,transcrib]'s notes3 7#a7 ,numb]s foll[+ lett]s 7)\t space7 l ,c#b 7 orig9ally pr9t$ 9 subscript4 ,lett] subscripts are prec$$ by an und]score1 l ,cn4 7#b7 ,*>act]s foll[+ a c>at 77 7 pr9t$ 9 sup]script4 7#c7 ,side-notes 7 relocat$ to function z titles ( _! respective p>agraphs4 7#d7 ,macrons & breves above lett]s & dots bel[ lett]s 7 n ins]t$4 7#e7 root /&s = ! root symbol2 alpha1 beta1 etc4 = greek lett]s4 7#f7 ,! foll[+ typographical errors h be5 correct$3 ,,article ,,ge(frey3 444 8 hi/ory 9 *iefly "o ( qu>rels1 ) ! see ( ,cant]bury1 ) ! *ancellor ,william ,long*amp1 ) 8 half-bro!rs ,ri*>d & ,john1 & especially ) 8 canons at ,york4 ',william' am5d$ f ',willian'4 ,,article ,,geology3 444 & at ! same "t great] appreciation has be5 %[n ( ! signification & /r5g? ( ! geological pro(s ( ! high antiquity ( our planet4 '/r5g?' am5d$ f '/5g?'4 ,,article ,,geology3 444 x c be demon/rat$ t sometimes an in* or two ( s$im5t might1 on c]ta9 horizons1 repres5t ! deposit ( an enorm\sly long] p]iod ?an a hundr$ or a ?\s& times ! same am\nt ( s$im5t on o!r horizons4 'might' am5d$ f 'mu*'4 ,,article ,,geology3 ,b no su* ext5sive di/urbance ( ! normal conditions ( ! di/ribution ( life c take place )\t c>ry+ ) x _m second>y effects1 & sett+ 9 motion a wide cycle ( *ange & ( reaction 9 ! animal & vegetable k+doms4 'k+doms' am5d$ f 'k9dgoms'4 ,,article ,,geometry3 ,! bases & altitudes ( equal solid p>allelepip$s are reciprocally proportional2 & if ! bases & altitudes be reciprocally proportional1 ! solid p>allelepip$s are equal4 'are' am5d$ f '&'4 ,,article ,,geometry3 ,an alt]native me?od ( te/+ a relation is illu/rat$ 9 ! foll[+ example3--,if ,a1 ,b1 ,c1 ,d1 ,e1 ,f be six coll9e> po9ts1 !n 'foll[+ example3--' am5d$ f 'example3 foll[+'4 ,,article ,,geometry3 #c4 ,9 an hyp]bolic involution any two conjugate po9ts are h>monic conjugates ) reg>d to ! two foci4 'an' am5d$ f 'a'4 ,,article ,,geometry3 ,if two l9es1 giv5 by _! projections1 int]sect1 ! int]section ( _! planes & ! int]section ( _! elevations mu/ lie 9 a l9e p]p5dicul> to ! axis1 because !y mu/ be ! projections ( ! po9t common to ! two l9es4 'planes' am5d$ f 'plans'4 ,,article ,,geometry3 ,:]e ?is is ! case1 if alpha be ! measure ( curvature1 ! l9e> elem5t c be put into ! =m 'if' am5d$ f 'x'4 ,,article ,,geometry3 ,! developm5t ( ! consequ5ces ( ^! metrical def9itions is ! subject ( non-,euclidean geometry4 'subject' am5d$ f 'subjct'4 ,,encyclopa$ia ,,britannica ,a ,,diction>y ,,( ,,arts1 ,,sci5ces1 ,,lit]ature ,,& ,,g5]al ,,in=mation ,,elev5? ,,edition ,,volume ,,xi1 ,,slice ,,vi ,,geodesy to ,,geometry ,,articles ,,9 ,,?is ,,slice3 ,,geodesy ,,ge(froy1 ,,eti5ne ,,francois ,,ge(frey 7,m>tel7 ,,ge(froy1 ,,juli5 ,,l\is ,,ge(frey 7,plantag5et7 ,,ge(froy ,,sa9t-,,hilaire1 ,,eti5ne ,,ge(frey 7duke ( ,brittany7 ,,ge(froy ,,sa9t-,,hilaire1 ,,isidore ,,ge(frey 7ar*bi%op ( ,york7 ,,geography ,,ge(frey ,,de ,,montbray ,,geoid ,,ge(frey ,,( ,,monm\? ,,geok-,,tepe ,,ge(frey ,,( ,,p>is ,,geology ,,ge(frey ,,! ,,bak] ,,geometrical ,,cont9uity ,,ge(fr91 ,,m>ie ,,!rese ,,rodet ,,geometry ,,geodesy 7f ! ,gr4 ,greek3 ge1 ! e>?1 & ,greek3 daie91 to divide71 ! sci5ce ( survey+ 7q4v47 ext5d$ to l>ge tracts ( c\ntry1 hav+ 9 view n only ! production ( a sy/em ( maps ( v great accuracy1 b ! det]m9ation ( ! curvature ( ! surface ( ! e>?1 & ev5tually ( ! figure & dim5sions ( ! e>?4 ,?is la/1 inde$1 may be ! sole object 9 view1 z 0 ! case 9 ! op]ations conduct$ 9 ,p]u & 9 ,lapl& by ! celebrat$ ,fr5* a/ronom]s ,p4 ,b\gu]1 ,c4,m4 de la ,condam9e1 ,p4,l4,m4 de ,maup]tuis1 ,a4,c4 ,clairault & o!rs2 & ! measurem5t ( ! m]idian arc ( ,france by ,p4,f4,a4 ,me*a9 & ,j4,b4,j4 ,delambre _h = its end ! det]m9ation ( ! true l5g? ( ! metre :i* 0 to be ! legal /&>d ( l5g? ( ,france 7see ,,e>?1 ,,figure ,,( ,,!74 ,! basis ( e ext5sive survey is an accurate triangulation1 & ! op]ations ( geodesy consi/ 9 ! measurem5t1 by !odolites1 ( ! angles ( ! triangles2 ! measurem5t ( "o or m sides ( ^! triangles on ! gr\nd2 ! det]m9ation by a/ronomical obs]vations ( ! azimu? ( ! :ole network ( triangles2 ! det]m9ation ( ! actual position ( ! same on ! surface ( ! e>? by obs]vations1 fir/ = latitude at "s ( ! /ations1 & secondly = longitude2 ! det]m9ation ( altitude = all /ations4 ,= ! computation1 ! po9ts ( ! actual surface ( ! e>? are imag9$ z project$ along _! plumb l9es on ! ma!matical figure1 :i* is giv5 by ! /ation>y sea-level1 & ! ext5sion ( ! sea ?r\gh ! cont95ts by a sy/em ( imag9>y canals4 ,= _m purposes ! ma!matical surface is assum$ to be a plane2 9 o!r cases a sph]e ( radius #fcga kilometres 7#bj1ijj1jjj ft474 ,9 ! case ( ext5sive op]ations ! surface mu/ be consid]$ z a compress$ ellipsoid ( rotation1 ^: m9or axis co9cides ) ! e>?'s axis1 & ^: compression1 flatt5+1 or ellipticity is ab\t #a_/bih4 ,measurem5t ( ,base ,l9es4 ,to det]m9e by actual measurem5t on ! gr\nd ! l5g? ( a side ( "o ( ! triangles 7base l9e71 :]efrom to inf] ! l5g?s ( all ! o!r sides 9 ! triangulation1 is n ! lea/ difficult op]ation ( a trigonometrical survey4 ,:5 ! problem is /at$ ?us--,to det]m9e ! numb] ( times t a c]ta9 /&>d or unit ( l5g? is conta9$ betwe5 two f9ely m>k$ po9ts on ! surface ( ! e>? at a di/ance ( "s miles asund]1 s t ! error ( ! result may be pron\nc$ to lie betwe5 c]ta9 v n>r[ limits1--!n ! "q dem&s v s]i\s consid]ation4 ,! repres5tation ( ! unit ( l5g? by means ( ! di/ance betwe5 two f9e l9es on ! surface ( a b> ( metal at a c]ta9 temp]ature is nev] itself free f unc]ta9ty & probable error1 ow+ to ! difficulty ( kn[+ at any mom5t ! precise temp]ature ( ! b>2 & ! transf]5ce ( ?is unit1 or a multiple ( x1 to a measur+ b> w be affect$ n only ) errors ( obs]vation1 b ) errors aris+ f unc]ta9ty ( temp]ature ( bo? b>s4 ,if ! measur+ b> be n self-comp5sat+ = temp]ature1 its expansion mu/ be det]m9$ by v c>eful exp]im5ts4 ,! !rmomet]s requir$ = ?is purpose mu/ be v c>efully /udi$1 & _! errors ( division & index error det]m9$4 ,9 ord] to avoid ! difficulty 9 exactly det]m9+ ! temp]ature ( a b> by ! m]cury !rmomet]1 ,f4,w4 ,bessel introduc$ 9 #ahcd ne> ,konigsb]g a comp\nd b> :i* con/itut$ a metallic !rmomet]4#a ,a z9c b> is laid on an iron b> two toises long1 bo? b>s be+ p]fectly plan$ & 9 free contact1 ! z9c b> be+ slightly %ort] & ! two b>s rigidly unit$ at "o end4 ,z ! temp]ature v>ies1 ! diff]5ce ( ! l5g?s ( ! b>s1 z p]ceiv$ by ! o!r end1 also v>ies1 & af=ds a quantitative correction = temp]ature v>iations1 :i* is appli$ to r$uce ! l5g? to /&>d temp]ature4 ,dur+ ! measurem5t ( ! base l9e ! b>s 7 n all[$ to come into contact1 ! int]val be+ measur$ by ! ins]tion ( glass w$ges4 ,! results ( ! comp>isons ( f\r measur+ rods ) "o ano!r & ) ! /&>ds 7 elaborately comput$ by ! me?od ( lea/-squ>es4 ,! probable error ( ! measur$ l5g? ( #ice toises 7ab\t #fjjj ft47 has be5 e/imat$ z #a_/hfcejj or #a4b mu 7mu d5ot+ a million?74 ,) ?is app>atus f\rte5 base l9es 7 measur$ 9 ,prussia & "s neighb\r+ /ates2 9 ^! cases a some:at high] degree ( accuracy 0 obta9$4 ,! pr9cipal triangulation ( ,great ,brita9 & ,irel& has sev5 base l9es3 five h be5 measur$ by /eel *a9s1 & two1 m exactly1 by ! comp5sation b>s ( ,g5]al ,t4,f4 ,colby1 an app>atus introduc$ 9 #ahbg-ahbh at ,l\gh ,foyle 9 ,irel&4 ,t5 base l9es 7 measur$ 9 ,india 9 #ahca-ahfi by ! same app>atus4 ,?is is a sy/em ( six comp\nd-b>s self-correct+ = temp]ature4 ,! b>s may be ?us describ$3 ,two b>s1 "o ( brass & ! o!r ( iron1 are laid 9 p>allelism side by side1 firmly unit$ at _! c5tres1 f :i* !y may freely exp& or contract2 at ! /&>d temp]ature !y are ( ! same l5g?4 ,let ,,ab be "o b>1 ,a',b' ! o!r2 draw l9es ?r\gh ! correspond+ extremities ,,aa' 7to ,p7 & ,,bb' 7to ,q71 & make ,a',p ,b',q1 ,,aa' be+ equal to ,,bb'4 ,if ! ratio ,a',p_/,,ap equals ! ratio ( ! coeffici5ts ( expansion ( ! b>s ,a',b' & ,,ab1 !n1 obvi\sly1 ! di/ance ,,pq is con/ant 7or ne>ly s74 ,9 ! actual in/rum5t ,p & ,q are f9ely engrav$ dots #aj ft4 ap>t4 ,9 practice ! b>s1 :5 align$1 are n 9 contact1 an int]val ( #f 94 be+ all[$ betwe5 ea* b> & its neighb\r4 ,?is di/ance is accurately measur$ by an +5i\s micrometrical arrangem5t con/ruct$ on exactly ! same pr9ciple z ! b>s !mselves4 ,! la/ base l9e measur$ 9 ,india _h a l5g? ( #hiac ft4 ,9 consequ5ce ( "s suspicion z to ! accuracy ( ! comp5sation app>atus1 ! measurem5t 0 repeat$ f\r times1 ! op]ations be+ conduct$ s z to det]m9e ! actual values ( ! probable errors ( ! app>atus4 ,! direction ( ! l9e 7:i* is at ,cape ,comor97 is nor? & s\?4 ,9 two ( ! measurem5ts ! brass compon5t 0 to ! we/1 9 ! o!rs to ! ea/2 ! diff]5ces betwe5 ! individual measurem5ts & ! mean ( ! f\r 7 #j4jjag1 -#j4jjdi1 -#j4jjae1 #j4jjde ft4 ,^! diff]5ces are v small2 an elaborate inve/igation ( all s\rces ( error %[s t ! probable error ( a base l9e 9 ,india is on ! av]age -#b4h mu4 ,^! comp5sation b>s 7 also us$ by ,sir ,?omas ,macle> 9 ! measurem5t ( ! base l9e 9 8 ext5sion ( ,lacaille's arc at ! ,cape4 ,! acc\nt ( ?is op]ation w be f\nd 9 a volume entitl$ ,v]ification & ,ext5sion ( ,lacaille's ,arc ( ,m]idian at ! ,cape ( ,good ,hope1 by ,sir ,?omas ,macle>1 publi%$ 9 #ahff4 ,a r$iscussion has be5 giv5 by ,sir ,david ,gill 9 8 ,report on ! ,geodetic ,survey ( ,s\? ,africa1 c41 #ahif4 ,a v simple base app>atus 0 employ$ by ,w4 ,/ruve 9 8 triangulations 9 ,russia f #ahag to #ahee4 ,?is consi/$ ( f\r wr\ght-iron b>s1 ea* two toises 7r m ?an #ac ft47 long2 "o end ( ea* b> is t]m9at$ 9 a small /eel cyl9d] pres5t+ a slightly convex surface = contact1 ! o!r end c>ries a contact lev] rigidly connect$ ) ! b>4 ,! %ort] arm ( ! lev] t]m9ates bel[ 9 a poli%$ hemisph]e1 ! upp] & long] arm trav]s+ a v]tical divid$ arc4 ,9 measur+1 ! plane end ( "o b> is br\ght into contact ) ! %ort arm ( ! contact lev] 7pu%$ =w>d by a weak spr+7 ( ! next b>4 ,ea* b> has two !rmomet]s1 & a level = det]m9+ ! incl9ation ( ! b> 9 measur+4 ,! mann] ( transf]r+ ! end ( a b> to ! gr\nd is simply ?is3 "u ! end ( ! b> a /ake is driv5 v firmly into ! gr\nd1 c>ry+ on its upp] surface a disk1 capable ( movem5t 9 ! direction ( ! measur$ l9e by means ( sl[-motion screws4 ,a f9e m>k on ?is disk is br\ght v]tically "u ! end ( ! b> by means ( a !odolite :i* is plant$ at a di/ance ( #be ft4 f ! /ake 9 a direction p]p5dicul> to ! base4 ,/ruve inve/igat$ = ea* base ! probable errors ( ! measurem5t aris+ f ea* ( ^! sev5 causes3 ,alignm5t1 incl9ation1 comp>isons ) /&>ds1 read+s ( index1 p]sonal errors1 unc]ta9ties ( temp]ature1 & ! probable errors ( adopt$ rates ( expansion4 ,he f\nd t -#j4h mu 0 ! mean ( ! probable errors ( ! sev5 bases measur$ by him4 ,! ,au/ro-,hung>ian app>atus is simil>2 ! di/ance ( ! rods is measur$ by a slid]1 :i* re/s on "o ( ! ends ( ea* rod4 ,tw5ty-two base l9es 7 measur$ 9 #ahdj-ahii4 ,g5]al ,c>los ,ibanez employ$ 9 #aheh-ahgi1 = ! measurem5t ( n9e base l9es 9 ,spa91 two app>atus simil> to ! app>atus previ\sly employ$ by ,porro 9 ,italy2 "o is complicat$1 ! o!r simplifi$4 ,! fir/1 an app>atus ( ! bro!rs ,brunn] ( ,p>is1 0 a !rmometric comb9ation ( two b>s1 "o ( plat9um & "o ( brass1 9 l5g? #d metres1 furni%$ ) ?ree levels & f\r !rmomet]s4 ,suppose ,a1 ,b1 ,c ?ree micromet] microscopes v firmly support$ at int]vals ( #d metres ) _! axes v]tical1 & align$ 9 ! plane ( ! base l9e by means ( a transit in/rum5t1 _! micromet] screws be+ 9 ! l9e ( measurem5t4 ,! measur+ b> is br\ght "u say ,a & ,b1 & ^? micromet]s read2 ! b> is !n %ift$ & br\ght "u ,b & ,c4 ,by repetition ( ?is process1 ! read+ ( a micromet] indicat+ ! end ( ea* position ( ! b>1 ! measurem5t is made4 ,q simil> app>atus 7among o!rs7 has be5 employ$ by ! ,fr5* & ,g]mans4 ,s9ce1 h[ev]1 x only p]mitt$ a di/ance ( ab\t #cjj m4 to be measur$ daily1 ,ibanez introduc$ a simplification2 ! measur+ rod be+ made simply ( /eel1 & provid$ ) inlaid m]cury !rmomet]s4 ,?is app>atus 0 us$ 9 ,switz]l& = ! measurem5t ( ?ree base l9es4 ,! accuracy is %[n by ! e/imat$ probable errors3 -#j4b mu to -#j4h mu4 ,! di/ance measur$ daily am\nts at lea/ to #hjj m4 ,a great] daily di/ance c be measur$ ) ! same accuracy by means ( ,bessel's app>atus2 ?is p]mits ! ready measurem5t ( #bjjj m4 daily4 ,= ?is1 h[ev]1 x is important to notice t a l>ge /aff & fav\rable gr\nd are necess>y4 ,an important improvem5t 0 introduc$ by ,edw>d ,jad]9 ( ,/ockholm1 :o measures ) /ret*$ wires ( ab\t #bd metres long2 ^! wires are ab\t #a4fe mm4 9 diamet]1 & :5 9 use are /ret*$ by an accurate spr+ balance ) a t5sion ( #aj kg4#b ,! nature ( ! gr\nd has a v trifl+ effect on ?is me?od4 ,! difficulty ( temp]ature det]m9ations is remov$ by employ+ wires made ( inv>1 an alloy ( /eel 7#fd7 & nickel 7#cf7 :i* has practically no l9e> expansion = small !rmal *anges at ord9>y temp]atures2 ?is alloy 0 discov]$ 9 #ahif by ,b5oit & ,guillaume ( ! ,int]national ,bureau ( ,weights & ,measures at ,breteuil4 ,app>5tly ! future ( base-l9e measurem5ts re/s ) ! inv> wires ( ! ,jad]9 app>atus2 next comes ,porro's app>atus ) inv> b>s #d to #e metres long4 ,results h be5 obta9$ 9 ! ,unit$ ,/ates1 ( great importance 9 view ( _! accuracy1 rapidity ( det]m9ation & economy4 ,= ! measurem5t ( ! arc ( m]idian 9 longitude #ih deg4 ,e41 9 #aijj1 n9e base l9es ( a total l5g? ( #fi4b km4 7 measur$ 9 six mon?s4 ,! total co/ ( "o base 0 #abca4 ,at ! beg9n+ & at ! end ( ! field-season a di/ance ( exactly #ajj m4 0 measur$ ) ,r4,s4 ,woodw>d's #e-m4 ice-b> 7inv5t$ 9 #ahia72 by means ( ! remeasurem5t ( ?is l5g? ! /&>dization ( ! app>atus 0 done "u ! same conditions z exi/$ 9 ! case ( ! base measurem5ts4 ,= ! measurem5ts !re 7 employ$ two /eel tapes ( #ajj m4 long1 provid$ ) supports at di/ances ( #be m41 two ( #ej m41 & ! duplex app>atus ( ,eimbeck1 consi/+ ( f\r #e-m4 rods4 ,ea* base 0 divid$ into sections ( ab\t #ajjj m42 "o ( ^!1 ! te/ kilometre1 0 measur$ ) all ! five app>atus1 ! o!rs only ) two app>atus1 mo/ly tapes4 ,! probable error 0 ab\t -#j4h mu1 & ! day's "w a di/ance ( ab\t #bjjj m4 ,ea* ( ! f\r rods ( ! duplex app>atus consi/s ( two b>s ( brass & /eel4 ,m]cury !rmomet]s are ins]t$ 9 bo? b>s2 ^! s]ve = ! measurem5t ( ! l5g? ( ! base l9es by ea* ( ! b>s1 z !y are br\ght into _! consecutive positions1 ! contact be+ made by an ela/ic-slid+ contact4 ,! l5g? ( ! base l9es may be calculat$ = ea* b> only1 & also by ! supposition t bo? b>s h ! same temp]ature4 ,! app>atus ?us af=ds ?ree sets ( results1 :i* mutually control !mselves1 & ! contact adju/m5ts p]mit rapid "w4 ,! same device has be5 appli$ to ! old] bimetallic-comp5sat+ app>atus ( ,ba*e-,wurdemann 7six bases1 #ahdg-aheg7 & ( ,s*ott4 ,!re 0 also employ$ a s+le rod bimetallic app>atus on ,f4 ,porro's pr9ciple1 con/ruct$ by ! bro!rs ,repsold = "s base l9es4 ,excell5t results h be5 m rec5tly obta9$ ) inv> tapes4 ,! foll[+ results %[ ! l5g?s ( ! same ,g]man base l9es z measur$ by diff]5t app>atus3 metres4 ,base at ,b]l9 #ahfd ,app>atus ( ,bessel #bccf4cibj #ahhj ,brunn] 4#cibd ,base at ,/rehl5 #ahed ,bessel #bgfb4ehbd #ahgi ,brunn] 4#eheb ,old base at ,bonn #ahdg ,bessel #bacc4ijie #ahib 4#ijig ,new base at ,bonn #ahib #beab4ifab #ahib ,brunn] 4#ifif ,x is necess>y t ! altitude above ! level ( ! sea ( e "p ( a base l9e be asc]ta9$ by _s levell+1 9 ord] t ! measur$ l5g? may be r$uc$ to :at x w\ld h be5 _h ! measurem5t be5 made on ! surface ( ! sea1 produc$ 9 imag9ation4 ,?us if l be ! l5g? ( a measur+ b>1 h its height at any giv5 position 9 ! measurem5t1 r ! radius ( ! e>?1 !n ! l5g? radially project$ on to ! level ( ! sea is l7#a - h_/r74 ,9 ! ,salisbury ,pla9 base l9e ! r$uction to ! level ( ! sea is -#j4fbid ft4 ,! total numb] ( base l9es measur$ 9 ,europe up to ! pres5t "t is ab\t "o hundr$ & t51 n9ete5 ( :i* d n exce$ 9 l5g? #bejj metres1 or ab\t #a-a_/b miles1 & ?ree--"o 9 ,france1 ! o!rs 9 ,bav>ia--exce$ #ai1jjj metres4 ,! "q has be5 frequ5tly discuss$ :e!r or n ! advantage ( a long base is suffici5tly great to w>rant ! exp5diture ( "t t x requires1 or :e!r z mu* precision is n obta9able 9 ! end by c>eful triangulation f a %ort base4 ,b ! answ] _c be giv5 g5]ally2 x mu/ dep5d on ! circum/ances ( ea* p>ticul> case4 ,) ,jad]9's app>atus1 provid$ ) inv> wires1 bases ( #bj to #cj km4 long are obta9$ )\t difficulty4 ,illu/ration3 ,,fig4 #a4 ,9 work+ away f a base l9e ab1 /ations c1 d1 e1 f are c>efully select$ s z to obta9 f well-%ap$ triangles gradually increas+ sides4 ,be=e1 h[ev]1 f9ally leav+ ! base l9e1 x is usual to v]ify x by triangulation ?us3 dur+ ! measurem5t two or m po9ts1 z p1 q 7fig4 #a71 are m>k$ 9 ! base 9 positions su* t ! l5g?s ( ! diff]5t segm5ts ( ! l9e are kn[n2 !n1 tak+ suitable ext]nal /ations1 z h1 k1 ! angles ( ! triangles bhp1 phq1 hqk1 kqa are measur$4 ,f ^! angles c be comput$ ! ratios ( ! segm5ts1 :i* mu/ agree1 if all op]ations are correctly p]=m$1 ) ! ratios result+ f ! measures4 ,leav+ ! base l9e1 ! sides increase up to #aj1 #cj or #ej miles occasionally1 b seldom rea*+ #ajj miles4 ,! triangulation po9ts may ei!r be natural objects pres5t+ !mselves 9 suitable positions1 su* z *ur* t[]s2 or !y may be objects specially con/ruct$ 9 /one or wood on m\nta9 tops or o!r prom95t gr\nd4 ,9 e case x is necess>y t ! precise c5tre ( ! /ation be m>k$ by "s p]man5t m>k4 ,9 ,india no exp5se is sp>$ 9 mak+ p]man5t ! pr9cipal trigonometrical /ations--co/ly t[]s 9 masonry be+ erect$4 ,x is ess5tial t e trigonometrical /ation %all pres5t a f9e object = obs]vation f surr\nd+ /ations4 ,horizontal ,angles4 ,9 plac+ ! !odolite ov] a /ation to be obs]v$ f1 ! fir/ po9t to be att5d$ to is t x %all re/ ^u a p]fectly solid f\ndation4 ,! me?od ( obta9+ ?is desid]atum mu/ dep5d entirely on ! nature ( ! gr\nd2 ! in/rum5t mu/ if possible be support$ on rock1 or if t be impossible a solid f\ndation mu/ be obta9$ by digg+4 ,:5 ! !odolite is requir$ to be rais$ above ! surface ( ! gr\nd 9 ord] to comm& p>ticul> po9ts1 x is necess>y to build two scaffolds1--! out] "o to c>ry ! obs]vatory1 ! inn] "o to c>ry ! in/rum5t1--& ^! two edifices mu/ h no po9t ( contact4 ,_m cases ( high scaffold+ h occurr$ on ! ,engli% ,ordnance ,survey1 z = in/ance at ,?axt$ *ur*1 :]e ! t[]1 #hj ft4 high1 is surm\nt$ by a spire ( #ij ft4 ,! scaffold = ! obs]vatory 0 c>ri$ f ! base to ! top ( ! spire2 t = ! in/rum5t 0 rais$ f a po9t ( ! spire #adj ft4 above ! gr\nd1 hav+ its be>+ ^u timb]s pass+ ?r\gh ! spire at t height4 ,?us ! in/rum5t1 at a height ( #agh ft4 above ! gr\nd1 0 insulat$1 & n affect$ by ! action ( ! w9d on ! obs]vatory4 ,at e /ation x is necess>y to exam9e & correct ! adju/m5ts ( ! !odolite1 :i* are ^!3 ! l9e ( collimation ( ! telescope mu/ be p]p5dicul> to its axis ( rotation2 ?is axis p]p5dicul> to ! v]tical axis ( ! in/rum5t2 & ! latt] p]p5dicul> to ! plane ( ! horizon4 ,! micromet] microscopes mu/ also measure correct quantities on ! divid$ circle or circles4 ,! me?od ( obs]v+ is ?is4 ,let ,a1 ,b1 ,c 444 be ! /ations to be obs]v$ tak5 9 ord] ( azimu?2 ! telescope is fir/ direct$ to ,a & ! cross-hairs ( ! telescope made to bisect ! object pres5t$ by ,a1 !n ! microscopes or v]ni]s ( ! horizontal circle 7also ( ! v]tical circle if necess>y7 are read & record$4 ,! telescope is !n turn$ to ,b1 :i* is obs]v$ 9 ! same mann]2 !n ,c & ! o!r /ations4 ,com+ r\nd by cont9u\s motion to ,a1 x is aga9 obs]v$1 & ! agreem5t ( ?is second read+ ) ! fir/ is "s te/ ( ! /ability ( ! in/rum5t4 ,9 tak+ ?is r\nd ( angles--or arc1 z x is call$ on ! ,ordnance ,survey--x is desirable t ! int]val ( "t betwe5 ! fir/ & second obs]vations ( ,a %\ld be z small z may be consi/5t ) due c>e4 ,be=e tak+ ! next arc ! horizontal circle is mov$ ?r\gh #bj deg4 or #cj deg42 ?us a diff]5t set ( divisions ( ! circle is us$ 9 ea* arc1 :i* t5ds to elim9ate ! errors ( division4 ,x is v desirable t all arcs at a /ation %\ld conta9 "o po9t 9 common1 to :i* all angul> measurem5ts are ?us ref]r$1--! obs]vations on ea* arc comm5c+ & end+ ) ?is po9t1 :i* is on ! ,ordnance ,survey call$ ! ref]r+ object4 ,x is usual = ?is purpose to select1 f among ! po9ts :i* h to be obs]v$1 t "o :i* af=ds ! be/ object = precise obs]vation4 ,= m\nta9 tops a ref]r+ object is con/ruct$ ( two rectangul> plates ( metal 9 ! same v]tical plane1 _! edges p>allel & plac$ at su* a di/ance ap>t t ! light ( ! sky se5 ?r\gh appe>s z a v]tical l9e ab\t #aj 9 wid?4 ,! be/ di/ance = ?is object is f #a to #b miles4 ,?is me?od seems at fir/ sight v advantage\s2 b if1 h[ev]1 x be desir$ to atta9 ! highe/ accuracy1 x is bett]1 z %[n by ,g5]al ,s*reib] ( ,b]l9 9 #ahgh1 to measure only s+le angles1 & z _m ( ^! z possible betwe5 ! directions to be det]m9$4 ,division-errors are ?us m p]fectly elim9at$1 & errors due to ! v>iation 9 ! /ability1 c41 ( ! in/rum5ts are dim9i%$4 ,?is me?od is rapidly ga9+ prec$5ce4 ,! !odolites us$ 9 geodesy v>y 9 patt]n & 9 size--! horizontal circles rang+ f #aj 94 to #cf 94 9 diamet]4 ,9 ,ramsd5's #cf-;94 !odolite ! telescope has a focal l5g? ( #cf 94 & an ap]ture ( #b4e 941 ! ord9>ily us$ magnify+ p[] be+ #ed2 ?is la/1 h[ev]1 c ( c\rse be *ang$ at ! requirem5ts ( ! obs]v] or ( ! wea!r4 ,! probable error ( a s+le obs]vation ( a f9e object ) ?is !odolite is ab\t #j4b4 ,fig4 #b repres5ts an altazimu? !odolite ( an improv$ patt]n us$ on ! ,ordnance ,survey4 ,! horizontal circle ( #ad-;94 diamet] is read by ?ree micromet] microscopes2 ! v]tical circle has a diamet] ( #ab 941 & is read by two microscopes4 ,9 ! great trigonometrical survey ( ,india ! !odolites us$ 9 ! m important p>ts ( ! "w h be5 ( #b & #c ft4 diamet]--! circle read by five equidi/ant microscopes4 ,e angle is measur$ twice 9 ea* position ( ! z]o ( ! horizontal circle1 ( :i* !re are g5]ally t52 ! entire numb] ( measures ( an angle is nev] less ?an #bj4 ,an exam9ation ( #adjg angles %[$ t ! probable error ( an obs]v$ angle is on ! av]age - #j4bh4 ,= ! obs]vations ( v di/ant /ations x is usual to employ a heliotrope 7f ! ,gr4 ,greek3 helios1 sun2 ,greek3 tropos1 a turn71 inv5t$ by ,gauss at ,gott+5 9 #ahba4 ,9 its simple/ =m ?is is a plane mirror1 #d1 #f1 or #h 94 9 diamet]1 capable ( rotation r\nd a horizontal & a v]tical axis4 ,?is mirror is plac$ at ! /ation to be obs]v$1 & 9 f9e wea!r x is kept s direct$ t ! rays ( ! sun reflect$ by x /rike ! di/ant obs]v+ telescope4 ,to ! obs]v] ! heliotrope pres5ts ! appe>ance ( a /> ( ! fir/ or second magnitude1 & is g5]ally a pleasant object = obs]v+4 ,obs]vations at night1 ) ! aid ( light-signals1 h be5 repeat$ly made1 & ) good results1 p>ticul>ly 9 ,france by ,g5]al ,francois ,p]ri]1 & m rec5tly 9 ! ,unit$ ,/ates by ! ,coa/ & ,geodetic ,survey2 ! signal employ$ be+ an acetyl5e bicycle-lamp1 ) a l5s #e 94 9 diamet]4 ,p>ticul>ly notewor?y are ! trigonometrical connexions ( ,spa9 & ,alg]ia1 :i* 7 c>ri$ out 9 #ahgi by ,g5]als ,ibanez & ,p]ri] 7ov] a di/ance ( #bgj km471 ( ,sicily & ,malta 9 #aijj1 & ( ! isl&s ( ,elba & ,s>d9ia 9 #aijb by ,dr ,gu>ducci 7ov] di/ances up to #bcj km472 9 ^! cases artificial light 0 employ$3 9 ! fir/ case electric light & 9 ! two o!rs acetyl5e lamps4 ,illu/ration3 ,,fig4 #b4--;,altazimu? ,!odolite4 ,a/ronomical ,obs]vations4 ,! direction ( ! m]idian is det]m9$ ei!r by a !odolite or a portable transit in/rum5t4 ,9 ! =m] case ! op]ation consi/s 9 obs]v+ ! angle betwe5 a t]re/rial object--g5]ally a m>k specially erect$ & capable ( illum9ation at night--& a close circumpol> /> at its greate/ ea/]n or we/]n azimu?1 or1 at any rate1 :5 v ne> t position4 ,if ! obs]vation be made t m9utes ( "t be=e or aft] ! "t ( greate/ azimu?1 ! azimu? !n w diff] f its maximum value by 7#dejt7#b s9 #a s9 #b;delta_/ s9 z1 9 seconds ( angle1 omitt+ small] t]ms1 delta be+ ! />'s decl9ation & z its z5i? di/ance4 ,! collimation & level errors are v c>efully det]m9$ be=e & aft] ^! obs]vations1 & x is usual to arrange ! obs]vations by ! rev]sal ( ! telescope s t collimation error %all disappe>4 ,if b1 c be ! level & collimation errors1 ! correction to ! circle read+ is b cot z - c cosec z1 b be+ positive :5 ! we/ end ( ! axis is high4 ,x is cle> t any unc]ta9ty z to ! real /ate ( ! level w produce a correspond+ unc]ta9ty 9 ! result+ value ( ! azimu?1--an unc]ta9ty :i* increases ) ! latitude & is v l>ge 9 high latitudes4 ,?is may be p>tly rem$i$ by obs]v+ 9 connexion ) ! /> its reflection 9 m]cury4 ,9 det]m9+ ! value ( "o division ( a level tube1 x is necess>y to be> 9 m9d t 9 "s ! value v>ies consid]ably ) ! temp]ature4 ,by exp]im5ts on ! level ( ,ramsd5's #c-;foot !odolite1 x 0 f\nd t ?\gh at ! ord9>y temp]ature ( #ff deg4 ! value ( a division 0 ab\t "o second1 yet at #cb deg4 x 0 ab\t five seconds4 ,9 a v excell5t portable transit us$ on ! ,ordnance ,survey1 ! uprights c>ry+ ! telescope are con/ruct$ ( mahogany1 ea* upright be+ built ( sev]al pieces glu$ & screw$ toge!r2 ! base1 :i* is a solid & heavy plate ( iron1 c>ries a rev]s+ app>atus = lift+ ! telescope out ( its be>+s1 rev]s+ x & lett+ x d[n aga94 ,?us is avoid$ ! *ange ( temp]ature :i* ! telescope w\ld incur by be+ lift$ by ! h&s ( ! obs]v]4 ,ano!r =m ( transit is ! ,g]man diagonal =m1 9 :i* ! rays ( light aft] pass+ ?r\gh ! object-glass are turn$ by a total reflection prism ?r\gh "o ( ! transv]se arms ( ! telescope1 at ! extremity ( :i* arm is ! eye-piece4 ,! unus$ half ( ! ord9>y telescope be+ cut away is replac$ by a c\nt]poise4 ,9 ?is in/rum5t !re is ! advantage t ! obs]v] )\t mov+ ! position ( 8 eye comm&s ! :ole m]idian1 & t ! level may rema9 on ! pivots :atev] be ! elevation ( ! telescope4 ,b !re is ! disadvantage t ! flexure ( ! transv]se axis causes a v>iable collimation error dep5d+ on ! z5i? di/ance ( ! /> to :i* x is direct$2 & moreov] x has be5 f\nd t 9 "s cases ! p]sonal error ( an obs]v] is n ! same 9 ! two positions ( ! telescope4 ,to det]m9e ! direction ( ! m]idian1 x is well to erect two m>ks at ne>ly equal angul> di/ances on ei!r side ( ! nor? m]idian l9e1 s t ! pole /> crosses ! v]tical ( ea* m>k a %ort "t be=e & aft] atta9+ its greate/ ea/]n & we/]n azimu?s4 ,if n[ ! in/rum5t1 p]fectly levell$1 is adju/$ to h its c5tre wire on "o ( ! m>ks1 !n :5 elevat$ to ! />1 ! /> w trav]se ! wire1 & its exact position 9 ! field at any mom5t c be measur$ by ! micromet] wire4 ,alt]nate obs]vations ( ! /> & ! t]re/rial m>k1 comb9$ ) c>eful level read+s & rev]sals ( ! in/rum5t1 w enable "o1 ev5 ) only "o m>k1 to det]m9e ! direction ( ! m]idian 9 ! c\rse ( an h\r ) a probable error ( less ?an a second4 ,! second m>k enables "o to complete ! /ation m rapidly & gives a *eck ^u ! "w4 ,z an in/ance1 at ,f9dlay ,seat1 9 latitude #eg deg4 #ce'1 ! result+ azimu?s ( ! two m>ks 7 #agg deg4 #de' #cg4bi - #j4bj & #ahb deg4 #ag' #ae4fa - #j4ac1 :ile ! angle betwe5 ! two m>ks directly measur$ by a !odolite 0 f\nd to be #d deg4 #ca' #cg4dc - #j4bc4 ,illu/ration3 ,,fig4 #c4 ,we n[ come to ! consid]ation ( ! det]m9ation ( "t ) ! transit in/rum5t4 ,let fig4 #c repres5t ! sph]e /]eographically project$ on ! plane ( ! horizon1--ns be+ ! m]idian1 we ! prime v]tical1 ,z1 ,p ! z5i? & ! pole4 ,let p be ! po9t 9 :i* ! production ( ! axis ( ! in/rum5t meets ! cele/ial sph]e1 ,s ! position ( a /> :5 obs]v$ on a wire ^: di/ance f ! collimation c5tre is c4 ,let a be ! azimu?al deviation1 namely1 ! angle w,zp1 b ! level error s t ,zp #ij deg4 - b4 ,let also ! h\r angle correspond+ to p be #ij deg4 - n1 & ! decl9ation ( ! same m1 ! />'s decl9ation be+ delta1 & ! latitude phi4 ,!n to f9d ! h\r angle ,,zps tau ( ! /> :5 obs]v$1 9 ! triangles p,p,s1 p,p,z we h1 s9ce p,p,s #ij tau - n1 -,s9 c s9 m s9 delta cos m cos delta s9 7n - tau71 ,s9 m s9 b s9 phi - cos b cos phi s9 a1 ,cos m s9 n s9 b cos phi cos b s9 phi s9 a4 ,& ^! equations solve ! problem1 h[ev] l>ge be ! errors ( ! in/rum5t4 ,suppos+1 z usual1 a1 b1 m1 n to be small1 we h at once tau n c sec delta m tan delta1 :i* is ! correction to ! obs]v$ "t ( transit4 ,or1 elim9at+ m & n by means ( ! second & ?ird equations1 & putt+ z = ! z5i? di/ance ( ! />1 t = ! obs]v$ "t ( transit1 ! correct$ "t is t 7a s9 z b cos z c7 _/ cos delta4 ,ano!r v conv5i5t =m = />s ne> ! z5i? is tau b sec phi c sec delta m 7tan delta - tan phi74 ,suppose t 9 comm5c+ to obs]ve at a /ation ! error ( ! *ronomet] is n kn[n2 !n hav+ secur$ = ! in/rum5t a v solid f\ndation1 remov$ z f> z possible level & collimation errors1 & plac$ x by e/imation ne>ly 9 ! m]idian1 let two />s diff]+ consid]ably 9 decl9ation be obs]v$--! in/rum5t n be+ rev]s$ betwe5 !m4 ,f ^! two />s1 nei!r ( :i* %\ld be a close circumpol> />1 a good approximation to ! *ronomet] error c be obta9$2 ?us let epsilon#a1 epsilon#b1 be ! app>5t clock errors giv5 by ^! />s if delta#a1 delta#b be _! decl9ations ! real error is epsilon epsilon#a 7epsilon#a - epsilon#b7 7tan phi - tan delta#a7 _/ 7tan delta#a - tan delta#b74 ,( c\rse ?is is /ill only approximate1 b x w enable ! obs]v] 7:o by ! help ( a table ( natural tang5ts c compute epsilon 9 a few m9utes7 to f9d ! m]idian by plac+ at ! prop] "t1 :i* he n[ kn[s approximately1 ! c5tre wire ( 8 in/rum5t on ! fir/ /> t passes--n ne> ! z5i?4 ,! transit in/rum5t is always rev]s$ at lea/ once 9 ! c\rse ( an ev5+'s obs]v+1 ! level be+ frequ5tly read & record$4 ,x is necess>y 9 mo/ in/rum5ts to add a correction = ! diff]5ce 9 size ( ! pivots4 ,! transit in/rum5t is also us$ 9 ! prime v]tical = ! det]m9ation ( latitudes4 ,9 ! prec$+ figure let q be ! po9t 9 :i* ! nor!rn extremity ( ! axis ( ! in/rum5t produc$ meets ! cele/ial sph]e4 ,let n,zq be ! azimu?al deviation a1 & b be+ ! level error1 ,zq #ij deg4 - b2 let also n,pq tau & ,pq psi4 ,let ,s' be ! position ( a /> :5 obs]v$ on a wire ^: di/ance f ! collimation c5tre is c1 positive :5 to ! s\?1 & let h be ! obs]v$ h\r angle ( ! />1 viz4 ,,zps'4 ,!n ! triangles q,p,s'1 g,p,z give -,s9 c s9 delta cos psi - cos delta s9 psi cos 7h tau71 ,cos psi s9 b s9 phi cos b cos phi cos a1 ,s9 psi s9 tau cos b s9 a4 ,n[ :5 a & b are v small1 we see f ! la/ two equations t psi phi - b1 a tau s9 psi1 & if we calculate phi' by ! =mula cot phi' cot delta cos h1 ! fir/ equation leads u to ?is result-- phi phi' 7a s9 z b cos z c7_/cos z1 ! correction = in/rum5tal error be+ v simil> to t appli$ to ! obs]v$ "t ( transit 9 ! case ( m]idian obs]vations4 ,:5 a is n v small & z is small1 ! =mulae requir$ are m complicat$4 ,illu/ration3 ,,fig4 #d4--,z5i? ,telescope con/ruct$ = ! ,int]national ,/ations at ,mizusawa1 ,c>l(orte1 ,gai!rsburg & ,ukiah1 by ,h]mann ,wans*aff1 ,b]l94 ,! me?od ( det]m9+ latitude by transits 9 ! prime v]tical has ! disadvantage ( be+ a some:at sl[ process1 & ( requir+ a v precise k ( ! "t1 a disadvantage f :i* ! z5i? telescope is free4 ,9 pr9ciple ?is in/rum5t is bas$ on ! proposition t :5 ! m]idian z5i? di/ances ( two />s at _! upp] culm9ations--"o be+ to ! nor? & ! o!r to ! s\? ( ! z5i?--are equal1 ! latitude is ! mean ( _! decl9ations2 or1 if ! z5i? di/ance ( a /> culm9at+ to ! s\? ( ! z5i? be ,z1 its decl9ation be+ delta1 & t ( ano!r culm9at+ to ! nor? ) z5i? di/ance ,z' & decl9ation delta'1 !n cle>ly ! latitude is #a_/b7;delta delta'7 #a_/b7,z - ,z'74 ,n[ ! z5i? telescope does away ) ! divid$ circle1 & sub/itutes ! measurem5t micrometrically ( ! quantity ,z' - ,z4 ,9 fig4 #d is %[n a z5i? telescope by ,h4 ,wans*aff ( ,b]l91 :i* is ! type us$ 7accord+ to ! ,c5tral ,bureau at ,potsdam7 s9ce ab\t #ahij = ! det]m9ation ( ! v>iations ( latitude due to diff]5t1 b z yet imp]fectly und]/ood1 influ5ces4 ,! in/rum5t is support$ on a /rong tripod1 fitt$ ) levell+ screws2 to ?is tripod is fix$ ! azimu? circle & a long v]tical /eel axis4 ,fitt+ on ?is axis is a holl[ axis :i* c>ries on its upp] end a %ort transv]se horizontal axis ) a level4 ,?is latt] c>ries ! telescope1 :i*1 support$ at ! c5tre ( its l5g?1 is free to rotate 9 a v]tical plane4 ,! telescope is ?us m\nt$ ecc5trically ) respect to ! v]tical axis ar\nd :i* x revolves4 ,two extremely s5sitive levels are atta*$ to ! telescope1 :i* latt] c>ries a micromet] 9 its eye-piece1 ) a screw ( long range = measur+ diff]5ces ( z5i? di/ance4 ,two levels are employ$ = controll+ & increas+ ! accuracy4 ,= ?is in/rum5t />s are select$ 9 pairs1 pass+ nor? & s\? ( ! z5i?1 culm9at+ )9 a few m9utes ( "t & )9 ab\t tw5ty m9utes 7angul>7 ( z5i? di/ance ( ea* o!r4 ,:5 a pair ( />s is to be obs]v$1 ! telescope is set to ! mean ( ! z5i? di/ances & 9 ! plane ( ! m]idian4 ,! fir/ /> on pass+ ! c5tral m]idional wire is bisect$ by ! micromet]2 !n ! telescope is rotat$ v c>efully ?r\gh #ahj deg4 r\nd ! v]tical axis1 & ! second /> on pass+ ?r\gh ! field is bisect$ by ! micromet] on ! c5tre wire4 ,! micromet] has ?us measur$ ! diff]5ce ( ! z5i? di/ances1 & ! calculation to get ! latitude is mo/ simple4 ,( c\rse x is necess>y to read ! level1 & ! obs]vations are n necess>ily conf9$ to ! c5tre wire4 ,9 fact if n1 s be ! nor? & s\? read+s ( ! level = ! s\? />1 n'1 s' ! same = ! nor? />1 l ! value ( "o division ( ! level1 m ! value ( "o division ( ! micromet]1 r1 r' ! refraction corrections1 mu1 mu' ! micromet] read+s ( ! s\? & nor? />1 ! micromet] be+ suppos$ to read f ! z5i?1 !n1 suppos+ ! obs]vation made on ! c5tre wire1-- phi #a_/b7;delta delta'7 #a_/b7mu - mu'7m #a_/d7n n' - s - s'7l #a_/b7r - r'74 ,x is ( c\rse ( ! highe/ importance t ! value m ( ! screw be well det]m9$4 ,?is is done mo/ effectually by obs]v+ ! v]tical movem5t ( a close circumpol> /> :5 at its greate/ azimu?4 ,9 a s+le night ) ?is in/rum5t a v accurate result1 say ) a probable error ( ab\t #j4b1 c\ld be obta9$ = latitude f1 say1 tw5ty pair ( />s2 b :5 ! latitude is requir$ to be obta9$ ) ! highe/ possible precision1 two nights at lea/ are necess>y4 ,! weak po9t ( ! z5i? telescope lies 9 ! circum/ance t its requirem5ts prev5t ! selection ( />s ^: positions are well fix$2 v frequ5tly x is necess>y to h ! decl9ations ( ! />s select$ = ?is in/rum5t specially obs]v$ at fix$ obs]vatories4 ,! z5i? telescope is made 9 v>i\s sizes f #cj to #ed 94 9 focal l5g?2 a #cj-;94 telescope is suffici5t = ! highe/ purposes & is v portable4 ,! net obs]vation probable-error = "o pair ( />s is only -#j4a4 ,! z5i? telescope is a p>ticul>ly pleasant in/rum5t to "w )1 & an obs]v] has be5 kn[n 7a s]geant ( ,royal ,eng9e]s1 on "o occasion7 to take e /> 9 8 li/ dur+ elev5 h\rs on a /ret*1 namely1 f #f o'clock ,p4,m4 until #e ,a4,m41 & ?is on a v cold ,novemb] night on "o ( ! highe/ po9ts ( ! ,grampians4 ,obs]v]s accu/om$ to geodetic op]ations atta9 consid]able p[]s ( endurance4 ,%ortly aft] ! comm5cem5t ( ! obs]vations on "o ( ! hills 9 ! ,isle ( ,skye a /orm c>ri$ away ! wood5 h\ses ( ! m5 & left ! obs]vatory ro(less4 ,?ree obs]vatory ro(s 7 subsequ5tly demoli%$1 & = "s "t ! obs]vatory 0 us$ )\t a ro(1 be+ fill$ ) sn[ e night & empti$ e morn+4 ,q diff]5t1 h[ev]1 0 ! exp]i5ce ( ! same p>ty :5 on ! top ( ,b5 ,nevis1 #ddjf ft4 high4 ,= ab\t a =tnight ! /ate ( ! atmosph]e 0 unusually calm1 s mu* s1 t a light$ c&le c\ld (t5 be c>ri$ betwe5 ! t5ts ( ! m5 & ! obs]vatory1 :il/ at ! foot ( ! hill ! wea!r 0 wild & /ormy4 ,! det]m9ation ( ! diff]5ce ( longitude betwe5 two /ations ,a & ,b resolves itself into ! det]m9ation ( ! local "t at ea* ( ! /ations1 & ! comp>ison by signals ( ! clocks at ,a & ,b4 ,:5ev] telegraphic l9es are available ^! comp>isons are made by telegraphy4 ,a small & delicately-made app>atus introduc$ into ! me*anism ( an a/ronomical clock or *ronomet] breaks or closes by ! action ( ! clock an electric circuit e second4 ,9 ord] to record ! m9utes z well z seconds1 "o second 9 ea* m9ute1 namely t numb]$ #j or #fj1 is omitt$4 ,! seconds are record$ on a *ronograph1 :i* consi/s ( a cyl9d] revolv+ uni=mly at ! rate ( "o revolution p] m9ute cov]$ ) :ite pap]1 on :i* a p5 hav+ a sl[ movem5t 9 ! direction ( ! axis ( ! cyl9d] describes a cont9u\s spiral4 ,?is p5 is deflect$ ?r\gh ! ag5cy ( an electromagnet e second1 & ?us ! seconds ( ! clock are record$ on ! *ronograph by (fsets f ! spiral curve4 ,an obs]v] hav+ 8 h& on a contact key 9 ! same circuit c record 9 ! same mann] 8 obs]v$ times ( transits ( />s4 ,! me?od ( det]m9ation ( diff]5ce ( longitude is1 !re=e1 virtually z foll[s4 ,aft] ! necess>y obs]vations = in/rum5tal corrections1 :i* are record$ only at ! /ation ( obs]vation1 ! clock at ,a is put 9 connexion ) ! circuit s z to write on bo? *ronographs1 namely1 t at ,a & t at ,b4 ,!n ! clock at ,b is made to write on bo? *ronographs4 ,x is cle> t by ?is d\ble op]ation "o c elim9ate ! effect ( ! small int]val ( "t consum$ 9 ! transmission ( signals1 = ! diff]5ce ( longitude obta9$ f ! "o *ronograph w be 9 excess by z mu* z t obta9$ f ! o!r w be 9 defect4 ,! det]m9ation ( ! p]sonal errors ( ! obs]v]s 9 ?is delicate op]ation is a matt] ( ! greate/ importance1 z !re9 lies probably ! *ief s\rce ( residual error4 ,^! errors c nev]!less be almo/ entirely avoid$ by us+ ! imp]sonal micromet] ( ,dr ,repsold 7,hamburg1 #ahhi74 ,9 ?is device !re is a movable micromet] wire :i* is br\ght by h& into co9cid5ce ) ! /> & mov$ along ) x2 at fix$ po9ts !re are electrical contacts1 :i* replace ! fix$ wires4 ,exp]im5ts at ! ,geodetic ,in/itute & ,c5tral ,bureau at ,potsdam 9 #ahia gave ! foll[+ p]sonal equations 9 ! case ( f\r obs]v]s3-- ,old] ,proc$ure4 ,new ,proc$ure4 ,a-,b -#js4#ajh -#js4#jjd ,a-,g -#js4#cad -#js4#jce ,a-,s -#js4#ahd -#js4#jbg ,b-,g -#js4#bbe #js4#jac ,b-,s -#js4#jhf -#js4#jbc ,g-,s #js4#aji -#js4#jjf ,^! results %[ t 9 ! lat] me?od ! p]sonal equation is small & n s v>iable2 & consequ5tly ! repetition ( longitude det]m9ations ) ex*ang$ obs]v]s & app>atus entirely elim9ates ! con/ant errors1 ! probable error ( su* det]m9ations on t5 nights be+ sc>cely -#js4#ja4 ,calculation ( ,triangulation4 ,! surface ( ,great ,brita9 & ,irel& is uni=mly cov]$ by triangulation1 ( :i* ! sides are ( v>i\s l5g?s f #aj to #aaa miles4 ,! l>ge/ triangle has "o angle at ,sn[don 9 ,wales1 ano!r on ,slieve ,don>d 9 ,irel&1 & a ?ird at ,scaw ,fell 9 ,cumb]l&2 ea* side is ov] a hundr$ miles & ! sph]ical excess is #fd4 ,! m ord9>y me?od ( triangulation is1 h[ev]1 t ( *a9s ( triangles1 9 ! direction ( ! m]idian & p]p5dicul> !reto4 ,! pr9cipal triangulations ( ,france1 ,spa91 ,au/ria & ,india are s arrang$4 ,oblique *a9s ( triangles are =m$ 9 ,italy1 ,sw$5 & ,norway1 also 9 ,g]many & ,russia1 & 9 ! ,unit$ ,/ates4 ,*a9s are compos$ sometimes m]ely ( consecutive pla9 triangles2 sometimes1 & m frequ5tly 9 ,india1 ( comb9ations ( triangles =m+ consecutive polygonal figures4 ,9 ?is me?od ( triangulat+1 ! sides ( ! triangles are g5]ally f #bj to #cj miles 9 l5g?--seldom exce$+ #dj4 ,! inevitable errors ( obs]vation1 :i* are insep>able f all angul> z well z o!r measurem5ts1 introduce a great difficulty into ! calculation ( ! sides ( a triangulation4 ,/>t+ f a giv5 base 9 ord] to get a requir$ di/ance1 x may g5]ally be obta9$ 9 sev]al diff]5t ways--t is1 by us+ diff]5t sets ( triangles4 ,! results w c]ta9ly diff] "o f ano!r1 & probably no two w agree4 ,! exp]i5ce ( ! comput] w !n come to 8 aid1 & enable him to say :i* is ! mo/ tru/wor?y result2 b no exp]i5ce or ability w c>ry him ?r\gh a l>ge network ( triangles ) any?+ l assurance4 ,! only way to obta9 tru/wor?y results is to employ ! me?od ( lea/ squ>es4 ,we _c "h give any illu/ration ( ?is me?od z appli$ to g5]al triangulation1 = x is mo/ labori\s1 ev5 = ! simple/ cases4 ,?ree /ations1 project$ on ! surface ( ! sea1 give a sph]ical or sph]oidal triangle accord+ to ! adoption ( ! sph]e or ! ellipsoid z ! =m ( ! surface4 ,a sph]oidal triangle diff]s f a sph]ical triangle1 n only 9 t ! curvatures ( ! sides are diff]5t "o f ano!r1 b m especially 9 ?is t1 :ile 9 ! sph]ical triangle ! normals to ! surface at ! angul> po9ts meet at ! c5tre ( ! sph]e1 9 ! sph]oidal triangle ! normals at ! angles ,a1 ,b1 ,c meet ! axis ( revolution ( ! sph]oid 9 ?ree diff]5t po9ts1 :i* we may designate alpha1 beta1 gamma respectively4 ,n[ ! angle ,a ( ! triangle z measur$ by a !odolite is ! incl9ation ( ! planes ,,baalpha & ,,caalpha1 & ! angle at ,b is t conta9$ by ! planes ,,abbeta & ,,cbbeta4 ,b ! planes ,,abalpha & ,,abbeta conta9+ ! l9e ,,ab 9 common cut ! surface 9 two di/9ct plane curves4 ,9 ord]1 !re=e1 t a sph]oidal triangle may be exactly def9$1 x is necess>y t ! nature ( ! l9es jo9+ ! ?ree v]tices be /at$4 ,9 a ma!matical po9t ( view ! mo/ natural def9ition is t ! sides be geodetic or %orte/ l9es4 ,c4,c4,g4 ,&rae1 ( ,cop5hag51 has also %[n t o!r l9es give a less conv5i5t computation4 ,k4,f4 ,gauss1 9 8 treatise1 ,disquisitiones g5]ales circa sup]ficies curvas1 ent]$ fully into ! subject ( geodetic 7or geodesic7 triangles1 & inve/igat$ expressions = ! angles ( a geodetic triangle ^: sides are giv51 n c]ta9ly f9ite expressions1 b approximations inclusive ( small quantities ( ! f\r? ord]1 ! side ( ! triangle or its ratio to ! radius ( ! ne>ly sph]ical surface be+ a small quantity ( ! fir/ ord]4 ,! t]ms ( ! f\r? ord]1 z giv5 by ,gauss = any surface 9 g5]al1 are v complicat$ ev5 :5 ! surface is a sph]oid4 ,if we reta9 small quantities ( ! second ord] only1 & put ,a1 ,b1 ,c = ! angles ( ! geodetic triangle1 :ile ,a1 ,b1 ,c are ^? ( a plane triangle hav+ sides equal respectively to ^? ( ! geodetic triangle1 !n1 sigma be+ ! area ( ! plane triangle & a1 b1 c ! measures ( curvature at ! angul> po9ts1 ,a ,a sigma7#b;a b c7_/#ab1 ,b ,b sigma7a #b;b c7_/#ab1 ,c ,c sigma7a b #b;c7_/#ab4 ,= ! sph]e a b r1 & mak+ ?is simplification1 we obta9 ! !orem previ\sly giv5 by ,a4,m4 ,leg5dre4 ,) ! t]ms ( ! f\r? ord]1 we h 7aft] ,&rae73 epsilon sigma _/m#b - a#b a - k ,a - ,a --------- -------k 7 ---------k ------- 71 #c #c #bj #dk _/ epsilon sigma _/m#b - b#b b - k ,b - ,b --------- -------k 7 ---------k -------- 71 #c #c #bj #dk _/ epsilon sigma _/m#b - c#b c - k ,c - ,c --------- -------k 7 ---------k -------- 71 #c #c #bj #dk _/ 9 :i* epsilon sigma k #a 7m#bk _/ #h71 #cm#b a#b b#b c#b1 #ck a b c4 ,= ! ellipsoid ( rotation ! measure ( curvature is equal to #a _/ rhon1 rho & n be+ ! radii ( curvature ( ! m]idian & p]p5dicul>4 ,x is r>ely t ! t]ms ( ! f\r? ord] are requir$4 ,z a rule sph]oidal triangles are calculat$ z sph]ical 7aft] ,leg5dre71 i4e4 l plane triangles ) a decrease ( ea* angle ( ab\t epsilon _/ #c2 epsilon mu/1 h[ev]1 be calculat$ = ea* triangle sep>ately ) its mean measure ( curvature k4 ,! geodetic l9e be+ ! %orte/ t c be drawn on any surface betwe5 two giv5 po9ts1 we may be conduct$ to its mo/ important *>act]i/ics by ! foll[+ consid]ations3 let p1 q be adjac5t po9ts on a curv$ surface2 ?r\gh s ! middle po9t ( ! *ord pq imag9e a plane drawn p]p5dicul> to pq1 & let ,s be any po9t 9 ! int]section ( ?is plane ) ! surface2 !n p,s ,sq is evid5tly lea/ :5 s,s is a m9imum1 :i* is :5 s,s is a normal to ! surface2 h5ce x foll[s t ( all plane curves on ! surface jo9+ p1 q1 :5 ^? po9ts are indef9itely ne> to "o ano!r1 t is ! %orte/ :i* is made by ! normal plane4 ,t is to say1 ! osculat+ plane at any po9t ( a geodetic l9e conta9s ! normal to ! surface at t po9t4 ,imag9e n[ ?ree po9ts 9 space1 ,a1 ,b1 ,c1 su* t ,,ab ,,bc c2 let ! direction cos9es ( ,,ab be l1 m1 n1 ^? ( ,,bc l'1 m'1 n'1 !n x1 y1 z be+ ! co-ord9ates ( ,b1 ^? ( ,a & ,c w be respectively-- x - cl 3 y - cm 3 z - cn x cl'3 y cm'3 z cn'4 ,h5ce ! co-ord9ates ( ! middle po9t ,m ( ,,ac are x #a_/b;c7l' - l71 y #a_/b;c7m' - m71 z #a_/b;c7n' - n71 & ! direction cos9es ( ,,bm are !re=e proportional to l' - l 3 m' - m 3 n' - n4 ,if ! angle made by ,,bc ) ,,ab be indef9itely small1 ! direction cos9es ( ,,bm are z deltal 3 deltam 3 deltan4 ,n[ if ,,ab1 ,,bc be two contigu\s elem5ts ( a geodetic1 !n ,,bm mu/ be a normal to ! surface1 & s9ce deltal1 deltam1 deltan are 9 ?is case repres5t$ by delta7dx_/ds71 delta7dy_/ds71 delta7dz_/ds71 & if ! equation ( ! surface be u #j1 we h d#bx _/ du d#by _/ du d#bz _/ du ---- _/ -- ---- _/ -- ---- _/ --1 ds#b _/ dx ds#b _/ dy ds#b _/ dz :i*1 h[ev]1 are equival5t to only "o equation4 ,9 ! case ( ! sph]oid ?is equation becomes d#bx d#by y ---- - x ---- #j1 ds#b ds#b :i* integrat$ gives ydx - xdy ,cds4 ,?is aga9 may be put 9 ! =m r s9 a ,c1 :]e a is ! azimu? ( ! geodetic at any po9t--! angle betwe5 its direction & t ( ! m]idian--& r ! di/ance ( ! po9t f ! axis ( revolution4 ,f ?is x may be %[n t ! azimu? at ,a ( ! geodetic jo9+ ,,ab is n ! same z ! a/ronomical azimu? at ,a ( ,b or t det]m9$ by ! v]tical plane ,aalpha,b4 ,g5]ally speak+1 ! geodetic lies betwe5 ! two plane section curves jo9+ ,a & ,b :i* are =m$ by ! two v]tical planes1 suppos+ ^! po9ts n f> ap>t4 ,if1 h[ev]1 ,a & ,b are ne>ly 9 ! same latitude1 ! geodetic may cross 7betwe5 ,a & ,b7 t plane curve :i* lies ne>e/ ! adjac5t pole ( ! sph]oid4 ,! condition ( cross+ is ?is4 ,suppose t = a mom5t we drop ! consid]ation ( ! e>?'s non-sph]icity1 & draw a p]p5dicul> f ! pole ,c on ,,ab1 meet+ x 9 ,s betwe5 ,a & ,b4 ,!n ,a be+ t po9t :i* is ne>e/ ! pole1 ! geodetic w cross ! plane curve if ,,z be betwe5 #a_/d;,,ab & #c_/h ,,ab4 ,if ,,z lie betwe5 ?is la/ value & #a_/b;,,ab1 ! geodetic w lie :olly to ! nor? ( bo? plane curves1 t is1 suppos+ bo? po9ts to be 9 ! nor!rn hemisph]e4 ,! diff]5ce ( ! azimu?s ( ! v]tical section ,,ab & ( ! geodetic ,,ab1 i4e4 ! a/ronomical & geodetic azimu?s1 is v small = all obs]vable di/ances1 be+ approximately3-- ,geod4 azimu? ,a/r4 azimu? -7#a_/ab7 e#b_/7a - e#b7 7s#b_/rhon7 7cos#bphi s9 #b;alpha 7s_/#d;a7s9 #bphi s9 alpha71 9 :i*3 e & a are ! num]ical ecc5tricity & semi-major axis respectively ( ! m]idian ellipse1 phi & alpha are ! latitude & azimu? at ,a1 s ,,ab1 & rho & n are ! radii ( curvature ( ! m]idian & p]p5dicul> at ,a4 ,= s #ajj kilometres1 only ! fir/ t]m is ( mom5t2 its value is #j4jbh cos#b phi s9 #b;alpha1 & x lies well )9 ! errors ( obs]vation4 ,if we imag9e ! geodetic ,,ab1 x w g5]ally trisect ! angles betwe5 ! v]tical sections at ,a & ,b1 s t ! geodetic at ,a is ne> ! v]tical section ,,ab1 & at ,b ne> ! section ,,ba4#c ,! greate/ di/ance ( ! v]tical sections "o f ano!r is e#bs#c cos#b phi#j s9 #b;alpha#j_/af;a#b1 9 :i* phi#j & alpha#j are ! mean latitude & azimu? respectively ( ! middle po9t ( ,,ab4 ,= ! value s #fd kilometres1 ! maximum di/ance is #c mm4 ,an idea ( ! c\rse ( a long] geodetic l9e may be ga!r$ f ! foll[+ example4 ,let ! l9e be t jo9+ ,cadiz & ,/ ,pet]sburg1 ^: approximate positions are-- ,cadiz4 ,/ ,pet]sburg4 ,lat4 #cf deg4 #bb' ,n4 #ei deg4 #ef' ,n4 ,long4 #f deg4 #ah' ,w4 #cj deg4 #ag' ,e4 ,if ,g be ! po9t on ! geodetic correspond+ to ,f on t "o ( ! plane curves :i* conta9s ! normal at ,cadiz 7by correspond+ we mean t ,f & ,g are on a m]idian7 !n ,g is to ! nor? ( ,f2 at a qu>t] ( ! :ole di/ance f ,cadiz ,,gf is #deh ft41 at half ! di/ance x is #fcg ft41 & at ?ree-qu>t]s x is #dgc ft4 ,! azimu? ( ! geodetic at ,cadiz diff]s #bj f t ( ! v]tical plane1 :i* is ! a/ronomical azimu?4 ,! azimu? ( a geodetic l9e _c be obs]v$1 s t ! l9e does n ent] ( necessity into practical geodesy1 al?\gh _m =mulae connect$ ) its use are ( great simplicity & elegance4 ,! geodetic l9e has always held a m important place 9 ! sci5ce ( geodesy among ! ma!maticians ( ,france1 ,g]many & ,russia ?an has be5 assign$ to x 9 ! op]ations ( ! ,engli% & ,indian triangulations4 ,al?\gh ! obs]v$ angles ( a triangulation are n geodetic angles1 yet 9 ! calculation ( ! di/ance & reciprocal be>+s ( two po9ts :i* are f> ap>t1 & are connect$ by a long *a9 ( triangles1 we may fall ^u ! geodetic l9e 9 ?is mann]3-- ,if ,a1 ,z be ! po9ts1 !n to />t ! calculation f ,a1 we obta9 by "s prelim9>y calculation ! approximate azimu? ( ,z1 or ! angle made by ! direction ( ,z ) ! side ,,ab or ,,ac ( ! fir/ triangle4 ,let ,p#a be ! po9t :]e ?is l9e int]sects ,,bc2 !n1 to f9d ,p#b1 :]e ! l9e cuts ! next triangle side ,,cd1 we make ! angle ,,bp#a,p#b su* t ,,bp#a,p#b ,,bp#a;,a #ahj deg4 ,?is fixes ,p#b1 & ,p#c is fix$ by a repetition ( ! same process2 s = ,p#d1 ,p#e 4444 ,n[ x is cle> t ! po9ts ,p#a1 ,p#b1 ,p#c s comput$ are ^? :i* w\ld be actually fix$ by an obs]v] ) a !odolite1 proce$+ 9 ! foll[+ mann]4 ,hav+ set ! in/rum5t up at ,a1 & turn$ ! telescope 9 ! direction ( ! comput$ be>+1 an assi/ant places a m>k ,p#a on ! l9e ,,bc1 adju/+ x till bisect$ by ! cross-hairs ( ! telescope at ,a4 ,! !odolite is !n plac$ ov] ,p#a1 & ! telescope turn$ to ,a2 ! horizontal circle is !n mov$ ?r\gh #ahj deg4 ,! assi/ant !n places a m>k ,p#b on ! l9e ,,cd1 s z to be bisect$ by ! telescope1 :i* is !n mov$ to ,p#b1 & 9 ! same mann] ,p#c is fix$4 ,n[ x is cle> t ! s]ies ( po9ts ,p#a1 ,p#b1 ,p#c approa*es to ! geodetic l9e1 = ! plane ( any two consecutive elem5ts ,p7n-#a7 ,pn1 ,pn ,p7n#a7 conta9s ! normal at ,pn4 ,if ! objection be rais$ t n ! geodetic azimu?s b ! a/ronomical azimu?s are obs]v$1 x is necess>y to consid] t ! obs]v$ v]tical sections d n correspond to po9ts on ! sea-level b to elevat$ po9ts4 ,s9ce ! normals ( ! ellipsoid ( rotation d n 9 g5]al int]sect1 !re consequ5tly arises an influ5ce ( ! height on ! azimu?4 ,9 ! case ( ! measurem5t ( ! azimu? f ,a to ,b1 ! in/rum5t is set to a po9t ,a' ov] ! surface ( ! ellipsoid 7! sea-level71 & x is !n adju/$ to a po9t ,b'1 also ov] ! surface1 say at a height h'4 ,! v]tical plane conta9+ ,a' & ,b' also conta9s ,a b n ,b3 x mu/ !re=e be rotat$ ?r\gh a small azimu? 9 ord] to conta9 ,b4 ,! correction am\nts approximately to -e#b;h' cos#bphi s9 #b;alpha_/#b;a2 9 ! case ( h' #ajjj m41 its value is #j4ajh cos#bphi s9 #b;alpha4 ,?is correction is !re=e ( great] importance 9 ! case ( obs]v$ azimu?s & horizontal angles ?an 9 ! previ\sly consid]$ case ( ! a/ronomical & ! geodetic azimu?s4 ,! obs]v$ azimu?s & horizontal angles mu/ !re=e also be correct$ 9 ! case1 :]e x is requir$ to disp5se ) geodetic l9es4 ,:5 ! angles ( a triangulation h be5 adju/$ by ! me?od ( lea/ squ>es1 & ! sides are calculat$1 ! next process is to calculate ! latitudes & longitudes ( all ! /ations />t+ f "o giv5 po9t4 ,! calculat$ latitudes1 longitudes & azimu?s1 :i* are designat$ geodetic latitudes1 longitudes & azimu?s1 are n to be conf\nd$ ) ! obs]v$ latitudes1 longitudes & azimu?s1 = ^! la/ are subject to some:at l>ge errors4 ,suppos+ ! latitudes ( a numb] ( /ations 9 ! triangulation to be obs]v$1 practically ! mean ( ^! det]m9es ! position 9 latitude ( ! network1 tak5 z a :ole4 ,s ! ori5tation or g5]al azimu? ( ! :ole is inf]r$ f all ! azimu? obs]vations4 ,! triangulation is !n suppos$ to be project$ on a sph]oid ( giv5 elem5ts1 repres5t+ z ne>ly z "o kn[s ! real figure ( ! e>?4 ,!n1 tak+ ! latitude ( "o po9t & ! direction ( ! m]idian !re z giv5--obta9$1 namely1 f ! a/ronomical obs]vations !re--"o c compute ! latitudes ( all ! o!r po9ts ) any degree ( precision t may be consid]$ desirable4 ,x is necess>y to employ = ?is purpose =mulae :i* w give results true ev5 = ! longe/ di/ances to ! second place ( decimals ( seconds1 o!rwise !re w arise an accumulation ( errors f imp]fect calculation :i* %\ld always be avoid$4 ,= v long di/ances1 eight places ( decimals %\ld be employ$ 9 log>i?mic calculations2 if sev5 places only are available v great c>e w be requir$ to keep ! la/ place true4 ,n[ let phi1 phi' be ! latitudes ( two /ations ,a & ,b2 alpha1 alpha99 _! mutual azimu?s c\nt$ f nor? by ea/ cont9u\sly f #j deg4 to #cfj deg42 omega _! diff]5ce ( longitude measur$ f we/ to ea/2 & s ! di/ance ,,ab4 ,fir/ compute a latitude phi#a by means ( ! =mula phi#a phi 7s cos alpha7 _/ rho1 :]e rho is ! radius ( curvature ( ! m]idian at ! latitude phi2 ?is w require b f\r places ( log>i?ms4 ,!n1 9 ! fir/ two ( ! foll[+1 five places are suffici5t-- s#b s#b epsilon ------- s9 alpha cos a1 eta ------- s9#b;alpha tanphi#a1 #brhon #brhon s phi' - phi ---- cos 7alpha - #b_/c;epsilon7 - eta1 rho#j s s9 7alpha - #a_/c;epsilon7 omega ----------------------------1 n cos 7phi' #a_/c;eta7 alpha99 - alpha omega s9 7phi' #b_/c;eta7 - epsilon #ahj deg4 ,"h n is ! normal or radius ( curvature p]p5dicul> to ! m]idian2 bo? n & rho correspond to latitude phi#a1 & rho#j to latitude #a_/b7phi phi'74 ,= calculations ( latitude & longitude1 tables ( ! log>i?mic values ( rho s9 #a1 n s9 #a1 & #b n rho s9 #a are necess>y4 ,! foll[+ table conta9s ^! log>i?ms = e t5 m9utes ( latitude f #eb deg4 to #ec deg4 comput$ ) ! elem5ts a #bjibfjfj & a 3 b #bie 3 #bid 3-- ---------------------------------------- ------------------ #a #a #a ,lat4 ,log4------------4 ,log4--------4 ,log4--------------4 rho s9 #a n s9 #a #brhon s9 #a ---------------------------------------- ------------------ deg4 ' #eb #j #g4iicidcd #g4iibhbca #j4cgaca #aj #icji #haij #bi #bj #iahe #hadh #bh #cj #ijfj #hajg #bf #dj #hicf #hjfe #bd #ej #hhab #hjbd #bc #ec #j #hfhh #gihb #bb ---------------------------------------- ------------------ ,! log>i?m 9 ! la/ column is t requir$ also = ! calculation ( sph]ical excesses1 ! sph]ical excess ( a triangle be+ express$ by a b s9 7,c_/#brhon7 s9 #a4 ,x is frequ5tly necess>y to obta9 ! co-ord9ates ( "o po9t ) ref]5ce to ano!r po9t2 t is1 let a p]p5dicul> arc be drawn f ,b to ! m]idian ( ,a meet+ x 9 ,p1 !n1 alpha be+ ! azimu? ( ,b at ,a1 ! co-ord9ates ( ,b ) ref]5ce to ,a are ,,ap s cos 7alpha - #b_/c;epsilon71 ,,bp s s9 7alpha - #a_/c;epsilon71 :]e epsilon is ! sph]ical excess ( ,,apb1 viz4 s#b s9 alpha cos alpha multipli$ by ! quantity ^: log>i?m is 9 ! f\r? column ( ! above table4 ,if x be necess>y to det]m9e ! geographical latitude & longitude z well z ! azimu?s to a great] degree ( accuracy ?an is giv5 by ! above =mulae1 we make use ( ! foll[+ =mula3 giv5 ! latitude phi ( ,a1 & ! azimu? alpha & ! di/ance s ( ,b1 to det]m9e ! latitude phi' & longitude omega ( ,b1 & ! back azimu? alpha'4 ,"h x is und]/ood t alpha' is symmetrical to alpha1 s t alpha99 alpha' #cfj deg4 ,let !ta s ,delta _/ a1 :]e ,delta 7#a - e#b s9#b phi7#a_/b & e#b !ta#b xi ------------- cos#b phi s9 #b;alpha1 7#d 7#a - e#b7 e#b !ta#c xi' ------------- cos#b phi cos#b alpha2 7#f 7#a - e#b7 xi1 xi' are always v m9ute quantities ev5 = ! longe/ di/ances2 !n1 putt+ kappa #ij deg4 - phi1 alpha' xi - omega s9 #a_/b7kappa - !ta - xi'7 alpha tan------------------------- ---------------------------------- cot ------- #b s9 #a_/b7kappa !ta xi'7 #b alpha' xi omega cos #a_/b7kappa - !ta - xi'7 alpha tan------------------------- ---------------------------------- cot ------- #b cos #a_/b7kappa !ta xi'7 #b s s9 #a_/b7;alpha' xi - alpha7 _/ !ta#b alpha' - alpha phi' - phi ---------------------------------------- - 7 #a ---------cos#b ------------------ 72 rho#j s9 #a_/b7;alpha' xi alpha7 #ab #b _/ "h rho#j is ! radius ( curvature ( ! m]idian = ! mean latitude #a_/b7phi phi'74 ,^! =mulae are approximate only1 b !y are suffici5tly precise ev5 = v long di/ances4 ,= l9es ( any l5g? ! =mulae ( ,f4,w4 ,bessel 7,a/r4 ,na*41 #ahbc1 iv4 #bda7 are suitable4 ,if ! two po9ts ,a & ,b be def9$ by _! geographical co-ord9ates1 we c accurately calculate ! correspond+ a/ronomical azimu?s1 i4e4 ^? ( ! v]tical section1 & !n proce$1 9 ! case ( n too great di/ances1 to det]m9e ! l5g? & ! azimu? ( ! %orte/ l9es4 ,= any di/ances rec\rse mu/ aga9 be made to ,bessel's =mula4#d ,let alpha1 alpha' be ! mutual azimu?s ( two po9ts ,a1 ,b on a sph]oid1 k ! *ord l9e jo9+ !m1 mu1 mu' ! angles made by ! *ord ) ! normals at ,a & ,b1 phi1 phi'1 omega _! latitudes & diff]5ce ( longitude1 & 7x#b y#b7_/;a#b z#b b#b #a ! equation ( ! surface2 !n if ! plane xz passes ?r\gh ,a ! co-ord9ates ( ,a & ,b w be x 7a_/,delta7 cos phi1 x' 7a_/,delta'7 cos phi' cos omega1 y #j y' 7a_/,delta'7 cos phi' s9 omega1 z 7a_/,delta7 7#a - e#b7 s9 phi1 z' 7a_/,delta'7 7#a - e#b7 s9 phi'1 :]e ,delta 7#a - e#b s9#b phi7#a_/b1 ,delta' 7#a - e#b s9#b phi'7#a_/b1 & e is ! ecc5tricity4 ,let f1 g1 h be ! direction cos9es ( ! normal to t plane :i* conta9s ! normal at ,a & ! po9t ,b1 & ^: incl9ations to ! m]idian plane ( ,a is alpha2 let also l1 m1 n & l'1 m'1 n' be ! direction cos9es ( ! normal at ,a1 & ( ! tang5t to ! surface at ,a :i* lies 9 ! plane pass+ ?r\gh ,b1 !n s9ce ! fir/ l9e is p]p5dicul> to ea* ( ! o!r two & to ! *ord k1 ^: direction cos9es are proportional to x' - x1 y' - y1 z' - z1 we h ^! ?ree equations f7x' - x7 gy' h7z' - z7 #j fl gm hn #j fl' gm' hn' #j4 ,elim9ate f1 g1 h f ^! equations1 & sub/itute l cos phi l' - s9 phi cos alpha m #j m' s9 alpha n s9 phi n' cos phi cos alpha1 & we get 7x' - x7 s9 phi y' cot alpha - 7z' - z7 cos phi #j4 ,! sub/itution ( ! values ( x1 z1 x'1 y'1 z' 9 ?is equation w give imm$iately ! value ( cot alpha2 & if we put zeta1 zeta' = ! correspond+ azimu?s on a sph]e1 or on ! supposition e #j1 ! foll[+ relations exi/ cos phi ,q cot alpha - cot zeta e#b ------------------ cos phi' ,delta cos phi' ,q cot alpha' - cot zeta' e#b ------------------ cos phi ,delta' ,delta' s9 phi - ,delta s9 phi' ,q s9 omega4 ,if f ,b we let fall a p]p5dicul> on ! m]idian plane ( ,a1 & f ,a let fall a p]p5dicul> on ! m]idian plane ( ,b1 !n ! foll[+ equations become geometrically evid5t3 k s9 mu s9 alpha 7a_/,delta'7 cos phi' s9 omega k s9 mu' s9 alpha' 7a_/,delta7 cos phi s9 omega4 ,n[ 9 any surface u #j we h k#b 7x' - x7#b 7y' - y7#b 7z' - z7#b du du du _/ _/ du#b du#b du#b #a_/b -cos mu 7x' - x7 -- 7y' - y7 -- 7z' - z7 -- _/ k 7 ---- ---- ---- 7 dx dy dz_/ dx#b dy#b dz#b _/ du du du _/ _/ du#b du#b du#b #a_/b -cos mu' 7x' - x7 --- 7y' - y7 --- 7z' - z7 --- _/ k 7 ----- ----- ----- 74 dx' dy' dz'_/ dx'#b dy'#b dz'#b _/ ,9 ! pres5t case1 if we put xx' zz' #a - --- - --- ,u1 a#b b#b !n k#b _/z' - z #b --- #b,u - e#b 7 ------ 7 a#b b _/ cos mu 7a_/k7 ,delta,u2 cos mu' 7a_/k7 ,delta',u4 ,let u be su* an angle t 7#a - e#b7a_/b s9 phi ,delta s9 u cos phi ,delta cos u1 !n on express+ x1 x'1 z1 z' 9 t]ms ( u & u'1 ,u #a - cos u cos u' cos omega - s9 u s9 u'2 also1 if v be ! ?ird side ( a sph]ical triangle1 ( :i* two sides are #a_/bpi - u & #a_/bpi - u' & ! includ$ angle omega1 us+ a subsidi>y angle psi su* t s9 psi s9 #a_/bv e s9 #a_/b7u' - u7 cos #a_/b7u' u71 we obta9 f9ally ! foll[+ equations3-- k #b;a cos psi s9 #a_/bv cos mu ,delta sec psi s9 #a_/bv cos mu' ,delta' sec psi s9 #a_/bv s9 mu s9 alpha 7a_/k7 cos u' s9 omega s9 mu' s9 alpha' 7a_/k7 cos u s9 omega4 ,^! det]m9e rigor\sly ! di/ance1 & ! mutual z5i? di/ances & azimu?s1 ( any two po9ts on a sph]oid ^: latitudes & diff]5ce ( longitude are giv54 ,by a s]ies ( r$uctions f ! equations conta9+ zeta1 zeta' x may be %[n t alpha alpha' zeta zeta' #a_/d;e#domega7phi' - phi7#b cos#d phi#j s9 phi#j 4441 :]e phi#j is ! mean ( phi & phi'1 & ! high] p[]s ( e are neglect$4 ,a %ort computation w %[ t ! small quantity on ! "r-h& side ( ?is equation _c am\nt ev5 to ! ?\s&? "p ( a second = k #j4a;a1 :i* is1 practically speak+1 z]o2 consequ5tly ! sum ( ! azimu?s alpha alpha' on ! sph]oid is equal to ! sum ( ! sph]ical azimu?s1 :5ce foll[s ?is v important !orem 7kn[n z ,dalby's !orem74 ,if phi1 phi' be ! latitudes ( two po9ts on ! surface ( a sph]oid1 omega _! diff]5ce ( longitude1 alpha1 alpha' _! reciprocal azimu?s1 tan #a_/bomega cot #a_/b7;alpha alpha'7 cos #a_/b7phi' - phi7_/ s9 #a_/b7phi' phi74 ,! computation ( ! geodetic f ! a/ronomical azimu?s has be5 giv5 above4 ,f k we c n[ compute ! l5g? s ( ! v]tical section1 & f ?is ! %orte/ l5g?4 ,! diff]5ce ( l5g? ( ! geodetic l9e & ei!r ( ! plane curves is e#d s#e cos#d phi#j s9#b #b;alpha#j_/cfj a#d4 ,at lea/ ?is is an approximate expression4 ,suppos+ s #j4a;a1 ?is quantity w\ld be less ?an "o-hundr$? ( a millimetre4 ,! l9e s is n[ to be calculat$ z a circul> arc ) a mean radius r along ,,ab4 ,if phi#j #a_/b7phi phi'71 alpha#j #a_/b7ahj deg4 alpha - alpha'71 ,delta#j 7#a - e#b s9#b phi#j7a_/b1 !n #a_/r ,delta#j_/;a #a e#b_/7a - e#b7 7cos#b phi#j cos#b alpha#j71 & approximately s9 7s_/#br7 k_/#br4 ,^! =mulae give1 9 ! case ( k #j4a;a1 values c]ta9 to eight log>i?mic decimal places4 ,an excell5t s]ies ( =mulae = ! solution ( ! problem1 to det]m9e ! azimu?s1 *ord & di/ance along ! surface f ! geographical co-ord9ates1 0 giv5 9 #ahhb by ,*4 ,m4 ,s*ols 7,ar*ives ,ne]l&aises1 vol4 xvii474 ,irregul>ities ( ! ,e>?'s ,surface4 ,9 consid]+ ! effect ( unequal di/ribution ( matt] 9 ! e>?'s cru/ on ! =m ( ! surface1 we may simplify ! matt] by disreg>d+ ! consid]ations ( rotation & ecc5tricity4 ,9 ! fir/ place1 suppos+ ! e>? a sph]e cov]$ ) a film ( wat]1 let ! d5sity rho be a function ( ! di/ance f ! c5tre s t surfaces ( equal d5sity are conc5tric sph]es4 ,let n[ a di/urbance ( ! arrangem5t ( matt] take place1 s t ! d5sity is no long] to be express$ by rho1 a function ( r only1 b is express$ by rho rho'1 :]e rho' is a function ( ?ree co-ord9ates !ta1 phi1 r4 ,!n rho' is ! d5sity ( :at may be designat$ di/urb+ matt]2 x is positive 9 "s places & negative 9 o!rs1 & ! :ole quantity ( matt] ^: d5sity is rho' is z]o4 ,! previ\sly sph]ical surface ( ! sea ( radius a n[ takes a new =m4 ,let ,p be a po9t on ! di/urb$ surface1 ,p' ! correspond+ po9t v]tically bel[ x on ! undi/urb$ surface1 ,,pp' ,n4 ,! k ( ,n ov] ! :ole surface gives u ! =m ( ! di/urb$ or actual surface ( ! sea2 x is an equipot5tial surface1 & if ,v be ! pot5tial at ,p ( ! di/urb+ matt] rho'1 ,m ! mass ( ! e>? 7! attraction-con/ant is assum$ equal to unity7 ,m ,m ,m ----- ,v ,c -- - --- ,n ,v4 a ,n a a#b ,z f> z we "k1 ,n is always a v small quantity1 & we h ) suffici5t approximation ,n #c,v_/#dpideltaa1 :]e delta is ! mean d5sity ( ! e>?4 ,?us we h ! di/urbance 9 elevation ( ! sea-level express$ 9 t]ms ( ! pot5tial ( ! di/urb+ matt]4 ,if at any po9t ,p ! value ( ,n rema9 con/ant :5 we pass to any adjac5t po9t1 !n ! actual surface is !re p>allel to ! ideal sph]ical surface2 z a rule1 h[ev]1 ! normal at ,p is incl9$ to t at ,p'1 & a/ronomical obs]vations h %[n t ?is incl9ation1 ! deflection or deviation1 am\nt+ ord9>ily to "o or two seconds1 may 9 "s cases exce$ #aj1 or1 z at ! foot ( ! ,himalayas1 ev5 #fj4 ,by ! expression ma!matical figure ( ! e>? we mean ! surface ( ! sea produc$ 9 imag9ation s z to p]colate ! cont95ts4 ,we see !n t ! effect ( ! unev5 di/ribution ( matt] 9 ! cru/ ( ! e>? is to produce small elevations & depressions on ! ma!matical surface :i* w\ld be o!rwise sph]oidal4 ,no geodesi/ c proce$ f> 9 8 "w )\t enc\nt]+ ! irregul>ities ( ! ma!matical surface1 & x is necess>y t he %\ld "k h[ !y affect 8 a/ronomical obs]vations4 ,! :ole ( ?is subject is dealt ) 9 8 usual elegant mann] by ,bessel 9 ! ,a/ronomis*e ,na*ri*t51 ,nos4 #cbi1 #ccj1 #cca1 9 a pap] entitl$ ,ueb] d5 ,e9fluss d] ,unregelmassigkeit5 d] ,figur d] ,erde auf geodatis*e ,arbeit51 c4 ,b )\t ent]+ into fur!r details x is n difficult to see h[ local attraction at any /ation affects ! det]m9ations ( latitude1 longitude & azimu? !re4 ,let !re be at ! /ation an attraction to ! nor?-ea/ ?r[+ ! z5i? to ! s\?-we/1 s t x takes 9 ! cele/ial sph]e a position ,z'1 its undi/urb$ position be+ ,z4 ,let ! rectangul> compon5ts ( ! displacem5t ,,zz' be xi measur$ s\?w>ds & eta measur$ we/w>ds4 ,n[ ! great circle jo9+ ,z' ) ! pole ( ! heav5s ,p makes !re an angle ) ! m]idian ,,pz eta cosec ,,pz' eta sec phi1 :]e phi is ! latitude ( ! /ation4 ,also ?is great circle meets ! horizon 9 a po9t ^: di/ance f ! great circle ,,pz is eta sec phi s9 phi eta tan phi4 ,t is1 a m]idian m>k1 fix$ by obs]vations ( ! pole />1 w be plac$ t am\nt to ! ea/ ( nor?4 ,h5ce ! obs]v$ latitude requires ! correction xi2 ! obs]v$ longitude a correction eta sec phi2 & any obs]v$ azimu? a correction eta tan phi4 ,"h x is suppos$ t azimu?s are measur$ f nor? by ea/1 & longitudes ea/w>ds4 ,! horizontal angles are also influ5c$ by ! deflections ( ! plumb-l9e1 9 fact1 j z if ! direction ( ! v]tical axis ( ! !odolite v>i$ by ! same am\nt4 ,?is influ5ce1 h[ev]1 is slight1 s long z ! sights po9t almo/ horizontally at ! objects1 :i* is always ! case 9 ! obs]vation ( di/ant po9ts4 ,! expression giv5 = ,n enables "o to =m an approximate e/imate ( ! effect ( a compact m\nta9 9 rais+ ! sea-level4 ,take1 = in/ance1 ,b5 ,nevis1 :i* conta9s ab\t a c\ple ( cubic miles2 a simple calculation %[s t ! elevation produc$ w\ld only am\nt to ab\t #c 94 ,9 ! case ( a m\nta9 mass l ! ,himalayas1 /ret*+ ov] "s #aejj miles ( c\ntry ) a bread? ( #cjj & an av]age height ( #c miles1 al?\gh x is difficult or impossible to f9d an expression = ,v1 yet we may asc]ta9 t an elevation am\nt+ to sev]al hundr$ feet may exi/ ne> _! base4 ,! geodetical op]ations1 h[ev]1 r negative ?is idea1 = x 0 %[n by ,colonel ,cl>ke 7,phil4 ,mag41 #ahgh7 t ! =m ( ! sea-level along ! ,indian arc dep>ts b slightly f t ( ! mean figure ( ! e>?4 ,if ?is be s1 ! action ( ! ,himalayas mu/ be c\nt]act$ by subt]ranean t5uity4 ,suppose n[ t ,a1 ,b1 ,c1 444 are ! /ations ( a network ( triangulation project$ on or ly+ on a sph]oid ( semiaxis major & ecc5tricity a1 e1 ?is sph]oid hav+ its axis p>allel to ! axis ( rotation ( ! e>?1 & its surface co9cid+ ) ! ma!matical surface ( ! e>? at ,a4 ,!n bas+ ! calculations on ! obs]v$ elem5ts at ,a1 ! calculat$ latitudes1 longitudes & directions ( ! m]idian at ! o!r po9ts w be ! true latitudes1 c41 ( ! po9ts z project$ on ! sph]oid4 ,on comp>+ ^! geodetic elem5ts ) ! correspond+ a/ronomical det]m9ations1 !re w appe> a sy/em ( diff]5ces :i* repres5t ! incl9ations1 at ! v>i\s po9ts1 ( ! actual irregul> surface to ! surface ( ! sph]oid ( ref]5ce4 ,^! diff]5ces w sugge/ two ?+s1--fir/1 t we may improve ! agreem5t ( ! two surfaces1 by n re/rict+ ! sph]oid ( ref]5ce by ! condition ( mak+ its surface co9cide ) ! ma!matical surface ( ! e>? at ,a2 & secondly1 by alt]+ ! =m & dim5sions ( ! sph]oid4 ,) respect to ! fir/ circum/ance1 we may all[ ! sph]oid two degrees ( fre$om1 t is1 ! normals ( ! surfaces at ,a may be all[$ to sep>ate a small quantity1 comp\nd$ ( a m]idional diff]5ce & a diff]5ce p]p5dicul> to ! same4 ,let ! sph]oid be s plac$ t its normal at ,a lies to ! nor? ( ! normal to ! e>?'s surface by ! small quantity xi & to ! ea/ by ! quantity eta4 ,!n 9 />t+ ! calculation ( geodetic latitudes1 longitudes & azimu?s f ,a1 we mu/ take1 n ! obs]v$ elem5ts phi1 alpha1 b = phi1 phi xi1 & = alpha1 alpha eta tan phi1 & z]o longitude mu/ be replac$ by eta sec phi4 ,at ! same "t suppose ! elem5ts ( ! sph]oid to be alt]$ f a1 e to a da1 e de4 ,conf9+ our att5tion at fir/ to ! two po9ts ,a1 ,b1 let 7phi'71 7alpha'71 7omega7 be ! num]ical elem5ts at ,b z obta9$ 9 ! fir/ calculation1 viz4 be=e ! %ift+ & alt]ation ( ! sph]oid2 !y w n[ take ! =m 7phi'7 fxi geta hda kde1 7alpha'7 f'xi g'eta h'da k'de1 omega fxi geta hda kde1 :]e ! coeffici5ts f1 g1 444 c4 c be num]ically calculat$4 ,n[ ^! elem5ts1 correspond+ to ! projection ( ,b on ! sph]oid ( ref]5ce1 mu/ be equal sev]ally to ! a/ronomically det]m9$ elem5ts at ,b1 correct$ = ! incl9ation ( ! surfaces !re4 ,if xi'1 eta' be ! compon5ts ( ! incl9ation at t po9t1 !n we h xi' 7phi'7 - phi' fxi geta hda kde1 eta' tan phi' 7alpha'7 - alpha' f'xi g'eta h'da k'de1 eta' sec phi' 7omega7 - omega fxi geta hda kde1 :]e phi'1 alpha'1 omega are ! obs]v$ elem5ts at ,b4 ,"h x appe>s t ! obs]vation ( longitude gives no additional in=mation1 b is available z a *eck ^u ! azimu?al obs]vations4 ,if n[ !re be a numb] ( a/ronomical /ations 9 ! triangulation1 & we =m equations su* z ! above = ea* po9t1 !n we c f !m det]m9e ^? values ( xi1 eta1 da1 de1 :i* make ! quantity xi#b eta#b xi'#b eta'#b 444 a m9imum4 ,?us we obta9 t sph]oid :i* be/ repres5ts ! surface cov]$ by ! triangulation4 ,9 ! ,acc\nt ( ! ,pr9cipal ,triangulation ( ,great ,brita9 & ,irel& w be f\nd ! det]m9ation1 f #ge equations1 ( ! sph]oid be/ repres5t+ ! surface ( ! ,briti% ,isles4 ,its elem5ts are a #bjibgjje - #bie ft41 b 3 a - b #bhj - #h2 & x is s plac$ t at ,gre5wi* ,obs]vatory xi #a4hfd1 eta -#j4edf4 ,tak+ ,durham ,obs]vatory z ! orig91 & ! tang5t plane to ! surface 7det]m9$ by xi -#j4ffd1 eta -#d4aag7 z ! plane ( x & y1 ! =m] measur$ nor?w>ds1 & z measur$ v]tically d[nw>ds1 ! equation to ! surface is 4#iiebdiec x#b 4#iibhhjje y#b 4#iigfcjeb z#b - #j4jjfgajjcxz - #dafeejgjz #j4 ,altitudes4 ,! precise det]m9ation ( ! altitude ( 8 /ation is a matt] ( second>y importance to ! geodesi/2 nev]!less x is usual to obs]ve ! z5i? di/ances ( all trigonometrical po9ts4 ,( great importance is a k ( ! height ( ! base = its r$uction to ! sea-level4 ,aga9 ! height ( a /ation does influ5ce a little ! obs]vation ( t]re/rial angles1 = a v]tical l9e at ,b does n lie g5]ally 9 ! v]tical plane ( ,a 7see above74 ,! height above ! sea-level also influ5ces ! geographical latitude1 inasmu* z ! c5trifugal =ce is increas$ & ! magnitude & direction ( ! attraction ( ! e>? are alt]$1 & ! effect ^u ! latitude is a v small t]m express$ by ! =mula h 7g'- g7 s9 #b phi _/ ag1 :]e g1 g' are ! values ( gravity at ! equator & at ! pole4 ,?is is h s9 #b phi _/ #ehbj seconds1 h be+ 9 metres1 a quantity :i* may be neglect$1 s9ce = ord9>y m\nta9 heights x am\nts to only a few hundr$?s ( a second4 ,we c assume ?is am\nt z jo9$ ) ! nor!rn compon5t ( ! plumb-l9e p]turbations4 ,! unc]ta9ties ( t]re/rial refraction r5d] x impossible to det]m9e accurately by v]tical angles ! heights ( di/ant po9ts4 ,g5]ally speak+1 refraction is greate/ at ab\t daybreak2 f t "t x dim9i%es1 be+ at a m9imum = a c\ple ( h\rs be=e & aft] mid-day2 lat] 9 ! aft]noon x aga9 increases4 ,?is at lea/ is ! g5]al m>* ( ! ph5om5on1 b x is by no means regul>4 ,! v]tical angles measur$ at ! /ation on ,h>t ,fell %[$ on "o occasion 9 ! mon? ( ,septemb] a refraction ( d\ble ! av]age am\nt1 la/+ f #a ,p4,m4 to #e ,p4,m4 ,! mean value ( ! coeffici5t ( refraction k det]m9$ f a v l>ge numb] ( obs]vations ( t]re/rial z5i? di/ances 9 ,great ,brita9 is 4#jgib - 4#jjdg2 & if we sep>ate ^? rays :i* = a consid]able portion ( _! l5g? cross ! sea f ^? :i* d n1 ! =m] give k 4#jhac & ! latt] k 4#jgec4 ,^! values are det]m9$ f high /ations & long di/ances2 :5 ! di/ance is %ort1 & ! rays graze ! gr\nd1 ! am\nt ( refraction is extremely unc]ta9 & v>iable4 ,a case is not$ 9 ! ,indian survey :]e ! z5i? di/ance ( a /ation #aj4e miles (f v>i$ f a depression ( #d' #eb4f at #d4cj ,p4,m4 to an elevation ( #b' #bd4j at #aj4ej ,p4,m4 ,if h1 h' be ! heights above ! level ( ! sea ( two /ations1 #ij deg4 delta1 #ij deg4 delta' _! mutual z5i? di/ances 7delta be+ t obs]v$ at h71 s _! di/ance ap>t1 ! e>? be+ reg>d$ z a sph]e ( radius a1 !n1 ) suffici5t precision1 _/ #a - #bk _/ #a - #bk h' - h s tan 7 s -------- - delta 71 h - h' s tan 7 -------- - delta' 74 #b;a _/ #b;a _/ ,if f a /ation ^: height is h ! horizon ( ! sea be obs]v$ to h a z5i? di/ance #ij deg4 delta1 !n ! above =mula gives = h ! value a tan#b delta h -- -------------4 #b #a - #bk ,suppose ! depression delta to be n m9utes1 !n h #a4jedn#b if ! ray be = ! great] "p ( its l5g? cross+ ! sea2 if o!rwise1 h #a4jdjn#b4 ,to take an example3 ! mean ( eight obs]vations ( ! z5i? di/ance ( ! sea horizon at ! top ( ,b5 ,nevis is #ia deg4 #d' #dh1 or delta #fd4h2 ! ray is pretty equally dispos$ ov] l& & wat]1 & h5ce h #a4jdgn#b #dcif ft4 ,! actual height ( ! hill by _s-levell+ is #ddjf ft41 s t ! error ( ! height ?us obta9$ is only #aj ft4 ,! det]m9ation ( altitudes by means ( _s-levell+ is und\bt$ly ! mo/ exact me?od1 p>ticul>ly 9 its pres5t developm5t z precise-levell+1 by :i* !re h be5 det]m9$ 9 all civiliz$ c\ntries close-me%$ nets ( elevat$ po9ts cov]+ ! entire l&4 7,a4 ,r4 ,c2 ,f4 ,r4 ,h47 ,,footnotes3 #a ,an arrangem5t act+ simil>ly _h be5 previ\sly introduc$ by ,borda4 #b ,geodetic ,survey ( ,s\? ,africa1 vol4 iii4 7#aije71 p4 viii2 ,les ,n\veaux ,app>eils p\r la mesure rapide des bases geod41 p> ,j4 ,r5e ,b5oit et ,*4 ,ed4 ,guillaume 7#aijf74 #c ,see a pap] ,on ! ,c\rse ( ,geodetic ,l9es on ! ,e>?'s ,surface 9 ! ,phil4 ,mag4 #ahgj2 ,helm]t1 ,!ori5 d] hoh]5 ,geodasie1 #a4 #cba4 #d ,helm]t1 ,!ori5 d] hoh]5 ,geodasie1 #a4 #bcb1 #bdg4 ,,ge(frey1 surnam$ ,,m>tel 7#ajjf-ajfj71 c\nt ( ,anj\1 son ( ! c\nt ,fulk ,n]ra 7q4v47 & ( ! c\ntess ,hildeg>de or ,audeg>de1 0 born on ! #ad? ( ,octob] #ajjf4 ,dur+ 8 fa!r's lifetime he 0 recogniz$ z suz]a9 by ,fulk l',oison 7! ,gosl+71 c\nt ( ,v5dome1 ! son ( 8 half-si/] ,adela4 ,fulk hav+ revolt$1 he confiscat$ ! c\nt%ip1 :i* he did n re/ore till #ajej4 ,on ! #a/ ( ,janu>y #ajcb he m>ri$ ,agnes1 wid[ ( ,william ! ,great1 duke ( ,aquita9e1 & tak+ arms aga9/ ,william ! ,fat1 elde/ son & successor ( ,william ! ,great1 defeat$ him & took him prison] at ,mont-,c\] ne> ,sa9t-,j\9-de-,m>nes on ! #bj? ( ,septemb] #ajcc4 ,he !n tri$ to w9 recognition z dukes ( ,aquita9e = ! sons ( 8 wife ,agnes by ,william ! ,great1 :o 7 /ill m9ors1 b ,fulk ,n]ra promptly took up arms to def5d 8 suz]a9 ,william ! ,fat1 f :om he held ! ,l\dunois & ,sa9tonge 9 fief aga9/ 8 son4 ,9 #ajcf ,ge(frey ,m>tel _h to lib]ate ,william ! ,fat1 on paym5t ( a heavy ransom1 b ! latt] hav+ di$ 9 #ajch1 & ! second son ( ,william ! ,great1 ,odo1 duke ( ,gascony1 hav+ fall5 9 8 turn at ! siege ( ,mauze 7#aj? ( ,m>* #ajci7 ,ge(frey made peace ) 8 fa!r 9 ! autumn ( #ajci1 & _h 8 wife's two sons recogniz$ z dukes4 ,ab\t ?is "t1 also1 he _h int]f]$ 9 ! affairs ( ,ma9e1 ?\gh )\t mu* result1 = hav+ sid$ aga9/ ,g]vais1 bi%op ( ,le ,mans1 :o 0 try+ to make himself gu>dian ( ! "y c\nt ( ,ma9e1 ,hugh1 he _h be5 beat5 & =c$ to make t]ms ) ,g]vais 9 #ajch4 ,9 #ajdj he succe$$ 8 fa!r 9 ,anj\ & 0 able to conqu] ,t\ra9e 7#ajdd7 & ass]t 8 au?ority ov] ,ma9e 7see ,,anj\74 ,ab\t #ajej he repudiat$ ,agnes1 8 fir/ wife1 & m>ri$ ,grecie1 ! wid[ ( ,bellay1 "l ( ,montreuil-,bellay 7be=e ,augu/ #ajeb71 :om he subsequ5tly left 9 ord] to m>ry ,adela1 daught] ( a c]ta9 ,c\nt ,odo4 ,lat] he return$ to ,grecie1 b aga9 left h] to m>ry ,adelaide ! ,g]man4 ,:51 h[ev]1 he di$ on ! #ad? ( ,novemb] #ajfj1 at ! mona/]y ( ,/ ,ni*olas at ,ang]s1 he left no *ildr51 & transmitt$ ! c\nt%ip to ,ge(frey ! ,be>d$1 ! elde/ ( 8 nephews 7see ,,anj\74 ,see ,l\is ,halph51 ,le ,comte d',anj\ au ,,xie siecle 7,p>is1 #aijf74 ,a summ>y biography is giv5 by ,cele/9 ,port1 ,dictionnaire hi/orique1 geographique et biographique de ,ma9e-et-,loire 7#c vols41 ,p>is-,ang]s1 #ahgd-ahgh71 vol4 ii4 pp4 #beb-bec1 & a sket* ( ! w>s by ,kate ,norgate1 ,engl& "u ! ,angev9 ,k+s 7#b vols41 ,london1 #ahhg71 vol4 i4 *s4 iii4 iv4 7,l4 ,h4997 ,,ge(frey1 surnam$ ,,plantag5et or ,,planteg5et 7#aaac-aaea71 c\nt ( ,anj\1 0 ! son ( ,c\nt ,fulk ! ,"y & ( ,eremburge 7or ,aremb\rg ( ,la ,fle*e72 he 0 born on ! #bd? ( ,augu/ #aaac4 ,he is also call$ le bel or ! h&some1 & receiv$ ! surname ( ,plantag5et f ! habit :i* he is said to h _h ( we>+ 9 8 cap a sprig ( broom 7g5et74 ,9 #aabg he 0 made a knight1 & on ! #bnd ( ,june #aabi m>ri$ ,matilda1 daught] ( ,h5ry ,i4 ( ,engl&1 & wid[ ( ! emp]or ,h5ry ,v4 ,"s mon?s aft]w>ds he succe$$ to 8 fa!r1 :o gave up ! c\nt%ip :5 he def9itively w5t to ! k+dom ( ,j]usalem4 ,! ye>s ( 8 gov]nm5t 7 sp5t 9 subdu+ ! ,angev9 b>ons & 9 conqu]+ ,norm&y 7see ,,anj\74 ,9 #aaea1 :ile return+ f ! siege ( ,montreuil-,bellay1 he took cold1 9 consequ5ce ( ba?+ 9 ! ,loir at ,*ateau-du-,loir1 & di$ on ! #g? ( ,septemb]4 ,he 0 buri$ 9 ! ca!dral ( ,le ,mans4 ,by 8 wife ,matilda he _h ?ree sons3 ,h5ry ,plantag5et1 born at ,le ,mans on ,sunday1 ! #e? ( ,m>* #aacc2 ,ge(frey1 born at ,arg5tan on ! #a/ ( ,june #aacd2 & ,william ,long-,sword1 born on ! #bbnd ( ,july #aacf4 ,see ,kate ,norgate1 ,engl& "u ! ,angev9 ,k+s 7#b vols41 ,london1 #ahhg71 vol4 i4 *s4 v4-viii42 ,cele/9 ,port1 ,dictionnaire hi/orique1 geographique et biographique de ,ma9e-et-,loire 7#c vols41 ,p>is-,ang]s1 #ahgd-ahgh71 vol4 ii4 pp4 #bed-bef4 ,a hi/ory ( ,ge(frey le ,bel has yet to be writt52 !re is a biography ( him writt5 9 ! #ab? c5tury by ,jean1 a monk ( ,m>m\ti]1 ,hi/oria ,gaufr$i1 ducis ,normannorum et comitis ,&egavorum1 publi%$ by ,m>*egay et ,salmon2 ,*roniques des comtes d',anj\ 7,societe de l'hi/oire de ,france1 ,p>is1 #ahef71 pp4 #bbi-caj4 7,l4 ,h4997 ,,ge(frey 7#aaeh-aahf71 duke ( ,brittany1 f\r? son ( ! ,engli% k+ ,h5ry ,,ii4 & 8 wife ,eleanor ( ,aquita9e1 0 born on ! #bcrd ( ,septemb] #aaeh4 ,9 #aafg ,h5ry sugge/$ a m>riage betwe5 ,ge(frey & ,con/ance 7d4 #abja71 daught] & heiress ( ,conan ,,iv41 duke ( ,brittany 7d4 #aaga72 & ,conan n only ass5t$1 p]haps "u compulsion1 to ?is proposal1 b surr5d]$ ! great] "p ( 8 unruly du*y to ! ,engli% k+4 ,hav+ receiv$ ! homage ( ! ,breton nobles1 ,ge(frey jo9$ 8 bro!rs1 ,h5ry & ,ri*>d1 :o1 9 alliance ) ,l\is ,,vii4 ( ,france1 7 9 revolt aga9/ _! fa!r2 b he made 8 peace 9 #aagd1 aft]w>ds help+ to re/ore ord] 9 ,brittany & ,norm&y1 & aid+ ! new ,fr5* k+1 ,philip ,augu/us1 to cru% "s rebelli\s vassals4 ,9 ,july #aaha 8 m>riage ) ,con/ance 0 celebrat$1 & practically ! :ole ( 8 subsequ5t life 0 sp5t 9 w>f>e ) 8 bro!r ,ri*>d4 ,9 #aahc he made peace ) 8 fa!r1 :o _h come to ,ri*>d's assi/ance2 b a fre% /ruggle soon broke out = ! possession ( ,anj\1 & ,ge(frey 0 9 ,p>is treat+ = aid ) ,philip ,augu/us1 :5 he di$ on ! #ai? ( ,augu/ #aahf4 ,he left a daught]1 ,eleanor1 & 8 wife bore a po/hum\s son1 ! un=tunate ,ar?ur4 ,,ge(frey 7c4 #aaeb-abab71 ar*bi%op ( ,york1 0 a ba/>d son ( ,h5ry ,,ii41 k+ ( ,engl&4 ,he 0 di/+ui%$ f 8 legitimate half-bro!rs by 8 consi/5t atta*m5t & fidelity to 8 fa!r4 ,he 0 made bi%op ( ,l9coln at ! age ( tw5ty-"o 7#aagc72 b ?\gh he enjoy$ ! temporalities he 0 nev] consecrat$ & resign$ ! see 9 #aahc4 ,he !n became 8 fa!r's *ancellor1 hold+ a l>ge numb] ( lucrative b5efices 9 plurality4 ,ri*>d nom9at$ him ar*bi%op ( ,york 9 #aahi1 b he 0 n consecrat$ till #aaia1 or en?ron$ till #aaid4 ,ge(frey1 ?\gh ( high "*1 0 a man ( unev5 temp]2 8 hi/ory 9 *iefly "o ( qu>rels1 ) ! see ( ,cant]bury1 ) ! *ancellor ,william ,long*amp1 ) 8 half-bro!rs ,ri*>d & ,john1 & especially ) 8 canons at ,york4 ,?is la/ dispute kept him 9 litigation be=e ,ri*>d & ! pope = _m ye>s4 ,he l$ ! cl]gy 9 _! refusal to be tax$ by ,john & 0 =c$ to fly ! k+dom 9 #abjg4 ,he di$ 9 ,norm&y on ! #ab? ( ,decemb] #abab4 ,see ,giraldus ,cambr5sis1 ,vita ,galfridi2 ,/ubbs's prefaces to ,rog] de ,hov$51 vols4 iii4 & iv4 7,rolls ,s]ies74 7,h4 ,w4 ,c4 ,d47 ,,ge(frey ,,de ,,montbray 7d4 #ajic71 bi%op ( ,c\tances 7,con/anti5sis71 a "r-h& man ( ,william ! ,conqu]or1 0 a type ( ! great feudal prelate1 w>rior & adm9i/rator at ne$4 ,he knew1 says ,ord]ic1 m ab\t m>%all+ mail$ knights ?an edify+ psalm-s++ cl]ks4 ,obta9+1 z a "y man1 9 #ajdh1 ! see ( ,c\tances1 by 8 bro!r's influ5ce 7see ,,m[bray71 he rais$ f 8 fell[ nobles & f _! ,sicilian spoils funds = complet+ 8 ca!dral1 :i* 0 consecrat$ 9 #ajef4 ,) bi%op ,odo1 a w>rior l himself1 he 0 on ! battle-field ( ,ha/+s1 exhort+ ! ,normans to victory2 & at ,william's coronation x 0 he :o call$ on !m to acclaim _! duke z k+4 ,8 rew>d 9 ,engl& 0 a mighty fief scatt]$ ov] twelve c\nties4 ,he accompani$ ,william on 8 visit to ,norm&y 7#ajfg71 b1 return+1 l$ a royal =ce to ! relief ( ,montacute 9 ,septemb] #ajfi4 ,9 #ajge he aga9 took ! field1 lead+ ) ,bi%op ,odo a va/ ho/ aga9/ ! rebel e>l ( ,norfolk1 ^: /ronghold at ,norwi* !y besieg$ & captur$4 ,mean:ile ! ,conqu]or _h inve/$ him ) important judicial functions4 ,9 #ajgb he _h presid$ ov] ! great ,k5ti% suit betwe5 ! primate & ,bi%op ,odo1 & ab\t ! same "t ov] ^? betwe5 ! abbot ( ,ely & 8 despoil]s1 & betwe5 ! bi%op ( ,worce/] & ! abbot ( ,ely1 & !re is "s reason to ?9k t he act$ z a ,domesday commission] 7#ajhf71 & 0 plac$ ab\t ! same "t 9 *>ge ( ,nor?umb]l&4 ,! bi%op1 :o att5d$ ! ,conqu]or's fun]al1 jo9$ 9 ! great ris+ aga9/ ,william ,rufus next ye> 7#ajhh71 mak+ ,bri/ol1 ) :i* 7z ,domesday %[s7 he 0 closely connect$ & :]e he _h built a /rong ca/le1 8 base ( op]ations4 ,he burn$ ,ba? & ravag$ ,som]set1 b _h submitt$ to ! k+ be=e ! end ( ! ye>4 ,he appe>s to h be5 at ,dov] ) ,william 9 ,janu>y #ajij1 b1 )draw+ to ,norm&y1 di$ at ,c\tances ?ree ye>s lat]4 ,9 8 fidelity to ,duke ,rob]t he seems to h !re held out = him aga9/ 8 bro!r ,h5ry1 :5 ! latt] obta9$ ! ,cot5t94 ,see ,e4,a4 ,freeman1 ,norman ,conque/ & ,william ,rufus2 ,j4,h4 ,r\nd1 ,feudal ,engl&2 &1 = orig9al au?orities1 ! works ( ,ord]ic ,vitalis & ,william ( ,poiti]s1 & ( ,flor5ce ( ,worce/]2 ! ,anglo-,saxon ,*ronicle2 ,william ( ,malmesbury's ,ge/a pontificum1 & ,lanfranc's works1 ed4 ,giles2 ,domesday ,book4 7,j4 ,h4 ,r47 ,,ge(frey ,,( ,,monm\? 7d4 #aaed71 bi%op ( ,/ ,asaph & writ] on e>ly ,briti% hi/ory1 0 born ab\t ! ye> #aajj4 ,( 8 e>ly life little is kn[n1 except t he receiv$ a lib]al education "u ! eye ( 8 pat]nal uncle1 ,u*tryd1 :o 0 at t "t ar*deacon1 & subsequ5tly bi%op1 ( ,ll&aff4 ,9 #aabi ,ge(frey appe>s at ,ox=d among ! witnesses ( an ,os5ey *>t]4 ,he subscribes himself ,ge(frey ,arturus2 f ?is we may p]haps inf] t he _h already begun 8 exp]im5ts 9 ! manufacture ( ,celtic my?ology4 ,a fir/ edition ( 8 ,hi/oria ,britonum 0 9 circulation by ! ye> #aaci1 al?\gh ! text :i* we possess appe>s to date f #aadg4 ,?is fam\s "w1 :i* ! au?or has ! audacity to place on ! same level ) ! hi/ories ( ,william ( ,malmesbury & ,h5ry ( ,hunt+don1 pr(esses to be a translation f a ,celtic s\rce2 a v old book 9 ! ,briti% tongue :i* ,walt]1 ar*deacon ( ,ox=d1 _h br\ght f ,brittany4 ,walt] ! ar*deacon is a hi/orical p]sonage2 :e!r 8 book has any real exi/5ce may be fairly que/ion$4 ,!re is no?+ 9 ! matt] or ! /yle ( ! ,hi/oria to preclude u f suppos+ t ,ge(frey drew p>tly ^u confus$ traditions1 p>tly on 8 own p[]s ( inv5tion1 & to a v slight degree ^u ! accept$ au?orities = e>ly ,briti% hi/ory4 ,8 *ronology is fanta/ic & incr$ible2 ,william ( ,newburgh ju/ly rem>ks t1 if we accept$ ! ev5ts :i* ,ge(frey relates1 we %\ld h to suppose t !y _h happ5$ 9 ano!r _w4 ,william ( ,newburgh wrote1 h[ev]1 9 ! reign ( ,ri*>d ,i4 :5 ! reputation ( ,ge(frey's "w 0 too well e/abli%$ to be %ak5 by su* criticisms4 ,! fe>less romanc] _h a*iev$ an imm$iate success4 ,he 0 patroniz$ by ,rob]t1 e>l ( ,gl\ce/]1 & by two bi%ops ( ,l9coln2 he obta9$1 ab\t #aadj1 ! ar*deaconry ( ,ll&aff on acc\nt ( 8 le>n+2 & 9 #aaea 0 promot$ to ! see ( ,/ ,asaph4 ,be=e 8 dea? ! ,hi/oria ,britonum _h already become a model & a qu>ry = poets & *ronicl]s4 ,! li/ ( imitators beg9s ) ,ge(frey ,gaim>1 ! au?or ( ! ,e/orie des ,engles 7c4 #aadg71 & ,wace1 ^: ,roman de ,brut 7#aaee7 is p>tly a translation & p>tly a free p>aphrase ( ! ,hi/oria4 ,9 ! next c5tury ! influ5ce ( ,ge(frey is unmi/akably atte/$ by ! ,brut ( ,layamon1 & ! rhym+ ,engli% *ronicle ( ,rob]t ( ,gl\ce/]4 ,among lat] hi/orians :o 7 deceiv$ by ! ,hi/oria ,britonum x is only ne$ful to m5tion ,higdon1 ,h>dyng1 ,fabyan 7#aeab71 ,hol9%$ 7#aehj7 & ,john ,milton4 ,/ill great] 0 ! influ5ce ( ,ge(frey ^u ^? writ]s :o1 l ,w>n] 9 ,albion's ,engl& 7#aehf71 & ,drayton 9 ,polyolbion 7#afac71 delib]ately made _! acc\nts ( ,engli% hi/ory z poetical z possible4 ,! /ories :i* ,ge(frey pres]v$ or inv5t$ 7 n infrequ5tly a s\rce ( inspiration to lit]>y arti/s4 ,! e>lie/ ,engli% trag$y1 ,gorboduc 7#aefe71 ! ,mirror = ,magi/rates 7#aehg71 & ,%akespe>e's ,le>1 are in/ances 9 po9t4 ,x 01 h[ev]1 ! ,ar?urian leg5d :i* ( all 8 fabrications atta9$ ! greate/ vogue4 ,9 ! "w ( exp&+ & elaborat+ ?is !me ! successors ( ,ge(frey w5t z f> beyond him z he _h gone beyond ,n5nius2 b he reta9s ! cr$it due to ! f\nd] ( a great s*ool4 ,m>ie de ,france1 :o wrote at ! c\rt ( ,h5ry ,,ii41 & ,*reti5 de ,troyes1 h] ,fr5* contempor>y1 7 ! e>lie/ ( ! av[$ romanc]s to take up ! !me4 ,! succe$+ age saw ! ,ar?urian /ory popul>iz$1 ?r\gh translations ( ! ,fr5* romances1 z f> afield z ,g]many & ,sc&9avia4 ,x produc$ 9 ,engl& ! ,roman du ,sa9t ,graal & ! ,roman de ,m]l91 bo? f ! p5 ( ,rob]t de ,borron2 ! ,roman de ,lancelot2 ! ,roman de ,tri/an1 :i* is attribut$ to a fictiti\s ,lucas de ,ga/4 ,9 ! reign ( ,edw>d ,,iv4 ,sir ,?omas ,malory p>aphras$ & arrang$ ! be/ episodes ( ^! romances 9 ,engli% prose4 ,8 ,morte d',ar?ur1 pr9t$ by ,caxton 9 #adhe1 epitomizes ! ri* my?ology :i* ,ge(frey's "w _h fir/ call$ into life1 & gave ! ,ar?urian /ory a la/+ place 9 ! ,engli% imag9ation4 ,! influ5ce ( ! ,hi/oria ,britonum may be illu/rat$ 9 ano!r way1 by enum]at+ ! m famili> ( ! leg5ds to :i* x fir/ gave popul>ity4 ,( ! twelve books into :i* x is divid$ only ?ree 7,bks4 ,,ix41 ,x41 ,,xi47 are conc]n$ ) ,ar?ur4 ,e>li] 9 ! "w1 h[ev]1 we h ! adv5tures ( ,brutus2 ( 8 foll[] ,cor9eus1 ! vanqui%] ( ! ,corni% giant ,goemagol 7,gogmagog72 ( ,locr9us & 8 daught] ,sabre 7immortaliz$ 9 ,milton's ,comus72 ( ,bladud ! build] ( ,ba?2 ( ,le> & 8 daught]s2 ( ! ?ree pairs ( bro!rs1 ,f]rex & ,porrex1 ,br5nius & ,bel9us1 ,elidure & ,p]idure4 ,! /ory ( ,vortig]n & ,r[5a takes its f9al =m 9 ! ,hi/oria ,britonum2 & ,m]l9 makes 8 fir/ appe>ance 9 ! prelude to ! ,ar?ur leg5d4 ,besides ! ,hi/oria ,britonum ,ge(frey is also cr$it$ ) a ,life ( ,m]l9 compos$ 9 ,lat9 v]se4 ,! au?or%ip ( ?is "w has1 h[ev]1 be5 disput$1 on ! gr\nd t ! /yle is di/9ctly sup]ior to t ( ! ,hi/oria4 ,a m9or composition1 ! ,prophecies ( ,m]l91 0 writt5 be=e #aacf1 & aft]w>ds incorporat$ ) ! ,hi/oria1 ( :i* x =ms ! sev5? book4 ,= a discussion ( ! manuscripts ( ,ge(frey's "w1 see ,sir ,t4,d4 ,h>dy's ,descriptive ,catalogue 7,rolls ,s]ies71 i4 pp4 #cda ff4 ,! ,hi/oria ,britonum has be5 critically edit$ by ,san ,m>te 7,halle1 #ahed74 ,!re is an ,engli% translation by ,j4,a4 ,giles 7,london1 #ahdb74 ,! ,vita ,m]l9i has be5 edit$ by ,f4 ,mi*el & ,t4 ,wright 7,p>is1 #ahcg74 ,see also ! ,dubl9 ,univ4 ,magaz9e = ,april #ahgf1 = an article by ,t4 ,gilray on ! lit]>y influ5ce ( ,ge(frey2 ,g4 ,heeg]'s ,trojan]sage d] ,britt5 7#ahhi72 & ,la ,bord]ie's ,etudes hi/oriques bretonnes 7#ahhc74 7,h4 ,w4 ,c4 ,d47 ,,ge(frey ,,( ,,p>is 7d4 c4 #acbj71 ,fr5* *ronicl]1 0 probably ! au?or ( ! ,*ronique metrique de ,philippe le ,bel1 or ,*ronique rimee de ,ge(froi de ,p>is4 ,?is "w1 :i* deals ) ! hi/ory ( ,france f #acjj to #acaf1 conta9s #giah v]ses1 & is valuable z t ( a writ] :o _h a p]sonal k ( _m ( ! ev5ts :i* he relates4 ,v>i\s %ort hi/orical poems h also be5 attribut$ to ,ge(frey1 b !re is no c]ta9 in=mation ab\t ei!r 8 life or 8 writ+s4 ,! ,*ronique 0 publi%$ by ,j4,a4 ,bu*on 9 8 ,collection des *roniques1 tome ix4 7,p>is1 #ahbg71 & x has also be5 pr9t$ 9 tome xxii4 ( ! ,recueil des hi/ori5s des ,gaules et de la ,france 7,p>is1 #ahfe74 ,see ,g4 ,p>is1 ,hi/oire de la litt]ature francaise au moy5 age 7,p>is1 #ahij72 & ,a4 ,mol9i]1 ,les ,s\rces de l'hi/oire de ,france1 tome iii4 7,p>is1 #aijc74 ,,ge(frey ,,! ,,bak] 7d4 c4 #acfj71 ,engli% *ronicl]1 is also call$ ,walt] ( ,sw9broke1 & 0 probably a secul> cl]k at ,sw9brook 9 ,ox=d%ire4 ,he wrote a ,*ronicon ,angliae temporibus ,edw>di ,,ii4 et ,edw>di ,,iii41 :i* deals ) ! hi/ory ( ,engl& f #acjc to #acef4 ,f ! beg9n+ until ab\t #acbd ?is "w is bas$ ^u ,adam ,murimu?'s ,cont9uatio *ronic>um1 b aft] ?is date x is valuable & int]e/+1 conta9+ in=mation n f\nd else:]e1 & clos+ ) a good acc\nt ( ! battle ( ,poiti]s4 ,! au?or obta9$ 8 k ab\t ! la/ days ( ,edw>d ,,ii4 f ,william ,biss*op1 a companion ( ! k+'s murd]]s1 ,?omas ,gurney & ,john ,maltrav]s4 ,ge(frey also wrote a ,*roniculum f ! creation ( ! _w until #accf1 ! value ( :i* is v slight4 ,8 writ+s h be5 edit$ ) notes by ,sir ,e4,m4 ,?ompson z ! ,*ronicon ,galfridi le ,bak] de ,swynebroke 7,ox=d1 #ahhi74 ,"s d\bt exi/s conc]n+ ,ge(frey's %>e 9 ! compilation ( ! ,vita et mors ,edw>di ,,ii41 usually attribut$ to ,sir ,?omas de la ,m1 or ,moor1 & pr9t$ by ,camd5 9 8 ,anglica scripta4 ,x has be5 ma9ta9$ by ,camd5 & o!rs t ,m wrote an acc\nt ( ,edw>d's reign 9 ,fr5*1 & t ?is 0 translat$ into ,lat9 by ,ge(frey & us$ by him 9 compil+ 8 ,*ronicon4 ,rec5t s*ol>%ip1 h[ev]1 ass]ts t ,m 0 no writ]1 & t ! ,vita et mors is an extract f ,ge(frey's ,*ronicon1 & 0 attribut$ to ,m1 :o 0 ! au?or's patron4 ,9 ! ma9 ?is conclusion sub/antiates ! v]dict ( ,/ubbs1 :o has publi%$ ! ,vita et mors 9 8 ,*ronicles ( ! reigns ( ,edw>d ,i4 & ,edw>d ,,ii4 7,london1 #ahhc74 ,! manuscripts ( ,ge(frey's works are 9 ! ,bodleian libr>y at ,ox=d4 ,,ge(fr91 ,,m>ie ,,!rese ,,rodet 7#afii-aggg71 a ,fr5*woman :o play$ an int]e/+ "p 9 ,fr5* lit]>y & arti/ic life1 0 born 9 ,p>is 9 #afii4 ,%e m>ri$1 on ! #ai? ( ,july #agac1 ,pi]re ,francois ,ge(fr91 a ri* manufactur] & lieut5ant-colonel ( ! ,national ,gu>d1 :o di$ 9 #agej4 ,x 0 n till ,mme ,ge(fr9 0 ne>ly fifty ye>s ( age t we beg9 to he> ( h] z a p[] 9 ,p>isian society4 ,%e _h le>n$ mu* f ,mme de ,t5c91 & ab\t #agdh began to ga!r r\nd h] a lit]>y & arti/ic circle4 ,%e _h e week two d9n]s1 on ,monday = arti/s1 & on ,w$nesday = h] fri5ds ! ,encyclopa$i/s & o!r m5 ( lett]s4 ,%e receiv$ _m =eign]s ( di/9ction1 ,hume & ,horace ,walpole among o!rs4 ,walpole sp5t mu* "t 9 h] society be=e he 0 f9ally atta*$ to ,mme du ,deff&1 & speaks ( h] 9 8 lett]s z a model ( common s5se4 ,%e 0 inde$ some:at ( a small tyrant 9 h] circle4 ,%e _h adopt$ ! pose ( an old woman e>li] ?an necess>y1 & h] coquetry1 if su* x c be call$1 took ! =m ( be+ "m & m5tor to h] gue/s1 _m ( :om 7 indebt$ to h] g5]osity = sub/antial help4 ,al?\gh h] aim appe>s to h be5 to h ! ,encyclop$ie 9 conv]sation & action ar\nd h]1 %e 0 extremely displeas$ ) any ( h] fri5ds :o 7 s ra% z to incur op5 disgrace4 ,m>montel lo/ h] fav\r aft] ! (ficial c5sure ( ,belisaire1 & h] advanc$ views did n prev5t h] f obs]v+ ! =ms ( religion4 ,a devot$ ,p>isian1 ,mme ,ge(fr9 r>ely left ! city1 s t h] j\rney to ,pol& 9 #agff to visit ! k+1 ,/anislas ,poniat[ski1 :om %e _h kn[n 9 8 e>ly days 9 ,p>is1 0 a great ev5t 9 h] life4 ,h] exp]i5ces induc$ a s5sible gratitude t %e _h be5 born ,francaise & p>ticuli]e4 ,9 h] la/ illness h] daught]1 ,!rese1 m>quise de la ,f]te ,imbault1 exclud$ h] "m's old fri5ds s t %e might die z a good ,*ri/ian1 a proce$+ wittily describ$ by ! old lady3 ,my daught] is l ,godfrey de ,b\illon1 %e wi%$ to def5d my tomb f ! infidels4 ,mme ,ge(fr9 di$ 9 ,p>is on ! #f? ( ,octob] #aggg4 ,see ,correspondance in$ite du roi ,/anislas ,augu/e ,poniat[ski et de ,madame ,ge(fr91 edit$ by ! comte de ,m\y 7#ahge72 ,p4 de ,segur1 ,le ,royaume de la rue ,sa9t-,honore1 ,madame ,ge(fr9 et sa fille 7#ahig72 ,a4 ,tornezy1 ,un ,bureau d'esprit au ,,xviiie siecle3 le salon de ,madame ,ge(fr9 7#ahie72 & ,janet ,aldis1 ,madame ,ge(fr91 h] ,salon & h] ,times1 #agej-aggg 7#aije74 ,,ge(froy1 ,,eti5ne ,,francois 7#afgb-agca71 ,fr5* *emi/1 born 9 ,p>is on ! #ac? ( ,febru>y #afgb1 0 fir/ an apo!c>y & !n practis$ m$ic9e4 ,aft] /udy+ at ,montpelli] he accompani$ ,m>%al ,tall>d on 8 embassy to ,london 9 #afih & !nce travell$ to ,holl& & ,italy4 ,return+ to ,p>is he became pr(essor ( *emi/ry at ! ,j>d9 du ,roi & ( ph>macy & m$ic9e at ! ,college de ,france1 & dean ( ! faculty ( m$ic9e4 ,he di$ 9 ,p>is on ! #f? ( ,janu>y #agca4 ,8 "n is be/ kn[n 9 connexion ) 8 tables ( aff9ities 7tables des rapports71 :i* he pres5t$ to ! ,fr5* ,academy 9 #agah & #agbj4 ,^! 7 li/s1 prep>$ by collat+ obs]vations on ! actions ( sub/ances "o ^u ano!r1 %[+ ! v>y+ degrees ( aff9ity exhibit$ by analog\s bodies = diff]5t reag5ts1 & !y reta9$ _! vogue = ! re/ ( ! c5tury1 until displac$ by ! pr(\nd] conceptions introduc$ by ,c4,l4 ,b]?ollet4 ,ano!r ( 8 pap]s dealt ) ! delusions ( ! philosoph]'s /one1 b nev]!less he believ$ t iron c\ld be artificially =m$ 9 ! combu/ion ( vegetable matt]4 ,8 ,tractatus de mat]ia m$ica1 publi%$ po/hum\sly 9 #agda1 0 long celebrat$4 ,8 bro!r ,,claude ,,joseph1 kn[n z ,ge(froy ! y\ng] 7#afhe-ageb71 0 also an apo!c>y & *emi/ :o1 hav+ a consid]able k ( botany1 devot$ himself especially to ! /udy ( ! ess5tial oils 9 plants4 ,,ge(froy1 ,,juli5 ,,l\is 7#agdc-ahad71 ,fr5* critic1 0 born at ,r5nes 9 #agdc4 ,he /udi$ 9 ! s*ool ( 8 native t[n & at ! ,college ,l\is le ,gr& 9 ,p>is4 ,he took ord]s & fulfill$ = "s "t ! humble functions ( an u%]1 ev5tually becom+ pr(essor ( rhetoric at ! ,college ,maz>94 ,a bad trag$y1 ,caton1 0 accept$ at ! ,!atre ,francais1 b 0 nev] act$4 ,on ! dea? ( ,elie ,fr]on 9 #aggf ! o!r collaborators 9 ! ,annee litt]aire ask$ ,ge(froy to succe$ him1 & he conduct$ ! j\rnal until 9 #agib x ceas$ to appe>4 ,ge(froy 0 a bitt] critic ( ,voltaire & 8 foll[]s1 & made = himself _m enemies4 ,an en?usia/ic royali/1 he publi%$ ) ,fr]on's bro!r-9-law1 ! abbe ,?omas ,roy\ 7#agda-agib71 a j\rnal1 ,l',ami du roi 7#agij-agib71 :i* possibly did m h>m ?an good to ! k+'s cause by its ill-advis$ p>tisan%ip4 ,dur+ ! ,t]ror ,ge(froy hid 9 ! neighb\rhood ( ,p>is1 only return+ 9 #agii4 ,an attempt to revive ! ,annee litt]aire fail$1 & ,ge(froy und]took ! dramatic feuilleton ( ! ,j\rnal des debats4 ,8 sca?+ criticisms _h a success ( notoriety1 b _! popul>ity 0 ephem]al1 & ! publication ( !m 7#e vols41 #ahai-ahbj7 z ,c\rs de litt]ature dramatique prov$ a failure4 ,he 0 also ! au?or ( a p]functory ,comm5taire on ! works ( ,rac9e prefix$ to ,l5ormant's edition 7#ahjh74 ,he di$ 9 ,p>is on ! #bg? ( ,febru>y #ahad4 ,,ge(froy ,,sa9t-,,hilaire1 ,,eti5ne 7#aggb-ahdd71 ,fr5* naturali/1 0 ! son ( ,jean ,g]>d ,ge(froy1 procurator & magi/rate ( ,etampes1 ,se9e-et-,oise1 :]e he 0 born on ! #ae? ( ,april #aggb4 ,de/9$ = ! *ur* he ent]$ ! college ( ,nav>re1 9 ,p>is1 :]e he /udi$ natural philosophy "u ,m4,j4 ,brisson2 & 9 #aghh he obta9$ "o ( ! canonicates ( ! *apt] ( ,sa9te ,croix at ,etampes1 & also a b5efice4 ,sci5ce1 h[ev]1 (f]$ him a m cong5ial c>e]1 & he ga9$ f 8 fa!r p]mission to rema9 9 ,p>is1 & to att5d ! lectures at ! ,college de ,france & ! ,j>d9 des ,plantes1 on ! condition t he %\ld also read law4 ,he accord+ly took up 8 resid5ce at ,c>d9al ,lemo9e's college1 & !re became ! pupil & soon ! e/eem$ associate ( ,brisson's fri5d1 ! abbe ,hauy1 ! m9]alogi/4 ,hav+1 be=e ! close ( ! ye> #agij1 tak5 ! degree ( ba*elor 9 law1 he became a /ud5t ( m$ic9e1 & att5d$ ! lectures ( ,a4,f4 de ,f\rcroy at ! ,j>d9 des ,plantes1 & ( ,l4,j4,m4 ,daub5ton at ! ,college de ,france4 ,8 /udies at ,p>is 7 at l5g? sudd5ly int]rupt$1 =1 9 ,augu/ #agib1 ,hauy & ! o!r pr(essors ( ,lemo9e's college1 z also ^? ( ! college ( ,nav>re1 7 arre/$ by ! revolutioni/s z prie/s1 & conf9$ 9 ! prison ( ,/ ,firm94 ,?r\gh ! influ5ce ( ,daub5ton & o!rs ,ge(froy on ! #ad? ( ,augu/ obta9$ an ord] = ! release ( ,hauy 9 ! "n ( ! ,academy2 /ill ! o!r pr(essors ( ! two colleges1 save ,c4,f4 ,lhomond1 :o _h be5 rescu$ by 8 pupil ,j4,l4 ,talli51 rema9$ 9 conf9em5t4 ,ge(froy1 =esee+ _! c]ta9 de/ruction if !y rema9$ 9 ! h&s ( ! revolutioni/s1 det]m9$ if possible to secure _! lib]ty by /ratagem4 ,by brib+ "o ( ! (ficials at ,/ ,firm91 & disguis+ himself z a commission] ( prisons1 he ga9$ admission to 8 fri5ds1 & entreat$ !m to effect _! escape by foll[+ him4 ,all1 h[ev]1 dread+ le/ _! deliv]ance %\ld r5d] ! doom ( _! fell[-captives ! m c]ta91 refus$ ! (f]1 & "o prie/ only1 :o 0 unkn[n to ,ge(froy1 left ! prison4 ,already on ! night ( ! #bnd ( ,septemb] ! massacre ( ! proscrib$ _h begun1 :5 ,ge(froy1 yet int5t on sav+ ! life ( 8 fri5ds & tea*]s1 repair$ to ,/ ,firm94 ,at #d o'clock on ! morn+ ( ! #crd ( ,septemb]1 aft] eight h\rs' wait+1 he by means ( a ladd] assi/$ ! escape ( twelve ecclesia/ics1 n ( ! numb] ( 8 acqua9tance1 & !n ! approa* ( dawn & ! dis*>ge ( a gun direct$ at him w>n$ him1 8 *ief purpose unaccompli%$1 to return to 8 lodg+s4 ,leav+ ,p>is he retir$ to ,etampes1 :]e1 9 consequ5ce ( ! anxieties ( :i* he _h lately be5 ! prey1 & ! horrors :i* he _h witness$1 he 0 = "s "t s]i\sly ill4 ,at ! beg9n+ ( ! w9t] ( #agib he return$ to 8 /udies 9 ,p>is1 & 9 ,m>* ( ! foll[+ ye> ,daub5ton1 ?r\gh ! int]e/ ( ,b]n>d9 de ,sa9t ,pi]re1 procur$ him ! (fice ( sub-keep] & assi/ant demon/rator ( ! cab9et ( natural hi/ory1 vacant by ! resignation ( ,b4,g4,e4 ,lacep$e4 ,by a law pass$ 9 ,june #agic1 ,ge(froy 0 appo9t$ "o ( ! twelve pr(essors ( ! newly con/itut$ museum ( natural hi/ory1 be+ assign$ ! *air ( zoology4 ,9 ! same ye> he busi$ himself ) ! =mation ( a m5ag]ie at t in/itution4 ,9 #agid ?r\gh ! introduction ( ,a4,h4 ,tessi] he ent]$ into correspond5ce ) ,georges ,cuvi]1 to :om1 aft] ! p]usal ( "s ( 8 manuscripts1 he wrote3 ,v5ez j\] p>mi n\s le role de ,l9ne1 d'un autre legislateur de l'hi/oire naturelle4 ,%ortly aft] ! appo9tm5t ( ,cuvi] z assi/ant at ! ,museum d',hi/oire ,naturelle1 ,ge(froy receiv$ him into 8 h\se4 ,! two fri5ds wrote toge!r five memoirs on natural hi/ory1 "o ( :i*1 on ! classification ( mammals1 puts =w>d ! idea ( ! subord9ation ( *>act]s ^u :i* ,cuvi] bas$ 8 zoological sy/em4 ,x 0 9 a pap] entitl$ ,hi/oire des ,makis1 ou s+es de ,madagasc>1 writt5 9 #agie1 t ,ge(froy fir/ gave expression to 8 views on ! unity ( organic composition1 ! influ5ce ( :i* is p]ceptible 9 all 8 subsequ5t writ+s2 nature1 he obs]ves1 pres5ts u ) only "o plan ( con/ruction1 ! same 9 pr9ciple1 b v>i$ 9 its accessory p>ts4 ,9 #agih ,ge(froy 0 *os5 a memb] ( ! great sci5tific exp$ition to ,egypt1 & on ! capitulation ( ,alex&ria 9 ,augu/ #ahja1 he took "p 9 resi/+ ! claim made by ! ,briti% g5]al to ! collections ( ! exp$ition1 decl>+ t1 7 t dem& p]si/$ 91 hi/ory w\ld h to record t he also _h burnt a libr>y 9 ,alex&ria4 ,e>ly 9 ,janu>y #ahjb ,ge(froy return$ to 8 accu/om$ lab\rs 9 ,p>is4 ,he 0 elect$ a memb] ( ! academy ( sci5ces ( t city 9 ,septemb] #ahjg4 ,9 ,m>* ( ! foll[+ ye> ! emp]or1 :o _h already recogniz$ 8 national s]vices by ! aw>d ( ! cross ( ! legion ( hon\r1 select$ him to visit ! museums ( ,portugal1 = ! purpose ( procur+ collections f !m1 & 9 ! face ( consid]able opposition f ! ,briti% he ev5tually 0 successful 9 reta9+ !m z a p]man5t possession = 8 c\ntry4 ,9 #ahji1 ! ye> aft] 8 return to ,france1 he 0 made pr(essor ( zoology at ! faculty ( sci5ces at ,p>is1 & f t p]iod he devot$ himself m exclusively ?an be=e to anatomical /udy4 ,9 #ahah he gave to ! _w ! fir/ "p ( 8 celebrat$ ,philosophie anatomique1 ! second volume ( :i*1 publi%$ 9 #ahbb1 & subsequ5t memoirs acc\nt = ! =mation ( mon/rosities on ! pr9ciple ( arre/ ( developm5t1 & ( ! attraction ( simil> p>ts4 ,:51 9 #ahcj1 ,ge(froy proce$$ to apply to ! inv]tebrata 8 views z to ! unity ( animal composition1 he f\nd a vigor\s oppon5t 9 ,georges ,cuvi]1 & ! discussion betwe5 !m1 cont9u$ up to ! "t ( ! dea? ( ! latt]1 soon attract$ ! att5tion ( ! sci5tific ?r\gh\t ,europe4 ,ge(froy1 a syn!si/1 cont5d$1 9 accordance ) 8 !ory ( unity ( plan 9 organic composition1 t all animals are =m$ ( ! same elem5ts1 9 ! same numb]1 & ) ! same connexions3 homolog\s p>ts1 h[ev] !y diff] 9 =m & size1 mu/ rema9 associat$ 9 ! same inv>iable ord]4 ,) ,goe! he held t !re is 9 nature a law ( comp5sation or balanc+ ( gr[?1 s t if "o organ take on an excess ( developm5t1 x is at ! exp5se ( "s o!r "p2 & he ma9ta9$ t1 s9ce nature takes no sudd5 leaps1 ev5 organs :i* are sup]flu\s 9 any giv5 species1 if !y h play$ an important "p 9 o!r species ( ! same family1 are reta9$ z rudim5ts1 :i* te/ify to ! p]man5ce ( ! g5]al plan ( creation4 ,x 0 8 conviction t1 ow+ to ! conditions ( life1 ! same =ms _h n be5 p]petuat$ s9ce ! orig9 ( all ?+s1 al?\gh x 0 n 8 belief t exi/+ species are becom+ modifi$4 ,cuvi]1 :o 0 an analytical obs]v] ( facts1 admitt$ only ! preval5ce ( laws ( co-exi/5ce or h>mony 9 animal organs1 & ma9ta9$ ! absolute inv>iability ( species1 :i* he decl>$ _h be5 creat$ ) a reg>d to ! circum/ances 9 :i* !y 7 plac$1 ea* organ contriv$ ) a view to ! function x _h to fulfil1 ?us putt+1 9 ,ge(froy's consid]ations1 ! effect = ! cause4 ,9 ,july #ahdj ,ge(froy became bl9d1 & "s mon?s lat] he _h a p>alytic attack4 ,f t "t 8 /r5g? gradually fail$ him4 ,he resign$ 8 *air at ! museum 9 #ahda1 & di$ at ,p>is on ! #ai? ( ,june #ahdd4 ,ge(froy wrote3 ,catalogue des mammif]es du ,museum ,national d',hi/oire ,naturelle 7#ahac71 n q complet$2 ,philosophie anatomique--t4 i41 ,des organes respiratoires 7#ahah71 & t4 ii41 ,des mon/ruosites huma9es 7#ahbb72 ,sy/eme d5taire des mammif]es et des oiseaux 7#a/ pt41 #ahbd72 ,sur le pr9cipe de l'unite de composition organique 7#ahbh72 ,c\rs de l'hi/oire naturelle des mammif]es 7#ahbi72 ,pr9cipes de philosophie zoologique 7#ahcj72 ,etudes progressives d'un naturali/e 7#ahce72 ,fragm5ts biographiques 7#ahcb72 ,notions syn!tiques1 hi/oriques et physiologiques de philosophie naturelle 7#ahch71 & o!r works2 also "p ( ! ,description de l',egypte p> la commission des sci5ces 7#ahba-ahcj72 &1 ) ,fr$]ic ,cuvi] 7#aggc-ahch71 a y\ng] bro!r ( ,g4 ,cuvi]1 ,hi/oire naturelle des mammif]es 7#d vols41 #ahbj-ahdb72 besides num]\s pap]s on su* subjects z ! anatomy ( m>supials1 rum9ants & electrical fi%es1 ! v]tebrate !ory ( ! skull1 ! op]cula ( fi%es1 t]atology1 palaeontology & ! influ5ce ( surr\nd+ conditions 9 modify+ animal =ms4 ,see ,vie1 travaux1 et doctr9e sci5tifique d',eti5ne ,ge(froy ,sa9t-,hilaire1 p> son fils ,m4 ,isidore ,ge(froy ,sa9t-,hilaire 7,p>is & ,/rasburg1 #ahdg71 to :i* is app5d$ a li/ ( ,ge(froy's works2 & ,joly1 9 ,biog4 univ]selle1 t4 xvi4 7#ahef74 ,,ge(froy ,,sa9t-,,hilaire1 ,,isidore 7#ahje-ahfa71 ,fr5* zoologi/1 son ( ! prec$+1 0 born at ,p>is on ! #af? ( ,decemb] #ahje4 ,9 8 e>li] ye>s he %[$ an aptitude = ma!matics1 b ev5tually he devot$ himself to ! /udy ( natural hi/ory & ( m$ic9e1 & 9 #ahbd he 0 appo9t$ assi/ant naturali/ to 8 fa!r4 ,on ! occasion ( 8 tak+ ! degree ( doctor ( m$ic9e 9 ,septemb] #ahbi1 he read a !sis entitl$ ,propositions sur la mon/ruosite1 consid]ee *ez l'homme et les animaux2 & 9 #ahcb-ahcg 0 publi%$ 8 great t]atological "w1 ,hi/oire g5]ale et p>ticuli]e des anomalies de l'organisation *ez l'homme et les animaux1 #c vols4 #hvo4 ) #bj plates4 ,9 #ahbi he deliv]$ = 8 fa!r ! second "p ( a c\rse ( lectures on orni?ology1 & dur+ ! ?ree foll[+ ye>s he taught zoology at ! ,a!nee1 & t]atology at ! ,ecole pratique4 ,he 0 elect$ a memb] ( ! academy ( sci5ces at ,p>is 9 #ahcc1 0 9 #ahcg appo9t$ to act z deputy = 8 fa!r at ! faculty ( sci5ces 9 ,p>is1 & 9 ! foll[+ ye> 0 s5t to ,bordeaux to organize a simil> faculty !re4 ,he became successively inspector ( ! academy ( ,p>is 7#ahdj71 pr(essor ( ! museum on ! retirem5t ( 8 fa!r 7#ahda71 inspector-g5]al ( ! univ]sity 7#ahdd71 a memb] ( ! royal c\ncil = public in/ruction 7#ahde71 & on ! dea? ( ,h4,m4,d4 de ,bla9ville1 pr(essor ( zoology at ! faculty ( sci5ces 7#ahej74 ,9 #ahed he f\nd$ ! ,acclimatization ,society ( ,p>is1 ( :i* he 0 presid5t4 ,he di$ at ,p>is on ! #aj? ( ,novemb] #ahfa4 ,besides ! above-m5tion$ works1 he wrote3 ,essais de zoologie g5]ale 7#ahda72 ,vie 444 d',eti5ne ,ge(froy ,sa9t-,hilaire 7#ahdg72 ,acclimatation et dome/ication des animaux utiles 7#ahdi2 #d? ed41 #ahfa72 ,lettres sur les sub/ances alim5taires et p>ticuli]em5t sur la vi&e de *eval 7#ahef72 & ,hi/oire naturelle g5]ale des regnes organiques 7#c vols41 #ahed-ahfb71 :i* 0 n q complet$4 ,he 0 ! au?or also ( v>i\s pap]s on zoology1 comp>ative anatomy & palaeontology4 ,,geography 7,gr4 ,greek3 ge1 e>?1 & ,greek3 graphe91 to write71 ! exact & organiz$ k ( ! di/ribution ( ph5om5a on ! surface ( ! e>?4 ,! fundam5tal basis ( geography is ! v]tical relief ( ! e>?'s cru/1 :i* controls all mobile di/ributions4 ,! gr&] features ( ! relief ( ! li?osph]e or /ony cru/ ( ! e>? control ! di/ribution ( ! hydrosph]e or collect$ wat]s :i* ga!r into ! holl[s1 fill+ !m up to a height correspond+ to ! volume1 & ?us produc+ ! important practical division ( ! surface into l& & wat]4 ,! di/ribution ( ! mass ( ! atmosph]e ov] ! surface ( ! e>? is also controll$ by ! relief ( ! cru/1 its great] or less] d5sity at ! surface correspond+ to ! less] or great] elevation ( ! surface4 ,! simplicity ( ! zonal di/ribution ( sol> en]gy on ! e>?'s surface1 :i* w\ld *>act]ize a uni=m globe1 is entirely de/roy$ by ! dissimil> action ( l& & wat] ) reg>d to radiant heat1 & by ! influ5ce ( cru/-=ms on ! direction ( ! result+ circulation4 ,! influ5ce ( physical environm5t becomes cle>] & /rong] :5 ! di/ribution ( plant & animal life is consid]$1 & if x is less di/9ct 9 ! case ( man1 ! reason is f\nd 9 ! modifications ( environm5t consci\sly produc$ by human ef=t4 ,geography is a syn!tic sci5ce1 dep5d5t = ! data ) :i* x deals on ! results ( specializ$ sci5ces su* z a/ronomy1 geology1 oceanography1 meteorology1 biology & an?ropology1 z well z on topographical description4 ,! physical & natural sci5ces are conc]n$ 9 geography only s f> z !y deal ) ! =ms ( ! e>?'s surface1 or z reg>ds ! di/ribution ( ph5om5a4 ,! di/9ctive task ( geography z a sci5ce is to inve/igate ! control ex]cis$ by ! cru/-=ms directly or indirectly ^u ! v>i\s mobile di/ributions4 ,?is gives to x unity & def9it5ess1 & r5d]s sup]flu\s ! attempts t h be5 made f "t to "t to def9e ! limits :i* divide geography f geology on ! "o h& & f hi/ory on ! o!r4 ,x is ess5tial to classify ! subject-matt] ( geography 9 su* a mann] z to give prom95ce n only to facts1 b to _! mutual relations & _! natural & inevitable ord]4 ,! fundam5tal conception ( geography is =m1 includ+ ! figure ( ! e>? & ! v>ieties ( cru/al relief4 ,h5ce ma!matical geography 7see ,,map71 includ+ c>tography z a practical application1 comes fir/4 ,x m]ges into physical geography1 :i* takes acc\nt ( ! =ms ( ! li?osph]e 7geomorphology71 & also ( ! di/ribution ( ! hydrosph]e & ! re>rangem5ts result+ f ! work+s ( sol> en]gy ?r\gh\t ! hydrosph]e & atmosph]e 7oceanography & climatology74 ,next foll[s ! di/ribution ( plants & animals 7biogeography71 & f9ally ! di/ribution ( mank9d & ! v>i\s artificial b\nd>ies & r$i/ributions 7an?ropogeography74 ,! applications ( an?ropogeography to human uses give rise to political & comm]cial geography1 9 ! elucidation ( :i* all ! e>li] dep>tm5ts or /ages h to be consid]$1 toge!r ) hi/orical & o!r purely human conditions4 ,! evolution>y idea has revolutioniz$ & unifi$ geography z x did biology1 break+ d[n ! old h>d-&-fa/ p>titions betwe5 ! v>i\s dep>tm5ts1 & sub/itut+ ! /udy ( ! nature & influ5ce ( actual t]re/rial environm5ts = ! e>li] motive1 ! discov]y & exploration ( new l&s4 ,,hi/ory ,,( ,,geographical ,,!ory ,! e>lie/ conceptions ( ! e>?1 l ^? held by ! primitive peoples ( ! pres5t day1 are difficult to discov] & almo/ impossible fully to grasp4 ,e>ly g5]alizations1 z f> z !y 7 made f kn[n facts1 7 usually express$ 9 symbolic language1 & = our pres5t purpose x is n pr(itable to speculate on ! und]ly+ tru?s :i* may sometimes be suspect$ 9 ! old my?ological cosmogonies4 ,e>ly ,greek ideas4 ,flat e>? ( ,hom]4 ,hecataeus4 ,h]odotus4 ,! idea ( symmetry4 ,! fir/ def9ite geographical !ories to affect ! we/]n _w 7 ^? evolv$1 or at lea/ fir/ express$1 by ! ,greeks4#a ,! e>lie/ !oretical problem ( geography 0 ! =m ( ! e>?4 ,! natural supposition t ! e>? is a flat disk1 circul> or elliptical 9 outl9e1 _h 9 ! "t ( ,hom] acquir$ a special def9it5ess by ! introduction ( ! idea ( ! ocean riv] b\nd+ ! :ole1 an application ( imp]fectly und]/ood obs]vations4 ,?ales ( ,miletus is claim$ z ! fir/ expon5t ( ! idea ( a sph]ical e>?2 b1 al?\gh ?is does n appe> to be w>rant$1 8 disciple ,anaxim&] 7c4 #ehj ,b4,c47 put =w>d ! !ory t ! e>? _h ! figure ( a solid body hang+ freely 9 ! c5tre ( ! holl[ sph]e ( ! />ry heav5s4 ,! ,py?agorean s*ool ( philosoph]s adopt$ ! !ory ( a sph]ical e>?1 b f metaphysical r ?an sci5tific reasons2 _! conv9c+ argum5t 0 t a sph]e be+ ! mo/ p]fect solid figure 0 ! only "o wor?y to circumscribe ! dwell+-place ( man4 ,! division ( ! sph]e into p>allel zones & "s ( ! consequ5ces ( ?is g5]alization seem to h pres5t$ !mselves to ,p>m5ides 7c4 #dej ,b4,c472 b ^! ideas did n influ5ce ! ,ionian s*ool ( philosoph]s1 :o 9 _! treatm5t ( geography pref]r$ to deal ) facts demon/rable by travel r ?an ) speculations4 ,?us ,hecataeus1 claim$ by ,h4,f4 ,toz]#b z ! fa!r ( geography on acc\nt ( 8 ,p]iodos1 or g5]al treatise on ! e>?1 did n advance beyond ! primitive conception ( a circul> disk4 ,he sy/ematiz$ ! =m ( ! l& )9 ! r+ ( ocean--! ,greek3 oik\m5e1 or habitable _w--by recogniz+ two cont95ts3 ,europe to ! nor?1 & ,asia to ! s\? ( ! midl& sea4 ,h]odotus1 equally oblivi\s ( ! sph]e1 criticiz$ & ridicul$ ! circul> outl9e ( ! oekum5e1 :i* he knew to be long] f ea/ to we/ ?an x 0 broad f nor? to s\?4 ,he also po9t$ out reasons = accept+ a division ( ! l& into ?ree cont95ts--,europe1 ,asia & ,africa4 ,beyond ! limits ( 8 p]sonal travels ,h]odotus appli$ ! *>act]i/ically ,greek !ory ( symmetry to complete1 9 ! unkn[n1 outl9es ( l&s & riv]s analog\s to ^? :i* _h be5 explor$4 ,symmetry 0 9 fact ! fir/ geographical !ory1 & ! effect ( ,h]odotus's hypo!sis t ! ,nile mu/ fl[ f we/ to ea/ be=e turn+ nor? 9 ord] to balance ! ,danube runn+ f we/ to ea/ be=e turn+ s\? l+]$ 9 ! maps ( ,africa d[n to ! "t ( ,mungo ,p>k4#c ,ari/otle & ! sph]e4 ,to ,ari/otle 7#chd-cbb ,b4,c47 mu/ be giv5 ! di/9ction ( f\nd+ sci5tific geography4 ,he demon/rat$ ! sph]icity ( ! e>? by ?ree argum5ts1 two ( :i* c\ld be te/$ by obs]vation4 ,^! 73 7#a7 t ! e>? mu/ be sph]ical1 because ( ! t5d5cy ( matt] to fall toge!r t[>ds a common c5tre2 7#b7 t only a sph]e c\ld always ?r[ a circul> %ad[ on ! moon dur+ an eclipse2 & 7#c7 t ! %ift+ ( ! horizon & ! appe>ance ( new con/ellations1 or ! disappe>ance ( famili> />s1 z "o travell$ f nor? to s\?1 c\ld only be expla9$ on ! hypo!sis t ! e>? 0 a sph]e4 ,ari/otle1 too1 gave great] def9it5ess to ! idea ( zones conceiv$ by ,p>m5ides1 :o _h pictur$ a torrid zone un9habitable by reason ( heat1 two frigid zones un9habitable by reason ( cold1 & two int]m$iate temp]ate zones fit = human occupation4 ,ari/otle def9$ ! temp]ate zone z ext5d+ f ! tropic to ! arctic circle1 b !re is "s unc]ta9ty z to ! precise mean+ he gave to ! t]m arctic circle4 ,soon aft] 8 "t1 h[ev]1 ?is conception 0 cle>ly e/abli%$1 & ) s l>ge a g5]alization ! m5tal horizon 0 wid5$ to conceive ( a geography :i* 0 a sci5ce4 ,ari/otle _h himself %[n t 9 ! s\!rn temp]ate zone w9ds simil> to ^? ( ! nor!rn temp]ate zone %\ld bl[1 b f ! opposite direction4 ,fitt+ ! oekum5e to ! sph]e4 ,:ile ! !ory ( ! sph]e 0 be+ elaborat$ ! ef=ts ( practical geograph]s 7 /eadily direct$ t[>ds asc]ta9+ ! outl9e & configuration ( ! oekum5e1 or habitable _w1 ! only portion ( ! t]re/rial surface kn[n to ! anci5ts & to ! m$ieval peoples1 & /ill reta9+ a %ad[ ( its old monopoly ( geographical att5tion 9 its mod]n "n ( ! ,old ,_w4 ,! fitt+ ( ! oekum5e to ! sph]e 0 ! second !oretical problem4 ,! circul> outl9e _h giv5 way 9 geographical op9ion to ! elliptical ) ! long axis ly+ ea/ & we/1 & ,ari/otle 0 incl9$ to view x z a v long & relatively n>r[ b& almo/ encircl+ ! globe 9 ! temp]ate zone4 ,8 argum5t z to ! n>r[ness ( ! sea betwe5 ,we/ ,africa & ,ea/ ,asia1 f ! occurr5ce ( elephants at bo? extremities1 is difficult to und]/&1 al?\gh x %[s t he look$ on ! di/ribution ( animals z a problem ( geography4 ,problem ( ! ,antipodes4 ,py?agoras _h speculat$ z to ! exi/5ce ( antipodes1 b x 0 n until ! fir/ approximately accurate measurem5ts ( ! globe & e/imates ( ! l5g? & bread? ( ! oekum5e 7 made by ,erato/h5es 7c4 #bej ,b4,c47 t ! fact t1 z !n kn[n1 x occupi$ less ?an a qu>t] ( ! surface ( ! sph]e 0 cle>ly recogniz$4 ,x 0 natural1 if n /rictly logical1 t ! ocean riv] %\ld be ext5d$ f a n>r[ /ream to a _w-embrac+ sea1 & "h aga9 ,greek !ory1 or r fancy1 gave its mod]n "n to ! greate/ feature ( ! globe4 ,! old in/9ctive idea ( symmetry mu/ (t5 h sugge/$ o!r oekum5e balanc+ ! kn[n _w 9 ! o!r qu>t]s ( ! globe4 ,! ,/oic philosoph]s1 especially ,crates ( ,mallus1 argu+ f ! love ( nature = life1 plac$ an oekum5e 9 ea* qu>t] ( ! sph]e1 ! ?ree unkn[n _w-isl&s be+ ^? ( ! ,antoeci1 ,p]ioeci & ,antipodes4 ,?is 0 a !ory n only attractive to ! philosophical m9d1 b em95tly adapt$ to promote exploration4 ,x _h its oppon5ts1 h[ev]1 = ,h]odotus %[$ t sea-bas9s exi/$ cut (f f ! ocean1 & x is /ill a matt] ( controv]sy h[ f> ! pre-,ptolemaic geograph]s believ$ 9 a wat]-connexion betwe5 ! ,atlantic & ,indian oceans4 ,x is q cle> t ,pomponius ,mela 7c4 ,a4,d4 #dj71 foll[+ ,/rabo1 held t ! s\!rn temp]ate zone conta9$ a habitable l&1 :i* he designat$ by ! "n ,anti*?ones4 ,ari/otle's geographical views4 ,ari/otle left no "w on geography1 s t x is impossible to "k :at facts he associat$ ) ! sci5ce ( ! e>?'s surface4 ,! ^w geography did n appe> be=e ,ari/otle1 ! fir/ use ( x be+ 9 ! ,greek3 ,p]i kosmon1 :i* is "o ( ! writ+s d\btfully ascrib$ to him1 & ,h4 ,b]g] consid]s t ! expression 0 introduc$ by ,erato/h5es4#d ,ari/otle 0 c]ta9ly conv]sant ) _m facts1 su* z ! =mation ( deltas1 coa/-erosion1 & to a c]ta9 ext5t ! dep5d5ce ( plants & animals on _! physical surr\nd+s4 ,he =m$ a compreh5sive !ory ( ! v>iations ( climate ) latitude & season1 & 0 conv9c$ ( ! necessity ( a circulation ( wat] betwe5 ! sea & riv]s1 ?\gh1 l ,plato1 he held t ?is took place by wat] ris+ f ! sea ?r\gh crevices 9 ! rocks1 los+ its dissolv$ salts 9 ! process4 ,he speculat$ on ! diff]5ces 9 ! "* ( races ( mank9d liv+ 9 diff]5t climates1 & correlat$ ! political =ms ( communities ) _! situation on a sea%ore1 or 9 ! neighb\rhood ( natural /rongholds4 ,/rabo4 ,/rabo 7c4 #ej ,b4,c4-,a4,d4 #bd7 foll[$ ,erato/h5es r ?an ,ari/otle1 b ) sympa?ies :i* w5t out m to ! human int]e/s ?an ! ma!matical basis ( geography4 ,he compil$ a v rem>kable "w deal+1 9 l>ge measure f p]sonal travel1 ) ! c\ntries surr\nd+ ! ,m$it]ranean4 ,he may be said to h set ! patt]n :i* 0 foll[$ 9 succe$+ ages by ! compil]s ( political geographies deal+ less ) !ories ?an ) facts1 & illu/rat+ r ?an =mulat+ ! pr9ciples ( ! sci5ce4 ,ptolemy4 ,claudius ,ptolemaeus 7c4 ,a4,d4 #aej7 conc5trat$ 9 8 writ+s ! f9al outcome ( all ,greek geographical le>n+1 & pass$ x across ! gulf ( ! middle ages by ! h&s ( ! ,arabs1 to =m ! />t+-po9t ( ! sci5ce 9 mod]n times4 ,8 geography 0 bas$ m imm$iately on ! "w ( 8 pr$ecessor1 ,m>9us ( ,tyre1 & on t ( ,hipp>*us1 ! foll[] & critic ( ,erato/h5es4 ,x 0 ! ambition ( ,ptolemy to describe & repres5t accurately ! surface ( ! oekum5e1 = :i* purpose he took imm5se tr\ble to collect all exi/+ det]m9ations ( ! latitude ( places1 all e/imates ( longitude1 & to make e possible rectification 9 ! e/imates ( di/ances by l& or sea4 ,8 "w 0 ma9ly c>tographical 9 its aim1 & !ory 0 z f> z possible exclud$4 ,! symmetrically plac$ hypo!tical isl&s 9 ! great cont9u\s ocean disappe>$1 & ! oekum5e acquir$ a new =m by ! repres5tation ( ! ,indian ,ocean z a l>g] ,m$it]ranean completely cut (f by l& f ! ,atlantic4 ,! t]ra incognita unit+ ,africa & ,f>!r ,asia 0 an un=tunate hypo!sis :i* help$ to ret>d exploration4 ,ptolemy us$ ! ^w geography to signify ! description ( ! :ole oekum5e on ma!matical pr9ciples1 :ile *orography signifi$ ! full] description ( a p>ticul> region1 & topography ! v detail$ description ( a small] locality4 ,he introduc$ ! simile t geography repres5t$ an arti/'s sket* ( a :ole portrait1 :ile *orography correspond$ to ! c>eful & detail$ draw+ ( an eye or an e>4#e ,! ,caliph al-,mam un 7c4 ,a4,d4 #hae71 ! son & successor ( ,h ar un al-,ra% id1 caus$ an ,arabic v]sion ( ,ptolemy's great a/ronomical "w 7,greek3 ,suntaxis megi/e7 to be made1 :i* is kn[n z ! ,almage/1 ! ^w be+ no?+ m ?an ! ,gr4 ,greek3 megi/e ) ! ,arabic article al prefix$4 ,! geography ( ,ptolemy 0 also kn[n & is con/antly ref]r$ to by ,arab writ]s4 ,! ,arab a/ronom]s measur$ a degree on ! pla9s ( ,mesopotamia1 !reby d$uc+ a fair approximation to ! size ( ! e>?4 ,! caliph's libr>ian1 ,abu ,jaf> ,muhammad ,b5 ,musa1 wrote a geographical "w1 n[ un=tunately lo/1 entitl$ ,rasm el ,arsi 7,a ,description ( ! ,_w71 :i* is (t5 ref]r$ to by subsequ5t writ]s z hav+ be5 compos$ on ! model ( t ( ,ptolemy4 ,geography 9 ! middle ages4 ,! middle ages saw geographical k die out 9 ,*ri/5dom1 al?\gh x reta9$1 ?r\gh ! ,arabic translations ( ,ptolemy1 a c]ta9 vitality 9 ,islam4 ,! v]bal int]pretation ( ,scripture l$ ,lactantius 7c4 ,a4,d4 #cbj7 & o!r ecclesia/ics to d5\nce ! sph]ical !ory ( ! e>? z h]etical4 ,! wret*$ subt]fuge ( ,cosmas 7c4 ,a4,d4 #eej7 to expla9 ! ph5om5a ( ! app>5t movem5ts ( ! sun by means ( an e>? modell$ on ! plan ( ! ,jewi% ,tab]nacle gave place ultimately to ! :eel-maps--! ,t 9 an ,o--:i* rev]t$ to ! primitive ignorance ( ! times ( ,hom] & ,hecataeus4#f ,! j\rney ( ,m>co ,polo1 ! increas+ trade to ! ,ea/ & ! voyages ( ! ,arabs 9 ! ,indian ,ocean prep>$ ! way = ! reacceptance ( ,ptolemy's ideas :5 ! seal$ books ( ! ,greek orig9al 7 translat$ into ,lat9 by ,angelus 9 #adaj4 ,revival ( geography4 ,! old argum5ts ( ,ari/otle & ! old measurem5ts ( ,ptolemy 7 us$ by ,toscanelli & ,columbus 9 urg+ a we/w>d voyage to ,india2 & ma9ly on ?is acc\nt did ! cross+ ( ! ,atlantic rank high] 9 ! hi/ory ( sci5tific geography ?an ! labori\s feel+ out ( ! coa/-l9e ( ,africa4 ,b n until ! voyage ( ,magellan %ook ! scales f ! eyes ( ,europe did mod]n geography beg9 to advance4 ,discov]y _h outrun !ory2 ! ru% ( new facts made ,ptolemy practically obsolete 9 a g5]ation1 aft] hav+ be5 ! f\nt & orig9 ( all geography = a mill5nium4 ,apianus4 ,! e>lie/ evid5ce ( ! re9c>nation ( a s\nd !oretical geography is to be f\nd 9 ! text-books by ,pet] ,apian & ,seba/ian ,mun/]4 ,apian 9 8 ,cosmographicus lib]1 publi%$ 9 #aebd1 & subsequ5tly edit$ & add$ to by ,gemma ,frisius "u ! title ( ,cosmographia1 bas$ ! :ole sci5ce on ma!matics & measurem5t4 ,he foll[$ ,ptolemy closely1 enl>g+ on 8 di/9ction betwe5 geography & *orography1 & express+ ! arti/ic analogy 9 a r\gh diagram4 ,?is sl5d] di/9ction 0 made mu* ( by mo/ subsequ5t writ]s until ,na?anael ,c>p5t] 9 #afbe po9t$ out t ! diff]5ce betwe5 geography & *orography 0 simply "o ( degree1 n ( k9d4 ,mun/]4 ,seba/ian ,mun/]1 on ! o!r h&1 9 8 ,cosmographia univ]salis ( #aedd1 paid no reg>d to ! ma!matical basis ( geography1 b1 foll[+ ! model ( ,/rabo1 describ$ ! _w accord+ to its diff]5t political divisions1 & ent]$ ) great ze/ into ! "q ( ! productions ( c\ntries1 & into ! mann]s & co/umes ( ! v>i\s peoples4 ,?us e>ly comm5c$ ! sep>ation betwe5 :at 7 long call$ ma!matical & political geography1 ! "o subject appeal+ ma9ly to ma!maticians1 ! o!r to hi/orians4 ,?r\gh\t ! #af? & #ag? c5turies ! rapidly accumulat+ /ore ( facts z to ! ext5t1 outl9e & m\nta9 & riv] sy/ems ( ! l&s ( ! e>? 7 put 9 ord] by ! g5]ation ( c>tograph]s ( :i* ,m]cator 0 ! *ief2 b ! writ+s ( ,apian & ,mun/] held ! field = a hundr$ ye>s )\t a s]i\s rival1 unless ! _m annotat$ editions ( ,ptolemy might be s consid]$4 ,mean:ile ! new facts 7 ! subject ( orig9al /udy by philosoph]s & by practical m5 )\t ref]5ce to classical traditions4 ,bacon argu$ ke5ly on geographical matt]s & 0 a lov] ( maps1 9 :i* he obs]v$ & reason$ ^u su* resemblances z t betwe5 ! outl9es ( ,s\? ,am]ica & ,africa4 ,cluv]ius4 ,philip ,cluv]'s ,introductio 9 geographiam univ]sam tam vet]em quam novam 0 publi%$ 9 #afbd4 ,geography he def9$ z ! description ( ! :ole e>?1 s f> z x is kn[n to u4 ,x is di/+ui%$ f cosmography by deal+ ) ! e>? alone1 n ) ! univ]se1 & f *orography & topography by deal+ ) ! :ole e>?1 n ) a c\ntry or a place4 ,! fir/ book1 ( f\rte5 %ort *apt]s1 is conc]n$ ) ! g5]al prop]ties ( ! globe2 ! rema9+ six books treat 9 consid]able detail ( ! c\ntries ( ,europe & ( ! o!r cont95ts4 ,ea* c\ntry is describ$ ) p>ticul> reg>d to its p z well z to its surface1 & ! prom95ce giv5 to ! human elem5t is ( special int]e/4 ,c>p5t]4 ,a little-kn[n book :i* appe>s to h escap$ ! att5tion ( mo/ writ]s on ! hi/ory ( mod]n geography 0 publi%$ at ,ox=d 9 #afbe by ,na?anael ,c>p5t]1 fell[ ( ,exet] ,college1 ) ! title ,geographie del9eat$ =? 9 ,two ,bookes1 conta9+ ! ,sph]icall & ,topicall p>ts !re(4 ,x is discursive 9 its /yle & v]bose2 b1 consid]+ ! p]iod at :i* x appe>$1 x is rem>kable = ! /rong common s5se display$ by ! au?or1 8 comp>ative fre$om f prejudice1 & 8 firm application ( ! me?ods ( sci5tific reason+ to ! int]pretation ( ph5om5a4 ,bas+ 8 "w on ! pr9ciples ( ,ptolemy1 he br+s toge!r illu/rations f ! mo/ rec5t travell]s1 & does n hesitate to take z illu/rative examples ! famili> city ( ,ox=d & 8 native c\nty ( ,devon4 ,he divides geography into ,! ,sph]ical ,"p1 or t = ! /udy ( :i* ma!matics alone is requir$1 & ,! ,topical ,"p1 or ! description ( ! physical relations ( p>ts ( ! e>?'s surface1 pref]r+ ?is division to t fav\r$ by ! anci5t geograph]s--into g5]al & special4 ,x is di/+ui%$ f o!r ,engli% geographical books ( ! p]iod by conf9+ att5tion to ! pr9ciples ( geography1 & n describ+ ! c\ntries ( ! _w4 ,v>5ius4 ,a mu* m important "w 9 ! hi/ory ( geographical me?od is ! ,geographia g5]alis ( ,b]nh>d ,v>5ius1 a ,g]man m$ical doctor ( ,leid51 :o di$ at ! age ( tw5ty-eight 9 #afej1 ! ye> ( ! publication ( 8 book4 ,al?\gh = a "t x 0 lo/ sight ( on ! cont95t1 ,sir ,isaac ,newton ?\ght s highly ( ?is book t he prep>$ an annotat$ edition :i* 0 publi%$ 9 ,cambridge 9 #afgb1 ) ! addition ( ! plates :i* _h be5 plann$ by ,v>5ius1 b n produc$ by ! orig9al publi%]s4 ,! reason :y ?is great man took s mu* c>e 9 correct+ & publi%+ our au?or 01 because he ?\ght him necess>y to be read by 8 audi5ce1 ! "y g5tlem5 ( ,cambridge1 :ile he 0 deliv]+ lectures on ! same subject f ! ,lucasian ,*air4#g ,! treatise ( ,v>5ius is a model ( logical arrangem5t & t]se expression2 x is a "w ( sci5ce & ( g5ius2 "o ( ! few ( t age :i* c /ill be /udi$ ) pr(it4 ,! ,engli% translation r5d]s ! def9ition ?us3 ,geography is t "p ( mix$ ma!matics :i* expla9s ! /ate ( ! e>? & ( its p>ts1 dep5d+ on quantity1 viz4 its figure1 place1 magnitude & motion1 ) ! cele/ial appe>ances1 c4 ,by "s x is tak5 9 too limit$ a s5se1 = a b>e description ( ! sev]al c\ntries2 & by o!rs too ext5sively1 :o along ) su* a description w\ld h _! political con/itution4 ,v>5ius 0 reluctant to include ! human side ( geography 9 8 sy/em1 & only all[$ x z a concession to cu/om1 & 9 ord] to attract read]s by imp>t+ int]e/ to ! /]n] details ( ! sci5ce4 ,8 division ( geography 0 into two p>ts--7i47 ,g5]al or univ]sal1 deal+ ) ! e>? 9 g5]al1 & expla9+ its prop]ties )\t reg>d to p>ticul> c\ntries2 & 7ii47 ,special or p>ticul>1 deal+ ) ea* c\ntry 9 turn f ! *orographical or topographical po9t ( view4 ,g5]al geography 0 divid$ into--7#a7 ! ,absolute "p1 deal+ ) ! =m1 dim5sions1 position & sub/ance ( ! e>?1 ! di/ribution ( l& & wat]1 m\nta9s1 woods & des]ts1 hydrography 7includ+ all ! wat]s ( ! e>?7 & ! atmosph]e2 7#b7 ! ,relative "p1 includ+ ! cele/ial prop]ties1 i4e4 latitude1 climate zones1 longitude1 c42 & 7#c7 ! ,comp>ative "p1 :i* consid]s ! p>ticul>s aris+ f comp>+ "o "p ) ano!r2 b "u ?is head ! que/ions discuss$ 7 longitude1 ! situation & di/ances ( places1 & navigation4 ,v>5ius does n treat ( special geography1 b gives a s*eme = x "u ?ree heads--7#a7 ,t]re/rial1 includ+ position1 outl9e1 b\nd>ies1 m\nta9s1 m9es1 woods & des]ts1 wat]s1 f]tility & fruits1 & liv+ creatures2 7#b7 ,cele/ial1 includ+ appe>ance ( ! heav5s & ! climate2 7#c7 ,human1 b ?is 0 add$ out ( def]5ce to popul> usage4 ,?is sy/em ( geography f\nd$ a new epo*1 & ! book--translat$ into ,engli%1 ,dut* & ,fr5*--0 ! un*all5g$ /&>d = m ?an a c5tury4 ,! framework 0 capable ( accommodat+ itself to new facts1 & 0 inde$ f> 9 advance ( ! k ( ! p]iod4 ,! me?od includ$ a recognition ( ! causes & effects ( ph5om5a z well z ! m]e fact ( _! occurr5ce1 & = ! fir/ "t ! importance ( ! v]tical relief ( ! l& 0 fairly recogniz$4 ,! physical side ( geography cont9u$ to be elaborat$ aft] ,v>5ius's me?ods1 :ile ! hi/orical side 0 develop$ sep>ately4 ,bo? bran*es1 al?\gh enri*$ by new facts1 rema9$ /ation>y s f> z me?od is conc]n$ until ne>ly ! end ( ! #ah? c5tury4 ,! compilation ( geography books by un9/ruct$ writ]s l$ to ! p]nici\s habit1 :i* is n yet :olly ov]come1 ( r$uc+ ! g5]al or physical "p to a few pages ( conc5trat$ in=mation1 & exp&+ ! p>ticul> or political "p by includ+ unrevis$ travell]s' /ories & uncritical descriptions ( ! v>i\s c\ntries ( ! _w4 ,su* books 7 9 fact n geography1 b m]ely compress$ travel4 ,b]gman4 ,! next m>k$ advance 9 ! !ory ( geography may be tak5 z ! ne>ly simultane\s /udies ( ! physical e>? c>ri$ out by ! ,sw$i% *emi/1 ,torb]n ,b]gman1 act+ "u ! impulse ( ,l9naeus1 & by ! ,g]man philosoph]1 ,immanuel ,kant4 ,b]gman's ,physical ,description ( ! ,e>? 0 publi%$ 9 ,sw$i% 9 #agff1 & translat$ into ,engli% 9 #aggb & into ,g]man 9 #aggd4 ,x is a pla91 /raight=w>d description ( ! globe1 & ( ! v>i\s ph5om5a ( ! surface1 deal+ only ) def9itely asc]ta9$ facts 9 ! natural ord] ( _! relation%ips1 b avoid+ any sy/ematic classification or ev5 def9itions ( t]ms4 ,kant4 ,! problems ( geography _h be5 light5$ by ! de/ructive criticism ( ! ,fr5* c>tograph] ,d',anville 7:o _h purg$ ! map ( ! _w ( ! la/ remnants ( traditional fact unv]ifi$ by mod]n obs]vations7 & r5d]$ ri*] by ! dawn ( ! new era ( sci5tific travel1 :5 ,kant br\ght 8 logical p[]s to be> ^u !m4 ,kant's lectures on physical geography 7 deliv]$ 9 ! univ]sity ( ,konigsb]g f #agfe onw>ds4#h ,geography appeal$ to him z a valuable educational discipl9e1 ! jo9t f\ndation ) an?ropology ( t k ( ! _w :i* 0 ! result ( reason & exp]i5ce4 ,9 ?is connexion he divid$ ! communication ( exp]i5ce f "o p]son to ano!r into two categories--! n>rative or hi/orical & ! descriptive or geographical2 bo? hi/ory & geography be+ view$ z descriptions1 ! =m] a description 9 ord] ( "t1 ! latt] a description 9 ord] ( space4 ,physical geography he view$ z a summ>y ( nature1 ! basis n only ( hi/ory b also ( all ! o!r possible geographies1 ( :i* he enum]ates five1 viz4 7#a7 ,ma!matical geography1 :i* deals ) ! =m1 size & movem5ts ( ! e>? & its place 9 ! sol> sy/em2 7#b7 ,moral geography1 or an acc\nt ( ! diff]5t cu/oms & *>act]s ( mank9d accord+ to ! region !y inhabit2 7#c7 ,political geography1 ! divisions accord+ to _! organiz$ gov]nm5ts2 7#d7 ,m]cantile geography1 deal+ ) ! trade 9 ! surplus products ( c\ntries2 7#e7 ,!ological geography1 or ! di/ribution ( religions4 ,"h !re is a cle> & =mal /atem5t ( ! int]action & causal relation ( all ! ph5om5a ( di/ribution on ! e>?'s surface1 includ+ ! influ5ce ( physical geography ^u ! v>i\s activities ( mank9d f ! l[e/ to ! highe/4 ,not)/&+ ! =m ( ?is classification1 ,kant himself treats ma!matical geography z prelim9>y to1 & !re=e n dep5d5t on1 physical geography4 ,physical geography itself is divid$ into two p>ts3 a g5]al1 :i* has to d ) ! e>? & all t belongs to x--wat]1 air & l&2 & a p>ticul>1 :i* deals ) special products ( ! e>?--mank9d1 animals1 plants & m9]als4 ,p>ticul> importance is giv5 to ! v]tical relief ( ! l&1 on :i* ! v>i\s bran*es ( human geography are %[n to dep5d4 ,humboldt4 ,alex&] von ,humboldt 7#agfi-ahei7 0 ! fir/ mod]n geograph] to become a great travell]1 & ?us to acquire an ext5sive /ock ( fir/-h& in=mation on :i* an improv$ sy/em ( geography might be f\nd$4 ,! impulse giv5 to ! /udy ( natural hi/ory by ! example ( ,l9naeus2 ! results br\ght back by ,sir ,joseph ,banks1 ,dr ,sol&] & ! two ,=/]s1 :o accompani$ ,cook 9 8 voyages ( discov]y2 ! /udies ( ,de ,saussure 9 ! ,alps1 & ! li/s ( desid]ata 9 physical geography drawn up by t inve/igator1 comb9$ to prep>e ! way = ,humboldt4 ,! !ory ( geography 0 advanc$ by ,humboldt ma9ly by 8 insi/5ce on ! great pr9ciple ( ! unity ( nature4 ,he br\ght all ! obs]vable ?+s1 :i* ! eag] collectors ( ! previ\s c5tury _h be5 heap+ toge!r reg>dless ( ord] or sy/em1 into relation ) ! v]tical relief & ! horizontal =ms ( ! e>?'s surface4 ,?us he demon/rat$ t ! =ms ( ! l& ex]cise a directive & det]m9+ influ5ce on climate1 plant life1 animal life & on man himself4 ,?is 0 no new idea2 x _h be5 famili> = c5turies 9 a less def9ite =m1 d$uc$ f a priori consid]ations1 & s f> z reg>ds ! influ5ce ( surr\nd+ circum/ances ^u man1 ,kant _h already giv5 x full expression4 ,humboldt's concrete illu/rations & ! rem>kable p[] ( 8 p]sonality enabl$ him to en=ce ^! pr9ciples 9 a way t produc$ an imm$iate & la/+ effect4 ,! treatises on physical geography by ,mrs ,m>y ,som]ville & ,sir ,john ,h]s*el 7! latt] writt5 = ! eigh? edition ( ! ,encyclopa$ia ,britannica7 %[$ ! effect produc$ 9 ,great ,brita9 by ! /imulus ( ,humboldt's "w4 ,ritt]4 ,humboldt's contempor>y1 ,c>l ,ritt] 7#aggi-ahei71 ext5d$ & dissem9at$ ! same views1 & 9 8 int]pretation ( ,comp>ative ,geography he laid /ress on ! importance ( =m+ conclusions1 n f ! /udy ( "o region by itself1 b f ! comp>ison ( ! ph5om5a ( _m places4 ,impress$ by ! influ5ce ( t]re/rial relief & climate on human movem5ts1 ,ritt] 0 l$ deep] & deep] into ! /udy ( hi/ory & ar*aeology4 ,8 monum5tal ,v]glei*5de ,geographie1 :i* 0 to h made ! :ole _w its !me1 di$ out 9 a wild]ness ( detail 9 tw5ty-"o volumes be=e x _h cov]$ m ( ! e>?'s surface ?an ,asia & a portion ( ,africa4 ,"s ( 8 foll[]s %[$ a t5d5cy to look on geography r z an auxili>y to hi/ory ?an z a /udy ( intr9sic wor?4 ,geography z a natural sci5ce4 ,dur+ ! rapid developm5t ( physical geography _m bran*es ( ! /udy ( nature1 :i* _h be5 includ$ 9 ! cosmography ( ! e>ly writ]s1 ! physiography ( ,l9naeus & ev5 ! ,erdkunde ( ,ritt]1 _h be5 s mu* advanc$ by ! lab\rs ( speciali/s t _! connexion 0 apt to be =gott54 ,?us geology1 meteorology1 oceanography & an?ropology develop$ into di/9ct sci5ces4 ,! absurd attempt 01 & sometimes is /ill1 made by geograph]s to include all natural sci5ce 9 geography2 b x is m common = speciali/s 9 ! v>i\s detail$ sci5ces to ?9k1 & sometimes to ass]t1 t ! gr\nd ( physical geography is n[ fully occupi$ by ^! sci5ces4 ,political geography has be5 too (t5 look$ on f bo? sides z a m]e summ>y ( guide-book k1 useful 9 ! s*oolroom1 a poor relation ( physical geography t x 0 r>ely necess>y to recognize4 ,! sci5ce ( geography1 pass$ on f antiquity by ,ptolemy1 re-e/abli%$ by ,v>5ius & ,newton1 & sy/ematiz$ by ,kant1 includ$ )9 itself def9ite aspects ( all ^? t]re/rial ph5om5a :i* are n[ treat$ exhau/ively "u ! heads ( geology1 meteorology1 oceanography & an?ropology2 & ! inclusion ( ! requisite portions ( ! p]fect$ results ( ^! sci5ces 9 geography is simply ! ga!r+ 9 ( fruit matur$ f ! se$ scatt]$ by geography itself4 ,! /udy ( geography 0 advanc$ by improvem5ts 9 c>tography 7see ,,map71 n only 9 ! me?ods ( survey & projection1 b 9 ! repres5tation ( ! ?ird dim5sion by means ( cont\r l9es introduc$ by ,philippe ,bua*e 9 #agcg1 & ! m rem>kable because less obvi\s inv5tion ( iso!rms introduc$ by ,humboldt 9 #ahag4 ,! teleological argum5t 9 geography4 ,! argum5t f design _h be5 a fav\rite =m ( reason+ among/ ,*ri/ian !ologians1 &1 z work$ out by ,paley 9 8 ,natural ,!ology1 x s]v$ ! useful purpose ( emphasiz+ ! fitness :i* exi/s betwe5 all ! inhabitants ( ! e>? & _! physical environm5t4 ,x 0 held t ! e>? _h be5 creat$ s z to fit ! wants ( man 9 e p>ticul>4 ,?is argum5t 0 tacitly accept$ or explicitly av[$ by almo/ e writ] on ! !ory ( geography1 & ,c>l ,ritt] di/9ctly recogniz$ & adopt$ x z ! unify+ pr9ciple ( 8 sy/em4 ,z a /ud5t ( nature1 h[ev]1 he did n fail to see1 & z pr(essor ( geography he always taught1 t man 0 9 v l>ge measure condition$ by 8 physical environm5t4 ,! app>5t opposition ( ! obs]v$ fact to ! assign$ !ory he ov]came by look+ ^u ! =ms ( ! l& & ! arrangem5t ( l& & sea z in/rum5ts ( ,div9e ,provid5ce = guid+ ! de/9y z well z = supply+ ! requirem5ts ( man4 ,?is 0 ! c5tral !me ( ,ritt]'s philosophy2 8 religion & 8 geography 7 "o1 & ! consequ5t f]v\r ) :i* he pursu$ 8 mission goes f> to acc\nt = ! imm5se influ5ce he acquir$ 9 ,g]many4 ,! !ory ( evolution 9 geography4 ,! evolution>y !ory1 m ?an h9t$ at 9 ,kant's ,physical ,geography1 has1 s9ce ! writ+s ( ,*>les ,d>w91 become ! unify+ pr9ciple 9 geography4 ,! conception ( ! developm5t ( ! plan ( ! e>? f ! fir/ cool+ ( ! surface ( ! planet ?r\gh\t ! long geological p]iods1 ! guid+ p[] ( environm5t on ! circulation ( wat] & ( air1 on ! di/ribution ( plants & animals1 & f9ally on ! movem5ts ( man1 give to geography a philosophical dignity & a sci5tific complet5ess :i* x nev] previ\sly possess$4 ,! influ5ce ( environm5t on ! organism may n be q s pot5t z x 0 once believ$ to be1 9 ! writ+s ( ,buckle1 = in/ance1#i & c]ta9ly man1 ! ultimate t]m 9 ! s]ies1 reacts ^u & greatly modifies 8 environm5t2 yet ! fact t environm5t does influ5ce all di/ributions is e/abli%$ beyond ! possibility ( d\bt4 ,9 ?is way also ! position ( geography1 at ! po9t :]e physical sci5ce meets & m+les ) m5tal sci5ce1 is expla9$ & ju/ifi$4 ,! *ange :i* took place dur+ ! #ai? c5tury 9 ! sub/ance & /yle ( geography may be well se5 by comp>+ ! eight volumes ( ,malte-,brun's ,geographie univ]selle 7,p>is1 #ahab-ahbi7 ) ! tw5ty-"o volumes ( ,reclus's ,geographie univ]selle 7,p>is1 #ahgf-ahie74 ,9 e/imat+ ! influ5ce ( rec5t writ]s on geography x is usual to assign to ,osc> ,pes*el 7#ahbf-ahge7 ! cr$it ( hav+ correct$ ! prepond]ance :i* ,ritt] gave to ! hi/orical elem5t1 & ( re/or+ physical geography to its old pre-em95ce4#aj ,z a matt] ( fact1 ea* ( ! lead+ mod]n expon5ts ( !oretical geography--su* z ,f]d9& von ,ri*?(51 ,h]mann ,wagn]1 ,fri$ri* ,ratzel1 ,william ,m4 ,davis1 ,a4 ,p5ck1 ,a4 de ,lapp>5t & ,elisee ,reclus--has 8 individual po9t ( view1 "o devot+ m att5tion to ! results ( geological processes1 ano!r to an?ropological conditions1 & ! re/ view+ ! subject 9 v>i\s bl5d+s ( ! extreme lights4 ,! two conceptions :i* may n[ be said to animate ! !ory ( geography are ! g5etic1 :i* dep5ds ^u processes ( orig91 & ! morphological1 :i* dep5ds on facts ( =m & di/ribution4 ,,progress ,,( ,,geographical ,,discov]y ,exploration & geographical discov]y mu/ h />t$ f m ?an "o c5tre1 & to deal ju/ly ) ! matt] "o ought to treat ( ^! sep>ately 9 ! e>ly ages be=e ! :ole civiliz$ _w 0 b\nd toge!r by ! bonds ( mod]n int]communication4 ,at ! lea/ !re %\ld be "s consid]ation ( f\r sep>ate sy/ems ( discov]y--! ,ea/]n1 9 :i* ,*9ese & ,japanese explor]s acquir$ k ( ! geography ( ,asia1 & felt _! way t[>ds ,europe & ,am]ica2 ! ,we/]n1 9 :i* ! dom9ant races ( ! ,mexican & ,s\? ,am]ican plateaus ext5d$ _! k ( ! ,am]ican cont95t be=e ,columbus2 ! ,polynesian1 9 :i* ! conqu]+ races ( ! ,pacific ,isl&s f\nd _! way f gr\p to gr\p2 & ! ,m$it]ranean4 ,= "s ( ^! we h no c]ta9 in=mation1 & reg>d+ o!rs ! tales n>rat$ 9 ! e>ly records are s h>d to reconcile ) pres5t k t !y are bett] fitt$ to be ! battle-gr\nd ( s*ol>s *ampion+ rival !ories ?an ! basis ( def9ite hi/ory4 ,s x has come ab\t t ! only practicable hi/ory ( geographical exploration />ts f ! ,m$it]ranean c5tre1 ! fir/ home ( t civilization :i* has come to be kn[n z ,european1 ?\gh its field ( activity has long s9ce ov]spread ! habitable l& ( bo? temp]ate zones1 ea/]n ,asia alone 9 "p except$4 ,f all c5tres ! lead+ motives ( exploration 7 probably ! same--comm]cial int]c\rse1 w>like op]ations1 :e!r result+ 9 conque/ or 9 flight1 religi\s zeal express$ 9 pilgrimages or mission>y j\rneys1 or1 f ! o!r side1 ! avoidance ( p]secution1 &1 m p>ticul>ly 9 lat] ye>s1 ! advancem5t ( k = its own sake4 ,at diff]5t times "o or ! o!r motive pr$om9at$4 ,be=e ! #ad? c5tury ,b4,c4 ! w>rior k+s ( ,egypt _h c>ri$ ! p[] ( _! arms s\?w>d f ! delta ( ! ,nile well-nigh to its s\rce1 & ea/w>d to ! conf9es ( ,assyria4 ,! hi]oglyphic inscriptions ( ,egypt & ! cunei=m inscriptions ( ,assyria are ri* 9 records ( ! movem5ts & a*ievem5ts ( armies1 ! conque/ ( t[ns & ! subjugation ( peoples2 b ?\gh _m ( ! record$ sites h be5 id5tifi$1 _! discov]y by w&]+ armies 0 isolat$ f _! subsequ5t hi/ory & ne$ n conc]n u "h4 ,! ,pho5icians4 ,! ,pho5icians are ! e>lie/ ,m$it]ranean p 9 ! consecutive *a9 ( geographical discov]y :i* jo9s pre-hi/oric "t ) ! pres5t4 ,f ,sidon1 & lat] f its m fam\s rival ,tyre1 ! m]*ant adv5tur]s ( ,pho5icia explor$ & coloniz$ ! coa/s ( ! ,m$it]ranean & f>$ =? into ! ocean beyond4 ,!y trad$ also on ! ,r$ sea1 & op5$ up regul> traffic ) ,india z well z ) ! ports ( ! s\? & we/1 s t x 0 natural = ,solomon to employ ! m]*ant navies ( ,tyre 9 8 ov]sea trade4 ,! we/]n emporium kn[n 9 ! scriptures z ,t>%i% 0 probably situat$ 9 ! s\? ( ,spa91 possibly at ,cadiz1 al?\gh "s writ]s cont5d t x 0 ,c>?age 9 ,nor? ,africa4 ,/ill m div]sity ( op9ion prevails z to ! s\!rn gold-export+ port ( ,ophir1 :i* "s s*ol>s place 9 ,arabia1 o!rs at "o or ano!r po9t on ! ea/ coa/ ( ,africa4 ,:e!r associat$ ) ! exploitation ( ,ophir 7q4v47 or n ! fir/ great voyage ( ,african discov]y appe>s to h be5 accompli%$ by ! ,pho5icians sail+ ! ,r$ ,sea4 ,h]odotus 7himself a notable travell] 9 ! #e? c5tury ,b4,c47 relates t ! ,egyptian k+ ,ne*o ( ! ,x,x,v,ith ,dyna/y 7c4 #fjj ,b4,c47 built a fleet on ! ,r$ ,sea1 & confid$ x to ,pho5ician sailors ) ! ord]s to sail s\?w>d & return to ,egypt by ! ,pill>s ( ,h]cules & ! ,m$it]ranean sea4 ,accord+ to ! tradition1 :i* ,h]odotus quotes sceptically1 ?is 0 accompli%$2 b ! /ory is too vague to be accept$ z m ?an a possibility4 ,! great ,pho5ician colony ( ,c>?age1 f\nd$ be=e #hjj ,b4,c41 p]petuat$ ! comm]cial ent]prise ( ! p>5t /ate1 & ext5d$ ! sph]e ( practical trade to ! ocean %ores ( ,africa & ,europe4 ,! mo/ celebrat$ voyage ( antiquity und]tak5 = ! express purpose ( discov]y 0 t fitt$ out by ! s5ate ( ,c>?age "u ! comm& ( ,hanno1 ) ! int5tion ( f\nd+ new colonies along ! we/ coa/ ( ,africa4 ,accord+ to ,pl9y1 ! only au?ority on ?is po9t1 ! p]iod ( ! voyage 0 t ( ! greate/ prosp]ity ( ,c>?age1 :i* may be tak5 z some:]e betwe5 #egj & #dhj ,b4,c4 ,! ext5t ( ?is voyage is d\btful1 b x seems probable t ! f>!