,,! ,,philosophy ,,( ,,ma!matics4 ,,! ,,sci5ce ,,( ,,ma!matics4 ---------------------------------------- - ,,ab/ract ,,ma!matics4 ,,concrete ,,ma!matics4 ------------- ,,analysis2 or1 ,! ,calculus4 ,,geometry4 ,,me*anics4 ----------------- ---------------- ,ord9>y ,transc5d5tal ,syn!tic ,analytic ,analysis2 ,analysis2 or ,special or ,g5]al or1 ,calculus or1 ,calculus ,geometry4 ,geometry4 ( ,direct ( ,indirect ,functions4 ,functions4 ---- ------- -------- ------ ,algebra4 ,calculus ,algebraic4 ,( ?ree ( ,trigonometry4 dim5sions4 ,v>iations4 ,ari?metic4 ,diff]5tial ,graphical4 ,( two & ,integral ,descriptive dim5sions4 ,calculus4 ,geometry4 ,,! ,,philosophy ,,( ,,ma!matics2 ,,translat$ ,,f ,,! ,,c\rs ,,de ,,philosophie ,,positive ,,( ,,augu/e ,,comte1 ,,by ,w4 ,m4 ,,gillespie1 ,,pr(essor ,,( ,,civil ,,eng9e]+ ,,adj4 ,,pr(4 ,,( ,,ma!matics ,,9 ,,union ,,college4 ,,new ,,york3 ,,h>p] ,,bro!rs1 ,,publi%]s1 #hb ,,cliff ,,/reet #ahea4 ,ent]$1 accord+ to ,act ( ,congress1 9 ! ye> "o ?\s& eight hundr$ & fifty-"o1 by ,,h>p] ,,bro!rs4 9 ! ,cl]k's ,(fice ( ! ,di/rict ,c\rt ( ! ,s\!rn ,di/rict ( ,new ,york4 ,,preface4 ,! pleasure & pr(it :i* ! translator has receiv$ f ! great "w "h pres5t$1 h induc$ him to lay x be=e 8 fell[-tea*]s & /ud5ts ( ,ma!matics 9 a m accessible =m ?an t 9 :i* x has hi!rto appe>$4 ,! want ( a compreh5sive map ( ! wide region ( ma!matical sci5ce--a bird's-eye view ( its lead+ features1 & ( ! true be>+s & relations ( all its p>ts--is felt by e ?\ghtful /ud5t4 ,he is l ! visitor to a great city1 :o gets no j idea ( its ext5t & situation till he has se5 x f "s comm&+ em95ce4 ,to h a panoramic view ( ! :ole di/rict--pres5t+ at "o glance all ! p>ts 9 due co-ord9ation1 & ! d>ke/ nooks cle>ly %[n--is invaluable to ei!r travell] or /ud5t4 ,x is ?is :i* has be5 mo/ p]fectly accompli%$ = ma!matical sci5ce by ! au?or ^: "w is "h pres5t$4 ,cle>ness & dep?1 compreh5siv5ess & precision1 h nev]1 p]haps1 be5 s rem>kably unit$ z 9 ,,augu/e ,,comte4 ,he views 8 subject f an elevation :i* gives to ea* "p ( ! complex :ole its true position & value1 :ile 8 telescopic glance loses none ( ! ne$ful details1 & n only itself pi]ces to ! he>t ( ! matt]1 b conv]ts its opaqu5ess into su* transp>5t cry/al1 t o!r eyes are enabl$ to see z deeply into x z 8 own4 ,any ma!matician :o p]uses ?is volume w ne$ no o!r ju/ification ( ! high op9ion "h express$2 b o!rs may appreciate ! foll[+ endorsem5ts ( well-kn[n au?orities4 ,mill1 9 8 ,logic1 calls ! "w ( ,m4 ,comte by f> ! greate/ yet produc$ on ! ,philosophy ( ! sci5ces2 & adds1 ( ?is admirable "w1 "o ( ! mo/ admirable portions is t 9 :i* he may truly be said to h creat$ ! ,philosophy ( ! high] ,ma!matics3 ,morell1 9 8 ,speculative ,philosophy ( ,europe1 says1 ,! classification giv5 ( ! sci5ces at l>ge1 & _! regul> ord] ( developm5t1 is unque/ionably a ma/]-piece ( sci5tific ?9k+1 z simple z x is compreh5sive2 & ,lewes1 9 8 ,biographical ,hi/ory ( ,philosophy1 names ,comte ! ,bacon ( ! n9ete5? c5tury1 & says1 ,i unhesitat+ly record my conviction t ?is is ! greate/ "w ( our age4 ,! complete "w ( ,m4 ,comte--8 ,c\rs de ,philosophie ,positive--fills six l>ge octavo volumes1 ( six or sev5 hundr$ pages ea*1 two ?irds ( ! fir/ volume compris+ ! purely ma!matical portion4 ,! great bulk ( ! ,c\rse is ! probable cause ( ! fewness ( ^? to :om ev5 ?is section ( x is kn[n4 ,its pres5tation 9 its pres5t =m is !re=e felt by ! translator to be a mo/ useful contribution to ma!matical progress 9 ?is c\ntry4 ,! compreh5siv5ess ( ! /yle ( ! au?or--grasp+ all possible =ms ( an idea 9 "o ,bri>ean s5t5ce1 arm$ at all po9ts aga9/ leav+ any op5+ = mi/ake or =getfulness--occasionally v]ges ^u cumb]som5ess & =mality4 ,! translator has1 !re=e1 sometimes tak5 ! lib]ty ( break+ up or cond5s+ a long s5t5ce1 & omitt+ a few passages n absolutely necess>y1 or ref]r+ to ! peculi> ,positive philosophy ( ! au?or2 b he has g5]ally aim$ at a consci5ti\s fidelity to ! orig9al4 ,x has (t5 be5 difficult to reta9 its f9e %ades & subtile di/9ctions ( mean+1 &1 at ! same "t1 replace ! peculi>ly appropriate ,fr5* idioms by correspond+ ,engli% ones4 ,! attempt1 h[ev]1 has always be5 made1 ?\gh1 :5 ! be/ c\rse has be5 at all d\btful1 ! language ( ! orig9al has be5 foll[$ z closely z possible1 &1 :5 necess>y1 smoo?ness & grace h be5 unhesitat+ly sacrific$ to ! high] attributes ( cle>ness & precision4 ,"s =ms ( expression may /rike ! read] z unusual1 b !y h be5 reta9$ because !y 7 *>act]i/ic1 n ( ! m]e language ( ! orig9al1 b ( its _s4 ,:5 a great ?9k] has clo!d 8 conceptions 9 phrases :i* are s+ul> ev5 9 8 own tongue1 he :o pr(esses to translate him is b\nd fai?fully to pres]ve su* =ms ( spee*1 z f> z is practicable2 & ?is has be5 "h done ) respect to su* peculi>ities ( expression z belong to ! au?or1 n z a =eign]1 b z an individual--n because he writes 9 ,fr5*1 b because he is ,augu/e ,comte4 ,! "y /ud5t ( ,ma!matics %\ld n attempt to read ! :ole ( ?is volume at once1 b %\ld p]use ea* portion ( x 9 connexion ) ! tempor>y subject ( 8 special /udy3 ! fir/ *apt] ( ! fir/ book1 = example1 :ile he is /udy+ ,algebra2 ! fir/ *apt] ( ! second book1 :5 he has made "s progress 9 ,geometry2 & s ) ! re/4 ,passages :i* are obscure at ! fir/ read+ w bright5 up at ! second2 & z 8 own /udies cov] a l>g] portion ( ! field ( ,ma!matics1 he w see m & m cle>ly _! relations to "o ano!r1 & to ^? :i* he is next to take up4 ,= ?is end he is urg5tly recomm5d$ to obta9 a p]fect famili>ity ) ! ,analytical ,table ( ,cont5ts1 :i* maps out ! :ole subject1 ! gr& divisions ( :i* are also indicat$ 9 ! ,tabul> ,view fac+ ! title-page4 ,correspond+ heads w be f\nd 9 ! body ( ! "w1 ! pr9cipal divisions be+ 9 ,,small ,,capitals1 & ! subdivisions 9 ,italics4 ,= ^! details ! translator alone is responsible4 ,,analytical ,,table ,,( ,,cont5ts4 ,,introduction4 ,page ,,g5]al ,,consid]ations ,,on ,,ma!matical ,,sci5ce #ag ,,! ,,object ,,( ,,ma!matics #ah ,measur+ ,magnitudes #ah ,difficulties #ai ,g5]al ,me?od #bj ,illu/rations #ba #a4 ,fall+ ,bodies #ba #b4 ,inaccessible ,di/ances #bc #c4 ,a/ronomical ,facts #bd ,,true ,,def9ition ,,( ,,ma!matics #be ,a ,sci5ce1 n an ,art #be ,,its ,,two ,,fundam5tal ,,divisions #bf ,_! diff]5t ,objects #bg ,_! diff]5t ,natures #bi ,concrete ,ma!matics #ca ,geometry & ,me*anics #cb ,ab/ract ,ma!matics #cc ,! ,calculus1 or ,analysis #cc ,,ext5t ,,( ,,its ,,field #ce ,its ,univ]sality #cf ,its ,limitations #cg ,,book ,i4 ,,analysis4 ,,*apt] ,i4 ,page ,,g5]al ,,view ,,( ,,ma!matical ,,analysis #de ,,! ,,true ,,idea ,,( ,,an ,,equation #df ,division ( ,functions into ,ab/ract & ,concrete #dg ,enum]ation ( ,ab/ract ,functions #ej ,,divisions ,,( ,,! ,,calculus #ec ,! ,calculus ( ,values1 or ,ari?metic #eg ,its ,ext5t #eg ,its true ,nature #ei ,! ,calculus ( ,functions #fa ,two ,modes ( obta9+ ,equations #fa #a4 ,by ! ,relations betwe5 ! giv5 ,quantities #fa #b4 ,by ! ,relations betwe5 auxili>y ,quantities #fd ,correspond+ ,divisions ( ! ,calculus ( ,functions #fg ,,*apt] ,,ii4 ,,ord9>y ,,analysis2 ,,or1 ,,algebra4 #fi ,its ,object #fi ,classification ( ,equations #gj ,,algebraic ,,equations #ga ,_! ,classification #ga ,,algebraic ,,resolution ,,( ,,equations #gb ,its ,limits #gb ,g5]al ,solution #gb ,:at we "k 9 ,algebra #gd ,,num]ical ,,resolution ,,( ,,equations #ge ,its limit$ ,usefulness #gf ,diff]5t ,divisions ( ! two ,sy/ems #gh ,,! ,,!ory ,,( ,,equations #gi ,,! ,,me?od ,,( ,,indet]m9ate ,,coeffici5ts #hj ,,imag9>y ,,quantities #ha ,,negative ,,quantities #ha ,,! ,,pr9ciple ,,( ,,homog5eity #hd ,,*apt] ,,iii4 ,,transc5d5tal ,,analysis3 ,page ,,its ,,diff]5t ,,conceptions #hh ,prelim9>y ,rem>ks #hh ,its e>ly ,hi/ory #hi ,,me?od ,,( ,,leibnitz #ia ,inf9itely small ,elem5ts #ia ,examples3 #a4 ,tang5ts #ic #b4 ,rectification ( an ,arc #id #c4 ,quadrature ( a ,curve #ie #d4 ,velocity 9 v>iable ,motion #ie #e4 ,di/ribution ( ,heat #if ,g5]ality ( ! ,=mulas #ig ,demon/ration ( ! ,me?od #ih ,illu/ration by ,tang5ts #ajb ,,me?od ,,( ,,newton #ajc ,me?od ( ,limits #ajc ,examples3 #a4 ,tang5ts #ajd #b4 ,rectifications #aje ,fluxions & ,flu5ts #ajf ,,me?od ,,( ,,lagrange #ajh ,d]iv$ ,functions #ajh ,an ext5sion ( ord9>y ,analysis #ajh ,example3 ,tang5ts #aji ,fundam5tal ,id5tity ( ! ?ree ,me?ods #aaj ,_! comp>ative ,value #aac ,t ( ,leibnitz #aac ,t ( ,newton #aae ,t ( ,lagrange #aag ,,*apt] ,,iv4 ,page ,,! ,,diff]5tial ,,& ,,integral ,,calculus #abj ,,its ,,two ,,fundam5tal ,,divisions #abj ,,_! ,,relations ,,to ,,ea* ,,o!r #aba #a4 ,use ( ! ,diff]5tial ,calculus z prep>atory to t ( ! ,integral #abc #b4 ,employm5t ( ! ,diff]5tial ,calculus alone #abe #c4 ,employm5t ( ! ,integral ,calculus alone #abe ,?ree ,classes ( ,que/ions h5ce result+ #abf ,,! ,,diff]5tial ,,calculus #abg ,two ,cases3 ,explicit & ,implicit ,functions #abg ,two sub-,cases3 a s+le ,v>iable or sev]al #abi ,two o!r ,cases3 ,functions sep>ate or comb9$ #acj ,r$uction ( all to ! ,diff]5tiation ( ! t5 elem5t>y ,functions #aca ,trans=mation ( d]iv$ ,functions = new ,v>iables #acb ,diff]5t ,ord]s ( ,diff]5tiation #acc ,analytical ,applications #acc ,,! ,,integral ,,calculus #ace ,its fundam5tal ,division3 ,explicit & ,implicit ,functions #ace ,subdivisions3 a s+le ,v>iable or sev]al #acf ,calculus ( p>tial ,diff]5ces #acg ,ano!r ,subdivision3 diff]5t ,ord]s ( ,diff]5tiation #ach ,ano!r equival5t ,di/9ction #adj ,quadratures #adb ,integration ( ,transc5d5tal ,functions #adc ,integration by ,p>ts #adc ,integration ( ,algebraic ,functions #adc ,s+ul> ,solutions #add ,def9ite ,integrals #adf ,prospects ( ! ,integral ,calculus #adh ,,*apt] ,v4 ,page ,,! ,,calculus ,,( ,,v>iations #aea ,,problems ,,giv+ ,,rise ,,to ,,x #aea ,ord9>y ,que/ions ( ,maxima & ,m9ima #aea ,a new ,class ( ,que/ions #aeb ,solid ( lea/ ,resi/ance2 ,bra*y/o*rone2 ,isop]imet]s #aec ,,analytical ,,nature ,,( ,,^! ,,que/ions #aed ,,me?ods ,,( ,,! ,,old] ,,geomet]s #aee ,,me?od ,,( ,,lagrange #aef ,two ,classes ( ,que/ions #aeg #a4 ,absolute ,maxima & ,m9ima #aeg ,equations ( ,limits #aei ,a m g5]al ,consid]ation #aei #b4 ,relative ,maxima & ,m9ima #afj ,o!r ,applications ( ! ,me?od ( ,v>iations #afb ,,its ,,relations ,,to ,,! ,,ord9>y ,,calculus #afc ,,*apt] ,,vi4 ,,! ,,calculus ,,( ,,f9ite ,,diff]5ces #afg ,its g5]al ,"* #afg ,its true ,nature #afh ,,g5]al ,,!ory ,,( ,,s]ies #agj ,its ,id5tity ) ?is ,calculus #agb ,,p]iodic ,,or ,,discont9u\s ,,functions #agc ,,applications ,,( ,,?is ,,calculus #agc ,s]ies #agc ,int]polation #agc ,approximate ,rectification1 c4 #agd ,,book ,,ii4 ,,geometry4 ,,*apt] ,i4 ,page ,a ,,g5]al ,,view ,,( ,,geometry #agi ,! true ,nature ( ,geometry #agi ,two fundam5tal ,ideas #aha #a4 ,! ,idea ( ,space #aha #b4 ,diff]5t k9ds ( ,ext5sion #ahb ,,! ,,f9al ,,object ,,( ,,geometry #ahd ,nature ( ,geometrical ,measurem5t #ahe ,( ,surfaces & ,volumes #ahe ,( curve ,l9es #ahg ,( "r ,l9es #ahi ,,! ,,inf9ite ,,ext5t ,,( ,,its ,,field #aij ,inf9ity ( ,l9es #aij ,inf9ity ( ,surfaces #aia ,inf9ity ( ,volumes #aib ,analytical ,inv5tion ( ,curves1 c4 #aic ,,expansion ,,( ,,orig9al ,,def9ition #aic ,prop]ties ( ,l9es & ,surfaces #aie ,necessity ( _! ,/udy #aie #a4 ,to f9d ! mo/ suitable ,prop]ty #aie #b4 ,to pass f ! ,concrete to ! ,ab/ract #aig ,illu/rations3 ,orbits ( ! ,planets #aih ,figure ( ! ,e>? #aii ,,! ,,two ,,g5]al ,,me?ods ,,( ,,geometry #bjb ,_! fundam5tal ,diff]5ce #bjc #a4 ,diff]5t ,que/ions ) respect to ! same ,figure #bjd #b4 ,simil> ,que/ions ) respect to diff]5t ,figures #bjd ,geometry ( ! ,anci5ts #bjd ,geometry ( ! ,mod]ns #bjf ,sup]iority ( ! ,mod]n #bjg ,! ,anci5t ! base ( ! ,mod]n #bji ,,*apt] ,,ii4 ,,anci5t ,,or ,,syn!tic ,,geometry ,page ,,its ,,prop] ,,ext5t #bab ,l9es2 ,polygons2 ,polyh$rons #bab ,n to be f>!r re/rict$ #bac ,improp] ,application ( ,analysis #bad ,attempt$ ,demon/rations ( ,axioms #baf ,,geometry ,,( ,,! ,,"r ,,l9e #bag ,,graphical ,,solutions #bah ,descriptive ,geometry #bbj ,,algebraical ,,solutions #bbd ,trigonometry #bbe ,two ,me?ods ( introduc+ ,angles #bbf #a4 ,by ,arcs #bbf #b4 ,by trigonometrical ,l9es #bbf ,advantages ( ! latt] #bbf ,its ,division ( trigonometrical ,que/ions #bbg #a4 ,relations betwe5 ,angles & trigonometrical ,l9es #bbh #b4 ,relations betwe5 trigonometrical ,l9es & ,sides #bbh ,increase ( trigonometrical ,l9es #bbh ,/udy ( ! ,relations betwe5 !m #bcj ,,*apt] ,,iii4 ,,mod]n ,,or ,,analytical ,,geometry ,page ,,! ,,analytical ,,repres5tation ,,( ,,figures #bcb ,r$uction ( ,figure to ,position #bcc ,det]m9ation ( ! position ( a ,po9t #bcd ,,plane ,,curves #bcg ,expression ( ,l9es by ,equations #bcg ,expression ( ,equations by ,l9es #bch ,any *ange 9 ! ,l9e *anges ! ,equation #bdj ,e ,def9ition ( a ,l9e is an ,equation #bda ,*oice ( ,co-ord9ates #bde ,two diff]5t po9ts ( ,view #bde #a4 ,repres5tation ( ,l9es by ,equations #bdf #b4 ,repres5tation ( ,equations by ,l9es #bdf ,sup]iority ( ! rectil9e> ,sy/em #bdh ,advantages ( p]p5dicul> ,axes #bdi ,,surfaces #bea ,det]m9ation ( a ,po9t 9 ,space #bea ,expression ( ,surfaces by ,equations #bec ,expression ( ,equations by ,surfaces #bec ,,curves ,,9 ,,space #bee ,imp]fections ( ,analytical ,geometry #beh ,relatively to ,geometry #beh ,relatively to ,analysis #beh ,,! ,,philosophy ,,( ,,ma!matics4 ,,introduction4 ,,g5]al ,,consid]ations4 ,al?\gh ,ma!matical ,sci5ce is ! mo/ anci5t & ! mo/ p]fect ( all1 yet ! g5]al idea :i* we ought to =m ( x has n yet be5 cle>ly det]m9$4 ,its def9ition & its pr9cipal divisions h rema9$ till n[ vague & unc]ta94 ,inde$ ! plural "n--,! ,ma!matics--by :i* we commonly designate x1 w\ld alone suffice to indicate ! want ( unity 9 ! common conception ( x4 ,9 tru?1 x 0 n till ! comm5cem5t ( ! la/ c5tury t ! diff]5t fundam5tal conceptions :i* con/itute ?is great sci5ce 7 ea* ( !m suffici5tly develop$ to p]mit ! true _s ( ! :ole to manife/ itself ) cle>ness4 ,s9ce t epo* ! att5tion ( geomet]s has be5 too exclusively absorb$ by ! special p]fect+ ( ! diff]5t bran*es1 & by ! application :i* !y h made ( !m to ! mo/ important laws ( ! univ]se1 to all[ !m to give due att5tion to ! g5]al sy/em ( ! sci5ce4 ,b at ! pres5t "t ! progress ( ! special dep>tm5ts is no long] s rapid z to =bid ! contemplation ( ! :ole4 ,! sci5ce ( ma!matics is n[ suffici5tly develop$1 bo? 9 itself & z to its mo/ ess5tial application1 to h arriv$ at t /ate ( consi/5cy 9 :i* we ought to /rive to arrange its diff]5t p>ts 9 a s+le sy/em1 9 ord] to prep>e = new advances4 ,we may ev5 obs]ve t ! la/ important improvem5ts ( ! sci5ce h directly pav$ ! way = ?is important philosophical op]ation1 by impress+ on its pr9cipal p>ts a "* ( unity :i* did n previ\sly exi/4 ,to =m a j idea ( ! object ( ma!matical sci5ce1 we may />t f ! indef9ite & mean+less def9ition ( x usually giv51 9 call+ x ,! sci5ce ( magnitudes1 or1 :i* is m def9ite1 ,! sci5ce :i* has = its object ! measurem5t ( magnitudes4 ,let u see h[ we c rise f ?is r\gh sket* 7:i* is s+ul>ly defici5t 9 precision & dep?1 ?\gh1 at bottom1 j7 to a v]itable def9ition1 wor?y ( ! importance1 ! ext5t1 & ! difficulty ( ! sci5ce4 ,,! ,,object ,,( ,,ma!matics4 ,measur+ ,magnitudes4 ,! "q ( measur+ a magnitude 9 itself pres5ts to ! m9d no o!r idea ?an t ( ! simple direct comp>ison ( ?is magnitude ) ano!r simil> magnitude1 suppos$ to be kn[n1 :i* x takes = ! unit ( comp>ison among all o!rs ( ! same k9d4 ,accord+ to ?is def9ition1 !n1 ! sci5ce ( ma!matics--va/ & pr(\nd z x is ) reason reput$ to be--in/ead ( be+ an imm5se concat5ation ( prolong$ m5tal lab\rs1 :i* (f] inexhau/ible occupation to our intellectual activity1 w\ld seem to consi/ ( a simple s]ies ( me*anical processes = obta9+ directly ! ratios ( ! quantities to be measur$ to ^? by :i* we wi% to measure !m1 by ! aid ( op]ations ( simil> "* to ! sup]position ( l9es1 z practic$ by ! c>p5t] ) 8 rule4 ,! error ( ?is def9ition consi/s 9 pres5t+ z direct an object :i* is almo/ always1 on ! contr>y1 v indirect4 ,! direct measurem5t ( a magnitude1 by sup]position or any simil> process1 is mo/ frequ5tly an op]ation q impossible = u to p]=m2 s t if we _h no o!r means = det]m9+ magnitudes ?an direct comp>isons1 we %\ld be oblig$ to r5\nce ! k ( mo/ ( ^? :i* int]e/ u4 ,difficulties4 ,! =ce ( ?is g5]al obs]vation w be und]/ood if we limit ourselves to consid] specially ! p>ticul> case :i* evid5tly (f]s ! mo/ facility--t ( ! measurem5t ( "o /raight l9e by ano!r4 ,?is comp>ison1 :i* is c]ta9ly ! mo/ simple :i* we c conceive1 c nev]!less sc>cely ev] be effect$ directly4 ,9 reflect+ on ! :ole ( ! conditions necess>y to r5d] a l9e susceptible ( a direct measurem5t1 we see t mo/ frequ5tly !y _c be all fulfill$ at ! same "t4 ,! fir/ & ! mo/ palpable ( ^! conditions--t ( be+ able to pass ov] ! l9e f "o end ( x to ! o!r1 9 ord] to apply ! unit ( measurem5t to its :ole l5g?--evid5tly excludes at once by f> ! great] "p ( ! di/ances :i* int]e/ u ! mo/2 9 ! fir/ place1 all ! di/ances betwe5 ! cele/ial bodies1 or f any "o ( !m to ! e>?2 & !n1 too1 ev5 ! great] numb] ( t]re/rial di/ances1 :i* are s frequ5tly inaccessible4 ,b ev5 if ?is fir/ condition be f\nd to be fulfill$1 x is /ill f>!r necess>y t ! l5g? be nei!r too great nor too small1 :i* w\ld r5d] a direct measurem5t equally impossible4 ,! l9e mu/ also be suitably situat$2 = let x be "o :i* we c\ld measure ) ! greate/ facility1 if x 7 horizontal1 b conceive x to be turn$ up v]tically1 & x becomes impossible to measure x4 ,! difficulties :i* we h indicat$ 9 ref]5ce to measur+ l9es1 exi/ 9 a v mu* great] degree 9 ! measurem5t ( surfaces1 volumes1 velocities1 times1 =ces1 c4 ,x is ?is fact :i* makes necess>y ! =mation ( ma!matical sci5ce1 z we are go+ to see2 = ! human m9d has be5 compell$ to r5\nce1 9 almo/ all cases1 ! direct measurem5t ( magnitudes1 & to seek to det]m9e !m indirectly1 & x is ?us t x has be5 l$ to ! creation ( ma!matics4 ,g5]al ,me?od4 ,! g5]al me?od :i* is con/antly employ$1 & evid5tly ! only "o conceivable1 to asc]ta9 magnitudes :i* d n admit ( a direct measurem5t1 consi/s 9 connect+ !m ) o!rs :i* are susceptible ( be+ det]m9$ imm$iately1 & by means ( :i* we succe$ 9 discov]+ ! fir/ ?r\gh ! relations :i* subsi/ betwe5 ! two4 ,su* is ! precise object ( ma!matical sci5ce view$ z a :ole4 ,9 ord] to =m a suffici5tly ext5d$ idea ( x1 we mu/ consid] t ?is indirect det]m9ation ( magnitudes may be indirect 9 v diff]5t degrees4 ,9 a great numb] ( cases1 :i* are (t5 ! mo/ important1 ! magnitudes1 by means ( :i* ! pr9cipal magnitudes s\ght are to be det]m9$1 _c !mselves be measur$ directly1 & mu/ !re=e1 9 _! turn1 become ! subject ( a simil> "q1 & s on2 s t on _m occasions ! human m9d is oblig$ to e/abli% a long s]ies ( int]m$iates betwe5 ! sy/em ( unkn[n magnitudes :i* are ! f9al objects ( its rese>*es1 & ! sy/em ( magnitudes susceptible ( direct measurem5t1 by ^: means we f9ally det]m9e ! fir/1 ) :i* at fir/ !y appe> to h no connexion4 ,illu/rations4 ,"s examples w make cle> any ?+ :i* may seem too ab/ract 9 ! prec$+ g5]alities4 #a4 ,fall+ ,bodies4 ,let u consid]1 9 ! fir/ place1 a natural ph5om5on1 v simple1 inde$1 b :i* may nev]!less give rise to a ma!matical "q1 really exi/+1 & susceptible ( actual applications--! ph5om5on ( ! v]tical fall ( heavy bodies4 ,! m9d ! mo/ unus$ to ma!matical conceptions1 9 obs]v+ ?is ph5om5on1 p]ceives at once t ! two quantities :i* x pres5ts--namely1 ! height f :i* a body has fall51 & ! "t ( its fall--are necess>ily connect$ ) ea* o!r1 s9ce !y v>y toge!r1 & simultane\sly rema9 fix$2 or1 9 ! language ( geomet]s1 t !y are functions ( ea* o!r4 ,! ph5om5on1 consid]$ "u ?is po9t ( view1 gives rise !n to a ma!matical "q1 :i* consi/s 9 sub/itut+ = ! direct measurem5t ( "o ( ^! two magnitudes1 :5 x is impossible1 ! measurem5t ( ! o!r4 ,x is ?us1 = example1 t we may det]m9e indirectly ! dep? ( a precipice1 by m]ely measur+ ! "t t a heavy body w\ld occupy 9 fall+ to its bottom1 & by suitable proc$ures ?is inaccessible dep? w be kn[n ) z mu* precision z if x 0 a horizontal l9e plac$ 9 ! mo/ fav\rable circum/ances = easy & exact measurem5t4 ,on o!r occasions x is ! height f :i* a body has fall5 :i* x w be easy to asc]ta91 :ile ! "t ( ! fall c\ld n be obs]v$ directly2 !n ! same ph5om5on w\ld give rise to ! inv]se "q1 namely1 to det]m9e ! "t f ! height2 z1 = example1 if we wi%$ to asc]ta9 :at w\ld be ! duration ( ! v]tical fall ( a body fall+ f ! moon to ! e>?4 ,9 ?is example ! ma!matical "q is v simple1 at lea/ :5 we d n pay att5tion to ! v>iation 9 ! int5sity ( gravity1 or ! resi/ance ( ! fluid :i* ! body passes ?r\gh 9 its fall4 ,b1 to ext5d ! "q1 we h only to consid] ! same ph5om5on 9 its greate/ g5]ality1 9 suppos+ ! fall oblique1 & 9 tak+ into ! acc\nt all ! pr9cipal circum/ances4 ,!n1 in/ead ( (f]+ simply two v>iable quantities connect$ ) ea* o!r by a relation easy to foll[1 ! ph5om5on w pres5t a mu* great] numb]2 namely1 ! space trav]s$1 :e!r 9 a v]tical or horizontal direction2 ! "t employ$ 9 trav]s+ x2 ! velocity ( ! body at ea* po9t ( its c\rse2 ev5 ! int5sity & ! direction ( its primitive impulse1 :i* may also be view$ z v>iables2 & f9ally1 9 c]ta9 cases 7to take e ?+ into ! acc\nt71 ! resi/ance ( ! m$ium & ! int5sity ( gravity4 ,all ^! diff]5t quantities w be connect$ ) "o ano!r1 9 su* a way t ea* 9 its turn may be indirectly det]m9$ by means ( ! o!rs2 & ?is w pres5t z _m di/9ct ma!matical que/ions z !re may be co-exi/+ magnitudes 9 ! ph5om5on "u consid]ation4 ,su* a v slight *ange 9 ! physical conditions ( a problem may cause 7z 9 ! above example7 a ma!matical rese>*1 at fir/ v elem5t>y1 to be plac$ at once 9 ! rank ( ! mo/ difficult que/ions1 ^: complete & rigor\s solution surpasses z yet ! utmo/ p[] ( ! human intellect4 #b4 ,inaccessible ,di/ances4 ,let u take a second example f geometrical ph5om5a4 ,let x be propos$ to det]m9e a di/ance :i* is n susceptible ( direct measurem5t2 x w be g5]ally conceiv$ z mak+ "p ( a figure1 or c]ta9 sy/em ( l9es1 *os5 9 su* a way t all its o!r p>ts may be obs]v$ directly2 ?us1 9 ! case :i* is mo/ simple1 & to :i* all ! o!rs may be f9ally r$uc$1 ! propos$ di/ance w be consid]$ z belong+ to a triangle1 9 :i* we c det]m9e directly ei!r ano!r side & two angles1 or two sides & "o angle4 ,!nce-=w>d1 ! k ( ! desir$ di/ance1 in/ead ( be+ obta9$ directly1 w be ! result ( a ma!matical calculation1 :i* w consi/ 9 d$uc+ x f ! obs]v$ elem5ts by means ( ! relation :i* connects x ) !m4 ,?is calculation w become successively m & m complicat$1 if ! p>ts :i* we h suppos$ to be kn[n _c !mselves be det]m9$ 7z is mo/ frequ5tly ! case7 except 9 an indirect mann]1 by ! aid ( new auxili>y sy/ems1 ! numb] ( :i*1 9 great op]ations ( ?is k9d1 f9ally becomes v consid]able4 ,! di/ance be+ once det]m9$1 ! k ( x w frequ5tly be suffici5t = obta9+ new quantities1 :i* w become ! subject ( new ma!matical que/ions4 ,?us1 :5 we "k at :at di/ance any object is situat$1 ! simple obs]vation ( its app>5t diamet] w evid5tly p]mit u to det]m9e indirectly its real dim5sions1 h[ev] inaccessible x may be1 &1 by a s]ies ( analog\s inve/igations1 its surface1 its volume1 ev5 its weight1 & a numb] ( o!r prop]ties1 a k ( :i* seem$ =bidd5 to u4 #c4 ,a/ronomical ,facts4 ,x is by su* calculations t man has be5 able to asc]ta91 n only ! di/ances f ! planets to ! e>?1 &1 consequ5tly1 f ea* o!r1 b _! actual magnitude1 _! true figure1 ev5 to ! inequalities ( _! surface2 &1 :at seem$ /ill m completely hidd5 f u1 _! respective masses1 _! mean d5sities1 ! pr9cipal circum/ances ( ! fall ( heavy bodies on ! surface ( ea* ( !m1 c4 ,by ! p[] ( ma!matical !ories1 all ^! diff]5t results1 & _m o!rs relative to ! diff]5t classes ( ma!matical ph5om5a1 h requir$ no o!r direct measurem5ts ?an ^? ( a v small numb] ( /raight l9es1 suitably *os51 & ( a great] numb] ( angles4 ,we may ev5 say1 ) p]fect tru?1 s z to indicate 9 a ^w ! g5]al range ( ! sci5ce1 t if we did n fe> to multiply calculations unnecess>ily1 & if we _h n1 9 consequ5ce1 to res]ve !m = ! det]m9ation ( ! quantities :i* c\ld n be measur$ directly1 ! det]m9ation ( all ! magnitudes susceptible ( precise e/imation1 :i* ! v>i\s ord]s ( ph5om5a c (f] u1 c\ld be f9ally r$uc$ to ! direct measurem5t ( a s+le /raight l9e & ( a suitable numb] ( angles4 ,,true ,,def9ition ,,( ,,ma!matics4 ,we are n[ able to def9e ma!matical sci5ce ) precision1 by assign+ to x z its object ! indirect measurem5t ( magnitudes1 & by say+ x con/antly proposes to det]m9e c]ta9 magnitudes f o!rs by means ( ! precise relations exi/+ betwe5 !m4 ,?is enunciation1 in/ead ( giv+ ! idea ( only an art1 z d all ! ord9>y def9itions1 *>act]izes imm$iately a true sci5ce1 & %[s x at once to be compos$ ( an imm5se *a9 ( intellectual op]ations1 :i* may evid5tly become v complicat$1 because ( ! s]ies ( int]m$iate l9ks :i* x w be necess>y to e/abli% betwe5 ! unkn[n quantities & ^? :i* admit ( a direct measurem5t2 ( ! numb] ( v>iables coexi/5t 9 ! propos$ "q2 & ( ! nature ( ! relations betwe5 all ^! diff]5t magnitudes furni%$ by ! ph5om5a "u consid]ation4 ,accord+ to su* a def9ition1 ! _s ( ma!matics consi/s 9 always reg>d+ all ! quantities :i* any ph5om5on c pres5t1 z connect$ & int]wov5 ) "o ano!r1 ) ! view ( d$uc+ !m f "o ano!r4 ,n[ !re is evid5tly no ph5om5on :i* _c give rise to consid]ations ( ?is k9d2 :5ce results ! naturally indef9ite ext5t & ev5 ! rigor\s logical univ]sality ( ma!matical sci5ce4 ,we %all seek f>!r on to circumscribe z exactly z possible its real ext5sion4 ,! prec$+ explanations e/abli% cle>ly ! propriety ( ! "n employ$ to designate ! sci5ce :i* we are consid]+4 ,?is d5om9ation1 :i* has tak5 to-day s def9ite a mean+ by itself signifies simply sci5ce 9 g5]al4 ,su* a designation1 rigor\sly exact = ! ,greeks1 :o _h no o!r real sci5ce1 c\ld be reta9$ by ! mod]ns only to indicate ! ma!matics z ! sci5ce1 beyond all o!rs--! sci5ce ( sci5ces4 ,inde$1 e true sci5ce has = its object ! det]m9ation ( c]ta9 ph5om5a by means ( o!rs1 9 accordance ) ! relations :i* exi/ betwe5 !m4 ,e sci5ce consi/s 9 ! co-ord9ation ( facts2 if ! diff]5t obs]vations 7 entirely isolat$1 !re w\ld be no sci5ce4 ,we may ev5 say1 9 g5]al t]ms1 t sci5ce is ess5tially de/9$ to disp5se1 s f> z ! diff]5t ph5om5a p]mit x1 ) all direct obs]vation1 by enabl+ u to d$uce f ! smalle/ possible numb] ( imm$iate data ! greate/ possible numb] ( results4 ,is n ?is ! real use1 :e!r 9 speculation or 9 action1 ( ! laws :i* we succe$ 9 discov]+ among natural ph5om5a8 ,ma!matical sci5ce1 9 ?is po9t ( view1 m]ely pu%es to ! highe/ possible degree ! same k9d ( rese>*es :i* are pursu$1 9 degrees m or less inf]ior1 by e real sci5ce 9 its respective sph]e4 ,,its ,,two ,,fundam5tal ,,divisions4 ,we h ?us f> view$ ma!matical sci5ce only z a :ole1 )\t pay+ any reg>d to its divisions4 ,we mu/ n[1 9 ord] to complete ?is g5]al view1 & to =m a j idea ( ! philosophical "* ( ! sci5ce1 consid] its fundam5tal division4 ,! second>y divisions w be exam9$ 9 ! foll[+ *apt]s4 ,?is pr9cipal division1 :i* we are ab\t to inve/igate1 c be truly rational1 & d]iv$ f ! real nature ( ! subject1 only s f> z x spontane\sly pres5ts itself to u1 9 mak+ ! exact analysis ( a complete ma!matical "q4 ,we w1 !re=e1 hav+ det]m9$ above :at is ! g5]al object ( ma!matical lab\rs1 n[ *>act]ize ) precision ! pr9cipal diff]5t ord]s ( inquiries1 ( :i* !y are con/antly compos$4 ,_! diff]5t ,objects4 ,! complete solution ( e ma!matical "q divides itself necess>ily into two p>ts1 ( natures ess5tially di/9ct1 & ) relations inv>iably det]m9ate4 ,we h se5 t e ma!matical inquiry has = its object to det]m9e unkn[n magnitudes1 accord+ to ! relations betwe5 !m & kn[n magnitudes4 ,n[ = ?is object1 x is evid5tly necess>y1 9 ! fir/ place1 to asc]ta9 ) precision ! relations :i* exi/ betwe5 ! quantities :i* we are consid]+4 ,?is fir/ bran* ( inquiries con/itutes t :i* ,i call ! concrete "p ( ! solution4 ,:5 x is f9i%$1 ! "q *anges2 x is n[ r$uc$ to a pure "q ( numb]s1 consi/+ simply 9 det]m9+ unkn[n numb]s1 :5 we "k :at precise relations connect !m ) kn[n numb]s4 ,?is second bran* ( inquiries is :at ,i call ! ab/ract "p ( ! solution4 ,h5ce foll[s ! fundam5tal division ( g5]al ma!matical sci5ce into two great sci5ces--,,ab/ract ,,ma!matics1 & ,,concrete ,,ma!matics4 ,?is analysis may be obs]v$ 9 e complete ma!matical "q1 h[ev] simple or complicat$ x may be4 ,a s+le example w suffice to make x intelligible4 ,tak+ up aga9 ! ph5om5on ( ! v]tical fall ( a heavy body1 & consid]+ ! simple/ case1 we see t 9 ord] to succe$ 9 det]m9+1 by means ( "o ano!r1 ! height :5ce ! body has fall51 & ! duration ( its fall1 we mu/ comm5ce by discov]+ ! exact relation ( ^! two quantities1 or1 to use ! language ( geomet]s1 ! equation :i* exi/s betwe5 !m4 ,be=e ?is fir/ rese>* is complet$1 e attempt to det]m9e num]ically ! value ( "o ( ^! two magnitudes f ! o!r w\ld evid5tly be premature1 = x w\ld h no basis4 ,x is n en\gh to "k vaguely t !y dep5d on "o ano!r--:i* e "o at once p]ceives--b x is necess>y to det]m9e 9 :at ?is dep5d5ce consi/s4 ,?is inquiry may be v difficult1 & 9 fact1 9 ! pres5t case1 con/itutes incomp>ably ! great] "p ( ! problem4 ,! true sci5tific _s is s mod]n1 t no "o1 p]haps1 be=e ,galileo1 _h ev] rem>k$ ! increase ( velocity :i* a body exp]i5ces 9 its fall3 a circum/ance :i* excludes ! hypo!sis1 t[>ds :i* our m9d 7always involunt>ily incl9$ to suppose 9 e ph5om5on ! mo/ simple functions1 )\t any o!r motive ?an its great] facility 9 conceiv+ !m7 w\ld be naturally l$1 t ! height 0 proportional to ! "t4 ,9 a ^w1 ?is fir/ inquiry t]m9at$ 9 ! discov]y ( ! law ( ,galileo4 ,:5 ?is concrete "p is complet$1 ! inquiry becomes "o ( q ano!r nature4 ,kn[+ t ! spaces pass$ ?r\gh by ! body 9 ea* successive second ( its fall increase z ! s]ies ( odd numb]s1 we h !n a problem purely num]ical & ab/ract2 to d$uce ! height f ! "t1 or ! "t f ! height2 & ?is consi/s 9 f9d+ t ! fir/ ( ^! two quantities1 accord+ to ! law :i* has be5 e/abli%$1 is a kn[n multiple ( ! second p[] ( ! o!r2 f :i*1 f9ally1 we h to calculate ! value ( ! "o :5 t ( ! o!r is giv54 ,9 ?is example ! concrete "q is m difficult ?an ! ab/ract "o4 ,! rev]se w\ld be ! case if we consid]$ ! same ph5om5on 9 its greate/ g5]ality1 z ,i h done above = ano!r object4 ,accord+ to ! circum/ances1 sometimes ! fir/1 sometimes ! second1 ( ^! two p>ts w con/itute ! pr9cipal difficulty ( ! :ole "q2 = ! ma!matical law ( ! ph5om5on may be v simple1 b v difficult to obta91 or x may be easy to discov]1 b v complicat$2 s t ! two great sections ( ma!matical sci5ce1 :5 we comp>e !m z :oles1 mu/ be reg>d$ z exactly equival5t 9 ext5t & 9 difficulty1 z well z 9 importance1 z we %all %[ f>!r on1 9 consid]+ ea* ( !m sep>ately4 ,_! diff]5t ,natures4 ,^! two p>ts1 ess5tially di/9ct 9 _! object1 z we h j se51 are no less s ) reg>d to ! nature ( ! inquiries ( :i* !y are compos$4 ,! fir/ %\ld be call$ concrete1 s9ce x evid5tly dep5ds on ! "* ( ! ph5om5a consid]$1 & mu/ necess>ily v>y :5 we exam9e new ph5om5a2 :ile ! second is completely indep5d5t ( ! nature ( ! objects exam9$1 & is conc]n$ ) only ! num]ical relations :i* !y pres5t1 = :i* reason x %\ld be call$ ab/ract4 ,! same relations may exi/ 9 a great numb] ( diff]5t ph5om5a1 :i*1 9 spite ( _! extreme div]sity1 w be view$ by ! geomet] z (f]+ an analytical "q susceptible1 :5 /udi$ by itself1 ( be+ resolv$ once = all4 ,?us1 = in/ance1 ! same law :i* exi/s betwe5 ! space & ! "t ( ! v]tical fall ( a body 9 a vacuum1 is f\nd aga9 9 _m o!r ph5om5a :i* (f] no analogy ) ! fir/ nor ) ea* o!r2 = x expresses ! relation betwe5 ! surface ( a sph]ical body & ! l5g? ( its diamet]2 x det]m9es1 9 l mann]1 ! decrease ( ! int5sity ( light or ( heat 9 relation to ! di/ance ( ! objects light$ or heat$1 c4 ,! ab/ract "p1 common to ^! diff]5t ma!matical que/ions1 hav+ be5 treat$ 9 ref]5ce to "o ( ^!1 w ?us h be5 treat$ = all2 :ile ! concrete "p w h necess>ily to be aga9 tak5 up = ea* "q sep>ately1 )\t ! solution ( any "o ( !m be+ able to give any direct aid1 9 t connexion1 = ! solution ( ! re/4 ,! ab/ract "p ( ma!matics is1 !n1 g5]al 9 its nature2 ! concrete "p1 special4 ,to pres5t ?is comp>ison "u a new po9t ( view1 we may say concrete ma!matics has a philosophical "*1 :i* is ess5tially exp]im5tal1 physical1 ph5om5al2 :ile t ( ab/ract ma!matics is purely logical1 rational4 ,! concrete "p ( e ma!matical "q is necess>ily f\nd$ on ! consid]ation ( ! ext]nal _w1 & c\ld nev] be resolv$ by a simple s]ies ( intellectual comb9ations4 ,! ab/ract "p1 on ! contr>y1 :5 x has be5 v completely sep>at$1 c consi/ only ( a s]ies ( logical d$uctions1 m or less prolong$2 = if we h once f\nd ! equations ( a ph5om5on1 ! det]m9ation ( ! quantities !re9 consid]$1 by means ( "o ano!r1 is a matt] = reason+ only1 :atev] ! difficulties may be4 ,x belongs to ! und]/&+ alone to d$uce f ^! equations results :i* are evid5tly conta9$ 9 !m1 al?\gh p]haps 9 a v involv$ mann]1 )\t !re be+ occasion to consult anew ! ext]nal _w2 ! consid]ation ( :i*1 hav+ become !nce=? =eign to ! subject1 ought ev5 to be c>efully set aside 9 ord] to r$uce ! lab\r to its true peculi> difficulty4 ,! ab/ract "p ( ma!matics is !n purely in/rum5tal1 & is only an imm5se & admirable ext5sion ( natural logic to a c]ta9 class ( d$uctions4 ,on ! o!r h&1 geometry & me*anics1 :i*1 z we %all see pres5tly1 con/itute ! concrete "p1 mu/ be view$ z real natural sci5ces1 f\nd$ on obs]vation1 l all ! re/1 al?\gh ! extreme simplicity ( _! ph5om5a p]mits an inf9itely great] degree ( sy/ematization1 :i* has sometimes caus$ a misconception ( ! exp]im5tal "* ( _! fir/ pr9ciples4 ,we see1 by ?is brief g5]al comp>ison1 h[ natural & pr(\nd is our fundam5tal division ( ma!matical sci5ce4 ,we h n[ to circumscribe1 z exactly z we c 9 ?is fir/ sket*1 ea* ( ^! two great sections4 ,,concrete ,,ma!matics4 ,concrete ,ma!matics hav+ = its object ! discov]y ( ! equations ( ph5om5a1 x w\ld seem at fir/ t x mu/ be compos$ ( z _m di/9ct sci5ces z we f9d really di/9ct categories among natural ph5om5a4 ,b we are yet v f> f hav+ discov]$ ma!matical laws 9 all k9ds ( ph5om5a2 we %all ev5 see1 pres5tly1 t ! great] "p w v probably always hide !mselves f our inve/igations4 ,9 reality1 9 ! pres5t condition ( ! human m9d1 !re are directly b two great g5]al classes ( ph5om5a1 ^: equations we con/antly "k2 ^! are1 fir/ly1 geometrical1 &1 secondly1 me*anical ph5om5a4 ,?us1 !n1 ! concrete "p ( ma!matics is compos$ ( ,,geometry & ,,rational ,,me*anics4 ,?is is suffici5t1 x is true1 to give to x a complete "* ( logical univ]sality1 :5 we consid] all ph5om5a f ! mo/ elevat$ po9t ( view ( natural philosophy4 ,9 fact1 if all ! p>ts ( ! univ]se 7 conceiv$ z immovable1 we %\ld evid5tly h only geometrical ph5om5a to obs]ve1 s9ce all w\ld be r$uc$ to relations ( =m1 magnitude1 & position2 !n1 hav+ reg>d to ! motions :i* take place 9 x1 we w\ld h also to consid] me*anical ph5om5a4 ,h5ce ! univ]se1 9 ! /atical po9t ( view1 pres5ts only geometrical ph5om5a2 &1 consid]$ dynamically1 only me*anical ph5om5a4 ,?us geometry & me*anics con/itute ! two fundam5tal natural sci5ces1 9 ?is s5se1 t all natural effects may be conceiv$ z simple necess>y results1 ei!r ( ! laws ( ext5sion or ( ! laws ( motion4 ,b al?\gh ?is conception is always logically possible1 ! difficulty is to specialize x ) ! necess>y precision1 & to foll[ x exactly 9 ea* ( ! g5]al cases (f]$ to u by ! /udy ( nature2 t is1 to effectually r$uce ea* pr9cipal "q ( natural philosophy1 = a c]ta9 det]m9ate ord] ( ph5om5a1 to ! "q ( geometry or me*anics1 to :i* we might rationally suppose x %\ld be br\ght4 ,?is trans=mation1 :i* requires great progress to h be5 previ\sly made 9 ! /udy ( ea* class ( ph5om5a1 has ?us f> be5 really execut$ only = ^? ( a/ronomy1 & = a "p ( ^? consid]$ by t]re/rial physics1 prop]ly s call$4 ,x is ?us t a/ronomy1 ac\/ics1 optics1 c41 h f9ally become applications ( ma!matical sci5ce to c]ta9 ord]s ( obs]vations4#a ,b ^! applications n be+ by _! nature rigor\sly circumscrib$1 to conf\nd !m ) ! sci5ce w\ld be to assign to x a vague & indef9ite doma92 & ?is is done 9 ! usual division1 s faulty 9 s _m o!r respects1 ( ! ma!matics into ,pure & ,appli$4 ,footnote #a3 ,! inve/igation ( ! ma!matical ph5om5a ( ! laws ( heat by ,b>on ,f\ri] has l$ to ! e/abli%m5t1 9 an entirely direct mann]1 ( ,!rmological equations4 ,?is great discov]y t5ds to elevate our philosophical hopes z to ! future ext5sions ( ! legitimate applications ( ma!matical analysis1 & r5d]s x prop]1 9 ! op9ion ( au?or1 to reg>d ,!rmology z a ?ird pr9cipal bran* ( concrete ma!matics4 ,,ab/ract ,,ma!matics4 ,! nature ( ab/ract ma!matics 7! g5]al division ( :i* w be exam9$ 9 ! foll[+ *apt]7 is cle>ly & exactly det]m9$4 ,x is compos$ ( :at is call$ ! ,calculus1#b tak+ ?is ^w 9 its greate/ ext5t1 :i* rea*es f ! mo/ simple num]ical op]ations to ! mo/ sublime comb9ations ( transc5d5tal analysis4 ,! ,calculus has ! solution ( all que/ions relat+ to numb]s = its peculi> object4 ,its />t+ po9t is1 con/antly & necess>ily1 ! k ( ! precise relations1 i4e41 ( ! equations1 betwe5 ! diff]5t magnitudes :i* are simultane\sly consid]$2 t :i* is1 on ! contr>y1 ! /opp+ po9t ( concrete ma!matics4 ,h[ev] complicat$1 or h[ev] indirect ^! relations may be1 ! f9al object ( ! calculus always is to obta9 f !m ! values ( ! unkn[n quantities by means ( ^? :i* are kn[n4 ,?is sci5ce1 al?\gh ne>] p]fection ?an any o!r1 is really little advanc$ z yet1 s t ?is object is r>ely atta9$ 9 a mann] completely satisfactory4 ,footnote #b3 ,! translator has felt ju/ifi$ 9 employ+ ?is v conv5i5t ^w 7= :i* our language has no precise equival5t7 z an ,engli% "o1 9 its mo/ ext5d$ s5se1 9 spite ( its be+ (t5 popul>ly conf\nd$ ) its ,diff]5tial & ,integral dep>tm5t4 ,ma!matical analysis is1 !n1 ! true rational basis ( ! entire sy/em ( our actual k4 ,x con/itutes ! fir/ & ! mo/ p]fect ( all ! fundam5tal sci5ces4 ,! ideas ) :i* x occupies itself are ! mo/ univ]sal1 ! mo/ ab/ract1 & ! mo/ simple :i* x is possible = u to conceive4 ,?is peculi> nature ( ma!matical analysis enables u easily to expla9 :y1 :5 x is prop]ly employ$1 x is su* a p[]ful in/rum5t1 n only to give m precision to our real k1 :i* is self-evid5t1 b especially to e/abli% an inf9itely m p]fect co-ord9ation 9 ! /udy ( ! ph5om5a :i* admit ( t application2 =1 our conceptions hav+ be5 s g5]aliz$ & simplifi$ t a s+le analytical "q1 ab/ractly resolv$1 conta9s ! implicit solution ( a great numb] ( div]se physical que/ions1 ! human m9d mu/ necess>ily acquire by ^! means a great] facility 9 p]ceiv+ relations betwe5 ph5om5a :i* at fir/ appe>$ entirely di/9ct f "o ano!r4 ,we ?us naturally see arise1 ?r\gh ! m$ium ( analysis1 ! mo/ frequ5t & ! mo/ unexpect$ approximations betwe5 problems :i* at fir/ (f]$ no app>5t connection1 & :i* we (t5 end 9 view+ z id5tical4 ,c\ld we1 = example1 )\t ! aid ( analysis1 p]ceive ! lea/ resemblance betwe5 ! det]m9ation ( ! direction ( a curve at ea* ( its po9ts & t ( ! velocity acquir$ by a body at e in/ant ( its v>iable motion8 & yet ^! que/ions1 h[ev] diff]5t !y may be1 compose b "o 9 ! eyes ( ! geomet]4 ,! high relative p]fection ( ma!matical analysis is z easily p]ceptible4 ,?is p]fection is n due1 z "s h ?\ght1 to ! nature ( ! signs :i* are employ$ z in/rum5ts ( reason+1 em95tly concise & g5]al z !y are4 ,9 reality1 all great analytical ideas h be5 =m$ )\t ! algebraic signs hav+ be5 ( any ess5tial aid1 except = work+ !m out aft] ! m9d _h conceiv$ !m4 ,! sup]ior p]fection ( ! sci5ce ( ! calculus is due pr9cipally to ! extreme simplicity ( ! ideas :i* x consid]s1 by :atev] signs !y may be express$2 s t !re is n ! lea/ hope1 by any artifice ( sci5tific language1 ( p]fect+ to ! same degree !ories :i* ref] to m complex subjects1 & :i* are necess>ily condemn$ by _! nature to a great] or less logical inf]iority4 ,,! ,,ext5t ,,( ,,its ,,field4 ,our exam9ation ( ! philosophical "* ( ma!matical sci5ce w\ld rema9 incomplete1 if1 aft] hav+ view$ its object & composition1 we did n exam9e ! real ext5t ( its doma94 ,its ,univ]sality4 ,= ?is purpose x is indisp5sable to p]ceive1 fir/ ( all1 t1 9 ! purely logical po9t ( view1 ?is sci5ce is by itself necess>ily & rigor\sly univ]sal2 = !re is no "q :atev] :i* may n be f9ally conceiv$ z consi/+ 9 det]m9+ c]ta9 quantities f o!rs by means ( c]ta9 relations1 & consequ5tly z admitt+ ( r$uction1 9 f9al analysis1 to a simple "q ( numb]s4 ,9 all our rese>*es1 inde$1 on :atev] subject1 our object is to arrive at numb]s1 at quantities1 ?\gh (t5 9 a v imp]fect mann] & by v unc]ta9 me?ods4 ,?us1 tak+ an example 9 ! class ( subjects ! lea/ accessible to ma!matics1 ! ph5om5a ( liv+ bodies1 ev5 :5 consid]$ 7to take ! mo/ complicat$ case7 9 ! /ate ( disease1 is x n manife/ t all ! que/ions ( !rapeutics may be view$ z consi/+ 9 det]m9+ ! quantities ( ! diff]5t ag5ts :i* modify ! organism1 & :i* mu/ act ^u x to br+ x to its normal /ate1 admitt+1 = "s ( ^! quantities 9 c]ta9 cases1 values :i* are equal to z]o1 or negative1 or ev5 contradictory8 ,! fundam5tal idea ( ,desc>tes on ! relation ( ! concrete to ! ab/ract 9 ma!matics1 has prov51 9 opposition to ! sup]ficial di/9ction ( metaphysics1 t all ideas ( quality may be r$uc$ to ^? ( quantity4 ,?is conception1 e/abli%$ at fir/ by its immortal au?or 9 relation to geometrical ph5om5a only1 has s9ce be5 effectually ext5d$ to me*anical ph5om5a1 & 9 our days to ^? ( heat4 ,z a result ( ?is gradual g5]alization1 !re are n[ no geomet]s :o d n consid] x1 9 a purely !oretical s5se1 z capable ( be+ appli$ to all our real ideas ( e sort1 s t e ph5om5on is logically susceptible ( be+ repres5t$ by an equation2 z mu* s1 inde$1 z is a curve or a motion1 except+ ! difficulty ( discov]+ x1 & !n ( resolv+ x1 :i* may be1 & (t5times are1 sup]ior to ! greate/ p[]s ( ! human m9d4 ,its ,limitations4 ,important z x is to compreh5d ! rigor\s univ]sality1 9 a logical po9t ( view1 ( ma!matical sci5ce1 x is no less indisp5sable to consid] n[ ! great real limitations1 :i*1 ?r\gh ! feebl5ess ( our intellect1 n>r[ 9 a rem>kable degree its actual doma91 9 proportion z ph5om5a1 9 becom+ special1 become complicat$4 ,e "q may be conceiv$ z capable ( be+ r$uc$ to a pure "q ( numb]s2 b ! difficulty ( effect+ su* a trans=mation increases s mu* ) ! complication ( ! ph5om5a ( natural philosophy1 t x soon becomes insurm\ntable4 ,?is w be easily se51 if we consid] t to br+ a "q )9 ! field ( ma!matical analysis1 we mu/ fir/ h discov]$ ! precise relations :i* exi/ betwe5 ! quantities :i* are f\nd 9 ! ph5om5on "u exam9ation1 ! e/abli%m5t ( ^! equations be+ ! necess>y />t+ po9t ( all analytical lab\rs4 ,?is mu/ evid5tly be s mu* ! m difficult z we h to d ) ph5om5a :i* are m special1 & !re=e m complicat$4 ,we %all ?us f9d t x is only 9 inorganic physics1 at ! mo/1 t we c ju/ly hope ev] to obta9 t high degree ( sci5tific p]fection4 ,! fir/ condition :i* is necess>y 9 ord] t ph5om5a may admit ( ma!matical laws1 susceptible ( be+ discov]$1 evid5tly is1 t _! diff]5t quantities %\ld admit ( be+ express$ by fix$ numb]s4 ,we soon f9d t 9 ?is respect ! :ole ( organic physics1 & probably also ! mo/ complicat$ p>ts ( inorganic physics1 are necess>ily inaccessible1 by _! nature1 to our ma!matical analysis1 by reason ( ! extreme num]ical v>iability ( ! correspond+ ph5om5a4 ,e precise idea ( fix$ numb]s is truly out ( place 9 ! ph5om5a ( liv+ bodies1 :5 we wi% to employ x o!rwise ?an z a means ( reliev+ ! att5tion1 & :5 we atta* any importance to ! exact relations ( ! values assign$4 ,we ought n1 h[ev]1 on ?is acc\nt1 to cease to conceive all ph5om5a z be+ necess>ily subject to ma!matical laws1 :i* we are condemn$ to be ignorant (1 only because ( ! too great complication ( ! ph5om5a4 ,! mo/ complex ph5om5a ( liv+ bodies are d\btless ess5tially ( no o!r special nature ?an ! simple/ ph5om5a ( unorganiz$ matt]4 ,if x 7 possible to isolate rigor\sly ea* ( ! simple causes :i* concur 9 produc+ a s+le physiological ph5om5on1 e ?