solved by ,huygens1 who showed that the curve called a cycloid1 previously discovered and described by ,galileo1 possessed the isochronal property2 that is1 that a body moving in it by the force of gravity1 would vibrate in the same time1 whatever be the length of the arc described4 ,let ,o ,a1 fig4 #gg41 be a horizontal line1 and let ,o ,b be a circle placed below this line1 and in contact with it4 ,if this circle be rolled upon the line from ,o towards ,a1 a point upon its circumference1 which at the beginning of the motion is placed at ,o1 will during the motion trace the curve ,o ,c ,a4 ,this curve is called a cycloid4 ,if the circle be supposed to roll in the opposite direction towards ,a1 the same point will trace another cycloid ,o ,c ,a4 ,the points ,c and ,c being the lowest points of the curves1 if the perpendiculars ,c ,d and ,c ,d be drawn1 they will respectively be equal to the diameter of the circle4 ,by a known property of this curve1 the arcs ,o ,c and ,o ,c are equal to twice the diameter of the circle4 ,from the point ,o suppose a flexible thread to be suspended1 whose length is twice the diameter of the circle1 and which sustains a pendulous body ,p at its extremity4 ,if the curves ,o ,c and ,o ,c1 from the plane of the paper1 be raised so as to form surfaces to which the thread may be applied1 the extremity ,p will extend to the points ,c and ,c1 when the entire thread has been applied to either of the curves4 ,as the thread is deflected on either side of its vertical position1 it is applied to a greater or lesser portion of either curve1 according to the quantity of its deflection from the vertical4 ,if it be deflected on each side until the point ,p reaches the points ,c and ,c1 the extremity would trace a cycloid ,c ,p ,c precisely equal and similar to those already mentioned4 ,availing himself of this property of the curve1 ,huygens constructed his cycloidal pendulum4 ,the time of vibration was subject to no variation1 however the arc of vibration might change1 provided only that the length of the string ,o ,p continued the same4 ,if small arcs of the cycloid be taken on either side of the point ,p1 they will not sensibly differ from arcs of a circle described with the centre ,o and the radius ,o ,p2 for1 in slight deflections from the vertical position1 the effect of the curves ,o ,c and ,o ,c on the thread ,o ,p is altogether inconsiderable4 ,it is for this reason that when the arcs of vibration of a circular pendulum are small1 they partake of the property of isochronism peculiar to those of a cycloid4 ,but when the deflection of ,p from the vertical is great1 the effect of the curves ,o ,c and ,o ,c on the thread produces a considerable deviation of the point ,p from the arc of the circle whose centre is ,o and whose radius is ,o ,p1 and consequently the property of isochronism will no longer be observed in the circular pendulum4 ,,chap4 ,,xii4 ,,of ,,simple ,,machines4 7#bbb47 ,a ,,machine is an instrument by which force or motion may be transmitted and modified as to its quantity and direction4 ,there are two ways in which a machine may be applied1 and which give rise to a division of mechanical science into parts denominated ,,statics and ,,dynamics2 the one including the theory of equilibrium1 and the other the theory of motion4 ,when a machine is considered statically1 it is viewed as an instrument by which forces of determinate quantities and direction are made to balance other forces of other quantities and other directions4 ,if it be viewed dynamically1 it is considered as a means by which certain motions of determinate quantity and direction may be made to produce other motions in other directions and quantities4 ,it will not be convenient1 however1 in the present treatise1 to follow this division of the subject4 ,we shall1 on the other hand1 as hitherto1 consider the phenomena of equilibrium and motion together4 ,the effects of machinery are too frequently described in such a manner as to invest them with the appearance of paradox1 and to excite astonishment at what appears to contradict the results of the most common experience4 ,it will be our object here to take a different course1 and to attempt to show that those effects which have been held up as matters of astonishment are the necessary1 natural1 and obvious results of causes adapted to produce them in a manner analogous to the objects of most familiar experience4 7#bbc47 ,in the application of a machine there are three things to be considered4 #a4 ,the force or resistance which is required to be sustained1 opposed1 or overcome4 #b4 ,the force which is used to sustain1 support1 or overcome that resistance4 #c4 ,the machine itself by which the effect of this latter force is transmitted to the former4 ,of whatever nature be the force or the resistance which is to be sustained or overcome1 it is technically called the weight1 since1 whatever it be1 a weight of equivalent effect may always be found4 ,the force which is employed to sustain or overcome it is technically called the power4 7#bbd47 ,in expressing the effect of machinery it is usual to say that the power sustains the weight2 but this1 in fact1 is not the case1 and hence arises that appearance of paradox which has already been alluded to4 ,if1 for example1 it is said that a power of one ounce sustains the weight of one ton1 astonishment is not unnaturally excited1 because the fact1 as thus stated1 if the terms be literally interpreted1 is physically impossible4 ,no power less than a ton can1 in the ordinary acceptation of the word1 support the weight of a ton4 ,it will1 however1 be asked how it happens that a machine appears to do this8 how it happens that by holding a silken thread1 which an ounce weight would snap1 many hundred weight may be sustained8 ,to explain this it will only be necessary to consider the effect of a machine1 when the power and weight are in equilibrium4 7#bbe47 ,in every machine there are some fixed points or props2 and the arrangement of the parts is always such1 that the pressure1 excited by the power or weight1 or both1 is distributed among these props4 ,if the weight amount to twenty hundred1 it is possible so to distribute it1 that any proportion1 however great1 of it may be thrown on the fixed points or props of the machine2 the remaining part only can properly be said to be supported by the power1 and this part can never be greater than the power4 ,considering the effect in this way1 it appears that the power supports just so much of the weight and no more as is equal to its own force1 and that all the remaining part of the weight is sustained by the machine4 ,the force of these observations will be more apparent when the nature and properties of the mechanic powers and other machines have been explained4 7#bbf47 ,when a machine is considered dynamically1 its effects are explained on different principles4 ,it is true that1 in this case1 a very small power may elevate a very great weight2 but nevertheless1 in so doing1 whatever be the machine used1 the total expenditure of power1 in raising the weight through any height1 is never less than that which would be expended if the power were immediately applied to the weight without the intervention of any machine4 ,this circumstance arises from an universal property of machines by which the velocity of the weight is always less than that of the power1 in exactly the same proportion as the power itself is less than the weight2 so that when a certain power is applied to elevate a weight1 the rate at which the elevation is effected is always slow in the same proportion as the weight is great4 ,from a due consideration of this remarkable law1 it will easily be understood1 that a machine can never diminish the total expenditure of power necessary to raise any weight or to overcome any resistance4 ,in such cases1 all that a machine ever does or ever can do1 is to enable the power to be expended at a slow rate1 and in a more advantageous direction than if it were immediately applied to the weight or the resistance4 ,let us suppose that ,p is a power amounting to an ounce1 and that ,w is a weight amounting to #ej ounces1 and that ,p elevates ,w by means of a machine4 ,in virtue of the property already stated1 it follows1 that while ,p moves through #ej feet1 ,w will be moved through #a foot2 but in moving ,p through #ej feet1 #ej distinct efforts are made1 by each of which #a ounce is moved through #a foot1 and by which collectively #ej distinct ounces might be successively raised through #a foot4 ,but the weight ,w is #ej ounces1 and has been raised through #a foot2 from whence it appears1 that the expenditure of power is equal to that which would be necessary to raise the weight without the intervention of any machine4 ,this important principle may be presented under another aspect1 which will perhaps render it more apparent4 ,suppose the weight ,w were actually divided into #ej equal parts1 or suppose it were a vessel of liquid weighing #ej ounces1 and containing #ej equal measures2 if these #ej measures were successively lifted through a height of #a foot2 the efforts necessary to accomplish this would be the same as those used to move the power ,p through #ej feet1 and it is obvious1 that the total expenditure of force would be the same as that which would be necessary to lift the entire contents of the vessel through #a foot4 ,when the nature and properties of the mechanic powers and other machines have been explained1 the force of these observations will be more distinctly perceived4 ,the effects of props and fixed points in sustaining a part of the weight1 and sometimes the whole1 both of the weight and power1 will then be manifest1 and every machine will furnish a verification of the remarkable proportion between the velocities of the weight and power1 which has enabled us to explain what might otherwise be paradoxical and difficult of comprehension4 7#bbg47 ,the most simple species of machines are those which are commonly denominated the ,,mechanic ,,powers4 ,these have been differently enumerated by different writers4 ,if1 however1 the object be to arrange in distinct classes1 and in the smallest possible number of them1 those machines which are alike in principle1 the mechanic powers may be reduced to three4 #a4 ,the lever4 #b4 ,the cord4 #c4 ,the inclined plane4 ,to one or other of these classes all simple machines whatever may be reduced1 and all complex machines may be resolved into simple elements which come under them4 7#bbh47 ,the first class includes every machine which is composed of a solid body revolving on a fixed axis1 although the name lever has been commonly confined to cases where the machine affects certain particular forms4 ,this is by far the most useful class of machines1 and will require in subsequent chapters very detailed development4 ,the general principle1 upon which equilibrium is established between the power and weight in machines of this class has been already explained in 7#ahc47 ,the power and weight are always supposed to be applied in directions at right angles to the axis4 ,if lines be drawn from the axis perpendicular to the directions of power and weight1 equilibrium will subsist1 provided the power multiplied by the perpendicular distance of its direction from the axis1 be equal to the weight multiplied by the perpendicular distance of its direction from the axis4 ,this is a principle to which we shall have occasion to refer in explaining the various machines of this class4 7#bbi47 ,if the moment of the power 7#ahd47 be greater than that of the weight1 the effect of the power will prevail over that of the weight1 and elevate it2 but if1 on the other hand1 the moment of the power be less than that of the weight1 the power will be insufficient to support the weight1 and will allow it to fall4 7#bcj47 ,the second class of simple machines includes all those cases in which force is transmitted by means of flexible threads1 ropes1 or chains4 ,the principle1 by which the effects of these machines are estimated1 is1 that the tension throughout the whole length of the same cord1 provided it be perfectly flexible1 and free from the effects of friction1 must be the same4 ,thus1 if a force acting at one end be balanced by a force acting at the other end1 however the cord may be bent1 or whatever course it may be compelled to take1 by any causes which may affect it between its ends1 these forces must be equal1 provided the cord be free to move over any obstacles which may deflect it4 ,within this class of machines are included all the various forms of pulleys4 7#bca47 ,the third class of simple machines includes all those cases in which the weight or resistance is supported or moved on a hard surface inclined to the vertical direction4 ,the effects of such machines are estimated by resolving the whole weight of the body into two elements by the parallelogram of forces4 ,one of these elements is perpendicular to the surface1 and supported by its resistance2 the other is parallel to the surface1 and supported by the power4 ,the proportion1 therefore1 of the power to the weight will always depend on the obliquity of the surface to the direction of the weight4 ,this will be easily understood by referring to what has been already explained in ,chapter ,,viii4 ,under this class of machines come the inclined plane1 commonly so called1 the wedge1 the screw1 and various others4 7#bcb47 ,in order to simplify the development of the elementary theory of machines1 it is expedient to omit the consideration of many circumstances1 of which1 however1 a strict account must be taken before any practically useful application of that theory can be attempted4 ,a machine1 as we must for the present contemplate it1 is a thing which can have no real or practical existence4 ,its various parts are considered to be free from friction3 all surfaces which move in contact are supposed to be infinitely smooth and polished4 ,the solid parts are conceived to be absolutely inflexible4 ,the weight and inertia of the machine itself are wholly neglected1 and we reason upon it as if it were divested of these qualities4 ,cords and ropes are supposed to have no stiffness1 to be infinitely flexible4 ,the machine1 when it moves1 is supposed to suffer no resistance from the atmosphere1 and to be in all respects circumstanced as if it were in vacuo4 ,it is scarcely necessary to state1 that1 all these suppositions being false1 none of the consequences deduced from them can be true4 ,nevertheless1 as it is the business of art to bring machines as near to this state of ideal perfection as possible1 the conclusions which are thus obtained1 though false in a strict sense1 yet deviate from the truth in but a small degree4 ,like the first outline of a picture1 they resemble in their general features that truth to which1 after many subsequent corrections1 they must finally approximate4 ,after a first approximation has been made on the several false suppositions which have been mentioned1 various effects1 which have been previously neglected1 are successively taken into account4 ,roughness1 rigidity1 imperfect flexibility1 the resistance of air and other fluids1 the effects of the weight and inertia of the machine1 are severally examined1 and their laws and properties detected4 ,the modifications and corrections1 thus suggested as necessary to be introduced into our former conclusions1 are applied1 and a second approximation1 but still only an approximation1 to truth is made4 ,for1 in investigating the laws which regulate the several effects just mentioned1 we are compelled to proceed upon a new group of false suppositions4 ,to determine the laws which regulate the friction of surfaces1 it is necessary to assume that every part of the surfaces of contact are uniformly rough2 that the solid parts which are imperfectly rigid1 and the cords which are imperfectly flexible1 are constituted throughout their entire dimensions of a uniform material2 so that the imperfection does not prevail more in one part than another4 ,thus1 all irregularity is left out of account1 and a general average of the effects taken4 ,it is obvious1 therefore1 that by these means we have still failed in obtaining a result exactly conformable to the real state of things2 but it is equally obvious1 that we have obtained one much more conformable to that state than had been previously accomplished1 and sufficiently near it for most practical purposes4 ,this apparent imperfection in our instruments and powers of investigation is not peculiar to mechanics3 it pervades all departments of natural science4 ,in astronomy1 the motions of the celestial bodies1 and their various changes and appearances as developed by theory1 assisted by observation and experience1 are only approximations to the real motions and appearances which take place in nature4 ,it is true that these approximations are susceptible of almost unlimited accuracy2 but still they are1 and ever will continue to be1 only approximations4 ,optics and all other branches of natural science are liable to the same observations4 ,,chap4 ,,xiii4 ,,of ,,the ,,lever4 7#bcc47 ,an inflexible1 straight bar1 turning on an axis1 is commonly called a lever4 ,the arms of the lever are those parts of the bar which extend on each side of the axis4 ,the axis is called the fulcrum or prop4 7#bcd47 ,levers are commonly divided into three kinds1 according to the relative positions of the power1 the weight1 and the fulcrum4 ,in a lever of the first kind1 as in fig4 #gh41 the fulcrum is between the power and weight4 ,in a lever of the second kind1 as in fig4 #gi41 the weight is between the fulcrum and power4 ,in a lever of the third kind1 as in fig4 #hj41 the power is between the fulcrum and weight4 7#bce47 ,in all these cases1 the power will sustain the weight in equilibrium1 provided its moment be equal to that of the weight4 7#ahd47 ,but the moment of the power is1 in this case1 equal to the product obtained by multiplying the power by its distance from the fulcrum2 and the moment of the weight by multiplying the weight by its distance from the fulcrum4 ,thus1 if the number of ounces in ,p1 being multiplied by the number of inches in ,p ,f1 be equal to the number of ounces in ,w1 multiplied by the number of inches in ,w ,f1 equilibrium will be established4 ,it is evident from this1 that as the distance of the power from the fulcrum increases in comparison to the distance of the weight from the fulcrum1 in the same degree exactly will the proportion of the power to the weight diminish4 ,in other words1 the proportion of the power to the weight will be always the same as that of their distances from the fulcrum taken in a reverse order4 ,in cases where a small power is required to sustain or elevate a great weight1 it will therefore be necessary either to remove the power to a great distance from the fulcrum1 or to bring the weight very near it4 7#bcf47 ,numerous examples of levers of the first kind may be given4 ,a crow-bar1 applied to elevate a stone or other weight1 is an instance4 ,the fulcrum is another stone placed near that which is to be raised1 and the power is the hand placed at the other end of the bar4 ,a handspike is a similar example4 ,a poker applied to raise fuel is a lever of the first kind1 the fulcrum being the bar of the grate4 ,scissors1 shears1 nippers1 pincers1 and other similar instruments are composed of two levers of the first kind2 the fulcrum being the joint or pivot1 and the weight the resistance of the substance to be cut or seized2 the power being the fingers applied at the other end of the levers4 ,the brake of a pump is a lever of the first kind2 the pump-rods and piston being the weight to be raised4 7#bcg47 ,examples of levers of the second kind1 though not so frequent as those just mentioned1 are not uncommon4 ,an oar is a lever of the second kind4 ,the reaction of the water against the blade is the fulcrum4 ,the boat is the weight1 and the hand of the boatman the power4 ,the rudder of a ship or boat is an example of this kind of lever1 and explained in a similar way4 ,the chipping knife is a lever of the second kind4 ,the end attached to the bench is the fulcrum1 and the weight the resistance of the substance to be cut1 placed beneath it4 ,a door moved upon its hinges is another example4 ,nut-crackers are two levers of the second kind2 the hinge which unites them being the fulcrum1 the resistance of the shell placed between them being the weight1 and the hand applied to the extremity being the power4 ,a wheelbarrow is a lever of the second kind2 the fulcrum being the point at which the wheel presses on the ground1 and the weight being that of the barrow and its load1 collected at their centre of gravity4 ,the same observation may be applied to all two-wheeled carriages1 which are partly sustained by the animal which draws them4 7#bch47 ,in a lever of the third kind1 the weight1 being more distant from the fulcrum than the power1 must be proportionably less than it4 ,in this instrument1 therefore1 the power acts upon the weight to a mechanical disadvantage1 inasmuch as a greater power is necessary to support or move the weight than would be required if the power were immediately applied to the weight1 without the intervention of a machine4 ,we shall1 however1 hereafter show that the advantage which is lost in force is gained in despatch1 and that in proportion as the weight is less than the power which moves it1 so will the speed of its motion be greater than that of the power4 ,hence a lever of the third kind is only used in cases where the exertion of great power is a consideration subordinate to those of rapidity and despatch4 ,the most striking example of levers of the third kind is found in the animal economy4 ,the limbs of animals are generally levers of this description4 ,the socket of the bone is the fulcrum2 a strong muscle attached to the bone near the socket is the power2 and the weight of the limb1 together with whatever resistance is opposed to its motion1 is the weight4 ,a slight contraction of the muscle in this case gives a considerable motion to the limb3 this effect is particularly conspicuous in the motion of the arms and legs in the human body2 a very inconsiderable contraction of the muscles at the shoulders and hips giving the sweep to the limbs from which the body derives so much activity4 ,the treddle of the turning lathe is a lever of the third kind4 ,the hinge which attaches it to the floor is the fulcrum1 the foot applied to it near the hinge is the power1 and the crank upon the axis of the fly-wheel1 with which its extremity is connected1 is the weight4 ,tongs are levers of this kind1 as also the shears used in shearing sheep4 ,in these cases the power is the hand placed immediately below the fulcrum or point where the two levers are connected4 7#bci47 ,when the power is said to support the weight by means of a lever or any other machine1 it is only meant that the power keeps the machine in equilibrium1 and thereby enables it to sustain the weight4 ,it is necessary to attend to this distinction1 to remove the difficulty which may arise from the paradox of a small power sustaining a great weight4 ,in a lever of the first kind1 the fulcrum ,f1 fig4 #gh41 or axis1 sustains the united forces of the power and weight4 ,in a lever of the second kind1 if the power be supposed to act over a wheel ,r1 fig4 #gi41 the fulcrum ,f sustains a pressure equal to the difference between the power and weight1 and the axis of the wheel ,r sustains a pressure equal to twice the power2 so that the total pressures on ,f and ,r are equivalent to the united forces of the power and weight4 ,in a lever of the third kind similar observations are applicable4 ,the wheel ,r1 fig4 #hj41 sustains a pressure equal to twice the power1 and the fulcrum ,f sustains a pressure equal to the difference between the power and weight4 ,these facts may be experimentally established by attaching a string to the lever immediately over the fulcrum1 and suspending the lever by that string from the arm of a balance4 ,the counterpoising weight1 when the fulcrum is removed1 will1 in the first case1 be equal to the sum of the weight and power1 and in the last two cases equal to their difference4 7#bdj47 ,we have hitherto omitted the consideration of the effect of the weight of the lever itself4 ,if the centre of gravity of the lever be in the vertical line through the axis1 the weight of the instrument will have no other effect than to increase the pressure on the axis by its own amount4 ,but if the centre of gravity be on the same side of the axis with the weight1 as at ,g1 it will oppose the effect of the power1 a certain part of which must therefore be allowed to support it4 ,to ascertain what part of the power is thus expended1 it is to be considered that the moment of the weight of the lever collected at ,g1 is found by multiplying that weight by the distance ,g ,f4 ,the moment of that part of the power which supports this must be equal to it2 therefore1 it is only necessary to find how much of the power multiplied by ,p ,f will be equal to the weight of the lever multiplied by ,g ,f4 ,this is a question in common arithmetic4 ,if the centre of gravity of the lever be at a different side of the axis from the weight1 as at ,g1 the weight of the instrument will co-operate with the power in sustaining the weight ,w4 ,to determine what portion of the weight ,w is thus sustained by the weight of the lever1 it is only necessary to find how much of ,w1 multiplied by the distance ,w ,f1 is equal to the weight of the lever multiplied by ,g ,f4 ,in these cases the pressure on the fulcrum1 as already estimated1 will always be increased by the weight of the lever4 7#bda47 ,the sense in which a small power is said to sustain a great weight1 and the manner of accomplishing this1 being explained1 we shall now consider how the power is applied in moving the weight4 ,let ,p ,w1 fig4 #ha41 be the places of the power and weight1 and ,f that of the fulcrum1 and let the power be depressed to ,p while the weight is raised to ,w4 ,the space ,p ,p evidently bears the same proportion to ,w ,w1 as the arm ,p ,f to ,w ,f4 ,thus if ,p ,f be ten times ,w ,f1 ,p ,p will be ten times ,w ,w4 ,a power of one pound at ,p being moved from ,p to ,p1 will carry a weight of ten pounds from ,w to ,w4 ,but in this case it ought not to be said1 that a lesser weight moves a greater1 for it is not difficult to show1 that the total expenditure of force in the motion of one pound from ,p to ,p is exactly the same as in the motion of ten pounds from ,w to ,w4 ,if the space ,p ,p be ten inches1 the space ,w ,w will be one inch4 ,a weight of one pound is therefore moved through ten successive inches1 and in each inch the force expended is that which would be sufficient to move one pound through one inch4 ,the total expenditure of force from ,p to ,p is ten times the force necessary to move one pound through one inch1 or what is the same1 it is that which would be necessary to move ten pounds through one inch4 ,but this is exactly what is accomplished by the opposite end ,w of the lever2 for the weight ,w is ten pounds1 and the space ,w ,w is one inch4 ,if the weight ,w of ten pounds could be conveniently divided into ten equal parts of one pound each1 each part might be separately raised through one inch1 without the intervention of the lever or any other machine4 ,in this case1 the same quantity of power would be expended1 and expended in the same manner as in the case just mentioned4 ,it is evident1 therefore1 that when a machine is applied to raise a weight or to overcome resistance1 as much force must be really used as if the power were immediately applied to the weight or resistance4 ,all that is accomplished by the machine is to enable the power to do that by a succession of distinct efforts which should be otherwise performed by a single effort4 ,these observations will be found to be applicable to all machines whatever4 7#bdb47 ,weighing machines of almost every kind1 whether used for commercial or philosophical purposes1 are varieties of the lever4 ,the common balance1 which1 of all weighing machines1 is the most perfect and best adapted for ordinary use1 whether in commerce or experimental philosophy1 is a lever with equal arms4 ,in the steel-yard one weight serves as a counterpoise and measure of others of different amount1 by receiving a leverage variable according to the varying amount of the weight against which it acts4 ,a detailed account of such instruments will be found in ,chapter ,,xxi4 7#bdc47 ,we have hitherto considered the power and weight as acting on the lever1 in directions perpendicular to its length and parallel to each other4 ,this does not always happen4 ,let ,a ,b1 fig4 #hc41 be a lever whose fulcrum is ,f1 and let ,a ,r be the direction of the power1 and ,b ,s the direction of the weight4 ,if the lines ,r ,a and ,s ,b be continued1 and perpendiculars ,f ,c and ,f ,d drawn from the fulcrum to those lines1 the moment of the power will be found by multiplying the power by the line ,f ,c1 and the moment of the weight by multiplying the weight by ,f ,d4 ,if these moments be equal1 the power will sustain the weight in equilibrium4 7#ahe74 ,it is evident1 that the same reasoning will be applicable when the arms of the lever are not in the same direction4 ,these arms may be of any figure or shape1 and may be placed relatively to each other in any position4 7#bdd47 ,in the rectangular lever the arms are perpendicular to each other1 and the fulcrum ,f1 fig4 #hd41 is at the right angle4 ,the moment of the power1 in this case1 is ,p multiplied by ,a ,f1 and that of the weight ,w multiplied by ,b ,f4 ,when the instrument is in equilibrium these moments must be equal4 ,when the hammer is used for drawing a nail1 it is a lever of this kind3 the claw of the hammer is the shorter arm2 the resistance of the nail is the weight2 and the hand applied to the handle the power4 7#bde47 ,when a beam rests on two props ,a ,b1 fig4 #he41 and supports1 at some intermediate place ,c1 a weight ,w1 this weight is distributed between the props in a manner which may be determined by the principles already explained4 ,if the pressure on the prop ,b be considered as a power sustaining the weight ,w1 by means of the lever of the second kind ,b ,a1 then this power multiplied by ,b ,a must be equal to the weight multiplied by ,c ,a4 ,hence the pressure on ,b will be the same fraction of the weight as the part ,a ,c is of ,a ,b4 ,in the same manner it may be proved1 that the pressure on ,a is the same fraction of the weight as ,b ,c is of ,b ,a4 ,thus1 if ,a ,c be one third1 and therefore ,b ,c two thirds of ,b ,a1 the pressure on ,b will be one third of the weight1 and the pressure on ,a two thirds of the weight4 ,it follows from this reasoning1 that if the weight be in the middle1 equally distant from ,b and ,a1 each prop will sustain half the weight4 ,the effect of the weight of the beam itself may be determined by considering it to be collected at its centre of gravity4 ,if this point1 therefore1 be equally distant from the props1 the weight of the beam will be equally distributed between them4 ,according to these principles1 the manner in which a load borne on poles between two bearers is distributed between them may be ascertained4 ,as the efforts of the bearers and the direction of the weight are always parallel2 the position of the poles relatively to the horizon makes no difference in the distribution of the weights between the bearers4 ,whether they ascend or descend1 or move on a level plane1 the weight will be similarly shared between them4 ,if the beam extend beyond the prop1 as in fig4 #hf41 and the weight be suspended at a point not placed between them1 the props must be applied at different sides of the beam4 ,the pressures which they sustain may be calculated in the same manner as in the former case4 ,the pressure of the prop ,b may be considered as a power sustaining the weight ,w by means of the lever ,b ,c4 ,hence1 the pressure of ,b1 multiplied by ,b ,a1 must be equal to the weight ,w multiplied by ,a ,c4 ,therefore1 the pressure on ,b bears the same proportion to the weight as ,a ,c does to ,a ,b4 ,in the same manner1 considering ,b as a fulcrum1 and the pressure of the prop ,a as the power1 it may be proved that the pressure of ,a bears the same proportion to the weight as the line ,b ,c does to ,a ,b4 ,it therefore appears1 that the pressure on the prop ,a is greater than the weight4 7#bdf47 ,when great power is required1 and it is inconvenient to construct a long lever1 a combination of levers may be used4 ,in fig4 #hg4 such a system of levers is represented1 consisting of three levers of the first kind4 ,the manner in which the effect of the power is transmitted to the weight may be investigated by considering the effect of each lever successively4 ,the power at ,p produces an upward force at ,p1 which bears to ,p the same proportion as ,p ,f to ,p ,f4 ,therefore1 the effect at ,p is as many times the power as the line ,p ,f is of ,p ,f4 ,thus1 if ,p ,f be ten times ,p ,f1 the upward force at ,p is ten times the power4 ,the arm ,p ,f of the second lever is pressed upwards by a force equal to ten times the power at ,p4 ,in the same manner this may be shown to produce an effect at ,p as many times greater than ,p as ,p ,f is greater than ,p ,f4 ,thus1 if ,p ,f be twelve times ,p ,f1 the effect at ,p will be twelve times that of ,p4 ,but this last was ten times the power1 and therefore the ,p will be one hundred and twenty times the power4 ,in the same manner it may be shown that the weight is as many times greater than the effect at ,p as ,p ,f is greater than ,w ,f4 ,if ,p ,f be five times ,w ,f1 the weight will be five times the effect at ,p4 ,but this effect is one hundred and twenty times the power1 and therefore the weight would be six hundred times the power4 ,in the same manner the effect of any compound system of levers may be ascertained by taking the proportion of the weight to the power in each lever separately1 and multiplying these numbers together4 ,in the example given1 these proportions are #aj1 #ab1 and #e1 which multiplied together give #fjj4 ,in fig4 #hg4 the levers composing the system are of the first kind2 but the principles of the calculation will not be altered if they be of the second or third kind1 or some of one kind and some of another4 7#bdg47 ,that number which expresses the proportion of the weight to the equilibrating power in any machine1 we shall call the power of the machine4 ,thus1 if1 in a lever1 a power of one pound support a weight of ten pounds1 the power of the machine is ten4 ,if a power of #blbs4 support a weight of #aalbs41 the power of the machine is #e-a_/b1 #b being contained in #aa #e-a_/b times4 7#bdh47 ,as the distances of the power and weight from the fulcrum of a lever may be varied at pleasure1 and any assigned proportion given to them1 a lever may always be conceived having a power equal to that of any given machine4 ,such a lever may be called1 in relation to that machine1 the equivalent lever4 ,as every complex machine consists of a number of simple machines acting one upon another1 and as each simple machine may be represented by an equivalent lever1 the complex machine will be represented by a compound system of equivalent levers4 ,from what has been proved in 7#bdf471 it therefore follows that the power of a complex machine may be calculated by multiplying together the powers of the several simple machines of which it is composed4 ,,chap4 ,,xiv4 ,,of ,,wheel-,,work4 7#bdi47 ,when a lever is applied to raise a weight1 or overcome a resistance1 the space through which it acts at any one time is small1 and the work must be accomplished by a succession of short and intermitting efforts4 ,in fig4 #ha41 after the weight has been raised from ,w to ,w1 the lever must again return to its first position1 to repeat the action4 ,during this return the motion of the weight is suspended1 and it will fall downwards unless some provision be made to sustain it4 ,the common lever is1 therefore1 only used in cases where weights are required to be raised through small spaces1 and under these circumstances its great simplicity strongly recommends it4 ,but where a continuous motion is to be produced1 as in raising ore from the mine1 or in weighing the anchor of a vessel1 some contrivance must be adopted to remove the intermitting action of the lever1 and render it continual4 ,the various forms given to the lever1 with a view to accomplish this1 are generally denominated the wheel and axle4 ,illustration3 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,in fig4 #hh41 ,a ,b is a horizontal axle1 which rests in pivots at its extremities1 or is supported in gudgeons1 and capable of revolving4 ,round this axis a rope is coiled1 which sustains the weight ,w4 ,on the same axis a wheel ,c is fixed1 round which a rope is coiled in a contrary direction1 to which is appended the power ,p4 ,the moment of the power is found by multiplying it by the radius of a wheel1 and the moment of the weight1 by multiplying it by the radius of its axle4 ,if these moments be equal 7#ahe471 the machine will be in equilibrium4 ,whence it appears that the power of the machine 7#bdg47 is expressed by the proportion which the radius of the wheel bears to the radius of the axle2 or1 what is the same1 of the diameter of the wheel to the diameter of the axle4 ,this principle is applicable to the wheel and axle in every variety of form under which it can be presented4 7#bej47 ,it is evident that as the power descends continually1 and the rope is uncoiled from the wheel1 the weight will be raised continually1 the rope by which it is suspended being at the same time coiled upon the axle4 ,when the machine is in equilibrium1 the forces of both the weight and power are sustained by the axle1 and distributed between its props1 in the manner explained in 7#bde47 ,when the machine is applied to raise a weight1 the velocity with which the power moves is as many times greater than that with which the weight rises1 as the weight itself is greater than the power4 ,this is a principle which has already been noticed1 and which is common to all machines whatsoever4 ,it may hence be proved1 that in the elevation of the weight a quantity of power is expended equal to that which would be necessary to elevate the weight if the power were immediately applied to it1 without the intervention of any machine4 ,this has been explained in the case of the lever in 7#bda471 and may be explained in the present instance in nearly the same words4 ,in one revolution of the machine the length of rope uncoiled from the wheel is equal to the circumference of the wheel1 and through this space the power must therefore move4 ,at the same time the length of rope coiled upon the axle is equal to the circumference of the axle1 and through this space the weight must be raised4 ,the spaces1 therefore1 through which the power and weight move in the same time1 are in the proportion of the circumferences of the wheel and axle2 but these circumferences are in the same proportion as their diameters4 ,therefore the velocity of the power will bear to the velocity of the weight the same proportion as the diameter of the wheel bears to the diameter of the axle1 or1 what is the same1 as the weight bears to the power 7#bdi74 7#bea47 ,we have here omitted the consideration of the thickness of the rope4 ,when this is considered1 the force must be conceived as acting in the direction of the centre of the rope1 and therefore the thickness of the rope which supports the power ought to be added to the diameter of the wheel1 and the thickness of the rope which supports the weight to the diameter of the axle4 ,it is the more necessary to attend to this circumstance1 as the strength of the rope necessary to support the weight causes its thickness to bear a considerable proportion to the diameter of the axle2 while the rope which sustains the power not requiring the same strength1 and being applied to a larger circle1 bears a very inconsiderable proportion to its diameter4 7#beb47 ,in numerous forms of the wheel and axle1 the weight or resistance is applied by a rope coiled upon the axle2 but the manner in which the power is applied is very various1 and not often by means of a rope4 ,the circumference of a wheel sometimes carries projecting pins1 as represented in fig4 #hh41 to which the hand is applied to turn the machine4 ,an instance of this occurs in the wheel used in the steerage of a vessel4 ,in the common windlass1 the power is applied by means of a winch1 which is a rectangular lever1 as represented in fig4 #hi4 ,the arm ,b ,c of the winch represents the radius of the wheel1 and the power is applied to ,c ,d at right angles to ,b ,c4 ,in some cases no wheel is attached to the axle2 but it is pierced with holes directed towards its centre1 in which long levers are incessantly inserted1 and a continuous action produced by several men working at the same time2 so that while some are transferring the levers from hole to hole1 others are working the windlass4 ,the axle is sometimes placed in a vertical position1 the wheel or levers being moved horizontally4 ,the capstan is an example of this3 a vertical axis is fixed in the deck of the ship2 the circumference is pierced with holes presented towards its centre4 ,these holes receive long levers1 as represented in fig4 #ij4 ,the men who work the capstan walk continually round the axle1 pressing forward the levers near their extremities4 ,in some cases the wheel is turned by the weight of animals placed at its circumference1 who move forward as fast as the wheel descends1 so as to maintain their position continually at the extremity of the horizontal diameter4 ,the treadmill1 fig4 #ia41 and certain cranes1 such as fig4 #ib41 are examples of this4 ,in water-wheels1 the power is the weight of water contained in buckets at the circumference1 as in fig4 #ic41 which is called an over-shot wheel3 and sometimes by the impulse of water against float-boards at the circumference1 as in the under-shot wheel1 fig4 #id4 ,both these principles act in the breast-wheel1 fig4 #ie4 ,in the paddle-wheel of a steam-boat1 the power is the resistance which the water offers to the motion of the paddle-boards4 ,in windmills1 the power is the force of the wind acting on various parts of the arms1 and may be considered as different powers simultaneously acting on different wheels having the same axle4 7#bec47 ,in most cases in which the wheel and axle is used1 the action of the power is liable to occasional suspension or intermission1 in which case some contrivance is necessary to prevent the recoil of the weight4 ,a ratchet wheel ,r1 fig4 #hh41 is provided for this purpose1 which is a contrivance which permits the wheel to turn in one direction2 but a catch which falls between the teeth of a fixed wheel prevents its motion in the other direction4 ,the effect of the power or weight is sometimes transmitted to the wheel or axle by means of a straight bar1 on the edge of which teeth are raised1 which engage themselves in corresponding teeth on the wheel or axle4 ,such a bar is called a rack2 and an instance of its use may be observed in the manner of working the pistons of an air-pump4 7#bed47 ,the power of the wheel and axle being expressed by the number of times the diameter of the axle is contained in that of the wheel1 there are obviously only two ways by which this power may be increased2 viz4 either by increasing the diameter of the wheel1 or diminishing that of the axle4 ,in cases where great power is required1 each of these methods is attended with practical inconvenience and difficulty4 ,if the diameter of the wheel be considerably enlarged1 the machine will become unwieldy1 and the power will work through an unmanageable space4 ,if1 on the other hand1 the power of the machine be increased by reducing the thickness of the axle1 the strength of the axle will become insufficient for the support of that weight1 the magnitude of which had rendered the increase of the power of the machine necessary4 ,to combine the requisite strength with moderate dimensions and great mechanical power is1 therefore1 impracticable in the ordinary form of the wheel and axle4 ,this has1 however1 been accomplished by giving different thicknesses to different parts of the axle1 and carrying a rope1 which is coiled on the thinner part1 through a wheel attached to the weight1 and coiling it in the opposite direction on the thicker part1 as in fig4 #if4 ,to investigate the proportion of the power to the weight in this case1 let fig4 #ig4 represent a section of the apparatus at right angles to the axis4 ,the weight is equally suspended by the two parts of the rope1 ,s and ,s1 and therefore each part is stretched by a force equal to half the weight4 ,the moment of the force1 which stretches the rope ,s1 is half the weight multiplied by the radius of the thinner part of the axle4 ,this force being at the same side of the centre with the power1 co-operates with it in supporting the force which stretches ,s1 and which acts at the other side of the centre4 ,by the principle established in 7#ahe471 the moments of ,p and ,s must be equal to that of ,s2 and therefore if ,p be multiplied by the radius of the wheel1 and added to half the weight multiplied by the radius of the thinner part of the axle1 we must obtain a sum equal to half the weight multiplied by the radius of the thicker part of the axle4 ,hence it is easy to perceive1 that the power multiplied by the radius of the wheel is equal to half the weight multiplied by the difference of the radii of the thicker and thinner parts of the axle2 or1 what is the same1 the power multiplied by the diameter of the wheel1 is equal to the weight multiplied by half the difference of the diameters of the thinner and thicker parts of the axle4 ,a wheel and axle constructed in this manner is equivalent to an ordinary one1 in which the wheel has the same diameter1 and whose axle has a diameter equal to half the difference of the diameters of the thicker and thinner parts4 ,the power of the machine is expressed by the proportion which the diameter of the wheel bears to half the difference of these diameters2 and therefore this power1 when the diameter of the wheel is given1 does not1 as in the ordinary wheel and axle1 depend on the smallness of the axle1 but on the smallness of the difference of the thinner and thicker parts of it4 ,the axle may1 therefore1 be constructed of such a thickness as to give it all the requisite strength1 and yet the difference of the diameters of its