/ po9t rea*$ 0 on ! ea/-runn+ coa/ :i* b\nds ! ,gulf ( ,gu9ea on ! nor?4 ,himilco1 a contempor>y ( ,hanno1 0 *>g$ ) an exp$ition along ! we/ coa/ ( ,ib]ia nor?w>d1 & z f> z ! unc]ta9 ref]5ces to ?is voyage c be und]/ood1 he seems to h pass$ ! ,bay ( ,biscay & possibly sight$ ! coa/ ( ,engl&4 ,! ,greeks4 ,! sea p[] ( ! ,greek communities on ! coa/ ( ,asia ,m9or & 9 ! ,ar*ipelago began to be a =midable rival to ! ,pho5ician soon aft] ! "t ( ,hanno & ,himilco1 & peculi> int]e/ atta*es to ! fir/ record$ ,greek voyage beyond ! ,pill>s ( ,h]cules4 ,py!as1 a navigator ( ! ,phocean colony ( ,massilia 7,m>seilles71 det]m9$ ! latitude ( t port ) consid]able precision by ! some:at clumsy me?od ( asc]ta9+ ! l5g? ( ! longe/ day1 & :51 ab\t #ccj ,b4,c41 he set out on exploration to ! nor?w>d 9 se>* ( ! l&s :5ce came gold1 t9 & amb]1 he foll[$ ?is sy/em ( asc]ta9+ 8 position f "t to "t4 ,if on ea* occasion he himself made ! obs]vations 8 voyage mu/ h ext5d$ ov] six ye>s2 b x is n impossible t he asc]ta9$ ! approximate l5g? ( ! longe/ day 9 "s cases by que/ion+ ! natives4 ,py!as1 ^: own n>rative is n pres]v$1 coa/$ ! ,bay ( ,biscay1 sail$ up ! ,engli% ,*annel & foll[$ ! coa/ ( ,brita9 to its mo/ nor!rly po9t4 ,beyond ?is he spoke ( a l& call$ ,?ule1 :i*1 if 8 e/imate ( ! l5g? ( ! longe/ day is correct1 may h be5 ,%etl&1 b 0 possibly ,icel&2 & f "s confus$ /atem5ts z to a sea :i* c\ld n be sail$ ?r\gh1 x has be5 assum$ t ,py!as 0 ! fir/ ( ! ,greeks to obta9 direct k ( ! ,arctic regions4 ,dur+ ?is or a second voyage ,py!as ent]$ ! ,baltic1 discov]$ ! coa/s :]e amb] is obta9$ & return$ to ! ,m$it]ranean4 ,x does n seem t any m>itime trade foll[$ ^! discov]ies1 & inde$ x is d\btful :e!r 8 contempor>ies accept$ ! tru? ( ,py!as's n>rative2 ,/rabo f\r hundr$ ye>s lat] c]ta9ly did n1 b ! critical /udies ( mod]n s*ol>s h rehabilitat$ ! ,massilian explor]4 ,alex&] ! ,great4 ,! ,greco-,p]sian w>s _h made ! remot] p>ts ( ,asia ,m9or m ?an a "n to ! ,greek geograph]s be=e ! "t ( ,alex&] ! ,great1 b ! campaigns ( t conqu]or f #cbi to #cbe ,b4,c4 op5$ up ! great] ,asia to ! k ( ,europe4 ,8 armies cross$ ! pla9s beyond ! ,caspian1 p5etrat$ ! wild m\nta9 passes nor?-we/ ( ,india1 & did n turn back until !y _h ent]$ on ! ,indo-,gangetic pla94 ,?is 0 "o ( ! few great epo*s ( geographical discov]y4 ,! _w 0 h5ce=? view$ z a v l>ge place /ret*+ f> on e side beyond ! ,midl& or ,m$it]ranean ,sea1 & ! l& j\rney ( ,alex&] result$ 9 a voyage ( discov]y 9 ! out] ocean f ! m\? ( ! ,indus to t ( ! ,tigris1 ?us op5+ direct int]c\rse betwe5 ,grecian & ,h9du civilization4 ,! ,greeks :o accompani$ ,alex&] describ$ ) c>e ! t[ns & villages1 ! products & ! aspect ( ! c\ntry4 ,! conqu]or also int5d$ to op5 up trade by sea betwe5 ,europe & ,india1 & ! n>rative ( 8 g5]al ,ne>*us records ?is fam\s voyage ( discov]y1 ! detail$ acc\nts ( ! *ief pilot ,onesicritus be+ lo/4 ,at ! beg9n+ ( ,octob] #cbf ,b4,c4 ,ne>*us left ! ,indus ) 8 fleet1 & ! an*orages s\ght = ea* night are c>efully record$4 ,he ent]$ ! ,p]sian ,gulf1 & rejo9$ ,alex&] at ,susa1 :5 he 0 ord]$ to prep>e ano!r exp$ition = ! circumnavigation ( ,arabia4 ,alex&] di$ at ,babylon 9 #cbc ,b4,c41 & ! fleet 0 disp]s$ )\t mak+ ! voyage4 ,! dyna/ies f\nd$ by ,alex&]'s g5]als1 ,seleucus1 ,antio*us & ,ptolemy1 enc\rag$ ! same _s ( ent]prise :i* _! ma/] _h fo/]$1 & ext5d$ geographical k 9 sev]al directions4 ,seleucus ,nicator e/abli%$ ! ,greco-,bactrian empire & cont9u$ ! int]c\rse ) ,india4 ,au!ntic in=mation respect+ ! great valley ( ! ,ganges 0 suppli$ by ,mega/h5es1 an ambassador s5t by ,seleucus1 :o rea*$ ! remote city ( ,patali-putra1 ! mod]n ,patna4 ,! ,ptolemies4 ,! ,ptolemies 9 ,egypt %[$ equal anxiety to ext5d ! b\nds ( geographical k4 ,ptolemy ,eu]getes 7#bdg-bbb ,b4,c47 r5d]$ ! greate/ s]vice to geography by ! protection & enc\ragem5t ( ,erato/h5es1 ^: lab\rs gave ! fir/ approximate k ( ! true size ( ! sph]ical e>?4 ,! second ,eu]getes & 8 successor ,ptolemy ,la?yrus 7#aah-aae ,b4,c47 furni%$ ,eudoxus ) a fleet to explore ! ,arabian sea4 ,aft] two successful voyages1 ,eudoxus1 impress$ ) ! idea t ,africa 0 surr\nd$ by ocean on ! s\?1 left ! ,egyptian s]vice1 & proce$$ to ,cadiz & o!r ,m$it]ranean c5tres ( trade seek+ a patron :o w\ld f9ance an exp$ition = ! purpose ( ,african discov]y2 & we le>n f ,/rabo t ! vet]an explor] made at lea/ two voyages s\?w>d along ! coa/ ( ,africa4 ,! ,ptolemies cont9u$ to s5d fleets annually f _! ,r$ ,sea ports ( ,b]5ice & ,myos ,hormus to ,arabia1 z well z to ports on ! coa/s ( ,africa & ,india4 ,! ,romans4 ,! ,romans did n enc\rage navigation & comm]ce ) ! same ard\r z _! pr$ecessors2 /ill ! luxury ( ,rome1 :i* gave rise to dem&s = ! v>i$ products ( all ! c\ntries ( ! kn[n _w1 l$ to an active trade bo? by %ips & c>avans4 ,b x 0 ! milit>y g5ius ( ,rome1 & ! ambition = univ]sal empire1 :i* l$1 n only to ! discov]y1 b also to ! survey ( ne>ly all ,europe1 & ( l>ge tracts 9 ,asia & ,africa4 ,e new w> produc$ a new survey & it9]>y ( ! c\ntries :i* 7 conqu]$1 & add$ "o m to ! imp]i%able roads t l$ f e qu>t] ( ! kn[n _w to ,rome4 ,9 ! height ( _! p[] ! ,romans _h survey$ & explor$ all ! coa/s ( ! ,m$it]ranean1 ,italy1 ,greece1 ! ,balkan ,p59sula1 ,spa91 ,gaul1 we/]n ,g]many & s\!rn ,brita94 ,9 ,africa _! empire includ$ ,egypt1 ,c>?age1 ,numidia & ,mauritania4 ,9 ,asia !y held ,asia ,m9or & ,syria1 _h s5t exp$itions into ,arabia1 & 7 acqua9t$ ) ! m di/ant c\ntries =m]ly invad$ by ,alex&]1 includ+ ,p]sia1 ,scy?ia1 ,bactria & ,india4 ,roman int]c\rse ) ,india especially l$ to ! ext5sion ( geographical k4 ,be=e ! ,roman legions 7 s5t into a new region to ext5d ! limits ( ! empire1 x 0 usual to s5d out explor+ exp$itions to report z to ! nature ( ! c\ntry4 ,x is n>rat$ by ,pl9y & ,s5eca t ! emp]or ,n]o s5t out two c5turions on su* a mission t[>ds ! s\rce ( ! ,nile 7probably ab\t ,a4,d4 #fj71 & t ! travell]s pu%$ s\?w>ds until !y rea*$ va/ m>%es ?r\gh :i* !y c\ld n make _! way ei!r on foot or 9 boats4 ,?is seems to indicate t !y _h p5etrat$ to ab\t #i deg4 ,n4 ,%ortly be=e ,a4,d4 #gi ,hippalus took advantage ( ! regul> alt]nation ( ! monsoons to make ! voyage f ! ,r$ ,sea to ,india across ! op5 ocean out ( sight ( l&4 ,ev5 ?\gh ?is sea-r\te 0 kn[n1 ! au?or ( ! ,p]iplus ( ! ,ery?raean ,sea1 publi%$ aft] ! "t ( ,pl9y1 recites ! old it9]>y ar\nd ! coa/ ( ! ,arabian ,gulf4 ,x 01 h[ev]1 9 ! reigns ( ,sev]us & 8 imm$iate successors t ,roman int]c\rse ) ,india 0 at its height1 & f ! writ+s ( ,pausanias 7c4 #agd7 x appe>s t direct communication betwe5 ,rome & ,*9a _h already tak5 place4 ,aft] ! division ( ! ,roman empire1 ,con/ant9ople became ! la/ refuge ( le>n+1 arts & ta/e2 :ile ,alex&ria cont9u$ to be ! emporium :5ce 7 import$ ! commodities ( ! ,ea/4 ,! emp]or ,ju/9ian 7#dhc-efe71 9 ^: reign ! greatness ( ! ,ea/]n empire culm9at$1 s5t two ,ne/orian monks to ,*9a1 :o return$ ) eggs ( ! silkworm conceal$ 9 a holl[ cane1 & ?us silk manufactures 7 e/abli%$ 9 ! ,peloponnesus & ! ,greek isl&s4 ,x 0 also 9 ! reign ( ,ju/9ian t ,cosmas ,indicopleu/es1 an ,egyptian m]*ant1 made sev]al voyages1 & aft]w>ds compos$ 8 ,greek3 ,*ri/ianike topographia 7,*ri/ian ,topography71 conta9+1 9 addition to 8 absurd cosmogony1 a tol]able description ( ,india4 ,! ,arabs4 ,! great outbur/ ( ,mahomm$an conque/ 9 ! #g? c5tury 0 foll[$ by ! ,arab civilization1 hav+ its c5tres at ,bagdad & ,cordova1 9 connexion ) :i* geography aga9 receiv$ a %>e ( att5tion4 ,! works ( ! anci5t ,greek geograph]s 7 translat$ into ,arabic1 & />t+ ) a s\nd basis ( !oretical k1 exploration once m made progress4 ,f ! #i? to ! #ac? c5tury intellig5t ,arab travell]s wrote acc\nts ( :at !y _h se5 & he>d 9 di/ant l&s4 ,! e>lie/ ,arabian travell] ^: obs]vations h come d[n to u is ! m]*ant ,sulaiman1 :o emb>k$ 9 ! ,p]sian ,gulf & made sev]al voyages to ,india & ,*9a1 9 ! middle ( ! #i? c5tury4 ,abu ,zaid also wrote on ,india1 & 8 "w is ! mo/ important t we possess be=e ! epo*-mak+ discov]ies ( ,m>co ,polo4 ,masudi1 a great travell] :o knew f p]sonal exp]i5ce all ! c\ntries betwe5 ,spa9 & ,*9a1 describ$ ! pla9s1 m\nta9s & seas1 ! dyna/ies & peoples1 9 8 ,mead[s ( ,gold1 an ab/ract made by himself ( 8 l>g] "w ,news ( ! ,"t4 ,he di$ 9 #ief1 & 0 kn[n1 f ! compreh5siv5ess ( 8 survey1 z ! ,pl9y ( ! ,ea/4 ,among/ 8 contempor>ies 7 ,i/akhri1 :o travell$ ?r\gh all ! ,mahomm$an c\ntries & wrote 8 ,book ( ,climates 9 #iej1 & ,ibn ,haukal1 ^: ,book ( ,roads & ,k+doms1 bas$ on ! "w ( ,i/akhri1 0 writt5 9 #igf4 ,idrisi1 ! be/ kn[n ( ! ,arabian geographical au?ors1 aft] travell+ f> & wide 9 ! fir/ half ( ! #ab? c5tury1 settl$ 9 ,sicily1 :]e he wrote a treatise descriptive ( an armill>y sph]e :i* he _h con/ruct$ = ,rog] ,,ii41 ! ,norman k+1 & 9 ?is "w he incorporat$ all accessible results ( contempor>y travel4 ,! ,nor?m54 ,! ,nor?m5 ( ,d5m>k & ,norway1 ^: piratical adv5tures 7 ! t]ror ( all ! coa/s ( ,europe1 & :o e/abli%$ !mselves 9 ,great ,brita9 & ,irel&1 9 ,france & ,sicily1 7 also geographical explor]s 9 _! r\gh b practical way dur+ ! d>ke/ p]iod ( ! middle ages4 ,all ,nor?m5 7 n b5t on rap9e & plund]2 _m 7 peaceful m]*ants4 ,alfr$ ! ,great1 k+ ( ! ,saxons 9 ,engl&1 n only educat$ 8 p 9 ! le>n+ ( ! pa/ ages2 he ins]t$ 9 ! geographical works he translat$ _m n>ratives ( ! travel ( 8 own "t4 ,?us he plac$ on record ! voyages ( ! m]*ant ,ulf/5 9 ! ,baltic1 includ+ p>ticul>s ( ! geography ( ,g]many4 ,& 9 p>ticul> he told ( ! rem>kable voyage ( ,o!r1 a ,norwegian ( ,helgel&1 :o 0 ! fir/ au!ntic ,arctic explor]1 ! fir/ to tell ( ! r\nd+ ( ! ,nor? ,cape & ! sight ( ! midnight sun4 ,?is voyage ( ! middle ( ! #i? c5tury des]ves to be held 9 happy memory1 = x unites ! fir/ ,norwegian pol> explor] ) ! fir/ ,engli% collector ( travels4 ,sc&9avian m]*ants br\ght ! products ( ,india to ,engl& & ,irel&4 ,f ! #h? to ! #aa? c5tury a comm]cial r\te f ,india pass$ ?r\gh ,novgorod to ! ,baltic1 & ,arabian co9s f\nd 9 ,sw$51 & p>ticul>ly 9 ! isl& ( ,gotl&1 prove h[ closely ! ent]prise ( ! ,nor?m5 & ( ! ,arabs int]tw9$4 ,five-six?s ( ^! co9s pres]v$ at ,/ockholm 7 f ! m9ts ( ! ,samanian dyna/y1 :i* reign$ 9 ,khorasan & ,transoxiana f ab\t ,a4,d4 #ijj to #ajjj4 ,x 0 ! trade ) ! ,ea/ t orig9ally gave importance to ! city ( ,visby 9 ,gotl&4 ,9 ! end ( ! #i? c5tury ,icel& 0 coloniz$ f ,norway2 & ab\t #ihe ! intrepid vik+1 ,eric ! ,r$1 discov]$ ,gre5l&1 & induc$ "s ( 8 ,icel&ic c\ntrym5 to settle on its inhospitable %ores4 ,8 son1 ,leif ,ericsson1 & o!rs ( 8 foll[]s 7 conc]n$ 9 ! discov]y ( ! ,nor? ,am]ican coa/ 7see ,,v9l&71 :i*1 b = ! isolation ( ,icel& f ! c5tres ( ,european awak5+1 w\ld h _h mom5t\s consequ5ces4 ,z ?+s 71 ! importance ( ?is discov]y pass$ unrecogniz$4 ,! /ory ( two ,v5etians1 ,nicolo & ,antonio ,z5o1 :o gave a vague acc\nt ( voyages 9 ! nor!rn seas 9 ! end ( ! #ac? c5tury1 is no long] to be accept$ z hi/ory4 ,close ( ! d>k ages4 ,at l5g? ! long p]iod ( b>b>ism :i* accompani$ & foll[$ ! fall ( ! ,roman empire drew to a close 9 ,europe4 ,! ,crusades _h a fav\rable influ5ce on ! intellectual /ate ( ! ,we/]n nations4 ,int]e/+ regions1 kn[n only by ! scant reports ( pilgrims1 7 made ! objects ( att5tion & /udy2 :ile religi\s zeal1 & ! hope ( ga91 comb9$ ) motives ( m]e curiosity1 induc$ sev]al p]sons to travel by l& into remote regions ( ! ,ea/1 f> beyond ! c\ntries to :i* ! op]ations ( ! crusad]s ext5d$4 ,among ^! 0 ,b5jam9 ( ,tudela1 :o set out f ,spa9 9 #aafj1 travell$ by l& to ,con/ant9ople1 & hav+ visit$ ,india & "s ( ! ea/]n isl&s1 return$ to ,europe by way ( ,egypt aft] an abs5ce ( ?irte5 ye>s4 ,asiatic j\rneys4 ,joannes de ,plano ,c>p9i1 a ,franciscan monk1 0 ! head ( "o ( ! missions despat*$ by ,pope ,innoc5t to call ! *ief & p ( ! ,tat>s to a bett] m9d4 ,he rea*$ ! headqu>t]s ( ,batu1 on ! ,volga1 9 ,febru>y #abdf2 &1 aft] "s /ay1 w5t on to ! camp ( ! great khan ne> ,k>akorum 9 c5tral ,asia1 & return$ safely 9 ! autumn ( #abdg4 ,a few ye>s aft]w>ds1 a ,flem+ nam$ ,rubruquis 0 s5t on a simil> mission1 & _h ! m]it ( be+ ! fir/ travell] ( ?is era :o gave a correct acc\nt ( ! ,caspian ,sea4 ,he asc]ta9$ t x _h no outlet4 ,at ne>ly ! same "t ,hayton1 k+ ( ,arm5ia1 made a j\rney to ,k>akorum 9 #abed1 by a r\te f> to ! nor? ( t foll[$ by ,c>p9i & ,rubruquis4 ,he 0 treat$ ) hon\r & hospitality1 & return$ by way ( ,sam>k& & ,tabriz1 to 8 own t]ritory4 ,! curi\s n>rative ( ,k+ ,hayton 0 translat$ by ,klapro?4 ,:ile ! republics ( ,italy1 & above all ! /ate ( ,v5ice1 7 engag$ 9 di/ribut+ ! ri* products ( ,india & ! ,f> ,ea/ ov] ! ,we/]n _w1 x 0 impossible t motives ( curiosity1 z well z a desire ( comm]cial advantage1 %\ld n be awak5$ to su* a degree z to impel "s ( ! m]*ants to visit ^? remote l&s4 ,among ^! 7 ! bro!rs ,polo1 :o trad$ ) ! ,ea/ & !mselves visit$ ,tat>y4 ,! recital ( _! travels fir$ ! y\?ful imag9ation ( "y ,m>co ,polo1 son ( ,nicolo1 & he set out = ! c\rt ( ,kublai ,khan1 ) 8 fa!r & uncle1 9 #abfe4 ,m>co rema9$ = sev5te5 ye>s 9 ! s]vice ( ! ,great ,khan1 & 0 employ$ on _m important missions4 ,besides :at he le>nt f 8 own obs]vation1 he collect$ mu* in=mation f o!rs conc]n+ c\ntries :i* he did n visit4 ,he return$ to ,europe possess$ ( a va/ /ore ( k respect+ ! ea/]n p>ts ( ! _w1 &1 be+ aft]w>ds made a prison] by ! ,g5oese1 he dictat$ ! n>rative ( 8 travels dur+ 8 captivity4 ,! "w ( ,m>co ,polo is ! mo/ valuable n>rative ( travels t appe>$ dur+ ! middle ages1 & despite a cold reception & _m d5ials ( ! accuracy ( ! record1 its sub/antial tru?fulness has be5 abundantly prov$4 ,mission>ies cont9u$ to d useful geographical "w4 ,among !m 7 ,john ( ,monte ,corv9o1 a ,franciscan monk1 ,&rew ( ,p]ugia1 ,john ,m>ignioli & ,fri> ,jordanus1 :o visit$ ! we/ coa/ ( ,india1 & above all ,fri> ,odoric ( ,pord5one4 ,odoric set out on 8 travels ab\t #acah1 & 8 j\rneys embrac$ p>ts ( ,india1 ! ,malay ,ar*ipelago1 ,*9a & ev5 ,tibet1 :]e he 0 ! fir/ ,european to ent] ,lhasa1 n yet a =bidd5 city4 ,ibn ,batuta1 ! great ,arab travell]1 is sep>at$ by a wide space ( "t f 8 c\ntrym5 already m5tion$1 & he f9ds 8 prop] place 9 a *ronological notice aft] ! days ( ,m>co ,polo1 = he did n beg9 8 w&]+s until #acbe1 8 c>e] ?us co9cid+ 9 "t ) ! fabl$ j\rney+s ( ,sir ,john ,m&eville4 ,:ile ,arab le>n+ fl\ri%$ dur+ ! d>ke/ ages ( ,european ignorance1 ! la/ ( ! ,arab geograph]s liv$ to see ! dawn ( ! great p]iod ( ! ,european awak5+4 ,ibn ,batuta w5t by l& f ,tangi] to ,cairo1 !n visit$ ,syria1 & p]=m$ ! pilgrimages to ,m$9a & ,mecca4 ,aft] explor+ ,p]sia1 & aga9 resid+ = "s "t at ,mecca1 he made a voyage d[n ! ,r$ sea to ,yem51 & travell$ ?r\gh t c\ntry to ,ad54 ,!nce he visit$ ! ,african coa/1 t\*+ at ,mombasa & ,quiloa1 & !n sail$ across to ,ormuz & ! ,p]sian ,gulf4 ,he cross$ ,arabia f ,bahre9 to ,jidda1 trav]s$ ! ,r$ sea & ! des]t to ,sy5e1 & desc5d$ ! ,nile to ,cairo4 ,aft] ?is he revisit$ ,syria & ,asia ,m9or1 & cross$ ! ,black sea1 ! des]t f ,a/rakhan to ,bokh>a1 & ! ,h9du ,ku%4 ,he 0 9 ! s]vice ( ,muhammad ,tughluk1 rul] ( ,delhi1 ab\t eight ye>s1 & 0 s5t on an embassy to ,*9a1 9 ! c\rse ( :i* ! ambassadors sail$ d[n ! we/ coa/ ( ,india to ,calicut1 & !n visit$ ! ,maldive ,isl&s & ,ceylon4 ,ibn ,batuta made ! voyage ?r\gh ! ,malay ,ar*ipelago to ,*9a1 & on 8 return he proce$$ f ,malab> to ,bagdad & ,damascus1 ultimately rea*+ ,fez1 ! capital ( 8 native c\ntry1 9 ,novemb] #acdi4 ,aft] a j\rney into ,spa9 he set out once m = ,c5tral ,africa 9 #aceb1 & rea*$ ,timbuktu & ! ,nig]1 return+ to ,fez 9 #acec4 ,8 n>rative 0 committ$ to writ+ f 8 dictation4 ,spani% exploration4 ,! ,european c\ntry :i* _h come ! mo/ completely "u ! influ5ce ( ,arab culture n[ began to s5d =? explor]s to di/ant l&s1 ?\gh ! impulse came n f ! ,moors b f ,italian m]*ant navigators 9 ,spani% s]vice4 ,! peaceful reign ( ,h5ry ,,iii4 ( ,ca/ile is fam\s = ! attempts ( t pr9ce to ext5d ! diplomatic relations ( ,spa9 to ! remote/ p>ts ( ! e>?4 ,he s5t embassies to all ! pr9ces ( ,*ri/5dom & to ! ,moors4 ,9 #adjc ! ,spani% k+ s5t a knight ( ,madrid1 ,ruy ,gonzalez de ,clavijo1 to ! di/ant c\rt ( ,timur1 at ,sam>k&4 ,he return$ 9 #adjf1 & wrote a valuable n>rative ( 8 travels4 ,italians cont9u$ to make important j\rneys 9 ! ,ea/ dur+ ! #ae? c5tury4 ,among !m 0 ,nicolo ,conti1 :o pass$ ?r\gh ,p]sia1 sail$ along ! coa/ ( ,malab>1 visit$ ,sumatra1 ,java & ! s\? ( ,*9a1 return$ by ! ,r$ sea1 & got home to ,v5ice 9 #addd aft] an abs5ce ( tw5ty-five ye>s4 ,he relat$ 8 adv5tures to ,poggio ,bracciol9i1 secret>y to ,pope ,eug5ius ,,iv42 & ! n>rative conta9s mu* int]e/+ in=mation4 ,"o ( ! mo/ rem>kable ( ! ,italian travell]s 0 ,ludovico di ,v>!ma1 :o left 8 native l& 9 #aejb4 ,he w5t to ,egypt & ,syria1 & = ! sake ( visit+ ! holy cities became a ,mahomm$an4 ,he 0 ! fir/ ,european :o gave an acc\nt ( ! int]ior ( ,yem54 ,he aft]w>ds visit$ & describ$ _m places 9 ,p]sia1 ,india & ! ,malay ,ar*ipelago1 return+ to ,europe 9 a ,portuguese %ip aft] an abs5ce ( five ye>s4 ,portuguese exploration--,pr9ce ,h5ry ! ,navigator4 ,9 ! #ae? c5tury ! "t 0 approa*+ :5 ! discov]y ( ! ,cape ( ,good ,hope 0 to wid5 ! scope ( geographical ent]prise4 ,?is great ev5t 0 prec$$ by ! g5]al utilization 9 ,europe ( ! pol>ity ( ! magnetic ne$le 9 ! con/ruction ( ! m>9]'s compass4 ,portugal took ! lead along ?is new pa?1 & =emo/ among h] pione]s /&s ,pr9ce ,h5ry ! ,navigator 7#acid-adfj71 :o 0 a patron bo? ( exploration & ( ! /udy ( geographical !ory4 ,! great we/w>d projection ( ! coa/ ( ,africa1 & ! isl&s to ! nor?-we/ ( t cont95t1 7 ! pr9cipal sc5e ( ! "w ( ! m>9]s s5t out at 8 exp5se2 b 8 object 0 to pu% onw>d & rea* ,india f ! ,atlantic4 ,! progress ( discov]y receiv$ a *eck on 8 dea?1 b only = a "t4 ,9 #adfb ,p$ro de ,c9tra ext5d$ ,portuguese exploration along ! ,african coa/ & discov]$ ,si]ra ,leone4 ,f]nan ,gomez foll[$ 9 #adfi1 & op5$ trade ) ! ,gold ,coa/2 & 9 #adhd ,diogo ,cao discov]$ ! m\? ( ! ,congo4 ,! k+ ( ,portugal next despat*$ ,b>tolomeu ,diaz 9 #adhf to cont9ue discov]ies s\?w>ds2 :ile1 9 ! foll[+ ye>1 he s5t ,p$ro de ,covilhao & ,affonso de ,payva to discov] ! c\ntry ( ,pre/] ,john4 ,diaz succe$$ 9 r\nd+ ! s\!rn po9t ( ,africa1 :i* he nam$ ,cabo ,torm5toso--! ,cape ( ,/orms--b ,k+ ,joao ,,ii41 =esee+ ! realization ( ! long-s\ght passage to ,india1 gave x ! /imulat+ & endur+ "n ( ! ,cape ( ,good ,hope4 ,payva di$ at ,cairo2 b ,covilhao1 hav+ he>d t a ,*ri/ian rul] reign$ 9 ! m\nta9s ( ,e?iopia1 p5etrat$ into ,abyss9ia 9 #adij4 ,he deliv]$ ! lett] :i* ,joao ,,ii4 _h address$ to ,pre/] ,john to ! ,negus ,alex&] ( ,abyss9ia1 b he 0 deta9$ by t pr9ce & nev] all[$ to leave ! c\ntry4 ,columbus4 ,! ,portuguese1 foll[+ ! lead ( ,pr9ce ,h5ry1 cont9u$ to look = ! road to ,india by ! ,cape ( ,good ,hope4 ,! same end 0 s\ght by ,*ri/oph] ,columbus1 foll[+ ! sugge/ion ( ,toscanelli1 & "u-e/imat+ ! diamet] ( ! globe1 by sail+ due we/4 ,! voyages ( ,columbus 7#adib-adih7 result$ 9 ! discov]y ( ! ,we/ ,indies & ,nor? ,am]ica :i* b>r$ ! way to ! ,f> ,ea/4 ,9 #adic ! pope1 ,alex&] ,,vi41 issu$ a bull in/itut+ ! fam\s l9e ( dem>cation runn+ f ,n4 to ,s4 #ajj leagues ,w4 ( ! ,azores1 to ! we/ ( :i* ! ,spani>ds 7 au?oriz$ to explore & to ! ea/ ( :i* ! ,portuguese receiv$ ! monopoly ( discov]y4 ,! direct l9e ( ,portuguese exploration result$ 9 ! discov]y ( ! ,cape r\te to ,india by ,vasco da ,gama 7#adih71 & 9 #aejj to ! indep5d5t discov]y ( ,s\? ,am]ica by ,p$ro ,alv>ez ,cabral4 ,! voyages ( ,columbus & ( ,vasco da ,gama 7 s important t x is unnecess>y to detail _! results 9 ?is place4 ,see ,,columbus1 ,,*ri/oph]2 ,,gama1 ,,vasco ,,da4 ,vasco da ,gama4 ,! ?ree voyages ( ,vasco da ,gama 7:o di$ on ! sc5e ( 8 lab\rs1 at ,co*91 9 #aebd7 revolutioniz$ ! comm]ce ( ! ,ea/4 ,until !n ! ,v5etians held ! c>ry+ trade ( ,india1 :i* 0 br\ght by ! ,p]sian ,gulf & ,r$ sea into ,syria & ,egypt1 ! ,v5etians receiv+ ! products ( ! ,ea/ at ,alex&ria & ,beirut & di/ribut+ !m ov] ,europe4 ,?is comm]ce 0 a great s\rce ( weal? to ,v5ice2 b aft] ! discov]y ( ! new passage r\nd ! ,cape1 & ! conque/s ( ! ,portuguese1 ! trade ( ! ,ea/ pass$ into o!r h&s4 ,spani>ds 9 ,am]ica4 ,! discov]ies ( ,columbus awak5$ a _s ( ent]prise 9 ,spa9 :i* cont9u$ 9 full =ce = a c5tury2 adv5tur]s flock$ eag]ly across ! ,atlantic1 & discov]y foll[$ discov]y 9 rapid succession4 ,_m ( ! companions ( ,columbus cont9u$ 8 "w4 ,vic5te ,yanez ,p9zon 9 #aejj rea*$ ! m\? ( ! ,amazon4 ,9 ! same ye> ,alonso de ,oj$a1 accompani$ by ,juan de la ,cosa1 f ^: maps we le>n mu* ( ! discov]ies ( ! #af? c5tury navigators1 & by a ,flor5t9e nam$ ,am]igo ,vespucci1 t\*$ ! coa/ ( ,s\? ,am]ica some:]e ne> ,sur9am1 foll[+ ! %ore z f> z ! ,gulf ( ,m>acaibo4 ,vespucci aft]w>ds made ?ree voyages to ! ,brazilian coa/2 & 9 #aejd he wrote an acc\nt ( 8 f\r voyages1 :i* 0 widely circulat$1 & became ! means ( procur+ = its au?or at ! h&s ( ! c>tograph] ,waldseemull] 9 #aejg ! disproportionate di/9ction ( giv+ 8 "n to ! :ole cont95t4 ,9 #aejh ,alonso de ,oj$a obta9$ ! gov]nm5t ( ! coa/ ( ,s\? ,am]ica f ,cabo de la ,vela to ! ,gulf ( ,d>i52 ,oj$a l&$ at ,c>tag5a 9 #aeaj1 & su/a9$ a defeat f ! natives1 9 :i* 8 lieut5ant1 ,juan de la ,cosa1 0 kill$4 ,aft] ano!r rev]se on ! ea/ side ( ! ,gulf ( ,d>i5 ,oj$a return$ to ,hispaniola & di$ !re4 ,! ,spani>ds 9 ! ,gulf ( ,d>i5 7 left by ,oj$a "u ! comm& ( ,francisco ,piz>ro1 ! future conqu]or ( ,p]u4 ,aft] suff]+ mu* f fam9e & disease1 ,piz>ro resolv$ to leave1 & emb>k$ ! survivors 9 small vessels1 b outside ! h>b\r !y met a %ip :i* prov$ to be t ( ,m>t9 ,f]n&ez ,enciso1 ,oj$a's p>tn]1 com+ ) provisions & re9=cem5ts4 ,"o ( ! crew ( ,enciso's %ip1 ,vasco ,nunez de ,balboa1 ! future discov]] ( ! ,pacific ,ocean1 induc$ 8 comm&] to =m a settlem5t on ! o!r side ( ! ,gulf ( ,d>i54 ,! soldi]s became discont5t$ & depos$ ,enciso1 :o 0 a man ( le>n+ & an accompli%$ cosmograph]4 ,8 "w ,suma de ,geografia1 :i* 0 pr9t$ 9 #aeai1 is ! fir/ ,spani% book :i* gives an acc\nt ( ,am]ica4 ,vasco ,nunez1 ! new comm&]1 ent]$ ^u a c>e] ( conque/ 9 ! neighb\rhood ( ,d>i51 :i* end$ 9 ! discov]y ( ! ,pacific ,ocean on ! #be? ( ,septemb] #aeac4 ,vasco ,nunez 0 behead$ 9 #aeag by ,p$r>ias de ,avila1 :o 0 s5t out to sup]s$e him4 ,?is 0 "o ( ! greate/ calamities t c\ld h happ5$ to ,s\? ,am]ica2 = ! discov]] ( ! ,s\? sea 0 on ! po9t ( sail+ ) a little fleet into 8 unkn[n ocean1 & a humane & judici\s man w\ld probably h be5 ! conqu]or ( ,p]u1 in/ead ( ! cruel & ignorant ,piz>ro4 ,9 ! ye> #aeai ,panama 0 f\nd$ by ,p$r>ias2 & ! conque/ ( ,p]u by ,piz>ro foll[$ a few ye>s aft]w>ds4 ,h]nan ,cortes ov]ran & conqu]$ ,mexico f #aeah to #aeba1 & ! discov]y & conque/ ( ,guatemala by ,alv>ado1 ! invasion ( ,florida by ,de ,soto1 & ( ,nueva ,granada by ,quesada1 foll[$ 9 rapid succession4 ,! fir/ detail$ acc\nt ( ! we/ coa/ ( ,s\? ,am]ica 0 writt5 by a ke5ly obs]vant old soldi]1 ,p$ro de ,cieza de ,leon1 :o 0 travell+ 9 ,s\? ,am]ica f #aecc to #aeej1 & publi%$ 8 /ory at ,seville 9 #aeec4 ,pacific ,ocean4 ,! great desire ( ! ,spani% gov]nm5t at t "t 0 to f9d a we/w>d r\te to ! ,moluccas4 ,= ?is purpose ,juan ,diaz de ,solis 0 despat*$ 9 ,octob] #aeae1 & 9 ,janu>y #aeaf he discov]$ ! m\? ( ! ,rio de la ,plata4 ,he 01 h[ev]1 kill$ by ! natives1 & 8 %ips return$4 ,9 ! foll[+ ye> ! ,portuguese ,f]d9&o ,magalhaes1 famili>ly kn[n z ,magellan1 laid be=e ,*>les ,v41 at ,valladolid1 a s*eme = rea*+ ! ,spice ,isl&s by sail+ we/w>d4 ,he />t$ on ! #ba/ ( ,septemb] #aeai1 ent]$ ! /rait :i* n[ be>s 8 "n 9 ,octob] #aebj1 work$ 8 way ?r\gh betwe5 ,patagonia & ,ti]ra del ,fuego1 & ent]$ on ! va/ ,pacific :i* he cross$ )\t sight+ any ( its innum]able isl& gr\ps4 ,?is 0 unque/ionably ! greate/ ( ! voyages :i* foll[$ f ! impulse ( ,pr9ce ,h5ry1 & x 0 r5d]$ possible only by ! magnific5t c\rage ( ! comm&] 9 spite ( rebellion1 mut9y & />vation4 ,x 0 ! #f? ( ,m>* #aeba :5 he rea*$ ! ,ladrone ,isl&s4 ,!nce ,magellan proce$$ to ! ,philipp9es1 & !re 8 c>e] end$ 9 an unimportant enc\nt] ) ho/ile natives4 ,ev5tually a ,biscayan nam$ ,seba/ian del ,cano1 sail+ home by way ( ! ,cape ( ,good ,hope1 rea*$ ,san ,luc> 9 comm& ( ! ,victoria on ! #f? ( ,septemb] #aebb1 ) eighte5 survivors2 ?is "o %ip ( ! squadron :i* sail$ on ! que/ succe$$ 9 accompli%+ ! fir/ circumnavigation ( ! globe4 ,del ,cano 0 receiv$ ) great di/9ction by ! emp]or1 :o grant$ him a globe = 8 cre/1 & ! motto ,primus circumd$i/i me4 ,portuguese 9 ,africa & ! ,ea/4 ,:ile ! ,spani>ds 7 circumnavigat+ ! _w & complet+ _! k ( ! coa/s ( ,c5tral & ,s\? ,am]ica1 ! ,portuguese 7 actively engag$ on simil> "w z reg>ds ,africa & ! ,ea/ ,indies4 ,) ,abyss9ia ! mission ( ,covilhao l$ to fur!r int]c\rse4 ,9 ,april #aebj ,vasco da ,gama1 z vic]oy ( ! ,indies1 took a fleet into ! ,r$ sea1 & l&$ an embassy consi/+ ( ,dom ,rodriguez de ,lima & ,fa!r ,francisco ,alv>ez1 a prie/ ^: detail$ n>rative is ! e>lie/ & n ! lea/ int]e/+ acc\nt we possess ( ,abyss9ia4 ,x 0 n until #aebf t ! embassy 0 dismiss$2 & n _m ye>s aft]w>ds ! negus entreat$ ! help ( ! ,portuguese aga9/ ,mahomm$an invad]s1 & ! vic]oy s5t an exp$ition>y =ce1 comm&$ by 8 bro!r ,cri/(oro da ,gama1 ) #dej muskete]s4 ,da ,gama 0 tak5 prison] & kill$1 b 8 foll[]s enabl$ ! ,*ri/ians ( ,abyss9ia to rega9 _! p[]1 & a ,jesuit mission rema9$ 9 ! c\ntry4 ,! ,portuguese also e/abli%$ a close connexion ) ! k+dom ( ,congo on ! we/ side ( ,africa1 & obta9$ mu* in=mation respect+ ! int]ior ( ! cont95t4 ,du>te ,lopez1 a ,portuguese settl$ 9 ! c\ntry1 0 s5t on a mission to ,rome by ! k+ ( ,congo1 & ,pope ,sixtus ,v4 caus$ him to rec\nt to 8 *amb]la91 ,felipe ,pigafetta1 all he _h le>n$ dur+ ! n9e ye>s he _h be5 9 ,africa1 f #aegh to #aehg4 ,?is n>rative1 "u ! title ( ,description ( ! ,k+dom ( ,congo1 0 publi%$ at ,rome by ,pigafetta 9 #aeia4 ,a map 0 atta*$ on :i* sev]al great equatorial lakes are %[n1 & ! empire ( ,monomwezi or ,unyamwezi is laid d[n4 ,! mo/ valuable "w on ,africa ab\t ?is "t is1 h[ev]1 t writt5 by ! ,moor ,leo ,africanus 9 ! e>ly "p ( ! #af? c5tury4 ,leo travell$ ext5sively 9 ! nor? & we/ ( ,africa1 & 0 ev5tually tak5 by pirates & sold to a ma/] :o pres5t$ him to ,pope ,leo ,x4 ,at ! pope's desire he translat$ 8 "w on ,africa into ,italian4 ,9 ,fur!r ,india & ! ,malay ,ar*ipelago ! ,portuguese acquir$ pr$om9at+ influ5ce at sea1 e/abli%+ factories on ! ,malab> coa/1 9 ! ,p]sian ,gulf1 at ,malacca1 & 9 ! ,spice ,isl&s1 & ext5d+ _! comm]cial ent]prises f ! ,r$ sea to ,*9a4 ,_! mission>ies 7 receiv$ at ! c\rt ( ,akb>1 & ,b5$ict ,goes1 a native ( ! ,azores1 0 despat*$ on a j\rney ov]l& f ,agra to ,*9a4 ,he />t$ 9 #afjc1 &1 aft] trav]s+ ! lea/-kn[n p>ts ( ,c5tral ,asia1 he rea*$ ! conf9es ( ,*9a4 ,he appe>s to h asc5d$ f ,kabul to ! plateau ( ! ,pamir1 & !nce onw>ds by ,y>k&1 ,khotan & ,aksu4 ,he di$ on ! j\rney 9 ,m>* #afjg2 & ?us1 z "o ( ! bre?r5 pron\nc$ 8 epitaph1 seek+ ,ca?ay he f\nd heav54 ,engli%1 ,dut* & ,fr5*4 ,! activity & love ( adv5ture1 :i* became a passion = two or ?ree g5]ations 9 ,spa9 & ,portugal1 spread to o!r c\ntries4 ,x 0 ! _s ( ! age2 & ,engl&1 ,holl& & ,france 7 fir$ by x4 ,engli% ent]prise 0 fir/ ar\s$ by ,john & ,seba/ian ,cabot1 fa!r & son1 :o came f ,v5ice & settl$ at ,bri/ol 9 ! "t ( ,h5ry ,,vii4 ,! ,cabots receiv$ a pat5t 9 #adif1 emp[]+ !m to seek unkn[n l&s2 & ,john ,cabot discov]$ ,newf\ndl& & "p ( ! coa/ ( ,am]ica4 ,seba/ian aft]w>ds made a voyage to ,rio de la ,plata 9 ! s]vice ( ,spa91 b he return$ to ,engl& 9 #aedh & receiv$ a p5sion f ,edw>d ,,vi4 ,at 8 sugge/ion a voyage 0 und]tak5 = ! discov]y ( a nor?-ea/ passage to ,ca?ay1 ) ,sir ,hugh ,will\ghby z capta9-g5]al ( ! fleet & ,ri*>d ,*ancellor z pilot-major4 ,!y sail$ 9 ,may #aeec1 b ,will\ghby & all 8 crew p]i%$ on ! ,lapl& coa/4 ,*ancellor1 h[ev]1 0 m =tunate4 ,he rea*$ ! ,:ite ,sea1 p]=m$ ! j\rney ov]l& to ,mosc[1 :]e he 0 well receiv$1 & may be said to h be5 ! f\nd] ( ! trade betwe5 ,russia & ,engl&4 ,he return$ to ,ar*angel & br\ght 8 %ip back 9 safety to ,engl&4 ,on a second voyage1 9 #aeef1 ,*ancellor 0 dr[n$2 & ?ree subsequ5t voyages1 l$ by ,/eph5 ,burr\gh1 ,ar?ur ,pet & ,*>les ,jackman1 9 small craft ( #ej tons & "u1 c>ri$ on an exam9ation ( ! /raits :i* lead into ! ,k>a sea4 ,! ,fr5* foll[$ closely on ! track ( ,john ,cabot1 & ,norman & ,breton fi%]m5 frequ5t$ ! banks ( ,newf\ndl& at ! beg9n+ ( ! #af? c5tury4 ,9 #aebd ,francis ,i4 s5t ,giovanni da ,v]azzano ( ,flor5ce on an exp$ition ( discov]y to ! coa/ ( ,nor? ,am]ica2 & ! details ( 8 voyage 7 embodi$ 9 a lett] address$ by him to ! k+ ( ,france f ,dieppe1 9 ,july #aebd4 ,9 #aecd ,jacques ,c>ti] set out to cont9ue ! discov]ies ( ,v]azzano1 & visit$ ,newf\ndl& & ! ,gulf ( ,/ ,lawr5ce4 ,9 ! foll[+ ye> he made ano!r voyage1 discov]$ ! isl& ( ,antico/i1 & asc5d$ ! ,/ ,lawr5ce to ,ho*elaga1 n[ ,montreal4 ,he return$1 aft] pass+ two w9t]s 9 ,canada2 & on ano!r occasion he also fail$ to e/abli% a colony4 ,admiral de ,coligny made sev]al unsuccessful endeav\rs to =m a colony 9 ,florida "u ,jean ,ribault ( ,dieppe1 ,r5e de ,laudonni]e & o!rs1 b ! settl]s 7 furi\sly assail$ by ! ,spani>ds & ! attempt 0 ab&on$4 ,! ,elizabe?an era4 ,! reign ( ,elizabe? is fam\s = ! gallant ent]prises t 7 und]tak5 by sea & l& to discov] & br+ to light ! unkn[n p>ts ( ! e>?4 ,! great promot] ( geographical discov]y 9 ! ,elizabe?an p]iod 0 ,ri*>d ,hakluyt 7#aeec-afaf71 :o 0 active 9 ! =mation ( ! two companies = coloniz+ ,virg9ia 9 #afjf2 & devot$ 8 life to enc\rag+ & record+ simil> und]tak+s4 ,he publi%$ mu*1 & left _m valuable pap]s at 8 dea?1 mo/ ( :i*1 toge!r ) _m o!r n>ratives1 7 publi%$ 9 #afbb 9 ! great "w ( ! ,rev4 ,samuel ,pur*as1 entitl$ ,hakluytus ,po/humus1 or ,pur*as 8 ,pilgrimes4 ,x is f ^! works t our k ( ! gallant de$s ( ! ,engli% & o!r explor]s ( ! ,elizabe?an age is ma9ly d]iv$4 ,! great & spl5didly illu/rat$ collections ( voyages & travels ( ,!odorus de ,bry & ,hulsius s]v$ a simil> useful purpose on ! cont95t ( ,europe4 ,"o important object ( ,engli% m>itime adv5tur]s ( ^? days 0 to discov] a r\te to ,ca?ay by ! nor?-we/1 a second 0 to settle ,virg9ia1 & a ?ird 0 to raid ! ,spani% settlem5ts 9 ! ,we/ ,indies4 ,nor 0 ! trade to ,muscovy & ,turkey neglect$2 :ile latt]ly a resolute & successful attempt 0 made to e/abli% direct comm]cial relations ) ,india4 ,! conception ( ! nor?-we/]n r\te to ,ca?ay n[ leads ! /ory ( exploration1 = ! fir/ "t z f> z important & su/a9$ ef=ts are conc]n$1 t[>ds ! ,arctic seas4 ,?is "p ( ! /ory is fully told "u ! head+ ( ,,pol> ,,regions1 & only ! names ( ,m>t9 ,frobi%] 7#aegf71 ,john ,davis 7#aehe71 ,h5ry ,hudson 7#afjg7 & ,william ,baff9 7#afaf7 ne$ be m5tion$ "h 9 ord] to pres]ve ! complete conspectus ( ! hi/ory ( discov]y4 ,! ,dut* emulat$ ! ,briti% 9 ! ,arctic seas dur+ ?is p]iod1 direct+ _! ef=ts ma9ly t[>ds ! discov]y ( a nor?-ea/ passage r\nd ! nor!rn end ( ,novaya ,zemlya2 & ,william ,b>5ts or ,b>5dsz 7#aeid-aeig7 is ! mo/ fam\s "n 9 ?is connexion1 8 boat voyage along ! coa/ ( ,novaya ,zemlya aft] los+ 8 %ip & w9t]+ 9 a high latitude1 be+ "o ( ! mo/ rem>kable a*ievem5ts 9 pol> annals4 ,_m ,engli% voyages 7 also made to ,gu9ea & ! ,we/ ,indies1 & twice ,engli% vessels foll[$ 9 ! track ( ,magellan1 & circumnavigat$ ! globe4 ,9 #aegg ,francis ,drake1 :o _h previ\sly s]v$ ) ,hawk9s 9 ! ,we/ ,indies1 und]took 8 celebrat$ voyage r\nd ! _w4 ,rea*+ ! ,pacific ?r\gh ! ,/rait ( ,magellan1 ,drake proce$$ nor?w>d along ! we/ coa/ ( ,am]ica1 resolv$ to attempt ! discov]y ( a nor!rn passage f ! ,pacific to ! ,atlantic4 ,! coa/ f ! s\!rn extremity ( ! ,cali=nian p59sula to ,cape ,m5doc9o _h be5 discov]$ by ,juan ,rodriguez ,cabrillo & ,francisco de ,ulloa 9 #aeci4 ,drake's discov]ies ext5d$ f ,cape ,m5doc9o to #dh deg4 ,n41 9 :i* latitude he gave up 8 que/1 sail$ across ! ,pacific & rea*$ ! ,philipp9e ,isl&s1 return+ home r\nd ! ,cape ( ,good ,hope 9 #aehj4 ,?omas ,cav5di%1 emul\s ( ,drake's example1 fitt$ out ?ree vessels = an exp$ition to ! ,s\? sea 9 #aehf4 ,he took ! same r\te z ,drake along ! we/ coa/ ( ,am]ica4 ,f ,cape ,san ,lucas ,cav5di% /e]$ across ! ,pacific1 see+ no l& until he rea*$ ! ,ladrone ,isl&s4 ,he return$ to ,engl& 9 #aehh4 ,! ?ird ,engli% voyage into ! ,pacific 0 n s =tunate4 ,sir ,ri*>d ,hawk9s 7#aeic7 on rea*+ ! bay ( ,atacames1 9 #a deg4,n4 9 #aeid1 0 attack$ by a ,spani% fleet1 &1 aft] a desp]ate naval engagem5t1 0 =c$ to surr5d]4 ,hawk9s decl>$ 8 object to be discov]y & ! survey ( unkn[n l&s1 & 8 voyage1 ?\gh t]m9at+ 9 disa/]1 bore good fruit4 ,! ,obs]vations ( ,sir ,ri*>d ,hawk9s 9 8 ,voyage into ! ,s\? ,sea1 publi%$ 9 #afbb1 are v valuable4 ,x 0 long be=e ano!r ,briti% %ip ent]$ ! ,pacific ,ocean4 ,sir ,john ,n>bor\gh took two %ips ?r\gh ! ,/rait ( ,magellan 9 #afgj & t\*$ on ! coa/ ( ,*ile1 b x 0 n until #afhe t ,dampi] sail$ ov] ! "p ( ! ,pacific :]e ,hawk9s met 8 defeat4 ,! explor+ ent]prise ( ! ,spani% nation did n wane aft] ! conque/ ( ,p]u & ,mexico1 & ! acquisition ( ! va/ empire ( ! ,indies4 ,x 0 spurr$ into r5ew$ activity by ! audacity ( ,sir ,john ,hawk9s 9 ! ,we/ ,indies1 & by ! appe>ance ( ,drake1 ,cav5di% & ,ri*>d ,hawk9s 9 ! ,pacific4 ,9 ! int]ior ( ,s\? ,am]ica ! ,spani% conqu]ors _h explor$ ! region ( ! ,&es f ! i/hmus ( ,panama to ,*ile4 ,p$ro de ,valdivia 9 #aedj made an exp$ition into ! c\ntry ( ! ,araucanian ,indians ( ,*ile1 & 0 ! fir/ to explore ! ea/]n base ( ! ,&es 9 :at is n[ ,arg5t9e ,patagonia4 ,9 #aeda ,francisco de ,orellana discov]$ ! :ole c\rse ( ! ,amazon f its s\rce 9 ! ,&es to ! ,atlantic4 ,a second voyage on ! ,amazon 0 made 9 #aefa by ! mad pirate ,lope de ,aguirre2 b x 0 n until #afci t a full acc\nt 0 writt5 ( ! great riv] by ,fa!r ,cri/oval de ,acuna1 :o asc5d$ x f its m\? & rea*$ ! city ( ,quito4 ,spani>ds 9 ! ,pacific4 ,! voyage ( ,drake across ! ,pacific 0 prec$$ by t ( ,alv>o de ,m5dana1 :o 0 despat*$ f ,p]u 9 #aefg to discov] ! great ,ant>ctic cont95t :i* 0 believ$ to ext5d f> nor?w>d into ! ,s\? sea1 ! se>* = :i* n[ became "o ( ! lead+ motives ( exploration4 ,aft] a voyage ( eighty days across ! ,pacific1 ,m5dana discov]$ ! ,solomon ,isl&s2 & ! exp$ition return$ 9 safety to ,callao4 ,! appe>ance ( ,drake on ! ,p]uvian coa/ l$ to an exp$ition be+ fitt$ out at ,callao1 to g 9 *ase ( him1 "u ! comm& ( ,p$ro ,s>mi5to4 ,he sail$ f ,callao 9 ,octob] #aegi1 & made a c>eful survey ( ! ,/rait ( ,magellan1 ) ! object ( =tify+ t entrance to ! ,s\? sea4 ,! colony :i* he aft]w>ds took out f ,spa9 0 a complete failure1 & is only rememb]$ n[ f ! "n ( ,port ,fam9e1 :i* ,cav5di% gave to ! site at :i* he f\nd ! />v+ remnant ( ,s>mi5to's settl]s4 ,9 ,june #aeie ,m5dana sail$ f ! coa/ ( ,p]u 9 comm& ( a second exp$ition to colonize ! ,solomon ,isl&s4 ,aft] discov]+ ! ,m>quesas1 he rea*$ ! isl& ( ,santa ,cruz ( evil memory1 :]e he & _m ( ! settl]s di$4 ,8 "y wid[ took comm& ( ! survivors & br\ght !m safely to ,manila4 ,! vic]oys ( ,p]u /ill p]sev]$ 9 _! attempts to plant a colony 9 ! hypo!tical s\!rn cont95t4 ,p$ro ,f]n&ez de ,quiros1 :o 0 pilot "u ,m5dana & ,luis ,vaez de ,torres1 7 s5t 9 comm& ( two %ips to cont9ue ! "w ( exploration4 ,!y sail$ f ,callao 9 ,decemb] #afje1 & discov]$ sev]al isl&s ( ! ,new ,hebrides gr\p4 ,!y an*or$ 9 a bay ( a l>ge isl& :i* ,quiros nam$ ,au/ralia del ,espiritu ,santo4 ,f ?is place ,quiros return$ to ,am]ica1 b ,torres cont9u$ ! voyage1 pass$ ?r\gh ! /rait betwe5 ,au/ralia & ,new ,gu9ea :i* be>s 8 "n1 & explor$ & mapp$ ! s\!rn & ea/]n coa/s ( ,new ,gu9ea4 ,! ,portuguese1 9 ! e>ly "p ( ! #ag? c5tury 7#aegh-afdj71 7 "u ! dom9ion ( ,spa91 & _! ent]prise 0 to "s ext5t damp$2 b _! mission>ies ext5d$ geographical k 9 ,africa4 ,fa!r ,francisco ,paez acquir$ great influ5ce 9 ,abyss9ia1 & explor$ its highl&s f #afjj to #afbb4 ,fa!rs ,m5dez & ,lobo trav]s$ ! des]ts betwe5 ! coa/ ( ! ,r$ sea & ! m\nta9s1 became acqua9t$ ) ,lake ,tsana1 & discov]$ ! s\rces ( ! ,blue ,nile 9 #afbd-afcc4 ,rivalry 9 ! ,ea/4 ,b ! att5tion ( ! ,portuguese 0 ma9ly devot$ to va9 attempts to ma9ta9 _! monopoly ( ! trade ( ,india aga9/ ! p[]ful rivalry ( ! ,engli% & ,dut*4 ,! ,engli% ent]prises 7 p]sev]+1 cont9u\s & successful4 ,james ,lanca/] made a voyage to ! ,indian ,ocean f #aeia to #aeid2 & 9 #aeii ! m]*ants & adv5tur]s ( ,london resolv$ to =m a company1 ) ! object ( e/abli%+ a trade ) ! ,ea/ ,indies4 ,on ! #ca/ ( ,decemb] #aeii ,que5 ,elizabe? grant$ ! *>t] ( incorporation to ! ,ea/ ,india ,company1 & ,sir ,james ,lanca/]1 "o ( ! directors1 0 appo9t$ g5]al ( _! fir/ fleet4 ,he 0 accompani$ by ,john ,davis1 ! great ,arctic navigator1 z pilot-major4 ,?is voyage 0 em95tly successful4 ,! %ips t\*$ at ,a*9 9 ,sumatra & at ,java1 return+ ) full lad+s ( pepp] 9 #afjc4 ,! second voyage 0 comm&$ by ,sir ,h5ry ,middleton2 b x 0 9 ! ?ird voyage1 "u ,keel+e & ,hawk9s1 t ! ma9l& ( ,india 0 fir/ rea*$ 9 #afjg4 ,capta9 ,hawk9s l&$ at ,surat & travell$ ov]l& to ,agra1 pass+ "s "t at ! c\rt ( ! ,great ,mogul4 ,9 ! voyage ( ,sir ,edw>d ,mi*elborne 9 #afje1 ,john ,davis lo/ 8 life 9 a fight ) a ,japanese junk4 ,! eigh? voyage1 l$ by ,capta9 ,s>is1 ext5d$ ! op]ations ( ! company to ,japan2 & 9 #afac ! ,japanese gov]nm5t grant$ privileges to ! company2 b ! ,briti% retir$ 9 #afbc1 giv+ up _! factory4 ,! *ief result ( ?is e>ly int]c\rse betwe5 ,great ,brita9 & ,japan 0 ! int]e/+ s]ies ( lett]s writt5 by ,william ,adams f #afaa to #afag4 ,f ! t5? voyage ( ! ,ea/ ,india ,company1 comm&$ by ,capta9 ,be/1 :o left ,engl& 9 #afab1 dates ! e/abli%m5t ( p]man5t ,briti% factories on ! coa/ ( ,india4 ,x 0 ,capta9 ,be/ :o secur$ a regul> firman = trade f ! ,great ,mogul4 ,f t "t a fleet 0 despat*$ e ye>1 & ! company's op]ations greatly increas$ geographical k ( ,india & ! ,ea/]n ,ar*ipelago4 ,briti% visits to ,ea/]n c\ntries1 at ?is "t1 7 n conf9$ to ! voyages ( ! company4 ,j\rneys 7 also made by l&1 &1 among o!rs1 ! ent]ta9+ au?or ( ! ,crudities1 ,?omas ,coryate1 ( ,odcombe 9 ,som]set%ire1 w&]$ on foot f ,france to ,india1 & di$ 7#afag7 9 ! company's factory at ,surat4 ,9 #aefa ,an?ony ,j5k9son arriv$ 9 ,p]sia ) a lett] f ,que5 ,elizabe? to ! %ah4 ,he travell$ ?r\gh ,russia to ,bokh>a1 & return$ by ! ,caspian & ,volga4 ,9 #aegi ,*ri/oph] ,burr\ghs built a %ip at ,nizhniy ,novgorod & trad$ across ! ,caspian to ,baku2 & 9 #aeih ,sir ,an?ony & ,rob]t ,%irley arriv$ 9 ,p]sia1 & ,rob]t 0 aft]w>ds s5t by ! %ah to ,europe z 8 ambassador4 ,he 0 foll[$ by a ,spani% mission "u ,g>cia de ,silva1 :o wrote an int]e/+ acc\nt ( 8 travels2 & to ,sir ,dorm] ,cotton's mission1 9 #afbh1 we are indebt$ = ,sir ,?omas ,h]b]t's *>m+ n>rative4 ,9 l mann] ,sir ,?omas ,roe's mission to ,india result$ n only 9 a l>ge collection ( valuable reports & lett]s ( 8 own1 b also 9 ! detail$ acc\nt ( 8 *apla9 ,t]ry4 ,b ! mo/ le>n$ & intellig5t travell] 9 ! ,ea/1 dur+ ! #ag? c5tury1 0 ! ,g]man1 ,engelbre*t ,kaempf]1 :o accompani$ an embassy to ,p]sia1 9 #afhd1 & 0 aft]w>ds a surgeon 9 ! s]vice ( ! ,dut* ,ea/ ,india ,company4 ,he 0 9 ! ,p]sian ,gulf1 ,india & ,java1 & resid$ = m ?an two ye>s 9 ,japan1 ( :i* he wrote a hi/ory4 ,dut* exploration1 #af?-#ag? c5turies4 ,! ,dut* nation1 z soon z x 0 emancipat$ f ,spani% tyranny1 display$ an am\nt ( ent]prise1 :i*1 = a long "t1 0 fully equal to t ( ! ,briti%4 ,! ,arctic voyages ( ,b>5ts 7 quickly foll[$ by ! e/abli%m5t ( a ,dut* ,ea/ ,india ,company2 & ! ,dut*1 ou/+ ! ,portuguese1 n only e/abli%$ factories on ! ma9l& ( ,india & 9 ,japan1 b acquir$ a prepond]at+ influ5ce ?r\gh\t ! ,malay ,ar*ipelago4 ,9 #aehc ,jan ,hug5 van ,l9s*ot5 made a voyage to ,india ) a ,portuguese fleet1 & 8 full & graphic descriptions ( ,india1 ,africa1 ,*9a & ! ,malay ,ar*ipelago mu/ h be5 ( no small use to 8 c\ntrym5 9 _! di/ant voyages4 ,! fir/ ( ! ,dut* ,indian voyages 0 p]=m$ by %ips :i* sail$ 9 ,april #aeie1 & r\nd$ ! ,cape ( ,good ,hope4 ,a second l>ge ,dut* fleet sail$ 9 #aeih2 &1 s eag] 0 ! republic to ext5d h] comm]ce ov] ! _w t ano!r fleet1 consi/+ ( five %ips ( ,rott]dam1 0 s5t 9 ! same ye> by way ( ,magellan's ,/rait1 "u ,jacob ,mahu z admiral1 ) ,william ,adams z pilot4 ,mahu di$ on ! passage out1 & 0 succe$$ by ,simon de ,cordes1 :o 0 kill$ on ! coa/ ( ,*ile4 ,9 ,septemb] #aeii ! fleet _h ent]$ ! ,pacific4 ,! %ips 7 !n /e]$ direct = ,japan1 & an*or$ (f ,bungo 9 ,april #afjj4 ,9 ! same ye>1 #aeih1 a ?ird exp$ition 0 despat*$ "u ,oliv] van ,noort1 a native ( ,utre*t1 b ! voyage contribut$ no?+ to geography4 ,! ,dut* ,company 9 #afad aga9 resolv$ to s5d a fleet to ! ,moluccas by ! we/w>d r\te1 & ,joris ,spilb]g5 0 appo9t$ to ! comm& z admiral1 ) a commission f ! ,/ates-,g5]al4 ,he 0 furni%$ ) f\r %ips ( ,am/]dam1 two ( ,rott]dam & "o f ,zeel&4 ,on ! #f? ( ,may #afae ,spilb]g5 ent]$ ! ,pacific ,ocean1 & t\*$ at sev]al places on ! coa/ ( ,*ile & ,p]u1 defeat+ ! ,spani% fleet 9 a naval engagem5t (f ,*ilca4 ,aft] plund]+ ,payta & mak+ requisitions at ,acapulco1 ! ,dut* fleet cross$ ! ,pacific & rea*$ ! ,moluccas 9 ,m>* #afaf4 ,! ,dut* n[ resolv$ to discov] a passage into ! ,pacific to ! s\? ( ,ti]ra del ,fuego1 ! insul> nature ( :i* _h be5 asc]ta9$ by ,sir ,francis ,drake4 ,! vessels fitt$ out = ?is purpose 7 ! ,e5dra*t1 ( #cfj tons1 comm&$ by ,jacob ,lemaire1 & ! ,hoorn1 ( #aaj tons1 "u ,willem ,s*\t54 ,!y sail$ f ! ,texel on ! #ad? ( ,june #afae1 & by ! #bj? ( ,janu>y #afaf !y 7 s\? ( ! entrance ( ,magellan's ,/rait4 ,pass+ ?r\gh ! /rait ( ,lemaire !y came to ! s\!rn extremity ( ,ti]ra del ,fuego1 :i* 0 nam$ ,cape ,horn1 9 hon\r ( ! t[n ( ,hoorn 9 ,we/ ,friesl&1 ( :i* ,s*\t5 0 a native4 ,!y pass$ ! cape on ! #ca/ ( ,janu>y1 enc\nt]+ ! usual we/]ly w9ds4 ,! great m]it ( ?is discov]y ( a second passage into ! ,s\? sea lies 9 ! fact t x 0 n accid5tal or un=ese51 b 0 due to ! sagacity ( ^? :o design$ ! voyage4 ,on ! #a/ ( ,m>* ! ,dut* fleet sight$ ! isl& ( ,juan ,f]n&ez2 &1 hav+ cross$ ! ,pacific1 ! explor]s sail$ along ! nor? coa/ ( ,new ,gu9ea & arriv$ at ! ,moluccas on ! #ag? ( ,septemb] #afaf4 ,!re 7 sev]al e>ly indications ( ! exi/5ce ( ! great ,au/ralian cont95t1 & ! ,dut* endeav\r$ to obta9 fur!r k conc]n+ ! c\ntry & its ext5t2 b only its nor!rn & we/]n coa/s _h be5 visit$ be=e ! "t ( ,gov]nor van ,diem54 ,dirk ,h>tog _h be5 on ! we/ coa/ 9 latitude #bf deg4 #cj' ,s4 9 #afaf4 ,pels]t /ruck on a reef call$ ,h\tman's ,abrolhos on ! #d? ( ,june #afbi4 ,9 #afig ! ,dut* capta9 ,vlam+h l&$ on ! we/ coa/ ( ,au/ralia1 !n call$ ,new ,holl&1 9 #ca deg4 #dc' ,s41 & nam$ ! ,swan riv] f ! black swans he discov]$ !re4 ,9 #afdb ! gov]nor & c\ncil ( ,batavia fitt$ out two %ips to prosecute ! discov]y ( ! s\? l&1 !n believ$ to be "p ( a va/ ,ant>ctic cont95t1 & entru/$ ! comm& to ,capta9 ,abel ,jans5 ,tasman4 ,?is voyage prov$ to be ! mo/ important to geography t _h be5 und]tak5 s9ce ! fir/ circumnavigation ( ! globe4 ,tasman sail$ f ,batavia 9 #afdb1 & on ! #bd? ( ,novemb] sight$ high l& 9 #db deg4 #cj' ,s41 :i* 0 nam$ van ,diem5's ,l&1 & aft] l&+ !re proce$$ to ! discov]y ( ! we/]n coa/ ( ,new ,zeal&2 at fir/ call$ ,/at5 ,l&1 & suppos$ to be connect$ ) ! ,ant>ctic cont95t f :i* ?is voyage prov$ ,new ,holl& to be sep>at$4 ,he !n rea*$ ,tongatabu1 "o ( ! ,fri5dly ,isl&s ( ,cook2 & return$ by ! nor? coa/ ( ,new ,gu9ea to ,batavia4 ,9 #afdd ,tasman made a second voyage to effect a full] discov]y ( ,new ,gu9ea4 ,fr5* 9 ,nor? ,am]ica4 ,! ,fr5* direct$ _! ent]prise m 9 ! direction ( ,nor? ,am]ica ?an ( ! ,indies4 ,"o ( _! mo/ di/+ui%$ explor]s 0 ,samuel ,*ampla91 a capta9 9 ! navy1 :o1 aft] a rem>kable j\rney ?r\gh ,mexico & ! ,we/ ,indies f #aeii to #afjb1 e/abli%$ 8 hi/oric connexion ) ,canada1 to ! geographical k ( :i* he made a v l>ge addition4 ,mission>ies 9 ! ,ea/4 ,! pr9ciples & me?ods ( survey+ & position f9d+ _h by ?is "t become well advanc$1 & ! mo/ rem>kable example ( ! e>ly application ( ^! improvem5ts is to be f\nd 9 ! survey ( ,*9a by ,jesuit mission>ies4 ,!y fir/ prep>$ a map ( ! c\ntry r\nd ,pek+1 :i* 0 submitt$ to ! emp]or ,kang-hi1 &1 be+ satisfi$ ) ! accuracy ( ! ,european me?od ( survey+1 he resolv$ to h a survey made ( ! :ole empire on ! same pr9ciples4 ,?is great "w 0 begun 9 ,july #agjh1 & ! complet$ maps 7 pres5t$ to ! emp]or 9 #agah4 ,! records pres]v$ 9 ea* city 7 exam9$1 topographical in=mation 0 dilig5tly collect$1 & ! ,jesuit fa!rs *eck$ _! triangulation by m]idian altitudes ( ! sun & pole /> & by a sy/em ( remeasurem5ts4 ,! result 0 a m accurate map ( ,*9a ?an exi/$1 at t "t1 ( any c\ntry 9 ,europe4 ,kang-hi next ord]$ a simil> map to be made ( ,tibet1 ! survey be+ execut$ by two lamas :o 7 c>efully tra9$ z surveyors by ! ,jesuits at ,pek+4 ,f ^! surveys 7 con/ruct$ ! well-kn[n maps :i* 7 =w>d$ to ,duhalde1 & :i* ,d',anville utiliz$ = 8 atlas4 ,! #ah? c5tury4 ,asia4 ,sev]al ,european mission>ies _h previ\sly f\nd _! way f ,india to ,tibet4 ,antonio ,&rada1 9 #afbd1 0 ! fir/ ,european to ent] ,tibet s9ce ! visit ( ,fri> ,odoric 9 #acbe4 ,! next j\rney 0 t ( ,fa!rs ,grueb] & ,dorville ab\t #affj1 :o succe$$ 9 pass+ f ,*9a1 ?r\gh ,tibet1 into ,india4 ,9 #agae ,fa!rs ,desid]i & ,freyre made _! way f ,agra1 across ! ,himalayas1 to ,lhasa1 & ! ,capu*9 ,fri> ,orazio della ,p5na resid$ 9 t city f #agce until #agdg4 ,b ! mo/ rem>kable j\rney 9 ?is direction 0 p]=m$ by a ,dut* travell] nam$ ,samuel van de ,putte4 ,he left ,holl& 9 #agah1 w5t by l& ?r\gh ,p]sia to ,india1 & ev5tually made 8 way to ,lhasa1 :]e he resid$ = a long "t4 ,he w5t !nce to ,*9a1 return$ to ,lhasa1 & 0 9 ,india 9 "t to be an eye-witness ( ! sack ( ,delhi by ,nadir ,%ah 9 #agcg4 ,9 #agdc he left ,india & di$ at ,batavia on ! #bg? ( ,septemb] #agde4 ,! premature dea? ( ?is illu/ri\s travell] is ! m to be lam5t$ because 8 va/ k di$ ) him4 ,two ,engli% missions s5t by ,w>r5 ,ha/+s to ,tibet1 "o l$ by ,george ,bogle 9 #aggd1 & ! o!r by ,capta9 ,turn] 9 #aghc1 complete ,tibetan exploration 9 ! #ah? c5tury4 ,f ,p]sia mu* new in=mation 0 suppli$ by ,jean ,*>d91 ,jean ,tav]ni]1 ,*>les ,hamilton1 ,jean de ,!v5ot & ,fa!r ,jude ,krus9ski1 & by ,engli% trad]s on ! ,caspian4 ,9 #agch ,john ,elton trad$ betwe5 ,a/rakhan & ! ,p]sian port ( ,enzeli on ! ,caspian1 & und]took to build a fleet = ,nadir ,%ah4 ,ano!r ,engli% m]*ant1 nam$ ,jonas ,hanway1 arriv$ at ,a/rabad f ,russia1 & travell$ to ! camp ( ,nadir at ,kazv94 ,"o la/+ & valuable result ( ,hanway's w&]+s 0 a *>m+ book ( travels4 ,9 #agjj ,guillaume ,delisle publi%$ 8 map ( ! cont95ts ( ! ,old ,_w2 & 8 successor ,d',anville produc$ 8 map ( ,india 9 #ageb4 ,d',anville's map conta9$ all t 0 !n kn[n1 b t5 ye>s aft]w>ds ,major ,r5nell began 8 survey+ lab\rs1 :i* ext5d$ ov] ! p]iod f #agfc to #aghb4 ,8 survey cov]$ an area #ijj m4 long by #cjj wide1 f ! ea/]n conf9es ( ,b5gal to ,agra1 & f ! ,himalayas to ,calpi4 ,r5nell 0 indefatigable 9 collect+ geographical in=mation2 8 ,b5gal atlas appe>$ 9 #agha1 8 fam\s map ( ,india 9 #aghh & ! memoir 9 #agib4 ,surveys 7 also made along ! ,indian coa/s4 ,arabia receiv$ v c>eful att5tion1 9 ! #ah? c5tury1 f ! ,dani% sci5tific mission1 :i* includ$ ,c>/5 ,niebuhr among its memb]s4 ,niebuhr l&$ at ,loheia1 on ! coa/ ( ,yem51 9 ,decemb] #agfb1 & w5t by l& to ,sana4 ,all ! o!r memb]s ( ! mission di$1 b he proce$$ f ,mokha to ,bombay4 ,he !n made a j\rney ?r\gh ,p]sia & ,syria to ,con/ant9ople1 return+ to ,cop5hag5 9 #agfg4 ,8 valuable "w1 ! ,description ( ,arabia1 0 publi%$ 9 #aggb1 & 0 foll[$ 9 #aggd-aggh by two volumes ( travels 9 ,asia4 ,! great travell] surviv$ until #ahae1 :5 he di$ at ! age ( eighty-two4 ,africa4 ,james ,bruce ( ,k9naird1 ! contempor>y ( ,niebuhr1 0 equally devot$ to ,ea/]n travel2 & 8 pr9cipal geographical "w 0 ! trac+ ( ! ,blue ,nile f its s\rce to its junction ) ! ,:ite ,nile4 ,be=e ! dea? ( ,bruce an ,african ,association 0 =m$1 9 #aghh1 = collect+ in=mation respect+ ! int]ior ( t cont95t1 ) ,major ,r5nell & ,sir ,joseph ,banks z lead+ memb]s4 ,! association fir/ employ$ ,john ,l$y>d 7:o _h previ\sly made an extraord9>y j\rney into ,sib]ia7 to cross ,africa f ea/ to we/ on ! p>allel ( ! ,nig]1 & ,william ,lucas to cross ! ,sah>a to ,fezzan4 ,lucas w5t f ,tripoli to ,mesurata1 obta9$ "s in=mation respect+ ,fezzan & return$ 9 #aghi4 ,"o ( ! *ief problems ! association wi%$ to solve 0 t ( ! exi/5ce & c\rse ( ! riv] ,nig]1 :i* 0 believ$ by "s au?orities to be id5tical ) ! ,congo4 ,mungo ,p>k1 !n an assi/ant surgeon ( an ,indiaman1 volunte]$ 8 s]vices1 :i* 7 accept$ by ! association1 & 9 #agie he succe$$ 9 rea*+ ! t[n ( ,segu on ! ,nig]1 b 0 prev5t$ f cont9u+ 8 j\rney to ,timbuktu4 ,five ye>s lat] he accept$ an (f] f ! gov]nm5t to comm& an exp$ition into ! int]ior ( ,africa1 ! plan be+ to cross f ! ,gambia to ! ,nig] & desc5d ! latt] riv] to ! sea4 ,aft] los+ mo/ ( 8 companions he himself & ! re/ p]i%$ 9 a rapid on ! ,nig] at ,busa1 hav+ be5 attack$ f ! %ore by ord] ( a *ief :o ?\ght he _h n receiv$ suitable pres5ts4 ,8 "w1 h[ev]1 _h e/abli%$ ! fact t ! ,nig] 0 n id5tical ) ! ,congo4 ,:ile ! ,briti% 7 at "w 9 ! direction ( ! ,nig]1 ! ,portuguese 7 n unm9dful ( _! old explor+ fame4 ,9 #agih ,dr ,f4,j4,m4 de ,lac]da1 an accompli%$ a/ronom]1 0 appo9t$ to comm& a sci5tific exp$ition ( discov]y to ! nor? ( ! ,zambesi4 ,he />t$ 9 ,july1 cross$ ! ,mu*5ja ,m\nta9s1 & rea*$ ! capital ( ! ,cazembe1 :]e he di$ ( fev]4 ,lac]da left a valuable record ( 8 adv5tur\s j\rney2 b ) ,mungo ,p>k & ,lac]da ! hi/ory ( ,african exploration 9 ! #ah? c5tury closes4 ,s\? ,am]ica4 ,9 ,s\? ,am]ica sci5tific exploration 0 active dur+ ?is p]iod4 ,! great geographical ev5t ( ! c5tury1 z reg>ds t cont95t1 0 ! measurem5t ( an arc ( ! m]idian4 ,! und]tak+ 0 propos$ by ! ,fr5* ,academy z "p ( an inve/igation ) ! object ( asc]ta9+ ! l5g? ( ! degree ne> ! equator & ne> ! pole respectively s z to det]m9e ! figure ( ! e>?4 ,a commission left ,p>is 9 #agce1 consi/+ ( ,*>les ,m>ie de la ,condam9e1 ,pi]re ,b\gu]1 ,l\is ,god9 & ,joseph de ,jussieu ! naturali/4 ,spa9 appo9t$ two accompli%$ naval (fic]s1 ! bro!rs ,ulloa1 z coadjutors4 ,! op]ations 7 c>ri$ on dur+ eight ye>s on a pla9 to ! s\? ( ,quito2 &1 9 addition to 8 memoir on ?is memorable measurem5t1 ,la ,condam9e collect$ mu* valuable geographical in=mation dur+ a voyage d[n ! ,amazon4 ,! arc measur$ 0 #c deg4 #g' #c 9 l5g?2 & ! "w consi/$ ( two measur$ bases connect$ by a s]ies ( triangles1 "o nor? & ! o!r s\? ( ! equator1 on ! m]idian ( ,quito4 ,contemporane\sly1 9 #agch1 ,pi]re ,l\is ,moreau de ,maup]tuis1 ,alexis ,claude ,clairaut1 ,*>les ,eti5ne ,l\is ,camus1 ,pi]re ,*>les ,lemonni] & ! ,sw$i% physici/ ,celsius measur$ an arc ( ! m]idian 9 ,lapl&4 ,! ,pacific ,ocean4 ,! ,briti% & ,fr5* gov]nm5ts despat*$ sev]al exp$itions ( discov]y into ! ,pacific & r\nd ! _w dur+ ! #ah? c5tury4 ,!y 7 prec$$ by ! wond]ful & romantic voyages ( ! buccane]s4 ,! n>ratives ( su* m5 z ,woodes ,rog]s1 ,edw>d ,davis1 ,george ,%elvocke1 ,clipp]ton & ,william ,dampi]1 c nev] fail to int]e/1 :ile !y are n )\t geographical value4 ,! works ( ,dampi] are especially valuable1 & ! n>ratives ( ,william ,funnell & ,lionel ,waf] furni%$ ! be/ acc\nts !n extant ( ! ,i/hmus ( ,d>i54 ,dampi]'s lit]>y ability ev5tually secur$ = him a commission 9 ! k+'s s]vice2 & he 0 s5t on a voyage ( discov]y1 dur+ :i* he explor$ "p ( ! coa/s ( ,au/ralia & ,new ,gu9ea1 & discov]$ ! /rait :i* be>s 8 "n betwe5 ,new ,gu9ea & ,new ,brita91 return+ 9 #agja4 ,9 #agba ,jacob ,roggewe9 0 despat*$ on a voyage ( "s importance across ! ,pacific by ! ,dut* ,we/ ,india ,company1 dur+ :i* he discov]$ ,ea/] ,isl& on ! #f? ( ,april #agbb4 ,! voyage ( ,"l ,anson to ! ,pacific 9 #agdj-agdd 0 ( a pr$atory "*1 & he lo/ m ?an half 8 m5 f scurvy2 :ile x is n pleasant to reflect t at ! v "t :5 ! ,fr5* & ,spani>ds 7 measur+ an arc ( ! m]idian at ,quito1 ! ,briti% "u ,anson 7 pillag+ along ! coa/ ( ! ,pacific & burn+ ! t[n ( ,payta4 ,b a romantic int]e/ atta*es to ! wreck ( ! ,wag]1 "o ( ,anson's fleet1 on a des]t isl& ne> ,*iloe1 = x bore fruit 9 ! *>m+ n>rative ( ,capta9 ,john ,byron1 :i* w endure = all "t4 ,9 #agfd ,byron himself 0 s5t on a voyage ( discov]y r\nd ! _w1 :i* l$ imm$iately aft] 8 return to ! despat* ( ano!r to complete 8 "w1 "u ! comm& ( ,capta9 ,samuel ,wallis4 ,! exp$ition1 consi/+ ( ! ,dolph9 comm&$ by ,wallis1 & ! ,swall[ "u ,capta9 ,philip ,c>t]et1 sail$ 9 ,septemb] #agff1 b ! %ips 7 sep>at$ on ent]+ ! ,pacific f ! ,/rait ( ,magellan4 ,wallis discov]$ ,tahiti on ! #ai? ( ,june #agfg1 & he gave a detail$ acc\nt ( t isl&4 ,he return$ to ,engl& 9 ,may #agfh4 ,c>t]et discov]$ ! ,*>lotte & ,gl\ce/] ,isl&s1 & ,pitcairn ,isl& on ! #bnd ( ,july #agfg2 revisit$ ! ,santa ,cruz gr\p1 :i* 0 discov]$ by ,m5dana & ,quiros2 & discov]$ ! /rait sep>at+ ,new ,brita9 f ,new ,irel&4 ,he rea*$ ,spi!ad aga9 9 ,febru>y #agfi4 ,wallis & ,c>t]et 7 foll[$ v closely by ! ,fr5* exp$ition ( ,b\ga9ville1 :i* sail$ f ,nantes 9 ,novemb] #agff4 ,b\ga9ville _h fir/ to p]=m ! unpleasant task ( deliv]+ up ! ,falkl& ,isl&s1 :]e he _h enc\rag$ ! =mation ( a ,fr5* settlem5t1 to ! ,spani>ds4 ,he !n ent]$ ! ,pacific1 & rea*$ ,tahiti 9 ,april #agfh4 ,pass+ ?r\gh ! ,new ,hebrides gr\p he t\*$ at ,batavia1 & arriv$ at ,/ ,malo aft] an abs5ce ( two ye>s & f\r mon?s4 ,capta9 ,cook4 ,! ?ree voyages ( ,capta9 ,james ,cook =m an era 9 ! hi/ory ( geographical discov]y4 ,9 #agfg he sail$ = ,tahiti1 ) ! object ( obs]v+ ! transit ( ,v5us1 accompani$ by two naturali/s1 ,sir ,joseph ,banks & ,dr ,sol&]1 a pupil ( ,l9naeus1 z well z by two a/ronom]s4 ,! transit 0 obs]v$ on ! #crd ( ,june #agfi4 ,aft] explor+ ,tahiti & ! ,society gr\p1 ,cook sp5t six mon?s survey+ ,new ,zeal&1 :i* he discov]$ to be an isl&1 & ! coa/ ( ,new ,s\? ,wales f latitude #ch deg4 ,s4 to ! nor!rn extremity4 ,! belief 9 a va/ ,ant>ctic cont95t /ret*+ f> into ! temp]ate zone _h nev] be5 ab&on$1 & 0 vehem5tly ass]t$ by ,*>les ,dalrymple1 a disappo9t$ c&idate nom9at$ by ! ,royal ,society = ! comm& ( ! ,transit exp$ition ( #agfi4 ,9 #aggb ! ,fr5* explor] ,yves ,k]guel5 de ,trem>ec _h discov]$ ! l& t be>s 8 "n 9 ! ,s\? ,indian ,ocean )\t recogniz+ x to be an isl&1 & naturally believ$ x to be "p ( ! s\!rn cont95t4 ,cook's second voyage 0 ma9ly int5d$ to settle ! "q ( ! exi/5ce ( su* a cont95t once = all1 & to def9e ! limits ( any l& t might exi/ 9 navigable seas t[>ds ! ,ant>ctic circle4 ,james ,cook at 8 fir/ attempt rea*$ a s\? latitude ( #eg deg4 #ae'4 ,on a second cruise f ! ,society ,isl&s1 9 #aggc1 he1 fir/ ( all m51 cross$ ! ,ant>ctic circle1 & 0 /opp$ by ice 9 #ga deg4 #aj' ,s4 ,dur+ ! second voyage ,cook visit$ ,ea/] ,isl&1 discov]$ sev]al isl&s ( ! ,new ,hebrides & ,new ,cal$onia2 & on 8 way home by ,cape ,horn1 9 ,m>* #aggd1 he discov]$ ! ,s&wi* ,isl& gr\p & describ$ ,s\? ,georgia4 ,he prov$ conclusively t any s\!rn cont95t t might exi/ lay "u ! pol> ice4 ,! ?ird voyage 0 int5d$ to attempt ! passage f ! ,pacific to ! ,atlantic by ! nor?-ea/4 ,! ,resolution & ,discov]y sail$ 9 #aggf1 & ,cook aga9 took ! r\te by ! ,cape ( ,good ,hope4 ,on rea*+ ! ,nor? ,am]ican coa/1 he proce$$ nor?w>d1 fix$ ! position ( ! we/]n extremity ( ,am]ica & survey$ ,b]+ ,/rait4 ,he 0 /opp$ by ! ice 9 #gj deg4 #da' ,n41 & nam$ ! f>!/ visible po9t on ! ,am]ican %ore ,icy ,cape4 ,he !n visit$ ! ,asiatic %ore & discov]$ ,cape ,nor?4 ,return+ to ,hawaii1 ,cook 0 murd]$ by ! natives4 ,on ! #ad? ( ,febru>y #aggi1 8 second1 ,capta9 ,edw>d ,cl]ke1 took comm&1 & proce$+ to ,petropavlovsk 9 ! foll[+ summ]1 he aga9 exam9$ ! edge ( ! ice1 b only got z f> z #gj deg4 #cc' ,n4 ,! %ips return$ to ,engl& 9 ,octob] #aghj4 ,9 #aghe ! ,fr5* gov]nm5t c>efully fitt$ out an exp$ition ( discov]y at ,bre/1 :i* 0 plac$ "u ! comm& ( ,francois ,la ,p]\se1 an accompli%$ & exp]i5c$ (fic]4 ,aft] t\*+ at ,concepcion 9 ,*ile & at ,ea/] ,isl&1 ,la ,p]\se proce$$ to ,hawaii & !nce to ! coa/ ( ,cali=nia1 ( :i* he has giv5 a v int]e/+ acc\nt4 ,he !n cross$ ! ,pacific to ,macao1 & 9 ,july #aghg he proce$$ to explore ! ,gulf ( ,t>t>y & ! %ores ( ,sakhal91 rema9+ "s "t at ,ca/ries ,bay1 s nam$ aft] ! ,fr5* m9i/] ( m>9e4 ,!nce he w5t to ! ,kurile ,isl&s & ,kam*atka1 & sail$ f ! f> nor? d[n ! m]idian to ! ,navigator & ,fri5dly ,isl&s4 ,he 0 9 ,botany ,bay 9 ,janu>y #aghh2 & sail+ !nce1 ! explor]1 8 %ip & crew 7 nev] se5 aga94 ,_! fate 0 long unc]ta94 ,9 ,septemb] #agia ,capta9 ,anto9e d',entreca/eaux sail$ f ,bre/ ) two vessels to seek = tid+s4 ,he visit$ ! ,new ,hebrides1 ,santa ,cruz1 ,new ,cal$onia & ,solomon ,isl&s1 & made c>eful ?\gh r\gh surveys ( ! ,l\isiade ,ar*ipelago1 isl&s nor? ( ,new ,brita9 & "p ( ,new ,gu9ea4 ,d',entreca/eaux di$ on bo>d 8 %ip on ! #bj? ( ,july #agic1 )\t asc]ta9+ ! fate ( ,la ,p]\se4 ,capta9 ,pet] ,dillon at l5g? asc]ta9$1 9 #ahbh1 t ! %ips ( ,la ,p]\se _h be5 wreck$ on ! isl& ( ,vanikoro dur+ a hurricane4 ,! "w ( ,capta9 ,cook bore fruit 9 _m ways4 ,8 ma/]1 ,capta9 ,william ,bligh1 0 s5t 9 ! ,b\nty to convey breadfruit plants f ,tahiti to ! ,we/ ,indies4 ,he rea*$ ,tahiti 9 ,octob] #aghh1 & 9 ,april #aghi a mut9y broke out1 & he1 ) sev]al (fic]s & m51 0 ?ru/ into an op5 boat 9 mid-ocean4 ,dur+ ! rem>kable voyage he !n made to ,timor1 ,bligh pass$ among/ ! nor!rn isl&s ( ! ,new ,hebrides1 :i* he nam$ ! ,banks ,gr\p1 & made sev]al runn+ surveys4 ,he rea*$ ,engl& 9 ,m>* #agij4 ,! ,p&ora1 "u ,capta9 ,edw>ds1 0 s5t out 9 se>* ( ! ,b\nty1 & discov]$ ! isl&s ( ,*]ry & ,mitre1 ea/ ( ! ,santa ,cruz gr\p1 b %e 0 ev5tually lo/ on a reef 9 ,torres ,/rait4 ,9 #agif-agig ,capta9 ,wilson1 9 ! mission>y %ip ,duff1 discov]$ ! ,gambi] & o!r isl&s1 & r$iscov]$ ! isl&s kn[n to & se5 by ,quiros1 b s9ce call$ ! ,duff ,gr\p4 ,ano!r result ( ,capta9 ,cook's "w 0 ! colonization ( ,au/ralia4 ,on ! #ah? ( ,janu>y #aghh ,admiral ,phillip & ,capta9 ,hunt] arriv$ 9 ,botany ,bay 9 ! ,supply & ,sirius1 foll[$ by six transports1 & e/abli%$ a colony at ,port ,jackson4 ,surveys 7 !n und]tak5 9 sev]al directions4 ,9 #agie & #agif ,mat!w ,fl9d]s & ,george ,bass 7 engag$ on explor+ "w 9 a small boat call$ ! ,tom ,?umb4 ,9 #agig ,bass1 :o _h be5 a surgeon1 made an exp$ition s\?w>ds1 cont9u$ ! "w ( ,cook f ,ram ,head1 & explor$ ! /rait :i* be>s 8 "n1 & 9 #agih he & ,fl9d]s 7 survey+ on ! ea/ coa/ ( ,van ,diem5's l&4 ,yet ano!r outcome ( ,capta9 ,cook's "w 0 ! voyage ( ,george ,vanc\v]1 :o _h s]v$ z a mid%ipman 9 ,cook's second & ?ird voyages4 ,! ,spani>ds "u ,quadra _h begun a survey ( nor?-we/]n ,am]ica & occupi$ ,nootka ,s\nd1 :i* _! gov]nm5t ev5tually agre$ to surr5d]4 ,capta9 ,vanc\v] 0 s5t out to receive ! cession1 & to survey ! coa/ f ,cape ,m5doc9o nor?w>ds4 ,he comm&$ ! old ,discov]y1 & 0 at "w dur+ ! seasons ( #agib1 #agic & #agid1 w9t]+ at ,hawaii4 ,return+ home 9 #agie1 he complet$ 8 n>rative & a valuable s]ies ( *>ts4 ,arctic regions4 ,! #ah? c5tury saw ! ,arctic coa/ ( ,nor? ,am]ica rea*$ at two po9ts1 z well z ! fir/ sci5tific attempt to rea* ! ,nor? ,pole4 ,! ,hudson ,bay ,company _h be5 incorporat$ 9 #afgj1 & its s]vants soon ext5d$ _! op]ations ov] a wide area to ! nor? & we/ ( ,canada4 ,9 #agda ,capta9 ,*ri/oph] ,middleton 0 ord]$ to solve ! "q ( a passage f ,hudson ,bay to ! we/w>d4 ,leav+ ,=t ,*ur*ill 9 ,july #agdb1 he discov]$ ! ,wag] riv] & ,repulse ,bay4 ,he 0 foll[$ by ,capta9 ,w4 ,moor 9 #agdf1 & ,capta9 ,coats 9 #agea1 :o exam9$ ! ,wag] ,inlet up to ! end4 ,9 ,novemb] #agfi ,samuel ,he>ne 0 s5t by ! ,hudson ,bay ,company to discov] ! sea on ! nor? side ( ,am]ica1 b 0 oblig$ to return4 ,9 ,febru>y #aggj he set out aga9 f ,=t ,pr9ce ( ,wales2 b1 aft] great h>d%ips1 he 0 aga9 =c$ to return to ! =t4 ,he />t$ once m 9 ,decemb] #agga1 & at l5g? rea*$ ! ,copp]m9e riv]1 :i* he survey$ to its m\?1 b 8 obs]vations are unreliable4 ,) ! same object ,alex&] ,mack5zie1 ) a p>ty ( ,canadians1 set out f ,=t ,*ippewyan on ! #crd ( ,june #aghi1 & desc5d+ ! great riv] :i* n[ be>s ! explor]'s "n rea*$ ! ,arctic sea4 ,9 ,febru>y #aggc ! ,royal ,society submitt$ a proposal to ! k+ = an exp$ition t[>ds ! ,nor? ,pole4 ,! exp$ition 0 fitt$ out "u ,capta9s ,con/ant9e ,phipps & ,skeff+ton ,lutwidge1 & ! highe/ latitude rea*$ 0 #hj deg4 #dh' ,n41 b no op5+ 0 discov]$ 9 ! heavy ,pol> pack4 ,! mo/ important ,arctic "w 9 ! #ah? c5tury 0 p]=m$ by ! ,russians1 = !y succe$$ 9 del9eat+ ! :ole ( ! nor!rn coa/ ( ,sib]ia4 ,"s ( ?is "w 0 possibly done at a /ill e>li] date4 ,! ,cossack ,simon ,dezhneff is ?\ght to h made a voyage1 9 ! summ] ( #afdh1 f ! riv] ,kolyma1 ?r\gh ,b]+ ,/rait 7:i* 0 r$iscov]$ by ,vitus ,b]+ 9 #agbh7 to ,anadyr4 ,betwe5 #agch & #agej ,man9 & ,/]leg(f made _! way 9 small sloops f ! m\? ( ! ,y5esei z f> nor? z #ge deg4 #ae' ,n4 ,! l& f ,taimyr to ,cape ,*elyusk91 ! mo/ nor!rn extremity ( ,sib]ia1 0 mapp$ 9 _m ye>s ( pati5t exploration by ,*elyusk91 :o rea*$ ! extreme po9t 7#gg deg4 #cd' ,n47 9 ,may #agdb4 ,to ! ea/ ( ,cape ,*elyusk9 ! ,russians enc\nt]$ great] difficulties4 ,!y built small vessels at ,yakutsk on ! ,l5a1 #ijj m4 f its m\?1 :5ce ! fir/ exp$ition 0 despat*$ "u ,lieut4 ,pronts*i*ev 9 #agce4 ,he sail$ f ! m\? ( ! ,l5a to ! m\? ( ! ,olonek1 :]e he w9t]$1 & on ! #a/ ( ,septemb] #agcf he got z f> z #gg deg4 #bi' ,n41 )9 #e m4 ( ,cape ,*elyusk94 ,bo? he & 8 "y wife di$ ( scurvy1 & ! vessel return$4 ,a second exp$ition1 "u ,lieut4 ,laptyev1 />t$ f ! ,l5a 9 #agci1 b enc\nt]$ masses ( drift ice 9 ,*atanga bay1 & ) ?is end$ ! voyages to ! we/w>d ( ! ,l5a4 ,sev]al attempts 7 also made to navigate ! sea f ! ,l5a to ! ,kolyma4 ,9 #agcf ,lieut4 ,laptyev sail$1 b 0 /opp$ by ! drift ice 9 ,augu/1 & 9 #agci1 dur+ ano!r trial1 he rea*$ ! m\? ( ! ,indigirka1 :]e he w9t]$4 ,9 ! season ( #agdj he cont9u$ 8 voyage to beyond ! ,kolyma1 w9t]+ at ,nizhni ,kolymsk4 ,9 ,septemb] #agdj ,vitus ,b]+ sail$ f ,okhotsk on a second ,arctic voyage ) ,george ,william ,/ell] on bo>d z naturali/4 ,9 ,june #agda he nam$ ! magnific5t peak on ! coa/ ( ,nor? ,am]ica ,m\nt ,/ ,elias & explor$ ! ,aleutian ,isl&s4 ,9 ,novemb] ! %ip 0 wreck$ on ,b]+ ,isl&2 & ! gallant ,dane1 worn out ) scurvy1 di$ !re on ! #h? ( ,decemb] #agda4 ,9 ,m>* #aggj a m]*ant nam$ ,liakhov saw a l>ge h]d ( re9de] com+ f ! nor? to ! ,sib]ian coa/1 :i* induc$ him to />t 9 a sl$ge 9 ! direction :5ce !y came4 ,?us he rea*$ ! ,new ,sib]ian or ,liakhov ,isl&s1 & = ye>s aft]w>ds ! seek]s = fossil ivory resort$ to !m4 ,! ,russian ,capta9 ,vassili ,*its*akov 9 #agfe & #agff made two p]sev]+ attempts to p5etrate ! ice nor? ( ,spitsb]g51 & rea*$ #hj deg4 #cj' ,n41 :ile ,russian p>ties twice w9t]$ at ,bell ,s\nd4 ,geographical societies4 ,9 review+ ! progress ( geographical discov]y ?us f>1 x has be5 possible to keep fairly closely to a *ronological ord]4 ,b 9 ! #ai? c5tury & aft] explor+ "w 0 s g5]ally & /eadily ma9ta9$ 9 all directions1 & 0 9 s _m cases n>r[$ d[n f long j\rneys to detail$ surveys )9 relatively small areas1 t x becomes desirable to cov] ! :ole p]iod at "o view = c]ta9 great divisions ( ! _w4 7,see ,,africa2 ,,asia2 ,,au/ralia2 ,,pol> ,,regions2 c47 ,"h1 h[ev]1 may be notic$ ! developm5t ( geographical societies devot$ to ! enc\ragem5t ( exploration & rese>*4 ,! fir/ ( ! exi/+ geographical societies 0 t ( ,p>is1 f\nd$ 9 #ahbe "u ! title ( ,la ,societe de ,geographie4 ,! ,b]l9 ,geographical ,society 7,gesells*aft fur ,erdkunde7 is second 9 ord] ( s5iority1 hav+ be5 f\nd$ 9 #ahbg4 ,! ,royal ,geographical ,society1 :i* 0 f\nd$ 9 ,london 9 #ahcj1 comes ?ird on ! li/2 b x may be view$ z a direct result ( ! e>li] ,african ,association f\nd$ 9 #aghh4 ,sir ,john ,b>r[1 ,sir ,john ,cam ,hobh\se 7,"l ,br\ghton71 ,sir ,rod]ick ,mur*ison1 ,mr ,rob]t ,br[n & ,mr ,b>tle ,fr]e =m$ ! f\ndation committee ( ! ,royal ,geographical ,society1 & ! fir/ presid5t 0 ,"l ,god]i*4 ,! action ( ! society 9 supply+ practical in/ruction to int5d+ travell]s1 9 a/ronomy1 survey+ & ! v>i\s bran*es ( sci5ce useful to collectors1 has _h mu* to d ) advancem5t ( discov]y4 ,s9ce ! w> ( #ahgj _m geographical societies h be5 e/abli%$ on ! cont95t ( ,europe4 ,at ! close ( ! #ai? c5tury !re 7 upw>ds ( #ajj su* societies 9 ! _w1 ) m ?an #ej1jjj memb]s1 & ov] #aej j\rnals 7 devot$ entirely to geographical subjects4#aa ,! great developm5t ( photography has be5 a notable aid to explor]s1 n only by plac+ at _! disposal a fai?ful & ready means ( record+ ! features ( a c\ntry & ! types ( inhabitants1 b by supply+ a me?od ( quick & accurate topographical survey+4 ,,! ,,pr9ciples ,,( ,,geography ,z reg>ds ! scope ( geography1 ! ord] ( ! v>i\s dep>tm5ts & _! int]-relation1 !re is little diff]5ce ( op9ion1 & ! pr9ciples ( geography#ab are n[ g5]ally accept$ by mod]n geograph]s4 ,! ord] 9 :i* ! v>i\s subjects are treat$ 9 ! foll[+ sket* is ! natural succession f fundam5tal to dep5d5t facts1 :i* corresponds also to ! evolution ( ! div]sities ( ! e>?'s cru/ & ( its inhabitants4 ,ma!matical geography4 ,! fundam5tal geographical conceptions are ma!matical1 ! relations ( space & =m4 ,! figure & dim5sions ( ! e>? are ! fir/ ( ^!4 ,!y are asc]ta9$ by a comb9ation ( actual measurem5t ( ! highe/ precision on ! surface & angul> obs]vations ( ! positions ( ! heav5ly bodies4 ,! sci5ce ( geodesy is "p ( ma!matical geography1 ( :i* ! arts ( survey+ & c>tography are applications4 ,! motions ( ! e>? z a planet mu/ be tak5 into acc\nt1 z !y r5d] possible ! det]m9ation ( position & direction by obs]vations ( ! heav5ly bodies4 ,! diurnal rotation ( ! e>? furni%es two fix$ po9ts or poles1 ! axis jo9+ :i* is fix$ or ne>ly s 9 its direction 9 space4 ,! rotation ( ! e>? ?us fixes ! directions ( nor? & s\? & def9es ^? ( ea/ & we/4 ,! angle :i* ! e>?'s axis makes ) ! plane 9 :i* ! planet revolves r\nd ! sun det]m9es ! v>y+ seasonal di/ribution ( sol> radiation ov] ! surface & ! ma!matical zones ( climate4 ,ano!r important consequ5ce ( rotation is ! deviation produc$ 9 mov+ bodies relatively to ! surface4 ,9 ! =m kn[n z ,f]rell's ,law ?is runs3 ,if a body moves 9 any direction on ! e>?'s surface1 !re is a deflect+ =ce :i* arises f ! e>?'s rotation :i* t5ds to deflect x to ! "r 9 ! nor!rn hemisph]e b to ! left 9 ! s\!rn hemisph]e4 ,! deviation is ( importance 9 ! movem5t ( air1 ( ocean curr5ts1 & to "s ext5t ( riv]s4#ac ,physical geography4 ,9 popul> usage ! words physical geography h come to mean geography view$ f a p>ticul> /&po9t r ?an any special dep>tm5t ( ! subject4 ,! popul> mean+ is bett] convey$ by ! ^w physiography1 a t]m :i* appe>s to h be5 introduc$ by ,l9naeus1 & 0 re9v5t$ z a sub/itute = ! cosmography ( ! middle ages by ,pr(essor ,huxley4 ,al?\gh ! t]m has s9ce be5 limit$ by "s writ]s to "o p>ticul> "p ( ! subject1 x seems be/ to ma9ta9 ! orig9al & lit]al mean+4 ,9 ! /rict] s5se1 physical geography is t "p ( geography :i* involves ! processes ( contempor>y *ange 9 ! cru/ & ! circulation ( ! fluid envelopes4 ,x ?us draws ^u physics = ! explanation ( ! ph5om5a ) ! space-relations ( :i* x is specially conc]n$4 ,physical geography naturally falls into ?ree divisions1 deal+ respectively ) ! surface ( ! li?osph]e--geomorphology2 ! hydrosph]e--oceanography2 & ! atmosph]e--climatology4 ,all ^! re/ ^u ! facts ( ma!matical geography1 & ! ?ree are s closely int]-relat$ t !y _c be rigidly sep>at$ 9 any discussion4 ,geomorphology4 ,geomorphology is ! "p ( geography :i* deals ) t]re/rial relief1 includ+ ! subm>9e z well z ! suba]ial portions ( ! cru/4 ,! hi/ory ( ! orig9 ( ! v>i\s =ms belongs to geology1 & c be completely /udi$ only by geological me?ods4 ,b ! relief ( ! cru/ is n a f9i%$ piece ( sculpture2 ! =ms are = ! mo/ "p transitional1 ow+ _! *>act]i/ic outl9es to ! process by :i* !y are produc$2 !re=e ! geograph] mu/1 = /rictly geographical purposes1 take "s acc\nt ( ! processes :i* are n[ 9 action modify+ ! =ms ( ! cru/4 ,op9ion /ill diff]s z to ! ext5t to :i* ! geograph]'s "w %\ld ov]lap t ( ! geologi/4 ,! prim>y di/9ction ( ! =ms ( ! cru/ is t betwe5 elevations & depressions4 ,grant+ t ! geoid or mean surface ( ! ocean is a uni=m sph]oid1 ! di/ribution ( l& & wat] approximately indicates a division ( ! surface ( ! globe into two areas1 "o ( elevation & "o ( depression4 ,! increas+ numb] ( measurem5ts ( ! height ( l& 9 all cont95ts & isl&s1 & ! v detail$ levell+s 9 ^? c\ntries :i* h be5 ?or\ghly survey$1 enable ! av]age elevation ( ! l& above sea-level to be fairly e/imat$1 al?\gh _m va/ gaps 9 accurate k rema91 & ! e/imate is n an exact "o4 ,! only "p ( ! sea-b$ ! configuration ( :i* is at all well kn[n is ! zone bord]+ ! coa/s :]e ! dep? is less ?an ab\t #ajj fa?oms or #bjj metres1 i4e4 ^? p>ts :i* sailors speak ( z 9 s\nd+s4 ,actual or project$ r\tes = telegraph cables across ! deep sea h also be5 s\nd$ ) extreme accuracy 9 _m cases2 b beyond ^! l9es ( s\nd+ ! va/ spaces ( ! ocean rema9 unplumb$ save = ! r>e rese>*es ( sci5tific exp$itions1 su* z ^? ( ! ,*all5g]1 ! ,valdivia1 ! ,albatross & ! ,scotia4 ,?us ! be/ approximation to ! av]age dep? ( ! ocean is little m ?an an exp]t guess2 yet a fair approximation is probable = ! features ( sub-oceanic relief are s mu* m uni=m ?an ^? ( ! l& t a small] numb] ( fix$ po9ts is requir$ to det]m9e !m4 ,cru/al relief4 ,! *ief elem5t ( unc]ta9ty z to ! l>ge/ features ( ! relief ( ! e>?'s cru/ is due to ! unexplor$ area 9 ! ,arctic region & ! l>g] regions ( ! ,ant>ctic1 ( :i* we "k no?+4 ,we "k t ! e>?'s surface if unveil$ ( wat] w\ld exhibit a great region ( elevation arrang$ ) a c]ta9 r\gh radiate symmetry r\nd ! nor? pole1 & ext5d+ s\?w>ds 9 ?ree unequal arms :i* tap] to po9ts 9 ! s\?4 ,a depression surr\nds ! little-kn[n s\? pol> region 9 a cont9u\s r+ & ext5ds nor?w>ds 9 ?ree va/ holl[s ly+ betwe5 ! arms ( ! elevat$ area4 ,s f> only is x possible to speak ) c]ta9ty1 b x is p]missible to take a few /eps into ! twilight ( dawn+ k & indicate ! *ief subdivisions :i* are likely to be e/abli%$ 9 ! great cru/-holl[ & ! great cru/-heap4 ,! b\nd>y betwe5 ^! %\ld obvi\sly be ! mean surface ( ! sph]e4 ,sir ,john ,murray d$uc$ ! mean height ( ! l& ( ! globe z ab\t #bbej ft4 above sea-level1 & ! mean dep? ( ! oceans z #bjhj fa?oms or #ab1dhj ft4 bel[ sea-level4#ad ,calculat+ ! area ( ! l& at #ee1jjj1jjj sq4 m4 7or #bh4f ( ! surface71 & t ( ! oceans z #acg1bjj1jjj sq4 m4 7or #ga4d ( ! surface71 he f\nd t ! volume ( ! l& above sea-level 0 #bc1dej1jjj cub4 m41 ! volume ( wat] bel[ sea-level #cbc1hjj1jjj1 & ! total volume ( ! wat] equal to ab\t #a_/fff? ( ! volume ( ! :ole globe4 ,f ^! data1 z revis$ by ,a4 ,supan1#ae ,h4,r4 ,mill calculat$ ! position ( mean sph]e-level at ab\t #aj1jjj ft4 or #agjj fa?oms bel[ sea-level4 ,he %[$ t an imag9>y sph]oidal %ell1 conc5tric ) ! e>? & cutt+ ! slope betwe5 ! elevat$ & depress$ areas at ! cont\r-l9e ( #agjj fa?oms1 w\ld n only leave above x a volume ( ! cru/ equal to ! volume ( ! holl[ left bel[ x1 b w\ld also divide ! surface ( ! e>? s t ! area ( ! elevat$ region 0 equal to t ( ! depress$ region4#af ,areas ( ! cru/ accord+ to ,murray4 ,a simil> obs]vation 0 made almo/ simultane\sly by ,romieux1#ag :o fur!r speculat$ on ! equilibrium betwe5 ! weight ( ! elevat$ l& mass & t ( ! total wat]s ( ! ocean1 & d$uc$ "s int]e/+ relations betwe5 !m4 ,murray1 z ! result ( 8 /udy1 divid$ ! e>?'s surface into ?ree zones--! cont95tal area conta9+ all dry l&1 ! transitional area includ+ ! subm>9e slopes d[n to #ajjj fa?oms1 & ! abysmal area consi/+ ( ! floor ( ! ocean beyond t dep?2 & ,mill propos$ to take ! l9e ( mean-sph]e level1 in/ead ( ! empirical dep? ( #ajjj fa?oms1 z ! b\nd>y betwe5 ! transitional & abysmal areas4 ,an elaborate criticism ( all ! exi/+ data reg>d+ ! volume relations ( ! v]tical relief ( ! globe 0 made 9 #ahid by ,pr(essor ,h]mann ,wagn]1 ^: recalculations ( volumes & mean heights--! be/ results :i* h yet be5 obta9$--l$ to ! foll[+ conclusions4#ah ,areas ( ! cru/ accord+ to ,wagn]4 ,! area ( ! dry l& 0 tak5 z #bh4c ( ! surface ( ! globe1 & t ( ! oceans z #ga4g4 ,! mean height d$uc$ = ! l& 0 #bcjj ft4 above sea-level1 ! mean dep? ( ! sea #aa1ejj ft4 bel[1 :ile ! position ( mean-sph]e level comes out z #gejj ft4 7#abej fa?oms7 bel[ sea-level4 ,f ?is x w\ld appe> t #dc ( ! e>?'s surface 0 above & #eg bel[ ! mean level4 ,x mu/ be not$1 h[ev]1 t s9ce #ahie ! s\nd+s ( ,nans5 9 ! nor? pol> area1 ( ! ,valdivia1 ,belgica1 ,gauss & ,scotia 9 ! ,s\!rn ,ocean1 & ( v>i\s survey+ %ips 9 ! ,nor? & ,s\? ,pacific1 h prov$ t ! mean dep? ( ! ocean is consid]ably great] ?an _h be5 suppos$1 & mean-sph]e level mu/ !re=e lie deep] ?an ! calculations ( #ahie %[2 possibly n f> f ! position d$uc$ f ! fre] e/imate ( #ahhh4 ,! :ole ( ! available data 7 utiliz$ by ! pr9ce ( ,monaco 9 #aije 9 ! prep>ation ( a complete ba?ymetrical map ( ! oceans on a uni=m scale1 :i* mu/ long rema9 ! /&>d "w = ref]5ce on ocean dep?s4 ,by ! device ( a hypsographic curve co-ord9at+ ! v]tical relief & ! areas ( ! e>?'s surface occupi$ by ea* zone ( elevation1 accord+ to ! sy/em introduc$ by ,supan1#ai ,wagn] %[$ 8 results graphically4 ,?is curve ) ! values r$uc$ f metres to feet is reproduc$ bel[4 ,wagn] subdivides ! e>?'s surface1 accord+ to elevation1 into ! foll[+ five regions3 ,wagn]'s ,divisions ( ! ,e>?'s ,cru/3 ---------------------------------------- ------------------ ,"n4 ,p] c5t ( ,f ,to ,surface4 ---------------------------------------- ------------------ ,depress$ area #c ,deepe/4 -#af1djj feet4 ,oceanic plateau #ed -#af1djj feet4- #g1djj ,cont95tal slope #i - #g1djj - #ffj ,cont95tal plateau #bh - #ffj #c1jjj ,culm9at+ area #f #c1cjj ,highe/4 ---------------------------------------- ------------------ ,illu/ration ,! cont95tal plateau might = purposes ( detail$ /udy be divid$ into ! cont95tal %elf f -#ffj ft4 to sea-level1 & l[l&s f sea-level to #ffj ft4 7correspond+ to ! mean level ( ! :ole globe74#bj ,upl&s rea*+ f #ffj ft4 to #bcjj 7! approximate mean level ( ! l&71 & highl&s1 f #bcjj upw>ds1 might also be di/+ui%$4 ,arrangem5t ( _w-ridges & holl[s4 ,a /rik+ fact 9 ! configuration ( ! cru/ is t ea* cont95t1 or elevat$ mass ( ! cru/1 is diametrically opposite to an ocean bas9 or great depression2 ! only p>tial exception be+ 9 ! case ( s\!rn ,s\? ,am]ica1 :i* is antipodal to ea/]n ,asia4 ,pr(essor ,c4 ,lapwor? has g5]aliz$ ! gr& features ( cru/al relief 9 a s*eme ( attractive simplicity4 ,he sees ?r\gh\t all ! *aos ( irregul> cru/-=ms ! recurr5ce ( a c]ta9 h>mony1 a succession ( folds or waves :i* build up all ! m9or features4#ba ,"o great s]ies ( cru/ waves f ea/ to we/ is cross$ by a second great s]ies ( cru/ waves f nor? to s\?1 giv+ rise by _! int]f]5ce to six great elevat$ masses 7! cont95ts71 arrang$ 9 ?ree gr\ps1 ea* consi/+ ( a nor!rn & a s\!rn memb] sep>at$ by a m9or depression4 ,^! elevat$ masses are divid$ f "o ano!r by simil> great depressions4 ,lapwor?'s fold-!ory4 ,he says3 ,! surface ( ea* ( our great cont95tal masses ( l& resembles t ( a long & broad ar*-l =m1 ( :i* we see ! simple/ type 9 ! ,new ,_w4 ,! surface ( ! ,nor? ,am]ican ar* is sagg$ d[nw>ds 9 ! middle into a c5tral depression :i* lies betwe5 two long m>g9al plateaus1 & ^! plateaus are f9ally cr[n$ by ! wr9kl$ cre/s :i* =m its two mod]n m\nta9 sy/ems4 ,! surface ( ea* ( our ocean floors exactly resembles t ( a cont95t turn$ upside d[n4 ,tak+ ! ,atlantic z our simple/ type1 we may say t ! surface ( an ocean bas9 resembles t ( a mighty tr\gh or syncl9e1 buckl$ up m or less c5trally 9 a m$ial ridge1 :i* is b\nd$ by two long & deep m>g9al holl[s1 9 ! cores ( :i* /ill deep] grooves s9k to ! pr(\nde/ dep?s4 ,?is complem5t>y relation%ip desc5ds ev5 to ! m9or features ( ! two4 ,:]e ! great cont95tal sag s9ks bel[ ! ocean level1 we h our gulfs & our ,m$it]raneans1 se5 9 our type cont95t1 z ! ,mexican ,gulf & ,hudson ,bay4 ,:]e ! c5tral oceanic buckle atta9s ! wat]-l9e we h our oceanic isl&s1 se5 9 our type ocean1 z ,/ ,hel5a & ! ,azores4 ,al?\gh ! app>5t cru/-waves are nei!r equal 9 size nor symmetrical 9 =m1 ?is complem5t>y relation%ip betwe5 !m is always disc]nible4 ,! broad ,pacific depression seems to answ] to ! broad elevation ( ! ,old ,_w--! n>r[ tr\gh ( ! ,atlantic to ! n>r[ cont95t ( ,am]ica4 ,suess's !ory4 ,! mo/ ?or\gh discussion ( ! great features ( t]re/rial relief 9 ! light ( _! orig9 is t by ,pr(essor ,e4 ,suess1#bb :o po9ts out t ! plan ( ! e>? is ! result ( two movem5ts ( ! cru/--"o1 subsid5ce ov] wide areas1 giv+ rise to oceanic depressions & leav+ ! cont95ts protub]ant2 ! o!r1 fold+ along comp>atively n>r[ belts1 giv+ rise to m\nta9 ranges4 ,?is !ory ( cru/ blocks dropp$ by subsid5ce is oppos$ to ,lapwor?'s !ory ( va/ cru/-folds1 b geology is ! sci5ce :i* has to decide betwe5 !m4 ,geomorphology is conc]n$1 h[ev]1 9 ! sugge/ions :i* h be5 made z to ! cause ( ! di/ribution ( heap & holl[ 9 ! l>g] features ( ! cru/4 ,elie de ,beaumont1 9 8 speculations on ! relation betwe5 ! direction ( m\nta9 ranges & _! geological age & "*1 0 feel+ t[>ds a compreh5sive !ory ( ! =ms ( cru/al relief2 b 8 ideas 7 too geometrical1 & 8 !ory t ! e>? is a sph]oid built up on a rhombic dodecah$ron1 ! p5tagonal faces ( :i* det]m9$ ! direction ( m\nta9 ranges1 c\ld n be prov$4#bc ,! tetrah$ral !ory br\ght =w>d by ,l[?ian ,gre51#bd t ! =m ( ! e>? is a sph]oid bas$ on a regul> tetrah$ron1 is m s]viceable1 because x acc\nts = ?ree v int]e/+ facts ( ! t]re/rial plan--7#a7 ! antipodal position ( cont95ts & ocean bas9s2 7#b7 ! triangul> outl9e ( ! cont95ts2 & 7#c7 ! excess ( sea 9 ! s\!rn hemisph]e4 ,rec5t inve/igations h recall$ att5tion to ! "w ( ,l[?ian ,gre51 b ! "q is /ill 9 ! controv]sial /age4#be ,! /udy ( tidal /ra9 9 ! e>?'s cru/ by ,sir ,george ,d>w9 has l$ t physici/ to indicate ! possibility ( ! triangul> =m & s\!rly direction ( ! cont95ts be+ a result ( ! diff]5tial or tidal attraction ( ! sun & moon4 ,m rec5tly ,pr(essor ,a4,e4,h4 ,love has %[n t ! great features ( ! relief ( ! li?osph]e may be express$ by sph]ical h>monics ( ! fir/1 second & ?ird degrees1 & _! =mation relat$ to gravitational action 9 a sph]e ( unequal d5sity4#bf ,9 any case x is fully recogniz$ t ! plan ( ! e>? is s cle> z to leave no d\bt z to its be+ due to "s g5]al cause :i* %\ld be capable ( detection4 ,! cont95ts4 ,if ! level ( ! sea 7 to become co9cid5t ) ! mean level ( ! li?osph]e1 !re w\ld result "o tri-radiate l&-mass ( ne>ly uni=m outl9e & "o cont9u\s %eet ( wat] brok5 by few isl&s4 ,! actual position ( sea-level lies s ne> ! summit ( ! cru/-heap t ! v>i$ relief ( ! upp] portion leads to ! =mation ( a complicat$ coa/-l9e & a great numb] ( deta*$ portions ( l&4 ,! hydrosph]e is1 9 fact1 cont9u\s1 & ! l& is all 9 insul> masses3 ! l>ge/ is ! ,old ,_w ( ,europe1 ,asia & ,africa2 ! next 9 size1 ,am]ica2 ! ?ird1 possibly1 ,ant>ctica2 ! f\r?1 ,au/ralia2 ! fif?1 ,gre5l&4 ,aft] ?is !re is a consid]able gap be=e ,new ,gu9ea1 ,borneo1 ,madagasc>1 ,sumatra & ! va/ multitude ( small] isl&s desc5d+ 9 size by regul> gradations to m]e rocks4 ,! contra/ betwe5 isl& & ma9l& 0 natural en\gh 9 ! days be=e ! discov]y ( ,au/ralia1 & ! ma9l& ( ! ,old ,_w 0 traditionally divid$ into ?ree cont95ts4 ,^! cont95ts1 p>ts ( ! e>?1 or qu>t]s ( ! globe1 prov$ to be conv5i5t divisions2 ,am]ica 0 add$ z a f\r?1 & subsequ5tly divid$ into two1 :ile ,au/ralia on its discov]y 0 class$ sometimes z a new cont95t1 sometimes m]ely z an isl&1 sometimes compromis+ly z an isl&-cont95t1 accord+ to individual op9ion4 ,! discov]y ( ! insul>ity ( ,gre5l& might aga9 give rise to ! argum5t z to ! di/9ction betwe5 isl& & cont95t4 ,al?\gh ! "n ( cont95t 0 n appli$ to l>ge portions ( l& = any physical reasons1 x s happ5s t !re is a c]ta9 physical simil>ity or homology betwe5 !m :i* is n %>$ by ! small] isl&s or p59sulas4 ,homology ( cont95ts4 ,! typical cont95tal =m is triangul> z reg>ds its sea-level outl9e4 ,! relief ( ! surface typically includes a c5tral pla91 sometimes dipp+ bel[ sea-level1 b\nd$ by lat]al highl&s or m\nta9 ranges1 l(ti] on "o side ?an on ! o!r1 ! high] enclos+ a plateau %ut 9 by m\nta9s4 ,s\? ,am]ica & ,nor? ,am]ica foll[ ?is type mo/ closely2 ,eurasia 7! l& mass ( ,europe & ,asia7 comes next1 :ile ,africa & ,au/ralia are f>!r remov$ f ! type1 & ! /ructure ( ,ant>ctica & ,gre5l& is unkn[n4 ,if ! cont9u\s1 unbrok51 horizontal ext5t ( l& 9 a cont95t is t]m$ its trunk1#bg & ! portions cut up by inlets or *annels ( ! sea into isl&s & p59sulas ! limbs1 x is possible to comp>e ! cont95ts 9 an in/ructive mann]4 ,! foll[+ table is f ! /ati/ics ( ,pr(essor ,h4 ,wagn]1#bh 8 metric measurem5ts be+ transpos$ into ,briti% units3 ,comp>ison ( ! ,cont95ts4 ---------------------------------------- ---------------------- ,area ,area ,mean ,area p59- ,area ,area ,area total height1 trunk1sulas1isl&s1limbs1limbs1 mil4 feet4 mil4 mil4 mil4 mil4 p] sq4 m4 sq4 m4sq4 m4 sq4 m4 sq4 m4 c5t4 ---------------------------------------- ---------------------- ,old ,_w #ce4h #bcfj ,new ,_w #af4b #bbcj ,eurasia #bj4he #bfbj #ae4db #d4ji #a4cd #e4dc #bf ,africa #aa4df #bacj #aa4bb 44 #j4bd #j4bd #b4a ,nor? ,am]ica #i4bf #bcjj #f4ib #j4gh #a4ef #b4cd #be ,s\? ,am]ica #f4hd #aigj #f4gf #j4jb #j4jf #j4jh #a4a ,au/ralia #c4dc #acaj #b4gg #j4af #j4ej #j4ff #ai ,asia #ag4jb #cabj #ab4ic #c4je #a4jd #d4ji #bd ,europe #c4hc #ihj #b4di #a4jd #j4cj #a4cd #ce ---------------------------------------- ---------------------- ,isl&s4 ,! usual classification ( isl&s is into cont95tal & oceanic4 ,! =m] class includes all ^? :i* rise f ! cont95tal %elf1 or %[ evid5ce 9 ! "* ( _! rocks ( hav+ at "o "t be5 cont9u\s ) a neighb\r+ cont95t4 ,! latt] rise abruptly f ! oceanic abysses4 ,oceanic isl&s are divid$ accord+ to _! geological "* into volcanic isl&s & ^? ( organic orig91 includ+ coral isl&s4 ,m elaborate subdivisions accord+ to /ructure1 orig9 & position h be5 propos$4#bi ,9 "s cases a piece ( l& is only an isl& at high wat]1 & by imp]ceptible gradation ! =m passes into a p59sula4 ,! typical p59sula is connect$ ) ! ma9l& by a relatively n>r[ i/hmus2 ! "n is1 h[ev]1 ext5d$ to any limb project+ f ! trunk ( ! ma9l&1 ev5 :51 z 9 ! ,indian p59sula1 x is connect$ by its wide/ "p4 ,coa/s4 ,small p59sulas are kn[n z promontories or headl&s1 & ! extremity z a cape4 ,! opposite =m1 an inlet ( ! sea1 is kn[n :5 wide z a gulf1 bay or bight1 accord+ to size & degree ( inflection1 or z a fjord or ria :5 long & n>r[4 ,x is conv5i5t to employ a specific "n = a projection ( a coa/-l9e less pron\nc$ ?an a p59sula1 & = an inlet less pron\nc$ ?an a bay or bight2 outcurve & incurve may s]ve ! turn4 ,! v>ieties ( coa/-l9es 7 r$uc$ to an exact classification by ,ri*?(51 :o gr\p$ !m accord+ to ! height & slope ( ! l& into cliff-coa/s 7,/eilku/57--n>r[ bea* coa/s ) cliffs1 wide bea* coa/s ) cliffs1 & l[ coa/s1 subdivid+ ea* gr\p accord+ z ! coa/-l9e runs p>allel to or crosses ! l9e ( /rike ( ! m\nta9s1 or is n relat$ to m\nta9 /ructure4 ,a fur!r subdivision dep5ds on ! "* ( ! int]-relation ( l& & sea along ! %ore produc+ su* types z a fjord-coa/1 ria-coa/ or lagoon-coa/4 ,?is extremely elaborate subdivision may be r$uc$1 z ,wagn] po9ts out1 to ?ree types--! cont95tal coa/ :]e ! sea comes up to ! solid rock-mat]ial ( ! l&2 ! m>9e coa/1 :i* is =m$ entirely ( s(t mat]ial sort$ out by ! sea2 & ! composite coa/1 9 :i* bo? =ms are comb9$4 ,coa/-l9es4 ,on l>ge-scale maps x is necess>y to %[ two coa/-l9es1 "o = ! highe/1 ! o!r = ! l[e/ tide2 b 9 small-scale maps a s+le l9e is usually wid] ?an is requir$ to repres5t ! :ole bread? ( ! int]-tidal zone4 ,! measurem5t ( a coa/-l9e is difficult1 because ! l5g? w necess>ily be great] :5 measur$ on a l>ge-scale map :]e m9ute irregul>ities c be tak5 into acc\nt4 ,x is usual to di/+ui% betwe5 ! g5]al coa/-l9e measur$ f po9t to po9t ( ! headl&s disreg>d+ ! small] bays1 & ! detail$ coa/-l9e :i* takes acc\nt ( e inflection %[n by ! map employ$1 & foll[s up riv] entrances to ! po9t :]e tidal action ceases4 ,! ratio betwe5 ^! two coa/-l9es repres5ts ! coa/al developm5t ( any region4 ,subm>9e =ms4 ,:ile ! =ms ( ! sea-b$ are n yet suffici5tly well kn[n to admit ( exact classification1 !y are recogniz$ to be z a rule di/9ct f ! =ms ( ! l&1 & ! importance ( us+ a di/9ctive t]m9ology is felt4 ,ef=ts h be5 made to arrive at a def9ite int]national agreem5t on ?is subject1 & c]ta9 t]ms sugge/$ by a committee 7 adopt$ by ! ,eigh? ,int]national ,geographical ,congress at ,new ,york 9 #aijd4cj ,! =ms ( ! ocean floor include ! %elf1 or %all[ sea m>g91 ! depression1 a g5]al t]m appli$ to all subm>9e holl[s1 & ! elevation4 ,a depression :5 ( great ext5t is t]m$ a bas91 :5 x is ( a m or less r\nd =m ) approximately equal diamet]s1 a tr\gh :5 x is wide & elongat$ ) g5tly slop+ bord]s1 & a tr5* :5 n>r[ & elongat$ ) /eeply slop+ bord]s1 "o ( :i* rises high] ?an ! o!r4 ,! ext5sion ( a tr\gh or bas9 p5etrat+ ! l& or an elevation is t]m$ an embaym5t :5 wide1 & a gully :5 long & n>r[2 & ! deepe/ "p ( a depression is t]m$ a deep4 ,a depression ( small ext5t :5 /eep-sid$ is t]m$ a caldron1 & a long n>r[ depression cross+ a "p ( ! cont95tal bord] is t]m$ a furr[4 ,an elevation ( great ext5t :i* rises at a v g5tle angle f a surr\nd+ depression is t]m$ a rise1 "o :i* is relatively n>r[ & /eep-sid$ a ridge1 & "o :i* is approximately equal 9 l5g? & bread? b /eep-sid$ a plateau1 :e!r x spr+s direct f a depression or f a rise4 ,an elevation ( small ext5t is di/+ui%$ z a dome :5 x is m ?an #ajj fa?oms f ! surface1 a bank :5 x is ne>] ! surface ?an #ajj fa?oms b deep] ?an #f fa?oms1 & a %oal :5 x comes )9 #f fa?oms ( ! surface & s becomes a s]i\s dang] to %ipp+4 ,! highe/ po9t ( an elevation is t]m$ a height1 if x does n =m an isl& or "o ( ! m9or =ms4 ,l& =ms4 ,! =ms ( ! dry l& are ( inf9ite v>iety1 & h be5 /udi$ 9 great detail4#ca ,f ! descriptive or topographical po9t ( view1 geometrical =m alone %\ld be consid]$2 b ! orig9 & geological /ructure ( l& =ms mu/ 9 _m cases be tak5 into acc\nt :5 deal+ ) ! function !y ex]cise 9 ! control ( mobile di/ributions4 ,! geograph]s :o h hi!rto giv5 mo/ att5tion to ! =ms ( ! l& h be5 tra9$ z geologi/s1 & consequ5tly !re is a g5]al t5d5cy to make orig9 or /ructure ! basis ( classification r ?an =m alone4 ,! six elem5t>y l& =ms4 ,! fundam5tal =m-elem5ts may be r$uc$ to ! six propos$ by ,pr(essor ,p5ck z ! basis ( 8 d\ble sy/em ( classification by =m & orig94#cb ,^! may be look$ ^u z be+ all d]iv$ by v>i\s modifications or arrangem5ts ( ! s+le =m-unit1 ! slope or incl9$ plane surface4 ,no "o =m occurs alone1 b always gr\p$ toge!r ) o!rs 9 v>i\s ways to make up di/ricts1 regions & l&s ( di/9ctive *>act]s4 ,! =m-elem5ts are3 #a4 ,! pla9 or g5tly incl9$ uni=m surface4 #b4 ,! sc>p or /eeply incl9$ slope2 ?is is necess>ily ( small ext5t except 9 ! direction ( its l5g?4 #c4 ,! valley1 compos$ ( two lat]al p>allel slopes incl9$ t[>ds a n>r[ /rip ( pla9 at a l[] level :i* itself slopes d[nw>ds 9 ! direction ( its l5g?4 ,_m v>ieties ( ?is fundam5tal =m may be di/+ui%$4 #d4 ,! m\nt1 compos$ ( a surface fall+ away on e side f a p>ticul> place4 ,?is place may ei!r be a po9t1 z 9 a volcanic cone1 or a l9e1 z 9 a m\nta9 range or ridge ( hills4 #e4 ,! holl[ or =m produc$ by a l& surface slop+ inw>ds f all sides to a p>ticul> l[e/ place1 ! conv]se ( a m\nt4 #f4 ,! cav]n or space entirely surr\nd$ by a l& surface4 ,geology & l& =ms4 ,^! =ms nev] occur scatt]$ haphaz>d ov] a region1 b always 9 an ord]ly subord9ation dep5d+ on _! mode ( orig94 ,! dom9ant =ms result f cru/al movem5ts1 ! subsidi>y f second>y reactions dur+ ! action ( ! primitive =ms on mobile di/ributions4 ,! geological /ructure & ! m9]al composition ( ! rocks are (t5 ! *ief causes det]m9+ ! "* ( ! l& =ms ( a region4 ,?us ! sc5]y ( a lime/one c\ntry dep5ds on ! solubility & p]meability ( ! rocks1 lead+ to ! typical ,k>/-=mations ( cav]ns1 swall[-holes & und]gr\nd /ream c\rses1 ) ! cont+5t ph5om5a ( dry valleys & natural bridges4 ,a s&y bea* or des]t owes its "* to ! mobility ( its con/itu5t s&-gra9s1 :i* are readily drift$ & pil$ up 9 ! =m ( dunes4 ,a region :]e volcanic activity has l$ to ! emb$d+ ( dykes or bosses ( h>d rock among/ s(t] /rata produces a pla9 brok5 by abrupt & isolat$ em95ces4#cc ,classification ( m\nta9s4 ,x w\ld be impracticable to g fully into ! v>ieties ( ea* specific =m2 b1 p>tly z an example ( mod]n geographical classification1 p>tly because ( ! exceptional importance ( m\nta9s among/ ! features ( ! l&1 "o exception may be made4 ,! classification ( m\nta9s into types has usually _h reg>d r to geological /ructure ?an to ext]nal =m1 s t "s geologi/s w\ld ev5 apply ! "n ( a m\nta9 range to a region n di/+ui%$ by relief f ! re/ ( ! c\ntry if x be> geological evid5ce ( hav+ once be5 a true range4 ,a m\nta9 may be describ$ 7x _c be def9$7 z an elevat$ region ( irregul> surface ris+ comp>atively abruptly f l[] gr\nd4 ,! actual elevation ( a summit above sea-level does n necess>ily affect its m\nta9\s "*2 a g5tle em95ce1 = in/ance1 ris+ a few hundr$ feet above a tablel&1 ev5 if at an elevation ( say #ae1jjj ft41 c\ld only be call$ a hill4#cd ,b x may be said t any abrupt slope ( #bjjj ft4 or m 9 v]tical height may ju/ly be call$ a m\nta91 :ile abrupt slopes ( less] height may be call$ hills4 ,exi/+ classifications1 h[ev]1 d n take acc\nt ( any diff]5ce 9 k9d betwe5 m\nta9 & hills1 al?\gh x is common 9 ! ,g]man language to speak ( ,hugell&1 ,mittelgebirge & ,ho*gebirge ) a def9ite significance4 ,! simple classification employ$ by ,pr(essor ,james ,geikie#ce into m\nta9s ( accumulation1 m\nta9s ( elevation & m\nta9s ( circumd5udation1 is n consid]$ suffici5tly ?or\gh by ,g]man geograph]s1 :o1 foll[+ ,ri*?(51 g5]ally adopt a classification dep5d5t on six prim>y divisions1 ea* ( :i* is subdivid$4 ,! t]ms employ$1 especially = ! subdivisions1 _c be easily translat$ into o!r languages1 & ! ,engli% equival5ts 9 ! foll[+ table are only put =w>d t5tatively3-- ,,ri*?(5',s ,,classification ,,( ,,m\nta9s#cf ,i4 ,tektonis*e ,gebirge--,tectonic m\nta9s4 7a7 ,bru*gebirge od] ,s*oll5gebirge--,block m\nta9s4 #a4 ,e9seitige ,s*oll5gebirge od] ,s*oll5r&gebirge-- ,sc>p or tilt$ block m\nta9s4 7i47 ,tafels*olle--,table blocks4 7ii47 ,abrasionss*olle--,abrad$ blocks4 7iii47 ,transgressionss*olle--,blocks ( uncon=mable /rata4 #b4 ,flexurgebirge--,flexure m\nta9s4 #c4 ,hor/gebirge--,symmetrical block m\nta9s4 7b7 ,faltungsgebirge--,fold m\nta9s4 #a4 ,homoomorphe ,faltungsgebirge--,homomorphic fold m\nta9s4 #b4 ,het]omorphe ,faltungsgebirge--,het]omorphic fold m\nta9s4 ,,ii4 ,rumpfgebirge od] ,abrasionsgebirge--,trunk or abrad$ m\nta9s4 ,,iii4 ,ausbru*sgebirge--,eruptive m\nta9s4 ,,iv4 ,aufs*uttungsgebirge--,m\nta9s ( accumulation4 ,v4 ,fla*bod5--,plateaux4 7a7 ,abrasionsplatt5--,abrad$ plateaux4 7b7 ,m>9es ,fla*l&--,pla9 ( m>9e erosion4 7c7 ,s*i*tung/afell&--,horizontally /ratifi$ tablel&4 7d7 ,ub]gus/afell&--,lava pla94 7e7 ,/romfla*l&--,riv] pla94 7f7 ,fla*bod5 d] atmosph>is*5 ,aufs*uttung--,pla9s ( aeolian =mation4 ,,vi4 ,erosionsgebirge--,m\nta9s ( erosion4 ,m\nta9 =ms4 ,f ! morphological po9t ( view x is m important to di/+ui% ! associations ( =ms1 su* z ! m\nta9 mass or gr\p ( m\nta9s radiat+ f a c5tre1 ) ! valleys furr[+ _! flanks spread+ t[>ds e direction2 ! m\nta9 *a9 or l9e ( heights1 =m+ a long n>r[ ridge or s]ies ( ridges sep>at$ by p>allel valleys2 ! dissect$ plateau or highl&1 divid$ into m\nta9s ( circumd5udation by a sy/em ( deeply-cut valleys2 & ! isolat$ peak1 usually a volcanic cone or a h>d rock mass left project+ aft] ! s(t] /rata :i* emb$d$ x h be5 worn away 7,monadnock ( ,pr(essor ,davis74 ,di/ribution ( m\nta9s4 ,! geographical di/ribution ( m\nta9s is intimately associat$ ) ! great /ructural l9es ( ! cont95ts ( :i* !y =m ! culm9at+ region4 ,l(ty l9es ( fold m\nta9s =m ! backbones ( ,nor? ,am]ica 9 ! ,rocky ,m\nta9s & ! we/ coa/ sy/ems1 ( ,s\? ,am]ica 9 ! ,cordill]a ( ! ,&es1 ( ,europe 9 ! ,pyr5ees1 ,alps1 ,c>pa?ians & ,caucasus1 & ( ,asia 9 ! m\nta9s ( ,asia ,m9or1 conv]g+ on ! ,pamirs & div]g+ !nce 9 ! ,himalaya & ! va/ m\nta9 sy/ems ( c5tral & ea/]n ,asia4 ,! rem>kable l9e ( volcanoes ar\nd ! :ole coa/ ( ! ,pacific & along ! m>g9 ( ! ,c>ibbean & ,m$it]ranean seas is "o ( ! mo/ conspicu\s features ( ! globe4 ,functions ( l& =ms4 ,l& wa/e4 ,glaci]s4 ,if l& =ms may be comp>$ to organs1 ! "p !y s]ve 9 ! economy ( ! e>? may1 )\t /ra9+ ! t]m1 be *>act]iz$ z functions4 ,! fir/ & simple/ function ( ! l& surface is t ( guid+ loose mat]ial to a l[] level4 ,! d[nw>d pull ( gravity suffices to br+ ab\t ! fall ( su* mat]ial1 b ! pa? x w foll[ & ! di/ance x w travel be=e com+ to re/ dep5d ^u ! l& =m4 ,! loose mat]ial may1 & 9 an arid region does1 consi/ only ( portions ( ! high] p>ts ( ! surface deta*$ by ! expansion & contraction produc$ by heat+ & cool+ due to radiation4 ,su* brok5 mat]ial roll+ d[n a uni=m sc>p w\ld t5d to r$uce its /eepness by ! loss ( mat]ial 9 ! upp] "p & by ! accumulation ( a m\nd or scree aga9/ ! l[] "p ( ! slope4 ,b :]e ! side is n a uni=m sc>p1 b made up ( a s]ies ( ridges & valleys1 ! t5d5cy w be to di/ribute ! detritus 9 an irregul> mann]1 direct+ x away f "o place & collect+ x 9 great masses 9 ano!r1 s t 9 "t ! l& =m assumes a new appe>ance4 ,sn[ accumulat+ on ! high] portions ( ! l&1 :5 compact$ into ice & caus$ to fl[ d[nw>ds by gravity1 gives rise1 on acc\nt ( its m coh]5t "*1 to cont9u\s glaci]s1 :i* m\ld !mselves to ! slopes d[n :i* !y are guid$1 diff]5t ice-/reams conv]g+ to s5d =w>d a great] volume4 ,gradually com+ to occupy def9ite b$s1 :i* are deep5$ & poli%$ by ! friction1 !y impress a *>act]i/ic appe>ance on ! l&1 :i* guides !m z !y trav]se x1 &1 al?\gh ! ice melts at l[] levels1 va/ quantities ( clay & brok5 /ones are br\ght d[n & deposit$ 9 t]m9al mora9es :]e ! glaci] ends4 ,ra94 ,riv] sy/ems4 ,adju/m5t ( riv]s to l&4 ,ra9 is by f> ! mo/ important ( ! inorganic mobile di/ributions ^u :i* l& =ms ex]cise _! function ( guidance & control4 ,! precipitation ( ra9 f ! aque\s vap\r ( ! atmosph]e is caus$ 9 "p by v]tical movem5ts ( ! atmosph]e involv+ heat *anges & app>5tly indep5d5t ( ! surface ^u :i* precipitation occurs2 b 9 great] "p x is dictat$ by ! =m & altitude ( ! l& surface & ! direction ( ! prevail+ w9ds1 :i* itself is l>gely influ5c$ by ! l&4 ,x is on ! w9dw>d faces ( ! highe/ gr\nd1 or j beyond ! summit ( less dom9ant heights ^u ! leew>d side1 t mo/ ra9 falls1 & all t does n evaporate or p]colate into ! gr\nd is conduct$ back to ! sea by a r\te :i* dep5ds only on ! =m ( ! l&4 ,m mobile & m se>*+ ?an ice or rock rubbi%1 ! trickl+ drops are guid$ by ! deepe/ l9es ( ! hillside 9 _! incipi5t fl[1 & z ^! l9es conv]ge1 ! /ream1 ga9+ /r5g?1 proce$s 9 its torr5tial c\rse to c>ve its *annel deep] & entr5* itself 9 p]man5t occupation4 ,?us ! /ream-b$1 f :i* at fir/ ! wat] might be bl[n away into a new *annel by a gale ( w9d1 ultimately gr[s to be ! /ronge/ l9e ( ! l&scape4 ,z ! ma9 valley deep5s1 ! tribut>y /ream-b$s are deep5$ also1 & gradually cut _! way headw>ds1 enl>g+ ! area :5ce !y draw _! supplies4 ,?us new l& =ms are creat$--valleys ( curi\s complexity1 = example--by ! capture & div]sion ( ! wat] ( "o riv] by ano!r1 lead+ to a *ange ( wat]%$4#cg ,! m9or tribut>ies become m num]\s & m con/ant1 until ! sy/em ( torr5ts has impress$ its own individuality on ! m\nta9 side4 ,z ! riv] leaves ! m\nta91 ev] gr[+ by ! accession ( tribut>ies1 x ceases1 save 9 flood "t1 to be a =midable in/rum5t ( de/ruction2 ! g5tl] slope ( ! l& surface gives to x only p[] suffici5t to transport small /ones1 gravel1 s& & ultimately mud4 ,its valley banks are cut back by ! erosion ( m9or tribut>ies1 or by ra9-wa% if ! climate be moi/1 or left /eep & %>p :ile ! riv] deep5s its b$ if ! climate be arid4 ,! outl9e ( ! curve ( a valley's sides ultimately dep5ds on ! angle ( repose ( ! detritus :i* cov]s !m1 if !re has be5 no subsequ5t *ange1 su* z ! passage ( a glaci] along ! valley1 :i* t5ds to de/roy ! regul>ity ( ! cross-section4 ,! slope ( ! riv] b$ dim9i%es until ! pla9 compels ! riv] to move sl[ly1 sw++ 9 me&]s proportion$ to its size1 & gradually1 controll$ by ! flatt5+ l&1 ceas+ to transport mat]ial1 b rais+ its banks & silt+ up its b$ by ! dropp$ s$im5t1 until1 split up & %oal$1 its di/ribut>ies /ruggle across its delta to ! sea4 ,?is is ! typical riv] ( :i* !re are inf9ite v>ieties1 yet e v>iety w\ld1 if "t 7 giv51 & ! l& rema9$ un*ang$ 9 level relatively to ! sea1 ultimately approa* to ! type4 ,movem5ts ( ! l& ei!r ( subsid5ce or elevation1 *anges 9 ! l& by ! action ( erosion 9 cutt+ back an esc>pm5t or cutt+ ?r\gh a col1 *anges 9 climate by affect+ ! ra9fall & ! volume ( wat]1 all t5d to ?r[ ! riv] valley out ( h>mony ) ! actual condition ( its /ream4 ,!re is no?+ m /rik+ 9 geography ?an ! p]fection ( ! adju/m5t ( a great riv] sy/em to its valleys :5 ! l& has rema9$ /able = a v l5g!n$ p]iod4 ,be=e full adju/m5t has be5 atta9$ ! riv] b$ may be brok5 9 places by wat]falls or int]rupt$ by lakes2 aft] adju/m5t ! b$ assumes a p]man5t outl9e1 ! slope dim9i%+ m & m gradually1 )\t a break 9 its symmetrical desc5t4 ,excell5t examples ( ! indecisive dra9age ( a new l& surface1 on :i* ! riv] sy/em has n _h "t to impress itself1 are to be se5 9 nor!rn ,canada & 9 ,f9l&1 :]e riv]s are sep>at$ by sc>cely p]ceptible divides1 & ! num]\s lakes frequ5tly belong to m ?an "o riv] sy/em4 ,! geographical cycle4 ,! action ( riv]s on ! l& is s important t x has be5 made ! basis ( a sy/em ( physical geography by ,pr(essor ,w4,m4 ,davis1 :o classifies l& surfaces 9 t]ms ( ! ?ree factors--/ructure1 process & "t4#ch ,( ^! "t1 dur+ :i* ! process is act+ on ! /ructure1 is ! mo/ important4 ,a l& may ?us be *>act]iz$ by its position 9 ! geographical cycle1 or cycle ( erosion1 z "y1 mature or old1 ! la/ t]m be+ rea*$ :5 ! base-level ( erosion is atta9$1 & ! l&1 h[ev] v>i$ its relief may h be5 9 y\? or maturity1 is r$uc$ to a ne>ly uni=m surface or p5epla94 ,by a re-elevation ( a p5epla9 ! riv]s ( an old l& surface may be re/or$ to y\?ful activity1 & resume _! %ap+ action1 deep5+ ! old valleys & initiat+ new ones1 />t+ afre% ! :ole c\rse ( ! geographical cycle4 ,x is1 h[ev]1 n ! action ( ! runn+ wat] on ! l&1 b ! function ex]cis$ by ! l& on ! runn+ wat]1 t is consid]$ "h to be ! special prov9ce ( geography4 ,at e /age ( ! geographical cycle ! l& =ms1 z !y exi/ at t /age1 are conc]n$ 9 guid+ ! cond5sation & fl[ ( wat] 9 c]ta9 def9ite ways4 ,?us1 = example1 9 a m\nta9 range at "r angles to a prevail+ sea-w9d1 x is ! l& =ms :i* det]m9e t "o side ( ! range %all be ri*ly wat]$ & deeply dissect$ by a complete sy/em ( valleys1 :ile ! o!r side is dry1 indef9ite 9 its valley sy/ems1 & s5ds none ( its scanty dra9age to ! sea4 ,! action ( ra91 ice & riv]s conspires ) ! movem5t ( l& wa/e to /rip ! lay] ( soil f /eep slopes z rapidly z x =ms1 & to cause x to accumulate on ! flat valley bottoms1 on ! graceful flatt5$ cones ( alluvial fans at ! outlet ( ! gorges ( tribut>ies1 or 9 ! smoo?ly-spread surface ( alluvial pla9s4 ,! :ole "q ( ! regime ( riv]s & lakes is sometimes treat$ "u ! "n hydrography1 a "n us$ by "s writ]s 9 ! s5se ( m>9e survey+1 & by o!rs z synonym\s ) oceanography4 ,= ! /udy ( riv]s alone ! "n potamology#ci has be5 sugge/$ by ,p5ck1 & ! subject be+ ( mu* practical importance has receiv$ a good deal ( att5tion4#dj ,lakes & int]nal dra9age4 ,! /udy ( lakes has also be5 specializ$ "u ! "n ( limnology 7see ,,lake74#da ,! exi/5ce ( lakes 9 holl[s ( ! l& dep5ds ^u ! balance betwe5 precipitation & evaporation4 ,a /ream fl[+ into a holl[ w t5d to fill x up1 & ! wat] w beg9 to escape z soon z its level rises high en\gh to rea* ! l[e/ "p ( ! rim4 ,9 ! case ( a l>ge holl[ 9 a v dry climate ! rate ( evaporation may be suffici5t to prev5t ! wat] f ev] ris+ to ! lip1 s t !re is no outfl[ to ! sea1 & a bas9 ( int]nal dra9age is ! result4 ,?is is ! case1 = in/ance1 9 ! ,caspian sea1 ! ,aral & ,balkha% lakes1 ! ,t>im bas91 ! ,sah>a1 inn] ,au/ralia1 ! great bas9 ( ! ,unit$ ,/ates & ! ,titicaca bas94 ,^! bas9s ( int]nal dra9age are calculat$ to am\nt to #bb ( ! l& surface4 ,! p]c5tages ( ! l& surface dra9+ to ! diff]5t oceans are approximately--,atlantic1 #cd4c2 ,arctic sea1 #af4e2 ,pacific1 #ad4d2 ,indian ,ocean1 #ab4h4db ,t]m9ology ( riv] sy/ems4 ,! p>ts ( a riv] sy/em h n be5 s cle>ly def9$ z is desirable1 h5ce ! exagg]at$ importance popul>ly atta*$ to ! s\rce ( a riv]4 ,a well-develop$ riv] sy/em has 9 fact _m equally important & widely-sep>at$ s\rces1 ! mo/ di/ant f ! m\?1 ! highe/1 or ev5 t ( l>ge/ initial volume n be+ necess>ily ( great] geographical int]e/ ?an ! re/4 ,! :ole ( ! l& :i* directs dra9age t[>ds "o riv] is kn[n z its bas91 cat*m5t area or dra9age area--sometimes1 by an incorrect expression1 z its valley or ev5 its wat]%$4 ,! b\nd>y l9e betwe5 "o dra9age area & o!rs is rightly t]m$ ! wat]%$1 b on acc\nt ( ! ambiguity :i* has be5 tol]at$ x is bett] to call x wat]-p>t+ or1 z 9 ,am]ica1 divide4 ,! only o!r important t]m :i* requires to be not$ "h is talweg1 a ^w introduc$ f ! ,g]man into ,fr5* & ,engli%1 & mean+ ! deepe/ l9e along ! valley1 :i* is necess>ily occupi$ by a /ream unless ! valley is dry4 ,! functions ( l& =ms ext5d beyond ! control ( ! circulation ( ! atmosph]e1 ! hydrosph]e & ! wat] :i* is cont9ually be+ int]*ang$ betwe5 !m2 !y are ex]cis$ ) increas$ effect 9 ! high] dep>tm5ts ( biogeography & an?ropogeography4 ,biogeography4 ,! sum ( ! organic life on ! globe is t]m$ by "s geograph]s ! biosph]e1 & x has be5 e/imat$ t ! :ole mass ( liv+ sub/ance 9 exi/5ce at "o "t w\ld cov] ! surface ( ! e>? to a dep? ( "o-fif? ( an in*4#dc ,! di/ribution ( liv+ organisms is a complex problem1 a function ( _m factors1 sev]al ( :i* are yet b little kn[n4 ,!y include ! biological nature ( ! organism & its physical environm5t1 ! latt] involv+ conditions 9 :i* geographical elem5ts1 direct or indirect1 prepond]ate4 ,! direct geographical elem5ts are ! arrangem5t ( l& & sea 7cont95ts & isl&s /&+ 9 %>p contra/7 & ! v]tical relief ( ! globe1 :i* int]poses b>ri]s ( a less absolute k9d betwe5 portions ( ! same l& area or oceanic depression4 ,! indirect geographical elem5ts1 :i*1 z a rule1 act ) & int5sify ! direct1 are ma9ly climatic2 ! prevail+ w9ds1 ra9fall1 mean & extreme temp]atures ( e locality dep5d+ on ! arrangem5t ( l& & sea & ( l& =ms4 ,climate ?us guid$ affects ! wea!r+ ( rocks1 & s det]m9es ! k9d & arrangem5t ( soil4 ,diff]5t species ( organisms come to p]fection 9 diff]5t climates2 & x may be /at$ z a g5]al rule t a species1 :e!r ( plant or animal1 once e/abli%$ at "o po9t1 w\ld spread ov] ! :ole zone ( ! climate cong5ial to x unless "s b>ri] 7 int]pos$ to its progress4 ,9 ! case ( l& & fre%-wat] organisms ! sea is ! *ief b>ri]2 9 ! case ( m>9e organisms1 ! l&4 ,diff]5ces 9 l& =ms d n ex]t great influ5ce on ! di/ribution ( liv+ creatures directly1 b indirectly su* l& =ms z m\nta9 ranges & int]nal dra9age bas9s are v pot5t ?r\gh _! action on soil & climate4 ,a sn[-capp$ m\nta9 ridge or an arid des]t =ms a b>ri] betwe5 diff]5t =ms ( life :i* is (t5 m effective ?an an equal bread? ( sea4 ,9 ?is way ! surface ( ! l& is divid$ into num]\s natural regions1 ! flora & fauna ( ea* ( :i* include "s di/9ctive species n %>$ by ! o!rs4 ,! di/ribution ( life is discuss$ 9 ! v>i\s articles 9 ?is ,encyclopa$ia deal+ ) biological1 botanical & zoological subjects4#dd ,floral zones4 ,! classification ( ! l& surface into areas inhabit$ by di/9ctive gr\ps ( plants has be5 attempt$ by _m phyto-geograph]s1 b )\t result+ 9 any s*eme ( g5]al acceptance4 ,! simple/ classification is p]haps t ( ,drude accord+ to climatic zones1 subdivid$ accord+ to cont95ts4 ,?is takes acc\nt (--7#a7 ! ,arctic-,alp9e zone1 includ+ all ! vegetation ( ! region bord]+ on p]petual sn[2 7#b7 ! ,boreal zone1 includ+ ! temp]ate l&s ( ,nor? ,am]ica1 ,europe & ,asia1 all ( :i* are sub/antially alike 9 botanical "*2 7#c7 ! ,tropical zone1 divid$ %>ply into 7a7 ! tropical zone ( ! ,new ,_w1 & 7b7 ! tropical zone ( ! ,old ,_w1 ! =ms ( :i* diff] 9 a significant degree2 7#d7 ! ,au/ral zone1 compris+ all cont95tal l& s\? ( ! equator1 & %>ply divid$ into ?ree regions ! floras ( :i* are /rik+ly di/9ct--7a7 ,s\? ,am]ican1 7b7 ,s\? ,african & 7c7 ,au/ralian2 7#e7 ! ,oceanic1 compris+ all oceanic isl&s1 ! flora ( :i* consi/s exclusively ( =ms ^: se$s c\ld be drift$ unde/roy$ by ocean curr5ts or c>ri$ by birds4 ,to ^! might be add$ ! ant>ctic1 :i* is /ill v imp]fectly kn[n4 ,_m subdivisions & transitional zones h be5 sugge/$ by diff]5t au?ors4 ,vegetation areas4 ,f ! po9t ( view ( ! economy ( ! globe ?is classification by species is p]haps less important ?an t by mode ( life & physiological "* 9 accordance ) environm5t4 ,! foll[+ are ! *ief areas ( vegetational activity usually recogniz$3 7#a7 ,! ice-des]ts ( ! arctic & ant>ctic & ! highe/ m\nta9 regions1 :]e !re is no vegetation except ! l[e/ =ms1 l t :i* causes r$ sn[4 7#b7 ,! tundra or region ( int5sely cold w9t]s1 =bidd+ tree-gr[?1 :]e mosses & li*5s cov] mo/ ( ! gr\nd :5 unfroz51 & %rubs occur ( species :i* 9 o!r conditions are trees1 "h /unt$ to ! height ( a few in*es4 ,a simil> zone surr\nds ! p]man5t sn[ on l(ty m\nta9s 9 all latitudes4 ,! tundra passes by imp]ceptible gradations into ! moor1 bog & hea? ( w>m] climates4 7#c7 ,! temp]ate =e/s ( ev]gre5 or decidu\s trees1 accord+ to circum/ances1 :i* occupy ^? p>ts ( bo? temp]ate zones :]e ra9fall & sunlight are bo? abundant4 7#d7 ,! grassy /eppes or prairies :]e ! ra9fall is dim9i%$ & temp]atures are extreme1 & grass is ! prevail+ =m ( vegetation4 ,^! pass imp]ceptibly into--7#e7 ! arid des]t1 :]e ra9fall is at a m9imum1 & ! only plants are ^? modifi$ to subsi/ ) ! smalle/ supply ( wat]4 7#f7 ,! tropical =e/1 :i* repres5ts ! maximum ( plant luxuriance1 /imulat$ by ! heavie/ ra9fall1 greate/ heat & /ronge/ light4 ,^! divisions m]ge "o into ! o!r1 & admit ( almo/ indef9ite subdivision1 :ile !y are subject to great modifications by human int]f]5ce 9 cle>+ & cultivat+4 ,plants exhibit ! controll+ p[] ( environm5t to a high degree1 & ?us vegetation is usually 9 close adju/m5t to ! bold] geographical features ( a region4 ,faunal realms4 ,! divisions ( ! e>? into faunal regions by ,dr ,p4,l4 ,sclat] h be5 f\nd to hold good = a l>ge numb] ( gr\ps ( animals z diff]5t 9 _! mode ( life z birds & mammals1 & !y may ?us be accept$ z bas$ on nature4 ,!y are six 9 numb]3 7#a7 ,palae>ctic1 includ+ ,europe1 ,asia nor? ( ! ,himalaya1 & ,africa nor? ( ! ,sah>a2 7#b7 ,e?iopian1 consi/+ ( ,africa s\? ( ! ,atlas range1 & ,madagasc>2 7#c7 ,ori5tal1 includ+ ,india1 ,indo-,*9a & ! ,malay ,ar*ipelago nor? ( ,wallace's l9e1 :i* runs betwe5 ,bali & ,lombok2 7#d7 ,au/ralian1 includ+ ,au/ralia1 ,new ,zeal&1 ,new ,gu9ea & ,polynesia2 7#e7 ,ne>ctic or ,nor? ,am]ica1 nor? ( ,mexico2 & 7#f7 ,neotropical or ,s\? ,am]ica4 ,ea* ( ^! divisions is ! home ( a special fauna1 _m species ( :i* are conf9$ to x alone2 9 ! ,au/ralian region1 inde$1 practically ! :ole fauna is peculi> & di/9ctive1 sugge/+ a prolong$ p]iod ( complete biological isolation4 ,9 "s cases1 su* z ! ,e?iopian & ,neotropical & ! ,palae>ctic & ,ne>ctic regions1 ! faunas1 al?\gh di/9ct1 are relat$1 sev]al =ms on opposite sides ( ! ,atlantic be+ analog\s1 e4g4 ! lion & puma1 o/ri* & rhea4 ,:]e two ( ! faunal realms meet !re is usually1 ?\gh n always1 a mix+ ( faunas4 ,^! facts h l$ "s naturali/s to include ! ,palae>ctic & ,ne>ctic regions 9 "o1 t]m$ ,hol>ctic1 & to sugge/ transitional regions1 su* z ! ,sonoran1 betwe5 ,nor? & ,s\? ,am]ica1 & ! ,m$it]ranean1 betwe5 ,europe & ,africa1 or to create sub-regions1 su* z ,madagasc> & ,new ,zeal&4 ,oceanic isl&s h1 z a rule1 di/9ctive faunas & floras :i* resemble1 b are n id5tical )1 ^? ( o!r isl&s 9 simil> positions4 ,biological di/ribution z a means ( geographical rese>*4 ,! /udy ( ! evolution ( faunas & ! comp>ison ( ! faunas ( di/ant regions h furni%$ a tru/wor?y in/rum5t ( pre-hi/oric geographical rese>*1 :i* enables e>li] geographical relations ( l& & sea to be trac$ out1 & ! approximate p]iod1 or at lea/ ! *ronological ord] ( ! l>g] *anges1 to be e/imat$4 ,9 ?is way1 = example1 x has be5 sugge/$ t a l&1 ,lemuria1 once connect$ ,madagasc> ) ! ,malay ,ar*ipelago1 & t a nor!rn ext5sion ( ! ant>ctic l& once unit$ ! ?ree s\!rn cont95ts4 ,! di/ribution ( fossils frequ5tly makes x possible to map out approximately ! g5]al features ( l& & sea 9 long-pa/ geological p]iods1 & s to enable ! hi/ory ( cru/al relief to be trac$4#de ,reaction ( organisms on environm5t4 ,:ile ! t5d5cy is = ! liv+ =ms to come into h>mony ) _! environm5t & to approa* ! /ate ( equilibrium by successive adju/m5ts if ! environm5t %\ld happ5 to *ange1 x is to be obs]v$ t ! action ( organisms !mselves (t5 t5ds to *ange _! environm5t4 ,corals & o!r quick-gr[+ calc>e\s m>9e organisms are ! mo/ p[]ful 9 ?is respect by creat+ new l& 9 ! ocean4 ,vegetation ( all sorts acts 9 a simil> way1 ei!r 9 =m+ soil & assi/+ 9 break+ up rocks1 9 fill+ up %all[ lakes1 & ev51 l ! mangrove1 9 reclaim+ wide /ret*es ( l& f ! sea4 ,plant life1 utiliz+ sol> light to comb9e ! inorganic elem5ts ( wat]1 soil & air into liv+ sub/ance1 is ! basis ( all animal life4 ,?is is n by ! supply ( food alone1 b also by ! )drawal ( c>bonic acid f ! atmosph]e1 by :i* vegetation ma9ta9s ! composition ( ! air 9 a /ate fit = ! support ( animal life4 ,man 9 ! primitive /ages ( culture is sc>cely to be di/+ui%$ f o!r animals z reg>ds 8 subjection to environm5t1 b 9 ! high] grades ( culture ! conditions ( control & reaction become mu* m complicat$1 & ! dep>tm5t ( an?ropogeography is devot$ to _! consid]ation4 ,an?ropogeography4 ,! fir/ requisites ( all human be+s are food & protection1 9 _! se>* = :i* m5 are br\ght into intimate relations ) ! =ms & productions ( ! e>?'s surface4 ,! degree ( dep5d5ce ( any p ^u environm5t v>ies inv]sely z ! degree ( culture or civilization1 :i* = ?is purpose may p]haps be def9$ z ! p[] ( an individual to ex]cise control ov] ! individual & ov] ! environm5t = ! b5efit ( ! community4 ,! developm5t ( culture is to a c]ta9 ext5t a "q ( race1 & al?\gh =m+ "o species1 ! v>ieties ( man diff] 9 almo/ imp]ceptible gradations ) a complexity defy+ classification 7see ,,an?ropology74 ,pr(essor ,keane gr\ps man r\nd f\r lead+ types1 :i* may be nam$ ! black1 yell[1 r$ & :ite1 or ! ,e?iopic1 ,mongolic1 ,am]ican & ,caucasic4 ,ea* may be subdivid$1 ?\gh n ) great exactness1 into small] gr\ps1 ei!r accord+ to physical *>act]i/ics1 ( :i* ! =m ( ! head is mo/ important1 or accord+ to language4 ,types ( man4 ,! black type is f\nd only 9 tropical or sub-tropical c\ntries1 & is usually 9 a primitive condition ( culture1 unless educat$ by contact ) p ( ! :ite type4 ,!y foll[ ! mo/ primitive =ms ( religion 7ma9ly feti%ism71 live on products ( ! woods or ( ! *ase1 ) ! m9imum ( "w1 & h only a loose political organization4 ,! r$ type is peculi> to ,am]ica1 inhabit+ e climate f pol> to equatorial1 & conta9+ repres5tatives ( _m /ages ( culture :i* _h app>5tly develop$ )\t ! aid or int]f]5ce ( p ( any o!r race until ! close ( ! #ae? c5tury4 ,! yell[ type is capable ( a high] culture1 *]i%es high] religi\s beliefs1 & inhabits z a rule ! temp]ate zone1 al?\gh ext5d+ to ! tropics on "o side & to ! arctic regions on ! o!r4 ,! :ite type1 orig9at+ 9 ! nor? temp]ate zone1 has spread ov] ! :ole _w4 ,!y h atta9$ ! highe/ culture1 pr(ess ! pure/ =ms ( mono!i/ic religion1 & h br\ght all ! p ( ! black type & _m ( ^? ( ! yell[ "u _! dom9ation4 ,! contra/ betwe5 ! yell[ & :ite types has be5 s(t5$ by ! rem>kable developm5t ( ! ,japanese foll[+ ! assimilation ( we/]n me?ods4 ,! actual numb] ( human inhabitants 9 ! _w has be5 calculat$ z foll[s3 ,by ,cont95ts4#df ,asia #hge1jjj1jjj ,europe #cib1jjj1jjj ,africa #agj1jjj1jjj ,am]ica #adc1jjj1jjj ,au/ralia & ,polynesia #g1jjj1jjj ------------- ,total #a1ehg1jjj1jjj ,by ,race4#dg ,:ite 7,caucasic7 #ggj1jjj1jjj ,yell[ 7,mong47 #edj1jjj1jjj ,black 7,e?iopic7 #age1jjj1jjj ,r$ 7,am]ican7 #bb1jjj1jjj ------------- ,total #a1ejg1jjj1jjj ,9 r\nd numb]s ! population ( ! _w is ab\t #a1fjj1jjj1jjj1 &1 accord+ to an e/imate by ,rav5/e91#dh ! maximum population :i* x w be possible = ! e>? to ma9ta9 is #fjjj millions1 a numb] :i*1 if ! av]age rate ( increase 9 #ahia cont9u$1 w\ld be rea*$ )9 #bjj ye>s4 ,:ile highly civiliz$ communities are able to evade _m ( ! re/rictions ( environm5t1 to ov]come ! b>ri]s to int]communication int]pos$ by l& or sea1 to c\nt]act ! adv]se influ5ce ( climate1 & by ! developm5t ( trade ev5 to inhabit c\ntries :i* _c yield a food-supply1 ! mass ( mank9d is /ill completely "u ! control ( ^? conditions :i* 9 ! pa/ det]m9$ ! di/ribution & ! mode ( life ( ! :ole human race4 ,influ5ce ( environm5t on man4 ,9 tropical =e/s primitive tribes dep5d on ! collection ( wild fruits1 & 9 a m9or degree on ! *ase ( wild animals1 = _! food4 ,clo?+ is unnecess>y2 h5ce !re is little occasion = ex]cis+ ! m5tal faculties beyond ! s5se ( p]ception to avoid enemies1 or ! inv5tive arts beyond :at is requir$ = ! simple/ weapons & ! mo/ primitive =tifications4 ,:5 ! pursuit ( game becomes ! *ief occupation ( a p !re is ( necessity a high] developm5t ( c\rage1 skill1 p[]s ( obs]vation & inv5tion2 & ^! qualities are /ill fur!r enhanc$ 9 pr$atory tribes :o take by =ce ! food1 clo?+ & o!r prop]ty prep>$ or collect$ by a feebl] p4 ,! fruit-eat+ savage _c /ray beyond 8 woods :i* b\nd 8 life z ! wat] b\nds t ( a fi%2 ! hunt] is free to live on ! m>g9 ( =e/s or 9 op5 c\ntry1 :ile ! robb] or w>rior f "s natural /ronghold ( ! m\nta9s sweeps ov] ! adjac5t pla9s & c>ries 8 raids into di/ant l&s4 ,wide grassy /eppes lead to ! organization ( ! p z nomads ^: weal? consi/s 9 flocks & h]ds1 & _! dwell+s are t5ts4 ,! nomad n only dome/icates & turns to 8 own use ! g5tl] & m p[]ful animals1 su* z %eep1 cattle1 horses1 camels1 b ev5 turns "s pr$atory creatures1 l ! dog1 into a means ( def5d+ _! natural prey4 ,!y hunt ! bea/s ( prey de/ructive to _! flocks1 & =m arm$ b&s = protection aga9/ m>aud]s or = purposes ( aggression on weak] s$5t>y neighb\rs4 ,on ! f]tile l[ gr\nds along ! m>g9s ( riv]s or 9 cle>+s ( =e/s1 agricultural communities naturally take _! rise1 dwell+ 9 villages & cultivat+ ! wild gra9s1 :i* by c>eful nurture & selection h be5 turn$ into ri* c]eals4 ,! agriculturi/ z a rule is root$ to ! soil4 ,! l& he tills he holds1 & acquires a clos] connexion ) a p>ticul> pat* ( gr\nd ?an ei!r ! hunt] or ! h]dsman4 ,9 ! temp]ate zone1 :]e ! seasons are %>ply contra/$1 b foll[ ea* o!r ) regul>ity1 =esight & self-d5ial 7 fo/]$1 because if m5 did n ex]cise ^! qualities se$-"t or h>ve/ might pass into lo/ opportunities & ! tribes w\ld suff]4 ,! m extreme climates ( arid regions on ! m>g9s ( ! tropics1 by ! unpr$ictable succession ( dr\ghts & floods1 conf\nd ! prevision ( un9/ruct$ p1 & make prud5ce & indu/ry qualities too unc]ta9 9 _! results to be wor? cultivat+4 ,?us ! civilization ( agricultural peoples ( ! temp]ate zone grew rapidly1 yet 9 ea* community a special type arose adapt$ to ! soil1 ! crop & ! climate4 ,on ! sea%ore fi%+ naturally became a means ( livelihood1 & dwell]s by ! sea1 9 virtue ( ! dang]s to :i* !y are expos$ f /orm & unseawor?y craft1 are /imulat$ to a high] degree ( =esight1 quick] obs]vation1 prompt] decision & m en]getic action 9 em]g5cies ?an ^? :o live inl&4 ,! build+ & h&l+ ( vessels also1 & ! utilization ( su* uncontrollable p[]s ( nature z w9d & tide1 help$ =w>d me*anical inv5tion4 ,to e type ( coa/ !re may be relat$ a special type ( occupation & ev5 ( "*2 ! deep & gloomy fjord1 back$ by almo/ impassable m\nta9s1 br$ bold m>9]s ^: only outlet = ent]prise 0 seaw>ds t[>ds o!r l&s--! viks creat$ ! vik+s4 ,on ! g5tly slop+ m>g9 ( ! e/u>y ( a great riv] a view ( tranquil inl& life 0 equally pres5t$ to ! %ore-dwell]1 & ! ocean did n pres5t ! only prospect ( a c>e]4 ,f9ally ! m\nta9 valley1 ) its pat*es ( cultivable soil on ! alluvial fans ( tribut>y torr5ts1 its n>r[ pa/ures on ! upl&s only left cle> ( sn[ 9 summ]1 its int5sifi$ extremes ( climates & its isolation1 almo/ equal to t ( an isl&1 has 9 all c\ntries produc$ a special type ( brave & h>dy p1 ^: utmo/ ef=t may br+ !m com=t1 b n weal?1 by hone/ toil1 :o "k little ( ! out] _w1 & to :om ! natural outlet = ambition is m>aud+ on ! f]tile pla9s4 ,! highl&] & vik+1 products ( ! valleys rais$ high amid ! m\nta9s or half-dr[n$ 9 ! sea1 are ev]y:]e ( k9dr$ _s4 ,x is 9 "s su* mann] z ^! t ! natural conditions ( regions1 :i* mu/ be con=m$ to by prud5ce & utiliz$ by lab\r to yield %elt] & food1 h l$ to ! gr[? ( peoples diff]+ 9 _! ways ( life1 ?\ght & spee*4 ,! initial diff]5ces s produc$ are confirm$ & p]petuat$ by ! same b>ri]s :i* divide ! faunal or floral regions1 ! sea1 m\nta9s1 des]ts & ! l1 & mu* ( ! c\rse ( pa/ hi/ory & pres5t politics becomes cle> :5 ! comb9$ results ( diff]+ race & diff]+ environm5t are tak5 into acc\nt4#di ,d5sity ( population4 ,! specialization :i* accompanies ! division ( lab\r has important geographical consequ5ces1 = x necessitates communication betwe5 communities & ! int]*ange ( _! products4 ,trade makes x possible to "w m9]al res\rces 9 localities :]e food c only be gr[n ) great difficulty & exp5se1 or :i* are ev5 totally b>r5 & wat]less1 entirely dep5d5t on supplies f di/ant s\rces4 ,! population :i* c be p]man5tly support$ by a giv5 area ( l& diff]s greatly accord+ to ! nature ( ! res\rces & ! requirem5ts ( ! p4 ,pa/oral communities are always scatt]$ v ?9ly ov] l>ge areas2 agricultural populations may be almo/ equally sp>se :]e advanc$ me?ods ( agriculture & lab\r-sav+ ma*9]y are employ$2 b :]e a frugal p are situat$ on a f]tile & inexhau/ible soil1 su* z ! deltas & riv] pla9s ( ,egypt1 ,india & ,*9a1 an enorm\s population may be support$ on a small area4 ,9 mo/ cases1 h[ev]1 a v d5se population c only be ma9ta9$ 9 regions :]e m9]al res\rces h fix$ ! site ( great manufactur+ indu/ries4 ,! maximum d5sity ( population :i* a giv5 region c support is v difficult to det]m9e2 x dep5ds p>tly on ! race & /&>d ( culture ( ! p1 p>tly on ! nature & orig9 ( ! res\rces on :i* !y dep5d1 p>tly on ! artificial burd5s impos$ & v l>gely on ! climate4 ,d5sity ( population is measur$ by ! av]age numb] ( p resid+ on a unit ( area2 b 9 ord] to comp>e "o "p ( ! _w ) ano!r ! av]age %\ld1 /rictly speak+1 be tak5 = regions ( equal size or ( equal population2 & ! portions ( ! c\ntry :i* are p]man5tly un9habitable ought to be exclud$ f ! calculation4#ej ,consid]+ ! av]age d5sity ( population )9 ! political limits ( c\ntries1 ! foll[+ li/ is ( "s value2 ! figures = a few small] divisions ( l>ge c\ntries are add$ 79 brackets7 = comp>ison3 ,av]age ,population on #a sq4 m4 7,= #aijj or #aija47 ---------------------------------------- ----------------- ,c\ntry4 ,d5sity ,c\ntry4 ,d5sity ( pop4 ( pop4 ---------------------------------------- ----------------- 7,saxony7 #gdc99 ,ceylon #ada9999 ,belgium #ehi99 ,greece #ig ,java #efh9999 ,european ,turkey #ij 7,engl& & ,wales7 #eeh ,spa9 #ig 7,b5gal7 #die9999 ,european ,russia #ee9999 ,holl& #dcf ,sw$5 #cj ,unit$ ,k+dom #cdd ,unit$ ,/ates #be ,japan #cag ,mexico #ah ,italy #bic ,norway #ah ,*9a prop] #bgj9999 ,p]sia #ae ,g]man ,empire #bgj ,new ,zeal& #g ,au/ria #bbf ,arg5t9a #e ,switz]l& #bjg ,brazil #d4e ,france #ahh ,ea/]n ,/ates ( ,indian ,empire #afg9999 ,au/ralia #c ,d5m>k #afj9999 ,dom9ion ( ,canada #a4e ,hung>y #aed9999 ,sib]ia #a ,portugal #adf ,we/ ,au/ralia #j4b ---------------------------------------- ----------------- 99 ,almo/ exclusively indu/rial4 9999 ,almo/ exclusively agricultural4 ,migration4 ,! movem5t ( p f "o place to ano!r )\t ! imm$iate int5tion ( return+ is kn[n z migration1 & accord+ to its orig9 x may be class$ z c5trifugal 7direct$ f a p>ticul> area7 & c5tripetal 7direct$ t[>ds a p>ticul> area74 ,c5trifugal migration is usually a matt] ( compulsion2 x may be necessitat$ by natural causes1 su* z a *ange ( climate lead+ to ! )]+ ( pa/ures or de/ruction ( agricultural l&1 to inundation1 e>?quake1 pe/il5ce or to an excess ( population ov] means ( support2 or to artificial causes1 su* z ! :olesale deportation ( a conqu]$ p2 or to political or religi\s p]secution4 ,9 any case ! p are driv5 out by "s adv]se *ange2 & :5 ! urg5cy is great !y may require to drive out 9 turn weak] p :o occupy a desirable t]ritory1 ?us propagat+ ! wave ( migration1 ! direction ( :i* is guid$ by ! =ms ( ! l& into inevitable *annels4 ,_m ( ! great hi/oric movem5ts ( peoples 7 d\btless due to ! gradual *ange ( geographical or climatic conditions2 & ! sl[ desiccation ( ,c5tral ,asia has be5 plausibly sugge/$ z ! real cause ( ! peopl+ ( mod]n ,europe & ( ! m$ieval w>s ( ! ,old ,_w1 ! !atres ( :i* 7 critical po9ts on ! great natural l9es ( communication betwe5 ea/ & we/4 ,9 ! case ( c5tripetal migrations p flock to "s p>ticul> place :]e exceptionally fav\rable conditions h be5 f\nd to exi/4 ,! ru%es to gold-fields & diamond-fields are typical in/ances2 ! gr[? ( t[ns on coal-fields & ne> o!r s\rces ( p[]1 & ! rapid settlem5t ( su* ri* agricultural di/ricts z ! :eat-l&s ( ! ,am]ican prairies & great pla9s are o!r examples4 ,!re is1 h[ev]1 a t5d5cy = p to rema9 root$ to ! l& ( _! bir?1 :5 n compell$ or induc$ by p[]ful ext]nal causes to seek a new home4 ,political geography4 ,?us arises ! _s ( patriotism1 a product ( purely geographical conditions1 !reby diff]+ f ! s5tim5t ( loyalty1 :i* is ( racial orig94 ,:]e race & soil conspire to evoke bo? loyalty & patriotism 9 a p1 ! moral qualities ( a great & p]man5t nation are secur$4 ,x is noticeable t ! patriotic _s is /ronge/ 9 ^? places :]e p are br\ght mo/ intimately into relation ) ! l&2 dwell]s 9 ! m\nta9 or by ! sea1 &1 above all1 ! p ( rugg$ coa/s & m\nta9\s ar*ipelagoes1 h always be5 r5[n$ = love ( c\ntry1 :ile ! inhabitants ( f]tile pla9s & trad+ communities are frequ5tly less /rongly atta*$ to _! own l&4 ,among/ nomads ! tribe is ! unit ( gov]nm5t1 ! political bond is p]sonal1 & !re is no def9ite t]ritorial association ( ! p1 :o may be loyal b _c be patriotic4 ,! idea ( a c\ntry arises only :5 a nation1 ei!r homog5e\s or compos$ ( sev]al races1 e/abli%es itself 9 a region ! b\nd>ies ( :i* may be def9$ & def5d$ aga9/ aggression f )\t4 ,political geography takes acc\nt ( ! p>tition ( ! e>? among/ organiz$ communities1 deal+ ) ! relation ( races to regions1 & ( nations to c\ntries1 & consid]+ ! conditions ( t]ritorial equilibrium & in/ability4 ,b\nd>ies4 ,! def9ition ( b\nd>ies & _! delimitation is "o ( ! mo/ important p>ts ( political geography4 ,natural b\nd>ies are always ! mo/ def9ite & ! /ronge/1 l5d+ !mselves mo/ readily to def5ce aga9/ aggression4 ,! sea is ! mo/ effective ( all1 & an isl& /ate is recogniz$ z ! mo/ /able4 ,next 9 importance comes a m\nta9 range1 b "h !re is (t5 difficulty z to ! def9ition ( ! actual cre/-l9e1 & m\nta9 ranges be+ broad regions1 x may happ5 t a small indep5d5t /ate1 l ,switz]l& or ,&orra1 occupies ! m\nta9 valleys betwe5 two or m great c\ntries4 ,riv]s d n =m effective int]national b\nd>ies1 al?\gh betwe5 dep5d5t self-gov]n+ communities !y are conv5i5t l9es ( dem>cation4 ,a des]t1 or a belt ( c\ntry left purposely )\t inhabitants1 l ! m>k1 m>*es or debatable l&s ( ! middle ages1 0 once a common means ( sep>at+ nations :i* n\ri%$ h]$it>y grievances4 ,! buff]-/ate ( mod]n diplomacy is ( ! same ineffectual type4 ,a less def9ite ?\gh v practical b\nd>y is t =m$ by ! meet+-l9e ( two languages1 or ! di/ricts inhabit$ by two races4 ,! l9e ( =tresses protect+ ,au/ria f ,italy lies 9 "s places well back f ! political b\nd>y1 b j inside ! l+ui/ic fronti]1 s z to sep>ate ! ,g]man & ,italian races occupy+ ,au/rian t]ritory4 ,arbitr>y l9es1 ei!r trac$ f po9t to po9t & m>k$ by po/s on ! gr\nd1 or def9$ z portions ( m]idians & p>allels1 are n[ ! mo/ common type ( b\nd>ies fix$ by treaty4 ,9 ,europe & ,asia fronti]s are usually /rongly =tifi$ & /rictly wat*$ 9 times ( peace z well z dur+ w>4 ,9 ,s\? ,am]ica /rictly def9$ b\nd>ies are /ill ! exception1 & ! claims ( neighb\r+ nations h v frequ5tly giv5 rise to w>1 ?\gh n[ m commonly to arbitration4#ea ,=ms ( gov]nm5t4 ,! modes ( gov]nm5t among/ civiliz$ peoples h little influ5ce on political geography2 "s republics are z arbitr>y & exact+ 9 _! fronti] regulations z "s absolute mon>*ies4 ,x is1 h[ev]1 to be notic$ t absolute mon>*ies are conf9$ to ! ea/ ( ,europe & to ,asia1 ,japan be+ ! only e/abli%$ con/itutional mon>*y ea/ ( ! ,c>pa?ians4 ,limit$ mon>*ies are 7) ! exception ( ,japan7 peculi> to ,europe1 & 9 ^! ! degree ( democratic control may be said to dim9i% z "o passes ea/w>ds f ! ,unit$ ,k+dom4 ,republics1 al?\gh repres5t$ 9 ,europe1 are ! peculi> =m ( gov]nm5t ( ,am]ica & are unkn[n 9 ,asia4 ,! =ms ( gov]nm5t ( colonies pres5t a s]ies ( transitional types f ! autocratic adm9i/ration ( a gov]nor appo9t$ by ! home gov]nm5t to complete democratic self-gov]nm5t4 ,! latt] occurs only 9 ! temp]ate possessions ( ! ,briti% empire1 9 :i* !re is no great prepond]ance ( a col\r$ native population4 ,new colonial =ms h be5 develop$ dur+ ! p>tition ( ,africa among/ ,european p[]s1 ! sph]e ( influ5ce be+ especially wor?y ( notice4 ,?is is a vagu] =m ( control ?an a protectorate1 & frequ5tly am\nts m]ely to an agreem5t among/ civiliz$ p[]s to respect ! "r ( "o ( _! numb] to ex]cise gov]nm5t )9 a c]ta9 area1 if x %\ld decide to d s at any future "t4 ,! c5tral gov]nm5ts ( all civiliz$ c\ntries conc]n$ ) ext]nal relations are closely simil> 9 _! modes ( action1 b ! int]nal adm9i/ration may be v v>i$4 ,9 ?is respect a c\ntry is ei!r c5traliz$1 l ! ,unit$ ,k+dom or ,france1 or f$]at$ ( di/9ct self-gov]n+ units l ,g]many 7:]e ! units include k+doms1 at lea/ ?ree m9or types ( mon>*ies1 municipalities & a cr[n l& "u a nom9at$ gov]nor71 or ! ,unit$ ,/ates1 :]e ! units are democratic republics4 ,! ultimate cause ( ! pr$om9ant =m ( f$]al gov]nm5t may be ! geographical div]sity ( ! c\ntry1 z 9 ! cantons occupy+ ! once isolat$ m\nta9 valleys ( ,switz]l&1 ! racial div]sity ( ! p1 z 9 ,au/ria-,hung>y1 or m]ely political exp$i5cy1 z 9 republics ( ! ,am]ican type4 ,! m9or subdivisions into prov9ces1 c\nties & p>i%es1 or analog\s areas1 may also be relat$ 9 _m cases to natural features or racial diff]5ces p]petuat$ by hi/orical causes4 ,! t]ritorial divisions & subdivisions (t5 survive ! conditions :i* l$ to _! orig92 h5ce ! /udy ( political geography is alli$ to hi/ory z closely z ! /udy ( physical geography is alli$ to geology1 & = ! same reason4 ,t[ns4 ,! aggregation ( population 9 t[ns 0 at "o "t ma9ly br\ght ab\t by ! necessity = def5ce1 a fact indicat$ by ! def5sive sites ( _m old t[ns4 ,9 lat] times1 t[ns h be5 m (t5 f\nd$ 9 proximity to valuable m9]al res\rces1 & at critical po9ts or nodes on l9es ( communication4 ,^! are places :]e ! mode ( travell+ or ( transport is *ang$1 su* z seaports1 riv] ports & railway t]m9i1 or natural re/+-places1 su* z a =d1 ! foot ( a /eep asc5t on a road1 ! entrance ( a valley lead+ up f a pla9 into ! m\nta9s1 or a cross+-place ( roads or railways4#eb ,! exi/5ce ( a good natural h>b\r is (t5 suffici5t to give orig9 to a t[n & to fix "o end ( a l9e ( l& communication4 ,l9es ( communication4 ,9 c\ntries ( uni=m surface or fa9t relief1 roads & railways may be con/ruct$ 9 any direction )\t reg>d to ! configuration4 ,9 places :]e ! l[ gr\nd is m>%y1 roads & railways (t5 foll[ ! ridge-l9es ( hills1 or1 z 9 ,f9l&1 ! old glacial esk]s1 :i* run p>allel to ! %ore4 ,:]ev] ! relief ( ! l& is pron\nc$1 roads & railways are oblig$ to occupy ! l[e/ gr\nd w9d+ along ! valleys ( riv]s & ?r\gh passes 9 ! m\nta9s4 ,9 exceptional cases ob/ructions :i* x w\ld be impossible or too co/ly to turn are ov]come by a bridge or tunnel1 ! magnitude ( su* works increas+ ) ! gr[? ( eng9e]+ skill & f9ancial ent]prise4 ,simil>ly ! ob/ructions (f]$ to wat] communication by int]ruption ?r\gh l& or %all[s are ov]come by cutt+ canals or dr$g+ out *annels4 ,! economy & success ( mo/ l9es ( communication dep5d on foll[+ z f> z possible exi/+ natural l9es & utiliz+ exi/+ natural s\rces ( p[]4#ec ,comm]cial geography4 ,comm]cial geography may be def9$ z ! description ( ! e>?'s surface ) special ref]5ce to ! discov]y1 production1 transport & ex*ange ( commodities4 ,! transport conc]ns l& r\tes & sea r\tes1 ! latt] be+ ! m important4 ,:ile /eam has be5 said to make a %ip indep5d5t ( w9d & tide1 x is /ill true t a long voyage ev5 by /eam mu/ be plann$ s z to enc\nt] ! lea/ resi/ance possible f prevail+ w9ds & p]man5t curr5ts1 & ?is involves ! application ( oceanographical & meteorological k4 ,! old] navigation by utiliz+ ! p[] ( ! w9d dem&s a v intimate k ( ^! conditions1 & x is probable t a revival ( sail+ %ips may 9 ! pres5t c5tury va/ly increase ! importance ( ! /udy ( m>itime meteorology4 ,! discov]y & production ( commodities require a k ( ! di/ribution ( geological =mations = m9]al products1 ( ! natural di/ribution1 life-conditions & cultivation or bre$+ ( plants & animals & ( ! lab\r m>ket4 ,att5tion mu/ also be paid to ! artificial re/rictions ( political geography1 to ! legislative re/rictions be>+ on lab\r & trade z impos$ 9 diff]5t c\ntries1 &1 above all1 to ! incessant fluctuations ( ! economic conditions ( supply & dem& & ! comb9ations ( capitali/s or work]s :i* affect ! m>ket4#ed ,! t]m appli$ geography has be5 employ$ to designate comm]cial geography1 ! fact be+ t e aspect ( sci5tific geography may be appli$ to practical purposes1 includ+ ! purposes ( trade4 ,b ap>t f ! appli$ sci5ce1 !re is an aspect ( pure geography :i* conc]ns ! !ory ( ! relation ( economics to ! surface ( ! e>?4 ,conclusion4 ,x w be se5 t z ea* successive aspect ( geographical sci5ce is consid]$ 9 its natural sequ5ce ! conditions become m num]\s1 complex1 v>iable & practically important4 ,f ! und]ly+ ab/ract ma!matical consid]ations all ?r\gh ! sup]impos$ physical1 biological1 an?ropological1 political & comm]cial developm5t ( ! subject runs ! det]m9+ control ex]cis$ by cru/-=ms act+ directly or indirectly on mobile di/ributions2 & ?is is ! ess5tial pr9ciple ( geography4 7,h4 ,r4 ,m47 ,,footnotes3 #a ,a concise sket* ( ! :ole hi/ory ( geographical me?od or !ory z di/+ui%$ f ! hi/ory ( geographical discov]y 7see lat] section ( ?is article7 is only to be f\nd 9 ! introduction to ,h4 ,wagn]'s ,lehrbu* d] ,geographie1 vol4 i4 7,leipzig1 #aijj71 :i* is 9 e way ! mo/ complete treatise on ! pr9ciples ( geography4 #b ,hi/ory ( ,anci5t ,geography 7,cambridge1 #ahig71 p4 #gj4 #c ,see ,j4,l4 ,myres1 ,an ,attempt to recon/ruct ! ,maps us$ by ,h]odotus1 ,geographical ,j\rnal1 viii4 7#ahif71 p4 #fje4 #d ,ges*i*te d] wiss5s*aftli*5 ,erdkunde d] ,grie*5 7,leipzig1 #ahia71 ,abt4 #c1 p4 #fj4 #e ,bunbury's ,hi/ory ( ,anci5t ,geography 7#b vols41 ,london1 #ahgi71 ,mull]'s ,geographi ,graeci m9ores 7#b vols41 ,p>is1 #ahee1 #ahfa7 & ,b]g]'s ,ges*i*te d] wiss5s*aftli*5 ,erdkunde d] ,grie*5 7#d vols41 ,leipzig1 #ahhg-ahic7 are /&>d au?orities on ! ,greek geograph]s4 #f ,! p]iod ( ! e>ly middle ages is dealt ) 9 ,beazley's ,dawn ( ,mod]n ,geography 7,london2 "p i41 #ahig2 "p ii41 #aija2 "p iii41 #aijf72 see also ,w9/$t1 ,cosmos ,indicopleu/es 7#aiaj74 #g ,f translator's preface to ! ,engli% v]sion by ,mr ,dugdale 7#agcc71 entitl$ ,a ,complete ,sy/em ( ,g5]al ,geography1 revis$ by ,dr ,pet] ,%aw 7,london1 #agef74 #h ,pr9t$ 9 ,s*rift5 zur physis*5 ,geographie1 vol4 vi4 ( ,s*ub]t's edition ( ! collect$ works ( ,kant 7,leipzig1 #ahci74 ,fir/ publi%$ ) notes by ,r9k 9 #ahjb4 #i ,hi/ory ( ,civilization1 vol4 i4 7#aheg74 #aj ,see ,h4,j4 ,mack9d] 9 ,briti% ,association ,report 7,ipswi*71 #ahie1 p4 #gch1 = a summ>y ( ,g]man op9ion1 :i* has be5 express$ by _m writ]s 9 a some:at volum9\s lit]ature4 #aa ,h4 ,wagn]'s ye>-book1 ,geographis*e ,jahrbu*1 publi%$ at ,go?a1 is ! be/ sy/ematic record ( ! progress ( geography 9 all dep>tm5ts2 & ,haack's ,geograph5 ,kal5d]1 also publi%$ annually at ,go?a1 gives complete li/s ( ! geographical societies & geograph]s ( ! _w4 #ab ,?is phrase is old1 appe>+ 9 "o ( ! e>lie/ ,engli% works on geography1 ,william ,cun+ham's ,cosmographical ,glasse conte9yng ! pleasant ,pr9ciples ( ,cosmographie1 ,geographie1 ,hydrographie or ,navigation 7,london1 #aeei74 #ac ,see also ,s4 ,gun!r1 ,h&bu* d] ma!matis*5 ,geographie 7,/uttg>t1 #ahij74 #ad ,on ! ,height ( ! ,l& & ! ,dep? ( ! ,ocean1 ,scot4 ,geog4 ,mag4 iv4 7#ahhh71 p4 #a4 ,e/imates _h be5 made previ\sly by ,humboldt1 ,de ,lapp>5t1 ,h4 ,wagn]1 & subsequ5tly by ,p5ck & ,heid]i*1 & = ! oceans by ,k>/5s4 #ae ,pet]manns ,mitteilung51 xxv4 7#ahhi71 p4 #ag4 #af ,proc4 ,roy4 ,soc4 ,ed94 xvii4 7#ahij7 p4 #ahe4 #ag ,comptes r5dus ,acad4 ,sci4 7,p>is1 #ahij71 vol4 iii4 p4 #iid4 #ah ,areal und mittl]e ,erhebung d] ,l&fla*5 s[ie d] ,erdkru/e 9 ,g]l&'s ,beitrage zur ,geophysik1 ii4 7#ahie7 p4 #ffg4 ,see also ,nature1 #ed 7#ahif71 p4 #aab4 #ai ,pet]manns ,mitteilung51 xxxv4 7#ahhi7 p4 #ai4 #bj ,! areas ( ! cont95tal %elf & l[l&s are approximately equal1 & x is an int]e/+ circum/ance t1 tak5 z a :ole1 ! actual coa/-l9e comes j midway on ! mo/ ne>ly level belt ( ! e>?'s surface1 except+ ! ocean floor4 ,! configuration ( ! cont95tal slope has be5 treat$ 9 detail by ,nans5 9 ,sci5tific ,results ( ,norwegian ,nor? ,pol> ,exp$ition1 vol4 iv4 7#aijd71 :]e full ref]5ces to ! lit]ature ( ! subject w be f\nd4 #ba ,briti% ,association ,report 7,ed9burgh1 #ahib71 p4 #fii4 #bb ,das ,antlitz d] ,erde 7#d vols41 ,leipzig1 #ahhe1 #ahhh1 #aija74 ,translat$ "u ! editor%ip ( ,e4 de ,m>g]ie1 ) mu* additional matt]1 z ,la ,face de la t]re1 vols4 i4 & ii4 7,p>is1 #ahig1 #aijj71 & into ,engli% by ,dr ,h]?a ,sollas z ,! ,face ( ! ,e>?1 vols4 i4 & ii4 7,ox=d1 #aijd1 #aijf74 #bc ,elie de ,beaumont1 ,notice sur les sy/emes de montagnes 7#c vols41 ,p>is1 #aheb74 #bd ,ve/iges ( ! ,molt5 ,globe 7,london1 #ahge74 #be ,see ,j4,w4 ,gregory1 ,! ,plan ( ! ,e>? & its ,causes1 ,geog4 ,j\rnal1 xiii4 7#ahii7 p4 #bbe2 ,"l ,avebury1 ibid4 xv4 7#aijj7 p4 #df2 ,m>cel ,b]tr&1 ,de=mation tetra$rique de la t]re et deplacem5t du pole1 ,comptes r5dus ,acad4 ,sci4 7,p>is1 #aijj71 vol4 cxxx4 p4 #ddi2 & ,a4 de ,lapp>5t1 ibid4 p4 #fad4 #bf ,see ,a4,e4,h4 ,love1 ,gravitational ,/ability ( ! ,e>?1 ,phil4 ,trans4 s]4 ,a4 vol4 ccvii4 7#aijg7 p4 #aga4 #bg ,rumpf1 9 ,g]man1 ! language 9 :i* ?is di/9ction 0 fir/ made4 #bh ,lehrbu* d] ,geographie 7,hanov] & ,leipzig1 #aijj71 ,bd4 i4 ,s4 #bde1 #bdi4 #bi ,see1 = example1 ,f4,g4 ,hahn's ,insel-,/udi5 7,leipzig1 #ahhc74 #cj ,see ,geographical ,j\rnal1 xxii4 7#aijc7 pp4 #aia-aid4 #ca ,! mo/ important works on ! classification ( l& =ms are ,f4 von ,ri*?(51 ,fuhr] fur ,=s*ungsreis5de 7,b]l91 #ahhf72 ,g4 de la ,noe & ,e4 de ,m>g]ie1 ,les ,=mes du t]ra9 7,p>is1 #ahhh72 & above all ,a4 ,p5ck1 ,morphologie d] ,erdob]fla*e 7#b vols41 ,/uttg>t1 #ahid74 ,comp>e also ,a4 de ,lapp>5t1 ,lecons de geographie physique 7#bnd ed41 ,p>is1 #ahih71 & ,w4,m4 ,davis1 ,physical ,geography 7,bo/on1 #ahii74 #cb ,geomorphologie als g5etis*e ,wiss5s*aft1 9 ,report ( ,six? ,int]national ,geog4 ,congress 7,london1 #ahie71 p4 #gce 7,engli% ,ab/ract1 p4 #gdh74 #cc ,on ?is subject see ,j4 ,geikie1 ,e>? ,sculpture 7,london1 #ahih72 ,j4,e4 ,m>r1 ,! ,sci5tific ,/udy ( ,sc5]y 7,london1 #aijj72 ,sir ,a4 ,geikie1 ,! ,sc5]y & ,geology ( ,scotl& 7,london1 #bnd ed41 #ahhg72 ,"l ,avebury 7,sir ,j4 ,lubbock71 ,! ,sc5]y ( ,switz]l& 7,london1 #ahif7 & ,! ,sc5]y ( ,engl& 7,london1 #aijb74 #cd ,"s geograph]s di/+ui% a m\nta9 f a hill by orig92 ?us ,pr(essor ,seeley says a m\nta9 implies elevation & a hill implies d5udation1 b ! ext]nal =ms ( bo? are (t5 id5tical4 ,report ,,vi4 ,int4 ,geog4 ,congress 7,london1 #ahie71 p4 #gea4 #ce ,m\nta9s1 9 ,scot4 ,geog4 ,mag4 ii4 7#ahif7 p4 #ade4 #cf ,fuhr] fur ,=s*ungsreis5de1 pp4 #feb-fhe4 #cg ,see1 = a summ>y ( riv]-action1 ,a4 ,phillipson1 ,/udi5 ub] ,wass]s*eid5 7,leipzig1 #ahhf72 also ,i4,c4 ,russell1 ,riv] ,developm5t1 7,london1 #ahih7 7publi%$ z ,! ,riv]s ( ,nor? ,am]ica1 ,new ,york1 #ahih74 #ch ,w4,m4 ,davis1 ,! ,geographical ,cycle1 ,geog4 ,j\rn4 xiv4 7#ahii7 p4 #dhd4 #ci ,a4 ,p5ck1 ,potamology z a ,bran* ( ,physical ,geography1 ,geog4 ,j\rn4 x4 7#ahig7 p4 #fai4 #dj ,see1 = in/ance1 ,e4 ,wisotzki1 ,hauptfluss und ,neb5fluss 7,/ett91 #ahhi74 ,= practical /udies see (ficial reports on ! ,mississippi1 ,rh9e1 ,se9e1 ,elbe & o!r great riv]s4 #da ,f4,a4 ,=el1 ,h&bu* d] ,se5kunde3 allgeme9e ,limnologie 7,/uttg>t1 #aija72 ,f4,a4 ,=el1 ,la ,limnologie1 bran*e de la geographie1 ,report ,,vi4 ,int4 ,geog4 ,congress 7,london1 #ahie71 p4 #eic2 also ,le ,leman 7#b vols41 ,lausanne1 #ahib1 #ahid72 ,h4 ,lullies1 ,/udi5 ub] ,se51 ,jubilaumss*rift d] ,alb]tus-,univ]sitat 7,konigsb]g1 #ahid72 & ,g4,r4 ,cr$n]1 ,die ,relikt5se51 ,pet]manns ,mitteilung51 ,erganzung%efte #hf & #hi 7,go?a41 #ahhg1 #ahhh74 #db ,j4 ,murray1 ,dra9age ,areas ( ! ,cont95ts1 ,scot4 ,geog4 ,mag4 ii4 7#ahhf7 p4 #edh4 #dc ,wagn]1 ,lehrbu* d] ,geographie 7#aijj71 i4 #ehf4 #dd ,= details1 see ,a4,r4 ,wallace1 ,geographical ,di/ribution ( ,animals & ,isl& ,life2 ,a4 ,heilpr91 ,geographical & ,geological ,di/ribution ( ,animals 7#ahhg72 ,o4 ,drude1 ,h&bu* d] ,pflanz5geographie2 ,a4 ,engl]1 ,entwickelungsges*i*te d] ,pflanz5welt2 also ,b$d>d1 ,zoogeography 7,cambridge1 #ahie72 & ,sclat]1 ,! ,geography ( ,mammals 7,london1 #ahii74 #de ,see p>ticul>ly ,a4 de ,lapp>5t1 ,traite de geologie 7#d? ed41 ,p>is1 #aijj74 #df ,e/imate = #aijj4 ,h4 ,wagn]1 ,lehrbu* d] ,geographie1 i4 ,p4 #feh4 #dg ,e/imate = ye> n /at$4 ,a4,h4 ,keane 9 ,int]national ,geography1 p4 #ajh4 #dh ,9 ,proc4 ,r4 ,g4 ,s4 xiii4 7#ahia7 p4 #bg4 #di ,on ! influ5ce ( l& on p see ,%al]1 ,nature & ,man 9 ,am]ica 7,new ,york & ,london1 #ahib72 & ,ell5 ,c4 ,semple's ,am]ican ,hi/ory & its ,geographic ,conditions 7,bo/on1 #aijc74 #ej ,see maps ( d5sity ( population 9 ,b>?olomew's great l>ge-scale atlases1 ,atlas ( ,scotl& & ,atlas ( ,engl&4 #ea ,= ! hi/ory ( t]ritorial *anges 9 ,europe1 see ,freeman1 ,hi/orical ,geography ( ,europe1 edit$ by ,bury 7,ox=d71 #aijc2 & = ! (ficial def9ition ( exi/+ b\nd>ies1 see ,h]tslet1 ,! ,map ( ,europe by ,treaty 7#d vols41 ,london1 #ahge1 #ahia72 ,! ,map ( ,africa by ,treaty 7#c vols41 ,london1 #ahif74 ,also ,"l ,curzon's ,ox=d address on ,fronti]s 7#aijg74 #eb ,= num]\s special in/ances ( ! det]m9+ causes ( t[n sites1 see ,g4,g4 ,*i%olm1 ,on ! ,di/ribution ( ,t[ns & ,villages 9 ,engl&1 ,geographical ,j\rnal 7#ahig71 ix4 #gf1 x4 #eaa4 #ec ,! :ole subject ( an?ropogeography is treat$ 9 a ma/]ly way by ,f4 ,ratzel 9 8 ,an?ropogeographie 7,/uttg>t1 vol4 i4 #bnd ed41 #ahii1 vol4 ii4 #ahia71 & 9 8 ,politis*e ,geographie 7,leipzig1 #ahig74 ,! special "q ( ! reaction ( man on 8 environm5t is h&l$ by ,g4,p4 ,m>% 9 ,man & ,nature1 or ,physical ,geography z modifi$ by ,human ,action 7,london1 #ahfd74 #ed ,= comm]cial geography see ,g4,g4 ,*i%olm1 ,manual ( ,comm]cial ,geography 7#ahij74 ,,geoid 7f ,gr4 ,greek3 ge1 ! e>?71 an imag9>y surface employ$ by geodesi/s :i* has ! prop]ty t e elem5t ( x is p]p5dicul> to ! plumb-l9e :]e t l9e cuts x4 ,comp>$ ) ! sph]oid ( ref]5ce ! surface ( ! geoid is 9 g5]al depress$ ov] ! oceans & rais$ ov] ! great l& masses4 7,see ,,e>?1 ,,figure ,,( ,,!47 ,,geok-,,tepe1 a =m] =tress ( ! ,turkomans1 9 ,russian ,transcaspia1 9 ! oasis ( ,akhal-tekke1 on ! ,transcaspian railway1 #bh m4 ,n4,w4 ( ,askabad4 ,x consi/$ ( a wall$ enclosure #a-c_/d m4 9 circuit1 ! wall be+ #ah ft4 high & #bj to #cj ft4 ?ick4 ,9 ,decemb] #ahhj ! place 0 attack$ by #fjjj ,russians "u ,g5]al ,skobelev1 & aft] a siege ( tw5ty-?ree days 0 c>ri$ by /orm1 al?