+ leads u to believe t x w\ld %[ itself end[$1 9 det]m9ate circum/ances1 ) a k9d ( influ5ce & ) a quantity ( action z exactly fix$ z we see x 9 univ]sal gravitation1 a v]itable type ( ! fundam5tal laws ( nature4 ,!re is a second reason :y we _c br+ complicat$ ph5om5a "u ! dom9ion ( ma!matical analysis4 ,ev5 if we c\ld asc]ta9 ! ma!matical law :i* gov]ns ea* ag5t1 tak5 by itself1 ! comb9ation ( s great a numb] ( conditions w\ld r5d] ! correspond+ ma!matical problem s f> above our feeble means1 t ! "q w\ld rema9 9 mo/ cases incapable ( solution4 ,to appreciate ?is difficulty1 let u consid] h[ complicat$ ma!matical que/ions become1 ev5 ^? relat+ to ! mo/ simple ph5om5a ( unorganiz$ bodies1 :5 we desire to br+ suffici5tly ne> toge!r ! ab/ract & ! concrete /ate1 hav+ reg>d to all ! pr9cipal conditions :i* c ex]cise a real influ5ce ov] ! effect produc$4 ,we "k1 = example1 t ! v simple ph5om5on ( ! fl[ ( a fluid ?r\gh a giv5 orifice1 by virtue ( its gravity alone1 has n z yet any complete ma!matical solution1 :5 we take into ! acc\nt all ! ess5tial circum/ances4 ,x is ! same ev5 ) ! /ill m simple motion ( a solid projectile 9 a resi/+ m$ium4 ,:y has ma!matical analysis be5 able to adapt itself ) su* admirable success to ! mo/ pr(\nd /udy ( cele/ial ph5om5a8 ,because !y are1 9 spite ( popul> appe>ances1 mu* m simple ?an any o!rs4 ,! mo/ complicat$ problem :i* !y pres5t1 t ( ! modification produc$ 9 ! motions ( two bodies t5d+ t[>ds ea* o!r by virtue ( _! gravitation1 by ! influ5ce ( a ?ird body act+ on bo? ( !m 9 ! same mann]1 is mu* less complex ?an ! mo/ simple t]re/rial problem4 ,&1 nev]!less1 ev5 x pres5ts difficulties s great t we yet possess only approximate solutions ( x4 ,x is ev5 easy to see t ! high p]fection to :i* sol> a/ronomy has be5 able to elevate itself by ! employm5t ( ma!matical sci5ce is1 besides1 ess5tially due to our hav+ skilfully pr(it$ by all ! p>ticul>1 &1 s to say1 accid5tal facilities pres5t$ by ! peculi>ly fav\rable con/itution ( our planet>y sy/em4 ,! planets :i* compose x are q few 9 numb]1 & _! masses are 9 g5]al v unequal1 & mu* less ?an t ( ! sun2 !y are1 besides1 v di/ant f "o ano!r2 !y h =ms almo/ sph]ical2 _! orbits are ne>ly circul>1 & only slightly incl9$ to ea* o!r1 & s on4 ,x results f all ^! circum/ances t ! p]turbations are g5]ally inconsid]able1 & t to calculate !m x is usually suffici5t to take into ! acc\nt1 9 connexion ) ! action ( ! sun on ea* p>ticul> planet1 ! influ5ce ( only "o o!r planet1 capable1 by its size & its proximity1 ( caus+ p]ceptible d]angem5ts4 ,if1 h[ev]1 in/ead ( su* a /ate ( ?+s1 our sol> sy/em _h be5 compos$ ( a great] numb] ( planets conc5trat$ into a less space1 & ne>ly equal 9 mass2 if _! orbits _h pres5t$ v diff]5t incl9ations1 & consid]able ecc5tricities2 if ^! bodies _h be5 ( a m complicat$ =m1 su* z v ecc5tric ellipsoids1 x is c]ta9 t1 suppos+ ! same law ( gravitation to exi/1 we %\ld n yet h succe$$ 9 subject+ ! /udy ( ! cele/ial ph5om5a to our ma!matical analysis1 & probably we %\ld n ev5 h be5 able to dis5tangle ! pres5t pr9cipal law4 ,^! hypo!tical conditions w\ld f9d !mselves exactly realiz$ 9 ! highe/ degree 9 *emical ph5om5a1 if we attempt$ to calculate !m by ! !ory ( g5]al gravitation4 ,on prop]ly weigh+ ! prec$+ consid]ations1 ! read] w be conv9c$1 ,i ?9k1 t 9 r$uc+ ! future ext5sion ( ! great applications ( ma!matical analysis1 :i* are really possible1 to ! field compris$ 9 ! diff]5t dep>tm5ts ( inorganic physics1 ,i h r exagg]at$ ?an contract$ ! ext5t ( its actual doma94 ,important z x 0 to r5d] app>5t ! rigor\s logical univ]sality ( ma!matical sci5ce1 x 0 equally s to indicate ! conditions :i* limit = u its real ext5sion1 s z n to contribute to lead ! human m9d a/ray f ! true sci5tific direction 9 ! /udy ( ! mo/ complicat$ ph5om5a1 by ! *im]ical se>* aft] an impossible p]fection4 99 99 99 99 99 ,hav+ ?us exhibit$ ! ess5tial object & ! pr9cipal composition ( ma!matical sci5ce1 z well z its g5]al relations ) ! :ole body ( natural philosophy1 we h n[ to pass to ! special exam9ation ( ! great sci5ces ( :i* x is compos$4 ,note4--,,analysis & ,,geometry are ! two great heads "u :i* ! subject is ab\t to be exam9$4 ,to ^! ,m4 ,comte adds ,rational ,,me*anics2 b z x is n compris$ 9 ! usual idea ( ,ma!matics1 & z its discussion w\ld be ( b limit$ utility & int]e/1 x is n includ$ 9 ! pres5t translation4 ,,book ,i4 ,,analysis4 ,,book ,i4 ,,analysis4 ,,*apt] ,i4 ,,g5]al ,,view ,,( ,,ma!matical ,,analysis4 ,9 ! hi/orical developm5t ( ma!matical sci5ce s9ce ! "t ( ,desc>tes1 ! advances ( its ab/ract portion h always be5 det]m9$ by ^? ( its concrete portion2 b x is none ! less necess>y1 9 ord] to conceive ! sci5ce 9 a mann] truly logical1 to consid] ! ,calculus 9 all its pr9cipal bran*es be=e proce$+ to ! philosophical /udy ( ,geometry & ,me*anics4 ,its analytical !ories1 m simple & m g5]al ?an ^? ( concrete ma!matics1 are 9 !mselves ess5tially indep5d5t ( ! latt]2 :ile ^!1 on ! contr>y1 h1 by _! nature1 a cont9ual ne$ ( ! =m]1 )\t ! aid ( :i* !y c\ld make sc>cely any progress4 ,al?\gh ! pr9cipal conceptions ( analysis reta9 at pres5t "s v p]ceptible traces ( _! geometrical or me*anical orig91 !y are n[1 h[ev]1 ma9ly fre$ f t primitive "*1 :i* no long] manife/s itself except 9 "s second>y po9ts2 s t x is possible 7especially s9ce ! lab\rs ( ,lagrange7 to pres5t !m 9 a dogmatic exposition1 by a purely ab/ract me?od1 9 a s+le & cont9u\s sy/em4 ,x is ?is :i* w be und]tak5 9 ! pres5t & ! five foll[+ *apt]s1 limit+ our inve/igations to ! mo/ g5]al consid]ations ^u ea* pr9cipal bran* ( ! sci5ce ( ! calculus4 ,! def9ite object ( our rese>*es 9 concrete ma!matics be+ ! discov]y ( ! equations :i* express ! ma!matical laws ( ! ph5om5on "u consid]ation1 & ^! equations con/itut+ ! true />t+ po9t ( ! calculus1 :i* has = its object to obta9 f !m ! det]m9ation ( c]ta9 quantities by means ( o!rs1 ,i ?9k x indisp5sable1 be=e proce$+ any f>!r1 to g m deeply ?an has be5 cu/om>y into t fundam5tal idea ( equation1 ! cont9ual subject1 ei!r z end or z beg9n+1 ( all ma!matical lab\rs4 ,besides ! advantage ( circumscrib+ m def9itely ! true field ( analysis1 !re w result f x ! important consequ5ce ( trac+ 9 a m exact mann] ! real l9e ( dem>cation betwe5 ! concrete & ! ab/ract "p ( ma!matics1 :i* w complete ! g5]al exposition ( ! fundam5tal division e/abli%$ 9 ! introductory *apt]4 ,,! ,,true ,,idea ,,( ,,an ,,equation4 ,we usually =m mu* too vague an idea ( :at an equation is1 :5 we give t "n to e k9d ( relation ( equality betwe5 any two functions ( ! magnitudes :i* we are consid]+4 ,=1 ?\gh e equation is evid5tly a relation ( equality1 x is f> f be+ true t1 reciprocally1 e relation ( equality is a v]itable equation1 ( ! k9d ( ^? to :i*1 by _! nature1 ! me?ods ( analysis are applicable4 ,?is want ( precision 9 ! logical consid]ation ( an idea :i* is s fundam5tal 9 ma!matics1 br+s ) x ! s]i\s inconv5i5ce ( r5d]+ x almo/ impossible to expla91 9 g5]al t]ms1 ! great & fundam5tal difficulty :i* we f9d 9 e/abli%+ ! relation betwe5 ! concrete & ! ab/ract1 & :i* /&s out s prom95tly 9 ea* great ma!matical "q tak5 by itself4 ,if ! mean+ ( ! ^w equation 0 truly z ext5d$ z we habitually suppose x to be 9 our def9ition ( x1 x is n app>5t :at great difficulty !re c\ld really be1 9 g5]al1 9 e/abli%+ ! equations ( any problem :atsoev]2 = ! :ole w\ld ?us appe> to consi/ 9 a simple "q ( =m1 :i* ought nev] ev5 to exact any great intellectual ef=ts1 see+ t we c h>dly conceive ( any precise relation :i* is n imm$iately a c]ta9 relation ( equality1 or :i* _c be readily br\ght !reto by "s v easy trans=mations4 ,?us1 :5 we admit e species ( functions into ! def9ition ( equations1 we d n at all acc\nt = ! extreme difficulty :i* we almo/ always exp]i5ce 9 putt+ a problem into an equation1 & :i* s (t5 may be comp>$ to ! ef=ts requir$ by ! analytical elaboration ( ! equation :5 once obta9$4 ,9 a ^w1 ! ord9>y ab/ract & g5]al idea ( an equation does n at all correspond to ! real mean+ :i* geomet]s atta* to t expression 9 ! actual developm5t ( ! sci5ce4 ,"h1 !n1 is a logical fault1 a defect ( correlation1 :i* x is v important to rectify4 ,division ( ,functions into ,ab/ract & ,concrete4 ,to succe$ 9 do+ s1 ,i beg9 by di/+ui%+ two sorts ( functions1 ab/ract or analytical functions1 & concrete functions4 ,! fir/ alone c ent] into v]itable equations4 ,we may1 !re=e1 h5ce=? def9e e equation1 9 an exact & suffici5tly pr(\nd mann]1 z a relation ( equality betwe5 two ab/ract functions ( ! magnitudes "u consid]ation4 ,9 ord] n to h to return aga9 to ?is fundam5tal def9ition1 ,i mu/ add "h1 z an indisp5sable complem5t1 )\t :i* ! idea w\ld n be suffici5tly g5]al1 t ^! ab/ract functions may ref] n only to ! magnitudes :i* ! problem pres5ts ( itself1 b also to all ! o!r auxili>y magnitudes :i* are connect$ ) x1 & :i* we w (t5 be able to introduce1 simply z a ma!matical artifice1 ) ! sole object ( facilitat+ ! discov]y ( ! equations ( ! ph5om5a4 ,i "h anticipate summ>ily ! result ( a g5]al discussion ( ! highe/ importance1 :i* w be f\nd at ! end ( ?is *apt]4 ,we w n[ return to ! ess5tial di/9ction ( functions z ab/ract & concrete4 ,?is di/9ction may be e/abli%$ 9 two ways1 ess5tially diff]5t1 b complem5t>y ( ea* o!r1 priori & po/]iori2 t is to say1 by *>act]iz+ 9 a g5]al mann] ! peculi> nature ( ea* species ( functions1 & !n by mak+ ! actual enum]ation ( all ! ab/ract functions at pres5t kn[n1 at lea/ s f> z relates to ! elem5ts ( :i* !y are compos$4 priori1 ! functions :i* ,i call ab/ract are ^? :i* express a mann] ( dep5d5ce betwe5 magnitudes1 :i* c be conceiv$ betwe5 numb]s alone1 )\t !re be+ ne$ ( indicat+ any ph5om5on :atev] 9 :i* x is realiz$4 ,i "n1 on ! o!r h&1 concrete functions1 ^? = :i* ! mode ( dep5d5ce express$ _c be def9$ or conceiv$ except by assign+ a det]m9ate case ( physics1 geometry1 me*anics1 c41 9 :i* x actually exi/s4 ,mo/ functions 9 _! orig91 ev5 ^? :i* are at pres5t ! mo/ purely ab/ract1 h begun by be+ concrete2 s t x is easy to make ! prec$+ di/9ction und]/ood1 by cit+ only ! successive diff]5t po9ts ( view "u :i*1 9 proportion z ! sci5ce has become =m$1 geomet]s h consid]$ ! mo/ simple analytical functions4 ,i w indicate p[]s1 = example1 :i* h 9 g5]al become ab/ract functions only s9ce ! lab\rs ( ,vieta & ,desc>tes4 ,! functions x1 x1 :i* 9 our pres5t analysis are s well conceiv$ z simply ab/ract1 71 = ! geomet]s ( antiquity1 p]fectly concrete functions1 express+ ! relation ( ! sup]ficies ( a squ>e1 or ! volume ( a cube to ! l5g? ( _! side4 ,^! _h 9 _! eyes su* a "* s exclusively1 t x 0 only by means ( ! geometrical def9itions t !y discov]$ ! elem5t>y algebraic prop]ties ( ^! functions1 relat+ to ! decomposition ( ! v>iable into two p>ts1 prop]ties :i* 7 at t epo* only real !orems ( geometry1 to :i* a num]ical mean+ 0 n atta*$ until long aft]w>d4 ,i %all h occasion to cite pres5tly1 = ano!r reason1 a new example1 v suitable to make app>5t ! fundam5tal di/9ction :i* ,i h j exhibit$2 x is t ( circul> functions1 bo? direct & inv]se1 :i* at ! pres5t "t are /ill sometimes concrete1 sometimes ab/ract1 accord+ to ! po9t ( view "u :i* !y are reg>d$4 po/]iori1 ! g5]al "* :i* r5d]s a function ab/ract or concrete hav+ be5 e/abli%$1 ! "q z to :e!r a c]ta9 det]m9ate function is v]itably ab/ract1 & !re=e susceptible ( ent]+ into true analytical equations1 becomes a simple "q ( fact1 inasmu* z we are go+ to enum]ate all ! functions ( ?is species4 ,enum]ation ( ,ab/ract ,functions4 ,at fir/ view ?is enum]ation seems impossible1 ! di/9ct analytical functions be+ inf9ite 9 numb]4 ,b :5 we divide !m into simple & comp\nd1 ! difficulty disappe>s2 =1 ?\gh ! numb] ( ! diff]5t functions consid]$ 9 ma!matical analysis is really inf9ite1 !y are1 on ! contr>y1 ev5 at ! pres5t day1 compos$ ( a v small numb] ( elem5t>y functions1 :i* c be easily assign$1 & :i* are evid5tly suffici5t = decid+ ! ab/ract or concrete "* ( any giv5 function2 :i* w be ( ! "o or ! o!r nature1 accord+ z x %all be compos$ exclusively ( ^! simple ab/ract functions1 or z x %all include o!rs4 ,we evid5tly h to consid]1 = ?is purpose1 only ! functions ( a s+le v>iable1 s9ce ^? relative to sev]al indep5d5t v>iables are con/antly1 by _! nature1 m or less comp\nd4 ,let x be ! indep5d5t v>iable1 y ! correlative v>iable :i* dep5ds ^u x4 ,! diff]5t simple modes ( ab/ract dep5d5ce1 :i* we c n[ conceive betwe5 y & x1 are express$ by ! t5 foll[+ elem5t>y =mulas1 9 :i* ea* function is c\pl$ ) its inv]se1 t is1 ) t :i* w\ld be obta9$ f ! direct function by ref]r+ x to y1 in/ead ( ref]r+ y to x4 ,,function4 ,,its ,,"n4 #a/ c\ple #a y a x ,sum4 #b y a - x ,diff]5ce4 #b;d c\ple #a y ax ,product4 #b y a_/x ,quoti5t4 #c;d c\ple #a y xa ,p[]4 #b y a? rootx ,root4 #d? c\ple #a y ax ,expon5tial4 #b y log ax ,log>i?mic4 #e? c\ple #a y s94 x ,direct ,circul>4 #b y arc7s94 x74 ,inv]se ,circul>4#c ,footnote #c3 ,) ! view ( increas+ z mu* z possible ! res\rces & ! ext5t 7n[ s insuffici5t7 ( ma!matical analysis1 geomet]s c\nt ?is la/ c\ple ( functions among ! analytical elem5ts4 ,al?\gh ?is inscription is /rictly legitimate1 x is important to rem>k t circul> functions are n exactly 9 ! same situation z ! o!r ab/ract elem5t>y functions4 ,!re is ?is v ess5tial diff]5ce1 t ! functions ( ! f\r fir/ c\ples are at ! same "t simple & ab/ract1 :ile ! circul> functions1 :i* may manife/ ea* "* 9 succession1 accord+ to ! po9t ( view "u :i* !y are consid]$ & ! mann] 9 :i* !y are employ$1 nev] pres5t ^! two prop]ties simultane\sly4 ,"s o!r concrete functions may be usefully introduc$ into ! numb] ( analytical elem5ts1 c]ta9 conditions be+ fulfill$4 ,x is ?us1 = example1 t ! lab\rs ( ,m4 ,leg5dre & ( ,m4 ,jacobi on elliptical functions h truly enl>g$ ! field ( analysis2 & ! same is true ( "s def9ite integrals obta9$ by ,m4 ,f\ri] 9 ! !ory ( heat4 ,su* are ! elem5ts1 v few 9 numb]1 :i* directly compose all ! ab/ract functions kn[n at ! pres5t day4 ,few z !y are1 !y are evid5tly suffici5t to give rise to an inf9ite numb] ( analytical comb9ations4 ,no rational consid]ation rigor\sly circumscribes1 priori1 ! prec$+ table1 :i* is only ! actual expression ( ! pres5t /ate ( ! sci5ce4 ,our analytical elem5ts are at ! pres5t day m num]\s ?an !y 7 = ,desc>tes1 & ev5 = ,newton & ,leibnitz3 x is only a c5tury s9ce ! la/ two c\ples h be5 introduc$ into analysis by ! lab\rs ( ,john ,b]n\illi & ,eul]4 ,d\btless new ones w be h]eaft] admitt$2 b1 z ,i %all %[ t[>ds ! end ( ?is *apt]1 we _c hope t !y w ev] be greatly multipli$1 _! real augm5tation giv+ rise to v great difficulties4 ,we c n[ =m a def9ite1 &1 at ! same "t1 suffici5tly ext5d$ idea ( :at geomet]s und]/& by a v]itable equation4 ,?is explanation is especially suit$ to make u und]/& h[ difficult x mu/ be really to e/abli% ! equations ( ph5om5a1 s9ce we h effectually succe$$ 9 s do+ only :5 we h be5 able to conceive ! ma!matical laws ( ^! ph5om5a by ! aid ( functions entirely compos$ ( only ! ma!matical elem5ts :i* ,i h j enum]at$4 ,x is cle>1 9 fact1 t x is !n only t ! problem becomes truly ab/ract1 & is r$uc$ to a pure "q ( numb]s1 ^! functions be+ ! only simple relations :i* we c conceive betwe5 numb]s1 consid]$ by !mselves4 ,up to ?is p]iod ( ! solution1 :atev] ! appe>ances may be1 ! "q is /ill ess5tially concrete1 & does n come )9 ! doma9 ( ! calculus4 ,n[ ! fundam5tal difficulty ( ?is passage f ! concrete to ! ab/ract 9 g5]al consi/s especially 9 ! insuffici5cy ( ?is v small numb] ( analytical elem5ts :i* we possess1 & by means ( :i*1 nev]!less1 9 spite ( ! little real v>iety :i* !y (f] u1 we mu/ succe$ 9 repres5t+ all ! precise relations :i* all ! diff]5t natural ph5om5a c manife/ to u4 ,consid]+ ! inf9ite div]sity :i* mu/ necess>ily exi/ 9 ?is respect 9 ! ext]nal _w1 we easily und]/& h[ f> bel[ ! true difficulty our conceptions mu/ frequ5tly be f\nd1 especially if we add t z ^! elem5ts ( our analysis h be5 9 ! fir/ place furni%$ to u by ! ma!matical consid]ation ( ! simple/ ph5om5a1 we h1 priori1 no rational gu>antee ( _! necess>y suitabl5ess to repres5t ! ma!matical law ( e o!r class ( ph5om5a4 ,i w expla9 pres5tly ! g5]al artifice1 s pr(\ndly +5i\s1 by :i* ! human m9d has succe$$ 9 dim9i%+1 9 a rem>kable degree1 ?is fundam5tal difficulty :i* is pres5t$ by ! relation ( ! concrete to ! ab/ract 9 ma!matics1 )\t1 h[ev]1 its hav+ be5 necess>y to multiply ! numb] ( ^! analytical elem5ts4 ,,! ,,two ,,pr9cipal ,,divisions ,,( ,,! ,,calculus4 ,! prec$+ explanations det]m9e ) precision ! true object & ! real field ( ab/ract ma!matics4 ,i mu/ n[ pass to ! exam9ation ( its pr9cipal divisions1 = ?us f> we h consid]$ ! calculus z a :ole4 ,! fir/ direct consid]ation to be pres5t$ on ! composition ( ! sci5ce ( ! calculus consi/s 9 divid+ x1 9 ! fir/ place1 into two pr9cipal bran*es1 to :i*1 = want ( m suitable d5om9ations1 ,i w give ! names ( ,algebraic calculus1 or ,algebra1 & ( ,ari?metical calculus1 or ,ari?metic2 b ) ! caution to take ^! two expressions 9 _! mo/ ext5d$ logical acceptation1 9 ! place ( ! by f> too re/rict$ mean+ :i* is usually atta*$ to !m4 ,! complete solution ( e "q ( ! calculus1 f ! mo/ elem5t>y up to ! mo/ transc5d5tal1 is necess>ily compos$ ( two successive p>ts1 ^: nature is ess5tially di/9ct4 ,9 ! fir/1 ! object is to trans=m ! propos$ equations1 s z to make app>5t ! mann] 9 :i* ! unkn[n quantities are =m$ by ! kn[n ones3 x is ?is :i* con/itutes ! algebraic "q4 ,9 ! second1 our object is to f9d ! values ( ! =mulas ?us obta9$2 t is1 to det]m9e directly ! values ( ! numb]s s\ght1 :i* are already repres5t$ by c]ta9 explicit functions ( giv5 numb]s3 ?is is ! ari?metical "q4#d ,x is app>5t t1 9 e solution :i* is truly rational1 x necess>ily foll[s ! algebraical "q1 ( :i* x =ms ! indisp5sable complem5t1 s9ce x is evid5tly necess>y to "k ! mode ( g5]ation ( ! numb]s s\ght = be=e det]m9+ _! actual values = ea* p>ticul> case4 ,?us ! /opp+-place ( ! algebraic "p ( ! solution becomes ! />t+ po9t ( ! ari?metical "p4 ,footnote #d3 ,suppose1 = example1 t a "q gives ! foll[+ equation betwe5 an unkn[n magnitude x1 & two kn[n magnitudes1 a & b1 x #c;ax #b;b1 z is ! case 9 ! problem ( ! trisection ( an angle4 ,we see at once t ! dep5d5ce betwe5 x on ! "o side1 & ab on ! o!r1 is completely det]m9$2 b1 s long z ! equation pres]ves its primitive =m1 we d n at all p]ceive 9 :at mann] ! unkn[n quantity is d]iv$ f ! data4 ,?is mu/ be discov]$1 h[ev]1 be=e we c ?9k ( det]m9+ its value4 ,su* is ! object ( ! algebraic "p ( ! solution4 ,:51 by a s]ies ( trans=mations :i* h successively r5d]$ t d]ivation m & m app>5t1 we h arriv$ at pres5t+ ! propos$ equation "u ! =m x 7b 7b a77 7b - 7b a771 ! "w ( algebra is f9i%$2 & ev5 if we c\ld n p]=m ! ari?metical op]ations indicat$ by t =mula1 we w\ld nev]!less h obta9$ a k v real1 & (t5 v important4 ,! "w ( ari?metic w n[ consi/ 9 tak+ t =mula = its />t+ po9t1 & f9d+ ! numb] x :5 ! values ( ! numb]s a & b are giv54 ,we ?us see t ! algebraic calculus & ! ari?metical calculus diff] ess5tially 9 _! object4 ,!y diff] no less 9 ! po9t ( view "u :i* !y reg>d quantities2 :i* are consid]$ 9 ! fir/ z to _! relations1 & 9 ! second z to _! values4 ,! true _s ( ! calculus1 9 g5]al1 requires ?is di/9ction to be ma9ta9$ ) ! mo/ sev]e exactitude1 & ! l9e ( dem>cation betwe5 ! two p]iods ( ! solution to be r5d]$ z cle> & di/9ct z ! propos$ "q p]mits4 ,! att5tive obs]vation ( ?is precept1 :i* is too mu* neglect$1 may be ( mu* assi/ance1 9 ea* p>ticul> "q1 9 direct+ ! ef=ts ( our m9d1 at any mom5t ( ! solution1 t[>ds ! real correspond+ difficulty4 ,9 tru?1 ! imp]fection ( ! sci5ce ( ! calculus obliges u v (t5 7z w be expla9$ 9 ! next *apt]7 to int]m+le algebraic & ari?metical consid]ations 9 ! solution ( ! same "q4 ,b1 h[ev] impossible x may be to sep>ate cle>ly ! two p>ts ( ! lab\r1 yet ! prec$+ indications w always enable u to avoid conf\nd+ !m4 ,9 endeav\r+ to sum up z succ9ctly z possible ! di/9ction j e/abli%$1 we see t ,,algebra may be def9$1 9 g5]al1 z hav+ = its object ! resolution ( equations2 tak+ ?is expression 9 its full logical mean+1 :i* signifies ! trans=mation ( implicit functions into equival5t explicit ones4 ,9 ! same way1 ,,ari?metic may be def9$ z de/9$ to ! det]m9ation ( ! values ( functions4 ,h5ce=?1 !re=e1 we w briefly say t ,,algebra is ! ,calculus ( ,functions1 & ,,ari?metic ! ,calculus ( ,values4 ,we c n[ p]ceive h[ insuffici5t & ev5 errone\s are ! ord9>y def9itions4 ,mo/ g5]ally1 ! exagg]at$ importance attribut$ to ,signs has l$ to ! di/+ui%+ ! two fundam5tal bran*es ( ! sci5ce ( ! ,calculus by ! mann] ( designat+ 9 ea* ! subjects ( discussion1 an idea :i* is evid5tly absurd 9 pr9ciple & false 9 fact4 ,ev5 ! celebrat$ def9ition giv5 by ,newton1 *>act]iz+ ,algebra z ,univ]sal ,ari?metic1 gives c]ta9ly a v false idea ( ! nature ( algebra & ( t ( ari?metic4#e ,footnote #e3 ,i h ?\ght t ,i ought to specially notice ?is def9ition1 because x s]ves z ! basis ( ! op9ion :i* _m intellig5t p]sons1 unacqua9t$ ) ma!matical sci5ce1 =m ( its ab/ract "p1 )\t consid]+ t at ! "t ( ?is def9ition ma!matical analysis 0 n suffici5tly develop$ to enable ! g5]al "* ( ea* ( its pr9cipal p>ts to be prop]ly appreh5d$1 :i* expla9s :y ,newton c\ld at t "t propose a def9ition :i* at ! pres5t day he w\ld c]ta9ly reject4 ,hav+ ?us e/abli%$ ! fundam5tal division ( ! calculus into two pr9cipal bran*es1 ,i h n[ to comp>e 9 g5]al t]ms ! ext5t1 ! importance1 & ! difficulty ( ^! two sorts ( calculus1 s z to h h]eaft] to consid] only ! ,calculus ( ,functions1 :i* is to be ! pr9cipal subject ( our /udy4 ,,! ,,calculus ,,( ,,values1 ,,or ,,ari?metic4 ,its ,ext5t4 ,! ,calculus ( ,values1 or ,ari?metic1 w\ld appe>1 at fir/ view1 to pres5t a field z va/ z t ( algebra1 s9ce x w\ld seem to admit z _m di/9ct que/ions z we c conceive diff]5t algebraic =mulas ^: values are to be det]m9$4 ,b a v simple reflection w %[ ! diff]5ce4 ,divid+ functions into simple & comp\nd1 x is evid5t t :5 we "k h[ to det]m9e ! value ( simple functions1 ! consid]ation ( comp\nd functions w no long] pres5t any difficulty4 ,9 ! algebraic po9t ( view1 a comp\nd function plays a v diff]5t "p f t ( ! elem5t>y functions ( :i* x consi/s1 & f ?is1 inde$1 proce$ all ! pr9cipal difficulties ( analysis4 ,b x is v diff]5t ) ! ,ari?metical ,calculus4 ,?us ! numb] ( truly di/9ct ari?metical op]ations is only t det]m9$ by ! numb] ( ! elem5t>y ab/ract functions1 ! v limit$ li/ ( :i* has be5 giv5 above4 ,! det]m9ation ( ! values ( ^! t5 functions necess>ily gives t ( all ! functions1 inf9ite 9 numb]1 :i* are consid]$ 9 ! :ole ( ma!matical analysis1 su* at lea/ z x exi/s at pres5t4 ,!re c be no new ari?metical op]ations )\t ! creation ( really new analytical elem5ts1 ! numb] ( :i* mu/ always be extremely small4 ,! field ( ari?metic is1 !n1 by its nature1 exce$+ly re/rict$1 :ile t ( algebra is rigor\sly indef9ite4 ,x is1 h[ev]1 important to rem>k1 t ! doma9 ( ! calculus ( values is1 9 reality1 mu* m ext5sive ?an x is commonly repres5t$2 = sev]al que/ions truly ari?metical1 s9ce !y consi/ ( det]m9ations ( values1 are n ord9>ily class$ z su*1 because we are accu/om$ to treat !m only z incid5tal 9 ! mid/ ( a body ( analytical rese>*es m or less elevat$1 ! too high op9ion commonly =m$ ( ! influ5ce ( signs be+ aga9 ! pr9cipal cause ( ?is confusion ( ideas4 ,?us n only ! con/ruction ( a table ( log>i?ms1 b also ! calculation ( trigonometrical tables1 are true ari?metical op]ations ( a high] k9d4 ,we may also cite z be+ 9 ! same class1 al?\gh 9 a v di/9ct & m elevat$ ord]1 all ! me?ods by :i* we det]m9e directly ! value ( any function = ea* p>ticul> sy/em ( values attribut$ to ! quantities on :i* x dep5ds1 :5 we _c express 9 g5]al t]ms ! explicit =m ( t function4 ,9 ?is po9t ( view ! num]ical solution ( que/ions :i* we _c resolve algebraically1 & ev5 ! calculation ( ,def9ite ,integrals1 ^: g5]al integrals we d n "k1 really make a "p1 9 spite ( all appe>ances1 ( ! doma9 ( ari?metic1 9 :i* we mu/ necess>ily comprise all t :i* has = its object ! det]m9ation ( ! values ( functions4 ,! consid]ations relative to ?is object are1 9 fact1 con/antly homog5e\s1 :atev] ! det]m9ations 9 "q1 & are always v di/9ct f truly algebraic consid]ations4 ,to complete a j idea ( ! real ext5t ( ! calculus ( values1 we mu/ include 9 x likewise t "p ( ! g5]al sci5ce ( ! calculus :i* n[ be>s ! "n ( ! ,!ory ( ,numb]s1 & :i* is yet s little advanc$4 ,?is bran*1 v ext5sive by its nature1 b ^: importance 9 ! g5]al sy/em ( sci5ce is n v great1 has = its object ! discov]y ( ! prop]ties inh]5t 9 diff]5t numb]s by virtue ( _! values1 & indep5d5t ( any p>ticul> sy/em ( num]ation4 ,x =ms1 !n1 a sort ( transc5d5tal ari?metic2 & to x w\ld really apply ! def9ition propos$ by ,newton = algebra4 ,! entire doma9 ( ari?metic is1 !n1 mu* m ext5d$ ?an is commonly suppos$2 b ?is calculus ( values w /ill nev] be m ?an a po9t1 s to speak1 9 comp>ison ) ! calculus ( functions1 ( :i* ma!matical sci5ce ess5tially consi/s4 ,?is comp>ative e/imate w be /ill m app>5t f "s consid]ations :i* ,i h n[ to indicate respect+ ! true nature ( ari?metical que/ions 9 g5]al1 :5 !y are m pr(\ndly exam9$4 ,its true ,nature4 ,9 seek+ to det]m9e ) precision 9 :at det]m9ations ( values prop]ly consi/1 we easily recognize t !y are no?+ else b v]itable trans=mations ( ! functions to be valu$2 trans=mations :i*1 9 spite ( _! special end1 are none ! less ess5tially ( ! same nature z all ^? taught by analysis4 ,9 ?is po9t ( view1 ! calculus ( values might be simply conceiv$ z an app5dix1 & a p>ticul> application ( ! calculus ( functions1 s t ari?metic w\ld disappe>1 s to say1 z a di/9ct section 9 ! :ole body ( ab/ract ma!matics4 ,9 ord] ?or\ghly to compreh5d ?is consid]ation1 we mu/ obs]ve t1 :5 we propose to det]m9e ! value ( an unkn[n numb] ^: mode ( =mation is giv51 x is1 by ! m]e enunciation ( ! ari?metical "q1 already def9$ & express$ "u a c]ta9 =m2 & t 9 det]m9+ its value we only put its expression "u ano!r det]m9ate =m1 to :i* we are accu/om$ to ref] ! exact notion ( ea* p>ticul> numb] by mak+ x re-ent] into ! regul> sy/em ( num]ation4 ,! det]m9ation ( values consi/s s completely ( a simple trans=mation1 t :5 ! primitive expression ( ! numb] is f\nd to be already con=m$ to ! regul> sy/em ( num]ation1 !re is no long] any det]m9ation ( value1 prop]ly speak+1 or1 r1 ! "q is answ]$ by ! "q itself4 ,let ! "q be to add ! two numb]s "o & tw5ty1 we answ] x by m]ely repeat+ ! enunciation ( ! "q1#f & nev]!less we ?9k t we h det]m9$ ! value ( ! sum4 ,?is signifies t 9 ?is case ! fir/ expression ( ! function _h no ne$ ( be+ trans=m$1 :ile x w\ld n be ?us 9 add+ tw5ty-?ree & f\rte51 = !n ! sum w\ld n be imm$iately express$ 9 a mann] con=m$ to ! rank :i* x occupies 9 ! fix$ & g5]al scale ( num]ation4 ,footnote #f3 ,?is is less /rictly true 9 ! ,engli% sy/em ( num]ation ?an 9 ! ,fr5*1 s9ce tw5ty-"o is our m usual mode ( express+ ?is numb]4 ,to sum up z compreh5sively z possible ! prec$+ views1 we may say1 t to det]m9e ! value ( a numb] is no?+ else ?an putt+ its primitive expression "u ! =m a bz cz dz ez 4 4 4 4 4 pzm1 z be+ g5]ally equal to #aj1 & ! coeffici5ts a1 b1 c1 d1 c41 be+ subject$ to ! conditions ( be+ :ole numb]s less ?an z2 capable ( becom+ equal to z]o2 b nev] negative4 ,e ari?metical "q may ?us be /at$ z consi/+ 9 putt+ "u su* a =m any ab/ract function :atev] ( diff]5t quantities1 :i* are suppos$ to h !mselves a simil> =m already4 ,we might !n see 9 ! diff]5t op]ations ( ari?metic only simple p>ticul> cases ( c]ta9 algebraic trans=mations1 except+ ! special difficulties belong+ to conditions relat+ to ! nature ( ! coeffici5ts4 ,x cle>ly foll[s t ab/ract ma!matics is ess5tially compos$ ( ! ,calculus ( ,functions1 :i* _h be5 already se5 to be its mo/ important1 mo/ ext5d$1 & mo/ difficult "p4 ,x w h5ce=? be ! exclusive subject ( our analytical inve/igations4 ,i w !re=e no long] delay on ! ,calculus ( ,values1 b pass imm$iately to ! exam9ation ( ! fundam5tal division ( ! ,calculus ( ,functions4 ,,! ,,calculus ,,( ,,functions1 ,,or ,,algebra4 ,pr9ciple ( its ,fundam5tal ,division4 ,we h det]m9$1 at ! beg9n+ ( ?is *apt]1 :]e9 prop]ly consi/s ! difficulty :i* we exp]i5ce 9 putt+ ma!matical que/ions into equations4 ,x is ess5tially because ( ! insuffici5cy ( ! v small numb] ( analytical elem5ts :i* we possess1 t ! relation ( ! concrete to ! ab/ract is usually s difficult to e/abli%4 ,let u endeav\r n[ to appreciate 9 a philosophical mann] ! g5]al process by :i* ! human m9d has succe$$1 9 s great a numb] ( important cases1 9 ov]com+ ?is fundam5tal ob/acle to ,! e/abli%m5t ( ,equations4 #a4 ,by ! ,creation ( new ,functions4 ,9 look+ at ?is important "q f ! mo/ g5]al po9t ( view1 we are l$ at once to ! conception ( "o means ( facilitat+ ! e/abli%m5t ( ! equations ( ph5om5a4 ,s9ce ! pr9cipal ob/acle 9 ?is matt] comes f ! too small numb] ( our analytical elem5ts1 ! :ole "q w\ld seem to be r$uc$ to creat+ new ones4 ,b ?is means1 ?\gh natural1 is really illusory2 & ?\gh x might be useful1 x is c]ta9ly insuffici5t4 ,9 fact1 ! creation ( an elem5t>y ab/ract function1 :i* %all be v]itably new1 pres5ts 9 itself ! greate/ difficulties4 ,!re is ev5 some?+ contradictory 9 su* an idea2 = a new analytical elem5t w\ld evid5tly n fulfil its ess5tial & appropriate conditions1 if we c\ld n imm$iately det]m9e its value4 ,n[1 on ! o!r h&1 h[ are we to det]m9e ! value ( a new function :i* is truly simple1 t is1 :i* is n =m$ by a comb9ation ( ^? already kn[n8 ,t appe>s almo/ impossible4 ,! introduction into analysis ( ano!r elem5t>y ab/ract function1 or r ( ano!r c\ple ( functions 7= ea* w\ld be always accompani$ by its inv]se71 supposes !n1 ( necessity1 ! simultane\s creation ( a new ari?metical op]ation1 :i* is c]ta9ly v difficult4 ,if we endeav\r to obta9 an idea ( ! means :i* ! human m9d employs = inv5t+ new analytical elem5ts1 by ! exam9ation ( ! proc$ures by ! aid ( :i* x has actually conceiv$ ^? :i* we already possess1 our obs]vations leave u 9 t respect 9 an entire unc]ta9ty1 = ! artifices :i* x has already made use ( = t purpose are evid5tly exhau/$4 ,to conv9ce ourselves ( x1 let u consid] ! la/ c\ple ( simple functions :i* has be5 introduc$ into analysis1 & at ! =mation ( :i* we h be5 pres5t1 s to speak1 namely1 ! f\r? c\ple2 =1 z ,i h expla9$1 ! fif? c\ple does n /rictly give v]itable new analytical elem5ts4 ,! function ax1 &1 consequ5tly1 its inv]se1 h be5 =m$ by conceiv+1 "u a new po9t ( view1 a function :i* _h be5 a long "t kn[n1 namely1 p[]s--:5 ! idea ( !m _h become suffici5tly g5]aliz$4 ,! consid]ation ( a p[] relatively to ! v>iation ( its expon5t1 in/ead ( to ! v>iation ( its base1 0 suffici5t to give rise to a truly novel simple function1 ! v>iation foll[+ !n an entirely diff]5t r\te4 ,b ?is artifice1 z simple z +5i\s1 c furni% no?+ m2 =1 9 turn+ ov] 9 ! same mann] all our pres5t analytical elem5ts1 we end 9 only mak+ !m return into "o ano!r4 ,we h1 !n1 no idea z to h[ we c\ld proce$ to ! creation ( new elem5t>y ab/ract functions :i* w\ld prop]ly satisfy all ! necess>y conditions4 ,?is is n to say1 h[ev]1 t we h at pres5t atta9$ ! effectual limit e/abli%$ 9 t respect by ! b\nds ( our intellig5ce4 ,x is ev5 c]ta9 t ! la/ special improvem5ts 9 ma!matical analysis h contribut$ to ext5d our res\rces 9 t respect1 by introduc+ )9 ! doma9 ( ! calculus c]ta9 def9ite integrals1 :i* 9 "s respects supply ! place ( new simple functions1 al?\gh !y are f> f fulfill+ all ! necess>y conditions1 :i* has prev5t$ me f ins]t+ !m 9 ! table ( true analytical elem5ts4 ,b1 on ! :ole1 ,i ?9k x unque/ionable t ! numb] ( ^! elem5ts _c increase except ) extreme sl[ness4 ,x is !re=e n f ^! s\rces t ! human m9d has drawn its mo/ p[]ful means ( facilitat+1 z mu* z is possible1 ! e/abli%m5t ( equations4 #b4 ,by ! ,conception ( ,equations betwe5 c]ta9 auxili>y ,quantities4 ,?is fir/ me?od be+ set aside1 !re rema9s evid5tly b "o o!r3 x is1 see+ ! impossibility ( f9d+ directly ! equations betwe5 ! quantities "u consid]ation1 to seek = correspond+ ones betwe5 o!r auxili>y quantities1 connect$ ) ! fir/ accord+ to a c]ta9 det]m9ate law1 & f ! relation betwe5 :i* we may return to t betwe5 ! primitive magnitudes4 ,su* is1 9 sub/ance1 ! em95tly fruitful conception1 :i* ! human m9d has succe$$ 9 e/abli%+1 & :i* con/itutes its mo/ admirable in/rum5t = ! ma!matical explanation ( natural ph5om5a2 ! analysis1 call$ transc5d5tal4 ,z a g5]al philosophical pr9ciple1 ! auxili>y quantities1 :i* are introduc$ 9 ! place ( ! primitive magnitudes1 or concurr5tly ) !m1 9 ord] to facilitate ! e/abli%m5t ( equations1 might be d]iv$ accord+ to any law :atev] f ! imm$iate elem5ts ( ! "q4 ,?is conception has ?us a mu* m ext5sive rea* ?an has be5 commonly attribut$ to x by ev5 ! mo/ pr(\nd geomet]s4 ,x is extremely important = u to view x 9 its :ole logical ext5t1 = x w p]haps be by e/abli%+ a g5]al mode ( d]ivation diff]5t f t to :i* we h ?us f> conf9$ ourselves 7al?\gh x is evid5tly v f> f be+ ! only possible "o7 t we %all "o day succe$ 9 ess5tially p]fect+ ma!matical analysis z a :ole1 & consequ5tly 9 e/abli%+ m p[]ful means ( inve/igat+ ! laws ( nature ?an our pres5t processes1 :i* are unque/ionably susceptible ( becom+ exhau/$4 ,b1 reg>d+ m]ely ! pres5t con/itution ( ! sci5ce1 ! only auxili>y quantities habitually introduc$ 9 ! place ( ! primitive quantities 9 ! ,transc5d5tal ,analysis are :at are call$1 #a1 inf9itely small elem5ts1 ! diff]5tials 7( diff]5t ord]s7 ( ^? quantities1 if we reg>d ?is analysis 9 ! mann] ( ,,leibnitz2 or1 #b1 ! fluxions1 ! limits ( ! ratios ( ! simultane\s increm5ts ( ! primitive quantities comp>$ ) "o ano!r1 or1 m briefly1 ! prime & ultimate ratios ( ^! increm5ts1 if we adopt ! conception ( ,,newton2 or1 #c1 ! d]ivatives1 prop]ly s call$1 ( ^? quantities1 t is1 ! coeffici5ts ( ! diff]5t t]ms ( _! respective increm5ts1 accord+ to ! conception ( ,,lagrange4 ,^! ?ree pr9cipal me?ods ( view+ our pres5t transc5d5tal analysis1 & all ! o!r less di/9ctly *>act]iz$ ones :i* h be5 successively propos$1 are1 by _! nature1 necess>ily id5tical1 :e!r 9 ! calculation or 9 ! application1 z w be expla9$ 9 a g5]al mann] 9 ! ?ird *apt]4 ,z to _! relative value1 we %all !re see t ! conception ( ,leibnitz has ?us f>1 9 practice1 an inconte/able sup]iority1 b t its logical "* is exce$+ly vici\s2 :ile t ! conception ( ,lagrange1 admirable by its simplicity1 by its logical p]fection1 by ! philosophical unity :i* x has e/abli%$ 9 ma!matical analysis 7till !n sep>at$ into two almo/ entirely indep5d5t worlds71 pres5ts1 z yet1 s]i\s inconv5i5ces 9 ! applications1 by ret>d+ ! progress ( ! m9d4 ,! conception ( ,newton occupies ne>ly middle gr\nd 9 ^! v>i\s relations1 be+ less rapid1 b m rational ?an t ( ,leibnitz2 less philosophical1 b m applicable ?an t ( ,lagrange4 ,?is is n ! place to expla9 ! advantages ( ! introduction ( ?is k9d ( auxili>y quantities 9 ! place ( ! primitive magnitudes4 ,! ?ird *apt] is devot$ to ?is subject4 ,at pres5t ,i limit myself to consid] ?is conception 9 ! mo/ g5]al mann]1 9 ord] to d$uce !refrom ! fundam5tal division ( ! calculus ( functions into two sy/ems ess5tially di/9ct1 ^: dep5d5ce1 = ! complete solution ( any "o ma!matical "q1 is inv>iably det]m9ate4 ,9 ?is connexion1 & 9 ! logical ord] ( ideas1 ! transc5d5tal analysis pres5ts itself z be+ necess>ily ! fir/1 s9ce its g5]al object is to facilitate ! e/abli%m5t ( equations1 an op]ation :i* mu/ evid5tly prec$e ! resolution ( ^? equations1 :i* is ! object ( ! ord9>y analysis4 ,b ?\gh x is exce$+ly important to conceive 9 ?is way ! true relations ( ^! two sy/ems ( analysis1 x is none ! less prop]1 9 con=mity ) ! regul> usage1 to /udy ! transc5d5tal analysis aft] ord9>y analysis2 = ?\gh ! =m] is1 at bottom1 by itself logically indep5d5t ( ! latt]1 or1 at lea/1 may be ess5tially dis5gag$ f x1 yet x is cle> t1 s9ce its employm5t 9 ! solution ( que/ions has always m or less ne$ ( be+ complet$ by ! use ( ! ord9>y analysis1 we w\ld be con/ra9$ to leave ! que/ions 9 susp5se if ?is latt] _h n be5 previ\sly /udi$4 ,correspond+ ,divisions ( ! ,calculus ( ,functions4 ,x foll[s f ! prec$+ consid]ations t ! ,calculus ( ,functions1 or ,algebra 7tak+ ?is ^w 9 its mo/ ext5d$ mean+71 is compos$ ( two di/9ct fundam5tal bran*es1 "o ( :i* has = its imm$iate object ! resolution ( equations1 :5 !y are directly e/abli%$ betwe5 ! magnitudes !mselves :i* are "u consid]ation2 & ! o!r1 />t+ f equations 7g5]ally mu* easi] to =m7 betwe5 quantities indirectly connect$ ) ^? ( ! problem1 has = its peculi> & con/ant de/9ation ! d$uction1 by inv>iable analytical me?ods1 ( ! correspond+ equations betwe5 ! direct magnitudes :i* we are consid]+2 :i* br+s ! "q )9 ! doma9 ( ! prec$+ calculus4 ,! =m] calculus be>s mo/ frequ5tly ! "n ( ,ord9>y ,analysis1 or ( ,algebra1 prop]ly s call$4 ,! second con/itutes :at is call$ ! ,transc5d5tal ,analysis1 :i* has be5 designat$ by ! diff]5t d5om9ations ( ,inf9itesimal ,calculus1 ,calculus ( ,fluxions & ( ,flu5ts1 ,calculus ( ,vani%+ ,quantities1 ! ,diff]5tial & ,integral ,calculus1 c41 accord+ to ! po9t ( view 9 :i* x has be5 conceiv$4 ,9 ord] to remove e =eign consid]ation1 ,i w propose to "n x ,,calculus ,,( ,,indirect ,,functions1 giv+ to ord9>y analysis ! title ( ,,calculus ,,( ,,direct ,,functions4 ,^! expressions1 :i* ,i =m ess5tially by g5]aliz+ & epitomiz+ ! ideas ( ,lagrange1 are simply int5d$ to indicate ) precision ! true g5]al "* belong+ to ea* ( ^! two =ms ( analysis4 ,hav+ n[ e/abli%$ ! fundam5tal division ( ma!matical analysis1 ,i h next to consid] sep>ately ea* ( its two p>ts1 comm5c+ ) ! ,calculus ( ,direct ,functions1 & res]v+ m ext5d$ developm5ts = ! diff]5t bran*es ( ! ,calculus ( ,indirect ,functions4 ,,*apt] ,,ii4 ,,ord9>y ,,analysis1 ,,or ,,algebra4 ,! ,calculus ( direct ,functions1 or ,algebra1 is 7z 0 %[n at ! end ( ! prec$+ *apt]7 entirely suffici5t = ! solution ( ma!matical que/ions1 :5 !y are s simple t we c =m directly ! equations betwe5 ! magnitudes !mselves :i* we are consid]+1 )\t its be+ necess>y to introduce 9 _! place1 or conjo9tly ) !m1 any sy/em ( auxili>y quantities d]iv$ f ! fir/4 ,x is true t 9 ! greate/ numb] ( important cases its use requires to be prec$$ & prep>$ by t ( ! ,calculus ( indirect ,functions1 :i* is int5d$ to facilitate ! e/abli%m5t ( equations4 ,b1 al?\gh algebra has !n only a second>y (fice to p]=m1 x has none ! less a necess>y "p 9 ! complete solution ( ! "q1 s t ! ,calculus ( direct ,functions mu/ cont9ue to be1 by its nature1 ! fundam5tal base ( all ma!matical analysis4 ,we mu/ !re=e1 be=e go+ any fur!r1 consid] 9 a g5]al mann] ! logical composition ( ?is calculus1 & ! degree ( developm5t to :i* x has at ! pres5t day arriv$4 ,its ,object4 ,! f9al object ( ?is calculus be+ ! resolution 7prop]ly s call$7 ( equations1 t is1 ! discov]y ( ! mann] 9 :i* ! unkn[n quantities are =m$ f ! kn[n quantities1 9 accordance ) ! equations :i* exi/ betwe5 !m1 x naturally pres5ts z _m diff]5t dep>tm5ts z we c conceive truly di/9ct classes ( equations4 ,its appropriate ext5t is consequ5tly rigor\sly indef9ite1 ! numb] ( analytical functions susceptible ( ent]+ into equations be+ 9 itself q unlimit$1 al?\gh !y are compos$ ( only a v small numb] ( primitive elem5ts4 ,classification ( ,equations4 ,! rational classification ( equations mu/ evid5tly be det]m9$ by ! nature ( ! analytical elem5ts ( :i* _! numb]s are compos$2 e o!r classification w\ld be ess5tially arbitr>y4 ,accord+ly1 analy/s beg9 by divid+ equations ) "o or m v>iables into two pr9cipal classes1 accord+ z !y conta9 functions ( only ! fir/ ?ree c\ples 7see ! table 9 *apt] i41 page #ea71 or z !y include also expon5tial or circul> functions4 ,! names ( ,algebraic functions & ,transc5d5tal functions1 commonly giv5 to ^! two pr9cipal gr\ps ( analytical elem5ts1 are und\bt$ly v inappropriate4 ,b ! univ]sally e/abli%$ division betwe5 ! correspond+ equations is none ! less v real 9 ?is s5se1 t ! resolution ( equations conta9+ ! functions call$ transc5d5tal necess>ily pres5ts m difficulties ?an ^? ( ! equations call$ algebraic4 ,h5ce ! /udy ( ! =m] is z yet exce$+ly imp]fect1 s t frequ5tly ! resolution ( ! mo/ simple ( !m is /ill unkn[n to u1#g & our analytical me?ods h almo/ exclusive ref]5ce to ! elaboration ( ! latt]4 ,footnote #g3 ,simple z may seem1 = example1 ! equation ax bx cx1 we d n yet "k h[ to resolve x1 :i* may give "s idea ( ! extreme imp]fection ( ?is "p ( algebra4 ,,algebraic ,,equations4 ,consid]+ n[ only ^! ,algebraic equations1 we mu/ obs]ve1 9 ! fir/ place1 t al?\gh !y may (t5 conta9 irrational functions ( ! unkn[n quantities z well z rational functions1 we c always1 by m or less easy trans=mations1 make ! fir/ case come "u ! second1 s t x is ) ?is la/ t analy/s h _h to occupy !mselves exclusively 9 ord] to resolve all sorts ( algebraic equations4 ,_! ,classification4 ,9 ! infancy ( algebra1 ^! equations 7 class$ accord+ to ! numb] ( _! t]ms4 ,b ?is classification 0 evid5tly faulty1 s9ce x sep>at$ cases :i* 7 really simil>1 & br\ght toge!r o!rs :i* _h no?+ 9 common besides ?is unimportant *>act]i/ic4#h ,x has be5 reta9$ only = equations ) two t]ms1 :i* are1 9 fact1 capable ( be+ resolv$ 9 a mann] peculi> to !mselves4 ,footnote #h3 ,! same error 0 aft]w>d committ$1 9 ! infancy ( ! inf9itesimal calculus1 9 relation to ! integration ( diff]5tial equations4 ,! classification ( equations by :at is call$ _! degrees1 is1 on ! o!r h&1 em95tly natural1 = ?is di/9ction rigor\sly det]m9es ! great] or less difficulty ( _! resolution4 ,?is gradation is app>5t 9 ! cases ( all ! equations :i* c be resolv$2 b x may be indicat$ 9 a g5]al mann] indep5d5tly ( ! fact ( ! resolution4 ,we ne$ only consid] t ! mo/ g5]al equation ( ea* degree necess>ily compreh5ds all ^? ( ! diff]5t inf]ior degrees1 z mu/ also ! =mula :i* det]m9es ! unkn[n quantity4 ,consequ5tly1 h[ev] slight we may suppose ! difficulty peculi> to ! degree :i* we are consid]+1 s9ce x is inevitably complicat$ 9 ! execution ) ^? pres5t$ by all ! prec$+ degrees1 ! resolution really (f]s m & m ob/acles1 9 proportion z ! degree ( ! equation is elevat$4 ,,algebraic ,,resolution ,,( ,,equations4 ,its ,limits4 ,! resolution ( algebraic equations is z yet kn[n to u only 9 ! f\r fir/ degrees1 su* is ! increase ( difficulty notic$ above4 ,9 ?is respect1 algebra has made no consid]able progress s9ce ! lab\rs ( ,desc>tes & ! ,italian analy/s ( ! sixte5? c5tury1 al?\gh 9 ! la/ two c5turies !re has be5 p]haps sc>cely a s+le geomet] :o has n busi$ himself 9 try+ to advance ! resolution ( equations4 ,! g5]al equation ( ! fif? degree itself has ?us f> resi/$ all attacks4 ,! con/antly increas+ complication :i* ! =mulas = resolv+ equations mu/ necess>ily pres5t1 9 proportion z ! degree increases 7! difficulty ( us+ ! =mula ( ! f\r? degree r5d]+ x almo/ inapplicable71 has det]m9$ analy/s to r5\nce1 by a tacit agreem5t1 ! pursuit ( su* rese>*es1 al?