different parts may be so small as to give it all the requisite power4 7#bee47 ,it often happens that a varying weight is to be raised1 or resistance overcome by a uniform power4 ,if1 in such a case1 the weight be raised by a rope coiled upon a uniform axle1 the action of the power would not be uniform1 but would vary with the weight4 ,it is1 however1 in most cases desirable or necessary that the weight or resistance1 even though it vary1 shall be moved uniformly4 ,this will be accomplished if by any means the leverage of the weight is made to increase in the same proportion as the weight diminishes1 and to diminish in the same proportion as the weight increases3 for in that case the moment of the weight will never vary1 whatever it gains by the increase of weight being lost by the diminished leverage1 and whatever it loses by the diminished weight being gained by the increased leverage4 ,an axle1 the surface of which is curved in such a manner1 that the thickness on which the rope is coiled continually increased or diminishes in the same proportion as the weight or resistance diminishes or increases1 will produce this effect4 ,it is obvious that all that has been said respecting a variable weight or resistance1 is also applicable to a variable power1 which1 therefore1 may1 by the same means1 be made to produce a uniform effect4 ,an instance of this occurs in a watch1 which is moved by a spiral spring4 ,when the watch has been wound up1 this spring acts with its greatest intensity1 and as the watch goes down1 the elastic force of the spring gradually loses its energy4 ,this spring is connected by a chain with an axle of varying thickness1 called a fusee4 ,when the spring is at its greatest intensity1 the chain acts upon the thinnest part of the fusee1 and as it is uncoiled it acts upon a part of the fusee which is continually increasing in thickness1 the spring at the same time losing its elastic power in exactly the same proportion4 ,a representation of the fusee1 and the cylindrical box which contains the spring1 is given in fig4 #ih41 and of the spring itself in fig4 #ii4 7#bef47 ,when great power is required1 wheels and axles may be combined in a manner analogous to a compound system of levers1 explained in 7#bdf47 ,in this case the power acts on the circumference of the first wheel1 and its effect is transmitted to the circumference of the first axle4 ,that circumference is placed in connection with the circumference of the second wheel1 and the effect is thereby transmitted to the circumference of the second axle1 and so on4 ,it is obvious from what was proved in 7#bdh471 that the power of such a combination of wheels and axles will be found by multiplying together the powers of the several wheels of which it is composed4 ,it is sometimes convenient to compute this power by numbers expressing the proportions of the circumferences or diameters of the several wheels1 to the circumferences or diameters of the several axles respectively4 ,this computation is made by first multiplying the numbers together which express the circumferences or diameters of the wheels1 and then multiplying together the numbers which express the circumferences or diameters of the several axles4 ,the proportion of the two products will express the power of the machine4 ,thus1 if the circumferences or diameters be as the numbers #aj1 #ad1 and #ae1 their product will be #bajj2 and if the circumferences or diameters of the axles be expressed by the numbers #c1 #d1 and #e1 their product will be #fj1 and the power of the machine will be expressed by the proportion of #bajj and #fj1 or #ce to #a4 ,illustration3 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 7#beg47 ,the manner in which the circumferences of the axles act upon the circumferences of the wheels in compound wheel-work is various4 ,sometimes a strap or cord is applied to a groove in the circumference of the axle1 and carried round a similar groove in the circumference of the succeeding wheel4 ,the friction of this cord or strap with the groove is sufficient to prevent its sliding and to communicate the force from the axle to the wheel1 or vice versa4 ,this method of connecting wheel-work is represented in fig4 #ajj4 ,numerous examples of wheels and axles driven by straps or cords occur in machinery applied to almost every department of the arts and manufactures4 ,in the turning lathe1 the wheel worked by the treddle is connected with the mandrel by a catgut cord passing through grooves in the wheel and axle4 ,in all great factories1 revolving shafts are carried along the apartments1 on which1 at certain intervals1 straps are attached passing round their circumferences and carried round the wheels which give motion to the several machines4 ,if the wheels1 connected by straps or cords1 are required to revolve in the same direction1 these cords are arranged as in fig4 #ajj42 but if they are required to revolve in contrary directions1 they are applied as in fig4 #aja4 ,one of the chief advantages of the method of transmitting motion between wheels and axles by straps or cords1 is that the wheel and axle may be placed at any distance from each other which may be found convenient1 and may be made to turn either in the same or contrary directions4 7#beh47 ,when the circumference of the wheel acts immediately on the circumference of the succeeding axle1 some means must necessarily be adopted to prevent the wheel from moving in contact with the axle without compelling the latter to turn4 ,if the surfaces of both were perfectly smooth1 so that all friction were removed1 it is obvious that either would slide over the surface of the other1 without communicating motion to it4 ,but1 on the other hand1 if there were any asperities1 however small1 upon these surfaces1 they would become mutually inserted among each other1 and neither the wheel nor axle could move without causing the asperities with which its edge is studded to encounter those asperities which project from the surface of the other2 and thus1 until these projections should be broken off1 both wheel and axle must be moved at the same time4 ,it is on this account that if the surfaces of the wheels and axles are by any means rendered rough1 and pressed together with sufficient force1 the motion of either will turn the other1 provided the load or resistance be not greater than the force necessary to break off these small projections which produce the friction4 ,in cases where great power is not required1 motion is communicated in this way through a train of wheel-work1 by rendering the surface of the wheel and axle rough1 either by facing them with buff leather1 or with wood cut across the grain4 ,this method is sometimes used in spinning machinery1 where one large buffed wheel1 placed in a horizontal position1 revolves in contact with several small buffed rollers1 each roller communicating motion to a spindle4 ,the position of the wheel ,w1 and the rollers ,r ,r1 c41 are represented in fig4 #ajb4 ,each roller can be thrown out of contact with the wheel1 and restored to it at pleasure4 ,the communication of motion between wheels and axles by friction has the advantage of great smoothness and evenness1 and of proceeding with little noise2 but this method can only be used in cases where the resistance is not very considerable1 and therefore is seldom adopted in works on a large scale4 ,dr4 ,gregory mentions an instance of a saw mill at ,southampton1 where the wheels act upon each other by the contact of the end grain of wood4 ,the machinery makes very little noise1 and wears very well1 having been used not less than #bj years4 7#bei47 ,the most usual method of transmitting motion through a train of wheel-work is by the formation of teeth upon their circumferences1 so that these indentures of each wheel fall between the corresponding ones of that in which it works1 and ensure the action so long as the strain is not so great as to fracture the tooth4 ,in the formation of teeth very minute attention must be given to their figure1 in order that the motion may be communicated from wheel to wheel with smoothness and uniformity4 ,this can only be accomplished by shaping the teeth according to curves of a peculiar kind1 which mathematicians have invented1 and assigned rules for drawing4 ,the ill consequences of neglecting this will be very apparent1 by considering the nature of the action which would be produced if the teeth were formed of square projecting pins1 as in fig4 #ajc4 ,when the tooth ,a comes into contact with ,b1 it acts obliquely upon it1 and1 as it moves1 the corner of ,b slides upon the plane surface of ,a in such a manner as to produce much friction1 and to grind away the side of ,a and the end of ,b4 ,as they approach the position ,c ,d1 they sustain a jolt the moment their surfaces come into full contact2 and after passing the position of ,c ,d1 the same scraping and grinding effect is produced in the opposite direction1 until by the revolution of the wheels the teeth become disengaged4 ,these effects are avoided by giving to the teeth the curved forms represented in fig4 #ajd4 ,by such means the surfaces of the teeth roll upon each other with very inconsiderable friction1 and the direction in which the pressure is excited is always that of a line ,m ,n1 touching the two wheels1 and at right angles to the radii4 ,thus the pressure being always the same1 and acting with the same leverage1 produces a uniform effect4 7#bfj47 ,when wheels work together1 their teeth must necessarily be of the same size1 and therefore the proportion of their circumferences may always be estimated by the number of teeth which they carry4 ,hence it follows1 that in computing the power of compound wheel-work1 the number of teeth may always be used to express the circumferences respectively1 or the diameters which are proportional to these circumferences4 ,when teeth are raised upon an axle1 it is generally called a pinion1 and in that case the teeth are called leaves4 ,the rule for computing the train of wheel-work given in 7#bef47 will be expressed as follows3 when the wheel and axle carry teeth1 multiply together the number of teeth in each of the wheels1 and next the number of leaves in each of the pinions2 the proportion of the two products will express the power of the machine4 ,if some of the wheels and axles carry teeth1 and others not1 this computation may be made by using for those circumferences which do not bear teeth the number of teeth which would fill them4 ,fig4 #aje4 represents a train of three wheels and pinions4 ,the wheel ,f which bears the power1 and the axle which bears the weight1 have no teeth2 but it is easy to find the number of teeth which they would carry4 7#bfa47 ,it is evident that each pinion revolves much more frequently in a given time than the wheel which it drives4 ,thus1 if the pinion ,c be furnished with ten teeth1 and the wheel ,e1 which it drives1 have sixty teeth1 the pinion ,c must turn six times1 in order to turn the wheel ,e once round4 ,the velocities of revolution of every wheel and pinion which work in one another will therefore have the same proportion as their number of teeth taken in a reverse order1 and by this means the relative velocity of wheels and pinions may be determined according to any proposed rate4 ,wheel-work1 like all other machinery1 is used to transmit and modify force in every department of the arts and manufactures2 but it is also used in cases where motion alone1 and not force1 is the object to be attained4 ,the most remarkable example of this occurs in watch and clock-work1 where the object is merely to produce uniform motions of rotation1 having certain proportions1 and without any regard to the elevation of weights1 or the overcoming of resistances4 7#bfb47 ,a crane is an example of combination of wheel-work used for the purpose of raising or lowering great weights4 ,fig4 #ajf4 represents a machine of this kind4 ,a ,b is a strong vertical beam1 resting on a pivot1 and secured in its position by beams in the floor4 ,it is capable1 however1 of turning on its axis1 being confined between rollers attached to the beams and fixed in the floor4 ,c ,d is a projecting arm called a gib1 formed of beams which are mortised into ,a ,b4 ,the wheel-work is mounted in two cast-iron crosses1 bolted on each side of the beams1 one of which appears at ,e ,f ,g ,h4 ,the winch at which the power is applied is at ,i4 ,this carries a pinion immediately behind ,h4 ,this pinion works in a wheel ,k1 which carries another pinion upon its axle4 ,this last pinion works in a larger wheel ,l1 which carries upon its axis a barrel ,m1 on which a chain or rope is coiled4 ,the chain passes over a pulley ,d at the top of the gib4 ,at the end of the chain a hook ,o is attached1 to support the weight ,w4 ,during the elevation of the weight it is convenient that its recoil should be hindered in case of any occasional suspension of the power4 ,this is accomplished by a ratchet wheel attached to the barrel ,m1 as explained in 7#bec472 but when the weight ,w is to be lowered1 the catch must be removed from this ratchet wheel4 ,in this case the too rapid descent of the weight is in some cases checked by pressure excited on some part of the wheel-work1 so as to produce sufficient friction to retard the descent in any required degree1 or even to suspend it1 if necessary4 ,the vertical beam at ,b resting on a pivot1 and being fixed between rollers1 allows the gib to be turned round in any direction2 so that a weight raised from one side of the crane may be carried round1 and deposited on another side1 at any distance within the range of the gib4 ,thus1 if a crane be placed upon a wharf near a vessel1 weights may be raised1 and when elevated1 the gib may be turned round so as to let them descend into the hold4 ,the power of this machine may be computed upon the principles already explained4 ,the magnitude of the circle1 in which the power at ,i moves1 may be determined by the radius of the winch1 and therefore the number of teeth which a wheel of that size would carry may be found4 ,in like manner we may determine the number of leaves in a pinion whose magnitude would be equal to the barrel ,m4 ,let the first number be multiplied by the number of teeth in the wheel ,k1 and that product by the number of teeth in the wheel ,l4 ,next let the number of leaves in the pinion ,h be multiplied by the number of leaves in the pinion attached to the axle of the wheel ,k1 and let that product be multiplied by the number of leaves in a pinion1 whose diameter is equal to that of the barrel ,m4 ,these two products will express the power of the machine4 7#bfc47 ,toothed wheels are of three kinds1 distinguished by the position which the teeth bear with respect to the axis of the wheel4 ,when they are raised upon the edge of the wheel as in fig4 #aje41 they are called spur wheels1 or spur gear4 ,when they are raised parallel to the axis1 as in fig4 #ajg41 it is called a crown wheel4 ,when the teeth are raised on a surface inclined to the plane of the wheel1 as in fig4 #ajh41 they are called bevelled wheels4 ,illustration3 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,if a motion round one axis is to be communicated to another axis parallel to it1 spur gear is generally used4 ,thus1 in fig4 #aje41 the three axes are parallel to each other4 ,if a motion round one axis is to be communicated to another at right angles to it1 a crown wheel1 working in a spur pinion1 as in fig4 #ajg41 will serve4 ,or the same object may be obtained by two bevelled wheels1 as in fig4 #ajh4 ,if a motion round one axis is required to be communicated to another inclined to it at any proposed angle1 two bevelled wheels can always be used4 ,in fig4 #aji4 let ,a ,b and ,a ,c be the two axles2 two bevelled wheels1 such as ,d ,e and ,e ,f1 on these axles will transmit the motion or rotation from one to the other1 and the relative velocity may1 as usual1 be regulated by the proportional magnitude of the wheels4 7#bfd47 ,in order to equalise the wear of the teeth of a wheel and pinion1 which work in one another1 it is necessary that every leaf of the pinion should work in succession through every tooth of the wheel1 and not continually act upon the same set of teeth4 ,if the teeth could be accurately shaped according to mathematical principles1 and the materials of which they are formed be perfectly uniform1 this precaution would be less necessary2 but as slight inequalities1 both of material and form1 must necessarily exist1 the effects of these should be as far as possible equalised1 by distributing them through every part of the wheel4 ,for this purpose it is usual1 especially in mill-work1 where considerable force is used1 so to regulate the proportion of the number of teeth in the wheel and pinion1 that the same leaf of the pinion shall not be engaged twice with any one tooth of the wheel1 until after the action of a number of teeth1 expressed by the product of the number of teeth in the wheel and pinion4 ,let us suppose that the pinion contains ten leaves1 which we shall denominate by the numbers #a1 #b1 #c1 c41 and that the wheel contains #fj teeth similarly denominated4 ,at the commencement of the motion suppose the leaf #a of the pinion engages the tooth #a of the wheel2 then after one revolution the leaf #a of the pinion will engage the tooth #aa of the wheel1 and after two revolutions the leaf #a of the pinion will engage the tooth #ba of the wheel2 and in like manner1 after #c1 #d1 and #e revolutions of the pinion1 the leaf #a will engage successively the teeth #ca1 #da1 and #ea of the wheel4 ,after the sixth revolution1 the leaf #a of the pinion will again engage the tooth #a of the wheel4 ,thus it is evident1 that in the case here supposed the leaf #a of the pinion will continually be engaged with the teeth #a1 #aa1 #ba1 #ca1 #da1 and #ea of the wheel1 and no others4 ,the like may be said of every leaf of the pinion4 ,thus the leaf #b of the pinion will be successively engaged with the teeth #b1 #ab1 #bb1 #cb1 #db1 and #eb of the wheel1 and no others4 ,any accidental inequalities of these teeth will therefore continually act upon each other1 until the circumference of the wheel be divided into parts of ten teeth each1 unequally worn4 ,this effect would be avoided by giving either the wheel or pinion one tooth more or one tooth less4 ,thus1 suppose the wheel1 instead of having sixty teeth1 had sixty-one1 then after six revolutions of the pinion the leaf #a of the pinion would be engaged with the tooth #fa of the wheel2 and after one revolution of the wheel1 the leaf #b of the pinion would be engaged with the tooth #a of the wheel4 ,thus1 during the first revolution of the wheel the leaf #a of the pinion would be successively engaged with the teeth #a1 #aa1 #ba1 #ca1 #da1 #ea1 and #fa of the wheel3 at the commencement of the second revolution of the wheel the leaf #b of the pinion would be engaged with the tooth #a of the wheel2 and during the second revolution of the wheel the leaf #a of the pinion would be successively engaged with the teeth #aj1 #bj1 #cj1 #dj1 #ej1 and #fj of the wheel4 ,in the same manner it may be shown1 that in the third revolution of the wheel the leaf #a of the pinion would be successively engaged with the teeth #i1 #ai1 #bi1 #ci1 #di1 and #ei of the wheel3 during the fourth revolution of the wheel the leaf #a of the pinion would be successively engaged with the teeth #h1 #ah1 #bh1 #ch1 #dh1 and #eh of the wheel4 ,by continuing this reasoning it will appear1 that during the tenth revolution of the wheel the leaf #a of the pinion will be engaged successively with the teeth #b1 #ab1 #bb1 #cb1 #db1 and #eb of the wheel4 ,at the commencement of the eleventh revolution of the wheel the leaf #a of the pinion will be engaged with the tooth #a of the wheel1 as at the beginning of the motion4 ,it is evident1 therefore1 that during the first ten revolutions of the wheel each leaf of the pinion has been successively engaged with every tooth of the wheel1 and that during these ten revolutions the pinion has revolved sixty-one times4 ,thus the leaves of the pinion have acted six hundred and ten times upon the teeth of the wheel1 before two teeth can have acted twice upon each other4 ,the odd tooth which produces this effect is called by millwrights the hunting cog4 7#bfe47 ,the most familiar case in which wheel-work is used to produce and regulate motion merely1 without any reference to weights to be raised or resistances to be overcome1 is that of chronometers4 ,in watch and clock work the object is to cause a wheel to revolve with a uniform velocity1 and at a certain rate4 ,the motion of this wheel is indicated by an index or hand placed upon its axis1 and carried round with it4 ,in proportion to the length of the hand the circle over which its extremity plays is enlarged1 and its motion becomes more perceptible4 ,this circle is divided1 so that very small fractions of a revolution of the hand may be accurately observed4 ,in most chronometers it is required to give motion to two hands1 and sometimes to three4 ,these motions proceed at different rates1 according to the subdivisions of time generally adopted4 ,one wheel revolves in a minute1 bearing a hand which plays round a circle divided into sixty equal parts2 the motion of the hand over each part indicating one second1 and a complete revolution of the hand being performed in one minute4 ,another wheel revolves once1 while the former revolves sixty times2 consequently the hand carried by this wheel revolves once in sixty minutes1 or one hour4 ,the circle on which it plays is1 like the former1 divided into sixty equal parts1 and the motion of the hand over each division is performed in one minute4 ,this is generally called the minute hand1 and the former the second hand4 ,a third wheel revolves once1 while that which carries the minute hand revolves twelve times2 consequently this last wheel1 which carries the hour hand1 revolves at a rate twelve times less than that of the minute hand1 and therefore seven hundred and twenty times less than the second hand4 ,we shall now endeavour to explain the manner in which these motions are produced and regulated4 ,let ,a1 ,b1 ,c1 ,d1 ,e1 fig4 #aaj41 represent a train of wheels1 and a1 b1 c1 d represent their pinions1 e being a cylinder on the axis of the wheel ,e1 round which a rope is coiled1 sustaining a weight ,w4 ,let the effect of this weight transmitted through the train of wheels be opposed by a power ,p acting upon the wheel ,a1 and let this power be supposed to be of such a nature as to cause the weight ,w to descend with a uniform velocity1 and at any proposed rate4 ,the wheel ,e carries on its circumference eighty-four teeth4 ,the wheel ,d carries eighty teeth2 the wheel ,c is also furnished with eighty teeth1 and the wheel ,b with seventy-five4 ,the pinions d and c are each furnished with twelve leaves1 and the pinions b and a with ten4 ,if the power at ,p be so regulated as to allow the wheel ,a to revolve once in a minute1 with a uniform velocity1 a hand attached to the axis of this wheel will serve as the second hand4 ,the pinion a carrying ten teeth must revolve seven times and a half to produce one revolution of ,b1 consequently fifteen revolutions of the wheel ,a will produce two revolutions of the wheel ,b2 the wheel ,b1 therefore1 revolves twice in fifteen minutes4 ,the pinion b must revolve eight times to produce one revolution of the wheel ,c1 and therefore the wheel ,c must revolve once in four quarters of an hour1 or in one hour4 ,if a hand be attached to the axis of this wheel1 it will have the motion necessary for the minute hand4 ,the pinion c must revolve six times and two thirds to produce one revolution of the wheel ,d1 and therefore this wheel must revolve once in six hours and two thirds4 ,the pinion d revolves seven times for one revolution of the wheel ,e1 and therefore the wheel ,e will revolve once in forty-six hours and two thirds4 ,on the axis of the wheel ,c a second pinion may be placed1 furnished with seven leaves1 which may lead a wheel of eighty-four teeth1 so that this wheel shall turn once during twelve turns of the wheel ,c4 ,if a hand be fixed upon the axis1 this hand will revolve once for twelve revolutions of the minute hand fixed upon the axis of the wheel ,c2 that is1 it will revolve once in twelve hours4 ,if it play upon a dial divided into twelve equal parts1 it will move over each part in an hour1 and will serve the purpose of the hour hand of the chronometer4 ,we have here supposed that the second hand1 the minute hand1 and the hour hand move on separate dials4 ,this1 however1 is not necessary4 ,the axis of the hour hand is commonly a tube1 inclosing within it that of the minute hand1 so that the same dial serves for both4 ,the second hand1 however1 is generally furnished with a separate dial4 7#bff47 ,we shall now explain the manner in which a power is applied to the wheel ,a1 so as to regulate and equalise the effect of the weight ,w4 ,suppose the wheel ,a furnished with thirty teeth1 as in fig4 #aaa42 if nothing check the motion1 the weight ,w would descend with an accelerated velocity1 and would communicate an accelerated motion to the wheel ,a4 ,this effect1 however1 is interrupted by the following contrivance3--,l ,m is a pendulum vibrating on the centre ,l1 and so regulated that the time of its oscillation is one second4 ,the pallets ,i and ,k are connected with the pendulum1 so as to oscillate with it4 ,in the position of the pendulum represented in the figure1 the pallet ,i stops the motion of the wheel ,a1 and entirely suspends the action of the weight ,w1 fig4 #aaj41 so that for a moment the entire machine is motionless4 ,the weight ,m1 however1 falls by its gravity towards the lowest position1 and disengages the pallet ,i from the tooth of the wheel4 ,the weight ,w begins then to take effect1 and the wheel ,a turns from ,a towards ,b4 ,meanwhile the pendulum ,m oscillates to the other side1 and the pallet ,k falls under a tooth of the wheel ,a1 and checks for a moment its further motion4 ,on the returning vibration the pallet ,k becomes again disengaged1 and allows the tooth of the wheel to escape1 and by the influence of the weight ,w another tooth passes before the motion of the wheel ,a is again checked by the interposition of the pallet ,i4 ,from this explanation it will appear that1 in two vibrations of the pendulum1 one tooth of the wheel ,a passes the pallet ,i1 and therefore1 if the wheel ,a be furnished with #cj teeth1 it will be allowed to make one revolution during #fj vibrations of the pendulum4 ,if1 therefore1 the pendulum be regulated so as to vibrate seconds1 this wheel will revolve once in a minute4 ,from the action of the pallets in checking the motion of the wheel ,a1 and allowing its teeth alternately to escape1 this has been called the escapement wheel2 and the wheel and pallets together are generally called the escapement1 or 'scapement4 ,we have already explained1 that by reason of the friction on the points of support1 and other causes1 the swing of the pendulum would gradually diminish1 and its vibration at length cease4 ,this1 however1 is prevented by the action of the teeth of the scapement wheel upon the pallets1 which is just sufficient to communicate that quantity of force to the pendulum which is necessary to counteract the retarding effects1 and to maintain its motion4 ,it thus appears1 that although the effect of the gravity of the weight ,w in giving motion to the machine is at intervals suspended1 yet this part of the force is not lost1 being1 during these intervals1 employed in giving to the pendulum all that motion which it would lose by the resistances to which it is inevitably exposed4 ,in stationary clocks1 and in other cases in which the bulk of the machine is not an objection1 a descending weight is used as the moving power4 ,but in watches and portable chronometers1 this would be attended with evident inconvenience4 ,in such cases1 a spiral spring1 called the mainspring1 is the moving power4 ,the manner in which this spring communicates rotation to an axis1 and the ingenious method of equalising the effect of its variable elasticity by giving to it a leverage1 which increases as the elastic force diminishes1 have been already explained4 7#bee47 ,a similar objection lies against the use of a pendulum in portable chronometers4 ,a spiral spring of a similar kind1 but infinitely more delicate1 called a hair spring1 is substituted in its place4 ,this spring is connected with a nicely-balanced wheel1 called the balance wheel1 which plays in pivots4 ,when this wheel is turned to a certain extent in one direction1 the hair spring is coiled up1 and its elasticity causes the wheel to recoil1 and return to a position in which the energy of the spring acts in the opposite direction4 ,the balance wheel then returns1 and continually vibrates in the same manner4 ,the axis of this wheel is furnished with pallets similar to those of the pendulum1 which are alternately engaged with the teeth of a crown wheel1 which takes the place of the scapement wheel already described4 ,a general view of the work of a common watch is represented in fig4 #aaa4 bis4 ,a is the balance wheel bearing pallets p p upon its axis2 ,c is the crown wheel1 whose teeth are suffered to escape alternately by those pallets in the manner already described in the scapement of a clock4 ,on the axis of the crown wheel is placed a pinion d1 which drives another crown wheel ,k4 ,on the axis of this is placed the pinion c1 which plays in the teeth of the third wheel ,l4 ,the pinion b on the axis of ,l is engaged with the wheel ,m1 called the centre wheel4 ,the axle of this wheel is carried up through the centre of the dial4 ,a pinion a is placed upon it1 which works in the great wheel ,n4 ,on this wheel the mainspring immediately acts4 ,o ,p is the mainspring stripped of its barrel4 ,the axis of the wheel ,m passing through the centre of the dial is squared at the end to receive the minute hand4 ,a second pinion ,q is placed upon this axle which drives a wheel ,t4 ,on the axle of this wheel a pinion g is placed1 which drives the hour wheel ,v4 ,this wheel is placed upon a tubular axis1 which incloses within it the axis of the wheel ,m4 ,this tubular axis passing through the centre of the dial1 carries the hour hand4 ,the wheels ,a1 ,b1 ,c1 ,d1 ,e1 fig4 #aaj41 correspond to the wheels ,c1 ,k1 ,l1 ,m1 ,n1 fig4 #aab42 and the pinions a1 b1 c1 d1 e1 fig4 #aji41 correspond to the pinions d1 c1 b1 a1 fig4 #aaa4 ,from what has already been explained of these wheels1 it will be obvious that the wheel ,m1 fig4 #aaa41 revolves once in an hour1 causing the minute hand to move round the dial once in that time4 ,this wheel at the same time turns the pinion ,q which leads the wheel ,t4 ,this wheel again turns the pinion g which leads the hour wheel ,v4 ,the leaves and teeth of these pinions and wheels are proportioned1 as already explained1 so that the wheel ,v revolves once during twelve revolutions of the wheel ,m4 ,the hour hand1 therefore1 which is carried by the tubular axle of the wheel ,v1 moves once round the dial in twelve hours4 ,our object here has not been to give a detailed account of watch and clock work1 a subject for which we must refer the reader to the proper department of this work4 ,such a general account has only been attempted as may explain how tooth and pinion work may be applied to regulate motion4 ,illustration3 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,,chap4 ,,xv4 ,,of ,,the ,,pulley4 7#bfg47 ,the next class of simple machines1 which present themselves to our attention1 is that which we have called the cord4 ,if a rope were perfectly flexible1 and were capable of being bent over a sharp edge1 and of moving upon it without friction1 we should be enabled by its means to make a force in any one direction overcome resistance1 or communicate motion in any other direction4 ,thus if ,p1 fig4 #aab41 be such an edge1 a perfectly flexible rope passing over it would be capable of transmitting a force ,s ,f to a resistance ,q ,r1 so as to support or overcome ,r1 or by a motion in the direction of ,s ,f to produce another motion in the direction ,r ,q4 ,but as no materials of which ropes can be constructed can give them perfect flexibility1 and as in proportion to the strength by which they are enabled to transmit force their rigidity increases1 it is necessary1 in practice1 to adopt means to remove or mitigate those effects which attend imperfect flexibility1 and which would otherwise render cords practically inapplicable as machines4 ,when a cord is used to transmit a force from one direction to another1 its stiffness renders some force necessary in bending it over the angle ,p1 which the two directions form2 and if the angle be sharp1 the exertion of such a force may be attended with the rupture of the cord4 ,if1 instead of bending the rope at one point over a single angle1 the change of direction were produced by successively deflecting it over several angles1 each of which would be less sharp than a single one could be1 the force requisite for the deflection1 as well as the liability of rupturing the cord1 would be considerably diminished4 ,but this end will be still more perfectly attained if the deflection of the cord be produced by bending it over the surface of a curve4 ,if a rope were applied only to sustain1 and not to move a weight1 this would be sufficient to remove the inconveniences arising from its rigidity4 ,but when motion is to be produced1 the rope1 in passing over the curved surface1 would be subject to excessive friction1 and consequently to rapid wear4 ,this inconvenience is removed by causing the surface on which the rope runs to move with it1 so that no more friction is produced than would arise from the curved surface rolling upon the rope4 7#bfh47 ,all these ends are attained by the common pulley1 which consists of a wheel called a sheave1 fixed in a block and turning on a pivot4 ,a groove is formed in the edge of the wheel in which the rope runs1 the wheel revolving with it4 ,such an apparatus is represented in fig4 #aac4 ,we shall1 for the present1 omit the consideration of that part of the effects of the stiffness and friction of the machine which is not removed by the contrivance just explained1 and shall consider the rope as perfectly flexible and moving without friction4 ,from the definition of a flexible cord1 it follows1 that its tension1 or the force by which it is stretched throughout its entire length1 must be uniform4 ,from this principle1 and this alone1 all the mechanical properties of pulleys may be derived4 ,although1 as already explained1 the whole mechanical efficacy of this machine depends on the qualities of the cord1 and not on those of the block and sheave1 which are only introduced to remove the accidental effects of stiffness and friction2 yet it has been usual to give the name pulley to the block and sheave1 and a combination of blocks1 sheaves1 and ropes is called a tackle4 7#bfi47 ,when the rope passes over a single wheel1 which is fixed in its position1 as in fig4 #aac41 the machine is called a fixed pulley4 ,since the tension of the cord is uniform throughout its length1 it follows1 that in this machine the power and weight are equal4 ,for the weight stretches that part of the cord which is between the weight and pulley1 and the power stretches that part between the power and the pulley4 ,and since the tension throughout the whole length is the same1 the weight must be equal to the power4 ,hence it appears that no mechanical advantage is gained by this machine4 ,nevertheless1 there is scarcely any engine1 simple or complex1 attended with more convenience4 ,in the application of power1 whether of men or animals1 or arising from natural forces1 there are always some directions in which it may be exerted to much greater convenience and advantage than others1 and in many cases the exertion of these powers is limited to a single direction4 ,a machine1 therefore1 which enables us to give the most advantageous direction to the moving power1 whatever be the direction of the resistance opposed to it1 contributes as much practical convenience as one which enables a small power to balance or overcome a great weight4 ,in directing the power against the resistance1 it is often necessary to use two fixed pulleys4 ,thus1 in elevating a weight ,a1 fig4 #aad41 to the summit of a building1 by the strength of a horse moving below1 two fixed pulleys ,b and ,c may be used4 ,the rope is carried from ,a over the pulley ,b2 and1 passing downwards1 is brought under ,c1 and finally drawn by the animal on the horizontal plane4 ,in the same manner sails are spread1 and flags hoisted on the yards and masts of a ship1 by sailors pulling a rope on the deck4 ,by means of the fixed pulley a man may raise himself to a considerable height1 or descend to any proposed depth4 ,if he be placed in a chair or bucket attached to one end of a rope which is carried over a fixed pulley1 by laying hold of this rope on the other side1 as represented in fig4 #aae41 he may1 at will1 descend to a depth equal to half of the entire length of the rope1 by continually yielding rope on the one side1 and depressing the bucket or chair by his weight on the other4 ,fire-escapes have been constructed on this principle1 the fixed pulley being attached to some part of the building4 7#bgj47 ,a single moveable pulley is represented in fig4 #aaf4 ,a cord is carried from a fixed point ,f1 and passing through a block ,b1 attached to a weight ,w1 passes over a fixed pulley ,c1 the power being applied at ,p4 ,we shall first suppose the parts of the cord on each side the wheel ,b to be parallel2 in this case1 the whole weight ,w being sustained by the parts of the cords ,b ,c and ,b ,f1 and these parts being equally stretched 7#bfh471 each must sustain half the weight1 which is therefore the tension of the cord4 ,this tension is resisted by the power at ,p1 which must1 therefore1 be equal to half the weight4 ,in this machine1 therefore1 the weight is twice the power4 7#bga47 ,if the parts of the cord ,b ,c and ,b ,f be not parallel1 as in fig4 #aag41 a greater power than half the weight is therefore necessary to sustain it4 ,to determine the power necessary to support a given weight1 in this case take the line ,b ,a in the vertical direction1 consisting of as many inches as the weight consists of ounces2 from ,a draw ,a ,d parallel to ,b ,c1 and ,a ,e parallel to ,b ,f2 the force of the weight represented by ,a ,b will be equivalent to two forces represented by ,b ,d and ,b ,e4 7#gd47 ,the number of inches in these lines respectively will represent the number of ounces which are equivalent to the tensions of the parts ,b ,f and ,b ,c of the cord4 ,but as these tensions are equal1 ,b ,d and ,b ,e must be equal1 and each will express the amount of the power ,p1 which stretches the cord at ,p ,c4 ,it is evident that the four lines1 ,a ,e1 ,e ,b1 ,b ,d1 and ,d ,a1 are equal4 ,and as each of them represents the power1 the weight which is represented by ,a ,b must be less than twice the power which is represented by ,a ,e and ,e ,b taken together4 ,it follows1 therefore1 that as parts of the ropes which support the weight depart from parallelism the machine becomes less and less efficacious2 and there are certain obliquities at which the equilibrating power would be much greater than the weight4 7#bgb47 ,the mechanical power of pulleys admits of being almost indefinitely increased by combination4 ,systems of pulleys may be divided into two classes2 those in which a single rope is used1 and those which consist of several distinct ropes4 ,fig4 #aah4 and #aai4 represent two systems of pulleys1 each having a single rope4 ,the weight is in each case attached to a moveable block1 ,b1 in which are fixed two or more wheels2 ,a is a fixed block1 and the rope is successively passed over the wheels above and below1 and1 after passing over the last wheel above1 is attached to the power4 ,the tension of that part of the cord to which the power is attached is produced by the power1 and therefore equivalent to it1 and the same tension must extend throughout its whole length4 ,the weight is sustained by all those parts of the cord which pass from the lower block1 and as the force which stretches them all is the same1 viz4 that of the power1 the effect of the weight must be equally distributed among them1 their directions being supposed to be parallel4 ,it will be evident1 from this reasoning1 that the weight will be as many times greater than the power as the number of cords which support the lower block4 ,thus1 if there be six cords1 each cord will support a sixth part of the weight1 that is1 the weight will be six times the tension of the cord1 or six times the power4 ,in fig4 #aah4 the cord is represented as being finally attached to a hook on the upper block4 ,but it may be carried over an additional wheel fixed in that block1 and finally attached to a hook in the lower block1 as in fig4 #aai41 by which one will be added to the power of the machine1 the number of cords at the lower block being increased by one4 ,in the system represented in fig4 #aah4 the wheels are placed in the blocks one above the other2 in fig4 #aai4 they are placed side by side4 ,in all systems of pulleys of this class1 the weight of the lower block is to be considered as a part of the weight to be raised1 and in estimating the power of the machine1 this should always be attended to4 7#bgc47 ,when the power of the machine1 and therefore the number of wheels1 is considerable1 some difficulty arises in the arrangement of the wheels and cords4 ,the celebrated ,smeaton contrived a tackle1 which takes its name from him1 in which there are ten wheels in each block3 five large wheels placed side by side1 and five smaller ones similarly placed above them in the lower block1 and below them in the upper4 ,fig4 #abj4 represents ,smeaton's blocks without the rope4 ,the wheels are marked with the numbers #a1 #b1 #c1 c41 in the order in which the rope is to be passed over them4 ,as in this pulley #bj distinct parts of the rope support the lower block1 the weight1 including the lower block1 will be #bj times the equilibrating power4 7#bgd47 ,in all these systems of pulleys1 every wheel has a separate axle1 and there is a distinct wheel for every turn of the rope at each block4 ,each wheel is attended with friction on its axle1 and also with friction between the sheave and block4 ,the machine is by this means robbed of a great part of its efficacy1 since1 to overcome the friction alone1 a considerable power is in most cases necessary4 ,an ingenious contrivance has been suggested1 by which all the advantage of a large number of wheels may be obtained without the multiplied friction of distinct sheaves and axles4 ,to comprehend the excellence of this contrivance1 it will be necessary to consider the rate at which the rope passes over the several wheels of such a system1 as fig4 #aah4 ,if one foot of the rope ,g ,f pass over the pulley ,f1 two feet must pass over the pulley ,e1 because the distance between ,f and ,e being shortened one foot1 the total length of the rope ,g ,f ,e must be shortened two feet4 ,these two feet of rope must pass in the direction ,e ,d1 and the wheel ,d1 rising one foot1 three feet of rope must consequently pass over it4 ,these three feet of rope passing in the direction ,d ,c1 and the rope ,d ,c being also shortened one foot by the ascent of the lower block1 four feet of rope must pass over the wheel ,c4 ,in the same way it may be shown that five feet must pass over ,b1 and six feet over ,a4 ,thus1 whatever be the number of wheels in the upper and lower blocks1 the parts of the rope which pass in the same time over the wheels in the lower block are in the proportion of the odd numbers #a1 #c1 #e1 c42 and those which pass over the wheels in the upper block in the same time1 are as the even numbers #b1 #d1 #f1 c4 ,if the wheels were all of equal size1 as in fig4 #aai41 they would revolve with velocities proportional to the rate at which the rope passes over them4 ,so that1 while the first wheel below revolves once1 the first wheel above will revolve twice2 the second wheel below three times2 the second wheel above1 four times1 and so on4 ,if1 however1 the wheels differed in size in proportion to the quantity of rope which must pass over them1 they would evidently revolve in the same time4 ,thus1 if the first wheel above were twice the size of the first wheel below1 one revolution would throw off twice the quantity of rope4 ,again1 if the second wheel below were thrice the size of the first wheel below1 it would throw off in one revolution thrice the quantity of rope1 and so on4 ,wheels thus proportioned1 revolving in exactly the same time1 might be all placed on one axle1 and would partake of one common motion1 or1 what is to the same effect1 several grooves might be cut upon the face of one solid wheel1 with diameters in the proportion of the odd numbers #a1 #c1 and #e1 c41 for the lower pulley1 and corresponding grooves on the face of another solid wheel represented by the even numbers #b1 #d1 #f1 c41 for the upper pulley4 ,the rope being passed successively over the grooves of such wheels1 would be thrown off exactly in the same manner as if every groove were upon a separate wheel1 and every wheel revolved independently of the others4 ,such is ,white's pulley1 represented in fig4 #aba4 ,the advantage of this machine1 when accurately constructed1 is very considerable4 ,the friction1 even when great resistances are to be opposed1 is very trifling2 but1 on the other hand1 it has corresponding disadvantages which greatly circumscribe its practical utility4 ,in the workmanship of the grooves great difficulty is found in giving them the exact proportions4 ,in doing which1 the thickness of the rope must be accurately allowed for2 and consequently it follows1 that the same pulley can never act except with a rope of a particular diameter4 ,a very slight deviation from the true proportion of the grooves will cause the rope to be unequally stretched1 and will throw on some parts of it an undue proportion of the weight1 while other parts become nearly1 and sometimes altogether slack4 ,besides these defects1 the rope is so liable to derangement by being thrown out of the grooves1 that the pulley can scarcely be considered portable4 ,for these and other reasons1 this machine1 ingenious as it unquestionably is1 has never been extensively used4 7#bge47 ,in the several systems of pulleys just explained1 the hook to which the fixed block is attached supports the entire of both