\gh ! def5d]s numb]$ #be1jjj4 ,a monum5t & a small museum commemorate ! ev5t4 ,,geology 7f ,gr4 ,greek3 ge1 ! e>?1 & ,greek3 logos1 sci5ce71 ! sci5ce :i* inve/igates ! physical hi/ory ( ! e>?4 ,its object is to trace ! /ructural progress ( our planet f ! e>lie/ beg9n+s ( its sep>ate exi/5ce1 ?r\gh its v>i\s /ages ( gr[?1 d[n to ! pres5t condition ( ?+s4 ,x seeks to det]m9e ! mann] 9 :i* ! evolution ( ! e>?'s great surface features has be5 effect$4 ,x unravels ! complicat$ processes by :i* ea* cont95t has be5 built up4 ,x foll[s1 ev5 into detail1 ! v>i$ sculpture ( m\nta9 & valley1 crag & rav9e4 ,nor does x conf9e itself m]ely to *anges 9 ! inorganic _w4 ,geology %[s t ! pres5t races ( plants & animals are ! desc5dants ( o!r & v diff]5t races :i* once peopl$ ! e>?4 ,x tea*es t !re has be5 a progressive developm5t ( ! inhabitants1 z well z "o ( ! globe on :i* !y h dwelt2 t ea* successive p]iod 9 ! e>?'s hi/ory1 s9ce ! introduction ( liv+ ?+s1 has be5 m>k$ by *>act]i/ic types ( ! animal & vegetable k+doms2 & t1 h[ev] imp]fectly ! rema9s ( ^! organisms h be5 pres]v$ or may be deciph]$1 mat]ials exi/ = a hi/ory ( life ^u ! planet4 ,! geographical di/ribution ( exi/+ faunas & floras is (t5 made cle> & intelligible by geological evid5ce2 & 9 ! same way light is ?r[n ^u "s ( ! remot] phases 9 ! hi/ory ( man himself4 ,a subject s compreh5sive z ?is mu/ require a wide & v>i$ basis ( evid5ce4 ,x is "o ( ! *>act]i/ics ( geology to ga!r evid5ce f s\rces :i* at fir/ sight seem f> remov$ f its scope1 & to seek aid f almo/ e o!r lead+ bran* ( sci5ce4 ,?us1 9 deal+ ) ! e>lie/ conditions ( ! planet1 ! geologi/ mu/ fully avail himself ( ! lab\rs ( ! a/ronom]4 ,:atev] is asc]ta9able by telescope1 spectroscope or *emical analysis1 reg>d+ ! con/itution ( o!r heav5ly bodies1 has a geological be>+4 ,! exp]im5ts ( ! physici/1 und]tak5 to det]m9e conditions ( matt] & ( en]gy1 may sometimes be tak5 z ! />t+-po9ts ( geological inve/igation4 ,! "w ( ! *emical laboratory =ms ! f\ndation ( a va/ & increas+ mass ( geological inquiry4 ,to ! botani/1 ! zoologi/1 ev5 to ! unsci5tific1 if obs]vant1 travell] by l& or sea1 ! geologi/ turns = in=mation & assi/ance4 ,b :ile ?us cull+ freely f ! dom9ions ( o!r sci5ces1 geology claims z its peculi> t]ritory ! rocky framework ( ! globe4 ,9 ! mat]ials compos+ t framework1 _! composition & arrangem5t1 ! processes ( _! =mation1 ! *anges :i* !y h und]gone1 & ! t]re/rial revolutions to :i* !y be> witness1 lie ! ma9 data ( geological hi/ory4 ,x is ! task ( ! geologi/ to gr\p ^! elem5ts 9 su* a way t !y may be made to yield up _! evid5ce z to ! m>* ( ev5ts 9 ! evolution ( ! planet4 ,he f9ds t !y h 9 l>ge measure arrang$ !mselves 9 *ronological sequ5ce1--! olde/ ly+ at ! bottom & ! newe/ at ! top4 ,relics ( an anci5t sea-floor are ov]la9 by traces ( a vani%$ l&-surface2 ^! are 9 turn cov]$ by ! deposits ( a =m] lake1 above :i* once m appe> pro(s ( ! return ( ! sea4 ,among ^! rocky records lie ! lavas & a%es ( long-ext9ct volcanoes4 ,! ripple left ^u ! %ore1 ! cracks =m$ by ! sun's heat ^u ! muddy bottom ( a dri$-up pool1 ! v impr9t ( ! drops ( a pass+ ra9%[]1 h all be5 accurately pres]v$1 & yield _! evid5ce z to geographical conditions (t5 widely diff]5t f ^? :i* exi/ :]e su* m>k+s are n[ f\nd4 ,b x is ma9ly by ! rema9s ( plants & animals imb$d$ 9 ! rocks t ! geologi/ is guid$ 9 unravell+ ! *ronological succession ( geological *anges4 ,he has f\nd t a c]ta9 ord] ( appe>ance *>act]izes ^! organic rema9s1 t ea* great gr\p ( rocks is m>k$ by its own special types ( life1 & t ^! types c be recogniz$1 & ! rocks 9 :i* !y occur c be correlat$ ev5 9 di/ant c\ntries1 & :]e no o!r means ( comp>ison w\ld be possible4 ,at "o mom5t he has to deal ) ! bones ( "s l>ge mammal scatt]$ ?r\gh a deposit ( sup]ficial gravel1 at ano!r "t ) ! m9ute =am9if]s & o/racods ( an uprais$ sea-bottom4 ,corals & cr9oids cr[d$ & cru%$ into a massive lime/one :]e !y liv$ & di$1 f]ns & t]re/rial plants matt$ toge!r into a b$ ( coal :]e !y orig9ally grew1 ! scatt]$ %ells ( a subm>9e s&-bank1 ! snails & liz>ds :i* liv$ & di$ )9 a holl[-tree1 ! insects :i* h be5 imprison$ )9 ! exud+ res9 ( old =e/s1 ! footpr9ts ( birds & quadrup$s1 ! trails ( worms left ^u =m] %ores--^!1 & innum]able o!r pieces ( evid5ce1 enable ! geologi/ to realize 9 "s measure :at ! faunas & floras ( successive p]iods h be51 & :at geographical *anges ! site ( e l& has und]gone4 ,x is evid5t t to deal successfully ) ^! v>i$ mat]ials1 a consid]able acqua9tance ) diff]5t bran*es ( sci5ce is ne$ful4 ,especially necess>y is a tol]ably wide k ( ! processes n[ at "w 9 *ang+ ! surface ( ! e>?1 & ( at lea/ ^? =ms ( plant & animal life ^: rema9s are apt to be pres]v$ 9 geological deposits1 or :i* 9 _! /ructure & habitat enable u to realize :at _! =]unn]s 74 ,x has (t5 be5 insi/$ t ! pres5t is ! key to ! pa/2 & 9 a wide s5se ?is ass]tion is em95tly true4 ,only 9 proportion z we und]/& ! pres5t1 :]e ev]y?+ is op5 on all sides to ! fulle/ inve/igation1 c we expect to deciph] ! pa/1 :]e s mu* is obscure1 imp]fectly pres]v$ or n pres]v$ at all4 ,a /udy ( ! exi/+ economy ( nature ought ?us to be ! f\ndation ( ! geologi/'s tra9+4 ,:ile1 h[ev]1 ! pres5t condition ( ?+s is ?us employ$1 we mu/ obvi\sly be on our gu>d aga9/ ! dang] ( unconsci\sly assum+ t ! phase ( nature's op]ations :i* we n[ witness has be5 ! same 9 all pa/ "t1 t geological *anges h always or g5]ally tak5 place 9 =m] ages 9 ! mann] & on ! scale :i* we behold to-day1 & t at ! pres5t "t all ! great geological processes1 :i* h produc$ *anges 9 ! pa/ eras ( ! e>?'s hi/ory1 are /ill exi/5t & active4 ,z a work+ hypo!sis we may suppose t ! nature ( geological processes has rema9$ con/ant f ! beg9n+2 b we _c po/ulate t ! action ( ^! processes has nev] v>i$ 9 en]gy4 ,! few c5turies :]e9 man has be5 obs]v+ nature obvi\sly =m mu* too brief an int]val by :i* to measure ! int5sity ( geological action 9 all pa/ "t4 ,= aught we c tell ! pres5t is an era ( quietude & sl[ *ange1 comp>$ ) "s ( ! eras :i* h prec$$ x4 ,nor p]haps c we be q sure t1 :5 we h explor$ e geological process n[ 9 progress1 we h exhau/$ all ! causes ( *ange :i*1 ev5 9 comp>atively rec5t times1 h be5 at "w4 ,9 deal+ ) ! geological record1 z ! accessible solid "p ( ! globe is call$1 we _c too vividly realize t at ! be/ x =ms b an imp]fect *ronicle4 ,geological hi/ory _c be compil$ f a full & cont9u\s s]ies ( docum5ts4 ,f ! v nature ( its orig9 ! record is necess>ily fragm5t>y1 & x has be5 fur!r mutilat$ & obscur$ by ! revolutions ( successive ages4 ,& ev5 :]e ! *ronicle ( ev5ts is cont9u\s1 x is ( v unequal value 9 diff]5t places4 ,9 "o case1 = example1 x may pres5t u ) an unbrok5 succession ( deposits _m ?\s&s ( feet 9 ?ickness1 f :i*1 h[ev]1 only a few meagre facts z to geological hi/ory c be glean$4 ,9 ano!r in/ance x br+s be=e u1 )9 ! compass ( a few y>ds1 ! evid5ce ( a mo/ v>i$ & complicat$ s]ies ( *anges 9 physical geography1 z well z an abundant & int]e/+ suite ( organic rema9s4 ,^! & o!r *>act]i/ics ( ! geological record become m app>5t & intelligible z we proce$ 9 ! /udy ( ! sci5ce4 ,classification4--,= sy/ematic treatm5t ! subject may be conv5i5tly arrang$ 9 ! foll[+ p>ts3-- #a4 ,! ,hi/orical ,developm5t ( ,geological ,sci5ce4--,"h a brief outl9e w be giv5 ( ! gradual gr[? ( geological conceptions f ! days ( ! ,greeks & ,romans d[n to mod]n times1 trac+ ! sep>ate progress ( ! m important bran*es ( inquiry & not+ "s ( ! /ages :i* 9 ea* case h l$ up to ! pres5t condition ( ! sci5ce4 #b4 ,! ,cosmical ,aspects ( ,geology4--,?is section embraces ! evid5ce suppli$ by a/ronomy & physics reg>d+ ! =m & motions ( ! e>?1 ! composition ( ! planets & sun1 & ! probable hi/ory ( ! sol> sy/em4 ,! subjects dealt ) "u ?is head are *iefly treat$ 9 sep>ate articles4 #c4 ,geognosy4--,an inquiry into ! mat]ials ( ! e>?'s sub/ance4 ,?is division1 :i* deals ) ! p>ts ( ! e>?1 its envelopes ( air & wat]1 its solid cru/ & ! probable condition ( its int]ior1 especially treats ( ! m important m9]als ( ! cru/1 & ! *ief rocks ( :i* t cru/ is built up4 ,geognosy ?us lays a f\ndation ( k reg>d+ ! nature ( ! mat]ials con/itut+ ! mass ( ! globe1 & prep>es ! way = an inve/igation ( ! processes by :i* ^! mat]ials are produc$ & alt]$4 #d4 ,dynamical ,geology /udies ! nature & work+ ( ! v>i\s geological processes :]eby ! rocks ( ! e>?'s cru/ are =m$ & metamorphos$1 & by :i* *anges are effect$ ^u ! di/ribution ( sea & l&1 & ^u ! =ms ( t]re/rial surfaces4 ,su* an inquiry necessitates a c>eful exam9ation ( ! exi/+ geological economy ( nature1 & =ms a fitt+ introduction to an inquiry into ! geological *anges ( =m] p]iods4 #e4 ,geotectonic or ,/ructural ,geology has = its object ! ar*itecture ( ! e>?'s cru/4 ,x embraces an inquiry into ! mann] 9 :i* ! v>i\s mat]ials compos+ ?is cru/ h be5 arrang$4 ,x %[s t "s h be5 =m$ 9 b$s or /rata ( s$im5t on ! floor ( ! sea1 t o!rs h be5 built up by ! sl[ aggregation ( organic =ms1 t o!rs h be5 p\r$ out 9 a molt5 condition or 9 %[]s ( loose du/ f subt]ranean s\rces4 ,x fur!r reveals t1 ?\gh orig9ally laid d[n 9 almo/ horizontal b$s1 ! rocks h subsequ5tly be5 crumpl$1 contort$ & dislocat$1 t !y h be5 incessantly worn d[n1 & h (t5 be5 depress$ & buri$ b5ea? lat] accumulations4 #f4 ,palaeontological ,geology4--,?is bran* ( ! subject1 />t+ f ! evid5ce suppli$ by ! organic =ms :i* are f\nd pres]v$ 9 ! cru/ ( ! e>?1 includes su* que/ions z ! relations betwe5 ext9ct & liv+ types1 ! laws :i* appe> to h gov]n$ ! di/ribution ( life 9 "t & 9 space1 ! relative importance ( diff]5t g5]a ( animals 9 geological inquiry1 ! nature & use ( ! evid5ce f organic rema9s reg>d+ =m] conditions ( physical geography4 ,"s ( ^! problems belong also to zoology & botany1 & are m fully discuss$ 9 ! articles ,,palaeontology & ,,palaeobotany4 #g4 ,/ratigraphical ,geology4--,?is section might be call$ geological hi/ory4 ,x works out ! *ronological succession ( ! great =mations ( ! e>?'s cru/1 & endeav\rs to trace ! sequ5ce ( ev5ts ( :i* !y conta9 ! record4 ,m p>ticul>ly1 x det]m9es ! ord] ( succession ( ! v>i\s plants & animals :i* 9 pa/ "t h peopl$ ! e>?1 & ?us asc]ta9s :at has be5 ! gr& m>* ( life ^u ?is planet4 #h4 ,physiographical ,geology1 proce$+ f ! basis ( fact laid d[n by /ratigraphical geology reg>d+ =m] geographical *anges1 embraces an inquiry into ! orig9 & hi/ory ( ! features ( ! e>?'s surface--cont95tal ridges & ocean bas9s1 pla9s1 valleys & m\nta9s4 ,x expla9s ! causes on :i* local diff]5ces ( sc5]y dep5d1 & %[s "u :at v diff]5t circum/ances1 & at :at widely sep>at$ int]vals1 ! hills & m\nta9s1 ev5 ( a s+le c\ntry1 h be5 produc$4 ,mo/ ( ! detail embrac$ 9 ^! sev]al sections is relegat$ to sep>ate articles1 to :i* ref]5ces are "h ins]t$4 ,! foll[+ pages ?us deal ma9ly ) ! g5]al pr9ciples & hi/orical developm5t ( ! sci5ce3-- ,,"p ,i4--,,hi/orical ,,developm5t ,geological ,ideas among ! ,greeks & ,romans4--,_m geological ph5om5a pres5t !mselves 9 s /rik+ a =m t !y c\ld h>dly fail to impress ! imag9ation ( ! e>lie/ & rude/ races ( mank9d4 ,su* incid5ts z e>?quakes & volcanic eruptions1 de/ructive /orms on l& & sea1 disa/r\s floods & l&slips sudd5ly /rew+ valleys ) ru91 mu/ h awak5$ ! t]ror ( ^? :o witness$ !m4 ,prom95t features ( l&scape1 su* z m\nta9-*a9s ) _! sn[s1 cl\ds & ?und]/orms1 d>k riv]-*asms t seem purposely cleft op5 9 ord] to give passage to ! torr5ts t ru% ?r\gh !m1 crags ) _! impressive array ( p9nacles & recesses mu/ h appeal$ ( old1 z !y /ill d1 to ! awe & wond] ( ^? :o = ! fir/ "t behold !m4 ,aga91 banks ( sea-%ells 9 f> inl& di/ricts w\ld1 9 c\rse ( "t1 arre/ ! att5tion ( ! m intellig5t & reflective obs]v]s1 & raise 9 _! m9ds "s k9d ( surmise z to h[ su* %ells c\ld ev] h come !re4 ,^! & o!r conspicu\s geological problems f\nd _! e>lie/ solution 9 leg5ds & my?s1 :]e9 ! m /rik+ t]re/rial features & ! elem5tal =ces ( nature 7 repres5t$ to be ! manife/ation ( ! p[] ( unse5 sup]natural be+s4 ,! bas9 ( ! ,m$it]ranean ,sea 0 especially well adapt$1 f its physical conditions1 to be ! bir?-place ( su* fables4 ,x is a region frequ5tly %ak5 by e>?quakes1 & conta9s two di/9ct c5tres ( volcanic activity1 "o 9 ! ,aegean ,sea & "o 9 ,italy4 ,x is b\nd$ on ! nor? by a long succession ( l(ty sn[-capp$ m\nta9-ranges1 :5ce copi\s riv]s1 (t5 swoll5 by heavy ra9s or melt$ sn[s1 c>ry ! dra9age into ! sea4 ,on ! s\? x boa/s ! ,nile1 once s full ( my/]y2 likewise wide tracts ( arid des]t ) _! dread$ du/ /orms4 ,! ,m$it]ranean itself1 ?\gh an inl& sea1 is subject to gales1 :i*1 on expos$ coa/s1 raise break]s q l>ge en\gh to give a vivid impression ( ! p[] ( ocean waves4 ,! c\ntries t surr\nd ?is great %eet ( wat] display 9 _m places widely-spread deposits full ( sea %ells1 l ^? t /ill live 9 ! neighb\r+ bays & gulfs4 ,su* a region 0 n only well fitt$ to supply subjects = my?ology1 b also to furni%1 on e side1 mat]ials :i*1 9 _! int]e/ & sugge/iv5ess1 w\ld appeal to ! reason ( obs]vant m54 ,x 0 natural1 !re=e1 t ! e>ly philosoph]s ( ,greece %\ld h not$ "s ( ^! geological features1 & %\ld h s\ght = o!r explanations ( !m ?an ^? to be f\nd 9 ! popul> my?s4 ,! op9ions ent]ta9$ 9 antiquity on ^! subjects may be conv5i5tly gr\p$ "u two heads3 7#a7 ,geological processes n[ 9 op]ation1 & 7#b7 geological *anges 9 ! pa/4 ,e>?quakes & volcanoes4 #a4 ,contempor>y ,processes4--,! geological processes ( ! pres5t "t are p>tly at "w und]gr\nd & p>tly on ! surface ( ! e>?4 ,! =m]1 f _! frequ5tly disa/r\s "*1 receiv$ mu* att5tion f ,greek & ,roman au?ors4 ,ari/otle1 9 8 ,meteorics1 cites ! speculations ( sev]al ( 8 pr$ecessors :i* he rejects 9 fav\r ( 8 own op9ion to ! effect t e>?quakes are due to ! g5]ation ( w9d )9 ! e>?1 "u ! influ5ce ( ! w>m? ( ! sun & ! int]nal heat4 ,w9d1 be+ ! lighte/ & mo/ rapidly mov+ body1 is ! cause ( motion 9 o!r bodies1 & fire1 unit$ ) w9d1 becomes flame1 :i* is end[$ ) great rapidity ( motion4 ,ari/otle look$ ^u e>?quakes & volcanic eruptions z closely connect$ ) ea* o!r1 ! dis*>ge ( hot mat]ials to ! surface be+ ! result ( a sev]e e>?quake1 :5 f9ally ! w9d ru%es out ) viol5ce1 & sometimes buries ! surr\nd+ c\ntry "u sp>ks & c9d]s1 z _h happ5$ at ,lip>i4 ,^! crude conceptions ( ! nature ( volcanic action1 & ! cause ( e>?quakes1 cont9u$ to prevail = _m c5turies4 ,!y are repeat$ by ,lucretius1 :o1 h[ev]1 foll[+ ,anaxim5es1 includes z "o ( ! causes ( e>?quakes ! fall ( m\nta9\s masses ( rock und]m9$ by "t1 & ! consequ5t propagation ( gigantic tremors f> & wide ?r\gh ! e>?4 ,/rabo1 hav+ travell$ ?r\gh ! volcanic di/ricts ( ,italy1 0 able to recognize t ,vesuvius _h once be5 an active volcano1 al?\gh no eruption _h tak5 place f x )9 human memory4 ,he cont9u$ to hold ! belief t volcanic en]gy arose f ! movem5t ( subt]ranean w9d4 ,he believ$ t ! di/rict ar\nd ! ,/rait ( ,mess9a1 :i* _h =m]ly suff]$ f de/ructive e>?quakes1 0 seldom visit$ by !m aft] ! volcanic v5ts ( t region _h be5 op5$1 s z to provide an escape = ! subt]ranean fire1 w9d1 wat] & burn+ masses4 ,he cites 9 8 ,geography a numb] ( examples ( widespread z well z local s9k+s ( l&1 & alludes also to ! uprise ( ! sea-bottom4 ,he likewise reg>ds "s isl&s z hav+ be5 ?r[n up by volcanic ag5cy1 & o!rs z torn f ! ma9l& by su* convulsions z e>?quakes4 ,! mo/ detail$ acc\nt ( e>?quake ph5om5a :i* has come d[n to u f antiquity is t ( ,s5eca 9 8 ,quae/iones ,naturales4 ,?is philosoph] _h be5 mu* int]e/$ 9 ! acc\nts giv5 him by survivors & witnesses ( ! e>?quake :i* convuls$ ! di/rict ( ,naples 9 ,febru>y ,a4,d4 #fc4 ,he di/+ui%$ sev]al di/9ct movem5ts ( ! gr\nd3 #a/1 ! up & d[n motion 7succussio72 #bnd1 ! oscillatory motion 7incl9atio72 & probably a ?ird1 t ( trembl+ or vibration4 ,:ile admitt+ t "s e>?quakes may arise f ! collapse ( ! walls ( subt]ranean cavities1 he adh]$ to ! old idea1 held by ! mo/ num]\s & important previ\s writ]s1 t ^! commotions are caus$ ma9ly by ! movem5ts ( w9d imprison$ )9 ! e>?4 ,z to ! orig9 ( volcanic outbur/s he suppos$ t ! subt]ranean w9d 9 /ruggl+ = an outlet1 & :irl+ ?r\gh ! *asms & passages1 meets ) great /ore ( sulphur & o!r combu/ible sub/ances1 :i* by m]e friction are set on fire4 ,! eld] ,pl9y reit]ates ! commonly accept$ op9ion z to ! efficacy ( w9d und]gr\nd4 ,9 discuss+ ! ph5om5a ( e>?quakes he rem>ks t t[ns ) _m culv]ts & h\ses ) cell>s suff] less ?an o!rs1 & t at ,naples ^? h\ses are mo/ %ak5 :i* /& on h>d gr\nd4 ,x ?us appe>s t ) reg>d to subt]ranean geological op]ations1 no advance 0 made dur+ ! "t ( ! ,greeks & ,romans z to ! !oretical explanation ( ^! ph5om5a2 b a consid]able body ( facts 0 collect$1 especially z to ! effects ( e>?quakes & ! occurr5ce ( volcanic eruptions4 ,action ( riv]s4 ,! sup]ficial processes ( geology1 be+ mu* less /rik+ ?an ^? ( subt]ranean en]gy1 naturally attract$ less att5tion 9 antiquity4 ,! op]ations ( riv]s1 h[ev]1 :i* s intimately affect a human population1 7 wat*$ ) m or less c>e4 ,h]odotus1 /ruck by ! am\nt ( alluvial silt br\ght d[n annually by ! ,nile & spread ov] ! flat inundat$ l&1 inf]r$ t ,egypt is ! gift ( ! riv]4 ,ari/otle1 9 discuss+ "s ( ! features ( riv]s1 displays consid]able acqua9tance ) ! v>i\s dra9age-sy/ems on ! nor? side ( ! ,m$it]ranean bas94 ,he ref]s to ! m\nta9s z cond5s]s ( ! atmosph]ic moi/ure1 & %[s t ! l>ge/ riv]s rise among ! l(tie/ high gr\nds4 ,he %[s h[ s5sibly ! alluvial deposits c>ri$ d[n to ! sea increase ! bread? ( ! l&1 & cites "s p>ts ( ! %ores ( ! ,black ,sea1 :]e1 9 sixty ye>s1 ! riv]s _h br\ght d[n su* a quantity ( mat]ial t ! vessels !n 9 use requir$ to be ( mu* small] draught ?an previ\sly1 ! wat] %all[+ s mu* t ! m>%y gr\nd w\ld1 9 c\rse ( "t1 become dry l&4 ,/rabo supplies fur!r int]e/+ in=mation z to ! "w ( riv]s 9 mak+ _! alluvial pla9s & 9 pu%+ _! deltas seaw>d4 ,he rem>ks t ^! deltas are prev5t$ f advanc+ f>!r outw>d by ! ebb & fl[ ( ! tides4 ,occurr5ces ( fossils4 #b4 ,pa/ ,processes4--,! abundant well-pres]v$ m>9e %ells expos$ among ! uprais$ ,t]ti>y & po/-,t]ti>y deposits 9 ! c\ntries bord]+ ! ,m$it]ranean are n infrequ5tly allud$ to 9 ,greek & ,lat9 lit]ature4 ,x5ophanes ( ,colophon 7#fad ,b4,c47 notic$ ! occurr5ce ( %ells & o!r m>9e productions inl& among ! m\nta9s1 & inf]r$ f !m t ! l& _h ris5 out ( ! sea4 ,a simil> conclusion 0 drawn by ,xan?us ! ,lydian 7#dfd ,b4,c47 f %ells l scallops & cockles1 :i* 7 f\nd f> f ! sea 9 ,arm5ia & ,l[] ,phrygia4 ,h]odotus1 ,erato/h5es1 ,/rato & ,/rabo not$ ! va/ quantities ( fossil %ells 9 diff]5t p>ts ( ,egypt1 toge!r ) b$s ( salt1 z evid5ce t ! sea _h once spread ov] ! c\ntry4 ,b by f> ! mo/ philosophical op9ions on ! pa/ mutations ( ! e>?'s surface are ^? express$ by ,ari/otle 9 ! treatise already cit$4 ,review+ ! evid5ce ( ^! *anges1 he recogniz$ t ! sea n[ cov]s tracts t 7 once dry l&1 & t l& w "o day reappe> :]e !re is n[ sea4 ,^! alt]nations are to be reg>d$ z foll[+ ea* o!r 9 a c]ta9 ord] & p]iodicity4 ,b !y are apt to escape our notice because !y require successive p]iods ( "t1 :i*1 comp>$ ) our brief exi/5ce1 are ( enorm\s duration1 & because !y are br\ght ab\t s imp]ceptibly t we fail to detect !m 9 progress4 ,9 a celebrat$ passage 9 8 ,metamorphoses1 ,ovid puts into ! m\? ( ! philosoph] ,py?agoras an acc\nt ( :at 0 probably reg>d$ z ! ,py?agorean view ( ! subject 9 ! ,augu/an age4 ,x affirms ! int]*ange ( l& & sea1 ! erosion ( valleys by desc5d+ riv]s1 ! wa%+ d[n ( m\nta9s into ! sea1 ! disappe>ance ( ! riv]s & ! subm]g5ce ( l& by e>?quake movem5ts1 ! sep>ation ( "s isl&s f1 & ! union ( o!rs )1 ! ma9l&1 ! uprise ( hills by volcanic action1 ! rise & ext9ction ( burn+ m\nta9s4 ,!re 0 a "t be=e ,etna began to gl[1 & ! "t is com+ :5 ! m\nta9 w cease to burn4 ,f ?is brief sket* x w be se5 t :ile ! anci5ts _h accumulat$ a good deal ( in=mation reg>d+ ! occurr5ce ( geological *anges1 _! int]pretations ( ! ph5om5a 7 to a consid]able ext5t m]e fanciful speculation4 ,!y _h acquir$ only a mo/ imp]fect conception ( ! nature & op]ation ( ! geological processes2 & ?\gh _m writ]s realiz$ t ! surface ( ! e>? has n always be51 & w n always rema91 z x is n[1 !y _h no glimpse ( ! va/ succession ( *anges ( t surface :i* h be5 reveal$ by geology4 ,!y built hypo!ses on ! sl5d]e/ basis ( fact1 & did n realize ! necessity ( te/+ or v]ify+ !m4 ,progress ( ,geological ,conceptions 9 ! ,middle ,ages4--,dur+ ! c5turies t succe$$ ! fall ( ! ,we/]n empire little progress 0 made 9 natural sci5ce4 ,! s*oolm5 9 ! mona/]ies & o!r sem9>ies 7 cont5t to take _! sci5ce f ! lit]ature ( ,greece & ,rome4 ,! ,arabs1 h[ev]1 n only collect$ & translat$ t lit]ature1 b 9 "s dep>tm5ts made orig9al obs]vations !mselves4 ,to "o ( ! mo/ illu/ri\s ( _! numb]1 ,avic5na1 ! translator ( ,ari/otle1 a treatise has be5 ascrib$1 9 :i* s+ul>ly mod]n ideas are express$ reg>d+ m\nta9s1 "s ( :i* are !re /at$ to h be5 produc$ by an uplift+ ( ! gr\nd1 :ile o!rs h be5 left prom95t1 ow+ to ! we>+ away ( ! s(t] rocks ar\nd !m4 ,9 ei!r case1 x is confess$ t ! process w\ld dem& long tracts ( "t = its completion4 ,aft] ! revival ( le>n+ ! anci5t problem pres5t$ by fossil %ells imb$d$ 9 ! rocks ( ! int]ior ( _m c\ntries receiv$ r5ew$ att5tion4 ,b ! conditions = its solution 7 no long] :at !y _h be5 9 ! days ( ! philosoph]s ( antiquity4 ,m5 7 n n[ free to adopt & tea* any doctr9e !y pleas$ on ! subject4 ,! ,*ri/ian *ur* _h mean:ile aris5 to p[] all ov] ,europe1 & adjudg$ z h]etics all :o v5tur$ to impugn any ( h] dogmas4 ,%e taught t ! l& & ! sea _h be5 sep>at$ on ! ?ird day ( creation1 be=e ! appe>ance ( any animal life1 :i* 0 n creat$ until ! fif? day4 ,to ass]t t ! dry l& is made up 9 great "p ( rocks t 7 =m$ 9 ! sea1 & are cr[d$ ) ! rema9s ( animals1 0 pla9ly to impugn ! v]acity ( ! ,bible4 ,aga91 x _h come to be ! or?odox belief t only some:]e ab\t #fjjj ye>s _h elaps$ s9ce ! "t ( ,adam & ,eve4 ,if any ?\ghtful obs]v]1 impress$ ) ! ov]:elm+ =ce ( ! evid5ce t ! fossilif]\s =mations ( ! e>?'s cru/ mu/ h tak5 long p]iods ( "t = _! accumulation1 v5tur$ to give public expression to 8 conviction1 he ran consid]able risk ( be+ proce$$ aga9/ z a h]etic4 ,x 0 ne$ful1 !re=e1 to f9d "s explanation ( ! facts ( nature1 :i* w\ld n run c\nt] to ! ecclesia/ical sy/em ( ! day4 ,v>i\s su* int]pretations 7 propos$1 d\btless 9 an hone/ endeav\r at reconciliation4 ,?ree ( ^! des]ve special notice3 7#a7 ,_m able obs]v]s & dilig5t collectors ( fossils p]suad$ !mselves t ^! objects nev] belong$ to organisms ( any k9d1 b %\ld be reg>d$ z m]e freaks ( nature1 hav+ no m connexion ) any once liv+ creature ?an ! fro/ patt]ns on a w9d[4 ,!y 7 /yl$ =m$ or figur$ /ones1 lapides sui g5]is1 & 7 ass]t$ to be due to "s inorganic imitative process )9 ! e>? or to ! influ5ce ( ! />s4 7#b7 ,obs]v]s :o c\ld n resi/ ! evid5ce ( _! s5ses t ! fossil %ells once belong$ to liv+ animals1 & :o1 at ! same "t1 felt ! necessity ( acc\nt+ = ! pres5ce ( m>9e organisms 9 ! rocks ( :i* ! dry l& is l>gely built up1 s\ght a way out ( ! difficulty by invok+ ! ,deluge ( ,noah4 ,"h 0 a cata/rophe :i*1 !y said1 ext5d$ ov] ! :ole globe1 & by :i* ! entire dry l& 0 subm]g$ ev5 up to ! tops ( ! high hills4 ,true1 x only la/$ "o hundr$ & fifty days1 b s little 7 ! facts !n appreciat$ t no difficulty seems to h be5 g5]ally felt 9 cr[d+ ! accumulation ( ! ?\s&s ( feet ( fossilif]\s =mations into t brief space ( "t4 7#c7 ,"s m intellig5t m5 9 ,italy1 recogniz+ t ^! int]pretations c\ld n be upheld1 fell back ^u ! idea t ! rocks 9 :i* fossil %ells are imb$d$ might h be5 heap$ up by repeat$ & vigor\s eruptions f volcanic c5tres4 ,c]ta9 mod]n eruptions 9 ! ,aegean ,sea & 9 ! ,bay ( ,naples _h drawn att5tion to ! rapidity ) :i* hills ( consid]able size c\ld be pil$ ar\nd an active crat]4 ,x 0 argu$ t if ,monte ,nuovo ne> ,naples c\ld h be5 accumulat$ to a height ( ne>ly #ejj ft4 9 two days1 !re seem$ to be no reason aga9/ believ+ t1 dur+ ! "t ( ! ,flood1 & 9 ! c\rse ( ! c5turies t h elaps$ s9ce t ev5t1 ! :ole ( ! fossilif]\s rocks might h be5 deposit$4 ,un=tunately = ?is hypo!sis x ignor$ ! fact t ^! rocks d n consi/ ( volcanic mat]ials4 ,leon>do da ,v9ci2 ,fraca/orio2 ,falloppio4 ,s long z ! fundam5tal "q rema9$ 9 dispute z to ! true "* & hi/ory ( ! /ratifi$ portion ( ! e>?'s cru/ conta9+ organic rema9s1 geology z a sci5ce c\ld n beg9 its exi/5ce4 ,! diluviali/s 7^? :o reli$ on ! hypo!sis ( ! ,flood7 held ! field dur+ ! #af?1 #ag? & a great "p ( ! #ah? c5tury4 ,!y 7 look$ on z ! *ampions ( or?odoxy2 &1 on t acc\nt1 !y d\btless wield$ mu* m influ5ce ?an w\ld h be5 ga9$ by !m f ! =ce ( _! argum5ts4 ,yet dur+ ^? ages !re 7 n want+ occasional obs]v]s :o did good s]vice 9 combat+ ! preval5t misconceptions1 & 9 prep>+ ! way = ! ultimate triumph ( tru?4 ,x 0 m especially 9 ,italy1 :]e _m ( ! m /rik+ ph5om5a ( geology are conspicu\sly display$1 t ! e>ly pione]s ( ! sci5ce arose1 & t = sev]al g5]ations ! mo/ m>k$ progress 0 made t[>ds plac+ ! inve/igations ( ! pa/ hi/ory ( ! e>? ^u a basis ( c>eful obs]vation & sci5tific d$uction4 ,"o ( ! fir/ ( ^! lead]s 0 ,leon>do da ,v9ci 7#adeb-aeai71 :o1 besides 8 a*ievem5ts 9 pa9t+1 sculpture1 ar*itecture & eng9e]+1 contribut$ "s notable obs]vations reg>d+ ! great problem ( ! orig9 ( fossil %ells4 ,he ridicul$ ! notion t ^! objects c\ld h be5 =m$ by ! influ5ce ( ! />s1 & ma9ta9$ t !y _h once belong$ to liv+ organisms1 & !re=e t :at is n[ l& 0 =m]ly cov]$ by ! sea4 ,girolamo ,fraca/orio 7#adhc-aeec7 claim$ t ! %ells c\ld nev] h be5 left by ! ,flood1 :i* 0 a m]e tempor>y inundation1 b t !y prov$ ! m\nta9s1 9 :i* !y occur1 to h be5 successively uplift$ out ( ! sea4 ,on ! o!r h&1 ev5 an accompli%$ anatomi/ l ,gabriello ,falloppio 7#aebc-aefb7 f\nd x easi] to believe t ! bones ( elephants1 tee? ( %>ks1 %ells & o!r fossils 7 m]e e>?y inorganic concretions1 ?an t ! wat]s ( ,noah's ,flood c\ld ev] nave rea*$ z f> z ,italy4 ,nicolas ,/5o4 ,by mu* ! mo/ important memb] ( ?is e>ly b& ( ,italian writ]s 0 und\bt$ly ,nicolas ,/5o 7#afca-afhg71 :o1 ?\gh born 9 ,cop5hag51 ultimately settl$ 9 ,flor5ce4 ,hav+ made a ,european reputation z an anatomi/1 8 att5tion 0 drawn to geological problems by f9d+ t ! rocks ( ! nor? ( ,italy conta9$ :at appe>$ to be %>ks' tee? closely resembl+ ^? ( a dog-fi%1 ( :i* he _h publi%$ ! anatomy4 ,cauti\sly at fir/1 = fe> ( (f5d+ or?odox op9ions1 b aft]w>ds m boldly1 he proclaim$ 8 conviction t ^? objects _h once be5 "p ( liv+ animals1 & t !y ?rew light on "s ( ! pa/ hi/ory ( ! e>?4 ,he publi%$ 9 #affi a small tract1 ,de solido intra solidum naturalit] cont5to1 9 :i* he develop$ ! ideas he _h =m$ ( ?is hi/ory f an att5tive /udy ( ! rocks4 ,he %[$ t ! /ratifi$ =mations ( ! hills & valleys consi/ ( su* mat]ials z w\ld be laid d[n 9 ! =m ( s$im5t 9 turbid wat]2 t :]e !y conta9 m>9e productions ?is wat] is prov$ to h be5 ! sea2 t div]sities 9 _! composition po9t to comm+l+ ( curr5ts1 c>ry+ diff]5t k9ds ( s$im5t ( :i* ! heavie/ w\ld fir/ s9k to ! bottom4 ,he made orig9al & important obs]vations on /ratification1 & laid d[n "s ( ! fundam5tal axioms 9 /ratigraphy4 ,he reason$ t z ! orig9al position ( /rata 0 approximately horizontal1 :5 !y are f\nd to be /eeply incl9$ or v]tical1 or b5t into ar*es1 !y h be5 disrupt$ by subt]ranean exhalations1 or by ! fall+ 9 ( ! ro(s ( und]gr\nd cav]n\s spaces4 ,x is to ?is alt]ation ( ! orig9al position ( ! /rata t ! inequalities ( ! e>?'s surface1 su* z m\nta9s1 are to be ascrib$1 ?\gh "s h be5 =m$ by ! outbur/ ( fire1 a%es & /ones f inside ! e>?4 ,ano!r effect ( ! dislocation has be5 to provide fissures1 :i* s]ve z outlets = spr+s4 ,/5o's anatomical tra9+ peculi>ly fitt$ him = deal+ au?oritatively ) ! "q ( ! nature & orig9 ( ! fossils conta9$ 9 ! rocks4 ,he _h no hesitation 9 affirm+ t1 ev5 if no %ells _h ev] be5 f\nd liv+ 9 ! sea1 ! int]nal /ructure ( ^! fossils w\ld demon/rate t !y once =m$ p>ts ( liv+ animals4 ,& n only %ells1 b tee?1 bones & skeletons ( _m k9ds ( fi%es _h be5 qu>ri$ out ( ! rocks1 :ile "s ( ! /rata _h skulls1 horns & tee? ( l&-animals4 ,illu/rat+ 8 g5]al pr9ciples by a sket* ( :at he suppos$ to h be5 ! pa/ hi/ory ( ,tuscany1 he add$ a s]ies ( diagrams :i* %[ h[ cle>ly he _h conceiv$ ! ess5tial elem5ts ( /ratigraphy4 ,he ?\ght he c\ld p]ceive ! records ( six successive phases 9 ! evolution ( ! framework ( t c\ntry1 & 0 incl9$ to believe t a simil> *ronological sequ5ce w\ld be f\nd all ov] ! _w4 ,he anticipat$ ! objections t w\ld be br\ght aga9/ 8 views on acc\nt ( ! insup]able difficulty 9 grant+ ! l5g? ( "t t w\ld be requir$ = all ! geographical vicissitudes :i* 8 int]pretation requir$4 ,he ?\ght t _m ( ! fossils mu/ be z old z ! "t ( ! g5]al deluge1 b he 0 c>eful n to indulge 9 any speculation z to ! antiquity ( ! e>?4 ,lazz>o ,moro4 ,to ! ,italian s*ool1 z especially typifi$ 9 ,/5o1 mu/ be assign$ ! hon\r ( hav+ ?us begun to lay firmly & truly ! fir/ f\ndation /ones ( ! mod]n sci5ce ( geology4 ,! same s*ool includ$ ,antonio ,vallisn]i 7#affa-agcj71 :o surpass$ 8 pr$ecessors 9 8 wid] & m exact k ( ! fossilif]\s rocks t =m ! backbone ( ! ,italian p59sula1 :i* he cont5d$ 7 =m$ dur+ a wide & prolong$ subm]g5ce ( ! region1 altoge!r diff]5t f ! brief deluge ( ,noah4 ,!re 0 likewise ,lazz>o ,moro 7#afhg-agdj71 :o did good s]vice aga9/ ! diluviali/s1 b ! fundam5tal feature ( 8 sy/em ( nature lay 9 ! prepond]ant "p :i*1 unaw>e ( ! great diff]5ce betwe5 volcanic mat]ials & ord9>y s$im5t1 he assign$ to volcanic action 9 ! production ( ! s$im5t>y rocks ( ! e>?'s cru/4 ,he suppos$ t 9 ! beg9n+ ! globe 0 completely surr\nd$ ) wat]1 b5ea? :i* ! solid e>? lay z a smoo? ball4 ,on ! ?ird day ( creation1 h[ev]1 va/ fires 7 k9dl$ inside ! globe1 :]eby ! smoo? surface ( /one 0 brok5 up1 & portions ( x1 appe>+ above ! wat]1 =m$ ! e>lie/ l&4 ,f t "t onw>d1 volcanic eruptions succe$$ ea* o!r1 n only on ! em]g$ l&1 b on ! sea-floor1 ov] :i* ! eject$ mat]ial spread 9 an ev] augm5t+ ?ickness ( s$im5t>y /rata4 ,9 ?is way ,moro c>ri$ ! hi/ory ( ! /ratifi$ rocks beyond ! "t ( ! ,flood back to ! ,creation1 :i* 0 suppos$ to h be5 "s #afjj ye>s e>li]2 & he br\ght x d[n to ! pres5t day1 :5 fre% s$im5t>y deposits are cont9ually accumulat+4 ,he ?us incurr$ no c5sure f ! ecclesia/ical gu>dians ( ! fai?1 & he succe$$ 9 attract+ increas$ public att5tion to ! problems ( geology4 ,! influ5ce ( 8 tea*+1 h[ev]1 0 subsequ5tly 9 great "p due to ! ,c>melite fri> ,g5]elli1 :o publi%$ an eloqu5t exposition ( ,moro's views4 ,! ,cosmogoni/s & ,!ories ( ! ,e>?4--,:ile 9 ,italy sub/antial progress 0 made 9 collect+ in=mation reg>d+ ! fossilif]\s =mations ( t c\ntry1 & 9 =m+ conclusions conc]n+ !m bas$ ^u m or less accurate obs]vations1 ! t5d5cy to m]e fanciful speculation1 :i* c\ld n be :olly repress$ 9 any c\ntry1 rea*$ a rem>kable extravagance 9 ,engl&4 ,9 proportion z mat]ials 7 yet lack+ f :i* to con/ruct a hi/ory ( ! evolution ( our planet 9 accordance ) ! tea*+ ( ! *ur*1 imag9ation suppli$ ! place ( asc]ta9$ fact1 & !re appe>$ dur+ ! la/ tw5ty ye>s ( ! #ah? c5tury a gr\p ( ,engli% cosmogoni/s1 :o1 by ! s5sational "* ( _! speculations1 ar\s$ g5]al att5tion bo? 9 ,brita9 & on ! cont95t4 ,x may be d\bt$1 h[ev]1 :e!r ! effect ( _! writ+s 0 n to h9d] ! advance ( true sci5ce by div]t+ m5 f ! obs]vation ( nature into b>r5 controv]sy ov] unrealities4 ,x is n ne$ful "h to d m ?an m5tion ! names ( ,?omas ,burnet1 ^: ,sacr$ ,!ory ( ! ,e>? appe>$ 9 #afha1 & ,william ,:i/on1 ^: ,new ,!ory ( ! ,e>? 0 publi%$ 9 #afif4 ,h>dly less fanciful ?an ^! writ]s1 ?\gh 8 practical acqua9tance ) rocks & fossils 0 inf9itely great]1 0 ,john ,woodw>d1 ^: ,essay t[>ds a ,natural ,hi/ory ( ! ,e>? dates f #afie4 ,m important z a contribution to sci5ce 0 ! catalogue ( ! l>ge collection ( fossils1 :i* he _h made f ! rocks ( ,engl& & :i* he bequea!d to ! univ]sity ( ,cambridge4 ,?is catalogue appe>$ 9 #agbh-agbi ) ! title ( ,an attempt t[>ds a ,natural ,hi/ory ( ! ,fossils ( ,engl&4 ,desc>tes4 ,a /rik+ contra/ to ^! cosmogoni/s is furni%$ by ano!r gr\p1 :i* arose 9 ,france & ,g]many1 & gave to ! _w ! fir/ rational ideas conc]n+ ! probable primeval evolution ( our globe4 ,! e>lie/ ( ^! pione]s 0 ! illu/ri\s philosoph] ,r5e ,desc>tes 7#aeif-afej74 ,he prop\nd$ a s*eme ( cosmical developm5t 9 :i* he repres5t$ ! e>?1 l ! o!r planets1 to h be5 orig9ally a mass ( gl[+ mat]ial l ! sun1 & to h gradually cool$ on ! outside1 :ile /ill reta9+ an inc&esc5t1 self-lum9\s nucleus4 ,yet ) ?is noble conception1 :i* mod]n sci5ce has accept$1 ,desc>tes c\ld n %ake himself free f ! "t-hon\r$ error 9 reg>d to ! orig9 ( volcanic action4 ,he ?\ght t c]ta9 exhalations )9 ! e>? cond5se into oil1 :i*1 :5 9 viol5t motion1 ent]s into ! subt]ranean cavities1 :]e x passes into a k9d ( smoke4 ,?is smoke is f "t to "t ignit$ by a sp>k ( fire &1 press+ viol5tly aga9/ its conta9+ walls1 gives rise to e>?quakes4 ,if ! flame breaks ?r\gh to ! surface at ! top ( a m\nta91 x may escape ) enorm\s en]gy1 hurl+ =? mu* e>? m+l$ ) sulphur or bitum51 & ?us produc+ a volcano4 ,! m\nta9 might burn = a long "t until at la/ its /ore ( fuel 9 ! %ape ( sulphur or bitum5 w\ld be exhau/$4 ,n only did ! philosoph] refra9 f avail+ himself ( ! high int]nal temp]ature ( ! globe z ! s\rce ( volcanic en]gy1 he ev5 did n make use ( x z ! cause ( ! ignition ( 8 suppos$ int]nal fuel1 b speculat$ on ! k9dl+ ( ! subt]ranean fires by ! spirits or gases sett+ fire to ! exhalations1 or by ! fall ( masses ( rock & ! sp>ks produc$ by _! friction or p]cussion4 ,leibnitz4 ,! ideas ( ,desc>tes reg>d+ planet>y evolution 7 enl>g$ & made m def9ite by ,wilhelm ,gottfri$ ,leibnitz 7#afdf-agaf71 ^: tea*+ has l>gely influ5c$ all subsequ5t speculation on ! subject4 ,9 8 great tract1 ! ,protogaea 7publi%$ 9 #agdi1 ?irty-?ree ye>s aft] 8 dea?71 he trac$ ! probable passage ( our e>? f an orig9al condition ( inc&esc5t vap\r into t ( a smoo? molt5 globe1 :i*1 by cont9u\s cool+1 acquir$ an ext]nal solid cru/ & rugose surface4 ,he ?\ght t ! m anci5t rocks1 su* z granite & gneiss1 might be portions ( ! e>lie/ out] cru/2 & t z ! ext]nal solidification advanc$1 imm5se subt]ranean cavities 7 left :i* 7 fill$ ) air & wat]4 ,by ! collapse ( ! ro(s ( ^! cav]ns1 valleys might be orig9at$ at ! surface1 :ile ! solid int]v5+ walls w\ld rema9 9 place & =m m\nta9s4 ,by ! disruption ( ! cru/1 enorm\s bodies ( wat] 7 laun*$ ov] ! surface ( ! e>?1 :i* swept va/ quantities ( s$im5t toge!r1 & ?us gave rise to s$im5t>y deposits4 ,aft] _m vicissitudes ( ?is k9d1 ! t]re/rial =ces calm$ d[n1 & a m /able condition ( ?+s 0 e/abli%$4 ,an important feature 9 ! cosmogony ( ,leibnitz is ! prom95t place :i* he assign$ to organic rema9s 9 ! /ratifi$ rocks ( ! cru/4 ,ridicul+ ! fooli% attempts to acc\nt = ! pres5ce ( ^! objects by call+ !m sports ( nature1 he %[$ t !y are to be reg>d$ z hi/orical monum5ts2 & he adduc$ a numb] ( in/ances :]e9 successive plat=ms ( /rata1 conta9+ organic rema9s1 be> witness to a s]ies ( advances & retreats ( ! sea4 ,he recogniz$ t "s ( ! fossils appe>$ to h no?+ l !m 9 ! liv+ _w ( to-day1 b "s analog\s =ms might yet be f\nd1 he ?\ght1 9 /ill unexplor$ p>ts ( ! e>?2 & ev5 if no liv+ repres5tatives %\ld ev] be discov]$1 _m types ( animals might h und]gone trans=mation dur+ ! great *anges :i* _h affect$ ! surface ( ! e>?4 ,9 spite ( 8 cle> realization ( ! va/ /ore ( pot5tial en]gy resid+ )9 ! highly heat$ int]ior ( ! e>?1 ,leibnitz cont9u$ to reg>d volcanic action z due to ! combu/ion ( inflammable sub/ances enclos$ )9 ! t]re/rial cru/1 su* z /one-coal1 naph?a & sulphur4 ,buffon4 ,appeal+ to a mu* wid] public ?an ,desc>tes or ,leibnitz1 & bas+ 8 speculations on a wid] acqua9tance ) ! organic & inorganic realms ( nature1 ,g4,l4,l4 de ,buffon 7#agjg-aghh7 0 und\bt$ly "o ( ! mo/ influ5tial =ces t 9 ,europe guid$ ! gr[? ( geological ideas dur+ ! #ah? c5tury4 ,he publi%$ 9 #agdi a ,!ory ( ! ,e>?1 9 :i* he adopt$ views simil> to ^? ( ,desc>tes & ,leibnitz z to planet>y evolution2 b ?\gh he realiz$ ! importance ( fossils z records ( =m] conditions ( ! e>?'s surface1 he acc\nt$ = !m by suppos+ t !y _h be5 deposit$ f a univ]sal ocean1 a l>ge "p ( :i* _h subsequ5tly be5 engulf$ into cav]ns 9 ! int]ior ( ! globe4 ,?irty ye>s lat]1 aft] hav+ lab\r$ ) skill & en?usiasm 9 all bran*es ( natural hi/ory1 he publi%$ ano!r "w1 8 fam\s ,epoques de la nature 7#aggh71 :i* is specially rem>kable z ! fir/ attempt to deal ) ! hi/ory ( ! e>? 9 a *ronological mann]1 & to compute1 on a basis ( exp]im5t1 ! antiquity ( ! sev]al /ages ( ?is hi/ory4 ,8 exp]im5ts 7 made ) globes ( ca/ iron1 & c\ld n h yield$ results ( any value = 8 purpose2 b 9 s f> z 8 calculations 7 n m]e r&om guesses b _h "s k9d ( f\ndation on exp]im5t1 !y des]ve respectful recognition4 ,he divid$ ! hi/ory ( our e>? into six p]iods ( unequal duration1 ! :ole compris+ a p]iod ( "s #gj1jjj or #ge1jjj ye>s4 ,he suppos$ t ! /age ( inc&esc5ce1 be=e ! globe _h consolidat$ to ! c5tre1 la/$ #bicf ye>s1 & t ab\t #ce1jjj ye>s elaps$ be=e ! surface _h cool$ suffici5tly to be t\*$1 & !re=e to be capable ( support+ liv+ ?+s4 ,t]re/rial animal life1 h[ev]1 0 n introduc$ until #ee1jjj or #fj1jjj ye>s aft] ! beg9n+ ( ! _w or ab\t #ae1jjj ye>s be=e our "t4 ,look+ into ! future1 he =esaw t1 by cont9u$ refrig]ation1 our globe w ev5tually become cold] ?an ice1 & ?is fair face ( nature1 ) its manifold v>ieties ( plant & animal life1 w p]i% aft] hav+ exi/$ = #acb1jjj ye>s4 ,buffon's conception ( ! op]ation ( ! geological ag5ts did n become broad] or m accurate 9 ! int]val betwe5 ! appe>ance ( 8 two treatises4 ,he /ill cont9u$ to believe 9 ! l[]+ ( ! ocean by subsid5ce into va/ subt]ranean cavities1 ) a consequ5t em]g5ce ( l&4 ,he /ill look$ on volcanoes z due to ! burn+ ( pyrit\s & combu/ible /ones1 ?\gh he n[ call$ 9 ! co-op]ation ( electricity4 ,he calculat$ t ! fir/ volcanoes c\ld n arise until "s #ej1jjj ye>s aft] ! beg9n+ ( ! _w1 by :i* "t a suffici5t ext5t ( d5se vegetation _h be5 buri$ 9 ! e>? to supply !m ) fuel4 ,he appe>s to h _h b an imp]fect acqua9tance ) ! lit]ature ( 8 own "t4 ,at lea/ !re c be little d\bt t _h he avail$ himself ( ! lab\rs ( 8 own c\ntryman1 ,jean ,eti5ne ,guett>d 7#agae-aghf71 ( ,giovanni ,ardu9o 7#agad-agie7 9 ,italy1 & ( ,johann ,gottlob ,lehmann 7d4 #agfg7 & ,george ,*ri/ian ,fu*sel 7#agbb-aggc7 9 ,g]many1 he w\ld h be5 able to give to 8 epo*s a m def9ite succession ( ev5ts & a great] correspond5ce ) ! facts ( nature4 ,james ,hutton4 ,among ! writ]s ( ! #ah? c5tury1 :o =m$ philosophical conceptions ( ! sy/em ( processes by :i* ! life ( our e>? z a habitable globe is c>ri$ on1 a =emo/ place mu/ be assign$ to ,james ,hutton 7#agbf-agig74 ,educat$ = ! m$ical pr(ession1 he /udi$ at ,ed9burgh & at ,p>is1 & took 8 doctor's degree at ,leid54 ,b hav+ inh]it$ a small l&$ prop]ty 9 ,b]wick%ire1 he took to agriculture1 & aft] putt+ 8 l& into excell5t ord]1 let 8 f>m & betook himself to ,ed9burgh1 !re to gratify ! sci5tific ta/es :i* he _h develop$ e>ly 9 life4 ,he _h be5 m especially l$ to /udy m9]als & rocks1 & to m$itate on ! problems :i* !y sugge/ z to ! con/itution & hi/ory ( ! e>?4 ,8 j\rneys 9 ,brita9 & on ! cont95t ( ,europe _h furni%$ him ) mat]ial = reflection2 & he _h gradually evolv$ a sy/em or !ory 9 :i* all ! scatt]$ facts c\ld be arrang$ s z to %[ _! mutual dep5d5ce & _! place 9 ! ord]ly me*anism ( ! _w4 ,he us$ to discuss 8 views ) "o or two ( 8 fri5ds1 b refra9$ f publi%+ !m to ! _w until1 on ! f\ndation ( ! ,royal ,society ( ,ed9burgh1 he communicat$ an outl9e ( 8 doctr9e to t le>n$ body 9 #aghe4 ,"s ye>s lat] he exp&$ ?is fir/ essay into a l>g] "w 9 two volumes1 :i* 7 publi%$ 9 #agie ) ! title ( ,!ory ( ! ,e>?1 ) ,pro(s & ,illu/rations4 ,john ,playfair4 ,hutton's tea*+ has ex]cis$ a pr(\nd influ5ce on mod]n geology4 ,?is influ5ce1 h[ev]1 has aris5 less f 8 own writ+s ?an f ! acc\nt ( 8 doctr9es giv5 by 8 fri5d ,john ,playfair 9 ! classic "w entitl$ ,illu/rations ( ! ,huttonian ,!ory1 publi%$ 9 #ahjb4 ,hutton wrote 9 s prolix & obscure a /yle z r to repel ?an attract read]s4 ,playfair1 on ! o!r h&1 express$ himself 9 su* cle> & graceful language z to comm& g5]al att5tion1 & to ga9 wide acceptance = 8 ma/]'s views4 ,unlike ! old] cosmogoni/s1 ,hutton refra9$ f try+ to expla9 ! orig9 ( ?+s1 & f speculations z to :at might possibly h be5 ! e>ly hi/ory ( our globe4 ,he det]m9$ f ! outset to int]pret ! pa/ by :at c be se5 to be ! pres5t ord] ( nature2 & he refus$ to admit ! op]ation ( causes :i* _c be %[n to be "p ( ! actual t]re/rial sy/em4 ,l o!r obs]v]s :o _h prec$$ him1 he recogniz$ 9 ! v>i\s rocks compos+ ! dry l& evid5ce ( =m] geographical conditions v diff]5t f ^? :i* n[ prevail4 ,he saw t ! va/ majority ( rocks consi/ ( h>d5$ s$im5ts & mu/ h be5 deposit$ 9 ! sea4 ,he c\ld di/+ui% among !m an old] or ,prim>y s]ies1 & a y\ng] or ,second>y s]ies2 & did n dispute ! exi/5ce ( a ,t]ti>y s]ies claim$ by ,pet] ,simon ,pallas 7#agda-ahaa74 ,he believ$ t ^! v>i\s aque\s accumulations _h be5 consolidat$ by subt]ranean heat1 t ! olde/ & l[e/ rocks _h suff]$ mo/ f ?is action1 t into ^! m deep-seat$ masses subsequ5t ve9s & l>g] bodies ( molt5 matt] 7 inject$ f bel[1 & ?us t :at 0 orig9ally loose detritus ev5tually became *ang$ 9 su* cry/all9e s*i/s z are n[ f\nd 9 m\nta9-*a9s4 ,9 ! c\rse ( ^! t]re/rial revolutions s$im5t>y /rata1 orig9ally m or less ne>ly horizontal1 h be5 pu%$ upw>d1 dislocat$1 crumpl$1 plac$ on end1 & ev5 elevat$ to =m ranges ( l(ty m\nta9s4 ,hutton look$ ^u ^! di/urbances z due to ! expansive p[] ( subt]ranean heat2 b he did n attempt to sket* ! me*anism ( ! process1 & he expressly decl9$ to (f] any conjecture z to h[ ! l& s elevat$ rema9s 9 t position4 ,he ?\ght t ! int]ior ( our planet may be a fluid mass1 melt$1 b un*ang$ by ! action ( heat2 &1 f> f connect+ volcanoes ) ! combu/ion ( inflammable sub/ances1 z _h be5 ! preval5t belief = s _m c5turies1 he look$ ^u !m z a b5efic5t provision ( spiracles to ! subt]ranean furnace1 9 ord] to prev5t ! unnecess>y elevation ( l& & fatal effects ( e>?quakes4 ,a di/+ui%+ feature ( ! ,huttonian philosophy is to be se5 9 ! bread? ( its conceptions reg>d+ ! geological op]ations cont9ually 9 progress on ! surface ( ! globe4 ,hutton saw t ! l& is und]go+ a ceaseless process ( degradation1 ?r\gh ! influ5ce ( ! air1 fro/1 ra91 riv]s & ! sea1 & t 9 c\rse ( "t1 if no c\nt]vail+ ag5cy %\ld int]v5e1 ! :ole ( ! dry l& w be wa%$ away into ! sea4 ,b he also p]ceiv$ t ?is univ]sal erosion is n ev]y:]e c>ri$ on at ! same rate2 t x is specially active along ! *annels ( torr5ts & riv]s1 & t1 ow+ to ?is diff]5ce ^! *annels are gradually deep5$ & wid5$1 until ! complicat$ valley-sy/em ( a c\ntry is c>v$ out4 ,he recogniz$ t ! detritus worn away f ! l& mu/ be spread out ov] ! floor ( ! sea1 s z to =m !re /rata simil> to ^? t compose mo/ ( ! dry l&4 ,z he c\ld detect 9 ! /ructure ( l& conv9c+ evid5ce t =m] sea floors _h be5 elevat$ to =m ! cont95ts & isl&s ( to-day1 he c\ld look =w>d to future ages1 :5 ! same subt]ranean ag5cy :i* _h rais$ up ! pres5t l& w\ld aga9 be employ$ to uplift ! b$ ( ! exi/+ ocean1 ?us to r5ew ! surface ( our e>? z a habitable globe1 & to />t a fre% cycle ( erosion & deposition4 ,lam>ck4 ,?\gh ,hutton 0 n unaw>e t organic rema9s ab\nd 9 _m ( ! /ratifi$ rocks1 he left !m out ( consid]ation 9 ! elaboration ( 8 !ory4 ,x 0 o!rwise ) "o ( 8 ,fr5* contempor>ies1 ! illu/ri\s ,j4,b4 ,lam>ck 7#agdd-ahbi71 :o1 aft] hav+ atta9$ great em95ce z a botani/1 turn$ to zoology :5 he 0 ne>ly fifty ye>s ( age1 & be=e long rose to ev5 great] di/9ction 9 t dep>tm5t ( sci5ce4 ,8 %>e 9 ! classification & description ( ! mollusca & 9 f\nd+ inv]tebrate palaeontology1 8 !ory ( organic evolution & 8 philosophical treatm5t ( _m biological que/ions h be5 t>dily recogniz$1 b 8 contributions to geology h be5 less g5]ally ackn[l$g$4 ,:5 he accept$ ! pr(essor%ip ( zoology2 ( insects1 ( worms & ( microscopic animals at ! ,museum ( ,natural ,hi/ory1 ,p>is1 9 #agic1 he at once ent]$ ) *>act]i/ic ard\r & capacity into ! new field ( rese>* !n op5$ to him4 ,9 deal+ ) ! mollusca he consid]$ n m]ely ! liv+ b also ! ext9ct =ms1 especially ! abundant1 v>i$ & well-pres]v$ g5]a & species furni%$ by ! ,t]ti>y deposits ( ! ,p>is bas91 ( :i* he publi%$ descriptions & plates t prov$ ( ess5tial s]vice 9 ! /ratigraphical "w ( ,cuvi] & ,alex&re ,brongni>t 7#aggj-ahdg74 ,8 lab\rs among ^! relics ( anci5t seas & lakes l$ him to pond] ov] ! pa/ hi/ory ( ! globe1 & z he 0 seldom dilatory 9 mak+ kn[n ! op9ions he _h =m$1 he communicat$ "s ( 8 conclusions to ! ,national ,in/itute 9 #agii4 ,^!1 includ+ a fur!r elaboration ( 8 views1 he publi%$ 9 #ahjb 9 a small volume entitl$ ,hydrogeologie4 ,?is treatise1 ?\gh x did n rea* a second edition & has nev] be5 repr9t$1 des]ves an hon\rable place 9 geological lit]ature4 ,its object1 ! au?or /ates1 0 to pres5t "s important & novel consid]ations1 :i* he ?\ght %\ld =m ! basis ( a true !ory ( ! e>?4 ,he entirely agre$ ) ! doctr9e ( ! suba]ial degradation ( ! l& & ! erosion ( valleys by runn+ wat]4 ,n ev5 ,playfair c\ld h /at$ ?is doctr9e m emphatically1 & x is wor?y ( notice t ,playfair's ,illu/rations ( ! ,huttonian ,!ory appe>$ 9 ! same ye> ) ,lam>ck's book4 ,! ,fr5* naturali/1 h[ev]1 c>ri$ 8 conclusions s f> z to take no acc\nt ( any great movem5ts ( ! t]re/rial cru/1 :i* might h produc$ or modifi$ ! ma9 physical features ( ! surface ( ! globe4 ,he ?\ght t all m\nta9s1 except su* z 7 ?r[n up by volcanic ag5cy or local accid5ts1 h be5 cut out ( pla9s1 ! orig9al surfaces ( :i* are indicat$ by ! cre/s & summits ( ^! elevations4 ,lam>ck1 9 reflect+ ^u ! wide diffusion ( fossil %ells & ! great height above ! sea at :i* !y are f\nd1 conceiv$ ! extraord9>y idea t ! ocean bas9 has be5 sc\r$ out by ! sea1 & t1 by an impulse communicat$ to ! wat]s ?r\gh ! influ5ce *iefly ( ! moon1 ! sea is sl[ly eat+ away ! ea/]n m>g9s ( ! cont95ts1 & ?r[+ up detritus on _! we/]n coa/s1 & is ?us gradually %ift+ its bas9 r\nd ! globe4 ,he w\ld n admit ! op]ation ( cataclysms2 b insi/$ z /rongly z ,hutton on ! cont9uity ( natural processes1 & on ! necessity ( expla9+ =m] *anges ( ! e>?'s surface by causes :i* c /ill be se5 to be 9 op]ation4 ,z might be anticipat$ f 8 previ\s /udies1 he br\ght liv+ ?+s & _! rema9s into ! =efront ( 8 !ory ( ! e>?4 ,he look$ ^u fossils z "o ( ! *ief means ( compreh5d+ ! revolutions :i* ! surface ( ! e>? has und]gone2 & 9 8 little volume he aga9 & aga9 dwells on ! va/ antiquity to :i* ^! revolutions be> witness4 ,he acutely argues1 f ! condition ( fossil %ells1 t !y mu/ h liv$ & di$ :]e _! rema9s are n[ f\nd4 ,9 ! la/ "p ( 8 treatise ,lam>ck advances "s peculi> op9ions 9 physics & *emi/ry1 :i* he _h broa*$ eighte5 ye>s be=e1 b :i* _h met ) no acceptance among ! sci5tific m5 ( 8 "t4 ,he believ$ t ! t5d5cy ( all comp\nd sub/ances is to decay1 & !reby to be resolv$ into _! compon5t con/itu5ts4 ,yet he saw t ! visible cru/ ( ! e>? consi/s almo/ :olly ( comp\nd bodies4 ,he !re=e set himself to solve ! problem ?us pres5t$4 ,p]ceiv+ t ! biological action ( liv+ organisms is con/antly =m+ comb9ations ( matt]1 :i* w\ld nev] h o!rwise come into exi/5ce1 he proce$$ to draw ! extraord9>y conclusion t ! action ( plant & animal life 7! ,p\voir de la vie7 ^u ! inorganic _w is s univ]sal & s pot5t1 t ! rocks & m9]als :i* =m ! out] "p ( ! e>?'s cru/ are all1 )\t exception1 ! result ( ! op]ations ( once liv+ bodies4 ,?\gh ?is sweep+ d$uction mu/ be all[$ to detract f ! value ( ,lam>ck's "w1 !re c be no d\bt t he realiz$1 m fully ?an any "o _h done be=e him1 ! efficacy ( plants & animals z ag5ts ( geological *ange4 ,cuvi]4 ,! la/ notable contributor to ! cosmological lit]ature ( geology 0 ano!r illu/ri\s ,fr5*man1 ! comp>ative anatomi/ ,cuvi] 7#agfi-ahcb74 ,he 0 contempor>y ) ,lam>ck1 b ( a v diff]5t type ( m9d4 ,! brilliance ( 8 speculations1 & ! *>m ) :i* he exp\nd$ !m1 e>ly ga9$ = him a prom95t place 9 ! society ( ,p>is4 ,he too 0 drawn by 8 zoological /udies to inve/igate fossil organic rema9s1 & to consid] ! =m] conditions ( ! e>?'s surface1 ( :i* !y are memorials4 ,x 0 among ! v]tebrate organisms ( ! ,p>is bas9 t he f\nd 8 *ief mat]ial1 & f !m t he prep>$ ! memoirs :i* l$ to him be+ reg>d$ z ! f\nd] ( v]tebrate palaeontology4 ,b beyond _! biological int]e/1 !y awak5$ 9 him a ke5 desire to asc]ta9 ! "* & sequ5ce ( ! geographical revolutions to :i* !y be> witness4 ,he approa*$ ! subject f an opposite & less philosophical po9t ( view ?an t ( ,lam>ck1 com+ to x ) c]ta9 preconceiv$ notions1 :i* affect$ all 8 subsequ5t writ+s4 ,:ile ,lam>ck 0 by in/9ct an evolutioni/1 :o s\ght to trace 9 ! hi/ory ( ! pa/ ! op]ation ( ! same natural processes z are /ill at "w1 ,cuvi]1 on ! o!r h&1 0 a cata/rophi/1 :o invok$ a succession ( va/ cataclysms to acc\nt = ! int]ruptions 9 ! cont9uity ( ! geological record4 ,9 a prelim9>y ,disc\rse prefix$ to 8 ,re*]*es sur les ossem5s fossiles 7#ahba7 ,cuvi] gave an outl9e ( :at he conceiv$ to h be5 ! pa/ hi/ory ( our globe1 s f> z he _h be5 able to compreh5d x f 8 inve/igations ( ! ,t]ti>y =mations ( ,france4 ,he believ$ t 9 t hi/ory evid5ce c be recogniz$ ( ! occurr5ce ( _m sudd5 & disa/r\s revolutions1 :i*1 to judge f _! effects on ! animal life ( ! "t1 mu/ h exce$$ 9 viol5ce any?+ we c conceive at ! pres5t day1 & mu/ h be5 br\ght ab\t by o!r ag5cies ?an ^? :i* are n[ 9 op]ation4 ,yet1 9 spite ( ^! cata/rophes1 he saw t !re has be5 an upw>d progress 9 ! animal =ms inhabit+ ! globe1 until ! s]ies end$ 9 ! adv5t ( man4 ,he c\ld n1 h[ev]1 f9d any evid5ce t "o species has be5 develop$ f ano!r1 = 9 t case !re %\ld h be5 traces ( int]m$iate =ms among ! /ratifi$ =mations1 :]e he affirm$ t !y _h nev] be5 f\nd4 ,a prom95t position 9 ! ,disc\rse is giv5 to a /r5u\s argum5t to disprove ! alleg$ antiquity ( "s nations1 & to %[ t ! la/ great cata/rophe occurr$ n m ?an "s #ejjj or #fjjj ye>s ago4 ,cuvi] ?us l9k$ himself ) ^? :o 9 previ\s g5]ations _h cont5d$ = ! efficacy ( ! ,deluge4 ,b 8 rese>*es among fossil animals _h giv5 him a f> wid] outlook into ! geological pa/1 & _h op5$ up to him a succession ( deeply int]e/+ problems 9 ! hi/ory ( life ^u ! e>?1 :i*1 ?\gh he _h n himself mat]ial = _! solution1 he c\ld =esee w\ld be cle>$ up 9 ! future4 ,gradual ,%ap+ ( ,geology into a ,di/9ct ,bran* ( ,sci5ce4--,x w be se5 f ! =ego+ hi/orical sket* t x 0 only aft] ! lapse ( long c5turies1 & f ! lab\rs ( _m successive g5]ations ( obs]v]s & writ]s1 t :at we n[ "k z ! sci5ce ( geology came to be recogniz$ z a di/9ct dep>tm5t ( natural k1 f\nd$ ^u c>eful & ext5d$ /udy ( ! /ructure ( ! e>?1 & ^u obs]vation ( ! natural processes1 :i* are n[ at "w 9 *ang+ ! e>?'s surface4 ,! t]m geology1#a descriptive ( ?is bran* ( ! inve/igation ( nature1 0 n propos$ until ! la/ qu>t] ( ! #ah? c5tury by ,jean ,&re ,de ,luc 7#agbg-ahag7 & ,horace ,b5$ict ,de ,saussure 7#agdj-agdi74 ,b ! sci5ce 0 !n 9 a m>k$ly half-=m$ condition1 !oretical speculation /ill 9 l>ge "p supply+ ! place ( d$uctions f a detail$ exam9ation ( actual fact4 ,9 #ahjg a few ent]pris+ spirits f\nd$ ! ,geological ,society ( ,london = ! special purpose ( c\nt]act+ ! preval5t t5d5cy & conf9+ _! int5tion to inve/igate ! m9]al /ructure ( ! e>?4 ,! cosmogoni/s & fram]s ( ,!ories ( ! ,e>? 7 succe$$ by o!r s*ools ( ?\ght4 ,! ,cata/rophi/s saw 9 ! composition ( ! cru/ ( ! e>? di/9ct evid5ce t ! =ces ( nature 7 once mu* m /up5d\s 9 _! op]ation ?an !y n[ are1 & t !y _h f "t to "t deva/at$ ! e>?'s surface2 extirpat+ ! races ( plants & animals1 & prep>+ ! gr\nd = new creations ( organiz$ life4 ,!n came ! ,uni=mit>ians1 :o1 pu%+ ! doctr9es ( ,hutton to an extreme :i* he did n propose1 saw no evid5ce t ! activity ( ! v>i\s geological causes has ev] s]i\sly diff]$ f :at x is at pres5t4 ,!y 7 incl9$ to disbelieve t ! /ratifi$ =mations ( ! e>?'s cru/ furni% conclusive evid5ce ( a gradual progression1 f simple types ( life 9 ! olde/ /rata to ! mo/ highly develop$ =ms 9 ! y\nge/2 & saw no reason :y rema9s ( ! high] v]tebrates %\ld n be met ) among ! ,palaeozoic =mations4 ,sir ,*>les ,lyell 7#agig-ahge7 0 ! great lead] ( ?is s*ool4 ,8 admirably cle> & philosophical pres5tations ( geological facts :i*1 ) unwe>i$ indu/ry1 he collect$ f ! writ+s ( obs]v]s 9 all p>ts ( ! _w1 impress$ 8 views ^u ! :ole ,engli%-speak+ _w1 & gave to geological sci5ce a coh]5ce & int]e/ :i* l>gely accel]at$ its progress4 ,9 8 lat] ye>s1 h[ev]1 he frankly accept$ ! views ( ,d>w9 9 reg>d to ! progressive "* ( ! geological record4 ,! y\nge/ ( ! s*ools ( geological ?\ght is t ( ! ,evolutioni/s4 ,po9t+ to ! :ole body ( evid5ce f inorganic & organic nature1 !y ma9ta9 t ! hi/ory ( our planet has be5 "o ( cont9ual & unbrok5 developm5t f ! e>lie/ cosmical beg9n+s d[n to ! pres5t "t1 & t ! cru/ ( ! e>? conta9s an abundant1 ?\gh incomplete1 record ( ! successive /ages ?r\gh :i* ! plant & animal k+doms h rea*$ _! exi/+ organization4 ,! publication ( ,d>w9's ,orig9 ( ,species 9 #ahei1 9 :i* evolution 0 made ! key to ! hi/ory ( ! animal & vegetable k+doms1 produc$ an extraord9>y revolution 9 geological op9ion4 ,! old] s*ools ( ?\ght rapidly di$ out1 & evolution became ! recogniz$ cre$ ( geologi/s all ov] ! _w4 ,w]n]4 ,developm5t ( ,op9ion reg>d+ ,igne\s ,rocks4--,s long z ! idea prevail$ t volcanoes are caus$ by ! combu/ion ( inflammable sub/ances und]gr\nd1 !re c\ld be no rational conception ( volcanic action & its products4 ,ev5 s late z ! middle ( ! #ah? c5tury1 z above rem>k$1 su* a good obs]v] z ,lazz>o ,moro drew s little di/9ction betwe5 volcanic & o!r rocks t he c\ld believe ! fossilif]\s =mations to h be5 ma9ly =m$ ( mat]ials eject$ f eruptive v5ts4 ,aft] 8 "t ! notion cont9u$ to prevail t all ! rocks :i* =m ! dry l& 7 laid d[n "u wat]4 ,ev5 /reams ( lava1 :i* 7 se5 to fl[ f an active crat]1 7 reg>d$ only z portions ( s$im5t>y or o!r rocks1 :i* _h be5 melt$ by ! f]v5t heat ( ! burn+ inflammable mat]ials t _h be5 k9dl$ und]gr\nd4 ,9 spite ( ! speculations ( ,desc>tes & ,leibnitz1 x 0 n yet g5]ally compreh5d$ t !re exi/s b5ea? ! t]re/rial cru/ a molt5 magma1 :i*1 f "t to "t1 has be5 inject$ into t cru/1 & has pi]c$ ?r\gh x1 s z to escape at ! surface ) all ! en]gy ( an active volcano4 ,:at we n[ recognize to be memorials ( ^! =m] injections & propulsions 7 all conf\nd$ ) ! rocks ( unque/ionably aque\s orig94 ,! la/ great tea*] by :om ^! antiquat$ doctr9es 7 =mulat$ into a sy/em & promulgat$ to ! _w 0 ,abraham ,gottlob ,w]n] 7#agdi-ahae71 ! mo/ illu/ri\s ,g]man m9]alogi/ & geogno/ ( ! second half ( ! #ah? c5tury4 ,:ile /ill "u tw5ty-six ye>s ( age1 he 0 appo9t$ tea*] ( m9+ & m9]alogy at ! ,m9+ ,academy ( ,freib]g 9 ,saxony--a po/ :i* he cont9u$ to fill up to ! end ( 8 life4 ,possess$ ( great en?usiasm = 8 subject1 cle>1 me?odical & eloqu5t 9 8 exposition ( x1 he soon drew ar\nd him m5 f all p>ts ( ! _w1 :o repair$ to /udy "u ! great oracle ( :at he call$ geognosy 7,gr4 ,greek3 ge1 ! e>?1 ,greek3 gnosis1 k7 or e>?-k4 ,reviv+ doctr9es t _h be5 curr5t long be=e 8 "t1 he taught t ! globe 0 once completely surr\nd$ ) an ocean1 f :i* ! rocks ( ! e>?'s cru/ 7 deposit$ z *emical precipitates1 9 a c]ta9 def9ite ord] ov] ! :ole planet4 ,among ^! univ]sal =mations ( aque\s orig9 7 includ$ _m rocks1 :i* h long be5 recogniz$ to h be5 once molt51 & to h ris5 f bel[ into ! upp] p>ts ( ! t]re/rial cru/4 ,w]n]1 foll[+ ! old tradition1 look$ ^u volcanoes z mod]n features 9 ! hi/ory ( ! planet1 :i* c\ld n h come into exi/5ce until a suffici5t am\nt ( vegetation _h be5 buri$ to furni% fuel = _! ma9t5ance4 ,h5ce he atta*$ b little importance to !m1 & did n include 9 8 sy/em ( rocks any division ( volcanic or igne\s mat]ials4 ,f ! pr$om9ant "p assign$ by him to ! sea 9 ! accumulation ( ! mat]ials ( ! visible "p ( ! e>?1 ,w]n] & 8 s*ool 7 kn[n z ,neptuni/s4 ,orig9 ( basalt4 ,b _m ye>s be=e ! ,saxon pr(essor began to tea*1 cle> evid5ce _h be5 produc$ f c5tral ,france t basalt1 "o ( ! rocks claim$ by him z a *emical precipitate & a univ]sal =mation1 is a lava :i* has be5 p\r$ out 9 a molt5 /ate at v>i\s widely sep>at$ p]iods ( "t & at _m diff]5t places4 ,s f> back z #ageb ,j4,e4 ,guett>d 7#agae-aghf7 _h %[n t ! basaltic rocks ( ,auv]gne are true lavas1 :i* h fl[$ out 9 /reams f gr\ps ( once active cones4 ,elev5 ye>s lat] ! obs]vation 0 confirm$ & greatly ext5d$ by ,ni*olas ,desm>e/ 7#agbe-ahae71 :o1 dur+ a long c\rse ( ye>s1 work$ out & mapp$ ! complicat$ volcanic records ( t int]e/+ region1 & demon/rat$ to all :o 7 will+ imp>tially to exam9e ! evid5ce ! true volcanic nature ( basalt4 ,^! views f\nd acceptance f "s obs]v]s1 b !y 7 vehem5tly oppos$ by ! foll[]s ( ,w]n]1 :o1 by ! =ce ( 8 g5ius1 made 8 !oretical conceptions pr$om9ate all ov] ,europe4 ,! controv]sy z to ! orig9 ( basalt 0 wag$ ) great vig\r dur+ ! lat] decades ( ! #ah? c5tury4 ,desm>e/ took no "p 9 x4 ,he _h accumulat$ su* conclusive pro( ( ! correctness ( 8 d$uctions1 & _h s fully exp\nd$ ! cle>ness ( ! evid5ce 9 _! fav\r furni%$ by ! region ( ,auv]gne1 t1 :5 any "o came to consult him on ! subject1 he cont5t$ himself ) giv+ ! advice to g & see4 ,:ile ! debate 0 9 progress on ! cont95t1 ! subject 0 approa*$ f a new & indep5d5t po9t ( view by ,hutton 9 ,scotl&4 ,?is illu/ri\s philosoph]1 z already /at$1 realiz$ ! importance ( ! int]nal heat ( ! globe 9 consolidat+ ! s$im5t>y rocks1 & believ$ t molt5 mat]ial f ! e>?'s int]ior has be5 protrud$ f bel[ into ! ov]ly+ cru/4 ,"s ( ! mat]ial ?us inject$ c\ld be recogniz$1 he ?\ght1 9 granite & 9 ! v>i\s d>k massive rocks :i*1 kn[n 9 ,scotl& "u ! "n ( :9/one1 7 aft]w>ds call$ ,trap1 & are n[ gr\p$ "u v>i\s names1 su* z basalt1 dol]ite & diorite4 ,s important a %>e did ,hutton ?us assign to ! int]nal heat 9 ! geological evolution ( ! planet1 t he & ^? :o adopt$ ! same op9ions 7 /yl$ ,plutoni/s1 or1 especially :]e !y conc]n$ !mselves ) ! volcanic orig9 ( basalt1 ,vulcani/s4 ,! geological _w 0 ?us divid$ into two ho/ile camps1 t ( ! ,neptuni/s or ,w]n]ians1 & t ( ! ,plutoni/s1 ,vulcani/s or ,huttonians4 ,aft] _m ye>s ( futile controv]sy ! fir/ s]i\s weak5+ ( ! position ( ! dom9ant ,neptuni/ s*ool arose f ! defection ( "s ( ! mo/ prom95t ( ,w]n]'s pupils4 ,9 p>ticul> ,jean ,francois ,d',aubuisson de ,vois9s 7#agfi-ahai71 :o _h writt5 a treatise on ! aque\s orig9 ( ! basalts ( ,saxony1 w5t aft]w>ds to ,auv]gne1 :]e he 0 spe$ily a conv]t to ! views exp\nd$ by ,desm>e/ z to ! volcanic nature ( basalt4 ,hav+ ?us to rel9qui% "o ( ! fundam5tal articles ( ! ,freib]g fai?1 he 0 subsequ5tly l$ to modify 8 adh]5ce to o!rs until1 z he himself confess$1 8 views came almo/ :olly to agree ) ^? ( ,hutton4 ,n less complete1 & ev5 m important1 0 ! conv]sion ( ! great ,leopold von ,bu* 7#aggd-ahec74 ,he1 too1 0 tra9$ by ,w]n] himself1 & prov$ to be ! mo/ illu/ri\s pupil ( ! ,saxon pr(essor4 ,full ( admiration = ! ,neptunism 9 :i* he _h be5 re>$1 he1 9 8 e>lie/ sep>ate "w1 ma9ta9$ ! aque\s orig9 ( basalt1 & contra/$ ! wide field op5$ up to ! _s ( obs]vation by 8 ma/]'s tea*+ ) ! n>r[] outlook (f]$ by ! volcanic !ory4 ,b a little fur!r acqua9tance ) ! facts ( nature l$ ,von ,bu* also to ab&on 8 e>li] prepossessions4 ,x 0 a p]sonal visit to ! volcanic region ( ,auv]gne t fir/ op5$ 8 eyes1 & l$ him to recant :at he _h believ$ & writt5 ab\t basalt4 ,b ! ab&onm5t ( s ess5tial a portion ( ! ,w]n]ian cre$ prep>$ ! way = fur!r rel9qui%m5ts4 ,:5 a few ye>s lat] he w5t to ,norway & f\nd to 8 a/oni%m5t t granite1 :i* he _h be5 taught to reg>d z ! olde/ *emical precipitate f ! univ]sal ocean1 c\ld !re be se5 to h brok5 ?r\gh & metamorphos$ fossilif]\s lime/ones1 & to h s5t ve9s into !m1 8 fai? 9 ,w]n]'s ord] ( ! succession ( ! rocks 9 ! e>?'s cru/ receiv$ a fur!r mom5t\s %ock4 ,:ile "o aft] ano!r ( ! ,freib]g doctr9es crumbl$ away be=e him1 he 0 n[ able to int]rogate nature on a wid] field ?an ! n>r[ limits ( ,saxony1 & he 0 ?us gradually l$ to embrace ! t5ets ( ! opposite s*ool4 ,8 comm&+ position1 z ! mo/ accompli%$ geologi/ on ! cont95t1 gave great importance to 8 recantation ( ! ,neptuni/ cre$4 ,8 defection inde$ 0 ! sev]e/ bl[ t ?is cre$ _h yet su/a9$4 ,x may be said to h rung ! knell ( ,w]n]ianism1 :i* !reaft] rapidly decl9$ 9 influ5ce1 :ile ,plutonism came /eadily to ! front1 :]e x has ev] s9ce rema9$4 ,al?\gh ,desm>e/ _h trac$ 9 ,auv]gne a long succession ( volcanic eruptions1 ( :i* ! olde/ w5t back to a remote p]iod ( "t1 & al?\gh he _h %[n t ?is succession1 c\pl$ ) ! records ( contemporane\s d5udation1 might be us$ 9 def9+ epo*s ( geological hi/ory1 x 0 n until _m ye>s aft] 8 day t volcanic action came to be recogniz$ z a normal "p ( ! me*anism ( our globe1 :i* _h be5 9 op]ation f ! remote/ pa/1 & :i* _h left num]\s records among ! rocks ( ! t]re/rial cru/4 ,dur+ ! progress ( ! controv]sy betwe5 ! two great oppos+ factions 9 ! lat] portion ( ! #ah? & ! fir/ ?ree decades ( ! #ai? c5tury1 ^? :o esp\s$ ! ,vulcani/ cause 7 int5t on prov+ t c]ta9 rocks1 :i* are int]calat$ among ! /ratifi$ =mations & :i* 7 claim$ by ! ,neptuni/s z obvi\sly =m$ by wat]1 are nev]!less ( truly igne\s orig94 ,^! obs]v]s fix$ _! eyes on ! evid5ce t ! mat]ial ( su* rocks1 in/ead ( hav+ be5 deposit$ f aque\s solution1 _h once be5 actually molt51 & _h 9 t condition be5 ?ru/ betwe5 ! /rata1 _h envelop$ portions ( !m1 & _h indurat$ or o!rwise alt]$ !m4 ,!y spoke ( ^! masses z un]upt$ lavas2 & und\bt$ly 9 innum]able in/ances !y 7 "r4 ,b _! zeal to e/abli% an intrusive orig9 l$ !m to ov]look ! pro(s t "s int]calat$ %eets ( igne\s mat]ial _h n be5 inject$ into ! /rata1 b _h be5 p\r$ out at ! surface z truly volcanic dis*>ges1 & !re=e belong$ to ! anci5t p]iods repres5t$ by ! /rata betwe5 :i* !y are int]pos$4 ,x may readily be suppos$ t any pro(s ( ! contemporane\s int]calation ( su* %eets w\ld be eag]ly seiz$ ^u by ! ,neptuni/s 9 fav\r ( _! aque\s !ory4 ,! influ5ce ( ! anci5t belief t burn+ m\nta9s c\ld only rise f ! combu/ion ( subt]ranean inflammable mat]ials ext5d$ ev5 into ! ranks ( ! ,vulcani/s1 s f> at lea/ z to lead to a g5]al acquiesc5ce 9 ! assumption t volcanoes appe>$ to belong to a late phase 9 ! hi/ory ( ! planet4 ,x 0 n until aft] consid]able progress _h be5 made 9 det]m9+ ! palaeontological di/9ctions & ord] ( succession ( ! /ratifi$ =mations ( ! e>?'s cru/ t x became possible to trace among ^! =mations a succession ( volcanic episodes :i* 7 contemporane\s ) !m4 ,9 no "p ( ! _w has an ampl] record ( su* episodes be5 pres]v$ ?an 9 ! ,briti% ,isles4 ,x 0 natural1 !re=e1 t ! subject %\ld !re receive mo/ att5tion4 ,z f> back z #ahbj ,ami ,b\e 7#agid-ahha7 %[$ t ! ,old ,r$ ,s&/one ( ,scotl& includes a great s]ies ( volcanic rocks1 & t o!r rocks ( volcanic orig9 are associat$ ) ! ,c>bonif]\s =mations4 ,h4,t4 de la ,be*e 7#agif-ahee7 aft]w>ds trac$ pro(s ( contemporane\s eruptions among ! ,devonian rocks ( ! s\?-we/ ( ,engl&4 ,adam ,s$gwick 7#aghe-ahgc7 %[$1 fir/ 9 ! ,lake ,di/rict1 & aft]w>ds 9 ,nor? ,wales1 ! pres5ce ( abundant volcanic %eets among ! olde/ divisions ( ! ,palaeozoic s]ies2 :ile ,rod]ick ,impey ,mur*ison 7#agib-ahga7 made simil> discov]ies among ! ,l[] ,silurian rocks4 ,f ! "t ( ^! pione]s ! volcanic hi/ory ( ! c\ntry has be5 work$ out by _m obs]v]s until x is n[ kn[n ) a fulness z yet unatta9$ 9 any o!r region4 ,gr[? ( ,op9ion reg>d+ ,e>?quakes4--,we h se5 h[ crude 7 ! conceptions ( ! anci5ts reg>d+ ! causes ( volcanic action1 & t !y connect$ volcanoes & e>?quakes z results ( ! commotion ( w9d imprison$ )9 subt]ranean cav]ns & passages4 ,"o ( ! e>lie/ treatises1 9 :i* ! ph5om5a ( t]re/rial movem5ts 7 discuss$ 9 ! _s ( mod]n sci5ce1 0 ! po/hum\s collection ( pap]s by ,rob]t ,hooke 7#afce-agjc71 entitl$ ,lectures & ,disc\rses ( ,e>?quakes & ,subt]ranean ,eruptions1 :]e ! probable ag5cy ( e>?quakes 9 upheav+ & depress+ l& is fully consid]$1 b )\t any def9ite pron\ncem5t z to ! au?or's conception ( its orig94 ,hooke /ill associat$ e>?quakes ) volcanic action1 & connect$ bo? ) :at he call$ ! g5]al congregation ( sulphur\s subt]rane\s vap\rs4 ,he conceiv$ t "s k9d ( f]m5tation takes place )9 ! e>?1 & t ! mat]ials :i* cat* fire & give rise to eruptions or e>?quakes are analog\s to ^? t con/itute gunp[d]4 ,! fir/ essay :]e9 e>?quakes are treat$ f ! mod]n po9t ( view z ! results ( a %ock t s5ds waves ?r\gh ! cru/ ( ! e>? 0 writt5 by ! ,rev4 ,john ,mi*ell1 & communicat$ to ! ,royal ,society 9 ! ye> #agfj4 ,/ill "u ! old misconception t volcanoes are due to ! combu/ion ( inflammable mat]ials1 :i* he ?\ght might be set on fire by ! spontane\s combu/ion ( pyrit\s /rata1 he suppos$ t1 by ! sudd5 access ( l>ge bodies ( wat] to ^! subt]ranean fires1 vap\r is produc$ 9 su* quantity & ) su* =ce z to give rise to ! %ock4 ,f ! c5tre ( orig9 ( ?is %ock waves1 he ?\ght1 are propagat$ ?r\gh ! e>?1 :i* are l>ge/ at ! />t & gradually dim9i% z !y travel outw>ds4 ,by draw+ l9es at diff]5t places 9 ! direction ( ! track ( ^! waves1 he believ$ t ! place ( common int]section ( ^! l9es w\ld be ne>ly ! c5tre ( ! di/urbance4 ,9 ?is way he %[$ t ! great ,lisbon e>?quake ( #agee _h its focus "u ! ,atlantic1 some:]e betwe5 ! latitudes ( ,lisbon & ,oporto1 & he e/imat$ t ! dep? at :i* x orig9at$ c\ld n be mu* less ?an #a m41 & probably did n exce$ #c m4 ,mi*ell1 h[ev]1 misconceiv$ ! "* ( ! waves :i* he describ$1 see+ t he believ$ !m to be due to ! actual propagation ( ! vap\r itself und]nea? ! surface ( ! e>?4 ,a c5tury _h almo/ pass$ aft] ! date ( 8 essay be=e mod]n sci5tific me?ods ( obs]vation & ! use ( record+ in/rum5ts began to be appli$ to ! /udy ( e>?quake ph5om5a4 ,9 #ahdf ,rob]t ,mallet 7#ahaj-ahha7 publi%$ an important pap] ,on ! ,dynamics ( ,e>?quakes 9 ! ,transactions ( ! ,royal ,iri% ,academy4 ,f t "t onw>d he cont9u$ to devote 8 en]gies to ! inve/igation1 /udy+ ! effects ( ! ,calabrian e>?quake ( #aheg1 exp]im5t+ on ! transmission ( waves ( %ock ?r\gh v>i\s mat]ials1 caus$ by explod+ *>ges ( gunp[d]1 & collect+ all ! in=mation to be obta9$ on ! subject4 ,8 writ+s1 & especially 8 "w 9 two volumes on ,! ,fir/ ,pr9ciples ( ,obs]vational ,seismology1 mu/ be reg>d$ z hav+ laid ! f\ndations ( ?is bran* ( mod]n geology 7see ,,e>?quake2 ,,seismomet]74 ,hi/ory ( ! ,evolution ( ,/ratigraphical ,geology4--,m5 _h long be5 famili> ) ! evid5ce t ! pres5t dry l& once lay "u ! sea1 be=e !y began to realize t ! rocks1 ( :i* ! l& consi/s1 conta9 a record ( _m alt]nations ( l& & sea1 & relics ( a long succession ( plants & animals f e>ly & simple types up to ! manifold & complex =ms ( to-day4 ,9 c\ntries :]e coal-m9+ _h be5 prosecut$ = g5]ations1 x _h be5 recogniz$ t ! rocks consi/ ( /rata sup]pos$ on ea* o!r 9 a def9ite ord]1 :i* 0 f\nd to ext5d ov] ! :ole ( a di/rict4 ,z f> back z #agai ,john ,/ra*ey drew att5tion to ?is fact 9 a communication publi%$ 9 ! ,philosophical ,transactions4 ,john ,mi*ell 7#agfj71 9 ! pap] on e>?quakes already cit$1 %[$ t he _h acquir$ a cle> und]/&+ ( ! ord] ( succession among /ratifi$ =mations1 & p]ceiv$ t to di/urbances ( ! t]re/rial cru/ mu/ be ascrib$ ! fact t ! l[] or old] & m incl9$ /rata =m ! m\nta9s1 :ile ! y\ng] & m horizontal /rata are spread ov] ! pla9s4 ,9 ,italy ,g4 ,ardu9o 7#agac-agie7 classifi$ ! rocks 9 ! nor? ( ! p59sula z ,primitive1 ,second>y1 ,t]ti>y & ,volcanic4 ,a simil> ?reefold ord] 0 ann\nc$ = ! ,h>z & ,erzgebirge by ,j4,g4 ,lehmann 9 #agef4 ,he recogniz$ 9 t region an anci5t s]ies ( rocks 9 incl9$ or v]tical /rata1 :i* rise to ! tops ( ! hills & desc5d to an unkn[n dep? into ! int]ior4 ,^! masses1 he ?\ght1 7 contemporane\s ) ! mak+ ( ! _w4 ,next came ! ,flotzgebirge1 consi/+ ( y\ng] s$im5ts1 dispos$ 9 flat or g5tly incl9$ %eets :i* ov]lie ! fir/ & m di/urb$ s]ies1 & are full ( petrifi$ rema9s ( plants & animals4 ,la/ly he includ$ ! m\nta9s :i* h f "t to "t be5 =m$ by local accid5ts4 ,/ill m advanc$ 7 ! conceptions ( ,g4,c4 ,fu*sel1 :o 9 ! ye> #agfb publi%$ 9 ,lat9 ,a ,hi/ory ( ! ,e>? & ! ,sea1 bas$ on a ,hi/ory ( ! ,m\nta9s ( ,?ur+ia2 & 9 #aggc1 9 ,g]man1 a ,sket* ( ! mo/ ,anci5t ,hi/ory ( ! ,e>? & ,man4 ,9 ^! works he describ$ ! /ratigraphical relations & g5]al *>act]s ( ! v>i\s geological =mations 9 8 little pr9cipality2 & tak+ !m z indicative ( a g5]al ord] ( succession1 he trac$ :at he believ$ to h be5 a s]ies ( revolutions ?r\gh :i* ! e>? has pass$4 ,9 int]pret+ ?is geological hi/ory1 he laid great /ress on ! evid5ce ( ! fossils conta9$ 9 ! rocks4 ,he recogniz$ t ! v>i\s =mations diff] f ea* o!r 9 _! enclos$ organic rema9s1 & t f ^! diff]5ces ! exi/5ce ( =m] sea-bottoms & l& surfaces c be det]m9$4 ,! lab\rs ( ^! pione]s pav$ ! way = ! adv5t ( ,w]n]4 ,?\gh ! sy/em evolv$ by ?is tea*] claim$ to disc>d !ory & to be e/abli%$ on a basis ( obs]v$ facts1 x re/$ on a succession ( hypo!ses1 = :i* no bett] f\ndation c\ld be %[n ?an ! belief ( _! au?or 9 _! validity4 ,/>t+ f ! extremely limit$ /ratigraphical range display$ 9 ! geological /ructure ( ,saxony1 he took x z a type = ! re/ ( ! globe1 p]suad+ himself & impress+ ^u 8 foll[]s t ! rocks ( t small k+dom 7 to be tak5 z examples ( 8 univ]sal =mations4 ,! olde/ portion ( ! s]ies1 class$ by him z ,primitive1 consi/$ ( rocks :i* he ma9ta9$ _h be5 deposit$ f *emical solution4 ,yet !y includ$ granite1 gneiss1 basalt1 porphyry & s]p5t9e1 :i*1 ev5 9 8 own day1 7 by _m obs]v]s correctly reg>d$ z ( igne\s orig94 ,a lat] gr\p ( rocks1 to :i* he gave ! "n ( ,transition1 compris$1 9 8 belief1 p>tly *emical1 p>tly me*anical s$im5ts1 & conta9$ ! e>lie/ fossil organic rema9s4 ,a ?ird gr\p1 = :i* he res]v$ ,lehmann's "n ,flotz1 0 made up *iefly ( me*anical detritus1 :ile y\nge/ ( all came ! ,alluvial s]ies ( loams1 clays1 s&s1 gravels & peat4 ,x 0 by ! gradual subsid5ce ( ! ocean t1 z he believ$1 ! g5]al mass ( ! dry l& em]g$1 ! fir/-=m$ rocks be+ left /&+ up1 sometimes on end1 to =m ! m\nta9s1 :ile ^? ( lat] date1 less /eeply incl9$1 occupi$ successively l[] levels d[n to ! flat alluvial accumulations ( ! pla9s4 ,nei!r ,w]n]1 nor any ( 8 foll[]s1 v5tur$ to acc\nt = :at became ( ! wat] z ! sea-level subsid$1 ?\gh1 9 despite ( _! antipa?y to any?+ l speculation1 !y c\ld n help sugge/+1 z an answ] to ! cog5t argum5ts ( _! oppon5ts1 t "o ( ! cele/ial bodies :i* sometimes approa* ne> to ! e>? may h be5 able to )draw a portion ( our atmosph]e & ( our ocean4 ,nor 0 any attempt made to expla9 ! extraord9>y nature ( ! suppos$ *emical precipitates ( ! univ]sal ocean4 ,! progress ( inquiry ev5 9 ,w]n]'s lifetime disprov$ "s ( ! fundam5tal portions ( 8 sy/em4 ,_m ( ! *emical precipitates 7 %[n to be masses t _h be5 erupt$ 9 a molt5 /ate f bel[4 ,8 ord] ( succession 0 f\nd n to hold good2 & ?\gh he tri$ to readju/ 8 sequ5ce & to introduce into x modifications to suit new facts1 its inh]5t artificiality l$ to its spe$y decl9e aft] 8 dea?4 ,x mu/ be conc$$1 h[ev]1 t ! /ress :i* he laid ^u ! fact t ! rocks ( ! e>?'s cru/ 7 deposit$ 9 a def9ite ord] _h an important influ5ce 9 direct+ att5tion to ?is subject1 & 9 prep>+ ! way = a m natural sy/em1 bas$ n on m]e m9]alogical *>act]s1 b hav+ reg>d to ! organic rema9s1 :i* 7 n[ be+ ga!r$ 9 ev]-increas+ numb]s & v>iety f /ratifi$ =mations ( _m diff]5t ages & f all p>ts ( ! globe4 ,x 0 9 ,france & 9 ,engl& t ! f\ndations ( /ratigraphy1 bas$ ^u a k ( organic rema9s1 7 fir/ successfully laid4 ,abbe ,j4,l4 ,giraud-,s\lavie 7#ageb-ahac71 9 8 ,hi/oire naturelle de la ,france m]idionale1 :i* appe>$ 9 sev5 volumes1 subdivid$ ! lime/ones ( ,viv>ais into five ages1 ea* m>k$ by a di/9ct assemblage ( %ells4 ,9 ! l[e/ /rata1 repres5t+ ! fir/ age1 none ( ! fossils 7 believ$ by him to h any liv+ repres5tatives1 & he call$ ^! rocks ,primordial4 ,9 ! next gr\p a m+l+ ( liv+ ) ext9ct =ms 0 obs]vable4 ,! ?ird age 0 m>k$ by ! pres5ce ( %ells ( /ill exi/+ species4 ,! /rata ( ! f\r? s]ies 7 *>act]iz$ by c>bonace\s %ales or slates1 conta9+ rema9s ( primordial vegetation1 & p]haps equival5ts ( ! fir/ ?ree calc>e\s s]ies4 ,! fif? age 0 m>k$ by rec5t deposits conta9+ rema9s ( t]re/rial vegetation & ( l& animals4 ,x is rem>kable t ^! sagaci\s conclusions %\ld h be5 =m$ & publi%$ at a "t :5 ! geologi/s ( ! ,cont95t 7 engag$ 9 ! controv]sy ab\t ! orig9 ( basalt1 or 9 disputes ab\t ! "* & /ratigraphical position ( ! suppos$ univ]sal =mations1 & :5 ! int]e/ & importance ( fossil organic rema9s /ill rema9$ unrecogniz$ by ! va/ majority ( ! combatants4 ,! rocks ( ! ,p>is bas9 display s cle>ly an ord]ly arrangem5t1 & are s di/+ui%$ = ! v>iety & p]fect pres]vation ( _! enclos$ organic rema9s1 t !y c\ld n fail to attract ! e>ly notice ( obs]v]s4 ,j4 ,e4 ,guett>d1 ,g4,f4 ,r\elle 7#agjc-aggj71 ,n4 ,desm>e/1 ,a4,l4 ,lavoisi] 7#agdc-agid7 & o!rs made obs]vations 9 ?is int]e/+ di/rict4 ,b x 0 res]v$ = ,cuvi] 7#agfi-ahcb7 & ,a4 ,brongni>t 7#aggj-ahdg7 to "w out ! detail$ succession ( ! ,t]ti>y =mations1 & to %[ h[ ea* ( ^! is *>act]iz$ by its own peculi> assemblage ( organic rema9s4 ,! lat] progress ( inve/igation has slightly correct$ & greatly amplifi$ ! tabul> arrangem5t e/abli%$ by ^! au?ors 9 #ahjh1 b ! broad outl9es ( ! ,t]ti>y /ratigraphy ( ! ,p>is bas9 rema9 /ill z ,cuvi] & ,brongni>t left !m4 ,! mo/ important subsequ5t *ange 9 ! classification ( ! ,t]ti>y =mations 0 made by ,sir ,*>les ,lyell1 :o1 conceiv+ 9 #ahbh ! idea ( a classification ( ^! rocks by ref]5ce to _! relative proportions ( liv+ & ext9ct species ( %ells1 e/abli%$1 9 collaboration ) ,g4,p4 ,de%ayes1 ! n[ univ]sally accept$ divisions ,eoc5e1 ,mioc5e & ,plioc5e4 ,long be=e ,cuvi] & ,brongni>t publi%$ an acc\nt ( _! rese>*es1 ano!r obs]v] _h be5 at "w among ! ,second>y =mations ( ! we/ ( ,engl&1 & _h indep5d5tly discov]$ t ! compon5t memb]s ( ^! =mations 7 ea* di/+ui%$ by a peculi> gr\p ( organic rema9s2 & t ?is di/9ction c\ld be us$ to discrim9ate !m ov] all ! region ?r\gh :i* he _h trac$ !m4 ,! rem>kable man :o arriv$ at ?is f>-rea*+ g5]alization 0 ,william ,smi? 7#agfi-ahci71 a l& surveyor :o1 9 ! prosecution ( 8 pr(essional bus9ess1 f\nd opportunities ( trav]s+ a great "p ( ,engl&1 & ( putt+ 8 d$uctions to ! te/4 ,z ! result ( ^! j\rneys he accumulat$ mat]ials en\gh to enable him to produce a geological map ( ! c\ntry1 on :i* ! di/ribution & succession ( ! rocks 7 = ! fir/ "t del9eat$4 ,smi?'s lab\rs laid ! f\ndation ( /ratigraphical geology 9 ,engl& & he 0 /yl$ ev5 9 8 lifetime ! ,fa!r ( ,engli% geology4 ,f 8 day onw>d ! significance ( fossil organic rema9s ga9$ rapidly increas+ recognition4 ,?us 9 ,engl& ! outl9es trac$ by him among ! ,second>y & ,t]ti>y =mations 7 admirably fill$ 9 by ,?omas ,web/] 7#aggc-ahdd72 :ile ! ,cretace\s s]ies 0 work$ out 9 /ill great] detail 9 ! classic memoirs ( ,william ,h5ry ,fitton 7#aghj-ahfa74 ,!re 0 "o /ratigraphical doma91 h[ev]1 into :i* ,william ,smi? did n ent]4 ,he trac$ 8 sequ5ce ( rocks d[n into ! ,coal ,measures1 b cont5t$ himself ) only a vague ref]5ce to :at lay und]nea? t =mation4 ,?\gh "s ( ^! und]ly+ rocks _h 9 v>i\s c\ntries yield$ abundant fossils1 !y _h g5]ally suff]$ s mu* f t]re/rial di/urbances1 & _! ord] ( succession 0 consequ5tly (t5 s mu* obscur$ ?r\gh\t we/]n ,europe1 t !y rema9$ b little kn[n = _m ye>s aft] ! /ratigraphy ( ! ,second>y & ,t]ti>y s]ies _h be5 e/abli%$4 ,at la/ 9 #ahca ,mur*ison began to attack ?is t]ra incognita on ! bord]s ( ,s\? ,wales1 work+ into x f ! ,old ,r$ ,s&/one1 ! /ratigraphical position ( :i* 0 well kn[n4 ,9 a few ye>s he succe$$ 9 demon/rat+ ! exi/5ce ( a succession ( =mations1 ea* di/+ui%$ by its own peculi> assemblage ( organic rema9s :i* 7 di/9ct f ^? 9 any ( ! ov]ly+ /rata4 ,to ^! =mations he gave ! "n ( ,silurian 7q4v474 ,f ! key :i* 8 rese>*es suppli$1 x 0 possible to recognize 9 o!r c\ntries ! same ord] ( =mations & ! same sequ5ce ( fossils1 s t1 9 ! c\rse ( a few ye>s1 repres5tatives ( ! ,silurian sy/em 7 f\nd f> & wide ov] ! globe4 ,:ile ,mur*ison 0 ?us engag$1 ,s$gwick devot$ himself to ! m difficult task ( unravell+ ! complicat$ /ructure ( ,nor? ,wales4 ,he ev5tually made out ! ord] ( ! sev]al =mations !re1 ) _! va/ int]calations ( volcanic mat]ial4 ,he nam$ !m ! ,cambrian sy/em 7q4v471 & f\nd !m to conta9 fossils1 :i*1 h[ev]1 lay = "s "t unexam9$ by him4 ,he at fir/ believ$1 z ,mur*ison also did1 t 8 rocks 7 all old] ?an any "p ( ! ,silurian s]ies4 ,x 0 ev5tually discov]$ t a portion ( !m 0 equival5t to ! l[] "p ( t s]ies4 ,! olde/ ( ,s$gwick's gr\ps1 conta9+ di/9ctive fossils1 reta9 ! "n ,cambrian1 & are ( high int]e/1 z !y enclose ! rema9s ( ! e>lie/ faunas :i* are yet well kn[n4 ,s$gwick & ,mur*ison r5d]$ yet ano!r signal s]vice to /ratigraphical geology by e/abli%+1 9 #ahci1 on a basis ( palaeontological evid5ce suppli$ by ,w4 ,lonsdale1 ! indep5d5ce ( ! ,devonian sy/em 7q4v474 ,= _m ye>s ! rocks bel[ ! olde/ fossilif]\s deposits receiv$ comp>atively little att5tion4 ,!y 7 vaguely describ$ z ! cry/all9e s*i/s & 7 (t5 ref]r$ to z p>ts ( ! primeval cru/ 9 :i* no *ronology 0 to be look$ =4 ,w4,e4 ,logan 7#agih-ahge7 l$ ! way1 9 ,canada1 by e/abli%+ !re sev]al va/ s]ies ( rocks1 p>tly ( cry/all9e s*i/s & gneisses 7,laur5tian7 & p>tly ( slates & conglom]ates 7,huronian74 ,lat] obs]v]s1 bo? 9 ,canada & ! ,unit$ ,/ates1 h greatly increas$ our k ( ^! rocks1 & h %[n _! /ructure to be mu* m complex ?an 0 at fir/ suppos$ 7see ,,ar*ean ,,sy/em74 ,dur+ ! latt] half ( ! #ai? c5tury ! mo/ important developm5t ( /ratigraphical geology 0 ! detail$ work+ out & application ( ! pr9ciple ( zonal classification to ! fossilif]\s =mations--t is1 ! det]m9ation ( ! sequ5ce & di/ribution ( organic rema9s 9 ^! =mations1 & ! arrangem5t ( ! /rata into zones1 ea* ( :i* is di/+ui%$ by a peculi> assemblage ( fossil species 7see "u ,"p ,,vi474 ,! zones are usually nam$ aft] "o especially *>act]i/ic species4 ,?is sy/em ( classification 0 begun 9 ,g]many ) ref]5ce to ! memb]s ( ! ,jurassic sy/em 7q4v47 by ,a4 ,oppel 7#ahef-aheh7 & ,f4,a4 von ,qu5/$t 7#aheh71 & x has s9ce be5 ext5d$ ?r\gh ! o!r ,mesozoic =mations4 ,x has ev5 be5 f\nd to be applicable to ! ,palaeozoic rocks1 :i* are n[ subdivid$ into palaeontological zones4 ,9 ! ,silurian sy/em1 = example1 ! graptolites h be5 %[n by ,c4 ,lapwor? to furni% a useful basis = zonal subdivisions4 ,! l[e/ fossilif]\s horizon 9 ! ,cambrian rocks ( ,europe & ,nor? ,am]ica is kn[n z ! ,ol5ellus zone1 f ! prom95ce 9 x ( t g5us ( trilobite4 ,ano!r conspicu\s feature 9 ! progress ( /ratigraphy dur+ ! second half ( ! #ai? c5tury 0 display$ by ! rise & rapid developm5t ( :at is kn[n z ,glacial geology4 ,! v>i\s deposits ( drift spread ov] nor!rn ,europe1 & ! b\ld]s scatt]$ across ! surface ( ! pla9s _h long attract$ notice1 & _h ev5 f\nd a place 9 popul> leg5d & sup]/ition4 ,:5 m5 began to exam9e !m ) a view to asc]ta9 _! orig91 !y 7 naturally reg>d$ z evid5ces ( ! ,noa*ian deluge4 ,! fir/ obs]v] :o drew att5tion to ! smoo!d & /riat$ surfaces ( rock t und]lie ! ,drifts 0 ,hutton's fri5d1 ,sir ,james ,hall1 :o /udi$ !m 9 ! l[l&s ( ,scotl& & ref]r$ !m to ! action ( great debacles ( wat]1 :i*1 9 ! c\rse ( "s anci5t t]re/rial convulsion1 _h be5 laun*$ across ! face ( ! c\ntry4 ,playfair1 h[ev]1 po9t$ out t ! mo/ pot5t geological ag5ts = ! transportation ( l>ge blocks ( /one are ! glaci]s4 ,b no "o 0 !n bold en\gh to connect ! travell$ b\ld]s ) glaci]s on ! pla9s ( ,g]many & ( ,brita94 ,yet ! transport+ ag5cy ( ice 0 invok$ 9 explanation ( _! diffusion4 ,x came to be ! preval5t belief among ! geologi/s ( ! fir/ half ( ! #ai? c5tury1 t ! fall ( temp]ature1 indicat$ by ! gradual increase 9 ! numb] ( nor!rn species ( %ells 9 ! ,engli% ,crag deposits1 rea*$ its climax dur+ ! "t ( ! ,drift1 & t mu* ( ! nor? & c5tre ( ,europe 0 !n subm]g$ b5ea? a sea1 across :i* float+ iceb]gs & floes transport$ ! mat]ials ( ! ,drift & dropp$ ! scatt]$ b\ld]s4 ,z ! ph5om5a are well develop$ ar\nd ! ,alps1 x 0 necess>y to suppose t ! subm]g5ce involv$ ! l[l&s ( ! ,cont95t up to ! foot ( t m\nta9 *a9--a geographical *ange s /up5d\s z to dem& mu* m evid5ce ?an 0 adduc$ 9 its support4 ,at la/ ,l\is ,agassiz 7#ahjg-ahgc71 :o _h v>i$ 8 palaeontological /udies at ,neu*atel by excursions into ! ,alps1 0 s mu* /ruck by ! pro(s ( ! =m] f> great] ext5sion ( ! ,swiss glaci]s1 t he pursu$ ! inve/igation & satisfi$ himself t ! ice _h =m]ly ext5d$ f ! ,alp9e valleys "r across ! great pla9 ( ,switz]l&1 & _h transport$ huge b\ld]s f ! c5tral m\nta9s to ! flanks ( ! ,jura4 ,9 ! ye> #ahdj he visit$ ,brita9 & soon f\nd evid5ce ( simil> conditions !re4 ,he %[$ t x 0 n by subm]g5ce 9 a sea cumb]$ ) float+ ice1 b by ! =m] pres5ce ( va/ glaci]s or %eets ( ice t ! ,drift & erratic blocks _h be5 di/ribut$4 ,! idea ?us prop\nd$ by him did n at once comm& complete approval1 ?\gh traces ( anci5t glaci]s 9 ,scotl& & ,wales 7 soon detect$ by native geologi/s1 p>ticul>ly by ,w4 ,buckl&1 ,lyell1 ,j4,d4 ,=bes & ,*>les ,macl>54 ,rob]t ,*amb]s 7#ahjb-ahga7 did good s]vice 9 ga!r+ additional evid5ce f ,scotl& & ,norway 9 fav\r ( ,agassiz's views1 :i* /eadily ga9$ adh]5ts until1 aft] "s qu>t] ( a c5tury1 !y 7 adopt$ by ! great majority ( geologi/s 9 ,brita91 & subsequ5tly 9 o!r c\ntries4 ,s9ce t "t ! lit]ature ( geology has be5 swoll5 by a va/ numb] ( contributions 9 :i* ! hi/ory ( ! ,glacial p]iod1 & its records bo? 9 ! ,old & ,new ,_w1 h be5 fully discuss$4 ,rise & ,progress ( ,palaeontological ,geology4--,z ?is bran* ( ! sci5ce deals ) ! evid5ce furni%$ by fossil organic rema9s z to =m] geographical conditions1 x e>ly attract$ obs]v]s :o1 9 ! sup]ficial b$s ( m>9e %ells f\nd at "s di/ance f ! coa/1 saw pro(s ( ! =m] subm]g5ce ( ! l& "u ! sea4 ,b ! occurr5ce ( fossils emb$d$ 9 ! he>t ( ! solid rocks ( ! m\nta9s (f]$ mu* great] difficulties ( explanation1 & fur!r progress 0 consequ5tly sl[4 ,especially baneful 0 ! belief t ^! objects 7 m]e sports ( nature1 & _h no connexion ) any once liv+ organisms4 ,s long z ! true organic orig9 ( ! fossil plants & animals conta9$ 9 ! rocks 0 9 dispute1 x 0 h>dly possible t mu* advance c\ld be made 9 _! sy/ematic /udy1 or 9 ! geological d$uctions to be drawn f !m4 ,"o good result ( ! controv]sy1 h[ev]1 0 to be se5 9 ! l>ge collections ( ^! =m$ /ones t 7 ga!r$ toge!r 9 ! cab9ets & museums ( ! #ag? & #ah? c5turies4 ,! accumulation & comp>ison ( ^! objects naturally l$ to ! production ( treatises 9 :i* !y 7 describ$ & n unfrequ5tly illu/rat$ by good engrav+s4 ,switz]l& 0 m p>ticul>ly not$ = ! numb] & m]it ( its works ( ?is k9d1 su* z t ( ,k4,n4 ,lang 7,hi/oria lapidum figuratorum ,helvetiae1 #agjh7 & ^? ( ,johann ,jacob ,s*eu*z] 7#afgb-agcc74 ,9 ,engl&1 also1 illu/rat$ treatises 7 publi%$ bo? by m5 :o look$ on fossils z m]e freaks ( nature1 & by ^? :o reg>d$ !m z pro(s ( ,noah's flood4 ,( ! =m] type 7 ! works ( ,m>t9 ,li/] 7#afch-agab7 & ,rob]t ,plot 7,natural ,hi/ory ( ,ox=d%ire1 #afgg74 ,! ,celtic s*ol> ,edw>d ,llwyd 7#affj-agji7 wrote a ,lat9 treatise conta9+ good plates ( a ?\s& fossils 9 ! ,a%molean ,museum1 ,ox=d1 & ,j4 ,woodw>d1 9 #agbh-agbi1 publi%$ 8 ,natural ,hi/ory ( ! ,fossils ( ,engl&1 already m5tion$1 :]e9 he describ$ 8 own ext5sive collection1 :i* he bequea!d to ! ,univ]sity ( ,cambridge1 :]e x is /ill c>efully pres]v$4 ,! mo/ volum9\s & important ( all ^! works1 h[ev]1 appe>$ at a lat] date at ,nuremb]g4 ,x 0 begun by ,g4,w4 ,knorr 7#agje-agfa71 :o himself engrav$ = x a s]ies ( plates1 :i* = beauty & accuracy h seldom be5 surpass$4 ,aft] 8 dea? ! "w 0 cont9u$ by ,j4,e4,i4 ,wal* 7#agbe-aggh71 & ultimately consi/$ ( f\r massive folio volumes & ne>ly #cjj plates "u ! title ( ,lapides diluvii univ]salis te/es4 ,al?\gh ! au?ors suppos$ _! fossils to be relics ( ,noah's flood1 _! "w mu/ be ackn[l$g$ to m>k a di/9ct onw>d /age 9 ! palaeontological dep>tm5t ( geology4 ,x 0 9 ,france t palaeontological geology began to be cultivat$ 9 a sci5tific _s4 ,! pott] ,b]n>d ,palissy1 z f> back z #aehj1 _h dwelt on ! importance ( fossil %ells z monum5ts ( revolutions ( ! e>?'s surface2 b ! obs]v] :o fir/ und]took ! detail$ /udy ( ! subject 0 ,jean ,eti5ne ,guett>d1 :o began 9 #agea to publi% 8 descriptions ( fossils 9 ! =m ( memoirs pres5t$ to ! ,academy ( ,sci5ces ( ,p>is4 ,to him !y 7 n only ( deep int]e/ z monum5ts ( =m] types ( exi/5ce1 b !y _h an especial value z records ( ! *anges :i* ! c\ntry _h und]gone f sea to l& & f l& to sea4 ,m especially notewor?y 0 a monograph by him :i* appe>$ 9 #agfe be>+ ! title ,on ! accid5ts t h befall5 ,fossil ,%ells comp>$ ) ^? :i* are f\nd to happ5 to %ells n[ liv+ 9 ! ,sea4 ,9 ?is treatise he %[$ t ! fossils h be5 encru/$ ) b>nacles & s]pulae1 h be5 bor$ into by o!r organisms1 & h (t5 be5 r\nd$ or brok5 be=e f9al entombm5t2 & he inf]r$ t ^! fossils mu/ h liv$ & di$ on ! sea-floor "u simil> conditions to ^? :i* obta9 on ! sea-floor to-day4 ,8 argum5t 0 ! mo/ triumphant t _h ev] be5 br\ght aga9/ ! doctr9e ( lusus naturae1 & t ( ! efficacy ( ,noah's flood--doctr9es :i* /ill held _! gr\nd 9 ,guett>d's day4 ,:5 ,s\lavie1 ,cuvi] & ,brongni>t 9 ,france1 & ,william ,smi? 9 ,engl&1 %[$ t ! rock =mations ( ! e>?'s cru/ c\ld be arrang$ 9 *ronological ord]1 & c\ld be recogniz$ f> & wide by means ( _! enclos$ organic rema9s1 ! va/ significance ( ^! rema9s 9 geological rese>* 0 spe$ily realiz$1 & palaeontological geology at once ent]$ on a new & enl>g$ phase ( developm5t4 ,b ap>t f _! value z *ronological monum5ts1 & z witnesses ( =m] conditions ( geography1 fossils pres5t$ 9 !mselves a wide field ( inve/igation z types ( life t _h =m]ly exi/$1 b _h n[ pass$ away4 ,x 0 9 ,france t ?is subject fir/ took def9ite %ape z an important bran* ( sci5ce4 ,! mollusca ( ! ,t]ti>y deposits ( ! ,p>is bas9 became1 9 ! h&s ( ,lam>ck1 ! basis on :i* inv]tebrate palaeontology 0 f\nd$4 ,! same s]ies ( /rata furni%$ to ,cuvi] ! rema9s ( ext9ct l& animals1 ( :i*1 by critical /udy ( _! fragm5t>y bones & skeletons1 he work$ out re/orations t may be look$ on z ! />t+-po9t ( v]tebrate palaeontology4 ,^! brilliant rese>*es1 r\s+ widespread int]e/ 9 su* /udies1 %[$ h[ great a flood ( light c\ld be ?r[n on ! pa/ hi/ory ( ! e>? & its inhabitants4 ,b ! full significance ( ^! ext9ct types ( life c\ld n be und]/ood s long z ! doctr9e ( ! immutability ( species1 s /r5u\sly upheld by ,cuvi]1 ma9ta9$ its sway among naturali/s4 ,lam>ck1 z f> back z ! ye> #ahjj1 _h begun to prop\nd 8 !ory ( evolution & ! trans=mation ( species2 b 8 views1 /rongly oppos$ by ,cuvi] & ! great body ( naturali/s ( ! day1 fell into neglect4 ,n until aft] ! publication 9 #ahei ( ! ,orig9 ( ,species by ,*>les ,d>w9 7 ! b>ri]s ( old prejudice 9 ?is matt] f9ally brok5 d[n4 ,! possibility ( trac+ ! ance/ry ( liv+ =ms back into ! remote/ ages 0 !n p]ceiv$2 ! "t-hon\r$ fiction t ! /ratifi$ =mations record a s]ies ( cata/rophes & re-creations 0 f9ally dissipat$2 & ! e>?'s cru/ 0 se5 to conta9 a noble1 ?\gh imp]fect1 record ( ! gr& evolution ( organic types ( :i* our planet has be5 ! !atre4 ,developm5t ( ,petrographical ,geology4--,!ophra/us1 ! fav\rite pupil ( ,ari/otle1 wrote a treatise ,on ,/ones1 :i* has come d[n to our own day1 & may be reg>d$ z ! e>lie/ "w on petrography4 ,at a subsequ5t p]iod ,pl9y1 9 8 ,natural ,hi/ory1 collect$ all t 0 kn[n 9 8 day reg>d+ ! occurr5ce & uses ( m9]als & rocks4 ,b nei!r ( ^! works is ( great sci5tific importance1 ?\gh conta9+ mu* int]e/+ in=mation4 ,m9]als f _! beauty & value attract$ notice be=e mu* att5tion 0 paid to rocks1 & _! /udy gave rise to ! sci5ce ( m9]alogy long be=e geology came into exi/5ce4 ,:5 rocks began to be m p>ticul>ly scrut9iz$1 x 0 *iefly f ! side ( _! usefulness = build+ & o!r economic purposes4 ,! occurr5ce ( m>9e %ells 9 _m ( !m _h e>ly attract$ att5tion to !m4 ,b _! v>ieties ( composition & orig9 did n become ! subject ( s]i\s /udy until aft] ,l9naeus & ,j4,g4 ,wall]ius 9 ! #ah? c5tury _h made a beg9n+4 ,! fir/ important contribution to ?is dep>tm5t ( ! sci5ce 0 t ( ,w]n]1 :o 9 #aghf publi%$ a classification & description ( rocks 9 :i* he arrang$ !m 9 two divisions1 simple & comp\nd1 & fur!r di/+ui%$ !m by v>i\s ext]nal *>act]s & by _! relative age4 ,! publication ( ?is s*eme may be said to m>k ! beg9n+ ( sci5tific petrography4 ,w]n]'s sy/em1 h[ev]1 _h ! s]i\s defect t ! *ronological ord] 9 :i* he gr\p$ ! rocks1 & ! hypo!sis by :i* he acc\nt$ = !m z *emical precipitates f ! orig9al ocean1 7 bo? alike contr>y to nature4 ,x 0 h>dly possible inde$ t mu* progress c\ld be made 9 ?is bran* ( geology until *emi/ry & m9]alogy _h made great] advances2 & especially until x 0 possible to asc]ta9 ! intimate *emical & m9]alogical composition1 & ! m9ute /ructure ( rocks4 ,! /udy1 h[ev]1 cont9u$ to be pursu$ 9 ,g]many1 :]e ! influ5ce ( ,w]n]'s en?usiasm /ill l$ m5 to ent] ! petrographical r ?an ! palaeontological doma94 ,! res\rces ( mod]n *emi/ry 7 press$ into ! s]vice1 & analyses 7 made & multipli$ to su* a degree t x seem$ z if ! ultimate *emical con/itution ( e type ( rock _h n[ be5 ?or\ghly reveal$4 ,! condition ( ! sci5ce 9 ! middle ( ! #ai? c5tury 0 well %[n by ,j4,l4,a4 ,ro?1 :o 9 #ahfa collect$ ab\t #ajjj tru/wor?y analyses :i* up to t "t _h be5 made4 ,b ?\gh ! *emical elem5ts ( ! rocks _h be5 fairly well det]m9$1 ! mann] 9 :i* !y 7 comb9$ 9 ! comp\nd rocks c\ld = ! mo/ "p be only m or less plausibly conjectur$4 ,z f> back z #ahca an acc\nt 0 publi%$ ( a process devis$ by ,william ,nicol ( ,ed9burgh1 :]eby sections ( fossil wood c\ld be cut1 m\nt$ on glass1 & r$uc$ to su* a degree ( transp>5cy z to be easily exam9$ "u a microscope4 ,h5ry ,sorby1 ( ,%effield1 hav+ se5 ,nicol's prep>ations1 p]ceiv$ h[ admirably adapt$ ! process 0 = ! /udy ( ! m9ute /ructure & composition ( rocks4 ,9 #aheh he publi%$ 9 ! ,qu>t]ly ,j\rnal ( ! ,geological ,society a pap] ,on ! ,microscopical ,/ructure ( ,cry/als4 ,?is essay l$ to a complete revolution ( petrographical me?ods & gave a va/ impetus to ! /udy ( rocks4 ,petrology ent]$ ^u a new & wid] field ( inve/igation4 ,n only 7 ! m9]alogical con/itu5ts ( ! rocks detect$1 b m9ute /ructures 7 reveal$ :i* %$ new light on ! orig9 & hi/ory ( ^! m9]al masses1 & op5$ up new pa?s 9 !oretical geology4 ,9 ! h&s ( ,h4 ,vogelsang1 ,f4 ,zirkel1 ,h4 ,ros5bus*1 & a ho/ ( o!r work]s 9 all civiliz$ c\ntries1 ! lit]ature ( ?is dep>tm5t ( ! sci5ce has gr[n to a rem>kable ext5t4 ,arm$ ) ! p[]ful aid ( mod]n optical in/rum5ts1 geologi/s are n[ able ) f> m prospect ( success to resume ! exp]im5ts begun a c5tury be=e by de ,saussure & ,hall4 ,g4,a4 ,daubree1 ,c4 ,fri$el1 ,e4 ,s>as91 ,f4 ,f\que & ,a4 ,mi*el ,levy 9 ,france1 ,c4 ,doelt] y ,ci/]i* & ,e4 ,hussak ( ,gratz1 ,j4 ,morozewicz ( ,w>saw & o!rs1 h greatly advanc$ our k by _! syn!tical analyses1 & !re is e reason to hope t fur!r advances w be made 9 ?is field ( rese>*4 ,rise ( ,physiographical ,geology4--,until /ratigraphical geology _h advanc$ s f> z to %[ ( :at a va/ succession ( rocks ! cru/ ( ! e>? is built up1 by :at a long & complicat$ s]ies ( revolutions ^! rocks h come to assume _! pres5t positions1 & h[ enorm\s has be5 ! lapse ( "t :i* all ^! *anges repres5t1 x 0 n possible to make a sci5tific /udy ( ! surface features ( our globe4 ,f anci5t times x _h be5 kn[n t _m p>ts ( ! l& _h once be5 "u ! sea2 b d[n ev5 to ! beg9n+ ( ! #ai? c5tury ! vague/ conceptions cont9u$ to prevail z to ! op]ations conc]n$ 9 ! subm]g5ce & elevation ( l&1 & z to ! processes :]eby ! pres5t outl9es ( t]re/rial topography 7 det]m9$4 ,we h se51 = in/ance1 t accord+ to ! tea*+ ( ,w]n] ! olde/ rocks 7 fir/ precipitat$ f solution 9 ! univ]sal ocean to =m ! m\nta9s1 t ! v]tical position ( _! /rata 0 orig9al1 t z ! wat]s subsid$ successive =mations 7 deposit$ & laid b>e1 & t f9ally ! sup]flu\s portion ( ! ocean 0 :isk$ away into space by "s unexpla9$ co-op]ation ( ano!r planet>y body4 ,desm>e/1 9 8 inve/igation ( ! volcanic hi/ory ( ,auv]gne1 0 ! fir/ obs]v] to p]ceive by :at a long process ( sculpture ! pres5t configuration ( ! l& has be5 br\ght ab\t4 ,he %[$ conclusively t ! valleys h be5 c>v$ out by ! /reams t fl[ 9 !m1 & t :ile !y h sunk deep] & deep] into ! framework ( ! l&1 ! spaces ( gr\nd betwe5 !m h be5 left z int]v5+ ridges & hills4 ,de ,saussure le>nt a simil> lesson f 8 /udies ( ! ,alps1 & ,hutton & ,playfair made x a c>d9al feature 9 _! !ory ( ! e>?4 ,nev]!less ! idea enc\nt]$ s mu* opposition t x made b little way until aft] ! middle ( ! #ai? c5tury4 ,geologi/s pref]r$ to believe 9 convulsions ( nature1 :]eby valleys 7 op5$ & m\nta9s 7 upheav$4 ,t ! ma9 features ( ! l&1 su* z ! great m\nta9-*a9s1 _h be5 produc$ by gigantic plication ( ! t]re/rial cru/ 0 n[ g5]ally admitt$1 & also t m9or fractures & folds _h probably initiat$ _m ( ! valleys4 ,b ^? :o realiz$ mo/ vividly ! mom5t\s results a*iev$ by ages ( suba]ial d5udation p]ceiv$ t1 z ,hutton %[$1 ev5 )\t ! aid ( und]gr\nd ag5cy1 ! m]e fl[ ( wat] 9 /reams across a mass ( l& mu/ 9 c\rse ( "t c>ve out j su* a sy/em ( valleys z may any:]e be se54 ,x 0 ,j4,b4 ,jukes :o1 9 #ahfb1 fir/ reviv$ ! ,huttonian doctr9e1 & %[$ h[ completely x expla9$ ! dra9age-l9es 9 ! s\? ( ,irel&4 ,o!r writ]s foll[$ 9 quick succession until1 9 a few ye>s1 ! doctr9e came to be widely recogniz$ z "o ( ! e/abli%$ pr9ciples ( mod]n geology4 ,mu* help 0 d]iv$ f ! admirable illu/rations ( l&-sculpture & riv]-erosion suppli$ f ! ,we/]n ,t]ritories & ,/ates ( ! ,am]ican ,union4 ,ano!r bran* ( physiographical geology :i* c\ld only come into exi/5ce aft] mo/ ( ! o!r dep>tm5ts ( ! sci5ce _h made l>ge progress1 deals ) ! evolution ( ! framework ( ea* c\ntry & ( ! sev]al cont95ts & oceans ( ! globe4 ,x is n[ possible1 ) m or less confid5ce1 to trace backw>d ! hi/ory ( e t]re/rial area1 to see h[ sea & l& h !re succe$$ ea* o!r1 h[ riv]s & lakes h come & gone1 h[ ! cru/ ( ! e>? has be5 ridg$ up at widely sep>at$ int]vals1 ea* movem5t det]m9+ "s l9e ( m\nta9s or pla9s1 h[ ! b\nd>ies ( ! oceans h %ift$ aga9 & aga9 9 ! pa/1 & ?us h[1 aft] s prolong$ a s]ies ( revolutions1 ! pres5t topography ( ea* c\ntry1 & ( ! globe z a :ole1 has be5 produc$4 ,9 ! prosecution ( ?is subject maps h be5 con/ruct$ to %[ :at is conjectur$ to h be5 ! di/ribution ( sea & l& dur+ ! v>i\s geological p]iods 9 diff]5t p>ts ( ! _w1 & ?us to indicate ! successive /ages ?r\gh :i* ! ar*itecture ( ! l& has be5 gradually evolv$4 ,! mo/ notewor?y contribution to ?is dep>tm5t ( ! sci5ce is ! ,antlitz d] ,erde ( ,pr(essor ,suess ( ,vi5na4 ,?is important & sugge/ive "w has be5 translat$ into ,fr5* & ,engli%4 ,,"p ,,ii4--,,cosmical ,,aspects ,be=e geology _h atta9$ to ! position ( an inductive sci5ce1 x 0 cu/om>y to beg9 inve/igations into ! hi/ory ( ! e>? by prop\nd+ or adopt+ "s m or less fanciful hypo!sis 9 explanation ( ! orig9 ( our planet1 or ev5 ( ! univ]se4 ,su* prelim9>y notions 7 look$ ^u z ess5tial to a "r und]/&+ ( ! mann] 9 :i* ! mat]ials ( ! globe _h be5 put toge!r4 ,"o ( ! di/+ui%+ features ( ,hutton's ,!ory ( ! ,e>? consi/$ 9 8 prote/ t x is no "p ( ! prov9ce ( geology to discuss ! orig9 ( ?+s4 ,he taught t 9 ! mat]ials f :i* geological evid5ce is to be compil$ !re c be f\nd no traces ( a beg9n+1 no prospect ( an end4 ,9 ,engl&1 ma9ly to ! influ5ce ( ! s*ool :i* he f\nd$1 & to ! subsequ5t rise ( ! ,geological ,society ( ,london1 :i* resolv$ to collect facts in/ead ( fight+ ov] hypo!ses1 is due ! disappe>ance ( ! crude & unsci5tific cosmologies by :i* ! writ+s ( ! e>li] geologi/s 7 di/+ui%$4 ,b !re c n[ be little d\bt t 9 ! reaction aga9/ ^? vision>y & (t5 grotesque speculations1 geologi/s 7 c>ri$ too f> 9 an opposite direction4 ,9 all[+ !mselves to believe t geology _h no?+ to d ) que/ions ( cosmogony1 !y gradually grew up 9 ! conviction t su* que/ions c\ld nev] be o!r ?an m]e speculation1 int]e/+ or amus+ z a !me = ! employm5t ( ! fancy1 b h>dly com+ )9 ! doma9 ( sob] & inductive sci5ce4 ,nor w\ld !y soon h be5 awak5$ out ( ?is belief by any?+ 9 _! own sci5ce4 ,x is /ill true t 9 ! data ) :i* !y are accu/om$ to deal1 z compris+ ! sum ( geological evid5ce1 !re c be f\nd no trace ( a beg9n+1 ?\gh ! evid5ce furni%$ by ! t]re/rial cru/ %[s a g5]al evolution ( organic =ms f "s />t+-po9t :i* _c be se54 ,! olde/ rocks :i* h be5 discov]$ on any "p ( ! globe h probably be5 d]iv$ f o!r rocks old] ?an !mselves4 ,geology by itself has n yet reveal$1 & is little likely ev] to reveal1 a trace ( ! fir/ solid cru/ ( our globe4 ,if1 !n1 geological hi/ory is to be compil$ f direct evid5ce furni%$ by ! rocks ( ! e>?1 x _c beg9 at ! beg9n+ ( ?+s1 b mu/ be cont5t to date its fir/ *apt] f ! e>lie/ p]iod ( :i* any record has be5 pres]v$ among ! rocks4 ,nev]!less1 ?\gh geology 9 its usual re/rict$ s5se has be51 & mu/ ev] be1 unable to reveal ! e>lie/ hi/ory ( our planet1 x no long] ignores1 z m]e speculation1 :at is attempt$ 9 ?is subject by its si/] sci5ces4 ,a/ronomy1 physics & *emi/ry h 9 late ye>s all contribut$ to ca/ light on ! e>li] /ages ( ! e>?'s exi/5ce1 previ\s to ! beg9n+ ( :at is commonly reg>d$ z geological hi/ory4 ,b :atev] ext5ds our k ( ! =m] conditions ( our globe may be legitimately claim$ z "p ( ! doma9 ( geology4 ,if ?is bran* ( inquiry1 !re=e1 is to cont9ue wor?y ( its "n z ! sci5ce ( ! e>?1 x mu/ take cognizance ( ^! rec5t contributions f o!r sci5ces4 ,x mu/ no long] be cont5t to beg9 its annals ) ! records ( ! olde/ rocks1 b mu/ endeav\r to grope its way ?r\gh ! ages :i* prec$$ ! =mation ( any rocks4 ,?anks to ! results a*iev$ ) ! telescope1 ! spectroscope & ! *emical laboratory1 ! /ory ( ^! e>lie/ ages ( our e>? is e ye> becom+ m def9ite & intelligible4 ,up to ! pres5t "t no def9ite light has be5 ?r[n by physics on ! orig9 & e>lie/ condition ( our globe4 ,! fam\s nebul> !ory 7q4v47 ( ,kant & ,laplace sket*$ ! suppos$ evolution ( ! sol> sy/em f a gase\s nebula1 sl[ly rotat+ r\nd a m cond5s$ c5tral portion ( its mass1 :i* ev5tually became ! sun4 ,z a consequ5ce ( increas$ rapidity ( rotation result+ f cool+ & contraction1 ! nebula acquir$ a m & m l5ticul> =m1 until at la/ x ?rew (f f its equatorial protub]ance a r+ ( matt]4 ,subsequ5tly ! same process 0 repeat$1 & o!r simil> r+s successively sep>at$ f ! p>5t mass4 ,ea* r+ w5t ?r\gh a correspond+ s]ies ( *anges until x ultimately became a planet1 ) or )\t "o or m att5dant satellites4 ,! intimate relation%ip ( our e>? to ! sun & ! o!r planets 01 9 ?is way1 %[n4 ,b !re are "s s]i\s physical difficulties 9 ! way ( ! acceptance ( ! nebul> hypo!sis4 ,ano!r explanation is giv5 by ! meteoritic hypo!sis1 accord+ to :i*1 out ( ! sw>ms ( meteorites ) :i* ! regions ( space are cr[d$1 ! sun & planets h be5 =m$ by gradual accretion4 ,accord+ to ^! !oretical views we %\ld expect to f9d a g5]al uni=mity ( composition 9 ! con/itu5t matt] ( ! sol> sy/em4 ,= _m ye>s ! only available evid5ce on ?is po9t 0 d]iv$ f ! meteorites 7q4v47 :i* s con/antly fall f out] space ^u ! surface ( ! e>?4 ,^! bodies 7 f\nd to consi/ ( elem5ts1 all ( :i* _h be5 recogniz$ z ent]+ into ! con/itution ( ! e>?4 ,b ! discov]ies ( spectroscopic rese>* h made kn[n a f> m widely s]viceable me?od ( inve/igation1 :i* c be appli$ ev5 to ! lum9\s />s & nebulae t lie f> beyond ! b\nds ( ! sol> sy/em4 ,by ?is me?od in=mation has be5 obta9$ reg>d+ ! con/itution ( ! sun1 & _m ( our t]re/rial metals1 su* z iron1 nickel & magnesium1 h be5 asc]ta9$ to exi/ 9 ! =m ( inc&esc5t vap\r 9 ! sol> atmosph]e4 ,! pres5t condition ( ! sun probably repres5ts "o ( ! phases ?r\gh :i* />s & planets pass 9 _! progress t[>ds becom+ cool & d>k bodies 9 space4 ,if our globe 0 at fir/1 l its p>5t sun1 an inc&esc5t mass ( probably gase\s matt]1 occupy+ mu* m space ?an x n[ fills1 we c conceive t x has ev] s9ce be5 cool+ & contract+ until x has rea*$ its pres5t =m & dim5sions1 & t x /ill reta9s a high int]nal temp]ature4 ,its oblately sph]oidal =m is su* z w\ld be assum$ by a rotat+ mass ( matt] 9 ! transition f a vapor\s & self-lum9\s or liquid condition to "o ( cool & d>k solidity4 ,b x has be5 claim$ t ev5 a solid sph]ical globe might develop1 "u ! influ5ce ( protract$ rotation1 su* a %ape z ! e>? at pres5t possesses4 ,! obs]v$ increase ( temp]ature d[nw>ds 9 our planet has hi!rto be5 g5]ally accept$ z a relic & pro( ( an orig9al high temp]ature & mobility ( sub/ance4 ,rec5tly1 h[ev]1 ! validity ( ?is pro( has be5 *all5g$ on ! gr\nd t ! asc]ta9$ am\nt ( radium 9 ! rocks ( ! out] cru/ is m ?an suffici5t to acc\nt = ! obs]v$ d[nw>d increase ( temp]ature4 ,too little1 h[ev]1 is kn[n ( ! hi/ory & prop]ties ( :at is call$ radium to af=d a satisfactory gr\nd on :i* to disc>d :at has be51 & /ill rema9s1 ! preval5t belief on ?is subject4 ,an important epo* 9 ! geological hi/ory ( ! e>? 0 m>k$ by ! sep>ation ( ! moon f its mass 7see ,,tide74 ,:e!r ! sev]ance arose f ! rupture ( a surr\nd+ r+ or ! gradual cond5sation ( matt] 9 su* a r+1 or f ! ejection ( a s+le mass ( matt] f ! rapidly rotat+ planet1 x has be5 %[n t our satellite 0 only a few ?\s& miles f ! e>?'s surface1 s9ce :5 x has retreat$ to its pres5t di/ance ( #bdj1jjj m4 ,h5ce ! influ5ce ( ! moon's attraction1 & all ! geological effects to :i* x gives rise1 atta9$ _! maximum f> back 9 ! developm5t ( ! globe1 & h be5 sl[ly dim9i%+ ?r\gh\t geological hi/ory4 ,! sun by virtue ( its va/ size has n yet pass$ out ( ! condition ( gl[+ gas1 & /ill cont9ues to radiate heat beyond ! f>!/ planet ( ! sol> sy/em4 ,! e>?1 h[ev]1 be+ s small a body 9 comp>ison1 w\ld cool d[n mu* m quickly4 ,und]nea? its hot atmosph]e a cru/ w\ld conceivably beg9 to =m ov] its molt5 surface1 ?\gh ! int]ior might /ill possess a high temp]ature &1 ow+ to ! feeble conduct+ p[] ( rocks1 w\ld rema9 int5sely hot = a protract$ s]ies ( ages4 ,full in=mation reg>d+ ! =m & size ( ! e>?1 & its relations to ! o!r planet>y memb]s ( ! sol> sy/em1 w be f\nd 9 ! articles ,,planet & ,,sol> ,,sy/em4 ,= ! purposes ( geological inquiry ! read] w be> 9 m9d t ! equatorial diamet] ( our globe is e/imat$ to be ab\t #gibe m41 & ! pol> diamet] ab\t #ghii m42 ! diff]5ce betwe5 ^! two sums repres5t+ ! am\nt ( flatt5+ at ! poles 7ab\t #bf-a_/b m474 ,! planet has be5 comp>$ 9 %ape to an orange1 b x resembles an orange :i* has be5 some:at squeez$1 = its equatorial circumf]5ce is n a regul> circle b an ellipse1 ( :i* ! major axis lies 9 long4 #h deg4 #ae' ,w4--on a m]idian :i* cuts ! nor?-we/ corn] ( ,am]ica1 pass+ ?r\gh ,portugal & ,irel&1 & ! nor?-ea/ corn] ( ,asia 9 ! opposite hemisph]e4 ,! rotation ( ! e>? on its axis ex]ts an important influ5ce on ! movem5ts ( ! atmosph]e1 & !reby affects ! geological op]ations connect$ ) ^! movem5ts4 ,! influ5ce ( rotation is mo/ m>k$ 9 ! great a]ial circulation betwe5 ! poles & ! equator4 ,curr5ts ( air1 :i* set out 9 a m]idional direction f high latitudes t[>ds ! equator1 come f regions :]e ! velocity due to rotation is small to :]e x is great]1 & !y consequ5tly fall beh9d4 ,?us1 9 ! nor!rn hemisph]e a nor? w9d1 z x moves away f its nor!rn s\rce ( orig91 is gradually deflect$ m & m t[>ds ! we/ & becomes a nor?-ea/ curr5t2 :ile 9 ! opposite hemisph]e a w9d mak+ f high s\!rn latitudes t[>ds ! equator becomes1 f ! same cause1 a s\?-ea/ curr5t4 ,:]e1 on ! o!r h&1 ! air moves f ! equatorial to ! pol> regions its high] velocity ( rotation c>ries x ea/w>d1 s t on ! s\? side ( ! equator x becomes a nor?-we/ curr5t & on ! nor? side a s\?-we/ curr5t4 ,x is to ?is cause t ! ea/+ & we/+ ( ! great atmosph]ic curr5ts are to be attribut$1 z is famili>ly exemplifi$ 9 ! trade w9ds4 ,! atmosph]ic circulation ?us deflect$ influ5ces ! circulation ( ! ocean4 ,! w9ds :i* p]si/5tly bl[ f ! nor?-ea/ on ! nor? side ( ! equator1 & f ! s\?-ea/ on ! s\? side1 drive ! sup]ficial wat]s onw>ds1 & give rise to conv]g+ oceanic curr5ts :i* unite to =m ! great we/]ly equatorial curr5t4 ,a m direct effect ( t]re/rial rotation has be5 claim$ 9 ! case ( riv]s :i* fl[ 9 a m]idional direction4 ,x has be5 ass]t$ t ^?1 :i* 9 ! nor!rn hemisph]e fl[ f nor? to s\?1 l ! ,volga1 by cont9ually pass+ into regions :]e ! velocity ( rotation is increas+ly great]1 are ?r[n m aga9/ _! we/]n ?an _! ea/]n banks1 :ile ^? ^: g5]al c\rse is 9 an opposite direction1 l ! ,irtis* & ,y5esei1 press m ^u _! ea/]n sides4 ,!re _c be any d\bt t ! t5d5cy ( ! /reams mu/ be 9 ! directions indicat$4 ,b :5 ! comp>atively sl[ curr5t & con/antly me&]+ c\rse ( mo/ riv]s are tak5 into consid]ation1 x may be d\bt$ :e!r ! influ5ce ( rotation is ( mu* practical acc\nt s f> z riv]-erosion is conc]n$4 ,"o ( ! cosmical relations ( our planet :i* has be5 m especially prom95t 9 geological speculations relates to ! position ( ! e>?'s axis ( rotation4 ,abundant evid5ce has n[ be5 obta9$ to prove t at a comp>atively late geological p]iod a ri* flora1 resembl+ t ( w>m climates at ! pres5t day1 exi/$ 9 high latitudes ev5 )9 less ?an #i deg4 ( ! nor? pole1 :]e1 ) an extremely l[ temp]ature & d>kness la/+ = half ( ! ye>1 no su* vegetation c\ld possibly n[ exi/4 ,x has accord+ly be5 ma9ta9$ by _m geologi/s t ! axis ( rotation mu/ h %ift$1 & t :5 ! rem>kable ,arctic assemblage ( fossil plants liv$ ! region ( _! gr[? mu/ h la9 9 latitudes mu* ne>] to ! equator ( ! "t4 ,! possibility ( any s]i\s displacem5t ( ! rotational axis s9ce a v e>ly p]iod 9 ! e>?'s hi/ory has be5 /r5u\sly d5i$ by a/ronom]s1 & _! argum5ts h be5 g5]ally1 b some:at reluctantly1 accept$ by geologi/s1 :o f9d !mselves confront$ ) a problem :i* has hi!rto seem$ insoluble4 ,t ! axis is n rigidly /able1 h[ev]1 has be5 po/ulat$ by "s physici/s1 & has n[ be5 demon/rat$ by actual obs]vation & measurem5t4 ,x is admitt$ t by ! movem5t ( l>ge bodies ( wat] ! air ov] ! surface ( ! globe1 & m p>ticul>ly by ! accumulation ( va/ masses ( sn[ & ice 9 diff]5t regions1 ! position ( ! axis might be to "s ext5t %ift$2 m s]i\s effects might foll[ f widespread upheavals or depressions ( ! surface ( ! li?osph]e4 ,on ! assumption ( ! extreme rigidity ( ! e>?'s int]ior1 h[ev]1 ! g5]al result ( ma!matical calculation is to negative ! supposition t 9 any ( ^! ways )9 ! p]iod repres5t$ by :at is kn[n z ! geological record1 t is1 s9ce ! "t ( ! olde/ kn[n s$im5t>y =mations1 ! rotational axis has ev] be5 s s]i\sly displac$ z to acc\nt = su* /up5d\s geological ev5ts z ! spread ( a luxuriant vegetation f> up into pol> latitudes4 ,if1 h[ev]1 ! inside ( ! globe possesses a great pla/icity ?an has be5 all[$1 ! %ift+ ( ! axis might n be impossible1 ev5 to su* an ext5t z w\ld satisfy ! geological requirem5ts4 ,?is "q is "o on :i* ! la/ ^w has n be5 said1 & reg>d+ :i* judgm5t mu/ rema9 9 susp5se4 ,9 rec5t ye>s fre% in=mation be>+ on ! m9or devagations ( ! pole has be5 obta9$ f a s]ies ( sev]al ?\s& c>eful obs]vations made 9 ,europe & ,nor? ,am]ica4 ,x has ?us be5 asc]ta9$ t ! pole w&]s ) a curi\sly irregul> b some:at spiral movem5t1 )9 an amplitude ( betwe5 #dj & #ej ft41 & completes its erratic circuit 9 ab\t #dbh days4 ,x 0 n suppos$ t its movem5t _h any geological int]e/1 b ,dr ,john ,milne has rec5tly po9t$ out t ! times ( %>pe/ curvature 9 ! pa? ( ! pole co9cide ) ! occurr5ce ( l>ge e>?quakes1 & has sugge/$ t1 al?\gh x c h>dly be assum$ t ?is co9cid5ce %[s any direct connexion betwe5 e>?quake frequ5cy & *anges 9 ! position ( ! e>?'s axis1 bo? effects may n improbably arise f ! same r$i/ribution ( surface mat]ial by ocean curr5ts & meteorological causes4 ,if = any reason ! e>?'s c5tre ( gravity 7 s5sibly displac$1 mom5t\s geological *anges w\ld necess>ily ensue4 ,t ! c5tre ( gravity does n co9cide ) ! c5tre ( figure ( ! globe1 b lies to ! s\? ( x1 has long be5 kn[n4 ,?is great] aggregation ( d5se mat]ial 9 ! s\!rn hemisph]e probably dates f ! e>ly ages ( ! e>?'s consolidation1 & x is difficult to believe t any readju/m5t ( ! di/ribution ( ?is mat]ial 9 ! e>?'s int]ior is n[ possible4 ,b c]ta9 re>rangem5ts ( ! hydrosph]e on ! surface ( ! globe may1 f "t to "t1 cause a %ift+ ( ! c5tre ( gravity1 :i* w affect ! level ( ! ocean4 ,! accumulation ( enorm\s masses ( ice ar\nd ! pole w give rise to su* a displacem5t1 & w ?us increase ! body ( oceanic wat] 9 ! glaciat$ hemisph]e4 ,v>i\s calculations h be5 made ( ! effect ( ! transf]5ce ( ! ice-cap f "o pole to ! o!r1 a revolution :i* may possibly h occurr$ m ?an once 9 ! pa/ hi/ory ( ! globe4 ,james ,croll e/imat$ t if ! mass ( ice 9 ! s\!rn hemisph]e be assum$ to be #ajjj ft4 ?ick d[n to lat4 #fj deg41 its removal to ! opposite hemisph]e w\ld raise ! level ( ! sea #hj ft4 at ! nor? pole1 :ile ! ,rev4 ,osmond ,fi%] made ! rise z mu* z #dji ft4 ,! melt+ ( ! ice w\ld /ill fur!r raise ! sea-level by ! addition ( s l>ge a volume ( wat] to ! ocean4 ,to :at ext5t sup]ficial *anges ( ?is k9d h op]at$ 9 geological hi/ory rema9s an unsolv$ problem1 b _! probable occurr5ce 9 ! pa/ has to be recogniz$ z "o ( ! factors t mu/ be consid]$ 9 trac+ ! revolutions ( ! e>?'s surface4 ,! ,age ( ! ,e>?4--,intimately connect$ ) ! relations ( our globe to ! sun & ! o!r memb]s ( ! sol> sy/em is ! "q ( ! planet's antiquity--a subject ( great geological importance1 reg>d+ :i* mu* discussion has tak5 place s9ce ! middle ( ! #ai? c5tury4 ,?\gh an acc\nt ( ?is discussion necess>ily involves allusion to dep>tm5ts ( geology :i* are m appropriately ref]r$ to 9 lat] p>ts ( ?is article1 x may p]haps be mo/ conv5i5tly includ$ "h4 ,geologi/s 7 = _m ye>s 9 ! habit ( believ+ t no limit c\ld be assign$ to ! antiquity ( ! planet1 & t !y 7 at lib]ty to make unlimit$ drafts on ! ages ( ! pa/4 ,9 #ahfb & subsequ5t ye>s1 h[ev]1 ,"l ,kelv9 7!n ,sir ,william ,?omson7 po9t$ out t ^! dem&s 7 oppos$ to kn[n physical facts1 & t ! am\nt ( "t requir$ = geological hi/ory 0 n only limit$1 b mu/ h be5 compris$ )9 a comp>atively n>r[ compass4 ,8 argum5t re/$ on ?ree k9ds ( evid5ce3 7#a7 ! int]nal heat & rate ( cool+ ( ! e>?2 7#b7 ! tidal ret>dation ( ! e>?'s rotation2 & 7#c7 ! orig9 & age ( ! sun's heat4 #a4 ,apply+ ,f\ri]'s !ory ( !rmal conductivity1 ,"l ,kelv9 cont5d$ t 9 ! kn[n rate ( increase ( temp]ature d[nw>d & b5ea? ! surface1 & ! rate ( loss ( heat f ! e>?1 we h a limit to ! antiquity ( ! planet4 ,he %[$1 f ! data available at ! "t1 t ! sup]ficial consolidation ( ! globe c\ld n h occurr$ less ?an #bj million ye>s ago1 or ! und]gr\nd heat w\ld h be5 great] ?an x is2 nor m ?an #djj million ye>s ago1 o!rwise ! und]gr\nd temp]ature w\ld h %[n no s5sible increase d[nw>ds4 ,he admitt$ t v wide limits 7 necess>y4 ,9 subsequ5tly discuss+ ! subject1 he incl9$ r t[>ds ! l[] ?an ! high] antiquity1 b conclud$ t ! limit1 f a consid]ation ( all ! evid5ce1 mu/ be plac$ )9 "s su* p]iod ( pa/ "t z #ajj millions ( ye>s4 #b4 ,! argum5t f tidal ret>dation proce$s on ! admitt$ fact t1 ow+ to ! friction ( ! tide-wave1 ! rotation ( ! e>? is ret>d$1 & is1 !re=e1 mu* sl[] n[ ?an x mu/ h be5 at "o "t4 ,"l ,kelv9 affirm$ t _h ! globe become solid "s #aj1jjj million ye>s ago1 or inde$ any high antiquity beyond #ajj million ye>s1 ! c5trifugal =ce due to ! m rapid rotation mu/ h giv5 ! planet a v mu* great] pol> flatt5+ ?an x actually possesses4 ,he admitt$1 h[ev]1 t1 ?\gh #ajj million ye>s ago t =ce mu/ h be5 ab\t #c great] ?an n[1 yet no?+ we "k reg>d+ ! figure ( ! e>?1 & ! disposition ( l& & wat]1 w\ld ju/ify u 9 say+ t a body consolidat$ :5 !re 0 m c5trifugal =ce by #c ?an n[1 might n n[ be 9 all respects l ! e>?1 s f> z we "k x at pres5t4 #c4 ,! ?ird argum5t1 bas$ ^u ! age ( ! sun's heat1 is confess$ly less to be reli$ on ?an ! two previ\s ones4 ,x proce$s ^u calculations z to ! am\nt ( heat :i* w\ld be available by ! fall+ toge!r ( masses f space1 :i* gave rise by _! impact to our sun4 ,! vagu5ess ( ! data on :i* ?is argum5t re/s may be inf]r$ f ! fact t 9 "o passage ,p4,g4 ,tait plac$ ! limit ( "t dur+ :i* ! sun has be5 illum9at+ ! e>? z1 on ! v highe/ computation1 n m ?an ab\t #ae or #bj millions ( ye>s2 :ile1 9 ano!r s5t5ce ( ! same volume1 he admitt$ t1 by calculations 9 :i* !re is no possibility ( l>ge error1 ?is hypo!sis ( ! orig9 ( ! sun's heat by ! fall+ toge!r ( masses ( matt] is ?or\ghly compet5t to expla9 #ajj millions ( ye>s' sol> radiation at ! pres5t rate1 p]haps m4 ,9 m rec5tly review+ 8 argum5t1 ,"l ,kelv9 express$ himself 9 fav\r ( m /rictly limit+ geological "t ?an he _h at fir/ be5 dispos$ to d4 ,he insi/s t ! "t 0 m ?an #bj & less ?an #dj millions ( ye>s & probably mu* ne>] #bj ?an #dj4 ,geologi/s appe> to h reluctantly br\ght !mselves to believe t p]haps1 aft] all1 #ajj millions ( ye>s might suffice = ! evolution ( geological hi/ory4 ,b :5 ! "t 0 cut d[n to #ae or #bj millions !y prote/$ t su* a re/rict$ p]iod 0 insuffici5t = t evolution1 & ?\gh !y did n (f] any effective criticism ( ! argum5ts ( ! physici/s !y felt conv9c$ t !re mu/ be "s flaw 9 ! premises on :i* ^! argum5ts 7 bas$4 ,by degrees1 h[ev]1 !re h aris5 among ! physici/s !mselves grave d\bts z to ! validity ( ! physical evid5ce on :i* ! limitation ( ! e>?'s age has be5 f\nd$1 & at ! same "t great] appreciation has be5 %[n ( ! signification & /r5g? ( ! geological pro(s ( ! high antiquity ( our planet4 ,9 an address f ! *air ( ! ,ma!matical ,section ( ! ,briti% ,association 9 #ahhf1 ,pr(essor 7aft]w>ds ,sir7 ,george ,d>w9 review$ ! controv]sy1 & pron\nc$ ! foll[+ delib]ate judgm5t 9 reg>d to x3 ,9 consid]+ ^! ?ree argum5ts ,i h adduc$ "s reasons aga9/ ! validity ( ! fir/ tidal friction1 & h endeav\r$ to %[ t !re are elem5ts ( unc]ta9ty surr\nd+ ! second secul> cool+ ( ! e>?2 nev]!less1 !y und\bt$ly con/itute a contribution ( ! fir/ importance to physical geology4 ,:il/1 !n1 we may prote/ aga9/ ! precision ) :i* ,pr(essor ,tait seeks to d$uce results f !m1 we are fully ju/ifi$ 9 foll[+ ,sir ,william ,?omson1 :o says t '! exi/+ /ate ( ?+s on ! e>?1 life on ! e>?--all geological hi/ory %[+ cont9uity ( life--mu/ be limit$ )9 "s su* p]iod ( pa/ "t z #ajj million ye>s'4 ,"l ,kelv9 has nev] dealt ) ! geological & palaeontological objections aga9/ ! limitation ( geological "t to a few millions ( ye>s4 ,b ,pr(essor ,d>w91 9 ! address j cit$1 utt]$ ! memorable w>n+3 ,at pres5t our k ( a def9ite limit to geological "t has s little precision t we %\ld d wrong summ>ily to reject !ories :i* appe> to dem& long] p]iods ( "t ?an ^? :i* n[ appe> all[able4 ,9 8 presid5tial address to ! ,briti% ,association at ,cape ,t[n 9 #aije he return$ to ! subject1 rem>k+ t ! argum5t d]iv$ f ! increase ( und]gr\nd temp]ature seems to be entirely de/roy$ by ! discov]y ( ! prop]ties ( radium4 ,he ?9ks t x does n seem extravagant to suppose t #ejj to #ajjj million ye>s may h elaps$ s9ce ! bir? ( ! moon4 ,he has always believ$ t ! geologi/s 7 m ne>ly correct ?an ! physici/s1 not)/&+ ! fact t appe>ances 7 s /rongly aga9/ !m1 & he concludes ?us3 ,x appe>s1 !n1 t ! physical argum5t is n susceptible ( a great] degree ( c]ta9ty ?an t ( ! geologi/s1 & ! scale ( geological "t rema9s 9 great measure unkn[n 7see also ,tide1 *ap4 viii474 ,9 an address to ! ma!matical section ( ! ,am]ican ,association = ! ,advancem5t ( ,sci5ce 9 #ahhi1 ! vice-presid5t ( ! section1 ,r4,s4 ,woodw>d1 ?us express$ himself ) reg>d to ! physical argum5ts br\ght =w>d by ,"l ,kelv9 & ,pr(essor ,tait 9 limitation ( geological "t3 ,hav+ be5 at "s pa9s to look into ?is matt]1 ,i feel b\nd to /ate t1 al?\gh ! hypo!sis appe>s to be ! be/ :i* c be =mulat$ at pres5t1 ! odds are aga9/ its correctness4 ,its weak l9ks are ! unv]ifi$ assumptions ( an initial uni=m temp]ature & a con/ant diffusivity4 ,v likely ^! are approximations1 b ( :at ord] we _c decide4 ,fur!rmore1 if we accept ! hypo!sis1 ! odds appe> to be aga9/ ! pres5t atta9m5t ( tru/wor?y num]ical results1 s9ce ! data = calculation1 obta9$ mo/ly f obs]vations on cont95tal areas1 are f> too meagre to give satisfactory av]age values = ! entire mass ( ! e>?4 ,/ill m emphatic is ! prote/ made f ! physical side by ,pr(essor ,john ,p]ry4 ,he has attack$ ea* ( ! ?ree l9es ( argum5t ( ,"l ,kelv91 & has impugn$ ! validity ( ! conclusions drawn f !m4 ,! argum5t f tidal ret>dation he dismisses z fallaci\s1 foll[+ 9 ?is cont5tion ! previ\s criticism ( ! ,rev4 ,maxwell ,close & ,sir ,george ,d>w94 ,9 deal+ ) ! argum5t bas$ on ! secul> cool+ ( ! e>?1 he holds x to be p]fectly all[able to assume a mu* high] conductivity = ! int]ior ( ! globe1 & t su* a reasonable assumption w\ld enable u greatly to increase our e/imate ( ! e>?'s antiquity4 ,z = ! ?ird argum5t1 f ! age ( ! sun's heat1 he po9ts out t ! sun may h be5 repeat$ly f$ by a supply ( meteorites f outside1 :ile ! e>? may h be5 protect$ f radiation1 & be5 able to reta9 mu* ( its heat by be+ envelop$ 9 a d5se atmosph]e4 ,rem>k+ t almo/ any?+ is possible z to ! pres5t int]nal /ate ( ! e>?1 he concludes ?us3 ,to sum up1 we c f9d no publi%$ record ( any l[] maximum age ( life on ! e>?1 z calculat$ by physici/s1 ?an #djj millions ( ye>s4 ,f ! ?ree physical argum5ts ,"l ,kelv9's high] limits are #ajjj1 #djj & #ejj million ye>s4 ,i h %[n t we h reasons = believ+ t ! age1 f all ^!1 may be v consid]ably und]e/imat$4 ,x is to be obs]v$ t if we exclude ev]y?+ b ! argum5ts f m]e physics1 ! probable age ( life on ! e>? is mu* less ?an any ( ! above e/imates2 b if ! palaeontologi/s h good reasons = dem&+ mu* great] times1 ,i see no?+ f ! physici/s' po9t ( view :i* d5ies !m f\r times ! greate/ ( ^! e/imates4 ,a fre% l9e ( argum5t aga9/ ,"l ,kelv9's limitation ( ! antiquity ( our globe has rec5tly be5 />t$ by ! rem>kable discov]ies 9 radio-activity4 ,f ! asc]ta9$ prop]ties ( radium x appe>s to be possible t our e/imates ( sol> heat1 z d]iv$ f ! !ory ( gravitation1 may h to be augm5t$ t5 or tw5ty times2 t /ores ( radium & simil> bodies )9 ! e>? may h indef9itely def]r$ ! e/abli%m5t ( ! pres5t temp]ature gradi5t f ! surface inw>d2 t consequ5tly ! e>? may h rema9$ = long ages at a temp]ature n greatly diff]5t f t :i* x n[ possesses1 & h5ce t ! times dur+ :i* our globe has support$ animal & vegetable life may be v mu* long] ?an t all[$ 9 ! e/imates previ\sly made by physici/s f o!r data 7see ,,radioactivity74 ,! argum5ts f ! geological side aga9/ ! physical cont5tion t w\ld limit ! age ( our globe to "s #aj or #bj millions ( ye>s are ma9ly bas$ on ! obs]v$ rates ( geological & biological *anges at ! pres5t "t ^u l& & sea1 & on ! nature1 physical hi/ory & organic cont5ts ( ! /ratifi$ cru/ ( ! e>?4 ,un=tunately1 actual num]ical data are n obta9able 9 _m dep>tm5ts ( geological activity1 & ev5 :]e !y c be procur$ !y d n yet re/ on a suffici5tly wide collection ( accurate & co-ord9at$ obs]vations4 ,b 9 "s bran*es ( dynamical geology1 mat]ial exi/s =1 at lea/1 a prelim9>y computation ( ! rate ( *ange4 ,?is is m especially ! case 9 respect ( ! wide doma9 ( d5udation4 ,! obs]vational records ( ! action ( ! sea1 ( spr+s1 riv]s & glaci]s are becom+ gradually full] & m tru/wor?y4 ,a me?od ( mak+ use ( ^! records = e/imat+ ! rate ( d5udation ( ! l& has be5 devis$4 ,tak+ ! ,mississippi z a g5]al type ( riv] action1 x has be5 %[n t ! am\nt ( mat]ial convey$ by ?is /ream into ! sea 9 "o ye> is equival5t to ! l[]+ ( ! g5]al surface ( ! dra9age bas9 ( ! riv] by #a_/fjjj ( a foot4 ,?is w\ld am\nt to "o foot 9 #fjjj ye>s & #ajjj ft4 9 #f million ye>s4 ,s t at ! pres5t rate ( wa/e 9 ! ,mississippi bas9 a :ole cont95t might be worn away 9 a few millions ( ye>s4 ,x is evid5t t z deposition & d5udation are simultane\s processes1 ! asc]ta9m5t ( ! rate at :i* solid mat]ial is remov$ f ! surface ( ! l& supplies "s necess>y in=mation = e/imat+ ! rate at :i* new s$im5t>y =mations are be+ accumulat$ on ! floor ( ! sea1 & = a computation ( ! l5g? ( "t t w\ld be requir$ at ! pres5t rate ( *ange = ! deposition ( all ! /ratifi$ rocks t ent] into ! composition ( ! cru/ ( our globe4 ,if ! ?ickness ( ^! rocks be assum$ to be #ajj1jjj ft41 & if we c\ld suppose !m to h be5 laid d[n ov] z wide an area z t ( ! dra9age bas9s f ! wa/e ( :i* !y 7 d]iv$1 !n at ! pres5t rate ( d5udation _! accumulation w\ld require "s #fjj millions ( ye>s4 ,b1 z ,dr ,a4,r4 ,wallace has ju/ly po9t$ out1 ! tract ( sea-floor ov] :i* ! mat]ial d]iv$ f ! wa/e ( ! t]re/rial surface is laid d[n is at pres5t mu* less ?an t f :i* ?is mat]ial is worn away4 ,we h no means1 h[ev]1 ( det]m9+ :at may h be5 ! ratio betwe5 ! two areas 9 pa/ "t4 ,c]ta9ly anci5t m>9e s$im5t>y rocks cov] at ! pres5t day a mu* m ext5sive area ?an t 9 :i* !y are n[ be+ elaborat$4 ,if we take ! ratio po/ulat$ by ,dr ,wallace--#a to #ai--! #ajj1jjj ft4 ( s$im5t>y /rata w\ld require #ca millions ( ye>s = _! accumulation4 ,x is q possible1 h[ev]1 t ?is ratio may be mu* too high4 ,!re are reasons = believ+ t ! proportion ( coa/-l9e to l& area has be5 dim9i%+ dur+ geological "t2 9 o!r words1 t 9 e>ly times ! l& 0 m insul> & is n[ m cont95tal4 ,s t ! #ca millions ( ye>s may be mu* less ?an ! p]iod t w\ld be requir$1 ev5 on ! supposition ( cont9u\s un9t]rupt$ d5udation & s$im5tation1 dur+ ! :ole ( ! "t repres5t$ by ! /ratifi$ =mations4 ,b no "o :o has made himself famili> ) ! actual composition ( ^! =mations & ! detail$ /ructure ( ! t]re/rial cru/ c fail to recognize h[ vague1 imp]fect & mislead+ are ! data on :i* su* computations are f\nd$4 ,x requires no prolong$ acqua9tance ) ! e>?'s cru/ to impress ^u ! m9d t "o all-important elem5t is omitt$1 & inde$ c h>dly be all[$ = f want ( suffici5tly precise data1 b ! neglect ( :i* mu/ ne$s s]i\sly impair ! value ( all num]ical calculations made )\t x4 ,! assumption t ! /ratifi$ =mations c be treat$ z if !y consi/$ ( a cont9u\s unbrok5 sequ5ce ( s$im5ts1 indicat+ a va/ & un9t]rupt$ process ( wa/e & deposition1 is "o t is beli$ on e h& by ! actual /ructure ( ^! =mations4 ,x c only give u a m9imum ( ! "t requir$2 =1 in/ead ( an unbrok5 s]ies1 ! s$im5t>y =mations are full ( uncon=mabilities--gaps 9 ! sequ5ce ( ! *ronological records--z if :ole *apt]s & gr\ps ( *apt]s _h be5 torn out ( a hi/orical "w4 ,x c (t5 be %[n t ^! breaks ( cont9uity mu/ h be5 ( va/ duration1 & actually exce$$ 9 *ronological importance ?ick gr\ps ( /rata ly+ bel[ & above !m 7see ,"p ,,vi474 ,moreov]1 ev5 among ! un9t]rupt$ /rata1 :]e no su* uncon=mabilities exi/1 b :]e ! s$im5ts foll[ ea* o!r 9 app>5tly un9t]rupt$ sequ5ce1 & might be ?\ght to h be5 deposit$ cont9u\sly at ! same g5]al rate1 & )\t ! int]v5tion ( any pause1 x c be demon/rat$ t sometimes an in* or two ( s$im5t might1 on c]ta9 horizons1 repres5t ! deposit ( an enorm\sly long] p]iod ?an a hundr$ or a ?\s& times ! same am\nt ( s$im5t on o!r horizons4 ,a prolong$ /udy ( ^! que/ions leads to a pr(\nd conviction t 9 _m p>ts ( ! geological record ! "t repres5t$ by s$im5t>y deposits may be va/ly less ?an ! "t :i* is n s repres5t$4 ,x has (t5 be5 object$ t ! pres5t rate ( geological *ange ought n to be tak5 z a measure ( ! rate 9 pa/ "t1 because ! total sum ( t]re/rial en]gy has be5 /eadily dim9i%+1 & geological processes mu/ consequ5tly h be5 m vigor\s 9 =m] ages ?an !y are n[4 ,geologi/s d n pret5d to ass]t t !re has be5 no v>iation or dim9ution 9 ! activities ( ! v>i\s processes :i* !y h to /udy4 ,:at !y d insi/ on is t ! pres5t rate ( *ange is ! only "o :i* we c wat* & measure1 & :i* w ?us supply a /ati/ical basis = any computations on ! subject4 ,b x has be5 dogmatically affirm$ t because t]re/rial en]gy has be5 dim9i%+ !re=e all k9ds ( geological "w mu/ h be5 m vigor\sly & m rapidly c>ri$ on 9 =m] times ?an n[2 t !re 7 f> m abundant & m /up5d\s volcanoes1 m frequ5t & m de/ructive e>?quakes1 m gigantic upheavals & subsid5ces1 m p[]ful oceanic waves & tides1 m viol5t atmosph]ic di/urbances ) heavi] ra9fall & m active d5udation4 ,x is easy to make ^! ass]tions1 & !y look plausible2 b1 aft] all1 !y re/ on no?+ /rong] ?an assumption4 ,!y c be te/$ by an appeal to ! cru/ ( ! e>?1 9 :i* ! geological hi/ory ( our planet has be5 s fully record$4 ,_h su* port5t\s manife/ations ( geological activity ev] be5 ! normal condition ( ?+s s9ce ! beg9n+ ( t hi/ory1 !re ought to be a record ( !m 9 ! rocks4 ,b no evid5ce = !m has be5 f\nd !re1 ?\gh x has be5 dilig5tly s\ght = 9 all qu>t]s ( ! globe4 ,we may confid5tly ass]t t :ile geological *anges may q possibly h tak5 place on a gigantic scale 9 ! e>lie/ ages ( ! e>?'s exi/5ce1 ( :i* no geological record rema9s1 !re is no pro( t !y h ev] done s s9ce ! "t :5 ! v olde/ ( ! /ratifi$ =mations 7 deposit$4 ,!re is no ne$ to ma9ta9 t !y h always be5 conduct$ precisely on ! same scale z n[1 or to d5y t !y may h gradually become less vigor\s z ! g5]al sum ( t]re/rial en]gy has dim9i%$4 ,b we may unhesitat+ly affirm t no actual evid5ce ( any su* progressive dim9ution ( activity has be5 adduc$ f ! geological record 9 ! cru/ ( ! e>?3 t1 on ! contr>y1 no appe>ances h be5 detect$ !re :i* necess>ily dem& ! assumption ( ^? m p[]ful op]ations po/ulat$ by physici/s1 or :i* are n satisfactorily explicable by ref]5ce to ! exi/+ scale ( nature's processes4 ,t ?is conclusion is w>rant$ ev5 ) reg>d to ! innate en]gy ( ! globe itself w be se5 if we in/itute a comp>ison betwe5 ! m anci5t & ! m rec5t manife/ations ( t en]gy4 ,take1 = example1 ! pro(s ( gigantic plication1 fracture & displacem5t )9 ! t]re/rial cru/4 ,^!1 z !y h affect$ ! mo/ anci5t rocks ( ,europe1 h be5 work$ out 9 great detail 9 ! nor?-we/ ( ,scotl&4 ,b !y are n ess5tially diff]5t f or on a great] scale ?an ^? :i* h be5 prov$ to h affect$ ! ,alps1 & to h involv$ /rata ( s rec5t a date z ! old] ,t]ti>y =mations4 ,on ! contr>y1 x may be d\bt$ :e!r any d5ud$ core ( an anci5t m\nta9-*a9 reveals traces ( su* /up5d\s di/urbances ( ! cru/ z ^? :i* h giv5 rise to ! y\ng] m\nta9-*a9s ( ! globe4 ,x may1 inde$1 q well h be5 ! rule t in/ead ( dim9i%+ 9 int5sity ( effect1 ! consequ5ces ( t]re/rial contraction h increas$ 9 magnitude1 ! augm5t+ ?ickness ( ! cru/ (f]+ great] resi/ance to ! /resses1 & giv+ rise to va/] plications1 faults1 ?ru/-planes & metamorphism1 z ?is gr[+ resi/ance _h to be ov]come4 ,! ass]tion t volcanic action mu/ h be5 m viol5t & m p]si/5t 9 anci5t times ?an x is n[ has assur$ly no geological evid5ce 9 its support4 ,x is q true t !re are va/ly m rema9s ( =m] volcanoes scatt]$ ov] ! surface ( ! globe ?an !re are active crat]s n[1 & t traces ( copi\s eruptions ( volcanic mat]ial c be foll[$ back into "s ( ! olde/ p>ts ( ! geological record4 ,b we h no pro( t ev] at any "o "t 9 geological hi/ory !re h be5 m or l>g] or m vigor\s volcanoes ?an ^? ( rec5t p]iods4 ,x may be said t ! abs5ce ( su* pro( ought n to invalidate ! ass]tion until a f> wid] area ( ! e>?'s surface has be5 geologically /udi$4 ,b mo/ assur$ly1 z f> z geological inve/igation has yet gone1 !re is an ov]:elm+ body ( evid5ce to %[ t f ! e>lie/ epo*s 9 geological hi/ory1 z regi/]$ 9 ! /ratifi$ rocks1 volcanic action has manife/$ itself v mu* z x does n[1 b on a less r ?an on a great] scale4 ,n[h]e c ?is subject be m exhau/ively /udi$ ?an 9 ! ,briti% ,isles1 :]e a rem>kably complete s]ies ( volcanic eruptions has be5 *ronicl$ rang+ f ! e>lie/ ,palaeozoic d[n to old] ,t]ti>y "t4 ,! result ( a prolong$ /udy ( ,briti% volcanic geology has demon/rat$ t1 ev5 to m9ute po9ts ( detail1 !re has be5 a s+ul> uni=mity 9 ! ph5om5a f beg9n+ to end4 ,! olde/ lavas & a%es diff] 9 no ess5tial respect f ! y\nge/4 ,nor h !y be5 erupt$ m copi\sly or m frequ5tly4 ,_m successive volcanic p]iods h foll[$ ea* o!r aft] prolong$ int]vals ( repose1 ea* display+ ! same g5]al sequ5ce ( ph5om5a & simil> evid5ce ( gradual dim9ution & ext9ction4 ,! y\nge/1 in/ead ( be+ ! feeble/1 7 ! mo/ ext5sive outbur/s 9 ! :ole ( ?is prolong$ s]ies4 ,if n[ we turn = evid5ce ( ! alleg$ great] activity ( all ! epig5e or sup]ficial =ces1 & especially = pro(s ( m rapid d5udation & deposition on ! e>?'s surface1 we se>* = x 9 va9 among ! /ratifi$ =mations ( ! t]re/rial cru/4 ,_h ! olde/ ( ^! rocks be5 accumulat$ 9 a "t ( great atmosph]ic p]turbation1 ( torr5tial ra9s1 colossal tides & viol5t /orms1 we might surely expect to f9d among ! s$im5ts "s pro( ( su* di/urb$ meteorological & geographical conditions4 ,we %\ld look1 on ! "o h&1 = tumultu\s accumulations ( co>se unworn detritus1 rapidly swept by ra9s1 floods & waves f l& to sea1 & on ! o!r h&1 = an abs5ce ( any evid5ce ( ! tranquil & cont9u\s deposit ( su* f9e lam9at$ silt z c\ld only settle 9 quiet wat]4 ,b an appeal to ! geological record is made 9 va9 = any su* pro(s4 ,! olde/ s$im5ts1 l ! y\nge/1 reveal ! op]ation only ( su* ag5ts & su* rates ( activity z are /ill to be witness$ 9 ! accumulation ( ! same k9d ( deposits4 ,if1 = in/ance1 we se>* ! mo/ anci5t ?ick s$im5t>y =mation 9 ,brita9--! ,torridon ,s&/one ( nor?-we/ ,scotl&1 :i* is old] ?an ! olde/ fossilif]\s deposits--we meet ) no?+ :i* might n be f\nd 9 any ,palaeozoic1 ,mesozoic or ,ca9ozoic gr\p ( simil> s$im5ts4 ,we see an accumulation1 at lea/ #hjjj or #aj1jjj ft4 ?ick1 ( consolidat$ s&1 gravel & mud1 su* z may be ga!r+ n[ on ! floor ( any l>ge m\nta9-girdl$ lake4 ,! conglom]ates ( ?is anci5t s]ies are n pell-mell heaps ( angul> detritus1 viol5tly swept away f ! l& & huddl$ promiscu\sly on ! sea-floor4 ,!y are1 9 g5]al1 built up ( pebbles t h be5 worn smoo?1 r\nd$ & poli%$ by prolong$ attrition 9 runn+ wat]1 & !y foll[ ea* o!r on successive plat=ms ) int]v5+ lay]s ( f9] s$im5t4 ,! s&/ones are compos$ ( well wat]-worn s&1 "s ( :i* has be5 laid d[n s tranquilly t its compon5t gra9s h be5 sep>at$ out 9 lay]s accord+ to _! specific gravity1 9 su* mann] t !y n[ pres5t d>k lam9ae 9 :i* p>ticles ( magnetic iron1 zircon & o!r heavy m9]als h be5 sift$ out toge!r1 j z iron-s& may be se5 ga!r$ into ?9 %eets on s&y bea*es at ! pres5t day4 ,aga91 ! same s]ies ( primeval s$im5ts includes int]calations ( f9e silt1 :i* has be5 deposit$ z regul>ly & int]mitt5tly !re z x has be5 among ! mo/ rec5t =mations4 ,^! b&s ( %ale h be5 dilig5tly se>*$ = fossils1 z yet )\t success2 b !y may ev5tually disclose organic rema9s old] ?an any hi!rto f\nd 9 ,europe4 ,we n[ come to ! consid]ation ( ! palaeontological evid5ce z to ! value ( geological "t4 ,"h ! conclusions d]iv$ f a /udy ( ! /ructure ( ! s$im5t>y =mations are va/ly /r5g!n$ & ext5d$4 ,9 ! fir/ place1 ! organization ( ! mo/ anci5t plants & animals furni%es no indication t !y _h to cont5d ) any great] viol5ce ( /orm1 flood1 wave or ocean-curr5t ?an is famili> to _! mod]n desc5dants4 ,! olde/ trees1 %rubs1 f]ns & club-mosses display no special /ructures t sugge/ a diff]5ce 9 ! g5]al conditions ( _! environm5t4 ,! mo/ anci5t cr9oids1 sponges1 cru/aceans1 ara*nids & molluscs 7 z delicately con/ruct$ z ^? ( to-day1 & _! rema9s are (t5 f\nd 9 su* p]fect pres]vation z to %[ t nei!r dur+ _! lifetime nor aft] _! dea? 7 !y subject to any great] viol5ce ( ! elem5ts ?an _! liv+ repres5tatives n[ exp]i5ce4 ,( mu* m cog5cy1 h[ev]1 is ! evid5ce suppli$ by ! gr& upw>d succession ( organic =ms1 f ! mo/ anci5t /ratifi$ rocks up to ! pres5t day4 ,no biologi/ n[ d\bts = a mom5t t ?is m>vell\s succession is ! result ( a gradual process ( evolution f l[] to high] types ( organization4 ,!re may be diff]5ces ( op9ion z to ! causes :i* h gov]n$ ?is process & ! ord] ( ! /eps ?r\gh :i* x has advanc$1 b no "o :o is conv]sant ) ! facts w n[ v5ture to d5y t x has tak5 place1 & t1 on any possible explanation ( its progress1 x mu/ h dem&$ an enorm\s lapse ( "t4 ,9 ! ,cambrian or olde/ fossilif]\s =mations !re is already a l>ge & v>i$ fauna1 9 :i* ! lead+ gr\ps ( inv]tebrate life are repres5t$4 ,on no t5able hypo!sis c ^! be reg>d$ z ! fir/ organisms t came into be+ on our planet4 ,!y mu/ h _h a long ance/ry1 & z ,d>w9 fir/ ma9ta9$1 ! "t requir$ = _! evolution may h be5 z long z1 or probably f> long] ?an1 ! :ole int]val f ! ,silurian ,cambrian age to ! pres5t day4 ,! records ( ^! e>lie/ eras ( organic developm5t h un=tunately n surviv$ ! geological revolutions ( ! pa/2 at lea/1 !y h n yet be5 recov]$4 ,b x _c be d\bt$ t !y once exi/$ & regi/]$ _! te/imony to ! prodigi\s lapse ( "t prior to ! deposition ( ! mo/ anci5t fossilif]\s =mations :i* h escap$ de/ruction4 ,! impressive "* ( ! evid5ce furni%$ by ! sequ5ce ( organic =ms ?r\gh\t ! great s]ies ( fossilif]\s /rata c h>dly be fully realiz$ )\t a detail$ & c>eful /udy ( ! subject4 ,pr(essor ,e4,b4 ,p\lton1 9 an address to ! zoological section ( ! ,briti% ,association at ! ,liv]pool ,meet+ 9 #ahif1 %[$ h[ ov]:elm+ are ! dem&s :i* ?is evid5ce makes = long p]iods ( "t1 & h[ impossible x is ( compreh5sion unless ^! dem&s be conc$$4 ,! hi/ory ( life ^u ! e>?1 ?\gh x w probably always be surr\nd$ ) great & ev5 insup]able difficulties1 becomes broadly compreh5sible 9 its g5]al progress :5 suffici5t "t is grant$ = ! evolution :i* x records2 b x rema9s un9telligible on any o!r conditions4 ,tak5 !n z a :ole1 ! body ( evid5ce1 geological & palaeontological1 9 fav\r ( ! high antiquity ( our globe is s great1 s manifold1 & bas$ on su* an ev]-increas+ bread? ( obs]vation & reflection1 t x may be confid5tly appeal$ to 9 answ] to ! physical argum5ts :i* w\ld seek to limit t antiquity to t5 or tw5ty millions ( ye>s4 ,9 ! pres5t /ate ( sci5ce x is out ( our p[] to /ate positively :at mu/ be ! l[e/ limit ( ! age ( ! e>?4 ,b we _c assume x to be mu* less1 & x may possibly h be5 mu* m1 ?an ! #ajj millions ( ye>s :i* ,"l ,kelv9 0 at "o "t will+ to conc$e4#b ,,"p ,,iii4--,,geognosy4 ,,! ,,inve/igation ,,( ,,! ,,nature ,,& ,,composition ,,( ,,! ,,mat]ials ,,( ,,:i* ,,! ,,e>? ,,consi/s ,?is division ( ! sci5ce is devot$ to a description ( ! p>ts ( ! e>?--( ! atmosph]e & ocean t surr\nd ! planet1 & m especially ( ! solid mat]ials t und]lie ^! envelopes & ext5d d[nw>ds to an unkn[n di/ance into ! int]ior4 ,^! v>i\s con/itu5ts ( ! globe are "h consid]$ z =ms ( matt] capable ( be+ analys$1 & arrang$ accord+ to _! composition & ! place !y take 9 ! g5]al composition ( ! globe4 ,view$ 9 ! simple/ way ! e>? may be reg>d$ z made up ( ?ree di/9ct p>ts1 ea* ( :i* ev] s9ce an e>ly p]iod ( planet>y hi/ory has be5 ! !atre ( important geological op]ations4 7#a7 ,an envelope ( air1 t]m$ ! atmosph]e1 :i* surr\nds ! :ole globe2 7#b7 ,a l[] & less ext5sive envelope ( wat]1 kn[n z ! hydrosph]e 7,gr4 ,greek3 hydor1 wat]7 :i*1 con/itut+ ! oceans & seas1 cov]s ne>ly ?ree-f\r?s ( ! und]ly+ solid surface ( ! planet2 7#c7 ,a globe1 call$ ! li?osph]e 7,gr4 ,greek3 li?os1 /one71 ! ext]nal "p ( :i*1 consi/+ ( solid /one1 =ms ! cru/1 :ile und]nea?1 & =m+ ! va/ mass ( ! int]ior1 lies ! nucleus1 reg>d+ ! true con/itution ( :i* we are /ill ignorant4 #a4 ,! ,atmosph]e4--,! g5]al *>act]s ( ! atmosph]e are describ$ 9 sep>ate articles 7see especially ,,atmosph]e2 ,,meteorology74 ,only its relations to geology h "h to be consid]$4 ,z ?is gase\s envelope encircles ! :ole globe x is ! mo/ univ]sally pres5t & active ( all ! ag5ts ( geological *ange4 ,its efficacy 9 ?is respect arises p>tly f its composition1 & ! *emical reactions :i* x effects ^u ! surface ( ! l&1 p>tly f its great v>iations 9 temp]ature & moi/ure1 & p>tly f its movem5ts4 ,_m speculations h be5 made reg>d+ ! *emical composition ( ! atmosph]e dur+ =m] geological p]iods4 ,!re c inde$ be little d\bt t x mu/ orig9ally h diff]$ greatly f its pres5t condition4 ,if ! :ole mass ( ! planet orig9ally exi/$ 9 a gase\s /ate1 !re w\ld be practically no atmosph]e4 ,! pres5t out] envelope ( air may be consid]$ to be ! surviv+ relic ( ?is condition1 aft] all ! o!r con/itu5ts h be5 incorporat$ into ! hydrosph]e & li?osph]e4 ,! oxyg51 :i* n[ =ms fully a half ( ! out] cru/ ( ! e>?1 0 d\btless orig9ally1 :e!r free or 9 comb9ation1 "p ( ! atmosph]e4 ,s1 too1 ! va/ b$s ( coal f\nd all ov] ! _w1 9 geological =mations ( _m diff]5t ages1 repres5t s mu* c>bonic acid once pres5t 9 ! air4 ,! *lorides & o!r salts 9 ! sea may likewise p>tly repres5t mat]ials c>ri$ d[n out ( ! atmosph]e 9 ! primitive cond5sation ( ! aque\s vap\r1 ?