\gh !y are f> f reg>d+ x z impossible to obta9 ! resolution ( equations ( ! fif? degree1 & ( sev]al o!r high] ones4 ,g5]al ,solution4 ,! only "q ( ?is k9d :i* w\ld be really ( great importance1 at lea/ 9 its logical relations1 w\ld be ! g5]al resolution ( algebraic equations ( any degree :atsoev]4 ,n[1 ! m we m$itate on ?is subject1 ! m we are l$ to ?9k1 ) ,lagrange1 t x really surpasses ! scope ( our intellig5ce4 ,we mu/ besides obs]ve t ! =mula :i* w\ld express ! root ( an equation ( ! m? degree w\ld necess>ily include radicals ( ! m? ord] 7or functions ( an equival5t multiplicity71 because ( ! m det]m9ations :i* x mu/ admit4 ,s9ce we h se51 besides1 t ?is =mula mu/ also embrace1 z a p>ticul> case1 t =mula :i* corresponds to e l[] degree1 x foll[s t x w\ld inevitably also conta9 radicals ( ! next l[] degree1 ! next l[] to t1 c41 s t1 ev5 if x 7 possible to discov] x1 x w\ld almo/ always pres5t too great a complication to be capable ( be+ usefully employ$1 unless we c\ld succe$ 9 simplify+ x1 at ! same "t reta9+ all its g5]ality1 by ! introduction ( a new class ( analytical elem5ts ( :i* we yet h no idea4 ,we h1 !n1 reason to believe t1 )\t hav+ already "h arriv$ at ! limits impos$ by ! feeble ext5t ( our intellig5ce1 we %\ld n be long 9 rea*+ !m if we actively & e>ne/ly prolong$ ?is s]ies ( inve/igations4 ,x is1 besides1 important to obs]ve t1 ev5 suppos+ we _h obta9$ ! resolution ( algebraic equations ( any degree :atev]1 we w\ld /ill h treat$ only a v small "p ( algebra1 prop]ly s call$1 t is1 ( ! calculus ( direct functions1 includ+ ! resolution ( all ! equations :i* c be =m$ by ! kn[n analytical functions4 ,f9ally1 we mu/ rememb] t1 by an und5iable law ( human nature1 our means = conceiv+ new que/ions be+ mu* m p[]ful ?an our res\rces = resolv+ !m1 or1 9 o!r words1 ! human m9d be+ mu* m ready to inquire ?an to reason1 we %all necess>ily always rema9 bel[ ! difficulty1 no matt] to :at degree ( developm5t our intellectual lab\r may arrive4 ,?us1 ev5 ?\gh we %\ld "s day discov] ! complete resolution ( all ! analytical equations at pres5t kn[n1 *im]ical z ! supposition is1 !re c be no d\bt t1 be=e atta9+ ?is end1 & probably ev5 z a subsidi>y means1 we w\ld h already ov]come ! difficulty 7a mu* small] "o1 ?\gh /ill v great7 ( conceiv+ new analytical elem5ts1 ! introduction ( :i* w\ld give rise to classes ( equations ( :i*1 at pres5t1 we are completely ignorant2 s t a simil> imp]fection 9 algebraic sci5ce w\ld be cont9ually reproduc$1 9 spite ( ! real & v important increase ( ! absolute mass ( our k4 ,:at we "k 9 ,algebra4 ,9 ! pres5t condition ( algebra1 ! complete resolution ( ! equations ( ! fir/ f\r degrees1 ( any b9omial equations1 ( c]ta9 p>ticul> equations ( ! high] degrees1 & ( a v small numb] ( expon5tial1 log>i?mic1 or circul> equations1 con/itute ! fundam5tal me?ods :i* are pres5t$ by ! calculus ( direct functions = ! solution ( ma!matical problems4 ,b1 limit$ z ^! elem5ts are1 geomet]s h nev]!less succe$$ 9 treat+1 9 a truly admirable mann]1 a v great numb] ( important que/ions1 z we %all f9d 9 ! c\rse ( ! volume4 ,! g5]al improvem5ts introduc$ )9 a c5tury into ! total sy/em ( ma!matical analysis1 h _h = _! pr9cipal object to make immeasurably useful ?is little k :i* we h1 in/ead ( t5d+ to increase x4 ,?is result has be5 s fully obta9$1 t mo/ frequ5tly ?is calculus has no real %>e 9 ! complete solution ( ! "q1 except by its mo/ simple p>ts2 ^? :i* h ref]5ce to equations ( ! two fir/ degrees1 ) "o or m v>iables4 ,,num]ical ,,resolution ,,( ,,equations4 ,! extreme imp]fection ( algebra1 ) respect to ! resolution ( equations1 has l$ analy/s to occupy !mselves ) a new class ( que/ions1 ^: true "* %\ld be "h not$4 ,!y h busi$ !mselves 9 fill+ up ! imm5se gap 9 ! resolution ( algebraic equations ( ! high] degrees1 by :at !y h nam$ ! num]ical resolution ( equations4 ,n be+ able to obta91 9 g5]al1 ! =mula :i* expresses :at explicit function ( ! giv5 quantities ! unkn[n "o is1 !y h s\ght 79 ! abs5ce ( ?is k9d ( resolution1 ! only "o really algebraic7 to det]m9e1 indep5d5tly ( t =mula1 at lea/ ! value ( ea* unkn[n quantity1 = v>i\s designat$ sy/ems ( p>ticul> values attribut$ to ! giv5 quantities4 ,by ! successive lab\rs ( analy/s1 ?is incomplete & illegitimate op]ation1 :i* pres5ts an intimate mixture ( truly algebraic que/ions ) o!rs :i* are purely ari?metical1 has be5 r5d]$ possible 9 all cases = equations ( any degree & ev5 ( any =m4 ,! me?ods = ?is :i* we n[ possess are suffici5tly g5]al1 al?\gh ! calculations to :i* !y lead are (t5 s complicat$ z to r5d] x almo/ impossible to execute !m4 ,we h no?+ else to d1 !n1 9 ?is "p ( algebra1 b to simplify ! me?ods suffici5tly to r5d] !m regul>ly applicable1 :i* we may hope h]eaft] to effect4 ,9 ?is condition ( ! calculus ( direct functions1 we endeav\r1 9 its application1 s to dispose ! propos$ que/ions z f9ally to require only ?is num]ical resolution ( ! equations4 ,its limit$ ,usefulness4 ,valuable z is su* a res\rce 9 ! abs5ce ( ! v]itable solution1 x is ess5tial n to misconceive ! true "* ( ^! me?ods1 :i* analy/s rightly reg>d z a v imp]fect algebra4 ,9 fact1 we are f> f be+ always able to r$uce our ma!matical que/ions to dep5d f9ally ^u only ! num]ical resolution ( equations2 t c be done only = que/ions q isolat$ or truly f9al1 t is1 = ! smalle/ numb]4 ,mo/ que/ions1 9 fact1 are only prep>atory1 & int5d$ to s]ve z an indisp5sable prep>ation = ! solution ( o!r que/ions4 ,n[1 = su* an object1 x is evid5t t x is n ! actual value ( ! unkn[n quantity :i* x is important to discov]1 b ! =mula1 :i* %[s h[ x is d]iv$ f ! o!r quantities "u consid]ation4 ,x is ?is :i* happ5s1 = example1 9 a v ext5sive class ( cases1 :5ev] a c]ta9 "q includes at ! same "t sev]al unkn[n quantities4 ,we h !n1 fir/ ( all1 to sep>ate !m4 ,by suitably employ+ ! simple & g5]al me?od s happily inv5t$ by analy/s1 & :i* consi/s 9 ref]r+ all ! o!r unkn[n quantities to "o ( !m1 ! difficulty w\ld always disappe> if we knew h[ to obta9 ! algebraic resolution ( ! equations "u consid]ation1 :ile ! num]ical solution w\ld !n be p]fectly useless4 ,x is only = want ( kn[+ ! algebraic resolution ( equations ) a s+le unkn[n quantity1 t we are oblig$ to treat ,elim9ation z a di/9ct "q1 :i* =ms "o ( ! greate/ special difficulties ( common algebra4 ,labori\s z are ! me?ods by ! aid ( :i* we ov]come ?is difficulty1 !y are n ev5 applicable1 9 an entirely g5]al mann]1 to ! elim9ation ( "o unkn[n quantity betwe5 two equations ( any =m :atev]4 ,9 ! mo/ simple que/ions1 & :5 we h really to resolve only a s+le equation ) a s+le unkn[n quantity1 ?is num]ical resolution is none ! less a v imp]fect me?od1 ev5 :5 x is /rictly suffici5t4 ,x pres5ts1 9 fact1 ?is s]i\s inconv5i5ce ( oblig+ u to repeat ! :ole s]ies ( op]ations = ! slighte/ *ange :i* may take place 9 a s+le "o ( ! quantities consid]$1 al?\gh _! relations to "o ano!r rema9 un*ang$2 ! calculations made = "o case n enabl+ u to disp5se ) any ( ^? :i* relate to a case v slightly diff]5t4 ,?is happ5s because ( our inability to ab/ract & treat sep>ately t purely algebraic "p ( ! "q :i* is common to all ! cases :i* result f ! m]e v>iation ( ! giv5 numb]s4 ,accord+ to ! prec$+ consid]ations1 ! calculus ( direct functions1 view$ 9 its pres5t /ate1 divides into two v di/9ct bran*es1 accord+ z its subject is ! algebraic resolution ( equations or _! num]ical resolution4 ,! fir/ dep>tm5t1 ! only "o truly satisfactory1 is unhappily v limit$1 & w probably always rema9 s2 ! second1 too (t5 insuffici5t1 has1 at lea/1 ! advantage ( a mu* great] g5]ality4 ,! necessity ( cle>ly di/+ui%+ ^! two p>ts is evid5t1 because ( ! ess5tially diff]5t object propos$ 9 ea*1 & consequ5tly ! peculi> po9t ( view "u :i* quantities are !re9 consid]$4 ,diff]5t ,divisions ( ! two ,me?ods ( ,resolution4 ,if1 moreov]1 we consid] ^! p>ts ) ref]5ce to ! diff]5t me?ods ( :i* ea* is compos$1 we f9d 9 _! logical di/ribution an entirely diff]5t arrangem5t4 ,9 fact1 ! fir/ "p mu/ be divid$ accord+ to ! nature ( ! equations :i* we are able to resolve1 & indep5d5tly ( e consid]ation relative to ! values ( ! unkn[n quantities4 ,9 ! second "p1 on ! contr>y1 x is n accord+ to ! degrees ( ! equations t ! me?ods are naturally di/+ui%$1 s9ce !y are applicable to equations ( any degree :atev]2 x is accord+ to ! num]ical "* ( ! values ( ! unkn[n quantities2 =1 9 calculat+ ^! numb]s directly1 )\t d$uc+ !m f g5]al =mulas1 diff]5t means w\ld evid5tly be employ$ :5 ! numb]s are n susceptible ( hav+ _! values det]m9$ o!rwise ?an by a s]ies ( approximations1 always incomplete1 or :5 !y c be obta9$ ) entire exactness4 ,?is di/9ction ( incomm5surable & ( comm5surable roots1 :i* require q diff]5t pr9ciples = _! det]m9ation1 important z x is 9 ! num]ical resolution ( equations1 is entirely insignificant 9 ! algebraic resolution1 9 :i* ! rational or irrational nature ( ! numb]s :i* are obta9$ is a m]e accid5t ( ! calculation1 :i* _c ex]cise any influ5ce ov] ! me?ods employ$2 x is1 9 a ^w1 a simple ari?metical consid]ation4 ,we may say z mu*1 ?\gh 9 a less degree1 ( ! division ( ! comm5surable roots !mselves into entire & fractional4 ,9 f9e1 ! case is ! same1 9 a /ill great] degree1 ) ! mo/ g5]al classification ( roots1 z real & imag9>y4 ,all ^! diff]5t consid]ations1 :i* are prepond]ant z to ! num]ical resolution ( equations1 & :i* are ( no importance 9 _! algebraic resolution1 r5d] m & m s5sible ! ess5tially di/9ct nature ( ^! two pr9cipal p>ts ( algebra4 ,,! ,,!ory ,,( ,,equations4 ,^! two dep>tm5ts1 :i* con/itute ! imm$iate object ( ! calculus ( direct functions1 are subord9ate to a ?ird "o1 purely speculative1 f :i* bo? ( !m borr[ _! mo/ p[]ful res\rces1 & :i* has be5 v exactly designat$ by ! g5]al "n ( ,!ory ( ,equations1 al?\gh x z yet relates only to ,algebraic equations4 ,! num]ical resolution ( equations1 because ( its g5]ality1 has special ne$ ( ?is rational f\ndation4 ,?is la/ & important bran* ( algebra is naturally divid$ into two ord]s ( que/ions1 viz41 ^? :i* ref] to ! composition ( equations1 & ^? :i* conc]n _! trans=mation2 ^! latt] hav+ = _! object to modify ! roots ( an equation )\t kn[+ !m1 9 accordance ) any giv5 law1 provid+ t ?is law is uni=m 9 relation to all ! p>ts4#i ,footnote #i3 ,! fundam5tal pr9ciple on :i* reposes ! !ory ( equations1 & :i* is s frequ5tly appli$ 9 all ma!matical analysis--! decomposition ( algebraic1 rational1 & entire functions1 ( any degree :atev]1 into factors ( ! fir/ degree--is nev] employ$ except = functions ( a s+le v>iable1 )\t any "o hav+ exam9$ if x ought to be ext5d$ to functions ( sev]al v>iables4 ,! g5]al impossibility ( su* a decomposition is demon/rat$ by ! au?or 9 detail1 b m prop]ly belongs to a special treatise4 ,,! ,,me?od ,,( ,,indet]m9ate ,,coeffici5ts4 ,to complete ?is rapid g5]al enum]ation ( ! diff]5t ess5tial p>ts ( ! calculus ( direct functions1 ,i mu/1 la/ly1 m5tion expressly "o ( ! mo/ fruitful & important !ories ( algebra prop]1 t relat+ to ! trans=mation ( functions into s]ies by ! aid ( :at is call$ ! ,me?od ( indet]m9ate ,coeffici5ts4 ,?is me?od1 s em95tly analytical1 & :i* mu/ be reg>d$ z "o ( ! mo/ rem>kable discov]ies ( ,desc>tes1 has und\bt$ly lo/ "s ( its importance s9ce ! inv5tion & ! developm5t ( ! inf9itesimal calculus1 ! place ( :i* x might s happily take 9 "s p>ticul> respects4 ,b ! increas+ ext5sion ( ! transc5d5tal analysis1 al?\gh x has r5d]$ ?is me?od mu* less necess>y1 has1 on ! o!r h&1 multipli$ its applications & enl>g$ its res\rces2 s t by ! useful comb9ation betwe5 ! two !ories1 :i* has f9ally be5 effect$1 ! use ( ! me?od ( indet]m9ate coeffici5ts has become at pres5t mu* m ext5sive ?an x 0 ev5 be=e ! =mation ( ! calculus ( indirect functions4 99 99 99 99 99 ,hav+ ?us sket*$ ! g5]al outl9es ( algebra prop]1 ,i h n[ to (f] "s consid]ations on sev]al lead+ po9ts 9 ! calculus ( direct functions1 our ideas ( :i* may be advantage\sly made m cle> by a philosophical exam9ation4 ,,imag9>y ,,quantities4 ,! difficulties connect$ ) sev]al peculi> symbols to :i* algebraic calculations sometimes lead1 & especially to ! expressions call$ imag9>y1 h be51 ,i ?9k1 mu* exagg]at$ ?r\gh purely metaphysical consid]ations1 :i* h be5 =c$ ^u !m1 9 ! place ( reg>d+ ^! abnormal results 9 _! true po9t ( view z simple analytical facts4 ,view+ !m ?us1 we readily see t1 s9ce ! _s ( ma!matical analysis consi/s 9 consid]+ magnitudes 9 ref]5ce to _! relations only1 & )\t any reg>d to _! det]m9ate value1 analy/s are oblig$ to admit indiff]5tly e k9d ( expression :i* c be eng5d]$ by algebraic comb9ations4 ,! int]diction ( ev5 "o expression because ( its app>5t s+ul>ity w\ld de/roy ! g5]ality ( _! conceptions4 ,! common emb>rassm5t on ?is subject seems to me to proce$ ess5tially f an unconsci\s confusion betwe5 ! idea ( function & ! idea ( value1 or1 :at comes to ! same ?+1 betwe5 ! algebraic & ! ari?metical po9t ( view4 ,a ?or\gh exam9ation w\ld %[ ma!matical analysis to be mu* m cle> 9 its nature ?an ev5 ma!maticians commonly suppose4 ,,negative ,,quantities4 ,z to negative quantities1 :i* h giv5 rise to s _m misplac$ discussions1 z irrational z useless1 we mu/ di/+ui% betwe5 _! ab/ract signification & _! concrete int]pretation1 :i* h be5 almo/ always conf\nd$ up to ! pres5t day4 ,"u ! fir/ po9t ( view1 ! !ory ( negative quantities c be e/abli%$ 9 a complete mann] by a s+le algebraical consid]ation4 ,! necessity ( admitt+ su* expressions is ! same z = imag9>y quantities1 z above indicat$2 & _! employm5t z an analytical artifice1 to r5d] ! =mulas m compreh5sive1 is a me*anism ( calculation :i* _c really give rise to any s]i\s difficulty4 ,we may !re=e reg>d ! ab/ract !ory ( negative quantities z leav+ no?+ ess5tial to desire2 x pres5ts no ob/acles b ^? inappropriately introduc$ by sophi/ical consid]ations4 ,x is f> f be+ s1 h[ev]1 ) _! concrete !ory4 ,?is consi/s ess5tially 9 t admirable prop]ty ( ! signs &-1 ( repres5t+ analytically ! oppositions ( directions ( :i* c]ta9 magnitudes are susceptible4 ,?is g5]al !orem on ! relation ( ! concrete to ! ab/ract 9 ma!matics is "o ( ! mo/ beautiful discov]ies :i* we owe to ! g5ius ( ,desc>tes1 :o obta9$ x z a simple result ( prop]ly direct$ philosophical obs]vation4 ,a great numb] ( geomet]s h s9ce /riv5 to e/abli% directly its g5]al demon/ration1 b ?us f> _! ef=ts h be5 illusory4 ,_! va9 metaphysical consid]ations & het]og5e\s m+l+s ( ! ab/ract & ! concrete h s confus$ ! subject1 t x becomes necess>y to "h di/9ctly enunciate ! g5]al fact4 ,x consi/s 9 ?is3 if1 9 any equation :atev]1 express+ ! relation ( c]ta9 quantities :i* are susceptible ( opposition ( directions1 "o or m ( ^? quantities come to be reckon$ 9 a direction contr>y to t :i* belong$ to !m :5 ! equation 0 fir/ e/abli%$1 x w n be necess>y to =m directly a new equation = ?is second /ate ( ! ph5om5a2 x w suffice to *ange1 9 ! fir/ equation1 ! sign ( ea* ( ! quantities :i* %all h *ang$ its direction2 & ! equation1 ?us modifi$1 w always rigor\sly co9cide ) t :i* we w\ld h arriv$ at 9 recomm5c+ to inve/igate1 = ?is new case1 ! analytical law ( ! ph5om5on4 ,! g5]al !orem consi/s 9 ?is con/ant & necess>y co9cid5ce4 ,n[1 z yet1 no "o has succe$$ 9 directly prov+ ?is2 we h assur$ ourselves ( x only by a great numb] ( geometrical & me*anical v]ifications1 :i* are1 x is true1 suffici5tly multipli$1 & especially suffici5tly v>i$1 to prev5t any cle> m9d f hav+ ! lea/ d\bt ( ! exactitude & ! g5]ality ( ?is ess5tial prop]ty1 b :i*1 9 a philosophical po9t ( view1 d n at all disp5se ) ! rese>* = s important an explanation4 ,! extreme ext5t ( ! !orem mu/ make u compreh5d bo? ! fundam5tal difficulties ( ?is rese>* & ! high utility = ! p]fect+ ( ma!matical sci5ce :i* w\ld belong to ! g5]al conception ( ?is great tru?4 ,?is imp]fection ( !ory1 h[ev]1 has n prev5t$ geomet]s f mak+ ! mo/ ext5sive & ! mo/ important use ( ?is prop]ty 9 all p>ts ( concrete ma!matics4 ,x foll[s f ! above g5]al enunciation ( ! fact1 indep5d5tly ( any demon/ration1 t ! prop]ty ( :i* we speak mu/ nev] be appli$ to magnitudes ^: directions are cont9ually v>y+1 )\t giv+ rise to a simple opposition ( direction2 9 t case1 ! sign ) :i* e result ( calculation is necess>ily affect$ is n susceptible ( any concrete int]pretation1 & ! attempts sometimes made to e/abli% "o are errone\s4 ,?is circum/ance occurs1 among o!r occasions1 9 ! case ( a radius vector 9 geometry1 & div]g+ =ces 9 me*anics4 ,,pr9ciple ,,( ,,homog5eity4 ,a second g5]al !orem on ! relation ( ! concrete to ! ab/ract is t :i* is ord9>ily designat$ "u ! "n ( ,pr9ciple ( ,homog5eity4 ,x is und\bt$ly mu* less important 9 its applications ?an ! prec$+1 b x p>ticul>ly m]its our att5tion z hav+1 by its nature1 a /ill great] ext5t1 s9ce x is applicable to all ph5om5a )\t di/9ction1 & because ( ! real utility :i* x (t5 possesses = ! v]ification ( _! analytical laws4 ,i c1 moreov]1 exhibit a direct & g5]al demon/ration ( x :i* seems to me v simple4 ,x is f\nd$ on ?is s+le obs]vation1 :i* is self-evid5t1 t ! exactitude ( e relation betwe5 any concrete magnitudes :atsoev] is indep5d5t ( ! value ( ! units to :i* !y are ref]r$ = ! purpose ( express+ !m 9 numb]s4 ,= example1 ! relation :i* exi/s betwe5 ! ?ree sides ( a "r-angl$ triangle is ! same1 :e!r !y are measur$ by y>ds1 or by miles1 or by in*es4 ,x foll[s f ?is g5]al consid]ation1 t e equation :i* expresses ! analytical law ( any ph5om5on mu/ possess ?is prop]ty ( be+ 9 no way alt]$1 :5 all ! quantities :i* are f\nd 9 x are made to und]go simultane\sly ! *ange correspond+ to t :i* _! respective units w\ld exp]i5ce4 ,n[ ?is *ange evid5tly consi/s 9 all ! quantities ( ea* sort becom+ at once m times small]1 if ! unit :i* corresponds to !m becomes m times great]1 or reciprocally4 ,?us e equation :i* repres5ts any concrete relation :atev] mu/ possess ?is *>act]i/ic ( rema9+ ! same1 :5 we make m times great] all ! quantities :i* x conta9s1 & :i* express ! magnitudes betwe5 :i* ! relation exi/s2 except+ always ! numb]s :i* designate simply ! mutual ratios ( ^! diff]5t magnitudes1 & :i* !re=e rema9 inv>iable dur+ ! *ange ( ! units4 ,x is ?is prop]ty :i* con/itutes ! law ( ,homog5eity 9 its mo/ ext5d$ signification1 t is1 ( :atev] analytical functions ! equations may be compos$4 ,b mo/ frequ5tly we consid] only ! cases 9 :i* ! functions are su* z are call$ algebraic1 & to :i* ! idea ( degree is applicable4 ,9 ?is case we c give m precision to ! g5]al proposition by det]m9+ ! analytical "* :i* mu/ be necess>ily pres5t$ by ! equation1 9 ord] t ?is prop]ty may be v]ifi$4 ,x is easy to see1 !n1 t1 by ! modification j expla9$1 all ! t]ms ( ! fir/ degree1 :atev] may be _! =m1 rational or irrational1 entire or fractional1 w become m times great]2 all ^? ( ! second degree1 m times2 ^? ( ! ?ird1 m times1 c4 ,?us ! t]ms ( ! same degree1 h[ev] diff]5t may be _! composition1 v>y+ 9 ! same mann]1 & ! t]ms ( diff]5t degrees v>y+ 9 an unequal proportion1 :atev] simil>ity !re may be 9 _! composition1 x w be necess>y1 to prev5t ! equation f be+ di/urb$1 t all ! t]ms :i* x conta9s %\ld be ( ! same degree4 ,x is 9 ?is t prop]ly consi/s ! ord9>y !orem ( ,homog5eity1 & x is f ?is circum/ance t ! g5]al law has d]iv$ its "n1 :i*1 h[ev]1 ceases to be exactly prop] = all o!r functions4 ,9 ord] to treat ?is subject 9 its :ole ext5t1 x is important to obs]ve an ess5tial condition1 to :i* att5tion mu/ be paid 9 apply+ ?is prop]ty :5 ! ph5om5on express$ by ! equation pres5ts magnitudes ( diff]5t natures4 ,?us x may happ5 t ! respective units are completely indep5d5t ( ea* o!r1 & !n ! !orem ( ,homog5eity w hold good1 ei!r ) ref]5ce to all ! correspond+ classes ( quantities1 or ) reg>d to only a s+le "o or m ( !m4 ,b x w happ5 on o!r occasions t ! diff]5t units w h fix$ relations to "o ano!r1 det]m9$ by ! nature ( ! "q2 !n x w be necess>y to pay att5tion to ?is subord9ation ( ! units 9 v]ify+ ! homog5eity1 :i* w n exi/ any long] 9 a purely algebraic s5se1 & ! precise =m ( :i* w v>y accord+ to ! nature ( ! ph5om5a4 ,?us1 = example1 to fix our ideas1 :51 9 ! analytical expression ( geometrical ph5om5a1 we are consid]+ at once l9es1 areas1 & volumes1 x w be necess>y to obs]ve t ! ?ree correspond+ units are necess>ily s connect$ ) ea* o!r t1 accord+ to ! subord9ation g5]ally e/abli%$ 9 t respect1 :5 ! fir/ becomes m times great]1 ! second becomes m times1 & ! ?ird m times4 ,x is ) su* a modification t homog5eity w exi/ 9 ! equations1 9 :i*1 if !y are algebraic1 we w h to e/imate ! degree ( ea* t]m by d\bl+ ! expon5ts ( ! factors :i* correspond to areas1 & tripl+ ^? ( ! factors relat+ to volumes4 99 99 99 99 99 ,su* are ! pr9cipal g5]al consid]ations relat+ to ! ,calculus ( ,direct ,functions4 ,we h n[ to pass to ! philosophical exam9ation ( ! ,calculus ( ,indirect ,functions1 ! mu* sup]ior importance & ext5t ( :i* claim a full] developm5t4 ,,*apt] ,,iii4 ,,transc5d5tal ,,analysis3 ,,diff]5t ,,modes ,,( ,,view+ ,,x4 ,we det]m9$1 9 ! second *apt]1 ! philosophical "* ( ! transc5d5tal analysis1 9 :atev] mann] x may be conceiv$1 consid]+ only ! g5]al nature ( its actual de/9ation z a "p ( ma!matical sci5ce4 ,?is analysis has be5 pres5t$ by geomet]s "u sev]al po9ts ( view1 really di/9ct1 al?\gh necess>ily equival5t1 & lead+ always to id5tical results4 ,!y may be r$uc$ to ?ree pr9cipal ones2 ^? ( ,,leibnitz1 ( ,,newton1 & ( ,,lagrange1 ( :i* all ! o!rs are only second>y modifications4 ,9 ! pres5t /ate ( sci5ce1 ea* ( ^! ?ree g5]al conceptions (f]s ess5tial advantages :i* p]ta9 to x exclusively1 )\t our hav+ yet succe$$ 9 con/ruct+ a s+le me?od unit+ all ^! diff]5t *>act]i/ic qualities4 ,?is comb9ation w probably be h]eaft] effect$ by "s me?od f\nd$ ^u ! conception ( ,lagrange :5 t important philosophical lab\r %all h be5 accompli%$1 ! /udy ( ! o!r conceptions w h only a hi/oric int]e/2 b1 until !n1 ! sci5ce mu/ be consid]$ z 9 only a provisional /ate1 :i* requires ! simultane\s consid]ation ( all ! v>i\s modes ( view+ ?is calculus4 ,illogical z may appe> ?is multiplicity ( conceptions ( "o id5tical subject1 /ill1 )\t !m all1 we c\ld =m b a v insuffici5t idea ( ?is analysis1 :e!r 9 itself1 or m especially 9 relation to its applications4 ,?is want ( sy/em 9 ! mo/ important "p ( ma!matical analysis w n appe> /range if we consid]1 on ! "o h&1 its great ext5t & its sup]ior difficulty1 &1 on ! o!r1 its rec5t =mation4 ,,its ,,e>ly ,,hi/ory4 ,if we _h to trace "h ! sy/ematic hi/ory ( ! successive =mation ( ! transc5d5tal analysis1 x w\ld be necess>y previ\sly to di/+ui% c>efully f ! calculus ( indirect functions1 prop]ly s call$1 ! orig9al idea ( ! inf9itesimal me?od1 :i* c be conceiv$ by itself1 indep5d5tly ( any calculus4 ,we %\ld see t ! fir/ g]m ( ?is idea is f\nd 9 ! proc$ure con/antly employ$ by ! ,greek geomet]s1 "u ! "n ( ! ,me?od ( ,exhau/ions1 z a means ( pass+ f ! prop]ties ( /raight l9es to ^? ( curves1 & consi/+ ess5tially 9 sub/itut+ = ! curve ! auxili>y consid]ation ( an inscrib$ or circumscrib$ polygon1 by means ( :i* !y rose to ! curve itself1 tak+ 9 a suitable mann] ! limits ( ! primitive ratios4 ,inconte/able z is ?is filiation ( ideas1 x w\ld be giv+ x a greatly exagg]at$ importance to see 9 ?is me?od ( exhau/ions ! real equival5t ( our mod]n me?ods1 z "s geomet]s h done2 = ! anci5ts _h no logical & g5]al means = ! det]m9ation ( ^! limits1 & ?is 0 commonly ! greate/ difficulty ( ! "q2 s t _! solutions 7 n subject$ to ab/ract & inv>iable rules1 ! uni=m application ( :i* w\ld lead ) c]ta9ty to ! k s\ght2 :i* is1 on ! contr>y1 ! pr9cipal *>act]i/ic ( our transc5d5tal analysis4 ,9 a ^w1 !re /ill rema9$ ! task ( g5]aliz+ ! conceptions us$ by ! anci5ts1 &1 m especially1 by consid]+ x 9 a mann] purely ab/ract1 ( r$uc+ x to a complete sy/em ( calculation1 :i* to !m 0 impossible4 ,! fir/ idea :i* 0 produc$ 9 ?is new direction goes back to ! great geomet] ,f]mat1 :om ,lagrange has ju/ly pres5t$ z hav+ block$ out ! direct =mation ( ! transc5d5tal analysis by 8 me?od = ! det]m9ation ( maxima & m9ima1 & = ! f9d+ ( tang5ts1 :i* consi/$ ess5tially 9 introduc+ ! auxili>y consid]ation ( ! correlative increm5ts ( ! propos$ v>iables1 increm5ts aft]w>d suppress$ z equal to z]o :5 ! equations _h und]gone c]ta9 suitable trans=mations4 ,b1 al?\gh ,f]mat 0 ! fir/ to conceive ?is analysis 9 a truly ab/ract mann]1 x 0 yet f> f be+ regul>ly =m$ into a g5]al & di/9ct calculus hav+ its own notation1 & especially fre$ f ! sup]flu\s consid]ation ( t]ms :i*1 9 ! analysis ( ,f]mat1 7 f9ally n tak5 into ! acc\nt1 aft] hav+ nev]!less greatly complicat$ all ! op]ations by _! pres5ce4 ,?is is :at ,leibnitz s happily execut$1 half a c5tury lat]1 aft] "s int]m$iate modifications ( ! ideas ( ,f]mat introduc$ by ,wallis1 & /ill m by ,b>r[2 & he has ?us be5 ! true creator ( ! transc5d5tal analysis1 su* z we n[ employ x4 ,?is admirable discov]y 0 s ripe 7l all ! great conceptions ( ! human intellect at ! mom5t ( _! manife/ation71 t ,newton1 on 8 side1 _h arriv$1 at ! same "t1 or a little e>li]1 at a me?od exactly equival5t1 by consid]+ ?is analysis "u a v diff]5t po9t ( view1 :i*1 al?\gh m logical 9 itself1 is really less adapt$ to give to ! common fundam5tal me?od all ! ext5t & ! facility :i* h be5 imp>t$ to x by ! ideas ( ,leibnitz4 ,f9ally1 ,lagrange1 putt+ aside ! het]og5e\s consid]ations :i* _h guid$ ,leibnitz & ,newton1 has succe$$ 9 r$uc+ ! transc5d5tal analysis1 9 its greate/ p]fection1 to a purely algebraic sy/em1 :i* only wants m aptitude = its practical applications4 ,aft] ?is summ>y glance at ! g5]al hi/ory ( ! transc5d5tal analysis1 we w proce$ to ! dogmatic exposition ( ! ?ree pr9cipal conceptions1 9 ord] to appreciate exactly _! *>act]i/ic prop]ties1 & to %[ ! necess>y id5tity ( ! me?ods :i* are !nce d]iv$4 ,let u beg9 ) t ( ,leibnitz4 ,,me?od ,,( ,,leibnitz4 ,inf9itely small ,elem5ts4 ,?is consi/s 9 introduc+ into ! calculus1 9 ord] to facilitate ! e/abli%m5t ( equations1 ! inf9itely small elem5ts ( :i* all ! quantities1 ! relations betwe5 :i* are s\ght1 are consid]$ to be compos$4 ,^! elem5ts or diff]5tials w h c]ta9 relations to "o ano!r1 :i* are con/antly & necess>ily m simple & easy to discov] ?an ^? ( ! primitive quantities1 & by means ( :i* we w be enabl$ 7by a special calculus hav+ = its peculi> object ! elim9ation ( ^! auxili>y inf9itesimals7 to g back to ! desir$ equations1 :i* x w\ld h be5 mo/ frequ5tly impossible to obta9 directly4 ,?is indirect analysis may h diff]5t degrees ( indirectness2 =1 :5 !re is too mu* difficulty 9 =m+ imm$iately ! equation betwe5 ! diff]5tials ( ! magnitudes "u consid]ation1 a second application ( ! same g5]al artifice w h to be made1 & ^! diff]5tials be treat$1 9 _! turn1 z new primitive quantities1 & a relation be s\ght betwe5 _! inf9itely small elem5ts 7:i*1 ) ref]5ce to ! f9al objects ( ! "q1 w be second diff]5tials71 & s on2 ! same trans=mation admitt+ ( be+ repeat$ any numb] ( times1 on ! condition ( f9ally elim9at+ ! con/antly increas+ numb] ( inf9itesimal quantities introduc$ z auxili>ies4 ,a p]son n yet famili> ) ^! consid]ations does n p]ceive at once h[ ! employm5t ( ^! auxili>y quantities c facilitate ! discov]y ( ! analytical laws ( ph5om5a2 = ! inf9itely small increm5ts ( ! propos$ magnitudes be+ ( ! same species ) !m1 x w\ld seem t _! relations %\ld n be obta9$ ) m ease1 inasmu* z ! great] or less value ( a quantity _c1 9 fact1 ex]cise any influ5ce on an inquiry :i* is necess>ily indep5d5t1 by its nature1 ( e idea ( value4 ,b x is easy1 nev]!less1 to expla9 v cle>ly1 & 9 a q g5]al mann]1 h[ f> ! "q mu/ be simplifi$ by su* an artifice4 ,= ?is purpose1 x is necess>y to beg9 by di/+ui%+ diff]5t ord]s ( inf9itely small quantities1 a v precise idea ( :i* may be obta9$ by consid]+ !m z be+ ei!r ! successive p[]s ( ! same primitive inf9itely small quantity1 or z be+ quantities :i* may be reg>d$ z hav+ f9ite ratios ) ^! p[]s2 s t1 to take an example1 ! second1 ?ird1 c41 diff]5tials ( any "o v>iable are class$ z inf9itely small quantities ( ! second ord]1 ! ?ird1 c41 because x is easy to discov] 9 !m f9ite multiples ( ! second1 ?ird1 c41 p[]s ( a c]ta9 fir/ diff]5tial4 ,^! prelim9>y ideas be+ e/abli%$1 ! _s ( ! inf9itesimal analysis consi/s 9 con/antly neglect+ ! inf9itely small quantities 9 comp>ison ) f9ite quantities1 & g5]ally ! inf9itely small quantities ( any ord] :atev] 9 comp>ison ) all ^? ( an inf]ior ord]4 ,x is at once app>5t h[ mu* su* a lib]ty mu/ facilitate ! =mation ( equations betwe5 ! diff]5tials ( quantities1 s9ce1 9 ! place ( ^! diff]5tials1 we c sub/itute su* o!r elem5ts z we may *oose1 & z w be m simple to consid]1 only tak+ c>e to con=m to ?is s+le condition1 t ! new elem5ts diff] f ! prec$+ ones only by quantities inf9itely small 9 comp>ison ) !m4 ,x is ?us t x w be possible1 9 geometry1 to treat curv$ l9es z compos$ ( an inf9ity ( rectil9e> elem5ts1 curv$ surfaces z =m$ ( plane elem5ts1 &1 9 me*anics1 v>iable motions z an inf9ite s]ies ( uni=m motions1 succe$+ "o ano!r at inf9itely small int]vals ( "t4 ,,examples4 ,consid]+ ! importance ( ?is admirable conception1 ,i ?9k t ,i ought "h to complete ! illu/ration ( its fundam5tal "* by ! summ>y indication ( "s lead+ examples4 #a4 ,tang5ts4 ,let x be requir$ to det]m9e1 = ea* po9t ( a plane curve1 ! equation ( :i* is giv51 ! direction ( its tang5t2 a "q ^: g5]al solution 0 ! primitive object ( ! inv5tors ( ! transc5d5tal analysis4 ,we w consid] ! tang5t z a secant jo9+ two po9ts inf9itely ne> to ea* o!r2 & !n1 designat+ by dy & dx ! inf9itely small diff]5ces ( ! co-ord9ates ( ^? two po9ts1 ! elem5t>y pr9ciples ( geometry w imm$iately give ! equation t dy_/dx = ! trigonometrical tang5t ( ! angle :i* is made ) ! axis ( ! abscissas by ! desir$ tang5t1 ?is be+ ! mo/ simple way ( fix+ its position 9 a sy/em ( rectil9e> co-ord9ates4 ,?is equation1 common to all curves1 be+ e/abli%$1 ! "q is r$uc$ to a simple analytical problem1 :i* w consi/ 9 elim9at+ ! inf9itesimals dx & dy1 :i* 7 introduc$ z auxili>ies1 by det]m9+ 9 ea* p>ticul> case1 by means ( ! equation ( ! propos$ curve1 ! ratio ( dy to dx1 :i* w be con/antly done by uni=m & v simple me?ods4 #b4 ,rectification ( an ,arc4 ,9 ! second place1 suppose t we wi% to "k ! l5g? ( ! arc ( any curve1 consid]$ z a function ( ! co-ord9ates ( its extremities4 ,x w\ld be impossible to e/abli% directly ! equation betwe5 ?is arc s & ^! co-ord9ates1 :ile x is easy to f9d ! correspond+ relation betwe5 ! diff]5tials ( ^! diff]5t magnitudes4 ,! mo/ simple !orems ( elem5t>y geometry w 9 fact give at once1 consid]+ ! inf9itely small arc ds z a "r l9e1 ! equations ds dy dx1 or ds dx dy dz1 accord+ z ! curve is ( s+le or d\ble curvature4 ,9 ei!r case1 ! "q is n[ entirely )9 ! doma9 ( analysis1 :i*1 by ! elim9ation ( ! diff]5tials 7:i* is ! peculi> object ( ! calculus ( indirect functions71 w c>ry u back f ?is relation to t :i* exi/s betwe5 ! f9ite quantities !mselves "u exam9ation4 #c4 ,quadrature ( a ,curve4 ,x w\ld be ! same ) ! quadrature ( curvil9e> areas4 ,if ! curve is a plane "o1 & ref]r$ to rectil9e> co-ord9ates1 we w conceive ! area ,a compris$ betwe5 ?is curve1 ! axis ( ! abscissas1 & two extreme co-ord9ates1 to increase by an inf9itely small quantity d,a1 z ! result ( a correspond+ increm5t ( ! abscissa4 ,! relation betwe5 ^! two diff]5tials c be imm$iately obta9$ ) ! greate/ facility by sub/itut+ = ! curvil9e> elem5t ( ! propos$ area ! rectangle =m$ by ! extreme ord9ate & ! elem5t ( ! abscissa1 f :i* x evid5tly diff]s only by an inf9itely small quantity ( ! second ord]4 ,?is w at once give1 :atev] may be ! curve1 ! v simple diff]5tial equation d,a ydx1 f :i*1 :5 ! curve is def9$1 ! calculus ( indirect functions w %[ h[ to d$uce ! f9ite equation1 :i* is ! imm$iate object ( ! problem4 #d4 ,velocity 9 ,v>iable ,motion4 ,9 l mann]1 9 ,dynamics1 :5 we desire to "k ! expression = ! velocity acquir$ at ea* in/ant by a body impress$ ) a motion v>y+ accord+ to any law1 we w consid] ! motion z be+ uni=m dur+ an inf9itely small elem5t ( ! "t t1 & we w ?us imm$iately =m ! diff]5tial equation de vdt1 9 :i* v designates ! velocity acquir$ :5 ! body has pass$ ov] ! space e2 & !nce x w be easy to d$uce1 by simple & inv>iable analytical proc$ures1 ! =mula :i* w\ld give ! velocity 9 ea* p>ticul> motion1 9 accordance ) ! correspond+ relation betwe5 ! "t & ! space2 or1 reciprocally1 :at ?is relation w\ld be if ! mode ( v>iation ( ! velocity 0 suppos$ to be kn[n1 :e!r ) respect to ! space or to ! "t4 #e4 ,di/ribution ( ,heat4 ,la/ly1 to indicate ano!r k9d ( que/ions1 x is by simil> /eps t we are able1 9 ! /udy ( !rmological ph5om5a1 accord+ to ! happy conception ( ,m4 ,f\ri]1 to =m 9 a v simple mann] ! g5]al diff]5tial equation :i* expresses ! v>iable di/ribution ( heat 9 any body :atev]1 subject$ to any influ5ces1 by means ( ! s+le & easily-obta9$ relation1 :i* repres5ts ! uni=m di/ribution ( heat 9 a "r-angl$ p>allelopip$on1 consid]+ 7geometrically7 e o!r body z decompos$ into inf9itely small elem5ts ( a simil> =m1 & 7!rmologically7 ! fl[ ( heat z con/ant dur+ an inf9itely small elem5t ( "t4 ,h5ce=?1 all ! que/ions :i* c be pres5t$ by ab/ract !rmology w be r$uc$1 z 9 geometry & me*anics1 to m]e difficulties ( analysis1 :i* w always consi/ 9 ! elim9ation ( ! diff]5tials introduc$ z auxili>ies to facilitate ! e/abli%m5t ( ! equations4 ,examples ( su* diff]5t natures are m ?an suffici5t to give a cle> g5]al idea ( ! imm5se scope ( ! fundam5tal conception ( ! transc5d5tal analysis z =m$ by ,leibnitz1 con/itut+1 z x und\bt$ly does1 ! mo/ l(ty ?\ght to :i* ! human m9d has z yet atta9$4 ,x is evid5t t ?is conception 0 indisp5sable to complete ! f\ndation ( ma!matical sci5ce1 by enabl+ u to e/abli%1 9 a broad & fruitful mann]1 ! relation ( ! concrete to ! ab/ract4 ,9 ?is respect x mu/ be reg>d$ z ! necess>y complem5t ( ! great fundam5tal idea ( ,desc>tes on ! g5]al analytical repres5tation ( natural ph5om5a3 an idea :i* did n beg9 to be wor?ily appreciat$ & suitably employ$ till aft] ! =mation ( ! inf9itesimal analysis1 )\t :i* x c\ld n produce1 ev5 9 geometry1 v important results4 ,g5]ality ( ! ,=mulas4 ,besides ! admirable facility :i* is giv5 by ! transc5d5tal analysis = ! inve/igation ( ! ma!matical laws ( all ph5om5a1 a second fundam5tal & inh]5t prop]ty1 p]haps z important z ! fir/1 is ! extreme g5]ality ( ! diff]5tial =mulas1 :i* express 9 a s+le equation ea* det]m9ate ph5om5on1 h[ev] v>i$ ! subjects 9 relation to :i* x is consid]$4 ,?us we see1 9 ! prec$+ examples1 t a s+le diff]5tial equation gives ! tang5ts ( all curves1 ano!r _! rectifications1 a ?ird _! quadratures2 & 9 ! same way1 "o inv>iable =mula expresses ! ma!matical law ( e v>iable motion2 &1 f9ally1 a s+le equation con/antly repres5ts ! di/ribution ( heat 9 any body & = any case4 ,?is g5]ality1 :i* is s exce$+ly rem>kable1 & :i* is = geomet]s ! basis ( ! mo/ elevat$ consid]ations1 is a =tunate & necess>y consequ5ce ( ! v _s ( ! transc5d5tal analysis1 especially 9 ! conception ( ,leibnitz4 ,?us ! inf9itesimal analysis has n only furni%$ a g5]al me?od = indirectly =m+ equations :i* x w\ld h be5 impossible to discov] 9 a direct mann]1 b x has also p]mitt$ u to consid]1 = ! ma!matical /udy ( natural ph5om5a1 a new ord] ( m g5]al laws1 :i* nev]!less pres5t a cle> & precise signification to e m9d habituat$ to _! int]pretation4 ,by virtue ( ?is second *>act]i/ic prop]ty1 ! entire sy/em ( an imm5se sci5ce1 su* z geometry or me*anics1 has be5 cond5s$ into a small numb] ( analytical =mulas1 f :i* ! human m9d c d$uce1 by c]ta9 & inv>iable rules1 ! solution ( all p>ticul> problems4 ,demon/ration ( ! ,me?od4 ,to complete ! g5]al exposition ( ! conception ( ,leibnitz1 !re rema9s to be consid]$ ! demon/ration ( ! logical proc$ure to :i* x leads1 & ?is1 un=tunately1 is ! mo/ imp]fect "p ( ?is beautiful me?od4 ,9 ! beg9n+ ( ! inf9itesimal analysis1 ! mo/ celebrat$ geomet]s rightly atta*$ m importance to ext5d+ ! immortal discov]y ( ,leibnitz & multiply+ its applications ?an to rigor\sly e/abli%+ ! logical bases ( its op]ations4 ,!y cont5t$ !mselves = a long "t by answ]+ ! objections ( second-rate geomet]s by ! unhop$-= solution ( ! mo/ difficult problems2 d\btless p]suad$ t 9 ma!matical sci5ce1 mu* m ?an 9 any o!r1 we may boldly welcome new me?ods1 ev5 :5 _! rational explanation is imp]fect1 provid$ !y are fruitful 9 results1 inasmu* z its mu* easi] & m num]\s v]ifications w\ld n p]mit any error to rema9 long undiscov]$4 ,b ?is /ate ( ?+s c\ld n long exi/1 & x 0 necess>y to g back to ! v f\ndations ( ! analysis ( ,leibnitz 9 ord] to prove1 9 a p]fectly g5]al mann]1 ! rigor\s exactitude ( ! proc$ures employ$ 9 ?is me?od1 9 spite ( ! app>5t infractions ( ! ord9>y rules ( reason+ :i* x p]mitt$4 ,leibnitz1 urg$ to answ]1 _h pres5t$ an explanation entirely errone\s1 say+ t he treat$ inf9itely small quantities z incomp>ables1 & t he neglect$ !m 9 comp>ison ) f9ite quantities1 l gra9s ( s& 9 comp>ison ) ! sea3 a view :i* w\ld h completely *ang$ ! nature ( 8 analysis1 by r$uc+ x to a m]e approximative calculus1 :i*1 "u ?is po9t ( view1 w\ld be radically vici\s1 s9ce x w\ld be impossible to =esee1 9 g5]al1 to :at degree ! successive op]ations might increase ^! fir/ errors1 :i* c\ld ?us evid5tly atta9 any am\nt4 ,leibnitz1 !n1 did n see1 except 9 a v confus$ mann]1 ! true logical f\ndations ( ! analysis :i* he _h creat$4 ,8 e>lie/ successors limit$ !mselves1 at fir/1 to v]ify+ its exactitude by %[+ ! con=mity ( its results1 9 p>ticul> applications1 to ^? obta9$ by ord9>y algebra or ! geometry ( ! anci5ts2 reproduc+1 accord+ to ! anci5t me?ods1 s f> z !y 7 able1 ! solutions ( "s problems aft] !y _h be5 once obta9$ by ! new me?od1 :i* alone 0 capable ( discov]+ !m 9 ! fir/ place4 ,:5 ?is great "q 0 consid]$ 9 a m g5]al mann]1 geomet]s1 in/ead ( directly attack+ ! difficulty1 pref]r$ to elude x 9 "s way1 z ,eul] & ,d',alemb]t1 = example1 h done1 by demon/rat+ ! necess>y & con/ant con=mity ( ! conception ( ,leibnitz1 view$ 9 all its applications1 ) o!r fundam5tal conceptions ( ! transc5d5tal analysis1 t ( ,newton especially1 ! exactitude ( :i* 0 free f any objection4 ,su* a g5]al v]ification is und\bt$ly /rictly suffici5t to dissipate any unc]ta9ty z to ! legitimate employm5t ( ! analysis ( ,leibnitz4 ,b ! inf9itesimal me?od is s important--x (f]s /ill1 9 almo/ all its applications1 su* a practical sup]iority ov] ! o!r g5]al conceptions :i* h be5 successively propos$--t !re w\ld be a real imp]fection 9 ! philosophical "* ( ! sci5ce if x c\ld n ju/ify itself1 & ne$$ to be logically f\nd$ on consid]ations ( ano!r ord]1 :i* w\ld !n cease to be employ$4 ,x 01 !n1 ( real importance to e/abli% directly & 9 a g5]al mann] ! necess>y rationality ( ! inf9itesimal me?od4 ,aft] v>i\s attempts m or less imp]fect1 a di/+ui%$ geomet]1 ,c>not1 pres5t$ at la/ ! true direct logical explanation ( ! me?od ( ,leibnitz1 by %[+ x to be f\nd$ on ! pr9ciple ( ! necess>y comp5sation ( errors1 ?is be+1 9 fact1 ! precise & lum9\s manife/ation ( :at ,leibnitz _h vaguely & confus$ly p]ceiv$4 ,c>not has ?us r5d]$ ! sci5ce an ess5tial s]vice1 al?\gh1 z we %all see t[>ds ! end ( ?is *apt]1 all ?is logical scaffold+ ( ! inf9itesimal me?od1 prop]ly s call$1 is v probably susceptible ( only a provisional exi/5ce1 inasmu* z x is radically vici\s 9 its nature4 ,/ill1 we %\ld n fail to notice ! g5]al sy/em ( reason+ propos$ by ,c>not1 9 ord] to directly legitimate ! analysis ( ,leibnitz4 ,"h is ! sub/ance ( x3 ,9 e/abli%+ ! diff]5tial equation ( a ph5om5on1 we sub/itute1 = ! imm$iate elem5ts ( ! diff]5t quantities consid]$1 o!r simpl] inf9itesimals1 :i* diff] f !m inf9itely little 9 comp>ison ) !m2 & ?is sub/itution con/itutes ! pr9cipal artifice ( ! me?od ( ,leibnitz1 :i* )\t x w\ld possess no real facility = ! =mation ( equations4 ,c>not reg>ds su* an hypo!sis z really produc+ an error 9 ! equation ?us obta9$1 & :i* = ?is reason he calls imp]fect2 only1 x is cle> t ?is error mu/ be inf9itely small4 ,n[1 on ! o!r h&1 all ! analytical op]ations1 :e!r ( diff]5tiation or ( integration1 :i* are p]=m$ ^u ^! diff]5tial equations1 9 ord] to raise !m to f9ite equations by elim9at+ all ! inf9itesimals :i* h be5 introduc$ z auxili>ies1 produce z con/antly1 by _! nature1 z is easily se51 o!r analog\s errors1 s t an exact comp5sation takes place1 & ! f9al equations1 9 ! words ( ,c>not1 become p]fect4 ,c>not views1 z a c]ta9 & inv>iable indication ( ! actual e/abli%m5t ( ?is necess>y comp5sation1 ! complete elim9ation ( ! v>i\s inf9itely small quantities1 :i* is always1 9 fact1 ! f9al object ( all ! op]ations ( ! transc5d5tal analysis2 = if we h committ$ no o!r infractions ( ! g5]al rules ( reason+ ?an ^? ?us exact$ by ! v nature ( ! inf9itesimal me?od1 ! inf9itely small errors ?us produc$ _c h eng5d]$ o!r ?an inf9itely small errors 9 all ! equations1 & ! relations are necess>ily ( a rigor\s exactitude z soon z !y exi/ betwe5 f9ite quantities alone1 s9ce ! only errors !n possible mu/ be f9ite ones1 :ile none su* c h ent]$4 ,all ?is g5]al reason+ is f\nd$ on ! conception ( inf9itesimal quantities1 reg>d$ z indef9itely decreas+1 :ile ^? f :i* !y are d]iv$ are reg>d$ z fix$4 ,illu/ration by ,tang5ts4 ,?us1 to illu/rate ?is ab/ract exposition by a s+le example1 let u take up aga9 ! "q ( tang5ts1 :i* is ! mo/ easy to analyze completely4 ,we w reg>d ! equation t dy_/dx1 obta9$ above1 z be+ affect$ ) an inf9itely small error1 s9ce x w\ld be p]fectly rigor\s only = ! secant4 ,n[ let u complete ! solution by seek+1 accord+ to ! equation ( ea* curve1 ! ratio betwe5 ! diff]5tials ( ! co-ord9ates4 ,if we suppose ?is equation to be y ax1 we %all evid5tly h dy #b;axdx adx4 ,9 ?is =mula we %all h to neglect ! t]m dx z an inf9itely small quantity ( ! second ord]4 ,!n ! comb9ation ( ! two imp]fect equations4 t dy_/dx1 dy #b;ax7dx71 be+ suffici5t to elim9ate entirely ! inf9itesimals1 ! f9ite result1 t #b;ax1 w necess>ily be rigor\sly correct1 f ! effect ( ! exact comp5sation ( ! two errors committ$2 s9ce1 by its f9ite nature1 x _c be affect$ by an inf9itely small error1 & ?is is1 nev]!less1 ! only "o :i* x c\ld h1 accord+ to ! _s ( ! op]ations :i* h be5 execut$4 ,x w\ld be easy to reproduce 9 a uni=m mann] ! same reason+ ) ref]5ce to all ! o!r g5]al applications ( ! analysis ( ,leibnitz4 ,?is +5i\s !ory is und\bt$ly m subtile ?an solid1 :5 we exam9e x m pr(\ndly2 b x has really no o!r radical logical fault ?an t ( ! inf9itesimal me?od itself1 ( :i* x is1 x seems to me1 ! natural developm5t & ! g5]al explanation1 s t x mu/ be adopt$ = z long a "t z x %all be ?