the power and weight4 ,when the machine is in equilibrium1 the power only supports so much of the weight as is equal to the tension of the cord1 all the remainder of the weight being thrown on the fixed point1 according to what was observed in 7#bbe47 ,if the power be moved so as to raise the weight1 it will move with a velocity as many times greater than that of the weight as the weight itself is greater than the power4 ,thus in fig4 #aah4 if the weight attached to the lower block ascend one foot1 six feet of line will pass over the pulley ,a1 according to what has been already proved4 ,thus1 the power will descend through six feet1 while the weight rises one foot4 ,but1 in this case1 the weight is six times the power4 ,all the observations in 7#bbf47 will therefore be applicable to the cases of great weights raised by small powers by means of the system of pulleys just described4 7#bgf47 ,when two or more ropes are used1 pulleys may be combined in various ways so as to produce any degree of mechanical effect4 ,if to any of the systems already described a single moveable pulley be added1 the power of the machine would be doubled4 ,in this case1 the second rope is attached to the hook of the lower block1 as in fig4 #abb41 and being carried through a moveable pulley attached to the weight1 it is finally brought up to a fixed point4 ,the tension of the second cord is equal to half the weight 7#bgj472 and therefore the power ,p1 by means of the first cord1 will have only half the tension which it would have if the weight were attached to the lower block4 ,a moveable pulley thus applied is called a runner4 ,illustration3 ,c4 ,varley1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 7#bgg47 ,two systems of pulleys1 called ,spanish bartons1 having each two ropes1 are represented in fig4 #abc4 ,the tension of the rope ,p ,a ,b ,c in the first system is equal to the power2 and therefore the parts ,b ,a and ,b ,c support a portion of the weight equal to twice the power4 ,the rope ,e ,a supports the tensions of ,a ,p and ,a ,b2 and therefore the tension of ,a ,e ,d is twice the power4 ,thus1 the united tensions of the ropes which support the pulley ,b is four times the power1 which is therefore the amount of the weight4 ,in the second system1 the rope ,p ,a ,d is stretched by the power4 ,the rope ,a ,e ,b ,c acts against the united tensions ,a ,p and ,a ,d2 and therefore the tension of ,a ,e or ,e ,b is twice the power4 ,thus1 the weight acts against three tensions2 two of which are equal to twice the power1 and the remaining one is equal to the power4 ,the weight is therefore equal to five times the power4 ,a single rope may be so arranged with one moveable pulley as to support a weight equal to three times the power4 ,in fig4 #abd4 this arrangement is represented1 where the numbers sufficiently indicate the tension of the rope1 and the proportion of the weight and power4 ,in fig4 #abe4 another method of producing the same effect with two ropes is represented4 7#bgh47 ,if several single moveable pulleys be made successively to act upon each other1 the effect is doubled by every additional pulley3 such a system as this is represented in fig4 #abf4 ,the tension of the first rope is equal to the power2 the second rope acts against twice the tension of the first1 and therefore it is stretched with a force equal to twice the power3 the third rope acts against twice this tension1 and therefore it is stretched with a force equal to four times the power1 and so on4 ,in the system represented in fig4 #abf4 there are three ropes1 and the weight is eight times the power4 ,another rope would render it sixteen times the power1 and so on4 ,in this system1 it is obvious that the ropes will require to have different degrees of strength1 since the tension to which they are subject increases in a double proportion from the power to the weight4 7#bgi47 ,if each of the ropes1 instead of being attached to fixed points at the top1 are carried over fixed pulleys1 and attached to the several moveable pulleys respectively1 as in fig4 #abg41 the power of the machine will be greatly increased2 for in that case the forces which stretch the successive ropes increase in a treble instead of a double proportion1 as will be evident by attending to the numbers which express the tensions in the figure4 ,one rope would render the weight three times the power1 two ropes nine times1 three ropes twenty-seven times1 and so on4 ,an arrangement of pulleys is represented in fig4 #abh41 by which each rope1 instead of being finally attached to a fixed point1 as in fig4 #abf41 is attached to the weight4 ,the weight is in this case supported by three ropes2 one stretched with a force equal to the power2 another with a force equal to twice the power2 and a third with a force equal to four times the power4 ,the weight is therefore1 in this case1 seven times the power4 7#bhj47 ,if the ropes1 instead of being attached to the weight1 pass through wheels1 as in fig4 #abi41 and are finally attached to the pulleys above1 the power of the machine will be considerably increased4 ,in the system here represented the weight is twenty-six times the power4 7#bha47 ,in considering these several combinations of pulleys1 we have omitted to estimate the effects produced by the weights of the sheaves and blocks4 ,without entering into the details of this computation1 it may be observed generally1 that in the systems represented in figs4 #abf41 #abg4 the weight of the wheel and blocks acts against the power2 but that in figs4 #abh4 and #abi4 they assist the powers in supporting the weight4 ,in the systems represented in fig4 #abc4 the weight of the pulleys1 to a certain extent1 neutralise each other4 7#bhb47 ,it will in all cases be found1 that that quantity by which the weight exceeds the power is supported by fixed points2 and therefore1 although it be commonly stated that a small power supports a great weight1 yet in the pulley1 as in all other machines1 the power supports no more of the weight than is exactly equal to its own amount4 ,it will not be necessary to establish this in each of the examples which have been given3 having explained it in one instance1 the student will find no difficulty in applying the same reasoning to others4 ,in fig4 #abf41 the fixed pulley sustains a force equal to twice the power1 and by it the power giving tension to the first rope sustains a part of the weight equal to itself4 ,the first hook sustains a portion of the weight equal to the tension of the first string1 or to the power4 ,the second hook sustains a force equal to twice the power2 and the third hook sustains a force equal to four times the power4 ,the three hooks therefore sustain a portion of the weight equal to seven times the power2 and the weight itself being eight times the power1 it is evident that the part of the weight which remains to be supported by the power is equal to the power itself4 7#bhc47 ,when a weight is raised by any of the systems of pulleys which have been last described1 the proportion between the velocity of the weight and the velocity of the power1 so frequently noticed in other machines1 will always be observed4 ,in the system of pulleys represented in fig4 #abf4 the weight being eight times the power1 the velocity of the power will be eight times that of the weight4 ,if the power be moved through eight feet1 that part of the rope between the fixed pulley and the first moveable pulley will be shortened by eight feet4 ,and since the two parts which lie above the first moveable pulley must be equally shortened1 each will be diminished by four feet2 therefore the first pulley will rise through four feet while the power moves through eight feet4 ,in the same way it may be shown1 that while the first pulley moves through four feet1 the second moves through two2 and while the second moves through two1 the third1 to which the weight is attached1 is raised through one foot4 ,while the power1 therefore1 is carried through eight feet1 the weight is moved through one foot4 ,by reasoning similar to this1 it may be shown that the space through which the power is moved in every case is as many times greater than the height through which the weight is raised1 as the weight is greater than the power4 7#bhd47 ,from its portable form1 cheapness of construction1 and the facility with which it may be applied in almost every situation1 the pulley is one of the most useful of the simple machines4 ,the mechanical advantage1 however1 which it appears in theory to possess is considerably diminished in practice1 owing to the stiffness of the cordage1 and the friction of the wheels and blocks4 ,by this means it is computed that in most cases so great a proportion as two thirds of the power is lost4 ,the pulley is much used in building1 where weights are to be elevated to great heights4 ,but its most extensive application is found in the rigging of ships1 where almost every motion is accomplished by its means4 7#bhe47 ,in all the examples of pulleys1 we have supposed the parts of the rope sustaining the weight and each of the moveable pulleys to be parallel to each other4 ,if they be subject to considerable obliquity1 the relative tensions of the different ropes must be estimated according to the principle applied in 7#bga47 ,,chap4 ,,xvi4 ,,on ,,the ,,inclined ,,plane1 ,,wedge1 ,,and ,,screw4 7#bhf47 ,the inclined plane is the most simple of all machines4 ,it is a hard plane surface forming some angle with a horizontal plane1 that angle not being a right angle4 ,when a weight is placed on such a plane1 a two-fold effect is produced4 ,a part of the effect of the weight is resisted by the plane1 and produces a pressure upon it2 and the remainder urges the weight down the plane1 and would produce a pressure against any surface resisting its motion placed in a direction perpendicular to the plane 7#aca47 ,let ,a ,b1 fig4 #acj41 be such a plane1 ,b ,c its horizontal base1 ,a ,c its height1 and ,a ,b ,c its angle of elevation4 ,let ,w be a weight placed upon it4 ,this weight acts in the vertical direction ,w ,d1 and is equivalent to two forces1 ,w ,f perpendicular to the plane1 and ,w ,e directed down the plane 7#gd47 ,if a plane be placed at right angles to the inclined plane below ,w1 it will resist the descent of the weight1 and sustain a pressure expressed by ,w ,e4 ,thus1 the weight ,w resting in the corner1 instead of producing one pressure in the direction ,w ,d1 will produce two pressures1 one expressed by ,w ,f upon the inclined plane1 and the other expressed by ,w ,e upon the resisting plane4 ,these pressures respectively have the same proportion to the entire weight as ,w ,f and ,w ,e have to ,w ,d1 or as ,d ,e and ,w ,e have to ,w ,d1 because ,d ,e is equal to ,w ,f4 ,now the triangle ,w ,e ,d is in all respects similar to the triangle ,a ,b ,c1 the one differing from the other only in the scale on which it is constructed4 ,therefore1 the three lines ,a ,c1 ,c ,b1 and ,b ,a1 are in the same proportion to each other as the lines ,w ,e1 ,e ,d1 and ,w ,d4 ,hence1 ,a ,b has to ,a ,c the same proportion as the whole weight has to the pressure directed toward ,b1 and ,a ,b has to ,b ,c the same proportion as the whole weight has to the pressure on the inclined plane4 ,we have here supposed the weight to be sustained upon the inclined plane by a hard plane fixed at right angles to it4 ,but the power necessary to sustain the weight will be the same in whatever way it is applied1 provided it act in the direction of the plane4 ,thus1 a cord may be attached to the weight1 and stretched towards ,a1 or the hands of men may be applied to the weight below it1 so as to resist its descent towards ,b4 ,but in whatever way it be applied1 the amount of the power will be determined in the same manner4 ,suppose the weight to consist of as many pounds as there are inches in ,a ,b1 then the power requisite to sustain it upon the plane will consist of as many pounds as there are inches in ,a ,c1 and the pressure on the plane will amount to as many pounds as there are inches in ,b ,c4 ,from what has been stated it may easily be inferred that the less the elevation of the plane is1 the less will be the power requisite to sustain a given weight upon it1 and the greater will be the pressure upon it4 ,suppose the inclined plane ,a ,b to turn upon a hinge at ,b1 and to be depressed so that its angle of elevation shall be diminished1 it is evident that as this angle decreases the height of the plane decreases1 and its base increases4 ,thus1 when it takes the position ,b ,a1 the height ,a ,c is less than the former height ,a ,c1 while the base ,b ,c is greater than the former base ,b ,c4 ,the power requisite to support the weight upon the plane in the position ,b ,a is represented by ,a ,c1 and is as much less than the power requisite to sustain it upon the plane ,a ,b1 as the height ,a ,c is less than the height ,a ,c4 ,on the other hand1 the pressure upon the plane in the position ,b ,a is as much greater than the pressure upon the plane ,b ,a1 as the base ,b ,c is greater than the base ,b ,c4 7#bhg47 ,the power of an inclined plane1 considered as a machine1 is therefore estimated by the proportion which its length bears to its height4 ,this power is always increased by diminishing the elevation of the plane4 ,roads which are not level may be regarded as inclined planes1 and loads drawn upon them in carriages1 considered in reference to the powers which impel them1 are subject to all the conditions which have been established for inclined planes4 ,the inclination of the road is estimated by the height corresponding to some proposed length4 ,thus it is said to rise one foot in fifteen1 one foot in twenty1 c41 meaning that if fifteen or twenty feet of the road be taken as the length of an inclined plane1 such as ,a ,b1 the corresponding height will be one foot4 ,or the same may be expressed thus3 that if fifteen or twenty feet be measured upon the road1 the difference of the levels of the two extremities of the distance measured is one foot4 ,according to this method of estimating the inclination of roads1 the power requisite to sustain a load upon them 7setting aside the effect of friction71 is always proportional to that elevation4 ,thus1 if a road rise one foot in twenty1 a power of one ton will be sufficient to sustain twenty tons1 and so on4 ,on a horizontal plane the only resistance which the power has to overcome is the friction of the load with the plane1 and the consideration of this being for the present omitted1 a weight once put in motion would continue moving for ever1 without any further action of the power4 ,but if the plane be inclined1 the power will be expended in raising the weight through the perpendicular height of the plane4 ,thus1 in a road which rises one foot in ten1 the power is expended in raising the weight through one perpendicular foot for every ten feet of the road over which it is moved4 ,as the expenditure of power depends upon the rate at which the weight is raised perpendicularly1 it is evident that the greater the inclination of the road is1 the slower the motion must be with the same force4 ,if the energy of the power be such as to raise the weight at the rate of one foot per minute1 the weight may be moved in each minute through that length of the road which corresponds to a rise of one foot4 ,thus1 if two roads rise one at the rate of a foot in fifteen feet1 and the other at the rate of one foot in twenty feet1 the same expenditure of power will move the weight through fifteen feet of the one1 and twenty feet of the other at the same rate4 ,from such considerations as these1 it will readily appear that it may often be more expedient to carry a road through a circuitous route than to continue it in the most direct course2 for though the measured length of road may be considerably greater than in the former case1 yet more may be gained in speed with the same expenditure of power than is lost by the increase of distance4 ,by attending to these circumstances1 modern road-makers have greatly facilitated and expedited the intercourse between distant places4 7#bhh47 ,if the power act obliquely to the plane1 it will have a twofold effect2 a part being expended in supporting or drawing the weight1 and a part in diminishing or increasing the pressure upon the plane4 ,let ,w ,p1 fig4 #acj41 be the power4 ,this will be equivalent to two forces1 ,w ,f1 perpendicular to the plane1 and ,w ,e in the direction of the plane4 7#gd47 ,in order that the power should sustain the weight1 it is necessary that that part ,w ,e of the power which acts in the direction of the plane should be equal to that part ,w ,e1 fig4 #acj41 of the weight which acts down the plane4 ,the other part ,w ,f of the power acting perpendicular to the plane is immediately opposed to that part ,w ,f of the weight which produces pressure4 ,the pressure upon the plane will therefore be diminished by the amount of ,w ,f4 ,the amount of the power which will equilibrate with the weight may1 in this case1 be found as follows4 ,take ,w ,e equal to ,w ,e1 and draw ,e ,p perpendicular to the plane1 and meeting the direction of the power4 ,the proportion of the power to the weight will be that of ,w ,p to ,w ,d4 ,and the proportion of the pressure to the weight will be that of the difference between ,w ,f and ,w ,f to ,w ,d4 ,if the amount of the power have a less proportion to the weight than ,w ,p has to ,w ,d1 it will not support the body on the plane1 but will allow it to descend4 ,and if it have a greater proportion1 it will draw the weight up the plane towards ,a4 7#bhi47 ,it sometimes happens that a weight upon one inclined plane is raised or supported by another weight upon another inclined plane4 ,thus1 if ,a ,b and ,a ,b1 fig4 #aca41 be two inclined planes forming an angle at ,a1 and ,w ,w be two weights placed upon these planes1 and connected by a cord passing over a pulley at ,a1 the one weight will either sustain the other1 or one will descend1 drawing the other up4 ,to determine the circumstances under which these effects will ensue1 draw the lines ,w ,d and ,w ,d in the vertical direction1 and take upon them as many inches as there are ounces in the weights respectively4 ,w ,d and ,w ,d being the lengths thus taken1 and therefore representing the weights1 the lines ,w ,e and ,w ,e will represent the effects of these weights respectively down the planes4 ,if ,w ,e and ,w ,e be equal1 the weights will sustain each other without motion4 ,but if ,w ,e be greater than ,w ,e1 the weight ,w will descend1 drawing the weight ,w up4 ,and if ,w ,e be greater than ,w ,e1 the weight ,w will descend1 drawing the weight ,w up4 ,in every case the lines ,w ,f and ,w ,f will represent the pressures upon the planes respectively4 ,it is not necessary1 for the effect just described1 that the inclined planes should1 as represented in the figure1 form an angle with each other4 ,they may be parallel1 or in any other position1 the rope being carried over a sufficient number of wheels placed so as to give it the necessary deflection4 ,this method of moving loads is frequently applied in great public works where rail-roads are used4 ,loaded waggons descend one inclined plane1 while other waggons1 either empty or so loaded as to permit the descent of those with which they are connected1 are drawn up the other4 7#bij47 ,in the application of the inclined plane which we have hitherto noticed1 the machine itself is supposed to be fixed in its position1 while the weight or load is moved upon it4 ,but it frequently happens that resistances are to be overcome which do not admit of being thus moved4 ,in such cases1 instead of moving the load upon the planes1 the plane is to be moved under or against the load4 ,let ,d ,e1 fig4 #acb41 be a heavy beam secured in a vertical position between guides ,f ,g and ,h ,i1 so that it is free to move upwards and downwards1 but not laterally4 ,let ,a ,b ,c be an inclined plane1 the extremity of which is placed beneath the end of the beam4 ,a force applied to the back of this plane ,a ,c1 in the direction ,c ,b1 will urge the plane under the beam so as to raise the beam to the position represented in fig4 #acc4 ,thus1 while the inclined plane is moved through the distance ,c ,b1 the beam is raised through the height ,c ,a4 7#bia47 ,when the inclined plane is applied in this manner1 it is called a wedge4 ,and if the power applied to the back were a continued pressure1 its proportion to the weight would be that of ,a ,c to ,c ,b4 ,it follows1 therefore1 that the more acute the angle ,b is1 the more powerful will be the wedge4 ,in some cases1 the wedge is formed of two inclined planes1 placed base to base1 as represented in fig4 #acd4 ,the theoretical estimation of the power of this machine is not applicable in practice with any degree of accuracy4 ,this is in part owing to the enormous proportion which the friction in most cases bears to the theoretical value of the power1 but still more to the nature of the power generally used4 ,the force of a blow is of a nature so wholly different from continued forces1 such as the pressure of weights1 or the resistance offered by the cohesion of bodies1 that it admits of no numerical comparison with them4 ,hence we cannot properly state the proportion which the force of a blow bears to the amount of a weight or resistance4 ,the wedge is almost invariably urged by percussion2 while the resistances which it has to overcome are as constantly forces of the other kind4 ,although1 however1 no exact numerical comparison can be made1 yet it may be stated in a general way that the wedge is more and more powerful as its angle is more acute4 ,illustration3 ,c4 ,varley1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,in the arts and manufactures1 wedges are used where enormous force is to be exerted through a very small space4 ,thus it is resorted to for splitting masses of timber or stone4 ,ships are raised in docks by wedges driven under their keels4 ,the wedge is the principal agent in the oil-mill4 ,the seeds from which the oil is to be extracted are introduced into hair bags1 and placed between planes of hard wood4 ,wedges inserted between the bags are driven by allowing heavy beams to fall on them4 ,the pressure thus excited is so intense1 that the seeds in the bags are formed into a mass nearly as solid as wood4 ,instances have occurred in which the wedge has been used to restore a tottering edifice to its perpendicular position4 ,all cutting and piercing instruments1 such as knives1 razors1 scissors1 chisels1 c41 nails1 pins1 needles1 awls1 c4 are wedges4 ,the angle of the wedge1 in these cases1 is more or less acute1 according to the purpose to which it is to be applied4 ,in determining this1 two things are to be considered--the mechanical power1 which is increased by diminishing the angle of the wedge2 and the strength of the tool1 which is always diminished by the same cause4 ,there is1 therefore1 a practical limit to the increase of the power1 and that degree of sharpness only is to be given to the tool which is consistent with the strength requisite for the purpose to which it is to be applied4 ,in tools intended for cutting wood1 the angle is generally about #cj4 ,for iron it is from #ej to #fj2 and for brass1 from #hj to #ij4 ,tools which act by pressure may be made more acute than those which are driven by a blow2 and in general the softer and more yielding the substance to be divided is1 and the less the power required to act upon it1 the more acute the wedge may be constructed4 ,in many cases the utility of the wedge depends on that which is entirely omitted in its theory1 viz4 the friction which arises between its surface and the substance which it divides4 ,this is the case when pins1 bolts1 or nails are used for binding the parts of structures together2 in which case1 were it not for the friction1 they would recoil from their places1 and fail to produce the desired effect4 ,even when the wedge is used as a mechanical engine1 the presence of friction is absolutely indispensable to its practical utility4 ,the power1 as has already been stated1 generally acts by successive blows1 and is therefore subject to constant intermission1 and but for the friction the wedge would recoil between the intervals of the blows with as much force as it had been driven forward4 ,thus the object of the labour would be continually frustrated4 ,the friction in this case is of the same use as a ratchet wheel1 but is much more necessary1 as the power applied to the wedge is more liable to intermission than in the cases where ratchet wheels are generally used4 7#bib47 ,when a road directly ascends the side of a hill1 it is to be considered as an inclined plane2 but it will not lose its mechanical character1 if1 instead of directly ascending towards the top of the hill1 it winds successively round it1 and gradually ascends so as after several revolutions to reach the top4 ,in the same manner a path may be conceived to surround a pillar by which the ascent may be facilitated upon the principle of the inclined plane4 ,winding stairs constructed in the interior of great columns partake of this character2 for although the ascent be produced by successive steps1 yet if a floor could be made sufficiently rough to prevent the feet from slipping1 the ascent would be accomplished with equal facility4 ,in such a case the winding path would be equivalent to an inclined plane1 bent into such a form as to accommodate it to the peculiar circumstances in which it would be required to be used4 ,it will not be difficult to trace the resemblance between such an adaptation of the inclined plane and the appearances presented by the thread of a screw3 and it may hence be easily understood that a screw is nothing more than an inclined plane constructed upon the surface of a cylinder4 ,this will1 perhaps1 be more apparent by the following contrivance3 ,let ,a ,b1 fig4 #ace41 be a common round ruler1 and let ,c ,d ,e be a piece of white paper cut in the form of an inclined plane1 whose height ,c ,d is equal to the length of the ruler ,a ,b1 and let the edge ,c ,e of the paper be marked with a broad black line3 let the edge ,c ,d be applied to the ruler ,a ,b1 and being attached thereto1 let the paper be rolled round the ruler2 the ruler will then present the appearance of a screw1 fig4 #acf4 the thread of the screw being marked by the black line ,c ,e1 winding continually round the ruler4 ,let ,d ,f1 fig4 #ace41 be equal to the circumference of the ruler1 and draw ,f ,g parallel to ,d ,c1 and ,g ,h parallel to ,d ,e1 the part ,c ,g ,f ,d of the paper will exactly surround the ruler once3 the part ,c ,g will form one convolution of the thread1 and may be considered as the length of one inclined plane surrounding the cylinder1 ,c ,h being the corresponding height1 and ,g ,h the base4 ,the power of the screw does not1 as in the ordinary cases of the inclined plane1 act parallel to the plane or thread1 but at right angles to the length of the cylinder ,a ,b1 or1 what is to the same effect1 parallel to the base ,h ,g2 therefore the proportion of the power to the weight will be1 according to principles already explained1 the same as that of ,c ,h to the space through which the power moves parallel to ,h ,g in one revolution of the screw4 ,h ,c is evidently the distance between the successive positions of the thread as it winds round the cylinder2 and it appears from what has been just stated1 that the less this distance is1 or1 in other words1 the finer the thread is1 the more powerful the machine will be4 7#bic47 ,in the application of the screw the weight or resistance is not1 as in the inclined plane and wedge1 placed upon the surface of the plane or thread4 ,the power is usually transmitted by causing the screw to move in a concave cylinder1 on the interior surface of which a spiral cavity is cut1 corresponding exactly to the thread of the screw1 and in which the thread will move by turning round the screw continually in the same direction4 ,this hollow cylinder is usually called the nut or concave screw4 ,the screw surrounded by its spiral thread is represented in fig4 #acg42 and a section of the same playing in the nut is represented in fig4 #ach4 ,there are several ways in which the effect of the power may be conveyed to the resistance by this apparatus4 ,first1 let us suppose that the nut ,a ,b is fixed4 ,if the screw be continually turned on its axis1 by a lever ,e ,f inserted in one end of it1 it will be moved in the direction ,c ,d1 advancing every revolution through a space equal to the distance between two contiguous threads4 ,by turning the lever in an opposite direction1 the screw will be moved in the direction ,d ,c4 ,if the screw be fixed1 so as to be incapable either of moving longitudinally or revolving on its axis1 the nut ,a ,b may be turned upon the screw by a lever1 and will move on the screw towards ,c or towards ,d1 according to the direction in which the lever is turned4 ,in the former case we have supposed the nut to be absolutely immoveable1 and in the latter case the screw to be absolutely immoveable4 ,it may happen1 however1 that the nut1 though capable of revolving1 is incapable of moving longitudinally2 and that the screw1 though incapable of revolving1 is capable of moving longitudinally4 ,in that case1 by turning the nut ,a ,b upon the screw by the lever1 the screw will be urged in the direction ,c ,d or ,d ,c1 according to the way in which the nut is turned4 ,the apparatus may1 on the contrary1 be so arranged1 that the nut1 though incapable of revolving1 is capable of moving longitudinally2 and the screw1 though capable of revolving1 is incapable of moving longitudinally4 ,in this case1 by turning the screw in the one direction or in the other1 the nut ,a ,b will be urged in the direction ,c ,d or ,d ,c4 ,all these various arrangements may be observed in different applications to the machine4 7#bid47 ,a screw may be cut upon a cylinder by placing the cylinder in a turning lathe1 and giving it a rotatory motion upon its axis4 ,the cutting point is then presented to the cylinder1 and moved in the direction of its length1 at such a rate as to be carried through the distance between the intended thread1 while the cylinder revolves once4 ,the relative motions of the cutting point and the cylinder being preserved with perfect uniformity1 the thread will be cut from one end to the other4 ,the shape of the threads may be either square1 as in fig4 #acg41 or triangular1 as in fig4 #aci4 7#bie47 ,the screw is generally used in cases where severe pressure is to be excited through small spaces2 it is therefore the agent in most presses4 ,in fig4 #adj41 the nut is fixed1 and by turning the lever1 which passes through the head of the screw1 a pressure is excited upon any substance placed upon the plate immediately under the end of the screw4 ,in fig4 #ada41 the screw is incapable of revolving1 but is capable of advancing in the direction of its length4 ,on the other hand1 the nut is capable of revolving1 but does not advance in the direction of the screw4 ,when the nut is turned by means of the screw inserted in it1 the screw advances in the direction of its length1 and urges the board which is attached to it upwards1 so as to press any substance placed between it and the fixed board above4 ,in cases where liquids or juices are to be expressed from solid bodies1 the screw is the agent generally employed4 ,it is also used in coining1 where the impression of a die is to be made upon a piece of metal1 and in the same way in producing the impression of a seal upon wax or other substance adapted to receive it4 ,when soft and light materials1 such as cotton1 are to be reduced to a convenient bulk for transportation1 the screw is used to compress them1 and they are thus reduced into hard dense masses4 ,in printing1 the paper is urged by a severe and sudden pressure upon the types1 by means of a screw4 7#bif47 ,as the mechanical power of the screw depends upon the relative magnitude of the circumference through which the power revolves1 and the distance between the threads1 it is evident1 that1 to increase the efficacy of the machine1 we must either increase the length of the lever by which the power acts1 or diminish the magnitude of the thread4 ,although there is no limit in theory to the increase of the mechanical efficacy by these means1 yet practical inconvenience arises which effectually prevents that increase being carried beyond a certain extent4 ,if the lever by which the power acts be increased1 the same difficulty arises as was already explained in the wheel and axle 7#bed472 the space through which the power should act would be so unwieldy1 that its application would become impracticable4 ,if1 on the other hand1 the power of the machine be increased by diminishing the size of the thread1 the strength of the thread will be so diminished1 that a slight resistance will tear it from the cylinder4 ,the cases in which it is necessary to increase the power of the machine1 being those in which the greatest resistances are to be overcome1 the object will evidently be defeated1 if the means chosen to increase that power deprive the machine of the strength which is necessary to sustain the force to which it is to be submitted4 7#big47 ,these inconveniences are removed by a contrivance of ,mr4 ,hunter1 which1 while it gives to the machine all the requisite strength and compactness1 allows it to have an almost unlimited degree of mechanical efficacy4 ,this contrivance consists in the use of two screws1 the threads of which may have any strength and magnitude1 but which have a very small difference of breadth4 ,while the working point is urged forward by that which has the greater thread1 it is drawn back by that which has the less2 so that during each revolution of the screw1 instead of being advanced through a space equal to the magnitude of either of the threads1 it moves through a space equal to their difference4 ,the mechanical power of such a machine will be the same as that of a single screw having a thread1 whose magnitude is equal to the difference of the magnitudes of the two threads just mentioned4 ,thus1 without inconveniently increasing the sweep of the power1 on the one hand1 or1 on the other1 diminishing the thread until the necessary strength is lost1 the machine will acquire an efficacy limited by nothing but the smallness of the difference between the two threads4 ,this principle was first applied in the manner represented in fig4 #adb4 ,a is the greater thread1 playing in the fixed nut2 ,b is the lesser thread1 cut upon a smaller cylinder1 and playing in a concave screw1 cut within the greater cylinder4 ,during every revolution of the screw1 the cylinder ,a descends through a space equal to the distance between its threads4 ,at the same time the smaller cylinder ,b ascends through a space equal to the distance between the threads cut upon it3 the effect is1 that the board ,d descends through a space equal to the difference between the threads upon ,a and the threads upon ,b1 and the machine has a power proportionate to the smallness of this difference4 ,thus1 suppose the screw ,a has twenty threads in an inch1 while the screw ,b has twenty-one2 during one revolution1 the screw ,a will descend through a space equal to the #bjth part of an inch4 ,if1 during this motion1 the screw ,b did not turn within ,a1 the board ,d would be advanced through the #bjth of an inch2 but because the hollow screw within ,a turns upon ,b1 the screw ,b will1 relatively to ,a1 be raised in one revolution through a space equal to the #bast part of an inch4 ,thus1 while the board ,d is depressed through the #bjth of an inch by the screw ,a1 it is raised through the #bast of an inch by the screw ,b4 ,it is1 therefore1 on the whole1 depressed through a space equal to the excess of the #bjth of an inch above the #bast of an inch1 that is1 through the #dbjth of an inch4 ,the power of this machine will1 therefore1 be expressed by the number of times the #dbjth of an inch is contained in the circumference through which the power moves4 7#bih47 ,in the practical application of this principle at present the arrangement is somewhat different4 ,the two threads are usually cut on different parts of the same cylinder4 ,if nuts be supposed to be placed upon these1 which are capable of moving in the direction of the length1 but not of revolving1 it is evident that by turning the screw once round1 each nut will be advanced through a space equal to the breadth of the respective threads4 ,by this means the two nuts will either approach each other1 or mutually recede1 according to the direction in which the screw is turned1 through a space equal to the difference of the breadth of the threads1 and they will exert a force either in compressing or extending any substance placed between them1 proportionate to the smallness of that difference4 7#bii47 ,a toothed wheel is sometimes used instead of a nut1 so that the same quality by which the revolution of the screw urges the nut forward is applied to make the wheel revolve4 ,the screw is in this case called an endless screw1 because its action upon the wheel may be continued without limit4 ,this application of the screw is represented in fig4 #adc4 ,p is the winch to which the power is applied2 and its effect at the circumference of the wheel is estimated in the same manner as the effect of the screw upon the nut4 ,this effect is to be considered as a power acting upon the circumference of the wheel2 and its proportion to the weight or resistance is to be calculated in the same manner as the proportion of the power to the weight in the wheel and axle4 7#cjj47 ,we have hitherto considered the screw as an engine used to overcome great resistances4 ,it is also eminently useful in several departments of experimental science1 for the measurement of very minute motions and spaces1 the magnitude of which could scarcely be ascertained by any other means4 ,the very slow motion which may be imparted to the end of a screw1 by a very considerable motion in the power1 renders it peculiarly well adapted for this purpose4 ,to explain the manner in which it is applied--suppose a screw to be so cut as to have fifty threads in an inch1 each revolution of the screw will advance its point through the fiftieth part of an inch4 ,now1 suppose the head of the screw to be a circle1 whose diameter is an inch1 the circumference of the head will be something more than three inches3 this may be easily divided into a hundred equal parts distinctly visible4 ,if a fixed index be presented to this graduated circumference1 the hundredth part of a revolution of the screw may be observed1 by noting the passage of one division of the head under the index4 ,since one entire revolution of the head moves the point through the fiftieth of an inch1 one division will correspond to the five thousandth of an inch4 ,in order to observe the motion of the point of the screw in this case1 a fine wire is attached to it1 which is carried across the field of view of a powerful microscope1 by which the motion is so magnified as to be distinctly perceptible4 ,a screw used for such purposes is called a micrometer screw4 ,such an apparatus is usually attached to the limbs of graduated instruments1 for the purposes of astronomical and other observation4 ,without the aid of this apparatus1 no observation could be taken with greater accuracy than the amount of the smallest division upon the limb4 ,thus1 if an instrument for measuring angles were divided into small arcs of one minute1 and an angle were observed which brought the index of the instrument to some point between two divisions1 we could only conclude that the observed angle must consist of a certain number of degrees and minutes1 together with an additional number of seconds1 which would be unknown1 inasmuch as there would be no means of ascertaining the fraction of a minute between the index and the adjacent division of the instrument4 ,but if a screw be provided1 the point of which moves through a space equal to one division of the instrument1 with sixty revolutions of the head1 and that the head itself be divided into one hundred equal parts1 each complete revolution of the screw will correspond to the sixtieth part of a minute1 or to one second1 and each division on the head of the screw will correspond to the hundredth part of a second4 ,the index being attached to this screw1 let the head be turned until the index be moved from its observed position to the adjacent division of the limb4 ,the number of complete revolutions of the screw necessary to accomplish this will be the number of seconds2 and the number of parts of a revolution over the complete number of revolutions will be the hundredth parts of a second necessary to be added to the degrees and minutes primarily observed4 ,it is not1 however1 only to such instruments that the micrometer screw is applicable2 any spaces whatever may be measured by it4 ,an instance of its mechanical application may be mentioned in a steel-yard1 an instrument for ascertaining the amount of weights by a given weight1 sliding on a long graduated arm of a lever4 ,the distance from the fulcrum1 at which this weight counterpoises the weight to be ascertained1 serves as a measure to the amount of that weight4 ,when the sliding weight happens to be placed between two divisions of the arm1 a micrometer screw is used to ascertain the fraction of the division4 ,hunter's screw1 already described1 seems to be well adapted to micrometrical purposes2 since the motion of the point may be rendered indefinitely slow1 without requiring an exquisitely fine thread1 such as in the single screw would be necessary4 ,,chap4 ,,xvii4 ,,on ,,the ,,regulation ,,and ,,accumulation ,,of ,,force4 7#cja47 ,it is frequently indispensable1 and always desirable1 that the operation of a machine should be regular and uniform4 ,sudden changes in its velocity1 and desultory variations in the effective energy of its power1 are often injurious or destructive to the apparatus itself1 and when applied to manufactures never fail to produce unevenness in the work4 ,to invent methods for insuring the regular motion of machinery1 by removing those causes of inequality which may be avoided1 and by compensating others1 has therefore been a problem to which much attention and ingenuity have been directed4 ,this is chiefly accomplished by controlling1 and1 as it were1 measuring out the power according to the exigencies of the machine1 and causing its effective energy to be always commensurate with the resistance which it has to overcome4 ,illustration3 ,c4 ,varley1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,irregularity in the motion of machinery may proceed from one or more of the following causes3--#a4 irregularity in the prime mover2 #b4 occasional variation in the amount of the load or resistance2 and1 #c4 because1 in the various positions which the parts of the machine assume during its motion1 the power may not be transmitted with equal effect to the working point4 ,the energy of the prime mover is seldom if ever regular4 ,the force of water varies with the copiousness of the stream4 ,the power which impels the windmill is proverbially capricious4 ,the pressure of steam varies with the intensity of the furnace4 ,animal power1 the result of will1 temper1 and health is difficult of control4 ,human labour is most of all unmanageable2 hence no machine works so irregularly as one which is manipulated4 ,in some cases the moving force is subject1 by the very conditions of its existence1 to constant variation1 as in the example of a spring1 which gradually loses its energy as it recoils4 7#bee47 ,in many instances the prime mover is liable to regular intermission1 and is actually suspended for certain intervals of time4 ,this is the case in the single acting steam-engine1 where the pressure of the steam urges the descent of the piston1 but is suspended during its ascent4 ,the load or resistance to which the machine is applied is not less fluctuating4 ,in mills there are a multiplicity of parts which are severally liable to be occasionally disengaged1 and to have their operation suspended4 ,in large factories for spinning1 weaving1 printing1 c4 a great number of separate spinning machines1 looms1 presses1 or other engines1 are usually worked by one common mover1 such as a water-wheel or steam-engine4 ,in these cases the number of machines employed from time to time necessarily varies with the fluctuating demand for the articles produced1 and from other causes4 ,under such circumstances the velocity with which every part of the machinery is moved would suffer corresponding changes1 increasing its rapidity with every augmentation of the moving power or diminution of the resistance1 or being retarded in its speed by the contrary circumstances4 ,but even when the prime mover and the resistance are both regular1 or rendered so by proper contrivances1 still it will rarely happen that the machine by which the energy of the one is transmitted to the other conveys this with unimpaired effect in all the phases of its operation4 ,to give a general notion of this cause of inequality to those who have not been familiar with machinery would not be easy1 without having recourse to an example4 ,for the present we shall merely state1 that the several moving parts of every machine assume in succession a variety of positions2 that at regular periods they return to their first position1 and again undergo the same succession of changes4 ,in the different positions through which they are carried in every period of motion1 the efficacy of the machine to transmit the power to the resistance is different1 and thus the effective energy of the machine in acting upon the resistance would be subject to continual fluctuation4 ,this will be more clearly understood when we come to explain the methods of counteracting the defect or equalising the action of the power upon the resistance4 ,such are the chief causes of the inequalities incidental to the motion of machinery1 and we now propose to describe a few of the many ingenious contrivances which the skill of engineers has produced to remove the consequent inconveniences4 7#cjb47 ,setting aside1 for the present1 the last cause of inequality1 and considering the machinery1 whatever it be1 to transmit the power to the resistance without irregular interruption1 it is evident that every contrivance1 having for its object to render the velocity uniform1 can only accomplish this by causing the variations of the power and resistance to be proportionate to each other4 ,this may be done either by increasing or