,,memorabilia ,,ma!matica ,,! ,,macmillan ,,company ,,new ,,york4 ,,bo/on4 ,,*icago4 ,,dallas ,,atlanta4 ,,san ,,francisco ,,macmillan ,,co41 ,,limit$ ,,london4 ,,bombay4 ,,calcutta ,,melb\rne ,,! ,,macmillan ,,co4 ,,( ,,canada1 ,,ltd4 ,,toronto ,,memorabilia ,,ma!matica ,,or ,,! ,,philoma?',s ,,quotation-,,book ,,by ,,rob]t ,,ed\>d ,,moritz1 ,,ph4 ,d41 ,,ph4 ,n4 ,d4 ,,pr(essor ,,( ,,ma!matics ,,9 ,,! ,,univ]sity ,,( ,,wa%+ton ,new ,york ,,! ,,macmillan ,,company #aiad ,all rights res]v$ ,,copyright1 #aiad1 ,,by ,,rob]t ,,ed\>d ,,moritz ,,preface ,e "o kn[s t ! f9e phrase 8,god geometrizes0 is attribut$ to ,plato1 b few "k :]e ?is fam\s passage is f\nd1 or ! exact words 9 :i* x 0 fir/ express$4 ,^? :o1 l ! au?or1 h sp5t h\rs & ev5 days 9 ! se>* ( ! exact /atem5ts1 or ! exact ref]5ces1 ( simil> fam\s passages1 w n "q ! timel9ess & usefulness ( a book ^: di/9ct purpose x is to br+ toge!r into a s+le volume exact quotations1 ) _! exact ref]5ces1 be>+ on "o ( ! mo/ "t-honor$1 & ev5 today ! mo/ active & mo/ fruitful ( all ! sci5ces1 ! que5-"m ( all ! sci5ces1 t is1 ma!matics4 ,x is hop$ t ! pres5t volume w prove indisp5sable to e tea*] ( ma!matics1 to e writ] on ma!matics1 & t ! /ud5t ( ma!matics & ! relat$ sci5ces w f9d its p]usal n only a s\rce ( pleasure b ( enc\ragem5t & inspiration z well4 ,! layman w f9d x a repository ( useful in=mation cov]+ a field ( k :i*1 ow+ to ! unfamili> & h5ce repellant "* ( ! language employ$ by ma!maticians1 is peculi>ly inaccessible to ! g5]al read]4 ,no te*nical processes or te*nical facility is requir$ to und]/& & appreciate ! weal? ( ideas "h set =? 9 ! words ( ! _w's great ?9k]s4 ,no labor has be5 sp>$ to make ! pres5t volume wor?y ( a place among collections ( a l k9d 9 o!r fields4 ,t5 ye>s h be5 devot$ to its prep>ation1 ye>s1 :i* if !y c\ld h be5 m pr(itably1 c\ld sc>cely h be5 m pleasurably employ$4 ,z a result !re h be5 br\ght toge!r ov] "o ?\s& m or less famili> passages p]ta9+ to ma!matics1 by poets1 philosoph]s1 hi/orians1 /atesm51 sci5ti/s1 & ma!maticians4 ,^! h be5 ga!r$ f ov] ?ree hundr$ au?ors1 & h be5 gr\p$ "u tw5ty heads1 & cross index$ "u ne>ly sev5 hundr$ topics4 ,! au?or's orig9al plan 0 to give =eign quotations bo? 9 ! orig9al & 9 translation1 b ) ! gr[? ( mat]ial ?is plan 0 ab&on$ z infeasible4 ,x 0 ?\ght to s]ve ! be/ int]e/ ( ! great] numb] ( ,engli% read]s to give translations only1 :ile pres]v+ ! ref]5ces to ! orig9al s\rces1 s t ! /ud5t or critical read] may readily consult ! orig9al ( any giv5 extract4 ,9 cases :]e ! translation is borr[$ ! translator's "n is ins]t$ 9 brackets imm$iately aft] ! au?or's "n4 ,brackets are also us$ to indicate ins]t$ words or phrases made necess>y to br+ out ! context4 ,! abs5ce ( simil> ,engli% works has made ! au?or's "w l>gely t ( ! pione]4 ,rebire's 8,ma?matiques et ,ma?matici5s0 & ,ahr5s' 8,s*]z und ,ern/ 9 d] ,ma!matik0 h inde$ be5 frequ5tly consult$ b r ) a view to avoid ov]lapp+ ?an to receive aid4 ,?us c]ta9 topics z ! correspond5ce ( ,g]man & ,fr5* ma!maticians1 s excell5tly treat$ by ,ahr5s1 h purposely be5 omitt$4 ,! repetitions are limit$ to a small numb] ( fam\s utt]ances ^: abs5ce f a "w ( ?is k9d c\ld sc>cely be def5d$ on any gr\nds4 ,no "o c be m ke5ly aw>e ( ! %ortcom+s ( a "w ?an its au?or1 = none c h s intimate an acqua9tance ) x4 ,among ^? ( ! pres5t "w is its incomplet5ess1 b x %\ld be borne 9 m9d t incomplet5ess is a necess>y concomitant ( e collection ( :atev] k9d4 ,mu* less c complet5ess be expect$ 9 a fir/ collection1 made by a s+le individual1 9 8 leisure h\rs1 & 9 a field :i* is already b\ndless & is yet exp&+ day by day4 ,a collection ( great ?\ghts1 ev5 if complete today1 w\ld be incomplete tomorr[4 ,aga91 if "s au?ors are quot$ m frequ5tly ?an o!rs ( great] fame & au?ority1 ! reason may be s\ght n only 9 ! fact t ! writ+s ( "s au?ors peculi>ly l5t !mselves to quotation1 a quality s+ul>ly abs5t 9 o!r writ]s ( ! greate/ m]it & au?ority1 b also 9 ?is1 t ! greate/ fre$om has be5 ex]cis$ 9 ! *oice ( selections4 ,! au?or has foll[$ ! b5t ( 8 own fancy 9 collect+ :atev] seem$ to him suffici5tly valuable because ( its cont5t1 its beauty1 its orig9ality1 or its t]s5ess1 to des]ve a place 9 a 8,memorabilia40 ,great pa9s has be5 tak5 to furni% exact read+s & ref]5ces4 ,9 "s cases :]e a passage c\ld n be trac$ to its fir/ s\rce1 ! second>y s\rce has be5 giv5 r ?an ! reput$ s\rce4 ,= ! same reason _m ref]5ces are to lat] editions r ?an to inaccessible fir/ editions4 ,! au?or feels confid5t t ?is "w w be ( assi/ance to 8 co-work]s 9 ! field ( ma!matics & alli$ fields4 ,if 9 addition x %\ld aid 9 a bett] appreciation ( ma!maticians & _! "w on ! "p ( laym5 & /ud5ts 9 o!r fields1 ! au?or's =emo/ aim 9 ! prep>ation ( ?is "w w h be5 a*iev$4 ,,rob]t ,,ed\>d ,,moritz1 ,septemb]1 #aiac4 ,,cont5ts ,,*apt] ,,page ,i4 ,,def9itions ,,& ,,object ,,( ,,ma!matics #a ,,ii4 ,,! ,,nature ,,( ,,ma!matics #aj ,,iii4 ,,e/imates ,,( ,,ma!matics #ci ,,iv4 ,,! ,,value ,,( ,,ma!matics #di ,v4 ,,! ,,tea*+ ,,( ,,ma!matics #gb ,,vi4 ,,/udy ,,& ,,rese>* ,,9 ,,ma!matics #hf ,,vii4 ,,mod]n ,,ma!matics #ajh ,,viii4 ,,! ,,ma!matician #aba ,,ix4 ,,p]sons ,,& ,,anecdotes 7,a-,m7 #ace ,x4 ,,p]sons ,,& ,,anecdotes 7,n-,z7 #aff ,,xi4 ,,ma!matics ,,z ,a ,,f9e ,,art #aha ,,xii4 ,,ma!matics ,,z ,a ,,language #aid ,,xiii4 ,,ma!matics ,,& ,,logic #bja ,,xiv4 ,,ma!matics ,,& ,,philosophy #bji ,,xv4 ,,ma!matics ,,& ,,sci5ce #bbd ,,xvi4 ,,ari?metic #bfa ,,xvii4 ,,algebra #bge ,,xviii4 ,,geometry #bib ,,xix4 ,,! ,,calculus ,,& ,,alli$ ,,topics #cbc ,,xx4 ,,! ,,fundam5tal ,,concepts ,,( ,,"t ,,& ,,space #cde ,,xxi4 ,,p>adoxes ,,& ,,curiosities #cfd ,,index #che ,alles ,ges*eite i/ s*on g$a*t word52 man muss nur v]su*51 es no* e9mal zu d5k54--,,goe!4 ,spr*e 9 ,prosa1 ,e?is*es1 ,i4 #a4 ,a great man quotes bravely1 & w n draw on 8 inv5tion :5 8 memory s]ves him ) a ^w z good4--,,em]son4 ,lett]s & ,social ,aims1 ,quotation & ,orig9ality4 ,,memorabilia ,,ma!matica ,,memorabilia ,,ma!matica ,,*apt] ,i ,,def9itions ,,& ,,object ,,( ,,ma!matics #aja4 ,i ?9k x w\ld be desirable t ?is =m ( ^w ma!matics %\ld be res]v$ = ! applications ( ! sci5ce1 & t we %\ld use ma!matic 9 ! s+ul> to d5ote ! sci5ce itself1 9 ! same way z we speak ( logic1 rhetoric1 or 7own si/] to algebra7 music4--,,sylve/]1 ,j4 ,j4 ,presid5tial ,address to ! ,briti% ,association1 ,exet] ,briti% ,association ,report 7#ahfi72 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b1 p4 #fei4 #ajb4 444 all ! sci5ces :i* h = _! end inve/igations conc]n+ ord] & measure1 are relat$ to ma!matics1 x be+ ( small importance :e!r ?is measure be s\ght 9 numb]s1 =ms1 />s1 s\nds1 or any o!r object2 t1 accord+ly1 !re ought to exi/ a g5]al sci5ce :i* %\ld expla9 all t c be kn[n ab\t ord] & measure1 consid]$ indep5d5tly ( any application to a p>ticul> subject1 & t1 inde$1 ?is sci5ce has its own prop] "n1 consecrat$ by long usage1 to wit1 ma!matics4 ,& a pro( t x f> surpasses 9 facility & importance ! sci5ces :i* dep5d ^u x is t x embraces at once all ! objects to :i* ^! are devot$ & a great _m o!rs besides2 4444 --,,desc>tes4 ,rules = ! ,direction ( ! ,m9d1 ,philosophy ( ,d4 ,torrey 7,new ,york1 #ahib71 p4 #gb4 #ajc4 ,ma!matics has = its object ! indirect measurem5t ( magnitudes1 & x purposes to det]m9e magnitudes by ea* o!r1 accord+ to ! precise relations :i* exi/ betwe5 !m4--,,comte4 ,positive ,philosophy ,m>t9eau1 ,bk4 #a1 *ap4 #a4 #ajd4 ,! bus9ess ( concrete ma!matics is to discov] ! equations :i* express ! ma!matical laws ( ! ph5om5on "u consid]ation2 & ^! equations are ! />t+-po9t ( ! calculus1 :i* mu/ obta9 f !m c]ta9 quantities by means ( o!rs4--,,comte4 ,positive ,philosophy ,m>t9eau1 ,bk4 #a1 *ap4 #b4 #aje4 ,ma!matics is ! sci5ce ( ! connection ( magnitudes4 ,magnitude is any?+ t c be put equal or unequal to ano!r ?+4 ,two ?+s are equal :5 9 e ass]tion ea* may be replac$ by ! o!r4--,,grassmann1 ,,h]mann4 ,/cke aus dem ,lehrbu*e d] ,ari?metik1 ,w]ke 7,leipzig1 #aijd71 ,bd4 #b1 p4 #bih4 #ajf4 ,ma!matic is ei!r ,pure or ,mix$3 ,to ,pure ,ma!matic belong ^? sci5ces :i* h&le ,quantity entirely sev]$ f matt] & f axioms ( natural philosophy4 ,^! are two1 ,geometry & ,ari?metic2 ! "o h&l+ quantity cont9u$1 ! o!r dissev]$4444 ,mix$ ,ma!matic has = its subject "s axioms & p>ts ( natural philosophy1 & consid]s quantity 9 s f> z x assi/s to expla91 demon/rate & actuate ^!4 --,,bacon1 ,,francis4 ,de ,augm5tis1 ,bk4 #c2 ,advancem5t ( ,le>n+1 ,bk4 #b4 #ajg4 ,! ideas :i* ^! sci5ces1 ,geometry1 ,!oretical ,ari?metic & ,algebra involve ext5d to all objects & *anges :i* we obs]ve 9 ! ext]nal _w2 & h5ce ! consid]ation ( ma!matical relations =ms a l>ge portion ( _m ( ! sci5ces :i* treat ( ! ph5om5a & laws ( ext]nal nature1 z ,a/ronomy1 ,optics1 & ,me*anics4 ,su* sci5ces are h5ce (t5 t]m$ ,mix$ ,ma!matics1 ! relations ( space & numb] be+1 9 ^! bran*es ( k1 comb9$ ) pr9ciples collect$ f special obs]vation2 :ile ,geometry1 ,algebra1 & ! l subjects1 :i* involve no result ( exp]i5ce1 are call$ ,pure ,ma!matics4--,,:ewell1 ,,william4 ,! ,philosophy ( ! ,inductive ,sci5ces1 ,"p #a1 ,bk4 #b1 *ap4 ,i1 sect4 #d4 7,london1 #aheh74 #ajh4 ,high] ,ma!matics is ! art ( reason+ ab\t num]ical relations betwe5 natural ph5om5a2 & ! sev]al sections ( ,high] ,ma!matics are diff]5t modes ( view+ ^! relations4--,,mellor1 ,j4 ,w4 ,high] ,ma!matics = ,/ud5ts ( ,*emi/ry & ,physics 7,new ,york1 #aijb71 ,prologue4 #aji4 ,numb]1 place1 & comb9ation 444 ! ?ree int]sect+ b di/9ct sph]es ( ?\ght to :i* all ma!matical ideas admit ( be+ ref]r$4--,,sylve/]1 ,j4 ,j4 ,philosophical ,magaz9e1 ,vol4 #bd 7#ahdd71 p4 #bhe2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #a1 p4 #ia4 #aaj4 ,!re are ?ree rul+ ideas1 ?ree s to say1 sph]es ( ?\ght1 :i* p]vade ! :ole body ( ma!matical sci5ce1 to "s "o or o!r ( :i*1 or to two or all ?ree ( !m comb9$1 e ma!matical tru? admits ( be+ ref]r$2 ^! are ! ?ree c>d9al notions1 ( ,numb]1 ,space & ,ord]4 ,ari?metic has = its object ! prop]ties ( numb] 9 ! ab/ract4 ,9 algebra1 view$ z a sci5ce ( op]ations1 ord] is ! pr$om9at+ idea4 ,! bus9ess ( geometry is ) ! evolution ( ! prop]ties ( space1 or ( bodies view$ z exi/+ 9 space4--,,sylve/]1 ,j4 ,j4 ,a ,probation>y ,lecture on ,geometry1 ,york ,briti% ,association ,report 7#ahdd71 ,"p #b2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b1 p4 #e4 #aaa4 ,! object ( pure ma!matics is ^? relations :i* may be conceptually e/abli%$ among any conceiv$ elem5ts :atsoev] by assum+ !m conta9$ 9 "s ord]$ manifold2 ! law ( ord] ( ?is manifold mu/ be subject to our *oice2 ! latt] is ! case 9 bo? ( ! only conceivable k9ds ( manifolds1 9 ! discrete z well z 9 ! cont9u\s4 --,,papp]itz1 ,e4 b] das ,sy/em d] re9 ma!matis*5 ,wiss5s*aft51 ,jahresb]i*t d] ,deuts*5 ,ma!matik]-,v]e9igung1 ,bd4 #a1 p4 #cf4 #aab4 ,pure ma!matics is n conc]n$ ) magnitude4 ,x is m]ely ! doctr9e ( notation ( relatively ord]$ ?\ght op]ations :i* h become me*anical4--,,novalis4 ,s*rift5 7,b]l91 #aija71 ,zweit] ,teil1 p4 #bhb4 #aac4 ,any conception :i* is def9itely & completely det]m9$ by means ( a f9ite numb] ( specifications1 say by assign+ a f9ite numb] ( elem5ts1 is a ma!matical conception4 ,ma!matics has = its function to develop ! consequ5ces involv$ 9 ! def9ition ( a gr\p ( ma!matical conceptions4 ,int]dep5d5ce & mutual logical consi/5cy among ! memb]s ( ! gr\p are po/ulat$1 o!rwise ! gr\p w\ld ei!r h to be treat$ z sev]al di/9ct gr\ps1 or w\ld lie beyond ! sph]e ( ma!matics4--,,*ry/al1 ,,george4 ,encyclop$ia ,britannica 7#i? edition71 ,article 8,ma!matics40 #aad4 ,! purely =mal sci5ces1 logic & ma!matics1 deal ) ^? relations :i* are1 or c be1 indep5d5t ( ! p>ticul> cont5t or ! sub/ance ( objects4 ,to ma!matics 9 p>ticul> fall ^? relations betwe5 objects :i* involve ! concepts ( magnitude1 ( measure & ( numb]4--,,hankel1 ,,h]mann4 ,!orie d] ,complex5 ,zahl5sy/eme1 7,leipzig1 #ahfg71 p4 #a4 #aae4 ,quantity is t :i* is op]at$ ) accord+ to fix$ mutually consi/5t laws4 ,bo? op]ator & op]& mu/ d]ive _! mean+ f ! laws ( op]ation4 ,9 ! case ( ord9>y algebra ^! are ! ?ree laws already indicat$ ! commutative1 associative1 & di/ributive laws1 9 ! algebra ( quat]nions ! same save ! law ( commutation = multiplication & division1 & s on4 ,x may be que/ion$ :e!r ?is def9ition is suffici5t1 & x may be object$ t x is vague2 b ! read] w d well to reflect t any def9ition mu/ include ! l9e> algebras ( ,peirce1 ! algebra ( logic1 & o!rs t may be easily imag9$1 al?\gh !y h n yet be5 develop$4 ,?is g5]al def9ition ( quantity enables u to see h[ op]ators may be treat$ z quantities1 & ?us to und]/& ! rationale ( ! s call$ symbolical me?ods4--,,*ry/al1 ,,george4 ,encyclop$ia ,britannica 7#i? edition71 ,article 8,ma!matics40 #aaf4 ,ma!matics--9 a /rict s5se--is ! ab/ract sci5ce :i* inve/igates d$uctively ! conclusions implicit 9 ! elem5t>y conceptions ( spatial & num]ical relations4 --,,murray1 ,j4 ,a4 ,h4 ,a ,new ,engli% ,diction>y4 #aag4 ,ev]y?+ t ! greate/ m9ds ( all times h accompli%$ t[>d ! compreh5sion ( =ms by means ( concepts is ga!r$ into "o great sci5ce1 ma!matics4 --,,h]b>t1 ,j4 ,f4 ,pe/alozzi's ,idee e9es ,a ,b ,c d] ,ans*auung1 ,w]ke ,kehrba*1 7,lang5salza1 #ahij71 ,bd4 #a1 p4 #afc4 #aah4 ,p]haps ! lea/ inadequate description ( ! g5]al scope ( mod]n ,pure ,ma!matics--,i w n call x a def9ition--w\ld be to say t x deals ) =m1 9 a v g5]al s5se ( ! t]m2 ?is w\ld include algebraic =m1 functional relation%ip1 ! relations ( ord] 9 any ord]$ set ( entities su* z numb]s1 & ! analysis ( ! peculi>ities ( =m ( gr\ps ( op]ations4--,,hobson1 ,e4 ,w4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#aiaj72 ,nature1 ,vol4 #hd1 p4 #bhg4 #aai4 ,! ideal ( ma!matics %\ld be to erect a calculus to facilitate reason+ 9 connection ) e prov9ce ( ?\ght1 or ( ext]nal exp]i5ce1 9 :i* ! succession ( ?\ghts1 or ( ev5ts c be def9itely asc]ta9$ & precisely /at$4 ,s t all s]i\s ?\ght :i* is n philosophy1 or inductive reason+1 or imag9ative lit]ature1 %all be ma!matics develop$ by means ( a calculus4 --,,:itehead1 ,a4 ,n4 ,univ]sal ,algebra 7,cambridge1 #ahih71 ,preface4 #abj4 ,ma!matics is ! sci5ce :i* draws necess>y conclusions4--,,peirce1 ,,b5jam94 ,l9e> ,associative ,algebra1 ,am]ican ,j\rnal ( ,ma!matics1 ,vol4 #d 7#ahha71 p4 #ig4 #aba4 ,ma!matics is ! univ]sal art apodictic4--,,smi?1 ,w4 ,b4 ,quot$ by ,keys]1 ,c4 ,j4 9 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #ac4 #abb4 ,ma!matics 9 its wide/ signification is ! developm5t ( all types ( =mal1 necess>y1 d$uctive reason+4 --,,:itehead1 ,a4 ,n4 ,univ]sal ,algebra 7,cambridge1 #ahih71 ,preface1 p4 vi4 #abc4 ,ma!matics 9 g5]al is fundam5tally ! sci5ce ( self-evid5t ?+s4--,,kle91 ,,felix4 ,anw5dung d] ,diff]5tial- und ,integralre*nung auf ,geometrie 7,leipzig1 #aijb71 p4 #bf4 #abd4 ,a ma!matical sci5ce is any body ( propositions :i* is capable ( an ab/ract =mulation & arrangem5t 9 su* a way t e proposition ( ! set aft] a c]ta9 "o is a =mal logical consequ5ce ( "s or all ! prec$+ propositions4 ,ma!matics consi/s ( all su* ma!matical sci5ces4--,,"y1 ,,*>les ,,wesley4 ,fundam5tal ,concepts ( ,algebra & ,geometry 7,new ,york1 #aiaa71 p4 #bbb4 #abe4 ,pure ma!matics is a collection ( hypo!tical1 d$uctive !ories1 ea* consi/+ ( a def9ite sy/em ( primitive1 undef9$1 concepts or symbols & primitive1 unprov$1 b self-consi/5t assumptions 7commonly call$ axioms7 toge!r ) _! logically d$ucible consequ5ces foll[+ by rigidly d$uctive processes )\t appeal to intuition4--,,fit*1 ,g4 ,d4 ,! ,f\r? ,dim5sion simply ,expla9$ 7,new ,york1 #aiaj71 p4 #eh4 #abf4 ,! :ole ( ,ma!matics consi/s 9 ! organization ( a s]ies ( aids to ! imag9ation 9 ! process ( reason+4 --,,:itehead1 ,a4 ,n4 ,univ]sal ,algebra 7,cambridge1 #ahih71 p4 #ab4 #abg4 ,pure ma!matics consi/s entirely ( su* assev]ations z t1 if su* & su* a proposition is true ( any?+1 !n su* & su* ano!r proposition is true ( t ?+4 ,x is ess5tial n to discuss :e!r ! fir/ proposition is really true1 & n to m5tion :at ! any?+ is ( :i* x is suppos$ to be true4444 ,if our hypo!sis is ab\t any?+ & n ab\t "s "o or m p>ticul> ?+s1 !n our d$uctions con/itute ma!matics4 ,?us ma!matics may be def9$ z ! subject 9 :i* we nev] "k :at we are talk+ ab\t1 nor :e!r :at we are say+ is true4--,,russell1 ,,b]tr&4 ,rec5t ,"w on ! ,pr9ciples ( ,ma!matics1 ,int]national ,mon?ly1 ,vol4 #d 7#aija71 p4 #hd4 #abh4 ,pure ,ma!matics is ! class ( all propositions ( ! =m 8p implies q10 :]e p & q are propositions conta9+ "o or m v>iables1 ! same 9 ! two propositions1 & nei!r p nor q conta9s any con/ants except logical con/ants4 ,& logical con/ants are all notions def9able 9 t]ms ( ! foll[+3 ,implication1 ! relation ( a t]m to a class ( :i* x is a memb]1 ! notion ( su* t1 ! notion ( relation1 & su* fur!r notions z may be involv$ 9 ! g5]al notion ( propositions ( ! above =m4 ,9 addition to ^!1 ,ma!matics uses a notion :i* is n a con/itu5t ( ! propositions :i* x consid]s--namely1 ! notion ( tru?4--,,russell1 ,,b]tr&4 ,pr9ciples ( ,ma!matics 7,cambridge1 #aijc71 p4 #a4 #abi4 ,! object ( pure ,physic is ! unfold+ ( ! laws ( ! intelligible _w2 ! object ( pure ,ma!matic t ( unfold+ ! laws ( human intellig5ce4--,,sylve/]1 ,j4 ,j4 ,on a !orem1 connect$ ) ,newton's ,rule1 etc41 ,collect$ ,ma!matical ,pap]s1 ,vol4 #c1 p4 #dbd4 #acj4 ,fir/ ( all1 we ought to obs]ve1 t ma!matical propositions1 prop]ly s call$1 are always judgm5ts a priori1 & n empirical1 because !y c>ry along ) !m necessity1 :i* c nev] be d$uc$ f exp]i5ce4 ,if p %\ld object to ?is1 ,i am q will+ to conf9e my /atem5ts to pure ma!matics1 ! v concept ( :i* implies t x does n conta9 empirical1 b only pure k a priori4--,,kant1 ,,immanuel4 ,critique ( ,pure ,reason ,mll]1 7,new ,york1 #aijj71 p4 #gbj4 #aca4 ,ma!matics1 ! sci5ce ( ! ideal1 becomes ! means ( inve/igat+1 und]/&+ & mak+ kn[n ! _w ( ! real4 ,! complex is express$ 9 t]ms ( ! simple4 ,f "o po9t ( view ma!matics may be def9$ z ! sci5ce ( successive sub/itutions ( simpl] concepts = m complex4444 --,,:ite1 ,,william ,f4 ,a ,scrap-book ( ,elem5t>y ,ma!matics1 7,*icago1 #aijh71 p4 #bae4 #acb4 ,! critical ma!matician has ab&on$ ! se>* = tru?4 ,he no long] flatt]s himself t 8 propositions are or c be kn[n to him or to any o!r human be+ to be true2 & he cont5ts himself ) aim+ at ! correct1 or ! consi/5t4 ,! di/9ction is n annull$ nor ev5 blurr$ by ! reflection t consi/5cy conta9s imman5tly a k9d ( tru?4 ,he is n absolutely c]ta91 b he believes pr(\ndly t x is possible to f9d v>i\s sets ( a few propositions ea* su* t ! propositions ( ea* set are compatible1 t ! propositions ( ea* su* set imply o!r propositions1 & t ! latt] c be d$uc$ f ! =m] ) c]ta9ty4 ,t is to say1 he believes t !re are sy/ems ( coh]5t or consi/5t propositions1 & he reg>ds x 8 bus9ess to discov] su* sy/ems4 ,any su* sy/em is a bran* ( ma!matics4 --,,keys]1 ,c4 ,j4 ,sci5ce1 ,new ,s]ies1 ,vol4 #ce1 p4 #ajg4 #acc4 ,ma!matics is ! /udy ( ideal con/ructions 7(t5 applicable to real problems71 & ! discov]y !reby ( relations betwe5 ! p>ts ( ^! con/ructions1 be=e unkn[n4--,,peirce1 ,c4 ,s4 ,c5tury ,diction>y1 ,article 8,ma!matics40 #acd4 ,ma!matics is t =m ( intellig5ce 9 :i* we br+ ! objects ( ! ph5om5al _w "u ! control ( ! conception ( quantity4 ,provisional def9ition4--,,h[ison1 ,g4 ,h4 ,! ,dep>tm5ts ( ,ma!matics1 & _! ,mutual ,relations2 ,j\rnal ( ,speculative ,philosophy1 ,vol4 #e1 p4 #afd4 #ace4 ,ma!matics is ! sci5ce ( ! functional laws & trans=mations :i* enable u to conv]t figur$ ext5sion & rat$ motion into numb]4--,,h[ison1 ,g4 ,h4 ,! ,dep>tm5ts ( ,ma!matics1 & _! ,mutual ,relations2 ,j\rnal ( ,speculative ,philosophy1 ,vol4 #e1 p4 #agj4 ,,*apt] ,,ii ,,! ,,nature ,,( ,,ma!matics #bja4 ,ma!matics1 f ! e>lie/ times to :i* ! hi/ory ( human reason c rea*1 has foll[$1 among t wond]ful p ( ! ,greeks1 ! safe way ( sci5ce4 ,b x mu/ n be suppos$ t x 0 z easy = ma!matics z = logic1 9 :i* reason is conc]n$ ) itself alone1 to f9d1 or r to make = itself t royal road4 ,i believe1 on ! contr>y1 t !re 0 a long p]iod ( t5tative "w 7*iefly /ill among ! ,egyptians71 & t ! *ange is to be ascrib$ to a revolution1 produc$ by ! happy ?\ght ( a s+le man1 ^: exp]im5ts po9t$ unmi/akably to ! pa? t _h to be foll[$1 & op5$ & trac$ out = ! mo/ di/ant times ! safe way ( a sci5ce4 ,! hi/ory ( t intellectual revolution1 :i* 0 f> m important ?an ! passage r\nd ! celebrat$ ,cape ( ,good ,hope1 & ! "n ( its =tunate au?or1 h n be5 pres]v$ to u4444 ,a new light fla%$ on ! fir/ man :o demon/rat$ ! prop]ties ( ! isosceles triangle 7:e!r 8 "n 0 ,?ales or any o!r "n71 = he f\nd t he _h n to inve/igate :at he saw 9 ! figure1 or ! m]e concepts ( t figure1 & ?us to le>n its prop]ties2 b t he _h to produce 7by con/ruction7 :at he _h himself1 accord+ to concepts a priori1 plac$ into t figure & repres5t$ 9 x1 s t1 9 ord] to "k any?+ ) c]ta9ty a priori1 he mu/ n attribute to t figure any?+ beyond :at necess>ily foll[s f :at he has himself plac$ into x1 9 accordance ) ! concept4 --,,kant1 ,,immanuel4 ,critique ( ,pure ,reason1 ,preface to ! ,second ,edition ,mll]1 7,new ,york1 #aijj71 p4 #fij4 #bjb4 ,:5 foll[$ 9 ! prop] _s1 !re is no /udy 9 ! _w :i* br+s into m h>moni\s action all ! faculties ( ! m9d ?an ! "o ma!matics ( :i* ,i /& "h z ! humble repres5tative & advocate4 ,!re is none o!r :i* prep>es s _m agreeable surprises = its foll[]s1 m wond]ful ?an ! trans=mation sc5e ( a pantomime1 or1 l ?is1 seems to raise !m1 by successive /eps ( initiation to high] & high] /ates ( consci\s intellectual be+4--,,sylve/]1 ,j4 ,j4 ,a ,plea = ! ,ma!matician1 ,nature1 ,vol4 #a1 p4 #bfa4 #bjc4 ,?\ght-economy is mo/ highly develop$ 9 ma!matics1 t sci5ce :i* has rea*$ ! highe/ =mal developm5t1 & on :i* natural sci5ce s frequ5tly calls = assi/ance4 ,/range z x may seem1 ! /r5g? ( ma!matics lies 9 ! avoidance ( all unnecess>y ?\ghts1 9 ! utmo/ economy ( ?\ght-op]ations4 ,! symbols ( ord]1 :i* we call numb]s1 =m already a sy/em ( wond]ful simplicity & economy4 ,:5 9 ! multiplication ( a numb] ) sev]al digits we employ ! multiplication table & ?us make use ( previ\sly accompli%$ results r ?an to repeat !m ea* "t1 :5 by ! use ( tables ( log>i?ms we avoid new num]ical calculations by replac+ !m by o!rs long s9ce p]=m$1 :5 we employ det]m9ants in/ead ( c>ry+ ?r\gh f ! beg9n+ ! solution ( a sy/em ( equations1 :5 we decompose new integral expressions into o!rs t are famili>1--we see 9 all ?is b a fa9t reflection ( ! intellectual activity ( a ,lagrange or ,cau*y1 :o ) ! ke5 disc]nm5t ( a milit>y comm&] m>%alls a :ole troop ( complet$ op]ations 9 ! execution ( a new "o4--,,ma*1 ,e4 ,populr-wiss5s*afli*e ,vorlesung5 7#aijh71 pp4 #bbd-bbe4 #bjd4 ,pure ma!matics proves itself a royal sci5ce bo? ?r\gh its cont5t & =m1 :i* conta9s )9 itself ! cause ( its be+ & its me?ods ( pro(4 ,= 9 complete indep5d5ce ma!matics creates = itself ! object ( :i* x treats1 its magnitudes & laws1 its =mulas & symbols4 --,,dillmann1 ,e4 ,die ,ma!matik die ,fackeltrg]9 e9] neu5 ,zeit 7,/uttg>t1 #ahhi71 p4 #id4 #bje4 ,! ess5ce ( ma!matics lies 9 its fre$om4 --,,cantor1 ,,george4 ,ma!matis*e ,annal51 ,bd4 #ba1 p4 #efd4 #bjf4 ,ma!matics pursues its own c\rse unre/ra9$1 n inde$ ) an unbridl$ lic5ce :i* submits to no laws1 b r ) ! fre$om :i* is det]m9$ by its own nature & 9 con=mity ) its own be+4--,,hankel1 ,,h]mann4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 p4 #af4 #bjg4 ,ma!matics is p]fectly free 9 its developm5t & is subject only to ! obvi\s consid]ation1 t its concepts mu/ be free f contradictions 9 !mselves1 z well z def9itely & ord]ly relat$ by means ( def9itions to ! previ\sly exi/+ & e/abli%$ concepts4--,,cantor1 ,,george4 ,grundlag5 e9] allgeme95 ,manigfaltigkeitslehre 7,leipzig1 #ahhc71 ,sect4 #h4 #bjh4 ,ma!maticians assume ! "r to *oose1 )9 ! limits ( logical contradiction1 :at pa? !y please 9 rea*+ _! results4--,,adams1 ,,h5ry4 ,a ,lett] to ,am]ican ,tea*]s ( ,hi/ory 7,wa%+ton1 #aiaj71 ,introduction1 p4 v4 #bji4 ,ma!matics is ! pr$om9ant sci5ce ( our "t2 its conque/s gr[ daily1 ?\gh )\t noise2 he :o does n employ x = himself1 w "s day f9d x employ$ aga9/ himself4--,,h]b>t1 ,j4 ,f4 ,w]ke ,kehrba* 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #aje4 #baj4 ,ma!matics is n ! discov]] ( laws1 = x is n induction2 nei!r is x ! fram] ( !ories1 = x is n hypo!sis2 b x is ! judge ov] bo?1 & x is ! arbit] to :i* ea* mu/ ref] its claims2 & nei!r law c rule nor !ory expla9 )\t ! sanction ( ma!matics4 --,,peirce1 ,,b5jam94 ,l9e> ,associative ,algebra1 ,am]ican ,j\rnal ( ,ma!matics1 ,vol4 #d 7#ahha71 p4 #ig4 #baa4 ,ma!matics is a sci5ce cont9ually exp&+2 & its gr[?1 unlike "s political & indu/rial ev5ts1 is att5d$ by univ]sal acclamation4--,,:ite1 ,h4 ,s4 ,congress ( ,arts & ,sci5ces 7,bo/on & ,new ,york1 #aije71 ,vol4 #a1 p4 #dee4 #bab4 ,ma!matics accompli%es really no?+ outside ( ! realm ( magnitude2 m>vell\s1 h[ev]1 is ! skill ) :i* x ma/]s magnitude :]ev] x f9ds x4 ,we recall at once ! network ( l9es :i* x has spun ab\t heav5s & e>?2 ! sy/em ( l9es to :i* azimu? & altitude1 decl9ation & "r asc5sion1 longitude & latitude are ref]r$2 ^? abscissas & ord9ates1 tang5ts & normals1 circles ( curvature & evolutes2 ^? trigonometric & log>i?mic functions :i* h be5 prep>$ 9 advance & await application4 ,a look at ?is app>atus is suffici5t to %[ t ma!maticians are n magicians1 b t ev]y?+ is accompli%$ by natural means2 "o is r impress$ by ! multitude ( skilful ma*9es1 num]\s witnesses ( a manifold & int5sely active indu/ry1 admirably fitt$ = ! acquisition ( true & la/+ treasures4--,,h]b>t1 ,j4 ,f4 ,w]ke ,kehrba* 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #aja4 #bac4 ,!y ma!maticians only take ^? ?+s into consid]ation1 ( :i* !y h cle> & di/9ct ideas1 designat+ !m by prop]1 adequate1 & inv>iable names1 & premis+ only a few axioms :i* are mo/ not$ & c]ta9 to inve/igate _! affections & draw conclusions f !m1 & agreeably lay+ d[n a v few hypo!ses1 su* z are 9 ! highe/ degree consonant ) reason & n to be d5i$ by anyone 9 8 "r m9d4 ,9 l mann] !y assign g5]ations or causes easy to be und]/ood & readily admitt$ by all1 !y pres]ve a mo/ accurate ord]1 e proposition imm$iately foll[+ f :at is suppos$ & prov$ be=e1 & reject all ?+s h[soev] speci\s & probable :i* c n be inf]r$ & d$uc$ aft] ! same mann]4--,,b>r[1 ,,isaac4 ,ma!matical ,lectures 7,london1 #agcd71 p4 #ff4 #bad4 ,! dext]\s managem5t ( t]ms & be+ able to f5d & prove ) !m1 ,i "k has & does pass 9 ! _w = a great "p ( le>n+2 b x is le>n+ di/9ct f k1 = k consi/s only 9 p]ceiv+ ! habitudes & relations ( ideas "o to ano!r1 :i* is done )\t words2 ! int]v5tion ( s\nds helps no?+ to x4 ,& h5ce we see t !re is lea/ use ( di/9ction :]e !re is mo/ k3 ,i mean 9 ma!matics1 :]e m5 h det]m9$ ideas ) kn[n names to !m2 & s1 !re be+ no room = equivocations1 !re is no ne$ ( di/9ctions4 --,,locke1 ,,john4 ,conduct ( ! ,und]/&+1 ,sect4 #ca4 #bae4 ,9 ma!matics x sophi/ry _h no place f ! beg9n+3 ,ma!maticians hav+ _h ! wisdom to def9e accurately ! t]ms !y use1 & to lay d[n1 z axioms1 ! fir/ pr9ciples on :i* _! reason+ is gr\nd$4 ,accord+ly we f9d no p>ties among ma!maticians1 & h>dly any disputes4 --,,reid1 ,,?omas4 ,essays on ! ,intellectual ,p[]s ( ,man1 ,essay #a1 *ap4 #a4 #baf4 ,9 mo/ sci5ces "o g5]ation te>s d[n :at ano!r has built & :at "o has e/abli%$ ano!r undoes4 ,9 ,ma!matics alone ea* g5]ation builds a new /ory to ! old /ructure4--,,hankel1 ,,h]mann4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 p4 #be4 #bag4 ,ma!matics1 ! prie/ess ( def9it5ess & cle>ness4--,,h]b>t1 ,j4 ,f4 ,w]ke ,kehrba* 7,lang5salza1 #ahij71 ,bd4 #a1 p4 #aga4 #bah4 444 ma!matical analysis is co-ext5sive ) nature itself1 x def9es all p]ceivable relations1 measures times1 spaces1 =ces1 temp]atures2 x is a difficult sci5ce :i* =ms b sl[ly1 b pres]ves c>efully e pr9ciple once acquir$2 x increases & becomes /rong] incessantly amid/ all ! *anges & errors ( ! human m9d4 ,its *ief attribute is cle>ness2 x has no means = express+ confus$ ideas4 ,x comp>es ! mo/ div]se ph5om5a & discov]s ! secret analogies :i* unite !m4 ,if matt] escapes u1 z t ( air & light because ( its extreme t5uity1 if bodies are plac$ f> f u 9 ! imm5sity ( space1 if man wi%es to "k ! aspect ( ! heav5s at successive p]iods sep>at$ by _m c5turies1 if gravity & heat act 9 ! int]ior ( ! solid e>? at dep?s :i* w =ev] be inaccessible1 ma!matical analysis is /ill able to trace ! laws ( ^! ph5om5a4 ,x r5d]s !m pres5t & measurable1 & appe>s to be ! faculty ( ! human m9d de/9$ to supplem5t ! brevity ( life & ! imp]fection ( ! s5ses1 & :at is ev5 m rem>kable1 x foll[s ! same c\rse 9 ! /udy ( all ph5om5a2 x expla9s !m 9 ! same language1 z if 9 witness to ! unity & simplicity ( ! plan ( ! univ]se1 & to make m manife/ ! un*angeable ord] :i* presides ov] all natural causes4 --,,f\ri]1 ,j4 ,?orie ,analytique de la ,*aleur1 ,disc\rs ,prlim9aire4 #bai4 ,let u n[ decl>e ! means :]eby our und]/&+ c rise to k )\t fe> ( error4 ,!re are two su* means3 intuition & d$uction4 ,by intuition ,i mean n ! v>y+ te/imony ( ! s5ses1 nor ! d$uctive judgm5t ( imag9ation naturally extravagant1 b ! conception ( an att5tive m9d s di/9ct & s cle> t no d\bt rema9s to x ) reg>d to t :i* x compreh5ds2 or1 :at am\nts to ! same ?+1 ! self-evid5c+ conception ( a s\nd & att5tive m9d1 a conception :i* spr+s f ! light ( reason alone1 & is m c]ta91 because m simple1 ?an d$uction itself4444 ,x may p]haps be ask$ :y to intuition we add ?is o!r mode ( kn[+1 by d$uction1 t is to say1 ! process :i*1 f some?+ ( :i* we h c]ta9 k1 draws consequ5ces :i* necess>ily foll[ !refrom4 ,b we are oblig$ to admit ?is second /ep2 = !re are a great _m ?+s :i*1 )\t be+ evid5t ( !mselves1 nev]!less be> ! m>ks ( c]ta9ty if only !y are d$uc$ f true & inconte/able pr9ciples by a cont9u\s & un9t]rupt$ movem5t ( ?\ght1 ) di/9ct intuition ( ea* ?+2 j z we "k t ! la/ l9k ( a long *a9 holds to ! fir/1 al?\gh we c n take 9 ) "o glance ( ! eye ! int]m$iate l9ks1 provid$ t1 aft] hav+ run ov] !m 9 succession1 we c recall !m all1 ea* z be+ jo9$ to its fell[s1 f ! fir/ up to ! la/4 ,?us we di/+ui% intuition f d$uction1 inasmu* z 9 ! latt] case !re is conceiv$ a c]ta9 progress or succession1 :ile x is n s 9 ! =m]2444 :5ce x foll[s t prim>y propositions1 d]iv$ imm$iately f pr9ciples1 may be said to be kn[n1 accord+ to ! way we view !m1 n[ by intuition1 n[ by d$uction2 al?\gh ! pr9ciples !mselves c be kn[n only by intuition1 ! remote consequ5ces only by d$uction4 --,,desc>tes4 ,rules = ! ,direction ( ! ,m9d1 ,philosophy ( ,d4 ,torrey 7,new ,york1 #ahib71 pp4 #fd1 #fe4 #bbj4 ,analysis & natural philosophy owe _! mo/ important discov]ies to ?is fruitful means1 :i* is call$ induction4 ,newton 0 indebt$ to x = 8 !orem ( ! b9omial & ! pr9ciple ( univ]sal gravity4--,,laplace4 ,a ,philosophical ,essay on ,probabilities ,truscott & ,emory 7,new ,york #aijb71 p4 #agf4 #bba4 ,!re is 9 e /ep ( an ari?metical or algebraical calculation a real induction1 a real inf]5ce f facts to facts1 & :at disguises ! induction is simply its compreh5sive nature1 & ! consequ5t extreme g5]ality ( its language4 --,,mill1 ,j4 ,s4 ,sy/em ( ,logic1 ,bk4 #b1 *ap4 #f1 #b4 #bbb4 ,x w\ld appe> t ,d$uctive & ,demon/rative ,sci5ces are all1 )\t exception1 ,inductive ,sci5ces3 t _! evid5ce is t ( exp]i5ce1 b t !y are also1 9 virtue ( ! peculi> "* ( "o indisp5sable portion ( ! g5]al =mulae accord+ to :i* _! inductions are made1 ,hypo!tical ,sci5ces4 ,_! conclusions are true only ^u c]ta9 suppositions1 :i* are1 or ought to be1 approximations to ! tru?1 b are seldom1 if ev]1 exactly true2 & to ?is hypo!tical "* is to be ascrib$ ! peculi> c]ta9ty1 :i* is suppos$ to be inh]5t 9 demon/ration4--,,mill1 ,j4 ,s4 ,sy/em ( ,logic1 ,bk4 #b1 *ap4 #f1 #a4 #bbc4 ,! peculi> "* ( ma!matical tru? is1 t x is necess>ily & inevitably true2 & "o ( ! mo/ important lessons :i* we le>n f our ma!matical /udies is a k t !re are su* tru?s1 & a famili>ity ) _! =m & "*4 ,?is lesson is n only lo/1 b read backw>d1 if ! /ud5t is taught t !re is no su* diff]5ce1 & t ma!matical tru?s !mselves are le>n$ by exp]i5ce4--,,:ewell1 ,w4 ,?\ghts on ! ,/udy ( ,ma!matics4 ,pr9ciples ( ,engli% ,univ]sity ,education 7,london1 #ahch74 #bbd4 ,^! sci5ces1 ,geometry1 ,!oretical ,ari?metic & ,algebra1 h no pr9ciples besides def9itions & axioms1 & no process ( pro( b d$uction2 ?is process1 h[ev]1 assum+ a mo/ rem>kable "*2 & exhibit+ a comb9ation ( simplicity & complexity1 ( rig\r & g5]ality1 q unp>allel$ 9 o!r subjects4--,,:ewell1 ,w4 ,! ,philosophy ( ! ,inductive ,sci5ces1 ,"p #a1 ,bk4 #b1 *ap4 #a1 sect4 #b 7,london1 #aheh74 #bbe4 ,! apodictic quality ( ma!matical ?\ght1 ! c]ta9ty & correctness ( its conclusions1 are due1 n to a special mode ( ratioc9ation1 b to ! "* ( ! concepts ) :i* x deals4 ,:at is t di/9ctive *>act]i/ic8 ,i answ]3 precision1 %>pness1 complet5ess1#a ( def9ition4 ,b h[ comes y\r ma!matician by su* complet5ess8 ,!re is no my/]i\s trick involv$2 "s ideas admit ( su* precision1 o!rs d n2 & ! ma!matician is "o :o deals ) ^? t d4--,,keys]1 ,c4 ,j4 ,! ,univ]se & ,beyond2 ,hibb]t ,j\rnal1 ,vol4 #c 7#aijd-aije71 p4 #cji4 #a i4e41 9 t]ms ( ! absolutely cle> & 9def9able4 #bbf4 ,! reason+ ( ma!maticians is f\nd$ on c]ta9 & infallible pr9ciples4 ,e ^w !y use conveys a det]m9ate idea1 & by accurate def9itions !y excite ! same ideas 9 ! m9d ( ! read] t 7 9 ! m9d ( ! writ]4 ,:5 !y h def9$ ! t]ms !y int5d to make use (1 !y premise a few axioms1 or self-evid5t pr9ciples1 t e "o mu/ ass5t to z soon z propos$4 ,!y !n take = grant$ c]ta9 po/ulates1 t no "o c d5y !m1 su* z1 t a "r l9e may be drawn f any giv5 po9t to ano!r1 & f ^! pla91 simple pr9ciples !y h rais$ mo/ a/oni%+ speculations1 & prov$ ! ext5t ( ! human m9d to be m spaci\s & capaci\s ?an any o!r sci5ce4--,,adams1 ,,john4 ,di>y1 ,works 7,bo/on1 #ahej71 ,vol4 #b1 p4 #ba4 #bbg4 ,x may be obs]v$ ( ma!maticians t !y only m$dle ) su* ?+s z are c]ta91 pass+ by ^? t are d\btful & unkn[n4 ,!y pr(ess n to "k all ?+s1 nei!r d !y affect to speak ( all ?+s4 ,:at !y "k to be true1 & c make good by inv9cible argum5ts1 t !y publi% & ins]t among _! !orems4 ,( o!r ?+s !y are sil5t & pass no judgm5t at all1 *oos+ r to ackn[l$ge _! ignorance1 ?an affirm any?+ ra%ly4 ,!y affirm no?+ among _! argum5ts or ass]tions :i* is n mo/ manife/ly kn[n & exam9$ ) utmo/ rig\r1 reject+ all probable conjectures & little witticisms4 ,!y submit no?+ to au?ority1 indulge no affection1 dete/ subt]fuges ( words1 & decl>e _! s5tim5ts1 z 9 a c\rt ( ju/ice1 )\t passion1 )\t apology2 kn[+ t _! reasons1 z ,s5eca te/ifies ( !m1 are n br\ght to p]suade1 b to compel4--,,b>r[1 ,,isaac4 ,ma!matical ,lectures 7,london1 #agcd71 p4 #fd4 #bbh4 ,:at is exact ab\t ma!matics b exactness8 ,& is n ?is a consequ5ce ( ! inn] s5se ( tru?8--,,goe!4 ,spr*e 9 ,prosa1 ,natur1 #f1 #idh4 #bbi4 444 ! ?ree positive *>act]i/ics t di/+ui% ma!matical k f o!r k 444 may be briefly express$ z foll[s3 fir/1 ma!matical k be>s m di/9ctly ! impr9t ( tru? on all its results ?an any o!r k9d ( k2 secondly1 x is always a sure prelim9>y /ep to ! atta9m5t ( o!r correct k2 ?irdly1 x has no ne$ ( o!r k4--,,s*ub]t1 ,h4 ,ma!matical ,essays & ,recreations 7,*icago1 #ahih71 p4 #ce4 #bcj4 ,x is n[ necess>y to indicate m def9itely ! reason :y ma!matics n only c>ries conviction 9 itself1 b also transmits conviction to ! objects to :i* x is appli$4 ,! reason is f\nd1 fir/ ( all1 9 ! p]fect precision ) :i* ! elem5t>y ma!matical concepts are det]m9$2 9 ?is respect ea* sci5ce mu/ look to its own salvation4444 ,b ?is is n all4 ,z soon z human ?\ght attempts long *a9s ( conclusions1 or difficult matt]s g5]ally1 !re arises n only ! dang] ( error b also ! suspicion ( error1 because s9ce all details _c be survey$ ) cle>ness at ! same in/ant "o mu/ 9 ! end be satisfi$ ) a belief t no?+ has be5 ov]look$ f ! beg9n+4 ,e "o kn[s h[ mu* ?is is ! case ev5 9 ari?metic1 ! mo/ elem5t>y use ( ma!matics4 ,no "o w\ld imag9e t ! high] p>ts ( ma!matics f>e bett] 9 ?is respect2 on ! contr>y1 9 m complicat$ conclusions ! unc]ta9ty & suspicion ( hidd5 errors increases 9 rapid progression4 ,h[ does ma!matics manage to rid itself ( ?is inconv5i5ce :i* atta*es to x 9 ! highe/ degree8 ,by mak+ pro(s m rigor\s8 ,by giv+ new rules accord+ to :i* ! old rules %all be appli$8 ,n 9 ! lea/4 ,a v great unc]ta9ty cont9ues to atta* to ! result ( ea* s+le computation4 ,b !re are *ecks4 ,9 ! realm ( ma!matics ea* po9t may be rea*$ by a hundr$ diff]5t ways2 & if ea* ( a hundr$ ways leads to ! same po9t1 "o may be sure t ! "r po9t has be5 rea*$4 ,a calculation )\t a *eck is z good z none4 ,j s x is ) e isolat$ pro( 9 any speculative sci5ce :atev]2 ! pro( may be ev] s +5i\s1 & ev] s p]fectly true & correct1 x w /ill fail to conv9ce p]man5tly4 ,he w !re=e be mu* deceiv$1 :o1 9 metaphysics1 or 9 psy*ology :i* dep5ds on metaphysics1 hopes to see 8 greate/ c>e 9 ! precise det]m9ation ( ! concepts & 9 ! logical conclusions rew>d$ by conviction1 mu* less by success 9 transmitt+ conviction to o!rs4 ,n only mu/ ! conclusions support ea* o!r1 )\t co]cion or suspicion ( subreption1 b 9 all matt]s orig9at+ 9 exp]i5ce1 or judg+ conc]n+ exp]i5ce1 ! results ( speculation mu/ be v]ifi$ by exp]i5ce1 n only sup]ficially1 b 9 c\ntless special cases4--,,h]b>t1 ,j4 ,f4 ,w]ke ,kehrba* 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #aje4 #bca4 ,9 ma!matics we behold ! consci\s logical activity ( ! human m9d 9 its pure/ & mo/ p]fect =m4 ,"h we le>n to realize ! labori\s nature ( ! process1 ! great c>e ) :i* x mu/ proce$1 ! accuracy :i* is necess>y to det]m9e ! exact ext5t ( ! g5]al propositions arriv$ at1 ! difficulty ( =m+ & compreh5d+ ab/ract concepts2 b "h we le>n also to place confid5ce 9 ! c]ta9ty1 scope & fruitfulness ( su* intellectual activity4--,,helmholtz1 ,h4 ,ueb] das ,v]hltniss d] ,naturwiss5s*aft5 zur ,gesamm!it d] ,wiss5s*aft1 ,vortrge und ,r$51 ,bd4 #a 7#ahif71 p4 #agf4 #bcb4 ,x is true t ma!matics1 ow+ to ! fact t its :ole cont5t is built up by means ( purely logical d$uction f a small numb] ( univ]sally compreh5d$ pr9ciples1 has n unfitt+ly be5 designat$ z ! sci5ce ( ! self-evid5t ,selb/v]/ndli*54 ,exp]i5ce h[ev]1 %[s t = ! majority ( ! cultur$1 ev5 ( sci5ti/s1 ma!matics rema9s ! sci5ce ( ! incompreh5sible ,unv]/ndli*54--,,pr+%eim1 ,,alfr$4 ,ueb] ,w]t und angebli*5 ,unw]t d] ,ma!matik1 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung 7#aijd71 p4 #ceg4 #bcc4 ,ma!matical reason+ is d$uctive 9 ! s5se t x is bas$ ^u def9itions :i*1 z f> z ! validity ( ! reason+ is conc]n$ 7ap>t f any exi/5tial import71 ne$s only ! te/ ( self-consi/5cy4 ,?us no ext]nal v]ification ( def9itions is requir$ 9 ma!matics1 z long z x is consid]$ m]ely z ma!matics4--,,:itehead1 ,a4 ,n4 ,univ]sal ,algebra 7,cambridge1 #ahih71 ,preface1 p4 vi4 #bcd4 ,! ma!matician pays n ! lea/ reg>d ei!r to te/imony or conjecture1 b d$uces ev]y?+ by demon/rative reason+1 f 8 def9itions & axioms4 ,inde$1 :atev] is built ^u conjecture1 is improp]ly call$ sci5ce2 = conjecture may beget op9ion1 b _c produce k4 --,,reid1 ,,?omas4 ,essays on ! ,intellectual ,p[]s ( ,man1 ,essay #a1 *ap4 #c4 #bce4 444 = ! sav+ ! long progression ( ! ?\ghts to remote & fir/ pr9ciples 9 e case1 ! m9d %\ld provide itself sev]al /ages2 t is to say1 int]m$iate pr9ciples1 :i* x might h rec\rse to 9 ! exam9+ ^? positions t come 9 its way4 ,^!1 ?\gh !y are n self-evid5t pr9ciples1 yet1 if !y h be5 made out f !m by a w>y & unque/ionable d$uction1 may be dep5d$ on z c]ta9 & infallible tru?s1 & s]ve z unque/ionable tru?s to prove o!r po9ts dep5d+ ^u !m1 by a ne>] & %ort] view ?an remote & g5]al maxims4444 ,& ?us ma!maticians d1 :o d n 9 e new problem run x back to ! fir/ axioms ?r\gh all ! :ole tra9 ( int]m$iate propositions4 ,c]ta9 !orems t !y h settl$ to !mselves ^u sure demon/ration1 s]ve to resolve to !m multitudes ( propositions :i* dep5d on !m1 & are z firmly made out f !nce z if ! m9d w5t afre% ov] e l9k ( ! :ole *a9 t tie !m to fir/ self-evid5t pr9ciples4--,,locke1 ,,john4 ,! ,conduct ( ! ,und]/&+1 ,sect4 #ba4 #bcf4 ,^? int]v5+ ideas1 :i* s]ve to %[ ! agreem5t ( any two o!rs1 are call$ pro(s2 & :]e ! agreem5t or disagreem5t is by ?is means pla9ly & cle>ly p]ceiv$1 x is call$ demon/ration2 x be+ %[n to ! und]/&+1 & ! m9d made to see t x is s4 ,a quickness 9 ! m9d to f9d out ^! int]m$iate ideas1 7t %all discov] ! agreem5t or disagreem5t ( any o!r7 & to apply !m "r1 is1 ,i suppose1 t :i* is call$ sagacity4--,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #f1 *aps4 #b1 #c4 #bcg4 444 ! speculative propositions ( ma!matics d n relate to facts2 444 all t we are conv9c$ ( by any demon/ration 9 ! sci5ce1 is ( a necess>y connection subsi/+ betwe5 c]ta9 suppositions & c]ta9 conclusions4 ,:5 we f9d ^! suppositions actually take place 9 a p>ticul> in/ance1 ! demon/ration =ces u to apply ! conclusion4 ,?us1 if ,i c\ld =m a triangle1 ! ?ree sides ( :i* 7 accurately ma!matical l9es1 ,i might affirm ( ?is individual figure1 t its ?ree angles are equal to two "r angles2 b1 z ! imp]fection ( my s5ses puts x out ( my p[] to be1 9 any case1 c]ta9 ( ! exact correspond5ce ( ! diagram :i* ,i del9eate1 ) ! def9itions giv5 9 ! elem5ts ( geometry1 ,i nev] c apply ) confid5ce to a p>ticul> figure1 a ma!matical !orem4 ,on ! o!r h&1 x appe>s f ! daily te/imony ( our s5ses t ! speculative tru?s ( geometry may be appli$ to mat]ial objects ) a degree ( accuracy suffici5t = ! purposes ( life2 & f su* applications ( !m1 advantages ( ! mo/ important k9d h be5 ga9$ to society4--,,/ew>t1 ,,dugald4 ,elem5ts ( ! ,philosophy ( ! ,human ,m9d1 ,"p #c1 *ap4 #a1 sect4 #c4 #bch4 ,no process ( s\nd reason+ c e/abli% a result n conta9$ 9 ! premises4--,,mellor1 ,j4 ,w4 ,high] ,ma!matics = ,/ud5ts ( ,*emi/ry & ,physics 7,new ,york1 #aijb71 p4 #b4 #bci4 444 we _c get m out ( ! ma!matical mill ?an we put into x1 ?\gh we may get x 9 a =m inf9itely m useful = our purpose4--,,hopk9son1 ,,john4 ,james ,=re/ ,lecture1 #ahid4 #bdj4 ,! iron labor ( consci\s logical reason+ dem&s great p]sev]ance & great caution2 x moves on b sl[ly1 & is r>ely illum9at$ by brilliant fla%es ( g5ius4 ,x kn[s little ( t facility ) :i* ! mo/ v>i$ in/ances come ?rong+ into ! memory ( ! philologi/ or hi/orian4 ,r is x an ess5tial condition ( ! me?odical progress ( ma!matical reason+ t ! m9d %\ld rema9 conc5trat$ on a s+le po9t1 undi/urb$ alike by collat]al ideas on ! "o h&1 & by wi%es & hopes on ! o!r1 & mov+ on /eadily 9 ! direction x has delib]ately *os54 --,,helmholtz1 ,h4 ,ueb] das ,v]hltniss d] ,naturwiss5s*aft5 zur ,gesamm!it d] ,wiss5s*aft1 ,vortrge und ,r$51 ,bd4 #a 7#ahif71 p4 #agh4 #bda4 ,if x 7 always necess>y to r$uce ev]y?+ to intuitive k1 demon/ration w\ld (t5 be insuff]ably prolix4 ,?is is :y ma!maticians h _h ! clev]ness to divide ! difficulties & to demon/rate sep>ately ! int]v5+ propositions4 ,& !re is art also 9 ?is2 = z ! m$iate tru?s 7:i* are call$ lemmas1 s9ce !y appe> to be a digression7 may be assign$ 9 _m ways1 x is well1 9 ord] to aid ! und]/&+ & memory1 to *oose ( !m ^? :i* greatly %ort5 ! process1 & appe> memorable & wor?y 9 !mselves ( be+ demon/rat$4 ,b !re is ano!r ob/acle1 viz43 t x is n easy to demon/rate all ! axioms1 & to r$uce demon/rations :olly to intuitive k4 ,& if we _h *os5 to wait = t1 p]haps we %\ld n yet h ! sci5ce ( geometry4--,,leibnitz1 ,g4 ,w4 ,new ,essay on ,human ,und]/&+ ,langley1 ,bk4 #d1 *aps4 #b1 #h4 #bdb4 ,9 ,pure ,ma!matics1 :]e all ! v>i\s tru?s are necess>ily connect$ ) ea* o!r1 7be+ all necess>ily connect$ ) ^? hypo!ses :i* are ! pr9ciples ( ! sci5ce71 an arrangem5t is beautiful 9 proportion z ! pr9ciples are few2 & :at we admire p]haps *iefly 9 ! sci5ce1 is ! a/oni%+ v>iety ( consequ5ces :i* may be demon/rably d$uc$ f s small a numb] ( premises4 --,,/ew>t1 ,,dugald4 ,! ,elem5ts ( ! ,philosophy ( ! ,human ,m9d1 ,"p #c1 *ap4 #a1 sect4 #c4 #bdc4 ,:5ev] 444 a controv]sy arises 9 ma!matics1 ! issue is n :e!r a ?+ is true or n1 b :e!r ! pro( might n be conduct$ m simply 9 "s o!r way1 or :e!r ! proposition demon/rat$ is suffici5tly important = ! advancem5t ( ! sci5ce z to des]ve especial enunciation & emphasis1 or f9ally1 :e!r ! proposition is n a special case ( "s o!r & m g5]al tru? :i* is z easily discov]$4--,,s*ub]t1 ,h4 ,ma!matical ,essays & ,recreations 7,*icago1 #ahih71 p4 #bh4 #bdd4 444 j z ! a/ronom]1 ! physici/1 ! geologi/1 or o!r /ud5t ( objective sci5ce looks ab\t 9 ! _w ( s5se1 s1 n metaphorically speak+ b lit]ally1 ! m9d ( ! ma!matician goes =? 9 ! univ]se ( logic 9 que/ ( ! ?+s t are !re2 explor+ ! heights & dep?s = facts--ideas1 classes1 relation%ips1 implications1 & ! re/2 obs]v+ ! m9ute & elusive ) ! p[]ful microscope ( 8 ,inf9itesimal ,analysis2 obs]v+ ! elusive & va/ ) ! limitless telescope ( 8 ,calculus ( ! ,inf9ite2 mak+ guesses reg>d+ ! ord] & int]nal h>mony ( ! data obs]v$ & collocat$2 te/+ ! hypo!ses1 n m]ely by ! complete induction peculi> to ma!matics1 b1 l 8 colleagues ( ! out] _w1 resort+ also to exp]im5tal te/s & incomplete induction2 frequ5tly f9d+ x necess>y1 9 view ( un=ese5 disclosures1 to ab&on "o hopeful hypo!sis or to trans=m x by retr5*m5t or by enl>gem5t3--?us1 9 8 own doma91 mat*+1 po9t = po9t1 ! processes1 me?ods & exp]i5ce famili> to ! devotee ( natural sci5ce4 --,,keys]1 ,,cassius ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #bf4 #bde4 ,t ma!matics 8d n cultivate ! p[] ( g5]alization10 444 w be admitt$ by no p]son ( compet5t k1 except 9 a v qualifi$ s5se4 ,! g5]alizations ( ma!matics1 are1 no d\bt1 a diff]5t ?+ f ! g5]alizations ( physical sci5ce2 b 9 ! difficulty ( seiz+ !m1 & ! m5tal t5sion !y require1 !y are no contemptible prep>ation = ! mo/ ardu\s ef=ts ( ! sci5tific m9d4 ,ev5 ! fundam5tal notions ( ! high] ma!matics1 f ^? ( ! diff]5tial calculus upw>ds are products ( a v high ab/raction4444 ,to p]ceive ! ma!matical laws common to ! results ( _m ma!matical op]ations1 ev5 9 s simple a case z t ( ! b9omial !orem1 involves a vigor\s ex]cise ( ! same faculty :i* gave u ,kepl]'s laws1 & rose ?r\gh ^? laws to ! !ory ( univ]sal gravitation4 ,e process ( :at has be5 call$ ,univ]sal ,geometry--! great creation ( ,desc>tes & 8 successors1 9 :i* a s+le tra9 ( reason+ solves :ole classes ( problems at once1 & o!rs common to l>ge gr\ps ( !m--is a practical lesson 9 ! managem5t ( wide g5]alizations1 & ab/raction ( ! po9ts ( agreem5t f ^? ( diff]5ce among objects ( great & confus+ div]sity1 to :i* ! purely inductive sci5ces _c furni% _m sup]ior4 ,ev5 s elem5t>y an op]ation z t ( ab/ract+ f ! p>ticul> configuration ( ! triangles or o!r figures1 & ! relative situation ( ! p>ticul> l9es or po9ts1 9 ! diagram :i* aids ! appreh5sion ( a common geometrical demon/ration1 is a v useful1 & f> f be+ always an easy1 ex]cise ( ! faculty ( g5]alization s /rangely imag9$ to h no place or "p 9 ! processes ( ma!matics4--,,mill1 ,,john ,,/u>t4 ,an ,exam9ation ( ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 pp4 #fab1 #fac4 #bdf4 ,:5 ! greate/ ( ,am]ican logicians1 speak+ ( ! p[]s t con/itute ! born geometrician1 _h nam$ ,conception1 ,imag9ation1 & ,g5]alization1 he paus$4 ,!reupon f "o ( ! audi5ce !re came ! *all5ge1 8,:at ( reason80 ,! in/ant response1 n less j ?an brilliant1 03 8,ratioc9ation--t is b ! smoo? pavem5t on :i* ! *>iot rolls40--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #ca4 #bdg4 444 ! reason+ process employ$ 9 ma!matics is n diff]5t f t ( any o!r bran* ( k1 444 b !re is requir$1 & 9 a great degree1 t att5tion ( m9d :i* is 9 "s "p necess>y = ! acquisition ( all k1 & 9 ?is bran* is indisp5sably necess>y4 ,?is mu/ be giv5 9 its fulle/ int5sity2 444 ! o!r elem5ts especially *>act]i/ic ( a ma!matical m9d are quickness 9 p]ceiv+ logical sequ5ce1 love ( ord]1 me?odical arrangem5t & h>mony1 di/9ctness ( conception4--,,price1 ,b4 ,treatise on ,inf9itesimal ,calculus 7,ox=d1 #ahfh71 ,vol4 #c1 p4 #f4 #bdh4 ,hi/ories make m5 wise2 poets1 witty2 ! ma!matics1 subtile2 natural philosophy1 deep2 moral1 grave2 logic & rhetoric1 able to cont5d4--,,bacon1 ,,francis4 ,essays1 ,( ,/udies4 #bdi4 ,! ,ma!matician deals ) two prop]ties ( objects only1 numb] & ext5sion1 & all ! inductions he wants h be5 =m$ & f9i%$ ages ago4 ,he is n[ occupi$ ) no?+ b d$uction & v]ification4--,,huxley1 ,t4 ,h4 ,on ! ,educational ,value ( ! ,natural ,hi/ory ,sci5ces2 ,lay ,s]mons1 ,addresses & ,reviews2 7,new ,york1 #ahgb71 p4 #hg4 #bej4 ,ma!matics is t subject :i* kn[s no?+ ( obs]vation1 no?+ ( exp]im5t1 no?+ ( induction1 no?+ ( causation4--,,huxley1 ,t4 ,h4 ,! ,sci5tific ,aspects ( ,positivism1 ,=tnightly ,review 7#ahih72 ,lay ,s]mons1 ,addresses & ,reviews1 7,new ,york1 #ahgb71 p4 #afi4 #bea4 ,we are told t 8,ma!matics is t /udy :i* kn[s no?+ ( obs]vation1 no?+ ( exp]im5t1 no?+ ( induction1 no?+ ( causation40 ,i ?9k no /atem5t c\ld h be5 made m opposite to ! facts ( ! case2 t ma!matical analysis is con/antly invok+ ! aid ( new pr9ciples1 new ideas1 & new me?ods1 n capable ( be+ def9$ by any =m ( words1 b spr++ direct f ! inh]5t p[]s & activities ( ! human m9d1 & f cont9ually r5ew$ introspection ( t inn] _w ( ?\ght ( :i* ! ph5om5a are z v>i$ & require z close att5tion to disc]n z ^? ( ! out] physical _w 7to :i* ! inn] "o 9 ea* individual man may1 ,i ?9k1 be conceiv$ to /& some:at 9 ! same relation ( correspond5ce z a %ad[ to ! object f :i* x is project$1 or z ! holl[ palm ( "o h& to ! clos$ fi/ :i* x grasps ( ! o!r71 t x is unceas+ly call+ =? ! faculties ( obs]vation & comp>ison1 t "o ( its pr9cipal weapons is induction1 t x has frequ5t rec\rse to exp]im5tal trial & v]ification1 & t x af=ds a b\ndless scope = ! ex]cise ( ! highe/ ef=ts ( ! imag9ation & inv5tion4 --,,sylve/]1 ,j4 ,j4 ,presid5tial ,address to ,briti% ,association1 ,exet] ,briti% ,association ,report 7#ahfi71 pp4 #a-i42 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b1 p4 #fed4 #beb4 ,! actual evolution ( ma!matical !ories proce$s by a process ( induction /rictly analog\s to ! me?od ( induction employ$ 9 build+ up ! physical sci5ces2 obs]vation1 comp>ison1 classification1 trial1 & g5]alisation are ess5tial 9 bo? cases4 ,n only are special results1 obta9$ indep5d5tly ( "o ano!r1 frequ5tly se5 to be really includ$ 9 "s g5]alisation1 b bran*es ( ! subject :i* h be5 develop$ q indep5d5tly ( "o ano!r are sometimes f\nd to h connections :i* enable !m to be syn!sis$ 9 "o s+le body ( doctr9e4 ,! ess5tial nature ( ma!matical ?\ght manife/s itself 9 ! disc]nm5t ( fundam5tal id5tity 9 ! ma!matical aspects ( :at are sup]ficially v diff]5t doma9s4 ,a /rik+ example ( ?is species ( imman5t id5tity ( ma!matical =m 0 exhibit$ by ! discov]y ( t di/+ui%$ ma!matician 444 ,major ,mac,mahon1 t all possible ,lat9 squ>es are capable ( enum]ation by ! consid]ation ( c]ta9 diff]5tial op]ators4 ,"h we h a case 9 :i* an enum]ation1 :i* appe>s to be n am5able to direct treatm5t1 c actually be c>ri$ out 9 a simple mann] :5 ! und]ly+ id5tity ( ! op]ation is recognis$ ) t involv$ 9 c]ta9 op]ations due to diff]5tial op]ators1 ! calculus ( :i* belongs sup]ficially to a :olly diff]5t region ( ?\ght f t relat+ to ,lat9 squ>es4 --,,hobson1 ,e4 ,w4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#aiaj72 ,nature1 ,vol4 #hd1 p4 #bij4 #bec4 ,x has be5 ass]t$ 444 t ! p[] ( obs]vation is n develop$ by ma!matical /udies2 :ile ! tru? is1 t2 f ! mo/ elem5t>y ma!matical notion t arises 9 ! m9d ( a *ild to ! f>!/ v]ge to :i* ma!matical inve/igation has be5 pu%$ & appli$1 ?is p[] is 9 con/ant ex]cise4 ,by obs]vation1 z "h us$1 c only be meant ! fix+ ( ! att5tion ^u objects 7physical or m5tal7 s z to note di/9ctive peculi>ities--to recognize resemblances1 diff]5ces1 & o!r relations4 ,n[ ! fir/ m5tal act ( ! *ild recogniz+ ! di/9ction betwe5 "o & m ?an "o1 betwe5 "o & two1 two & ?ree1 etc41 is exactly ?is4 ,s1 aga91 ! fir/ geometrical notions are z pure an ex]cise ( ?is p[] z c be giv54 ,to "k a /raight l9e1 to di/+ui% x f a curve2 to recognize a triangle & di/+ui% ! sev]al =ms--:at are ^!1 & all p]ception ( =m1 b a s]ies ( obs]vations8 ,nor is x alone 9 secur+ ^! fundam5tal conceptions ( numb] & =m t obs]vation plays s important a "p4 ,! v g5ius ( ! common geometry z a me?od ( reason+--a sy/em ( inve/igation--is1 t x is b a s]ies ( obs]vations4 ,! figure be+ be=e ! eye 9 actual repres5tation1 or be=e ! m9d 9 conception1 is s closely scrut9iz$1 t all its di/9ctive features are p]ceiv$2 auxili>y l9es are drawn 7! imag9ation lead+ 9 ?is71 & a new s]ies ( inspections is made2 & ?us1 by means ( direct1 simple obs]vations1 ! inve/igation proce$s4 ,s *>act]i/ic ( common geometry is ?is me?od ( inve/igation1 t ,comte1 p]haps ! able/ ( all writ]s ^u ! philosophy ( ma!matics1 is dispos$ to class geometry1 z to its me?od1 ) ! natural sci5ces1 be+ bas$ ^u obs]vation4 ,moreov]1 :5 we consid] appli$ ma!matics1 we ne$ only to notice t ! ex]cise ( ?is faculty is s ess5tial1 t ! basis ( all su* reason+1 ! v mat]ial ) :i* we build1 h receiv$ ! "n obs]vations4 ,?us we might proce$ to consid] ! :ole range ( ! human faculties1 & f9d = ! mo/ ( !m ample scope = ex]cise 9 ma!matical /udies4 ,c]ta9ly1 ! memory w n be f\nd to be neglect$4 ,! v fir/ /eps 9 numb]--c\nt+1 ! multiplication table1 etc41 make heavy dem&s on ?is p[]2 :ile ! high] bran*es require ! memoriz+ ( =mulas :i* are simply appall+ to ! un9itiat$4 ,s ! imag9ation1 ! creative faculty ( ! m9d1 has con/ant ex]cise 9 all orig9al ma!matical inve/igations1 f ! solution ( ! simple/ problems to ! discov]y ( ! mo/ recondite pr9ciple2 = x is n by sure1 consecutive /eps1 z _m suppose1 t we advance f ! kn[n to ! unkn[n4 ,! imag9ation1 n ! logical faculty1 leads 9 ?is advance4 ,9 fact1 practical obs]vation is (t5 9 advance ( logical exposition4 ,?us1 9 ! discov]y ( tru?1 ! imag9ation habitually pres5ts hypo!ses1 & obs]vation supplies facts1 :i* x may require ages = ! t>dy reason to connect logically ) ! kn[n4 ,( ?is tru?1 ma!matics1 z well z all o!r sci5ces1 af=ds abundant illu/rations4 ,s rem>kably true is ?is1 t today x is s]i\sly que/ion$ by ! majority ( ?9k]s1 :e!r ! sublime/ bran* ( ma!matics1--! inf9itesimal calculus--has any?+ m ?an an empirical f\ndation1 ma!maticians !mselves n be+ agre$ z to its logical basis4 ,t ! imag9ation1 & n ! logical faculty1 leads 9 all orig9al inve/igation1 no "o :o has ev] succe$$ 9 produc+ an orig9al demon/ration ( "o ( ! simpl] propositions ( geometry1 c h any d\bt4 ,nor are induction1 analogy1 ! scrut9ization ( premises or ! se>* = !m1 or ! balanc+ ( probabilities1 sph]es ( m5tal op]ations =eign to ma!matics4 ,no "o1 inde$1 c claim pre-em95ce = ma!matical /udies 9 all ^! dep>tm5ts ( intellectual culture1 b x may1 p]haps1 be claim$ t sc>cely any dep>tm5t ( sci5ce af=ds discipl9e to s great a numb] ( faculties1 & t none pres5ts s complete a gradation 9 ! ex]cise ( ^! faculties1 f ! fir/ pr9ciples ( ! sci5ce to ! f>!/ ext5t ( its applications1 z ma!matics4--,,olney1 ,,edw>d4 ,kiddle & ,s*em's ,encyclop$ia ( ,education1 7,new ,york1 #ahgg71 ,article 8,ma!matics40 #bed4 ,! op9ion appe>s to be ga9+ gr\nd t ?is v g5]al conception ( functionality1 born on ma!matical gr\nd1 is de/9$ to sup]s$e ! n>r[] notion ( causation1 traditional 9 connection ) ! natural sci5ces4 ,z an ab/ract =mulation ( ! idea ( det]m9ation 9 its mo/ g5]al s5se1 ! notion ( functionality includes & transc5ds ! m special notion ( causation z a "o-sid$ det]m9ation ( future ph5om5a by means ( pres5t conditions2 x c be us$ to express ! fact ( ! subsumption "u a g5]al law ( pa/1 pres5t1 & future alike1 9 a sequ5ce ( ph5om5a4 ,f ?is po9t ( view ! rem>k ( ,huxley t ,ma!matics 8kn[s no?+ ( causation0 c\ld only be tak5 to express ! :ole tru?1 if by ! t]m 8causation0 is und]/ood 8effici5t causation40 ,! latt] notion has1 h[ev]1 9 rec5t times be5 to an increas+ ext5t reg>d$ z j z irrelevant 9 ! natural sci5ces z x is 9 ,ma!matics2 ! idea ( ?or\gh-go+ det]m9ancy1 9 accordance ) =mal law1 be+ ?\ght to be alone significant 9 ei!r doma94--,,hobson1 ,e4 ,w4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#aiaj72 ,nature1 ,vol4 #hd1 p4 #bij4 #bee4 ,mo/1 if n all1 ( ! great ideas ( mod]n ma!matics h _h _! orig9 9 obs]vation4 ,take1 = in/ance1 ! ari?metical !ory ( =ms1 ( :i* ! f\ndation 0 laid 9 ! diophant9e !orems ( ,f]mat1 left )\t pro( by _! au?or1 :i* resi/$ all ef=ts ( ! myriad-m9d$ ,eul] to r$uce to demon/ration1 & only yield$ up _! cause ( be+ :5 turn$ ov] 9 ! bl[-pipe flame ( ,gauss's transc5d5t g5ius2 or ! doctr9e ( d\ble p]iodicity1 :i* result$ f ! obs]vation ( ,jacobi ( a purely analytical fact ( trans=mation2 or ,leg5dre's law ( reciprocity2 or ,/urm's !orem ab\t ! roots ( equations1 :i*1 z he in=m$ me ) 8 own lips1 />$ him 9 ! face 9 ! mid/ ( "s me*anical inve/igations connect$ 7if my memory s]ves me "r7 ) ! motion ( comp\nd p5dulums2 or ,huygh5's me?od ( cont9u$ fractions1 *>act]iz$ by ,lagrange z "o ( ! pr9cipal discov]ies ( t great ma!matician1 & to :i* he appe>s to h be5 l$ by ! con/ruction ( 8 ,planet>y ,automaton2 or ! new algebra1 speak+ ( :i* "o ( my pr$ecessors 7,mr4 ,spottiswoode7 has said1 n )\t j reason & au?ority1 f ?is *air1 8t x rea*es out & indissolubly connects itself ea* ye> ) fre% bran*es ( ma!matics1 t ! !ory ( equations has become almo/ new ?r\gh x1 algebraic geometry transfigur$ 9 its light1 t ! calculus ( v>iations1 molecul> physics1 & me*anics0 7he might1 if speak+ at ! pres5t mom5t1 g on to add ! !ory ( ela/icity & ! developm5t ( ! integral calculus7 8h all felt its influ5ce40--,,sylve/]1 ,j4 ,j4 ,a ,plea = ! ,ma!matician1 ,nature1 ,vol4 #a1 p4 #bch2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b1 pp4 #fee1 #fef4 #bef4 ,! ability to imag9e relations is "o ( ! mo/ indisp5sable conditions ( all precise ?9k+4 ,no subject c be nam$1 9 ! inve/igation ( :i* x is n imp]atively ne$$2 b x c be n[h]e else s ?or\ghly acquir$ z 9 ! /udy ( ma!matics4--,,fiske1 ,,john4 ,d>w9ism & o!r ,essays 7,bo/on1 #ahic71 p4 #bif4 #beg4 ,! great sci5ce ma!matics occupies itself at lea/ j z mu* ) ! p[] ( imag9ation z ) ! p[] ( logical conclusion4--,,h]b>t1 ,f4 ,j4 ,pe/alozzi's ,idee e9es ,,abc d] ,ans*auung4 ,w]ke ,kehrba* 7,lang5saltza1 #ahij71 ,bd4 #a1 p4 #agd4 #beh4 ,! mov+ p[] ( ma!matical inv5tion is n reason+ b imag9ation4--,,de ,,morgan1 ,a4 ,quot$ 9 ,graves' ,life ( ,sir ,w4 ,r4 ,hamilton1 ,vol4 #c 7#ahhi71 p4 #bai4 #bei4 ,!re is an a/oni%+ imag9ation1 ev5 9 ! sci5ce ( ma!matics4444 ,we repeat1 !re 0 f> m imag9ation 9 ! head ( ,ar*im$es ?an 9 t ( ,hom]4--,,voltaire4 ,a ,philosophical ,diction>y 7,bo/on1 #ahha71 ,vol4 #c1 p4 #dj4 ,article 8,imag9ation40 #bfj4 ,z ! pr]ogative ( ,natural ,sci5ce is to cultivate a ta/e = obs]vation1 s t ( ,ma!matics is1 almo/ f ! />t+ po9t1 to /imulate ! faculty ( inv5tion4 --,,sylve/]1 ,j4 ,j4 ,a ,plea = ! ,ma!matician1 ,nature1 ,vol4 #a1 p4 #bfa2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b 7,cambridge1 #aijh71 p4 #gag4 #bfa4 ,a m>veil\s newtrality h ^! ?+s ma!maticall1 & also a /range p>ticipation betwe5 ?+s sup]naturall1 immortall1 intellectuall1 simple & indivisible1 & ?+s naturall1 mortall1 s5sible1 compond$ & divisible4--,,dee1 ,,john4 ,euclid 7#aegj71 ,preface4 #bfb4 ,ma!matics /&s =? z t :i* unites1 m$iates betwe5 ,man & ,nature1 inn] & out] _w1 ?