\gh !y h be5 cont9ually increas$ ev] s9ce by contributions f ! dra9age ( ! l&4 ,x has (t5 be5 sugge/$ t1 dur+ ! ,c>bonif]\s p]iod1 ! atmosph]e mu/ h be5 w>m] & m *>g$ ) aque\s vap\r & c>bon dioxide ?an at ! pres5t day1 to admit ( s luxuriant a flora z t f :i* ! coal-seams 7 =m$4 ,!re seems1 h[ev]1 to be at pres5t no me?od ( arriv+ at any c]ta9ty on ?is subject4 ,la/ly1 ! am\nt ( c>bonic acid absorb$ 9 ! wea!r+ ( rocks at ! surface1 & ! consequ5t production ( c>bonates1 repres5ts an enorm\s ab/raction ( ?is gas4 ,z at pres5t con/itut$1 ! atmosph]e is reg>d$ z a me*anical mixture ( ne>ly f\r volumes ( nitrog5 & "o ( oxyg51 toge!r ) an av]age ( #c4e p>ts ( c>bon dioxide 9 e #aj1jjj p>ts ( air1 & m9ute quantities ( v>i\s o!r gases & solid p>ticles4 ,( ! vap\rs conta9$ 9 x by f> ! mo/ important is t ( wat] :i*1 al?\gh always pres5t1 v>ies greatly 9 am\nt accord+ to v>iations 9 temp]ature4 ,by cond5sation ! wat] vap\r appe>s 9 visible =m z dew1 mi/1 cl\d1 ra91 hail1 sn[ & ice1 & 9 ^! =ms includes & c>ries d[n "s ( ! o!r vap\rs1 gases & solid p>ticles pres5t 9 ! air4 ,! circulation ( wat] f ! atmosph]e to ! l&1 f ! l& to ! sea1 & aga9 f ! sea to ! l&1 =ms ! great geological process :]eby ! habitable condition ( ! planet is ma9ta9$ & ! surface ( ! l& is sculptur$ 7,"p ,,iv474 #b4 ,! ,hydrosph]e4--,! wat] envelope cov]s ne>ly ?ree-f\r?s ( ! surface ( ! e>?1 & =ms ! v>i\s oceans & seas :i*1 ?\gh = conv5i5ce ( ref]5ce di/+ui%$ by sep>ate names1 are all l9k$ toge!r 9 "o great body4 ,! physical *>act]s ( ?is va/ envelope are discuss$ 9 sep>ate articles 7see ,,ocean & ,,oceanography74 ,view$ f ! geological /&po9t1 ! features ( ! sea t specially des]ve att5tion are fir/ ! composition ( its wat]s1 & secondly its movem5ts4 ,sea-wat] is di/+ui%$ f t ( ord9>y lakes & riv]s by its great] specific gravity & its sal9e ta/e4 ,its av]age d5sity is ab\t #a4jbf1 b x v>ies ev5 )9 ! same ocean1 be+ lea/ :]e l>ge quantities ( fre% wat] are add$ f ra9 or melt+ sn[ & ice1 & greate/ :]e evaporation is mo/ active4 ,t sea-wat] is heavi] ?an fre% arises f ! great] proportion ( salts :i* x conta9s 9 solution4 ,^! salts con/itute ab\t ?ree & a half p>ts 9 e hundr$ ( wat]4 ,!y consi/ ma9ly ( *lorides ( sodium & magnesium1 ! sulphates ( magnesium1 calcium & potassium1 ) m9ut] quantities ( magnesium bromide & calcium c>bonate4 ,/ill small] proportions ( o!r sub/ances h be5 detect$1 gold = example hav+ be5 f\nd 9 ! proportion ( #a "p 9 #ae1ahj1jjj4 ,t _m ( ! salts h exi/$ 9 ! sea f ! "t ( its fir/ cond5sation out ( ! primeval atmosph]e appe>s to be probable4 ,x is manife/1 h[ev]1 t1 :atev] may h be5 ! orig9al composition ( ! oceans1 !y h = a va/ section ( geological "t be5 con/antly receiv+ m9]al matt] 9 solution f ! l&4 ,e spr+1 brook & riv] removes v>i\s salts f ! rocks ov] :i* x moves1 & ^! sub/ances1 ?us dissolv$1 ev5tually f9d _! way into ! sea4 ,consequ5tly sea-wat] ought to conta9 m or less traceable proportions ( e sub/ance :i* ! t]re/rial wat]s c remove f ! l&1 9 %ort1 ( probably e elem5t pres5t 9 ! out] %ell ( ! globe1 = !re seems to be no con/itu5t ( ?is e>? :i* may n1 "u c]ta9 circum/ances1 be held 9 solution 9 wat]4 ,moreov]1 unless !re be "s c\nt]act+ process to remove ^! m9]al +r$i5ts1 ! ocean wat] ought to be gr[+1 ins5sibly p]haps1 b /ill assur$ly1 salti]1 = ! supply ( sal9e matt] f ! l& is incessant4 ,to ! geologi/ ! pres5ce ( m9]al solutions 9 sea-wat] is a fact ( mu* importance1 = x expla9s ! orig9 ( a consid]able "p ( ! /ratifi$ rocks ( ! e>?'s cru/4 ,by evaporation ! wat] has giv5 rise to deposits ( rock-salt1 gypsum & o!r mat]ials4 ,! lime conta9$ 9 solution1 :e!r z sulphate or c>bonate1 has be5 extract$ by _m tribes ( m>9e animals1 :i* h ?us built up out ( _! rema9s va/ masses ( solid lime/one1 ( :i* _m m\nta9-*a9s l>gely consi/4 ,ano!r important geological feature ( ! sea is to be se5 9 ! fact t its bas9s =m ! great receptacles = ! detritus worn away f ! l&4 ,besides ! lime/ones1 ! visible p>ts ( ! t]re/rial cru/ are1 9 l>ge measure1 compos$ ( s$im5t>y rocks :i* 7 orig9ally laid d[n on ! sea-bottom4 ,moreov]1 by its v>i\s movem5ts1 ! sea occupies a prom95t place among ! epig5e or sup]ficial ag5ts :i* produce geological *anges on ! surface ( ! globe4 #c4 ,! ,li?osph]e4--,b5ea? ! gase\s & liquid envelopes lies ! solid "p ( ! planet1 :i* is conv5i5tly reg>d$ z consi/+ ( two p>ts1--7a7 ! cru/1 & 7b7 ! int]ior or nucleus4 ,! cru/4 ,x 0 = a long "t a preval5t belief t ! int]ior ( ! globe is a molt5 mass r\nd :i* an out] %ell has gradually =m$ ?r\gh cool+4 ,h5ce ! t]m cru/ 0 appli$ to ?is ext]nal solid envelope1 :i* 0 v>i\sly comput$ to be #aj1 #bj1 or m miles 9 ?ickness4 ,! portion ( ?is cru/ accessible to human obs]vation 0 se5 to af=d abundant evid5ce ( va/ plications & corrugations ( its sub/ance1 :i* 7 reg>d$ z only explicable on ! supposition ( a ?9 solid collapsible %ell float+ on a d5s] liquid int]ior4 ,:51 h[ev]1 physical argum5ts 7 adduc$ to %[ ! great rigidity ( ! e>? z a :ole1 ! idea ( a ?9 cru/ enclos+ a molt5 nucleus 0 reluctantly ab&on$ by geologi/s1 :o f\nd ! problem ( ! e>?'s int]ior to be incapable ( solution by any evid5ce :i* _! sci5ce c\ld produce4 ,!y cont9u$1 h[ev]1 to use ! t]m cru/ z a conv5i5t ^w to d5ote ! cool out] lay] ( ! e>?'s mass1 ! /ructure & hi/ory ( :i* =m ! ma9 subjects ( geological inve/igation4 ,m rec5tly1 h[ev]1 v>i\s l9es ( rese>* h concurr$ 9 sugge/+ t1 :atev] may be ! condition ( ! int]ior1 its sub/ance mu/ diff] greatly f t ( ! out] %ell1 & t !re may be m reason ?an appe>$ = ! ret5tion ( ! "n ( cru/4 ,obs]vations on e>?quake motion by ,dr ,john ,milne & o!rs1 %[ t ! rate & "* ( ! waves transmitt$ ?r\gh ! int]ior ( ! e>? diff] 9 a m>k$ degree f ^? propagat$ along ! cru/4 ,?is diff]5ce indicates t rocky mat]ial1 su* z we "k at ! surface1 may ext5d inw>ds = "s #cj m41 bel[ :i* ! e>?'s int]ior rapidly becomes fairly homog5e\s & possesses a high rigidity4 ,f measurem5ts ( ! =ce ( gravity 9 ,india by ,colonel ,s4,g4 ,burr>d1 x has be5 inf]r$ t ! v>iations 9 d5sity ( ! out] p>ts ( ! e>? d n desc5d f>!r ?an #cj or #dj m41 :i* might be assum$ to be ! limit ( ! ?ickness ( ! cru/4 ,rec5t rese>*es 9 reg>d to ! radio-active sub/ances pres5t 9 rocks sugge/ t ! cru/ is n m ?an #ej m4 ?ick1 & t ! int]ior diff]s f x 9 possess+ little or no radio-active mat]ial4 ,! int]ior4 ,?\gh we _c hope ev] to h direct acqua9tance ) m ?an ! m]e outside sk9 ( our planet1 we may be l$ to inf] ! irregul> di/ribution ( mat]ials )9 ! cru/ f ! pres5t di/ribution ( l& & wat]1 & ! obs]v$ diff]5ces 9 ! am\nt ( deflection ( ! plumb-l9e ne> ! sea & ne> m\nta9-*a9s4 ,! fact t ! s\!rn hemisph]e is almo/ :olly cov]$ ) wat] appe>s explicable only on ! assumption ( an excess ( d5sity 9 ! mass ( t portion ( ! planet4 ,! exi/5ce ( su* a va/ %eet ( wat] z t ( ! ,pacific ,ocean is to be acc\nt$ =1 z ,ar*deacon ,j4,h4 ,pratt po9t$ out1 by ! pres5ce ( "s excess ( matt] 9 ! solid p>ts ( ! e>? betwe5 ! ,pacific ,ocean & ! e>?'s c5tre1 :i* reta9s ! wat] 9 its place1 o!rwise ! ocean w\ld fl[ away to ! o!r p>ts ( ! e>?4 ,a deflection ( ! plumb-l9e t[>ds ! sea1 :i* has 9 a numb] ( cases be5 obs]v$1 indicates t ! d5sity ( ! cru/ b5ea? ! m\nta9s mu/ be less ?an t bel[ ! pla9s1 & /ill less ?an t bel[ ! ocean-b$4 ,ap>t !re=e f ! depression ( ! e>?'s surface 9 :i* ! oceans lie1 we mu/ reg>d ! int]nal d5sity1 :e!r ( cru/ or nucleus1 to be some:at irregul>ly arrang$1 !re be+ an excess ( heavy mat]ials 9 ! wat] hemisph]e1 & b5ea? ! ocean-b$s1 z comp>$ ) ! cont95tal masses4 ,9 our ignorance reg>d+ ! *emical con/itution ( ! nucleus ( our planet1 an argum5t has sometimes be5 bas$ ^u ! kn[n fact t ! specific gravity ( ! globe z a :ole is ab\t d\ble t ( ! cru/4 ,?is has be5 held by "s writ]s to prove t ! int]ior mu/ consi/ ( mu* heavi] mat]ial & is !re=e probably metallic4 ,b ! effect ( pressure ought to make ! d5sity ( ! nucleus mu* high]1 ev5 if ! int]ior consi/$ ( matt] no heavi] ?an ! cru/4 ,t ! total d5sity ( ! planet does n greatly exce$ its obs]v$ am\nt seems only explicable on ! supposition t "s antagoni/ic =ce c\nt]acts ! effects ( pressure4 ,! only =ce we c suppose capable ( s act+ is heat4 ,b comp>atively little is yet kn[n reg>d+ ! compression ( gases1 liquids & solids "u su* va/ pressures z mu/ exi/ )9 ! nucleus4 ,t ! int]ior ( ! e>? possesses a high temp]ature is inf]r$ f ! evid5ce ( v>i\s s\rces4 7#a7 ,volcanoes1 :i* are op5+s t con/antly1 or int]mitt5tly1 give out hot vap\rs & molt5 lava f res]voirs b5ea? ! cru/4 ,besides active volcanoes1 x is kn[n t =m] eruptive v5ts h be5 abundantly & widely di/ribut$ ov] ! globe f ! e>lie/ geological p]iods d[n to our own day4 7#b7 ,hot spr+s are f\nd 9 _m p>ts ( ! globe1 ) temp]atures v>y+ up to ! boil+ po9t ( wat]4 7#c7 ,f m9es1 tunnels & deep bor+s into ! e>? x has be5 asc]ta9$ t 9 all qu>t]s ( ! globe bel[ ! sup]ficial zone ( inv>iable temp]ature1 !re is a progressive increase ( heat t[>ds ! int]ior4 ,! rate ( ?is increase v>ies1 be+ influ5c$1 among o!r causes1 by ! v>y+ conductivity ( ! rocks4 ,b ! av]age appe>s to be ab\t #a deg4 ,fahr4 = e #ej or #fj ft4 ( desc5t1 z f> d[n z obs]vations h ext5d$4 ,?\gh ! increase may n advance 9 ! same proportion at great dep?s1 ! inf]5ce has be5 confid5tly drawn t ! temp]ature ( ! nucleus mu/ be exce$+ly high4 ,! probable condition ( ! e>?'s int]ior has be5 a fruitful s\rce ( speculation ev] s9ce geology came into exi/5ce2 b no g5]al agreem5t has be5 arriv$ at on ! subject4 ,?ree *ief hypo!ses h be5 prop\nd$3 7#a7 t ! nucleus is a molt5 mass enclos$ )9 a solid %ell2 7#b7 t1 save 9 local vesicul> spaces :i* may be fill$ ) molt5 or gase\s mat]ial1 ! globe is solid & rigid to ! c5tre2 7#c7 t ! great body ( ! nucleus consi/s ( inc&esc5t vap\rs & gases1 especially vapor\s iron1 :i* "u ! gigantic pressure )9 ! e>? are s compress$ z to conf] practical rigidity on ! globe z a :ole1 & t outside ?is ma9 "p ( ! nucleus ! gases pass into a %ell ( molt5 magma1 :i*1 9 turn1 %ades (f outw>ds into ! comp>atively ?91 cool solidifi$ cru/4 ,rec5t seismological obs]vations h l$ to ! inf]5ce t ! out] cru/1 "s #cj to #de m4 ?ick1 mu/ rapidly m]ge into a fairly homog5e\s nucleus :i*1 :atev] be its con/itution1 transmits undulatory movem5ts ?r\gh its sub/ance ) uni=m velocity & is believ$ to possess a high rigidity4 ,! orig9 ( ! e>?'s high int]nal temp]ature has be5 v>i\sly acc\nt$ =4 ,mo/ usually x has be5 assum$ to be ! residue ( ! orig9al tracts ( flu5t heat out ( :i* ! planet %ap$ itself into a globe4 ,accord+ to ano!r supposition ! effects ( ! gradual gravitational compression ( ! e>?'s mass h be5 ! ma9 s\rce ( ! high temp]ature4 ,rec5t rese>*es 9 radio-activity1 to :i* ref]5ce has already be5 made1 h indicat$ ano!r possible s\rce ( ! int]nal heat 9 ! pres5ce ( radium 9 ! rocks ( ! cru/4 ,?is sub/ance has be5 detect$ 9 all igne\s rocks1 especially among ! granites1 9 quantity suffici5t1 accord+ to ! ,hon4 ,r4,j4 ,/rutt1 to acc\nt = ! obs]v$ temp]ature-gradi5t 9 ! cru/1 & to indicate t ?is cru/ _c be m ?an #de m4 ?ick1 o!rwise ! outfl[ ( heat w\ld be great] ?an ! am\nt actually asc]ta9$4 ,inside ?is ext]nal cru/ conta9+ radio-active sub/ances1 x is suppos$1 z already /at$1 t ! nucleus consi/s ( "s totally diff]5t matt] conta9+ little or no radium4 ,con/itution ( ! ,e>?'s ,cru/4--,z ! cru/ ( ! e>? conta9s ! geological record1 or /ony *ronicle f :i* geology int]prets ! hi/ory ( our globe1 x =ms ! ma9 subject ( /udy to ! geologi/4 ,! mat]ials ( :i* ?is cru/ consi/s are kn[n z m9]als & rocks4 ,f _m *emical analyses1 :i* h be5 made ( ^! mat]ials1 ! g5]al *emical con/itution (1 at lea/1 ! accessible portion ( ! cru/ has be5 satisfactorily asc]ta9$4 ,?is in=mation becomes ( mu* importance 9 speculations reg>d+ ! e>ly hi/ory ( ! globe4 ,( ! elem5ts kn[n to ! *emi/ ! great majority =m b a small proportion ( ! composition ( ! cru/1 :i* is ma9ly built up ( ab\t tw5ty ( !m4 ,( ^! by f> ! mo/ important are ! non-metallic elem5ts oxyg5 & silicon4 ,! =m] =ms ab\t #dg & ! latt] r m ?an #bh ( ! orig9al cru/1 s t ^! two elem5ts make up ab\t ?ree-f\r?s ( ! :ole4 ,next aft] !m come ! metals alum9ium 7#h4af71 iron 7#d4fd71 calcium 7#c4ej71 magnesium 7#b4fb71 sodium 7#b4fc71 & potassium 7#b4ce74 ,! o!r twelve elem5ts includ$ 9 ! tw5ty v>y 9 am\nt f a proportion ( #j4da 9 ! case ( titanium1 to n m ?an #j4ja ( *lor9e1 fluor9e1 *romium1 nickel & li?ium4 ,! o!r fifty or m elem5ts exi/ 9 su* m9ute proportions 9 ! cru/ t1 probably1 n "o ( !m am\nts to z mu* z #j4ja1 ?\gh !y include ! useful metals1 except iron4 ,tak+ ! cru/1 & ! ext]nal envelopes ( ! ocean & ! air1 we ?us p]ceive t ^! out] p>ts ( our planet consi/ ( m ?an ?ree-f\r?s ( non-metals & less ?an "o-f\r? ( metals4 ,! comb9ations ( ! elem5ts :i* are ( mo/ importance 9 ! con/itution ( ! t]re/rial cru/ consi/ ( oxides4 ,f ! mean ( a l>ge numb] ( analyses ( ! rocks ( ! l[] or primitive portion ( ! cru/1 x has be5 asc]ta9$ t silica 7,si,o#b7 =ms almo/ #fj & alum9a 7,al#b,o#c7 upw>ds ( #ae ( ! :ole4 ,! o!r comb9ations 9 ord] ( importance are lime 7,ca,o7 #d4ij1 magnesia 7,mg,o7 #d4cf1 soda 7,na#b,o7 #c4ee1 f]r\s oxide 7,fe,o7 #c4eb1 pota% 7,k#b,o7 #b4hj1 f]ric oxide 7,fe#b,o#c7 #b4fc1 wat] 7,h#b,o7 #a4eb1 titanium oxide 7,ti,o#b7 #j4fj1 phosphoric acid 7,p#b,o#e7 #j4bb2 ! o!r comb9ations ( elem5ts ?us =m less ?an #a ( ! cru/4 ,^! diff]5t comb9ations ( ! elem5ts ent] into fur!r comb9ations ) ea* o!r s z to produce ! wide assortm5t ( simple m9]als 7see ,,m9]alogy74 ,?us1 silica & alum9a are comb9$ to =m ! alum9\s silicates1 :i* ent] s l>gely into ! composition ( ! cru/ ( ! e>?4 ,! silicates ( magnesia1 pota% & soda con/itute o!r important families ( m9]als4 ,a mass ( mat]ial compos$ ( "o1 b m usually ( m ?an "o m9]al1 is kn[n z a rock4 ,"u ?is t]m geologi/s are accu/om$ to class n only solid /one1 su* z granite & lime/one1 b also less coh]5t mat]ials su* z clay1 peat & ev5 loose s&4 ,! accessible portion ( ! e>?'s cru/ consi/s ( v>i\s k9ds ( rocks1 :i* diff] f ea* o!r 9 /ructure1 composition & orig91 & are !re=e susceptible ( div]se classifications accord+ to ! po9t ( view f :i* !y are consid]$4 ,! details ( ?is subject w be f\nd 9 ! article ,,petrology4 ,classification ( ,rocks4--,v>i\s sy/ems ( classification ( rocks h be5 propos$1 b none ( !m is :olly satisfactory4 ,! mo/ useful arrangem5t = mo/ purposes ( ! geologi/ is "o bas$ on ! broad diff]5ces betwe5 !m 9 reg>d to _! mode ( orig94 ,f ?is po9t ( view !y may be rang$ 9 ?ree divisions3 #a4 ,9 ! fir/ place1 a l>ge numb] ( rocks may be describ$ z orig9al or und]iv$1 = x is n possible to trace !m back to any e>li] s\rce4 ,!y belong to ! primitive con/itution ( ! planet1 &1 z !y h all come up f bel[ ?r\gh ! cru/1 !y s]ve to %[ ! nature ( ! mat]ial :i* lies imm$iately bel[ ! out] p>ts ( t cru/4 ,!y include ! num]\s v>ieties ( lava1 :i* h be5 p\r$ out 9 a molt5 /ate f volcanic v5ts1 also a great s]ies ( o!r rocks :i*1 ?\gh !y may nev] h be5 erupt$ to ! surface1 h be5 =c$ upw>d 9 a melt$ condition into ! o!r rocks ( ! cru/ & h solidifi$ !re4 ,f _! mode ( orig9 ?is great class ( rocks has be5 call$ igne\s or eruptive4 ,z !y g5]ally %[ no def9ite int]nal /ructure save su* z may result f jo9ts1 !y h be5 t]m$ massive or un/ratifi$1 to di/+ui% !m f ^? ( ! second division :i* are /rongly m>k$ out by ! pres5ce ( a /ratifi$ /ructure4 ,! igne\s rocks pres5t a consid]able range ( composition4 ,= ! mo/ "p !y consi/ ma9ly ( alum9\s silicates1 "s ( !m be+ highly acid comp\nds ) #ge or m ( silica4 ,b !y also include highly basic v>ieties :]e9 ! proportion ( silica s9ks to #dj1 & :]e magnesia greatly pr$om9ates ov] alum9a4 ,! textures ( igne\s rocks likewise comprise a wide s]ies ( v>ieties4 ,on ! "o h&1 "s are completely vitre\s1 l obsidian1 :i* is a natural glass4 ,f ?is extreme e gradation may be trac$ ?r\gh gradual increase ( ! products ( devitrification1 until ! mass may become completely cry/all9e4 ,aga91 "s cry/all9e igne\s rocks are s f9e 9 gra9 z n to %[ _! compon5t cry/als save "u ! microscope1 :ile 9 o!rs ! texture is s co>se z to pres5t ! compon5t m9]als 9 sep>ate cry/als an in* or m 9 l5g?4 ,^! diff]5ces indicate t1 at fir/1 ! mat]ials ( ! rock may h be5 z completely molt5 z artificial glass1 & t ! cry/all9e condition has be5 subsequ5tly develop$ by cool+1 & ! sep>ation ( ! *emical con/itu5ts into def9ite cry/all9e m9]als4 ,_m ( ! *>act]s ( igne\s rocks h be5 reproduc$ exp]im5tally by fus+ toge!r _! m9]als1 or ! con/itu5ts ( _! m9]als1 9 ! prop] proportion4 ,b x has n yet be5 f\nd possible to imitate ! /ructure ( su* rocks z granite4 ,d\btless ^! rocks consolidat$ ) extreme sl[ness at great dep?s bel[ ! surface1 "u va/ pressures & probably 9 ! pres5ce ( wat] or wat]-vap\r--conditions :i* _c be adequately imitat$ 9 a laboratory4 ,?\gh ! igne\s rocks occupy ext5sive areas 9 "s c\ntries1 !y nev]!less cov] a mu* small] "p ( ! :ole surface ( ! l& ?an is tak5 up by ! second division or /ratifi$ rocks4 ,b !y increase 9 quantity d[nw>ds & probably ext5d cont9u\sly r\nd ! globe bel[ ! o!r rocks4 ,?is important s]ies br+s be=e u ! relations ( ! molt5 magma )9 ! e>? to ! ov]ly+ cru/ & to ! out] surface4 ,on ! "o h&1 x includes ! olde/ & mo/ deep-seat$ extravasations ( t magma1 :i* h be5 br\ght to light by ruptures & upheavals ( ! cru/ & prolong$ d5udation4 ,on ! o!r1 x pres5ts to our /udy ! v>i$ outp\r+s ( molt5 & fragm5t>y mat]ials 9 ! dis*>ges ( mod]n & anci5t volcanoes4 ,betwe5 ^! two extremes ( position & age1 we f9d t ! cru/ has be51 z x 71 riddl$ ) injections ( ! magma f bel[4 ,^! features w be fur!r notic$ 9 ,"p ,v4 ( ?is article4 #b4 ,! s$im5t>y or /ratifi$ rocks =m by mu* ! l>g] "p ( ! dry l& ( ! globe1 & !y are prolong$ to an unkn[n di/ance f ! %ores "u ! b$ ( ! sea4 ,!y include ^? masses ( m9]al matt] :i*1 unlike ! igne\s rocks1 c be trac$ back to a def9ite orig9 on ! surface ( ! e>?4 ,?ree di/9ct types may be recogniz$ among !m3 7a7 ,by f> ! l>ge/ proportion ( !m consi/s ( diff]5t k9ds ( s$im5t d]iv$ f ! dis9tegration ( pre-exi/+ rocks4 ,9 ?is fragm5tal gr\p are plac$ all ! v>ieties ( %+le1 gravel1 s&1 clay & mud1 :e!r ^! mat]ials rema9 9 a loose incoh]5t condition1 or h be5 compact$ into solid /one4 7b7 ,ano!r gr\p consi/s ( mat]ials t h be5 deposit$ by *emical precipitation f solution 9 wat]4 ,! :ite s9t] laid d[n by calc>e\s spr+s is a famili> example on a small scale4 ,b$s ( rock-salt1 gypsum & dolomite h1 9 "s regions1 be5 accumulat$ to a ?ickness ( _m ?\s& feet1 by successive precipitations ( ! salt conta9$ 9 ! wat] ( inl& seas4 7c7 ,an abundant & highly important s]ies ( s$im5t>y =mations has be5 =m$ f ! rema9s ( plants & animals4 ,su* accumulations may arise ei!r f ! transport & deposit ( ^! rema9s1 z 9 ! case ( %eets ( drift-wood1 & banks ( drift$ sea-%ells1 or f ! gr[? & decay ( ! organisms on ! spot1 z happ5s 9 peat bogs & 9 coral-reefs4 ,z ! s$im5t>y rocks h = ! mo/ "p be5 laid d[n "u wat]1 & m especially on ! sea-floor1 !y are (t5 spok5 ( z aque\s1 9 contradi/9ction to ! igne\s rocks4 ,"s ( !m1 h[ev]1 are accumulat$ by ! drift+ action ( w9d ^u loose mat]ials1 & are kn[n z aeolian =mations4 ,famili> in/ances ( su* w9d-=m$ deposits are ! s&-dunes along _m p>ts ( ! sea coa/4 ,mu* m ext5sive 9 area are ! s&s ( ! great des]ts 9 ! arid regions ( ! globe4 ,x is f ! s$im5t>y rocks t ! ma9 portion ( geological hi/ory is d]iv$4 ,!y h be5 deposit$ "o ov] ano!r 9 successive /rata f a remote p]iod 9 ! developm5t ( ! globe d[n to ! pres5t "t4 ,f ?is arrangem5t !y h be5 t]m$ /ratifi$1 9 contra/ to ! un/ratifi$ or igne\s s]ies4 ,!y h pres]v$ memorials ( ! geographical revolutions :i* ! surface ( ! e>? has und]gone2 & above all1 9 ! abundant fossils :i* !y h enclos$1 !y furni% a mom5t\s record ( ! v>i\s tribes ( plants & animals :i* h successively fl\ri%$ on l& & sea4 ,_! inve/igation is ?us ! mo/ important task :i* devolves ^u ! geologi/4 #c4 ,9 ! ?ird place comes a s]ies ( rocks :i* are n n[ 9 _! orig9al condition1 b h und]gone su* alt]ation z to h acquir$ new *>act]s t m or less conceal _! fir/ /ructures4 ,"s ( !m c be readily recogniz$ z alt]$ igne\s masses2 o!rs are z manife/ly ( s$im5t>y orig92 :ile ( _m x is difficult to decide :at may h be5 _! pri/9e "*4 ,to ?is s]ies ! t]m metamorphic has be5 appli$4 ,its memb]s are specially di/+ui%$ by a prevail+ fissile1 or s*i/ose1 /ructure :i* !y did n at fir/ possess1 & :i* diff]s f any?+ f\nd 9 unalt]$ igne\s or s$im5t>y rocks4 ,?is fissility is comb9$ ) a m or less pron\nc$ cry/all9e /ructure4 ,^! *anges are believ$ to be ! result ( movem5ts )9 ! cru/ ( ! e>?1 :]eby ! mo/ solid rocks 7 cru%$ & %e>$1 :ile1 at ! same "t1 "u ! influ5ce ( a high temp]ature & ! pres5ce ( wat]1 !y und]w5t int]nal *emical reactions1 :i* l$ to a re>rangem5t & recomposition ( _! m9]al con/itu5ts & ! production ( a cry/all9e /ructure 7see ,,metamorphism74 ,among ! less alt]$ metamorphic rocks ( s$im5t>y orig91 ! successive lam9ae ( deposit ( ! orig9al s$im5t c be easily obs]v$2 b !y are also trav]s$ by a new set ( divisional planes1 along :i* !y split across ! orig9al b$d+4 ,toge!r ) ?is sup]9duc$ cleavage !re h be5 develop$ 9 !m m9ute hairs1 scales & rudim5t>y cry/als4 ,fur!r /ages ( alt]ation are m>k$ by ! increase ( micace\s scales1 g>nets & o!r m9]als1 especially along ! planes ( cleavage1 until ! :ole rock becomes cry/all9e1 & displays its *ief compon5t m9]als 9 successive discont9u\s folia :i* m]ge into ea* o!r1 & are (t5 crumpl$ & puck]$4 ,massive igne\s rocks c be obs]v$ to h und]gone int5se cru%+ & cleavage1 & to h ultimately assum$ a cry/all9e foliat$ "*4 ,rocks :i* pres5t ?is aspect are kn[n z s*i/s 7q4v474 ,!y range f ! f9e/ silky slates1 or phyllites1 up to ! co>se/ gneisses1 :i* 9 h&-specim5s c h>dly be di/+ui%$ f granites4 ,!re is inde$ e reason to believe t su* gneisses 7 probably orig9ally true granites1 & t _! foliation & recry/allization h be5 ! result ( metamorphism4 ,! s*i/s are m especially to be f\nd 9 ! he>t ( m\nta9-*a9s1 & 9 regions :]e ! l[e/ & olde/ p>ts ( ! e>?'s cru/ h1 9 ! c\rse ( geological revolutions1 be5 expos$ to ! light ( day4 ,!y h be5 claim$ by "s writ]s to be "p ( ! orig9al or primitive surface ( our globe t fir/ consolidat$ on ! molt5 nucleus4 ,b ! progress ( inve/igation all ov] ! _w has %[n t ?is supposition _c be su/a9$4 ,! olde/ kn[n rocks pres5t none ( ! *>act]s ( molt5 mat]ial t has cool$ & h>d5$ 9 ! air1 l ! v>i\s =ms ( rec5t lava4 ,on ! contr>y1 !y possess _m ( ! features *>act]i/ic ( bodies ( eruptive mat]ial t h be5 inject$ into ! cru/ at "s dep? und]gr\nd1 & are n[ visible at ! surface1 ow+ to ! removal by d5udation ( ! rocks "u :i* !y consolidat$4 ,9 _! less foliat$ portions !y c be recogniz$ z true eruptive rocks4 ,9 _m places gneisses t possess a ?or\ghly typical foliation h be5 f\nd to pi]ce anci5t s$im5t>y =mations z intrusive bosses & ve9s4 ,,"p ,,iv4--,,dynamical ,,geology ,?is section ( ! sci5ce includes ! inve/igation ( ^? processes ( *ange :i* are at pres5t 9 progress ^u ! e>?1 :]eby modifications are made on ! /ructure & composition ( ! cru/1 on ! relations betwe5 ! int]ior & ! surface1 z %[n by volcanoes1 e>?quakes & o!r t]re/rial di/urbances1 on ! di/ribution ( oceans & cont95ts1 on ! outl9es ( ! l&1 on ! =m & dep? ( ! sea-bottom1 on climate1 & on ! races ( plants & animals by :i* ! e>? is t5ant$4 ,x br+s be=e u1 9 %ort1 ! :ole range ( activities :i* x is ! prov9ce ( geology to /udy1 & leads u to precise notions reg>d+ _! relations to ea* o!r & ! results :i* !y a*ieve4 ,a k ( ?is bran* ( ! subject is ?us ! ess5tial gr\ndwork ( a true & fruitful acqua9tance ) ! pr9ciples ( geology1 see+ t x necessitates a /udy ( ! pres5t ord] ( nature1 & ?us provides a key = ! int]pretation ( ! pa/4 ,! :ole range ( op]ations includ$ )9 ! scope ( inquiry 9 ?is bran* ( ! sci5ce may be reg>d$ z a va/ cycle ( *ange1 into :i* we may break at any po9t1 & r\nd :i* we may travel1 only to f9d ourselves br\ght back to our />t+-po9t4 ,x is a matt] ( comp>atively small mom5t at :at "p ( ! cycle we beg9 our inquiries4 ,we %all always f9d t ! *anges we see 9 action h result$ f "s t prec$$1 & give place to o!rs :i* foll[ !m4 ,at an e>ly "t 9 ! e>?'s hi/ory1 ant]ior to any ( ! p]iods ( :i* a record rema9s 9 ! visible rocks1 ! *ief s\rces ( geological action probably lay )9 ! e>? itself4 ,if1 z is g5]ally suppos$1 ! planet /ill reta9$ a great /ore ( its initial heat1 x 0 d\btless ! !atre ( great *emical *anges1 giv+ rise1 p]haps1 to manife/ations ( volcanic en]gy some:at l ^? :i* h s m>vell\sly r\gh5$ ! surface ( ! moon4 ,z ! out] lay]s ( ! globe cool$1 & ! di/urbances due to int]nal heat & *emical action became less m>k$1 ! conditions w\ld arise 9 :i* ! mat]ials = geological hi/ory 7 accumulat$4 ,! influ5ce ( ! sun1 :i* mu/ always h op]at$1 w\ld !n /& out m cle>ly1 giv+ rise to t wide circle ( sup]ficial *anges :]e9 v>iations ( temp]ature & ! circulation ( air & wat] ov] ! surface ( ! e>? come into play4 ,9 ! pursuit ( 8 inquiries into ! pa/ hi/ory & into ! pres5t regime ( ! e>?1 ! geologi/ mu/ ne$s keep 8 m9d ev] op5 to ! reception ( evid5ce = k9ds & especially = degrees ( action :i* he _h n be=e imag9$4 ,human exp]i5ce has be5 too %ort to all[ him to assume t all ! causes & modes ( geological *ange h be5 def9itively asc]ta9$4 ,on ! e>? itself !re may rema9 = future discov]y evid5ce ( =m] op]ations by heat1 magnetism1 *emical *ange or o!rwise1 :i* may expla9 _m ( ! ph5om5a ) :i* geology has to deal4 ,( ! influ5ces1 s _m & pr(\nd1 :i* ! sun ex]ts ^u our planet1 we c z yet only p]ceive a little4 ,nor c we tell :at o!r cosmical influ5ces may h l5t _! aid 9 ! evolution ( geological *anges4 ,mu* useful in=mation reg>d+ _m geological processes has be5 obta9$ f exp]im5tal rese>* 9 laboratories & else:]e1 & mu* m may be confid5tly look$ = f future ext5sions ( ?is me?od ( inquiry4 ,! e>ly exp]im5ts ( ,sir ,james ,hall1 already notic$1 =m$ ! />t+-po9t = num]\s subsequ5t rese>*es1 :i* h elucidat$ _m po9ts 9 ! orig9 & hi/ory ( rocks4 ,x is true t we _c hope to imitate ^? op]ations ( nature :i* dem& enorm\s pressures & excessively high temp]atures comb9$ ) a long lapse ( "t4 ,b exp]i5ce has %[n t 9 reg>d to a l>ge numb] ( processes1 x is possible to imitate nature's work+ ) suffici5t accuracy to enable u to und]/& !m1 & s to modify & control ! results z to obta9 a satisfactory solution ( "s geological problems4 ,9 ! pres5t /ate ( our k1 all ! geological en]gy ^u & )9 ! e>? mu/ ultimately be trac$ back to ! primeval en]gy ( ! p>5t nebula or sun4 ,!re is1 h[ev]1 a c]ta9 propriety & conv5i5ce 9 di/+ui%+ betwe5 t "p ( x :i* is due to ! survival ( "s ( ! orig9al en]gy ( ! planet & t "p :i* arises f ! pres5t supply ( en]gy receiv$ day by day f ! sun4 ,9 ! =m] case we h to deal ) ! int]ior ( ! e>?1 & its reaction ^u ! surface2 9 ! latt]1 we deal ) ! surface ( ! e>? & to "s ext5t ) its reaction on ! int]ior4 ,?is di/9ction all[s ( a broad treatm5t ( ! subject "u two divisions3 ,i4 ,hypog5e or ,plutonic ,action3 ,! *anges )9 ! e>? caus$ by int]nal heat1 me*anical movem5t & *emical re>rangem5ts4 ,,ii4 ,epig5e or ,surface ,action3 ,! *anges produc$ on ! sup]ficial p>ts ( ! e>?1 *iefly by ! circulation ( air & wat] set 9 motion by ! sun's heat4 ,,division ,i4--,,hypog5e ,,or ,,plutonic ,,action ,9 ! discussion ( ?is bran* ( ! subject we mu/ c>ry 9 our m9ds ! conception ( a globe /ill possess+ a high int]nal temp]ature1 radiat+ heat into space & consequ5tly contract+ 9 bulk4 ,portions ( molt5 rocks f inside are f "t to "t p\r$ out at ! surface4 ,sudd5 %ocks are g5]at$ by :i* de/ructive e>?quakes are propagat$ ?r\gh ! diamet] ( ! globe z well z to & along its surface4 ,wide geographical areas are pu%$ up or s9k d[n4 ,9 ! mid/ ( ^! movem5ts rem>kable *anges are produc$ ^u ! rocks ( ! cru/2 !y are plicat$1 fractur$1 cru%$1 r5d]$ cry/all9e & ev5 fus$4 7,a7 ,volcanoes & ,volcanic ,action4 ,?is subject is discuss$ 9 ! article ,,volcano1 & only a g5]al view ( its ma9 features w be giv5 "h4 ,"u ! t]m volcanic action 7vulcanism1 vulcanicity7 are embrac$ all ! ph5om5a connect$ ) ! expulsion ( heat$ mat]ials f ! int]ior ( ! e>? to ! surface4 ,a volcano may be def9$ z a conical hill or m\nta91 built up :olly or ma9ly ( mat]ials :i* h be5 eject$ f bel[1 & :i* h accumulat$ ar\nd ! c5tral v5t ( eruption4 ,z a rule its truncat$ summit pres5ts a cup-%ap$ cavity1 t]m$ ! crat]1 at ! bottom ( :i* is ! op5+ ( ! ma9 funnel or pipe :]eby communication is ma9ta9$ ) ! heat$ int]ior4 ,f "t to "t1 h[ev]1 9 l>ge volcanoes r5ts are =m$ on ! sides ( ! cone1 :5ce /eam & o!r hot vap\rs & also /reams ( molt5 lava are p\r$ =?4 ,on su* r5ts small] or p>asitic cones are (t5 =m$1 :i* imitate ! op]ations ( ! p>5t cone &1 aft] repeat$ eruptions1 may rise to hills hundr$s ( feet 9 height4 ,9 c\rse ( c5turies ! result ( ! con/ant outp\r+ ( volcanic mat]ials may be to build up a l>ge m\nta9 l ,etna1 :i* t[]s above ! sea to a height ( #aj1hdj feet1 & has "s #bjj m9or cones along its flanks4 ,b all volcanic eruptions d n proce$ f c5tral orifices4 ,9 ,icel& x has be5 obs]v$ t1 f fissures op5$ 9 ! gr\nd & ext5d+ = long di/ances1 molt5 mat]ial has issu$ 9 su* abundance z to be spread ov] ! surr\nd+ c\ntry = _m miles1 :ile along ! l9es ( fissure small cones or hillocks ( fragm5t>y mat]ial h accumulat$ r\nd m active p>ts ( ! r5t4 ,!re is reason to believe t 9 ! geological pa/ ?is fissure-type ( eruption has repeat$ly be5 develop$1 z well z ! m common =m ( c5tral cones l ,vesuvius or ,etna4 ,9 ! op]ations ( exi/+ volcanoes only ! sup]ficial manife/ations ( volcanic action are obs]vable4 ,b :5 ! rocks ( ! e>?'s cru/ are /udi$1 !y are f\nd to enclose ! relics ( =m] volcanic eruptions4 ,! roots ( anci5t volcanoes h ?us be5 laid b>e by geological revolutions2 & "s ( ! subt]ranean phases ( volcanic action are !reby reveal$ :i* are :olly conceal$ 9 an active volcano4 ,h5ce to obta9 z complete a conception z possible ( ! nature & hi/ory ( volcanic action1 reg>d mu/ be _h1 n m]ely to mod]n volcanoes1 b to ! records ( anci5t eruptions :i* h be5 pres]v$ )9 ! cru/4 ,! sub/ances dis*>g$ f volcanic v5ts consi/ (--7#a7 ,gases & vap\rs3 :i*1 dissolv$ 9 ! molt5 magma ( ! int]ior1 take ! *ief %>e 9 volcanic activity4 ,!y include 9 greate/ abundance wat]-gas1 :i* cond5ses into ! cl\ds ( /eam s conspicu\s 9 volcanic eruptions4 ,hydro*loric acid & sulphurett$ hydrog5 are likewise pl5tiful1 toge!r ) _m o!r sub/ances :i*1 sublim$ by ! high int]nal temp]ature1 take a solid =m on cool+ at ! surface4 7#b7 ,molt5 rock or lava3 :i* ranges f ! extremely acid type ( ! obsidians & rhyolites ) #gj or m ( silica1 to ! m basic & heavy v>ieties su* z basalts & leucite-lavas ) mu* iron1 & sometimes no m ?an #de ( silica4 ,! specific gravity ( lavas v>ies betwe5 #b4cg & #c4bb1 & ! texture ranges f ne>ly pure glass1 l obsidian1 to a co>se granitoid comp\nd1 z 9 "s rhyolites4 7#c7 ,fragm5t>y mat]ials1 :i* are sometimes dis*>g$ 9 enorm\s quantity & disp]s$ ov] a wide ext5t ( c\ntry1 ! f9] p>ticles be+ transport$ by upp] air-curr5ts = hundr$s ( miles4 ,^! mat]ials arise ei!r f ! explosion ( lava by ! sudd5 expansion ( ! dissolv$ vap\rs & gases1 z ! molt5 rock rises to ! surface1 or f ! break+ up & expulsion ( portions ( ! walls ( ! v5t1 or ( ! lava1 :i* happ5s to h solidifi$ )9 ^! walls4 ,!y v>y f ! f9e/ impalpable du/ & a%es1 ?r\gh increas+ /ages ( co>s5ess up to huge bombs torn f ! upp] surface ( ! molt5 rock 9 ! v5t1 & l>ge blocks ( already solidifi$ lava1 or ( non-volcanic rock deta*$ f ! sides ( ! pipe up :i* ! eruptions take place4 ,no?+ is yet kn[n z to ! det]m9+ cause ( any p>ticul> volcanic eruption4 ,"s v5ts1 l t ( ,/romboli1 9 ! ,m$it]ranean1 are cont9ually active1 & h be5 s ev] s9ce man has obs]v$ !m4 ,o!rs aga9 h be5 only int]mitt5tly 9 eruption1 ) int]vals ( c5turies betwe5 _! p]iods ( activity4 ,we are equally 9 ! d>k z to :at has det]m9$ ! sites on :i* volcanic action has manife/$ itself4 ,!re is reason1 inde$1 to believe t ext5sive fractures ( ! t]re/rial cru/ h (t5 provid$ passages up :i* ! vap\rs1 imprison$ 9 ! int]nal magma1 h be5 able to make _! way1 accompani$ by o!r products4 ,:]e *a9s ( volcanoes rise along def9ite l9es1 l ^? ( ,sumatra1 ,java1 & _m o!r tracts bo? 9 ! ,old & ! ,new ,_w1 !re appe>s to be little d\bt t _! l9e> di/ribution %\ld be attribut$ to ?is cause4 ,b :]e a volcano has appe>$ by itself1 9 a region previ\sly exempt f volcanic action1 ! exi/5ce ( a contribut+ fissure _c be s confid5tly presum$4 ,! /udy ( c]ta9 anci5t volcanoes1 ! roots ( :i* h be5 expos$ by long d5udation1 has %[n an abs5ce ( any visible trace ( _! hav+ avail$ !mselves ( fractures 9 ! cru/4 ,! inf]5ce has be5 drawn t volcanic en]gy is capable ( itself drill+ an orifice ?r\gh ! cru/1 probably at "s weak] "p1 & eject+ its products at ! surface4 ,! s\rce ( ?is en]gy is to be s\ght 9 ! enorm\s expansive =ce ( ! vap\rs & gases dissolv$ 9 ! magma4 ,!y are kept 9 solution by ! enorm\s pressure )9 ! e>?2 b z ! lava approa*es ! surface & ?is pressure is reliev$ ^! dissolv$ vap\rs & gases ru% out ) explosive viol5ce1 bl[+ ! upp] "p ( ! lava column into du/1 & all[+ portions ( ! liquid mass bel[ to rise & escape1 ei!r f ! crat] or f "s fissure :i* ! vig\r ( explosion has op5$ on ! side ( ! cone4 ,s gigantic is ! en]gy ( ^! p5t-up vap\rs1 t1 aft] a long p]iod ( volcanic quiesc5ce1 !y sometimes bur/ =? ) su* viol5ce z to bl[ (f ! :ole ( ! upp] "p or ev5 "o side ( a l>ge cone4 ,! hi/ory ( ,vesuvius1 & ! great eruptions ( ,krakatoa 9 #ahhc & ( ,b&aizan 9 #ahhh furni% memorable examples ( great volcanic convulsions4 ,x has be5 obs]v$ t su* /up5d\s dis*>ges ( a]i=m & fragm5t>y matt] may be att5d$ ) ! emission ( little or no lava4 ,on ! o!r h&1 "s ( ! l>ge/ outfl[s ( lava h be5 accompani$ by comp>atively little fragm5t>y mat]ial4 ,?us1 ! great lava-floods ( ,icel& 9 #aghc spread = #dj m4 away f _! p>5t fissure1 :i* 0 m>k$ only by a l9e ( little cones ( slag4 ,! temp]ature ( lava z x issues f und]gr\nd has be5 measur$ m or less satisfactorily1 & af=ds an indication ( t exi/+ )9 ! e>?4 ,at ,vesuvius x has be5 asc]ta9$ to be m ?an #bjjj deg4 ,fahr4 ,at fir/ ! molt5 rock gl[s ) a :ite light1 :i* rapidly r$d5s1 & disappe>s "u ! rugg$ br[n & black cru/ t =ms on ! surface4 ,und]nea? ?is badly conduct+ cru/1 ! lava cools s sl[ly t columns ( /eam h be5 notic$ ris+ f its surface m ?an #hj ye>s aft] its eruption4 ,consid]able alt]ation 9 ! topography ( volcanic regions may be produc$ by successive eruptions4 ,! fragm5t>y mat]ials are sometimes dis*>g$ 9 su* abundance z to cov] ! gr\nd = _m miles ar\nd ) a deposit ( loose a%es1 c9d]s & slag4 ,su* a deposit accumulat+ to a dep? ( _m feet may completely bury valleys & wat]-c\rses1 & ?us greatly affect ! dra9age4 ,! co>se/ mat]ials accumulate ne>e/ to ! v5t t emits !m4 ,! f9] du/ is n infrequ5tly hurl$ =? ) su* an impetus z to be c>ri$ = ?\s&s ( feet into ! tracks ( upp] air-curr5ts1 :]eby x may be borne = hundr$s ( miles away f ! v5t s z ultimately to fall to ! gr\nd 9 c\ntries f> remov$ f any active volcano4 ,outfl[s ( lava1 f _! great] solidity & durability1 produce /ill m s]i\s & la/+ *anges 9 ! ext]nal features ( ! gr\nd ov] :i* !y fl[4 ,z !y naturally seek ! l[e/ levels1 !y f9d _! way into ! *annels ( /reams4 ,if !y keep along ! *annels1 !y seal !m up "u a mass ( compact /one :i* ! runn+ wat]1 if n :olly div]t$ else:]e1 w take _m long c5turies to cut ?r\gh4 ,if1 on ! o!r h&1 ! lava crosses a /ream1 x =ms a massive dam1 above :i* ! wat] is pond$ back s z to =m a lake4 ,z ! result ( prolong$ activity a volcanic cone is gradually built up by successive outfl[s ( lava & %[]s ( du/ & /ones4 ,^! mat]ials are arrang$ 9 b$s1 or %eets1 incl9$ outw>ds f ! c5tral v5t4 ,on surr\nd+ level gr\nd ! alt]nat+ b$s are flat4 ,9 c\rse ( "t1 deep gullies are cut on ! out] slopes ( ! cone by ra91 & by ! heavy %[]s t arise f ! cond5sation ( ! copi\s dis*>ges ( /eam dur+ eruptions4 ,along ! sides ( ^! rav9es in/ructive sections may be /udi$ ( ! volcanic /rata4 ,! l>g] riv]s ( "s volcanic regions h likewise erod$ va/ gorges 9 ! m horizontal lavas & a%es ( ! flatt] c\ntry1 & h ?us laid b>e /up5d\s cliffs1 along :i* ! successive volcanic %eets c be se5 pil$ above ea* o!r = _m hundr$ feet4 ,on a small scale1 "s ( ^! features are well display$ among ! riv]s t dra9 ! volcanic tracts ( c5tral ,france2 on a great scale1 !y are pres5t$ 9 ! c\rse ( ! ,snake riv]1 & o!r /reams t trav]se ! great volcanic c\ntry ( we/]n ,nor? ,am]ica4 ,simil> volcanic sc5]y has be5 produc$ 9 we/]n ,europe by ! action ( d5udation 9 dissect+ ! flat ,t]ti>y lavas ( ,scotl&1 ! ,fa]oe ,isles & ,icel&4 ,( special int]e/ to ! geologi/ are ^? volcanoes :i* h tak5 _! rise on ! sea-bottom2 = ! volcanic int]calations among ! /ratifi$ =mations ( ! e>?'s cru/ are almo/ entirely ( subm>9e orig94 ,_m active volcanoes situat$ on isl&s h begun _! eruptions bel[ sea-level4 ,bo? ,vesuvius & ,etna sprang up on ! floor ( ! ,m$it]ranean sea1 & h gradually built up _! cones into conspicu\s p>ts ( ! dry l&4 ,examples ( a simil> hi/ory are to be f\nd among ! volcanic isl&s ( ! ,pacific ,ocean4 ,9 "s ( ^! cases a movem5t ( elevation has c>ri$ ! subm>9e lavas1 tuffs & agglom]ates above sea-level1 & has furni%$ opportunities ( comp>+ ^! mat]ials ) ^? ( rec5t suba]ial orig91 & also ) ! anci5t records ( subm>9e eruptions :i* h be5 pres]v$ among ! /ratifi$ =mations4 ,f ! evid5ce ?us suppli$1 x c be %[n t ! mat]ials eject$ f mod]n subm>9e volcanic v5ts closely resemble ^? accumulat$ by suba]ial volcanoes2 t ! du/1 a%es & /ones become int]m+l$ or int]/ratifi$ ) coral-mud1 or o!r non-volcanic deposit ( ! sea-bottom1 t vesicul> lavas may be int]calat$ among !m z on l&1 & t betwe5 ! successive %eets ( volcanic orig91 lay]s ( lime/one may be laid d[n :i* are compos$ *iefly1 or :olly1 ( ! rema9s ( calc>e\s m>9e organisms4 ,?\gh active volcanoes are widely di/ribut$ ov] ! globe1 & are especially abundant ar\nd ! va/ bas9 ( ! ,pacific ,ocean1 !y af=d an incomplete picture ( ! ext5t to :i* volcanic action has display$ itself on ! surface ( our planet4 ,:5 ! rocks ( ! l& are att5tively /udi$ !y disclose pro(s ( t action 9 _m di/ricts :]e !re is n[ no outw>d sign ( x4 ,n only s1 b !y reveal t volcanoes h be5 9 eruption 9 "s ( ^! di/ricts dur+ _m diff]5t p]iods ( ! pa/1 back to ! beg9n+s ( geological hi/ory4 ,! ,briti% ,isl&s furni% a rem>kable example ( su* a s]ies ( anci5t eruptions4 ,f ! ,cambrian p]iod all ?r\gh ,palaeozoic times !re rose at int]vals 9 t c\ntry a succession ( volcanic c5tres f "s ( :i* ?\s&s ( feet ( lavas & tuffs 7 dis*>g$4 ,aga9 9 old] ,t]ti>y times ! same region witness$ a /up5d\s outp\r+ ( basalt1 ! surviv+ relics ( :i* are m ?an #cjjj ft4 ?ick1 & cov] _m hundr$s ( squ>e miles4 ,simil> evid5ce is suppli$ 9 o!r c\ntries bo? 9 ! ,old & ! ,new _w4 ,h5ce x is prov$ t1 9 ! geological pa/1 volcanic action has be5 vigor\s at long int]vals on ! same sites dur+ a va/ s]ies ( ages1 ?\gh no active v5ts are to be se5 !re n[4 ,! volcanoes n[ active =m b a small proportion ( ! total numb] :i* has appe>$ on ! surface ( ! e>?4 ,) reg>d to ! cause ( volcanic action mu* has be5 speculat$1 b little c be confid5tly affirm$4 ,t wat] 9 ! =m ( occlud$ gas plays ! *ief "p 9 =c+ ! lava column up a volcanic *imney1 & 9 ! viol5t explosions t accompany ! rise ( ! molt5 mat]ial1 is g5]ally admitt$4 ,b op9ions diff] z to ! s\rce ( ?is wat]4 ,accord+ to "s inve/igators1 x %\ld be reg>d$ z 9 l>ge measure ( meteoric orig91 d]iv$ f ! desc5t ( ra9 into ! e>?1 & its absorption by ! molt5 magma 9 ! int]ior4 ,o!rs1 cont5d+ t ! supply s furni%$1 ev5 if x c\ld rea* & be dissolv$ 9 ! magma1 w\ld yet be insuffici5t to furni% ! prodigi\s quantity ( aque\s vap\r dis*>g$ dur+ an eruption1 ma9ta9 t ! wat] belongs to ! magma itself4 ,!y po9t to ! admitt$ fact t _m sub/ances1 p>ticul>ly metals 9 a /ate ( fusion1 c absorb l>ge quantities ( vap\rs & gases )\t *emical comb9ation1 & on cool+ dis*>ge !m ) eruptive ph5om5a some:at l ^? ( volcanoes4 ,?is "q mu/ be reg>d$ z "o ( ! /ill unsolv$ problems ( geology4 7,b7 ,movem5ts ( ! ,e>?'s ,cru/4 ,among ! hypog5e =ces 9 geological dynamics an important place mu/ be assign$ to movem5ts ( ! t]re/rial cru/4 ,?\gh ! expression ! solid e>? has become prov]bial1 x appe>s s+ul>ly inappropriate 9 ! light ( ! results obta9$ 9 rec5t ye>s by ! use ( delicate in/rum5ts ( obs]vation4 ,) ! facilities suppli$ by ^! in/rum5ts 7see ,,seismomet]71 x has be5 asc]ta9$ t ! gr\nd b5ea? our feet is subject to cont9ual slight tremors1 & feeble pulsations ( long] duration1 "s ( :i* may be due to daily or seasonal v>iations ( temp]ature1 atmosph]ic pressure or o!r meteorological causes4 ,! e/abli%m5t ( self-record+ seismomet]s all ov] ! _w has l$ to ! detection ( _m o!rwise imp]ceptible %ocks1 ov] & above ! appreciable e>?-waves propagat$ f e>?quake c5tres ( di/urbance4 ,moreov]1 x has be5 asc]ta9$ t "s p>ts ( ! surface ( ! l& are sl[ly ris+1 :ile o!rs are fall+ ) ref]5ce to ! sea-level4 ,f "t to "t ! surface suff]s calamit\s deva/ation f e>?quakes1 :5 portions ( ! cru/ "u great /ra9 sudd5ly give way4 ,la/ly1 at int]vals1 probably sep>at$ f ea* o!r by va/ p]iods ( "t1 ! t]re/rial cru/ und]goes int5se plication & fracture1 & is consequ5tly ridg$ up into m\nta9-*a9s4 ,no ev5t ( ?is k9d has be5 witness$ s9ce man began to record 8 exp]i5ces4 ,b f ! /ructure ( m\nta9s1 z laid op5 by prolong$ d5udation1 x is possible to =m a vivid conception ( ! nature & effects ( ^! mo/ /up5d\s ( all geological revolutions4 ,9 consid]+ ?is dep>tm5t ( geological inquiry x w be conv5i5t to treat x "u ! foll[+ heads3 7#a7 ,sl[ depression & upheaval2 7#b7 ,e>?quakes2 7#c7 ,m\nta9-mak+2 7#d7 ,metamorphism ( rocks4 #a4 ,sl[ ,depression & ,upheaval4--,on ! we/ side ( ,japan ! l& is believ$ to be s9k+ bel[ ! sea1 = fields are replac$ by bea*es ( s& or %+le1 :ile ! dep? ( ! sea (f %ore has p]ceptibly increas$4 ,a subsid5ce ( ! s\? ( ,sw$5 has tak5 place 9 comp>atively rec5t times1 = /reets & f\ndations ( h\ses at successive levels are f\nd bel[ high-wat] m>k4 ,! we/ coa/ ( ,gre5l& ov] an ext5t ( m ?an #fjj m4 is s9k+1 & old settlem5ts are n[ subm]g$4 ,pro(s ( subm]g5ce ( l& are furni%$ by subm]g$ =e/s1 & b$s ( t]re/rial peat n[ ly+ at v>i\s dep?s bel[ ! level ( ! sea1 ( :i* _m examples h be5 collect$ along ! %ores ( ! ,briti% ,isles1 ,holl& & ,france4 ,int]e/+ evid5ce t ! we/ ( ,europe n[ /&s at a l[] level ?an x did at a late geological p]iod is suppli$ 9 ! *>ts ( ! ,nor? ,sea & ,atlantic1 :i* %[ t ! valleys ( ! l& are prolong$ "u ! sea4 ,^! valleys h be5 erod$ out ( ! rocks by ! /reams :i* fl[ 9 !m1 & ! dep? ( _! subm]g$ portions bel[ ! sea level af=ds an indication ( ! ext5t ( ! subsid5ce4 ,! uprise ( l& has be5 detect$ 9 v>i\s p>ts ( ! _w4 ,"o ( ! mo/ celebrat$ in/ances is t ( ! %ores ( ! ,gulf ( ,bo?nia1 :]e1 at ,/ockholm1 ! elevation1 betwe5 ! ye>s #aggd & #ahge1 appe>s to h be5 #dh c5timetres 7#ah-a_/b 947 9 a c5tury4 ,b on ! we/ side ( ,sw$51 front+ ! ,skag] ,rak1 ! coa/1 betwe5 ! ye>s #ahbj & #ahgj1 rose #cj c5timetres1 :i* is at ! rate ( #fj c5timetres1 or ne>ly #b ft4 9 a c5tury4 ,9 ! region ( ! ,great ,lakes 9 ! int]ior ( ,canada & ! ,unit$ ,/ates x has be5 asc]ta9$ t ! l& is und]go+ a sl[ tilt t[>ds ! s\?-we/1 ( :i* ! mean rate appe>s to be r less ?an #f 94 9 a c5tury4 ,if ?is rate ( *ange %\ld cont9ue ! wat]s ( ,lake ,mi*igan1 ow+ to ! progress ( ! tilt1 w1 9 "s #ejj or #fjj ye>s1 subm]ge ! city ( ,*icago1 & ev5tually ! dra9age ( ! lakes w be div]t$ into ! bas9 ( ! ,mississippi4 ,pro( ( rec5t em]g5ce ( l& is suppli$ by :at are call$ rais$ bea*es or /r&-l9es1 t is1 l9es ( =m] %ores m>k$ by %eets ( littoral deposits1 or plat=ms cut by %ore-waves 9 rock & flank$ by old sea-cliffs & l9es ( sea-worn caves4 ,admirable examples ( ^! features are to be se5 along ! we/ coa/ ( ,europe f ! s\? ( ,engl& to ! nor? ( ,norway4 ,^! l9es ( old %ores become fa9t] 9 proportion to _! antiquity4 ,9 ,brita9 !y occur at v>i\s heights1 ! plat=ms at #be1 #ej & #ajj ft4 be+ well m>k$4 ,! cause ( ^! sl[ upw>d & d[nw>d movem5ts ( ! cru/ ( ! e>? is /ill imp]fectly und]/ood4 ,upheaval might conceivably be produc$ by an asc5t ( ! int]nal magma1 & ! consequ5t expansion ( ! ov]ly+ cru/ by heat2 :ile depression might foll[ any subsid5ce ( ! magma1 or its displacem5t to ano!r di/rict4 ,if1 z is g5]ally believ$1 ! globe is /ill contract+1 ! %r9kage ( ! surface may cause bo? ^! movem5ts4 ,subsid5ce w be 9 excess1 b betwe5 subsid+ tracts lat]al ?ru/ may suffice to pu% upw>d int]v5+ m solid & /able gr\nd2 b no solution ( ! problem yet propos$ is :olly satisfactory4 #b4 ,e>?quakes4--,z ?is subject is discuss$ 9 a sep>ate article x w be suffici5t "h to take note ( its m important geological be>+s4 ,x 0 = _m c5turies tak5 = grant$ t e>?quakes & volcanoes are due to a common cause4 ,we h se5 t 9 classical antiquity !y 7 look$ on z ! results ( ! movem5ts ( w9d imprison$ )9 ! e>?4 ,long aft] ?is notion 0 disc>d$1 & a m sci5tific appreciation ( volcanic action 0 rea*$1 x 0 /ill ?\ght t e>?quakes %\ld be reg>d$ z manife/ations ( ! same s\rce ( en]gy z t :i* displays itself 9 volcanic eruptions4 ,x is true t e>?quakes are frequ5t 9 di/ricts ( active volcanoes1 & !y may und\bt$ly be (t5 due !re to ! explosions ( ! magma1 or to ! rupture ( rocks caus$ by its asc5t t[>ds ! surface4 ,b su* %ocks are comp>atively local 9 _! range & feeble 9 _! effects4 ,!re is n[ a g5]al agreem5t t betwe5 ! great _w-%ak+ e>?quakes & volcanic ph5om5a1 no imm$iate & intimate relation%ip c be trac$1 ?\gh !y may be connect$ 9 ways :i* are n yet p]ceiv$4 ,"s ( ! m rec5t great e>?quakes on l& h prov$ t ! waves ( %ock are produc$ by ! sudd5 rupture or collapse ( rocks "u great /ra91 ei!r along l9es ( previ\s fracture or ( new r5ts 9 ! t]re/rial cru/2 & t su* ruptures may occur at a remote di/ance f any volcano4 ,?us ! rec5t disa/r\s ,san ,francisco e>?quake has be5 recogniz$ to h result$ f a slipp+ ( gr\nd along ! l9e ( an old fault1 :i* has be5 trac$ = a long di/ance 9 ,cali=nia g5]ally p>allel to ! coa/4 ,! position ( ?is fault at ! surface has long be5 cle>ly foll[$ by its *>act]i/ic topography4 ,aft] ! e>?quake ^! sup]ficial features 7 f\nd to h be5 remov$ by ! same cause t _h orig9at$ !m4 ,= "s #cjj m4 on ! track ( ?is old fault-l9e a r5ew$ slipp+ 0 se5 to h tak5 place along "o or bo? sides1 & ! gr\nd at ! surface 0 ruptur$ z well z displac$ horizontally4 ,obvi\sly1 ! j> occasion$ by ! sudd5 & simultane\s subsid5ce ( a portion ( ! e>?'s cru/ sev]al hundr$ miles long1 mu/ be f> m s]i\s ?an c\ld be produc$ by an e>?quake radiat+ f a s+le local volcanic focus4 ,f _! disa/r\s effects on build+s & human lives1 an exagg]at$ importance has be5 imput$ to e>?quakes z ag5ts ( geological *ange4 ,exp]i5ce %[s t ev5 aft] a sev]e %ock :i* may h de/roy$ num]\s t[ns & villages1 toge!r ) ?\s&s ( _! inhabitants1 ! face ( ! c\ntry has suff]$ sc>cely any p]ceptible *ange1 & t1 9 ! c\rse ( a ye> or two1 :5 ! ru9$ h\ses & pro/rate trees h be5 cle>$ away1 little or no obvi\s trace ( ! cata/rophe may rema94 ,among ! m endur+ records ( a great e>?quake may be enum]at$ 7a7 l&slips1 :i* lay b>e hillsides1 & sometimes pond back ! dra9age ( valleys s z to give rise to lakes2 7b7 alt]ations ( ! topography1 z 9 fissur+ ( ! gr\nd1 or 9 ! production ( inequalities :]eby ! dra9age is affect$2 new valleys & new lakes may ?us be =m$1 :ile previ\sly exi/+ lakes may be empti$2 7c7 p]man5t *anges ( level1 ei!r 9 an upw>d or d[nw>d direction4 #c4 ,m\nta9-mak+4--,?is subject may be ref]r$ to "h = ! /rik+ evid5ce :i* x supplies ( ! importance ( movem5ts ( ! e>?'s cru/ among geological processes4 ,! /ructure ( a great m\nta9-*a9 su* z ! ,alps proves t ! cru/ ( ! e>? has be5 int5sely plicat$1 crumpl$ & fractur$4 ,va/ piles ( s$im5t>y /rata h be5 fold$ to su* an ext5t z to occupy n[ only half ( _! orig9al horizontal ext5t4 ,?is compression 9 ! case ( ! ,alps has be5 comput$ to am\nt to z mu* z #abj1jjj metres or #gd ,engli% miles1 s t two po9ts on ! opposite sides ( t *a9 h be5 br\ght by s mu* ne>] to ea* o!r ?an !y 7 orig9ally be=e ! movem5ts4 ,besides su* int5se plication1 ext5sive ruptur+ ( ! cru/ has tak5 place 9 ! same range ( m\nta9s4 ,n only h ! mo/ anci5t rocks be5 squeez$ up into ! c5tral axis ( ! *a91 b huge slices ( !m h be5 torn away f ! ma9 body1 & ?ru/ =w>d = _m miles1 s z n[ actually to =m ! summits ( m\nta9s1 :i* are almo/ entirely compos$ ( mu* y\ng] =mations4 ,if ^! colossal di/urbances occurr$ rapidly1 !y w\ld give rise to cataclysms ( inconceivable magnitude ov] ! surface ( ! globe4 ,no record has be5 discov]$ ( su* accompany+ deva/ation4 ,b :e!r sudd5 & viol5t1 or prolong$ & gradual1 su* /up5d\s upturn+s ( ! cru/ did und\bt$ly take place1 z is cle>ly reveal$ 9 innum]able natural sections1 :i* h be5 laid op5 by ! d5udation ( ! cre/s & sides ( ! m\nta9s4 #d4 ,metamorphism ( ,rocks 7see ,,metamorphism74--,dur+ ! movem5ts to :i* ! cru/ ( ! e>? has be5 subject1 n only h ! rocks be5 fold$ & fractur$1 b !y h likewise1 9 _m regions1 acquir$ new int]nal /ructures1 & h ?us und]gone a process ( regional metamorphism4 ,?is re>rangem5t ( _! sub/ance has be5 gov]n$ by conditions :i* are probably n yet all recogniz$1 b among !m we %\ld d\btless include a high temp]ature1 int5se pressure1 me*anical movem5t result+ 9 cru%+1 %e>+ & foliation1 & ! pres5ce ( wat] 9 _! pores4 ,x is among igne\s rocks t ! progressive /ages ( metamorphism c be mo/ easily trac$4 ,_! def9ite orig9al /ructure & m9]al composition af=d a />t+-po9t f :i* ! inve/igation may be begun & pursu$4 ,:]e an igne\s rock has be5 invad$ by metamorphic *anges1 x may be obs]v$ to h be5 fir/ brok5 d[n into sep>ate l5ticles1 ! cores ( :i* may /ill reta91 ) little or no alt]ation1 ! orig9al *>act]i/ic m9]als & cry/all9e /ructure ( ! rock4 ,betwe5 ^! l5ticles1 ! int]v5+ portions h be5 cru%$ d[n into a p[d] or pa/e1 :i* seems to h be5 squeez$ r\nd & pa/ !m1 & %[s a lam9at$ arrangem5t t resembles ! fl[-/ructure 9 lavas4 ,z ! degree ( metamorphism increases1 ! l5ticles dim9i% 9 size1 & ! int]v5+ cru%$ & foliat$ matrix increases 9 am\nt1 until at la/ x may =m ! entire mass ( ! rock4 ,:ile ! orig9al m9]als are ?us brok5 d[n1 new v>ieties make _! appe>ance4 ,( ^!1 among ! e>lie/ to pres5t !mselves are usually ! micas1 t imp>t _! *>act]i/ic silv]y %e5 to ! surfaces ( ! folia along :i* !y spread4 ,y\ng] felsp>s1 z well z mica1 are develop$1 & !re arise also sillimanite1 g>net1 &alusite & _m o!rs4 ,! texture becomes m co>sely cry/all9e1 & ! segregation ( ! con/itu5t m9]als m def9ite along ! l9es ( foliation4 ,f ! f9e/ silky phyllites a graduation may be trac$ ?r\gh successively co>s] mica-s*i/s1 until we rea* ! almo/ granitic texture ( ! co>se/ gneisses4 ,regional metamorphism has aris5 9 ! he>t ( m\nta9-*a9s1 & 9 any o!r di/rict :]e ! de=mation ( ! cru/ has be5 suffici5tly int5se4 ,!re is ano!r type ( alt]ation t]m$ contact-metamorphism1 :i* is develop$ ar\nd masses ( igne\s rock1 especially :]e ^! h be5 intrud$ 9 l>ge bosses among /ratifi$ =mations4 ,x is p>ticul>ly display$ ar\nd masses ( granite1 :]e s&/ones are f\nd alt]$ into qu>tzite1 %ales & grits into s*i/ose comp\nds1 & :]e sometimes fossils are /ill recognizable among ! metamorphic m9]als4 ,,division ,,ii4--,,epig5e ,,or ,,sup]ficial ,,action ,x is on ! surface ( ! globe1 & by ! op]ation ( ag5ts work+ !re1 t at pres5t ! *ief am\nt ( visible geological *ange is effect$4 ,9 consid]+ ?is bran* ( inquiry1 we are n involv$ 9 a prelim9>y difficulty reg>d+ ! v nature ( ! ag5cies z is ! case 9 ! inve/igation ( plutonic action4 ,on ! contr>y1 ! surface ag5ts are c>ry+ on _! "w "u our v eyes4 ,we c wat* x 9 all its /ages1 measure its progress1 & m>k 9 _m ways h[ accurately x repres5ts simil> *anges :i*1 = long ages previ\sly1 mu/ h be5 effect$ by ! same means4 ,b 9 ! sy/ematic treatm5t ( ?is subject we enc\nt] a difficulty ( ano!r k9d4 ,we discov] t :ile ! op]ations to be discuss$ are num]\s & readily obs]vable1 !y are s int]wov5 into "o great network t any sep>ation ( !m "u diff]5t subdivisions is sure to be m or less artificial & to convey an errone\s impression4 ,:ile1 !re=e1 "u ! unavoidable necessity ( mak+ use ( su* a classification ( subjects1 we mu/ always be> 9 m9d t x is employ$ m]ely = conv5i5ce1 & t 9 nature sup]ficial geological action mu/ be cont9ually view$ z a :ole1 s9ce ! "w ( ea* ag5t has con/ant ref]5ce to t ( ! o!rs1 & is n prop]ly intelligible unless t connexion be kept 9 view4 ,! movem5ts ( ! air2 ! evaporation f l& & sea2 ! fall ( ra91 hail & sn[2 ! fl[ ( riv]s & glaci]s2 ! tides1 curr5ts & waves ( ! ocean2 ! gr[? & decay ( organiz$ exi/5ce1 alike on l& & 9 ! dep?s ( ! sea2--9 %ort1 ! :ole circle ( movem5t1 :i* is cont9ually 9 progress ^u ! surface ( our planet1 are ! subjects n[ to be exam9$4 ,x is desirable to adopt "s g5]al t]m to embrace ! :ole ( ?is range ( inquiry4 ,= ?is end ! ^w epig5e 7,gr4 ,greek3 epi1 ^u7 has be5 sugge/$ z a conv5i5t t]m1 & anti!tical to hypog5e 7,gr4 ,greek3 hypo1 "u71 or subt]ranean action4 ,a simple arrangem5t ( ?is "p ( ,geological ,dynamics is 9 ?ree sections3 ,a4 ,air4--,! influ5ce ( ! atmosph]e 9 de/roy+ & =m+ rocks4 ,b4 ,wat]4--,! geological functions ( ! circulation ( wat] ?r\gh ! air & betwe5 sea & l&1 & ! action ( ! sea4 ,c4 ,life4--,! "p tak5 by plants & animals 9 pres]v+1 de/roy+ or reproduc+ geological =mations4 ,! words de/ructive1 reproductive & cons]vative1 employ$ 9 describ+ ! op]ations ( ! epig5e ag5ts1 d n necess>ily imply t any?+ useful to man is de/roy$1 reproduc$ or pres]v$4 ,on ! contr>y1 ! de/ructive action ( ! atmosph]e may turn b>r5 rock into ri* soil1 :ile its reproductive effects sometimes turn ri* l& into b>r5 des]t4 ,aga91 ! cons]vative influ5ce ( vegetation has sometimes = c5turies reta9$ z b>r5 morass :at might o!rwise h become ri* mead[ or luxuriant woodl&4 ,! t]ms1 !re=e1 are us$ 9 a /rictly geological s5se1 to d5ote ! removal & re-deposition ( mat]ial1 & its ag5cy 9 pres]v+ :at lies b5ea? x4 7,a7 ,! ,air4 ,z a geological ag5t1 ! air br+s ab\t *anges p>tly by its compon5t gases & p>tly by its movem5ts4 ,its de/ructive action is bo? *emical & me*anical4 ,! *emical *anges are probably ma9ly1 if n entirely1 due to ! moi/ure ( ! air1 & p>ticul>ly to ! gases1 vap\rs & organic matt] :i* ! moi/ure conta9s4 ,dry air seems to h little or no appreciable influ5ce 9 promot+ ^! reactions4 ,z ! *anges 9 "q are simil> to ^? mu* m abundantly br\ght ab\t by ra9 !y are describ$ 9 ! foll[+ section "u ! division on ra94 ,among ! m recognizable me*anical *anges effect$ 9 ! atmosph]e1 "o ( consid]able importance is to be se5 9 ! result ( great & rapid *anges ( temp]ature4 ,heat exp&s rocks1 :ile cold contracts !m4 ,9 c\ntries ) a great annual range ( temp]ature1 consid]able difficulty is sometimes exp]i5c$ 9 select+ build+ mat]ials liable to be little affect$ by ! alt]nate expansion & contraction1 :i* prev5ts ! jo9ts ( masonry f rema9+ close & tight4 ,9 dry tropical climates1 :]e ! days are int5sely hot & ! nights extremely cold1 ! rapid nocturnal contraction produces a /ra9 s great z to rival fro/ 9 its influ5ce ^u ! surface ( expos$ rocks1 dis9tegrat+ !m into s&1 or caus+ !m to crack or peel (f 9 sk9s or irregul> pieces4 ,dr ,liv+/one f\nd 9 ,africa 7#ab deg4 ,s4 lat41 #cd deg4 ,e4 long47 t surfaces ( rock :i* dur+ ! day 7 heat$ up to #acg deg4 ,fahr41 cool$ s rapidly by radiation at night t1 unable to su/a9 ! /ra9 ( contraction1 !y split & ?rew (f %>p angul> fragm5ts f a few ounces to #ajj or #bjj lb 9 weight4 ,9 temp]ate regions ?is action1 ?\gh mu* less pron\nc$1 /ill makes itself felt4 ,9 ^! climates1 h[ev]1 & /ill m 9 high latitudes1 some:at simil> results are br\ght ab\t by fro/4 ,by its motion 9 w9d ! air drives loose s& ov] rocks1 & 9 c\rse ( "t abrades & smoo!s !m4 ,des]t poli% is ! "n giv5 to ! *>act]i/ic lu/r\s surface ?us imp>t$4 ,holes are said to be drill$ 9 w9d[ glass at ,cape ,cod by ! same ag5cy4 ,cavities are n[ & !n holl[$ out ( rocks by ! gyration 9 !m ( little fragm5ts ( /one or gra9s ( s& kept 9 motion by ! w9d4 ,hurricanes =m important geological ag5ts ^u l& 9 uproot+ trees1 & ?us sometimes imp$+ ! dra9age ( a c\ntry & giv+ rise to ! =mation ( peat mosses4 ,! reproductive action ( ! air arises p>tly f ! effect ( ! *emical & me*anical dis9tegration involv$ 9 ! process ( wea!r+1 & p>tly f ! transport+ p[] ( w9d & ( a]ial curr5ts4 ,! lay] ( soil1 :i* cov]s s mu* ( ! surface ( ! l&1 is ! result ( ! decay ( ! und]ly+ rocks1 m+l$ ) m9]al matt] bl[n ov] ! gr\nd by w9d1 or wa%$ ?i!r by ra91 & ) ! m\ld]+ rema9s ( plants & animals4 ,! ext5t to :i* f9e du/ may be transport$ ov] ! surface ( ! l& c h>dly be realiz$ 9 c\ntries clo!d ) a cov]+ ( vegetation1 ?\gh ev5 !re1 9 dry wea!r dur+ spr+1 cl\ds ( du/ may (t5 be se5 bl[n away by w9d f b>e pl\gh$ fields4 ,int]cept$ by ! leaves ( plants & wa%$ d[n to _! roots by ra91 ?is du/ goes to increase ! soil bel[4 ,9 arid climates1 :]e du/ cl\ds are d5se & frequ5t1 enorm\s quantities ( f9e m9]al p>ticles are ?us borne along & accumulat$4 ,! rem>kable deposit ( ,loess1 :i* is sometimes m ?an #aejj ft4 ?ick & cov]s ext5sive areas 9 ,*9a & o!r c\ntries1 is reg>d$ z due to ! drift+ ( du/ by w9d4 ,aga9 ! dunes ( s& s abundant along ! inn] side ( s&y sea-bea*es 9 _m diff]5t p>ts ( ! _w are attributable to ! same action4 7,b7 ,wat]4 ,9 treat+ ( ! epig5e action ( wat] 9 geological processes x w be conv5i5t to deal fir/ ) its op]ations 9 trav]s+ ! l&1 & !n ) ^? :i* x p]=ms 9 ! sea4 ,! circulation ( wat] f l& to sea & aga9 f sea to l& con/itutes ! fundam5tal cause ( mo/ ( ! daily *anges by :i* ! surface ( ! l& is affect$4 #a4 ,ra94--,ra9 effects two k9ds ( *anges ^u ! surface ( ! l&4 ,x acts *emically ^u soils & /ones1 & s9k+ "u gr\nd cont9ues a great s]ies ( simil> reactions !re4 ,x acts me*anically1 by wa%+ away loose mat]ials1 & ?us p[]fully affect+ ! cont\rs ( ! l&4 ,its *emical action dep5ds ma9ly ^u ! nature & proportion ( ! sub/ances :i*1 9 desc5d+ to ! e>?1 x ab/racts f ! atmosph]e4 ,ra9 always absorbs a little air1 :i*1 9 addition to its nitrog5 & oxyg51 conta9s c>bonic acid1 & 9 m9ute proportions1 sodium *loride1 sulphuric acid & o!r +r$i5ts1 especially inorganic du/1 organic p>ticles & liv+ g]ms4 ,probably ! mo/ g5]ally effici5t ( ^! con/itu5ts are oxyg51 c>bonic acid & organic matt]4 ,arm$ ) ^! reag5ts1 ra9 effects a *emical decomposition ( ! rocks on :i* x falls1 & ?r\gh :i* x s9ks und]gr\nd4 ,! pr9cipal *anges ?us produc$ are z foll[s3 7a7 ,oxidation4--,ow+ to ! prom95ce ( oxyg5 9 ra9-wat]1 & its read9ess to unite ) any sub/ance :i* c conta9 m ( x1 a ?9 oxidiz$ pellicle is =m$ on ! surface ( _m rocks on :i* ra9 falls1 & ?is oxidiz$ lay] if n at once wa%$ (f1 s9ks deep] until a cru/ is =m$ ov] ! /one4 ,a famili> illu/ration ( ?is action is af=d$ by ! ru/1 or oxide1 :i* =ms on iron :5 expos$ to moi/ure1 ?\gh ?is iron may be kept long bright if all[$ to rema9 scre5$ f moi/ air & ra94 7b7 ,deoxidation4--,organic matt] hav+ an aff9ity = m oxyg5 decomposes p]oxides by depriv+ !m ( "s "p ( _! %>e ( t elem5t & r$uc+ !m to protoxides4 ,^! *anges are especially noticeable among ! iron oxides s abundantly diffus$ among rocks4 ,h5ce ra9-wat]1 9 s9k+ ?r\gh soil & obta9+ su* organic matt]1 becomes !reby a r$uc+ ag5t4 7c7 ,solution4--,?is may take place ei!r by ! simple action ( ! wat]1 z 9 ! solution ( rock-salt1 or by ! influ5ce ( ! c>bonic acid pres5t 9 ! ra94 7d7 ,=mation ( ,c>bonates4--,a famili> example ( ! action ( c>bonic acid 9 ra9 is to be se5 9 ! corrosion ( expos$ m>ble slabs4 ,! c>bonic acid dissolves "s ( ! lime1 :i*1 z a bic>bonate1 is held 9 solution 9 ! c>bonat$ wat]1 b is deposit$ aga9 :5 ! wat] loses its c>bonic acid or evaporates4 ,x is n m]ely c>bonates1 h[ev]1 :i* are liable to ?is k9d ( de/ruction4 ,ev5 silicates ( lime1 pota% & soda1 comb9ations exi/+ abundantly z con/itu5ts ( rocks1 are attack$2 _! silica is lib]at$1 & _! alkalis or alkal9e e>?s1 becom+ c>bonates1 are remov$ 9 solution4 7e7 ,hydration4--,"s m9]als1 conta9+ little or no wat]1 & !re=e call$ anhydr\s1 :5 expos$ to ! action ( ! atmosph]e1 absorb wat]1 or become hydr\s1 & are !n usually m prone to fur!r *ange4 ,h5ce ! rocks ( :i* !y =m "p become dis9tegrat$4 ,besides ! reactions "h enum]at$1 a consid]able am\nt ( decay may be obs]v$ z ! result ( ! pres5ce ( sulphuric & nitric acid 9 ! air1 especially 9 t ( l>ge t[ns & manufactur+ di/ricts1 :]e mu* coal is consum$4 ,metallic surfaces1 z well z v>i\s k9ds ( /one1 are !re corrod$1 :ile ! mort> ( walls may (t5 be obs]v$ to be sl[ly swell+ out & dropp+ (f1 ow+ to ! conv]sion ( ! lime into sulphate4 ,great injury is likewise done f a simil> cause to m>ble monum5ts 9 expos$ gravey>ds4 ,! g5]al result ( ! dis9tegrat+ action ( ! air & ( ra91 includ+ also t ( plants & animals1 to be notic$ 9 ! sequel1 is d5ot$ by ! t]m wea!r+4 ,! am\nt ( decay dep5ds p>tly on conditions ( climate1 especially ! range ( temp]ature1 ! abundance ( moi/ure1 height above ! sea & exposure to preval5t w9ds4 ,_m rocks liable to be saturat$ ) ra9 & rapidly dri$ "u a w>m sun are apt to dis9tegrate at ! surface ) comp>ative rapidity4 ,! nature & progress ( ! wea!r+ are ma9ly gov]n$ by ! composition & texture ( ! rocks expos$ to x4 ,rocks compos$ ( p>ticles liable to little *emical *ange f ! influ5ce ( moi/ure are be/ fitt$ to resi/ wea!r+1 provid$ !y possess suffici5t cohesion to )/& ! me*anical processes ( dis9tegration4 ,silice\s s&/ones are excell5t examples ( ?is p]man5ce4 ,consi/+ :olly or ma9ly ( ! durable m9]al qu>tz1 !y are sometimes able s to )/& decay t build+s made ( !m /ill reta91 aft] ! lapse ( c5turies1 ! *isel-m>ks ( ! build]s4 ,"s rocks1 :i* yield ) comp>ative rapidity to ! *emical attacks ( moi/ure1 may %[ little or no m>k ( dis9tegration on _! surface4 ,?is is p>ticul>ly ! case ) c]ta9 calc>e\s rocks4 ,lime/one :5 pure is :olly soluble 9 acidulat$ wat]4 ,ra9 fall+ on su* a rock removes "s ( x 9 solution1 & w cont9ue to d s until ! :ole is dissolv$ away4 ,b :]e a lime/one is full ( impurities1 a wea!r$ cru/ ( m or less insoluble p>ticles rema9s aft] ! solution ( ! calc>e\s "p ( ! /one4 ,h5ce ! relative purity ( lime/ones may be r\ghly det]m9$ by exam9+ _! wea!r$ surfaces1 :]e1 if !y conta9 mu* s&1 ! gra9s w be se5 project+ f ! calc>e\s matrix1 & :]e1 %\ld ! rock be v f]rug9\s1 ! yell[ hydr\s p]oxide1 or o*re1 w be f\nd z a p[d]y cru/4 ,9 lime/ones conta9+ abundant encr9ites1 %ells1 or o!r organic rema9s1 ! wea!r$ surface commonly pres5ts ! fossils /&+ out 9 relief4 ,! cry/all9e arrangem5t ( ! lime 9 ! organic /ructures enables !m to resi/ dis9tegration bett] ?an ! g5]al me*anically aggregat$ matrix ( ! rock4 ,an exp]i5c$ fossil collector w always se>* well su* wea!r$ surfaces1 = he (t5 f9ds !re1 delicately pick$ out by ! wea!r1 m9ute & frail fossils :i* are :olly invisible on a fre%ly brok5 surface ( ! /one4 ,_m rocks wea!r ) a ?ick cru/1 or ev5 decay inw>ds = _m feet or y>ds4 ,basalt1 = example1 (t5 %[s a yell[i%-br[n f]rug9\s lay] on its surface1 =m$ by ! conv]sion ( its felsp> into kaol91 & ! removal ( its calcium silicate z c>bonate1 by ! hydration ( its oliv9e & augite & _! conv]sion into s]p5t9e1 or "s o!r hydr\s magnesian silicate1 & by ! conv]sion ( its magnetite into limonite4 ,granite sometimes %[s 9 a mo/ rem>kable way ! di/ance to :i* wea!r+ c rea*4 ,x may occasionally be dug into = a dep? ( #bj or #cj ft41 ! qu>tz cry/als & ve9s reta9+ _! orig9al positions1 :ile ! felsp> is completely kaol9iz$4 ,x is to ! endlessly v>i$ effects ( wea!r+ t ! abundant fanta/ic %apes assum$ by crags & o!r rocky masses are due4 ,mo/ v>ieties ( rock h _! own *>act]i/ic modes ( wea!r+1 :]eby !y may be recogniz$ ev5 f a di/ance4 ,to "s ( ^! features ref]5ce w be made 9 ,"p ,,viii4 ,! me*anical action ( ra91 :i* is intimately b\nd up ) its *emical action1 consi/s 9 wa%+ (f ! f9e sup]ficial p>ticles ( rocks :i* h be5 corrod$ & loos5$ by ! process ( wea!r+1 & 9 ?us lay+ op5 fre% portions to ! same influ5ces ( decay4 ,! detritus s remov$ is p>tly c>ri$ d[n into ! soil :i* is !reby enri*$1 p>tly held 9 susp5sion 9 ! little runnels into :i* ! ra9-drops ga!r z !y beg9 to fl[ ov] ! l&1 p>tly pu%$ d[nw>ds along ! surface ( slop+ gr\nd4 ,a good deal ( x f9ds its way into ! ne>e/ brooks & riv]s1 :i* are consequ5tly made muddy by heavy ra94 ,x is natural t a casual consid]ation ( ! subject %\ld lead to an impression t1 ?\gh ! g5]al result ( ! fall ( ra9 ^u a l&-surface mu/ lead to "s am\nt ( dis9tegration & l[]+ ( t surface1 ! process mu/ be s sl[ & slight z h>dly to be consid]$ ( mu* importance among geological op]ations4 ,b fur!r att5tion w %[ su* an impression to be s+ul>ly errone\s4 ,x loses sight ( ! fact t a *ange :i* may be h>dly appreciable )9 a human lifetime1 or ev5 )9 ! comp>atively brief span ( geological "t embrac$ 9 ! compass ( human hi/ory1 may nev]!less become gigantic 9 its results 9 ! c\rse ( imm5sely protract$ p]iods4 ,an in/ructive lesson 9 ! erosive action ( ra9 may be f\nd 9 ! pitt$ & *annell$ surface ( gr\nd ly+ "u ! drip ( ! eaves ( a cottage4 ,! fragm5ts ( /one & pebbles ( gravel t =m "p ( ! soil c !re be se5 /ick+ out ( ! gr\nd1 because be+ h>d !y resi/ ! impetus ( ! fall+ drops1 protect+ = a "t ! e>? b5ea? !m1 :ile t :i* surr\nd$ & cov]$ !m is wa%$ away4 ,f ?is famili> illu/ration ! obs]v] may advance ?r\gh e /age 9 ! disappe>ance ( mat]ial :i* once cov]$ ! surface1 until he comes to examples :]e once cont9u\s & ?ick %eets ( solid rock h be5 r$uc$ to a few fragm5ts or h be5 entirely remov$4 ,s9ce ! :ole l& surface ov] :i* ra9 falls is expos$ to ?is wa/e1 ! sup]ficial cov]+ ( decay$ rock or soil1 z ,hutton insi/$1 is con/antly1 ?\gh imp]ceptibly1 travell+ outw>d & d[nw>d to ! sea4 ,9 ?is process ( transport ra9 is an important c>ry+ ag5t1 :ile at ! same "t x s]ves to connect ! "w ( ! o!r dis9tegrat+ =ces1 & to make x conducive to ! g5]al degradation ( ! l&4 ,?\gh ?is decay is g5]al & con/ant1 x is obvi\sly n uni=m4 ,9 "s places :]e1 f ! nature ( ! rock1 f ! flatness ( ! gr\nd1 or f o!r causes1 ra9 works "u great difficulties1 ! rate ( wa/e may be extremely sl[4 ,9 o!r places x may be rapid en\gh to be appreciable f ye> to ye>4 ,a survey ( ?is dep>tm5t ( geological activity %[s h[ unequal wa/+ by ra91 comb9$ ) ! op]ations ( brooks & riv]s1 has produc$ ! details ( ! pres5t relief ( ! l&1 ^? tracts :]e ! de/ruction has be5 greate/ =m+ holl[s & valleys1 o!rs1 :]e x has be5 less1 ris+ into ridges & hills 7,"p ,,viii474 ,ra9-action is n m]ely de/ructive1 b is accompani$ ) reproductive effects1 *ief ( :i* is ! =mation ( soil4 ,9 fav\rable situations x has ga!r$ toge!r accumulations ( loam & e>? f neighb\r+ high] gr\nd1 su* z ! brick-e>?1 head1 & ra9-wa% ( ! s\? ( ,engl&--e>?y deposits1 sometimes full ( angul> /ones1 d]iv$ f ! suba]ial wa/e ( ! rocks ( ! neighb\rhood4 #b4 ,und]gr\nd ,wat]4--,( ! ra9 :i* falls ^u ! l& "o portion fl[s (f into brooks & riv]s by :i* ! wat] is conduct$ back to ! ocean2 ! l>g] "p1 h[ev]1 s9ks into ! gr\nd & disappe>s4 ,x is ?is latt] "p :i* has n[ to be consid]$4 ,ov] & above ! proportion ( ! ra9fall :i* is absorb$ by liv+ vegetation & by ! soil1 !re is a cont9ual filt]+ d[n ( ! wat] f ! surface into ! rocks t lie bel[1 :]e x p>tly lodges 9 pores & int]/ices1 & p>tly f9ds its way into subt]ranean jo9ts & fissures1 9 :i* x p]=ms an und]gr\nd circulation1 & ultimately issues once m at ! surface 9 ! =m ( spr+s 7q4v474 ,9 ! c\rse ( ?is circulation ! wat] p]=ms an important geological task4 ,n only c>ry+ d[n ) x ! sub/ances :i* ! ra9 has ab/ract$ f ! air1 b obta9+ m acids & organic matt] f ! soil1 x is enabl$ to effect *emical *anges 9 ! rocks und]nea?1 & especially to dissolve lime/one & o!r calc>e\s =mations4 ,s consid]able is ! ext5t ( ?is solution 9 "s places t ! spr+s :i* come to ! surface1 & beg9 !re to evaporate & lose "s ( _! c>bonic acid1 conta9 m dissolv$ lime ?an !y c hold4 ,!y consequ5tly deposit x 9 ! =m ( calc>e\s tuff or s9t] 7q4v474 ,o!r subt]ranean wat]s issue ) a l>ge proportion ( iron-salts 9 solution :i* =m deposits ( o*re4 ,! v>i\s m9]al spr+s s l>gely made use ( = ! mitigation or cure ( diseases owe _! prop]ties to ! v>i\s salts :i* !y h dissolv$ out ( rocks und]gr\nd4 ,z ! result ( prolong$ subt]ranean solution 9 lime/one di/ricts1 passages & caves 7q4v471 sometimes ( great wid? & l5g?1 are =m$4 ,:5 ^! lie ne> ! surface _! ro(s sometimes fall 9 & engulf brooks & riv]s1 :i* !n fl[ = "s way und]gr\nd until ! tunnels conduct !m back aga9 to daylight on "s l[] gr\nd4 ,besides its *emical activity wat] ex]ts among subt]ranean rocks a me*anical influ5ce :i* leads to important *anges 9 ! topography ( ! surface4 ,9 remov+ ! m9]al matt]1 ei!r 9 solution or z f9e s$im5t1 x sometimes loos5s ! support ( ov]ly+ masses ( rock :i* may ultimately give way on slop+ gr\nd1 & ru% d[n ! declivities 9 ! =m ( l&slips4 ,^! de/ructive effects are specially frequ5t on ! sides ( valleys 9 m\nta9\s c\ntries & on l9es ( sea-cliff4 #c4 ,brooks & ,riv]s4--,z geological ag5ts ! runn+ wat]s on ! face ( ! l& play an important "p 9 epig5e *anges4 ,l ra9 & spr+s !y h bo? a *emical & a me*anical action4 ,! latt] receives mo/ att5tion1 z x und\bt$ly is ! m important2 b ! =m] ought n to be omitt$ 9 any survey ( ! g5]al wa/e ( ! e>?'s surface4 ,! wat] ( riv]s mu/ possess ! p[]s ( a *emical solv5t l ra9 & spr+s1 ?\gh its actual "w 9 ?is respect c be less easily measur$1 see+ t riv] wat] is directly d]iv$ f ra9 & spr+s1 & necess>ily conta9s 9 solution m9]al sub/ances suppli$ to x by !m & n by its own op]ation4 ,nev]!less1 x is sometimes easy to prove t /reams dissolve *emically ! rocks ( _! *annels4 ,?us1 9 lime/one di/ricts ! base ( ! cliffs ( riv] rav9es may be f\nd eat5 away into tunnels1 ar*es1 & ov]hang+ projections1 pres5t+ 9 _! smoo? surfaces a great contra/ to ! angul> jo9t$ faces ( ! same rock1 :]e n[ expos$ to ! influ5ce only ( ! wea!r on ! high] p>ts ( ! cliff4 ,! me*anical action ( riv]s consi/s 7a7 9 transport+ mud1 s&1 gravel & blocks ( /one f high] to l[] levels2 7b7 9 us+ ^! loose mat]ials to wid5 & deep5 _! *annels by erosion2 7c7 9 deposit+ _! load ( detritus :]ev] possible & ?us to make new geological =mations4 7a7 ,transport+ ,p[]4--,riv]-wat] is di/+ui%$ f t ( spr+s by be+ less transp>5t1 because x conta9s m or less m9]al matt] 9 susp5sion1 d]iv$ ma9ly f :at is wa%$ d[n by ra91 or c>ri$ 9 by brooks1 b p>tly also f ! abrasion ( ! wat]-*annels by ! erosive action ( ! riv]s !mselves4 ,! progress ( ?is burd5 ( detritus may be in/ructively foll[$ f ! m\nta9-tribut>ies ( a riv] d[n to ! m\? ( ! ma9 /ream4 ,9 ! high gr\nds ! wat]-c\rses may be obs]v$ to be *ok$ ) l>ge fragm5ts ( rock dis5gag$ f ! cliffs & crags on ei!r side4 ,trac$ d[nw>ds ! blocks are se5 to become gradually small] & m r\nd$4 ,!y are gr\nd aga9/ ea* o!r1 & ^u ! rocky sides & bottom ( ! *annel1 gett+ m & m r$uc$ z !y desc5d1 & at ! same "t abrad+ ! rocks ov] or aga9/ :i* !y are driv54 ,h5ce a great deal ( debris is produc$1 & is swept along by ! onw>d & d[nw>d movem5t ( ! wat]4 ,! f9] portions1 su* z mud & f9e s&1 are c>ri$ 9 susp5sion1 & imp>t ! *>act]i/ic turbidity to riv]-wat]2 ! co>s] s& & gravel are driv5 along ! riv]-bottom4 ,! proportion ( susp5d$ m9]al matt] has be5 asc]ta9$ ) m or less precision = a numb] ( riv]s4 ,z an illu/rative example ( a riv] dra9+ a va/ area ) diff]5t climates1 =ms ( surface & geological /ructure ! ,mississippi may be cit$4 ,! av]age proportion ( s$im5t 9 its wat] 0 asc]ta9$ by ,humphreys & ,abbot to be #a_/aejj by weight or #a_/bijj by volume4 ,^! eng9e]s f\nd t1 9 addition to ?is susp5d$ mat]ial1 co>se detritus is con/antly be+ pu%$ =w>d along ! b$ ( ! riv] into ! ,gulf ( ,mexico1 to an am\nt :i* !y e/imat$ at ab\t #gej1jjj1jjj cubic ft4 ( s&1 e>? & gravel2 !y conclud$ t ! ,mississippi c>ries into ! gulf e ye> an am\nt ( me*anically transport$ s$im5t suffici5t to make a prism "o squ>e mile 9 area & #bfh ft4 9 height4 7b7 ,excavat+ ,p[]4--,x is by means ( ! s&1 gravel & /ones :i* !y drive aga9/ ! sides & bottoms ( _! *annels t /reams h holl[$ out ! b$s 9 :i* !y fl[4 ,n only is ! co>se detritus r$uc$ 9 size by ! friction ( ! /ones aga9/ ea* o!r1 b1 at ! same "t1 ^! mat]ials abrade ! rocks aga9/ :i* !y are driv5 by ! curr5t4 ,:]e1 ow+ to ! %ape ( ! bottom ( ! *annel1 ! /ones are caught 9 eddies1 & are kept :irl+ r\nd !re1 !y become m & m worn d[n !mselves1 & at ! same "t sc\r out bas9-%ap$ cavities1 or pot-holes1 9 ! solid rock bel[4 ,! unev5 b$ ( a swiftly fl[+ /ream may 9 ?is way be honeycomb$ ) su* erod$ bas9s :i* coalesce & ?us appreciably l[] ! surface ( ! b$4 ,! /eep] ! *annel1 o!r conditions be+ equal1 ! m rapid w be ! erosion4 ,geological /ructure also affects ! "* & rate ( ! excavation4 ,:]e ! rocks are s arrang$ z to fav\r ! =mation & p]si/5ce ( a wat]fall1 a long *asm may be holl[$ out l t ( ! ,niag>a bel[ ! falls1 :]e a h>d ?ick b$ ( ne>ly flat lime/one lies on s(t] & m easily erod$ %ales4 ,! latt] are scoop$ out f und]nea? ! lime/one1 :i* f "t to "t breaks (f 9 l>ge masses & ! wat]fall gradually retreats up /ream1 :ile ! rav9e is proportionately l5g!n$4 ,to ! excavat+ p[] ( riv]s ! orig9 ( ! valley sy/ems ( ! dry l& mu/ be ma9ly assign$ 7see ,"p ,,viii474 7c7 ,reproductive ,p[]4--,s long z a /ream fl[s ov] a /eep declivity its velocity suffices to keep ! s$im5t 9 susp5sion1 b :5 f any cause1 su* z a dim9ution ( slope1 ! velocity is *eck$1 ! transport+ p[] is less5$ & ! s$im5t beg9s to fall to ! bottom & to rema9 !re4 ,h5ce v>i\s riv]-=m$ or alluvial deposits are laid d[n4 ,^! sometimes cov] consid]able spaces at ! foot ( m\nta9s4 ,! floors ( valleys are /rewn ) detritus1 & _! level may !reby be s5sibly rais$4 ,9 floods ! gr\nd inundat$ on ei!r side ( a /ream int]cepts "s "p ( ! detritus1 :i* is !n spread ov] ! flood-pla9 & gradually height5s x4 ,at ! same "t ! /ream cont9ues to erode ! *annel1 & ultimately is unable to rea* ! old flood-pla94 ,x consequ5tly =ms a new pla9 at a l[] level1 & ?us1 by degrees1 x comes to be flank$ on ei!r side by a s]ies ( successive t]races or plat=ms1 ea* ( :i* m>ks "o ( its =m] levels4 ,:]e a riv] ent]s a l>ge body ( wat] its curr5t is *eck$4 ,"s ( its s$im5t is consequ5tly dropp$1 & by sl[ accumulation =ms a delta 7q4v474 ,on l&1 e lake 9 m\nta9 di/ricts furni%es in/ances ( ?is k9d ( alluvium4 ,b ! mo/ important deltas are ^? =m$ 9 ! sea at ! m\?s ( ! l>g] riv]s ( ! globe4 ,(f _m coa/-l9es ! detritus wa%$ f ! l& ga!rs into b>s1 :i* enclose long /rips ( wat] m or less completely sep>at$ f ! sea outside & kn[n z lagoons4 ,a *a9 ( su* lagoon-b>ri]s /ret*es = hundr$s ( miles r\nd ! ,gulf ( ,mexico & ! ea/]n %ores ( ! ,unit$ ,/ates4 #d4 ,lakes4--,^! %eets ( wat]1 consid]$ z a :ole1 d n belong to ! normal sy/em ( dra9age on ! l& :]eby valleys are excavat$4 ,on ! contr>y !y are exceptional to x2 = ! con/ant t5d5cy ( runn+ wat] is to fill !m up1 or to dra9 !m by we>+ d[n ! b>ri]s t conta9 !m at _! outfl[4 ,"s ( !m are ref]able to movem5ts ( ! t]re/rial cru/ :]eby depressions arise on ! surface ( ! l&1 z has be5 not$ aft] e>?quakes4 ,o!rs h aris5 f solution su* z t ( rock-salt or ( lime/one1 ! removal ( :i* by und]gr\nd wat] causes a subsid5ce ( ! gr\nd above4 ,a ?ird type ( lake-bas9 occurs 9 regions t are n[ or h once be5 subject to ! erosive action ( glaci]s 7see "u next subdivision1 ,t]re/rial ,ice74 ,_m small lakes or t>ns h be5 caus$ by ! deposit ( debris across a valley z by l&slips or mora9es4 ,consid]$ f a geological po9t ( view1 lakes p]=m an important function 9 regulat+ ! dra9age ( ! gr\nd bel[ _! outfall & dim9i%+ ! de/ructive effects ( floods1 9 filt]+ ! wat] receiv$ f _! afflu5t /reams1 & 9 provid+ undi/urb$ areas ( deposit 9 :i* ?ick & ext5sive lacu/r9e =mations may be accumulat$4 ,9 ! inl& bas9s ( "s dry climates ! lakes are salt1 ow+ to excess ( evaporation1 & _! bottoms become ! sites ( *emical deposits1 p>ticul>ly ( *lorides ( sodium & magnesium1 & calcium sulphate & c>bonate4 #e4 ,t]re/rial ,ice4--,ea* ( ! =ms assum$ by froz5 wat] has its own *>act]i/ic action 9 geological processes4 ,fro/ has a p[]ful influ5ce 9 break+ up damp soils & surfaces ( /one 9 ! pores or cracks ( :i* moi/ure has lodg$4 ,! wat] 9 freez+ exp&s1 & 9 s do+ pu%es asund] ! compon5t p>ticles ( soil or /one1 or wid5s ! space betwe5 ! walls ( jo9ts or crevices4 ,:5 ! ice melts ! loos5$ gra9s rema9 ap>t ready to be wa%$ away by ra9 or bl[n (f by w9d1 :ile by ! wid5+ ( jo9ts l>ge blocks ( rock are deta*$ f ! faces ( cliffs4 ,:]e riv]s or lakes are froz5 ov] ! ice ex]ts a m>k$ pressure on _! banks2 & :5 x breaks up l>ge %eets ( x are driv5 a%ore1 pu%+ up quantities ( gravel & /ones above ! level ( ! wat]4 ,! pil+ up ( ! disrupt$ ice aga9/ ob/ructions 9 riv]s ponds back ! wat]1 & (t5 leads to de/ructive floods :5 ! ice b>ri]s break4 ,:]e ! ice has =m$ r\nd b\ld]s 9 %all[ wat]1 or at ! bottom 7an*or-ice71 x may lift ^! up :5 ! fro/ gives way1 & may transport !m = "s di/ance4 ,ice =m$ 9 ! atmosph]e1 & desc5d+ to ! gr\nd 9 ! =m ( hail1 (t5 causes great de/ruction to vegetation & n infrequ5tly to animal life4 ,:]e ! froz5 moi/ure rea*es ! e>? z sn[1 x s]ves to protect rock1 soil & vegetation f ! effects ( fro/2 b on slop+ gr\nd x is apt to give rise to de/ructive avalan*es or l&slips1 :ile indirectly1 by its rapid melt+1 x may cause s]i\s floods 9 riv]s4 ,b ! mo/ /rik+ geological "w p]=m$ by t]re/rial ice is t a*iev$ by glaci]s 7q4v47 & ice-%eets4 ,^! va/ masses ( mov+ ice1 :5 !y desc5d f m\nta9s :]e ! /eep] rocks are cle> ( sn[1 receive on _! surface ! debris deta*$ by fro/ f ! declivities above1 & be> ^! mat]ials to l[] levels or to ! sea4 ,enorm\s quantities ( rock-rubbi% are ?us transport$ 9 ! ,alps & o!r high m\nta9 ranges4 ,:5 ! ice retreats ! b\ld]s c>ri$ by x are dropp$ :]e x melts1 & left !re z memorials ( ! =m] ext5sion ( ! glaci]s4 ,evid5ce ( ?is nature proves ! mu* wid] ext5t ( ! ,alp9e ice at a comp>atively rec5t geological date4 ,x c also be %[n t detritus f ,sc&9avia has be5 ice-borne to ! s\?-ea/ ( ,engl& & f> into ! he>t ( ,europe4 ,! ice1 by means ( gra9s ( s& & pieces ( /one :i* x drags along1 scores1 scrat*es & poli%es ! surfaces ( rock und]nea? x1 &1 9 ?is way1 produces ! abundant f9e s$im5t t gives ! *>act]i/ic milky appe>ance to ! riv]s t issue f ! l[] ends ( glaci]s4 ,by su* long-cont9u$ attrition ! rocks are worn d[n1 portions ( !m ( s(t] nature1 or :]e ! ice acts ) especial vig\r1 are holl[$ out into cavities :i*1 on ! disappe>ance ( ! ice1 may be fill$ ) wat] & become t>ns or lakes4 ,rocks ov] :i* l&-ice has pass$ are m>k$ by a peculi> smoo?1 fl[+ outl9e1 :i* =ms a contra/ to ! m rugg$ surface produc$ by ord9>y wea!r+4 ,!y are cov]$ ) groov+s1 :i* range f ! f9e/ /riae left by %>p gra9s ( s& to deep ruts gr\nd out by blocks ( /one4 ,! tr5d ( ^! m>k+s %[s ! direction 9 :i* ! ice fl[$4 ,by _! evid5ce ! position & movem5t ( =m] glaci]s 9 c\ntries f :i* ! ice has entirely vani%$ may be cle>ly det]m9$ 7see ,,glacial ,,p]iod74 #f4 ,! ,sea4--,! physical features ( ! sea are discuss$ 9 sep>ate articles 7see ,,ocean ,,& ,,oceanography74 ,! sea mu/ be reg>d$ z ! great regulator ( temp]ature & climate ov] ! globe1 & z ?us ex]t+ a pr(\nd influ5ce on ! di/ribution ( plant & animal life4 ,its di/9ctly geological "w is p>tly erosive & p>tly reproductive4 ,z an erod+ ag5t x mu/ to "s ext5t effect *emical decompositions 9 ! rocks & s$im5ts ov] :i* x spreads2 b ^! *anges h n yet be5 satisfactorily /udi$4 ,und\bt$ly1 its *ief de/ructive p[] is ( a me*anical k9d1 & arises f ! action ( its waves 9 beat+ ^u %ore-cliffs4 ,by ! alt]nate compression & expansion ( ! air 9 crevices ( ! rocks on :i* heavy break]s fall1 & by ! hydraulic pressure :i* ^! masses ( sea-wat] ex]t on ! walls ( ! fissures into :i* !y ru%1 l>ge masses ( rock are loos5$ & deta*$1 & caves & tunnels are drill$ along ! base ( sea-cliffs4 ,probably /ill m efficaci\s are ! bl[s ( ! loose %+le1 :i*1 caught up & hurl$ =w>d by ! waves1 falls ) great =ce ^u ! %ore rocks1 batt]+ !m z ) a k9d ( artill]y until !y are worn away4 ,! smoo? surfaces ( ! rocks )9 rea* ( ! waves contra/$ ) _! angul> =ms above t limit be> witness to ! am\nt ( wa/e1 :ile ! r\nd$ =ms ( ! b\ld]s & %+le %[ t !y too are be+ cont9ually r$uc$ 9 size4 ,?us ! sea1 by its action on ! coa/s1 produces mu* s$im5t1 :i* is swept away by its waves & curr5ts & /rewn ov] its floor4 ,besides ?is mat]ial1 x is con/antly receiv+ ! f9e silt & s& c>ri$ d[n by riv]s4 ,z ! floor ( ! ocean is ?us ! f9al receptacle = ! wa/e ( ! l&1 x becomes ! *ief era on ! surface ( ! globe = ! accumulation ( new /ratifi$ =mations4 ,& su* has be5 "o ( its great functions s9ce ! beg9n+ ( geological "t1 z is prov$ by ! rocks t =m ! visible "p ( ! e>?'s cru/1 & consi/ 9 great "p ( m>9e deposits4 ,*emical precipitates take place m especially 9 enclos$ p>ts ( ! sea1 :]e conc5tration ( ! wat] by evaporation c take place1 & :]e lay]s ( sodium *loride1 calcium sulphate & c>bonate1 & o!r salts are laid d[n4 ,b ! *ief m>9e accumulations are ( detrital orig94 ,ne> ! l& & = a v>iable di/ance ext5d+ sometimes to #bjj or #cjj m4 f %ore ! deposits consi/ *iefly ( s$im5ts d]iv$ f ! wa/e ( ! l&1 ! f9] silts be+ transport$ f>!/ f _! s\rce4 ,at great] dep?s & di/ances ! ocean floor receives a sl[ deposit ( exce$+ly f9e clay1 :i* is believ$ to be d]iv$ f ! decomposition ( pumice & volcanic du/ f insul> or subm>9e volcanoes4 ,wide tracts ( ! bottom are cov]$ ) v>i\s =ms ( ooze d]iv$ f ! accumulation ( ! rema9s ( m9ute organisms4 7,c7 ,life4 ,among ! ag5ts by :i* geological *anges are c>ri$ on ^u ! surface ( ! globe liv+ organisms mu/ be enum]at$4 ,bo? plants & animals co-op]ate ) ! inorganic ag5ts 9 promot+ ! degradation ( ! l&4 ,9 "s cases1 on ! o!r h&1 !y protect rocks f decay1 :ile1 by ! accumulation ( _! rema9s1 !y give rise to ext5sive =mations bo? ^u ! l& & 9 ! sea4 ,_! op]ations may h5ce be describ$ z alike de/ructive1 cons]vative & reproductive4 ,"u ?is head+ also ! influ5ce ( ,man z a geological ag5t des]ves notice4 7a7 ,plants4--,vegetation promotes ! dis9tegration ( rocks & soil 9 ! foll[+ ways3 7#a7 ,by keep+ ! surfaces ( /one moi/1 & ?us promot+ bo? me*anical & *emical dissolution1 z is especially %[n by liv]worts1 mosses & o!r moi/ure-lov+ plants4 7#b7 ,by produc+ ?r\gh _! decay c>bonic & o!r acids1 :i*1 toge!r ) decay+ organic matt] tak5 up by pass+ moi/ure1 become pot5t 9 effect+ ! *emical decomposition ( rocks & 9 promot+ ! dis9tegration ( soils4 7#c7 ,by ins]t+ _! roots or bran*es betwe5 jo9ts ( rock1 :i* are !reby loos5$1 s t l>ge slices may be ev5tually w$g$ (f4 7#d7 ,by attract+ ra91 z ?ick woods1 =e/s & peat-mosses d1 & ?us accel]at+ ! g5]al wa/e ( a c\ntry by runn+ wat]4 7#e7 ,by promot+ ! decay ( diseas$ & dead plants & animals1 z :5 fungi ov]spread a damp rott+ tree or ! c>case ( a dead animal4 ,t plants also ex]t a cons]vative influ5ce on ! surface ( ! l& is %[n 9 v>i\s ways4 7#a7 ,! =mation ( a /ratum ( turf protects ! soil & rocks und]nea? f be+ rapidly dis9tegrat$ & wa%$ away by atmosph]ic action4 7#b7 ,_m plants1 ev5 )\t =m+ a lay] ( turf1 s]ve by _! roots or bran*es to protect ! loose s& or soil on :i* !y gr[ f be+ remov$ by w9d4 ,! common s&-c>ex & o!r ar5ace\s plants b9d ! loose s&-dunes ( our coa/s1 & give !m a p]man5ce1 :i* w\ld at once be de/roy$ 7 ! s& laid b>e aga9 to /orms4 ,! gr[? ( %rubs & bru%wood along ! c\rse ( a /ream n only keeps ! alluvial banks f be+ s easily und]m9$ & remov$ z w\ld o!rwise be ! case1 b s]ves to arre/ ! s$im5t 9 floods1 filt]+ ! wat] & !reby add+ to ! height ( ! flood pla94 7#c7 ,"s m>9e plants1 l ! calc>e\s nullipores1 af=d protection to %ore rocks by cov]+ !m ) a h>d incru/ation4 ,! tangles & small] ,fuci :i* gr[ abundantly on ! littoral zone break ! =ce ( ! waves or dim9i% ! effects ( gr\nd swell4 7#d7 ,=e/s & bru%wood protect ! soil1 especially on slopes1 f be+ wa%$ away by ra9 or pl\gh$ up by avalan*es4 ,plants contribute by ! aggregation ( _! rema9s to ! =mation ( /ratifi$ deposits4 ,"s m>9e algae :i* secrete c>bonate ( lime n only encru/ rocks b give rise to %eets ( subm>9e lime/one4 ,an analog\s "p is play$ 9 fre%-wat] lakes by v>i\s lime-secret+ plants1 su* z ,*>a4 ,long-cont9u$ gr[? ( vegetation has1 9 "s regions1 produc$ ?ick accumulations ( a d>k loam1 z 9 ! black cotton soil 7regur7 ( ,india1 & ! black e>? 7t*]nozom7 ( ,russia4 ,peat-mosses are =m$ 9 temp]ate & arctic climates by ! gr[? ( m>%-lov+ plants1 sometimes to a ?ickness ( #dj or #ej ft4 ,9 tropical regions ! mangrove swamps on l[ moi/ %ores =m a d5se jungle1 sometimes #bj m4 broad1 :i* protects ^! %ores f ! sea until1 by ! arre/ ( s$im5t & ! con/ant contribution ( decay$ vegetation1 ! spongy gr\nd is at la/ turn$ into firm soil4 ,"s plants 7diatoms7 c ab/ract silica & build x into _! framework1 s t _! rema9s =m a silice\s deposit or ooze :i* cov]s spaces ( ! deep sea-floor e/imat$ at m ?an t5 millions ( squ>e miles 9 ext5t4 7b7 ,animals4--,^! ex]t a de/ructive influ5ce 9 ! foll[+ ways3 7#a7 ,by s]i\sly affect+ ! composition & arrangem5t ( ! vegetable soil4 ,worms br+ up ! l[] portions ( ! soil to ! surface1 & :ile ?us promot+ its f]tility increase its liability to be wa%$ away by ra94 ,burr[+ animals1 by ?r[+ up ! soil & subsoil1 expose ^! to be dri$ & bl[n away by ! w9d4 ,at ! same "t _! subt]ranean passages s]ve to dra9 (f ! sup]ficial wat] & to injure ! /ability ( ! surface ( ! gr\nd above !m4 ,9 ,brita9 ! mole & rabbit are famili> examples4 7#b7 ,by int]f]+ ) or ev5 div]t+ ! fl[ ( /reams4 ,?us beav]-dams *eck ! curr5t ( wat]-c\rses1 int]cept float+ mat]ials1 & sometimes turn /reams into new *annels4 ,! embankm5ts ( ! ,mississippi are sometimes weak5$ to su* an ext5t by ! burr[+s ( ! cray-fi% z to give way & all[ ! riv] to inundate ! surr\nd+ c\ntry4 ,simil> results h happ5$ 9 ,europe f subt]ranean op]ations ( rats4 7#c7 ,"s mollusca bore into /one or wood & by ! numb] ( contigu\s p]=ations greatly weak5 ! mat]ial4 7#d7 ,_m animals ex]cise a ru9\sly de/ructive influ5ce ^u vegetation4 ,( ! num]\s plagues ( ?is k9d ! locu/1 phyllox]a & ,colorado beetle may be cit$4 ,! mo/ important geological function p]=m$ by animals is ! =mation ( new deposits out ( _! rema9s4 ,x is *iefly by ! l[] grades ( ! animal k+dom t ?is "w is accompli%$1 especially by molluscs1 corals & =am9if]a4 ,%ell-banks are =m$ abundantly 9 su* comp>atively %all[ & enclos$ bas9s z t ( ! ,nor? ,sea1 & on a mu* m ext5sive scale on ! floor ( ! ,we/ ,indian seas4 ,by ! coral polyps ?ick masses ( lime/ones h be5 built up 9 ! w>m] seas ( ! globe 7see ,,coral ,,reefs74 ,! floor ( ! ,atlantic & o!r oceans is cov]$ ) a f9e calc>e\s ooze d]iv$ ma9ly f ! rema9s ( =am9if]a1 :ile 9 o!r regions ! bottom %[s a silice\s ooze =m$ almo/ entirely ( radiol>ia4 ,v]tebrate animals give rise to phosphatic deposits =m$ sometimes ( _! excrem5t1 z 9 guano & coprolites1 sometimes ( an accumulation ( _! bones4 7c7 ,man4--,no survey ( ! geological work+s ( plant & animal life ^u ! surface ( ! globe c be complete :i* does n take acc\nt ( ! influ5ce ( man--an influ5ce ( enorm\s & increas+ consequ5ce 9 physical geography1 = man has introduc$1 z x 71 an elem5t ( antagonism to nature4 ,8 int]f]5ce %[s itself 9 8 relations to climate1 :]e he has affect$ ! meteorological conditions ( diff]5t c\ntries3 7#a7 ,by remov+ =e/s1 & lay+ b>e to ! sun & w9ds areas :i* 7 previ\sly kept cool & damp "u trees1 or :i*1 ly+ on ! lee side1 7 protect$ f tempe/s4 ,x is suppos$ t ! :olesale de/ruction ( ! woodl&s =m]ly exi/+ 9 c\ntries bord]+ ! ,m$it]ranean has be5 9 "p ! cause ( ! pres5t desiccation ( ^! di/ricts4 7#b7 ,by dra9age1 :]eby ! dis*>g$ ra9fall is rapidly remov$1 & ! evaporation is less5$1 ) a consequ5t dim9ution ( ra9fall & "s increase 9 ! g5]al temp]ature ( a c\ntry4 7#c7 ,by ! o!r processes ( agriculture1 su* z ! trans=mation ( moor & bog into cultivat$ l&1 & ! clo?+ ( b>e hillsides ) gre5 crops or plantations ( conif]\s & h>dwood trees4 ,/ill m obvi\s are ! results ( human int]f]5ce ) ! fl[ ( wat]3 7#a7 ,by increas+ or dim9i%+ ! ra9fall man directly affects ! volume ( riv]s4 7#b7 ,by 8 dra9age op]ations he makes ! ra9 to run (f m rapidly ?an be=e1 & !reby increases ! magnitude ( floods & ( ! de/ruction caus$ by !m4 7#c7 ,by wells1 bores1 m9es1 or o!r subt]ranean works he int]f]es ) ! und]gr\nd wat]s1 & consequ5tly ) ! dis*>ge ( spr+s4 7#d7 ,by embank+ riv]s he conf9es !m to n>r[ *annels1 sometimes increas+ _! sc\r1 & enabl+ !m to c>ry _! s$im5t fur!r seaw>d1 sometimes caus+ !m to deposit x ov] ! pla9s & raise _! level4 7#e7 ,by 8 eng9e]+ op]ations = wat]-supply he ab/racts wat] f its natural bas9s & depletes ! /reams4 ,9 _m ways man alt]s ! aspect ( a c\ntry3 7#a7 ,by *ang+ =e/ into b>e m\nta91 or clo?+ b>e m\nta9s ) =e/4 7#b7 ,by promot+ ! gr[? or caus+ ! removal ( peat-mosses4 7#c7 ,by he$lessly uncov]+ s&-dunes1 & !reby sett+ 9 motion a process ( de/ruction :i* may conv]t hundr$s ( acres ( f]tile l& into wa/e s&1 or by prud5tly plant+ ! dunes ) s&-lov+ vegetation & ?us arre/+ _! l&w>d progress4 7#d7 ,by s guid+ ! c\rse ( riv]s z to make !m aid him 9 reclaim+ wa/e l&1 & br++ x "u cultivation4 7#e7 ,by pi]s & bulw>ks1 :]eby ! ravages ( ! sea are /ay$1 or by ! ?\ghtless removal f ! bea* ( /ones :i* ! waves _h !mselves ?r[n up1 & :i* w\ld h s]v$ = a "t to protect ! l&4 7#f7 ,by =m+ new deposits ei!r design$ly or incid5tally4 ,! roads1 bridges1 canals1 railways1 tunnels1 villages & t[ns ) :i* man has cov]$ ! surface ( ! l& w 9 _m cases =m a p]man5t record ( 8 pres5ce4 ,"u 8 h& ! :ole surface ( civiliz$ c\ntries is v sl[ly cov]$ ) a /ratum1 ei!r =m$ :olly by him or due 9 great measure to 8 op]ations & conta9+ _m relics ( 8 pres5ce4 ,! soil ( anci5t t[ns has be5 increas$ to a dep? ( _m feet by _! successive de/ructions & r5ovations4 ,p]haps ! mo/ subtle ( human influ5ces are to be se5 9 ! di/ribution ( plant & animal life ^u ! globe4 ,"s ( man's do+s 9 ?is doma9 are inde$ pla9 en\gh1 su* z ! extirpation ( wild animals1 ! dim9ution or de/ruction ( "s =ms ( vegetation1 ! introduction ( plants & animals useful to himself1 & especially ! enorm\s pr$om9ance giv5 by him to ! c]eals & to ! spread ( %eep & cattle4 ,b no su* ext5sive di/urbance ( ! normal conditions ( ! di/ribution ( life c take place )\t c>ry+ ) x _m second>y effects1 & sett+ 9 motion a wide cycle ( *ange & ( reaction 9 ! animal & vegetable k+doms4 ,= example1 ! incessant w>f>e wag$ by man aga9/ birds & bea/s ( prey 9 di/ricts giv5 up to ! *ase leads sometimes to un=ese5 results4 ,! weak game is all[$ to live1 :i* w\ld o!rwise be kill$ (f & give m room = ! heal?y rema9d]4 ,o!r animals :i* fe$ p]haps on ! same mat]ials z ! game are by ! same cause p]mitt$ to live un*eck$1 & !reby to act z a fur!r h9drance to ! spread ( ! protect$ species4 ,b ! indirect results ( man's int]f]5ce ) ! regime ( plants & animals /ill require mu* prolong$ obs]vation4 ,,"p ,v4--,,geotectonic ,,or ,,/ructural ,,geology ,f a /udy ( ! nature & composition ( m9]als & rocks1 & an inve/igation ( ! diff]5t ag5cies by :i* !y are =m$ & modifi$1 ! geologi/ proce$s to inquire h[ ^! mat]ials h be5 put toge!r s z to build up ! visible "p ( ! e>?'s cru/4 ,he soon asc]ta9s t !y h n be5 ?r[n toge!r :olly at r&om1 b t !y %[ a recognizable ord] ( arrangem5t4 ,"s ( !m1 especially ^? ( mo/ rec5t gr[?1 rema9 9 _! orig9al condition & position1 b1 9 proportion to _! antiquity1 !y g5]ally pres5t increas+ alt]ation1 until x may no long] be possible to tell :at 0 _! pri/9e /ate4 ,z by f> ! l>ge/ accessible portion ( ! t]re/rial cru/ consi/s ( /ratifi$ rocks1 & z ^! furni% cle> evid5ce ( mo/ ( ! modifications to :i* !y h be5 subject$ 9 ! long c\rse ( geological hi/ory1 x is conv5i5t to take !m into consid]ation fir/4 ,!y possess a numb] ( /ructures :i* belong to ! orig9al conditions 9 :i* !y 7 accumulat$4 ,!y pres5t 9 addition o!r /ructures :i* h be5 sup]9duc$ ^u !m1 & :i* !y %>e ) ! un/ratifi$ or igne\s rocks4 #a4 ,,orig9al ,,/ructures 7a7 ,/ratifi$ ,rocks4--,?is ext5sive & important s]ies is above all di/+ui%$ by possess+ a prevail+ /ratifi$ arrangem5t4 ,_! mat]ials h be5 laid d[n 9 lam9ae1 lay]s & /rata1 or b$s1 po9t+ g5]ally to ! int]mitt5t deposition ( ! s$im5ts ( :i* !y consi/4 ,z ?is /ratification 01 z a rule1 orig9ally ne>ly or q horizontal1 x s]ves z a base f :i* to measure any subsequ5t di/urbance :i* ! rocks h und]gone4 ,! occurr5ce ( false-b$d+1 i4e4 b&s ( incl9$ lay]s betwe5 ! normal planes ( /ratification1 does n =m any real exception2 b indicates ! action ( %ift+ curr5ts :]eby ! s$im5t 0 transport$ & ?r[n d[n4 ,o!r important records ( ! orig9al conditions ( deposit are suppli$ by ripple-m>ks1 sun-cracks1 ra9-pr9ts & concretions4 ,f ! nature ( ! mat]ial fur!r light is ca/ on ! geographical conditions 9 :i* ! /rata 7 accumulat$4 ,?us1 conglom]ates indicate ! proximity ( old %ore-l9es1 s&/ones m>k deposits 9 comp>atively %all[ wat]1 clays & %ales po9t to ! tranquil accumulation ( f9e silt at a great] dep? & fur!r f l&1 :ile fossilif]\s lime/ones be> witness to cle>] wat] 9 :i* organisms fl\ri%$ at "s di/ance f deposits ( s& & mud4 ,aga91 ! alt]nation ( diff]5t k9ds ( s$im5t sugge/s a v>iability 9 ! conditions ( deposition1 su* z a %ift+ ( ! s$im5t-be>+ curr5ts & ( ! areas ( muddy & cle> wat]4 ,a ?ick gr\p ( con=mable /rata1 t is1 a s]ies ( deposits :i* %[ no discordance 9 _! /ratification1 may usually be reg>d$ z hav+ be5 laid d[n on a sea-floor t 0 g5tly s9k+4 ,"h & !re evid5ce is obta9able ( ! limits or ( ! progress ( ! subsid5ce by :at is call$ ov]lap4 ,( ! absolute l5g? ( "t repres5t$ by any /rata or gr\ps ( /rata no satisfactory e/imates c yet be =m$4 ,c]ta9 g5]al conclusions may inde$ be drawn1 & comp>isons may be made betwe5 diff]5t s]ies ( rocks4 ,s&/ones full ( false-b$d+ 7 probably accumulat$ m rapidly ?an f9ely-lam9at$ %ales or clays4 ,x is n uncommon 9 c]ta9 ,c>bonif]\s =mations to f9d conif]\s & o!r trunks emb$d$ 9 s&/one4 ,"s ( ^! trees seem to h be5 c>ri$ along & to h sunk1 _! heavi] or root end t\*+ ! bottom & _! upp] end slant+ upw>d 9 ! direction ( ! curr5t1 exactly z 9 ! case ( ! snags ( ! ,mississippi4 ,9 o!r cases ! trees h be5 subm]g$ :ile /ill 9 _! positions ( gr[?4 ,! cont9u\s deposit ( s& at la/ rose above ! level ( ! trunks & buri$ !m4 ,x is cle> !n t ! rate ( deposit mu/ h be5 sometimes suffici5tly rapid to all[ s& to accumulate to a dep? ( #cj ft4 or m be=e ! decay ( ! wood4 ,mod]n in/ances are kn[n :]e1 "u c]ta9 circum/ances1 subm]g$ trees may la/ = "s c5turies1 b ev5 ! mo/ durable mu/ decay 9 :at1 aft] all1 is a brief space ( geological "t4 ,s9ce cont9u\s lay]s ( ! same k9d ( deposit sugge/ a p]si/5ce ( geological conditions1 :ile num]\s alt]nations ( diff]5t k9ds ( s$im5t>y matt] po9t to vicissitudes or alt]nations ( conditions1 x may be suppos$ t ! "t repres5t$ by a giv5 ?ickness ( simil> /rata 0 less ?an t %[n by ! same ?ickness ( dissimil> /rata1 because ! *anges ne$$ to br+ new v>ieties ( s$im5t into ! area ( deposit w\ld usually require ! lapse ( "s "t = _! completion4 ,b ?is conclusion may (t5 be errone\s4 ,x w be be/ support$ :51 f ! v nature ( ! rocks1 wide v>iations 9 ! "* ( ! wat]-bottom c be e/abli%$4 ,?us a gr\p ( %ales foll[$ by a fossilif]\s lime/one w\ld almo/ always m>k ! lapse ( a mu* long] p]iod ?an an equal dep? ( s&y /rata4 ,a ?ick mass ( lime/one1 made up ( organic rema9s :i* liv$ & di$ ^u ! spot1 & ^: rema9s are cr[d$ toge!r g5]ation above g5]ation1 mu/ h dem&$ _m ye>s or c5turies = its =mation4 ,b 9 all speculations ( ?is k9d we mu/ be> 9 m9d t ! l5g? ( "t repres5t$ by a giv5 dep? ( /rata is n to be e/imat$ m]ely f _! ?ickness or li?ological "*4 ,! int]val betwe5 ! deposit ( two successive lam9ae ( %ale may h be5 z long z1 or ev5 long] ?an1 t requir$ = ! =mation ( "o ( ! lam9ae4 ,9 l mann] ! int]val ne$$ = ! transition f "o /ratum or k9d ( /rata to ano!r may (t5 h be5 m ?an equal to ! "t requir$ = ! =mation ( ! /rata on ei!r side4 ,b ! relative *ronological importance ( ! b>s or l9es 9 ! geological record c seldom be satisfactorily discuss$ m]ely on li?ological gr\nds4 ,?is mu/ ma9ly be decid$ on ! evid5ce ( organic rema9s1 z %[n 9 ,"p ,,vi41 :]e ! gr\p+ ( ! /ratifi$ rocks into =mations & sy/ems is describ$4 7b7 ,igne\s ,rocks4--,z "p ( ! e>?'s cru/ ^! rocks pres5t *>act]s by :i* !y are /rongly diff]5tiat$ f ! /ratifi$ s]ies4 ,:ile ! broad petrographical di/9ctions ( _! sev]al v>ieties rema9 p]si/5t1 !y pres5t suffici5t local v>iations ( type to po9t to ! exi/5ce ( :at h be5 call$ petrographic prov9ces1 9 ea* ( :i* ! eruptive masses are connect$ by a g5]al family relation%ip1 diff]+ m or less f t ( a neighb\r+ prov9ce4 ,9 ea* region pres5t+ a long *ronological s]ies ( eruptive rocks a petrographical sequ5ce c be trac$1 :i* is obs]v$ to be n absolutely ! same ev]y:]e1 ?\gh its g5]al features may be p]si/5t4 ,! e>lie/ manife/ations ( eruptive mat]ial 9 any di/rict appe> to h be5 mo/ frequ5tly ( an int]m$iate type betwe5 acid & basic1 pass+ !nce into a ?or\ghly acid s]ies & conclud+ ) an effusion ( basic mat]ial4 ,consid]$ z "p ( ! ar*itecture ( ! cru/ ( ! e>?1 igne\s rocks are conv5i5tly divisible into two great s]ies3 7#a7 ^? bodies ( mat]ial :i* h be5 inject$ into ! cru/ & h solidifi$ !re1 & 7#b7 ^? :i* h rea*$ ! surface & h be5 eject$ !re1 ei!r 9 a molt5 /ate z lava or 9 a fragm5tal =m z du/1 a%es & scoriae4 ,! fir/ ( ^! divisions repres5ts ! plutonic1 intrusive or subsequ5t phase ( eruptivity2 ! second m>ks ! volcanic1 int]/ratifi$ or contemporane\s phase4 #a4 ,! plutonic or intrusive rocks1 :i* h be5 =c$ into ! cru/ & h consolidat$ !re1 pres5t a wide range ( texture f ! mo/ co>se-gra9$ granites to ! mo/ p]fect natural glass4 ,see+ t !y h usually cool$ ) extreme sl[ness und]gr\nd1 !y are z a g5]al rule m l>gely cry/all9e ?an ! volcanic s]ies4 ,! =m assum$ by ea* individual body ( intrusive mat]ial has dep5d$ ^u ! %ape ( ! space into :i* x has be5 inject$1 & :]e x has cool$ & become solid4 ,?is %ape has be5 det]m9$ by ! local /ructure ( ! e>?'s cru/ on ! "o h& & by ! en]gy ( ! eruptive =ce on ! o!r4 ,x (f]s a conv5i5t basis = ! classification ( ! intrusive rocks1 :i*1 z "p ( ! framework ( ! cru/1 may ?us be gr\p$ accord+ to ! %ape ( ! cavity :i* receiv$ !m1 z bosses1 sills1 dikes & necks4 ,bosses1 or /ocks1 are ! l>ge/ & mo/ %apeless extravasations ( erupt$ mat]ial4 ,!y include ! great bodies ( granite :i*1 9 mo/ c\ntries ( ! _w1 h ris5 = _m miles ?r\gh ! /ratifi$ =mations & h alt]$ ! rocks ar\nd !m by contact-metamorphism4 ,sills1 or intrusive %eets1 are b$-l masses :i* h be5 ?ru/ betwe5 ! planes ( s$im5t>y or ev5 ( igne\s rocks4 ,! t]m laccolite has be5 appli$ to sills :i* are connect$ ) bosses4 ,intrusive %eets are di/+ui%able f true contemporane\sly int]calat$ lavas by n keep+ always to ! same plat=m1 b break+ across & alt]+ ! contigu\s /rata1 & by ! clos5ess ( _! texture :]e !y come 9 contact ) ! contigu\s rocks1 :i*1 be+ cold1 *ill$ ! molt5 mat]ial & caus$ x to consolidate on its out] m>g9s m rapidly ?an 9 its int]ior4 ,dikes or ve9s are v]tical walls or ramify+ bran*es ( intrusive mat]ial :i* has consolidat$ 9 fissures or irregul> clefts ( ! cru/4 ,necks are volcanic *imneys :i* h be5 fill$ up ) erupt$ mat]ial1 & h n[ be5 expos$ at ! surface aft] prolong$ d5udation has remov$ n only ! sup]ficial volcanic masses orig9ally associat$ ) !m1 b also m or less ( ! upp] "p ( ! v5ts4 ,plutonic rocks d n pres5t evid5ce ( _! precise geological age4 ,all t c be c]ta9ly affirm$ f !m is t !y mu/ be y\ng] ?an ! rocks into :i* !y h be5 intrud$4 ,f _! int]nal /ructure1 h[ev]1 & f ! evid5ce ( ! rocks associat$ ) !m1 "s m or less def9ite conjectures may be made z to ! limits ( "t )9 :i* !y 7 probably inject$4 #b4 ,! int]/ratifi$ or volcanic s]ies is ( special importance 9 geology1 inasmu* z x conta9s ! records ( volcanic action dur+ ! pa/ hi/ory ( ! globe4 ,x 0 po9t$ out 9 ,"p ,i4 t :ile t[>ds ! end ( ! #ah? & 9 ! beg9n+ ( ! #ai? c5tury mu* att5tion 0 paid by ,hutton & 8 foll[]s to ! pro(s ( intrusion af=d$ by :at !y call$ ! un]upt$ lavas )9 ! e>?'s cru/1 ^! obs]v]s lo/ sight ( ! possibility t "s ( ^! rocks might h be5 erupt$ at ! surface1 & might ?us be *ronicles ( volcanic action 9 =m] geological p]iods4 ,x is n always possible to satisfactorily discrim9ate betwe5 ! two types ( contemporane\sly int]calat$ & subsequ5tly inject$ mat]ial4 ,b rocks ( ! =m] type h n brok5 into or involv$ ! ov]ly+ /rata1 & !y are usually m>k$ by ! *>act]i/ic /ructures ( sup]ficial lavas & by _! association ) volcanic tuffs4 ,by means ( ! evid5ce :i* !y supply1 x has be5 asc]ta9$ t volcanic action has be5 manife/$ 9 ! globe s9ce ! e>lie/ geological p]iods4 ,9 ! ,briti% ,isles1 = example1 ! volcanic record is rem>kably full = ! long s]ies ( ages f ,cambrian to ,p]mian "t1 & aga9 = ! old] ,t]ti>y p]iod4 #b4 ,,subsequ5tly ,,induc$ ,,/ructures ,aft] _! accumulation1 :e!r z /ratifi$ or eruptive masses1 all k9ds ( rocks h be5 subject to v>i\s *anges1 & h acquir$ 9 consequ5ce a v>iety ( sup]9duc$ /ructures4 ,x has be5 po9t$ out 9 ! "p ( ?is article deal+ ) dynamical geology t "o ( ! mo/ important =ms ( en]gy 9 ! evolution ( geological processes is to be f\nd 9 ! movem5ts t take place )9 ! cru/ ( ! e>?4 ,"s ( ^! movem5ts are s slight z to be only recognizable by means ( delicate in/rum5ts2 b f ?is inf]ior limit !y range up to gigantic convulsions by :i* m\nta9-*a9s are upheav$4 ,! cru/ mu/ be reg>d$ z 9 a p]petual /ate ( /ra91 & its compon5t mat]ials are !re=e subject to all ! effects :i* fl[ f t condition4 ,x is ! "o great object ( ! geotectonic division ( geology to /udy ! /ructures :i* h be5 develop$ 9 consequ5ce ( e>?-movem5ts1 & to discov] f ?is inve/igation ! nature ( ! processes :]eby ! rocks ( ! cru/ h be5 br\ght into ! condition & ! positions 9 :i* we n[ f9d !m4 ,! details ( ?is subject w be f\nd 9 sep>ate articles descriptive ( ea* ( ! te*nical t]ms appli$ to ! sev]al k9ds ( sup]9duc$ /ructures4 ,all t ne$ be (f]$ "h is a g5]al outl9e connect+ ! sev]al portions ( ! subject toge!r4 ,"o ( ! mo/ univ]sal ( ^! lat] /ructures is to be se5 9 ! divisional planes1 usually v]tical or highly incl9$1 by :i* rocks are split into quadrangul> or irregul>ly %ap$ blocks4 ,to ^! planes ! "n ( jo9ts has be5 giv54 ,!y are ( prime importance f an indu/rial po9t ( view1 see+ t ! art ( qu>ry+ consi/s ma9ly 9 detect+ & mak+ prop] use ( !m4 ,_! abundance 9 all k9ds ( rocks1 f ^? ( rec5t date up to ^? ( ! highe/ antiquity1 af=ds a rem>kable te/imony to ! /ra9s :i* ! t]re/rial cru/ has suff]$4 ,!y h aris5 sometimes f t5sion1 su* z t caus$ by contraction f ! dry+ & consolidation ( an aque\s s$im5t or f ! cool+ ( a molt5 mass2 sometimes f torsion dur+ movem5ts ( ! cru/4 ,al?\gh ! /ratifi$ rocks 7 orig9ally deposit$ 9 a m or less ne>ly horizontal position on ! floor ( ! sea1 :]e n[ visible on ! dry l& !y are seldom f\nd to h reta9$ _! flatness4 ,on ! contr>y1 !y are se5 to h be5 g5]ally tilt$ up at v>i\s angles1 sometimes ev5 plac$ on end 7crop1 dip1 /rike74 ,:5 a suffici5tly l>ge area ( gr\nd is exam9$1 ! incl9ation into :i* ! /rata h be5 ?r[n may be obs]v$ n to cont9ue f> 9 ! same direction1 b to turn ov] to ! opposite or ano!r qu>t]4 ,x c !n be se5 t 9 reality ! rocks h be5 ?r[n into undulations4 ,f ! l[e/ & flatte/ ar*es :]e ! dep>ture f horizontality may be only trifl+1 e /ep may be foll[$ up to int5se curvature1 :]e ! /rata h be5 compress$ & plicat$ z if !y _h be5 piles ( s(t c>pets 7anticl9e1 syncl9e1 monocl9e1 geo-anticl9e1 geo-syncl9e1 isocl9al1 plication1 curvature1 quaquav]sal74 ,x has fur!r happ5$ abundantly all ov] ! surface ( ! globe t relief f int]nal /ra9 9 ! cru/ has be5 obta9$ by fracture1 & ! consequ5t subsid5ce or elevation ( "o or bo? sides ( ! fissure4 ,! diff]5tial movem5t betwe5 ! two sides may be sc>cely p]ceptible 9 ! feeble/ dislocation1 b 9 ! extreme cases x may am\nt to _m ?\s& feet 7fault1 fissure1 dislocation1 hade1 slick5sides74 ,! great faults 9 a c\ntry are among its mo/ important /ructural features1 & z !y n infrequ5tly cont9ue to be l9es ( weakness 9 ! cru/ along :i* sudd5 slipp+ may f "t to "t take place1 !y become ! l9es ( orig9 ( e>?quakes4 ,! ,san ,francisco e>?quake ( #aijf1 already cit$1 af=ds a memorable illu/ration ( ?is connexion4 ,x is 9 a great m\nta9-*a9 t ! extraord9>y complication ( plicat$ & fault$ /ructures 9 ! cru/ ( ! e>? c be mo/ impressively beheld4 ,! comb9ation ( ov]turn$ folds ) rupture has be5 already ref]r$ to z a *>act]i/ic feature 9 ! ,alps 7,"p ,,iv474 ,! gigantic folds h 9 _m places be5 pu%$ ov] ea* o!r s z to lie almo/ flat1 :ile ! upp] limb has n infrequ5tly be5 driv5 = _m miles beyond ! l[] by a rupture along ! axis4 ,9 ?is way successive slices ( a ?ick s]ies ( =mations h be5 c>ri$ nor?w>ds on ! nor!rn slope ( ! ,alps1 & h be5 pil$ s abnormally above ea* o!r t "s ( _! olde/ memb]s recur sev]al times on diff]5t ?ru/-planes1 ! :ole be+ und]la9 by ,t]ti>y /rata 7see ,,alps74 ,fur!r pro( ( ! colossal compression to :i* ! rocks h be5 subject$ is af=d$ by _! int5se crumpl+ & corrugation1 & by ! abundantly fault$ & cru%$ condition to :i* !y h be5 r$uc$4 ,simil> evid5ce z to /resses 9 ! t]re/rial cru/ & ! important *anges :i* !y produce among ! rocks may also be obta9$ on a small] scale 9 _m non-m\nta9\s c\ntries4 ,ano!r m>k$ result ( ! compression ( ! t]re/rial cru/ has be5 induc$ 9 "s rocks by ! production ( ! fissile /ructure :i* is typically %[n 9 ro(+-slate 7cleavage74 ,closely connect$ ) ?is int]nal re>rangem5t has be5 ! developm5t ( microscopic microlites or cry/als 7rutile1 mica1 c47 9 argillace\s slates :i* 7 und\bt$ly orig9ally f9e m>9e mud & silt4 ,f ?is incipi5t =m ( metamorphism successive /ages may be trac$ ?r\gh ! v>i\s k9ds ( argillite & phyllite into mica-s*i/1 & !nce into m cry/all9e gneissoid v>ieties 7foliation1 slate1 mica-s*i/1 gneiss74 ,! ,alps af=d excell5t illu/rations ( ^! trans=mations4 ,! fissures produc$ 9 ! cru/ are sometimes clean1 %>ply def9$ divisional planes1 l cracks across a pane ( glass4 ,mu* m usually1 h[ev]1 ! rocks on ei!r side h be5 brok5 up by ! friction ( movem5t1 & ! fault is m>k$ by a v>iable bread? ( ?is brok5 mat]ial4 ,sometimes ! walls h sep>at$ & molt5 rock has ris5 f bel[ & solidifi$ betwe5 !m z a dike4 ,occasionally ! fissures h op5$ to ! surface1 & h be5 fill$ 9 f above ) detritus1 z 9 ! s&/one-dikes ( ,colorado & ,cali=nia4 ,9 m9]al di/ricts ! fissures h be5 fill$ ) v>i\s sp>s & ores1 =m+ :at are kn[n z m9]al ve9s4 ,:]e "o s]ies ( rocks is cov]$ by ano!r )\t any break or discordance 9 ! /ratification !y are said to be con=mable4 ,b :]e ! old] s]ies has be5 tilt$ up or visibly d5ud$ be=e be+ ov]la9 by ! y\ng]1 ! latt] is t]m$ uncon=mable4 ,?is relation is "o ( ! greate/ value 9 /ructural geology1 = x m>ks a gap 9 ! geological record1 :i* may repres5t a va/ lapse ( "t n !re record$ by /rata4 ,,"p ,,vi4--,,paleontological ,,geology ,?is division ( ! sci5ce deals ) fossils1 or ! traces ( plants & animals pres]v$ 9 ! rocks ( ! e>?'s cru/1 & endeav\rs to ga!r f !m in=mation z to ! hi/ory ( ! globe & its inhabitants4 ,! t]m fossil 7,lat4 fossilis1 f fod]e1 to dig up71 mean+ lit]ally any?+ dug up1 0 =m]ly appli$ indiscrim9ately to any m9]al sub/ance tak5 out ( ! e>?'s cru/1 :e!r organiz$ or n4 ,s9ce ! "t ( ,lam>ck1 h[ev]1 ! mean+ ( ! ^w has be5 re/rict$1 s z to include only ! rema9s or traces ( plants & animals pres]v$ 9 any natural =mation :e!r h>d rock or sup]ficial deposit4 ,x includes n m]ely ! petrifi$ /ructures ( organisms1 b :atev] 0 directly connect$ ) or produc$ by ^! organisms4 ,?us ! res9 :i* 0 exud$ f trees ( long-p]i%$ =e/s is z mu* a fossil z any portion ( ! /em1 leaves1 fl[]s or fruit1 & 9 "s respects is ev5 m valuable to ! geologi/ ?an m det]m9able rema9s ( its p>5t trees1 because x has (t5 pres]v$ 9 admirable p]fection ! insects :i* flitt$ ab\t 9 ! woodl&s4 ,! burr[s & trails ( a worm pres]v$ 9 s&/one & %ale claim recognition z fossils1 & inde$ are commonly ! only indications to be met ) ( ! exi/5ce ( annelid life among old geological =mations4 ,! dropp+s ( fi%es & reptiles1 call$ coprolites1 are excell5t fossils1 & tell _! tale z to ! pres5ce & food ( v]tebrate life 9 anci5t wat]s4 ,! little agglut9at$ cases ( ! caddis-worm rema9 z fossils 9 =mations f :i*1 p]*ance1 mo/ o!r traces ( life may h pass$ away4 ,nay1 ! v h&iwork ( man1 :5 pres]v$ 9 any natural mann]1 is entitl$ to rank among fossils2 z :]e 8 fl9t-implem5ts h be5 dropp$ into ! pre-hi/oric gravels ( riv]-valleys or :]e 8 canoes h be5 buri$ 9 ! silt ( lake-bottoms4 ,a /udy ( ! l&-surfaces & sea-floors ( ! pres5t "t %[s t !re are s _m *ances aga9/ ! cons]vation ( ! rema9s ( ei!r t]re/rial or m>9e animals & plants t if1 z is probable1 ! same conditions exi/$ 9 =m] geological p]iods1 we %\ld reg>d ! occurr5ce ( organic rema9s among ! /ratifi$ =mations ( ! e>?'s cru/ z g5]ally ! result ( v>i\s =tunate accid5ts4 ,let u consid]1 9 ! fir/ place1 ! *ances = ! pres]vation ( rema9s ( ! pres5t fauna & flora ( a c\ntry4 ,! surface ( ! l& may be d5sely clo!d ) =e/ & abundantly peopl$ ) animal life4 ,b ! trees die & m\ld] into soil4 ,! animals1 too1 disappe>1 g5]ation aft] g5]ation1 & leave few or no p]ceptible traces ( _! exi/5ce4 ,if we 7 n aw>e f au!ntic records t c5tral & nor!rn ,europe 7 cov]$ ) va/ =e/s at ! beg9n+ ( our era1 h[ c\ld we "k ?is fact8 ,:at has become ( ! h]ds ( wild ox51 ! be>s1 wolves & o!r d5iz5s ( primeval ,europe8 ,h[ c\ld we prove f ! exam9ation ( ! surface soil ( any c\ntry t ^? creatures _h once ab\nd$ !re8 ,! conditions = ! pres]vation ( any relics ( ! plant & animal life ( a t]re/rial surface mu/ obvi\sly be always exceptional4 ,!y are suppli$ only :]e ! organic rema9s c be protect$ f ! air & sup]ficial decay4 ,h5ce !y may be obs]v$ 9 7#a7 ! deposits on ! floors ( lakes2 7#b7 9 peat-mosses2 7#c7 9 deltas at riv]-m\?s2 & 7#d7 "u ! /alagmite ( cav]ns 9 lime/one di/ricts4 ,b 9 ^! & o!r fav\rable places a m]e inf9itesimal fraction ( ! fauna or flora ( a l&-surface is likely to be entomb$ or pres]v$4 ,9 ! second place1 al?\gh 9 ! sea ! conditions = ! pres]vation ( organic rema9s are 9 _m respects m fav\rable ?an on l&1 !y are apt to be fru/rat$ by _m adv]se circum/ances4 ,:ile ! level ( ! l& rema9s /ation>y1 !re c be b little effective entombm5t ( m>9e organisms 9 littoral deposits2 = only a limit$ accumulation ( s$im5t w be =m$ until subsid5ce ( ! sea-floor takes place4 ,9 ! trifl+ b$s ( s& or gravel ?r[n up on a /ation>y %ore1 only ! h>d] & m durable =ms ( life1 su* z ga/ropods & lamellibran*s1 :i* c )/& ! triturat+ effects ( ! bea* waves1 are likely to rema9 uneffac$4 ,bel[ tide-m>ks1 along ! m>g9 ( ! l& :]e s$im5t is gradually deposit$1 ! conditions are m fav\rable = ! pres]vation ( m>9e organisms4 ,9 ! %eets ( s& & mud !re laid d[n ! h>d] p>ts ( _m =ms ( life may be entomb$ & protect$ f decay4 ,b only a small proportion ( ! total m>9e fauna may be expect$ to appe> 9 su* deposits4 ,at ! be/1 m]ely littoral & %all[-wat] =ms w occur1 &1 ev5 "u ! mo/ fav\rable conditions1 !y w repres5t b a fraction ( ! :ole assemblage ( life 9 ^! juxta-t]re/rial p>ts ( ! ocean4 ,z we rec$e f ! l& ! rate ( deposition ( s$im5t on ! sea-floor mu/ become feebl]1 until1 9 ! remote c5tral abysses1 x rea*es a h>dly appreciable m9imum4 ,except1 !re=e1 :]e "s k9d ( ooze or o!r deposit is accumulat+ 9 ^! m pelagic regions1 ! conditions mu/ be on ! :ole unfav\rable = ! pres]vation ( any adequate repres5tation ( ! deep-sea fauna4 ,h>d durable objects1 su* z tee? & bones1 may sl[ly accumulate1 & be protect$ by a coat+ ( p]oxide ( manganese1 or ( "s ( ! silicates n[ =m+ "h & !re ov] ! deep-sea bottom2 or ! rate ( gr[? ( ! abysmal deposit may be s t>dy t mo/ ( ! rema9s ( at lea/ ! l>g] animals w disappe>1 ow+ to decay1 be=e !y c be cov]$ up & pres]v$4 ,any su* deep-sea =mation1 if rais$ into l&1 w\ld supply b a meagre picture ( ! :ole life ( ! sea4 ,x w\ld ?us appe> t ! portion ( ! sea-floor be/ suit$ = receiv+ & pres]v+ ! mo/ v>i$ assemblage ( m>9e organic rema9s is ! area 9 front ( ! l&1 to :i* riv]s & curr5ts br+ cont9ual supplies ( s$im5t4 ,! mo/ fav\rable conditions = ! accumulation ( a ?ick mass ( m>9e fossilif]\s /rata w arise :5 ! area ( deposit is und]go+ a gradual subsid5ce4 ,if ! rate ( depression & t ( deposit 7 equal1 or ne>ly s1 ! movem5t might proce$ = a va/ p]iod )\t produc+ any great app>5t *ange 9 m>9e geography1 & ev5 )\t s]i\sly affect+ ! di/ribution ( life ov] ! sea-floor )9 ! area ( subsid5ce4 ,hundr$s or ?\s&s ( feet ( s$im5t>y /rata might 9 ?is way be heap$ up r\nd ! cont95ts1 conta9+ a fragm5t>y s]ies ( organic rema9s belong+ to ^? =ms ( comp>atively %all[-wat] life :i* _h h>d p>ts capable ( pres]vation4 ,!re c be little d\bt t su* has1 9 fact1 be5 ! hi/ory ( ! ma9 mass ( /ratifi$ =mations 9 ! e>?'s cru/4 ,by f> ! l>ge/ proportion ( ^! piles ( m>9e /rata has unque/ionably be5 laid d[n 9 wat] ( no great dep? )9 ! area ( deposit ( t]re/rial s$im5t4 ,! enorm\s ?ickness to :i* !y atta9 seems only explicable by prolong$ & repeat$ movem5ts ( subsid5ce1 int]rupt$1 h[ev]1 z we "k1 by o!r movem5ts ( a contr>y k9d4 ,s9ce ! conditions = ! pres]vation ( organic rema9s exi/ m fav\rably "u ! sea ?an on l&1 m>9e organisms mu/ be f> m abundantly cons]v$ ?an ^? ( ! l&4 ,?is is true to-day1 & has1 z f> z kn[n1 be5 true 9 all pa/ geological "t4 ,h5ce = ! purposes ( ! geologi/ ! fossil rema9s ( m>9e =ms ( life f> surpass all o!rs 9 value4 ,among !m !re w necess>ily be a gradation ( importance1 regulat$ *iefly by _! relative abundance4 ,n[1 ( all ! m>9e tribes :i* live )9 ! juxta-t]re/rial belt ( s$im5tation1 unque/ionably ! ,mollusca /& 9 ! place ( pre-em95ce z reg>ds _! aptitude = becom+ fossils4 ,!y almo/ all possess a h>d1 durable %ell1 capable ( resi/+ consid]able abrasion & readily pass+ into a m9]aliz$ condition4 ,!y are extremely abundant bo? z to individuals & g5]a4 ,!y occur on ! %ore )9 tide m>k1 & range !nce d[n into ! abysses4 ,moreov]1 !y appe> to h possess$ ^! qualifications f e>ly geological times4 ,9 ! m>9e ,mollusca1 !re=e1 we h a common gr\nd ( comp>ison betwe5 ! /ratifi$ =mations ( diff]5t p]iods4 ,!y h be5 /yl$ ! alphabet ( palaeontological inquiry4 ,!re are two ma9 purposes to :i* fossils may be put 9 geological rese>*3 7#a7 to ?r[ light ^u =m] conditions ( physical geography1 su* z ! pres5ce ( l&1 riv]s1 lakes & seas1 9 places :]e !y d n n[ exi/1 *anges ( climate1 & ! =m] di/ribution ( plants & animals2 & 7#b7 to furni% a guide 9 geological *ronology :]eby rocks may be classifi$ accord+ to relative date1 & ! facts ( geological hi/ory may be arrang$ & int]pret$ z a connect$ record ( ! e>?'s progress4 #a4 ,z examples ( ! fir/ ( ^! two directions ( inquiry ref]5ce may be made to 7a7 =m] l&-surfaces reveal$ by ! occurr5ce ( lay]s ( soil ) tree-/umps & roots /ill 9 ! position ( gr[? 7see ,,purbeckian72 7b7 anci5t lakes prov$ by b$s ( m>l or lime/one full ( lacu/r9e %ells2 7c7 old sea-bottoms m>k$ by ! occurr5ce ( m>9e organisms2 7d7 v>iations 9 ! quality ( ! wat]1 su* z fre%ness or saltness1 indicat$ by *anges 9 ! size & %ape ( ! fossils2 7e7 proximity to =m] l&1 sugge/$ by ! occurr5ce ( abundant drift-wood 9 ! /rata2 7f7 =m] conditions ( climate1 diff]5t f ! pres5t1 z evid5c$ by su* organisms z tropical types ( plants & animals int]calat$ among ! /rata ( temp]ate or nor!rn c\ntries4 #b4 ,9 apply+ fossils to ! det]m9ation ( geological *ronology x is fir/ necess>y to asc]ta9 ! ord] ( sup]position ( ! rocks4 ,obvi\sly1 9 a cont9u\s s]ies ( undi/urb$ s$im5t>y deposits ! l[e/ mu/ necess>ily be ! olde/1 & ! plants or animals :i* !y conta9 mu/ h liv$ & di$ be=e any ( ! organisms t occur 9 ! ov]ly+ /rata4 ,?is ord] ( sup]position hav+ be5 settl$ 9 a s]ies ( =mations1 x is f\nd t ! fossils at ! bottom are n q ! same z ^? at ! top ( ! s]ies4 ,trac+ ! b$s upw>d1 we discov] t species aft] species ( ! l[e/ plat=ms disappe>s1 until p]haps n "o ( !m is f\nd4 ,) ! cessation ( ^! old] species o!rs make _! entrance4 ,^!1 9 turn1 are f\nd to die out1 & to be replac$ by new] =ms4 ,aft] pati5t exam9ation ( ! rocks1 x has be5 asc]ta9$ t e well-m>k$ =mation1 or gr\p ( /rata1 is *>act]iz$ by its own species or g5]a1 or by a g5]al assemblage1 or facies1 ( organic =ms4 ,su* a g5]alization c only1 ( c\rse1 be det]m9$ by actual practical exp]i5ce ov] an area ( "s size4 ,:5 ! typical fossils ( a =mation are kn[n1 !y s]ve to id5tify t =mation 9 its progress across a c\ntry4 ,?us1 9 tracts :]e ! true ord] ( sup]position _c be det]m9$1 ow+ to ! want ( sections or to ! di/urb$ condition ( ! rocks1 fossils s]ve z a means ( id5tification & furni% a guide to ! succession ( ! rocks4 ,!y ev5 demon/rate t 9 "s m\nta9\s gr\nd ! b$s h be5 turn$ completely upside d[n1 :]e x c be %[n t ! fossils 9 :at are n[ ! upp]mo/ /rata ought prop]ly to lie und]nea? ^? 9 ! b$s bel[ !m4 ,x is by _! *>act]i/ic fossils t ! /ratifi$ rocks ( ! e>?'s cru/ c be mo/ satisfactorily subdivid$ into conv5i5t gr\ps ( /rata & class$ 9 *ronological ord]4 ,ea* =mation is di/+ui%$ by its own peculi> assemblage ( organic rema9s1 by means ( :i* x c be foll[$ & recogniz$1 ev5 amid ! crumpl+s & dislocations ( a di/urb$ region4 ,! same g5]al succession ( organic types c be obs]v$ ov] a l>ge "p ( ! _w1 ?\gh1 ( c\rse1 ) important modifications 9 diff]5t c\ntries4 ,?is simil>ity ( succession has be5 t]m$ homotaxis1 a t]m :i* expresses ! fact t ! ord] 9 :i* ! lead+ types ( organiz$ exi/5ce h appe>$ ^u ! e>? has be5 simil> ev5 9 widely sep>at$ regions4 ,x is evid5t t1 9 ?is way1 a reliable me?od ( comp>ison is furni%$1 :]eby ! /ratifi$ =mations ( diff]5t p>ts ( ! e>?'s cru/ c be br\ght into relation ) ea* o!r4 ,_h ! geologi/ cont9u$ to rema91 z 9 ! days ( ,w]n]1 hamp]$ by ! limitations impos$ by a reliance on m]e li?ological *>act]s1 he w\ld h made little or no progress 9 deciph]+ ! record ( ! successive phases ( ! hi/ory ( ! globe *ronicl$ 9 ! cru/4 ,j z1 at ! pres5t "t1 %eets ( gravel 9 "o place are contemporane\s ) %eets ( mud at ano!r1 s 9 ! pa/ all k9ds ( s$im5tation h be5 9 progress simultane\sly1 & ^? ( "o p]iod may n be di/+ui%able 9 !mselves f ^? ( ano!r4 ,little or no reliance c be plac$ ^u li?ological resemblances or diff]5ces 9 comp>+ ! s$im5t>y =mations ( diff]5t c\ntries4 ,9 mak+ use ( fossil evid5ce = ! purpose ( subdivid+ ! /ratifi$ rocks ( ! e>?'s cru/1 x is f\nd to be applicable to ! small] details ( /ratigraphy z well z to ! def9ition ( l>ge gr\ps ( /rata4 ,?us a p>ticul> /ratum may be m>k$ by ! occurr5ce 9 x ( v>i\s fossils1 "o or m ( :i* may be di/9ctive1 ei!r f occurr+ 9 no o!r b$ above & bel[ or f special abundance 9 t /ratum4 ,"o or m ( ^! species is !re=e us$ z a guide to ! occurr5ce ( ! b$ 9 "q1 :i* is call$ by ! "n ( ! mo/ abundant species4 ,9 ?is way :at is call$ a geological horizon1 or zone1 is m>k$ (f1 & its exact position 9 ! s]ies ( =mations is fix$4 ,p]haps ! mo/ di/9ctive feature 9 ! progress ( palaeontological geology dur+ ! la/ half c5tury has be5 ! recognition & wide application ( ?is me?od ( zonal /ratigraphy1 :i*1 9 itself1 0 only a fur!r developm5t ( ,william ,smi?'s fam\s idea1 ,/rata id5tifi$ by ,organiz$ ,fossils4 ,x 0 fir/ c>ri$ out 9 detail by v>i\s palaeontologi/s 9 ref]5ce to ! ,jurassic =mations1 notably by ,f4,a4 von ,qu5/$t & ,c4,a4 ,oppel 9 ,g]many & ,a4,d4 d',orbigny 9 ,france4 ,! publication ( ,oppel's classic "w ,die ,jura=mation ,engl&s1 ,frankrei*s und des sudwe/li*5 ,deuts*l&s 7#ahef-aheh7 m>k$ an epo* 9 ! developm5t ( /ratigraphical geology4 ,comb9+ :at _h be5 done by v>i\s obs]v]s ) 8 own labori\s rese>*es 9 ,france1 ,engl&1 ,wurttemb]g & ,bav>ia1 he drew up a classification ( ! ,jurassic sy/em1 gr\p+ its sev]al =mations into zones1 ea* *>act]iz$ by "s di/9ctly pr$om9ant fossil aft] :i* x 0 nam$ 7see ,,lias74 ,! same me?od ( classification 0 aft]w>ds ext5d$ to ! ,cretace\s s]ies by ,a4,d4 d',orbigny1 ,e4 ,heb]t & o!rs1 until ! :ole ,mesozoic rocks f ! ,trias to ! top ( ! ,*alk has n[ be5 p>tition$ into zones1 ea* nam$ aft] "s *>act]i/ic species or g5us ( fossils4 ,m rec5tly ! pr9ciple has be5 ext5d$ to ! ,palaeozoic =mations1 ?\gh z yet less fully ?an to ! y\ng] p>ts ( ! geological record4 ,x has be5 successfully appli$ by ,pr(essor ,c4 ,lapwor? to ! inve/igation ( ! ,silurian s]ies 7see ,,silurian2 ,,ordovician ,,sy/em74 ,he f\nd t ! species ( graptolites h ea* a comp>atively n>r[ v]tical range1 & !y may consequ5tly be us$ = /ratigraphical purposes4 ,apply+ ! me?od1 9 ! fir/ in/ance1 to ! highly plicat$ ,silurian rocks ( ! s\? ( ,scotl&1 he f\nd t by means ( graptolites he 0 able to "w out ! /ructure ( ! gr\nd4 ,ea* great gr\p ( /rata 0 se5 to possess its own graptolitic zones1 & by _! means c\ld be id5tifi$ n only 9 ! orig9al complex ,scotti% area1 b 9 ,engl& & ,wales & 9 ,irel&4 ,x 0 ev5tually asc]ta9$ t ! succession ( zones 9 ,great ,brita9 c\ld be recogniz$ on ! ,cont95t1 9 ,nor? ,am]ica & ev5 9 ,au/ralia4 ,! bra*iopods & trilobites h likewise be5 made use ( = zonal purposes among ! olde/ s$im5t>y =mations4 ,! mo/ anci5t ( ! ,palaeozoic sy/ems has z its fitt+ base ! ,ol5ellus zone4 ,)9 undef9$ & no d\bt v>iable geographical limits palaeontological zones h be5 f\nd to be rem>kably p]si/5t4 ,!y foll[ ea* o!r 9 ! same g5]al ord]1 b n always ) equal def9it5ess4 ,! type fossil may appe> 9 "s di/ricts on a high] or a l[] plat=m ?an x does 9 o!rs4 ,only to a limit$ degree is !re any co9cid5ce betwe5 li?ological v>iations 9 ! /rata & ! sequ5ce ( ! zones4 ,9 ! ,jurassic =mations1 inde$1 :]e frequ5t alt]nations ( diff]5t s$im5t>y mat]ials are to be met )1 x is 9 "s cases possible to trace a def9ite upw>d or d[nw>d limit = a zone by "s abrupt *ange 9 ! s$im5tation1 su* z f lime/one to %ale4 ,b su* a precise dem>cation is impossible :]e no di/9ct b&s ( diff]5t s$im5ts are to be se54 ,! zones c !n only be vaguely det]m9$ by f9d+ _! *>act]i/ic fossils1 & not+ :]e ^! beg9 to appe> 9 ! /rata & :]e !y cease4 ,x w\ld seem1 !re=e1 t ! sequ5ce ( palaeontological zones1 or life-horizons1 has n dep5d$ m]ely ^u *anges 9 ! nature ( ! conditions "u :i* ! organisms liv$4 ,we %\ld naturally expect t ^! *anges w\ld h _h a m>k$ influ5ce2 t1 = in/ance1 a diff]5ce %\ld be p]ceptible betwe5 ! "* ( ! fossils 9 a lime/one & t ( ^? 9 a %ale or a s&/one4 ,! environm5t1 :5 a lime/one 0 9 c\rse ( deposition1 w\ld g5]ally be "o ( cle> wat]1 fav\rable = a m vigor\s & m v>i$ fauna ?an :]e a %ale s]ies 0 accumulat+1 :5 ! wat] w\ld be discol\r$1 & only su* animals w\ld cont9ue to live 9 x1 or on ! bottom1 z c\ld ma9ta9 !mselves 9 ! mid/ ( mud4 ,b no su* li?ological reason1 betok5+ geographical *anges t w\ld affect liv+ creatures1 c be adduc$ z a univ]sally applicable explanation ( ! occurr5ce & limitation ( palaeontological zones4 ,"o ( ^! zones may be only a few in*es1 or feet or y>ds 9 v]tical ext5t1 & no obvi\s li?ological or o!r cause c be se5 :y its specially *>act]i/ic fossils %\ld n be f\nd j z frequ5tly 9 ! simil> /rata above & bel[4 ,!re is (t5 little or no evid5ce ( any s]i\s *ange 9 ! conditions ( s$im5tation1 /ill less ( any widespread physical di/urbance1 su* z ! cata/rophes by :i* ! old] geologi/s expla9$ ! ext9ction ( successive types ( life4 ,x has be5 sugge/$ t1 :]e ! life-zones are well def9$1 s$im5tation has be5 extremely sl[1 & t ?\gh ^! zones foll[ ea* o!r ) no break 9 ! s$im5tation1 !y 7 really sep>at$ by prolong$ int]vals ( "t dur+ :i* organic evolution c\ld come effectively into play4 ,b x is n easy to expla9 h[1 = example 9 ! ,l[] ,lias1 !re c\ld h be5 a succession ( prodigi\s int]vals1 :5 practically no s$im5t 0 laid d[n1 & yet t ! /rata %\ld %[ no sign ( contemporane\s di/urbance or d5udation1 b succe$ ea* o!r z if !y _h be5 accumulat$ by "o cont9u\s process ( deposit4 ,x mu/ be admitt$ t ! problem ( life-zones 9 /ratigraphical geology has n yet be5 solv$4 ,z ,d>w9 fir/ cog5tly %[$1 ! hi/ory ( life has be5 v imp]fectly regi/]$ 9 ! /ratifi$ p>ts ( ! e>?'s cru/4 ,ap>t f ! fact t1 ev5 "u ! mo/ fav\rable conditions1 only a small proportion ( ! total flora & fauna ( any p]iod w\ld be pres]v$ 9 ! fossil /ate1 enorm\s gaps occur :]e no record has surviv$ at all4 ,x is z if :ole *apt]s & books 7 miss+ f a hi/orical "w4 ,"s ( ^! lacunae are suffici5tly obvi\s4 ,?us1 9 "s cases1 p[]ful dislocations h ?r[n consid]able portions ( ! rocks out ( sight4 ,sometimes ext5sive metamorphism has s affect$ !m t _! orig9al *>act]s1 includ+ _! organic cont5ts1 h be5 de/roy$4 ,(t5e/ ( all1 d5udation has come into play1 & va/ masses ( fossilif]\s rock h be5 entirely worn away1 z is demon/rat$ by ! abundant uncon=mabilities 9 ! /ructure ( ! e>?'s cru/4 ,:ile ! m]e fact t "o s]ies ( rocks lies uncon=mably on ano!r proves ! lapse ( a consid]able int]val betwe5 _! respective dates1 ! relative l5g? ( ?is int]val may sometimes be prov$ by means ( fossil evid5ce1 & by ?is alone4 ,let u suppose1 = example1 t a c]ta9 gr\p ( =mations has be5 di/urb$1 uprais$1 d5ud$ & cov]$ uncon=mably by a second gr\p4 ,9 li?ological *>act]s ! two may closely resemble ea* o!r1 & !re may be no?+ to %[ t ! gap repres5t$ by _! uncon=mability is ( an important "*4 ,9 _m cases1 inde$1 x w\ld be q impossible to pron\nce any well-gr\nd$ judgm5t z to ! am\nt ( int]val1 ev5 measur$ by ! vague relative /&>ds ( geological *ronology4 ,b if ea* gr\p conta9s a well-pres]v$ suite ( organic rema9s1 x may n only be possible1 b easy1 to say exactly h[ mu* ( ! geological record has be5 left out betwe5 ! two sets ( =mations4 ,by comp>+ ! fossils ) ^? obta9$ f regions :]e ! geological record is m complete1 x may be asc]ta9$1 p]haps1 t ! l[] rocks belong to a c]ta9 plat=m or /age 9 geological hi/ory :i* = our pres5t purpose we may call ,d1 & t ! upp] rocks c 9 l mann] be p>allel$ ) /age ,h4 ,x w\ld be !n app>5t t at ?is locality ! *ronicles ( ?ree great geological p]iods ,e1 ,f1 & ,g 7 want+1 :i* are else:]e f\nd to be int]calat$ betwe5 ,d & ,h4 ,! lapse ( "t repres5t$ by ?is uncon=mability w\ld ?us be equival5t to t requir$ = ! accumulation ( ! ?ree miss+ =mations 9 ^? regions :]e s$im5tation 0 m cont9u\s4 ,fossil evid5ce may be made to prove ! exi/5ce ( gaps :i* are n o!rwise app>5t4 ,z has be5 already rem>k$1 *anges 9 organic =ms mu/1 on ! :ole1 h be5 extremely sl[ 9 ! geological pa/4 ,! :ole species ( a sea-floor c\ld n pass entirely away1 & be replac$ by o!r =ms1 )\t ! lapse ( long p]iods ( "t4 ,if !n among ! con=mable /ratifi$ =mations ( =m] ages we enc\nt] sudd5 & abrupt *anges 9 ! facies ( ! fossils1 we may be c]ta9 t ^! mu/ m>k omissions 9 ! record1 :i* we may hope to fill 9 f a m p]fect s]ies else:]e4 ,! complete biological contra/s betwe5 ! fossil cont5ts ( uncon=mable /rata are suffici5tly explicable4 ,x is n s easy to give a satisfactory acc\nt ( ^? :i* occur :]e ! b$s are /rictly con=mable1 & :]e no evid5ce c be obs]v$ ( any consid]able *ange ( physical conditions at ! "t ( deposit4 ,a gr\p ( /rata hav+ ! same g5]al li?ological *>act]s ?r\gh\t may be m>k$ by a great discrepance betwe5 ! fossils above & bel[ a c]ta9 l9e4 ,a few species may pass f ! "o into ! o!r1 or p]haps e species may be diff]5t4 ,9 cases ( ?is k9d1 :5 prov$ to be n m]ely local b p]si/5t ov] wide areas1 we mu/ admit1 not)/&+ ! app>5tly undi/urb$ & cont9u\s "* ( ! orig9al deposition ( ! /rata1 t ! abrupt transition f ! "o facies ( fossils to ! o!r repres5ts a long int]val ( "t :i* has n be5 record$ by ! deposit ( /rata4 ,a4,c4 ,ramsay1 :o call$ att5tion to ^! gaps1 t]m$ !m breaks 9 ! succession ( organic rema9s4 ,he %[$ t !y occur abundantly among ! ,palaeozoic & ,second>y rocks ( ,engl&4 ,x is obvi\s1 ( c\rse1 t su* breaks1 ev5 ?\gh traceable ov] wide regions1 7 n g5]al ov] ! :ole globe4 ,!re h nev] be5 any univ]sal int]ruptions 9 ! cont9uity ( ! *a9 ( be+1 s f> z geological evid5ce c %[4 ,b ! physical *anges :i* caus$ ! breaks may h be5 g5]al ov] a zoological di/rict or m9or region4 ,!y no d\bt (t5 caus$ ! complete ext9ction ( g5]a & species :i* _h a small geographical range4 ,f all ^! facts x is cle> t ! geological record1 z x n[ exi/s1 is at ! be/ b an imp]fect *ronicle ( geological hi/ory4 ,9 no c\ntry is x complete4 ,! lacunae ( "o region mu/ be suppli$ f ano!r4 ,yet 9 proportion to ! geographical di/ance betwe5 ! localities :]e ! gaps occur & ^? :5ce ! miss+ int]vals are suppli$1 ! elem5t ( unc]ta9ty 9 our read+ ( ! record is increas$4 ,! mo/ desirable me?od ( rese>* is to exhau/ ! evid5ce = ea* area or prov9ce1 & to comp>e ! g5]al ord] ( its succession z a :ole ) t :i* c be e/abli%$ = o!r prov9ces4 ,,"p ,,vii4--,,/ratigraphical ,,geology ,?is bran* ( ! sci5ce arranges ! rocks ( ! e>?'s cru/ 9 ! ord] ( _! appe>ance1 & int]prets ! sequ5ce ( ev5ts ( :i* !y =m ! records4 ,its prov9ce is to cull f ! o!r dep>tm5ts ( geology ! facts :i* may be ne$$ to %[ :at has be5 ! progress ( our planet1 & ( ea* cont95t & c\ntry1 f ! e>lie/ times ( :i* ! rocks h pres]v$ any memorial4 ,?us f m9]alogy & petrography x conta9s in=mation reg>d+ ! orig9 & subsequ5t mutations ( m9]als & rocks4 ,f dynamical geology x le>ns by :at ag5cies ! mat]ials ( ! e>?'s cru/ h be5 =m$1 alt]$1 brok51 upheav$ & melt$4 ,f geotectonic geology x und]/&s ! v>i\s processes :]eby ^! mat]ials 7 put toge!r s z to build up ! complicat$ cru/ ( ! e>?4 ,f palaeontological geology x receives 9 well-det]m9$ fossil rema9s a clue by :i* to discrim9ate ! diff]5t /ratifi$ =mations1 & to trace ! gr& onw>d m>* ( organiz$ exi/5ce ^u ?is planet4 ,/ratigraphical geology ?us ga!rs up ! sum ( all t is made kn[n by ! o!r dep>tm5ts ( ! sci5ce1 & makes x subs]vi5t to ! int]pretation ( ! geological hi/ory ( ! e>?4 ,! lead+ pr9ciples ( /ratigraphy may be summ$ up z foll[s3 #a4 ,9 e /ratigraphical rese>* ! fundam5tal requisite is to e/abli% ! ord] ( sup]position ( ! /rata4 ,until ?is is accompli%$ x is impossible to arrange ! dates1 & make out ! sequ5ce ( geological hi/ory4 #b4 ,! /ratifi$ portion ( ! e>?'s cru/1 or :at has be5 call$ ! geological record1 c be subdivid$ into natural gr\ps1 or s]ies ( /rata1 *>act]iz$ by di/9ctive organic rema9s & recognizable by ^! rema9s1 9 spite ( great *anges 9 li?ological "* f place to place4 ,a b$1 or a numb] ( b$s1 l9k$ toge!r by conta9+ "o or m di/9ctive species or g5]a ( fossils is t]m$ a zone or horizon1 & usually be>s ! "n ( "o ( its m *>act]i/ic fossils1 z ! ,planorbis-zone ( ! ,l[] ,lias1 :i* is s call$ f ! preval5ce 9 x ( ! ammonite ,psiloc]as planorbis4 ,two or m su* zones relat$ to ea* o!r by ! possession ( a numb] ( ! same *>act]i/ic species or g5]a h be5 designat$ b$s or an assise4 ,two or m sets ( b$s or assises simil>ly relat$ =m a gr\p or /age2 a numb] ( gr\ps or /ages make a s]ies1 =mation or section1 & a succession ( =mations may be unit$ into a sy/em4 #c4 ,"s liv+ species ( plants & animals c be trac$ d[nw>ds ?r\gh ! m rec5t geological =mations2 b ! numb] :i* c be s foll[$ gr[s small] z ! exam9ation is pursu$ into m anci5t deposits4 ,) _! disappe>ance o!r species or g5]a pres5t !mselves :i* are no long] liv+4 ,^! 9 turn may be trac$ backw>d into e>li] =mations1 till !y too cease & _! places are tak5 by yet old] =ms4 ,x is ?us %[n t ! /ratifi$ rocks conta9 ! records ( a gradual progression ( organic =ms4 ,a species :i* has once di$ out does n seem ev] to h reappe>$4 #d4 ,:5 ! ord] ( succession ( organic rema9s among ! /ratifi$ rocks has be5 det]m9$1 !y become an invaluable guide 9 ! inve/igation ( ! relative age ( rocks & ! /ructure ( ! l&4 ,ea* zone & =mation1 be+ *>act]iz$ by its own species or g5]a1 may be recogniz$ by _! means1 & ! true succession ( /rata may ?us be confid5tly e/abli%$ ev5 9 a c\ntry :]e9 ! rocks h be5 %att]$ by dislocation1 fold$1 inv]t$ or metamorphos$4 #e4 ,?\gh local diff]5ces exi/ 9 reg>d to ! precise zone 9 :i* a giv5 species ( organism may make its fir/ appe>ance1 ! g5]al ord] ( succession ( ! organic =ms f\nd 9 ! rocks is nev] inv]t$4 ,! record is n[h]e complete 9 any region1 b ! portions repres5t$1 ev5 ?\gh extremely imp]fect1 always foll[ ea* o!r 9 _! prop] *ronological ord]1 unless :]e di/urbance ( ! cru/ has int]v5$ to de/roy ! orig9al sequ5ce4 #f4 ,! relative *ronological value ( ! divisions ( ! geological record is n to be measur$ by m]e dep? ( /rata4 ,:ile x may be reasonably assum$ t1 9 g5]al1 a great ?ickness ( /ratifi$ rock mu/ m>k ! passage ( a long p]iod ( "t1 x _c safely be affirm$ t a mu* less ?ickness else:]e mu/ repres5t a correspond+ly dim9i%$ p]iod4 ,! ne$ = ?is caution may sometimes be made evid5t by an uncon=mability betwe5 two sets ( rocks1 z has already be5 expla9$4 ,! total dep? ( bo? gr\ps toge!r may be1 say #ajjj ft4 ,else:]e we may f9d a s+le unbrok5 =mation rea*+ a dep? ( #aj1jjj ft42 b x w\ld be unw>rantable to assume t ! latt] repres5ts t5 times ! l5g? ( "t indicat$ by ! =m] two4 ,s f> f ?is be+ ! case1 x might n be difficult to %[ t ! m9or ?ickness ( rock really d5otes by f> ! long] geological int]val4 ,if1 = in/ance1 x c\ld be prov$ t ! upp] "p ( bo? ! sections lies on "o & ! same geological plat=m1 b t ! l[] uncon=mable s]ies 9 ! "o locality belongs to a f> l[] & old] sy/em ( rocks ?an ! base ( ! ?ick con=mable s]ies 9 ! o!r1 !n x w\ld be cle> t ! gap m>k$ by ! uncon=mability really indicates a long] p]iod ?an ! massive succession ( deposits4 #g4 ,fossil evid5ce furni%es ! *ief means ( comp>+ ! relative value ( =mations & gr\ps ( rock4 ,a break 9 ! succession ( organic rema9s1 z already expla9$1 m>ks an int]val ( "t (t5 unrepres5t$ by /rata at ! place :]e ! break is f\nd4 ,! relative importance ( ^! breaks1 & !re=e1 probably1 ! comp>ative int]vals ( "t :i* !y m>k1 may be e/imat$ by ! diff]5ce ( ! facies or g5]al "* ( ! fossils on ea* side4 ,if1 = example1 9 "o case we f9d e species to be dissimil> above & bel[ a c]ta9 horizon1 :ile 9 ano!r locality only half ( ! species on ea* side are peculi>1 we naturally inf]1 if ! total numb] ( species seems l>ge en\gh to w>rant ! inf]5ce1 t ! int]val m>k$ by ! =m] break 0 mu* long] ?an t m>k$ by ! second4 ,b we may g fur!r & comp>e by means ( fossil evid5ce ! relation betwe5 breaks 9 ! succession ( organic rema9s & ! dep? ( /rata betwe5 !m4 ,?ree =mations ( fossilif]\s /rata1 ,a1 ,c1 & ,h1 may occur con=mably above ea* o!r4 ,by a comp>ison ( ! fossil cont5ts ( all p>ts ( ,a1 x may be asc]ta9$ t1 :ile "s species are peculi> to its l[]1 o!rs to its high] portions1 yet ! majority ext5d ?r\gh\t ! =mation4 ,if n[ x is f\nd t ( ! total numb] ( species 9 ! upp] portion ( ,a only "o-?ird passes up into ,c1 x may be inf]r$ ) "s plausibility t ! "t repres5t$ by ! break betwe5 ,a & ,c 0 really long] ?an t requir$ = ! accumulation ( ! :ole ( ! =mation ,a4 ,x might ev5 be possible to discov] else:]e a ?ick int]m$iate =mation ,b fill+ up ! gap betwe5 ,a & ,c4 ,9 l mann] 7 x to be discov]$ t1 :ile ! :ole ( ! =mation ,c is *>act]iz$ by a common suite ( fossils1 n "o ( ! species & only "o half ( ! g5]a pass up into ,h1 ! inf]5ce c\ld h>dly be resi/$ t ! gap betwe5 ! two =mations m>ks ! passage ( a f> long] int]val ?an 0 ne$$ = ! deposition ( ! :ole ( ,c4 ,& ?us we rea* ! rem>kable conclusion t1 ?ick ?\gh ! /ratifi$ =mations ( a c\ntry may be1 9 "s cases !y may n repres5t s long a total p]iod ( "t z d ! gaps 9 _! succession1--9 o!r words1 t non-deposition 0 m frequ5t & prolong$ ?an deposition1 or t ! int]vals ( "t :i* h be5 record$ by /rata h n be5 s long z ^? :i* h n be5 s record$4 ,9 all speculations ( ?is nature1 h[ev]1 x is necess>y to reason f z wide a basis ( obs]vation z possible1 see+ t s mu* ( ! evid5ce is negative4 ,especially ne$ful is x to be> 9 m9d t ! cessation ( "o or m species at a c]ta9 l9e among ! rocks ( a p>ticul> di/rict may mean no?