\ght prop] to employ ?is me?od directly4 99 99 99 99 99 ,i pass n[ to ! g5]al exposition ( ! two o!r fundam5tal conceptions ( ! transc5d5tal analysis1 limit+ myself 9 ea* to its pr9cipal idea1 ! philosophical "* ( ! analysis hav+ be5 suffici5tly det]m9$ above 9 ! exam9ation ( ! conception ( ,leibnitz1 :i* ,i h specially dwelt ^u because x admits ( be+ mo/ easily grasp$ z a :ole1 & mo/ rapidly describ$4 ,,me?od ,,( ,,newton4 ,newton has successively pres5t$ 8 own me?od ( conceiv+ ! transc5d5tal analysis "u sev]al diff]5t =ms4 ,t :i* is at pres5t ! mo/ commonly adopt$ 0 designat$ by ,newton1 sometimes "u ! "n ( ! ,me?od ( prime & ultimate ,ratios1 sometimes "u t ( ! ,me?od ( ,limits4 ,me?od ( ,limits4 ,! g5]al _s ( ! transc5d5tal analysis1 f ?is po9t ( view1 consi/s 9 introduc+ z auxili>ies1 9 ! place ( ! primitive quantities1 or concurr5tly ) !m1 9 ord] to facilitate ! e/abli%m5t ( equations1 ! limits ( ! ratios ( ! simultane\s increm5ts ( ^! quantities2 or1 9 o!r words1 ! f9al ratios ( ^! increm5ts2 limits or f9al ratios :i* c be easily %[n to h a det]m9ate & f9ite value4 ,a special calculus1 :i* is ! equival5t ( ! inf9itesimal calculus1 is !n employ$ to pass f ! equations betwe5 ^! limits to ! correspond+ equations betwe5 ! primitive quantities !mselves4 ,! p[] :i* is giv5 by su* an analysis1 ( express+ ) m ease ! ma!matical laws ( ph5om5a1 dep5ds 9 g5]al on ?is1 t s9ce ! calculus applies1 n to ! increm5ts !mselves ( ! propos$ quantities1 b to ! limits ( ! ratios ( ^? increm5ts1 we c always sub/itute = ea* increm5t any o!r magnitude m easy to consid]1 provid$ t _! f9al ratio is ! ratio ( equality1 or1 9 o!r words1 t ! limit ( _! ratio is unity4 ,x is cle>1 inde$1 t ! calculus ( limits w\ld be 9 no way affect$ by ?is sub/itution4 ,/>t+ f ?is pr9ciple1 we f9d ne>ly ! equival5t ( ! facilities (f]$ by ! analysis ( ,leibnitz1 :i* are !n m]ely conceiv$ "u ano!r po9t ( view4 ,?us curves w be reg>d$ z ! limits ( a s]ies ( rectil9e> polygons1 v>iable motions z ! limits ( a collection ( uni=m motions ( con/antly dim9i%+ durations1 & s on4 ,,examples4 #a4 ,tang5ts4 ,suppose1 = example1 t we wi% to det]m9e ! direction ( ! tang5t to a curve2 we w reg>d x z ! limit t[>ds :i* w\ld t5d a secant1 :i* %\ld turn ab\t ! giv5 po9t s t its second po9t ( int]section %\ld indef9itely approa* ! fir/4 ,repres5t+ ! diff]5ces ( ! co-ord9ates ( ! two po9ts by y & x1 we w\ld h at ea* in/ant1 = ! trigonometrical tang5t ( ! angle :i* ! secant makes ) ! axis ( abscissas1 t y_/x2 f :i*1 tak+ ! limits1 we w obta91 relatively to ! tang5t itself1 ?is g5]al =mula ( transc5d5tal analysis1 t ,l7y_/x71 ! *>act]i/ic ,l be+ employ$ to designate ! limit4 ,! calculus ( indirect functions w %[ h[ to d$uce f ?is =mula 9 ea* p>ticul> case1 :5 ! equation ( ! curve is giv51 ! relation betwe5 t & x1 by elim9at+ ! auxili>y quantities :i* h be5 introduc$4 ,if we suppose1 9 ord] to complete ! solution1 t ! equation ( ! propos$ curve is y ax1 we %all evid5tly h y #b;axx a7x71 f :i* we %all obta9 y_/x #b;ax ax4 ,n[ x is cle> t ! limit t[>ds :i* ! second numb] t5ds1 9 proportion z x dim9i%es1 is #b;ax4 ,we %all !re=e f9d1 by ?is me?od1 t #b;ax1 z we obta9$ x = ! same case by ! me?od ( ,leibnitz4 #b4 ,rectifications4 ,9 l mann]1 :5 ! rectification ( a curve is desir$1 we mu/ sub/itute = ! increm5t ( ! arc s ! *ord ( ?is increm5t1 :i* evid5tly has su* a connexion ) x t ! limit ( _! ratio is unity2 & !n we f9d 7pursu+ 9 o!r respects ! same plan z ) ! me?od ( ,leibnitz7 ?is g5]al equation ( rectifications3 7,ls_/x7 #a 7,ly_/x71 or 7,ls_/x7 #a 7,ly_/x7 7,lz_/x71 accord+ z ! curve is plane or ( d\ble curvature4 ,x w n[ be necess>y1 = ea* p>ticul> curve1 to pass f ?is equation to t betwe5 ! arc & ! abscissa1 :i* dep5ds on ! transc5d5tal calculus prop]ly s call$4 ,we c\ld take up1 ) ! same facility1 by ! me?od ( limits1 all ! o!r g5]al que/ions1 ! solution ( :i* has be5 already indicat$ accord+ to ! inf9itesimal me?od4 ,su* is1 9 sub/ance1 ! conception :i* ,newton =m$ = ! transc5d5tal analysis1 or1 m precisely1 t :i* ,maclaur9 & ,d',alemb]t h pres5t$ z ! mo/ rational basis ( t analysis1 9 seek+ to fix & to arrange ! ideas ( ,newton ^u t subject4 ,fluxions & ,flu5ts4 ,ano!r di/9ct =m "u :i* ,newton has pres5t$ ?is same me?od %\ld be "h notic$1 & des]ves p>ticul>ly to fix our att5tion1 z mu* by its +5i\s cle>ness 9 "s cases z by its hav+ furni%$ ! notation be/ suit$ to ?is mann] ( view+ ! transc5d5tal analysis1 &1 moreov]1 z hav+ be5 till lately ! special =m ( ! calculus ( indirect functions commonly adopt$ by ! ,engli% geomet]s4 ,i ref] to ! calculus ( fluxions & ( flu5ts1 f\nd$ on ! g5]al idea ( velocities4 ,to facilitate ! conception ( ! fundam5tal idea1 let u consid] e curve z g5]at$ by a po9t impress$ ) a motion v>y+ accord+ to any law :atev]4 ,! diff]5t quantities :i* ! curve c pres5t1 ! abscissa1 ! ord9ate1 ! arc1 ! area1 c41 w be reg>d$ z simultane\sly produc$ by successive degrees dur+ ?is motion4 ,! velocity ) :i* ea* %all h be5 describ$ w be call$ ! fluxion ( t quantity1 :i* w be inv]sely nam$ its flu5t4 ,h5ce=? ! transc5d5tal analysis w consi/1 accord+ to ?is conception1 9 =m+ directly ! equations betwe5 ! fluxions ( ! propos$ quantities1 9 ord] to d$uce !refrom1 by a special calculus1 ! equations betwe5 ! flu5ts !mselves4 ,:at has be5 /at$ respect+ curves may1 moreov]1 evid5tly be appli$ to any magnitudes :atev]1 reg>d$1 by ! aid ( suitable images1 z produc$ by motion4 ,x is easy to und]/& ! g5]al & necess>y id5tity ( ?is me?od ) t ( limits complicat$ ) ! =eign idea ( motion4 ,9 fact1 resum+ ! case ( ! curve1 if we suppose1 z we evid5tly always may1 t ! motion ( ! describ+ po9t is uni=m 9 a c]ta9 direction1 t ( ! abscissa1 = example1 !n ! fluxion ( ! abscissa w be con/ant1 l ! elem5t ( ! "t2 = all ! o!r quantities g5]at$1 ! motion _c be conceiv$ to be uni=m1 except = an inf9itely small "t4 ,n[ ! velocity be+ 9 g5]al accord+ to its me*anical conception1 ! ratio ( ea* space to ! "t employ$ 9 trav]s+ x1 & ?is "t be+ "h proportional to ! increm5t ( ! abscissa1 x foll[s t ! fluxions ( ! ord9ate1 ( ! arc1 ( ! area1 c41 are really no?+ else 7reject+ ! int]m$iate consid]ation ( "t7 ?an ! f9al ratios ( ! increm5ts ( ^! diff]5t quantities to ! increm5t ( ! abscissa4 ,?is me?od ( fluxions & flu5ts is1 !n1 9 reality1 only a mann] ( repres5t+1 by a comp>ison borr[$ f me*anics1 ! me?od ( prime & ultimate ratios1 :i* alone c be r$uc$ to a calculus4 ,x evid5tly1 !n1 (f]s ! same g5]al advantages 9 ! v>i\s pr9cipal applications ( ! transc5d5tal analysis1 )\t its be+ necess>y to pres5t special pro(s ( ?is4 ,,me?od ,,( ,,lagrange4 ,d]iv$ ,functions4 ,! conception ( ,lagrange1 9 its admirable simplicity1 consi/s 9 repres5t+ ! transc5d5tal analysis z a great algebraic artifice1 by :i*1 9 ord] to facilitate ! e/abli%m5t ( equations1 we introduce1 9 ! place ( ! primitive functions1 or concurr5tly ) !m1 _! d]iv$ functions2 t is1 accord+ to ! def9ition ( ,lagrange1 ! coeffici5t ( ! fir/ t]m ( ! increm5t ( ea* function1 arrang$ accord+ to ! asc5d+ p[]s ( ! increm5t ( its v>iable4 ,! special calculus ( indirect functions has = its con/ant object1 "h z well z 9 ! conceptions ( ,leibnitz & ( ,newton1 to elim9ate ^! d]ivatives :i* h be5 ?us employ$ z auxili>ies1 9 ord] to d$uce f _! relations ! correspond+ equations betwe5 ! primitive magnitudes4 ,an ,ext5sion ( ord9>y ,analysis4 ,! transc5d5tal analysis is1 !n1 no?+ b a simple ?\gh v consid]able ext5sion ( ord9>y analysis4 ,geomet]s h long be5 accu/om$ to introduce 9 analytical inve/igations1 9 ! place ( ! magnitudes !mselves :i* !y wi%$ to /udy1 _! diff]5t p[]s1 or _! log>i?ms1 or _! s9es1 c41 9 ord] to simplify ! equations1 & ev5 to obta9 !m m easily4 ,?is successive d]ivation is an artifice ( ! same nature1 only ( great] ext5t1 & procur+1 9 consequ5ce1 mu* m important res\rces = ?is common object4 ,b1 al?\gh we c readily conceive1 priori1 t ! auxili>y consid]ation ( ^! d]ivatives may facilitate ! e/abli%m5t ( equations1 x is n easy to expla9 :y ?is mu/ necess>ily foll[ f ?is mode ( d]ivation r ?an f any o!r trans=mation4 ,su* is ! weak po9t ( ! great idea ( ,lagrange4 ,! precise advantages ( ?is analysis _c z yet be grasp$ 9 an ab/ract mann]1 b only %[n by consid]+ sep>ately ea* pr9cipal "q1 s t ! v]ification is (t5 exce$+ly labori\s4 ,,example4 ,tang5ts4 ,?is mann] ( conceiv+ ! transc5d5tal analysis may be be/ illu/rat$ by its application to ! mo/ simple ( ! problems above exam9$--t ( tang5ts4 ,in/ead ( conceiv+ ! tang5t z ! prolongation ( ! inf9itely small elem5t ( ! curve1 accord+ to ! notion ( ,leibnitz--or z ! limit ( ! secants1 accord+ to ! ideas ( ,newton--,lagrange consid]s x1 accord+ to its simple geometrical "*1 analog\s to ! def9itions ( ! anci5ts1 to be a "r l9e su* t no o!r "r l9e c pass ?r\gh ! po9t ( contact betwe5 x & ! curve4 ,!n1 to det]m9e its direction1 we mu/ seek ! g5]al expression ( its di/ance f ! curve1 measur$ 9 any direction :atev]--9 t ( ! ord9ate1 = example--& dispose ( ! arbitr>y con/ant relat+ to ! incl9ation ( ! "r l9e1 :i* w necess>ily ent] into t expression1 9 su* a way z to dim9i% t sep>ation z mu* z possible4 ,n[ ?is di/ance1 be+ evid5tly equal to ! diff]5ce ( ! two ord9ates ( ! curve & ( ! "r l9e1 :i* correspond to ! same new abscissa x h1 w be repres5t$ by ! =mula 7f'7x7 - t7h qh rh etc41 9 :i* t designates1 z above1 ! unkn[n trigonometrical tang5t ( ! angle :i* ! requir$ l9e makes ) ! axis ( abscissas1 & f'7x7 ! d]iv$ function ( ! ord9ate f7x74 ,?is be+ und]/ood1 x is easy to see t1 by dispos+ ( t s z to make ! fir/ t]m ( ! prec$+ =mula equal to z]o1 we w r5d] ! int]val betwe5 ! two l9es ! lea/ possible1 s t any o!r l9e = :i* t did n h ! value ?us det]m9$ w\ld necess>ily dep>t f>!r f ! propos$ curve4 ,we h1 !n1 = ! direction ( ! tang5t s\ght1 ! g5]al expression t f'7x71 a result exactly equival5t to ^? furni%$ by ! ,inf9itesimal ,me?od & ! ,me?od ( ,limits4 ,we h yet to f9d f'7x7 9 ea* p>ticul> curve1 :i* is a m]e "q ( analysis1 q id5tical ) ^? :i* are pres5t$1 at ?is /age ( ! op]ations1 by ! o!r me?ods4 ,aft] ^! consid]ations ^u ! pr9cipal g5]al conceptions1 we ne$ n /op to exam9e "s o!r !ories propos$1 su* z ,eul]'s ,calculus ( ,vani%+ ,quantities1 :i* are really modifications--m or less important1 &1 moreov]1 no long] us$--( ! prec$+ me?ods4 ,i h n[ to e/abli% ! comp>ison & ! appreciation ( ^! ?ree fundam5tal me?ods4 ,_! p]fect & necess>y con=mity is fir/ to be prov5 9 a g5]al mann]4 ,,fundam5tal ,,id5tity ,,( ,,! ,,?ree ,,me?ods4 ,x is1 9 ! fir/ place1 evid5t f :at prec$es1 consid]+ ^! ?ree me?ods z to _! actual de/9ation1 indep5d5tly ( _! prelim9>y ideas1 t !y all consi/ 9 ! same g5]al logical artifice1 :i* has be5 *>act]iz$ 9 ! fir/ *apt]2 to wit1 ! introduction ( a c]ta9 sy/em ( auxili>y magnitudes1 hav+ uni=m relations to ^? :i* are ! special objects ( ! inquiry1 & sub/itut$ = !m expressly to facilitate ! analytical expression ( ! ma!matical laws ( ! ph5om5a1 al?\gh !y h f9ally to be elim9at$ by ! aid ( a special calculus4 ,x is ?is :i* has det]m9$ me to regul>ly def9e ! transc5d5tal analysis z ! calculus ( indirect functions1 9 ord] to m>k its true philosophical "*1 at ! same "t avoid+ any discussion ^u ! be/ mann] ( conceiv+ & apply+ x4 ,! g5]al effect ( ?is analysis1 :atev] ! me?od employ$1 is1 !n1 to br+ e ma!matical "q mu* m promptly )9 ! p[] ( ! calculus1 & ?us to dim9i% consid]ably ! s]i\s difficulty :i* is usually pres5t$ by ! passage f ! concrete to ! ab/ract4 ,:atev] progress we may make1 we c nev] hope t ! calculus w ev] be able to grasp e "q ( natural philosophy1 geometrical1 or me*anical1 or !rmological1 c41 imm$iately ^u its bir?1 :i* w\ld evid5tly involve a contradiction4 ,e problem w con/antly require a c]ta9 prelim9>y lab\r to be p]=m$1 9 :i* ! calculus c be ( no assi/ance1 & :i*1 by its nature1 _c be subject$ to ab/ract & inv>iable rules2 x is t :i* has = its special object ! e/abli%m5t ( equations1 :i* =m ! indisp5sable />t+ po9t ( all analytical rese>*es4 ,b ?is prelim9>y lab\r has be5 rem>kably simplifi$ by ! creation ( ! transc5d5tal analysis1 :i* has ?us ha/5$ ! mom5t at :i* ! solution admits ( ! uni=m & precise application ( g5]al & ab/ract me?ods2 by r$uc+1 9 ea* case1 ?is special lab\r to ! inve/igation ( equations betwe5 ! auxili>y magnitudes2 f :i* ! calculus !n leads to equations directly ref]r+ to ! propos$ magnitudes1 :i*1 be=e ?is admirable conception1 x _h be5 necess>y to e/abli% directly & sep>ately4 ,:e!r ^! indirect equations are diff]5tial equations1 accord+ to ! idea ( ,leibnitz1 or equations ( limits1 con=mably to ! conception ( ,newton1 or1 la/ly1 d]iv$ equations1 accord+ to ! !ory ( ,lagrange1 ! g5]al proc$ure is evid5tly always ! same4 ,b ! co9cid5ce ( ^! ?ree pr9cipal me?ods is n limit$ to ! common effect :i* !y produce2 x exi/s1 besides1 9 ! v mann] ( obta9+ x4 ,9 fact1 n only d all ?ree consid]1 9 ! place ( ! primitive magnitudes1 c]ta9 auxili>y ones1 b1 /ill f>!r1 ! quantities ?us introduc$ z subsidi>y are exactly id5tical 9 ! ?ree me?ods1 :i* consequ5tly diff] only 9 ! mann] ( view+ !m4 ,?is c be easily %[n by tak+ = ! g5]al t]m ( comp>ison any "o ( ! ?ree conceptions1 especially t ( ,lagrange1 :i* is ! mo/ suitable to s]ve z a type1 z be+ ! free/ f =eign consid]ations4 ,is x n evid5t1 by ! v def9ition ( d]iv$ functions1 t !y are no?+ else ?an :at ,leibnitz calls diff]5tial coeffici5ts1 or ! ratios ( ! diff]5tial ( ea* function to t ( ! correspond+ v>iable1 s9ce1 9 det]m9+ ! fir/ diff]5tial1 we w be oblig$1 by ! v nature ( ! inf9itesimal me?od1 to limit ourselves to tak+ ! only t]m ( ! increm5t ( ! function :i* conta9s ! fir/ p[] ( ! inf9itely small increm5t ( ! v>iable8 ,9 ! same way1 is n ! d]iv$ function1 by its nature1 likewise ! necess>y limit t[>ds :i* t5ds ! ratio betwe5 ! increm5t ( ! primitive function & t ( its v>iable1 9 proportion z ?is la/ indef9itely dim9i%es1 s9ce x evid5tly expresses :at t ratio becomes :5 we suppose ! increm5t ( ! v>iable to equal z]o8 ,t :i* is designat$ by dx_/dy 9 ! me?od ( ,leibnitz2 t :i* ought to be not$ z ,l7y_/x7 9 t ( ,newton2 & t :i* ,lagrange has indicat$ by f'7x71 is con/antly "o same function1 se5 f ?ree diff]5t po9ts ( view1 ! consid]ations ( ,leibnitz & ,newton prop]ly consi/+ 9 mak+ kn[n two g5]al necess>y prop]ties ( ! d]iv$ function4 ,! transc5d5tal analysis1 exam9$ ab/ract$ly & 9 its pr9ciple1 is !n always ! same1 :atev] may be ! conception :i* is adopt$1 & ! proc$ures ( ! calculus ( indirect functions are necess>ily id5tical 9 ^! diff]5t me?ods1 :i* 9 l mann] mu/1 = any application :atev]1 lead con/antly to rigor\sly uni=m results4 ,,comp>ative ,,value ,,( ,,! ,,?ree ,,me?ods4 ,if n[ we endeav\r to e/imate ! comp>ative value ( ^! ?ree equival5t conceptions1 we %all f9d 9 ea* advantages & inconv5i5ces :i* are peculi> to x1 & :i* /ill prev5t geomet]s f conf9+ !mselves to any "o ( !m1 consid]$ z f9al4 ,t ( ,leibnitz4 ,! conception ( ,leibnitz pres5ts inconte/ably1 9 all its applications1 a v m>k$ sup]iority1 by lead+ 9 a mu* m rapid mann]1 & ) mu* less m5tal ef=t1 to ! =mation ( equations betwe5 ! auxili>y magnitudes4 ,x is to its use t we owe ! high p]fection :i* has be5 acquir$ by all ! g5]al !ories ( geometry & me*anics4 ,:atev] may be ! diff]5t speculative op9ions ( geomet]s ) respect to ! inf9itesimal me?od1 9 an ab/ract po9t ( view1 all tacitly agree 9 employ+ x by pref]5ce1 z soon z !y h to treat a new "q1 9 ord] n to complicate ! necess>y difficulty by ?is purely artificial ob/acle proce$+ f a misplac$ ob/9acy 9 adopt+ a less exp$iti\s c\rse4 ,lagrange himself1 aft] hav+ recon/ruct$ ! transc5d5tal analysis on new f\ndations1 has 7) t noble frankness :i* s well suit$ 8 g5ius7 r5d]$ a /rik+ & decisive homage to ! *>act]i/ic prop]ties ( ! conception ( ,leibnitz1 by foll[+ x exclusively 9 ! entire sy/em ( 8 ,m*anique ,analytique4 ,su* a fact r5d]s any comm5ts unnecess>y4 ,b :5 we consid] ! conception ( ,leibnitz 9 itself & 9 its logical relations1 we _c escape admitt+1 ) ,lagrange1 t x is radically vici\s 9 ?is1 t1 adopt+ its own expressions1 ! notion ( inf9itely small quantities is a false idea1 ( :i* x is 9 fact impossible to obta9 a cle> conception1 h[ev] we may deceive ourselves 9 t matt]4 ,ev5 if we adopt ! +5i\s idea ( ! comp5sation ( errors1 z above expla9$1 ?is involves ! radical inconv5i5ce ( be+ oblig$ to di/+ui% 9 ma!matics two classes ( reason+s1 ^? :i* are p]fectly rigor\s1 & ^? 9 :i* we design$ly commit errors :i* subsequ5tly h to be comp5sat$4 ,a conception :i* leads to su* /range consequ5ces is und\bt$ly v unsatisfactory 9 a logical po9t ( view4 ,to say1 z d "s geomet]s1 t x is possible 9 e case to r$uce ! inf9itesimal me?od to t ( limits1 ! logical "* ( :i* is irreproa*able1 w\ld evid5tly be to elude ! difficulty r ?an to remove x2 besides1 su* a trans=mation almo/ entirely /rips ! conception ( ,leibnitz ( its ess5tial advantages ( facility & rapidity4 ,f9ally1 ev5 disreg>d+ ! prec$+ important consid]ations1 ! inf9itesimal me?od w\ld no less evid5tly pres5t by its nature ! v s]i\s defect ( break+ ! unity ( ab/ract ma!matics1 by creat+ a transc5d5tal analysis f\nd$ on pr9ciples s diff]5t f ^? :i* =m ! basis ( ! ord9>y analysis4 ,?is division ( analysis into two worlds almo/ entirely indep5d5t ( ea* o!r1 t5ds to h9d] ! =mation ( truly g5]al analytical conceptions4 ,to fully appreciate ! consequ5ces ( ?is1 we %\ld h to g back to ! /ate ( ! sci5ce be=e ,lagrange _h e/abli%$ a g5]al & complete h>mony betwe5 ^! two great sections4 ,t ( ,newton4 ,pass+ n[ to ! conception ( ,newton1 x is evid5t t by its nature x is n expos$ to ! fundam5tal logical objections :i* are call$ =? by ! me?od ( ,leibnitz4 ,! notion ( limits is1 9 fact1 rem>kable = its simplicity & its precision4 ,9 ! transc5d5tal analysis pres5t$ 9 ?is mann]1 ! equations are reg>d$ z exact f _! v orig91 & ! g5]al rules ( reason+ are z con/antly obs]v$ z 9 ord9>y analysis4 ,b1 on ! o!r h&1 x is v f> f (f]+ su* p[]ful res\rces = ! solution ( problems z ! inf9itesimal me?od4 ,! obligation :i* x imposes1 ( nev] consid]+ ! increm5ts ( magnitudes sep>ately & by !mselves1 nor ev5 9 _! ratios1 b only 9 ! limits ( ^? ratios1 ret>ds consid]ably ! op]ations ( ! m9d 9 ! =mation ( auxili>y equations4 ,we may ev5 say t x greatly emb>rasses ! purely analytical trans=mations4 ,?us ! transc5d5tal analysis1 consid]$ sep>ately f its applications1 is f> f pres5t+ 9 ?is me?od ! ext5t & ! g5]ality :i* h be5 impr9t$ ^u x by ! conception ( ,leibnitz4 ,x is v difficult1 = example1 to ext5d ! !ory ( ,newton to functions ( sev]al indep5d5t v>iables4 ,b x is especially ) ref]5ce to its applications t ! relative inf]iority ( ?is !ory is mo/ /rongly m>k$4 ,sev]al ,cont95tal geomet]s1 9 adopt+ ! me?od ( ,newton z ! m logical basis ( ! transc5d5tal analysis1 h p>tially disguis$ ?is inf]iority by a s]i\s inconsi/5cy1 :i* consi/s 9 apply+ to ?is me?od ! notation inv5t$ by ,leibnitz = ! inf9itesimal me?od1 & :i* is really appropriate to x alone4 ,9 designat+ by dy_/dx t :i* logically ought1 9 ! !ory ( limits1 to be d5ot$ by ,l7y_/x71 & 9 ext5d+ to all ! o!r analytical conceptions ?is displacem5t ( signs1 !y int5d$1 und\bt$ly1 to comb9e ! special advantages ( ! two me?ods2 b1 9 reality1 !y h only succe$$ 9 caus+ a vici\s confusion betwe5 !m1 a famili>ity ) :i* h9d]s ! =mation ( cle> & exact ideas ( ei!r4 ,x w\ld c]ta9ly be s+ul>1 consid]+ ?is usage 9 itself1 t1 by ! m]e means ( signs1 x c\ld be possible to effect a v]itable comb9ation betwe5 two !ories s di/9ct z ^? "u consid]ation4 ,f9ally1 ! me?od ( limits pres5ts also1 ?\gh 9 a less degree1 ! great] inconv5i5ce1 :i* ,i h above not$ 9 ref]5ce to ! inf9itesimal me?od1 ( e/abli%+ a total sep>ation betwe5 ! ord9>y & ! transc5d5tal analysis2 = ! idea ( limits1 ?\gh cle> & rigor\s1 is none ! less 9 itself1 z ,lagrange has rem>k$1 a =eign idea1 ^u :i* analytical !ories ought n to be dep5d5t4 ,t ( ,lagrange4 ,?is p]fect unity ( analysis1 & ?is purely ab/ract "* ( its fundam5tal notions1 are f\nd 9 ! highe/ degree 9 ! conception ( ,lagrange1 & are f\nd !re alone2 x is1 = ?is reason1 ! mo/ rational & ! mo/ philosophical ( all4 ,c>efully remov+ e het]og5e\s consid]ation1 ,lagrange has r$uc$ ! transc5d5tal analysis to its true peculi> "*1 t ( pres5t+ a v ext5sive class ( analytical trans=mations1 :i* facilitate 9 a rem>kable degree ! expression ( ! conditions ( v>i\s problems4 ,at ! same "t1 ?is analysis is ?us necess>ily pres5t$ z a simple ext5sion ( ord9>y analysis2 x is only a high] algebra4 ,all ! diff]5t p>ts ( ab/ract ma!matics1 previ\sly s incoh]5t1 h f t mom5t admitt$ ( be+ conceiv$ z =m+ a s+le sy/em4 ,unhappily1 ?is conception1 :i* possesses su* fundam5tal prop]ties1 indep5d5tly ( its s simple & s lucid notation1 & :i* is und\bt$ly de/9$ to become ! f9al !ory ( transc5d5tal analysis1 because ( its high philosophical sup]iority ov] all ! o!r me?ods propos$1 pres5ts 9 its pres5t /ate too _m difficulties 9 its applications1 z comp>$ ) ! conception ( ,newton1 & /ill m ) t ( ,leibnitz1 to be z yet exclusively adopt$4 ,lagrange himself has succe$$ only ) great difficulty 9 r$iscov]+1 by 8 me?od1 ! pr9cipal results already obta9$ by ! inf9itesimal me?od = ! solution ( ! g5]al que/ions ( geometry & me*anics2 we may judge f t :at ob/acles w\ld be f\nd 9 treat+ 9 ! same mann] que/ions :i* 7 truly new & important4 ,x is true t ,lagrange1 on sev]al occasions1 has %[n t difficulties call =?1 f m5 ( g5ius1 sup]ior ef=ts1 capable ( lead+ to ! greate/ results4 ,x 0 ?us t1 9 try+ to adapt 8 me?od to ! exam9ation ( ! curvature ( l9es1 :i* seem$ s f> f admitt+ its application1 he arriv$ at t beautiful !ory ( contacts :i* has s greatly p]fect$ t important "p ( geometry4 ,b1 9 spite ( su* happy exceptions1 ! conception ( ,lagrange has nev]!less rema9$1 z a :ole1 ess5tially unsuit$ to applications4 ,! f9al result ( ! g5]al comp>ison :i* ,i h too briefly sket*$1 is1 !n1 z already sugge/$1 t1 9 ord] to really und]/& ! transc5d5tal analysis1 we %\ld n only consid] x 9 its pr9ciples accord+ to ! ?ree fundam5tal conceptions ( ,leibnitz1 ( ,newton1 & ( ,lagrange1 b %\ld besides accu/om ourselves to c>ry out almo/ indiff]5tly1 accord+ to ^! ?ree pr9cipal me?ods1 & especially accord+ to ! fir/ & ! la/1 ! solution ( all important que/ions1 :e!r ( ! pure calculus ( indirect functions or ( its applications4 ,?is is a c\rse :i* ,i c\ld n too /rongly recomm5d to all ^? :o desire to judge philosophically ( ?is admirable creation ( ! human m9d1 z well z to ^? :o wi% to le>n to make use ( ?is p[]ful in/rum5t ) success & ) facility4 ,9 all ! o!r p>ts ( ma!matical sci5ce1 ! consid]ation ( diff]5t me?ods = a s+le class ( que/ions may be useful1 ev5 indep5d5tly ( its hi/orical int]e/1 b x is n indisp5sable2 "h1 on ! contr>y1 x is /rictly necess>y4 ,hav+ det]m9$ ) precision1 9 ?is *apt]1 ! philosophical "* ( ! calculus ( indirect functions1 accord+ to ! pr9cipal fundam5tal conceptions ( :i* x admits1 we h next to consid]1 9 ! foll[+ *apt]1 ! logical division & ! g5]al composition ( ?is calculus4 ,,*apt] ,,iv4 ,,! ,,diff]5tial ,,& ,,integral ,,calculus4 ,,its ,,two ,,fundam5tal ,,divisions4 ,! calculus ( indirect functions1 9 accordance ) ! consid]ations expla9$ 9 ! prec$+ *apt]1 is necess>ily divid$ into two p>ts 7or1 m prop]ly1 is decompos$ into two diff]5t calculi entirely di/9ct1 al?\gh intimately connect$ by _! nature71 accord+ z x is propos$ to f9d ! relations betwe5 ! auxili>y magnitudes 7! introduction ( :i* con/itutes ! g5]al _s ( ?is calculus7 by means ( ! relations betwe5 ! correspond+ primitive magnitudes2 or1 conv]sely1 to try to discov] ^! direct equations by means ( ! indirect equations orig9ally e/abli%$4 ,su* is1 9 fact1 con/antly ! d\ble object ( ! transc5d5tal analysis4 ,^! two sy/ems h receiv$ diff]5t names1 accord+ to ! po9t ( view "u :i* ?is analysis has be5 reg>d$4 ,! inf9itesimal me?od1 prop]ly s call$1 hav+ be5 ! mo/ g5]ally employ$ = ! reasons :i* h be5 giv51 almo/ all geomet]s employ habitually ! d5om9ations ( ,diff]5tial ,calculus & ( ,integral ,calculus1 e/abli%$ by ,leibnitz1 & :i* are1 9 fact1 v rational consequ5ces ( 8 conception4 ,newton1 9 accordance ) 8 me?od1 nam$ ! fir/ ! ,calculus ( ,fluxions1 & ! second ! ,calculus ( ,flu5ts1 expressions :i* 7 commonly employ$ 9 ,engl&4 ,f9ally1 foll[+ ! em95tly philosophical !ory f\nd$ by ,lagrange1 "o w\ld be call$ ! ,calculus ( ,d]iv$ ,functions1 & ! o!r ! ,calculus ( ,primitive ,functions4 ,i w cont9ue to make use ( ! t]ms ( ,leibnitz1 z be+ m conv5i5t = ! =mation ( second>y expressions1 al?\gh ,i ought1 9 accordance ) ! sugge/ions made 9 ! prec$+ *apt]1 to employ concurr5tly all ! diff]5t conceptions1 approa*+ z ne>ly z possible to t ( ,lagrange4 ,,_! ,,relations ,,to ,,ea* ,,o!r4 ,! diff]5tial calculus is evid5tly ! logical basis ( ! integral calculus2 = we d n & _c "k h[ to integrate directly any o!r diff]5tial expressions ?an ^? produc$ by ! diff]5tiation ( ! t5 simple functions :i* con/itute ! g5]al elem5ts ( our analysis4 ,! art ( integration consi/s1 !n1 ess5tially 9 br++ all ! o!r cases1 z f> z is possible1 to f9ally dep5d on only ?is small numb] ( fundam5tal integrations4 ,9 consid]+ ! :ole body ( ! transc5d5tal analysis1 z ,i h *>act]iz$ x 9 ! prec$+ *apt]1 x is n at fir/ app>5t :at c be ! peculi> utility ( ! diff]5tial calculus1 indep5d5tly ( ?is necess>y relation ) ! integral calculus1 :i* seems z if x mu/ be1 by itself1 ! only "o directly indisp5sable4 ,9 fact1 ! elim9ation ( ! inf9itesimals or ( ! d]ivatives1 introduc$ z auxili>ies to facilitate ! e/abli%m5t ( equations1 con/itut+1 z we h se51 ! f9al & inv>iable object ( ! calculus ( indirect functions1 x is natural to ?9k t ! calculus :i* tea*es h[ to d$uce f ! equations betwe5 ^! auxili>y magnitudes1 ^? :i* exi/ betwe5 ! primitive magnitudes !mselves1 ought /rictly to suffice = ! g5]al wants ( ! transc5d5tal analysis )\t our p]ceiv+1 at ! fir/ glance1 :at special & con/ant "p ! solution ( ! inv]se "q c h 9 su* an analysis4 ,x w\ld be a real error1 ?\gh a common "o1 to assign to ! diff]5tial calculus1 9 ord] to expla9 its peculi>1 direct1 & necess>y influ5ce1 ! de/9ation ( =m+ ! diff]5tial equations1 f :i* ! integral calculus !n enables u to arrive at ! f9ite equations2 = ! primitive =mation ( diff]5tial equations is n & _c be1 prop]ly speak+1 ! object ( any calculus1 s9ce1 on ! contr>y1 x =ms by its nature ! indisp5sable />t+ po9t ( any calculus :atev]4 ,h[1 9 p>ticul>1 c\ld ! diff]5tial calculus1 :i* 9 itself is r$uc$ to tea*+ ! means ( diff]5tiat+ ! diff]5t equations1 be a g5]al proc$ure = e/abli%+ !m8 ,t :i* 9 e application ( ! transc5d5tal analysis really facilitates ! =mation ( equations1 is ! inf9itesimal me?od1 & n ! inf9itesimal calculus1 :i* is p]fectly di/9ct f x1 al?\gh x is its indisp5sable complem5t4 ,su* a consid]ation w\ld1 !n1 give a false idea ( ! special de/9ation :i* *>act]izes ! diff]5tial calculus 9 ! g5]al sy/em ( ! transc5d5tal analysis4 ,b we %\ld nev]!less v imp]fectly conceive ! real peculi> importance ( ?is fir/ bran* ( ! calculus ( indirect functions1 if we saw 9 x only a simple prelim9>y lab\r1 hav+ no o!r g5]al & ess5tial object ?an to prep>e indisp5sable f\ndations = ! integral calculus4 ,z ! ideas on ?is matt] are g5]ally confus$1 ,i ?9k t ,i ought "h to expla9 9 a summ>y mann] ?is important relation z ,i view x1 & to %[ t 9 e application ( ! transc5d5tal analysis a prim>y1 direct1 & necess>y "p is con/antly assign$ to ! diff]5tial calculus4 #a4 ,use ( ! ,diff]5tial ,calculus z prep>atory to t ( ! ,integral4 ,9 =m+ ! diff]5tial equations ( any ph5om5on :atev]1 x is v seldom t we limit ourselves to introduce diff]5tially only ^? magnitudes ^: relations are s\ght4 ,to impose t condition w\ld be to uselessly dim9i% ! res\rces pres5t$ by ! transc5d5tal analysis = ! expression ( ! ma!matical laws ( ph5om5a4 ,mo/ frequ5tly we introduce into ! primitive equations1 ?r\gh _! diff]5tials1 o!r magnitudes ^: relations are already kn[n or suppos$ to be s1 & )\t ! consid]ation ( :i* x w\ld be frequ5tly impossible to e/abli% equations4 ,?us1 = example1 9 ! g5]al problem ( ! rectification ( curves1 ! diff]5tial equation1 ds dy dx1 or ds dx dy dz1 is n only e/abli%$ betwe5 ! desir$ function s & ! indep5d5t v>iable x1 to :i* x is ref]r$1 b1 at ! same "t1 !re h be5 introduc$1 z indisp5sable int]m$i>ies1 ! diff]5tials ( "o or two o!r functions1 y & z1 :i* are among ! data ( ! problem2 x w\ld n h be5 possible to =m directly ! equation betwe5 ds & dx1 :i* w\ld1 besides1 be peculi> to ea* curve consid]$4 ,x is ! same = mo/ que/ions4 ,n[ 9 ^! cases x is evid5t t ! diff]5tial equation is n imm$iately suitable = integration4 ,x is previ\sly necess>y t ! diff]5tials ( ! functions suppos$ to be kn[n1 :i* h be5 employ$ z int]m$i>ies1 %\ld be entirely elim9at$1 9 ord] t equations may be obta9$ betwe5 ! diff]5tials ( ! functions :i* alone are s\ght & ^? ( ! really indep5d5t v>iables1 aft] :i* ! "q dep5ds on only ! integral calculus4 ,n[ ?is prep>atory elim9ation ( c]ta9 diff]5tials1 9 ord] to r$uce ! inf9itesimals to ! smalle/ numb] possible1 belongs simply to ! diff]5tial calculus2 = x mu/ evid5tly be done by det]m9+1 by means ( ! equations betwe5 ! functions suppos$ to be kn[n1 tak5 z int]m$i>ies1 ! relations ( _! diff]5tials1 :i* is m]ely a "q ( diff]5tiation4 ,?us1 = example1 9 ! case ( rectifications1 x w be fir/ necess>y to calculate dy1 or dy & dz1 by diff]5tiat+ ! equation or ! equations ( ea* curve propos$2 aft] elim9at+ ^! expressions1 ! g5]al diff]5tial =mula above enunciat$ w !n conta9 only ds & dx2 hav+ arriv$ at ?is po9t1 ! elim9ation ( ! inf9itesimals c be complet$ only by ! integral calculus4 ,su* is1 !n1 ! g5]al (fice necess>ily belong+ to ! diff]5tial calculus 9 ! complete solution ( ! que/ions :i* exact ! employm5t ( ! transc5d5tal analysis2 to produce1 z f> z is possible1 ! elim9ation ( ! inf9itesimals1 t is1 to r$uce 9 ea* case ! primitive diff]5tial equations s t !y %all conta9 only ! diff]5tials ( ! really indep5d5t v>iables1 & ^? ( ! functions s\ght1 by caus+ to disappe>1 by elim9ation1 ! diff]5tials ( all ! o!r kn[n functions :i* may h be5 tak5 z int]m$i>ies at ! "t ( ! =mation ( ! diff]5tial equations ( ! problem :i* is "u consid]ation4 #b4 ,employm5t ( ! ,diff]5tial ,calculus alone4 ,= c]ta9 que/ions1 :i*1 al?\gh few 9 numb]1 h none ! less1 z we %all see h]eaft]1 a v great importance1 ! magnitudes :i* are s\ght ent] directly1 & n by _! diff]5tials1 into ! primitive diff]5tial equations1 :i* !n conta9 diff]5tially only ! diff]5t kn[n functions employ$ z int]m$i>ies1 9 accordance ) ! prec$+ explanation4 ,^! cases are ! mo/ fav\rable ( all2 = x is evid5t t ! diff]5tial calculus is !n entirely suffici5t = ! complete elim9ation ( ! inf9itesimals1 )\t ! "q giv+ rise to any integration4 ,?is is :at occurs1 = example1 9 ! problem ( tang5ts 9 geometry2 9 t ( velocities 9 me*anics1 c4 #c4 ,employm5t ( ! ,integral ,calculus alone4 ,f9ally1 "s o!r que/ions1 ! numb] ( :i* is also v small1 b ! importance ( :i* is no less great1 pres5t a second exceptional case1 :i* is 9 its nature exactly ! conv]se ( ! prec$+4 ,!y are ^? 9 :i* ! diff]5tial equations are f\nd to be imm$iately ready = integration1 because !y conta91 at _! fir/ =mation1 only ! inf9itesimals :i* relate to ! functions s\ght1 or to ! really indep5d5t v>iables1 )\t its be+ necess>y to introduce1 diff]5tially1 o!r functions z int]m$i>ies4 ,if 9 ^! new cases we introduce ^! la/ functions1 s9ce1 by hypo!sis1 !y w ent] directly & n by _! diff]5tials1 ord9>y algebra w suffice to elim9ate !m1 & to br+ ! "q to dep5d on only ! integral calculus4 ,! diff]5tial calculus w !n h no special "p 9 ! complete solution ( ! problem1 :i* w dep5d entirely ^u ! integral calculus4 ,! g5]al "q ( quadratures (f]s an important example ( ?is1 = ! diff]5tial equation be+ !n d,a ydx1 w become imm$iately fit = integration z soon z we %all h elim9at$1 by means ( ! equation ( ! propos$ curve1 ! int]m$i>y function y1 :i* does n ent] into x diff]5tially4 ,! same circum/ances exi/ 9 ! problem ( cubatures1 & 9 "s o!rs equally important4 ,?ree classes ( ,que/ions h5ce result+4 ,z a g5]al result ( ! previ\s consid]ations1 x is !n necess>y to divide into ?ree classes ! ma!matical que/ions :i* require ! use ( ! transc5d5tal analysis2 ! fir/ class comprises ! problems susceptible ( be+ entirely resolv$ by means ( ! diff]5tial calculus alone1 )\t any ne$ ( ! integral calculus2 ! second1 ^? :i* are1 on ! contr>y1 entirely dep5d5t ^u ! integral calculus1 )\t ! diff]5tial calculus hav+ any "p 9 _! solution2 la/ly1 9 ! ?ird & ! mo/ ext5sive1 :i* con/itutes ! normal case1 ! two o!rs be+ only exceptional1 ! diff]5tial & ! integral calculus h ea* 9 _! turn a di/9ct & necess>y "p 9 ! complete solution ( ! problem1 ! =m] mak+ ! primitive diff]5tial equations und]go a prep>ation :i* is indisp5sable = ! application ( ! latt]4 ,su* are exactly _! g5]al relations1 ( :i* too indef9ite & inexact ideas are g5]ally =m$4 99 99 99 99 99 ,let u n[ take a g5]al survey ( ! logical composition ( ea* calculus1 beg9n+ ) ! diff]5tial4 ,,! ,,diff]5tial ,,calculus4 ,9 ! exposition ( ! transc5d5tal analysis1 x is cu/om>y to int]m+le ) ! purely analytical "p 7:i* r$uces itself to ! treatm5t ( ! ab/ract pr9ciples ( diff]5tiation & integration7 ! /udy ( its diff]5t pr9cipal applications1 especially ^? :i* conc]n geometry4 ,?is confusion ( ideas1 :i* is a consequ5ce ( ! actual mann] 9 :i* ! sci5ce has be5 develop$1 pres5ts1 9 ! dogmatic po9t ( view1 s]i\s inconv5i5ces 9 ?is respect1 t x makes x difficult prop]ly to conceive ei!r analysis or geometry4 ,hav+ to consid] "h ! mo/ rational co-ord9ation :i* is possible1 ,i %all include1 9 ! foll[+ sket*1 only ! calculus ( indirect functions prop]ly s call$1 res]v+ = ! portion ( ?is volume :i* relates to ! philosophical /udy ( concrete ma!matics ! g5]al exam9ation ( its great geometrical & me*anical applications4 ,two ,cases3 explicit & implicit ,functions4 ,! fundam5tal division ( ! diff]5tial calculus1 or ( ! g5]al subject ( diff]5tiation1 consi/s 9 di/+ui%+ two cases1 accord+ z ! analytical functions :i* are to be diff]5tiat$ are explicit or implicit2 f :i* fl[ two p>ts ord9>ily designat$ by ! names ( diff]5tiation ( =mulas & diff]5tiation ( equations4 ,x is easy to und]/&1 priori1 ! importance ( ?is classification4 ,9 fact1 su* a di/9ction w\ld be illusory if ! ord9>y analysis 0 p]fect2 t is1 if we knew h[ to resolve all equations algebraically1 = !n x w\ld be possible to r5d] e implicit function explicit2 &1 by diff]5tiat+ x 9 t /ate alone1 ! second "p ( ! diff]5tial calculus w\ld be imm$iately compris$ 9 ! fir/1 )\t giv+ rise to any new difficulty4 ,b ! algebraical resolution ( equations be+1 z we h se51 /ill almo/ 9 its infancy1 & z yet impossible = mo/ cases1 x is pla9 t ! case is v diff]5t1 s9ce we h1 prop]ly speak+1 to diff]5tiate a function )\t kn[+ x1 al?\gh x is det]m9ate4 ,! diff]5tiation ( implicit functions con/itutes !n1 by its nature1 a "q truly di/9ct f t pres5t$ by explicit functions1 & necess>ily m complicat$4 ,x is ?us evid5t t we mu/ comm5ce ) ! diff]5tiation ( =mulas1 & r$uce ! diff]5tiation ( equations to ?is prim>y case by c]ta9 inv>iable analytical consid]ations1 :i* ne$ n be "h m5tion$4 ,^! two g5]al cases ( diff]5tiation are also di/9ct 9 ano!r po9t ( view equally necess>y1 & too important to be left unnotic$4 ,! relation :i* is obta9$ betwe5 ! diff]5tials is con/antly m indirect1 9 comp>ison ) t ( ! f9ite quantities1 9 ! diff]5tiation ( implicit functions ?an 9 t ( explicit functions4 ,we "k1 9 fact1 f ! consid]ations pres5t$ by ,lagrange on ! g5]al =mation ( diff]5tial equations1 t1 on ! "o h&1 ! same primitive equation may give rise to a great] or less numb] ( d]iv$ equations ( v diff]5t =ms1 al?\gh at bottom equival5t1 dep5d+ ^u :i* ( ! arbitr>y con/ants is elim9at$1 :i* is n ! case 9 ! diff]5tiation ( explicit =mulas2 & t1 on ! o!r h&1 ! unlimit$ sy/em ( ! diff]5t primitive equations1 :i* correspond to ! same d]iv$ equation1 pres5ts a mu* m pr(\nd analytical v>iety ?an t ( ! diff]5t functions1 :i* admit ( "o same explicit diff]5tial1 & :i* are di/+ui%$ f ea* o!r only by a con/ant t]m4 ,implicit functions mu/ !re=e be reg>d$ z be+ 9 reality /ill m modifi$ by diff]5tiation ?an explicit functions4 ,we %all aga9 meet ) ?is consid]ation relatively to ! integral calculus1 :]e x acquires a prepond]ant importance4 ,two ,sub-cases3 ,a s+le ,v>iable or sev]al ,v>iables4 ,ea* ( ! two fundam5tal p>ts ( ! ,diff]5tial ,calculus is subdivid$ into two v di/9ct !ories1 accord+ z we are requir$ to diff]5tiate functions ( a s+le v>iable or functions ( sev]al indep5d5t v>iables4 ,?is second case is1 by its nature1 q di/9ct f ! fir/1 & evid5tly pres5ts m complication1 ev5 9 consid]+ only explicit functions1 & /ill m ^? :i* are implicit4 ,z to ! re/1 "o ( ^! cases is d$uc$ f ! o!r 9 a g5]al mann]1 by ! aid ( an inv>iable & v simple pr9ciple1 :i* consi/s 9 reg>d+ ! total diff]5tial ( a function :i* is produc$ by ! simultane\s increm5ts ( ! diff]5t indep5d5t v>iables :i* x conta9s1 z ! sum ( ! p>tial diff]5tials :i* w\ld be produc$ by ! sep>ate increm5t ( ea* v>iable 9 turn1 if all ! o!rs 7 con/ant4 ,x is necess>y1 besides1 c>efully to rem>k1 9 connection ) ?is subject1 a new idea :i* is introduc$ by ! di/9ction ( functions into ^? ( "o v>iable & ( sev]al2 x is ! consid]ation ( ^! diff]5t special d]iv$ functions1 relat+ to ea* v>iable sep>ately1 & ! numb] ( :i* increases m & m 9 proportion z ! ord] ( ! d]ivation becomes high]1 & also :5 ! v>iables become m num]\s4 ,x results f ?is t ! diff]5tial relations belong+ to functions ( sev]al v>iables are1 by _! nature1 bo? mu* m indirect1 & especially mu* m indet]m9ate1 ?an ^? relat+ to functions ( a s+le v>iable4 ,?is is mo/ app>5t 9 ! case ( implicit functions1 9 :i*1 9 ! place ( ! simple arbitr>y con/ants :i* elim9ation causes to disappe> :5 we =m ! prop] diff]5tial equations = functions ( a s+le v>iable1 x is ! arbitr>y functions ( ! propos$ v>iables :i* are !n elim9at$2 :5ce mu/ result special difficulties :5 ^! equations come to be integrat$4 ,f9ally1 to complete ?is summ>y sket* ( ! diff]5t ess5tial p>ts ( ! diff]5tial calculus prop]1 ,i %\ld add1 t 9 ! diff]5tiation ( implicit functions1 :e!r ( a s+le v>iable or ( sev]al1 x is necess>y to make ano!r di/9ction2 t ( ! case 9 :i* x is requir$ to diff]5tiate at once diff]5t functions ( ?is k9d1 comb9$ 9 c]ta9 primitive equations1 f t 9 :i* all ^! functions are sep>ate4 ,! functions are evid5tly1 9 fact1 /ill m implicit 9 ! fir/ case ?an 9 ! second1 if we consid] t ! same imp]fection ( ord9>y analysis1 :i* =bids our conv]t+ e implicit function into an equival5t explicit function1 9 l mann] r5d]s u unable to sep>ate ! functions :i* ent] simultane\sly into any sy/em ( equations4 ,x is !n necess>y to diff]5tiate1 n only )\t kn[+ h[ to resolve ! primitive equations1 b ev5 )\t be+ able to effect ! prop] elim9ations among !m1 ?us produc+ a new difficulty4 ,r$uction ( ! :ole to ! ,diff]5tiation ( ! t5 elem5t>y ,functions4 ,su*1 !n1 are ! natural connection & ! logical di/ribution ( ! diff]5t pr9cipal !ories :i* compose ! g5]al sy/em ( diff]5tiation4 ,s9ce ! diff]5tiation ( implicit functions is d$uc$ f t ( explicit functions by a s+le con/ant pr9ciple1 & ! diff]5tiation ( functions ( sev]al v>iables is r$uc$ by ano!r fix$ pr9ciple to t ( functions ( a s+le v>iable1 ! :ole ( ! diff]5tial calculus is f9ally f\nd to re/ ^u ! diff]5tiation ( explicit functions ) a s+le v>iable1 ! only "o :i* is ev] execut$ directly4 ,n[ x is easy to und]/& t ?is fir/ !ory1 ! necess>y basis ( ! entire sy/em1 consi/s simply 9 ! diff]5tiation ( ! t5 simple functions1 :i* are ! uni=m elem5ts ( all our analytical comb9ations1 & ! li/ ( :i* has be5 giv5 9 ! fir/ *apt]1 on page #ea2 = ! diff]5tiation ( comp\nd functions is evid5tly d$uc$1 9 an imm$iate & necess>y mann]1 f t ( ! simple functions :i* compose !m4 ,x is1 !n1 to ! k ( ^! t5 fundam5tal diff]5tials1 & to t ( ! two g5]al pr9ciples j m5tion$1 :i* br+ "u x all ! o!r possible cases1 t ! :ole sy/em ( diff]5tiation is prop]ly r$uc$4 ,we see1 by ! comb9ation ( ^! diff]5t consid]ations1 h[ simple & h[ p]fect is ! entire sy/em ( ! diff]5tial calculus4 ,x c]ta9ly con/itutes1 9 its logical relations1 ! mo/ int]e/+ spectacle :i* ma!matical analysis c pres5t to our und]/&+4 ,trans=mation ( d]iv$ ,functions = new ,v>iables4 ,! g5]al sket* :i* ,i h j summ>ily drawn w\ld nev]!less pres5t an important defici5cy1 if ,i did n "h di/9ctly indicate a f9al !ory1 :i* =ms1 by its nature1 ! indisp5sable complem5t ( ! sy/em ( diff]5tiation4 ,x is t :i* has = its object ! con/ant trans=mation ( d]iv$ functions1 z a result ( det]m9ate *anges 9 ! indep5d5t v>iables1 :5ce results ! possibility ( ref]r+ to new v>iables all ! g5]al diff]5tial =mulas primitively e/abli%$ = o!rs4 ,?is "q is n[ resolv$ 9 ! mo/ complete & ! mo/ simple mann]1 z are all ^? ( :i* ! diff]5tial calculus is compos$4 ,x is easy to conceive ! g5]al importance :i* x mu/ h 9 any ( ! applications ( ! transc5d5tal analysis1 ! fundam5tal res\rces ( :i* x may be consid]$ z augm5t+1 by p]mitt+ u to *oose 79 ord] to =m ! diff]5tial equations1 9 ! fir/ place1 ) m ease7 t sy/em ( indep5d5t v>iables :i* may appe> to be ! mo/ advantage\s1 al?\gh x is n to be f9ally reta9$4 ,x is ?us1 = example1 t mo/ ( ! pr9cipal que/ions ( geometry are resolv$ mu* m easily by ref]r+ ! l9es & surfaces to rectil9e> co-ord9ates1 & t we may1 nev]!less1 h occasion to express ^! l9es1 etc41 analytically by ! aid ( pol> co-ord9ates1 or 9 any o!r mann]4 ,we w !n be able to comm5ce ! diff]5tial solution ( ! problem by employ+ ! rectil9e> sy/em1 b only z an int]m$iate /ep1 f :i*1 by ! g5]al !ory "h ref]r$ to1 we c pass to ! f9al sy/em1 :i* sometimes c\ld n h be5 consid]$ directly4 ,diff]5t ,ord]s ( ,diff]5tiation4 ,9 ! logical classification ( ! diff]5tial calculus :i* has j be5 giv51 "s may be incl9$ to sugge/ a s]i\s omission1 s9ce ,i h n subdivid$ ea* ( its f\r ess5tial p>ts accord+ to ano!r g5]al consid]ation1 :i* seems at fir/ view v important2 namely1 t ( ! high] or l[] ord] ( diff]5tiation4 ,b x is easy to und]/& t ?is di/9ction has no real influ5ce 9 ! diff]5tial calculus1 inasmu* z x does n give rise to any new difficulty4 ,if1 inde$1 ! diff]5tial calculus 0 n rigor\sly complete1 t is1 if we did n "k h[ to diff]5tiate at w any function :atev]1 ! diff]5tiation to ! second or high] ord] ( ea* det]m9ate function might eng5d] special difficulties4 ,b ! p]fect univ]sality ( ! diff]5tial calculus pla9ly gives u ! assurance ( be+ able to diff]5tiate1 to any ord] :atev]1 all kn[n functions :atev]1 ! "q r$uc+ itself to a con/antly repeat$ diff]5tiation ( ! fir/ ord]4 ,?is di/9ction1 unimportant z x is = ! diff]5tial calculus1 acquires1 h[ev]1 a v great importance 9 ! integral calculus1 on acc\nt ( ! extreme imp]fection ( ! latt]4 ,analytical ,applications4 ,f9ally1 ?\gh ?is is n ! place to consid] ! v>i\s applications ( ! diff]5tial calculus1 yet an exception may be made = ^? :i* consi/ 9 ! solution ( que/ions :i* are purely analytical1 :i* ought1 inde$1 to be logically treat$ 9 cont9uation ( a sy/em ( diff]5tiation1 because ( ! evid5t homog5eity ( ! consid]ations involv$4 ,^! que/ions may be r$uc$ to ?ree ess5tial ones4 ,fir/ly1 ! developm5t into s]ies ( functions ( "o or m v>iables1 or1 m g5]ally1 ! trans=mation ( functions1 :i* con/itutes ! mo/ beautiful & ! mo/ important application ( ! diff]5tial calculus to g5]al analysis1 & :i* comprises1 besides ! fundam5tal s]ies discov]$ by ,taylor1 ! rem>kable s]ies discov]$ by ,maclaur91 ,john ,b]n\illi1 ,lagrange1 c43 ,secondly1 ! g5]al !ory ( maxima & m9ima values = any functions :atev]1 ( "o or m v>iables2 "o ( ! mo/ int]e/+ problems :i* analysis c pres5t1 h[ev] elem5t>y x may n[ h become1 & to ! complete solution ( :i* ! diff]5tial calculus naturally applies3 ,?irdly1 ! g5]al det]m9ation ( ! true value ( functions :i* pres5t !mselves "u an indet]m9ate appe>ance = c]ta9 hypo!ses made on ! values ( ! correspond+ v>iables2 :i* is ! lea/ ext5sive & ! lea/ important ( ! ?ree4 ,! fir/ "q is c]ta9ly ! pr9cipal "o 9 all po9ts ( view2 x is also ! mo/ susceptible ( receiv+ a new ext5sion h]eaft]1 especially by conceiv+1 9 a broad] mann] ?an has yet be5 done1 ! employm5t ( ! diff]5tial calculus 9 ! trans=mation ( functions1 on :i* subject ,lagrange has left "s valuable h9ts4 99 99 99 99 99 ,hav+ ?us summ>ily1 ?