\ght & p]ception1 z no o!r subject does4--,,froebel4 ,h]=d translation 7,london1 #ahic71 ,vol4 #a1 p4 #hd4 #bfc4 ,! intr9sic "* ( ma!matical rese>* & k is bas$ ess5tially on ?ree prop]ties3 fir/1 on its cons]vative attitude t[>ds ! old tru?s & discov]ies ( ma!matics2 secondly1 on its progressive mode ( developm5t1 due to ! incessant acquisition ( new k on ! basis ( ! old2 & ?irdly1 on its self-suffici5cy & its consequ5t absolute indep5d5ce4--,,s*ub]t1 ,h4 ,ma!matical ,essays & ,recreations 7,*icago1 #ahih71 p4 #bg4 #bfd4 ,our sci5ce1 9 contra/ ) o!rs1 is n f\nd$ on a s+le p]iod ( human hi/ory1 b has accompani$ ! developm5t ( culture ?r\gh all its /ages4 ,ma!matics is z mu* int]wov5 ) ,greek culture z ) ! mo/ mod]n problems 9 ,eng9e]+4 ,%e n only l5ds a h& to ! progressive natural sci5ces b p>ticipates at ! same "t 9 ! ab/ract inve/igations ( logicians & philosoph]s4 --,,kle91 ,f4 ,kle9 und ,riecke3 ,ueb] angew&te ,ma!matik und ,physik 7#aijj71 p4 #bbh4 #bfe4 ,!re is probably no o!r sci5ce :i* pres5ts su* diff]5t appe>ances to "o :o cultivates x & to "o :o does n1 z ma!matics4 ,to ?is p]son x is anci5t1 v5]able1 & complete2 a body ( dry1 irrefutable1 unambigu\s reason+4 ,to ! ma!matician1 on ! o!r h&1 8 sci5ce is yet 9 ! purple bloom ( vigor\s y\?1 ev]y:]e /ret*+ out aft] ! 8atta9able b unatta9$0 & full ( ! excitem5t ( nasc5t ?\ghts2 its logic is beset ) ambiguities1 & its analytic processes1 l ,bunyan's road1 h a quagmire on "o side & a deep dit* on ! o!r & bran* (f into innum]able by-pa?s t end 9 a wild]ness4 --,,*apman1 ,c4 ,h4 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #b 7,fir/ s]ies71 p4 #fa4 #bff4 ,ma!matical sci5ce is 9 my op9ion an indivisible :ole1 an organism ^: vitality is condition$ ^u ! connection ( its p>ts4 ,= ) all ! v>iety ( ma!matical k1 we are /ill cle>ly consci\s ( ! simil>ity ( ! logical devices1 ! relation%ip ( ! ideas 9 ma!matics z a :ole & ! num]\s analogies 9 its diff]5t dep>tm5ts4 ,we also notice t1 ! f>!r a ma!matical !ory is develop$1 ! m h>moni\sly & uni=mly does its con/ruction proce$1 & unsuspect$ relations are disclos$ betwe5 hi!rto sep>at$ bran*es ( ! sci5ce4 ,s x happ5s t1 ) ! ext5sion ( ma!matics1 its organic "* is n lo/ b manife/s itself ! m cle>ly4--,,hilb]t1 ,d4 ,ma!matical ,problems1 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #h1 p4 #dgh4 #bfg4 ,! ma!matics h always be5 ! implacable enemies ( sci5tific romances4--,,arago4 ,oeuvres 7#ahee71 t4 #c1 p4 #dih4 #bfh4 ,^? skill$ 9 ma!matical analysis "k t its object is n simply to calculate numb]s1 b t x is also employ$ to f9d ! relations betwe5 magnitudes :i* _c be express$ 9 numb]s & betwe5 functions ^: law is n capable ( algebraic expression4--,,c\rnot1 ,,augu/94 ,ma!matical ,!ory ( ! ,pr9ciples ( ,weal? ,bacon1 ,n4 ,t41 7,new ,york1 #ahig71 p4 #c4 #bfi4 ,cot]m9\s ) space & coeval ) "t is ! ,k+dom ( ,ma!matics2 )9 ?is range h] dom9ion is supreme2 o!rwise ?an accord+ to h] ord] no?+ c exi/2 9 contradiction to h] laws no?+ takes place4 ,on h] my/]i\s scroll is to be f\nd writt5 = ^? :o c read x t :i* has be51 t :i* is1 & t :i* is to come4 ,ev]y?+ mat]ial :i* is ! subject ( k has numb]1 ord]1 or position2 & ^! are h] fir/ outl9es = a sket* ( ! univ]se4 ,if our feeble h&s _c foll[ out ! details1 /ill h] "p has be5 drawn ) an un]r+ p51 & h] "w _c be ga9said4 ,s wide is ! range ( ma!matical sci5ces1 s indef9itely may x ext5d beyond our actual p[]s ( manipulation t at "s mom5ts we are incl9$ to fall d[n ) ev5 m ?an rev]5ce be=e h] maje/ic pres5ce4 ,b s /rictly limit$ are h] promises & p[]s1 ab\t s mu* t we might wi% to "k does %e (f] no in=mation :atev]1 t at o!r mom5ts we are fa9 to call h] results b a va9 ?+1 & to reject !m z a /one :]e we _h ask$ = bread4 ,if "o aspect ( ! subject enc\rages our hopes1 s does ! o!r t5d to *a/5 our desires1 & he is p]haps ! wise/1 & 9 ! long run ! happie/1 among 8 fell[s1 :o has le>n$ n only ?is sci5ce1 b also ! l>g] lesson :i* x directly tea*es1 namely1 to temp] our aspirations to t :i* is possible1 to mod]ate our desires to t :i* is atta9able1 to re/rict our hopes to t ( :i* accompli%m5t1 if n imm$iately practicable1 is at lea/ di/9ctly )9 ! range ( conception4--,,spottiswoode1 ,w4 ,quot$ 9 ,sonn5s*e9's ,encyclop$ia ( ,education 7,london1 #aijf71 p4 #bjh4 #bgj4 ,b x is precisely ma!matics1 & ! pure sci5ce g5]ally1 f :i* ! g5]al educat$ public & indep5d5t /ud5ts h be5 deb>r$1 & into :i* !y h only r>ely atta9$ m ?an a v meagre insight4 ,! reason ( ?is is tw(old4 ,9 ! fir/ place1 ! asc5dant & consecutive "* ( ma!matical k r5d]s its results absolutely insusceptible ( pres5tation to p]sons :o are unacqua9t$ ) :at has gone be=e1 & s necessitates on ! "p ( its devotees a ?or\gh & pati5t exploration ( ! field f ! v beg9n+1 z di/+ui%$ f ^? sci5ces :i* may1 s to speak1 be begun at ! end1 & :i* are consequ5tly cultivat$ ) ! greate/ zeal4 ,! second reason is t1 p>tly ?r\gh ! exig5cies ( academic in/ruction1 b ma9ly ?r\gh ! m>t9et traditions ( antiquity & ! influ5ce ( m$ival logic-mong]s1 ! great bulk ( ! elem5t>y text-books ( ma!matics h unconsci\sly assum$ a v repellant =m1--some?+ simil> to :at is t]m$ 9 ! !ory ( protective mimicry 9 biology 8! t]rify+ =m40 ,& x is ma9ly to ?is =midabl5ess & t\*-me-n "* ( ext]ior1 conceal+ )al a h>mless body1 t ! undue neglect ( typical ma!matical /udies is to be attribut$4--,,mccormack1 ,t4 ,j4 ,preface to ,de ,morgan's ,elem5t>y ,illu/rations ( ! ,diff]5tial & ,integral ,calculus 7,*icago1 #ahii74 #bga4 ,ma!matics 9 gross1 x is pla91 are a grievance 9 natural philosophy1 & ) reason3 = ma!matical pro(s1 l diamonds1 are h>d z well z cle>1 & w be t\*$ ) no?+ b /rict reason+4 ,ma!matical pro(s are out ( ! rea* ( topical argum5ts2 & are n to be attack$ by ! equivocal use ( words or decl>ation1 t make s great a "p ( o!r disc\rses1--nay1 ev5 ( controv]sies4--,,locke1 ,,john4 ,second ,reply to ! ,bi%op ( ,worce/]4 #bgb4 ,! belief t ma!matics1 because x is ab/ract1 because x is /atic & cold & gray1 is deta*$ f life1 is a mi/ak5 belief4 ,ma!matics1 ev5 9 its pure/ & mo/ ab/ract e/ate1 is n deta*$ f life4 ,x is j ! ideal h&l+ ( ! problems ( life1 z sculpture may idealize a human figure or z poetry or pa9t+ may idealize a figure or a sc5e4 ,ma!matics is precisely ! ideal h&l+ ( ! problems ( life1 & ! c5tral ideas ( ! sci5ce1 ! great concepts ab\t :i* its /ately doctr9es h be5 built up1 are precisely ! *ief ideas ) :i* life mu/ always deal & :i*1 z x tumbles & rolls ab\t !m ?r\gh "t & space1 give x its int]e/s & problems1 & its ord] & rationality4 ,t su* is ! case a few indications w suffice to %[4 ,! ma!matical concepts ( con/ant & v>iable are repres5t$ famili>ly 9 life by ! notions ( fix$ness & *ange4 ,! concept ( equation or t ( an equational sy/em1 impos+ re/riction ^u v>iability1 is mat*$ 9 life by ! concept ( natural & spiritual law1 giv+ ord] to :at 7 else *aotic *ange & provid+ p>tial fre$om 9 lieu ( none at all4 ,:at is kn[n 9 ma!matics "u ! "n ( limit is ev]y:]e pres5t 9 life 9 ! guise ( "s ideal1 "s excell5ce high-dwell+ among ! rocks1 an 8ev] fly+ p]fect0 z ,em]son calls x1 unto :i* we may approximate ne>] & ne>]1 b :i* we c nev] q atta91 save 9 aspiration4 ,! supreme concept ( functionality f9ds its correlate 9 life 9 ! all-p]vasive s5se ( int]dep5d5ce & mutual det]m9ation among ! elem5ts ( ! _w4 ,:at is kn[n 9 ma!matics z trans=mation--t is1 lawful transf] ( att5tion1 s]v+ to mat* 9 ord]ly fa%ion ! ?+s ( "o sy/em ) ^? ( ano!r--is conceiv$ 9 life z a process ( transmutation by :i*1 9 ! flux ( ! _w1 ! cont5t ( ! pres5t has come out ( ! pa/ & 9 its turn1 9 ceas+ to be1 gives bir? to its successor1 z ! boy is fa!r to ! man & z ?+s1 9 g5]al1 become :at !y are n4 ,! ma!matical concept ( inv>iance & t ( inf9itude1 especially ! impos+ doctr9es t expla9 _! mean+s & be> _! names--,:at are !y b ma!maticizations ( t :i* has ev] be5 ! *ief ( life's hopes & dreams1 ( t :i* has ev] be5 ! object ( its deepe/ passion & ( its dom9ant ent]prise1 ,i mean ! f9d+ ( ! wor? t abides1 ! f9d+ ( p]man5ce 9 ! mid/ ( *ange1 & ! discov]y ( a pres5ce1 9 :at has seem$ to be a f9ite _w1 ( be+ t is inf9ite8 ,x is ne$less fur!r to multiply examples ( a correlation t is s ab\nd+ & complete z inde$ to sugge/ a d\bt :e!r x be ju/] to view ma!matics z ! ab/ract idealization ( life ?an to reg>d life z ! concrete realization ( ma!matics4--,,keys]1 ,c4 ,j4 ,! ,humanization ( ! ,tea*+ ( ,ma!matics2 ,sci5ce1 ,new ,s]ies1 ,vol4 #ce1 pp4 #fde-fdf4 #bgc4 ,ma!matics1 l dialectics1 is an organ ( ! inn] high] s5se2 9 its execution x is an art l eloqu5ce4 ,bo? alike c>e no?+ = ! cont5t1 to bo? no?+ is ( value b ! =m4 ,x is immat]ial to ma!matics :e!r x computes p5nies or gu9eas1 to rhetoric :e!r x def5ds tru? or error4--,,goe!4 ,wilhelm ,mei/]s ,w&]jahre1 ,zweites ,bu*4 #bgd4 ,! g5u9e _s ( ,ma!sis is dev\t4 ,no intellectual pursuit m truly leads to pr(\nd impressions ( ! exi/5ce & attributes ( a ,creator1 & to a deep s5se ( our filial relations to him1 ?an ! /udy ( ^! ab/ract sci5ces4 ,:o c und]/& s well h[ feeble are our conceptions ( ,almighty ,p[]1 z he :o has calculat$ ! attraction ( ! sun & ! planets1 & weigh$ 9 8 balance ! irresi/ible =ce ( ! lightn+8 ,:o c s well und]/& h[ confus$ is our e/imate ( ! ,et]nal ,wisdom1 z he :o has trac$ out ! secret laws :i* guide ! ho/s ( heav51 & comb9e ! atoms on e>?8 ,:o c s well und]/& t man is made 9 ! image ( 8 ,creator1 z he :o has s\ght to frame new laws & conditions to gov]n imag9>y worlds1 & f\nd 8 own ?\ghts simil> to ^? on :i* 8 ,creator has act$8 --,,hill1 ,,?omas4 ,! ,imag9ation 9 ,ma!matics2 ,nor? ,am]ican ,review1 ,vol4 #he1 p4 #bbf4 #bge4 444 :at is physical is subject to ! laws ( ma!matics1 & :at is spiritual to ! laws ( ,god1 & ! laws ( ma!matics are b ! expression ( ! ?\ghts ( ,god4--,,hill1 ,,?omas4 ,! ,uses ( ,ma!sis2 ,biblio!ca ,sacra1 ,vol4 #cb1 p4 #ebc4 #bgf4 ,x is 9 ! inn] _w ( pure ?\ght1 :]e all entia dwell1 :]e is e type ( ord] & mann] ( correlation & v>iety ( relation%ip1 x is 9 ?is inf9ite ensemble ( et]nal v]ities :5ce1 if !re be "o cosmos or _m ( !m1 ea* d]ives its "* & mode ( be+1--x is !re t ! _s ( ma!sis has its home & its life4 ,is x a re/rict$ home1 a n>r[ life1 /atic & cold & grey ) logic1 )\t arti/ic int]e/1 devoid ( emotion & mood & s5tim5t8 ,t _w1 x is true1 is n a _w ( sol> light1 n clad 9 ! col\rs t liv5 & glorify ! ?+s ( s5se1 b x is an illum9at$ _w1 & ov] x all & ev]y:]e ?r\gh\t are hues & t9ts transc5d+ s5se1 pa9t$ !re by radiant p5cils ( psy*ic light1 ! light 9 :i* x lies4 ,x is a sil5t _w1 &1 nev]!less1 9 respect to ! highe/ pr9ciple ( art--! int]p5etration ( cont5t & =m1 ! p]fect fusion ( mode & mean+--x ev5 surpasses music4 ,9 a s5se1 x is a /atic _w1 b s1 too1 are ! worlds ( ! sculptor & ! ar*itect4 ,! figures1 h[ev]1 :i* reason con/ructs & ! ma!matic vision beholds1 transc5d ! temple & ! /atue1 alike 9 simplicity & 9 intricacy1 9 delicacy & 9 grace1 9 symmetry & 9 poise4 ,n only are ?is home & ?is life ?us ri* 9 /hetic int]e/s1 really controll$ & su/a9$ by motives ( a sublim$ & sup]s5su\s art1 b ! religi\s aspiration1 too1 f9ds !re1 especially 9 ! beautiful doctr9e ( inv>iants1 ! mo/ p]fect symbols ( :at x seeks--! *angeless 9 ! mid/ ( *ange1 abid+ ?+s 9 a _w ( flux1 configurations t rema9 ! same despite ! swirl & /ress ( c\ntless ho/s ( curi\s trans=mations4 ,! doma9 ( ma!matics is ! sole doma9 ( c]ta9ty4 ,!re & !re alone prevail ! /&>ds by :i* e hypo!sis respect+ ! ext]nal univ]se & all obs]vation & all exp]im5t mu/ be f9ally judg$4 ,x is ! realm to :i* all speculation & all ?\ght mu/ repair = *a/5+ & sanitation--! c\rt ( la/ resort1 ,i say x rev]5tly1 = all intellection :atsoev]1 :e!r ( demon or man or deity4 ,x is !re t m9d z m9d atta9s its highe/ e/ate1 & ! condition ( k !re is ! ultimate object1 ! tantalis+ goal ( ! aspiration1 ! ,&]s-,/reb51 ( all o!r k ( e k9d4--,,keys]1 ,c4 ,j4 ,! ,univ]se & ,beyond2 ,hibb]t ,j\rnal1 ,vol4 #c 7#aijd-aije71 pp4 #cac-cad4 ,,*apt] ,,iii ,,e/imates ,,( ,,ma!matics #cja4 ,! _w ( ideas :i* x ma!matics discloses or illum9ates1 ! contemplation ( div9e beauty & ord] :i* x induces1 ! h>moni\s connection ( its p>ts1 ! inf9ite hi]>*y & absolute evid5ce ( ! tru?s ) :i* ma!matical sci5ce is conc]n$1 ^!1 & su* l1 are ! sure/ gr\nds ( its title ( human reg>d1 & w\ld rema9 unimpair$ 7 ! plan ( ! univ]se unroll$ l a map at our feet1 & ! m9d ( man qualifi$ to take 9 ! :ole s*eme ( creation at a glance4--,,sylve/]1 ,j4 ,j4 ,a ,plea = ! ,ma!matician1 ,nature1 #a1 p4 #bfb2 ,collect$ ,ma!matical ,pap]s 7,cambridge1 #aijh71 #b1 p4 #fei4 #cjb4 ,x may well be d\bt$ :e!r1 9 all ! range ( ,sci5ce1 !re is any field s fasc9at+ to ! explor]--s ri* 9 hidd5 treasures--s fruitful 9 delightful surprises--z t ( ,pure ,ma!matics4 ,! *>m lies *iefly 444 9 ! absolute c]ta9ty ( its results3 = t is :at1 beyond all m5tal treasures1 ! human intellect craves =4 ,let u only be sure ( some?+6 ,m light1 m light6 8,& if our fate be dea?1 give light & let u die60 ,?is is ! cry t1 ?r\gh all ! ages1 is go+ up f p]plex$ ,humanity1 & ,sci5ce has little else to (f]1 t w really meet ! dem&s ( its vot>ies1 ?an ! conclusions ( ,pure ,ma!matics4 --,,dodgson1 ,c4 ,l4 ,a ,new ,!ory ( ,p>allels 7,london1 #ahie71 ,introduction4 #cjc4 ,9 e case ! awak5+ t\* has be5 ! ma!matical _s1 ! attempt to c\nt1 to measure1 or to calculate4 ,:at to ! poet or ! se] may appe> to be ! v dea? ( all 8 poetry & all 8 visions--! cold t\* ( ! calculat+ m9d1--?is has prov$ to be ! spell by :i* k has be5 born1 by :i* new sci5ces h be5 creat$1 & hundr$s ( def9ite problems put be=e ! m9ds & into ! h&s ( dilig5t /ud5ts4 ,x is ! geometrical figure1 ! dry algebraical =mula1 :i* trans=ms ! vague reason+ ( ! philosoph] into a tangible & manageable conception2 :i* repres5ts1 ?\gh x does n fully describe1 :i* corresponds to1 ?\gh x does n expla91 ! ?+s & processes ( nature3 ?is clo!s ! fruitful1 b o!rwise indef9ite1 ideas 9 su* a =m t ! /rict logical me?ods ( ?\ght c be appli$1 t ! human m9d c 9 its inn] *amb] evolve a tra9 ( reason+ ! result ( :i* corresponds to ! ph5om5a ( ! out] _w4--,,m]z1 ,j4 ,t4 ,a ,hi/ory ( ,european ,?\ght 9 ! ,n9ete5? ,c5tury 7,ed9burgh & ,london1 #aijd71 ,vol4 #a1 p4 #cad4 #cjd4 ,ma!matics 444 ! ideal & norm ( all c>eful ?9k+4--,,hall1 ,g4 ,,/anley4 ,educational ,problems 7,new ,york1 #aiaa71 p4 #cic4 #cje4 ,ma!matics is ! only true metaphysics4 --,,?omson1 ,w4 7,,"l ,,kelv974 ,?ompson1 ,s4 ,p43 ,life ( ,"l ,kelv9 7,london1 #aiaj71 p4 #aj4 #cjf4 ,he :o kn[s n ma!matics & ! results ( rec5t sci5tific inve/igation dies )\t kn[+ tru?4 --,,s*ellba*1 ,c4 ,h4 ,quot$ 9 ,"y's ,tea*+ ( ,ma!matics 7,london1 #aijg71 p4 #dd4 #cjg4 ,! reason+ ( ma!matics is a type ( p]fect reason+4--,,b>nett1 ,p4 ,a4 ,common ,s5se 9 ,education & ,tea*+ 7,new ,york1 #aije71 p4 #bbb4 #cjh4 ,ma!matics1 once fairly e/abli%$ on ! f\ndation ( a few axioms & def9itions1 z ^u a rock1 has gr[n f age to age1 s z to become ! mo/ solid fabric t human reason c boa/4--,,reid1 ,,?omas4 ,essays on ! ,intellectual ,p[]s ( ,man1 #d?4 ,ed41 p4 #dfa4 #cji4 ,! analytical geometry ( ,desc>tes & ! calculus ( ,newton & ,leibniz h exp&$ into ! m>vel\s ma!matical me?od--m d>+ ?an any?+ t ! hi/ory ( philosophy records--( ,loba*evsky & ,riemann1 ,gauss & ,sylve/]4 ,inde$1 ma!matics1 ! indisp5sable tool ( ! sci5ces1 defy+ ! s5ses to foll[ its spl5did flights1 is demon/rat+ today1 z x nev] has be5 demon/rat$ be=e1 ! supremacy ( ! pure reason4--,,butl]1 ,,ni*olas ,,murray4 ,! ,mean+ ( ,education & o!r ,essays & ,addresses 7,new ,york1 #aije71 p4 #de4 #caj4 ,ma!matics is ! gate & key ( ! sci5ces4444 ,neglect ( ma!matics works injury to all k1 s9ce he :o is ignorant ( x _c "k ! o!r sci5ces or ! ?+s ( ?is _w4 ,& :at is worse1 m5 :o are ?us ignorant are unable to p]ceive _! own ignorance & s d n seek a rem$y4--,,bacon1 ,,rog]4 ,opus ,majus1 ,"p #d1 ,di/9ctia ,prima1 cap4 #a4 #caa4 ,j z x w nev] be successfully *all5g$ t ! ,fr5* language1 progressively develop+ & gr[+ m p]fect day by day1 has ! bett] claim to s]ve z a develop$ c\rt & _w language1 s no "o w v5ture to e/imate lightly ! debt :i* ! _w owes to ma!maticians1 9 t !y treat 9 _! own language matt]s ( ! utmo/ importance1 & gov]n1 det]m9e & decide :atev] is subject1 us+ ! ^w 9 ! highe/ s5se1 to numb] & measurem5t4 --,,goe!4 ,spr*e 9 ,prosa1 ,natur1 ,,iii1 #hfh4 #cab4 ,d n imag9e t ma!matics is h>d & crabb$1 & repulsive to common s5se4 ,x is m]ely ! e!realization ( common s5se4--,,?omson1 ,w4 7,,"l ,,kelv974 ,?ompson1 ,s4 ,p43 ,life ( ,"l ,kelv9 7,london1 #aiaj71 p4 #aaci4 #cac4 ,! advancem5t & p]fection ( ma!matics are intimately connect$ ) ! prosp]ity ( ! ,/ate4--,,napoleon ,i4 ,correspondance de ,napolon1 t4 #bd 7#ahfh71 p4 #aab4 #cad4 ,! love ( ma!matics is daily on ! increase1 n only ) u b 9 ! army4 ,! result ( ?is 0 unmi/akably app>5t 9 our la/ campaigns4 ,bonap>te himself has a ma!matical head1 & ?\gh all :o /udy ?is sci5ce may n become geometricians l ,laplace or ,lagrange1 or h]oes l ,bonap>te1 !re is yet left an influ5ce ^u ! m9d :i* enables !m to accompli% m ?an !y c\ld possibly h a*iev$ )\t ?is tra9+4--,,lal&e4 ,quot$ 9 ,bruhns' ,alex&] von ,humboldt 7#ahgb71 ,bd4 #a1 p4 #bcb4 #cae4 ,9 ,pure ,ma!matics1 :]e all ! v>i\s tru?s are necess>ily connect$ ) ea* o!r1 7be+ all necess>ily connect$ ) ^? hypo!ses :i* are ! pr9ciples ( ! sci5ce71 an arrangem5t is beautiful 9 proportion z ! pr9ciples are few2 & :at we admire p]haps *iefly 9 ! sci5ce1 is ! a/oni%+ v>iety ( consequ5ces :i* may be demon/rably d$uc$ f s small a numb] ( premises4 --,,/ew>t1 ,,dugald4 ,philosophy ( ! ,human ,m9d1 ,"p #c1 *ap4 #a1 sect4 #c2 ,collect$ ,works ,hamilton 7,ed9burgh1 #ahed71 ,vol4 #d4 #caf4 ,x is curi\s to obs]ve h[ diff]5tly ^! great m5 ,plato & ,bacon e/imat$ ! value ( e k9d ( k4 ,take ,ari?metic = example4 ,plato1 aft] speak+ slightly ( ! conv5i5ce ( be+ able to reckon & compute 9 ! ord9>y transactions ( life1 passes to :at he consid]s z a f> m important advantage4 ,! /udy ( ! prop]ties ( numb]s1 he tells u1 habituates ! m9d to ! contemplation ( pure tru?1 & raises u above ! mat]ial univ]se4 ,he w\ld h 8 disciples apply !mselves to ?is /udy1 n t !y may be able to buy or sell1 n t !y may qualify !mselves to be %op-keep]s or travell+ m]*ants1 b t !y may le>n to )draw _! m9ds f ! ev]-%ift+ spectacle ( ?is visible & tangible _w1 & to fix !m on ! immutable ess5ces ( ?+s4 ,bacon1 on ! o!r h&1 valu$ ?is bran* ( k only on acc\nt ( its uses ) ref]5ce to t visible & tangible _w :i* ,plato s mu* despis$4 ,he speaks ) scorn ( ! my/ical ari?metic ( ! lat] ,platoni/s1 & lam5ts ! prop5sity ( mank9d to employ1 on m]e matt]s ( curiosity1 p[]s ! :ole ex]tion ( :i* is requir$ = purposes ( solid advantage4 ,he advises ari?meticians to leave ^! trifles1 & employ !mselves 9 fram+ conv5i5t expressions :i* may be ( use 9 physical rese>*es4--,,macaulay4 ,"l ,bacon3 ,ed9burgh ,review1 ,july1 #ahcg4 ,critical & ,miscellane\s ,essays 7,new ,york1 #ahgi71 ,vol4 #a1 p4 #cig4 #cag4 ,a?4 ,!re /ill rema9 ?ree /udies suitable = freem54 ,calculation 9 ari?metic is "o ( !m2 ! measurem5t ( l5g?1 surface1 & dep? is ! second2 & ! ?ird has to d ) ! revolutions ( ! />s 9 ref]5ce to "o ano!r 444 !re is 9 !m some?+ t is necess>y & _c be set aside1 444 if ,i am n mi/ak51 some?+ ( div9e necessity2 = z to ! human necessities ( :i* m5 (t5 speak :5 !y talk 9 ?is mann]1 no?+ c be m ridicul\s ?an su* an application ( ! words4 ,cle4 ,& :at necessities ( k are !re1 ,/rang]1 :i* are div9e & n human8 ,a?4 ,i conceive !m to be ^? ( :i* he :o has no use nor any k at all _c be a god1 or demi-god1 or h]o to mank9d1 or able to take any s]i\s ?\ght or *>ge ( !m4 --,,plato4 ,republic1 ,bk4 #g4 ,j[ett's ,dialogues ( ,plato 7,new ,york1 #ahig71 ,vol4 #d1 p4 #ccd4 #cah4 ,^? :o ass]t t ! ma!matical sci5ces make no affirmation ab\t :at is fair or good make a false ass]tion2 = !y d speak ( ^! & frame demon/rations ( !m 9 ! mo/ em95t s5se ( ! ^w4 ,= if !y d n actually employ ^! names1 !y d n exhibit ev5 ! results & ! reasons ( ^!1 & !re=e c be h>dly said to make any ass]tion ab\t !m4 ,( :at is fair1 h[ev]1 ! mo/ important species are ord] & symmetry1 & t :i* is def9ite1 :i* ! ma!matical sci5ces make manife/ 9 a mo/ em95t degree4 ,& s9ce1 at lea/1 ^! appe> to be ! causes ( _m ?+s--n[1 ,i mean1 = example1 ord]1 & t :i* is a def9ite ?+1 x is evid5t t !y w\ld ass]t1 also1 ! exi/5ce ( a cause ( ?is description1 & its subsi/5ce aft] ! same mann] z t :i* is fair subsi/s 94--,,ari/otle4 ,metaphysics ,mac,mahon ,bk4 #ab1 *ap4 #c4 #cai4 ,_m arts !re are :i* beautify ! m9d ( man2 ( all o!r none d m g>ni% & beautify x ?an ^? arts :i* are call$ ma!matical4--,,bill+sley1 ,h4 ,! ,elem5ts ( ,geometrie ( ! mo/ anci5t ,philosoph] ,euclide ( ,meg>a 7,london1 #aegj71 ,note to ! ,read]4 #cbj4 ,z ! sun eclipses ! />s by 8 brilliancy1 s ! man ( k w eclipse ! fame ( o!rs 9 assemblies ( ! p if he proposes algebraic problems1 & /ill m if he solves !m4--,,brahmagupta4 ,quot$ 9 ,cajori's ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #ib4 #cba4 ,s highly did ! anci5ts e/eem ! p[] ( figures & numb]s1 t ,democritus ascrib$ to ! figures ( atoms ! fir/ pr9ciples ( ! v>iety ( ?+s2 & ,py?agoras ass]t$ t ! nature ( ?+s consi/$ ( numb]s4 --,,bacon1 ,,"l4 ,de ,augm5tis1 ,bk4 #c2 ,advancem5t ( ,le>n+1 ,bk4 #b4 #cbb4 ,!re has n be5 any sci5ce s mu* e/eem$ & honor$ z ?is ( ma!matics1 nor ) s mu* indu/ry & vigilance become ! c>e ( great m51 & labor$ 9 by ! pot5tates ( ! _w1 viz4 emp]ors1 k+s1 pr9ces1 etc4 --,,frankl91 ,,b5jam94 ,on ! ,usefulness ( ,ma!matics1 ,works 7,bo/on1 #ahdj71 ,vol4 #b1 p4 #bh4 #cbc4 ,:atev] may h be5 imput$ to "s o!r /udies "u ! notion ( insignificancy & loss ( "t1 yet ^! ma!matics1 ,i believe1 nev] caus$ rep5tance 9 any1 except x 0 = _! remissness 9 ! prosecution ( !m4 --,,frankl91 ,,b5jam94 ,on ! ,usefulness ( ,ma!matics1 ,works 7,bo/on1 #ahdj71 ,vol4 #b1 p4 #fi4 #cbd4 ,:at sci5ce c !re be m noble1 m excell5t1 m useful = m51 m admirably high & demon/rative1 ?an ?is ( ! ma!matics8--,,frankl91 ,,b5jam94 ,on ! ,usefulness ( ,ma!matics1 ,works 7,bo/on1 #ahdj71 ,vol4 #b1 p4 #fi4 #cbe4 ,! great tru?s ) :i* x ma!matics deals1 are clo!d ) au/]e gr&eur1 f> above all purposes ( imm$iate conv5i5ce or pr(it4 ,x is 9 !m t our limit$ und]/&+s approa* ne>e/ to ! conception ( t absolute & inf9ite1 t[>ds :i* 9 mo/ o!r ?+s !y aspire 9 va94 ,9 ! pure ma!matics we contemplate absolute tru?s1 :i* exi/$ 9 ! div9e m9d be=e ! morn+ />s sang toge!r1 & :i* w cont9ue to exi/ !re1 :5 ! la/ ( _! radiant ho/ %all h fall5 f heav54 ,!y exi/$ n m]ely 9 metaphysical possibility1 b 9 ! actual contemplation ( ! supreme reason4 ,! p5 ( inspiration1 rang+ all nature & life = imag]y to set =? ! ,creator's p[] & wisdom1 f9ds !m be/ symboliz$ 9 ! skill ( ! surveyor4 8,he met$ out heav5 z ) a span20 & an anci5t sage1 nei!r falsely nor irrev]5tly1 v5tur$ to say1 t 8,god is a geomet]40--,,ev]ett1 ,,edw>d4 ,orations & ,spee*es 7,bo/on1 #ahgj71 ,vol4 #c1 p4 #ead4 #cbf4 ,!re is no sci5ce :i* tea*es ! h>monies ( nature m cle>ly ?an ma!matics1 4444--,,c>us1 ,,paul4 ,&rews3 ,magic ,squ>es & ,cubes 7,*icago1 #aijh71 ,introduction4 #cbg4 ,= x be+ ! nature ( ! m9d ( man 7to ! extreme prejudice ( k7 to delight 9 ! spaci\s lib]ty ( g5]alities1 z 9 a *ampion region1 & n 9 ! enclosures ( p>ticul>ity2 ! ,ma!matics 7 ! goodlie/ fields to satisfy t appetite4--,,bacon1 ,,"l4 ,de ,augm5tis1 ,bk4 #c2 ,advancem5t ( ,le>n+1 ,bk4 #b4 #cbh4 ,i w\ld h my son m9d & und]/& bus9ess1 read little hi/ory1 /udy ! ma!matics & cosmography2 ^! are good1 ) subord9ation to ! ?+s ( ,god4444 ,^! fit = public s]vices = :i* man is born4--,,cromwell1 ,,oliv]4 ,lett]s & ,spee*es ( ,oliv] ,cromwell 7,new ,york1 #ahii71 ,vol4 #a1 p4 #cga4 #cbi4 ,ma!matics is ! life supreme4 ,! life ( ! gods is ma!matics4 ,all div9e mess5g]s are ma!maticians4 ,pure ma!matics is religion4 ,its atta9m5t requires a !ophany4 --,,novalis4 ,s*rift5 7,b]l91 #aija71 ,bd4 #b1 p4 #bbc4 #ccj4 ,! ,ma!matics :i* effectually ex]cises1 n va9ly deludes or vexati\sly torm5ts /udi\s ,m9ds ) obscure ,subtilties1 p]plex$ ,difficulties1 or cont5ti\s ,disquisitions2 :i* ov]comes )\t ,opposition1 triumphs )\t ,pomp1 compels )\t ,=ce1 & rules absolutely )\t ,loss ( ,lib]ty2 :i* does n privately ov]rea* a weak ,fai?1 b op5ly assaults an arm$ ,reason1 obta9s a total ,victory1 & puts on inevitable ,*a9s2 ^: ,words are s _m ,oracles1 & ,works z _m ,miracles2 :i* blabs out no?+ ra%ly1 nor designs any?+ f ! ,purpose1 b pla9ly demon/rates & readily p]=ms all ,?+s )9 its ,v]ge2 :i* obtrudes no false ,%ad[ ( ,sci5ce1 b ! v ,sci5ce itself1 ! ,m9d firmly adh]es to x1 z soon z possess$ ( x1 & c nev] aft] des]t x ( its own ,accord1 or be depriv$ ( x by any ,=ce ( o!rs3 ,la/ly ! ,ma!matics1 :i* dep5d ^u ,pr9ciples cle> to ! ,m9d1 & agreeable to ,exp]i5ce2 :i* draws c]ta9 ,conclusions1 in/ructs by pr(itable ,rules1 unfolds pleasant ,que/ions2 & produces wond]ful ,effects2 :i* is ! fruitful ,p>5t (1 ,i _h almo/ said all1 ,arts1 ! un%ak5 ,f\ndation ( ,sci5ces1 & ! pl5tiful ,f\nta9 ( ,advantage to human ,affairs4--,,b>r[1 ,,isaac4 ,oration be=e ! ,univ]sity ( ,cambridge on be+ elect$ ,lucasian ,pr(essor ( ,ma!matics1 ,ma!matical ,lectures 7,london1 #agcd71 p4 #bh4 #cca4 ,d\btless ! reason+ faculty1 ! m9d1 is ! lead+ & *>act]i/ic attribute ( ! human race4 ,by ! ex]cise ( ?is1 man arrives at ! prop]ties ( ! natural bodies4 ,?is is sci5ce1 prop]ly & emphatically s call$4 ,x is ! sci5ce ( pure ma!matics2 & 9 ! high bran*es ( ?is sci5ce lies ! truly sublime ( human acquisition4 ,if any atta9m5t des]ves t epi!t1 x is ! k1 :i*1 f ! m5suration ( ! m9ute/ du/ ( ! balance1 proce$s on ! ris+ scale ( mat]ial bodies1 ev]y:]e weigh+1 ev]y:]e measur+1 ev]y:]e detect+ & expla9+ ! laws ( =ce & motion1 p5etrat+ into ! secret pr9ciples :i* hold ! univ]se ( ,god toge!r1 & balanc+ worlds aga9/ worlds1 & sy/em aga9/ sy/em4 ,:5 we seek to accompany ^? :o pursue /udies at once s high1 s va/1 & s exact2 :5 we arrive at ! discov]ies ( ,newton1 :i* p\r 9 day on ! works ( ,god1 z if a second fiat _h gone =? f 8 own m\?2 :51 fur!r1 we attempt to foll[ ^? :o set out :]e ,newton paus$1 mak+ 8 goal _! />t+-place1 &1 proce$+ ) demon/ration ^u demon/ration1 & discov]y ^u discov]y1 br+ new worlds & new sy/ems ( worlds )9 ! limits ( ! kn[n univ]se1 fail+ to le>n all only because all is inf9ite2 h[ev] we may say ( man1 9 admiration ( 8 physical /ructure1 t 89 =m & mov+ he is express & admirable10 x is "h1 & "h )\t irrev]5ce1 we may exclaim1 8,9 appreh5sion h[ l a god60 ,! /udy ( ! pure ma!matics w ( c\rse n be ext5sively pursu$ 9 an in/itution1 :i*1 l ?is ,bo/on ,me*anics' ,in/itute1 has a direct practical t5d5cy & aim4 ,b x is /ill to be rememb]$1 t pure ma!matics lie at ! f\ndation ( me*anical philosophy1 & t x is ignorance only :i* c speak or ?9k ( t sublime sci5ce z useless rese>* or b>r5 speculation4--,,web/]1 ,,daniel4 ,works 7,bo/on1 #ahgb71 ,vol4 #a1 p4 #ahj4 #ccb4 ,! s*ool ( ,plato has advanc$ ! int]e/s ( ! race z mu* ?r\gh geometry z ?r\gh philosophy4 ,! mod]n eng9e]1 ! navigator1 ! a/ronom]1 built on ! tru?s :i* ^? e>ly ,greeks discov]$ 9 _! purely speculative inve/igations4 ,& if ! poetry1 /atesman%ip1 oratory1 & philosophy ( our day owe mu* to ,plato's div9e ,dialogues1 our comm]ce1 our manufactures1 & our sci5ce are equally indebt$ to 8 ,conic ,sections4 ,lat] in/ances may be abundantly quot$1 to %[ t ! labors ( ! ma!matician h outla/$ ^? ( ! /atesman1 & wr\ght mighti] *anges 9 ! condition ( ! _w4 ,n t we w\ld rank ! geomet] above ! patriot1 b we claim t he is wor?y ( equal honor4 --,,hill1 ,,?omas4 ,imag9ation 9 ,ma!matics2 ,nor? ,am]ican ,review1 ,vol4 #he1 p4 #bbh4 #ccc4 ,! discov]ies ( ,newton h done m = ,engl& & = ! race1 ?an has be5 done by :ole dyna/ies ( ,briti% mon>*s2 & we d\bt n t 9 ! great ma!matical bir? ( #ahec1 ! ,quat]nions ( ,hamilton1 !re is z mu* real promise ( b5efit to mank9d z 9 any ev5t ( ,victoria's reign4--,,hill1 ,,?omas4 ,imag9ation 9 ,ma!matics2 ,nor? ,am]ican ,review1 ,vol4 #he1 p4 #bbh4 #ccd4 ,geometrical & ,me*anical ph5om5a are ! mo/ g5]al1 ! mo/ simple1 ! mo/ ab/ract ( all1--! mo/ irr$ucible to o!rs4 ,x foll[s t ! /udy ( !m is an indisp5sable prelim9>y to t ( all o!rs4 ,!re=e mu/ ,ma!matics hold ! fir/ place 9 ! hi]>*y ( ! sci5ces1 & be ! po9t ( dep>ture ( all ,education1 :e!r g5]al or special4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,introduction1 *ap4 #b4 ,,*apt] ,,iv ,,! ,,value ,,( ,,ma!matics #dja4 ,ma!matics because ( its nature & /ructure is peculi>ly fitt$ = high s*ool in/ruction ,gymnasiallehrfa*4 ,especially ! high] ma!matics1 ev5 if pres5t$ only 9 its elem5ts1 comb9es )9 itself all ^? qualities :i* are dem&$ ( a second>y subject4 ,x engages1 x fructifies1 x quick5s1 compels att5tion1 is z circumspect z inv5tive1 induces c\rage & self-confid5ce z well z mode/y & submission to tru?4 ,x yields ! ess5ce & k]nel ( all ?+s1 is brief 9 =m & ov]fl[s ) its weal? ( cont5t4 ,x discloses ! dep? & bread? ( ! law & spiritual elem5t beh9d ! surface ( ph5om5a2 x impels f po9t to po9t & c>ries )9 itself ! inc5tive t[>d progress2 x /imulates ! arti/ic p]ception1 good ta/e 9 judgm5t & execution1 z well z ! sci5tific compreh5sion ( ?+s4 ,ma!matics1 !re=e1 above all o!r subjects1 makes ! /ud5t lu/ aft] k1 fills him1 z x 71 ) a long+ to fa?om ! cause ( ?+s & to employ 8 own p[]s indep5d5tly2 x collects 8 m5tal =ces & conc5trates !m on a s+le po9t & ?us awak5s ! _s ( individual inquiry1 self-confid5ce & ! joy ( do+2 x fasc9ates because ( ! view-po9ts :i* x (f]s & creates c]ta9ty & assurance1 ow+ to ! univ]sal validity ( its me?ods4 ,?us1 bo? :at he receives & :at he himself contributes t[>d ! prop] conception & solution ( a problem1 comb9e to mature ! /ud5t & to make him skillful1 to lead him away f ! surface ( ?+s & to ex]cise him 9 ! p]ception ( _! ess5ce4 ,a /ud5t ?us prep>$ ?ir/s aft] k & is ready = ! univ]sity & its sci5ces4 ,?us x appe>s1 t high] ma!matics is ! be/ guide to philosophy & to ! philosophic conception ( ! _w 7consid]$ z a self-conta9$ :ole7 & ( "o's own be+4--,,dillmann1 ,e4 ,die ,ma!matik die ,fackeltrg]9 e9] neu5 ,zeit 7,/uttg>t1 #ahhi71 p4 #dj4 #djb4 ,^! ,discipl9es ma!matics s]ve to inure & corroborate ! ,m9d to a con/ant ,dilig5ce 9 ,/udy2 to und]go ! ,tr\ble ( an att5tive ,m$itation1 & *e]fully cont5d ) su* ,difficulties z lie 9 ! ,way4 ,!y :olly deliv] u f a cr$ul\s ,simplicity1 mo/ /rongly =tify u aga9/ ! ,vanity ( ,scepticism1 effectually re/ra9 f a ra% ,presumption1 mo/ easily incl9e u to a due ,ass5t1 p]fectly subject u to ! ,gov]nm5t ( "r ,reason1 & inspire u ) ,resolution to wre/le aga9/ ! unju/ ,tyranny ( false ,prejudices4 ,if ! ,fancy be un/able & fluctuat+1 x is to be pois$ by ?is ,balla/1 & /eadi$ by ?is ,an*or1 if ! ,wit be blunt x is %>p5$ ^u ?is ,:et/one2 if luxuriant x is p>$ by ?is ,knife2 if head/rong x is re/ra9$ by ?is ,bridle2 & if dull x is r\s$ by ?is ,spur4 ,! ,/eps are guid$ by no ,lamp m cle>ly ?r\gh ! d>k ,mazes ( ,nature1 by no ,?read m surely ?r\gh ! intricate ,labyr9?s ( ,philosophy1 nor la/ly is ! ,bottom ( ,tru? s\nd$ m happily by any o!r ,l9e4 ,i w n m5tion h[ pl5tiful a ,/ock ( ,k ! ,m9d is furni%$ f ^!1 ) :at :olesome ,food x is n\ri%$1 & :at s9c]e ,pleasure x enjoys4 ,b if ,i speak f>!r1 ,i %all nei!r be ! only ,p]son1 nor ! fir/1 :o affirms x2 t :ile ! ,m9d is ab/ract$ & elevat$ f s5sible ,matt]1 di/9ctly views pure ,=ms1 conceives ! ,beauty ( ,ideas1 & inve/igates ! ,h>mony ( ,proportions2 ! ,mann]s !mselves are s5sibly correct$ & improv$1 ! ,affections compos$ & rectifi$1 ! ,fancy calm$ & settl$1 & ! ,und]/&+ rais$ & excit$ to m div9e ,contemplation4 ,all :i* ,i might def5d by ,au?ority1 & confirm by ! ,suffrages ( ! greate/ ,philosoph]s4--,,b>r[1 ,,isaac4 ,prefatory ,oration3 ,ma!matical ,lectures 7,london1 #agcd71 p4 #ca4 #djc4 ,no s*ool subject s readily furni%es tasks ^: purpose c be made s cle>1 s imm$iate & s appeal+ to ! sob] second-?\ght ( ! immature le>n] z ! "r sort ( elem5t>y s*ool ma!matics4--,,my]s1 ,,george4 ,ari?metic 9 ,public ,s*ool ,education 7,*icago1 #aiaa71 p4 #h4 #djd4 ,ma!matics is a type ( ?\ght :i* seems +ra9$ 9 ! human m9d1 :i* manife/s itself to "s ext5t ) ev5 ! primitive races1 & :i* is develop$ to a high degree ) ! gr[? ( civilization4444 ,a type ( ?\ght1 a body ( results1 s ess5tially *>act]i/ic ( ! human m9d1 s little influ5c$ by environm5t1 s uni=mly pres5t 9 e civilization1 is "o ( :i* no well-in=m$ m9d today c be ignorant4--,,"y1 ,j4 ,w4 ,a4 ,! ,tea*+ ( ,ma!matics 7,london1 #aijg71 p4 #ad4 #dje4 ,probably among all ! pursuits ( ! ,univ]sity1 ma!matics pre-em95tly dem& self-d5ial1 pati5ce1 & p]sev]ance f y\?1 precisely at t p]iod :5 !y h lib]ty to act = !mselves1 & :5 on acc\nt ( obvi\s temptations1 habits ( re/ra9t & application are peculi>ly valuable4--,,todhunt]1 ,,isaac4 ,! ,conflict ( ,/udies & o!r ,essays 7,london1 #ahgc71 p4 #ab4 #djf4 ,ma!matics r5d]s its be/ s]vice ?r\gh ! imm$iate fur!r+ ( rigor\s ?\ght & ! _s ( inv5tion4 --,,h]b>t ,j4 ,f4 ,ma!matis*] ,lehrplan fr ,reals*ul53 ,w]ke ,kehrba* 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #agj4 #djg4 ,x seems to me t ! old] subjects1 classics & ma!matics1 are /rongly to be recomm5d$ on ! gr\nd ( ! accuracy ) :i* we c comp>e ! relative p]=mance ( ! /ud5ts4 ,9 fact ! def9it5ess ( ^! subjects is obvi\s1 & is commonly admitt$4 ,!re is h[ev] ano!r advantage1 :i* ,i ?9k belongs 9 g5]al to ^! subjects1 t ! exam9ations c be br\ght to be> on :at is really mo/ valuable 9 ^! subjects4--,,todhunt]1 ,,isaac4 ,conflict ( ,/udies & o!r ,essays 7,london1 #ahgc71 pp4 #f1 #g4 #djh4 ,x is bett] to tea* ! *ild ari?metic & ,lat9 gramm> ?an rhetoric & moral philosophy1 because !y require exactitude ( p]=mance x is made c]ta9 t ! lesson is ma/]$1 & t p[] ( p]=mance is wor? m ?an k4--,,em]son1 ,r4 ,w4 ,lecture on ,education4 #dji4 ,besides accu/om+ ! /ud5t to dem& complete pro(1 & to "k :5 he has n obta9$ x1 ma!matical /udies are ( imm5se b5efit to 8 education by habituat+ him to precision4 ,x is "o ( ! peculi> excell5cies ( ma!matical discipl9e1 t ! ma!matician is nev] satisfi$ ) peu prs4 ,he requires ! exact tru?4 ,h>dly any ( ! non-ma!matical sci5ces1 except *emi/ry1 has ?is advantage4 ,"o ( ! commone/ modes ( loose ?\ght1 & s\rces ( error bo? 9 op9ion & 9 practice1 is to ov]look ! importance ( quantities4 ,ma!maticians & *emi/s are taught by ! :ole c\rse ( _! /udies1 t ! mo/ fundam5tal diff]5ce ( quality dep5ds on "s v slight diff]5ce 9 proportional quantity2 & t f ! qualities ( ! influ5c+ elem5ts1 )\t c>eful att5tion to _! quantities1 false expectation w\ld con/antly be =m$ z to ! v nature & ess5tial "* ( ! result produc$4--,,mill1 ,j4 ,s4 ,an ,exam9ation ( ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #faa4 #daj4 ,9 ma!matics ,i c report no defici5ce1 except x be t m5 d n suffici5tly und]/& ! excell5t use ( ! ,pure ,ma!matics1 9 t !y d rem$y & cure _m defects 9 ! wit & faculties intellectual4 ,= if ! wit be too dull1 !y %>p5 x2 if too w&]+1 !y fix x2 if too inh]5t 9 ! s5ses1 !y ab/ract x4 ,s t z t5nis is a game ( no use 9 itself1 b ( great use 9 respect x make? a quick eye & a body ready to put itself into all positions2 s 9 ! ,ma!matics1 t use :i* is collat]al & int]v5i5t is no less wor?y ?an t :i* is pr9cipal & int5d$4--,,bacon1 ,,"l4 ,de ,augm5tis1 ,bk4 #c2 ,advancem5t ( ,le>n+1 ,bk4 #b4 #daa4 ,if a man's wit be w&]+1 let him /udy ma!matics2 = 9 demon/rations1 if 8 wit be call$ away nev] s little1 he mu/ beg9 aga94--,,bacon1 ,,"l4 ,essays3 ,on ,/udies4 #dab4 ,if "o be bird-witt$1 t is easily di/ract$ & unable to keep 8 att5tion z long z he %\ld1 ma!matics provides a rem$y2 = 9 !m if ! m9d be caught away b a mom5t1 ! demon/ration has to be comm5c$ anew4--,,bacon1 ,,"l4 ,de ,augm5tis1 ,bk4 #f2 ,advancem5t ( ,le>n+1 ,bk4 #b4 #dac4 ,! metaphysical philosoph] f 8 po9t ( view recognizes ma!matics z an in/rum5t ( education1 :i* /r5g!ns ! p[] ( att5tion1 develops ! s5se ( ord] & ! faculty ( con/ruction1 & enables ! m9d to grasp "u ! simple =mulae ! quantitative diff]5ces ( physical ph5om5a4--,,j[ett1 ,b4 ,dialogues ( ,plato 7,new ,york1 #ahig71 ,vol4 #b1 p4 #gh4 #dad4 ,nor d ,i "k any /udy :i* c compete ) ma!matics 9 g5]al 9 furni%+ matt] = sev]e & cont9u$ ?\ght4 ,metaphysical problems may be ev5 m difficult2 b !n !y are f> less def9ite1 &1 z !y r>ely lead to any precise conclusion1 we miss ! p[] ( *eck+ our own op]ations1 & ( discov]+ :e!r we are ?9k+ & reason+ or m]ely fancy+ & dream+4--,,todhunt]1 ,,isaac4 ,conflict ( ,/udies 7,london1 #ahgc71 p4 #ac4 #dae4 ,ano!r great & special excell5ce ( ma!matics is t x dem&s e>ne/ volunt>y ex]tion4 ,x is simply impossible = a p]son to become a good ma!matician by ! happy accid5t ( hav+ be5 s5t to a good s*ool2 ?is may give him a prep>ation & a />t1 b by 8 own individual ef=ts alone c he rea* an em95t position4--,,todhunt]1 ,,isaac4 ,conflict ( ,/udies 7,london1 #ahgc71 p4 #b4 #daf4 ,! faculty ( resolution is possibly mu* invigorat$ by ma!matical /udy1 & especially by t highe/ bran* ( x :i*1 unju/ly1 m]ely on acc\nt ( its retrograde op]ations1 has be5 call$1 z if p> excell5ce1 analysis4--,,poe1 ,e4 ,a4 ,! ,murd]s 9 ,rue ,morgue4 #dag4 ,he :o gives a portion ( 8 "t & tal5t to ! inve/igation ( ma!matical tru? w come to all o!r que/ions ) a decid$ advantage ov] 8 oppon5ts4 ,he w be 9 argum5t :at ! anci5t ,romans 7 9 ! field3 to !m ! day ( battle 0 a day ( comp>ative recreation1 because !y 7 ev] accu/om$ to ex]cise ) arms mu* heavi] ?an !y f\ght2 & reviews diff]$ f a real battle 9 two respects3 !y enc\nt]$ m fatigue1 b ! victory 0 bloodless4--,,colton1 ,c4 ,c4 ,lacon 7,new ,york1 #ahff74 #dah4 ,ma!matics is ! /udy :i* =ms ! f\ndation ( ! c\rse ,we/ ,po9t ,milit>y ,academy4 ,?is is necess>y1 bo? to imp>t to ! m9d t comb9$ /r5g? & v]satility1 ! peculi> vigor & rapidity ( comp>ison necess>y = milit>y action1 & to pave ! way = progress 9 ! high] milit>y sci5ces4 ,congressional ,committee on ,milit>y ,affairs1 #ahcd2 ,u4 ,s4 ,bureau ( ,education1 ,bullet9 #aiab1 ,no4 #b1 p4 #aj4 #dai4 ,ma!matics1 among all s*ool subjects1 is especially adapt$ to fur!r cle>ness1 def9ite brevity & precision 9 expression1 al?\gh x (f]s no ex]cise 9 flights ( rhetoric4 ,?is is due 9 ! fir/ place to ! logical rig\r ) :i* x develops ?\ght1 avoid+ e dep>ture f ! %orte/1 mo/ direct way1 nev] all[+ empty phrases to ent]4 ,o!r subjects excel 9 ! developm5t ( expression 9 o!r respects3 translation f =eign languages into ! "m tongue gives ex]cise 9 f9d+ ! prop] ^w = ! giv5 =eign ^w & gives k ( laws ( syntax1 ! /udy ( poetry & prose furni% fit patt]ns = connect$ pres5tation & elegant =m ( expression1 composition is to ex]cise ! pupil 9 a l pres5tation ( 8 own or borr[$ ?\ghts & _! developm5t1 ! natural sci5ces tea* description ( natural objects1 app>atus & processes1 z well z ! /atem5t ( laws on ! gr\nds ( imm$iate s5se-p]ception4 ,b all ^! aids = ex]cise 9 ! use ( ! "m tongue1 ea* 9 its way valuable & indisp5sable1 d n gu>antee1 9 ! same mann] z ma!matical tra9+1 ! exclusion ( words ^: concepts1 if n entirely want+1 are n suffici5tly cle>4 ,!y d n furni% 9 ! same measure t :i* ! ma!matician dem&s p>ticul>ly z reg>ds precision ( expression4--,,reidt1 ,f4 ,anleitung zum ma!matis*5 ,unt]ri*t 9 hh]5 ,s*ul5 7,b]l91 #aijf71 p4 #ag4 #dbj4 ,"o r>ely he>s ( ! ma!matical recitation z a prep>ation = public speak+4 ,yet ma!matics %>es ) ^! /udies =eign languages1 draw+ & natural sci5ce _! advantages1 & has ano!r 9 a high] degree ?an ei!r ( !m4 ,mo/ read]s w agree t a prime requisite = heal?ful exp]i5ce 9 public speak+ is t ! att5tion ( ! speak] & he>]s alike be drawn :olly away f ! speak] & conc5trat$ ^u ! ?\ght4 ,9 p]haps no o!r classroom is ?is s easy z 9 ! ma!matical1 :]e ! close reason+1 ! rigor\s demon/ration1 ! trac+ ( necess>y conclusions f giv5 hypo!ses1 comm&s & secures ! entire m5tal p[] ( ! /ud5t :o is expla9+1 & ( 8 classmates4 ,9 :at o!r circum/ances d /ud5ts feel s in/9ctively t mann] c\nts = s little & m9d = s mu*8 ,9 :at o!r circum/ances1 !re=e1 is a simple1 unaffect$1 easy1 graceful mann] s naturally & s heal?fully cultivat$8 ,mann]isms t are m]e affectation or ! result ( bad lit]>y habit rec$e to ! backgr\nd & f9ally disappe>1 :ile ^? peculi>ities t are ! expression ( p]sonality & are insep>able f its activity cont9ually develop1 :]e ! /ud5t frequ5tly pres5ts1 to an audi5ce ( 8 intellectual pe]s1 a connect$ tra9 ( reason+4444 ,"o w\ld almo/ wi% t our in/itutions ( ! sci5ce & art ( public speak+ w\ld put ov] _! doors ! motto t ,plato _h ov] ! entrance to 8 s*ool ( philosophy3 8,let no "o :o is unacqua9t$ ) geometry ent] "h40--,,:ite1 ,w4 ,f4 ,a ,scrap-book ( ,elem5t>y ,ma!matics 7,*icago1 #aijh71 p4 #baj4 #dba4 ,! tra9+ :i* ma!matics gives 9 work+ ) symbols is an excell5t prep>ation = o!r sci5ces2 444 ! _w's "w requires con/ant ma/]y ( symbols4--,,"y1 ,j4 ,w4 ,a4 ,! ,tea*+ ( ,ma!matics 7,new ,york1 #aijg71 p4 #db4 #dbb4 ,"o /rik+ peculi>ity ( ma!matics is its unlimit$ p[] ( evolv+ examples & problems4 ,a /ud5t may read a book ( ,euclid1 or a few *apt]s ( ,algebra1 & )9 t limit$ range ( k x is possible to set him ex]cises z real & z int]e/+ z ! propositions !mselves :i* he has /udi$2 d$uctions :i* might h pleas$ ! ,greek geomet]s1 & algebraic propositions :i* ,pascal & ,f]mat w\ld n h disda9$ to inve/igate4--,,todhunt]1 ,,isaac4 ,private ,/udy ( ,ma!matics3 ,conflict ( ,/udies & o!r ,essays 7,london1 #ahgc71 p4 #hb4 #dbc4 ,w\ld y h a man reason well1 y mu/ use him to x betimes2 ex]cise 8 m9d 9 obs]v+ ! connection betwe5 ideas1 & foll[+ !m 9 tra94 ,no?+ does ?is bett] ?an ma!matics1 :i* !re=e1 ,i ?9k %\ld be taught to all :o h ! "t & opportunity1 n s mu* to make !m ma!maticians1 z to make !m reasonable creatures2 = ?\gh we all call ourselves s1 because we are born to x if we please1 yet we may truly say t nature gives u b ! se$s ( x1 & we are c>ri$ no f>!r ?an indu/ry & application h c>ri$ u4--,,locke1 ,,john4 ,conduct ( ! ,und]/&+1 ,sect4 #f4 #dbd4 ,secondly1 ! /udy ( ma!matics w\ld %[ !m ! necessity !re is 9 reason+1 to sep>ate all ! di/9ct ideas1 & to see ! habitudes t all ^? conc]n$ 9 ! pres5t inquiry h to "o ano!r1 & to lay by ^? :i* relate n to ! proposition 9 h&1 & :olly to leave !m out ( ! reckon+4 ,?is is t :i*1 9 o!r respects besides quantity is absolutely requisite to j reason+1 ?\gh 9 !m x is n s easily obs]v$ & s c>efully practis$4 ,9 ^? p>ts ( k :]e x is ?\ght demon/ration has no?+ to d1 m5 reason z x 7 9 a lump2 & if ^u a summ>y & confus$ view1 or ^u a p>tial consid]ation1 !y c raise ! appe>ance ( a probability1 !y usually re/ cont5t2 especially if x be 9 a dispute :]e e little /raw is laid hold on1 & ev]y?+ t c b be drawn 9 any way to give color to ! argum5t is advanc$ ) o/5tation4 ,b t m9d is n 9 a po/ure to f9d tru? t does n di/9ctly take all ! p>ts asund]1 &1 omitt+ :at is n at all to ! po9t1 draws a conclusion f ! result ( all ! p>ticul>s :i* 9 any way influ5ce x4 --,,locke1 ,,john4 ,conduct ( ! ,und]/&+1 ,sect4 #g4 #dbe4 ,i h be=e m5tion$ ma!matics1 :]e9 algebra gives new helps & views to ! und]/&+4 ,if ,i propose ^! x is n to make e man a ?or\gh ma!matician or deep algebrai/2 b yet ,i ?9k ! /udy ( !m is ( inf9ite use ev5 to gr[n m52 fir/ by exp]im5tally conv9c+ !m1 t to make anyone reason well1 x is n en\gh to h p>ts :]e) he is satisfi$1 & t s]ve him well en\gh 9 8 ord9>y c\rse4 ,a man 9 ^? /udies w see1 t h[ev] good he may ?9k 8 und]/&+1 yet 9 _m ?+s1 & ^? v visible1 x may fail him4 ,?is w\ld take (f t presumption t mo/ m5 h ( !mselves 9 ?is "p2 & !y w\ld n be s apt to ?9k _! m9ds want$ no helps to enl>ge !m1 t !re c\ld be no?+ add$ to ! acut5ess & p5etration ( _! und]/&+4--,,locke1 ,,john4 ,! ,conduct ( ! ,und]/&+1 ,sect4 #g4 #dbf4 ,i h m5tion$ ma!matics z a way to settle 9 ! m9d a habit ( reason+ closely & 9 tra92 n t ,i ?9k x necess>y t all m5 %\ld be deep ma!maticians1 b t1 hav+ got ! way ( reason+ :i* t /udy necess>ily br+s ! m9d to1 !y might be able to transf] x to o!r p>ts ( k1 z !y %all h occasion4 ,= 9 all sorts ( reason+1 e s+le argum5t %\ld be manag$ z a ma!matical demon/ration2 ! connection & dep5d5ce ( ideas %\ld be foll[$ till ! m9d is br\ght to ! s\rce on :i* x bottoms1 & obs]ves ! coh]5ce all along2 4444--,,locke1 ,,john4 ,! ,conduct ( ! ,und]/&+1 ,sect4 #g4 #dbg4 ,z an ex]cise ( ! reason+ faculty1 pure ma!matics is an admirable ex]cise1 because x consi/s ( reason+ alone1 & does n encumb] ! /ud5t ) an ex]cise ( judgm5t3 & x is well to beg9 ) le>n+ "o ?+ at a "t1 & to def] a comb9ation ( m5tal ex]cises to a lat] p]iod4--,,:ately1 ,r4 ,annotations to ,bacon's ,essays 7,bo/on1 #ahgc71 ,essay #a1 p4 #dic4 #dbh4 ,x ha? be5 an old rem>k1 t ,geometry is an excell5t ,logic4 ,& x mu/ be own$ t :5 ! def9itions are cle>2 :5 ! po/ulata _c be refus$1 nor ! axioms d5i$2 :5 f ! di/9ct contemplation & comp>ison ( figures1 _! prop]ties are d]iv$1 by a p]petual well-connect$ *a9 ( consequ5ces1 ! objects be+ /ill kept 9 view1 & ! att5tion ev] fix$ ^u !m2 !re is acquir$ a habit ( reason+1 close & exact & me?odical2 :i* habit /r5g!ns & %>p5s ! m9d1 & be+ transf]r$ to o!r subjects is ( g5]al use 9 ! inquiry aft] tru?4--,,b]kely1 ,,george4 ,! ,analy/1 #b2 ,works 7,london1 #ahih71 ,vol4 #c1 p4 #aj4 #dbi4 ,suppose !n ,i want to give myself a little tra9+ 9 ! art ( reason+2 suppose ,i want to get out ( ! region ( conjecture & probability1 free myself f ! difficult task ( weigh+ evid5ce1 & putt+ in/ances toge!r to arrive at g5]al propositions1 & simply desire to "k h[ to deal ) my g5]al propositions :5 ,i get !m1 & h[ to d$uce "r inf]5ces f !m2 x is cle> t ,i %all obta9 ?is sort ( discipl9e be/ 9 ^? dep>tm5ts ( ?\ght 9 :i* ! fir/ pr9ciples are unque/ionably true4 ,= 9 all our ?9k+1 if we come to errone\s conclusions1 we come to !m ei!r by accept+ false premises to />t )--9 :i* case our reason+1 h[ev] good1 w n save u f error2 or by reason+ badly1 9 :i* case ! data we />t f may be p]fectly s\nd1 & yet our conclusions may be false4 ,b 9 ! ma!matical or pure sci5ces1--geometry1 ari?metic1 algebra1 trigonometry1 ! calculus ( v>iations or ( curves1--we "k at lea/ t !re is n1 & _c be1 error 9 our fir/ pr9ciples1 & we may !re=e fa/5 our :ole att5tion ^u ! processes4 ,z m]e ex]cises 9 logic1 !re=e1 ^! sci5ces1 bas$ z !y all are on prim>y tru?s relat+ to space & numb]1 h always be5 suppos$ to furni% ! mo/ exact discipl9e4 ,:5 ,plato wrote ov] ! portal ( 8 s*ool4 8,let no "o ignorant ( geometry ent] "h10 he did n mean t que/ions relat+ to l9es & surfaces w\ld be discuss$ by 8 disciples4 ,on ! contr>y1 ! topics to :i* he direct$ _! att5tion 7 "s ( ! deepe/ problems1--social1 political1 moral1--on :i* ! m9d c\ld ex]cise itself4 ,plato & 8 foll[]s tri$ to ?9k out toge!r conclusions respect+ ! be+1 ! duty1 & ! de/9y ( man1 & ! relation 9 :i* he /ood to ! gods & to ! unse5 _w4 ,:at _h geometry to d ) ^! ?+s8 ,simply ?is3 ,t a man ^: m9d has n und]gone a rigor\s tra9+ 9 sy/ematic ?9k+1 & 9 ! art ( draw+ legitimate inf]5ces f premises1 0 unfitt$ to ent] on ! discussion ( ^! high topics2 & t ! sort ( logical discipl9e :i* he ne$$ 0 mo/ likely to be obta9$ f geometry--! only ma!matical sci5ce :i* 9 ,plato's "t _h be5 =mulat$ & r$uc$ to a sy/em4 ,& we 9 ?is c\ntry ,engl& h long act$ on ! same pr9ciple4 ,our future lawy]s1 cl]gy1 & /atesm5 are expect$ at ! ,univ]sity to le>n a good deal ab\t curves1 & angles1 & numb]s & proportions2 n because ^! subjects h ! smalle/ relation to ! ne$s ( _! lives1 b because 9 ! v act ( le>n+ !m !y are likely to acquire t habit ( /eadfa/ & accurate ?9k+1 :i* is indisp5sable to success 9 all ! pursuits ( life4 --,,fit*1 ,j4 ,c4 ,lectures on ,tea*+ 7,new ,york1 #aijf71 pp4 #bia-bib4 #dcj4 ,x is admitt$ by all t a f9i%$ or ev5 a compet5t reason] is n ! "w ( nature alone2 ! exp]i5ce ( e day makes x evid5t t education develops faculties :i* w\ld o!rwise nev] h manife/$ _! exi/5ce4 ,x is1 !re=e1 z necess>y to le>n to reason be=e we c expect to be able to reason1 z x is to le>n to swim or f5ce1 9 ord] to atta9 ei!r ( ^? arts4 ,n[1 some?+ mu/ be reason$ ^u1 x matt]s n mu* :at x is1 provid$ x c be reason$ ^u ) c]ta9ty4 ,! prop]ties ( m9d or matt]1 or ! /udy ( languages1 ma!matics1 or natural hi/ory1 may be *os5 = ?is purpose4 ,n[ ( all ^!1 x is desirable to *oose ! "o :i* admits ( ! reason+ be+ v]ifi$1 t is1 9 :i* we c f9d out by o!r means1 su* z measurem5t & ocul> demon/ration ( all sorts1 :e!r ! results are true or n4 ,:5 ! guid+ prop]ty ( ! load/one 0 fir/ asc]ta9$1 & x 0 necess>y to le>n h[ to use ?is new discov]y1 & to f9d out h[ f> x might be reli$ on1 x w\ld h be5 ?\ght advisable to make _m passages betwe5 ports t 7 well kn[n be=e attempt+ a voyage ( discov]y4 ,s x is ) our reason+ faculties3 x is desirable t _! p[]s %\ld be ex]t$ ^u objects ( su* a nature1 t we c tell by o!r means :e!r ! results :i* we obta9 are true or false1 & ?is be=e x is safe to tru/ entirely to reason4 ,n[ ! ma!matics are peculi>ly well adapt$ = ?is purpose1 on ! foll[+ gr\nds3 #a4 ,e t]m is di/9ctly expla9$1 & has b "o mean+1 & x is r>ely t two words are employ$ to mean ! same ?+4 #b4 ,! fir/ pr9ciples are self-evid5t1 &1 ?\gh d]iv$ f obs]vation1 d n require m ( x ?an has be5 made by *ildr5 9 g5]al4 #c4 ,! demon/ration is /rictly logical1 tak+ no?+ = grant$ except self-evid5t fir/ pr9ciples1 re/+ no?+ ^u probability1 & entirely indep5d5t ( au?ority & op9ion4 #d4 ,:5 ! conclusion is obta9$ by reason+1 its tru? or falsehood c be asc]ta9$1 9 geometry by actual measurem5t1 9 algebra by common ari?metical calculation4 ,?is gives confid5ce1 & is absolutely necess>y1 if1 z 0 said be=e1 reason is n to be ! in/ructor1 b ! pupil4 #e4 ,!re are no words ^: mean+s are s mu* alike t ! ideas :i* !y /& = may be conf\nd$4 ,betwe5 ! mean+ ( t]ms !re is no di/9ction1 except a total di/9ction1 & all adjectives & adv]bs express+ diff]5ce ( degrees are avoid$4--,,de ,,morgan1 ,,augu/us4 ,on ! ,/udy & ,difficulties ( ,ma!matics 7,*icago1 #ahih71 *ap4 #a4 #dca4 ,! in/ruction ( *ildr5 %\ld aim gradually to comb9e kn[+ & do+ ,wiss5 und ,knn54 ,among all sci5ces ma!matics seems to be ! only "o ( a k9d to satisfy ?is aim mo/ completely4--,,kant1 ,,immanuel4 ,w]ke ,ros5kranz und ,s*ub]t1 ,bd4 #i 7,leipzig1 #ahch71 p4 #dji4 #dcb4 ,e discipl9e mu/ be honor$ = reason o!r ?an its utility1 o!rwise x yields no en?usiasm = indu/ry4 ,= bo? reasons1 ,i consid] ma!matics ! *ief subject = ! common s*ool4 ,no m highly honor$ ex]cise = ! m9d c be f\nd2 ! buoyancy ,spannkraft :i* x produces is ev5 great] ?an t produc$ by ! anci5t languages1 :ile its utility is unque/ion$4--,,h]b>t1 ,j4 ,f4 ,ma!matis*] ,lehrplan fr ,realgymnasi51 ,w]ke ,kehrba*1 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #afg4 #dcc4 ,! motive = ! /udy ( ma!matics is insight into ! nature ( ! univ]se4 ,/>s & /rata1 heat & electricity1 ! laws & processes ( becom+ & be+1 incorporate ma!matical tru?s4 ,if language imitates ! voice ( ! ,creator1 reveal+ ,8 he>t1 ma!matics discloses ,8 intellect1 repeat+ ! /ory ( h[ ?+s came into be+4 ,& ! value ( ma!matics1 appeal+ z x does to our en]gy & to our honor1 to our desire to "k ! tru? & !reby to live z ( "r 9 ! h\sehold ( ,god1 is t x e/abli%es u 9 l>g] & l>g] c]ta9ties4 ,z lit]ature develops emotion1 und]/&+1 & sympa?y1 s ma!matics develops obs]vation1 imag9ation1 & reason4--,,*ancellor1 ,w4 ,e4 ,a ,!ory ( ,motives1 ,ideals & ,values 9 ,education 7,bo/on & ,new ,york1 #aijg71 p4 #djf4 #dcd4 ,ma!matics 9 its pure =m1 z ari?metic1 algebra1 geometry1 & ! applications ( ! analytic me?od1 z well z ma!matics appli$ to matt] & =ce1 or /atics & dynamics1 furni%es ! peculi> /udy t gives to u1 :e!r z *ildr5 or z m51 ! comm& ( nature 9 ?is its quantitative aspect2 ma!matics furni%es ! in/rum5t1 ! tool ( ?\ght1 :i* we wield 9 ?is realm4--,,h>ris1 ,w4 ,t4 ,psy*ologic ,f\ndations ( ,education 7,new ,york1 #ahih71 p4 #cbe4 #dce4 ,little c be und]/ood ( ev5 ! simple/ ph5om5a ( nature )\t "s k ( ma!matics1 & ! attempt to p5etrate deep] into ! my/]ies ( nature compels simultane\s developm5t ( ! ma!matical processes4--,,"y1 ,j4 ,w4 ,a4 ,! ,tea*+ ( ,ma!matics 7,new ,york1 #aijg71 p4 #af4 #dcf4 ,= _m p>ts ( nature c nei!r be inv5t$ ) suffici5t subtility nor demon/rat$ ) suffici5t p]spicuity nor accommodat$ unto use ) suffici5t dext]ity1 )\t ! aid & int]v5+ ( ma!matics4--,,bacon1 ,,"l4 ,de ,augm5tis1 ,bk4 #b2 ,advancem5t ( ,le>n+1 ,bk4 #c4 #dcg4 ,i confess1 t aft] ,i began 444 to disc]n h[ useful ma!maticks may be made to physicks1 ,i h (t5 wi%$ t ,i _h employ$ ab\t ! speculative "p ( geometry1 & ! cultivation ( ! speci\s ,algebra ,i _h be5 taught v "y1 a good "p ( t "t & indu/ry1 t ,i _h sp5t ab\t survey+ & =tification 7( :i* ,i rememb] ,i once wrote an entire treatise7 & o!r p>ts ( practick ma!maticks4 --,,boyle1 ,,rob]t4 ,! ,usefulness ( ,ma!maticks to ,natural ,philosophy2 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbf4 #dch4 ,ma!matics gives ! "y man a cle> idea ( demon/ration & habituates him to =m long tra9s ( ?\ght & reason+ me?odically connect$ & su/a9$ by ! f9al c]ta9ty ( ! result2 & x has ! fur!r advantage1 f a purely moral po9t ( view1 ( inspir+ an absolute & fanatical respect = tru?4 ,9 addition to all ?is1 ma!matics1 & *iefly algebra & inf9itesimal calculus1 excite to a high degree ! conception ( ! signs & symbols--necess>y in/rum5ts to ext5d ! p[] & rea* ( ! human m9d by summ>iz+ an aggregate ( relations 9 a cond5s$ =m & 9 a k9d ( me*anical way4 ,^! auxili>ies are ( special value 9 ma!matics because !y are !re adequate to _! def9itions1 a *>act]i/ic :i* !y d n possess to ! same degree 9 ! physical & ma!matical natural8 sci5ces4 ,!re are1 9 fact1 a mass ( m5tal & moral faculties t c be put 9 full play only by in/ruction 9 ma!matics2 & !y w\ld be made /ill m available if ! tea*+ 0 direct$ s z to leave free play to ! p]sonal "w ( ! /ud5t4 --,,b]!lot1 ,m4 ,p4 ,e4 ,m4 ,sci5ce z an ,in/rum5t ( ,education2 ,popul> ,sci5ce ,mon?ly 7#ahig71 p4 #bec4 #dci4 ,ma!matical k1 !re=e1 appe>s to u ( value n only 9 s f> z x s]ves z means to o!r ends1 b = its own sake z well1 & we behold1 bo? 9 its sy/ematic ext]nal & int]nal developm5t1 ! mo/ complete & pure/ logical m9d-activity1 ! embodim5t ( ! highe/ intellect-e/hetics4--,,pr+%eim1 ,,alfr$4 ,ueb] ,w]t und angebli*5 ,unw]t d] ,ma!matik2 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #ac1 p4 #cha4 #ddj4 ,! advantages :i* ma!matics d]ives f ! peculi> nature ( ^? relations ab\t :i* x is conv]sant1 f its simple & def9ite phraseology1 & f ! sev]e logic s admirably display$ 9 ! concat5ation ( its innum]able !orems1 are inde$ imm5se1 & well entitl$ to sep>ate & ample illu/ration4--,,/ew>t1 ,,dugald4 ,philosophy ( ! ,human ,m9d1 ,"p #b1 *ap4 #b1 sect4 #c4 #dda4 ,i d n int5d to g deeply into ! "q h[ f> ma!matical /udies1 z ! repres5tatives ( consci\s logical reason+1 %\ld take a m important place 9 s*ool education4 ,b x is1 9 reality1 "o ( ! que/ions ( ! day4 ,9 proportion z ! range ( sci5ce ext5ds1 its sy/em & organization mu/ be improv$1 & x mu/ inevitably come ab\t t individual /ud5ts w f9d !mselves compell$ to g ?r\gh a /rict] c\rse ( tra9+ ?an gramm> is 9 a position to supply4 ,:at /rikes me 9 my own exp]i5ce ) /ud5ts :o pass f our classical s*ools to sci5tific & m$ical /udies1 is fir/1 a c]ta9 laxity 9 ! application ( /rictly univ]sal laws4 ,! grammatical rules1 9 :i* !y h be5 ex]cis$1 are = ! mo/ "p foll[$ by long li/s ( exceptions2 accord+ly !y are n 9 ! habit ( rely+ implicitly on ! c]ta9ty ( a legitimate d$uction f a /rictly univ]sal law4 ,secondly1 ,i f9d !m = ! mo/ "p too mu* incl9$ to tru/ to au?ority1 ev5 9 cases :]e !y might =m an indep5d5t judgm5t4 ,9 fact1 9 philological /udies1 inasmu* z x is seldom possible to take 9 ! :ole ( ! premises at a glance1 & inasmu* z ! decision ( disput$ que/ions (t5 dep5ds on an /hetic feel+ = beauty ( expression1 or = ! g5ius ( ! language1 atta9able only by long tra9+1 x mu/ (t5 happ5 t ! /ud5t is ref]r$ to au?orities ev5 by ! be/ tea*]s4 ,bo? faults are traceable to c]ta9 indol5ce & vagu5ess ( ?\ght1 ! sad effects ( :i* are n conf9$ to subsequ5t sci5tific /udies4 ,b c]ta9ly ! be/ rem$y = bo? is to be f\nd 9 ma!matics1 :]e !re is absolute c]ta9ty 9 ! reason+1 & no au?ority is recogniz$ b t ( "o's own intellig5ce4--,,helmholtz1 ,h4 ,on ! ,relation ( ,natural ,sci5ce to ,sci5ce 9 g5]al2 ,popul> ,lectures on ,sci5tific ,subjects2 ,atk9son 7,new ,york1 #aijj71 pp4 #be-bf4 #ddb4 ,:at r5d]s a problem def9ite1 & :at leaves x indef9ite1 may be/ be und]/ood f ma!matics4 ,! v important idea ( solv+ a problem )9 limits ( error is an elem5t ( rational culture1 com+ f ! same s\rce4 ,! art ( totaliz+ fluctuations by curves is capable ( be+ c>ri$1 9 conception1 f> beyond ! ma!matical doma91 :]e x is fir/ le>n$4 ,! di/9ction betwe5 laws & coeffici5ts applies 9 e dep>tm5t ( causation4 ,! !ory ( ,probable ,evid5ce is ! ma!matical contribution to ,logic1 & is ( p>am\nt importance4--,,ba91 ,,alex&]4 ,education z a ,sci5ce 7,new ,york1 #ahih71 pp4 #aea-aeb4 #ddc4 ,we receive x z a fact1 t "s m9ds are s con/itut$ z absolutely to require = _! nurture ! sev]e logic ( ! ab/ract sci5ces2 t rigor\s sequ5ce ( ideas :i* leads f ! premises to ! conclusion1 by a pa?1 ardu\s & n>r[1 x may be1 & :i* ! y\?ful reason may f9d x h>d to m\nt1 b :]e x _c /ray2 & on :i*1 if x move at all1 x mu/ move onw>d & upw>d4444 ,ev5 = intellects ( a diff]5t "*1 ^: natural aptitude is = moral evid5ce & ^? relations ( ideas :i* are p]ceiv$ & appreciat$ by ta/e1 ! /udy ( ! exact sci5ces may be recomm5d$ z ! be/ protection aga9/ ! errors into :i* !y are mo/ likely to fall4 ,al?\gh ! /udy ( language is 9 _m respects no mean ex]cise 9 logic1 yet x mu/ be admitt$ t an em95tly practical m9d is h>dly to be =m$ )\t ma!matical tra9+4 --,,ev]ett1 ,,edw>d4 ,orations & ,spee*es 7,bo/on1 #ahgj71 ,vol4 #b1 p4 #eaj4 #ddd4 ,! value ( ma!matical in/ruction z a prep>ation = ^? m difficult inve/igations1 consi/s 9 ! applicability n ( its doctr9es b ( its me?ods4 ,ma!matics w ev] rema9 ! pa/ p]fect type ( ! d$uctive me?od 9 g5]al2 & ! applications ( ma!matics to ! simpl] bran*es ( physics furni% ! only s*ool 9 :i* philosoph]s c effectually le>n ! mo/ difficult & important ( _! art1 ! employm5t ( ! laws ( simpl] ph5om5a = expla9+ & pr$ict+ ^? ( ! m complex4 ,^! gr\nds are q suffici5t = deem+ ma!matical tra9+ an indisp5sable basis ( real sci5tific education1 & reg>d+ ) ,plato1 "o :o is 1 z want+ 9 "o ( ! mo/ ess5tial qualifications = ! successful cultivation ( ! high] bran*es ( philosophy4--,,mill1 ,j4 ,s4 ,sy/em ( ,logic1 ,bk4 #c1 *ap4 #bd1 sect4 #i4 #dde4 ,?is sci5ce1 ,geometry1 is "o ( indisp5sable use & con/ant ref]5ce1 = e /ud5t ( ! laws ( nature2 = ! relations ( space & numb] are ! alphabet 9 :i* ^? laws are writt54 ,b besides ! int]e/ & importance ( ?is k9d :i* geometry possesses1 x has a great & peculi> value = all :o wi% to und]/& ! f\ndations ( human k1 & ! me?ods by :i* x is acquir$4 ,= ! /ud5t ( geometry acquires1 ) a degree ( insight & cle>ness :i* ! unma!matical read] c b feebly imag9e1 a conviction t !re are necess>y tru?s1 _m ( !m ( a v complex & /rik+ "*2 & t a few ( ! mo/ simple & self-evid5t tru?s :i* x is possible = ! m9d ( man to appreh5d1 may1 by sy/ematic d$uction1 lead to ! mo/ remote & unexpect$ results4--,,:ewell1 ,,william4 ,! ,philosophy ( ! ,inductive ,sci5ces1 ,"p #a1 ,bk4 #b1 *ap4 #d1 sect4 #h 7,london1 #aheh74 #ddf4 ,ma!matics1 :ile giv+ no quick remun]ation1 l ! art ( /5ography or ! craft ( bricklay+1 does furni% ! p[] = delib]ate ?\ght & accurate /atem5t1 & to speak ! tru? is "o ( ! mo/ social qualities a p]son c possess4 ,gossip1 flatt]y1 sl&]1 deceit1 all spr+ f a slov5ly m9d t has n be5 tra9$ 9 ! p[] ( tru?ful /atem5t1 :i* is "o ( ! highe/ utilities4--,,dutton1 ,s4 ,t4 ,social ,phases ( ,education 9 ! ,s*ool & ! ,home 7,london1 #aijj71 p4 #cj4 #ddg4 ,x is f ?is absolute indiff]5ce & tranquility ( ! m9d1 t ma!matical speculations d]ive "s ( _! mo/ consid]able advantages2 because !re is no?