+ m ?an t1 onw>d f ! "t m>k$ by t l9e1 ^! species1 ow+ to "s *ange 9 ! conditions ( life1 7 compell$ to migrate or became locally ext9ct or1 f "s alt]ation 9 ! conditions ( fossilization1 7 no long] imb$d$ & pres]v$ z fossils4 ,!y may h cont9u$ to fl\ri% abundantly 9 neighb\r+ di/ricts = a long p]iod aft]w>d4 ,_m examples ( ?is obvi\s tru? might be cit$4 ,?us 9 a great succession ( m+l$ m>9e1 bracki%-wat] & t]re/rial /rata1 l t ( ! ,c>bonif]\s ,lime/one s]ies ( ,scotl&1 corals1 cr9oids & bra*iopods ab\nd 9 ! lime/ones & accompany+ %ales1 b disappe> z ! s&/ones1 iron/ones1 clays1 coals & bitum9\s %ales sup]v5e4 ,an obs]v] meet+ = ! fir/ "t ) an in/ance ( ?is disappe>ance1 & rememb]+ :at he _h read ab\t breaks 9 succession1 might be tempt$ to speculate ab\t ! ext9ction ( ^! organisms1 & _! replacem5t by o!r & lat] =ms ( life1 su* z ! f]ns1 lycopods1 e/u>9e or fre%-wat] %ells1 ganoid fi%es & o!r fossils s abundant 9 ! ov]ly+ /rata4 ,b fur!r rese>* w\ld %[ him t high above ! plant-be>+ s&/ones & coals o!r lime/ones & %ales might be obs]v$1 once m *>g$ ) ! same m>9e fossils z be=e1 & /ill f>!r ov]ly+ gr\ps ( s&/ones1 coals & c>bonace\s b$s foll[$ by yet high] m>9e lime/ones4 ,he w\ld ?us le>n t ! same organisms1 aft] be+ locally ext]m9at$1 return$ aga9 & aga9 to ! same area4 ,aft] su* a lesson he w\ld probably pause be=e too confid5tly ass]t+ t ! highe/ b$ 9 :i* we c detect c]ta9 fossils m>ks _! f9al appe>ance 9 ! hi/ory ( life4 ,"s breaks 9 ! succession may ?us be extremely local1 "o set ( organisms hav+ be5 driv5 to a diff]5t "p ( ! same region1 :ile ano!r set occupi$ _! place until ! fir/ 0 enabl$ to return4 #h4 ,! geological record is at ! be/ b an imp]fect *ronicle ( ! geological hi/ory ( ! e>?4 ,x ab\nds 9 gaps1 "s ( :i* h be5 caus$ by ! de/ruction ( /rata ow+ to metamorphism1 d5udation or o!rwise1 o!rs by orig9al non-deposition1 z above expla9$4 ,nev]!less f ?is record alone c ! progress ( ! e>? be trac$4 ,x conta9s ! regi/]s ( ! appe>ance & disappe>ance ( tribes ( plants & animals :i* h f "t to "t fl\ri%$ on ! e>?4 ,only a small proportion ( ! total numb] ( species :i* h liv$ 9 pa/ "t h be5 ?us *ronicl$1 yet by collect+ ! brok5 fragm5ts ( ! record an outl9e at lea/ ( ! hi/ory ( life ^u ! e>? c be deciph]$4 ,x _c be too frequ5tly /at$1 nor too prom95tly kept 9 view1 t1 al?\gh gaps occur 9 ! succession ( organic rema9s z record$ 9 ! rocks1 !y d n w>rant ! conclusion t any su* blank int]vals ev] int]rupt$ ! progress ( plant & animal life ^u ! globe4 ,!re is e reason to believe t ! m>* ( life has be5 unbrok51 onw>d & upw>d4 ,geological hi/ory1 !re=e1 if its records 9 ! /ratifi$ =mations 7 p]fect1 ought to %[ a bl5d+ & gradation ( epo* ) epo*4 ,b ! progress has be5 con/antly int]rupt$1 n[ by upheaval1 n[ by volcanic outbur/s1 n[ by depression4 ,^! int]ruptions s]ve z natural divisions 9 ! *ronicle1 & enable ! geologi/ to arrange 8 hi/ory into p]iods4 ,z ! ord] ( succession among /ratifi$ rocks 0 fir/ made out 9 ,europe1 & z _m ( ! gaps 9 t succession 7 f\nd to be widespread ov] ! ,european area1 ! divisions :i* exp]i5ce e/abli%$ = t portion ( ! globe came to be reg>d$ z typical1 & ! names adopt$ = !m 7 appli$ to ! rocks ( o!r & f> di/ant regions4 ,?is application has br\ght out ! fact t "s ( ! mo/ m>k$ breaks 9 ! ,european s]ies d n exi/ else:]e1 &1 on ! o!r h&1 t "s portions ( t s]ies are mu* m complete ?an ! correspond+ sections 9 o!r regions4 ,h5ce1 :ile ! g5]al simil>ity ( succession may rema91 diff]5t subdivisions & nom5clature are requir$ z we pass f cont95t to cont95t4 ,! nom5clature adopt$ = ! subdivisions ( ! geological record be>s witness to ! rapid gr[? ( geology4 ,x is a pat*-"w 9 :i* no sy/em nor language has be5 adh]$ to1 b :]e ! influ5ces by :i* ! progress ( ! sci5ce has be5 m\ld$ may be di/9ctly trac$4 ,"s ( ! e>lie/ names are li?ological1 & rem9d u ( ! fact t m9]alogy & petrography prec$$ geology 9 ! ord] ( bir?--,*alk1 ,oolite1 ,gre5s&1 ,mill/one ,grit4 ,o!rs are topographical1 & (t5 recall ! lab\rs ( ! e>ly geologi/s ( ,engl&--,london ,clay1 ,ox=d ,clay1 ,purbeck1 ,portl&1 ,kimm]idge b$s4 ,o!rs are tak5 f local ,engli% prov9cial names1 & rem9d u ( ! debt we owe to ,william ,smi?1 by :om s _m ( !m 7 fir/ us$--,lias1 ,gault1 ,crag1 ,cornbra%4 ,o!rs ( lat] date recognize an ord] ( sup]position z already e/abli%$ among =mations--,old ,r$ ,s&/one1 ,new ,r$ ,s&/one4 ,by common cons5t x is admitt$ t names tak5 f ! region :]e a =mation or gr\p ( rocks is typically develop$ are be/ adapt$ = g5]al use4 ,cambrian1 ,silurian1 ,devonian1 ,p]mian1 ,jurassic are ( ?is class1 & h be5 adopt$ all ov] ! globe4 ,b :atev] be ! "n *os5 to designate a p>ticul> gr\p ( /rata1 x soon comes to be us$ z a *ronological or homotaxial t]m1 ap>t altoge!r f ! /ratigraphical "* ( ! /rata to :i* x is appli$4 ,?us we speak ( ! ,*alk or ,cretace\s sy/em1 & embrace "u t t]m =mations :i* may conta9 no *alk2 & we may describe z ,silurian a s]ies ( /rata utt]ly unlike 9 li?ological *>act]s to ! =mations 9 ! typical ,silurian c\ntry4 ,9 us+ ^! t]ms we unconsci\sly all[ ! idea ( relative date to arise prom95tly be=e u4 ,h5ce su* a ^w z *alk or cretace\s does n sugge/ s mu* to u ! gr\p ( /rata s call$ z ! int]val ( geological hi/ory :i* ^! /rata repres5t4 ,we speak ( ! ,cretace\s1 ,jurassic1 & ,cambrian p]iods1 & ( ! ,cretace\s fauna1 ! ,jurassic flora1 ! ,cambrian trilobites1 z if ^! adjectives d5ot$ simply epo*s ( geological "t4 ,! /ratifi$ =mations ( ! e>?'s cru/1 or geological record1 are classifi$ into five ma9 divisions1 :i* 9 _! ord] ( antiquity are z foll[s3 7#a7 ,ar*ean or ,pre-,cambrian1 call$ also sometimes ,azoic 7lifeless7 or ,eozoic 7dawn ( life72 7#b7 ,palaeozoic 7anci5t life7 or ,prim>y2 7#c7 ,mesozoic 7middle life7 or ,second>y2 7#d7 ,ca9ozoic 7rec5t life7 or ,t]ti>y2 7#e7 ,quat]n>y or ,po/-,t]ti>y4 ,^! divisions are fur!r rang$ into sy/ems1 =mations1 gr\ps or /ages1 assises & zones4 ,acc\nts ( ! v>i\s subdivisions nam$ are giv5 9 sep>ate articles "u _! own head+s4 ,9 ord]1 h[ev]1 t ! sequ5ce ( ! =mations & _! p>allelism 9 ,europe & ,nor? ,am]ica may be pres5t$ toge!r a /ratigraphical table is giv5 on next page4 ,,"p ,,viii4--,,physiographical ,,geology ,?is dep>tm5t ( geological inquiry inve/igates ! orig9 & hi/ory ( ! pres5t topographical features ( ! l&4 ,z ^! features mu/ obvi\sly be relat$ to ^? ( e>li] "t :i* are record$ 9 ! rocks ( ! e>?'s cru/1 !y _c be satisfactorily /udi$ until at lea/ ! ma9 outl9es ( ! hi/ory ( ^! rocks h be5 trac$4 ,h5ce physiographical rese>* comes appropriately aft] ! o!r bran*es ( ! sci5ce h be5 consid]$4 ,f ! /ratigraphy ( ! t]re/rial cru/ we le>n t by f> ! l>ge/ "p ( ! area ( dry l& is built up ( m>9e =mations2 & !re=e t ! pres5t l& is n an aborig9al portion ( ! e>?'s surface1 b has be5 ov]spread by ! sea 9 :i* its rocks 7 ma9ly accumulat$4 ,we fur!r discov] t ?is subm]g5ce ( ! l& did n happ5 once only1 b aga9 & aga9 9 pa/ ages & 9 all p>ts ( ! _w4 ,yet al?\gh ! t]re/rial areas v>i$ mu* f age to age 9 _! ext5t & 9 _! di/ribution1 be+ at "o "t m cont95tal1 at ano!r m insul>1 !re is reason to believe t ^! successive dim9utions & expansions h on ! :ole be5 effect$ )91 or n f> outside1 ! limits ( ! exi/+ cont95ts4 ,!re is no evid5ce t any portion ( ! pres5t l& ev] lay "u ! deep] p>ts ( ! ocean4 ,! abysmal deposits ( ! ocean-floor h no true repres5tatives among ! s$im5t>y =mations any:]e visible on ! l&4 ,nor1 on ! o!r h&1 c x be %[n t any "p ( ! exi/+ ocean abysses ev] rose above sea-level into dry l&4 ,h5ce geologi/s h drawn ! inf]5ce t ! ocean bas9s h probably be5 always :]e !y n[ are2 & t al?\gh ! cont95tal areas h (t5 be5 n>r[$ by subm]g5ce & by d5udation1 !re has probably seldom or nev] be5 a complete disappe>ance ( l&4 ,! fact t ! s$im5t>y =mations ( ea* successive geological p]iod consi/ to s l>ge an ext5t ( me*anically =m$ t]rig5\s detritus1 af=ds good evid5ce ( ! coexi/5ce ( tracts ( l& z well z ( ext5sive d5udation4 ,! ,geological ,record or ,ord] ( ,succession ( ! ,/ratifi$ ,=mations ( ! ,e>?'s ,cru/4 ---------------------------------------- ---------------------------------------- --- ,europe4 ,nor? ,am]ica4 ---------------------------------------- ---------------------------------------- --- ,q ,hi/oric1 up to ! pres5t "t4 ,simil> to ! ,european u ,prehi/oric1 compris+ deposits ( developm5t1 b ) scanti] a ! ,iron1 ,bronze1 & lat] traces ( ! pres5ce ( man4 t ,/one ,ages4 e ,neoli?ic--alluvium1 peat1 lake- r ,rec5t1 dwell+s1 loess1 c4 n ,po/- ,palaeoli?ic--riv]-gravels1 cave- a glacial deposits1 c4 r or y ,human4 o r ---------------------------------------- ---------------------------------------- ,plei/- ,old] ,loess & valley-gravels2 ,z 9 ,europe1 x is h>dly ,p oc5e cave-deposits4 possible to assign a def9ite o or ,/r&-l9es or rais$ bea*es2 *ronological place to ea* ( s ,glacial4 y\nge/ mora9es4 ! v>i\s deposits ( ?is t ,upp] ,b\ld]-clays2 esk]s2 m>9e p]iod1 t]re/rial & m>9e4 s&s & clays4 ,!y g5]ally resemble ! ,t ,int]glacial deposits4 ,european s]ies4 ,! e ,l[] b\ld]-clay or ,till1 ) *>act]i/ic m>9e1 r /riat$ rock-surfaces bel[4 fluviatile & lacu/r9e t t]races1 :i* ov]lie ! i _/ old] drifts1 h be5 a _/ class$ z ! ,*ampla9 ,gr\p4 r _/ y _/ 4 _/ ---------------------------------------- ---------------------------------------- --- ,p ,new]3--,engli% ,=e/-,b$ ,gr\p2 ,r$ & ,on ! ,atlantic bord] l ,norwi* ,crag2 ,am/elian & ,scaldesian repres5t$ by ! m>9e i gr\ps ( ,belgium & ,holl&2 ,sicilian ,floridian s]ies2 9 ! o & ,a/ian ( ,france & ,italy4 int]ior by a suba]ial & c ,old]3--,engli% ,corall9e ,crag2 ,die/ian lacu/r9e s]ies2 & on ! e ( ,belgium2 ,plaisancian ( s\!rn ,pacific bord] by ! ?ick n ,france & ,italy4 m>9e s]ies ( ,san ,francisco4 e 4 ---------------------------------------- ---------------------------------------- ,m ,want+ 9 ,brita92 well develop$ 9 ,repres5t$ 9 ! ,ea/]n ,/ates i ,france1 ,s4 ,e4 ,europe & ,italy2 by a m>9e s]ies 7,yorkt[n or ,c o divisible into ! foll[+ gr\ps 9 ,*esapeake1 ,*ipola & a c desc5d+ ord]3 7#a7 ,pontian2 7#b7 ,*attahoo*ee gr\ps71 & 9 i e ,s>matian2 7#c7 ,tortonian2 7#d7 ,helvetian2 ! int]ior by ! lacu/r9e n n 7#e7 ,langhian 7,burdigalian74 ,l\p ,=k 7,nebraska71 ,deep o e ,riv]1 & ,john ,day gr\ps4 z 4 o ---------------------------------------- ---------------------------------------- i ,9 ,brita9 ! fluvio-m>9e s]ies ( ,on ! ,atlantic bord] no c ,o ! ,isle ( ,wight2 also ! volcanic equival5ts h be5 l plateaux ( ,antrim & ,inn] ,hebrides satisfactorily recognis$1 b o i & ^? ( ! ,fa]oe ,isles & on ! ,pacific side !re are r g ,icel&4 ,9 cont95tal ,europe ! m>9e deposits 9 ,n4 ,w4 o foll[+ subdivisions h be5 ,oregon1 :i* may repres5t ,t c e/abli%$ 9 desc5d+ ord]3 7#a7 ?is division4 ,9 ! int]ior e e ,aquitanian1 7#b7 ,/ampian 7,rupelian71 ! equival5t is believ$ to r n 7#c7 ,tongra9 7,sannoisian74 be ! fre%-wat] ,:ite ,riv] t e s]ies1 includ+ 7#a7 i 4 ,protoc]as b$s1 7#b7 a ,oreodon b$s1 & 7#c7 r ,titano!rvum b$s4 y ---------------------------------------- ---------------------------------------- 4 ,b>ton s&s & clays2 ,ludian s]ies ( ,wood/ock & ,aquia ,creek gr\ps ,france4 ( ,potomac ,riv]2 ,vicksburg1 ,brackle%am ,b$s2 ,lutetian 7,calcaire ,jackson1 ,claiborne1 ,buhr/one1 ,e grossi] & ,caillasses7 ( ,p>is & ,lignitic gr\ps ( o bas94 ,mississippi4 c ,london clay1 ,woolwi* & ,read+ ,b$s2 ,9 ! int]ior a ?ick s]ies ( e ,?anet s&s2 ,ypresian or ,lond9ian ( fre%-wat] =mations1 n ,n4 ,france & ,belgium2 ,sp>nacian & compris+1 9 desc5d+ e ,?anetian gr\ps4 ord]1 ! ,u9ta1 ,bridg]1 4 ,w9d ,riv]1 ,wasat*1 ,torrejon1 & ,pu]co gr\ps4 ,on ! ,pacific side ! m>9e ,tejon s]ies ( ,oregon & ,cali=nia4 ---------------------------------------- ---------------------------------------- --- ,upp] ,on ! ,atlantic bord] bo? m>9e /rata & o!rs ,danian--want+ 9 ,brita92 upp]mo/ conta9+ a t]re/rial flora lime/one ( ,d5m>k4 repres5t ! ,cretace\s s]ies ,s5onian--,upp] ,*alk ) ,fl9ts ( ( =mations4 ,engl&2 ,aturian & ,ems*]ian /ages ,9 ! int]ior !re is also a on ! ,european cont95t4 comm+l+ ( m>9e ) ,turonian--,middle ,*alk ) few fl9ts1 lacu/r9e deposits4 ,at ! top & compris+ ! ,ang\mian & /ages4 lies ! ,l>amie or ,lignitic ,c ,c5omanian--,l[] ,*alk & ,*alk ,m>l4 s]ies ) an abundant r t]re/rial flora1 pass+ d[n e ,l[] into ! lacu/r9e & t bracki%-wat] ,montana s]ies4 a ,albian--,upp] ,gre5s& & ,gault4 ,( old] date1 ! ,colorado c ,aptian--,l[] ,gre5s&2 ,m>ls & s]ies conta9s an abundant e lime/ones ( ,prov5ce1 c4 m>9e fauna1 yet includes also o ,urgonian 7,b>remian7--,a!rfield clay2 "s ,niobr>a m>ls & u massive ,hippurite lime/ones ( lime/ones are likewise ( s s\!rn ,france4 m>9e orig91 b ! l[] 4 ,neocomian--,weald clay & ,ha/+s s&2 memb]s ( ! s]ies 7,b5ton ,haut]ivian & ,valang9ian sub-/ages & ,dakota7 %[ ano!r great ( ,switz]l& & ,france4 repres5tation ( fre%-wat] ,m s$im5tation ) lignites & e coals4 s ,9 ,cali=nia a va/ succession o ( m>9e deposits 7,%a/a- z ,*ico7 repres5ts ! o ,cretace\s sy/em2 & 9 i we/]n ,briti% ,n4 ,am]ica c coal-seams also occur4 ---------------------------------------- ---------------------------------------- o ,purbeckian--,purbeck b$s2 ,mund] ,m]gel2 ,repres5tatives ( ! ,middle & r l>gely pres5t 9 ,we/phalia4 l[] ,jurassic =mations h ,portl&ian--,portl& gr\p ( ,engl&1 be5 f\nd 9 ,cali=nia & ,s repres5t$ 9 ,s4 ,france by ! ?ick ,oregon1 & f>!r nor? among e ,ti?onian lime/ones4 ! ,arctic isl&s4 c ,kimm]idgian--,kimm]idge ,clay ( ,engl&2 ,/rata conta9+ ,l[] ,jurassic o ,virgulian & ,pt]oc]ian gr\ps ( ,n4 m>9e fossils appe> 9 n ,j ,france2 repres5t$ by ?ick lime/ones ,wyom+ & ,dakota2 & above d u 9 ! ,m$it]ranean bas94 !m come ! ,atlantosaurus a r ,corallian--,coral ,rag1 ,corall9e ,oolite2 & ,baptanodon b$s1 :i* r a ,sequanian /ages ( ! ,cont95t1 h yield$ s l>ge a y s compris+ ! sub-/ages ( ,a/>tian v>iety ( de9osaurs & o!r 4 s & ,rauracian4 v]tebrates1 & especially ! i ,ox=dian--,ox=d ,clay2 ,axgovian & rema9s ( a numb] ( g5]a c ,neuvizyan /ages4 ( small mammals4 4 ,callovian--,kellaways ,rock1 ,divesian sub-/age ( ,n4 ,france4 ,ba?onian--s]ies ( ,engli% /rata f ,cornbra% d[n to ,full]'s ,e>?4 ,bajocian--,inf]ior ,oolite ( ,engl&4 ,lassic--divisible into 7#a7 ,upp] ,lias or ,to>cian1 7#b7 ,middle ,lias1 ,m>l/one or ,*>m\?ian1 7#c7 ,l[] ,lias ( ,s9emurian & ,hettangian4 ---------------------------------------- ---------------------------------------- ,9 ,g]many & we/]n ,europe ?is ,9 ,new ,york1 ,connecticut1 ,new ,t division repres5ts ! deposits ( ,brunswick1 & ,nova ,scotia r inl& seas or lagoons1 & is divisible a s]ies ( r$ s&/one i into ! foll[+ /ages 9 desc5d+ 7,new>k s]ies7 conta9s l&- a ord]3 7#a7 ,rhaetic1 7#b7 ,keup]1 7#c7 plants & labyr9?odonts s ,mus*elkalk1 7#d7 ,bunt]4 ,9 ! l ! lagoon type ( c5tral s ea/]n ,alps & ! ,m$it]ranean & we/]n ,europe4 ,on ! i bas9 ! contemporane\s s$im5t>y ,pacific slope1 h[ev]1 m>9e c =mations are ^? ( op5 cle> equival5ts occur1 repres5t+ 4 sea1 9 :i* a ?ickness ( _m ! pelagic type ( s\?- ?\s& feet ( /rata 0 accumulat$4 ea/]n ,europe4 ---------------------------------------- ---------------------------------------- --- ,p ,?ur+ian--,ze*/e91 ,magnesian ,to ?is division ( ! geologi- e ,lime/one2 nam$ f its developm5t cal record ! ,upp] ,b>r5 r 9 ,?ur+ia2 well repres5t$ ,measures ( ! coal-fields ( m also 9 ,saxony1 ,bav>ia & ,bohemia4 ,p5nsylvania1 ,pr9ce ,edw>d i ,saxonian--,ro?lieg5des ,gr\p2 ,r$ ,isl&1 ,nova ,scotia & a ,s&/ones1 c4 ,new ,brunswick h be5 n ,autunian--:]e ! /rata pres5t ! assign$4 4 lagoon facies1 well display$ at ,autun ,f>!r s\? 9 ,kansas1 ,texas1 9 ,france2 :]e ! m>9e type is & ,nebraska ! repres5ta- pr$om9ant1 z 9 ,russia1 ! gr\p tives ( ! division h an has be5 t]m$ ,art9skian4 abundant m>9e fauna4 ---------------------------------------- ---------------------------------------- ,c ,/ephanian or ,uralian--repres5t$ 9 ,upp] productive ,coal-measures4 a ,russia by m>9e =mations1 & 9 ,l[] ,b>r5 measures4 r c5tral & we/]n ,europe by num]\s ,l[] productive ,coal-measures4 b small bas9s conta9+ a peculi> ,pottsville conglom]ate4 o flora & 9 "s places a great v>iety ,mau* ,*unk %ales2 lime/ones n ( insects4 ( ,*e/]1 ,/ ,l\is1 c4 i ,we/phalian or ,moscovian--,coal-measures1 ,pocono s]ies2 ,k9d]hook f ,mill/one ,grit4 lime/one4 e ,culm or ,d9antian--,c>bonif]\s ,lime/one r & ,calcif]\s ,s&/one s]ies4 o u s 4 ---------------------------------------- ---------------------------------------- ,devonian & ,old ,r$ ,s&/one4 ,p ---------------------------------------- ---------------------------------------- a ,devonian type4 ,old ,r$ ,s&/one l type4 a ---------------------------------------- ------ _/ ,catskill r$ s&/one2 ,old e _/ ,fam5nian4 ,yell[ & r$ ,r$ ,s&/one type3 ! o ,upp] s&/one ) /rata bel[ %[ ! z ,frasnian4 ,holopty*ius1 ,devonian type4 o ,bo?riolepis1c4 ,*emung ,gr\p4 i ,g5esee c _/ ,givetian4 ,cai?ness ,flag/ones o ,middle ) ,o/eolepus1 _/ ,hamilton ,gr\p4 r ,eifelian4 ,dipt]us1 ,m>cellus ,homo/eus1 c4 ,p r ,r$ & purple _/ ,cornif]\s ,lime/one4 _/ ,upp] i _/,cobl5tizian4 s&/ones & ,held- m ,l[] conglom]ates ) ,onondaga ,lime/one4 erb]g a ,g$9nian4 ,cephalaspis1 ,gr\p4 r ,pt]aspis1 c4 ,oriskany ,s&/one4 y ---------------------------------------- --------------------------------------- 4 _/ ,l[] ,held]b]g ,gr\p4 ,s _/ ,ludl[ ,gr\p4 ,wat]-,lime4 i ,upp] ,w5lock ,niag>a ,%ale & ,lime/one4 l ,ll&ov]y ,cl9ton ,gr\p4 u ,m$9a r i _/ ,c9c9nati ,gr\p4 a ,l[] _/ ,c>adoc or ,bala ,gr\p4 ,utica n 7,ordovician7 ,ll&eilo ,tr5ton 4 ,ar5ig ,*azy ,calcif]\s ---------------------------------------- ---------------------------------------- ,c ,upp] or ,ol5us s]ies--,tremadoc ,upp] or ,potsdam s]ies ) a slates & ,l+ula ,flags4 ,ol5us & ,dicelocephalus m ,middle or ,p>doxides s]ies--,m5evian fauna4 b ,gr\p4 ,middle or ,acadian s]ies ) r ,l[] or ,ol5ellus s]ies--,llanb]is ,p>adoxides fauna4 i & ,h>le* ,gr\p1 & ,ol5ellus- ,l[] or ,georgian s]ies ) a zone4 ,ol5ellus fauna4 n 4 ---------------------------------------- ---------------------------------------- --- ,ar*ean1 ,pre-,cambrian1 ,eozoic4 ---------------------------------------- ---------------------------------------- --- ,9 ,scotl&1 und]nea? ! ,cambrian ,9 ,canada & ! ,lake ,sup]ior ,ol5ellus gr\p1 lies uncon=mably region ( ! ,unit$ ,/ates a mass ( r$ s&/one & con- a va/ succession ( rocks ( glom]ate 7,torridonian7 #hjjj or #aj1jjj ,pre-,cambrian age has be5 ft4 ?ick1 :i* re/s ) a /rong gr\p$ into ! foll[+ gneisses & s*i/s 7,lewisian74 ,a subdivisions 9 desc5d+ ?ick s]ies ( slates & phyllites ord]3 7#a7 ,kewe5wan1 ly+ lies bel[ ! olde/ ,palaeozoic rocks uncon=mably on 7#b7 ,animikie1 9 c5tral ,europe1 ) co>se sep>at$ by a /rong gneisses bel[4 uncon=mability f 7#c7 ,upp] ,huronian1 7#d7 ,l[] ,huronian ) an uncon=mable base1 7#e7 ,g\t*i*+1 7#f7 ,laur5tian4 ,9 ! ea/]n "p ( ,canada1 ,newf\ndl&1 c41 & also 9 ,montana1 s$im5t>y =mations ( great ?ickness bel[ ! l[e/ ,cambrian zone h be5 f\nd to conta9 "s obscure organisms4 ---------------------------------------- ---------------------------------------- --- ,f ^! g5]al consid]ations we proce$ to inquire h[ ! exi/+ topographical features ( ! l& arose4 ,obvi\sly ! co-op]ation ( ! two great geological ag5cies ( hypog5e & epig5e en]gy1 :i* h be5 at "w f ! beg9n+ ( our globe's deciph]able hi/ory1 mu/ h be5 ! cause to :i* ^! features are to be assign$2 & ! task ( ! geologi/ is to asc]ta91 if possible1 ! "p t has be5 tak5 by ea*4 ,!re is a natural t5d5cy to see 9 a /up5d\s piece ( sc5]y1 su* z a deep rav9e1 a range ( hills1 a l9e ( precipice or a *a9 ( m\nta9s1 evid5ce only ( subt]ranean convulsion2 & be=e ! subject 0 tak5 up z a matt] ( /rict sci5tific induction1 an appeal to =m] cataclysms 0 consid]$ a suffici5t solution ( ! problems pres5t$ by su* features ( l&scape4 ,! rise ( ! mod]n ,huttonian s*ool1 h[ev]1 l$ to a m c>eful exam9ation ( ^! problems4 ,! important %>e tak5 by erosion 9 ! det]m9ation ( ! pres5t features ( l&scape 0 !n recogniz$1 :ile a full] appreciation ( ! relative p>ts play$ by ! hypog5e & epig5e causes has gradually be5 rea*$4 #a4 ,! /udy ( ! progress ( d5udation at ! pres5t "t has l$ to ! conclusion t ev5 if ! rate ( wa/e 7 n m rapid ?an x is to-day1 x w\ld yet suffice 9 a comp>atively brief geological p]iod to r$uce ! dry l& to bel[ ! sea-level4 ,b n only w\ld ! area ( ! l& be dim9i%$ by d5udation1 x c\ld h>dly fail to be m or less involv$ 9 ^? widespread movem5ts ( subsid5ce1 dur+ :i* ! ?ick s$im5t>y =mations ( ! cru/ appe> to h be5 accumulat$4 ,x is ?us manife/ t !re mu/ h be5 f "t to "t dur+ ! hi/ory ( our globe upw>d movem5ts ( ! cru/1 :]eby ! balance betwe5 l& & sea 0 r$ress$4 ,pro(s ( su* movem5ts h be5 abundantly pres]v$ among ! /ratifi$ =mations4 ,we !re le>n t ! uplifts h usually foll[$ ea* o!r at long int]vals betwe5 :i* subsid5ce prevail$1 & ?us t !re has be5 a prolong$ oscillation ( ! cru/ ov] ! great cont95tal areas ( ! e>?'s surface4 ,an exam9ation ( t surface leads to ! recognition ( two great types ( upheaval4 ,9 ! "o1 ! sea-floor1 ) all its ?ick accumulations ( s$im5t1 has be5 c>ri$ upw>ds1 sometimes = sev]al ?\s& feet1 s equably t ! /rata reta9 _! orig9al flatness ) h>dly any s5sible di/urbance = hundr$s ( squ>e miles4 ,9 ! o!r type ! solid cru/ has be5 plicat$1 corrugat$ & dislocat$1 especially along p>ticul> l9es1 & has atta9$ its mo/ /up5d\s disruption 9 l(ty *a9s ( m\nta9s4 ,betwe5 ^! two phases ( uplift _m int]m$iate /ages h be5 develop$1 accord+ to ! direction & int5sity ( ! subt]ranean =ce & ! v>y+ nature & disposition ( ! rocks ,( ! cru/4 7a7 ,:]e ! uplift has ext5d$ ov] wide spaces1 )\t appreciable de=mation ( ! cru/1 ! flat /rata h giv5 rise to l[ pla9s1 or if ! am\nt ( uprise has be5 great en\gh1 to high pla9s1 plateaux or tablel&s4 ,! pla9s ( ,russia1 = example1 lie = ! mo/ "p on su* tracts ( equably uplift$ /rata4 ,! great pla9s ( ! we/]n int]ior ( ! ,unit$ ,/ates =m a great plateau or tablel&1 #ejjj or #fjjj ft4 above ! sea1 & _m ?\s&s ( squ>e miles 9 ext5t1 on :i* ! ,rocky ,m\nta9s h be5 ridg$ up4 7b7 ,x is 9 a great m\nta9-*a9 t ! complicat$ /ructures develop$ dur+ di/urbances ( ! e>?'s cru/ c be/ be /udi$ 7see ,p>ts ,,iv4 & ,v4 ( ?is article71 & :]e ! influ5ce ( ^! /ructures on ! topography ( ! surface is mo/ effectively display$4 ,su* a *a9 may be ! result ( "o colossal di/urbance2 b ^? ( high geological antiquity usually furni% pro(s ( successive uplifts ) m or less int]v5+ d5udation4 ,=m$ along l9es ( cont95tal displacem5t 9 ! cru/1 !y h aga9 & aga9 giv5 relief f ! /ra9 ( compression by fre% crumpl+1 fracture & uprise4 ,! *ief guide 9 trac+ ^! successive /ages ( gr[? is suppli$ by uncon=mability4 ,if1 = example1 a m\nta9-range consi/s ( uprais$ ,silurian rocks1 ^u ! upturn$ & d5ud$ edges ( :i* ! ,c>bonif]\s ,lime/one lies transgressively1 x is cle> t its orig9al upheaval mu/ h tak5 place 9 ! p]iod ( geological "t repres5t$ by ! int]val betwe5 ! ,silurian & ! ,c>bonif]\s ,lime/one =mations4 ,if1 z ! range is foll[$ along its c\rse1 ! ,c>bonif]\s ,lime/one is f\nd to be also highly incl9$ & cov]$ uncon=mably by ! ,upp] ,coal-measures1 a second uplift ( t portion ( ! gr\nd c be prov$ to h tak5 place betwe5 ! "t ( ! ,lime/one & t ( ! ,upp] ,coal-measures4 ,by ?is simple & obvi\s k9d ( evid5ce ! relative ages ( diff]5t m\nta9-*a9s may be comp>$4 ,9 mo/ great *a9s1 h[ev]1 ! rocks h be5 s int5sely crumpl$1 & ev5 inv]t$1 t mu* lab\r may be requir$ be=e _! true relations c be det]m9$4 ,! ,alps furni% an in/ructive example ( ! long s]ies ( revolutions ?r\gh :i* a great m\nta9-sy/em may h pass$ be=e rea*+ its pres5t developm5t4 ,! fir/ beg9n+s ( ! *a9 may h be5 uprais$ be=e ! olde/ ,palaeozoic =mations 7 laid d[n4 ,!re are at lea/ traces ( l& & %ore-l9es 9 ! ,c>bonif]\s p]iod4 ,subsequ5t subm]g5ces & uplifts appe> to h occurr$ dur+ ! ,mesozoic p]iods4 ,!re is evid5ce t !reaft] ! :ole region sank deep "u ! sea1 9 :i* ! old] ,t]ti>y s$im5ts 7 accumulat$1 & :i* seems to h spread "r across ! he>t ( ! ,old ,_w4 ,b aft] ! deposition ( ! ,eoc5e =mations came ! gigantic disruptions :]eby all ! rocks ( ! ,alp9e region 7 fold$ ov] ea* o!r1 cru%$1 corrugat$1 fractur$ & displac$1 "s ( _! old] portions1 includ+ ! fundam5tal gneisses & s*i/s1 be+ squeez$ up1 torn (f1 & pu%$ horizontally = _m miles ov] ! y\ng] rocks4 ,b ?is upheaval1 ?\gh ! mo/ mom5t\s1 0 n ! la/ :i* ! *a9 has und]gone1 = at a lat] epo* 9 ,t]ti>y "t r5ew$ di/urbance gave rise to a fur!r s]ies ( ruptures & plications4 ,! *a9 ?us successively upheav$ has be5 cont9u\sly expos$ to d5udation & has consequ5tly lo/ mu* ( its orig9al height4 ,t x has be5 left 9 a /ate ( in/ability is indicat$ by ! frequ5t e>?quakes ( ! ,alp9e region1 :i* d\btless arise f ! sudd5 snapp+ ( rocks "u int5se /ra94 ,a di/9ct type ( m\nta9 due to direct hypog5e action is to be se5 9 a volcano4 ,x has be5 already po9t$ out 7,"p ,,iv4 sect4 #a7 t at ! v5ts :i* ma9ta9 a communication betwe5 ! molt5 magma ( ! e>?'s int]ior & ! surface1 eruptions take place :]eby quantities ( lava & fragm5t>y mat]ials are heap$ r\nd ea* orifice ( dis*>ge4 ,a typical volcanic m\nta9 takes ! =m ( a p]fect cone1 b z x gr[s 9 size & its ma9 v5t is *ok$1 :ile ! sides ( ! cone are unable to )/& ! =ce ( ! explosions or ! pressure ( ! asc5d+ column ( lava1 eruptions take place lat]ally1 & num]\s p>asitic cones arise on ! flanks ( ! p>5t m\nta94 ,:]e lava fl[s out f long fissures1 x may pile up va/ %eets ( rock1 & bury ! surr\nd+ c\ntry "u sev]al ?\s& feet ( solid /one1 cov]+ _m hundr$s ( squ>e miles4 ,9 ?is way volcanic tablel&s h be5 =m$ :i*1 attack$ by ! d5ud+ =ces1 are gradually tr5*$ by valleys & rav9es1 until ! orig9al level surface ( ! lava-field may be almo/ or :olly lo/4 ,z /rik+ examples ( ?is physiographical type ref]5ce may be made to ! plateau ( ,abyss9ia1 ! ,ghats ( ,india1 ! plateaux ( ,antrim1 ! ,inn] ,hebrides & ,icel&1 & ! great lava-pla9s ( ! we/]n t]ritories ( ! ,unit$ ,/ates4 #b4 ,b :ile ! subt]ranean movem5ts h uprais$ portions ( ! surface ( ! li?osph]e above ! level ( ! ocean1 & h ?us be5 in/rum5tal 9 produc+ ! exi/+ tracts ( l&1 ! detail$ topographical features ( a l&scape are n solely1 nor 9 g5]al ev5 *iefly1 attributable to ^! movem5ts4 ,f ! "t t any portion ( ! sea-floor appe>s above sea-level1 x und]goes erosion by ! v>i\s epig5e ag5ts4 ,ea* climate & geological region has its own developm5t ( ^! ag5ts1 :i* include air1 aridity1 rapid & frequ5t alt]nations ( wetness & dryness or ( heat & cold1 ra91 spr+s1 fro/s1 riv]s1 glaci]s1 ! sea1 plant & animal life4 ,9 a dry climate subject to great extremes ( temp]ature ! "* & rate ( decay w diff] f ^? ( a moi/ or an arctic climate4 ,b x mu/ be rememb]$ t1 h[ev] mu* !y may v>y 9 activity & 9 ! results :i* !y effect1 ! epig5e =ces "w )\t int]mission1 :ile ! hypog5e =ces br+ ab\t ! upheaval ( l& only aft] long int]vals4 ,h5ce1 trifl+ z ! results dur+ a human life may appe>1 if we realize ! multiply+ influ5ce ( "t we are l$ to p]ceive t ! app>5tly feeble sup]ficial ag5ts c1 9 ! c\rse ( ages1 a*ieve /up5d\s trans=mations 9 ! aspect ( ! l&4 ,if ?is efficacy may be d$uc$ f :at c be se5 to be 9 progress n[1 x may n less conv9c+ly be %[n1 f ! nature ( ! s$im5t>y rocks ( ! e>?'s cru/1 to h be5 9 progress f ! e>ly beg9n+s ( geological hi/ory4 ,side by side ) ! v>i\s upheavals & subsid5ces1 !re has be5 a cont9u\s removal ( mat]ials f ! l&1 & an equally p]si/5t deposit ( ^! mat]ials "u wat]1 ) ! consequ5t gr[? ( new rocks4 ,d5udation has be5 aptly comp>$ to a process ( sculptur+ :]e91 :ile ea* ( ! implem5ts employ$ by nature1 l a special k9d ( grav+ tool1 produces its own *>act]i/ic impress on ! l&1 !y all comb9e h>moni\sly t[>ds ! a*ievem5t ( _! "o common task4 ,h5ce ! pres5t cont\rs ( ! l& dep5d p>tly on ! orig9al configuration ( ! gr\nd1 & ! influ5ce x may h _h 9 guid+ ! op]ations ( ! erosive ag5ts1 p>tly on ! vig\r ) :i* ^! ag5ts p]=m _! "w1 & p>tly on ! v>y+ /ructure & p[]s ( resi/ance possess$ by ! rocks on :i* ! erosion is c>ri$ on4 ,:]e a new tract ( l& has be5 rais$ out ( ! sea by su* an en]getic movem5t z broke up ! cru/ & produc$ ! complicat$ /ructure & tumultu\s ext]nal =ms ( a great m\nta9 *a91 ! influ5ce ( ! hypog5e =ces on ! topography atta9s its highe/ developm5t4 ,b ev5 ! y\nge/ exi/+ *a9 has suff]$ s greatly f d5udation t ! aspect :i* x pres5t$ at ! "t ( its uplift c only be dimly p]ceiv$4 ,no m /rik+ illu/ration ( ?is feature c be f\nd ?an t suppli$ by ! ,alps1 nor "o :]e ! geotectonic /ructures h be5 s fully /udi$ 9 detail4 ,on ! out] flanks ( ^! m\nta9s ! longitud9al ridges & valleys ( ! ,jura correspond ) l9es ( anticl9e & syncl9e4 ,yet ?\gh ! dom9ant topographical elem5ts ( ! region h obvi\sly be5 produc$ by ! plication ( ! /ratifi$ =mations1 ea* ridge has suff]$ s l>ge an am\nt ( erosion t ! y\ng] rocks h be5 remov$ f its cre/ :]e ! old] memb]s ( ! s]ies are n[ expos$ to view1 :ile on e slope pro(s may be se5 ( ext5sive d5udation4 ,if f ^! long wave-l undulations ( ! gr\nd1 :]e ! relations betwe5 ! disposition ( ! rocks bel[ & ! =ms ( ! surface are s cle>ly traceable1 ! obs]v] proce$s inw>ds to ! ma9 *a91 he f9ds t ! plications & displacem5ts ( ! v>i\s =mations assume an increas+ly complicat$ "*2 & t al?\gh pro(s ( great d5udation cont9ue to ab\nd1 x becomes increas+ly difficult to =m any satisfactory conjecture z to ! %ape ( ! gr\nd :5 ! upheaval end$ or any reliable e/imate ( ! am\nt ( mat]ial :i* has s9ce !n be5 remov$4 ,along ! c5tral heights ! m\nta9s lift !mselves t[>ds ! sky l ! /orm-swept cre/s ( va/ e>?-bill[s4 ,! :ole aspect ( ! gr\nd sugge/s int5se commotion1 & ! impression ?us giv5 is (t5 mu* int5sifi$ by ! twi/$ & crumpl$ /rata1 visible f a long di/ance1 on ! crags & cre/s4 ,on ?is brok5-up surface ! v>i\s ag5ts ( d5udation h be5 ceaselessly engag$ s9ce x em]g$ f ! sea4 ,!y h excavat$ valleys1 sometimes along depressions provid$ = !m by ! subt]ranean di/urbances1 sometimes d[n ! slopes ( ! disrupt$ blocks ( gr\nd4 ,s p[]ful has be5 ?is erosion t valleys cut out along l9es ( anticl9e1 :i* 7 natural ridges1 h sometimes become m important ?an ^? 9 l9es ( syncl9e1 :i* 7 /ructurally depressions4 ,! same suba]ial =ces h erod$ lake-bas9s1 dug out corries or cirques1 not*$ ! ridges1 spl9t]$ ! cre/s & furr[$ ! slopes1 leav+ no "p ( ! orig9al surface ( ! uplift$ *a9 unmodifi$4 ,x has (t5 be5 not$ ) surprise t features ( und]gr\nd /ructure :i*1 x might h be5 confid5tly anticipat$1 %\ld h ex]cis$ a m>k$ influ5ce on ! topography ( ! surface h n be5 able to resi/ ! levell+ action ( ! d5ud+ ag5ts1 & d n n[ affect ! surface at all4 ,?is result is conspicu\sly se5 9 coal-fields :]e ! /rata are abundantly trav]s$ by faults4 ,^! dislocations1 hav+ sometimes a displacem5t ( sev]al hundr$ feet1 might h be5 expect$ to break up ! surface into a network ( cliffs & pla9s2 yet 9 g5]al !y d n modify ! level "* ( ! gr\nd above4 ,"o ( ! mo/ rem>kable faults 9 ,europe is ! great ?ru/ :i* b\nds ! s\!rn edge ( ! ,belgian coal-field & br+s ! ,devonian rocks above ! ,coal-measures4 ,x c be trac$ across ,belgium into ! ,b\lonnais1 & may n improbably run b5ea? ! ,second>y & ,t]ti>y rocks ( ! s\? ( ,engl&4 ,x is cross$ by ! valleys ( ! ,meuse & o!r nor!rly-fl[+ /reams4 ,yet s indi/9ctly is x m>k$ 9 ! ,meuse valley t no "o w\ld suspect its exi/5ce f any peculi>ity 9 ! g5]al =m ( ! gr\nd1 & ev5 an exp]i5c$ geologi/1 until he _h le>n$ ! /ructure ( ! di/rict1 w\ld sc>cely detect any fault at all4 ,:]e faults h influ5c$ ! sup]ficial topography1 x is usually by giv+ rise to a holl[ along :i* ! suba]ial ag5ts & especially runn+ wat] c act effectively4 ,su* a holl[ may be ev5tually wid5$ & deep5$ into a valley4 ,on b>e crags & cre/s1 l9es ( fault are apt to be m>k$ by not*es or clefts1 & !y ?us help to produce ! p9nacles & s]rat$ outl9es ( ^! expos$ upl&s4 ,x 0 cog5tly en=c$ by ,hutton & ,playfair1 & indep5d5tly by ,lam>ck1 t no co-op]ation ( und]gr\nd ag5cy is ne$$ to produce su* topography z may be se5 9 a great "p ( ! _w1 b t if a tract ( sea-floor 7 uprais$ into a wide pla91 ! fall ( ra9 & ! circulation ( wat] ov] its surface w\ld 9 ! end c>ve out su* a sy/em ( hills & valleys z may be se5 on ! dry l& n[4 ,no su* pla9 w\ld be a dead-level4 ,x w\ld h inequalities on its surface :i* w\ld s]ve z *annels to guide ! dra9age f ! fir/ %[]s ( ra94 ,& ^! *annels w\ld be sl[ly wid5$ & deep5$ until !y w\ld become rav9es & valleys1 :ile ! gr\nd betwe5 !m w\ld be left project+ z ridges & hills4 ,nor w\ld ! erosion ( su* a sy/em ( wat]-c\rses require a long s]ies ( geological p]iods = its accompli%m5t4 ,f measurem5ts & e/imates ( ! am\nt ( erosion n[ tak+ place 9 ! bas9 ( ! ,mississippi riv] x has be5 comput$ t valleys #hjj ft4 deep might be c>v$ out 9 less ?an a million ye>s4 ,9 ! va/ tablel&s ( ,colorado & o!r we/]n regions ( ! ,unit$ ,/ates an impressive picture is pres5t$ ( ! results ( m]e suba]ial erosion on undi/urb$ & ne>ly level /rata4 ,sy/ems ( /ream-c\rses & valleys1 riv] gorges unexampl$ else:]e 9 ! _w = dep? & l5g?1 va/ w9d+ l9es ( esc>pm5t1 l ranges ( sea-cliffs1 t]rac$ slopes ris+ f plateau to plateau1 huge buttresses & solit>y /acks /&+ l isl&s out ( ! pla9s1 great m\nta9-masses t[]+ into picturesque peaks & p9nacles cleft by innum]able gullies1 yet ev]y:]e m>k$ by ! p>allel b>s ( ! horizontal /rata out ( :i* !y h be5 c>v$--^! are ! ord]ly symmetrical *>act]i/ics ( a c\ntry :]e ! sc5]y is due entirely to ! action ( suba]ial ag5ts on ! "o h& & ! v>y+ resi/ance ( p]fectly regul> /ratifi$ rocks on ! o!r4 ,! details ( ! sculpture ( ! l& h ma9ly dep5d$ on ! nature ( ! mat]ials on :i* nature's erosive tools h be5 employ$4 ,! jo9ts by :i* all rocks are trav]s$ h be5 especially s]viceable z dom9ant l9es d[n :i* ! ra9 has filt]$1 up :i* ! spr+s h ris5 & into :i* ! fro/ w$ges h be5 driv54 ,on ! high b>e sc>ps ( a l(ty m\nta9 ! inn] /ructure ( ! mass is laid op51 & !re ! sy/em ( jo9ts ev5 m ?an faults is se5 to h det]m9$ ! l9es ( cre/1 ! v]tical walls ( cliff & precipice1 ! =ms ( buttress & recess1 ! position ( cleft & *asm1 ! outl9e ( spire & p9nacle4 ,on ! l[] slopes1 ev5 "u ! tape/ry ( v]dure :i* nature delights to hang :]e %e c ov] h] nak$ rocks1 we may detect ! same p]vad+ influ5ce ( ! jo9ts ^u ! =ms assum$ by rav9es & crags4 ,ea* k9d ( /one1 too1 gives rise to its own *>act]i/ic =m ( sc5]y4 ,massive cry/all9e rocks1 su* z granite1 break up along _! jo9ts & (t5 decay into s& or e>? along _! expos$ surfaces1 giv+ rise to rugg$ crags ) long talus slopes at _! base4 ,! /ratifi$ rocks besides splitt+ at _! jo9ts are especially di/+ui%$ by p>allel l$ges1 cornices & recesses1 produc$ by ! irregul> decay ( _! compon5t /rata1 s t !y (t5 assume curi\sly ar*itectural types ( sc5]y4 ,b besides ?is family feature !y display _m m9or v>ieties ( aspect accord+ to _! li?ological composition4 ,a range ( s&/one hills1 = example1 pres5ts a m>k$ contra/ to "o ( lime/one1 & a l9e ( *alk d[ns to ! esc>pm5ts =m$ by alt]nat+ b&s ( h>d] & s(t] clays & %ales4 ,x may suffice "h m]ely to allude to a few ( ! m important p>ts ( ! topography ( ! l& 9 _! relation to physiographical geology4 ,a true m\nta9-*a91 view$ f ! geological side1 is a mass ( high gr\nd :i* owes its prom95ce to a ridg+-up ( ! e>?'s cru/1 & ! int5se plication & rupture ( ! rocks ( :i* x is compos$4 ,b ranges ( hills almo/ m\nta9\s 9 _! bulk may be =m$ by ! gradual erosion ( valleys out ( a mass ( orig9al high gr\nd1 su* z a high plateau or tablel&4 ,em95ces :i* h be5 isolat$ by d5udation f ! ma9 mass ( ! =mations ( :i* !y orig9ally =m$ "p are kn[n z outli]s or hills ( circumd5udation4 ,tablel&s1 z already po9t$ out1 may be produc$ ei!r by ! upheaval ( tracts ( horizontal /rata f ! sea-floor into l&2 or by ! uprise ( pla9s ( d5udation1 :]e rocks ( v>i\s composition1 /ructure & age h be5 levell$ d[n to ne> or bel[ ! level ( ! sea by ! co-op]ation ( ! v>i\s erosive ag5ts4 ,mo/ ( ! great tablel&s ( ! globe are plat=ms ( little-di/urb$ /rata :i* h be5 uprais$ bodily to a consid]able elevation4 ,no soon]1 h[ev]1 are !y plac$ 9 t position ?an !y are attack$ by runn+ wat]1 & beg9 to be holl[$ out into sy/ems ( valleys4 ,z ! valleys s9k1 ! plat=ms betwe5 !m gr[ into n>r[] & m def9ite ridges1 until ev5tually ! level tablel& is conv]t$ into a complicat$ network ( hills & valleys1 :]e91 nev]!less1 ! key to ! :ole arrangem5t is furni%$ by a k ( ! disposition & effects ( ! fl[ ( wat]4 ,! examples ( ?is process br\ght to light 9 ,colorado1 ,wyom+1 ,nevada & ! o!r we/]n regions by ,newb]ry1 ,k+1 ,hayd51 ,p[ell & o!r explor]s1 are among ! mo/ /rik+ monum5ts ( geological op]ations 9 ! _w4 ,examples ( anci5t & mu* decay$ tablel&s =m$ by ! d5udation ( mu* di/urb$ rocks are furni%$ by ! ,highl&s ( ,scotl& & ( ,norway4 ,ea* ( ^! tracts ( high gr\nd consi/s ( "s ( ! olde/ & mo/ dislocat$ =mations ( ,europe1 :i* at a remote p]iod 7 worn d[n into a pla91 & 9 t condition may h la9 long subm]g$ "u ! sea & may possibly h be5 ov]spread !re ) y\ng] =mations4 ,hav+ at a mu* lat] "t be5 rais$ sev]al ?\s& feet above sea-level ! anci5t plat=ms ( ,brita9 & ,sc&9avia h be5 s9ce expos$ to d5udation1 :]eby ea* ( !m has be5 s deeply *annel$ into gl5s & fjords t x pres5ts to-day a surface ( rugg$ hills1 ei!r isolat$ or connect$ along ! flanks1 :ile only fragm5ts ( ! g5]al surface ( ! tablel& c "h & !re be recogniz$ amid/ ! g5]al de/ruction4 ,valleys h 9 g5]al be5 holl[$ out by ! great] erosive action ( runn+ wat] along ! *annels ( dra9age4 ,_! direction has be5 probably det]m9$ 9 ! great majority ( cases by irregul>ities ( ! surface along :i* ! dra9age fl[$ on ! fir/ em]g5ce ( ! l&4 ,sometimes ^! irregul>ities h be5 produc$ by folds ( ! t]re/rial cru/1 sometimes by faults1 sometimes by ! irregul>ities on ! surface ( an uplift$ plat=m ( deposition or ( d5udation4 ,two dom9ant tr5ds may be obs]v$ among !m4 ,"s are longitud9al & run along ! l9e ( flexures 9 ! uprais$ tract ( l&1 o!rs are transv]se :]e ! dra9age has fl[$ d[n ! slopes ( ! ridges into ! longitud9al valleys or into ! sea4 ,! =ms ( valleys h be5 gov]n$ p>tly by ! /ructure & composition ( ! rocks1 & p>tly by ! relative pot5cy ( ! diff]5t d5ud+ ag5ts4 ,:]e ! influ5ce ( ra9 & fro/ has be5 slight1 & ! /reams1 suppli$ f di/ant s\rces1 h _h suffici5t declivity1 deep1 n>r[1 precipit\s rav9es or gorges h be5 excavat$4 ,! canyons ( ! arid region ( ! ,colorado are a magnific5t example ( ?is result4 ,:]e1 on ! o!r h&1 ord9>y atmosph]ic action has be5 m rapid1 ! sides ( ! riv] *annels h be5 attack$1 & op5 slop+ gl5s & valleys h be5 holl[$ out4 ,a gorge or defile is usually due to ! action ( a wat]fall1 :i*1 beg9n+ ) "s abrupt declivity or precipice 9 ! c\rse ( ! riv] :5 x fir/ comm5c$ to fl[1 or caus$ by "s h>d rock cross+ ! *annel1 has eat5 its way backw>d4 ,lakes h be5 already ref]r$ to1 & _! modes ( orig9 h be5 m5tion$4 ,z !y are cont9ually be+ fill$ up ) ! detritus wa%$ into !m f ! surr\nd+ regions !y _c be ( any great geological antiquity1 unless :]e by "s unkn[n process _! bas9s are f "t to "t wid5$ & deep5$4 ,9 ! g5]al suba]ial d5udation ( a c\ntry1 innum]able m9or features are work$ out z ! /ructure ( ! rocks controls ! op]ations ( ! erod+ ag5ts4 ,?us1 among comp>atively undi/urb$ /rata1 a h>d b$ re/+ ^u o!rs ( a s(t] k9d is apt to =m along its outcrop a l9e ( cliff or esc>pm5t4 ,?\gh a long range ( su* cliffs resembles a coa/ t has be5 worn by ! sea1 x may be entirely due to m]e atmosph]ic wa/e4 ,aga91 ! m resi/+ portions ( a rock may be se5 project+ z crags or knolls4 ,an igne\s mass w /& out z a bold hill f amid/ ! m decomposable /rata ?r\gh :i* x has ris54 ,^! features1 (t5 s m>k$ on ! l[] gr\nds1 atta9 _! mo/ conspicu\s developm5t among ! high] & b>] p>ts ( ! m\nta9s1 :]e suba]ial dis9tegration is mo/ rapid4 ,! torr5ts te> out deep gullies f ! sides ( ! declivities4 ,corries or cirques are scoop$ out on ! "o h& & nak$ precipices are left on ! o!r4 ,! h>d] b&s ( rock project z massive ribs d[n ! slopes1 %oot up into prom95t aiguilles1 or help to give to ! summits ! not*$ saw-l outl9es !y s (t5 pres5t4 ,! mat]ials worn f ! surface ( ! high] are spread out ov] ! l[] gr\nds4 ,! /reams z !y desc5d beg9 to drop _! freight ( s$im5t :51 by ! less5+ ( _! declivity1 _! c>ry+ p[] is dim9i%$4 ,! great pla9s ( ! e>?'s surface are due to ?is deposit ( gravel1 s& & loam4 ,!y are ?us monum5ts at once ( ! de/ructive & reproductive processes :i* h be5 9 progress unceas+ly s9ce ! fir/ l& rose above ! sea & ! fir/ %[] ( ra9 fell4 ,e pebble & p>ticle ( _! soil1 once "p ( ! di/ant m\nta9s1 has travell$ sl[ly & fitfully to l[] levels4 ,aga9 & aga9 h ^! mat]ials be5 %ift$1 ev] mov+ d[nw>d & sea-w>d4 ,= c5turies1 p]haps1 !y h tak5 _! %>e 9 ! f]tility ( ! pla9s & h m9i/]$ to ! nurture ( fl[] & tree1 ( ! bird ( ! air1 ! bea/ ( ! field & ( man himself4 ,b _! de/9y is /ill ! great ocean4 ,9 t b\rne alone c !y f9d undi/urb$ repose1 & !re1 sl[ly accumulat+ 9 massive b$s1 !y w rema9 until1 9 ! c\rse ( ages1 r5ew$ upheaval %all raise !m into future l&1 !re once m to pass ?r\gh ! same cycle ( *ange4 7,a4 ,ge47 ,,lit]ature4--,hi/orical3 ,! /&>d "w is ,k>l ,a4 von ,zittel's ,ges*i*te d] ,geologie und ,palaontologie 7#ahii71 ( :i* !re is an abbreviat$1 b /ill valuable1 ,engli% translation2 ,d',ar*iac1 ,hi/oire des progres de la geologie1 deals especially ) ! p]iod #ahcd-ahej2 ,kef]/e91 ,ges*i*te und ,lit]atur d] ,geognosie1 gives a summ>y up to #ahdj2 :ile ,sir ,a4 ,geikie's ,f\nd]s ( ,geology 7#ahig2 #bnd ed41 #aijf7 deals m p>ticul>ly ) ! p]iod #agej-ahbj4 ,g5]al treatises3 ,sir ,*>les ,lyell's ,pr9ciples ( ,geology is a classic4 ,( mod]n ,engli% works1 ,sir ,a4 ,geikie's ,text ,book ( ,geology 7#d? ed41 #aijc7 occupies ! fir/ place2 ! "w ( ,t4,c4 ,*amb]l9 & ,r4,d4 ,salisbury1 ,geology2 ,e>? ,hi/ory 7#c vols41 #aije-aijf71 is especially valuable = ,am]ican geology4 ,a4 de ,lapp>5t's ,traite de geologie 7#e? ed41 #aijf71 is ! /&>d ,fr5* "w4 ,h4 ,cr$n]'s ,elem5te d] ,geologie has gone ?r\gh sev]al editions 9 ,g]many4 ,dynamical & physiographical geology are elaborately treat$ by ,e4 ,suess1 ,das ,antlitz d] ,erde1 translat$ into ,engli%1 ) ! title ,! ,face ( ! ,e>?4 ,! practical /udy ( ! sci5ce is treat$ ( by ,f4 von ,ri*?(51 ,fuhr] fur ,=s*ungsreis5de 7#ahhf72 ,g4,a4 ,cole1 ,aids 9 ,practical ,geology 7#e? ed41 #aijf72 ,a4 ,geikie1 ,outl9es ( ,field ,geology 7#e? ed41 #aijj74 ,! practical applications ( ,geology are discuss$ by ,j4,v4 ,elsd51 ,appli$ ,geology 7#ahih-ahii74 ,! relations ( ,geology to sc5]y are dealt ) by ,sir ,a4 ,geikie1 ,sc5]y ( ,scotl& 7#crd ed41 #aija72 ,j4,e4 ,m>r1 ,! ,sci5tific ,/udy ( ,sc5]y 7#aijj72 ,"l ,avebury1 ,! ,sc5]y ( ,switz]l& 7#ahif72 ,! ,sc5]y ( ,engl& 7#aijb72 & ,j4 ,geikie1 ,e>? ,sculpture 7#ahih74 ,a detail$ bibliography is giv5 9 ,sir ,a4 ,geikie's ,text ,book ( ,geology4 ,see also ! sep>ate articles on geological subjects = special ref]5ces to au?orities4 ,,footnotes3 #a ,9 ,de ,luc's ,lettres physiques et morales sur les montagnes 7#aggh71 ! ^w cosmology is us$ = our sci5ce1 ! au?or /at+ t geology is m appropriate1 b x 0 n a ^w 9 use4 ,9 a complet$ edition1 publi%$ 9 #aggi1 ! same /atem5t is made1 b geology occurs 9 ! text2 9 ! same ye> ,de ,saussure us$ ! ^w )\t any explanation1 z if x 7 well kn[n4 #b ,! subject ( ! age ( ! e>? has also be5 discuss$ by ,pr(essor ,j4 ,joly & ,pr(essor ,w4,j4 ,sollas4 ,! =m] geologi/1 approa*+ ! "q f a novel po9t ( view1 has e/imat$ ! total quantity ( sodium 9 ! wat] ( ! ocean & ! quantity ( t elem5t receiv$ annually by ! ocean f ! d5udation ( ! l&4 ,divid+ ! "o sum by ! o!r1 he arrives at ! result t ! probable age ( ! e>? is betwe5 #ij & #ajj millions ( ye>s 7,trans4 ,roy4 ,dubl9 ,soc4 s]4 ii4 vol4 vii41 #ahii1 p4 #bc3 ,geol4 ,mag41 #aijj1 p4 #bbj74 ,pr(essor ,sollas believes t ?is limit exce$s :at is requir$ = ! evolution ( geological hi/ory1 t ! l[] limit assign$ by ,"l ,kelv9 falls %ort ( :at ! facts dem&1 & t geological "t w probably be f\nd to h be5 compris$ )9 "s indet]m9ate p]iod betwe5 ^! limits4 7,address to ,section ,c1 ,brit4 ,assoc4 ,report1 #aijj2 ,age ( ! ,e>?1 ,london1 #aije47 ,,geometrical ,,cont9uity4 ,9 a report ( ! ,in/itute prefix$ to ,jean ,victor ,poncelet's ,traite des proprietes projectives des figures 7,p>is1 #ahbb71 x is said t he employ$ ce qu'il appelle le pr9cipe de cont9uite4 ,! law or pr9ciple ?us nam$ by him _h1 he tells u1 be5 tacitly assum$ z axiomatic by les plus savans geometres4 ,x _h 9 fact be5 enunciat$ z lex cont9uationis1 & la loi de la cont9uite1 by ,gottfri$ ,wilhelm ,leibnitz 7,oxf4 ,n4,e4,d471 & previ\sly "u ano!r "n by ,johann ,kepl] 9 cap4 iv4 #d ( 8 ,ad ,vitellionem p>alipom5a quibus a/ronomiae p>s optica traditur 7,franc(urti1 #afjd74 ,( sections ( ! cone1 he says1 !re are five species f ! recta l9ea or l9e-pair to ! circle4 ,f ! l9e-pair we pass ?r\gh an inf9ity ( hyp]bolas to ! p>abola1 & !nce ?r\gh an inf9ity ( ellipses to ! circle4 ,relat$ to ! sections are c]ta9 rem>kable po9ts :i* h no "n4 ,kepl] calls !m foci4 ,! circle has "o focus at ! c5tre1 an ellipse or hyp]bola two foci equidi/ant f ! c5tre4 ,! p>abola has "o focus )9 x1 & ano!r1 ! caecus focus1 :i* may be imag9$ to be at inf9ity on ! axis )9 or )\t ! curve4 ,! l9e f x to any po9t ( ! section is p>allel to ! axis4 ,to c>ry out ! analogy we mu/ speak p>adoxically1 & say t ! l9e-pair likewise has foci1 :i* 9 ?is case coalesce z 9 ! circle & fall ^u ! l9es !mselves2 = our geometrical t]ms %\ld be subject to analogy4 ,kepl] de>ly loves analogies1 8 mo/ tru/y tea*]s1 acqua9t$ ) all ! secrets ( nature1 omnium naturae arcanorum conscios4 ,& !y are to be especially reg>d$ 9 geometry z1 by ! use ( h[ev] absurd expressions1 class+ extreme limit+ =ms ) an inf9ity ( int]m$iate cases1 & plac+ ! :ole ess5ce ( a ?+ cle>ly be=e ! eyes4 ,"h1 !n1 we f9d =mulat$ by ,kepl] ! doctr9e ( ! concurr5ce ( p>allels at a s+le po9t at inf9ity & ! pr9ciple ( cont9uity 7"u ! "n analogy7 9 relation to ! inf9itely great4 ,su* conceptions s /rik+ly prop\nd$ 9 a fam\s "w c\ld n escape ! notice ( contempor>y ma!maticians4 ,h5ry ,briggs1 9 a lett] to ,kepl] f ,m]ton ,college1 ,ox=d1 dat$ #aj ,cal4 ,m>tiis #afbe1 sugge/s improvem5ts 9 ! ,ad ,vitellionem p>alipom5a1 & gives ! foll[+ con/ruction3 ,draw a l9e ,,cbadc1 & let an ellipse1 a p>abola1 & a hyp]bola h ,b & ,a = focus & v]tex4 ,let ,,cc be ! o!r foci ( ! ellipse & ! hyp]bola4 ,make ,,ad equal to ,,ab1 & ) c5tres ,,cc & radius 9 ea* case equal to ,,cd describe circles4 ,!n any po9t ( ! ellipse is equidi/ant f ! focus ,b & "o circle1 & any po9t ( ! hyp]bola f ! focus ,b & ! o!r circle4 ,any po9t ,p ( ! p>abola1 9 :i* ! second focus is miss+ or inf9itely di/ant1 is equidi/ant f ! focus ,b & ! l9e ?r\gh ,d :i* we call ! directrix1 ?is tak+ ! place ( ei!r circle :5 its c5tre ,c is at inf9ity1 & e l9e ,,cp be+ !n p>allel to ! axis4 ,?us ,briggs1 & we "k n h[ _m savans geometres :o h left no record1 _h already tak5 up ! new doctr9e 9 geometry 9 its au?or's lifetime4 ,six ye>s aft] ,kepl]'s dea? 9 #afcj ,gir>d ,des>gues1 ! ,monge ( 8 age1 br\ght out ! fir/ ( 8 rem>kable works f\nd$ on ! same pr9ciples1 a %ort tract entitl$ ,me?ode univ]selle de mettre en p]spective les objets donnes reellem5t ou en devis 7,p>is1 #afcf72 b ,le privilege etoit de #afcj4 7,p\dra1 ,,oeuvres de ,des41 i4 #ee74 ,kepl] z a mod]n geomet] is be/ kn[n by 8 ,new ,/]eometry ( ,w9e ,casks 7,l9cii1 #afae71 9 :i* he replaces ! circuit\s ,ar*im$ean me?od ( exhau/ion by a direct royal road ( inf9itesimals1 treat+ a vani%+ arc z a /raight l9e & reg>d+ a curve z made up ( a succession ( %ort *ords4 ,"s #bjjj ye>s previ\sly "o ,antipho1 probably ! well-kn[n oppon5t ( ,socrates1 has reg>d$ a circle 9 l mann] z ! limit+ =m ( a _m-sid$ inscrib$ rectil9e> figure4 ,antipho's notion 0 reject$ by ! m5 ( 8 day z uns\nd1 & :5 reproduc$ by ,kepl] x 0 aga9 /\tly oppos$ z incapable ( any sort ( geometrical demon/ration--n altoge!r )\t reason1 = x re/$ on an assum$ law ( cont9uity r ?an on palpable pro(4 ,to complete ! !ory ( cont9uity1 ! "o ?+ ne$ful 0 ! idea ( imag9>y po9ts impli$ 9 ! algebraical geometry ( ,r5e ,desc>tes1 9 :i* equations betwe5 v>iables repres5t+ co-ord9ates 7 f\nd (t5 to h imag9>y roots4 ,newton1 9 8 two sections on ,inv5tio orbium 7,pr9cipia i4 #d1 #e71 %[s 9 8 brief way t he is famili> ) ! pr9ciples ( mod]n geometry4 ,9 two propositions he uses an auxili>y l9e :i* is suppos$ to cut ! conic 9 ,x & ,y1 b1 z he rem>ks at ! end ( ! second 7prop4 #bd71 x may n cut x at all4 ,= ! sake ( brevity he passes on at once ) ! obs]vation t ! requir$ con/ructions are evid5t f ! case 9 :i* ! l9e cuts ! trajectory4 ,9 ! s*olium app5d$ to prop4 #bg1 aft] say+ t an asymptote is a tang5t at inf9ity1 he gives an unexpla9$ g5]al con/ruction = ! axes ( a conic1 :i* seems to imply t x has asymptotes4 ,9 all su* cases1 hav+ equations to 8 loci 9 ! backgr\nd1 he may h ?\ght ( elem5ts ( ! figure z pass+ into ! imag9>y /ate 9 su* mann] z n to vitiate conclusions arriv$ at on ! hypo!sis ( _! reality4 ,rog] ,joseph ,boscovi*1 a c>eful /ud5t ( ,newton's works1 has a full & ?or\gh discussion ( geometrical cont9uity 9 ! ?ird & la/ volume ( 8 ,elem5ta univ]sae ma!seos 7ed4 prim4 ,v5et1 #ageg71 :i* conta9s ,sectionum conic>um elem5ta nova quadam me?odo conc9nata et diss]tationem de trans=matione locorum geometricorum1 ubi de cont9uitatis lege1 et de quibusdam inf9iti my/]iis4 ,8 fir/ pr9ciple is t all v>ieties ( a def9$ locus h ! same prop]ties1 s t :at is demon/rable ( "o %\ld be demon/rable 9 l mann] ( all1 al?\gh "s artifice may be requir$ to br+ out ! und]ly+ analogy betwe5 !m4 ,! opposite extremities ( an inf9ite /raight l9e1 he says1 are to be reg>d$ z jo9$1 z if ! l9e 7 a circle hav+ its c5tre at ! inf9ity on ei!r side ( x4 ,?is leads up to ! idea ( a veluti plus quam inf9ita ext5sio1 a l9e-circle conta9+1 z we say1 ! l9e inf9ity4 ,*ange f ! real to ! imag9>y /ate is cont+5t ^u ! passage ( "s elem5t ( a figure ?r\gh z]o or inf9ity & nev] takes place p] saltum4 ,l9es be+ "s positive & "s negative1 !re mu/ be negative rectangles & negative squ>es1 su* z ^? ( ! ext]ior diamet]s ( a hyp]bola4 ,boscovi*'s fir/ pr9ciple 0 t ( ,kepl]1 by ^: quantumvis absurdis locutionibus ! bolde/ applications ( x are cov]$1 z :5 we say ) ,poncelet t all conc5tric circles 9 a plane t\* "o ano!r 9 two imag9>y fix$ po9ts at inf9ity4 ,9 ,g4,k4 ,*4 von ,/audt's ,geometrie d] ,lage & ,beitrage zur ,g4 d] ,l4 7,nurnb]g1 #ahdg1 #ahef-ahfj7 ! geometry ( position1 includ+ ! ext5sion ( ! field ( pure geometry to ! inf9ite & ! imag9>y1 is pres5t$ z an indep5d5t sci5ce1 wel*e des ,mess5s ni*t b$>f4 7,see ,,geometry3 ,projective47 ,ocul> illusions due to di/ance1 su* z ,rog] ,bacon notices 9 ! ,opus majus 7i4 #abf1 ii4 #ajh1 #dig2 ,ox=d1 #ahig71 lead up to or illu/rate ! ma!matical uses ( ! inf9ite & its reciprocal ! inf9itesimal4 ,speci\s objections c1 ( c\rse1 be made to ! anomalies ( ! law ( cont9uity1 b !y are inh]5t 9 ! high] geometry1 :i* has taught u s mu* ( ! secrets ( nature4 ,kepl]'s excursus on ! analogy betwe5 ! conic sections h]e9be=e ref]r$ to is giv5 at l5g? 9 an article on ,! ,geometry ( ,kepl] & ,newton 9 vol4 xviii4 ( ! ,transactions ( ! ,cambridge ,philosophical ,society 7#aijj74 ,x _h be5 g5]ally ov]look$1 until att5tion 0 call$ to x by ! pres5t writ] 9 a note read 9 #ahhj 7,proc4 ,c4,p4,s4 iv4 #ad-ag71 & %ortly aft]w>ds 9 ,! ,anci5t & ,mod]n ,geometry ( ,conics1 ) ,hi/orical ,notes & ,prolegom5a 7,cambridge #ahha74 7,c4 ,t4997 ,,geometry1 ! g5]al t]m = ! bran* ( ma!matics :i* has = its prov9ce ! /udy ( ! prop]ties ( space4 ,f exp]i5ce1 or possibly intuitively1 we *>act]ize exi/5t space by c]ta9 fundam5tal qualities1 t]m$ axioms1 :i* are insusceptible ( pro(2 & ^! axioms1 9 conjunction ) ! ma!matical entities ( ! po9t1 /raight l9e1 curve1 surface & solid1 appropriately def9$1 are ! premises f :i* ! geomet] draws conclusions4 ,! geometrical axioms are m]ely conv5tions2 on ! "o h&1 ! sy/em may be bas$ ^u inductions f exp]i5ce1 9 :i* case ! d$uc$ geometry may be reg>d$ z a bran* ( physical sci5ce2 or1 on ! o!r h&1 ! sy/em may be =m$ by purely logical me?ods1 9 :i* case ! geometry is a phase ( pure ma!matics4 ,obvi\sly ! geometry ) :i* we are mo/ famili> is t ( exi/5t space--! ?ree-dim5sional space ( exp]i5ce2 ?is geometry may be t]m$ ,euclidean1 aft] its mo/ fam\s expositor4 ,b o!r geometries exi/1 = x is possible to frame sy/ems ( axioms :i* def9itely *>act]ize "s o!r k9d ( space1 & f ^! axioms to d$uce a s]ies ( non-contradictory propositions2 su* geometries are call$ non-,euclidean4 ,x is conv5i5t to discuss ! subject-matt] ( geometry "u ! foll[+ head+s3 ,i4 ,euclidean ,geometry3 a discussion ( ! axioms ( exi/5t space & ( ! geometrical entities1 foll[$ by a synoptical acc\nt ( ,euclid's ,elem5ts4 ,,ii4 ,projective ,geometry3 prim>ily ,euclidean1 b diff]+ f ,i4 9 employ+ ! notion ( geometrical cont9uity 7q4v47--po9ts & l9es at inf9ity4 ,,iii4 ,descriptive ,geometry3 ! me?ods = repres5t+ ^u planes figures plac$ 9 space ( ?ree dim5sions4 ,,iv4 ,analytical ,geometry3 ! repres5tation ( geometrical figures & _! relations by algebraic equations4 ,v4 ,l9e ,geometry3 an analytical treatm5t ( ! l9e reg>d$ z ! space elem5t4 ,,vi4 ,non-,euclidean ,geometry3 a discussion ( geometries o!r ?an t ( ! space ( exp]i5ce4 ,,vii4 ,axioms ( ,geometry3 a critical analysis ( ! f\ndations ( geometry4 ,special subjects are treat$ "u _! own head+s3 e4g4 ,,projection1 ,,p]spective2 ,,curve1 ,,surface2 ,,circle1 ,,conic ,,section2 ,,triangle1 ,,polygon1 ,,polyh$ron2 !re are also articles on special curves & figures1 e4g4 ,,ellipse1 ,,p>abola1 ,,hyp]bola2 ,,tetrah$ron1 ,,cube1 ,,octah$ron1 ,,dodecah$ron1 ,,icosah$ron2 ,,c>dioid1 ,,cat5>y1 ,,cissoid1 ,,con*oid1 ,,cycloid1 ,,epicycloid1 ,,limacon1 ,,oval1 ,,quadratrix1 ,,spiral1 c4 ,hi/ory4--,! orig9 ( geometry 7,gr4 ,greek3 ge1 e>?1 ,greek3 metron1 a measure7 is1 accord+ to ,h]odotus1 to be f\nd 9 ! etymology ( ! ^w4 ,its bir?place 0 ,egypt1 & x arose f ! ne$ ( survey+ ! l&s inundat$ by ! ,nile floods4 ,9 its infancy x !re=e consi/$ ( a few rules1 v r\gh & approximate1 = comput+ ! areas ( triangles & quadrilat]als2 &1 ) ! ,egyptians1 x proce$$ no fur!r1 ! geometrical entities--! po9t1 l9e1 surface & solid--be+ only discuss$ 9 s f> z !y 7 involv$ 9 practical affairs4 ,! po9t 0 realiz$ z a m>k or position1 a /raight l9e z a /ret*$ /r+ or ! trac+ ( a pole1 a surface z an area2 b ^! units 7 n ab/ract$2 & = ! ,egyptians geometry 0 only an art--an auxili>y to survey+4#a ,! fir/ /ep t[>ds its elevation to ! rank ( a sci5ce 0 made by ,?ales 7q4v47 ( ,miletus1 :o transplant$ ! elem5t>y ,egyptian m5suration to ,greece4 ,?ales cle>ly ab/ract$ ! notions ( po9ts & l9es1 f\nd+ ! geometry ( ! latt] unit1 & discov]+ p] saltum _m propositions conc]n+ areas1 ! circle1 c4 ,! empirical rules ( ! ,egyptians 7 correct$ & develop$ by ! ,ionic ,s*ool :i* he f\nd$1 especially by ,anaxim&] & ,anaxagoras1 & 9 ! #f? c5tury ,b4,c4 pass$ into ! c>e ( ! ,py?agoreans4 ,f ?is "t geometry ex]cis$ a p[]ful influ5ce on ,greek ?\ght4 ,py?agoras 7q4v471 seek+ ! key ( ! univ]se 9 ari?metic & geometry1 inve/igat$ logically ! pr9ciples und]ly+ ! kn[n propositions2 & ?is result$ 9 ! =mulation ( def9itions1 axioms & po/ulates :i*1 9 addition to f\nd+ a sci5ce ( geometry1 p]mitt$ a cry/allization1 fractional1 x is true1 ( ! amorph\s collection ( mat]ial at h&4 ,py?agorean geometry 0 ess5tially a geometry ( areas & solids2 its goal 0 ! regul> solids--! tetrah$ron1 cube1 octah$ron1 dodecah$ron & icosah$ron--:i* symboliz$ ! five elem5ts ( ,greek cosmology4 ,! geometry ( ! circle1 previ\sly /udi$ 9 ,egypt & mu* m s]i\sly by ,?ales1 0 some:at neglect$1 al?\gh ?is curve 0 reg>d$ z ! mo/ p]fect ( all plane figures & ! sph]e ! mo/ p]fect ( all solids4 ,! circle1 h[ev]1 0 tak5 up by ! ,sophi/s1 :o made mo/ ( _! discov]ies 9 attempts to solve ! classical problems ( squ>+ ! circle1 d\bl+ ! cube & trisect+ an angle4 ,^! problems1 besides /imulat+ pure geometry1 i4e4 ! geometry ( con/ructions made by ! rul] & compasses1 ex]cis$ consid]able influ5ce 9 o!r directions4 ,! fir/ problem l$ to ! discov]y ( ! me?od ( exhau/ion = det]m9+ areas4 ,antiphon inscrib$ a squ>e 9 a circle1 & on ea* side an isosceles triangle hav+ its v]tex on ! circle2 on ! sides ( ! octagon s obta9$1 isosceles triangles 7 aga9 con/ruct$1 ! process lead+ to inscrib$ polygons ( #h1 #af & #cb sides2 & ! areas ( ^! polygons1 :i* are easily det]m9$1 are successive approximations to ! area ( ! circle4 ,bryson ( ,h]aclea took an important /ep :5 he circumscrib$1 9 addition to inscrib+1 polygons to a circle1 b he committ$ an error 9 treat+ ! circle z ! mean ( ! two polygons4 ,! me?od ( ,antiphon1 9 assum+ t by cont9u$ division a polygon c be con/ruct$ co9cid5t ) ! circle1 dem&$ t magnitudes are n inf9itely divisible4 ,mu* controv]sy rang$ ab\t ?is po9t2 ,ari/otle support$ ! doctr9e ( inf9ite divisibility2 ,z5o attempt$ to %[ its absurdity4 ,! me*anical trac+ ( loci1 a pr9ciple initiat$ by ,ar*ytas ( ,t>5tum to solve ! la/ two problems1 0 a frequ5t subject = /udy1 & sev]al me*anical curves 7 ?us discov]$ at subsequ5t dates 7cissoid1 con*oid1 quadratrix74 ,m5tion may be made ( ,hippocrates1 :o1 besides develop+ ! kn[n me?ods1 made a /udy ( simil> figures1 &1 z a consequ5ce1 ( proportion4 ,?is /ep is important z br++ into l9e discont9u\s numb] & cont9u\s magnitude4 ,a fre% /imulus 0 giv5 by ! succe$+ ,platoni/s1 :o1 accept+ 9 "p ! ,py?agorean cosmology1 made ! /udy ( geometry prelim9>y to t ( philosophy4 ,! _m discov]ies made by ?is s*ool 7 facilitat$ 9 no small measure by ! cl>ification ( ! axioms & def9itions1 ! logical sequ5ce ( propositions :i* 0 adopt$1 &1 m especially1 by ! =mulation ( ! analytic me?od1 i4e4 ( assum+ ! tru? ( a proposition & !n reason+ to a kn[n tru?4 ,! ma9 /r5g? ( ! ,platoni/ geomet]s lies 9 /]eometry or ! geometry ( solids4 ,! ,py?agoreans _h dealt ) ! sph]e & regul> solids1 b ! pyramid1 prism1 cone & cyl9d] 7 b little kn[n until ! ,platoni/s took !m 9 h&4 ,eudoxus e/abli%$ _! m5suration1 prov+ ! pyramid & cone to h "o-?ird ! cont5t ( a prism & cyl9d] on ! same base & ( ! same height1 & 0 probably ! discov]] ( a pro( t ! volumes ( sph]es are z ! cubes ( _! radii4 ,! discussion ( sections ( ! cone & cyl9d] l$ to ! discov]y ( ! ?ree curves nam$ ! p>abola1 ellipse & hyp]bola 7see ,,conic ,,section72 x is difficult to ov]-e/imate ! importance ( ?is discov]y2 its inve/igation m>ks ! cr[n+ a*ievem5t ( ,greek geometry1 & l$ 9 lat] ye>s to ! fundam5tal !orems & me?ods ( mod]n geometry4 ,! pres5tation ( ! subject-matt] ( geometry z a connect$ & logical s]ies ( propositions1 prefac$ by ,greek3 ,horoi or f\ndations1 _h be5 attempt$ by _m2 b x is to ,euclid t we owe a complete exposition4 ,little inde$ 9 ! ,elem5ts is probably orig9al except ! arrangem5t2 b 9 ?is ,euclid surpass$ su* pr$ecessors z ,hippocrates1 ,leon1 pupil ( ,neocleides1 & ,!udius ( ,magnesia1 devis+ an apt logical model1 al?\gh :5 scrut9iz$ 9 ! light ( mod]n ma!matical conceptions ! pro(s are riddl$ ) fallacies4 ,accord+ to ! comm5tator ,proclus1 ! ,elem5ts 7 writt5 ) a tw(old object1 fir/1 to introduce ! novice to geometry1 & secondly1 to lead him to ! regul> solids2 conic sections f\nd no place !re94 ,:at ,euclid did = ! l9e & circle1 ,apollonius did = ! conic sections1 b !re we h a discov]] z well z editor4 ,^! two works1 :i* conta9 ! greate/ contributions to anci5t geometry1 are treat$ 9 detail 9 ,section ,i4 ,euclidean ,geometry & ! articles ,,euclid2 ,,conic ,,section2 ,,appolonius4 ,betwe5 ,euclid & ,apollonius !re fl\ri%$ ! illu/ri\s ,ar*im$es1 ^: geometrical discov]ies are ma9ly conc]n$ ) ! m5suration ( ! circle & conic sections1 & ( ! sph]e1 cone & cyl9d]1 & ^: greate/ contribution to geometrical me?od is ! elevation ( ! me?od ( exhau/ion to ! dignity ( an in/rum5t ( rese>*4 ,apollonius 0 foll[$ by ,nicom$es1 ! inv5tor ( ! con*oid2 ,diocles1 ! inv5tor ( ! cissoid2 ,z5odorus1 ! f\nd] ( ! /udy ( isop]imetrical figures2 ,hipp>*us1 ! f\nd] ( trigonometry2 & ,h]on ! eld]1 :o wrote aft] ! mann] ( ! ,egyptians1 & prim>ily direct$ att5tion to problems ( practical survey+4 ,( ! _m isolat$ discov]ies made by ! lat] ,alex&rian ma!maticians1 ^? ( ,m5elaus are ( importance4 ,he %[$ h[ to treat sph]ical triangles1 e/abli%+ _! prop]ties & det]m9+ _! congru5ce2 8 !orem on ! products ( ! segm5ts 9 :i* ! sides ( a triangle are cut by a l9e 0 ! f\ndation on :i* ,c>not erect$ 8 !ory ( transv]sals4 ,^! propositions1 & also ^? ( ,hipp>*us1 7 utiliz$ & develop$ by ,ptolemy 7q4v471 ! expositor ( trigonometry & discov]] ( _m isolat$ propositions4 ,m5tion may be made ( ! comm5tator ,pappus1 ^: ,ma!matical ,collections is valuable = its weal? ( hi/orical matt]2 ( ,!on1 an editor ( ,euclid's ,elem5ts & comm5tator ( ,ptolemy's ,almage/2 ( ,proclus1 a comm5tator ( ,euclid2 & ( ,eutocius1 a comm5tator ( ,apollonius & ,ar*im$es4 ,! ,romans1 ess5tially practical & hav+ no incl9ation to /udy sci5ce qua sci5ce1 only _h a geometry :i* suffic$ = survey+2 & ev5 "h !re 7 abundant inaccuracies1 ! empirical rules employ$ be+ ak9 to ^? ( ! ,egyptians & ,h]on4 ,! ,h9dus1 likewise1 gave m att5tion to computation1 & _! geometry 0 ei!r ( ,greek orig9 or 9 ! =m pres5t$ 9 trigonometry1 m p>ticul>ly connect$ ) ari?metic4 ,x _h no logical f\ndations2 ea* proposition /ood alone2 & ! results 7 empirical4 ,! ,arabs m closely foll[$ ! ,greeks1 a plan adopt$ z a sequel to ! translation ( ! works ( ,euclid1 ,apollonius1 ,ar*im$es & _m o!rs into ,arabic4 ,_! *ief contribution to geometry is exhibit$ 9 _! solution ( algebraic equations by int]sect+ conics1 a /ep already tak5 by ! ,greeks 9 isolat$ cases1 b only elevat$ into a me?od by ,om> al ,hayyami1 :o fl\ri%$ 9 ! #aa? c5tury4 ,dur+ ! middle ages little 0 add$ to ,greek & ,arabic geometry4 ,leon>do ( ,pisa wrote a ,practica geometriae 7#abbj71 :]e9 ,euclidean me?ods are employ$2 b x 0 n until ! #ad? c5tury t geometry1 g5]ally ,euclid's ,elem5ts1 became an ess5tial item 9 univ]sity curricula4 ,!re 01 h[ev]1 no sign ( orig9al developm5t1 o!r bran*es ( ma!matics1 ma9ly algebra & trigonometry1 ex]cis+ a great] fasc9ation until ! #af? c5tury1 :5 ! subject aga9 came into fav\r4 ,! extraord9>y ma!matical tal5t :i* came into be+ 9 ! #af? & #ag? c5turies react$ on geometry & gave rise to all ^? *>act]s :i* di/+ui% mod]n f anci5t geometry4 ,! fir/ innovation ( mom5t 0 ! =mulation ( ! pr9ciple ( geometrical cont9uity by ,kepl]4 ,! notion ( inf9ity :i* x involv$ p]mitt$ g5]alizations & sy/ematizations hi!rto un?\ght ( 7see ,,geometrical ,,cont9uity72 & ! me?od ( indef9ite division appli$ to rectification1 & quadrature & cubature problems avoid$ ! cumbr\s me?od ( exhau/ion & provid$ m accurate results4 ,fur!r progress 0 made by ,bonav5tura ,cavali]i1 :o1 9 8 ,geometria indivisibilibus cont9uorum 7#afbj71 devis$ a me?od int]m$iate betwe5 t ( exhau/ion & ! inf9itesimal calculus ( ,leibnitz & ,newton4 ,! logical basis ( 8 sy/em 0 correct$ by ,rob]val & ,pascal2 & _! discov]ies1 tak5 9 conjunction ) ^? ( ,leibnitz1 ,newton1 & _m o!rs 9 ! fluxional calculus1 culm9at$ 9 ! bran* ( our subject kn[n z diff]5tial geometry 7see ,,inf9itesimal ,,calculus2 ,,curve2 ,,surface74 ,a second important advance foll[$ ! recognition t conics c\ld be reg>d$ z projections ( a circle1 a conception :i* l$ at ! h&s ( ,des>gues & ,pascal to mod]n projective geometry & p]spective4 ,a ?ird1 & p]haps ! mo/ important1 advance att5d$ ! application ( algebra to geometry by ,desc>tes1 :o !reby f\nd$ analytical geometry4 ,! new fields ?us op5$ up 7 dilig5tly explor$1 b ! calculus ex]cis$ ! greate/ attraction & relatively little progress 0 made 9 geometry until ! beg9n+ ( ! #ai? c5tury1 :5 a new era op5$4 ,gasp>d ,monge 0 ! fir/ important contributor1 /imulat+ analytical & diff]5tial geometry & f\nd+ descriptive geometry 9 a s]ies ( pap]s & especially 9 8 lectures at ! ,ecole polyte*nique4 ,projective geometry1 f\nd$ by ,des>gues1 ,pascal1 ,monge & ,l4,n4,m4 ,c>not1 0 cry/alliz$ by ,j4,v4 ,poncelet1 ! creator ( ! mod]n me?ods4 ,9 8 ,traite des proprietes des figures 7#ahbb7 ! l9e & circul> po9ts at inf9ity1 imag9>ies1 pol> reciprocation1 homology1 cross-ratio & projection are sy/ematically employ$4 ,9 ,g]many1 ,a4,f4 ,mobius1 ,j4 ,pluck] & ,j4 ,/e9] 7 mak+ f>-rea*+ contributions4 ,mobius1 9 8 ,b>yc5tris*e ,calcul 7#ahbg71 introduc$ homog5e\s co-ord9ates1 & also ! p[]ful notion ( geometrical trans=mation1 includ+ ! special cases ( coll9eation & duality2 ,pluck]1 9 8 ,analytis*-geometris*e ,entwickelung5 7#ahbh-ahca71 & 8 ,sy/em d] analytis*5 ,geometrie 7#ahce71 introduc$ ! abridg$ notation1 l9e & plane co-ord9ates1 & ! conception ( g5]aliz$ space elem5ts2 :ile ,/e9]1 besides enri*+ geometry 9 num]\s directions1 0 ! fir/ to sy/ematically g5]ate figures by projective p5cils4 ,we may also notice ,m4 ,*asles1 ^: ,ap]cu hi/orique 7#ahcg7 is a classic4 ,syn!tic geometry1 *>act]iz$ by its fruitfulness & beauty1 attract$ mo/ att5tion1 & x s happ5$ t its orig9ally weak logical f\ndations became replac$ by a m sub/antial set ( axioms4 ,^! 7 f\nd 9 ! anh>monic ratio1 a device lead+ to ! lib]ation ( syn!tic geometry f metrical relations1 & 9 involution1 :i* yield$ rigor\s def9itions ( imag9>ies4 ,^! innovations 7 made by ,k4,j4,c4 von ,/audt4 ,analytical geometry 0 /imulat$ by ! algebra ( inv>iants1 a subject mu* develop$ by ,a4 ,cayley1 ,g4 ,salmon1 ,s4,h4 ,aronhold1 ,l4,o4 ,hesse1 & m p>ticul>ly by ,r4,f4,a4 ,clebs*4 ,! introduction ( ! l9e z a space elem5t1 initiat$ by ,h4 ,grassmann 7#ahdd7 & ,cayley 7#ahei71 yield$ at ! h&s ( ,pluck] a new geometry1 t]m$ l9e geometry1 a subject develop$ m notably by ,f4 ,kle91 ,clebs*1 ,c4,t4 ,reye & ,f4,o4,r4 ,/urm 7see ,section ,v41 ,l9e ,geometry74 ,non-euclidean geometries1 hav+ prim>ily _! orig9 9 ! discussion ( ,euclidean p>allels1 & treat$ by ,wallis1 ,sac*]i & ,lamb]t1 h be5 especially develop$ dur+ ! #ai? c5tury4 ,f\r l9es ( inve/igation may be di/+ui%$3--! naive-syn!tic1 associat$ ) ,lobats*ewski1 ,bolyai1 ,gauss2 ! metric diff]5tial1 /udi$ by ,riemann1 ,helmholtz1 ,beltrami2 ! projective1 develop$ by ,cayley1 ,kle91 ,clif=d2 & ! critical-syn!tic1 promot$ *iefly by ! ,italian ma!maticians ,peano1 ,v]onese1 ,burali-,=te1 ,levi ,civitta1 & ! ,g]mans ,pas* & ,hilb]t4 7,c4 ,e4997 ,i4 ,,euclidean ,,geometry ,?is bran* ( ! sci5ce ( geometry is s nam$ s9ce its me?ods & arrangem5t are ^? laid d[n 9 ,euclid's ,elem5ts4 ,s #a4 ,axioms4--,! object ( geometry is to inve/igate ! prop]ties ( space4 ,! fir/ /ep mu/ consi/ 9 e/abli%+ ^? fundam5tal prop]ties f :i* all o!rs foll[ by processes ( d$uctive reason+4 ,!y are laid d[n 9 ! ,axioms1 & ^! ought to =m su* a sy/em t no?+ ne$ be add$ to !m 9 ord] fully to *>act]ize space1 & t no?+ may be omitt$ )\t mak+ ! sy/em incomplete4 ,!y mu/1 9 fact1 completely def9e space4 ,s #b4 ,def9itions4--,! axioms ( ,euclidean ,geometry are obta9$ f inspection ( exi/5t space & ( solids 9 exi/5t space1--h5ce f exp]i5ce4 ,! same s\rce gives u ! notions ( ! geometrical entities to :i* ! axioms relate1 viz4 solids1 surfaces1 l9es or curves1 & po9ts4 ,a solid is directly giv5 by exp]i5ce2 we h only to ab/ract all mat]ial f x 9 ord] to ga9 ! notion ( a geometrical solid4 ,?is has %ape1 size1 position1 & may be mov$4 ,its b\nd>y or b\nd>ies are call$ surfaces4 ,!y sep>ate "o "p ( space f ano!r1 & are said to h no ?ickness4 ,_! b\nd>ies are curves or l9es1 & ^! h l5g? only4 ,_! b\nd>ies1 aga91 are po9ts1 :i* h no magnitude b only position4 ,we ?us come 9 ?ree /eps f solids to po9ts :i* h no magnitude2 9 ea* /ep we lose "o ext5sion4 ,h5ce we say a solid has ?ree dim5sions1 a surface two1 a l9e "o1 & a po9t none4 ,space itself1 ( :i* a solid =ms only a "p1 is also said to be ( ?ree dim5sions4 ,! same ?+ is int5d$ to be express$ by say+ t a solid has l5g?1 bread? & ?ickness1 a surface l5g? & bread?1 a l9e l5g? only1 & a po9t no ext5sion :atsoev]4 ,euclid gives ! ess5ce ( ^! /atem5ts z def9itions3-- ,def4 #a1 ,i4 ,a po9t is t :i* has no p>ts1 or :i* has no magnitude4 ,def4 #b1 ,i4 ,a l9e is l5g? )\t bread?4 ,def4 #e1 ,i4 ,a sup]ficies is t :i* has only l5g? & bread?4 ,def4 #a1 ,,xi4 ,a solid is t :i* has l5g?1 bread? & ?ickness4 ,x is to be not$ t ! syn!tic me?od is adopt$ by ,euclid2 ! analytical d]ivation ( ! successive ideas ( surface1 l9e1 & po9t f ! exp]im5tal realization ( a solid does n f9d a place 9 8 sy/em1 al?\gh possess+ m advantages4 ,if we all[ motion 9 geometry1 we may g5]ate ^! entities by mov+ a po9t1 a l9e1 or a surface1 ?us3-- ,! pa? ( a mov+ po9t is a l9e4 ,! pa? ( a mov+ l9e is1 9 g5]al1 a surface4 ,! pa? ( a mov+ surface is1 9 g5]al1 a solid4 ,& we may !n assume t ! l9es1 surfaces & solids1 z def9$ be=e1 c all be g5]at$ 9 ?is mann]4 ,f ?is g5]ation ( ! entities x foll[s aga9 t ! b\nd>ies--! fir/ & la/ position ( ! mov+ elem5t--( a l9e are po9ts1 & s on2 & ?us we come back to ! consid]ations ) :i* we />t$4 ,euclid po9ts ?is out 9 8 def9itions1--,def4 #c1 ,i41 ,def4 #f1 ,i41 & ,def4 #b1 ,,xi4 ,he does n1 h[ev]1 %[ ! connexion :i* ^! def9itions h ) ^? m5tion$ be=e4 ,:5 po9ts & l9es h be5 def9$1 a /atem5t l ,def4 #c1 ,i41 ,! extremities ( a l9e are po9ts1 is a proposition :i* ei!r has to be prov$1 & !n x is a !orem1 or :i* has to be tak5 = grant$1 9 :i* case x is an axiom4 ,& s ) ,def4 #f1 ,i41 & ,def4 #b1 ,,xi4 ,s #c4 ,euclid's def9itions m5tion$ above are attempts to describe1 9 a few words1 notions :i* we h obta9$ by inspection ( & ab/raction f solids4 ,a few m notions h to be add$ to ^!1 pr9cipally ^? ( ! simple/ l9e--! /raight l9e1 & ( ! simple/ surface--! flat surface or plane4 ,^! notions we possess1 b to def9e !m accurately is difficult4 ,euclid's ,def9ition #d1 ,i41 ,a /raight l9e is t :i* lies ev5ly betwe5 its extreme po9ts1 mu/ be mean+less to any "o :o has n ! notion ( /raightness 9 8 m9d4 ,nei!r does x /ate a prop]ty ( ! /raight l9e :i* c be us$ 9 any fur!r inve/igation4 ,su* a prop]ty is giv5 9 ,axiom #aj1 ,i4 ,x is really ?is axiom1 toge!r ) ,po/ulates #b & #c1 :i* *>act]izes ! /raight l9e4 ,:il/ = ! /raight l9e ! v]bal def9ition & axiom are kept ap>t1 ,euclid mixes !m up 9 ! case ( ! plane4 ,"h ! ,def9ition #g1 ,i41 includes an axiom4 ,x def9es a plane z a surface :i* has ! prop]ty t e /raight l9e :i* jo9s any two po9ts 9 x lies altoge!r 9 ! surface4 ,b if we take a /raight l9e & a po9t 9 su* a surface1 & draw all /raight l9es :i* jo9 ! latt] to all po9ts 9 ! fir/ l9e1 ! surface w be fully det]m9$4 ,?is con/ruction is !re=e suffici5t z a def9ition4 ,t e o!r /raight l9e :i* jo9s any two po9ts 9 ?is surface lies altoge!r 9 x is a fur!r prop]ty1 & to assume x gives ano!r axiom4 ,?us a numb] ( ,euclid's axioms are hidd5 among 8 fir/ def9itions4 ,a /ill great] confusion exi/s 9 ! pres5t editions ( ,euclid betwe5 ! po/ulates & axioms s call$1 b ?is is due to lat] editors & n to ,euclid himself4 ,! latt] _h ! la/ ?ree axioms put toge!r ) ! po/ulates ,greek3 7aitemata71 s t ^! 7 meant to include all assumptions relat+ to space4 ,! rema9+ assumptions1 :i* relate to magnitudes 9 g5]al1 viz4 ! fir/ eight axioms 9 mod]n editions1 7 call$ common notions ,greek3 7koivai ennoiai74 ,( ! latt] a few may be said to be def9itions4 ,?us ! eigh? might be tak5 z a def9ition ( equal1 & ! sev5? ( halves4 ,if we wi% to collect ! axioms us$ 9 ,euclid's ,elem5ts1 we h !re=e to take ! ?ree po/ulates1 ! la/ ?ree axioms z g5]ally giv51 a few axioms hidd5 9 ! def9itions1 & an axiom us$ by ,euclid 9 ! pro( ( ,prop4 #d1 ,i1 & on a few o!r occasions1 viz4 t figures may be mov$ 9 space )\t *ange ( %ape or size4 ,s #d4 ,po/ulates4--,! assumptions actually made by ,euclid may be /at$ z foll[s3-- 7#a7 ,/raight l9es exi/ :i* h ! prop]ty t any "o ( !m may be produc$ bo? ways )\t limit1 t ?r\gh any two po9ts 9 space su* a l9e may be drawn1 & t any two ( !m co9cide ?r\gh\t _! indef9ite ext5sions z soon z two po9ts 9 ! "o co9cide ) two po9ts 9 ! o!r4 7,?is gives ! cont5ts ( ,def4 #d1 "p ( ,def4 #ce1 ! fir/ two ,po/ulates1 & ,axiom #aj47 7#b7 ,plane surfaces or planes exi/ hav+ ! prop]ty laid d[n 9 ,def4 #g1 t e /raight l9e jo9+ any two po9ts 9 su* a surface lies altoge!r 9 x4 7#c7 ,"r angles1 z def9$ 9 ,def4 #aj1 are possible1 & all "r angles are equal2 t is to say1 :]ev] 9 space we take a plane1 & :]ev] 9 t plane we con/ruct a "r angle1 all angles ?us con/ruct$ w be equal1 s t any "o ( !m may be made to co9cide ) any o!r4 7,axiom #aa47 7#d7 ,! #ab? ,axiom ( ,euclid4 ,?is we %all n /ate n[1 b only introduce x :5 we _c proce$ any fur!r )\t x4 7#e7 ,figures maybe freely mov$ 9 space )\t *ange ( %ape or size4 ,?is is assum$ by ,euclid1 b n /at$ z an axiom4 7#f7 ,9 any plane a circle may be describ$1 hav+ any po9t 9 t plane z c5tre1 & its di/ance f any o!r po9t 9 t plane z radius4 7,po/ulate #c47 ,! def9itions :i* h n be5 m5tion$ are all nom9al def9itions1 t is to say1 !y fix a "n = a ?+ describ$4 ,_m ( !m ov]det]m9e a figure4 ,s #e4 ,euclid's ,elem5ts 7see ,,euclid7 are conta9$ 9 ?irte5 books4 ,( ^! ! fir/ f\r & ! six? are devot$ to plane geometry1 z ! inve/igation ( figures 9 a plane is g5]ally call$4 ,! #e? book conta9s ! !ory ( proportion :i* is us$ 9 ,book ,,vi4 ,! #g?1 #h? & #i? books are purely ari?metical1 :il/ ! #aj? conta9s a mo/ +5i\s treatm5t ( geometrical irrational quantities4 ,^! f\r books w be exclud$ f our survey4 ,! rema9+ ?ree books relate to figures 9 space1 or1 z x is g5]ally call$1 to solid geometry4 ,! #g?1 #h?1 #i?1 #aj?1 #ac? & "p ( ! #aa? & #ab? books are n[ g5]ally omitt$ f ! s*ool editions ( ! ,elem5ts4 ,9 ! fir/ f\r & 9 ! #f? book x is to be und]/ood t all figures are drawn 9 a plane4 ,,book ,i4 ,,( ,,euclid',s ,,elem5ts4 ,s #f4 ,accord+ to ! ?ird po/ulate x is possible to draw 9 any plane a circle :i* has its c5tre at any giv5 po9t1 & its radius equal to ! di/ance ( ?is po9t f any o!r po9t giv5 9 ! plane4 ,?is makes x possible 7,prop4 #a7 to con/ruct on a giv5 l9e ,,ab an equilat]al triangle1 by draw+ fir/ a circle ) ,a z c5tre & ,,ab z radius1 & !n a circle ) ,b z c5tre & ,,ba z radius4 ,! po9t :]e ^! circles int]sect--t !y int]sect ,euclid quietly assumes--is ! v]tex ( ! requir$ triangle4 ,euclid does n suppose1 h[ev]1 t a circle may be drawn :i* has its radius equal to ! di/ance betwe5 any two po9ts unless "o ( ! po9ts be ! c5tre4 ,?is implies also t we are n suppos$ to be able to make any /raight l9e equal to any o!r /raight l9e1 or to c>ry a di/ance ab\t 9 space4 ,euclid !re=e next solves ! problem3 ,x is requir$ along a giv5 /raight l9e f a po9t 9 x to set (f a di/ance equal to ! l5g? ( ano!r /raight l9e giv5 any:]e 9 ! plane4 ,?is is done 9 two /eps4 ,x is %[n 9 ,prop4 #b h[ a /raight l9e may be drawn f a giv5 po9t equal 9 l5g? to ano!r giv5 /raight l9e n drawn f t po9t4 ,& !n ! problem itself is solv$ 9 ,prop4 #c1 by draw+ fir/ ?r\gh ! giv5 po9t "s /raight l9e ( ! requir$ l5g?1 & !n ab\t ! same po9t z c5tre a circle hav+ ?is l5g? z radius4 ,?is circle w cut (f f ! giv5 /raight l9e a l5g? equal to ! requir$ "o4 ,n[adays1 in/ead ( go+ ?r\gh ?is long process1 we take a pair ( compasses & set (f ! giv5 l5g? by its aid4 ,?is assumes t we may move a l5g? ab\t )\t *ang+ x4 ,b ,euclid has n assum$ x1 & ?is proce$+ w\ld be fully ju/ifi$ by 8 desire n to take = grant$ m ?an 0 necess>y1 if he 7 n oblig$ at 8 v next /ep actually to make ?is assumption1 ?\gh )\t /at+ x4 ,s #g4 ,we n[ come 79 ,prop4 #d7 to ! fir/ !orem4 ,x is ! fundam5tal !orem ( ,euclid's :ole sy/em1 !re be+ only a v few propositions 7l ,props4 #ac1 #ad1 #ae1 ,i471 except ^? 9 ! #e? book & ! fir/ half ( ! #aa?1 :i* d n dep5d ^u x4 ,x is /at$ v accurately1 ?\gh some:at clumsily1 z foll[s3-- ,if two triangles h two sides ( ! "o equal to two sides ( ! o!r1 ea* to ea*1 & h also ! angles conta9$ by ^? sides equal to "o ano!r1 !y %all also h _! bases or ?ird sides equal2 & ! two triangles %all be equal2 & _! o!r angles %all be equal1 ea* to ea*1 namely1 ^? to :i* ! equal sides are opposite4 ,t is to say1 ! triangles are id5tically equal1 & "o may be consid]$ z a copy ( ! o!r4 ,! pro( is v simple4 ,! fir/ triangle is tak5 up & plac$ on ! second1 s t ! p>ts ( ! triangles :i* are kn[n to be equal fall ^u ea* o!r4 ,x is !n easily se5 t also ! rema9+ p>ts ( "o co9cide ) ^? ( ! o!r1 & t !y are !re=e equal4 ,?is process ( apply+ "o figure to ano!r ,euclid sc>cely uses aga91 ?\gh _m pro(s w\ld be simplifi$ by do+ s4 ,! process introduces motion into geometry1 & includes1 z already /at$1 ! axiom t figures may be mov$ )\t *ange ( %ape or size4 ,if ! la/ proposition be appli$ to an isosceles triangle1 :i* has two sides equal1 we obta9 ! !orem 7,prop4 #e71 if two sides ( a triangle are equal1 !n ! angles opposite ^! sides are equal4 ,euclid's pro( is some:at complicat$1 & a /umbl+-block to _m s*oolboys4 ,! pro( becomes mu* simpl] if we consid] ! isosceles triangle ,,abc 7,,ab ,,ac7 twice ov]1 once z a triangle ,,bac1 & once z a triangle ,,cab2 & n[ rememb] t ,,ab1 ,,ac 9 ! fir/ are equal respectively to ,,ac1 ,,ab 9 ! second1 & ! angles includ$ by ^! sides are equal4 ,h5ce ! triangles are equal1 & ! angles 9 ! "o are equal to ^? 9 ! o!r1 viz4 ^? :i* are opposite equal sides1 i4e4 angle ,,abc 9 ! fir/ equals angle ,,acb 9 ! second1 z !y are opposite ! equal sides ,,ac & ,,ab 9 ! two triangles4 ,!re foll[s ! conv]se !orem 7,prop4 #f74 ,if two angles 9 a triangle are equal1 !n ! sides opposite !m are equal1--i4e4 ! triangle is isosceles4 ,! pro( giv5 consi/s 9 :at is call$ a r$uctio ad absurdum1 a k9d ( pro( (t5 us$ by ,euclid1 & pr9cipally 9 prov+ ! conv]se ( a previ\s !orem4 ,x assumes t ! !orem to be prov$ is wrong1 & !n %[s t ?is assumption leads to an absurdity1 i4e4 to a conclusion :i* is 9 contradiction to a proposition prov$ be=e--t !re=e ! assumption made _c be true1 & h5ce t ! !orem is true4 ,x is (t5 /at$ t ,euclid inv5t$ ?is k9d ( pro(1 b ! me?od is mo/ likely mu* old]4 ,s #h4 ,x is next prov$ t two triangles :i* h ! ?ree sides ( ! "o equal respectively to ^? ( ! o!r are id5tically equal1 h5ce t ! angles ( ! "o are equal respectively to ^? ( ! o!r1 ^? be+ equal :i* are opposite equal sides4 ,?is is ,prop4 #h1 ,prop4 #g conta9+ only a fir/ /ep t[>ds its pro(4 ,^! !orems all[ n[ ( ! solution ( a numb] ( problems1 viz43-- ,to bisect a giv5 angle 7,prop4 #i74 ,to bisect a giv5 f9ite /raight l9e 7,prop4 #aj74 ,to draw a /raight l9e p]p5dicul>ly to a giv5 /raight l9e ?r\gh a giv5 po9t 9 x 7,prop4 #aa71 & also ?r\gh a giv5 po9t n 9 x 7,prop4 #ab74 ,! solutions all dep5d ^u prop]ties ( isosceles triangles4 ,s #i4 ,! next ?ree !orems relate to angles only1 & might h be5 prov$ be=e ,prop4 #d1 or ev5 at ! v beg9n+4 ,! fir/ 7,prop4 #ac7 says1 ,! angles :i* "o /raight l9e makes ) ano!r /raight l9e on "o side ( x ei!r are two "r angles or are toge!r equal to two "r angles4 ,?is !orem w\ld h be5 unnecess>y if ,euclid _h admitt$ ! notion ( an angle su* t its two limits are 9 ! same /raight l9e1 & _h besides def9$ ! sum ( two angles4 ,its conv]se 7,prop4 #ad7 is ( great use1 inasmu* z x enables u 9 _m cases to prove t two /raight l9es drawn f ! same po9t are "o ! cont9uation ( ! o!r4 ,s also is ,prop4 #ae4 ,if two /raight l9es cut "o ano!r1 ! v]tical or opposite angles %all be equal4 ,s #aj4 ,euclid returns n[ to prop]ties ( triangles4 ,( great importance = ! next /eps 7?\gh aft]w>ds sup]s$$ by a m complete !orem7 is ,prop4 #af4 ,if "o side ( a triangle be produc$1 ! ext]ior angle %all be great] ?an ei!r ( ! int]ior opposite angles4 ,prop4 #ag4 ,any two angles ( a triangle are toge!r less ?an two "r angles1 is an imm$iate consequ5ce ( x4 ,by ! aid ( ^! two1 ! foll[+ fundam5tal prop]ties ( triangles are easily prov$3-- ,prop4 #ah4 ,! great] side ( e triangle has ! great] angle opposite to x2 ,its conv]se1 ,prop4 #ai4 ,! great] angle ( e triangle is subt5d$ by ! great] side1 or has ! great] side opposite to x2 ,prop4 #bj4 ,any two sides ( a triangle are toge!r great] ?an ! ?ird side2 ,& also ,prop4 #ba4 ,if f ! ends ( ! side ( a triangle !re be drawn two /raight l9es to a po9t )9 ! triangle1 ^! %all be less ?an ! o!r two sides ( ! triangle1 b %all conta9 a great] angle4 ,s #aa4 ,hav+ solv$ two problems 7,props4 #bb1 #bc71 he returns to two triangles :i* h two sides ( ! "o equal respectively to two sides ( ! o!r4 ,x is kn[n 7,prop4 #d7 t if ! includ$ angles are equal !n ! ?ird sides are equal2 & conv]sely 7,prop4 #h71 if ! ?ird sides are equal1 !n ! angles includ$ by ! fir/ sides are equal4 ,f ?is x foll[s t if ! includ$ angles are n equal1 ! ?ird sides are n equal2 & conv]sely1 t if ! ?ird sides are n equal1 ! includ$ angles are n equal4 ,euclid n[ completes ?is k by prov+1 t if ! includ$ angles are n equal1 !n ! ?ird side 9 t triangle is ! great] :i* conta9s ! great] angle2 & conv]sely1 t if ! ?ird sides are unequal1 t triangle conta9s ! great] angle :i* conta9s ! great] side4 ,^! are ,prop4 #bd & ,prop4 #be4 ,s #ab4 ,! next !orem 7,prop4 #bf7 says t if two triangles h "o side & two angles ( ! "o equal respectively to "o side & two angles ( ! o!r1 viz4 9 bo? triangles ei!r ! angles adjac5t to ! equal side1 or "o angle adjac5t & "o angle opposite x1 !n ! two triangles are id5tically equal4 ,?is !orem belongs to a gr\p ) ,prop4 #d & ,prop4 #h4 ,its fir/ case might h be5 giv5 imm$iately aft] ,prop4 #d1 b ! second case requires ,prop4 #af = its pro(4 ,s #ac4 ,we come n[ to ! inve/igation ( p>allel /raight l9es1 i4e4 ( /raight l9es :i* lie 9 ! same plane1 & _c be made to meet h[ev] f> !y be produc$ ei!r way4 ,! inve/igation :i* />ts f ,prop4 #af1 w become cle>] if a few names be expla9$ :i* are n all us$ by ,euclid4 ,if two /raight l9es be cut by a ?ird1 ! latt] is n[ g5]ally call$ a transv]sal ( ! figure4 ,x =ms at ! two po9ts :]e x cuts ! giv5 l9es f\r angles ) ea*4 ,^? ( ! angles :i* lie betwe5 ! giv5 l9es are call$ int]ior angles1 & ( ^!1 aga91 any two :i* lie on opposite sides ( ! transv]sal b "o at ea* ( ! two po9ts are call$ alt]nate angles4 ,we may n[ /ate ,prop4 #af ?us3--,if two /raight l9es :i* meet are cut by a transv]sal1 _! alt]nate angles are unequal4 ,= ! l9es w =m a triangle1 & "o ( ! alt]nate angles w be an ext]ior angle to ! triangle1 ! o!r int]ior & opposite to x4 ,f ?is foll[s at once ! !orem conta9$ 9 ,prop4 #bg4 ,if two /raight l9es :i* are cut by a transv]sal make alt]nate angles equal1 ! l9es _c meet1 h[ev] f> !y be produc$1 h5ce !y are p>allel4 ,?is proves ! exi/5ce ( p>allel l9es4 ,prop4 #bh /ates ! same fact 9 diff]5t =ms4 ,if a /raight l9e1 fall+ on two o!r /raight l9es1 make ! ext]ior angle equal to ! int]ior & opposite angle on ! same side ( ! l9e1 or make ! int]ior angles on ! same side toge!r equal to two "r angles1 ! two /raight l9es %all be p>allel to "o ano!r4 ,h5ce we "k t1 if two /raight l9es :i* are cut by a transv]sal meet1 _! alt]nate angles are n equal2 & h5ce t1 if alt]nate angles are equal1 !n ! l9es are p>allel4 ,! "q n[ arises1 ,are ! propositions conv]se to ^! true or n8 ,t is to say1 ,if alt]nate angles are unequal1 d ! l9es meet8 ,& if ! l9es are p>allel1 are alt]nate angles necess>ily equal8 ,! answ] to ei!r ( ^! two que/ions implies ! answ] to ! o!r4 ,b x has be5 f\nd impossible to prove t ! negation or ! affirmation ( ei!r is true4 ,! difficulty :i* ?us arises is ov]come by ,euclid assum+ t ! fir/ "q has to be answ]$ 9 ! affirmative4 ,?is gives 8 la/ axiom 7#ab71 :i* we quote 9 8 own words4 ,axiom #ab4--;,if a /raight l9e meet two /raight l9es1 s z to make ! two int]ior angles on ! same side ( x tak5 toge!r less ?an two "r angles1 ^! /raight l9es1 be+ cont9ually produc$1 %all at l5g? meet on t side on :i* are ! angles :i* are less ?an two "r angles4 ,! answ] to ! second ( ! above que/ions foll[s f ?is1 & gives ! !orem ,prop4 #bi3--;,if a /raight l9e fall on two p>allel /raight l9es1 x makes ! alt]nate angles equal to "o ano!r1 & ! ext]ior angle equal to ! int]ior & opposite angle on ! same side1 & also ! two int]ior angles on ! same side toge!r equal to two "r angles4 ,s #ad4 ,) ?is a new "p ( elem5t>y geometry beg9s4 ,! e>li] propositions are indep5d5t ( ?is axiom1 & w\ld be true ev5 if a wrong assumption _h be5 made 9 x4 ,!y all relate to figures 9 a plane4 ,b a plane is only "o among an inf9ite numb] ( conceivable surfaces4 ,we may draw figures on any "o ( !m & /udy _! prop]ties4 ,we may1 = in/ance1 take a sph]e in/ead ( ! plane1 & obta9 sph]ical 9 ! place ( plane geometry4 ,if on "o ( ^! surfaces l9es & figures c\ld be drawn1 answ]+ to all ! def9itions ( our plane figures1 & if ! axioms ) ! exception ( ! la/ all hold1 !n all propositions up to ! #bh? w be true = ^! figures4 ,?is is ! case 9 sph]ical geometry if we sub/itute %orte/ l9e or great circle = /raight l9e1 small circle = circle1 & if1 besides1 we limit all figures to a "p ( ! sph]e :i* is less ?an a hemisph]e1 s t two po9ts on x _c be opposite ends ( a diamet]1 & !re=e det]m9e always "o & only "o great circle4 ,= sph]ical triangles1 !re=e1 all ! important propositions #d1 #h1 #bf2 #e & #f2 & #ah1 #ai & #bj w hold good4 ,?is rem>k w be suffici5t to %[ ! impossibility ( prov+ ,euclid's la/ axiom1 :i* w\ld mean prov+ t ?is axiom is a consequ5ce ( ! o!rs1 & h5ce t ! !ory ( p>allels w\ld hold on a sph]ical surface1 :]e ! o!r axioms d hold1 :il/ p>allels d n ev5 exi/4 ,x foll[s t ! axiom 9 "q /ates an inh]5t diff]5ce betwe5 ! plane & o!r surfaces1 & t ! plane is only fully *>act]iz$ :5 ?is axiom is add$ to ! o!r assumptions4 ,s #ae4 ,! introduction ( ! new axiom & ( p>allel l9es leads to a new class ( propositions4 ,aft] prov+ 7,prop4 #cj7 t two l9es :i* are ea* p>allel to a ?ird are p>allel to ea* o!r1 we obta9 ! new prop]ties ( triangles conta9$ 9 ,prop4 #cb4 ,( ^! ! second "p is ! mo/ important1 viz4 ! !orem1 ,! ?ree int]ior angles ( e triangle are toge!r equal to two "r angles4 ,z easy d$uctions n giv5 by ,euclid b add$ by ,simson foll[ ! propositions ab\t ! angles 9 polygons1 !y are giv5 9 ,engli% editions z coroll>ies to ,prop4 #cb4 ,^! !orems d n hold = sph]ical figures4 ,! sum ( ! int]ior angles ( a sph]ical triangle is always great] ?an two "r angles1 & increases ) ! area4 ,s #af4 ,! !ory ( p>allels z su* may be said to be f9i%$ ) ,props4 #cc & #cd1 :i* /ate prop]ties ( ! p>allelogram1 i4e4 ( a quadrilat]al =m$ by two pairs ( p>allels4 ,!y are-- ,prop4 #cc4 ,! /raight l9es :i* jo9 ! extremities ( two equal & p>allel /raight l9es t[>ds ! same p>ts are !mselves equal & p>allel2 & ,prop4 #cd4 ,! opposite sides & angles ( a p>allelogram are equal to "o ano!r1 & ! diamet] 7diagonal7 bisects ! p>allelogram1 t is1 divides x into two equal p>ts4 ,s #ag4 ,! re/ ( ! fir/ book relates to areas ( figures4 ,! !ory is made to dep5d ^u ! !orems-- ,prop4 #ce4 ,p>allelograms on ! same base & betwe5 ! same p>allels are equal to "o ano!r2 & ,prop4 #cf4 ,p>allelograms on equal bases & betwe5 ! same p>allels are equal to "o ano!r4 ,z ea* p>allelogram is bisect$ by a diagonal1 ! la/ !orems hold also if ! ^w p>allelogram be replac$ by triangle1 z is done 9 ,props4 #cg & #ch4 ,x is to be rem>k$ t ,euclid proves ^! propositions only 9 ! case :5 ! p>allelograms or triangles h _! bases 9 ! same /raight l9e4 ,! !orems conv]se to ! la/ =m ! cont5ts ( ! next ?ree propositions1 viz43 ,props1 #dj & #da4--;,equal triangles1 on ! same or on equal bases1 9 ! same /raight l9e1 & on ! same side ( x1 are betwe5 ! same p>allels4 ,t ! two cases "h /at$ are giv5 by ,euclid 9 two sep>ate propositions prov$ sep>ately is *>act]i/ic ( 8 me?od4 ,s #ah4 ,to comp>e areas ( o!r figures1 ,euclid %[s fir/1 9 ,prop4 #db1 h[ to draw a p>allelogram :i* is equal 9 area to a giv5 triangle1 & has "o ( its angles equal to a giv5 angle4 ,if ! giv5 angle is "r1 !n ! problem is solv$ to draw a rectangle equal 9 area to a giv5 triangle4 ,next ?is p>allelogram is trans=m$ into ano!r p>allelogram1 :i* has "o ( its sides equal to a giv5 /raight l9e1 :il/ its angles rema9 unalt]$4 ,?is may be done by aid ( ! !orem 9 ,prop4 #dc4 ,! complem5ts ( ! p>allelograms :i* are ab\t ! diamet] ( any p>allelogram are equal to "o ano!r4 ,?us ! problem 7,prop4 #dd7 is solv$ to con/ruct a p>allelogram on a giv5 l9e1 :i* is equal 9 area to a giv5 triangle1 & :i* has "o angle equal to a giv5 angle 7g5]ally a "r angle74 ,z e polygon c be divid$ into a numb] ( triangles1 we c n[ con/ruct a p>allelogram hav+ a giv5 angle1 say a "r angle1 & be+ equal 9 area to a giv5 polygon4 ,= ea* ( ! triangles into :i* ! polygon has be5 divid$1 a p>allelogram may be con/ruct$1 hav+ "o side equal to a giv5 /raight l9e & "o angle equal to a giv5 angle4 ,if ^! p>allelograms be plac$ side by side1 !y may be add$ toge!r to =m a s+le p>allelogram1 hav+ /ill "o side ( ! giv5 l5g?4 ,?is is done 9 ,prop4 #de4 ,h]e) a means is f\nd to comp>e areas ( diff]5t polygons4 ,we ne$ only con/ruct two rectangles equal 9 area to ! giv5 polygons1 & hav+ ea* "o side ( giv5 l5g?4 ,by comp>+ ! unequal sides we are enabl$ to judge :e!r ! areas are equal1 or :i* is ! great]4 ,euclid does n /ate ?is consequ5ce1 b ! problem is tak5 up aga9 at ! end ( ! second book1 :]e x is %[n h[ to con/ruct a squ>e equal 9 area to a giv5 polygon4 ,prop4 #df is3 ,to describe a squ>e on a giv5 /raight l9e4 ,s #ai4 ,! fir/ book concludes ) "o ( ! mo/ important !orems 9 ! :ole ( geometry1 & "o :i* has be5 celebrat$ s9ce ! e>lie/ times4 ,x is /at$1 b on d\btful au?ority1 t ,py?agoras discov]$ x1 & x has be5 call$ by 8 "n4 ,if we call t side 9 a "r-angl$ triangle :i* is opposite ! "r angle ! hypot5use1 we may /ate x z foll[s3-- ,!orem ( ,py?agoras 7,prop4 #dg74--;,9 e "r-angl$ triangle ! squ>e on ! hypot5use is equal to ! sum ( ! squ>es ( ! o!r sides4 ,& conv]sely-- ,prop4 #dh4 ,if ! squ>e describ$ on "o ( ! sides ( a triangle be equal to ! squ>es describ$ on ! o!r sides1 !n ! angle conta9$ by ^! two sides is a "r angle4 ,on ?is !orem 7,prop4 #dg7 almo/ all geometrical measurem5t dep5ds1 :i* _c be directly obta9$4 ,,book ,,ii4 ,s #bj4 ,! propositions 9 ! second book are v diff]5t 9 "* f ^? 9 ! fir/2 !y all relate to areas ( rectangles & squ>es4 ,_! true significance is be/ se5 by /at+ !m 9 an algebraic =m4 ,?is is (t5 done by express+ ! l5g?s ( l9es by aid ( numb]s1 :i* tell h[ _m times a *os5 unit is conta9$ 9 ! l9es4 ,if !re is a unit to be f\nd :i* is conta9$ an exact numb] ( times 9 ea* side ( a rectangle1 x is easily se51 & g5]ally %[n 9 ! tea*+ ( ari?metic1 t ! rectangle conta9s a numb] ( unit squ>es equal to ! product ( ! numb]s :i* measure ! sides1 a unit squ>e be+ ! squ>e on ! unit l9e4 ,if1 h[ev]1 no su* unit c be f\nd1 ?is process requires t connexion betwe5 l9es & numb]s :i* is only e/abli%$ by aid ( ratios ( l9es1 & :i* is !re=e at ?is /age altoge!r inadmissible4 ,b !re exi/s ano!r way ( connect+ ^! propositions ) algebra1 bas$ on mod]n notions :i* seem de/9$ greatly to *ange & to simplify ma!matics4 ,we %all introduce "h z mu* ( x z is requir$ = our pres5t purpose4 ,at ! beg9n+ ( ! second book we f9d a def9ition accord+ to :i* a rectangle is said to be 'conta9$' by ! two sides :i* conta9 "o ( its "r angles2 9 ! text ?is phraseology is ext5d$ by speak+ ( rectangles conta9$ by any two /raight l9es1 mean+ ! rectangle :i* has two adjac5t sides equal to ! two /raight l9es4 ,we %all d5ote a f9ite /raight l9e by a s+le small lett]1 a1 b1 c1 444 x1 & ! area ( ! rectangle conta9$ by two l9es a & b by ab1 & ?is we %all call ! product ( ! two l9es a & b4 ,x w be und]/ood t ?is def9ition has no?+ to d ) ! def9ition ( a product ( numb]s4 ,we def9e z foll[s3-- ,! sum ( two /raight l9es a & b means a /raight l9e c :i* may be divid$ 9 two p>ts equal respectively to a & b4 ,?is sum is d5ot$ by a b4 ,! diff]5ce ( two l9es a & b 79 symbols1 a-b7 means a l9e c :i* :5 add$ to b gives a2 t is1 a - b c if b c a4 ,! product ( two l9es a & b 79 symbols1 ab7 means ! area ( ! rectangle conta9$ by ! l9es a & b4 ,= aa1 :i* means ! squ>e on ! l9e a1 we write a#b4 ,s #ba4 ,! fir/ t5 ( ! f\rte5 propositions ( ! second book may !n be writt5 9 ! =m ( =mulae z foll[s3-- ,prop4 #a4 a7b c d 444 7 ab ac ad 444 #b4 ab ac a#b if b c a4 #c4 a7a b7 a#b ab4 #d4 7a b7#b a#b #b;ab b#b4 #e4 7a b77a - b7 b#b a#b4 #f4 7a b77a - b7 b#b a#b4 #g4 a#b 7a - b7#b #b;a7a - b7 b#b4 #h4 #d7;a b7a b#b 7#b;a b7#b4 #i4 7a b7#b 7a - b7#b #b;a#b #b;b#b4 #aj4 7a b7#b 7a - b7#b #b;a#b #b;b#b4 ,x w be se5 t #e & #f1 & also #i & #aj1 are id5tical4 ,9 ,euclid's /atem5t !y d n look ! same1 ! figures be+ arrang$ diff]5tly4 ,if ! lett]s a1 b1 c1 444 d5ot$ numb]s1 x foll[s f algebra t ea* ( ^! =mulae is true4 ,b ?is does n prove !m 9 our case1 :]e ! lett]s d5ote l9es1 & _! products areas )\t any ref]5ce to numb]s4 ,to prove !m we h to discov] ! laws :i* rule ! op]ations introduc$1 viz4 addition & multiplication ( segm5ts4 ,?is we %all d n[2 & we %all f9d t ^! laws are ! same ) ^? :i* hold 9 algebraical addition & multiplication4 ,s #bb4 ,9 a sum ( numb]s we may *ange ! ord] 9 :i* ! numb]s are add$1 & we may also add ! numb]s toge!r 9 gr\ps & !n add ^! gr\ps4 ,b ?is also holds = ! sum ( segm5ts & = ! sum ( rectangles1 z a little consid]ation %[s4 ,t ! sum ( rectangles has always a mean+ foll[s f ! ,props4 #dc-de 9 ! fir/ book4 ,^! laws ab\t addition are r$ucible to ! two-- a b b a 7#a71 a 7b c7 a b c 7#b72 or1 :5 express$ = rectangles1 ab ed ed ab 7#c71 ab 7cd ef7 ab cd ef 7#d74 ,! brackets mean t ! t]ms 9 ! bracket h be5 add$ toge!r be=e !y are add$ to ano!r t]m4 ,! m g5]al cases = m t]ms may be d$uc$ f ! above4 ,= ! product ( two numb]s we h ! law t x rema9s unalt]$ if ! factors be int]*ang$4 ,?is also holds = our geometrical product4 ,= if ab d5otes ! area ( ! rectangle :i* has a z base & b z altitude1 !n ba w d5ote ! area ( ! rectangle :i* has b z base & a z altitude4 ,b 9 a rectangle we may take ei!r ( ! two l9es :i* conta9 x z base1 & !n ! o!r w be ! altitude4 ,?is gives ab ba 7#e74 ,9 ord] fur!r to multiply a sum by a numb]1 we h 9 algebra ! rule3--,multiply ea* t]m ( ! sum1 & add ! products ?us obta9$4 ,t ?is holds = our geometrical products is %[n by ,euclid 9 8 fir/ proposition ( ! second book1 :]e he proves t ! area ( a rectangle ^: base is ! sum ( a numb] ( segm5ts is equal to ! sum ( rectangles :i* h ^! segm5ts sep>ately z bases4 ,9 symbols ?is gives1 9 ! simple/ case1 a7b c7 ab ac 7#f74 & 7b c7a ba ca _/ ,to ^! laws1 :i* h be5 inve/igat$ by ,sir ,william ,hamilton & by ,h]mann ,grassmann1 ! =m] has giv5 special names4 ,he calls ! laws express$ 9 7#a7 & 7#c7 ! commutative law = addition2 7#e7 multiplication2 7#b7 & 7#d7 ! associative laws = addition2 7#f7 ! di/ributive law4 ,s #bc4 ,hav+ prov$ t ^! six laws hold1 we c at once prove e "o ( ! above propositions 9 _! algebraical =m4 ,! fir/ is prov$ geometrically1 x be+ "o ( ! fundam5tal laws4 ,! next two propositions are only special cases ( ! fir/4 ,( ! o!rs we %all prove "o1 viz4 ! f\r?3-- 7a b7#b 7a b77a b7 7a b7a 7a b7b by 7#f74 ,b 7a b7a aa ba by 7#f71 aa ab by 7#e72 & 7a b7b ab bb by 7#f74 ,!re=e 7a b7#b aa ab 7ab bb7 aa 7ab ab7 bb by 7#d74 aa #b;ab bb _/ ,?is gives ! !orem 9 "q4 ,9 ! same mann] e "o ( ! fir/ t5 propositions is prov$4 ,x w be se5 t ! op]ations p]=m$ are exactly ! same z if ! lett]s d5ot$ numb]s4 ,props4 #e & #f may also be writt5 ?us-- 7a b77a - b7 a#b - b#b4 ,prop4 #g1 :i* is an easy consequ5ce ( ,prop4 #d1 may be trans=m$4 ,if we d5ote by c ! l9e a b1 s t c a b1 a c - b1 we get c#b 7c - b7#b #b;c7c - b7 b#b #b;c#b - #b;bc b#b4 ,subtract+ c#b f bo? sides1 & writ+ a = c1 we get 7a - b7#b a#b - #b;ab b#b4 ,9 ,euclid's ,elem5ts ?is =m ( ! !orem does n appe>1 all propositions be+ s /at$ t ! notion ( subtraction does n ent] into !m4 ,s #bd4 ,! rema9+ two !orems 7,props4 #ab & #ac7 connect ! squ>e on "o side ( a triangle ) ! sum ( ! squ>es on ! o!r sides1 9 case t ! angle betwe5 ! latt] is acute or obtuse4 ,!y are important !orems 9 trigonometry1 :]e x is possible to include !m 9 a s+le !orem4 ,s #be4 ,!re are 9 ! second book two problems1 ,props4 #aa & #ad4 ,if writt5 9 ! above symbolic language1 ! =m] requires to f9d a l9e x su* t a7a - x7 x#b4 ,prop4 #aa conta9s1 !re=e1 ! solution ( a quadratic equation1 :i* we may write x#b ax a#b4 ,! solution is requir$ lat] on 9 ! con/ruction ( a regul> decagon4 ,m important is ! problem 9 ! la/ proposition 7,prop4 #ad74 ,x requires ! con/ruction ( a squ>e equal 9 area to a giv5 rectangle1 h5ce a solution ( ! equation x#b ab4 ,9 ,book ,i41 #db-de1 x has be5 %[n h[ a rectangle may be con/ruct$ equal 9 area to a giv5 figure b\nd$ by /raight l9es4 ,by aid ( ! new proposition we may !re=e n[ det]m9e a l9e su* t ! squ>e on t l9e is equal 9 area to any giv5 rectil9e> figure1 or we c squ>e any su* figure4 ,z ( two squ>es t is ! great] :i* has ! great] side1 x foll[s t n[ ! comp>ison ( two areas has be5 r$uc$ to ! comp>ison ( two l9es4 ,! problem ( r$uc+ o!r areas to squ>es is frequ5tly met ) among ,greek ma!maticians4 ,we ne$ only m5tion ! problem ( squ>+ ! circle 7see ,,circle74 ,9 ! pres5t day ! comp>ison ( areas is p]=m$ 9 a simpl] way by r$uc+ all areas to rectangles hav+ a common base4 ,_! altitudes give !n a measure ( _! areas4 ,! con/ruction ( a rectangle hav+ ! base u1 & be+ equal 9 area to a giv5 rectangle1 dep5ds ^u ,prop4 #dc1 ,i4 ,?is !re=e gives a solution ( ! equation ab ux1 :]e x d5otes ! unkn[n altitude4 ,,book ,,iii4 ,s #bf4 ,! ?ird book ( ! ,elem5ts relates exclusively to prop]ties ( ! circle4 ,a circle & its circumf]5ce h be5 def9$ 9 ,book ,i41 ,def4 #ae4 ,we re/ate x "h 9 slightly diff]5t words3-- ,def9ition4--,! circumf]5ce ( a circle is a plane curve su* t all po9ts 9 x h ! same di/ance f a fix$ po9t 9 ! plane4 ,?is po9t is call$ ! c5tre ( ! circle4 ,( ! new def9itions1 ( :i* elev5 are giv5 at ! beg9n+ ( ! ?ird book1 a few only require special m5tion4 ,! fir/1 :i* says t circles ) equal radii are equal1 is 9 "p a !orem1 b easily prov$ by apply+ ! "o circle to ! o!r4 ,or x may be consid]$ prov$ by aid ( ,prop4 #bd1 equal circles n be+ us$ till aft] ?is !orem4 ,9 ! second def9ition is expla9$ :at is meant by a l9e :i* t\*es a circle4 ,su* a l9e is n[ g5]ally call$ a tang5t to ! circle4 ,! introduction ( ?is "n all[s u to /ate _m ( ,euclid's propositions 9 a mu* %ort] =m4 ,= ! same reason we %all call a /raight l9e jo9+ two po9ts on ! circumf]5ce ( a circle a *ord4 ,def9itions #d & #e may be replac$ ) a slight g5]alization by ! foll[+3-- ,def9ition4--,by ! di/ance ( a po9t f a l9e is meant ! l5g? ( ! p]p5dicul> drawn f ! po9t to ! l9e4 ,s #bg4 ,f ! def9ition ( a circle x foll[s t e circle has a c5tre4 ,prop4 #a requires to f9d x :5 ! circle is giv51 i4e4 :5 its circumf]5ce is drawn4 ,to solve ?is problem a *ord is drawn 7t is1 any two po9ts 9 ! circumf]5ce are jo9$71 & ?r\gh ! po9t :]e ?is is bisect$ a p]p5dicul> to x is erect$4 ,euclid !n proves1 fir/1 t no po9t (f ?is p]p5dicul> c be ! c5tre1 h5ce t ! c5tre mu/ lie 9 ?is l9e2 &1 secondly1 t ( ! po9ts on ! p]p5dicul> "o only c be ! c5tre1 viz4 ! "o :i* bisects ! p>ts ( ! p]p5dicul> b\nd$ by ! circle4 ,9 ! second "p ,euclid sil5tly assumes t ! p]p5dicul> !re us$ does cut ! circumf]5ce 9 two1 & only 9 two po9ts4 ,! pro( !re=e is incomplete4 ,! pro( ( ! fir/ "p1 h[ev]1 is exact4 ,by draw+ two non-p>allel *ords1 & ! p]p5dicul>s :i* bisect !m1 ! c5tre w be f\nd z ! po9t :]e ^! p]p5dicul>s int]sect4 ,s #bh4 ,9 ,prop4 #b x is prov$ t a *ord ( a circle lies altoge!r )9 ! circle4 ,:at we h call$ ! fir/ "p ( ,euclid's solution ( ,prop4 #a may be /at$ z a !orem3-- ,e /raight l9e :i* bisects a *ord1 & is at "r angles to x1 passes ?r\gh ! c5tre ( ! circle4 ,! conv]se to ?is gives ,prop4 #c1 :i* may be /at$ ?us3-- ,if a /raight l9e ?r\gh ! c5tre ( a circle bisect a *ord1 !n x is p]p5dicul> to ! *ord1 & if x be p]p5dicul> to ! *ord x bisects x4 ,an easy consequ5ce ( ?is is ! foll[+ !orem1 :i* is ess5tially ! same z ,prop4 #d3-- ,two *ords ( a circle1 ( :i* nei!r passes ?r\gh ! c5tre1 _c bisect ea* o!r4 ,^! la/ ?ree !orems are fundam5tal = ! !ory ( ! circle4 ,x is to be rem>k$ t ,euclid nev] proves t a /raight l9e _c h m ?an two po9ts 9 common ) a circumf]5ce4 ,s #bi4 ,! next two propositions 7#e & #f7 might be replac$ by a s+le & a simpl] !orem1 viz3-- ,two circles :i* h a common c5tre1 & ^: circumf]5ces h "o po9t 9 common1 co9cide4 ,or1 m 9 agreem5t ) ,euclid's =m3-- ,two diff]5t circles1 ^: circumf]5ces h a po9t 9 common1 _c h ! same c5tre4 ,t ,euclid treats ( two cases is *>act]i/ic ( ,greek ma!matics4 ,! next two propositions 7#g & #h7 aga9 belong toge!r4 ,!y may be comb9$ ?us3-- ,if f a po9t 9 a plane ( a circle1 :i* is n ! c5tre1 /raight l9es be drawn to ! diff]5t po9ts ( ! circumf]5ce1 !n ( all ^! l9es "o is ! %orte/1 & "o ! longe/1 & ^! lie bo? 9 t /raight l9e :i* jo9s ! giv5 po9t to ! c5tre4 ,( all ! rema9+ l9es ea* is equal to "o & only "o o!r1 & ^! equal l9es lie on opposite sides ( ! %orte/ or longe/1 & make equal angles ) !m4 ,euclid di/+ui%es ! two cases :]e ! giv5 po9t lies )9 or )\t ! circle1 omitt+ ! case :]e x lies 9 ! circumf]5ce4 ,f ! la/ proposition x foll[s t if f a po9t m ?an two equal /raight l9es c be drawn to ! circumf]5ce1 ?is po9t mu/ be ! c5tre4 ,?is is ,prop4 #i4 ,z a consequ5ce ( ?is we get ,if ! circumf]5ces ( ! two circles h ?ree po9ts 9 common !y co9cide4 ,= 9 ?is case ! two circles h a common c5tre1 because f ! c5tre ( ! "o ?ree equal l9es c be drawn to po9ts on ! circumf]5ce ( ! o!r4 ,b two circles :i* h a common c5tre1 & ^: circumf]5ces h a po9t 9 common1 co9cide4 7,comp>e above /atem5t ( ,props4 #e & #f47 ,?is !orem may also be /at$ ?us3-- ,?r\gh ?ree po9ts only "o circumf]5ce may be drawn2 or1 ,?ree po9ts det]m9e a circle4 ,euclid does n give ! !orem 9 ?is =m4 ,he proves1 h[ev]1 t ! two circles _c cut ano!r 9 m ?an two po9ts 7,prop4 #aj71 & t two circles _c t\* "o ano!r 9 m po9ts ?an "o 7,prop4 #ac74 ,s #cj4 ,propositions #aa & #ab ass]t t if two circles t\*1 !n ! po9t ( contact lies on ! l9e jo9+ _! c5tres4 ,?is gives two propositions1 because ! circles may t\* ei!r int]nally or ext]nally4 ,s #ca4 ,propositions #ad & #ae relate to ! l5g? ( *ords4 ,! fir/ says t equal *ords are equidi/ant f ! c5tre1 & t *ords :i* are equidi/ant f ! c5tre are equal2 ,:il/ ,prop4 #ae comp>es unequal *ords1 viz4 ,( all *ords ! diamet] is ! greate/1 & ( o!r *ords t is ! great] :i* is ne>] to ! c5tre2 & conv]sely1 ! great] *ord is ne>] to ! c5tre4 ,s #cb4 ,9 ,prop4 #af ! tang5t to a circle is = ! fir/ "t introduc$4 ,! proposition is meant to %[ t ! /raight l9e at ! end po9t ( ! diamet] & at "r angles to x is a tang5t4 ,! proposition itself does n /ate ?is4 ,x runs ?us3-- ,prop4 #af4 ,! /raight l9e drawn at "r angles to ! diamet] ( a circle1 f ! extremity ( x1 falls )\t ! circle2 & no /raight l9e c be drawn f ! extremity1 betwe5 t /raight l9e & ! circumf]5ce1 s z n to cut ! circle4 ,coroll>y4--,! /raight l9e at "r angles to a diamet] drawn ?r\gh ! end po9t ( x t\*es ! circle4 ,! /atem5t ( ! proposition & its :ole treatm5t %[ ! difficulties :i* ! tang5ts pres5t$ to ,euclid4 ,prop4 #ag solves ! problem ?r\gh a giv5 po9t1 ei!r 9 ! circumf]5ce or )\t x1 to draw a tang5t to a giv5 circle4 ,closely connect$ ) ,prop4 #af are ,props4 #ah & #ai1 :i* /ate 7,prop4 #ah71 t ! l9e jo9+ ! c5tre ( a circle to ! po9t ( contact ( a tang5t is p]p5dicul> to ! tang5t2 & conv]sely 7,prop4 #ai71 t ! /raight l9e ?r\gh ! po9t ( contact (1 & p]p5dicul> to1 a tang5t to a circle passes ?r\gh ! c5tre ( ! circle4 ,s #cc4 ,! re/ ( ! book relates to angles connect$ ) a circle1 viz4 angles :i* h ! v]tex ei!r at ! c5tre or on ! circumf]5ce1 & :i* are call$ respectively angles at ! c5tre & angles at ! circumf]5ce4 ,betwe5 ^! two k9ds ( angles exi/s ! important relation express$ z foll[s3-- ,prop4 #bj4 ,! angle at ! c5tre ( a circle is d\ble ( ! angle at ! circumf]5ce on ! same base1 t is1 on ! same arc4 ,?is is ( great importance = its consequ5ces1 ( :i* ! two foll[+ are ! pr9cipal3-- ,prop4 #ba4 ,! angles 9 ! same segm5t ( a circle are equal to "o ano!r2 ,prop4 #bb4 ,! opposite angles ( any quadrilat]al figure inscrib$ 9 a circle are toge!r equal to two "r angles4 ,fur!r consequ5ces are3-- ,prop4 #bc4 ,on ! same /raight l9e1 & on ! same side ( x1 !re _c be two simil> segm5ts ( circles1 n co9cid+ ) "o ano!r2 ,prop4 #bd4 ,simil> segm5ts ( circles on equal /raight l9es are equal to "o ano!r4 ,! problem ,prop4 #be4 ,a segm5t ( a circle be+ giv5 to describe ! circle ( :i* x is a segm5t1 may be solv$ mu* m easily by aid ( ! con/ruction describ$ 9 relation to ,prop4 #a1 ,,iii41 9 ,s #bg4 ,s #cd4 ,!re foll[ f\r !orems connect+ ! angles at ! c5tre1 ! arcs into :i* !y divide ! circumf]5ce1 & ! *ords subt5d+ ^! arcs4 ,!y are express$ = angles1 arcs & *ords 9 equal circles1 b !y hold also = angles1 arcs & *ords 9 ! same circle4 ,! !orems are3-- ,prop4 #bf4 ,9 equal circles equal angles /& on equal arcs1 :e!r !y be at ! c5tres or circumf]5ces2 ,prop4 #bg4 7conv]se to ,prop4 #bf74 ,9 equal circles ! angles :i* /& on equal arcs are equal to "o ano!r1 :e!r !y be at ! c5tres or ! circumf]5ces2 ,prop4 #bh4 ,9 equal circles equal /raight l9es 7equal *ords7 cut (f equal arcs1 ! great] equal to ! great]1 & ! less equal to ! less2 ,prop4 #bi 7conv]se to ,prop4 #bh74 ,9 equal circles equal arcs are subt5d$ by equal /raight l9es4 ,s #ce4 ,o!r important consequ5ces ( ,props4 #bj-bb are3-- ,prop4 #ca4 ,9 a circle ! angle 9 a semicircle is a "r angle2 b ! angle 9 a segm5t great] ?an a semicircle is less ?an a "r angle2 & ! angle 9 a segm5t less ?an a semicircle is great] ?an a "r angle2 ,prop4 #cb4 ,if a /raight l9e t\* a circle1 & f ! po9t ( contact a /raight l9e be drawn cutt+ ! circle1 ! angles :i* ?is l9e makes ) ! l9e t\*+ ! circle %all be equal to ! angles :i* are 9 ! alt]nate segm5ts ( ! circle4 ,s #cf4 ,propositions #cj1 #cc1 #cd1 conta9 problems :i* are solv$ by aid ( ! propositions prec$+ !m3-- ,prop4 #cj4 ,to bisect a giv5 arc1 t is1 to divide x into two equal p>ts2 ,prop4 #cc4 ,on a giv5 /raight l9e to describe a segm5t ( a circle conta9+ an angle equal to a giv5 rectil9eal angle2 ,prop4 #cd4 ,f a giv5 circle to cut (f a segm5t conta9+ an angle equal to a giv5 rectil9eal angle4 ,s #cg4 ,if we draw *ords ?r\gh a po9t ,a )9 a circle1 !y w ea* be divid$ by ,a into two segm5ts4 ,betwe5 ^! segm5ts ! law holds t ! rectangle conta9$ by !m has ! same area on :atev] *ord ?r\gh ,a ! segm5ts are tak54 ,! value ( ?is rectangle *anges1 ( c\rse1 ) ! position ( ,a4 ,a simil> !orem holds if ! po9t ,a be tak5 )\t ! circle4 ,on e /raight l9e ?r\gh ,a1 :i* cuts ! circle 9 two po9ts ,b & ,c1 we h two segm5ts ,,ab & ,,ac1 & ! rectangles conta9$ by !m are aga9 equal to "o ano!r1 & equal to ! squ>e on a tang5t drawn f ,a to ! circle4 ,! fir/ ( ^! !orems gives ,prop4 #ce1 & ! second ,prop4 #cf1 ) its coroll>y1 :il/ ,prop4 #cg1 ! la/ ( ,book ,,iii41 gives ! conv]se to ,prop4 #cf4 ,! fir/ two !orems may be comb9$ 9 "o3-- ,if ?r\gh a po9t ,a 9 ! plane ( a circle a /raight l9e be drawn cutt+ ! circle 9 ,b & ,c1 !n ! rectangle ,,ab4,,ac has a con/ant value s long z ! po9t ,a be fix$2 & if f ,a a tang5t ,,ad c be drawn to ! circle1 t\*+ at ,d1 !n ! above rectangle equals ! squ>e on ,,ad4 ,prop4 #cg may be /at$ ?us3-- ,if f a po9t ,a )\t a circle a l9e be drawn cutt+ ! circle 9 ,b & ,c1 & ano!r l9e to a po9t ,d on ! circle1 & ,,ab4,,ac ,,ad#b1 !n ! l9e ,,ad t\*es ! circle at ,d4 ,x is n difficult to prove also ! conv]se to ! g5]al proposition z above /at$4 ,?is proposition & its conv]se may be express$ z foll[s3-- ,if f\r po9ts ,,abcd be tak5 on ! circumf]5ce ( a circle1 & if ! l9es ,,ab1 ,,cd1 produc$ if necess>y1 meet at ,e1 !n ,,ea4,,eb ,,ec4,,ed2 & conv]sely1 if ?is relation holds !n ! f\r po9ts lie on a circle1 t is1 ! circle drawn ?r\gh ?ree ( !m passes ?r\gh ! f\r?4 ,t a circle may always be drawn ?r\gh ?ree po9ts1 provid$ t !y d n lie 9 a /raight l9e1 is prov$ only lat] on 9 ,book ,,iv4 ,,book ,,iv4 ,s #ch4 ,! f\r? book conta9s only problems1 all relat+ to ! con/ruction ( triangles & polygons inscrib$ 9 & circumscrib$ ab\t circles1 & ( circles inscrib$ 9 or circumscrib$ ab\t triangles & polygons4 ,!y are ne>ly all giv5 = _! own sake1 & n = future use 9 ! con/ruction ( figures1 z are mo/ ( ^? 9 ! =m] books4 ,9 sev5 def9itions at ! beg9n+ ( ! book x is expla9$ :at is und]/ood by figures inscrib$ 9 or describ$ ab\t o!r figures1 ) special ref]5ce to ! case :]e "o figure is a circle4 ,in/ead1 h[ev]1 ( say+ t "o figure is describ$ ab\t ano!r1 x is n[ g5]ally said t ! "o figure is circumscrib$ ab\t ! o!r4 ,we may !n /ate ! def9itions #c or #d ?us3-- ,def9ition4--,a polygon is said to be inscrib$ 9 a circle1 & ! circle is said to be circumscrib$ ab\t ! polygon1 if ! v]tices ( ! polygon lie 9 ! circumf]5ce ( ! circle4 ,& def9itions #e & #f ?us3-- ,def9ition4--,a polygon is said to be circumscrib$ ab\t a circle1 & a circle is said to be inscrib$ 9 a polygon1 if ! sides ( ! polygon are tang5ts to ! circle4 ,s #ci4 ,! fir/ problem is m]ely con/ructive4 ,x requires to draw 9 a giv5 circle a *ord equal to a giv5 /raight l9e1 :i* is n great] ?an ! diamet] ( ! circle4 ,! problem is n a det]m9ate "o1 inasmu* z ! *ord may be drawn f any po9t 9 ! circumf]5ce4 ,?is may be said ( almo/ all problems 9 ?is book1 especially ( ! next two4 ,!y are3-- ,prop4 #b4 ,9 a giv5 circle to inscribe a triangle equiangul> to a giv5 triangle2 ,prop4 #c4 ,ab\t a giv5 circle to circumscribe a triangle equiangul> to a giv5 triangle4 ,s #dj4 ,( some:at great] int]e/ are ! next problems1 :]e ! triangles are giv5 & ! circles to be f\nd4 ,prop4 #d4 ,to inscribe a circle 9 a giv5 triangle4 ,! result is t ! problem has always a solution1 viz4 ! c5tre ( ! circle is ! po9t :]e ! bisectors ( two ( ! int]ior angles ( ! triangle meet4 ,! solution %[s1 ?\gh ,euclid does n /ate ?is1 t ! problem has b "o solution2 & also1 ,! ?ree bisectors ( ! int]ior angles ( any triangle meet 9 a po9t1 & ?is is ! c5tre ( ! circle inscrib$ 9 ! triangle4 ,! solutions ( mo/ ( ! o!r problems conta9 also !orems4 ,( ^! we %all /ate ^? :i* are ( special int]e/2 ,euclid does n /ate any "o ( !m4 ,s #da4 ,prop4 #e4 ,to circumscribe a circle ab\t a giv5 triangle4 ,! "o solution :i* always exi/s conta9s ! foll[+3-- ,! ?ree /raight l9es :i* bisect ! sides ( a triangle at "r angles meet 9 a po9t1 & ?is po9t is ! c5tre ( ! circle circumscrib$ ab\t ! triangle4 ,euclid adds 9 a coroll>y ! foll[+ prop]ty3-- ,! c5tre ( ! circle circumscrib$ ab\t a triangle lies )91 on a side (1 or )\t ! triangle1 accord+ z ! triangle is acute-angl$1 "r-angl$ or obtuse-angl$4 ,s #db4 ,:il/ x is always possible to draw a circle :i* is inscrib$ 9 or circumscrib$ ab\t a giv5 triangle1 ?is is n ! case ) quadrilat]als or polygons ( m sides4 ,( ^? = :i* ?is is possible ! regul> polygons1 i4e4 polygons :i* h all _! sides & angles equal1 are ! mo/ int]e/+4 ,9 ea* ( !m a circle may be inscrib$1 & ano!r may be circumscrib$ ab\t x4 ,euclid does n use ! ^w regul>1 b he describes ! polygons 9 "q z equiangul> & equilat]al4 ,we %all use ! "n regul> polygon4 ,! regul> triangle is equilat]al1 ! regul> quadrilat]al is ! squ>e4 ,euclid consid]s ! regul> polygons ( #d1 #e1 #f & #ae sides4 ,= ea* ( ! fir/ ?ree he solves ! problems--7#a7 to inscribe su* a polygon 9 a giv5 circle2 7#b7 to circumscribe x ab\t a giv5 circle2 7#c7 to inscribe a circle 91 & 7#d7 to circumscribe a circle ab\t1 su* a polygon4 ,= ! regul> triangle ! problems are n repeat$1 because m g5]al problems h be5 solv$4 ,props4 #f1 #g1 #h & #i solve ^! problems = ! squ>e4 ,! g5]al problem ( inscrib+ 9 a giv5 circle a regul> polygon ( n sides dep5ds ^u ! problem ( divid+ ! circumf]5ce ( a circle into n equal p>ts1 or :at comes to ! same ?+1 ( draw+ f ! c5tre ( ! circle n radii su* t ! angles betwe5 consecutive radii are equal1 t is1 to divide ! space ab\t ! c5tre into n equal angles4 ,?us1 if x is requir$ to inscribe a squ>e 9 a circle1 we h to draw f\r l9es f ! c5tre1 mak+ ! f\r angles equal4 ,?is is done by draw+ two diamet]s at "r angles to "o ano!r4 ,! ends ( ^! diamet]s are ! v]tices ( ! requir$ squ>e4 ,if1 on ! o!r h&1 tang5ts be drawn at ^! ends1 we obta9 a squ>e circumscrib$ ab\t ! circle4 ,s #dc4 ,to con/ruct a regul> p5tagon1 we f9d x conv5i5t fir/ to con/ruct a regul> decagon4 ,?is requires to divide ! space ab\t ! c5tre into t5 equal angles4 ,ea* w be #a_/aj? ( a "r angle1 or #a_/e? ( two "r angles4 ,if we suppose ! decagon con/ruct$1 & if we jo9 ! c5tre to ! end ( "o side1 we get an isosceles triangle1 :]e ! angle at ! c5tre equals #a_/e? ( two "r angles2 h5ce ea* ( ! angles at ! base w be #b_/e?s ( two "r angles1 z all ?ree angles toge!r equal two "r angles4 ,?us we h to con/ruct an isosceles triangle1 hav+ ! angle at ! v]tex equal to half an angle at ! base4 ,?is is solv$ 9 ,prop4 #aj1 by aid ( ! problem 9 ,prop4 #aa ( ! second book4 ,if we make ! sides ( ?is triangle equal to ! radius ( ! giv5 circle1 !n ! base w be ! side ( ! regul> decagon inscrib$ 9 ! circle4 ,?is side be+ kn[n ! decagon c be con/ruct$1 & if ! v]tices are jo9$ alt]nately1 leav+ out half _! numb]1 we obta9 ! regul> p5tagon4 7,prop4 #aa47 ,euclid does n proce$ ?us4 ,he wants ! p5tagon be=e ! decagon4 ,?is1 h[ev]1 does n *ange ! real nature ( 8 solution1 nor does 8 solution become simpl] by n m5tion+ ! decagon4 ,once ! regul> p5tagon is inscrib$1 x is easy to circumscribe ano!r by draw+ tang5ts at ! v]tices ( ! inscrib$ p5tagon4 ,?is is %[n 9 ,prop4 #ab4 ,props4 #ac & #ad tea* h[ a circle may be inscrib$ 9 or circumscrib$ ab\t any giv5 regul> p5tagon4 ,s #dd4 ,! regul> hexagon is m easily con/ruct$1 z %[n 9 ,prop4 #ae4 ,! result is t ! side ( ! regul> hexagon inscrib$ 9 a circle is equal to ! radius ( ! circle4 ,= ?is polygon ! o!r ?ree problems m5tion$ are n solv$4 ,s #de4 ,! book closes ) ,prop4 #af4 ,to inscribe a regul> qu9decagon 9 a giv5 circle4 ,if we inscribe a regul> p5tagon & a regul> hexagon 9 ! circle1 hav+ "o v]tex 9 common1 !n ! arc f ! common v]tex to ! next v]tex ( ! p5tagon is #a_/e? ( ! circumf]5ce1 & to ! next v]tex ( ! hexagon is #a_/f? ( ! circumf]5ce4 ,! diff]5ce betwe5 ^! arcs is1 !re=e1 #a_/e - #a_/f #a_/cj? ( ! circumf]5ce4 ,! latt] may1 !re=e1 be divid$ into ?irty1 & h5ce also 9 fifte5 equal p>ts1 & ! regul> qu9decagon be describ$4 ,s #df4 ,we conclude ) a few !orems ab\t regul> polygons :i* are n giv5 by ,euclid4 ,! /raight l9es p]p5dicul> to & bisect+ ! sides ( any regul> polygon meet 9 a po9t4 ,! /raight l9es bisect+ ! angles 9 ! regul> polygon meet 9 ! same po9t4 ,?is po9t is ! c5tre ( ! circles circumscrib$ ab\t & inscrib$ 9 ! regul> polygon4 ,we c bisect any giv5 arc 7,prop4 #cj1 ,,iii474 ,h5ce we c divide a circumf]5ce into #bn equal p>ts z soon z x has be5 divid$ into n equal p>ts1 or z soon z a regul> polygon ( n sides has be5 con/ruct$4 ,h5ce-- ,if a regul> polygon ( n sides has be5 con/ruct$1 !n a regul> polygon ( #bn sides1 ( #dn1 ( #hn sides1 c41 may also be con/ruct$4 ,euclid %[s h[ to con/ruct regul> polygons ( #c1 #d1 #e & #ae sides4 ,x foll[s t we c con/ruct regul> polygons ( #c1 #f1 #ab1 #bd sides #d1 #h1 #af1 #cb #e1 #aj1 #bj1 #dj #ae1 #cj1 #fj1 #abj ,! con/ruction ( any new regul> polygon n includ$ 9 "o ( ^! s]ies w give rise to a new s]ies4 ,till ! beg9n+ ( ! #ai? c5tury no?+ 0 add$ to ! k ( regul> polygons z giv5 by ,euclid4 ,!n ,gauss1 9 8 celebrat$ ,ari?metic1 prov$ t e regul> polygon ( #bn #a sides may be con/ruct$ if ?is numb] #bn #a be prime1 & t no o!rs except ^? ) #bm7#bn #a7 sides c be con/ruct$ by elem5t>y me?ods4 ,?is %[s t regul> polygons ( #g1 #i1 #ac sides _c ?us be con/ruct$1 b t a regul> polygon ( #ag sides is possible2 = #ag #bd #a4 ,! next polygon is "o ( #beg sides4 ,! con/ruction becomes already r complicat$ = #ag sides4 ,,book ,v4 ,s #dg4 ,! fif? book ( ! ,elem5ts is n exclusively geometrical4 ,x conta9s ! !ory ( ratios & proportion ( quantities 9 g5]al4 ,! treatm5t1 z "h giv51 is admirable1 & 9 e respect sup]ior to ! algebraical me?od by :i* ,euclid's !ory is n[ g5]ally replac$4 ,we %all treat ! subject 9 ord] to %[ :y ! usual algebraical treatm5t ( proportion is n really s\nd4 ,we beg9 by quot+ ^? def9itions at ! beg9n+ ( ,book ,v4 :i* are mo/ important4 ,^! def9itions h giv5 rise to mu* discussion4 ,! only def9itions :i* are ess5tial = ! fif? book are ,defs4 #a1 #b1 #d1 #e1 #f & #g4 ,( ! rema9d] #c1 #h & #i are m ?an useless1 & probably n ,euclid's1 b additions ( lat] editors1 ( :om ,!on ( ,alex&ria 0 ! mo/ prom95t4 ,defs4 #aj & #aa belong r to ! six? book1 :il/ all ! o!rs are m]ely nom9al4 ,! really important ones are #d1 #e1 #f & #g4 ,s #dh4 ,to def9e a magnitude is n attempt$ by ,euclid4 ,! fir/ two def9itions /ate :at is meant by a "p1 t is1 a submultiple or measure1 & by a multiple ( a giv5 magnitude4 ,! mean+ ( ,def4 #d is t two giv5 quantities c h a ratio to "o ano!r only 9 case t !y are comp>able z to _! magnitude1 t is1 if !y are ( ! same k9d4 ,def4 #c1 :i* is probably due to ,!on1 pr(esses to def9e a ratio1 b is z mean+less z x is uncall$ =1 = all t is want$ is giv5 9 ,defs4 #e & #g4 ,9 ,def4 #e x is expla9$ :at is meant by say+ t two magnitudes h ! same ratio to "o ano!r z two o!r magnitudes1 & 9 ,def4 #g :at we h to und]/& by a great] or a less ratio4 ,! #f? def9ition is only nom9al1 expla9+ ! mean+ ( ! ^w proportional4 ,euclid repres5ts magnitudes by l9es1 & (t5 d5otes !m ei!r by s+le lett]s or1 l l9es1 by two lett]s4 ,we %all use only s+le lett]s = ! purpose4 ,if a & b d5ote two magnitudes ( ! same k9d1 _! ratio w be d5ot$ by a 3 b2 if c & d are two o!r magnitudes ( ! same k9d1 b possibly ( a diff]5t k9d f a & b1 !n if c & d h ! same ratio to "o ano!r z a & b1 ?is w be express$ by writ+-- a 3 b 33 c 3 d4 ,fur!r1 if m is a 7:ole7 numb]1 ma %all d5ote ! multiple ( a :i* is obta9$ by tak+ x m times4 ,s #di4 ,! :ole !ory ( ratios is bas$ on ,def4 #e4 ,def4 #e4 ,! fir/ ( f\r magnitudes is said to h ! same ratio to ! second t ! ?ird has to ! f\r? :51 any equimultiples :atev] ( ! fir/ & ! ?ird be+ tak51 & any equimultiples :atev] ( ! second & ! f\r?1 if ! multiple ( ! fir/ be less ?an t ( ! second1 ! multiple ( ! ?ird is also less ?an t ( ! f\r?2 & if ! multiple ( ! fir/ is equal to t ( ! second1 ! multiple ( ! ?ird is also equal to t ( ! f\r?2 & if ! multiple ( ! fir/ is great] ?an t ( ! second1 ! multiple ( ! ?ird is also great] ?an t ( ! f\r?4 ,x w be well to %[ at once 9 an example h[ ?is def9ition c be us$1 by prov+ ! fir/ "p ( ! fir/ proposition 9 ! six? book4 ,triangles ( ! same altitude are to "o ano!r z _! bases1 or if a & b are ! bases1 & alpha & beta ! areas1 ( two triangles :i* h ! same altitude1 !n a 3 b 33 alpha 3 beta4 ,to prove ?is1 we h1 accord+ to ,def9ition #e1 to %[-- if ma nb1 !n malpha nbeta1 if ma nb1 !n malpha nbeta1 if ma nb1 !n malpha nbeta4 ,t ?is is true is 9 our case easily se54 ,we may suppose t ! triangles h a common v]tex1 & _! bases 9 ! same l9e4 ,we set (f ! base a along ! l9e conta9+ ! bases m times2 we !n jo9 ! diff]5t p>ts ( division to ! v]tex1 & get m triangles all equal to alpha4 ,! triangle on ma z base equals1 !re=e1 malpha4 ,if we proce$ 9 ! same mann] ) ! base b1 sett+ x (f n times1 we f9d t ! area ( ! triangle on ! base nb equals nbeta1 ! v]tex ( all triangles be+ ! same4 ,b if two triangles h ! same altitude1 !n _! areas are equal if ! bases are equal2 h5ce malpha nbeta if ma nb1 & if _! bases are unequal1 !n t has ! great] area :i* is on ! great] base2 9 o!r words1 malpha is great] ?an1 equal to1 or less ?an nbeta1 accord+ z ma is great] ?an1 equal to1 or less ?an nb1 :i* 0 to be prov$4 ,s #ej4 ,x w be se5 t ev5 9 ?is example x does n become evid5t :at a ratio really is4 ,x is /ill an op5 "q :e!r ratios are magnitudes :i* we c comp>e4 ,we d n "k :e!r ! ratio ( two l9es is a magnitude ( ! same k9d z ! ratio ( two areas4 ,?\gh we might say t ,def4 #e def9es equal ratios1 /ill we d n "k :e!r !y are equal 9 ! s5se ( ! axiom1 t two ?+s :i* are equal to a ?ird are equal to "o ano!r4 ,t ?is is ! case requires a pro(1 & until ?is pro( is giv5 we %all use ! 33 in/ead ( ! sign 1 :i*1 h[ev]1 we %all aft]w>ds introduce4 ,z soon z x has be5 e/abli%$ t all ratios are l magnitudes1 x becomes easy to %[ t1 9 "s cases at lea/1 !y are numb]s4 ,?is /ep 0 nev] made by ,greek ma!maticians4 ,!y di/+ui%$ always mo/ c>efully betwe5 cont9u\s magnitudes & ! discrete s]ies ( numb]s4 ,9 mod]n times x has become ! cu/om to ignore ?is diff]5ce4 ,if1 9 det]m9+ ! ratio ( two l9es1 a common measure c be f\nd1 :i* is conta9$ m times 9 ! fir/1 & n times 9 ! second1 !n ! ratio ( ! two l9es equals ! ratio ( ! two numb]s m 3 n4 ,?is is %[n by ,euclid 9 ,prop4 #e1 ,x4 ,b ! ratio ( two numb]s is1 z a rule1 a fraction1 & ! ,greeks did n1 z we d1 consid] fractions z numb]s4 ,f> less _h !y any notion ( introduc+ irrational numb]s1 :i* are nei!r :ole nor fractional1 z we are oblig$ to d if we wi% to say t all ratios are numb]s4 ,! incomm5surable numb]s :i* are ?us introduc$ z ratios ( incomm5surable quantities are n[adays z famili> to u z fractions2 b a pro( is g5]ally omitt$ t we may apply to !m ! rules :i* h be5 e/abli%$ = rational numb]s only4 ,euclid's treatm5t ( ratios avoids ?is difficulty4 ,8 def9itions hold = comm5surable z well z = incomm5surable quantities4 ,ev5 ! notion ( incomm5surable quantities is avoid$ 9 ,book ,v4 ,b he proves t ! m elem5t>y rules ( algebra hold = ratios4 ,we %all /ate all 8 propositions 9 t algebraical =m to :i* we are n[ accu/om$4 ,?is may1 ( c\rse1 be done )\t *ang+ ! "* ( ,euclid's me?od4 ,s4 #ea4 ,us+ ! notation expla9$ above we express ! fir/ propositions z foll[s3-- ,prop4 #a4 ,if a ma'1 b mb'1 c mc'1 !n a b c m7a' b' c'74 ,prop4 #b4 ,if a mb1 & c md1 e nb1 & f nd1 !n a e is ! same multiple ( b z c f is ( d1 viz43-- a e 7m n7b1 & c f 7m n7d4 ,prop4 #c4 ,if a mb1 c md1 !n is na ! same multiple ( b t nc is ( d1 viz4 na nmb1 nc nmd4 ,prop4 #d4 ,if a 3 b 33 c 3 d1 !n ma 3 nb 33 mc 3 nd4 ,prop4 #e4 ,if a mb1 & c md1 !n a - c m7b - d74 ,prop4 #f4 ,if a mb1 c md1 !n are a - nb & c - nd ei!r equal to1 or equimultiples (1 b & d1 viz4 a - nb 7m - n7b & c - nd 7m - n7d1 :]e m - n may be unity4 ,all ^! propositions relate to equimultiples4 ,n[ foll[ propositions ab\t ratios :i* are comp>$ z to _! magnitude4 ,s #eb4 ,prop4 #g4 ,if a b1 !n a 3 c 33 b 3 c & c 3 a 33 c 3 b4 ,! pro( is simply ?is4 ,z a b we "k t ma mb2 !re=e if ma nc1 !n mb nc1 if ma nc1 !n mb nc1 if ma nc1 !n mb nc1 !re=e ! fir/ proportion holds by ,def9ition #e4 ,prop4 #h4 ,if a b1 !n a 3 c b 3 c1 & c 3 a c 3 b4 ,! pro( dep5ds on ,def9ition #g4 ,prop4 #i 7conv]se to ,prop4 #g74 ,if a 3 c 33 b 3 c1 or if c 3 a 33 c 3 b1 !n a b4 ,prop4 #aj 7conv]se to ,prop4 #h74 ,if a 3 c b 3 c1 !n a b1 & if c 3 a c 3 b1 !n a b4 ,prop4 #aa4 ,if a 3 b 33 c 3 d1 & a 3 b 33 e 3 f1 !n c 3 d 33 e 3 f4 ,9 words1 if too ratios are equal to a ?ird1 !y are equal to "o ano!r4 ,aft] ^! propositions h be5 prov$1 we h a "r to consid] a ratio z a magnitude1 = only n[ c we consid] a ratio z some?+ = :i* ! axiom ab\t magnitudes holds3 ?+s :i* are equal to a ?ird are equal to "o ano!r4 ,we %all indicate ?is by writ+ 9 future ! sign in/ead ( 334 ,! rema9+ propositions1 :i* expla9 !mselves1 may !n be /at$ z foll[s3 ,s #ec4 ,prop4 #ab4 ,if a 3 b c 3 d e 3 f1 !n a c e 3 b d f a 3 b4 ,prop4 #ac4 ,if a 3 b c 3 d & c 3 d e 3 f1 !n a 3 b e 3 f4 ,prop4 #ad4 ,if a 3 b c 3 d1 & a c1 !n b d4 ,prop4 #ae4 ,magnitudes h ! same ratio to "o ano!r t _! equimultiples h-- ma 3 mb a 3 b4 ,prop4 #af4 ,if a1 b1 c1 d are magnitudes ( ! same k9d1 & if a 3 b c 3 d1 !n a 3 c b 3 d4 ,prop4 #ag4 ,if a b 3 b c d 3 d1 !n a 3 b c 3 d4 ,prop4 #ah 7conv]se to #ag74 ,if a 3 b c 3 d !n a b 3 b c d 3 d4 ,prop4 #ai4 ,if a1 b1 c1 d are quantities ( ! same k9d1 & if a 3 b c 3 d1 !n a - c 3 b - d a 3 b4 ,s #ed4 ,prop4 #bj4 ,if !re be ?ree magnitudes1 & ano!r ?ree1 :i* h ! same ratio1 tak5 two & two1 !n if ! fir/ be great] ?an ! ?ird1 ! f\r? %all be great] ?an ! six?3 & if equal1 equal2 & if less1 less4 ,if we und]/& by a 3 b 3 c 3 d 3 e 3 444 a' 3 b' 3 c' 3 d' 3 e' 3 444 t ! ratio ( any two consecutive magnitudes on ! fir/ side equals t ( ! correspond+ magnitudes on ! second side1 we may write ?is !orem 9 symbols1 ?us3-- ,if a1 b1 c be quantities ( "o1 & d1 e1 f magnitudes ( ! same or any o!r k9d1 su* t a 3 b 3 c d 3 e 3 f1 & if a c1 !n d f1 b if a c1 !n d f1 & if a c1 !n d f4 ,prop4 #ba4 ,if a 3 b e 3 f & b 3 c d 3 e1 or if a 3 b 3 c #a_/;f 3 #a_/;e 3 #a_/;d1 & if a c1 !n d f1 b if a c1 !n d f1 & if a c1 !n d f4 ,by aid ( ^! two propositions ! foll[+ two are prov$4 ,s #ee4 ,prop4 #bb4 ,if !re be any numb] ( magnitudes1 & z _m o!rs1 :i* h ! same ratio1 tak5 two & two 9 ord]1 ! fir/ %all h to ! la/ ( ! fir/ magnitudes ! same ratio :i* ! fir/ ( ! o!rs has to ! la/4 ,we may /ate x m g5]ally1 ?us3 ,if a 3 b 3 c 3 d 3 e3 444 a' 3 b' 3 c' 3 d' 3 e' 3 444 1 !n n only h two consecutive1 b any two magnitudes on ! fir/ side1 ! same ratio z ! correspond+ magnitudes on ! o!r4 ,= in/ance-- a 3 c a' 3 c'2 b 3 e b' 3 e'1 c4 ,prop4 #bc we /ate only 9 symbols1 viz43-- ,if a 3 b 3 c 3 d 3 e 3 444 #a_/;a' 3 #a_/;b' 3 #a_/;c' 3 #a_/;d' 3 #a_/;e' 4441 !n a 3 c c' 3 a'1 b 3 e e' 3 b'1 & s on4 ,prop4 #bd comes to ?is3 ,if a 3 b c 3 d & e 3 b f 3 d1 !n a e 3 b c f 3 d4 ,"s ( ! proportions :i* are consid]$ 9 ! above propositions h special names4 ,^! we h omitt$1 z be+ ( no use1 s9ce algebra has enabl$ u to br+ ! diff]5t op]ations conta9$ 9 ! propositions "u a common po9t ( view4 ,s #ef4 ,! la/ proposition 9 ! fif? book is ( a diff]5t "*4 ,prop4 #be4 ,if f\r magnitudes ( ! same k9d be proportional1 ! greate/ & lea/ ( !m toge!r %all be great] ?an ! o!r two toge!r4 ,9 symbols-- ,if a1 b1 c1 d be magnitudes ( ! same k9d1 & if a 3 b c 3 d1 & if a is ! greate/1 h5ce d ! lea/1 !n a d b c4 ,s #eg4 ,we return once aga9 to ! "q4 ,:at is a ratio8 ,we h se5 t we may treat ratios z magnitudes1 & t all ratios are magnitudes ( ! same k9d1 = we may comp>e any two z to _! magnitude4 ,x w pres5tly be %[n t ratios ( l9es may be consid]$ z quoti5ts ( l9es1 s t a ratio appe>s z answ] to ! "q1 ,h[ (t5 is "o l9e conta9$ 9 ano!r8 ,b ! answ] to ?is "q is giv5 by a numb]1 at lea/ 9 "s cases1 & 9 all cases if we admit incomm5surable numb]s4 ,consid]$ f ?is po9t ( view1 we may say ! fif? book ( ! ,elem5ts %[s t "s ( ! simpl] algebraical op]ations hold = incomm5surable numb]s4 ,9 ! ord9>y algebraical treatm5t ( numb]s ?is pro( is altoge!r omitt$1 or giv5 by a process ( limits :i* does n seem to be natural to ! subject4 ,,book ,,vi4 ,s #eh4 ,! six? book conta9s ! !ory ( simil> figures4 ,aft] a few def9itions expla9+ t]ms1 ! fir/ proposition gives ! fir/ application ( ! !ory ( proportion4 ,prop4 #a4 ,triangles & p>allelograms ( ! same altitude are to "o ano!r z _! bases4 ,! pro( has already be5 consid]$ 9 ,s #di4 ,f ?is foll[s easily ! important !orem ,prop4 #b4 ,if a /raight l9e be drawn p>allel to "o ( ! sides ( a triangle x %all cut ! o!r sides1 or ^? sides produc$1 proportionally2 & if ! sides or ! sides produc$ be cut proportionally1 ! /raight l9e :i* jo9s ! po9ts ( section %all be p>allel to ! rema9+ side ( ! triangle4 ,s #ei4 ,! next proposition1 toge!r ) "o add$ by ,simson z ,prop4 ,a1 may be express$ m conv5i5tly if we introduce a mod]n phraseology1 viz4 if 9 a l9e ,,ab we assume a po9t ,c betwe5 ,a & ,b1 we %all say t ,c divides ,,ab int]nally 9 ! ratio ,,ac 3 ,,cb2 b if ,c be tak5 9 ! l9e ,,ab produc$1 we %all say t ,,ab is divid$ ext]nally 9 ! ratio ,,ac 3 ,,cb4 ,! two propositions !n come to ?is3 ,prop4 #c4 ,! bisector ( an angle 9 a triangle divides ! opposite side int]nally 9 a ratio equal to ! ratio ( ! two sides includ+ t angle2 & conv]sely1 if a l9e ?r\gh ! v]tex ( a triangle divide ! base int]nally 9 ! ratio ( ! two o!r sides1 !n t l9e bisects ! angle at ! v]tex4 ,simson's ,prop4 ,a4 ,! l9e :i* bisects an ext]ior angle ( a triangle divides ! opposite side ext]nally 9 ! ratio ( ! o!r sides2 & conv]sely1 if a l9e ?r\gh ! v]tex ( a triangle divide ! base ext]nally 9 ! ratio ( ! sides1 !n x bisects an ext]ior angle at ! v]tex ( ! triangle4 ,if we comb9e bo? we h-- ,! two l9es :i* bisect ! int]ior & ext]ior angles at "o v]tex ( a triangle divide ! opposite side int]nally & ext]nally 9 ! same ratio1 viz4 9 ! ratio ( ! o!r two sides4 ,s #fj4 ,! next f\r propositions conta9 ! !ory ( simil> triangles1 ( :i* f\r cases are consid]$4 ,!y may be /at$ toge!r4 ,two triangles are simil>1-- #a4 7,prop4 #d74 ,if ! triangles are equiangul>3 #b4 7,prop4 #e74 ,if ! sides ( ! "o are proportional to ^? ( ! o!r2 #c4 7,prop4 #f74 ,if two sides 9 "o are proportional to two sides 9 ! o!r1 & if ! angles conta9$ by ^! sides are equal2 #d4 7,prop4 #g74 ,if two sides 9 "o are proportional to two sides 9 ! o!r1 if ! angles opposite homolog\s sides are equal1 & if ! angles opposite ! o!r homolog\s sides are bo? acute1 bo? "r or bo? obtuse2 homolog\s sides be+ 9 ea* case ^? :i* are opposite equal angles4 ,an important application ( ^! !orems is at once made to a "r-angl$ triangle1 viz43-- ,prop4 #h4 ,9 a "r-angl$ triangle1 if a p]p5dicul> be drawn f ! "r angle to ! base1 ! triangles on ea* side ( x are simil> to ! :ole triangle1 & to "o ano!r4 ,coroll>y4--,f ?is x is manife/ t ! p]p5dicul> drawn f ! "r angle ( a "r-angl$ triangle to ! base is a mean proportional betwe5 ! segm5ts ( ! base1 & also t ea* ( ! sides is a mean proportional betwe5 ! base & ! segm5t ( ! base adjac5t to t side4 ,s #fa4 ,!re foll[ f\r propositions conta9+ problems1 9 language slightly diff]5t f ,euclid's1 viz43-- ,prop4 #i4 ,to divide a /raight l9e into a giv5 numb] ( equal p>ts4 ,prop4 #aj4 ,to divide a /raight l9e 9 a giv5 ratio4 ,prop4 #aa4 ,to f9d a ?ird proportional to two giv5 /raight l9es4 ,prop4 #ab4 ,to f9d a f\r? proportional to ?ree giv5 /raight l9es4 ,prop4 #ac4 ,to f9d a mean proportional betwe5 two giv5 /raight l9es4 ,! la/ ?ree may be writt5 z equations ) "o unkn[n quantity--viz4 if we call ! giv5 /raight l9es a1 b1 c1 & ! requir$ l9e x1 we h to f9d a l9e x s t ,prop4 #aa4 a 3 b b 3 x2 ,prop4 #ab4 a 3 b c 3 x2 ,prop4 #ac4 a 3 x x 3 b4 ,we %all see pres5tly h[ ^! may be writt5 )\t ! signs ( ratios4 ,s #fb4 ,euclid consid]s next proportions connect$ ) p>allelograms & triangles :i* are equal 9 area4 ,prop4 #ad4 ,equal p>allelograms :i* h "o angle ( ! "o equal to "o angle ( ! o!r h _! sides ab\t ! equal angles reciprocally proportional2 & p>allelograms :i* h "o angle ( ! "o equal to "o angle ( ! o!r1 & _! sides ab\t ! equal angles reciprocally proportional1 are equal to "o ano!r4 ,prop4 #ae4 ,equal triangles :i* h "o angle ( ! "o equal to "o angle ( ! o!r1 h _! sides ab\t ! equal angles reciprocally proportional2 & triangles :i* h "o angle ( ! "o equal to "o angle ( ! o!r1 & _! sides ab\t ! equal angles reciprocally proportional1 are equal to "o ano!r4 ,illu/ration ,! latt] proposition is really ! same z ! =m]1 = if1 z 9 ! accompany+ diagram1 9 ! figure belong+ to ! =m] ! two equal p>allelograms ,,ab & ,,bc be bisect$ by ! l9es ,,df & ,,eg1 & if ,,ef be drawn1 we get ! figure belong+ to ! latt]4 ,x is wor? notic+ t ! l9es ,,fe & ,,dg are p>allel4 ,we may /ate !re=e ! !orem-- ,if two triangles are equal 9 area1 & h "o angle 9 ! "o v]tically opposite to "o angle 9 ! o!r1 !n ! two /raight l9es :i* jo9 ! rema9+ two v]tices ( ! "o to ^? ( ! o!r triangle are p>allel4 ,s #fc4 ,a mo/ important !orem is ,prop4 #af4 ,if f\r /raight l9es be proportionals1 ! rectangle conta9$ by ! extremes is equal to ! rectangle conta9$ by ! means2 & if ! rectangle conta9$ by ! extremes be equal to ! rectangle conta9$ by ! means1 ! f\r /raight l9es are proportionals4 ,9 symbols1 if a1 b1 c1 d are ! f\r l9es1 & if a 3 b c 3 d1 !n ad bc2 & conv]sely1 if ad bc1 !n a 3 b c 3 d1 :]e ad & bc d5ote 7z 9 ,s #bj71 ! areas ( ! rectangles conta9$ by a & d & by b & c respectively4 ,?is all[s u to trans=m e proportion betwe5 f\r l9es into an equation betwe5 two products4 ,x %[s fur!r t ! op]ation ( =m+ a product ( two l9es1 & ! op]ation ( =m+ _! ratio are ea* ! inv]se ( ! o!r4 ,if we n[ def9e a quoti5t a_/b ( two l9es z ! numb] :i* multipli$ into b gives a1 s t a -- b a1 b we see t f ! equality ( two quoti5ts a c -- -- b d foll[s1 if we multiply bo? sides by bd1 a c -- b4d -- d4b1 b d ad cb4 ,b f ?is x foll[s1 accord+ to ! la/ !orem1 t a 3 b c 3 d4 ,h5ce we conclude t ! quoti5t a_/b & ! ratio a 3 b are diff]5t =ms ( ! same magnitude1 only ) ?is important diff]5ce t ! quoti5t a_/b w\ld h a mean+ only if a & b h a common measure1 until we introduce incomm5surable numb]s1 :ile ! ratio a 3 b has always a mean+1 & ?us gives rise to ! introduction ( incomm5surable numb]s4 ,?us x is really ! !ory ( ratios 9 ! fif? book :i* enables u to ext5d ! geometrical calculus giv5 be=e 9 connexion ) ,book ,,ii4 ,x w also be se5 t if we write ! ratios 9 ,book ,v4 z quoti5ts1 or r z fractions1 !n mo/ ( ! !orems /ate prop]ties ( quoti5ts or ( fractions4 ,s #fd4 ,prop4 #ag4 ,if ?ree /raight l9es are proportional ! rectangle conta9$ by ! extremes is equal to ! squ>e on ! mean2 & conv]sely1 is only a special case ( #af4 ,aft] ! problem1 ,prop4 #ah1 ,on a giv5 /raight l9e to describe a rectil9eal figure simil> & simil>ly situat$ to a giv5 rectil9eal figure1 !re foll[s ano!r fundam5tal !orem3 ,prop4 #ai4 ,simil> triangles are to "o ano!r 9 ! duplicate ratio ( _! homolog\s sides4 ,9 o!r words1 ! areas ( simil> triangles are to "o ano!r z ! squ>es on homolog\s sides4 ,?is is g5]aliz$ 93 ,prop4 #bj4 ,simil> polygons may be divid$ into ! same numb] ( simil> triangles1 hav+ ! same ratio to "o ano!r t ! polygons h2 & ! polygons are to "o ano!r 9 ! duplicate ratio ( _! homolog\s sides4 ,s #fe4 ,prop4 #ba4 ,rectil9eal figures :i* are simil> to ! same rectil9eal figure are also simil> to ea* o!r1 is an imm$iate consequ5ce ( ! def9ition ( simil> figures4 ,z simil> figures may be said to be equal 9 %ape b n 9 size1 we may /ate x also ?us3 ,figures :i* are equal 9 %ape to a ?ird are equal 9 %ape to ea* o!r4 ,prop4 #bb4 ,if f\r /raight l9es be proportionals1 ! simil> rectil9eal figures simil>ly describ$ on !m %all also be proportionals2 & if ! simil> rectil9eal figures simil>ly describ$ on f\r /raight l9es be proportionals1 ^? /raight l9es %all be proportionals4 ,?is is ess5tially ! same z ! foll[+3-- ,if a 3 b c 3 d1 !n a#b 3 b#b c#b 3 d#b4 ,s #ff4 ,n[ foll[s a proposition :i* has be5 mu* discuss$ ) reg>d to ,euclid's exact mean+ 9 say+ t a ratio is comp\nd$ ( two o!r ratios1 viz43 ,prop4 #bc4 ,p>allelograms :i* are equiangul> to "o ano!r1 h to "o ano!r ! ratio :i* is comp\nd$ ( ! ratios ( _! sides4 ,! pro( ( ! proposition makes its mean+ cle>4 ,9 symbols ! ratio a 3 c is comp\nd$ ( ! two ratios a 3 b & b 3 c1 & if a 3 b a' 3 b'1 b 3 c b 3 c1 !n a 3 c is comp\nd$ ( a' 3 b' & b 3 c4 ,if we consid] ! ratios z numb]s1 we may say t ! "o ratio is ! product ( ^? ( :i* x is comp\nd$1 or 9 symbols1 a a b a' b a a' b b -- -- 4 -- -- 4 --1 if -- -- & -- --4 c b c b' c b b' c c ,! !orem 9 ,prop4 #bc is ! f\ndation ( all m5suration ( areas4 ,f x we see at once t two rectangles h ! ratio ( _! areas comp\nd$ ( ! ratios ( _! sides4 ,if ,a is ! area ( a rectangle conta9$ by a & b1 & ,b t ( a rectangle conta9$ by c & d1 s t ,a ab1 ,b cd1 !n ,a 3 ,b ab 3 cd1 & ?is is1 ! !orem says1 comp\nd$ ( ! ratios a 3 c & b 3 d4 ,9 =ms ( quoti5ts1 a b ab -- 4 -- --4 c d cd ,?is %[s h[ to multiply quoti5ts 9 our geometrical calculus4 ,fur!r1 ,two triangles h ! ratios ( _! areas comp\nd$ ( ! ratios ( _! bases & _! altitude4 ,= a triangle is equal 9 area to half a p>allelogram :i* has ! same base & ! same altitude4 ,s #fg4 ,to br+ ^! !orems to ! =m 9 :i* !y are usually giv51 we assume a /raight l9e u z our unit ( l5g? 7g5]ally an in*1 a foot1 a mile1 c471 & det]m9e ! numb] alpha :i* expresses h[ (t5 u is conta9$ 9 a l9e a1 s t alpha d5otes ! ratio a 3 u :e!r comm5surable or n1 & t a alphau4 ,we call ?is numb] alpha ! num]ical value ( a4 ,if 9 ! same mann] beta be ! num]ical value ( a l9e b we h a 3 b alpha 3 beta2 9 words3 ,! ratio ( two l9es 7& ( two l quantities 9 g5]al7 is equal to t ( _! num]ical values4 ,?is is easily prov$ by obs]v+ t a alphau1 b betau1 !re=e a 3 b alphau 3 betau1 & ?is may )\t difficulty be %[n to equal alpha 3 beta4 ,if n[ a1 b be base & altitude ( "o1 a'1 b' ^? ( ano!r p>allelogram1 alpha1 beta & alpha'1 beta' _! num]ical values respectively1 & ,a1 ,a' _! areas1 !n ,a a b alpha beta alphabeta -- -- 4 -- -------- 4 ------ ---------------4 ,a' a' b' alpha' beta' alpha'beta' ,9 words3 ,! areas ( two p>allelograms are to ea* o!r z ! products ( ! num]ical values ( _! bases & altitudes4 ,if especially ! second p>allelogram is ! unit squ>e1 i4e4 a squ>e on ! unit ( l5g?1 !n alpha' beta' #a1 ,a' u#b1 & we h ,a -- alphabeta or ,a alphabeta 4 u#b4 ,a' ,?is gives ! !orem3 ,! numb] ( unit squ>es conta9$ 9 a p>allelogram equals ! product ( ! num]ical values ( base & altitude1 & simil>ly ! numb] ( unit squ>es conta9$ 9 a triangle equals half ! product ( ! num]ical values ( base & altitude4 ,?is is (t5 /at$ by say+ t ! area ( a p>allelogram is equal to ! product ( ! base & ! altitude1 mean+ by ?is product ! product ( ! num]ical values1 & n ! product z def9$ above 9 ,s #bj4 ,s #fh4 ,propositions #bd & #bf relate to p>allelograms ab\t diagonals1 su* z are consid]$ 9 ,book ,i41 #dc4 ,!y are-- ,prop4 #bd4 ,p>allelograms ab\t ! diamet] ( any p>allelogram are simil> to ! :ole p>allelogram & to "o ano!r2 & its conv]se 7,prop4 #bf71 ,if two simil> p>allelograms h a common angle1 & be simil>ly situat$1 !y are ab\t ! same diamet]4 ,betwe5 ^! is ins]t$ a problem4 ,prop4 #be4 ,to describe a rectil9eal figure :i* %all be simil> to "o giv5 rectil9e> figure1 & equal to ano!r giv5 rectil9eal figure4 ,s #fi4 ,prop4 #bg conta9s a !orem relat+ to ! !ory ( maxima & m9ima4 ,we may /ate x ?us3 ,prop4 #bg4 ,if a p>allelogram be divid$ into two by a /raight l9e cutt+ ! base1 & if on half ! base ano!r p>allelogram be con/ruct$ simil> to "o ( ^? p>ts1 !n ?is ?ird p>allelogram is great] ?an ! o!r "p4 ,( f> great] int]e/ ?an ?is g5]al !orem is a special case ( x1 :]e ! p>allelograms are *ang$ into rectangles1 & :]e "o ( ! p>ts into :i* ! p>allelogram is divid$ is made a squ>e2 = !n ! !orem *anges into "o :i* is easily recogniz$ to be id5tical ) ! foll[+3-- ,( all rectangles :i* h ! same p]imet] ! squ>e has ! greate/ area4 ,?is may also be /at$ ?us3-- ,( all rectangles :i* h ! same area ! squ>e has ! lea/ p]imet]4 ,s #gj4 ,! next ?ree propositions conta9 problems :i* may be said to be solutions ( quadratic equations4 ,! fir/ two are1 l ! la/1 involv$ 9 some:at obscure language4 ,we transcribe !m z foll[s3 ,problem4--,to describe on a giv5 base a p>allelogram1 & to divide x ei!r int]nally 7,prop4 #bh7 or ext]nally 7,prop4 #bi7 f a po9t on ! base into two p>allelograms1 ( :i* ! "o has a giv5 size 7is equal 9 area to a giv5 figure71 :il/ ! o!r has a giv5 %ape 7is simil> to a giv5 p>allelogram74 ,if we express ?is aga9 9 symbols1 call+ ! giv5 base a1 ! "o "p x1 & ! altitude y1 we h to det]m9e x & y 9 ! fir/ case f ! equations 7a - x7y k#b1 x p -- --1 y q k#b be+ ! giv5 size ( ! fir/1 & p & q ! base & altitude ( ! p>allelogram :i* det]m9e ! %ape ( ! second ( ! requir$ p>allelograms4 ,if we sub/itute ! value ( y1 we get pk#b 7a - x7x ----1 q or1 ax - x#b b#b1 :]e a & b are kn[n quantities1 tak+ b#b pk#b_/q4 ,! second case 7,prop4 #bi7 gives rise1 9 ! same mann]1 to ! quadratic ax x#b b#b4 ,! next problem-- ,prop4 #cj4 ,to cut a giv5 /raight l9e 9 extreme & mean ratio1 leads to ! equation ax x#b a#b4 ,?is is1 !re=e1 only a special case ( ! la/1 & is1 besides1 an old acqua9tance1 be+ ess5tially ! same problem z t propos$ 9 ,,ii4 #aa4 ,prop4 #cj may !re=e be solv$ 9 two ways1 ei!r by aid ( ,prop4 #bi or by aid ( ,,ii4 #aa4 ,euclid gives bo? solutions4 ,s #ga4 ,prop4 #ca 7,!orem74 ,9 any "r-angl$ triangle1 any rectil9eal figure describ$ on ! side subt5d+ ! "r angle is equal to ! simil> & simil>ly-describ$ figures on ! sides conta9+ ! "r angle1--is a pretty g5]alization ( ! !orem ( ,py?agoras 7,i4 #dg74 ,leav+ out ! next proposition1 :i* is ( little int]e/1 we come to ! la/ 9 ?is book4 ,prop4 #cc4 ,9 equal circles angles1 :e!r at ! c5tres or ! circumf]5ces1 h ! same ratio :i* ! arcs on :i* !y /& h to "o ano!r2 s also h ! sectors4 ,( ?is1 ! "p relat+ to angles at ! c5tre is ( special importance2 x enables u to measure angles by arcs4 ,) ?is closes t "p ( ! ,elem5ts :i* is devot$ to ! /udy ( figures 9 a plane4 ,,book ,,xi4 ,s #gb4 ,9 ?is book figures are consid]$ :i* are n conf9$ to a plane1 viz4 fir/ relations betwe5 l9es & planes 9 space1 & aft]w>ds prop]ties ( solids4 ,( new def9itions we m5tion ^? :i* relate to ! p]p5dicul>ity & ! incl9ation ( l9es & planes4 ,def4 #c4 ,a /raight l9e is p]p5dicul>1 or at "r angles1 to a plane :5 x makes "r angles ) e /raight l9e meet+ x 9 t plane4 ,! def9ition ( p]p5dicul> planes 7,def4 #d7 (f]s no difficulty4 ,euclid def9es ! incl9ation ( l9es to planes & ( planes to planes 7,defs4 #e & #f7 by aid ( plane angles1 includ$ by /raight l9es1 ) :i* we h be5 made famili> 9 ! fir/ books4 ,! o!r important def9itions are ^? ( p>allel planes1 :i* nev] meet 7,def4 #h71 & ( solid angles =m$ by ?ree or m planes meet+ 9 a po9t 7,def4 #i74 ,to ^! we add ! def9ition ( a l9e p>allel to a plane z a l9e :i* does n meet ! plane4 ,s #gc4 ,be=e we inve/igate ! cont5ts ( ,book ,,xi41 x w be well to recapitulate %ortly :at we "k ( planes & l9es f ! def9itions & axioms ( ! fir/ book4 ,!re a plane has be5 def9$ z a surface :i* has ! prop]ty t e /raight l9e :i* jo9s two po9ts 9 x lies altoge!r 9 x4 ,?is is equival5t to say+ t a /raight l9e :i* has two po9ts 9 a plane has all po9ts 9 ! plane4 ,h5ce1 a /raight l9e :i* does n lie 9 ! plane _c h m ?an "o po9t 9 common ) ! plane4 ,?is is virtually ! same z ,euclid's ,prop4 #a1 viz43-- ,prop4 #a4 ,"o "p ( a /raight l9e _c be 9 a plane & ano!r "p )\t x4 ,x also foll[s1 z 0 po9t$ out 9 ,s #c1 9 discuss+ ! def9itions ( ,book ,i41 t a plane is det]m9$ already by "o /raight l9e & a po9t )\t x1 viz4 if all l9es be drawn ?r\gh ! po9t1 & cutt+ ! l9e1 !y w =m a plane4 ,?is may be /at$ ?us3-- ,a plane is det]m9$-- #a/1 ,by a /raight l9e & a po9t :i* does n lie on x2 #bnd1 ,by ?ree po9ts :i* d n lie 9 a /raight l9e2 = if two ( ^! po9ts be jo9$ by a /raight l9e we h case #a2 #crd1 ,by two int]sect+ /raight l9es2 = ! po9t ( int]section & two o!r po9ts1 "o 9 ea* l9e1 give case #b2 #d?1 ,by two p>allel l9es 7,def4 #ce1 ,i474 ,! ?ird case ( ?is !orem is ,euclid's ,prop4 #b4 ,two /raight l9es :i* cut "o ano!r are 9 "o plane1 & ?ree /raight l9es :i* meet "o ano!r are 9 "o plane4 ,& ! f\r? is ,euclid's ,prop4 #g4 ,if two /raight l9es be p>allel1 ! /raight l9e drawn f any po9t 9 "o to any po9t 9 ! o!r is 9 ! same plane ) ! p>allels4 ,f ! def9ition ( a plane fur!r foll[s ,prop4 #c4 ,if two planes cut "o ano!r1 _! common section is a /raight l9e4 ,s #gd4 ,:il/ ^! propositions are virtually conta9$ 9 ! def9ition ( a plane1 ! next gives u a new & fundam5tal prop]ty ( space1 %[+ at ! same "t t x is possible to h a /raight l9e p]p5dicul> to a plane1 accord+ to ,def4 #c4 ,x /ates-- ,prop4 #d4 ,if a /raight l9e is p]p5dicul> to two /raight l9es 9 a plane :i* x meets1 !n x is p]p5dicul> to all l9es 9 ! plane :i* x meets1 & h5ce x is p]p5dicul> to ! plane4 ,def4 #c may be /at$ ?us3 ,if a /raight l9e is p]p5dicul> to a plane1 !n x is p]p5dicul> to e l9e 9 ! plane :i* x meets4 ,! conv]se to ?is w\ld be ,all /raight l9es :i* meet a giv5 /raight l9e 9 ! same po9t1 & are p]p5dicul> to x1 lie 9 a plane :i* is p]p5dicul> to t l9e4 ,?is ,euclid /ates ?us3 ,prop4 #e4 ,if ?ree /raight l9es meet all at "o po9t1 & a /raight l9e /&s at "r angles to ea* ( !m at t po9t1 ! ?ree /raight l9es %all be 9 "o & ! same plane4 ,s #ge4 ,!re foll[ !orems relat+ to ! !ory ( p>allel l9es 9 space1 viz43-- ,prop4 #f4 ,any two l9es :i* are p]p5dicul> to ! same plane are p>allel to ea* o!r2 & conv]sely ,prop4 #h4 ,if ( two p>allel /raight l9es "o is p]p5dicul> to a plane1 ! o!r is s also4 ,prop4 #g4 ,if two /raight l9es are p>allel1 ! /raight l9e :i* jo9s any po9t 9 "o to any po9t 9 ! o!r is 9 ! same plane z ! p>allels4 7,see above1 ,s #gc47 ,prop4 #i4 ,two /raight l9es :i* are ea* ( !m p>allel to ! same /raight l9e1 & n 9 ! same plane ) x1 are p>allel to "o ano!r2 :]e ! words1 & n 9 ! same plane ) x1 may be omitt$1 = !y exclude ! case ( ?ree p>allels 9 a plane1 :i* has be5 prov$ be=e2 & ,prop4 #aj4 ,if two angles 9 diff]5t planes h ! two limits ( ! "o p>allel to ^? ( ! o!r1 !n ! angles are equal4 ,t _! planes are p>allel is %[n lat] on 9 ,prop4 #ae4 ,?is !orem is n necess>ily true1 = ! angles 9 "q may be supplem5t>y2 b !n ! "o angle w be equal to t :i* is adjac5t & supplem5t>y to ! o!r1 & ?is latt] angle w also h its limits p>allel to ^? ( ! fir/4 ,f ?is !orem x foll[s t if we take any two /raight l9es 9 space :i* d n meet1 & if we draw ?r\gh any po9t ,p 9 space two l9es p>allel to !m1 !n ! angle includ$ by ^! l9es w always be ! same1 :atev] ! position ( ! po9t ,p may be4 ,?is angle has 9 mod]n times be5 call$ ! angle betwe5 ! giv5 l9es3-- ,by ! angles betwe5 two n int]sect+ l9es we und]/& ! angles :i* two int]sect+ l9es include t are p>allel respectively to ! two giv5 l9es4 ,s #gf4 ,x is n[ possible to solve ! foll[+ two problems3-- ,to draw a /raight l9e p]p5dicul> to a giv5 plane f a giv5 po9t :i* lies #a4 ,n 9 ! plane 7,prop4 #aa74 #b4 ,9 ! plane 7,prop4 #ab74 ,! second case is easily r$uc$ to ! fir/--viz4 if by aid ( ! fir/ we h drawn any p]p5dicul> to ! plane f "s po9t )\t x1 we ne$ only draw ?r\gh ! giv5 po9t 9 ! plane a l9e p>allel to x1 9 ord] to h ! requir$ p]p5dicul> giv54 ,! solution ( ! fir/ "p is ( int]e/ 9 itself4 ,x dep5ds ^u a con/ruction :i* may be express$ z a !orem4 ,if f a po9t ,a )\t a plane a p]p5dicul> ,,ab be drawn to ! plane1 & if f ! foot ,b ( ?is p]p5dicul> ano!r p]p5dicul> ,,bc be drawn to any /raight l9e 9 ! plane1 !n ! /raight l9e jo9+ ,a to ! foot ,c ( ?is second p]p5dicul> w also be p]p5dicul> to ! l9e 9 ! plane4 ,! !ory ( p]p5dicul>s to a plane is conclud$ by ! !orem-- ,prop4 #ac4 ,?r\gh any po9t 9 space1 :e!r 9 or )\t a plane1 only "o /raight l9e c be drawn p]p5dicul> to ! plane4 ,s #gg4 ,! next f\r propositions treat ( p>allel planes4 ,x is %[n t planes :i* h a common p]p5dicul> are p>allel 7,prop4 #ad72 t two planes are p>allel if two int]sect+ /raight l9es 9 ! "o are p>allel respectively to two /raight l9es 9 ! o!r plane 7,prop4 #ae72 t p>allel planes are cut by any plane 9 p>allel /raight l9es 7,prop4 #af72 & la/ly1 t any two /raight l9es are cut proportionally by a s]ies ( p>allel planes 7,prop4 #ag74 ,?is !ory is made m complete by add+ ! foll[+ !orems1 :i* are easy d$uctions f ! la/3 ,two p>allel planes h common p]p5dicul>s 7conv]se to #ad72 & ,two planes :i* are p>allel to a ?ird plane are p>allel to ea* o!r4 ,x w be not$ t ,prop4 #ae at once all[s ( ! solution ( ! problem3 ,?r\gh a giv5 po9t to draw a plane p>allel to a giv5 plane4 ,& x is also easily prov$ t ?is problem all[s always ( "o1 & only ( "o1 solution4 ,s #gh4 ,we come n[ to planes :i* are p]p5dicul> to "o ano!r4 ,two !orems relate to !m4 ,prop4 #ah4 ,if a /raight l9e be at "r angles to a plane1 e plane :i* passes ?r\gh x %all be at "r angles to t plane4 ,prop4 #ai4 ,if two planes :i* cut "o ano!r be ea* ( !m p]p5dicul> to a ?ird plane1 _! common section %all be p]p5dicul> to ! same plane4 ,s #gi4 ,if ?ree planes pass ?r\gh a common po9t1 & if !y b\nd ea* o!r1 a solid angle ( ?ree faces1 or a trih$ral angle1 is =m$1 & simil>ly by m planes a solid angle ( m faces1 or a polyh$ral angle4 ,^! h _m prop]ties :i* are q analog\s to ^? ( triangles & polygons 9 a plane4 ,euclid /ates "s1 viz43-- ,prop4 #bj4 ,if a solid angle be conta9$ by ?ree plane angles1 any two ( !m are toge!r great] ?an ! ?ird4 ,b ! next-- ,prop4 #ba4 ,e solid angle is conta9$ by plane angles1 :i* are toge!r less ?an f\r "r angles--has no analog\s !orem 9 ! plane4 ,we may m5tion1 h[ev]1 t ! !orems ab\t triangles conta9$ 9 ! propositions ( ,book ,i41 :i* d n dep5d ^u ! !ory ( p>allels 7t is all up to ,prop4 #bg71 h _! correspond+ !orems ab\t trih$ral angles4 ,! latt] are =m$1 if = side ( a triangle we write plane angle or face ( trih$ral angle1 & = angle ( triangle we sub/itute angle betwe5 two faces :]e ! planes conta9+ ! solid angle are call$ its faces4 ,we get1 = in/ance1 f ,i4 #d1 ! !orem1 ,if two trih$ral angles h ! angles ( two faces 9 ! "o equal to ! angles ( two faces 9 ! o!r1 & h likewise ! angles includ$ by ^! faces equal1 !n ! angles 9 ! rema9+ faces are equal1 & ! angles betwe5 ! o!r faces are equal ea* to ea*1 viz4 ^? :i* are opposite equal faces4 ,! solid angles !mselves are n necess>ily equal1 = !y may be only symmetrical l ! "r h& & ! left4 ,! connexion indicat$ betwe5 triangles & trih$ral angles w also be recogniz$ 9 ,prop4 #bb4 ,if e two ( ?ree plane angles be great] ?an ! ?ird1 & if ! /raight l9es :i* conta9 !m be all equal1 a triangle may be made ( ! /raight l9es t jo9 ! extremities ( ^? equal /raight l9es4 ,& ,prop4 #bc solves ! problem1 ,to con/ruct a trih$ral angle hav+ ! angles ( its faces equal to ?ree giv5 plane angles1 any two ( !m be+ great] ?an ! ?ird4 ,x is1 ( c\rse1 analog\s to ! problem ( con/ruct+ a triangle hav+ its sides ( giv5 l5g?4 ,two o!r !orems ( ?is k9d are add$ by ,simson 9 8 edition ( ,euclid's ,elem5ts4 ,s #hj4 ,^! are ! pr9cipal prop]ties ( l9es & planes 9 space1 b be=e we g on to _! applications x w be well to def9e ! ^w di/ance4 ,9 geometry di/ance means always %orte/ di/ance2 viz4 ! di/ance ( a po9t f a /raight l9e1 or f a plane1 is ! l5g? ( ! p]p5dicul> f ! po9t to ! l9e or plane4 ,! di/ance betwe5 two non-int]sect+ l9es is ! l5g? ( _! common p]p5dicul>1 !re be+ b "o4 ,! di/ance betwe5 two p>allel l9es or betwe5 two p>allel planes is ! l5g? ( ! common p]p5dicul> betwe5 ! l9es or ! planes4 ,s #ha4 ,p>allelepip$s4--,! re/ ( ! book is devot$ to ! /udy ( ! p>allelepip$4 ,9 ,prop4 #bd ! possibility ( su* a solid is prov$1 viz43-- ,prop4 #bd4 ,if a solid be conta9$ by six planes two & two ( :i* are p>allel1 ! opposite planes are simil> & equal p>allelograms4 ,euclid calls ?is solid h5ce=? a p>allelepip$1 ?