\gh p]haps too briefly1 consid]$ ! *ief po9ts 9 ! diff]5tial calculus1 ,i n[ proce$ to an equally rapid exposition ( a sy/ematic outl9e ( ! ,integral ,calculus1 prop]ly s call$1 t is1 ! ab/ract subject ( integration4 ,,! ,,integral ,,calculus4 ,its ,fundam5tal ,division4 ,! fundam5tal division ( ! ,integral ,calculus is f\nd$ on ! same pr9ciple z t ( ! ,diff]5tial ,calculus1 9 di/+ui%+ ! integration ( explicit diff]5tial =mulas1 & ! integration ( implicit diff]5tials or ( diff]5tial equations4 ,! sep>ation ( ^! two cases is ev5 mu* m pr(\nd 9 relation to integration ?an to diff]5tiation4 ,9 ! diff]5tial calculus1 9 fact1 ?is di/9ction re/s1 z we h se51 only on ! extreme imp]fection ( ord9>y analysis4 ,b1 on ! o!r h&1 x is easy to see t1 ev5 ?\gh all equations c\ld be algebraically resolv$1 diff]5tial equations w\ld none ! less con/itute a case ( integration q di/9ct f t pres5t$ by ! explicit diff]5tial =mulas2 =1 limit+ ourselves1 = ! sake ( simplicity1 to ! fir/ ord]1 & to a s+le function y ( a s+le v>iable x1 if we suppose any diff]5tial equation betwe5 x1 y1 & dy_/dx1 to be resolv$ ) ref]5ce to dy_/dx1 ! expression ( ! d]iv$ function be+ !n g5]ally f\nd to conta9 ! primitive function itself1 :i* is ! object ( ! inquiry1 ! "q ( integration w n h at all *ang$ its nature1 & ! solution w n really h made any o!r progress ?an t ( hav+ br\ght ! propos$ diff]5tial equation to be ( only ! fir/ degree relatively to ! d]iv$ function1 :i* is 9 itself ( little importance4 ,! diff]5tial w\ld n !n be det]m9$ 9 a mann] mu* less implicit ?an be=e1 z reg>ds ! integration1 :i* w\ld cont9ue to pres5t ess5tially ! same *>act]i/ic difficulty4 ,! algebraic resolution ( equations c\ld n make ! case :i* we are consid]+ come )9 ! simple integration ( explicit diff]5tials1 except 9 ! special cases 9 :i* ! propos$ diff]5tial equation did n conta9 ! primitive function itself1 :i* w\ld consequ5tly p]mit u1 by resolv+ x1 to f9d dy_/dx 9 t]ms ( x only1 & ?us to r$uce ! "q to ! class ( quadratures4 ,/ill great] difficulties w\ld evid5tly be f\nd 9 diff]5tial equations ( high] ord]s1 or conta9+ simultane\sly diff]5t functions ( sev]al indep5d5t v>iables4 ,! integration ( diff]5tial equations is !n necess>ily m complicat$ ?an t ( explicit diff]5tials1 by ! elaboration ( :i* la/ ! integral calculus has be5 creat$1 & ^u :i* ! o!rs h be5 made to dep5d z f> z x has be5 possible4 ,all ! v>i\s analytical me?ods :i* h be5 propos$ = integrat+ diff]5tial equations1 :e!r x be ! sep>ation ( ! v>iables1 ! me?od ( multipli]s1 c41 h 9 fact = _! object to r$uce ^! integrations to ^? ( diff]5tial =mulas1 ! only "o :i*1 by its nature1 c be und]tak5 directly4 ,un=tunately1 imp]fect z is /ill ?is necess>y base ( ! :ole integral calculus1 ! art ( r$uc+ to x ! integration ( diff]5tial equations is /ill less advanc$4 ,subdivisions3 "o v>iable or sev]al4 ,ea* ( ^! two fundam5tal bran*es ( ! integral calculus is next subdivid$ into two o!rs 7z 9 ! diff]5tial calculus1 & = precisely analog\s reasons71 accord+ z we consid] functions ) a s+le v>iable1 or functions ) sev]al indep5d5t v>iables4 ,?is di/9ction is1 l ! prec$+ "o1 /ill m important = integration ?an = diff]5tiation4 ,?is is especially rem>kable 9 ref]5ce to diff]5tial equations4 ,inde$1 ^? :i* dep5d on sev]al indep5d5t v>iables may evid5tly pres5t ?is *>act]i/ic & mu* m s]i\s difficulty1 t ! desir$ function may be diff]5tially def9$ by a simple relation betwe5 its diff]5t special d]ivatives relative to ! diff]5t v>iables tak5 sep>ately4 ,h5ce results ! mo/ difficult & also ! mo/ ext5sive bran* ( ! integral calculus1 :i* is commonly nam$ ! ,integral ,calculus ( p>tial diff]5ces1 creat$ by ,d',alemb]t1 & 9 :i*1 accord+ to ! j appreciation ( ,lagrange1 geomet]s ought to h se5 a really new calculus1 ! philosophical "* ( :i* has n yet be5 det]m9$ ) suffici5t exactness4 ,a v /rik+ diff]5ce betwe5 ?is case & t ( equations ) a s+le indep5d5t v>iable consi/s1 z has be5 already obs]v$1 9 ! arbitr>y functions :i* take ! place ( ! simple arbitr>y con/ants1 9 ord] to give to ! correspond+ integrals all ! prop] g5]ality4 ,x is sc>cely necess>y to say t ?is high] bran* ( transc5d5tal analysis is /ill entirely 9 its infancy1 s9ce1 ev5 9 ! mo/ simple case1 t ( an equation ( ! fir/ ord] betwe5 ! p>tial d]ivatives ( a s+le function ) two indep5d5t v>iables1 we are n yet completely able to r$uce ! integration to t ( ! ord9>y diff]5tial equations4 ,! integration ( functions ( sev]al v>iables is mu* f>!r advanc$ 9 ! case 7inf9itely m simple inde$7 9 :i* x has to d ) only explicit diff]5tial =mulas4 ,we c !n1 9 fact1 :5 ^! =mulas fulfil ! necess>y conditions ( integrability1 always r$uce _! integration to quadratures4 ,o!r ,subdivisions3 diff]5t ,ord]s ( ,diff]5tiation4 ,a new g5]al di/9ction1 applicable z a subdivision to ! integration ( explicit or implicit diff]5tials1 ) "o v>iable or sev]al1 is drawn f ! high] or l[] ord] ( ! diff]5tials3 a di/9ction :i*1 z we h above rem>k$1 does n give rise to any special "q 9 ! diff]5tial calculus4 ,relatively to explicit diff]5tials1 :e!r ( "o v>iable or ( sev]al1 ! necessity ( di/+ui%+ _! diff]5t ord]s belongs only to ! extreme imp]fection ( ! integral calculus4 ,9 fact1 if we c\ld always integrate e diff]5tial =mula ( ! fir/ ord]1 ! integration ( a =mula ( ! second ord]1 or ( any o!r1 w\ld evid5tly n =m a new "q1 s9ce1 by integrat+ x at fir/ 9 ! fir/ degree1 we w\ld arrive at ! diff]5tial expression ( ! imm$iately prec$+ ord]1 f :i*1 by a suitable s]ies ( analog\s integrations1 we w\ld be c]ta9 ( f9ally arriv+ at ! primitive function1 ! f9al object ( ^! op]ations4 ,b ! little k :i* we possess on integration ( ev5 ! fir/ ord] causes q ano!r /ate ( affairs1 s t a high] ord] ( diff]5tials produces new difficulties2 =1 hav+ diff]5tial =mulas ( any ord] above ! fir/1 x may happ5 t we may be able to integrate !m1 ei!r once1 or sev]al times 9 succession1 & t we may /ill be unable to g back to ! primitive functions1 if ^! prelim9>y lab\rs h produc$1 = ! diff]5tials ( a l[] ord]1 expressions ^: integrals are n kn[n4 ,?is circum/ance mu/ occur s mu* ! (t5] 7! numb] ( kn[n integrals be+ /ill v small71 see+ t ^! successive integrals are g5]ally v diff]5t functions f ! d]ivatives :i* h produc$ !m4 ,) ref]5ce to implicit diff]5tials1 ! di/9ction ( ord]s is /ill m important2 =1 besides ! prec$+ reason1 ! influ5ce ( :i* is evid5tly analog\s 9 ?is case1 & is ev5 great]1 x is easy to p]ceive t ! high] ord] ( ! diff]5tial equations necess>ily gives rise to que/ions ( a new nature4 ,9 fact1 ev5 if we c\ld integrate e equation ( ! fir/ ord] relat+ to a s+le function1 t w\ld n be suffici5t = obta9+ ! f9al integral ( an equation ( any ord] :atev]1 inasmu* z e diff]5tial equation is n r$ucible to t ( an imm$iately inf]ior ord]4 ,?us1 = example1 if we h giv5 any relation betwe5 x1 y1 dx_/dy1 & dy_/dx1 to det]m9e a function y ( a v>iable x1 we %all n be able to d$uce f x at once1 aft] effect+ a fir/ integration1 ! correspond+ diff]5tial relation betwe5 x1 y1 & dy_/dx1 f :i*1 by a second integration1 we c\ld asc5d to ! primitive equations4 ,?is w\ld n necess>ily take place1 at lea/ )\t introduc+ new auxili>y functions1 unless ! propos$ equation ( ! second ord] did n conta9 ! requir$ function y1 toge!r ) its d]ivatives4 ,z a g5]al pr9ciple1 diff]5tial equations w h to be reg>d$ z pres5t+ cases :i* are m & m implicit1 z !y are ( a high] ord]1 & :i* _c be made to dep5d on "o ano!r except by special me?ods1 ! inve/igation ( :i* consequ5tly =ms a new class ( que/ions1 ) respect to :i* we z yet "k sc>cely any ?+1 ev5 = functions ( a s+le v>iable4#aj ,footnote #aj3 ,! only important case ( ?is class :i* has ?us f> be5 completely treat$ is ! g5]al integration ( l9e> equations ( any ord] :atev]1 ) con/ant coeffici5ts4 ,ev5 ?is case f9ally dep5ds on ! algebraic resolution ( equations ( a degree equal to ! ord] ( diff]5tiation4 ,ano!r equival5t di/9ction4 ,/ill f>!r1 :5 we exam9e m pr(\ndly ?is di/9ction ( diff]5t ord]s ( diff]5tial equations1 we f9d t x c be always made to come "u a f9al g5]al di/9ction1 relative to diff]5tial equations1 :i* rema9s to be notic$4 ,diff]5tial equations ) "o or m indep5d5t v>iables may conta9 simply a s+le function1 or 79 a case evid5tly m complicat$ & m implicit1 :i* corresponds to ! diff]5tiation ( simultane\s implicit functions7 we may h to det]m9e at ! same "t sev]al functions f ! diff]5tial equations 9 :i* !y are f\nd unit$1 toge!r ) _! diff]5t d]ivatives4 ,x is cle> t su* a /ate ( ! "q necess>ily pres5ts a new special difficulty1 t ( sep>at+ ! diff]5t functions desir$1 by =m+ = ea*1 f ! propos$ diff]5tial equations1 an isolat$ diff]5tial equation :i* does n conta9 ! o!r functions or _! d]ivatives4 ,?is prelim9>y lab\r1 :i* is analog\s to ! elim9ation ( algebra1 is evid5tly indisp5sable be=e attempt+ any direct integration1 s9ce we _c und]take g5]ally 7except by special artifices :i* are v r>ely applicable7 to det]m9e directly sev]al di/9ct functions at once4 ,n[ x is easy to e/abli% ! exact & necess>y co9cid5ce ( ?is new di/9ction ) ! prec$+ "o respect+ ! ord] ( diff]5tial equations4 ,we "k1 9 fact1 t ! g5]al me?od = isolat+ functions 9 simultane\s diff]5tial equations consi/s ess5tially 9 =m+ diff]5tial equations1 sep>ately 9 relation to ea* function1 & ( an ord] equal to ! sum ( all ^? ( ! diff]5t propos$ equations4 ,?is trans=mation c always be effect$4 ,on ! o!r h&1 e diff]5tial equation ( any ord] 9 relation to a s+le function might evid5tly always be r$uc$ to ! fir/ ord]1 by introduc+ a suitable numb] ( auxili>y diff]5tial equations1 conta9+ at ! same "t ! diff]5t ant]ior d]ivatives reg>d$ z new functions to be det]m9$4 ,?is me?od has1 inde$1 sometimes be5 actually employ$ ) success1 ?\gh x is n ! natural "o4 ,"h1 !n1 are two necess>ily equival5t ord]s ( conditions 9 ! g5]al !ory ( diff]5tial equations2 ! simultane\sness ( a great] or small] numb] ( functions1 & ! high] or l[] ord] ( diff]5tiation ( a s+le function4 ,by augm5t+ ! ord] ( ! diff]5tial equations1 we c isolate all ! functions2 &1 by artificially multiply+ ! numb] ( ! functions1 we c r$uce all ! equations to ! fir/ ord]4 ,!re is1 consequ5tly1 9 bo? cases1 only "o & ! same difficulty f two diff]5t po9ts ( sight4 ,b1 h[ev] we may conceive x1 ?is new difficulty is none ! less real1 & con/itutes none ! less1 by its nature1 a m>k$ sep>ation betwe5 ! integration ( equations ( ! fir/ ord] & t ( equations ( a high] ord]4 ,i pref] to indicate ! di/9ction "u ?is la/ =m z be+ m simple1 m g5]al1 & m logical4 ,quadratures4 ,f ! diff]5t consid]ations :i* h be5 indicat$ respect+ ! logical dep5d5ce ( ! v>i\s pr9cipal p>ts ( ! integral calculus1 we see t ! integration ( explicit diff]5tial =mulas ( ! fir/ ord] & ( a s+le v>iable is ! necess>y basis ( all o!r integrations1 :i* we nev] succe$ 9 effect+ b s f> z we r$uce !m to ?is elem5t>y case1 evid5tly ! only "o :i*1 by its nature1 is capable ( be+ treat$ directly4 ,?is simple fundam5tal integration is (t5 designat$ by ! conv5i5t expression ( quadratures1 see+ t e integral ( ?is k9d1 ,sf7x7dx1 may1 9 fact1 be reg>d$ z repres5t+ ! area ( a curve1 ! equation ( :i* 9 rectil9e> co-ord9ates w\ld be y f7x74 ,su* a class ( que/ions corresponds1 9 ! diff]5tial calculus1 to ! elem5t>y case ( ! diff]5tiation ( explicit functions ( a s+le v>iable4 ,b ! integral "q is1 by its nature1 v diff]5tly complicat$1 & especially mu* m ext5sive ?an ! diff]5tial "q4 ,?is latt] is1 9 fact1 necess>ily r$uc$1 z we h se51 to ! diff]5tiation ( ! t5 simple functions1 ! elem5ts ( all :i* are consid]$ 9 analysis4 ,on ! o!r h&1 ! integration ( comp\nd functions does n necess>ily foll[ f t ( ! simple functions1 ea* comb9ation ( :i* may pres5t special difficulties ) respect to ! integral calculus4 ,h5ce results ! naturally indef9ite ext5t1 & ! s v>i$ complication ( ! "q ( quadratures1 ^u :i*1 9 spite ( all ! ef=ts ( analy/s1 we /ill possess s little complete k4 ,9 decompos+ ?is "q1 z is natural1 accord+ to ! diff]5t =ms :i* may be assum$ by ! d]ivative function1 we di/+ui% ! case ( algebraic functions & t ( transc5d5tal functions4 ,integration ( ,transc5d5tal ,functions4 ,! truly analytical integration ( transc5d5tal functions is z yet v little advanc$1 :e!r = expon5tial1 or = log>i?mic1 or = circul> functions4 ,b a v small numb] ( cases ( ^! ?ree diff]5t k9ds h z yet be5 treat$1 & ^? *os5 f among ! simple/2 & /ill ! necess>y calculations are 9 mo/ cases extremely labori\s4 ,a circum/ance :i* we ought p>ticul>ly to rem>k 9 its philosophical connection is1 t ! diff]5t proc$ures ( quadrature h no relation to any g5]al view ( integration1 & consi/ ( simple artifices v incoh]5t ) ea* o!r1 & v num]\s1 because ( ! v limit$ ext5t ( ea*4 ,"o ( ^! artifices %\ld1 h[ev]1 "h be notic$1 :i*1 )\t be+ really a me?od ( integration1 is nev]!less rem>kable = its g5]ality2 x is ! proc$ure inv5t$ by ,john ,b]n\illi1 & kn[n "u ! "n ( integration by p>ts1 by means ( :i* e integral may be r$uc$ to ano!r :i* is sometimes f\nd to be m easy to be obta9$4 ,?is +5i\s relation des]ves to be notic$ = ano!r reason1 z hav+ sugge/$ ! fir/ idea ( t trans=mation ( integrals yet unkn[n1 :i* has lately receiv$ a great] ext5sion1 & ( :i* ,m4 ,f\ri] especially has made s new & important a use 9 ! analytical que/ions produc$ by ! !ory ( heat4 ,integration ( ,algebraic ,functions4 ,z to ! integration ( algebraic functions1 x is f>!r advanc$4 ,h[ev]1 we "k sc>cely any ?+ 9 relation to irrational functions1 ! integrals ( :i* h be5 obta9$ only 9 extremely limit$ cases1 & p>ticul>ly by r5d]+ !m rational4 ,! integration ( rational functions is ?us f> ! only !ory ( ! integral calculus :i* has admitt$ ( be+ treat$ 9 a truly complete mann]2 9 a logical po9t ( view1 x =ms1 !n1 its mo/ satisfactory "p1 b p]haps also ! lea/ important4 ,x is ev5 ess5tial to rem>k1 9 ord] to h a j idea ( ! extreme imp]fection ( ! integral calculus1 t ?is case1 limit$ z x is1 is n entirely resolv$ except = :at prop]ly conc]ns integration view$ 9 an ab/ract mann]2 =1 9 ! execution1 ! !ory f9ds its progress mo/ frequ5tly q /opp$1 indep5d5tly ( ! complication ( ! calculations1 by ! imp]fection ( ord9>y analysis1 see+ t x makes ! integration f9ally dep5d ^u ! algebraic resolution ( equations1 :i* greatly limits its use4 ,to grasp 9 a g5]al mann] ! _s ( ! diff]5t proc$ures :i* are employ$ 9 quadratures1 we mu/ obs]ve t1 by _! nature1 !y c be primitively f\nd$ only on ! diff]5tiation ( ! t5 simple functions4 ,! results ( ?is1 conv]sely consid]$1 e/abli% z _m direct !orems ( ! integral calculus1 ! only ones :i* c be directly kn[n4 ,all ! art ( integration aft]w>ds consi/s1 z has be5 said 9 ! beg9n+ ( ?is *apt]1 9 r$uc+ all ! o!r quadratures1 s f> z is possible1 to ?is small numb] ( elem5t>y ones1 :i* unhappily we are 9 mo/ cases unable to effect4 ,s+ul> ,solutions4 ,9 ?is sy/ematic enum]ation ( ! v>i\s ess5tial p>ts ( ! integral calculus1 consid]$ 9 _! logical relations1 ,i h design$ly neglect$ 79 ord] n to break ! *a9 ( sequ5ce7 to consid] a v important !ory1 :i* =ms implicitly a portion ( ! g5]al !ory ( ! integration ( diff]5tial equations1 b :i* ,i ought "h to notice sep>ately1 z be+1 s to speak1 outside ( ! integral calculus1 & be+ nev]!less ( ! greate/ int]e/1 bo? by its logical p]fection & by ! ext5t ( its applications4 ,i ref] to :at are call$ ,s+ul> ,solutions ( diff]5tial equations1 call$ sometimes1 b improp]ly1 p>ticul> solutions1 :i* h be5 ! subject ( v rem>kable inve/igations by ,eul] & ,laplace1 & ( :i* ,lagrange especially has pres5t$ su* a beautiful & simple g5]al !ory4 ,clairaut1 :o fir/ _h occasion to rem>k _! exi/5ce1 saw 9 !m a p>adox ( ! integral calculus1 s9ce ^! solutions h ! peculi>ity ( satisfy+ ! diff]5tial equations )\t be+ compris$ 9 ! correspond+ g5]al integrals4 ,lagrange has s9ce expla9$ ?is p>adox 9 ! mo/ +5i\s & mo/ satisfactory mann]1 by %[+ h[ su* solutions are always d]iv$ f ! g5]al integral by ! v>iation ( ! arbitr>y con/ants4 ,he 0 also ! fir/ to suitably appreciate ! importance ( ?is !ory1 & x is ) good reason t he devot$ to x s full a developm5t 9 8 ,calculus ( ,functions4 ,9 a logical po9t ( view1 ?is !ory des]ves all our att5tion by ! "* ( p]fect g5]ality :i* x admits (1 s9ce ,lagrange has giv5 inv>iable & v simple proc$ures = f9d+ ! s+ul> solution ( any diff]5tial equation :i* is susceptible ( x2 &1 :at is no less rem>kable1 ^! proc$ures require no integration1 consi/+ only ( diff]5tiations1 & are !re=e always applicable4 ,diff]5tiation has ?us become1 by a happy artifice1 a means ( comp5sat+1 9 c]ta9 circum/ances1 = ! imp]fection ( ! integral calculus4 ,inde$1 c]ta9 problems especially require1 by _! nature1 ! k ( ^! s+ul> solutions2 su*1 = example1 9 geometry1 are all ! que/ions 9 :i* a curve is to be det]m9$ f any prop]ty ( its tang5t or its osculat+ circle4 ,9 all cases ( ?is k9d1 aft] hav+ express$ ?is prop]ty by a diff]5tial equation1 x w be1 9 its analytical relations1 ! s+ul> equation :i* w =m ! mo/ important object ( ! inquiry1 s9ce x alone w repres5t ! requir$ curve2 ! g5]al integral1 :i* !nce=? x becomes unnecess>y to "k1 designat+ only ! sy/em ( ! tang5ts1 or ( ! osculat+ circles ( ?is curve4 ,we may h5ce easily und]/& all ! importance ( ?is !ory1 :i* seems to me to be n z yet suffici5tly appreciat$ by mo/ geomet]s4 ,def9ite ,integrals4 ,f9ally1 to complete our review ( ! va/ collection ( analytical rese>*es ( :i* is compos$ ! integral calculus1 prop]ly s call$1 !re rema9s to be m5tion$ "o !ory1 v important 9 all ! applications ( ! transc5d5tal analysis1 :i* ,i h _h to leave outside ( ! sy/em1 z n be+ really de/9$ = v]itable integration1 & propos+1 on ! contr>y1 to supply ! place ( ! k ( truly analytical integrals1 :i* are mo/ g5]ally unkn[n4 ,i ref] to ! det]m9ation ( def9ite integrals4 ,! expression1 always possible1 ( integrals 9 inf9ite s]ies1 may at fir/ be view$ z a happy g5]al means ( comp5sat+ = ! extreme imp]fection ( ! integral calculus4 ,b ! employm5t ( su* s]ies1 because ( _! complication1 & ( ! difficulty ( discov]+ ! law ( _! t]ms1 is commonly ( only mod]ate utility 9 ! algebraic po9t ( view1 al?\gh sometimes v ess5tial relations h be5 !nce d$uc$4 ,x is p>ticul>ly 9 ! ari?metical po9t ( view t ?is proc$ure acquires a great importance1 z a means ( calculat+ :at are call$ def9ite integrals1 t is1 ! values ( ! requir$ functions = c]ta9 det]m9ate values ( ! correspond+ v>iables4 ,an inquiry ( ?is nature exactly corresponds1 9 transc5d5tal analysis1 to ! num]ical resolution ( equations 9 ord9>y analysis4 ,be+ g5]ally unable to obta9 ! v]itable integral--nam$ by opposition ! g5]al or indef9ite integral2 t is1 ! function :i*1 diff]5tiat$1 has produc$ ! propos$ diff]5tial =mula--analy/s h be5 oblig$ to employ !mselves 9 det]m9+ at lea/1 )\t kn[+ ?is function1 ! p>ticul> num]ical values :i* x w\ld take on assign+ c]ta9 designat$ values to ! v>iables4 ,?is is evid5tly resolv+ ! ari?metical "q )\t hav+ previ\sly resolv$ ! correspond+ algebraic "o1 :i* mo/ g5]ally is ! mo/ important "o4 ,su* an analysis is1 !n1 by its nature1 z imp]fect z we h se5 ! num]ical resolution ( equations to be4 ,x pres5ts1 l ?is la/1 a vici\s confusion ( ari?metical & algebraic consid]ations1 :5ce result analog\s inconv5i5ces bo? 9 ! purely logical po9t ( view & 9 ! applications4 ,we ne$ n "h repeat ! consid]ations sugge/$ 9 our ?ird *apt]4 ,b x w be und]/ood t1 unable z we almo/ always are to obta9 ! true integrals1 x is ( ! highe/ importance to h be5 able to obta9 ?is solution1 incomplete & necess>ily insuffici5t z x is4 ,n[ ?is has be5 =tunately atta9$ at ! pres5t day = all cases1 ! det]m9ation ( ! value ( def9ite integrals hav+ be5 r$uc$ to entirely g5]al me?ods1 :i* leave no?+ to desire1 9 a great numb] ( cases1 b less complication 9 ! calculations1 an object t[>ds :i* are at pres5t direct$ all ! special trans=mations ( analy/s4 ,reg>d+ n[ ?is sort ( transc5d5tal ari?metic z p]fect1 ! difficulty 9 ! applications is ess5tially r$uc$ to mak+ ! propos$ rese>* dep5d1 f9ally1 on a simple det]m9ation ( def9ite integrals1 :i* evid5tly _c always be possible1 :atev] analytical skill may be employ$ 9 effect+ su* a trans=mation4 ,prospects ( ! ,integral ,calculus4 ,f ! consid]ations indicat$ 9 ?is *apt]1 we see t1 :ile ! diff]5tial calculus con/itutes by its nature a limit$ & p]fect sy/em1 to :i* no?+ ess5tial rema9s to be add$1 ! integral calculus1 or ! simple sy/em ( integration1 pres5ts necess>ily an inexhau/ible field = ! activity ( ! human m9d1 indep5d5tly ( ! indef9ite applications ( :i* ! transc5d5tal analysis is evid5tly susceptible4 ,! g5]al argum5t by :i* ,i h endeav\r$1 9 ! second *apt]1 to make app>5t ! impossibility ( ev] discov]+ ! algebraic solution ( equations ( any degree & =m :atsoev]1 has und\bt$ly inf9itely m =ce ) reg>d to ! se>* = a s+le me?od ( integration1 inv>iably applicable to all cases4 ,x is1 says ,lagrange1 "o ( ^? problems ^: g5]al solution we _c hope =4 ,! m we m$itate on ?is subject1 ! m we w be conv9c$ t su* a rese>* is utt]ly *im]ical1 z be+ f> above ! feeble rea* ( our intellig5ce2 al?\gh ! lab\rs ( geomet]s mu/ c]ta9ly augm5t h]eaft] ! am\nt ( our k respect+ integration1 & ?us create me?ods ( great] g5]ality4 ,! transc5d5tal analysis is /ill too ne> its orig9--!re is especially too little "t s9ce x has be5 conceiv$ 9 a truly rational mann]--= u n[ to be able to h a correct idea ( :at x w h]eaft] become4 ,b1 :atev] %\ld be our legitimate hopes1 let u n =get to consid]1 be=e all1 ! limits :i* are impos$ by our intellectual con/itution1 & :i*1 ?\gh n susceptible ( a precise det]m9ation1 h none ! less an inconte/able reality4 ,i am induc$ to ?9k t1 :5 geomet]s %all h exhau/$ ! mo/ important applications ( our pres5t transc5d5tal analysis1 in/ead ( /riv+ to impress ^u x1 z n[ conceiv$1 a *im]ical p]fection1 !y w r create new res\rces by *ang+ ! mode ( d]ivation ( ! auxili>y quantities introduc$ 9 ord] to facilitate ! e/abli%m5t ( equations1 & ! =mation ( :i* might foll[ an inf9ity ( o!r laws besides ! v simple relation :i* has be5 *os51 accord+ to ! conception sugge/$ 9 ! fir/ *apt]4 ,! res\rces ( ?is nature appe> to me susceptible ( a mu* great] fecundity ?an ^? :i* w\ld consi/ ( m]ely pu%+ f>!r our pres5t calculus ( indirect functions4 ,x is a sugge/ion :i* ,i submit to ! geomet]s :o h turn$ _! ?\ghts t[>ds ! g5]al philosophy ( analysis4 ,f9ally1 al?\gh1 9 ! summ>y exposition :i* 0 ! object ( ?is *apt]1 ,i h _h to exhibit ! condition ( extreme imp]fection :i* /ill belongs to ! integral calculus1 ! /ud5t w\ld h a false idea ( ! g5]al res\rces ( ! transc5d5tal analysis if he gave t consid]ation too great an importance4 ,x is ) x1 inde$1 z ) ord9>y analysis1 9 :i* a v small am\nt ( fundam5tal k respect+ ! resolution ( equations has be5 employ$ ) an imm5se degree ( utility4 ,little advanc$ z geomet]s really are z yet 9 ! sci5ce ( integrations1 !y h nev]!less obta9$1 f _! scanty ab/ract conceptions1 ! solution ( a multitude ( que/ions ( ! fir/ importance 9 geometry1 9 me*anics1 9 !rmology1 c4 ,! philosophical explanation ( ?is d\ble g5]al fact results f ! necess>ily prepond]at+ importance & grasp ( ab/ract bran*es ( k1 ! lea/ ( :i* is naturally f\nd to correspond to a cr[d ( concrete rese>*es1 man hav+ no o!r res\rce = ! successive ext5sion ( 8 intellectual means ?an 9 ! consid]ation ( ideas m & m ab/ract1 & /ill positive4 99 99 99 99 99 ,9 ord] to f9i% ! complete exposition ( ! philosophical "* ( ! transc5d5tal analysis1 !re rema9s to be consid]$ a f9al conception1 by :i* ! immortal ,lagrange has r5d]$ ?is analysis /ill bett] adapt$ to facilitate ! e/abli%m5t ( equations 9 ! mo/ difficult problems1 by consid]+ a class ( equations /ill m indirect ?an ! ord9>y diff]5tial equations4 ,x is ! ,calculus1 or1 r1 ! ,me?od ( ,v>iations2 ! g5]al appreciation ( :i* w be our next subject4 ,,*apt] ,v4 ,,! ,,calculus ,,( ,,v>iations4 ,9 ord] to grasp ) m ease ! philosophical "* ( ! ,me?od ( ,v>iations1 x w be well to beg9 by consid]+ 9 a summ>y mann] ! special nature ( ! problems1 ! g5]al resolution ( :i* has r5d]$ necess>y ! =mation ( ?is hyp]-transc5d5tal analysis4 ,x is /ill too ne> its orig91 & its applications h be5 too few1 to all[ u to obta9 a suffici5tly cle> g5]al idea ( x f a purely ab/ract exposition ( its fundam5tal !ory4 ,,problems ,,giv+ ,,rise ,,to ,,x4 ,! ma!matical que/ions :i* h giv5 bir? to ! ,calculus ( ,v>iations consi/ g5]ally 9 ! inve/igation ( ! maxima & m9ima ( c]ta9 indet]m9ate integral =mulas1 :i* express ! analytical law ( su* or su* a ph5om5on ( geometry or me*anics1 consid]$ indep5d5tly ( any p>ticul> subject4 ,geomet]s = a long "t designat$ all ! que/ions ( ?is "* by ! common "n ( ,isop]imetrical ,problems1 :i*1 h[ev]1 is really suitable to only ! smalle/ numb] ( !m4 ,ord9>y ,que/ions ( ,maxima & ,m9ima4 ,9 ! common !ory ( maxima & m9ima1 x is propos$ to discov]1 ) ref]5ce to a giv5 function ( "o or m v>iables1 :at p>ticul> values mu/ be assign$ to ^! v>iables1 9 ord] t ! correspond+ value ( ! propos$ function may be a maximum or a m9imum ) respect to ^? values :i* imm$iately prec$e & foll[ x2 t is1 prop]ly speak+1 we seek to "k at :at in/ant ! function ceases to increase & comm5ces to decrease1 or reciprocally4 ,! diff]5tial calculus is p]fectly suffici5t1 z we "k1 = ! g5]al resolution ( ?is class ( que/ions1 by %[+ t ! values ( ! diff]5t v>iables1 :i* suit ei!r ! maximum or m9imum1 mu/ always r$uce to z]o ! diff]5t fir/ d]ivatives ( ! giv5 function1 tak5 sep>ately ) ref]5ce to ea* indep5d5t v>iable1 & by indicat+1 moreov]1 a suitable *>act]i/ic = di/+ui%+ ! maximum f ! m9imum2 consi/+1 9 ! case ( a function ( a s+le v>iable1 = example1 9 ! d]iv$ function ( ! second ord] tak+ a negative value = ! maximum1 & a positive value = ! m9imum4 ,su* are ! well-kn[n fundam5tal conditions belong+ to ! greate/ numb] ( cases4 ,a new ,class ( ,que/ions4 ,! con/ruction ( ?is g5]al !ory hav+ necess>ily de/roy$ ! *ief int]e/ :i* que/ions ( ?is k9d _h = geomet]s1 !y almo/ imm$iately rose to ! consid]ation ( a new ord] ( problems1 at once mu* m important & ( mu* great] difficulty--^? ( isop]imet]s4 ,x is1 !n1 no long] ! values ( ! v>iables belong+ to ! maximum or ! m9imum ( a giv5 function t x is requir$ to det]m9e4 ,x is ! =m ( ! function itself :i* is requir$ to be discov]$1 f ! condition ( ! maximum or ( ! m9imum ( a c]ta9 def9ite integral1 m]ely indicat$1 :i* dep5ds ^u t function4 ,solid ( lea/ ,resi/ance4 ,! olde/ "q ( ?is nature is t ( ! solid ( lea/ resi/ance1 treat$ by ,newton 9 ! second book ( ! ,pr9cipia1 9 :i* he det]m9es :at ought to be ! m]idian curve ( a solid ( revolution1 9 ord] t ! resi/ance exp]i5c$ by t body 9 ! direction ( its axis may be ! lea/ possible4 ,b ! c\rse pursu$ by ,newton1 f ! nature ( 8 special me?od ( transc5d5tal analysis1 _h n a "* suffici5tly simple1 suffici5tly g5]al1 & especially suffici5tly analytical1 to attract geomet]s to ?is new ord] ( problems4 ,to effect ?is1 ! application ( ! inf9itesimal me?od 0 ne$$2 & ?is 0 done1 9 #afie1 by ,john ,b]n\illi1 9 propos+ ! celebrat$ problem ( ! ,bra*y/o*rone4 ,?is problem1 :i* aft]w>ds sugge/$ su* a long s]ies ( analog\s que/ions1 consi/s 9 det]m9+ ! curve :i* a heavy body mu/ foll[ 9 ord] to desc5d f "o po9t to ano!r 9 ! %orte/ possible "t4 ,limit+ ! conditions to ! simple fall 9 a vacuum1 ! only case :i* 0 at fir/ consid]$1 x is easily f\nd t ! requir$ curve mu/ be a rev]s$ cycloid ) a horizontal base1 & ) its orig9 at ! highe/ po9t4 ,b ! "q may become s+ul>ly complicat$1 ei!r by tak+ into acc\nt ! resi/ance ( ! m$ium1 or ! *ange 9 ! int5sity ( gravity4 ,isop]imet]s4 ,al?\gh ?is new class ( problems 0 9 ! fir/ place furni%$ by me*anics1 x is 9 geometry t ! pr9cipal inve/igations ( ?is "* 7 subsequ5tly made4 ,?us x 0 propos$ to discov] :i*1 among all ! curves ( ! same cont\r trac$ betwe5 two giv5 po9ts1 is t ^: area is a maximum or m9imum1 :5ce has come ! "n ( ,problem ( ,isop]imet]s2 or x 0 requir$ t ! maximum or m9imum %\ld belong to ! surface produc$ by ! revolution ( ! requir$ curve ab\t an axis1 or to ! correspond+ volume2 9 o!r cases1 x 0 ! v]tical height ( ! c5t] ( gravity ( ! unkn[n curve1 or ( ! surface & ( ! volume :i* x might g5]ate1 :i* 0 to become a maximum or m9imum1 c4 ,f9ally1 ^! problems 7 v>i$ & complicat$ almo/ to inf9ity by ! ,b]n\illis1 by ,taylor1 & especially by ,eul]1 be=e ,lagrange r$uc$ _! solution to an ab/ract & entirely g5]al me?od1 ! discov]y ( :i* has put a /op to ! en?usiasm ( geomet]s = su* an ord] ( inquiries4 ,?is is n ! place = trac+ ! hi/ory ( ?is subject4 ,i h only enum]at$ "s ( ! simple/ pr9cipal que/ions1 9 ord] to r5d] app>5t ! orig9al g5]al object ( ! me?od ( v>iations4 ,analytical ,nature ( ^! ,problems4 ,we see t all ^! problems1 consid]$ 9 an analytical po9t ( view1 consi/1 by _! nature1 9 det]m9+ :at =m a c]ta9 unkn[n function ( "o or m v>iables ought to h1 9 ord] t su* or su* an integral1 dep5d5t ^u t function1 %all h1 )9 assign$ limits1 a value :i* is a maximum or a m9imum ) respect to all ^? :i* x w\ld take if ! requir$ function _h any o!r =m :atev]4 ,?us1 = example1 9 ! problem ( ! bra*y/o*rone1 x is well kn[n t if y f7z71 x 7z71 are ! rectil9e> equations ( ! requir$ curve1 suppos+ ! axes ( x & ( y to be horizontal1 & ! axis ( z to be v]tical1 ! "t ( ! fall ( a heavy body 9 t curve f ! po9t ^: ord9ate is z1 to t ^: ord9ate is z1 is express$ 9 g5]al t]ms by ! def9ite integral zz7#a 7f'7z77 7'7z77_/7#b;gz77dz4 ,x is1 !n1 necess>y to f9d :at ! two unkn[n functions f & mu/ be1 9 ord] t ?is integral may be a m9imum4 ,9 ! same way1 to dem& :at is ! curve among all plane isop]imetrical curves1 :i* includes ! greate/ area1 is ! same ?+ z to propose to f9d1 among all ! functions f7x7 :i* c give a c]ta9 con/ant value to ! integral dx7#a 7f'7x7 771 t "o :i* r5d]s ! integral f7x7dx1 tak5 betwe5 ! same limits1 a maximum4 ,x is evid5tly always s 9 o!r que/ions ( ?is class4 ,me?ods ( ! old] ,geomet]s4 ,9 ! solutions :i* geomet]s be=e ,lagrange gave ( ^! problems1 !y propos$1 9 sub/ance1 to r$uce !m to ! ord9>y !ory ( maxima & m9ima4 ,b ! means employ$ to effect ?is trans=mation consi/$ 9 special simple artifices peculi> to ea* case1 & ! discov]y ( :i* did n admit ( inv>iable & c]ta9 rules1 s t e really new "q con/antly reproduc$ analog\s difficulties1 )\t ! solutions previ\sly obta9$ be+ really ( any ess5tial aid1 o!rwise ?an by _! discipl9e & tra9+ ( ! m9d4 ,9 a ^w1 ?is bran* ( ma!matics pres5t$1 !n1 ! necess>y imp]fection :i* always exi/s :5 ! "p common to all que/ions ( ! same class has n yet be5 di/9ctly grasp$ 9 ord] to be treat$ 9 an ab/ract & !nce=? g5]al mann]4 ,,me?od ,,( ,,lagrange4 ,lagrange1 9 endeav\r+ to br+ all ! diff]5t problems ( isop]imet]s to dep5d ^u a common analysis1 organiz$ into a di/9ct calculus1 0 l$ to conceive a new k9d ( diff]5tiation1 to :i* he has appli$ ! *>act]i/ic 1 res]v+ ! *>act]i/ic d = ! common diff]5tials4 ,^! diff]5tials ( a new species1 :i* he has designat$ "u ! "n ( ,v>iations1 consi/ ( ! inf9itely small increm5ts :i* ! integrals receive1 n by virtue ( analog\s increm5ts on ! "p ( ! correspond+ v>iables1 z 9 ! ord9>y transc5d5tal analysis1 b by suppos+ t ! =m ( ! function plac$ "u ! sign ( integration und]goes an inf9itely small *ange4 ,?is di/9ction is easily conceiv$ ) ref]5ce to curves1 9 :i* we see ! ord9ate1 or any o!r v>iable ( ! curve1 admit ( two sorts ( diff]5tials1 evid5tly v diff]5t1 accord+ z we pass f "o po9t to ano!r inf9itely ne> x on ! same curve1 or to ! correspond+ po9t ( ! inf9itely ne> curve produc$ by a c]ta9 det]m9ate modification ( ! fir/ curve4#aa ,x is moreov] cle>1 t ! relative v>iations ( diff]5t magnitudes connect$ ) ea* o!r by any laws :atev] are calculat$1 all b ! *>act]i/ic1 almo/ exactly 9 ! same mann] z ! diff]5tials4 ,f9ally1 f ! g5]al notion ( v>iations are 9 l mann] d$uc$ ! fundam5tal pr9ciples ( ! algori?m prop] to ?is me?od1 consi/+ simply 9 ! evid5tly p]missible lib]ty ( transpos+ at w ! *>act]i/ics specially appropriat$ to v>iations1 be=e or aft] ^? :i* correspond to ! ord9>y diff]5tials4 ,footnote #aa3 ,leibnitz _h already consid]$ ! comp>ison ( "o curve ) an o!r inf9itely ne> to x1 call+ x ,diff]5tiatio de curva 9 curvam4 ,b ?is comp>ison _h no analogy ) ! conception ( ,lagrange1 ! curves ( ,leibnitz be+ embrac$ 9 ! same g5]al equation1 f :i* !y 7 d$uc$ by ! simple *ange ( an arbitr>y con/ant4 ,?is ab/ract conception hav+ be5 once =m$1 ,lagrange 0 able to r$uce ) ease1 & 9 ! mo/ g5]al mann]1 all ! problems ( ,isop]imet]s to ! simple ord9>y !ory ( maxima & m9ima4 ,to obta9 a cle> idea ( ?is great & happy trans=mation1 we mu/ previ\sly consid] an ess5tial di/9ction :i* arises 9 ! diff]5t que/ions ( isop]imet]s4 ,two ,classes ( ,que/ions4 ,^! inve/igations mu/1 9 fact1 be divid$ into two g5]al classes1 accord+ z ! maxima & m9ima dem&$ are absolute or relative1 to employ ! abridg$ expressions ( geomet]s4 ,que/ions ( ! fir/ ,class4 ,! fir/ case is t 9 :i* ! indet]m9ate def9ite integrals1 ! maximum or m9imum ( :i* is s\ght1 are n subject$1 by ! nature ( ! problem1 to any condition2 z happ5s1 = example1 9 ! problem ( ! bra*y/o*rone1 9 :i* ! *oice is to be made betwe5 all imag9able curves4 ,! second case takes place :51 on ! contr>y1 ! v>iable integrals c v>y only accord+ to c]ta9 conditions1 :i* usually consi/ 9 o!r def9ite integrals 7:i* dep5d1 9 l mann]1 ^u ! requir$ functions7 always reta9+ ! same giv5 value2 z1 = example1 9 all ! geometrical que/ions relat+ to real isop]imetrical figures1 & 9 :i*1 by ! nature ( ! problem1 ! integral relat+ to ! l5g? ( ! curve1 or to ! area ( ! surface1 mu/ rema9 con/ant dur+ ! v>iation ( t integral :i* is ! object ( ! propos$ inve/igation4 ,! ,calculus ( ,v>iations gives imm$iately ! g5]al solution ( que/ions ( ! =m] class2 = x evid5tly foll[s1 f ! ord9>y !ory ( maxima & m9ima1 t ! requir$ relation mu/ r$uce to z]o ! v>iation ( ! propos$ integral ) ref]5ce to ea* indep5d5t v>iable2 :i* gives ! condition common to bo? ! maximum & ! m9imum3 &1 z a *>act]i/ic = di/+ui%+ ! "o f ! o!r1 t ! v>iation ( ! second ord] ( ! same integral mu/ be negative = ! maximum & positive = ! m9imum4 ,?us1 = example1 9 ! problem ( ! bra*y/o*rone1 we w h1 9 ord] to det]m9e ! nature ( ! curve s\ght1 ! equation ( condition zz7#a 7f'7z77 7'7z77_/7#b;gz77dz #j1 :i*1 be+ decompos$ into two1 ) respect to ! two unkn[n functions f & 1 :i* are indep5d5t ( ea* o!r1 w completely express ! analytical def9ition ( ! requir$ curve4 ,! only difficulty peculi> to ?is new analysis consi/s 9 ! elim9ation ( ! *>act]i/ic 1 = :i* ! calculus ( v>iations furni%es inv>iable & complete rules1 f\nd$1 9 g5]al1 on ! me?od ( integration by p>ts1 f :i* ,lagrange has ?us d]iv$ imm5se advantage4 ,! con/ant object ( ?is fir/ analytical elaboration 7:i* ?is is n ! place = treat+ 9 detail7 is to arrive at real diff]5tial equations1 :i* c always be done2 & !reby ! "q comes "u ! ord9>y transc5d5tal analysis1 :i* furni%es ! solution1 at lea/ s f> z to r$uce x to pure algebra if ! integration c be effect$4 ,! g5]al object ( ! me?od ( v>iations is to effect ?is trans=mation1 = :i* ,lagrange has e/abli%$ rules1 :i* are simple1 inv>iable1 & c]ta9 ( success4 ,equations ( ,limits4 ,among ! greate/ special advantages ( ! me?od ( v>iations1 comp>$ ) ! previ\s isolat$ solutions ( isop]imetrical problems1 is ! important consid]ation ( :at ,lagrange calls ,equations ( ,limits1 :i* 7 entirely neglect$ be=e him1 ?\gh )\t !m ! great] "p ( ! p>ticul> solutions rema9$ necess>ily incomplete4 ,:5 ! limits ( ! propos$ integrals are to be fix$1 _! v>iations be+ z]o1 !re is no occasion = notic+ !m4 ,b x is no long] s :5 ^! limits1 in/ead ( be+ rigor\sly inv>iable1 are only subject$ to c]ta9 conditions2 z1 = example1 if ! two po9ts betwe5 :i* ! requir$ curve is to be trac$ are n fix$1 & h only to rema9 ^u giv5 l9es or surfaces4 ,!n x is necess>y to pay att5tion to ! v>iation ( _! co-ord9ates1 & to e/abli% betwe5 !m ! relations :i* correspond to ! equations ( ^! l9es or ( ^! surfaces4 ,a m g5]al consid]ation4 ,?is ess5tial consid]ation is only ! f9al complem5t ( a m g5]al & m important consid]ation relative to ! v>iations ( diff]5t indep5d5t v>iables4 ,if ^! v>iables are really indep5d5t ( "o ano!r1 z :5 we comp>e toge!r all ! imag9able curves susceptible ( be+ trac$ betwe5 two po9ts1 x w be ! same ) _! v>iations1 &1 consequ5tly1 ! t]ms relat+ to ea* ( ^! v>iations w h to be sep>ately equal to z]o 9 ! g5]al equation :i* expresses ! maximum or ! m9imum4 ,b if1 on ! contr>y1 we suppose ! v>iables to be subject$ to any fix$ conditions1 x w be necess>y to take notice ( ! result+ relation betwe5 _! v>iations1 s t ! numb] ( ! equations into :i* ?is g5]al equation is !n decompos$ is always equal to only ! numb] ( ! v>iables :i* rema9 truly indep5d5t4 ,x is ?us1 = example1 t in/ead ( seek+ = ! %orte/ pa? betwe5 any two po9ts1 9 *oos+ x f among all possible ones1 x may be propos$ to f9d only :at is ! %orte/ among all ^? :i* may be tak5 on any giv5 surface2 a "q ! g5]al solution ( :i* =ms c]ta9ly "o ( ! mo/ beautiful applications ( ! me?od ( v>iations4 ,que/ions ( ! second ,class4 ,problems 9 :i* su* modify+ conditions are consid]$ approa* v ne>ly1 9 _! nature1 to ! second g5]al class ( applications ( ! me?od ( v>iations1 *>act]iz$ above z consi/+ 9 ! inve/igation ( relative maxima & m9ima4 ,!re is1 h[ev]1 ?is ess5tial diff]5ce betwe5 ! two cases1 t 9 ?is la/ ! modification is express$ by an integral :i* dep5ds ^u ! function s\ght1 :ile 9 ! o!r x is designat$ by a f9ite equation :i* is imm$iately giv54 ,x is h5ce app>5t t ! inve/igation ( relative maxima & m9ima is con/antly & necess>ily m complicat$ ?an t ( absolute maxima & m9ima4 ,luckily1 a v important g5]al !ory1 discov]$ by ! g5ius ( ! great ,eul] be=e ! inv5tion ( ! ,calculus ( ,v>iations1 gives a uni=m & v simple means ( mak+ "o ( ^! two classes ( que/ions dep5d5t on ! o!r4 ,x consi/s 9 ?is1 t if we add to ! integral :i* is to be a maximum or a m9imum1 a con/ant & indet]m9ate multiple ( t "o :i*1 by ! nature ( ! problem1 is to rema9 con/ant1 x w be suffici5t to seek1 by ! g5]al me?od ( ,lagrange above indicat$1 ! absolute maximum or m9imum ( ?is :ole expression4 ,x c be easily conceiv$1 inde$1 t ! "p ( ! complete v>iation :i* w\ld proce$ f ! la/ integral mu/ be equal to z]o 7because ( ! con/ant "* ( ?is la/7 z well z ! portion due to ! fir/ integral1 :i* disappe>s by virtue ( ! maximum or m9imum /ate4 ,^! two conditions evid5tly unite to produce1 9 t respect1 effects exactly alike4 ,su* is a sket* ( ! g5]al mann] 9 :i* ! me?od ( v>iation is appli$ to all ! diff]5t que/ions :i* compose :at is call$ ! ,!ory ( ,isop]imet]s4 ,x w und\bt$ly h be5 rem>k$ 9 ?is summ>y exposition h[ mu* use has be5 made 9 ?is new analysis ( ! second fundam5tal prop]ty ( ! transc5d5tal analysis notic$ 9 ! ?ird *apt]1 namely1 ! g5]ality ( ! inf9itesimal expressions = ! repres5tation ( ! same geometrical or me*anical ph5om5on1 9 :atev] body x may be consid]$4 ,^u ?is g5]ality1 inde$1 are f\nd$1 by _! nature1 all ! solutions due to ! me?od ( v>iations4 ,if a s+le =mula c\ld n express ! l5g? or ! area ( any curve :atev]2 if ano!r fix$ =mula c\ld n designate ! "t ( ! fall ( a heavy body1 accord+ to :atev] l9e x may desc5d1 c41 h[ w\ld x h be5 possible to resolve que/ions :i* unavoidably require1 by _! nature1 ! simultane\s consid]ation ( all ! cases :i* c be det]m9$ 9 ea* ph5om5on by ! diff]5t subjects :i* exhibit x4 ,o!r ,applications ( ?is ,me?od4 ,not)/&+ ! extreme importance ( ! !ory ( isop]imet]s1 & ?\gh ! me?od ( v>iations _h at fir/ no o!r object ?an ! logical & g5]al solution ( ?is ord] ( problems1 we %\ld /ill h b an incomplete idea ( ?is beautiful analysis if we limit$ its de/9ation to ?is4 ,9 fact1 ! ab/ract conception ( two di/9ct natures ( diff]5tiation is evid5tly applicable n only to ! cases = :i* x 0 creat$1 b also to all ^? :i* pres5t1 = any reason :atev]1 two diff]5t mann]s ( mak+ ! same magnitudes v>y4 ,x is 9 ?is way t ,lagrange himself has made1 9 8 ,m*anique ,analytique1 an ext5sive & important application ( 8 calculus ( v>iations1 by employ+ x to di/+ui% ! two sorts ( *anges :i* are naturally pres5t$ by ! que/ions ( rational me*anics = ! diff]5t po9ts :i* are consid]$1 accord+ z we comp>e ! successive positions :i* are occupi$1 9 virtue ( its motion1 by ! same po9t ( ea* body 9 two consecutive in/ants1 or z we pass f "o po9t ( ! body to ano!r 9 ! same in/ant4 ,"o ( ^! comp>isons produces ord9>y diff]5tials2 ! o!r gives rise to v>iations1 :i*1 !re z e :]e1 are only diff]5tials tak5 "u a new po9t ( view4 ,su* is ! g5]al acceptation 9 :i* we %\ld conceive ! ,calculus ( ,v>iations1 9 ord] suitably to appreciate ! importance ( ?is admirable logical in/rum5t1 ! mo/ p[]ful t ! human m9d has z yet con/ruct$4 ,! me?od ( v>iations be+ only an imm5se ext5sion ( ! g5]al transc5d5tal analysis1 ,i h no ne$ ( prov+ specially t x is susceptible ( be+ consid]$ "u ! diff]5t fundam5tal po9ts ( view :i* ! calculus ( indirect functions1 consid]$ z a :ole1 admits (4 ,lagrange inv5t$ ! ,calculus ( ,v>iations 9 accordance ) ! inf9itesimal conception1 &1 inde$1 long be=e he und]took ! g5]al recon/ruction ( ! transc5d5tal analysis4 ,:5 he _h execut$ ?is important re=mation1 he easily %[$ h[ x c\ld also be appli$ to ! ,calculus ( ,v>iations1 :i* he exp\nd$ ) all ! prop] developm5t1 accord+ to 8 !ory ( d]ivative functions4 ,b ! m t ! use ( ! me?od ( v>iations is difficult ( compreh5sion1 because ( ! high] degree ( ab/raction ( ! ideas consid]$1 ! m necess>y is x1 9 its application1 to economize ! ex]tions ( ! m9d1 by adopt+ ! mo/ direct & rapid analytical conception1 namely1 t ( ,leibnitz4 ,accord+ly1 ,lagrange himself has con/antly pref]r$ x 9 ! important use :i* he has made ( ! ,calculus ( ,v>iations 9 8 ,analytical ,me*anics4 ,9 fact1 !re does n exi/ ! lea/ hesitation 9 ?is respect among geomet]s4 ,,its ,,relations ,,to ,,! ,,ord9>y ,,calculus4 ,9 ord] to make z cle> z possible ! philosophical "* ( ! ,calculus ( ,v>iations1 ,i ?9k t ,i %\ld1 9 conclusion1 briefly indicate a consid]ation :i* seems to me important1 & by :i* ,i c approa* x to ! ord9>y transc5d5tal analysis 9 a high] degree ?an ,lagrange seems to me to h done4#ab ,footnote #ab3 ,i propose h]eaft] to develop ?is new consid]ation1 9 a special "w ^u ! ,calculus ( ,v>iations1 int5d$ to pres5t ?is hyp]-transc5d5tal analysis 9 a new po9t ( view1 :i* ,i ?9k adapt$ to ext5d its g5]al range4 ,we notic$ 9 ! prec$+ *apt] ! =mation ( ! calculus ( p>tial diff]5ces1 creat$ by ,d',alemb]t1 z hav+ introduc$ into ! transc5d5tal analysis a new elem5t>y idea2 ! notion ( two k9ds ( increm5ts1 di/9ct & indep5d5t ( "o ano!r1 :i* a function ( two v>iables may receive by virtue ( ! *ange ( ea* v>iable sep>ately4 ,x is ?us t ! v]tical ord9ate ( a surface1 or any o!r magnitude :i* is ref]r$ to x1 v>ies 9 two mann]s :i* are q di/9ct1 & :i* may foll[ ! mo/ diff]5t laws1 accord+ z we increase ei!r ! "o or ! o!r ( ! two horizontal co-ord9ates4 ,n[ su* a consid]ation seems to me v ne>ly alli$1 by its nature1 to t :i* s]ves z ! g5]al basis ( ! me?od ( v>iations4 ,?is la/1 inde$1 has 9 reality done no?