+ to int]e/ ! imag9ation2 because ! judgm5t sits free & unbias$ to exam9e ! po9t4 ,all proportions1 e arrangem5t ( quantity1 is alike to ! und]/&+1 because ! same tru?s result to x f all2 f great] f less]1 f equality & inequality4--,,burke1 ,,edmund4 ,on ! ,sublime & ,beautiful1 ,"p #c1 sect4 #b4 #ddh4 ,out ( ! int]action ( =m & cont5t 9 ma!matics gr[s an acqua9tance ) me?ods :i* enable ! /ud5t to produce indep5d5tly )9 c]ta9 ?\gh mod]ate limits1 & to ext5d 8 k ?r\gh 8 own reflection4 ,! deep5+ ( ! consci\sness ( ! intellectual p[]s connect$ ) ?is k9d ( activity1 & ! gradual awak5+ ( ! feel+ ( intellectual self-reliance may well be consid]$ z ! mo/ beautiful & highe/ result ( ma!matical tra9+4 --,,pr+%eim1 ,,alfr$4 ,ueb] ,w]t und angebli*5 ,unw]t d] ,ma!matik2 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung 7#aijd71 p4 #cgd4 #ddi4 ,he :o w\ld "k :at geometry is1 mu/ v5ture boldly into its dep?s & le>n to ?9k & feel z a geomet]4 ,i believe t x is impossible to d ?is1 & to /udy geometry z x admits ( be+ /udi$ & am consci\s x c be taught1 )\t f9d+ ! reason invigorat$1 ! inv5tion quick5$1 ! s5tim5t ( ! ord]ly & beautiful awak5$ & enhanc$1 & rev]5ce = tru?1 ! f\ndation ( all integrity ( "*1 conv]t$ into a fix$ pr9ciple ( ! m5tal & moral con/itution1 accord+ to ! old & expressive adage 8abeunt /udia 9 mores40--,,sylve/]1 ,j4 ,j4 ,a probation>y ,lecture on ,geometry2 ,collect$ ,ma!matical ,pap]s 7,cambridge1 #aijh71 ,vol4 #b1 p4 #i4 #dej4 ,ma!matical k adds vig\r to ! m9d1 frees x f prejudice1 cr$ulity1 & sup]/ition4--,,arbu?not1 ,,john4 ,usefulness ( ,ma!matical ,le>n+4 #dea4 ,:5 ! boy beg9s to und]/& t ! visible po9t is prec$$ by an invisible po9t1 t ! %orte/ di/ance betwe5 two po9ts is conceiv$ z a /raight l9e be=e x is ev] drawn ) ! p5cil on pap]1 he exp]i5ces a feel+ ( pride1 ( satisfaction4 ,& ju/ly s1 = ! f\nta9 ( all ?\ght has be5 op5$ to him1 ! diff]5ce betwe5 ! ideal & ! real1 pot5tia et actu1 has become cle> to him2 h5ce=? ! philosoph] c reveal him no?+ new1 z a geometrician he has discov]$ ! basis ( all ?\ght4--,,goe!4 ,spr*e 9 ,prosa1 ,e?is*es1 ,,vi1 #dee4 #deb4 ,9 ma!matics1 444 & 9 natural philosophy s9ce ma!matics 0 appli$ to x1 we see ! noble/ in/ance ( ! =ce ( ! human m9d1 & ( ! sublime heights to :i* x may rise by cultivation4 ,an acqua9tance ) su* sci5ces naturally leads u to ?9k well ( our faculties1 & to indulge sangu9e expectations conc]n+ ! improvem5t ( o!r p>ts ( k4 ,to ?is ,i may add1 t1 z ma!matical & physical tru?s are p]fectly un9t]e/+ 9 _! consequ5ces1 ! und]/&+ readily yields its ass5t to ! evid5ce :i* is pres5t$ to x2 & 9 ?is way may be expect$ to acquire ! habit ( tru/+ to its own conclusions1 :i* w contribute to =tify x aga9/ ! weaknesses ( scepticism1 9 ! m int]e/+ inquiries aft] moral tru? 9 :i* x may aft]w>ds engage4--,,/ew>t1 ,,dugald4 ,philosophy ( ! ,human ,m9d1 ,"p #c1 *ap4 #a1 sect4 #c4 #dec4 ,^? t c readily ma/] ! difficulties ( ,ma!matics f9d a consid]able *>m 9 ! /udy1 sometimes am\nt+ to fasc9ation4 ,?is is f> f univ]sal2 b ! subject conta9s elem5ts ( /rong int]e/ ( a k9d t con/itutes ! pleasures ( k4 ,! m>vell\s devices = solv+ problems elate ! m9d ) ! feel+ ( intellectual p[]2 & ! innum]able con/ructions ( ! sci5ce leave u lo/ 9 wond]4--,,ba91 ,,alex&]4 ,education z a ,sci5ce 7,new ,york1 #ahih71 p4 #aec4 #ded4 ,?9k+ is m]ely ! comp>+ ( ideas1 disc]n+ relations ( lik5ess & ( diff]5ce betwe5 ideas1 & draw+ inf]5ces4 ,x is seiz+ g5]al tru?s on ! basis ( cle>ly appreh5d$ p>ticul>s4 ,x is b g5]aliz+ & p>ticul>iz+4 ,:o w d5y t a *ild c deal pr(itably ) sequ5ces ( ideas l3 ,h[ _m m>bles are #b m>bles & #c m>bles8 #b p5cils & #c p5cils8 #b balls & #c balls8 #b *ildr5 & #c *ildr58 #b in*es & #c in*es8 #b feet & #c feet8 #b & #c8 ,:o has n se5 ! c\nt5ance ( "s little le>n] light up at ! end ( su* a s]ies ( que/ions ) ! exclamation1 8,:y x's always t way4 ,isn't x80 ,?is is ! gl[ ( pleasure t ! g5]aliz+ /ep always af=ds him :o takes ! /ep himself4 ,?is is ! g5u9e life-giv+ joy :i* comes f feel+ t "o c successfully take ?is /ep4 ,! reality ( su* a discov]y is z great1 & ! la/+ effect ^u ! m9d ( him t makes x is z sure z 0 t by :i* ! great ,newton hit ^u ! g5]alization ( ! law ( gravitation4 ,x is ?r\gh ^! ?rills ( discov]y t love to le>n & intellectual pleasure are begott5 & fo/]$4 ,good ari?metic tea*+ ab\nds 9 su* opportunities4 --,,my]s1 ,,george4 ,ari?metic 9 ,public ,education 7,*icago71 p4 #ac4 #dee4 ,a g5]al c\rse 9 ma!matics %\ld be requir$ ( all (fic]s = its practical value1 b no less = its educational value 9 tra9+ ! m9d to logical =ms ( ?\ght1 9 develop+ ! s5se ( absolute tru?fulness1 toge!r ) a confid5ce 9 ! accompli%m5t ( def9ite results by def9ite means4--,,e*ols1 ,c4 ,p4 ,ma!matics at ,we/ ,po9t & ,annapolis2 ,u4 ,s4 ,bureau ( ,education1 ,bullet9 #aiab1 ,no4 #b1 p4 #aa4 #def4 ,ex]cise 9 ! mo/ rigor\s ?9k+ t is possible w ( its own accord /r5g!n ! s5se ( tru? & "r1 = ea* advance 9 ! ability to di/+ui% betwe5 correct & false ?\ghts1 ea* habit mak+ = rig\r 9 ?\ght developm5t w increase 9 ! s\nd pupil ! ability & ! wi% to asc]ta9 :at is "r 9 life & to def5d x4 --,,reidt1 ,f4 ,anleitung zum ma!matis*5 ,unt]ri*t 9 d5 hh]5 ,s*ul5 7,b]l91 #aijf71 p4 #bh4 #deg4 ,i d n ma9ta9 t ! *ief value ( ! /udy ( ari?metic consi/s 9 ! lessons ( morality t arise f ?is /udy4 ,i claim only t1 to be impress$ f day to day1 t !re is some?+ t is "r z an answ] to ! que/ions ) :i* "o is able to grapple1 & t !re is a wrong answ]--t !re are ways 9 :i* ! "r answ] c be e/abli%$ z "r1 t ^! ways automatically reject error & slov5l9ess1 & t ! le>n] is able himself to manipulate ^! ways & to arrive at ! e/abli%m5t ( ! true z oppos$ to ! untrue1 ?is rel5tless hew+ to ! l9e & /opp+ at ! l9e1 mu/ color di/9ctly ! ?\ght life ( ! pupil ) m ?an a t+e ( morality4444 ,to be neighborly ) tru?1 to feel "o's self some:at facile 9 ways ( recogniz+ & e/abli%+ :at is "r1 :at is correct1 to f9d ! wrong p]si/5tly & unfail+ly reject$ z ( no value1 to feel t "o c apply ^! ways = himself1 t "o c ?9k & "w indep5d5tly1 h a real1 a positive1 & a purify+ effect ^u moral "*4 ,!y are ! quiet1 /eady und]tones ( ! "w t always appeal to ! le>n] = ! sanction ( 8 be/ judgm5t1 & ^! are ! really significant matt]s 9 s*ool "w4 ,x is n ! noise & blu/]1 n ev5 ! dramatics or ! polemics f ! tea*]'s desk1 t abide longe/ & leave ! deepe/ & /able/ impr9t ^u "*4 ,x is ^! /ill1 small voices t speak unmi/akably = ! "r & aga9/ ! wrong & ! errone\s t really =m human "*4 ,:5 ! s*ool subjects are arrang$ on ! basis ( ! degree to :i* !y contribute to ! moral upbuild+ ( human "* good ari?metic w be well up ! li/4--,,my]s1 ,,george4 ,ari?metic 9 ,public ,education 7,*icago71 p4 #ah4 #deh4 ,9 de/roy+ ! pr$isposition to ang]1 sci5ce ( all k9d is useful2 b ! ma!matics possess ?is prop]ty 9 ! mo/ em95t degree4--,,dr4 ,,ru%4 ,quot$ 9 ,day's ,collacon 7,london1 no date74 #dei4 ,! ma!matics are ! fri5ds to religion1 inasmu* z !y *>m ! passions1 re/ra9 ! impetuosity ( ! imag9ation1 & purge ! m9d f error & prejudice4 ,vice is error1 confusion & false reason+2 & all tru? is m or less opposite to x4 ,besides1 ma!matical tru? may s]ve = a pleasant ent]ta9m5t = ^? h\rs :i* "y m5 are apt to ?r[ away ^u _! vices2 ! delightfulness ( !m be+ su* z to make solitude n only easy b desirable4 --,,arbu?not1 ,,john4 ,usefulness ( ,ma!matical ,le>n+4 #dfj4 ,!re is no prophet :i* prea*es ! sup]p]sonal ,god m pla9ly ?an ma!matics4--,,c>us1 ,,paul4 ,reflections on ,magic ,squ>es2 ,moni/ 7#aijf71 p4 #adg4 #dfa4 ,ma!matics mu/ subdue ! flights ( our reason2 !y are ! /aff ( ! bl9d2 no "o c take a /ep )\t !m2 & to !m & exp]i5ce is due all t is c]ta9 9 physics4 --,,voltaire4 ,oeuvres ,compltes 7,p>is1 #ahhj71 t4 #ce1 p4 #bai4 ,,*apt] ,v ,,! ,,tea*+ ,,( ,,ma!matics #eja4 ,9 ma!matics two ends are con/antly kept 9 view3 ,fir/1 /imulation ( ! inv5tive faculty1 ex]cise ( judgm5t1 developm5t ( logical reason+1 & ! habit ( concise /atem5t2 second1 ! association ( ! bran*es ( pure ma!matics ) ea* o!r & ) appli$ sci5ce1 t ! pupil may see cle>ly ! true relations ( pr9ciples & ?+s4 ,int]national ,commission on ! ,tea*+ ( ,ma!matics1 ,am]ican ,report2 ,u4 ,s4 ,bureau ( ,education1 ,bullet9 #aiab1 ,no4 #d1 p4 #g4 #ejb4 ,! ends to be atta9$ 9 ! tea*+ ( ma!matics 9 ! second>y s*ools are ! k ( a body ( geometrical tru?s1 ! p[] to draw correct inf]5ces f giv5 premises1 ! p[] to use algebraic processes z a means ( f9d+ results 9 practical problems1 & ! awak5+ ( int]e/ 9 ! sci5ce ( ma!matics4 ,int]national ,commission on ! ,tea*+ ( ,ma!matics1 ,am]ican ,report2 ,u4 ,s4 ,bureau ( ,education1 ,bullet9 #aiab1 ,no4 #d1 p4 #g4 #ejc4 ,g5]al prep>atory in/ruction mu/ cont9ue to be ! aim 9 ! in/ruction at ! high] in/itutions ( le>n+4 ,exclusive selection & treatm5t ( subject matt] ) ref]5ce to specific avocations is disadvantage\s4 ,resolution adopt$ by ! ,g]man ,association = ! ,advancem5t ( ,sci5tific & ,ma!matical ,in/ruction2 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung 7#ahif71 p4 #da4 #ejd4 ,9 ! second>y s*ools ma!matics %\ld be a "p ( g5]al culture & n contributory to te*nical tra9+ ( any k9d2 x %\ld cultivate space intuition1 logical ?9k+1 ! p[] to rephrase 9 cle> language ?\ghts recogniz$ z correct1 & e?ical & e/hetic effects2 s treat$1 ma!matics is a q indisp5sable factor ( g5]al education 9 s f> z ! latt] %[s its traces 9 ! compreh5sion ( ! developm5t ( civilization & ! ability to p>ticipate 9 ! fur!r tasks ( civilization4 ,unt]ri*tsbltt] fr ,ma!matik und ,naturwiss5s*aft 7#aijd71 p4 #abh4 #eje4 ,inde$1 ! aim ( tea*+ ma!matics %\ld be r to /r5g!n 8 ! pupil's faculties1 & to supply a me?od ( reason+ applicable to o!r subjects1 ?an to furni% him ) an in/rum5t = solv+ practical problems4--,,magnus1 ,,philip4 ,p]ry's ,tea*+ ( ,ma!matics 7,london1 #aijb71 p4 #hd4 #ejf4 ,! p>ticipation 9 ! g5]al developm5t ( ! m5tal p[]s )\t special ref]5ce to 8 future vocation mu/ be recogniz$ z ! ess5tial aim ( ma!matical in/ruction4--,,reidt1 ,f4 ,anleitung zum ,ma!matis*5 ,unt]ri*t an hh]5 ,s*ul5 7,b]l91 #aijf71 p4 #ab4 #ejg4 ,i am ( ! decid$ op9ion1 t ma!matical in/ruction mu/ h = its fir/ aim a deep p5etration & complete comm& ( ab/ract ma!matical !ory toge!r ) a cle> insight into ! /ructure ( ! sy/em1 & d\bt n t ! in/ruction :i* accompli%es ?is is valuable & int]e/+ ev5 if x neglects practical applications4 ,if ! in/ruction %>p5s ! und]/&+1 if x ar\ses ! sci5tific int]e/1 :e!r ma!matical or philosophical1 if f9ally x calls into life an e/hetic feel+ = ! beauty ( a sci5tific edifice1 ! in/ruction w take on an e?ical value z well1 provid$ t ) ! int]e/ x awak5s also ! impulse t[>d sci5tific activity4 ,i cont5d1 !re=e1 t ev5 )\t ref]5ce to its applications ma!matics 9 ! high s*ools has a value equal to t ( ! o!r subjects ( in/ruction4--,,goett+1 ,e4 ,ueb] das ,lehrziel im ma!matis*5 ,unt]ri*t d] hh]5 ,realan/alt52 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #b1 p4 #aib4 #ejh4 ,ma!matics w n be prop]ly e/eem$ 9 wid] circles until m ?an ! a b c ( x is taught 9 ! s*ools1 & until ! un=tunate impression is gott5 rid ( t ma!matics s]ves no o!r purpose 9 in/ruction ?an ! =mal tra9+ ( ! m9d4 ,! aim ( ma!matics is its cont5t1 its =m is a second>y consid]ation & ne$ n necess>ily be t hi/oric =m :i* is due to ! circum/ance t ma!matics took p]man5t %ape "u ! influ5ce ( ,greek logic4--,,hankel1 ,h4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 p4 #f4 #eji4 ,! idea t aptitude = ma!matics is r>] ?an aptitude = o!r subjects is m]ely an illusion :i* is caus$ by belat$ or neglect$ beg9n]s4--,,h]b>t1 ,j4 ,f4 ,umriss pdagogis*] ,vorlesung52 ,w]ke ,kehrba* 7,lang5salza1 #aijb71 ,bd4 #aj1 p4 #aja4 #eaj4 ,i believe t ! useful me?ods ( ma!matics are easily to be le>n$ by q "y p]sons1 j z languages are easily le>n$ 9 y\?4 ,:at a wondr\s philosophy & hi/ory und]lie ! use ( almo/ e ^w 9 e language--yet ! *ild le>ns to use ! ^w unconsci\sly4 ,no d\bt :5 su* a ^w 0 fir/ inv5t$ x 0 /udi$ ov] & lectur$ ^u1 j z "o might lecture n[ ^u ! idea ( a rate1 or ! use ( ,c>tesian co-ord9ates1 & we may dep5d ^u x t *ildr5 ( ! future w use ! idea ( ! calculus1 & use squ>$ pap] z readily z !y n[ ciph]4444 ,:5 ,egyptian & ,*aldean philosoph]s sp5t ye>s 9 difficult calculations1 :i* w\ld n[ be ?\ght easy by "y *ildr51 d\btless !y _h ! same notions ( ! dep? ( _! k t ,sir ,william ,?omson might n[ h ( 84 ,h[ is x1 !n1 t ,?omson ga9$ 8 imm5se k 9 ! "t tak5 by a ,*aldean philosoph] to acquire a simple k ( ari?metic8 ,! reason is pla94 ,?omson1 :5 a *ild1 0 taught 9 a few ye>s m ?an all t 0 kn[n ?ree ?\s& ye>s ago ( ! prop]ties ( numb]s4 ,:5 x is f\nd ess5tial to a boy's future t ma*9]y %\ld be giv5 to 8 bra91 x is giv5 to him2 he is taught to use x1 & 8 bright memory makes ! use ( x a second nature to him2 b x is n till aft]-life t he makes a close inve/igation ( :at !re actually is 9 8 bra9 :i* has enabl$ him to d s mu*4 ,x is tak5 because ! *ild has mu* fai?4 ,9 aft] ye>s he w accept no?+ )\t c>eful consid]ation4 ,! ma*9]y giv5 to ! bra9 ( *ildr5 is gett+ m & m complicat$ z "t goes on2 b !re is really no reason :y x %\ld n be tak5 9 z e>ly1 & us$ z readily1 z 7 ! axioms ( *ildi% education 9 anci5t ,*aldea4--,,p]ry1 ,,john4 ,! ,tea*+ ( ,ma!matics 7,london1 #aijb71 p4 #ad4 #eaa4 ,! anci5ts devot$ a lifetime to ! /udy ( ari?metic2 x requir$ days to extract a squ>e root or to multiply two numb]s toge!r4 ,is !re any h>m 9 skipp+ all t1 9 lett+ ! s*ool boy le>n multiplication sums1 & 9 />t+ 8 m ab/ract reason+ at a m advanc$ po9t8 ,:]e w\ld be ! h>m 9 lett+ ! boy assume ! tru? ( _m propositions ( ! fir/ f\r books ( ,euclid1 lett+ him assume _! tru? p>tly by fai?1 p>tly by trial8 ,giv+ him ! :ole fif? book ( ,euclid by simple algebra8 ,lett+ him assume ! six? z axiomatic8 ,lett+ him1 9 fact1 beg9 8 sev]] /udies :]e he is n[ 9 ! habit ( leav+ (f8 ,we d mu* less or?odox ?+s4 ,e "h & !re 9 "o's ma!matical /udies "o makes exce$+ly l>ge assumptions1 because ! me?odical /udy w\ld be ridicul\s ev5 9 ! eyes ( ! mo/ p$antic ( tea*]s4 ,i c imag9e a :ole ye> devot$ to ! philosophical /udy ( _m ?+s t a /ud5t n[ takes 9 8 /ride )\t tr\ble4 ,! pres5t me?od ( tra9+ ! m9d ( a ma!matical tea*] causes x to /ra9 at gnats & to swall[ camels4 ,su* gnats are mo/ ( ! propositions ( ! six? book ( ,euclid2 propositions g5]ally ab\t incomm5surables2 ! use ( ari?metic 9 geometry2 ! p>allelogram ( =ces1 etc41 decimals4--,,p]ry1 ,,john4 ,! ,tea*+ ( ,ma!matics 7,london1 #aijd71 p4 #ab4 #eab4 ,! tea*+ ( elem5t>y ma!matics %\ld be conduct$ s t ! way %\ld be prep>$ = ! build+ ^u !m ( ! high] ma!matics4 ,! tea*] %\ld always be> 9 m9d & look =w>d to :at is to come aft]4 ,! pupil %\ld n be taught :at may be suffici5t = ! "t1 b w lead to difficulties 9 ! future4444 ,i ?9k ! fault 9 tea*+ ari?metic is t ( n att5d+ to g5]al pr9ciples & tea*+ in/ead ( p>ticul> rules4444 ,i am incl9$ to attack ! tea*+ ( ma!matics on ! gr\nds t x does n dwell suffici5tly on a few g5]al axiomatic pr9ciples4 --,,hudson1 ,w4 ,h4 ,h4 ,p]ry's ,tea*+ ( ,ma!matics 7,london1 #aijd71 p4 #cc4 #eac4 8,ma!matics 9 ,prussia6 ,ah1 sir1 !y tea* ma!matics 9 ,prussia z y tea* y\r boys r[+ 9 ,engl&3 !y are tra9$ by m5 :o h be5 tra9$ by m5 :o h !mselves be5 tra9$ = g5]ations back40--,,langley1 ,e4 ,m4 ,p]ry's ,tea*+ ( ,ma!matics 7,london1 #aijd71 p4 #dc4 #ead4 ,a sup]ficial k ( ma!matics may lead to ! belief t ?is subject c be taught incid5tally1 & t ex]cises ak9 to c\nt+ ! petals ( fl[]s or ! legs ( a gras%opp] are ma!matical4 ,su* "w ignores ! fundam5tal idea out ( :i* quantitative reason+ gr[s--! equality ( magnitudes4 ,x leaves ! pupil unaw>e ( t relativity :i* is ! ess5ce ( ma!matical sci5ce4 ,num]ical /atem5ts are frequ5tly requir$ 9 ! /udy ( natural hi/ory1 b to repeat ^! z a drill ^u numb]s w sc>cely l5d *>m to ^! /udies1 & c]ta9ly w n result 9 ma!matical k4--,,spe]1 ,w4 ,w4 ,prim>y ,ari?metic 7,bo/on1 #ahig71 pp4 #bf-bg4 #eae4 ,ma!matics is no m ! art ( reckon+ & computation ?an ar*itecture is ! art ( mak+ bricks or hew+ wood1 no m ?an pa9t+ is ! art ( mix+ colors on a palette1 no m ?an ! sci5ce ( geology is ! art ( break+ rocks1 or ! sci5ce ( anatomy ! art ( but*]+4 --,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #bi4 #eaf4 ,! /udy ( ma!matics--f ord9>y reckon+ up to ! high] processes--mu/ be connect$ ) k ( nature1 & at ! same "t ) exp]i5ce1 t x may ent] ! pupil's circle ( ?\ght4--,,h]b>t1 ,j4 ,f4 ,lett]s & ,lectures on ,education ,felk9 7,london1 #aijh71 p4 #aag4 #eag4 ,fir/1 z conc]ns ! success ( tea*+ ma!matics4 ,no in/ruction 9 ! high s*ools is z difficult z t ( ma!matics1 s9ce ! l>ge majority ( /ud5ts are at fir/ decid$ly dis9cl9$ to be h>ness$ into ! rigid framework ( logical conclusions4 ,! int]e/ ( "y p is won mu* m easily1 if s5se-objects are made ! />t+ po9t & ! transition to ab/ract =mulation is br\ght ab\t gradually4 ,= ?is reason x is psy*ologically q correct to foll[ ?is c\rse4 ,n less to be recomm5d$ is ?is c\rse if we inquire into ! ess5tial purpose ( ma!matical in/ruction4 ,=m]ly x 0 too exclusively held t ?is purpose is to %>p5 ! und]/&+4 ,surely ano!r important end is to implant 9 ! /ud5t ! conviction t correct ?9k+ bas$ on true premises secures ma/]y ov] ! out] _w4 ,to accompli% ?is ! out] _w mu/ receive its %>e ( att5tion f ! v beg9n+4 ,d\btless ?is is true b !re is a dang] :i* ne$s po9t+ out4 ,x is z 9 ! case ( language tea*+ :]e ! mod]n t5d5cy is to secure 9 addition to gramm> also an und]/&+ ( ! au?ors4 ,! dang] lies 9 gramm> be+ completely set aside leav+ ! subject )\t its indisp5sable solid basis4 ,j s 9 ! tea*+ ( ma!matics x is possible to accumulate int]e/+ applications to su* an ext5t z to /unt ! ess5tial logical developm5t4 ,?is %\ld 9 no wise be p]mitt$1 = ?us ! k]nel ( ! :ole matt] is lo/4 ,!re=e3 ,we d want ?r\gh\t a quick5+ ( ma!matical in/ruction by ! introduction ( applications1 b we d n want t ! p5dulum1 :i* 9 =m] decades may h incl9$ too mu* t[>d ! ab/ract side1 %\ld n[ sw+ to ! o!r extreme2 we w\ld r pursue ! prop] middle c\rse4--,,kle91 ,,felix4 ,ueb] d5 ,ma!matis*5 ,unt]ri*t an d5 hh]5 ,s*ul52 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #aa1 p4 #aca4 #eah4 ,x is above all ! duty ( ! me?odical text-book to adapt itself to ! pupil's p[] ( compreh5sion1 only *all5g+ 8 high] ef=ts ) ! increas+ developm5t ( 8 imag9ation1 8 logical p[] & ! ability ( ab/raction4 ,?is inde$ con/itutes a te/ ( ! art ( tea*+1 x is "h :]e p$agogic tact becomes manife/4 ,9 ref]5ce to ! axioms1 caution is necess>y4 ,x %\ld be po9t$ out comp>atively e>ly1 9 h[ f> ! ma!matical body diff]s f ! mat]ial body4 ,fur!rmore1 s9ce ma!matical bodies are really portions ( space1 ?is space is to be conceiv$ z ma!matical space & to be cle>ly di/+ui%$ f real or physical space4 ,gradually ! /ud5t w become consci\s t ! portion ( ! real space :i* lies beyond ! visible /ell> univ]se is n cognizable ?r\gh ! s5ses1 t we "k no?+ ( its prop]ties & consequ5tly h no basis = judgm5ts conc]n+ x4 ,ma!matical space1 on ! o!r h&1 may be subject$ to conditions1 = in/ance1 we may condition its prop]ties at inf9ity1 & ^! conditions con/itute ! axioms1 say ! ,euclidean axioms4 ,b e /ud5t w require ye>s be=e ! conviction ( ! tru? ( ?is la/ /atem5t w =ce itself ^u him4 --,,holzm,,ll]1 ,,gu/av4 ,me?odis*es ,lehrbu* d] ,elem5t>-,ma!matik 7,leipzig1 #aijd71 ,teil #a1 ,vorwort1 pp4 #d-e4 #eai4 ,l almo/ e subject ( human int]e/1 ?is "o ma!matics is j z easy or z difficult z we *oose to make x4 ,a lifetime may be sp5t by a philosoph] 9 discuss+ ! tru? ( ! simple/ axiom4 ,! simple/ fact z to our exi/5ce may fill u ) su* wond] t our m9ds w rema9 ov]:elm$ ) wond] all ! "t4 ,a ,scot* pl\ghman makes a work+ religion out ( a sy/em :i* appalls a m5tal philosoph]4 ,"s boys ( t5 ye>s ( age /udy ! me?ods ( ! diff]5tial calculus2 o!r mu* clev]] boys work+ at ma!matics to ! age ( n9ete5 h a difficulty 9 compreh5d+ ! fundam5tal ideas ( ! calculus4--,,p]ry1 ,,john4 ,! ,tea*+ ( ,ma!matics 7,london1 #aijb71 pp4 #ai-bj4 #ebj4 ,poor tea*+ leads to ! inevitable idea t ! subject ma!matics is only adapt$ to peculi> m9ds1 :5 x is ! "o univ]sal sci5ce & ! "o ^: f\r gr\nd-rules are taught u almo/ 9 infancy & reappe> 9 ! motions ( ! univ]se4--,,saf=d1 ,t4 ,h4 ,ma!matical ,tea*+ 7,bo/on1 #aijg71 p4 #ai4 #eba4 ,! numb] ( ma!matical /ud5ts 444 w\ld be mu* augm5t$ if ^? :o hold ! highe/ rank 9 sci5ce w\ld condesc5d to give m effective assi/ance 9 cle>+ ! elem5ts ( ! difficulties :i* !y pres5t4--,,de ,,morgan1 ,a4 ,/udy & ,difficulties ( ,ma!matics 7,*icago1 #aijb71 ,preface4 #ebb4 ,he t c\ld tea* ma!matics well1 w\ld n be a bad tea*] 9 any ( ! re/ physics1 *emi/ry1 biology1 psy*ology unless by ! accid5t ( total inaptitude = exp]im5tal illu/ration2 :ile ! m]e exp]im5tali/ is likely to fall into ! error ( miss+ ! ess5tial condition ( sci5ce z reason$ tru?2 n to speak ( ! dang] ( mak+ ! in/ruction an affair ( s5sation1 glitt]1 or pyrote*nic %[4--,,ba91 ,,alex&]4 ,education z a ,sci5ce 7,new ,york1 #ahih71 p4 #bih4 #ebc4 ,i %\ld l to draw att5tion to ! inexhau/ible v>iety ( ! problems & ex]cises :i* x ma!matics furni%es2 ^! may be graduat$ to precisely ! am\nt ( atta9m5t :i* may be possess$1 :ile yet reta9+ an int]e/ & value4 ,x seems to me t no o!r bran* ( /udy at all comp>es ) ma!matics 9 ?is4 ,:5 we propose a d$uction to a beg9n] we give him an ex]cise 9 _m cases t w\ld h be5 admir$ 9 ! vigor\s days ( ,greek geometry4 ,al?\gh grammatical ex]cises are well suit$ to insure ! great b5efits connect$ ) ! /udy ( languages1 yet ^! ex]cises seem to me /iff & artificial 9 comp>ison ) ! problems ( ma!matics4 ,x is n absurd to ma9ta9 t ,euclid & ,apollonius w\ld h reg>d$ ) int]e/ _m ( ! elegant d$uctions :i* are inv5t$ = ! use ( our /ud5ts 9 geometry2 b x seems sc>cely conceivable t ! great ma/]s 9 any o!r l9e ( /udy c\ld condesc5d to give a mom5t's att5tion to ! elem5t>y books ( ! beg9n]4 --,,todhunt]1 ,,isaac4 ,conflict ( ,/udies 7,london1 #ahgc71 pp4 #aj-aa4 #ebd4 ,! visible figures by :i* pr9ciples are illu/rat$ %\ld1 s f> z possible1 h no accessories4 ,!y %\ld be magnitudes pure & simple1 s t ! ?\ght ( ! pupil may n be di/ract$1 & t he may "k :at features ( ! ?+ repres5t$ he is to pay att5tion to4 ,report ( ! ,committee ( ,t5 on ,second>y ,s*ool ,subjects1 7,new ,york1 #ahid71 p4 #aji4 #ebe4 ,geometrical reason+1 & ari?metical process1 h ea* its own (fice3 to mix ! two 9 elem5t>y in/ruction1 is injuri\s to ! prop] acquisition ( bo?4--,,de ,,morgan1 ,a4 ,trigonometry & ,d\ble ,algebra 7,london1 #ahdi71 p4 #ib4 #ebf4 ,equations are ,expressions ( ,ari?metical ,computation1 & prop]ly h no place 9 ,geometry1 except z f> z ,quantities truly ,geometrical 7t is1 ,l9es1 ,surfaces1 ,solids1 & ,proportions7 may be said to be "s equal to o!rs4 ,multiplications1 ,divisions1 & su* sort ( ,computations1 are newly receiv$ into ,geometry1 & t unw>ily1 & contr>y to ! fir/ ,design ( ?is ,sci5ce4 ,= :osoev] consid]s ! ,con/ruction ( a ,problem by a "r ,l9e & a ,circle1 f\nd out by ! fir/ ,geometricians1 w easily p]ceive t ,geometry 0 inv5t$ t we might exp$iti\sly avoid1 by draw+ ,l9es1 ! ,t$i\sness ( ,computation4 ,!re=e ^! two ,sci5ces ought n to be conf\nd$4 ,! ,anci5ts did s indu/ri\sly di/+ui% !m f "o ano!r1 t !y nev] introduc$ ,ari?metical ,t]ms into ,geometry4 ,& ! ,mod]ns1 by conf\nd+ bo?1 h lo/ ! ,simplicity 9 :i* all ! ,elegance ( ,geometry consi/s4 ,:]e=e t is ,ari?metically m simple :i* is det]m9$ by ! m simple ,equation1 b t is ,geometrically m simple :i* is det]m9$ by ! m simple draw+ ( ,l9es2 & 9 ,geometry1 t ought to be reckon$ be/ :i* is geometrically mo/ simple4--,,newton4 ,on ! ,l9e> ,con/ruction ( ,equations2 ,univ]sal ,ari?metic 7,london1 #agfi71 ,vol4 #b1 p4 #dgj4 #ebg4 ,z long z algebra & geometry proce$$ along sep>ate pa?s1 _! advance 0 sl[ & _! applications limit$4 ,b :5 ^! sci5ces jo9$ company1 !y drew f ea* o!r fre% vitality & !nce=w>d m>*$ on at a rapid pace t[>d p]fection4--,,lagrange4 ,leons lm5taires sur les ,ma!matiques1 ,leon c9quime4 ,mc,cormack4 #ebh4 ,! greate/ enemy to true ari?metic "w is f\nd 9 s-call$ practical or illu/rative problems1 :i* are freely giv5 to our pupils1 ( a degree ( difficulty & complexity altoge!r unsuit$ to _! age & m5tal developm5t4444 ,i am1 myself1 no bad ma!matician1 & all ! reason+ p[]s ) :i* nature end[$ me h long be5 z fully develop$ z !y are ev] likely to be2 b ,i h1 n infrequ5tly1 be5 puzzl$1 & at times foil$1 by ! subtle logical difficulty runn+ ?r\gh "o ( ^! problems1 giv5 to my own *ildr54 ,! head-ma/] ( "o ( our ,bo/on high s*ools confess$ to me t he _h sometimes be5 unable to unravel "o ( ^! tangl$ ske9s1 9 try+ to help 8 own daught] ?r\gh h] ev5+'s "w4 ,dur+ ?is summ]1 ,dr4 ,fairbairn1 ! di/+ui%$ head ( "o ( ! colleges ( ,ox=d1 ,engl&1 told me t n only _h he himself enc\nt]$ a simil> difficulty1 9 ! case ( 8 own *ildr51 b t1 on "o occasion1 hav+ z 8 gue/ "o ( ! fir/ ma!maticians ( ,engl&1 ! two toge!r _h be5 completely puzzl$ by "o ( ^! ari?metical conundrums4 --,,walk]1 ,f4 ,a4 ,discussions 9 ,education 7,new ,york1 #ahii71 pp4 #bec-bed4 #ebi4 ,x is (t5 assum$ t because ! "y *ild is n compet5t to /udy geometry sy/ematically he ne$ be taught no?+ geometrical2 t because x w\ld be fooli% to pres5t to him physics & me*anics z sci5ces x is useless to pres5t to him any physical or me*anical pr9ciples4 ,an error ( l orig91 :i* has wr\ght incalculable mis*ief1 d5ies to ! s*ol> ! use ( ! symbols & me?ods ( algebra 9 connection ) 8 e>ly essays 9 numb]s because1 =soo?1 he is n z yet capable ( ma/]+ quadratics6444 ,! :ole infant g5]ation1 wre/l+ ) ari?metic1 seek = a sign & groan & travail toge!r 9 pa9 = ! want ( x2 b no sign is giv5 !m save ! sign ( ! prophet ,jonah1 ! )]$ g\rd1 fruitless endeavor1 wa/$ /r5g?4 --,,walk]1 ,f4 ,a4 ,indu/rial ,education2 ,discussions 9 ,education 7,new ,york1 #ahii71 p4 #acb4 #ecj4 ,p>ticul> & cont+5t inv5tions 9 ! solution ( problems1 :i*1 ?\gh _m times m concise ?an a g5]al me?od w\ld all[1 yet1 9 my judgm5t1 are less prop] to in/ruct a le>n]1 z acro/ics1 & su* k9d ( artificial poetry1 ?\gh nev] s excell5t1 w\ld be b improp] examples to in/ruct "o t aims at ,ovidean poetry4--,,newton1 ,,isaac4 ,lett] to ,coll9s1 #afgj2 ,macclesfield1 ,correspond5ce ( ,sci5tific ,m5 7,ox=d1 #ahda71 ,vol4 #b1 p4 #cjg4 #eca4 ,! logic ( ! subject algebra1 :i*1 bo? educationally & sci5tifically speak+1 is ! mo/ important "p ( x1 is :olly neglect$4 ,! :ole tra9+ consi/s 9 example gr9d+4 ,:at %\ld h be5 m]ely ! help to atta9 ! end has become ! end itself4 ,! result is t algebra1 z we tea* x1 is nei!r an art nor a sci5ce1 b an ill-dige/$ f>rago ( rules1 ^: object is ! solution ( exam9ation problems4444 ,! result1 s f> z problems work$ 9 exam9ations g1 is1 aft] all1 v mis]able1 z ! reit]at$ compla9ts ( exam9]s %[2 ! effect on ! exam9ee is a well-kn[n en]vation ( m9d1 an almo/ incurable sup]ficiality1 :i* might be call$ ,problematic ,p>alysis--a disease :i* unfits a man to foll[ an argum5t ext5d+ beyond ! l5g? ( a pr9t$ octavo page4 --,,*ry/al1 ,,george4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 #ahhe2 ,nature1 ,vol4 #cb1 pp4 #ddg-ddh4 #ecb4 ,x is a s]i\s "q :e!r ,am]ica1 foll[+ ,engl&'s lead1 has n gone into problem-solv+ too ext5sively4 ,c]ta9 x is t we are produc+ no text-books 9 :i* ! !ory is pres5t$ 9 ! delightful /yle :i* *>act]izes _m ( ! ,fr5* works 4441 or ^? ( ! rec5t ,italian s*ool1 or1 inde$1 ^? ( ! cont95tal writ]s 9 g5]al4 --,,smi?1 ,d4 ,e4 ,! ,tea*+ ( ,elem5t>y ,ma!matics 7,new ,york1 #aijb71 p4 #bai4 #ecc4 ,! problem = a writ] ( a text-book has come n[1 9 fact1 to be ?is--to write a book s neatly trimm$ & compact$ t no coa*1 on look+ ?r\gh x1 c m>k a s+le passage :i* ! c&idate = a m9imum pass c safely omit4 ,"s ( ^! text-books ,i h se51 :]e ! sci5tific matt] has be51 l ! lady's wai/ 9 ! nurs]y song1 compress$ 8s g5t & sma'10 t ! ?ickness b>ely1 if at all1 surpasses :at is devot$ to ! publi%]'s adv]tisem5ts4 ,we %all return1 ,i v]ily believe1 to ! ,comp5dium ( ,m>tianus ,capella4 ,! result ( all ?is is t sci5ce1 9 ! h&s ( speciali/s1 so>s high] & high] into ! light ( day1 :ile educators & ! educat$ are left m & m to w&] 9 primeval d>kness4--,,*ry/al1 ,,george4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 #ahhe2 ,nature1 ,vol4 #cb1 p4 #ddh4 #ecd4 ,"s p]sons h cont5d$ t ma!matics ought to be taught by mak+ ! illu/rations obvi\s to ! s5ses4 ,no?+ c be m absurd or injuri\s3 x ought to be our nev]-ceas+ ef=t to make p ?9k1 n feel4--,,col]idge1 ,s4 ,t4 ,lectures on ,%akesp]e 7,bohn ,libr>y71 p4 #eb4 #ece4 ,i h come to ! conclusion t ! ex]tion1 )\t :i* a k ( ma!matics _c be acquir$1 is n mat]ially increas$ by logical rigor 9 ! me?od ( in/ruction4 --,,pr+%eim1 ,,alfr$4 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung 7#ahih71 p4 #adc4 #ecf4 ,! only way 9 :i* to treat ! elem5ts ( an exact & rigor\s sci5ce is to apply to !m all ! rigor & exactness possible4--,d',,alemb]t4 ,quot$ by ,de ,morgan3 ,trigonometry & ,d\ble ,algebra 7,london1 #ahdi71 ,title page4 #ecg4 ,x is an error to believe t rigor 9 pro( is an enemy ( simplicity4 ,on ! contr>y we f9d x confirm$ by num]\s examples t ! rigor\s me?od is at ! same "t ! simpl] & ! m easily compreh5d$4 ,! v ef=t = rigor =ces u to f9d out simpl] me?ods ( pro(4--,,hilb]t1 ,d4 ,ma!matical ,problems2 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #h1 p4 #dda4 #ech4 ,few w d5y t ev5 9 ! fir/ sci5tific in/ruction 9 ma!matics ! mo/ rigor\s me?od is to be giv5 pref]5ce ov] all o!rs4 ,especially w e tea*] pref] a consi/5t pro( to "o :i* is bas$ on fallacies or proce$s 9 a vici\s circle1 inde$ x w be morally impossible = ! tea*] to pres5t a pro( ( ! latt] k9d consci\sly & ?us 9 a s5se deceive 8 pupils4 ,not)/&+ ^! objectionable s-call$ pro(s1 s f> z ! f\ndation & ! developm5t ( ! sy/em is conc]n$1 pr$om9ate 9 our textbooks to ! pres5t "t4 ,p]haps x w be answ]$1 t rigor\s pro( is f\nd too difficult = ! pupil's p[] ( compreh5sion4 ,%\ld ?is be any:]e ! case1--:i* w\ld only indicate "s defect 9 ! plan or treatm5t ( ! :ole1--! only rem$y w\ld be to m]ely /ate ! !orem 9 a hi/oric way1 & =ego a pro( ) ! frank confession t no pro( has be5 f\nd :i* c\ld be compreh5d$ by ! pupil2 a rem$y :i* is ev] d\btful & %\ld only be appli$ 9 ! case ( extreme necessity4 ,b ?is rem$y is to be pref]r$ to a pro( :i* is no pro(1 & is !re=e ei!r :olly un9telligible to ! pupil1 or deceives him ) an appe>ance ( k :i* op5s ! door to all sup]ficiality & lack ( sci5tific me?od4--,,grassmann1 ,,h]mann4 ,/cke aus dem ,lehrbu*e d] ,ari?metik2 ,w]ke1 ,bd4 #b 7,leipsig1 #aijd71 p4 #bif4 #eci4 ,! av]age ,engli% au?or ( ma!matical texts leaves "o "u ! impression t he has made a b>ga9 ) 8 read] to put be=e him ! tru?1 ! great] "p ( ! tru?1 & no?+ b ! tru?2 & t if he has put ! facts ( 8 subject into 8 book1 h[ev] difficult x may be to une>? !m1 he has fulfill$ 8 contract ) 8 read]4 ,?is is a v mu* mi/ak5 view1 because effective tea*+ requires a great deal m ?an a b>e recitation ( facts1 ev5 if ^! are duly set =? 9 logical ord]--z 9 ,engli% books !y (t5 are n4 ,! probable difficulties :i* w occur to ! /ud5t1 ! objections :i* ! intellig5t /ud5t w naturally & necess>ily raise to "s /atem5t ( fact or !ory--^! ?+s our au?ors seldom or nev] notice1 & yet a recognition & anticipation ( !m by ! au?or w\ld be (t5 ( priceless value to ! /ud5t4 ,aga91 a t\* ( hum\r 7/range z ! cont5tion may seem7 9 ma!matical works is n only possible ) p]fect propriety1 b v helpful2 & ,i c\ld give in/ances ( ?is ev5 f ! pure ma!matics ( ,salmon & ! physics ( ,cl]k ,maxwell4 --,,m9*91 ,g4 ,m4 ,p]ry's ,tea*+ ( ,ma!matics 7,london1 #aijb71 pp4 #ei-fa4 #edj4 ,rememb] ?is1 ! rule = giv+ an extempore lecture is--let ! m9d re/ f ! subject entirely = an int]val prec$+ ! lecture1 aft] ! notes are prep>$2 ! ?\ghts w f]m5t )\t y\r kn[+ x1 & ent] into new comb9ations2 b if y keep ! m9d active ^u ! subject up to ! mom5t1 ! subject w n f]m5t b /upefy4 --,,de ,,morgan1 ,a4 ,lett] to ,hamilton2 ,graves3 ,life ( ,w4 ,r4 ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #dhg4 ,,*apt] ,,vi ,,/udy ,,& ,,rese>* ,,9 ,,ma!matics #fja4 ,! fir/ ?+ to be att5d$ to 9 read+ any algebraic treatise is ! ga9+ a p]fect und]/&+ ( ! diff]5t processes !re exhibit$1 & ( _! connection ) "o ano!r4 ,?is _c be atta9$ by ! m]e read+ ( ! book1 h[ev] great ! att5tion :i* may be giv54 ,x is impossible 9 a ma!matical "w to fill up e process 9 ! mann] 9 :i* x mu/ be fill$ up 9 ! m9d ( ! /ud5t be=e he c be said to h completely ma/]$ x4 ,_m results mu/ be giv5 ( :i* ! details are suppress$1 su* are ! additions1 multiplications1 extractions ( squ>e roots1 etc41 ) :i* ! inve/igations ab\nd4 ,^! mu/ n be tak5 on tru/ by ! /ud5t1 b mu/ be work$ out by 8 own p51 :i* mu/ nev] be out ( 8 own h& :ile engag$ 9 any ma!matical process4--,,de ,,morgan1 ,a4 ,/udy & ,difficulties ( ,ma!matics 7,*icago1 #aijb71 *ap4 #ab4 #fjb4 ,! /ud5t %\ld n lose any opportunity ( ex]cis+ himself 9 num]ical calculation & p>ticul>ly 9 ! use ( log>i?mic tables4 ,8 p[] ( apply+ ma!matics to que/ions ( practical utility is 9 direct proportion to ! facility :i* he possesses 9 computation4--,,de ,,morgan1 ,a4 ,/udy & ,difficulties ( ,ma!matics 7,*icago1 #aijb71 *ap4 #ab4 #fjc4 ,! examples :i* a beg9n] %\ld *oose = practice %\ld be simple & %\ld n conta9 v l>ge numb]s4 ,! p[]s ( ! m9d _c be direct$ to two ?+s at once2 if ! complexity ( ! numb]s us$ requires all ! /ud5t's att5tion1 he _c obs]ve ! pr9ciple ( ! rule :i* he is foll[+4--,,de ,,morgan1 ,a4 ,/udy & ,difficulties ( ,ma!matics 7,*icago1 #aijb71 *ap4 #c4 #fjd4 ,euclid & ,ar*im$es are all[$ to be kn[+1 & to h demon/rat$ :at !y say3 & yet :osoev] %all read ov] _! writ+s )\t p]ceiv+ ! connection ( _! pro(s1 & see+ :at !y %[1 ?\gh he may und]/& all _! words1 yet he is n ! m kn[+4 ,he may believe1 inde$1 b does n "k :at !y say1 & s is n advanc$ "o jot 9 ma!matical k by all 8 read+ ( ^? approv$ ma!maticians4--,,locke1 ,,john4 ,conduct ( ! ,und]/&+1 sect4 #bd4 #fje4 ,! /ud5t %\ld read 8 au?or ) ! mo/ su/a9$ att5tion1 9 ord] to discov] ! mean+ ( e s5t5ce4 ,if ! book is well writt51 x w endure & repay 8 close att5tion3 ! text ought to be fairly intelligible1 ev5 )\t illu/rative examples4 ,(t51 f> too (t51 a read] hurries ov] ! text )\t any s9c]e & vigor\s ef=t to und]/& x2 & ru%es to "s example to cle> up :at ought n to h be5 obscure1 if x _h be5 adequately consid]$4 ,! habit ( scrupul\sly inve/igat+ ! text seems to me important on sev]al gr\nds4 ,! close scrut9y ( language is a v valuable ex]cise bo? = /udi\s & practical life4 ,9 ! high] dep>tm5ts ( ma!matics ! habit is indisp5sable3 9 ! long inve/igations :i* occur !re x w\ld be impossible to int]pose illu/rative examples at e /age1 ! /ud5t mu/ !re=e enc\nt] & ma/]1 s5t5ce by s5t5ce1 an ext5sive & complicat$ argum5t4--,,todhunt]1 ,,isaac4 ,private ,/udy ( ,ma!matics2 ,conflict ( ,/udies & o!r ,essays 7,london1 #ahgc71 p4 #fg4 #fjf4 ,x mu/ happ5 t 9 "s cases ! au?or is n und]/ood1 or is v imp]fectly und]/ood2 & ! "q is :at is to be done4 ,aft] giv+ a reasonable am\nt ( att5tion to ! passage1 let ! /ud5t pass on1 res]v+ ! obscurity = future ef=ts4444 ,! natural t5d5cy ( solit>y /ud5ts1 ,i believe1 is n to hurry away prematurely f a h>d passage1 b to hang f> too long ov] x2 ! j pride t does n l to ackn[l$ge defeat1 & ! /rong w t _c endure to be ?w>t$1 bo? urge to a cont9uance ( ef=t ev5 :5 success seems hopeless4 ,x is only by exp]i5ce we ga9 ! conviction t :5 ! m9d is ?or\ghly fatigu$ x has nei!r ! p[] to cont9ue ) advantage its c\rse 9 an assign$ direction1 nor ela/icity to /rike out a new pa?2 b t1 on ! o!r h&1 aft] be+ )drawn = a "t f ! pursuit1 x may return & ga9 ! desir$ end4--,,todhunt]1 ,,isaac4 ,private ,/udy ( ,ma!matics2 ,conflict ( ,/udies & o!r ,essays 7,london1 #ahgc71 p4 #fh4 #fjg4 ,e ma!matical book t is wor? read+ mu/ be read 8backw>ds & =w>ds10 if ,i may use ! expression4 ,i w\ld modify ,lagrange's advice a little & say1 8,g on1 b (t5 return to /r5g!n y\r fai?40 ,:5 y come on a h>d or dre>y passage1 pass x ov]2 & come back to x aft] y h se5 its importance or f\nd ! ne$ = x fur!r on4 --,,*ry/al1 ,,george4 ,algebra1 ,"p #b 7,ed9burgh1 #ahhi71 ,preface1 p4 #h4 #fjh4 ,! l>ge collection ( problems :i* our mod]n ,cambridge books supply w be f\nd to be almo/ an exclusive peculi>ity ( ^! books2 su* collections sc>cely exi/ 9 =eign treatises on ma!matics1 nor ev5 9 ,engli% treatises ( an e>li] date4 ,?is fact %[s1 ,i ?9k1 t a k ( ma!matics may be ga9$ )\t ! p]petual work+ ( examples4444 ,d n tr\ble y\rselves ) ! examples1 make x y\r ma9 bus9ess1 ,i might almo/ say y\r exclusive bus9ess1 to und]/& ! text ( y\r au?or4--,,todhunt]1 ,,isaac4 ,private ,/udy ( ,ma!matics2 ,conflict ( ,/udies & o!r ,essays 7,london1 #ahgc71 p4 #gd4 #fji4 ,9 my op9ion ! ,engli% excel 9 ! art ( writ+ text-books = ma!matical tea*+2 z reg>ds ! cle> exposition ( !ories & ! abundance ( excell5t examples1 c>efully select$1 v few books exi/ 9 o!r c\ntries :i* c compete ) ^? ( ,salmon & _m o!r di/+ui%$ ,engli% au?ors t c\ld be nam$4--,,cremona1 ,l4 ,projective ,geometry ,leudesdorf 7,ox=d1 #ahhe71 ,preface4 #faj4 ,! solution ( fallacies1 :i* give rise to absurdities1 %\ld be to him :o is n a fir/ beg9n] 9 ma!matics an excell5t means ( te/+ = a prop] intelligible insight into ma!matical tru?1 ( %>p5+ ! wit1 & ( conf9+ ! judgm5t & reason )9 /rictly ord]ly limits4--,,viola1 ,j4 ,ma!matis*e ,sophism5 7,wi51 #ahfd71 ,vorwort4 #faa4 ,success 9 ! solution ( a problem g5]ally dep5ds 9 a great measure on ! selection ( ! mo/ appropriate me?od ( approa*+ x2 _m prop]ties ( conic sections 7= in/ance7 be+ demon/rable by a few /eps ( pure geometry :i* w\ld involve ! mo/ labori\s op]ations ) tril9e> co-ord9ates1 :ile o!r prop]ties are almo/ self-evid5t "u ! me?od ( tril9e> co-ord9ates1 :i* x w\ld p]haps be actually impossible to prove by ! old geometry4 --,,:itwor?1 ,w4 ,a4 ,mod]n ,analytic ,geometry 7,cambridge1 #ahff71 p4 #aed4 #fab4 ,! deep /udy ( nature is ! mo/ fruitful s\rce ( ma!matical discov]ies4 ,by (f]+ to rese>* a def9ite end1 ?is /udy has ! advantage ( exclud+ vague que/ions & useless calculations2 besides x is a sure means ( =m+ analysis itself & ( discov]+ ! elem5ts :i* x mo/ conc]ns u to "k1 & :i* natural sci5ce ought always to cons]ve4--- ,,f\ri]1 ,j4 ,?orie ,analytique de la ,*aleur1 ,disc\rs ,prlim9aire4 #fac4 ,x is c]ta9ly true t all physical ph5om5a are subject to /rictly ma!matical conditions1 & ma!matical processes are unassailable 9 !mselves4 ,! tr\ble arises f ! data employ$4 ,mo/ ph5om5a are s highly complex t "o c nev] be q sure t he is deal+ ) all ! factors until ! exp]im5t proves x4 ,s t exp]im5t is r ! crit]ion ( ma!matical conclusions & mu/ lead ! way4 --,,dolbe>1 ,a4 ,e4 ,matt]1 ,e!r1 ,motion 7,bo/on1 #ahid71 p4 #hi4 #fad4 ,/ud5ts %\ld le>n to /udy at an e>ly /age ! great works ( ! great ma/]s in/ead ( mak+ _! m9ds /]ile ?r\gh ! ev]la/+ ex]cises ( college1 :i* are ( no use :atev]1 except to produce a new ,arcadia :]e indol5ce is veil$ "u ! =m ( useless activity4444 ,h>d /udy on ! great models has ev] br\ght out ! /rong2 & ( su* mu/ be our new sci5tific g5]ation if x is to be wor?y ( ! era to :i* x is born & ( ! /ruggles to :i* x is de/9$4 --,,beltrami4 ,giornale di matemati*e1 ,vol4 #aa1 p4 #aec4 ,"y1 ,j4 ,w4 #fae4 ,! hi/ory ( ma!matics may be in/ructive z well z agreeable2 x may n only rem9d u ( :at we h1 b may also tea* u to increase our /ore4 ,says ,de ,morgan1 8,! e>ly hi/ory ( ! m9d ( m5 ) reg>ds to ma!matics leads u to po9t out our own errors2 & 9 ?is respect x is well to pay att5tion to ! hi/ory ( ma!matics40 ,x w>ns u aga9/ ha/y conclusions2 x po9ts out ! importance ( a good notation ^u ! progress ( ! sci5ce2 x disc\rages excessive specialization on ! "p ( ! inve/igator1 by %[+ h[ app>5tly di/9ct bran*es h be5 f\nd to possess unexpect$ connect+ l9ks2 x saves ! /ud5t f wa/+ "t & en]gy ^u problems :i* 71 p]haps1 solv$ long s9ce2 x disc\rages him f attack+ an unsolv$ problem by ! same me?od :i* has l$ o!r ma!maticians to failure2 x tea*es t =tifications c be tak5 by o!r ways ?an by direct attack1 t :5 repuls$ f a direct assault x is well to reconnoitre & occupy ! surr\nd+ gr\nd & to discov] ! secret pa?s by :i* ! app>5tly unconqu]able position c be tak54--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 pp4 #a-b4 #faf4 ,! hi/ory ( ma!matics is important also z a valuable contribution to ! hi/ory ( civilization4 ,human progress is closely id5tifi$ ) sci5tific ?\ght4 ,ma!matical & physical rese>*es are a reliable record ( intellectual progress4--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #d4 #fag4 ,x w\ld be ra% to say t no?+ rema9s = discov]y or improvem5t ev5 9 elem5t>y ma!matics1 b x may be safely ass]t$ t ! gr\nd has be5 s long & s ?or\ghly explor$ z to hold out little hope ( pr(itable return = a casual adv5tur]4--,,todhunt]1 ,,isaac4 ,private ,/udy ( ,ma!matics2 ,conflict ( ,/udies & o!r ,essays 7,london1 #ahgc71 p4 #gc4 #fah4 ,we d n live 9 a "t :5 k c be ext5d$ along a pa?way smoo? & free f ob/acles1 z at ! "t ( ! discov]y ( ! inf9itesimal calculus1 & 9 a measure also :5 9 ! developm5t ( projective geometry ob/acles 7 sudd5ly remov$ :i*1 hav+ hemm$ progress = a long "t1 p]mitt$ a /ream ( inve/igators to p\r 9 ^u virg9 soil4 ,!re is no long] any br[s+ along ! beat5 pa?s2 & into ! primeval =e/ only ^? may v5ture :o are equipp$ ) ! %>pe/ tools4--,,burkh>dt1 ,h4 ,ma!matis*es und wiss5s*aftli*es ,d5k52 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #aa1 p4 #ee4 #fai4 ,?\gh we mu/ n )\t fur!r consid]ation condemn a body ( reason+ m]ely because x is easy1 nev]!less we mu/ n all[ ourselves to be lur$ on m]ely by eas9ess2 & we %\ld take c>e t e problem :i* we *oose = attack1 :e!r x be easy or difficult1 %all h a useful purpose1 t x %all contribute 9 "s measure to ! up-build+ ( ! great edifice4--,,segre1 ,,corradi4 ,"s ,rec5t ,t5d5cies 9 ,geometric ,inve/igation2 ,rivi/a di ,matematica 7#ahia71 p4 #fc4 ,bullet9 ,am]ican ,ma!matical ,society1 #aijd1 p4 #dfe4 ,"y1 ,j4 ,w44 #fbj4 ,no ma!matician n[-a-days sets any /ore on ! discov]y ( isolat$ !orems1 except z af=d+ h9ts ( an unsuspect$ new sph]e ( ?\ght1 l meteorites deta*$ f "s undiscov]$ planet>y orb ( speculation4--,,sylve/]1 ,j4 ,j4 ,notes to ! ,exet] ,association ,address2 ,collect$ ,ma!matical ,pap]s 7,cambridge1 #aijh71 ,vol4 #b1 p4 #gae4 #fba4 ,isolat$1 s-call$ 8pretty !orems0 h ev5 less value 9 ! eyes ( a mod]n ma!matician ?an ! discov]y ( a new 8pretty fl[]0 has to ! sci5tific botani/1 ?\gh ! layman f9ds 9 ^! ! *ief *>m ( ! respective sci5ces4 --,,hankel1 ,,h]mann4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 p4 #ae4 #fbb4 ,x is1 s to speak1 a sci5tific tact1 :i* mu/ guide ma!maticians 9 _! inve/igations1 & gu>d !m f sp5d+ _! =ces on sci5tifically wor?less problems & ab/ruse realms1 a tact :i* is closely relat$ to e/hetic tact & :i* is ! only ?+ 9 our sci5ce :i* _c be taught or acquir$1 & is yet ! indisp5sable end[m5t ( e ma!matician4--,,hankel1 ,,h]mann4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 p4 #ba4 #fbc4 ,! ma!matician requires tact & good ta/e at e /ep ( 8 "w1 & he has to le>n to tru/ to 8 own in/9ct to di/+ui% betwe5 :at is really wor?y ( 8 ef=ts & :at is n2 he mu/ take c>e n to be ! slave ( 8 symbols1 b always to h be=e 8 m9d ! realities :i* !y m]ely s]ve to express4 ,= ^! & o!r reasons x seems to me ( ! highe/ importance t a ma!matician %\ld be tra9$ 9 no n>r[ s*ool2 a wide c\rse ( read+ 9 ! fir/ few ye>s ( 8 ma!matical /udy _c fail to influ5ce = good ! "* ( ! :ole ( 8 subsequ5t "w4--,,glai%]1 ,j4 ,w4 ,l4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a1 7#ahij72 ,nature1 ,vol4 #db1 p4 #dfg4 #fbd4 ,z long z a bran* ( sci5ce (f]s an abundance ( problems1 s long x is alive2 a lack ( problems =e%ad[s ext9ction or ! cessation ( indep5d5t developm5t4 --,,hilb]t1 ,d4 ,ma!matical ,problems2 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #h1 p4 #dch4 #fbe4 ,9 ma!matics z 9 o!r fields1 to f9d "o self lo/ 9 wond] at "s manife/ation is frequ5tly ! half ( a new discov]y4--,,diri*let1 ,p4 ,g4 ,l4 ,w]ke1 ,bd4 #b 7,b]l91 #ahig71 p4 #bcc4 #fbf4 ,! /ud5t ( ma!matics (t5 f9ds x h>d to ?r[ (f ! uncom=table feel+ t 8 sci5ce1 9 ! p]son ( 8 p5cil1 surpasses him 9 intellig5ce1--an impression :i* ! great ,eul] confess$ he (t5 c\ld n get rid (4 ,?is feel+ f9ds a sort ( ju/ification :5 we reflect t ! majority ( ! ideas we deal ) 7 conceiv$ by o!rs1 (t5 c5turies ago4 ,9 a great measure x is really ! intellig5ce ( o!r p t confronts u 9 sci5ce4 --,,ma*1 ,,ern/4 ,popul> ,sci5tific ,lectures 7,*icago1 #aiaj71 p4 #aif4 #fbg4 ,x is probably ?is fact ref]r+ to ! circum/ance t ! problems ( ! p>allel axiom1 ! squ>+ ( ! circle1 ! solution ( ! equation ( ! fif? degree1 h f9ally f\nd fully satisfactory & rigor\s solutions along ) o!r philosophical reasons t gives rise to ! conviction 7:i* e ma!matician %>es1 b :i* no "o has yet support$ by pro(7 t e def9ite ma!matical problem mu/ necess>ily be susceptible ( an exact settlem5t1 ei!r 9 ! =m ( an actual answ] to ! "q ask$1 or by ! pro( ( ! impossibility ( its solution & !re) ! necess>y failure ( all attempts4444 ,?is conviction ( ! solvability ( e ma!matical problem is a p[]ful inc5tive to ! work]4 ,we he> )9 u ! p]petual call3 ,!re is ! problem4 ,seek its solution4 ,y c f9d x by pure reason1 = 9 ma!matics !re is no ignorabimus4--,,hilb]t1 ,d4 ,ma!matical ,problems2 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #h1 pp4 #ddd-dde4 #fbh4 ,he :o seeks = me?ods )\t hav+ a def9ite problem 9 m9d seeks = ! mo/ "p 9 va94--,,hilb]t1 ,d4 ,ma!matical ,problems2 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #h1 p4 #ddd4 #fbi4 ,a ma!matical problem %\ld be difficult 9 ord] to entice u1 yet n completely inaccessible1 le/ x mock at our ef=ts4 ,x %\ld be to u a guide po/ on ! mazy pa?s to hidd5 tru?s1 & ultimately a rem9d] ( our pleasure 9 ! successful solution4--,,hilb]t1 ,d4 ,ma!matical ,problems2 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #h1 p4 #dch4 #fcj4 ,! great ma!maticians h act$ on ! pr9ciple 8,div9ez avant de demontr]10 & x is c]ta9ly true t almo/ all important discov]ies are made 9 ?is fa%ion4 --,,kasn]1 ,,edw>d4 ,! ,pres5t ,problems 9 ,geometry2 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #aa1 p4 #bhe4 #fca4 8,divide et imp]a0 is z true 9 algebra z 9 /atecraft2 b no less true & ev5 m f]tile is ! maxim 8auge et imp]a40 ,! m to d or to prove1 ! easi] ! do+ or ! pro(4--,,sylve/]1 ,j4 ,j4 ,pro( ( ! ,fundam5tal ,!orem ( ,inv>iants2 ,philosophic ,magaz9e 7#ahgh71 p4 #ahf2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #c1 p4 #abf4 #fcb4 ,z 9 ! doma9s ( practical life s likewise 9 sci5ce !re has come ab\t a division ( labor4 ,! individual c no long] control ! :ole field ( ma!matics3 x is only possible = him to ma/] sep>ate p>ts ( x 9 su* a mann] z to enable him to ext5d ! b\nd>ies ( k by creative rese>*4--,,lampe1 ,e4 ,die re9e ,ma!matik 9 d5 ,jahr5 #ahhd-ahii1 p4 #aj4 #fcc4 ,) ! ext5sion ( ma!matical k w x n f9ally become impossible = ! s+le inve/igator to embrace all dep>tm5ts ( ?is k8 ,9 answ] let me po9t out h[ ?or\ghly x is +ra9$ 9 ma!matical sci5ce t e real advance goes h& 9 h& ) ! inv5tion ( %>p] tools & simpl] me?ods :i* at ! same "t assi/ 9 und]/&+ e>li] !ories & to ca/ aside "s m complicat$ developm5ts4 ,x is !re=e possible = ! individual inve/igator1 :5 he makes ^! %>p] tools & simpl] me?ods 8 own1 to f9d 8 way m easily 9 ! v>i\s bran*es ( ma!matics ?an is possible 9 any o!r sci5ce4--,,hilb]t1 ,d4 ,ma!matical ,problems2 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #h1 p4 #dgi4 #fcd4 ,x w\ld seem at fir/ sight z if ! rapid expansion ( ! region ( ma!matics mu/ be a s\rce ( dang] to its future progress4 ,n only does ! area wid5 b ! subjects ( /udy increase rapidly 9 numb]1 & ! "w ( ! ma!matician t5ds to become m & m specializ$4 ,x is1 ( c\rse1 m]ely a brilliant exagg]ation to say t no ma!matician is able to und]/& ! "w ( any o!r ma!matician1 b x is c]ta9ly true t x is daily becom+ m & m difficult = a ma!matician to keep himself acqua9t$1 ev5 9 a g5]al way1 ) ! progress ( any ( ! bran*es ( ma!matics except ^? :i* =m ! field ( 8 own lab\rs4 ,i believe1 h[ev]1 t ! increas+ ext5t ( ! t]ritory ( ma!matics w always be c\nt]act$ by increas$ facilities 9 ! means ( communication4 ,additional k op5s to u new pr9ciples & me?ods :i* may conduct u ) ! greate/ ease to results :i* previ\sly 7 mo/ difficult ( access2 & improvem5ts 9 notation may ex]cise ! mo/ p[]ful effects bo? 9 ! simplification & accessibility ( a subject4 ,x re/s ) ! work] 9 ma!matics n only to explore new tru?s1 b to devise ! language by :i* !y may be discov]$ & express$2 & ! g5ius ( a great ma!matician displays itself no less 9 ! notation he inv5ts = deciph]+ 8 subject ?an 9 ! results atta9$4444 ,i h great fai? 9 ! p[] ( well-*os5 notation to simplify complicat$ !ories & to br+ remote ones ne> & ,i ?9k x is safe to pr$ict t ! increas$ k ( pr9ciples & ! result+ improvem5ts 9 ! symbolic language ( ma!matics w always enable u to grapple satisfactorily ) ! difficulties aris+ f ! m]e ext5t ( ! subject4--,,glai%]1 ,j4 ,w4 ,l4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a41 7#ahij71 ,nature1 ,vol4 #db1 p4 #dff4 #fce4 ,q di/9ct f ! !oretical "q ( ! mann] 9 :i* ma!matics w rescue itself f ! p]ils to :i* x is expos$ by its own prolific nature is ! practical problem ( f9d+ means ( r5d]+ available = ! /ud5t ! results :i* h be5 already accumulat$1 & mak+ x possible = ! le>n] to obta9 "s idea ( ! pres5t /ate ( ! v>i\s dep>tm5ts ( ma!matics4444 ,! great mass ( ma!matical lit]ature w be always conta9$ 9 ,j\rnals & ,transactions1 b !re is no reason :y x %\ld n be r5d]$ f> m useful & accessible ?an at pres5t by means ( treatises or high] text-books4 ,! :ole sci5ce suff]s f want ( av5ues ( approa*1 & _m beautiful bran*es ( ma!matics are reg>d$ z difficult & te*nical m]ely because !y are n easily accessible4444 ,i feel v /rongly t any introduction to a new subject writt5 by a compet5t p]son conf]s a real b5efit on ! :ole sci5ce4 ,! numb] ( excell5t text-books ( an elem5t>y k9d t are publi%$ 9 ?is c\ntry makes x all ! m to be regrett$ t we h s few t are int5d$ = ! advanc$ /ud5t4 ,z an example ( ! high] k9d ( text-book1 ! want ( :i* is s badly felt 9 _m subjects1 ,i may m5tion ! second "p ( ,pr(4 ,*ry/al's 8,algebra0 publi%$ la/ ye>1 :i* 9 a small compass gives a great mass ( valuable & fundam5tal k t has hi!rto be5 beyond ! rea* ( an ord9>y /ud5t1 ?\gh 9 reality ly+ s close at h&4 ,i may add t 9 any treatise or high] text-book x is always desirable t ref]5ces to ! orig9al memoirs %\ld be giv51 &1 if possible1 %ort hi/oric notices also4 ,i am sure t no subject loses m ?an ma!matics by any attempt to dissociate x f its hi/ory4--,,glai%]1 ,j4 ,w4 ,l4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a 7#ahij72 ,nature1 ,vol4 #db1 p4 #dff4 #fcf4 ,! m a sci5ce advances1 ! m w x be possible to und]/& imm$iately results :i* =m]ly c\ld be demon/rat$ only by means ( l5g?y int]m$iate consid]ations3 a ma!matical subject _c be consid]$ z f9ally complet$ until ?is end has be5 atta9$4--,,gordan1 ,,paul4 ,=m5sy/em b9r] ,=m5 7,leipzig1 #ahge71 p4 #b4 #fcg4 ,an old ,fr5* geomet] us$ to say t a ma!matical !ory 0 nev] to be consid]$ complete till y _h made x s cle> t y c\ld expla9 x to ! fir/ man y met 9 ! /reet4--,,smi?1 ,h4 ,j4 ,s4 ,nature1 ,vol4 #h 7#ahgc71 p4 #deb4 #fch4 ,9 ord] to compreh5d & fully control ari?metical concepts & me?ods ( pro(1 a high degree ( ab/raction is necess>y1 & ?is condition has at times be5 *>g$ aga9/ ari?metic z a fault4 ,i am ( ! op9ion t all o!r fields ( k require at lea/ an equally high degree ( ab/raction z ma!matics1--provid$1 t 9 ^! fields ! f\ndations are also ev]y:]e exam9$ ) ! rig\r & complet5ess :i* is actually necess>y4--,,hilb]t1 ,d4 ,die ,!orie d] algebrais*5 ,zahlkorp]1 ,vorwort2 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #d4 #fci4 ,! anxi\s precision ( mod]n ma!matics is necess>y = accuracy1 444 x is necess>y = rese>*4 ,x makes = cle>ness ( ?\ght & = f]tility 9 try+ new comb9ations ( ideas4 ,:5 ! initial /atem5ts are vague & slip%od1 at e subsequ5t /age ( ?\ght1 common s5se has to /ep 9 to limit applications & to expla9 mean+s4 ,n[ 9 creative ?\ght common s5se is a bad ma/]4 ,its sole crit]ion = judgm5t is t ! new ideas %all look l ! old ones1 9 o!r words x c only act by suppress+ orig9ality4 --,,:itehead1 ,a4 ,n4 ,introduction to ,ma!matics 7,new ,york1 #aiaa71 p4 #aeg4 #fdj4 ,ma!maticians atta* great importance to ! elegance ( _! me?ods & _! results4 ,?is is n pure dilettantism4 ,:at is x inde$ t gives u ! feel+ ( elegance 9 a solution1 9 a demon/ration8 ,x is ! h>mony ( ! div]se p>ts1 _! symmetry1 _! happy balance2 9 a ^w x is all t introduces ord]1 all t gives unity1 t p]mits u to see cle>ly & to compreh5d at once bo? ! ensemble & ! details4 ,b ?is is exactly :at yields great results1 9 fact ! m we see ?is aggregate cle>ly & at a s+le glance1 ! bett] we p]ceive its analogies ) o!r neighbor+ objects1 consequ5tly ! m *ances we h ( div9+ ! possible g5]alizations4 ,elegance may produce ! feel+ ( ! un=ese5 by ! unexpect$ meet+ ( objects we are n accu/om$ to br+ toge!r2 !re aga9 x is fruitful1 s9ce x ?us unveils = u k9%ips be=e unrecogniz$4 ,x is fruitful ev5 :5 x results only f ! contra/ betwe5 ! simplicity ( ! means & ! complexity ( ! problem set2 x makes u !n ?9k ( ! reason = ?is contra/ & v (t5 makes u see t *ance is n ! reason2 t x is to be f\nd 9 "s unexpect$ law4 ,9 a ^w1 ! feel+ ( ma!matical elegance is only ! satisfaction due to any adaptation ( ! solution to ! ne$s ( our m9d1 & x is because ( ?is v adaptation t ?is solution c be = u an in/rum5t4 ,consequ5tly ?is e/hetic satisfaction is b\nd up ) ! economy ( ?\ght4--,,po9c>1 ,h4 ,! ,future ( ,ma!matics2 ,moni/1 ,vol4 #bj1 p4 #hj4 ,hal/$4 #fda4 ,! importance ( a result is l>gely relative1 is judg$ diff]5tly by diff]5t m51 & *anges ) ! times & circum/ances4 ,x has (t5 happ5$ t great importance has be5 atta*$ to a problem m]ely on acc\nt ( ! difficulties :i* x pres5t$2 & inde$ if = its solution x has be5 necess>y to inv5t new me?ods1 notewor?y artifices1 etc41 ! sci5ce has ga9$ m p]haps ?r\gh ^! ?an ?r\gh ! f9al result4 ,9 g5]al we may call important all inve/igations relat+ to ?+s :i* 9 !mselves are important2 all ^? :i* h a l>ge degree ( g5]ality1 or :i* unite "u a s+le po9t ( view subjects app>5tly di/9ct1 simplify+ & elucidat+ !m2 all ^? :i* lead to results t promise to be ! s\rce ( num]\s consequ5ces2 etc4 --,,segre1 ,,corradi4 ,"s ,rec5t ,t5d5cies 9 ,geometric ,inve/igations4 ,rivi/a di ,matematica1 ,vol4 #a1 p4 #dd4 ,bullet9 ,am]ican ,ma!matical ,society1 #aijd1 p4 #ddd4 ,"y1 ,j4 ,w44 #fdb4 ,geometric writ+s are n r>e 9 :i* "o w\ld seek 9 va9 = an idea at all novel1 = a result :i* soon] or lat] might be ( s]vice1 = any?+ 9 fact :i* might be de/9$ to survive 9 ! sci5ce2 & "o f9ds in/ead treatises on trivial problems or inve/igations on special =ms :i* h absolutely no use1 no importance1 :i* h _! orig9 n 9 ! sci5ce itself b 9 ! caprice ( ! au?or2 or "o f9ds applications ( kn[n me?ods :i* h already be5 made ?\s&s ( times2 or g5]alizations f kn[n results :i* are s easily made t ! k ( ! latt] suffices to give at once ! =m]4 ,n[ su* "w is n m]ely useless2 x is actually h>mful because x produces a real incumbrance 9 ! sci5ce & an emb>rassm5t = ! m s]i\s inve/igators2 & because (t5 x cr[ds out c]ta9 l9es ( ?\ght :i* might well h des]v$ to be /udi$4 --,,segre1 ,,corradi4 ,on "s ,rec5t ,t5d5cies 9 ,geometric ,inve/igations2 ,rivi/a di ,matematica1 #ahia1 p4 #dc4 ,bullet9 ,am]ican ,ma!matical ,society1 #aijd1 p4 #ddc ,"y1 ,j4 ,w44 #fdc4 ,a /ud5t :o wi%es n[-a-days to /udy geometry by divid+ x %>ply f analysis1 )\t tak+ acc\nt ( ! progress :i* ! latt] has made & is mak+1 t /ud5t no matt] h[ great 8 g5ius1 w nev] be a :ole geomet]4 ,he w n possess ^? p[]ful in/rum5ts ( rese>* :i* mod]n analysis puts into ! h&s ( mod]n geometry4 ,he w rema9 ignorant ( _m geometrical results :i* are to be f\nd1 p]haps implicitly1 9 ! writ+s ( ! analy/4 ,& n only w he be unable to use !m 9 8 own rese>*es1 b he w probably toil to discov] !m himself1 &1 z happ5s v (t51 he w publi% !m z new1 :5 really he has only r$iscov]$ !m4--,,segre1 ,,corradi4 ,on "s rec5t ,t5d5cies 9 ,geometrical ,inve/igations2 ,rivi/a di ,matematica1 #ahia1 p4 #dc4 ,bullet9 ,am]ican ,ma!matical ,society1 #aijd1 p4 #ddc ,"y1 ,j4 ,w44 #fdd4 ,rese>* may />t f def9ite problems ^: importance x recognizes & ^: solution is s\ght m or less directly by all =ces4 ,b equally legitimate is ! o!r me?od ( rese>* :i* only selects ! field ( its activity &1 contr>y to ! fir/ me?od1 freely reconnoitres 9 ! se>* = problems :i* are capable ( solution4 ,diff]5t individuals w hold diff]5t views z to ! relative value ( ^! two me?ods4 ,if ! fir/ me?od leads to great] p5etration x is also easily expos$ to ! dang] ( unproductivity4 ,to ! second me?od we owe ! acquisition ( l>ge & new fields1 9 :i* ! details ( _m ?+s rema9 to be det]m9$ & explor$ by ! fir/ me?od4--,,clebs*1 ,a4 ,zum ,g$*tniss an ,julius ,plck]2 ,gtt+] ,abh&lung51 #af1 #ahga1 ,ma!matis*e ,classe1 p4 #f4 #fde4 ,dur+ a conv]sation ) ! writ] 9 ! la/ weeks ( 8 life1 ,sylve/] rem>k$ z curi\s t not)/&+ he _h always consid]$ ! b5t ( 8 m9d to be r analytical ?an geometrical1 he f\nd 9 ne>ly e case t ! solution ( an analytical problem turn$ ^u "s q simple geometrical notion1 & t he 0 nev] satisfi$ until he c\ld pres5t ! argum5t 9 geometrical language4 --,,macmahon1 ,p4 ,a4 ,proce$+s ,london ,royal ,society1 ,vol4 #fc1 p4 #ag4 #fdf4 ,! orig9 ( a sci5ce is usually to be s\ght = n 9 any sy/ematic treatise1 b 9 ! inve/igation & solution ( "s p>ticul> problem4 ,?is is especially ! case 9 ! ord9>y hi/ory ( ! great improvem5ts 9 any dep>tm5t ( ma!matical sci5ce4 ,"s problem1 ma!matical or physical1 is propos$1 :i* is f\nd to be insoluble by kn[n me?ods4 ,?is condition ( insolubility may arise f "o ( two causes3 ,ei!r !re exi/s no ma*9]y p[]ful en\gh to effect ! requir$ r$uction1 or ! workm5 are n suffici5tly exp]t to employ _! tools 9 ! p]=mance ( an entirely new piece ( "w4 ,! problem propos$ is1 h[ev]1 f9ally solv$1 & 9 its solution "s new pr9ciple1 or new application ( old pr9ciples1 is necess>ily introduc$4 ,if a pr9ciple is br\ght to light x is soon f\nd t 9 its application x is n necess>ily limit$ to ! p>ticul> "q :i* occasion$ its discov]y1 & x is !n /at$ 9 an ab/ract =m & appli$ to problems ( gradually increas+ g5]ality4 ,o!r pr9ciples1 simil> 9 _! nature1 are add$1 & ! orig9al pr9ciple itself receives su* modifications & ext5sions z are f "t to "t deem$ necess>y4 ,! same is true ( new applications ( old pr9ciples2 ! application is fir/ ?\ght to be m]ely conf9$ to a p>ticul> problem1 b x is soon recogniz$ t ?is problem is b "o1 & g5]ally a v simple "o1 out ( a l>ge class1 to :i* ! same process ( inve/igation & solution are applicable4 ,! result 9 bo? ( ^! cases is ! same4 ,a "t comes :5 ^! sev]al problems1 solutions1 & pr9ciples are gr\p$ toge!r & f\nd to produce an entirely new & consi/5t me?od2 a nom5clature & uni=m sy/em ( notation is adopt$1 & ! pr9ciples ( ! new me?od become entitl$ to rank z a di/9ct sci5ce4--,,craig1 ,,?omas4 ,a ,treatise on ,projection1 ,preface4 ,u4 ,s4 ,coa/ & ,geodetic ,survey1 ,treasury ,dep>tm5t ,docum5t1 ,no4 #fa4 #fdg4 ,! aim ( rese>* is ! discov]y ( ! equations :i* subsi/ betwe5 ! elem5ts ( ph5om5a4--,,ma*1 ,,ern/4 ,popul> ,sci5tific ,lectures 7,*icago1 #aiaj71 p4 #bje4 #fdh4 ,let him ! au?or be p]mitt$ also 9 all humility to add 444 t 9 consequ5ce ( ! l>ge arre>s ( algebraical & ari?metical speculations wait+ 9 8 m9d _! turn to be call$ into outw>d exi/5ce1 he is driv5 to ! alt]native ( leav+ ! fruits ( 8 m$itations to p]i% 7z has be5 ! fate ( too _m =egone !ories1 ! /ill-born prog5y ( 8 bra91 n[ =ev] resolv$ back aga9 into ! primordial matt] ( ?