\gh he nev] def9es ! ^w4 ,ei!r face ( x may be tak5 z base1 & its di/ance f ! opposite face z altitude4 ,prop4 #be4 ,if a solid p>allelepip$ be cut by a plane p>allel to two ( its opposite planes1 x divides ! :ole into two solids1 ! base ( "o ( :i* %all be to ! base ( ! o!r z ! "o solid is to ! o!r4 ,?is !orem corresponds to ! !orem 7,,vi4 #a7 t p>allelograms betwe5 ! same p>allels are to "o ano!r z _! bases4 ,a simil> analogy is to be obs]v$ among a numb] ( ! rema9+ propositions4 ,s #hb4 ,aft] solv+ a few problems we come to ,prop4 #bh4 ,if a solid p>allelepip$ be cut by a plane pass+ ?r\gh ! diagonals ( two ( ! opposite planes1 x %all be cut 9 two equal p>ts4 ,9 ! pro( ( ?is1 z ( sev]al o!r propositions1 ,euclid neglects ! diff]5ce betwe5 solids :i* are symmetrical l ! "r h& & ! left4 ,prop4 #ca4 ,solid p>allelepip$s1 :i* are ^u equal bases1 & ( ! same altitude1 are equal to "o ano!r4 ,props4 #bi & #cj conta9 special cases ( ?is !orem lead+ up to ! pro( ( ! g5]al !orem4 ,z consequ5ces ( ?is fundam5tal !orem we get ,prop4 #cb4 ,solid p>allelepip$s1 :i* h ! same altitude1 are to "o ano!r z _! bases2 & ,prop4 #cc4 ,simil> solid p>allelepip$s are to "o ano!r 9 ! triplicate ratio ( _! homolog\s sides4 ,if we consid]1 z 9 ,s #fg1 ! ratios ( l9es z numb]s1 we may also say-- ,! ratio ( ! volumes ( simil> p>allelepip$s is equal to ! ratio ( ! ?ird p[]s ( homolog\s sides4 ,p>allelepip$s :i* are n simil> b equal are comp>$ by aid ( ! !orem ,prop4 #cd4 ,! bases & altitudes ( equal solid p>allelepip$s are reciprocally proportional2 & if ! bases & altitudes be reciprocally proportional1 ! solid p>allelepip$s are equal4 ,s #hc4 ,( ! foll[+ propositions ! #cg? & #dj? are ( special int]e/4 ,prop4 #cg4 ,if f\r /raight l9es be proportionals1 ! simil> solid p>allelepip$s1 simil>ly describ$ f !m1 %all also be proportionals2 & if ! simil> p>allelepip$s simil>ly describ$ f f\r /raight l9es be proportionals1 ! /raight l9es %all be proportionals4 ,9 symbols x says-- ,if a 3 b c 3 d1 !n a#c 3 b#c c#c 3 d#c4 ,prop4 #dj tea*es h[ to comp>e ! volumes ( triangul> prisms ) ^? ( p>allelepip$s1 by prov+ t a triangul> prism is equal 9 volume to a p>allelepip$1 :i* has its altitude & its base equal to ! altitude & ! base ( ! triangul> prism4 ,s #hd4 ,f ^! propositions foll[ all results relat+ to ! m5suration ( volumes4 ,we %all /ate ^! z we did 9 ! case ( areas4 ,! />t+-po9t is ! rectangul> p>allelepip$1 :i* has e edge p]p5dicul> to ! planes x meets1 & :i* takes ! place ( ! rectangle 9 ! plane4 ,if ?is has all its edges equal we obta9 ! cube4 ,if we take a c]ta9 l9e u z unit l5g?1 !n ! squ>e on u is ! unit ( area1 & ! cube on u ! unit ( volume1 t is to say1 if we wi% to measure a volume we h to det]m9e h[ _m unit cubes x conta9s4 ,a rectangul> p>allelepip$ has1 z a rule1 ! ?ree edges unequal1 :i* meet at a po9t4 ,e o!r edge is equal to "o ( !m4 ,if a1 b1 c be ! ?ree edges meet+ at a po9t1 !n we may take ! rectangle conta9$ by two ( !m1 say by b & c1 z base & ! ?ird z altitude4 ,let ,v be its volume1 ,v' t ( ano!r rectangul> p>allelepip$ :i* has ! edges a'1 b1 c1 h5ce ! same base z ! fir/4 ,x foll[s !n easily1 f ,prop4 #be or #cb1 t ,v 3 ,v' a 3 a'2 or 9 words1 ,rectangul> p>allelepip$s on equal bases are proportional to _! altitudes4 ,if we h two rectangul> p>allelepip$s1 ( :i* ! fir/ has ! volume ,v & ! edges a1 b1 c1 & ! second1 ! volume ,v' & ! edges a'1 b'1 c'1 we may comp>e !m by aid ( two new ones :i* h respectively ! edges a'1 b1 c & a'1 b'1 c1 & ! volumes ,v#a & ,v#b4 ,we !n h ,v 3 ,v#a a 3 a'2 ,v#a 3 ,v#b b 3 b'1 ,v#b 3 ,v' c 3 c'4 ,comp\nd+ ^!1 we h ,v 3 ,v' 7a 3 a'77b 3 b'77c 3 c'71 or ,v a b c -- -- 4 -- 4 --4 ,v' a' b' c' ,h5ce1 z a special case1 mak+ ,v' equal to ! unit cube ,u on u we get ,v a b c -- -- 4 -- 4 -- alpha4beta4gamma1 ,u u u u :]e alpha1 beta1 gamma are ! num]ical values ( a1 b1 c2 t is1 ,! numb] ( unit cubes 9 a rectangul> p>allelepip$ is equal to ! product ( ! num]ical values ( its ?ree edges4 ,?is is g5]ally express$ by say+ ! volume ( a rectangul> p>allelepip$ is measur$ by ! product ( its sides1 or by ! product ( its base into its altitude1 :i* 9 ?is case is ! same4 ,prop4 #ca all[s u to ext5d ?is to any p>allelepip$s1 & ,props4 #bh or #dj1 to triangul> prisms4 ,! volume ( any p>allelepip$1 or ( any triangul> prism1 is measur$ by ! product ( base & altitude4 ,! consid]ation t any polygonal prism may be divid$ into a numb] ( triangul> prisms1 :i* h ! same altitude & ! sum ( _! bases equal to ! base ( ! polygonal prism1 %[s fur!r t ! same holds = any prism :atev]4 ,,book ,,xii4 ,s #he4 ,9 ! la/ "p ( ,book ,,xi4 we h le>nt h[ to comp>e ! volumes ( p>allelepip$s & ( prisms4 ,9 ord] to det]m9e ! volume ( any solid b\nd$ by plane faces we mu/ det]m9e ! volume ( pyramids1 = e su* solid may be decompos$ into a numb] ( pyramids4 ,z e pyramid may aga9 be decompos$ into triangul> pyramids1 x becomes only necess>y to det]m9e _! volume4 ,?is is done by ! ,!orem4--,e triangul> pyramid is equal 9 volume to "o ?ird ( a triangul> prism hav+ ! same base & ! same altitude z ! pyramid4 ,?is is an imm$iate consequ5ce ( ,euclid's ,prop4 #g4 ,e prism hav+ a triangul> base may be divid$ into ?ree pyramids t h triangul> bases1 & are equal to "o ano!r4 ,! pro( ( ?is !orem is difficult1 because ! ?ree triangul> pyramids into :i* ! prism is divid$ are by no means equal 9 %ape1 & _c be made to co9cide4 ,x has fir/ to be prov$ t two triangul> pyramids h equal volumes1 if !y h equal bases & equal altitudes4 ,?is ,euclid does 9 ! foll[+ mann]4 ,he fir/ %[s 7,prop4 #c7 t a triangul> pyramid may be divid$ into f\r p>ts1 ( :i* two are equal triangul> pyramids simil> to ! :ole pyramid1 :il/ ! o!r two are equal triangul> prisms1 & fur!r1 t ^! two prisms toge!r are great] ?an ! two pyramids1 h5ce m ?an half ! giv5 pyramid4 ,he next %[s 7,prop4 #d7 t if two triangul> pyramids are giv51 hav+ equal bases & equal altitudes1 & if ea* be divid$ z above1 !n ! two triangul> prisms 9 ! "o are equal to ^? 9 ! o!r1 & ea* ( ! rema9+ pyramids 9 ! "o has its base & altitude equal to ! base & altitude ( ! rema9+ pyramids 9 ! o!r4 ,h5ce to ^! pyramids ! same process is aga9 applicable4 ,we are ?us enabl$ to cut out ( ! two giv5 pyramids equal p>ts1 ea* great] ?an half ! orig9al pyramid4 ,( ! rema9d] we c aga9 cut out equal p>ts great] ?an half ^! rema9d]s1 & s on z f> z we l4 ,?is process may be cont9u$ till ! la/ rema9d] is small] ?an any assignable quantity1 h[ev] small4 ,x foll[s1 s we %\ld conclude at pres5t1 t ! two volumes mu/ be equal1 = !y _c diff] by any assignable quantity4 ,to ,greek ma!maticians ?is conclusion (f]s f> great] difficulties4 ,!y prove elaborately1 by a r$uctio ad absurdum1 t ! volumes _c be unequal4 ,?is pro( mu/ be read 9 ! ,elem5ts4 ,we mu/1 h[ev]1 /ate t we h 9 ! above n prov$ ,euclid's ,prop4 #e1 b only a special case ( x4 ,euclid does n suppose t ! bases ( ! two pyramids to be comp>$ are equal1 & h5ce he proves t ! volumes are z ! bases4 ,! reason+ ( ! pro( becomes cle>] 9 ! special case1 f :i* ! g5]al "o may be easily d$uc$4 ,s #hf4 ,prop4 #f ext5ds ! result to pyramids ) polygonal bases4 ,f ^! results foll[ aga9 ! rules at pres5t giv5 = ! m5suration ( solids1 viz4 a pyramid is ! ?ird "p ( a triangul> prism hav+ ! same base & ! same altitude4 ,b a triangul> prism is equal 9 volume to a p>allelepip$ :i* has ! same base & altitude4 ,h5ce if ,b is ! base & h ! altitude1 we h ,volume ( prism ,bh1 ,volume ( pyramid #a_/c;,bh1 /atem5ts :i* h to be tak5 9 ! s5se t ,b means ! numb] ( squ>e units 9 ! base1 h ! numb] ( units ( l5g? 9 ! altitude1 or t ,b & h d5ote ! num]ical values ( base & altitude4 ,s #hg4 ,a me?od simil> to t us$ 9 prov+ ,prop4 #e leads to ! foll[+ results relat+ to solids b\nd$ by simple curv$ surfaces3-- ,prop4 #aj4 ,e cone is ! ?ird "p ( a cyl9d] :i* has ! same base1 & is ( an equal altitude ) x4 ,prop4 #aa4 ,cones or cyl9d]s ( ! same altitude are to "o ano!r z _! bases4 ,prop4 #ab4 ,simil> cones or cyl9d]s h to "o ano!r ! triplicate ratio ( t :i* ! diamet]s ( _! bases h4 ,prop4 #ac4 ,if a cyl9d] be cut by a plane p>allel to its opposite planes or bases1 x divides ! cyl9d] into two cyl9d]s1 "o ( :i* is to ! o!r z ! axis ( ! fir/ to ! axis ( ! o!r2 :i* may also be /at$ ?us3-- ,cyl9d]s on ! same base are proportional to _! altitudes4 ,prop4 #ad4 ,cones or cyl9d]s ^u equal bases are to "o ano!r z _! altitudes4 ,prop4 #ae4 ,! bases & altitudes ( equal cones or cyl9d]s are reciprocally proportional1 & if ! bases & altitudes be reciprocally proportional1 ! cones or cyl9d]s are equal to "o ano!r4 ,^! !orems aga9 lead to =mulae 9 m5suration1 if we comp>e a cyl9d] ) a prism hav+ its base & altitude equal to ! base & altitude ( ! cyl9d]4 ,?is may be done by ! me?od ( exhau/ion4 ,we get1 !n1 ! result t _! bases are equal1 & h1 if ,b d5otes ! num]ical value ( ! base1 & h t ( ! altitude1 ,volume ( cyl9d] ,bh1 ,volume ( cone #a_/c;,bh4 ,s #hh4 ,! rema9+ propositions relate to circles & sph]es4 ,( ! sph]e only "o prop]ty is prov$1 viz43-- ,prop4 #ah4 ,sph]es h to "o ano!r ! triplicate ratio ( t :i* _! diamet]s h4 ,! m5suration ( ! sph]e1 l t ( ! circle1 ! cyl9d] & ! cone1 _h n be5 settl$ 9 ! "t ( ,euclid4 ,x 0 done by ,ar*im$es4 ,,book ,,xiii4 ,s #hi4 ,! #ac? & la/ book ( ,euclid's ,elem5ts is devot$ to ! regul> solids 7see ,,polyh$ron74 ,x is %[n t !re are five ( !m1 viz43-- #a4 ,! regul> tetrah$ron1 ) #d triangul> faces & #d v]tices2 #b4 ,! cube1 ) #h v]tices & #f squ>e faces2 #c4 ,! octah$ron1 ) #f v]tices & #h triangul> faces2 #d4 ,! dodecah$ron1 ) #ab p5tagonal faces1 #c at ea* ( ! #bj v]tices2 #e4 ,! icosah$ron1 ) #bj triangul> faces1 #e at ea* ( ! #ab v]tices4 ,x is %[n h[ to inscribe ^! solids 9 a giv5 sph]e1 & h[ to det]m9e ! l5g?s ( _! edges4 ,s #ij4 ,! #ac? book1 & !re=e ! ,elem5ts1 conclude ) ! s*olium1 t no o!r regul> solid exi/s besides ! five ones enum]at$4 ,! pro( is v simple4 ,ea* face is a regul> polygon1 h5ce ! angles ( ! faces at any v]tex mu/ be angles 9 equal regul> polygons1 mu/ be toge!r less ?an f\r "r angles 7,,xi4 #ba71 & mu/ be ?ree or m 9 numb]4 ,ea* angle 9 a regul> triangle equals two-?irds ( "o "r angle4 ,h5ce x is possible to =m a solid angle ) ?ree1 f\r or five regul> triangles or faces4 ,^! give ! solid angles ( ! tetrah$ron1 ! octah$ron & ! icosah$ron4 ,! angle 9 a squ>e 7! regul> quadrilat]al7 equals "o "r angle4 ,h5ce ?ree w =m a solid angle1 t ( ! cube1 & f\r w n4 ,! angle 9 ! regul> p5tagon equals #f_/e ( a "r angle4 ,h5ce ?ree ( !m equal #ah_/e 7i4e4 less ?an #d7 "r angles1 & =m ! solid angle ( ! dodecah$ron4 ,?ree regul> polygons ( six or m sides _c =m a solid angle4 ,!re=e no o!r regul> solids are possible4 7,o4 ,h47 ,,ii4 ,,projective ,,geometry ,x is difficult1 at ! outset1 to *>act]ize projective geometry z comp>$ ) ,euclidean4 ,b a few examples w at lea/ indicate ! practical diff]5ces betwe5 ! two4 ,9 ,euclid's ,elem5ts almo/ all propositions ref] to ! magnitude ( l9es1 angles1 areas or volumes1 & !re=e to measurem5t4 ,! /atem5t t an angle is "r1 or t two /raight l9es are p>allel1 ref]s to measurem5t4 ,on ! o!r h&1 ! fact t a /raight l9e does or does n cut a circle is indep5d5t ( measurem5t1 x be+ dep5d5t only ^u ! mutual position ( ! l9e & ! circle4 ,?is diff]5ce becomes cle>] if we project any figure f "o plane to ano!r 7see ,,projection74 ,by ?is ! l5g? ( l9es1 ! magnitude ( angles & areas1 is alt]$1 s t ! projection1 or %ad[1 ( a squ>e on a plane w n be a squ>e2 x w1 h[ev]1 be "s quadrilat]al4 ,aga91 ! projection ( a circle w n be a circle1 b "s o!r curve m or less resembl+ a circle4 ,b "o prop]ty may be /at$ at once--no /raight l9e c cut ! projection ( a circle 9 m ?an two po9ts1 because no /raight l9e c cut a circle 9 m ?an two po9ts4 ,!re are1 !n1 "s prop]ties ( figures :i* d n alt] by projection1 :il/ o!rs d4 ,to ! latt] belong ne>ly all prop]ties relat+ to measurem5t1 at lea/ 9 ! =m 9 :i* !y are g5]ally giv54 ,! o!rs are said to be projective prop]ties1 & _! inve/igation =ms ! subject ( projective geometry4 ,diff]5t z are ! k9ds ( prop]ties inve/igat$ 9 ! old & ! new sci5ces1 ! me?ods foll[$ diff] 9 a /ill great] degree4 ,9 ,euclid ea* proposition /&s by itself2 its connexion ) o!rs is nev] indicat$2 ! lead+ ideas conta9$ 9 its pro( are n /at$2 g5]al pr9ciples d n exi/4 ,9 ! mod]n me?ods1 on ! o!r h&1 ! greate/ importance is atta*$ to ! lead+ ?\ghts :i* p]vade ! :ole2 & g5]al pr9ciples1 :i* br+ :ole gr\ps ( !orems "u "o aspect1 are giv5 r ?an sep>ate propositions4 ,! :ole t5d5cy is t[>ds g5]alization4 ,a /raight l9e is consid]$ z giv5 9 its entirety1 ext5d+ bo? ways to inf9ity1 :ile ,euclid nev] admits any?+ b f9ite quantities4 ,! treatm5t ( ! inf9ite is 9 fact ano!r fundam5tal diff]5ce betwe5 ! two me?ods3 ,euclid avoids x2 9 mod]n geometry x is sy/ematically introduc$4 ,( ! diff]5t mod]n me?ods ( geometry1 we %all treat pr9cipally ( ! me?ods ( projection & correspond5ce :i* h prov$ to be ! mo/ p[]ful4 ,^! h become indep5d5t ( ,euclidean ,geometry1 especially ?r\gh ! ,geometrie d] ,lage ( ,v4 ,/audt & ! ,ausdehnungslehre ( ,grassmann4 ,= ! sake ( brevity we %all presuppose a k ( ,euclid's ,elem5ts1 al?\gh we %all use only a few ( 8 propositions4 ,s #a4 ,geometrical ,elem5ts4 ,we consid] space z fill$ ) po9ts1 l9es & planes1 & ^! we call ! elem5ts out ( :i* our figures are to be =m$1 call+ any comb9ation ( ^! elem5ts a figure4 ,by a l9e we mean a /raight l9e 9 its entirety1 ext5d+ bo? ways to inf9ity2 & by a plane1 a plane surface1 ext5d+ 9 all directions to inf9ity4 ,we accept ! ?ree-dim5sional space ( exp]i5ce--! space assum$ by ,euclid--:i* has = its prop]ties 7among o!rs73-- ,?r\gh any two po9ts 9 space "o & only "o l9e may be drawn2 ,?r\gh any ?ree po9ts :i* are n 9 a l9e1 "o & only "o plane may be plac$2 ,! int]section ( two planes is a l9e2 ,a l9e :i* has two po9ts 9 common ) a plane lies 9 ! plane1 h5ce ! int]section ( a l9e & a plane is a s+le po9t2 & ,?ree planes :i* d n meet 9 a l9e h "o s+le po9t 9 common4 ,^! results may be /at$ diff]5tly 9 ! foll[+ =m3-- ,i4 ,a plane is det]m9$-- ,a po9t is det]m9$-- #a4 ,by ?ree po9ts :i* d #a4 ,by ?ree planes :i* d n lie 9 a l9e2 n pass ?r\gh a l9e2 #b4 ,by two int]sect+ l9es2 #b4 ,by two int]sect+ l9es2 #c4 ,by a l9e & a po9t #c4 ,by a plane & a l9e :i* does n lie 9 x4 :i* does n lie 9 x4 ,,ii4 ,a l9e is det]m9$-- #a4 ,by two po9ts2 #b4 ,by two planes4 ,x w be obs]v$ t n only are planes det]m9$ by po9ts1 b also po9ts by planes2 t !re=e ! planes may be consid]$ z elem5ts1 l po9ts2 & also t 9 any "o ( ! above /atem5ts we may int]*ange ! words po9t & plane1 & we obta9 aga9 a correct /atem5t1 provid$ t ^! /atem5ts !mselves are true4 ,z !y /&1 we ought1 9 sev]al cases1 to add if !y are n p>allel1 or "s su* words1 p>allel l9es & planes be+ evid5tly left altoge!r out ( consid]ation4 ,to correct ?is we h to reconsid] ! !ory ( p>allels4 ,illu/ration3 ,,fig4 #a4 ,s #b4 ,p>allels4 ,po9t at ,inf9ity4--,let u take 9 a plane a l9e p 7fig4 #a71 a po9t ,s n 9 ?is l9e1 & a l9e q drawn ?r\gh ,s4 ,!n ?is l9e q w meet ! l9e p 9 a po9t ,a4 ,if we turn ! l9e q ab\t ,s t[>ds q'1 its po9t ( int]section ) p w move along p t[>ds ,b1 pass+1 on cont9u$ turn+1 to a great] & great] di/ance1 until x is mov$ out ( our rea*4 ,if we turn q /ill f>!r1 its cont9uation w meet p1 b n[ at ! o!r side ( ,a4 ,! po9t ( int]section has disappe>$ to ! "r & reappe>$ to ! left4 ,!re is "o int]m$iate position :]e q is p>allel to p--t is :]e x does n cut p4 ,9 e o!r position x cuts p 9 "s f9ite po9t4 ,if1 on ! o!r h&1 we move ! po9t ,a to an inf9ite di/ance 9 p1 !n ! l9e q :i* passes ?r\gh ,a w be a l9e :i* does n cut p at any f9ite po9t4 ,?us we are l$ to say3 ,e l9e ?r\gh ,s :i* jo9s x to any po9t at an inf9ite di/ance 9 p is p>allel to p4 ,b by ,euclid's #ab? axiom !re is b "o l9e p>allel to p ?r\gh ,s4 ,! difficulty 9 :i* we are ?us involv$ is due to ! fact t we try to reason ab\t inf9ity z if we1 ) our f9ite capabilities1 c\ld compreh5d ! inf9ite4 ,to ov]come ?is difficulty1 we may say t all po9ts at inf9ity 9 a l9e appe> to u z "o1 & may be replac$ by a s+le ideal po9t4 ,we may !re=e n[ give ! foll[+ def9itions & axiom3-- ,def9ition4--,l9es :i* meet at inf9ity are call$ p>allel4 ,axiom4--,all po9ts at an inf9ite di/ance 9 a l9e may be consid]$ z "o s+le po9t4 ,def9ition4--,?is ideal po9t is call$ ! po9t at inf9ity 9 ! l9e4 ,! axiom is equival5t to ,euclid's ,axiom #ab1 = x foll[s f ei!r t ?r\gh any po9t only "o l9e may be drawn p>allel to a giv5 l9e4 ,?is po9t at inf9ity 9 a l9e is rea*$ :e!r we move a po9t 9 ! "o or 9 ! opposite direction ( a l9e to inf9ity4 ,a l9e ?us appe>s clos$ by ?is po9t1 & we speak z if we c\ld move a po9t along ! l9e f "o position ,a to ano!r ,b 9 two ways1 ei!r ?r\gh ! po9t at inf9ity or ?r\gh f9ite po9ts only4 ,x mu/ nev] be =gott5 t ?is po9t at inf9ity is ideal2 9 fact1 ! :ole notion ( inf9ity is only a ma!matical conception1 & owes its introduction 7z a me?od ( rese>*7 to ! work+ g5]alizations :i* x p]mits4 ,s #c4 ,l9e & ,plane at ,inf9ity4--,hav+ arriv$ at ! notion ( replac+ all po9ts at inf9ity 9 a l9e by "o ideal po9t1 !re is no difficulty 9 replac+ all po9ts at inf9ity 9 a plane by "o ideal l9e4 ,to make ?is cle>1 let u suppose t a l9e p1 :i* cuts two fix$ l9es a & b 9 ! po9ts ,a & ,b1 moves p>allel to itself to a great] & great] di/ance4 ,x w at la/ cut bo? a & b at _! po9ts at inf9ity1 s t a l9e :i* jo9s ! two po9ts at inf9ity 9 two int]sect+ l9es lies altoge!r at inf9ity4 ,e o!r l9e 9 ! plane w meet x !re=e at inf9ity1 & ?us x conta9s all po9ts at inf9ity 9 ! plane4 ,all po9ts at inf9ity 9 a plane lie 9 a l9e1 :i* is call$ ! l9e at inf9ity 9 ! plane4 ,x foll[s t p>allel planes mu/ be consid]$ z planes hav+ a common l9e at inf9ity1 = any o!r plane cuts !m 9 p>allel l9es :i* h a po9t at inf9ity 9 common4 ,if we next take two int]sect+ planes1 !n ! po9t at inf9ity 9 _! l9e ( int]section lies 9 bo? planes1 s t _! l9es at inf9ity meet4 ,h5ce e l9e at inf9ity meets e o!r l9e at inf9ity1 & !y are !re=e all 9 "o plane4 ,all po9ts at inf9ity 9 space may be consid]$ z ly+ 9 "o ideal plane1 :i* is call$ ! plane at inf9ity4 ,s #d4 ,p>allelism4--,we h n[ ! foll[+ def9itions3-- ,p>allel l9es are l9es :i* meet at inf9ity2 ,p>allel planes are planes :i* meet at inf9ity2 ,a l9e is p>allel to a plane if x meets x at inf9ity4 ,!orems l ?is--,l9es 7or planes7 :i* are p>allel to a ?ird are p>allel to ea* o!r--foll[ at once4 ,?is view ( p>allels leads !re=e to no contradiction ( ,euclid's ,elem5ts4 ,z imm$iate consequ5ces we get ! propositions3-- ,e l9e meets a plane 9 "o po9t1 or x lies 9 x2 ,e plane meets e o!r plane 9 a l9e2 ,any two l9es 9 ! same plane meet4 ,s #e4 ,aggregates ( ,geometrical ,elem5ts4--,we h call$ po9ts1 l9es & planes ! elem5ts ( geometrical figures4 ,we also say t an elem5t ( "o k9d conta9s "o ( ! o!r if x lies 9 x or passes ?r\gh x4 ,all ! elem5ts ( "o k9d :i* are conta9$ 9 "o or two elem5ts ( a diff]5t k9d =m aggregates :i* h to be enum]at$4 ,!y are ! foll[+3-- ,i4 ,( "o dim5sion4 #a4 ,! r[1 or range1 ( po9ts =m$ by all po9ts 9 a l9e1 :i* is call$ its base4 #b4 ,! flat p5cil =m$ by all ! l9es ?r\gh a po9t 9 a plane4 ,its base is ! po9t 9 ! plane4 #c4 ,! axial p5cil =m$ by all planes ?r\gh a l9e :i* is call$ its base or axis4 ,,ii4 ,( two dim5sions4 #a4 ,! field ( po9ts & l9es--t is1 a plane ) all its po9ts & all its l9es4 #b4 ,! p5cil ( l9es & planes--t is1 a po9t 9 space ) all l9es & all planes ?r\gh x4 ,,iii4 ,( ?ree dim5sions4 ,! space ( po9ts--t is1 all po9ts 9 space4 ,! space ( planes--t is1 all planes 9 space4 ,,iv4 ,( f\r dim5sions4 ,! space ( l9es1 or all l9es 9 space4 ,s #f4 ,mean+ ( ,dim5sions4--,! ^w dim5sion 9 ! above ne$s explanation4 ,if 9 a plane we take a r[ p & a p5cil ) c5tre ,q1 !n ?r\gh e po9t 9 p "o l9e 9 ! p5cil w pass1 & e ray 9 ,q w cut p 9 "o po9t1 s t we are entitl$ to say a r[ conta9s z _m po9ts z a flat p5cil l9es1 &1 we may add1 z an axial p5cil planes1 because an axial p5cil is cut by a plane 9 a flat p5cil4 ,! numb] ( elem5ts 9 ! r[1 9 ! flat p5cil1 & 9 ! axial p5cil is1 ( c\rse1 inf9ite & indef9ite too1 b ! same 9 all4 ,?is numb] may be d5ot$ by inf9ity4 ,!n a plane conta9s inf9ity#b po9ts & z _m l9es4 ,to see ?is1 take a flat p5cil 9 a plane4 ,x conta9s inf9ity l9es1 & ea* l9e conta9s inf9ity po9ts1 :il/ ea* po9t 9 ! plane lies on "o ( ^! l9es4 ,simil>ly1 9 a plane ea* l9e cuts a fix$ l9e 9 a po9t4 ,b ?is l9e is cut at ea* po9t by inf9ity l9es & conta9s inf9ity po9ts2 h5ce !re are inf9ity#b l9es 9 a plane4 ,a p5cil 9 space conta9s z _m l9es z a plane conta9s po9ts & z _m planes z a plane conta9s l9es1 = any plane cuts ! p5cil 9 a field ( po9ts & l9es4 ,h5ce a p5cil conta9s inf9ity#b l9es & inf9ity#b planes4 ,! field & ! p5cil are ( two dim5sions4 ,to c\nt ! numb] ( po9ts 9 space we obs]ve t ea* po9t lies on "s l9e 9 a p5cil4 ,b ! p5cil conta9s inf9ity#b l9es1 & ea* l9e inf9ity po9ts2 h5ce space conta9s inf9ity#c po9ts4 ,ea* plane cuts any fix$ plane 9 a l9e4 ,b a plane conta9s inf9ity#b l9es1 & ?r\gh ea* pass inf9ity planes2 !re=e space conta9s inf9ity#c planes4 ,h5ce space conta9s z _m planes z po9ts1 b x conta9s an inf9ite numb] ( times m l9es ?an po9ts or planes4 ,to c\nt !m1 notice t e l9e cuts a fix$ plane 9 "o po9t4 ,b inf9ity#b l9es pass ?r\gh ea* po9t1 & !re are inf9ity#b po9ts 9 ! plane4 ,h5ce !re are inf9ity#d l9es 9 space4 ,! space ( po9ts & planes is ( ?ree dim5sions1 b ! space ( l9es is ( f\r dim5sions4 ,a field ( po9ts or l9es conta9s an inf9ite numb] ( r[s & flat p5cils2 a p5cil conta9s an inf9ite numb] ( flat p5cils & ( axial p5cils2 space conta9s a triple inf9ite numb] ( p5cils & ( fields1 inf9ity#d r[s & axial p5cils & inf9ity#e flat p5cils--or1 9 o!r words1 ea* po9t is a c5tre ( inf9ity#b flat p5cils4 ,s #g4 ,! above enum]ation all[s a classification ( figures4 ,figures 9 a r[ consi/ ( gr\ps ( po9ts only1 & figures 9 ! flat or axial p5cil consi/ ( gr\ps ( l9es or planes4 ,9 ! plane we may draw polygons2 & 9 ! p5cil or 9 ! po9t1 solid angles1 & s on4 ,we may also di/+ui% ! diff]5t measurem5ts ,we h-- ,9 ! r[1 l5g? ( segm5t2 ,9 ! flat p5cil1 angles2 ,9 ! axial p5cil1 dih$ral angles betwe5 two planes2 ,9 ! plane1 areas2 ,9 ! p5cil1 solid angles2 ,9 ! space ( po9ts or planes1 volumes4 ,,segm5ts ,,( ,a ,,l9e ,s #h4 ,any two po9ts ,a & ,b 9 space det]m9e on ! l9e ?r\gh !m a f9ite "p1 :i* may be consid]$ z be+ describ$ by a po9t mov+ f ,a to ,b4 ,?is we %all d5ote by ,,ab1 & di/+ui% x f ,,ba1 :i* is suppos$ z be+ describ$ by a po9t mov+ f ,b to ,a1 & h5ce 9 a direction or 9 a s5se opposite to ,,ab4 ,su* a f9ite l9e1 :i* has a def9ite s5se1 we %all call a segm5t1 s t ,,ab & ,,ba d5ote diff]5t segm5ts1 :i* are said to be equal 9 l5g? b ( opposite s5se4 ,! "o s5se is (t5 call$ positive & ! o!r negative4 ,9 introduc+ ! ^w s5se = direction 9 a l9e1 we h ! ^w direction res]v$ = direction ( ! l9e itself1 s t diff]5t l9es h diff]5t directions1 unless !y be p>allel1 :il/ 9 ea* l9e we h a positive & negative s5se4 ,we may also say1 ) ,clif=d1 t ,,ab d5otes ! /ep ( go+ f ,a to ,b4 ,illu/ration3 ,,fig4 #b4 ,s #i4 ,if we h ?ree po9ts ,a1 ,b1 ,c 9 a l9e 7fig4 #b71 ! /ep ,,ab w br+ u f ,a to ,b1 & ! /ep ,,bc f ,b to ,c4 ,h5ce bo? /eps are equival5t to ! "o /ep ,,ac4 ,?is is express$ by say+ t ,,ac is ! sum ( ,,ab & ,,bc2 9 symbols-- ,,ab ,,bc ,,ac1 :]e acc\nt is to be tak5 ( ! s5se4 ,?is equation is true :atev] be ! position ( ! ?ree po9ts on ! l9e4 ,z a special case we h ,,ab ,,ba #j1 7#a7 & simil>ly ,,ab ,,bc ,,ca #j1 7#b7 :i* aga9 is true = any ?ree po9ts 9 a l9e4 ,we fur!r write ,,ab -,,ba4 :]e - d5otes negative s5se4 ,we c !n1 j z 9 algebra1 *ange subtraction ( segm5ts into addition by *ang+ ! s5se1 s t ,,ab - ,,cb is ! same z ,,ab 7-,,cb7 or ,,ab ,,bc4 ,a figure w at once %[ ! tru? ( ?is4 ,! s5se is1 9 fact1 9 e respect equival5t to ! sign ( a numb] 9 algebra4 ,s #aj4 ,( ! _m =mulae :i* exi/ betwe5 po9ts 9 a l9e we %all h to use only "o m1 :i* connects ! segm5ts betwe5 any f\r po9ts ,a1 ,b1 ,c1 ,d 9 a l9e4 ,we h ,,bc ,,bd ,,dc1 ,,ca ,,cd ,,da1 ,,ab ,,ad ,,db2 or multiply+ ^! by ,,ad1 ,,bd1 ,,cd respectively1 we get ,,bc4,,ad ,,bd4,,ad ,,dc4,,ad ,,bd4,,ad - ,,cd4,,ad ,,ca4,,bd ,,cd4,,bd ,,da4,,bd ,,cd4,,bd - ,,ad4,,bd ,,ab4,,cd ,,ad4,,cd ,,db4,,cd ,,ad4,,cd - ,,bd4,,cd4 ,x w be se5 t ! sum ( ! "r-h& sides vani%es1 h5ce t ,,bc4,,ad ,,ca4,,bd ,,ab4,,cd #j 7#c7 = any f\r po9ts on a l9e4 ,illu/ration3 ,,fig4 #c4 ,s #aa4 ,if ,c is any po9t 9 ! l9e ,,ab1 !n we say t ,c divides ! segm5t ,,ab 9 ! ratio ,,ac_/,,cb1 acc\nt be+ tak5 ( ! s5se ( ! two segm5ts ,,ac & ,,cb4 ,if ,c lies betwe5 ,a & ,b ! ratio is positive1 z ,,ac & ,,cb h ! same s5se4 ,b if ,c lies )\t ! segm5t ,,ab1 i4e4 if ,c divides ,,ab ext]nally1 !n ! ratio is negative4 ,to see h[ ! value ( ?is ratio *anges ) ,c1 we w move ,c along ! :ole l9e 7fig4 #c71 :il/ ,a & ,b rema9 fix$4 ,if ,c lies at ! po9t ,a1 !n ,,ac #j1 h5ce ! ratio ,,ac 3 ,,cb vani%es4 ,z ,c moves t[>ds ,b1 ,,ac increases & ,,cb decreases1 s t our ratio increases4 ,at ! middle po9t ,m ( ,,ab x assumes ! value #a1 & !n increases till x rea*es an inf9itely l>ge value1 :5 ,c arrives at ,b4 ,on pass+ beyond ,b ! ratio becomes negative4 ,if ,c is at ,p we h ,,ac ,,ap ,,ab ,,bp1 h5ce ,,ac ,,ab ,,bp ,,ab -- -- -- - -- - #a4 ,,cb ,,pb ,,pb ,,bp ,9 ! la/ expression ! ratio ,,ab 3 ,,bp is positive1 has its greate/ value inf9ity :5 ,c co9cides ) ,b1 & vani%es :5 ,,bc becomes inf9ite4 ,h5ce1 z ,c moves f ,b to ! "r to ! po9t at inf9ity1 ! ratio ,,ac 3 ,,cb v>ies f -inf9ity to -#a4 ,if1 on ! o!r h&1 ,c is to ! left ( ,a1 say at ,q1 we h ,,ac ,,aq ,,ab ,,bq ,,ab - ,,qb1 h5ce ,,ac_/,,cb ,,ab_/,,qb - #a4 ,"h ,,ab ,,qb1 h5ce ! ratio ,,ab 3 ,,qb is positive & always less ?an "o1 s t ! :ole is negative & #a4 ,if ,c is at ! po9t at inf9ity x is -#a1 & !n increases z ,c moves to ! "r1 till = ,c at ,a we get ! ratio #j4 ,h5ce-- ,z ,c moves along ! l9e f an inf9ite di/ance to ! left to an inf9ite di/ance at ! "r1 ! ratio always increases2 x />ts ) ! value -#a1 rea*es #j at ,a1 #a at ,m1 inf9ity at ,b1 n[ *anges sign to -inf9ity1 & increases till at an inf9ite di/ance x rea*es aga9 ! value -#a4 ,x assumes !re=e all possible values f -inf9ity to inf9ity1 & ea* value only once1 s t n only does e position ( ,c det]m9e a def9ite value ( ! ratio ,,ac 3 ,,cb1 b also1 conv]sely1 to e positive or negative value ( ?is ratio belongs "o s+le po9t 9 ! l9e ,,ab4 ,relations betwe5 segm5ts ( l9es are int]e/+ z %[+ an application ( algebra to geometry4 ,! g5esis ( su* relations f algebraic id5tities is v simple4 ,= example1 if a1 b1 c1 x be any f\r quantities1 !n a b --------------------- --------------------- 7a - b77a - c77x - a7 7b - c77b - a77x - b7 c x --------------------- ---------------------2 7c - a77c - b77x - c7 7x - a77x - b77x - c7 ?is may be prov$1 cumbr\sly1 by multiply+ up1 or1 simply1 by decompos+ ! "r-h& memb] ( ! id5tity into p>tial fractions4 ,n[ take a l9e ,,abcdx1 & let ,,ab a1 ,,ac b1 ,,ad c1 ,,ax x4 ,!n obvi\sly 7a - b7 ,,ab - ,,ac -,,bc1 pay+ reg>d to signs2 7a - c7 ,,ab - ,,ad ,,db1 & s on4 ,sub/itut+ ^! values 9 ! id5tity we obta9 ! foll[+ relation connect+ ! segm5ts =m$ by five po9ts on a l9e3-- ,,ab ,,ac ,,ad ,,ax -------- -------- -------- --------4 ,,bc4,,bd4,,bx ,,cd4,,cb4,,cx ,,db4,,dc4,,dx ,,bx4,,cx4,,dx ,conv]sely1 if a metrical relation be giv51 its validity may be te/$ by r$uc+ to an algebraic equation1 :i* is an id5tity if ! relation be true4 ,= example1 if ,,abcdx be five coll9e> po9ts1 prove ,,ad4,,ax ,,bd4,,bx ,,cd4,,cx ----- ----- ----- #a4 ,,ab4,,ac ,,bc4,,ba ,,ca4,,cb ,cle>+ ( fractions by multiply+ ?r\gh\t by ,,ab4,,bc4,,ca1 we h to prove -,,ad4,,ax4,,bc - ,,bd4,,bx4,,ca - ,,cd4,,cx4,,ab ,,ab4,,bc4,,ca4 ,take ,a z orig9 & let ,,ab a1 ,,ac b1 ,,ad c1 ,,ax x4 ,sub/itut+ = ! segm5ts 9 t]ms ( a1 b1 c1 x1 we obta9 on simplification a#b;b - ab#b -ab#b a#b;b1 an obvi\s id5tity4 ,an alt]native me?od ( te/+ a relation is illu/rat$ 9 ! foll[+ example3-- ,if ,a1 ,b1 ,c1 ,d1 ,e1 ,f be six coll9e> po9ts1 !n ,,ae4,,af ,,be4,,bf ,,ce4,,cf ,,de4,,df -------- -------- -------- -------- #j4 ,,ab4,,ac4,,ad ,,bc4,,bd4,,ba ,,cd4,,ca4,,cb ,,da4,,db4,,dc ,cle>+ ( fractions by multiply+ ?r\gh\t by ,,ab4,,bc4,,cd4,,da1 & r$uc+ to a common orig9 ,o 7call+ ,,oa a1 ,,ob b1 c471 an equation conta9+ ! second & l[] p[]s ( ,,oa 7 a71 c41 is obta9$4 ,call+ ,,oa x1 x is f\nd t x b1 x c1 x d are solutions4 ,h5ce ! quadratic has ?ree roots2 consequ5tly x is an id5tity4 ,! relations connect+ five po9ts :i* we h in/anc$ above may be readily d$uc$ f ! six-po9t relation2 ! fir/ by tak+ ,d at inf9ity1 & ! second by tak+ ,f at inf9ity1 & !n mak+ ! obvi\s p]mutations ( ! po9ts4 ,,projection ,,& ,,cross-,,ratios ,s #ab4 ,if we jo9 a po9t ,a to a po9t ,s1 !n ! po9t :]e ! l9e ,,sa cuts a fix$ plane pi is call$ ! projection ( ,a on ! plane pi f ,s z c5tre ( projection4 ,if we h two planes pi & pi' & a po9t ,s1 we may project e po9t ,a 9 pi to ! o!r plane4 ,if ,a' is ! projection ( ,a1 !n ,a is also ! projection ( ,a'1 s t ! relations are reciprocal4 ,to e figure 9 pi we get z its projection a correspond+ figure 9 pi'4 ,we %all det]m9e su* prop]ties ( figures z rema9 true = ! projection1 & :i* are call$ projective prop]ties4 ,= ?is purpose x w be suffici5t to consid] at fir/ only con/ructions 9 "o plane4 ,illu/ration3 ,,fig4 #d4 ,illu/ration3 ,,fig4 #e4 ,let u suppose we h giv5 9 a plane two l9es p & p' & a c5tre ,s 7fig4 #d72 we may !n project ! po9ts 9 p f ,s to p'4 ,let ,a'1 ,b' 444 be ! projections ( ,a1 ,b 4441 ! po9t at inf9ity 9 p :i* we %all d5ote by ,i w be project$ into a f9ite po9t ,i' 9 p'1 viz4 into ! po9t :]e ! p>allel to p ?r\gh ,s cuts p'4 ,simil>ly "o po9t ,j 9 p w be project$ into ! po9t ,j' at inf9ity 9 p'4 ,?is po9t ,j is ( c\rse ! po9t :]e ! p>allel to p' ?r\gh ,s cuts p4 ,we ?us see t e po9t 9 p is project$ into a s+le po9t 9 p'4 ,fig4 #e %[s t a segm5t ,,ab w be project$ into a segm5t ,a',b' :i* is n equal to x1 at lea/ n z a rule2 & also t ! ratio ,,ac 3 ,,cb is n equal to ! ratio ,a',c' 3 ,c',b' =m$ by ! projections4 ,^! ratios w become equal only if p & p' are p>allel1 = 9 ?is case ! triangle ,,sab is simil> to ! triangle ,,sa',b'4 ,betwe5 ?ree po9ts 9 a l9e & _! projections !re exi/s !re=e 9 g5]al no relation4 ,b betwe5 f\r po9ts a relation does exi/4 ,s #ac4 ,let ,a1 ,b1 ,c1 ,d be f\r po9ts 9 p1 ,a'1 ,b'1 ,c1 ,d' _! projections 9 p'1 !n ! ratio ( ! two ratios ,,ac 3 ,,cb & ,,ad 3 ,,db into :i* ,c & ,d divide ! segm5t ,,ab is equal to ! correspond+ expression betwe5 ,a'1 ,b'1 ,c'1 ,d'4 ,9 symbols we h ,,ac ,,ad ,a',c' ,a',d' -- 3 -- ---- 3 ----4 ,,cb ,,db ,c',b' ,d',b' ,?is is easily prov$ by aid ( simil> triangles4 ,illu/ration3 ,,fig4 #f4 ,?r\gh ! po9ts ,a & ,b on p draw p>allels to p'1 :i* cut ! project+ rays 9 ,c#b1 ,d#b1 ,b#b & ,a#a1 ,c#a1 ,d#a1 z indicat$ 9 fig4 #f4 ,! two triangles ,,acc#b & ,,bcc#a w be simil>1 z w also be ! triangles ,,add#b & ,,bdd#a4 ,! pro( is left to ! read]4 ,?is result is ( fundam5tal importance4 ,! expression ,,ac_/,,cb 3 ,,ad_/,,db has be5 call$ by ,*asles ! anh>monic ratio ( ! f\r po9ts ,a1 ,b1 ,c1 ,d4 ,pr(essor ,clif=d propos$ ! %ort] "n ( cross-ratio4 ,we %all adopt ! latt]4 ,we h !n ! ,,fundam5tal ,,!orem4--,! cross-ratio ( f\r po9ts 9 a l9e is equal to ! cross-ratio ( _! projections on any o!r l9e :i* lies 9 ! same plane ) x4 ,s #ad4 ,be=e we draw conclusions f ?is result1 we mu/ inve/igate ! mean+ ( a cross-ratio some:at m fully4 ,if f\r po9ts ,a1 ,b1 ,c1 ,d are giv51 & we wi% to =m _! cross-ratio1 we h fir/ to divide !m into two gr\ps ( two1 ! po9ts 9 ea* gr\p be+ tak5 9 a def9ite ord]4 ,?us1 let ,a1 ,b be ! fir/1 ,c1 ,d ! second pair1 ,a & ,c be+ ! fir/ po9ts 9 ea* pair4 ,! cross-ratio is !n ! ratio ,,ac 3 ,,cb divid$ by ,,ad 3 ,,db4 ,?is w be d5ot$ by 7,,ab1 ,,cd71 s t ,,ac ,,ad 7,,ab1 ,,cd7 -- 3 --4 ,,cb ,,db ,?is is easily rememb]$4 ,9 ord] to write x out1 make fir/ ! two l9es = ! fractions1 & put above & bel[ ^! ! lett]s ,a & ,b 9 _! places1 ?us1 ,a99_/,b 3 ,a99_/,b2 & !n fill up1 crosswise1 ! fir/ by ,c & ! o!r by ,d4 ,s #ae4 ,if we take ! po9ts 9 a diff]5t ord]1 ! value ( ! cross-ratio w *ange4 ,we c d ?is 9 tw5ty-f\r diff]5t ways by =m+ all p]mutations ( ! lett]s4 ,b ( ^! tw5ty-f\r cross-ratios gr\ps ( f\r are equal1 s t !re are really only six diff]5t ones1 & ^! six are reciprocals 9 pairs4 ,we h ! foll[+ rules3-- ,i4 ,if 9 a cross-ratio ! two gr\ps be int]*ang$1 its value rema9s unalt]$1 i4e4 7,,ab1 ,,cd7 7,,cd1 ,,ab7 7,,ba1 ,,dc7 7,,dc1 ,,ba74 ,,ii4 ,if 9 a cross-ratio ! two po9ts belong+ to "o ( ! two gr\ps be int]*ang$1 ! cross-ratio *anges into its reciprocal1 i4e4 7,,ab1 ,,cd7 #a_/7;,,ab1 ,,dc7 #a_/7;,,ba1 ,,cd7 #a_/7;,,cd1 ,,ba7 #a_/7;,,dc1 ,,ab74 ,f ,i4 & ,,ii4 we see t eight cross-ratios are associat$ ) 7,,ab1 ,,cd74 ,,iii4 ,if 9 a cross-ratio ! two middle lett]s be int]*ang$1 ! cross-ratio alpha *anges into its complem5t #a - alpha1 i4e4 7,,ab1 ,,cd7 #a - 7,,ac1 ,,bd74 ,s #af4 ,if lambda 7,,ab1 ,,cd71 mu 7,,ac1 ,,db71 nu 7,,ad1 ,,bc71 !n lambda1 mu1 nu & _! reciprocals #a_/lambda1 #a_/mu1 #a_/nu are ! values ( ! total numb] ( tw5ty-f\r cross-ratios4 ,moreov]1 lambda1 mu1 nu are connect$ by ! relations lambda #a_/mu mu #a_/nu nu #a_/lambda -lambdamunu #a2 ?is proposition may be prov$ by sub/itut+ = lambda1 mu1 nu & r$uc+ to a common orig94 ,!re are !re=e f\r equations betwe5 ?ree unkn[ns2 h5ce if "o cross-ratio be giv51 ! rema9+ tw5ty-?ree are det]m9ate4 ,moreov]1 two ( ! quantities lambda1 mu1 nu are positive1 & ! rema9+ "o negative4 ,! foll[+ s*eme %[s ! tw5ty-f\r cross-ratios express$ 9 t]ms ( lambda1 mu1 nu4 ---------------------------------------- ---------------------- 7,,ab1 ,,cd7 7,,ba1 ,,dc7 lambda #a - mu #a_/7a - nu7 7,,cd1 ,,ab7 7,,dc1 ,,ba7 ---------------------------------------- ---------------------- 7,,ac1 ,,db7 7,,bd1 ,,ca7 #a_/7a - lambda7 #a_/mu 7nu - #a7_/nu 7,,ca1 ,,bd7 7,,db1 ,,ac7 ---------------------------------------- ---------------------- 7,,ab1 ,,dc7 7,,ba1 ,,cd7 #a_/lambda #a_/7a - mu7 #a - nu 7,,cd1 ,,ba7 7,,dc1 ,,ab7 ---------------------------------------- ---------------------- 7,,ad1 ,,bc7 7,,bc1 ,,ad7 7lambda - #a7_/lambdamu_/7mu - #a7 nu 7,,cb1 ,,da7 7,,da1 ,,cb7 ---------------------------------------- ---------------------- 7,,ac1 ,,bd7 7,,bd1 ,,ac7 #a - lambda mu nu_/7nu - #a7 7,,ca1 ,,db7 7,,db1 ,,ca7 ---------------------------------------- ---------------------- 7,,ad1 ,,cb7 7,,bc1 ,,da7 lambda_/7lambda - #a77mu - #a7_/mu #a_/nu 7,,cb1 ,,ad7 7,,da1 ,,bc7 ---------------------------------------- ---------------------- ,s #ag4 ,if "o ( ! po9ts ( :i* a cross-ratio is =m$ is ! po9t at inf9ity 9 ! l9e1 ! cross-ratio *anges into a simple ratio4 ,x is conv5i5t to let ! po9t at inf9ity occupy ! la/ place 9 ! symbolic expression = ! cross-ratio4 ,?us if ,i is a po9t at inf9ity1 we h 7,,ab1 ,,ci7 -,,ac_/,,cb1 because ,,ai 3 ,,ib -#a4 ,e common ratio ( ?ree po9ts 9 a l9e may ?us be express$ z a cross-ratio1 by add+ ! po9t at inf9ity to ! gr\p ( po9ts4 ,,h>monic ,,ranges ,s #ah4 ,if ! po9ts h special positions1 ! cross-ratios may h su* a value t1 ( ! six diff]5t ones1 two & two become equal4 ,if ! fir/ two %all be equal1 we get lambda #a_/lambda1 or lambda#b #a1 lambda -#a4 ,if we take lambda #a1 we h 7,,ab1 ,,cd7 #a1 or ,,ac_/,,cb ,,ad_/,,db2 t is1 ! po9ts ,c & ,d co9cide1 provid$ t ,a & ,b are diff]5t4 ,if we take lambda -#a1 s t 7,,ab1 ,,cd7 -#a1 we h ,,ac_/,,cb -,,ad_/,,db4 ,h5ce ,c & ,d divide ,,ab int]nally & ext]nally 9 ! same ratio4 ,! f\r po9ts are 9 ?is case said to be h>monic po9ts1 & ,c & ,d are said to be h>monic conjugates ) reg>d to ,a & ,b4 ,b we h also 7,,cd1 ,,ab7 -#a1 s t ,a & ,b are h>monic conjugates ) reg>d to ,c & ,d4 ,! pr9cipal prop]ty ( h>monic po9ts is t _! cross-ratio rema9s unalt]$ if we int]*ange ! two po9ts belong+ to "o pair1 viz4 7,,ab1 ,,cd7 7,,ab1 ,,dc7 7,,ba1 ,,cd74 ,= f\r h>monic po9ts ! six cross-ratios become equal two & two3 lambda lambda -#a1 #a - lambda #b1 ------------ #a_/b1 lambda - #a #a #a lambda - #a -------- -#a1 ------------ #a_/b1 ------------ #b4 lambda #a - lambda lambda ,h5ce if we get f\r po9ts ^: cross-ratio is #b or #a_/b1 !n !y are h>monic1 b n arrang$ s t conjugates are pair$4 ,if ?is is ! case ! cross-ratio -#a4 ,s #ai4 ,if we equate any two ( ! above six values ( ! cross-ratios1 we get ei!r lambda #a1 #j1 inf9ity1 or lambda -#a1 #b1 #a_/b1 or else lambda becomes a root ( ! equation lambda#b - lambda #a #j1 t is1 an imag9>y cube root ( -#a4 ,9 ?is case ! six values become ?ree & ?ree equal1 s t only two diff]5t values rema94 ,?is case1 ?\gh important 9 ! !ory ( cubic curves1 is = our purposes ( no int]e/1 :il/ h>monic po9ts are all-important4 ,s #bj4 ,f ! def9ition ( h>monic po9ts1 & by aid ( ,s #aa1 ! foll[+ prop]ties are easily d$uc$4 ,if ,c & ,d are h>monic conjugates ) reg>d to ,a & ,b1 !n "o ( !m lies 91 ! o!r )\t ,,ab2 x is impossible to move f ,a to ,b )\t pass+ ei!r ?r\gh ,c or ?r\gh ,d2 ! "o blocks ! f9ite way1 ! o!r ! way ?r\gh inf9ity4 ,?is is express$ by say+ ,a & ,b are sep>at$ by ,c & ,d4 ,= e position ( ,c !re w be "o & only "o po9t ,d :i* is its h>monic conjugate ) reg>d to any po9t pair ,a1 ,b4 ,if ,a & ,b are diff]5t po9ts1 & if ,c co9cides ) ,a or ,b1 ,d does4 ,b if ,a & ,b co9cide1 "o ( ! po9ts ,c or ,d1 ly+ betwe5 !m1 co9cides ) !m1 & ! o!r may be any:]e 9 ! l9e4 ,x foll[s t1 if ( f\r h>monic conjugates two co9cide1 !n a ?ird co9cides ) !m1 & ! f\r? may be any po9t 9 ! l9e4 ,if ,c is ! middle po9t betwe5 ,a & ,b1 !n ,d is ! po9t at inf9ity2 = ,,ac 3 ,,cb #a1 h5ce ,,ad 3 ,,db mu/ be equal to -#a4 ,! h>monic conjugate ( ! po9t at inf9ity 9 a l9e ) reg>d to two po9ts ,a1 ,b is ! middle po9t ( ,,ab4 ,?is important prop]ty gives a fir/ example h[ metric prop]ties are connect$ ) projective ones4 ,s #ba4 ,h>monic prop]ties ( ! complete quadrilat]al & quadrangle4 ,illu/ration3 ,,fig4 #g4 ,illu/ration3 ,,fig4 #h4 ,a figure =m$ by f\r l9es 9 a plane is call$ a complete quadrilat]al1 or1 %ort]1 a f\r-side4 ,! f\r sides meet 9 six po9ts1 nam$ ! v]tices1 :i* may be jo9$ by ?ree l9es 7o!r ?an ! sides71 nam$ ! diagonals or h>monic l9es4 ,! diagonals enclose ! h>monic triangle ( ! quadrilat]al4 ,9 fig4 #g1 ,a',b',c'1 ,b',,ac1 ,c',,ab1 ,,cba' are ! sides1 ,a1 ,a'1 ,b1 ,b'1 ,c1 ,c' ! v]tices1 ,,aa'1 ,,bb'1 ,,cc' ! h>monic l9es1 & alphabetagamma ! h>monic triangle ( ! quadrilat]al4 ,a figure =m$ by f\r coplan> po9ts is nam$ a complete quadrangle1 or1 %ort]1 a f\r-po9t4 ,! f\r po9ts may be jo9$ by six l9es1 nam$ ! sides1 :i* int]sect 9 ?ree o!r po9ts1 t]m$ ! diagonal or h>monic po9ts4 ,! h>monic po9ts are ! v]tices ( ! h>monic triangle ( ! complete quadrangle4 ,9 fig4 #h1 ,,aa'1 ,,bb' are ! po9ts1 ,,aa'1 ,,bb'1 ,a',b'1 ,b',a1 ,,ab1 ,,ba' are ! sides1 ,l1 ,m1 ,n are ! diagonal po9ts1 & ,,lmn is ! h>monic triangle ( ! quadrangle4 ,! h>monic prop]ty ( ! complete quadrilat]al is3 ,any diagonal or h>monic l9e is h>monically divid$ by ! o!r two2 & ( a complete quadrangle3 ,! angle at any h>monic po9t is divid$ h>monically by ! jo9s to ! o!r h>monic po9ts4 ,to prove ! fir/ !orem1 we h to prove 7,,aa'1 betagamma71 7,,bb'1 gammaalpha71 7,,cc'1 betaalpha7 are h>monic4 ,consid] ! cross-ratio 7,,cc'1 alphabeta74 ,!n project+ f ,a on ,,bb' we h ,a7,,cc'1 alphabeta7 ,a7,b',b1 alphagamma74 ,project+ f ,a' on ,,bb'1 ,a'7,,cc'1 alphabeta7 ,a'7,,bb'1 alphagamma74 ,h5ce 7,b',b1 alphagamma7 7,,bb'1 alphagamma71 i4e4 ! cross-ratio 7,,bb'1 alphagamma7 equals t ( its reciprocal2 h5ce ! range is h>monic4 ,! second !orem /ates t ! p5cils ,l7,,ba1 ,,nm71 ,m7,b',a1 ,,ln71 ,n7,,ba1 ,,lm7 are h>monic4 ,def]r+ ! subject ( h>monic p5cils to ! next section1 x w suffice to /ate "h t any transv]sal int]sects an h>monic p5cil 9 an h>monic range4 ,consid] ! p5cil ,l7,,ba1 ,,nm71 !n x is suffici5t to prove 7,,ba'1 ,,nm'7 is h>monic4 ,?is foll[s f ! previ\s !orem by consid]+ ,a',b z a diagonal ( ! quadrilat]al ,,alb',m4 ,?is prop]ty ( ! complete quadrilat]al all[s ! solution ( ! problem3 ,to con/ruct ! h>monic conjugate ,d to a po9t ,c ) reg>d to two giv5 po9ts ,a & ,b4 ,?r\gh ,a draw any two l9es1 & ?r\gh ,c "o cutt+ ! =m] two 9 ,g & ,h4 ,jo9 ^! po9ts to ,b1 cutt+ ! =m] two l9es 9 ,e & ,f4 ,! po9t ,d :]e ,,ef cuts ,,ab w be ! h>monic conjugate requir$4 ,?is rem>kable con/ruction requires no?+ b ! draw+ ( l9es1 & is !re=e indep5d5t ( measurem5t4 ,9 a simil> mann] ! h>monic conjugate ( ! l9e ,,va = two l9es ,,vc1 ,,vd is con/ruct$ ) ! aid ( ! prop]ty ( ! complete quadrangle4 ,s #bb4 ,h>monic ,p5cils4--,! !ory ( cross-ratios may be ext5d$ f po9ts 9 a r[ to l9es 9 a flat p5cil & to planes 9 an axial p5cil4 ,we h se5 7,s #ac7 t if ! l9es :i* jo9 f\r po9ts ,a1 ,b1 ,c1 ,d to any po9t ,s be cut by any o!r l9e 9 ,a'1 ,b'1 ,c'1 ,d'1 !n 7,,ab1 ,,cd7 7,a',b'1 ,c',d'74 ,9 o!r words1 f\r l9es 9 a flat p5cil are cut by e o!r l9e 9 f\r po9ts ^: cross-ratio is con/ant4 ,def9ition4--,by ! cross-ratio ( f\r rays 9 a flat p5cil is meant ! cross-ratio ( ! f\r po9ts 9 :i* ! rays are cut by any l9e4 ,if a1 b1 c1 d be ! l9es1 !n ?is cross-ratio is d5ot$ by 7ab1 cd74 ,def9ition4--,by ! cross-ratio ( f\r planes 9 an axial p5cil is und]/ood ! cross-ratio ( ! f\r po9ts 9 :i* any l9e cuts ! planes1 or1 :at is ! same ?+1 ! cross-ratio ( ! f\r rays 9 :i* any plane cuts ! f\r planes4 ,9 ord] t ?is def9ition may h a mean+1 x has to be prov$ t all l9es cut ! p5cil 9 po9ts :i* h ! same cross-ratio4 ,?is is se5 at once = two int]sect+ l9es1 z _! plane cuts ! axial p5cil 9 a flat p5cil1 :i* is itself cut by ! two l9es4 ,! cross-ratio ( ! f\r po9ts on "o l9e is !re=e equal to t on ! o!r1 & equal to t ( ! f\r rays 9 ! flat p5cil4 ,if two non-int]sect+ l9es p & q cut ! f\r planes 9 ,a1 ,b1 ,c1 ,d & ,a'1 ,b'1 ,c'1 ,d'1 draw a l9e r to meet bo? p & q1 & let ?is l9e cut ! planes 9 ,a1 ,b1 ,c1 ,d4 ,!n 7,,ab1 ,,cd7 7,a',b'1 ,c',d'71 = ea* is equal to 7,a,b1 ,c,d74 ,s #bc4 ,we may n[ also ext5d ! notion ( h>monic elem5ts1 viz4 ,def9ition4--,f\r rays 9 a flat p5cil & f\r planes 9 an axial p5cil are said to be h>monic if _! cross-ratio equals -#a1 t is1 if !y are cut by a l9e 9 f\r h>monic po9ts4 ,if we und]/& by a m$ian l9e ( a triangle a l9e :i* jo9s a v]tex to ! middle po9t ( ! opposite side1 & by a m$ian l9e ( a p>allelogram a l9e jo9+ middle po9ts ( opposite sides1 we get z special cases ( ! la/ !orem3 ,! diagonals & m$ian l9es ( a p>allelogram =m an h>monic p5cil2 & ,at a v]tex ( any triangle1 ! two sides1 ! m$ian l9e1 & ! l9e p>allel to ! base =m an h>monic p5cil4 ,tak+ ! p>allelogram a rectangle1 or ! triangle isosceles1 we get3 ,any two l9es & ! bisections ( _! angles =m an h>monic p5cil4 ,or3 ,9 an h>monic p5cil1 if two conjugate rays are p]p5dicul>1 !n ! o!r two are equally incl9$ to !m2 &1 conv]sely1 if "o ray bisects ! angle betwe5 conjugate rays1 x is p]p5dicul> to its conjugate4 ,?is connects p]p5dicul>ity & bisection ( angles ) projective prop]ties4 ,s #bd4 ,we add a few !orems & problems :i* are easily prov$ or solv$ by aid ( h>monics4 ,an h>monic p5cil is cut by a l9e p>allel to "o ( its rays 9 ?ree equidi/ant po9ts4 ,?r\gh a giv5 po9t to draw a l9e su* t ! segm5t det]m9$ on x by a giv5 angle is bisect$ at t po9t4 ,hav+ giv5 two p>allel l9es1 to bisect on ei!r any giv5 segm5t )\t us+ a pair ( compasses4 ,hav+ giv5 9 a l9e a segm5t & its middle po9t1 to draw ?r\gh any giv5 po9t 9 ! plane a l9e p>allel to ! giv5 l9e4 ,to draw a l9e :i* jo9s a giv5 po9t to ! int]section ( two giv5 l9es :i* meet (f ! draw+ pap] 7by aid ( ,s #ba74 ,,correspond5ce4 ,,homographic ,,& ,,p]spective ,,ranges ,s #be4 ,two r[s1 p & p'1 :i* are "o ! projection ( ! o!r 7z 9 fig4 #e71 /& 9 a def9ite relation to ea* o!r1 *>act]iz$ by ! foll[+ prop]ties4 #a4 ,to ea* po9t 9 ei!r corresponds "o po9t 9 ! o!r2 t is1 ^? po9ts are said to correspond :i* are projections ( "o ano!r4 #b4 ,! cross-ratio ( any f\r po9ts 9 "o equals t ( ! correspond+ po9ts 9 ! o!r4 #c4 ,! l9es jo9+ correspond+ po9ts all pass ?r\gh ! same po9t4 ,if we suppose correspond+ po9ts m>k$1 & ! r[s br\ght into any o!r position1 !n ! l9es jo9+ correspond+ po9ts w no long] meet 9 a common po9t1 & h5ce ! ?ird ( ! above prop]ties w n hold any long]2 b we h /ill a correspond5ce betwe5 ! po9ts 9 ! two r[s possess+ ! fir/ two prop]ties4 ,su* a correspond5ce has be5 call$ a "o-"o correspond5ce1 :il/ ! two r[s betwe5 :i* su* correspond5ce has be5 e/abli%$ are said to be projective or homographic4 ,two r[s :i* are ea* ! projection ( ! o!r are !re=e projective4 ,we %all pres5tly see1 also1 t any two projective r[s may always be plac$ 9 su* a position t "o appe>s z ! projection ( ! o!r4 ,if !y are 9 su* a position ! r[s are said to be 9 p]spective position1 or simply to be 9 p]spective4 ,s #bf4 ,! notion ( a "o-"o correspond5ce betwe5 r[s may be ext5d$ to flat & axial p5cils1 viz4 a flat p5cil w be said to be projective to a flat p5cil if to ea* ray 9 ! fir/ corresponds "o ray 9 ! second1 & if ! cross-ratio ( f\r rays 9 "o equals t ( ! correspond+ rays 9 ! second4 ,simil>ly an axial p5cil may be projective to an axial p5cil4 ,b a flat p5cil may also be projective to an axial p5cil1 or ei!r p5cil may be projective to a r[4 ,! def9ition is ! same 9 ea* case3 !re is a "o-"o correspond5ce betwe5 ! elem5ts1 & f\r elem5ts h ! same cross-ratio z ! correspond+ ones4 ,s #bg4 ,!re is also 9 ea* case a special position :i* is call$ p]spective1 viz4 #a4 ,two projective r[s are p]spective if !y lie 9 ! same plane1 & if ! "o r[ is a projection ( ! o!r4 #b4 ,two projective flat p5cils are p]spective--7#a7 if !y lie 9 ! same plane1 & h a r[ z a common section2 7#b7 if !y lie 9 ! same p5cil 79 space71 & are bo? sections ( ! same axial p5cil2 7#c7 if !y are 9 space & h a r[ z common section1 or are bo? sections ( ! same axial p5cil1 "o ( ! conditions involv+ ! o!r4 #c4 ,two projective axial p5cils1 if _! axes meet1 & if !y h a flat p5cil z a common section4 #d4 ,a r[ & a projective flat p5cil1 if ! r[ is a section ( ! p5cil1 ea* po9t ly+ 9 its correspond+ l9e4 #e4 ,a r[ & a projective axial p5cil1 if ! r[ is a section ( ! p5cil1 ea* po9t ly+ 9 its correspond+ l9e4 #f4 ,a flat & a projective axial p5cil1 if ! =m] is a section ( ! o!r1 ea* ray ly+ 9 its correspond+ plane4 ,t 9 ea* case ! correspond5ce e/abli%$ by ! position indicat$ is su* z has be5 call$ projective foll[s at once f ! def9ition4 ,x is n s evid5t t ! p]spective position may always be obta9$4 ,we %all %[ 9 ,s #cj ?is = ! fir/ ?ree cases4 ,fir/1 h[ev]1 we %all give a few !orems :i* relate to ! g5]al correspond5ce1 n to ! p]spective position4 ,s #bh4 ,two r[s or p5cils1 flat or axial1 :i* are projective to a ?ird are projective to ea* o!r2 ?is foll[s at once f ! def9itions4 ,s #bi4 ,if two r[s1 or two p5cils1 ei!r flat or axial1 or a r[ & a p5cil1 be projective1 we may assume to any ?ree elem5ts 9 ! "o ! ?ree correspond+ elem5ts 9 ! o!r1 & !n ! correspond5ce is uniquely det]m9$4 ,= if 9 two projective r[s we assume t ! po9ts ,a1 ,b1 ,c 9 ! fir/ correspond to ! giv5 po9ts ,a'1 ,b'1 ,c' 9 ! second1 !n to any f\r? po9t ,d 9 ! fir/ w correspond a po9t ,d' 9 ! second1 s t 7,,ab1 ,,cd7 7,a',b'1 ,c',d'74 ,b !re is only "o po9t1 ,d'1 :i* makes ! cross-ratio 7,a',b'1 ,c',d'7 equal to ! giv5 numb] 7,,ab1 ,,cd74 ,! same reason+ holds 9 ! o!r cases4 ,s #cj4 ,if two r[s are p]spective1 !n ! l9es jo9+ correspond+ po9ts all meet 9 a po9t1 ! c5tre ( projection2 & ! po9t 9 :i* ! two bases ( ! r[s int]sect z a po9t 9 ! fir/ r[ co9cides ) its correspond+ po9t 9 ! second4 ,?is foll[s f ! def9ition4 ,! conv]se also holds1 viz4 ,if two projective r[s h su* a position t "o po9t 9 ! "o co9cides ) its correspond+ po9t 9 ! o!r1 !n !y are p]spective1 t is1 ! l9es jo9+ correspond+ po9ts all pass ?r\gh a common po9t1 & =m a flat p5cil4 ,= let ,a1 ,b1 ,c1 ,d 444 be po9ts 9 ! "o1 & ,a'1 ,b'1 ,c'1 ,d' 444 ! correspond+ po9ts 9 ! o!r r[1 & let ,a be made to co9cide ) its correspond+ po9t ,a'4 ,let ,s be ! po9t :]e ! l9es ,,bb' & ,,cc' meet1 & let u jo9 ,s to ! po9t ,d 9 ! fir/ r[4 ,?is l9e w cut ! second r[ 9 a po9t ,d1 s t ,a1 ,b1 ,c1 ,d are project$ f ,s into ! po9ts ,a1 ,b'1 ,c'1 ,d4 ,! cross-ratio 7,,ab1 ,,cd7 is !re=e equal to 7,,ab'1 ,c',d71 & by hypo!sis x is equal to 7,a',b'1 ,c',d'74 ,h5ce 7,a',b'1 ,c',d7 7,a',b'1 ,c',d'71 t is1 ,d is ! same po9t z ,d'4 ,s #ca4 ,if two project$ flat p5cils 9 ! same plane are 9 p]spective1 !n ! int]sections ( correspond+ l9es =m a r[1 & ! l9e jo9+ ! two c5tres z a l9e 9 ! fir/ p5cil corresponds to ! same l9e z a l9e 9 ! second4 ,& conv]sely1 ,if two projective p5cils 9 ! same plane1 b ) diff]5t c5tres1 h "o l9e 9 ! "o co9cid5t ) its correspond+ l9e 9 ! o!r1 !n ! two p5cils are p]spective1 t is1 ! int]section ( correspond+ l9es lie 9 a l9e4 ,! pro( is ! same z 9 ,s #cj4 ,s #cb4 ,if two projective flat p5cils 9 ! same po9t 7p5cil 9 space71 b n 9 ! same plane1 are p]spective1 !n ! planes jo9+ correspond+ rays all pass ?r\gh a l9e 7!y =m an axial p5cil71 & ! l9e common to ! two p5cils 79 :i* _! planes int]sect7 corresponds to itself4 ,& conv]sely3-- ,if two flat p5cils :i* h a common c5tre1 b d n lie 9 a common plane1 are plac$ s t "o ray 9 ! "o co9cides ) its correspond+ ray 9 ! o!r1 !n !y are p]spective1 t is1 ! planes jo9+ correspond+ l9es all pass ?r\gh a l9e4 ,s #cc4 ,if two projective axial p5cils are p]spective1 !n ! int]section ( correspond+ planes lie 9 a plane1 & ! plane common to ! two p5cils 79 :i* ! two axes lie7 corresponds to itself4 ,& conv]sely3-- ,if two projective axial p5cils are plac$ 9 su* a position t a plane 9 ! "o co9cides ) its correspond+ plane1 !n ! two p5cils are p]spective1 t is1 correspond+ planes meet 9 l9es :i* lie 9 a plane4 ,! pro( aga9 is ! same z 9 ,s #cj4 ,s #cd4 ,^! !orems relat+ to p]spective position become illusory if ! projective r[s ( p5cils h a common base4 ,we !n h3-- ,9 two projective r[s on ! same l9e--& also 9 two projective & conc5tric flat p5cils 9 ! same plane1 or 9 two projective axial p5cils ) a common axis--e elem5t 9 ! "o co9cides ) its correspond+ elem5t 9 ! o!r z soon z ?ree elem5ts 9 ! "o co9cide ) _! correspond+ elem5ts 9 ! o!r4 ,pro( 79 case ( two r[s74--,betwe5 f\r elem5ts ,a1 ,b1 ,c1 ,d & _! correspond+ elem5ts ,a'1 ,b'1 ,c'1 ,d' exi/s ! relation 7,,abcd7 7,a',b',c',d'74 ,if n[ ,a'1 ,b'1 ,c' co9cide respectively ) ,a1 ,b1 ,c1 we get 7,,ab1 ,,cd7 7,,ab1 ,,cd'71 h5ce ,d & ,d' co9cide4 ,! la/ !orem may also be /at$ ?us3-- ,9 two projective r[s or p5cils1 :i* h a common base b are n id5tical1 n m ?an two elem5ts 9 ! "o c co9cide ) _! correspond+ elem5ts 9 ! o!r4 ,?us two projective r[s on ! same l9e _c h m ?an two pairs ( co9cid5t po9ts unless e po9t co9cides ) its correspond+ po9t4 ,x is easy to con/ruct two projective r[s on ! same l9e1 :i* h two pairs ( correspond+ po9ts co9cid5t4 ,let ! po9ts ,a1 ,b1 ,c z po9ts belong+ to ! "o r[ correspond to ,a1 ,b1 & ,c' z po9ts 9 ! second4 ,!n ,a & ,b co9cide ) _! correspond+ po9ts1 b ,c does n4 ,x is1 h[ev]1 n necess>y t two su* r[s h twice a po9t co9cid5t ) its correspond+ po9t2 x is possible t ?is happ5s only once or n at all4 ,( ?is we %all see examples lat]4 ,illu/ration3 ,,fig4 #i4 ,s #ce4 ,if two projective r[s or p5cils are 9 p]spective position1 we "k at once :i* elem5t 9 "o corresponds to any giv5 elem5t 9 ! o!r4 ,if p & q 7fig4 #i7 are two projective r[s1 s t ,k corresponds to itself1 & if we "k t to ,a & ,b 9 p correspond ,a' & ,b' 9 q1 !n ! po9t ,s1 :]e ,,aa' meets ,,bb'1 is ! c5tre ( projection1 & h5ce1 9 ord] to f9d ! po9t ,c' correspond+ to ,c1 we h only to jo9 ,c to ,s2 ! po9t ,c'1 :]e ?is l9e cuts q1 is ! po9t requir$4 ,illu/ration3 ,,fig4 #aj4 ,if two flat p5cils1 ,s#a & ,s#b1 9 a plane are p]spective 7fig4 #aj71 we ne$ only to "k two pairs1 a1 a' & b1 b'1 ( correspond+ rays 9 ord] to f9d ! axis s ( projection4 ,?is be+ kn[n1 a ray c' 9 ,s#b1 correspond+ to a giv5 ray c 9 ,s#a1 is f\nd by jo9+ ,s#b to ! po9t :]e c cuts ! axis s4 ,a simil> con/ruction holds 9 ! o!r cases ( p]spective figures4 ,on ?is dep5ds ! solution ( ! foll[+ g5]al problem4 ,s #cf4 ,?ree pairs ( correspond+ elem5ts 9 two projective r[s or p5cils be+ giv51 to det]m9e = any elem5t 9 "o ! correspond+ elem5t 9 ! o!r4 ,we solve ?is 9 ! two cases ( two projective r[s & ( two projective flat p5cils 9 a plane4 ,problem ,i4--,let ,a1 ,b1 ,c be ,problem ,,ii4--,let a1 b1 c be ?ree po9ts 9 a r[ s1 ,a'1 ,b'1 ?ree rays 9 a p5cil ,s1 a'1 ,c' ! correspond+ po9ts 9 a b'1 c' ! correspond+ rays 9 projective r[ s'1 bo? be+ 9 a projective p5cil ,s'1 bo? a plane2 x is requir$ to f9d be+ 9 ! same plane2 x is = any po9t ,d 9 s ! requir$ to f9d = any ray d correspond+ po9t ,d' 9 s'4 9 ,s ! correspond+ ray d' 9 ,s'4 ,! solution is made to dep5d on ! con/ruction ( an auxili>y r[ or p5cil :i* is p]spective to bo? ! giv5 ones4 ,?is is f\nd z foll[s3-- ,illu/ration3 ,,fig4 #aa4 ,solution ( ,problem ,i4--,on ! l9e jo9+ two correspond+ po9ts1 say ,,aa' 7fig4 #aa71 take any two po9ts1 ,s & ,s'1 z c5tres ( auxili>y p5cils4 ,jo9 ! int]section ,b#a ( ,,sb & ,s',b' to ! int]section ,c#a ( ,,sc & ,s',c' by ! l9e s#a4 ,!n a r[ on s#a w be p]spective to s ) ,s z c5tre ( projection1 & to s' ) ,s' z c5tre4 ,to f9d n[ ! po9t ,d' on s' correspond+ to a po9t ,d on s we h only to det]m9e ! po9t ,d#a1 :]e ! l9e ,,sd cuts s#a1 & to draw ,s',d#a2 ! po9t :]e ?is l9e cuts s' w be ! requir$ po9t ,d'4 ,pro(4--,! r[s s & s' are bo? p]spective to ! r[ s#a1 h5ce !y are projective to "o ano!r4 ,to ,a1 ,b1 ,c1 ,d on s correspond ,a#a1 ,b#a1 ,c#a1 ,d#a on s#a1 & to ^! correspond ,a'1 ,b'1 ,c'1 ,d' on s'2 s t ,d & ,d' are correspond+ po9ts z requir$4 ,illu/ration3 ,,fig4 #ab4 ,solution ( ,problem ,,ii4--,?r\gh ! int]section ,a ( two correspond+ rays a & a' 7fig4 #ab71 take two l9es1 s & s'1 z bases ( auxili>y r[s4 ,let ,s#a be ! po9t :]e ! l9e b#a1 :i* jo9s ,b & ,b'1 cuts ! l9e c#a1 :i* jo9s ,c & ,c'4 ,!n a p5cil ,s#a w be p]spective to ,s ) s z axis ( projection4 ,to f9d ! ray d' 9 ,s' correspond+ to a giv5 ray d 9 ,s1 cut d by s at ,d2 project ?is po9t f ,s#a to ,d' on s' & jo9 ,d' to ,s'4 ,?is w be ! requir$ ray4 ,pro(4--,t ! p5cil ,s#a is p]spective to ,s & also to ,s' foll[s f con/ruction4 ,to ! l9es a#a1 b#a1 c#a1 d#a 9 ,s#a correspond ! l9es a1 b1 c1 d 9 ,s & ! l9es a'1 b'1 c'1 d' 9 ,s'1 s t d & d' are correspond+ rays4 ,9 ! fir/ solution ! two c5tres1 ,s1 ,s'1 are any two po9ts on a l9e jo9+ any two correspond+ po9ts1 s t ! solution ( ! problem all[s ( a great _m diff]5t con/ructions4 ,b :atev] con/ruction be us$1 ! po9t ,d'1 correspond+ to ,d1 mu/ be always ! same1 accord+ to ! !orem 9 ,s #bi4 ,?is gives rise to a numb] ( !orems1 into :i*1 h[ev]1 we %all n ent]4 ,! same rem>ks hold = ! second problem4 ,s #cg4 ,homological ,triangles4--,z a fur!r application ( ! !orems ab\t p]spective r[s & p5cils we %all prove ! foll[+ important !orem4 ,!orem4--,if ,,abc & ,a',b',c' 7fig4 #ac7 be two triangles1 su* t ! l9es ,,aa'1 ,,bb'1 ,,cc' meet 9 a po9t ,s1 !n ! int]sections ( ,,bc & ,b',c'1 ( ,,ca & ,c',a'1 & ( ,,ab & ,a',b' w lie 9 a l9e4 ,su* triangles are said to be homological1 or 9 p]spective4 ,! triangles are co-axial 9 virtue ( ! prop]ty t ! meets ( correspond+ sides are coll9e> & copol>1 s9ce ! l9es jo9+ correspond+ v]tices are concurr5t4 ,pro(4--,let a1 b1 c d5ote ! l9es ,,aa'1 ,,bb'1 ,,cc'1 :i* meet at ,s4 ,!n ^! may be tak5 z bases ( projective r[s1 s t ,a1 ,a'1 ,s on a correspond to ,b1 ,b'1 ,s on b1 & to ,c1 ,c'1 ,s on c4 ,z ! po9t ,s is common to all1 any two ( ^! r[s w be p]spective4 ,if ,s#a be ! c5tre ( projection ( r[s b & c1 ,s#b c & a1 ,s#c a & b1 & if ! l9e ,s#a,s#b cuts a 9 ,a#a1 & b 9 ,b#a1 & c 9 ,c#a1 !n ,a#a1 ,b#a w be correspond+ po9ts 9 a & b1 bo? correspond+ to ,c#a 9 c4 ,b a & b are p]spective1 !re=e ! l9e ,a#a;,b#a1 t is ,s#a,s#b1 jo9+ correspond+ po9ts mu/ pass ?r\gh ! c5tre ( projection ,s#c ( a & b4 ,9 o!r words1 ,s#a1 ,s#b1 ,s#c lie 9 a l9e4 ,?is is ,des>gues' celebrat$ !orem if we /ate x ?us3-- ,illu/ration3 ,,fig4 #ac4 ,!orem ( ,des>gues4--,if ea* ( two triangles has "o v]tex on ea* ( ?ree concurr5t l9es1 !n ! int]sections ( correspond+ sides lie 9 a l9e1 ^? sides be+ call$ correspond+ :i* are opposite to v]tices on ! same l9e4 ,! conv]se !orem holds also1 viz4 ,!orem4--,if ! sides ( "o triangle meet ^? ( ano!r 9 ?ree po9ts :i* lie 9 a l9e1 !n ! v]tices lie on ?ree l9es :i* meet 9 a po9t4 ,! pro( is almo/ ! same z be=e4 ,s #ch4 ,metrical ,relations betwe5 ,projective ,r[s4--,e r[ conta9s "o po9t :i* is di/+ui%$ f all o!rs1 viz4 ! po9t at inf9ity4 ,9 two projective r[s1 to ! po9t ,i at inf9ity 9 "o corresponds a po9t ,i' 9 ! o!r1 & to ! po9t ,j' at inf9ity 9 ! second corresponds a po9t ,j 9 ! fir/4 ,! po9ts ,i' & ,j are 9 g5]al f9ite4 ,if n[ ,a & ,b are any two po9ts 9 ! "o1 ,a'1 ,b' ! correspond+ po9ts 9 ! o!r r[1 !n 7,,ab1 ,,ji7 7,a',b'1 ,j',i'71 or ,,aj_/,,jb 3 ,,ai_/,,ib ,a',j'_/,j',b' 3 ,a',i'_/,i',b'4 ,b1 by ,s #ag1 ,,ai_/,,ib ,a',j'_/,j',b' -#a2 !re=e ! la/ equation *anges into ,,aj4,a',i' ,,bj4,b',i'1 t is to say-- ,!orem4--,! product ( ! di/ances ( any two correspond+ po9ts 9 two projective r[s f ! po9ts :i* correspond to ! po9ts at inf9ity 9 ! o!r is con/ant1 viz4 ,,aj4,a',i' k4 ,/e9] has call$ ?is numb] k ! ,p[] ( ! correspond5ce4 ,! relation ,,aj 4 ,a',i' k %[s t if ,j1 ,i' be giv5 !n ! po9t ,a' correspond+ to a specifi$ po9t ,a is readily f\nd2 h5ce ,a1 ,a' g5]ate homographic ranges ( :i* ,i & ,j' correspond to ! po9ts at inf9ity on ! ranges4 ,if we take any two orig9s ,o1 ,o'1 on ! ranges & r$uce ! expression ,,aj 4 ,a',i' k to its algebraic equival5t1 we d]ive an equation ( ! =m alphaxx' betax gammax' delta #j4 ,conv]sely1 if a relation ( ?is nature holds1 !n po9ts correspond+ to solutions 9 x1 x' =m homographic ranges4 ,s #ci4 ,simil> ,r[s4--,if ! po9ts at inf9ity 9 two projective r[s correspond s t ,i' & ,j are at inf9ity1 ?is result loses its mean+4 ,b if ,a1 ,b1 ,c be any ?ree po9ts 9 "o1 ,a'1 ,b'1 ,c' ! correspond+ ones on ! o!r r[1 we h 7,,ab1 ,,ci7 7,a',b'1 ,c',i'71 :i* r$uces to ,,ac_/,,cb ,a',c'_/,c',b' or ,,ac_/,a',c' ,,bc_/,b',c'1 t is1 correspond+ segm5ts are proportional4 ,conv]sely1 if correspond+ segm5ts are proportional1 !n to ! po9t at inf9ity 9 "o corresponds ! po9t at inf9ity 9 ! o!r4 ,if we call su* r[s simil>1 we may /ate ! result ?us-- ,!orem4--,two projective r[s are simil> if to ! po9t at inf9ity 9 "o corresponds ! po9t at inf9ity 9 ! o!r1 & conv]sely1 if two r[s are simil> !n !y are projective1 & ! po9ts at inf9ity are correspond+ po9ts4 ,f ?is ! well-kn[n propositions foll[3-- ,two l9es are cut proportionally 79 simil> r[s7 by a s]ies ( p>allels4 ,! r[s are p]spective1 ) c5tre ( projection at inf9ity4 ,if two simil> r[s are plac$ p>allel1 !n ! l9es jo9+ homolog\s po9ts pass ?r\gh a common po9t4 ,s #dj4 ,if two flat p5cils be projective1 !n !re exi/s 9 ei!r1 "o s+le pair ( l9es at "r angles to "o ano!r1 su* t ! correspond+ l9es 9 ! o!r p5cil are aga9 at "r angles4 ,illu/ration3 ,,fig4 #ad4 ,to prove ?is1 we place ! p5cils 9 p]spective position 7fig4 #ad7 by mak+ "o ray co9cid5t ) its correspond+ ray4 ,correspond+ rays meet !n on a l9e p4 ,& n[ we draw ! circle :i* has its c5tre ,o on p1 & :i* passes ?r\gh ! c5tres ,s & ,s' ( ! two p5cils4 ,?is circle cuts p 9 two po9ts ,h & ,k4 ,! two pairs ( rays1 h1 k1 & h'1 k'1 jo9+ ^! po9ts to ,s & ,s' w be pairs ( correspond+ rays at "r angles4 ,! con/ruction gives 9 g5]al b "o circle1 b if ! l9e p is ! p]p5dicul> bisector ( ,,ss'1 !re exi/s an inf9ite numb]1 & to e "r angle 9 ! "o p5cil corresponds a "r angle 9 ! o!r4 ,,pr9ciple ,,( ,,duality ,s #da4 ,x has be5 /at$ 9 ,s #a t n only po9ts1 b also planes & l9es1 are tak5 z elem5ts out ( :i* figures are built up4 ,we %all n[ see t ! con/ruction ( "o figure :i* possesses c]ta9 prop]ties gives rise 9 _m cases to ! con/ruction ( ano!r figure1 by replac+1 accord+ to def9ite rules1 elem5ts ( "o k9d by ^? ( ano!r4 ,! new figure ?us obta9$ w !n possess prop]ties :i* may be /at$ z soon z ^? ( ! orig9al figure are kn[n4 ,we obta9 ?us a pr9ciple1 kn[n z ! pr9ciple ( duality or ( reciprocity1 :i* enables u to con/ruct to any figure n conta9+ any measurem5t 9 its con/ruction a reciprocal figure1 z x is call$1 & to d$uce f any !orem a reciprocal !orem1 = :i* no fur!r pro( is ne$$4 ,x is conv5i5t to pr9t reciprocal propositions on opposite sides ( a page brok5 into two columns1 & ?is plan w occasionally be adopt$4 ,we beg9 by repeat+ 9 ?is =m a few ( our =m] /atem5ts3-- ,two po9ts det]m9e a l9e4 ,two planes det]m9e a l9e4 ,?ree po9ts :i* are n 9 a ,?ree planes :i* d n pass l9e det]m9e a plane4 ?r\gh a l9e det]m9e a po9t4 ,a l9e & a po9t )\t x ,a l9e & a plane n ?r\gh det]m9e a plane4 x det]m9e a po9t4 ,two l9es 9 a plane det]m9e ,two l9es ?r\gh a po9t a po9t4 det]m9e a plane4 ,^! propositions %[ t x w be possible1 :5 any figure is giv51 to con/ruct a second figure by tak+ planes in/ead ( po9ts1 & po9ts in/ead ( planes1 b l9es :]e we _h l9es4 ,= in/ance1 if 9 ! fir/ figure we take a plane & ?ree po9ts 9 x1 we h to take 9 ! second figure a po9t & ?ree planes ?r\gh x4 ,! ?ree po9ts 9 ! fir/1 toge!r ) ! ?ree l9es jo9+ !m two & two1 =m a triangle2 ! ?ree planes 9 ! second & _! ?ree l9es ( int]section =m a trih$ral angle4 ,a triangle & a trih$ral angle are !re=e reciprocal figures4 ,simil>ly1 to any figure 9 a plane consi/+ ( po9ts & l9es w correspond a figure consi/+ ( planes & l9es pass+ ?r\gh a po9t ,s1 & h5ce belong+ to ! p5cil :i* has ,s z c5tre4 ,! figure reciprocal to f\r po9ts 9 space :i* d n lie 9 a plane w consi/ ( f\r planes :i* d n meet 9 a po9t4 ,9 ?is case ea* figure =ms a tetrah$ron4 ,s #db4 ,z o!r examples we h ! foll[+3-- ,to a r[ is reciprocal an axial p5cil1 a flat p5cil a flat p5cil1 a field ( po9ts & l9es a p5cil ( planes & l9es1 ! space ( po9ts ! space ( planes4 ,= ! r[ consi/s ( a l9e & all ! po9ts 9 x1 reciprocal to x !re=e w be a l9e ) all planes ?r\gh x1 t is1 an axial p5cil2 & s = ! o!r cases4 ,?is correspond5ce ( reciprocity breaks d[n1 h[ev]1 if we take figures :i* conta9 measurem5t 9 _! con/ruction4 ,= in/ance1 !re is no figure reciprocal to two planes at "r angles1 because !re is no segm5t 9 a r[ :i* has a magnitude z def9ite z a "r angle4 ,we add a few examples ( reciprocal propositions :i* are easily prov$4 ,!orem4--,if ,a1 ,b1 ,c1 ,d are ,!orem4--,if alpha1 beta1 any f\r po9ts 9 space1 & if gamma1 delta are f\r planes ! l9es ,,ab & ,,cd meet1 !n 9 space1 & if ! l9es all f\r po9ts lie 9 a plane1 alphabeta & gammadelta h5ce also ,,ac & ,,bd1 z well meet1 !n all f\r planes lie z ,,ad & ,,bc1 meet4 9 a po9t 7p5cil71 h5ce also alphagamma & betadelta1 well z alphadelta & z betagamma1 meet4 ,!orem4--,if ( any numb] ( l9es e "o meets e o!r1 :il/ all d n lie 9 a po9t1 !n all lie 9 lie 9 a plane1 !n all lie 9 a plane4 a po9t 7p5cil74 ,s #dc4 ,reciprocal figures z expla9$ lie bo? 9 space ( ?ree dim5sions4 ,if ! "o is conf9$ to a plane 7is =m$ ( elem5ts :i* lie 9 a plane71 !n ! reciprocal figure is conf9$ to a p5cil 7is =m$ ( elem5ts :i* pass ?r\gh a po9t74 ,b !re is also a m special pr9ciple ( duality1 accord+ to :i* figures are reciprocal :i* lie bo? 9 a plane or bo? 9 a p5cil4 ,9 ! plane we take po9ts & l9es z reciprocal elem5ts1 = !y h ?is fundam5tal prop]ty 9 common1 t two elem5ts ( "o k9d det]m9e "o ( ! o!r4 ,9 ! p5cil1 on ! o!r h&1 l9es & planes h to be tak5 z reciprocal1 & "h x holds aga9 t two l9es or planes det]m9e "o plane or l9e4 ,?us1 to "o plane figure we c con/ruct "o reciprocal figure 9 ! plane1 & to ea* "o reciprocal figure 9 a p5cil4 ,we m5tion a few ( ^!4 ,at fir/ we expla9 a few names3-- ,a figure consi/+ ( n po9ts ,a figure consi/+ ( n l9es 9 a plane w be call$ an 9 a plane w be call$ an n-po9t4 n-side4 ,a figure consi/+ ( n planes ,a figure consi/+ ( n l9es 9 a p5cil w be call$ an 9 a p5cil w be call$ an n-flat4 n-edge4 ,x w be und]/ood t an n-side is diff]5t f a polygon ( n sides4 ,! latt] has sides ( f9ite l5g? & n v]tices1 ! =m] has sides all ( inf9ite ext5sion1 & e po9t :]e two ( ! sides meet w be a v]tex4 ,a simil> diff]5ce exi/s betwe5 a solid angle & an n-edge or an n-flat4 ,we notice p>ticul>ly-- ,a f\r-po9t has six sides1 ( ,a f\r-side has six v]tices1 ( :i* two & two are opposite1 :i* two & two are opposite1 & ?ree diagonal po9ts1 :i* & ?ree diagonals1 :i* jo9 are int]sections ( opposite opposite v]tices4 sides4 ,a f\r-flat has six edges1 ( ,a f\r-edge has six faces1 ( :i* two & two are opposite1 :i* two & two are opposite1 & ?ree diagonal planes1 :i* & ?ree diagonal edges1 :i* pass ?r\gh opposite edges4 are int]sections ( opposite faces4 ,a f\r-side is usually call$ a complete quadrilat]al1 & a f\r-po9t a complete quadrangle4 ,! above notation1 h[ev]1 seems bett] adapt$ = ! /atem5t ( reciprocal propositions4 ,s #dd4 ,if a po9t moves 9 a plane x ,if a l9e moves 9 a plane x describes a plane curve4 envelopes a plane curve 7fig4 #ae74 ,if a plane moves 9 a p5cil x ,if a l9e moves 9 a p5cil x envelopes a cone4 describes a cone4 ,a curve ?us appe>s z g5]at$ ei!r by po9ts1 & !n we call x a locus1 or by l9es1 & !n we call x an envelope4 ,9 ! same mann] a cone1 :i* means "h a surface1 appe>s ei!r z ! locus ( l9es pass+ ?r\gh a fix$ po9t1 ! v]tex ( ! cone1 or z ! envelope ( planes pass+ ?r\gh ! same po9t4 ,illu/ration3 ,,fig4 #ae4 ,to a surface z locus ( po9ts corresponds1 9 ! same mann]1 a surface z envelope ( planes2 & to a curve 9 space z locus ( po9ts corresponds a developable surface z envelope ( planes4 ,x w be se5 f ! above t we may1 by aid ( ! pr9ciple ( duality1 con/ruct = e figure a reciprocal figure1 & t to any prop]ty ( ! "o a reciprocal prop]ty ( ! o!r w exi/1 z long z we consid] only prop]ties :i* dep5d ^u no?+ b ! positions & int]sections ( ! diff]5t elem5ts & n ^u measurem5t4 ,= su* propositions x w !re=e be unnecess>y to prove m ?an "o ( two reciprocal !orems4 ,,g5]ation ,,( ,,curves ,,& ,,cones ,,( ,,second ,,ord] ,,or ,,second ,,class ,s #de4 ,conics4--,if we h two projective p5cils 9 a plane1 correspond+ rays w meet1 & _! po9t ( int]section w con/itute "s locus :i* we h to inve/igate4 ,reciprocally1 if two projective r[s 9 a plane are giv51 !n ! l9es :i* jo9 correspond+ po9ts w envelope "s curve4 ,we prove fir/3-- ,!orem4--,if two projective ,!orem4--,if two projective flat p5cils lie 9 a plane1 b r[s lie 9 a plane1 b are are nei!r 9 p]spective nor nei!r 9 p]spective nor on a conc5tric1 !n ! locus ( common base1 !n ! envelope int]sections ( correspond+ ( l9es jo9+ correspond+ rays is a curve ( ! second po9ts is a curve ( ! second ord]1 t is1 no l9e conta9s class1 t is1 ?r\gh no po9t m ?an two po9ts ( ! pass m ?an two ( ! locus4 envelop+ l9es4 ,pro(4--,we draw any l9e t4 ,pro(4--,we take any po9t ,t ,?is cuts ea* ( ! p5cils 9 & jo9 x to all po9ts 9 ea* a r[1 s t we h on t two r[4 ,?is gives two conc5tric r[s1 & ^! are projective p5cils1 :i* are projective because ! p5cils are because ! r[s are projective4 projective4 ,if correspond+ rays ,if a l9e jo9+ correspond+ ( ! two p5cils meet on ! po9ts 9 ! two r[s passes l9e t1 _! int]section w ?r\gh ,t1 x w be a l9e 9 be a po9t 9 ! "o r[ :i* ! "o p5cil :i* co9cides co9cides ) its correspond+ ) its correspond+ l9e 9 po9t 9 ! o!r4 ,b two ! o!r4 ,b two projective projective r[s on ! same base conc5tric flat p5cils 9 ! _c h m ?an two same plane _c h m ?an po9ts ( "o co9cid5t ) two l9es ( "o co9cid5t ) _! correspond+ po9ts 9 _! correspond+ l9e 9 ! ! o!r 7,s #cd74 o!r 7,s #cd74 ,x w be se5 t ! pro(s are reciprocal1 s t ! "o may be copi$ f ! o!r by simply int]*ang+ ! words po9t & l9e1 locus & envelope1 r[ & p5cil1 & s on4 ,we %all !re=e 9 future prove seldom m ?an "o ( two reciprocal !orems1 & (t5 /ate "o !orem only1 ! read] be+ recomm5d$ to g ?r\gh ! reciprocal pro( by himself1 & to supply ! reciprocal !orems :5 n giv54 ,s #df4 ,we /ate ! !orems 9 ! p5cil reciprocal to ! la/1 )\t prov+ !m3-- ,!orem4--,if two projective ,!orem4--,if two projective flat p5cils are conc5tric1 b axial p5cils lie 9 ! same are nei!r p]spective nor p5cil 7_! axes meet 9 a coplan>1 !n ! envelope ( po9t71 b are nei!r p]spective ! planes jo9+ correspond+ nor co-axial1 !n ! locus rays is a cone ( ! second ( l9es jo9+ correspond+ class2 t is1 no l9e ?r\gh planes is a cone ( ! second ! common c5tre conta9s m ord]2 t is1 no plane 9 ! ?an two ( ! envelop+ p5cil conta9s m ?an two planes4 ,s #dg4 ,( !orems ab\t cones ( second ord] & cones ( second class we %all /ate only v few4 ,we po9t out1 h[ev]1 ! foll[+ connexion betwe5 ! curves & cones "u consid]ation3 ,! l9es :i* jo9 any po9t ,e plane section ( a cone 9 space to ! po9ts on a curve ( ! second ord] is a curve ( ( ! second ord] =m a cone ! second ord]4 ( ! second ord]4 ,! planes :i* jo9 any ,e plane section ( a cone po9t 9 space to ! l9es ( ! second class is a curve ( envelop+ a curve ( ! ! second class4 second class envelope !mselves a cone ( ! second class4 ,by its aid1 or by ! pr9ciple ( duality1 x w be easy to obta9 !orems ab\t !m f ! !orems ab\t ! curves4 ,we prove ! fir/4 ,a curve ( ! second ord] is g5]at$ by two projective p5cils4 ,^! p5cils1 :5 jo9$ to ! po9t 9 space1 give rise to two projective axial p5cils1 :i* g5]ate ! cone 9 "q z ! locus ( ! l9es :]e correspond+ planes meet4 ,s#dh4 ,!orem4--,! curve ( second ,!orem4--,! envelope ( ord] :i* is g5]at$ by two second class :i* is g5]at$ projective flat p5cils passes by two projective r[s conta9s ?r\gh ! c5tres ( ! two ! bases ( ^! r[s z p5cils4 envelop+ l9es or tang5ts4 ,pro(4--,if ,s & ,s' are ! ,pro(4--,if s & s' are ! two p5cils1 !n to ! ray ,,ss' two r[s1 !n to ! po9t ss' or p' 9 ! p5cil ,s' or ,p' z a po9t 9 s' corresponds 9 ! p5cil ,s a corresponds 9 s a po9t ,p1 ray p1 :i* is diff]5t f :i* is n co9cid5t ) ,p'1 p'1 = ! p5cils are n = ! r[s are n p]spective4 ,b p & p' meet p]spective4 ,b ,p & ,p' are at ,s1 s t ,s is a po9t on jo9$ by s1 s t s is "o ( ! curve1 & simil>ly ,s'4 ! envelop+ l9es1 & simil>ly s'4 ,x foll[s t e l9e 9 "o ( ! two p5cils cuts ! curve 9 two po9ts1 viz4 once at ! c5tre ,s ( ! p5cil1 & once :]e x cuts its correspond+ ray 9 ! o!r p5cil4 ,^! two po9ts1 h[ev]1 co9cide1 if ! l9e is cut by its correspond+ l9e at ,s itself4 ,! l9e p 9 ,s1 :i* corresponds to ! l9e ,,ss' 9 ,s'1 is !re=e ! only l9e ?r\gh ,s :i* has b "o po9t 9 common ) ! curve1 or :i* cuts ! curve 9 two co9cid5t po9ts4 ,su* a l9e is call$ a tang5t to ! curve1 t\*+ ! latt] at ! po9t ,s1 :i* is call$ ! po9t ( contact4 ,9 ! same mann] we get 9 ! reciprocal inve/igation ! result t ?r\gh e po9t 9 "o ( ! r[s1 say 9 s1 two tang5ts may be drawn to ! curve1 ! "o be+ s1 ! o!r ! l9e jo9+ ! po9t to its correspond+ po9t 9 s'4 ,!re is1 h[ev]1 "o po9t ,p 9 s = :i* ^! two l9es co9cide4 ,su* a po9t 9 "o ( ! tang5ts is call$ ! po9t ( contact ( ! tang5t4 ,we ?us get-- ,!orem4--,to ! l9e jo9+ ,!orem4--,to ! po9t ( ! c5tres ( ! projective int]section ( ! bases ( two p5cils z a l9e 9 "o p5cil projective r[s z a po9t 9 corresponds 9 ! o!r ! "o r[ corresponds 9 ! o!r tang5t at its c5tre4 ! po9t ( contact ( its base4 ,s #di4 ,two projective p5cils are det]m9$ if ?ree pairs ( correspond+ l9es are giv54 ,h5ce if a#a1 b#a1 c#a are ?ree l9es 9 a p5cil ,s#a1 & a#b1 b#b1 c#b ! correspond+ l9es 9 a projective p5cil ,s#b1 ! correspond5ce & !re=e ! curve ( ! second ord] g5]at$ by ! po9ts ( int]section ( correspond+ rays is det]m9$4 ,( ?is curve we "k ! two c5tres ,s#a & ,s#b1 & ! ?ree po9ts a#a;a#b1 b#a;b#b1 c#a;c#b1 h5ce five po9ts 9 all4 ,?is & ! reciprocal consid]ations enable u to solve ! foll[+ two problems3 ,problem4--,to con/ruct a curve ,problem4--,to con/ruct a curve ( ! second ord]1 ( :i* ( ! second class1 ( :i* five po9ts ,s#a1 ,s#b1 ,a1 ,b1 ,c are five tang5ts u#a1 u#b1 a1 b1 c giv54 are giv54 ,9 ord] to solve ! left-h& problem1 we take two ( ! giv5 po9ts1 say ,s#a & ,s#b1 z c5tres ( p5cils4 ,^! we make projective by tak+ ! rays a#a1 b#a1 c#a1 :i* jo9 ,s#a to ,a1 ,b1 ,c respectively1 z correspond+ to ! rays a#b1 b#b1 c#b1 :i* jo9 ,s#b to ,a1 ,b1 ,c respectively1 s t ?ree rays meet _! correspond+ rays at ! giv5 po9ts ,a1 ,b1 ,c4 ,?is det]m9es ! correspond5ce ( ! p5cils :i* w g5]ate a curve ( ! second ord] pass+ ?r\gh ,a1 ,b1 ,c & ?r\gh ! c5tres ,s#a & ,s#b1 h5ce ?r\gh ! five giv5 po9ts4 ,to f9d m po9ts on ! curve we h to con/ruct = any ray 9 ,s#a ! correspond+ ray 9 ,s#b4 ,?is has be5 done 9 ,s #cf4 ,b we repeat ! con/ruction 9 ord] to d$uce fur!r prop]ties f x4 ,we also solve ! "r-h& problem4 ,"h we select two1 viz4 u#a1 u#b ( ! five giv5 l9es1 u#a1 u#b1 a1 b1 c1 z bases ( two r[s1 & ! po9ts ,a#a1 ,b#a1 ,c#a :]e a1 b1 c cut u#a z correspond+ to ! po9ts ,a#b1 ,b#b1 ,c#b :]e a1 b1 c cut u#b4 ,we get !n ! foll[+ solutions ( ! two problems3 ,solution4--,?r\gh ! po9t ,a ,solution4--,9 ! l9e a take draw any two l9es1 u#a & u#b any two po9ts ,s#a & ,s#b z 7fig4 #af71 ! fir/ u#a to cut c5tres ( p5cils 7fig4 #ag71 ! p5cil ,s#a 9 a r[ ,,ab#a;,c#a1 ! fir/ ,s#a 7,a#a;,b#a;,c#a7 to project ! o!r u#b to cut ! p5cil ! r[ u#a1 ! o!r ,s#b ,s#b 9 a r[ ,,ab#b;,c#b4 ,^! two 7,a#b;,b#b;,c#b7 to project ! r[ u#b4 r[s w be p]spective1 z ! ,^! two p5cils w be po9t ,a corresponds to itself1 p]spective1 ! l9e ,s#a;,a#a be+ & ! c5tre ( projection ! same z ! correspond+ w be ! po9t ,s1 :]e ! l9e ,s#b;,a#b1 & ! axis ( l9es ,b#a;,b#b & ,c#a;,c#b meet4 ,to projection w be ! l9e u1 f9d n[ = any ray d#a 9 ,s#a :i* jo9s ! int]section ,b its correspond+ ray d#b 9 ,s#b1 ( ,s#a;,b#a & ,s#b;,b#b to ! we det]m9e ! po9t ,d#a :]e int]section ,c ( ,s#a;,c#a & ,s#b;,c#b4 d#a cuts u#a1 project ?is po9t ,to f9d n[ = any po9t ,d#a 9 f ,s to ,d#b on u#b & jo9 ,s#b u#a ! correspond+ po9t ,d#b 9 to ,d#b4 ,?is w be ! requir$ u#b1 we draw ,s#a;,d#a & project ! ray d#b :i* cuts d#a at "s po9t ,d :]e ?is l9e cuts u po9t ,d on ! curve4 f ,s#b to u#b4 ,?is w give ! requir$ po9t ,d#b1 & ! l9e d jo9+ ,d#a to ,d#b w be a new tang5t to ! curve4 ,s #ej4 ,^! con/ructions prove1 :5 rightly int]pret$1 v important prop]ties ( ! curves 9 "q4 ,illu/ration3 ,,fig4 #af4 ,if 9 fig4 #af we draw 9 ! p5cil ,s#a ! ray k#a :i* passes ?r\gh ! auxili>y c5tre ,s1 x w be f\nd t ! correspond+ ray k#b cuts x on u#b4 ,h5ce-- ,!orem4--,9 ! above ,!orem4--,9 ! above con/ruction ! bases ( ! con/ruction 7fig4 #ag7 ! auxili>y r[s u#a & u#b cut ! tang5ts to ! curve f ! curve :]e !y cut ! rays c5tres ( ! auxili>y p5cils ,s#b,s & ,s#a,s respectively4 ,s#a & ,s#b are ! l9es :i* pass ?r\gh u#bu & u#au respectively4 ,z ,a is any giv5 po9t on ! curve1 & u#a any l9e ?r\gh x1 we h solv$ ! problems3 ,problem4--,to f9d ! second ,problem4--,to f9d ! second po9t 9 :i* any l9e ?r\gh tang5t :i* c be drawn f a kn[n po9t on ! curve cuts any po9t 9 a giv5 tang5t to ! curve4 ! curve4 ,if we det]m9e 9 ,s#a 7fig4 #af7 ! ray correspond+ to ! ray ,s#b,s#a 9 ,s#b1 we get ! tang5t at ,s#a4 ,simil>ly1 we c det]m9e ! po9t ( contact ( ! tang5ts u#a or u#b 9 fig4 #ag4 ,illu/ration3 ,,fig4 #ag4 ,s #ea4 ,if five po9ts are giv51 ( :i* n ?ree are 9 a l9e1 !n we c1 z has j be5 %[n1 always draw a curve ( ! second ord] ?r\gh !m2 we select two ( ! po9ts z c5tres ( projective p5cils1 & !n "o su* curve is det]m9$4 ,x w be pres5tly %[n t we get always ! same curve if two o!r po9ts are tak5 z c5tres ( p5cils1 t !re=e five po9ts det]m9e "o curve ( ! second ord]1 & reciprocally1 t five tang5ts det]m9e "o curve ( ! second class4 ,six po9ts tak5 at r&om w !re=e n lie on a curve ( ! second ord]4 ,9 ord] t ?is may be ! case a c]ta9 condition has to be satisfi$1 & ?is condition is easily obta9$ f ! con/ruction 9 ,s #di1 fig4 #af4 ,if we consid] ! conic det]m9$ by ! five po9ts ,a1 ,s#a1 ,s#b1 ,k1 ,l1 !n ! po9t ,d w be on ! curve if1 & only if1 ! po9ts on ,d#a1 ,s1 ,d#b be 9 a l9e4 ,illu/ration3 ,,fig4 #ah4 ,?is may be /at$ diff]5tly if we take ,,aks#a;,,ds#b,l 7figs4 #af & #ah7 z a hexagon inscrib$ 9 ! conic1 !n ,,ak & ,,ds#b w be opposite sides1 s w be ,,ks#a & ,s#b,l1 z well z ,s#a;,d & ,,la4 ,! fir/ two meet 9 ,d#b1 ! o!rs 9 ,s & ,d#a respectively4 ,we may !re=e /ate ! requir$ condition1 toge!r ) ! reciprocal "o1 z foll[s3-- ,pascal's ,!orem4--,if a hexagon ,brian*on's ,!orem4--,if a be inscrib$ 9 a curve ( ! hexagon be circumscrib$ ab\t second ord]1 !n ! a curve ( ! second class1 !n int]sections( opposite sides ! l9es jo9+ opposite v]tices are ?ree po9ts 9 a l9e4 are ?ree l9es meet+ 9 a po9t4 ,^! celebrat$ !orems1 :i* are kn[n by ! names ( _! discov]]s1 are p]haps ! mo/ fruitful 9 ! :ole !ory ( conics4 ,be=e we g ov] to _! applications we h to %[ t we obta9 ! same curve if we take1 in/ead ( ,s#a1 ,s#b1 any two o!r po9ts on ! curve z c5tres ( projective p5cils4 ,s #eb4 ,we "k t ! curve dep5ds only ^u ! correspond5ce betwe5 ! p5cils ,s#a & ,s#b1 & n ^u ! special con/ruction us$ = f9d+ new po9ts on ! curve4 ,! po9t ,a 7fig4 #af or #ah71 ?r\gh :i* ! two auxili>y r[s u#a1 u#b 7 drawn1 may !re=e be *ang$ to any o!r po9t on ! curve4 ,let u n[ suppose ! curve drawn1 & keep ! po9ts ,s#a1 ,s#b1 ,k1 ,l & ,d1 & h5ce also ! po9t ,s fix$1 :il/ we move ,a along ! curve4 ,!n ! l9e ,,al w describe a p5cil ab\t ,l z c5tre1 & ! po9t ,d#a a r[ on ,s#a;,d p]spective to ! p5cil ,l4 ,at ! same "t ,,ak describes a p5cil ab\t ,k & ,d#b a r[ p]spective to x on ,s#b;,d4 ,b by ,pascal's !orem ,d#a & ,d#b w always lie 9 a l9e ) ,s1 s t ! r[s describ$ by ,d#a & ,d#b are p]spective4 ,x foll[s t ! p5cils ,k & ,l w !mselves be projective1 correspond+ rays meet+ on ! curve4 ,?is proves t we get ! same curve :atev] pair ( ! five giv5 po9ts we take z c5tres ( projective p5cils4 ,h5ce-- ,only "o curve ( ! second ,only "o curve ( ! second ord] c be drawn :i* passes class c be drawn :i* t\*es ?r\gh five giv5 po9ts4 five giv5 l9es4 ,we h se5 t if on a curve ( ! second ord] two po9ts co9cide at ,a1 ! l9e jo9+ !m becomes ! tang5t at ,a4 ,if1 !re=e1 a po9t on ! curve & its tang5t are giv51 ?is w be equival5t to hav+ giv5 two po9ts on ! curve4 ,simil>ly1 if on ! curve ( second class a tang5t & its po9t ( contact are giv51 ?is w be equival5t to two giv5 tang5ts4 ,we may !re=e ext5d ! la/ !orem3 ,only "o curve ( ! second ,only "o curve ( ! second ord] c be drawn1 ( :i* class c be drawn1 ( :i* f\r f\r po9ts & ! tang5t at tang5ts & ! po9t ( contact one( !m1 or ?ree po9ts at "o ( !m1 or ?ree tang5ts & ! tang5ts at two ( & ! po9ts ( contact at two !m1 are giv54 ( !m1 are giv54 ,s #ec4 ,at ! same "t x has be5 prov$3 ,if all po9ts on a curve ( ! ,all tang5ts to a curve ( second second ord] be jo9$ to any class are cut by any two ( two ( !m1 !n ! two !m 9 projective r[s1 ^? p5cils ?us =m$ are be+ correspond+ po9ts :i* projective1 ^? rays be+ lie on ! same tang5t4 ,h5ce-- correspond+ :i* meet on ! curve4 ,h5ce-- ,! cross-ratio ( f\r rays ,! cross-ratio ( ! f\r jo9+ a po9t ,s on a curve ( po9ts 9 :i* any tang5t u is second ord] to f\r fix$ cut by f\r fix$ tang5ts a1 b1 c1 po9ts ,a1 ,b1 ,c1 ,d 9 ! curve d is indep5d5t ( ! position ( is indep5d5t ( ! position u1 & is call$ ! cross-ratio ( ( ,s1 & is call$ ! cross- ! f\r tang5ts a1 b1 c1 d4 ratio ( ! f\r po9ts ,a1 ,b1 ,c1 ,d4 ,if ?is cross-ratio equals -#a ,if ?is cross-ratio equals -#a ! f\r po9ts are said to be ! f\r tang5ts are said to be f\r h>monic po9ts4 f\r h>monic tang5ts4 ,we h se5 t a curve ( second ord]1 z g5]at$ by projective p5cils1 has at ! c5tre ( ea* p5cil "o tang5t2 & fur!r1 t any po9t on ! curve may be tak5 z c5tre ( su* p5cil4 ,h5ce-- ,a curve ( second ord] has ,a curve ( second class has on at e po9t "o tang5t4 e tang5t a po9t ( contact4 ,s #ed4 ,we return to ,pascal's & ,brian*on's !orems & _! applications1 & %all1 z be=e1 /ate ! results bo? = curves ( ! second ord] & curves ( ! second class1 b prove !m only = ! =m]4 ,pascal's !orem may be us$ :5 five po9ts are giv5 to f9d m po9ts on ! curve1 viz4 x enables u to f9d ! po9t :]e any l9e ?r\gh "o ( ! giv5 po9ts cuts ! curve aga94 ,x is conv5i5t1 9 mak+ use ( ,pascal's !orem1 to numb] ! po9ts1 to indicate ! ord] 9 :i* !y are to be tak5 9 =m+ a hexagon1 :i*1 by ! way1 may be done 9 #fj diff]5t ways4 ,x w be se5 t #a #b 7leav+ out #c7 #d #e are opposite sides1 s are #b #c & 7leav+ out #d7 #e #f1 & also #c #d & 7leav+ out #e7 #f #a4 ,if ! po9ts #a #b #c #d #e are giv51 & we want a #f? po9t on a l9e drawn ?r\gh #a1 we "k all ! sides ( ! hexagon ) ! exception ( #e #f1 & ?is is f\nd by ,pascal's !orem4 ,if ?is l9e %\ld happ5 to pass ?r\gh #a1 !n #f & #a co9cide1 or ! l9e #f #a is ! tang5t at #a4 ,& always if two consecutive v]tices ( ! hexagon approa* ne>] & ne>]1 !n ! side jo9+ !m w ultimately become a tang5t4 ,we may !re=e consid] a p5tagon inscrib$ 9 a curve ( second ord] & ! tang5t at "o ( its v]tices z a hexagon1 & ?us get ! !orem3 ,e p5tagon inscrib$ 9 a ,e p5tagon circumscrib$ curve ( second ord] has ! ab\t a curve ( ! second class prop]ty t ! int]sections has ! prop]ty t ! l9es ( two pairs ( non-consecutive :i* jo9 two pairs ( non- sides lie 9 a l9e ) ! consecutive v]tices meet on t po9t :]e ! fif? side cuts l9e :i* jo9s ! fif? v]tex ! tang5t at ! opposite to ! po9t ( contact ( ! v]tex4 opposite side4 ,?is enables u also to solve ! foll[+ problems4 ,giv5 five po9ts on a curve ( ,giv5 five tang5ts to a curve second ord] to con/ruct ! ( second class to con/ruct ! tang5t at any "o ( !m4 po9t ( contact ( any "o ( !m4 ,illu/ration3 ,,fig4 #ai4 ,if two pairs ( adjac5t v]tices co9cide1 ! hexagon becomes a quadrilat]al1 ) tang5ts at two v]tices4 ,^! we take to be opposite1 & get ! foll[+ !orems3 ,if a quadrilat]al be inscrib$ ,if a quadrilat]al be circumscrib$ 9 a curve ( second ord]1 ! ab\t a curve ( second int]sections ( opposite sides1 class1 ! l9es jo9+ opposite & also ! int]sections ( v]tices1 & also ! l9es jo9+ ! tang5ts at opposite po9ts ( contact ( opposite v]tices1 lie 9 a l9e 7fig4 sides1 meet 9 a po9t4 #ai74 ,illu/ration3 ,,fig4 #bj4 ,if we consid] ! hexagon made up ( a triangle & ! tang5ts at its v]tices1 we get-- ,if a triangle is inscrib$ 9 a ,if a triangle be circumscrib$ curve ( ! second ord]1 ! ab\t a curve ( second class1 po9ts 9 :i* ! sides are ! l9es :i* jo9 ! v]tices cut by ! tang5ts at ! to ! po9ts ( contact ( ! opposite v]tices meet 9 a opposite sides meet 9 a po9t po9t4 7fig4 #bj74 ,s #ee4 ,( ^! !orems1 ^? ab\t ! quadrilat]al give rise to a numb] ( o!rs4 ,f\r po9ts ,a1 ,b1 ,c1 ,d may 9 ?ree diff]5t ways be =m$ into a quadrilat]al1 = we may take !m 9 ! ord] ,,abcd1 or ,,acbd1 or ,,acdb1 s t ei!r ( ! po9ts ,b1 ,c1 ,d may be tak5 z ! v]tex opposite to ,a4 ,accord+ly we may apply ! !orem 9 ?ree diff]5t ways4 ,let ,a1 ,b1 ,c1 ,d be f\r po9ts on a curve ( second ord] 7fig4 #ba71 & let u take !m z =m+ a quadrilat]al by tak+ ! po9ts 9 ! ord] ,,abcd1 s t ,a1 ,c & also ,b1 ,d are pairs ( opposite v]tices4 ,!n ,p1 ,q w be ! po9ts :]e opposite sides meet1 & ,e1 ,f ! int]sections ( tang5ts at opposite v]tices4 ,! f\r po9ts ,p1 ,q1 ,e1 ,f lie !re=e 9 a l9e4 ,! quadrilat]al ,,acbd gives u 9 ! same way ! f\r po9ts ,q1 ,r1 ,g1 ,h 9 a l9e1 & ! quadrilat]al ,,abdc a l9e conta9+ ! f\r po9ts ,r1 ,p1 ,i1 ,k4 ,^! ?ree l9es =m a triangle ,,pqr4 ,! relation betwe5 ! po9ts & l9es 9 ?is figure may be express$ m cle>ly if we consid] ,,abcd z a f\r-po9t inscrib$ 9 a conic1 & ! tang5ts at ^! po9ts z a f\r-side circumscrib$ ab\t x1--viz4 x w be se5 t ,p1 ,q1 ,r are ! diagonal po9ts ( ! f\r-po9t ,,abcd1 :il/ ! sides ( ! triangle ,,pqr are ! diagonals ( ! circumscrib+ f\r-side4 ,h5ce ! !orem-- ,any f\r-po9t on a curve ( ! second ord] & ! f\r-side =m$ by ! tang5ts at ^! po9ts /& 9 ?is relation t ! diagonal po9ts ( ! f\r-po9t lie 9 ! diagonals ( ! f\r-side4 ,& conv]sely1 ,if a f\r-po9t & a circumscrib$ f\r-side /& 9 ! above relation1 !n a curve ( ! second ord] may be describ$ :i* passes ?r\gh ! f\r po9ts & t\*es !re ! f\r sides ( ^! figures4 ,t ! la/ "p ( ! !orem is true foll[s f ! fact t ! f\r po9ts ,a1 ,b1 ,c1 ,d & ! l9e a1 z tang5t at ,a1 det]m9e a curve ( ! second ord]1 & ! tang5ts to ?is curve at ! o!r po9ts ,b1 ,c1 ,d are giv5 by ! con/ruction :i* leads to fig4 #ba4 ,illu/ration3 ,,fig4 #ba4 ,! !orem reciprocal to ! la/ is-- ,any f\r-side circumscrib$ ab\t a curve ( second class & ! f\r-po9t =m$ by ! po9ts ( contact /& 9 ?is relation t ! diagonals ( ! f\r-side pass ?r\gh ! diagonal po9ts ( ! f\r-po9t4 ,& conv]sely1 ,if a f\r-side & an inscrib$ f\r-po9t /& 9 ! above relation1 !n a curve ( ! second class may be describ$ :i* t\*es ! sides ( ! f\r-side at ! po9ts ( ! f\r-po9t4 ,s #ef4 ,! f\r-po9t & ! f\r-side 9 ! two reciprocal !orems are alike4 ,h5ce if we h a f\r-po9t ,,abcd & a f\r-side abcd relat$ 9 ! mann] describ$1 !n n only may a curve ( ! second ord] be drawn1 b also a curve ( ! second class1 :i* bo? t\* ! l9es a1 b1 c1 d at ! po9ts ,a1 ,b1 ,c1 ,d4 ,! curve ( second ord] is already m ?an det]m9$ by ! po9ts ,a1 ,b1 ,c & ! tang5ts a1 b1 c at ,a1 ,b & ,c4 ,! po9t ,d may !re=e be any po9t on ?is curve1 & d any tang5t to ! curve4 ,on ! o!r h& ! curve ( ! second class is m ?an det]m9$ by ! ?ree tang5ts a1 b1 c & _! po9ts ( contact ,a1 ,b1 ,c1 s t d is any tang5t to ?is curve4 ,x foll[s t e tang5t to ! curve ( second ord] is a tang5t ( a curve ( ! second class hav+ ! same po9t ( contact4 ,9 o!r words1 ! curve ( second ord] is a curve ( second class1 & vice v]sa4 ,h5ce ! important !orems-- ,e curve ( second ord] is ,e curve ( second class is a a curve ( second class4 curve ( second ord]4 ,! curves ( second ord] & ( second class1 hav+ ?us be5 prov$ to be id5tical1 %all h5ce=? be call$ by ! common "n ( ,conics4 ,= ^! curves hold1 !re=e1 all prop]ties :i* h be5 prov$ = curves ( second ord] or ( second class4 ,we may !re=e n[ /ate ,pascal's & ,brian*on's !orem ?us-- ,pascal's ,!orem4--,if a hexagon be inscrib$ 9 a conic1 !n ! int]sections ( opposite sides lie 9 a l9e4 ,brian*on's ,!orem4--,if a hexagon be circumscrib$ ab\t a conic1 !n ! diagonals =m+ opposite c5tres meet 9 a po9t4 ,s #eg4 ,if we suppose 9 fig4 #ba t ! po9t ,d toge!r ) ! tang5t d moves along ! curve1 :il/ ,a1 ,b1 ,c & _! tang5ts a1 b1 c rema9 fix$1 !n ! ray ,,da w describe a p5cil ab\t ,a1 ! po9t ,q a projective r[ on ! fix$ l9e ,,bc1 ! po9t ,f ! r[ b1 & ! ray ,,ef a p5cil ab\t ,e4 ,b ,,ef passes always ?r\gh ,q4 ,h5ce ! p5cil describ$ by ,,ad is projective to ! p5cil describ$ by ,,ef1 & !re=e to ! r[ describ$ by ,f on b4 ,at ! same "t ! l9e ,,bd describes a p5cil ab\t ,b projective to t describ$ by ,,ad 7,s #ec74 ,!re=e ! p5cil ,,bd & ! r[ ,f on b are projective4 ,h5ce-- ,if on a conic a po9t ,a be tak5 & ! tang5t a at ?is po9t1 !n ! cross-ratio ( ! f\r rays :i* jo9 ,a to any f\r po9ts on ! curve is equal to ! cross-ratio ( ! po9ts 9 :i* ! tang5ts at ^! po9ts cut ! tang5t at ,a4 ,s #eh4 ,!re are !orems ab\t cones ( second ord] & second class 9 a p5cil :i* are reciprocal to ! above1 accord+ to ,s #dc4 ,we m5tion only a few ( ! m important ones4 ,! locus ( int]sections ( correspond+ planes 9 two projective axial p5cils ^: axes meet is a cone ( ! second ord]4 ,! envelope ( planes :i* jo9 correspond+ l9es 9 two projective flat p5cils1 n 9 ! same plane1 is a cone ( ! second class4 ,cones ( second ord] & cones ( second class are id5tical4 ,e plane cuts a cone ( ! second ord] 9 a conic4 ,a cone ( second ord] is uniquely det]m9$ by five ( its edges or by five ( its tang5t planes1 or by f\r edges & ! tang5t plane at "o ( !m1 c4 c4 ,pascal's ,!orem4--,if a solid angle ( six faces be inscrib$ 9 a cone ( ! second ord]1 !n ! int]sections ( opposite faces are ?ree l9es 9 a plane4 ,brian*on's ,!orem4--,if a solid angle ( six edges be circumscrib$ ab\t a cone ( ! second ord]1 !n ! planes ?r\gh opposite edges meet 9 a l9e4 ,ea* ( ! o!r !orems ab\t conics may be /at$ = cones ( ! second ord]4 ,s #ei4 ,projective ,def9itions ( ! ,conics4--,we n[ consid] ! %ape ( ! conics4 ,we "k t any l9e 9 ! plane ( ! conic1 & h5ce t ! l9e at inf9ity1 ei!r has no po9t 9 common ) ! curve1 or "o 7c\nt+ = two co9cid5t po9ts7 or two di/9ct po9ts4 ,if ! l9e at inf9ity has no po9t on ! curve ! latt] is altoge!r f9ite1 & is call$ an ,ellipse 7fig4 #ba74 ,if ! l9e at inf9ity has only "o po9t 9 common ) ! conic1 ! latt] ext5ds to inf9ity1 & has ! l9e at inf9ity a tang5t4 ,x is call$ a ,p>abola 7fig4 #bb74 ,if1 la/ly1 ! l9e at inf9ity cuts ! curve 9 two po9ts1 x consi/s ( two sep>ate p>ts :i* ea* ext5d 9 two bran*es to ! po9ts at inf9ity :]e !y meet4 ,! curve is 9 ?is case call$ an ,hyp]bola 7see fig4 #bj74 ,! tang5ts at ! two po9ts at inf9ity are f9ite because ! l9e at inf9ity is n a tang5t4 ,!y are call$ ,asymptotes4 ,! bran*es ( ! hyp]bola approa* ^! l9es indef9itely z a po9t on ! curves moves to inf9ity4 ,illu/ration3 ,,fig4 #bb4 ,s #fj4 ,t ! circle belongs to ! curves ( ! second ord] is se5 at once if we /ate 9 a slightly diff]5t =m ! !orem t 9 a circle all angles at ! circumf]5ce /&+ ^u ! same arc are equal4 ,if two po9ts ,s#a1 ,s#b on a circle be jo9$ to any o!r two po9ts ,a & ,b on ! circle1 !n ! angle includ$ by ! rays ,s#a;,a & ,s#a;,b is equal to t betwe5 ! rays ,s#b;,a & ,s#b;,b1 s t z ,a moves along ! circumf]5ce ! rays ,s#a;,a & ,s#b;,a describe equal & !re=e projective p5cils4 ,! circle c ?us be g5]at$ by two projective p5cils1 & is a curve ( ! second ord]4 ,if we jo9 a po9t 9 space to all po9ts on a circle1 we get a 7circul>7 cone ( ! second ord] 7,s #dc74 ,e plane section ( ?is cone is a conic4 ,?is conic w be an ellipse1 a p>abola1 or an hyp]bola1 accord+ z ! l9e at inf9ity 9 ! plane has no1 "o or two po9ts 9 common ) ! conic 9 :i* ! plane at inf9ity cuts ! cone4 ,x foll[s t our curves ( second ord] may be obta9$ z sections ( a circul> cone1 & t !y are id5tical ) ! ,conic ,sections ( ! ,greek ma!maticians4 ,s #fa4 ,any two tang5ts to a p>abola are cut by all o!rs 9 projective r[s2 b ! l9e at inf9ity be+ "o ( ! tang5ts1 ! po9ts at inf9ity on ! r[s are correspond+ po9ts1 & ! r[s !re=e simil>4 ,h5ce ! !orem-- ,! tang5ts to a p>abola cut ea* o!r proportionally4 ,,pole ,,& ,,pol> ,s #fb4 ,we return once aga9 to fig4 #ba1 :i* we obta9$ 9 ,s #ee4 ,if a f\r-side be circumscrib$ ab\t & a f\r-po9t inscrib$ 9 a conic1 s t ! v]tices ( ! second are ! po9ts ( contact ( ! sides ( ! fir/1 !n ! triangle =m$ by ! diagonals ( ! fir/ is ! same z t =m$ by ! diagonal po9ts ( ! o!r4 ,su* a triangle w be call$ a pol>-triangle ( ! conic1 s t ,,pqr 9 fig4 #ba is a pol>-triangle4 ,x has ! prop]ty t on ! side p opposite ,p meet ! tang5ts at ,a & ,b1 & also ^? at ,c & ,d4 ,f ! h>monic prop]ties ( f\r-po9ts & f\r-sides x foll[s fur!r t ! po9ts ,l1 ,m1 :]e x cuts ! l9es ,,ab & ,,cd1 are h>monic conjugates ) reg>d to ,,ab & ,,cd respectively4 ,if ! po9t ,p is giv51 & we draw a l9e ?r\gh x1 cutt+ ! conic 9 ,a & ,b1 !n ! po9t ,q h>monic conjugate to ,p ) reg>d to ,,ab1 & ! po9t ,h :]e ! tang5ts at ,a & ,b meet1 are det]m9$4 ,b !y lie bo? on p1 & !re=e ?is l9e is det]m9$4 ,if we n[ draw a second l9e ?r\gh ,p1 cutt+ ! conic 9 ,c & ,d1 !n ! po9t ,m h>monic conjugate to ,p ) reg>d to ,,cd1 & ! po9t ,g :]e ! tang5ts at ,c & ,d meet1 mu/ also lie on p4 ,z ! fir/ l9e ?r\gh ,p already det]m9es p1 ! second may be any l9e ?r\gh ,p4 ,n[ e two l9es ?r\gh ,p det]m9e a f\r-po9t ,,abcd on ! conic1 & !re=e a pol>-triangle :i* has "o v]tex at ,p & its opposite side at p4 ,?is result1 toge!r ) its reciprocal1 gives ! !orems-- ,all pol>-triangles :i* h "o v]tex 9 common h also ! opposite side 9 common4 ,all pol>-triangles :i* h "o side 9 common h also ! opposite v]tex 9 common4 ,s #fc4 ,to any po9t ,p 9 ! plane (1 b n on1 a conic corresponds ?us "o l9e p z ! side opposite to ,p 9 all pol>-triangles :i* h "o v]tex at ,p1 & reciprocally to e l9e p corresponds "o po9t ,p z ! v]tex opposite to p 9 all triangles :i* h p z "o side4 ,we call ! l9e p ! pol> ( ,p1 & ! po9t ,p ! pole ( ! l9e p ) reg>d to ! conic4 ,if a po9t lies on ! conic1 we call ! tang5t at t po9t its pol>2 & reciprocally we call ! po9t ( contact ! pole ( tang5t4 ,s #fd4 ,f ^! def9itions & =m] results foll[-- ,! pol> ( any po9t ,p n ,! pole ( any l9e p n a on ! conic is a l9e p1 :i* tang5t to ! conic is a po9t has ! foll[+ prop]ties3-- ,p1 :i* has ! foll[+ prop]ties3-- #a4 ,on e l9e ?r\gh ,p #a4 ,( all l9es ?r\gh a po9t :i* cuts ! conic1 ! pol> on p f :i* two tang5ts ( ,p conta9s ! h>monic may be drawn to ! conic1 ! conjugate ( ,p ) reg>d to pole ,p conta9s ! l9e :i* is ^? po9ts on ! conic4 h>monic conjugate to p1 ) reg>d to ! two tang5ts4 #b4 ,if tang5ts c be drawn #b4 ,if p cuts ! conic1 ! f ,p1 _! po9ts ( contact tang5ts at ! int]sections lie on p4 meet at ,p4 #c4 ,tang5ts drawn at ! #c4 ,! po9t ( contact ( po9ts :]e any l9e ?r\gh ,p tang5ts drawn f any po9t cuts ! conic meet on p2 & on p to ! conic lie 9 a l9e conv]sely1 ) ,p2 & conv]sely1 #d4 ,if f any po9t on p1 #d4 ,tang5ts drawn at po9ts tang5ts be drawn1 _! po9ts :]e any l9e ?r\gh ,p cuts ! ( contact w lie 9 a l9e conic meet on p4 ) ,p4 #e4 ,any f\r-po9t on ! conic #e4 ,any f\r-side circumscrib$ :i* has "o diagonal po9t at ab\t a conic :i* has "o ,p has ! o!r two ly+ on p4 diagonal on p has ! o!r two meet+ at ,p4 ,! tru? ( #b foll[s f #a4 ,if ,t be a po9t :]e p cuts ! conic1 !n "o ( ! po9ts :]e ,,pt cuts ! conic1 & :i* are h>monic conjugates ) reg>d to ,,pt1 co9cides ) ,t2 h5ce ! o!r does--t is1 ,,pt t\*es ! curve at ,t4 ,t #d is true foll[s ?us3 ,if we draw f a po9t ,h on ! pol> "o tang5t a to ! conic1 jo9 its po9t ( contact ,a to ! pole ,p1 det]m9e ! second po9t ( int]section ,b ( ?is l9e ) ! conic1 & draw ! tang5t at ,b1 x w pass ?r\gh ,h1 & w !re=e be ! second tang5t :i* may be drawn f ,h to ! curve4 ,s #fe4 ,! second prop]ty ( ! pol> or pole gives rise to ! !orem-- ,f a po9t 9 ! plane ( a ,a l9e 9 ! plane ( a conic conic1 two1 "o or no tang5ts has two1 "o or no po9ts 9 may be drawn to ! conic1 common ) ! conic1 accord+ z its pol> has two1 z two1 "o or no tang5ts "o1 or no po9ts 9 common c be drawn f its pole to ! ) ! curve4 conic4 ,( any po9t 9 ! plane ( a conic we say t x 0 )\t1 on or )9 ! curve accord+ z two1 "o or no tang5ts to ! curve pass ?r\gh x4 ,! po9ts on ! conic sep>ate ^? )9 ! conic f ^? )\t4 ,t ?is is true = a circle is kn[n f elem5t>y geometry4 ,t x also holds = o!r conics foll[s f ! fact t e conic may be consid]$ z ! projection ( a circle1 :i* w be prov$ lat] on4 ,! fif? prop]ty ( pole & pol> /at$ 9 ,s #fd %[s h[ to f9d ! pol> ( any po9t & ! pole ( any l9e by aid ( ! /raight-edge only4 ,practically x is (t5 conv5i5t to draw ?ree secants ?r\gh ! pole1 & to det]m9e only "o ( ! diagonal po9ts = two ( ! f\r-po9ts =m$ by pairs ( ^! l9es & ! conic 7fig4 #bb74 ,^! con/ructions also solve ! problem-- ,f a po9t )\t a conic1 to draw ! two tang5ts to ! conic by aid ( ! /raight-edge only4 ,= we ne$ only draw ! pol> ( ! po9t 9 ord] to f9d ! po9ts ( contact4 ,s #ff4 ,! prop]ty ( a pol>-triangle may n[ be /at$ ?us-- ,9 a pol>-triangle ea* side is ! pol> ( ! opposite v]tex1 & ea* v]tex is ! pole ( ! opposite side4 ,illu/ration3 ,,fig4 #bc4 ,if ,p is "o v]tex ( a pol>-triangle1 !n ! o!r v]tices1 ,q & ,r1 lie on ! pol> p ( ,p4 ,"o ( ^! v]tices we may *oose arbitr>ily4 ,= if f any po9t ,q on ! pol> a secant be drawn cutt+ ! conic 9 ,a & ,d 7fig4 #bc71 & if ! l9es jo9+ ^! po9ts to ,p cut ! conic aga9 at ,b & ,c1 !n ! l9e ,,bc w pass ?r\gh ,q4 ,h5ce ,p & ,q are two ( ! v]tices on ! pol>-triangle :i* is det]m9$ by ! f\r-po9t ,,abcd4 ,! ?ird v]tex ,r lies also on ! l9e p4 ,x foll[s1 !re=e1 also-- ,if ,q is a po9t on ! pol> ( ,p1 !n ,p is a po9t on ! pol> ( ,q2 & reciprocally1 ,if q is a l9e ?r\gh ! pole ( p1 !n p is a l9e ?r\gh ! pole ( q4 ,?is is a v important !orem4 ,x may also be /at$ ?us-- ,if a po9t moves along a l9e describ+ a r[1 its pol> turns ab\t ! pole ( ! l9e describ+ a p5cil4 ,?is p5cil is projective to ! r[1 s t ! cross-ratio ( f\r poles 9 a r[ equals ! cross-ratio ( its f\r pol>s1 :i* pass ?r\gh ! pole ( ! r[4 ,to prove ! la/ "p1 let u suppose t ,p1 ,a & ,b 9 fig4 #bc rema9 fix$1 :il/ ,q moves along ! pol> p ( ,p4 ,?is w make ,,cd turn ab\t ,p & move ,r along p1 :il/ ,,qd & ,,rd describe projective p5cils ab\t ,a & ,b4 ,h5ce ,q & ,r describe projective r[s1 & h5ce ,,pr1 :i* is ! pol> ( ,q1 describes a p5cil projective to ei!r4 ,s #fg4 ,two po9ts1 ( :i* "o1 & !re=e ea*1 lies on ! pol> ( ! o!r1 are said to be conjugate ) reg>d to ! conic2 & two l9es1 ( :i* "o1 & !re=e ea*1 passes ?r\gh ! pole ( ! o!r1 are said to be conjugate ) reg>d to ! conic4 ,h5ce all po9ts conjugate to a po9t ,p lie on ! pol> ( ,p2 all l9es conjugate to a l9e p pass ?r\gh ! pole ( p4 ,if ! l9e jo9+ two conjugate poles cuts ! conic1 !n ! poles are h>monic conjugates ) reg>d to ! po9ts ( int]section2 h5ce "o lies )9 ! o!r )\t ! conic1 & all po9ts conjugate to a po9t )9 a conic lie )\t x4 ,( a pol>-triangle any two v]tices are conjugate poles1 any two sides conjugate l9es4 ,if1 !re=e1 "o side cuts a conic1 !n "o ( ! two v]tices :i* lie on ?is side is )9 & ! o!r )\t ! conic4 ,! v]tex opposite ?is side lies also )\t1 = x is ! pole ( a l9e :i* cuts ! curve4 ,9 ?is case !re=e "o v]tex lies )91 ! o!r two )\t4 ,if1 on ! o!r h&1 we beg9 ) a side :i* does n cut ! conic1 !n its pole lies )9 & ! o!r v]tices )\t4 ,h5ce-- ,e pol>-triangle has "o & only "o v]tex )9 ! conic4 ,we add1 )\t a pro(1 ! !orem-- ,! f\r po9ts 9 :i* a conic is cut by two conjugate pol>s are f\r h>monic po9ts 9 ! conic4 ,s #fh4 ,if two conics int]sect 9 f\r po9ts 7!y _c h m po9ts 9 common1 ,s #eb71 !re exi/s "o & only "o f\r-po9t :i* is inscrib$ 9 bo?1 & !re=e "o pol>-triangle common to bo?4 ,!orem4--,two conics :i* int]sect 9 f\r po9ts h always "o & only "o common pol>-triangle2 & reciprocally1 ,two conics :i* h f\r common tang5ts h always "o & only "o common pol>-triangle4 ,,diamet]s ,,& ,,axes ,,( ,,conics ,s #fi4 ,diamet]s4--,! !orems ab\t ! h>monic prop]ties ( poles & pol>s conta91 z special cases1 a numb] ( important metrical prop]ties ( conics4 ,^! are obta9$ if ei!r ! pole or ! pol> is mov$ to inf9ity1--x be+ rememb]$ t ! h>monic conjugate to a po9t at inf9ity1 ) reg>d to two po9ts ,a1 ,b1 is ! middle po9t ( ! segm5t ,,ab4 ,! mo/ important prop]ties are /at$ 9 ! foll[+ !orems3-- ,! middle po9ts ( p>allel *ords ( a conic lie 9 a l9e--viz4 on ! pol> to ! po9t at inf9ity on ! p>allel *ords4 ,?is l9e is call$ a diamet]4 ,! pol> ( e po9t at inf9ity is a diamet]4 ,! tang5ts at ! end po9ts ( a diamet] are p>allel1 & are p>allel to ! *ords bisect$ by ! diamet]4 ,all diamet]s pass ?r\gh a common po9t1 ! pole ( ! l9e at inf9ity4 ,all diamet]s ( a p>abola are p>allel1 ! pole to ! l9e at inf9ity be+ ! po9t :]e ! curve t\*es ! l9e at inf9ity4 ,9 case ( ! ellipse & hyp]bola1 ! pole to ! l9e at inf9ity is a f9ite po9t call$ ! c5tre ( ! curve4 ,a c5tre ( a conic bisects e *ord ?r\gh x4 ,! c5tre ( an ellipse is )9 ! curve1 = ! l9e at inf9ity does n cut ! ellipse4 ,! c5tre ( an hyp]bola is )\t ! curve1 because ! l9e at inf9ity cuts ! curve4 ,h5ce also-- ,f ! c5tre ( an hyp]bola two tang5ts c be drawn to ! curve :i* h _! po9t ( contact at inf9ity4 ,^! are call$ ,asymptotes 7,s #ei74 ,to con/ruct a diamet] ( a conic1 draw two p>allel *ords & jo9 _! middle po9ts4 ,to f9d ! c5tre ( a conic1 draw two diamet]s2 _! int]section w be ! c5tre4 ,s #gj4 ,conjugate ,diamet]s4--,a pol>-triangle ) "o v]tex at ! c5tre w h ! opposite side at inf9ity4 ,! o!r two sides pass ?r\gh ! c5tre1 & are call$ conjugate diamet]s1 ea* be+ ! pol> ( ! po9t at inf9ity on ! o!r4 ,( two conjugate diamet]s ea* bisects ! *ords p>allel to ! o!r1 & if "o cuts ! curve1 ! tang5ts at its ends are p>allel to ! o!r diamet]4 ,fur!r-- ,e p>allelogram inscrib$ 9 a conic has its sides p>allel to two conjugate diamet]s2 & ,e p>allelogram circumscrib$ ab\t a conic has z diagonals two conjugate diamet]s4 ,?is w be se5 by consid]+ ! p>allelogram 9 ! fir/ case z an inscrib$ f\r-po9t1 9 ! o!r z a circumscrib$ f\r-side1 & det]m9+ 9 ea* case ! correspond+ pol>-triangle4 ,! fir/ may also be enunciat$ ?us-- ,! l9es :i* jo9 any po9t on an ellipse or an hyp]bola to ! ends ( a diamet] are p>allel to two conjugate diamet]s4 ,s #ga4 ,if e diamet] is p]p5dicul> to its conjugate ! conic is a circle4 ,= ! l9es :i* jo9 ! ends ( a diamet] to any po9t on ! curve include a "r angle4 ,a conic :i* has m ?an "o pair ( conjugate diamet]s at "r angles to ea* o!r is a circle4 ,illu/ration3 ,,fig4 #bd4 ,let ,,aa' & ,,bb' 7fig4 #bd7 be "o pair ( conjugate diamet]s at "r angles to ea* o!r1 ,,cc & ,,dd' a second pair4 ,if we draw ?r\gh ! end po9t ,a ( "o diamet] a *ord ,,ap p>allel to ,,dd'1 & jo9 ,p to ,a'1 !n ,,pa & ,,pa' are1 accord+ to ,s #gj1 p>allel to two conjugate diamet]s4 ,b ,,pa is p>allel to ,,dd'1 h5ce ,,pa' is p>allel to ,,cc1 & !re=e ,,pa & ,,pa' are p]p5dicul>4 ,if we fur!r draw ! tang5ts to ! conic at ,a & ,a'1 ^! w be p]p5dicul> to ,,aa'1 !y be+ p>allel to ! conjugate diamet] ,,bb'4 ,we "k ?us five po9ts on ! conic1 viz4 ! po9ts ,a & ,a' ) _! tang5ts1 & ! po9t ,p4 ,?r\gh ^! a circle may be drawn hav+ ,,aa' z diamet]2 & z ?r\gh five po9ts "o conic only c be drawn1 ?is circle mu/ co9cide ) ! giv5 conic4 ,s #gb4 ,axes4--,conjugate diamet]s p]p5dicul> to ea* o!r are call$ axes1 & ! po9ts :]e !y cut ! curve v]tices ( ! conic4 ,9 a circle e diamet] is an axis1 e po9t on x is a v]tex2 & any two l9es at "r angles to ea* o!r may be tak5 z a pair ( axes ( any circle :i* has its c5tre at _! int]section4 ,illu/ration3 ,,fig4 #be4 ,if we describe on a diamet] ,,ab ( an ellipse or hyp]bola a circle conc5tric to ! conic1 x w cut ! latt] 9 ,a & ,b 7fig4 #be74 ,ea* ( ! semicircles 9 :i* x is divid$ by ,,ab w be p>tly )91 p>tly )\t ! curve1 & mu/ cut ! latt] !re=e aga9 9 a po9t4 ,! circle & ! conic h ?us f\r po9ts ,a1 ,b1 ,c1 ,d1 & !re=e "o pol>-triangle1 9 common 7,s #fh74 ,( ?is ! c5tre is "o v]tex1 = ! l9e at inf9ity is ! pol> to ?is po9t1 bo? ) reg>d to ! circle & ! o!r conic4 ,! o!r two sides are conjugate diamet]s ( bo?1 h5ce p]p5dicul> to ea* o!r4 ,?is gives-- ,an ellipse z well z an hyp]bola has "o pair ( axes4 ,?is reason+ %[s at ! same "t h[ to con/ruct ! axis ( an ellipse or ( an hyp]bola4 ,a p>abola has "o axis1 if we def9e an axis z a diamet] p]p5dicul> to ! *ords :i* x bisects4 ,x is easily con/ruct$4 ,! l9e :i* bisects any two p>allel *ords is a diamet]4 ,*ords p]p5dicul> to x w be bisect$ by a p>allel diamet]1 & ?is is ! axis4 ,s #gc4 ,! fir/ "p ( ! "r-h& !orem 9 ,s #fd may be /at$ ?us3 any two conjugate l9es ?r\gh a po9t ,p )\t a conic are h>monic conjugates ) reg>d to ! two tang5ts t may be drawn f ,p to ! conic4 ,if we take in/ead ( ,p ! c5tre ,c ( an hyp]bola1 !n ! conjugate l9es become conjugate diamet]s1 & ! tang5ts asymptotes4 ,h5ce-- ,any two conjugate diamet]s ( an hyp]bola are h>monic conjugates ) reg>d to ! asymptotes4 ,z ! axes are conjugate diamet]s at "r angles to "o ano!r1 x foll[s 7,s #bc7-- ,! axes ( an hyp]bola bisect ! angles betwe5 ! asymptotes4 ,illu/ration3 ,,fig4 #bf4 ,let ,o be ! c5tre ( ! hyp]bola 7fig4 #bf71 t any secant :i* cuts ! hyp]bola 9 ,c1 ,d & ! asymptotes 9 ,e1 ,f1 !n ! l9e ,,om :i* bisects ! *ord ,,cd is a diamet] conjugate to ! diamet] ,,ok :i* is p>allel to ! secant t1 s t ,,ok & ,,om are h>monic ) reg>d to ! asymptotes4 ,! po9t ,m !re=e bisects ,,ef4 ,b by con/ruction ,m bisects ,,cd4 ,x foll[s t ,,df ,,ec1 & ,,ed ,,cf2 or ,on any secant ( an hyp]bola ! segm5ts betwe5 ! curve & ! asymptotes are equal4 ,if ! *ord is *ang$ into a tang5t1 ?is gives-- ,! segm5t betwe5 ! asymptotes on any tang5t to an hyp]bola is bisect$ by ! po9t ( contact4 ,! fir/ "p all[s a simple solution ( ! problem to f9d any numb] ( po9ts on an hyp]bola1 ( :i* ! asymptotes & "o po9t are giv54 ,?is is equival5t to ?ree po9ts & ! tang5ts at two ( !m4 ,?is con/ruction requires measurem5t4 ,s #gd4 ,= ! p>abola1 too1 foll[ "s metrical prop]ties4 ,a diamet] ,,pm 7fig4 #bg7 bisects e *ord conjugate to x1 & ! pole ,p ( su* a *ord ,,bc lies on ! diamet]4 ,b a diamet] cuts ! p>abola once at inf9ity4 ,h5ce-- ,! segm5t ,,pm :i* jo9s ! middle po9t ,m ( a *ord ( a p>abola to ! pole ,p ( ! *ord is bisect$ by ! p>abola at ,a4 ,s #ge4 ,two asymptotes & any two tang5ts to an hyp]bola may be consid]$ z a quadrilat]al circumscrib$ ab\t ! hyp]bola4 ,b 9 su* a quadrilat]al ! int]sections ( ! diagonals & ! po9ts ( contact ( opposite sides lie 9 a l9e 7,s #ed74 ,if !re=e ,,defg 7fig4 #bh7 is su* a quadrilat]al1 !n ! diagonals ,,df & ,,ge w meet on ! l9e :i* jo9s ! po9ts ( contact ( ! asymptotes1 t is1 on ! l9e at inf9ity2 h5ce !y are p>allel4 ,f ?is ! foll[+ !orem is a simple d$uction3 ,all triangles =m$ by a tang5t & ! asymptotes ( an hyp]bola are equal 9 area4 ,illu/ration3 ,,fig4 #bg4 ,illu/ration3 ,,fig4 #bh4 ,if we draw at a po9t ,p 7fig4 #bh7 on an hyp]bola a tang5t1 ! "p ,,hk betwe5 ! asymptotes is bisect$ at ,p4 ,! p>allelogram ,,pqoq' =m$ by ! asymptotes & l9es p>allel to !m ?r\gh ,p w be half ! triangle ,,ohk1 & w !re=e be con/ant4 ,if we n[ take ! asymptotes ,,ox & ,,oy z oblique axes ( co-ord9ates1 ! l9es ,,oq & ,,qp w be ! co-ord9ates ( ,p1 & w satisfy ! equation xy con/4 a#b4 ,= ! asymptotes z axes ( co-ord9ates ! equation ( ! hyp]bola is xy con/4 ,,involution ,illu/ration3 ,,fig4 #bi4 ,s #gf4 ,if we h two projective r[s1 ,,abc on u & ,a',b',c' on u'1 & place _! bases on ! same l9e1 !n ea* po9t 9 ?is l9e c\nts twice1 once z a po9t 9 ! r[ u & once z a po9t 9 ! r[ u'4 ,9 fig4 #bi we d5ote ! po9ts z po9ts 9 ! "o r[ by lett]s above ! l9e ,a1 ,b1 ,c 4441 & z po9ts 9 ! second r[ by ,a'1 ,b'1 ,c' 444 bel[ ! l9e4 ,let n[ ,a & ,b' be ! same po9t1 !n to ,a w correspond a po9t ,a' 9 ! second1 & to ,b' a po9t ,b 9 ! fir/ r[4 ,9 g5]al ^! po9ts ,a' & ,b w be diff]5t4 ,x may1 h[ev]1 happ5 t !y co9cide4 ,!n ! correspond5ce is a peculi> "o1 z ! foll[+ !orem %[s3 ,if two projective r[s lie on ! same base1 & if x happ5s t to "o po9t 9 ! base ! same po9t corresponds1 :e!r we consid] ! po9t z belong+ to ! fir/ or to ! second r[1 !n ! same w happ5 = e po9t 9 ! base--t is to say1 to e po9t 9 ! l9e corresponds ! same po9t 9 ! fir/ z 9 ! second r[4 ,illu/ration3 ,,fig4 #cj4 ,9 ord] to det]m9e ! correspond5ce1 we may assume ?ree pairs ( correspond+ po9ts 9 two projective r[s4 ,let !n ,a'1 ,b'1 ,c'1 9 fig4 #cj1 correspond to ,a1 ,b1 ,c1 s t ,a & ,b'1 & also ,b & ,a'1 d5ote ! same po9t4 ,let u fur!r d5ote ! po9t ,c' :5 consid]$ z a po9t 9 ! fir/ r[ by ,d2 !n x is to be prov$ t ! po9t ,d'1 :i* corresponds to ,d1 is ! same po9t z ,c4 ,we "k t ! cross-ratio ( f\r po9ts is equal to t ( ! correspond+ r[4 ,h5ce 7,,ab1 ,,cd7 7,a',b'1 ,c',d'7 b replac+ ! da%$ lett]s by ^? unda%$ ones :i* d5ote ! same po9ts1 ! second cross-ratio equals 7,,ba1 ,,dd'71 :i*1 accord+ to ,s #ae1 equals 7,,ab1 ,d',d72 s t ! equation becomes 7,,ab1 ,,cd7 7,,ab1 ,d',d74 ,?is requires t ,c & ,d' co9cide4 ,s #gg4 ,two projective r[s on ! same base1 :i* h ! above prop]ty1 t to e po9t1 :e!r x be consid]$ z a po9t 9 ! "o or 9 ! o!r r[1 corresponds ! same po9t1 are said to be 9 involution1 or to =m an involution ( po9ts on ! l9e4 ,we m5tion1 b )\t prov+ x1 t any two projective r[s may be plac$ s z to =m an involution4 ,an involution may be said to consi/ ( a r[ ( pairs ( po9ts1 to e po9t ,a correspond+ a po9t ,a'1 & to ,a' aga9 ! po9t ,a4 ,^! po9ts are said to be conjugate1 or1 bett]1 "o po9t is t]m$ ! mate ( ! o!r4 ,f ! def9ition1 accord+ to :i* an involution may be consid]$ z made up ( two projective r[s1 foll[ at once ! foll[+ important prop]ties3 #a4 ,! cross-ratio ( f\r po9ts equals t ( ! f\r conjugate po9ts4 #b4 ,if we call a po9t :i* co9cides ) its mate a focus or d\ble po9t ( ! involution1 we may say3 ,an involution has ei!r two foci1 or "o1 or none1 & is call$ respectively a hyp]bolic1 p>abolic or elliptic involution 7,s #cd74 #c4 ,9 an hyp]bolic involution any two conjugate po9ts are h>monic conjugates ) reg>d to ! two foci4 ,= if ,a1 ,a' be two conjugate po9ts1 ,f#a1 ,f#b ! two foci1 !n to ! po9ts ,f#a1 ,f#b1 ,a1 ,a' 9 ! "o r[ correspond ! po9ts ,f#a1 ,f#b1 ,a'1 ,a 9 ! o!r1 ea* focus correspond+ to itself4 ,h5ce 7,f#a;,f#b1 ,,aa'7 7,f#a;,f#b1 ,a',a7--t is1 we may int]*ange ! two po9ts ,,aa' )\t alt]+ ! value ( ! cross-ratio1 :i* is ! *>act]i/ic prop]ty ( h>monic conjugates 7,s #ah74 #d4 ,! po9t conjugate to ! po9t at inf9ity is call$ ! c5tre ( ! involution4 ,e involution has a c5tre1 unless ! po9t at inf9ity be a focus1 9 :i* case we may say t ! c5tre is at inf9ity4 ,9 an hyp]bolic involution ! c5tre is ! middle po9t betwe5 ! foci4 #e4 ,! product ( ! di/ances ( two conjugate po9ts ,a1 ,a' f ! c5tre ,o is con/ant3 ,,oa 4 ,,oa' c4 ,= let ,a1 ,a' & ,b1 ,b' be two pairs ( conjugate po9ts1 ! c5tre1 ,i ! po9t at inf9ity1 !n 7,,ab1 ,,oi7 7,a',b'1 ,,io71 or ,,oa 4 ,,oa' ,,ob 4 ,,ob'4 ,9 ord] to det]m9e ! di/ances ( ! foci f ! c5tre1 we write ,f = ,a & ,a' & get ,,(#b c2 ,,( -rootc4 ,h5ce if c is positive ,,( is real1 & has two values1 equal & opposite4 ,! involution is hyp]bolic4 ,if c #j1 ,,( #j1 & ! two foci bo? co9cide ) ! c5tre4 ,if c is negative1 rootc becomes imag9>y1 & !re are no foci4 ,h5ce we may write-- ,9 an hyp]bolic involution1 ,,oa4,,oa' k#b1 ,9 a p>abolic involution1 ,,oa4,,oa' #j1 ,9 an elliptic involution1 ,,oa4,,oa' -k#b4 ,f ^! expressions x foll[s t conjugate po9ts ,a1 ,a' 9 an hyp]bolic involution lie on ! same side ( ! c5tre1 & 9 an elliptic involution on opposite sides ( ! c5tre1 & t 9 a p>abolic involution "o co9cides ) ! c5tre4 ,9 ! fir/ case1 = in/ance1 ,,oa4,,oa' is positive2 h5ce ,,oa & ,,oa' h ! same sign4 ,x also foll[s t two segm5ts1 ,,aa' & ,,bb'1 betwe5 pairs ( conjugate po9ts h ! foll[+ positions3 9 an hyp]bolic involution !y lie ei!r "o altoge!r )9 or altoge!r )\t ea* o!r2 9 a p>abolic involution !y h "o po9t 9 common2 & 9 an elliptic involution !y ov]lap1 ea* be+ p>tly )9 & p>tly )\t ! o!r4 ,pro(4--,we h ,,oa4,,oa' ,,ob4,,ob' k#b 9 case ( an hyp]bolic involution4 ,let ,a & ,b be ! po9ts 9 ea* pair :i* are ne>] to ! c5tre ,o4 ,if n[ ,a1 ,a' & ,b1 ,b' lie on ! same side ( ,o1 & if ,b is ne>] to ,o ?an ,a1 s t ,,ob ,,oa1 !n ,,ob' ,,oa'2 h5ce ,b' lies f>!r away f ,o ?an ,a'1 or ! segm5t ,,aa' lies )9 ,,bb'4 ,& s on = ! o!r cases4 #f4 ,an involution is det]m9$-- 7alpha7 ,by two pairs ( conjugate po9ts4 ,h5ce also 7beta7 ,by "o pair ( conjugate po9ts & ! c5tre2 7gamma7 ,by ! two foci2 7delta7 ,by "o focus & "o pair ( conjugate po9ts2 7epsilon7 ,by "o focus & ! c5tre4 #g4 ,! condition t ,a1 ,b1 ,c & ,a'1 ,b'1 ,c' may =m an involution may be writt5 9 "o ( ! =ms-- 7,,ab1 ,,cc'7 7,a',b'1 ,c',c71 or 7,,ab1 ,,ca'7 7,a',b'1 ,c',a71 or 7,,ab1 ,c',a'7 7,a',b'1 ,,ca71 = ea* expresses t 9 ! two projective r[s 9 :i* ,a1 ,b1 ,c & ,a'1 ,b'1 ,c' are conjugate po9ts two conjugate elem5ts may be int]*ang$4 #h4 ,any ?ree pairs4 ,a1 ,a'1 ,b1 ,b'1 ,c1 ,c'1 ( conjugate po9ts are connect$ by ! relations3 ,,ab'4,,bc'4,,ca' ,,ab'4,,bc4,c',a' ,,ab4,b',c'4,,ca' ,,ab4,b',c4,c',a' ----------- ----------- ----------- ----------- -#a4 ,a',b4,b',c4,c',a ,a',b4,b',c'4,,ca ,a',b'4,,bc4,c',a ,a',b'4,,bc'4,,ca ,^! relations readily foll[ by work+ out ! relations 9 7#g7 7above74 ,s #gh4 ,involution ( a quadrangle4--,! sides ( any f\r-po9t are cut by any l9e 9 six po9ts 9 involution1 opposite sides be+ cut 9 conjugate po9ts4 ,let ,a#a;,b#a;,c#a;,d#a 7fig4 #ca7 be ! f\r-po9t4 ,if its sides be cut by ! l9e p 9 ! po9ts ,a1 ,a'1 ,b1 ,b'1 ,c1 ,c'1 if fur!r1 ,c#a;,d#a cuts ! l9e ,a#a;,b#a 9 ,c#b1 & if we project ! r[ ,a#a;,b#a;,c#b;,c to p once f ,d#a & once f ,c#a1 we get 7,a',b'1 ,c',c7 7,,ba1 ,c',c74 ,int]*ang+ 9 ! la/ cross-ratio ! lett]s 9 ea* pair we get 7,a',b'1 ,c',c7 7,,ab1 ,,cc'74 ,h5ce by ,s #gg 7#g7 ! po9ts are 9 involution4 ,! !orem may also be /at$ ?us3 ,! ?ree po9ts 9 :i* any l9e cuts ! sides ( a triangle & ! projections1 f any po9t 9 ! plane1 ( ! v]tices ( ! triangle on to ! same l9e are six po9ts 9 involution4 ,illu/ration3 ,,fig4 #ca4 ,or aga9-- ,! projections f any po9t on to any l9e ( ! six v]tices ( a f\r-side are six po9ts 9 involution1 ! projections ( opposite v]tices be+ conjugate po9ts4 ,?is prop]ty gives a simple means to con/ruct1 by aid ( ! /raight edge only1 9 an involution ( :i* two pairs ( conjugate po9ts are giv51 to any po9t its conjugate4 ,s #gi4 ,p5cils 9 ,involution4--,! !ory ( involution may at once be ext5d$ f ! r[ to ! flat & ! axial p5cil--viz4 we say t !re is an involution 9 a flat or 9 an axial p5cil if any l9e cuts ! p5cil 9 an involution ( po9ts4 ,an involution 9 a p5cil consi/s ( pairs ( conjugate rays or planes2 x has two1 "o or no focal rays 7d\ble l9es7 or planes1 b no?