+ b transf] to ! indep5d5t v>iables !mselves ! peculi> conception :i* _h be5 already adopt$ = ! functions ( ^! v>iables2 a modification :i* has rem>kably enl>g$ its use4 ,i ?9k1 !re=e1 t s f> z reg>ds m]ely ! fundam5tal conceptions1 we may consid] ! calculus creat$ by ,d',alemb]t z hav+ e/abli%$ a natural & necess>y transition betwe5 ! ord9>y inf9itesimal calculus & ! calculus ( v>iations2 su* a d]ivation ( :i* seems to be adapt$ to make ! g5]al notion m cle> & simple4 ,accord+ to ! diff]5t consid]ations indicat$ 9 ?is *apt]1 ! me?od ( v>iations pres5ts itself z ! highe/ degree ( p]fection :i* ! analysis ( indirect functions has yet atta9$4 ,9 its primitive /ate1 ?is la/ analysis pres5t$ itself z a p[]ful g5]al means ( facilitat+ ! ma!matical /udy ( natural ph5om5a1 by introduc+1 = ! expression ( _! laws1 ! consid]ation ( auxili>y magnitudes1 *os5 9 su* a mann] t _! relations are necess>ily m simple & m easy to obta9 ?an ^? ( ! direct magnitudes4 ,b ! =mation ( ^! diff]5tial equations 0 n suppos$ to admit ( any g5]al & ab/ract rules4 ,n[ ! ,analysis ( ,v>iations1 consid]$ 9 ! mo/ philosophical po9t ( view1 may be reg>d$ z ess5tially de/9$1 by its nature1 to br+ )9 ! rea* ( ! calculus ! actual e/abli%m5t ( ! diff]5tial equations2 =1 9 a great numb] ( important & difficult que/ions1 su* is ! g5]al effect ( ! v>i$ equations1 :i*1 /ill m indirect ?an ! simple diff]5tial equations ) respect to ! special objects ( ! inve/igation1 are also mu* m easy to =m1 & f :i* we may !n1 by inv>iable & complete analytical me?ods1 ! object ( :i* is to elim9ate ! new ord] ( auxili>y inf9itesimals :i* h be5 introduc$1 d$uce ^? ord9>y diff]5tial equations :i* x w\ld (t5 h be5 impossible to e/abli% directly4 ,! me?od ( v>iations =ms1 !n1 ! mo/ sublime "p ( t va/ sy/em ( ma!matical analysis1 :i*1 sett+ out f ! mo/ simple elem5ts ( algebra1 organizes1 by an un9t]rupt$ succession ( ideas1 g5]al me?ods m & m p[]ful1 = ! /udy ( natural philosophy1 & :i*1 9 its :ole1 pres5ts ! mo/ incomp>ably impos+ & unequivocal monum5t ( ! p[] ( ! human intellect4 ,we mu/1 h[ev]1 also admit t ! conceptions :i* are habitually consid]$ 9 ! me?od ( v>iations be+1 by _! nature1 m indirect1 m g5]al1 & especially m ab/ract ?an all o!rs1 ! employm5t ( su* a me?od exacts necess>ily & cont9u\sly ! highe/ kn[n degree ( intellectual ex]tion1 9 ord] nev] to lose sight ( ! precise object ( ! inve/igation1 9 foll[+ reason+s :i* (f] to ! m9d su* unc]ta9 re/+-places1 & 9 :i* signs are ( sc>cely any assi/ance4 ,we mu/ und\bt$ly attribute 9 a great degree to ?is difficulty ! little real use :i* geomet]s1 ) ! exception ( ,lagrange1 h z yet made ( su* an admirable conception4 ,,*apt] ,,vi4 ,,! ,,calculus ,,( ,,f9ite ,,diff]5ces4 ,! diff]5t fundam5tal consid]ations indicat$ 9 ! five prec$+ *apt]s con/itute1 9 reality1 all ! ess5tial bases ( a complete exposition ( ma!matical analysis1 reg>d$ 9 ! philosophical po9t ( view4 ,nev]!less1 9 ord] n to neglect any truly important g5]al conception relat+ to ?is analysis1 ,i ?9k t ,i %\ld "h v summ>ily expla9 ! v]itable "* ( a k9d ( calculus :i* is v ext5d$1 & :i*1 ?\gh at bottom x really belongs to ord9>y analysis1 is /ill reg>d$ z be+ ( an ess5tially di/9ct nature4 ,i ref] to ! ,calculus ( ,f9ite ,diff]5ces1 :i* w be ! special subject ( ?is *apt]4 ,its g5]al ,"*4 ,?is calculus1 creat$ by ,taylor1 9 8 celebrat$ "w entitl$ ,me?odus ,increm5torum1 consi/s ess5tially 9 ! consid]ation ( ! f9ite increm5ts :i* functions receive z a consequ5ce ( analog\s increm5ts on ! "p ( ! correspond+ v>iables4 ,^! increm5ts or diff]5ces1 :i* take ! *>act]i/ic 1 to di/+ui% !m f diff]5tials1 or inf9itely small increm5ts1 may be 9 _! turn reg>d$ z new functions1 & become ! subject ( a second simil> consid]ation1 & s on2 f :i* results ! notion ( diff]5ces ( v>i\s successive ord]s1 analog\s1 at lea/ 9 appe>ance1 to ! consecutive ord]s ( diff]5tials4 ,su* a calculus evid5tly pres5ts1 l ! calculus ( indirect functions1 two g5]al classes ( que/ions3 #a4 ,to det]m9e ! successive diff]5ces ( all ! v>i\s analytical functions ( "o or m v>iables1 z ! result ( a def9ite mann] ( increase ( ! indep5d5t v>iables1 :i* are g5]ally suppos$ to augm5t 9 ari?metical progression4 #b4 ,reciprocally1 to />t f ^! diff]5ces1 or1 m g5]ally1 f any equations e/abli%$ betwe5 !m1 & g back to ! primitive functions !mselves1 or to _! correspond+ relations4 ,h5ce foll[s ! decomposition ( ?is calculus into two di/9ct ones1 to :i* are usually giv5 ! names ( ! ,direct1 & ! ,inv]se ,calculus ( ,f9ite ,diff]5ces1 ! latt] be+ also sometimes call$ ! ,integral ,calculus ( ,f9ite ,diff]5ces4 ,ea* ( ^! w\ld1 also1 evid5tly admit ( a logical di/ribution simil> to t giv5 9 ! f\r? *apt] = ! diff]5tial & ! integral calculus4 ,its true ,nature4 ,!re is no d\bt t ,taylor ?\ght t by su* a conception he _h f\nd$ a calculus ( an entirely new nature1 absolutely di/9ct f ord9>y analysis1 & m g5]al ?an ! calculus ( ,leibnitz1 al?\gh re/+ on an analog\s consid]ation4 ,x is 9 ?is way1 also1 t almo/ all geomet]s h view$ ! analysis ( ,taylor2 b ,lagrange1 ) 8 usual pr(undity1 cle>ly p]ceiv$ t ^! prop]ties belong$ mu* m to ! =ms & to ! notations employ$ by ,taylor ?an to ! sub/ance ( 8 !ory4 ,9 fact1 t :i* con/itutes ! peculi> "* ( ! analysis ( ,leibnitz1 & makes ( x a truly di/9ct & sup]ior calculus1 is ! circum/ance t ! d]iv$ functions are 9 g5]al ( an entirely diff]5t nature f ! primitive functions1 s t !y may give rise to m simple & m easily =m$ relations3 :5ce result ! admirable fundam5tal prop]ties ( ! transc5d5tal analysis1 :i* h be5 already expla9$4 ,b x is n s ) ! diff]5ces consid]$ by ,taylor2 = ^! diff]5ces are1 by _! nature1 functions ess5tially simil> to ^? :i* h produc$ !m1 a circum/ance :i* r5d]s !m unsuitable to facilitate ! e/abli%m5t ( equations1 & prev5ts _! lead+ to m g5]al relations4 ,e equation ( f9ite diff]5ces is truly1 at bottom1 an equation directly relat+ to ! v magnitudes ^: successive /ates are comp>$4 ,! scaffold+ ( new signs1 :i* produce an illusion respect+ ! true "* ( ^! equations1 disguises x1 h[ev]1 9 a v imp]fect mann]1 s9ce x c\ld always be easily made app>5t by replac+ ! diff]5ces by ! equival5t comb9ations ( ! primitive magnitudes1 ( :i* !y are really only ! abridg$ designations4 ,?us ! calculus ( ,taylor nev] has (f]$1 & nev] c (f]1 9 any "q ( geometry or ( me*anics1 t p[]ful g5]al aid :i* we h se5 to result necess>ily f ! analysis ( ,leibnitz4 ,lagrange has1 moreov]1 v cle>ly prov5 t ! pret5d$ analogy obs]v$ betwe5 ! calculus ( diff]5ces & ! inf9itesimal calculus 0 radically vici\s1 9 ?is way1 t ! =mulas belong+ to ! =m] calculus c nev] furni%1 z p>ticul> cases1 ^? :i* belong to ! latt]1 ! nature ( :i* is ess5tially di/9ct4 ,f ^! consid]ations ,i am l$ to ?9k t ! calculus ( f9ite diff]5ces is1 9 g5]al1 improp]ly class$ ) ! transc5d5tal analysis prop]1 t is1 ) ! calculus ( indirect functions4 ,i consid] x1 on ! contr>y1 9 accordance ) ! views ( ,lagrange1 to be only a v ext5sive & v important bran* ( ord9>y analysis1 t is to say1 ( t :i* ,i h nam$ ! calculus ( direct functions1 ! equations :i* x consid]s be+ always1 9 spite ( ! notation1 simple direct equations4 ,,g5]al ,,!ory ,,( ,,s]ies4 ,to sum up z briefly z possible ! prec$+ explanation1 ! calculus ( ,taylor ought to be reg>d$ z hav+ con/antly = its true object ! g5]al !ory ( ,s]ies1 ! mo/ simple cases ( :i* _h alone be5 consid]$ be=e t illu/ri\s geomet]4 ,i ought1 prop]ly1 to h m5tion$ ?is important !ory 9 treat+1 9 ! second *apt]1 ( ,algebra prop]1 ( :i* x is su* an ext5sive bran*4 ,b1 9 ord] to avoid a d\ble ref]5ce to x1 ,i h pref]r$ to notice x only 9 ! consid]ation ( ! calculus ( f9ite diff]5ces1 :i*1 r$uc$ to its mo/ simple g5]al expression1 is no?+ b a complete logical /udy ( que/ions relat+ to s]ies4 ,e ,s]ies1 or succession ( numb]s d$uc$ f "o ano!r accord+ to any con/ant law1 necess>ily gives rise to ^! two fundam5tal que/ions3 #a4 ,! law ( ! s]ies be+ suppos$ kn[n1 to f9d ! expression = its g5]al t]m1 s z to be able to calculate imm$iately any t]m :atev] )\t be+ oblig$ to =m successively all ! prec$+ t]ms4 #b4 ,9 ! same circum/ances1 to det]m9e ! sum ( any numb] ( t]ms ( ! s]ies by means ( _! places1 s t x c be kn[n )\t ! necessity ( cont9ually add+ ^! t]ms toge!r4 ,^! two fundam5tal que/ions be+ consid]$ to be resolv$1 x may be propos$1 reciprocally1 to f9d ! law ( a s]ies f ! =m ( its g5]al t]m1 or ! expression ( ! sum4 ,ea* ( ^! diff]5t problems has s mu* ! m ext5t & difficulty1 z !re c be conceiv$ a great] numb] ( diff]5t laws = ! s]ies1 accord+ to ! numb] ( prec$+ t]ms on :i* ea* t]m directly dep5ds1 & accord+ to ! function :i* expresses t dep5d5ce4 ,we may ev5 consid] s]ies ) sev]al v>iable indices1 z ,laplace has done 9 8 ,analytical ,!ory ( ,probabilities1 by ! analysis to :i* he has giv5 ! "n ( ,!ory ( ,g5]at+ ,functions1 al?\gh x is really only a new & high] bran* ( ! calculus ( f9ite diff]5ces or ( ! g5]al !ory ( s]ies4 ,^! g5]al views :i* ,i h indicat$ give only an imp]fect idea ( ! truly inf9ite ext5t & v>iety ( ! que/ions to :i* geomet]s h ris5 by means ( ?is s+le consid]ation ( s]ies1 s simple 9 appe>ance & s limit$ 9 its orig94 ,x necess>ily pres5ts z _m diff]5t cases z ! algebraic resolution ( equations1 consid]$ 9 its :ole ext5t2 & x is1 by its nature1 mu* m complicat$1 s mu*1 inde$1 t x always ne$s ?is la/ to conduct x to a complete solution4 ,we may1 !re=e1 anticipate :at mu/ /ill be its extreme imp]fection1 9 spite ( ! successive lab\rs ( sev]al geomet]s ( ! fir/ ord]4 ,we d n1 inde$1 possess z yet ! complete & logical solution ( any b ! mo/ simple que/ions ( ?is nature4 ,its id5tity ) ?is ,calculus4 ,x is n[ easy to conceive ! necess>y & p]fect id5tity1 :i* has be5 already ann\nc$1 betwe5 ! calculus ( f9ite diff]5ces & ! !ory ( s]ies consid]$ 9 all its be>+s4 ,9 fact1 e diff]5tiation aft] ! mann] ( ,taylor evid5tly am\nts to f9d+ ! law ( =mation ( a s]ies ) "o or ) sev]al v>iable indices1 f ! expression ( its g5]al t]m2 9 ! same way1 e analog\s integration may be reg>d$ z hav+ = its object ! summation ( a s]ies1 ! g5]al t]m ( :i* w\ld be express$ by ! propos$ diff]5ce4 ,9 ?is po9t ( view1 ! v>i\s problems ( ! calculus ( diff]5ces1 direct or inv]se1 resolv$ by ,taylor & 8 successors1 h really a v great value1 z treat+ ( important que/ions relat+ to s]ies4 ,b x is v d\btful if ! =m & ! notation introduc$ by ,taylor really give any ess5tial facility 9 ! solution ( que/ions ( ?is k9d4 ,x w\ld be1 p]haps1 m advantage\s = mo/ cases1 & c]ta9ly m logical1 to replace ! diff]5ces by ! t]ms !mselves1 c]ta9 comb9ations ( :i* !y repres5t4 ,z ! calculus ( ,taylor does n re/ on a truly di/9ct fundam5tal idea1 & has no?+ peculi> to x b its sy/em ( signs1 !re c\ld nev] really be any important advantage 9 consid]+ x z deta*$ f ord9>y analysis1 ( :i* x is1 9 reality1 only an imm5se bran*4 ,?is consid]ation ( diff]5ces1 mo/ g5]ally useless1 ev5 if x does n cause complication1 seems to me to reta9 ! "* ( an epo* 9 :i*1 analytical ideas n be+ suffici5tly famili> to geomet]s1 !y 7 naturally l$ to pref] ! special =ms suitable = simple num]ical comp>isons4 ,,p]iodic ,,or ,,discont9u\s ,,functions4 ,h[ev] t may be1 ,i mu/ n f9i% ?is g5]al appreciation ( ! calculus ( f9ite diff]5ces )\t notic+ a new conception to :i* x has giv5 bir?1 & :i* has s9ce acquir$ a great importance4 ,x is ! consid]ation ( ^? p]iodic or discont9u\s functions :i* pres]ve ! same value = an inf9ite s]ies ( values ( ! correspond+ v>iables1 subject$ to a c]ta9 law1 & :i* mu/ be necess>ily add$ to ! integrals ( ! equations ( f9ite diff]5ces 9 ord] to r5d] !m suffici5tly g5]al1 z simple arbitr>y con/ants are add$ to all quadratures 9 ord] to complete _! g5]ality4 ,?is idea1 primitively introduc$ by ,eul]1 has s9ce be5 ! subject ( ext5d$ inve/igation by ,m4 ,f\ri]1 :o has made new & important applications ( x 9 8 ma!matical !ory ( heat4 ,,applications ,,( ,,?is ,,calculus4 ,s]ies4 ,among ! pr9cipal g5]al applications :i* h be5 made ( ! calculus ( f9ite diff]5ces1 x w\ld be prop] to place 9 ! fir/ rank1 z ! mo/ ext5d$ & ! mo/ important1 ! solution ( que/ions relat+ to s]ies2 if1 z has be5 %[n1 ! g5]al !ory ( s]ies ought n to be consid]$ z con/itut+1 by its nature1 ! actual f\ndation ( ! calculus ( ,taylor4 ,int]polations4 ,?is great class ( problems be+ !n set aside1 ! mo/ ess5tial ( ! v]itable applications ( ! analysis ( ,taylor is1 und\bt$ly1 ?us f>1 ! g5]al me?od ( int]polations1 s frequ5tly & s usefully employ$ 9 ! inve/igation ( ! empirical laws ( natural ph5om5a4 ,! "q consi/s1 z is well kn[n1 9 int]calat+ betwe5 c]ta9 giv5 numb]s o!r int]m$iate numb]s1 subject$ to ! same law :i* we suppose to exi/ betwe5 ! fir/4 ,we c abundantly v]ify1 9 ?is pr9cipal application ( ! calculus ( ,taylor1 h[ truly =eign & (t5 inconv5i5t is ! consid]ation ( diff]5ces ) respect to ! que/ions :i* dep5d on t analysis4 ,inde$1 ,lagrange has replac$ ! =mulas ( int]polation1 d$uc$ f ! ord9>y algori?m ( ! calculus ( f9ite diff]5ces1 by mu* simpl] g5]al =mulas1 :i* are n[ almo/ always pref]r$1 & :i* h be5 f\nd directly1 )\t mak+ any use ( ! notion ( diff]5ces1 :i* only complicates ! "q4 ,approximate ,rectification1 c4 ,a la/ important class ( applications ( ! calculus ( f9ite diff]5ces1 :i* des]ves to be di/+ui%$ f ! prec$+1 consi/s 9 ! em95tly useful employm5t made ( x 9 geometry = det]m9+ by approximation ! l5g? & ! area ( any curve1 & 9 ! same way ! cubature ( a body ( any =m :atev]4 ,?is proc$ure 7:i* may besides be conceiv$ ab/ractly z dep5d+ on ! same analytical inve/igation z ! "q ( int]polation7 frequ5tly (f]s a valuable supplem5t to ! entirely logical geometrical me?ods :i* (t5 lead to integrations1 :i* we d n yet "k h[ to effect1 or to calculations ( v complicat$ execution4 99 99 99 99 99 ,su* are ! v>i\s pr9cipal consid]ations to be notic$ ) respect to ! calculus ( f9ite diff]5ces4 ,?is exam9ation completes ! propos$ philosophical outl9e ( ,,ab/ract ,,ma!matics4 ,,concrete ,,ma!matics w n[ be ! subject ( a simil> lab\r4 ,9 x we %all p>ticul>ly devote ourselves to exam9+ h[ x has be5 possible 7suppos+ ! g5]al sci5ce ( ! calculus to be p]fect71 by inv>iable proc$ures1 to r$uce to pure que/ions ( analysis all ! problems :i* c be pres5t$ by ,geometry & ,me*anics1 & ?us to impress on ^! two fundam5tal bases ( natural philosophy a degree ( precision & especially ( unity2 9 a ^w1 a "* ( high p]fection1 :i* c\ld be communicat$ to !m by su* a c\rse alone4 ,,book ,,ii4 ,,geometry4 ,,book ,,ii4 ,,geometry4 ,,*apt] ,i4 ,,g5]al ,,view ,,( ,,geometry4 ,its true ,nature4 ,aft] ! g5]al exposition ( ! philosophical "* ( concrete ma!matics1 comp>$ ) t ( ab/ract ma!matics1 giv5 9 ! introductory *apt]1 x ne$ n "h be %[n 9 a special mann] t geometry mu/ be consid]$ z a true natural sci5ce1 only mu* m simple1 & !re=e mu* m p]fect1 ?an any o!r4 ,?is necess>y p]fection ( geometry1 obta9$ ess5tially by ! application ( ma!matical analysis1 :i* x s em95tly admits1 is apt to produce errone\s views ( ! real nature ( ?is fundam5tal sci5ce1 :i* mo/ m9ds at pres5t conceive to be a purely logical sci5ce q indep5d5t ( obs]vation4 ,x is nev]!less evid5t1 to any "o :o exam9es ) att5tion ! "* ( geometrical reason+s1 ev5 9 ! pres5t /ate ( ab/ract geometry1 t1 al?\gh ! facts :i* are consid]$ 9 x are mu* m closely unit$ ?an ^? relat+ to any o!r sci5ce1 /ill !re always exi/s1 ) respect to e body /udi$ by geomet]s1 a c]ta9 numb] ( primitive ph5om5a1 :i*1 s9ce !y are n e/abli%$ by any reason+1 mu/ be f\nd$ on obs]vation alone1 & :i* =m ! necess>y basis ( all ! d$uctions4 ,! sci5tific sup]iority ( geometry arises f ! ph5om5a :i* x consid]s be+ necess>ily ! mo/ univ]sal & ! mo/ simple ( all4 ,n only may all ! bodies ( nature give rise to geometrical inquiries1 z well z me*anical ones1 b /ill f>!r1 geometrical ph5om5a w\ld /ill exi/1 ev5 ?\gh all ! p>ts ( ! univ]se %\ld be consid]$ z immovable4 ,geometry is !n1 by its nature1 m g5]al ?an me*anics4 ,at ! same "t1 its ph5om5a are m simple1 = !y are evid5tly indep5d5t ( me*anical ph5om5a1 :ile ^! latt] are always complicat$ ) ! =m]4 ,! same relations hold good 9 comp>+ geometry ) ab/ract !rmology4 ,= ^! reasons1 9 our classification we h made geometry ! fir/ "p ( concrete ma!matics2 t "p ! /udy ( :i*1 9 addition to its own importance1 s]ves z ! indisp5sable basis ( all ! re/4 ,be=e consid]+ directly ! philosophical /udy ( ! diff]5t ord]s ( inquiries :i* con/itute our pres5t geometry1 we %\ld obta9 a cle> & exact idea ( ! g5]al de/9ation ( t sci5ce1 view$ 9 all its be>+s4 ,su* is ! object ( ?is *apt]4 ,def9ition4 ,geometry is commonly def9$ 9 a v vague & entirely improp] mann]1 z be+ ! sci5ce ( ext5sion4 ,an improvem5t on ?is w\ld be to say t geometry has = its object ! measurem5t ( ext5sion2 b su* an explanation w\ld be v insuffici5t1 al?\gh at bottom correct1 & w\ld be f> f giv+ any idea ( ! true g5]al "* ( geometrical sci5ce4 ,to d ?is1 ,i ?9k t ,i %\ld fir/ expla9 two fundam5tal ideas1 :i*1 v simple 9 !mselves1 h be5 s+ul>ly obscur$ by ! employm5t ( metaphysical consid]ations4 ,! ,idea ( ,space4 ,! fir/ is t ( ,space4 ,?is conception prop]ly consi/s simply 9 ?is1 t1 in/ead ( consid]+ ext5sion 9 ! bodies !mselves1 we view x 9 an indef9ite m$ium1 :i* we reg>d z conta9+ all ! bodies ( ! univ]se4 ,?is notion is naturally sugge/$ to u by obs]vation1 :5 we ?9k ( ! impression :i* a body w\ld leave 9 a fluid 9 :i* x _h be5 plac$4 ,x is cle>1 9 fact1 t1 z reg>ds its geometrical relations1 su* an impression may be sub/itut$ = ! body itself1 )\t alt]+ ! reason+s respect+ x4 ,z to ! physical nature ( ?is indef9ite space1 we are spontane\sly l$ to repres5t x to ourselves1 z be+ entirely analog\s to ! actual m$ium 9 :i* we live2 s t if ?is m$ium 0 liquid in/ead ( gase\s1 our geometrical space w\ld und\bt$ly be conceiv$ z liquid also4 ,?is circum/ance is1 moreov]1 only v second>y1 ! ess5tial object ( su* a conception be+ only to make u view ext5sion sep>ately f ! bodies :i* manife/ x to u4 ,we c easily und]/& 9 advance ! importance ( ?is fundam5tal image1 s9ce x p]mits u to /udy geometrical ph5om5a 9 !mselves1 ab/raction be+ made ( all ! o!r ph5om5a :i* con/antly accompany !m 9 real bodies1 )\t1 h[ev]1 ex]t+ any influ5ce ov] !m4 ,! regul> e/abli%m5t ( ?is g5]al ab/raction mu/ be reg>d$ z ! fir/ /ep :i* has be5 made 9 ! rational /udy ( geometry1 :i* w\ld h be5 impossible if x _h be5 necess>y to consid]1 toge!r ) ! =m & ! magnitude ( bodies1 all _! o!r physical prop]ties4 ,! use ( su* an hypo!sis1 :i* is p]haps ! mo/ anci5t philosophical conception creat$ by ! human m9d1 has n[ become s famili> to u1 t we h difficulty 9 exactly e/imat+ its importance1 by try+ to appreciate ! consequ5ces :i* w\ld result f its suppression4 ,diff]5t ,k9ds ( ,ext5sion4 ,! second prelim9>y geometrical conception :i* we h to exam9e is t ( ! diff]5t k9ds ( ext5sion1 designat$ by ! words volume1 surface1 l9e1 & ev5 po9t1 & ( :i* ! ord9>y explanation is s unsatisfactory4#ac ,footnote #ac3 ,lacroix has ju/ly criticis$ ! expression ( solid1 commonly us$ by geomet]s to designate a volume4 ,x is c]ta91 9 fact1 t :5 we wi% to consid] sep>ately a c]ta9 portion ( indef9ite space1 conceiv$ z gase\s1 we m5tally solidify its ext]ior envelope1 s t a l9e & a surface are habitually1 to our m9ds1 j z solid z a volume4 ,x may also be rem>k$ t mo/ g5]ally1 9 ord] t bodies may p5etrate "o ano!r ) m facility1 we are oblig$ to imag9e ! int]ior ( ! volumes to be holl[1 :i* r5d]s /ill m s5sible ! impropriety ( ! ^w solid4 ,al?\gh x is evid5tly impossible to conceive any ext5sion absolutely depriv$ ( any "o ( ! ?ree fundam5tal dim5sions1 x is no less inconte/able t1 9 a great numb] ( occasions1 ev5 ( imm$iate utility1 geometrical que/ions dep5d on only two dim5sions1 consid]$ sep>ately f ! ?ird1 or on a s+le dim5sion1 consid]$ sep>ately f ! two o!rs4 ,aga91 indep5d5tly ( ?is direct motive1 ! /udy ( ext5sion ) a s+le dim5sion1 & aft]w>ds ) two1 cle>ly pres5ts itself z an indisp5sable prelim9>y = facilitat+ ! /udy ( complete bodies ( ?ree dim5sions1 ! imm$iate !ory ( :i* w\ld be too complicat$4 ,su* are ! two g5]al motives :i* oblige geomet]s to consid] sep>ately ext5sion ) reg>d to "o or to two dim5sions1 z well z relatively to all ?ree toge!r4 ,! g5]al notions ( surface & ( l9e h be5 =m$ by ! human m9d1 9 ord] t x may be able to ?9k1 9 a p]man5t mann]1 ( ext5sion 9 two directions1 or 9 "o only4 ,! hyp]bolical expressions habitually employ$ by geomet]s to def9e ^! notions t5d to convey false ideas ( !m2 b1 exam9$ 9 !mselves1 !y h no o!r object ?an to p]mit u to reason ) facility respect+ ^! two k9ds ( ext5sion1 mak+ complete ab/raction ( t :i* ought n to be tak5 into consid]ation4 ,n[ = ?is x is suffici5t to conceive ! dim5sion :i* we wi% to elim9ate z becom+ gradually small] & small]1 ! two o!rs rema9+ ! same1 until x arrives at su* a degree ( t5uity t x c no long] fix ! att5tion4 ,x is ?us t we naturally acquire ! real idea ( a surface1 &1 by a second analog\s op]ation1 ! idea ( a l9e1 by repeat+ = bread? :at we _h at fir/ done = ?ickness4 ,f9ally1 if we aga9 repeat ! same op]ation1 we arrive at ! idea ( a po9t1 or ( an ext5sion consid]$ only ) ref]5ce to its place1 ab/raction be+ made ( all magnitude1 & design$ consequ5tly to det]m9e positions4 ,surfaces evid5tly h1 moreov]1 ! g5]al prop]ty ( exactly circumscrib+ volumes2 & 9 ! same way1 l9es1 9 _! turn1 circumscribe surfaces & are limit$ by po9ts4 ,b ?is consid]ation1 to :i* too mu* importance is (t5 giv51 is only a second>y "o4 ,surfaces & l9es are1 !n1 9 reality1 always conceiv$ ) ?ree dim5sions2 x w\ld be1 9 fact1 impossible to repres5t to "o's self a surface o!rwise ?an z an extremely ?9 plate1 & a l9e o!rwise ?an z an inf9itely f9e ?read4 ,x is ev5 pla9 t ! degree ( t5uity attribut$ by ea* individual to ! dim5sions ( :i* he wi%es to make ab/raction is n con/antly id5tical1 = x mu/ dep5d on ! degree ( subtilty ( 8 habitual geometrical obs]vations4 ,?is want ( uni=mity has1 besides1 no real inconv5i5ce1 s9ce x is suffici5t1 9 ord] t ! ideas ( surface & ( l9e %\ld satisfy ! ess5tial condition ( _! de/9ation1 = ea* "o to repres5t to himself ! dim5sions :i* are to be neglect$ z be+ small] ?an all ^? ^: magnitude 8 daily exp]i5ce gives him occasion to appreciate4 ,we h5ce see h[ devoid ( all mean+ are ! fanta/ic discussions ( metaphysicians ^u ! f\ndations ( geometry4 ,x %\ld also be rem>k$ t ^! primordial ideas are habitually pres5t$ by geomet]s 9 an unphilosophical mann]1 s9ce1 = example1 !y expla9 ! notions ( ! diff]5t sorts ( ext5t 9 an ord] absolutely ! inv]se ( _! natural dep5d5ce1 :i* (t5 produces ! mo/ s]i\s inconv5i5ces 9 elem5t>y in/ruction4 ,,! ,,f9al ,,object ,,( ,,geometry4 ,^! prelim9>ies be+ e/abli%$1 we c proce$ directly to ! g5]al def9ition ( geometry1 cont9u+ to conceive ?is sci5ce z hav+ = its f9al object ! measurem5t ( ext5sion4 ,x is necess>y 9 ?is matt] to g into a ?or\gh explanation1 f\nd$ on ! di/9ction ( ! ?ree k9ds ( ext5sion1 s9ce ! notion ( measurem5t is n exactly ! same ) ref]5ce to surfaces & volumes z to l9es4 ,nature ( ,geometrical ,measurem5t4 ,if we take ! ^w measurem5t 9 its direct & g5]al ma!matical acceptation1 :i* signifies simply ! det]m9ation ( ! value ( ! ratios betwe5 any homog5e\s magnitudes1 we mu/ consid]1 9 geometry1 t ! measurem5t ( surfaces & ( volumes1 unlike t ( l9es1 is nev] conceiv$1 ev5 9 ! mo/ simple & ! mo/ fav\rable cases1 z be+ effect$ directly4 ,! comp>ison ( two l9es is reg>d$ z direct2 t ( two surfaces or ( two volumes is1 on ! contr>y1 always indirect4 ,?us we conceive t two l9es may be sup]pos$2 b ! sup]position ( two surfaces1 or1 /ill m s1 ( two volumes1 is evid5tly impossible 9 mo/ cases2 &1 ev5 :5 x becomes rigor\sly practicable1 su* a comp>ison is nev] ei!r conv5i5t or exact4 ,x is1 !n1 v necess>y to expla9 :]e9 prop]ly consi/s ! truly geometrical measurem5t ( a surface or ( a volume4 ,measurem5t ( ,surfaces & ( ,volumes4 ,= ?is we mu/ consid] t1 :atev] may be ! =m ( a body1 !re always exi/s a c]ta9 numb] ( l9es1 m or less easy to be assign$1 ! l5g? ( :i* is suffici5t to def9e exactly ! magnitude ( its surface or ( its volume4 ,geometry1 reg>d+ ^! l9es z alone susceptible ( be+ directly measur$1 proposes to d$uce1 f ! simple det]m9ation ( !m1 ! ratio ( ! surface or ( ! volume s\ght1 to ! unity ( surface1 or to ! unity ( volume4 ,?us ! g5]al object ( geometry1 ) respect to surfaces & to volumes1 is prop]ly to r$uce all comp>isons ( surfaces or ( volumes to simple comp>isons ( l9es4 ,besides ! v great facility :i* su* a trans=mation evid5tly (f]s = ! measurem5t ( volumes & ( surfaces1 !re results f x1 9 consid]+ x 9 a m ext5d$ & m sci5tific mann]1 ! g5]al possibility ( r$uc+ to que/ions ( l9es all que/ions relat+ to volumes & to surfaces1 consid]$ ) ref]5ce to _! magnitude4 ,su* is (t5 ! mo/ important use ( ! geometrical expressions :i* det]m9e surfaces & volumes 9 functions ( ! correspond+ l9es4 ,x is true t direct comp>isons betwe5 surfaces or betwe5 volumes are sometimes employ$2 b su* measurem5ts are n reg>d$ z geometrical1 b only z a supplem5t sometimes necess>y1 al?\gh too r>ely applicable1 to ! insuffici5cy or to ! difficulty ( truly rational me?ods4 ,x is ?us t we (t5 det]m9e ! volume ( a body1 & 9 c]ta9 cases its surface1 by means ( its weight4 ,9 ! same way1 on o!r occasions1 :5 we c sub/itute = ! propos$ volume an equival5t liquid volume1 we e/abli% directly ! comp>ison ( ! two volumes1 by pr(it+ by ! prop]ty possess$ by liquid masses1 ( assum+ any desir$ =m4 ,b all means ( ?is nature are purely me*anical1 & rational geometry necess>ily rejects !m4 ,to r5d] m s5sible ! diff]5ce betwe5 ^! modes ( det]m9ation & true geometrical measurem5ts1 ,i w cite a s+le v rem>kable example2 ! mann] 9 :i* ,galileo det]m9$ ! ratio ( ! ord9>y cycloid to t ( ! g5]at+ circle4 ,! geometry ( 8 "t 0 z yet insuffici5t = ! rational solution ( su* a problem4 ,galileo conceiv$ ! idea ( discov]+ t ratio by a direct exp]im5t4 ,hav+ weigh$ z exactly z possible two plates ( ! same mat]ial & ( equal ?ickness1 "o ( !m hav+ ! =m ( a circle & ! o!r t ( ! g5]at$ cycloid1 he f\nd ! weight ( ! latt] always triple t ( ! =m]2 :5ce he inf]r$ t ! area ( ! cycloid is triple t ( ! g5]at+ circle1 a result agree+ ) ! v]itable solution subsequ5tly obta9$ by ,pascal & ,wallis4 ,su* a success evid5tly dep5ds on ! extreme simplicity ( ! ratio s\ght2 & we c und]/& ! necess>y insuffici5cy ( su* exp$i5ts1 ev5 :5 !y are actually practicable4 ,we see cle>ly1 f :at prec$es1 ! nature ( t "p ( geometry relat+ to volumes & t relat+ to surfaces4 ,b ! "* ( ! geometry ( l9es is n s app>5t1 s9ce1 9 ord] to simplify ! exposition1 we h consid]$ ! measurem5t ( l9es z be+ made directly4 ,!re is1 !re=e1 ne$$ a complem5t>y explanation ) respect to !m4 ,measurem5t ( curv$ ,l9es4 ,= ?is purpose1 x is suffici5t to di/+ui% betwe5 ! "r l9e & curv$ l9es1 ! measurem5t ( ! fir/ be+ alone reg>d$ z direct1 & t ( ! o!r z always indirect4 ,al?\gh sup]position is sometimes /rictly practicable = curv$ l9es1 x is nev]!less evid5t t truly rational geometry mu/ necess>ily reject x1 z n admitt+ ( any precision1 ev5 :5 x is possible4 ,! geometry ( l9es has1 !n1 = its g5]al object1 to r$uce 9 e case ! measurem5t ( curv$ l9es to t ( "r l9es2 & consequ5tly1 9 ! mo/ ext5d$ po9t ( view1 to r$uce to simple que/ions ( "r l9es all que/ions relat+ to ! magnitude ( any curves :atev]4 ,to und]/& ! possibility ( su* a trans=mation1 we mu/ rem>k1 t 9 e curve !re always exi/ c]ta9 "r l9es1 ! l5g? ( :i* mu/ be suffici5t to det]m9e t ( ! curve4 ,?us1 9 a circle1 x is evid5t t f ! l5g? ( ! radius we mu/ be able to d$uce t ( ! circumf]5ce2 9 ! same way1 ! l5g? ( an ellipse dep5ds on t ( its two axes2 ! l5g? ( a cycloid ^u ! diamet] ( ! g5]at+ circle1 c42 & if1 in/ead ( consid]+ ! :ole ( ea* curve1 we dem&1 m g5]ally1 ! l5g? ( any arc1 x w be suffici5t to add to ! diff]5t rectil9e> p>amet]s1 :i* det]m9e ! :ole curve1 ! *ord ( ! propos$ arc1 or ! co-ord9ates ( its extremities4 ,to discov] ! relation :i* exi/s betwe5 ! l5g? ( a curv$ l9e & t ( simil> "r l9es1 is ! g5]al problem ( ! "p ( geometry :i* relates to ! /udy ( l9es4 ,comb9+ ?is consid]ation ) ^? previ\sly sugge/$ ) respect to volumes & to surfaces1 we may =m a v cle> idea ( ! sci5ce ( geometry1 conceiv$ 9 all its p>ts1 by assign+ to x1 = its g5]al object1 ! f9al r$uction ( ! comp>isons ( all k9ds ( ext5t1 volumes1 surfaces1 or l9es1 to simple comp>isons ( "r l9es1 ! only comp>isons reg>d$ z capable ( be+ made directly1 & :i* inde$ c\ld n be r$uc$ to any o!rs m easy to effect4 ,su* a conception1 at ! same "t1 indicates cle>ly ! v]itable "* ( geometry1 & seems suit$ to %[ at a s+le glance its utility & its p]fection4 ,measurem5t ( "r ,l9es4 ,9 ord] to complete ?is fundam5tal explanation1 ,i h yet to %[ h[ !re c be1 9 geometry1 a special section relat+ to ! "r l9e1 :i* seems at fir/ incompatible ) ! pr9ciple t ! measurem5t ( ?is class ( l9es mu/ always be reg>d$ z direct4 ,x is s1 9 fact1 z comp>$ ) t ( curv$ l9es1 & ( all ! o!r objects :i* geometry consid]s4 ,b x is evid5t t ! e/imation ( a "r l9e _c be view$ z direct except s f> z ! l9e> unit c be appli$ to x4 ,n[ ?is (t5 pres5ts insurm\ntable difficulties1 z ,i _h occasion to %[1 = ano!r reason1 9 ! introductory *apt]4 ,we mu/1 !n1 make ! measurem5t ( ! propos$ "r l9e dep5d on o!r analog\s measurem5ts capable ( be+ effect$ directly4 ,!re is1 !n1 necess>ily a prim>y di/9ct bran* ( geometry1 exclusively devot$ to ! "r l9e2 its object is to det]m9e c]ta9 "r l9es f o!rs by means ( ! relations belong+ to ! figures result+ f _! assemblage4 ,?is prelim9>y "p ( geometry1 :i* is almo/ imp]ceptible 9 view+ ! :ole ( ! sci5ce1 is nev]!less susceptible ( a great developm5t4 ,x is evid5tly ( especial importance1 s9ce all o!r geometrical measurem5ts are ref]r$ to ^? ( "r l9es1 & if !y c\ld n be det]m9$1 ! solution ( e "q w\ld rema9 unf9i%$4 ,su*1 !n1 are ! v>i\s fundam5tal p>ts ( rational geometry1 arrang$ accord+ to _! natural dep5d5ce2 ! geometry ( l9es be+ fir/ consid]$1 beg9n+ ) ! "r l9e2 !n ! geometry ( surfaces1 &1 f9ally1 t ( solids4 ,,inf9ite ,,ext5t ,,( ,,its ,,field4 ,hav+ det]m9$ ) precision ! g5]al & f9al object ( geometrical inquiries1 ! sci5ce mu/ n[ be consid]$ ) respect to ! field embrac$ by ea* ( its ?ree fundam5tal sections4 ,?us consid]$1 geometry is evid5tly susceptible1 by its nature1 ( an ext5sion :i* is rigor\sly inf9ite2 = ! measurem5t ( l9es1 ( surfaces1 or ( volumes pres5ts necess>ily z _m di/9ct que/ions z we c conceive diff]5t figures subject$ to exact def9itions2 & _! numb] is evid5tly inf9ite4 ,geomet]s limit$ !mselves at fir/ to consid] ! mo/ simple figures :i* 7 directly furni%$ !m by nature1 or :i* 7 d$uc$ f ^! primitive elem5ts by ! lea/ complicat$ comb9ations4 ,b !y h p]ceiv$1 s9ce ,desc>tes1 t1 9 ord] to con/itute ! sci5ce 9 ! mo/ philosophical mann]1 x 0 necess>y to make x apply to all imag9able figures4 ,?is ab/ract geometry w !n inevitably compreh5d z p>ticul> cases all ! diff]5t real figures :i* ! ext]ior _w c\ld pres5t4 ,x is !n a fundam5tal pr9ciple 9 truly rational geometry to consid]1 z f> z possible1 all figures :i* c be rigor\sly conceiv$4 ,! mo/ sup]ficial exam9ation is en\gh to conv9ce u t ^! figures pres5t a v>iety :i* is q inf9ite4 ,inf9ity ( ,l9es4 ,) respect to curv$ l9es1 reg>d+ !m z g5]at$ by ! motion ( a po9t gov]n$ by a c]ta9 law1 x is pla9 t we %all h1 9 g5]al1 z _m diff]5t curves z we conceive diff]5t laws = ?is motion1 :i* may evid5tly be det]m9$ by an inf9ity ( di/9ct conditions2 al?\gh x may sometimes accid5tally happ5 t new g5]ations produce curves :i* h be5 already obta9$4 ,?us1 among plane curves1 if a po9t moves s z to rema9 con/antly at ! same di/ance f a fix$ po9t1 x w g5]ate a circle2 if x is ! sum or ! diff]5ce ( its di/ances f two fix$ po9ts :i* rema9s con/ant1 ! curve describ$ w be an ellipse or an hyp]bola2 if x is _! product1 we %all h an entirely diff]5t curve2 if ! po9t dep>ts equally f a fix$ po9t & f a fix$ l9e1 x w describe a p>abola2 if x revolves on a circle at ! same "t t ?is circle rolls along a /raight l9e1 we %all h a cycloid2 if x advances along a /raight l9e1 :ile ?is l9e1 fix$ at "o ( its extremities1 turns 9 any mann] :atev]1 !re w result :at 9 g5]al t]ms are call$ spirals1 :i* ( !mselves evid5tly pres5t z _m p]fectly di/9ct curves z we c suppose diff]5t relations betwe5 ^! two motions ( translation & ( rotation1 c4 ,ea* ( ^! diff]5t curves may !n furni% new ones1 by ! diff]5t g5]al con/ructions :i* geomet]s h imag9$1 & :i* give rise to evolutes1 to epicycloids1 to cau/ics1 c4 ,f9ally1 !re exi/s a /ill great] v>iety among curves ( d\ble curvature4 ,inf9ity ( ,surfaces4 ,z to surfaces1 ! figures are necess>ily m diff]5t /ill1 consid]+ !m z g5]at$ by ! motion ( l9es4 ,inde$1 ! figure may !n v>y1 n only 9 consid]+1 z 9 curves1 ! diff]5t inf9itely num]\s laws to :i* ! motion ( ! g5]at+ l9e may be subject$1 b also 9 suppos+ t ?is l9e itself may *ange its nature2 a circum/ance :i* has no?+ analog\s 9 curves1 s9ce ! po9ts :i* describe !m _c h any di/9ct figure4 ,two classes ( v diff]5t conditions may !n cause ! figures ( surfaces to v>y1 :ile !re exi/s only "o = l9es4 ,x is useless to cite examples ( ?is d\bly inf9ite multiplicity ( surfaces4 ,x w\ld be suffici5t to consid] ! extreme v>iety ( ! s+le gr\p ( surfaces :i* may be g5]at$ by a "r l9e1 & :i* compreh5ds ! :ole family ( cyl9drical surfaces1 t ( conical surfaces1 ! mo/ g5]al class ( developable surfaces1 c4 ,inf9ity ( ,volumes4 ,) respect to volumes1 !re is no occasion = any special consid]ation1 s9ce !y are di/+ui%$ f ea* o!r only by ! surfaces :i* b\nd !m4 ,9 ord] to complete ?is sket*1 x %\ld be add$ t surfaces !mselves furni% a new g5]al means ( conceiv+ new curves1 s9ce e curve may be reg>d$ z produc$ by ! int]section ( two surfaces4 ,x is 9 ?is way1 inde$1 t ! fir/ l9es :i* we may reg>d z hav+ be5 truly inv5t$ by geomet]s 7 obta9$1 s9ce nature gave directly ! /raight l9e & ! circle4 ,we "k t ! ellipse1 ! p>abola1 & ! hyp]bola1 ! only curves completely /udi$ by ! anci5ts1 7 9 _! orig9 conceiv$ only z result+ f ! int]section ( a cone ) circul> base by a plane 9 diff]5t positions4 ,x is evid5t t1 by ! comb9$ employm5t ( ^! diff]5t g5]al means = ! =mation ( l9es & ( surfaces1 we c\ld produce a rigor\sly inf9itely s]ies ( di/9ct =ms 9 />t+ f only a v small numb] ( figures directly furni%$ by obs]vation4 ,analytical inv5tion ( ,curves1 c4 ,f9ally1 all ! v>i\s direct means = ! inv5tion ( figures h sc>cely any f>!r importance1 s9ce rational geometry has assum$ its f9al "* 9 ! h&s ( ,desc>tes4 ,inde$1 z we %all see m fully 9 *apt] iii41 ! inv5tion ( figures is n[ r$uc$ to ! inv5tion ( equations1 s t no?+ is m easy ?an to conceive new l9es & new surfaces1 by *ang+ at w ! functions introduc$ into ! equations4 ,?is simple ab/ract proc$ure is1 9 ?is respect1 inf9itely m fruitful ?an all ! direct res\rces ( geometry1 develop$ by ! mo/ p[]ful imag9ation1 :i* %\ld devote itself exclusively to t ord] ( conceptions4 ,x also expla9s1 9 ! mo/ g5]al & ! mo/ /rik+ mann]1 ! necess>ily inf9ite v>iety ( geometrical =ms1 :i* ?us corresponds to ! div]sity ( analytical functions4 ,la/ly1 x %[s no less cle>ly t ! diff]5t =ms ( surfaces mu/ be /ill m num]\s ?an ^? ( l9es1 s9ce l9es are repres5t$ analytically by equations ) two v>iables1 :ile surfaces give rise to equations ) ?ree v>iables1 :i* necess>ily pres5t a great] div]sity4 ,! prec$+ consid]ations are suffici5t to %[ cle>ly ! rigor\sly inf9ite ext5t ( ea* ( ! ?ree g5]al sections ( geometry4 ,,expansion ,,( ,,orig9al ,,def9ition4 ,to complete ! =mation ( an exact & suffici5tly ext5d$ idea ( ! nature ( geometrical inquiries1 x is n[ indisp5sable to return to ! g5]al def9ition above giv51 9 ord] to pres5t x "u a new po9t ( view1 )\t :i* ! complete sci5ce w\ld be only v imp]fectly conceiv$4 ,:5 we assign z ! object ( geometry ! measurem5t ( all sorts ( l9es1 surfaces1 & volumes1 t is1 z has be5 expla9$1 ! r$uction ( all geometrical comp>isons to simple comp>isons ( "r l9es1 we h evid5tly ! advantage ( indicat+ a g5]al de/9ation v precise & v easy to compreh5d4 ,b if we set aside e def9ition1 & exam9e ! actual composition ( ! sci5ce ( geometry1 we w at fir/ be induc$ to reg>d ! prec$+ def9ition z mu* too n>r[2 = x is c]ta9 t ! great] "p ( ! inve/igations :i* con/itute our pres5t geometry d n at all appe> to h = _! object ! measurem5t ( ext5sion4 ,9 spite ( ?is fundam5tal objection1 ,i w p]si/ 9 reta9+ ?is def9ition2 =1 9 fact1 if1 in/ead ( conf9+ ourselves to consid]+ ! diff]5t que/ions ( geometry isolat$ly1 we endeav\r to grasp ! lead+ que/ions1 9 comp>ison ) :i* all o!rs1 h[ev] important !y may be1 mu/ be reg>d$ z only second>y1 we w f9ally recognize t ! measurem5t ( l9es1 ( surfaces1 & ( volumes1 is ! inv>iable object1 sometimes direct1 ?\gh mo/ (t5 indirect1 ( all geometrical lab\rs4 ,?is g5]al proposition be+ fundam5tal1 s9ce x c alone give our def9ition all its value1 x is indisp5sable to ent] into "s developm5ts ^u ?is subject4 ,,prop]ties ,,( ,,l9es ,,& ,,surfaces4 ,:5 we exam9e ) att5tion ! geometrical inve/igations :i* d n seem to relate to ! measurem5t ( ext5t1 we f9d t !y consi/ ess5tially 9 ! /udy ( ! diff]5t prop]ties ( ea* l9e or ( ea* surface2 t is1 9 ! k ( ! diff]5t modes ( g5]ation1 or at lea/ ( def9ition1 peculi> to ea* figure consid]$4 ,n[ we c easily e/abli% 9 ! mo/ g5]al mann] ! necess>y relation ( su* a /udy to ! "q ( measurem5t1 = :i* ! mo/ complete k ( ! prop]ties ( ea* =m is an indisp5sable prelim9>y4 ,?is is concurr5tly prov5 by two consid]ations1 equally fundam5tal1 al?\gh q di/9ct 9 _! nature4 ,,necessity ,,( ,,_! ,,/udy3 #a4 ,to f9d ! mo/ suitable ,prop]ty4 ,! fir/1 purely sci5tific1 consi/s 9 rem>k+ t1 if we did n "k any o!r *>act]i/ic prop]ty ( ea* l9e or surface ?an t "o accord+ to :i* geomet]s _h fir/ conceiv$ x1 9 mo/ cases x w\ld be impossible to succe$ 9 ! solution ( que/ions relat+ to its measurem5t4 ,9 fact1 x is easy to und]/& t ! diff]5t def9itions :i* ea* figure admits ( are n all equally suitable = su* an object1 & t !y ev5 pres5t ! mo/ complete oppositions 9 t respect4 ,besides1 s9ce ! primitive def9ition ( ea* figure 0 evid5tly n *os5 ) ?is condition 9 view1 x is cle> t we mu/ n expect1 9 g5]al1 to f9d x ! mo/ suitable2 :5ce results ! necessity ( discov]+ o!rs1 t is1 ( /udy+ z f> z is possible ! prop]ties ( ! propos$ figure4 ,let u suppose1 = example1 t ! circle is def9$ to be ! curve :i*1 ) ! same cont\r1 conta9s ! greate/ area4 ,?is is c]ta9ly a v *>act]i/ic prop]ty1 b we w\ld evid5tly f9d insurm\ntable difficulties 9 try+ to d$uce f su* a />t+ po9t ! solution ( ! fundam5tal que/ions relat+ to ! rectification or to ! quadrature ( ?is curve4 ,x is cle>1 9 advance1 t ! prop]ty ( hav+ all its po9ts equally di/ant f a fix$ po9t mu/ evid5tly be mu* bett] adapt$ to inquiries ( ?is nature1 ev5 ?\gh x be n precisely ! mo/ suitable4 ,9 l mann]1 w\ld ,ar*im$es ev] h be5 able to discov] ! quadrature ( ! p>abola if he _h kn[n no o!r prop]ty ( t curve ?an t x 0 ! section ( a cone ) a circul> base1 by a plane p>allel to its g5]atrix8 ,! purely speculative lab\rs ( prec$+ geomet]s1 9 trans=m+ ?is fir/ def9ition1 7 evid5tly indisp5sable prelim9>ies to ! direct solution ( su* a "q4 ,! same is true1 9 a /ill great] degree1 ) respect to surfaces4 ,to =m a j idea ( ?is1 we ne$ only comp>e1 z to ! "q ( cubature or quadrature1 ! common def9ition ( ! sph]e ) t "o1 no less *>act]i/ic c]ta9ly1 :i* w\ld consi/ 9 reg>d+ a sph]ical body1 z t "o :i*1 ) ! same area1 conta9s ! greate/ volume4 ,no m examples are ne$$ to %[ ! necessity ( kn[+1 s f> z is possible1 all ! prop]ties ( ea* l9e or ( ea* surface1 9 ord] to facilitate ! inve/igation ( rectifications1 ( quadratures1 & ( cubatures1 :i* con/itutes ! f9al object ( geometry4 ,we may ev5 say t ! pr9cipal difficulty ( que/ions ( ?is k9d consi/s 9 employ+ 9 ea* case ! prop]ty :i* is be/ adapt$ to ! nature ( ! propos$ problem4 ,?us1 :ile we cont9ue to indicate1 = m precision1 ! measurem5t ( ext5sion z ! g5]al de/9ation ( geometry1 ?is fir/ consid]ation1 :i* goes to ! v bottom ( ! subject1 %[s cle>ly ! necessity ( includ+ 9 x ! /udy1 z ?or\gh z possible1 ( ! diff]5t g5]ations or def9itions belong+ to ! same =m4 #b4 ,to pass f ! ,concrete to ! ,ab/ract4 ,a second consid]ation1 ( at lea/ equal importance1 consi/s 9 su* a /udy be+ indisp5sable = organiz+ 9 a rational mann] ! relation ( ! ab/ract to ! concrete 9 geometry4 ,! sci5ce ( geometry hav+ to consid] all imag9able figures :i* admit ( an exact def9ition1 x necess>ily results f ?is1 z we h rem>k$1 t que/ions relat+ to any figures pres5t$ by nature are always implicitly compris$ 9 ?is ab/ract geometry1 suppos$ to h atta9$ its p]fection4 ,b :5 x is necess>y to actually pass to concrete geometry1 we con/antly meet ) a fundam5tal difficulty1 t ( kn[+ to :i* ( ! diff]5t ab/ract types we are to ref]1 ) suffici5t approximation1 ! real l9es or surfaces :i* we h to /udy4 ,n[ x is = ! purpose ( e/abli%+ su* a relation t x is p>ticul>ly indisp5sable to "k ! greate/ possible numb] ( prop]ties ( ea* figure consid]$ 9 geometry4 ,9 fact1 if we always conf9$ ourselves to ! s+le primitive def9ition ( a l9e or ( a surface1 suppos+ ev5 t we c\ld !n measure x 7:i*1 accord+ to ! fir/ ord] ( consid]ations1 w\ld g5]ally be impossible71 ?is k w\ld rema9 almo/ necess>ily b>r5 9 ! application1 s9ce we %\ld n ord9>ily "k h[ to recognize t figure 9 nature :5 x pres5t$ itself !re2 to ensure t1 x w\ld be necess>y t ! s+le *>act]i/ic1 accord+ to :i* geomet]s _h conceiv$ x1 %\ld be precisely t "o ^: v]ification ext]nal circum/ances w\ld admit3 a co9cid5ce :i* w\ld be purely =tuit\s1 & on :i* we c\ld n c\nt1 al?\gh x might sometimes take place4 ,x is1 !n1 only by multiply+ z mu* z possible ! *>act]i/ic prop]ties ( ea* ab/ract figure1 t we c be assur$1 9 advance1 ( recogniz+ x 9 ! concrete /ate1 & ( ?us turn+ to acc\nt all our rational lab\rs1 by v]ify+ 9 ea* case ! def9ition :i* is susceptible ( be+ directly prov54 ,?is def9ition is almo/ always ! only "o 9 giv5 circum/ances1 & v>ies1 on ! o!r h&1 = ! same figure1 ) diff]5t circum/ances2 a d\ble reason = its previ\s det]m9ation4 ,illu/ration3 ,orbits ( ! ,planets4 ,! geometry ( ! heav5s furni%es u ) a v memorable example 9 ?is matt]1 well suit$ to %[ ! g5]al necessity ( su* a /udy4 ,we "k t ! ellipse 0 discov]$ by ,kepl] to be ! curve :i* ! planets describe ab\t ! sun1 & ! satellites ab\t _! planets4 ,n[ w\ld ?is fundam5tal discov]y1 :i* re-creat$ a/ronomy1 ev] h be5 possible1 if geomet]s _h be5 always conf9$ to conceiv+ ! ellipse only z ! oblique section ( a circul> cone by a plane8 ,no su* def9ition1 x is evid5t1 w\ld admit ( su* a v]ification4 ,! mo/ g5]al prop]ty ( ! ellipse1 t ! sum ( ! di/ances f any ( its po9ts to two fix$ po9ts is a con/ant quantity1 is und\bt$ly mu* m susceptible1 by its nature1 ( caus+ ! curve to be recogniz$ 9 ?is case1 b /ill is n directly suitable4 ,! only *>act]i/ic :i* c "h be imm$iately v]ifi$ is t :i* is d]iv$ f ! relation :i* exi/s 9 ! ellipse betwe5 ! l5g? ( ! focal di/ances & _! direction2 ! only relation :i* admits ( an a/ronomical int]pretation1 z express+ ! law :i* connects ! di/ance f ! planet to ! sun1 ) ! "t elaps$ s9ce ! beg9n+ ( its revolution4 ,x 01 !n1 necess>y t ! purely speculative lab\rs ( ! ,greek geomet]s on ! prop]ties ( ! conic sections %\ld h previ\sly pres5t$ _! g5]ation "u a multitude ( diff]5t po9ts ( view1 be=e ,kepl] c\ld ?us pass f ! ab/ract to ! concrete1 9 *oos+ f among all ^! diff]5t *>act]i/ics t "o :i* c\ld be mo/ easily prov5 = ! planet>y orbits4 ,illu/ration3 ,figure ( ! ,e>?4 ,ano!r example ( ! same ord]1 b relat+ to surfaces1 occurs 9 consid]+ ! important "q ( ! figure ( ! e>?4 ,if we _h nev] kn[n any o!r prop]ty ( ! sph]e ?an its primitive "* ( hav+ all its po9ts equally di/ant f an int]ior po9t1 h[ w\ld we ev] h be5 able to discov] t ! surface ( ! e>? 