\ght71 or v5tur+ to produce f "t to "t su* imp]fect sket*es z ! pres5t1 calculat$ to evoke ! m5tal co-op]ation ( 8 read]s1 9 :om ! algebraical in/9ct has be5 to "s ext5t develop$1 r ?an to satisfy ! /rict dem&s ( rigor\sly sy/ematic exposition4 --,,sylve/]1 ,j4 ,j4 ,philosophic ,magaz9e 7#ahfc71 p4 #dfj4 #fdi4 ,9 o!r bran*es ( sci5ce1 :]e quick publication seems to be s mu* desir$1 !re may possibly be "s excuse = giv+ to ! _w slov5ly or ill-dige/$ "w1 b !re is no su* excuse 9 ma!matics4 ,! =m ought to be z p]fect z ! sub/ance1 & ! demon/rations z rigor\s z ^? ( ,euclid4 ,! ma!matician has to deal ) ! mo/ exact facts ( ,nature1 & he %\ld sp>e no ef=t to r5d] 8 int]pretation wor?y ( 8 subject1 & to give to 8 "w its highe/ degree ( p]fection4 8,pauca s$ matura0 0 ,gauss's motto4--,,glai%]1 ,j4 ,w4 ,l4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a1 7#ahij72 ,nature1 ,vol4 #db1 p4 #dfg4 #fej4 ,x is ! man n ! me?od t solves ! problem4 --,,mas*ke1 ,h4 ,pres5t ,problems ( ,algebra & ,analysis2 ,congress ( ,arts & ,sci5ces 7,new ,york & ,bo/on1 #aije71 ,vol4 #a1 p4 #ecj4 #fea4 ,today x is no long] que/ion$ t ! pr9ciples ( ! analy/s are ! m f>-rea*+4 ,inde$1 ! syn!si/s lack two ?+s 9 ord] to engage 9 a g5]al !ory ( algebraic configurations3 ^! are on ! "o h& a def9ition ( imag9>y elem5ts1 on ! o!r an int]pretation ( g5]al algebraic concepts4 ,bo? ( ^! h subsequ5tly be5 develop$ 9 syn!tic =m1 b to d ?is ! ess5tial pr9ciple ( syn!tic geometry _h to be set aside4 ,?is pr9ciple :i* manife/s itself s brilliantly 9 ! !ory ( l9e> =ms & ! =ms ( ! second degree1 is ! possibility ( imm$iate pro( by means ( visualiz$ con/ructions4--,,kle91 ,,felix4 ,riemanns*e ,fl*5 7,leipzig1 #aijf71 ,bd4 #a1 p4 #bcd4 #feb4 ,ab/ruse ma!matical rese>*es 444 are 444 (t5 abus$ = hav+ no obvi\s physical application4 ,! fact is t ! mo/ useful p>ts ( sci5ce h be5 inve/igat$ = ! sake ( tru?1 & n = _! usefulness4 ,a new bran* ( ma!matics1 :i* has sprung up 9 ! la/ tw5ty ye>s1 0 d5\nc$ by ! ,a/ronom] ,royal be=e ! ,univ]sity ( ,cambridge z doom$ to be =gott51 on acc\nt ( its uselessness4 ,n[ x turns out t ! reason :y we _c g fur!r 9 our inve/igations ( molecul> action is t we d n "k en\gh ( ?is bran* ( ma!matics4--,,clif=d1 ,w4 ,k4 ,conditions ( ,m5tal ,developm5t2 ,lectures & ,essays 7,london1 #aija71 ,vol4 #a1 p4 #aae4 #fec4 ,9 geometry1 z 9 mo/ sci5ces1 x is v r>e t an isolat$ proposition is ( imm$iate utility4 ,b ! !ories mo/ p[]ful 9 practice are =m$ ( propositions :i* curiosity alone br\ght to light1 & :i* long rema9$ useless )\t its be+ able to div9e 9 :at way !y %\ld "o day cease to be s4 ,9 ?is s5se x may be said1 t 9 real sci5ce1 no !ory1 no rese>*1 is 9 effect useless4--,,voltaire4 ,a ,philosophical ,diction>y1 ,article 8,geometry02 7,bo/on1 #ahha71 ,vol4 #a1 p4 #cgd4 #fed4 ,sci5tific subjects d n progress necess>ily on ! l9es ( direct usefulness4 ,v _m applications ( ! !ories ( pure ma!matics h come _m ye>s1 sometimes c5turies1 aft] ! actual discov]ies !mselves4 ,! weapons 7 at h&1 b ! m5 7 n able to use !m4 --,,=sy?1 ,a4 ,r4 ,p]ry's ,tea*+ ( ,ma!matics 7,london1 #aijb71 p4 #ce4 #fee4 ,x is no p>adox to say t 9 our mo/ !oretical moods we may be ne>e/ to our mo/ practical applications4 --,,:itehead1 ,a4 ,n4 ,introduction to ,ma!matics 7,new ,york71 p4 #ajj4 #fef4 ,al?\gh ) ! majority ( ^? :o /udy & practice 9 ^! capacities eng9e]s1 build]s1 surveyors1 geograph]s1 navigators1 hydrograph]s1 a/ronom]s1 second-h& acquirem5ts1 trite =mulas1 & appropriate tables are suffici5t = ord9>y purposes1 yet ^! trite =mulas & famili> rules 7 orig9ally or gradually d$uc$ f ! pr(\nd inve/igations ( ! mo/ gift$ m9ds1 f ! dawn ( sci5ce to ! pres5t day4444 ,! fur!r developm5ts ( ! sci5ce1 ) its possible applications to l>g] purposes ( human utility & gr&] !oretical g5]alizations1 is an a*ievem5t res]v$ = a few ( ! *oice/ spirits1 t\*$ f "t to "t by ,heav5 to ^! highe/ issues4 ,! intellectual _w is fill$ ) lat5t & undiscov]$ tru? z ! mat]ial _w is fill$ ) lat5t electricity4--,,ev]ett1 ,,edw>d4 ,orations & ,spee*es1 ,vol4 #c 7,bo/on1 #ahgj71 p4 #eac4 #feg4 ,if we view ma!matical speculations ) ref]5ce to _! use1 x appe>s t !y %\ld be divid$ into two classes4 ,to ! fir/ belong ^? :i* furni% "s m>k$ advantage ei!r to common life or to "s art1 & ! value ( su* is usually det]m9$ by ! magnitude ( ?is advantage4 ,! o!r class embraces ^? speculations :i*1 ?\gh (f]+ no direct advantage1 are nev]!less valuable 9 t !y ext5d ! b\nd>ies ( analysis & increase our res\rces & skill4 ,n[ s9ce _m inve/igations1 f :i* great advantage may be expect$1 mu/ be ab&on$ solely because ( ! imp]fection ( analysis1 no small value %\ld be assign$ to ^? speculations :i* promise to enl>ge ! field ( analysis4--,,eul]4 ,novi ,comm4 ,petr41 ,vol4 #d1 ,preface4 #feh4 ,! discov]y ( ! conic sections1 attribut$ to ,plato1 fir/ ?rew op5 ! high] species ( =m to ! contemplation ( geomet]s4 ,b = ?is discov]y1 :i* 0 probably reg>d$ 9 ,plato's "t & long aft] him1 z ! unpr(itable amusem5t ( a speculative bra91 ! :ole c\rse ( practical philosophy ( ! pres5t day1 ( ! sci5ce ( a/ronomy1 ( ! !ory ( projectiles1 ( ! art ( navigation1 might h run 9 a diff]5t *annel2 & ! greate/ discov]y t has ev] be5 made 9 ! hi/ory ( ! _w1 ! law ( univ]sal gravitation1 ) its innum]able direct & indirect consequ5ces & applications to e dep>tm5t ( human rese>* & indu/ry1 might nev] to ?is h\r h be5 elicit$4 --,,sylve/]1 ,j4 ,j4 ,a ,probation>y ,lecture on ,geometry2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b 7,cambridge1 #aijh71 p4 #g4 #fei4 ,no m impressive w>n+ c be giv5 to ^? :o w\ld conf9e k & rese>* to :at is app>5tly useful1 ?an ! reflection t conic sections 7 /udi$ = eighte5 hundr$ ye>s m]ely z an ab/ract sci5ce1 )\t reg>d to any utility o!r ?an to satisfy ! crav+ = k on ! "p ( ma!maticians1 & t !n at ! end ( ?is long p]iod ( ab/ract /udy1 !y 7 f\nd to be ! necess>y key ) :i* to atta9 ! k ( ! mo/ important laws ( nature4--,,:itehead1 ,a4 ,n4 ,introduction to ,ma!matics 7,new ,york1 ,york1 #aiaa71 pp4 #acf-acg4 #ffj4 ,! ,greeks 9 ! fir/ vig\r ( _! pursuit ( ma!matical tru?1 at ! "t ( ,plato & soon aft]1 _h by no means conf9$ !mselves to ^? propositions :i* _h a visible be>+ on ! ph5om5a ( nature2 b _h foll[$ out _m beautiful tra9s ( rese>* conc]n+ v>i\s k9ds ( figures1 = ! sake ( _! beauty alone2 z = in/ance 9 _! doctr9e ( ,conic ,sections1 ( :i* curves !y _h discov]$ all ! pr9cipal prop]ties4 ,b x is curi\s to rem>k1 t ^! inve/igations1 ?us pursu$ at fir/ z m]e matt]s ( curiosity & intellectual gratification1 7 de/9$1 two ?\s& ye>s lat]1 to play a v important "p 9 e/abli%+ t sy/em ( cele/ial motions :i* succe$$ ! ,platonic s*eme ( cycles & epicycles4 ,if ! prop]ties ( conic sections _h n be5 demon/rat$ by ! ,greeks & ?us r5d]$ famili> to ! ma!maticians ( succe$+ ages1 ,kepl] w\ld probably n h be5 able to discov] ^? laws respect+ ! orbits & motions ( planets :i* 7 ! occasion ( ! greate/ revolution t ev] happ5$ 9 ! hi/ory ( sci5ce4--,,:ewell1 ,w4 ,hi/ory ( ,sci5tific ,ideas1 ,bk4 #b1 *ap4 #ad1 sect4 #c4 #ffa4 ,! greate/ ma!maticians1 z ,ar*im$es1 ,newton1 & ,gauss1 always unit$ !ory & applications 9 equal measure4 --,,kle91 ,,felix4 ,elem5t>ma!matik vom hh]5 ,/&punkte aus 7,leipzig1 #aiji71 ,bd4 #b1 p4 #cib4 #ffb4 ,we may see h[ unexpect$ly recondite p>ts ( pure ma!matics may be> ^u physical sci5ce1 by call+ to m9d ! circum/ance t ,fresnel obta9$ "o ( ! mo/ curi\s confirmations ( ! !ory 7! laws ( ,circul> ,pol>ization by reflection7 ?r\gh an int]pretation ( an algebraical expression1 :i*1 accord+ to ! orig9al conv5tional mean+ ( ! symbols1 involv$ an impossible quantity4--,,:ewell1 ,w4 ,hi/ory ( ,sci5tific ,ideas1 ,bk4 #b1 *ap4 #ad1 sect4 #h4 #ffc4 ,a great dep>tm5t ( ?\ght mu/ h its own inn] life1 h[ev] transc5d5t may be ! importance ( its relations to ! outside4 ,no dep>tm5t ( sci5ce1 lea/ ( all "o requir+ s high a degree ( m5tal conc5tration z ,ma!matics1 c be develop$ entirely1 or ev5 ma9ly1 ) a view to applications outside its own range4 ,! increas$ complexity & specialisation ( all bran*es ( k makes x true 9 ! pres5t1 h[ev] x may h be5 9 =m] times1 t important advances 9 su* a dep>tm5t z ,ma!matics c be expect$ only f m5 :o are int]e/$ 9 ! subject = its own sake1 & :o1 :il/ keep+ an op5 m9d = sugge/ions f outside1 all[ _! ?\ght to range freely 9 ^? l9es ( advance :i* are indicat$ by ! pres5t /ate ( _! subject1 untrammell$ by any preoccupation z to applications to o!r dep>tm5ts ( sci5ce4 ,ev5 ) a view to applications1 if ,ma!matics is to be adequately equipp$ = ! purpose ( cop+ ) ! intricate problems :i* w be pres5t$ to x 9 ! future by ,physics1 ,*emi/ry & o!r bran*es ( physical sci5ce1 _m ( ^! problems probably ( a "* :i* we _c at pres5t =eca/1 x is ess5tial t ,ma!matics %\ld be all[$ to develop freely on its own l9es4--,,hobson1 ,e4 ,w4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a1 7#aiaj72 ,nature1 ,vol4 #hd1 p4 #bhf4 #ffd4 ,to emphasize ?is op9ion t ma!maticians w\ld be unwise to accept practical issues z ! sole guide or ! *ief guide 9 ! curr5t ( _! inve/igations1 444 let me take "o m in/ance1 by *oos+ a subject 9 :i* ! purely ma!matical int]e/ is deem$ supreme1 ! !ory ( functions ( a complex v>iable4 ,t at lea/ is a !ory 9 pure ma!matics1 initiat$ 9 t region1 & develop$ 9 t region2 x is built up 9 scores ( pap]s1 & its plan c]ta9ly has n be51 & is n n[1 dom9at$ or guid$ by consid]ations ( applicability to natural ph5om5a4 ,yet :at has turn$ out to be its relation to practical issues8 ,! inve/igations ( ,lagrange & o!rs ^u ! con/ruction ( maps appe> z a portion ( ! g5]al prop]ty ( con=mal repres5tation2 :i* is m]ely ! g5]al geometrical me?od ( reg>d+ functional relations 9 t !ory4 ,aga91 ! int]e/+ & important inve/igations ^u discont9u\s two-dim5sional fluid motion 9 hydrodynamics1 made 9 ! la/ tw5ty ye>s1 c all be1 & n[ are all1 ,i believe1 d$uc$ f simil> consid]ations by int]pret+ functional relations betwe5 complex v>iables4 ,9 ! dynamics ( a rotat+ heavy body1 ! only sub/antial ext5sion ( our k s9ce ! "t ( ,lagrange has accru$ f associat+ ! g5]al prop]ties ( functions ) ! discussion ( ! equations ( motion4 ,fur!r1 "u ! title ( conjugate functions1 ! !ory has be5 appli$ to v>i\s que/ions 9 electro/atics1 p>ticul>ly 9 connection ) cond5sors & electromet]s4 ,&1 la/ly1 9 ! doma9 ( physical a/ronomy1 "s ( ! mo/ conspicu\s advances made 9 ! la/ few ye>s h be5 a*iev$ by introduc+ into ! discussion ! ideas1 ! pr9ciples1 ! me?ods1 & ! results ( ! !ory ( functions 444 ! ref9$ & extremely difficult "w ( ,po9c> & o!rs 9 physical a/ronomy has be5 possible only by ! use ( ! mo/ elaborate developm5ts ( "s purely ma!matical subjects1 developm5ts :i* 7 made )\t a ?\ght ( su* applications4--,,=sy?1 ,a4 ,r4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a1 7#ahig72 ,nature1 ,vol4 #ef1 p4 #cgg4 ,,*apt] ,,vii ,,mod]n ,,ma!matics #gja4 ,surely ?is is ! gold5 age ( ma!matics4 --,,pi]pont1 ,,james4 ,hi/ory ( ,ma!matics 9 ! ,n9ete5? ,c5tury2 ,congress ( ,arts & ,sci5ces 7,bo/on & ,new ,york1 #aije71 ,vol4 #a1 p4 #dic4 #gjb4 ,! gold5 age ( ma!matics--t 0 n ! age ( ,euclid1 x is ours4 ,ours is ! age :5 no less ?an six int]national congresses h be5 held 9 ! c\rse ( n9e ye>s4 ,x is 9 our day t m ?an a doz5 ma!matical societies conta9 a gr[+ memb]%ip ( m ?an two ?\s& m5 repres5t+ ! c5t]s ( sci5tific light ?r\gh\t ! great culture nations ( ! _w4 ,x is 9 our "t t ov] five hundr$ sci5tific j\rnals are ea* devot$ 9 "p1 :ile m ?an two score o!rs are devot$ exclusively1 to ! publication ( ma!matics4 ,x is 9 our "t t ! ,jahrbu* b] die ,=ts*ritte d] ,ma!matik1 ?\gh admitt+ only cond5s$ ab/racts ) titles1 & n report+ on all ! j\rnals1 has1 nev]!less1 gr[n to ne>ly =ty huge volumes 9 z _m ye>s4 ,x is 9 our "t t z _m z two ?\s& books & memoirs drop f ! ma!matical press ( ! _w 9 a s+le ye>1 ! e/imat$ numb] m\nt+ up to fifty ?\s& 9 ! la/ g5]ation4 ,f9ally1 to adduce yet ano!r evid5ce ( a simil> k9d1 x requires n less ?an sev5 pond]\s tomes ( ! =?com+ ,encyclopa$ie d] ,ma!matis*5 ,wiss5s*aft5 to conta91 n expositions1 n demon/rations1 b m]ely compact reports & bibliographic notices sket*+ developm5ts t h tak5 place s9ce ! beg9n+ ( ! n9ete5? c5tury4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #h4 #gjc4 ,i h said t ma!matics is ! olde/ ( ! sci5ces2 a glance at its m rec5t hi/ory w %[ t x has ! en]gy ( p]petual y\?4 ,! output ( contributions to ! advance ( ! sci5ce dur+ ! la/ c5tury & m has be5 s enorm\s t x is difficult to say :e!r pride 9 ! greatness ( a*ievem5t 9 ?is subject1 or despair at 8 inability to cope ) ! multiplicity ( its detail$ developm5ts1 %\ld be ! dom9ant feel+ ( ! ma!matician4 ,few p outside ( ! small circle ( ma!matical speciali/s h any idea ( ! va/ gr[? ( ma!matical lit]ature4 ,! ,royal ,society ,catalogue conta9s a li/ ( ne>ly ?irty-n9e ?\s& pap]s on subjects ( ,pure ,ma!matics alone1 :i* h appe>$ 9 sev5 hundr$ s]ials dur+ ! n9ete5? c5tury4 ,?is repres5ts only a portion ( ! total output1 ! v l>ge numb] ( treatises1 diss]tations1 & monographs publi%$ dur+ ! c5tury be+ omitt$4--,,hobson1 ,e4 ,w4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a1 7#aiaj72 ,nature1 ,vol4 #hd1 p4 #bhe4 #gjd4 ,ma!matics is "o ( ! olde/ ( ! sci5ces2 x is also "o ( ! mo/ active1 = its /r5g? is ! vig\r ( p]petual y\?4--,,=sy?1 ,a4 ,r4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a1 7#ahig72 ,nature1 ,vol4 #ef1 p4 #cgh4 #gje4 ,! n9ete5? c5tury :i* prides itself ^u ! inv5tion ( /eam & evolution1 might h d]iv$ a m legitimate title to fame f ! discov]y ( pure ma!matics4 --,,russell1 ,,b]tr&4 ,int]national ,mon?ly1 ,vol4 #d 7#aija71 p4 #hc4 #gjf4 ,"o ( ! *iefe/ triumphs ( mod]n ma!matics consi/s 9 hav+ discov]$ :at ma!matics really is4 --,,russell1 ,,b]tr&4 ,int]national ,mon?ly1 ,vol4 #d 7#aija71 p4 #hd4 #gjg4 ,mod]n ma!matics1 t mo/ a/\nd+ ( intellectual creations1 has project$ ! m9d's eye ?r\gh inf9ite "t & ! m9d's h& into b\ndless space4--,,butl]1 ,n4 ,m4 ,! ,mean+ ( ,education & o!r ,essays & ,addresses 7,new ,york1 #aije71 p4 #dd4 #gjh4 ,! extraord9>y developm5t ( ma!matics 9 ! la/ c5tury is q unp>allel$ 9 ! long hi/ory ( ?is mo/ anci5t ( sci5ces4 ,n only h ^? bran*es ( ma!matics :i* 7 tak5 ov] f ! eighte5? c5tury /eadily gr[n1 b entirely new ones h sprung up 9 almo/ bewild]+ pr(usion1 & _m ( !m h promptly assum$ proportions ( va/ ext5t4--,,pi]pont1 ,j4 ,! ,hi/ory ( ,ma!matics 9 ! ,n9ete5? ,c5tury2 ,congress ( ,arts & ,sci5ces 7,bo/on & ,new ,york1 #aije71 ,vol4 #a1 p4 #dgd4 #gji4 ,! ,mod]n ,!ory ( ,functions--t /atelie/ ( all ! pure creations ( ! human intellect4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #af4 #gaj4 ,if a ma!matician ( ! pa/1 an ,ar*im$es or ev5 a ,desc>tes1 c\ld view ! field ( geometry 9 its pres5t condition1 ! fir/ feature to impress him w\ld be its lack ( concret5ess4 ,!re are :ole classes ( geometric !ories :i* proce$ n only )\t models & diagrams1 b )\t ! slighte/ 7app>5t7 use ( spatial intuition4 ,9 ! ma9 ?is is due1 to ! p[] ( ! analytic in/rum5ts ( inve/igations z comp>$ ) ! purely geometric4 --,,kasn]1 ,,edw>d4 ,! ,pres5t ,problems 9 ,geometry2 ,bullet9 ,am]ican ,ma!matical ,society1 #aije1 p4 #bhe4 #gaa4 ,9 ,euclid ea* proposition /&s by itself2 its connection ) o!rs is nev] indicat$2 ! lead+ ideas conta9$ 9 its pro( are n /at$2 g5]al pr9ciples d n exi/4 ,9 mod]n me?ods1 on ! o!r h&1 ! greate/ importance is atta*$ to ! lead+ ?\ghts :i* p]vade ! :ole2 & g5]al pr9ciples1 :i* br+ :ole gr\ps ( !orems "u "o aspect1 are giv5 r ?an sep>ate propositions4 ,! :ole t5d5cy is t[>d g5]alization4 ,a /raight l9e is consid]$ z giv5 9 its entirety1 ext5d+ bo? ways to inf9ity1 :ile ,euclid is v c>eful nev] to admit any?+ b f9ite quantities4 ,! treatm5t ( ! inf9ite is 9 fact ano!r fundam5tal diff]5ce betwe5 ! two me?ods4 ,euclid avoids x1 9 mod]n ma!matics x is sy/ematically introduc$1 = only ?us is g5]ality obta9$4 --,,cayley1 ,,ar?ur4 ,encyclop$ia ,britannica 7#i? edition71 ,article 8,geometry40 #gab4 ,?is is "o ( ! greate/ advantages ( mod]n geometry ov] ! anci5t1 to be able1 ?r\gh ! consid]ation ( positive & negative quantities1 to include 9 a s+le enunciation ! sev]al cases :i* ! same !orem may pres5t by a *ange 9 ! relative position ( ! diff]5t p>ts ( a figure4 ,?us 9 our day ! n9e pr9cipal problems & ! num]\s p>ticul> cases1 :i* =m ! object ( eighty-?ree !orems 9 ! two books ,de sectione det]m9ata ( ,appolonius con/itute only "o problem :i* is resolv$ by a s+le equation4--,,*asles1 ,m4 ,hi/oire de la ,gomtrie1 *ap4 #a1 sect4 #ce4 #gac4 ,euclid always contemplates a /raight l9e z drawn betwe5 two def9ite po9ts1 & is v c>eful to m5tion :5 x is to be produc$ beyond ?is segm5t4 ,he nev] ?9ks ( ! l9e z an entity giv5 once = all z a :ole4 ,?is c>eful def9ition & limitation1 s z to exclude an inf9ity n imm$iately app>5t to ! s5ses1 0 v *>act]i/ic ( ! ,greeks 9 all _! _m activities4 ,x is en%r9$ 9 ! diff]5ce betwe5 ,greek ar*itecture & ,go?ic ar*itecture1 & betwe5 ,greek religion & mod]n religion4 ,! spire ( a ,go?ic ca!dral & ! importance ( ! unb\nd$ /raight l9e 9 mod]n ,geometry are bo? emblematic ( ! trans=mation ( ! mod]n _w4--,,:itehead1 ,a4 ,n4 ,introduction to ,ma!matics 7,new ,york1 #aiaa71 p4 #aai4 #gad4 ,! geometrical problems & !orems ( ! ,greeks always ref] to def9ite1 (t5times to r complicat$ figures4 ,n[ frequ5tly ! po9ts & l9es ( su* a figure may assume v _m diff]5t relative positions2 ea* ( ^! possible cases is !n consid]$ sep>ately4 ,on ! contr>y1 pres5t day ma!maticians g5]ate _! figures "o f ano!r1 & are accu/om$ to consid] !m subject to v>iation2 9 ?is mann] !y unite ! v>i\s cases & comb9e !m z mu* z possible by employ+ negative & imag9>y magnitudes4 ,= example1 ! problems :i* ,appolonius treats 9 8 two books ,de sectione rationis1 are solv$ today by means ( a s+le1 univ]sally applicable con/ruction2 ,apollonius1 on ! contr>y1 sep>ates x into m ?an eighty diff]5t cases v>y+ only 9 position4 ,?us1 z ,h]mann ,hankel has fitt+ly rem>k$1 ! anci5t geometry sacrifices to a seem+ simplicity ! true simplicity :i* consi/s 9 ! unity ( pr9ciples2 x atta9$ a trivial s5sual pres5tability at ! co/ ( ! recognition ( ! relations ( geometric =ms 9 all _! *anges & 9 all ! v>iations ( _! s5sually pres5table positions4 --,,reye1 ,,!odore4 ,die syn!tis*e ,geometrie im ,alt]tum und 9 d] ,neuzeit2 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #b1 pp4 #cdf-cdg4 #gae4 ,x is kn[n t ! ma!matics prescrib$ = ! high s*ool ,gymnasi5 is ess5tially ,euclidean1 :ile x is mod]n ma!matics1 ! !ory ( functions & ! inf9itesimal calculus1 :i* has secur$ = u an insight into ! me*anism & laws ( nature4 ,euclidean ma!matics is inde$1 a pr]equisite = ! !ory ( functions1 b j z "o1 ?\gh he has le>n$ ! inflections ( ,lat9 n\ns & v]bs1 w n !reby be enabl$ to read a ,lat9 au?or mu* less to appreciate ! beauties ( a ,horace1 s ,euclidean ma!matics1 t is ! ma!matics ( ! high s*ool1 is unable to unlock nature & h] laws4 ,euclidean ma!matics assumes ! complet5ess & inv>iability ( ma!matical =ms2 ^! =ms x describes ) appropriate accuracy & enum]ates _! inh]5t & relat$ prop]ties ) p]fect cle>ness1 ord]1 & complet5ess1 t is1 ,euclidean ma!matics op]ates on =ms aft] ! mann] t anatomy op]ates on ! dead body & its memb]s4 ,on ! o!r h&1 ! ma!matics ( v>iable magnitudes--function !ory or analysis--consid]s ma!matical =ms 9 _! g5esis4 ,by writ+ ! equation ( ! p>abola1 we express its law ( g5]ation1 ! law accord+ to :i* ! v>iable po9t moves4 ,! pa?1 produc$ be=e ! eyes ( ! /ud5t by a po9t mov+ 9 accordance to ?is law1 is ! p>abola4 ,if1 !n1 ,euclidean ma!matics treats space & numb] =ms aft] ! mann] 9 :i* anatomy treats ! dead body1 mod]n ma!matics deals1 z x 71 ) ! liv+ body1 ) gr[+ & *ang+ =ms1 & ?us furni%es an insight1 n only into nature z %e is & appe>s1 b also into nature z %e g5]ates & creates1--reveals h] transition /eps & 9 s do+ creates a m9d = & und]/&+ ( ! laws ( becom+4 ,?us mod]n ma!matics be>s ! same relation to ,euclidean ma!matics t physiology or biology 444 be>s to anatomy4 ,b x is exactly 9 ?is respect t our view ( nature is s f> above t ( ! anci5ts2 t we no long] look on nature z a quiesc5t complete :ole1 :i* compels admiration by its sublimity & weal? ( =ms1 b t we conceive ( h] z a vigor\s gr[+ organism1 unfold+ accord+ to def9ite1 z delicate z f>-rea*+1 laws2 t we are able to lay hold ( ! p]man5t amid/ ! transitory1 ( law amid/ fleet+ ph5om5a1 & to be able to give ^! _! simple/ & true/ expression ?r\gh ! ma!matical =mulas4 --,,dillmann1 ,e4 ,die ,ma!matik die ,fackeltrg]9 e9] neu5 ,zeit 7,/uttg>t1 #ahhi71 p4 #cg4 #gaf4 ,! ,excell5ce ( ,mod]n ,geometry is 9 no?+ m evid5t1 ?an 9 ^? full & adequate ,solutions x gives to ,problems2 repres5t+ all possible ,cases 9 "o view1 & 9 "o g5]al ,!orem _m times compreh5d+ :ole ,sci5ces2 :i* d$uc$ at l5g? into ,propositions1 & demon/rat$ aft] ! mann] ( ! ,anci5ts1 might well become ! subjects ( l>ge ,treatises3 ,= :atsoev] ,!orem solves ! mo/ complicat$ ,problem ( ! k9d1 does ) a due ,r$uction rea* all ! subord9ate ,cases4--,,halley1 ,e4 ,an ,in/ance ( ! ,excell5ce ( ,mod]n ,algebra1 etc42 ,philosophical ,transactions1 #afid1 p4 #ifj4 #gag4 ,"o ( ! mo/ conspicu\s & di/9ctive features ( ma!matical ?\ght 9 ! n9ete5? c5tury is its critical _s4 ,beg9n+ ) ! calculus1 x soon p]meates all analysis1 & t[>d ! close ( ! c5tury x ov]hauls & reca/s ! f\ndations ( geometry & aspires to fur!r conque/s 9 me*anics & 9 ! imm5se doma9s ( ma!matical physics4444 ,a se>*+ exam9ation ( ! f\ndations ( ari?metic & ! calculus has br\ght to light ! insuffici5cy ( mu* ( ! reason+ =m]ly consid]$ z conclusive4--,,pi]pont1 ,j4 ,hi/ory ( ,ma!matics 9 ! ,n9ete5? ,c5tury2 ,congress ( ,arts & ,sci5ces 7,bo/on & ,new ,york1 #aije71 ,vol4 #a1 p4 #dhb4 #gah4 ,if we comp>e a ma!matical problem ) an imm5se rock1 ^: int]ior we wi% to p5etrate1 !n ! "w ( ! ,greek ma!maticians appe>s to u l t ( a robu/ /onecutt]1 :o1 ) indefatigable p]sev]ance1 attempts to demoli% ! rock gradually f ! outside by means ( hamm] & *isel2 b ! mod]n ma!matician resembles an exp]t m9]1 :o fir/ con/ructs a few passages ?r\gh ! rock & !n explodes x ) a s+le bla/1 br++ to light its inn] treasures4 --,,hankel1 ,,h]mann4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 p4 #i4 #gai4 ,all ! mod]n high] ma!matics is bas$ on a calculus ( op]ations1 on laws ( ?\ght4 ,all ma!matics1 f ! fir/1 0 s 9 reality2 b ! evolv]s ( ! mod]n high] calculus h kn[n t x is s4 ,!re=e elem5t>y tea*]s :o1 at ! pres5t day1 p]si/ 9 ?9k+ ab\t algebra & ari?metic z deal+ ) laws ( numb]1 & ab\t geometry z deal+ ) laws ( surface & solid cont5t1 are do+ ! be/ t 9 !m lies to put _! pupils on ! wrong track = rea*+ 9 ! future any true und]/&+ ( ! high] algebras4 ,algebras deal n ) laws ( numb]1 b ) su* laws ( ! human ?9k+ ma*9]y z h be5 discov]$ 9 ! c\rse ( inve/igations on numb]s4 ,plane geometry deals ) su* laws ( ?\ght z 7 discov]$ by m5 int5t on f9d+ out h[ to measure surface2 & solid geometry ) su* additional laws ( ?\ght z 7 discov]$ :5 m5 began to ext5d geometry into ?ree dim5sions4--,,boole ,m4 ,e4 ,logic ( ,ari?metic 7,ox=d1 #aijc71 ,preface1 pp4 #ah-ai4 #gbj4 ,x is n only a decid$ pref]5ce = syn!sis & a complete d5ial ( g5]al me?ods :i* *>act]izes ! anci5t ma!matics z aga9/ our new] sci5ce mod]n ma!matics3 besides ?is ext]nal =mal diff]5ce !re is ano!r real1 m deeply seat$1 contra/1 :i* arises f ! diff]5t attitudes :i* ! two assum$ relative to ! use ( ! concept ( v>iability4 ,= :ile ! anci5ts1 on acc\nt ( consid]ations :i* _h be5 transmitt$ to !m f ! philosophic s*ool ( ! ,eleatics1 nev] employ$ ! concept ( motion1 ! spatial expression = v>iability1 9 _! rigor\s sy/em1 & made incid5tal use ( x only 9 ! treatm5t ( phonoromically g5]at$ curves1 mod]n geometry dates f ! in/ant t ,desc>tes left ! purely algebraic treatm5t ( equations & proce$$ to inve/igate ! v>iations :i* an algebraic expression und]goes :5 "o ( its v>iables assumes a cont9u\s succession ( values4--,,hankel1 ,,h]mann4 ,unt]su*ung5 b] die un5dli* (t oszilli]5d5 und un/etig5 ,function52 ,o/wald's ,klassik] d] exact5 ,wiss5s*aft51 ,no4 #aec1 pp4 #dd-de4 #gba4 ,)\t d\bt "o ( ! mo/ *>act]i/ic features ( ma!matics 9 ! la/ c5tury is ! sy/ematic & univ]sal use ( ! complex v>iable4 ,mo/ ( its great !ories receiv$ invaluable aid f x1 & _m owe _! v exi/5ce to x4 --,,pi]pont1 ,j4 ,hi/ory ( ,ma!matics 9 ! ,n9ete5? ,c5tury2 ,congress ( ,arts & ,sci5ces 7,bo/on & ,new ,york1 #aije71 ,vol4 #a1 p4 #dgd4 #gbb4 ,! notion1 :i* is really ! fundam5tal "o 7& ,i _c too /rongly emphasise ! ass]tion71 und]ly+ & p]vad+ ! :ole ( mod]n analysis & geometry1 is t ( imag9>y magnitude 9 analysis & ( imag9>y space 9 geometry4--,,cayley1 ,,ar?ur4 ,presid5tial ,address2 ,collect$ ,works1 ,vol4 #aa1 p4 #dcd4 #gbc4 ,! solution ( ! difficulties :i* =m]ly surr\nd$ ! ma!matical inf9ite is probably ! greate/ a*ievem5t ( :i* our age has to boa/4--,,russell1 ,,b]tr&4 ,! ,/udy ( ,ma!matics2 ,philosophical ,essays 7,london1 #aiaj71 p4 #gg4 #gbd4 ,induction & analogy are ! special *>act]i/ics ( mod]n ma!matics1 9 :i* !orems h giv5 place to !ories & no tru? is reg>d$ o!rwise ?an z a l9k 9 an inf9ite *a94 8,omne exit 9 inf9itum0 is _! favorite motto & accept$ axiom4--,,sylve/]1 ,j4 ,j4 ,a ,plea = ! ,ma!matician2 ,nature1 ,vol4 #a1 p4 #bfa4 #gbe4 ,! conception ( correspond5ce plays a great "p 9 mod]n ma!matics4 ,x is ! fundam5tal notion 9 ! sci5ce ( ord] z di/+ui%$ f ! sci5ce ( magnitude4 ,if ! old] ma!matics 7 mo/ly dom9at$ by ! ne$s ( m5suration1 mod]n ma!matics are dom9at$ by ! conception ( ord] & arrangem5t4 ,x may be t ?is t5d5cy ( ?\ght or direction ( reason+ goes h& 9 h& ) ! mod]n discov]y 9 physics1 t ! *anges 9 nature dep5d n only or n s mu* on ! quantity ( mass & en]gy z on _! di/ribution or arrangem5t4--,,m]z1 ,j4 ,t4 ,hi/ory ( ,european ,?\ght 9 ! ,n9ete5? ,c5tury 7,ed9burgh & ,london1 #aijc71 p4 #gcf4 #gbf4 ,n[ ?is e/abli%m5t ( correspond5ce betwe5 two aggregates & inve/igation ( ! propositions t are c>ri$ ov] by ! correspond5ce may be call$ ! c5tral idea ( mod]n ma!matics4--,,clif=d1 ,w4 ,k4 ,philosophy ( ! ,pure ,sci5ces2 ,lectures & ,essays 7,london1 #aija71 ,vol4 #a1 p4 #djb4 #gbg4 ,9 our c5tury ! conceptions sub/itution & sub/itution gr\p1 trans=mation & trans=mation gr\p1 op]ation & op]ation gr\p1 inv>iant1 diff]5tial inv>iant & diff]5tial p>amet]1 appe> m & m cle>ly z ! mo/ important conceptions ( ma!matics4--,,lie1 ,,sophus4 ,leipzig] ,b]i*te1 ,no4 #dg 7#ahie71 p4 #bfa4 #gbh4 ,g5]ality ( po9ts ( view & ( me?ods1 precision & elegance 9 pres5tation1 h become1 s9ce ,lagrange1 ! common prop]ty ( all :o w\ld lay claim to ! rank ( sci5tific ma!maticians4 ,&1 ev5 if ?is g5]ality leads at times to ab/rus5ess at ! exp5se ( intuition & applicability1 s t g5]al !orems are =mulat$ :i* fail to apply to a s+le special case1 if fur!rmore precision at times deg5]ates into a /udi$ brevity :i* makes x m difficult to read an article ?an x 0 to write x2 if1 f9ally1 elegance ( =m has well-nigh become 9 our day ! crit]ion ( ! wor? or wor?lessness ( a proposition1--yet are ^! conditions ( ! highe/ importance to a :olesome developm5t1 9 t !y keep ! sci5tific mat]ial )9 ! limits :i* are necess>y bo? intr9sically & extr9sically if ma!matics is n to sp5d itself 9 trivialities or smo!r 9 pr(usion4 --,,hankel1 ,,h]mann4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 pp4 #ad-ae4 #gbi4 ,! developm5t ( ab/ract me?ods dur+ ! pa/ few ye>s has giv5 ma!matics a new & vital pr9ciple :i* furni%es ! mo/ p[]ful in/rum5t = exhibit+ ! ess5tial unity ( all its bran*es4--,,"y1 ,j4 ,w4 ,fundam5tal ,concepts ( ,algebra & ,geometry 7,new ,york1 #aiaa71 p4 #bbe4 #gcj4 ,ev]ybody praises ! incomp>able p[] ( ! ma!matical me?od1 b s is ev]ybody aw>e ( its incomp>able unpopul>ity4--,,rosanes1 ,j4 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #ac1 p4 #ag4 #gca4 ,inde$ ! mod]n developm5ts ( ma!matics con/itute n only "o ( ! mo/ impressive1 b "o ( ! mo/ *>act]i/ic1 ph5om5a ( our age4 ,x is a ph5om5on1 h[ev]1 ( :i* ! boa/$ intellig5ce ( a 8univ]saliz$0 daily press seems /rangely unaw>e2 & !re is no o!r great human int]e/1 :e!r ( sci5ce or ( art1 reg>d+ :i* ! m9d ( ! educat$ public is p]mitt$ to hold s _m fallaci\s op9ions & inf]ior e/imates4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,arts 7,new ,york1 #aijh71 p4 #h4 #gcb4 ,x may be ass]t$ )\t exagg]ation t ! doma9 ( ma!matical k is ! only "o ( :i* our o!rwise omnisci5t j\rnalism has n yet possess$ itself4 --,,pr+%eim1 ,,alfr$4 ,ueb] ,w]t und angebli*5 ,unw]t d] ,ma!matik2 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 7#aijd7 p4 #ceg4 #gcc4 ,! inaccessibility ( special fields ( ma!matics1 except by ! regul> way ( logically antec$5t acquirem5ts1 r5d]s ! /udy disc\rag+ or hateful to weak or indol5t m9ds4--,,lefevre1 ,,ar?ur4 ,numb] & its ,algebra 7,bo/on1 #aijc71 sect4 #bbc4 #gcd4 ,! majority ( ma!matical tru?s n[ possess$ by u presuppose ! intellectual toil ( _m c5turies4 ,a ma!matician1 !re=e1 :o wi%es today to acquire a ?or\gh und]/&+ ( mod]n rese>* 9 ?is dep>tm5t1 mu/ ?9k ov] aga9 9 quick5$ tempo ! ma!matical labors ( sev]al c5turies4 ,?is con/ant dep5d5ce ( new tru?s on old ones /amps ma!matics z a sci5ce ( uncommon exclusiv5ess & r5d]s x g5]ally impossible to lay op5 to un9itiat$ read]s a spe$y pa? to ! appreh5sion ( ! high] ma!matical tru?s4 ,= ?is reason1 too1 ! !ories & results ( ma!matics are r>ely adapt$ = popul> pres5tation4444 ,?is same inaccessibility ( ma!matics1 al?\gh x secures = x a l(ty & ari/ocratic place among ! sci5ces1 also r5d]s x odi\s to ^? :o h nev] le>n$ x1 & :o dread ! great labor involv$ 9 acquir+ an und]/&+ ( ! que/ions ( mod]n ma!matics4 ,nei!r 9 ! languages nor 9 ! natural sci5ces are ! inve/igations & results s closely int]dep5d5t z to make x impossible to acqua9t ! un9itiat$ /ud5t ) s+le bran*es or ) p>ticul> results ( ^! sci5ces1 )\t caus+ him to g ?r\gh a long c\rse ( prelim9>y /udy4 --,,s*ub]t1 ,h4 ,ma!matical ,essays & ,recreations 7,*icago1 #ahih71 p4 #cb4 #gce4 ,su* is ! "* ( ma!matics 9 its pr(\nd] dep?s & 9 its high] & remot] zones t x is well nigh impossible to convey to "o :o has n devot$ ye>s to its exploration a j impression ( ! scope & magnitude ( ! exi/+ body ( ! sci5ce4 ,an imag9ation =m$ by o!r discipl9es & accu/om$ to ! int]e/s ( ano!r field may sc>cely receive sudd5ly an apocalyptic vision ( t inf9ite int]ior _w4 ,b h[ amaz+ & h[ edify+ 7 su* a revelation1 if x only c\ld be made4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #f4 #gcf4 ,x is n s long s9ce1 dur+ "o ( ! meet+s ( ! ,association1 "o ( ! lead+ ,engli% newspap]s briefly describ$ a sitt+ ( ?is ,section 9 ! words1 8,saturday morn+ 0 devot$ to pure ma!matics1 & s !re 0 no?+ ( any g5]al int]e/30 /ill1 su* tol]ation is bett] ?an undisguis$ & ill-in=m$ ho/ility4--,,=sy?1 ,a4 ,r4 ,report ( ! #fg? meet+ ( ! ,briti% ,association = ! ,advancem5t ( ,sci5ce4 #gcg4 ,! sci5ce ( ma!matics has gr[n to su* va/ proportion t probably no liv+ ma!matician c claim to h a*iev$ its ma/]y z a :ole4--,,:itehead1 ,a4 ,n4 ,an ,introduction to ,ma!matics 7,new ,york1 #aiaa71 p4 #beb4 #gch4 ,!re is p]haps no sci5ce ( :i* ! developm5t has be5 c>ri$ s f>1 :i* requires great] conc5tration & w p[]1 & :i* by ! ab/ract height ( ! qualities requir$ t5ds m to sep>ate "o f daily life4 ,provisional ,report ( ! ,am]ican ,subcommittee ( ! ,int]national ,commission on ! ,tea*+ ( ,ma!matics2 ,bullet9 ,am]ican ,society 7#aiaj71 p4 #ig4 #gci4 ,angl+ may be said to be s l ! ma!matics1 t x c nev] be fully le>nt4--,,walton1 ,,isaac4 ,! ,complete ,angl]1 ,preface4 #gdj4 ,! flights ( ! imag9ation :i* occur to ! pure ma!matician are 9 g5]al s mu* bett] describ$ 9 8 =mul ?an 9 words1 t x is n rem>kable to f9d ! subject treat$ by outsid]s z some?+ ess5tially cold & un9t]e/+--444 ! only successful attempt to inve/ ma!matical reason+ ) a halo ( glory--t made 9 ?is section by ,pr(4 ,sylve/]--is kn[n to a comp>ative few1 4444 --,,tait1 ,p4 ,g4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#ahga72 ,nature ,vol4 #d1 p4 #bga4 ,,*apt] ,,viii ,,! ,,ma!matician #hja4 ,! real ma!matician is an en?usia/ p] se4 ,)\t en?usiasm no ma!matics4--,,novalis4 ,s*rift5 7,b]l91 #aija71 ,zweit] ,teil1 p4 #bbc4 #hjb4 ,x is true t a ma!matician1 :o is n some:at ( a poet1 w nev] be a p]fect ma!matician4--,,wei]/rass4 ,quot$ by ,mittag-,leffl]2 ,compte r5du du deuxime congrs int]national des ma?matici5s 7,p>is1 #aijb71 p4 #adi4 #hjc4 ,! ma!matician is p]fect only 9 s f> z he is a p]fect be+1 9 s f> z he p]ceives ! beauty ( tru?2 only !n w 8 "w be ?or\gh1 transp>5t1 compreh5sive1 pure1 cle>1 attractive & ev5 elegant4 ,all ?is is necess>y to resemble ,lagrange4--,,goe!4 ,wilhelm ,mei/]'s ,w&]jahre1 ,zweites ,bu*2 ,spr*e 9 ,prosa2 ,natur1 ,,vi1 #iej4 #hjd4 ,a ?or\gh advocate 9 a j cause1 a p5etrat+ ma!matician fac+ ! />ry heav5s1 bo? alike be> ! semblance ( div9ity4--,,goe!4 ,wilhelm ,mei/]'s ,w&]jahre1 ,zweites ,bu*2 ,spr*e 9 ,prosa2 ,natur1 ,,vi1 #idg4 #hje4 ,ma!maticians practice absolute fre$om4--,,adams1 ,,h5ry4 ,a ,lett] to ,am]ican ,tea*]s ( ,hi/ory 7,wa%+ton1 #aiaj71 p4 #afi4 #hjf4 ,! ma!matical me?od is ! ess5ce ( ma!matics4 ,he :o fully compreh5ds ! me?od is a ma!matician4--,,novalis4 ,s*rift5 7,b]l91 #aija71 ,zweit] ,teil1 p4 #aij4 #hjg4 ,he :o is unfamili> ) ma!matics lit]ally1 he :o is a layman 9 ma!matics rema9s m or less a /rang] to our "t4--,,dillmann1 ,e4 ,die ,ma!matik die ,fackeltrg]9 e9] neu5 ,zeit 7,/uttg>t1 #ahhi71 p4 #ci4 #hjh4 ,enli/ a great ma!matician & a di/+ui%$ ,grecian2 y\r problem w be solv$4 ,su* m5 c tea* 9 a dwell+-h\se z well z 9 a palace4 ,"p ( ! app>atus !y w br+2 "p we w furni%4 ,advice giv5 to ! ,tru/ees ( ,johns ,hopk9s ,univ]sity on ! *oice ( a pr(essorial /aff4--,,gilman1 ,d4 ,c4 ,report ( ! ,presid5t ( ,johns ,hopk9s ,univ]sity 7#ahhh71 p4 #bi4 #hji4 ,p]sons1 :o h a decid$ ma!matical tal5t1 con/itute1 z x 71 a favor$ class4 ,!y be> ! same relation to ! re/ ( mank9d t ^? :o are academically tra9$ be> to ^? :o are n4--,,moebius1 ,p4 ,j4 ,ueb] die ,anlage zur ,ma!matik 7,leipzig1 #aijj71 p4 #d4 #haj4 ,"o may be a ma!matician ( ! fir/ rank )\t be+ able to compute4 ,x is possible to be a great comput] )\t hav+ ! slighte/ idea ( ma!matics4--,,novalis4 ,s*rift51 ,zweit] ,teil 7,b]l91 #aija71 p4 #bbc4 #haa4 ,x has long be5 a compla9t aga9/ ma!maticians t !y are h>d to conv9ce3 b x is a f> great] disqualification bo? = philosophy1 & = ! affairs ( life1 to be too easily conv9c$2 to h too l[ a /&>d ( pro(4 ,! only s\nd intellects are ^? :i*1 9 ! fir/ in/ance1 set _! /&>ds ( pro( high4 ,practice 9 concrete affairs soon tea*es !m to make ! necess>y abatem5t3 b !y reta9 ! consci\sness1 )\t :i* !re is no s\nd practical reason+1 t 9 accept+ inf]ior evid5ce because !re is no bett] to be _h1 !y d n by t acceptance raise x to complet5ess4--,,mill1 ,j4 ,s4 ,an ,exam9ation ( ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #faa4 #hab4 ,x is easi] to squ>e ! circle ?an to get r\nd a ma!matician4--,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #ij4 #hac4 ,ma!maticians are l ,fr5*m53 :atev] y say to !m !y translate into _! own language & =?) x is some?+ entirely diff]5t4--,,goe!4 ,maxim5 und ,reflexion51 ,se*/e ,ab!ilung4 #had4 ,:at ,i *iefly admir$1 & ?\ght altoge!r unacc\ntable1 0 ! /rong disposition ,i obs]v$ 9 !m ! ma!maticians ( ,laputa t[>ds news & politics2 p]petually inquir+ into public affairs2 giv+ _! judgm5ts 9 matt]s ( /ate2 & passionately disput+ e in* ( p>ty op9ion4 ,i h inde$ obs]v$ ! same disposition among mo/ ( ! ma!maticians ,i h kn[n 9 ,europe1 al?\gh ,i c\ld nev] discov] ! lea/ analogy betwe5 ! two sci5ces4 --,,swift1 ,,jona?an4 ,gulliv]'s ,travels1 ,"p #c1 *ap4 #b4 #hae4 ,! great ma!matician1 l ! great poet or naturali/ or great adm9i/rator1 is born4 ,my cont5tion %all be t :]e ! ma!matic end[m5t is f\nd1 !re w usually be f\nd associat$ ) x1 z ess5tial implications 9 x1 o!r end[m5ts 9 g5]\s measure1 & t ! appeal ( ! sci5ce is to ! :ole m9d1 direct no d\bt to ! c5tral p[]s ( ?\ght1 b indirectly ?r\gh sympa?y ( all1 r\s+1 enl>g+1 develop+1 emancipat+ all1 s t ! faculties ( w1 ( intellect & feel+ le>n to respond1 ea* 9 its appropriate ord] & degree1 l ! p>ts ( an or*e/ra to ! 8urge & ardor0 ( its lead] & "l4 --,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #bb4 #haf4 ,:oev] limits 8 ex]tions to ! gratification ( o!rs1 :e!r by p]sonal exhibition1 z 9 ! case ( ! actor & ( ! mimic1 or by ^? k9ds ( lit]>y composition :i* are calculat$ = no end b to please or to ent]ta91 r5d]s himself1 9 "s measure1 dep5d5t on _! caprices & hum\rs4 ,! div]sity among m51 9 _! judgm5ts conc]n+ ! objects ( ta/e1 is incomp>ably great] ?an 9 _! speculative conclusions2 & accord+ly1 a ma!matician w publi% to ! _w a geometrical demon/ration1 or a philosoph]1 a process ( ab/ract reason+1 ) a confid5ce v diff]5t f :at a poet w\ld feel1 9 communicat+ "o ( 8 productions ev5 to a fri5d4--,,/ew>t1 ,,dugald4 ,elem5ts ( ! ,philosophy ( ! ,human ,m9d1 ,"p #c1 *ap4 #a1 sect4 #c4 #hag4 ,consid]+ t1 among all ^? :o up to ?is "t made discov]ies 9 ! sci5ces1 x 0 ! ma!maticians alone :o _h be5 able to arrive at demon/rations--t is to say1 at pro(s c]ta9 & evid5t--,i did n d\bt t ,i %\ld beg9 ) ! same tru?s t !y h inve/igat$1 al?\gh ,i _h look$ = no o!r advantage f !m ?an to accu/om my m9d to n\ri% itself ^u tru?s & n to be satisfi$ ) false reasons4--,,desc>tes4 ,disc\rse ^u ,me?od1 ,"p #b2 ,philosophy ( ,desc>tes ,torrey 7,new ,york1 #ahib71 p4 #dh4 #hah4 ,:5 ! late ,sophus ,lie 444 0 ask$ to "n ! *>act]i/ic end[m5t ( ! ma!matician1 8 answ] 0 ! foll[+ quat]nion3 ,phantasie1 ,en]gie1 ,selb/v]trau51 ,selb/kritik4--,,keys]1 ,c4 ,j4 ,lectures on ,philosophy1 ,sci5ce & ,art 7,new ,york1 #aijh71 p4 #ca4 #hai4 ,! exi/5ce ( an ext5sive ,sci5ce ( ,ma!matics1 requir+ ! highe/ sci5tific g5ius 9 ^? :o contribut$ to its creation1 & call+ = ! mo/ cont9u$ & vigor\s ex]tion ( intellect 9 ord] to appreciate x :5 creat$1 etc4--,,mill1 ,j4 ,s4 ,sy/em ( ,logic1 ,bk4 #b1 *ap4 #d1 sect4 #d4 #hbj4 ,x may be true1 t m51 :o are m]e ma!maticians1 h c]ta9 specific %ortcom+s1 b t is n ! fault ( ma!matics1 = x is equally true ( e o!r exclusive occupation4 ,s !re are m]e philologi/s1 m]e juri/s1 m]e soldi]s1 m]e m]*ants1 etc4 ,to su* idle talk x might fur!r be add$3 t :5ev] a c]ta9 exclusive occupation is c\pl$ ) specific %ortcom+s1 x is likewise almo/ c]ta9ly divorc$ f c]ta9 o!r %ortcom+s4--,,gauss4 ,gauss-,s*uma*] ,briefwe*sel1 ,bd4 #d1 7,altona1 #ahfb71 p4 #chg4 #hba4 ,ma!matical /udies 444 :5 comb9$1 z !y n[ g5]ally are1 ) a ta/e = physical sci5ce1 enl>ge inf9itely our views ( ! wisdom & p[] display$ 9 ! univ]se4 ,! v intimate connexion inde$1 :i*1 s9ce ! date ( ! ,newtonian philosophy1 has exi/$ betwe5 ! diff]5t bran*es ( ma!matical & physical k1 r5d]s su* a "* z t ( a m]e ma!matician a v r>e & sc>cely possible occurr5ce4--,,/ew>t1 ,,dugald4 ,elem5ts ( ! ,philosophy ( ! ,human ,m9d1 "p #c1 *ap4 #a1 sect4 #c4 #hbb4 ,once :5 lectur+ to a class he ,"l ,kelv9 us$ ! ^w 8ma!matician10 & !n int]rupt+ himself ask$ 8 class3 8,d y "k :at a ma!matician is80 ,/epp+ to ! blackbo>d he wrote ^u x3-- _/ -x e dx _/- ,!n putt+ 8 f+] on :at he _h writt51 he turn$ to 8 class & said3 8,a ma!matician is "o to :om t is z obvi\s z t twice two makes f\r is to y4 ,li\ville 0 a ma!matician4--,,?ompson1 ,s4 ,p4 ,life ( ,"l ,kelv9 7,london1 #aiaj71 p4 #aaci4 #hbc4 ,x is n surpris+1 9 view ( ! polydynamic con/itution ( ! g5u9ely ma!matical m9d1 t _m ( ! major h]oes ( ! sci5ce1 m5 l ,des>gues & ,pascal1 ,desc>tes & ,leibnitz1 ,newton1 ,gauss & ,bolzano1 ,helmholtz & ,clif=d1 ,riemann & ,salmon & ,plck] & ,po9c>1 h atta9$ to high di/9ction 9 o!r fields n only ( sci5ce b ( philosophy & lett]s too4 ,& :5 we reflect t ! v greate/ ma!matical a*ievem5ts h be5 due1 n alone to ! pe]+1 microscopic1 hi/ologic vision ( m5 l ,wei]/rass1 illum9at+ ! hidd5 recesses1 ! m9ute & intimate /ructure ( logical reality1 b to ! l>g] vision also ( m5 l ,kle9 :o survey ! k+doms ( geometry & analysis = ! endless v>iety ( ?+s t n\ri% !re1 z ! eye ( ,d>w9 rang$ ov] ! flora & fauna ( ! _w1 or z a comm]cial mon>* contemplates its indu/ry1 or z a /atesman beholds an empire2 :5 we reflect n only t ! ,calculus ( ,probability is a creation ( ma!matics b t ! ma/] ma!matician is con/antly requir$ to ex]cise judgm5t--judgm5t1 t is1 9 matt]s n admitt+ ( c]ta9ty--balanc+ probabilities n yet r$uc$ nor ev5 r$ucible p]haps to calculation2 :5 we reflect t he is call$ ^u to ex]cise a function analog\s to t ( ! comp>ative anatomi/ l ,cuvi]1 comp>+ !ories & doctr9es ( e degree ( simil>ity & dissimil>ity ( /ructure2 :51 f9ally1 we reflect t he seldom deals ) a s+le idea at a "t1 b is = ! mo/ "p engag$ 9 wield+ organiz$ ho/s ( !m1 z a g5]al wields at once ! division ( an army or z a great civil adm9i/rator directs f 8 c5tral (fice div]se & scatt]$ b relat$ gr\ps ( int]e/s & op]ations2 !n1 ,i say1 ! curr5t op9ion t devotion to ma!matics unfits ! devotee = practical affairs %\ld be kn[n = false on a priori gr\nds4 ,& "o %\ld be ?us prep>$ to f9d t z a fact ,gasp>d ,monge1 creator ( descriptive geometry1 au?or ( ! classic 8,applications de l'analyse la gomtrie02 ,laz>e ,c>not1 au?or ( ! celebrat$ works1 8,gomtrie de position10 & 8,rflections sur la ,mtaphysique du ,calcul inf9itesimal02 ,f\ri]1 immortal creator ( ! 8,?orie analytique de la *aleur02 ,arago1 rightful inh]itor ( ,monge's *air ( geometry2 ,poncelet1 creator ( pure projective geometry2 "o %\ld n be surpris$1 ,i say1 to f9d t ^! & o!r ma!maticians 9 a l& sagaci\s en\gh to invoke _! aid1 r5d]$1 alike 9 peace & 9 w>1 em95t public s]vice4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 pp4 #cb-cc4 #hbd4 ,if 9 ,g]many ! goddess ,ju/itia _h n ! un=tunate habit ( deposit+ ! m9i/]ial portfolios only 9 ! cradles ( h] own prog5y1 :o kn[s h[ _m a ,g]man ma!matician might n also h made an excell5t m9i/]4 --,,pr+%eim1 ,a4 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #ac 7#aijd71 p4 #cgb4 #hbe4 ,we pass ) admiration along ! great s]ies ( ma!maticians1 by :om ! sci5ce ( !oretical me*anics has be5 cultivat$1 f ! "t ( ,newton to our own4 ,!re is no gr\p ( m5 ( sci5ce ^: fame is high] or bright]4 ,! great discov]ies ( ,cop]nicus1 ,galileo1 ,newton1 _h fix$ all eyes on ^? portions ( human k on :i* _! successors employ$ _! labors4 ,! c]ta9ty belong+ to ?is l9e ( speculation seem$ to elevate ma!maticians above ! /ud5ts ( o!r subjects2 & ! beauty ( ma!matical relations & ! subtlety ( intellect :i* may be %[n 9 deal+ ) !m1 7 fitt$ to w9 unb\nd$ applause4 ,! successors ( ,newton & ! ,b]n\llis1 z ,eul]1 ,clairaut1 ,d',alemb]t1 ,lagrange1 ,laplace1 n to introduce liv+ names1 h be5 "s ( ! mo/ rem>kable m5 ( tal5t :i* ! _w has se54--,,:ewell1 ,w4 ,hi/ory ( ! ,inductive ,sci5ces1 ,vol4 #a1 ,bk4 #d1 *ap4 #f1 sect4 #f4 #hbf4 ,! p]sons :o h be5 employ$ on ^! problems ( apply+ ! prop]ties ( matt] & ! laws ( motion to ! explanation ( ! ph5om5a ( ! _w1 & :o h br\ght to !m ! high & admirable qualities :i* su* an (fice requires1 h ju/ly excit$ 9 a v em95t degree ! admiration :i* mank9d feels = great intellectual p[]s4 ,_! names occupy a di/+ui%$ place 9 lit]>y hi/ory2 & probably !re are no sci5tific reputations ( ! la/ c5tury high]1 & none m m]it$1 ?an ^? e>n$ by great ma!maticians :o h lab\r$ ) su* wond]ful success 9 unfold+ ! me*anism ( ! heav5s2 su* = in/ance z ,d',alemb]t1 ,clairaut1 ,eul]1 ,lagrange1 ,laplace4--,,:ewell1 ,w4 ,a/ronomy & ,g5]al ,physics 7,london1 #ahcc71 ,bk4 #c1 *ap4 #d1 p4 #cbg4 #hbg4 ,two extreme views h always be5 held z to ! use ( ma!matics4 ,to "s1 ma!matics is only measur+ & calculat+ in/rum5ts1 & _! int]e/ ceases z soon z discussions arise :i* _c b5efit ^? :o use ! in/rum5ts = ! purposes ( application 9 me*anics1 a/ronomy1 physics1 /ati/ics1 & o!r sci5ces4 ,at ! o!r extreme we h ^? :o are animat$ exclusively by ! love ( pure sci5ce4 ,to !m pure ma!matics1 ) ! !ory ( numb]s at ! head1 is ! only real & g5u9e sci5ce1 & ! applications h only an int]e/ 9 s f> z !y conta9 or sugge/ problems 9 pure ma!matics4 ,( ! two greate/ ma!maticians ( mod]n times1 ,newton & ,gauss1 ! =m] c be consid]$ z a repres5tative ( ! fir/1 ! latt] ( ! second class2 nei!r ( !m 0 exclusively s1 & ,newton's inv5tions 9 ! sci5ce ( pure ma!matics 7 probably equal to ,gauss's "w 9 appli$ ma!matics4 ,newton's reluctance to publi% ! me?od ( fluxions inv5t$ & us$ by him may p]haps be attribut$ to ! fact t he 0 n satisfi$ ) ! logical f\ndations ( ! ,calculus2 & ,gauss is kn[n to h ab&on$ 8 electro-dynamic speculations1 z he c\ld n f9d a satisfy+ physical basis4444 ,newton's greate/ "w1 ! 8,pr9cipia01 laid ! f\ndation ( ma!matical physics2 ,gauss's greate/ "w1 ! 8,disquisitiones ,ari?meticae01 t ( high] ari?metic z di/+ui%$ f algebra4 ,bo? works1 writt5 9 ! syn!tic /yle ( ! anci5ts1 are difficult1 if n det]r5t1 9 _! =m1 nei!r ( !m lead+ ! read] by easy /eps to ! results4 ,x took tw5ty or m ye>s be=e ei!r ( ^! works receiv$ due recognition2 nei!r f\nd fav\r at once be=e t great tribunal ( ma!matical ?\ght1 ! ,p>is ,academy ( ,sci5ces4444 ,! c\ntry ( ,newton is /ill pre-em95t = its culture ( ma!matical physics1 t ( ,gauss = ! mo/ ab/ract "w 9 ma!matics4--,,m]z1 ,j4 ,t4 ,hi/ory ( ,european ,?\ght 9 ! ,n9ete5? ,c5tury 7,ed9burgh & ,london1 #aijc71 p4 #fcj4 #hbh4 ,z !re is no /udy :i* may be s advantage\sly ent]$ ^u ) a less /ock ( prep>atory k ?an ma!matics1 s !re is none 9 :i* a great] numb] ( un$ucat$ m5 h rais$ !mselves1 by _! own ex]tions1 to di/9ction & em95ce4444 ,_m ( ! intellectual defects :i*1 9 su* cases1 are commonly plac$ to ! acc\nt ( ma!matical /udies1 ought to be ascrib$ to ! want ( a lib]al education 9 e>ly y\?4--,,/ew>t1 ,,dugald4 ,elem5ts ( ! ,philosophy ( ! ,human ,m9d1 ,"p #c1 *ap4 #a1 sect4 #c4 #hbi4 ,i "k1 inde$1 & c conceive ( no pursuit s antagoni/ic to ! cultivation ( ! oratorical faculty 444 z ! /udy ( ,ma!matics4 ,an eloqu5t ma!matician mu/1 f ! nature ( ?+s1 ev] rema9 z r>e a ph5om5on z a talk+ fi%1 & x is c]ta9 t ! m anyone gives himself up to ! /udy ( oratorical effect ! less w he f9d himself 9 a fit /ate to ma!maticize4 ,x is ! con/ant aim ( ! ma!matician to r$uce all 8 expressions to _! l[e/ t]ms1 to retr5* e sup]flu\s ^w & phrase1 & to cond5se ! ,maximum ( mean+ into ! ,m9imum ( language4 ,he has to turn 8 eye ev] inw>ds1 to see ev]y?+ 9 its drye/ light1 to tra9 & inure himself to a habit ( int]nal & imp]sonal reflection & elaboration ( ab/ract ?\ght1 :i* makes x mo/ difficult = him to t\* or enl>ge ^u any ( ^? !mes :i* appeal to ! emotional nature ( 8 fell[-m54 ,:5 call$ ^u to speak 9 public he feels z a man might d :o has pass$ all 8 life 9 pe]+ ?r\gh a microscope1 & is sudd5ly call$ ^u to take *>ge ( a a/ronomical obs]vatory4 ,he has to get out ( himself1 z x 71 & *ange ! habitual focus ( 8 vision4 --,,sylve/]1 ,j4 ,j4 ,baltimore ,address2 ,ma!matical ,pap]s1 ,vol4 #c1 pp4 #gb-gc4 #hcj4 ,an accompli%$ ma!matician1 i4e4 a mo/ wret*$ orator4--,,b>r[1 ,,isaac4 ,ma!matical ,lectures 7,london1 #agcd71 p4 #cb4 #hca4 ,nemo ma!maticus g5ium indemnatus habebit4 ,no ma!matician#b is e/eem$ a g5ius until condemn$4 ,juv5al1 ,lib]ii1 ,satura ,,vi1 #efb4 #b ,us$ "h 9 ! s5se ( a/rolog]1 or soo?say]4 #hcb4 ,tak+ 444 ! ma!matical faculty1 probably few] ?an "o 9 a hundr$ really possess x1 ! great bulk ( ! population hav+ no natural ability = ! /udy1 or feel+ ! slighte/ int]e/ 9 x4#c ,& if we attempt to measure ! am\nt ( v>iation 9 ! faculty itself betwe5 a fir/-class ma!matician & ! ord9>y run ( p :o f9d any k9d ( calculation confus+ & altoge!r devoid ( int]e/1 x is probable t ! =m] c\ld n be e/imat$ at less ?an a hundr$ times ! latt]1 & p]haps a ?\s& times w\ld m ne>ly measure ! diff]5ce betwe5 !m4--,,wallace1 ,a4 ,r4 ,d>w9ism1 *ap4 #ae4 #c ,?is is ! e/imate furni%$ me by two ma!matical ma/]s 9 "o ( our great public s*ools ( ! proportion ( boys :o h any special ta/e or capacity = ma!matical /udies4 ,_m m1 ( c\rse1 c be drill$ into a fair k ( elem5t>y ma!matics1 b only ?is small proportion possess ! natural faculty :i* r5d]s x possible = !m ev] to rank high z ma!maticians1 to take any pleasure 9 x1 or to d any orig9al ma!matical "w4 #hcc4 444 ! pres5t gigantic developm5t ( ! ma!matical faculty is :olly unexpla9$ by ! !ory ( natural selection1 & mu/ be due to "s altoge!r di/9ct cause4--,,wallace1 ,a4 ,r4 ,d>w9ism1 *ap4 #ae4 #hcd4 ,dr4 ,wallace1 9 8 8,d>w9ism01 decl>es t he c f9d no gr\nd = ! exi/5ce ( pure sci5ti/s1 especially ma!maticians1 on ! hypo!sis ( natural selection4 ,if we put aside ! fact t great p[] 9 !oretical sci5ce is correlat$ ) o!r developm5ts ( increas+ bra9-activity1 we may1 ,i ?9k1 /ill acc\nt = ! exi/5ce ( pure sci5ti/s z ,dr4 ,wallace w\ld himself acc\nt = t ( work]-bees4 ,_! function may n fit !m individually to survive 9 ! /ruggle = exi/5ce1 b !y are a s\rce ( /r5g? & effici5cy to ! society :i* produces !m4 --,,pe>son1 ,,k>l4 ,gramm> ( ,sci5ce 7,london1 #aiaa71 ,"p #a1 p4 #bba4 #hce4 ,x is only 9 ma!matics1 & to "s ext5t 9 poetry1 t orig9ality may be atta9$ at an e>ly age1 b ev5 !n x is v r>e 7,newton & ,keats are examples71 & x is n notable until adolesc5ce is complet$4--,,ellis1 ,,havelock4 ,a ,/udy ( ,briti% ,g5ius 7,london1 #aijd71 p4 #adb4 #hcf4 ,! ,anglo-,dane appe>s to possess an aptitude = ma!matics :i* is n %>$ by ! native ( any o!r ,engli% di/rict z a :ole1 & x is 9 ! exact sci5ces t ! ,anglo-,dane triumphs4#d--;,,ellis1 ,,havelock4 ,a ,/udy ( ,briti% ,g5ius 7,london1 #aijd71 p4 #fi4 #d ,! ma!matical t5d5cies ( ,cambridge are due to ! fact t ,cambridge dra9s ! ability ( ne>ly ! :ole ,anglo-,dani% di/rict4 #hcg4 ,9 ! :ole hi/ory ( ! _w !re 0 nev] a race ) less lik+ = ab/ract reason+ ?an ! ,anglo-,saxon4444 ,common-s5se & compromise are believ$ 91 logical d$uctions f philosophical pr9ciples are look$ ^u ) suspicion1 n only by legislators1 b by all our mo/ le>n$ pr(essional m54--,,p]ry1 ,,john4 ,! ,tea*+ ( ,ma!matics 7,london1 #aijb71 pp4 #bj-ba4 #hch4 ,! degree ( exactness ( ! intuition ( space may be diff]5t 9 diff]5t individuals1 p]haps ev5 9 diff]5t races4 ,x w\ld seem z if a /rong nave space-intuition 7 an attribute pre-em95tly ( ! ,teutonic race1 :ile ! critical1 purely logical s5se is m fully develop$ 9 ! ,lat9 & ,hebrew races4 ,a full inve/igation ( ?is subject1 some:at on ! l9es sugge/$ by ,francis ,galton 9 8 rese>*es on h]$ity1 might be int]e/+4--,,kle91 ,,felix4 ,! ,evan/on ,colloquium ,lectures 7,new ,york1 #ahid71 p4 #df4 #hci4 ,?is ! fact t ! pursuit ( ma!matics br+s into h>moni\s action all ! faculties ( ! human m9d acc\nts = ! extraord9>y longevity ( all ! greate/ ma/]s ( ! ,analytic art1 ! ,dii ,majores ( ! ma!matical ,pan!on4 ,leibnitz liv$ to ! age ( #gj2 ,eul] to #gf2 ,lagrange to #gg2 ,laplace to #gh2 ,gauss to #gh2 ,plato1 ! suppos$ inv5tor ( ! conic sections1 :o made ma!matics 8 /udy & delight1 :o call$ !m ! h&les or aids to philosophy1 ! m$ic9e ( ! s\l1 & is said nev] to h let a day g by )\t inv5t+ "s new !orems1 liv$ to #hb2 ,newton1 ! cr[n & glory ( 8 race1 to #he2 ,ar*im$es1 ! ne>e/ ak91 probably1 to ,newton 9 g5ius1 0 #ge1 & might h liv$ on to be #ajj1 = aught we c guess to ! contr>y1 :5 he 0 sla9 by ! impati5t & ill-mann]$ s]geant1 s5t to br+ him be=e ! ,roman g5]al1 9 ! full vig\r ( 8 faculties1 & 9 ! v act ( work+ out a problem2 ,py?agoras1 9 ^: s*ool1 ,i believe1 ! ^w ma!matician 7us$1 h[ev]1 9 a some:at wid] ?an its pres5t s5se7 orig9at$1 ! second f\nd] ( geometry1 ! inv5tor ( ! mat*less !orem :i* goes by 8 "n1 ! pre-cogniz] ( ! und\bt$ly mis-call$ ,cop]nican !ory1 ! discov]] ( ! regul> solids & ! musical canon :o /&s at ! v apex ( ?is pyramid ( fame1 7if we may cr$it ! tradition7 aft] sp5d+ #bb ye>s /udy+ 9 ,egypt1 & #ab 9 ,babylon1 op5$ s*ool :5 #ef or #eg ye>s old 9 ,magna ,grcia1 m>ri$ a "y wife :5 pa/ #fj1 & di$1 c>ry+ on 8 "w ) en]gy unsp5t to ! la/1 at ! age ( #ii4 ,! ma!matician lives long & lives "y2 ! w+s ( 8 s\l d n e>ly drop (f1 nor d its pores become clogg$ ) ! e>?y p>ticles bl[n f ! du/y highways ( vulg> life4--,,sylve/]1 ,j4 ,j4 ,presid5tial ,address to ! ,briti% ,association2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b 7#aijh71 p4 #feh4 #hdj4 ,! game ( *ess has always fasc9at$ ma!maticians1 & !re is reason to suppose t ! possession ( great p[]s ( play+ t game is 9 _m features v mu* l ! possession ( great ma!matical ability4 ,!re are ! diff]5t pieces to le>n1 ! pawns1 ! knights1 ! bi%ops1 ! ca/les1 & ! que5 & k+4 ,! bo>d possesses c]ta9 possible comb9ations ( squ>es1 z 9 r[s1 diagonals1 etc4 ,! pieces are subject to c]ta9 rules by :i* _! motions are gov]n$1 & !re are o!r rules gov]n+ ! play]s4444 ,"o has only to increase ! numb] ( pieces1 to enl>ge ! field ( ! bo>d1 & to produce new rules :i* are to gov]n ei!r ! pieces or ! play]1 to h a pretty good idea ( :at ma!matics consi/s4--,,%aw1 ,j4 ,b4 ,:at is ,ma!matics8 ,bullet9 ,am]ican ,ma!matical ,society ,vol4 #ah 7#aiab71 pp4 #chf-chg4 #hda4 ,e man is ready to jo9 9 ! approval or condemnation ( a philosoph] or a /atesman1 a poet or an orator1 an arti/ or an ar*itect4 ,b :o c judge ( a ma!matician8 ,:o w write a review ( ,hamilton's ,quat]nions1 & %[ u :]e9 x is sup]ior to ,newton's ,fluxions8--,,hill1 ,,?omas4 ,imag9ation 9 ,ma!matics2 ,nor? ,am]ican ,review1 ,vol4 #he1 p4 #bbd4 #hdb4 ,! pursuit ( ma!matical sci5ce makes its vot>y appe> s+ul>ly indiff]5t to ! ord9>y int]e/s & c>es ( m54 ,seek+ et]nal tru?s1 & f9d+ 8 pleasures 9 ! realities ( =m & numb]1 he has little int]e/ 9 ! disputes & cont5tions ( ! pass+ h\r4 ,8 views on social & political que/ions p>take ( ! gr&eur ( 8 favorite contemplations1 &1 :ile c>eful to ?r[ 8 mite ( influ5ce on ! side ( "r & tru?1 he is cont5t to abide ! work+s ( ^? g5]al laws by :i* he d\bts n t ! fluctuations ( human hi/ory are z un]r+ly guid$ z are ! p]turbations ( ! planet>y ho/s4--,,hill1 ,,?omas4 ,imag9ation 9 ,ma!matics2 ,nor? ,am]ican ,review1 ,vol4 #he1 p4 #bbg4 #hdc4 ,!re is some?+ sublime 9 ! secrecy 9 :i* ! really great de$s ( ! ma!matician are done4 ,no popul> applause foll[s ! act2 nei!r contempor>y nor succe$+ g5]ations ( ! p und]/& x4 ,! geomet] mu/ be tri$ by 8 pe]s1 & ^? :o truly des]ve ! title ( geomet] or analy/ h usually be5 unable to f9d s _m z twelve liv+ pe]s to =m a jury4 ,ar*im$es s f> out/ripp$ 8 competitors 9 ! race1 t m ?an a ?\s& ye>s elaps$ be=e any man appe>$1 able to sit 9 judgm5t on 8 "w1 & to say h[ f> he _h really gone4 ,& 9 judg+ ( ^? m5 ^: names are wor?y ( be+ m5tion$ 9 connection ) 81--,galileo1 ,desc>tes1 ,leibnitz1 ,newton1 & ! ma!maticians creat$ by ,leibnitz & ,newton's calculus1--we are =c$ to dep5d ^u _! te/imony ( "o ano!r4 ,!y are too f> above our rea* = u to judge ( !m4--,,hill1 ,,?omas4 ,imag9ation 9 ,ma!matics2 ,nor? ,am]ican ,review1 ,vol4 #he1 p4 #bbc4 #hdd4 ,to ?9k ! ?9kable--t is ! ma!matician's aim4 --,,keys]1 ,c4 ,j4 ,! ,univ]se & ,beyond2 ,hibb]t ,j\rnal1 ,vol4 #c 7#aijd-aije71 p4 #cab4 #hde4 ,e common me*anic has some?+ to say 9 8 craft ab\t good & evil1 useful & useless1 b ^! practical consid]ations nev] ent] into ! purview ( ! ma!matician4 --,,ari/ippus ,,! ,,cyr5aic4 ,quot$ 9 ,hicks1 ,r4 ,d41 ,/oic & ,epicurean1 7,new ,york1 #aiaj7 p4 #baj4 ,,*apt] ,,ix ,,p]sons ,,& ,,anecdotes 7,a-,m7 #ija4 ,alex&] is said to h ask$ ,m5*mus to tea* him geometry concisely1 b ,m5*mus repli$3 8,o k+1 ?r\gh ! c\ntry !re are royal roads & roads = common citiz5s1 b 9 geometry !re is "o road = all40 ,/obu 7,edition ,wa*smu?1 ,b]l91 #ahhd71 ,ecl4 #b1 p4 #cj4 #ijb4 ,alex&] ! k+ ( ! ,mac$onians1 began l a wret* to le>n geometry1 t he might "k h[ little ! e>? 01 :]e( he _h possess$ v little4 ,?us1 ,i say1 l a wret* = ?is1 because he 0 to und]/& t he did be> a false surname4 ,= :o c be great 9 s small a ?+8 ,^? ?+s t 7 deliv]$ 7 subtile1 & to be le>n$ by dilig5t att5tion3 n :i* t mad man c\ld p]ceive1 :o s5t 8 ?\ghts beyond ! ocean sea4 ,tea* me1 sai? he1 easy ?+s4 ,to :om 8 ma/] said3 ,^! ?+s be ! same1 & alike difficult unto all4 ,?9k ?\ t ! nature ( ?+s sai? ?is4 ,^! ?+s :]e( ?\ compla9e/1 !y are ! same unto all3 m easy ?+s c be giv5 unto none2 b :osoev] w1 %all make ^? ?+s m easy unto himself4 ,h[8 ,) uprightness ( m9d4--,,s5eca4 ,epi/le #ia ,?omas ,lodge4 #ijc4 ,ar*im$es 444 _h /at$ t giv5 ! =ce1 any giv5 weight might be mov$1 & ev5 boa/$1 we are told1 rely+ on ! /r5g? ( demon/ration1 t if !re 7 ano!r e>?1 by go+ into x he c\ld remove ?is4 ,hi]o be+ /ruck ) amazem5t at ?is1 & entreat+ him to make good ?is problem by actual exp]im5t1 & %[ "s great weight mov$ by a small eng9e1 he fix$ accord+ly ^u a %ip ( burd5 out ( ! k+'s ars5al1 :i* c\ld n be drawn out ( ! dock )\t great labor & _m m52 &1 load+ h] ) _m pass5g]s & a full freight1 sitt+ himself ! :ile f> (f ) no great endeavor1 b only hold+ ! head ( ! pulley 9 8 h& & draw+ ! cords by degrees1 he drew ! %ip 9 a /raight l9e1 z smoo?ly & ev5ly1 z if %e _h be5 9 ! sea4 ,! k+1 a/oni%$ at ?is1 & conv9c$ ( ! p[] ( ! art1 prevail$ ^u ,ar*im$es to make him eng9es accommodat$ to all ! purposes1 (f5sive & def5sive1 ( a siege 444 ! app>atus 01 9 mo/ opportune "t1 ready at h& = ! ,syracusans1 & ) x also ! eng9e] himself4 --,,plut>*4 ,life ( ,m>cellus ,dryd54 #ijd4 ,^! ma*9es us$ 9 ! def5se ( ! ,syracusans aga9/ ! ,romans "u ,m>cellus he ,ar*im$es _h design$ & contriv$1 n z matt]s ( any importance1 b z m]e amusem5ts 9 geometry2 9 compliance ) k+ ,hi]o's desire & reque/1 "s "t be=e1 t he %\ld r$uce to practice "s "p ( 8 admirable speculation 9 sci5ce1 & by accommodat+ ! !oretic tru? to s5sation & ord9>y use1 br+ x m )9 ! appreciation ( p 9 g5]al4 ,eudoxus & ,ar*ytas _h be5 ! fir/ orig9ators ( ?is f>-fam$ & highly-priz$ art ( me*anics1 :i* !y employ$ z an elegant illu/ration ( geometrical tru?s1 & z means ( su/a9+ exp]im5tally1 to ! satisfaction ( ! s5ses conclusions too intricate = pro( by words & diagrams4 ,z1 = example1 to solve ! problem1 s (t5 requir$ 9 con/ruct+ geometrical figures1 giv5 ! two extremes1 to f9d ! two mean l9es ( a proportion1 bo? ^! ma!maticians _h rec\rse to ! aid ( in/rum5ts1 adapt+ to _! purpose c]ta9 curves & sections ( l9es4 ,b :at ) ,plato's indignation at x1 & 8 invectives aga9/ x z ! m]e corruption & annihilation ( ! "o good ( geometry1--:i* 0 ?us %amefully turn+ its back ^u ! unembodi$ objects ( pure intellig5ce to recur to s5sation1 & to ask help 7n to be obta9$ )\t base sup]visions & depravation7 f matt]2 s x 0 t me*anics came to be sep>at$ f geometry1 &1 repudiat$ & neglect$ by philosoph]s1 took its place z a milit>y art4--,,plut>*4 ,life ( ,m>cellus ,dryd54 #ije4 ,ar*im$es 0 n free f ! prevail+ notion t geometry 0 degrad$ by be+ employ$ to produce any?+ useful4 ,x 0 ) difficulty t he 0 induc$ to /oop f speculation to practice4 ,he 0 half a%am$ ( ^? inv5tions :i* 7 ! wond] ( ho/ile nations1 & always spoke ( !m slight+ly z m]e amusem5ts1 z trifles 9 :i* a ma!matician might be suff]$ to relax 8 m9d aft] int5se application to ! high] p>ts ( 8 sci5ce4--,,macaulay4 ,"l ,bacon2 ,ed9burgh ,review1 ,july #ahcg2 ,critical & ,miscellane\s ,essays 7,new ,york1 #ahgi71 ,vol4 #a1 p4 #chj4 #ijf4 ,call ,ar*im$es f 8 buri$ tomb ,^u ! pla9 ( vani%$ ,syracuse1 ,& feel+ly ! sage %all make report ,h[ insecure1 h[ baseless 9 itself1 ,is ! philosophy1 ^: sway dep5ds ,on m]e mat]ial in/rum5ts--h[ weak ,^? arts1 & high inv5tions1 if unpropp$ ,by virtue4 --,,wordswor?4 ,! ,excursion4 #ijg4 ,zu ,ar*im$es kam e9/ e9 wissbegi]ig] ,jngl+4 8,weihe mi*10 spra* er zu ihm1 8e9 9 die gttli*e ,kun/1 ,die s h]rli*e ,fru*t dem ,vat]l&e getrag51 ,und die ,mau]n d] ,/adt vor d] ,sambuca bes*tzt60 8,gttli* n5n/ du die ,kun/8 ,sie i/s10 v]setzte d] ,weise2 8,ab] das w> sie1 me9 ,sohn1 eh sie dem ,/aat no* g$i5t4 ,will/ du nur ,fr*te von ihr1 die kann au* die ,/]bli*e zeug52 ,w] um die ,gtt9 freit1 su*e 9 ihr ni*t das ,weib40 --,,s*ill]4 ,ar*im$es und d] ,s*l]4 ,to ,ar*im$es once came a y\? int5t ^u k4 ,said he 8,initiate me into ! ,sci5ce div9e1 ,:i* to our c\ntry has borne glori\s fruits 9 abundance1 ,& :i* ! walls ( ! t[n 'ga9/ ! ,sambuca protects40 8,call/ ?