diminishing the power as the resistance increases or diminishes2 or by increasing or diminishing the resistance as the power increases or diminishes4 ,according to the facilities or convenience presented by the peculiar circumstances of the case either of these methods is adopted4 ,the contrivances for effecting this are called regulators4 ,most regulators act upon that part of the machine which commands the supply of the power by means of levers1 or some other mechanical contrivance1 so as to check the quantity of the moving principle conveyed to the machine when the velocity has a tendency to increase2 and1 on the other hand1 to increase that supply upon any undue abatement of its speed4 ,in a water-mill this is done by acting upon the shuttle2 in a wind-mill1 by an adjustment of the sail-cloth2 and in a steam-engine1 by opening or closing1 in a greater or less degree1 the valve by which the cylinder is supplied with steam4 7#cjc47 ,of all the contrivances for regulating machinery1 that which is best known and most commonly used is the governor4 ,this regulator1 which had been long in use in mill-work and other machinery1 has of late years attracted more general notice by its beautiful adaptation in the steam-engines of ,watt4 ,it consists of heavy balls ,b ,b1 fig4 #add41 attached to the extremities of rods ,b ,f4 ,these rods play upon a joint at ,e1 passing through a mortise in the vertical stem ,d ,d4 ,at ,f they are united by joints to the short rods ,f ,h1 which are again connected by joints at ,h to a ring which slides upon the vertical shaft ,d ,d4 ,from this description it will be apparent that when the balls ,b are drawn from the axis1 their upper arms ,e ,f are caused to increase their divergence in the same manner as the blades of a scissors are opened by separating the handles4 ,these1 acting upon the ring by means of the short links ,f ,h1 draw it down the vertical axis from ,d towards ,e4 ,a contrary effect is produced when the balls ,b are brought closer to the axis1 and the divergence of the rods ,b ,e diminished4 ,a horizontal wheel ,w is attached to the vertical axis ,d ,d1 having a groove to receive a rope or strap upon its rim4 ,this strap passes round the wheel or axis by which motion is transmitted to the machinery to be regulated1 so that the spindle or shaft ,d ,d will always be made to revolve with a speed proportionate to that of the machinery4 ,as the shaft ,d ,d revolves1 the balls ,b are carried round it with a circular motion1 and consequently acquire a centrifugal force which causes them to recede from the axle1 and therefore to depress the ring ,h4 ,on the edge or rim of this ring is formed a groove1 which is embraced by the prongs of a fork ,i1 at the extremity of one arm of a lever whose fulcrum is at ,g4 ,the extremity ,k of the other arm is connected by some means with the part of the machine which supplies the power4 ,in the present instance we shall suppose it a steam-engine1 in which case the rod ,k ,i communicates with a flat circular valve ,v1 placed in the principal steam-pipe1 and so arranged that1 when ,k is elevated as far as by their divergence the balls ,b have power over it1 the passage of the pipe will be closed by the valve ,v1 and the passage of steam entirely stopped2 and1 on the other hand1 when the balls subside to their lowest position1 the valve will be presented with its edge in the direction of the tube1 so as to intercept no part of the steam4 ,the property which renders this instrument so admirably adapted to the purpose to which it is applied is1 that when the divergence of the balls is not very considerable1 they must always revolve with the same velocity1 whether they move at a greater or lesser distance from the vertical axis4 ,if any circumstance increases that velocity1 the balls instantly recede from the axis1 and closing the valve ,v1 check the supply of steam1 and thereby diminishing the speed of the motion1 restore the machine to its former rate4 ,if1 on the contrary1 that fixed velocity be diminished1 the centrifugal force being no longer sufficient to support the balls1 they descend towards the axle1 open the valve ,v1 and1 increasing the supply of steam1 restore the proper velocity of the machine4 ,when the governor is applied to a water-wheel it is made to act upon the shuttle through which the water flows1 and controls its quantity as effectually1 and upon the same principle1 as has just been explained in reference to the steam-engine4 ,when applied to a windmill it regulates the sail-cloth so as to diminish the efficacy of the power upon the arms as the force of the wind increases1 or vice vers4 ,in cases where the resistance admits of easy and convenient change1 the governor may act so as to accommodate it to the varying energy of the power4 ,this is often done in corn-mills1 where it acts upon the shuttle which metes out the corn to the millstones4 ,when the power which drives the mill increases1 a proportionally increased feed of corn is given to the stones1 so that the resistance being varied in the ratio of the power1 the same velocity will be maintained4 7#cjd47 ,in some cases the centrifugal force of the revolving balls is not sufficiently great to control the power or the resistance1 and regulators of a different kind must be resorted to4 ,the following contrivance is called the water-regulator3-- ,a common pump is worked by the machine1 whose motion is to be regulated1 and water is thus raised and discharged into a cistern4 ,it is allowed to flow from this cistern through a pipe of a given magnitude4 ,when the water is pumped up with the same velocity as it is discharged by this pipe1 it is evident that the level of the water in the cistern will be stationary1 since it receives from the pump the exact quantity which it discharges from the pipe4 ,but if the pump throw in more water in a given time than is discharged by the pipe1 the cistern will begin to be filled1 and the level of the water will rise4 ,if1 on the other hand1 the supply from the pump be less than the discharge from the pipe1 the level of the water in the cistern will subside4 ,since the rate at which water is supplied from the pump will always be proportional to the velocity of the machine1 it follows that every fluctuation in this velocity will be indicated by the rising or subsiding of the level of the water in the cistern1 and that level never can remain stationary1 except at that exact velocity which supplies the quantity of water discharged by the pipe4 ,this pipe may be constructed so as by an adjustment to discharge the water at any required rate2 and thus the cistern may be adapted to indicate a constant velocity of any proposed amount4 ,if the cistern were constantly watched by an attendant1 the velocity of the machine might be abated by regulating the power when the level of the water is observed to rise1 or increased when it falls2 but this is much more effectually and regularly performed by causing the surface of the water itself to perform the duty4 ,a float or large hollow metal ball is placed upon the surface of the water in the cistern4 ,this ball is connected with a lever acting upon some part of the machinery1 which controls the power or regulates the amount of resistance1 as already explained in the case of the governor4 ,when the level of the water rises1 the buoyancy of the ball causes it to rise also with a force equal to the difference between its own weight and the weight of as much water as it displaces4 ,by enlarging the floating ball1 a force may be obtained sufficiently great to move those parts of the machinery which act upon the power or resistance1 and thus either to diminish the supply of the moving principle or to increase the amount of the resistance1 and thereby retard the motion and reduce the velocity to its proper limit4 ,when the level of the water in the cistern falls1 the floating ball being no longer supported on the liquid surface1 descends with the force of its own weight1 and producing an effect upon the power or resistance contrary to the former1 increases the effective energy of the one1 or diminishes that of the other1 until the velocity proper to the machine be restored4 ,the sensibility of these regulators is increased by making the surface of water in the cistern as small as possible2 for then a small change in the rate at which the water is supplied by the pump will produce a considerable change in the level of the water in the cistern4 ,instead of using a float1 the cistern itself may be suspended from the lever which controls the supply of the power1 and in this case a sliding weight may be placed on the other arm1 so that it will balance the cistern when it contains that quantity of water which corresponds to the fixed level already explained4 ,if the quantity of water in the cistern be increased by an undue velocity of the machine1 the weight of the cistern will preponderate1 draw down the arm of the lever1 and check the supply of the power4 ,if1 on the other hand1 the supply of water be too small1 the cistern will no longer balance the counterpoise1 the arm by which it is suspended will be raised1 and the energy of the power will be increased4 7#cje47 ,in the steam-engine the self-regulating principle is carried to an astonishing pitch of perfection4 ,the machine itself raises in due quantity the cold water necessary to condense the steam4 ,it pumps off the hot water produced by the steam1 which has been cooled1 and lodges it in a reservoir for the supply of the boiler4 ,it carries from this reservoir exactly that quantity of water which is necessary to supply the wants of the boiler1 and lodges it therein according as it is required4 ,it breathes the boiler of redundant steam1 and preserves that which remains fit1 both in quantity and quality1 for the use of the engine4 ,it blows its own fire1 maintaining its intensity1 and increasing or diminishing it according to the quantity of steam which it is necessary to raise2 so that when much work is expected from the engine1 the fire is proportionally brisk and vivid4 ,it breaks and prepares its own fuel1 and scatters it upon the bars at proper times and in due quantity4 ,it opens and closes its several valves at the proper moments1 works its own pumps1 turns its own wheels1 and is only not alive4 ,among so many beautiful examples of the self-regulating principle1 it is difficult to select4 ,we shall1 however1 mention one or two1 and for others refer the reader to our treatise on this subject4#c #c ,lardner on the ,steam-,engine1 ,steam-,navigation1 ,roads1 and ,railways4 #hth edition4 #ahea4 ,it is necessary in this machine that the water in the boiler be maintained constantly at the same level1 and1 therefore1 that as much be supplied1 from time to time1 as is consumed by evaporation4 ,a pump which is wrought by the engine itself supplies a cistern ,c1 fig4 #ade41 with hot water4 ,at the bottom of this cistern is a valve ,v opening into a tube which descends into the boiler4 ,this valve is connected by a wire with the arm of a lever on the fulcrum ,d1 the other arm ,e of which is also connected by a wire with a stone float ,f1 which is partially immersed in the water of the boiler1 and is balanced by a sliding weight ,a4 ,the weight ,a only counterpoises the stone float ,f by the aid of its buoyance in the water2 for if the water be removed1 the stone ,f will preponderate1 and raise the weight ,a4 ,when the water in the boiler is at its proper level1 the length of the wire connecting the valve ,v with the lever is so adjusted that this valve shall be closed1 the wire at the same time being fully extended4 ,when1 by evaporation1 the water in the boiler begins to be diminished1 the level falls1 and the stone weight ,f1 being no longer supported1 overcomes the counterpoise ,a1 raises the arm of the lever1 and1 pulling the wire1 opens the valve ,v4 ,the water in the cistern ,c then flows through the tube into the boiler1 and continues to flow until the level be so raised that the stone weight ,f is again elevated1 the valve ,v closed1 and the further supply of water from the cistern ,c suspended4 ,in order to render the operation of this apparatus easily intelligible1 we have here supposed an imperfection which does not exist4 ,according to what has just been stated1 the level of the water in the boiler descends from its proper height1 and subsequently returns to it4 ,but1 in fact1 this does not happen4 ,the float ,f and valve ,v adjust themselves1 so that a constant supply of water passes through the valve1 which proceeds exactly at the same rate as that at which the water in the boiler is consumed4 7#cjf47 ,in the same machine there occurs a singularly happy example of self-adjustment1 in the method by which the strength of the fire is regulated4 ,the governor regulates the supply of steam to the engine1 and proportions it to the work to be done4 ,with this work1 therefore1 the demands upon the boiler increase or diminish1 and with these demands the production of steam in the boiler ought to vary4 ,in fact1 the rate at which steam is generated in the boiler1 ought to be equal to that at which it is consumed in the engine1 otherwise one of two effects must ensue3 either the boiler will fail to supply the engine with steam1 or steam will accumulate in the boiler1 being produced in undue quantity1 and1 escaping at the safety valve1 will thus be wasted4 ,it is1 therefore1 necessary to control the agent which generates the steam1 namely1 the fire1 and to vary its intensity from time to time1 proportioning it to the demands of the engine4 ,to accomplish this1 the following contrivance has been adopted3--,let ,t1 fig4 #adf41 be a tube inserted in the top of the boiler1 and descending nearly to the bottom4 ,the pressure of the steam confined in the boiler1 acting upon the surface of the water1 forces it to a certain height in the tube ,t4 ,a weight ,f1 half immersed in the water in the tube1 is suspended by a chain1 which passes over the wheels ,p ,p1 and is balanced by a metal plate ,d1 in the same manner as the stone float1 fig4 #ade41 is balanced by the weight ,a4 ,the plate ,d passes through the mouth of the flue ,e as it issues finally from the boiler2 so that when the plate ,d falls it stops the flue1 suspending thereby the draught of air through the furnace1 mitigating the intensity of the fire1 and checking the production of steam4 ,if1 on the contrary1 the plate ,d be drawn up1 the draught is increased1 the fire is rendered more active1 and the production of steam in the boiler is stimulated4 ,now1 suppose that the boiler produces steam faster than the engine consumes it1 either because the load on the engine has been diminished1 and1 therefore1 its consumption of steam reduced1 or because the fire has become too intense2 the consequence is1 that the steam1 beginning to accumulate in the boiler1 will press upon the surface of the water with increased force1 and the water will be raised in the tube ,t4 ,the weight ,f will1 therefore1 be lifted1 and the plate ,d will descend1 diminish1 or stop the draught1 mitigate the fire1 and retard the production of steam1 and will continue to do so until the rate at which steam is produced shall be commensurate to the wants of the engine4 ,if1 on the other hand1 the production of steam be inadequate to the exigency of the machine1 either because of an increased load1 or of the insufficient force of the fire1 the steam in the boiler will lose its elasticity1 and the surface of the water not sustaining its wonted pressure1 the water in the tube ,t will fall2 consequently the weight ,f will descend1 and the plate ,d will be raised4 ,the flue being thus opened1 the draught will be increased1 and the fire rendered more intense4 ,thus the production of steam becomes more rapid1 and is rendered sufficiently abundant for the purposes of the engine4 ,this apparatus is called the self-acting damper4 7#cjg47 ,when a perfectly uniform rate of motion has not been attained1 it is often necessary to indicate small variations of velocity4 ,the following contrivance1 called a tachometer#d1 has been invented to accomplish this4 ,a cup1 fig4 #adg41 is filled to the level ,c ,d with quicksilver1 and is attached to a spindle1 which is whirled by the machine in the same manner as the governor already described4 ,it is well known that the centrifugal force produced by this whirling motion will cause the mercury to recede from the centre and rise upon the sides of the cup1 so that its surface will assume the concave appearance represented in fig4 #adh4 ,in this case the centre of the surface will obviously have fallen below its original level1 fig4 #adg41 and the edges will have risen above that level4 ,as this effect is produced by the velocity of the machine1 so it is proportionate to that velocity1 and subject to corresponding variations4 ,any method of rendering visible small changes in the central level of the surface of the quicksilver will indicate minute variations in the velocity of the machine4 #d ,from the ,greek words tachos speed1 and metron measure4 ,a glass tube ,a1 open at both ends1 and expanding at one extremity into a bell ,b1 is immersed with its wider end in the mercury1 the surface of which will stand at the same level in the bell ,b1 and in the cup ,c ,d4 ,the tube is so suspended as to be unconnected with the cup4 ,this tube is then filled to a certain height ,a1 with spirits tinged with some colouring matter1 to render it easily observable4 ,when the cup is whirled by the machine to which it is attached1 the level of the quicksilver in the bell falls1 leaving more space for the spirits1 which1 therefore1 descends in the tube4 ,as the motion is continued1 every change of velocity causes a corresponding change in the level of the mercury1 and1 therefore1 also in the level ,a of the spirits4 ,it will be observed1 that1 in consequence of the capacity of the bell ,b being much greater than that of the tube ,a1 a very small change in the level of the quicksilver in the bell will produce a considerable change in the height of the spirits in the tube4 ,thus this ingenious instrument becomes a very delicate indicator of variations in the motion of machinery4 7#cjh47 ,the governor1 and other methods of regulating the motion of machinery which have been just described1 are adapted principally to cases in which the proportion of the resistance to the load is subject to certain fluctuations or gradual changes1 or at least to cases in which the resistance is not at any time entirely withdrawn1 nor the energy of the power actually suspended4 ,circumstances1 however1 frequently occur in which1 while the power remains in full activity1 the resistance is at intervals suddenly removed and as suddenly again returns4 ,on the other hand1 cases also present themselves1 in which1 while the resistance is continued1 the impelling power is subject to intermission at regular periods4 ,in the former case1 the machine would be driven with a ruinous rapidity during those periods at which it is relieved from its load1 and on the return of the load every part would suffer a violent strain1 from its endeavour to retain the velocity which it had acquired1 and the speedy destruction of the engine could not fail to ensue4 ,in the latter case1 the motion would be greatly retarded or entirely suspended during those periods at which the moving power is deprived of its activity1 and1 consequently1 the motion which it would communicate would be so irregular as to be useless for the purposes of manufactures4 ,it is also frequently desirable1 by means of a weak but continued power1 to produce a severe but instantaneous effect4 ,thus a blow may be required to be given by the muscular action of a man's arm with a force to which1 unaided by mechanical contrivance1 its strength would be entirely inadequate4 ,in all these cases1 it is evident that the object to be attained is1 an effectual method of accumulating the energy of the power so as to make it available after the action by which it has been produced has ceased4 ,thus1 in the case in which the load is at periodical intervals withdrawn from the machine1 if the force of the power could be imparted to something by which it would be preserved1 so as to be brought against the load when it again returned1 the inconvenience would be removed4 ,in like manner1 in the case where the power itself is subject to intermission1 if a part of the force which it exerts in its intervals of action could be accumulated and preserved1 it might be brought to bear upon the machine during its periods of suspension4 ,by the same means of accumulating force1 the strength of an infant1 by repeated efforts1 might produce effects which would be vainly attempted by the single and momentary action of the strongest man4 7#cji47 ,the property of inertia1 explained and illustrated in the third and fourth chapters of this volume furnishes an easy and effectual method of accomplishing this4 ,a mass of matter retains1 by virtue of its inertia1 the whole of any force which may have been given to it1 except that part of which friction and the atmospheric resistance deprives it4 ,by contrivances which are well known and present no difficulty1 the part of the moving force thus lost may be rendered comparatively small1 and the moving mass may be regarded as retaining nearly the whole of the force impressed upon it4 ,to render this method of accumulating force fully intelligible1 let us first imagine a polished level plane on which a heavy globe of metal1 also polished1 is placed4 ,it is evident that the globe will remain at rest on any part of the plane without a tendency to move in any direction4 ,as the friction is nearly removed by the polish of the surfaces1 the globe will be easily moved by the least force applied to it4 ,suppose a slight impulse given to it1 which will cause it to move at the rate of one foot in a second4 ,setting aside the effects of friction1 it will continue to move at this rate for any length of time4 ,the same impulse repeated will increase its speed to two feet per second4 ,a third impulse to three feet1 and so on4 ,thus #aj1jjj repetitions of the impulse will cause it to move at the rate of #aj1jjj feet per second4 ,if the body to which these impulses were communicated were a cannon ball1 it might1 by a constant repetition of the impelling force1 be at length made to move with as much force as if it were projected from the most powerful piece of ordnance4 ,the force with which the ball in such a case would strike a building might be sufficient to reduce it to ruins1 and yet such force would be nothing more than the accumulation of a number of weak efforts not beyond the power of a child to exert1 which are stored up1 and preserved1 as it were1 by the moving mass1 and thereby brought to bear1 at the same moment1 upon the point to which the force is directed4 ,it is the sum of a number of actions exerted successively1 and1 during a long interval1 brought into operation at one and the same moment4 ,but the case which is here supposed cannot actually occur2 because we have not usually any practical means of moving a body for any considerable time in the same direction without much friction1 and without encountering numerous obstacles which would impede its progress4 ,it is not1 however1 essential to the effect which is to be produced1 that the motion should be in a straight line4 ,if a leaden weight be attached to the end of a light rod or cord1 and be whirled by the force of the arm in a circle1 it will gradually acquire increased speed and force1 and at length may receive an impetus which would cause it to penetrate a piece of board as effectually as if it were discharged from a musket4 ,the force of a hammer or sledge depends partly on its weight1 but much more on the principle just explained4 ,were it allowed merely to fall by the force of its weight upon the head of a nail1 or upon a bar of heated iron which is to be flattened1 an inconsiderable effect would be produced4 ,but when it is wielded by the arm of a man1 it receives at every moment of its motion increased force1 which is finally expended in a single instant on the head of the nail1 or on the bar of iron4 ,the effects of flails in threshing1 of clubs1 whips1 canes1 and instruments for striking1 axes1 hatchets1 cleavers1 and all instruments which cut by a blow1 depend on the same principle1 and are similarly explained4 ,the bow-string which impels the arrow does not produce its effect at once4 ,it continues to act upon the shaft until it resumes its straight position1 and then the arrow takes flight with the force accumulated during the continuance of the action of the string1 from the moment it was disengaged from the finger of the bow-man4 ,fire-arms themselves act upon a similar principle1 as also the air-gun and steam-gun4 ,in these instruments the ball is placed in a tube1 and suddenly exposed to the pressure of a highly elastic fluid1 either produced by explosion as in fire-arms1 by previous condensation as in the air-gun1 or by the evaporation of highly heated liquids as in the steam-gun4 ,but in every case this pressure continues to act upon it until it leaves the mouth of the tube1 and then it departs with the whole force communicated to it during its passage along the tube4 7#caj47 ,from all these considerations it will easily be perceived that a mass of inert matter may be regarded as a magazine in which force may be deposited and accumulated1 to be used in any way which may be necessary4 ,for many reasons1 which will be sufficiently obvious1 the form commonly given to the mass of matter used for this purpose in machinery is that of a wheel1 in the rim of which it is principally collected4 ,conceive a massive ring of metal1 fig4 #adi41 connected with a central box or nave by light spokes1 and turning on an axis with little friction4 ,such an apparatus is called a fly-wheel4 ,if any force be applied to it1 with that force 7making some slight deduction for friction7 it will move1 and will continue to move until some obstacle be opposed to its motion1 which will receive from it a part of the force it has acquired4 ,the uses of this apparatus will be easily understood by examples of its application4 ,suppose that a heavy stamper or hammer is to be raised to a certain height1 and thence to be allowed to fall1 and that the power used for this purpose is a water-wheel4 ,while the stamper ascends1 the power of the wheel is nearly balanced by its weight1 and the motion of the machine is slow4 ,but the moment the stamper is disengaged and allowed to fall1 the power of the wheel1 having no resistance1 nor any object on which to expend itself1 suddenly accelerates the machine1 which moves with a speed proportioned to the amount of the power1 until it again engages the stamper1 when its velocity is as suddenly checked4 ,every part suffers a strain1 and the machine moves again slowly until it discharges its load1 when it is again accelerated1 and so on4 ,in this case1 besides the certainty of injury and wear1 and the probability of fracture from the sudden and frequent changes of velocity1 nearly the whole force exerted by the power in the intervals between the commencement of each descent of the stamper and the next ascent is lost4 ,these defects are removed by a fly-wheel4 ,when the stamper is discharged1 the energy of the power is expended in moving the wheel1 which1 by reason of its great mass1 will not receive an undue velocity4 ,in the interval between the descent and ascent of the stamper1 the force of the power is lodged in the heavy rim of the fly-wheel4 ,when the stamper is again taken up by the machine1 this force is brought to bear upon it1 combined with the immediate power of the water-wheel1 and the stamper is elevated with nearly the same velocity as that with which the machine moved in the interval of its descent4 7#caa47 ,in many cases1 when the moving power is not subject to variation1 the efficacy of the machine to transmit it to the working point is subject to continual change4 ,the several parts of every machine have certain periods of motion1 in which they pass through a variety of positions1 to which they continually return after stated intervals4 ,in these different positions the effect of the power transmitted to the working point is different2 and cases even occur in which this effect is altogether annihilated1 and the machine is brought into a predicament in which the power loses all influence over the weight4 ,in such cases the aid of a fly-wheel is effectual and indispensable4 ,in those phases of the machine1 which are most favourable to the transmission of force1 the fly-wheel shares the effect of the power with the load1 and retaining the force thus received directs it upon the load at the moments when the transmission of power by the machine is either feeble or altogether suspended4 ,these general observations will1 perhaps1 be more clearly apprehended by an example of an application of the fly-wheel1 in a case such as those now alluded to4 ,let ,a ,b ,c ,d ,e ,f1 fig4 #aej41 be a crank1 which is a double winch 77#beb47 and fig4 #hi471 by which an axle1 ,a ,b ,e ,f1 is to be turned4 ,attached to the middle of ,c ,d by a joint is a rod1 which is connected with a beam1 worked with an alternate motion on a centre1 like the brake of a pump1 and driven by any constant power1 such as a steam-engine4 ,the bar ,c ,d is to be carried with a circular motion round the axis ,a ,e4 ,let the machine1 viewed in the direction ,a ,b ,e ,f of the axis1 be conceived to be represented in fig4 #aea41 where ,a represents the centre round which the motion is to be produced1 and ,g the point where the connecting rod ,g ,h is attached to the arm of the crank4 ,the circle through which ,g is to be urged by the rod is represented by the dotted line4 ,in the position represented in fig4 #aea41 the rod acting in the direction ,h ,g has its full power to turn the crank ,g ,a round the centre ,a4 ,as the crank comes into the position represented in fig4 #aeb41 this power is diminished1 and when the point ,g comes immediately below ,a1 as in fig4 #aec41 the force in the direction ,h ,g has no effect in turning the crank round ,a1 but1 on the contrary1 is entirely expended in pulling the crank in the direction ,a ,g1 and1 therefore1 only acts upon the pivots or gudgeons which support the axle4 ,at this crisis of the motion1 therefore1 the whole effective energy of the power is annihilated4 ,after the crank has passed to the position represented in fig4 #aed41 the direction of the force which acts upon the connecting rod is changed1 and now the crank is drawn upward in the direction ,g ,h4 ,in this position the moving force has some efficacy to produce rotation round ,a1 which efficacy continually increases until the crank attains the position shown in fig4 #aee41 when its power is greatest4 ,passing from this position its efficacy is continually diminished1 until the point ,g comes immediately above the axis ,a1 fig4 #aef4 ,here again the power loses all its efficacy to turn the axle4 ,the force in the direction ,g ,h or ,h ,g can obviously produce no other effect than a strain upon the pivots or gudgeons4 ,in the critical situations represented in fig4 #aec41 and fig4 #aef41 the machine would be incapable of moving1 were the immediate force of the power the only impelling principle4 ,but having been previously in motion by virtue of the inertia of its various parts1 it has a tendency to continue in motion2 and if the resistance of the load and the effects of friction be not too great1 this disposition to preserve its state of motion will extricate the machine from the dilemma in which it is involved in the cases just mentioned1 by the peculiar arrangement of its parts4 ,in many cases1 however1 the force thus acquired during the phases of the machine1 in which the power is active1 is insufficient to carry it through the dead points 7fig4 #aec4 and fig4 #aef472 and in all cases the motion would be very unequal1 being continually retarded as it approached these points1 and continually accelerated after it passed them4 ,a fly-wheel attached to the axis ,a1 or to some other part of the machinery1 will effectually remove this defect4 ,when the crank assumes the positions in fig4 #aea4 and fig4 #aee41 the power is in full play upon it1 and a share of the effect is imparted to the massive rim of the fly-wheel4 ,when the crank gets into the predicament exhibited in fig4 #aec4 and fig4 #aef41 the momentum which the fly-wheel received when the crank acted with most advantage1 immediately extricates the machine1 and1 carrying the crank beyond the dead point1 brings the power again to bear upon it4 ,the astonishing effects of a fly-wheel1 as an accumulator of force1 have led some into the error of supposing that such an apparatus increases the actual power of a machine4 ,it is hoped1 however1 that after what has been explained respecting the inertia of matter and the true effects of machines1 the reader will not be liable to a similar mistake4 ,on the contrary1 as a fly cannot act without friction1 and as the amount of the friction1 like that of inertia1 is in proportion to the weight1 a portion of the actual moving force must unavoidably be lost by the use of a fly4 ,in cases1 however1 where a fly is properly applied this loss of power is inconsiderable1 compared with the advantageous distribution of what remains4 ,illustration3 ,c4 ,varley1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,as an accumulator of force1 a fly can never have more force than has been applied to put it in motion4 ,in this respect it is analogous to an elastic spring1 or the force of condensed air1 or any other power which derives its existence from causes purely mechanical4 ,in bending a spring a gradual expenditure of power is necessary4 ,on the recoil this power is exerted in a much shorter time than that consumed in its production1 but its total amount is not altered4 ,air is condensed by a succession of manual efforts1 one of which alone would be incapable of projecting a leaden ball with any considerable force1 and all of which could not be immediately applied to the ball at the same instant4 ,but the reservoir of condensed air is a magazine in which a great number of such efforts are stored up1 so as to be brought at once into action4 ,if a ball be exposed to their effect1 it may be projected with a destructive force4 ,in mills for rolling metal the fly-wheel is used in this way4 ,the water-wheel or other moving power is allowed for some time to act upon the fly-wheel alone1 no load being placed upon the machine4 ,a force is thus gained which is sufficient to roll a large piece of metal1 to which without such means the mill would be quite inadequate4 ,in the same manner a force may be gained by the arm of a man acting on a fly for a few seconds1 sufficient to impress an image on a piece of metal by an instantaneous stroke4 ,the fly is1 therefore1 the principal agent in coining presses4 7#cab47 ,the power of a fly is often transmitted to the working point by means of a screw4 ,at the extremities of the cross arm ,a ,b1 fig4 #aeg41 which works the screw1 two heavy balls of metal are placed4 ,when the arm ,a ,b is whirled round1 those masses of metal acquire a momentum1 by which the screw1 being driven downward1 urges the die with an immense force against the substance destined to receive the impression4 ,some engines used in coining have flies with arms four feet long1 bearing one hundred weight at each of their extremities4 ,by turning such an arm at the rate of one entire circumference in a second1 the die will be driven against the metal with the same force as that with which #gejj pounds weight would fall from the height of #af feet2 an enormous power1 if the simplicity and compactness of the machine be considered4 ,the place to be assigned to a fly-wheel relatively to the other parts of the machinery is determined by the purpose for which it is used4 ,if it be intended to equalise the action1 it should be near the working point4 ,thus1 in a steam-engine1 it is placed on the crank which turns the axle by which the power of the engine is transmitted to the object it is finally designed to affect4 ,on the contrary1 in handmills1 such as those commonly used for grinding coffee1 c41 it is placed upon the axis of the winch by which the machine is worked4 ,the open work of fenders1 fire-grates1 and similar ornamental articles constructed in metal1 is produced by the action of a fly1 in the manner already described4 ,the cutting tool1 shaped according to the pattern to be executed1 is attached to the end of the screw2 and the metal being held in a proper position beneath it1 the fly is made to urge the tool downwards with such force as to stamp out pieces of the required figure4 ,when the pattern is complicated1 and it is necessary to preserve with exactness the relative situation of its different parts1 a number of punches are impelled together1 so as to strike the entire piece of metal at the same instant1 and in this manner the most elaborate open work is executed by a single stroke4 ,,chap4 ,,xviii4 ,,mechanical ,,contrivances ,,for ,,modifying ,,motion4 7#cac47 ,the classes of simple machines denominated mechanic powers1 have relation chiefly to the peculiar principle which determines the action of the power on the weight or resistance4 ,in explaining this arrangement various other reflections have been incidentally mixed up with our investigations2 yet still much remains to be unfolded before the student can form a just notion of those means by which the complex machinery used in the arts and manufactures so effectually attains the ends1 to the accomplishment of which it is directed4 ,by a power of a given energy to oppose a resistance of a different energy1 or by a moving principle having a given velocity to generate another velocity of a different amount1 is only one of the many objects to be effected by a machine4 ,in the arts and manufactures the kind of motion produced is generally of greater importance than its rate4 ,the latter may affect the quantity of work done in a given time1 but the former is essential to the performance of the work in any quantity whatever4 ,in the practical application of machines1 the object to be attained is generally to communicate to the working point some peculiar sort of motion suitable to the uses for which the machine is intended2 but it rarely happens that the moving power has this sort of motion4 ,hence1 the machine must be so contrived that1 while that part on which this power acts is capable of moving in obedience to it1 its connection with the other parts shall be such that the working point may receive that motion which is necessary for the purposes to which the machine is applied4 ,to give a perfect solution of this problem it would be necessary to explain1 first1 all the varieties of moving powers which are at our disposal2 secondly1 all the variety of motions which it may be necessary to produce2 and1 thirdly1 to show all the methods by which each variety of prime mover may be made to produce the several species of motion in the working point4 ,it is obvious that such an enumeration would be impracticable1 and even an approximation to it would be unsuitable to the present treatise4 ,nevertheless1 so much ingenuity has been displayed in many of the contrivances for modifying motion1 and an acquaintance with some of them is so essential to a clear comprehension of the nature and operation of complex machines1 that it would be improper to omit some account of those at least which most frequently occur in machinery1 or which are most conspicuous for elegance and simplicity4 ,the varieties of motion which most commonly present themselves in the practical application of mechanics may be divided into rectilinear and rotatory4 ,in rectilinear motion the several parts of the moving body proceed in parallel straight lines with the same speed4 ,in rotatory motion the several points revolve round an axis1 each performing a complete circle1 or similar parts of a circle1 in the same time4 ,each of these may again be resolved into continued and reciprocating4 ,in a continued motion1 whether rectilinear or rotatory1 the parts move constantly in the same direction1 whether that be in parallel straight lines1 or in rotation on an axis4 ,in reciprocating motion the several parts move alternately in opposite directions1 tracing the same spaces from end to end continually4 ,thus1 there are four principal species of motion which more frequently than any others act upon1 or are required to be transmitted by1 machines3-- #a4 ,continued rectilinear motion4 #b4 ,reciprocating rectilinear motion4 #c4 ,continued circular motion4 #d4 ,reciprocating circular motion4 ,these will be more clearly understood by examples of each kind4 ,continued rectilinear motion is observed in the flowing of a river1 in a fall of water1 in the blowing of the wind1 in the motion of an animal upon a straight road1 in the perpendicular fall of a heavy body1 in the motion of a body down an inclined plane4 ,reciprocating rectilinear motion is seen in the piston of a common syringe1 in the rod of a common pump1 in the hammer of a pavier1 the piston of a steam-engine1 the stampers of a fulling mill4 ,continued circular motion is exhibited in all kinds of wheel-work1 and is so common1 that to particularise it is needless4 ,reciprocating circular motion is seen in the pendulum of a clock1 and in the balance-wheel of a watch4 ,we shall now explain some of the contrivances by which a power having one of these motions may be made to communicate either the same species of motion changed in its velocity or direction1 or any of the other three kinds of motion4 7#cad47 ,by a continued rectilinear motion another continued rectilinear motion in a different direction may be produced1 by one or more fixed pulleys4 ,a cord passed over these1 one end of it being moved by the power1 will transmit the same motion unchanged to the other end4 ,if the directions of the two motions cross each other1 one fixed pulley will be sufficient2 see fig4 #aac41 where the hand takes the direction of the one motion1 and the weight that of the other4 ,in this case the pulley must be placed in the angle at which the directions of the two motions cross each other4 ,if this angle be distant from the places at which the objects in motion are situate1 an inconvenient length of rope may be necessary4 ,in this case the same may be effected by the use of two pulleys1 as in fig4 #aeh4 ,if the directions of the two motions be parallel1 two pulleys must be used as in fig4 #aeh41 where ,p ,a is one motion1 and ,b ,w the other4 ,in these cases the axles of the two wheels are parallel4 ,it may so happen that the directions of the two motions neither cross each other nor are parallel4 ,this would happen1 for example1 if the direction of one were upon the paper in the line ,p ,a1 while the other were perpendicular to the paper from the point ,o4 ,in this case two pulleys should be used1 the axle of one ,o being perpendicular to the paper1 while the axle of the other ,o should be on the paper4 ,this will be evident by a little reflection4 ,in general1 the axle of each pulley must be perpendicular to the two directions in which the rope passes from its groove2 and by due attention to this condition it will be perceived1 that a continued rectilinear motion may be transferred from any one direction to any other direction1 by means of a cord and two pulleys1 without changing its velocity4 ,if it be necessary to change the velocity1 any of the systems of pulleys described in chap4 ,,xv4 may be used in addition to the fixed pulleys4 ,by the wheel and axle any one continued rectilinear motion may be made to produce another in any other direction1 and with any other velocity4 ,it has been already explained 7#bej47 that the proportion of the velocity of the power to that of the weight is as the diameter of the wheel to the diameter of the axle4 ,the thickness of the axle being therefore regulated in relation to the size of the wheel1 so that their diameters shall have that proportion which subsists between the proposed velocities1 one condition of the problem will be fulfilled4 ,the rope coiled upon the axle may be carried1 by means of one or more fixed pulleys1 into the direction of one of the proposed motions1 while that which surrounds the wheel is carried into the direction of the other by similar means4 7#cae47 ,by the wheel and axle a continued rectilinear motion may be made to produce a continued rotatory motion1 or vice vers4 ,if the power be applied by a rope coiled upon the wheel1 the continued motion of the power in a straight line will cause the machine to have a rotatory motion4 ,again1 if the weight be applied by a rope coiled upon the axle1 a power having a rotatory motion applied to the wheel will cause the continued ascent of the weight in a straight line4 ,continued rectilinear and rotatory motions may be made to produce each other1 by causing a toothed wheel to work in a straight bar1 called a rack1 carrying teeth upon its edge4 ,such an apparatus is represented in fig4 #aei4 ,in some cases the teeth of the wheel work in the links of a chain4 ,the wheel is then called a rag-wheel1 fig4 #afj4 ,straps1 bands1 or ropes1 may communicate rotation to a wheel1 by their friction in a groove upon its edge4 ,a continued rectilinear motion is produced by a continued circular motion in the case of a screw4 ,the lever which turns the screw has a continued circular motion1 while the screw itself advances with a continued rectilinear motion4 ,the continued rectilinear motion of a stream of water acting upon a wheel produces continued circular motion in the wheel1 fig4 #ic1 #id1 #ie4 ,in like manner the continued rectilinear motion of the wind produces a continued circular motion in the arms of a windmill4 ,cranes for raising and lowering heavy weights convert a circular motion of the power into a continued rectilinear motion of the weight4 7#caf47 ,continued circular motion may produce reciprocating rectilinear motion1 by a great variety of ingenious contrivances4 ,reciprocating rectilinear motion is used when heavy stampers are to be raised to a certain height1 and allowed to fall upon some object placed beneath them4 ,this may be accomplished by a wheel bearing on its edge curved teeth1 called wipers4 ,the stamper is furnished with a projecting arm or peg1 beneath which the wipers are successively brought by the revolution of the wheel4 ,as the wheel revolves the wiper raises the stamper1 until its extremity passes the extremity of the projecting arm of the stamper1 when the latter