0 sph]ical8 ,= ?is1 x 0 necess>y previ\sly to d$uce f ?is def9ition ( ! sph]e "s prop]ties capable ( be+ v]ifi$ by obs]vations made ^u ! surface alone1 su* z ! con/ant ratio :i* exi/s betwe5 ! l5g? ( ! pa? trav]s$ 9 ! direction ( any m]idian ( a sph]e go+ t[>ds a pole1 & ! angul> height ( ?is pole above ! horizon at ea* po9t4 ,ano!r example1 b involv+ a mu* long] s]ies ( prelim9>y speculations1 is ! subsequ5t pro( t ! e>? is n rigor\sly sph]ical1 b t its =m is t ( an ellipsoid ( revolution4 ,aft] su* examples1 x w\ld be ne$less to give any o!rs1 :i* any "o besides may easily multiply4 ,all ( !m prove t1 )\t a v ext5d$ k ( ! diff]5t prop]ties ( ea* figure1 ! relation ( ! ab/ract to ! concrete1 9 geometry1 w\ld be purely accid5tal1 & t ! sci5ce w\ld consequ5tly want "o ( its mo/ ess5tial f\ndations4 ,su*1 !n1 are two g5]al consid]ations :i* fully demon/rate ! necessity ( introduc+ into geometry a great numb] ( inve/igations :i* h n ! measurem5t ( ext5sion = _! direct object2 :ile we cont9ue1 h[ev]1 to conceive su* a measurem5t z be+ ! f9al de/9ation ( all geometrical sci5ce4 ,9 ?is way we c reta9 ! philosophical advantages ( ! cle>ness & precision ( ?is def9ition1 & /ill include 9 x1 9 a v logical ?\gh indirect mann]1 all kn[n geometrical rese>*es1 9 consid]+ ^? :i* d n seem to relate to ! measurem5t ( ext5sion1 z int5d$ ei!r to prep>e = ! solution ( ! f9al que/ions1 or to r5d] possible ! application ( ! solutions obta9$4 ,hav+ ?us recogniz$1 z a g5]al pr9ciple1 ! close & necess>y connexion ( ! /udy ( ! prop]ties ( l9es & surfaces ) ^? rese>*es :i* con/itute ! f9al object ( geometry1 x is evid5t t geomet]s1 9 ! progress ( _! lab\rs1 mu/ by no means con/ra9 !mselves to keep su* a connexion always 9 view4 ,kn[+1 once = all1 h[ important x is to v>y z mu* z possible ! mann] ( conceiv+ ea* figure1 !y %\ld pursue t /udy1 )\t consid]+ ( :at imm$iate use su* or su* a special prop]ty may be = rectifications1 quadratures1 & cubatures4 ,!y w\ld uselessly fett] _! inquiries by atta*+ a pu]ile importance to ! cont9u$ e/abli%m5t ( t co-ord9ation4 ,?is g5]al exposition ( ! g5]al object ( geometry is s mu* ! m indisp5sable1 s9ce1 by ! v nature ( ! subject1 ?is /udy ( ! diff]5t prop]ties ( ea* l9e & ( ea* surface necess>ily composes by f> ! great] "p ( ! :ole body ( geometrical rese>*es4 ,inde$1 ! que/ions directly relat+ to rectifications1 to quadratures1 & to cubatures1 are evid5tly1 by !mselves1 v few 9 numb] = ea* figure consid]$4 ,on ! o!r h&1 ! /udy ( ! prop]ties ( ! same figure pres5ts an unlimit$ field to ! activity ( ! human m9d1 9 :i* x may always hope to make new discov]ies4 ,?us1 al?\gh geomet]s h occupi$ !mselves = tw5ty c5turies1 ) m or less activity und\bt$ly1 b )\t any real int]ruption1 9 ! /udy ( ! conic sections1 !y are f> f reg>d+ t s simple subject z be+ exhau/$2 & x is c]ta91 inde$1 t 9 cont9u+ to devote !mselves to x1 !y w\ld n fail to f9d /ill unkn[n prop]ties ( ^? diff]5t curves4 ,if lab\rs ( ?is k9d h slack5$ consid]ably = a c5tury pa/1 x is n because !y are complet$1 b only1 z w be pres5tly expla9$1 because ! philosophical revolution 9 geometry1 br\ght ab\t by ,desc>tes1 has s+ul>ly dim9i%$ ! importance ( su* rese>*es4 ,x results f ! prec$+ consid]ations t n only is ! field ( geometry necess>ily inf9ite because ( ! v>iety ( figures to be consid]$1 b also 9 virtue ( ! div]sity ( ! po9ts ( view "u ! same figure may be reg>d$4 ,?is la/ conception is1 inde$1 t :i* gives ! broade/ & mo/ complete idea ( ! :ole body ( geometrical rese>*es4 ,we see t /udies ( ?is k9d consi/ ess5tially1 = ea* l9e or = ea* surface1 9 connect+ all ! geometrical ph5om5a :i* x c pres5t1 ) a s+le fundam5tal ph5om5on1 reg>d$ z ! primitive def9ition4 ,,! ,,two ,,g5]al ,,me?ods ,,( ,,geometry4 ,hav+ n[ expla9$ 9 a g5]al & yet precise mann] ! f9al object ( geometry1 & %[n h[ ! sci5ce1 ?us def9$1 compreh5ds a v ext5sive class ( rese>*es :i* did n at fir/ appe> necess>ily to belong to x1 !re rema9s to be consid]$ ! me?od to be foll[$ = ! =mation ( ?is sci5ce4 ,?is discussion is indisp5sable to complete ?is fir/ sket* ( ! philosophical "* ( geometry4 ,i %all "h conf9e myself to indicat+ ! mo/ g5]al consid]ation 9 ?is matt]1 develop+ & summ+ up ?is important fundam5tal idea 9 ! foll[+ *apt]s4 ,geometrical que/ions may be treat$ accord+ to two me?ods s diff]5t1 t !re result f !m two sorts ( geometry1 s to say1 ! philosophical "* ( :i* does n seem to me to h yet be5 prop]ly appreh5d$4 ,! expressions ( ,syn!tical ,geometry & ,analytical ,geometry1 habitually employ$ to designate !m1 give a v false idea ( !m4 ,i w\ld mu* pref] ! purely hi/orical d5om9ations ( ,geometry ( ! ,anci5ts & ,geometry ( ! ,mod]ns1 :i* h at lea/ ! advantage ( n caus+ _! true "* to be misund]/ood4 ,b ,i propose to employ h5ce=? ! regul> expressions ( ,special ,geometry & ,g5]al ,geometry1 :i* seem to me suit$ to *>act]ize ) precision ! v]itable nature ( ! two me?ods4 ,_! fundam5tal ,diff]5ce4 ,! fundam5tal diff]5ce betwe5 ! mann] 9 :i* we conceive ,geometry s9ce ,desc>tes1 & ! mann] 9 :i* ! geomet]s ( antiquity treat$ geometrical que/ions1 is n ! use ( ! ,calculus 7or ,algebra71 z is commonly ?\ght to be ! case4 ,on ! "o h&1 x is c]ta9 t ! use ( ! calculus 0 n entirely unkn[n to ! anci5t geomet]s1 s9ce !y us$ to make cont9ual & v ext5sive applications ( ! !ory ( proportions1 :i* 0 = !m1 z a means ( d$uction1 a sort ( real1 ?\gh v imp]fect & especially extremely limit$ equival5t = our pres5t algebra4 ,! calculus may ev5 be employ$ 9 a mu* m complete mann] ?an !y h us$ x1 9 ord] to obta9 c]ta9 geometrical solutions1 :i* w /ill reta9 all ! ess5tial "* ( ! anci5t geometry2 ?is occurs v frequ5tly ) respect to ^? problems ( geometry ( two or ( ?ree dim5sions1 :i* are commonly designat$ "u ! "n ( det]m9ate4 ,on ! o!r h&1 important z is ! influ5ce ( ! calculus 9 our mod]n geometry1 v>i\s solutions obta9$ )\t algebra may sometimes manife/ ! peculi> "* :i* di/+ui%es x f ! anci5t geometry1 al?\gh analysis is g5]ally indisp5sable4 ,i w cite1 z an example1 ! me?od ( ,rob]val = tang5ts1 ! nature ( :i* is ess5tially mod]n1 & :i*1 h[ev]1 leads 9 c]ta9 cases to complete solutions1 )\t any aid f ! calculus4 ,x is n1 !n1 ! in/rum5t ( d$uction employ$ :i* is ! pr9cipal di/9ction betwe5 ! two c\rses :i* ! human m9d c take 9 geometry4 ,! real fundam5tal diff]5ce1 z yet imp]fectly appreh5d$1 seems to me to consi/ 9 ! v nature ( ! que/ions consid]$4 ,9 tru?1 geometry1 view$ z a :ole1 & suppos$ to h atta9$ entire p]fection1 mu/1 z we h se5 on ! "o h&1 embrace all imag9able figures1 &1 on ! o!r1 discov] all ! prop]ties ( ea* figure4 ,x admits1 f ?is d\ble consid]ation1 ( be+ treat$ accord+ to two ess5tially di/9ct plans2 ei!r1 #a1 by gr\p+ toge!r all ! que/ions1 h[ev] diff]5t !y may be1 :i* relate to ! same figure1 & isolat+ ^? relat+ to diff]5t bodies1 :atev] analogy !re may exi/ betwe5 !m2 or1 #b1 on ! contr>y1 by unit+ "u "o po9t ( view all simil> inquiries1 to :atev] diff]5t figures !y may relate1 & sep>at+ ! que/ions relat+ to ! really diff]5t prop]ties ( ! same body4 ,9 a ^w1 ! :ole body ( geometry may be ess5tially arrang$ ei!r ) ref]5ce to ! bodies /udi$ or to ! ph5om5a to be consid]$4 ,! fir/ plan1 :i* is ! mo/ natural1 0 t ( ! anci5ts2 ! second1 inf9itely m rational1 is t ( ! mod]ns s9ce ,desc>tes4 ,geometry ( ! ,anci5ts4 ,inde$1 ! pr9cipal *>act]i/ics ( ! anci5t geometry is t !y /udi$1 "o by "o1 ! diff]5t l9es & ! diff]5t surfaces1 n pass+ to ! exam9ation ( a new figure till !y ?\ght !y _h exhau/$ all t !re 0 int]e/+ 9 ! figures already kn[n4 ,9 ?is way ( proce$+1 :5 !y und]took ! /udy ( a new curve1 ! :ole ( ! lab\r be/[$ on ! prec$+ ones c\ld n (f] directly any ess5tial assi/ance1 o!rwise ?an by ! geometrical practice to :i* x _h tra9$ ! m9d4 ,:atev] might be ! real simil>ity ( ! que/ions propos$ z to two diff]5t figures1 ! complete k acquir$ = ! "o c\ld n at all disp5se ) tak+ up aga9 ! :ole ( ! inve/igation = ! o!r4 ,?us ! progress ( ! m9d 0 nev] assur$2 s t !y c\ld n be c]ta91 9 advance1 ( obta9+ any solution :atev]1 h[ev] analog\s ! propos$ problem might be to que/ions :i* _h be5 already resolv$4 ,?us1 = example1 ! det]m9ation ( ! tang5ts to ! ?ree conic sections did n furni% any rational assi/ance = draw+ ! tang5t to any o!r new curve1 su* z ! con*oid1 ! cissoid1 c4 ,9 a ^w1 ! geometry ( ! anci5ts 01 accord+ to ! expression propos$ above1 ess5tially special4 ,geometry ( ! ,mod]ns4 ,9 ! sy/em ( ! mod]ns1 geometry is1 on ! contr>y1 em95tly g5]al1 t is to say1 relat+ to any figures :atev]4 ,x is easy to und]/&1 9 ! fir/ place1 t all geometrical expressions ( any int]e/ may be propos$ ) ref]5ce to all imag9able figures4 ,?is is se5 directly 9 ! fundam5tal problems--( rectifications1 quadratures1 & cubatures--:i* con/itute1 z has be5 %[n1 ! f9al object ( geometry4 ,b ?is rem>k is no less inconte/able1 ev5 = inve/igations :i* relate to ! diff]5t prop]ties ( l9es & ( surfaces1 & ( :i* ! mo/ ess5tial1 su* z ! "q ( tang5ts or ( tang5t planes1 ! !ory ( curvatures1 c41 are evid5tly common to all figures :atev]4 ,! v few inve/igations :i* are truly peculi> to p>ticul> figures h only an extremely second>y importance4 ,?is be+ und]/ood1 mod]n geometry consi/s ess5tially 9 ab/ract+1 9 ord] to treat x by itself1 9 an entirely g5]al mann]1 e "q relat+ to ! same geometrical ph5om5on1 9 :atev] bodies x may be consid]$4 ,! application ( ! univ]sal !ories ?us con/ruct$ to ! special det]m9ation ( ! ph5om5on :i* is treat$ ( 9 ea* p>ticul> body1 is n[ reg>d$ z only a subalt]n lab\r1 to be execut$ accord+ to inv>iable rules1 & ! success ( :i* is c]ta9 9 advance4 ,?is lab\r is1 9 a ^w1 ( ! same "* z ! num]ical calculation ( an analytical =mula4 ,!re c be no o!r m]it 9 x ?an t ( pres5t+ 9 ea* case ! solution :i* is necess>ily furni%$ by ! g5]al me?od1 ) all ! simplicity & elegance :i* ! l9e or ! surface consid]$ c admit (4 ,b no real importance is atta*$ to any ?+ b ! conception & ! complete solution ( a new "q belong+ to any figure :atev]4 ,lab\rs ( ?is k9d are alone reg>d$ z produc+ any real advance 9 sci5ce4 ,! att5tion ( geomet]s1 ?us reliev$ f ! exam9ation ( ! peculi>ities ( diff]5t figures1 & :olly direct$ t[>ds g5]al que/ions1 has be5 !reby able to elevate itself to ! consid]ation ( new geometrical conceptions1 :i*1 appli$ to ! curves /udi$ by ! anci5ts1 h l$ to ! discov]y ( important prop]ties :i* !y _h n be=e ev5 suspect$4 ,su* is geometry1 s9ce ! radical revolution produc$ by ,desc>tes 9 ! g5]al sy/em ( ! sci5ce4 ,! ,sup]iority ( ! mod]n ,geometry4 ,! m]e indication ( ! fundam5tal "* ( ea* ( ! two geometries is und\bt$ly suffici5t to make app>5t ! imm5se necess>y sup]iority ( mod]n geometry4 ,we may ev5 say t1 be=e ! great conception ( ,desc>tes1 rational geometry 0 n truly con/itut$ ^u def9itive bases1 :e!r 9 its ab/ract or concrete relations4 ,9 fact1 z reg>ds sci5ce1 consid]$ speculatively1 x is cle> t1 9 cont9u+ indef9itely to foll[ ! c\rse ( ! anci5ts1 z did ! mod]ns be=e ,desc>tes1 & ev5 = a little :ile aft]w>ds1 by add+ "s new curves to ! small numb] ( ^? :i* !y _h /udi$1 ! progress ?us made1 h[ev] rapid x might h be51 w\ld /ill be f\nd1 aft] a long s]ies ( ages1 to be v inconsid]able 9 comp>ison ) ! g5]al sy/em ( geometry1 see+ ! inf9ite v>iety ( ! =ms :i* w\ld /ill h rema9$ to be /udi$4 ,on ! contr>y1 at ea* "q resolv$ accord+ to ! me?od ( ! mod]ns1 ! numb] ( geometrical problems to be resolv$ is !n1 once = all1 dim9i%$ by s mu* ) respect to all possible bodies4 ,ano!r consid]ation is1 t x result$1 f _! complete want ( g5]al me?ods1 t ! anci5t geomet]s1 9 all _! inve/igations1 7 entirely ab&on$ to _! own /r5g?1 )\t ev] hav+ ! c]ta9ty ( obta9+1 soon] or lat]1 any solution :atev]4 ,?\gh ?is imp]fection ( ! sci5ce 0 em95tly suit$ to call =? all _! admirable sagacity1 x necess>ily r5d]$ _! progress extremely sl[2 we c =m "s idea ( ?is by ! consid]able "t :i* !y employ$ 9 ! /udy ( ! conic sections4 ,mod]n geometry1 mak+ ! progress ( our m9d c]ta91 p]mits u1 on ! contr>y1 to make ! greate/ possible use ( ! =ces ( our intellig5ce1 :i* ! anci5ts 7 (t5 oblig$ to wa/e on v unimportant que/ions4 ,a no less important diff]5ce betwe5 ! two sy/ems appe>s :5 we come to consid] geometry 9 ! concrete po9t ( view4 ,inde$1 we h already rem>k$ t ! relation ( ! ab/ract to ! concrete 9 geometry c be f\nd$ ^u rational bases only s f> z ! inve/igations are made to be> directly ^u all imag9able figures4 ,9 /udy+ l9es1 only "o by "o1 :atev] may be ! numb]1 always necess>ily v small1 ( ^? :i* we %all h consid]$1 ! application ( su* !ories to figures really exi/+ 9 nature w nev] h any o!r ?an an ess5tially accid5tal "*1 s9ce !re is no?+ to assure u t ^! figures c really be br\ght "u ! ab/ract types consid]$ by geomet]s4 ,?us1 = example1 !re is c]ta9ly some?+ =tuit\s 9 ! happy relation e/abli%$ betwe5 ! speculations ( ! ,greek geomet]s ^u ! conic sections & ! det]m9ation ( ! true planet>y orbits4 ,9 cont9u+ geometrical rese>*es ^u ! same plan1 !re 0 no good reason = hop+ = simil> co9cid5ces2 & x w\ld h be5 possible1 9 ^! special /udies1 t ! rese>*es ( geomet]s %\ld h be5 direct$ to ab/ract figures entirely incapable ( any application1 :ile !y neglect$ o!rs1 susceptible p]haps ( an important & imm$iate application4 ,x is cle>1 at lea/1 t no?+ positively gu>ante$ ! necess>y applicability ( geometrical speculations4 ,x is q ano!r ?+ 9 ! mod]n geometry4 ,f ! s+le circum/ance t 9 x we proce$ by g5]al que/ions relat+ to any figures :atev]1 we h 9 advance ! evid5t c]ta9ty t ! figures really exi/+ 9 ! ext]nal _w c\ld 9 no case escape ! appropriate !ory if ! geometrical ph5om5on :i* x consid]s pres5ts itself 9 !m4 ,f ^! diff]5t consid]ations1 we see t ! anci5t sy/em ( geometry we>s ess5tially ! "* ( ! infancy ( ! sci5ce1 :i* did n beg9 to become completely rational till aft] ! philosophical resolution produc$ by ,desc>tes4 ,b x is evid5t1 on ! o!r h&1 t geometry c\ld n be at fir/ conceiv$ except 9 ?is special mann]4 ,g5]al geometry w\ld n h be5 possible1 & its necessity c\ld n ev5 h be5 felt1 if a long s]ies ( special lab\rs on ! mo/ simple figures _h n previ\sly furni%$ bases = ! conception ( ,desc>tes1 & r5d]$ app>5t ! impossibility ( p]si/+ indef9itely 9 ! primitive geometrical philosophy4 ,! ,anci5t ! ,base ( ! ,mod]n4 ,f ?is la/ consid]ation we mu/ inf] t1 al?\gh ! geometry :i* ,i h call$ g5]al mu/ be n[ reg>d$ z ! only true dogmatical geometry1 & t to :i* we %all *iefly conf9e ourselves1 ! o!r hav+ no long] mu* m ?an an hi/orical int]e/1 nev]!less x is n possible to entirely disp5se ) special geometry 9 a rational exposition ( ! sci5ce4 ,we und\bt$ly ne$ n borr[ directly f anci5t geometry all ! results :i* x has furni%$2 b1 f ! v nature ( ! subject1 x is necess>ily impossible entirely to disp5se ) ! anci5t me?od1 :i* w always s]ve z ! prelim9>y basis ( ! sci5ce1 dogmatically z well z hi/orically4 ,! reason ( ?is is easy to und]/&4 ,9 fact1 g5]al geometry be+ ess5tially f\nd$1 z we %all soon e/abli%1 ^u ! employm5t ( ! calculus 9 ! trans=mation ( geometrical into analytical consid]ations1 su* a mann] ( proce$+ c\ld n take possession ( ! subject imm$iately at its orig94 ,we "k t ! application ( ma!matical analysis1 f its nature1 c nev] comm5ce any sci5ce :atev]1 s9ce evid5tly x _c be employ$ until ! sci5ce has already be5 suffici5tly cultivat$ to e/abli%1 ) respect to ! ph5om5a consid]$1 "s equations :i* c s]ve z />t+ po9ts = ! analytical op]ations4 ,^! fundam5tal equations be+ once discov]$1 analysis w enable u to d$uce f !m a multitude ( consequ5ces :i* x w\ld h be5 previ\sly impossible ev5 to suspect2 x w p]fect ! sci5ce to an imm5se degree1 bo? ) respect to ! g5]ality ( its conceptions & to ! complete co-ord9ation e/abli%$ betwe5 !m4 ,b m]e ma!matical analysis c\ld nev] be suffici5t to =m ! bases ( any natural sci5ce1 n ev5 to demon/rate !m anew :5 !y h once be5 e/abli%$4 ,no?+ c disp5se ) ! direct /udy ( ! subject1 pursu$ up to ! po9t ( ! discov]y ( precise relations4 ,we ?us see t ! geometry ( ! anci5ts w always h1 by its nature1 a prim>y "p1 absolutely necess>y & m or less ext5sive1 9 ! complete sy/em ( geometrical k4 ,x =ms a rigor\sly indisp5sable introduction to g5]al geometry4 ,b x is to ?is t x mu/ be limit$ 9 a completely dogmatic exposition4 ,i w consid]1 !n1 directly1 9 ! foll[+ *apt]1 ?is special or prelim9>y geometry re/rict$ to exactly its necess>y limits1 9 ord] to occupy myself !nce=? only ) ! philosophical exam9ation ( g5]al or def9itive geometry1 ! only "o :i* is truly rational1 & :i* at pres5t ess5tially composes ! sci5ce4 ,,*apt] ,,ii4 ,,anci5t ,,or ,,syn!tic ,,geometry4 ,! geometrical me?od ( ! anci5ts necess>ily con/itut+ a prelim9>y dep>tm5t 9 ! dogmatical sy/em ( geometry1 design$ to furni% g5]al geometry ) indisp5sable f\ndations1 x is n[ prop] to beg9 ) det]m9+ :]e9 /rictly consi/s ?is prelim9>y function ( special geometry1 ?us r$uc$ to ! n>r[e/ possible limits4 ,,its ,,prop] ,,ext5t4 ,l9es2 ,polygons2 ,polyh$rons4 ,9 consid]+ x "u ?is po9t ( view1 x is easy to recognize t we might re/rict x to ! /udy ( ! "r l9e alone = :at conc]ns ! geometry ( l9es2 to ! quadrature ( rectil9e> plane areas2 &1 la/ly1 to ! cubature ( bodies t]m9at$ by plane faces4 ,! elem5t>y propositions relat+ to ^! ?ree fundam5tal que/ions =m1 9 fact1 ! necess>y />t+ po9t ( all geometrical inquiries2 !y alone _c be obta9$ except by a direct /udy ( ! subject2 :ile1 on ! contr>y1 ! complete !ory ( all o!r figures1 ev5 t ( ! circle1 & ( ! surfaces & volumes :i* are connect$ ) x1 may at ! pres5t day be completely compreh5d$ 9 ! doma9 ( g5]al or analytical geometry2 ^! primitive elem5ts at once furni%+ equations :i* are suffici5t to all[ ( ! application ( ! calculus to geometrical que/ions1 :i* w\ld n h be5 possible )\t ?is previ\s condition4 ,x results f ?is consid]ation t1 9 common practice1 we give to elem5t>y geometry m ext5t ?an w\ld be rigor\sly necess>y to x2 s9ce1 besides ! "r l9e1 polygons1 & polyh$rons1 we also include 9 x ! circle & ! r\nd bodies2 ! /udy ( :i* might1 h[ev]1 be z purely analytical z t1 = example1 ( ! conic sections4 ,an unreflect+ v5]ation = antiquity contributes to ma9ta9 ?is defect 9 me?od2 b ! be/ reason :i* c be giv5 = x is ! s]i\s inconv5i5ce = ord9>y in/ruction :i* !re w\ld be 9 po/pon+1 to s di/ant an epo* ( ma!matical education1 ! solution ( sev]al ess5tial que/ions1 :i* are susceptible ( a direct & cont9ual application to a great numb] ( important uses4 ,9 fact1 to proce$ 9 ! mo/ rational mann]1 we %\ld employ ! integral calculus 9 obta9+ ! int]e/+ results relat+ to ! l5g? or ! area ( ! circle1 or to ! quadrature ( ! sph]e1 c41 :i* h be5 det]m9$ by ! anci5ts f extremely simple consid]ations4 ,?is inconv5i5ce w\ld be ( little importance ) reg>d to ! p]sons de/9$ to /udy ! :ole ( ma!matical sci5ce1 & ! advantage ( proce$+ 9 a p]fectly logical ord] w\ld h a mu* great] comp>ative value4 ,b ! contr>y case be+ ! m frequ5t1 !ories s ess5tial h necess>ily be5 reta9$ 9 elem5t>y geometry4 ,p]haps ! conic sections1 ! cycloid1 c41 might be advantage\sly add$ 9 su* cases4 ,n to be f>!r re/rict$4 ,:ile ?is prelim9>y portion ( geometry1 :i* _c be f\nd$ on ! application ( ! calculus1 is r$uc$ by its nature to a v limit$ s]ies ( fundam5tal rese>*es1 relat+ to ! "r l9e1 polygonal areas1 & polyh$rons1 x is c]ta91 on ! o!r h&1 t we _c re/rict x any m2 al?\gh1 by a v]itable abuse ( ! _s ( analysis1 x has be5 rec5tly attempt$ to pres5t ! e/abli%m5t ( ! pr9cipal !orems ( elem5t>y geometry "u an algebraical po9t ( view4 ,?us "s h pret5d$ to demon/rate1 by simple ab/ract consid]ations ( ma!matical analysis1 ! con/ant relation :i* exi/s betwe5 ! ?ree angles ( a rectil9e> triangle1 ! fundam5tal proposition ( ! !ory ( simil> triangles1 t ( p>allelopip$ons1 c42 9 a ^w1 precisely ! only geometrical propositions :i* _c be obta9$ except by a direct /udy ( ! subject1 )\t ! calculus be+ susceptible ( hav+ any "p 9 x4 ,su* ab]rations are ! unreflect+ exagg]ations ( t natural & philosophical t5d5cy :i* leads u to ext5d f>!r & f>!r ! influ5ce ( analysis 9 ma!matical /udies4 ,9 me*anics1 ! pret5d$ analytical demon/rations ( ! p>allelogram ( =ces are ( simil> "*4 ,! vici\sness ( su* a mann] ( proce$+ foll[s f ! pr9ciples previ\sly pres5t$4 ,we h already1 9 fact1 recogniz$ t1 s9ce ! calculus is n1 & _c be1 any ?+ b a means ( d$uction1 x w\ld indicate a radically false idea ( x to wi% to employ x 9 e/abli%+ ! elem5t>y f\ndations ( any sci5ce :atev]2 = on :at w\ld ! analytical reason+s 9 su* an op]ation repose8 ,a lab\r ( ?is nature1 v f> f really p]fect+ ! philosophical "* ( a sci5ce1 w\ld con/itute a return t[>ds ! metaphysical age1 9 pres5t+ real facts z m]e logical ab/ractions4 ,:5 we exam9e 9 !mselves ^! pret5d$ analytical demon/rations ( ! fundam5tal propositions ( elem5t>y geometry1 we easily v]ify _! necess>y want ( mean+4 ,!y are all f\nd$ on a vici\s mann] ( conceiv+ ! pr9ciple ( homog5eity1 ! true g5]al idea ( :i* 0 expla9$ 9 ! second *apt] ( ! prec$+ book4 ,^! demon/rations suppose t ?is pr9ciple does n all[ u to admit ! coexi/5ce 9 ! same equation ( numb]s obta9$ by diff]5t concrete comp>isons1 :i* is evid5tly false1 & contr>y to ! con/ant practice ( geomet]s4 ,?us x is easy to recognize t1 by employ+ ! law ( homog5eity 9 ?is arbitr>y & illegitimate acceptation1 we c\ld succe$ 9 demon/rat+1 ) q z mu* app>5t rig\r1 propositions ^: absurdity is manife/ at ! fir/ glance4 ,9 exam9+ att5tively1 = example1 ! proc$ure by ! aid ( :i* x has be5 attempt$ to prove analytically t ! sum ( ! ?ree angles ( any rectil9e> triangle is con/antly equal to two "r angles1 we see t x is f\nd$ on ?is prelim9>y pr9ciple t1 if two triangles h two ( _! angles respectively equal1 ! ?ird angle ( ! "o w necess>ily be equal to ! ?ird angle ( ! o!r4 ,?is fir/ po9t be+ grant$1 ! propos$ relation is imm$iately d$uc$ f x 9 a v exact & simple mann]4 ,n[ ! analytical consid]ation by :i* ?is previ\s proposition has be5 attempt$ to be e/abli%$1 is ( su* a nature t1 if x c\ld be correct1 we c\ld rigor\sly d$uce f x1 9 reproduc+ x conv]sely1 ?is palpable absurdity1 t two sides ( a triangle are suffici5t1 )\t any angle1 = ! entire det]m9ation ( ! ?ird side4 ,we may make analog\s rem>ks on all ! demon/rations ( ?is sort1 ! sophisms ( :i* w be ?us v]ifi$ 9 a p]fectly app>5t mann]4 ,! m reason t we h "h to consid] geometry z be+ at ! pres5t day ess5tially analytical1 ! m necess>y 0 x to gu>d aga9/ ?is abusive exagg]ation ( ma!matical analysis1 accord+ to :i* all geometrical obs]vation w\ld be disp5s$ )1 9 e/abli%+ ^u pure algebraical ab/ractions ! v f\ndations ( ?is natural sci5ce4 ,attempt$ ,demon/rations ( ,axioms1 c4 ,ano!r indication t geomet]s h too mu* ov]look$ ! "* ( a natural sci5ce :i* is necess>ily inh]5t 9 geometry1 appe>s f _! va9 attempts1 s long made1 to demon/rate rigor\sly1 n by ! aid ( ! calculus1 b by means ( c]ta9 con/ructions1 sev]al fundam5tal propositions ( elem5t>y geometry4 ,:atev] may be effect$1 x w evid5tly be impossible to avoid sometimes recurr+ to simple & direct obs]vation 9 geometry z a means ( e/abli%+ v>i\s results4 ,:ile1 9 ?is sci5ce1 ! ph5om5a :i* are consid]$ are1 by virtue ( _! extreme simplicity1 mu* m closely connect$ ) "o ano!r ?an ^? relat+ to any o!r physical sci5ce1 "s mu/ /ill be f\nd :i* _c be d$uc$1 & :i*1 on ! contr>y1 s]ve z />t+ po9ts4 ,x may be admitt$ t ! greate/ logical p]fection ( ! sci5ce is to r$uce ^! to ! smalle/ numb] possible1 b x w\ld be absurd to pret5d to make !m completely disappe>4 ,i av[1 moreov]1 t ,i f9d few] real inconv5i5ces 9 ext5d+1 a little beyond :at w\ld be /rictly necess>y1 ! numb] ( ^! geometrical notions ?us e/abli%$ by direct obs]vation1 provid$ !y are suffici5tly simple1 ?an 9 mak+ !m ! subjects ( complicat$ & indirect demon/rations1 ev5 :5 ^! demon/rations may be logically irreproa*able4 ,! true dogmatic de/9ation ( ! geometry ( ! anci5ts1 r$uc$ to its lea/ possible indisp5sable developm5ts1 hav+ ?us be5 *>act]iz$ z exactly z possible1 x is prop] to consid] summ>ily ea* ( ! pr9cipal p>ts ( :i* x mu/ be compos$4 ,i ?9k t ,i may "h limit myself to consid]+ ! fir/ & ! mo/ ext5sive ( ^! p>ts1 t :i* has = its object ! /udy ( ! "r l9e2 ! two o!r sections1 namely1 ! quadrature ( polygons & ! cubature ( polyh$rons1 f _! limit$ ext5t1 n be+ capable ( giv+ rise to any philosophical consid]ation ( any importance1 di/9ct f ^? indicat$ 9 ! prec$+ *apt] ) respect to ! measure ( areas & ( volumes 9 g5]al4 ,,geometry ,,( ,,! ,,"r ,,l9e4 ,! f9al "q :i* we always h 9 view 9 ! /udy ( ! "r l9e1 prop]ly consi/s 9 det]m9+1 by means ( "o ano!r1 ! diff]5t elem5ts ( any "r-l9$ figure :atev]2 :i* enables u always to "k indirectly ! l5g? & position ( a "r l9e1 9 :atev] circum/ances x may be plac$4 ,?is fundam5tal problem is susceptible ( two g5]al solutions1 ! nature ( :i* is q di/9ct1 ! "o graphical1 ! o!r algebraic4 ,! fir/1 ?\gh v imp]fect1 is t :i* mu/ be fir/ consid]$1 because x is spontane\sly d]iv$ f ! direct /udy ( ! subject2 ! second1 mu* m p]fect 9 ! mo/ important respects1 _c be /udi$ till aft]w>ds1 because x is f\nd$ ^u ! previ\s k ( ! o!r4 ,,graphical ,,solutions4 ,! graphical solution consi/s 9 con/ruct+ at w ! propos$ figure1 ei!r ) ! same dim5sions1 or1 m usually1 ) dim5sions *ang$ 9 any ratio :atev]4 ,! fir/ mode ne$ m]ely be m5tion$ z be+ ! mo/ simple & ! "o :i* w\ld fir/ occur to ! m9d1 = x is evid5tly1 by its nature1 almo/ entirely incapable ( application4 ,! second is1 on ! contr>y1 susceptible ( be+ mo/ ext5sively & mo/ usefully appli$4 ,we /ill make an important & cont9ual use ( x at ! pres5t day1 n only to repres5t ) exactness ! =ms ( bodies & _! relative positions1 b ev5 = ! actual det]m9ation ( geometrical magnitudes1 :5 we d n ne$ great precision4 ,! anci5ts1 9 consequ5ce ( ! imp]fection ( _! geometrical k1 employ$ ?is proc$ure 9 a mu* m ext5sive mann]1 s9ce x 0 = a long "t ! only "o :i* !y c\ld apply1 ev5 9 ! mo/ important precise det]m9ations4 ,x 0 ?us1 = example1 t ,ari/>*us ( ,samos e/imat$ ! relative di/ance f ! sun & f ! moon to ! e>?1 by mak+ measurem5ts on a triangle con/ruct$ z exactly z possible1 s z to be simil> to ! "r-angl$ triangle =m$ by ! ?ree bodies at ! in/ant :5 ! moon is 9 quadrature1 & :5 an obs]vation ( ! angle at ! e>? w\ld consequ5tly be suffici5t to def9e ! triangle4 ,ar*im$es himself1 al?\gh he 0 ! fir/ to introduce calculat$ det]m9ations into geometry1 sev]al times employ$ simil> means4 ,! =mation ( trigonometry did n cause ?is me?od to be entirely ab&on$1 al?\gh x greatly dim9i%$ its use2 ! ,greeks & ! ,arabians cont9u$ to employ x = a great numb] ( rese>*es1 9 :i* we n[ reg>d ! use ( ! calculus z indisp5sable4 ,?is exact reproduction ( any figure :atev] on a diff]5t scale _c pres5t any great !oretical difficulty :5 all ! p>ts ( ! propos$ figure lie 9 ! same plane4 ,b if we suppose1 z mo/ frequ5tly happ5s1 t !y are situat$ 9 diff]5t planes1 we see1 !n1 a new ord] ( geometrical consid]ations arise4 ,! artificial figure1 :i* is con/antly plane1 n be+ capable1 9 t case1 ( be+ a p]fectly fai?ful image ( ! real figure1 x is necess>y previ\sly to fix ) precision ! mode ( repres5tation1 :i* gives rise to diff]5t sy/ems ( ,projection4 ,x !n rema9s to be det]m9$ accord+ to :at laws ! geometrical ph5om5a correspond 9 ! two figures4 ,?is consid]ation g5]ates a new s]ies ( geometrical inve/igations1 ! f9al object ( :i* is prop]ly to discov] h[ we c replace con/ructions 9 relief by plane con/ructions4 ,! anci5ts _h to resolve sev]al elem5t>y que/ions ( ?is k9d = v>i\s cases 9 :i* we n[ employ sph]ical trigonometry1 pr9cipally = diff]5t problems relat+ to ! cele/ial sph]e4 ,su* 0 ! object ( _! analemmas1 & ( ! o!r plane figures :i* = a long "t suppli$ ! place ( ! calculus4 ,we see by ?is t ! anci5ts really knew ! elem5ts ( :at we n[ "n ,descriptive ,geometry1 al?\gh !y did n conceive x 9 a di/9ct & g5]al mann]4 ,i ?9k x prop] briefly to indicate 9 ?is place ! true philosophical "* ( ?is ,descriptive ,geometry2 al?\gh1 be+ ess5tially a sci5ce ( application1 x ought n to be includ$ )9 ! prop] doma9 ( ?is "w4 ,,descriptive ,,geometry4 ,all que/ions ( geometry ( ?ree dim5sions necess>ily give rise1 :5 we consid] _! graphical solution1 to a common difficulty :i* is peculi> to !m2 t ( sub/itut+ = ! diff]5t con/ructions 9 relief1 :i* are necess>y to resolve !m directly1 & :i* x is almo/ always impossible to execute1 simple equival5t plane con/ructions1 by means ( :i* we f9ally obta9 ! same results4 ,)\t ?is indisp5sable trans=mation1 e solution ( ?is k9d w\ld be evid5tly incomplete & really inapplicable 9 practice1 al?\gh !oretically ! con/ructions 9 space are usually pref]able z be+ m direct4 ,x 0 9 ord] to furni% g5]al means = always effect+ su* a trans=mation t ,descriptive ,geometry 0 creat$1 & =m$ into a di/9ct & homog5e\s sy/em1 by ! illu/ri\s ,,monge4 ,he inv5t$1 9 ! fir/ place1 a uni=m me?od ( repres5t+ bodies by figures trac$ on a s+le plane1 by ! aid ( projections on two diff]5t planes1 usually p]p5dicul> to ea* o!r1 & "o ( :i* is suppos$ to turn ab\t _! common int]section s z to co9cide ) ! o!r produc$2 9 ?is sy/em1 or 9 any o!r equival5t to x1 x is suffici5t to reg>d po9ts & l9es z be+ det]m9$ by _! projections1 & surfaces by ! projections ( _! g5]at+ l9es4 ,?is be+ e/abli%$1 ,monge--analyz+ ) pr(\nd sagacity ! v>i\s p>tial lab\rs ( ?is k9d :i* _h be=e be5 execut$ by a numb] ( incongru\s proc$ures1 & consid]+ also1 9 a g5]al & direct mann]1 9 :at any que/ions ( t nature mu/ consi/--f\nd t !y c\ld always be r$uc$ to a v small numb] ( inv>iable ab/ract problems1 capable ( be+ resolv$ sep>ately1 once = all1 by uni=m op]ations1 relat+ ess5tially "s to ! contacts & o!rs to ! int]sections ( surfaces4 ,simple & entirely g5]al me?ods = ! graphical solution ( ^! two ord]s ( problems hav+ be5 =m$1 all ! geometrical que/ions :i* may arise 9 any ( ! v>i\s arts ( con/ruction--/one-cutt+1 c>p5try1 p]spective1 diall+1 =tification1 c4--c h5ce=? be treat$ z simple p>ticul> cases ( a s+le !ory1 ! inv>iable application ( :i* w always necess>ily lead to an exact solution1 :i* may be facilitat$ 9 practice by pr(it+ by ! peculi> circum/ances ( ea* case4 99 99 99 99 99 ,?is important creation des]ves 9 a rem>kable degree to fix ! att5tion ( ^? philosoph]s :o consid] all t ! human species has yet effect$ z a fir/ /ep1 & ?us f> ! only really complete "o1 t[>ds t g5]al r5ovation ( human lab\rs1 :i* mu/ impr9t ^u all our arts a "* ( precision & ( rationality1 s necess>y to _! future progress4 ,su* a revolution mu/1 9 fact1 inevitably comm5ce ) t class ( indu/rial lab\rs1 :i* is ess5tially connect$ ) t sci5ce :i* is ! mo/ simple1 ! mo/ p]fect1 & ! mo/ anci5t4 ,x _c fail to ext5d h]eaft]1 ?\gh ) less facility1 to all o!r practical op]ations4 ,inde$ ,monge himself1 :o conceiv$ ! true philosophy ( ! arts bett] ?an any "o else1 endeav\r$ to sket* out a correspond+ sy/em = ! me*anical arts4 ,ess5tial z ! conception ( descriptive geometry really is1 x is v important n to deceive ourselves ) respect to its true de/9ation1 z did ^? :o1 9 ! excitem5t ( its fir/ discov]y1 saw 9 x a means ( enl>g+ ! g5]al & ab/ract doma9 ( rational geometry4 ,! result has 9 no way answ]$ to ^! mi/ak5 hopes4 ,&1 inde$1 is x n evid5t t descriptive geometry has no special value except z a sci5ce ( application1 & z =m+ ! true special !ory ( ! geometrical arts8 ,consid]$ 9 its ab/ract relations1 x c\ld n introduce any truly di/9ct ord] ( geometrical speculations4 ,we mu/ n =get t1 9 ord] t a geometrical "q %\ld fall )9 ! peculi> doma9 ( descriptive geometry1 x mu/ necess>ily h be5 previ\sly resolv$ by speculative geometry1 ! solutions ( :i* !n1 z we h se51 always ne$ to be prep>$ = practice 9 su* a way z to supply ! place ( con/ructions 9 relief by plane con/ructions2 a sub/itution :i* really con/itutes ! only *>act]i/ic function ( descriptive geometry4 ,x is prop]1 h[ev]1 to rem>k "h1 t1 ) reg>d to intellectual education1 ! /udy ( descriptive geometry possesses an important philosophical peculi>ity1 q indep5d5t ( its high indu/rial utility4 ,?is is ! advantage :i* x s pre-em95tly (f]s--9 habituat+ ! m9d to consid] v complicat$ geometrical comb9ations 9 space1 & to foll[ ) precision _! cont9ual correspond5ce ) ! figures :i* are actually trac$--( ?us ex]cis+ to ! utmo/1 9 ! mo/ c]ta9 & precise mann]1 t important faculty ( ! human m9d :i* is prop]ly call$ imag9ation1 & :i* consi/s1 9 its elem5t>y & positive acceptation1 9 repres5t+ to ourselves1 cle>ly & easily1 a va/ & v>iable collection ( ideal objects1 z if !y 7 really be=e u4 ,f9ally1 to complete ! indication ( ! g5]al nature ( descriptive geometry by det]m9+ its logical "*1 we h to obs]ve t1 :ile x belongs to ! geometry ( ! anci5ts by ! "* ( its solutions1 on ! o!r h& x approa*es ! geometry ( ! mod]ns by ! nature ( ! que/ions :i* compose x4 ,^! que/ions are 9 fact em95tly rem>kable = t g5]ality :i*1 z we saw 9 ! prec$+ *apt]1 con/itutes ! true fundam5tal "* ( mod]n geometry2 = ! me?ods us$ are always conceiv$ z applicable to any figures :atev]1 ! peculi>ity ( ea* hav+ only a purely second>y influ5ce4 ,! solutions ( descriptive geometry are !n graphical1 l mo/ ( ^? ( ! anci5ts1 & at ! same "t g5]al1 l ^? ( ! mod]ns4 99 99 99 99 99 ,aft] ?is important digression1 we w pursue ! philosophical exam9ation ( special geometry1 always consid]$ z r$uc$ to its lea/ possible developm5t1 z an indisp5sable introduction to g5]al geometry4 ,we h n[ suffici5tly consid]$ ! graphical solution ( ! fundam5tal problem relat+ to ! "r l9e--t is1 ! det]m9ation ( ! diff]5t elem5ts ( any "r-l9$ figure by means ( "o ano!r--& h n[ to exam9e 9 a special mann] ! algebraic solution4 ,,algebraic ,,solutions4 ,?is k9d ( solution1 ! evid5t sup]iority ( :i* ne$ n "h be dwelt ^u1 belongs necess>ily1 by ! v nature ( ! "q1 to ! sy/em ( ! anci5t geometry1 al?\gh ! logical me?od :i* is employ$ causes x to be g5]ally1 b v improp]ly1 sep>at$ f x4 ,we h ?us an opportunity ( v]ify+1 9 a v important respect1 :at 0 e/abli%$ g5]ally 9 ! prec$+ *apt]1 t x is n by ! employm5t ( ! calculus t ! mod]n geometry is ess5tially to be di/+ui%$ f ! anci5t4 ,! anci5ts are 9 fact ! true inv5tors ( ! pres5t trigonometry1 sph]ical z well z rectil9e>2 x be+ only mu* less p]fect 9 _! h&s1 on acc\nt ( ! extreme inf]iority ( _! algebraical k4 ,x is1 !n1 really 9 ?is *apt]1 & n1 z x might at fir/ be ?\ght1 9 ^? :i* we %all aft]w>ds devote to ! philosophical exam9ation ( g5]al geometry1 t x is prop] to consid] ! "* ( ?is important prelim9>y !ory1 :i* is usually1 ?\gh improp]ly1 includ$ 9 :at is call$ analytical geometry1 b :i* is really only a complem5t ( elem5t>y geometry prop]ly s call$4 ,s9ce all "r-l9$ figures c be decompos$ into triangles1 x is evid5tly suffici5t to "k h[ to det]m9e ! diff]5t elem5ts ( a triangle by means ( "o ano!r1 :i* r$uces polygonometry to simple trigonometry4 ,,trigonometry4 ,! difficulty 9 resolv+ algebraically su* a "q z ! above1 consi/s ess5tially 9 =m+1 betwe5 ! angles & ! sides ( a triangle1 ?ree di/9ct equations2 :i*1 :5 once obta9$1 w evid5tly r$uce all trigonometrical problems to m]e que/ions ( analysis4 ,h[ to introduce ,angles4 ,9 consid]+ ! e/abli%m5t ( ^! equations 9 ! mo/ g5]al mann]1 we imm$iately meet ) a fundam5tal di/9ction ) respect to ! mann] ( introduc+ ! angles into ! calculation1 accord+ z !y are made to ent] directly1 by !mselves or by ! circul> arcs :i* are proportional to !m2 or indirectly1 by ! *ords ( ^! arcs1 :i* are h5ce call$ _! trigonometrical l9es4 ,( ^! two sy/ems ( trigonometry ! second 0 ( necessity ! only "o orig9ally adopt$1 z be+ ! only practicable "o1 s9ce ! condition ( geometry made x easy en\gh to f9d exact relations betwe5 ! sides ( ! triangles & ! trigonometrical l9es :i* repres5t ! angles1 :ile x w\ld h be5 absolutely impossible at t epo* to e/abli% equations betwe5 ! sides & ! angles !mselves4 ,advantages ( introduc+ ,trigonometrical ,l9es4 ,at ! pres5t day1 s9ce ! solution c be obta9$ by ei!r sy/em indiff]5tly1 t motive = pref]5ce no long] exi/s2 b geomet]s h none ! less p]si/$ 9 foll[+ f *oice ! sy/em primitively admitt$ f necessity2 =1 ! same reason :i* enabl$ ^! trigonometrical equations to be obta9$ ) mu* m facility1 mu/1 9 l mann]1 z x is /ill m easy to conceive priori1 r5d] ^! equations mu* m simple1 s9ce !y !n exi/ only betwe5 "r l9es1 in/ead ( be+ e/abli%$ betwe5 "r l9es & arcs ( circles4 ,su* a consid]ation has s mu* ! m importance1 z ! "q relates to =mulas :i* are em95tly elem5t>y1 & de/9$ to be cont9ually employ$ 9 all p>ts ( ma!matical sci5ce1 z well z 9 all its v>i\s applications4 ,x may be object$1 h[ev]1 t :5 an angle is giv51 x is1 9 reality1 always giv5 by itself1 & n by its trigonometrical l9es2 & t :5 x is unkn[n1 x is its angul> value :i* is prop]ly to be det]m9$1 & n t ( any ( its trigonometrical l9es4 ,x seems1 accord+ to ?is1 t su* l9es are only useless int]m$i>ies betwe5 ! sides & ! angles1 :i* h to be f9ally elim9at$1 & ! introduction ( :i* does n appe> capable ( simplify+ ! propos$ rese>*4 ,x is inde$ important to expla91 ) m g5]ality & precision ?an is cu/om>y1 ! great real utility ( ?is mann] ( proce$+4 ,division ( ,trigonometry into two ,p>ts4 ,x consi/s 9 ! fact t ! introduction ( ^! auxili>y magnitudes divides ! entire "q ( trigonometry into two o!rs ess5tially di/9ct1 "o ( :i* has = its object to pass f ! angles to _! trigonometrical l9es1 or ! conv]se1 & ! o!r ( :i* proposes to det]m9e ! sides ( ! triangles by ! trigonometrical l9es ( _! angles1 or ! conv]se4 ,n[ ! fir/ ( ^! two fundam5tal que/ions is evid5tly susceptible1 by its nature1 ( be+ entirely treat$ & r$uc$ to num]ical tables once = all1 9 consid]+ all possible angles1 s9ce x dep5ds only ^u ^? angles1 & n at all ^u ! p>ticul> triangles 9 :i* !y may ent] 9 ea* case2 :ile ! solution ( ! second "q mu/ necess>ily be r5ew$1 at lea/ 9 its ari?metical relations1 = ea* new triangle :i* x is necess>y to resolve4 ,?is is ! reason :y ! fir/ portion ( ! complete "w1 :i* w\ld be precisely ! mo/ labori\s1 is no long] tak5 into ! acc\nt1 be+ always done 9 advance2 :ile1 if su* a decomposition _h n be5 p]=m$1 we w\ld evid5tly h f\nd ourselves "u ! obligation ( recomm5c+ ! entire calculation 9 ea* p>ticul> case4 ,su* is ! ess5tial prop]ty ( ! pres5t trigonometrical sy/em1 :i* 9 fact w\ld really pres5t no actual advantage1 if x 0 necess>y to calculate cont9ually ! trigonometrical l9e ( ea* angle to be consid]$1 or ! conv]se2 ! int]m$iate ag5cy introduc$ w\ld !n be m tr\blesome ?an conv5i5t4 ,9 ord] to cle>ly compreh5d ! true nature ( ?is conception1 x w be useful to comp>e x ) a /ill m important "o1 design$ to produce an analog\s effect ei!r 9 its algebraic1 or1 /ill m1 9 its ari?metical relations--! admirable !ory ( log>i?ms4 ,9 exam9+ 9 a philosophical mann] ! influ5ce ( ?is !ory1 we see 9 fact t its g5]al result is to decompose all imag9able ari?metical op]ations into two di/9ct p>ts4 ,! fir/ & mo/ complicat$ ( ^! is capable ( be+ execut$ 9 advance once = all 7s9ce x dep5ds only ^u ! numb]s to be consid]$1 & n at all ^u ! inf9itely diff]5t comb9ations into :i* !y c ent]71 & consi/s 9 consid]+ all numb]s z assignable p[]s ( a con/ant numb]4 ,! second "p ( ! calculation1 :i* mu/ ( necessity be recomm5c$ = ea* new =mula :i* is to h its value det]m9$1 is !nce=? r$uc$ to execut+ ^u ^! expon5ts correlative op]ations :i* are inf9itely m simple4 ,i conf9e myself "h to m]ely indicat+ ?is resemblance1 :i* any "o c c>ry out = himself4 ,we mu/ besides obs]ve1 z a prop]ty 7second>y at ! pres5t day1 b all-important at its orig97 ( ! trigonometrical sy/em adopt$1 ! v rem>kable circum/ance t ! det]m9ation ( angles by _! trigonometrical l9es1 or ! conv]se1 admits ( an ari?metical solution 7! only "o :i* is directly indisp5sable = ! special de/9ation ( trigonometry7 )\t ! previ\s resolution ( ! correspond+ algebraic "q4 ,x is d\btless to su* a peculi>ity t ! anci5ts ow$ ! possibility ( kn[+ trigonometry4 ,! inve/igation conceiv$ 9 ?is way 0 s mu* ! m easy1 inasmu* z tables ( *ords 7:i* ! anci5ts naturally took z ! trigonometrical l9es7 _h be5 previ\sly con/ruct$ = q a diff]5t object1 9 ! c\rse ( ! lab\rs ( ,ar*im$es on ! rectification ( ! circle1 f :i* result$ ! actual det]m9ation ( a c]ta9 s]ies ( *ords2 s t :5 ,hipp>*us subsequ5tly inv5t$ trigonometry1 he c\ld conf9e himself to complet+ t op]ation by suitable int]calations2 :i* %[s cle>ly ! connexion ( ideas 9 t matt]4 ,! ,increase ( su* ,trigonometrical ,l9es4 ,to complete ?is philosophical sket* ( trigonometry1 x is prop] n[ to obs]ve t ! ext5sion ( ! same consid]ations :i* lead u to replace angles or arcs ( circles by /raight l9es1 ) ! view ( simplify+ our equations1 mu/ also lead u to employ concurr5tly sev]al trigonometrical l9es1 in/ead ( conf9+ ourselves to "o only 7z did ! anci5ts71 s z to p]fect ?is sy/em by *oos+ t "o :i* w be algebraically ! mo/ conv5i5t on ea* occasion4 ,9 ?is po9t ( view1 x is cle> t ! numb] ( ^! l9es is 9 itself no ways limit$2 provid$ t !y are det]m9$ by ! arc1 & t !y det]m9e x1 :atev] may be ! law accord+ to :i* !y are d]iv$ f x1 !y are suitable to be sub/itut$ = x 9 ! equations4 ,! ,arabians1 & subsequ5tly ! mod]ns1 9 conf9+ !mselves to ! mo/ simple con/ructions1 h c>ri$ to f\r or five ! numb] ( direct trigonometrical l9es1 :i* might be ext5d$ mu* f>!r4 ,b in/ead ( recurr+ to geometrical =mations1 :i* w\ld f9ally become v complicat$1 we conceive ) ! utmo/ facility z _m new trigonometrical l9es z ! analytical trans=mations may require1 by means ( a rem>kable artifice1 :i* is n usually appreh5d$ 9 a suffici5tly g5]al mann]4 ,x consi/s 9 n directly multiply+ ! trigonometrical l9es appropriate to ea* arc consid]$1 b 9 introduc+ new ones1 by consid]+ ?is arc z indirectly det]m9$ by all l9es relat+ to an arc :i* is a v simple function ( ! fir/4 ,x is ?us1 = example1 t1 9 ord] to calculate an angle ) m facility1 we w det]m9e1 in/ead ( its s9e1 ! s9e ( its half1 or ( its d\ble1 c4 ,su* a creation ( indirect trigonometrical l9es is evid5tly mu* m fruitful ?an all ! direct geometrical me?ods = obta9+ new ones4 ,we may accord+ly say t ! numb] ( trigonometrical l9es actually employ$ at ! pres5t day by geomet]s is 9 reality unlimit$1 s9ce at e in/ant1 s to say1 ! trans=mations ( analysis may lead u to augm5t x by ! me?od :i* ,i h j indicat$4 ,special names1 h[ev]1 h be5 giv5 to ^? only ( ^! indirect l9es :i* ref] to ! complem5t ( ! primitive arc1 ! o!rs n occurr+ suffici5tly (t5 to r5d] su* d5om9ations necess>y2 a circum/ance :i* has caus$ a common misconception ( ! true ext5t ( ! sy/em ( trigonometry4 ,/udy ( _! ,mutual ,relations4 ,?is multiplicity ( trigonometrical l9es evid5tly gives rise to a ?ird fundam5tal "q 9 trigonometry1 ! /udy ( ! relations :i* exi/ betwe5 ^! diff]5t l9es2 s9ce1 )\t su* a k1 we c\ld n make use1 = our analytical necessities1 ( ?is v>iety ( auxili>y magnitudes1 :i*1 h[ev]1 h no o!r de/9ation4 ,x is cle>1 besides1 f ! consid]ation j indicat$1 t ?is ess5tial "p ( trigonometry1 al?