\ ! sci5ce div9e8 ,x is s10 ! wise man respond$2 8,b s x 01 my son1 ere ! /ate by h] s]vice 0 ble/4 ,w\ld'/ ?\ h fruit ( h] only8 ,mortals ) t c provide !e1 ,he :o ! goddess w\ld woo1 seek n ! woman 9 h]40 #ijh4 ,ar*im$es possess$ s high a _s1 s pr(\nd a s\l1 & su* treasures ( highly sci5tific k1 t ?\gh ^! inv5tions us$ to def5d ,syracuse aga9/ ! ,romans _h n[ obta9$ him ! r5[n ( m ?an human sagacity1 he yet w\ld n deign to leave beh9d him any comm5t>y or writ+ on su* subjects2 b1 repudiat+ z sordid & ignoble ! :ole trade ( eng9e]+1 & e sort ( art t l5ds itself to m]e use & pr(it1 he plac$ 8 :ole affection & ambition 9 ^? pur] speculations :]e !re c be no ref]5ce to ! vulg> ne$s ( life2 /udies1 ! sup]iority ( :i* to all o!rs is unque/ion$1 & 9 :i* ! only d\bt c be :e!r ! beauty & gr&eur ( ! subjects exam9$1 or ! precision & cog5cy ( ! me?ods & means ( pro(1 mo/ des]ve our admiration4--,,plut>*4 ,life ( ,m>cellus ,dryd54 #iji4 ,no?+ afflict$ ,m>cellus s mu* z ! dea? ( ,ar*im$es1 :o 0 !n1 z fate w\ld h x1 int5t ^u work+ out "s problem by a diagram1 & hav+ fix$ 8 m9d alike & 8 eyes ^u ! subject ( 8 speculation1 he nev] notic$ ! incursion ( ! ,romans1 nor t ! city 0 tak54 ,9 ?is transport ( /udy & contemplation1 a soldi]1 unexpect$ly com+ up to him1 comm&$ him to foll[ to ,m>cellus1 :i* he decl9$ to d be=e he _h work$ out 8 problem to a demon/ration2 ! soldi]1 enrag$1 drew 8 sword & ran him ?r\gh4 ,o!rs write1 t a ,roman soldi]1 runn+ ^u him ) a drawn sword1 (f]$ to kill him2 & t ,ar*im$es1 look+ back1 e>ne/ly bes\ght him to hold 8 h& a little :ile1 t he might n leave :at he 0 at "w ^u inconclusive & imp]fect2 b ! soldi]1 no?+ mov$ by 8 entreaty1 in/antly kill$ him4 ,o!rs aga9 relate1 t z ,ar*im$es 0 c>ry+ to ,m>cellus ma!matical in/rum5ts1 dials1 sph]es1 & angles1 by :i* ! magnitude ( ! sun might be measur$ to ! sight1 "s soldi]s see+ him1 & ?9k+ t he c>ri$ gold 9 a vessel1 slew him4 ,c]ta9 x is1 t 8 dea? 0 v afflict+ to ,m>cellus2 & t ,m>cellus ev] aft] reg>d$ him t kill$ him z a murd]]2 & t he s\ght = 8 k9dr$ & hon\r$ !m ) signal fav\rs4--,,plut>*4 ,life ( ,m>cellus ,dryd54 #iaj4 ,ar*im$es is said to h reque/$ 8 fri5ds & relations t :5 he 0 dead1 !y w\ld place ov] 8 tomb a sph]e conta9+ a cyl9d]1 inscrib+ x ) ! ratio :i* ! conta9+ solid be>s to ! conta9$4--,,plut>*4 ,life ( ,m>cellus ,dryd54 #iaa4 ,ar*im$es1 :o comb9$ a g5ius = ma!matics ) a physical insight1 mu/ rank ) ,newton1 :o liv$ ne>ly two ?\s& ye>s lat]1 z "o ( ! f\nd]s ( ma!matical physics4444 ,! day 7:5 hav+ discov]$ 8 fam\s pr9ciple ( hydro/atics he ran ?r\gh ! /reets %\t+ ,eureka6 ,eureka67 ought to be celebrat$ z ! bir?day ( ma!matical physics2 ! sci5ce came ( age :5 ,newton sat 9 8 or*>d4 --,,:itehead1 ,a4 ,n4 ,an ,introduction to ,ma!matics 7,new ,york1 #aiaa71 p4 #ch4 #iab4 ,x is n possible to f9d 9 all geometry m difficult & m intricate que/ions or m simple & lucid explanations ?an ^? giv5 by ,ar*im$es4 ,"s ascribe ?is to 8 natural g5ius2 :ile o!rs ?9k t incr$ible ef=t & toil produc$ ^!1 to all appe>ance1 easy & unlab\r$ results4 ,no am\nt ( inve/igation ( y\rs w\ld succe$ 9 atta9+ ! pro(1 & yet1 once se51 y imm$iately believe y w\ld h discov]$ x2 by s smoo? & s rapid a pa? he leads y to ! conclusion requir$4--,,plut>*4 ,life ( ,m>cellus ,dryd54 #iac4 ,"o feature :i* w probably mo/ impress ! ma!matician accu/om$ to ! rapidity & directness secur$ by ! g5]ality ( mod]n me?ods is ! delib]ation ) :i* ,ar*im$es approa*es ! solution ( any "o ( 8 ma9 problems4 ,yet ?is v *>act]i/ic1 ) its incid5tal effects1 is calculat$ to excite ! m admiration because ! me?od sugge/s ! tactics ( "s great /rategi/ :o =esees ev]y?+1 elim9ates ev]y?+ n imm$iately conducive to ! execution ( 8 plan1 ma/]s e position 9 its ord]1 & !n sudd5ly 7:5 ! v elaboration ( ! s*eme has almo/ obscur$1 9 ! m9d ( ! spectator1 its ultimate object7 /rikes ! f9al bl[4 ,?us we read 9 ,ar*im$es proposition aft] proposition ! be>+ ( :i* is n imm$iately obvi\s b :i* we f9d infallibly us$ lat] on2 & we are l$ by su* easy /ages t ! difficulties ( ! orig9al problem1 z pres5t$ at ! outset1 are sc>cely appreciat$4 ,z ,plut>* says3 8,x is n possible to f9d 9 geometry m difficult & tr\blesome que/ions1 or m simple & lucid explanations40 ,b x is decid$ly a rhetorical exagg]ation :5 ,plut>* goes on to say t we are deceiv$ by ! eas9ess ( ! successive /eps into ! belief t anyone c\ld h discov]$ !m = himself4 ,on ! contr>y1 ! /udi$ simplicity & ! p]fect f9i% ( ! treatises involve at ! same "t an elem5t ( my/]y4 ,?\gh ea* /ep dep5ds on ! prec$+ ones1 we are left 9 ! d>k z to h[ !y 7 sugge/$ to ,ar*im$es4 ,!re is1 9 fact1 mu* tru? 9 a rem>k by ,wallis to ! effect t he seems 8z x 7 ( set purpose to h cov]$ up ! traces ( 8 inve/igation z if he _h grudg$ po/]ity ! secret ( 8 me?od ( inquiry :ile he wi%$ to extort f !m ass5t to 8 results40 ,wallis adds ) equal reason t n only ,ar*im$es b ne>ly all ! anci5ts s hid away f po/]ity _! me?od ( ,analysis 7?\gh x is c]ta9 t !y _h "o7 t m mod]n ma!maticians f\nd x easi] to inv5t a new ,analysis ?an to seek out ! old4--,,hea?1 ,t4 ,l4 ,! ,works ( ,ar*im$es 7,cambridge1 #ahig71 ,preface4 #iad4 ,x is a great pity ,ari/otle _h n und]/ood ma!matics z well z ,mr4 ,newton1 & made use ( x 9 8 natural philosophy ) good success3 8 example _h !n au?oriz$ ! accommodat+ ( x to mat]ial ?+s4 --,,locke1 ,,john4 ,second ,reply to ! ,bi%op ( ,worce/]4 #iae4 ,! op9ion ( ,bacon on ?is subject geometry 0 diametrically oppos$ to t ( ! anci5t philosoph]s4 ,he valu$ geometry *iefly1 if n solely1 on acc\nt ( ^? uses1 :i* to ,plato appe>$ s base4 ,& x is rem>kable t ! long] ,bacon liv$ ! /rong] ?is feel+ became4 ,:5 9 #afje he wrote ! two books on ! ,advancem5t ( ,le>n+1 he dwelt on ! advantages :i* mank9d d]iv$ f mix$ ma!matics2 b he at ! same "t admitt$ t ! b5eficial effect produc$ by ma!matical /udy on ! intellect1 ?\gh a collat]al advantage1 0 8no less wor?y ?an t :i* 0 pr9cipal & int5d$40 ,b x is evid5t t 8 views und]w5t a *ange4 ,:5 ne> tw5ty ye>s lat]1 he publi%$ ! ,de ,augm5tis1 :i* is ! ,treatise on ! ,advancem5t ( ,le>n+1 greatly exp&$ & c>efully correct$1 he made important alt]ations 9 ! "p :i* relat$ to ma!matics4 ,he condemn$ ) sev]ity ! pret5sions ( ! ma!maticians1 8delicias et fa/um ma!maticorum40 ,assum+ ! well-be+ ( ! human race to be ! end ( k1 he pron\nc$ t ma!matical sci5ce c\ld claim no high] rank ?an t ( an app5dage or an auxili>y to o!r sci5ces4 ,ma!matical sci5ce1 he says1 is ! h&maid ( natural philosophy2 %e ought to demean h]self z su*2 & he decl>es t he _c conceive by :at ill *ance x has happ5$ t %e presumes to claim prec$5ce ov] h] mi/ress4--,,macaulay4 ,"l ,bacon3 ,ed9burgh ,review1 ,july1 #ahcg2 ,critical & ,miscellane\s ,essays 7,new ,york1 #ahgi71 ,vol4 #a1 p4 #chj4 #iaf4 ,if ,bacon err$ "h 9 valu+ ma!matics only = its uses1 we mu/ ackn[l$ge t we greatly pref] 8 error to ! opposite error ( ,plato4 ,we h no pati5ce ) a philosophy :i*1 l ^? ,roman matrons :o swall[$ abortives 9 ord] to pres]ve _! %apes1 takes pa9s to be b>r5 = fe> ( be+ homely4--,,macaulay4 ,"l ,bacon1 ,ed9burgh ,review1 ,july1 #ahcg2 ,critical & ,miscellane\s ,essays 7,new ,york1 #ahgi71 ,vol4 #b1 p4 #cha4 #iag4 ,he ,"l ,bacon appe>s to h be5 utt]ly ignorant ( ! discov]ies :i* _h j be5 made by ,kepl]'s calculations 444 he does n say a ^w ab\t ,napi]'s ,log>i?ms1 :i* _h be5 publi%$ only n9e ye>s be=e & repr9t$ m ?an once 9 ! int]val4 ,he compla9$ t no consid]able advance _h be5 made 9 ,geometry beyond ,euclid1 )\t tak+ any notice ( :at _h be5 done by ,ar*im$es & ,apollonius4 ,he saw ! importance ( det]m9+ accurately ! specific gravities ( diff]5t sub/ances1 & himself attempt$ to =m a table ( !m by a rude process ( 8 own1 )\t kn[+ ( ! m sci5tific ?\gh /ill imp]fect me?ods previ\sly employ$ by ,ar*im$es1 ,ghetaldus & ,porta4 ,he speaks ( ! ( ,ar*im$es 9 a mann] :i* implies t he did n cle>ly appreciate ei!r ! problem to be solv$ or ! pr9ciples ^u :i* ! solution dep5d$4 ,9 review+ ! progress ( ,me*anics1 he makes no m5tion ei!r ( ,ar*im$es1 or ,/ev9us1 ,galileo1 ,guld9us1 or ,ghetaldus4 ,he makes no allusion to ! !ory ( ,equilibrium4 ,he obs]ves t a ball ( "o p\nd weight w fall ne>ly z fa/ ?r\gh ! air z a ball ( two1 )\t allud+ to ! !ory ( accel]ation ( fall+ bodies1 :i* _h be5 made kn[n by ,galileo m ?an ?irty ye>s be=e4 ,he propos$ an inquiry ) reg>d to ! lev]1--namely1 :e!r 9 a balance ) arms ( diff]5t l5g? b equal weight ! di/ance f ! fulcrum has any effect ^u ! incl9ation--?\gh ! !ory ( ! lev] 0 z well und]/ood 9 8 own "t z x is n[4444 ,he speaks ( ! poles ( ! e>? z fix$1 9 a mann] :i* seems to imply t he 0 n acqua9t$ ) ! precession ( ! equ9oxes2 & 9 ano!r place1 ( ! nor? pole be+ above & ! s\? pole bel[1 z a reason :y 9 our hemisph]e ! nor? w9ds pr$om9ate ov] ! s\?4--,,sp$d+1 ,j4 ,works ( ,francis ,bacon 7,bo/on71 ,preface to ,de ,int]pretatione ,naturae ,prooemium4 #iah4 ,bacon himself 0 v ignorant ( all t _h be5 done by ma!matics2 &1 /range to say1 he especially object$ to a/ronomy be+ h&$ ov] to ! ma!maticians4 ,lev]ri] & ,adams1 calculat+ an unkn[n planet into a visible exi/5ce by enorm\s heaps ( algebra1 furni% ! la/ comm5t ( note on ?is specim5 ( ! goodness ( ,bacon's view4444 ,ma!matics 0 beg9n+ to be ! great in/rum5t ( exact inquiry3 ,bacon ?rew ! sci5ce aside1 f ignorance1 j at ! "t :5 8 enorm\s sagacity1 appli$ to k1 w\ld h made him see ! "p x 0 to play4 ,if ,newton _h tak5 ,bacon = 8 ma/]1 n he1 b somebody else1 w\ld h be5 ,newton4 --,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 pp4 #ec-ed4 #iai4 ,daniel ,b]n\lli us$ to tell two little adv5tures1 :i* he said _h giv5 him m pleasure ?an all ! o!r hon\rs he _h receiv$4 ,travell+ ) a le>n$ /rang]1 :o1 be+ pleas$ ) 8 conv]sation1 ask$ 8 "n2 8,i am ,daniel ,b]n\lli10 answ]$ he ) great mode/y2 8& ,i10 said ! /rang] 7:o ?\ght he meant to laugh at him7 8am ,isaac ,newton40 ,ano!r "t1 hav+ to d9e ) ! celebrat$ ,ko5ig1 ! ma!matician1 :o boa/$1 ) "s degree ( self-complac5cy1 ( a difficult problem he _h solv$ ) mu* tr\ble1 ,b]n\lli w5t on do+ ! hon\rs ( 8 table1 & :5 !y w5t to dr9k c(fee he pres5t$ ,ko5ig ) a solution ( ! problem m elegant ?an 8 own4 --,,hutton1 ,,*>les4 ,a ,philosophical & ,ma!matical ,diction>y 7,london1 #ahae71 ,vol4 #a1 p4 #bbf4 #ibj4 ,foll[+ ! example ( ,ar*im$es :o wi%$ 8 tomb decorat$ ) 8 mo/ beautiful discov]y 9 geometry & ord]$ x inscrib$ ) a cyl9d] circumscrib$ by a sph]e1 ,james ,b]n\lli reque/$ t 8 tomb be inscrib$ ) 8 log>i?mic spiral toge!r ) ! words1 8,eadem mutata resurgo10 a happy allusion to ! hope ( ! ,*ri/ians1 :i* is 9 a way symboliz$ by ! prop]ties ( t curve4 --,,font5elle4 ,eloge de ,m4 ,b]n\lli2 ,oeuvres de ,font5elle1 t4 #e 7#ageh71 p4 #aab4 #iba4 ,?is =mula = comput+ ,b]n\lli's numb]s 0 fir/ giv5 by ,james ,b]n\lli4 ,he gave no g5]al demon/ration2 b 0 q aw>e ( ! importance ( 8 !orem1 = he boa/s t by means ( x he calculat$ intra semi-quadrantem horae6 ! sum ( ! #aj? p[]s ( ! fir/ ?\s& integ]s1 & f\nd x to be #ia1dji1ibd1bda1dbd1bdc1dbd1bda1ibd1bdb1 ejj4 --,,*ry/al1 ,g4 ,algebra1 ,"p #b 7,ed9burgh1 #ahgi71 p4 #bji4 #ibb4 ,9 ! ye> #afib1 ,james ,b]n\lli1 discuss+ ! log>i?mic spiral or equiangul> spiral1 444 %[s t x reproduces itself 9 its evolute1 its involute1 & its cau/ics ( bo? reflection & refraction1 & !n adds3 8,b s9ce ?is m>vell\s spiral1 by su* a s+ul> & wond]ful peculi>ity1 pleases me s mu* t ,i c sc>ce be satisfi$ ) ?9k+ ab\t x1 ,i h ?\ght t x might n be inelegantly us$ = a symbolic repres5tation ( v>i\s matt]s4 ,= s9ce x always produces a spiral simil> to itself1 inde$ precisely ! same spiral1 h[ev] x may be involv$ or evolv$1 or reflect$ or refract$1 x may be tak5 z an emblem ( a prog5y always 9 all ?+s l ! p>5t1 simillima filia matri4 ,or1 if x is n =bidd5 to comp>e a !orem ( et]nal tru? to ! my/]ies ( our fai?1 x may be tak5 z an emblem ( ! et]nal g5]ation ( ! ,son1 :o z an image ( ! ,fa!r1 emanat+ f him1 z light f light1 rema9s ) him1 h[soev] ov]%ad[$4 ,or1 if y pref]1 s9ce our spira mirabilis rema9s1 amid all *anges1 mo/ p]si/5tly itself1 & exactly ! same z ev]1 x may be us$ z a symbol1 ei!r ( =titude & con/ancy 9 adv]sity1 or1 ( ! human body1 :i* aft] all its *anges1 ev5 aft] dea?1 w be re/or$ to its exact & p]fect self1 s t1 inde$1 if ! fa%ion ( ,ar*im$es 7 all[$ 9 ^! days1 ,i %\ld gladly h my tomb/one be> ?is spiral1 ) ! motto1 8,?\gh *ang$1 ,i arise aga9 exactly ! same1 ,eadem num]o mutata resurgo40--,,hill1 ,,?omas4 ,! ,uses ( ,ma!sis2 ,biblio!ca ,sacra1 ,vol4 #cb1 pp4 #eae-eaf4 #ibc4 ,babbage 0 "o ( ! f\nd]s ( ! ,cambridge ,analytical ,society ^: purpose he /at$ 0 to advocate 8! pr9ciples ( pure d-ism z oppos$ to ! dot-age ( ! univ]sity40--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #dea4 #ibd4 ,bolyai ,janos :5 9 g>rison ) cavalry (fic]s1 0 provok$ by ?irte5 ( !m & accept$ all _! *all5ges on condition t he be p]mitt$ aft] ea* duel to play a bit on 8 viol94 ,he came out victor f 8 ?irte5 duels1 leav+ 8 ?irte5 adv]s>ies on ! squ>e4--,,hal/$1 ,g4 ,b4 ,bolyai's ,sci5ce ,absolute ( ,space 7,au/91 #ahif71 ,introduction1 p4 #bi4 #ibe4 ,bolyai ,janos project$ a univ]sal language = spee* z we h x = music & ma!matics4--,,hal/$1 ,g4 ,b4 ,bolyai's ,sci5ce ,absolute ( ,space 7,au/91 #ahif71 ,introduction1 p4 #bi4 #ibf4 ,bolyai's ,sci5ce ,absolute ( ,space--! mo/ extraord9>y two doz5 pages 9 ! hi/ory ( ?\ght6 --,,hal/$1 ,g4 ,b4 ,bolyai's ,sci5ce ,absolute ( ,space 7,au/91 #ahif71 ,introduction1 p4 #ah4 #ibg4 ,wolfgang ,bolyai 0 extremely mode/4 ,no monum5t1 said he1 %\ld /& ov] 8 grave1 only an apple-tree1 9 memory ( ! ?ree apples3 ! two ( ,eve & ,p>is1 :i* made hell out ( e>?1 & t ( ,newton1 :i* elevat$ ! e>? aga9 into ! circle ( ! heav5ly bodies4--,,cajori1 ,f4 ,hi/ory ( ,elem5t>y ,ma!matics 7,new ,york1 #aiaj71 p4 #bgc4 #ibh4 ,b]n>d ,bolzano dispell$ ! cl\ds t ?r\gh\t all ! =egone c5turies _h envelop$ ! notion ( ,inf9itude 9 d>kness1 completely %e>$ ! great t]m ( its vagu5ess )\t %e>+ x ( its /r5g?1 & ?us r5d]$ x =ev] available = ! purposes ( logical disc\rse4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #db4 #ibi4 ,let me tell y h[ at "o "t ! fam\s ma!matician ,euclid became a physician4 ,x 0 dur+ a vacation1 :i* ,i sp5t 9 ,prague z ,i mo/ always did1 :5 ,i 0 attack$ by an illness nev] be=e exp]i5c$1 :i* manife/$ itself 9 *ill9ess & pa9ful we>9ess ( ! :ole body4 ,9 ord] to ease my condition ,i took up ,euclid's ,elem5ts & read = ! fir/ "t 8 doctr9e ( ratio1 :i* ,i f\nd treat$ !re 9 a mann] entirely new to me4 ,! +5uity display$ 9 ,euclid's pres5tation fill$ me ) su* vivid pleasure1 t =?) ,i felt z well z ev]4--,,bolzano1 ,,b]n>d4 ,selb/biographie 7,wi51 #ahge71 p4 #bj4 #icj4 ,mr4 ,cayley1 ( :om x may be s truly said1 :e!r ! matt] he takes 9 h& be great or small1 8nihil tetigit quod non ornavit104444--,,sylve/]1 ,j4 ,j4 ,philosophic ,transactions ( ! ,royal ,society1 ,vol4 #ag 7#ahfd71 p4 #fje4 #ica4 ,x is n ,cayley's way to analyze concepts into _! ultimate elem5ts4444 ,b he is ma/] ( ! empirical utilization ( ! mat]ial3 9 ! way he comb9es x to =m a s+le ab/ract concept :i* he g5]alizes & !n subjects to computative te/s1 9 ! way ! newly acquir$ data are made to yield at a s+le /roke ! g5]al compreh5sive idea to ! subsequ5t num]ical v]ification ( :i* ye>s ( labor are devot$4 ,cayley is ?us ! natural philosoph] among ma!maticians4--,,noe!r1 ,m4 ,ma!matis*e ,annal51 ,bd4 #df 7#ahie71 p4 #dgi4 #icb4 ,:5 ,cayley _h rea*$ 8 mo/ advanc$ g5]alizations he proce$$ to e/abli% !m directly by "s me?od or o!r1 ?\gh he seldom gave ! clue by :i* !y _h fir/ be5 obta9$3 a proce$+ :i* does n t5d to make 8 pap]s easy read+4444 ,8 lit]>y /yle is direct1 simple & cle>4 ,8 legal tra9+ _h an influ5ce1 n m]ely ^u 8 mode ( arrangem5t b also ^u 8 expression2 ! result is t 8 pap]s are sev]e & pres5t a curi\s contra/ to ! luxuriant en?usiasm :i* p]vades s _m ( ,sylve/]'s pap]s4 ,he us$ to prep>e 8 "w = publication z soon z he c>ri$ 8 inve/igations 9 any subject f> en\gh = 8 imm$iate purpose4444 ,a pap] once writt5 out 0 promptly s5t = publication2 ?is practice he ma9ta9$ ?r\gh\t life4444 ,! consequ5ce is t he has left few arre>s ( unf9i%$ or unpubli%$ pap]s2 8 "w has be5 giv5 by himself to ! _w4--,,=sy?1 ,a4 ,r4 ,proce$+s ( ,london ,royal ,society1 ,vol4 #eh 7#ahie71 pp4 #bc-bd4 #icc4 ,cayley 0 s+ul>ly le>n$ 9 ! "w ( o!r m51 & ca?olic 9 8 range ( k4 ,yet he did n read a memoir completely ?r\gh3 8 cu/om 0 to read only s mu* z w\ld enable him to grasp ! mean+ ( ! symbols & und]/& its scope4 ,! ma9 result w\ld !n become to him a subject ( inve/igation3 he w\ld e/abli% x 7or te/ x7 by algebraic analysis &1 n infrequ5tly1 develop x s to obta9 o!r results4 ,?is faculty ( grasp+ & te/+ rapidly ! "w ( o!rs1 toge!r ) 8 great k1 made him an invaluable ref]ee2 8 s]vices 9 ?is capacity 7 us$ ?r\gh a long s]ies ( ye>s by a numb] ( societies to :i* he 0 almo/ 9 ! position ( /&+ ma!matical advisor4 --,,=sy?1 ,a4 ,r4 ,proce$+s ,london ,royal ,society1 ,vol4 #eh 7#ahie71 pp4 #aa-ab4 #icd4 ,b]tr&1 ,d>b\x1 & ,glai%] h comp>$ ,cayley to ,eul]1 alike = 8 range1 8 analytical p[]1 &1 n lea/1 = 8 prolific production ( new views & f]tile !ories4 ,!re is h>dly a subject 9 ! :ole ( pure ma!matics at :i* he has n work$4--,,=sy?1 ,a4 ,r4 ,proce$+s ,london ,royal ,society1 ,vol4 #eh 7#ahie71 p4 #ba4 #ice4 ,! ma!matical tal5t ( ,cayley 0 *>act]iz$ by cle>ness & extreme elegance ( analytical =m2 x 0 re-en=c$ by an incomp>able capacity = "w :i* has caus$ ! di/+ui%$ s*ol> to be comp>$ ) ,cau*y4 --,,h]mite1 ,c4 ,comptes ,r5dus1 t4 #abj 7#ahie71 p4 #bcd4 #icf4 ,j4 ,j4 ,sylve/] 0 an en?usia/ic support] ( re=m 9 ! tea*+ ( geometry4 ,! diff]5ce 9 attitude on ?is "q betwe5 ! two =emo/ ,briti% ma!maticians1 ,j4 ,j4 ,sylve/]1 ! algebrai/1 & ,ar?ur ,cayley1 ! algebrai/ & geomet]1 0 grotesque4 ,sylve/] wi%$ to bury ,euclid 8deep] ?an e'er plummet s\nd$0 out ( ! s*oolboy's rea*2 ,cayley1 an ard5t admir] ( ,euclid1 desir$ ! ret5tion ( ,simson's ,euclid4 ,:5 rem9d$ t ?is treatise 0 a mixture ( ,euclid & ,simson1 ,cayley sugge/$ /rik+ out ,simson's additions & keep+ /rictly to ! orig9al treatise4 --,,cajori1 ,f4 ,hi/ory ( ,elem5t>y ,ma!matics 7,new ,york1 #aiaj71 p4 #bhe4 #icg4 ,tait once urg$ ! advantage ( ,quat]nions on ,cayley 7:o nev] us$ !m71 say+3 8,y "k ,quat]nions are j l a pocket-map40 8,t may be10 repli$ ,cayley1 8b y've got to take x out ( y\r pocket1 & unfold x1 be=e x's ( any use40 ,& he dismiss$ ! subject ) a smile4 --,,?ompson1 ,s4 ,p4 ,life ( ,"l ,kelv9 7,london1 #aiaj71 p4 #aacg4 #ich4 ,z he ,clif=d spoke he appe>$ n to be work+ out a "q1 b simply tell+ :at he saw4 ,)\t any diagram or symbolic aid he describ$ ! geometrical conditions on :i* ! solution dep5d$1 & !y seem$ to /& out visibly 9 space4 ,!re 7 no long] consequ5ces to be d$uc$1 b real & evid5t facts :i* only requir$ to be se54444 ,s :ole & complete 0 8 vision t = ! "t ! only /range ?+ 0 t anybody %\ld fail to see x 9 ! same way4 ,:5 "o endeavor$ to call x up aga91 & n till !n1 x became cle> t ! magic ( g5ius _h be5 at "w1 & t ! common sight _h be5 rais$ to t high] p]ception by ! p[] t makes & trans=ms ideas1 ! conqu]+ & ma/]ful quality ( ! human m9d :i* ,goe! call$ 9 "o ^w das ,dmonis*e4--,,pollock1 ,f4 ,clif=d's ,lectures & ,essays 7,new ,york1 #aija71 ,vol4 #a1 ,introduction1 pp4 #e-f4 #ici4 ,mu* ( 8 ,clif=d's be/ "w 0 actually spok5 be=e x 0 writt54 ,he gave mo/ ( 8 public lectures ) no visible prep>ation beyond v %ort notes1 & ! outl9e seem$ to be fill$ 9 )\t ef=t or hesitation4 ,aft]w>ds he w\ld revise ! lecture f a %or?& writ]'s report1 or sometimes write d[n f memory almo/ exactly :at he _h said4 ,x fell out n[ & !n1 h[ev]1 t nei!r ( ^! ?+s 0 done2 9 su* cases !re is n[ no record ( ! lecture at all4--,,pollock1 ,f4 ,clif=d's ,lectures & ,essays 7,new ,york1 #aija71 ,vol4 #a1 ,introduction1 p4 #aj4 #idj4 ,i _c f9d any?+ %[+ e>ly aptitude = acquir+ languages2 b t he ,clif=d _h x & 0 fond ( ex]cis+ x 9 lat] life is c]ta94 ,"o practical reason = x 0 ! desire ( be+ able to read ma!matical pap]s 9 =eign j\rnals2 b ?is w\ld n acc\nt = 8 tak+ up ,spani%1 ( :i* he acquir$ a compet5t k 9 ! c\rse ( a t\r to ! ,pyr5ees4 ,:5 he 0 at ,algi]s 9 #ahgf he began ,arabic1 & made progress en\gh to foll[ 9 a g5]al way a c\rse ( lessons giv5 9 t language4 ,he read mod]n ,greek flu5tly1 & at "o "t he 0 furi\s ab\t ,sanskrit4 ,he ev5 sp5t "s "t on hi]oglyphics4 ,a new language is a riddle be=e x is conqu]$1 a p[] 9 ! h& aft]w>ds3 to ,clif=d e riddle 0 a *all5ge1 & e *ance ( new p[] a div9e opportunity to be seiz$4 ,h5ce he 0 likewise int]e/$ 9 ! v>i\s modes ( convey+ & express+ language inv5t$ = special purposes1 su* z ! ,morse alphabet & %or?&4444 ,i h =gott5 to m5tion 8 comm& ( ,fr5* & ,g]man1 ! =m] ( :i* he knew v well1 & ! latt] q suffici5tly24444--,,pollock1 ,f4 ,clif=d's ,lectures & ,essays 7,new ,york1 #aija71 ,vol4 #a1 ,introduction1 pp4 #aa-ab4 #ida4 ,! mo/ rem>kable ?+ 0 8 ,clif=d's great /r5g? z comp>$ ) 8 weight1 z %[n 9 "s ex]cises4 ,at "o "t he c\ld pull up on ! b> ) ei!r h&1 :i* is well kn[n to be "o ( ! greate/ feats ( /r5g?4 ,8 n]ve at dang]\s heights 0 extraord9>y4 ,i am appall$ n[ to ?9k t he climb$ up & sat on ! cross b>s ( ! wea!rcock on a *ur* t[]1 & :5 by way ( do+ some?+ worse ,i w5t up & hung by my toes to ! b>s he did ! same4 ,quot$ f a lett] by "o ( ,clif=d's fri5ds to ,pollock1 ,f43 ,clif=d's ,lectures & ,essays 7,new ,york1 #aija71 ,vol4 #a1 ,introduction1 p4 #h4 #idb4 ,comte may truly be said to h creat$ ! philosophy ( high] ma!matics4--,,mill1 ,j4 ,s4 ,sy/em ( ,logic 7,new ,york1 #ahdf71 p4 #cfi4 #idc4 ,^! specim5s1 :i* ,i c\ld easily multiply1 may suffice to ju/ify a pr(\nd di/ru/ ( ,augu/e ,comte1 :]ev] he may v5ture to speak z a ma!matician4 ,b 8 va/ g5]al ability1 & t p]sonal intimacy ) ! great ,f\ri]1 :i* ,i mo/ will+ly take 8 own ^w = hav+ enjoy$1 mu/ always give an int]e/ to 8 views on any subject ( pure or appli$ ma!matics4--,,hamilton1 ,w4 ,r4 ,graves' ,life ( ,w4 ,r4 ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #dge4 #idd4 ,! mann] ( ,demoivre's dea? has a c]ta9 int]e/ = psy*ologi/s4 ,%ortly be=e x1 he decl>$ t x 0 necess>y = him to sleep "s t5 m9utes or a qu>t] ( an h\r long] ea* day ?an ! prec$+ "o3 ! day aft] he _h ?us rea*$ a total ( some?+ ov] tw5ty-?ree h\rs he slept up to ! limit ( tw5ty-f\r h\rs1 & !n di$ 9 8 sleep4--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aiaa71 p4 #cid4 #ide4 ,de ,morgan 0 expla9+ to an actu>y :at 0 ! *ance t a c]ta9 proportion ( "s gr\p ( p w\ld at ! end ( a giv5 "t be alive2 & quot$ ! actu>ial =mula1 involv+ 1 :i*1 9 answ] to a "q1 he expla9$ /ood = ! ratio ( ! circumf]5ce ( a circle to its diamet]4 ,8 acqua9tance1 :o _h s f> li/5$ to ! explanation ) int]e/1 int]rupt$ him & exclaim$1 8,my de> fri5d1 t mu/ be a delusion1 :at c a circle h to d ) ! numb] ( p alive at a giv5 "t80 --,,ball1 ,w4 ,w4 ,r4 ,ma!matical ,recreations & ,problems 7,london1 #ahif71 p4 #ahj2 ,see also ,de ,morgan's ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #agb4 #idf4 ,a few days aft]w>ds1 ,i w5t to him ! same actu>y ref]r$ to 9 #ide & v gravely told him t ,i _h discov]$ ! law ( human mortality 9 ! ,c>lisle ,table1 ( :i* he ?\ght v highly4 ,i told him t ! law 0 involv$ 9 ?is circum/ance4 ,take ! table ( ! expectation ( life1 *oose any age1 take its expectation & make ! ne>e/ integ] a new age1 d ! same ) t1 & s on2 beg9 at :at age y l1 y are sure to end at ! place :]e ! age pa/ is equal1 or mo/ ne>ly equal1 to ! expectation to come4 8,y don't mean t ?is always happ5s80--8,try x40 ,he did try1 aga9 & aga92 & f\nd x z ,i said4 8,?is is1 inde$1 a curi\s ?+2 ?is is a discov]y60 ,i might h s5t him ab\t trumpet+ ! law ( life3 b ,i cont5t$ myself ) in=m+ him t ! same ?+ w\ld happ5 ) any table :atsoev] 9 :i* ! fir/ column goes up & ! second goes d[n24444--,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #agb4 #idg4 ,de ,morgan relates t "s p]son _h made up #hjj anagrams on 8 "n1 ( :i* he _h se5 ab\t #fej4 ,comm5t+ on ^! he says3 ,two ( ^! ,i h jo9$ 9 ! title-page3 ,ut ag5do surgamus argu5do gu/amus4 ,a few ( ! o!rs are p]sonal rem>ks4 ,great gun6 d u a sum6 is a sne] at my pursuit2 b1 _/ n u ,g6 great sum6 a du _/ is m dignifi$4444 ,adsum1 nugator1 suge6 is address$ to a /ud5t :o cont9ues talk+ aft] ! lecture has comm5c$3 4444 ,graduatus sum6 nego applies to "o :o decl9$ to subscribe = an ,m4 ,a4 degree4 --,,de ,,morgan1 ,,augu/us4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #hb4 #idh4 ,desc>tes is ! complete/ type :i* hi/ory pres5ts ( ! purely ma!matical type ( m9d--t 9 :i* ! t5d5cies produc$ by ma!matical cultivation reign unbalanc$ & supreme4--,,mill1 ,j4 ,s4 ,an ,exam9ation ( ,sir ,w4 ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #fbf4 #idi4 ,to ,desc>tes1 ! great philosoph] ( ! #ag? c5tury1 is due ! undy+ cr$it ( hav+ remov$ ! bann :i* until !n re/$ ^u geometry4 ,! analytical geometry1 z ,desc>tes' me?od 0 call$1 soon l$ to an abundance ( new !orems & pr9ciples1 :i* f> transc5d$ ev]y?+ t ev] c\ld h be5 rea*$ ^u ! pa? pursu$ by ! anci5ts4--,,hankel1 ,h4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 p4 #aj4 #iej4 ,! application ( algebra has f> m ?an any ( 8 metaphysical speculations1 immortaliz$ ! "n ( ,desc>tes1 & con/itutes ! greate/ s+le /ep ev] made 9 ! progress ( ! exact sci5ces4--,,mill1 ,j4 ,s4 ,an ,exam9ation ( ,sir ,w4 ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #fag4 #iea4 444 1 4 444 !y say t ,ptolemy once ask$ him 7,euclid7 :e!r !re 0 9 geometry no %ort] way ?an t ( ! elem5ts1 & he repli$1 8,!re is no royal road to geometry40--,,proclus4 7,edition ,fri$le91 #ahgc71 ,prol4 ,,ii1 #ci4 #ieb4 ,someone :o _h begun to read geometry ) ,euclid1 :5 he _h le>n$ ! fir/ proposition1 ask$ ,euclid1 8,b :at %all ,i get by le>n+ ^! ?+s80 :]eupon ,euclid call$ 8 slave & said1 8,give him ?ree-p5ce1 s9ce he mu/ make ga9 out ( :at he le>ns40--,,/ob,,u4 7,edition ,wa*smu?1 #ahhd71 ,ecl4 ,,ii4 #iec4 ,! sacr$ writ+s except$1 no ,greek has be5 s mu* read & s v>i\sly translat$ z ,euclid4#e--;,,de ,,morgan1 ,a4 ,smi?'s ,diction>y ( ,greek & ,roman ,biology & ,my?ology 7,london1 #aijb71 ,article1 8,eucleides40 #e ,ricc>di's ,bibliografia ,euclidea 7,bologna1 #ahhg71 li/s ne>ly two ?\s& editions4 #ied4 ,! ?irte5 books ( ,euclid mu/ h be5 a trem5d\s advance1 probably ev5 great] ?an t conta9$ 9 ! 8,pr9cipia0 ( ,newton4--,,de ,,morgan1 ,a4 ,smi?'s ,diction>y ( ,greek & ,roman ,biography & ,my?ology 7,london1 #aijb71 ,article1 8,eucleides40 #iee4 ,to suppose t s p]fect a sy/em z t ( ,euclid's ,elem5ts 0 produc$ by "o man1 )\t any prec$+ model or mat]ials1 w\ld be to suppose t ,euclid 0 m ?an man4 ,we ascribe to him z mu* z ! weakness ( human und]/&+ w p]mit1 if we suppose t ! inv5tions 9 geometry1 :i* _h be5 made 9 a tract ( prec$+ ages1 7 by him n only c>ri$ mu* fur!r1 b dige/$ into s admirable a sy/em1 t 8 "w obscur$ all t w5t be=e x1 & made !m be =got & lo/4--,,reid1 ,,?omas4 ,essay on ! ,p[]s ( ! ,human ,m9d 7,ed9burgh1 #ahab71 ,vol4 #b1 p4 #cfh4 #ief4 ,x is ! invaluable m]it ( ! great ,basle ma!matician ,leonh>d ,eul]1 to h fre$ ! analytical calculus f all geometrical bonds1 & ?us to h e/abli%$ analysis z an indep5d5t sci5ce1 :i* f 8 "t on has ma9ta9$ an un*all5g$ lead]%ip 9 ! field ( ma!matics4--,,hankel1 ,h4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 p4 #ab4 #ieg4 ,we may safely say1 t ! :ole =m ( mod]n ma!matical ?9k+ 0 creat$ by ,eul]4 ,x is only ) ! greate/ difficulty t "o is able to foll[ ! writ+s ( any au?or imm$iately prec$+ ,eul]1 because x 0 n yet kn[n h[ to let ! =mulas speak = !mselves4 ,?is art ,eul] 0 ! fir/ "o to tea*4--,,rudio1 ,f4 ,quot$ by ,ahr5s ,w43 ,s*]z und ,ern/ 9 d] ,ma!matik 7,leipzig1 #aijd71 p4 #bea4 #ieh4 ,! g5]al k ( our au?or ,leonh>d ,eul] 0 m ext5sive ?an c\ld well be expect$1 9 "o :o _h pursu$1 ) su* unremitt+ ardor1 ma!matics & a/ronomy z 8 favorite /udies4 ,he _h made a v consid]able progress 9 m$ical1 botanical1 & *emical sci5ce4 ,:at 0 /ill m extraord9>y1 he 0 an excell5t s*ol>1 & possess$ 9 a high degree :at is g5]ally call$ erudition4 ,he _h att5tively read ! mo/ em95t writ]s ( anci5t ,rome2 ! civil & lit]>y hi/ory ( all ages & all nations 0 famili> to him2 & =eign]s1 :o 7 only acqua9t$ ) 8 works1 7 a/oni%$ to f9d 9 ! conv]sation ( a man1 ^: long life seem$ solely occupi$ 9 ma!matical & physical rese>*es & discov]ies1 su* an ext5sive acqua9tance ) ! mo/ int]e/+ bran*es ( lit]ature4 ,9 ?is respect1 no d\bt1 he 0 mu* indebt$ to an uncommon memory1 :i* seem$ to reta9 e idea t 0 convey$ to x1 ei!r f read+ or f m$itation4--,,hutton1 ,,*>les4 ,philosophical & ,ma!matical ,diction>y 7,london1 #ahae71 pp4 #dic-did4 #iei4 ,eul] c\ld repeat ! ,a5eid f ! beg9n+ to ! end1 & he c\ld ev5 tell ! fir/ & la/ l9es 9 e page ( ! edition :i* he us$4 ,9 "o ( 8 works !re is a le>n$ memoir on a "q 9 me*anics1 ( :i*1 z he himself in=ms u1 a v]se ( ,a5eid#f gave him ! fir/ idea4 --,,brew/]1 ,,david4 ,lett]s ( ,eul] 7,new ,york1 #ahgb71 ,vol4 #a1 p4 #bd4 #f ,! l9e ref]r$ to is3 8,! an*or drops1 ! ru%+ keel is /aid40 #ifj4 ,mo/ ( 8 ,eul]'s memoirs are conta9$ 9 ! transactions ( ! ,academy ( ,sci5ces at ,/4 ,pet]sburg1 & 9 ^? ( ! ,academy at ,b]l94 ,f #agbh to #aghc a l>ge portion ( ! ,petropolitan transactions 7 fill$ by 8 writ+s4 ,he _h engag$ to furni% ! ,pet]sburg ,academy ) memoirs 9 suffici5t numb] to enri* its acts = tw5ty ye>s--a promise m ?an fulfill$1 = d[n to #ahah ,eul] di$ 9 #agic ! volumes usually conta9$ "o or m pap]s ( 84 ,x has be5 said t an edition ( ,eul]'s complete works w\ld fill #af1jjj qu>to pages4--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 pp4 #bec-bed4 #ifa4 ,eul] :o c\ld h be5 call$ almo/ )\t metaphor1 & c]ta9ly )\t hyp]bole1 analysis inc>nate4--,,arago4 ,oeuvres1 t4 #b 7#ahed71 p4 #dcc4 #ifb4 ,eul] calculat$ )\t any app>5t ef=t1 j z m5 brea!1 z eagles su/a9 !mselves 9 ! air4--,,arago4 ,oeuvres1 t4 #b 7#ahed71 p4 #acc4 #ifc4 ,two ( 8 ,eul]'s pupils hav+ comput$ to ! #ag? t]m1 a complicat$ conv]g+ s]ies1 _! results diff]$ "o unit 9 ! fiftie? ciph]2 & an appeal be+ made to ,eul]1 he w5t ov] ! calculation 9 8 m9d1 & 8 decision 0 f\nd correct4--,,brew/]1 ,,david4 ,lett]s ( ,eul] 7,new ,york1 #ahgb71 ,vol4 #b1 p4 #bb4 #ifd4 ,9 #agce ! solv+ ( an a/ronomical problem1 propos$ by ! ,academy1 = :i* sev]al em95t ma!maticians _h dem&$ sev]al mon?s' "t1 0 a*iev$ 9 ?ree days by ,eul] ) aid ( improv$ me?ods ( 8 own4444 ,) /ill sup]ior me?ods ?is same problem 0 solv$ by ! illu/ri\s ,gauss 9 "o h\r4--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #bdh4 #ife4 ,eul]'s ,t5tam5 novae !orae musicae _h no great success1 z x conta9$ too mu* geometry = musicians1 & too mu* music = geomet]s4--,,fuss1 ,n4 ,quot$ by ,brew/]3 ,lett]s ( ,eul] 7,new ,york1 #ahgb71 ,vol4 #a1 p4 #bf4 #iff4 ,eul] 0 a believ] 9 ,god1 d[nright & /raight-=w>d4 ,! foll[+ /ory is told by ,?iebault1 9 8 ,s\v5irs de v+t ans de sj\r ,b]l91 4444 ,?iebault says t he has no p]sonal k ( ! tru? ( ! /ory1 b t x 0 believ$ ?r\gh\t ! :ole ( ! nor? ( ,europe4 ,did]ot paid a visit to ! ,russian ,c\rt at ! invitation ( ! ,empress4 ,he conv]s$ v freely1 & gave ! y\ng] memb]s ( ! ,c\rt circle a good deal ( lively a!ism4 ,! ,empress 0 mu* amus$1 b "s ( h] c\nsellors sugge/$ t x might be desirable to *eck ^! expositions ( doctr9e4 ,! ,empress did n l to put a direct muzzle on h] gue/'s tongue1 s ! foll[+ plot 0 contriv$4 ,did]ot 0 in=m$ t a le>n$ ma!matician 0 9 possession ( an algebraical demon/ration ( ! exi/5ce ( ,god1 & w\ld give x him be=e all ! ,c\rt1 if he desir$ to he> x4 ,did]ot gladly cons5t$3 ?\gh ! "n ( ! ma!matician is n giv51 x 0 ,eul]4 ,he advanc$ t[>d ,did]ot1 & said gravely1 & 9 a tone ( p]fect conviction3 a bn ,monsieur1 ------- x1 donc ,dieu exi/e2 repondez6 n ,did]ot1 to :om algebra 0 ,hebrew1 0 emb>rass$ & disconc]t$2 :ile peals ( laught] rose on all sides4 ,he ask$ p]mission to return to ,france at once1 :i* 0 grant$4 --,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #bea4 #ifg4 ,f]mat di$ ) ! belief t he _h f\nd a long-s\ght-= law ( prime numb]s 9 ! =mula #bbn #a a prime1 b he admitt$ t he 0 unable to prove x rigor\sly4 ,! law is n true1 z 0 po9t$ out by ,eul] 9 ! example #bbe #a #d1bid1ifg1big #f1gjj1dag times #fda4 ,! ,am]ican lightn+ calculator ,z]ah ,colburn1 :5 a boy1 readily f\nd ! factors b 0 unable to expla9 ! me?od by :i* he made 8 m>vell\s m5tal computation4--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #ahj4 #ifh4 ,i crave ! lib]ty to conceal my "n1 n to suppress x4 ,i h compos$ ! lett]s ( x writt5 9 ,lat9 9 ?is s5t5ce-- ,9 ,ma!si a sole fundes4#g --,,flam/e$1 ,j4 ,macclesfield3 ,correspond5ce ( ,sci5tific ,m5 7,ox=d1 #ahda71 ,vol4 #b1 p4 #ij4 #g ,johannes ,flam/e$ius4 #ifi4 ,to ! ,memory ( ,f\ri] ,f\ri]6 ) solemn & pr(\nd delight1 ,joy born ( awe1 b k9dl+ mom5tly ,to an int5se & ?rill+ ec/acy1 ,i gaze ^u ?y glory & gr[ bright3 ,z if irradiate ) behold5 light2 ,z if ! immortal t rema9s ( !e ,attun$ me to ?y _s's h>mony1 ,brea?+ s]5e resolve & tranquil might4 ,reveal$ appe> ?y sil5t ?\ghts ( y\?1 ,z if to consci\sness1 & all t view ,prophetic1 ( ! h]itage ( tru? ,to ?y maje/ic ye>s ( manhood due3 ,d>kness & error flee+ f> away1 ,& ! pure m9d en?ron$ 9 p]fect day4 --,,hamilton1 ,w4 ,r4 ,graves' ,life ( ,w4 ,r4 ,hamilton1 7,new ,york1 #ahhb71 ,vol4 #a1 p4 #eif4 #igj4 ,a/ronomy & ,pure ,ma!matics are ! magnetic poles t[>d :i* ! compass ( my m9d ev] turns4--,,gauss ,,to ,,bolyai4 ,briefwe*sel 7,s*midt-,/akel71 7#ahii71 p4 #ee4 #iga4 ,gauss calculat$ ! elem5ts ( ! planet ,c]es & 8 analysis prov$ him to be ! fir/ ( !oretical a/ronom]s no less ?an ! greate/ ( 8ari?meticians40--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #deh4 #igb4 ,! ma!matical giant ,gauss1 :o f 8 l(ty heights embraces 9 "o view ! />s & ! abysses4444--,,bolyai1 ,w4 ,kurz] ,grundriss e9es ,v]su*s 7,m>os ,vas>hely1 #ahea71 p4 #dd4 #igc4 ,almo/ ev]y?+1 :i* ! ma!matics ( our c5tury has br\ght =? 9 ! way ( orig9al sci5tific ideas1 atta*es to ! "n ( ,gauss4--,,kroneck]1 ,l4 ,zahl5!orie1 ,teil #a 7,leipzig1 #aija71 p4 #dc4 #igd4 ,i am giv+ ?is w9t] two c\rses ( lectures to ?ree /ud5ts1 ( :i* "o is only mod]ately prep>$1 ! o!r less ?an mod]ately1 & ! ?ird lacks bo? prep>ation & ability4 ,su* are ! on]a ( a ma!matical pr(ession4 --,,gauss ,,to ,,bessel1 #ahaj4 ,gauss-,bessel ,briefwe*sel 7#ahhj71 p4 #ajg4 #ige4 ,gauss once said 8,ma!matics is ! que5 ( ! sci5ces & numb]-!ory ! que5 ( ma!matics40 ,if ?is be true we may add t ! ,disquisitiones is ! ,magna ,*>ta ( numb]-!ory4 ,! advantage :i* sci5ce ga9$ by ,gauss' long-l+]+ me?od ( publication is ?is3 ,:at he put into pr9t is z true & important today z :5 fir/ publi%$2 8 publications are /atutes1 sup]ior to o!r human /atutes 9 ?is1 t n[h]e & nev] has a s+le error be5 detect$ 9 !m4 ,?is ju/ifies & makes intelligible ! pride ) :i* ,gauss said 9 ! ev5+ ( 8 life ( ! fir/ l>g] "w ( 8 y\?3 8,! ,disquisitiones ari?meticae belong to hi/ory40 --,,cantor1 ,m4 ,allgeme9e ,deuts*e ,biographie1 ,bd4 #h 7#ahgh71 p4 #dce4 #igf4 ,"h ,i am at ! limit :i* ,god & nature has assign$ to my individuality4 ,i am compell$ to dep5d ^u ^w1 language & image 9 ! mo/ precise s5se1 & am :olly unable to op]ate 9 any mann] :atev] ) symbols & numb]s :i* are easily intelligible to ! mo/ highly gift$ m9ds4--,,goe!4 ,lett] to ,naumann 7#ahbf72 ,vogel3 ,goe!'s ,selb/zeugnisse 7,leipzig1 #aijc71 p4 #ef4 #igg4 ,diri*let 0 n satisfi$ to /udy ,gauss' 8,disquisitiones ari?meticae0 once or sev]al times1 b cont9u$ ?r\gh\t life to keep 9 close t\* ) ! weal? ( deep ma!matical ?\ghts :i* x conta9s by p]us+ x aga9 & aga94 ,= ?is reason ! book 0 nev] plac$ on ! %elf b _h an abid+ place on ! table at :i* he work$4444 ,diri*let 0 ! fir/ "o1 :o n only fully und]/ood ?is "w1 b made x also accessible to o!rs4 --,,kumm]1 ,e4 ,e4 ,diri*let3 ,w]ke1 ,bd4 #b1 p4 #cae4 #igh4 ,! fam\s attack ( ,sir ,william ,hamilton on ! t5d5cy ( ma!matical /udies af=ds ! mo/ express evid5ce ( ^? fatal lacunae 9 ! circle ( 8 k1 :i* unfitt$ him = tak+ a compreh5sive or ev5 an accurate view ( ! processes ( ! human m9d 9 ! e/abli%m5t ( tru?4 ,if !re is any pre-requisite :i* all mu/ see to be indisp5sable 9 "o :o attempts to give laws to ! human intellect1 x is a ?or\gh acqua9tance ) ! modes by :i* human intellect has proce$$1 9 ! case :]e1 by univ]sal ackn[l$gm5t1 gr\nd$ on subsequ5t direct v]ification1 x has succe$$ 9 asc]ta9+ ! greate/ numb] ( important & recondite tru?s4 ,?is requisite ,sir ,w4 ,hamilton _h n1 9 any tol]able degree1 fulfill$4 ,ev5 ( pure ma!matics he app>5tly knew little b ! rudim5ts4 ,( ma!matics z appli$ to inve/igat+ ! laws ( physical nature2 ( ! mode 9 :i* ! prop]ties ( numb]1 ext5sion1 & figure1 are made in/rum5tal to ! asc]ta9m5t ( tru?s o!r ?an ari?metical or geometrical--x is too mu* to say t he _h ev5 a sup]ficial k3 !re is n a l9e 9 8 works :i* %[s him to h _h any k at all4--,,mill1 ,j4 ,s4 ,exam9ation ( ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #fjg4 #igi4 ,helmholtz--! physiologi/ :o le>n$ physics = ! sake ( 8 physiology1 & ma!matics = ! sake ( 8 physics1 & is n[ 9 ! fir/ rank ( all ?ree4 --,,clif=d1 ,w4 ,k4 ,aims & ,in/rum5ts ( ,sci5tific ,?\ght2 ,lectures & ,essays1 ,vol4 #a 7,london1 #aija71 p4 #afe4 #ihj4 ,x is said ( ,jacobi1 t he attract$ ! p>ticul> att5tion & fri5d%ip ( ,bckh1 ! director ( ! philological sem9>y at ,b]l91 by ! great tal5t he display$ = philology1 & only at ! end ( two ye>s' /udy at ! ,univ]sity1 & aft] a sev]e m5tal /ruggle1 0 able to make 8 f9al *oice 9 favor ( ma!matics4--,,sylve/]1 ,j4 ,j4 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b 7,cambridge1 #aijh71 p4 #fea4 #iha4 ,:5 ,dr4 ,johnson felt1 or fanci$ he felt1 8 fancy disord]$1 8 con/ant recurr5ce 0 to ! /udy ( ari?metic4--,,boswell1 ,j4 ,life ( ,johnson 7,h>p]'s ,edition1 #ahga71 ,vol4 #b1 p4 #bfd4 #ihb4 ,end[$ ) two qualities1 :i* seem$ incompatible ) ea* o!r1 a volcanic imag9ation & a p]t9acity ( intellect :i* ! mo/ t$i\s num]ical calculations c\ld n daunt1 ,kepl] conjectur$ t ! movem5ts ( ! cele/ial bodies mu/ be connect$ toge!r by simple laws1 or1 to use 8 own expression1 by h>monic laws4 ,^! laws he und]took to discov]4 ,a ?\s& fruitless attempts1 errors ( calculation insep>able f a colossal und]tak+1 did n prev5t him a s+le in/ant f advanc+ resolutely t[>d ! goal ( :i* he imag9$ he _h obta9$ a glimpse4 ,tw5ty-two ye>s 7 employ$ by him 9 ?is inve/igation1 & /ill he 0 n we>y ( x6 ,:at1 9 reality1 are tw5ty-two ye>s ( labor to him :o is ab\t to become ! legislator ( worlds2 :o %all inscribe 8 "n 9 ineffaceable *>act]s ^u ! frontispiece ( an immortal code2 :o %all be able to exclaim 9 di?yrambic language1 & )\t incurr+ ! reproa* ( anyone1 8,! die is ca/2 ,i h writt5 my book2 x w be read ei!r 9 ! pres5t age or by po/]ity1 x matt]s n :i*2 x may well await a read]1 s9ce ,god has wait$ six ?\s& ye>s = an int]pret] ( 8 words40--,,arago4 ,eulogy on ,laplace3 ,bad5 ,p[ell ,smi?sonian ,report1 #ahgd1 p4 #acb4 #ihc4 ,! great ma/]s ( mod]n analysis are ,lagrange1 ,laplace1 & ,gauss1 :o 7 contempor>ies4 ,x is int]e/+ to note ! m>k$ contra/ 9 _! /yles4 ,lagrange is p]fect bo? 9 =m & matt]1 he is c>eful to expla9 8 proc$ure1 & ?\gh 8 argum5ts are g5]al !y are easy to foll[4 ,laplace on ! o!r h& expla9s no?+1 is indiff]5t to /yle1 &1 if satisfi$ t 8 results are correct1 is cont5t to leave !m ei!r ) no pro( or ) a faulty "o4 ,gauss is z exact & elegant z ,lagrange1 b ev5 m difficult to foll[ ?an ,laplace1 = he removes e trace ( ! analysis by :i* he rea*$ 8 results1 & /udies to give a pro( :i* :ile rigor\s %all be z concise & syn!tical z possible4--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #dfc4 #ihd4 ,lagrange1 9 "o ( ! lat] ye>s ( 8 life1 imag9$ t he _h ov]come ! difficulty ( ! p>allel axiom4 ,he w5t s f> z to write a pap]1 :i* he took ) him to ! ,in/itute1 & began to read x4 ,b 9 ! fir/ p>agraph some?+ /ruck him :i* he _h n obs]v$3 he mutt]$ ,il faut que j'y songe encore1 & put ! pap] 9 8 pocket4 --,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #agc4 #ihe4 ,i nev] come across "o ( ,laplace's 8,?us x pla9ly appe>s0 )\t feel+ sure t ,i h h\rs ( h>d "w be=e me to fill up ! *asm & f9d out & %[ h[ x pla9ly appe>s4--,,b[dit*1 ,n4 ,quot$ by ,cajori3 ,tea*+ & ,hi/ory ( ,ma!matics 9 ! ,u4 ,s4 7,wa%+ton1 #ahif71 p4 #ajd4 #ihf4 ,biot1 :o assi/$ ,laplace 9 revis+ x ,! ,mecnique ,cle/e = ! press1 says t ,laplace himself 0 frequ5tly unable to recov] ! details 9 ! *a9 ( reason+1 & if satisfi$ t ! conclusions 7 correct1 he 0 cont5t to ins]t ! con/antly recurr+ =mula1 8,il e/ is a voir40 --,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p #dbg4 #ihg4 ,x w\ld be difficult to "n a man m rem>kable = ! greatness & ! univ]sality ( 8 intellectual p[]s ?an ,leibnitz4--,,mill1 ,j4 ,s4 ,sy/em ( ,logic1 ,bk4 #b1 *ap4 #e1 sect4 #f4 #ihh4 ,! influ5ce ( 8 ,leibnitz's g5ius 9 =m+ t peculi> ta/e bo? 9 pure & 9 mix$ ma!matics :i* has prevail$ 9 ,france1 z well z 9 ,g]many1 = a c5tury pa/1 w be f\nd1 ^u exam9ation1 to h be5 incomp>ably great] ?an t ( any o!r individual4--,,/ew>t1 ,,dugald4 ,philosophy ( ! ,human ,m9d1 ,"p #c1 *ap4 #a1 sect4 #c4 #ihi4 ,leibnitz's discov]ies lay 9 ! direction 9 :i* all mod]n progress 9 sci5ce lies1 9 e/abli%+ ord]1 symmetry1 & h>mony1 i4e41 compreh5siv5ess & p]spicuity1--r ?an 9 deal+ ) s+le problems1 9 ! solution ( :i* foll[]s soon atta9$ great] dext]ity ?an himself4 --,,m]z1 ,j4 ,t4 ,leibnitz1 ,*ap4 #f4 #iij4 ,x 0 8 ,leibnitz's love ( me?od & ord]1 & ! conviction t su* ord] & h>mony exi/$ 9 ! real _w1 & t our success 9 und]/&+ x dep5d$ ^u ! degree & ord] :i* we c\ld atta9 9 our own ?\ghts1 t orig9ally 0 probably no?+ m ?an a habit :i* by degrees grew into a =mal rule4#h ,?is habit 0 acquir$ by e>ly occupation ) legal & ma!matical que/ions4 ,we h se5 h[ ! !ory ( comb9ations & arrangem5ts ( elem5ts _h a special int]e/ = him4 ,we also saw h[ ma!matical calculations s]v$ him z a type & model ( cle> & ord]ly reason+1 & h[ he tri$ to introduce me?od & sy/em into logical discussions1 by r$uc+ to a small numb] ( t]ms ! multitude ( comp\nd notions he _h to deal )4 ,?is t5d5cy increas$ 9 /r5g?1 & ev5 9 ^? e>ly ye>s he elaborat$ ! idea ( a g5]al ari?metic1 ) a univ]sal language ( symbols1 or a *>act]i/ic :i* w\ld be applicable to all reason+ processes1 & r$uce philosophical inve/igations to t simplicity & c]ta9ty :i* ! use ( algebraic symbols _h introduc$ into ma!matics4 #h ,?is s5t5ce has be5 reword$ = ! purpose ( ?is quotation4 ,a m5tal attitude su* z ?is is always highly favorable = ma!matical z well z = philosophical inve/igations4 ,:]ev] progress dep5ds ^u precision & cle>ness ( ?\ght1 & :]ev] su* c be ga9$ by r$uc+ a v>iety ( inve/igations to a g5]al me?od1 by br++ a multitude ( notions "u a common t]m or symbol1 x proves ine/imable4 ,x necess>ily imports ! special qualities ( numb]--viz41 _! cont9uity1 inf9ity & inf9ite divisibility--l ma!matical quantities--& de/roys ! notion t irreconcilable contra/s exi/ 9 nature1 or gaps :i* _c be bridg$ ov]4 ,?us1 9 8 lett] to ,arnaud1 ,leibnitz expresses x z 8 op9ion t geometry1 or ! philosophy ( space1 =ms a /ep to ! philosophy ( motion--i4e41 ( corporeal ?+s--& ! philosophy ( motion a /ep to ! philosophy ( m9d4 --,,m]z1 ,j4 ,t4 ,leibnitz 7,philadelphia71 pp4 #dd-de4 #iia4 ,leibnitz believ$ he saw ! image ( creation 9 8 b9>y ari?metic 9 :i* he employ$ only two *>act]s1 unity & z]o4 ,s9ce ,god may be repres5t$ by unity1 & no?+ by z]o1 he imag9$ t ! ,supreme ,be+ might h drawn all ?+s f no?+1 j z 9 ! b9>y ari?metic all numb]s are express$ by unity ) z]o4 ,?is idea 0 s pleas+ to ,leibnitz1 t he communicat$ x to ! ,jesuit ,grimaldi1 ,presid5t ( ! ,ma!matical ,bo>d ( ,*9a1 ) ! hope t ?is emblem ( ! creation might conv]t to ,*ri/ianity ! reign+ emp]or :o 0 p>ticul>ly atta*$ to ! sci5ces4 --,,laplace4 ,essai ,philosophique sur les ,probabilits2 ,oeuvres 7,p>is1 #ahif71 t4 #g1 p4 #aai4 #iib4 ,sophus ,lie1 great comp>ative anatomi/ ( geometric !ories4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #ca4 #iic4 ,x has be5 ! f9al aim ( ,lie f ! beg9n+ to make progress 9 ! !ory ( diff]5tial equations2 z subsidi>y to ?is may be reg>d$ bo? 8 geometrical developm5ts & ! !ory ( cont9u\s gr\ps4--,,kle91 ,f4 ,lectures on ,ma!matics 7,new ,york1 #aiaa71 p4 #bd4 #iid4 ,to fully und]/& ! ma!matical g5ius ( ,sophus ,lie1 "o mu/ n turn to books rec5tly publi%$ by him 9 collaboration ) ,dr4 ,engel1 b to 8 e>li] memoirs1 writt5 dur+ ! fir/ ye>s ( 8 sci5tific c>e]4 ,!re ,lie %[s himself ! true geomet] t he is1 :ile 9 8 lat] publications1 f9d+ t he 0 b imp]fectly und]/ood by ! ma!maticians accu/om$ to ! analytic po9t ( view1 he adopt$ a v g5]al analytic =m ( treatm5t t is n always easy to foll[4--,,kle91 ,f4 ,lectures on ,ma!matics 7,new ,york1 #aiaa71 p4 #i4 #iie4 ,x is said t ! compos+ ( ! ,lilawati 0 occasion$ by ! foll[+ circum/ance4 ,lilawati 0 ! "n ( ! au?or's ,bhasc>a daught]1 conc]n+ :om x appe>$1 f ! qualities ( ! asc5dant at h] bir?1 t %e 0 de/9$ to pass h] life unm>ri$1 & to rema9 )\t *ildr54 ,! fa!r asc]ta9$ a lucky h\r = contract+ h] 9 m>riage1 t %e might be firmly connect$ & h *ildr54 ,x is said t :5 t h\r approa*$1 he br\ght 8 daught] & 8 int5d$ son ne> him4 ,he left ! h\r cup on ! vessel ( wat] & kept 9 att5dance a "t-kn[+ a/rolog]1 9 ord] t :5 ! cup %\ld subside 9 ! wat]1 ^? two preci\s jewels %\ld be unit$4 ,b1 z ! int5d$ arrangem5t 0 n accord+ to de/9y1 x happ5$ t ! girl1 f a curiosity natural to *ildr51 look$ into ! cup1 to obs]ve ! wat] com+ 9 at ! hole1 :5 by *ance a pe>l sep>at$ f h] bridal dress1 fell into ! cup1 &1 roll+ d[n to ! hole1 /opp$ ! influx ( wat]4 ,s ! a/rolog] wait$ 9 expectation ( ! promis$ h\r4 ,:5 ! op]ation ( ! cup _h ?us be5 delay$ beyond all mod]ate "t1 ! fa!r 0 9 con/]nation1 & exam9+1 he f\nd t a small pe>l _h /opp$ ! c\rse ( ! wat]1 & t ! long-expect$ h\r 0 pass$4 ,9 %ort1 ! fa!r1 ?us disappo9t$1 said to 8 un=tunate daught]1 ,i w write a book ( y\r "n1 :i* %all rema9 to ! late/ times--= a good "n is a second life1 & ! gr\nd-"w ( et]nal exi/5ce4--,,fizi4 ,preface to ! ,lilawati4 ,quot$ by ,a4 ,hutton3 ,a ,philosophical & ,ma!matical ,diction>y1 ,article 8,algebra0 7,london1 #ahae74 #iif4 ,is !re anyone ^: "n _c be twi/$ into ei!r praise or satire8 ,i h _h giv5 to me1 ,?omas ,bab+ton ,macaulay ,m\?s big3 a ,cantab anomaly4 --,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #hc4 ,,*apt] ,x ,,p]sons ,,& ,,anecdotes 7,n-,z7 #ajja4 ,:5 he _h a few mom5ts = div]sion1 he ,napoleon n unfrequ5tly employ$ !m ov] a book ( log>i?ms1 9 :i* he always f\nd recreation4--,,abbott1 ,j4 ,s4 ,c4 ,napoleon ,bonap>te 7,new ,york1 #aijd71 ,vol4 #a1 *ap4 #aj4 #ajjb4 ,! "n ( ,sir ,isaac ,newton has by g5]al cons5t be5 plac$ at ! head ( ^? great m5 :o h be5 ! ornam5ts ( _! species4444 ,! philosoph] ,laplace1 inde$1 to :om po/]ity w probably assign a place next to ,newton1 has *>act]iz$ ! ,pr9cipia z pre-em95t above all ! productions ( human intellect4--,,brew/]1 ,d4 ,life ( ,sir ,isaac ,newton 7,london1 #ahca71 pp4 #a1 #b4 #ajjc4 ,newton & ,laplace ne$ myriads ( ages & ?ick-/rewn cele/ial areas4 ,"o may say a gravitat+ sol> sy/em is already prophesi$ 9 ! nature ( ,newton's m9d4--,,em]son4 ,essay on ,hi/ory4 #ajjd4 ,! law ( gravitation is indisputably & incomp>ably ! greate/ sci5tific discov]y ev] made1 :e!r we look at ! advance :i* x involv$1 ! ext5t ( tru? disclos$1 or ! fundam5tal & satisfactory nature ( ?is tru?4 --,,:ewell1 ,w4 ,hi/ory ( ! ,inductive ,sci5ces1 ,bk4 #g1 *ap4 #b1 sect4 #e4 #ajje4 ,newton's !ory is ! circle ( g5]alization :i* includes all ! o!rs z ,kepl]'s laws1 ,ptolemy's !ory1 etc42--! highe/ po9t ( ! inductive asc5t2--! cata/rophe ( ! philosophic drama to :i* ,plato _h prologiz$2--! po9t to :i* m5's m9ds _h be5 j\rney+ = two ?\s& ye>s4--,,:ewell1 ,w4 ,hi/ory ( ! ,inductive ,sci5ces1 ,bk4 #g1 *ap4 #b1 sect4 #e4 #ajjf4 ,! ef=ts ( ! great philosoph] ,newton 7 always sup]human2 ! que/ions :i* he did n solve 7 incapable ( solution 9 8 "t4--,,arago4 ,eulogy on ,laplace1 ,bad5 ,p[ell ,smi?sonian ,report1 #ahgd1 p4 #acc4 #ajjg4 ,nature & ,nature's laws lay hid 9 night3 ,god said1 8,let ,newton be60 & all 0 light4 --,,pope1 ,a4 ,epitaph int5d$ = ,sir ,isaac ,newton4 #ajjh4 ,!re ,prie/ ( ,nature6 do/ ?\ %9e1 ,newton6 a ,k+ among ! ,k+s div9e4 --,,s\!y4 ,translation ( a ,greek ,ode on ,a/ronomy4 #ajji4 ,o'er ,nature's laws ,god ca/ ! veil ( night1 ,out-blaz'd a ,newton's s\l--& all 0 light4 --,,hill1 ,,a>on4 ,on ,sir ,isaac ,newton4 #ajaj4 ,tak+ ma!matics f ! beg9n+ ( ! _w to ! "t :5 ,newton liv$1 :at he _h done 0 mu* ! bett] half4--,,leibnitz4 ,quot$ by ,f4 ,r4 ,m\lton3 ,introduction to ,a/ronomy 7,new ,york1 #aijf71 p4 #aii4 #ajaa4 ,newton 0 ! greate/ g5ius t ev] exi/$1 & ! mo/ =tunate1 = we _c f9d m ?an once a sy/em ( ! _w to e/abli%4--,,lagrange4 ,quot$ by ,f4 ,r4 ,m\lton3 ,introduction to ,a/ronomy 7,new ,york1 #aijf71 p4 #aii4 #ajab4 ,a monum5t to ,newton6 a monum5t to ,%akespe>e6 ,look up to ,heav5--look into ! ,human ,he>t4 ,till ! planets & ! passions--! affections & ! fix$ />s are ext+ui%$--_! names _c die4--,,wilson1 ,,john4 ,noctes ,ambrosianae4 #ajac4 ,su* m5 z ,newton & ,l9naeus are incid5tal1 b augu/1 tea*]s ( religion4--,,wilson1 ,,john4 ,essays3 ,education ( ! ,p4 #ajad4 ,sir ,isaac ,newton1 ! supreme repres5tative ( ,anglo-,saxon g5ius4--,,ellis1 ,,havelock4 ,/udy ( ,briti% ,g5ius 7,london1 #aijd71 p4 #di4 #ajae4 ,?r\gh\t 8 life ,newton mu/ h devot$ at lea/ z mu* att5tion to *emi/ry & !ology z to ma!matics4444 --,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #cce4 #ajaf4 ,!re 0 a "t :5 he ,newton 0 possess$ ) ! old fool]ies ( a/rology2 & ano!r :5 he 0 s f> gone 9 ^? ( *emi/ry1 z to be ^u ! hunt aft] ! philosoph]'s /one4--,,rev4 ,j4 ,,sp5ce4 ,anecdotes1 ,obs]vations1 & ,*>act]s ( ,books & ,m5 7,london1 #ahfh71 p4 #ed4 #ajag4 ,= sev]al ye>s ?is great man ,newton 0 int5sely occupi$ 9 endeavor+ to discov] a way ( *ang+ ! base metals into gold4444 ,!re 7 p]iods :5 8 furnace fires 7 n all[$ to g out = six weeks2 he & 8 secret>y sitt+ up alt]nate nights to repl5i% !m4--,,p>ton1 ,,james4 ,sir ,isaac ,newton4 #ajah4 ,on ! day ( ,cromwell's dea?1 :5 ,newton 0 sixte51 a great /orm rag$ all ov] ,engl&4 ,he us$ to say1 9 8 old age1 t on t day he made 8 fir/ purely sci5tific exp]im5t4 ,to asc]ta9 ! =ce ( ! w9d1 he fir/ jump$ ) ! w9d & !n aga9/ x2 &1 by comp>+ ^! di/ances ) ! ext5t ( 8 own jump on a calm day1 he 0 enabl$ to compute ! =ce ( ! /orm4 ,:5 ! w9d blew !reaft]1 he us$ to say x 0 s _m feet /rong4 --,,p>ton1 ,,james4 ,sir ,isaac ,newton4 #ajai4 ,newton lectur$ n[ & !n to ! few /ud5ts :o *ose to he> him2 & x is record$ t v frequ5tly he came to ! lecture-room & f\nd x empty4 ,on su* occasions he w\ld rema9 fifte5 m9utes1 & !n1 if no "o came1 return to 8 ap>tm5ts4--,,p>ton1 ,,james4 ,sir ,isaac ,newton4 #ajbj4 ,sir ,isaac ,newton1 ?\gh s deep 9 algebra & fluxions1 c\ld n readily make up a common acc\nt3 &1 :5 he 0 ,ma/] ( ! ,m9t1 us$ to get somebody else to make up 8 acc\nts = him4--,,rev4 ,j4 ,,sp5ce4 ,anecdotes1 ,obs]vations1 & ,*>act]s ( ,books & ,m5 7,london1 #aheh71 p4 #acb4 #ajba4 ,we h "o ( 8 ,newton's college memor&um-books1 :i* is highly int]e/+4 ,! foll[+ are "s ( ! entries3 8,drills1 grav]s1 a hone1 a hamm]1 & a m&ril1 #es420 8a magnet1 #afs420 8compasses1 #bs420 8glass bubbles1 #ds420 8at ! tav]n sev]al o!r times1 #a20 8sp5t on my c\s91 #abs420 8on o!r acqua9tances1 #ajs420 8,philosophical ,intellig5ces1 #is4 #f;d420 8lo/ at c>ds twice1 #aes420 8at ! tav]n twice1 #cs4 #f;d420 8to ?ree prisms1 #c20 8f\r ounces ( putty1 #as4 #d;d420 8,bacon's ,miscellanies1 #as4 #f;d420 8a bible b9d+1 #cs420 8= oranges to my si/]1 #ds4 #b;d420 8= aqua=tis1 sublimate1 oyle p9k1 f9e silv]1 antimony1 v9eg>1 _s ( w9e1 :ite lead1 salt ( t>t>1 #b20 8,!atrum *emicum1 #a #hs40--,,p>ton1 ,,james4 ,sir ,isaac ,newton4 #ajbb4 ,on "o occasion1 :5 he 0 giv+ a d9n] to "s fri5ds at ! univ]sity1 he left ! table to get !m a bottle ( w9e2 b1 on 8 way to ! cell>1 he fell into reflection1 =got 8 err& & 8 company1 w5t to 8 *amb]1 put on 8 surplice1 & proce$$ to ! *apel4 ,sometimes he w\ld g into ! /reet half dress$1 & on discov]+ 8 condition1 run back 9 great ha/e1 mu* aba%$4 ,(t51 :ile /roll+ 9 8 g>d51 he w\ld sudd5ly /op1 & !n run rapidly to 8 room1 & beg9 to write1 /&+1 on ! fir/ piece ( pap] t pres5t$ itself4 ,int5d+ to d9e 9 ! public hall1 he w\ld g out 9 a br[n /udy1 take ! wrong turn1 walk a :ile1 & !n return to 8 room1 hav+ totally =gott5 ! d9n]4 ,once hav+ dism\nt$ f 8 horse to lead him up a hill1 ! horse slipp$ 8 head out ( ! bridle2 b ,newton1 oblivi\s1 nev] discov]$ x till1 on rea*+ a tollgate at ! top ( ! hill1 he turn$ to rem\nt & p]ceiv$ t ! bridle :i* he held 9 8 h& _h no horse atta*$ to x4 ,8 secret>y records t 8 =getfulness ( 8 d9n] 0 an excell5t ?+ = 8 old h\sekeep]1 :o 8sometimes f\nd bo? d9n] & supp] sc>cely ta/$ (1 :i* ! old woman has v pleasantly & mump+ly gone away )40 ,on gett+ out ( b$ 9 ! morn+1 he has be5 discov]$ to sit on 8 b$side = h\rs )\t dress+ himself1 utt]ly absorb$ 9 ?\ght4--,,p>ton1 ,,james4 ,sir ,isaac ,newton4 #ajbc4 ,i don't "k :at ,i may seem to ! _w1 b1 z to myself1 ,i seem to h be5 only z a boy play+ on ! sea%ore1 & div]t+ myself 9 n[ & !n f9d+ a smoo!r pebble or a pretti] %ell ?an ord9>y1 :il/ ! great ocean ( tru? lay all undiscov]$ be=e me4--,,newton1 ,i4 ,quot$ by ,rev4 ,j4 ,sp5ce3 ,anecdotes1 ,obs]vations1 & ,*>act]s ( ,books & ,m5 7,london1 #aheh71 p4 #dj4 #ajbd4 ,if ,i h se5 f>!r ?an ,desc>tes1 x is by /&+ on ! %\ld]s ( giants4--,,newton1 ,i4 ,quot$ by ,james ,p>ton3 ,sir ,isaac ,newton4 #ajbe4 ,newton c\ld n admit t !re 0 any diff]5ce betwe5 him & o!r m51 except 9 ! possession ( su* habits z 444 p]sev]ance & vigilance4 ,:5 he 0 ask$ h[ he made 8 discov]ies1 he answ]$1 8by always ?9k+ ab\t !m20 & at ano!r "t he decl>$ t if he _h done any?+1 x 0 due to no?+ b indu/ry & pati5t ?\ght3 8,i keep ! subject ( my inquiry con/antly be=e me1 & wait till ! fir/ dawn+ op5s gradually1 by little & little1 into a full & cle> light40--,,:ewell1 ,w4 ,hi/ory ( ! ,inductive ,sci5ces1 ,bk4 #g1 *ap4 #b1 sect4 #e4 #ajbf4 ,newton took no ex]cise1 indulg$ 9 no amusem5ts1 & work$ incessantly1 (t5 sp5d+ eighte5 or n9ete5 h\rs out ( ! tw5ty-f\r 9 writ+4--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #ceh4 #ajbg4 ,=e%ad[+s ( ! pr9ciples & ev5 ( ! language ( ! inf9itesimal calculus c be f\nd 9 ! writ+s ( ,napi]1 ,kepl]1 ,cavali]i1 ,pascal1 ,f]mat1 ,wallis1 & ,b>r[4 ,x 0 ,newton's good luck to come at a "t :5 ev]y?+ 0 ripe = ! discov]y1 & 8 ability enabl$ him to con/ruct almo/ at once a complete calculus4--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #cef4 #ajbh4 ,kepl]'s sugge/ion ( gravitation ) ! inv]se di/ance1 & ,b\illaud's propos$ sub/itution ( ! inv]se squ>e ( ! di/ance1 are ?+s :i* ,newton knew bett] ?an 8 mod]n read]s4 ,i h discov]$ two anagrams on 8 "n1 :i* are q conclusive3 ! notion ( gravitation 0 n new2 b ,newton w5t on4--,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #hb4 #ajbi4 ,= o!r great ma!maticians or philosoph]s1 he ,gauss us$ ! epi!ts magnus1 or cl>us1 or cl>issimus2 = ,newton alone he kept ! prefix summus4--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #cfb4 #ajcj4 ,to "k him ,sylve/] 0 to "k "o ( ! hi/oric figures ( all "t1 "o ( ! immortals2 & :5 he 0 really mov$ to speak1 8 eloqu5ce equall$ 8 g5ius4--,,hal/$1 ,g4 ,b4 ,f4 ,cajori's ,tea*+ & ,hi/ory ( ,ma!matics 9 ! ,u4 ,s4 7,wa%+ton1 #ahij71 p4 #bfe4 #ajca4 ,pr(essor ,sylve/]'s fir/ high class at ! new univ]sity ,johns ,hopk9s consi/$ ( only "o /ud5t1 ,g4 ,b4 ,hal/$1 :o _h p]si/$ 9 urg+ ,sylve/] to lecture on ! mod]n algebra4 ,! attempt to lecture on ?is subject l$ him into new inve/igations 9 quantics4--,,cajori1 ,f4 ,tea*+ & ,hi/ory ( ,ma!matics 9 ! ,u4 ,s4 7,wa%+ton1 #ahij71 p4 #bfd4 #ajcb4 ,b = ! p]si/5ce ( a /ud5t ( ?is univ]sity 9 urg+ ^u me 8 desire to /udy ) me ! mod]n algebra ,i %\ld nev] h be5 l$ into ?is inve/igation2 & ! new facts & pr9ciples :i* ,i h discov]$ 9 reg>d to x 7important facts1 ,i believe71 w\ld1 s f> z ,i am conc]n$1 h rema9$ /ill hidd5 9 ! womb ( "t4 ,9 va9 ,i repres5t$ to ?is inquisitive /ud5t t he w\ld d bett] to take up "s o!r subject ly+ less (f ! beat5 track ( /udy1 su* z ! high] p>ts ( ! calculus or elliptic functions1 or ! !ory ( sub/itutions1 or ,i wot n :at besides4 ,he /uck ) p]fect respectfulness1 b ) inv9cible p]t9acity1 to 8 po9t4 ,he w\ld h ! new algebra 7,heav5 kn[s :]e he _h he>d ab\t x1 = x is almo/ unkn[n 9 ?is cont95t71 t or no?+4 ,i 0 oblig$ to yield1 & :at 0 ! consequ5ce8 ,9 try+ to ?r[ light ^u an obscure explanation 9 our text-book1 my bra9 took fire1 ,i plung$ ) re-quick5$ zeal into a subject :i* ,i _h = ye>s ab&on$1 & f\nd food = ?\ghts :i* h engag$ my att5tion = a consid]able "t pa/1 & w probably occupy all my p[]s ( contemplation advantage\sly = sev]al mon?s to come4--,,sylve/]1 ,j4 ,j4 ,johns ,hopk9s ,commemoration ,day ,address2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #c1 p4 #gf4 #ajcc4 ,sylve/] 0 incapable ( read+ ma!matics 9 a purely receptive way4 ,app>5tly a subject ei!r fir$ 9 8 bra9 a tra9 ( active & re/less ?\ght1 or x w\ld n reta9 8 att5tion at all4 ,to a man ( su* a temp]am5t1 x w\ld h be5 peculi>ly helpful to live 9 an atmosph]e 9 :i* 8 human associations w\ld h suppli$ ! /imulus :i* he c\ld n f9d 9 m]e read+4 ,! great mod]n "w 9 ! !ory ( functions & 9 alli$ discipl9es1 he nev] became acqua9t$ )4444 ,:at w\ld h be5 ! effect if1 9 ! prime ( 8 p[]s1 he _h be5 surr\nd$ by ! influ5ces :i* prevail 9 ,b]l9 or 9 ,gtt+58 ,x may be confid5tly tak5 = grant$ t he w\ld h done spl5did "w 9 ^? doma9s ( analysis1 :i* h furni%$ ! laurels ( ! great ma!maticians ( ,g]many & ,france 9 ! second half ( ! pres5t c5tury4--,,frankl91 ,f4 ,johns ,hopk9s ,univ]sity ,circul>s #af 7#ahig71 p4 #ed4 #ajcd4 ,if we survey ! ma!matical works ( ,sylve/]1 we recognize inde$ a consid]able abundance1 b 9 contradi/9ction to ,cayley--n a v]satility t[>d sep>ate fields1 b1 ) few exceptions--a conf9em5t to ari?metic-algebraic bran*es4444 ,! concept ( ,function ( a cont9u\s v>iable1 ! fundam5tal concept ( mod]n ma!matics1 plays no role1 is inde$ sc>cely m5tion$ 9 ! entire "w ( ,sylve/]--,sylve/] 0 comb9atori/ comb9atorik]4--,,noe!r1 ,m4 ,ma!matis*e ,annal51 ,bd4 #ej 7#ahih71 pp4 #acd-ace4 #ajce4 ,sylve/]'s me?ods6 ,he _h none4 8,?ree lectures w be deliv]$ on a ,new ,univ]sal ,algebra10 he w\ld say2 !n1 8,! c\rse mu/ be ext5d$ to twelve40 ,x did la/ all ! re/ ( t ye>4 ,! foll[+ ye> ! c\rse 0 to be ,sub/itutions-,!orie1 by ,netto4 ,we all got ! text4 ,he lectur$ ab\t ?ree times1 foll[+ ! text closely & /opp+ %>p at ! end ( ! h\r4 ,!n he began to ?9k ab\t matrices aga94 8,i mu/ give "o lecture a week on ^?10 he said4 ,he c\ld n conf9e himself to ! h\r1 nor to ! "o lecture a week4 ,two weeks 7 pass$1 & ,netto 0 =gott5 entirely & nev] m5tion$ aga94 ,/atem5ts l ! foll[+ 7 n unfrequ5t 9 8 lectures3 8,i hav5't prov$ ?is1 b ,i am z sure z ,i c be ( any?