immediately falls by its own weight4 ,it is then taken up by the next wiper1 and so the process is continued4 ,a similar effect is produced if the wheel be partially furnished with teeth1 and the stamper carry a rack in which these teeth work4 ,such an apparatus is represented in fig4 #afa4 ,it is sometimes necessary that the reciprocating rectilinear motion shall be performed at a certain varying rate in both directions4 ,this may be accomplished by the machine represented in fig4 #afb4 ,a wheel upon the axle ,c turns uniformly in the direction ,a ,b ,d ,e4 ,a rod mn moves in guides1 which only permit it to ascend and descend perpendicularly4 ,its extremity m rests upon a path or groove raised from the face of the wheel1 and shaped into such a curve that as the wheel revolves the rod mn shall be moved alternately in opposite directions through the guides1 with the required velocity4 ,the manner in which the velocity varies will depend on the form given to the groove or channel raised upon the face of the wheel1 and this may be shaped so as to give any variation to the motion of the rod mn which may be required for the purpose to which it is to be applied4 ,the rose-engine in the turning-lathe is constructed on this principle4 ,it is also used in spinning machinery4 ,it is often necessary that the rod to which reciprocating motion is communicated shall be urged by the same force in both directions4 ,a wheel partially furnished with teeth1 acting on two racks placed on different sides of it1 and both connected with the bar or rod to which the reciprocating motion is to be communicated1 will accomplish this4 ,such an apparatus is represented in fig4 #afc41 and needs no further explanation4 ,another contrivance for the same purpose is shown in fig4 #afd41 where ,a is a wheel turned by a winch ,h1 and connected with a rod or beam moving in guides by the joint ab4 ,as the wheel ,a is turned by the winch ,h the beam is moved between the guides alternately in opposite directions1 the extent of its range being governed by the length of the diameter of the wheel4 ,such an apparatus is used for grinding and polishing plane surfaces1 and also occurs in silk machinery4 ,an apparatus applied by ,m4 ,zureda in a machine for pricking holes in leather is represented in fig4 #afe4 ,the wheel ,a ,b has its circumference formed into teeth1 the shape of which may be varied according to the circumstances under which it is to be applied4 ,one extremity of the rod ab rests upon the teeth of the wheel upon which it is pressed by a spring at the other extremity4 ,when the wheel revolves1 it communicates to this rod a reciprocating rectilinear motion4 ,leupold has applied this mechanism to move the pistons of pumps4#e ,upon the vertical axis of a horizontal hydraulic wheel is fixed another horizontal wheel1 which is furnished with seven teeth in the manner of a crown wheel 7#bfc474 ,these teeth are shaped like inclined planes1 the intervals between them being equal to the length of the planes4 ,projecting arms attached to the piston rods rest upon the crown of this wheel2 and1 as it revolves1 the inclined surfaces of the teeth1 being forced under the arm1 raise the rod upon the principle of the wedge4 ,to diminish the obstruction arising from friction1 the projecting arms of the piston rods are provided with rollers1 which run upon the teeth of the wheel4 ,in one revolution of the wheel each piston makes as many ascents and descents as there are teeth4 #e ,theatrum ,machinarum1 tom4 ii4 pl4 #cf4 fig4 #c4 7#cag47 ,wheel-work furnishes numerous examples of continued circular motion round one axis1 producing continued circular motion round another4 ,if the axles be in parallel directions1 and not too distant1 rotation may be transmitted from one to the other by two spur-wheels 7#bfc472 and the relative velocities may be determined by giving a corresponding proportion to the diameter of the wheels4 ,if a rotary motion is to be communicated from one axis to another parallel to it1 and at any considerable distance1 it cannot in practice be accomplished by wheels alone1 for their diameters would be too large4 ,in this case a strap or chain is carried round the circumferences of both wheels4 ,if they are intended to turn in the same direction1 the strap is arranged as in fig4 #ajj42 but if in contrary directions it is crossed1 as in fig4 #aja4 ,in this case1 as with toothed wheels1 the relative velocities are determined by the proportion of the diameters of the wheels4 ,if the axles be distant and not parallel1 the cord1 by which the motion is transmitted1 must be passed over grooved wheels1 or fixed pulleys1 properly placed between the two axles4 ,it may happen that the strain upon the wheel1 to which the motion is to be transmitted1 is too great to allow of a strap or cord being used4 ,in this case1 a shaft extending from the one axis to another1 and carrying two bevelled wheels 7#bfc471 will accomplish the object4 ,one of these bevelled wheels is placed upon the shaft near to1 and in connection with1 the wheel from which the motion is to be taken1 and the other at a part of it near to1 and in connection with1 that wheel to which the motion is to be conveyed1 fig4 #aff4 ,the methods of transmitting rotation from one axis to another perpendicular to it1 by crown and by bevelled wheels1 have been explained in 7#bfc474 ,the endless screw 7#bii47 is a machine by which a rotatory motion round one axis may communicate a rotatory motion round another perpendicular to it4 ,the power revolves round an axis coinciding with the length of the screw1 and the axis of the wheel driven by the screw is at right angles to this4 ,the axis to which rotation is to be given1 or from which it is to be taken1 is sometimes variable in its position4 ,in such cases1 an ingenious contrivance1 called a universal joint1 invented by the celebrated ,dr4 ,hooke1 may be used4 ,the two shafts or axles ,a ,b1 fig4 #afg41 between which the motion is to be communicated1 terminate in semicircles1 the diameters of which1 ,c ,d and ,e ,f1 are fixed in the form of a cross1 their extremities moving freely in bushes placed in the extremities of the semicircles4 ,thus1 while the central cross remains unmoved1 the shaft ,a and its semicircular end may revolve round ,c ,d as an axis2 and the shaft ,b and its semicircular end may revolve round ,e ,f as an axis4 ,if the shaft ,a be made to revolve without changing its direction1 the points ,c ,d will move in a circle whose centre is at the middle of the cross4 ,the motion thus given to the cross will cause the points ,e ,f to move in another circle round the same centre1 and hence the shaft ,b will be made to revolve4 ,illustration3 ,c4 ,varley1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,this instrument will not transmit the motion if the angle under the directions of the shafts be less than #adj4 ,in this case a double joint1 as represented in fig4 #afh41 will answer the purpose4 ,this consists of four semicircles united by two crosses1 and its principle and operation is the same as in the last case4 ,universal joints are of great use in adjusting the position of large telescopes1 where1 while the observer continues to look through the tube1 it is necessary to turn endless screws or wheels whose axes are not in an accessible position4 ,the cross is not indispensably necessary in the universal joint4 ,a hoop1 with four pins projecting from it at four points equally distant from each other1 or dividing the circle of the hoop into four equal arcs1 will answer the purpose4 ,these pins play in the bushes of the semicircles in the same manner as those of the cross4 ,the universal joint is much used in cotton-mills1 where shafts are carried to a considerable distance from the prime mover1 and great advantage is gained by dividing them into convenient lengths1 connected by a joint of this kind4 7#cah47 ,in the practical application of machinery1 it is often necessary to connect a part having a continued circular motion with another which has a reciprocating or alternate motion1 so that either may move the other4 ,there are many contrivances by which this may be effected4 ,one of the most remarkable examples of it is presented in the scapements of watches and clocks4 ,in this case1 however1 it can scarcely be said with strict propriety that it is the rotation of the scapement-wheel 7#bff47 which communicates the vibration to the balance-wheel or pendulum4 ,that vibration is produced in the one case by the peculiar nature of the spiral spring fixed upon the axis of the balance-wheel1 and in the other case by the gravity of the pendulum4 ,the force of the scapement-wheel only maintains the vibration1 and prevents its decay by friction and atmospheric resistance4 ,nevertheless1 between the two parts thus moving there exists a mechanical connection1 which is generally brought within the class of contrivances now under consideration4 ,a beam vibrating on an axis1 and driven by the piston of a steam-engine1 or any other power1 may communicate rotary motion to an axis by a connector and a crank4 ,this apparatus has been already described in 7#caa474 ,every steam-engine which works by a beam affords an example of this4 ,the working beam is generally placed over the engine1 the piston rod being attached to one end of it1 while the connecting rod unites the other end with the crank4 ,in boat-engines1 however1 this position would be inconvenient1 requiring more room than could easily be spared4 ,the piston rod1 in these cases1 is1 therefore1 connected with the end of the beam by long rods1 and the beam is placed beside and below the engine4 ,the use of a fly-wheel here would also be objectionable4 ,the effect of the dead points explained in 7#caa47 is avoided without the aid of a fly1 by placing two cranks upon the revolving axle1 and working them by two pistons4 ,the cranks are so placed that when either is at its dead point1 the other is in its most favourable position4 ,a wheel ,a1 fig4 #afi41 armed with wipers1 acting upon a sledge-hammer ,b1 fixed upon a centre or axle ,c1 will1 by a continued rotatory motion1 give the hammer the reciprocating motion necessary for the purposes to which it is applied4 ,the manner in which this acts must be evident on inspecting the figure4 ,the treddle of the lathe furnishes an obvious example of a vibrating circular motion producing a continued circular one4 ,the treddle acts upon a crank1 which gives motion to the principal wheel1 in the same manner as already described in reference to the working beam and crank in the steam-engine4 ,by the following ingenious mechanism an alternate or vibrating force may be made to communicate a circular motion continually in the same direction4 ,let ,a ,b1 fig4 #agj41 be an axis receiving an alternate motion from some force applied to it1 such as a swinging weight4 ,two ratchet wheels 7#bec47 m and n are fixed on this axle1 their teeth being inclined in opposite directions4 ,two toothed wheels ,c and ,d are likewise placed upon it1 but so arranged that they turn upon the axle with a little friction4 ,these wheels carry two catches p1 q1 which fall into the teeth of the ratchet wheels m1 n1 but fall on opposite sides conformably to the inclination of the teeth already mentioned4 ,the effect of these catches is1 that if the axis be made to revolve in one direction1 one of the two toothed wheels is always compelled 7by the catch against which the motion is directed7 to revolve with it1 while the other is permitted to remain stationary in obedience to any force sufficiently great to overcome its friction with the axle on which it is placed4 ,the wheels ,c and ,d are both engaged by bevelled teeth 7#bfc47 with the wheel ,e4 ,according to this arrangement1 in whichever direction the axis ,a ,b is made to revolve1 the wheel ,e will continually turn in the same direction1 and1 therefore1 if the axle ,a ,b be made to turn alternately in the one direction and the other1 the wheel ,e will not change the direction of its motion4 ,let us suppose that the axle ,a ,b is turned against the catch p4 ,the wheel ,c will then be made to turn with the axle4 ,this will drive the wheel ,e in the same direction4 ,the teeth on the opposite side of the wheel ,e being engaged with those of the wheel ,d1 the latter will be turned upon the axle1 the friction1 which alone resists its motion in that direction1 being overcome4 ,let the motion of the axle ,a ,b be now reversed4 ,since the teeth of the ratchet wheel n are moved against the catch q1 the wheel ,d will be compelled to revolve with the axle4 ,the wheel ,e will be driven in the same direction as before1 and the wheel ,c will be moved on the axle ,a ,b1 and in a contrary direction to the motion of the axle1 the friction being overcome by the force of the wheel ,e4 ,thus1 while the axle ,a ,b is turned alternately in the one direction and the other1 the wheel ,e is constantly moved in the same direction4 ,it is evident that the direction in which the wheel ,e moves may be reversed by changing the position of the ratchet wheels and catches4 7#cai47 ,it is often necessary to communicate an alternate circular motion1 like that of a pendulum1 by means of an alternate motion in a straight line4 ,a remarkable instance of this occurs in the steam engine4 ,the moving force in this machine is the pressure of steam1 which impels a piston from end to end alternately in a cylinder4 ,the force of this piston is communicated to the working beam by a strong rod1 which passes through a collar in one end of the piston4 ,since it is necessary that the steam included in the cylinder should not escape between the piston rod and the collar through which it moves1 and yet1 that it should move as freely and be subject to as little resistance as possible1 the rod is turned so as to be truly cylindrical1 and is well polished4 ,it is evident that1 under these circumstances1 it must not be subject to any lateral or cross strain1 which would bend it towards one side or the other of the cylinder4 ,but the end of the beam to which it communicates motion1 if connected immediately with the rod by a joint1 would draw it alternately to the one side and the other1 since it moves in the arc of a circle1 the centre of which is at the centre of the beam4 ,it is necessary1 therefore1 to contrive some method of connecting the rod and the end of the beam1 so that while the one shall ascend and descend in a straight line1 the other may move in the circular arc4 ,the method which first suggests itself to accomplish this is1 to construct an arch-head upon the end of the beam1 as in fig4 #aga4 ,let ,c be the centre on which the beam works1 and let ,b ,d be an arch attached to the end of the beam1 being a part of a circle having ,c for its centre4 ,to the highest point ,b of the arch a chain is attached1 which is carried upon the face of the arch ,b ,a1 and the other end of which is attached to the piston rod4 ,under these circumstances it is evident1 that when the force of the steam impels the piston downwards1 the chain ,p ,a ,b will draw the end of the beam down1 and will1 therefore1 elevate the other end4 ,when the steam-engine is used for certain purposes1 such as pumping1 this arrangement is sufficient4 ,the piston in that case is not forced upwards by the pressure of steam4 ,during its ascent it is not subject to the action of any force of steam1 and the other end of the beam falls by the weight of the pump-rods drawing the piston1 at the opposite end ,a1 to the top of the cylinder4 ,thus the machine is in fact passive during the ascent of the piston1 and exerts its power only during the descent4 ,if the machine1 however1 be applied to purposes in which a constant action of the moving force is necessary1 as is always the case in manufactures1 the force of the piston must drive the beam in its ascent as well as in its descent4 ,the arrangement just described cannot effect this2 for although a chain is capable of transmitting any force1 by which its extremities are drawn in opposite directions1 yet it is1 from its flexibility1 incapable of communicating a force which drives one extremity of it towards the other4 ,in the one case the piston first pulls down the beam1 and then the beam pulls up the piston4 ,the chain1 because it is inextensible1 is perfectly capable of both these actions2 and being flexible1 it applies itself to the arch-head of the beam1 so as to maintain the direction of its force upon the piston continually in the same straight line4 ,but when the piston acts upon the beam in both ways1 in pulling it down and pushing it up1 the chain becomes inefficient1 being from its flexibility incapable of the latter action4 ,the problem might be solved by extending the length of the piston rod1 so that its extremity shall be above the beam1 and using two chains2 one connecting the highest point of the rod with the lowest point of the arch-head1 and the other connecting the highest point of the arch-head with a point on the rod below the point which meets the arch-head when the piston is at the top of the cylinder1 fig4 #agb4 ,the connection required may also be made by arming the arch-head with teeth1 fig4 #agc41 and causing the piston rod to terminate in a rack4 ,in cases where1 as in the steam-engine1 smoothness of motion is essential1 this method is objectionable2 and under any circumstances such an apparatus is liable to rapid wear4 ,the method contrived by ,watt1 for connecting the motion of the piston with that of the beam1 is one of the most ingenious and elegant solutions ever proposed for a mechanical problem4 ,he conceived the motion of two straight rods ,a ,b1 ,c ,d1 fig4 #agd41 moving on centres or pivots ,a and ,c1 so that the extremities ,b and ,d would move in the arcs of circles having their centres at ,a and ,c4 ,the extremities ,b and ,d of these rods he conceived to be connected with a third rod ,b ,d united with them by pivots on which it could turn freely4 ,to the system of rods thus connected let an alternate motion on the centres ,a and ,c be communicated3 the points ,b and ,d will move upwards and downwards in the arcs expressed by the dotted lines1 but the middle point ,p of the connecting rod ,b ,d will move upwards and downwards without any sensible deviation from a straight line4 ,to prove this demonstratively would require some abstruse mathematical investigation4 ,it may1 however1 be rendered in some degree apparent by reasoning of a looser and more popular nature4 ,as the point ,b is raised to ,e1 it is also drawn aside towards the right4 ,at the same time the other extremity ,d of the rod ,b ,d is raised to ,e1 and is drawn aside towards the left4 ,the ends of the rod ,b ,d being thus at the same time drawn equally towards opposite sides1 its middle point ,p will suffer no lateral derangement1 and will move directly upwards4 ,on the other hand1 if ,b be moved downwards to ,f1 it will be drawn laterally to the right1 while ,d being moved to ,f will be drawn to the left4 ,hence1 as before1 the middle point ,p sustains no lateral derangement1 but merely descends4 ,thus1 as the extremities ,b and ,d move upwards and downwards in circles1 the middle point ,p moves upwards and downwards in a straight line4#f #f ,in a strictly mathematical sense1 the path of the point ,p is a curve1 and not a straight line2 but in the play given to it in its application to the steam-engine1 it moves through a part only of its entire locus1 and this part extending equally on each side of a point of inflection1 the radius of curvature is infinite1 so that in practice the deviation from a straight line1 when proper proportions are observed in the rods1 is imperceptible4 ,the application of this geometrical principle in the steam-engine evinces much ingenuity4 ,the same arm of the beam usually works two pistons1 that of the cylinder and that of the air-pump4 ,the apparatus is represented on the arm of the beam in fig4 #age4 ,the beam moves alternately upwards and downwards on its axis ,a4 ,every point of it1 therefore1 describes a part of a circle of which ,a is the centre4 ,let ,b be the point which divides the arm ,a ,g into two equal parts ,a ,b and ,b ,g2 and let ,c ,d be a straight rod1 equal in length to ,g ,b1 and fixed on a centre or pivot ,c on which it is at liberty to play4 ,the end ,d of this rod is connected by a straight bar with the point ,b1 by pivots on which the rod ,b ,d turns freely4 ,if the beam be now supposed to rise and fall alternately1 the points ,b and ,d will move upwards and downwards in circular arcs1 and1 as already explained with respect to the points ,b ,d1 fig4 #agd41 the middle point ,p of the connecting rod ,b ,d will move upwards and downwards without lateral deflection4 ,to this point one of the piston rods which are to be worked is attached4 ,to comprehend the method of working the other piston1 conceive a rod ,g ,p1 equal in length to ,b ,d1 to be attached to the end ,g of the beam by a pivot on which it moves freely2 and let its extremity ,p be connected with ,d by another rod ,p ,d1 equal in length to ,g ,b1 and playing on points at ,p and ,d4 ,the piston rod of the cylinder is attached to the point ,p1 and this point has a motion precisely similar to that of ,p1 without any lateral derangement1 but with a range in the perpendicular direction twice as great4 ,this will be apparent by conceiving a straight line drawn from the centre ,a of the beam to ,p1 which will also pass through ,p4 ,since ,g ,p is always parallel to ,b ,p1 it is evident that the triangle ,p ,g ,a is always similar to ,p ,b ,a1 and has its sides and angles similarly placed1 but those sides are each twice the magnitude of the corresponding sides of the other triangle4 ,hence the point ,p must be subject to the same changes of position as the point ,p1 with this difference only1 that in the same time it moves over a space of twice the magnitude4 ,in fact1 the line traced by ,p is the same as that traced by ,p1 but on a scale twice as large4 ,this contrivance is usually called the parallel motion1 but the same name is generally applied to all contrivances by which a circular motion is made to produce a rectilinear one4 ,,chap4 ,,xix4 ,,of ,,friction ,,and ,,the ,,rigidity ,,of ,,cordage4 7#cbj47 ,with a view to the simplification of the elementary theory of machines1 the consideration of several mechanical effects of great practical importance has been postponed1 and the attention of the student has been directed exclusively to the way in which the moving power is modified in being transmitted to the resistance independently of such effects4 ,a machine has been regarded as an instrument by which a moving principle1 inapplicable in its existing state to the purpose for which it is required1 may be changed either in its velocity or direction1 or in some other character1 so as to be adapted to that purpose4 ,but in accomplishing this1 the several parts of the machine have been considered as possessing in a perfect degree qualities which they enjoy only in an imperfect degree2 and accordingly the conclusions to which by such reasoning we are conducted are infected with errors1 the amount of which will depend on the degree in which the machinery falls short of perfection in those qualities which theoretically are imputed to it4 ,illustration3 ,c4 ,varley1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,of the several parts of a machine1 some are designed to move1 while others are fixed2 and of those which move1 some have motions differing in quantity and direction from those of others4 ,the several parts1 whether fixed or movable1 are subject to various strains and pressures1 which they are intended to resist4 ,these forces not only vary according to the load which the machine has to overcome1 but also according to the peculiar form and structure of the machine itself4 ,during the operation the surfaces of the movable parts move in immediate contact with the surfaces either of fixed parts or of parts having other motions4 ,if these surfaces were endued with perfect smoothness or polish1 and the several parts subject to strains possessed perfect inflexibility and infinite strength1 then the effects of machinery might be practically investigated by the principles already explained4 ,but the materials of which every machine is formed are endued with limited strength1 and therefore the load which is placed upon it must be restricted accordingly1 else it will be liable to be distorted by the flexure1 or even to be destroyed by the fracture of those parts which are submitted to an undue strain4 ,the surfaces of the movable parts1 and those surfaces with which they move in contact1 cannot in practice be rendered so smooth but that such roughness and inequality will remain as sensibly to impede the motion4 ,to overcome such an impediment requires no inconsiderable part of the moving power4 ,this part is1 therefore1 intercepted before its arrival at the working point1 and the resistance to be finally overcome is deprived of it4 ,the property thus depending on the imperfect smoothness of surfaces1 and impeding the motion of bodies whose surfaces are in immediate contact1 is called friction4 ,before we can form a just estimate of the effects of machinery1 it is necessary to determine the force lost by this impediment1 and the laws which under different circumstances regulate that loss4 ,when cordage is engaged in the formation of any part of a machine1 it has hitherto been considered as possessing perfect flexibility4 ,this is not the case in practice2 and the want of perfect flexibility1 which is called rigidity1 renders a certain quantity of force necessary to bend a cord or rope over the surface of an axle or the groove of a wheel4 ,during the motion of the rope a different part of it must thus be continually bent1 and the force which is expended in producing the necessary flexure must be derived from the moving power1 and is thus intercepted on its way to the working point4 ,in calculating the effects of cordage1 due regard must be had to this waste of power2 and therefore it is necessary to enquire into the laws which govern the flexure of imperfectly flexible ropes1 and the way in which these affect the machines in which ropes are commonly used4 ,to complete1 therefore1 the elementary theory of machinery1 we propose in the present and following chapter to explain the principal laws which determine the effects of friction1 the rigidity of cordage1 and the strength of materials4 7#cba47 ,if a horizontal plane surface were perfectly smooth1 and free from the smallest inequalities1 and a body having a flat surface also perfectly smooth were placed upon it1 any force applied to the latter would put it in motion1 and that motion would continue undiminished as long as the body would remain upon the smooth horizontal surface4 ,but if this surface1 instead of being every where perfectly even1 had in particular places small projecting eminences1 a certain quantity of force would be necessary to carry the moving body over these1 and a proportional diminution in its rate of motion would ensue4 ,thus1 if such eminences were of frequent occurrence1 each would deprive the body of a part of its speed1 so that between that and the next it would move with a less velocity than it had between the same and the preceding one4 ,this decrease being continued by a sufficient number of such eminences encountering the body in succession1 the velocity would at last be so much diminished that the body would not have sufficient force to carry it over the next eminence1 and its motion would thus altogether cease4 ,now1 instead of the eminences being at a considerable distance asunder1 suppose them to be contiguous1 and to be spread in every direction over the horizontal plane1 and also suppose corresponding eminences to be upon the surface of the moving body2 these projections incessantly encountering one another will continually obstruct the motion of the body1 and will gradually diminish its velocity1 until it be reduced to a state of rest4 ,such is the cause of friction4 ,the amount of this resisting force increases with the magnitude of these asperities1 or with the roughness of the surfaces2 but it does not solely depend on this4 ,the surfaces remaining the same1 a little reflection on the method of illustration just adopted1 will show that the amount of friction ought also to depend upon the force with which the surfaces moving one upon the other are pressed together4 ,it is evident1 that as the weight of the body supposed to move upon the horizontal plane is increased1 a proportionally greater force will be necessary to carry it over the obstacles which it encounters1 and therefore it will the more speedily be deprived of its velocity and reduced to a state of rest4 7#cbb47 ,thus we might predict with probability1 that which accurate experimental enquiry proves to be true1 that the resistance from friction depends conjointly on the roughness of the surfaces and the force of the pressure4 ,when the surfaces are the same1 a double pressure will produce a double amount of friction1 a treble pressure a treble amount of friction1 and so on4 ,experiment also1 however1 gives a result which1 at least at first view1 might not have been anticipated from the mode of illustration we have adopted4 ,it is found that the resistance arising from friction does not at all depend on the magnitude of the surface of contact2 but provided the nature of the surfaces and the amount of pressure remain the same1 this resistance will be equal1 whether the surfaces which move one upon the other be great or small4 ,thus1 if the moving body be a flat block of wood1 the face of which is equal to a square foot in magnitude1 and the edge of which does not exceed a square inch1 it will be subject to the same amount of friction1 whether it move upon its broad face or upon its narrow edge4 ,if we consider the effect of the pressure in each case1 we shall be able to perceive why this must be the case4 ,let us suppose the weight of the block to be #add ounces4 ,when it rests upon its face1 a pressure to this amount acts upon a surface of #add square inches1 so that a pressure of one ounce acts upon each square inch4 ,the total resistance arising from friction will1 therefore1 be #add times that resistance which would be produced by a surface of one square inch under a pressure of one ounce4 ,now1 suppose the block placed upon its edge1 there is then a pressure of #add ounces upon a surface equal to one square inch4 ,but it has been already shown1 that when the surface is the same1 the friction must increase in proportion to the pressure4 ,hence we infer that the friction produced in the present case is #add times the friction which would be produced by a pressure of one ounce acting on one square inch of surface1 which is the same resistance as that which the body was proved to be subject to when resting on its face4 ,these two laws1 that friction is independent of the magnitude of the surface1 and is proportional to the pressure when the quality of the surfaces is the same1 are useful in practice1 and generally true4 ,in very extreme cases they are1 however1 in error4 ,when the pressure is very intense1 in proportion to the surface1 the friction is somewhat less than it would be by these laws2 and when it is very small in proportion to the surface1 it is somewhat greater4 7#cbc47 ,there are two methods of establishing by experiment the laws of friction1 which have been just explained4 ,first4 ,the surfaces between which the friction is to be determined being rendered perfectly flat1 let one be fixed in the horizontal position on a table ,t ,t1 fig4 #agf42 and let the other be attached to the bottom of a box ,b ,c1 adapted to receive weights1 so as to vary the pressure4 ,let a silken cord ,s ,p1 attached to the box1 be carried parallel to the table over a wheel at ,p1 and let a dish ,d be suspended from it4 ,if no friction existed between the surfaces1 the smallest weight appended to the cord would draw the box towards ,p with a continually increasing speed4 ,but the friction which always exists interrupts this effect1 and a small weight may act upon the string without moving the box at all4 ,let weights be put in the dish ,d1 until a sufficient force is obtained to overcome the friction without giving the box an accelerated motion4 ,such a weight is equivalent to the amount of the friction4 ,the amount of the weight of the box being previously ascertained1 let this weight be now doubled by placing additional weights in the box4 ,the pressure will thus be doubled1 and it will be found that the weight of the dish ,d and its load1 which before was able to overcome the friction1 is now altogether inadequate to it4 ,let additional weights be placed in the dish until the friction be counteracted as before1 and it will be observed1 that the whole weight necessary to produce this effect is exactly twice the weight which produced it in the former case4 ,thus it appears that a double amount of pressure produces a double amount of friction2 and in a similar way it may be proved1 that any proposed increase or decrease of the pressure will be attended with a proportionate variation in the amount of the friction4 ,second4 ,let one of the surfaces be attached to a flat plane ,a ,b1 fig4 #agg41 which can be placed at any inclination with an horizontal plane ,b ,c1 the other surface being1 as before1 attached to the box adapted to receive weights4 ,the box being placed upon the plane1 let the latter be slightly elevated4 ,the tendency of the box to descend upon ,a ,b1 will bear the same proportion to its entire weight as the perpendicular ,a ,e bears to the length of the plane ,a ,b 7#bhf474 ,thus if the length ,a ,b be #cf inches1 and the height ,a ,e be three inches1 that is a twelfth part of the length1 then the tendency of the weight to move down the plane is equal to a twelfth part of its whole amount4 ,if the weight were twelve ounces1 and the surfaces perfectly smooth1 a force of one ounce acting up the plane would be necessary to prevent the descent of the weight4 ,in this case also the pressure on the plane will be represented by the length of the base ,b ,e 7#bhf471 that is1 it will bear the same proportion to the whole weight as ,b ,e bears to ,b ,a4 ,the relative amounts of the weight1 the tendency to descend1 and the pressure1 will always be exhibited by the relative lengths of ,a ,b1 ,a ,e1 and ,b ,e4 ,this being premised1 let the elevation of the plane ,a ,b be gradually increased until the tendency of the weight to descend just overcomes the friction1 but not so much as to allow the box to descend with accelerated speed4 ,the proportion of the whole weight1 which then acts down the plane1 will be found by measuring the height ,a ,e1 and the pressure will be determined by measuring the base ,b ,e4 ,now let the weight in the box be increased1 and it will be found that the same elevation is necessary to overcome the friction2 nor will this elevation suffer any change1 however the pressure or the magnitude of the surfaces which move in contact may be varied4 ,since1 therefore1 in all these cases1 the height ,a ,e and the base ,b ,e remain the same1 it follows that the proportion between the friction and pressure is undisturbed4 7#cbd47 ,the law that friction is proportional to the pressure1 has been questioned by the late professor ,vince of ,cambridge1 who deduced from a series of experiments1 that although the friction increases with the pressure1 yet that it increases in a somewhat less ratio2 and from this it would follow1 that the variation of the surface of contact must produce some effect upon the amount of friction4 ,the law1 as we have explained it1 however1 is sufficiently near the truth for most practical purposes4 7#cbe47 ,there are several circumstances regarding the quality of the surfaces which produce important effects on the quantity of friction1 and which ought to be noticed here4 ,this resistance is different in the surfaces of different substances4 ,when the surfaces are those of wood newly planed1 it amounts to about half the pressure1 but is different in different kinds of wood4 ,the friction of metallic surfaces is about one fourth of the pressure4 ,in general the friction between the surfaces of bodies of different kinds is less than between those of the same kind4 ,thus1 between wood and metal the friction is about one fifth of the pressure4 ,it is evident that the smoother the surfaces are the less will be the friction4 ,on this account1 the friction of surfaces1 when first brought into contact1 is often greater than after their attrition has been continued for a certain time1 because that process has a tendency to remove and rub off those minute asperities and projections on which the friction depends4 ,but this has a limit1 and after a certain quantity of attrition the friction ceases to decrease4 ,newly planed surfaces of wood have at first a degree of friction which is equal to half the entire pressure1 but after they are worn by attrition it is reduced to a third4 ,if the surfaces in contact be placed with their grains in the same direction1 the friction will be greater than if the grains cross each other4 ,smearing the surfaces with unctuous matter diminishes the friction1 probably by filling the cavities between the minute projections which produce the friction4 ,when the surfaces are first placed in contact1 the friction is less than when they are suffered to rest so for some time2 this is proved by observing the force which in each case is necessary to move the one upon the other1 that force being less if applied at the first moment of contact than when the contact has continued4 ,this1 however1 has a limit4 ,there is a certain time1 different in different substances1 within which this resistance attains its greatest amount4 ,in surfaces of wood this takes place in about two minutes2 in metals the time is imperceptibly short2 and when a surface of wood is placed upon a surface of metal1 it continues to increase for several days4 ,the limit is larger when the surfaces are great1 and belong to substances of different kinds4 ,the velocity with which the surfaces move upon one another produces but little effect upon the friction4 7#cbf47 ,there are several ways in which bodies may move one upon the other1 in which friction will produce different effects4 ,the principal of these are1 first1 the case where one body slides over another2 the second1 where a body having a round form rolls upon another2 and1 thirdly1 where an axis revolves within a hollow cylinder1 or the hollow cylinder revolves upon the axis4 ,with the same amount of pressure and a like quality of surface1 the quantity of friction is greatest in the first case and least in the second4 ,the friction in the second case also depends on the diameter of the body which rolls1 and is small in proportion as that diameter is great4 ,thus a carriage with large wheels is less impeded by the friction of the road than one with small wheels4 ,in the third case1 the leverage of the wheel aids the power in overcoming the friction4 ,let fig4 #agh4 represent a section of the wheel and axle2 let ,c be the centre of the axle1 and let ,b ,e be the hollow cylinder in the nave of the wheel in which the axle is inserted4 ,if ,b be the part on which the axle presses1 and the wheel turn in the direction ,n ,d ,m1 the friction will act at ,b in the direction ,b ,f1 and with the leverage ,b ,c4 ,the power acts against this at ,d in the direction ,d ,a1 and with the leverage ,d ,c4 ,it is therefore evident1 that as ,d ,c is greater than ,b ,c1 in the same proportion does the power act with mechanical advantage on the friction4 7#cbg47 ,contrivances for diminishing the effects of friction depend on the properties just explained1 the motion of rolling being as much as possible substituted for that of sliding2 and where the motion of rolling cannot be applied1 that of a wheel upon its axle is used4 ,in some cases both these motions are combined4 ,if a heavy load be drawn upon a plane in the manner of a sledge1 the motion will be that of sliding1 the species which is attended with the greatest quantity of friction2 but if the load be placed upon cylindrical rollers1 the nature of the motion is changed1 and becomes that in which there is the least quantity of friction4 ,thus large blocks of stone1 or heavy beams of timber1 which would require an enormous power to move them on a level road1 are easily advanced when rollers are put under them4 ,when very heavy weights are to be moved through small spaces1 this method is used with advantage2 but when loads ore to be transported to considerable distances1 the process is inconvenient and slow1 owing to the necessity of continually replacing the rollers in front of the load as they are left behind by its progressive advancement4 ,the wheels of carriages may be regarded as rollers which are continually carried forward with the load4 ,in addition to the friction of the rolling motion on the road1 they have1 it is true1 the friction of the axle in the nave2 but1 on the other hand1 they are free from the friction of the rollers with the under surface of the load1 or the carriage in which the load is transported4 ,the advantages of wheel carriages in diminishing the effects of friction is sometimes attributed to the slowness with which that axle moves within the box1 compared with the rate at which the wheel moves over the road2 but this is erroneous4 ,the quantity of friction does not in any case vary considerably with the velocity of the motion1 but least of all does it in that particular kind of motion here considered4 ,in certain cases1 where it is of great importance to remove the effects of friction1 a contrivance called friction-wheels1 or friction-rollers1 is used4 ,the axle of a friction-wheel1 instead of revolving within a hollow cylinder1 which is fixed1 rests upon the edges of wheels which revolve with it2 the species of motion thus becomes that in which the friction is of least amount4 ,let ,a ,b and ,d ,c1 fig4 #agi41 be two wheels revolving on pivots ,p ,q with as little friction as possible1 and so placed that the axle ,o of a third wheel ,e ,f may rest between their edges4 ,as the wheel ,e ,f revolves1 the axle ,o1 instead of grinding its surface on the surface on which it presses1 carries that surface with it1 causing the wheels ,a ,b1 ,c ,d1 to revolve4 ,in wheel carriages1 the roughness of the road is more easily overcome by large wheels than by small ones4 ,the cause of this arises partly from the large wheels not being so liable to sink into holes as small ones1 but more because1 in surmounting obstacles1 the load is elevated less abruptly4 ,this will be easily understood by observing the curves in fig4 #ahj41 which represent the elevation of the axle in each case4 7#cbh47 ,if a carriage were capable of moving on a road without friction1 the most advantageous direction in which a force could be applied to draw it would be parallel to the road4 ,when the motion is impeded by friction1 it is better1 however1 that the line of draught should be inclined to the road1 so that the drawing force may be expended partly in lessening the pressure on the road1 and partly in advancing the load4 ,let ,w1 fig4 #aha41 be a load which is to be moved upon the plane surface ,a ,b4 ,if the drawing force be applied in the direction ,c ,d1 parallel to the plane ,a ,b1 it will have to overcome the friction produced by the pressure of the whole weight of the load upon the plane2 but if it be inclined upwards in the direction ,c ,e1 it will be equivalent to two forces expressed 7#gd47 by ,c ,g and ,c ,f4 ,the part ,c ,g has the effect of lightening the pressure of the carriage upon the road1 and therefore of diminishing the friction in the same proportion4 ,the part ,c ,f draws the load along the plane4 ,since ,c ,f is less than ,c ,e or ,c ,d the whole moving force1 it is evident that a part of the force of draught is lost by this obliquity2 but1 on the other hand1 a part of the opposing resistance is also removed4 ,if the latter exceed the former1 an advantage will be gained by the obliquity2 but if the former exceed the latter1 force will be lost4 ,by mathematical reasoning1 founded on these considerations1 it is proved that the best angle of draught is exactly that obliquity which should be given to the road in order to enable the carriage to move of itself4 ,this obliquity is sometimes called the angle of repose1 and is that angle which determines the proportion of the friction to the pressure in the second method1 explained in 7#cbc474 ,the more rough the road is1 the greater will this angle be2 and therefore it follows1 that on bad roads the obliquity of the traces to the road should be greater than on good ones4 ,on a smooth ,macadamised way1 a very slight declivity would cause a carriage to roll by its own weight3 hence1 in this case1 the traces should be nearly parallel to the road4 ,in rail roads1 for like reasons1 the line of draught should be parallel to the road1 or nearly so4 7#cbi47 ,when ropes or cords form a part of machinery1 the effects of their imperfect flexibility are in a certain degree counteracted by bending them over the grooves of wheels4 ,but although this so far diminishes these effects as to render ropes practically useful1 yet still1 in calculating the powers of machinery1 it is necessary to take into account some consequences of the rigidity of cordage which even by these means are not removed4 ,to explain the way in which the stiffness of a rope modifies the operation of a machine1 we shall suppose it bent over a wheel and stretched by weights ,a ,b1 fig4 #ahb41 at its extremities4 ,the weights ,a and ,b being equal1 and acting at ,c and ,d in opposite ways1 balance the wheel4 ,if the weight ,a receive an addition1 it will overcome the resistance of ,b1 and turn the wheel in the direction ,d ,e ,c4 ,now1 for the present1 let us suppose that the rope is perfectly inflexible2 the wheel and weights will be turned into the position represented in fig4 #ahc4 ,the leverage by which ,a acts will be diminished1 and will become ,o ,f1 having been before ,o ,c2 and the leverage by which ,b acts will be increased to ,o ,g1 having been before ,o ,d4 ,but the rope not being inflexible will yield partially to the effects of the weights ,a and ,b1 and the parts ,a ,c and ,b ,d will be bent into the forms represented in fig4 #ahd4 ,the form of the curvature which the rope on each side of the wheel receives is still such that the descending weight ,a works with a diminished leverage ,f ,o1 while the ascending weight resists it with an increased leverage ,g ,o4 ,thus so much of the moving power is lost1 by the stiffness of the rope1 as is necessary to compensate this disadvantageous change in the power of the machine4 ,,chap4 ,,xx4 ,,on ,,the ,,strength ,,of ,,materials4 7#ccj47 ,experimental enquiries into the laws which regulate the strength of solid bodies1 or their power to resist forces variously applied to tear or break them1 are obstructed by practical difficulties1 the nature and extent of which are so discouraging that few have ventured to encounter them at all1 and still fewer have had the steadiness to persevere until any result showing a general law has been obtained4 ,these difficulties arise1 partly from the great forces which must be applied1 but more from the peculiar nature of the objects of those experiments4 ,the end to which such an enquiry must be directed is the development of a general law2 that is1 such a rule as would be rigidly observed if the materials1 the strength of which is the object of enquiry1 were perfectly uniform in their texture1 and subject to no casual inequalities4 ,in proportion as these inequalities are frequent1 experiments must be multiplied1 that a long average may embrace cases varying in both extremes1 so as to eliminate each other's effects in the final result4 ,the materials of which structures and works of art are composed are liable to so many and so considerable inequalities of texture1 that any rule which can be deduced1 even by the most extensive series of experiments1 must be regarded as a mean result1 from which individual examples will be found to vary in so great a degree1 that more than usual caution must be observed in its practical application4 ,the details of this subject belong to engineering1 more properly than to the elements of mechanics4 ,nevertheless1 a general view of the most important principles which have been established respecting the strength of materials will not be misplaced in this treatise4 ,a piece of solid matter may be submitted to the action of a force tending to separate its parts in several ways2 the principal of which are1-- #a4 ,to a direct pull1--as when a rope or wire is stretched by a weight4 ,when a tie-beam resists the separation of the sides of a structure1 c4 #b4 ,to a direct pressure or thrust1--as when a weight rests upon a pillar4 #c4 ,to a transverse strain1--as when weights on the ends of a lever press it on the fulcrum4 7#cca47 ,if a solid be submitted to a force which draws it in the direction of its length1 having a tendency to pull its ends in opposite directions1 its strength or power to resist such a force is proportional to the magnitude of its transverse section4 ,thus1 suppose a square rod of metal ,a ,b1 fig4 #ahe41 of the breadth and thickness of one inch1 be pulled by a force in the direction ,a ,b1 and that a certain force is found sufficient to tear it2 a rod of the same metal of twice the breadth and the same thickness will require double the force to break it2 one of treble the breadth and the same thickness will require treble the force to break it1 and so on4 ,the reason of this is evident4 ,a rod of double or treble the thickness1 in this case1 is equivalent to two or three equal and similar rods which equally and separately resist the drawing force1 and therefore possess a degree of strength proportionate to their number4 ,it will easily be perceived1 that whatever be the section1 the same reasoning will be applicable1 and the power of resistance will1 in general1 be proportional to its magnitude or area4 ,if the material were perfectly uniform throughout its dimensions1 the resistance to a direct pull would not be affected by the length of the rod4 ,in practice1 however1 the increase of length is found to lessen the strength4 ,this is to be attributed to the increased chance of inequality4 7#ccb47 ,no satisfactory results have been obtained either by theory or experiment respecting the laws by which solids resist compression4 ,the power of a perpendicular pillar to support a weight placed upon it evidently depends on its thickness1 or the magnitude of its base1 and on its height4 ,it is certain that when the height is the same1 the strength increases with every increase of the base1 but it seems doubtful whether the strength be exactly proportional to the base4 ,that is1 if two columns of the same material have equal heights1 and the base of one be double the base of the other1 the strength of one will be greater1 but it is not certain whether it will exactly double that of the other4 ,according to the theory of ,euler1 which is in a certain degree verified by the experiments of ,musschenbrock1 the strength will be increased in a greater proportion than the base1 so that1 if the base be doubled1 the strength will be more than doubled4 ,when the base is the same1 the strength is diminished by increasing the height1 and this decrease of strength is proportionally greater than the increase of height4 ,according to ,euler's theory1 the decrease of strength is proportional to the square of the height2 that is1 when the height is increased in a two-fold proportion1 the strength is diminished in a four-fold proportion4 7#ccc47 ,the strain to which solids forming the parts of structures of every kind are most commonly exposed is the lateral or transverse strain1 or that which acts at right angles to their lengths4 ,if any strain act obliquely to the direction of their length it may be resolved into two forces 7#gf471 one in the direction of the length1 and the other at right angles to the length4 ,that part which acts in the direction of the length will produce either compression or a direct pull1 and its effect must be investigated accordingly4 ,although the results of theory1 as well as those of experimental investigations1 present great discordances respecting the transverse strength of solids1 yet there are some particulars1 in which they1 for the most part1 agree2 to this it is our object here to confine our observations1 declining all details relating to disputed points4 ,let ,a ,b ,c ,d1 fig4 #ahf41 be a beam1 supported at its ends ,a and ,b4 ,its strength to support a weight at ,e pressing downwards at right angles to its length is evidently proportional to its breadth1 the other things being the same4 ,for a beam of double or treble breadth1 and of the same thickness1 is equivalent to two or three equal and similar beams placed side by side4 ,since each of these would possess the same strength1 the whole taken together would possess double or treble the strength of any one of them4 ,when the breadth and length are the same the strength obviously increases with the depth1 but not in the same proportion4 ,the increase of strength is found to be much greater in proportion than the increase of depth4 ,by the theory of ,galileo1 a double or treble thickness ought to increase the strength in a four-fold or nine-fold proportion1 and experiments in most cases do not materially vary from this rule4 ,if while the breadth and depth remain the same1 the length of the beam1 or rather1 the distance between the points of support1 vary1 the strength will vary accordingly1 decreasing in the same proportion as the length increases4 ,from these observations it appears1 that the transverse strength of a beam depends more on its thickness than its breadth4 ,hence we find that a broad thin board is much stronger when its edge is presented upwards4 ,on this principle the joists or rafters of floors and roofs are constructed4 ,if two beams be in all respects similar1 their strengths will be in the proportion of the squares of their lengths4 ,let the length1 breadth1 and depth of the one be respectively double the length1 breadth1 and depth of the other4 ,by the double breadth the beam doubles its strength1 but by doubling the length half this strength is lost4 ,thus the increase of length and breadth counteract each other's effects1 and as far as they are concerned the strength of the beam is not changed4 ,but by doubling the thickness the strength is increased in a four-fold proportion1 that is1 as the square of the length4 ,in the same manner it may be shown1 that when all the dimensions are trebled1 the strength is increased in a nine-fold proportion1 and so on4 7#ccd47 ,in all structures the materials have to support their own weight1 and therefore their available strength is to be estimated by the excess of their absolute strength above that degree of strength which is just sufficient to support their own weight4 ,this consideration leads to some conclusions1 of which numerous and striking illustrations are presented in the works of nature and art4 ,we have seen that the absolute strength with which a lateral strain is resisted is in the proportion of the square of the linear dimensions of similar parts of a structure1 and therefore the amount of this strength increases rapidly with every increase of the dimensions of a body4 ,but at the same time the weight of the body increases in a still more rapid proportion4 ,thus1 if the several dimensions be doubled1 the strength will be increased in a four-fold but the weight in an eight-fold proportion4 ,if the dimensions be trebled1 the strength will be multiplied nine times1 but the weight twenty-seven times4 ,again1 if the dimensions be multiplied four times1 the strength will be multiplied sixteen times1 and the weight sixty-four times1 and so on4 ,hence it is obvious1 that although the strength of a body of small dimensions may greatly exceed its weight1 and1 therefore1 it may be able to support a load many times its own weight2 yet by a great increase in the dimensions the weight increasing in a much greater degree the available strength may be much diminished1 and such a magnitude may be assigned1 that the weight of the body must exceed its strength1 and it not only would be unable to support any load1 but would actually fall to pieces by its own weight4 ,the strength of a structure of any kind is not1 therefore1 to be determined by that of its model1 which will always be much stronger in proportion to its size4 ,all works natural and artificial have limits of magnitude which1 while their materials remain the same1 they cannot surpass4 ,in conformity with what has just been explained1 it has been observed1 that small animals are stronger in proportion than large ones2 that the young plant has more available strength in proportion than the large forest tree2 that children are less liable to injury from accident than men1 c4 ,but although to a certain extent these observations are just1 yet it ought not to be forgotten1 that the mechanical conclusions which they are brought to illustrate are founded on the supposition1 that the smaller and greater bodies which are compared are composed of precisely similar materials4 ,this is not the case in any of the examples here adduced4 ,,chap4 ,,xxi4 ,,on ,,balances ,,and ,,pendulums4 7#cce47 ,the preceding chapters have been confined almost wholly to the consideration of the laws of mechanics1 without entering into a particular description of the machinery and instruments dependant upon those laws4 ,such descriptions would have interfered too much with the regular progress of the subject1 and it therefore appeared preferable to devote a chapter exclusively to this portion of the work4 ,perhaps there are no ideas which man receives through the medium of sense which may not be referred ultimately to matter and motion4 ,in proportion1 therefore1 as he becomes acquainted with the properties of the one and the laws of the other1 his knowledge is extended1 his comforts are multiplied2 he is enabled to bend the powers of nature to his will1 and to construct machinery which effects with ease that which the united labour of thousands would in vain be exerted to accomplish4 ,of the properties of matter1 one of the most important is its weight1 and the element which mingles inseparably with the laws of motion is time4 ,in the present chapter it is our intention to describe such instruments as are usually employed for determining the weight of bodies4 ,to attempt a description of the various machines which are used for the measurement of time1 would lead us into too wide a field for the present occasion1 and we shall1 therefore1 confine ourselves to an account of the methods which have been practised to perfect1 to perfect that instrument which affords the most correct means of measuring time1 the pendulum4 ,the instrument by which we are enabled to determine1 with greater accuracy than by any other means1 the relative weight of a body1 compared with the weight of another body assumed as a standard1 is the balance4 ,illustration3 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,of the ,balance4 ,the balance may be described as consisting of an inflexible rod or lever1 called the beam1 furnished with three axes2 one1 the fulcrum or centre of motion situated in the middle1 upon which the beam turns1 and the other two near the extremities1 and at equal distances from the middle4 ,these last are called the points of support1 and serve to sustain the pans or scales4 ,the points of support and the fulcrum are in the same right line1 and the centre of gravity of the whole should be a little below the fulcrum when the position of the beam is horizontal4 ,the arms of the lever being equal1 it follows that if equal weights be put into the scales no effect will be produced on the position of the balance1 and the beam will remain horizontal4 ,if a small addition be made to the weight in one of the scales1 the horizontality of the beam will be disturbed2 and after oscillating for some time1 it will1 on attaining a state of rest1 form an angle with the horizon1 the extent of which is a measure of the delicacy or sensibility of the balance4 ,as the sensibility of a balance is of the utmost importance in nice scientific enquiries1 we shall enter somewhat at large into a consideration of the circumstances by which this property is influenced4 ,in fig4 #ahg4 let ,a ,b represent the beam drawn from the horizontal position by a very small weight placed in the scale suspended from the point of support ,b2 then the force tending to draw the beam from the horizontal position may be expressed by ,p ,b1 multiplied by such very small weight acting upon the point ,b4 ,let the centre of gravity of the whole be at ,g2 then the force acting against the former will be ,g ,p multiplied into the weight of the beam and scales1 and when these forces are equal1 the beam will rest in an inclined position4 ,hence we may perceive that as the centre of gravity is nearer to or further from the fulcrum ,s1 7every thing else remaining the same7 the sensibility of the balance will be increased or diminished4 ,for1 suppose the centre of gravity were removed to g1 then to produce an opposing force equal to that acting upon the extremity of the beam1 the distance g p from the perpendicular line must be increased until it becomes nearly equal to ,g ,p2 but for this purpose the end of the beam ,b must descend1 which will increase the angle ,h ,s ,b4 ,as all weights placed in the scales are referred to the line joining the points of support1 and as this line is above the centre of gravity of the beam when not loaded1 such weights will raise the centre of gravity2 but it will be seen that the sensibility of the balance1 as far as it depends upon this cause1 will remain unaltered4 ,for1 calling the distance ,s ,g unity1 the distance of the centre of gravity from the point ,s 7to which the weight which has been added is referred7 will be expressed by the reciprocal of the weight of the beam so increased2 that is1 if the weight of the beam be doubled by weights placed in the scales1 ,s g will be one half of ,s ,g2 and if the weight of the beam be in like manner trebled1 ,s g will be one third of ,s ,g1 and so on4 ,and as ,g ,p varies as ,s ,g1 g p will be inversely proportionate to the increased weight of the beam1 and consequently1 the product obtained by multiplying g p by the weight of the beam and its load will be a constant quantity1 and the sensibility of the balance1 as before stated1 will suffer no alteration4 ,we will now suppose that the fulcrum ,s1 fig4 #ahh41 is situated below the line joining the points of support1 and that the centre of gravity of the beam when not loaded is at ,g4 ,also that when a very small weight is placed in the scale suspended from the point ,b1 the beam is drawn from its horizontal position1 the deviation being a measure of the sensibility of the balance4 ,then1 as before stated1 ,g ,p multiplied by the weight of the beam will be equal to ,p ,b multiplied by the very small additional weight acting on the point ,b4 ,now if we place equal weights in both scales1 such additional weights will be referred to the point ,w1 and the resulting distance of the centre of gravity from the point ,w1 calling ,w ,g unity1 will be expressed as before by the reciprocal of the increased weight of the loaded beam4 ,but ,g ,p will decrease in a greater proportion than ,w ,g3 thus1 supposing the weight of the beam to be doubled1 ,w g would be one half of ,w ,g2 but g p1 as will be evident on an inspection of the figure1 will be less than half of ,g ,p2 and the same small weight which was before applied to the point ,b1 if now added1 would depress the point ,b1 until the distance g p became such as that1 when multiplied by the weight of the whole1 the product would be as before equal to ,p ,b1 multiplied by the before mentioned very small added weight4 ,the sensibility of the balance1 therefore1 in this case would be increased4 ,if the beam be sufficiently loaded1 the centre of gravity will at length be raised to the fulcrum ,s1 and the beam will rest indifferently in any position4 ,if more weight be then added1 the centre of gravity will be raised above the fulcrum1 and the beam will turn over4 ,lastly1 if the fulcrum ,s1 fig4 #ahi41 is above the line joining the two points of support1 as any additional weights placed in the scales will be referred to the point ,w1 in the line joining ,a and ,b1 if the weight of the beam be doubled by such added weights1 and the centre of gravity be consequently raised to g1 ,w g will become equal to half of ,w ,g4 ,but g p1 being greater than one half of ,g ,p1 the end of the beam ,b will rise until g p becomes such as to be equal1 when multiplied by the whole increased weight of the beam1 to ,p ,b1 multiplied by the small weight1 which we suppose to have been placed as in the preceding examples1 in the scale4 ,from what has been said it will be seen that there are three positions of the fulcrum which influence the sensibility of the balance3 first1 when the fulcrum and the points of support are in a right line1 when the sensibility of the balance will remain the same1 though the weight with which the beam is loaded should be varied3 secondly1 when the fulcrum is below the line joining the two points of support1 in which case the sensibility of the balance will be increased by additional weights1 until at length the centre of gravity is raised above the fulcrum1 when the beam will turn over2 and1 thirdly1 when the fulcrum is above the line joining the two points of support1 in which case the sensibility of the balance will be diminished as the weight with which the beam is loaded is increased4 ,the sensibility of a balance1 as here defined1 is the angular deviation of the beam occasioned by placing an additional constant small weight in one of the scales2 but it is frequently expressed by the proportion which such small additional weight bears to the weight of the beam and its load1 and sometimes to the weight the value of which is to be determined4 ,this proportion1 however1 will evidently vary with different weights1 except in the case where the centre of gravity of the beam is in the line joining the points supporting the scales1 the fulcrum being above this line1 and it is therefore necessary1 in every other case1 when speaking of the sensibility of the balance1 to designate the weight with which it is loaded3 thus1 if a balance has a troy pound in each scale1 and the horizontality of the beam varies a certain small quantity1 just perceptible on the addition of one hundredth of a grain1 we say that the balance is sensible to #a_/aaebjjj part of its load with a pound in each scale1 or that it will determine the weight of a troy pound within #a_/egfjjj part of the whole4 ,the nearer the centre of gravity of a balance is to its fulcrum the slower will be the oscillations of the beam4 ,the number of oscillations1 therefore1 made by the beam in a given time 7a minute for example71 affords the most accurate method of judging of the sensibility of the balance1 which will be the greater as the oscillations are fewer4 ,balances of the most perfect kind1 and of such only it is our present object to treat1 are usually furnished with adjustments1 by means of which the length of the arms1 or the distances of the fulcrum from the points of support1 may be equalised1 and the fulcrum and the two points of support be placed in a right line2 but these adjustments1 as will hereafter be seen1 are not absolutely necessary4 ,the beam is variously constructed1 according to the purposes to which the balance is to be applied4 ,sometimes it is made of a rod of solid steel2 sometimes of two hollow cones joined at their bases2 and1 in some balances1 the beam is a frame in the form of a rhombus3 the principal object in all1 however1 is to combine strength and inflexibility with lightness4 ,a balance of the best kind1 made by ,troughton1 is so contrived as to be contained1 when not in use1 in a drawer below the case2 and when in use1 it is protected from any disturbance from currents of air1 by being enclosed in the case above the drawer1 the back and front of which are of plate glass4 ,there are doors in the sides1 through which the scale-pans are loaded1 and there is a door at the top through which the beam may be taken out4 ,a strong brass pillar1 in the centre of the box1 supports a square piece1 on the front and back of which rise two arches1 nearly semicircular1 on which are fixed two horizontal planes of agate1 intended to support the fulcrum4 ,within the pillar is a cylindrical tube1 which slides up and down by means of a handle on the outside of the case4 ,to the top of this interior tube is fixed an arch1 the terminations of which pass beneath and outside of the two arches before described4 ,these terminations are formed into ,y s1 destined to receive the ends of the fulcrum1 which are made cylindrical for this purpose1 when the interior tube is elevated in order to relieve the axis when the balance is not in use4 ,on depressing the interior tube1 the ,y s quit the axis1 and leave it in its proper position on the agate planes4 ,the beam is about eighteen inches long1 and is formed of two hollow cones of brass1 joined at their bases4 ,the thickness of the brass does not exceed #jjb of an inch1 but by means of circular rings driven into the cones at intervals they are rendered almost inflexible4 ,across the middle of the beam passes a cylinder of steel1 the lower side of which is formed into an edge1 having an angle of about thirty degrees1 which1 being hardened and well polished1 constitutes the fulcrum1 and rests upon the agate planes for the length of about #jje of an inch4 ,each point of suspension is formed of an axis having two sharp concave edges1 upon which rest at right angles two other sharp concave edges formed in the spur-shaped piece to which the strings carrying the scale-pan are attached4 ,the two points are adjustable1 the one horizontally1 for the purpose of equalising the arms of the beam1 and the other vertically1 for bringing the points of suspension and the fulcrum into a right line4 ,such is the form of ,troughton's balance3 we shall now give the description of a balance as constructed by ,mr4 ,robinson of ,devonshire ,street1 ,portland ,place3-- ,the beam of this balance is only ten inches long4 ,it is a frame of bell-metal in the form of a rhombus4 ,the fulcrum is an equilateral triangular prism of steel one inch in length2 but the edge on which the beam vibrates is formed to an angle of #abj1 in order to prevent any injury from the weight with which it may be loaded4 ,the chief peculiarity in this balance consists in the knife-edge which forms the fulcrum bearing upon an agate plane throughout its whole length1 whereas we have seen in the balance before described that the whole weight is supported by portions only of the knife-edge1 amounting together to one tenth of an inch4 ,the supports for the scales are knife-edges each six tenths of an inch long4 ,these are each furnished with two pressing screws1 by means of which they may be made parallel to the central knife-edge4 ,each end of the beam is sprung obliquely upwards and towards the middle1 so as to form a spring through which a pushing screw passes1 which serves to vary the distance of the point of support from the fulcrum1 and1 at the same time1 by its oblique action to raise or depress it1 so as to furnish a means of bringing the points of support and the fulcrum into a right line4 ,a piece of wire1 four inches long1 on which a screw is cut1 proceeds from the middle of the beam downwards4 ,this is pointed to serve as an index1 and a small brass ball moves on the screw1 by changing the situation of which the place of the centre of gravity may be varied at pleasure4 ,the fulcrum1 as before remarked1 rests upon an agate plane throughout its whole length1 and the scale-pans are attached to planes of agate which rest upon the knife-edges forming the points of support4 ,this method of supporting the scale-pans1 we have reason to believe1 is due to ,mr4 ,cavendish4 ,upon the lower half of the pillar to which the agate plane is fixed1 a tube slides up and down by means of a lever which passes to the outside of the case4 ,from the top of this tube arms proceed obliquely towards the ends of the balance1 serving to support a horizontal piece1 carrying at each extremity two sets of ,y s1 one a little above the other4 ,the upper ,y s are destined to receive the agate planes to which the scale-pans are attached1 and thus to relieve the knife-edges from their pressure2 the lower to receive the knife-edges which1 form the points of support1 consequently these latter ,y s1 when in action1 sustain the whole beam4 ,when the lever is freed from a notch in which it is lodged1 a spring is allowed to act upon the tube we have mentioned1 and to elevate it4 ,the upper ,y s first meet the agate planes carrying the scale-pans and free them from the knife-edges4 ,the lower ,y s then come into action and raise the whole beam1 elevating the central knife-edge above the agate plane4 ,this is the usual state of the balance when not in use3 when it is to be brought into action1 the reverse of what we have described takes place4 ,on pressing down the lever1 the central knife-edge first meets the agate plane1 and afterwards the two agate planes carrying the scale-pans are deposited upon their supporting knife-edges4 ,a balance of this construction was employed by the writer of this article in adjusting the national standard pound4 ,with a pound troy in each scale1 the addition of one hundredth of a grain caused the index to vary one division1 equal to one tenth of an inch1 and ,mr4 ,robinson adjusts these balances so that with one thousand grains in each scale1 the index varies perceptibly on the addition of one thousandth of a grain1 or of one-millionth part of the weight to be determined4 ,it may not be uninteresting to subjoin1 from the ,philosophical ,transactions for #ahbf1 the description of a balance perhaps the most sensible that has yet been made1 constructed for verifying the national standard bushel4 ,the author says1-- 8,the weight of the bushel measure1 together with the #hj lbs4 of water it should contain1 was about #bej lbs42 and as ,i could find no balance capable of determining so large a weight with sufficient accuracy1 ,i was under the necessity of constructing one for this express purpose4 8,i first tried cast iron2 but though the beam was made as light as was consistent with the requisite degree of strength1 the inertia of such a mass appeared to be so considerable1 that much time must have been lost before the balance would have answered to the small differences ,i wished to ascertain4 ,lightness was a property essentially necessary1 and bulk was very desirable1 in order to preclude such errors as might arise from the beam being partially affected by sudden alterations of temperature4 ,i therefore determined to employ wood1 a material in which the requisites ,i sought were combined4 ,the beam was made of a plank of mahogany1 about #g,o inches long1 #bb inches wide1 and #b-a_/d thick1 tapering from the middle to the extremities4 ,an opening was cut in the centre1 and strong blocks screwed to each side of the plank1 to form a bearing for the back of a knife-edge which passed through the centre4 ,blocks were also screwed to each side at the extremities of the beam on which rested the backs of the knife-edges for supporting the pans4 ,the opening in the centre was made sufficiently large to admit the support hereafter to be described1 upon which the knife-edge rested4 8,in all beams which ,i have seen1 with the exception of those made by ,mr4 ,robinson1 the whole weight is sustained by short portions at the extremities of the knife-edge2 and the weight being thus thrown upon a few points1 the knife-edge becomes more liable to change its figure and to suffer injury4 8,to remedy this defect1 the central knife-edge of the beam ,i am describing was made #f inches1 and the two others #e inches long4 ,they were triangular prisms with equal sides of three fourths of an inch1 very carefully finished1 and the edges ultimately formed to an angle of #abj4 8,each knife-edge was screwed to a thick plate of brass1 the surfaces in contact having been previously ground together2 and these plates were screwed to the beam1 the knife-edges being placed in the same plane1 and as nearly equidistant and parallel to each other as could be done by construction4 8,the support upon which the central knife-edge rested throughout its whole length was formed of a plate of polished hard steel1 screwed to a block of cast iron4 ,this block was passed through the opening before mentioned in the centre of the beam1 and properly attached to a frame of cast iron4 8,the stirrups to which the scales were hooked rested upon plates of polished steel to which they were attached1 and the under surfaces of which were formed by careful grinding into cylindrical segments4 ,these were in contact with the knife-edges their whole length1 and were known to be in their proper position by the correspondence of their extremities with those of the knife-edges4 ,a well imagined contrivance was applied by ,mr4 ,bate for raising the beam when loaded1 in order to prevent unnecessary wear of the knife-edge1 and for the purpose of adjusting the place of the centre of gravity1 when the beam was loaded with the weight required to be determined1 a screw carrying a movable ball projected vertically from the middle of die beam4 8,the performance of this balance fully equalled my expectations4 ,with two hundred and fifty pounds in each scale1 the addition of a single grain occasioned an immediate variation in the index of one twentieth of an inch1 the radius being fifty inches40 ,from the preceding account it appears that this balance is sensible to #a_/agejjjj part of the weight which was to be determined4 ,we shall now describe the method to be pursued in adjusting a balance4 #a4 ,to bring the points of suspension and the fulcrum into a right line4 ,make the vibrations of the balance very slow by moving the weight which influences the centre of gravity1 and bring the beam into a horizontal position1 by means of small bits of paper thrown into the scales4 ,then load the scales with nearly the greatest weight the beam is fitted to carry4 ,if the vibrations are performed in the same time as before1 no further adjustment is necessary2 but if the beam vibrates quicker1 or if it oversets1 cause it to vibrate in the same time as at first1 by moving the adjusting weight1 and note the distance through which the weight has passed4 ,move the weight then in the contrary direction through double this distance1 and then produce the former slow motion by means of the screw acting vertically on the point of support4 ,repeat this operation until the adjustment is perfect4 #b4 ,to make the arms of the beam of an equal length4 ,put weights in the scales as before2 bring the beam as nearly as possible to a horizontal position1 and note the division at which the index stands2 unhook the scales1 and transfer them with their weights to the other ends of the beam1 when1 if the index points to the same division1 the arms are of an equal length2 but if not1 bring the index to the division which had been noted1 by placing small weights in one or the other scale4 ,take away half these weights1 and bring the index again to the observed division by the adjusting screw1 which acts horizontally on the point of support4 ,if the scale-pans are known to be of the same weight1 it will not be necessary to change the scales1 but merely to transfer the weights from one scale-pan to the other4 ,of the ,use of the ,balance4 ,though we have described the method of adjusting the balance1 these adjustments1 as we have before remarked1 may be dispensed with4 ,indeed1 in all delicate scientific operations1 it is advisable never to rely upon adjustments1 which1 after every care has been employed in effecting them1 can only be considered as approximations to the truth4 ,we shall1 therefore1 now describe the best method of ascertaining the weight of a body1 and which does not depend on the accuracy of these adjustments4 ,having levelled the case which contains the balance1 and thrown the beam out of action1 place a weight in each scale-pan nearly equal to the weight which is to be determined4 ,lower the beam very gently till it is in action1 and by means of the adjustment for raising or lowering the centre of gravity1 cause the beam to vibrate very slowly4 ,remove these weights1 and place the substance1 the weight of which is to be determined in one of the scale-pans2 carefully counterpoise it by means of any convenient substances put into the other scale-pan1 and observe the division at which the index stands2 remove the body1 the weight of which is to be ascertained1 and substitute standard weights for it so as to bring the index to the same division as before4 ,these weights will be equal to the weight of the body4 ,if it be required to compare two weights together which are intended to be equal1 and to ascertain their difference1 if any1 the method of proceeding will be nearly the same4 ,the standard weight is to be carefully counterpoised1 and the division at which the index stands1 noted4 ,and now it will be convenient to add in either of the scales some small weight1 such as one or two hundredths of a grain1 and mark the number of divisions passed over in consequence by the index1 by which the value of one division of the scale will be known4 ,this should be repeated a few times1 and the mean taken for greater certainty4 ,having noted the division at which the index rests1 the standard weight is to be removed1 and the weight which is to be compared with it substituted for it4 ,the index is then again to be noted1 and the difference between this and the former indication will give the difference between the weights in parts of a grain4 ,if the balance is adjusted so as to be very sensible1 it will be long before it comes to a state of rest4 ,it may1 therefore1 sometimes be advisable to take the mean of the extent of the vibrations of the index as the point where it would rest1 and this may be repeated several times for greater accuracy4 ,it must1 however1 be remembered1 that it is not safe to do this when the extent of the vibrations is beyond one or two divisions of the scale2 but with this limitation it is1 perhaps1 as good a method as can be pursued4 ,many precautions are necessary to ensure a satisfactory result4 ,the weights should never be touched by the hand2 for not only would this oxydate the weight1 but by raising its temperature it would appear lighter1 when placed in the scale-pan1 than it should do1 in consequence of the ascent of the heated air4 ,for the larger weights a wooden fork or tongs1 according to the form of the weight1 should be employed2 and for the smaller1 a pair of forceps made of copper will be found the most convenient4 ,this metal possessing sufficient elasticity to open the forceps on their being released from pressure1 and yet not opposing a resistance sufficient to interfere with that delicacy of touch which is desirable in such operations4 ,of ,weights4 ,it must be obvious1 that the excellence of the balance would be of little use1 unless the weights employed were equally to be depended upon4 ,the weights may either be accurately adjusted1 or the difference between each weight and the standard may be determined1 and1 consequently1 its true value ascertained4 ,it has been already shown how the latter may be effected1 in the instructions which have been given for comparing two weights together2 and we shall now show the readiest mode of adjusting weights to an exact equality with a given standard4 ,the material of the weight may be either brass or platina1 and its form may be cylindrical3 the diameter being nearly twice the height4 ,a small spherical knob is screwed into the centre1 a space being left under the screw to receive the portions of fine wire used in the adjustment4 ,it will be convenient to form a cavity in the bottom of each weight to receive the knob of the weight upon which it may be placed4 ,each weight is now to be compared with the standard1 and should it be too heavy1 it is to be reduced till it becomes in a very small degree too light1 when the amount of the deficiency is to be carefully determined4 ,some very fine silver wire is now to be taken1 and the weight of three or four feet of it ascertained4 ,from this it will be known what length of the wire is equal to the error of the weight to be adjusted2 and this length being cut off is to be enclosed under the screw4 ,to guard against any possible error1 it will be advisable before the screw is firmly fixed in its place1 again to compare the weight with the standard4 ,the most approved method of making weights expressing the decimal parts of a grain1 is to determine1 as before1 with great care1 the weight of a certain length of fine wire1 and then to cut off such portions as are equal to the weights required4 ,before we conclude this article we shall give a description1 from the ,annals of ,philosophy for #ahbe1 of 8a very sensible balance10 used by the late ,dr4 ,black3-- 8,a thin piece of fir wood1 not thicker than a shilling1 and a foot long1 three tenths of an inch broad in the middle1 and one tenth and a half at each end1 is divided by transverse lines into twenty parts2 that is1 ten parts on each side of the middle4 ,these are the principal divisions1 and each of them is subdivided into halves and quarters4 ,across the middle is fixed one of the smallest needles ,i could procure1 to serve as an axis1 and it is fixed in its place by means of a little sealing wax4 ,the numeration of the divisions is from the middle to each end of the beam4 ,the fulcrum is a bit of plate brass1 the middle of which lies flat on my table when ,i use the balance1 and the two ends are bent up to a right angle so as to stand upright4 ,these two ends are ground at the same time on a flat hone1 that the extreme surfaces of them may be in the same plane2 and their distance is such that the needle1 when laid across them1 rests on them at a small distance from the sides of the beam4 ,they rise above the surface of the table only one tenth and a half or two tenths of an inch1 so that the beam is very limited in its play4 ,see fig4 #aij4 8,the weights ,i use are one globule of gold1 which weighs one grain1 and two or three others which weigh one tenth of a grain each2 and also a number of small rings of fine brass wire1 made in the manner first mentioned by ,mr4 ,lewis1 by appending a weight to the wire1 and coiling it with the tension of that weight round a thicker brass wire in a close spiral1 after which1 the extremity of the spiral being tied hard with waxed thread1 ,i put the covered wire into a vice1 and applying a sharp knife1 which is struck with a hammer1 ,i cut through a great number of the coils at one stroke1 and find them as exactly equal to one another as can be desired4 ,those ,i use happen to be the #a_/cj part of a grain each1 or #cjj of them weigh ten grains2 but ,i have others much lighter4 8,you will perceive that by means of these weights placed on different parts of the beam1 ,i can learn the weight of any little mass from one grain1 or a little more1 to the #a_/abjj of a grain4 ,for if the thing to be weighed weighs one grain1 it will1 when placed on one extremity of the beam1 counterpoise the large gold weight at the other extremity4 ,if it weighs half a grain it will counterpoise the heavy gold weight placed at #e4 ,if it weigh #f_/aj of a grain1 you must place the heavy gold weight at #e1 and one of the lighter ones at the extremity to counterpoise it1 and if it weighs only one or two1 or three or four hundredths of a grain1 it will be counterpoised by one of the small gold weights placed at the first or second1 or third or fourth division4 ,if1 on the contrary1 it weighs one grain and a fraction1 it will be counterpoised by the heavy gold weight at the extremity1 and one or more of the lighter ones placed in some other part of the beam4 8,this beam has served me hitherto for every purpose2 but had ,i occasion for a more delicate one1 ,i could make it easily by taking a much thinner and lighter slip of wood1 and grinding the needle to give it an edge4 ,it would also be easy to make it carry small scales of paper for particular purposes40 ,the writer of this article has used a balance of this kind1 and finds that it is sensible to #a_/ajjj of a grain when loaded with ten grains4 ,it is necessary1 however1 where accuracy is required1 to employ a scale-pan4 ,this may be made of thin card paper1 shaped as in fig4 #aia4 ,a thread is to be passed through the two ends1 by tightening which they may be brought near each other4 ,the most convenient weights for this beam appear to be two of one grain each1 and one of one tenth of a grain4 ,they should be made of straight wire2 and if the beam be notched at the divisions1 they may be lodged in these notches very conveniently4 ,ten divisions on each side of the middle will be sufficient4 ,the weight of the scale-pan must first be carefully ascertained1 in order that it may be deducted from the weight1 afterwards determined1 of the scale-pan and the substance it may contain4 ,if the scale-pan be placed at the tenth division of the beam1 it is evident that by means of the two grain weights1 a greater weight cannot be determined than one grain and nine tenths2 but if the scale-pan be placed at any other division of the beam1 the resulting apparent weight must be increased by multiplying it by ten1 and dividing by the number of the division at which the scale-pan is placed2 and in this manner it is evident that if the scale-pan be placed at the division numbered #a1 a weight amounting to nineteen grains may be determined4 ,we have been tempted to describe this little apparatus1 because it is extremely simple in its construction1 may be easily made1 and may be very usefully employed on many occasions where extreme accuracy is not necessary4 ,description of the ,steelyard4 ,the steelyard is a lever1 having unequal arms2 and in its most simple form it is so arranged1 that one weight alone serves to determine a great variety of others1 by sliding it along the longer arm of the lever1 and thus varying its distance from the fulcrum4 ,it has been demonstrated1 chapter xiii41 that in the lever the proportion of the power to the weight will be always the same as that of their distances from the fulcrum1 taken in a reverse order2 consequently1 when a constant weight is used1 and an equilibrium established by sliding this weight on the longer arm of the lever1 the relative weight of the substance weighed1 to the constant weight1 will be in the same proportion as the distance of the constant weight from the fulcrum is to the length of the shorter arm4 ,thus1 suppose the length of the shorter arm1 or the distance of the fulcrum from the point from which the weight to be determined is suspended1 to be one inch2 let the longer arm of the lever be divided into parts of one inch each1 beginning at the fulcrum4 ,now let the constant weight be equal to one pound1 and let the steelyard be so constructed that the shorter arm shall be sufficiently heavy to counterpoise the longer when the bar is unloaded4 ,then suppose a substance1 the weight of which is five pounds1 to be suspended from the shorter arm4 ,it will be found that when the constant weight is placed at the distance of five inches from the fulcrum1 the weights will be in equilibrium1 and the bar consequently horizontal4 ,in this steelyard1 therefore1 the distance of each inch from the fulcrum indicates a weight of one pound4 ,an instrument of this form was used by the ,romans1 and it is usually described as the ,roman statera or steelyard4 ,a representation of it is given at fig4 #aib4 ,the steelyard is in very general use for the coarser purposes of commerce1 but constructed differently from that which we have described4 ,the beam with the scales or hooks is seldom in equilibrium upon the point ,f1 when the weight ,p is removed2 but the longer arm usually preponderates1 and the commencement of the graduations1 therefore1 is not at ,f1 but at some point between ,b and ,f4 ,the common steelyard1 which we have represented at fig4 #aic41 is usually furnished with two points1 from either of which the substance1 the weight of which is to be determined1 may be suspended4 ,the value of the divisions is in this case increased in proportion as the length of the shorter arm is decreased4 ,thus1 in the steelyard which we have described1 if there be a second point of suspension at the distance of half an inch from the fulcrum1 each division of the longer arm will indicate two pounds instead of one1 and these divisions are usually marked upon the opposite edge of the steelyard1 which is made to turn over4 ,this instrument is very convenient1 because it requires but one weight2 and the pressure on the fulcrum is