+ correspond+ to a c5tre4 ,an involution 9 a flat p5cil conta9s always "o1 & 9 g5]al only "o1 pair ( conjugate rays :i* are p]p5dicul> to "o ano!r4 ,= 9 two projective flat p5cils exi/ always two correspond+ "r angles 7,s #dj74 ,ea* involution 9 an axial p5cil conta9s 9 ! same mann] "o pair ( conjugate planes at "r angles to "o ano!r4 ,z a rule1 !re exi/s b "o pair ( conjugate l9es or planes at "r angles to ea* o!r4 ,b x is possible t !re are m1 & !n !re is an inf9ite numb] ( su* pairs4 ,an involution 9 a flat p5cil1 9 :i* e ray is p]p5dicul> to its conjugate ray1 is said to be circul>4 ,t su* involution is possible is easily se5 ?us3 if 9 two conc5tric flat p5cils ea* ray on "o is made to correspond to t ray on ! o!r :i* is p]p5dicul> to x1 !n ! two p5cils are projective1 = if we turn ! "o p5cil ?r\gh a "r angle ea* ray 9 "o co9cides ) its correspond+ ray 9 ! o!r4 ,b ^! two projective p5cils are 9 involution4 ,a circul> involution has no focal rays1 because no ray 9 a p5cil co9cides ) ! ray p]p5dicul> to x4 ,s #hj4 ,e elliptical involution 9 a r[ may be consid]$ z a section ( a circul> involution4 ,9 an elliptical involution any two segm5ts ,,aa' & ,,bb' lie p>tly )9 & p>tly )\t ea* o!r 7fig4 #cb74 ,h5ce two circles describ$ on ,,aa' & ,,bb' z diamet]s w int]sect 9 two po9ts ,e & ,e'4 ,! l9e ,,ee' cuts ! base ( ! involution at a po9t ,o1 :i* has ! prop]ty t ,,oa4,,oa' ,,ob 4 ,,ob'1 = ea* is equal to ,,oe 4 ,,oe'4 ,! po9t ,o is !re=e ! c5tre ( ! involution4 ,if we wi% to con/ruct to any po9t ,c ! conjugate po9t ,c'1 we may draw ! circle ?r\gh ,,cee'4 ,?is w cut ! base 9 ! requir$ po9t ,c' = ,,oc4,,oc' ,,oa4,,oa'4 ,b ,,ec & ,,ec' are at "r angles4 ,h5ce ! involution :i* is obta9$ by jo9+ ,e or ,e' to ! po9ts 9 ! giv5 involution is circul>4 ,?is may also be express$ ?us3 ,illu/ration3 ,,fig4 #cb4 ,e elliptical involution has ! prop]ty t !re are two def9ite po9ts 9 ! plane f :i* any two conjugate po9ts are se5 "u a "r angle4 ,at ! same "t ! foll[+ problem has be5 solv$3 ,to det]m9e ! c5tre & also ! po9t correspond+ to any giv5 po9t 9 an elliptical involution ( :i* two pairs ( conjugate po9ts are giv54 ,s #ha4 ,involution ,range on a ,conic4--,by ! aid ( ,s #ec1 ! po9ts on a conic may be made to correspond to ^? on a l9e1 s t ! r[ ( po9ts on ! conic is projective to a r[ ( po9ts on a l9e4 ,we may also h two projective r[s on ! same conic1 & ^! w be 9 involution z soon z "o po9t on ! conic has ! same po9t correspond+ to x all ! same to :atev] r[ x belongs4 ,an involution ( po9ts on a conic w h ! prop]ty 7z foll[s f its def9ition1 & f ,s #ec7 t ! l9es :i* jo9 conjugate po9ts ( ! involution to any po9t on ! conic are conjugate l9es ( an involution 9 a p5cil1 & t a fix$ tang5t is cut by ! tang5ts at conjugate po9ts on ! conic 9 po9ts :i* are aga9 conjugate po9ts ( an involution on ! fix$ tang5t4 ,= su* involution on a conic ! foll[+ !orem holds3 ,! l9es :i* jo9 correspond+ po9ts 9 an involution on a conic all pass ?r\gh a fix$ po9t2 & reciprocally1 ! po9ts ( int]section ( conjugate l9es 9 an involution among tang5ts to a conic lie on a l9e4 ,illu/ration3 ,,fig4 #cc ,we prove ! fir/ "p only4 ,! involution is det]m9$ by two pairs ( conjugate po9ts1 say by ,a1 ,a' & ,b1 ,b' 7fig4 #cc74 ,let ,,aa' & ,,bb' meet 9 ,p4 ,if we jo9 ! po9ts 9 involution to any po9t on ! conic1 & ! conjugate po9ts to ano!r po9t on ! conic1 we obta9 two projective p5cils4 ,we take ,a & ,a' z c5tres ( ^! p5cils1 s t ! p5cils ,a7,a',,bb'7 & ,a'7,,ab',b7 are projective1 & 9 p]spective position1 because ,,aa' corresponds to ,a',a4 ,h5ce correspond+ rays meet 9 a l9e1 ( :i* two po9ts are f\nd by jo9+ ,,ab' to ,a',b & ,,ab to ,a',b'4 ,x foll[s t ! axis ( p]spective is ! pol> ( ! po9t ,p1 :]e ,,aa' & ,,bb' meet4 ,if we n[ wi% to con/ruct to any o!r po9t ,c on ! conic ! correspond+ po9t ,c'1 we jo9 ,c to ,a' & ! po9t :]e ?is l9e cuts p to ,a4 ,! latt] l9e cuts ! conic aga9 9 ,c'4 ,b we "k f ! !ory ( pole & pol> t ! l9e ,,cc' passes ?r\gh ,p4 ,! po9t ( concurr5ce is call$ ! pole ( ! involution1 & ! l9e ( coll9e>ity ( ! meets is call$ ! axis ( ! involution4 ,,involution ,,det]m9$ ,,by ,a ,,conic ,,on ,a ,,l9e4--,,foci ,s #hb4 ,! pol>s1 ) reg>d to a conic1 ( po9ts 9 a r[ p =m a p5cil ,p projective to ! r[ 7,s #ff74 ,?is p5cil cuts ! base ( ! r[ p 9 a projective r[4 ,if ,a is a po9t 9 ! giv5 r[1 ,a' ! po9t :]e ! pol> ( ,a cuts p1 !n ,a & ,a' w be correspond+ po9ts4 ,if we take ,a' a po9t 9 ! fir/ r[1 !n ! pol> ( ,a' w pass ?r\gh ,a1 s t ,a corresponds to ,a'--9 o!r words1 ! r[s are 9 involution4 ,! conjugate po9ts 9 ?is involution are conjugate po9ts ) reg>d to ! conic4 ,conjugate po9ts co9cide only if ! pol> ( a po9t ,a passes ?r\gh ,a--t is1 if ,a lies on ! conic4 ,h5ce-- ,a conic det]m9es on e l9e 9 its plane an involution1 9 :i* ^? po9ts are conjugate :i* are also conjugate ) reg>d to ! conic4 ,if ! l9e cuts ! conic ! involution is hyp]bolic1 ! po9ts ( int]section be+ ! foci4 ,if ! l9e t\*es ! conic ! involution is p>abolic1 ! two foci co9cid+ at ! po9t ( contact4 ,if ! l9e does n cut ! conic ! involution is elliptic1 hav+ no foci4 ,if1 on ! o!r h&1 we take a po9t ,p 9 ! plane ( a conic1 we get to ea* l9e a ?r\gh ,p "o conjugate l9e :i* jo9s ,p to ! pole ( a4 ,^! pairs ( conjugate l9es ?r\gh ,p =m an involution 9 ! p5cil at ,p4 ,! focal rays ( ?is involution are ! tang5ts drawn f ,p to ! conic4 ,?is gives ! !orem reciprocal to ! la/1 viz3-- ,a conic det]m9es 9 e p5cil 9 its plane an involution1 correspond+ l9es be+ conjugate l9es ) reg>d to ! conic4 ,if ! po9t is )\t ! conic ! involution is hyp]bolic1 ! tang5ts f ! po9ts be+ ! focal rays4 ,if ! po9t lies on ! conic ! involution is p>abolic1 ! tang5t at ! po9t c\nt+ = co9cid5t focal rays4 ,if ! po9t is )9 ! conic ! involution is elliptic1 hav+ no focal rays4 ,x w fur!r be se5 t ! involution det]m9$ by a conic on any l9e p is a section ( ! involution1 :i* is det]m9$ by ! conic at ! pole ,p ( p4 ,s #hc4 ,foci4--,! c5tre ( a p5cil 9 :i* ! conic det]m9es a circul> involution is call$ a focus ( ! conic4 ,9 o!r words1 a focus is su* a po9t t e l9e ?r\gh x is p]p5dicul> to its conjugate l9e4 ,! pol> to a focus is call$ a directrix ( ! conic4 ,f ! def9ition x foll[s t e focus lies on an axis1 = ! l9e jo9+ a focus to ! c5tre ( ! conic is a diamet] to :i* ! conjugate l9es are p]p5dicul>2 & e l9e jo9+ two foci is an axis1 = ! p]p5dicul>s to ?is l9e ?r\gh ! foci are conjugate to x4 ,^! conjugate l9es pass ?r\gh ! pole ( ! l9e1 ! pole lies !re=e at inf9ity1 & ! l9e is a diamet]1 h5ce by ! la/ prop]ty an axis4 ,x foll[s t all foci lie on "o axis1 = no l9e jo9+ a po9t 9 "o axis to a po9t 9 ! o!r c be an axis4 ,z ! conic det]m9es 9 ! p5cil :i* has its c5tre at a focus a circul> involution1 no tang5ts c be drawn f ! focus to ! conic4 ,h5ce ea* focus lies )9 a conic2 & a directrix does n cut ! conic4 ,fur!r prop]ties are f\nd by ! foll[+ consid]ations3 ,s #hd4 ,?r\gh a po9t ,p "o l9e p c be drawn1 :i* is ) reg>d to a giv5 conic conjugate to a giv5 l9e q1 viz4 t l9e :i* jo9s ! po9t ,p to ! pole ( ! l9e q4 ,if ! l9e q is made to describe a p5cil ab\t a po9t ,q1 !n ! l9e p w describe a p5cil ab\t ,p4 ,^! two p5cils w be projective1 = ! l9e p passes ?r\gh ! pole ( q1 & :il/ q describes ! p5cil ,q1 its pole describes a projective r[1 & ?is r[ is p]spective to ! p5cil ,p4 ,we n[ take ! po9t ,p on an axis ( ! conic1 draw any l9e p ?r\gh x1 & f ! pole ( p draw a p]p5dicul> q to p4 ,let q cut ! axis 9 ,q4 ,!n1 9 ! p5cils ( conjugate l9es1 :i* h _! c5tres at ,p & ,q1 ! l9es p & q are conjugate l9es at "r angles to "o ano!r4 ,besides1 to ! axis z a ray 9 ei!r p5cil w correspond 9 ! o!r ! p]p5dicul> to ! axis 7,s #gb74 ,! conic g5]at$ by ! int]section ( correspond+ l9es 9 ! two p5cils is !re=e ! circle on ,,pq z diamet]1 s t e l9e 9 ,p is p]p5dicul> to its correspond+ l9e 9 ,q4 ,to e po9t ,p on an axis ( a conic corresponds ?us a po9t ,q1 su* t conjugate l9es ?r\gh ,p & ,q are p]p5dicul>4 ,we %all %[ t ^! po9t-pairs ,p1 ,q =m an involution4 ,to d ?is let u move ,p along ! axis1 & ) x ! l9e p1 keep+ ! latt] p>allel to itself4 ,!n ,p describes a r[1 p a p]spective p5cil 7( p>allels71 & ! pole ( p a projective r[4 ,at ! same "t ! l9e q describes a p5cil ( p>allels p]p5dicul> to p1 & p]spective to ! r[ =m$ by ! pole ( p4 ,! po9t ,q1 !re=e1 :]e q cuts ! axis1 describes a r[ projective to ! r[ ( po9ts ,p4 ,! two po9ts ,p & ,q describe ?us two projective r[s on ! axis2 & n only does ,p z a po9t 9 ! fir/ r[ correspond to ,q1 b also ,q z a po9t 9 ! fir/ corresponds to ,p4 ,! two r[s !re=e =m an involution4 ,! c5tre ( ?is involution1 x is easily se51 is ! c5tre ( ! conic4 ,a focus ( ?is involution has ! prop]ty t any two conjugate l9es ?r\gh x are p]p5dicul>2 h5ce1 x is a focus to ! conic4 ,su* involution exi/s on ea* axis4 ,b only "o ( ^! c h foci1 because all foci lie on ! same axis4 ,! involution on "o ( ! axes is elliptic1 & appe>s 7,s #hj7 !re=e z ! section ( two circul> involutions 9 two p5cils ^: c5tres lie 9 ! o!r axis4 ,^! c5tres are foci1 h5ce ! "o axis conta9s two foci1 ! o!r axis none2 or e c5tral conic has two foci :i* lie on "o axis equidi/ant f ! c5tre4 ,! axis :i* conta9s ! foci is call$ ! pr9cipal axis2 9 case ( an hyp]bola x is ! axis :i* cuts ! curve1 because ! foci lie )9 ! conic4 ,9 case ( ! p>abola !re is b "o axis4 ,! involution on ?is axis has its c5tre at inf9ity4 ,"o focus is !re=e at inf9ity1 ! "o focus only is f9ite4 ,a p>abola has only "o focus4 ,illu/ration3 ,,fig4 #cd4 ,s #he4 ,if ?r\gh any po9t ,p 7fig4 #cd7 on a conic ! tang5t ,,pt & ! normal ,,pn 7i4e4 ! p]p5dicul> to ! tang5t ?r\gh ! po9t ( contact7 be drawn1 ^! w be conjugate l9es ) reg>d to ! conic1 & at "r angles to ea* o!r4 ,!y w !re=e cut ! pr9cipal axis 9 two po9ts1 :i* are conjugate 9 ! involution consid]$ 9 ,s #hd2 h5ce !y are h>monic conjugates ) reg>d to ! foci4 ,if !re=e ! two foci ,f#a & ,f#b be jo9$ to ,p1 ^! l9es w be h>monic ) reg>d to ! tang5t & normal4 ,z ! latt] are p]p5dicul>1 !y w bisect ! angles betwe5 ! o!r pair4 ,h5ce-- ,! l9es jo9+ any po9t on a conic to ! two foci are equally incl9$ to ! tang5t & normal at t po9t4 ,9 case ( ! p>abola ?is becomes-- ,! l9e jo9+ any po9t on a p>abola to ! focus & ! diamet] ?r\gh ! po9t1 are equally incl9$ to ! tang5t & normal at t po9t4 ,f ! def9ition ( a focus x foll[s t-- ,! segm5t ( a tang5t betwe5 ! directrix & ! po9t ( contact is se5 f ! focus belong+ to ! directrix "u a "r angle1 because ! l9es jo9+ ! focus to ! ends ( ?is segm5t are conjugate ) reg>d to ! conic1 & !re=e p]p5dicul>4 ,) equal ease ! foll[+ !orem is prov$3 ,! two l9es :i* jo9 ! po9ts ( contact ( two tang5ts ea* to "o focus1 b n bo? to ! same1 are se5 f ! int]section ( ! tang5ts "u equal angles4 ,s #hf4 ,o!r focal prop]ties ( a conic are obta9$ by ! foll[+ consid]ations3 ,illu/ration3 ,,fig4 #ce4 ,let ,f 7fig4 #ce7 be a focus to a conic1 f ! correspond+ directrix1 ,a & ,b ! po9ts ( contact ( two tang5ts meet+ at ,t1 & ,p ! po9t :]e ! l9e ,,ab cuts ! directrix4 ,!n ,,tf w be ! pol> ( ,p 7because pol>s ( ,f & ,t meet at ,p74 ,h5ce ,,tf & ,,pf are conjugate l9es ?r\gh a focus1 & !re=e p]p5dicul>4 ,!y are fur!r h>monic conjugates ) reg>d to ,,fa & ,,fb 7,,ss #fd & #ac71 s t !y bisect ! angles =m$ by ^! l9es4 ,?is by ! way proves-- ,! segm5ts betwe5 ! po9t ( int]section ( two tang5ts to a conic & _! po9ts ( contact are se5 f a focus "u equal angles4 ,if we next draw ?r\gh ,a & ,b l9es p>allel to ,,tf1 !n ! po9ts ,a#a1 ,b#a :]e ^! cut ! directrix w be h>monic conjugates ) reg>d to ,p & ! po9t :]e ,,ft cuts ! directrix4 ,! l9es ,,ft & ,,fp bisect !re=e also ! angles betwe5 ,,fa#a & ,,fb#a4 ,f ?is x foll[s easily t ! triangles ,,faa#a & ,,fbb#a are equiangul>1 & !re=e simil>1 s t ,,fa 3 ,,aa#a ,,fb 3 ,,bb#a4 ,! triangles ,,aa#a;,a#b & ,,bb#a;,b#b =m$ by draw+ p]p5dicul>s f ,a & ,b to ! directrix are also simil>1 s t ,,aa#a 3 ,,aa#b ,,bb#a 3 ,,bb#b4 ,?is1 comb9$ ) ! above proportion1 gives ,,fa 3 ,,aa#b ,,fb 3 ,,bb#b4 ,h5ce ! !orem3 ,! ratio ( ! di/ances ( any po9t on a conic f a focus & ! correspond+ directrix is con/ant4 ,to det]m9e ?is ratio we consid] its value = a v]tex on ! pr9cipal axis4 ,9 an ellipse ! focus lies betwe5 ! two v]tices on ?is axis1 h5ce ! focus is ne>] to a v]tex ?an to ! correspond+ directrix4 ,simil>ly1 9 an hyp]bola a v]tex is ne>] to ! directrix ?an to ! focus4 ,9 a p>abola ! v]tex lies halfway betwe5 directrix & focus4 ,x foll[s 9 an ellipse ! ratio betwe5 ! di/ance ( a po9t f ! focus to t f ! directrix is less ?an unity1 9 ! p>abola x equals unity1 & 9 ! hyp]bola x is great] ?an unity4 ,x is "h ! same :i* focus we take1 because ! two foci lie symmetrical to ! axis ( ! conic4 ,if n[ ,p is any po9t on ! conic hav+ ! di/ances r#a & r#b f ! foci & ! di/ances d#a & d#b f ! correspond+ directrices1 !n r#a_/;d#a r#b_/;d#b e1 :]e e is con/ant4 ,h5ce also r#a - r#b _/ d#a - d#b e4 ,9 ! ellipse1 :i* lies betwe5 ! directrices1 d#a d#b is con/ant1 !re=e also r#a r#b4 ,9 ! hyp]bola on ! o!r h& d#a - d#b is con/ant1 equal to ! di/ance betwe5 ! directrices1 !re=e 9 ?is case r#a - r#b is con/ant4 ,if we call ! di/ances ( a po9t on a conic f ! focus its focal di/ances we h ! !orem3 ,9 an ellipse ! sum ( ! focal di/ances is con/ant2 & 9 an hyp]bola ! diff]5ce ( ! focal di/ances is con/ant4 ,?is con/ant sum or diff]5ce equals 9 bo? cases ! l5g? ( ! pr9cipal axis4 ,,p5cil ,,( ,,conics ,s #hg4 ,?r\gh f\r po9ts ,a1 ,b1 ,c1 ,d 9 a plane1 ( :i* no ?ree lie 9 a l9e1 an inf9ite numb] ( conics may be drawn1 viz4 ?r\gh ^! f\r po9ts & any fif? "o s+le conic4 ,?is sy/em ( conics is call$ a p5cil ( conics4 ,simil>ly1 all conics t\*+ f\r fix$ l9es =m a sy/em su* t any fif? tang5t det]m9es "o & only "o conic4 ,we h "h ! !orems3 ,! pairs ( po9ts 9 :i* ,! pairs ( tang5ts :i* any l9e is cut by a sy/em ( c be drawn f a po9t to conics ?r\gh f\r fix$ po9ts a sy/em ( conics t\*+ f\r are 9 involution4 fix$ l9es are 9 involution4 ,illu/ration3 ,,fig4 #cf4 ,we prove ! fir/ !orem only4 ,let ,,abcd 7fig4 #cf7 be ! f\r-po9t1 !n any l9e t w cut two opposite sides ,,ac1 ,,bd 9 ! po9ts ,e1 ,e'1 ! pair ,,ad1 ,,bc 9 po9ts ,f1 ,f'1 & any conic ( ! sy/em 9 ,m1 ,n1 & we h ,a7,,cd1 ,,mn7 ,b7,,cd1 ,,mn74 ,if we cut ^! p5cils by t we get 7,,ef1 ,,mn7 7,f',e'1 ,,mn7 or 7,,ef1 ,,mn7 7,e',f'1 ,,nm74 ,b ?is is1 accord+ to ,s #gg 7#g71 ! condition t ,m1 ,n are correspond+ po9ts 9 ! involution det]m9$ by ! po9t pairs ,e1 ,e'1 ,f1 ,f' 9 :i* ! l9e t cuts pairs ( opposite sides ( ! f\r-po9t ,,abcd4 ,?is involution is indep5d5t ( ! p>ticul> conic *os54 ,s #hh4 ,!re foll[ sev]al important !orems3 ,?r\gh f\r po9ts two1 "o1 or no conics may be drawn :i* t\* any giv5 l9e1 accord+ z ! involution det]m9$ by ! giv5 f\r-po9t on ! l9e has real1 co9cid5t or imag9>y foci4 ,two1 "o1 or no conics may be drawn :i* t\* f\r giv5 l9es & pass ?r\gh a giv5 po9t1 accord+ z ! involution det]m9$ by ! giv5 f\r-side at ! po9t has real1 co9cid5t or imag9>y focal rays4 ,= ! conic ?r\gh f\r po9ts :i* t\*es a giv5 l9e has its po9t ( contact at a focus ( ! involution det]m9$ by ! f\r-po9t on ! l9e4 ,z a special case we get1 by tak+ ! l9e at inf9ity3 ,?r\gh f\r po9ts ( :i* none is at inf9ity ei!r two or no p>abolas may be drawn4 ,! problem ( draw+ a conic ?r\gh f\r po9ts & t\*+ a giv5 l9e is solv$ by det]m9+ ! po9ts ( contact on ! l9e1 t is1 by det]m9+ ! foci ( ! involution 9 :i* ! l9e cuts ! sides ( ! f\r-po9t4 ,! correspond+ rem>k holds = ! problem ( draw+ ! conics :i* t\* f\r l9es & pass ?r\gh a giv5 po9t4 ,,rul$ ,,quadric ,,surfaces ,s #hi4 ,we h consid]$ hi!rto projective r[s :i* lie 9 ! same plane1 9 :i* case l9es jo9+ correspond+ po9ts envelop a conic4 ,we %all n[ consid] projective r[s ^: bases d n meet4 ,9 ?is case1 correspond+ po9ts w be jo9$ by l9es :i* d n lie 9 a plane1 b on "s surface1 :i* l e surface g5]at$ by l9es is call$ a rul$ surface4 ,?is surface cle>ly conta9s ! bases ( ! two r[s4 ,if ! po9ts 9 ei!r r[ be jo9$ to ! base ( ! o!r1 we obta9 two axial p5cils :i* are also projective1 ^? planes be+ correspond+ :i* pass ?r\gh correspond+ po9ts 9 ! giv5 r[s4 ,if ,a'1 ,a be two correspond+ po9ts1 alpha1 alpha' ! planes 9 ! axial p5cils pass+ ?r\gh !m1 !n ,,aa' w be ! l9e ( int]section ( ! correspond+ planes alpha1 alpha' & also ! l9e jo9+ correspond+ po9ts 9 ! r[s4 ,if we cut ! :ole figure by a plane ?is w cut ! axial p5cils 9 two projective flat p5cils1 & ! curve ( ! second ord] g5]at$ by ^! w be ! curve 9 :i* ! plane cuts ! surface4 ,h5ce ,! locus ( l9es jo9+ correspond+ po9ts 9 two projective r[s :i* d n lie 9 ! same plane is a surface :i* conta9s ! bases ( ! r[s1 & :i* c also be g5]at$ by ! l9es ( int]section ( correspond+ planes 9 two projective axial p5cils4 ,?is surface is cut by e plane 9 a curve ( ! second ord]1 h5ce ei!r 9 a conic or 9 a l9e-pair4 ,no l9e :i* does n lie altoge!r on ! surface c h m ?an two po9ts 9 common ) ! surface1 :i* is !re=e said to be ( ! second ord] or is call$ a rul$ quadric surface4 ,t no l9e :i* does n lie on ! surface c cut ! surface 9 m ?an two po9ts is se5 at once if a plane be drawn ?r\gh ! l9e1 = ?is w cut ! surface 9 a conic4 ,x foll[s also t a l9e :i* conta9s m ?an two po9ts ( ! surface lies altoge!r on ! surface4 ,s #ij4 ,?r\gh any po9t 9 space "o l9e c always be drawn cutt+ two giv5 l9es :i* d n !mselves meet4 ,if !re=e ?ree l9es 9 space be giv5 ( :i* no two meet1 !n ?r\gh e po9t 9 ei!r "o l9e may be drawn cutt+ ! o!r two4 ,if a l9e moves s t x always cuts ?ree giv5 l9es ( :i* no two meet1 !n x g5]ates a rul$ quadric surface4 ,let a1 b1 c be ! giv5 l9es1 & p1 q1 r 444 l9es cutt+ !m 9 ! po9ts ,a1 ,a'1 ,a 4442 ,b1 ,b'1 ,b 4442 ,c1 ,c'1 ,c 444 respectively2 !n ! planes ?r\gh a conta9+ p1 q1 r1 & ! planes ?r\gh b conta9+ ! same l9es1 may be tak5 z correspond+ planes 9 two axial p5cils :i* are projective1 because bo? p5cils cut ! l9e c 9 ! same r[1 ,c1 ,c'1 ,c 4442 ! surface c !re=e be g5]at$ by projective axial p5cils4 ,( ! l9es p1 q1 r 444 no two c meet1 = o!rwise ! l9es a1 b1 c :i* cut !m w\ld also lie 9 _! plane4 ,!re is a s+le inf9ite numb] ( !m1 = "o passes ?r\gh ea* po9t ( a4 ,^! l9es are said to =m a set ( l9es on ! surface4 ,if n[ ?ree ( ! l9es p1 q1 r be tak51 !n e l9e d cutt+ !m w h ?ree po9ts 9 common ) ! surface1 & w !re=e lie altoge!r on x4 ,?is gives rise to a second set ( l9es on ! surface4 ,f :at has be5 said ! !orem foll[s3 ,a rul$ quadric surface conta9s two sets ( /raight l9es4 ,e l9e ( "o set cuts e l9e ( ! o!r1 b no two l9es ( ! same set meet4 ,any two l9es ( ! same set may be tak5 z bases ( two projective r[s1 or ( two projective p5cils :i* g5]ate ! surface4 ,!y are cut by ! l9es ( ! o!r set 9 two projective r[s4 ,! plane at inf9ity l e o!r plane cuts ! surface ei!r 9 a conic prop] or 9 a l9e-pair4 ,9 ! fir/ case ! surface is call$ an ,hyp]boloid ( "o %eet1 9 ! second an ,hyp]bolic ,p>aboloid4 ,! latt] may be g5]at$ by a l9e cutt+ ?ree l9es ( :i* "o lies at inf9ity1 t is1 cutt+ two l9es & rema9+ p>allel to a giv5 plane4 ,,quadric ,,surfaces ,s #ia4 ,! conics1 ! cones ( ! second ord]1 & ! rul$ quadric surfaces complete ! figures :i* c be g5]at$ by projective r[s or flat & axial p5cils1 t is1 by ^? aggregates ( elem5ts :i* are ( "o dim5sion 7,,ss #e1 #f74 ,we %all n[ consid] ! simpl] figures :i* are g5]at$ by aggregates ( two dim5sions4 ,! space at our disposal w n1 h[ev]1 all[ u to d m ?an indicate a few ( ! results4 ,s #ib4 ,we e/abli% a correspond5ce betwe5 ! l9es & planes 9 p5cils 9 space1 or reciprocally betwe5 ! po9ts & l9es 9 two or m planes1 b consid] pr9cipally p5cils4 ,9 two p5cils we may ei!r make planes correspond to planes & l9es to l9es1 or else planes to l9es & l9es to planes4 ,if h]eby ! condition be satisfi$ t to a flat1 or axial1 p5cil corresponds 9 ! fir/ case a projective flat1 or axial1 p5cil1 & 9 ! second a projective axial1 or flat1 p5cil1 ! p5cils are said to be projective 9 ! fir/ case & reciprocal 9 ! second4 ,= in/ance1 two p5cils :i* jo9 two po9ts ,s#a & ,s#b to ! diff]5t po9ts & l9es 9 a giv5 plane pi are projective 7& 9 p]spective position71 if ^? l9es & planes be tak5 z correspond+ :i* meet ! plane pi 9 ! same po9t or 9 ! same l9e4 ,9 ?is case e plane ?r\gh bo? c5tres ,s#a & ,s#b ( ! two p5cils w correspond to itself4 ,if ^! p5cils are br\ght into any o!r position !y w be projective 7b n p]spective74 ,! correspond5ce betwe5 two projective p5cils is uniquely det]m9$1 if to f\r rays 7or planes7 9 ! "o ! correspond+ rays 7or planes7 9 ! o!r are giv51 provid$ t no ?ree rays ( ei!r set lie 9 a plane4 ,let a1 b1 c1 d be f\r rays 9 ! "o1 a'1 b'1 c'1 d' ! correspond+ rays 9 ! o!r p5cil4 ,we %all %[ t we c f9d = e ray e 9 ! fir/ a s+le correspond+ ray e' 9 ! second4 ,to ! axial p5cil a 7b1 c1 d 4447 =m$ by ! planes :i* jo9 a to b1 c1 d 4441 respectively corresponds ! axial p5cil a' 7b'1 c'1 d' 444 71 & ?is correspond5ce is det]m9$4 ,h5ce1 ! plane a'e' :i* corresponds to ! plane ae is det]m9$4 ,simil>ly ! plane b'e' may be f\nd & bo? toge!r det]m9e ! ray e'4 ,simil>ly ! correspond5ce betwe5 two reciprocal p5cils is det]m9$ if = f\r rays 9 ! "o ! correspond+ planes 9 ! o!r are giv54 ,s #ic4 ,we may n[ comb9e-- #a4 ,two reciprocal p5cils4 ,ea* ray cuts its correspond+ plane 9 a po9t1 ! locus ( ^! po9ts is a quadric surface4 #b4 ,two projective p5cils4 ,ea* plane cuts its correspond+ plane 9 a l9e1 b a ray z a rule does n cut its correspond+ ray4 ,! locus ( po9ts :]e a ray cuts its correspond+ ray is a twi/$ cubic4 ,! l9es :]e a plane cuts its correspond+ plane are secants4 #c4 ,?ree projective p5cils4 ,! locus ( int]section ( correspond+ planes is a cubic surface4 ,( ^! we consid] only ! fir/ two cases4 ,s #id4 ,if two p5cils are reciprocal1 !n to a plane 9 ei!r corresponds a l9e 9 ! o!r1 to a flat p5cil an axial p5cil1 & s on4 ,e l9e cuts its correspond+ plane 9 a po9t4 ,if ,s#a & ,s#b be ! c5tres ( ! two p5cils1 & ,p be a po9t :]e a l9e a#a 9 ! fir/ cuts its correspond+ plane alpha#b1 !n ! l9e b#b 9 ! p5cil ,s#b :i* passes ?r\gh ,p w meet its correspond+ plane beta#a 9 ,p4 ,= b#b is a l9e 9 ! plane alpha#b4 ,! correspond+ plane beta#a mu/ !re=e pass ?r\gh ! l9e a#a1 h5ce ?r\gh ,p4 ,! po9ts 9 :i* ! l9es 9 ,s#a cut ! planes correspond+ to !m 9 ,s#b are !re=e ! same z ! po9ts 9 :i* ! l9es 9 ,s#b cut ! planes correspond+ to !m 9 ,s#a4 ,! locus ( ^! po9ts is a surface :i* is cut by a plane 9 a conic or 9 a l9e-pair & by a l9e 9 n m ?an two po9ts unless x lies altoge!r on ! surface4 ,! surface itself is !re=e call$ a quadric surface1 or a surface ( ! second ord]4 ,to prove ?is we consid] any l9e p 9 space4 ,! flat p5cil 9 ,s#a :i* lies 9 ! plane drawn ?r\gh p & ! correspond+ axial p5cil 9 ,s#b det]m9e on p two projective r[s1 & ^? po9ts 9 ^! :i* co9cide ) _! correspond+ po9ts lie on ! surface4 ,b !re exi/ only two1 or "o1 or no su* po9ts1 unless e po9t co9cides ) its correspond+ po9t4 ,9 ! latt] case ! l9e lies altoge!r on ! surface4 ,?is proves also t a plane cuts ! surface 9 a curve ( ! second ord]1 z no l9e c h m ?an two po9ts 9 common ) x4 ,to %[ t ?is is a curve ( ! same k9d z ^? consid]$ be=e1 we h to %[ t x c be g5]at$ by projective flat p5cils4 ,we prove fir/ t ?is is true = any plane ?r\gh ! c5tre ( "o ( ! p5cils1 & aft]w>ds t e po9t on ! surface may be tak5 z ! c5tre ( su* p5cil4 ,let !n alpha#a be a plane ?r\gh ,s#a4 ,to ! flat p5cil 9 ,s#a :i* x conta9s corresponds 9 ,s#b a projective axial p5cil ) axis a#b & ?is cuts alpha#a 9 a second flat p5cil4 ,^! two flat p5cils 9 alpha#a are projective1 &1 9 g5]al1 nei!r conc5tric nor p]spective4 ,!y g5]ate !re=e a conic4 ,b if ! l9e a#b passes ?r\gh ,s#a ! p5cils w h ,s#a z common c5tre1 & may !re=e h two1 or "o1 or no l9es unit$ ) _! correspond+ l9es4 ,! section ( ! surface by ! plane alpha#a w be accord+ly a l9e-pair or a s+le l9e1 or else ! plane alpha#a w h only ! po9t ,s#a 9 common ) ! surface4 ,e l9e l#a ?r\gh ,s#a cuts ! surface 9 two po9ts1 viz4 fir/ 9 ,s#a & !n at ! po9t :]e x cuts its correspond+ plane4 ,if n[ ! correspond+ plane passes ?r\gh ,s#a1 z 9 ! case j consid]$1 !n ! two po9ts :]e l#a cuts ! surface co9cide at ,s#a1 & ! l9e is call$ a tang5t to ! surface ) ,s#a z po9t ( contact4 ,h5ce if l#a be a tang5t1 x lies 9 t plane tau#a :i* corresponds to ! l9e ,s#b,s#a z a l9e 9 ! p5cil ,s#b4 ,! section ( ?is plane has j be5 consid]$4 ,x foll[s t-- ,all tang5ts to quadric surface at ! c5tre ( "o ( ! reciprocal p5cils lie 9 a plane :i* is call$ ! tang5t plane to ! surface at t po9t z po9t ( contact4 ,to ! l9e jo9+ ! c5tres ( ! two p5cils z a l9e 9 "o corresponds 9 ! o!r ! tang5t plane at its c5tre4 ,! tang5t plane to a quadric surface ei!r cuts ! surface 9 two l9es1 or x has only a s+le l9e1 or else only a s+le po9t 9 common ) ! surface4 ,9 ! fir/ case ! po9t ( contact is said to be hyp]bolic1 9 ! second p>abolic1 9 ! ?ird elliptic4 ,s #ie4 ,x rema9s to be prov$ t e po9t ,s on ! surface may be tak5 z c5tre ( "o ( ! p5cils :i* g5]ate ! surface4 ,let ,s be any po9t on ! surface ,phi' g5]at$ by ! reciprocal p5cils ,s#a & ,s#b4 ,we h to e/abli% a reciprocal correspond5ce betwe5 ! p5cils ,s & ,s#a1 s t ! surface g5]at$ by !m is id5tical ) ,phi4 ,to d ?is we draw two planes alpha#a & beta#a ?r\gh ,s#a1 cutt+ ! surface ,phi 9 two conics :i* we also d5ote by alpha#a & beta#a4 ,^! conics meet at ,s#a1 & at "s o!r po9t ,t :]e ! l9e ( int]section ( alpha#a & beta#a cuts ! surface4 ,9 ! p5cil ,s we draw "s plane sigma :i* passes ?r\gh ,t1 b n ?r\gh ,s#a or ,s#b4 ,x w cut ! two conics fir/ at ,t1 & !re=e ea* at "s o!r po9t :i* we call ,a & ,b respectively4 ,^! we jo9 to ,s by l9es a & b1 & n[ e/abli% ! requir$ correspond5ce betwe5 ! p5cils ,s#a & ,s z foll[s3--,to ,s#a,t %all correspond ! plane sigma1 to ! plane alpha#a ! l9e a1 & to beta#a ! l9e b1 h5ce to ! flat p5cil 9 alpha#a ! axial p5cil a4 ,^! p5cils are made projective by aid ( ! conic 9 alpha#a4 ,9 ! same mann] ! flat p5cil 9 beta#a is made projective to ! axial p5cil b by aid ( ! conic 9 beta#a1 correspond+ elem5ts be+ ^? :i* meet on ! conic4 ,?is det]m9es ! correspond5ce1 = we "k = m ?an f\r rays 9 ,s#a ! correspond+ planes 9 ,s4 ,! two p5cils ,s & ,s#a ?us made reciprocal g5]ate a quadric surface ,phi'1 :i* passes ?r\gh ! po9t ,s & ?r\gh ! two conics alpha#a & beta#a4 ,! two surfaces ,phi & ,phi' h !re=e ! po9ts ,s & ,s#a & ! conics alpha#a & beta#a 9 common4 ,to %[ t !y are id5tical1 we draw a plane ?r\gh ,s & ,s#b1 cutt+ ea* ( ! conics alpha#a & beta#a 9 two po9ts1 :i* w always be possible4 ,?is plane cuts ,phi & ,phi' 9 two conics :i* h ! po9t ,s & ! po9ts :]e x cuts alpha#a & beta#a 9 common1 t is five po9ts 9 all4 ,! conics !re=e co9cide4 ,?is proves t all ^? po9ts ,p on ,phi' lie on ,phi :i* h ! prop]ty t ! plane ,,ss#b,p cuts ! conics alpha#a1 beta#a 9 two po9ts ea*4 ,if ! plane ,,ss#b,p has n ?is prop]ty1 !n we draw a plane ,,ss#a,p4 ,?is cuts ea* surface 9 a conic1 & ^! conics h 9 common ! po9ts ,s1 ,s#a1 "o po9t on ea* ( ! conics alpha#a1 beta#a1 & "o po9t on "o ( ! conics ?r\gh ,s & ,s#b :i* lie on bo? surfaces1 h5ce five po9ts4 ,!y are !re=e co9cid5t1 & our !orem is prov$4 ,s #if4 ,! foll[+ propositions foll[3-- ,a quadric surface has at e po9t a tang5t plane4 ,e plane section ( a quadric surface is a conic or a l9e-pair4 ,e l9e :i* has ?ree po9ts 9 common ) a quadric surface lies on ! surface4 ,e conic :i* has five po9ts 9 common ) a quadric surface lies on ! surface4 ,?r\gh two conics :i* lie 9 diff]5t planes1 b h two po9ts 9 common1 & ?r\gh "o ext]nal po9t always "o quadric surface may be drawn4 ,s #ig4 ,e plane :i* cuts a quadric surface 9 a l9e-pair is a tang5t plane4 ,= e l9e 9 ?is plane ?r\gh ! c5tre ( ! l9e-pair 7! po9t ( int]section ( ! two l9es7 cuts ! surface 9 two co9cid5t po9ts & is !re=e a tang5t to ! surface1 ! c5tre ( ! l9e-pair be+ ! po9t ( contact4 ,if a quadric surface conta9s a l9e1 !n e plane ?r\gh ?is l9e cuts ! surface 9 a l9e-pair 7or 9 two co9cid5t l9es74 ,= ?is plane _c cut ! surface 9 a conic4 ,h5ce3-- ,if a quadric surface conta9s "o l9e p !n x conta9s an inf9ite numb] ( l9es1 & ?r\gh e po9t ,q on ! surface1 "o l9e q c be drawn :i* cuts p4 ,= ! plane ?r\gh ! po9t ,q & ! l9e p cuts ! surface 9 a l9e-pair :i* mu/ pass ?r\gh ,q & ( :i* p is "o l9e4 ,no two su* l9es q on ! surface c meet4 ,= z bo? meet p _! plane w\ld conta9 p & !re=e cut ! surface 9 a triangle4 ,e l9e :i* cuts ?ree l9es q w be on ! surface2 = x has ?ree po9ts 9 common ) x4 ,h5ce ! quadric surfaces :i* conta9 l9es are ! same z ! rul$ quadric surfaces consid]$ 9 ,,ss #hi-ic1 b ) "o important exception4 ,9 ! la/ inve/igation we h left out ( consid]ation ! possibility ( a plane hav+ only "o l9e 7two co9cid5t l9es7 9 common ) a quadric surface4 ,s #ih4 ,to inve/igate ?is case we suppose fir/ t !re is "o po9t ,a on ! surface ?r\gh :i* two diff]5t l9es a1 b c be drawn1 :i* lie altoge!r on ! surface4 ,if ,p is any o!r po9t on ! surface :i* lies nei!r on a nor b1 !n ! plane ?r\gh ,p & a w cut ! surface 9 a second l9e a' :i* passes ?r\gh ,p & :i* cuts a4 ,simil>ly !re is a l9e b' ?r\gh ,p :i* cuts b4 ,^! two l9es a' & b' may co9cide1 b !n !y mu/ co9cide ) ,,pa4 ,if ?is happ5s = "o po9t ,p1 x happ5s = e o!r po9t ,q4 ,= if two diff]5t l9es c\ld be drawn ?r\gh ,q1 !n by ! same reason+ ! l9e ,,pq w\ld be altoge!r on ! surface1 h5ce two l9es w\ld be drawn ?r\gh ,p aga9/ ! assumption4 ,f ?is foll[s3-- ,if !re is "o po9t on a quadric surface ?r\gh :i* "o1 b only "o1 l9e c be drawn on ! surface1 !n ?r\gh e po9t "o l9e c be drawn1 & all ^! l9es meet 9 a po9t4 ,! surface is a cone ( ! second ord]4 ,if ?r\gh "o po9t on a quadric surface1 two1 & only two1 l9es c be drawn on ! surface1 !n ?r\gh e po9t two l9es may be drawn1 & ! surface is rul$ quadric surface4 ,if ?r\gh "o po9t on a quadric surface no l9e on ! surface c be drawn1 !n ! surface conta9s no l9es4 ,us+ ! def9itions at ! end ( ,s #ie1 we may also say3-- ,on a quadric surface ! po9ts are all hyp]bolic1 or all p>abolic1 or all elliptic4 ,z an example ( a quadric surface ) elliptical po9ts1 we m5tion ! sph]e :i* may be g5]at$ by two reciprocal p5cils1 :]e to ea* l9e 9 "o corresponds ! plane p]p5dicul> to x 9 ! o!r4 ,s #ii4 ,poles & ,pol> ,planes4--,! !ory ( poles & pol>s ) reg>d to a conic is easily ext5d$ to quadric surfaces4 ,let ,p be a po9t 9 space n on ! surface1 :i* we suppose n to be a cone4 ,on e l9e ?r\gh ,p :i* cuts ! surface 9 two po9ts we det]m9e ! h>monic conjugate ,q ( ,p ) reg>d to ! po9ts ( int]section4 ,?r\gh "o ( ^! l9es we draw two planes alpha & beta4 ,! locus ( ! po9ts ,q 9 alpha is a l9e a1 ! pol> ( ,p ) reg>d to ! conic 9 :i* alpha cuts ! surface4 ,simil>ly ! locus ( po9ts ,q 9 beta is a l9e b4 ,?is cuts a1 because ! l9e ( int]section ( alpha & beta conta9s b "o po9t ,q4 ,! locus ( all po9ts ,q !re=e is a plane4 ,?is plane is call$ ! pol> plane ( ! po9t ,p1 ) reg>d to ! quadric surface4 ,if ,p lies on ! surface we take ! tang5t plane ( ,p z its pol>4 ,! foll[+ propositions hold3-- #a4 ,e po9t has a pol> plane1 :i* is con/ruct$ by draw+ ! pol>s ( ! po9t ) reg>d to ! conics 9 :i* two planes ?r\gh ! po9t cut ! surface4 #b4 ,if ,q is a po9t 9 ! pol> ( ,p1 !n ,p is a po9t 9 ! pol> ( ,q1 because ?is is true ) reg>d to ! conic 9 :i* a plane ?r\gh ,,pq cuts ! surface4 #c4 ,e plane is ! pol> plane ( "o po9t1 :i* is call$ ! ,pole ( ! plane4 ,! pole to a plane is f\nd by con/ruct+ ! pol> planes ( ?ree po9ts 9 ! plane4 ,_! int]section w be ! pole4 #d4 ,! po9ts 9 :i* ! pol> plane ( ,p cuts ! surface are po9ts ( contact ( tang5ts drawn f ,p to ! surface1 z is easily se54 ,h5ce3-- #e4 ,! tang5ts drawn f a po9t ,p to a quadric surface =m a cone ( ! second ord]1 = ! pol> plane ( ,p cuts x 9 a conic4 #f4 ,if ! pole describes a l9e a1 its pol> plane w turn ab\t ano!r l9e a'1 z foll[s f #b4 ,^! l9es a & a' are said to be conjugate ) reg>d to ! surface4 ,s #ajj4 ,! pole ( ! l9e at inf9ity is call$ ! c5tre ( ! surface4 ,if x lies at ! inf9ity1 ! plane at inf9ity is a tang5t plane1 & ! surface is call$ a p>aboloid4 ,! pol> plane to any po9t at inf9ity passes ?r\gh ! c5tre1 & is call$ a diametrical plane4 ,a l9e ?r\gh ! c5tre is call$ a diamet]4 ,x is bisect$ at ! c5tre4 ,! l9e conjugate to x lies at inf9ity4 ,if a po9t moves along a diamet] its pol> plane turns ab\t ! conjugate l9e at inf9ity2 t is1 x moves p>allel to itself1 its c5tre mov+ on ! fir/ l9e4 ,! middle po9ts ( p>allel *ords lie 9 a plane1 viz4 9 ! pol> plane ( ! po9t at inf9ity ?r\gh :i* ! *ords are drawn4 ,! c5tres ( p>allel sections lie 9 a diamet] :i* is a l9e conjugate to ! l9e at inf9ity 9 :i* ! planes meet4 ,,twi/$ ,,cubics ,s #aja4 ,if two p5cils ) c5tres ,s#a & ,s#b are made projective1 !n to a ray 9 "o corresponds a ray 9 ! o!r1 to a plane a plane1 to a flat or axial p5cil a projective flat or axial p5cil1 & s on4 ,!re is a d\ble inf9ite numb] ( l9es 9 a p5cil4 ,we %all see t a s+le inf9ite numb] ( l9es 9 "o p5cil meets its correspond+ ray1 & t ! po9ts ( int]section =m a curve 9 space4 ,( ! d\ble inf9ite numb] ( planes 9 ! p5cils ea* w meet its correspond+ plane4 ,?is gives a sy/em ( a d\ble inf9ite numb] ( l9es 9 space4 ,we "k 7,s #e7 t !re is a quadruple inf9ite numb] ( l9es 9 space4 ,f among ^! we may select ^? :i* satisfy "o or m giv5 conditions4 ,! sy/ems ( l9es ?us obta9$ 7 fir/ sy/ematically inve/igat$ & classifi$ by ,pluck]1 9 8 ,geometrie des ,raumes4 ,he uses ! foll[+ names3-- ,a treble inf9ite numb] ( l9es1 t is1 all l9es :i* satisfy "o condition1 are said to =m a complex ( l9es2 e4g4 all l9es cutt+ a giv5 l9e1 or all l9es t\*+ a surface4 ,a d\ble inf9ite numb] ( l9es1 t is1 all l9es :i* satisfy two conditions1 or :i* are common to two complexes1 are said to =m a congru5ce ( l9es2 e4g4 all l9es 9 a plane1 or all l9es cutt+ two curves1 or all l9es cutt+ a giv5 curve twice4 ,a s+le inf9ite numb] ( l9es1 t is1 all l9es :i* satisfy ?ree conditions1 or :i* belong to ?ree complexes1 =m a rul$ surface2 e4g4 "o set ( l9es on a rul$ quadric surface1 or developable surfaces :i* are =m$ by ! tang5ts to a curve4 ,x foll[s t all l9es 9 :i* correspond+ planes 9 two projective p5cils meet =m a congru5ce4 ,we %all see ?is congru5ce consi/s ( all l9es :i* cut a twi/$ cubic twice1 or ( all secants to a twi/$ cubic4 ,s #ajb4 ,let l#a be ! l9e ,s#a,s#b z a l9e 9 ! p5cil ,s#a4 ,to x corresponds a l9e l#b 9 ,s#b4 ,at ea* ( ! c5tres two correspond+ l9es meet4 ,! two axial p5cils ) l#a & l#b z axes are projective1 &1 z1 _! axes meet at ,s#b1 ! int]sections ( correspond+ planes =m a cone ( ! second ord] 7,s #eh71 ) ,s#b z c5tre4 ,if pi#a & pi#b be correspond+ planes1 !n _! int]section w be a l9e p#b :i* passes ?r\gh ,s#b4 ,correspond+ to x 9 ,s#a w be a l9e p#a :i* lies 9 ! plane pi#a1 & :i* !re=e meets p#b at "s po9t ,p4 ,conv]sely1 if p#b be any l9e 9 ,s#b :i* meets its correspond+ l9e p#a at a po9t ,p1 !n to ! plane l#bp#b w correspond ! plane l#ap#a1 t is1 ! plane ,s#a,s#b,p4 ,^! planes int]sect 9 p#b1 s t p#b is a l9e on ! quadric cone g5]at$ by ! axial p5cils l#a & l#b4 ,h5ce3-- ,all l9es 9 "o p5cil :i* meet _! correspond+ l9es 9 ! o!r =m a cone ( ! second ord] :i* has its c5tre at ! c5tre ( ! fir/ p5cil1 & passes ?r\gh ! c5tre ( ! second4 ,f ?is foll[s t ! po9ts 9 :i* correspond+ rays meet lie on two cones ( ! second ord] :i* h ! ray jo9+ _! c5tres 9 common1 & =m !re=e1 toge!r ) ! l9e ,s#a,s#b or l#a1 ! int]section ( ^! cones4 ,any plane cuts ea* ( ! cones 9 a conic4 ,^! two conics h necess>ily t po9t 9 common 9 :i* x cuts ! l9e l#a1 & !re=e besides ei!r "o or ?ree o!r po9ts4 ,x foll[s t ! curve is ( ! ?ird ord] z a plane may cut x 9 ?ree1 b n 9 m ?an ?ree1 po9ts4 ,h5ce3-- ,! locus ( po9ts 9 :i* correspond+ l9es on two projective p5cils meet is a curve ( ! ?ird ord] or a twi/$ cubic k1 :i* passes ?r\gh ! c5tres ( ! p5cils1 & :i* appe>s z ! int]section ( two cones ( ! second ord]1 :i* h "o l9e 9 common4 ,a l9e belong+ to ! congru5ce det]m9$ by ! p5cils is a secant ( ! cubic2 x has two1 or "o1 or no po9ts 9 common ) ?is cubic1 & is call$ accord+ly a secant prop]1 a tang5t1 or a secant improp] ( ! cubic4 ,a secant improp] may be consid]$1 to use ! language ( coord9ate geometry1 z a secant ) imag9>y po9ts ( int]section4 ,s #ajc4 ,if a#a & a#b be any two correspond+ l9es 9 ! two p5cils1 !n correspond+ planes 9 ! axial p5cils hav+ a#a & a#b z axes g5]ate a rul$ quadric surface4 ,if ,p be any po9t on ! cubic k1 & if p#a1 p#b be ! correspond+ rays 9 ,s#a & ,s#b :i* meet at ,p1 !n to ! plane a#ap#a 9 ,s#a corresponds a#bp#b 9 ,s#b4 ,^! !re=e meet 9 a l9e ?r\gh ,p4 ,?is may be /at$ ?us3-- ,^? secants ( ! cubic :i* cut a ray a#a1 drawn ?r\gh ! c5tre ,s#a ( "o p5cil1 =m a rul$ quadric surface :i* passes ?r\gh bo? c5tres1 & :i* conta9s ! twi/$ cubic k4 ,( su* surfaces an inf9ite numb] exi/s4 ,e ray ?r\gh ,s#a or ,s#b :i* is n a secant det]m9es "o ( !m4 ,if1 h[ev]1 ! rays a#a & a#b are secants meet+ at ,a1 !n ! rul$ quadric surface becomes a cone ( ! second ord]1 hav+ ,a z c5tre4 ,or all l9es ( ! congru5ce :i* pass ?r\gh a po9t on ! twi/$ cubic k =m a cone ( ! second ord]4 ,9 o!r words1 ! projection ( a twi/$ cubic f any po9t 9 ! curve on to any plane is a conic4 ,if a#a is n a secant1 b made to pass ?r\gh any po9t ,q 9 space1 ! rul$ quadric surface det]m9$ by a#a w pass ?r\gh ,q4 ,!re w !re=e be "o l9e ( ! congru5ce pass+ ?r\gh ,q1 & only "o4 ,= if two su* l9es pass ?r\gh ,q1 !n ! l9es ,s#a,q & ,s#b,q w be correspond+ l9es2 h5ce ,q w be a po9t on ! cubic k1 & an inf9ite numb] ( secants w pass ?r\gh x4 ,h5ce3-- ,?r\gh e po9t 9 space n on ! twi/$ cubic "o & only "o secant to ! cubic c be drawn4 ,s #ajd4 ,! fact t all ! secants ?r\gh a po9t on ! cubic =m a quadric cone %[s t ! c5tres ( ! projective p5cils g5]at+ ! cubic are n di/+ui%$ f any o!r po9ts on ! cubic4 ,if we take any two po9ts ,s1 ,s' on ! cubic1 & draw ! secants ?r\gh ea* ( !m1 we obta9 two quadric cones1 :i* h ! l9e ,,ss' 9 common1 & :i* int]sect besides along ! cubic4 ,if we make ^! two p5cils hav+ ,s & ,s' z c5tres projective by tak+ f\r rays on ! "o cone z correspond+ to ! f\r rays on ! o!r :i* meet ! fir/ on ! cubic1 ! correspond5ce is det]m9$4 ,^! two p5cils w g5]ate a cubic1 & ! two cones ( secants hav+ ,s & ,s' z c5tres w be id5tical ) ! above cones1 = ea* has five rays 9 common ) "o ( ! fir/1 viz4 ! l9e ,,ss' & ! f\r l9es det]m9$ = ! correspond5ce2 !re=e ^! two cones int]sect 9 ! orig9al cubic4 ,?is gives ! !orem3-- ,on a twi/$ cubic any two po9ts may be tak5 z c5tres ( projective p5cils :i* g5]ate ! cubic1 correspond+ planes be+ ^? :i* meet on ! same secant4 ,( ! two projective p5cils at ,s & ,s' we may keep ! fir/ fix$1 & move ! c5tre ( ! o!r along ! curve4 ,! p5cils w h]eby rema9 projective1 & a plane alpha 9 ,s w be cut by its correspond+ plane alpha' always 9 ! same secant a4 ,:il/ ,s' moves along ! curve ! plane alpha' w turn ab\t a1 describ+ an axial p5cil4 ,,au?orities4--,9 ?is article we h giv5 a purely geometrical !ory ( conics1 cones ( ! second ord]1 quadric surfaces1 c4 ,9 do+ s we h foll[$1 to a great ext5t1 ,reye's ,geometrie d] ,lage1 & to ?is excell5t "w ^? read]s are ref]r$ :o wi% = a m exhau/ive treatm5t ( ! subject4 ,o!r works especially valuable z %[+ ! developm5t ( ! subject are3 ,monge1 ,geometrie descriptive3 ,c>not1 ,geometrie de position 7#ahjc71 conta9+ a !ory ( transv]sals2 ,poncelet's great "w ,traite des proprietes projectives des figures 7#ahbb72 ,mob9s1 ,b>yc5tris*] ,calcul 7#ahbf72 ,/e9]1 ,abhangigkeit geometris*] ,ge/alt5 7#ahcb71 conta9+ ! fir/ full discussion ( ! projective relations betwe5 r[s1 p5cils1 c42 ,von ,/audt1 ,geometrie d] ,lage 7#ahdg7 & ,beitrage zur ,geometrie d] ,lage 7#ahef-ahfj71 9 :i* a sy/em ( geometry is built up f ! beg9n+ )\t any ref]5ce to numb]1 s t ultimately a numb] itself gets a geometrical def9ition1 & 9 :i* imag9>y elem5ts are sy/ematically introduc$ into pure geometry2 ,*asles1 ,ap]cu hi/orique 7#ahcg71 9 :i* ! au?or gives a brilliant acc\nt ( ! progress ( mod]n geometrical me?ods1 po9t+ out ! advantages ( ! diff]5t purely geometrical me?ods z comp>$ ) ! analytical ones1 b )\t tak+ z mu* acc\nt ( ! ,g]man z ( ! ,fr5* au?ors2 ,id41 ,rapport sur les progres de la geometrie 7#ahgj71 a cont9uation ( ! ,ap]cu2 ,id41 ,traite de geometrie sup]ieure 7#aheb72 ,cremona1 ,introduzione ad una teoria geometrica delle curve piane 7#ahfb7 & its cont9uation ,prelim9>i di una teoria geometrica delle sup]ficie 7,g]man translations by ,curtze74 ,z m elem5t>y books1 we m5tion3 ,cremona1 ,elem5ts ( ,projective ,geometry1 translat$ f ! ,italian by ,c4 ,leudesdorf 7#bnd ed41 #ahid72 ,j4,w4 ,russell1 ,pure ,geometry 7#bnd ed41 #aije74 7,o4 ,h47 ,,iii4 ,,descriptive ,,geometry ,?is bran* ( geometry is conc]n$ ) ! me?ods = repres5t+ solids & o!r figures 9 ?ree dim5sions by draw+s 9 "o plane4 ,! mo/ important me?od is t :i* 0 inv5t$ by ,monge t[>ds ! end ( ! #ah? c5tury4 ,x is bas$ on p>allel projections to a plane by rays p]p5dicul> to ! plane4 ,su* a projection is call$ or?ographic 7see ,,projection1 ,s #ah74 ,if ! plane is horizontal ! projection is call$ ! plan ( ! figure1 & if ! plane is v]tical ! elevation4 ,9 ,monge's me?od a figure is repres5t$ by its plan & elevation4 ,x is !re=e (t5 call$ draw+ 9 plan & elevation1 & sometimes simply or?ographic projection4 ,s #a4 ,we suppose !n t we h two planes1 "o horizontal1 ! o!r v]tical1 & ^! we call ! planes ( plan & ( elevation respectively1 or ! horizontal & ! v]tical plane1 & d5ote !m by ! lett]s pi#a & pi#b4 ,_! l9e ( int]section is call$ ! axis1 & w be d5ot$ by xy4 ,if ! surface ( ! draw+ pap] is tak5 z ! plane ( ! plan1 !n ! v]tical plane w be ! plane p]p5dicul> to x ?r\gh ! axis xy4 ,to br+ ?is also into ! plane ( ! draw+ pap] we turn x ab\t ! axis till x co9cides ) ! horizontal plane4 ,?is process ( turn+ "o plane d[n till x co9cides ) ano!r is call$ rabatt+ "o to ! o!r4 ,( c\rse !re is no necessity to h "o ( ! two planes horizontal1 b ev5 :5 ?is is n ! case x is conv5i5t to reta9 ! above names4 ,illu/ration3 ,,fig4 #cg4 ,illu/ration3 ,,fig4 #ch4 ,! :ole arrangem5t w be bett] und]/ood by ref]r+ to fig4 #cg4 ,a po9t ,a 9 space is !re project$ by ! p]p5dicul> ,,aa#a & ,,aa#b to ! planes pi#a & pi#b s t ,a#a & ,a#b are ! horizontal & v]tical projections ( ,a4 ,if we rememb] t a l9e is p]p5dicul> to a plane t is p]p5dicul> to e l9e 9 ! plane if only x is p]p5dicul> to any two int]sect+ l9es 9 ! plane1 we see t ! axis :i* is p]p5dicul> bo? to ,,aa#a & to ,,aa#b is also p]p5dicul> to ,a#a;,a#j & to ,a#b;,a#j because ^! f\r l9es are all 9 ! same plane4 ,h5ce1 if ! plane pi#b be turn$ ab\t ! axis till x co9cides ) ! plane pi#a1 !n ,a#b;,a#j w be ! cont9uation ( ,a#a;,a#j4 ,?is position ( ! planes is repres5t$ 9 fig4 #ch1 9 :i* ! l9e ,a#a;,a#b is p]p5dicul> to ! axis x4 ,conv]sely any two po9ts ,a#a1 ,a#b 9 a l9e p]p5dicul> to ! axis w be ! projections ( "s po9t 9 space :5 ! plane pi#b is turn$ ab\t ! axis till x is p]p5dicul> to ! plane pi#a1 because 9 ?is position ! two p]p5dicul>s to ! planes pi#a & pi#b ?r\gh ! po9ts ,a#a & ,a#b w be 9 a plane & !re=e meet at "s po9t ,a4 ,repres5tation ( ,po9ts4--,we h ?us ! foll[+ me?od ( repres5t+ 9 a s+le plane ! position ( po9ts 9 space3--we take 9 ! plane a l9e xy z ! axis1 & !n any pair ( po9ts ,a#a1 ,a#b 9 ! plane on a l9e p]p5dicul> to ! axis repres5t a po9t ,a 9 space4 ,if ! l9e ,a#a;,a#b cuts ! axis at ,a#j1 & if at ,a#a a p]p5dicul> be erect$ to ! plane1 !n ! po9t ,a w be 9 x at a height ,a#a;,a ,a#j;,a#b above ! plane4 ,?is gives ! position ( ! po9t ,a relative to ! plane pi#a4 ,9 ! same way1 if 9 a p]p5dicul> to pi#b ?r\gh ,a#b a po9t ,a be tak5 su* t ,a#b;,a ,a#j;,a#a1 !n ?is w give ! po9t ,a relative to ! plane pi#b4 ,illu/ration3 ,,fig4 #ci4 ,s #b4 ,! two planes pi#a1 pi#b 9 _! orig9al position divide space into f\r p>ts4 ,^! are call$ ! f\r quadrants4 ,we suppose t ! plane pi#b is turn$ z indicat$ 9 fig4 #cg1 s t ! po9t ,p comes to ,q & ,r to ,s1 !n ! quadrant 9 :i* ! po9t ,a lies is call$ ! fir/1 & we say t 9 ! fir/ quadrant a po9t lies above ! horizontal & 9 front ( ! v]tical plane4 ,n[ we g r\nd ! axis 9 ! s5se 9 :i* ! plane pi#b is turn$ & come 9 succession to ! second1 ?ird & f\r? quadrant4 ,9 ! second a po9t lies above ! plane ( ! plan & beh9d ! plane ( elevation1 & s on4 ,9 fig4 #ci1 :i* repres5ts a side view ( ! planes 9 fig4 #cg ! quadrants are m>k$1 & 9 ea* a po9t ) its projection is tak54 ,fig4 #ch %[s h[ ^! are repres5t$ :5 ! plane pi#b is turn$ d[n4 ,we see t ,a po9t lies 9 ! fir/ quadrant if ! plan lies bel[1 ! elevation above ! axis2 9 ! second if plan & elevation bo? lie above2 9 ! ?ird if ! plan lies above1 ! elevation bel[2 9 ! f\r? if plan & elevation bo? lie bel[ ! axis4 ,if a po9t lies 9 ! horizontal plane1 its elevation lies 9 ! axis & ! plan co9cides ) ! po9t itself4 ,if a po9t lies 9 ! v]tical plane1 its plan lies 9 ! axis & ! elevation co9cides ) ! po9t itself4 ,if a po9t lies 9 ! axis1 bo? its plan & elevation lie 9 ! axis & co9cide ) x4 ,( ea* ( ^! propositions1 :i* w easily be se5 to be true1 ! conv]se holds also4 ,s #c4 ,repres5tation ( a ,plane4--,z we are ?us enabl$ to repres5t po9ts 9 a plane1 we c repres5t any f9ite figure by repres5t+ its sep>ate po9ts4 ,x is1 h[ev]1 n possible to repres5t a plane 9 ?is way1 = ! projections ( its po9ts completely cov] ! planes pi#a & pi#b1 & no plane w\ld appe> diff]5t f any o!r4 ,b any plane alpha cuts ea* ( ! planes pi#a1 pi#b 9 a l9e4 ,^! are call$ ! traces ( ! plane4 ,!y cut ea* o!r 9 ! axis at ! po9t :]e ! latt] cuts ! plane alpha4 ,a plane is det]m9$ by its two traces1 :i* are two l9es t meet on ! axis1 &1 conv]sely1 any two l9es :i* meet on ! axis det]m9e a plane4 ,if ! plane is p>allel to ! axis its traces are p>allel to ! axis4 ,( ^! "o may be at inf9ity2 !n ! plane w cut "o ( ! planes ( projection at inf9ity & w be p>allel to x4 ,?us a plane p>allel to ! horizontal plane ( ! plan has only "o f9ite trace1 viz4 t ) ! plane ( elevation4 ,illu/ration3 ,,fig4 #dj4 ,if ! plane passes ?r\gh ! axis bo? its traces co9cide ) ! axis4 ,?is is ! only case 9 :i* ! repres5tation ( ! plane by its two traces fails4 ,a ?ird plane ( projection is !re=e introduc$1 :i* is be/ tak5 p]p5dicul> to ! o!r two4 ,we call x simply ! ?ird plane & d5ote x by pi#c4 ,z x is p]p5dicul> to pi#a1 x may be tak5 z ! plane ( elevation1 its l9e ( int]section gamma ) pi#a be+ ! axis1 & be turn$ d[n to co9cide ) pi#a4 ,?is is repres5t$ 9 fig4 #dj4 ,,oc is ! axis xy :il/ ,,oa & ,,ob are ! traces ( ! ?ird plane4 ,!y lie 9 "o l9e gamma4 ,! plane is rabatt$ ab\t gamma to ! horizontal plane4 ,a plane alpha ?r\gh ! axis xy w !n %[ 9 x a trace alpha#c4 ,9 fig4 #dj ! l9es ,,oc & ,,op w ?us be ! traces ( a plane ?r\gh ! axis xy1 :i* makes an angle ,,poq ) ! horizontal plane4 ,we c also f9d ! trace :i* any o!r plane makes ) pi#c4 ,9 rabatt+ ! plane pi#c its trace ,,ob ) ! plane pi#b w come to ! position ,,od4 ,h5ce a plane beta hav+ ! traces ,,ca & ,,cb w h ) ! ?ird plane ! trace beta#c1 or ,,ad if ,,od ,,ob4 ,x also foll[s imm$iately t-- ,if a plane alpha is p]p5dicul> to ! horizontal plane1 !n e po9t 9 x has its horizontal projection 9 ! horizontal trace ( ! plane1 z all ! rays project+ ^! po9ts lie 9 ! plane itself4 ,any plane :i* is p]p5dicul> to ! horizontal plane has its v]tical trace p]p5dicul> to ! axis4 ,any plane :i* is p]p5dicul> to ! v]tical plane has its horizontal trace p]p5dicul> to ! axis & ! v]tical projections ( all po9ts 9 ! plane lie 9 ?is trace4 ,s #d4 ,repres5tation ( a ,l9e4--,a l9e is det]m9$ ei!r by two po9ts 9 x or by two planes ?r\gh x4 ,we get accord+ly two repres5tations ( x ei!r by projections or by traces4 ,fir/4--,a l9e a is repres5t$ by its projections a#a & a#b on ! two planes pi#a & pi#b4 ,^! may be any two l9es1 =1 br++ ! planes pi#a1 pi#b into _! orig9al position1 ! planes ?r\gh ^! l9es p]p5dicul> to pi#a & pi#b respectively w int]sect 9 "s l9e a :i* has a#a1 a#b z its projections4 ,secondly4--,a l9e a is repres5t$ by its traces--t is1 by ! po9ts 9 :i* x cuts ! two planes pi#a1 pi#b4 ,any two po9ts may be tak5 z ! traces ( a l9e 9 space1 = x is det]m9$ :5 ! planes are 9 _! orig9al position z ! l9e jo9+ ! two traces4 ,?is repres5tation becomes undet]m9$ if ! two traces co9cide 9 ! axis4 ,9 ?is case we aga9 use a ?ird plane1 or else ! projections ( ! l9e4 ,! fact t !re are diff]5t me?ods ( repres5t+ po9ts & planes1 & h5ce two me?ods ( repres5t+ l9es1 sugge/s ! pr9ciple ( duality 7section ii41 ,projective ,geometry1 ,s #da74 ,x is wor? :ile to keep ?is 9 m9d4 ,x is also wor? rememb]+ t traces ( planes or l9es always lie 9 ! planes or l9es :i* !y repres5t4 ,projections d n z a rule d ?is except+ :5 ! po9t or l9e project$ lies 9 "o ( ! planes ( projection4 ,hav+ n[ %[n h[ to repres5t po9ts1 planes & l9es1 we h to /ate ! conditions :i* mu/ hold 9 ord] t ^! elem5ts may lie "o 9 ! o!r1 or else t ! figure =m$ by !m may possess c]ta9 metrical prop]ties4 ,x w be f\nd t ! =m] are v mu* simpl] ?an ! latt]4 ,be=e we d ?is1 h[ev]1 we %all expla9 ! notation us$2 = x is ( great importance to h a sy/ematic notation4 ,we %all d5ote po9ts 9 space by capitals ,a1 ,b1 ,c2 planes 9 space by ,greek lett]s alpha1 beta1 gamma2 l9es 9 space by small lett]s a1 b1 c2 horizontal projections by suffixes #a1 l ,a#a1 a#a2 v]tical projections by suffixes #b1 l ,a#b1 a#b2 traces by s+le & d\ble da%es alpha' alpha1 a'1 a4 ,h5ce ,p#a w be ! horizontal projection ( a po9t ,p 9 space2 a l9e a w h ! projections a#a1 a#b & ! traces a' & a2 a plane alpha has ! traces alpha' & alpha4 ,s #e4 ,if a po9t lies 9 a l9e1 ! projections ( ! po9t lie 9 ! projections ( ! l9e4 ,if a l9e lies 9 a plane1 ! traces ( ! l9e lie 9 ! traces ( ! plane4 ,^! propositions foll[ at once f ! def9itions ( ! projections & ( ! traces4 ,if a po9t lies 9 two l9es its projections mu/ lie 9 ! projections ( bo?4 ,h5ce ,if two l9es1 giv5 by _! projections1 int]sect1 ! int]section ( _! planes & ! int]section ( _! elevations mu/ lie 9 a l9e p]p5dicul> to ! axis1 because !y mu/ be ! projections ( ! po9t common to ! two l9es4 ,simil>ly--,if two l9es giv5 by _! traces lie 9 ! same plane or int]sect1 !n ! l9es jo9+ _! horizontal & v]tical traces respectively mu/ meet on ! axis1 because !y mu/ be ! traces ( ! plane ?r\gh !m4 ,s #f4 ,to f9d ! projections ( a l9e :i* jo9s two po9ts ,a1 ,b giv5 by _! projections ,a#a1 ,a#b & ,b#a1 ,b#b1 we jo9 ,a#a1 ,b#a & ,a#b1 ,b#b2 ^! w be ! projections requir$4 ,= example1 ! traces ( a l9e are two po9ts 9 ! l9e ^: projections are kn[n or at all ev5ts easily f\nd4 ,!y are ! traces !mselves & ! feet ( ! p]p5dicul>s f !m to ! axis4 ,h5ce if a' a 7fig4 #da7 are ! traces ( a l9e a1 & if ! p]p5dicul>s f !m cut ! axis 9 ,p & ,q respectively1 !n ! l9e a',q w be ! horizontal & a,p ! v]tical projection ( ! l9e4 ,illu/ration3 ,,fig4 #da4 ,conv]sely1 if ! projections a#a1 a#b ( a l9e are giv51 & if ^! cut ! axis 9 ,q & ,p respectively1 !n ! p]p5dicul>s ,pa' & ,qa to ! axis drawn ?r\gh ^! po9ts cut ! projections a#a & a#b 9 ! traces a' & a4 ,to f9d ! l9e ( int]section ( two planes1 we obs]ve t ?is l9e lies 9 bo? planes2 its traces mu/ !re=e lie 9 ! traces ( bo?4 ,h5ce ! po9ts :]e ! horizontal traces ( ! giv5 planes meet w be ! horizontal1 & ! po9t :]e ! v]tical traces meet ! v]tical trace ( ! l9e requir$4 ,s #g4 ,to decide :e!r a po9t ,a1 giv5 by its projections1 lies 9 a plane alpha1 giv5 by its traces1 we draw a l9e p by jo9+ ,a to "s po9t 9 ! plane alpha & det]m9e its traces4 ,if ^! lie 9 ! traces ( ! plane1 !n ! l9e1 & !re=e ! po9t ,a1 lies 9 ! plane2 o!rwise n4 ,?is is conv5i5tly done by jo9+ ,a#a to "s po9t p' 9 ! trace alpha'2 ?is gives p#a2 & ! po9t :]e ! p]p5dicul> f p' to ! axis cuts ! latt] we jo9 to ,a#b2 ?is gives p#b4 ,if ! v]tical trace ( ?is l9e lies 9 ! v]tical trace ( ! plane1 !n1 & !n only1 does ! l9e p1 & ) x ! po9t ,a1 lie 9 ! plane alpha4 ,s #h4 ,p>allel planes h p>allel traces1 because p>allel planes are cut by any plane1 h5ce also by pi#a & by pi#b1 9 p>allel l9es4 ,p>allel l9es h p>allel projections1 because po9ts at inf9ity are project$ to inf9ity4 ,if a l9e is p>allel to a plane1 !n l9es ?r\gh ! traces ( ! l9e & p>allel to ! traces ( ! plane mu/ meet on ! axis1 because ^! l9es are ! traces ( a plane p>allel to ! giv5 plane4 ,s #i4 ,to draw a plane ?r\gh two int]sect+ l9es or ?r\gh two p>allel l9es1 we det]m9e ! traces ( ! l9es2 ! l9es jo9+ _! horizontal & v]tical traces respectively w be ! horizontal & v]tical traces ( ! plane4 ,!y w meet1 at a f9ite po9t or at inf9ity1 on ! axis if ! l9es d int]sect4 ,to draw a plane ?r\gh a l9e & a po9t )\t ! l9e1 we jo9 ! giv5 po9t to any po9t 9 ! l9e & det]m9e ! plane ?r\gh ?is & ! giv5 l9e4 ,to draw a plane ?r\gh ?ree po9ts :i* are n 9 a l9e1 we draw two ( ! l9es :i* ea* jo9 two ( ! giv5 po9ts & draw ! plane ?r\gh !m4 ,if ! traces ( all ?ree l9es ,,ab1 ,,bc1 ,,ca be f\nd1 ^! mu/ lie 9 two l9es :i* meet on ! axis4 ,s #aj4 ,we h 9 ! la/ example got m po9ts1 or c easily get m po9ts1 ?an are necess>y = ! det]m9ation ( ! figure requir$--9 ?is case ! traces ( ! plane4 ,?is w happ5 9 a great _m con/ructions & is ( consid]able importance4 ,x may happ5 t "s ( ! po9ts or l9es obta9$ are n conv5i5t 9 ! actual con/ruction4 ,! horizontal traces ( ! l9es ,,ab & ,,ac may1 = in/ance1 fall v ne> toge!r1 9 :i* case ! l9e jo9+ !m is n well def9$4 ,or1 "o or bo? ( !m may fall beyond ! draw+ pap]1 s t !y are practically non-exi/5t = ! con/ruction4 ,9 ?is case ! traces ( ! l9e ,,bc may be us$4 ,or1 if ! v]tical traces ( ,,ab & ,,ac are bo? 9 conv5i5t position1 s t ! v]tical trace ( ! requir$ plane is f\nd & "o ( ! horizontal traces is got1 !n we may jo9 ! latt] to ! po9t :]e ! v]tical trace cuts ! axis4 ,! draughtsman mu/ rememb] t ! l9es :i* he draws are n ma!matical l9es )\t ?ickness1 & !re=e e draw+ is affect$ by "s errors4 ,x is !re=e v desirable to be able con/antly to *eck ! latt]4 ,su* *ecks always pres5t !mselves :5 ! same result c be obta9$ by diff]5t con/ructions1 or :51 z 9 ! above case1 "s l9es mu/ meet on ! axis1 or if ?ree po9ts mu/ lie 9 a l9e4 ,a c>eful draughtsman w always avail himself ( ^! *ecks4 ,s #aa4 ,to draw a plane ?r\gh a giv5 po9t p>allel to a giv5 plane alpha1 we draw ?r\gh ! po9t two l9es :i* are p>allel to ! plane alpha1 & det]m9e ! plane ?r\gh !m2 or1 z we "k t ! traces ( ! requir$ plane are p>allel to ^? ( ! giv5 "o 7,s #h71 we ne$ only draw "o l9e l ?r\gh ! po9t p>allel to ! plane & f9d "o ( its traces1 say ! v]tical trace l2 a l9e ?r\gh ?is p>allel to ! v]tical trace ( alpha w be ! v]tical trace beta ( ! requir$ plane beta1 & a l9e p>allel to ! horizontal trace ( alpha meet+ beta on ! axis w be ! horizontal trace beta'4 ,illu/ration3 ,,fig4 #db4 ,let ,a#a ,a#b 7fig4 #db7 be ! giv5 po9t1 alpha' alpha ! giv5 plane1 a l9e l#a ?r\gh ,a#a1 p>allel to alpha' & a horizontal l9e l#b ?r\gh ,a#b w be ! projections ( a l9e l ?r\gh ,a p>allel to ! plane1 because ! horizontal plane ?r\gh ?is l9e w cut ! plane alpha 9 a l9e c :i* has its horizontal projection c#a p>allel to alpha'4 ,s #ab4 ,we n[ come to ! metrical prop]ties ( figures4 ,a l9e is p]p5dicul> to a plane if ! projections ( ! l9e are p]p5dicul> to ! traces ( ! plane4 ,we prove x = ! horizontal projection4 ,if a l9e p is p]p5dicul> to a plane alpha1 e plane ?r\gh p is p]p5dicul> to alpha2 h5ce also ! v]tical plane :i* projects ! l9e p to p#a4 ,z ?is plane is p]p5dicul> bo? to ! horizontal plane & to ! plane alpha1 x is also p]p5dicul> to _! int]section--t is1 to ! horizontal trace ( alpha4 ,x foll[s t e l9e 9 ?is project+ plane1 !re=e also p#a1 ! plan ( p1 is p]p5dicul> to ! horizontal trace ( alpha4 ,to draw a plane ?r\gh a giv5 po9t ,a p]p5dicul> to a giv5 l9e p1 we fir/ draw ?r\gh "s po9t ,o 9 ! axis l9es gamma'1 gamma p]p5dicul> respectively to ! projections p#a & p#b ( ! giv5 l9e4 ,^! w be ! traces ( a plane gamma :i* is p]p5dicul> to ! giv5 l9e4 ,we next draw ?r\gh ! giv5 po9t ,a a plane p>allel to ! plane gamma2 ?is w be ! plane requir$4 ,o!r metrical prop]ties dep5d on ! det]m9ation ( ! real size or %ape ( a figure4 ,9 g5]al ! projection ( a figure diff]s bo? 9 size & %ape f ! figure itself4 ,b figures 9 a plane p>allel to a plane ( projection w be id5tical ) _! projections1 & w ?us be giv5 9 _! true dim5sions4 ,9 o!r cases !re is ! problem1 con/antly recurr+1 ei!r to f9d ! true %ape & size ( a plane figure :5 plan & elevation are giv51 or1 conv]sely1 to f9d ! latt] f ! kn[n true %ape ( ! figure itself4 ,to d ?is1 ! plane is turn$ ab\t "o ( its traces till x is laid d[n into t plane ( projection to :i* ! trace belongs4 ,?is is te*nically call$ rabatt+ ! plane respectively into ! plane ( ! plan or ! elevation4 ,z !re is no diff]5ce 9 ! treatm5t ( ! two cases1 we %all consid] only ! case ( rabatt+ a plane alpha into ! plane ( ! plan4 ,! plan ( ! figure is a p>allel 7or?ographic7 projection ( ! figure itself4 ,! results ( p>allel projection 7see ,,projection1 ,,ss #ag & #ah7 may !re=e n[ be us$4 ,! trace alpha' w h]eby take ! place ( :at =m]ly 0 call$ ! axis ( projection4 ,h5ce we see t correspond+ po9ts 9 ! plan & 9 ! rabatt$ plane are jo9$ by l9es :i* are p]p5dicul> to ! trace alpha' & t correspond+ l9es meet on ?is trace4 ,we also see t ! correspond5ce is completely det]m9$ if we "k = "o po9t or "o l9e 9 ! plan ! correspond+ po9t or l9e 9 ! rabatt$ plane4 ,be=e1 h[ev]1 we treat ( ?is we consid] "s special cases4 ,s #ac4 ,to det]m9e ! di/ance betwe5 two po9ts ,a1 ,b giv5 by _! projections ,a#a1 ,b#a & ,a#b1 ,b#b1 or1 9 o!r words1 to det]m9e ! true l5g? ( a l9e ! plan & elevation ( :i* are giv54 ,illu/ration3 ,,fig4 #dc4 ,solution4--,! two po9ts ,a1 ,b 9 space lie v]tically above _! plans ,a#a1 ,b#a 7fig4 #dc7 & ,a#a;,a ,a#j;,a#b1 ,b#a;,b ,b#j;,b#b4 ,! f\r po9ts ,a1 ,b1 ,a#a1 ,b#a !re=e =m a plane quadrilat]al on ! base ,a#a;,b#a & hav+ "r angles at ! base4 ,?is plane we rabatt ab\t ,a#a;,b#a by draw+ ,a#a;,a & ,b#a;,b p]p5dicul> to ,a#a;,b#a & mak+ ,a#a;,a ,a#j;,a#b1 ,b#a;,b ,b#j;,b#b4 ,!n ,,ab w give ! l5g? requir$4 ,! con/ruction might h be5 p]=m$ 9 ! elevation by mak+ ,a#b;,a ,a#j;,a#a & ,b#b;,b ,b#j;,b#a on l9es p]p5dicul> to ,a#b;,b#b4 ,( c\rse ,,ab mu/ h ! same l5g? 9 bo? cases4 ,?is figure may be turn$ into a model4 ,cut ! pap] along ,a#a;,a1 ,,ab & ,,bb#a1 & fold ! piece ,a#a;,,abb#a ov] along ,a#a;,b#a till x /&s upright at "r angles to ! horizontal plane4 ,! po9ts ,a1 ,b w !n be 9 _! true position 9 space relative to pi#a4 ,simil>ly if ,b#b;,,baa#b be cut out & turn$ along ,a#b;,b#b ?r\gh a "r angle we %all get ,,ab 9 its true position relative to ! plane pi#b4 ,la/ly we fold ! :ole plane ( ! pap] along ! axis x till ! plane pi#b is at "r angles to pi#a4 ,9 ?is position ! two sets ( po9ts ,,ab w co9cide if ! draw+ has be5 accurate4 ,models ( ?is k9d c be made 9 _m cases & _! con/ruction _c be too highly recomm5d$ 9 ord] to realize or?ographic projection4 ,s #ad4 ,to f9d ! angle betwe5 two giv5 l9es a1 b ( :i* ! projections a#a1 b#a & a#b1 b#b are giv54 ,illu/ration3 ,,fig4 #dd4 ,solution4--,let a#a1 b#a 7fig4 #dd7 meet 9 ,p#a1 a#b1 b#b 9 ,t1 !n if ! l9e ,p#a,t is n p]p5dicul> to ! axis ! two l9es w n meet4 ,9 ?is case we draw a l9e p>allel to b to meet ! l9e a4 ,?is is easie/ done by draw+ fir/ ! l9e ,p#a,p#b p]p5dicul> to ! axis to meet a#b 9 ,p#b1 & !n draw+ ?r\gh ,p#b a l9e c#b p>allel to b#b2 !n b#a1 c#b w be ! projections ( a l9e c :i* is p>allel to b & meets a 9 ,p4 ,! plane alpha :i* ^! two l9es det]m9e we rabatt to ! plan4 ,we det]m9e ! traces a' & c' ( ! l9es a & c2 !n a'c' is ! trace alpha' ( _! plane4 ,on rabatt+ ! po9t ,p comes to a po9t ,s on ! l9e ,p#a,q p]p5dicul> to a'c'1 s t ,,qs ,,qp4 ,b ,,qp is ! hypot5use ( a triangle ,,pp#a,q ) a "r angle ,p#a4 ,?is we con/ruct by mak+ ,,qr ,p#j,p#b2 !n ,p#a,r ,,pq4 ,! l9es a',s & c',s w !re=e include angles equal to ^? made by ! giv5 l9es4 ,x is to be rememb]$ t two l9es include two angles :i* are supplem5t>y4 ,:i* ( ^! is to be tak5 9 any special case dep5ds ^u ! circum/ances4 ,to det]m9e ! angle betwe5 a l9e & a plane1 we draw ?r\gh any po9t 9 ! l9e a p]p5dicul> to ! plane 7,s #ab7 & det]m9e ! angle betwe5 x & ! giv5 l9e4 ,! complem5t ( ?is angle is ! requir$ "o4 ,to det]m9e ! angle betwe5 two planes1 we draw ?r\gh any po9t two l9es p]p5dicul> to ! two planes & det]m9e ! angle betwe5 ! latt] z above4 ,9 special cases x is simpl] to det]m9e at once ! angle betwe5 ! two planes by tak+ a plane section p]p5dicul> to ! int]section ( ! two planes & rabatt ?is4 ,?is is especially ! case if "o ( ! planes is ! horizontal or v]tical plane ( projection4 ,?us 9 fig4 #de ! angle ,p#a,,qr is ! angle :i* ! plane alpha makes ) ! horizontal plane4 ,s #ae4 ,we return to ! g5]al case ( rabatt+ a plane alpha ( :i* ! traces alpha' alpha are giv54 ,illu/ration3 ,,fig4 #de4 ,"h x w be conv5i5t to det]m9e fir/ ! position :i* ! trace alpha--:i* is a l9e 9 alpha--assumes :5 rabatt$4 ,po9ts 9 ?is l9e co9cide ) _! elevations4 ,h5ce x is giv5 9 its true dim5sion1 & we c measure (f along x ! true di/ance betwe5 two po9ts 9 x4 ,if !re=e 7fig4 #de7 ,p is any po9t 9 alpha orig9ally co9cid5t ) its elevation ,p#b1 & if ,o is ! po9t :]e alpha cuts ! axis xy1 s t ,o is also 9 alpha'1 !n ! po9t ,p w aft] rabatt+ ! plane assume su* a position t ,,op ,,op#b4 ,at ! same "t ! plan is an or?ographic projection ( ! plane alpha4 ,h5ce ! l9e jo9+ ,p to ! plan ,p#a w aft] rabatt+ be p]p5dicul> to alpha'4 ,b ,p#a is kn[n2 x is ! foot ( ! p]p5dicul> f ,p#b to ! axis xy4 ,we draw !re=e1 to f9d ,p1 f ,p#a a p]p5dicul> ,p#a,q to alpha' & f9d on x a po9t ,p su* t ,,op ,,op#b4 ,!n ! l9e ,,op w be ! position ( alpha :5 rabatt$4 ,?is l9e corresponds !re=e to ! plan ( alpha--t is1 to ! axis xy1 correspond+ po9ts on ^! l9es be+ ^? :i* lie on a p]p5dicul> to alpha'4 ,we h ?us "o pair ( correspond+ l9es & c n[ f9d = any po9t ,b#a 9 ! plan ! correspond+ po9t ,b 9 ! rabatt$ plane4 ,we draw a l9e ?r\gh ,b#a1 say ,b#a,p#a1 cutt+ alpha' 9 ,c4 ,to x corresponds ! l9e ,,cp1 & ! po9t :]e ?is is cut by ! project+ ray ?r\gh ,b#a1 p]p5dicul> to alpha'1 is ! requir$ po9t ,b4 ,simil>ly any figure 9 ! rabatt$ plane c be f\nd :5 ! plan is kn[n2 b ?is is usually f\nd 9 a diff]5t mann] )\t any ref]5ce to ! g5]al !ory ( p>allel projection4 ,z ?is me?od & ! reason+ employ$ = x h _! peculi> advantages1 we give x also4 ,suppos+ ! planes pi#a & pi#b to be 9 _! positions 9 space p]p5dicul> to ea* o!r1 we take a section ( ! :ole figure by a plane p]p5dicul> to ! trace alpha' ab\t :i* we are go+ to rabatt ! plane alpha4 ,let ?is section pass ?r\gh ! po9t ,q 9 alpha'4 ,its traces w !n be ! l9es ,,qp#a & ,p#a,p#b 7fig4 #i74 ,^! w be at "r angles1 & w !re=e1 toge!r ) ! section ,,qp#b ( ! plane alpha1 =m a "r-angl$ triangle ,,qp#a,p#b ) ! "r angle at ,p#a1 & hav+ ! sides ,p#a,q & ,p#a,p#b :i* bo? are giv5 9 _! true l5g?s4 ,?is triangle we rabatt ab\t its base ,p#a,q1 mak+ ,p#a,r ,p#a,p#b4 ,! l9e ,,qr w !n give ! true l5g? ( ! l9e ,,qp 9 space4 ,if n[ ! plane alpha be turn$ ab\t alpha' ! po9t ,p w describe a circle ab\t ,q z c5tre ) radius ,,qp ,,qr1 9 a plane p]p5dicul> to ! trace alpha'4 ,h5ce :5 ! plane alpha has be5 rabatt$ into ! horizontal plane ! po9t ,p w lie 9 ! p]p5dicul> ,p#a,q to alpha'1 s t ,,qp ,,qr4 ,if ,a#a is ! plan ( a po9t ,a 9 ! plane alpha1 & if ,a#a lies 9 ,,qp#a1 !n ! po9t ,a w lie v]tically above ,a#a 9 ! l9e ,,qp4 ,on turn+ d[n ! triangle ,,qp#a,p#b1 ! po9t ,a w come to ,a#j1 ! l9e ,a#a;,a#j be+ p]p5dicul> to ,,qp#a4 ,h5ce ,a w be a po9t 9 ,,qp su* t ,,qa ,,qa#j4 ,if ,b#a is ! plan ( ano!r po9t1 b su* t ,a#a;,b#a is p>allel to alpha'1 !n ! correspond+ l9e ,,ab w also be p>allel to alpha'4 ,h5ce1 if ?r\gh ,a a l9e ,,ab be drawn p>allel to alpha'1 & ,b#a;,b p]p5dicul> to alpha'1 !n _! int]section gives ! po9t ,b4 ,?us ( any po9t giv5 9 plan ! real position 9 ! plane alpha1 :5 rabatt$1 c be f\nd by ?is second me?od4 ,?is is ! "o mo/ g5]ally giv5 9 books on geometrical draw+4 ,! fir/ me?od expla9$ is1 h[ev]1 9 mo/ cases pref]able z x gives ! draughtsman a great] v>iety ( con/ructions4 ,x requires a some:at great] am\nt ( !oretical k4 ,if in/ead ( our kn[+ ! plan ( a figure ! latt] is itself giv51 !n ! process ( f9d+ ! plan is ! rev]se ( ! above & ne$s little explanation4 ,we give an example4 ,s #af4 ,x is requir$ to draw ! plan & elevation ( a polygon ( :i* ! real %ape & position 9 a giv5 plane alpha are kn[n4 ,we fir/ rabatt ! plane alpha 7fig4 #df7 z be=e s t ,p#a comes to ,p1 h5ce ,,op#a to ,,op4 ,let ! giv5 polygon 9 alpha be ! figure ,,abcde4 ,we project1 n ! v]tices1 b ! sides4 ,to project ! l9e ,,ab1 we produce x to cut alpha' 9 ,f & ,,op 9 ,g1 & draw ,,gg#a p]p5dicul> to alpha'2 !n ,g#a corresponds to ,g1 !re=e ,,fg#a to ,,fg4 ,9 ! same mann] we might project all ! o!r sides1 at lea/ ^? :i* cut ,,( & ,,op 9 conv5i5t po9ts4 ,x w be be/1 h[ev]1 fir/ to produce all ! sides to cut ,,op & alpha' & !n to draw all ! project+ rays ?r\gh ,a1 ,b1 ,c 444 p]p5dicul> to alpha'1 & 9 ! same direction ! l9es ,g1 ,g#a1 c4 ,by draw+ ,,fg we get ! po9ts ,a#a1 ,b#a on ! project+ ray ?r\gh ,a & ,b4 ,we !n jo9 ,b to ! po9t ,m :]e ,,bc produc$ meets ! trace alpha'4 ,?is gives ,c#a4 ,s we g on till we h f\nd ,e#a4 ,! l9e ,a#a ,e#a mu/ !n meet ,,ae 9 alpha'1 & ?is gives a *eck4 ,if "o ( ! sides cuts alpha' or ,,op beyond ! draw+ pap] ?is me?od fails1 b !n we may easily f9d ! projection ( "s o!r l9e1 say ( a diagonal1 or directly ! projection ( a po9t1 by ! =m] me?ods4 ,! diagonals may also s]ve to *eck ! draw+1 = two correspond+ diagonals mu/ meet 9 ! trace alpha'4 ,illu/ration3 ,,fig4 #df4 ,hav+ got ! plan we easily f9d ! elevation4 ,! elevation ( ,g is above ,g#a 9 alpha1 & t ( ,f is at ,f#b 9 ! axis4 ,?is gives ! elevation ,f#b;,g#b ( ,,fg & 9 x we get ,a#b;,b#b 9 ! v]ticals ?r\gh ,a#a & ,b#a4 ,z a *eck we h ,,og ,,og#b4 ,simil>ly ! elevation ( ! o!r sides & v]tices are f\nd4 ,s #ag4 ,we proce$ to give "s applications ( ! above pr9ciples to ! repres5tation ( solids & ( ! solution ( problems connect$ ) !m4 ,( a pyramid are giv5 its base1 ! l5g? ( ! p]p5dicul> f ! v]tex to ! base1 & ! po9t :]e ?is p]p5dicul> cuts ! base2 x is requir$ fir/ to develop ! :ole surface ( ! pyramid into "o plane1 & second to det]m9e its section by a plane :i* cuts ! plane ( ! base 9 a giv5 l9e & makes a giv5 angle ) x4 #a4 ,z ! planes ( projection are n giv5 we c take !m z we l1 & we select !m 9 su* a mann] t ! solution becomes z simple z possible4 ,we take ! plane ( ! base z ! horizontal plane & ! v]tical plane p]p5dicul> to ! plane ( ! section4 ,let !n 7fig4 #dg7 ,,abcd be ! base ( ! pyramid1 ,v#a ! plan ( ! v]tex1 !n ! elevations ( ,a1 ,b1 ,c1 ,d w be 9 ! axis at ,a#b1 ,b#b1 ,c#b1 ,d#b1 & ! v]tex at "s po9t ,v#b above ,v#a at a kn[n di/ance f ! axis4 ,! l9es ,v#a;,a1 ,v#a;,b1 c41 w be ! plans & ! l9es ,v#b;,a#b1 ,v#b;,b#b1 c41 ! elevations ( ! edges ( ! pyramid1 ( :i* ?us plan & elevation are kn[n4 ,we develop ! surface into ! plane ( ! base by turn+ ea* lat]al face ab\t its l[] edge into ! horizontal plane by ! me?od us$ 9 ,s #ad4 ,if "o face has be5 turn$ d[n1 say ,,abv to ,,abp1 !n ! po9t ,q to :i* ! v]tex ( ! next face ,,bcv comes c be got m simply by f9d+ on ! l9e ,v#a,q p]p5dicul> to ,,bc ! po9t ,q su* t ,,bq ,,bp1 = ^! l9es repres5t ! same edge ,,bv ( ! pyramid4 ,next ,r is f\nd by mak+ ,,cr ,,cq1 & s on till we h got ! la/ v]tex--9 ?is case ,s4 ,! fact t ,,z mu/ equal ,,ap gives a conv5i5t *eck4 #b4 ,! plane alpha ^: section we h to det]m9e has its horizontal trace giv5 p]p5dicul> to ! axis1 & its v]tical trace makes ! giv5 angle ) ! axis4 ,?is det]m9es x4 ,to f9d ! section ( ! pyramid by ?is plane !re are two me?ods applicable3 we f9d ! sections ( ! plane ei!r ) ! faces or ) ! edges ( ! pyramid4 ,we use ! latt]4 ,z ! plane alpha is p]p5dicul> to ! v]tical plane1 ! trace alpha conta9s ! projection ( e figure 9 x2 ! po9ts ,e#b1 ,f#b1 ,g#b1 ,h#b :]e ?is trace cuts ! elevations ( ! edges w !re=e be ! elevations ( ! po9ts :]e ! edges cut alpha4 ,f ^! we f9d ! plans ,e#a1 ,f#a1 ,g#a1 ,h#a1 & by jo9+ !m ! plan ( ! section4 ,if f ,e#a1 ,f#a l9es be drawn p]p5dicul> to ,,ab1 ^! w det]m9e ! po9ts ,e1 ,f on ! develop$ face 9 :i* ! plane alpha cuts x2 h5ce also ! l9e ,,ef4 ,simil>ly on ! o!r faces4 ,( c\rse ,,bf mu/ be ! same l5g? on ,,bp & on ,,bq4 ,if ! plane alpha be rabatt$ to ! plan1 we get ! real %ape ( ! section z %[n 9 ! figure 9 ,,efgh4 ,?is is done easily by mak+ ,f#j;,f ,,(#b1 c4 ,if ! figure repres5t+ ! developm5t ( ! pyramid1 or bett] a copy ( x1 is cut out1 & if ! lat]al faces be b5t along ! l9es ,,ab1 ,,bc1 c41 we get a model ( ! pyramid ) ! section m>k$ on its faces4 ,?is may be plac$ on its plan ,,abcd & ! plane ( elevation b5t ab\t ! axis x4 ,! pyramid /&s !n 9 front ( its elevations4 ,if next ! plane alpha ) a hole cut out repres5t+ ! true section be b5t along ! trace alpha' till its edge co9cides ) alpha1 ! edges ( ! hole ought to co9cide ) ! l9es ,,ef1 ,,fg1 c41 on ! faces4 ,s #ah4 ,polyh$ra l ! pyramid 9 ,s #ag are repres5t$ by ! projections ( _! edges & v]tices4 ,b solids b\nd$ by curv$ surfaces1 or surfaces !mselves1 _c be ?us repres5t$4 ,= a surface we may use1 z 9 case ( ! plane1 its traces--t is1 ! curves 9 :i* x cuts ! planes ( projection4 ,we may also project po9ts & curves on ! surface4 ,a ray cuts ! surface g5]ally 9 m ?an "o po9t2 h5ce x w happ5 t "s ( ! rays t\* ! surface1 if two ( ^! po9ts co9cide4 ,! po9ts ( contact ( ^! rays w =m "s curve on ! surface1 & ?is w appe> f ! c5tre ( projection z ! b\nd>y ( ! surface or ( "p ( ! surface4 ,! outl9es ( all surfaces ( solids :i* we see ab\t u are =m$ by ! po9ts at :i* rays ?r\gh our eye t\* ! surface4 ,! projections ( ^! cont\rs are !re=e be/ adapt$ to give an idea ( ! %ape ( a surface4 ,illu/ration3 ,,fig4 #dg4 ,?us ! tang5ts drawn f any f9ite c5tre to a sph]e =m a "r circul> cone1 & ?is w be cut by any plane 9 a conic4 ,x is (t5 call$ ! projection ( a sph]e1 b x is bett] call$ ! cont\r-l9e ( ! sph]e1 z x is ! b\nd>y ( ! projections ( all po9ts on ! sph]e4 ,if ! c5tre is at inf9ity ! tang5t cone becomes a "r circul> cyl9d] t\*+ ! sph]e along a great circle1 & if ! projection is1 z 9 our case1 or?ographic1 !n ! section ( ?is cone by a plane ( projection w be a circle equal to ! great circle ( ! sph]e4 ,we get su* a circle 9 ! plan & ano!r 9 ! elevation1 _! c5tres be+ plan & elevation ( ! c5tre ( ! sph]e4 ,simil>ly ! rays t\*+ a cone ( ! second ord] w lie 9 two planes :i* pass ?r\gh ! v]tex ( ! cone1 ! cont\r-l9e ( ! projection ( ! cone consi/s !re=e ( two l9es meet+ 9 ! projection ( ! v]tex4 ,^! may1 h[ev]1 be invisible if no real tang5t rays c be drawn f ! c5tre ( projection2 & ?is happ5s :5 ! ray project+ ! c5tre ( ! v]tex lies )9 ! cone4 ,9 ?is case ! traces ( ! cone are ( importance4 ,?us 9 repres5t+ a cone ( revolution ) a v]tical axis we get 9 ! plan a circul> trace ( ! surface ^: c5tre is ! plan ( ! v]tex ( ! cone1 & 9 ! elevation ! cont\r1 consi/+ ( a pair ( l9es int]sect+ 9 ! elevation ( ! v]tex ( ! cone4 ,! circle 9 ! plan & ! pair ( l9es 9 ! elevation d n det]m9e ! surface1 = an inf9ite numb] ( surfaces might be conceiv$ :i* pass ?r\gh ! circul> trace & t\* two planes ?r\gh ! cont\r l9es 9 ! v]tical plane4 ,! surface becomes only completely def9$ if we write d[n to ! figure t x %all repres5t a cone4 ,! same holds = all surfaces4 ,ev5 a plane is fully repres5t$ by its traces only "u ! sil5t und]/&+ t ! traces are ^? ( a plane4 ,s #ai4 ,"s ( ! simpl] problems connect$ ) ! repres5tation ( surfaces are ! det]m9ation ( plane sections & ( ! curves ( int]section ( two su* surfaces4 ,! =m] is con/antly us$ 9 ne>ly all problems conc]n+ surfaces4 ,its solution dep5ds ( c\rse on ! nature ( ! surface4 ,to det]m9e ! curve ( int]section ( two surfaces1 we take a plane & det]m9e its section ) ea* ( ! two surfaces1 rabatt+ ?is plane if necess>y4 ,?is gives two curves :i* lie 9 ! same plane & ^: int]sections w give u po9ts on bo? surfaces4 ,x mu/ "h be rememb]$ t two curves 9 space d n necess>ily int]sect1 h5ce t ! po9ts 9 :i* _! projections int]sect are n necess>ily ! projections ( po9ts common to ! two curves4 ,?is w1 h[ev]1 be ! case if ! two curves lie 9 a common plane4 ,by tak+ !n a numb] ( plane sections ( ! surfaces we c get z _m po9ts on _! curve ( int]section z we l4 ,^! planes h1 ( c\rse1 to be select$ 9 su* a way t ! sections are curves z simple z ! case p]mits (1 & su* t !y c be easily & accurately drawn4 ,?us :5 possible ! sections %\ld be /raight l9es or circles4 ,?is n only saves "t 9 draw+ b det]m9es all po9ts on ! sections1 & !re=e also ! po9ts :]e ! two curves meet1 ) equal accuracy4 ,s #bj4 ,we give a few examples h[ ^! sections h to be select$4 ,a cone is cut by e plane ?r\gh ! v]tex 9 l9es1 & if x is a cone ( revolution by planes p]p5dicul> to ! axis 9 circles4 ,a cyl9d] is cut by e plane p>allel to ! axis 9 l9es1 & if x is a cyl9d] ( revolution by planes p]p5dicul> to ! axis 9 circles4 ,a sph]e is cut by e plane 9 a circle4 ,h5ce 9 case ( two cones situat$ any:]e 9 space we take sections ?r\gh bo? v]tices4 ,^! w cut bo? cones 9 l9es4 ,simil>ly 9 case ( two cyl9d]s we may take sections p>allel to ! axis ( bo?4 ,9 case ( a sph]e & a cone ( revolution ) v]tical axis1 horizontal sections w cut bo? surfaces 9 circles ^: plans are circles & ^: elevations are l9es1 :il/ v]tical sections ?r\gh ! v]tex ( ! cone cut ! latt] 9 l9es & ! sph]e 9 circles4 ,to avoid draw+ ! projections ( ^! circles1 :i* w\ld 9 g5]al be ellipses1 we rabatt ! plane & !n draw ! circles 9 _! real %ape4 ,& s on 9 o!r cases4 ,special att5tion %\ld 9 all cases be paid to ^? po9ts 9 :i* ! tang5ts to ! projection ( ! curve ( int]section are p>allel or p]p5dicul> to ! axis x1 or :]e ^! projections t\* ! cont\r ( "o ( ! surfaces4 7,o4 ,h47 ,,iv4 ,,analytical ,,geometry #a4 ,9 ! "n geometry !re is a la/+ record t ! sci5ce _h its orig9 9 ! k t two di/ances may be comp>$ by measurem5t1 & 9 ! idea t measurem5t mu/ be effectual 9 ! dissociation ( diff]5t directions z well z 9 ! comp>ison ( di/ances 9 ! same direction4 ,! di/ance f an obs]v]'s eye ( an object se5 w\ld be specifi$ z soon z x 0 asc]ta9$ t a rod1 /raight to ! eye & ( l5g? tak5 z kn[n1 c\ld be giv5 ! direction ( ! l9e ( vision1 & _h to be mov$ along x a c]ta9 numb] ( times ?r\gh l5g?s equal to its own 9 ord] to rea* ! object f ! eye4 ,moreov]1 if a field _h = two ( its b\nd>ies l9es /raight to ! eye1 "o runn+ f s\? to nor? & ! o!r f we/ to ea/1 ! position ( a po9t 9 ! field w\ld be specifi$ if ! rod1 :5 direct$ we/1 _h to be %ift$ f ! po9t "o obs]v$ numb] ( times we/w>d to meet ! =m] b\nd>y1 & also1 :5 direct$ s\?1 _h to be %ift$ ano!r obs]v$ numb] ( times s\?w>d to meet ! latt]4 ,comp>ison by measurem5t1 ! beg9n+ ( geometry1 involv$ c\nt+1 ! basis ( ari?metic2 & ! sci5ce ( numb] 0 m>k$ out f ! fir/ z ( geometrical importance4 ,b ! ari?metic ( ! anci5ts 0 inadequate z a sci5ce ( numb]4 ,?\gh a l5g? might be recogniz$ z kn[n :5 measurem5t c]tifi$ t x 0 s _m times a /&>d l5g?1 x 0 n e l5g? :i* c\ld be ?us specifi$ 9 t]ms ( ! same /&>d l5g?1 ev5 by an ari?metic enri*$ ) ! notion ( fractional numb]4 ,! idea ( possible incomm5surability ( l5g?s 0 introduc$ into ,europe by ,py?agoras2 & ! correspond+ idea ( irrationality ( numb] 0 abs5t f a crude ari?metic1 :ile !re 7 great practical difficulties 9 ! way ( its introduction4 ,h5ce p]haps x arose t1 till comp>atively mod]n times1 appeal to ari?metical aid 9 geometrical reason+ 0 9 all possible ways re/ra9$4 ,geometry figur$ r z ! help] ( ! m difficult sci5ce ( ari?metic4 #b4 ,x 0 res]v$ = algebra to remove ! disabilities ( ari?metic1 & to re/ore ! e>lie/ ideas ( ! l&-measur] to ! position ( controll+ ideas 9 geometrical inve/igation4 ,?is unifi$ sci5ce ( pure numb] made comp>atively little headway 9 ! h&s ( ! anci5ts1 b began to receive due att5tion %ortly aft] ! revival ( le>n+4 ,x expresses :ole classes ( ari?metical facts 9 s+le /atem5ts1 gives to ari?metical laws ! =m ( equations involv+ symbols :i* may mean any kn[n or s\ght numb]s1 & provides processes :i* enable u to analyse ! in=mation giv5 by an equation & d]ive f t equation o!r equations1 :i* express laws t are 9 effect consequ5ces or causes ( a law />t$ f1 b diff] greatly f x 9 =m4 ,above all1 = pres5t purposes1 x deals n only ) integral & fractional numb]1 b ) numb] reg>d$ z capable ( cont9u\s gr[?1 j z di/ance is capable ( cont9u\s gr[?4 ,! difficulty ( ! ari?metical expression ( irrational numb]1 a difficulty consid]$ by ! mod]n s*ool ( analy/s to h be5 at l5g? surm\nt$ 7see ,,function71 is n vital to x4 ,x c call ! ratio ( ! diagonal ( a squ>e to a side1 = in/ance1 or t ( ! circumf]5ce ( a circle to a diamet]1 a numb]1 & let a or x d5ote t numb]1 j z prop]ly z x may all[ ei!r lett] to d5ote any rational numb] :i* may be great] or less ?an ! ratio 9 "q by a diff]5ce less ?an any m9ute "o we *oose to assign4 ,c\nt+ only1 & n ! c\nt+ ( objects1 is ( ! ess5ce ( ari?metic1 & ( algebra4 ,b x is lawful to c\nt objects1 & 9 p>ticul> to c\nt equal l5g?s by measure4 ,! wid5$ idea is t ev5 :5 a or x is an irrational numb] we may speak ( a or x unit l5g?s by measure4 ,we may give concrete int]pretation to an algebraical equation by all[+ its t]ms all to mean numb]s ( times ! same unit l5g?1 or ! same unit area1 or c4 & 9 any equation lawfully d]iv$ f ! fir/ by algebraical processes we may d ! same4 ,desc>tes 9 8 ,geometrie 7#afcg7 0 ! fir/ to sy/ematize ! application ( ?is pr9ciple to ! inh]5t fir/ notions ( geometry2 & ! me?ods :i* he in/itut$ h become ! mo/ pot5t me?ods ( all 9 geometrical rese>*4 ,x is h>dly too mu* to say t1 :5 kn[n facts z to a geometrical figure h once be5 express$ 9 algebraical t]ms1 all /rictly consequ5tial facts z to ! figure c be d$uc$ by almo/ me*anical processes4 ,"s may well be unexpect$ consequ5ces2 & 9 obta9+ ^? ( :i* !re has be5 sugge/ion be=eh& ! (t5 bewild]+ lab\r ( con/ant att5tion to ! figure is obviat$4 ,^! are ! me?ods ( :at is n[ call$ analytical1 or sometimes algebraical1 geometry4 #c4 ,! mod]n use ( ! t]m analytical 9 geometry has obscur$1 b n made obsolete1 an e>li] use1 "o z old z ,plato4 ,!re is no?+ algebraical 9 ?is analysis1 z di/+ui%$ f syn!sis1 ( ! ,greeks1 & ( ! expositors ( pure geometry4 ,x has ref]5ce to an ord] ( ideas 9 demon/ration1 or1 m frequ5tly1 9 discov]+ means to effect ! geometrical con/ruction ( a figure ) an assign$ special prop]ty4 ,we h to suppose hypo!tically t ! con/ruction has be5 p]=m$1 draw+ a r\gh figure :i* exhibits x z ne>ly z is practicable4 ,we !n analyse or critically exam9e ! figure1 treat$ z correct1 & asc]ta9 o!r prop]ties :i* x c only possess 9 association ) ! "o 9 "q4 ,pres5tly "o ( ^! prop]ties w (t5 be f\nd :i* is ( su* a "* t ! con/ruction ( a figure possess+ x is simple4 ,! means ( effect+ syn!tically a con/ruction su* z 0 desir$ is ?us br\ght to light by :at ,plato call$ analysis4 ,or aga91 be+ ask$ to prove a !orem ,a1 we asc]ta9 t x mu/ be true if ano!r !orem ,b is1 t ,b mu/ be if ,c is1 & s on1 ?us ev5tually f9d+ t ! !orem ,a is ! consequ5ce1 ?r\gh a *a9 ( int]m$i>ies1 ( a !orem ,z ( :i* ! e/abli%m5t is easy4 ,?is geometrical analysis is n ! subject ( ! pres5t article2 b 9 ! reason+ f =m to =m ( an equation or sy/em ( equations1 ) ! object ( bas+ ! algebraical pro( ( a geometrical fact on o!r facts ( a m obvi\s "*1 ! same logic is utiliz$1 & ! "n analytical geometry is ?us 9 "p expla9$4 #d4 ,9 algebra real positive numb] 0 alone at fir/ dealt )1 & 9 geometry actual signless di/ance4 ,b 9 algebra x became ( importance to say t e equation ( ! fir/ degree has a root1 & ! notion ( negative numb] 0 introduc$4 ,! negative unit _h to be def9$ z :at c be add$ to ! positive unit & produce ! sum z]o4 ,! correspond+ notion 0 readily at h& 9 geometry1 :]e x 0 cle> t a unit di/ance c be measur$ to ! left or d[n f ! f>!r end ( a unit di/ance already measur$ to ! "r or up f a po9t ,o1 ) ! result ( rea*+ ,o aga94 ,?us1 to give full int]pretation 9 geometry to ! algebraically negative1 x 0 only necess>y to associate di/9ctness ( sign ) opposit5ess ( direction4 ,lat] x 0 discov]$ t algebraical reason+ w\ld be mu* facilitat$1 & t conclusions z to ! real w\ld reta9 all _! s\ndness1 if a pair ( imag9>y units -root-#a ( :at might be call$ numb] 7 all[$ to be contemplat$1 ! pair be+ def9$1 ?\gh n sep>ately1 by ! two prop]ties ( hav+ ! real sum #j & ! real product #a4 ,only 9 ^! two real comb9ations d !y ent] 9 conclusions z to ! real4 ,an advantage ga9$ 0 t e quadratic equation1 & n "s quadratics only1 c\ld be spok5 ( z hav+ two roots4 ,^! admissions ( new units into algebra 7 f9al1 z x admitt$ ( pro( t all equations ( degrees high] ?an two h ! full numb]s ( roots possible = _! respective degrees 9 any case1 & t e root has a value includ$ 9 ! =m a b root-#a1 ) a1 b1 real4 ,! correspond+ enri*m5t c\ld be giv5 to geometry1 ) correspond+ advantages & ! same abs5ce ( dang]1 & ?is 0 done4 ,on a l9e ( measurem5t ( di/ance we contemplate z exi/+1 n only an inf9ite cont9uum ( po9ts at real di/ances f an orig9 ( measurem5t ,o1 b a d\bly inf9ite cont9uum ( po9ts1 all b ! s+ly inf9ite cont9uum ( real ones imag9>y1 & imag9>y 9 conjugate pairs1 a conjugate pair be+ at imag9>y di/ances f ,o1 :i* h a real ari?metic & a real geometric mean4 ,to geometry enri*$ ) ?is conception all algebra has its application4 #e4 ,actual geometry is "o1 two or ?ree-dim5sional1 i4e4 l9eal1 plane or solid4 ,9 "o-dim5sional geometry positions & measurem5ts 9 a s+le l9e only are admitt$4 ,n[ descriptive con/ructions = po9ts 9 a l9e are impossible )\t go+ out ( ! l9e4 ,x has !re=e be5 held t !re is a s5se 9 :i* no sci5ce ( geometry /rictly conf9$ to "o dim5sion exi/s4 ,b an algebra ( "o v>iable c be appli$ to ! /udy ( di/ances along a l9e measur$ f a *os5 po9t on x1 s t ! idea ( con/ruction z di/9ct f measurem5t is n ess5tial to a "o-dim5sional geometry aid$ by algebra4 ,9 geometry ( two dim5sions1 ! flat ( ! l&-measur]1 ! passage f "o po9t ,o to any o!r po9t1 c be effect$ by two successive m>*es1 "o ea/ or we/ & "o nor? or s\?1 &1 z w be se51 an algebra ( two v>iables suffices = geometrical exploitation4 ,9 geometry ( ?ree dim5sions1 t ( space1 any po9t c be rea*$ f a *os5 "o by ?ree m>*es1 "o ea/ or we/1 "o nor? or s\?1 & "o up or d[n2 & we %all see t an algebra ( ?ree v>iables is all t is necess>y4 ,) ?ree dim5sions actual geometry /ops2 b algebra c supply any numb] ( v>iables4 ,f\r or m v>iables h be5 us$ 9 ways analog\s to ^? 9 :i* "o1 two & ?ree v>iables are us$ = ! purposes ( "o1 two & ?ree-dim5sional geometry1 & ! results h be5 express$ 9 quasi-geometrical language on ! supposition t a high] space c be conceiv$ (1 ?\gh n realiz$1 9 :i* f\r indep5d5t directions exi/1 su* t no succession ( m>*es along ?ree ( !m c effect ! same displacem5t ( a po9t z a m>* along ! f\r?2 & simil>ly = high] numb]s ?an f\r4 ,?us analytical1 ?\gh n actual1 geometries exi/ = f\r & m dim5sions4 ,!y are 9 fact algebras furni%$ ) nom5clature ( a geometrical ca/1 sugge/$ by conv5i5t =ms ( expression :i* actual geometry has1 9 return = b5efits receiv$1 conf]r$ on algebras ( "o1 two & ?ree v>iables4 ,we w conf9e ourselves to ! dim5sions ( actual geometry1 & w devote no space to ! "o-dim5sional1 except incid5tally z exi/+ )9 ! two-dim5sional4 ,! analytical me?od w n[ be expla9$ = ! cases ( two & ?ree dim5sions 9 succession4 ,! =m ( x orig9at$ by ,desc>tes1 & !nce kn[n z ,c>tesian1 w alone be consid]$ 9 mu* detail4 ,i4 ,plane ,analytical ,geometry4 ,illu/ration3 ,,fig4 #dh4 ,illu/ration3 ,,fig4 #di4 #f4 ,coord9ates4--,x is assum$ t ! po9ts1 l9es & figures consid]$ lie 9 "o & ! same plane1 :i* plane !re=e ne$ n be 9 any way ref]r$ to4 ,9 ! plane a po9t ,o1 & two l9es x',ox1 y',oy1 int]sect+ 9 ,o1 are tak5 once = all1 & reg>d$ z fix$4 ,o is call$ ! orig91 & x',ox1 y',oy ! axes ( x & y respectively4 ,o!r positions 9 ! plane are specifi$ 9 relation to ?is fix$ orig9 & ^! fix$ axes4 ,f any po9t ,p we suppose ,,pm drawn p>allel to ! axis ( y to meet ! axis ( x 9 ,m1 & may also suppose ,,pn drawn p>allel to ! axis ( x to meet ! axis ( y 9 ,n1 s t ,,ompn is a p>allelogram4 ,! position ( ,p is det]m9$ :5 we "k ,,om 7 ,,np7 & ,,mp 7 ,,on74 ,if ,,om is x times ! unit ( a scale ( measurem5t *os5 at pleasure1 & ,,mp is y times ! unit1 s t x & y h num]ical values1 we call x & y ! 7,c>tesian7 coord9ates ( ,p4 ,to di/+ui% !m we (t5 speak ( y z ! ord9ate1 & ( x z ! abscissa4 ,x is necess>y to att5d to signs2 x has "o sign or ! o!r accord+ z ! po9t ,p is on "o side or ! o!r ( ! axis ( y1 & y "o sign or ! o!r accord+ z ,p is on "o side or ! o!r ( ! axis ( x4 ,us+ ! lett]s ,n1 ,e1 ,s1 ,w1 z 9 a map1 & consid]+ ! plane z divid$ into f\r quadrants by ! axes1 ! signs are usually tak5 to be3 x y ,= quadrant ,n ,e - ,s ,e - ,n ,w - - ,s ,w ,a po9t is ref]r$ to z ! po9t 7a1 b71 :5 its coord9ates are x a1 y b4 ,a po9t may be fix$1 or x may be v>iable1 i4e4 be reg>d$ = ! "t be+ z free to move 9 ! plane4 ,! coord9ates 7x1 y7 ( a v>iable po9t are algebraic v>iables1 & are said to be curr5t coord9ates4 ,! axes ( x & y are usually 7z 9 fig4 #dh7 tak5 at "r angles to "o ano!r1 & we !n speak ( !m z rectangul> axes1 & ( x & y z rectangul> coord9ates ( a po9t ,p2 ,,ompn is !n a rectangle4 ,sometimes1 h[ev]1 x is conv5i5t to use axes :i* are oblique to "o ano!r1 s t 7z 9 fig4 #di7 ! angle x,oy betwe5 _! positive directions is "s kn[n angle omega di/9ct f a "r angle1 & ,,ompn is always an oblique p>allelogram ) giv5 angles2 & we !n speak ( x & y z oblique coord9ates4 ,! coord9ates are z a rule tak5 to be rectangul> 9 :at foll[s4 #g4 ,equations & loci4 ,if 7x1 y7 is ! po9t ,p1 & if we are giv5 t x #j1 we are told t1 9 fig4 #dh or fig4 #di1 ! po9t ,m lies at ,o1 :atev] value y may h1 i4e4 we are told ! "o fact t ,p lies on ! axis ( y4 ,conv]sely1 if ,p lies any:]e on ! axis ( y1 we h always ,,om #j1 i4e4 x #j4 ,?us ! equation x #j is "o satisfi$ by ! coord9ates 7x1 y7 ( e po9t 9 ! axis ( y1 & n by ^? ( any o!r po9t4 ,we say t x #j is ! equation ( ! axis ( y1 & t ! axis ( y is ! locus repres5t$ by ! equation x #j4 ,simil>ly y #j is ! equation ( ! axis ( x4 ,an equation x a1 :]e a is a con/ant1 expresses t ,p lies on a p>allel to ! axis ( y ?r\gh a po9t ,m on ! axis ( x su* t ,,om a4 ,e l9e p>allel to ! axis ( y has an equation ( ?is =m4 ,simil>ly1 e l9e p>allel to ! axis ( x has an equation ( ! =m y b1 :]e b is "s def9ite con/ant4 ,^! are simple cases ( ! fact t a s+le equation 9 ! curr5t coord9ates ( a v>iable po9t 7x1 y7 imposes "o limitation on ! fre$om ( t po9t to v>y4 ,! coord9ates ( a po9t tak5 at r&om 9 ! plane w1 z a rule1 n satisfy ! equation1 b inf9itely _m po9ts1 & 9 mo/ cases inf9itely _m real ones1 h coord9ates :i* d satisfy x1 & ^! po9ts are exactly ^? :i* lie ^u "s locus ( "o dim5sion1 a /raight l9e or m frequ5tly a curve1 :i* is said to be repres5t$ by ! equation4 ,take1 = in/ance1 ! equation y mx1 :]e m is a giv5 con/ant4 ,x is satisfi$ by ! coord9ates ( e po9t ,p1 :i* is su* t1 9 fig4 #dh1 ! di/ance ,,mp1 ) its prop] sign1 is m times ! di/ance ,,om1 ) its prop] sign1 i4e4 by ! coord9ates ( e po9t 9 ! /raight l9e ?r\gh ,o :i* we arrive at by mak+ a l9e1 orig9ally co9cid5t ) x',ox1 revolve ab\t ,o 9 ! direction opposite to t ( ! h&s ( a wat* ?r\gh an angle ( :i* m is ! tang5t1 & by ^? ( no o!r po9ts4 ,t l9e is ! locus :i* x repres5ts4 ,take1 m g5]ally1 ! equation y phi7x71 :]e phi7x7 is any giv5 non-ambigu\s function ( x4 ,*oos+ any po9t ,m on x',ox 9 fig4 #a1 & giv+ to x ! value ( ! num]ical measure ( ,,om1 ! equation det]m9es a s+le correspond+ y1 & s det]m9es a s+le po9t ,p on ! l9e ?r\gh ,m p>allel to y',oy4 ,?is is "o po9t ^: coord9ates satisfy ! equation4 ,n[ let ,m move f ! extreme left to ! extreme "r ( ! l9e x',ox1 reg>d$ z ext5d$ bo? ways z f> z we l1 i4e4 let x take all real values f -oo to oo4 ,) e value goes a po9t ,p1 z above1 on ! p>allel to y',oy ?r\gh ! correspond+ ,m2 & we ?us f9d t !re is a pa? f ! extreme left to ! extreme "r ( ! figure1 all po9ts ,p along :i* are di/+ui%$ f o!r po9ts by ! exceptional prop]ty ( satisfy+ ! equation by _! coord9ates4 ,?is pa? is a locus2 & ! equation y phi7x7 repres5ts x4 ,m g5]ally /ill1 take an equation f7x1 y7 #j :i* involves bo? x & y "u a functional =m4 ,any p>ticul> value giv5 to x 9 x produces f x an equation = ! det]m9ation ( a value or values ( y1 :i* g ) t value ( x 9 specify+ a po9t or po9ts 7x1 y71 ( :i* ! coord9ates satisfy ! equation f7x1 y7 #j4 ,"h aga91 z x takes all values1 ! po9t or po9ts describe a pa? or pa?s1 :i* con/itute a locus repres5t$ by ! equation4 ,except :5 y ent]s to ! fir/ degree only 9 f7x1 y71 x is n to be expect$ t all ! values ( y1 det]m9$ z go+ ) a *os5 value ( x1 w be necess>ily real2 inde$ x is n uncommon = all to be imag9>y = "s ranges ( values ( x4 ,! locus may l>gely consi/ ( cont9ua ( imag9>y po9ts2 b ! real p>ts ( x con/itute a real curve or real curves4 ,note t we h to all[ x to admit ( all imag9>y1 z well z ( all real1 values1 9 ord] to obta9 all imag9>y p>ts ( ! locus4 ,a locus or curve may be algebraically specifi$ 9 ano!r way2 viz4 we may be giv5 two equations x f7!ta71 y ,f7!ta71 :i* express ! coord9ates ( any po9t ( x z two functions ( ! same v>iable p>amet] !ta to :i* all values are op54 ,z !ta takes all values 9 turn1 ! po9t 7x1 y7 trav]ses ! curve4 ,x is a good ex]cise to trace a numb] ( curves1 tak5 z def9$ by ! equations :i* repres5t !m4 ,?is1 9 simple cases1 c be done approximately by plott+ ! values ( y giv5 by ! equation ( a curve z go+ ) a consid]able numb] ( values ( x1 & connect+ ! v>i\s po9ts 7x1 y7 ?us obta9$4 ,b me?ods exi/ = dim9i%+ ! lab\r ( ?is t5tative process4 ,ano!r problem1 :i* w be m att5d$ to "h1 is t ( det]m9+ ! equations ( curves ( kn[n int]e/1 tak5 z def9$ by geometrical prop]ties4 ,x is n a matt] = surprise t ! curves :i* h be5 mo/ & longe/ /udi$ geometrically are among ^? repres5t$ by equations ( ! simple/ "*4 #h4 ,! ,/raight ,l9e4--,?is is ! simple/ type ( locus4 ,also ! simple/ type ( equation 9 x & y is ,ax ,by ,c #j1 "o ( ! fir/ degree4 ,"h ! coeffici5ts ,a1 ,b1 ,c are con/ants4 ,!y are1 l ! curr5t coord9ates1 x1 y1 num]ical4 ,b1 9 giv+ int]pretation to su* an equation1 we mu/ ( c\rse ref] to numb]s ,ax1 ,by1 ,c ( unit magnitudes ( ! same k9d1 ( units ( c\nt+ = in/ance1 or unit l5g?s or unit squ>es4 ,x w n[ be se5 t e /raight l9e has an equation ( ! fir/ degree1 & t e equation ( ! fir/ degree repres5ts a /raight l9e4 ,illu/ration3 ,,fig4 #ej4 ,x has be5 se5 7,s #g7 t l9es p>allel to ! axes h equations ( ! fir/ degree1 free f "o ( ! v>iables4 ,take n[ a /raight l9e ,,abc incl9$ to bo? axes4 ,let x make a giv5 angle alpha ) ! positive direction ( ! axis ( x1 i4e4 9 fig4 #ej let ?is be ! angle ?r\gh :i* ,ax mu/ be revolv$ c\nt]-clockwise ab\t ,a 9 ord] to be made co9cid5t ) ! l9e4 ,let ,c1 ( coord9ates 7h1 k71 be a fix$ po9t on ! l9e1 & ,p7x1 y7 any o!r po9t ^u x4 ,draw ! ord9ates ,,cd1 ,,pm ( ,c & ,p1 & let ! p>allel to ! axis ( x ?r\gh ,c meet ,,pm1 produc$ if necess>y1 9 ,r4 ,! "r-angl$ triangle ,,crp tells u t1 ) ! signs appropriate to _! directions atta*$ to ,,cr & ,,rp1 ,,rp ,,cr tan alpha1 i4e4 ,,mp - ,,dc 7,,om - ,,od7 tan alpha1 & ?is gives t y - k tan alpha 7x - h71 an equation ( ! fir/ degree satisfi$ by x & y4 ,no po9t n on ! l9e satisfies ! same equation2 = ! l9e f ,c to any po9t (f ! l9e w\ld make ) ,,cr "s angle beta diff]5t f alpha1 & ! po9t 9 "q w\ld satisfy an equation y - k tan beta7x - h71 :i* is inconsi/5t ) ! above equation4 ,! equation ( ! l9e may also be writt5 y mx b1 :]e m tan alpha1 & b k - h tan alpha4 ,"h b is ! value obta9$ = y f ! equation :5 #j is put = x1 i4e4 x is ! num]ical measure1 ) prop] sign1 ( ,,ob1 ! int]cept made by ! l9e on ! axis ( y1 measur$ f ! orig94 ,= diff]5t /raight l9es1 m & b may h any con/ant values we l4 ,n[ ! g5]al equation ( ! fir/ degree ,ax ,by ,c #j may be writt5 y -7,a_/,b7x - ,c_/,b1 unless ,b #j1 9 :i* case x repres5ts a l9e p>allel to ! axis ( y2 & -,a_/,b1 -,c_/,b are values :i* c be giv5 to m & b1 s t e equation ( ! fir/ degree repres5ts a /raight l9e4 ,x is important to notice t ! g5]al equation1 :i* 9 appe>ance conta9s ?ree con/ants ,a1 ,b1 ,c1 9 effect dep5ds on two only1 ! ratios ( two ( !m to ! ?ird4 ,9 virtue ( ?is la/ rem>k1 we see t two di/9ct conditions suffice to det]m9e a /raight l9e4 ,= in/ance1 x is easy f ! above to see t x y -- -- #a a b is ! equation ( a /raight l9e det]m9$ by ! two conditions t x makes int]cepts ,,oa1 ,,ob on ! two axes1 ( :i* a & b are ! num]ical measures ) prop] signs3 note t 9 fig4 #ej a is negative4 ,aga91 y#b - y#a y - y#a ------- 7x - x#a71 x#b - x#a i4e4 7y#a - y#b7x - 7x#a - x#b7y x#ay#b - x#by#a #j1 repres5ts ! l9e det]m9$ by ! data t x passes ?r\gh two giv5 po9ts 7x#a1 y#a7 & 7x#b1 y#b74 ,to prove ?is f9d m 9 ! equation y - y#a m7x - x#a7 ( a l9e ?r\gh 7x#a1 y#a71 f ! condition t 7x#b1 y#b7 lies on ! l9e4 ,9 ?is p>agraph ! coord9ates h be5 assum$ rectangul>4 ,_h !y be5 oblique1 ! doctr9e ( simil> triangles w\ld h giv5 ! same results1 except t 9 ! =ms ( equation y - k m7x - h71 y mx b1 we %\ld n h _h m tan alpha4 #i4 ,! ,circle4--,x is easy to write d[n ! equation ( a giv5 circle4 ,let 7h1 k7 be its giv5 c5tre ,c1 & rho ! num]ical measure ( its giv5 radius4 ,take ,p 7x1 y7 any po9t on its circumf]5ce1 & con/ruct ! triangle ,,crp1 9 fig4 #ej z above4 ,! fact t ?is is "r-angl$ tells u t ,,cr#b ,,rp#b ,,cp#b1 & ?is at once gives ! equation 7x - h7#b 7y - k7#b rho#b4 ,a po9t n ^u ! circumf]5ce ( ! p>ticul> circle is at "s di/ance f 7h1 k7 diff]5t f rho1 & satisfies an equation inconsi/5t ) ?is "o2 :i* accord+ly repres5ts ! circumf]5ce1 or1 z we say1 ! circle4 ,! equation is ( ! =m x#b y#b #b;,ax #b;,by ,c #j4 ,conv]sely e equation ( ?is =m repres5ts a circle3 we h only to take -,a1 -,b1 ,a#b ,b#b - ,c = h1 k1 rho#b respectively1 to obta9 its c5tre & radius4 ,b ?is /atem5t mu/ appe> too unre/rict$4 ,ought we n to require ,a#b ,b#b - ,c to be positive8 ,c]ta9ly1 if by circle we are only to mean ! visible r\nd circumf]5ce ( ! geometrical def9ition4 ,yet1 analytically1 we contemplate altoge!r imag9>y circles1 = :i* rho#b is negative1 & circles1 = :i* rho #j1 ) all _! reality cond5s$ into _! c5tres4 ,ev5 :5 rho#b is positive1 s t a visible r\nd circumf]5ce exi/s1 we d n reg>d ?is z con/itut+ ! :ole ( ! circle4 ,giv+ to x any value :atev] 9 7x - h7#b 7y - k7#b rho#b1 we obta9 two values ( y1 real1 co9cid5t or imag9>y1 ea* ( :i* goes ) ! abscissa x z ! ord9ate ( a po9t1 real or imag9>y1 on :at is repres5t$ by ! equation ( ! circle4 ,! doctr9e ( ! imag9>y on a circle1 & 9 geometry g5]ally1 is ( purely algebraical inception2 b x has be5 9 its entirety accept$ by mod]n pure geomet]s1 & signal success has att5d$ ! ef=ts ( ^? :o1 l ,k4,g4,c4 von ,/audt1 h /riv5 to base its conclusions on pr9ciples n at all algebraical 9 =m1 ?\gh ( c\rse cognate to ^? adopt$ 9 introduc+ ! imag9>y into algebra4 ,a circle ) its c5tre at ! orig9 has an equation x#b y#b rho#b4 ,9 oblique coord9ates ! g5]al equation ( a circle is x#b #bxy cos omega y#b #b;,ax #b;,by ,c #j4 #aj4 ,! conic sections are ! next simple/ loci2 & x w be se5 lat] t !y are ! loci repres5t$ by equations ( ! second degree4 ,circles are p>ticul> cases ( conic sections2 & !y h j be5 se5 to h = _! equations a p>ticul> class ( equations ( ! second degree4 ,ano!r p>ticul> class ( su* equations is t includ$ 9 ! =m 7,ax ,by ,c77,a'x ,b'y ,c'7 #j1 :i* repres5ts two /raight l9es1 because ! product on ! left vani%es if1 & only if1 "o ( ! two factors does1 i4e4 if1 & only if1 7x1 y7 lies on "o or o!r ( two /raight l9es4 ,! condition t ax#b #b;hxy by#b #b;gx #b;fy c #j1 :i* is (t5 writt5 7a1 b1 c1 f1 g1 h77x1 y1 ,i7#b #j1 takes ?is =m is abc #b;fgh-af#b-;bg#b - *#b #j4 ,note t ! two l9es may1 9 p>ticul> cases1 be p>allel or co9cid5t4 ,any equation l ,f#a7x1 y7 ,f#b7x1 y7 444 ,fn7x1 y7 #j1 ( :i* ! left-h& side breaks up into factors1 repres5ts all ! loci sep>ately repres5t$ by ,f#a7x1 y7 #j1 ,f#b7x1 y7 #j1 444 ,fn7x1 y7 #j4 ,9 p>ticul> an equation ( degree n :i* is free f x repres5ts n /raight l9es p>allel to ! axis ( x1 & "o ( degree n :i* is homog5e\s 9 x & y1 i4e4 "o :i* ^u division by xn1 becomes an equation 9 ! ratio y_/x1 repres5ts n /raight l9es ?r\gh ! orig94 ,curves repres5t$ by equations ( ! ?ird degree are call$ cubic curves4 ,! g5]al equation ( ?is degree w be writt5 79977x1 y1 ,i7#c #j4 #aa4 ,descriptive ,geometry4--,a geometrical proposition is ei!r descriptive or metrical3 9 ! =m] case ! /atem5t ( x is indep5d5t ( ! idea ( magnitude 7l5g?1 incl9ation1 c471 & 9 ! latt] x has ref]5ce to ?is idea4 ,! me?od ( coord9ates seems to be by its inception ess5tially metrical4 ,yet 9 deal+ by ?is me?od ) descriptive propositions we are em95tly free f metrical consid]ations1 because ( our p[] to use g5]al equations1 & to avoid all assumption t measurem5ts impli$ are any p>ticul> measurem5ts4 ,illu/ration3 ,,fig4 #ea4 #ab4 ,x is wor? :ile to illu/rate ?is by ! in/ance ( ! well-kn[n !orem ( ! radical c5tre ( ?ree circles4 ,! !orem is t1 giv5 any ?ree circles ,a1 ,b1 ,c 7fig4 #ea71 ! common *ords alphaalpha'1 betabeta'1 gammagamma' ( ! ?ree pairs ( circles meet 9 a po9t4 ,! geometrical pro( is metrical ?r\gh\t3-- ,take ,o ! po9t ( int]section ( alphaalpha'1 betabeta'1 & jo9+ ?is ) gamma'1 suppose t gamma',o does n pass ?r\gh gamma1 b t x meets ! circles ,a1 ,b 9 two di/9ct po9ts gamma#b1 gamma#a respectively4 ,we h !n ! kn[n metrical prop]ty ( int]sect+ *ords ( a circle2 viz4 9 circle ,c1 :]e alphaalpha'1 betabeta'1 are *ords meet+ at a po9t ,o1 ,oalpha4,oalpha' ,obeta4,obeta'1 :]e1 z well z 9 :at imm$iately foll[s1 ,oalpha1 c4 d5ote1 ( c\rse1 l5g?s or di/ances4 ,simil>ly 9 circle ,a1 ,obeta4,obeta' ,ogamma#b4,ogamma'1 & 9 circle ,b1 ,oalpha4,oalpha' ,ogamma#a4,ogamma'4 ,consequ5tly ,ogamma#a4,ogamma' ,ogamma#b4,ogamma'1 t is1 ,ogamma#a ,ogamma#b1 or ! po9ts gamma#a & gamma#b co9cide2 t is1 !y ea* co9cide ) gamma4 ,we contra/ ?is ) ! analytical me?od3-- ,"h x only requires to be kn[n t an equation ,ax ,by ,c #j repres5ts a l9e1 & an equation x#b y#b ,ax ,by ,c #j repres5ts a circle4 ,a1 ,b1 ,c h1 9 ! two cases respectively1 metrical significations2 b ^! we are n conc]n$ )4 ,us+ ,s to d5ote ! function x#b y#b ,ax ,by ,c1 ! equation ( a circle is ,s o4 ,let ! equation ( any o!r circle be ,s'1 x#b y#b ,a'x ,b'y ,c' #j2 ! equation ,s - ,s' #j is a l9e> equation 7,s - ,s' is 9 fact 7,a - ,a'7x 7,b - ,b'7y ,c - ,c71 & x ?us repres5ts a l9e2 ?is equation is satisfi$ by ! coord9ates ( ea* ( ! po9ts ( int]section ( ! two circles 7= at ea* ( ^! po9ts ,s #j & ,s' #j1 !re=e also ,s - ,s' #j72 h5ce ! equation ,s - ,s' #j is t ( ! l9e jo9+ ! two po9ts ( int]section ( ! two circles1 or say x is ! equation ( ! common *ord ( ! two circles4 ,consid]+ !n a ?ird circle ,s1 x#b y#b ,ax ,by ,c #j1 ! equations ( ! common *ords are ,s-,s' #j1 ,s - ,s #j1 ,s' - ,s #j 7ea* ( ^! a l9e> equation72 at ! int]section ( ! fir/ & second ( ^! l9es ,s ,s' & ,s ,s1 !re=e also ,s' ,s1 or ! equation ( ! ?ird l9e is satisfi$ by ! coord9ates ( ! po9t 9 "q2 t is1 ! ?ree *ords int]sect 9 a po9t ,o1 ! coord9ates ( :i* are det]m9$ by ! equations ,s ,s' ,s4 ,x fur!r appe>s t if ! two circles ,s #j1 ,s' #j d n int]sect 9 any real po9ts1 !y mu/ be reg>d$ z int]sect+ 9 two imag9>y po9ts1 su* t ! l9e jo9+ !m is ! real l9e repres5t$ by ! equation ,s - ,s' #j2 or t two circles1 :e!r _! int]sections be real or imag9>y1 h always a real common *ord 7or radical axis71 & t = any ?ree circles ! common *ords int]sect 9 a po9t 7( c\rse real7 :i* is ! radical c5tre4 ,& by ?is v !orem1 giv5 two circles ) imag9>y int]sections1 we c1 by draw+ circles :i* meet ea* ( !m 9 real po9ts1 con/ruct ! radical axis ( ! fir/-m5tion$ two circles4 #ac4 ,! pr9ciple employ$ 9 %[+ t ! equation ( ! common *ord ( two circles is ,s - ,s' #j is "o ( v ext5sive application1 & "s m illu/rations ( x may be giv54 ,suppose ,s #j1 ,s' #j are l9es 7t is1 let ,s1 ,s' n[ d5ote l9e> functions ,ax ,by ,c1 ,a'x ,b'y ,c'71 !n ,s - k,s' #j 7k an arbitr>y con/ant7 is ! equation ( any l9e pass+ ?r\gh ! po9t ( int]section ( ! two giv5 l9es4 ,su* a l9e may be made to pass ?r\gh any giv5 po9t1 say ! po9t 7x#j1 y#j72 if ,s#j1 ,s'#j are :at ,s1 ,s' respectively become on writ+ = 7x1 y7 ! values 7x#j1 y#j71 !n ! value ( k is k ,s#j 3 ,s'#j4 ,! equation 9 fact is ,,ss'#j - ,s#j,s' #j2 & />t+ f ?is equation we at once v]ify x a po/]iori2 ! equation is a l9e> equation satisfi$ by ! values ( 7x1 y7 :i* make ,s #j1 ,s' #j2 & satisfi$ also by ! values 7x#j1 y#j72 & x is ?us ! equation ( ! l9e 9 "q4 ,if1 z be=e1 ,s #j1 ,s' #j repres5t circles1 !n 7k be+ arbitr>y7 ,s - k,s' #j is ! equation ( any circle pass+ ?r\gh ! two po9ts ( int]section ( ! two circles2 & to make ?is pass ?r\gh a giv5 po9t 7x#j1 y#j7 we h aga9 k ,s#j 3 ,s'#j4 ,9 ! p>ticul> case k #a1 ! circle becomes ! common *ord 7m accurately x becomes ! common *ord toge!r ) ! l9e inf9ity2 see ,s #bc bel[74 ,if ,s d5ote ! g5]al quadric function1 ,s ax#b #b;hxy by#b #b;fy #b;gx c1 !n ! equation ,s #j repres5ts a conic2 assum+ ?is1 !n1 if ,s' #j repres5ts ano!r conic1 ! equation ,s - k,s' #j repres5ts any conic ?r\gh ! f\r po9ts ( int]section ( ! two conics4 ,illu/ration3 ,,fig4 #eb4 #ad4 ,! object /ill be+ to illu/rate ! mode ( work+ ) coord9ates = descriptive purposes1 we consid] ! !orem ( ! pol> ( a po9t 9 reg>d to a circle4 ,giv5 a circle & a po9t ,o 7fig4 #eb71 we draw ?r\gh ,o any two l9es meet+ ! circle 9 ! po9ts ,a1 ,a' & ,b1 ,b' respectively1 & !n tak+ ,q z ! int]section ( ! l9es ,,ab' & ,a',b1 ! !orem is t ! locus ( ! po9t ,q is a "r l9e dep5d+ only ^u ,o & ! circle1 b indep5d5t ( ! p>ticul> l9es ,,oaa' & ,,obb'4 ,tak+ ,o z ! orig91 & = ! axes any two l9es ?r\gh ,o at "r angles to ea* o!r1 ! equation ( ! circle w be x#b y#b #b;,ax #b;,by ,c #j2 & if ! equation ( ! l9e ,,oaa' is tak5 to be y mx1 !n ! po9ts ,a1 ,a' are f\nd z ! int]sections ( ! /raight l9e ) ! circle2 or to det]m9e x we h x#b7a m#b7 #bx7,a ,bm7 ,c #j4 ,if7x#a1 y#a7 are ! coord9ates ( ,a1 & 7x#b1 y#b7 ( ,a'1 !n ! roots ( ?is equation are x#a1 x#b1 :5ce easily #a #a ,a ,bm -- -- -#b ------4 x#a x#b ,c ,& simil>ly1 if ! equation ( ! l9e ,,obb' is tak5 to be y m'x#a & ! coord9ates ( ,b1 ,b' to be 7x#c1 y#c7 & 7x#d1 y#d7 respectively1 !n #a #a ,a ,bm' -- -- -#b -------4 x#c x#d ,c' ,we h !n by ,s #h x7y#a - y#d7 - y7x#a - x#d7 x#ay#d - x#dy#a #j1 x7y#b - y#c7 - y7x#b - x#c7 x#by#c - x#cy#b #j1 z ! equations ( ! l9es ,,ab' & ,a',b respectively4 ,r$uc+ by means ( ! relations y#a - mx#a #j1 y#b - mx#b #j1 y#c - m'x#c #j1 y#d - m'x#d #j1 ! two equations become x7mx#a - m'x#d7 - y7x#a - x#d7 7m'- m7x#ax#d #j1 x7mx#b - m'x#c7 - y7x#b - x#c7 7m'- m7x#bx#c #j1 & if we divide ! fir/ ( ^! equations by x#ax#d1 & ! second by x#bx#c & !n add1 we obta9 _/ #a #a _/ #a #a #a #a _/ #a #a x m7 -- -- 7 - m'7 -- -- 7 - y -- -- - 7 -- -- 7 x#c x#d _/ x#a x#b_/ x#c x#d x#a x#b_/ #bm' - #bm #j1 or1 :at is ! same ?+1 _/ #a #a _/ #a #a 7 -- -- 77y - m'x7 - 7 -- -- 77y - mx7 #bm' - #bm #j1 x#a x#b _/ x#c x#d _/ :i* by :at prec$es is ! equation ( a l9e ?r\gh ! po9t ,q4 ,sub/itut+ h]e9 = #a_/x#a #a_/x#b1 #a_/x#c #a_/x#d _! =ego+ values1 ! equation becomes -7,a ,bm77y - m'x7 7,a ,bm'77y - mx7 ,c7m' - m7 #j2 t is1 7m - m'77,ax ,by ,c7 #j2 or f9ally x is ,ax ,by ,c #j1 %[+ t ! po9t ,q lies 9 a l9e ! position ( :i* is indep5d5t ( ! p>ticul> l9es ,,oaa'1 ,,obb' us$ 9 ! con/ruction4 ,x is prop] to notice t !re is no correspond5ce to ea* o!r ( ! po9ts ,a1 ,a' & ,b1 ,b'2 ! gr\p+ might z well h be5 ,a1 ,a' & ,b'1 ,b2 & x !nce appe>s t ! l9e ,ax ,by ,c #j j obta9$ is 9 fact ! l9e jo9+ ! po9t ,q ) ! po9t ,r :i* is ! int]section ( ,,ab & ,a',b'4 #ae4 ,9 ,s #h x has be5 se5 t two conditions det]m9e ! equation ( a /raight l9e1 because 9 ,ax ,by ,c #j "o ( ! coeffici5ts may be divid$ out1 leav+ only two p>amet]s to be det]m9$4 ,simil>ly five conditions in/ead ( six det]m9e an equation ( ! second degree 7a1 b1 c1 f1 g1 h77x1 y1 #a7b #j1 & n9e in/ead ( t5 det]m9e a cubic 79977x1 y1 #a7c #j4 ,x ?us appe>s t a cubic c be made to pass ?r\gh #i giv5 po9ts1 & t ! cubic s pass+ ?r\gh #i giv5 po9ts is completely det]m9$4 ,!re is1 h[ev]1 a rem>kable exception4 ,consid]+ two giv5 cubic curves ,s #j1 ,s' #j1 ^! int]sect 9 #i po9ts1 & ?r\gh ^! #i po9ts we h ! :ole s]ies ( cubics ,s - k,s' #j1 :]e k is an arbitr>y con/ant3 k may be det]m9$ s t ! cubic %all pass ?r\gh a giv5 t5? po9t 7k ,s#j 3 ,s'#j1 if ! coord9ates are 7x#j1 y#j71 & ,s#j1 ,s'#j d5ote ! correspond+ values ( ,s1 ,s'74 ,! result+ curve ,,ss'#j - ,s',s#j #j may be reg>d$ z ! cubic det]m9$ by ! conditions ( pass+ ?r\gh #h ( ! #i po9ts & ?r\gh ! giv5 po9t 7x#j1 y#j72 & f ! equation x !nce appe>s t ! curve passes ?r\gh ! rema9+ "o ( ! #i po9ts4 ,9 o!r words1 we ?us h ! !orem1 any cubic curve :i* passes ?r\gh #h ( ! #i int]sections ( two giv5 cubic curves passes ?r\gh ! #i? int]section4 ,! applications ( ?is !orem are v num]\s2 = in/ance1 we d]ive f x ,pascal's !orem ( ! inscrib$ hexagon4 ,consid] a hexagon inscrib$ 9 a conic4 ,! ?ree alt]nate sides con/itute a cubic1 & ! o!r ?ree alt]nate sides ano!r cubic4 ,! cubics int]sect 9 #i po9ts1 be+ ! #f v]tices ( ! hexagon1 & ! #c ,pascalian po9ts1 or int]sections ( ! pairs ( opposite sides ( ! hexagon4 ,draw+ a l9e ?r\gh two ( ! ,pascalian po9ts1 ! conic & ?is l9e con/itute a cubic pass+ ?r\gh #h ( ! #i po9ts ( int]section1 & x !re=e passes ?r\gh ! rema9+ po9t ( int]section--t is1 ! ?ird ,pascalian po9t2 & s9ce obvi\sly ?is does n lie on ! conic1 x mu/ lie on ! l9e--t is1 we h ! !orem t ! ?ree ,pascalian po9ts 7or po9ts ( int]section ( ! pairs ( opposite sides7 lie on a l9e4 #af4 ,metrical ,!ory resum$4 ,projections & ,p]p5dicul>s4--,x is a metrical fact ( fundam5tal importance1 already us$ 9 ,s #h1 t1 if a f9ite l9e ,,pq be project$ on any o!r l9e ,,oo' by p]p5dicul>s ,,pp'1 ,,qq' to ,,oo'1 ! l5g? ( ! projection ,p',q' is equal to t ( ,,pq multipli$ by ! cos9e ( ! acute angle betwe5 ! two l9es4 ,also ! algebraical sum ( ! projections ( ! sides ( any clos$ polygon ^u any l9e is z]o1 because z a po9t goes r\nd ! polygon1 f any v]tex ,a to ,a aga91 ! po9t :i* is its projection on ! l9e passes f ,a' ! projection ( ,a to ,a' aga91 i4e4 trav]ses equal di/ances along ! l9e 9 positive & negative s5ses4 ,if we consid] ! polygon z consi/+ ( two brok5 l9es1 ea* ext5d+ f ! same initial to ! same t]m9al po9t1 ! sum ( ! projections ( ! l9es :i* compose ! "o is equal1 9 sign & magnitude1 to ! sum ( ! projections ( ! l9es compos+ ! o!r4 ,obs]ve t ! projection on a l9e ( a l5g? p]p5dicul> to ! l9e is z]o4 ,let u h5ce f9d ! equation ( a /raight l9e su* t ! p]p5dicul> ,,od on x f ! orig9 is ( l5g? rho tak5 z positive1 & is incl9$ to ! axis ( x at an angle x,o,d alpha1 measur$ c\nt]-clockwise f ,ox4 ,take any po9t ,p 7x1 y7 on ! l9e1 & con/ruct ,,om & ,,mp z 9 fig4 #dh4 ,! sum ( ! projections ( ,,om & ,,mp on ,,od is ,,od itself2 & ?is gives ! equation ( ! l9e x cos alpha y s9 alpha rho4 ,obs]ve t cos alpha & s9 alpha "h are ! s9 alpha & -cos alpha1 or ! -s9 alpha & cos alpha ( ,s #h accord+ to circum/ances4 ,we c write d[n an expression = ! p]p5dicul> di/ance f ?is l9e ( any po9t 7x'1 y'7 :i* does n lie ^u x4 ,if ! p>allel ?r\gh 7x'1 y'7 to ! l9e meet ,,od 9 ,e1 we h x' cos alpha y' s9 alpha ,,oe1 & ! p]p5dicul> di/ance requir$ is ,,od - ,,oe1 i4e4 rho - x' cos alpha - y' s9 alpha2 x is ! p]p5dicul> di/ance tak5 positively or negatively accord+ z 7x'1 y'7 lies on ! same side ( ! l9e z ! orig9 or n4 ,! g5]al equation ,ax ,by ,c #j may be giv5 ! =m x cos alpha y s9 alpha - rho #j by divid+ x by root7,a#b ,b#c74 ,?us 7,ax' ,by' ,c7 _/ root7,a#b ,b#b7 is 9 absolute value ! p]p5dicul> di/ance ( 7x'1 y'7 f ! l9e ,ax ,by ,c #j4 ,rememb]1 h[ev]1 t !re is an ess5tial ambiguity ( sign atta*$ to a squ>e root4 ,! expression f\nd gives ! di/ance tak5 positively :5 7x'1 y'7 is on ! orig9 side ( ! l9e1 if ! sign ( ,c is giv5 to root7,a#b ,b#b74 #ag4 ,trans=mation ( ,coord9ates4--,we (t5 ne$ to adopt new axes ( ref]5ce 9 place ( old ones2 & ! above pr9ciple ( projections readily expresses ! old coord9ates ( any po9t 9 t]ms ( ! new4 ,illu/ration3 ,,fig4 #ec4 ,suppose1 = in/ance1 t we want to take = new orig9 ! po9t ,o' ( old coord9ates ,,oa h1 ,,ao' k1 & = new axes ( ,x & ,y l9es ?r\gh ,o' obta9$ by rotat+ p>allels to ! old axes ( x & y ?r\gh an angle !ta c\nt]-clockwise4 ,con/ruct 7fig4 #ec7 ! old & new coord9ates ( any po9t ,p4 ,express+ t ! projections1 fir/ on ! old axis ( x & secondly on ! old axis ( y1 ( ,,op are equal to ! sums ( ! projections1 on ^? axes respectively1 ( ! p>ts ( ! brok5 l9e ,,oo',m',p1 we obta93 x h ,x cos !ta ,y cos 7!ta #a_/bpi7 h ,x cos !ta - ,y s9 !ta1 & y k ,x cos 7#a_/bpi - !ta7 ,y cos !ta k ,x s9 !ta ,y cos !ta4 ,be c>eful to obs]ve t ^! =mulae d n apply to e conceivable *ange ( ref]5ce f "o set ( rectangul> axes to ano!r4 ,x might h be5 requir$ to take ,o',x1 ,o',y' = ! positive directions ( ! new axes1 s t ! *ange ( directions ( ! axes c\ld n be effect$ by rotation4 ,we mu/ !n write -,y = ,y 9 ! above4 ,7 ! new axes oblique1 mak+ angles alpha1 beta respectively ) ! old axis ( x1 & s incl9$ at ! angle beta - alpha1 ! same me?od w\ld give ! =mulae x h ,x cos alpha ,y cos beta1 y k ,x s9 alpha ,y s9 beta4 #ah4 ,! ,conic ,sections4--,! conics1 z !y are n[ call$1 7 at fir/ def9$ z curves ( int]section ( planes & a cone2 b ,apollonius sub/itut$ a def9ition free f ref]5ce to space ( ?ree dim5sions4 ,?is1 9 effect1 is t a conic is ! locus ( a po9t ! di/ance ( :i* f a giv5 po9t1 call$ ! focus1 has a giv5 ratio to its di/ance f a giv5 l9e1 call$ ! directrix 7see ,,conic ,,section74 ,if e 3 #a is ! ratio1 e is call$ ! ecc5tricity4 ,! di/ances are consid]$ signless4 ,take 7h1 k7 = ! focus1 & x cos alpha y s9 alpha - p #j = ! directrix4 ,! absolute values ( root 7x - h7#b 7y - k7#b & p - x cos alpha -y s9 alpha are to h ! ratio e 3 #a2 & ?is gives 7x - h7#b 7y - k7#b e#b7p - x cos alpha - y s9 alpha7#b z ! g5]al equation1 9 rectangul> coord9ates1 ( a conic4 ,x is ( ! second degree1 & is ! g5]al equation ( t degree4 ,if1 9 fact1 we multiply x by an unkn[n lambda1 we c1 by solv+ six simultane\s equations 9 ! six unkn[ns lambda1 h1 k1 e1 p1 alpha1 s *oose values = ^! z to make ! coeffici5ts 9 ! equation equal to ^? 9 any equation ( ! second degree :i* may be giv54 ,!re is no failure ( ?is /atem5t 9 ! special case :5 ! giv5 equation repres5ts two /raight l9es1 z 9 ,s #aj1 b !re is speciality3 if ! two l9es int]sect1 ! int]section & ei!r bisector ( ! angle betwe5 !m are a focus & directrix2 if !y are unit$ 9 "o l9e1 any po9t on ! l9e & a p]p5dicul> to x ?r\gh ! po9t are3 if !y are p>allel1 ! case is a limit+ "o 9 :i* e & h#b k#b h become inf9ite :ile e7-#b77;h#b k#b7 rema9s f9ite4 ,9 ! case 7,s #i7 ( an equation su* z repres5ts a circle !re is ano!r in/ance ( proce$+ to a limit3 e has to become #j1 :ile ep rema9s f9ite3 moreov] alpha is indet]m9ate4 ,! c5tre ( a circle is its focus1 & its directrix has gone to inf9ity1 hav+ no special direction4 ,?is la/ fact illu/rates ! necessity1 :i* is also =c$ on plane geometry by ?ree-dim5sional consid]ations1 ( treat+ all po9ts at inf9ity 9 a plane z ly+ on a s+le /raight l9e4 ,sometimes1 9 r$uc+ an equation to ! above focus & directrix =m1 we f9d = h1 k1 e1 p1 tan alpha1 or "s ( !m1 only imag9>y values1 z quadratic equations h to be solv$2 & we h 9 fact to contemplate ! exi/5ce ( entirely imag9>y conics4 ,= in/ance1 no real values ( x & y satisfy x#b #by#b #c #j4 ,ev5 :5 ! locus repres5t$ is real1 we obta91 z a rule1 f\r sets ( values ( h1 k1 e1 p1 ( :i* two sets are imag9>y2 a real conic has1 besides two real foci & correspond+ directrices1 two o!rs t are imag9>y4 ,9 oblique z well z rectangul> coord9ates equations ( ! second degree repres5t conics4 #ai4 ,! ?ree ,species ( ,conics4--,a real conic1 :i* does n deg5]ate into /raight l9es1 is call$ an ellipse1 p>abola or hyp]bola accord+ z e 1 1 or #a4 ,to trace ! ?ree =ms x is be/ s to *oose ! axes ( ref]5ce z to simplify _! equations4 ,9 ! case ( a p>abola1 let #b;c be ! di/ance betwe5 ! giv5 focus & directrix1 & take axes ref]r$ to :i* ^! are ! po9t 7c1 #j7 & ! l9e x - c4 ,! equation becomes 7x - c7#b y#b 7x c7#b1 i4e4 y#b #d;cx4 ,9 ! o!r cases1 take a su* t a7e e7-#a77 is ! di/ance ( focus f directrix1 & s *oose axes t ^! are 7ae1 #j7 & x ae7-#a71 ?us gett+ ! equation7x - ae7#b y#b e#b7x - ae7-#a77b1 i4e4 7#a - e#b7x#b y#b a#b7a - e#b74 ,:5 e #a1 i4e4 9 ! case ( an ellipse1 ?is may be writt5 x#b_/;a#b y#b_/;b#b #a1 :]e b#b a#b7a - e#b72 & :5 e #a1 i4e4 9 ! case ( an hyp]bola1 x#b_/;a#b - y#b_/;b#b #a1 :]e b#b a#b7;e#b - #a74 ,! axes ?us *os5 = ! ellipse & hyp]bola are call$ ! pr9cipal axes4 ,9 figs4 #ed1 #ee1 #ef 9 ord]1 conics ( ! ?ree species1 ?us ref]r$1 are depict$4 ,illu/ration3 ,,fig4 #ed ,illu/ration3 ,,fig4 #ee ,illu/ration3 ,,fig4 #ef4 ,! oblique /raight l9es 9 fig4 #ef are ! asymptotes x_/a -y_/b ( ! hyp]bola1 l9es to :i* ! curve t5ds ) unlimit$ clos5ess z x goes to inf9ity4 ,! hyp]bola w\ld h an equation ( ! =m xy c if ref]r$ to its asymptotes z axes1 ! coord9ates be+ !n oblique1 unless a b1 9 :i* case ! hyp]bola is call$ rectangul>4 ,an ellipse has two imag9>y asymptotes4 ,9 p>ticul> a circle x#b y#b a#b1 a p>ticul> ellipse1 has = asymptotes ! imag9>y l9es x -y root-#a4 ,^! run f ! c5tre to ! s-call$ circul> po9ts at inf9ity4 #bj4 ,tang5ts & ,curvature4--,let 7x'1 y'7 & 7x' h1 y' k7 be two neighb\r+ po9ts ,p1 ,p' on a curve4 ,! equation ( ! l9e on :i* bo? lie is h7y - y'7 k7x - x'74 ,n[ keep ,p fix$1 & let ,p' move t[>ds co9cid5ce ) x along ! curve4 ,! connect+ l9e w t5d t[>ds a limit+ position1 to :i* x c nev] atta9 z long z ,p & ,p' are di/9ct4 ,! l9e :i* occupies ?is limit+ position is ! tang5t at ,p4 ,n[ if we subtract ! equation ( ! curve1 ) 7x'1 y'7 = ! coord9ates 9 x1 f ! l equation 9 7x' h1 y' k71 we obta9 a relation 9 h & k1 :i* w1 z a rule1 be ( ! =m #j ,ah ,bk t]ms ( high] degrees 9 h & k1 :]e ,a1 ,b & ! o!r coeffici5ts involve x' & y'4 ,?is gives k_/h -,a_/,b t]ms :i* t5d to vani% z h & k d1 s t -,a 3 ,b is ! limit+ value t5d$ to by k 3 h4 ,h5ce ! equation ( ! tang5t is ,b7y - y'7 ,a7x - x'7 #j4 ,! normal at 7x'1 y'7 is ! l9e ?r\gh x at "r angles to ! tang5t1 & its equation is ,a7y - y'7 - ,b7x - x'7 #j4 ,9 ! case ( ! conic 7a1 b1 c1 f1 g1 h7 7x1 y1 #a7b #j we f9d t ,a_/,b 7ax' hy' g7_/7hx' by' f74 ,we c obta9 ! coord9ates ( ,q1 ! int]section ( ! normals ,,qp1 ,,qp' at 7x'1 y'7 & 7x' h1 y' k71 & !n1 us+ ! limit+ value ( k 3 h1 d$uce ^? ( its limit+ position z ,p' moves up to ,p4 ,?is is ! c5tre ( curvature ( ! curve at ,p 7x'1 y'71 & is s call$ because x is ! c5tre ( ! circle ( close/ contact ) ! curve at t po9t4 ,t x is s foll[s f ! facts t ! close/ circle is ! limit t5d$ to by ! circle :i* t\*es ! curve at ,p & passes ?r\gh ,p'1 & t ! arc f ,p to ,p' ( ?is circle lies betwe5 ! circles ( c5tre ,q & radii ,,qp1 ,,qp'1 :i* circles t5d1 n to diff]5t limits z ,p' moves up to ,p1 b to "o4 ,! di/ance f ,p to ! c5tre ( curvature is ! radius ( curvature4 #ba4 ,diff]5tial ,plane ,geometry4--,! language & notation ( ! diff]5tial calculus are v useful 9 ! /udy ( tang5ts & curvature4 ,d5ot+ by 7xi1 eta7 ! curr5t coord9ates1 we f9d1 z above1 t ! tang5t at a po9t 7x1 y7 ( a curve is eta - y 7xi - x7dy_/dx1 :]e dy_/dx is f\nd f ! equation ( ! curve4 ,if ?is be f7x1 y7 #j ! tang5t is 7xi - x7 7d,pf_/d,px7 7eta - y7 7d,pf_/d,py7 #j4 ,if rho & 7alpha1 beta7 are ! radius & c5tre ( curvature at 7x1 y71 we f9d t q7alpha - x7 -p7#a p#b71 q7beta - y7 #a p#b1 q#brho#b 7#a p#b7c1 :]e p1 q d5ote dy_/dx1 d#by_/dx#b respectively4 7,see ,,inf9itesimal ,,calculus47 ,9 any giv5 case we c1 at all ev5ts 9 !ory1 elim9ate x1 y betwe5 ! above equations = alpha - x & beta - y1 & ! equation ( ! curve4 ,! result+ equation 9 7alpha1 beta7 repres5ts ! locus ( ! c5tre ( curvature4 ,?is is ! evolute ( ! curve4 #bb4 ,pol> ,coord9ates4--,9 plane geometry ! di/ance ( any po9t ,p f a fix$ orig9 7or pole7 ,o1 & ! incl9ation x,o,p ( ,,op to a fix$ l9e ,ox1 det]m9e ! po9t3 r1 ! num]ical measure ( ,,op1 ! radius vector1 & !ta1 ! circul> measure ( x,o,p1 ! incl9ation1 are call$ pol> coord9ates ( ,p4 ,! =mulae x r cos !ta1 y r s9 !ta connect ,c>tesian & pol> coord9ates1 & make transition f ei!r sy/em to ! o!r easy4 ,9 pol> coord9ates ! equations ( a circle ?r\gh ,o1 & ( a conic ) ,o z focus1 take ! simple =ms r #b;a cos 7!ta-alpha71 r #a - e cos 7!ta-alpha7 l4 ,! use ( pol> coord9ates is v conv5i5t 9 discuss+ curves :i* h prop]ties ( symmetry ak9 to t ( a regul> polygon1 su* curves = in/ance z r a cos m !ta1 ) m integral1 & also ! curves call$ spirals1 :i* h equations giv+ r z functions ( !ta itself1 & n m]ely ( s9 !ta & cos !ta4 ,9 ! geometry ( motion "u c5tral =ces ! advantage ( work+ ) pol> coord9ates is great4 #bc4 ,tril9e> & ,areal ,coord9ates4--,consid] a fix$ triangle ,,abc1 & reg>d its sides z produc$ )\t limit4 ,d5ote1 z 9 trigonometry1 by a1 b1 c ! positive numb]s ( units ( a *os5 scale conta9$ 9 ! l5g?s ,,bc1 ,,ca1 ,,ab1 by ,a1 ,b1 ,c ! angles1 & by ,delta ! area1 ( ! triangle4 ,we might1 z 9 ,s #f1 take ,,ca1 ,,cb z axes ( x & y1 incl9$ at an angle ,c4 ,any po9t ,p 7x1 y7 9 ! plane is at p]p5dicul> di/ances y s9 ,c & x s9 ,c f ,,ca & ,,cb4 ,call ^! beta & alpha respectively4 ,! signs ( beta & alpha are ^? ( y & x1 i4e4 beta is positive or negative accord+ z ,p lies on ! same side ( ,,ca z ,b does or ! opposite1 & simil>ly = alpha4 ,an equation 9 7x1 y7 ( any degree may1 ^u replac+ 9 x x & y by alpha cosec ,c & beta cosec ,c1 be writt5 z "o ( ! same degree 9 7alpha1 beta74 ,n[ let gamma be ! p]p5dicul> di/ance ( ,p f ! ?ird side ,,ab1 tak5 z positive or negative z ,p is on ! ,c side ( ,,ab or n4 ,! geometry ( ! figure tells u t aalpha bbeta cgamma #b;,delta4 ,by means ( ?is relation 9 alpha1 beta1 gamma we c give an equation consid]$ c\ntless o!r =ms1 involv+ two or all ( alpha1 beta1 gamma4 ,9 p>ticul> we may make x homog5e\s 9 alpha1 beta1 gamma3 to d ?is we h only to multiply ! t]ms ( e degree less ?an ! highe/ pres5t 9 ! equation by a p[] ( 7aalpha bbeta cgamma7_/#b;,delta j suffici5t to raise !m1 9 ea* case1 to ! highe/ degree4 ,we call 7alpha1 beta1 gamma7 tril9e> coord9ates1 & an equation 9 !m ! tril9e> equation ( ! locus repres5t$4 ,tril9e> equations are1 z a rule1 dealt ) 9 _! homog5e\s =ms4 ,an advantage ?us ga9$ is t we ne$ n mean by 7alpha1 beta1 gamma7 ! actual measures ( ! p]p5dicul> di/ances1 b any prop]ly sign$ numb]s :i* h ! same ratio two & two z ^! di/ances4 ,9 place ( alpha1 beta1 gamma x is lawful to use1 z coord9ates specify+ ! position ( a po9t 9 ! plane ( a triangle ( ref]5ce ,,abc1 any giv5 multiples ( ^!4 ,= in/ance1 we may use x aalpha_/#b;,delta1 y bbeta_/#b;,delta1 z cgamma_/#b;,delta1 ! prop]ly sign$ ratios ( ! triangul> areas ,,pbc1 ,,pca1 ,,pab to ! triangul> area ,,abc4 ,^! are call$ ! areal coord9ates ( ,p4 ,9 areal coord9ates ! relation :i* enables u to make any equation homog5e\s takes ! simple =m x y z #a2 &1 z be=e1 we ne$ mean by x1 y1 z1 9 a homog5e\s equation1 only sign$ numb]s 9 ! "r ratios4 ,/raight l9es & conics are repres5t$ 9 tril9e> & 9 areal1 because 9 ,c>tesian1 coord9ates by equations ( ! fir/ & second degrees respectively1 & ^! degrees are pres]v$ :5 ! equations are made homog5e\s4 ,:at mu/ be said ab\t po9ts inf9itely f> (f 9 ord] to make univ]sal ! /atem5t1 to :i* !re is no exception z long z f9ite di/ances alone are consid]$1 t e homog5e\s equation ( ! fir/ degree repres5ts a /raight l9e8 ,let ! po9t ( areal coord9ates 7x'1 y'1 z'7 move inf9itely f> (f1 & mean by x1 y1 z f9ite quantities 9 ! ratios :i* x'1 y'1 z' t5d to assume z !y become inf9ite4 ,! relation x' y' z' #a gives t ! limit+ /ate ( ?+s t5d$ to is express$ by x y z #j4 ,?is p>ticul> equation ( ! fir/ degree is satisfi$ by no po9t at a f9ite di/ance2 b we see ! propriety ( say+ t x has to be tak5 z satisfi$ by all ! po9ts conceiv$ ( z actually at inf9ity4 ,accord+ly ! special prop]ty ( ^! po9ts is express$ by say+ t !y lie on a special /raight l9e1 ( :i* ! areal equation is x y z #j4 ,9 tril9e> coord9ates ?is l9e at inf9ity has = equation aalpha bbeta cgamma #j4 ,on ! "o special l9e at inf9ity p>allel l9es are treat$ z meet+4 ,!re are on x two special 7imag9>y7 po9ts1 ! circul> po9ts at inf9ity ( ,s #ai1 ?r\gh :i* all circles pass 9 ! same s5se4 ,9 fact if ,s ,o be "o circle1 9 areal coord9ates1 ,s 7x y z77lx my nz7 #j may1 by prop] *oice ( l1 m1 n1 be made any o!r2 s9ce ! add$ t]ms are once lx my nz1 & h ! g5]ality ( any expression l a'x b'y c' 9 ,c>tesian coord9ates4 ,n[ ^! two circles int]sect 9 ! two po9ts :]e ei!r meets x y z #j z well z 9 two po9ts on ! radical axis lx my nz #j4 #bd4 ,let u consid] ! p]p5dicul> di/ance ( a po9t 7alpha'1 beta'1 gamma'7 f a l9e lalpha mbeta ngamma4 ,we c take rectangul> axes ( ,c>tesian coord9ates 7= cle>ness z to equalities ( angle x is be/ to *oose an orig9 inside ,,abc71 & ref] to !m1 by putt+ expressions p - x cos!ta - y s9!ta1 c41 = alpha c42 we c !n apply ,s #af to get ! p]p5dicul> di/ance2 & f9ally rev]t to ! tril9e> notation4 ,! result is to f9d t ! requir$ di/ance is 7lalpha' mbeta' ngamma'7_/l1 m1 n1 :]e l1 m1 n#b l#b m#b n#b - #bmn cos ,a - #bnl cos ,b - #blm cos ,c4 ,9 areal coord9ates ! p]p5dicul> di/ance f 7x'1 y'1 z'7 to lx my nz #j is #b;,delta7lx' my' nz'7_/al1 bm1 cn4 ,9 bo? cases ! coord9ates are ( c\rse actual values4 ,n[ let xi1 eta1 zeta be ! p]p5dicul>s on ! l9e f ! v]tices ,a1 ,b1 ,c1 i4e4 ! po9ts 7#a1 #j1 #j71 7#j1 #a1 #j71 7#j1 #j1 #a71 ) signs 9 accord ) a conv5tion t opposit5ess ( sign implies di/9ction betwe5 "o side ( ! l9e & ! o!r4 ,?ree applications ( ! result above give xi_/l #b;,delta_/al1 bm1 cn eta_/m zeta_/n2 & we ?us h ! important fact t xix' etay' zetaz' is ! p]p5dicul> di/ance betwe5 a po9t ( areal coord9ates 7x'y'z'7 & a l9e on :i* ! p]p5dicul>s f ,a1 ,b1 ,c are xi1 eta1 zeta respectively4 ,we h also t xix etay zetaz #j is ! areal equation ( ! l9e on :i* ! p]p5dicul>s are xi1 eta1 zeta2 &1 by equat+ ! two expressions = ! p]p5dicul>s f 7x'1 y'1 z'7 on ! l9e1 t 9 all cases axi1 beta1 czeta#b #d;,delta#b4 #be4 ,l9e-coord9ates4 ,duality4--,a q diff]5t ord] ( ideas may be foll[$ 9 apply+ analysis to geometry4 ,! notion ( a /raight l9e specifi$ may prec$e t ( a po9t1 & po9ts may be dealt ) z ! int]sections ( l9es4 ,! specification ( a l9e may be by means ( coord9ates1 & t ( a po9t by an equation1 satisfi$ by ! coord9ates ( l9es :i* pass ?r\gh x4 ,sy/ems ( l9e-coord9ates w "h be only briefly consid]$4 ,e su* sy/em is alli$ to "s sy/em ( po9t-coord9ates2 & space w be sav$ by giv+ prom95ce to ?is fact1 & n recomm5c+ ab initio4 ,suppose t any p>ticul> sy/em ( po9t-coord9ates1 9 :i* lx my nz #j may repres5t any /raight l9e1 is be=e u3 notice t n only are tril9e> & areal coord9ates su* sy/ems1 b ,c>tesian coord9ates also1 s9ce we may write x_/z1 y_/z = ! ,c>tesian x1 y1 & multiply ?r\gh by z4 ,! l9e is exactly assign$ if l1 m1 n1 or _! mutual ratios1 are kn[n4 ,call 7l1 m1 n7 ! coord9ates ( ! l9e4 ,n[ keep x1 y1 z con/ant1 & let ! coord9ates ( ! l9e v>y1 b always s z to satisfy ! equation4 ,?is equation1 :i* we n[ write xl ym zn #j1 is satisfi$ by ! coord9ates ( e l9e ?r\gh a c]ta9 fix$ po9t1 & by ^? ( no o!r l9e2 x is ! equation ( t po9t 9 ! l9e-coord9ates l1 m1 n4 ,l9e-coord9ates are also call$ tang5tial coord9ates4 ,a curve is ! envelope ( l9es :i* t\* x1 z well z ! locus ( po9ts :i* lie on x4 ,a homog5e\s equation ( degree above ! fir/ 9 l1 m1 n is a relation connect+ ! coord9ates ( e l9e :i* t\*es "s curve1 & repres5ts t curve1 reg>d$ z an envelope4 ,= in/ance1 ! condition t ! l9e ( coord9ates 7l1 m1 n71 i4e4 ! l9e ( :i* ! alli$ po9t-coord9ate equation is lx my nz #j1 may t\* a conic 7a1 b1 c1 f1 g1 h7 7x1 y1 z7#b #j1 is readily f\nd to be ( ! =m 7,a1 ,b1 ,c1 ,f1 ,g1 ,h7 7l1 m1 n7#b #j1 i4e4 to be ( ! second degree 9 ! l9e-coord9ates4 ,x is n h>d to %[ t ! g5]al equation ( ! second degree 9 l1 m1 n ?us repres5ts a conic2 b ! deg5]ate conics ( l9e-coord9ates are n l9e-pairs1 z 9 po9t-coord9ates1 b po9t-pairs4 ,! degree ( ! po9t-coord9ate equation ( a curve is ! ord] ( ! curve1 ! numb] ( po9ts 9 :i* x cuts a /raight l9e4 ,t ( ! l9e-coord9ate equation is its class1 ! numb] ( tang5ts to x f a po9t4 ,! ord] & class ( a curve are g5]ally diff]5t :5 ei!r exce$s two4 #bf4 ,! sy/em ( l9e-coord9ates alli$ to ! areal sy/em ( po9t-coord9ates has special int]e/4 ,! l1 m1 n ( ?is sy/em are ! p]p5dicul>s xi1 eta1 zeta ( ,s #bd2 & x'xi y'eta z'zeta #j is ! equation ( ! po9t ( areal coord9ates 7x'1 y'1 z'71 i4e4 is a relation :i* ! p]p5dicul>s f ! v]tices ( ! triangle ( ref]5ce on e l9e ?r\gh ! po9t1 b no o!r l9e1 satisfy4 ,notice t a non-homog5e\s equation ( ! fir/ degree 9 xi1 eta1 zeta does n1 z a homog5e\s "o does1 repres5t a po9t1 b a circle4 ,9 fact x'xi y'eta z'zeta ,r expresses ! con/ancy ( ! p]p5dicul> di/ance ( ! fix$ po9t x'xi y'eta z'zeta #j f ! v>iable l9e 7xi1 eta1 zeta71 i4e4 ! fact t 7xi1 eta1 zeta7 t\*es a circle ) ! fix$ po9t = c5tre4 ,! relation 9 any xi1 eta1 zeta :i* enables u to make an equation homog5e\s is n l9e>1 z 9 po9t-coord9ates1 b quadratic1 viz4 x is ! relation axi1 beta1 czeta#b #d;,delta#b ( ,s #bd4 ,accord+ly ! homog5e\s equation ( ! above circle is #d;,delta#b7x'xi y'eta z'zeta7#b ,r#b;axi1 beta1 czeta#b4 ,e circle has an equation ( ?is =m 9 ! pres5t sy/em ( l9e-coord9ates4 ,notice t ! equation ( any circle is satisfi$ by ^? coord9ates ( l9es :i* satisfy bo? x'xi y'eta z'zeta #j1 ! equation ( its c5tre1 & axi1 beta1 czeta#b #j4 ,?is la/ equation1 ( :i* ! left-h& side satisfies ! condition = break+ up into two factors1 repres5ts ! two imag9>y circul> po9ts at inf9ity1 ?r\gh :i* all circles & _! asymptotes pass4 ,!re is /rict duality 9 descriptive geometry betwe5 po9t-l9e-locus & l9e-po9t-envelope !orems4 ,b 9 metrical geometry duality is encumb]$ by ! fact t !re is 9 a plane "o special l9e only1 associat$ ) di/ance1 :ile ( special po9ts1 associat$ ) direction1 !re are two3 moreov] ! l9e is real1 & ! po9ts bo? imag9>y4 ,,ii4 ,solid ,analytical ,geometry4 #bg4 ,any po9t 9 space may be specifi$ by ?ree coord9ates4 ,we consid] ?ree fix$ planes ( ref]5ce1 & g5]ally1 z 9 all t foll[s1 ?ree :i* are at "r angles two & two4 ,!y int]sect1 two & two1 9 l9es x',ox1 y',oy1 z',oz1 call$ ! axes ( x1 y1 z respectively1 & divide all space into eight p>ts call$ octants4 ,if f any po9t ,p 9 space we draw ,,pn p>allel to z,oz' to meet ! plane x,oy 9 ,n1 & !n f ,n draw ,,nm p>allel to y,oy' to meet x',ox 9 ,m1 ! coord9ates 7x1 y1 z7 ( ,p are ! num]ical measures ( ,,om1 ,,mn1 ,,np2 9 ! case ( rectangul> coord9ates ^! are ! p]p5dicul> di/ances ( ,p f ! ?ree planes ( ref]5ce4 ,! sign ( ea* coord9ate is positive or negative z ,p lies on "o side or ! o!r ( ! correspond+ plane4 ,9 ! octant del9eat$ ! signs are tak5 all positive4 ,illu/ration3 ,,fig4 #eg4 ,illu/ration3 ,,fig4 #eh4 ,9 fig4 #eg ! del9eation is on a plane ( ! pap] tak5 p>allel to ! plane z,ox1 ! po9ts ( a solid figure be+ project$ on t plane by p>allels to "s *os5 l9e ?r\gh ,o 9 ! positive octant4 ,sometimes x is cle>] to del9eate1 z 9 fig4 #eh1 by projection p>allel to t l9e 9 ! octant :i* is equally incl9$ to ,ox1 ,oy1 ,oz ^u a plane ( ! pap] p]p5dicul> to x4 ,x is possible by p>allel projection to del9eate equal scales along ,ox1 ,oy1 ,oz by scales hav+ any ratios we l along l9es 9 a plane hav+ any mutual incl9ations we l4 ,illu/ration3 ,,fig4 #ei4 ,= ! del9eation ( a surface ( simple =m x frequ5tly suffices to del9eate ! sections by ! coord9ate planes2 &1 9 p>ticul>1 :5 ! surface has symmetry ab\t ea* coord9ate plane1 to del9eate ! qu>t]-sections belong+ to a s+le octant4 ,?us fig4 #ei conv5i5tly repres5ts an octant ( ! wave surface1 :i* cuts ea* coord9ate plane 9 a circle & an ellipse4 ,or we may del9eate a s]ies ( cont\r l9es1 i4e4 sections by planes p>allel to x,oy1 or "s o!r *os5 plane2 ( c\rse o!r sections may be indicat$ too = great] cle>ness4 ,= ! del9eation ( a curve a good me?od is to repres5t1 z above1 a s]ies ( po9ts ,p !re(1 ea* accompani$ by its ord9ate ,,pn1 :i* s]ves to ref] x to ! plane ( xy4 ,! employm5t ( /]eographic projection is also int]e/+4 #bh4 ,9 plane geometry1 reckon+ ! l9e z a curve ( ! fir/ ord]1 we h only ! po9t & ! curve4 ,9 solid geometry1 reckon+ a l9e z a curve ( ! fir/ ord]1 & ! plane z a surface ( ! fir/ ord]1 we h ! po9t1 ! curve & ! surface2 b ! increase ( complexity is f> great] ?an w\ld h5ce at fir/ sight appe>4 ,9 plane geometry a curve is consid]$ 9 connexion ) l9es 7its tang5ts72 b 9 solid geometry ! curve is consid]$ 9 connexion ) l9es & planes 7its tang5ts & osculat+ planes71 & ! surface also 9 connexion ) l9es & planes 7its tang5t l9es & tang5t planes72 !re are surfaces aris+ out ( ! l9e--cones1 skew surfaces1 developables1 d\bly & triply inf9ite sy/ems ( l9es1 & :ole classes ( !ories :i* h no?+ analog\s to !m 9 plane geometry3 x is ?us a v small "p inde$ ( ! subject :i* c be ev5 ref]r$ to 9 ! pres5t article4 ,9 ! case ( a surface we h betwe5 ! coord9ates 7x1 y1 z7 a s+le1 or say a onefold relation1 :i* c be repres5t$ by a s+le relation f7x1 y1 z7 #j2 or we may consid] ! coord9ates express$ ea* ( !m z a giv5 function ( two v>iable p>amet]s p1 q2 ! =m z f7x1 y7 is a p>ticul> case ( ea* ( ^! modes ( repres5tation2 9 o!r words1 we h 9 ! fir/ mode f7x1 y1 z7 z - f7x1 y71 & 9 ! second mode x p1 y q = ! expression ( two ( ! coord9ates 9 t]ms ( ! p>amet]s4 ,9 ! case ( a curve we h betwe5 ! coord9ates 7x1 y1 z7 a tw(old relation3 two equations f7x1 y1 z7 #j1 phi7x1 y1 z7 #j give su* a relation2 i4e4 ! curve is "h consid]$ z ! int]section ( two surfaces 7b ! curve is n always ! complete int]section ( two surfaces1 & !re are h5ce difficulties72 or1 aga91 ! coord9ates may be giv5 ea* ( !m z a function ( a s+le v>iable p>amet]4 ,! =m y phi7x71 z psi7x71 :]e two ( ! coord9ates are giv5 9 t]ms ( ! ?ird1 is a p>ticul> case ( ea* ( ^! modes ( repres5tation4 #bi4 ,! rem>ks "u plane geometry z to descriptive & metrical propositions1 & z to ! non-metrical "* ( ! me?od ( coord9ates :5 us$ = ! pro( ( a descriptive proposition1 apply also to solid geometry2 & !y might be illu/rat$ 9 l mann] by ! in/ance ( ! !orem ( ! radical c5tre ( f\r sph]es4 ,! pro( is obta9$ f ! consid]ation t ,s & ,s' be+ ea* ( !m a function ( ! =m x#b y#b z#b ax by cz d1 ! diff]5ce ,s-,s' is a m]e l9e> function ( ! coord9ates1 & consequ5tly t ,s-,s' #j is ! equation ( ! plane conta9+ ! circle ( int]section ( ! two sph]es ,s #j & ,s' #j4 ,illu/ration3 ,,fig4 #fj4 #cj4 ,metrical ,!ory4--,! f\ndation 9 solid geometry ( ! metrical !ory is 9 fact ! be=e-m5tion$ !orem t if a f9ite "r l9e ,,pq be project$ ^u any o!r l9e ,,oo' by l9es p]p5dicul> to ,,oo'1 !n ! l5g? ( ! projection ,p',q' is equal to ! l5g? ( ,,pq into ! cos9e ( its incl9ation to ,p',q'--or 79 ! =m 9 :i* x is n[ conv5i5t to /ate ! !orem7 ! p]p5dicul> di/ance ,p',q' ( two p>allel planes is equal to ! incl9$ di/ance ,,pq into ! cos9e ( ! incl9ation4 ,! pr9ciple ( ,s #af1 t ! algebraical sum ( ! projections ( ! sides ( any clos$ polygon on any l9e is z]o1 or t ! two sets ( sides ( ! polygon :i* connect a v]tex ,a & a v]tex ,b h ! same sum ( projections on ! l9e1 9 sign & magnitude1 z we pass f ,a to ,b1 is applicable :5 ! sides d n all lie 9 "o plane4 #ca4 ,consid] ! skew quadrilat]al ,,qmnp1 ! sides ,,qm1 ,,mn1 ,,np be+ respectively p>allel to ! ?ree rectangul> axes ,ox1 ,oy1 ,oz2 let ! l5g?s ( ^! sides be xi1 eta1 zeta1 & t ( ! side ,,qp be rho2 & let ! cos9es ( ! incl9ations 7or say ! cos9e-incl9ations7 ( rho to ! ?ree axes be alpha1 beta1 gamma2 !n project+ successively on ! ?ree sides & on ,,qp we h xi1 eta1 zeta rhoalpha1 rhobeta1 rhogamma1 & rho alphaxi betaeta gammazeta1 :5ce rho#b xi#b eta#b zeta#b1 :i* is ! relation betwe5 a di/ance rho & its projections xi1 eta1 zeta ^u ?ree rectangul> axes4 ,& f ! same equations we obta9 alpha#b beta#b gamma#b #a1 :i* is a relation connect+ ! cos9e-incl9ations ( a l9e to ?ree rectangul> axes4 ,suppose we h ?r\gh ,q any o!r l9e ,,qt1 & let ! cos9e-incl9ations ( ?is to ! axes be alpha'1 beta'1 gamma'1 & delta be its cos9e-incl9ation to ,,qp2 also let rho be ! l5g? ( ! projection ( ,,qp ^u ,,qt2 !n project+ on ,,qt we h rho alpha'xi beta'eta gamma'zeta rhodelta4 ,& 9 ! la/ equation sub/itut+ = xi1 eta1 zeta _! values rhoalpha1 rhobeta1 rhogamma we f9d delta alphaalpha' betabeta' gammagamma'1 :i* is an expression = ! mutual cos9e-incl9ation ( two l9es1 ! cos9e-incl9ations ( :i* to ! axes are alpha1 beta1 gamma & alpha'1 beta'1 gamma' respectively4 ,we h ( c\rse alpha#b beta#b gamma#b #a & alpha'#b beta'#b gamma'#b #a2 & h5ce also #a - delta#b 7alpha#b beta#b gamma#b77;alpha'#b beta'#b gamma'#b7 - 7alphaalpha' betabeta' gammagamma'7#b1 7betagamma' - beta'gamma7#b 7gammaalpha' - gamma'alpha7#b 7alphabeta' - alpha'beta7#b2 s t ! s9e ( ! incl9ation c only be express$ z a squ>e root4 ,^! =mulae are ! f\ndation ( sph]ical trigonometry4 #cb4 ,/raight ,l9es1 ,planes & ,sph]es4--,! =ego+ =mulae give at once ! equations ( ^! loci4 ,= fir/1 tak+ ,q to be a fix$ po9t1 coord9ates 7a1 b1 c71 & ! cos9e-incl9ations 7alpha1 beta1 gamma7 to be con/ant1 !n ,p w be a po9t 9 ! l9e ?r\gh ,q 9 ! direction ?us det]m9$2 or1 tak+ 7x1 y1 z7 = its coord9ates1 ^! w be ! curr5t coord9ates ( a po9t 9 ! l9e4 ,! values ( xi1 eta1 zeta !n are x - a1 y - b1 z - c1 & we ?us h x - a y - b z - c ------- ----- ------- 7 rho71 alpha beta gamma :i* 7omitt+ ! la/ equation1 rho7 are ! equations ( ! l9e ?r\gh ! po9t 7a1 b1 c71 ! cos9e-incl9ations to ! axes be+ alpha1 beta1 gamma1 & ^! quantities be+ connect$ by ! relation alpha#b beta#b gamma#b #a4 ,?is equation may be omitt$1 & !n alpha1 beta1 gamma1 in/ead ( be+ equal1 w only be proportional1 to ! cos9e-incl9ations4 ,us+ ! la/ equation1 & writ+ x1 y1 z a alpharho1 b betarho1 c gammarho1 ^! are expressions = ! curr5t coord9ates 9 t]ms ( a p>amet] rho1 :i* is 9 fact ! di/ance f ! fix$ po9t 7a1 b1 c74 ,x is easy to see t1 if ! coord9ates 7x1 y1 z7 are connect$ by any two l9e> equations1 ^! equations c always be br\ght into ! =ego+ =m1 & h5ce t ! two l9e> equations repres5t a l9e4 ,secondly1 tak+ = great] simplicity ! po9t ,q to be co9cid5t ) ! orig91 & alpha'1 beta'1 gamma'1 p to be con/ant1 !n p is ! p]p5dicul> di/ance ( a plane f ! orig91 & alpha'1 beta'1 gamma' are ! cos9e-incl9ations ( ?is di/ance to ! axes 7alpha'#b beta'#b gamma'#b #a74 ,p is any po9t 9 ?is plane1 & tak+ its coord9ates to be 7x1 y1 z7 !n 7xi1 eta1 zeta7 are 7x1 y1 z71 & ! =ego+ equation p alpha'xi beta'eta gamma'zeta becomes alpha'x beta'y gamma'z p1 :i* is ! equation ( ! plane 9 "q4 ,if1 m g5]ally1 ,q is n co9cid5t ) ! orig91 !n1 tak+ its coord9ates to be 7a1 b1 c71 & writ+ p#a in/ead ( p1 ! equation is alpha'7x - a7 beta'7y - b7 gamma'7z - c7 p#a2 & we !nce h p#a p - 7aalpha' bbeta' cgamma'71 :i* is an expression = ! p]p5dicul> di/ance ( ! po9t 7a1 b1 c7 f ! plane 9 "q4 ,x is obvi\s t any l9e> equation ,ax ,by ,cz ,d ,o betwe5 ! coord9ates c always be br\ght into ! =ego+ =m1 & h5ce t su* an equation repres5ts a plane4 ,?irdly1 suppos+ ,q to be a fix$ po9t1 coord9ates 7a1 b1 c71 & ! di/ance ,,qp rho1 to be con/ant1 say ?is is d1 !n1 z be=e1 ! values ( xi1 eta1 zeta are x - a1 y - b1 z - c1 & ! equation xi#b eta#b zeta#b rho#b becomes 7x - a7#b 7y - b7#b 7z - c7#b d#b1 :i* is ! equation ( ! sph]e1 coord9ates ( ! c5tre 7a1 b1 c71 & radius d4 ,a quadric equation :]e9 ! t]ms ( ! second ord] are x#b y#b z#b1 viz4 an equation x#b y#b z#b ,ax ,by ,cz ,d #j1 c always1 x is cle>1 be br\ght into ! =ego+ =m2 & x ?us appe>s t ?is is ! equation ( a sph]e1 coord9ates ( ! c5tre -#a_/b;,a1 -#a_/b;,b1 -#a_/b;,c1 & squ>$ radius #a_/d7;,a#b ,b#b ,c#b7 - ,d4 #cc4 ,cyl9d]s1 ,cones1 rul$ ,surfaces4--,if ! two equations ( a /raight l9e involve a p>amet] to :i* any value may be giv51 we h a s+ly inf9ite sy/em ( l9es4 ,!y cov] a surface1 & ! equation ( ! surface is obta9$ by elim9at+ ! p>amet] betwe5 ! two equations4 ,if ! l9es all pass ?r\gh a giv5 po9t1 !n ! surface is a cone2 &1 9 p>ticul>1 if ! l9es are all p>allel to a giv5 l9e1 !n ! surface is a cyl9d]4 ,beg9n+ ) ?is la/ case1 suppose ! l9es are p>allel to ! l9e x mz1 y nz1 ! equations ( a l9e ( ! sy/em are x mz a1 y nz b1--:]e a1 b are suppos$ to be functions ( ! v>iable p>amet]1 or1 :at is ! same ?+1 !re is betwe5 !m a relation f7a1 b7 #j3 we h a x - mz1 b y - nz1 & ! result ( ! elim9ation ( ! p>amet] !re=e is f7x - mz1 y - nz7 #j1 :i* is ?us ! g5]al equation ( ! cyl9d] ! g5]at+ l9es :]e( are p>allel to ! l9e x mz1 y nz4 ,! equation ( ! section by ! plane z #j is f7x1 y7 #j1 & conv]sely if ! cyl9d] be det]m9$ by means ( its curve ( int]section ) ! plane z #j1 !n1 tak+ ! equation ( ?is curve to be f7x1 y7 #j1 ! equation ( ! cyl9d] is f7x - mz1 y - nz7 #j4 ,?us1 if ! curve ( int]section be ! circle 7x - alpha7#b 7y - beta7#b gamma#b1 we h 7x - mz - alpha7#b 7y - nz - beta7#b gamma#b z ! equation ( an oblique cyl9d] on ?is base1 & ?us also 7x - alpha7#b 7y - beta7#b gamma#b z ! equation ( ! "r cyl9d]4 ,if ! l9es all pass ?r\gh a giv5 po9t 7a1 b1 c71 !n ! equations ( a l9e are x - a alpha7z - c71 y - b beta7z - c71 :]e alpha1 beta are functions ( ! v>iable p>amet]1 or1 :at is ! same ?+1 !re exi/s betwe5 !m an equation f7alpha1 beta7 #j2 ! elim9ation ( ! p>amet] gives1 !re=e1 f7x - a7_/7x - c'71 7y - b7_/7z - c7 #j2 & ?is equation1 or1 :at is ! same ?+1 any homog5e\s equation f7x - a1 y - b1 z - c7 #j1 or1 tak+ f to be a rational & integral function ( ! ord] n1 say 79977x - a1 y - b1 z - c7n #j1 is ! g5]al equation ( ! cone hav+ ! po9t 7a1 b1 c7 = its v]tex4 ,tak+ ! v]tex to be at ! orig91 ! equation is 79977x1 y1 z7n #j2 &1 9 p>ticul>1 79977x1 y1 z7#b #j is ! equation ( a cone ( ! second ord]1 or quadricone1 hav+ ! orig9 = its v]tex4 #cd4 ,9 ! g5]al case ( a s+ly inf9ite sy/em ( l9es1 ! locus is a rul$ surface 7or regulus74 ,n[1 :5 a l9e is *ang+ its position 9 space1 x may be look$ ^u z 9 a /ate ( turn+ ab\t "s po9t 9 itself1 :ile t po9t is1 z a rule1 9 a /ate ( mov+ out ( ! plane 9 :i* ! turn+ takes place4 ,if in/antane\sly x is only 9 a /ate ( turn+1 x is usual1 ?\gh n /rictly accurate1 to say t x int]sects its consecutive position4 ,a regulus su* t consecutive l9es on x d n int]sect1 9 ?is s5se1 is call$ a skew surface1 or scroll2 "o on :i* !y d is call$ a developable surface or torse4 ,suppose1 = in/ance1 t ! equations ( a l9e 7dep5d+ on ! v>iable p>amet] !ta7 are x_/a y_/c !ta 7#a y_/b71 x_/a - z_/c #a_/!ta 7#a - y_/b72 !n1 elim9at+ !ta we h x#b_/;a#b - z#b_/;c#b #a - y#b_/;b#b1 or say1 x#b_/;a#b z#b_/;b#b - z#b_/;c#b #a1 ! equation ( a quadric surface1 aft]w>ds call$ ! hyp]boloid ( "o %eet2 ?is surface is consequ5tly a scroll4 ,x is to be rem>k$ t we h ^u ! surface a second s+ly inf9ite s]ies ( l9es2 ! equations ( a l9e ( ?is second sy/em 7dep5d+ on ! v>iable p>amet] phi7 are x z _/ y x z #a _/ y -- -- phi7 #a - -- 71 -- - -- ----- 7 #a -- 74 a c b _/ a c phi b _/ ,x is easily %[n t any l9e ( ! "o sy/em int]sects e l9e ( ! o!r sy/em4 ,consid]+ any curve 7( d\ble curvature7 :atev]1 ! tang5t l9es ( ! curve =m a s+ly inf9ite sy/em ( l9es1 ea* l9e int]sect+ ! consecutive l9e ( ! sy/em1--t is1 !y =m a developable1 or torse2 ! curve & torse are ?us insep>ably connect$ toge!r1 =m+ a s+le geometrical figure4 ,an osculat+ plane ( ! curve 7see ,s #ch bel[7 is a tang5t plane ( ! torse all along a g5]at+ l9e4 #ce4 ,trans=mation ( ,coord9ates4--,!re is no difficulty 9 *ang+ ! orig91 & x is = brevity assum$ t ! orig9 rema9s unalt]$4 ,we h1 !n1 two sets ( rectangul> axes1 ,ox1 ,oy1 ,oz1 & ,ox#a1 ,oy#a1 ,ozx#a1 ! mutual cos9e-incl9ations be+ %[n by ! diagram-- x y z ------------------------------ x#a alpha beta gamma ------------------------------ y#a alpha beta' gamma' ------------------------------ z#a alpha beta gamma ------------------------------ t is1 alpha1 beta1 gamma are ! cos9e-incl9ations ( ,ox#a to ,ox1 ,oy1 ,oz2 alpha'1 beta'1 gamma' ^? ( ,oy#a1 c4 ,& ?is diagram gives also ! l9e> expressions ( ! coord9ates 7x#a1 y#a1 z#a7 or 7x1 y1 z7 ( ei!r set 9 t]ms ( ^? ( ! o!r set2 we ?us h x#a alpha x beta y gamma z1 x alphax#a alpha'y#a alphaz#a1 y#a alpha'x beta'y gamma'z1 y betax#a beta'y#a betaz#a1 z#a alphax betay gammaz1 z gammax#a gamma'y#a gammaz#a1 :i* are obta9$ by projection1 z above expla9$4 ,ea* ( ^! equations is1 9 fact1 no?