+ t x mu/ be s4 ,f ?is x w foll[1 etc40 ,at ! next lecture x turn$ out t :at he 0 s sure ( 0 false4 ,nev] m9d1 he kept on =ev] guess+ & try+1 & pres5tly a wond]ful discov]y foll[$1 !n ano!r & ano!r4 ,aft]w>d he w\ld g back & "w x all ov] aga91 & surprise u ) all sorts ( side lights4 ,he !n made ano!r leap 9 ! d>k1 m treasures 7 discov]$1 & s on =ev]4--,,davis1 ,e4 ,w4 ,cajori's ,tea*+ & ,hi/ory ( ,ma!matics 9 ! ,u4,s4 7,wa%+ton1 #ahij71 pp4 #bfe-bff4 #ajcf4 ,i c see him ,sylve/] n[1 ) 8 :ite be>d & few locks ( gray hair1 8 =ehead wr9kl$ o'er ) ?\ghts1 writ+ rapidly 8 figures & =mulae on ! bo>d1 sometimes expla9+ z he wrote1 :ile we1 8 li/5]s1 caught ! reflect$ s\nds f ! bo>d4 ,b /op1 some?+ is n "r1 he pauses1 8 h& goes to 8 =ehead to help 8 ?\ght1 he goes ov] ! "w aga91 emphasizes ! lead+ po9ts1 & f9ally discov]s 8 difficulty4 ,p]haps x is "s error 9 8 figures1 p]haps an ov]sight 9 ! reason+4 ,sometimes1 h[ev]1 ! difficulty is n elucidat$1 & !n !re is n mu* to ! re/ ( ! lecture4 ,b at ! next lecture we w\ld he> ( "s new discov]y t 0 ! outcome ( t difficulty1 & ( "s article = ! ,j\rnal1 :i* he _h begun4 ,if a text-book _h be5 tak5 up at ! beg9n+1 ) ! int5tion ( foll[+ x1 t text-book 0 mo/ likely doom$ to oblivion = ! re/ ( ! t]m1 or until ! class _h be5 made li/5]s to e new ?\ght & pr9ciple t _h sprung f ! laboratory ( 8 m9d1 9 consequ5ce ( t fir/ difficulty4 ,o!r difficulties w\ld soon appe>1 s t no text-book c\ld la/ m ?an half ( ! t]m4 ,9 ?is way 8 class li/5$ to almo/ all ( ! "w t subsequ5tly appe>$ 9 ! ,j\rnal4 ,x seem$ to be ! quality ( 8 m9d t he mu/ adh]e to "o subject4 ,he w\ld ?9k ab\t x1 talk ab\t x to 8 class1 & f9ally write ab\t x = ! ,j\rnal4 ,! m]e/ accid5t might />t him1 b once />t$1 e mom5t1 e ?\ght 0 giv5 to x1 &1 z mu* z possible1 he read :at o!rs _h done 9 ! same direction2 b ?is la/ seem$ to be 8 real po9t2 he c\ld n read )\t f9d+ difficulties 9 ! way ( und]/&+ ! au?or4 ,?us1 (t5 8 own "w reproduc$ :at _h be5 done by o!rs1 & he did n f9d x out until too late4 ,a notable example ( ?is is 9 8 !ory ( cyclotomic functions1 :i* he _h reproduc$ 9 sev]al =eign j\rnals1 only to f9d t he _h be5 greatly anticipat$ by =eign au?ors4 ,x 0 manife/1 "o ( ! critics said1 t ! le>n$ pr(essor _h n read ,kumm]'s elem5t>y results 9 ! !ory ( ideal primes4 ,yet ,pr(essor ,smi?'s report on ! !ory ( numb]s1 :i* conta9$ a full synopsis ( ,kumm]'s !ory1 0 ,pr(essor ,sylve/]'s con/ant companion4 ,?is weakness ( ,pr(essor ,sylve/]1 9 n be+ able to read :at o!rs _h done1 is p]haps a concomitant ( 8 peculi> g5ius4 ,o!r m9ds c\ld pass ov] little difficulties & n be tr\bl$ by !m1 & s g on to a f9al und]/&+ ( ! results ( ! au?or4 ,b n s ) him4 ,a difficulty1 h[ev] small1 worri$ him1 & he 0 sure to h difficulties until ! subject _h be5 work$ ov] 9 8 own way1 to correspond ) 8 own mode ( ?\ght4 ,to read ! "w ( o!rs1 meant !re=e to him an almo/ indep5d5t developm5t ( x4 ,l ! man ^: pleasure 9 life is to pione] ! way = society into ! =e/s1 8 rugg$ m9d c\ld d]ive satisfaction only 9 hew+ out its own pa?s2 & only :5 8 ef=ts br\ght him into ! uncle>$ fields ( ma!matics did he f9d 8 place 9 ! ,univ]se4--,,ha?away1 ,a4 ,s4 ,f4 ,cajori's ,tea*+ & ,hi/ory ( ,ma!matics 9 ! ,u4 ,s4 7,wa%+ton1 #ahij71 pp4 #bff-bfg4 #ajcg4 ,pr(essor ,cayley has s9ce in=m$ me t ! !orem ab\t ^: orig9 ,i 0 9 d\bt1 w be f\nd 9 ,s*lfli's 8,de ,elim9atione40 ,?is is n ! fir/ unconsci\s plagi>ism ,i h be5 guilty ( t[>ds ?is em95t man ^: fri5d%ip ,i am pr\d to claim4 ,a m gl>+ case occurs 9 a note by me 9 ! 8,comptes ,r5dus10 on ! tw5ty-sev5 /raight l9es ( cubic surfaces1 :]e ,i believe ,i h foll[$ 7l "o walk+ 9 8 sleep71 d[n to ! v nom5clature & notation1 ! sub/ance ( a portion ( a pap] ins]t$ by ,s*lfli 9 ! 8,ma!matical ,j\rnal10 :i* be>s my "n z "o ( ! editors ^u ! face4--,,sylve/]1 ,j4 ,j4 ,philosophical ,transactions ( ! ,royal ,society 7#ahfd71 p4 #fdb4 #ajch4 ,he ,sylve/] _h "o rem>kable peculi>ity4 ,he seldom rememb]$ !orems1 propositions1 etc41 b _h always to d$uce !m :5 he wi%$ to use !m4 ,9 ?is he 0 ! v anti!sis ( ,cayley1 :o 0 ?or\ghly conv]sant ) ev]y?+ t _h be5 done 9 e bran* ( ma!matics4 ,i rememb] once submitt+ to ,sylve/] "s inve/igations t ,i _h be5 engag$ on1 & he imm$iately d5i$ my fir/ /atem5t1 say+ t su* a proposition _h nev] be5 he>d (1 let alone prov$4 ,to 8 a/oni%m5t1 ,i %[$ him a pap] ( 8 own 9 :i* he _h prov$ ! proposition2 9 fact1 ,i believe ! object ( 8 pap] _h be5 ! v pro( :i* 0 s /range to him4--,,durfee1 ,w4 ,p4 ,f4 ,cajori's ,tea*+ & ,hi/ory ( ,ma!matics 9 ! ,u4 ,s4 7,wa%+ton1 #ahij71 p4 #bfh4 #ajci4 ,a %ort1 broad man ( trem5d\s vitality1 ! physical type ( ,h]ew>d1 ! la/ ( ! ,engli%1 & 8 bro!r-9-arms1 ,w9t]1 ,sylve/]'s capaci\s head 0 ev] lo/ 9 ! highe/ cl\d-l&s ( pure ma!matics4 ,(t5 9 ! dead ( night he w\ld get 8 favorite pupil1 t he might communicate ! v la/ product ( 8 creative ?\ght4 ,ev]y?+ he saw sugge/$ to him some?+ new 9 ! high] algebra4 ,?is transmutation ( ev]y?+ into new ma!matics 0 a revelation to ^? :o knew him intimately4 ,!y began to d x !mselves4 ,8 ease & f]tility ( inv5tion prov$ a con/ant enc\ragem5t1 :ile 8 contempt = prov9cial /upidities1 su* z ! ,am]ican hi]oglyphics = & e1 :i* h ev5 f\nd _! way into ,web/]'s ,diction>y1 made ea* "y work] apply to himself ! /ricte/ te/s4--,,hal/$1 ,g4 ,b4 ,f4 ,cajori's ,tea*+ & ,hi/ory ( ,ma!matics 9 ! ,u4 ,s4 7,wa%+ton1 #ahij71 p4 #bfe4 #ajdj4 ,sylve/]'s writ+s are fl[]y & eloqu5t4 ,he 0 able to make ! dulle/ subject bright1 fre% & int]e/+4 ,8 en?usiasm is evid5t 9 e l9e4 ,he w\ld get q close up to 8 subject1 s t ev]y?+ else look$ small 9 comp>ison1 & = ! "t w\ld ?9k & make o!rs ?9k t ! _w conta9$ no f9] matt] = contemplation4 ,8 h&writ+ 0 bad1 & a tr\ble to 8 pr9t]s4 ,8 pap]s 7 f9i%$ ) difficulty4 ,no soon] 0 ! manuscript 9 ! editor's h&s ?an alt]ations1 corrections1 ameliorations & g5]alizations w\ld sugge/ !mselves to 8 m9d1 & e po/ w\ld c>ry fur!r directions to ! editors & pr9t]s4--,,macmahon4 ,p4 ,a4 ,nature1 ,vol4 #ee 7#ahig71 p4 #did4 #ajda4 ,! en?usiasm ( ,sylve/] = 8 own "w1 :i* manife/s itself "h z always1 indicates "o ( 8 *>act]i/ic qualities3 a high degree ( subjectivity 9 8 productions & publications4 ,sylve/] 0 s fully possess$ by ! matt] :i* = ! "t be+ engag$ 8 att5tion1 t x appe>$ to him & 0 designat$ by him z ! summit ( all t is important1 rem>kable & full ( future promise4 ,x w\ld excite 8 phantasy & p[] ( imag9ation 9 ev5 a great] measure ?an 8 p[] ( reflection1 s mu* s t he c\ld nev] m>%al ! ability to ma/] 8 subject-matt]1 mu* less to pres5t x 9 an ord]ly mann]4 ,consid]+ t he 0 also some:at ( a poet1 x w be easi] to ov]look ! poetic flights :i* p]vade 8 writ+1 (t5 bomba/ic1 sometimes furni%+ apt illu/rations2 m damag+ is ! complete lack ( =m & ord]l9ess ( 8 publications & _! sket*like "*1 444 :i* mu/ be accr$it$ at lea/ z mu* to lack ( objectivity z to a sup]fluity ( ideas4 ,aga91 ! text is p]meat$ ) associat$ emotional expressions1 biz>re utt]ances & p>adoxes & is ev]y:]e accompani$ by notes1 :i* con/itute an ess5tial "p ( ,sylve/]'s me?od ( pres5tation1 embody+ relations1 :e!r proximate or remote1 :i* mom5t>ily sugge/$ !mselves4 ,^! notes1 full ( inspiration & occasional fla%es ( g5ius1 are ! m /imulat+ ow+ to _! incomplet5ess4 ,b none ( 8 works manife/ a desire to p5etrate ! subject f all sides & to all[ x to mature2 ea* m]e surmise1 conceptions :i* arose dur+ publication1 immature ?\ghts & ev5 errors 7 u%]$ into publicity at ! mom5t ( _! inception1 ) utmo/ c>elessness1 & always ) complete unfamili>ity ( ! lit]ature ( ! subject4 ,n[h]e is !re ! lea/ trace ( self-criticism4 ,no "o c be expect$ to read ! treatises entire1 = 9 ! =m 9 :i* !y are available !y fail to give a cle> view ( ! matt] "u contemplation4 ,sylve/]'s 0 n a h>moni\sly gift$ or well-balanc$ m9d1 b r an in/9ctively active & creative m9d1 free f egotism4 ,8 reason+ mov$ 9 g5]alizations1 0 frequ5tly influ5c$ by analysis & at times 0 guid$ ev5 by my/ical num]ical relations4 ,8 reason+ consi/s less frequ5tly ( pure intelligible conclusions ?an ( inductions1 or r conjectures incit$ by individual obs]vations & v]ifications4 ,9 ?is he 0 guid$ by an algebraic s5se1 develop$ ?r\gh long occupation ) processes ( =ms1 & ?is l$ him luckily to g5]al fundam5tal tru?s :i* 9 "s in/ances rema9 veil$4 ,8 lack ( sy/em is "h (fset by ! advantage ( fre$om f purely me*anical logical activity4 ,! expon5ts ( 8 ess5tial *>act]i/ics are an intuitive tal5t & a faculty ( inv5tion to :i* we owe a s]ies ( ideas ( la/+ value & be>+ ! g]ms ( fruitful me?ods4 ,to no "o m fitt+ly ?an to ,sylve/] c be appli$ "o ( ! mottos ( ! ,philosophic ,magaz9e3 8,admiratio g5]at quae/ionem1 quae/io inve/igationem inve/igatio inv5tionem40--,,noe!r1 ,m4 ,ma!matis*e ,annal51 ,bd4 #ej 7#ahih71 pp4 #aee-afj4 #ajdb4 ,p]haps ,i may )\t immode/y lay claim to ! appellation ( ,ma!matical ,adam1 z ,i believe t ,i h giv5 m names 7pass$ into g5]al circulation7 ( ! creatures ( ! ma!matical reason ?an all ! o!r ma!maticians ( ! age comb9$4--,,sylve/]1 ,j4 ,j4 ,nature1 ,vol4 #cg 7#ahhg-ahhh71 p4 #afb4 #ajdc4 ,tait dubb$ ,maxwell dp_/dt1 = accord+ to !rmodynamics dp_/dt ,,jcm 7:]e ,c d5otes ,c>not's function7 ! initials ( 7,j4 ,c47 ,maxwell's "n4 ,on ! o!r h& ,maxwell d5ot$ ,?omson by ,t & ,tait by ,t2 s t x became cu/om>y to quote ,?omson & ,tait's ,treatise on ,natural ,philosophy z ,t & ,t4--,,macf>lane1 ,a4 ,biblio!ca ,ma!matica1 ,bd4 #c 7#aijc71 p4 #ahi4 #ajdd4 ,9 future times ,tait w be be/ kn[n = 8 "w 9 ! quat]nion analysis4 ,_h x n be5 = 8 expositions1 developm5ts & applications1 ,hamilton's inv5tion w\ld be today1 9 all probability1 a ma!matical curiosity4 --,,macf>lane1 ,a4 ,biblio!ca ,ma!matica1 ,bd4 #c 7#aijc71 p4 #ahi4 #ajde4 ,n seldom did he ,sir ,william ,?omson1 9 8 writ+s1 set d[n "s ma!matical /atem5t ) ! prefac+ rem>k 8x is obvi\s t0 to ! p]plexity ( ma!matical read]s1 to :om ! /atem5t 0 any?+ b obvi\s f su* ma!matics z prec$$ x on ! page4 ,to him x 0 obvi\s = physical reasons t might n sugge/ !mselves at all to ! ma!matician1 h[ev] compet5t4--,,?ompson1 ,s4 ,p4 ,life ( ,"l ,kelv9 7,london1 #aiaj71 p4 #aacf4 #ajdf4 ,! foll[+ is "o ( ! _m /ories told ( 8old ,donald ,mc,farlane0 ! fai?ful assi/ant ( ,sir ,william ,?omson4 ,! fa!r ( a new /ud5t :5 br++ him to ! ,univ]sity1 aft] call+ to see ! ,pr(essor ,?omson drew 8 assi/ant to "o side & bes\ght him to tell him :at 8 son mu/ d t he might /& well ) ! ,pr(essor4 8,y want y\r son to /& weel ) ! ,pr(eessorr80 ask$ ,mc,farlane4 8,yes40 8,weel1 !n1 he mu/ j h a guid bellyful o' ma!matics60 --,,?ompson1 ,s4 ,p4 ,life ( ,"l ,kelv9 7,london1 #aiaj71 p4 #dbj4 #ajdg4 ,! foll[+ /ory 7"h a little s(t5$ f ! v]nacul>7 0 n>rat$ by ,"l ,kelv9 himself :5 d9+ at ,tr9ity ,hall3-- ,a c]ta9 r\gh ,highl& lad at ! univ]sity _h done exce$+ly well1 & at ! close ( ! session ga9$ prizes bo? 9 ma!matics & 9 metaphysics4 ,8 old fa!r came up f ! f>m to see 8 son receive ! prizes1 & visit$ ! ,college4 ,?omson 0 deput$ to %[ him r\nd ! place4 8,weel1 ,mr4 ,?omson10 ask$ ! old man1 8& :at may ^! ma!matics be1 = :i* my son has gett5 a prize80 8,i told him10 repli$ ,?omson1 8t ma!matics meant reckon+ ) figures1 & calculat+40 8,oo ay10 said ! old man1 8he'll ha' gett5 t fra' me3 ,i 7 ev] a braw h& at ! c\nt9'40 ,aft] a pause he resum$3 8,& :at1 ,mr4 ,?omson1 might ^! metapheesics be80 8,i endeav\r$10 repli$ ,?omson1 8to expla9 h[ metaphysics 0 ! attempt to express 9 language ! indef9ite40 ,! old ,highl&] /ood /ill & scrat*$ 8 head4 8,oo ay3 may be he'll ha' gett5 t fra' 8 mi!r4 ,%e 7 aye a ble!r9' body40--,,?ompson1 ,s4 ,p4 ,life ( ,"l ,kelv9 7,london1 #aiaj71 p4 #aabd4 #ajdh4 ,"l ,kelv91 unable to meet 8 classes "o day1 po/$ ! foll[+ notice on ! door ( 8 lecture room1-- 8,pr(essor ,?omson w n meet 8 classes today40 ,! disappo9t$ class decid$ to play a joke on ! pr(essor4 ,eras+ ! 8c0 !y left ! leg5d to read1-- 8,pr(essor ,?omson w n meet 8 lasses today40 ,:5 ! class assembl$ ! next day 9 anticipation ( ! effect ( _! joke1 !y 7 a/oni%$ & *agr9$ to f9d t ! pr(essor _h outwitt$ !m4 ,! leg5d ( ye/]day 0 n[ f\nd to read1-- 8,pr(essor ,?omson w n meet 8 asses today40#i --,,nor?rup1 ,,cyrus4 ,univ]sity ( ,wa%+ton ,address1 ,novemb] #b1 #aijh4 #i ,au?or's note4 ,my colleague1 ,dr4 ,e4 ,t4 ,bell1 in=ms me t ?is same anecdote is associat$ ) ! "n ( ,j4 ,s4 ,blackie1 ,pr(essor ( ,greek at ,ab]de5 & ,ed9burgh4 #ajdi4 ,"o morn+ a great noise proce$$ f "o ( ! classrooms ( ! ,braunsb]g] gymnasium & on inve/igation x 0 f\nd t ,wei]/rass1 :o 0 to give ! recitation1 _h n appe>$4 ,! director w5t 9 p]son to ,wei]/rass' dwell+ & on knock+ 0 told to come 94 ,!re sat ,wei]/rass by a glimm]+ lamp 9 a d>k5$ room ?\gh x 0 daylight outside4 ,he _h work$ ! night ?r\gh & _h n notic$ ! approa* ( daylight4 ,:5 ! director rem9d$ him ( ! noisy ?rong ( /ud5ts :o 7 wait+ = him1 8 only reply 0 t he c\ld impossibly int]rupt 8 "w2 t he 0 ab\t to make an important discov]y :i* w\ld attract att5tion 9 sci5tific circles4--,,lampe1 ,e4 ,k>l ,wei]/rass3 ,jahrbu* d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #f 7#ahig71 pp4 #ch-ci4 #ajej4 ,wei]/rass relat$ 444 t he foll[$ ,sylve/]'s pap]s on ! !ory ( algebraic =ms v att5tively until ,sylve/] began to employ ,hebrew *>act]s4 ,t 0 m ?an he c\ld /& & aft] t he quit him4--,,lampe1 ,e4 ,naturwiss5s*aftli*e ,runds*au1 ,bd4 #ab 7#ahig71 p4 #cfa4 ,,*apt] ,,xi ,,ma!matics ,,z ,a ,,f9e ,,art #aaja4 ,! _w ( idea :i* x discloses or illum9ates1 ! contemplation ( div9e beauty & ord] :i* x induces1 ! h>moni\s connexion ( its p>ts1 ! inf9ite hi]>*y & absolute evid5ce ( ! tru?s ) :i* x is conc]n$1 ^!1 & su* l1 are ! sure/ gr\nds ( ! title ( ma!matics to human reg>d1 & w\ld rema9 unimpea*$ & unimpair$ 7 ! plan ( ! univ]se unroll$ l a map at our feet1 & ! m9d ( man qualifi$ to take 9 ! :ole s*eme ( creation at a glance4--,,sylve/]1 ,j4 ,j4 ,presid5tial ,address1 ,briti% ,association ,report 7#ahfi72 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b1 p4 #fei4 #aajb4 ,ma!matics has a triple end4 ,x %\ld furni% an in/rum5t = ! /udy ( nature4 ,fur!rmore x has a philosophic end1 &1 ,i v5ture to say1 an end e/hetic4 ,x ought to incite ! philosoph] to se>* into ! notions ( numb]1 space1 & "t2 &1 above all1 adepts f9d 9 ma!matics delights analog\s to ^? t pa9t+ & music give4 ,!y admire ! delicate h>mony ( numb] & ( =ms2 !y are amaz$ :5 a new discov]y discloses = !m an unlook$ = p]spective2 & ! joy !y ?us exp]i5ce1 has x n ! e/hetic "* al?\gh ! s5ses take no "p 9 x8 ,only ! privileg$ few are call$ to enjoy x fully1 x is true2 b is x n ! same ) all ! noble/ arts8 ,h5ce ,i d n hesitate to say t ma!matics des]ves to be cultivat$ = its own sake1 & t ! !ories n admitt+ ( application to physics des]ve to be /udi$ z well z o!rs4--,,po9c>1 ,,h5ri4 ,! ,relation ( ,analysis & ,ma!matical ,physics2 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #d 7#ahii71 p4 #bdh4 #aajc4 ,i l to look at ma!matics almo/ m z an art ?an z a sci5ce2 = ! activity ( ! ma!matician1 con/antly creat+ z he is1 guid$ ?\gh n controll$ by ! ext]nal _w ( ! s5ses1 be>s a resemblance1 n fanciful ,i believe b real1 to ! activity ( an arti/1 ( a pa9t] let u say4 ,rigor\s d$uctive reason+ on ! "p ( ! ma!matician may be lik5$ "h to te*nical skill 9 draw+ on ! "p ( ! pa9t]4 ,j z no "o c become a good pa9t] )\t a c]ta9 am\nt ( skill1 s no "o c become a ma!matician )\t ! p[] to reason accurately up to a c]ta9 po9t4 ,yet ^! qualities1 fundam5tal ?\gh !y are1 d n make a pa9t] or ma!matician wor?y ( ! "n1 nor inde$ are !y ! mo/ important factors 9 ! case4 ,o!r qualities ( a f> m subtle sort1 *ief among :i* 9 bo? cases is imag9ation1 g to ! mak+ ( a good arti/ or good ma!matician4 --,b,,*]1 ,,maxime4 ,fundam5tal ,conceptions & ,me?ods 9 ,ma!matics2 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #i 7#aijd71 p4 #acc4 #aajd4 ,ma!matics1 rightly view$1 possesses n only tru?1 b supreme beauty--a beauty cold & au/]e1 l t ( sculpture1 )\t appeal to any "p ( our weak] nature1 )\t ! gorge\s trapp+s ( pa9t+ or music1 yet sublimely pure1 & capable ( a /]n p]fection su* z only ! greate/ art c %[4 ,! true _s ( delight1 ! exaltation1 ! s5se ( be+ m ?an man1 :i* is ! t\*/one ( ! highe/ excell5ce1 is to be f\nd 9 ma!matics z surely z 9 poetry4 ,:at is be/ 9 ma!matics des]ves n m]ely to be le>n$ z a task1 b to be assimilat$ z a "p ( daily ?\ght1 & br\ght aga9 & aga9 be=e ! m9d ) ev]-r5ew$ enc\ragem5t4 ,real life is1 to mo/ m51 a long second-be/1 a p]petual compromise betwe5 ! real & ! possible2 b ! _w ( pure reason kn[s no compromise1 no practical limitations1 no b>ri] to ! creative activity embody+ 9 spl5did edifices ! passionate aspiration aft] ! p]fect f :i* all great "w spr+s4 ,remote f human passions1 remote ev5 f ! pitiful facts ( nature1 ! g5]ations h gradually creat$ an ord]$ cosmos1 :]e pure ?\ght c dwell z 9 its natural home1 & :]e "o1 at lea/1 ( our nobl] impulses c escape f ! dre>y exile ( ! natural _w4 --,,russell1 ,,b]tr&4 ,! ,/udy ( ,ma!matics3 ,philosophical ,essays 7,london1 #aiaj71 p4 #gc4 #aaje4 ,x 0 n alone ! /riv+ = univ]sal culture :i* attract$ ! great ma/]s ( ! ,r5aissance1 su* z ,brunellesco1 ,leon>do de ,v9ci1 ,raphael1 ,mi*ael ,angelo & especially ,albre*t ,dr]1 ) irresi/ible p[] to ! ma!matical sci5ces4 ,!y 7 consci\s t1 ) all ! fre$om ( ! individual phantasy1 art is subject to necess>y laws1 & conv]sely1 ) all its rigor ( logical /ructure1 ma!matics foll[s e/hetic laws4--,,rudio1 ,f4 ,vir*[-,holtz5dorf3 ,sammlung geme9v]/ndli*e wiss5s*aftli*e ,vortrge1 ,heft #adb1 p4 #ai4 #aajf4 ,surely ! claim ( ma!matics to take a place among ! lib]al arts mu/ n[ be admitt$ z fully made good4 ,:e!r we look at ! advances made 9 mod]n geometry1 9 mod]n integral calculus1 or 9 mod]n algebra1 9 ea* ( ^! ?ree a free h&l+ ( ! mat]ial employ$ is n[ possible1 & an almo/ unlimit$ scope is left to ! regulat$ play ( fancy4 ,x seems to me t ! :ole ( ae/hetic 7s f> z at pres5t reveal$7 may be reg>d$ z a s*eme hav+ f\r c5tres1 :i* may be treat$ z ! f\r apices ( a tetrah$ron1 namely ,epic1 ,music1 ,pla/ic1 & ,ma!matic4 ,!re w be f\nd a common plane to e ?ree ( ^!1 outside ( :i* lies ! f\r?2 & ?r\gh e two may be drawn a common axis opposite to ! axis pass+ ?r\gh ! o!r two4 ,s f> is c]ta9 & demon/rable4 ,i ?9k x also possible t !re is a c5tre ( gravity to ea* set ( ?ree1 & t ! l9e jo9+ ea* su* c5tre ) ! outside apex w int]sect 9 a common po9t--! c5tre ( gravity ( ! :ole body ( ae/hetic2 b :at t c5tre is or mu/ be ,i h n _h "t to ?9k out4 --,,sylve/]1 ,j4 ,j4 ,pro( ( ! hi!rto undemon/rat$ ,fundam5tal ,!orem ( ,inv>iants3 ,collect$ ,ma!matical ,pap]s1 ,vol4 #c1 p4 #abc4 #aajg4 ,x is ) ma!matics n o!rwise ?an x is ) music1 pa9t+ or poetry4 ,anyone c become a lawy]1 doctor or *emi/1 & z su* may succe$ well1 provid$ he is clev] & indu/ri\s1 b n e "o c become a pa9t]1 or a musician1 or a ma!matician3 g5]al clev]ness & indu/ry alone c\nt "h = no?+4--,,moebius1 ,p4 ,j4 ,ueb] die ,anlage zur ,ma!matik 7,leipzig1 #aijj71 p4 #e4 #aajh4 ,! true ma!matician is always a good deal ( an arti/1 an ar*itect1 yes1 ( a poet4 ,beyond ! real _w1 ?\gh p]ceptibly connect$ ) x1 ma!maticians h intellectually creat$ an ideal _w1 :i* !y attempt to develop into ! mo/ p]fect ( all worlds1 & :i* is be+ explor$ 9 e direction4 ,none has ! fa9te/ conception ( ?is _w1 except he :o kn[s x4--,,pr+%eim1 ,a4 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #cb1 p4 #cha4 #aaji4 ,:o has /udi$ ! works ( su* m5 z ,eul]1 ,lagrange1 ,cau*y1 ,riemann1 ,sophus ,lie1 & ,wei]/rass1 c d\bt t a great ma!matician is a great arti/8 ,! faculties possess$ by su* m51 v>y+ greatly 9 k9d & degree ) ! individual1 are analog\s ) ^? requisite = con/ructive art4 ,n e ma!matician possesses 9 a specially high degree t critical faculty :i* f9ds its employm5t 9 ! p]fection ( =m1 9 con=mity ) ! ideal ( logical complet5ess2 b e great ma!matician possesses ! r>] faculty ( con/ructive imag9ation4--,,hobson1 ,e4 ,w4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#aiaj7 ,nature1 ,vol4 #hd1 p4 #bij4 #aaaj4 ,ma!matics has beauties ( its own--a symmetry & proportion 9 its results1 a lack ( sup]fluity1 an exact adaptation ( means to ends1 :i* is exce$+ly rem>kable & to be f\nd else:]e only 9 ! works ( ! greate/ beauty4 ,x 0 a felicit\s expression ( ,goe!'s to call a noble ca!dral 8froz5 music10 b x might ev5 bett] be call$ 8petrifi$ ma!matics40 ,! beauties ( ma!matics--( simplicity1 ( symmetry1 ( complet5ess--c & %\ld be exemplifi$ ev5 to "y *ildr54 ,:5 ?is subject is prop]ly & concretely pres5t$1 ! m5tal emotion %\ld be t ( enjoym5t ( beauty1 n t ( repulsion f ! ugly & ! unpleasant4--,,"y1 ,j4 ,w4 ,a4 ,! ,tea*+ ( ,ma!matics 7,new ,york1 #aijg71 p4 #dd4 #aaaa4 ,a peculi> beauty reigns 9 ! realm ( ma!matics1 a beauty :i* resembles n s mu* ! beauty ( art z ! beauty ( nature & :i* affects ! reflective m9d1 :i* has acquir$ an appreciation ( x1 v mu* l ! latt]4 --,,kumm]1 ,e4 ,e4 ,b]l9] ,monatsb]i*te 7#ahfg71 p4 #cie4 #aaab4 ,ma!matics make ! m9d att5tive to ! objects :i* x consid]s4 ,?is !y d by ent]ta9+ x ) a great v>iety ( tru?s1 :i* are delightful & evid5t1 b n obvi\s4 ,tru? is ! same ?+ to ! und]/&+ z music to ! e> & beauty to ! eye4 ,! pursuit ( x does really z mu* gratify a natural faculty implant$ 9 u by our wise ,creator z ! pleas+ ( our s5ses3 only 9 ! =m] case1 z ! object & faculty are m spiritual1 ! delight is m pure1 free f regret1 turpitude1 lassitude1 & intemp]ance t commonly att5d s5sual pleasures4--,,arbu?not1 ,,john4 ,usefulness ( ,ma!matical ,le>n+4 #aaac4 ,h[ev] f> ! calculat+ reason ( ! ma!matician may seem sep>at$ f ! bold flight ( ! arti/'s phantasy1 x mu/ be rememb]$ t ^! expressions are b mom5t>y images snat*$ arbitr>ily f among ! activities ( bo?4 ,9 ! projection ( new !ories ! ma!matician ne$s z bold & creative a phantasy z ! productive arti/1 & 9 ! execution ( ! details ( a composition ! arti/ too mu/ calculate dispassionately ! means :i* are necess>y = ! successful consummation ( ! p>ts4 ,common to bo? is ! creation1 ! g5]ation1 ( =ms out ( m9d4--,,lampe1 ,e4 ,die ,entwickelung d] ,ma!matik1 etc4 7,b]l91 #ahic71 p4 #d4 #aaad4 ,z pure tru? is ! pol> /> ( our sci5ce ma!matics1 s x is ! great advantage ( our sci5ce ov] o!rs t x awak5s m easily ! love ( tru? 9 our pupils4444 ,if ,hegel ju/ly said1 8,:oev] does n "k ! works ( ! anci5ts1 has liv$ )\t kn[+ beauty10 ,s*ellba* responds ) equal "r1 8,:o does n "k ma!matics1 & ! results ( rec5t sci5tific inve/igation1 dies )\t kn[+ tru?40--,,simon1 ,,max4 ,quot$ 9 ,j4 ,w4 ,a4 ,"y3 ,tea*+ ( ,ma!matics 7,new ,york1 #aijg71 p4 #dd4 #aaae4 ,b*sel 9 8 rem9isc5ces f ! life ( a c\ntry p>son relates t he s\ght 8 recreation 9 ,lacroix's ,diff]5tial ,calculus & ?us f\nd intellectual refre%m5t = 8 call+4 ,in/ances l ?is make manife/ ! great advantage :i* occupation ) ma!matics af=ds to "o :o lives remote f ! city & is compell$ to =ego ! pleasures ( art4 ,! entranc+ *>m ( ma!matics1 :i* captivates e "o :o devotes himself to x1 & :i* is comp>able to ! f9e fr5zy "u ^: ban ! poet completes 8 "w1 has ev] be5 incompreh5sible to ! spectator & has (t5 caus$ ! en?usia/ic ma!matician to be held 9 d]ision4 ,a classic illu/ration is ! example ( ,ar*im$es1 4444--,,lampe1 ,e4 ,die ,entwickelung d] ,ma!matik1 etc4 7,b]l9 #ahic71 p4 #bb4 #aaaf4 ,among ! memoirs ( ,kir*h(f are "s ( uncommon beauty4 ,beauty1 ,i he> y ask1 d n ! ,graces flee :]e integrals /ret* =? _! necks8 ,c any?+ be beautiful1 :]e ! au?or has no "t = ! slighte/ ext]nal embelli%m5t8444 ,yet x is ?is v simplicity1 ! indisp5sabl5ess ( ea* ^w1 ea* lett]1 ea* little da%1 t among all arti/s raises ! ma!matician ne>e/ to ! ,_w-creator2 x e/abli%es a sublimity :i* is equall$ 9 no o!r art1--some?+ l x exi/s at mo/ 9 symphonic music4 ,! ,py?agoreans recogniz$ already ! simil>ity betwe5 ! mo/ subjective & ! mo/ objective ( ! arts4444 ,ultima se tangunt4 ,h[ expressive1 h[ nicely *>act]iz+ )al is ma!matics6 ,z ! musician recognizes ,moz>t1 ,bee?ov51 ,s*ub]t 9 ! fir/ *ords1 s ! ma!matician w\ld di/+ui% 8 ,cau*y1 ,gauss1 ,jacobi1 ,helmholtz 9 a few pages4 ,extreme ext]nal elegance1 sometimes a some:at weak skeleton ( conclusions *>act]izes ! ,fr5*2 ! ,engli%1 above all ,maxwell1 are di/+ui%$ by ! greate/ dramatic bulk4 ,:o does n "k ,maxwell's dynamic !ory ( gases8 ,at fir/ !re is ! maje/ic developm5t ( ! v>iations ( velocities1 !n ent] f "o side ! equations ( condition & f ! o!r ! equations ( c5tral motions1--high] & high] surges ! *aos ( =mulas1--sudd5ly f\r words bur/ =?3 8,put n #e40 ,! evil demon ,v disappe>s l ! sudd5 ceas+ ( ! basso p>ts 9 music1 :i* hi!rto wildly p]meat$ ! piece2 :at be=e seem$ beyond control is n[ ord]$ z by magic4 ,!re is no "t to /ate :y ?is or t sub/itution 0 made1 he :o _c feel ! reason may z well lay ! book aside2 ,maxwell is no program-musician :o expla9s ! notes ( 8 composition4 ,=?) ! =mulas yield ob$i5tly result aft] result1 until ! temp]ature-equilibrium ( a heavy gas is rea*$ z a surpris+ f9al climax & ! curta9 drops4444 ,kir*h(f's :ole t5d5cy1 & its true c\nt]p>t1 ! =m ( 8 pres5tation1 0 diff]5t4444 ,he is *>act]iz$ by ! extreme precision ( 8 hypo!ses1 m9ute execution1 a quiet r ?an epic developm5t ) utmo/ rigor1 nev] conceal+ a difficulty1 always dispell+ ! fa9te/ obscurity4 ,to return once m to my allegory1 he resembl$ ,bee?ov51 ! ?9k] 9 tones4--,he :o d\bts t ma!matical compositions c be beautiful1 let him read 8 memoir on ,absorption & ,emission 7,gesammelte ,abh&lung51 ,leipzig1 #ahhb1 p4 #ega-eih7 or ! *apt] ( 8 me*anics devot$ to ,hydrodynamics4--,,boltzmann1 ,l4 ,gu/av ,rob]t ,kir*h(f 7,leipzig #ahhh71 pp4 #bh-cj4 #aaag4 ,on poetry & geometric tru?1 ,& _! high privilege ( la/+ life1 ,f all int]nal injury exempt1 ,i mus$2 ^u ^! *iefly3 & at l5g?1 ,my s5ses yield+ to ! sultry air1 ,sleep seiz$ me1 & ,i pass$ into a dream4 --,,wordswor?4 ,! ,prelude1 ,bk4 #e4 #aaah4 ,geometry seems to /& = all t is practical1 poetry = all t is vision>y1 b 9 ! k+dom ( ! imag9ation y w f9d !m close ak91 & !y %\ld g toge!r z a preci\s h]itage to e y\?4--,,miln]1 ,,flor5ce4 ,s*ool ,review1 #ahih1 p4 #aad4 #aaai4 ,! beautiful has its place 9 ma!matics z else:]e4 ,! prose ( ord9>y int]c\rse & ( bus9ess correspond5ce might be held to be ! mo/ practical use to :i* language is put1 b we %\ld be poor inde$ )\t ! lit]ature ( imag9ation4 ,ma!matics too has its triumphs ( ! creative imag9ation1 its beautiful !orems1 its pro(s & processes ^: p]fection ( =m has made !m classic4 ,he mu/ be a 8practical0 man :o c see no poetry 9 ma!matics4 --,,:ite1 ,w4 ,f4 ,a ,scrap-book ( ,elem5t>y ,ma!matics 7,*icago1 #aijh71 p4 #bjh4 #aabj4 ,i v5ture to ass]t t ! feel+s "o has :5 ! beautiful symbolism ( ! inf9itesimal calculus fir/ gets a mean+1 or :5 ! delicate analysis ( ,f\ri] has be5 ma/]$1 or :ile "o foll[s ,cl]k ,maxwell or ,?omson into ! /range _w ( electricity1 n[ gr[+ s rapidly 9 =m & be+1 or c almo/ feel ) ,/okes ! pulsations ( light t gives nature to our eyes1 or track ) ,clausius ! c\rses ( molecules we c measure1 ev5 if we "k ) c]ta9ty t we c nev] see !m--,i v5ture to ass]t t ^! feel+s are altoge!r comp>able to ^? ar\s$ 9 u by an exquisite poem or a l(ty ?\ght4--,,workman1 ,w4 ,p4 ,f4 ,sp5c]3 ,aim & ,practice ( ,tea*+ 7,new ,york1 #ahig71 p4 #aid4 #aaba4 ,x is an op5 secret to ! few :o "k x1 b a my/]y & /umbl+ block to ! _m1 t ,sci5ce & ,poetry are own si/]s2 insomu* t 9 ^? bran*es ( sci5tific inquiry :i* are mo/ ab/ract1 mo/ =mal1 & mo/ remote f ! grasp ( ! ord9>y s5sible imag9ation1 a high] p[] ( imag9ation ak9 to ! creative insight ( ! poet is mo/ ne$$ & mo/ fruitful ( la/+ "w4--,,pollock1 ,f4 ,clif=d's ,lectures & ,essays 7,new ,york1 #aija71 ,vol4 #a1 ,introduction1 p4 #a4 #aabb4 ,x is z great a mi/ake to ma9ta9 t a high developm5t ( ! imag9ation is n ess5tial to progress 9 ma!matical /udies z to hold ) ,rusk9 & o!rs t sci5ce & poetry are antagoni/ic pursuits4--,,h(fman1 ,f4 ,s4 ,sph]e ( ,sci5ce 7,london1 #ahih71 p4 #ajg4 #aabc4 ,we h he>d mu* ab\t ! poetry ( ma!matics1 b v little ( x has z yet be5 sung4 ,! anci5ts _h a ju/] notion ( _! poetic value ?an we4 ,! mo/ di/9ct & beautiful /atem5ts ( any tru? mu/ take at la/ ! ma!matical =m4 ,we might s simplify ! rules ( moral philosophy1 z well z ( ari?metic1 t "o =mula w\ld express !m bo?4--,,?oreau1 ,h4 ,d4 ,a ,week on ! ,concord & ,m]rimac ,riv]s 7,bo/on1 #ahic71 p4 #dgg4 #aabd4 ,we d n li/5 ) ! be/ reg>d to ! v]ses ( a man :o is only a poet1 nor to 8 problems if he is only an algebrai/2 b if a man is at once acqua9t$ ) ! geometric f\ndation ( ?+s & ) _! fe/al spl5dor1 8 poetry is exact & 8 ari?metic musical4--,,em]son1 ,r4 ,w4 ,society & ,solitude1 ,*ap4 #g1 ,works & ,days4 #aabe4 ,ma!sis & ,poetry are 444 ! utt]ance ( ! same p[] ( imag9ation1 only t 9 ! "o case x is address$ to ! head1 & 9 ! o!r1 to ! he>t4--,,hill1 ,,?omas4 ,nor? ,am]ican ,review1 ,vol4 #he1 p4 #bcj4 #aabf4 ,! ,ma!matics are usually consid]$ z be+ ! v antipodes ( ,poesy4 ,yet ,ma!sis & ,poesy are ( ! close/ k9dr$1 = !y are bo? works ( ! imag9ation4 ,poesy is a creation1 a mak+1 a fiction2 & ! ,ma!matics h be5 call$1 by an admir] ( !m1 ! sublime/ & mo/ /up5d\s ( fictions4 ,x is true1 !y are n only 1 le>n+1 b 1 a creation4--,,hill1 ,,?omas4 ,nor? ,am]ican ,review1 ,vol4 #he1 p4 #bbi4 #aabg4 ,music & poesy us$ to quick5 y3 ,! ma!matics1 & ! metaphysics1 ,fall to !m z y f9d y\r /oma* s]ves y4 ,no pr(it gr[s1 :]e is no pleasure ta'en3-- ,9 brief1 sir1 /udy :at y mo/ affect4 --,,%akespe>e4 ,tam+ ( ! ,%rew1 ,act #a1 ,sc5e #a4 #aabh4 ,music has mu* resemblance to algebra4--,,novalis4 ,s*rift51 ,teil #b 7,b]l91 #aija71 p4 #edi4 #aabi4 ,i d pres5t y ) a man ( m9e1 ,cunn+ 9 music & 9 ma!matics1 ,to in/ruct h] fully 9 ^? sci5ces1 ,:]e(1 ,i "k1 %e is n ignorant4 --,,%akespe>e4 ,tam+ ( ! ,%rew1 ,act #b1 ,sc5e #a4 #aacj4 ,saturat$ ) t speculative _s !n p]vad+ ! ,greek m9d1 he ,py?agoras endeav\r$ to discov] "s pr9ciple ( homog5eity 9 ! univ]se4 ,be=e him1 ! philosoph]s ( ! ,ionic s*ool _h s\ght x 9 ! matt] ( ?+s2 ,py?agoras look$ = x 9 ! /ructure ( ?+s4 ,he obs]v$ ! v>i\s num]ical relations or analogies betwe5 numb]s & ! ph5om5a ( ! univ]se4 ,be+ conv9c$ t x 0 9 numb]s & _! relations t he 0 to f9d ! f\ndation to true philosophy1 he proce$$ to trace ! orig9 ( all ?+s to numb]s4 ,?us he obs]v$ t musical /r+s ( equal l5g?s /ret*$ by weights hav+ ! proportion ( #a_/b1 #b_/c1 #c_/d1 produc$ int]vals :i* 7 an octave1 a fif? & a f\r?4 ,h>mony1 !re=e1 dep5ds on musical proportion2 x is no?+ b a my/]i\s num]ical relation4 ,:]e h>mony is1 !re are numb]s4 ,h5ce ! ord] & beauty ( ! univ]se h _! orig9 9 numb]s4 ,!re are sev5 int]vals 9 ! musical scale1 & also sev5 planets cross+ ! heav5s4 ,! same num]ical relations :i* und]lie ! =m] mu/ und]lie ! latt]4 ,b :]e numb] is1 !re is h>mony4 ,h5ce 8 spiritual e> disc]n$ 9 ! planet>y motions a wond]ful 8,h>mony ( sph]es40--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #fg4 #aaca4 ,may n ,music be describ$ z ! ,ma!matic ( s5se1 ,ma!matic z ,music ( ! reason8 ! s\l ( ea* ! same6 ,?us ! musician feels ,ma!matic1 ! ma!matician ?9ks ,music1--,music ! dream1 ,ma!matic ! work+ life--ea* to receive its consummation f ! o!r :5 ! human intellig5ce1 elevat$ to its p]fect type1 %all %9e =? glorifi$ 9 "s future ,moz>t-,diri*let or ,bee?ov5-,gauss--a union already n indi/9ctly =e%ad[$ 9 ! g5ius & lab\rs ( a ,helmholtz6--,,sylve/]1 ,j4 ,j4 ,on ,newton's ,rule = ! ,discov]y ( ,imag9>y ,roots2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b1 p4 #dai4 #aacb4 ,j z ! musician is able to =m an ac\/ic image ( a composition :i* he has nev] he>d play$ by m]ely look+ at its score1 s ! equation ( a curve1 :i* he has nev] se51 furni%es ! ma!matician ) a complete picture ( its c\rse4 ,yea1 ev5 m3 z ! score frequ5tly reveals to ! musician niceties :i* w\ld escape 8 e> because ( ! complication & rapid *ange ( ! auditory impressions1 s ! insight :i* ! ma!matician ga9s f ! equation ( a curve is mu* deep] ?an t :i* is br\ght ab\t by a m]e inspection ( ! curve4--,,pr+%eim1 ,a4 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #ac1 p4 #cfd4 #aacc4 ,ma!matics & music1 ! mo/ %>ply contra/$ fields ( sci5tific activity :i* c be f\nd1 & yet relat$1 support+ ea* o!r1 z if to %[ =? ! secret connection :i* ties toge!r all ! activities ( our m9d1 & :i* leads u to surmise t ! manife/ations ( ! arti/'s g5ius are b ! unconsci\s expressions ( a my/]i\sly act+ rationality4--,,helmholtz1 ,h4 ,vortrge und ,r$51 ,bd4 #a 7,brauns*weig1 #ahhd71 p4 #hb4 #aacd4 ,among all highly civiliz$ peoples ! gold5 age ( art has always be5 closely co9cid5t ) ! gold5 age ( ! pure sci5ces1 p>ticul>ly ) ma!matics1 ! mo/ anci5t among !m4 ,?is co9cid5ce mu/ n be look$ ^u z accid5tal1 b z natural1 due to an inn] necessity4 ,j z art c ?rive only :5 ! arti/1 reliev$ ( ! anxieties ( exi/5ce1 c li/5 to ! inspirations ( 8 _s & foll[ 9 _! lead1 s ma!matics1 ! mo/ ideal ( ! sci5ces1 w yield its *oice/ blossoms only :5 life's dismal phantom dissolves & fades away1 :5 ! /riv+ aft] nak$ tru? alone pr$om9ates1 conditions :i* prevail only 9 nations :ile 9 ! prime ( _! developm5t4--,,lampe1 ,e4 ,die ,entwickelung d] ,ma!matik etc4 7,b]l91 #ahic71 p4 #d4 #aace4 ,till ! fifte5? c5tury little progress appe>s to h be5 made 9 ! sci5ce or practice ( music2 b s9ce t era x has advanc$ ) m>vel\s rapidity1 its progress be+ curi\sly p>allel ) t ( ma!matics1 inasmu* z great musical g5iuses appe>$ sudd5ly among diff]5t nations1 equal 9 _! possession ( ?is special faculty to any t h s9ce aris54 ,z ) ! ma!matical s ) ! musical faculty--x is impossible to trace any connection betwe5 its possession & survival 9 ! /ruggle = exi/5ce4 --,,wallace1 ,a4 ,r4 ,d>w9ism1 ,*ap4 #ae4 #aacf4 ,9 my op9ion1 !re is absolutely no tru/wor?y pro( t tal5ts h be5 improv$ by _! ex]cise ?r\gh ! c\rse ( a long s]ies ( g5]ations4 ,! ,ba* family %[s t musical tal5t1 & ! ,b]n\lli family t ma!matical p[]1 c be transmitt$ f g5]ation to g5]ation1 b ?is tea*es u no?+ z to ! orig9 ( su* tal5ts4 ,9 bo? families ! high-wat]m>k ( tal5t lies1 n at ! end ( ! s]ies ( g5]ations1 z x %\ld d if ! results ( practice are transmitt$1 b 9 ! middle4 ,aga91 tal5ts frequ5tly appe> 9 "s memb] ( a family :i* has n be5 previ\sly di/+ui%$4 ,gauss 0 n ! son ( a ma!matician2 ,h&el's fa!r 0 a surgeon1 ( ^: musical p[]s no?+ is kn[n2 ,titian 0 ! son & also ! nephew ( a lawy]1 :ile he & 8 bro!r1 ,francesco ,vecellio1 7 ! fir/ pa9t]s 9 a family :i* produc$ a succession ( sev5 o!r arti/s ) dim9i%+ tal5ts4 ,^! facts d n1 h[ev]1 prove t ! condition ( ! n]ve-tracts & c5tres ( ! bra91 :i* det]m9e ! specific tal5t1 appe>$ = ! fir/ "t 9 ^! m53 ! appropriate condition surely exi/$ previ\sly 9 _! p>5ts1 al?\gh x did n a*ieve expression4 ,!y prove1 z x seems to me1 t a high degree ( end[m5t 9 a special direction1 :i* we call tal5t1 _c h aris5 f ! exp]i5ce ( previ\s g5]ations1 t is1 by ! ex]cise ( ! bra9 9 ! same specific direction4--,,weismann1 ,,augu/4 ,essays ^u ,h]$ity ,a4 ,e4 ,%ipley1 7,ox=d1 #ahia71 ,vol4 #a1 p4 #ig4 ,,*apt] ,,xii ,,ma!matics ,,z ,a ,,language #abja4 ,! new ma!matics is a sort ( supplem5t to language1 af=d+ a means ( ?\ght ab\t =m & quantity & a means ( expression1 m exact1 compact1 & ready ?an ord9>y language4 ,! great body ( physical sci5ce1 a great deal ( ! ess5tial facts ( f9ancial sci5ce1 & endless social & political problems are only accessible & only ?9kable to ^? :o h _h a s\nd tra9+ 9 ma!matical analysis1 & ! "t may n be v remote :5 x w be und]/ood t = complete initiation z an effici5t citiz5 ( "o ( ! new great complex _w wide /ates t are n[ develop+1 x is z necess>y to be able to compute1 to ?9k 9 av]ages & maxima & m9ima1 z x is n[ to be able to read & to write4 --,,wells1 ,h4 ,g4 ,mank9d 9 ! ,mak+ 7,london1 #aijd71 pp4 #aia-aib4 #abjb4 ,ma!matical language is n only ! simple/ & mo/ easily und]/ood ( any1 b ! %orte/ also4--,,br\gham1 ,h4 ,l4 ,works 7,ed9burgh1 #ahgb71 ,vol4 #g1 p4 #cag4 #abjc4 ,ma!matics is ! sci5ce ( def9it5ess1 ! necess>y vocabul>y ( ^? :o "k4--,,:ite1 ,w4 ,f4 ,a ,scrap-book ( ,elem5t>y ,ma!matics 7,*icago1 #aijh71 p4 #g4 #abjd4 ,ma!matics1 too1 is a language1 & z conc]ns its /ructure & cont5t x is ! mo/ p]fect language :i* exi/s1 sup]ior to any v]nacul>2 inde$1 s9ce x is und]/ood by e p1 ma!matics may be call$ ! language ( languages4 ,?r\gh x1 z x 71 nature h]self speaks2 ?r\gh x ! ,creator ( ! _w has spok51 & ?r\gh x ! ,pres]v] ( ! _w cont9ues to speak4 --,,dillmann1 ,c4 ,die ,ma!matik die ,fackeltrg]9 e9] neu5 ,zeit 7,/uttg>t1 #ahhi71 p4 #e4 #abje4 ,w\ld x s\nd too presumptu\s to speak ( p]ception z a qu9tess5ce ( s5sation1 language 7t is1 communicable ?\ght7 ( p]ception1 ma!matics ( language8 ,we %\ld !n h f\r t]ms diff]5tiat+ f inorganic matt] & f ea* o!r ! ,vegetable1 ,animal1 ,rational1 & ,sup]-s5sual modes ( exi/5ce4--,,sylve/]1 ,j4 ,j4 ,presid5tial ,address1 ,briti% ,association2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b1 p4 #feb4 #abjf4 ,little c\ld ,plato h imag9$1 :51 indulg+ 8 in/9ctive love ( ! true & beautiful = _! own sakes1 he ent]$ ^u ^! ref9$ speculations & revell$ 9 a _w ( 8 own creation1 t he 0 writ+ ! gramm> ( ! language 9 :i* x w\ld be demon/rat$ 9 aft] ages t ! pages ( ! univ]se are writt54--,,sylve/]1 ,j4 ,j4 ,a ,probation>y ,lecture on ,geometry2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b1 p4 #g4 #abjg4 ,x is ! symbolic language ( ma!matics only :i* has yet prov$ suffici5tly accurate & compreh5sive to dem& famili>ity ) ?is conception ( an inv]se process4 --,,v5n1 ,,john4 ,symbolic ,logic 7,london & ,new ,york1 #ahid71 p4 #gd4 #abjh4 ,)\t ?is language ma!matics mo/ ( ! intimate analogies ( ?+s w\ld h rema9$ =ev] unkn[n to u2 & we %\ld =ev] h be5 ignorant ( ! int]nal h>mony ( ! _w1 :i* is ! only true objective reality4444 ,?is h>mony 444 is ! sole objective reality1 ! only tru? we c atta92 & :5 ,i add t ! univ]sal h>mony ( ! _w is ! s\rce ( all beauty1 x w be und]/ood :at price we %\ld atta* to ! sl[ & difficult progress :i* little by little enables u to "k x bett]4--,,po9c>1 ,h4 ,! ,value ( ,sci5ce ,hal/$ ,popul> ,sci5ce ,mon?ly1 #aijf1 pp4 #aie-aif4 #abji4 ,! mo/ /rik+ *>act]i/ic ( ! writt5 language ( algebra & ( ! high] =ms ( ! calculus is ! %>pness ( def9ition1 by :i* we are enabl$ to reason ^u ! symbols by ! m]e laws ( v]bal logic1 dis*>g+ our m9ds entirely ( ! mean+ ( ! symbols1 until we h rea*$ a /age ( ! process :]e we desire to int]pret our results4 ,! ability to att5d to ! symbols1 & to p]=m ! v]bal1 visible *anges 9 ! position ( !m p]mitt$ by ! logical rules ( ! sci5ce1 )\t all[+ ! m9d to be p]plex$ ) ! mean+ ( ! symbols until ! result is rea*$ :i* y wi% to int]pret1 is a fundam5tal "p ( :at is call$ analytical p[]4 ,_m /ud5ts f9d !mselves p]plex$ by a p]petual attempt to int]pret n only ! result1 b ea* /ep ( ! process4 ,!y ?us lose mu* ( ! b5efit ( ! labor-sav+ ma*9]y ( ! calculus & are1 inde$1 frequ5tly incapacitat$ = us+ x4--,,hill1 ,,?omas4 ,uses ( ,ma!sis2 ,biblio!ca ,sacra1 ,vol4 #cb1 p4 #eje4 #abaj4 ,! prom95t reason :y a ma!matician c be judg$ by none b ma!maticians1 is t he uses a peculi> language4 ,! language ( ma!sis is special & untranslatable4 ,9 its simple/ =ms x c be translat$1 z1 = in/ance1 we say a "r angle to mean a squ>e corn]4 ,b y g a little high] 9 ! sci5ce ( ma!matics1 & x is impossible to disp5se ) a peculi> language4 ,x w\ld defy all ! p[] ( ,m]cury himself to expla9 to a p]son ignorant ( ! sci5ce :at is meant by ! s+le phrase 8functional expon5t40 ,h[ mu* m impossible1 if we may say s1 w\ld x be to expla9 a :ole treatise l ,hamilton's ,quat]nions1 9 su* a wise z to make x possible to judge ( its value6 ,b to "o :o has le>n$ ?is language1 x is ! mo/ precise & cle> ( all modes ( expression4 ,x discloses ! ?\ght exactly z conceiv$ by ! writ]1 ) m or less beauty ( =m1 b nev] ) obscurity4 ,x may be prolix1 z x (t5 is among ,fr5* writ]s2 may delight 9 m]e v]bal metamorphoses1 z 9 ! ,cambridge ,univ]sity ( ,engl&2 or adopt ! briefe/ & cle>e/ =ms1 z "u ! p5s ( ! geomet]s ( our ,cambridge2 b x always reveals to u precisely ! writ]'s ?\ght4 --,,hill1 ,,?omas4 ,nor? ,am]ican ,review1 ,vol4 #he1 pp4 #bbd-bbe4 #abaa4 ,! doma91 ov] :i* ! language ( analysis ext5ds its sway1 is1 inde$1 relatively limit$1 b )9 ?is doma9 x s inf9itely excels ord9>y language t its attempt to foll[ ! =m] mu/ be giv5 up aft] a few /eps4 ,! ma!matician1 :o kn[s h[ to ?9k 9 ?is m>vel\sly cond5s$ language1 is z diff]5t f ! me*anical comput] z heav5 f e>?4--,,pr+%eim1 ,a4 ,jahresb]i*te d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #ac1 p4 #cfg4 #abab4 ,! results ( sy/ematic symbolical reason+ mu/ always express g5]al tru?s1 by _! nature2 & d n1 = _! ju/ification1 require ea* ( ! /eps ( ! process to repres5t "s def9ite op]ation ^u quantity4 ,! absolute univ]sality ( ! int]pretation ( symbols is ! fundam5tal pr9ciple ( _! use4--,,:ewell1 ,,william4 ,! ,philosophy ( ! ,inductive ,sci5ces1 ,"p ,i1 ,bk4 #b1 *ap4 #ab1 sect4 #b 7,london1 #aheh74 #abac4 ,anyone :o und]/&s algebraic notation1 reads at a glance 9 an equation results rea*$ ari?metically only ) great lab\r & pa9s4--,,c\rnot1 ,a4 ,!ory ( ,weal? ,n4 ,t4 ,bacon1 7,new ,york1 #ahig71 p4 #d4 #abad4 ,z ari?metic & algebra are sci5ces ( great cle>ness1 c]ta9ty1 & ext5t1 :i* are imm$iately conv]sant ab\t signs1 ^u ! skilful use :]e( !y entirely dep5d1 s a little att5tion to !m may possibly help u to judge ( ! progress ( ! m9d 9 o!r sci5ces1 :i*1 ?\gh diff]+ 9 nature1 design1 & object1 may yet agree 9 ! g5]al me?ods ( pro( & inquiry4--,,b]keley1 ,,george4 ,alciphron1 or ! ,m9ute ,philosoph]1 ,dialogue #g1 sect4 #ab4 #abae4 ,9 g5]al ! position z reg>ds all su* new calculi is ?is--,t "o _c accompli% by !m any?+ t c\ld n be accompli%$ )\t !m4 ,h[ev]1 ! advantage is1 t1 provid$ su* a calculus corresponds to ! inmo/ nature ( frequ5t ne$s1 anyone :o ma/]s x ?or\ghly is able--)\t ! unconsci\s inspiration ( g5ius :i* no "o c comm&--to solve ! respective problems1 yea1 to solve !m me*anically 9 complicat$ cases 9 :i*1 )\t su* aid1 ev5 g5ius becomes p[]less4 ,su* is ! case ) ! inv5tion ( g5]al algebra1 ) ! diff]5tial calculus1 & 9 a m limit$ region ) ,lagrange's calculus ( v>iations1 ) my calculus ( congru5ces1 & ) ,mbius's calculus4 ,su* conceptions unite1 z x 71 into an organic :ole c\ntless problems :i* o!rwise w\ld rema9 isolat$ & require = _! sep>ate solution m or less application ( inv5tive g5ius4--,,gauss1 ,c4 ,j4 ,w]ke1 ,bd4 #h1 p4 #bih4 #abaf4 ,! inv5tion ( :at we may call prim>y or fundam5tal notation has be5 b little indebt$ to analogy1 evid5tly ow+ to ! small ext5t ( ideas 9 :i* comp>ison c be made useful4 ,b at ! same "t analogy %\ld be att5d$ to1 ev5 if = no o!r reason ?an t1 by mak+ ! inv5tion ( notation an art1 ! ex]tion ( individual caprice ceases to be all[able4 ,no?+ is m easy ?an ! inv5tion ( notation1 & no?+ ( worse example & consequ5ce ?an ! confusion ( ma!matical expressions by unkn[n symbols4 ,if new notation be advisable1 p]man5tly or tempor>ily1 x %\ld c>ry ) x "s m>k ( di/9ction f t :i* is already 9 use1 unless x be a demon/rable ext5sion ( ! latt]4 --,,de ,,morgan1 ,a4 ,calculus ( ,functions2 ,encyclop$ia ,metropolitana1 ,addition to ,article #bf4 #abag4 ,be=e ! introduction ( ! ,arabic notation1 multiplication 0 difficult1 & ! division ev5 ( integ]s call$ into play ! highe/ ma!matical faculties4 ,probably no?+ 9 ! mod]n _w c\ld h m a/oni%$ a ,greek ma!matician ?an to le>n t1 "u ! influ5ce ( compulsory education1 ! :ole population ( ,we/]n ,europe1 f ! highe/ to ! l[e/1 c\ld p]=m ! op]ation ( division = ! l>ge/ numb]s4 ,?is fact w\ld h seem$ to him a %e] impossibility4444 ,our mod]n p[] ( easy reckon+ ) decimal fractions is ! mo/ miracul\s result ( a p]fect notation4--,,:itehead1 ,a4 ,n4 ,introduction to ,ma!matics 7,new ,york1 #aiaa71 p4 #ei4 #abah4 ,ma!matics is (t5 consid]$ a difficult & my/]i\s sci5ce1 because ( ! num]\s symbols :i* x employs4 ,( c\rse1 no?+ is m incompreh5sible ?an a symbolism :i* we d n und]/&4 ,also a symbolism1 :i* we only p>tially und]/& & are unaccu/om$ to use1 is difficult to foll[4 ,9 exactly ! same way ! te*nical t]ms ( any pr(ession or trade are incompreh5sible to ^? :o h nev] be5 tra9$ to use !m4 ,b ?is is n because !y are difficult 9 !mselves4 ,on ! contr>y !y h inv>iably be5 introduc$ to make ?+s easy4 ,s 9 ma!matics1 grant$ t we are giv+ any s]i\s att5tion to ma!matical ideas1 ! symbolism is inv>iably an imm5se simplification4 --,,:itehead1 ,a4 ,n4 ,introduction to ,ma!matics 7,new ,york1 #aiaa71 pp4 #ei-fj4 #abai4 ,symbolism is useful because x makes ?+s difficult4 ,n[ 9 ! beg9n+ ev]y?+ is self-evid5t1 & x is h>d to see :e!r "o self-evid5t proposition foll[s f ano!r or n4 ,obvi\sness is always ! enemy to correctness4 ,h5ce we mu/ inv5t a new & difficult symbolism 9 :i* no?+ is obvi\s4444 ,?us ! :ole ( ,ari?metic & ,algebra has be5 %[n to require ?ree indef9able notions & five indemon/rable propositions4--,,russell1 ,,b]tr&4 ,int]national ,mon?ly1 #aija1 p4 #he4 #abbj4 ,! employm5t ( ma!matical symbols is p]fectly natural :5 ! relations betwe5 magnitudes are "u discussion2 & ev5 if !y are n rigor\sly necess>y1 x w\ld h>dly be reasonable to reject !m1 because !y are n equally famili> to all read]s & because !y h sometimes be5 wrongly us$1 if !y are able to facilitate ! exposition ( problems1 to r5d] x m concise1 to op5 ! way to m ext5d$ developm5ts1 & to avoid ! digressions ( vague argum5tation4--,,c\rnot1 ,a4 ,!ory ( ,weal? ,n4 ,t4 ,bacon1 7,new ,york1 #ahig71 pp4 #c-d4 #abba4 ,an all-inclusive geometrical symbolism1 su* z ,hamilton & ,grassmann conceiv$ (1 is impossible4--,,burkh>dt1 ,h4 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #e1 p4 #eb4 #abbb4 ,! language ( analysis1 mo/ p]fect ( all1 be+ 9 itself a p[]ful in/rum5t ( discov]ies1 its notations1 especially :5 !y are necess>y & happily conceiv$1 are s _m g]ms ( new calculi4--,,laplace4 ,oeuvres1 t4 #g 7,p>is1 #ahif71 p4 xl4 ,,*apt] ,,xiii ,,ma!matics ,,& ,,logic #acja4 ,ma!matics belongs to e inquiry1 moral z well z physical4 ,ev5 ! rules ( logic1 by :i* x is rigidly b\nd1 c\ld n be d$uc$ )\t its aid4 ,! laws ( argum5t admit ( simple /atem5t1 b !y mu/ be curi\sly transpos$ be=e !y c be appli$ to ! liv+ spee* & v]ifi$ by obs]vation4 ,9 its pure & simple =m ! syllogism _c be directly comp>$ ) all exp]i5ce1 or x w\ld n h requir$ an ,ari/otle to discov] x4 ,x mu/ be transmut$ into all ! possible %apes 9 :i* reason+ loves to clo! itself4 ,! transmutation is ! ma!matical process 9 ! e/abli%m5t ( ! law4--,,peirce1 ,,b5jam94 ,l9e> ,associative ,algebra2 ,am]ican ,j\rnal ( ,ma!matics1 ,vol4 #d 7#ahha71 p4 #ig4 #acjb4 ,9 ma!matics we see ! consci\s logical activity ( our m9d 9 its pure/ & mo/ p]fect =m2 "h is made manife/ to u all ! labor & ! great c>e ) :i* x progresses1 ! precision :i* is necess>y to det]m9e exactly ! s\rce ( ! e/abli%$ g5]al !orems1 & ! difficulty ) :i* we =m & compreh5d ab/ract conceptions2 b we also le>n "h to h confid5ce 9 ! c]ta9ty1 bread?1 & fruitfulness ( su* intellectual labor4 --,,helmholtz1 ,h4 ,vortrge und ,r$51 ,bd4 #a 7,brauns*weig1 #ahif71 p4 #agf4 #acjc4 ,ma!matical demon/rations are a logic ( z mu* or m use1 ?an t commonly le>n$ at s*ools1 s]v+ to a j =mation ( ! m9d1 enl>g+ its capacity1 & /r5g!n+ x s z to r5d] ! same capable ( exact reason+1 & disc]n+ tru? f falsehood 9 all occurr5ces1 ev5 9 subjects n ma!matical4 ,= :i* reason x is said1 ! ,egyptians1 ,p]sians1 & ,lac$aemonians seldom elect$ any new k+s1 b su* z _h "s k 9 ! ma!matics1 imag9+ ^?1 :o _h n1 m5 ( imp]fect judgm5ts1 & unfit to rule & gov]n4--,,frankl91 ,,b5jam94 ,usefulness ( ,ma!matics2 ,works 7,bo/on1 #ahdj71 ,vol4 #b1 p4 #fh4 #acjd4 ,! ma!matical conception is1 f its v nature1 ab/ract2 inde$ its ab/ractness is usually ( a high] ord] ?an ! ab/ractness ( ! logician4--,,*ry/al1 ,,george4 ,encyclop$ia ,britannica 7,n9? ,edition71 ,article 8,ma!matics40 #acje4 ,ma!matics1 t giant p9c]s ( sci5tific logic4444 --,,hal/$1 ,g4 ,b4 ,sci5ce 7#aije71 p4 #afa4 #acjf4 ,logic has borr[$ ! rules ( geometry )\t und]/&+ its p[]4444 ,i am f> f plac+ logicians by ! side ( geomet]s :o tea* ! true way to guide ! reason4444 ,! me?od ( avoid+ error is s\ght by e "o4 ,! logicians pr(ess to lead ! way1 ! geomet]s alone rea* x1 & aside f _! sci5ce !re is no true demon/ration4 --,,pascal4 ,quot$ by ,a4 ,rebire3 ,ma?matiques et ,ma?matici5s 7,p>is1 #ahih71 pp4 #afb-afc4 #acjg4 ,ma!matics1 l dialectics1 is an organ ( ! high] s5se1 9 its execution x is an art l eloqu5ce4 ,to bo? no?+ b ! =m is ( value2 nei!r c>es any?+ = cont5t4 ,:e!r ma!matics consid]s p5nies or gu9eas1 :e!r rhetoric def5ds tru? or error1 is p]fectly immat]ial to ei!r4--,,goe!4 ,spr*e 9 ,prosa1 ,natur ,,iv1 #idf4 #acjh4 ,conf9$ to its true doma91 ma!matical reason+ is admirably adapt$ to p]=m ! univ]sal (fice ( s\nd logic3 to induce 9 ord] to d$uce1 9 ord] to con/ruct4444 ,x cont5ts itself to furni%1 9 ! mo/ favorable doma91 a model ( cle>ness1 ( precision1 & consi/5cy1 ! close contemplation ( :i* is alone able to prep>e ! m9d to r5d] o!r conceptions also z p]fect z _! nature p]mits4 ,its g5]al reaction1 m negative ?an positive1 mu/ consi/1 above all1 9 inspir+ u ev]y:]e ) an inv9cible av]sion = vagu5ess1 inconsi/5cy1 & obscurity1 :i* may always be really avoid$ 9 any reason+ :atsoev]1 if we make suffici5t ef=t4--,,comte1 ,a4 ,subjective ,syn!sis4 #acji4 ,=mal ?\ght1 consci\sly recogniz$ z su*1 is ! means ( all exact k2 & a correct und]/&+ ( ! ma9 =mal sci5ces1 ,logic & ,ma!matics1 is ! prop] & only safe f\ndation = a sci5tific education4--,,lefevre1 ,,ar?ur4 ,numb] & its ,algebra 7,bo/on1 ,sect4 #bbb47 #acaj4 ,x has come to pass1 ,i "k n h[1 t ,ma!matics & ,logic1 :i* ought to be b ! h&maids ( ,physic1 nev]!less presume on ! /r5g? ( ! c]ta9ty :i* !y possess to ex]cise dom9ion ov] x4--,,bacon1 ,,francis4 ,de ,augm5tis1 ,bk4 #c4 #acaa4 ,we may reg>d geometry z a practical logic1 = ! tru?s :i* x consid]s1 be+ ! mo/ simple & mo/ s5sible ( all1 are1 = ?is reason1 ! mo/ susceptible to easy & ready application ( ! rules ( reason+4--,d',,alemb]t4 ,quot$ 9 ,a4 ,rebire3 ,ma?matiques et ,ma?matici5s 7,p>is1 #ahih71 pp4 #aea-aeb4 #acab4 ,!re are notable examples en\gh ( demon/ration outside ( ma!matics1 & x may be said t ,ari/otle has already giv5 "s 9 8 8,prior ,analytics40 ,9 fact logic is z susceptible ( demon/ration z geometry1 4444 ,ar*im$es is ! fir/1 ^: works we h1 :o has practis$ ! art ( demon/ration ^u an occasion :]e he is treat+ ( physics1 z he has done 9 8 book on ,equilibrium4 ,fur!rmore1 juri/s may be said to h _m good demon/rations2 especially ! anci5t ,roman juri/s1 ^: fragm5ts h be5 pres]v$ to u 9 ! ,p&ects4--,,leibnitz1 ,g4 ,w4 ,new ,essay on ,human ,und]/&+ ,langley1 ,bk4 #d1 *ap4 #b1 sect4 #ab4 #acac4 ,x is commonly consid]$ t ma!matics owes its c]ta9ty to its reliance on ! immutable pr9ciples ( =mal logic4 ,?is 444 is only half ! tru? imp]fectly express$4 ,! o!r half w\ld be t ! pr9ciples ( =mal logic owe su* a degree ( p]man5ce z !y h l>gely to ! fact t !y h be5 temp]$ by long & v>i$ use by ma!maticians4 8,a vici\s circle60 y w p]haps say4 ,i %\ld r describe x z an example ( ! process kn[n by ma!maticians z ! me?od ( successive approximation4--,b,,*]1 ,,maxime4 ,bullet9 ( ! ,am]ican ,ma!matical ,society1 ,vol4 #aa1 p4 #abj4 #acad4 ,:atev] advantage c be attribut$ to logic 9 direct+ & /r5g!n+ ! action ( ! und]/&+ is f\nd 9 a high] degree 9 ma!matical /udy1 ) ! imm5se add$ advantage ( a det]m9ate subject1 di/9ctly circumscrib$1 admitt+ ( ! utmo/ precision1 & free f ! dang] :i* is inh]5t 9 all ab/ract logic1--( lead+ to useless & pu]ile rules1 or to va9 ontological speculations4 ,! positive me?od1 be+ ev]y:]e id5tical1 is z mu* at home 9 ! art ( reason+ z any:]e else3 & ?is is :y no sci5ce1 :e!r biology or any o!r1 c (f] any k9d ( reason+1 ( :i* ma!matics does n supply a simpl] & pur] c\nt]p>t4 ,?us1 we are enabl$ to elim9ate ! only rema9+ portion ( ! old philosophy :i* c\ld ev5 appe> to (f] any real utility2 ! logical "p1 ! value ( :i* is irrevocably absorb$ by ma!matical sci5ce4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 7,london1 #ahge71 ,vol4 #a1 pp4 #cba-cbb4 #acae4 ,we "k t ma!maticians c>e no m = logic ?an logicians = ma!matics4 ,! two eyes ( exact sci5ce are ma!matics & logic3 ! ma!matical sect puts out ! logical eye1 ! logical sect puts out ! ma!matical eye2 ea* believ+ t x c see bett] ) "o eye ?an ) two4 --,,de ,,morgan1 ,a4 ,quot$ 9 ,f4 ,cajori3 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #caf4 #acaf4 ,! progress ( ! art ( rational discov]y dep5ds 9 a great "p ^u ! art ( *>act]i/ic 7ars *>act]i/ica74 ,! reason :y p usually seek demon/rations only 9 numb]s & l9es & ?+s repres5t$ by ^! is none o!r ?an t !re are n1 outside ( numb]s1 conv5i5t *>act]s correspond+ to ! notions4--,,leibnitz1 ,g4 ,w4 ,philosophis*e ,s*rift5 ,g]h>dt ,bd4 #h1 p4 #aih4 #acag4 ,! influ5ce ( ! ma!matics ( ,leibnitz ^u 8 philosophy appe>s *iefly 9 connection ) 8 law ( cont9uity & 8 prolong$ ef=ts to e/abli% a ,logical ,calculus4444 ,to f9d a ,logical ,calculus 7imply+ a univ]sal philosophical language or sy/em ( signs7 is an attempt to apply 9 !ological & philosophical inve/igations an analytic me?od analog\s to t :i* _h prov$ s successful 9 ,geometry & ,physics4 ,x seem$ to ,leibnitz t if all ! complex & app>5tly disconnect$ ideas :i* make up our k c\ld be analys$ into _! simple elem5ts1 & if ^! elem5ts c\ld ea* be repres5t$ by a def9ite sign1 we %\ld h a k9d ( 8alphabet ( human ?\ghts40 ,by ! comb9ation ( ^! signs 7lett]s ( ! alphabet ( ?\ght7 a sy/em ( true k w\ld be built up1 9 :i* reality w\ld be m & m adequately repres5t$ or symboliz$4444 ,9 _m cases ! analysis may result 9 an inf9ite s]ies ( elem5ts2 b ! pr9ciples ( ! ,inf9itesimal ,calculus 9 ma!matics h %[n t ?is does n necess>ily r5d] calculation impossible or inaccurate4 ,?us x seem$ to ,leibnitz t a syn!tic calculus1 bas$ ^u a ?or\gh analysis1 w\ld be ! mo/ effective in/rum5t ( k t c\ld be devis$4 8,i feel10 he says1 8t controv]sies c nev] be f9i%$1 nor sil5ce impos$ ^u ! ,sects1 unless we give up complicat$ reason+s 9 favor ( simple calculations1 words ( vague & unc]ta9 mean+ 9 favor ( fix$ symbols40 ,?us x w appe> t 8e p>alogism is no?+ b an error ( calculation40 8,:5 controv]sies arise1 !re w be no m necessity ( disputation betwe5 two philosoph]s ?an betwe5 two acc\ntants4 ,no?+ w be ne$$ b t !y %\ld take p5 9 h&1 sit d[n ) _! c\nt+-tables1 & 7hav+ summon$ a fri5d1 if !y l7 say to "o ano!r3 ,let u calculate40--,,latta1 ,,rob]t4 ,leibnitz1 ,! ,monadology1 etc4 7,ox=d1 #ahih71 p4 #he4 #acah4 ,pure ma!matics 0 discov]$ by ,boole 9 a "w :i* he call$ 8,! ,laws ( ,?\ght04444 ,8 "w 0 conc]n$ ) =mal logic1 & ?is is ! same ?+ z ma!matics4 --,,russell1 ,,b]tr&4 ,int]national ,mon?ly1 #aija1 p4 #hc4 #acai4 ,ma!matics is b ! high] developm5t ( ,symbolic ,logic4--,,:e?am1 ,w4 ,c4 ,d4 ,rec5t ,developm5t ( ,physical ,sci5ce 7,philadelphia1 #aijd71 p4 #cd4 #acbj4 ,symbolic ,logic has be5 dis[n$ by _m logicians on ! plea t its int]e/ is ma!matical1 & by _m ma!maticians on ! plea t its int]e/ is logical4 --,,:itehead1 ,a4 ,n4 ,univ]sal ,algebra 7,cambridge1 #ahih71 ,preface1 p4 #f4 #acba4 444 ! two great compon5ts ( ! critical movem5t1 ?\gh di/9ct 9 orig9 & foll[+ sep>ate pa?s1 are f\nd to conv]ge at la/ 9 ! !sis3 ,symbolic ,logic is ,ma!matics1 ,ma!matics is ,symbolic ,logic1 ! twa9 are "o4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #ai4 #acbb4 ,! emancipation ( logic f ! yoke ( ,ari/otle v mu* resembles ! emancipation ( geometry f ! bondage ( ,euclid2 &1 by its subsequ5t gr[? & div]sification1 logic1 less abundantly p]haps b n less c]ta9ly ?an geometry1 has illu/rat$ ! bless+s ( fre$om4--,,keys]1 ,c4 ,j4 ,sci5ce1 ,vol4 #ce 7#aiab71 p4 #ajh4 #acbc4 ,i w\ld express x z my p]sonal view1 :i* is probably n yet %>$ g5]ally1 t pure ma!matics seems to me m]ely a bran* ( g5]al logic2 t bran* :i* is bas$ on ! concept ( numb]s1 to ^: economic advantages is to be attribut$ ! trem5d\s developm5t :i* ?is p>ticul> bran* has und]gone z comp>$ ) ! rema9+ bran*es ( logic1 :i* until ! mo/ rec5t times h rema9$ almo/ /ation>y4--,,s*r,,d]1 ,e4 ,ueb] ,pasigraphie etc42 ,v]h&lung5 des #a4 ,int]national5 ,ma!matik]-,kongresses 7,leipzig1 #ahih71 p4 #adi4 #acbd4 ,if logical tra9+ is to consi/1 n 9 repeat+ b>b>\s s*ola/ic =mulas or me*anically tack+ toge!r empty majors & m9ors1 b 9 acquir+ dext]ity 9 ! use ( tru/wor?y me?ods ( advanc+ f ! kn[n to ! unkn[n1 !n ma!matical inve/igation mu/ ev] rema9 "o ( its mo/ indisp5sable in/rum5ts4 ,once inur$ to ! habit ( accurately imag9+ ab/ract relations1 recogniz+ ! true value ( symbolic conceptions1 & famili>iz$ ) a fix$ /&>d ( pro(1 ! m9d is equipp$ = ! consid]ation ( q o!r objects ?an l9es & angles4 ,! tw9 treatises ( ,adam ,smi? on social sci5ce1 :]e91 by d$uc+ all human ph5om5a fir/ f ! un*eck$ action ( selfi%ness & !n f ! un*eck$ action ( sympa?y1 he arrives at mutually-limit+ conclusions ( transc5d5t practical importance1 furni% = all "t a brilliant illu/ration ( ! value ( ma!matical me?ods & ma!matical discipl9e4--,,fiske1 ,,john4 ,d>w9ism & o!r ,essays 7,bo/on1 #ahic71 pp4 #big-bih4 #acbe4 ,no irrational exagg]ation ( ! claims ( ,ma!matics c ev] deprive t "p ( philosophy ( ! prop]ty ( be+ ! natural basis ( all logical education1 ?r\gh its simplicity1 ab/ractness1 g5]ality1 & fre$om f di/urbance by human passion4 ,!re1 & !re alone1 we f9d 9 full developm5t ! art ( reason+1 all ! res\rces ( :i*1 f ! mo/ spontane\s to ! mo/ sublime1 are cont9ually appli$ ) f> m v>iety & fruitfulness ?an else:]e24444 ,! m ab/ract portion ( ma!matics may 9 fact be reg>d$ z an imm5se repository ( logical res\rces1 ready = use 9 sci5tific d$uction & co-ord9ation4 --,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 7,london1 #ahge71 ,vol4 #b1 p4 #dci4 #acbf4 ,logic x is call$ ref]r+ to ,:itehead & ,russell's ,pr9cipia ,ma!matica & logic x is1 ! logic ( propositions & functions & classes & relations1 by f> ! greate/ 7n m]ely ! bigge/7 logic t our planet has produc$1 s mu* t is new 9 matt] & 9 mann]2 b x is also ma!matics1 a prolegom5on to ! sci5ce1 yet itself ma!matics 9 its mo/ g5u9e s5se1 diff]+ f o!r p>ts ( ! sci5ce only 9 ! respects t x surpasses ^! 9 fundam5tality1 g5]ality & precision1 & lacks traditionality4 ,few w read x1 b all w feel its effect1 = beh9d x is ! urg5ce & pu% ( a magnific5t pa/3 two ?\s& five hundr$ ye>s ( record & yet long] tradition ( human endeavor to ?9k aright4--,,keys]1 ,c4 ,j4 ,sci5ce1 ,vol4 #ce 7#aiab71 p4 #aaj4 ,,*apt] ,,xiv ,,ma!matics ,,& ,,philosophy #adja4 ,socrates is prais$ by all ! c5turies = hav+ call$ philosophy f heav5 to m5 on e>?2 b if1 kn[+ ! condition ( our sci5ce1 he %\ld come aga9 & %\ld look once m to heav5 = a means ( cur+ m51 he w\ld !re f9d t to ma!matics1 r ?an to ! philosophy ( today1 _h be5 giv5 ! cr[n because ( its indu/ry & its mo/ happy & brilliant successes4--,,h]b>t1 ,j4 ,f4 ,w]ke ,kehrba*1 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #ie4 #adjb4 ,x is ! emb>rassm5t ( metaphysics t x is able to accompli% s little ) ! _m ?