less than in the balance1 when the substance to be weighed is heavier than the constant weight4 ,but1 on the contrary1 when the constant weight exceeds the substance to be weighed1 the pressure on the fulcrum is greater in the steelyard than in the balance1 and the balance is1 therefore1 preferable in determining small weights4 ,there is also an advantage in the balance1 because the subdivision of weights can be effected with a greater degree of precision than the subdivision of the arm of the steelyard4 ,c4 ,paul's ,steelyard4 ,a steelyard has been constructed by ,mr4 ,c4 ,paul1 inspector of weights and measures at ,geneva1 which is much to be preferred to that in common use4 ,mr4 ,c4 ,paul states1 that steelyards have two advantages over balances3 #a4 ,that their axis of suspension is not loaded with any other weight than that of the merchandise1 the constant weight of the apparatus itself excepted2 while the axis of the balance1 besides the weight of the instrument1 sustains a weight double to that of the merchandise4 #b4 ,the use of the balance requires a considerable assortment of weights1 which causes a proportional increase in the price of the apparatus1 independently of the chances of error which it multiplies1 and of the time employed in producing an equilibrium4 #a4 ,in ,c4 ,paul's steelyard the centres of the movement of suspension1 or the two constant centres1 are placed on the exact line of the divisions of the beam2 an elevation almost imperceptible in the axis of the beam1 destined to compensate for the very slight flexion of the bar1 alone excepted4 #b4 ,the apparatus1 by the construction of the beam1 is balanced below its centre of motion1 so that when no weight is suspended the beam naturally remains horizontal1 and resumes that position when removed from it1 as also when the steelyard is loaded1 and the weight is at the division which ought to show how much the merchandise weighs4 ,the horizontal situation in this steelyard1 as well as in the others1 is known by means of the tongue which rises vertically above the axis of suspension4 #c4 ,it may be discovered1 that the steelyard is deranged if1 when not loaded1 the beam does not remain horizontal4 #d4 ,the advantage of a great and a small side 7which in the other augments the extent of their power of weighing7 is supplied by a very simple process1 which accomplishes the same end with some additional advantages4 ,this process is to employ on the same division different weights4 ,the numbers of the divisions on the bar1 point out the degree of heaviness expressed by the corresponding weights4 ,for example1 when the large weight of the large steelyard weighs #af lbs41 each division it passes over on the bar is equivalent to a pound2 the small weight1 weighing sixteen times less than the large one1 will represent on each of these divisions the sixteenth part of a pound1 or one ounce2 and the opposite face of the bar is marked by pounds at each sixteenth division4 ,in this construction1 therefore1 we have the advantage of being able1 by employing both weights at once1 to ascertain1 for example1 almost within an ounce1 the weight of #ejj pounds of merchandise4 ,it will be sufficient to add what is indicated by the small weight in ounces1 to that of the large one in pounds1 after an equilibrium has been obtained by the position of the two weights1 viz4 the large one placed at the next pound below its real weight1 and the small one at the division which determines the number of ounces to be added4 #e4 ,as the beam is graduated only on one edge1 it may have the form of a thin bar1 which renders it much less susceptible of being bent by the action of the weight1 and affords room for making the figures more visible on both the faces4 #f4 ,in these steelyards the disposition of the axes is not only such that the beam represents a mathematical lever without weight1 but in the principle of its division1 the interval between every two divisions is a determined and aliquot part of the distance between the two fixed points of suspension2 and each of the two weights employed has for its absolute weight the unity of the weight it represents1 multiplied by the number of the divisions contained in the interval between the two centres of motion4 ,thus1 supposing the arms of the steelyard divided in such a manner that ten divisions are exactly contained in the distance between the two constant centres of motion1 a weight to express the pounds on each division of the beam must really weigh ten pounds2 that to point out the ounces on the same divisions must weigh ten ounces1 c4 ,so that the same steelyard may be adapted to any system of measures whatever1 and in particular to the decimal system1 by varying the absolute heaviness of the weights1 and their relation with each other4 ,but to trace out1 in a few words1 the advantages of the steelyards constructed by ,c4 ,paul for commercial purposes1 we shall only observe1-- #a4 ,that the buyer and seller are certain of the correctness of the instrument1 if the beam remains horizontal when it is unloaded and in its usual position4 #b4 ,that these steelyards have one suspension less than the old ones1 and are so much more simple4 #c4 ,that by these means we obtain1 with the greatest facility1 by employing two weights1 the exact weight of merchandise1 with all the approximation that can be desired1 and even with a greater precision than that given by common balances4 ,there are few of these which1 when loaded with #ejj pounds at each end1 give decided indication of an ounce variation2 and the steelyards of ,c4 ,paul possess that advantage1 and cost one half less than balances of equal dominion4 #d4 ,in the last place1 we may verify at pleasure the justness of the weights1 by the transposition which their ratio to each other will permit2 for example1 by observing whether1 when the weight of one pound is brought back one division1 and the weight of one ounce carried forward sixteen divisions1 the equilibrium still remains4 ,it is on this simple and advantageous principle that ,c4 ,paul has constructed his universal steelyard4 ,it serves for weighing in the usual manner1 and according to any system of weights1 all ponderable bodies to the precision of half a grain in the weight of a hundred ounces2 that is to say1 of a ten-thousandth part4 ,it is employed1 besides1 for ascertaining the specific gravity of solids1 of liquids1 and of air1 by processes extremely simple1 and which do not require many subdivisions in the weights4 ,we think the description above given will be sufficiently intelligible without a representation of this instrument4 ,an account of its application to the determination of specific gravities will be found in vol4 iii4 of the ,philosophical ,magazine4 ,the ,chinese ,steelyard4 ,this instrument is used in ,china and the ,east ,indies for weighing gems1 precious metals1 c4 ,the beam is a small rod of ivory1 about a foot in length4 ,upon this are three lines of divisions1 marked by fine silver studs1 all beginning from the end of the beam1 whence the first is extended #h inches1 the second #f-a_/b1 and the third #h-a_/b4 ,the first is ,european weight1 and the other two ,chinese4 ,at the other end of the beam hangs a round scale1 and at three several distances from this end are holes1 through which pass so many fine strings1 serving as different points of suspension4 ,the first distance makes #a-c_/e inches1 the second #c-a_/e1 or double the former1 and the third #d-d_/e1 or triple the same4 ,the instrument1 when used1 is held by one of the strings1 and a sealed weight of about #a-a_/d oz4 troy1 is slid upon the beam until an equilibrium is produced2 the weight of the body is then indicated by the graduated scale above mentioned4 ,the ,danish ,balance4 ,the ,danish balance is a straight bar or lever1 having a heavy weight fixed to one end1 and a hook or scale-pan to receive the substance1 the weight of which is to be determined1 suspended from the other end4 ,the fulcrum is moveable1 and is made to slide upon the bar1 till the beam rests in a horizontal position1 when the place of the fulcrum indicates the weight required4 ,in order to construct a balance of this kind1 let the distance of the centre of gravity from that point to which the substance to be weighed is suspended be found by experiment1 when the beam is unloaded4 ,multiply this distance by the weight of the whole apparatus1 and divide the product by the weight of the apparatus increased by the weight of the body4 ,this will give the distance from the point of suspension1 at which the fulcrum being placed1 the whole will be in equilibrio3 for example1 supposing the distance of the centre of gravity from the point of suspension to be #aj inches1 and the weight of the whole apparatus to be ten pounds2 suppose1 also1 it were required to mark the divisions which should indicate weights of one1 two1 or three pounds1 c4 ,first1 for the place of the division indicating one pound we have 7#aj #aj7_/7aj #a7 #ajj_/7aj #a7 #i-a_/aa inches1 the place of the division marking one pound4 ,for two pounds we have #ajj_/7aj #b7 #h-a_/c inches1 the place of the division indicating two pounds2 and for three pounds #ajj_/7aj #c7 #g-i_/ac inches for the place of the division indicating three pounds1 and so on4 ,this balance is subject to the inconvenience of the divisions becoming much shorter as the weight increases4 ,the distance between the divisions indicating one and two pounds being1 in the example we have given1 about seven tenths of an inch1 whilst that between #bj and #ba pounds is only one tenth of an inch2 consequently a very small error in the place of the divisions indicating the larger weights would occasion very inaccurate results4 ,the ,danish balance is represented at fig4 #aid4 ,the ,bent ,lever ,balance4 ,this instrument is represented at fig4 #aie4 ,the weight at ,c1 is fixed at the end of the bent lever ,a ,b ,c1 which is supported by its axis ,b on the pillar ,i ,h4 ,a scale-pan ,e1 is suspended from the other end of the lever at ,a4 ,through the centre of motion ,b draw the horizontal line ,k ,b ,g1 upon which1 from ,a and ,c let fall the perpendiculars ,a ,k and ,c ,d4 ,then if ,b ,k and ,b ,d are reciprocally proportional to the weights at ,a and ,c1 they will be in equilibrio1 but if not1 the weight ,c will move upwards or downwards along the arc ,f ,g till that ratio is obtained4 ,if the lever be so bent that when ,a coincides with the line ,g ,k1 ,c coincides with the vertical ,b ,h1 then as ,c moves from ,f to ,g1 its momentum will increase while that of the weight in the scale-pan ,e will decrease4 ,hence the weight in ,e1 corresponding to different positions of the balance1 may be expressed on the graduated arc ,f ,g4 ,brady's ,balance1 or ,weighing ,apparatus4 ,this partakes of the properties both of the bent lever balance and of the steelyard4 ,it is represented1 at fig4 #aif4 ,a ,b ,c is a frame of cast iron having a great part of its weight towards ,a4 ,f is a fulcrum1 and ,e a moveable suspender1 having a scale and hook at its lower extremity4 ,e ,k ,g are three distinct places1 to which the suspender ,e may be applied1 and to which belong respectively the three graduated scales of division expressing weights1 f ,c1 c d1 and a b4 ,when the scale and suspender are applied at ,g1 the apparatus is in equilibrio1 with the edge ,a ,b horizontal1 and the suspender cuts the zero on the scale a b4 ,now1 any substance1 the weight of which is to be ascertained1 being put into the scale1 the whole apparatus turns about ,f1 and the part towards ,b descends till the equilibrium is again established1 when the weight of the body is read off from the scale a b1 which registers to ounces and extends to two pounds4 ,if the weight of the body exceed two pounds1 and be less than eleven pounds1 the suspender is placed at ,k2 and when the scale is empty1 the number #b is found to the right of the index of the suspender4 ,if now weights exceeding two pounds be placed in the scale1 the whole again turns about ,f1 and the weight of the body is shown on the graduated arc c d1 which extends to eleven pounds1 and registers to every two ounces4 ,if the weight of the body exceed eleven pounds1 the suspender is hung on at ,e1 and the weights are ascertained in the same manner on the scale f ,c to thirty pounds1 the subdivisions being on this scale quarters of pounds4 ,the same principles would obviously apply to weights greater or less than the above4 ,to prevent mistake1 the three points of support ,g1 ,k1 ,e1 are numbered #a1 #b1 #c2 and the corresponding arcs are respectively numbered in the same manner4 ,when the hook is used instead of the scale1 the latter is turned upwards1 there being a joint at m for that purpose4 ,the ,weighing ,machine for ,turnpike ,roads4 ,this machine is for the purpose of ascertaining the weight of heavy bodies1 such as wheel carriages4 ,it consists of a wooden platform placed over a pit made in the line of the road1 and which contains the machinery4 ,the pit is walled withinside1 and the platform is fitted to the walls of the pit1 but without touching them1 and it is therefore at liberty to move freely up and down4 ,the platform is supported by levers placed beneath it1 and is exactly level with the surface of the road1 so that a carriage is easily drawn on it1 the wheels being upon the platform whilst the horses are upon the solid ground beyond it4 ,the construction of this machine will be readily understood by reference to fig4 #aig41 in which the platform is supposed to be transparent so as to allow of the levers being seen below it4 ,a1 ,b1 ,c1 ,d1 represent four levers tending towards the centre of the platform1 and each moveable on its fulcrum at ,a1 ,b1 ,c1 ,d2 the fulcrum of each rests upon a piece securely fixed in the corner of the pit4 ,the platform is supported upon the cross pins a1 b1 c1 d1 by means of pieces of iron which project from it near its corners1 and which are represented in the plate by the short dark lines crossing the pins a1 b1 c1 d4 ,the four levers are connected under the centre of the platform1 but not so as to prevent their free motion1 and are supported by a long lever at the point ,f1 the fulcrum of which rests upon a piece of masonry at ,e3 the end of this last lever passes below the surface of the road into the turnpike house1 and is there attached to one arm of a balance1 or1 as in ,salmon's patent weighing machine1 to a strap passing round a cylinder which winds up a small weight round a spiral1 and indicates1 by means of an index1 the weight placed upon the platform4 ,illustration3 ,captn4 ,kater1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,suppose the distance from ,a to ,f to be ten times as great as that from ,a to a1 then a force of one pound applied beneath ,f would balance ten pounds applied at a1 or upon the platform4 ,again3 let the distance from ,e to ,g be also ten times greater than the distance from the fulcrum ,e to ,f2 then a force of one pound applied to raise up the end of the lever ,g would counterpoise a weight of ten pounds placed upon ,f4 ,now1 as we gain ten times the power by the first levers1 and ten times more by the lever ,e ,g1 it follows1 that a force of one pound tending to elevate ,g1 would balance #ajj lbs4 placed on the platform2 so that if the end of the lever ,g be attached to one arm of a balance1 a weight of #aj lbs4 placed in a scale suspended from the other arm1 will express the value of #ajjj lbs4 placed upon the platform4 ,the levers are counterpoised1 when the platform is not loaded1 by a weight ,h applied to the end of the last lever1 continued beyond the fulcrum for that purpose4 ,of ,instruments for weighing by means of a ,spring4 ,the spring is well adapted to the construction of a weighing machine1 from the property it possesses of yielding in proportion to the force impressed1 and consequently giving a scale of equal parts for equal additions of weight4 ,it is liable1 however1 to suffer injury1 unless the steel of which it is composed be very well tempered1 from a want of perfect elasticity1 and1 consequently1 from not returning to its original place after it has been forcibly compressed4 ,this1 however1 must be considered to arise1 in a great measure1 from imperfection of workmanship1 or of the material employed1 or from its having been subjected to too great a force4 ,the ,spring ,steelyard4 ,the little instrument known by this name is in very general use1 and is particularly convenient where great accuracy is not necessary1 as a spring which will ascertain weights from one pound to fifty1 is contained in a cylinder only #d inches long and #c_/d inch diameter4 ,this instrument is represented at fig4 #aih4 ,it consists of a tube of iron1 of the dimensions just stated1 closed at the bottom1 to which is attached an iron hook for supporting the substance to be weighed2 a rod of iron a b1 four tenths of an inch wide and one tenth thick1 is firmly fixed in the circular plate c d1 which slides smoothly in the iron tube4 ,a strong steel spring is also fastened to this plate1 and passed round the rod a b without touching it1 and without coming in contact with the interior of the cylindrical tube4 ,the tube is closed at the top by a circular piece of iron through which the piece a b passes4 ,upon the face of a b the weight is expressed by divisions1 each of which indicates one pound1 and five of such divisions in the instrument now before us occupy two tenths of an inch4 ,the divisions1 notwithstanding1 are of sufficient size to enable them to be subdivided by the eye4 ,to use this instrument1 the substance to be weighed is suspended by the hook1 the instrument being held by a ring passing through the rod at the other end4 ,the spring then suffers a compression proportionate to the weight1 and the number of pounds is indicated by the division on the rod which is cut by the top of the cylindrical tube4 ,salter's improved ,spring ,balance4 ,a very neat form of the instrument last described has been recently brought before the public by ,mr4 ,salter1 under the name of the ,improved ,spring ,balance4 ,it is represented at fig4 #aii4 ,the spring is contained in the upper half of a cylinder behind the brass plate forming the face of the instrument2 and the rod is fixed to the lower extremity of the spring1 which is consequently extended1 instead of being compressed1 by the application of the weight4 ,the divisions1 each indicating half a pound1 are engraved upon the face of the brass plate1 and are pointed out by an index attached to the rod4 ,marriott's ,patent ,dial ,weighing ,machine4 ,the exterior of this instrument is represented at fig4 #bjj41 and the interior at fig4 #bja4 ,a ,b ,c is a shallow brass box1 having a solid piece as represented at ,a1 to which the spring ,d ,e ,f is firmly fixed by a nut at ,d4 ,the other end of the spring at ,f is pinned to the brass piece ,g ,h1 to the part of which at ,g is also fixed the iron racked plate ,i4 ,a screw ,l serves as a stop to keep this rack in its place4 ,the teeth of the rack fit into those of the pinion ,m1 the axis of which passes through the centre of the dial-plate1 and carries an index which points out the weight4 ,the brass piece ,g ,h is merely a plate where it passes over the spring1 and the tail piece ,h1 to which the weight is suspended1 passes through an opening in the side of the box4 ,of the ,dynamometer4 ,this is an important instrument in mechanics1 calculated to measure the muscular strength exerted by men and animals4 ,it consists essentially of a spring steelyard1 such as that we first described4 ,this is sometimes employed alone1 and sometimes in combination with various levers1 which allow of the spring being made more delicate1 and consequently increase the extent of the divisions indicating the weight4 ,the first instrument of this kind appears to have been invented by ,mr4 ,graham1 but it was too bulky and inconvenient for use4 ,m4 le ,roy made one of a more simple construction4 ,it consisted of a metal tube1 about a foot long1 placed vertically upon a stand1 and containing in the inside a spiral spring1 having above it a graduated rod terminating in a globe4 ,this rod entered the tube more or less in proportion to the force applied to the globe1 and the divisions indicated the quantity of this force4 ,therefore1 when a man pressed upon the globe with all his strength1 the divisions upon the rod showed the number of pounds weight to which it was equal4 ,an instrument of this kind for determining the force of a blow struck by a man with his fist was lately exhibited at the ,national ,repository4 ,it was fixed to a wall1 from which it projected horizontally4 ,in place of the globe there was a cushion to receive the blow1 and as the suddenness with which the spring returned rendered it impossible to read the division upon the rod1 another rod similarly divided was forced in by the plate forming the basis of the cushion1 and remained stationary when the spring returned4 ,the common spring steelyard1 however1 which we first described1 is in principle the same as ,m4 le ,roy's dynamometer1 and is much more conveniently constructed for the purpose we are considering4 ,the ring at one end may be fixed to an immovable object1 and the hook at the other attached to a man1 or to an animal1 and the extent to which the graduated rod is drawn out of the cylinder shows at once the force which is applied4 ,though this is perhaps the best1 and certainly the most simple dynamometer1 others have been contrived1 which are1 however1 but modifications of the spring steelyard4 ,one of these is represented at fig4 #bjb4 ,the spiral spring acts in the manner before described1 but its divisions are increased in size1 and therefore rendered more perceptible by means of a rack fixed to the plate1 acting against the spiral spring1 the teeth of which move a pinion upon which the arm ,i is fixed1 pointing to the graduated arc ,k4 ,another dynamometer has been invented by ,mr4 ,salmon2 it is represented at fig4 #bjc4 and is a combination of levers with the spring4 ,by means of these levers a much more delicate spring1 and which is therefore more sensible1 may be employed than in the dynamometer last described4 ,the manner in which these levers and spring act will be readily understood by an inspection of the figure4 ,like the weighing machine for carriages1 the fulcrum of each lever is at one end1 and the force is diminished in passing to the spring1 in the ratio of the length of its arms4 ,the spring moves a pinion by means of a rack1 upon which pinion a hand is placed1 indicating by divisions upon a circular dial-plate1 the amount of the force employed4 ,the spring used in this machine is calculated to weigh only about #ej lbs4 instead of about #e cwt41 as in the last described2 but by means of the levers which intervene between it and the force applied1 it will serve to estimate a force equal to #f cwt41 and might obviously be made to go to a much greater extent1 by varying the ratio of the length of the arms of the levers4 ,,on ,,compensation ,,pendulums4 7#ccf47 ,it is said of ,galileo that1 when very young1 he observed a lamp suspended from the roof of a church at ,pisa1 swinging backwards and forwards with a pendulous motion4 ,this1 if it had been remarked at all by an uneducated mind1 would1 most probably1 have been passed by as a common occurrence1 unworthy of the slightest notice2 but to the mind imbued with science no incident is insignificant2 and a circumstance apparently the most trivial1 when subjected to the giant force of expanded intellect1 may become of immense importance to the improvement and to the well-being of man4 ,the fall of an apple1 it is said1 suggested to ,newton the theory of gravitation1 and his powerful mind speedily extended to all creation that great law which brings an apple to the ground4 ,the swinging of a lamp in a church at ,pisa1 viewed by the piercing intellect of ,galileo1 gave rise to an instrument which affords the most perfect measure of time1 which serves to determine the figure of the earth1 and which is inseparably connected with all the refinements of modern astronomy4 ,the properties of the pendulum1 and the manner in which it serves to measure time1 have been fully explained in chapter xi42 and if a substance could be found not susceptible of any change in its dimensions from a change of temperature1 nothing more would be necessary1 as the centre of oscillation would always remain at the same distance from the point of suspension4 ,as every known substance1 however1 expands with heat1 and contracts with cold1 the length of the pendulum will vary with every alteration of temperature1 and thus the time of its vibration will suffer a corresponding change4 ,the effect of a difference of temperature of #be1 or that which usually occurs between winter and summer1 would occasion a clock furnished with a pendulum having an iron rod to gain or lose six seconds in twenty-four hours4 ,it became1 then1 highly important to discover some means of counteracting this variation to which the length of the pendulum was liable1 or1 in other words1 to devise a method by which the centre of oscillation should1 under every change of temperature1 remain at the same distance from the point of suspension3 happily1 the difference in the rate of expansion of different metals presented a ready means of effecting this4 ,graham1 in the year #agae1 made several experiments to ascertain the relative expansions of various metals1 with a view of availing himself of the difference of the expansions of two or more of them when opposed to each other1 to construct a compensating pendulum4 ,but the difference he found was so small1 that he gave up all hope of being able to accomplish his object in that way4 ,knowing1 however1 that mercury was much more affected by a given change of temperature than any other substance1 he saw that if the mercury could be made to ascend while the rod of the pendulum became longer1 and vice vers1 the centre of oscillation might always be kept at the same distance from the point of suspension4 ,this idea happily gave birth to the mercurial pendulum1 which is now in very general use4 ,illustration3 ,captn4 ,kater1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,in the mean time1 ,graham's suggestion excited the ingenuity of ,harrison1 originally a carpenter at ,barton in ,lincolnshire1 who1 in #agbf1 produced a pendulum formed of parallel brass and steel rods1 known by the name of the gridiron pendulum4 ,in the mercurial pendulum1 the bob or weight is the material affording the compensation2 but in the gridiron pendulum the object is attained by the greater expansion of the brass rods1 which raise the bob upwards towards the point of suspension as much as the steel rods elongate downwards4 ,in the present article1 we shall describe such compensation pendulums as appear to us likely to answer best in practice2 and we trust we shall be able to simplify the subject so as to render a knowledge of mathematics in the construction of this important instrument unnecessary4 ,the following table contains the linear expansion of various substances in parts of their length1 occasioned by a change of temperature amounting to one degree4 ,we have taken the liberty of extracting it from a very valuable paper by ,f4 ,bailey1 ,esq41 on the mercurial compensation pendulum1 published in the ,memoirs of the ,astronomical ,society of ,london for #ahbd4 ,,table ,i4 ,linear ,expansion of various ,substances for ,one ,degree of ,fahrenheit's ,thermometer4 ---------------------------------------- ------------------ ,substances4 ,expansions4 ,authors4 ---------------------------------------- ------------------ ,white ,deal1 #jjjjjbbfhe ,captain ,kater4 #jjjjjbhddd ,dr4 ,struve4 ,english ,flint ,glass1 #jjjjjdghhg ,dulong and ,petit4 ,iron 7cast71 #jjjjjfagjj ,general ,roy4 #jjjjjfeffh ,dulong and ,petit4 ,iron 7wire71 #jjjjjfhfac ,lavoisier and ,l4 ,iron 7bar71 #jjjjjfihdd ,hasslar4 ,steel 7rod71 #jjjjjfceif ,general ,roy4 ,commissioners of ,brass1 #jjjjajddjj ,weights and ,measures --mean of several experiments4 ,lead1 #jjjjaeibei ,smeaton4 ,zinc1 #jjjjafcdbf ,ditto4 ,zinc 7hammered71 #jjjjagbfhe ,ditto4 ,mercury in bulk1 #jjjajjaj ,dulong and ,petit4 ---------------------------------------- ------------------ ,from this table it is easy to determine the length of a rod of any substance the expansion of which shall be equal to that of a rod of given length of any other substance4 ,the lengths of such rods will be inversely proportionate to their expansions4 ,if1 therefore1 we divide the lesser expansion by the greater 7supposing the rod the length of which is given to be made of the lesser expansible material71 and multiply the given length by this quotient1 we shall have the required length of a rod1 the expansion of which will be equal to that of the rod given4 ,for example3--,the expansion of a rod of steel being1 from the above table1 #jjjjjfceif1 and that of brass1 #jjjjajddjj2 if it were required to determine the length of a rod of brass which should expand as much as a rod of steel of #ci inches in length1 we have #jjjjjfceif_/jjjjajddjj #fjia1 which1 multiplied by #ci1 gives #bcge inches for the length of brass required4 ,we shall here1 in order to facilitate calculation1 give the ratio of the lengths of such substances as may be employed in the construction of compensation pendulums4 ,,table ,,ii4 ---------------------------------------- ----------- ,steel rod and brass compensation1 as #a3 #fjia ,iron wire rod and lead compensation1 #dcjh ,steel rod and lead compensation1 #ciic ,iron wire rod and zinc compensation1 #cigc ,steel rod and zinc compensation1 #cfhb ,glass rod and lead compensation1 #cjjg ,glass rod and zinc compensation1 #bggc ,deal rod and lead compensation1 #adbg ,deal rod and zinc compensation1 #acac ,steel rod and mercury in a steel cylinder1 #jgbh ,steel rod and mercury in a glass cylinder1 #jgjc ,glass rod and mercury in a glass cylinder1 #jebi ---------------------------------------- ----------- ,it is evident that in this table the decimals express the length of a rod of the compensating material1 the expansion of which is equal to that of a pendulum rod whose length is unity4 ,as we are not aware of the existence of any work which contains instructions that might enable an artist or an amateur to make a compensation pendulum1 we shall endeavour to give such detailed information as may free the subject from every difficulty4 ,the pendulum of a clock is generally suspended by a spring1 fixed to its upper extremity1 and passing through a slit made in a piece which is called the cock of the pendulum4 ,the point of suspension is1 therefore1 that part of the spring which meets the lower surface of the cock4 ,now the distance of the centre of oscillation of the pendulum from this point may be varied in two ways2 the one by drawing up the spring through this slit1 and the other by raising the bob of the pendulum4 ,either of these methods may be practised in the compensation pendulum1 but the former is subject to objections from which the latter is exempt4 ,suppose it were required to compensate a pendulum of #ci inches in length1 of steel1 by means of the expansion of a brass rod4 ,here1 referring to fig4 #bjd41 we have ,s ,c #ci inches 7which is to remain constant7 of steel2 the pendulum spring1 passing through the cock at ,s1 is attached to another rod of steel1 which is fixed to the cross piece ,r ,a at ,a4 ,the other end of the cross piece at ,r is fastened to a brass rod1 the lower extremity of which is fixed to the cock of the pendulum at ,b4 ,now the brass rod ,b ,r must expand upwards1 as much as the steel rod ,a ,c expands downwards2 and the length of the brass must be such as to effect this1 leaving #ci inches of the steel rod below the cock of the pendulum4 ,let us first try #hj inches of steel4 ,multiplying this by #fjia1 we have #dhgc inches for the length of brass1 which compensates #hj inches of steel4 ,but as #dhgc inches of the steel1 equal in length to the brass1 would in this case be above the cock of the pendulum1 it would leave only #cabg inches below it1 instead of #ci inches4 ,let us now try #ajj inches of steel4 ,this1 multiplied as before by #fjia1 gives #fjia inches1 according to the expansions which we have used1 for the length of the brass rod1 and leaves #ciji inches below the cock of the pendulum1 which is sufficiently near for our present purpose4 ,from what has been said we may perceive that the total length of the material of which the pendulum rod is composed must be always equal to the length of the pendulum added to the length of the compensation4 ,in this instance we have effected our object1 by drawing the pendulum-spring through the slit2 but we will now show how the same thing may be done by moving the bob of the pendulum4 ,at fig4 #bje41 let ,s ,c1 as before1 be equal to #ci inches4 ,let the steel rod ,s ,d turn off at right angles at ,d1 and let a rod of brass ,b ,r1 of #fa inches in length1 ascend perpendicularly from this cross piece to ,r4 ,to the upper part of the brass rod fix another cross piece ,r ,a1 and from the extremity ,a let a steel rod descend to ,e1 bending it as in the figure till it reaches ,c4 ,now the total length of the pieces of steel expanding downwards is equal to ,s ,d1 ,d ,f1 and ,f ,c 7amounting together to #ci inches71 to which must be added a length of steel equal to that of the brass rod ,b ,r1 7#fa inches71 making together #ajj inches of steel as before1 the expansion of which downwards is compensated by that of the brass rod1 of #fa inches in length1 expanding upwards4 ,this form1 however1 is evidently inconvenient1 from the great length of brass and steel which is carried above the cock of the pendulum2 but it is the same thing whether the brass and steel be each in one piece1 or divided into several1 provided the pieces of steel be all so arranged as to expand downwards1 and those of brass upwards4 ,thus1 at fig4 #bjf41 the portions of steel expanding downwards are together equal1 as before1 to #ajj inches1 and the two brass pieces expanding upwards are together equal to #fa inches4 ,so that1 in fact1 the two last forms of compensation which we have described differ in no respect from each other in principle1 but only in the arrangement of the materials4 ,the last is the half of the gridiron pendulum1 the remaining bars being merely duplicates of those we have described1 and serving no other purpose but to form a secure frame-work4 ,harrison's ,gridiron ,pendulum4 ,after what has been said1 little more is necessary than to give a representation of this pendulum4 ,this is done at fig4 #bjg41 in which the darker lines represent the steel rods1 and the lighter those of brass4 ,the central rod is fixed at its lower extremity to the middle of the third cross piece from the bottom1 and passes freely through holes in the cross pieces which are above1 whilst the other rods are secured near their extremities to the cross pieces by pins passing through them4 ,in order to render the whole more secure1 the bars pass freely through holes made in two other cross pieces1 the extremities of which are fixed to the exterior steel wires4 ,as different kinds of the same metal vary in their rate of expansion1 the pendulum when finished may be found upon trial to be not duly compensated4 ,in this case one or more of the cross pieces is shifted higher or lower upon the bars1 and secured by pins passed through fresh holes4 ,troughton's ,tubular ,pendulum4 ,this is an admirable modification of ,harrison's gridiron pendulum4 ,it is represented at fig4 #bjh41 where it may be seen that it has the appearance of a simple pendulum1 as the whole compensation is concealed within a tube six tenths of an inch in diameter4 ,a steel wire1 about one tenth of an inch in diameter1 is fixed in the usual manner to the spring by which the pendulum is suspended4 ,this wire passes to the bottom of an interior brass tube1 in the centre of which it is firmly screwed4 ,the top of this tube is closed1 the steel rod passing freely through a hole in the centre4 ,into the top of this interior tube two steel wires1 of one tenth of an inch in diameter1 are screwed into holes made in that diameter1 which is at right angles to the motion of the pendulum4 ,these wires pass down the tube without touching either it or the central rod1 through holes made in the piece which closes the bottom of the interior tube4 ,the lower extremities of these wires1 which project a little beyond the inner tube1 are securely fixed in a piece which closes the bottom of an exterior brass tube1 which is of such a diameter as just to allow the interior tube to pass freely through it1 and of a sufficient length to extend a little above it4 ,the top of the exterior tube is closed like that of the interior1 having also a hole in its centre1 to allow the first steel rod to pass freely through it4 ,into the top of the exterior tube1 in that diameter which coincides with the motion of the pendulum1 a second pair of steel wires of the same diameter as the former are screwed1 their distance from the central rod being equal to the distance of each from the first pair4 ,they consequently pass down within the interior tube1 and through holes made in the pieces closing the lower ends of both the interior and exterior tubes4 ,the lower ends of these wires are fastened to a short cylindrical piece of brass of the same diameter as the exterior tube1 to which the bob is suspended by its centre4 ,illustration3 ,captn4 ,kater1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,fig4 #bji4 is a full sized section of the rod2 the three concentric circles represent the two tubes1 and the rectangular position of the two pair of wires round the middle one is shown by the five small circles4 ,fig4 #baj4 is the part which closes the upper end of the interior tube4 ,the two small circles are the two wires which proceed from it1 and the three large circles show the holes through which the middle wire and the other pair of wires pass4 ,fig4 #baa4 is the bottom of the interior tube4 ,the small circle in the centre is where the central rod is fastened to it1 the others the holes for the other four wires to pass through4 ,fig4 #bab4 is the part which closes the top of the external tube4 ,in the large circle in the centre a small brass tube is fixed1 which serves as a covering for the upper part of the middle wire1 and the two small circles are to receive the wires of the last expansion4 ,fig4 #bac4 represents the bottom of the exterior tube1 in which the small circles show the places where the wires of the second expansion are fastened1 and the larger ones the holes for the other pair of wires to pass through4 ,fig4 #bad4 is a cylindrical piece of brass1 showing the manner in which the lower ends of the wires of the last expansion are fastened to it1 and the hole in the middle is that by which it is pinned to the centre of the bob4 ,the upper ends of the two pair of wires are1 as we have observed1 fastened by screwing them into the pieces which stop up the ends of the tubes1 but at the lower ends they are all fixed as represented in fig4 #bad4 ,the pieces represented by figs4 #bac4 and #bad4 have each a jointed motion1 by means of which the fellow wires of each pair would be equally stretched1 although they were not exactly of the same length4 ,the action of this pendulum is evidently the same as that of the gridiron pendulum1 as we have three lengths of steel expanding downwards1 and two of brass expanding upwards4 ,the weight of the pendulum has a tendency to straighten the steel rods1 and the tubular form of the brass compensation effectually precludes the fear of its bending2 an advantage not possessed by the gridiron pendulum1 in which brass rods are employed4 ,mr4 ,troughton1 to the account he has given of this pendulum in ,nicholson's ,journal1 for ,december1 #ahjd1 has added the lengths of the different parts of which it was composed1 and the expansions of brass and steel from which these lengths were computed4 ,the length of the interior tube was #cai inches1 and that of the exterior one #cbh inches1 to which must be added #jd1 the quantity by which in this pendulum the centre of oscillation is higher than the centre of the bob4 ,these are all of brass4 ,the parts which are of steel are1--the middle wire1 which1 including #jf1 the length of the suspension spring1 is #cic inches4 ,the first pair of wires #cbe inches2 and the second pair1 #ccb inches4 ,the expansions used were1 for brass #jjjjafff1 and for steel #jjjjjffa1 in parts of their length for one degree of temperature4 ,benzenberg's ,pendulum4 ,this pendulum is mentioned in ,nicholson's ,journal for ,april1 #ahjd1 and is taken from ,voigt's ,magazin fr den ,neuesten ,zustande der ,naturkunde1 vol4 iv4 p4 #ghg4 ,the compensation appears to have been effected by a single rod of lead in the centre1 of about half an inch thick2 the descending rods were made of the best thick iron wire4 ,as this pendulum deserves attention from the ease with which it may be made1 and as others which have since been produced resemble it in principle1 we have given a representation of it at fig4 #bae41 where ,a ,b ,c ,d are two rods of iron wire riveted into the cross pieces ,a ,c ,b ,d4 ,e ,f is a rod of lead pinned to the middle of the piece ,b ,d1 and also at its upper extremity to the cross piece ,g ,h1 into which the second pair of iron wires are fixed1 which pass downwards freely through holes made in the cross piece ,b ,d4 ,the lower extremities of these last iron wires are fastened into the piece ,k ,l1 which carries the bob of the pendulum4 ,to determine the length of lead necessary for the compensation1 we must recollect1 as before1 that the distance from the point of suspension to the centre of the bob 7speaking always of a pendulum intended to vibrate seconds7 must be #ci inches4 ,let us suppose the total length of the iron wire to be #fj inches2 then1 from the table which we have given1 we have #dcjh for the length of a rod of lead1 the expansion of which is equivalent to that of an iron rod whose length is unity4 ,multiplying #fj inches by #dcjh1 we have #behd inches of lead1 which would compensate #fj inches of iron2 but this1 taken from #fj inches1 leaves only #cdaf instead of #ci inches4 ,trying again1 in like manner1 #fhe inches of iron1 we find #bie inches of lead for the length1 affording an equivalent compensation1 and which1 taken from #fhe inches1 leaves #ci inches4 ,the length of the rod of lead then required as a compensation in this pendulum is about #bi-a_/b inches4 ,the writer of this article would suggest another form for this pendulum1 which has the advantage of greater simplicity of construction4 ,s ,a1 fig4 #baf41 is a rod of iron wire1 to which the pendulum spring is attached4 ,upon this passes a cylindrical tube of lead1 #bi-a_/b inches long1 which is either pinned at its lower extremity to the end of the iron rod ,s ,a1 or rests upon a nut firmly screwed upon the extremity of this rod4 ,a tube of sheet iron passes over the tube of lead1 and is furnished at top with a flanche1 by which it is supported upon the leaden tube2 or it may be fastened to the top of this tube in any manner that may be thought convenient4 ,the bob of the pendulum may be either passed upon the iron tube 7continued to a sufficient length7 and secured by a pin passing through the centre of the bob1 or the iron tube may be terminated by an iron wire serving the same purpose4 ,here we have evidently the same expansions upwards and downwards as in the gridiron form1 given to this pendulum by ,mr4 ,benzenberg1 joined to the compactness of ,troughton's tubular pendulum4 ,ward's ,compensation ,pendulum4 ,in the year #ahjf1 ,mr4 ,henry ,ward1 of ,blandford in ,dorsetshire1 received the silver medal of the ,society of ,arts for the compensation pendulum which we are about to describe4 ,fig4 #bag4 is a side view of the pendulum rod when together4 ,h ,h and ,i ,i are two flat rods of iron about an eighth of an inch thick4 ,k ,k is a bar of zinc placed between them1 and is nearly a quarter of an inch thick4 ,the corners of the iron bars are bevelled off1 which gives them a much lighter appearance4 ,these bars are kept together by means of three screws1 ,o ,o ,o1 which pass through oblong holes in the bars ,h ,h and ,k ,k1 and screw into the rod ,i ,i4 ,the bar ,h ,h is fastened to the bar of zinc ,k ,k1 by the screw m1 which is called the adjusting screw4 ,this screw is tapped into ,h ,h1 and passes just through ,k ,k2 but that part of the screw which passes ,k ,k has its threads turned off4 ,the iron bar ,i ,i has a shoulder at its upper end1 and rests on the top of the zinc bar ,k ,k and is wholly supported by it4 ,there are several holes for the screw m1 in order to adjust the compensation4 ,the action of this pendulum is similar to that last described1 the zinc expanding upwards as much as the iron rods expand downwards1 and consequently the instance from the point of suspension to the centre of oscillation remains the same4 ,illustration3 ,captn4 ,kater1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,mr4 ,ward states that the expansion of the zinc he used 7hammered zinc7 was greater than that given in the tables4 ,he found that the true length of the zinc bar should be about #bc inches2 our computation would make it nearly #bf4 ,the ,compensation ,tube of ,julien le ,roy4 ,we mention this merely to state that it is similar in principal to the apparatus represented at fig4 #bjd41 with merely this difference1 that1 instead of the steel rod being fixed to a cross piece proceeding from the brass bar ,b ,r1 it is attached to a cap fixed upon a brass tube 7through which it passes7 of the same length as that of the brass rod ,b ,r4 ,cassini spoke well of this pendulum1 and it was used in the observatory of ,cluny about the year #agdh4 ,deparcieux's ,compensation4 ,this was contrived in the same year as that invented by ,julien le ,roy4 ,it is represented at fig4 #bah41 where ,a ,b ,d ,f is a steel bar1 the ends of which are to be fixed to the lower sides of pieces forming a part of the cock of the pendulum4 ,g ,e ,i ,h is of brass1 and stands with its extremities resting on the horizontal part ,b ,d of the steel frame4 ,the upper part ,e ,i of the brass frame passes above the cock of the pendulum1 and admits the tapped wire ,k1 to which the pendulum spring is fixed through a squared hole in the middle4 ,a nut upon this tapped wire gives the adjustment for time4 ,the spring passes through the slit in the cock in the usual manner4 ,it may be easily perceived that this pendulum is in principle