+ else ?an ! be=e-m5tion$ equation p alpha'xi beta'eta gamma'zeta1 adapt$ to ! problem 9 h&4 ,b we h to consid] ! relations betwe5 ! n9e coeffici5ts4 ,by :at prec$es1 or by ! consid]ation t we mu/ h id5tically x#b y#b z#b x#ab y#ab z#ab1 x appe>s t ^! satisfy ! relations-- a#b beta#b gamma#b #a1 alpha#b alpha'#b alpha#b #a1 alpha'#b beta'#b gamma'#b #a1 beta#b beta'#b beta#b #a1 alpha#b beta#b gamma#b #a1 gamma#b gamma'#b gamma#b #a1 a'a beta'beta gamma'gamma #j1 betagamma beta'gamma' betagamma #j1 alphaalpha betabeta gammagamma #j1 gammaalpha gamma'alpha' gammaalpha #j1 alphaalpha' betabeta' gammagamma' #j1 alphabeta alpha'beta' alphabeta #j1 ei!r set ( six equations be+ impli$ 9 ! o!r set4 ,x foll[s t ! squ>e ( ! det]m9ant alpha1 beta1 gamma alpha'1 beta'1 gamma' alpha1 beta1 gamma is #a2 & h5ce t ! det]m9ant itself is - #a4 ,! di/9ction ( ! two cases is an important "o3 if ! det]m9ant is #a1 !n ! axes ,ox#a1 ,oy#a1 ,oz#a are su* t !y c by a rotation ab\t ,o be br\ght to co9cide ) ,ox1 ,oy1 ,oz respectively2 if x is -#a1 !n !y _c4 ,b 9 ! latt] case1 by measur+ x#a1 y#a1 z#a 9 ! opposite directions we *ange ! signs ( all ! coeffici5ts & s make ! det]m9ant to be #a2 h5ce ! =m] case ne$ alone be consid]$1 & x is accord+ly assum$ t ! det]m9ant is #a4 ,?is be+ s1 x is f\nd t we h ! equality alpha beta'gamma - betagamma'1 & eight l ones1 obta9$ f ?is by cyclical int]*anges ( ! lett]s alpha1 beta1 gamma1 & ( unacc5t$1 s+ly & d\bly acc5t$ lett]s4 #cf4 ,! n9e cos9e-incl9ations above are1 z has be5 se51 connect$ by six equations4 ,x ought !n to be possible to express !m all 9 t]ms ( ?ree p>amet]s4 ,an elegant means ( do+ ?is has be5 giv5 by ,rodrigues1 :o has %[n t ! tabul> expression ( ! =mulae ( trans=mation may be writt5 x y z ---------------------------------------- ---------------------------------------- -------------------- x#a #a lambda#b - mu#b - nu#b #b7lambdamu - nu7 #b7nulambda mu7 ---------------------------------------- ---------------------------------------- ------------------- y#a #b7lambdamu nu7 #a - lambda#b mu#b - nu#b #b7munu lambda7 ---------------------------------------- ---------------------------------------- ------------------- z#a #b7nulambda - mu7 #b7munu lambda7 #a - lambda#b - mu#b nu#b ---------------------------------------- ---------------------------------------- -------------------- _/7#a lambda#b mu#b nu#b71 ! mean+ be+ t ! coeffici5ts 9 ! trans=mation are fractions1 ) num]ators express$ z 9 ! table1 & ! common d5om9ator4 #cg4 ,! ,species ( ,quadric ,surfaces4--,surfaces repres5t$ by equations ( ! second degree are call$ quadric surfaces4 ,quadric surfaces are ei!r prop] or special4 ,! special ones arise :5 ! coeffici5ts 9 ! g5]al equation are limit$ to satisfy c]ta9 special equations2 !y comprise 7#a7 plane-pairs1 includ+ 9 p>ticul> "o plane twice repeat$1 & 7#b7 cones1 includ+ 9 p>ticul> cyl9d]s2 !re is b "o =m ( cone1 b cyl9d]s may be elliptic1 p>abolic or hyp]bolic4 ,a discussion ( ! g5]al equation ( ! second degree %[s t ! prop] quadric surfaces are ( five k9ds1 repres5t$ respectively1 :5 ref]r$ to ! mo/ conv5i5t axes ( ref]5ce1 by equations ( ! five types 7a & b positive73 x#b y#b 7#a7 z --- ---1 elliptic p>aboloid4 #b;a #b;b x#b y#b 7#b7 z --- - ---1 hyp]bolic p>aboloid4 #b;a #b;b x#b y#b z#b 7#c7 --- --- --- #a1 ellipsoid4 a#b b#b c#b x#b y#b z#b 7#d7 --- --- - --- #a1 hyp]boloid ( "o %eet4 a#b b#b c#b x#b y#b z#b 7#e7 --- --- - --- -#a1 hyp]boloid ( two %eets4 a#b b#b c#b ,x is at once se5 t ^! are di/9ct surfaces2 & ! equations also %[ v readily ! g5]al =m & mode ( g5]ation ( ! sev]al surfaces4 ,illu/ration3 ,,fig4 #fa4 ,9 ! elliptic p>aboloid 7fig4 #fa7 ! sections by ! planes ( zx & zy are ! p>abolas x#b y#b z ---1 z --- #b;a #b;b hav+ ! common axes ,oz2 & ! section by any plane z gamma p>allel to t ( xy is ! ellipse x#b y#b gamma --- ---2 #b;a #b;b s t ! surface is g5]at$ by a v>iable ellipse mov+ p>allel to itself along ! p>abolas z directrices4 ,illu/ration3 ,,fig4 #fb4 ,illu/ration3 ,,fig4 #fc4 ,9 ! hyp]bolic p>aboloid 7figs4 #fb & #fc7 ! sections by ! planes ( zx1 zy are ! p>abolas z x#b_/b;a1 z - y#b_/b;b1 hav+ ! opposite axes ,oz1 ,oz'1 & ! section by a plane z gamma p>allel to t ( xy is ! hyp]bola gamma x#b_/b;a - y#b_/b;b1 :i* has its transv]se axis p>allel to ,ox or ,oy accord+ z gamma is positive or negative4 ,! surface is ?us g5]at$ by a v>iable hyp]bola mov+ p>allel to itself along ! p>abolas z directrices4 ,! =m is be/ se5 f fig4 #fc1 :i* repres5ts ! sections by planes p>allel to ! plane ( xy1 or say ! cont\r l9es2 ! cont9u\s l9es are ! sections above ! plane ( xy1 & ! dott$ l9es ! sections bel[ ?is plane4 ,! =m is1 9 fact1 t ( a saddle4 ,illu/ration3 ,,fig4 #fd4 ,9 ! ellipsoid 7fig4 #fd7 ! sections by ! planes ( zx1 zy1 & xy are ea* ( !m an ellipse1 & ! section by any p>allel plane is also an ellipse4 ,! surface may be consid]$ z g5]at$ by an ellipse mov+ p>allel to itself along two ellipses z directrices4 ,9 ! hyp]boloid ( "o %eet 7fig4 #fe71 ! sections by ! planes ( zx1 zy are ! hyp]bolas x#b z#b y#b z#b --- - --- #a1 --- - --- #a1 c#b c#b b#b c#b hav+ a common conjugate axis z,oz'2 ! section by ! plane ( x1 y1 & t by any p>allel plane1 is an ellipse2 & ! surface may be consid]$ z g5]at$ by a v>iable ellipse mov+ p>allel to itself along ! two hyp]bolas z directrices4 ,if we imag9e two equal & p>allel circul> disks1 _! po9ts connect$ by /r+s ( equal l5g?s1 s t ^! are ! g5]ators ( a "r circul> cyl9d]1 & if we turn "o ( ! disks ab\t its c5tre ?r\gh an angle 9 its plane1 ! /r+s 9 _! new positions w be "o sy/em ( g5]ators ( a hyp]boloid ( "o %eet1 = :i* a b2 & if we turn x ?r\gh ! same angle 9 ! opposite direction1 we get 9 l mann] ! g5]ators ( ! o!r sy/em2 !re w be ! same g5]al configuration :5 a b4 ,! hyp]bolic p>aboloid is also cov]$ by two sy/ems ( rectil9e> g5]ators z a me?od l t us$ 9 ,s #cd e/abli%es )\t difficulty4 ,! figures %\ld be /udi$ to see h[ !y c lie4 ,illu/ration3 ,,fig4 #fe4 ,illu/ration3 ,,fig4 #ff4 ,9 ! hyp]boloid ( two %eets 7fig4 #ff7 ! sections by ! planes ( zx & zy are ! hyp]bolas z#b x#b z#b y#b --- - --- #a1 --- - --- #a1 c#b a#b c#b b#b hav+ a common transv]se axis along z',oz2 ! section by any plane z -gamma p>allel to t ( xy is ! ellipse x#b y#b gamma#b --- --- --------- - #a1 a#b b#b c#b provid$ gamma#b c#b1 & ! surface1 consi/+ ( two di/9ct portions or %eets1 may be consid]$ z g5]at$ by a v>iable ellipse mov+ p>allel to itself along ! hyp]bolas z directrices4 #ch4 ,diff]5tial ,geometry ( ,curves4--,= conv5i5ce consid] ! coord9ates 7x1 y1 z7 ( a po9t on a curve 9 space to be giv5 z functions ( a v>iable p>amet] !ta1 :i* may 9 p>ticul> be "o ( !mselves4 ,use ! notation x'1 x = dx_/d!ta1 d#bx_/d!ta#b1 & simil>ly z to y & z4 ,only a few =mulae w be giv54 ,call ! curr5t coord9ates 7xi1 eta1 zeta74 ,! tang5t at 7x1 y1 z7 is ! l9e t5d$ to z a limit by ! connector ( 7x1 y1 z7 & a neighb\r+ po9t ( ! curve :5 ! latt] moves up to ! =m]3 its equations are 7xi - x7_/x' 7eta - y7_/y' 7zeta - z7_/z'4 ,! osculat+ plane at 7x1 y1 z7 is ! plane t5d$ to z a limit by t ?r\gh 7x1 y1 z7 & two neighb\r+ po9ts ( ! curve z ^!1 rema9+ di/9ct1 bo? move up to 7x1 y1 z73 its "o equation is 7xi - x77y'z - yz'7 7eta - y77z'x - zx'7 7zeta - z7 7x'y - xy'7 #j4 ,! normal plane is ! plane ?r\gh 7x1 y1 z7 at "r angles to ! tang5t l9e1 i4e4 ! plane x'7xi - x7 y'7eta - y7 z'7zeta - z7 #j4 ,x cuts ! osculat+ plane 9 a l9e call$ ! pr9cipal normal4 ,e l9e ?r\gh 7x1 y1 z7 9 ! normal plane is a normal4 ,! normal p]p5dicul> to ! osculat+ plane is call$ ! b9ormal4 ,a tang5t1 pr9cipal normal1 & b9ormal are a conv5i5t set ( rectangul> axes to use z ^? ( ref]5ce1 :5 ! nature ( a curve ne> a po9t on x is to be discuss$4 ,?r\gh 7x1 y1 z7 & ?ree neighb\r+ po9ts1 all on ! curve1 passes a s+le sph]e2 & z ! ?ree po9ts all move up to 7x1 y1 z7 cont9u+ di/9ct1 ! sph]e t5ds to a limit+ size & position4 ,! limit t5d$ to is ! sph]e ( close/ contact ) ! curve at 7x1 y1 z72 its c5tre & radius are call$ ! c5tre & radius ( sph]ical curvature4 ,x cuts ! osculat+ plane 9 a circle1 call$ ! circle ( absolute curvature2 & ! c5tre & radius ( ?is circle are ! c5tre & radius ( absolute curvature4 ,! c5tre ( absolute curvature is ! limit+ position ( ! po9t :]e ! pr9cipal normal at 7x1 y1 z7 is cut by ! normal plane at a neighb\r+ po9t1 z t po9t moves up to 7x1 y1 z74 #ci4 ,diff]5tial ,geometry ( ,surfaces4--,let 7x1 y1 z7 be any *os5 po9t on a surface f7x1 y1 z7 #j4 ,z a second po9t ( ! surface moves up to 7x1 y1 z71 its connector ) 7x1 y1 z7 t5ds to a limit+ position1 a tang5t l9e to ! surface at 7x1 y1 z74 ,all ^! tang5t l9es at 7x1 y1 z71 obta9$ by approa*+ 7x1 y1 z7 f diff]5t directions on a surface1 lie 9 "o plane d,pf d,pf d,pf ----- 7xi - x7 ----- 7eta - y7 ----- 7zeta - z7 #j4 d,px d,py d,pz ,?is plane is call$ ! tang5t plane at 7x1 y1 z74 ,"o l9e ?r\gh 7x1 y1 z7 is at "r angles to ! tang5t plane4 ,?is is ! normal _/d,pf _/d,pf _/d,pf 7xi - x7 _/ ----- 7eta - y7 _/ ----- 7zeta - z7 _/ -----4 _/ d,px _/ d,py _/ d,pz ,! tang5t plane is cut by ! surface 9 a curve1 real or imag9>y1 ) a node or d\ble po9t at 7x1 y1 z74 ,two ( ! tang5t l9es t\* ?is curve at ! node4 ,!y are call$ ! *ief tang5ts 7,haupt-tang5t57 at 7x1 y1 z72 !y h clos] contact ) ! surface ?an any o!r tang5ts4 ,9 ! case ( a quadric surface ! curve ( int]section ( a tang5t & ! surface is ( ! second ord] & has a node1 x mu/ !re=e consi/ ( two /raight l9es4 ,consequ5tly a quadric surface is cov]$ by two sets ( /raight l9es1 a pair ?r\gh e po9t on x2 ^! are imag9>y = ! ellipsoid1 hyp]boloid ( two %eets1 & elliptic p>aboloid4 ,a surface ( any ord] is cov]$ by two s+ly inf9ite sy/ems ( curves1 a pair ?r\gh e po9t1 ! tang5ts to :i* are all *ief tang5ts at _! respective po9ts ( contact4 ,^! are call$ *ief-tang5t curves2 on a quadric surface !y are ! above /raight l9es4 #dj4 ,! tang5ts at a po9t ( a surface :i* bisect ! angles betwe5 ! *ief tang5ts are call$ ! pr9cipal tang5ts at ! po9t4 ,!y are at "r angles1 & toge!r ) ! normal con/itute a conv5i5t set ( rectangul> axes to :i* to ref] ! surface :5 its prop]ties ne> ! po9t are "u discussion4 ,at a special po9t :i* is su* t ! *ief tang5ts !re run to ! circul> po9ts at inf9ity 9 ! tang5t plane1 ! pr9cipal tang5ts are indet]m9ate2 su* a special po9t is call$ an umbilic ( ! surface4 ,!re are two s+ly inf9ite sy/ems ( curves on a surface1 a pair cutt+ "o ano!r at "r angles ?r\gh e po9t ^u x1 all tang5ts to :i* are pr9cipal tang5ts ( ! surface at _! respective po9ts ( contact4 ,^! are call$ l9es ( curvature1 because ( a prop]ty next to be m5tion$4 ,z a po9t ,q moves 9 an arbitr>y direction on a surface f co9cid5ce ) a *os5 po9t ,p1 ! normal at x1 z a rule1 at once fails to meet ! normal at ,p2 b1 if x takes ! direction ( a l9e ( curvature ?r\gh ,p1 ?is is in/antane\sly n ! case4 ,we h ?us on ! normal two c5tres ( curvature1 & ! di/ances ( ^! f ! po9t on ! surface are ! two pr9cipal radii ( curvature ( ! surface at t po9t2 ^! are also ! radii ( curvature ( ! sections ( ! surface by planes ?r\gh ! normal & ! two pr9cipal tang5ts respectively2 or say !y are ! radii ( curvature ( ! normal sections ?r\gh ! two pr9cipal tang5ts respectively4 ,take at ! po9t ! axis ( z 9 ! direction ( ! normal1 & ^? ( x & y 9 ! directions ( ! pr9cipal tang5ts respectively1 !n1 if ! radii ( curvature be a1 b 7! signs be+ su* t ! coord9ates ( ! two c5tres ( curvature are z a & z b respectively71 ! surface has 9 ! neighb\rhood ( ! po9t ! =m ( ! p>aboloid x#b y#b z --- ---1 #b;a #b;b & ! *ief-tang5ts are det]m9$ by ! equation #j x#b_/b;a y#b_/b;b4 ,! two c5tres ( curvature may be on ! same side ( ! po9t or on opposite sides2 9 ! =m] case a & b h ! same sign1 ! p>aboloid is elliptic1 & ! *ief-tang5ts are imag9>y2 9 ! latt] case a & b h opposite signs1 ! p>aboloid is hyp]bolic1 & ! *ief-tang5ts are real4 ,! normal sections ( ! surface & ! p>aboloid by ! same plane h ! same radius ( curvature2 & x !nce readily foll[s t ! radius ( curvature ( a normal section ( ! surface by a plane incl9$ at an angle !ta to t ( zx is giv5 by ! equation #a cos#b !ta s9#b !ta ----- ------------- -------------4 rho a b ,! section 9 "q is t by a plane ?r\gh ! normal & a l9e 9 ! tang5t plane incl9$ at an angle !ta to ! pr9cipal tang5t along ! axis ( x4 ,to complete ! !ory1 consid] ! section by a plane hav+ ! same trace ^u ! tang5t plane1 b incl9$ to ! normal at an angle phi2 !n x is %[n )\t difficulty 7,meuni]'s !orem7 t ! radius ( curvature ( ?is incl9$ section ( ! surface is rho cos phi4 ,,au?orities4--,! above article is l>gely bas$ on t by ,ar?ur ,cayley 9 ! #i? edition ( ?is "w4 ,( e>ly & important rec5t publications on analytical geometry1 special m5tion is to be made ( ,r4 ,desc>tes1 ,geometrie 7,leyd51 #afcg72 ,john ,wallis1 ,tractatus de sectionibus conicis nova me?odo expositis 7#afee1 ,op]a ma!matica1 i41 ,ox=d1 #afie72 de l',hospital1 ,traite analytique des sections coniques 7,p>is1 #agbj72 ,leonh>d ,eul]1 ,introductio 9 analys9 inf9itorum1 ii4 7,lausanne1 #agdh72 ,gasp>d ,monge1 ,application d'algebre a la geometrie 7,j\rn4 ,ecole ,polyte*41 #ahja72 ,julius ,pluck]1 ,analytis*-geometris*e ,entwickelung51 #c ,bde4 7,ess51 #ahbh-ahca72 ,sy/em d] analytis*5 ,geometrie 7,b]l91 #ahce72 ,g4 ,salmon1 ,a ,treatise on ,conic ,sections 7,dubl91 #ahdh2 #f? ed41 ,london1 #ahgi72 ,*4 ,briot & ,j4 ,b\quet1 ,lecons de geometrie analytique 7,p>is1 #ahea2 #af? ed41 #ahig72 ,m4 ,*asles1 ,traite de geometrie sup]ieure 7,p>is1 #aheb72 ,wilhelm ,fi$l]1 ,analytis*e ,geometrie d] ,kegels*nitte na* ,g4 ,salmon frei be>beitet 7,leipzig1 #ete ,aufl41 #ahhg-ahhh72 ,n4,m4 ,f]r]s1 ,an ,elem5t>y ,treatise on ,tril9e> ,coord9ates 7,london1 #ahfa72 ,otto ,hesse1 ,vorlesung5 aus d] analytis*5 ,geometrie 7,leipzig1 #ahfe1 #ahha72 ,w4,a4 ,:itwor?1 ,tril9e> ,coord9ates & o!r ,me?ods ( ,mod]n ,analytical ,geometry 7,cambridge1 #ahff72 ,j4 ,boo?1 ,a ,treatise on ,"s ,new ,geometrical ,me?ods 7,london1 i41 #ahgc2 ii41 #ahgg72 ,a4 ,clebs*-,f4 ,l9demann1 ,vorlesung5 ub] ,geometrie1 ,bd4 i4 7,leipzig1 #ahgf1 #bte ,aufl41 #ahia72 ,r4 ,balts]1 ,analytis*e ,geometrie 7,leipzig1 #ahhb72 ,*>lotte ,a4 ,scott1 ,mod]n ,me?ods ( ,analytical ,geometry 7,london1 #ahid72 ,g4 ,salmon1 ,a ,treatise on ! ,analytical ,geometry ( ?ree ,dim5sions 7,dubl91 #ahfb2 #d? ed41 #ahhb72 ,salmon-,fi$l]1 ,analytis*e ,geometrie des ,raumes 7,leipzig1 #ahfc2 #dte ,aufl41 #ahih72 ,p4 ,fro/1 ,solid ,geometry 7,london1 #crd ed41 #ahhf2 #a/ ed41 ,fro/ & ,j4 ,wol/5holme74 ,see also ,e4 ,pascal1 ,rep]torio di matemati*e sup]iori1 ,,ii4 ,geometria 7,milan1 #aijj71 & articles n[ appe>+ 9 ! ,encyklopadie d] ma!matis*5 ,wiss5s*aft51 ,bd4 iii4 #a1 #b4 7,e4 ,b4 ,el47 ,v4 ,,l9e ,,geometry ,l9e geometry is ! "n appli$ to ^? geometrical inve/igations 9 :i* ! /raight l9e replaces ! po9t z elem5t4 ,j z ord9>y geometry deals prim>ily ) po9ts & sy/ems ( po9ts1 ?is !ory deals 9 ! fir/ in/ance ) /raight l9es & sy/ems ( /raight l9es4 ,9 two dim5sions !re is no necessity = a special l9e geometry1 inasmu* z ! /raight l9e & ! po9t are int]*angeable by ! pr9ciple ( duality2 b 9 ?ree dim5sions ! /raight l9e is its own reciprocal1 & = ! bett] discussion ( sy/ems ( l9es we require "s new app>atus1 e4g41 a sy/em ( coord9ates applicable to /raight l9es r ?an to po9ts4 ,! ess5tial features ( ! subject are mo/ easily elucidat$ by analytical me?ods3 we %all !re=e beg9 ) ! notion ( l9e coord9ates1 & 9 ord] to emphasize ! m]its ( ! sy/em ( coord9ates ultimately adopt$1 we fir/ notice a sy/em )\t ^! advantages1 b (t5 useful 9 special inve/igations4 ,9 ord9>y ,c>tesian coord9ates ! two equations ( a /raight l9e may be r$uc$ to ! =m y rx s1 z tx u1 & r1 s1 t1 u may be reg>d$ z ! f\r coord9ates ( ! l9e4 ,^! coord9ates lack symmetry3 moreov]1 9 *ang+ f "o base ( ref]5ce to ano!r ! trans=mation is n l9e>1 s t ! degree ( an equation is depriv$ ( real significance4 ,= purposes ( ! g5]al !ory we employ homog5e\s coord9ates2 if x#ay#az#aw#a & x#by#bz#bw#b are two po9ts on ! l9e1 x is easily v]ifi$ t ! six det]m9ants ( ! array x#ay#az#aw#a x#by#bz#bw#b are 9 ! same ratios = all po9t-pairs on ! l9e1 & fur!r1 t :5 ! po9t coord9ates und]go a l9e> trans=mation s also d ^! six det]m9ants4 ,we !re=e adopt ^! six det]m9ants = ! coord9ates ( ! l9e1 & express !m by ! symbols l1 lambda1 m1 mu1 n1 nu :]e l x#aw#b - x#bw#a1 lambda y#az#b - y#bz#a1 c4 ,!re is ! fur!r advantage t if a#a;b#a;c#a;d#a & a#b;b#b;c#b;d#b be two planes ?r\gh ! l9e1 ! six det]m9ants a#a;b#a;c#a;d#a a#b;b#b;c#b;d#b are 9 ! same ratios z ! =ego+1 s t except z reg>ds a factor ( proportionality we h lambda b#a;c#b - b#b;c#a1 l c#a;d#b - c#b;d#a1 c4 ,! id5tical relation llambda mmu nnu o r$uces ! numb] ( indep5d5t con/ants 9 ! six coord9ates to f\r1 = we are only conc]n$ ) _! mutual ratios2 & ! quadratic "* ( ?is relation m>ks an ess5tial diff]5ce betwe5 po9t geometry & l9e geometry4 ,! condition ( int]section ( two l9es is llambda' l'lambda mmu' m'mu nnu' n'nu #j :]e ! acc5t$ lett]s ref] to ! second l9e4 ,if ! coord9ates are ,c>tesian & l1 m1 n are direction cos9es1 ! quantity on ! left is ! mutual mom5t ( ! two l9es4 ,s9ce a l9e dep5ds on f\r con/ants1 !re are ?ree di/9ct types ( configurations aris+ 9 l9e geometry--^? conta9+ a triply-inf9ite1 a d\bly-inf9ite & a s+ly-inf9ite numb] ( l9es2 !y are call$ ,complexes1 ,congru5ces1 & ,rul$ ,surfaces or ,skews respectively4 ,a ,complex is ?us a sy/em ( l9es satisfy+ "o condition--t is1 ! coord9ates are connect$ by a s+le relation2 & ! degree ( ! complex is ! degree ( ?is equation suppos+ x to be algebraic4 ,! l9es ( a complex ( ! n? degree :i* pass ?r\gh any po9t lie on a cone ( ! n? degree1 ^? :i* lie 9 any plane envelop a curve ( ! n? class & !re are n l9es ( ! complex 9 any plane p5cil2 ! la/ /atem5t comb9es ! =m] two1 = x %[s t ! cone is ( ! n? degree & ! curve is ( ! n? class4 ,to f9d ! l9es common to f\r complexes ( degrees n#a1 n#b1 n#c1 n#d1 we h to solve five equations1 viz4 ! f\r complex equations toge!r ) ! quadratic equation connect+ ! l9e coord9ates1 !re=e ! numb] ( common l9es is #bn#an#bn#cn#d4 ,z an example ( complexes we h ! l9es meet+ a twi/$ curve ( ! n? degree1 :i* =m a complex ( ! n? degree4 ,a ,congru5ce is ! set ( l9es satisfy+ two conditions3 ?us a f9ite numb] m ( ! l9es pass ?r\gh any po9t1 & a f9ite numb] n lie 9 any plane2 ^! numb]s are call$ ! degree & class respectively1 & ! congru5ce is symbolically writt5 7m1 n74 ,! simple/ example ( a congru5ce is ! sy/em ( l9es con/itut$ by all ^? t pass ?r\gh m po9ts & ^? t lie 9 n planes2 ?r\gh any o!r po9t !re pass m ( ^! l9es1 & 9 any o!r plane !re lie n1 !re=e ! congru5ce is ( degree m & class n4 ,x has be5 %[n by ,g4,h4 ,halph5 t ! numb] ( l9es common to two congru5ces is mm' nn'1 :i* may be v]ifi$ by tak+ "o ( !m to be ( ?is simple type4 ,! l9es meet+ two fix$ l9es =m ! g5]al 7#a1 #a7 congru5ce2 & ! *ords ( a twi/$ cubic =m ! g5]al type ( a 7#a1 #c7 congru5ce2 ,halph5's result %[s t two twi/$ cubics h 9 g5]al t5 common *ords4 ,z reg>ds ! analytical treatm5t1 ! difficulty is ( ! same nature z t aris+ 9 ! !ory ( curves 9 space1 = a congru5ce is n 9 g5]al ! complete int]section ( two complexes4 ,a ,rul$ ,surface1 ,regulus or ,skew is a configuration ( l9es :i* satisfy ?ree conditions1 & !re=e dep5d on only "o p>amet]4 ,su* l9es all lie on a surface1 = we _c draw "o ?r\gh an arbitr>y po9t2 only "o l9e passes ?r\gh a po9t ( ! surface2 ! simple/ example1 t ( a quadric surface1 is really two skews on ! same surface4 ,! degree ( a rul$ surface qua l9e geometry is ! numb] ( its g5]at+ l9es conta9$ 9 a l9e> complex4 ,n[ ! numb] :i* meets a giv5 l9e is ! degree ( ! surface qua po9t geometry1 & z ! l9es meet+ a giv5 l9e =m a p>ticul> case ( l9e> complex1 x foll[s t ! degree is ! same f :i*ev] po9t ( view we reg>d x4 ,! l9es common to ?ree complexes ( degrees1 n#an#bn#c1 =m a rul$ surface ( degree #bn#an#bn#c2 b n e rul$ surface is ! complete int]section ( ?ree complexes4 ,l9e> complex4 ,9 ! case ( a complex ( ! fir/ degree 7or l9e> complex7 ! l9es ?r\gh a fix$ po9t lie 9 a plane call$ ! pol> plane or nul-plane ( t po9t1 & ^? ly+ 9 a fix$ plane pass ?r\gh a po9t call$ ! nul-po9t or pole ( ! plane4 ,if ! nul-plane ( ,a pass ?r\gh ,b1 !n ! nul-plane ( ,b w pass ?r\gh ,a2 ! nul-planes ( all po9ts on "o l9e l#a pass ?r\gh ano!r l9e l#b4 ,! relation betwe5 l#a & l#b is reciprocal2 any l9e ( ! complex t meets "o w also meet ! o!r1 & e l9e meet+ bo? belongs to ! complex4 ,!y are call$ conjugate or pol> l9es ) respect to ! complex4 ,on ^! pr9ciples c be f\nd$ a !ory ( reciprocation ) respect to a l9e> complex4 ,?is may be aptly illu/rat$ by an elegant example due to ,a4 ,voss4 ,s9ce a twi/$ cubic c be made to satisfy twelve conditions1 x might be suppos$ t a f9ite numb] c\ld be drawn to t\* f\r giv5 l9es1 b ?is is n ! case4 ,=1 suppose "o su* c be drawn1 !n its reciprocal ) respect to any l9e> complex conta9+ ! f\r l9es is a curve ( ! ?ird class1 i4e4 ano!r twi/$ cubic1 t\*+ ! same f\r l9es1 :i* are unalt]$ 9 ! process ( reciprocation2 z !re is an inf9ite numb] ( complexes conta9+ ! f\r l9es1 !re is an inf9ite numb] ( cubics t\*+ ! f\r l9es1 & ! problem is pori/ic4 ,! foll[+ are "s geometrical con/ructions relat+ to ! unique l9e> complex t c be drawn to conta9 five arbitr>y l9es3 ,to con/ruct ! nul-plane ( any po9t ,o1 we obs]ve t ! two l9es :i* meet any f\r ( ! giv5 five are conjugate l9es ( ! complex1 & ! l9e drawn ?r\gh ,o to meet !m is !re=e a ray ( ! complex2 simil>ly1 by *oos+ ano!r f\r we c f9d ano!r ray ?r\gh ,o3 ^! rays lie 9 ! nul-plane1 & !re is cle>ly a result involv$ t ! five l9es s obta9$ all lie 9 "o plane4 ,a reciprocal con/ruction w enable u to f9d ! nul-po9t ( any plane4 ,proce$+ n[ to ! metrical prop]ties & ! /atical & dynamical applications1 we rem>k t !re is j "o l9e su* t ! nul-plane ( any po9t on x is p]p5dicul> to x4 ,?is is call$ ! c5tral axis2 if d be ! %orte/ di/ance1 !ta ! angle betwe5 x & a ray ( ! complex1 !n d tan !ta p1 :]e p is a con/ant call$ ! pit* or p>amet]4 ,any sy/em ( =ces c be r$uc$ to a =ce ,r along a c]ta9 l9e1 & a c\ple ,g p]p5dicul> to t l9e2 ! l9es ( nul-mom5t = ! sy/em =m a l9e> complex ( :i* ! giv5 l9e is ! c5tral axis & ! quoti5t ,g_/,r is ! pit*4 ,any motion ( a rigid body c be r$uc$ to a screw motion ab\t a c]ta9 l9e1 i4e4 to an angul> velocity omega ab\t t l9e comb9$ ) a l9e> velocity u along ! l9e4 ,! plane drawn ?r\gh any po9t p]p5dicul> to ! direction ( its motion is its nul-plane ) respect to a l9e> complex hav+ ?is l9e = c5tral axis1 & ! quoti5t u_/omega = pit* 7cf4 ,sir ,r4,s4 ,ball1 ,!ory ( ,screws74 ,! foll[+ are "s prop]ties ( a configuration ( two l9e> complexes3 ,! l9es common to ! two-complexes also belong to an inf9ite numb] ( l9e> complexes1 ( :i* two r$uce to s+le /raight l9es4 ,^! two l9es are conjugate l9es ) respect to ea* ( ! complexes1 b !y may co9cide1 & !n "s simple modifications are requir$4 ,! locus ( ! c5tral axis ( ?is sy/em ( complexes is a surface ( ! ?ird degree call$ ! cyl9droid1 :i* plays a lead+ "p 9 ! !ory ( screws z develop$ syn!tically by ,ball4 ,s9ce a l9e> complex has an inv>iant ( ! second degree 9 its coeffici5ts1 x foll[s t two l9e> complexes h a l9eo-l9e> inv>iant4 ,?is inv>iant is fundam5tal3 if ! complexes be bo? /raight l9es1 its vani%+ is ! condition ( _! int]section z giv5 above2 if only "o ( !m be a /raight l9e1 its vani%+ is ! condition t ?is l9e %\ld belong to ! o!r complex4 ,:5 x vani%es = any two complexes !y are said to be 9 involution or apol>2 ! nul-po9ts ,p1 ,q ( any plane !n divide h>monically ! po9ts 9 :i* ! plane meets ! common conjugate l9es1 & ea* complex is its own reciprocal ) respect to ! o!r4 ,z reg>ds a configuration ( ^! l9e> complexes1 ! common l9es f "o sy/em ( g5]ators ( a quadric1 & ! d\bly inf9ite sy/em ( complexes conta9+ ! common l9es1 include an inf9ite numb] ( /raight l9es :i* =m ! o!r sy/em ( g5]ators ( ! same quadric4 ,g5]al l9e coord9ates4 ,if ! equation ( a l9e> complex is ,al ,bm ,cn ,dlambda ,emu ,fnu #j1 !n = a l9e n belong+ to ! complex we may reg>d ! expression on ! left-h& side z a multiple ( ! mom5t ( ! l9e ) respect to ! complex1 ! ^w mom5t be+ us$ 9 ! /atical s5se2 & we inf] t :5 ! coord9ates are replac$ by l9e> functions ( !mselves ! new coord9ates are multiples ( ! mom5ts ( ! l9e ) respect to six fix$ complexes4 ,! ess5tial features ( ?is coord9ate sy/em are ! same z ^? ( ! orig9al "o1 viz4 !re are six coord9ates connect$ by a quadratic equation1 b ?is relation has 9 g5]al a diff]5t =m4 ,by suitable *oice ( ! six fundam5tal complexes1 z !y may be call$1 ?is connect+ relation may be br\ght into o!r simple =ms ( :i* we m5tion two3 7i47 ,:5 ! six are mutually 9 involution x c be r$uc$ to x#ab x#bb x#cb x#db x#eb x#fb #j2 7ii47 ,:5 ! fir/ f\r are 9 involution & ! o!r two are ! l9es common to ! fir/ f\r x is x#ab x#bb x#cb x#db - #bx#ex#f #j4 ,^! g5]aliz$ coord9ates might be expla9$ )\t ref]5ce to actual magnitude1 j z homog5e\s po9t coord9ates c be2 ! ess5tial rem>k is t ! equation ( any coord9ate to z]o repres5ts a l9e> complex1 a po9t ( view :i* includes our orig9al sy/em1 = ! equation ( a coord9ate to z]o repres5ts all ! l9es meet+ an edge ( ! fundam5tal tetrah$ron4 ,! sy/em ( coord9ates ref]r$ to six complexes mutually 9 involution 0 introduc$ by ,felix ,kle91 & 9 _m cases is m useful ?an t d]iv$ directly f po9t coord9ates2 e4g4 9 ! discussion ( quadratic complexes3 by means ( x ,kle9 has develop$ an analogy betwe5 l9e geometry & ! geometry ( sph]es z treat$ by ,g4 ,d>b\x & o!rs4 ,9 fact1 9 t geometry a po9t is repres5t$ by five coord9ates1 connect$ by a relation ( ! same type z ! "o j m5tion$ :5 ! five fundam5tal sph]es are mutually at "r angles & ! equation ( a sph]e is ( ! fir/ degree4 ,ext5d+ ?is to f\r dim5sions ( space1 we obta9 an exact analogue ( l9e geometry1 9 :i* 7i47 a po9t corresponds to a l9e2 7ii47 a l9e> complex to a hyp]sph]e2 7iii47 two l9e> complexes 9 involution to two or?ogonal hyp]sph]es2 7iv47 a l9e> complex & two conjugate l9es to a hyp]sph]e & two inv]se po9ts4 ,_m results may be obta9$ by ?is pr9ciple1 & m /ill are sugge/$ by try+ to ext5d ! prop]ties ( circles to sph]es 9 ?ree & f\r dim5sions4 ,?us ! elem5t>y !orem1 t1 giv5 f\r l9es1 ! circles circumscrib$ to ! f\r triangles =m$ by !m are concurr5t1 may be ext5d$ to six hyp]planes 9 f\r dim5sions2 & !n we c d]ive a result 9 l9e geometry by translat+ ! inv]se ( ?is !orem4 ,aga91 j z !re is an inf9ite numb] ( sph]es t\*+ a surface at a giv5 po9t1 two ( !m hav+ contact ( a clos] nature1 s !re is an inf9ite numb] ( l9e> complexes t\*+ a non-l9e> complex at a giv5 l9e1 & ?ree ( ^! h contact ( a clos] nature 7cf4 ,kle91 ,ma?4 ,ann4 v474 ,sophus ,lie has po9t$ out a diff]5t analogy ) sph]e geometry4 ,suppose1 9 fact1 t ! equation ( a sph]e ( radius r is x#b y#b z#b #b;ax #b;by #b;cz d #j1 s t r#b a#b b#b c#b - d2 !n introduc+ ! quantity e to make ?is equation homog5e\s1 we may reg>d ! sph]e z giv5 by ! six coord9ates a1 b1 c1 d1 e1 r connect$ by ! equation a#b b#b c#b - r#b - de #j1 & x is easy to see t two sph]es t\*1 if ! pol> =m #b;aa#a #b;bb#a #b;cc#a - #brr#a - de#a - d#a;e vani%es4 ,comp>+ ?is ) ! equation x#ab x#bb x#cb x#db - #bx#ex#f #j giv5 above1 x appe>s t ?is sph]e geometry & l9e geometry are id5tical1 = we may write a x#a1 b x#b1 c x#c1 r x#d7_/;delta - #a71 d x#e1 e #a_/bx#f2 b x is to be notic$ t a sph]e is really replac$ by two l9es ^: coord9ates only diff] 9 ! sign ( x#d1 s t !y are pol> l9es ) respect to ! complex x#d #j4 ,two sph]es :i* t\* correspond to two l9es :i* int]sect1 or m accurately to two pairs ( l9es 7p1 p'7 & 7q1 q'71 ( :i* ! pairs 7p1 q7 & 7p'1 q'7 bo? int]sect4 ,by ?is means ! problem ( describ+ a sph]e to t\* f\r giv5 sph]es is r$uc$ to t ( draw+ a pair ( l9es 7t1 t'7 7( :i* t int]sects "o l9e ( ! f\r pairs 7pp'71 7qq'71 7rr'71 7ss'71 & t' int]sects ! rema9+ f\r74 ,we may1 h[ev]1 ignore ! acc5t$ lett]s 9 translat+ !orems1 = a configuration ( l9es & its pol> ) respect to a l9e> complex h ! same projective prop]ties4 ,9 ,lie's trans=mation a l9e> complex corresponds to ! totality ( sph]es cutt+ a giv5 sph]e at a giv5 angle4 ,a mo/ rem>kable result is t l9es ( curvature 9 ! sph]e geometry become asymptotic l9es 9 ! l9e geometry4 ,"s ( ! pr9ciples ( l9e geometry may be br\ght into cle>] light by admitt+ ! ideas ( space ( f\r & five dim5sions4 ,?us1 reg>d+ ! coord9ates ( a l9e z homog5e\s coord9ates 9 five dim5sions1 we may say t l9e geometry is equival5t to geometry on a quadric surface 9 five dim5sions4 ,a l9e> complex is repres5t$ by a hyp]plane section2 & if two su* complexes are 9 involution1 ! correspond+ hyp]planes are conjugate ) respect to ! fundam5tal quadric4 ,by project+ ?is quadric /]eographically into space ( f\r dim5sions we obta9 ,kle9's analogy4 ,9 ! same way geometry 9 a l9e> complex is equival5t to geometry on a quadric 9 f\r dim5sions2 :5 two l9es int]sect ! repres5tative po9ts are on ! same g5]ator ( ?is quadric4 ,/]eographic projection1 !re=e1 conv]ts a curve 9 a l9e> complex1 i4e4 "o ^: tang5ts all belong to ! complex1 into "o ^: tang5ts int]sect a fix$ conic3 :5 ?is conic is ! imag9>y circle at inf9ity ! curve is :at ,lie calls a m9imal curve4 ,curves 9 a l9e> complex h be5 ext5sively /udi$4 ,! osculat+ plane at any po9t ( su* a curve is ! nul-plane ( ! po9t ) respect to ! complex1 & po9ts ( sup]osculation always co9cide 9 pairs at ! po9ts ( contact ( /ation>y tang5ts4 ,:5 a po9t ( su* a curve is giv51 ! osculat+ plane is det]m9$1 h5ce all ! curves ?r\gh a giv5 po9t ) ! same tang5t h ! same torsion4 ,non-l9e> complexes4 ,! l9es ?r\gh a giv5 po9t t belong to a complex ( ! n? degree lie on a cone ( ! n? degree3 if ?is cone has a d\ble l9e ! po9t is said to be a s+ul> po9t4 ,simil>ly1 a plane is said to be s+ul> :5 ! envelope ( ! l9es 9 x has a d\ble tang5t4 ,x is v rem>kable t ! same surface is ! locus ( ! s+ul> po9ts & ! envelope ( ! s+ul> planes3 ?is surface is call$ ! s+ul> surface1 & bo? its degree & class are 9 g5]al #bn7n - #a7b1 :i* is equal to f\r = ! quadratic complex4 ,! s+ul> l9es ( a complex ,f #j are ! l9es common to ,f & ! complex delta,f delta,f delta,f delta,f delta,f delta,f -------- --------------- -------- ----------- -------- ----------- #j4 deltal deltalambda deltam deltamu deltan deltanu ,z already m5tion$1 at ea* l9e l ( a complex !re is an inf9ite numb] ( tang5t l9e> complexes1 & !y all conta9 ! l9es adjac5t to l4 ,if n[ l be a s+ul> l9e1 ^! complexes all r$uce to /raight l9es :i* =m a plane p5cil conta9+ ! l9e l4 ,suppose ! v]tex ( ! p5cil is ,a1 its plane a1 & "o ( its l9es xi1 !n l' be+ a complex l9e ne> l1 meets xi1 or m accurately ! mutual mom5t ( l'1 & is ( ! second ord] ( small quantities4 ,if ,p be a po9t on l1 a l9e ?r\gh ,p q ne> l 9 ! plane a w meet xi & is !re=e a l9e ( ! complex2 h5ce ! complex-cones ( all po9ts on l t\* a & ! complex-curves ( all planes ?r\gh l t\* l at ,a4 ,x foll[s t l is a d\ble l9e ( ! complex-cone ( ,a1 & a d\ble tang5t ( ! complex-curve ( a4 ,conv]sely1 a d\ble l9e ( a cone or curve is a s+ul> l9e1 & a s+ul> l9e cle>ly t\*es ! curves ( all planes ?r\gh x 9 ! same po9t4 ,suppose n[ t ! consecutive l9e l' is also a s+ul> l9e1 ,a' be+ ! alli$ s+ul> po9t1 a' ! s+ul> plane & xi' any l9e ( ! p5cil 7,a'1 a'7 s t xi' is a tang5t l9e at l' to ! complex3 ! mutual mom5ts ( ! pairs l'1 xi & l1 xi are ea* ( ! second ord]2 h5ce ! plane a' meets ! l9es l & xi' 9 two po9ts v ne> ,a4 ,?is be+ true = all s+ul> planes1 ne> a ! po9t ( contact ( a ) its envelope is 9 ,a1 i4e4 ! locus ( s+ul> po9ts is ! same z ! envelope ( s+ul> planes4 ,fur!r1 :5 a l9e t\*es a complex x t\*es ! s+ul> surface1 = x belongs to a plane p5cil l 7,aa71 & ?us 9 ,kle9's analogy ! analogue ( a focus ( a hyp]-surface be+ a bitang5t l9e ( ! complex is also a bitang5t l9e ( ! s+ul> surface4 ,! !ory ( cos+ul> complexes is ?us br\ght into l9e ) t ( confocal surfaces 9 f\r dim5sions1 & guid$ by ^! pr9ciples ! exi/5ce ( cos+ul> quadratic complexes c easily be e/abli%$1 ! analysis requir$ be+ almo/ ! same z t inv5t$ = confocal cyclides by ,d>b\x & o!rs4 ,( cos+ul> complexes ( high] degree no?+ is kn[n4 ,foll[+ ,j4 ,pluck]1 we give an acc\nt ( ! l9es ( a quadratic complex t meet a giv5 l9e4 ,! cones ^: v]tices are on ! giv5 l9e all pass ?r\gh eight fix$ po9ts & envelop a surface ( ! f\r? degree2 ! conics ^: planes conta9 ! giv5 l9e all lie on a surface ( ! f\r? class & t\* eight fix$ planes4 ,x is easy to see by elem5t>y geometry t ^! two surfaces are id5tical4 ,fur!r1 ! giv5 l9e conta9s f\r s+ul> po9ts ,a#a1 ,a#b1 ,a#c1 ,a#d1 & ! planes into :i* _! cones deg5]ate are ! eight common tang5t planes m5tion$ above2 simil>ly1 !re are f\r s+ul> planes1 a#a1 a#b1 a#c1 a#d1 ?r\gh ! l9e1 & ! eight po9ts into :i* _! conics deg5]ate are ! eight common po9ts above4 ,! locus ( ! pole ( ! l9e ) respect to all ! conics 9 planes ?r\gh x is a /raight l9e call$ ! pol> l9e ( ! giv5 "o2 & ?r\gh ?is l9e passes ! pol> plane ( ! giv5 l9e ) respect to ea* ( ! cones4 ,! "n pol> is appli$ 9 ! ord9>y analytical s5se2 any l9e has an inf9ite numb] ( pol> complexes ) respect to ! giv5 complex1 = ! equation ( ! latt] c be writt5 9 an inf9ite numb] ( ways2 "o ( ^! pol>s is a /raight l9e1 & is ! pol> l9e already introduc$4 ,! surface on :i* lie all ! conics ?r\gh a l9e l is call$ ! ,pluck] surface ( t l9e3 f ! kn[n prop]ties ( 7#b1 #b7 correspond5ces x c be %[n t ! ,pluck] surface ( l cuts l#a 9 a range ( ! same cross ratio z t ( ! range 9 :i* ! ,pluck] surface ( l#a cuts l4 ,apply+ ?is to ! case 9 :i* l#a is ! pol> ( l1 we f9d t ! cross ratios ( 7,a#a1 ,a#b1 ,a#c1 ,a#d7 & 7a#a1 a#b1 a#c1 a#d7 are equal4 ,! id5tity ( ! locus ( ! ,a's ) ! envelope ( ! a's foll[s at once2 moreov]1 a l9e meets ! s+ul> surface 9 f\r po9ts hav+ ! same cross ratio z t ( ! f\r tang5t planes drawn ?r\gh ! l9e to t\* ! surface4 ,! ,pluck] surface has eight nodes1 eight s+ul> tang5t planes1 & is a d\ble l9e4 ,! relation betwe5 a l9e & its pol> l9e is n a reciprocal "o ) respect to ! complex2 b ,w4 ,/ahl has po9t$ out t ! relation is reciprocal z f> z ! s+ul> surface is conc]n$4 ,quadratic complexes4 ,to facilitate ! discussion ( ! g5]al quadratic complex we introduce ,kle9's canonical =m4 ,we h1 9 fact1 to deal ) two quadratic equations 9 six v>iables2 & by suitable l9e> trans=mations ^! c be r$uc$ to ! =m a#ax#ab a#bx#bb a#cx#cb a#dx#db a#ex#eb a#fx#fb #j x#ab x#bb x#cb x#db x#eb x#fb #j subject to c]ta9 exceptions1 :i* w be m5tion$ lat]4 ,tak+ ! fir/ equation to be t ( ! complex1 we rem>k t bo? equations are unalt]$ by *ang+ ! sign ( any coord9ate2 ! geometrical mean+ ( ?is is1 t ! quadratic complex is its own reciprocal ) respect to ea* ( ! six fundam5tal complexes1 = *ang+ ! sign ( a coord9ate is equival5t to tak+ ! pol> ( a l9e ) respect to ! correspond+ fundam5tal complex4 ,x is easy to e/abli% ! exi/5ce ( six sy/ems ( bitang5t l9e> complexes1 = ! complex l#ax#a l#bx#b l#cx#c l#dx#d l#ex#e l#fx#f #j is a bitang5t :5 l#bb l#cb l#db l#eb l#fb l#a #j1 & ------- ------- ------- ------- ------- #j a#b - a#a a#c - a#a a#d - a#a a#e - a#a a#f - a#a & its l9es ( contact are conjugate l9es ) respect to ! fir/ fundam5tal complex4 ,we !re=e inf] ! exi/5ce ( six sy/ems ( bitang5t l9es ( ! complex1 ( :i* ! fir/ is giv5 by x#bb x#cb x#db x#eb x#fb x#a #j1 ------- ------- ------- ------- ------- #j4 a#b - a#a a#c - a#a a#d - a#a a#e - a#a a#f - a#a ,ea* ( ^! l9es is a bitang5t ( ! s+ul> surface1 :i* is !re=e completely det]m9$ z be+ ! focal surface ( ! 7#b1 #b7 congru5ce above4 ,x is !nce easy to v]ify t ! two complexes ,sigmaax#b #j & ,sigmabx#b #j are cos+ul> if br arlambda mu_/arnu rho4 ,! s+ul> surface ( ! g5]al quadratic complex is ! fam\s qu>tic1 ) sixte5 nodes & sixte5 s+ul> tang5t planes1 fir/ discov]$ by ,e4,e4 ,kumm]4 ,we _c give a full acc\nt ( its prop]ties "h1 b we d$uce at once f ! above t its bitang5ts break up into six 7#b1 #b7 congru5ces1 & ! six l9e> complexes conta9+ ^! are mutually 9 involution4 ,! nodes ( ! s+ul> surface are po9ts ^: complex cones are co9cid5t planes1 & ! complex conic 9 a s+ul> tang5t plane consi/s ( two co9cid5t po9ts4 ,?is configuration ( sixte5 po9ts & planes has _m int]e/+ prop]ties2 ?us ea* plane conta9s six po9ts :i* lie on a conic1 :ile ?r\gh ea* po9t !re pass six planes :i* t\* a quadric cone4 ,9 _m respects ! ,kumm] qu>tic plays a "p 9 ?ree dim5sions analog\s to ! g5]al qu>tic curve 9 two2 x fur!r gives a natural repres5tation ( c]ta9 relations betwe5 hyp]elliptic functions 7cf4 ,r4,w4,h4,t4 ,hudson1 ,kumm]'s ,qu>tic1 #aije74 ,classification ( quadratic complexes4 ,z might be expect$ f ! magnitude ( a =m 9 six v>iables1 ! numb] ( projectivally di/9ct v>ieties ( quadratic complexes is v great2 & 9 fact ,adolf ,weil]1 by :om ! "q 0 fir/ sy/ematically /udi$ on l9es indicat$ by ,kle91 enum]at$ no few] ?an =ty-n9e diff]5t types4 ,b ! pr9ciple ( ! classification is s important1 & )al s simple1 t we give a brief sket* :i* indicates its ess5tial features4 ,we h practically to /udy ! int]section ( two quadrics ,f & ,f' 9 six v>iables1 & to classify ! diff]5t cases aris+ we make use ( ! results ( ,k>l ,wei]/rass on ! equival5ce conditions ( two pairs ( quadratics4 ,z f> z at pres5t requir$1 !y are z foll[s3 ,suppose t ! factoriz$ =m ( ! det]m9antal equation ,disct 7,f lambda,f'7 #j is 7lambda - alpha77s#a s#b s#c 4447 7lambda - beta77t#a t#b t#c 4447 444 :]e ! root alpha occurs s#a s#b s#c 444 times 9 ! det]m9ant1 s#b s#c 444 times 9 e fir/ m9or1 s#c 444 times 9 e second m9or1 & s on2 ! mean+ ( ea* expon5t is !n p]fectly def9ite4 ,e factor ( ! type 7lambda - alpha7s is call$ an elem5t>!il 7elem5t>y divisor7 ( ! det]m9ant1 & ! condition ( equival5ce ( two pairs ( quadratics is simply t _! det]m9ants h ! same elem5t>y divisors4 ,we write ! pair ( =ms symbolically ?us 7s#as#b 44471 7t#at#b 44471 4441 lett]s 9 ! inn] brackets ref]r+ to ! same factor4 ,return+ n[ to ! two quadratics repres5t+ ! complex1 ! sum ( ! expon5ts w be six1 & two complexes are put 9 ! same class if !y h ! same symbolical expression2 i4e4 ! actual values ( ! roots ( ! det]m9antal equation ne$ n be ! same = bo?1 b _! mann] ( occurr5ce1 z f> z "h indicat$1 mu/ be id5tical 9 ! two4 ,! enum]ation ( all possible cases is ?us r$uc$ to a simple "q 9 comb9atorial analysis1 & ! actual /udy ( any p>ticul> case is mu* facilitat$ by a useful rule ( ,kle9's = writ+ d[n 9 a simple =m two quadratics belong+ to a giv5 class--"o ( :i*1 ( c\rse1 repres5ts ! equation connect+ l9e coord9ates1 & ! o!r ! equation ( ! complex4 ,! g5]al complex is naturally #aaaaaa2 ! complex ( tang5ts to a quadric is 7#aaa71 7#aaa7 & t ( l9es meet+ a conic is 7#bbb74 ,full in=mation w be f\nd 9 ,weil]'s memoir1 ,ma?4 ,ann4 vol4 vii4 ,! detail$ /udy ( ea* v>iety ( complex op5s up a va/ subject2 we only m5tion two special cases1 ! h>monic complex & ! tetrah$ral complex4 ,! h>monic complex1 fir/ /udi$ by ,battagl9i1 is g5]at$ 9 an inf9ite numb] ( ways by ! l9es cutt+ two quadrics h>monically4 ,tak+ ! mo/ g5]al case1 & ref]r+ ! quadrics to _! common self-conjugate tetrah$ron1 we c f9d its equation 9 a simple =m1 & v]ify t ?is complex really dep5ds only on sev5te5 con/ants1 s t x is n ! mo/ g5]al quadratic complex4 ,x belongs to ! g5]al type 9 s f> z x is discuss$ above1 b ! roots ( ! det]m9ant are 9 involution4 ,! s+ul> surface is ! tetrah$roid discuss$ by ,cayley4 ,z a p>ticul> case1 f a metrical po9t ( view1 we h ,l4,f4 ,pa9v9's complex g5]at$ by ! l9es ( int]section ( p]p5dicul> tang5t planes ( a quadric1 ! s+ul> surface n[ be+ ,fresnel's wave surface4 ,! tetrah$ral or ,reye complex is ! simple/ & be/ kn[n ( prop] quadratic complexes4 ,x is g5]at$ by ! l9es :i* cut ! faces ( a tetrah$ron 9 a con/ant cross ratio1 & !re=e by ^? subt5d+ ! same cross ratio at ! f\r v]tices4 ,! s+ul> surface is made up ( ! faces or ! v]tices ( ! fundam5tal tetrah$ron1 & ea* edge ( ?is tetrah$ron is a d\ble l9e ( ! complex4 ,! complex 0 fir/ discuss$ by ,k4,t4 ,reye z ! assemblage ( l9es jo9+ correspond+ po9ts 9 a homographic trans=mation ( space1 & ?is po9t ( view leads to _m important & elegant prop]ties4 ,a 7metrically7 p>ticul> case ( great int]e/ is ! complex g5]at$ by ! normals to a family ( confocal quadrics1 & = _m inve/igations x is conv5i5t to deal ) ?is complex ref]r$ to ! pr9cipal axes4 ,= example1 ,lie has develop$ ! !ory ( curves 9 a ,reye complex 7i4e4 curves ^: tang5ts belong to ! complex7 z solutions ( a diff]5tial equation ( ! =m 7b - c7xdydz 7c - a7ydzdx 7a - b7zdxdy #j1 & we c simplify ?is equation by a log>i?mic trans=mation4 ,_m !orems connect+ complexes ) diff]5tial equations h be5 giv5 by ,lie & 8 s*ool4 ,a l9e complex1 9 fact1 corresponds to a ,mongian equation hav+ oo#c l9e integrals4 ,congru5ces4 ,z ! coord9ates ( a l9e belong+ to a congru5ce are functions ( two indep5d5t p>amet]s1 ! !ory ( congru5ces is analog\s to t ( surfaces1 & we may reg>d x z a fundam5tal inquiry to f9d ! simple/ =m ( surface into :i* a giv5 congru5ce c be trans=m$4 ,mo/ ( ^? ^: prop]ties h be5 ext5sively discuss$ c be repres5t$ on a plane by a birational trans=mation4 ,b 9 addition to ! difficulties ( ! !ory ( algebraic surfaces1 a subject /ill 9 its infancy1 ! !ory ( congru5ces has o!r difficulties 9 t a congru5ce is seldom completely repres5t$1 ev5 by two equations4 ,a fundam5tal !orem is t ! l9es ( a congru5ce are 9 g5]al bitang5ts ( a surface2 9 fact1 s9ce ! condition ( int]section ( two consecutive /raight l9es is ldlambda dmdmu dndnu #j1 a l9e l ( ! congru5ce meets two adjac5t l9es1 say l#a & l#b4 ,suppose l1 l#a lie 9 ! plane p5cil 7,a#a;a#a7 & l1 l#b 9 ! plane p5cil 7,a#b;a#b71 !n ! locus ( ! ,a's is ! same z ! envelope ( ! a's1 b a#b is ! tang5t plane at ,a#a & a#a at ,a#b4 ,?is surface is call$ ! focal surface ( ! congru5ce1 & to x all ! l9es l are bitang5t4 ,! di/9ctive prop]ty ( ! po9ts ,a is t two ( ! congru5ce l9es ?r\gh !m co9cide1 & 9 l mann] ! planes a ea* conta9 two co9cid5t l9es4 ,! focal surface consi/s ( two %eets1 b "o or bo? may deg5]ate into curves2 ?us1 = example1 ! normals to a surface are bitang5ts ( ! surface ( c5tres1 & 9 ! case ( ,dup9's cyclide ?is surface deg5]ates into two conics4 ,9 ! discussion ( congru5ces x soon becomes necess>y to introduce ano!r numb] r1 call$ ! rank1 :i* expresses ! numb] ( plane p5cils ea* ( :i* conta9s an arbitr>y l9e & two l9es ( ! congru5ce4 ,! ord] ( ! focal surface is #bm7n - #a7 - #br1 & its class is m7m - #a7 - #br4 ,our k ( congru5ces is almo/ exclusively conf9$ to ^? 9 :i* ei!r m or n does n exce$ two4 ,we give a brief acc\nt ( ^? ( ! second ord] )\t s+ul> l9es1 ^? ( ord] unity n be+ especially int]e/+4 ,a congru5ce g5]ally has s+ul> po9ts ?r\gh :i* an inf9ite numb] ( l9es pass2 a s+ul> po9t is said to be ( ord] r :5 ! l9es ?r\gh x lie on a cone ( ! r? degree4 ,by means ( =mulae connect+ ! numb] ( s+ul> po9ts & _! ord]s ) ! class m ( quadratic congru5ce ,kumm] prov$ t ! class _c exce$ sev54 ,! focal surface is ( degree f\r & class #bm2 ?is k9d ( qu>tic surface has be5 ext5sively /udi$ by ,kumm]1 ,cayley1 ,rohn & o!rs4 ,! v>ieties 7#b1 #b71 7#b1 #c71 7#b1 #d71 7#b1 #e7 all belong to at lea/ "o ,reye complex2 & s also does ! mo/ important class ( 7#b1 #f7 congru5ces :i* includes all ! above z special cases4 ,! congru5ce 7#b1 #b7 belongs to a l9e> complex & =ty diff]5t ,reye complexes2 z above rem>k$1 ! s+ul> surface is ,kumm]'s sixte5-nodal qu>tic1 & ! same surface is focal = six diff]5t congru5ces ( ?is v>iety4 ,! !ory ( 7#b1 #b7 congru5ces is completely analog\s to t ( ! surfaces call$ cyclides 9 ?ree dim5sions4 ,fur!r p>ticul>s reg>d+ quadratic congru5ces w be f\nd 9 ,kumm]'s memoir ( #ahff1 & ! second volume ( ,/urm's treatise4 ,! prop]ties ( quadratic congru5ces hav+ s+ul> l9es1 i4e4 deg5]ate focal surfaces1 are n s int]e/+ z ^? ( ! above class2 !y h be5 discuss$ by ,kumm]1 ,/urm & o!rs4 ,rul$ surfaces4 ,s9ce a rul$ surface conta9s only inf9ity#a elem5ts1 ?is !ory is practically ! same z t ( curves4 ,if a l9e> complex conta9s m ?an n g5]ators ( a rul$ surface ( ! n? degree1 x conta9s all ! g5]ators1 h5ce = n #b !re are ?ree l9e>ly indep5d5t complexes1 conta9+ all ! g5]ators1 & ?is is a well-kn[n prop]ty ( quadric surfaces4 ,9 rul$ cubics ! g5]ators all meet two l9es :i* may or may n co9cide2 ^! two cases correspond to ! two ma9 classes ( cubics discuss$ by ,cayley & ,cremona4 ,z reg>ds rul$ qu>tics1 ! g5]ators mu/ lie 9 "o & may lie 9 two l9e> complexes4 ,! fir/ class is equival5t to a qu>tic 9 f\r dim5sions & is always rational1 b ! latt] class has to be subdivid$ into ! elliptic & ! rational1 j l twi/$ qu>tic curves4 ,a qu9tic skew may n lie 9 a l9e> complex1 & !n x is unicursal1 :ile ( sextics we h two classes n 9 a l9e> complex1 viz4 ! elliptic v>iety1 hav+ ?irty-six places :]e a l9e> complex conta9s six consecutive g5]ators1 & ! rational1 hav+ six su* places4 ,! g5]al !ory ( skews 9 two l9e> complexes is id5tical ) t ( curves on a quadric 9 ?ree dim5sions & is kn[n4 ,b = skews ly+ 9 only "o l9e> complex !re are difficulties2 ! curve n[ lies 9 f\r dim5sions1 & we repres5t x 9 ?ree by /]eographic projection z a curve meet+ a giv5 plane 9 n po9ts on a conic4 ,to f9d ! maximum defici5cy = a giv5 degree w\ld probably be difficult1 b z f> z degree eight ! space-curve !ory ( ,halph5 & ,no!r c be translat$ into l9e geometry at once4 ,:5 ! skew does n lie 9 a l9e> complex at all ! !ory is m difficult /ill1 & ! g5]al !ory cle>ly _c advance until fur!r progress is made 9 ! /udy ( twi/$ curves4 ,,ref]5ces4--,! e>lie/ works ( a g5]al nature are ,pluck]1 ,neue ,geometrie des ,raumes 7,leipzig1 #ahfh72 & ,kumm]1 ,ub] die algebrais*5 ,/rahl5sy/eme1 ,b]l9 ,academy 7#ahff74 ,sy/ematic developm5t on purely syn!tic l9es w be f\nd 9 ! ?ree volumes ( ,/urm1 ,l9i5geometrie 7,leipzig1 #ahib1 #ahic1 #ahif72 vol4 i4 deals ) ! l9e> & ,reye complexes1 vols4 ii4 & iii4 ) quadratic congru5ces & complexes respectively4 ,= a highly sugge/ive review by ,g9o ,loria see ,bullet9 des sci5ces ma!matiques 7#ahic1 #ahig74 ,a %ort] treatise1 giv+ a v int]e/+ acc\nt ( ,kle9's coord9ates1 is ! "w ( ,ko5igs1 ,la ,geometrie reglee et ses applications 7,p>is1 #ahih74 ,engli% treatises are ,c4,m4 ,jessop1 ,treatise on ! ,l9e ,complex 7#aijc72 ,r4,w4,h4,t4 ,hudson1 ,kumm]'s ,qu>tic 7#aije74 ,_m ref]5ces to memoirs on l9e geometry w be f\nd 9 ,hag51 ,synopsis d] hoh]5 ,ma!matik1 ii4 7,b]l91 #ahid72 ,loria1 ,il passato ed il pres5te delle pr9cipali teorie geometri*e 7,milan1 #ahig72 a cle> resume ( ! pr9cipal results is conta9$ 9 ! v elegant volume ( ,pascal1 ,rep]torio di ma!mati*e sup]iori1 ii4 7,milan1 #aijj74 ,ano!r treatise deal+ ext5sively ) l9e geometry is ,lie1 ,geometrie d] ,b]uhrung/rans=mation5 7,leipzig1 #ahif74 ,_m memoirs on ! subject h appe>$ 9 ! ,ma!matis*e ,annal52 a full li/ ( ^! w be f\nd 9 ! index to ! fir/ fifty volumes1 p4 #aae4 ,p]haps ! two memoirs :i* h left mo/ impression on ! subsequ5t developm5t ( ! subject are ,kle91 ,zur ,!orie d] ,l9i5complexe des er/5 und zweit5 ,grades1 ,ma?4 ,ann4 ii42 & ,lie1 ,ub] ,complexe1 insbesond]e ,l9i5- und ,kugelcomplexe1 ,ma?4 ,ann4 v4 7,j4 ,h4 ,gr47 ,,vi4 ,,non-,,euclidean ,,geometry ,! v>i\s metrical geometries are conc]n$ ) ! prop]ties ( ! v>i\s types ( congru5ce-gr\ps1 :i* are def9$ 9 ! /udy ( ! axioms ( geometry & ( _! imm$iate consequ5ces4 ,b ?is po9t ( view ( ! subject is ! outcome ( rec5t rese>*1 & hi/orically ! subject has a diff]5t orig94 ,non-,euclidean geometry arose f ! discussion1 ext5d+ f ! ,greek p]iod to ! pres5t day1 ( ! v>i\s assumptions :i* are implicit 9 ! traditional ,euclidean sy/em ( geometry4 ,9 ! c\rse ( ^! inve/igations x became evid5t t metrical geometries1 ea* int]nally consi/5t b inconsi/5t 9 _m respects ) ea* o!r & ) ! ,euclidean sy/em1 c\ld be develop$4 ,a %ort hi/orical sket* w expla9 ?is orig9 ( ! subject1 & describe ! fam\s & int]e/+ progress ( ?\ght on ! subject4 ,b previ\sly a description ( ! *ief *>act]i/ic prop]ties ( elliptic & ( hyp]bolic geometries w be giv51 assum+ ! /&po9t arriv$ at bel[ "u ,,vii4 ,axioms ( ,geometry4 ,fir/ assume ! equation to ! absolute 7cf4 loc4 cit47 to be w#b - x#b - y#b - z#b #j4 ,! absolute is !n real1 & ! geometry is hyb]bolic4 ,! di/ance 7d#ab7 betwe5 ! two po9ts 7x#a1 y#a1 z#a1 w#a7 & 7x#b1 y#b1 z#b1 w#b7 is giv5 by co% 7d#ab_/;gamma7 7w#aw#b - x#ax#b - y#ay#b - z#az#b7_/7w#ab - x#ab - y#ab - ,z#ab7 7w#bb - x#bb - y#bb - z#bb7a_/b 7#a7 ,! only po9ts to :i* ! metrical geometry applies are ^? )9 ! region enclos$ by ! quadric2 ! o!r po9ts are improp] ideal po9ts4 ,! angle 7!ta#ab7 betwe5 two planes1 l#ax m#ay n#az r#aw #j & l#bx m#by n#bz r#bw #j1 is giv5 by cos !ta#ab 7l#al#b m#am#b n#an#b - r#ar#b7_/7l#ab m#ab n#ab - r#ab7 7l#bb m#bb n#bb - r#bb7a_/b 7#b7 ,^! planes only h a real angle ( incl9ation if !y possess a l9e ( int]section )9 ! actual space1 i4e4 if !y int]sect4 ,planes :i* d n int]sect possess a %orte/ di/ance along a l9e :i* is p]p5dicul> to bo? ( !m4 ,if ?is %orte/ di/ance is delta#ab1 we h co% 7delta#ab_/;gamma7 7l#al#b m#am#b n#an#b - r#ar#b7_/7l#ab m#ab n#ab - r#ab7 7l#bb m#bb n#bb - r#bb7a_/b 7#c7 ,illu/ration3 ,,fig4 #fg4 ,?us 9 ! case ( ! two planes "o & only "o ( ! two1 !ta#ab & delta#ab1 is real4 ,! same consid]ations hold = coplan> /raight l9es 7see ,,vii4 ,axioms ( ,geometry74 ,let ,o 7fig4 #fg7 be ! po9t 7#j1 #j1 #j1 #a71 ,,ox ! l9e y #j1 z #j1 ,,oy ! l9e z #j1 x #j1 & ,,oz ! l9e x #j1 y #j4 ,^! are ! coord9ate axes & are at "r angles to ea* o!r4 ,let ,p be any po9t1 & let rho be ! di/ance ,,op1 !ta ! angle ,,poz1 & phi ! angle betwe5 ! planes ,,zox & ,,zop4 ,!n ! coord9ates ( ,p c be tak5 to be s9h 7rho_/gamma7 s9 !ta cos phi1 s9h 7rho_/gamma7 s9 !ta s9 phi1 s9h 7rho_/gamma7 cos!ta1 co% 7rho_/gamma74 ,illu/ration3 ,,fig4 #fh4 ,if ,,abc is a triangle1 & ! sides & angles are nam$ accord+ to ! usual conv5tion1 we h s9h 7a_/gamma7_/s9 ,a s9h 7b_/gamma7_/s9 ,b s9h 7c_/gamma7_/s9 ,c1 7#d7 & also co% 7a_/gamma7 co% 7b_/gamma7 co% 7c_/gamma7 - s9h 7b_/gamma7 s9h 7c_/gamma7 cos ,a1 7#e7 ) two simil> equations4 ,! sum ( ! ?ree angles ( a triangle is always less ?an two "r angles4 ,! area ( ! triangle ,,abc is lambda#b7pi - ,a - ,b - ,c74 ,if ! base ,,bc ( a triangle is kept fix$ & ! v]tex ,a moves 9 ! fix$ plane ,,abc s t ! area ,,abc is con/ant1 !n ! locus ( ,a is a l9e ( equal di/ance f ,,bc4 ,?is locus is n a /raight l9e4 ,! :ole !ory ( simil>ity is inapplicable2 two triangles are ei!r congru5t1 or _! angles are n equal two by two4 ,?us ! elem5ts ( a triangle are det]m9$ :5 its ?ree angles are giv54 ,by keep+ ,a & ,b & ! l9e ,,bc fix$1 b by mak+ ,c move (f to inf9ity along ,,bc1 ! l9es ,,bc & ,,ac become p>allel1 & ! sides a & b become inf9ite4 ,h5ce f equation 7#e7 above1 x foll[s t two p>allel l9es 7cf4 ,section ,,vii4 ,axioms ( ,geometry7 mu/ be consid]$ z mak+ a z]o angle ) ea* o!r4 ,also if ,b be a "r angle1 f ! equation 7#e71 rememb]+ t1 9 ! limit1 co% 7a_/gamma7_/co% 7b_/gamma7 co% 7a_/gamma7_/s9h 7b_/gamma7 #a1 we h cos ,a tanh 7c_/#b;gamma7 4444 7#f74 ,! angle ,a is call$ by ,n4,i4 ,lobat*ewsky ! angle ( p>allelism4 ,! :ole !ory ( l9es & planes at "r angles to ea* o!r is simply ! !ory ( conjugate elem5ts ) respect to ! absolute1 :]e ideal l9es & planes are introduc$4 ,?us if l & l' be any two conjugate l9es ) respect to ! absolute 7( :i* "o ( ! two mu/ be improp]1 say l'71 !n any plane ?r\gh l' & conta9+ prop] po9ts is p]p5dicul> to l4 ,also if p is any plane conta9+ prop] po9ts1 & ,p is its pole1 :i* is necess>ily improp]1 !n ! l9es ?r\gh ,p are ! normals to ,p4 ,! equation ( ! sph]e1 c5tre 7x#a1 y#a1 z#a1 w#a7 & radius rho1 is 7w#ab- x#ab- y#ab- z#ab77w#b - x#b - y#b - z#b7 co%#b7rho_/gamma7 7w#aw - x#ax - y#ay -z#az7#b 7#g74 ,! equation ( ! surface ( equal di/ance 7sigma7 f ! plane lx my nz rw #j is 7l#b m#b n#b - r#b77w#b - x#b - y#b - z#b7 s9h#b7sigma_/gamma7 7rw lx my nz7#b 7#h74 ,a surface ( equal di/ance is a sph]e ^: c5tre is improp]2 & bo? types ( surface are includ$ 9 ! family k#b7w#b - x#b - y#b - z#b7 7ax by cz dw7#b 7#i74 ,b ?is family also includes a ?ird type ( surfaces1 :i* c be look$ on ei!r z ! limits ( sph]es ^: c5tres h approa*$ ! absolute1 or z ! limits ( surfaces ( equal di/ance ^: c5tral planes h approa*$ a position tang5tial to ! absolute4 ,^! surfaces are call$ limit-surfaces4 ,?us 7#i7 d5otes a limit-surface1 if d#b - a#b - b#b - c#b #j4 ,two limit-surfaces only diff] 9 position4 ,?us ! two limit-surfaces :i* t\* ! plane ,,yoz at ,o1 b h _! concavities turn$ 9 opposite directions1 h z _! equations w#b - x#b - y#b - z#b 7w - x7#b4 ,! geodesic geometry ( a sph]e is elliptic1 t ( a surface ( equal di/ance is hyp]bolic1 & t ( a limit-surface is p>abolic 7i4e4 ,euclidean74 ,! equation ( ! surface 7cyl9d]7 ( equal di/ance 7delta7 f ! l9e ,,ox is 7w#b - x#b7 tanh#b7;delta_/gamma7 - y#b - z#b #j4 ,?is is n a rul$ surface4 ,h5ce 9 ?is geometry x is n possible = two /raight l9es to be at a con/ant di/ance f ea* o!r4 ,secondly1 let ! equation ( ! absolute be x#b y#b z#b w#b #j4 ,! absolute is n[ imag9>y & ! geometry is elliptic4 ,! di/ance 7d#ab7 betwe5 ! two po9ts 7x#a1 y#a1 z#a1 w#a7 & 7x#b1 y#b1 z#b1 w#b7 is giv5 by cos 7d#ab_/;gamma7 -7x#ax#b y#ay#b z#az#b w#aw#b7 _/ 7x#ab y#ab z#ab w#ab7 7x#bb y#bb z#bb w#bb7a_/b 7#aj74 ,?us !re are two di/ances betwe5 ! po9ts1 & if "o is d#ab1 ! o!r is pigamma-d#ab4 ,e /raight l9e returns into itself1 =m+ a clos$ s]ies4 ,?us !re are two segm5ts betwe5 any two po9ts1 toge!r =m+ ! :ole l9e :i* conta9s !m2 "o di/ance is associat$ ) "o segm5t1 & ! o!r di/ance ) ! o!r segm5t4 ,! complete l5g? ( e /raight l9e is pigamma4 ,! angle betwe5 ! two planes l#ax m#ay n#az r #aw #j & l#bx m#by n#bz r#bw #j is cos !ta#ab 7l#al#b m#am#b n#an#b r#ar#b7_/ 7l#ab m#ab n#ab r#ab7 7l#bb m#bb n#bb r#bb7a_/b 7#aa74 ,! pol> plane ) respect to ! absolute ( ! po9t 7x#a1 y#a1 z#a1 w#a7 is ! real plane x#ax y#ay z#az w#aw #j1 & ! pole ( ! plane l#ax m#ay n#az r#aw #j is ! po9t 7l#a1 m#a1 n#a1 r#a74 ,?us 7f equations #aj & #aa7 x foll[s t ! angle betwe5 ! pol> planes ( ! po9ts 7x#a1 4447 & 7x#b1 4447 is d#ab_/;gamma1 & t ! di/ance betwe5 ! poles ( ! planes 7l#a1 4447 & 7l#b1 4447 is gamma!ta#ab4 ,?us !re is complete reciprocity betwe5 po9ts & planes 9 respect to all prop]ties4 ,?is complete reign ( ! pr9ciple ( duality is "o ( ! great beauties ( ?is geometry4 ,! !ory ( l9es & planes at "r angles is simply ! !ory ( conjugate elem5ts ) respect to ! absolute4 ,a tetrah$ron self-conjugate ) respect to ! absolute has all its int]sect+ elem5ts 7edges & planes7 at "r angles4 ,if l & l' are two conjugate l9es1 ! planes ?r\gh "o are ! planes p]p5dicul> to ! o!r4 ,if ,p is ! pole ( ! plane p1 ! l9es ?r\gh ,p are ! normals to ! plane p4 ,! di/ance f ,p to p is #a_/bpigamma4 ,?us e sph]e is also a surface ( equal di/ance f ! pol> ( its c5tre1 & conv]sely4 ,a plane does n divide space2 = ! l9e jo9+ any two po9ts ,p & ,q only cuts ! plane once1 9 ,l say1 !n x is always possible to g f ,p to ,q by ! segm5t ( ! l9e ,,pq :i* does n conta9 ,l4 ,b ,p & ,q may be said to be sep>at$ by a plane p1 if ! po9t 9 :i* ,,pq cuts p lies on ! %orte/ segm5t betwe5 ,p & ,q4 ,) ?is s5se ( sep>ation1 x is possible#b to f9d ?ree po9ts ,p1 ,q1 ,r su* t ,p & ,q are sep>at$ by ! plane p1 b ,p & ,r are n sep>at$ by p1 nor are ,q & ,r4 ,let ,a1 ,b1 ,c be any ?ree non-coll9e> po9ts1 !n f\r triangles are def9$ by ^! po9ts4 ,?us if a1 b1 c & ,a1 ,b1 ,c are ! elem5ts ( any "o triangle1 !n ! f\r triangles h z _! elem5ts3 7#a7 a1 b1 c1 ,a1 ,b1 ,c4 7#b7 a1 pigamma - b1 pigamma - c1 ,a1 pi - ,b1 pi - ,c4 7#c7 pigamma - a1 b1 pigamma - c1 pi - ,a1 ,b1 pi - ,c4 7#d7 pigamma - a1 pigamma - b1 c1 pi - ,a1 pi - ,b1 ,c4 ,! =mulae connect+ ! elem5ts are s9 ,a_/s9 7a_/gamma7 s9 ,b_/s9 7b_/gamma7 s9 ,c_/s9 7c_/gamma71 7#ab7 & cos 7a_/gamma7 cos 7b_/gamma7 cos 7c_/gamma7 s9 7b_/gamma7 s9 7c_/gamma7 cos ,a1 7#ac7 ) two simil> equations4 ,two cases arise1 namely 7,i47 accord+ z "o ( ! f\r triangles has z its sides ! %orte/ segm5ts betwe5 ! angul> po9ts1 or 7,,ii47 accord+ z ?is is n ! case4 ,:5 case ,i4 holds !re is said to be a pr9cipal triangle4#c ,if all ! figures consid]$ lie )9 a sph]e ( radius #a_/dpigamma only case ,i4 c hold1 & ! pr9cipal triangle is ! triangle :olly )9 ?is sph]e1 also ! peculi>ities 9 respect to ! sep>ation ( po9ts by a plane _c !n arise4 ,! sum ( ! ?ree angles ( a triangle ,,abc is always great] ?an two "r angles1 & ! area ( ! triangle is gamma#b7;,a ,b ,c--pi74 ,?us z 9 hyp]bolic geometry ! !ory ( simil>ity does n hold1 & ! elem5ts ( a triangle are det]m9$ :5 its ?ree angles are giv54 ,! coord9ates ( a po9t c be writt5 9 ! =m s9 7rho_/gamma7 s9 ,phi cos phi1 s9 7rho_/gamma7 s9 ,phi s9 phi1 s9 7rho_/gamma7 cos ,phi1 cos 7rho_/gamma71 :]e rho1 ,phi & phi h ! same mean+s z 9 ! correspond+ =mulae 9 hyp]bolic geometry4 ,aga91 suppose a wat* is laid on ! plane ,,oxy1 face upw>ds ) its c5tre at ,o1 & ! l9e #ab to #f 7z m>k$ on dial7 along ! l9e ,,yoy4 ,let ! wat* be cont9ually pu%$ along ! plane along ! l9e ,,ox1 t is1 9 ! direction #i to #c4 ,!n ! l9e ,,xox be+ ( f9ite l5g?1 ! wat* w return to ,o1 b at its fir/ return x w be f\nd to be face d[nw>ds on ! o!r side ( ! plane1 ) ! l9e #ab to #f rev]s$ 9 direction along ! l9e ,,yoy4 ,?is peculi>ity 0 fir/ po9t$ out by ,felix ,kle94 ,! !ory ( p>allels z x exi/s 9 hyp]bolic space has no application 9 elliptic geometry4 ,b ano!r prop]ty ( ,euclidean p>allel l9es holds 9 elliptic geometry1 & by ! use ( x p>allel l9es are def9$4 ,= ! equation ( ! surface 7cyl9d]7 ( equal di/ance 7delta7 f ! l9e ,,xox is 7x#b w#b7 tan#b7;delta_/gamma7 - 7y#b z#b7 #j4 ,?is is also ! surface ( equal di/ance1 #a_/bpigamma-delta1 f ! l9e conjugate to ,,xox4 ,n[ f ! =m ( ! above equation ?is is a rul$ surface1 & ?r\gh e po9t ( x two g5]ators pass4 ,b ^! g5]ators are l9es ( equal di/ance f ,,xox4 ,?us ?r\gh\t e po9t ( space two l9es c be drawn :i* are l9es ( equal di/ance f a giv5 l9e l4 ,?is prop]ty 0 discov]$ by ,w4,k4 ,clif=d4 ,! two l9es are call$ ,clif=d's "r & left p>allels to l ?r\gh ! po9t4 ,?is prop]ty ( p>allelism is reciprocal1 s t if m is a left p>allel to l1 !n l is a left p>allel to m4 ,note also t two p>allel l9es l & m are n coplan>4 ,_m ( ^? prop]ties ( ,euclidean p>allels1 :i* d n hold = ,lobat*ewsky's p>allels 9 hyp]bolic geometry1 d hold = ,clif=d's p>allels 9 elliptic geometry4 ,! geodesic geometry ( sph]es is elliptic1 ! geodesic geometry ( surfaces ( equal di/ance f l9es 7cyl9d]s7 is ,euclidean1 & surfaces ( revolution c be f\nd#d ( :i* ! geodesic geometry is hyp]bolic4 ,b x is to be notic$ t ! connectivity ( ^! surfaces is diff]5t to t ( a ,euclidean plane4 ,= in/ance !re are only inf9#b congru5ce trans=mations ( ! cyl9drical surfaces ( equal di/ance into !mselves1 in/ead ( ! inf9#c = ! ord9>y plane4 ,x w\ld obvi\sly be possible to /ate axioms :i* ^! geodesics satisfy1 & ?us to def9e indep5d5tly1 & n z loci1 quasi-spaces ( ^! peculi> types4 ,! exi/5ce ( su* ,euclidean quasi-geometries 0 fir/ po9t$ out by ,clif=d4#e ,9 bo? elliptic & hyp]bolic geometry ! sph]ical geometry1 i4e4 ! relations betwe5 ! angles =m$ by l9es & planes pass+ ?r\gh ! same po9t1 is ! same z ! sph]ical trigonometry 9 ,euclidean geometry4 ,! con/ant gamma1 :i* appe>s 9 ! =mulae bo? ( hyp]bolic & elliptic geometry1 does n by its v>iation produce diff]5t types ( geometry4 ,!re is only "o type ( elliptic geometry & "o type ( hyp]bolic geometry2 & ! magnitude ( ! con/ant gamma 9 ea* case simply dep5ds ^u ! magnitude ( ! arbitr>y unit ( l5g? 9 comp>ison ) ! natural unit ( l5g? :i* ea* p>ticul> in/ance ( ei!r geometry pres5ts4 ,! exi/5ce ( a natural unit ( l5g? is a peculi>ity common bo? to hyp]bolic & elliptic geometries1 & diff]5tiates !m f ,euclidean geometry4 ,x is ! reason = ! failure ( ! !ory ( simil>ity 9 !m4 ,if gamma is v l>ge1 t is1 if ! natural unit is v l>ge comp>$ to ! arbitr>y unit1 & if ! l5g?s involv$ 9 ! figures consid]$ are n l>ge comp>$ to ! arbitr>y unit1 !n bo? ! elliptic & hyp]bolic geometries approximate to ! ,euclidean4 ,= f =mulae 7#d7 & 7#e7 & also f 7#ab7 & 7#ac7 we f9d1 aft] reta9+ only ! l[e/ p[]s ( small quantities1 z ! =mulae = any triangle ,,abc1 a_/ s9 ,a b_/ s9 ,b c_/ s9 ,c1 & a#b b#b c#b - #b;bc cos ,a1 ) two simil> equations4 ,?us ! geometries ( small figures are 9 bo? types ,euclidean4 ,!ory ( p>allels be=e ,gauss4 ,hi/ory4--,9 pul*]rimo ,geometriae corpore1 wrote ,sir ,h5ry ,savile 9 #afba1 duo sunt naevi1 duae labes nec quod sciam plures1 9 quibus elu5dis et emacul5dis cum vet]um tum rec5tiorum 444 vigilavit indu/ria4 ,^! two blemi%es are ! !ory ( p>allels & ! !ory ( proportion4 ,! indu/ry ( ! mod]ns1 9 bo? respects1 has giv5 rise to important bran*es ( ma!matics1 :ile at ! same "t %[+ t ,euclid is 9 ^! respects m free f blemi% ?an _h be5 previ\sly cr$ible4 ,x 0 f endeav\rs to improve ! !ory ( p>allels t non-,euclidean geometry arose2 & ?\gh x has n[ acquir$ a f> wid] scope1 its hi/orical orig9 rema9s in/ructive & int]e/+4 ,euclid's axiom ( p>allels appe>s z ,po/ulate ,v4 to ! fir/ book ( 8 ,elem5ts1 & is /at$ ?us1 ,& t1 if a /raight l9e fall+ on two /raight l9es make ! angles1 int]nal & on ! same side1 less ?an two "r angles1 ! two /raight l9es1 be+ produc$ indef9itely1 meet on ! side on :i* are ! angles less ?an two "r angles4 ,! orig9al ,greek is ,greek3 kai ean eis duo eu!ias eu!ia empipt\sa tas entos kai epi ta auta m]e gonias duo or?on elassonas poie1 ekballom5as tas duo eu!ias ep' apeiron sympipte91 eph' ha m]e eis9 hai ton duo or?on elassones4 ,to ,euclid's successors ?is axiom _h signally fail$ to appe> self-evid5t1 & _h fail$ equally to appe> indemon/rable4 ,)\t ! use ( ! po/ulate its conv]se is prov$ 9 ,euclid's #bh? proposition1 & x 0 hop$ t by fur!r ef=ts ! po/ulate itself c\ld be also prov$4 ,! fir/ /ep consi/$ 9 ! discov]y ( equival5t axioms4 ,*ri/oph ,clavius 9 #aegd d$uc$ ! axiom f ! assumption t a l9e ^: po9ts are all equidi/ant f a /raight l9e is itself /raight4 ,john ,wallis 9 #affc %[$ t ! po/ulate foll[s f ! possibility ( simil> triangles on diff]5t scales4 ,girolamo ,sac*]i 7#agcc7 %[$ t x is suffici5t to h a s+le triangle1 ! sum ( ^: angles is two "r angles4 ,o!r equival5t =ms may be obta9$1 b none %[s any ess5tial sup]iority to ,euclid's4 ,inde$ plausibility1 :i* is *iefly aim$ at1 becomes a positive dem]it :]e x conceals a real assumption4 ,sac*]i4 ,a new me?od1 :i*1 ?\gh x fail$ to lead to ! desir$ goal1 prov$ 9 ! end imm5sely fruitful1 0 inv5t$ by ,sac*]i1 9 a "w entitl$ ,euclides ab omni naevo v9dicatus 7,milan1 #agcc74 ,if ! po/ulate ( p>allels is involv$ 9 ,euclid's o!r assumptions1 contradictions mu/ em]ge :5 x is d5i$ :ile ! o!rs are ma9ta9$4 ,?is l$ ,sac*]i to attempt a r$uctio ad absurdum1 9 :i* he mi/ak5ly believ$ himself to h succe$$4 ,:at is int]e/+1 h[ev]1 is n 8 fallaci\s conclusion1 b ! non-,euclidean results :i* he obta9s 9 ! process4 ,sac*]i di/+ui%es ?ree hypo!ses 7correspond+ to :at are n[ kn[n z ,euclidean or p>abolic1 elliptic & hyp]bolic geometry71 & proves t "s "o ( ! ?ree mu/ be univ]sally true4 ,8 ?ree hypo!ses are ?us obta9$3 equal p]p5dicul>s ,,ac1 ,,bd are drawn f a /raight l9e ,,ab1 & ,,cd are jo9$4 ,x is %[n t ! angles ,,acd1 ,,bdc are equal4 ,! fir/ hypo!sis is t ^! are bo? "r angles2 ! second1 t !y are bo? obtuse2 & ! ?ird1 t !y are bo? acute4 ,_m ( ! results aft]w>ds obta9$ by ,lobat*ewsky & ,bolyai are "h develop$4 ,sac*]i fails to be ! f\nd] ( non-,euclidean geometry only because he does n p]ceive ! possible tru? ( 8 non-,euclidean hypo!ses4 ,lamb]t4 ,"s advance is made by ,johann ,he9ri* ,lamb]t 9 8 ,!orie d] ,p>allell9i5 7writt5 #agff2 po/hum\sly publi%$ #aghf74 ,?\gh he /ill believ$ 9 ! necess>y tru? ( ,euclidean geometry1 he confess$ t1 9 all 8 attempt$ pro(s1 some?+ rema9$ undemon/rat$4 ,he deals ) ! same ?ree hypo!ses z ,sac*]i1 %[+ t ! second holds on a sph]e1 :ile ! ?ird w\ld hold on a sph]e ( purely imag9>y radius4 ,! second hypo!sis he succe$s 9 condemn+1 s9ce1 l all :o prec$$ ,b]nh>d ,riemann1 he is unable to conceive ( ! /raight l9e z f9ite & clos$4 ,b ! ?ird hypo!sis1 :i* is ! same z ,lobat*ewsky's1 is n ev5 pr(ess$ly refut$4#f ,?ree p]iods ( non-,euclidean geometry4 ,non-,euclidean geometry prop] beg9s ) ,k>l ,fri$ri* ,gauss4 ,! advance :i* he made 0 r philosophical ?an ma!matical3 x 0 he 7probably7 :o fir/ recogniz$ t ! po/ulate ( p>allels is possibly false1 & %\ld be empirically te/$ by measur+ ! angles ( l>ge triangles4 ,! hi/ory ( non-,euclidean geometry has be5 aptly divid$ by ,felix ,kle9 into ?ree v di/9ct p]iods4 ,! fir/--:i* conta9s only ,gauss1 ,lobat*ewsky & ,bolyai--is *>act]iz$ by its syn!tic me?od & by its close relation to ,euclid4 ,! attempt at indirect pro( ( ! disput$ po/ulate w\ld seem to h be5 ! s\rce ( ^! ?ree m5's discov]ies2 b :5 ! po/ulate _h be5 d5i$1 !y f\nd t ! results1 s f> f %[+ contradictions1 7 j z self-consi/5t z ,euclid4 ,!y inf]r$ t ! po/ulate1 if true at all1 c only be prov$ by obs]vations & measurem5ts4 ,only "o k9d ( non-,euclidean space is kn[n to !m1 namely1 t :i* is n[ call$ hyp]bolic4 ,! second p]iod is analytical1 & is *>act]iz$ by a close relation to ! !ory ( surfaces4 ,x beg9s ) ,riemann's inaugural diss]tation1 :i* reg>ds space z a p>ticul> case ( a manifold2 b ! *>act]i/ic /&po9t ( ! p]iod is *iefly emphasiz$ by ,eug5io ,beltrami4 ,! conception ( measure ( curvature is ext5d$ by ,riemann f surfaces to spaces1 & a new k9d ( space1 f9ite b unb\nd$ 7correspond+ to ! second hypo!sis ( ,sac*]i & ,lamb]t71 is %[n to be possible4 ,z oppos$ to ! second p]iod1 :i* is purely metrical1 ! ?ird p]iod is ess5tially projective 9 its me?od4 ,x beg9s ) ,ar?ur ,cayley1 :o %[$ t metrical prop]ties are projective prop]ties relative to a c]ta9 fundam5tal quadric1 & t diff]5t geometries arise accord+ z ?is quadric is real1 imag9>y or deg5]ate4 ,kle91 to :om ! developm5t ( ,cayley's "w is due1 %[$ fur!r t !re are two =ms ( ,riemann's space1 call$ by him ! elliptic & ! sph]ical4 ,f9ally1 x has be5 %[n by ,sophus ,lie1 t if figures are to be freely movable ?r\gh\t all space 9 oo#f ways1 no o!r ?ree-dim5sional spaces ?an ! above f\r are possible4 ,gauss4 ,gauss publi%$ no?+ on ! !ory ( p>allels1 & x 0 n g5]ally kn[n until aft] 8 dea? t he _h int]e/$ himself 9 t !ory f a v e>ly date4 ,9 #agii he ann\nces t ,euclidean geometry w\ld foll[ f ! assumption t a triangle c be drawn great] ?an any giv5 triangle4 ,?\gh unwill+ to assume ?is1 we f9d him 9 #ahjd /ill hop+ to prove ! po/ulate ( p>allels4 ,9 #ahcj he ann\nces 8 conviction t geometry is n an a priori sci5ce2 9 ! foll[+ ye> he expla9s t non-,euclidean geometry is free f contradictions1 & t1 9 ?is sy/em1 ! angles ( a triangle dim9i% )\t limit :5 all ! sides are increas$4 ,he also gives = ! circumf]5ce ( a circle ( radius r ! =mula pik7e7r_/k7 - e7r-_/k771 :]e k is a con/ant dep5d+ ^u ! nature ( ! space4 ,9 #ahcb1 9 reply to ! receipt ( ,bolyai's ,app5dix1 he gives an elegant pro( t ! am\nt by :i* ! sum ( ! angles ( a triangle falls %ort ( two "r angles is proportional to ! area ( ! triangle4 ,f ^! & a few o!r rem>ks x appe>s t ,gauss possess$ ! f\ndations ( hyp]bolic geometry1 :i* he 0 probably ! fir/ to reg>d z p]haps true4 ,x is n kn[n ) c]ta9ty :e!r he influ5c$ ,lobat*ewsky & ,bolyai1 b ! evid5ce we possess is aga9/ su* a view4#g ,lobat*ewsky4 ,! fir/ to publi% a non-,euclidean geometry 0 ,ni*olas ,lobat*ewsky1 pr(essor ( ma!matics 9 ! new univ]sity ( ,kazan4#h ,9 ! place ( ! disput$ po/ulate he puts ! foll[+3 ,all /raight l9es :i*1 9 a plane1 radiate f a giv5 po9t1 c1 ) respect to any o!r /raight l9e 9 ! same plane1 be divid$ into two classes1 ! int]sect+ & ! non-int]sect+4 ,! b\nd>y l9e ( ! "o & ! o!r class is call$ p>allel to ! giv5 l9e4 ,x foll[s t !re are two p>allels to ! giv5 l9e ?r\gh any po9t1 ea* meet+ ! l9e at inf9ity1 l a ,euclidean p>allel4 7,h5ce a l9e has two di/9ct po9ts at inf9ity1 & n "o only z 9 ord9>y geometry47 ,! two p>allels to a l9e ?r\gh a po9t make equal acute angles ) ! p]p5dicul> to ! l9e ?r\gh ! po9t4 ,if p be ! l5g? ( ! p]p5dicul>1 ei!r ( ^! angles is d5ot$ by ,pi7p74 ,! det]m9ation ( ,pi7p7 is ! *ief problem 7cf4 equation 7#f7 above72 x appe>s f9ally t1 ) a suitable *oice ( ! unit ( l5g?1 tan #a_/b ,pi7p7 e7-p74 ,be=e obta9+ ?is result x is %[n t sph]ical trigonometry is un*ang$1 & t ! normals to a circle or a sph]e /ill pass ?r\gh its c5tre4 ,:5 ! radius ( ! circle or sph]e becomes inf9ite all ^! normals become p>allel1 b ! circle or sph]e does n become a /raight l9e or plane4 ,x becomes :at ,lobat*ewsky calls a limit-l9e or limit-surface4 ,! geometry on su* a surface is %[n to be ,euclidean1 limit-l9es replac+ ,euclidean /raight l9es4 7,x is1 9 fact1 a surface ( z]o measure ( curvature47 ,by ! help ( ^! propositions ,lobat*ewsky obta9s ! above value ( ,pi7p71 & !nce ! solution ( triangles4 ,he po9ts out t 8 =mulae result f ^? ( sph]ical trigonometry by sub/itut+ ia1 ib1 ic1 = ! sides a1 b1 c4 ,bolyai4 ,john ,bolyai1 a ,hung>ian1 obta9$ results closely correspond+ to ^? ( ,lobat*ewsky4 ,^! he publi%$ 9 an app5dix to a "w by 8 fa!r1 entitl$ ,app5dix ,sci5tiam spatii absolute v]am exhib5s3 a v]itate aut falsitate ,axiomatis ,,xi4 ,euclidei 7a priori haud unquam decid5da7 indep5d5tem3 adjecta ad casum falsitatis1 quadratura circuli geometrica4#i ,?is "w 0 publi%$ 9 #ahca1 b its conception dates f #ahbc4 ,x reveals a pr(\nd] appreciation ( ! importance ( ! new ideas1 b o!rwise diff]s little f ,lobat*ewsky's4 ,bo? m5 po9t out t ,euclidean geometry z a limit+ case ( _! own m g5]al sy/em1 t ! geometry ( v small spaces is always approximately ,euclidean1 t no a priori gr\nds exi/ = a decision1 & t obs]vation c only give an approximate answ]4 ,bolyai gives also1 z 8 title indicates1 a geometrical con/ruction1 9 hyp]bolic space1 = ! quadrature ( ! circle1 & %[s t ! area ( ! greate/ possible triangle1 :i* has all its sides p>allel & all its angles z]o1 is piiota#b1 :]e i is :at we %\ld n[ call ! space-con/ant4 ,riemann4 ,! works ( ,lobat*ewsky & ,bolyai1 ?\gh kn[n & valu$ by ,gauss1 rema9$ obscure & ineffective until1 9 #ahff1 !y 7 translat$ into ,fr5* by ,j4 ,h\el4 ,b at ?is "t ,riemann's diss]tation1 ,ub] die ,hypo!s51 wel*e d] ,geometrie zu ,grunde lieg51#aj 0 already ab\t to be publi%$4 ,9 ?is "w ,riemann1 )\t any k ( 8 pr$ecessors 9 ! same field1 inaugurat$ a f> m pr(\nd discussion1 bas$ on a f> m g5]al /&po9t2 & by its publication 9 #ahfg ! att5tion ( ma!maticians & philosoph]s 0 at la/ secur$4 7,! diss]tation dates f #ahed1 b ow+ to *anges :i* ,riemann wi%$ to make 9 x1 x rema9$ unpubli%$ until aft] 8 dea?47 ,def9ition ( a manifold4 ,riemann's "w conta9s two fundam5tal conceptions1 t ( a manifold & t ( ! measure ( curvature ( a cont9u\s manifold possess$ ( :at he calls flatness 9 ! smalle/ p>ts4 ,by means ( ^! conceptions space is made to appe> at ! end ( a gradual s]ies ( m & m specializ$ conceptions4 ,conceptions ( magnitude1 he expla9s1 are only possible :]e we h a g5]al conception capable ( det]m9ation 9 v>i\s ways4 ,! manifold consi/s ( all ^! v>i\s det]m9ations1 ea* ( :i* is an elem5t ( ! manifold4 ,! passage f "o elem5t to ano!r may be discrete or cont9u\s2 ! manifold is call$ discrete or cont9u\s accord+ly4 ,:]e x is discrete two portions ( x c be comp>$1 z to magnitude1 by c\nt+2 :]e cont9u\s1 by measurem5t4 ,b measurem5t dem&s sup]position1 & consequ5tly "s magnitude indep5d5t ( its place 9 ! manifold4 ,9 pass+1 9 a cont9u\s manifold1 f "o elem5t to ano!r 9 a det]m9ate way1 we pass ?r\gh a s]ies ( int]m$iate t]ms1 :i* =m a "o-dim5sional manifold4 ,if ?is :ole manifold be simil>ly caus$ to pass ov] into ano!r1 ea* ( its elem5ts passes ?r\gh a "o-dim5sional manifold1 & ?us on ! :ole a two-dim5sional manifold is g5]at$4 ,9 ?is way we c proce$ to n dim5sions4 ,conv]sely1 a manifold ( n dim5sions c be analys$ into "o ( "o dim5sion & "o ( 7n - #a7 dim5sions4 ,by repetitions ( ?is process ! position ( an elem5t may be at la/ det]m9$ by n magnitudes4 ,we may "h /op to obs]ve t ! above conception ( a manifold is ak9 to t due to ,h]mann ,grassmann 9 ! fir/ edition 7#ahdg7 ( 8 ,ausdehnungslehre4#aa ,measure ( curvature4 ,bo? concepts h be5 elaborat$ & sup]s$$ by ! mod]n proc$ure 9 respect to ! axioms ( geometry1 & by ! conception ( ab/ract geometry involv$ !re94 ,riemann proce$s to specialize ! manifold by consid]ations z to measurem5t4 ,if measurem5t is to be possible1 "s magnitude1 we saw1 mu/ be indep5d5t ( position2 let u consid] manifolds 9 :i* l5g?s ( l9es are su* magnitudes1 s t e l9e is measurable by e o!r4 ,! coord9ates ( a po9t be+ x#a1 x#b1 444 xn1 let u conf9e ourselves to l9es along :i* ! ratios dx#a 3 dx#b 3 444 3 dxn alt] cont9u\sly4 ,let u also assume t ! elem5t ( l5g?1 ds1 is un*ang$ 7to ! fir/ ord]7 :5 all its po9ts und]go ! same inf9itesimal motion4 ,!n if all ! increm5ts dx be alt]$ 9 ! same ratio1 ds is also alt]$ 9 ?is ratio4 ,h5ce ds is a homog5e\s function ( ! fir/ degree ( ! increm5ts dx4 ,moreov]1 ds mu/ be un*ang$ :5 all ! dx *ange sign4 ,! simple/ possible case is1 !re=e1 t 9 :i* ds is ! squ>e root ( a quadratic function ( ! dx4 ,?is case includes space1 & is alone consid]$ 9 :at foll[s4 ,x is call$ ! case ( flatness 9 ! smalle/ p>ts4 ,its fur!r discussion dep5ds ^u ! measure ( curvature1 ! second ( ,riemann's fundam5tal conceptions4 ,?is conception1 d]iv$ f ! !ory ( surfaces1 is appli$ z foll[s4 ,any "o ( ! %orte/ l9es :i* issue f a giv5 po9t 7say ! orig97 is completely det]m9$ by ! initial ratios ( ! dx4 ,two su* l9es1 def9$ by dx & deltax say1 det]m9e a p5cil1 or "o-dim5sional s]ies1 ( %orte/ l9es1 any "o ( :i* is def9$ by lambdadx mudeltax1 :]e ! p>amet] lambda 3 mu may h any value4 ,?is p5cil g5]ates a two-dim5sional s]ies ( po9ts1 :i* may be reg>d$ z a surface1 & = :i* we may apply ,gauss's =mula = ! measure ( curvature at any po9t4 ,?us at e po9t ( our manifold !re is a measure ( curvature correspond+ to e su* p5cil2 b all ^! c be f\nd :5 n4_/7n-#a7_/b ( !m are kn[n4 ,if figures are to be freely movable1 x is necess>y & suffici5t t ! measure ( curvature %\ld be ! same = all po9ts & all directions at ea* po9t4 ,:]e ?is is ! case1 if alpha be ! measure ( curvature1 ! l9e> elem5t c be put into ! =m ds root7,sigmadx#b7_/7a #a_/d;alpha,sigmax#b74 ,if alpha be positive1 space is f9ite1 ?\gh /ill unb\nd$1 & e /raight l9e is clos$--a possibility fir/ recogniz$ by ,riemann4 ,x is po9t$ out t1 s9ce ! possible values ( a =m a cont9u\s s]ies1 obs]vations _c prove t our space is /rictly ,euclidean4 ,x is also reg>d$ z possible t1 9 ! inf9itesimal1 ! measure ( curvature ( our space %\ld be v>iable4 ,!re are f\r po9ts 9 :i* ?is pr(\nd & epo*-mak+ "w is op5 to criticism or developm5t--7#a7 ! idea ( a manifold requires m precise det]m9ation2 7#b7 ! introduction ( coord9ates is entirely unexpla9$ & ! requisite presuppositions are unanalys$2 7#c7 ! assumption t ds is ! squ>e root ( a quadratic function ( dx#a1 dx#b1 444 is arbitr>y2 7#d7 ! idea ( sup]position1 or congru5ce1 is n adequately analys$4 ,! mod]n solution ( ^! difficulties is prop]ly consid]$ 9 connexion ) ! g5]al subject ( ! axioms ( geometry4 ,helmholtz4 ,! publication ( ,riemann's diss]tation 0 closely foll[$ by two works ( ,h]mann von ,helmholtz1#ab aga9 und]tak5 9 ignorance ( ! "w ( pr$ecessors4 ,9 ^! a pro( is attempt$ t ds mu/ be a rational integral quadratic function ( ! increm5ts ( ! coord9ates4 ,?is pro( has s9ce be5 %[n by ,lie to /& 9 ne$ ( correction 7see ,,vii4 ,axioms ( ,geometry74 ,helmholtz's rema9+ works on ! subject#ac are ( almo/ exclusively philosophical int]e/4 ,we %all return to !m lat]4 ,beltrami4 ,! only o!r writ] ( importance 9 ! second p]iod is ,eug5io ,beltrami1 by :om ,riemann's "w 0 br\ght into connexion ) t ( ,lobat*ewsky & ,bolyai4 ,z he gave1 by an elegant me?od1 a conv5i5t ,euclidean int]pretation ( hyp]bolic plane geometry1 8 results w be /at$ at "s l5g?#ad4 ,! ,saggio %[s t ,lobat*ewsky's plane geometry holds 9 ,euclidean geometry on surfaces ( con/ant negative curvature1 /raight l9es be+ replac$ by geodesics4 ,su* surfaces are capable ( a con=mal repres5tation on a plane1 by :i* geodesics are repres5t$ by /raight l9es4 ,h5ce if we take1 z coord9ates on ! surface1 ! ,c>tesian coord9ates ( correspond+ po9ts on ! plane1 ! geodesics mu/ h l9e> equations4 ,h5ce x foll[s t ds#b ,r#bw7-#d77;alpha#b - v#b7;du#b #buvdudv 7alpha#b - u#b7;dv#b :]e w#b alpha#b - u#b - v#b1 & 7-#a7_/,r#b is ! measure ( curvature ( our surface 7note t k gamma z us$ above74 ,! angle betwe5 two geodesics u con/41 v con/4 is !ta1 :]e cos !ta uv_/root7alpha#b - u#b77;alpha#b - v#b71 s9 !ta aw_/root7a#b - u#b77;a#b - v#b74 ,?us u #j is or?ogonal to all geodesies v con/41 & vice v]sa4 ,9 ord] t s9 !ta may be real1 w#b mu/ be positive2 ?us geodesics h no real int]section :5 ! correspond+ /raight l9es int]sect outside ! circle u#b v#b alpha#b4 ,:5 !y int]sect on ?is circle1 !ta #j4 ,?us ,lobat*ewsky's p>allels are repres5t$ by /raight l9es int]sect+ on ! circle4 ,aga91 trans=m+ to pol> coord9ates u r cos mu1 v r s9 mu1 & call+ rho ! geodesic di/ance ( u1 v f ! orig91 we h1 = a geodesic ?r\gh ! orig91 drho ,radr_/7a#b - r#b71 rho #a_/b,r log7a r7_/7a - r71 r a tan h 7rho_/,r74 ,?us po9ts on ! surface correspond+ to po9ts 9 ! plane on ! limit+ circle r a1 are all at an inf9ite di/ance f ! orig94 ,aga91 consid]+ r con/ant1 ! arc ( a geodesic circle subt5d+ an angle mu at ! orig9 is sigma ,rrmu_/root7a#b - r#b7 mu,r s9 h 7rho_/,r71 :5ce ! circumf]5ce ( a circle ( radius rho is #bpi,r s9 h 7rho_/,r74 ,aga91 if alpha be ! angle betwe5 any two geodesics ,v - v m7,u - u71 ,v - v n7,u - u71 !n tan alpha a7n - m7w_/7#a mn7a#b - 7v - mu7 7v - nu74 ,?us alpha is imag9>y :5 u1 v is outside ! limit+ circle1 & is z]o :51 & only :51 u1 v is on ! limit+ circle4 ,all ^! results agree ) ^? ( ,lobat*ewsky & ,bolyai4 ,! maximum triangle1 ^: angles are all z]o1 is repres5t$ 9 ! auxili>y plane by a triangle inscrib$ 9 ! limit+ circle4 ,! angle ( p>allelism is also easily obta9$4 ,! p]p5dicul> to v #j at a di/ance delta f ! orig9 is u a tan h 7delta_/,r71 & ! p>allel to ?is ?r\gh ! orig9 is u v s9 h 7delta_/,r74 ,h5ce ,pi 7delta71 ! angle :i* ?is p>allel makes ) v #j1 is giv5 by tan ,pi7delta7 4 s9 h 7delta_/,r7 #a1 or tan #a_/b,pi7delta7 e7-delta_/,r7 :i* is ,lobat*ewsky's =mula4 ,we also obta9 easily = ! area ( a triangle ! =mula ,r#b7pi - ,a - ,b - ,c74 ,beltrami's treatm5t connects two curves :i*1 9 ! e>li] treatm5t1 _h no connexion4 ,^! are limit-l9es & curves ( con/ant di/ance f a /raight l9e4 ,bo? may be reg>d$ z circles1 ! fir/ hav+ an inf9ite1 ! second an imag9>y radius4 ,! equation to a circle ( radius rho & c5tre u#jv#j is 7a#b - uu#j - vv#j7b cos h#b 7rho_/,r7w#jbw#b ,c#bw#b 7say74 ,?is equation rema9s real :5 rho is a pure imag9>y1 & rema9s f9ite :5 w#j #j1 provid$ rho becomes inf9ite 9 su* a way t w#j cos h 7rho_/,r7 rema9s f9ite4 ,9 ! latt] case ! equation repres5ts a limit-l9e4 ,9 ! =m] case1 by giv+ diff]5t values to ,c1 we obta9 conc5tric circles ) ! imag9>y c5tre u#jv#j4 ,"o ( ^!1 obta9$ by putt+ ,c #j1 is ! /raight l9e a#b - uu#j - vv#j #j4 ,h5ce ! o!rs are ea* ?r\gh\t at a con/ant di/ance f ?is l9e4 7,x may be %[n t all motions 9 a hyp]bolic plane consi/1 9 a g5]al s5se1 ( rotations2 b ?ree types mu/ be di/+ui%$ accord+ z ! c5tre is real1 imag9>y or at inf9ity4 ,all po9ts describe1 accord+ly1 "o ( ! ?ree types ( circles47 ,! above ,euclidean int]pretation fails = ?ree or m dim5sions4 ,9 ! ,teoria fondam5tale1 accord+ly1 :]e n dim5sions are consid]$1 ,beltrami treats hyp]bolic space 9 a purely analytical _s4 ,! pap] %[s t ,lobat*ewsky's space ( any numb] ( dim5sions has1 9 ,riemann's s5se1 a con/ant negative measure ( curvature4 ,beltrami />ts ) ! =mula 7analog\s to t ( ! ,saggio7 ds#b ,r#bx7-#b77;dx#b dx#ab dx#bb 444 dxn#b7 :]e x#b x#ab x#bb 444 xn#b a#b4 ,he %[s t geodesics are repres5t$ by l9e> equations betwe5 x#a1 x#b1 4441 xn1 & t ! geodesic di/ance rho betwe5 two po9ts x & x' is giv5 by rho a#b - x#ax'#a - x#bx'#b - 444 - xn x'n co% ----- ---------------------------------------- ----------------------------------- ,r 7a#b - x#ab - x#bb - 444 - xn#b77;a#b - x'#ab - x'#bb - 444 - x'n#b7a_/b 7a =mula practically id5tical ) ,cayley's1 ?\gh obta9$ by a v diff]5t me?od74 ,9 ord] to %[ t ! measure ( curvature is con/ant1 we make ! sub/itutions x#a rlambda#a1 x#b rlambda#b 444 xn rlambdan1 :]e ,sigmalambda#b #a4 ,h5ce ds#b 7,radr_/7a#b - r#b77b ,r#br#b;d,delta#b_/7;a#b - r#b74 :]e d,delta#b ,sigmadlambda#b4 ,also call+ rho ! geodesic di/ance f ! orig91 we h rho a rho r co% ----- -----------------1 s9h ----- -----------------4 ,r root7a#b - r#b7 ,r root7a#b - r#b7 ,h5ce ds#b drho#b 7,r s9 h 7rho_/,r77#b;d,delta#b4 ,putt+ z#a rholambda#a1 z#b rholambda#b1 444 zn rholambdan1 we obta9 #a _/ ,r rho#b ds#b ,sigmadz#b ------ 7 ----- s9h ----- 7 - #a ,sigma7zi dzk - zk dzi7#b4 rho#b rho ,r _/ ,h5ce :5 rho is small1 we h approximately #a ds#b ,sigmadz#b ----,sigma7zi dzk - zk dzi7#b 7#a74 #c,r#b ,consid]+ a surface elem5t ?r\gh ! orig91 we may *oose our axes s t1 = ?is elem5t1 z#c ,z#d 444 zn #j4 ,?us #a ds#b dz#ab dz#bb ----7z#a;dz#b - z#b;dz#a7b 7#b74 #c,r#b ,n[ ! area ( ! triangle ^: v]tices are 7#j1 #j71 7z#a1 z#b71 7dz#a1 dz#b7 is #a_/b7z#a1 dz#b - z#b;dz#a74 ,h5ce ! quoti5t :5 ! t]ms ( ! f\r? ord] 9 7#b7 are divid$ by ! squ>e ( ?is triangle is #d_/c,r#b2 h5ce1 return+ to g5]al axes1 ! same is ! quoti5t :5 ! t]ms ( ! f\r? ord] 9 7#a7 are divid$ by ! squ>e ( ! triangle ^: v]tices are 7#j1 #j1 444 #j71 7z#a1 z#b1 z#c1 444 zn71 7dz#a1 dz#b1 dz#c 444 dzn74 ,b -#c_/d ( ?is quoti5t is def9$ by ,riemann z ! measure ( curvature4#ae ,h5ce ! measure ( curvature is -#a_/,r#b1 i4e4 is con/ant & negative4 ,! prop]ties ( p>allels1 triangles1 c41 are z 9 ! ,saggio4 ,x is also %[n t ! analogues ( limit surfaces h z]o curvature2 & t sph]es ( radius rho h con/ant positive curvature #a_/,r#b s9h#b 7rho_/,r71 s t sph]ical geometry may be reg>d$ z conta9$ 9 ! pseudo-sph]ical 7z ,beltrami calls ,lobat*ewsky's sy/em74 ,transition to ! projective me?od4 ,! ,saggio1 z we saw1 gives a ,euclidean int]pretation conf9$ to two dim5sions4 ,b a consid]ation ( ! auxili>y plane sugge/s a diff]5t int]pretation1 :i* may be ext5d$ to any numb] ( dim5sions4 ,if1 in/ead ( ref]r+ to ! pseudosph]e1 we m]ely def9e di/ance & angle1 9 ! ,euclidean plane1 z ^? functions ( ! coord9ates :i* gave u di/ance & angle on ! pseudosph]e1 we f9d t ! geometry ( our plane has become ,lobat*ewsky's4 ,all ! po9ts ( ! limit+ circle are n[ at inf9ity1 & po9ts beyond x are imag9>y4 ,if we give our circle an imag9>y radius ! geometry on ! plane becomes elliptic4 ,replac+ ! circle by a sph]e1 we obta9 an analog\s repres5tation = ?ree dim5sions4 ,in/ead ( a circle or sph]e we may take any conic or quadric4 ,) ?is def9ition1 if ! fundam5tal quadric be ,sigma7xx7 #j1 & if ,sigma7xx7' be ! pol> =m ( ,sigma7xx71 ! di/ance rho betwe5 x & x' is giv5 by ! projective =mula cos7rho_/k7 ,sigmaxx'_/,sigma7xx74,sigmax'x'#a_/b4 ,t ?is =mula is projective is r5d]$ evid5t by obs]v+ t e7-#b;irho_/k7 is ! anh>monic ratio ( ! range consi/+ ( ! two po9ts & ! int]sections ( ! l9e jo9+ !m ) ! fundam5tal quadric4 ,) ?is we are br\ght to ! ?ird or projective p]iod4 ,! me?od ( ?is p]iod is due to ,cayley2 its application to previ\s non-,euclidean geometry is due to ,kle94 ,! projective me?od conta9s a g5]alization ( discov]ies already made by ,lagu]re#af 9 #ahec z reg>ds ,euclidean geometry4 ,! arbitr>9ess ( ?is proc$ure ( d]iv+ metrical geometry f ! prop]ties ( conics is remov$ by ,lie's !ory ( congru5ce4 ,we !n arrive at ! /age ( ?\ght :i* f9ds its expression 9 ! mod]n treatm5t ( ! axioms ( geometry4 ,! two k9ds ( elliptic space4 ,! projective me?od leads to a discrim9ation1 fir/ made by ,kle91#ag ( two v>ieties ( ,riemann's space2 ,kle9 calls ^! elliptic & sph]ical4 ,!y are also call$ ! pol> & antipodal =ms ( elliptic space4 ,! latt] names w "h be us$4 ,! diff]5ce is /rictly analog\s to t betwe5 ! diamet]s & ! po9ts ( a sph]e4 ,9 ! pol> =m two /raight l9es 9 a plane always int]sect 9 "o & only "o po9t2 9 ! antipodal =m !y int]sect always 9 two po9ts1 :i* are antipodes4 ,accord+ to ! def9ition ( geometry adopt$ 9 section ,,vii4 7,axioms ( ,geometry71 ! antipodal =m is n to be t]m$ geometry1 s9ce any pair ( coplan> /raight l9es int]sect ea* o!r 9 two po9ts4 ,x may be call$ a quasi-geometry4 ,simil>ly 9 ! antipodal =m two diamet]s always det]m9e a plane1 b two po9ts on a sph]e d n det]m9e a great circle :5 !y are antipodes1 & two great circles always int]sect 9 two po9ts4 ,aga91 a plane does n =m a b\nd>y among l9es ?r\gh a po9t3 we c pass f any "o su* l9e to any o!r )\t pass+ ?r\gh ! plane4 ,b a great circle does divide ! surface ( a sph]e4 ,s1 9 ! pol> =m1 a complete /raight l9e does n divide a plane1 & a plane does n divide space1 & does n1 l a ,euclidean plane1 h two sides4#ah ,b1 9 ! antipodal =m1 a plane is1 9 ^! respects1 l a ,euclidean plane4 ,x is expla9$ 9 section ,,vii4 9 :at s5se ! metrical geometry ( ! mat]ial _w c be consid]$ to be det]m9ate & n a matt] ( arbitr>y *oice4 ,! sci5tific "q z to ! be/ available evid5ce conc]n+ ! nature ( ?is geometry is "o beset ) difficulties ( a peculi> k9d4 ,we are ob/ruct$ by ! fact t all exi/+ physical sci5ce assumes ! ,euclidean hypo!sis4 ,?is hypo!sis has be5 involv$ 9 all actual measurem5ts ( l>ge di/ances1 & 9 all ! laws ( a/ronomy & physics4 ,! pr9ciple ( simplicity w\ld !re=e lead u1 9 g5]al1 :]e an obs]vation conflict$ ) "o or m ( ^? laws1 to ascribe ?is anomaly1 n to ! falsity ( ,euclidean geometry1 b to ! falsity ( ! laws 9 "q4 ,?is applies especially to a/ronomy4 ,on ! e>? our means ( measurem5t are _m & direct1 & s long z no great accuracy is s\ght !y involve few sci5tific laws4 ,?us we acquire1 f su* direct measurem5ts1 a v high degree ( probability t ! space-con/ant1 if n inf9ite1 is yet l>ge z comp>$ ) t]re/rial di/ances4 ,b a/ronomical di/ances & triangles c only be measur$ by means ( ! receiv$ laws ( a/ronomy & optics1 all ( :i* h be5 e/abli%$ by assum+ ! tru? ( ! ,euclidean hypo!sis4 ,x !re=e rema9s possible 7until a detail$ pro( ( ! contr>y is =?com+7 t a l>ge b f9ite space-con/ant1 ) diff]5t laws ( a/ronomy & optics1 w\ld h equally expla9$ ! ph5om5a4 ,we _c1 !re=e1 accept ! measurem5ts ( /ell> p>allaxes1 c41 z conclusive evid5ce t ! space-con/ant is l>ge z comp>$ ) /ell> di/ances4 ,= ! pres5t1 on gr\nds ( simplicity1 we may rightly adopt ?is view2 b x mu/ rema9 possible t1 9 view ( "s hi!rto undiscov]$ discrepancy1 a slight correction ( ! sort sugge/$ might prove ! simple/ alt]native4 ,b conv]sely1 a f9ite p>allax = v di/ant />s1 or a negative p>allax = any />1 c\ld n be accept$ z conclusive evid5ce t our geometry is non-,euclidean1 unless x 7 %[n--& ?is seems sc>cely possible--t no modification ( a/ronomy or optics c\ld acc\nt = ! ph5om5on4 ,?us al?\gh we may admit a probability t ! space-con/ant is l>ge 9 comp>ison ) /ell> di/ances1 a conclusive pro( or dispro( seems sc>cely possible4 ,f9ally1 x is ( int]e/ to note t1 ?\gh x is !oretically possible to prove1 by sci5tific me?ods1 t our geometry is non-,euclidean1 x is :olly impossible to prove by su* me?ods t x is accurately ,euclidean4 ,= ! unavoidable errors ( obs]vation mu/ always leave a slight m>g9 9 our measurem5ts4 ,a triangle might be f\nd ^: angles 7 c]ta9ly great]1 or c]ta9ly less1 ?an two "r angles2 b to prove !m exactly equal to two "r angles mu/ always be beyond our p[]s4 ,if1 !re=e1 any man *]i%es a hope ( prov+ ! exact tru? ( ,euclid1 su* a hope mu/ be bas$1 n ^u sci5tific1 b ^u philosophical consid]ations4 ,,bibliography4--,! bibliography app5d$ to section ,,vii4 %\ld be consult$ 9 ?is connexion4 ,also1 9 addition to ! citations already made1 ! foll[+ works may be m5tion$4 ,= ,lobat*ewsky's writ+s1 cf4 ,urkund5 zur ,ges*i*te d] ni*teuklidis*5 ,geometrie1 i41 ,nikolaj ,iwan[its* ,lobats*efsky1 by ,f4 ,engel & ,p4 ,/ackel 7,leipzig1 #ahih74 ,= ,john ,bolyai's ,app5dix1 cf4 ,absolute ,geometrie na* ,johann ,bolyai1 by ,j4 ,fris*auf 7,leipzig1 #ahgb71 & also ! new edition ( 8 fa!r's l>ge "w1 ,t5tam5 4441 publi%$ by ! ,ma!matical ,society ( ,budape/2 ! second volume conta9s ! app5dix4 ,cf4 also ,j4 ,fris*auf1 ,elem5te d] absolut5 ,geometrie 7,leipzig1 #ahgf72 ,m4,l4 ,g]>d1 ,sur la geometrie non-,euclidi5ne 7!sis = doctorate7 7,p>is1 #ahib72 de ,tilly1 ,essai sur les pr9cipes fondam5tales de la geometrie et de la mecanique 7,bordeaux1 #ahgi72 ,sir ,r4,s4 ,ball1 ,on ! ,!ory ( ,cont5t1 ,trans4 ,roy4 ,iri% ,acad4 vol4 xxix4 7#ahhi72 ,f4 ,l9demann1 ,me*anik bei projectiv] ,maasbe/immung1 ,ma?4 ,annal4 vol4 vii42 ,w4,k4 ,clif=d1 ,prelim9>y ,sket* ( ,biquat]nions1 ,proc4 ( ,lond4 ,ma?4 ,soc4 7#ahgc71 & ,coll4 ,works2 ,a4 ,bu*heim1 ,on ! ,!ory ( ,screws 9 ,elliptic ,space1 ,proc4 ,lond4 ,ma?4 ,soc4 vols4 xv41 xvi41 xvii42 ,h4 ,cox1 ,on ! ,application ( ,quat]nions & ,grassmann's ,algebra to diff]5t ,k9ds ( ,uni=m ,space1 ,trans4 ,camb4 ,phil4 ,soc4 7#ahhb72 ,m4 ,dehn1 ,die ,leg5d>is*5 ,satze ub] die ,w9kelsumme im ,dreieck1 ,ma?4 ,ann4 vol4 #ec 7#aijj71 & ,ub] d5 ,raum9halt1 ,ma?4 ,annal4 vol4 #ee 7#aijb74 ,= expositions ( ! :ole subject1 cf4 ,f4 ,kle91 ,ni*t-,euklidis*e ,geometrie 7,gott+51 #ahic72 ,r4 ,bonola1 ,la ,geometria non-,euclidea 7,bologna1 #aijf72 ,p4 ,b>b>91 ,la ,geometrie non-,euclidi5ne 7,p>is1 #aijb72 ,w4 ,kill+1 ,die ni*t-,euklidis*5 ,raum=m5 9 analytis*] ,beh&lung 7,leipzig1 #ahhe74 ,! la/-nam$ "w also deals ) geometry ( m ?an ?ree dim5sions2 9 ?is connexion cf4 also ,g4 ,v]onese1 ,fondam5ti di geometria a piu dim5sioni ed a piu specie di unita rettil9ee 444 7,padua1 #ahia1 ,g]man translation1 ,leipzig1 #ahid72 ,g4 ,font5e1 ,l',hyp]espace a 7n-#a7 dim5sions 7,p>is1 #ahib72 & ,a4,n4 ,:itehead1 loc4 cit4 ,cf4 also ,e4 ,/udy1 ,ub] ni*t-,euklidis*e und ,l9i5geometrie1 ,jahr4 d4 ,deuts*4 ,ma?4 ,v]4 vol4 xv4 7#aijf72 ,w4 ,burnside1 ,on ! ,k9ematics ( non-,euclidean ,space1 ,proc4 ,lond4 ,ma?4 ,soc4 vol4 xxvi4 7#ahid74 ,a bibliography on ! subject up to #ahgh has be5 publi%$ by ,g4,b4 ,hal/$1 ,am]4 ,j\rn4 ( ,ma?4 vols4 i4 & ii42 & "o up to #aijj by ,r4 ,bonola1 ,index op]um ad geometriam absolutam spectantium 444 7#aijb1 & ,leipzig1 #aijc74 7,b4 ,a4 ,w4 ,r42 ,a4 ,n4 ,w47 ,,vii4 ,,axioms ,,( ,,geometry ,!ories ( space4 ,until ! discov]y ( ! non-,euclidean geometries 7,lobat*ewsky1 #ahbf & #ahbi2 ,j4 ,bolyai1 #ahcb2 ,b4 ,riemann1 #ahed71 geometry 0 univ]sally consid]$ z be+ exclusively ! sci5ce ( exi/5t space4 7,see section ,,vi4 ,non-,euclidean ,geometry47 ,9 respect to ! sci5ce1 z ?us conceiv$1 two controv]sies may be notic$4 ,fir/1 !re is ! controv]sy respect+ ! absolute & relational !ories ( space4 ,accord+ to ! absolute !ory1 :i* is ! traditional view 7held explicitly by ,newton71 space has an exi/5ce1 9 "s s5se :atev] x may be1 indep5d5t ( ! bodies :i* x conta9s4 ,! bodies occupy space1 & x is n intr9sically unmean+ to say t any def9ite body occupies ?is "p ( space1 & n t "p ( space1 )\t ref]5ce to o!r bodies occupy+ space4 ,accord+ to ! relational !ory ( space1 ( :i* ! *ief expon5t 0 ,leibnitz1#ai space is no?+ b a c]ta9 assemblage ( ! relations betwe5 ! v>i\s p>ticul> bodies 9 space4 ,! idea ( space ) no bodies 9 x is absurd4 ,accord+ly !re c be no mean+ 9 say+ t a body is "h & n !re1 ap>t f a ref]5ce to ! o!r bodies 9 ! univ]se4 ,?us1 on ?is !ory1 absolute motion is intr9sically unmean+4 ,x is admitt$ on all h&s t 9 practice only relative motion is directly measurable4 ,newton1 h[ev]1 ma9ta9s 9 ! ,pr9cipia 7s*olium to ! #h? def9ition7 t x is indirectly measurable by means ( ! effects ( c5trifugal =ce z x occurs 9 ! ph5om5a ( rotation4 ,?is irrelevance ( absolute motion 7if !re be su* a ?+7 to sci5ce has l$ to ! g5]al adoption ( ! relational !ory by mod]n m5 ( sci5ce4 ,b no decisive argum5t = ei!r view has at pres5t be5 elaborat$4#bj ,kant's view ( space z be+ a =m ( p]ception at fir/ sight appe>s to cut across ?is controv]sy4 ,b he1 saturat$ z he 0 ) ! _s ( ! ,newtonian physics1 mu/ 7at lea/ 9 bo? editions ( ! ,critique7 be class$ ) ! uphold]s ( ! absolute !ory4 ,! =m ( p]ception has a type ( exi/5ce prop] to itself indep5d5tly ( ! p>ticul> bodies :i* x conta9s4 ,= example he writes3#ba ,space does n repres5t any quality ( objects by !mselves1 or objects 9 _! relation to "o ano!r1 i4e4 space does n repres5t any det]m9ation :i* is inh]5t 9 ! objects !mselves1 & w\ld rema91 ev5 if all subjective conditions ( intuition 7 remov$4 ,axioms4 ,! second controv]sy is t betwe5 ! view t ! axioms applicable to space are kn[n only f exp]i5ce1 & ! view t 9 "s s5se ^! axioms are giv5 a priori4 ,bo? ^! views1 ?us broadly /at$1 are capable ( v>i\s subtle modifications1 & a discussion ( !m w\ld m]ge into a g5]al treatise on epi/emology4 ,! crud] =ms ( ! a priori view h be5 made q unt5able by ! mod]n ma!matical discov]ies4 ,geomet]s n[ pr(ess ignorance 9 _m respects ( ! exact axioms :i* apply to exi/5t space1 & x seems unlikely t a pr(\nd /udy ( ! "q %\ld ?us oblit]ate a priori intuitions4 ,ano!r "q irrelevant to ?is article1 b ) "s relevance to ! above controv]sy1 is t ( ! d]ivation ( our p]ception ( exi/5t space f our v>i\s types ( s5sation4 ,?is is a "q = psy*ology4#bb ,def9ition ( ,ab/ract ,geometry4--,exi/5t space is ! subject matt] ( only "o ( ! applications ( ! mod]n sci5ce ( ab/ract geometry1 view$ z a bran* ( pure ma!matics4 ,geometry has be5 def9$#bc z ! /udy ( s]ies ( two or m dim5sions4 ,x has also be5 def9$#bd z ! sci5ce ( cross classification4 ,^! def9itions are f\nd$ ^u ! actual practice ( ma!maticians 9 respect to _! use ( ! t]m ,geometry4 ,ei!r ( !m br+s out ! fact t geometry is n a sci5ce ) a det]m9ate subject matt]4 ,x is conc]n$ ) any subject matt] to :i* ! =mal axioms may apply4 ,geometry is n peculi> 9 ?is respect4 ,all bran*es ( pure ma!matics deal m]ely ) types ( relations4 ,?us ! fundam5tal ideas ( geometry 7e4g4 ^? ( po9ts & ( /raight l9es7 are n ideas ( det]m9ate entities1 b ( any entities = :i* ! axioms are true4 ,& a set ( =mal geometrical axioms _c 9 !mselves be true or false1 s9ce !y are n det]m9ate propositions1 9 t !y d n ref] to a det]m9ate subject matt]4 ,! axioms are propositional functions4#be ,:5 a set ( axioms is giv51 we c ask 7#a7 :e!r !y are consi/5t1 7#b7 :e!r _! exi/5ce !orem is prov$1 7#c7 :e!r !y are indep5d5t4 ,axioms are consi/5t :5 ! contradictory ( any axiom _c be d$uc$ f ! rema9+ axioms4 ,_! exi/5ce !orem is ! pro( t !y are true :5 ! fundam5tal ideas are consid]$ z d5ot+ "s det]m9ate subject matt]1 s t ! axioms are develop$ into det]m9ate propositions4 ,x foll[s f ! logical law ( contradiction t ! pro( ( ! exi/5ce !orem proves also ! consi/5cy ( ! axioms4 ,?is is ! only me?od ( pro( ( consi/5cy4 ,! axioms ( a set are indep5d5t ( ea* o!r :5 no axiom c be d$uc$ f ! rema9+ axioms ( ! set4 ,! indep5d5ce ( a giv5 axiom is prov$ by e/abli%+ ! consi/5cy ( ! rema9+ axioms ( ! set1 toge!r ) ! contradictory ( ! giv5 axiom4 ,! enum]ation ( ! axioms is simply ! enum]ation ( ! hypo!ses#bf 7) respect to ! undet]m9$ subject matt]7 ( :i* "s at lea/ occur 9 ea* ( ! subsequ5t propositions4 ,any sci5ce is call$ a geometry if x inve/igates ! !ory ( ! classification ( a set ( entities 7! po9ts7 into classes 7! /raight l9es71 su* t 7#a7 !re is "o & only "o class :i* conta9s any giv5 pair ( ! entities1 & 7#b7 e su* class conta9s m ?an two memb]s4 ,9 ! two geometries1 important f _! relevance to exi/5t space1 axioms :i* secure an ord] ( ! po9ts on any l9e also occur4 ,^! geometries w be call$ ,projective ,geometry & ,descriptive ,geometry4 ,9 projective geometry any two /raight l9es 9 a plane int]sect1 & ! /raight l9es are clos$ s]ies :i* return into !mselves1 l ! circumf]5ce ( a circle4 ,9 descriptive geometry two /raight l9es 9 a plane d n necess>ily int]sect1 & a /raight l9e is an op5 s]ies )\t beg9n+ or end4 ,ord9>y ,euclidean geometry is a descriptive geometry2 x becomes a projective geometry :5 ! s-call$ po9ts at inf9ity are add$4 ,projective ,geometry4 ,projective geometry may be develop$ f two undef9$ fundam5tal ideas1 namely1 t ( a po9t & t ( a /raight l9e4 ,^! undet]m9$ ideas take diff]5t specific mean+s = ! v>i\s specific subject matt]s to :i* projective geometry c be appli$4 ,! numb] ( ! axioms is always to "s ext5t arbitr>y1 be+ dep5d5t ^u ! v]bal =ms ( /atem5t :i* are adopt$4 ,!y w be pres5t$#bg "h z twelve 9 numb]1 eight be+ axioms ( classification1 & f\r be+ axioms ( ord]4 ,axioms ( ,classification4--,! eight axioms ( classification are z foll[s3 #a4 ,po9ts =m a class ( entities ) at lea/ two memb]s4 #b4 ,any /raight l9e is a class ( po9ts conta9+ at lea/ ?ree memb]s4 #c4 ,any two di/9ct po9ts lie 9 "o & only "o /raight l9e4 #d4 ,!re is at lea/ "o /raight l9e :i* does n conta9 all ! po9ts4 #e4 ,if ,a1 ,b1 ,c are non-coll9e> po9ts1 & ,a' is on ! /raight l9e ,,bc1 & ,b' is on ! /raight l9e ,,ca1 !n ! /raight l9es ,,aa' & ,,bb' possess a po9t 9 common4 ,def9ition4--,if ,a1 ,b1 ,c are any ?ree non-coll9e> po9ts1 ! plane ,,abc is ! class ( po9ts ly+ on ! /raight l9es jo9+ ,a ) ! v>i\s po9ts on ! /raight l9e ,,bc4 #f4 ,!re is at lea/ "o plane :i* does n conta9 all ! po9ts4 #g4 ,!re exi/s a plane alpha1 & a po9t ,a n incid5t 9 alpha1 su* t any po9t lies 9 "s /raight l9e :i* conta9s bo? ,a & a po9t 9 alpha4 ,def9ition4--,h>m4 7,,abcd7 symbolizes ! foll[+ conjo9t /atem5ts3 7#a7 t ! po9ts ,a1 ,b1 ,c1 ,d are coll9e>1 & 7#b7 t a quadrilat]al c be f\nd ) "o pair ( opposite sides int]sect+ at ,a1 ) ! o!r pair int]sect+ at ,c1 & ) its diagonals pass+ ?r\gh ,b & ,d respectively4 ,!n ,b & ,d are said to be h>monic conjugates ) respect to ,a & ,c4 #h4 ,h>m4 7,,abcd7 implies t ,b & ,d are di/9ct po9ts4 ,9 ! above axioms #d secures at lea/ two dim5sions1 axiom #e is ! fundam5tal axiom ( ! plane1 axiom #f secures at lea/ ?ree dim5sions1 & axiom #g secures at mo/ ?ree dim5sions4 ,f axioms #a-e x c be prov$ t any two di/9ct po9ts 9 a /raight l9e det]m9e t l9e1 t any ?ree non-coll9e> po9ts 9 a plane det]m9e t plane1 t ! /raight l9e conta9+ any two po9ts 9 a plane lies :olly 9 t plane1 & t any two /raight l9es 9 a plane int]sect4 ,f axioms #a-f ,des>gue's well-kn[n !orem on triangles 9 p]spective c be prov$4 ,! enunciation ( ?is !orem is z foll[s3 ,if ,,abc & ,a',b',c' are two coplan> triangles su* t ! l9es ,,aa'1 ,,bb'1 ,,cc' are concurr5t1 !n ! ?ree po9ts ( int]section ( ,,bc & ,b',c' ( ,,ca & ,c',a'1 & ( ,,ab & ,a',b' are coll9e>2 & conv]sely if ! ?ree po9ts ( int]section are coll9e>1 ! ?ree l9es are concurr5t4 ,! pro( :i* c be appli$ is ! usual projective pro( by :i* a ?ird triangle ,a,b,c is con/ruct$ n coplan> ) ! o!r two1 b 9 p]spective ) ea* ( !m4 ,x has be5 prov$#bh t ,des>gues's !orem _c be d$uc$ f axioms #a-e1 t is1 if ! geometry be conf9$ to two dim5sions4 ,all ! pro(s proce$ by ! me?od ( produc+ a specification ( po9ts & /raight l9es :i* satisfies axioms #a-e1 & su* t ,des>gues's !orem does n hold4 ,x foll[s f axioms #a-e t ,h>m4 7,,abcd7 implies ,h>m4 7,,adcb7 & ,h>m4 7,,cbad71 & t1 if ,a1 ,b1 ,c be any ?ree di/9ct coll9e> po9ts1 !re exi/s at lea/ "o po9t ,d su* t ,h>m4 7,,abcd74 ,b x requires ,des>gues's !orem1 & h5ce axiom #f1 to prove t ,h>m4 7,,abcd7 & ,h>m4 7,,abcd'7 imply ! id5tity ( ,d & ,d'4 ,! necessity = axiom #h has be5 prov$ by ,g4 ,fano1#bi :o has produc$ a ?ree dim5sional geometry ( fifte5 po9ts1 i4e4 a me?od ( cross classification ( fifte5 entities1 9 :i* ea* /raight l9e conta9s ?ree po9ts1 & ea* plane conta9s sev5 /raight l9es4 ,9 ?is geometry axiom #h does n hold4 ,also f axioms #a-f & #h x foll[s t ,h>m4 7,,abcd7 implies ,h>m4 7,,bcda74 ,def9itions4--,:5 two plane figures c be d]iv$ f "o ano!r by a s+le projection1 !y are said to be 9 p]spective4 ,:5 two plane figures c be d]iv$ "o f ! o!r by a f9ite s]ies ( p]spective relations betwe5 int]m$iate figures1 !y are said to be projectively relat$4 ,any prop]ty ( a plane figure :i* necess>ily also belongs to any projectively relat$ figure1 is call$ a projective prop]ty4 ,! foll[+ !orem1 kn[n f its importance z ! fundam5tal !orem ( projective geometry1 _c be prov$#cj f axioms #a-h4 ,! enunciation is3 ,a projective correspond5ce betwe5 ! po9ts on two /raight l9es is completely det]m9$ :5 ! correspond5ts ( ?ree di/9ct po9ts on "o l9e are det]m9$ on ! o!r4 ,?is !orem is equival5t#ca 7assum+ axioms #a-h7 to ano!r !orem1 kn[n z ,pappus's ,!orem1 namely3 ,if l & l' are two di/9ct coplan> l9es1 & ,a1 ,b1 ,c are ?ree di/9ct po9ts on l1 & ,a'1 ,b'1 ,c' are ?ree di/9ct po9ts on l'1 !n ! ?ree po9ts ( int]section ( ,,aa' & ,b',c1 ( ,a',b & ,,cc'1 ( ,,bb' & ,c',a1 are coll9e>4 ,?is !orem is obvi\sly ,pascal's well-kn[n !orem respect+ a hexagon inscrib$ 9 a conic1 = ! special case :5 ! conic has deg5]at$ into ! two l9es l & l'4 ,ano!r !orem also equival5t 7assum+ axioms #a-h7 to ! fundam5tal !orem is ! foll[+3#cb ,if ! ?ree coll9e> pairs ( po9ts1 ,a & ,a'1 ,b & ,b'1 ,c & ,c'1 are su* t ! ?ree pairs ( opposite sides ( a complete quadrangle pass respectively ?r\gh !m1 i4e4 "o pair ?r\gh ,a & ,a' respectively1 & s on1 & if also ! ?ree sides ( ! quadrangle :i* pass ?r\gh ,a1 ,b1 & ,c1 are concurr5t 9 "o ( ! corn]s ( ! quadrangle1 !n ano!r quadrangle c be f\nd ) ! same relation to ! ?ree pairs ( po9ts1 except t its ?ree sides :i* pass ?r\gh ,a1 ,b1 & ,c1 are n concurr5t4 ,?us1 if we *oose to take any "o ( ^! ?ree !orems z an axiom1 all ! !orems ( projective geometry :i* d n require ord9al or metrical ideas = _! enunciation c be prov$4 ,also a conic c be def9$ z ! locus ( ! po9ts f\nd by ! usual con/ruction1 bas$ ^u ,pascal's !orem1 = po9ts on ! conic ?r\gh five giv5 po9ts4 ,b x is unnecess>y to assume "h any "o ( ! sugge/$ axioms2 = ! fundam5tal !orem c be d$uc$ f ! axioms ( ord] toge!r ) axioms #a-h4 ,axioms ( ,ord]4--,x is possible to def9e 7cf4 ,pi]i1 loc4 cit47 ! prop]ty ^u :i* ! ord] ( po9ts on a /raight l9e dep5ds4 ,b to secure t ?is prop]ty does 9 fact range ! po9ts 9 a s]ial ord]1 "s axioms are requir$4 ,a /raight l9e is to be a clos$ s]ies2 ?us1 :5 ! po9ts are 9 ord]1 x requires two po9ts on ! l9e to divide x into two di/9ct complem5t>y segm5ts1 :i* d n ov]lap1 & toge!r =m ! :ole l9e4 ,accord+ly ! problem ( ! def9ition ( ord] r$uces itself to ! def9ition ( ^! two segm5ts =m$ by any two po9ts on ! l9e2 & ! axioms are /at$ relatively to ^! segm5ts4 ,def9ition4--,if ,a1 ,b1 ,c are ?ree coll9e> po9ts1 ! po9ts on ! segm5t ,,abc are def9$ to be ^? po9ts su* z ,x1 = :i* !re exi/ two po9ts ,y & ,y' ) ! prop]ty t ,h>m4 7,,aycy'7 & ,h>m4 7,,byxy'7 bo? hold4 ,! supplem5t>y segm5t ,,abc is def9$ to be ! re/ ( ! po9ts on ! l9e4 ,?is def9ition is elucidat$ by notic+ t ) our ord9>y geometrical ideas1 if ,b & ,x are any two po9ts betwe5 ,a & ,c1 !n ! two pairs ( po9ts1 ,a & ,c1 ,b & ,x1 def9e an involution ) real d\ble po9ts1 namely1 ! ,y & ,y' ( ! above def9ition4 ,! prop]ty ( belong+ to a segm5t ,,abc is projective1 s9ce ! h>monic relation is projective4 ,! fir/ ?ree axioms ( ord] 7cf4 ,pi]i1 loc4 cit47 are3 #i4 ,if ,a1 ,b1 ,c are ?ree di/9ct coll9e> po9ts1 ! supplem5t>y segm5t ,,abc is conta9$ )9 ! segm5t ,,bca4 #aj4 ,if ,a1 ,b1 ,c are ?ree di/9ct coll9e> po9ts1 ! common "p ( ! segm5ts ,,bca & ,,cab is conta9$ 9 ! supplem5t>y segm5t ,,abc4 #aa4 ,if ,a1 ,b1 ,c are ?ree di/9ct coll9e> po9ts1 & ,d lies ,9 ! segm5t ,,abc1 !n ! segm5t ,,adc is conta9$ )9 ! segm5t ,,abc4 ,f ^! axioms all ! usual prop]ties ( a clos$ ord] foll[4 ,x w be notic$ t1 if ,a1 ,b1 ,c are any ?ree coll9e> po9ts1 ,c is necess>ily trav]s$ 9 pass+ f ,a to ,b by "o r\te along ! l9e1 & is n trav]s$ 9 pass+ f ,a to ,b along ! o!r r\te4 ,?us !re is no mean+1 z ref]r$ to clos$ /raight l9es1 9 ! simple /atem5t t ,c lies betwe5 ,a & ,b4 ,b !re may be a relation ( sep>ation betwe5 two pairs ( coll9e> po9ts1 su* z ,a & ,c1 & ,b & ,d4 ,! c\ple ,b & ,d is said to sep>ate ,a & ,c1 if ! f\r po9ts are coll9e> & ,d lies 9 ! segm5t complem5t>y to ! segm5t ,,abc4 ,! prop]ty ( ! sep>ation ( pairs ( po9ts by pairs ( po9ts is projective4 ,also x c be prov$ t ,h>m4 7,,abcd7 implies t ,b & ,d sep>ate ,a & ,c4 ,def9itions4--,a s]ies ( entities arrang$ 9 a s]ial ord]1 op5 or clos$1 is said to be compact1 if ! s]ies conta9s no imm$iately consecutive entities1 s t 9 trav]s+ ! s]ies f any "o entity to any o!r entity x is necess>y to pass ?r\gh entities di/9ct f ei!r4 ,x 0 ! m]it ( ,r4 ,d$ek9d & ( ,g4 ,cantor explicitly to =mulate ano!r fundam5tal prop]ty ( s]ies4 ,! ,d$ek9d prop]ty#cc z appli$ to an op5 s]ies c be def9$ ?us3 ,an op5 s]ies possesses ! ,d$ek9d prop]ty1 if1 h[ev]1 x be divid$ into two mutually exclusive classes u & v1 :i* 7#a7 conta9 betwe5 !m ! :ole s]ies1 & 7#b7 are su* t e memb] ( u prec$es 9 ! s]ial ord] e memb] ( v1 !re is always a memb] ( ! s]ies1 belong+ to "o ( ! two1 u or v1 :i* prec$es e memb] ( v 7o!r ?an itself if x belong to v71 & also succe$s e memb] ( u 7o!r ?an itself if x belong to u74 ,accord+ly 9 an op5 s]ies ) ! ,d$ek9d prop]ty !re is always a memb] ( ! s]ies m>k+ ! junction ( two classes su* z u & v4 ,an op5 s]ies is cont9u\s if x is compact & possesses ! ,d$ek9d prop]ty4 ,a clos$ s]ies c always be trans=m$ into an op5 s]ies by tak+ any arbitr>y memb] z ! fir/ t]m & by tak+ "o ( ! two ways r\nd z ! asc5d+ ord] ( ! s]ies4 ,?us ! def9itions ( compactness & ( ! ,d$ek9d prop]ty c be at once transf]r$ to a clos$ s]ies4 #ab4 ,! la/ axiom ( ord] is t !re exi/s at lea/ "o /raight l9e = :i* ! po9t ord] possesses ! ,d$ek9d prop]ty4 ,x foll[s f axioms #a-ab by projection t ! ,d$ek9d prop]ty is true = all l9es4 ,aga9 ! h>monic sy/em ,,abc1 :]e ,a1 ,b1 ,c are coll9e> po9ts1 is def9$#cd ?us3 take ! h>monic conjugates ,a'1 ,b'1 ,c' ( ea* po9t ) respect to ! o!r two1 aga9 take ! h>monic conjugates ( ea* ( ! six po9ts ,a1 ,b1 ,c1 ,a'1 ,b'1 ,c' ) respect to ea* pair ( ! rema9+ five1 & proce$ 9 ?is way by an un5d+ s]ies ( /eps4 ,! set ( po9ts ?us obta9$ is call$ ! h>monic sy/em ,,abc4 ,x c be prov$ t a h>monic sy/em is compact1 & t e segm5t ( ! l9e conta9+ x possesses memb]s ( x4 ,fur!rmore1 x is easy to prove t ! fundam5tal !orem holds = h>monic sy/ems1 9 ! s5se t1 if ,a1 ,b1 ,c are ?ree po9ts on a l9e l1 & ,a'1 ,b'1 ,c' are ?ree po9ts on a l9e l'1 & if by any two di/9ct s]ies ( projections ,a1 ,b1 ,c are project$ into ,a'1 ,b'1 ,c'1 !n any po9t ( ! h>monic sy/em ,,abc corresponds to ! same po9t ( ! h>monic sy/em ,a',b',c' accord+ to bo? ! projective relations :i* are ?us e/abli%$ betwe5 l & l'4 ,x n[ foll[s imm$iately t ! fundam5tal !orem mu/ hold = all ! po9ts on ! l9es l & l'1 s9ce 7z has be5 po9t$ out7 h>monic sy/ems are ev]y:]e d5se on _! conta9+ l9es4 ,?us ! fundam5tal !orem foll[s f ! axioms ( ord]4 ,a sy/em ( num]ical coord9ates c n[ be introduc$1 possess+ ! prop]ty t l9e> equations repres5t planes & /raight l9es4 ,! outl9e ( ! argum5t by :i* ?is rem>kable problem 79 t di/ance is z yet undef9$7 is solv$1 w n[ be giv54 ,x is fir/ prov$ t ! po9ts on any l9e c 9 a c]ta9 way be def9itely associat$ ) all ! positive & negative real numb]s1 s z to =m ) !m a "o-"o correspond5ce4 ,! arbitr>y elem5ts 9 ! e/abli%m5t ( ?is relation are ! po9ts on ! l9e associat$ ) #j1 #a & oo4 ,?is association#ce is mo/ easily effect$ by consid]+ a class ( projective relations ( ! l9e ) itself1 call$ by ,f4 ,s*ur 7loc4 cit47 prospectivities4 ,let l 7fig4 #fi7 be ! giv5 l9e1 m & n any two l9es int]sect+ at ,u on l1 ,s & ,s' two po9ts on n4 ,!n a projective relation betwe5 l & itself is =m$ by project+ l f ,s on to m1 & !n by project+ m f ,s' back on to l4 ,all su* projective relations1 h[ev] m1 n1 ,s & ,s' be v>i$1 are call$ prospectivities1 & ,u is ! d\ble po9t ( ! prospectivity4 ,if a po9t ,o on l is relat$ to ,a by a prospectivity1 !n all prospectivities1 :i* 7#a7 h ! same d\ble po9t ,u1 & 7#b7 relate ,o to ,a1 give ! same correspond5t 7,q1 9 figure7 to any po9t ,p on ! l9e l2 9 fact !y are all ! same prospectivity1 h[ev] m1 n1 ,s1 & ,s' may h be5 v>i$ subject to ^! conditions4 ,su* a prospectivity w be d5ot$ by 7,,oau#b74 ,illu/ration3 ,,fig4 #fi4 ,! sum ( two prospectivities1 writt5 7,,oau#b7 7,,obu#b71 is def9$ to be t trans=mation ( ! l9e l into itself :i* is obta9$ by fir/ apply+ ! prospectivity 7,,oau#b7 & !n apply+ ! prospectivity 7,,obu#b74 ,su* a trans=mation1 :5 ! two summ&s h ! same d\ble po9t1 is itself a prospectivity ) t d\ble po9t4 ,illu/ration3 ,,fig4 #gj ,) ?is def9ition ( addition x c be prov$ t prospectivities ) ! same d\ble po9t satisfy all ! axioms ( magnitude4 ,accord+ly !y c be associat$ 9 a "o-"o correspond5ce ) ! positive & negative real numb]s4 ,let ,e 7fig4 #gj7 be any po9t on l1 di/9ct f ,o & ,u4 ,!n ! prospectivity 7,,oeu#b7 is associat$ ) unity1 ! prospectivity 7,,o\#b7 is associat$ ) z]o1 & 7,,ouu#b7 ) inf9ity4 ,! prospectivities ( ! type 7,,opu#b71 :]e ,p is any po9t on ! segm5t ,,oeu1 correspond to ! positive numb]s2 also if ,p' is ! h>monic conjugate ( ,p ) respect to ,o & ,u1 ! prospectivity 7,,op',u#b7 is associat$ ) ! correspond+ negative numb]4 7,! subjo9$ figure expla9s ?is relation ( ! positive & negative prospectivities47 ,!n any po9t ,p on l is associat$ ) ! same numb] z is ! prospectivity 7,,opu#b74 ,illu/ration3 ,,fig4 #ga4 ,x c be prov$ t ! ord] ( ! numb]s 9 algebraic ord] ( magnitude agrees ) ! ord] on ! l9e ( ! associat$ po9ts4 ,let ! numb]s1 assign$ accord+ to ! prec$+ specification1 be said to be associat$ ) ! po9ts accord+ to ! num]ation-sy/em 7,,oeu74 ,! introduction ( a coord9ate sy/em = a plane is n[ manag$ z foll[s3 ,take any triangle ,,ouv 9 ! plane1 & on ! l9es ,,ou & ,,ov e/abli% ! num]ation sy/ems 7,,oe#a,u7 & 7,,oe#b,v71 :]e ,e#a & ,e#b are arbitr>ily *os54 ,!n 7cf4 fig4 #ga7 if ,m & ,n are associat$ ) ! numb]s x & y accord+ to ^! sy/ems1 ! coord9ates ( ,p are x & y4 ,x !n foll[s t ! equation ( a /raight l9e is ( ! =m ax by c #j4 ,bo? coord9ates ( any po9t on ! l9e ,,uv are inf9ite4 ,?is c be avoid$ by introduc+ homog5e\s coord9ates ,x1 ,y1 ,z1 :]e x ,x_/,z1 & y ,y_/,z1 & ,z #j is ! equation ( ,,uv4 ,illu/ration3 ,,fig4 #gb4 ,! proc$ure = ?ree dim5sions is simil>4 ,let ,,ouvw 7fig4 #gb7 be any tetrah$ron1 & associate po9ts on ,,ou1 ,,ov1 ,,ow ) numb]s accord+ to ! num]ation sy/ems 7,,oe#a,u71 7,,oe#b,v71 & 7,,oe#c,w74 ,let ! planes ,,vwp1 ,,wup1 ,,uvp cut ,,ou1 ,,ov1 ,,ow 9 ,l1 ,m1 ,n respectively2 & let x1 y1 z be ! numb]s associat$ ) ,l1 ,m1 ,n respectively4 ,!n ,p is ! po9t 7x1 y1 z74 ,also homog5e\s coord9ates c be introduc$ z be=e1 ?us avoid+ ! inf9ities on ! plane ,,uvw4 ,! cross ratio ( a range ( f\r coll9e> po9ts c n[ be def9$ z a numb] *>act]i/ic ( t range4 ,let ! coord9ates ( any po9t ,pr ( ! range ,p#a ,p#b ,p#c ,p#d be lambdar a mur a' lambdar b mur b' -------------------------1 ------------------------1 lambdar mur lambdar mur lambdar c mur c' ------------------------1 7r #a1 #b1 #c1 #d7 lambdar mur & let 7lambdar mus7 be writt5 = lambdar mus -lambdas mur4 ,!n ! cross ratio ,p#a ,p#b ,p#c ,p#d is def9$ to be ! numb] 7lambda#amu#b7 7lambda#cmu#d7 _/ 7lambda#amu#d7 7lambda#cmu#b74 ,! equality ( ! cross ratios ( ! ranges 7,p#a ,p#b ,p#c ,p#d7 & 7,q#a ,q#b ,q#c ,q#d7 is prov$ to be ! necess>y & suffici5t condition = _! mutual projectivity4 ,! cross ratios ( all h>monic ranges are !n easily se5 to be all equal to -#a1 by comp>+ ) ! range 7,,oe#a,,ue'#a7 on ! axis ( x4 ,?us all ! ord9>y propositions ( geometry 9 :i* di/ance & angul> measure d n ent] o!rwise ?an 9 cross ratios c n[ be enunciat$ & prov$4 ,accord+ly ! great] "p ( ! analytical !ory ( conics & quadrics belongs to geometry at ?is /age ,! !ory ( di/ance w be consid]$ aft] ! pr9ciples ( descriptive geometry h be5 develop$4 ,descriptive ,geometry4 ,descriptive geometry is ess5tially ! sci5ce ( multiple ord] = op5 s]ies4 ,! fir/ satisfactory sy/em ( axioms 0 giv5 by ,m4 ,pas*4#cf ,an improv$ v]sion is due to ,g4 ,peano4#cg ,bo? ^! au?ors treat ! idea ( ! class ( po9ts con/itut+ ! segm5t ly+ betwe5 two po9ts z an undef9$ fundam5tal idea4 ,?us 9 fact !re are 9 ?is sy/em two fundam5tal ideas1 namely1 ( po9ts & ( segm5ts4 ,x is !n easy en\gh to def9e ! prolongations ( ! segm5ts1 s z to =m ! complete /raight l9es4 ,d4 ,hilb]t's#ch =mulation ( ! axioms is 9 ?is respect practically bas$ on ! same fundam5tal ideas4 ,8 "w is ju/ly fam\s = "s ( ! ma!matical inve/igations conta9$ 9 x1 b 8 exposition ( ! axioms is di/9ctly inf]ior to t ( ,peano4 ,descriptive geometry c also be consid]$#ci z ! sci5ce ( a class ( relations1 ea* relation be+ a two-t]m$ s]ial relation1 z consid]$ 9 ! logic ( relations1 rang+ ! po9ts betwe5 :i* x holds into a l9e> op5 ord]4 ,?us ! relations are ! /raight l9es1 & ! t]ms betwe5 :i* !y hold are ! po9ts4 ,b a comb9ation ( ^! two po9ts ( view yields#dj ! simple/ /atem5t ( all4 ,descriptive geometry is !n conceiv$ z ! inve/igation ( an undef9$ fundam5tal relation betwe5 ?ree t]ms 7po9ts72 & :5 ! relation holds betwe5 ?ree po9ts ,a1 ,b1 ,c1 ! po9ts are said to be 9 ! l9e> ord] ,,abc4 ,o4 ,vebl5's axioms & def9itions1 slightly modifi$1 are z foll[s3-- #a4 ,if ! po9ts ,a1 ,b1 ,c are 9 ! ord] ,,abc1 !y are 9 ! ord] ,,cba4 #b4 ,if ! po9ts ,a1 ,b1 ,c are 9 ! ord] ,,abc1 !y are n 9 ! ord] ,,bca4 #c4 ,if ! po9ts ,a1 ,b1 ,c are 9 ! ord] ,,abc1 ,a is di/9ct f ,c4 #d4 ,if ,a & ,b are any two di/9ct po9ts1 !re exi/s a po9t ,c su* t ,a1 ,b1 ,c are 9 ! ord] ,,abc4 ,def9ition4--,! l9e ,,ab 7,a ,b7 consi/s ( ,a & ,b1 & ( all po9ts ,x 9 "o ( ! possible ord]s1 ,,abx1 ,,axb1 ,,xab4 ,! po9ts ,x 9 ! ord] ,,axb con/itute ! segm5t ,,ab4 #e4 ,if po9ts ,c & ,d 7,c ,d7 lie on ! l9e ,,ab1 !n ,a lies on ! l9e ,,cd4 #f4 ,!re exi/ ?ree di/9ct po9ts ,a1 ,b1 ,c n 9 any ( ! ord]s ,,abc1 ,,bca1 ,,cab4 #g4 ,if ?ree di/9ct po9ts ,a1 ,b1 ,c 7fig4 #gc7 d n lie on ! same l9e1 & ,d & ,e are two di/9ct po9ts 9 ! ord]s ,,bcd & ,,cea1 !n a po9t ,f exi/s 9 ! ord] ,,afb1 & su* t ,d1 ,e1 ,f are coll9e>4 ,illu/ration3 ,,fig4 #gc4 ,def9ition4--,if ,a1 ,b1 ,c are ?ree non-coll9e> po9ts1 ! plane ,,abc is ! class ( po9ts :i* lie on any "o ( ! l9es jo9+ any two ( ! po9ts belong+ to ! b\nd>y ( ! triangle ,,abc1 ! b\nd>y be+ =m$ by ! segm5ts ,,bc1 ,,ca & ,,ab4 ,! int]ior ( ! triangle ,,abc is =m$ by ! po9ts 9 segm5ts su* z ,,pq1 :]e ,p & ,q are po9ts respectively on two ( ! segm5ts ,,bc1 ,,ca1 ,,ab4 #h4 ,!re exi/s a plane ,,abc1 :i* does n conta9 all ! po9ts4 ,def9ition4--,if ,a1 ,b1 ,c1 ,d are f\r non-coplan> po9ts1 ! space ,,abcd is ! class ( po9ts :i* lie on any ( ! l9es conta9+ two po9ts on ! surface ( ! tetrah$ron ,,abcd1 ! surface be+ =m$ by ! int]iors ( ! triangles ,,abc1 ,,bcd1 ,,dca1 ,,dab4 #i4 ,!re exi/s a space ,,abcd :i* conta9s all ! po9ts4 #aj4 ,! ,d$ek9d prop]ty holds = ! ord] ( ! po9ts on any /raight l9e4 ,x foll[s f axioms #a-i t ! po9ts on any /raight l9e are arrang$ 9 an op5 s]ial ord]4 ,also all ! ord9>y !orems respect+ a po9t divid+ a /raight l9e into two p>ts1 a /raight l9e divid+ a plane into two p>ts1 & a plane divid+ space into two p>ts1 foll[4 ,aga91 9 any plane alpha consid] a l9e l & a po9t ,a 7fig4 #gd74 ,illu/ration3 ,,fig4 #gd4 ,let any po9t ,b divide l into two half-l9es l#a & l#b4 ,!n x c be prov$ t ! set ( half-l9es1 emanat+ f ,a & int]sect+ l#a 7su* z m71 are b\nd$ by two half-l9es1 ( :i* ,,abc is "o4 ,let r be ! o!r4 ,!n x c be prov$ t r does n int]sect l#a4 ,simil>ly = ! half-l9e1 su* z n1 int]sect+ l#b4 ,let s be its b\nd+ half-l9e4 ,!n two cases are possible4 7#a7 ,! half-l9es r & s are coll9e>1 & toge!r =m "o complete l9e4 ,9 ?is case1 !re is "o & only "o l9e 7viz4 r s7 ?r\gh ,a & ly+ 9 alpha :i* does n int]sect l4 ,?is is ! ,euclidean case1 & ! assumption t ?is case holds is ! ,euclidean p>allel axiom4 ,b 7#b7 ! half-l9es r & s may n be coll9e>4 ,9 ?is case !re w be an inf9ite numb] ( l9es1 su* z k = in/ance1 conta9+ ,a & ly+ 9 alpha1 :i* d n int]sect l4 ,!n ! l9es ?r\gh ,a 9 alpha are divid$ into two classes by ref]5ce to l1 namely1 ! secant l9es :i* int]sect l1 & ! non-secant l9es :i* d n int]sect l4 ,! two b\nd>y non-secant l9es1 ( :i* r & s are respectively halves1 may be call$ ! two p>allels to l ?r\gh ,a4 ,! p]ception ( ! possibility ( case #b con/itut$ ! />t+-po9t f :i* ,lobat*ewsky con/ruct$ ! fir/ explicit coh]5t !ory ( non-,euclidean geometry1 & ?us creat$ a revolution 9 ! philosophy ( ! subject4 ,= _m c5turies ! speculations ( ma!maticians on ! f\ndations ( geometry 7 almo/ conf9$ to hopeless attempts to prove ! p>allel axiom )\t ! introduction ( "s equival5t axiom4#da ,associat$ ,projective & ,descriptive ,spaces4--,a region ( a projective space1 su* t "o1 & only "o1 ( ! two supplem5t>y segm5ts betwe5 any pair ( po9ts )9 x lies entirely )9 x1 satisfies ! above axioms 7#a-aj7 ( descriptive geometry1 :]e ! po9ts ( ! region are ! descriptive po9ts1 & ! portions ( /raight l9es )9 ! region are ! descriptive l9es4 ,if ! exclud$ "p ( ! orig9al projective space is a s+le plane1 ! ,euclidean p>allel axiom also holds1 o!rwise x does n hold = ! descriptive space ( ! limit$ region4 ,aga91 conv]sely1 />t+ f an orig9al descriptive space an associat$ projective space c be con/ruct$ by means ( ! concept ( ideal po9ts4#db ,^! are also call$ projective po9ts1 :]e x is und]/ood t ! simple po9ts are ! po9ts ( ! orig9al descriptive space4 ,an ideal po9t is ! class ( /raight l9es :i* is compos$ ( two coplan> l9es a & b1 toge!r ) ! l9es ( int]section ( all pairs ( int]sect+ planes :i* respectively conta9 a & b1 toge!r ) ! l9es ( int]section ) ! plane ab ( all planes conta9+ any "o ( ! l9es 7o!r ?an a or b7 already specifi$ z belong+ to ! ideal po9t4 ,x is evid5t t1 if ! two orig9al l9es a & b int]sect1 ! correspond+ ideal po9t is no?+ else ?an ! :ole class ( l9es :i* are concurr5t at ! po9t ab4 ,b ! ess5ce ( ! def9ition is t an ideal po9t has an exi/5ce :5 ! l9es a & b d n int]sect1 s long z !y are coplan>4 ,an ideal po9t is t]m$ prop]1 if ! l9es compos+ x int]sect2 o!rwise x is improp]4 ,a !orem ess5tial to ! :ole !ory is ! foll[+3 if any two ( ! ?ree l9es a1 b1 c are coplan>1 b ! ?ree l9es are n all coplan>1 & simil>ly = ! l9es a1 b1 d1 !n c & d are coplan>4 ,x foll[s t any two l9es belong+ to an ideal po9t c be us$ z ! pair ( guid+ l9es 9 ! def9ition4 ,an ideal po9t is said to be coh]5t ) a plane1 if any ( ! l9es compos+ x lie 9 ! plane4 ,an ideal l9e is ! class ( ideal po9ts ea* ( :i* is coh]5t ) two giv5 planes4 ,if ! planes int]sect1 ! ideal l9e is t]m$ prop]1 o!rwise x is improp]4 ,x c be prov$ t any two planes1 ) :i* any two ( ! ideal po9ts are bo? coh]5t1 w s]ve z ! guid+ planes us$ 9 ! def9ition4 ,! ideal planes are def9$ z 9 projective geometry1 & all ! o!r def9itions 7= segm5ts1 ord]1 c47 ( projective geometry are appli$ to ! ideal elem5ts4 ,if an ideal plane conta9s "s prop] ideal po9ts1 x is call$ prop]1 o!rwise x is improp]4 ,e ideal plane conta9s "s improp] ideal po9ts4 ,x c n[ be prov$ t all ! axioms ( projective geometry hold ( ! ideal elem5ts z ?us obta9$2 & also t ! ord] ( ! ideal po9ts z obta9$ by ! projective me?od agrees ) ! ord] ( ! prop] ideal po9ts z obta9$ f t ( ! associat$ po9ts ( ! descriptive geometry4 ,?us a projective space has be5 con/ruct$ out ( ! ideal elem5ts1 & ! prop] ideal elem5ts correspond elem5t by elem5t ) ! associat$ descriptive elem5ts4 ,?us ! prop] ideal elem5ts =m a region 9 ! projective space )9 :i* ! descriptive axioms hold4 ,accord+ly1 by sub/itut+ ideal elem5ts1 a descriptive space c always be consid]$ z a region )9 a projective space4 ,?is is ! ju/ification = ! ord9>y use ( ! po9ts at inf9ity 9 ! ord9>y ,euclidean geometry2 ! reason+ has be5 transf]r$ f ! orig9al descriptive space to ! associat$ projective space ( ideal elem5ts2 & ) ! ,euclidean p>allel axiom ! improp] ideal elem5ts r$uce to ! ideal po9ts on a s+le improp] ideal plane1 namely1 ! plane at inf9ity4#dc ,congru5ce & ,measurem5t4--,! prop]ty ( physical space :i* is express$ by ! t]m measurability has n[ to be consid]$4 ,?is prop]ty has (t5 be5 consid]$ z ess5tial to ! v idea ( space4 ,= example1 ,kant writes1#dd ,space is repres5t$ z an inf9ite giv5 quantity4 ,?is quantitative aspect ( space arises f ! measurability ( di/ances1 ( angles1 ( surfaces & ( volumes4 ,^! f\r types ( quantity dep5d ^u ! two fir/ among !m z fundam5tal4 ,! measurability ( space is ess5tially connect$ ) ! idea ( congru5ce1 ( :i* ! simple/ examples are to be f\nd 9 ! pro(s ( equality by ! me?od ( sup]position1 z us$ 9 elem5t>y plane geometry4 ,! m]e concepts ( "p & ( :ole mu/ ( necessity be inadequate z ! f\ndation ( measurem5t1 s9ce we require ! comp>ison z to quantity ( regions ( space :i* h no portions 9 common4 ,! idea ( congru5ce1 z exemplifi$ by ! me?od ( sup]position 9 geometrical reason+1 appe>s to be f\nd$ ^u t ( ! rigid body1 :i* moves f "o position to ano!r ) its int]nal spatial relations un*ang$4 ,b unless !re is a previ\s concept ( ! metrical relations betwe5 ! p>ts ( ! body1 !re c be no basis f :i* to d$uce t !y are un*ang$4 ,x w\ld !re=e appe> z if ! idea ( ! congru5ce1 or metrical equality1 ( two portions ( space 7z empirically sugge/$ by ! motion ( rigid bodies7 mu/ be consid]$ z a fundam5tal idea incapable ( def9ition 9 t]ms ( ^? geometrical concepts :i* h already be5 enum]at$4 ,?is 0 9 effect ! po9t ( view ( ,pas*4#de ,x has1 h[ev]1 be5 prov$ by ,sophus ,lie#df t congru5ce is capable ( def9ition )\t rec\rse to a new fundam5tal idea4 ,?is he does by means ( 8 !ory ( f9ite cont9u\s gr\ps 7see ,,gr\ps1 ,,!ory ,,(71 ( :i* ! def9ition is possible 9 t]ms ( our e/abli%$ geometrical ideas1 rememb]+ t coord9ates h already be5 introduc$4 ,! displacem5t ( a rigid body is simply a mode ( def9+ to ! s5ses a "o-"o trans=mation ( all space into itself4 ,= at any po9t ( space a p>ticle may be conceiv$ to be plac$1 & to be rigidly connect$ ) ! rigid body2 & ?us !re is a def9ite correspond5ce ( any po9t ( space ) ! new po9t occupi$ by ! associat$ p>ticle aft] displacem5t4 ,aga9 two successive displacem5ts ( a rigid body f position ,a to position ,b1 & f position ,b to position ,c1 are ! same 9 effect z "o displacem5t f ,a to ,c4 ,b ?is is ! *>act]i/ic gr\p prop]ty4 ,?us ! trans=mations ( space into itself def9$ by displacem5ts ( rigid bodies =m a gr\p4 ,call ?is gr\p ( trans=mations a congru5ce-gr\p4 ,n[ accord+ to ,lie a congru5ce-gr\p is def9$ by ! foll[+ *>act]i/ics3-- #a4 ,a congru5ce-gr\p is a f9ite cont9u\s gr\p ( "o-"o trans=mations1 conta9+ ! id5tical trans=mation4 #b4 ,x is a sub-gr\p ( ! g5]al projective gr\p1 i4e4 ( ! gr\p ( :i* any trans=mation conv]ts planes into planes1 & /raight l9es into /raight l9es4 #c4 ,an inf9itesimal trans=mation c always be f\nd satisfy+ ! condition t1 at lea/ ?r\gh\t a c]ta9 enclos$ region1 any def9ite l9e & any def9ite po9t on ! l9e are lat5t1 i4e4 correspond to !mselves4 #d4 ,no inf9itesimal trans=mation ( ! gr\p exi/s1 su* t1 at lea/ 9 ! region = :i* 7#c7 holds1 a /raight l9e1 a po9t on x1 & a plane ?r\gh x1 %all all be lat5t4 ,! prop]ty enunciat$ by conditions 7#c7 & 7#d71 tak5 toge!r1 is nam$ by ,lie ,free mobility 9 ! inf9itesimal4 ,lie proves ! foll[+ !orems = a projective space3-- #a4 ,if ! above f\r conditions are only satisfi$ by a gr\p ?r\gh\t "p ( projective space1 ?is "p ei!r 7alpha7 mu/ be ! region enclos$ by a real clos$ quadric1 or 7beta7 mu/ be ! :ole ( ! projective space ) ! exception ( a s+le plane4 ,9 case 7alpha7 ! correspond+ congru5ce gr\p is ! cont9u\s gr\p = :i* ! enclos+ quadric is lat5t2 & 9 case 7beta7 an imag9>y conic 7) a real equation7 ly+ 9 ! lat5t plane is also lat5t1 & ! congru5ce gr\p is ! cont9u\s gr\p = :i* ! plane & conic are lat5t4 #b4 ,if ! above f\r conditions are satisfi$ by a gr\p ?r\gh\t ! :ole ( projective space1 ! congru5ce gr\p is ! cont9u\s gr\p = :i* "s imag9>y quadric 7) a real equation7 is lat5t4 ,by a prop] *oice ( non-homog5e\s co-ord9ates ! equation ( any quadrics ( ! types consid]$1 ei!r 9 !orem #a 7alpha71 or 9 !orem #b1 c be writt5 9 ! =m #a c7x#b y#b z#b7 #j1 :]e c is negative = a real clos$ quadric1 & positive = an imag9>y quadric4 ,!n ! g5]al inf9itesimal trans=mation is def9$ by ! ?ree equations3 dx_/dt u - omega#cy omega#bz cx7ux vy wz71 dy_/dt v - omega#az omega#cx cy7ux vy wz71 7,a7 dz_/dt w - omega#bx omega#ay cz7ux vy wz74 _/ ,9 ! ease consid]$ 9 !orem #a 7beta71 ) ! prop] *oice ( co-ord9ates ! ?ree equations def9+ ! g5]al inf9itesimal trans=mation are3 dx_/dt u - omega#cy omega#bz1 dy_/dt v - omega#az omega#cx1 7,b7 dz_/dt w - omega#bx omega#ay4 _/ ,9 ?is case ! lat5t plane is ! plane = :i* at lea/ "o ( x1 y1 z are inf9ite1 t is1 ! plane #j4x #j4y #j4z a #j2 & ! lat5t conic is ! conic 9 :i* ! cone x#b y#b z#b #j int]sects ! lat5t plane4 ,x foll[s f !orems #a & #b t !re is n "o unique congru5ce-gr\p1 b an indef9ite numb] ( !m4 ,!re is "o congru5ce-gr\p correspond+ to ea* clos$ real quadric1 "o to ea* imag9>y quadric ) a real equation1 & "o to ea* imag9>y conic 9 a real plane & ) a real equation4 ,! quadric ?us associat$ ) ea* congru5ce-gr\p is call$ ! absolute = t gr\p1 & 9 ! deg5]ate case ( #a 7beta7 ! absolute is ! lat5t plane toge!r ) ! lat5t imag9>y conic4 ,if ! absolute is real1 ! congru5ce-gr\p is hyp]bolic2 if imag9>y1 x is elliptic2 if ! absolute is a plane & imag9>y conic1 ! gr\p is p>abolic4 ,metrical geometry is simply ! !ory ( ! prop]ties ( "s p>ticul> congru5ce-gr\p select$ = /udy4 ,! def9ition ( di/ance is connect$ ) ! correspond+ congru5ce-gr\p by two consid]ations 9 respect to a range ( five po9ts 7,a#a1 ,a#b1 ,p#a1 ,p#b1 ,p#c71 ( :i* ,a#a & ,a#b are on ! absolute4 ,let ,a#a,p#a;,a#b,p#b /& = ! cross ratio 7z def9$ above7 ( ! range 7,a#a,p#a;,a#b,p#b71 ) a simil> notation = ! o!r ranges4 ,!n 7#a7 log,a#a,p#a;,a#b,p#b log,a#a,p#b;,a#b,p#c log,a#a,p#a;,a#b,p#c1 & 7#b71 if ! po9ts ,a#a1 ,a#b1 ,p#a1 ,p#b are trans=m$ into ,a'#a1 ,a'#b1 ,p'#a1 ,p'#b by any trans=mation ( ! congru5ce-gr\p1 7alpha7 ,a#a,p7#a;,a#b,p#b ,a'#a,p'#a;,a'#b,p'#b1 s9ce ! trans=mation is projective1 & 7beta7 ,a'#a1 ,a'#b are on ! absolute s9ce ,a#a & ,a#b are on x4 ,?us if we def9e ! di/ance ,p#a,p#b to be #a_/bk log ,a#a,p#a;,a#b,p#b1 :]e ,a#a & ,a#b are ! po9ts 9 :i* ! l9e ,p#a,p#b cuts ! absolute1 & k is "s con/ant1 ! two *>act]i/ic prop]ties ( di/ance1 namely1 7#a7 ! addition ( consecutive l5g?s on a /raight l9e1 & 7#b7 ! inv>iability ( di/ances dur+ a trans=mation ( ! congru5ce-gr\p1 are satisfi$4 ,?is is ! well-kn[n ,cayley-,kle9 projective def9ition#dg ( di/ance1 :i* 0 elaborat$ 9 view ( ! addition prop]ty alone1 previ\sly to ,lie's discov]y ( ! !ory ( congru5ce-gr\ps4 ,= a hyp]bolic gr\p :5 ,p#a & ,p#b are 9 ! region enclos$ by ! absolute1 log ,a#a,p#a;,a#b,p#b is real1 & !re=e k mu/ be real4 ,= an elliptic gr\p ,a#a & ,a#b are conjugate imag9>ies1 & log ,a#a,p#a;,a#b,p#b is a pure imag9>y1 & k is *os5 to be kappa_/iota1 :]e kappa is real & iota root-4 ,simil>ly ! angle betwe5 two planes1 p#a & p#b1 is def9$ to be 7#a_/b;iota7 log 7t#ap#at#bp#b71 :]e t#a & t#b are tang5t planes to ! absolute ?r\gh ! l9e p#ap#b4 ,! planes t#a & t#b are imag9>y = an elliptic gr\p1 & also = an hyp]bolic gr\p :5 ! planes p#a & p#b int]sect at po9ts )9 ! region enclos$ by ! absolute4 ,! developm5t ( ! consequ5ces ( ^! metrical def9itions is ! subject ( non-,euclidean geometry4 ,! def9itions = ! p>abolic case c be arriv$ at z limits ( ^? obta9$ 9 ei!r ( ! o!r two cases by mak+ k ultimately to vani%4 ,x is also obvi\s t1 if ,p#a & ,p#b be ! po9ts 7x#a1 y#a1 z#a7 & 7x#b1 y#b1 z#b71 x foll[s f equations 7,b7 above t 7x#a - x#b7b 7y#a - y#b7b 7z#a - z#b7ba_/b is unalt]$ by a congru5ce trans=mation & also satisfies ! addition prop]ty = coll9e> di/ances4 ,also ! previ\s def9ition ( an angle c be adapt$ to ?is case1 by mak+ t#a & t#b to be ! tang5t planes ?r\gh ! l9e p#ap#b to ! imag9>y conic4 ,simil>ly if p#a & p#b are int]sect+ l9es1 ! same def9ition ( an angle holds1 :]e t#a & t#b are n[ ! l9es f ! po9t p#ap#b to ! two po9ts :]e ! plane p#ap#b cuts ! imag9>y conic4 ,^! po9ts are 9 fact ! circul> po9ts at inf9ity on ! plane4 ,! developm5t ( ! consequ5ces ( ^! def9itions = ! p>abolic case gives ! ord9>y ,euclidean metrical geometry4 ,?us ! only metrical geometry = ! :ole ( projective space is ( ! elliptic type4 ,b ! actual measure-relations 7?\gh n _! g5]al prop]ties7 diff] accord+ to ! elliptic congru5ce-gr\p select$ = /udy4 ,9 a descriptive space a congru5ce-gr\p %\ld possess ! f\r *>act]i/ics ( su* a gr\p ?r\gh\t ! :ole ( ! space4 ,!n =m ! associat$ ideal projective space4 ,! associat$ congru5ce-gr\p = ?is ideal space mu/ satisfy ! f\r conditions ?r\gh\t ! region ( ! prop] ideal po9ts4 ,?us ! b\nd>y ( ?is region is ! absolute4 ,accord+ly !re c be no metrical geometry = ! :ole ( a descriptive space unless its b\nd>y 79 ! associat$ ideal space7 is a clos$ quadric or a plane4 ,if ! b\nd>y is a clos$ quadric1 !re is "o possible congru5ce-gr\p ( ! hyp]bolic type4 ,if ! b\nd>y is a plane 7! plane at inf9ity71 ! possible congru5ce-gr\ps are p>abolic2 & !re is a congru5ce-gr\p correspond+ to ea* imag9>y conic 9 ?is plane1 toge!r ) a ,euclidean metrical geometry correspond+ to ea* su* gr\p4 ,ow+ to ^! alt]native possibilities1 x w\ld appe> to be m accurate to say t sy/ems ( quantities c be f\nd 9 a space1 r ?an t space is a quantity4 ,lie has also d$uc$#dh ! same results ) respect to congru5ce-gr\ps f ano!r set ( def9+ prop]ties1 :i* explicitly assume ! exi/5ce ( a quantitative relation 7! di/ance7 betwe5 any two po9ts1 :i* is inv>iant = any trans=mation ( ! congru5ce-gr\p4#di ,! above results1 9 respect to congru5ce & metrical geometry1 consid]$ 9 relation to exi/5t space1 h l$ to ! doctr9e#ej t x is intr9sically unmean+ to ask :i* sy/em ( metrical geometry is true ( ! physical _w4 ,any "o ( ^! sy/ems c be appli$1 & 9 an indef9ite numb] ( ways4 ,! only "q be=e u is "o ( conv5i5ce 9 respect to simplicity ( /atem5t ( ! physical laws4 ,?is po9t ( view seems to neglect ! consid]ation t sci5ce is to be relevant to ! def9ite p]ceiv+ m9ds ( m52 & t 7neglect+ ! ambiguity introduc$ by ! inv>iable slight inexactness ( obs]vation :i* is n relevant to ?is special doctr9e7 we h1 9 fact1 pres5t$ to our s5ses a def9ite set ( trans=mations =m+ a congru5ce-gr\p1 result+ 9 a set ( measure relations :i* are 9 no respect arbitr>y4 ,accord+ly our sci5tific laws are to be /at$ relevantly to t p>ticul> congru5ce-gr\p4 ,?us ! inve/igation ( ! type 7elliptic1 hyp]bolic or p>abolic7 ( ?is special congru5ce-gr\p is a p]fectly def9ite problem1 to be decid$ by exp]im5t4 ,! consid]ation ( exp]im5ts adapt$ to ?is object requires "s developm5t ( non-,euclidean geometry 7see section ,,vi41 ,non-,euclidean ,geometry74 ,b if ! doctr9e means t1 assum+ "s sort ( objective reality = ! mat]ial univ]se1 be+s c be imag9$1 to :om ei!r all congru5ce-gr\ps are equally important1 or "s o!r congru5ce-gr\p is specially important1 ! doctr9e appe>s to be an imm$iate d$uction f ! ma!matical facts4 ,assum+ a def9ite congru5ce-gr\p1 ! inve/igation ( surfaces 7or ?ree-dim5sional loci 9 space ( f\r dim5sions7 ) geodesic geometries ( ! =m ( metrical geometries ( o!r types ( congru5ce-gr\ps =ms an important *apt] ( non-,euclidean geometry4 ,aris+ f ?is inve/igation !re is a widely-spread fallacy1 :i* has f\nd its way into _m philosophic writ+s1 namely1 t ! possibility ( ! geometry ( exi/5t ?ree-dim5sional space be+ o!r ?an ,euclidean dep5ds on ! physical exi/5ce ( ,euclidean space ( f\r or m dim5sions4 ,! =ego+ exposition %[s ! baselessness ( ?is idea4 ,,bibliography4--,= an acc\nt ( ! inve/igations on ! axioms ( geometry dur+ ! ,greek p]iod1 see ,m4 ,cantor1 ,vorlesung5 ub] die ,ges*i*te d] ,ma!matik1 ,bd4 i4 & iii42 ,t4,l4 ,hea?1 ,! ,?irte5 ,books ( ,euclid's ,elem5ts1 a ,new ,translation f ! ,greek1 ) ,introductory ,essays & ,comm5t>y1 ,hi/orical1 ,critical1 & ,explanatory 7,cambridge1 #aijh7--?is "w is ! /&>d s\rce ( in=mation2 ,w4,b4 ,frankl&1 ,euclid1 ,book ,i41 ) a ,comm5t>y 7,cambridge1 #aije7--! comm5t>y conta9s copi\s extracts f ! anci5t comm5tators4 ,! next p]iod ( really sub/antive importance is t ( ! #ah? c5tury4 ,! lead+ au?ors are3 ,g4 ,sac*]i1 ,s4,j41 ,euclides ab omni naevo v9dicatus 7,milan1 #agcc74 ,sac*]i 0 an ,italian ,jesuit :o unconsci\sly discov]$ non-,euclidean geometry 9 ! c\rse ( 8 ef=ts to prove its impossibility4 ,j4,h4 ,lamb]t1 ,!orie d] ,p>allell9i5 7#agff72 ,a4,m4 ,leg5dre1 ,elem5ts de geometrie 7#agid74 ,an adequate acc\nt ( ! above au?ors is giv5 by ,p4 ,/ackel & ,f4 ,engel1 ,die ,!orie d] ,p>allell9i5 von ,euklid bis auf ,gauss 7,leipzig1 #ahie74 ,! next p]iod ( "t 7r\ghly f #ahjj to #ahgj7 conta9s two /reams ( ?\ght1 bo? ( :i* are ess5tial to ! mod]n analysis ( ! subject4 ,! fir/ /ream is t :i* produc$ ! discov]y & inve/igation ( non-,euclidean geometries1 ! second /ream is t :i* has produc$ ! geometry ( position1 compris+ bo? projective & descriptive geometry n v accurately discrim9at$4 ,! lead+ au?ors on non-,euclidean geometry are ,k4,f4 ,gauss1 9 private lett]s to ,s*uma*]1 cf4 ,/ackel & ,engel1 loc4 cit42 ,n4 ,lobat*ewsky1 rector ( ! univ]sity ( ,kazan1 to :om ! hon\r ( ! effective discov]y ( non-,euclidean geometry mu/ be assign$4 ,8 fir/ publication 0 at ,kazan 9 #ahbf4 ,8 v>i\s memoirs h be5 re-edit$ by ,engel2 cf4 ,urkund5 zur ,ges*i*te d] ni*teuklidis*5 ,geometrie by ,/ackel & ,engel1 vol4 i4 ,lobat*ewsky4 ,j4 ,bolyai discov]$ non-,euclidean geometry app>5tly 9 indep5d5ce ( ,lobat*ewsky4 ,8 memoir 0 publi%$ 9 #ahca z an app5dix to a "w by 8 fa!r ,w4 ,bolyai1 ,t5tam5 juv5tutem4444 ,?is memoir has be5 sep>ately edit$ by ,j4 ,fris*auf1 ,absolute ,geometrie na* ,j4 ,bolyai 7,leipzig1 #ahgb72 ,b4 ,riemann1 ,ub] die ,hypo!s51 wel*e d] ,geometrie zu ,grunde lieg5 7#ahed72 cf4 ,gesamte ,w]ke1 a translation 9 ,! ,collect$ ,pap]s ( ,w4,k4 ,clif=d4 ,?is is a fundam5tal memoir on ! subject & mu/ rank ) ! "w ( ,lobat*ewsky4 ,riemann discov]$ elliptic metrical geometry1 & ,lobat*ewsky hyp]bolic geometry4 ,a full acc\nt ( ,riemann's ideas1 ) ! subsequ5t developm5ts due to ,clif=d1 ,f4 ,kle9 & ,w4 ,kill+1 w be f\nd 9 ,! ,bo/on ,colloquium = #aijc 7,new ,york1 #aije71 article ,=ms ( ,non-,euclidean ,space1 by ,f4,s4 ,woods4 ,a4 ,cayley1 loc4 cit4 7#ahei71 & ,f4 ,kle91 ,ub] die sog5annte ni*teuklidis*e ,geometrie1 ,ma?4 ,annal4 vols4 iv4 & vi4 7#ahga & #ahgb71 betwe5 !m elaborat$ ! projective !ory ( di/ance2 ,h4 ,helmholtz1 ,ub] die ?atsa*li*5 ,grundlag5 d] ,geometrie 7#ahff71 & ,ub] die ,?atsa*51 die d] ,geometrie zu ,grunde lieg5 7#ahfh71 bo? 9 8 ,wiss5s*aftli*e ,abh&lung51 vol4 ii41 & ,s4 ,lie1 loc4 cit4 7#ahij & #ahic71 betwe5 !m elaborat$ ! gr\p !ory ( congru5ce4 ,! numb]less works :i* h be5 writt5 to sugge/ equival5t alt]natives to ,euclid's p>allel axioms may be neglect$ z be+ ( trivial importance1 ?\gh _m ( !m are m>vels ( geometric +5uity4 ,! second /ream ( ?\ght conf9$ itself )9 ! circle ( ideas ( ,euclidean geometry4 ,its orig9 0 ma9ly due to a succession ( great ,fr5* ma!maticians1 = example1 ,g4 ,monge1 ,geometrie descriptive 7#ahjj72 ,j4,v4 ,poncelet1 ,traite des proprietes projectives des figures 7#ahbb72 ,m4 ,*asles1 ,ap]cu hi/orique sur l'orig9e et le developpem5t des me?odes en geometrie 7,bruxelles1 #ahcg71 & ,traite de geometrie sup]ieure 7,p>is1 #aheb72 & _m o!rs4 ,b ! works :i* h be51 & are /ill1 ( decisive influ5ce on ?\ght z a /ore-h\se ( ideas relevant to ! f\ndations ( geometry are ,k4,g4,c4 von ,/audt's two works1 ,geometrie d] ,lage 7,nurnb]g1 #ahdg72 & ,beitrage zur ,geometrie d] ,lage 7,nurnb]g1 #ahef1 #crd ed4 #ahfj74 ,! f9al p]iod is *>act]iz$ by ! successful production ( exact sy/ems ( axioms1 & by ! f9al solution ( problems :i* h occupi$ ma!maticians = two ?\s& ye>s4 ,! successful analysis ( ! ideas involv$ 9 s]ial cont9uity is due to ,r4 ,d$ek9d1 ,/etigkeit und irrationale ,zahl5 7#ahgb71 & to ,g4 ,cantor1 ,grundlag5 e9] allgeme95 ,mannigfaltigkeitslehre 7,leipzig1 #ahhc71 & ,acta ma?4 vol4 #b4 ,complete sy/ems ( axioms h be5 /at$ by ,m4 ,pas*1 loc4 cit42 ,g4 ,peano1 loc4 cit42 ,m4 ,pi]i1 loc4 cit42 ,b4 ,russell1 ,pr9ciples ( ,ma!matics2 ,o4 ,vebl51 loc4 cit42 & by ,g4 ,v]onese 9 8 treatise1 ,fondam5ti di geometria 7,padua1 #ahia2 ,g]man transl4 by ,a4 ,s*epp1 ,grundzuge d] ,geometrie1 ,leipzig1 #ahid74 ,mo/ ( ! lead+ memoirs on special que/ions involv$ h be5 cit$ 9 ! text2 9 addition !re may be m5tion$ ,m4 ,pi]i1 ,nuovi pr9cipii di geometria projettiva complessa1 ,trans4 ,accad4 ,r4 d4 ,sci4 7,tur91 #aije72 ,e4,h4 ,moore1 ,on ! ,projective ,axioms ( ,geometry1 ,trans4 ,am]4 ,ma?4 ,soc41 #aijb2 ,o4 ,vebl5 & ,w4,h4 ,bussey1 ,f9ite ,projective ,geometries1 ,trans4 ,am]4 ,ma?4 ,soc41 #aije2 ,a4,b4 ,kempe1 ,on ! ,relation betwe5 ! ,logical ,!ory ( ,classes & ! ,geometrical ,!ory ( ,po9ts1 ,proc4 ,lond4 ,ma?4 ,soc41 #ahij2 ,j4 ,royce1 ,! ,relation ( ! ,pr9ciples ( ,logic to ! ,f\ndations ( ,geometry1 ,trans4 ( ,am]4 ,ma?4 ,soc41 #aije2 ,a4 ,s*o5flies1 ,ub] die ,mogli*keit e9] projectiv5 ,geometrie bei transf9it] 7ni*t>*im$is*]7 ,massbe/immung1 ,deuts*4 ,m4-,v4 ,jahresb41 #aijf4 ,= g5]al expositions ( ! be>+s ( ! above inve/igations1 cf4 ,hon4 ,b]tr& ,russell1 loc4 cit42 ,l4 ,c\turat1 ,les ,pr9cipes des ma!matiques 7,p>is1 #aije72 ,h4 ,po9c>e1 loc4 cit42 ,russell & ,:itehead1 ,pr9cipia ma!matica 7,cambridge1 ,univ4 ,press74 ,! philosoph]s ^: views on space & geometric tru? des]ve especial /udy are ,desc>tes1 ,leibnitz1 ,hume1 ,kant & ,j4,s4 ,mill4 7,a4 ,n4 ,w47 ,,footnotes3 #a ,= ,egyptian geometry see ,,egypt1 ,s ,sci5ce & ,ma!matics4 #b ,cf4 ,a4,n4 ,:itehead1 ,univ]sal ,algebra1 ,bk4 vi4 7,cambridge1 #ahih74 #c ,cf4 ,a4,n4 ,:itehead1 loc4 cit4 #d ,cf4 ,a4,n4 ,:itehead1 ,! ,geodesic ,geometry ( ,surfaces 9 non-,euclidean ,space1 ,proc4 ,lond4 ,ma?4 ,soc4 vol4 xxix4 #e ,cf4 ,kle91 ,zur ni*t-,euklidis*5 ,geometrie1 ,ma?4 ,annal4 vol4 xxxvii4 #f ,on ! !ory ( p>allels be=e ,lobat*ewsky1 see ,/ackel und ,engel1 ,!orie d] ,p>allell9i5 von ,euklid bis auf ,gauss 7,leipzig1 #ahie74 ,! =ego+ rem>ks are bas$ ^u ! mat]ials collect$ 9 ?is "w4 #g ,see ,/ackel und ,engel1 op4 cit41 & ,gauss1 die beid5 ,bolyai1 und die ni*t-,euklidis*e ,geometrie1 ,ma?4 ,annal51 ,bd4 xlix42 also ,engel's translation ( ,lobat*ewsky 7,leipzig1 #ahih71 pp4 #cgh ff4 #h ,lobat*ewsky's works on ! subject are ! foll[+3--,on ! ,f\ndations ( ,geometry1 ,kazan ,mess5g]1 #ahbi-ahcj2 ,new ,f\ndations ( ,geometry1 ) a complete ,!ory ( ,p>allels1 ,proce$+s ( ! ,univ]sity ( ,kazan1 #ahce 7bo? 9 ,russian1 b translat$ into ,g]man by ,engel1 ,leipzig1 #ahih72 ,geometrie imag9aire1 ,crelle's ,j\rnal1 #ahcg2 ,!orie d] ,p>allell9i5 7,b]l91 #ahdj2 #bnd ed41 #ahhg2 translat$ by ,hal/$1 ,au/91 ,texas1 #ahia74 ,8 results appe> to h be5 set =? 9 a pap] 7n[ lo/7 :i* he read at ,kazan 9 #ahbf4 #i ,translat$ by ,hal/$ 7,au/91 ,texas1 #d? ed41 #ahif47 #aj ,abh&lung5 d4 ,konigl4 ,ges4 d4 ,wiss4 zu ,gott+51 ,bd4 xiii42 ,ges4 ma?4 ,w]ke1 pp4 #bed-bfi2 translat$ by ,clif=d1 ,collect$ ,ma!matical ,pap]s4 #aa ,cf4 ,gesamm4 ma?4 und phys4 ,w]ke1 vol4 i4 7,leipzig1 #ahid74 #ab ,wiss4 ,abh4 vol4 ii4 pp4 #faj1 #fah 7#ahff1 #ahfh74 #ac ,m9d1 ,o4,s41 vols4 i4 & iii42 ,vortrage und ,r$51 vol4 ii4 pp4 #a1 #bef4 #ad ,8 pap]s are ,saggio di int]pretazione della geometria non-,euclidea1 ,giornale di matemati*e1 vol4 vi4 7#ahfh72 ,teoria fondam5tale degli spazii di curvatura co/ante1 ,annali di matematica1 vol4 ii4 7#ahfh-ahfi74 ,bo? 7 translat$ into ,fr5* by ,j4 ,h\el1 ,annales sci5tifiques de l',ecole ,normale sup]ieure1 vol4 vi4 7#ahfi74 #ae ,beltrami %[s also t ?is def9ition agrees ) t ( ,gauss4 #af ,sur la !orie des foy]s1 ,n\v4 ,ann4 vol4 xii4 #ag ,ma?4 ,annal51 iv4 vi41 #ahga-ahgb4 #ah ,= an inve/igation ( ^! & simil> prop]ties1 see ,:itehead1 ,univ]sal ,algebra 7,cambridge1 #ahih71 bk4 vi4 *4 ii4 ,! pol> =m 0 indep5d5tly discov]$ by ,simon ,newcomb 9 #ahgg4 #ai ,= an analysis ( ,leibnitz's ideas on space1 cf4 ,b4 ,russell1 ,! ,philosophy ( ,leibnitz1 *s4 viii4-x4 #bj ,cf4 ,hon4 ,b]tr& ,russell1 ,is ,position 9 ,"t & ,space ,absolute or ,relative8 ,m9d1 n4s4 vol4 #aj 7#aija71 & ,a4,n4 ,:itehead1 ,ma!matical ,concepts ( ! ,mat]ial ,_w1 ,phil4 ,trans4 7#aijf71 p4 #bje4 #ba ,cf4 ,critique ( ,pure ,reason1 #a/ section3 ,( ,space1 conclusion ,a1 ,max ,mull]'s translation4 #bb ,cf4 ,ern/ ,ma*1 ,erk5ntniss und ,irrtum 7,leipzig72 ! relevant *apt]s are translat$ by ,t4,j4 ,mc,cormack1 ,space & ,geometry 7,london1 #aijf72 also ,a4 ,me9ong1 ,ub] die ,/ellung d] ,geg5/&/heorie im ,sy/em d] ,wiss5s*aft5 7,leipzig1 #aijg74 #bc ,cf4 ,russell1 ,pr9ciples ( ,ma!matics1 ,s #ceb 7,cambridge1 #aijc74 #bd ,cf4 ,a4,n4 ,:itehead1 ,! ,axioms ( ,projective ,geometry1 ,s #c 7,cambridge1 #aijf74 #be ,cf4 ,russell1 ,pr9c4 ( ,ma?41 *4 i4 #bf ,cf4 ,russell1 loc4 cit41 & ,g4 ,frege1 ,ub] die ,grundlag5 d] ,geometrie1 ,jahresb]4 d] ,deuts*4 ,ma?4 ,v]4 7#aijf74 #bg ,?is =mulation--?\gh n 9 respect to numb]--is 9 all ess5tials t ( ,m4 ,pi]i1 cf4 ,i pr9cipii della ,geometria di ,posizione1 ,accad4 ,r4 di ,tor9o 7#ahih72 also cf4 ,:itehead1 loc4 cit4 #bh ,cf4 ,g4 ,peano1 ,sui fondam5ti della ,geometria1 p4 #gc1 ,rivi/a di matematica1 vol4 iv4 7#ahid71 & ,d4 ,hilb]t1 ,grundlag5 d] ,geometrie 7,leipzig1 #ahii72 & ,r4,f4 ,m\lton1 ,a ,simple non-,des>guesian ,plane ,geometry1 ,trans4 ,am]4 ,ma?4 ,soc41 vol4 iii4 7#aijb74 #bi ,cf4 ,sui po/ulati fondam5tali della geometria projettiva1 ,giorn4 di matematica1 vol4 xxx4 7#ahia72 also ( ,pi]i1 loc4 cit41 & ,:itehead1 loc4 cit4 #cj ,cf4 ,hilb]t1 loc4 cit42 = a full] exposition ( ,hilb]t's pro( cf4 ,k4,t4 ,vahl51 ,ab/rakte ,geometrie 7,leipzig1 #aije71 also ,:itehead1 loc4 cit4 #ca ,cf4 ,h4 ,wi5]1 ,jahresb]4 d] ,deuts*4 ,ma?4 ,v]4 vol4 i4 7#ahij72 & ,f4 ,s*ur1 ,ub] d5 ,fundam5talsatz d] projectiv5 ,geometrie1 ,ma?4 ,ann4 vol4 li4 7#ahii74 #cb ,cf4 ,hilb]t1 loc4 cit41 & ,:itehead1 loc4 cit4 #cc ,cf4 ,d$ek9d1 ,/etigkeit und irrationale ,zahl5 7#ahgb74 #cd ,cf4 v4 ,/audt1 ,geometrie d] ,lage 7#ahdg74 #ce ,cf4 ,pas*1 ,vorlesung5 ub] neu]e ,geometrie 7,leipzig1 #ahhb71 a classic "w2 also ,fi$l]1 ,die d>/ell5de ,geometrie 7#a/ ed41 #ahga1 #crd ed41 #ahhh72 ,clebs*1 ,vorlesung5 ub] ,geometrie1 vol4 iii42 ,hilb]t1 loc4 cit42 ,f4 ,s*ur1 ,ma?4 ,ann4 ,bd4 lv4 7#aijb72 ,vahl51 loc4 cit42 ,:itehead1 loc4 cit4 #cf ,cf4 loc4 cit4 #cg ,cf4 ,i ,pr9cipii di geometria 7,tur91 #ahhi7 & ,sui fondam5ti della geometria1 ,rivi/a di mat4 vol4 iv4 7#ahid74 #ch ,cf4 loc4 cit4 #ci ,cf4 ,vailati1 ,rivi/a di mat4 vol4 iv4 & ,russell1 loc4 cit4 ,s #cgf4 #dj ,cf4 ,o4 ,vebl51 ,on ! ,projective ,axioms ( ,geometry1 ,trans4 ,am]4 ,ma?4 ,soc4 vol4 iii4 7#aijb74 #da ,cf4 ,p4 ,/ackel & ,f4 ,engel1 ,die ,!orie d] ,p>allell9i5 von ,euklid bis auf ,gauss 7,leipzig1 #ahie74 #db ,cf4 ,pas*1 loc4 cit41 & ,r4 ,bonola1 ,sulla introduzione degli enti improprii 9 geometria projettive1 ,giorn4 di mat4 vol4 xxxviii4 7#aijj72 & ,:itehead1 ,axioms ( ,descriptive ,geometry 7,cambridge1 #aijg74 #dc ,! orig9al idea 7conf9$ to ?is p>ticul> case7 ( ideal po9ts is due to von ,/audt 7loc4 cit474 #dd ,cf4 ,critique1 ,trans4 ,ae/h4 ,sect4 ,i4 #de ,cf4 loc4 cit4 #df ,cf4 ,ub] die ,grundlag5 d] ,geometrie 7,leipzig1 ,b]41 #ahij72 & ,!orie d] ,trans=mationsgrupp5 7,leipzig1 #ahic71 vol4 iii4 #dg ,cf4 ,a4 ,cayley1 ,a ,six? ,memoir on ,quantics1 ,trans4 ,roy4 ,soc41 #ahei1 & ,coll4 ,pap]s1 vol4 ii42 & ,f4 ,kle91 ,ma?4 ,ann4 vol4 iv41 #ahga4 #dh ,cf4 loc4 cit4 #di ,= simil> d$uctions f a ?ird set ( axioms1 sugge/$ 9 ess5ce by ,peano1 ,riv4 mat4 vol4 iv4 loc4 cit4 cf4 ,:itehead1 ,desc4 ,geom4 loc4 cit4 #ej ,cf4 ,h4 ,po9c>e1 ,la ,sci5ce et l'hypo!se1 *4 iii4