+s t ma!matics (f]s h]4--,,kant1 ,e4 ,metaphysis*e ,anfangsgrnde d] ,naturwiss5s*aft1 ,vorr$e4 #adjc4 ,philosoph]s1 :5 !y h possess$ a ?or\gh k ( ma!matics1 h be5 among ^? :o h enri*$ ! sci5ce ) "s ( its be/ ideas4 ,on ! o!r h& x mu/ be said t1 ) h>dly an exception1 all ! rem>ks on ma!matics made by ^? philosoph]s :o h possess$ b a slight or ha/y or late-acquir$ k ( x are entirely wor?less1 be+ ei!r trivial or wrong4--,,:itehead1 ,a4 ,n4 ,introduction to ,ma!matics 7,new ,york1 #aiaa71 p4 #aac4 #adjd4 ,! union ( philosophical & ma!matical productivity1 :i* besides 9 ,plato we f9d only 9 ,py?agoras1 ,desc>tes & ,leibnitz1 has always yield$ ! *oice/ fruits to ma!matics3 ,to ! fir/ we owe sci5tific ma!matics 9 g5]al1 ,plato discov]$ ! analytic me?od1 by means ( :i* ma!matics 0 elevat$ above ! view-po9t ( ! elem5ts1 ,desc>tes creat$ ! analytical geometry1 our own illu/ri\s c\ntryman discov]$ ! inf9itesimal calculus--& j ^! are ! f\r greate/ /eps 9 ! developm5t ( ma!matics4 --,,hankel1 ,,h]mann4 ,ges*i*te d] ,ma!matik im ,alt]tum und im ,mittelalt] 7,leipzig1 #ahgd71 pp4 #adi-aej4 #adje4 ,)\t ma!matics "o _c fa?om ! dep?s ( philosophy2 )\t philosophy "o _c fa?om ! dep?s ( ma!matics2 )\t ! two "o _c fa?om any?+4 --,,bordas-,,dem\l9s4 ,quot$ 9 ,a4 ,rebire3 ,ma?matiques et ,ma?matici5s 7,p>is1 #ahih71 p4 #adg4 #adjf4 ,9 ! end ma!matics is b simple philosophy1 & philosophy1 high] ma!matics 9 g5]al4--,,novalis4 ,s*rift5 7,b]l91 #aija71 ,teil #b1 p4 #ddc4 #adjg4 ,x is a safe rule to apply t1 :5 a ma!matical or philosophical au?or writes ) a mi/y pr(undity1 he is talk+ nons5se4--,,:itehead1 ,a4 ,n4 ,introduction to ,ma!matics 7,new ,york1 #aiaa71 p4 #bbg4 #adjh4 ,! real f9i%] ( our education is philosophy1 b x is ! (fice ( ma!matics to w>d (f ! dang]s ( philosophy4--,,h]b>t1 ,j4 ,f4 ,pe/alozzi's ,idee e9es ,,abc d] ,ans*auung2 ,w]ke ,kehrba*1 7,lang5salza1 #ahij71 ,bd4 #a1 p4 #afh4 #adji4 ,s9ce antiquity ma!matics has be5 reg>d$ z ! mo/ indisp5sable s*ool = philosophic ?\ght & 9 its highe/ sph]es ! rese>* ( ! ma!matician is inde$ mo/ closely relat$ to pure speculation4 ,ma!matics is ! mo/ p]fect union betwe5 exact k & !oretical ?\ght4--,,curtius1 ,e4 ,b]l9] ,monatsb]i*te 7#ahgc71 p4 #eag4 #adaj4 ,geometry has be51 ?r\gh\t1 ( supreme importance 9 ! hi/ory ( k4--,,russell1 ,,b]tr&4 ,f\ndations ( ,geometry 7,cambridge1 #ahig71 p4 #ed4 #adaa4 ,he is unwor?y ( ! "n ( man :o is ignorant ( ! fact t ! diagonal ( a squ>e is incomm5surable ) its side4--,,plato4 ,quot$ by ,sophie ,g]ma93 ,mmoire sur les surfaces la/iques4 #adab4 ,ma!matics1 consid]$ z a sci5ce1 owes its orig9 to ! ideali/ic ne$s ( ! ,greek philosoph]s1 & n z fable has x1 to ! practical dem&s ( ,egyptian economics4444 ,adam 0 no zologi/ :5 he gave names to ! bea/s ( ! field1 nor 7 ! ,egyptian surveyors ma!maticians4--,,hankel1 ,h4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 p4 #g4 #adac4 ,!re are only two ways op5 to man = atta9+ a c]ta9 k ( tru?3 cle> intuition & necess>y d$uction4--,,desc>tes4 ,rules = ! ,direction ( ! ,m9d2 ,torrey's ,! ,philosophy ( ,desc>tes 7,new ,york1 #ahib71 p4 #ajd4 #adad4 ,ma!maticians h1 9 _m cases1 prov$ "s ?+s to be possible & o!rs to be impossible1 :i*1 )\t demon/ration1 w\ld n h be5 believ$4444 ,ma!matics af=d _m in/ances ( impossibilities 9 ! nature ( ?+s1 :i* no man w\ld h believ$1 if !y _h n be5 /rictly demon/rat$4 ,p]haps1 if we 7 able to reason demon/ratively 9 o!r subjects1 to z great ext5t z 9 ma!matics1 we might f9d _m ?+s to be impossible1 :i* we conclude1 )\t hesitation1 to be possible4--,,reid1 ,,?omas4 ,essay on ! ,intellectual ,p[]s ( ,man1 ,essay #d1 *ap4 #c4 #adae4 ,if philosoph]s und]/ood ma!matics1 !y w\ld "k t indef9ite spee*1 :i* p]mits ea* "o to ?9k :at he pleases & produces a con/antly increas+ diff]5ce ( op9ion1 is utt]ly unable1 9 spite ( all f9e words & ev5 9 spite ( ! magnitude ( ! objects :i* are "u contemplation1 to ma9ta9 a balance aga9/ a sci5ce :i* in/ructs & advances ?r\gh e ^w :i* x utt]s & :i* at ! same "t w9s = itself endless a/oni%m5t1 n ?r\gh its survey ( imm5se spaces1 b ?r\gh ! exhibition ( ! mo/ prodigi\s human +5uity :i* surpasses all p[] ( description4--,,h]b>t1 ,j4 ,f4 ,w]ke ,kehrba* 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #aje4 #adaf4 ,g]man intellect is an excell5t ?+1 b :5 a ,g]man product is pres5t$ x mu/ be analys$4 ,mo/ probably x is a comb9ation ( intellect 7,i7 & tobacco-smoke 7,t74 ,c]ta9ly ,i,t1 & ,i,t1 occur2 b ,i,t is m common1 & ,i,t & ,i,t occur4 ,9 _m cases metaphysics 7,m7 occurs & ,i hold t ,ia,tb,mc nev] occurs )\t b c #b;a4 ,n4 ,b4--,be c>eful1 9 analys+ ! comp\nds ( ! ?ree1 n to conf\nd ,t & ,m1 :i* are /rongly suspect$ to be isomorphic4 ,?us1 ,i,t,m may easily be conf\nd$ ) ,i,t4 ,z f> z ,i d>e say any?+1 ^? :o h plac$ ,hegel1 ,fi*te1 etc41 9 ! rank ( ! ext5d]s ( ,kant h imag9$ ,t & ,m to be id5tical4--,,de ,,morgan1 ,a4 ,graves' ,life ( ,w4 ,r4 ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #ac1 p4 #ddf4 #adag4 ,! discov]y ( ,c]es 0 made by ,g4 ,piazzi ( ,pal]mo2 & x 0 ! m int]e/+ z its ann\ncem5t occurr$ simultane\sly ) a publication by ,hegel 9 :i* he sev]ely criticiz$ a/ronom]s = n pay+ m att5tion to philosophy1 a sci5ce1 said he1 :i* w\ld at once h %[n !m t !re c\ld n possibly be m ?an sev5 planets1 & a /udy ( :i* w\ld !re=e h prev5t$ an absurd wa/e ( "t 9 look+ = :at 9 ! nature ( ?+s c\ld nev] be f\nd4--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #deh4 #adah4 ,b :o %all p>cel out ,8 intellect by geometric rules1 ,split l a prov9ce into r\nd & squ>e8 --,,wordswor?4 ,! ,prelude1 ,bk4 #b4 #adai4 ,& ,proposition1 g5tle maid1 ,:o soo?ly ask'd /]n ,demon/ration's aid1 4444 --,,col]idge1 ,s4 ,t4 ,a ,ma!matical ,problem4 #adbj4 ,ma!matics connect !mselves on ! "o side ) common life & physical sci5ce2 on ! o!r side ) philosophy 9 reg>d to our notions ( space & "t1 & 9 ! que/ions :i* h aris5 z to ! univ]sality & necessity ( ! tru?s ( ma!matics & ! f\ndation ( our k ( !m4--,,cayley1 ,,ar?ur4 ,briti% ,association ,address 7#ahhh72 ,collect$ ,ma!matical ,pap]s1 ,vol4 #aa1 p4 #dcj4 #adba4 ,ma!matical tea*+ 444 tra9s ! m9d to capacities1 :i* 444 are ( ! close/ k9 to ^? ( ! greate/ metaphysician & philosoph]4 ,!re is "s color ( tru? = ! opposite doctr9e 9 ! case ( elem5t>y algebra4 ,! resolution ( a common equation c be r$uc$ to almo/ z me*anical a process z ! work+ ( a sum 9 ari?metic4 ,! r$uction ( ! "q to an equation1 h[ev]1 is no me*anical op]ation1 b "o :i*1 accord+ to ! degree ( its difficulty1 requires ne>ly e possible grade ( +5uity3 n to speak ( ! new1 & 9 ! pres5t /ate ( ! sci5ce insoluble1 equations1 :i* />t up at e fre% /ep attempt$ 9 ! application ( ma!matics to o!r bran*es ( k4--,,mill1 ,j4 ,s4 ,an ,exam9ation ( ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #fae4 #adbb4 ,! value ( ma!matical in/ruction z a prep>ation = ^? m difficult inve/igations1 consi/s 9 ! applicability n ( its doctr9es1 b ( its me?ods4 ,ma!matics w ev] rema9 ! mo/ p]fect type ( ! ,d$uctive ,me?od 9 g5]al2 & ! applications ( ma!matics to ! simpl] bran*es ( physics1 furni% ! only s*ool 9 :i* philosoph]s c effectually le>n ! mo/ difficult & important portion ( _! art1 ! employm5t ( ! laws ( ! simpl] ph5om5a = expla9+ & pr$ict+ ^? ( ! m complex4 ,^! gr\nds are q suffici5t = deem+ ma!matical tra9+ an indisp5sable basis ( real sci5tific education1 & reg>d+1 ) ,plato1 "o :o is 1 z want+ 9 "o ( ! mo/ ess5tial qualifications = ! successful cultivation ( ! high] bran*es ( philosophy4--,,mill1 ,j4 ,s4 ,sy/em ( ,logic1 ,bk4 #c1 *ap4 #bd1 sect4 #i4 #adbc4 ,9 metaphysical reason+1 ! process is always %ort4 ,! conclusion is b a /ep or two1 seldom m1 f ! fir/ pr9ciples or axioms on :i* x is gr\nd$1 & ! diff]5t conclusions dep5d n "o ^u ano!r4 ,x is o!rwise 9 ma!matical reason+4 ,"h ! field has no limits4 ,"o proposition leads on to ano!r1 t to a ?ird1 & s on )\t end4 ,if x %\ld be ask$1 :y demon/rative reason+ has s wide a field 9 ma!matics1 :ile1 9 o!r ab/ract subjects1 x is conf9$ )9 v n>r[ limits1 ,i conceive ?is is *iefly ow+ to ! nature ( quantity1 444 ma!matical quantities be+ made up ( p>ts )\t numb]1 c t\* 9 innum]able po9ts1 & be comp>$ 9 innum]able diff]5t ways4--,,reid1 ,,?omas4 ,essays on ! ,p[]s ( ! ,human ,m9d 7,ed9burgh1 #ahab71 ,vol4 #b1 pp4 #dbb-dbc4 #adbd4 ,! p[] ( ,reason 444 is unque/ionably ! mo/ important by f> ( ^? :i* are compreh5d$ "u ! g5]al title ( ,intellectual4 ,x is on ! "r use ( ?is p[] t our success 9 ! pursuit ( bo? k & ( happ9ess dep5ds2 & x is by ! exclusive possession ( x t man is di/+ui%$1 9 ! mo/ ess5tial respects1 f ! l[] animals4 ,x is1 inde$1 f _! subs]vi5cy to its op]ations1 t ! o!r faculties 444 d]ive _! *ief value4--,,/ew>t1 ,,dugald4 ,philosophy ( ! ,human ,m9d2 ,collect$ ,works 7,ed9burgh1 #ahed71 ,vol4 #h1 p4 #e4 #adbe4 ,:5 444 ,i ask$ myself :y 0 x !n t ! e>lie/ philosoph]s w\ld admit to ! /udy ( wisdom only ^? :o _h /udi$ ma!matics1 z if ?is sci5ce 0 ! easie/ ( all & ! "o mo/ necess>y = prep>+ & discipl9+ ! m9d to compreh5d ! m advanc$1 ,i suspect$ t !y _h k ( a ma!matical sci5ce diff]5t f t ( our "t4444 ,i believe ,i f9d "s traces ( ^! true ma!matics 9 ,pappus & ,diophantus1 :o1 al?\gh !y 7 n ( extreme antiquity1 liv$ nev]!less 9 times long prec$+ ours4 ,b ,i will+ly believe t ^! writ]s !mselves1 by a culpable ruse1 suppress$ ! k ( !m2 l "s artisans :o conceal _! secret1 !y fe>$1 p]haps1 t ! ease & simplicity ( _! me?od1 if become popul>1 w\ld dim9i% its importance1 & !y pref]r$ to make !mselves admir$ by leav+ to u1 z ! product ( _! art1 c]ta9 b>r5 tru?s d$uc$ ) subtlety1 r ?an to tea* u t art itself1 ! k ( :i* w\ld end our admiration4--,,desc>tes4 ,rules = ! ,direction ( ! ,m9d2 ,philosophy ( ,desc>tes ,torrey1 7,new ,york1 #ahib71 pp4 #gj-ga4 #adbf4 ,if we rightly adh]e to our rule t is1 t we %\ld occupy ourselves only ) ^? subjects 9 ref]5ce to :i* ! m9d is capable ( acquir+ c]ta9 & indubitable k !re w rema9 b few ?+s to ! /udy ( :i* we c devote ourselves4 ,!re exi/s 9 ! sci5ces h>dly a s+le "q ^u :i* m5 ( intellectual ability h n held diff]5t op9ions4 ,b :5ev] two m5 pass contr>y judgm5t on ! same ?+1 x is c]ta9 t "o ( ! two is wrong4 ,m ?an t1 nei!r ( !m has ! tru?2 = if "o ( !m _h a cle> & precise insight into x1 he c\ld s exhibit x to 8 oppon5t z to end ! discussion by compell+ 8 conviction4444 ,x foll[s f ?is1 if we reckon rightly1 t among exi/+ sci5ces !re rema9 only geometry & ari?metic1 to :i* ! obs]vance ( our rule w\ld br+ u4 --,,desc>tes4 ,rules = ! ,direction ( ! ,m9d2 ,philosophy ( ,desc>tes ,torrey1 7,new ,york1 #ahib71 p4 #fb4 #adbg4 ,! same reason :i* l$ ,plato to recomm5d ! /udy ( ari?metic l$ him to recomm5d also ! /udy ( geometry4 ,! vulg> cr[d ( geometricians1 he says1 w n und]/& him4 ,!y h practice always 9 view4 ,!y d n "k t ! real use ( ! sci5ce is to lead m5 to ! k ( ab/ract1 ess5tial1 et]nal tru?4 7,plato's ,republic1 ,book #g74 ,inde$ if we are to believe ,plut>*1 ,plato c>ri$ 8 feel+ s f> t he consid]$ geometry z degrad$ by be+ appli$ to any purpose ( vulg> utility4 ,ar*ytas1 x seems1 _h fram$ ma*9es ( extraord9>y p[] on ma!matical pr9ciples4 7,plut>*1 ,sympos41 ,,viii41 & ,life ( ,m>cellus4 ,! ma*9es ( ,ar*ytas are also m5tion$ by ,aulus ,gellius & ,diog5es ,la]tius74 ,plato remon/rat$ ) 8 fri5d1 & decl>$ t ?is 0 to degrade a noble intellectual ex]cise into a l[ craft1 fit only = c>p5t]s & :eelwrights4 ,! (fice ( geometry1 he said1 0 to discipl9e ! m9d1 n to m9i/] to ! base wants ( ! body4 ,8 int]f]5ce 0 successful2 & f t "t accord+ to ,plut>*1 ! sci5ce ( me*anics 0 consid]$ unwor?y ( ! att5tion ( a philosoph]4 --,,macaulay4 ,"l ,bacon2 ,ed9burgh ,review1 ,july1 #ahcg4 #adbh4 ,! intellectual habits ( ! ,ma!maticians are1 9 "s respects1 ! same ) ^? ( ! ,metaphysicians we h be5 n[ consid]+2 b1 9 o!r respects1 !y diff] widely4 ,bo? are fav\rable to ! improvem5t ( ! p[] ( att5tion1 b n 9 ! same mann]1 nor 9 ! same degree4 ,^? ( ! metaphysician give capacity ( fix+ ! att5tion on ! subjects ( our consci\sness1 )\t be+ di/ract$ by ?+s ext]nal2 b !y af=d little or no ex]cise to t species ( att5tion :i* enables u to foll[ long processes ( reason+1 & to keep 9 view all ! v>i\s /eps ( an inve/igation till we arrive at ! conclusion4 ,9 ma!matics1 su* processes are mu* long] ?an 9 any o!r sci5ce2 & h5ce ! /udy ( x is peculi>ly calculat$ to /r5g!n ! p[] ( /eady & concat5at$ ?9k+1--a p[] :i*1 9 all ! pursuits ( life1 :e!r speculative or active1 is "o ( ! mo/ valuable end[m5ts we c possess4 ,?is comm& ( att5tion1 h[ev]1 x may be prop] to add1 is to be acquir$1 n by ! practice ( mod]n me?ods1 b by ! /udy ( ,greek geometry1 m p>ticul>ly1 by accu/om+ ourselves to pursue long tra9s ( demon/ration1 )\t avail+ ourselves ( ! aid ( any s5sible diagrams2 ! ?\ghts be+ direct$ solely by ^? ideal del9eations :i* ! p[]s ( conception & ( memory enable u to =m4--,,/ew>t1 ,,dugald4 ,philosophy ( ! ,human ,m9d1 ,"p #c1 *ap4 #a1 sect4 #c4 #adbi4 ,!y ! ,greeks speculat$ & !oriz$ "u a lively p]suasion t a ,sci5ce ( e "p ( nature 0 possible1 & 0 a fit object = ! ex]cise ( a man's be/ faculties2 & !y 7 spe$ily l$ to ! conviction t su* a sci5ce mu/ clo! its conclusions 9 ! language ( ma!matics4 ,?is conviction is em95tly conspicu\s 9 ! writ+s ( ,plato4444 ,probably no succe$+ /ep 9 ! discov]y ( ! ,laws ( ,nature 0 ( s mu* importance z ! full adoption ( ?is p]vad+ conviction1 t !re mu/ be ,ma!matical ,laws ( ,nature1 & t x is ! bus9ess ( ,philosophy to discov] ^! ,laws4 ,?is conviction cont9ues1 ?r\gh all ! succe$+ ages ( ! hi/ory ( ! sci5ce1 to be ! animat+ & support+ pr9ciple ( sci5tific inve/igation & discov]y4 --,,:ewell1 ,w4 ,hi/ory ( ! ,inductive ,sci5ces1 ,vol4 #a1 bk4 #b1 *ap4 #c4 #adcj4 ,= to pass by ^? ,anci5ts1 ! wond]ful ,py?agoras1 ! sagaci\s ,democritus1 ! div9e ,plato1 ! mo/ subtle & v le>n$ ,ari/otle1 ,m5 :om e ,age has hi!rto ackn[l$g$ z des]v$ly honor$1 z ! greate/ ,philosoph]s1 ! ,r+-lead]s ( ,arts2 9 ^: ,judgm5ts h[ mu* ^! ,/udies ma!matics 7 e/eem$1 is abundantly proclaim$ 9 ,hi/ory & confirm$ by _! fam\s ,monum5ts1 :i* are ev]y:]e int]sp]s$ & bespangl$ ) ,ma!matical ,reason+s & ,examples1 z ) s _m ,/>s2 & consequ5tly anyone n 9 "s ,degree conv]sant 9 ^! ,/udies w 9 va9 expect to und]/&1 or unlock _! hidd5 ,mean+s1 )\t ! ,help ( a ,ma!matical ,key3 ,= :o c play well on ,ari/otle's ,in/rum5t b ) a ,ma!matical ,quill2 or n be altoge!r deaf to ! ,lessons ( natural ,philosophy1 :ile ignorant ( ,geometry8 ,:o void ( 7,geometry %all ,i say1 or7 ,ari?metic c compreh5d ,plato's ,socrates lisp+ ) ,*ildr5 conc]n+ ,squ>e ,numb]s2 or c conceive ,plato himself treat+ n only ( ! ,univ]se1 b ! ,polity ( ,commonweal?s regulat$ by ! ,laws ( ,geometry1 & =m$ accord+ to a ,ma!matical ,plan8--,,b>r[1 ,,isaac4 ,ma!matical ,lectures 7,london1 #agcd71 pp4 #bf-bg4 #adca4 ,& ,reason n[ ?r\gh numb]1 "t1 & space ,d>ts ! ke5 lu/re ( h] s]i\s eye2 ,& le>ns f facts comp>'d ! laws to trace ,^: long procession leads to ,deity --,,beattie1 ,,james4 ,! ,m9/rel1 ,bk4 #b1 /anza #dg4 #adcb4 ,t ,egyptian & ,*aldean wisdom ma!matical :]e) ,moses & ,daniel 7 furni%$1 4444--,,hook]1 ,,ri*>d4 ,ecclesia/ical ,polity1 ,bk4 #c1 sect4 #h4 #adcc4 ,g5]al & c]ta9 tru?s are only f\nd$ 9 ! habitudes & relations ( ab/ract ideas4 ,a sagaci\s & me?odical application ( our ?\ghts1 = ! f9d+ out ( ^! relations1 is ! only way to discov] all t c be put ) tru? & c]ta9ty conc]n+ !m into g5]al propositions4 ,by :at /eps we are to proce$ 9 ^!1 is to be le>n$ 9 ! s*ools ( ma!maticians1 :o1 f v pla9 & easy beg9n+s1 by g5tle degrees1 & a cont9u$ *a9 ( reason+s1 proce$ to ! discov]y & demon/ration ( tru?s t appe> at fir/ sight beyond human capacity4 ,! art ( f9d+ pro(s1 & ! admirable me?od !y h inv5t$ = ! s+l+ out & lay+ 9 ord] ^? int]m$iate ideas t demon/ratively %[ ! equality or inequality ( unapplicable quantities1 is t :i* has c>ri$ !m s f> & produc$ su* wond]ful & unexpect$ discov]ies2 b :e!r some?+ l ?is1 9 respect ( o!r ideas1 z well z ^? ( magnitude1 may n 9 "t be f\nd out1 ,i w n det]m9e4 ,?is1 ,i ?9k1 ,i may say1 t if o!r ideas t are ! real z well z ! nom9al ess5ces ( _! species1 7 pursu$ 9 ! way famili> to ma!maticians1 !y w\ld c>ry our ?\ghts fur!r1 & ) great] evid5ce & cle>ness ?an possibly we are apt to imag9e4--,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #d1 *ap4 #ab1 sect4 #g4 #adcd4 ,^? long *a9s ( reason+1 q simple & easy1 :i* geomet]s are wont to employ 9 ! accompli%m5t ( _! mo/ difficult demon/rations1 l$ me to ?9k t ev]y?+ :i* might fall "u ! cognizance ( ! human m9d might be connect$ toge!r 9 a simil> mann]1 & t1 provid$ only t "o %\ld take c>e n to receive any?+ z true :i* 0 n s1 & if "o 7 always c>eful to pres]ve ! ord] necess>y = d$uc+ "o tru? f ano!r1 !re w\ld be none s remote at :i* he might n at la/ arrive1 nor s conceal$ :i* he might n discov]4--,,desc>tes4 ,disc\rse ^u ,me?od1 "p #b2 ,! ,philosophy ( ,desc>tes ,torrey1 7,new ,york1 #ahib71 p4 #dg4 #adce4 ,if anyone wi%$ to write 9 ma!matical fa%ion 9 metaphysics or e?ics1 no?+ w\ld prev5t him f s do+ ) vigor4 ,"s h pr(ess$ to d ?is1 & we h a promise ( ma!matical demon/rations outside ( ma!matics2 b x is v r>e t !y h be5 successful4 ,?is is1 ,i believe1 because !y are disgu/$ ) ! tr\ble x is necess>y to take = a small numb] ( read]s :]e !y w\ld ask z 9 ,p]sius3 ,quis leget haec1 & reply3 ,vel duo vel nemo4 --,,leibnitz4 ,new ,essay conc]n+ ,human ,und]/&+1 ,langley1 ,bk #b1 *ap4 #bi1 sect4 #ab4 #adcf4 ,x is commonly ass]t$ t ma!matics & philosophy diff] f "o ano!r accord+ to _! objects1 ! =m] treat+ ( quantity1 ! latt] ( quality4 ,all ?is is false4 ,! diff]5ce betwe5 ^! sci5ces _c dep5d on _! object2 = philosophy applies to ev]y?+1 h5ce also to quanta1 & s does ma!matics 9 "p1 inasmu* z ev]y?+ has magnitude4 ,x is only ! diff]5t k9d ( rational k or application ( reason 9 ma!matics & philosophy :i* con/itutes ! specific diff]5ce betwe5 ^! two sci5ces4 ,= philosophy is rational k f m]e concepts1 ma!matics1 on ! contr>y1 is rational k f ! con/ruction ( concepts4 ,we con/ruct concepts :5 we repres5t !m 9 intuition a priori1 )\t exp]i5ce1 or :5 we repres5t 9 intuition ! object :i* corresponds to our concept ( x4--,! ma!matician c nev] apply 8 reason to m]e concepts1 nor ! philosoph] to ! con/ruction ( concepts4--,9 ma!matics ! reason is employ$ 9 concreto1 h[ev]1 ! intuition is n empirical1 b ! object ( contemplation is some?+ a priori4 ,9 ?is1 z we see1 ma!matics has an advantage ov] philosophy1 ! k 9 ! =m] be+ intuitive1 9 ! latt]1 on ! contr>y1 only discursive4 ,b ! reason :y 9 ma!matics we deal m ) quantity lies 9 ?is1 t magnitudes c be con/ruct$ 9 intuition a priori1 :ile qualities1 on ! contr>y1 d n p]mit ( be+ repres5t$ 9 intuition4--,,kant1 ,e4 ,logik2 ,w]ke ,h>t5/e91 7,leipzig1 #ahfh71 ,bd4 #h1 pp4 #bc-bd4 #adcg4 ,kant has divid$ human ideas into ! two categories ( quantity & quality1 :i*1 if true1 w\ld de/roy ! univ]sality ( ,ma!matics2 b ,desc>tes' fundam5tal conception ( ! relation ( ! concrete to ! ab/ract 9 ,ma!matics aboli%es ?is division1 & proves t all ideas ( quality are r$ucible to ideas ( quantity4 ,he _h 9 view geometrical ph5om5a only2 b 8 successors h includ$ 9 ?is g5]alization1 fir/1 me*anical ph5om5a1 &1 m rec5tly1 ^? ( heat4 ,!re are n[ no geomet]s :o d n consid] x ( univ]sal application1 & admit t e ph5om5on may be z logically capable ( be+ repres5t$ by an equation z a curve or a motion1 if only we 7 always capable 7:i* we are v f> f be+7 ( fir/ discov]+1 & !n resolv+ x4 ,! limitations ( ,ma!matical sci5ce are n1 !n1 9 its nature4 ,! limitations are 9 our intellig5ce3 & by ^! we f9d ! doma9 ( ! sci5ce rem>kably re/rict$1 9 proportion z ph5om5a1 9 becom+ special1 become complex4 --,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #a1 *ap4 #a4 #adch4 ,! great advantage ( ! ma!matical sci5ces ov] ! moral consi/s 9 ?is1 t ! ideas ( ! =m]1 be+ s5sible1 are always cle> & det]m9ate1 ! smalle/ di/9ction betwe5 !m be+ imm$iately p]ceptible1 & ! same t]ms are /ill expressive ( ! same ideas1 )\t ambiguity or v>iation4 ,an oval is nev] mi/ak5 = a circle1 nor an hyp]bola = an ellipsis4 ,! isosceles & scal5um are di/+ui%$ by b\nd>ies m exact ?an vice & virtue1 "r or wrong4 ,if any t]m be def9$ 9 geometry1 ! m9d readily1 ( itself1 sub/itutes on all occasions1 ! def9ition = ! ?+ def9$3 ,or ev5 :5 no def9ition is employ$1 ! object itself may be repres5t$ to ! s5ses1 & by t means be /eadily & cle>ly appreh5d$4 ,b ! f9] s5tim5ts ( ! m9d1 ! op]ations ( ! und]/&+1 ! v>i\s agitations ( ! passions1 ?\gh really 9 !mselves di/9ct1 easily escape u1 :5 survey$ by reflection2 nor is x 9 our p[] to recall ! orig9al object1 s (t5 z we h occasion to contemplate x4 ,ambiguity1 by ?is means1 is gradually introduc$ into our reason+s3 ,simil> objects are readily tak5 to be ! same3 ,& ! conclusion becomes at la/ v wide (f ! premises4--,,hume1 ,,david4 ,an ,inquiry conc]n+ ,human ,und]/&+1 sect4 #g1 "p #a4 #adci4 ,"o "p ( ^! disadvantages 9 moral ideas :i* has made !m be ?\ght n capable ( demon/ration1 may 9 a good measure be rem$i$ by def9itions1 sett+ d[n t collection ( simple ideas1 :i* e t]m %all /& =2 & !n us+ ! t]ms /eadily & con/antly = t precise collection4 ,& :at me?ods algebra1 or some?+ ( t k9d1 may h]eaft] sugge/1 to remove ! o!r difficulties1 x is n easy to =etell4 ,confid5t1 ,i am1 t if m5 w\ld 9 ! same me?od1 & ) ! same indiff]5cy1 se>* aft] moral z !y d ma!matical tru?s1 !y w\ld f9d !m h a /rong] connexion "o ) ano!r1 & a m necess>y consequ5ce f our cle> & di/9ct ideas1 & to come ne>] p]fect demon/ration ?an is commonly imag9$4--,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #d1 *ap4 #c1 sect4 #bj4 #addj4 ,t :i* 9 ?is respect has giv5 ! advantage to ! ideas ( quantity1 & made !m ?\ght m capable ( c]ta9ty & demon/ration ?an moral ideas1 is1 ,fir/1 ,t !y c be set d[n & repres5t$ by s5sible m>ks1 :i* h a great] & ne>] correspond5ce ) !m ?an any words or s\nds :atsoev]4 ,diagrams drawn on pap] are copies ( ! ideas 9 ! m9d1 & n liable to ! unc]ta9ty t words c>ry 9 _! signification4 ,an angle1 circle1 or squ>e1 drawn 9 l9es1 lies op5 to ! view1 & _c be mi/ak53 x rema9s un*angeable1 & may at leisure be consid]$ & exam9$1 & ! demon/ration be revis$1 & all ! p>ts ( x may be gone ov] m ?an once1 )\t any dang] ( ! lea/ *ange 9 ! ideas4 ,?is _c be done 9 moral ideas3 we h no s5sible m>ks t resemble !m1 :]eby we c set !m d[n2 we h no?+ b words to express !m by2 :i*1 ?\gh :5 writt5 !y rema9 ! same1 yet ! ideas !y /& = may *ange 9 ! same man2 & x is seldom t !y are n diff]5t 9 diff]5t p]sons4 ,secondly1 ,ano!r ?+ t makes ! great] difficulty 9 e?ics is1 ,t moral ideas are commonly m complex ?an ^? ( ! figures ord9>ily consid]$ 9 ma!matics4 ,f :5ce ^! two inconv5i5ces foll[3--,fir/1 t _! names are ( m unc]ta9 signification1 ! precise collection ( simple ideas !y /& = n be+ s easily agre$ on2 & s ! sign t is us$ = !m 9 communication always1 & 9 ?9k+ (t51 does n /eadily c>ry ) x ! same idea4 ,^u :i* ! same disord]1 confusion1 & error foll[1 z w\ld if a man1 go+ to demon/rate some?+ ( an heptagon1 %\ld1 9 ! diagram he took to d x1 leave out "o ( ! angles1 or by ov]sight make ! figure ) an angle m ?an ! "n ord9>ily import$1 or he int5d$ x %\ld :5 at fir/ he ?\ght ( 8 demon/ration4 ,?is (t5 happ5s1 & is h>dly avoidable 9 v complex moral ideas1 :]e ! same "n be+ reta9$1 an angle1 i4e4 "o simple idea is left out1 or put 9 ! complex "o 7/ill call$ by ! same "n7 m at "o "t ?an ano!r4 ,secondly1 ,f ! complex$ness ( ^! moral ideas !re foll[s ano!r inconv5i5ce1 viz41 t ! m9d _c easily reta9 ^? precise comb9ations s exactly & p]fectly z is necess>y 9 ! exam9ation ( ! habitudes & correspond5ces1 agreem5ts or disagreem5ts1 ( sev]al ( !m "o ) ano!r2 especially :]e x is to be judg$ ( by long d$uctions & ! int]v5tion ( sev]al o!r complex ideas to %[ ! agreem5t or disagreem5t ( two remote ones4 --,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #d1 *ap4 #c1 sect4 #ai4 #adda4 ,x has be5 g5]ally tak5 = grant$1 t ma!matics alone are capable ( demon/rative c]ta9ty3 b to h su* an agreem5t or disagreem5t z may be intuitively p]ceiv$1 be+1 z ,i imag9e1 n ! privileges ( ! ideas ( numb]1 ext5sion1 & figure alone1 x may possibly be ! want ( due me?od & application 9 u1 & n ( suffici5t evid5ce 9 ?+s1 t demon/ration has be5 ?\ght to h s little to d 9 o!r p>ts ( k1 & be5 sc>ce s mu* z aim$ at by any b ma!maticians4 ,= :atev] ideas we h :]e9 ! m9d c p]ceive ! imm$iate agreem5t or disagreem5t t is betwe5 !m1 !re ! m9d is capable ( intuitive k1 & :]e x c p]ceive ! agreem5t or disagreem5t ( any two ideas1 by an intuitive p]ception ( ! agreem5t or disagreem5t !y h ) any int]m$iate ideas1 !re ! m9d is capable ( demon/ration3 :i* is n limit$ to ! idea ( ext5sion1 figure1 numb]1 & _! modes4--,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #d1 *ap4 #b1 sect4 #i4 #addb4 ,n[ ,i %all rem>k aga9 :at ,i h already t\*$ ^u m ?an once1 t x is a common op9ion t only ma!matical sci5ces are capable ( a demon/rative c]ta9ty2 b z ! agreem5t & disagreem5t :i* may be kn[n intuitively is n a privilege belong+ only to ! ideas ( numb]s & figures1 x is p]haps = want ( application on our "p t ma!matics alone h atta9$ to demon/rations4 --,,leibnitz4 ,new ,essay conc]n+ ,human ,und]/&+1 ,bk4 #d1 *ap4 #b1 sect4 #i ,langley4 ,,*apt] ,,xv ,,ma!matics ,,& ,,sci5ce #aeja4 ,h[ comes x ab\t t ! k ( o!r sci5ces1 :i* dep5d ^u ?is ma!matics1 is pa9fully s\ght1 & t no "o puts himself to ! tr\ble ( /udy+ ?is sci5ce itself8 ,i %\ld c]ta9ly be surpris$1 if ,i did n "k t ev]ybody reg>d$ x z be+ v easy1 & if ,i _h n long ago obs]v$ t ! human m9d1 neglect+ :at x believes to be easy1 is always 9 ha/e to run aft] :at is novel & advanc$4--,,desc>tes4 ,rules = ! ,direction ( ! ,m9d2 ,philosophy ( ,desc>tes ,torrey1 7,new ,york1 #ahib71 p4 #gb4 #aejb4 ,all quantitative det]m9ations are 9 ! h&s ( ma!matics1 & x at once foll[s f ?is t all speculation :i* is he$less ( ma!matics1 :i* does n ent] into p>tn]%ip ) x1 :i* does n seek its aid 9 di/+ui%+ betwe5 ! manifold modifications t mu/ ( necessity arise by a *ange ( quantitative det]m9ations1 is ei!r an empty play ( ?\ghts1 or at mo/ a fruitless ef=t4 ,9 ! field ( speculation _m ?+s gr[ :i* d n />t f ma!matics nor give x any c>e1 & ,i am f> f ass]t+ t all t ?us gr[ are useless we$s1 among !m may be _m noble plants1 b )\t ma!matics none w develop to complete maturity4--,,h]b>t1 ,j4 ,f4 ,w]ke 7,kehrba*71 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #ajf4 #aejc4 ,!re are few ?+s :i* we "k1 :i* are n capable ( be+ r$uc'd to a ,ma!matical ,reason+1 & :5 !y _c1 x's a sign our k ( !m is v small & confus'd2 & :]e a ma!matical reason+ c be _h1 x's z great folly to make use ( any o!r1 z to grope = a ?+ 9 ! d>k1 :5 y h a c&le /&+ by y4--,,arbu?not4 ,quot$ 9 ,todhunt]'s ,hi/ory ( ! ,!ory ( ,probability 7,cambridge & ,london1 #ahfe71 p4 #ea4 #aejd4 ,ma!matical ,analysis is 444 ! true rational basis ( ! :ole sy/em ( our positive k4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #a1 *ap4 #a4 #aeje4 ,x is only ?r\gh ,ma!matics t we c ?or\ghly und]/& :at true sci5ce is4 ,"h alone we c f9d 9 ! highe/ degree simplicity & sev]ity ( sci5tific law1 & su* ab/raction z ! human m9d c atta94 ,any sci5tific education sett+ =? f any o!r po9t1 is faulty 9 its basis4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #a1 *ap4 #a4 #aejf4 ,9 ! pres5t /ate ( our k we mu/ reg>d ,ma!matics less z a con/itu5t "p ( natural philosophy ?an z hav+ be51 s9ce ! "t ( ,desc>tes & ,newton1 ! true basis ( ! :ole ( natural philosophy2 ?\gh x is1 exactly speak+1 bo? ! "o & ! o!r4 ,to u x is ( less use = ! k ( :i* x consi/s1 sub/antial & valuable z t k is1 ?an z be+ ! mo/ p[]ful in/rum5t t ! human m9d c employ 9 ! inve/igation ( ! laws ( natural ph5om5a4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,introduction1 *ap4 #b4 #aejg4 ,! concept ( ma!matics is ! concept ( sci5ce 9 g5]al4--,,novalis4 ,s*rift5 7,b]l91 #aija71 ,teil #b1 p4 #bbb4 #aejh4 ,i cont5d1 t ea* natural sci5ce is real sci5ce only 9 s f> z x is ma!matical4444 ,x may be t a pure philosophy ( nature 9 g5]al 7t is1 a philosophy :i* conc]ns itself only ) ! g5]al concepts ( nature7 is possible )\t ma!matics1 b a pure sci5ce ( nature deal+ ) def9ite objects 7physics or psy*ology71 is possible only by means ( ma!matics1 & s9ce ea* natural sci5ce conta9s only z mu* real sci5ce z x conta9s a priori k1 ea* natural sci5ce becomes real sci5ce only to ! ext5t t x p]mits ! application ( ma!matics4 --,,kant1 ,e4 ,metaphysis*e ,anfangsgrnde d] ,naturwiss5s*aft1 ,vorr$e4 #aeji4 ,! !ory mo/ preval5t among tea*]s is t ma!matics af=ds ! be/ tra9+ = ! reason+ p[]s2444 ,! mod]n1 & to my m9d true1 !ory is t ma!matics is ! ab/ract =m ( ! natural sci5ces2 & t x is valuable z a tra9+ ( ! reason+ p[]s1 n because x is ab/ract1 b because x is a repres5tation ( actual ?+s4--,,saf=d1 ,t4 ,h4 ,ma!matical ,tea*+ etc4 7,bo/on1 #ahhf71 p4 #i4 #aeaj4 ,x seems to me t no "o sci5ce c s well s]ve to co-ord9ate &1 z x 71 b9d toge!r all ( ! sci5ces z ! que5 ( !m all1 ma!matics4--,,davis1 ,e4 ,w4 ,proce$+s ,nebraska ,academy ( ,sci5ces = #ahif 7,l9coln1 #ahig71 p4 #bhb4 #aeaa4 ,& z = ,mix$ ,ma!matics1 ,i may only make ?is pr$iction1 t !re _c fail to be m k9ds ( !m1 z nature gr[s fur!r disclos$4--,,bacon1 ,,francis4 ,advancem5t ( ,le>n+1 ,bk4 #b2 ,de ,augm5tis1 ,bk4 #c4 #aeab4 ,besides ! ex]cise 9 ke5 compreh5sion & ! c]ta9 discov]y ( tru?1 ma!matics has ano!r =mative function1 t ( equipp+ ! m9d = ! survey ( a sci5tific sy/em4--,,grassmann1 ,h4 ,/cke aus dem ,lehrbu*e d] ,ari?metik2 ,w]ke 7,leipzig1 #aijd71 ,bd4 #b1 p4 #bih4 #aeac4 ,ma!maticks may help ! naturali/s1 bo? to frame hypo!ses1 & to judge ( ^? t are propos$ to !m1 especially su* z relate to ma!matical subjects 9 conjunction ) o!rs4--,,boyle1 ,,rob]t4 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbi4 #aead4 ,! m progress physical sci5ces make1 ! m !y t5d to ent] ! doma9 ( ma!matics1 :i* is a k9d ( c5tre to :i* !y all conv]ge4 ,we may ev5 judge ( ! degree ( p]fection to :i* a sci5ce has arriv$ by ! facility ) :i* x may be submitt$ to calculation4--,,quetelet4 ,quot$ 9 ,e4 ,mailly's ,eulogy on ,quetelet2 ,smi?sonian ,report1 #ahgd1 p4 #agc4 #aeae4 ,! ma!matical =mula is ! po9t ?r\gh :i* all ! light ga9$ by sci5ce passes 9 ord] to be ( use to practice2 x is also ! po9t 9 :i* all k ga9$ by practice1 exp]im5t1 & obs]vation mu/ be conc5trat$ be=e x c be sci5tifically grasp$4 ,! m di/ant & m>k$ ! po9t1 ! m conc5trat$ w be ! light com+ f x1 ! m unmi/akable ! insight convey$4 ,all sci5tific ?\ght1 f ! simple gravitation =mula ( ,newton1 ?r\gh ! m complicat$ =mulae ( physics & *emi/ry1 ! vagu] s call$ laws ( organic & animat$ nature1 d[n to ! unc]ta9 /atem5ts ( psy*ology & ! data ( our social & hi/orical k1 alike p>takes ( ?is *>act]i/ic1 t x is an attempt to ga!r up ! scatt]$ rays ( light1 ! diff]5t p>ts ( k1 9 a focus1 f :5ce x c be aga9 spread out & analyz$1 accord+ to ! ab/ract processes ( ! ?9k+ m9d4 ,b only :5 ?is c be done ) a ma!matical precision & accuracy is ! image %>p & well-def9$1 & ! d$uctions cle> & unmi/akable4 ,z we desc5d f ! me*anical1 ?r\gh ! physical1 *emical1 & biological1 to ! m5tal1 moral1 & social sci5ces1 ! process ( focalization becomes less & less p]fect1--! %>p po9t1 ! focus1 is replac$ by a l>g] or small] circle1 ! cont\rs ( ! image become less & less di/9ct1 & ) ! possible light :i* we ga9 !re is m+l$ mu* d>kness1 ! s\rces ( _m mi/akes & errors4 ,b ! t5d5cy ( all sci5tific ?\ght is t[>d cle>] & cle>] def9ition2 x lies 9 ! direction ( a m & m ext5d$ use ( ma!matical measurem5ts1 ( ma!matical =mulae4--,,m]z1 ,j4 ,t4 ,hi/ory ( ,european ,?\ght 9 ! #ai? ,c5tury 7,ed9burgh & ,london1 #aijd71 ,vol4 #a1 p4 #ccc4 #aeaf4 ,f ! v outset ( 8 inve/igations ! physici/ has to rely con/antly on ! aid ( ! ma!matician1 = ev5 9 ! simple/ cases1 ! direct results ( 8 measur+ op]ations are entirely )\t mean+ until !y h be5 submitt$ to m or less ( ma!matical discussion4 ,& :5 9 ?is way "s int]pretation ( ! exp]im5tal results has be5 arriv$ at1 & x has be5 prov$ t two or m physical quantities /& 9 a def9ite relation to ea* o!r1 ! ma!matician is v (t5 able to inf]1 f ! exi/5ce ( ?is relation1 t ! quantities 9 "q also fulfill "s o!r relation1 t 0 previ\sly unsuspect$4 ,?us :5 ,c\lomb1 comb9+ ! functions ( exp]im5tali/ & ma!matician1 _h discov]$ ! law ( ! =ce ex]t$ betwe5 two p>ticles ( electricity1 x became a purely ma!matical problem1 n requir+ any fur!r exp]im5t1 to asc]ta9 h[ electricity is di/ribut$ ^u a *>g$ conductor & ?is problem has be5 solv$ by ma!maticians 9 sev]al cases4--,,fo/]1 ,g4 ,c4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a 7#ahgg72 ,nature1 ,vol4 #af1 p4 #cab-cac4 #aeag4 ,)\t consummate ma!matical skill1 on ! "p ( "s inve/igators at any rate1 all ! high] physical problems w\ld be seal$ to u2 & )\t compet5t skill on ! "p ( ! ord9>y /ud5t no idea c be =m$ ( ! nature & cog5cy ( ! evid5ce on :i* ! solutions re/4 ,ma!matics are n m]ely a gate ?r\gh :i* we may approa* if we please1 b !y are ! only mode ( approa* to l>ge & important di/ricts ( ?\ght4--,,v5n1 ,,john4 ,symbolic ,logic 7,london & ,new ,york1 #ahid71 ,introduction1 p4 xix4 #aeah4 ,mu* ( ! skill ( ! true ma!matical physici/ & ( ! ma!matical a/ronom] consi/s 9 ! p[] ( adapt+ me?ods & results c>ri$ out on an exact ma!matical basis to obta9 approximations suffici5t = ! purposes ( physical measurem5ts4 ,x might p]haps be ?\ght t a s*eme ( ,ma!matics on a frankly approximative basis w\ld be suffici5t = all ! practical purposes ( application 9 ,physics1 ,eng9e]+ ,sci5ce1 & ,a/ronomy1 & no d\bt x w\ld be possible to develop1 to "s ext5t at lea/1 a species ( ,ma!matics on ^! l9es4 ,su* a sy/em w\ld1 h[ev]1 involve an intol]able awkw>dness & prolixity 9 ! /atem5ts ( results1 especially 9 view ( ! fact t ! degree ( approximation necess>y = v>i\s purposes is v diff]5t1 & ?us t unassign$ grades ( approximation w\ld h to be provid$ =4 ,moreov]1 ! ma!matician work+ on ^! l9es w\ld be cut (f f ! *ief s\rces ( inspiration1 ! ideals ( exactitude & logical rig\r1 z well z f "o ( 8 mo/ indisp5sable guides to discov]y1 symmetry1 & p]man5ce ( ma!matical =m4 ,! hi/ory ( ! actual movem5ts ( ma!matical ?\ght ?r\gh ! c5turies %[s t ^! ideals are ! v life-blood ( ! sci5ce1 & w>rants ! conclusion t a con/ant /riv+ t[>d _! atta9m5t is an absolutely ess5tial condition ( vigor\s gr[?4 ,^! ideals h _! roots 9 irresi/ible impulses & deep-seat$ ne$s ( ! human m9d1 manife/$ 9 its ef=ts to introduce intelligibility 9 c]ta9 great doma9s ( ! _w ( ?\ght4--,,hobson1 ,e4 ,w4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a 7#aiaj72 ,nature1 ,vol4 #hd1 pp4 #bhe-bhf4 #aeai4 ,! imm5se "p :i* ^? laws laws ( numb] & ext5sion take 9 giv+ a d$uctive "* to ! o!r dep>tm5ts ( physical sci5ce1 is well kn[n2 & is n surpris+1 :5 we consid] t all causes op]ate accord+ to ma!matical laws4 ,! effect is always dep5d5t ^u1 or 9 ma!matical language1 is a function (1 ! quantity ( ! ag5t2 & g5]ally ( its position also4 ,we _c1 !re=e1 reason respect+ causation1 )\t introduc+ consid]ations ( quantity & ext5sion at e /ep2 & if ! nature ( ! ph5om5a admits ( our obta9+ num]ical data ( suffici5t accuracy1 ! laws ( quantity become ! gr& in/rum5ts = calculat+ =w>d to an effect1 or backw>d to a cause4 --,,mill1 ,j4 ,s4 ,sy/em ( ,logic1 ,bk4 #c1 *ap4 #bd1 sect4 #i4 #aebj4 ,! ord9>y ma!matical treatm5t ( any appli$ sci5ce sub/itutes exact axioms = ! approximate results ( exp]i5ce1 & d$uces f ^! axioms ! rigid ma!matical conclusions4 ,9 apply+ ?is me?od x mu/ n be =gott5 t ! ma!matical developm5ts transc5d+ ! limits ( exactness ( ! sci5ce are ( no practical value4 ,x foll[s t a l>ge portion ( ab/ract ma!matics rema9s )\t f9d+ any practical application1 ! am\nt ( ma!matics t c be usefully employ$ 9 any sci5ce be+ 9 proportion to ! degree ( accuracy atta9$ 9 ! sci5ce4 ,?us1 :ile ! a/ronom] c put to use a wide range ( ma!matical !ory1 ! *emi/ is only j beg9n+ to apply ! fir/ d]ivative1 i4e4 ! rate ( *ange at :i* c]ta9 processes are go+ on2 = second d]ivatives he does n seem to h f\nd any use z yet4--,,kle91 ,f4 ,lectures on ,ma!matics 7,new ,york1 #aiaa71 p4 #dg4 #aeba4 ,! bond ( union among ! physical sci5ces is ! ma!matical _s & ! ma!matical me?od :i* p]vades !m4444 ,our k ( nature1 z x advances1 cont9u\sly resolves diff]5ces ( quality into diff]5ces ( quantity4 ,all exact reason+--inde$ all reason+--ab\t quantity is ma!matical reason+2 & ?us z our k increases1 t portion ( x :i* becomes ma!matical increases at a /ill m rapid rate4--,,smi?1 ,h4 ,j4 ,s4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a 7#ahgc72 ,nature1 ,vol4 #h1 p4 #ddi4 #aebb4 ,ano!r way ( conv9c+ ourselves h[ l>gely ?is process ( assimilation ( ma!matics by physics has gone on w\ld be to try to conceive ! effect ( "s intellectual cata/rophe1 suppos+ su* a ?+ possible1 :]eby all k ( ma!matics %\ld be swept away f m5's m9ds4 ,w\ld x n be t ! dep>ture ( ma!matics w\ld be ! de/ruction ( physics8 ,objective physical ph5om5a w\ld1 inde$1 rema9 z !y are n[1 b physical sci5ce w\ld cease to exi/4 ,we %\ld no d\bt see ! same col\rs on look+ into a spectroscope or pol>iscope1 vibrat+ /r+s w\ld produce ! same s\nds1 electrical ma*9es w\ld give sp>ks1 & galvanomet] ne$les w\ld be deflect$2 b all ^! ?+s w\ld h lo/ _! mean+2 !y w\ld be b z ! dry bones--! disjecta membra--( :at is n[ a liv+ & gr[+ sci5ce4 ,to foll[ ?is conception fur!r1 & to try to image to ourselves 9 "s detail :at w\ld be ! k9d ( k ( physics :i* w\ld rema9 possible1 suppos+ all ma!matical ideas to be blott$ out1 w\ld be extremely int]e/+1 b x w\ld lead u directly into a dim & entangl$ region :]e ! subjective seems to be always pass+ itself (f = ! objective1 & :]e ,i at lea/ c\ld n attempt to lead ! way1 gladly z ,i w\ld foll[ any "o :o c\ld %[ :]e a firm foot+ is to be f\nd4 ,b )\t v5tur+ to d m ?an to look f a safe di/ance ov] ?is puzzl+ gr\nd1 we may see cle>ly en\gh t ma!matics is ! connective tissue ( physics1 b9d+ :at w\ld else be m]ely a li/ ( deta*$ obs]vations into an organiz$ body ( sci5ce4--,,fo/]1 ,g4 ,c4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a 7#ahgg72 ,nature1 ,vol4 #af1 p4 #cac4 #aebc4 ,9 ,plato's "t ma!matics 0 purely a play ( ! free intellect2 ! ma!matic-my/ical rev]ies ( a ,py?agoras =e%ad[$ a f>-rea*+ significance1 b su* a significance 7except 9 ! case ( music7 0 z yet entirely a matt] ( fancy2 yet ev5 9 t "t ma!matics 0 ! pr]equisite to all o!r /udies6 ,b today1 :5 ma!matics furni%es ! only language by means ( :i* we may =mulate ! mo/ compreh5sive laws ( nature1 laws :i* ! anci5ts sc>cely dream$ (1 :5 moreov] ma!matics is ! only means by :i* ^! laws may be und]/ood1--h[ few le>n today any?+ ( ! real ess5ce ( our ma!matics6444 ,9 ! s*ools ( today ma!matics s]ves only z a discipl9>y /udy1 a m5tal gymna/ic2 t x includes ! highe/ ideal value = ! compreh5sion ( ! univ]se1 "o d>es sc>cely to ?9k ( 9 view ( our pres5t day in/ruction4--,,l9deman1 ,f4 ,lehr5 und ,l]n5 9 d] ,ma!matik 7,mn*51 #aijd71 p4 #ad4 #aebd4 ,all applications ( ma!matics consi/ 9 ext5d+ ! empirical k :i* we possess ( a limit$ numb] or region ( accessible ph5om5a into ! region ( ! unkn[n & inaccessible2 & mu* ( ! progress ( pure analysis consi/s 9 inv5t+ def9ite conceptions1 m>k$ by symbols1 ( complicat$ op]ations2 9 asc]ta9+ _! prop]ties z indep5d5t objects ( rese>*2 & 9 ext5d+ _! mean+ beyond ! limits !y 7 orig9ally inv5t$ =1--?us op5+ out new & l>g] regions ( ?\ght4--,,m]z1 ,j4 ,t4 ,hi/ory ( ,european ,?\ght 9 ! #ai? ,c5tury 7,ed9burgh & ,london1 #aijc71 ,vol4 #a1 p4 #fih4 #aebe4 ,all ! effects ( nature are only ma!matical results ( a small numb] ( immutable laws4--,,laplace4 ,a ,philosophical ,essay on ,probabilities ,truscott & ,emory 7,new ,york1 #aijb71 p4 #agg2 ,oeuvres1 t4 #g1 p4 #aci4 #aebf4 ,:at log>i?ms are to ma!matics t ma!matics are to ! o!r sci5ces4--,,novalis4 ,s*rift5 7,b]l91 #aija71 ,teil #b1 p4 #bbb4 #aebg4 ,any intellig5t man may n[1 by resolutely apply+ himself = a few ye>s to ma!matics1 le>n m ?an ! great ,newton knew aft] half a c5tury ( /udy & m$itation4 --,,macaulay4 ,milton2 ,critical & ,miscellane\s ,essays 7,new ,york1 #ahgi71 ,vol4 #a1 p4 #ac4 #aebh4 ,9 que/ions ( sci5ce ! au?ority ( a ?\s& is n wor? ! humble reason+ ( a s+le individual4--,,galileo4 ,quot$ 9 ,arago's ,eulogy on ,laplace2 ,smi?sonian ,report1 #ahgd1 p4 #afd4 #aebi4 ,beh9d ! artisan is ! *emi/1 beh9d ! *emi/ a physici/1 beh9d ! physici/ a ma!matician4--,,:ite1 ,w4 ,f4 ,scrap-book ( ,elem5t>y ,ma!matics 7,*icago1 #aijh71 p4 #bag4 #aecj4 ,! advance 9 our k ( physics is l>gely due to ! application to x ( ma!matics1 & e ye> x becomes m difficult = an exp]im5t] to make any m>k 9 ! subject unless he is also a ma!matician4--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #ejc4 #aeca4 ,9 v _m cases ! mo/ obvi\s & direct exp]im5tal me?od ( inve/igat+ a giv5 problem is extremely difficult1 or = "s reason or o!r untru/wor?y4 ,9 su* cases ! ma!matician c (t5 po9t out "s o!r problem m accessible to exp]im5tal treatm5t1 ! solution ( :i* involves ! solution ( ! =m] "o4 ,= example1 if we try to d$uce f direct exp]im5ts ! law accord+ to :i* "o pole ( a magnet attracts or repels a pole ( ano!r magnet1 ! obs]v$ action is s mu* complicat$ ) ! effects ( ! mutual induction ( ! magnets & ( ! =ces due to ! second pole ( ea* magnet1 t x is next to impossible to obta9 results ( any great accuracy4 ,gauss1 h[ev]1 %[$ h[ ! law :i* appli$ 9 ! case m5tion$ c be d$uc$ f ! deflections und]gone by a small susp5d$ magnetic ne$le :5 x is act$ ^u by a small fix$ magnet plac$ successively 9 two det]m9ate positions relatively to ! ne$le2 & be+ an exp]im5tali/ z well z a ma!matician1 he %[$ likewise h[ ^! deflections c be measur$ v easily & ) great precision4--,,fo/]1 ,g4 ,c4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a 7#ahgg72 ,nature1 ,vol4 #af1 p4 #cac4 #aecb4 ,give me to le>n ea* secret cause2 ,let ,numb]'s1 ,figure's1 ,motion's laws ,reveal'd be=e me /&2 ,^! to great ,nature's sc5es apply1 ,& r\nd ! globe1 & ?r\gh ! sky1 ,disclose h] work+ h&4 --,,ak5side1 ,m4 ,hymn to ,sci5ce4 #aecc4 ,n[ !re are sev]al scores1 ^u :i* skill 9 ma!maticks may be useful to ! exp]im5tal philosoph]4 ,= !re are "s g5]al advantages1 :i* ma!maticks may br+ to ! m9ds ( m51 to :atev] /udy !y apply !mselves1 & consequ5tly to ! /ud5t ( natural philosophy2 namely1 t ^! discipl9es are wont to make m5 accurate1 & v att5tive to ! employm5t t !y are ab\t1 keep+ _! ?\ghts f w&]+1 & inur+ !m to pati5ce 9 go+ ?r\gh ) t$i\s & intricate demon/rations2 besides1 t !y mu* improve reason1 by accu/om+ ! m9d to d$uce successive consequ5ces1 & judge ( !m )\t easily acquiesc+ 9 any?+ b demon/ration4--,,boyle1 ,,rob]t4 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbf4 #aecd4 ,x is n easy to anatomize ! con/itution & ! op]ations ( a m9d l ,newton's :i* makes su* an advance 9 k4 ,yet we may obs]ve t !re mu/ exi/ 9 x1 9 an em95t degree1 ! elem5ts :i* compose ! ma!matical tal5t4 ,x mu/ possess di/9ctness ( intuition1 t5acity & facility 9 trac+ logical connection1 f]tility ( inv5tion1 & a /rong t5d5cy to g5]alization4--,,:ewell1 ,w4 ,hi/ory ( ! ,inductive ,sci5ces 7,new ,york1 #ahid71 ,vol4 #a1 p4 #daf4 #aece4 ,! doma9 ( physics is no prop] field = ma!matical pa/imes4 ,! be/ security w\ld be 9 giv+ a geometrical tra9+ to physici/s1 :o ne$ n !n h rec\rse to ma!maticians1 ^: t5d5cy is to despise exp]im5tal sci5ce4 ,by ?is me?od w t union betwe5 ! ab/ract & ! concrete be effect$ :i* w p]fect ! uses ( ma!matical1 :ile ext5d+ ! positive value ( physical sci5ce4 ,meantime1 ! use ( analysis 9 physics is cle> en\gh4 ,)\t x we %\ld h no precision1 & no co-ord9ation2 & :at acc\nt c\ld we give ( our /udy ( heat1 weight1 light1 etc48 ,we %\ld h m]ely s]ies ( unconnect$ facts1 9 :i* we c\ld =esee no?+ b by con/ant rec\rse to exp]im5t2 :]eas1 !y n[ h a "* ( rationality :i* fits !m = purposes ( prevision4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #c1 *ap4 #a4 #aecf4 ,x mu/ ev] be rememb]$ t ! true positive _s fir/ came =? f ! pure s\rces ( ma!matical sci5ce2 & x is only ! m9d t has imbib$ x !re1 & :i* has be5 face to face ) ! lucid tru?s ( geometry & me*anics1 t c br+ into full action its natural positivity1 & apply x 9 br++ ! mo/ complex /udies into ! reality ( demon/ration4 ,no o!r discipl9e c fitly prep>e ! intellectual organ4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #c1 *ap4 #a4 #aecg4 ,dur+ ! la/ two c5turies & a half1 physical k has be5 gradually made to re/ ^u a basis :i* x _h n be=e4 ,x has become ma!matical4 ,! "q n[ is1 n :e!r ?is or t hypo!sis is bett] or worse to ! pure ?\ght1 b :e!r x accords ) obs]v$ ph5om5a 9 ^? consequ5ces :i* c be %[n necess>ily to foll[ f x1 if x be true4 ,ev5 9 ^? sci5ces :i* are n yet "u ! dom9ion ( ma!matics1 & p]haps nev] w be1 a work+ copy ( ! ma!matical process has be5 made4 ,?is is n kn[n to ! foll[]s ( ^? sci5ces :o are n !mselves ma!maticians1 & :o v (t5 exalt _! horns aga9/ ! ma!matics 9 consequ5ce4 ,!y might z well be squ>+ ! circle1 = any s5se !y %[ 9 ?is p>ticul>4--,,de ,,morgan1 ,a4 ,a ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #b4 #aech4 ,among ! m]e talk]s s f> z ma!matics are conc]n$1 are to be rank$ ?ree out ( f\r ( ^? :o apply ma!matics to physics1 :o1 want+ a tool only1 are v impati5t ( ev]y?+ :i* is n ( direct aid to ! actual me?ods :i* are 9 _! h&s4--,,de ,,morgan1 ,a4 ,graves' ,life ( ,sir ,william ,r[an ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #cdh4 #aeci4 ,some?+ has be5 said ab\t ! use ( ma!matics 9 physical sci5ce1 ! ma!matics be+ reg>d$ z a weapon =g$ by o!rs1 & ! /udy ( ! weapon be+ completely set aside4 ,i c only say t !re is dang] ( obta9+ untru/wor?y results 9 physical sci5ce1 if only ! results ( ma!matics are us$2 = ! p]son s us+ ! weapon c rema9 unacqua9t$ ) ! conditions "u :i* x c be rightly appli$4444 ,! results are (t5 correct1 sometimes are incorrect2 ! consequ5ce ( ! latt] class ( cases is to ?r[ d\bt ^u all ! applications ( su* a work] until a result has be5 o!rwise te/$4 ,moreov]1 su* a practice 9 ! use ( ma!matics leads a work] to a m]e repetition 9 ! use ( famili> weapons2 he is unable to adapt !m ) any confid5ce :5 "s new set ( conditions arise ) a dem& = a new me?od3 = want ( adequate in/ruction 9 ! =g+ ( ! weapon1 he may f9d himself1 soon] or lat] 9 ! progress ( 8 subject1 )\t any weapon wor? hav+4 --,,=sy?1 ,a4 ,r4 ,p]ry's ,tea*+ ( ,ma!matics 7,london1 #aijb71 p4 #cf4 #aedj4 ,if 9 ! range ( human endeavor aft] s\nd k !re is "o subject t ne$s to be practical1 x surely is ,m$ic9e4 ,yet 9 ! field ( ,m$ic9e x has be5 f\nd t bran*es su* z biology & pa?ology mu/ be /udi$ = !mselves & be develop$ by !mselves ) ! s+le aim ( increas+ k2 & x is !n t !y c be be/ appli$ to ! conduct ( liv+ processes4 ,s also 9 ! pursuit ( ma!matics1 ! pa? ( practical utility is too n>r[ & irregul>1 n always lead+ f>4 ,! witness ( hi/ory %[s t1 9 ! field ( natural philosophy1 ma!matics w furni% ! m effective assi/ance if1 9 its sy/ematic developm5t1 its c\rse c freely pass beyond ! ev]-%ift+ doma9 ( use & application4--,,=sy?1 ,a4 ,r4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a2 ,nature1 ,vol4 #ef 7#ahig71 p4 #cgg4 #aeda4 ,if ! ,greeks _h n cultivat$ ,conic ,sections1 ,kepl] c\ld n h sup]s$$ ,ptolemy2 if ! ,greeks _h cultivat$ ,dynamics1 ,kepl] might h anticipat$ ,newton4--,,:ewell1 ,w4 ,hi/ory ( ! ,inductive ,sci5ce 7,new ,york1 #ahid71 ,vol4 #a1 p4 #caa4 #aedb4 ,if we may use ! great names ( ,kepl] & ,newton to signify /ages 9 ! progress ( human discov]y1 x is n too mu* to say t )\t ! treatises ( ! ,greek geomet]s on ! conic sections !re c\ld h be5 no ,kepl]1 )\t ,kepl] no ,newton1 & )\t ,newton no sci5ce 9 ! mod]n s5se ( ! t]m1 or at lea/ no su* conception ( nature z n[ lies at ! basis ( all our sci5ce1 ( nature z subject 9 ! smalle/ z well z 9 its greate/ ph5om5a1 to exact quantitative relations1 & to def9ite num]ical laws4 --,,smi?1 ,h4 ,j4 ,s4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a2 ,nature1 ,vol4 #h 7#ahgc71 p4 #dej4 #aedc4 ,! sil5t "w ( ! great ,regiomontanus 9 8 *amb] at ,nuremb]g comput$ ! ,ephem]ides :i* made possible ! discov]y ( ,am]ica by ,columbus4--,,rudio1 ,f4 ,quot$ 9 ,max ,simon's ,ges*i*te d] ,ma!matik im ,alt]tum 7,b]l91 #aiji71 ,e9leitung1 p4 xi4 #aedd4 ,! calculation ( ! eclipses ( ,jupit]'s satellites1 _m a man might h be5 dispos$1 orig9ally1 to reg>d z a mo/ unpr(itable /udy4 ,b ! utility ( x to navigation 79 ! det]m9ation ( longitudes7 is n[ well kn[n4--,,:ately1 ,r4 ,annotations to ,bacon's ,essays 7,bo/on1 #aghc71 p4 #dib4 #aede4 ,:o c\ld h imag9$1 :5 ,galvani obs]v$ ! twit*+ ( ! frog muscles z he br\ght v>i\s metals 9 contact ) !m1 t eighty ye>s lat] ,europe w\ld be ov]spun ) wires :i* transmit messages f ,madrid to ,/4 ,pet]sburg ) ! rapidity ( lightn+1 by means ( ! same pr9ciple ^: fir/ manife/ations ?is anatomi/ !n obs]v$6444 ,he :o seeks = imm$iate practical use 9 ! pursuit ( sci5ce1 may be reasonably sure1 t he w seek 9 va94 ,complete k & complete und]/&+ ( ! action ( =ces ( nature & ( ! m9d1 is ! only ?+ t sci5ce c aim at4 ,! individual inve/igator mu/ f9d 8 rew>d 9 ! joy ( new discov]ies1 z new victories ( ?\ght ov] resi/+ matt]1 9 ! e/hetic beauty :i* a well-ord]$ doma9 ( k af=ds1 :]e all p>ts are intellectually relat$1 :]e "o ?+ evolves f ano!r1 & all %[ ! m>ks ( ! m9d's supremacy2 he mu/ f9d 8 rew>d 9 ! consci\sness ( hav+ contribut$ to ! gr[+ capital ( k on :i* dep5ds ! supremacy ( man ov] ! =ces ho/ile to ! _s4--,,helmholtz1 ,h4 ,vortrge und ,r$5 7,brauns*weig1 #ahhd71 ,bd4 #a1 p4 #adb4 #aedf4 ,:5 ! "t comes t k w n be s\ght = its own sake1 & m5 w n press =w>d simply 9 a desire ( a*ievem5t1 )\t hope ( ga91 to ext5d ! limits ( human k & in=mation1 !n1 inde$1 w ! race ent] ^u its decad5ce4--,,hughes1 ,c4 ,e4 ,quot$ 9 ,d4 ,e4 ,smi?'s ,tea*+ ( ,geometry 7,bo/on1 #aiaa71 p4 #i4 #aedg4 ,9 ! ,opus ,majus ( ,rog] ,bacon !re is a *apt]1 9 :i* x is prov$ by reason1 t all sci5ces require ma!matics4 ,& ! argum5ts :i* are us$ to e/abli% ?is doctr9e1 %[ a mo/ j appreciation ( ! (fice ( ma!matics 9 sci5ce4 ,!y are su* z foll[s3 ,t o!r sci5ces use examples tak5 f ma!matics z ! mo/ evid5t3--,t ma!matical k is1 z x 71 innate to u1 on :i* po9t he ref]s to ! well-kn[n dialogue ( ,plato1 z quot$ by ,cic]o3--,t ?is sci5ce1 be+ ! easie/1 (f]s ! be/ introduction to ! m difficult3--,t 9 ma!matics1 ?+s z kn[n to u are id5tical ) ?+s z kn[n to nature3--,t we c "h entirely avoid d\bt & error1 & obta9 c]ta9ty & tru?3--,t ma!matics is prior to o!r sci5ces 9 nature1 because x takes cognizance ( quantity1 :i* is appreh5d$ by intuition 7intuitu intellectus74 8,moreov]10 he adds1 8!re h be5 f\nd fam\s m51 z ,rob]t1 bi%op ( ,l9coln1 & ,bro!r ,adam ,m>%man 7de ,m>isco71 & _m o!rs1 :o by ! p[] ( ma!matics h be5 able to expla9 ! causes ( ?+s2 z may be se5 9 ! writ+s ( ^! m51 = in/ance1 conc]n+ ! ,ra9b[ & ,comets1 & ! g5]ation ( heat1 & climates1 & ! cele/ial bodies40--,,:ewell1 ,w4 ,hi/ory ( ! ,inductive ,sci5ces 7,new ,york1 #ahid71 ,vol4 #a1 p4 #eai4 ,bacon1 ,rog]3 ,opus ,majus1 ,"p #d1 ,di/9ctia ,prima1 cap4 #c4 #aedh4 ,! analysis :i* is bas$ ^u ! conception ( function discloses to ! a/ronom] & physici/ n m]ely ! =mulae = ! computation ( :atev] desir$ di/ances1 times1 velocities1 physical con/ants2 x moreov] gives him insight into ! laws ( ! processes ( motion1 tea*es him to pr$ict future occurr5ces f pa/ exp]i5ces & supplies him ) means to a sci5tific k ( nature1 i4e4 x enables him to trace back :ole gr\ps ( v>i\s1 sometimes extremely het]og5e\s1 ph5om5a to a m9imum ( simple fundam5tal laws4 --,,pr+%eim1 ,a4 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #ac1 p4 #cff4 #aedi4 8,z is kn[n1 sci5tific physics dates its exi/5ce f ! discov]y ( ! diff]5tial calculus4 ,only :5 x 0 le>n$ h[ to foll[ cont9u\sly ! c\rse ( natural ev5ts1 attempts1 to con/ruct by means ( ab/ract conceptions ! connection betwe5 ph5om5a1 met ) success4 ,to d ?is two ?+s are necess>y3 ,fir/1 simple fundam5tal concepts ) :i* to con/ruct2 second1 "s me?od by :i* to d$uce1 f ! simple fundam5tal laws ( ! con/ruction :i* relate to in/ants ( "t & po9ts 9 space1 laws = f9ite int]vals & di/ances1 :i* alone are accessible to obs]vation 7c be comp>$ ) exp]i5ce740 ,riemann4 ,! fir/ ( ! two problems "h indicat$ by ,riemann consi/s 9 sett+ up ! diff]5tial equation1 bas$ ^u physical facts & hypo!ses4 ,! second is ! integration ( ?is diff]5tial equation & its application to ea* sep>ate concrete case1 ?is is ! task ( ma!matics4--,,web]1 ,,he9ri*4 ,die p>tiell5 ,diff]5tialglei*ung5 d] ma!matis*5 ,physik 7,brauns*weig1 #ahhb71 ,bd4 #a1 ,vorr$e4 #aeej4 ,ma!matics is ! mo/ p[]ful in/rum5t :i* we possess = ?is purpose to trace into _! f>!/ results ^? g5]al laws :i* an inductive philosophy has suppli$3 9 _m sci5ces a pr(\nd k ( ma!matics is indisp5sable = a successful inve/igation4 ,9 ! mo/ delicate rese>*es into ! !ories ( light1 heat1 & s\nd x is ! only in/rum5t2 !y h prop]ties :i* no o!r language c express2 & _! argum5tative processes are beyond ! rea* ( o!r symbols4--,,price1 ,b4 ,treatise on ,inf9itesimal ,calculus 7,ox=d1 #aheh71 ,vol4 #c1 p4 #e4 #aeea4 ,not)/&+ ! em95t difficulties ( ! ma!matical !ory ( sonor\s vibrations1 we owe to x su* progress z has yet be5 made 9 ac\/ics4 ,! =mation ( ! diff]5tial equations prop] to ! ph5om5a is1 indep5d5t ( _! integration1 a v important acquisition1 on acc\nt ( ! approximations :i* ma!matical analysis all[s betwe5 que/ions1 o!rwise het]og5e\s1 :i* lead to simil> equations4 ,?is fundam5tal prop]ty1 ^: value we h s (t5 to recognize1 applies rem>kably 9 ! pres5t case2 & especially s9ce ! creation ( ma!matical !rmology1 ^: pr9cipal equations are /rongly analog\s to ^? ( vibratory motion4--,?is means ( inve/igation is all ! m valuable on acc\nt ( ! difficulties 9 ! way ( direct inquiry into ! ph5om5a ( s\nd4 ,we may decide ! necessity ( ! atmosph]ic m$ium = ! transmission ( sonor\s vibrations2 & we may conceive ( ! possibility ( det]m9+ by exp]im5t ! duration ( ! propagation1 9 ! air1 & !n ?r\gh o!r m$ia2 b ! g5]al laws ( ! vibrations ( sonor\s bodies escape imm$iate obs]vation4 ,we %\ld "k almo/ no?+ ( ! :ole case if ! ma!matical !ory did n come 9 to connect ! diff]5t ph5om5a ( s\nd1 enabl+ u to sub/itute = direct obs]vation an equival5t exam9ation ( m favorable cases subject$ to ! same law4 ,= in/ance1 :5 ! analysis ( ! problem ( vibrat+ *ords has %[n u t1 o!r ?+s be+ equal1 ! numb] ( oscillations is 9 inv]se proportion to ! l5g? ( ! *ord1 we see t ! mo/ rapid vibrations ( a v %ort *ord may be c\nt$1 s9ce ! law enables u to direct our att5tion to v sl[ vibrations4 ,! same sub/itution is at our comm& 9 _m cases 9 :i* x is less direct4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #c1 *ap4 #d4 #aeeb4 ,problems relative to ! uni=m propagation1 or to ! v>i$ movem5ts ( heat 9 ! int]ior ( solids1 are r$uc$ 444 to problems ( pure analysis1 & ! progress ( ?is "p ( physics w dep5d 9 consequ5ce ^u ! advance :i* may be made 9 ! art ( analysis4 ,! diff]5tial equations 444 conta9 ! *ief results ( ! !ory2 !y express1 9 ! mo/ g5]al & concise mann]1 ! necess>y relations ( num]ical analysis to a v ext5sive class ( ph5om5a2 & !y connect =ev] ) ma!matical sci5ce "o ( ! mo/ important bran*es ( natural philosophy4--,,f\ri]1 ,j4 ,!ory ( ,heat ,freeman1 7,cambridge1 #ahgh71 ,*ap4 #c1 p4 #aca4 #aeec4 ,! effects ( heat are subject to con/ant laws :i* _c be discov]$ )\t ! aid ( ma!matical analysis4 ,! object ( ! !ory is to demon/rate ^! laws2 x r$uces all physical rese>*es on ! propagation ( heat1 to problems ( ! integral calculus1 ^: elem5ts are giv5 by exp]im5t4 ,no subject has m ext5sive relations ) ! progress ( indu/ry & ! natural sci5ces2 = ! action ( heat is always pres5t1 x influ5ces ! processes ( ! arts1 & occurs 9 all ! ph5om5a ( ! univ]se4--,,f\ri]1 ,j4 ,!ory ( ,heat ,freeman1 7,cambridge1 #ahgh71 ,*ap4 #a1 p4 #ab4 #aeed4 ,deal+ ) any & e am\nt ( /atic electricity1 ! ma!matical m9d has balanc$ & adju/$ !m ) wond]ful advantage1 & has =etold results :i* ! exp]im5tali/ c d no m ?an v]ify4444 ,s 9 respect ( ! =ce ( gravitation1 x has calculat$ ! results ( ! p[] 9 su* a wond]ful mann] z to trace ! kn[n planets ?r\gh _! c\rses & p]turbations1 & 9 s do+ has discov]$ a planet be=e unkn[n4--,,f>aday4 ,"s ,?\ghts on ! ,cons]vation ( ,=ce4 #aeee4 ,c]ta9 bran*es ( natural philosophy 7su* z physical a/ronomy & optics71 444 are1 9 a great measure1 inaccessible to ^? :o h n receiv$ a regul> ma!matical education4444 --,,/ew>t1 ,,dugald4 ,philosophy ( ! ,human ,m9d1 ,"p #c1 *ap4 #a1 sect4 #c4 #aeef4 ,s intimate is ! union betwe5 ma!matics & physics t probably by f> ! l>g] "p ( ! accessions to our ma!matical k h be5 obta9$ by ! ef=ts ( ma!maticians to solve ! problems set to !m by exp]im5t1 & to create 8= ea* successive class ( ph5om5a1 a new calculus or a new geometry1 z ! case might be1 :i* might prove n :olly inadequate to ! subtlety ( nature40 ,sometimes1 inde$1 ! ma!matician has be5 be=e ! physici/1 & x has happ5$ t :5 "s great & new "q has occurr$ to ! exp]im5tali/ or ! obs]v]1 he has f\nd 9 ! arm\ry ( ! ma!matician ! weapons :i* he has ne$$ ready made to 8 h&4 ,b1 mu* (t5]1 ! que/ions propos$ by ! physici/ h transc5d$ ! utmo/ p[]s ( ! ma!matics ( ! "t1 & a fre% ma!matical creation has be5 ne$$ to supply ! logical in/rum5t requisite to int]pret ! new enigma4--,,smi?1 ,h4 ,j4 ,s4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a2 ,nature1 ,vol4 #h 7#ahgc71 p4 #dej4 #aeeg4 ,( all ! great subjects :i* belong to ! prov9ce ( 8 section1 take t :i* at fir/ sight is ! lea/ )9 ! doma9 ( ma!matics--,i mean meteorology4 ,yet ! "p :i* ma!matics plays 9 meteorology increases e ye>1 & seems de/9$ to increase4 ,n only is ! !ory ( ! simple/ in/rum5ts ess5tially ma!matical1 b ! discussions ( ! obs]vations--^u :i*1 be x rememb]$1 dep5d ! hopes :i* are already ent]ta9$ ) increas+ confid5ce1 ( r$uc+ ! mo/ v>iable & complex ( all kn[n ph5om5a to exact laws--is a problem :i* n only belongs :olly to ma!matics1 b :i* taxes to ! utmo/ ! res\rces ( ! ma!matics :i* we n[ possess4--,,smi?1 ,h4 ,j4 ,s4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce1 ,section ,a2 ,nature1 ,vol4 #h 7#ahgc71 p4 #ddi4 #aeeh4 ,y "k t if y make a dot on a piece ( pap]1 & !n hold a piece ( ,icel& sp> ov] x1 y w see n "o dot b two4 ,a m9]alogi/1 by measur+ ! angles ( a cry/al1 c tell y :e!r or no x possesses ?is prop]ty )\t look+ ?r\gh x4 ,he requires no sci5tific ?\ght to d t4 ,b ,sir ,william ,roman ,hamilton 444 kn[+ ^! facts & also ! explanation ( !m :i* ,fresnel _h giv51 ?\ght ab\t ! subject1 & he pr$ict$ t by look+ ?r\gh c]ta9 cry/als 9 a p>ticul> direction we %\ld see n two dots b a cont9u\s circle4 ,mr4 ,lloyd made ! exp]im5t1 & saw ! circle1 a result :i* _h nev] be5 ev5 suspect$4 ,?is has always be5 consid]$ "o ( ! mo/ signal in/ances ( sci5tific ?