the same as that of ,le ,roy2 the expansion of the total length of steel ,a ,b ,s ,c downwards being compensated by the equivalent expansion of the brass bar ,g ,e upwards4 ,it is1 however1 preferable to ,le ,roy's1 because the compensation is contained in the clock case4 ,deparcieux had previously published1 in the year #agci1 an improvement of an imperfectly compensating pendulum1 proposed in the year #agcc by ,regnauld1 a clockmaker of ,chalons4 ,in this pendulum ,deparcieux employed a lever with unequal arms to increase the effect of the expansion of the brass rod1 which was too short4 ,we may here remark1 that all fixed compensations are liable to the same objection1 namely1 that of not moving with the pendulum1 and therefore not taking precisely the same temperature4 ,captain ,kater's ,compensation ,pendulum4 ,in ,nicholson's ,journal1 for ,july1 #ahjh1 is the description of a compensation pendulum by the writer of this article4 ,in this pendulum the rod is of white deal1 three quarters of an inch wide1 and a quarter of an inch thick4 ,it was placed in an oven1 and suffered to remain there for a long time until it became a little charred4 ,the ends were then soaked in melted sealing-wax2 and the rod1 being cleaned1 was coated several times with copal varnish4 ,to the lower extremity of the rod a cap of brass was firmly fixed1 from which a strong steel screw proceeded for the purpose of regulating the pendulum for time in the usual manner4 ,a square tube of zinc was cast1 seven inches long and three quarters of an inch square2 the internal dimensions being four tenths of an inch4 ,the lower part of the pendulum rod was cut away on the two sides1 so as to slide with perfect freedom within the tube of zinc4 ,to the bottom of this zinc tube a piece of brass a quarter of an inch thick was soldered1 in which a circular hole was made nearly four tenths of an inch in diameter1 having a screw on the inside4 ,a cylinder of zinc1 furnished with a corresponding screw on its surface1 fitted into this aperture1 and a thin plate of brass screwed upon the cylinder1 served as a clamp to prevent any shake after the length of zinc necessary for compensation should have been determined4 ,a hole was made through the axis of the cylinder1 through which passed the steel screw terminating the pendulum rod4 ,an opening was made through the bob of the pendulum1 extending to its centre1 to admit the square tube of zinc which was fixed at its upper extremity to the centre of the bob4 ,the pendulum rod passed through the bob in the usual manner1 and the whole was supported by a nut on the steel screw at the extremity4 ,in this form the compensation acts immediately upon the centre of the bob1 elevating it along the rod as much as the rod elongates downwards3 the method of calculating the length of the required compensation is precisely the same as that we have before given4 ,assuming the length of the deal rod to be #dc inches1 and multiplying this by #acac from ,table ,,ii41 we have #efd inches for the length of the zinc necessary to counteract the expansion of the deal4 ,the length of the steel screw between the termination of the pendulum rod and the nut was two inches1 and that of the suspension spring one inch4 ,now1 #c inches of steel multiplied by #cfhb would give #aaj inches for the length of zinc which would compensate the steel1 and1 adding this to #efd inches1 we have #fgd inches for the whole length of zinc required4 ,in this pendulum1 the length of the compensating part may be varied by means of the zinc cylinder furnished with a screw for that purpose4 ,the bob of this pendulum and its compensation are represented at fig4 #bai4 ,it has been objected to the use of wooden pendulum rods1 that it is difficult1 if not impossible1 to secure them from the action of moisture1 which would at once be fatal to their correct performance4 ,the pendulum now before us has1 however1 been going with but little intermission since it was first constructed3 it is attached to a sidereal clock1 not of a superior description1 and exposed to very considerable variations of moisture and dryness2 yet the change in its rate has been so very trifling as to authorize the belief that moisture has little or no effect upon a wooden rod prepared in the manner we have described4 ,its rate1 under different temperatures1 shows that it is over-compensated2 the length of the zinc remaining1 as stated in ,nicholson's ,journal #gdb inches1 instead of which it appears1 by our present compensation1 that it should be #fgh inches4 ,reid's ,compensation ,pendulum4 ,mr4 ,adam ,reid of ,woolwich presented to the ,society of ,arts1 in #ahji1 a compensation pendulum1 for which he was rewarded with fifteen guineas4 ,this pendulum is the same in principle with that last described2 the rod1 however1 is of steel instead of wood1 and the compensation possesses no means of adjustment4 ,this pendulum is represented at fig4 #bbj41 where ,s ,b is the steel rod1 a little thicker where it enters the bob ,c1 and of a lozenge shape to prevent the bob turning1 but above and below it is cylindrical4 ,a tube of zinc ,d passes to the centre of the bob from below1 and the bob is supported upon it by a piece which crosses its centre1 and which meets the upper end of the tube4 ,the rod being passed through the bob and zinc tube1 a nut is applied upon a screw at the lower extremity of the rod in the usual manner4 ,if the compensation should be too much1 the zinc tube is to be shortened until it is correct4 ,the length of the zinc tube will be the same in this pendulum as in that of ,mr4 ,ward--about #bc inches1 if his experiments are to be relied upon4 ,the objection to this pendulum appears to be its great length1 which amounts to #fb inches4 ,we conceive it would be preferable to place the zinc above the bob1 as in the modification which we have suggested of ,benzenberg's pendulum4 ,ellicott's ,pendulum4 ,it appears that the idea of combining the expansions of different metals with a lever1 so as to form a compensation pendulum1 originated with ,mr4 ,graham2 for ,mr4 ,short1 in the ,philosophical ,transactions for #ageb1 states that he was informed by ,mr4 ,shelton1 that ,mr4 ,graham1 in the year #agcg1 made a pendulum1 consisting of three bars1 one of steel between two of brass2 and that the steel bar acted upon a lever so as to raise the pendulum when lengthened by heat1 and to let it down when shortened by cold4 ,illustration3 ,captn4 ,kater1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,this pendulum1 however1 was found upon trial to move by jerks1 and was therefore laid aside by the inventor to make way for the mercurial pendulum4 ,mr4 ,short also says that ,mr4 ,fotheringham1 a quaker of ,lincolnshire1 caused a pendulum to be made1 in the year #agch or #agci1 consisting of two bars1 one of brass and the other of steel1 fastened together by screws with levers to raise or let down the bob1 and that these levers were placed above the bob4 ,mr4 ,john ,ellicott of ,london had made very1 accurate experiments on the relative expansions of seven different metals1 which1 however1 will be found to differ more or less from the results of the experiments of others4 ,it is not1 however1 from this to be concluded that ,ellicott's determinations were erroneous2 for the expansion of a metal will suffer considerable change even by the processes to which it is necessarily subjected in the construction of a pendulum4 ,it is therefore desirable1 whenever a compensation pendulum is to be made1 that the expansions of the materials employed should be determined after the processes of drilling1 filing1 and hammering have been gone through4 ,it had been objected to ,harrison's gridiron pendulum1 that the adjustments of the rods was inconvenient1 and that the expansion of the bob supported at its lower edge would1 unless taken into the account1 vitiate the compensation4 ,these considerations1 it is supposed1 gave rise to ,ellicott's pendulum1 which is nearly similar to those we have just mentioned4 ,ellicott's pendulum is thus constructed3--,a bar of brass and a bar of iron are firmly fixed together at their upper ends1 the bar of brass lying upon the bar of iron1 which is the rod of the pendulum4 ,these bars are held near each other by screws passing through oblong holes in the brass1 and tapped into the iron1 and thus the brass is allowed to expand or contract freely upon the iron with any change of temperature4 ,the brass bar passes to the centre of the bob of the pendulum1 a little above and below which the iron is left broader for the purpose of attaching the levers to it1 and the iron is made of a sufficient length to pass quite through the bob of the pendulum4 ,the pivots of two strong steel levers turn in two holes drilled in the broad part of the iron bar4 ,the short arms of these levers are in contact with the lower extremity of the brass bar1 and their longer arms support the bob of the pendulum by meeting the heads of two screws which pass horizontally from each side of the bob towards its centre4 ,by advancing these screws towards the centre of the bob1 the longer arms of the lever are shortened1 and thus the compensation may be readily adjusted4 ,at the lower end of the iron rod1 under the bob1 a strong double spring is fixed1 to support the greater part of the weight of the bob by its pressure upwards against two points at equal distances from the pendulum rod4 ,mr4 ,ellicott gave a description of this pendulum to the ,royal ,society in #ageb1 but he says the thought was executed in #agch4 ,as this pendulum is very seldom met with1 we think it unnecessary to give a representation of it4 ,compensation by means of a ,compound ,bar of ,steel and ,brass4 ,several compensations for pendulums have been proposed1 by means of a compound bar formed of steel and brass soldered together4 ,in a bar of this description1 the brass expanding more than the steel1 the bar becomes curved by a change of temperature1 the brass side becoming convex and the steel concave with heat4 ,now1 if a bar of this description have its ends resting on supports on each side the cock of the pendulum1 the bar passing above the cock with the brass uppermost1 if the pendulum spring be attached to the middle of the bar1 and it pass in the usual manner through the slit of the cock1 it is evident that1 by an increase of temperature1 the bar will become curved upwards1 and the pendulum spring be drawn upwards through the slit1 and thus the elongation of the pendulum downwards will be compensated4 ,the compensation may be adjusted by varying the distance of the points of support from the middle of the bar4 ,such was one of the modes of compensation proposed by ,nicholson4 ,others of the same description 7that is1 with compound bars7 have been brought before the public by ,mr4 ,thomas ,doughty and ,mr4 ,david ,ritchie2 but as they are supposed to be liable to many practical objections1 we do not think it requisite to describe them more particularly4 ,there is1 however1 a mode of compensation by means of a compound bar1 described by ,m4 ,biot in the first volume of his ,trait de ,physique1 which appears to possess considerable merit1 of which he mentions having first witnessed the successful employment by the inventor1 a clockmaker named ,martin4 ,at fig4 #bba41 ,s ,c1 is the rod of the pendulum1 made1 in the usual manner1 of iron or steel2 this rod passes through the middle of a compound bar of brass and steel 7the brass being undermost71 which should be furnished with a short tube and screws1 by means of which1 or by passing a pin through the tube and rod1 it may be securely fixed at any part of the pendulum rod4 ,two small equal weights ,w ,w slide along the compound bar1 and1 when their proper position has been determined1 may be securely clamped4 ,the manner in which this compensation acts is thus3--,suppose the temperature to increase1 the brass expanding more than the steel1 the bar becomes curved1 and its extremities carrying the weights ,w and ,w are elevated1 and thus the place of the centre of oscillation is made to approach the point of suspension as much1 when the compensation is properly adjusted1 as it had receded from it by the elongation of the pendulum rod4 ,there are three methods of adjusting this compensation3 the first1 by increasing or diminishing the weights ,w and ,w2 the second1 by varying the distance of the weights ,w and ,w from the middle of the bar2 and the third1 by varying the distance of the bar from the bob of the pendulum1 taking care not to pass the middle of the rod4 ,the effect of the compensation is greater as the weights ,w and ,w are greater or more distant from the centre of the bar1 and also as the bar is nearer to the bob of the pendulum4 ,m4 ,biot says that he and ,m4 ,matthieu employed a pendulum of this kind for a long time in making astronomical observations in which they were desirous of attaining an extreme degree of precision1 and that they found its rate to be always perfectly regular4 ,in all the pendulums which we have described1 the bob is supposed to be fixed to the rod by a pin passing through its centre1 and the adjustment for time is to be made by means of a small weight sliding upon the rod4 ,of the ,mercurial ,pendulum4 ,we have been guided1 in our arrangement of the pendulums which we have described1 by the similarity in the mode of compensation employed2 and we have now to treat of that method of compensation which is effected by the expansion of the material of which the bob itself of the pendulum is composed4 ,on this subject1 as we have before observed1 an admirable paper1 from the pen of ,mr4 ,francis ,baily1 may be found in the ,memoirs of the ,astronomical ,society of ,london1 which leaves nothing to be desired by the mathematical reader4 ,but as our object is to simplify1 and to render our subjects as popular as may be1 we must endeavour to substitute for the perfect accuracy which ,mr4 ,baily's paper presents1 such rules as may be found not only readily intelligible1 but practically applicable1 within the limits of those inevitable errors which arise from a want of knowledge of the exact expansion of the materials employed4 ,at fig4 #bbb41 let ,s ,b represent the rod of a pendulum1 and ,f ,c ,b a metallic tube or cylinder1 supported by a nut at the extremity of the pendulum rod1 in the usual manner1 and having a greater expansibility than that of the rod4 ,now ,c1 the centre of gravity1 supposing the rod to be without weight1 will be in the middle of the cylinder2 and if ,c ,b1 or half the cylinder1 be of such a length as to expand upwards as much as the pendulum rod ,s ,b expands downwards1 it is evident that the centre of gravity ,c will remain1 under any change of temperature1 at the same distance from the point of suspension ,s4 ,m4 ,biot imagined that1 in effecting this1 a compensation sufficiently accurate would be obtained2 but ,mr4 ,baily has shown that this is by no means the fact4 ,let us suppose the place of the centre of oscillation to be at ,o1 about three or four tenths of an inch1 in a pendulum of the usual construction1 below the centre of gravity4 ,now1 the object of the compensation is to preserve the distance from ,s to ,o invariable1 and not the distance from ,s to ,c4 ,the distance of the centre of oscillation varies with the length of the cylinder ,f ,b1 and hence suffers an alteration in its distance from the point of suspension by the elongation of the cylinder1 although the distance of the centre of gravity ,c from the point of suspension remains unaltered4 ,we shall endeavour to render this perfectly familiar4 ,suppose a metallic cylinder1 #f inches long1 to be suspended by a thread #cf inches long1 thus forming a pendulum in which the distance of the centre of gravity from the point of suspension is #ci inches3 the centre of oscillation in such a pendulum will be nearly one tenth of an inch below the centre of gravity4 ,now let us imagine cylindrical portions of equal lengths to be added to each end of the cylinder1 until it reaches the point of suspension2 we shall then have a cylinder of #gh inches in length1 the centre of gravity of which will still be at the distance of #ci inches from the point of suspension4 ,but it is well known that the centre of oscillation of such a cylinder is at the distance of about two thirds of its length from the point of suspension4 ,the centre of oscillation1 therefore1 has been removed1 by the elongation of the cylinder1 about #ac inches below the centre of gravity1 whilst the centre of gravity has remained stationary4 ,now the same thing as that which we have just described takes place1 though in a very minor degree1 with our former cylinder1 employed as a compensating bob to a pendulum4 ,the rod expands downwards1 the centre of gravity remains at the same distance from the point of suspension1 and the cylinder elongates both above and below this point2 the consequence of which is1 that though the centre of gravity has remained stationary1 the distance of the centre of oscillation from the point of suspension has increased4 ,it is1 therefore1 evident that the length of the compensation must be such as to carry the centre of gravity a little nearer to the point of suspension than it was before the expansion took place2 by which means the centre of oscillation will be restored to its former distance from the point of suspension4 ,let us suppose the expansions to have taken place1 and that the centre of gravity1 remaining at the same distance from the point of suspension1 the centre of oscillation is removed to a greater distance1 as we have before explained4 ,it is well known that the product obtained by multiplying the distance from the point of suspension to the centre of gravity1 by the distance from the centre of gravity to the centre of oscillation1 is a constant quantity2 if1 therefore1 the distance from the centre of gravity to the point of suspension be lessened1 the distance from the centre of gravity to the centre of oscillation will be proportionally1 though not equally1 increased1 and the centre of oscillation will1 therefore1 be elevated4 ,we see1 then1 if we elevate the centre of gravity precisely the requisite quantity1 by employing a sufficient length of the compensating material1 that although the distance from the centre of gravity to the point of suspension is lessened1 yet the distance from the point of suspension to the centre of oscillation will suffer no change4 ,the following rule for finding the length of the compensating material in a pendulum of the kind we have been considering will be found sufficiently accurate for all practical purposes3-- ,find in the manner before directed the length of the compensating material1 the expansion of which will be equal to that of the rod of the pendulum4 ,double this length1 and increase the product by its one-tenth part1 which will give the total length required4 ,we shall give examples of this as we proceed4 ,graham's ,mercurial ,pendulum4 ,it was in the year #agba that ,graham first put up a pendulum of this description1 and subjected it to the test of experiment2 but it appears to have been afterwards set aside to make way for ,harrison's gridiron pendulum1 or for others of a similar description4 ,for some years past1 however1 its merits have been more generally known1 and it is not surprising that it should be considered as preferable to others1 both from the simplicity of its construction1 and the perfect ease with which the compensation may be adjusted4 ,we have already alluded to ,mr4 ,baily's very able paper on this pendulum1 and we shall take the liberty of extracting from it the following description3-- ,at fig4 #bbc4 is a drawing of the mercurial pendulum1 as constructed in the manner proposed by ,mr4 ,baily4 8,the rod ,s ,f is made of steel1 and perfectly straight2 its form may be either cylindrical1 of about a quarter of an inch in diameter1 or a flat bar1 three eighths of an inch wide1 and one eighth of an inch thick3 its length from ,s to ,f1 that is1 from the bottom of the spring to the bottom of the rod at ,f1 should be #cd inches4 ,the lower part of this rod1 which passes through the top of the stirrup1 and about half an inch above and below the same1 must be formed into a coarse and deep screw1 about two tenths of an inch in diameter1 and having about thirty turns in an inch4 ,a steel nut with a milled head must be placed at the end of the rod1 in order to support the stirrup2 and a similar nut must also be placed on the rod above the head of the stirrup1 in order to screw firmly down on the same1 and thus secure it in its position1 after it has been adjusted nearly to the required rate4 ,these nuts are represented at ,b and ,c4 ,a small slit is cut in the rod1 where it passes through the head of the stirrup1 through which a steel pin ,e is screwed1 in order to keep the stirrup from turning round on the rod4 ,the stirrup itself is also made of steel1 and the side pieces should be of the same form as the rod1 in order that they may readily acquire the same temperature4 ,the top of the stirrup consists of a flat piece of steel1 shaped as in the drawing1 somewhat more than three eighths of an inch thick4 ,through the middle of the top 7which at this part is about one inch deep7 a hole must be drilled sufficiently large to enable the screw of the rod to pass freely1 but without shaking4 ,the inside height of the stirrup from ,a to ,d may be #h-a_/b inches1 and the inside width between the bars about three inches4 ,the bottom piece should be about three eighths of an inch thick1 and hollowed out nearly a quarter of an inch deep1 so as to admit the glass cylinder freely4 ,this glass cylinder should have a brass or iron cover ,g1 which should fit the mouth of it freely1 with a shoulder projecting on each side1 by means of which it should be screwed to the side bars of the stirrup1 and thus be secured always in the same position4 ,this cap should not press on the glass cylinder1 so as to prevent its expansion4 ,the measures above given may require a slight modification1 according to the weight of the mercury employed1 and the magnitude of the cylinder3 the final adjustment1 however1 may be safely left to the artist4 ,some persons have recommended that a circular piece of thick plate glass should float on the mercury1 in order to preserve its surface uniformly level4#g ,the part at the bottom marked ,h is a piece of brass fastened with screws to the front of the bottom of the stirrup1 through a small hole1 in which a steel wire or common needle is passed1 in order to indicate 7on a scale affixed to the case of the clock7 the arc of vibration4 ,this wire should merely rest in the hole1 whereby it may be easily removed when it is required to detach the pendulum from the clock1 in order that the stirrup might then stand securely on its base4 ,one of the screw holes should be rather larger than the body of the screw1 in order to admit of a small adjustment1 in case the steel wire should not stand exactly perpendicular to the axis of motion4 ,the scale should be divided into degrees1 and not inches1 observing that with a radius of #dd inches 7the estimated distance from the bend of the spring to the end of the steel wire7 the length of each degree on the scale must be #jgfh inch40 #g ,the variation produced in the height of the column of mercury 7supposed to be #f-a_/b inches high7 by an alteration of #af in the temperature will be only #a_/ajj of an inch1 or in other words1 #a_/ajj of an inch will be the total variation from its mean state1 by an alteration of #cb in the temperature4 ,it is therefore probable that1 in most cases of moderate alteration in the temperature1 the centre only of the column of mercury is subject to elevation and depression1 whilst the exterior parts remain attached to the sides of the glass vessel4 ,it was with a view to obviate this inconvenience that ,henry ,browne1 ,esq4 of ,portland ,place 7,i believe7 first suggested the piece of floating glass4 ,in order to determine the length of the mercurial column necessary to form the compensation for this pendulum1 we must proceed in the following manner3-- ,let us suppose the length of the steel rod and stirrup together to be #db inches4 ,the absolute expansion of the mercury is #jjjajjaj2 but it is not the absolute expansion1 but the vertical expansion in a glass cylinder1 which is required1 and this will evidently be influenced by the expansion of the base of this cylinder4 ,it is easily demonstrable that1 if we multiply the linear expansion of any substance 7always supposed to be a very small part of its length7 by #c1 we may in all cases take the result for the cubical or absolute expansion of such substance4 ,in like manner1 if we multiply the linear expansion by #b1 we shall have the superficial expansion4 ,if we want the apparent expansion of mercury1 the absolute or cubical expansion of the glass vessel must be deducted from the absolute expansion of the mercury1 which will leave its excess or apparent expansion4 ,in like manner1 deducting the superficial expansion of glass from the absolute expansion of mercury1 we shall have its relative vertical expansion4 ,now1 taking the rate of expansion of glass to be #jjjjjdgi1 and multiplying it by #b1 the relative vertical expansion of the mercury in the glass cylinder will be #jjjajjaj - #jjjjjieh #jjjjijeb4 ,the expansion of a steel rod1 according to our table1 is #jjjjjfceif2 which1 divided by #jjjjijeb1 gives #jgjc for the length of a column of mercury1 the expansion of which is equal to that of a steel rod whose length is unity4 ,we have now to multiply #db inches by #jgjc1 which gives #bie inches2 and this1 deducted from #db1 leaves #cia inches2 so that the length of rod we have chosen is sufficiently near the truth4 ,now1 double #bie inches1 and add one tenth of its product1 and we shall have #fdi inches for the length of the mercurial column forming the requisite compensation4 ,mr4 ,baily's more accurate calculation gives #fca inches4 ,a mercurial compensation pendulum may be formed1 having a cylinder of steel or iron1 with its top constructed in the same manner as the top of the stirrup1 so as to receive the screw of the rod4 ,to find the length of the mercurial column necessary in a pendulum of this description 7that is1 with a cylinder made of steel71 we must double the linear expansion of steel1 and take it from the absolute expansion of mercury to obtain the relative vertical expansion of the mercury4 ,this will be #jjjajjaj - #jjjjabgb #jjjjhgch2 and1 proceeding as before1 we have #jjjjjfceif_/jjjjhgch #jgbgi4 ,let the length of the steel rod be1 as before1 #db inches4 ,multiplying this by #jgbgi1 we have #cjeg1 which being doubled1 and one tenth of the product added1 we obtain #fgb inches for the length of the compensating mercurial column2 which ,mr4 ,baily states to be #fei4 ,a mercurial compensation pendulum having a rod of glass has been employed by the writer of this article1 who has had reason to think well of its performance4 ,its cheapness and simplicity much recommend it4 ,it is merely a cylinder of glass of about #g inches in depth1 and #b-a_/b inches diameter1 terminated by a long neck1 which forms the rod of the pendulum1 the whole blown in one piece4 ,a cap of brass is clamped by means of screws to the top of the rod1 and to this the pendulum spring is pinned4 ,we have unquestionable authority for saying1 that the mercurial pendulum of the usual construction1 that is1 with a steel rod and glass cylinder1 is not affected by a change of temperature simultaneously in all its parts4 ,now1 the pendulum of which we are treating being formed throughout of the same material in a single piece1 and in every part of the same thickness1 it is presumed it cannot expand in a linear direction1 until the temperature has penetrated to the whole interior surface of the glass1 when it is rapidly diffused through the mass of mercury4 ,m4 ,biot mentions that a pendulum of this kind was formerly used in ,france1 and expresses his surprise that it was no longer employed1 as he had heard it very highly spoken of4 ,the writer of this article has also used a pendulum with a glass rod1 which differs from that we have just mentioned1 in having the lower end of the rod firmly fixed in a socket attached to the centre of a circular iron plate1 on the circumference of which a screw is cut1 which fits into a collar of iron1 supporting the cylinder 7to which it is cemented7 by means of a circular lip4 ,this arrangement1 though perhaps less perfect than that we have just described1 the pendulum not being in one piece1 has the advantage of allowing a circular plate of glass to be placed upon the surface of the mercury1 as practised by ,mr4 ,browne4 ,to determine the length of a column of mercury for a glass pendulum1 let us suppose the glass1 including the cylinder1 to be #da inches in length4 ,multiplying this by #jebi1 the number taken from ,table ,,ii4 for a glass rod and mercury in a glass cylinder1 we have #bag inches for the uncorrected length of mercury1 which compensates #da inches of glass4 ,suppose the steel spring to be one inch and a half long3 multiplying this by #jgjc1 the appropriate decimal taken from ,table ,,ii41 we have #ja1 the length of mercury due to the steel1 making with the former #bbg inches1 which1 being doubled1 and the product increased by its one-tenth part1 we obtain five inches for the length of the required column of mercury4 ,compensation ,pendulum of ,wood and ,lead1 on the ,principle of the ,mercurial ,pendulum4 ,if by any contrivance wood could be rendered impervious to moisture1 it would afford one of the most convenient substances known for a compensation pendulum4 ,it does not appear that sufficient experiments have been made upon this subject to decide the question4 ,mr4 ,browne of ,portland ,place1 who has devoted much of his time and attention to the most delicate enquiries of this kind1 has1 we believe1 found that if a teak rod is well gilded1 it will not afterwards be affected by moisture4 ,at all events1 it makes a far superior pendulum1 when thus prepared1 to what it does when such preparation is omitted4 ,mr4 ,baily1 in the paper we have before alluded to1 proposes an economical pendulum to be constructed by means of a leaden cylinder and a deal rod4 ,he prefers lead to zinc1 on account of its inferior price1 and the ease with which it may be formed into the required shape2 and as there is no considerable difference in their rates of expansion1 it is equally applicable to the purpose4 ,let the length of the deal rod be taken at #df inches4 ,then1 to find the length of the cylinder of lead to compensate this1 we have1 in ,table ,,ii41 #adbg for such a pendulum2 which1 being multiplied by #df1 the product doubled1 and one tenth of the result added to it1 gives #addd inches for the length of the leaden cylinder4 ,mr4 ,baily's compensation gives #adc inches4 ,illustration3 ,captn4 ,kater1 del4 ,h4 ,adlard1 sc4 ,london1 ,pubd4 by ,longman ,co4 ,the rod is recommended to be made of about three eighths of an inch in diameter3 the leaden cylinder is to be cast with a hole through its centre1 which will admit with perfect freedom the cylindrical end of the rod4 ,the cylinder is supported upon a nut1 which screws on the end of the rod in the usual manner4 ,this pendulum is represented at fig4 #bbd4 ,mr4 ,baily proposes that the pendulum should be adjusted nearly to the given rate by means of the screw at the bottom1 and that the final adjustment be made by means of a slider moving along the rod4 ,indeed1 this is a means of adjustment which we would recommend to be employed in every pendulum4 ,smeaton's ,pendulum4 ,we shall conclude our account of compensation pendulums with a description of that invented by ,mr4 ,smeaton4 ,the compensation for temperature in this pendulum is effected by combining the two modes1 which have been so fully described in the preceding part of this article4 ,the pendulum rod is of solid glass1 and is furnished with a steel screw and nut at the bottom in the usual manner4 ,upon the glass rod a hollow cylinder of zinc1 about the eighth of an inch thick1 and about #ab inches long1 passes freely1 and rests upon the nut at the bottom of the pendulum rod4 ,over the zinc cylinder passes a tube made of sheet-iron4 ,the edge of this tube at the top is turned inwards1 and is notched so as to allow of this being effected4 ,a flanche is thus formed1 by which the iron tube is supported1 upon the zinc cylinder4 ,the lower edge of the iron tube is turned outwards1 so as to form a base destined to support a leaden cylinder1 which we are about to describe4 ,a cylinder of lead1 rather more than #ab inches long1 is cast with a hole through its axis1 of such a diameter as to allow of its sliding freely1 but without shake1 upon the iron tube over which it passes1 and by the lower extremity of which it is supported4 ,now the zinc1 resting upon the nut and expanding upwards1 will raise the whole of the remaining part of the compensation4 ,this expansion upwards will be slightly counteracted by the lesser expansion downwards of the iron tube1 which carries with it the leaden cylinder4 ,the cylinder of lead now acts upon the principle of the mercurial pendulum1 and1 expanding upwards1 contributes that which was wanting to restore the centre of oscillation to its proper distance from the point of suspension4 ,this pendulum1 we have been informed1 does well in practice1 and we are not aware that any description of it has been before published4 ,the method of calculating the length of the tubes required to form the compensation is very simple2 nothing more is necessary than to find the length of zinc1 the expansion of which is equal to that of the pendulum rod4 ,let the pendulum rod be composed of #dc inches of glass1 the spring being an inch and a half long1 and the screw between the end of the glass rod and the nut half an inch1 making in the whole two inches of steel and #dc inches of glass4 ,now to find the length of zinc that will compensate the glass1 we have1 from ,table ,,ii41 for glass and zinc #bggc1 which1 multiplied by #dc1 gives #aaib inches4 ,in like manner we obtain as a compensation for two inches of steel #jgd of zinc1 which1 added to #aaib1 gives #abff inches for the total length of the zinc cylinder4 ,now if the iron tube and the lead cylinder be each made of the same length as the zinc1 and arranged as we have described1 the compensation will be perfect4 ,to prove this1 find1 by means of the expansions given in ,table ,i41 the actual expansion of each of the substances employed in the pendulum1 and we shall have the following results3-- ,the expansion of #abff inches of zinc expanding upwards is #jjjbahf ,deduct that of #abff inches of iron expanding downwards #jjjjhfi -------- ,remaining effect of expansion upwards1 referred to the lower extremity of the iron tube #jjjacag ,now1 for the lead4--,on the principle of the mercurial compensation1 subtract one tenth part of the length of the cylinder1 and take half the remainder1 and we shall have six inches of lead1 the expansion of which upwards is #jjjjiee -------- ,total expansion of the compensation upwards #jjjbbgb -------- ,to find the expansion of the rod1 we have the expansion of #dc inches of glass #jjjbjei ,of two inches of steel #jjjjabg -------- ,total expansion of the pendulum rod #jjjbahf ,agreeing near enough with that of the compensation before found4 ,as we conceive we have been sufficiently explicit in our description of this pendulum1 in the construction of which no difficulty presents itself1 we think an engraved representation of it would be superfluous4 ,we have hitherto treated only of compensations for temperature2 but there is another kind of error1 which has been sometimes insisted upon1 arising from a variation in the density of the atmosphere4 ,if the density of the atmosphere be increased1 the pendulum will experience a greater resistance1 the arc of vibration will in consequence be diminished1 and the pendulum will vibrate faster4 ,this1 however1 is in some measure counteracted by the increased buoyancy of the atmosphere1 which1 acting in opposition to gravity1 occasions the pendulum to vibrate slower4 ,if the one effect exactly equalled the other1 it is evident no error would arise2 and in a paper by ,mr4 ,davies ,gilbert1 ,president of the ,royal ,society of ,london1 published in the ,quarterly ,journal for #ahbf1 he has proved that1 by a happy chance1 the arc in which pendulums of clocks are usually made to vibrate is the arc at which this compensation of error takes place4 ,this arc1 for a pendulum having a brass bob1 is #a #ef #cj on each side of the perpendicular2 and for a mercurial pendulum1 #a #ca #dd1 or about one degree and a half4 ,it is well known that1 if a pendulum vibrates in a circular arc1 the times of vibration will vary nearly as the squares of the arcs2 but if the pendulum could be made to vibrate in a cycloid1 the time of its vibration in arcs of different extent would then remain the same4 ,huygens and others1 therefore1 endeavoured to effect this by placing the spring of the pendulum between cheeks of a cycloidal form4 ,when escapements are employed which do not insure an unvarying impulse to the pendulum1 the force may be unequally transmitted through the train of the clock in consequence of unavoidable imperfections of workmanship1 and the arc of vibration may suffer some increase or diminution from this cause4 ,to discover a remedy for this is certainly desirable4 ,the writer of this article some years ago imagined a mode1 which he believes has also been suggested by others1 by which he conceived a pendulum might be made to describe an arc approaching in form to that of a cycloid4 ,the pendulum spring was of a triangular form1 and the point or vertex was pinned into the top of the pendulum rod1 the base of the triangle forming the axis of suspension4 ,now it is evident that when the pendulum is in motion1 the spring will resist bending at the axis of suspension1 with a force in some sort proportionate to the base of the triangle4 ,suppose the pendulum to have arrived at the extent of its vibrations2 the spring will present a curved appearance2 and if the distance from the point of suspension to the centre of oscillation be then measured1 it will evidently1 in consequence of the curvature of the spring1 be shorter than the distance from the point of suspension to the centre of oscillation1 measured when the pendulum is in a perpendicular position1 and consequently when the spring is perfectly straight4 ,the base of the triangle may be diminished1 or the spring be made thinner2 either of which will lessen its effect4 ,we cannot say how this plan might answer upon further trial1 as sufficient experiments were not made at the time to authorize a decisive conclusion4 ,we have thus completed our account of compensation pendulums2 but before we conclude1 it may not be unacceptable if we offer a few remarks on some points which may be found of practical utility4 ,the cock of the pendulum should be firmly fixed either to the wall or to the case of the clock1 and not to the clock itself1 as is sometimes done1 and which has occasioned much irregularity in its rate1 from the motion communicated to the point of suspension4 ,we prefer a bracket or shelf of cast iron or brass1 upon which the clock may be fixed1 and the cock carrying the pendulum attached to its perpendicular back4 ,this bracket may either be screwed to the back of the clock-case1 or1 which is the better mode1 securely fixed to the wall2 and if the latter be adopted1 the whole may be defended from the atmosphere1 or from dust1 by the clock-case1 which thus has no connection either with the clock or with the pendulum4 ,the point of suspension should be distinctly defined and immovable4 ,this may be readily effected1 after the pendulum shall have taken the direction of gravity1 by means of a strong screw entering the cock 7which should be very stout7 on one side1 and pressing a flat piece of brass into firm contact with the spring4 ,the impulse should be given in that plane of the rod which coincides with the plane of vibration passing through the axis of the rod4 ,if the impulse be given at any point either before or behind this plane1 the probable result will be a tremulous unsteady motion of the pendulum4 ,a few rough trials1 and moving the weight1 will bring the pendulum near its intended time of vibration1 which should be left a little too slow2 when the bob should be firmly fixed to the rod1 if the form of the pendulum will admit of it1 by a pin or screw passing through its centre4 ,the more delicate adjustment may be completed by shifting the place of the slider with which the pendulum is supposed to be furnished on the rod4 ,mr4 ,browne 7of whom we have before spoken7 practises the following very delicate mode of adjustment for rate1 which will be found extremely convenient1 as it is not necessary to stop the pendulum in order to make the required alteration4 ,having ascertained1 by experiment1 the effect produced on the rate of the clock1 by placing a weight upon the bob equal to a given number of grains1 he prepares certain smaller weights of sheet-lead1 which are turned up at the corners1 that they may be conveniently laid hold of by a pair of forceps1 and the effect of these small weights on the rate of the clock will be1 of course1 known by proportion4 ,the rate being supposed to be in defect1 the weights necessary to correct this may be deposited1 without difficulty1 upon the bob of the pendulum1 or upon some convenient plane surface1 placed in order to receive them3 and should it be necessary to remove any one of the weights1 this may readily be done by employing a delicate pair of forceps1 without producing the slightest disturbance in the motion of the pendulum4 ,,index4 ,a4 ,action and reaction1 #cd4 ,aeriform fluids1 #bf4 ,animalcules1 #ab4 ,atmosphere1 impenetrability of1 #bb4 ,compressibility and elasticity of1 #bc4 ,atoms1 #f4 ,coherence of1 #g4 ,attraction1 magnetic1 of gravitation1 #h1 #ej1 #fd4 ,molecular or atomic1 #fi4 ,cohesion1 #gj4 ,attwood1 machine of1 #ib4 ,axes1 principal1 #ach4 ,axis1 mechanical properties of1 #abh4 ,b4 ,balance1 #bgi4 ,of ,bates1 #bhh4 ,use of1 #bhi4 ,danish1 #bii4 ,bent-lever of ,brady1 #cja4 ,bodies1 #b4 ,lines1 surfaces1 edges1 area1 length of1 #d4 ,figure1 volume1 shape of1 #e4 ,porosity of1 #ag4 ,compressibility of1 #ah4 ,elasticity1 dilatibility of1 #ai4 ,inertia of1 #bg4 ,rule for determining velocity of2 motion of two bodies after impact1 #ch4 ,c4 ,capillary attraction1 #gc4 ,capstan1 #agi4 ,cause and effect1 #g4 ,circle of curvature1 #ii4 ,cog1 hunting1 #aia4 ,components1 #ea4 ,cord1 #afc4 ,cordage1 friction and rigidity of1 #bfj4 ,crank1 #bda4 ,crystallisation1 #ad4 ,cycloid1 #aeh4 ,d4 ,damper1 self-acting1 #bcd4 ,deparcieux's compensation pendulum1 #cai4 ,diagonal1 #ea4 ,dynamics1 #afj4 ,dynamometer1 #cje4 ,e4 ,electricity1 #gf4 ,electro-magnetism1 #gf4 ,equilibrium1 neutral1 instable1 and stable1 #aah4 ,f4 ,figure1 #e4 ,fly-wheel1 #bci4 ,force1 #f4 ,composition and resolution of1 #di4 ,centrifugal1 #ih4 ,moment of2 leverage of1 #ace4 ,regulation and accumulation of1 #bbd4 ,friction1 effects of1 #if4 ,laws of1 #bfd4 ,g4 ,governor1 #bbg4 ,gravitation1 attraction of1 #gg4 ,terrestrial1 #hd4 ,gravity1 centre of1 #ajg4 ,gyration1 radius of1 centre of1 #acg4 ,h4 ,hooke's universal joint1 #beb4 ,hydrophane1 porosity of1 #ah4 ,i4 ,impact1 #ci4 ,impulse1 #fe4 ,inclined plane1 #afc-bji4 ,inclined roads1 #baa4 ,inertia1 #bg4 ,laws of1 #cb4 ,moment of1 #acg4 ,j4 ,julien le ,roy1 compensation tube of1 #cai4 ,l4 ,lever1 #afc4 ,fulcrum of2 three kinds of1 #afg4 ,equivalent1 #agf4 ,line of direction1 #aaj4 ,liquids1 compressibility of1 #bd4 ,loadstone1 #fh4 ,m4 ,machines1 simple1 #afj4 ,power of1 #age4 ,regulation of1 #bbe4 ,magnet1 #fh4 ,magnetic attraction1 #h4 ,magnetism1 #gf4 ,magnitude1 #d4 ,marriott's patent weighing machine1 #cje4 ,materials1 strength of1 #bgb4 ,matter1 properties of1 #b4 ,impenetrability of1 #d4 ,atoms of2 molecules of1 #f4 ,divisibility of1 #i4 ,examples of the subtilty of1 #ab4 ,limit to the divisibility of1 #ac4 ,porosity of2 density of1 #ag4 ,compressibility of1 #ah4 ,elasticity and dilatability of1 #ai4 ,impenetrability of1 #bb4 ,inertia of1 #bg4 ,mechanical science1 foundation of1 #af4 ,metronomes1 principles of1 #aec4 ,molecules1 #f4 ,motion1 laws of1 #df4 ,uniformly accelerated1 #hg4 ,table illustrative of1 #ij4 ,retarded2 of bodies on inclined planes and curves1 #id4 ,rotary and progressive1 #abg4 ,mechanical contrivances for the modification of1 #bde4 ,continued rectilinear2 reciprocatory rectilinear2 continued circular2 reciprocating circular1 #bdf4 ,n4 ,newton1 method of1 for determining the thickness of transparent substances1 #aj4 ,laws of motion of1 #df4 ,o4 ,oscillation1 #abi4 ,of the pendulum1 #ade4 ,centre of1 #aeb4 ,p4 ,parallelogram1 #ea4 ,particle1 #f4 ,pendulum1 oscillation or vibration of1 #ade4 ,isochronism of1 #adg4 ,centre of oscillation of1 #aeb4 ,of ,troughton1 #bhd4 ,compensation1 #cjg4 ,of ,harrison1 #cac4 ,tubular1 of ,troughton1 #cad4 ,of ,benzenberg1 #caf4 ,ward's compensation1 #cah4 ,captain ,kater's compensation1 #cbj4 ,reid's2 ,ellicott's compensation1 #cbb4 ,steel and brass compensation1 #cbd4 ,mercurial1 #cbf4 ,graham's mercurial1 #cbi4 ,wood and lead1 #ccd4 ,smeaton's1 #cce4 ,percussion1 #acj4 ,centre of1 #add4 ,planes of cleavage1 #ae4 ,porosity1 #ag4 ,power1 #afa4 ,properties1 #b4 ,projectiles1 curvilinear path of1 #hb4 ,pulley1 #afd4 ,tackle2 fixed1 #aih4 ,single moveable1 #bjj4 ,called a runner2 ,spanish bartons1 #bje4 ,r4 ,rail-roads1 #bac4 ,regulating damper1 #bcc4 ,regulators1 #bbg4 ,repulsion1 #h4 ,molecular1 #gd4 ,resultant1 #ea4 ,rose-engine1 #bej4 ,s4 ,salters1 spring balance of1 #cje4 ,screw1 #bji4 ,concave1 #bag4 ,micrometer1 #bbc4 ,shape1 #e4 ,siphon1 capillary1 #gc4 ,spring1 #cjd4 ,statics1 #afj4 ,steelyard1 #bid4 ,c4 ,paul's1 #bif4 ,chinese1 #bii4 ,t4 ,table1 whirling1 #ii4 ,tachometer1 #bcd4 ,tread-mill1 #agi4 ,v4 ,velocity1 angular1 #ii4 ,vibration1 #abi4 ,of the pendulum1 #ade4 ,centre of1 #aeb4 ,volume1 #e-ag4 ,w4 ,watch1 mainspring of2 balance wheel of1 #aie4 ,water regulator1 #bbi4 ,wedge1 #bji4 ,use of1 #bae4 ,weight1 #afa-bia4 ,weighing machines1 #bgh4 ,for turnpike roads1 #cjb4 ,by means of a spring1 #cjc4 ,wheels1 spur1 crown1 bevelled1 #ahi4 ,escapement1 #aid4 ,wheel and axle1 #agg4 ,wheel-work1 #agf4 ,winch1 #agi4 ,windlass1 #agh4 ,wollaston's wire1 #aj4 ,z4 ,zureda1 apparatus of2 ,leupold's application of1 #bea4 ,,end ,,of ,,mechanics4 ,,london3 ,,spottiswoodes and ,,shaw ,new-street-,square4