\gh simply prep>atory1 is1 by its nature1 susceptible ( an indef9ite ext5sion :5 we view x 9 its entire g5]ality1 :ile ! two o!rs are circumscrib$ )9 rigor\sly def9$ limits4 ,x is ne$less to add t ^! ?ree pr9cipal p>ts ( trigonometry h to be /udi$ 9 precisely ! inv]se ord] f t 9 :i* we h se5 !m necess>ily d]iv$ f ! g5]al nature ( ! subject2 = ! ?ird is evid5tly indep5d5t ( ! two o!rs1 & ! second1 ( t :i* 0 fir/ pres5t$--! resolution ( triangles1 prop]ly s call$--:i* mu/ = t reason be treat$ 9 ! la/ place2 :i* r5d]$ s mu* ! m important ! consid]ation ( _! natural succession & logical relations to "o ano!r4 ,x is useless to consid] "h sep>ately sph]ical trigonometry1 :i* _c give rise to any special philosophical consid]ation2 s9ce1 ess5tial z x is by ! importance & ! multiplicity ( its uses1 x c be treat$ at ! pres5t day only z a simple application ( rectil9e> trigonometry1 :i* furni%es directly its fundam5tal equations1 by sub/itut+ = ! sph]ical triangle ! correspond+ trih$ral angle4 ,?is summ>y exposition ( ! philosophy ( trigonometry has be5 "h giv5 9 ord] to r5d] app>5t1 by an important example1 t rigor\s dep5d5ce & ^? successive ramifications :i* are pres5t$ by :at are app>5tly ! mo/ simple que/ions ( elem5t>y geometry4 99 99 99 99 99 ,hav+ ?us exam9$ ! peculi> "* ( special geometry r$uc$ to its only dogmatic de/9ation1 t ( furni%+ to g5]al geometry an indisp5sable prelim9>y basis1 we h n[ to give all our att5tion to ! true sci5ce ( geometry1 consid]$ z a :ole1 9 ! mo/ rational mann]4 ,= t purpose1 x is necess>y to c>efully exam9e ! great orig9al idea ( ,desc>tes1 ^u :i* x is entirely f\nd$4 ,?is w be ! object ( ! foll[+ *apt]4 ,,*apt] ,,iii4 ,,mod]n ,,or ,,analytical ,,geometry4 ,g5]al 7or ,analytical7 geometry be+ entirely f\nd$ ^u ! trans=mation ( geometrical consid]ations into equival5t analytical consid]ations1 we mu/ beg9 ) exam9+ directly & 9 a ?or\gh mann] ! beautiful conception by :i* ,desc>tes has e/abli%$ 9 a uni=m mann] ! con/ant possibility ( su* a co-relation4 ,besides its own extreme importance z a means ( highly p]fect+ geometrical sci5ce1 or1 r1 ( e/abli%+ ! :ole ( x on rational bases1 ! philosophical /udy ( ?is admirable conception mu/ h s mu* ! great] int]e/ 9 our eyes f its *>act]iz+ ) p]fect cle>ness ! g5]al me?od to be employ$ 9 organiz+ ! relations ( ! ab/ract to ! concrete 9 ma!matics1 by ! analytical repres5tation ( natural ph5om5a4 ,!re is no conception1 9 ! :ole philosophy ( ma!matics :i* bett] des]ves to fix all our att5tion4 ,,analytical ,,repres5tation ,,( ,,figures4 ,9 ord] to succe$ 9 express+ all imag9able geometrical ph5om5a by simple analytical relations1 we mu/ evid5tly1 9 ! fir/ place1 e/abli% a g5]al me?od = repres5t+ analytically ! subjects !mselves 9 :i* ^! ph5om5a are f\nd1 t is1 ! l9es or ! surfaces to be consid]$4 ,! subject be+ ?us habitually consid]$ 9 a purely analytical po9t ( view1 we see h[ x is !nce=? possible to conceive 9 ! same mann] ! v>i\s accid5ts ( :i* x is susceptible4 ,9 ord] to organize ! repres5tation ( geometrical figures by analytical equations1 we mu/ previ\sly surm\nt a fundam5tal difficulty2 t ( r$uc+ ! g5]al elem5ts ( ! v>i\s conceptions ( geometry to simply num]ical ideas2 9 a ^w1 t ( sub/itut+ 9 geometry pure consid]ations ( quantity = all consid]ations ( quality4 ,r$uction ( ,figure to ,position4 ,= ?is purpose let u obs]ve1 9 ! fir/ place1 t all geometrical ideas relate necess>ily to ^! ?ree univ]sal categories3 ! magnitude1 ! figure1 & ! position ( ! ext5sions to be consid]$4 ,z to ! fir/1 !re is evid5tly no difficulty2 x ent]s at once into ! ideas ( numb]s4 ,) relation to ! second1 x mu/ be rem>k$ t x w always admit ( be+ r$uc$ to ! ?ird4 ,= ! figure ( a body evid5tly results f ! mutual position ( ! diff]5t po9ts ( :i* x is compos$1 s t ! idea ( position necess>ily compreh5ds t ( figure1 & e circum/ance ( figure c be translat$ by a circum/ance ( position4 ,x is 9 ?is way1 9 fact1 t ! human m9d has proce$$ 9 ord] to arrive at ! analytical repres5tation ( geometrical figures1 _! conception relat+ directly only to positions4 ,all ! elem5t>y difficulty is !n prop]ly r$uc$ to t ( ref]r+ ideas ( situation to ideas ( magnitude4 ,su* is ! direct de/9ation ( ! prelim9>y conception ^u :i* ,desc>tes has e/abli%$ ! g5]al sy/em ( analytical geometry4 ,8 philosophical lab\r1 9 ?is relation1 has consi/$ simply 9 ! entire g5]alization ( an elem5t>y op]ation1 :i* we may reg>d z natural to ! human m9d1 s9ce x is p]=m$ spontane\sly1 s to say1 9 all m9ds1 ev5 ! mo/ uncultivat$4 ,?us1 :5 we h to indicate ! situation ( an object )\t directly po9t+ x out1 ! me?od :i* we always adopt1 & evid5tly ! only "o :i* c be employ$1 consi/s 9 ref]r+ t object to o!rs :i* are kn[n1 by assign+ ! magnitude ( ! v>i\s geometrical elem5ts1 by :i* we conceive x connect$ ) ! kn[n objects4 ,^! elem5ts con/itute :at ,desc>tes1 & aft] him all geomet]s1 h call$ ! co-ord9ates ( ea* po9t consid]$4 ,!y are necess>ily two 9 numb]1 if x is kn[n 9 advance 9 :at plane ! po9t is situat$2 & ?ree1 if x may be f\nd indiff]5tly 9 any region ( space4 ,z _m diff]5t con/ructions z c be imag9$ = det]m9+ ! position ( a po9t1 :e!r on a plane or 9 space1 s _m di/9ct sy/ems ( co-ord9ates may be conceiv$2 !y are consequ5tly susceptible ( be+ multipli$ to inf9ity4 ,b1 :atev] may be ! sy/em adopt$1 we %all always h r$uc$ ! ideas ( situation to simple ideas ( magnitude1 s t we w consid] ! *ange 9 ! position ( a po9t z produc$ by m]e num]ical v>iations 9 ! values ( its co-ord9ates4 ,det]m9ation ( ! ,position ( a ,po9t4 ,consid]+ at fir/ only ! lea/ complicat$ case1 t ( plane geometry1 x is 9 ?is way t we usually det]m9e ! position ( a po9t on a plane1 by its di/ances f two fix$ "r l9es consid]$ z kn[n1 :i* are call$ axes1 & :i* are commonly suppos$ to be p]p5dicul> to ea* o!r4 ,?is sy/em is t mo/ frequ5tly adopt$1 because ( its simplicity2 b geomet]s employ occasionally an inf9ity ( o!rs4 ,?us ! position ( a po9t on a plane may be det]m9$1 #a1 by its di/ances f two fix$ po9ts2 or1 #b1 by its di/ance f a s+le fix$ po9t1 & ! direction ( t di/ance1 e/imat$ by ! great] or less angle :i* x makes ) a fix$ "r l9e1 :i* con/itutes ! sy/em ( :at are call$ pol> co-ord9ates1 ! mo/ frequ5tly us$ aft] ! sy/em fir/ m5tion$2 or1 #c1 by ! angles :i* ! "r l9es drawn f ! v>iable po9t to two fix$ po9ts make ) ! "r l9e :i* jo9s ^! la/2 or1 #d1 by ! di/ances f t po9t to a fix$ "r l9e & a fix$ po9t1 c4 ,9 a ^w1 !re is no geometrical figure :atev] f :i* x is n possible to d$uce a c]ta9 sy/em ( co-ord9ates m or less susceptible ( be+ employ$4 ,a g5]al obs]vation1 :i* x is important to make 9 ?is connexion1 is1 t e sy/em ( co-ord9ates is equival5t to det]m9+ a po9t1 9 plane geometry1 by ! int]section ( two l9es1 ea* ( :i* is subject$ to c]ta9 fix$ conditions ( det]m9ation2 a s+le "o ( ^! conditions rema9+ v>iable1 sometimes ! "o1 sometimes ! o!r1 accord+ to ! sy/em consid]$4 ,we c\ld n1 inde$1 conceive any o!r means ( con/ruct+ a po9t ?an to m>k x by ! meet+ ( two l9es4 ,?us1 9 ! mo/ common sy/em1 t ( rectil9e> co-ord9ates1 prop]ly s call$1 ! po9t is det]m9$ by ! int]section ( two "r l9es1 ea* ( :i* rema9s con/antly p>allel to a fix$ axis1 at a great] or less di/ance f x2 9 ! pol> sy/em1 ! position ( ! po9t is m>k$ by ! meet+ ( a circle1 ( v>iable radius & fix$ c5tre1 ) a movable "r l9e compell$ to turn ab\t ?is c5tre3 9 o!r sy/ems1 ! requir$ po9t might be designat$ by ! int]section ( two circles1 or ( any o!r two l9es1 c4 ,9 a ^w1 to assign ! value ( "o ( ! co-ord9ates ( a po9t 9 any sy/em :atev]1 is always necess>ily equival5t to det]m9+ a c]ta9 l9e on :i* t po9t mu/ be situat$4 ,! geomet]s ( antiquity _h already made ?is ess5tial rem>k1 :i* s]v$ z ! base ( _! me?od ( geometrical loci1 ( :i* !y made s happy a use to direct _! rese>*es 9 ! resolution ( det]m9ate problems1 9 consid]+ sep>ately ! influ5ce ( ea* ( ! two conditions by :i* 0 def9$ ea* po9t con/itut+ ! object1 direct or indirect1 ( ! propos$ "q4 ,x 0 ! g5]al sy/ematization ( ?is me?od :i* 0 ! imm$iate motive ( ! lab\rs ( ,desc>tes1 :i* l$ him to create analytical geometry4 ,aft] hav+ cle>ly e/abli%$ ?is prelim9>y conception--by means ( :i* ideas ( position1 & !nce1 implicitly1 all elem5t>y geometrical conceptions are capable ( be+ r$uc$ to simple num]ical consid]ations--x is easy to =m a direct conception1 9 its entire g5]ality1 ( ! great orig9al idea ( ,desc>tes1 relative to ! analytical repres5tation ( geometrical figures3 x is ?is :i* =ms ! special object ( ?is *apt]4 ,i w cont9ue to consid] at fir/1 = m facility1 only geometry ( two dim5sions1 :i* alone 0 treat$ by ,desc>tes2 & w aft]w>ds exam9e sep>ately1 "u ! same po9t ( view1 ! !ory ( surfaces & curves ( d\ble curvature4 ,,plane ,,curves4 ,expression ( ,l9es by ,equations4 ,9 accordance ) ! mann] ( express+ analytically ! position ( a po9t on a plane1 x c be easily e/abli%$ t1 by :atev] prop]ty any l9e may be def9$1 t def9ition always admits ( be+ replac$ by a correspond+ equation betwe5 ! two v>iable co-ord9ates ( ! po9t :i* describes ?is l9e2 an equation :i* w be !nce=? ! analytical repres5tation ( ! propos$ l9e1 e ph5om5on ( :i* w be translat$ by a c]ta9 algebraic modification ( its equation4 ,?us1 if we suppose t a po9t moves on a plane )\t its c\rse be+ 9 any mann] det]m9$1 we %all evid5tly h to reg>d its co-ord9ates1 to :atev] sy/em !y may belong1 z two v>iables entirely indep5d5t ( "o ano!r4 ,b if1 on ! contr>y1 ?is po9t is compell$ to describe a c]ta9 l9e1 we %all necess>ily be compell$ to conceive t its co-ord9ates1 9 all ! positions :i* x c take1 reta9 a c]ta9 p]man5t & precise relation to ea* o!r1 :i* is consequ5tly susceptible ( be+ express$ by a suitable equation2 :i* w become ! v cle> & v rigor\s analytical def9ition ( ! l9e "u consid]ation1 s9ce x w express an algebraical prop]ty belong+ exclusively to ! co-ord9ates ( all ! po9ts ( ?is l9e4 ,x is cle>1 inde$1 t :5 a po9t is n subject$ to any condition1 its situation is n det]m9$ except 9 giv+ at once its two co-ord9ates1 indep5d5tly ( ea* o!r2 :ile1 :5 ! po9t mu/ cont9ue ^u a def9$ l9e1 a s+le co-ord9ate is suffici5t = completely fix+ its position4 ,! second co-ord9ate is !n a det]m9ate function ( ! fir/2 or1 9 o!r words1 !re mu/ exi/ betwe5 !m a c]ta9 equation1 ( a nature correspond+ to t ( ! l9e on :i* ! po9t is compell$ to rema94 ,9 a ^w1 ea* ( ! co-ord9ates ( a po9t requir+ x to be situat$ on a c]ta9 l9e1 we conceive reciprocally t ! condition1 on ! "p ( a po9t1 ( hav+ to belong to a l9e def9$ 9 any mann] :atev]1 is equival5t to assign+ ! value ( "o ( ! two co-ord9ates2 :i* is f\nd 9 t case to be entirely dep5d5t on ! o!r4 ,! analytical relation :i* expresses ?is dep5d5ce may be m or less difficult to discov]1 b x mu/ evid5tly be always conceiv$ to exi/1 ev5 9 ! cases 9 :i* our pres5t means may be insuffici5t to make x kn[n4 ,x is by ?is simple consid]ation t we may demon/rate1 9 an entirely g5]al mann]--indep5d5tly ( ! p>ticul> v]ifications on :i* ?is fundam5tal conception is ord9>ily e/abli%$ = ea* special def9ition ( a l9e--! necessity ( ! analytical repres5tation ( l9es by equations4 ,expression ( ,equations by ,l9es4 ,tak+ up aga9 ! same reflections 9 ! inv]se direction1 we c\ld %[ z easily ! geometrical necessity ( ! repres5tation ( e equation ( two v>iables1 9 a det]m9ate sy/em ( co-ord9ates1 by a c]ta9 l9e2 ( :i* su* a relation w\ld be1 9 ! abs5ce ( any o!r kn[n prop]ty1 a v *>act]i/ic def9ition1 ! sci5tific de/9ation ( :i* w be to fix ! att5tion directly ^u ! g5]al c\rse ( ! solutions ( ! equation1 :i* w ?us be not$ 9 ! mo/ /rik+ & ! mo/ simple mann]4 ,?is pictur+ ( equations is "o ( ! mo/ important fundam5tal advantages ( analytical geometry1 :i* has !reby react$ 9 ! highe/ degree ^u ! g5]al p]fect+ ( analysis itself2 n only by assign+ to purely ab/ract rese>*es a cle>ly det]m9$ object & an inexhau/ible c>e]1 b1 9 a /ill m direct relation1 by furni%+ a new philosophical m$ium = analytical m$itation :i* c\ld n be replac$ by any o!r4 ,9 fact1 ! purely algebraic discussion ( an equation und\bt$ly makes kn[n its solutions 9 ! mo/ precise mann]1 b 9 consid]+ !m only "o by "o1 s t 9 ?is way no g5]al view ( !m c\ld be obta9$1 except z ! f9al result ( a long & labori\s s]ies ( num]ical comp>isons4 ,on ! o!r h&1 ! geometrical locus ( ! equation1 be+ only design$ to repres5t di/9ctly & ) p]fect cle>ness ! summ+ up ( all ^! comp>isons1 p]mits x to be directly consid]$1 )\t pay+ any att5tion to ! details :i* h furni%$ x4 ,x c !reby sugge/ to our m9d g5]al analytical views1 :i* we %\ld h arriv$ at ) mu* difficulty 9 any o!r mann]1 = want ( a means ( cle>ly *>act]iz+ _! object4 ,x is evid5t1 = example1 t ! simple inspection ( ! log>i?mic curve1 or ( ! curve y s94 x1 makes u p]ceive mu* m di/9ctly ! g5]al mann] ( ! v>iations ( log>i?ms ) respect to _! numb]s1 or ( s9es ) respect to _! arcs1 ?an c\ld ! mo/ att5tive /udy ( a table ( log>i?ms or ( natural s9es4 ,x is well kn[n t ?is me?od has become entirely elem5t>y at ! pres5t day1 & t x is employ$ :5ev] x is desir$ to get a cle> idea ( ! g5]al "* ( ! law :i* reigns 9 a s]ies ( precise obs]vations ( any k9d :atev]4 ,any ,*ange 9 ! ,l9e causes a ,*ange 9 ! ,equation4 ,return+ to ! repres5tation ( l9es by equations1 :i* is our pr9cipal object1 we see t ?is repres5tation is1 by its nature1 s fai?ful1 t ! l9e c\ld n exp]i5ce any modification1 h[ev] slight x might be1 )\t caus+ a correspond+ *ange 9 ! equation4 ,?is p]fect exactitude ev5 gives rise (t5times to special difficulties2 = s9ce1 9 our sy/em ( analytical geometry1 ! m]e displacem5ts ( l9es affect ! equations1 z well z _! real v>iations 9 magnitude or =m1 we %\ld be liable to conf\nd !m ) "o ano!r 9 our analytical expressions1 if geomet]s _h n discov]$ an +5i\s me?od design$ expressly to always di/+ui% !m4 ,?is me?od is f\nd$ on ?is pr9ciple1 t al?\gh x is impossible to *ange analytically at w ! position ( a l9e ) respect to ! axes ( ! co-ord9ates1 we c *ange 9 any mann] :atev] ! situation ( ! axes !mselves1 :i* evid5tly am\nts to ! same2 !n1 by ! aid ( ! v simple g5]al =mula by :i* ?is trans=mation ( ! axes is produc$1 x becomes easy to discov] :e!r two diff]5t equations are ! analytical expressions ( only ! same l9e diff]5tly situat$1 or ref] to truly di/9ct geometrical loci2 s9ce1 9 ! =m] case1 "o ( !m w pass into ! o!r by suitably *ang+ ! axes or ! o!r con/ants ( ! sy/em ( co-ord9ates employ$4 ,x mu/1 moreov]1 be rem>k$ on ?is subject1 t g5]al inconv5i5ces ( ?is nature seem to be absolutely inevitable 9 analytical geometry2 =1 s9ce ! ideas ( position are1 z we h se51 ! only geometrical ideas imm$iately r$ucible to num]ical consid]ations1 & ! conceptions ( figure _c be ?us r$uc$1 except by see+ 9 !m relations ( situation1 x is impossible = analysis to escape conf\nd+1 at fir/1 ! ph5om5a ( figure ) simple ph5om5a ( position1 :i* alone are directly express$ by ! equations4 ,e ,def9ition ( a ,l9e is an ,equation4 ,9 ord] to complete ! philosophical explanation ( ! fundam5tal conception :i* s]ves z ! base ( analytical geometry1 ,i ?9k t ,i %\ld "h indicate a new g5]al consid]ation1 :i* seems to me p>ticul>ly well adapt$ = putt+ 9 ! cle>e/ po9t ( view ?is necess>y repres5tation ( l9es by equations ) two v>iables4 ,x consi/s 9 ?is1 t n only1 z we h %[n1 mu/ e def9$ l9e necess>ily give rise to a c]ta9 equation betwe5 ! two co-ord9ates ( any "o ( its po9ts1 b1 /ill f>!r1 e def9ition ( a l9e may be reg>d$ z be+ already ( itself an equation ( t l9e 9 a suitable sy/em ( co-ord9ates4 ,x is easy to e/abli% ?is pr9ciple1 fir/ mak+ a prelim9>y logical di/9ction ) respect to diff]5t k9ds ( def9itions4 ,! rigor\sly indisp5sable condition ( e def9ition is t ( di/+ui%+ ! object def9$ f all o!rs1 by assign+ to x a prop]ty :i* belongs to x exclusively4 ,b ?is end may be g5]ally atta9$ 9 two v diff]5t ways2 ei!r by a def9ition :i* is simply *>act]i/ic1 t is1 indicative ( a prop]ty :i*1 al?\gh truly exclusive1 does n make kn[n ! mode ( g5]ation ( ! object2 or by a def9ition :i* is really explanatory1 t is1 :i* *>act]izes ! object by a prop]ty :i* expresses "o ( its modes ( g5]ation4 ,= example1 9 consid]+ ! circle z ! l9e1 :i*1 "u ! same cont\r1 conta9s ! greate/ area1 we h evid5tly a def9ition ( ! fir/ k9d2 :ile 9 *oos+ ! prop]ty ( its hav+ all its po9ts equally di/ant f a fix$ po9t1 we h a def9ition ( ! second k9d4 ,x is1 besides1 evid5t1 z a g5]al pr9ciple1 t ev5 :5 any object :atev] is kn[n at fir/ only by a *>act]i/ic def9ition1 we ought1 nev]!less1 to reg>d x z susceptible ( explanatory def9itions1 :i* ! f>!r /udy ( ! object w\ld necess>ily lead u to discov]4 ,?is be+ premis$1 x is cle> t ! g5]al obs]vation above made1 :i* repres5ts e def9ition ( a l9e z be+ necess>ily an equation ( t l9e 9 a c]ta9 sy/em ( co-ord9ates1 _c apply to def9itions :i* are simply *>act]i/ic2 x is to be und]/ood only ( def9itions :i* are truly explanatory4 ,b1 9 consid]+ only ?is class1 ! pr9ciple is easy to prove4 ,9 fact1 x is evid5tly impossible to def9e ! g5]ation ( a l9e )\t specify+ a c]ta9 relation betwe5 ! two simple motions ( translation or ( rotation1 into :i* ! motion ( ! po9t :i* describes x w be decompos$ at ea* in/ant4 ,n[ if we =m ! mo/ g5]al conception ( :at con/itutes a sy/em ( co-ord9ates1 & admit all possible sy/ems1 x is cle> t su* a relation w be no?+ else b ! equation ( ! propos$ l9e1 9 a sy/em ( co-ord9ates ( a nature correspond+ to t ( ! mode ( g5]ation consid]$4 ,?us1 = example1 ! common def9ition ( ! circle may evid5tly be reg>d$ z be+ imm$iately ! pol> equation ( ?is curve1 tak+ ! c5tre ( ! circle = ! pole4 ,9 ! same way1 ! elem5t>y def9ition ( ! ellipse or ( ! hyp]bola--z be+ ! curve g5]at$ by a po9t :i* moves 9 su* a mann] t ! sum or ! diff]5ce ( its di/ances f two fix$ po9ts rema9s con/ant--gives at once1 = ei!r ! "o or ! o!r curve1 ! equation y x c1 tak+ = ! sy/em ( co-ord9ates t 9 :i* ! position ( a po9t w\ld be det]m9$ by its di/ances f two fix$ po9ts1 & *oos+ = ^! poles ! two giv5 foci4 ,9 l mann]1 ! common def9ition ( any cycloid w\ld furni% directly1 = t curve1 ! equation y mx2 adopt+ z ! co-ord9ates ( ea* po9t ! arc :i* x m>ks ^u a circle ( inv>iable radius1 measur+ f ! po9t ( contact ( t circle ) a fix$ l9e1 & ! rectil9e> di/ance f t po9t ( contact to a c]ta9 orig9 tak5 on t "r l9e4 ,we c make analog\s & equally easy v]ifications ) respect to ! cu/om>y def9itions ( spirals1 ( epicycloids1 c4 ,we %all con/antly f9d t !re exi/s a c]ta9 sy/em ( co-ord9ates1 9 :i* we imm$iately obta9 a v simple equation ( ! propos$ l9e1 by m]ely writ+ algebraically ! condition impos$ by ! mode ( g5]ation consid]$4 ,besides its direct importance z a means ( r5d]+ p]fectly app>5t ! necess>y repres5tation ( e l9e by an equation1 ! prec$+ consid]ation seems to me to possess a true sci5tific utility1 9 *>act]iz+ ) precision ! pr9cipal g5]al difficulty :i* occurs 9 ! actual e/abli%m5t ( ^! equations1 & 9 consequ5tly furni%+ an int]e/+ indication ) respect to ! c\rse to be pursu$ 9 inquiries ( ?is k9d1 :i*1 by _! nature1 c\ld n admit ( complete & inv>iable rules4 ,9 fact1 s9ce any def9ition :atev] ( a l9e1 at lea/ among ^? :i* indicate a mode ( g5]ation1 furni%es directly ! equation ( t l9e 9 a c]ta9 sy/em ( co-ord9ates1 or1 r1 ( itself con/itutes t equation1 x foll[s t ! difficulty :i* we (t5 exp]i5ce 9 discov]+ ! equation ( a curve1 by means ( c]ta9 ( its *>act]i/ic prop]ties1 a difficulty :i* is sometimes v great1 mu/ proce$ ess5tially only f ! commonly impos$ condition ( express+ ?is curve analytically by ! aid ( a designat$ sy/em ( co-ord9ates1 in/ead ( admitt+ indiff]5tly all possible sy/ems4 ,^! diff]5t sy/ems _c be reg>d$ 9 analytical geometry z be+ all equally suitable2 = v>i\s reasons1 ! mo/ important ( :i* w be h]eaft] discuss$1 geomet]s ?9k t curves %\ld almo/ always be ref]r$1 z f> z is possible1 to rectil9e> co-ord9ates1 prop]ly s call$4 ,n[ we see1 f :at prec$es1 t 9 _m cases ^! p>ticul> co-ord9ates w n be ^? ) ref]5ce to :i* ! equation ( ! curve w be f\nd to be directly e/abli%$ by ! propos$ def9ition4 ,! pr9cipal difficulty pres5t$ by ! =mation ( ! equation ( a l9e really consi/s1 !n1 9 g5]al1 9 a c]ta9 trans=mation ( co-ord9ates4 ,x is und\bt$ly true t ?is consid]ation does n subject ! e/abli%m5t ( ^! equations to a truly complete g5]al me?od1 ! success ( :i* is always c]ta92 :i*1 f ! v nature ( ! subject1 is evid5tly *im]ical3 b su* a view may ?r[ mu* useful light ^u ! c\rse :i* x is prop] to adopt1 9 ord] to arrive at ! end propos$4 ,?us1 aft] hav+ 9 ! fir/ place =m$ ! prep>atory equation1 :i* is spontane\sly d]iv$ f ! def9ition :i* we are consid]+1 x w be necess>y1 9 ord] to obta9 ! equation belong+ to ! sy/em ( co-ord9ates :i* mu/ be f9ally admitt$1 to endeav\r to express 9 a function ( ^! la/ co-ord9ates ^? :i* naturally correspond to ! giv5 mode ( g5]ation4 ,x is ^u ?is la/ lab\r t x is evid5tly impossible to give inv>iable & precise precepts4 ,we c only say t we %all h s _m m res\rces 9 ?is matt] z we %all "k m ( true analytical geometry1 t is1 z we %all "k ! algebraical expression ( a great] numb] ( diff]5t algebraical ph5om5a4 ,,*oice ,,( ,,co-,,ord9ates4 ,9 ord] to complete ! philosophical exposition ( ! conception :i* s]ves z ! base ( analytical geometry1 ,i h yet to notice ! consid]ations relat+ to ! *oice ( ! sy/em ( co-ord9ates :i* is 9 g5]al ! mo/ suitable4 ,!y w give ! rational explanation ( ! pref]5ce unanim\sly accord$ to ! ord9>y rectil9e> sy/em2 a pref]5ce :i* has hi!rto be5 r ! effect ( an empirical s5tim5t ( ! sup]iority ( ?is sy/em1 ?an ! exact result ( a direct & ?or\gh analysis4 ,two diff]5t ,po9ts ( ,view4 ,9 ord] to decide cle>ly betwe5 all ! diff]5t sy/ems ( co-ord9ates1 x is indisp5sable to di/+ui% ) c>e ! two g5]al po9ts ( view1 ! conv]se ( "o ano!r1 :i* belong to analytical geometry2 namely1 ! relation ( algebra to geometry1 f\nd$ ^u ! repres5tation ( l9es by equations2 &1 reciprocally1 ! relation ( geometry to algebra1 f\nd$ on ! repres5tation ( equations by l9es4 ,x is evid5t t 9 e inve/igation ( g5]al geometry ^! two fundam5tal po9ts ( view are ( necessity always f\nd comb9$1 s9ce we h always to pass alt]nately1 & at ins5sible int]vals1 s to say1 f geometrical to analytical consid]ations1 & f analytical to geometrical consid]ations4 ,b ! necessity ( "h tempor>ily sep>at+ !m is none ! less real2 = ! answ] to ! "q ( me?od :i* we are exam9+ is1 9 fact1 z we %all see pres5tly1 v f> f be+ ! same 9 bo? ^! relations1 s t )\t ?is di/9ction we c\ld n =m any cle> idea ( x4 #a4 ,repres5tation ( ,l9es by ,equations4 ,"u ! fir/ po9t ( view--! repres5tation ( l9es by equations--! only reason :i* c\ld lead u to pref] "o sy/em ( co-ord9ates to ano!r w\ld be ! great] simplicity ( ! equation ( ea* l9e1 & great] facility 9 arriv+ at x4 ,n[ x is easy to see t !re does n exi/1 & c\ld n be expect$ to exi/1 any sy/em ( co-ord9ates des]v+ 9 t respect a con/ant pref]5ce ov] all o!rs4 ,9 fact1 we h above rem>k$ t = ea* geometrical def9ition propos$ we c conceive a sy/em ( co-ord9ates 9 :i* ! equation ( ! l9e is obta9$ at once1 & is necess>ily f\nd to be also v simple2 & ?is sy/em1 moreov]1 inevitably v>ies ) ! nature ( ! *>act]i/ic prop]ty "u consid]ation4 ,! rectil9e> sy/em c\ld n1 !re=e1 be con/antly ! mo/ advantage\s = ?is object1 al?\gh x may (t5 be v fav\rable2 !re is probably no sy/em :i*1 9 c]ta9 p>ticul> cases1 %\ld n be pref]r$ to x1 z well z to e o!r4 #b4 ,repres5tation ( ,equations by ,l9es4 ,x is by no means s1 h[ev]1 "u ! second po9t ( view4 ,we c1 inde$1 easily e/abli%1 z a g5]al pr9ciple1 t ! ord9>y rectil9e> sy/em mu/ necess>ily be bett] adapt$ ?an any o!r to ! repres5tation ( equations by ! correspond+ geometrical loci2 t is to say1 t ?is repres5tation is con/antly m simple & m fai?ful 9 x ?an 9 any o!r4 ,let u consid]1 = ?is object1 t1 s9ce e sy/em ( co-ord9ates consi/s 9 det]m9+ a po9t by ! int]section ( two l9es1 ! sy/em adapt$ to furni% ! mo/ suitable geometrical loci mu/ be t 9 :i* ^! two l9es are ! simple/ possible2 a consid]ation :i* conf9es our *oice to ! rectil9e> sy/em4 ,9 tru?1 !re is evid5tly an inf9ite numb] ( sy/ems :i* des]ve t "n1 t is to say1 :i* employ only "r l9es to det]m9e po9ts1 besides ! ord9>y sy/em :i* assigns ! di/ances f two fix$ l9es z co-ord9ates2 su*1 = example1 w\ld be t 9 :i* ! co-ord9ates ( ea* po9t %\ld be ! two angles :i* ! "r l9es1 :i* g f t po9t to two fix$ po9ts1 make ) ! "r l9e1 :i* jo9s ^! la/ po9ts3 s t ?is fir/ consid]ation is n rigor\sly suffici5t to expla9 ! pref]5ce unanim\sly giv5 to ! common sy/em4 ,b 9 exam9+ 9 a m ?or\gh mann] ! nature ( e sy/em ( co-ord9ates1 we also p]ceive t ea* ( ! two l9es1 ^: meet+ det]m9es ! po9t consid]$1 mu/ necess>ily (f] at e in/ant1 among its diff]5t conditions ( det]m9ation1 a s+le v>iable condition1 :i* gives rise to ! correspond+ co-ord9ate1 all ! re/ be+ fix$1 & con/itut+ ! axes ( ! sy/em1 tak+ ?is t]m 9 its mo/ ext5d$ ma!matical acceptation4 ,! v>iation is indisp5sable1 9 ord] t we may be able to consid] all possible positions2 & ! fixity is no less s1 9 ord] t !re may exi/ means ( comp>ison4 ,?us1 9 all rectil9e> sy/ems1 ea* ( ! two "r l9es w be subject$ to a fix$ condition1 & ! ord9ate w result f ! v>iable condition4 ,sup]iority ( rectil9e> ,co-ord9ates4 ,f ^! consid]ations x is evid5t1 z a g5]al pr9ciple1 t ! mo/ fav\rable sy/em = ! con/ruction ( geometrical loci w necess>ily be t 9 :i* ! v>iable condition ( ea* "r l9e %all be ! simple/ possible2 ! fix$ condition be+ left free to be made complex1 if necess>y to atta9 t object4 ,n[1 ( all possible mann]s ( det]m9+ two movable "r l9es1 ! easie/ to foll[ geometrically is c]ta9ly t 9 :i*1 ! direction ( ea* "r l9e rema9+ inv>iable1 x only approa*es or rec$es1 m or less1 to or f a con/ant axis4 ,x w\ld be1 = example1 evid5tly m difficult to figure to "o's self cle>ly ! *anges ( place ( a po9t :i* is det]m9$ by ! int]section ( two "r l9es1 :i* ea* turn ar\nd a fix$ po9t1 mak+ a great] or small] angle ) a c]ta9 axis1 z 9 ! sy/em ( co-ord9ates previ\sly notic$4 ,su* is ! true g5]al explanation ( ! fundam5tal prop]ty possess$ by ! common rectil9e> sy/em1 ( be+ bett] adapt$ ?an any o!r to ! geometrical repres5tation ( equations1 inasmu* z x is t "o 9 :i* x is ! easie/ to conceive ! *ange ( place ( a po9t result+ f ! *ange 9 ! value ( its co-ord9ates4 ,9 ord] to feel cle>ly all ! =ce ( ?is consid]ation1 x w\ld be suffici5t to c>efully comp>e ?is sy/em ) ! pol> sy/em1 9 :i* ?is geometrical image1 s simple & s easy to foll[1 ( two "r l9es mov+ p>allel1 ea* "o ( !m1 to its correspond+ axis1 is replac$ by ! complicat$ picture ( an inf9ite s]ies ( conc5tric circles1 cut by a "r l9e compell$ to turn ab\t a fix$ po9t4 ,x is1 moreov]1 easy to conceive 9 advance :at mu/ be ! extreme importance to analytical geometry ( a prop]ty s pr(\ndly elem5t>y1 :i*1 = t reason1 mu/ be recurr+ at e in/ant1 & take a progressively increas+ value 9 all lab\rs ( ?is k9d4 ,p]p5dicul>ity ( ! ,axes4 ,9 pursu+ f>!r ! consid]ation :i* demon/rates ! sup]iority ( ! ord9>y sy/em ( co-ord9ates ov] any o!r z to ! repres5tation ( equations1 we may also take notice ( ! utility = ?is object ( ! common usage ( tak+ ! two axes p]p5dicul> to ea* o!r1 :5ev] possible1 r ?an ) any o!r incl9ation4 ,z reg>ds ! repres5tation ( l9es by equations1 ?is second>y circum/ance is no m univ]sally prop] ?an we h se5 ! g5]al nature ( ! sy/em to be2 s9ce1 accord+ to ! p>ticul> occasion1 any o!r incl9ation ( ! axes may des]ve our pref]5ce 9 t respect4 ,b1 9 ! inv]se po9t ( view1 x is easy to see t rectangul> axes con/antly p]mit u to repres5t equations 9 a m simple & ev5 m fai?ful mann]2 =1 ) oblique axes1 space be+ divid$ by !m into regions :i* no long] h a p]fect id5tity1 x foll[s t1 if ! geometrical locus ( ! equation ext5ds into all ^! regions at once1 !re w be pres5t$1 by reason m]ely ( ?is inequality ( ! angles1 diff]5ces ( figure :i* d n correspond to any analytical div]sity1 & w necess>ily alt] ! rigor\s exactness ( ! repres5tation1 by be+ conf\nd$ ) ! prop] results ( ! algebraic comp>isons4 ,= example1 an equation l3 xm ym c1 :i*1 by its p]fect symmetry1 %\ld evid5tly give a curve compos$ ( f\r id5tical qu>t]s1 w be repres5t$1 on ! contr>y1 if we take axes n rectangul>1 by a geometric locus1 ! f\r p>ts ( :i* w be unequal4 ,x is pla9 t ! only means ( avoid+ all inconv5i5ces ( ?is k9d is to suppose ! angle ( ! two axes to be a "r angle4 ,! prec$+ discussion cle>ly %[s t1 al?\gh ! ord9>y sy/em ( rectil9e> co-ord9ates has no con/ant sup]iority ov] all o!rs 9 "o ( ! two fundam5tal po9ts ( view :i* are cont9ually comb9$ 9 analytical geometry1 yet z1 on ! o!r h&1 x is n con/antly inf]ior1 its necess>y & absolute great] aptitude = ! repres5tation ( equations mu/ cause x to g5]ally receive ! pref]5ce2 al?\gh x may evid5tly happ51 9 "s p>ticul> cases1 t ! necessity ( simplify+ equations & ( obta9+ !m m easily may det]m9e geomet]s to adopt a less p]fect sy/em4 ,! rectil9e> sy/em is1 !re=e1 ! "o by means ( :i* are ord9>ily con/ruct$ ! mo/ ess5tial !ories ( g5]al geometry1 int5d$ to express analytically ! mo/ important geometrical ph5om5a4 ,:5 x is ?\ght necess>y to *oose "s o!r1 ! pol> sy/em is almo/ always ! "o :i* is fix$ ^u1 ?is sy/em be+ ( a nature suffici5tly opposite to t ( ! rectil9e> sy/em to cause ! equations1 :i* are too complicat$ ) respect to ! latt]1 to become1 9 g5]al1 suffici5tly simple ) respect to ! o!r4 ,pol> co-ord9ates1 moreov]1 h (t5 ! advantage ( admitt+ ( a m direct & natural concrete signification2 z is ! case 9 me*anics1 = ! geometrical que/ions to :i* ! !ory ( circul> movem5t gives rise1 & 9 almo/ all ! cases ( cele/ial geometry4 99 99 99 99 99 ,9 ord] to simplify ! exposition1 we h ?us f> consid]$ ! fundam5tal conception ( analytical geometry only ) respect to plane curves1 ! g5]al /udy ( :i* 0 ! only object ( ! great philosophical r5ovation produc$ by ,desc>tes4 ,to complete ?is important explanation1 we h n[ to %[ summ>ily h[ ?is elem5t>y idea 0 ext5d$ by ,clairaut1 ab\t a c5tury aft]w>ds1 to ! g5]al /udy ( surfaces & curves ( d\ble curvature4 ,! consid]ations :i* h be5 already giv5 w p]mit me to limit myself on ?is subject to ! rapid exam9ation ( :at is /rictly peculi> to ?is new case4 ,,surfaces4 ,det]m9ation ( a ,po9t 9 ,space4 ,! complete analytical det]m9ation ( a po9t 9 space evid5tly requires ! values ( ?ree co-ord9ates to be assign$2 z1 = example1 9 ! sy/em :i* is g5]ally adopt$1 & :i* corresponds to ! rectil9e> sy/em ( plane geometry1 di/ances f ! po9t to ?ree fix$ planes1 usually p]p5dicul> to "o ano!r2 :i* pres5ts ! po9t z ! int]section ( ?ree planes ^: direction is inv>iable4 ,we might also employ ! di/ances f ! movable po9t to ?ree fix$ po9ts1 :i* w\ld det]m9e x by ! int]section ( ?ree sph]es ) a common c5tre4 ,9 l mann]1 ! position ( a po9t w\ld be def9$ by giv+ its di/ance f a fix$ po9t1 & ! direction ( t di/ance1 by means ( ! two angles :i* ?is "r l9e makes ) two inv>iable axes2 ?is is ! pol> sy/em ( geometry ( ?ree dim5sions2 ! po9t is !n con/ruct$ by ! int]section ( a sph]e hav+ a fix$ c5tre1 ) two "r cones ) circul> bases1 ^: axes & common summit d n *ange4 ,9 a ^w1 !re is evid5tly1 9 ?is case at lea/1 ! same inf9ite v>iety among ! v>i\s possible sy/ems ( co-ord9ates :i* we h already obs]v$ 9 geometry ( two dim5sions4 ,9 g5]al1 we h to conceive a po9t z be+ always det]m9$ by ! int]section ( any ?ree surfaces :atev]1 z x 0 9 ! =m] case by t ( two l9es3 ea* ( ^! ?ree surfaces has1 9 l mann]1 all its conditions ( det]m9ation con/ant1 except+ "o1 :i* gives rise to ! correspond+ co-ord9ates1 ^: peculi> geometrical influ5ce is ?us to con/ra9 ! po9t to be situat$ ^u t surface4 ,?is be+ premis$1 x is cle> t if ! ?ree co-ord9ates ( a po9t are entirely indep5d5t ( "o ano!r1 t po9t c take successively all possible positions 9 space4 ,b if ! po9t is compell$ to rema9 ^u a c]ta9 surface def9$ 9 any mann] :atev]1 !n two co-ord9ates are evid5tly suffici5t = det]m9+ its situation at ea* in/ant1 s9ce ! propos$ surface w take ! place ( ! condition impos$ by ! ?ird co-ord9ate4 ,we mu/ !n1 9 ?is case1 "u ! analytical po9t ( view1 necess>ily conceive ?is la/ co-ord9ate z a det]m9ate function ( ! two o!rs1 ^! latt] rema9+ p]fectly indep5d5t ( ea* o!r4 ,?us !re w be a c]ta9 equation betwe5 ! ?ree v>iable co-ord9ates1 :i* w be p]man5t1 & :i* w be ! only "o1 9 ord] to correspond to ! precise degree ( indet]m9ation 9 ! position ( ! po9t4 ,expression ( ,surfaces by ,equations4 ,?is equation1 m or less easy to be discov]$1 b always possible1 w be ! analytical def9ition ( ! propos$ surface1 s9ce x mu/ be v]ifi$ = all ! po9ts ( t surface1 & = !m alone4 ,if ! surface und]goes any *ange :atev]1 ev5 a simple *ange ( place1 ! equation mu/ und]go a m or less s]i\s correspond+ modification4 ,9 a ^w1 all geometrical ph5om5a relat+ to surfaces w admit ( be+ translat$ by c]ta9 equival5t analytical conditions appropriate to equations ( ?ree v>iables2 & 9 ! e/abli%m5t & int]pretation ( ?is g5]al & necess>y h>mony w ess5tially consi/ ! sci5ce ( analytical geometry ( ?ree dim5sions4 ,expression ( ,equations by ,surfaces4 ,consid]+ next ?is fundam5tal conception 9 ! inv]se po9t ( view1 we see 9 ! same mann] t e equation ( ?ree v>iables may1 9 g5]al1 be repres5t$ geometrically by a det]m9ate surface1 primitively def9$ by ! v *>act]i/ic prop]ty1 t ! co-ord9ates ( all its po9ts always reta9 ! mutual relation enunciat$ 9 ?is equation4 ,?is geometrical locus w evid5tly *ange1 = ! same equation1 accord+ to ! sy/em ( co-ord9ates :i* may s]ve = ! con/ruction ( ?is repres5tation4 ,9 adopt+1 = example1 ! rectil9e> sy/em1 x is cle> t 9 ! equation betwe5 ! ?ree v>iables1 x1 y1 z1 e p>ticul> value attribut$ to z w give an equation betwe5 at x & y1 ! geometrical locus ( :i* w be a c]ta9 l9e situat$ 9 a plane p>allel to ! plane ( x & y1 & at a di/ance f ?is la/ equal to ! value ( z2 s t ! complete geometrical locus w pres5t itself z compos$ ( an inf9ite s]ies ( l9es sup]impos$ 9 a s]ies ( p>allel planes 7except+ ! int]ruptions :i* may exi/71 & w consequ5tly =m a v]itable surface4 ,x w\ld be ! same 9 consid]+ any o!r sy/em ( co-ord9ates1 al?\gh ! geometrical con/ruction ( ! equation becomes m difficult to foll[4 ,su* is ! elem5t>y conception1 ! complem5t ( ! orig9al idea ( ,desc>tes1 on :i* is f\nd$ g5]al geometry relative to surfaces4 ,x w\ld be useless to take up "h directly ! o!r consid]ations :i* h be5 above indicat$1 ) respect to l9es1 & :i* any "o c easily ext5d to surfaces2 :e!r to %[ t e def9ition ( a surface by any me?od ( g5]ation :atev] is really a direct equation ( t surface 9 a c]ta9 sy/em ( co-ord9ates1 or to det]m9e among all ! diff]5t sy/ems ( possible co-ord9ates t "o :i* is g5]ally ! mo/ conv5i5t4 ,i w only add1 on ?is la/ po9t1 t ! necess>y sup]iority ( ! ord9>y rectil9e> sy/em1 z to ! repres5tation ( equations1 is evid5tly /ill m m>k$ 9 analytical geometry ( ?ree dim5sions ?an 9 t ( two1 because ( ! incomp>ably great] geometrical complication :i* w\ld result f ! *oice ( any o!r sy/em4 ,?is c be v]ifi$ 9 ! mo/ /rik+ mann] by consid]+ ! pol> sy/em 9 p>ticul>1 :i* is ! mo/ employ$ aft] ! ord9>y rectil9e> sy/em1 = surfaces z well z = plane curves1 & = ! same reasons4 ,9 ord] to complete ! g5]al exposition ( ! fundam5tal conception relative to ! analytical /udy ( surfaces1 a philosophical exam9ation %\ld be made ( a f9al improvem5t ( ! highe/ importance1 :i* ,monge has introduc$ into ! v elem5ts ( ?is !ory1 = ! classification ( surfaces 9 natural families1 e/abli%$ accord+ to ! mode ( g5]ation1 & express$ algebraically by common diff]5tial equations1 or by f9ite equations conta9+ arbitr>y functions4 ,,curves ,,( ,,d\ble ,,curvature4 ,let u n[ consid] ! la/ elem5t>y po9t ( view ( analytical geometry ( ?ree dim5sions2 t relat+ to ! algebraic repres5tation ( curves consid]$ 9 space1 9 ! mo/ g5]al mann]4 ,9 cont9u+ to foll[ ! pr9ciple :i* has be5 con/antly employ$1 t ( ! degree ( indet]m9ation ( ! geometrical locus1 correspond+ to ! degree ( indep5d5ce ( ! v>iables1 x is evid5t1 z a g5]al pr9ciple1 t :5 a po9t is requir$ to be situat$ ^u "s c]ta9 curve1 a s+le co-ord9ate is en\gh = completely det]m9+ its position1 by ! int]section ( ?is curve ) ! surface :i* results f ?is co-ord9ate4 ,?us1 9 ?is case1 ! two o!r co-ord9ates ( ! po9t mu/ be conceiv$ z functions necess>ily det]m9ate & di/9ct f ! fir/4 ,x foll[s t e l9e1 consid]$ 9 space1 is !n repres5t$ analytically1 no long] by a s+le equation1 b by ! sy/em ( two equations betwe5 ! ?ree co-ord9ates ( any "o ( its po9ts4 ,x is cle>1 inde$1 f ano!r po9t ( view1 t s9ce ea* ( ^! equations1 consid]$ sep>ately1 expresses a c]ta9 surface1 _! comb9ation pres5ts ! propos$ l9e z ! int]section ( two det]m9ate surfaces4 ,su* is ! mo/ g5]al mann] ( conceiv+ ! algebraic repres5tation ( a l9e 9 analytical geometry ( ?ree dim5sions4 ,?is conception is commonly consid]$ 9 too re/rict$ a mann]1 :5 we conf9e ourselves to consid]+ a l9e z det]m9$ by ! sy/em ( its two projections ^u two ( ! co-ord9ate planes2 a sy/em *>act]iz$1 analytically1 by ?is peculi>ity1 t ea* ( ! two equations ( ! l9e !n conta9s only two ( ! ?ree co-ord9ates1 in/ead ( simultane\sly includ+ ! ?ree v>iables4 ,?is consid]ation1 :i* consi/s 9 reg>d+ ! l9e z ! int]section ( two cyl9drical surfaces p>allel to two ( ! ?ree axes ( ! co-ord9ates1 besides ! inconv5i5ce ( be+ conf9$ to ! ord9>y rectil9e> sy/em1 has ! fault1 if we /rictly conf9e ourselves to x1 ( introduc+ useless difficulties into ! analytical repres5tation ( l9es1 s9ce ! comb9ation ( ^! two cyl9d]s w\ld evid5tly n be always ! mo/ suitable = =m+ ! equations ( a l9e4 ,?us1 consid]+ ?is fundam5tal notion 9 its entire g5]ality1 x w be necess>y 9 ea* case to *oose1 f among ! inf9ite numb] ( c\ples ( surfaces1 ! int]section ( :i* might produce ! propos$ curve1 t "o :i* w l5d itself ! be/ to ! e/abli%m5t ( equations1 z be+ compos$ ( ! be/ kn[n surfaces4 ,?us1 if ! problem is to express analytically a circle 9 space1 x w evid5tly be pref]able to consid] x z ! int]section ( a sph]e & a plane1 r ?an z proce$+ f any o!r comb9ation ( surfaces :i* c\ld equally produce x4 ,9 tru?1 ?is mann] ( conceiv+ ! repres5tation ( l9es by equations1 9 analytical geometry ( ?ree dim5sions1 produces1 by its nature1 a necess>y inconv5i5ce1 t ( a c]ta9 analytical confusion1 consi/+ 9 ?is3 t ! same l9e may ?us be express$1 ) ! same sy/em ( co-ord9ates1 by an inf9ite numb] ( diff]5t c\ples ( equations1 on acc\nt ( ! inf9ite numb] ( c\ples ( surfaces :i* c =m x2 a circum/ance :i* may cause "s difficulties 9 recogniz+ ?is l9e "u all ! algebraical disguises ( :i* x admits4 ,b !re exi/s a v simple me?od = caus+ ?is inconv5i5ce to disappe>2 x consi/s 9 giv+ up ! facilities :i* result f ?is v>iety ( geometrical con/ructions4 ,x suffices1 9 fact1 :atev] may be ! analytical sy/em primitively e/abli%$ = a c]ta9 l9e1 to be able to d$uce f x ! sy/em correspond+ to a s+le c\ple ( surfaces uni=mly g5]at$2 z1 = example1 to t ( ! two cyl9drical surfaces :i* project ! propos$ l9e ^u two ( ! co-ord9ate planes2 surfaces :i* w evid5tly be always id5tical1 9 :atev] mann] ! l9e may h be5 obta9$1 & :i* w n v>y except :5 t l9e itself %all *ange4 ,n[1 9 *oos+ ?is fix$ sy/em1 :i* is actually ! mo/ simple1 we %all g5]ally be able to d$uce f ! primitive equations ^? :i* correspond to !m 9 ?is special con/ruction1 by trans=m+ !m1 by two successive elim9ations1 into two equations1 ea* conta9+ only two ( ! v>iable co-ord9ates1 & !reby correspond+ to ! two surfaces ( projection4 ,su* is really ! pr9cipal de/9ation ( ?is sort ( geometrical comb9ation1 :i* ?us (f]s to u an inv>iable & c]ta9 means ( recogniz+ ! id5tity ( l9es 9 spite ( ! div]sity ( _! equations1 :i* is sometimes v great4 ,,imp]fections ,,( ,,analytical ,,geometry4 ,hav+ n[ consid]$ ! fundam5tal conception ( analytical geometry "u its pr9cipal elem5t>y aspects1 x is prop]1 9 ord] to make ! sket* complete1 to notice "h ! g5]al imp]fections yet pres5t$ by ?is conception ) respect to bo? geometry & to analysis4 ,relatively to geometry1 we mu/ rem>k t ! equations are z yet adapt$ to repres5t only entire geometrical loci1 & n at all det]m9ate portions ( ^? loci4 ,x w\ld1 h[ev]1 be necess>y1 9 "s circum/ances1 to be able to express analytically a "p ( a l9e or ( a surface1 or ev5 a discont9u\s l9e or surface1 compos$ ( a s]ies ( sections belong+ to di/9ct geometrical figures1 su* z ! cont\r ( a polygon1 or ! surface ( a polyh$ron4 ,!rmology1 especially1 (t5 gives rise to su* consid]ations1 to :i* our pres5t analytical geometry is necess>ily inapplicable4 ,! lab\rs ( ,m4 ,f\ri] on discont9u\s functions h1 h[ev]1 begun to fill up ?is great gap1 & h !reby introduc$ a new & ess5tial improvem5t into ! fundam5tal conception ( ,desc>tes4 ,b ?is mann] ( repres5t+ het]og5e\s or p>tial figures1 be+ f\nd$ on ! employm5t ( trigonometrical s]ies proce$+ accord+ to ! s9es ( an inf9ite s]ies ( multiple arcs1 or on ! use ( c]ta9 def9ite integrals equival5t to ^? s]ies1 & ! g5]al integral ( :i* is unkn[n1 pres5ts z yet too mu* complication to admit ( be+ imm$iately introduc$ into ! sy/em ( analytical geometry4 ,relatively to analysis1 we mu/ beg9 by obs]v+ t our inability to conceive a geometrical repres5tation ( equations conta9+ f\r1 five1 or m v>iables1 analog\s to ^? repres5tations :i* all equations ( two or ( ?ree v>iables admit1 mu/ n be view$ z an imp]fection ( our sy/em ( analytical geometry1 = x evid5tly belongs to ! v nature ( ! subject4 ,analysis be+ necess>ily m g5]al ?an geometry1 s9ce x relates to all possible ph5om5a1 x w\ld be v unphilosophical to desire always to f9d among geometrical ph5om5a alone a concrete repres5tation ( all ! laws :i* analysis c express4 ,!re exi/s1 h[ev]1 ano!r imp]fection ( less importance1 :i* mu/ really be view$ z proce$+ f ! mann] 9 :i* we conceive analytical geometry4 ,x consi/s 9 ! evid5t incomplet5ess ( our pres5t repres5tation ( equations ( two or ( ?ree v>iables by l9es or surfaces1 inasmu* z 9 ! con/ruction ( ! geometric locus we pay reg>d only to ! real solutions ( equations1 )\t at all notic+ any imag9>y solutions4 ,! g5]al c\rse ( ^! la/ %\ld1 h[ev]1 by its nature1 be q z susceptible z t ( ! o!rs ( a geometrical repres5tation4 ,x foll[s f ?is omission t ! graphic picture ( ! equation is con/antly imp]fect1 & sometimes ev5 s mu* s t !re is no geometric repres5tation at all :5 ! equation admits ( only imag9>y solutions4 ,b1 ev5 9 ?is la/ case1 we evid5tly ought to be able to di/+ui% betwe5 equations z diff]5t 9 !mselves z ^!1 = example1 x y #a #j1 x y #a #j1 y ex #j4 ,we "k1 moreov]1 t ?is pr9cipal imp]fection (t5 br+s ) x1 9 analytical geometry ( two or ( ?ree dim5sions1 a numb] ( second>y inconv5i5ces1 aris+ f sev]al analytical modifications n correspond+ to any geometrical ph5om5a4 99 99 99 99 99 ,our philosophical exposition ( ! fundam5tal conception ( analytical geometry %[s u cle>ly t ?is sci5ce consi/s ess5tially 9 det]m9+ :at is ! g5]al analytical expression ( su* or su* a geometrical ph5om5on belong+ to l9es or to surfaces2 &1 reciprocally1 9 discov]+ ! geometrical int]pretation ( su* or su* an analytical consid]ation4 ,a detail$ exam9ation ( ! mo/ important g5]al que/ions w\ld %[ u h[ geomet]s h succe$$ 9 actually e/abli%+ ?is beautiful h>mony1 & 9 ?us impr9t+ on geometrical sci5ce1 reg>d$ z a :ole1 its pres5t em95tly p]fect "* ( rationality & ( simplicity4 ,note4--,! au?or devotes ! two foll[+ *apt]s ( 8 c\rse to ! m detail$ exam9ation ( ,analytical ,geometry ( two & ( ?ree dim5sions2 b 8 subsequ5t publication ( a sep>ate "w ^u ?is bran* ( ma!matics has be5 ?\ght to r5d] unnecess>y ! reproduction ( ^! two *apt]s 9 ! pres5t volume4 ,,! ,,end4