\ght 9 ! doma9 ( physics4 --,,clif=d1 ,w4 ,k4 ,lectures & ,essays 7,new ,york1 #aija71 ,vol4 #a1 p4 #add4 #aeei4 ,! discov]y ( ?is planet ,neptune is ju/ly reckon$ z ! greate/ triumph ( ma!matical a/ronomy4 ,uranus fail$ to move precisely 9 ! pa? :i* ! comput]s pr$ict$ = x1 & 0 misguid$ by "s unkn[n influ5ce to an ext5t :i* a ke5 eye might almo/ see )\t telescopic aid4444 ,^! m9ute discrepancies con/itut$ ! data :i* 7 f\nd suffici5t = calculat+ ! position ( a hi!rto unkn[n planet1 & br++ x to light4 ,lev]ri] wrote to ,galle1 9 sub/ance3 8,direct y\r telescope to a po9t on ! ecliptic 9 ! con/ellation ( ,aqu>ius1 9 longitude #cbf1 & y w f9d )9 a degree ( t place a new planet1 look+ l a /> ( ab\t ! n9? magnitude1 & hav+ a p]ceptible disc40 ,! planet 0 f\nd at ,b]l9 on ! night ( ,sept4 #bf1 #ahdf1 9 exact accordance ) ?is pr$iction1 )9 half an h\r aft] ! a/ronom]s began look+ = x1 & only ab\t #eb di/ant f ! precise po9t t ,lev]ri] _h indicat$4 --,,"y1 ,c4 ,a4 ,g5]al ,a/ronomy 7,bo/on1 #ahia71 ,art4 #fec4 #aefj4 ,i am conv9c$ t ! future progress ( *emi/ry z an exact sci5ce dep5ds v mu* inde$ ^u ! alliance ) ma!matics4--,,frankl&1 ,a4 ,am]ican ,j\rnal ( ,ma!matics1 ,vol4 #a1 p4 #cdi4 #aefa4 ,x is almo/ impossible to foll[ ! lat] developm5ts ( physical or g5]al *emi/ry )\t a work+ k ( high] ma!matics4--,,mellor1 ,j4 ,w4 ,high] ,ma!matics 7,new ,york1 #aijb71 ,preface4 #aefb4 444 ,m\nt :]e sci5ce guides2 ,g measure e>?1 weigh air1 & /ate ! tides2 ,in/ruct ! planets 9 :at orb to run1 ,correct old "t1 & regulate ! sun4 --,,?omson1 ,w4 ,on ! ,figure ( ! ,e>?1 ,title page4 #aefc4 ,admission to its sanctu>y ref]r+ to a/ronomy & to ! privileges & feel+s ( a vot>y1 is only to be ga9$ by "o means1--s\nd & suffici5t k ( ma!matics1 ! great in/rum5t ( all exact inquiry1 )\t :i* no man c ev] make su* advances 9 ?is or any o!r ( ! high] dep>tm5ts ( sci5ce z c entitle him to =m an indep5d5t op9ion on any subject ( discussion )9 _! range4 --,,h]s*el1 ,j4 ,outl9es ( ,a/ronomy1 ,introduction1 sect4 #g4 #aefd4 ,! long s]ies ( connect$ tru?s :i* compose ! sci5ce ( a/ronomy1 h be5 evolv$ f ! appe>ances & obs]vations by calculation1 & a process ( reason+ entirely geometrical4 ,x 0 n )\t reason t ,plato call$ geometry & ari?metic ! w+s ( a/ronomy2 = x is only by means ( ^! two sci5ces t we c give a rational acc\nt ( any ( ! appe>ances1 or connect any fact ) !ory1 or ev5 r5d] a s+le obs]vation available to ! mo/ common a/ronomical purpose4 ,x is by geometry t we are enabl$ to reason our way up ?r\gh ! app>5t motions to ! real orbits ( ! planets1 & to assign _! positions1 magnitudes & ecc5tricities4 ,& x is by application ( geometry--a sublime geometry1 inde$1 inv5t$ = ! purpose--to ! g5]al laws ( me*anics1 t we demon/rate ! law ( gravitation1 trace x ?r\gh its remote/ effects on ! diff]5t planets1 &1 comp>+ ^! effects ) :at we obs]ve1 det]m9e ! d5sities & weights ( ! m9ute/ bodies belong+ to ! sy/em4 ,! :ole sci5ce ( a/ronomy is 9 fact a tissue ( geometrical reason+1 appli$ to ! data ( obs]vation2 & x is f ?is circum/ance t x d]ives its peculi> "* ( precision & c]ta9ty4 ,to disconnect x f geometry1 !re=e1 & to sub/itute famili> illu/rations & vague description = close & logical reason+1 is to deprive x ( its pr9cipal advantages1 & to r$uce x to ! condition ( an ord9>y prov9ce ( natural hi/ory4 ,ed9burgh ,review1 ,vol4 #eh 7#ahcc-ahcd71 p4 #afh4 #aefe4 ,b geometry is n only ! in/rum5t ( a/ronomical inve/igation1 & ! bond by :i* ! tru?s are en*a9$ toge!r1--x is also ! in/rum5t ( explanation1 af=d+1 by ! peculi> brevity & p]spicuity ( its te*nical processes1 n only aid to ! le>n]1 b also su* facilities to ! tea*] z he w f9d x v difficult to supply1 if he volunt>ily und]takes to =ego its assi/ance4 ,few und]tak+s1 inde$1 are att5d$ ) great] difficulty ?an t ( attempt+ to exhibit ! connect+ l9ks ( a *a9 ( ma!matical reason+1 :5 we lay aside ! te*nical symbols & notation :i* relieve ! memory1 & speak at once to ! eyes & ! und]/&+34444 ,ed9burgh ,review1 ,vol4 #eh 7#ahcc-ahcd71 p4 #afi4 #aeff4 ,) an ord9>y acqua9tance ( trigonometry1 & ! simple/ elem5ts ( algebra1 "o may take up any well-writt5 treatise on plane a/ronomy1 & "w 8 way ?r\gh x1 f beg9n+ to end1 ) p]fect ease2 & he w acquire1 9 ! c\rse ( 8 progress1 f ! m]e examples put be=e him1 an inf9itely m correct & precise idea ( a/ronomical me?ods & !ories1 ?an he c\ld obta9 9 a lifetime f ! mo/ eloqu5t g5]al descriptions t ev] 7 writt54 ,at ! same "t he w be /r5g!n+ himself = f>!r advances1 & accu/om+ 8 m9d to habits ( close comp>ison & rigid demon/ration1 :i* are ( inf9itely m importance ?an ! acquisition ( /ores ( undige/$ facts4 ,ed9burgh ,review1 ,vol4 #eh 7#ahcc-ahcd71 p4 #agj4 #aefg4 ,:ile ! telescope s]ves z a means ( p5etrat+ space1 & ( br++ its remote/ regions ne>] u1 ma!matics1 by inductive reason+1 h l$ u onw>ds to ! remote/ regions ( heav51 & br\ght a portion ( !m )9 ! range ( our possibilities2 nay1 9 our own times--s propiti\s to ! ext5sion ( k--! application ( all ! elem5ts yield$ by ! pres5t conditions ( a/ronomy has ev5 reveal$ to ! intellectual eyes a heav5ly body1 & assign$ to x its place1 orbit1 mass1 be=e a s+le telescope has be5 direct$ t[>ds x4--,,humboldt1 ,a4 ,cosmos ,otte1 ,vol4 #b1 "p #b1 sect4 #c4 #aefh4 ,mighty are numb]s1 jo9$ ) art resi/less4--,,euripides4 ,hecuba1 ,l9e #hhd4 #aefi4 ,no s+le in/rum5t ( y\?ful education has su* mighty p[]1 bo? z reg>ds dome/ic economy & politics1 & 9 ! arts1 z ! /udy ( ari?metic4 ,above all1 ari?metic /irs up him :o is by nature sleepy & dull1 & makes him quick to le>n1 ret5tive1 %rewd1 & aid$ by art div9e he makes progress q beyond 8 natural p[]s4--,,plato4 ,laws ,j[ett1 ,bk4 #e1 p4 #gdg4 #aegj4 ,= all ! high] arts ( con/ruction "s acqua9tance ) ma!matics is indisp5sable4 ,! village c>p5t]1 :o1 lack+ rational in/ruction1 lays out 8 "w by empirical rules le>n$ 9 8 appr5tice%ip1 equally ) ! build] ( a ,britannia ,bridge1 makes h\rly ref]5ce to ! laws ( quantitative relations4 ,! surveyor on ^: survey ! l& is pur*as$2 ! ar*itect 9 design+ a mansion to be built on x2 ! build] 9 prep>+ 8 e/imates2 8 =eman 9 lay+ out ! f\ndations2 ! masons 9 cutt+ ! /ones2 & ! v>i\s artisans :o put up ! fitt+s2 are all guid$ by geometrical tru?s4 ,railway-mak+ is regulat$ f beg9n+ to end by ma!matics3 alike 9 ! prep>ation ( plans & sections2 9 /ak+ out ! l9es2 9 ! m5suration ( cutt+s & embankm5ts2 9 ! design+1 e/imat+1 & build+ ( bridges1 culv]ts1 viaducts1 tunnels1 /ations4 ,& simil>ly ) ! h>bors1 docks1 pi]s1 & v>i\s eng9e]+ & ar*itectural works t fr+e ! coa/s & ov]spread ! face ( ! c\ntry1 z well z ! m9es t run und]nea? x4 ,out ( geometry1 too1 z appli$ to a/ronomy1 ! art ( navigation has gr[n2 & s1 by ?is sci5ce1 has be5 made possible t enorm\s =eign comm]ce :i* supports a l>ge "p ( our population1 & supplies u ) _m necess>ies & mo/ ( our luxuries4 ,& n[adays ev5 ! f>m]1 = ! correct lay+ out ( 8 dra9s1 has rec\rse to ! level--t is1 to geometrical pr9ciples4--,,sp5c]1 ,,h]b]t4 ,education1 *ap4 #a4 #aega4 ,ari?metic is ano!r ( ! great ma/]-keys ( life4 ,) x ! a/ronom] op5s ! dep?s ( ! heav5s2 ! eng9e]1 ! gates ( ! m\nta9s2 ! navigator1 ! pa?ways ( ! deep4 ,! skillful arrangem5t1 ! rapid h&l+ ( figures1 is a p]fect magician's w&4 ,! mighty comm]ce ( ! ,unit$ ,/ates1 =eign & dome/ic1 passes ?r\gh ! books kept by "s ?\s&s ( dilig5t & fai?ful cl]ks4 ,eight hundr$ bookkeep]s1 9 ! ,bank ( ,engl&1 /rike ! monet>y balance ( half ! civiliz$ _w4 ,_! skill & accuracy 9 apply+ ! common rules ( ari?metic are z important z ! ent]prise & capital ( ! m]*ant1 or ! indu/ry & c\rage ( ! navigator4 ,i look ^u a well-kept l$g] ) some?+ ( ! pleasure ) :i* ,i gaze on a picture or a /atue4 ,x is a beautiful "w ( art4--,,ev]ett1 ,,edw>d4 ,orations & ,spee*es 7,bo/on1 #ahgj71 ,vol4 #c1 p4 #dg4 #aegb4 ,ma!matics is ! fruitful ,p>5t (1 ,i _h almo/ said all1 ,arts1 ! un%ak5 ,f\ndation ( ,sci5ces1 & ! pl5tiful ,f\nta9 ( ,advantage to ,human ,affairs4 ,9 :i* la/ ,respect1 we may be said to receive f ! ,ma!matics1 ! pr9cipal ,delights ( ,life1 ,securities ( ,heal?1 ,increase ( ,=tune1 & ,conv5i5ces ( ,lab\r3 ,t we dwell elegantly & commodi\sly1 build dec5t ,h\ses = ourselves1 erect /ately ,temples to ,god1 & leave wond]ful ,monum5ts to ,po/]ity3 ,t we are protect$ by ^? ,rampires f ! ,incursions ( ! ,enemy2 rightly use ,arms1 skillfully range an ,army1 & manage ,w> by ,art1 & n by ! ,madness ( wild ,bea/s3 ,t we h safe ,traffick ?r\gh ! deceitful ,bill[s1 pass 9 a direct ,road ?r\gh ! tractless ,ways ( ! ,sea1 & come to ! design$ ,ports by ! unc]ta9 ,impulse ( ! ,w9ds3 ,t we rightly ca/ up our ,acc\nts1 d ,bus9ess exp$iti\sly1 dispose1 tabulate1 & calculate scatt]$ ,ranks ( ,numb]s1 & easily compute !m1 ?\gh expressive ( huge ,heaps ( ,s&1 nay imm5se ,hills ( ,atoms3 ,t we make pacifick ,sep>ations ( ! ,b\nds ( ,l&s1 exam9e ! ,mom5ts ( ,weights 9 an equal ,balance1 & di/ribute e "o 8 own by a j ,measure3 ,t ) a light ,t\* we ?ru/ =w>d va/ ,bodies :i* way we w1 & /op a huge ,resi/ance ) a v small ,=ce3 ,t we accurately del9eate ! ,face ( ?is ,e>?ly ,orb1 & subject ! ,oeconomy ( ! ,univ]se to our ,sight3 ,t we aptly dige/ ! fl[+ ,s]ies ( ,"t1 di/+ui% :at is act$ by due ,int]vals1 rightly acc\nt & disc]n ! v>i\s ,returns ( ! ,seasons1 ! /at$ ,p]iods ( ,ye>s & ,mon?s1 ! alt]nate ,increm5ts ( ,days & ,nights1 ! d\btful ,limits ( ,light & ,%ad[1 & ! exact ,diff]5ces ( ,h\rs & ,m9utes3 ,t we d]ive ! subtle ,virtue ( ! ,sol> ,rays to our ,uses1 inf9itely ext5d ! ,sph]e ( ,sight1 enl>ge ! ne> ,appe>ances ( ,?+s1 br+ to ,h& ,?+s remote1 discov] ,?+s hidd51 se>* ,nature out ( h] ,concealm5ts1 & unfold h] d>k ,my/]ies3 ,t we delight our ,eyes ) beautiful ,images1 cunn+ly imitate ! ,devices & portray ! ,works ( ,nature2 imitate did ,i say8 nay excel1 :ile we =m to ourselves ,?+s n 9 be+1 exhibit ,?+s abs5t1 & repres5t ,?+s pa/3 ,t we recreate our ,m9ds & delight our ,e>s ) melodi\s ,s\nds1 attemp]ate ! incon/ant ,undulations ( ! ,air to musical ,tunes1 add a pleasant ,voice to a sapless ,log & draw a sweet ,eloqu5ce f a rigid ,metal2 celebrate our ,mak] ) an h>moni\s ,praise1 & n unaptly imitate ! bless$ ,*oirs ( ,heav53 ,t we approa* & exam9e ! inaccessible ,seats ( ! ,cl\ds1 ! di/ant ,tracts ( ,l&1 unfrequ5t$ ,pa?s ( ! ,sea2 l(ty ,tops ( ! ,m\nta9s1 l[ ,bottoms ( ! ,valleys1 & deep ,gulphs ( ! ,ocean3 ,t 9 ,he>t we advance to ! ,sa9ts !mselves above1 yea draw !m to u1 scale ! e!rial ,t[]s1 freely range ?r\gh ! cele/ial ,fields1 measure ! ,magnitudes1 & det]m9e ! ,int]/ices ( ! ,/>s1 prescribe inviolable ,laws to ! ,heav5s !mselves1 & conf9e ! w&]+ ,circuits ( ! ,/>s )9 fix$ ,b\nds3 ,la/ly1 t we compreh5d ! va/ ,fabrick ( ! ,univ]se1 admire & contemplate ! wond]ful ,beauty ( ! ,div9e ,workman%ip1 & to le>n ! incr$ible ,=ce & ,sagacity ( our own ,m9ds1 by c]ta9 ,exp]im5ts1 & to ackn[l$ge ! ,bless+s ( ,heav5 ) pi\s ,affection4--,,b>r[1 ,,isaac4 ,ma!matical ,lectures 7,london1 #agcd71 pp4 #bg-cj4 #aegc4 ,analytical & graphical treatm5t ( /ati/ics is employ$ by ! economi/1 ! philan?ropi/1 ! bus9ess exp]t1 ! actu>y1 & ev5 ! physician1 ) ! mo/ surpris+ly valuable results2 :ile symbolic language involv+ ma!matical me?ods has become a "p ( wellnigh e l>ge bus9ess4 ,! h&l+ ( pig-iron does n seem to (f] any opportunity = ma!matical application4 ,yet graphical & analytical treatm5t ( ! data f long-cont9u$ exp]im5ts ) ?is mat]ial at ,be?lehem1 ,p5nsylvania1 result$ 9 ! discov]y ( ! law t fatigue v>i$ 9 proportion to a c]ta9 relation betwe5 ! load & ! p]iods ( re/4 ,practical application ( ?is law increas$ ! am\nt h&l$ by ea* man f twelve & a half to =ty-sev5 tons p] day4 ,su* /udy w\ld h be5 impossible )\t prelim9>y acqua9tance ) ! simple inv>iable elem5ts ( ma!matics4--,,k>p9sky1 ,l4 ,high ,s*ool ,education 7,new ,york1 #aiab71 *ap4 #f1 p4 #acd4 #aegd4 ,!y computation & ari?metic belong !n1 x seems1 to ! bran*es ( le>n+ :i* we are n[ inve/igat+2--= a milit>y man mu/ necess>ily le>n !m ) a view to ! m>%all+ ( 8 troops1 & s mu/ a philosoph] ) ! view ( und]/&+ real be+1 aft] hav+ em]g$ f ! un/able condition ( becom+1 or else he c nev] become an apt reason]4 ,t is ! fact he repli$4 ,b ! gu>dian ( ours happ5s to be bo? a milit>y man & a philosoph]4 ,unque/ionably s4 ,x w\ld be prop] !n1 ,glaucon1 to lay d[n laws = ?is bran* ( sci5ce & p]suade ^? ab\t to engage 9 ! mo/ important /ate-matt]s to apply !mselves to computation1 & /udy x1 n 9 ! common vulg> fa%ion1 b ) ! view ( arriv+ at ! contemplation ( ! nature ( numb]s by ! intellect itself1--n = ! sake ( buy+ & sell+ z anxi\s m]*ants & retail]s1 b = w> also1 & t ! s\l may acquire a facility 9 turn+ itself f :at is 9 ! c\rse ( g5]ation to tru? & real be+4--,,plato4 ,republic ,davis1 ,bk4 #g1 p4 #ebe4 #aege4 ,! sci5tific "p ( ,ari?metic & ,geometry w\ld be ( m use = regulat+ ! ?\ghts & op9ions ( m5 ?an all ! great advantage :i* ,society receives f ! g5]al application ( !m3 & ?is use _c be spread ?r\gh ! ,society by ! practice2 = ! ,practition]s1 h[ev] dextr\s1 h no m k ( ! ,sci5ce ?an ! v in/rum5ts ) :i* !y "w4 ,!y h tak5 up ! ,rules z !y f\nd !m deliv]$ d[n to !m by sci5tific m51 )\t ! lea/ inquiry aft] ! ,pr9ciples f :i* !y are d]iv$3 & ! m accurate ! ,rules1 ! less occasion !re is = inquir+ aft] ! ,pr9ciples1 & consequ5tly1 ! m difficult x is to make !m turn _! att5tion to ! ,fir/ ,pr9ciples2 &1 !re=e1 a ,nation ought to h bo? ,sci5tific & ,practical ,ma!maticians4--,,williamson1 ,,james4 ,elem5ts ( ,euclid ) ,diss]tations 7,ox=d1 #agha74 #aegf4 ,:]e !re is no?+ to measure !re is no?+ to calculate1 h5ce x is impossible to employ ma!matics 9 psy*ological inve/igations4 ,?us runs ! syllogism comp\nd$ ( an adh]5ce to usage & an app>5t tru?4 ,z to ! latt]1 x is :olly untrue t we may calculate only :]e we h measur$4 ,exactly ! opposite is true4 ,e hypo!tically assum$ law ( quantitative comb9ation1 ev5 su* z is recogniz$ z invalid1 is subject to calculation2 & 9 case ( deeply hidd5 b important matt]s x is imp]ative to try on hypo!ses & to subject ! consequ5ces :i* fl[ f !m to precise computation until x is f\nd :i* "o ( ! v>i\s hypo!ses co9cides ) exp]i5ce4 ,?us ! anci5t a/ronom]s tri$ ecc5tric circles1 & ,kepl] tri$ ! ellipse to acc\nt = ! motion ( ! planets1 ! latt] also comp>$ ! squ>es ( ! times ( revolution ) ! cubes ( ! mean di/ances be=e he discov]$ _! agreem5t4 ,9 l mann] ,newton tri$ :e!r a gravitation1 v>y+ inv]sely z ! squ>e ( ! di/ance1 suffic$ to keep ! moon 9 its orbit ab\t ! e>?2 if ?is supposition _h fail$ him1 he w\ld h tri$ "s o!r p[] ( ! di/ance1 z ! f\r? or fif?1 & d$uc$ ! correspond+ consequ5ces to comp>e !m ) ! obs]vations4 ,j ?is is ! greate/ b5efit ( ma!matics1 t x enables u to survey ! possibilities ^: range includes ! actual1 long be=e we h adequate def9ite exp]i5ce2 ?is makes x possible to employ v incomplete indications ( exp]i5ce to avoid at lea/ ! crude/ errors4 ,long be=e ! transit ( ,v5us 0 employ$ 9 ! det]m9ation ( ! sun's p>allax1 x 0 attempt$ to det]m9e ! in/ant at :i* ! sun illum9es exactly "o-half ( ! moon's disk1 9 ord] to compute ! sun's di/ance f ! kn[n di/ance ( ! moon f ! e>?4 ,?is 0 n possible1 =1 ow+ to psy*ological reasons1 our me?od ( measur+ "t is too crude to give u ! desir$ in/ant ) suffici5t accuracy2 yet ! attempt gave u ! k t ! sun's di/ance f u is at lea/ sev]al hundr$ times z great z t ( ! moon4 ,?is illu/ration %[s cle>ly t ev5 a v imp]fect e/imate ( a magnitude 9 a case :]e no precise obs]vation is possible1 may become v in/ructive1 if we "k h[ to exploit x4 ,0 x necess>y to "k ! scale ( our sol> sy/em 9 ord] to le>n ( its ord] 9 g5]al8 ,or1 tak+ an illu/ration f ano!r field1 0 x impossible to inve/igate ! laws ( motion until x 0 kn[n exactly h[ f> a body falls 9 a second at "s def9ite place8 ,n at all4 ,su* det]m9ations ( fundam5tal measures are 9 !mselves exce$+ly difficult1 b =tunately1 su* inve/igations =m a class ( _! own2 our k ( fundam5tal laws does n ne$ to wait on ^!4 ,to be sure1 computation invites measurem5t1 & e easily obs]v$ regul>ity ( c]ta9 magnitudes is an inc5tive to ma!matical inve/igation4 --,,h]b>t1 ,j4 ,f4 ,w]ke ,kehrba*1 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #ig4 #aegg4 ,^? :o pass = naturali/s1 h1 = ! mo/ "p1 be5 v little1 or n at all1 v]s$ 9 ma!maticks1 if n also jeal\s ( !m4--,,boyle1 ,,rob]t4 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbf4 #aegh4 ,h[ev] hurtful may h be5 ! incursions ( ! geomet]s1 direct & indirect1 into a doma9 :i* x is n = !m to cultivate1 ! physiologi/s are n ! less wrong 9 turn+ away f ma!matics altoge!r4 ,x is n only t )\t ma!matics !y c\ld n receive _! due prelim9>y tra9+ 9 ! int]v5+ sci5ces3 x is fur!r necess>y = !m to h geometrical & me*anical k1 to und]/& ! /ructure & ! play ( ! complex app>atus ( ! liv+1 & especially ! animal organism4 ,animal me*anics1 /atical & dynamical1 mu/ be un9telligible to ^? :o are ignorant ( ! g5]al laws ( rational me*anics4 ,! laws ( equilibrium & motion are 444 absolutely univ]sal 9 _! action1 dep5d+ :olly on ! en]gy1 & n at all on ! nature ( ! =ces consid]$3 & ! only difficulty is 9 _! num]ical application 9 cases ( complexity4 ,?us1 disc>d+ all idea ( a num]ical application 9 biology1 we p]ceive t ! g5]al !orems ( /atics & dynamics mu/ be /eadily v]ifi$ 9 ! me*anism ( liv+ bodies1 on ! rational /udy ( :i* !y ca/ an indisp5sable light4 ,! highe/ ord]s ( animals act 9 repose & motion1 l any o!r me*anical app>atus ( a simil> complexity1 ) ! "o diff]5ce ( ! mov]1 :i* has no p[] to alt] ! laws ( motion & equilibrium4 ,! p>ticipation ( rational me*anics 9 positive biology is ?us evid5t4 ,me*anics _c disp5se ) geometry2 & beside1 we see h[ anatomical & physiological speculations involve consid]ations ( =m & position1 & require a famili> k ( ! pr9cipal geometrical laws :i* may ca/ light ^u ^! complex relations4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #e1 *ap4 #a4 #aegi4 ,9 ma!matics we f9d ! primitive s\rce ( rationality2 & to ma!matics mu/ ! biologi/s resort = means to c>ry on _! rese>*es4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #e1 *ap4 #a4 #aehj4 ,9 ?is s*ool ( ma!matics mu/ !y biologi/s le>n famili>ly ! real *>act]s & conditions ( sci5tific evid5ce1 9 ord] to transf] x aft]w>ds to ! prov9ce ( _! own !ories4 ,! /udy ( x "h1 9 ! mo/ simple & p]fect cases1 is ! only s\nd prep>ation = its recognition 9 ! mo/ complex4 ,! /udy is equally necess>y = ! =mation ( intellectual habits2 = obta9+ an aptitude 9 =m+ & su/a9+ positive ab/ractions1 )\t :i* ! comp>ative me?od _c be us$ 9 ei!r anatomy or physiology4 ,! ab/raction :i* is to be ! /&>d ( comp>ison mu/ be fir/ cle>ly =m$1 & !n /eadily ma9ta9$ 9 its integrity1 or ! analysis becomes abortive3 & ?is is s completely 9 ! _s ( ma!matical comb9ations1 t practice 9 !m is ! be/ prep>ation = x4 ,a /ud5t :o _c accompli% ! process 9 ! m simple case may be assur$ t he is n qualifi$ = ! high] ord] ( biological rese>*es1 & mu/ be satisfi$ ) ! humbl] (fice ( collect+ mat]ials = ! use ( m9ds ( ano!r ord]4 ,h5ce arises ano!r use ( ma!matical tra9+2--t ( te/+ & classify+ m9ds1 z well z prep>+ & guid+ !m4 ,probably z mu* good w\ld be done by exclud+ ! /ud5ts :o only encumb] ! sci5ce by aimless & desultory inquiries1 z by fitly in/itut+ ^? :o c bett] fulfill its conditions4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #e1 *ap4 #a4 #aeha4 ,!re seems no suffici5t reason :y ! use ( sci5tific fictions1 s common 9 ! h&s ( geomet]s1 %\ld n be introduc$ into biology1 if sy/ematically employ$1 & adopt$ ) suffici5t sobriety4 ,9 ma!matical /udies1 great advantages h aris5 f imag9+ a s]ies ( hypo!tical cases1 ! consid]ation ( :i*1 ?\gh artificial1 may aid ! cle>+ up ( ! real subject1 or its fundam5tal elaboration4 ,?is art is usually conf\nd$ ) t ( hypo!ses2 b x is entirely diff]5t2 inasmu* z 9 ! latt] case ! solution alone is imag9>y2 :]eas 9 ! =m]1 ! problem itself is radically ideal4 ,its use c nev] be 9 biology comp>able to :at x is 9 ma!matics3 b x seems to me t ! ab/ract "* ( ! high] conceptions ( comp>ative biology r5d]s !m susceptible ( su* treatm5t4 ,! process w be to int]calate1 among diff]5t kn[n organisms1 c]ta9 purely fictiti\s organisms1 s imag9$ z to facilitate _! comp>ison1 by r5d]+ ! biological s]ies m homog5e\s & cont9u\s3 & x might be t sev]al might h]eaft] meet ) m or less ( a realization among organisms hi!rto unexplor$4 ,x may be possible1 9 ! pres5t /ate ( our k ( liv+ bodies1 to conceive ( a new organism capable ( fulfill+ c]ta9 giv5 conditions ( exi/5ce4 ,h[ev] t may be1 ! collocation ( real cases ) well-imag9$ ones1 aft] ! mann] ( geomet]s1 w d\btless be practis$ h]eaft]1 to complete ! g5]al laws ( comp>ative anatomy & physiology1 & possibly to anticipate occasionally ! direct exploration4 ,ev5 n[1 ! rational use ( su* an artifice might greatly simplify & cle> up ! ord9>y sy/em ( biological in/ruction4 ,b x is only ! highe/ ord] ( inve/igators :o c be tru/$ ) x4 ,:5ev] x is adopt$1 x w con/itute ano!r gr\nd ( relation betwe5 biology & ma!matics4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #e1 *ap4 #a4 #aehb4 ,i ?9k x may safely en\gh be affirm$1 t he1 t is n s mu* z indiff]5tly skill$ 9 ma!maticks1 c h>dly be m ?an indiff]5tly skill$ 9 ! fundam5tal pr9ciples ( physiology4--,,boyle1 ,,rob]t4 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dcj4 #aehc4 ,x is n only possible b necess>y t ma!matics be appli$ to psy*ology2 ! reason = ?is necessity lies briefly 9 ?is3 t by no o!r means c be rea*$ t :i* is ! ultimate aim ( all speculation1 namely conviction4--,,h]b>t1 ,j4 ,f4 ,w]ke ,kehrba*1 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #ajd4 #aehd4 ,all m def9ite k mu/ />t ) computation2 & ?is is ( mo/ important consequ5ces n only = ! !ory ( memory1 ( imag9ation1 ( und]/&+1 b z well = ! doctr9e ( s5sations1 ( desires1 & affections4 --,,h]b>t1 ,j4 ,f4 ,w]ke ,kehrba*1 7,lang5salza1 #ahij71 ,bd4 #e1 p4 #ajc4 #aehe4 ,9 ! ne> future ma!matics w play an important "p 9 m$ic9e3 already !re are increas+ indications t physiology1 descriptive anatomy1 pa?ology & !rapeutics _c escape ma!matical legitimation4--,,dessoir1 ,,max4 ,we/]mann's ,monatsb]i*te1 ,bd4 #gg1 p4 #chj2 ,ahr5s3 ,s*]z und ,ern/ 9 d] ,ma!matik 7,leipzig1 #aijd71 p4 #cie4 #aehf4 ,! social sci5ces ma!matically develop$ are to be ! controll+ factors 9 civilization4--,,:ite1 ,w4 ,f4 ,a ,scrap-book ( ,elem5t>y ,ma!matics 7,*icago1 #aijh71 p4 #bjh4 #aehg4 ,x is cle> t ?is education ref]r+ to education prep>atory to ! sci5ce ( sociology mu/ re/ on a basis ( ma!matical philosophy1 ev5 ap>t f ! necessity ( ma!matics to ! /udy ( inorganic philosophy4 ,x is only 9 ! region ( ma!matics t sociologi/s1 or anybody else1 c obta9 a true s5se ( sci5tific evid5ce1 & =m ! habit ( rational & decisive argum5tation2 c1 9 %ort1 le>n to fulfill ! logical conditions ( all positive speculation1 by /udy+ univ]sal positivism at its s\rce4 ,?is tra9+1 obta9$ & employ$ ) ! m c>e on acc\nt ( ! em95t difficulty ( social sci5ce1 is :at sociologi/s h to seek 9 ma!matics4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #f1 *ap4 #d4 #aehh4 ,x is cle> t ! individual z a social unit & ! /ate z a social aggregate require a c]ta9 modicum ( ma!matics1 "s ari?metic & algebra1 to conduct _! affairs4 ,"u ?is head w\ld fall ! !ory ( int]e/1 simple & comp\nd1 matt]s ( disc\nt & amortization1 &1 if lott]ies hold a prom95t place 9 rais+ moneys1 z 9 "s /ates1 que/ions ( probability mu/ be add$4 ,z ! /ate becomes m highly organiz$ & m int]e/$ 9 ! sci5tific analysis ( its life1 !re appe>s an urg5t necessity = v>i\s /ati/ical in=mation1 & ?is c be prop]ly obta9$1 r$uc$1 correlat$1 & int]pret$ only :5 ! guid+ _s 9 ! "w h ! necess>y ma!matical tra9+ 9 ! !ory ( /ati/ics4 7,figures may n lie1 b /ati/ics compil$ unsci5tifically & analyz$ incompet5tly are almo/ sure to be mislead+1 & :5 ?is condition is unnecess>ily *ronic ! s-call$ /ati/icians may well be call$ li>s47 ,! dep5d5ce ( insurance ( v>i\s k9ds on /ati/ical in=mation & ! v great place :i* insurance occupies 9 ! mod]n /ate1 albeit (t5 controll$ by private corporations in/ead ( by ! gov]nm5t1 makes ! !ories ( p>am\nt importance to our social life4--,,wilson1 ,e4 ,b4 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #ah 7#aiab71 p4 #dfc4 #aehi4 ,! !ory ( probabilities & ! !ory ( errors n[ con/itute a =midable body ( k ( great ma!matical int]e/ & ( great practical importance4 ,?\gh develop$ l>gely ?r\gh ! applications to ! m precise sci5ces ( a/ronomy1 geodesy1 & physics1 _! range ( applicability ext5ds to all ! sci5ces2 & !y are pla9ly de/9$ to play an increas+ly important rle 9 ! developm5t & 9 ! applications ( ! sci5ces ( ! future4 ,h5ce _! /udy is n only a comm5dable elem5t 9 a lib]al education1 b "s k ( !m is ess5tial to a correct und]/&+ ( daily ev5ts4--,,woodw>d1 ,r4 ,s4 ,probability & ,!ory ( ,errors 7,new ,york1 #aijf71 ,preface4 #aeij4 ,x 0 n to be anticipat$ t a new sci5ce ! sci5ce ( probabilities :i* took its rise 9 games ( *ance1 & :i* _h long to enc\nt] an obloquy1 h>dly yet ext9ct1 due to ! prevail+ idea t its only end 0 to facilitate & enc\rage ! calculations ( gambl]s1 c\ld ev] h atta9$ its pres5t /atus--t its aid %\ld be call$ = 9 e dep>tm5t ( natural sci5ce1 bo? to assi/ 9 discov]y1 :i* x has repeat$ly done 7ev5 9 pure ma!matics71 to m9imize ! unavoidable errors ( obs]vation1 & to detect ! pres5ce ( causes z reveal$ by obs]v$ ev5ts4 ,nor are comm]cial & o!r practical int]e/s ( life less indebt$ to x3 :]ev] ! future has to be =eca/$1 risk to be provid$ aga9/1 or ! true lessons to be d$uc$ f /ati/ics1 x corrects = u ! r\gh conjectures ( common s5se1 & decides :i* c\rse is really1 accord+ to ! lights ( :i* we are 9 possession1 ! wise/ = u to pursue4--,,cr(ton1 ,m4 ,w4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,probability40 #aeia4 ,! calculus ( probabilities1 :5 conf9$ )9 j limits1 ought to int]e/1 9 an equal degree1 ! ma!matician1 ! exp]im5tali/1 & ! /atesman4 ,f ! "t :5 ,pascal & ,f]mat e/abli%$ its fir/ pr9ciples1 x has r5d]$1 & cont9ues daily to r5d]1 s]vices ( ! mo/ em95t k9d4 ,x is ! calculus ( probabilities1 :i*1 aft] hav+ sugge/$ ! be/ arrangem5ts ( ! tables ( population & mortality1 tea*es u to d$uce f ^? numb]s1 9 g5]al s errone\sly int]pret$1 conclusions ( a precise & useful "*2 x is ! calculus ( probabilities :i* alone c regulate ju/ly ! premiums to be paid = assurances2 ! res]ve funds = ! disbursem5ts ( p5sions1 annuities1 disc\nts1 etc4 ,x is "u its influ5ce t lott]ies & o!r %ameful sn>es cunn+ly laid = av>ice & ignorance h def9itely disappe>$4--,,arago4 ,eulogy on ,laplace ,bad5-,p[ell1 ,smi?sonian ,report1 #ahgd1 p4 #afd4 #aeib4 ,m5 7 surpris$ to he> t n only bir?s1 dea?s1 & m>riages1 b ! decisions ( tribunals1 ! results ( popul> elections1 ! influ5ce ( puni%m5ts 9 *eck+ crime1 ! comp>ative values ( m$ical rem$ies1 ! probable limits ( error 9 num]ical results 9 e dep>tm5t ( physical inquiry1 ! detection ( causes1 physical1 social1 & moral1 nay1 ev5 ! weight ( evid5ce & ! validity ( logical argum5t1 might come to be survey$ ) ! lynx-ey$ scrut9y ( a dispassionate analysis4--,,h]s*el1 ,j4 ,quot$ 9 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,probability40 #aeic4 ,if economi/s expect ( ! application ( ! ma!matical me?od any ext5sive concrete num]ical results1 & x is to be fe>$ t l o!r non-ma!maticians all too _m ( !m ?9k ( ma!matics z m]ely an ari?metical sci5ce1 !y are b\nd to be disappo9t$ & to f9d a paucity ( results 9 ! works ( ! few ( _! colleagues :o use t me?od4 ,b !y %\ld r le>n1 z ! ma!maticians among !m "k full well1 t ma!matics is mu* broad]1 t x has an ab/ract quantitative 7or ev5 qualitative7 side1 t x deals ) relations z well z numb]s1 4444--,,wilson1 ,e4 ,b4 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #ah 7#aiab71 p4 #dfd4 #aeid4 ,! ef=t ( ! economi/ is to see1 to picture ! int]-play ( economic elem5ts4 ,! m cle>ly cut ^! elem5ts appe> 9 8 vision1 ! bett]2 ! m elem5ts he c grasp & hold 9 8 m9d at once1 ! bett]4 ,! economic _w is a mi/y region4 ,! fir/ explor]s us$ unaid$ vision4 ,ma!matics is ! lant]n by :i* :at be=e 0 dimly visible n[ looms up 9 firm1 bold outl9es4 ,! old phantasmagoria disappe>4 ,we see bett]4 ,we also see fur!r4--,,fi%]1 ,,irv+4 ,transactions ( ,connecticut ,academy1 ,vol4 #i 7#ahib71 p4 #aai4 #aeie4 ,9 ! great inquiries ( ! moral & social sci5ces 444 ma!matics 7,i always mean ,appli$ ,ma!matics7 af=ds ! only suffici5t type ( d$uctive art4 ,up to ?is "t1 ,i may v5ture to say t no "o ev] knew :at d$uction is1 z a means ( inve/igat+ ! laws ( nature1 :o _h n le>n$ x f ma!matics1 nor c any "o hope to und]/& x ?or\ghly1 :o has n1 at "s "t 9 8 life1 kn[n en\gh ( ma!matics to be famili> ) ! in/rum5t at "w4 --,,mill1 ,j4 ,s4 ,an ,exam9ation ( ,sir ,william ,hamilton's ,philosophy 7,london1 #ahgh71 p4 #fbb4 #aeif4 ,let me pass on to say a ^w or two ab\t ! tea*+ ( ma!matics z an academic tra9+ = g5]al pr(essional life4 ,x has imm5se capabilities 9 t respect4 ,if y consid] h[ mu* ( ! effectiv5ess ( an adm9i/rator dep5ds ^u ! capacity = co-ord9at+ appropriately a numb] ( diff]5t ideas1 precise accuracy ( def9ition1 rigidity ( pro(1 & su/a9$ reason+1 /rict 9 e /ep1 & :5 y consid] :at sub/itutes = ^! ?+s n9e m5 out ( e t5 )\t special tra9+ h to put up )1 x is cle> t a man ) a ma!matical tra9+ has incalculable advantages4--,,%aw1 ,w4 ,h4 ,p]ry's ,tea*+ ( ,ma!matics 7,london1 #aijb71 p4 #gc4 #aeig4 ,be=e y ent] on ! /udy ( law a suffici5t gr\nd "w mu/ be laid4444 ,ma!matics & natural philosophy are s useful 9 ! mo/ famili> occurr5ces ( life & are s peculi>ly engag+ & delightful z w\ld induce ev]yone to wi% an acqua9tance ) !m4 ,besides ?is1 ! faculties ( ! m9d1 l ! memb]s ( a body1 are /r5g!n$ & improv$ by ex]cise4 ,ma!matical reason+ & d$uctions are1 !re=e1 a f9e prep>ation = inve/igat+ ! ab/ruse speculations ( ! law4--,,jeff]son1 ,,?omas4 ,quot$ 9 ,cajori's ,tea*+ & ,hi/ory ( ,ma!matics 9 ! ,u4 ,s4 7,wa%+ton1 #ahij71 p4 #ce4 #aeih4 ,x has be5 obs]v$ 9 ,engl& ( ! /udy ( law1--?\gh ! acquisition ( ! mo/ difficult p>ts ( its le>n+1 ! int]pretation ( laws1 ! comp>ison ( au?orities1 & ! con/ruction ( in/rum5ts1 w\ld seem to require philological & critical tra9+2 ?\gh ! weigh+ ( evid5ce & ! inve/igation ( probable tru? belong to ! prov9ce ( ! moral sci5ces1 & ! peculi> duties ( ! advocate require rhetorical skill1--yet t a l>ge proportion ( ! mo/ di/+ui%$ memb]s ( ! pr(ession has proce$$ f ! univ]sity 7t ( ,cambridge7 mo/ celebrat$ = ! cultivation ( ma!matical /udies4--,,ev]ett1 ,,edw>d4 ,orations & ,spee*es 7,bo/on1 #ahgj71 ,vol4 #b1 p4 #eaa4 #aeii4 ,all hi/oric sci5ce t5ds to become ma!matical4 ,ma!matical p[] is classify+ p[]4--,,novalis4 ,s*rift5 7,b]l91 #aija71 ,teil #b1 p4 #aib4 #aeii;a4 ,hi/ory has nev] reg>d$ itself z a sci5ce ( /ati/ics4 ,x 0 ! ,sci5ce ( ,vital ,en]gy 9 relation ) "t2 & ( late ?is radiat+ c5tre ( its life has be5 /eadily t5d+1--toge!r ) e =m ( physical & me*anical en]gy1--t[>d ma!matical expression4--,,adam1 ,,h5ry4 ,a ,lett] to ,am]ican ,tea*]s ( ,hi/ory 7,wa%+ton1 #aiaj71 p4 #aae4 #aeii;b4 ,ma!matics c be %[n to su/a9 a c]ta9 relation to rhetoric & may aid 9 det]m9+ its laws4--,,%]man ,l4 ,a4 ,univ]sity ( ,nebraska ,/udies1 ,vol4 #a1 p4 #acj4 ,,*apt] ,,xvi ,,ari?metic #afja4 ,!re is no problem 9 all ma!matics t _c be solv$ by direct c\nt+4 ,b ) ! pres5t implem5ts ( ma!matics _m op]ations c be p]=m$ 9 a few m9utes :i* )\t ma!matical me?ods w\ld take a lifetime4 --,,ma*1 ,,ern/4 ,popul> ,sci5tific ,lectures ,mc,cormack 7,*icago1 #ahih71 p4 #aig4 #afjb4 ,!re is no inquiry :i* is n f9ally r$ucible to a "q ( ,numb]s2 = !re is none :i* may n be conceiv$ ( z consi/+ 9 ! det]m9ation ( quantities by ea* o!r1 accord+ to c]ta9 relations4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #a1 *ap4 #a4 #afjc4 ,py?agoras says t numb] is ! orig9 ( all ?+s1 & c]ta9ly ! law ( numb] is ! key t unlocks ! secrets ( ! univ]se4 ,b ! law ( numb] possesses an imman5t ord]1 :i* is at fir/ sight my/ify+1 b on a m intimate acqua9tance we easily und]/& x to be intr9sically necess>y2 & ?is law ( numb] expla9s ! wondr\s consi/5cy ( ! laws ( nature4--,,c>us1 ,,paul4 ,reflections on ,magic ,squ>es2 ,moni/1 ,vol4 #af 7#aijf71 p4 #aci4 #afjd4 ,an anci5t writ] said t ari?metic & geometry are ! w+s ( ma!matics2 ,i believe "o c say )\t speak+ metaphorically t ^! two sci5ces are ! f\ndation & ess5ce ( all ! sci5ces :i* deal ) quantity4 ,n only are !y ! f\ndation1 !y are also1 z x 71 ! cap/ones2 =1 :5ev] a result has be5 arriv$ at1 9 ord] to use t result1 x is necess>y to translate x into numb]s or into l9es2 to translate x into numb]s requires ! aid ( ari?metic1 to translate x into l9es necessitates ! use ( geometry4--,,lagrange4 ,leons ,elm5taires sur les ,ma?matiques1 ,leon seconde4 #afje4 ,x is numb] :i* regulates ev]y?+ & x is measure :i* e/abli%es univ]sal ord]4444 ,a quiet peace1 an inviolable ord]1 an inflexible security amid/ all *ange & turmoil *>act]ize ! _w :i* ma!matics discloses & ^: dep?s x unlocks4--,,dillmann1 ,e4 ,die ,ma!matik die ,fackeltrg]9 e9] neu5 ,zeit 7,/uttg>t1 #ahhi71 p4 #ab4 #afjf4 ,numb]1 ! induc] ( philosophies1 ,! syn!sis ( lett]s1 4444 --,,aes*ylus4 ,quot$ 91 ,?omson1 ,j4 ,a41 ,introduction to ,sci5ce1 *ap4 #a 7,london74 #afjg4 ,among/ all ! ideas we h1 z !re is none sugge/$ to ! m9d by m ways1 s !re is none m simple1 ?an t ( unity1 or "o3 x has no %ad[ ( v>iety or composition 9 x2 e object our s5ses are employ$ ab\t2 e idea 9 our und]/&+2 e ?\ght ( our m9ds1 br+s ?is idea along ) x4 ,& !re=e x is ! mo/ intimate to our ?\ghts1 z well z x is1 9 its agreem5t to all o!r ?+s1 ! mo/ univ]sal idea we h4--,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #b1 *ap4 #af1 sect4 #a4 #afjh4 ,! simple modes ( numb] are ( all o!r ! mo/ di/9ct2 e ! lea/ v>iation1 :i* is an unit1 mak+ ea* comb9ation z cle>ly diff]5t f t :i* approa*e? ne>e/ to x1 z ! mo/ remote2 two be+ z di/9ct f "o1 z two hundr$2 & ! idea ( two z di/9ct f ! idea ( ?ree1 z ! magnitude ( ! :ole e>? is f t ( a mite4--,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #b1 *ap4 #af1 sect4 #c4 #afji4 ,! numb] ( a class is ! class ( all classes simil> to ! giv5 class4--,,russell1 ,,b]tr&4 ,pr9ciples ( ,ma!matics 7,cambridge1 #aijc71 p4 #aae4 #afaj4 ,numb] is t prop]ty ( a gr\p ( di/9ct ?+s :i* rema9s un*ang$ dur+ any *ange to :i* ! gr\p may be subject$ :i* does n de/roy ! di/9ctness ( ! individual ?+s4--,,f9e1 ,h4 ,b4 ,numb]-sy/em ( ,algebra 7,bo/on & ,new ,york1 #ahij71 p4 #c4 #afaa4 ,! sci5ce ( ari?metic may be call$ ! sci5ce ( exact limitation ( matt] & ?+s 9 space1 =ce1 & "t4 --,,p>k]1 ,f4 ,w4 ,talks on ,p$agogics 7,new ,york1 #ahid71 p4 #fd4 #afab4 ,ari?metic is ! sci5ce ( ! ,evaluation ( ,functions1 ,algebra is ! sci5ce ( ! ,trans=mation ( ,functions4 --,,h[ison1 ,g4 ,h4 ,j\rnal ( ,speculative ,philosophy1 ,vol4 #e1 p4 #age4 #afac4 ,t ari?metic re/s on pure intuition ( "t is n s obvi\s z t geometry is bas$ on pure intuition ( space1 b x may be readily prov$ z foll[s4 ,all c\nt+ consi/s 9 ! repeat$ posit+ ( unity2 only 9 ord] to "k h[ (t5 x has be5 posit$1 we m>k x ea* "t ) a diff]5t ^w3 ^! are ! num]als4 ,n[ repetition is possible only ?r\gh succession3 b succession re/s on ! imm$iate intuition ( "t1 x is intelligible only by means ( ?is latt] concept3 h5ce c\nt+ is possible only by means ( "t4--,?is dep5d5ce ( c\nt+ on "t is evid5c$ by ! fact t 9 all languages multiplication is express$ by 8times0 mal1 t is1 by a concept ( "t2 sexies1 1 six fois1 six times4 --,,s*op5hau]1 ,a4 ,die ,welt als ,vor/ellung und ,wille2 ,w]ke 7,frau5/a$t7 7,leipzig1 #ahgg71 ,bd4 #c1 p4 #ci4 #afad4 ,! miracul\s p[]s ( mod]n calculation are due to ?ree inv5tions3 ! ,arabic ,notation1 ,decimal ,fractions & ,log>i?ms4--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #afa4 #afae4 ,! gr&e/ a*ievem5t ( ! ,h9doos & ! "o :i*1 ( all ma!matical inve/igations1 has contribut$ mo/ to ! g5]al progress ( intellig5ce1 is ! inv5tion ( ! pr9ciple ( position 9 writ+ numb]s4--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #hg4 #afaf4 ,! inv5tion ( log>i?ms & ! calculation ( ! e>li] tables =m a v /rik+ episode 9 ! hi/ory ( exact sci5ce1 &1 ) ! exception ( ! ,pr9cipia ( ,newton1 !re is no ma!matical "w publi%$ 9 ! c\ntry :i* has produc$ su* important consequ5ces1 or to :i* s mu* int]e/ atta*es z to ,napi]'s ,descriptio4 --,,glai%]1 ,j4 ,w4 ,l4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,log>i?ms40 #afag4 ,all m9ds are equally capable ( atta9+ ! sci5ce ( numb]s3 yet we f9d a prodigi\s diff]5ce 9 ! p[]s ( diff]5t m51 9 t respect1 aft] !y h gr[n up1 because _! m9ds h be5 m or less ex]cis$ 9 x4 --,,johnson1 ,,samuel4 ,boswell's ,life ( ,johnson1 ,h>p]'s ,edition 7#ahga71 ,vol4 #b1 p4 #cc4 #afah4 ,! me?od ( ari?metical tea*+ is p]haps ! be/ und]/ood ( any ( ! me?ods conc]n$ ) elem5t>y /udies4--,,ba91 ,,alex&]4 ,education z a ,sci5ce 7,new ,york1 #ahih71 p4 #bhh4 #afai4 ,:at a b5efite t onely ?yng is1 to haue ! witte :ett$ & %>p5$1 ,i neade n trauell to decl>e1 si? all m5 confesse x to be z greate z maie be4 ,excepte any witlesse p]sone ?9ke he maie bee to wise4 ,b he t mo/ fe>e? t1 is lea/e 9 daung] ( x4 ,:]e=e to conclude1 ,i see mo>e m5ne to ackn[l$ge ! b5efite ( nomb]1 ?an ,i c espie willy+ to /udie1 to atta9e ! b5efites ( x4 ,_m praise x1 b fewe dooe greatly practise x3 onlesse x bee = ! vulg>e practice1 conc]ny+ ,m]*aundes trade4 ,:]e9 ! desire & hope ( ga91 make? _m willy+ to su/a9e "s trauell4 ,= aide ( :om1 ,i did sette =? ! fir/ p>te ( ,ari?metike4 ,b if !i knewe h[ f>ee ?is seconde p>te1 doee? excell ! fir/e p>te1 !i w\ld n acc\mpte any tyme lo/e1 t 7 emploi$ 9 x4 ,yea !i w\ld n ?9ke any tyme well be/[$ till !i _h gott5 so*e habilitie by x1 t x might be _! aide 9 al o!r /udies4 --,,recorde1 ,,rob]t4 ,:et/one ( ,witte 7,london1 #aeeg74 #afbj4 ,y see !n1 my fri5d1 ,i obs]v$1 t our real ne$ ( ?is bran* ( sci5ce ari?metic is probably because x seems to compel ! s\l to use our intellig5ce 9 ! se>* aft] pure tru?4 ,aye1 rem>k$ he1 x does ?is to a rem>kable ext5t4 ,h y ev] notic$ t ^? :o h a turn = ari?metic are1 ) sc>cely an exception1 naturally quick 9 all sci5ces2 & t m5 ( sl[ intellect1 if !y be tra9$ & ex]cis$ 9 ?is /udy 444 become inv>iably quick] ?an !y 7 be=e8 ,exactly s1 he repli$4 ,&1 moreov]1 ,i ?9k y w n easily f9d t _m ?+s give ! le>n] & /ud5t m tr\ble ?an ?is4 ,( c\rse n4 ,on all ^! acc\nts1 !n1 we mu/ n omit ?is bran* ( sci5ce1 b ^? ) ! be/ ( tal5ts %\ld be in/ruct$ !re94--,,plato4 ,republic ,davis1 ,bk4 #g1 *ap4 #h4 #afba4 ,ari?metic has a v great & elevat+ effect1 compell+ ! s\l to reason ab\t ab/ract numb]1 & if visible or tangible objects are obtrud+ ^u ! argum5t1 refus+ to be satisfi$4--,,plato4 ,republic ,j[ett1 ,bk4 #g1 p4 #ebe4 #afbb4 ,good ari?metic contributes p[]fully to purposive ef=t1 to conc5tration1 to t5acity ( purpose1 to g5]al%ip1 to fai? 9 "r1 & to ! joy ( a*ievem5t1 :i* are ! elem5ts t make up effici5t citiz5%ip4444 ,good ari?metic exalts ?9k+1 furni%es intellectual pleasure1 adds appreciably to love ( "r1 & subord9ates pure memory4--,,my]s1 ,,george4 ,monograph on ,ari?metic 9 ,public ,education 7,*icago71 p4 #ba4 #afbc4 ,on ! "o side we may say t ! purpose ( numb] "w is to put a *ild 9 possession ( ! ma*9]y ( calculation2 on ! o!r side x is to give him a bett] ma/]y ( ! _w ?r\gh a cle> 7ma!matical7 insight into ! v>i$ physical objects & activities4 ,! :ole _w1 f "o po9t ( view1 c be def9itely int]pret$ & appreciat$ by ma!matical measurem5ts & e/imates4 ,ari?metic 9 ! common s*ool %\ld give a *ild ?is po9t ( view1 ! ability to see & e/imate ?+s ) a ma!matical eye4--,,mcmurray1 ,c4 ,a4 ,special ,me?od 9 ,ari?metic 7,new ,york1 #aijf71 p4 #ah4 #afbd4 ,we are s accu/om$ to he> ari?metic spok5 ( z "o ( ! ?ree fundam5tal +r$i5ts 9 all s*emes ( in/ruction1 t x seems l inquir+ too curi\sly to ask :y ?is %\ld be4 ,read+1 ,writ+1 & ,ari?metic--^! ?ree are assum$ to be ( co-ord9ate rank4 ,are !y inde$ co-ord9ate1 & if s on :at gr\nds8 ,9 ?is mod]n 8trivium0 ! art ( read+ is put fir/4 ,well1 !re is no d\bt z to its "r to ! =emo/ place4 ,= read+ is ! in/rum5t ( all our acquisition4 ,x is indisp5sable4 ,!re is n an h\r 9 our lives 9 :i* x does n make a great diff]5ce to u :e!r we c read or n4 ,& ! art ( ,writ+1 too2 t is ! in/rum5t ( all communication1 & x becomes1 9 "o =m or o!r1 useful to u e day4 ,b ,c\nt+--do+ sums1--h[ (t5 9 life does ?is accompli%m5t come into ex]cise8 ,beyond ! simple/ additions1 & ! p[] to *eck ! items ( a bill1 ! ari?metical k requir$ ( any well-in=m$ p]son 9 private life is v limit$4 ,= all practical purposes1 :atev] ,i may h le>n$ at s*ool ( fractions1 or proportion1 or decimals1 is1 unless ,i happ5 to be 9 bus9ess1 f> less available to me 9 life ?an a k1 say1 ( hi/ory ( my own c\ntry1 or ! elem5t>y tru?s ( physics4 ,! tru? is1 t reg>d$ z practical arts1 read+1 writ+1 & ari?metic h no "r to be class$ toge!r z co-ord9ate elem5ts ( education2 = ! la/ ( ^! is consid]ably less useful to ! av]age man or woman n only ?an ! o!r two1 b ?an _m o!rs t might be nam$4 ,b read+1 writ+1 & su* ma!matical or logical ex]cise z may be ga9$ 9 connection ) ! manife/ation ( numb]s1 h a "r to con/itute ! prim>y elem5ts ( in/ruction4 ,& ,i believe t ari?metic1 if x des]ves ! high place t x conv5tionally holds 9 our educational sy/em1 des]ves x ma9ly on ! gr\nd t x is to be treat$ z a logical ex]cise4 ,x is ! only bran* ( ma!matics :i* has f\nd its way into prim>y & e>ly education2 o!r dep>tm5ts ( pure sci5ce be+ res]v$ = :at is call$ high] or univ]sity in/ruction4 ,b all ! argum5ts 9 favor ( tea*+ algebra & trigonometry to advanc$ /ud5ts1 apply equally to ! tea*+ ( ! pr9ciples or !ory ( ari?metic to s*oolboys4 ,x is calculat$ to d = !m exactly ! same k9d ( s]vice1 to educate "o side ( _! m9ds1 to br+ into play "o set ( faculties :i* _c be s sev]ely or prop]ly ex]cis$ 9 any o!r dep>tm5t ( le>n+4 ,9 %ort1 relatively to ! ne$s ( a beg9n]1 ,ari?metic1 z a sci5ce1 is j z valuable--x is c]ta9ly q z intelligible--z ! high] ma!matics to a univ]sity /ud5t4--,,fit*1 ,j4 ,g4 ,lectures on ,tea*+ 7,new ,york1 #aijf71 pp4 #bfg-bfh4 #afbe4 ,:at ma!matics1 !re=e are expect$ to d = ! advanc$ /ud5t at ! univ]sity1 ,ari?metic1 if taught demon/ratively1 is capable ( do+ = ! *ildr5 ev5 ( ! humble/ s*ool4 ,x furni%es tra9+ 9 reason+1 & p>ticul>ly 9 d$uctive reason+4 ,x is a discipl9e 9 clos5ess & cont9uity ( ?\ght4 ,x reveals ! nature ( fallacies1 & refuses to avail itself ( unv]ifi$ assumptions4 ,x is ! "o dep>tm5t ( s*ool-/udy 9 :i* ! sceptical & inquisitive _s has ! mo/ legitimate scope2 9 :i* au?ority goes = no?+4 ,9 o!r dep>tm5ts ( in/ruction y h a "r to ask = ! s*ol>'s confid5ce1 & to expect _m ?+s to be receiv$ on y\r te/imony ) ! und]/&+ t !y w be expla9$ & v]ifi$ aft]w>ds4 ,b "h y are ju/ifi$ 9 say+ to y\r pupil 8,believe no?+ :i* y _c und]/&4 ,take no?+ = grant$40 ,9 %ort1 ! prop] (fice ( ari?metic is to s]ve z elem5t>y tra9+ 9 logic4 ,all ?r\gh y\r "w z tea*]s y w be> 9 m9d ! fundam5tal diff]5ce betwe5 kn[+ & ?9k+2 & w feel h[ mu* m important relatively to ! heal? ( ! intellectual life ! habit ( ?9k+ is ?an ! p[] ( kn[+1 or ev5 facility ( a*iev+ visible results4 ,b "h ?is pr9ciple has special significance4 ,x is by ,ari?metic m ?an by any o!r subject 9 ! s*ool c\rse t ! art ( ?9k+--consecutively1 closely1 logically--c be effectually taught4--,,fit*1 ,j4 ,g4 ,lectures on ,tea*+ 7,new ,york1 #aijf71 pp4 #bib-bic4 #afbf4 ,ari?metic & geometry1 ^? w+s on :i* ! a/ronom] so>s z high z heav54--,,boyle1 ,,rob]t4 ,usefulness ( ,ma!matics to ,natural ,philosophy2 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbi4 #afbg4 ,ari?metical symbols are writt5 diagrams & geometrical figures are graphic =mulas4--,,hilb]t1 ,d4 ,ma!matical ,problems2 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #h 7#aijb71 p4 #ddc4 #afbh4 ,ari?metic & geometry are mu* m c]ta9 ?an ! o!r sci5ces1 because ! objects ( !m are 9 !mselves s simple & s cle> t !y ne$ n suppose any?+ :i* exp]i5ce c call 9 "q1 & bo? proce$ by a *a9 ( consequ5ces :i* reason d$uces "o f ano!r4 ,!y are also ! easie/ & cle>e/ ( all ! sci5ces1 & _! object is su* z we desire2 =1 except = want ( att5tion1 x is h>dly supposable t a man %\ld g a/ray 9 !m4 ,we mu/ n be surpris$1 h[ev]1 t _m m9ds apply !mselves by pref]5ce to o!r /udies1 or to philosophy4 ,inde$ ev]yone all[s himself m freely ! "r to make 8 guess if ! matt] be d>k ?an if x be cle>1 & x is mu* easi] to h on any "q "s vague ideas ?an to arrive at ! tru? itself on ! simple/ ( all4--,,desc>tes4 ,rules = ! ,direction ( ! ,m9d2 ,torrey's ,philosophy ( ,desc>tes 7,new ,york1 #ahib71 p4 #fc4 #afbi4 ,:y are wise few1 fools num]\s 9 ! excesse8 ',cause1 want+ numb]1 !y are numb]lesse4 --,,lovelace4 ,noah ,bridges3 ,vulg> ,ari?metike 7,london1 #afei71 p4 #abg4 #afcj4 ,! cle>ness & di/9ctness ( ea* mode ( numb] f all o!rs1 ev5 ^? t approa* ne>e/1 makes me apt to ?9k t demon/rations 9 numb]s1 if !y are n m evid5t & exact ?an 9 ext5sion1 yet !y are m g5]al 9 _! use1 & m det]m9ate 9 _! application4 ,because ! ideas ( numb]s are m precise & di/+ui%able ?an 9 ext5sion2 :]e e equality & excess are n s easy to be obs]v$ or measur$2 because our ?\ghts _c 9 space arrive at any det]m9$ smallness beyond :i* x _c g1 z an unit2 & !re=e ! quantity or proportion ( any ! lea/ excess _c be discov]$4--,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #b1 *ap4 #af1 sect4 #d4 #afca4 ,battalions ( figures are l battalions ( m51 n always z /rong z is suppos$4--,,sage1 ,m4 ,mrs4 ,pip] & ! ,society = ,psy*ical ,rese>* ,rob]tson 7,new ,york1 #aiji71 p4 #aea4 #afcb4 ,numb] 0 born 9 sup]/ition & re>$ 9 my/]y1444 numb]s 7 once made ! f\ndation ( religion & philosophy1 & ! tricks ( figures h _h a m>vell\s effect on a cr$ul\s p4--,,p>k]1 ,f4 ,w4 ,talks on ,p$agogics 7,new ,york1 #ahid71 ,p4 #fd4 #afcc4 ,a rule to trick ?' ari?metic4--,,kipl+1 ,r4 ,to ! ,true ,romance4 #afcd4 ,god made integ]s1 all else is ! "w ( man4 --,,kroneck]1 ,l4 ,jahresb]i*te d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #b1 p4 #ai4 #afce4 ,plato said 8 40 ,jacobi *ang$ ?is to 8 40 ,!n came ,kroneck] & creat$ ! memorable expression 8,die ganz5 ,zahl5 hat ,gott gema*t1 alles &]e i/ ,m5s*5w]k40--,,kle91 ,f4 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #f1 p4 #acf4 #afcf4 ,integral numb]s are ! f\nta9head ( all ma!matics4 --,,m9k[ski1 ,h4 ,diophantis*e ,approximation5 7,leipzig1 #aijg71 ,vorr$e4 #afcg4 ,! 8,disquisitiones ,ari?meticae0 t great book ) sev5 seals4--,,m]z1 ,j4 ,t4 ,a ,hi/ory ( ,european ,?\ght 9 ! ,n9ete5? ,c5tury 7,ed9burgh & ,london1 #aijh71 p4 #gba4 #afch4 ,x may fairly be said t ! g]ms ( ! mod]n algebra ( l9e> sub/itutions & concomitants are to be f\nd 9 ! fif? section ( ! ,disquisitiones ,ari?meticae2 & inv]sely1 e advance 9 ! algebraic !ory ( =ms is an acquisition to ! ari?metical !ory4--,,ma!ws1 ,g4 ,b4 ,!ory ( ,numb]s 7,cambridge1 #ahib71 ,"p #a1 sect4 #dh4 #afci4 ,/rictly speak+1 ! !ory ( numb]s has no?+ to d ) negative1 or fractional1 or irrational quantities1 z su*4 ,no !orem :i* _c be express$ )\t ref]5ce to ^! notions is purely ari?metical3 & no pro( ( an ari?metical !orem1 c be consid]$ f9ally satisfactory if x intr9sically dep5ds ^u extrane\s analytical !ories4 --,,ma!ws1 ,g4 ,b4 ,!ory ( ,numb]s 7,cambridge1 #ahib71 ,"p #a1 sect4 #a4 #afdj4 ,_m ( ! greate/ ma/]s ( ! ma!matical sci5ces 7 fir/ attract$ to ma!matical inquiry by problems relat+ to numb]s1 & no "o c glance at ! p]iodicals ( ! pres5t day :i* conta9 que/ions = solution )\t notic+ h[ s+ul> a *>m su* problems /ill cont9ue to ex]t4 ,! int]e/ 9 numb]s seems implant$ 9 ! human m9d1 & x is a pity t x %\ld n h fre] scope 9 ?is c\ntry4 ,! me?ods ( ! !ory ( numb]s are peculi> to itself1 & are n readily acquir$ by a /ud5t ^: m9d has = ye>s be5 famili>iz$ ) ! v diff]5t treatm5t :i* is appropriate to ! !ory ( cont9u\s magnitude2 x is !re=e extremely desirable t "s portion ( ! !ory %\ld be includ$ 9 ! ord9>y c\rse ( ma!matical in/ruction at our ,univ]sity4 ,f ! mom5t t ,gauss1 9 8 wond]ful treatise ( #ahja1 laid d[n ! true l9es ( ! !ory1 x ent]$ ^u a new day1 & no "o is likely to be able to d useful "w 9 any "p ( ! subject :o is unacqua9t$ ) ! pr9ciples & conceptions ) :i* he end[$ x4--,,glai%]1 ,j4 ,w4 ,l4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#ahij72 ,nature1 ,vol4 #db1 p4 #dfg4 #afda4 ,let u look = a mom5t at ! g5]al significance ( ! fact t calculat+ ma*9es actually exi/1 :i* relieve ma!maticians ( ! purely me*anical "p ( num]ical computations1 & :i* accompli% ! "w m quickly & ) a great] degree ( accuracy2 = ! ma*9e is n subject to ! slips ( ! human calculator4 ,! exi/5ce ( su* a ma*9e proves t computation is n conc]n$ ) ! significance ( numb]s1 b t x is conc]n$ ess5tially only ) ! =mal laws ( op]ation2 = x is only ^! t ! ma*9e c obey--hav+ be5 ?us con/ruct$--an intuitive p]ception ( ! significance ( numb]s be+ out ( ! "q4--,,kle91 ,f4 ,elem5t>ma!matik vom hh]5 ,/&punkte aus4 7,leipzig1 #aijh71 ,bd4 #a1 p4 #ec4 #afdb4 ,ma!matics is ! que5 ( ! sci5ces & ari?metic ! que5 ( ma!matics4 ,%e (t5 condesc5ds to r5d] s]vice to a/ronomy & o!r natural sci5ces1 b 9 all relations %e is entitl$ to ! fir/ rank4--,,gauss4 ,s>torius von ,walt]%aus53 ,gauss zum ,g$*tniss4 7,leipzig1 #ahff71 p4 #gi4 #afdc4 ,zu ,ar*im$es kam e9 wissbegi]ig] ,jngl+1 ,weihe mi*1 spra* er zu ihm1 e9 9 die gttli*e ,kun/1 ,die s h]rli*e ,di5/e d] ,/]n5kunde gelei/et1 ,h9t] dem ,uranos no* e95 ,planet5 entdeckt4 ,gttli* n5n/ ,du die ,kun/1 sie i/'s1 v]setzte d] ,weise1 ,ab] sie w> es1 bevor no* sie d5 ,kosmos er=s*t1 ,ehe sie h]rli*e ,di5/e d] ,/]n5kunde gelei/et1 ,h9t] dem ,uranos no* e95 ,planet5 entdeckt4 ,0 ,du im ,kosmos erblick/1 i/ nur d] ,gttli*5 ,abglanz1 ,9 d] ,olympi] ,s*a> ?ronet die ewige ,zahl4 --,,jacobi1 ,c4 ,g4 ,j4 ,j\rnal fr ,ma!matik1 ,bd4 #aja 7#ahhg71 p4 #cch4 ,to ,ar*im$es came a y\? int5t ^u k1 ,quo? he1 8,initiate me into ! sci5ce div9e ,:i* to a/ronomy1 lo6 su* excell5t s]vice has r5d]$1 ,& beyond ,uranus' orb a hidd5 planet reveal$40 8,call'/ ?\ ! sci5ce div9e8 ,s x is10 ! wise man respond$1 8,b s x 0 long be=e its light on ! ,cosmos x %$1 ,ere 9 a/ronomy's realm su* excell5t s]vice x r5d]$1 ,& beyond ,uranus' orb a hidd5 planet reveal$4 ,only reflection div9e is t :i* ,cosmos discloses1 ,numb] h]self sits en?ron$ among ,olympia's ho/s40 #afdd4 ,! high] ari?metic pres5ts u ) an inexhau/ible /ore ( int]e/+ tru?s1--( tru?s too1 :i* are n isolat$1 b /& 9 a close int]nal connexion1 & betwe5 :i*1 z our k increases1 we are cont9ually discov]+ new & sometimes :olly unexpect$ ties4 ,a great "p ( its !ories d]ives an additional *>m f ! peculi>ity t important propositions1 ) ! impress ( simplicity ^u !m1 are (t5 easily discov]able by induction1 & yet are ( s pr(\nd a "* t we _c f9d _! demon/ration till aft] _m va9 attempts2 & ev5 !n1 :5 we d succe$1 x is (t5 by "s t$i\s & artificial process1 :ile ! simpl] me?ods may long rema9 conceal$4--,,gauss1 ,c4 ,f4 ,preface to ,eis5/e9's ,ma!matis*e ,abh&lung5 7,b]l91 #ahdg71 ,h4 ,j4 ,s4 ,smi?4 #afde4 ,! ,!ory ( ,numb]s has acquir$ a great & increas+ claim to ! att5tion ( ma!maticians4 ,x is equally rem>kable = ! numb] & importance ( its results1 = ! precision & rigor\sness ( its demon/rations1 = ! v>iety ( its me?ods1 = ! intimate relations betwe5 tru?s app>5tly isolat$ :i* x sometimes discloses1 & = ! num]\s applications ( :i* x is susceptible 9 o!r p>ts ( analysis4--,,smi?1 ,h4 ,j4 ,s4 ,report on ! ,!ory ( ,numb]s1 ,briti% ,association1 #ahei2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #a1 p4 #ch4 #afdf4 ,! inv5tion ( ! symbol by ,gauss af=ds a /rik+ example ( ! advantage :i* may be d]iv$ f an appropriate notation1 & m>ks an epo* 9 ! developm5t ( ! sci5ce ( ari?metic4--,,ma!ws1 ,g4 ,b4 ,!ory ( ,numb]s 7,cambridge1 #ahib71 ,"p #a1 sect4 #bi4 #afdg4 ,z ,gauss fir/ po9t$ out1 ! problem ( cyclotomy1 or division ( ! circle into a numb] ( equal p>ts1 dep5ds 9 a v rem>kable way ^u ari?metical consid]ations4 ,we h "h ! e>lie/ & simple/ example ( ^? relations ( ! !ory ( numb]s to transc5d5tal analysis1 & ev5 to pure geometry1 :i* s (t5 unexpect$ly pres5t !mselves1 & :i*1 at fir/ sight1 are s my/]i\s4--,,ma!ws1 ,g4 ,b4 ,!ory ( ,numb]s 7,cambridge1 #ahib71 ,"p #a1 sect4 #afg4 #afdh4 ,i h sometimes ?\ght t ! pr(\nd my/]y :i* envelops our conceptions relative to prime numb]s dep5ds ^u ! limitations ( our faculties 9 reg>d to "t1 :i* l space may be 9 its ess5ce poly-dim5sional1 & t ?is & su* sort ( tru?s w\ld become self-evid5t to a be+ ^: mode ( p]ception is accord+ to sup]ficially z di/+ui%$ f our own limitation to l9e>ly ext5d$ "t4--,,sylve/]1 ,j4 ,j4 ,collect$ ,ma!matical ,pap]s1 ,vol4 #d1 p4 #fjj1 footnote4 ,,*apt] ,,xvii ,,algebra #agja4 ,! sci5ce ( algebra1 indep5d5tly ( any ( its uses1 has all ! advantages :i* belong to ma!matics 9 g5]al z an object ( /udy1 & :i* x is n necess>y to enum]ate4 ,view$ ei!r z a sci5ce ( quantity1 or z a language ( symbols1 x may be made ( ! greate/ s]vice to ^? :o are suffici5tly acqua9t$ ) ari?metic1 & :o h suffici5t p[] ( compreh5sion to ent] fairly ^u its difficulties4 --,,de ,,morgan1 ,a4 ,elem5ts ( ,algebra 7,london1 #ahcg71 ,preface4 #agjb4 ,algebra is g5]\s1 %e (t5 gives m ?an is ask$ ( h]4--,d',,alemb]t4 ,quot$ 9 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #b 7#aije71 p4 #bhe4 #agjc4 ,! op]ations ( symbolic ari?metick seem to me to af=d m5 "o ( ! cle>e/ ex]cises ( reason t ,i ev] yet met )1 no?+ be+ !re to be p]=m$ )\t /rict & wat*ful ratioc9ation1 & ! :ole me?od & progress ( t appe>+ at once ^u ! pap]1 :5 ! op]ation is f9i%$1 & af=d+ ! analy/ a la/+1 &1 z x 71 visible ratioc9ation4--,,boyle1 ,,rob]t4 ,works 7,london1 #aggb71 ,vol4 #c1 p4 #dbf4 #agjd4 ,! human m9d has nev] inv5t$ a labor-sav+ ma*9e equal to algebra4-- ,! ,nation1 ,vol4 #cc1 p4 #bcg4 #agje4 ,!y t are ignorant ( ,algebra _c imag9e ! wond]s 9 ?is k9d are to be done by x3 & :at fur!r improvem5ts & helps advantage\s to o!r p>ts ( k ! sagaci\s m9d ( man may yet f9d out1 x is n easy to det]m9e4 ,?is at lea/ ,i believe1 t ! ideas ( quantity are n ^? alone t are capable ( demon/ration & k2 & t o!r1 & p]haps m useful1 p>ts ( contemplation1 w\ld af=d u c]ta9ty1 if vices1 passions1 & dom9e]+ int]e/ did n oppose & m5ace su* endeav\rs4 --,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #d1 *ap4 #c1 sect4 #ah4 #agjf4 ,algebra is b writt5 geometry & geometry is b figur$ algebra4--,,g]ma91 ,,sophie4 ,mmoire sur la surfaces la/iques4 #agjg4 ,s long z algebra & geometry proce$$ sep>ately _! progress 0 sl[ & _! application limit$1 b :5 ^! two sci5ces 7 unit$1 !y mutually /r5g!n$ ea* o!r1 & m>*$ toge!r at a rapid pace t[>d p]fection4 --,,lagrange4 ,leons lm5taires sur les ,ma?matiques1 ,leon ,c9quime4 #agjh4 ,! laws ( algebra1 ?\gh sugge/$ by ari?metic1 d n dep5d on x4 ,!y dep5d entirely on ! conv5tions by :i* x is /at$ t c]ta9 modes ( gr\p+ ! symbols are to be consid]$ z id5tical4 ,?is assigns c]ta9 prop]ties to ! m>ks :i* =m ! symbols ( algebra4 ,! laws regulat+ ! manipulation ( algebraic symbols are id5tical ) ^? ( ari?metic4 ,x foll[s t no algebraic !orem c ev] contradict any result :i* c\ld be arriv$ at by ari?metic2 = ! reason+ 9 bo? cases m]ely applies ! same g5]al laws to diff]5t classes ( ?+s4 ,if an algebraic !orem c be int]pret$ 9 ari?metic1 ! correspond+ ari?metical !orem is !re=e true4--,,:itehead1 ,a4 ,n4 ,univ]sal ,algebra 7,cambridge1 #ahih71 p4 #b4 #agji4 ,t a =mal sci5ce l algebra1 ! creation ( our ab/ract ?\ght1 %\ld ?us1 9 a s5se1 dictate ! laws ( its own be+1 is v rem>kable4 ,x has requir$ ! exp]i5ce ( c5turies = u to realize ! full =ce ( ?is appeal4 --,,ma!ws1 ,g4 ,b4 ,f4 ,sp5c]3 ,*apt]s on ,aims & ,practice ( ,tea*+ 7,london1 #ahii71 p4 #ahd4 #agaj4 ,! rules ( algebra may be inve/igat$ by its own pr9ciples1 )\t any aid f geometry2 & al?\gh 9 _m cases ! two sci5ces may s]ve to illu/rate ea* o!r1 !re is n n[ ! lea/ necessity 9 ! m elem5t>y p>ts to call 9 ! aid ( ! latt] 9 exp\nd+ ! =m]4 --,,*ry/al1 ,,george4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,algebra40 #agaa4 ,algebra1 z an art1 c be ( no use to any "o 9 ! bus9ess ( life2 c]ta9ly n z taught 9 ! s*ools4 ,i appeal to e man :o has be5 ?r\gh ! s*ool r\t9e :e!r ?is be n ! case4 ,taught z an art x is ( little use 9 ! high] ma!matics1 z ^? are made to feel :o attempt to /udy ! diff]5tial calculus )\t kn[+ m ( ! pr9ciples ?an is conta9$ 9 books ( rules4 --,,de ,,morgan1 ,a4 ,elem5ts ( ,algebra 7,london1 #ahcg71 ,preface4 #agab4 ,we may always dep5d ^u x t algebra1 :i* _c be translat$ into good ,engli% & s\nd common s5se1 is bad algebra4--,,clif=d1 ,w4 ,k4 ,common ,s5se 9 ! ,exact ,sci5ces 7,london1 #ahhe71 *ap4 #a1 sect4 #g4 #agac4 ,! be/ review ( ari?metic consi/s 9 ! /udy ( algebra4--,,cajori1 ,f4 ,tea*+ & ,hi/ory ( ,ma!matics 9 ,u4 ,s4 7,wa%+ton1 #ahif71 p4 #aaj4 #agad4 ,algebra has = 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 way ari?metic may be def9$ z de/9$ to ! det]m9ation ( ! values ( functions4444 ,we w briefly say t ,algebra is ! ,calculus ( ,functions1 & ,ari?metic ! ,calculus ( ,values4--,,comte1 ,a4 ,philosophy ( ,ma!matics ,gillespie 7,new ,york1 #ahea71 p4 #ee4 #agae4 444 ! subject matt] ( algebraic sci5ce is ! ab/ract notion ( "t2 dive/$ (1 or n yet clo!d )1 any actual k :i* we may possess ( ! real ,ev5ts ( ,hi/ory1 or any conception :i* we may frame ( ,cause & ,effect 9 ,nature2 b involv+1 :at inde$ x _c be dive/$ (1 ! ?\ght ( possible ,succession1 or ( pure1 ideal ,progression4--,,hamilton1 ,w4 ,r4 ,graves' ,life ( ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #fcc4 #agaf4 444 in/ead ( seek+ to atta9 consi/5cy & uni=mity ( sy/em1 z "s mod]n writ]s h attempt$1 by bani%+ ?is ?\ght ( "t f ! high] ,algebra1 ,i seek to atta9 ! same object1 by sy/ematically introduc+ x into ! l[] or e>li] p>ts ( ! sci5ce4--,,hamilton1 ,w4 ,r4 ,graves' ,life ( ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #fcd4 #agag4 ,! circum/ances t algebra has its orig9 9 ari?metic1 h[ev] widely x may 9 ! end diff] f t sci5ce1 l$ ,sir ,isaac ,newton to designate x 8,univ]sal ,ari?metic10 a designation :i*1 vague z x is1 indicates its "* bett] ?an any o!r by :i* x has be5 attempt$ to express its functions--bett] c]ta9ly1 to ord9>y m9ds1 ?an ! designation :i* has be5 appli$ to x by ,sir ,william ,r[an ,hamilton1 "o ( ! greate/ ma!maticians ! _w has se5 s9ce ! days ( ,newton--8! ,sci5ce ( ,pure ,"t20 or ev5 ?an ! title by :i* ,de ,morgan w\ld p>aphrase ,hamilton's words--8! ,calculus ( ,succession40--,,*ry/al1 ,,george4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,algebra40 #agah4 ,"t is said to h only "o dim5sion1 & space to h ?ree dim5sions4444 ,! ma!matical quat]nion p>takes ( bo? ^! elem5ts2 9 te*nical language x may be said to be 8"t plus space10 or 8space plus "t30 & 9 ?is s5se x has1 or at lea/ involves a ref]5ce to1 f\r dim5sions4444 ,& h[ ! ,"o ( ,"t1 ( ,space ! ,?ree1 ,might 9 ! ,*a9 ( ,symbols girdl$ be4 --,,hamilton1 ,w4 ,r4 ,graves' ,life ( ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #fce4 #agai4 ,x is confid5tly pr$ict$1 by ^? be/ qualifi$ to judge1 t 9 ! com+ c5turies ,hamilton's ,quat]nions w /& out z ! great discov]y ( our n9ete5? c5tury4 ,yet h[ sil5tly has ! book tak5 its place ^u ! %elves ( ! ma!matician's libr>y6 ,p]haps n fifty m5 on ?is side ( ! ,atlantic h se5 x1 c]ta9ly n five h read x4 --,,hill1 ,,?omas4 ,nor? ,am]ican ,review1 ,vol4 #he1 p4 #bbc4 #agbj4 ,i ?9k ! "t may come :5 d\ble algebra w be ! beg9n]'s tool2 & quat]nions w be :]e d\ble algebra is n[4 ,! ,"l only kn[s :at w come above ! quat]nions4 --,,de ,,morgan1 ,a4 ,graves' ,life ( ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #dic4 #agba4 ,quat]nions came f ,hamilton aft] 8 really good "w _h be5 done2 & ?\gh beautifully +5i\s1 h be5 an unmix$ evil to ^? :o h t\*$ !m 9 any way1 includ+ ,cl]k ,maxwell4--,,?omson1 ,,william4 ,?ompson1 ,s4 ,p43 ,life ( ,"l ,kelv9 7,london1 #aiaj71 p4 #aach4 #agbb4 ,! :ole affair quat]nions has 9 respect to ma!matics a value n inf]ior to t ( 8,volapuk0 9 respect to language4--,,?omson1 ,,william4 ,?ompson1 ,s4 ,p43 ,life ( ,"l ,kelv9 7,london1 #aiaj71 p4 #aach4 #agbc4 ,a quat]nion ( maladies6 ,d s5d me "s =mula by help ( :i* ,i may s doctor !m t !y may all become imag9>y or positively equal to no?+4--,,s$gwick4 ,graves' ,life ( ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #b4 #agbd4 ,if no?+ m c\ld be said ( ,quat]nions ?an t !y enable u to exhibit 9 a s+ul>ly compact & elegant =m1 ^: mean+ is obvi\s at a glance on acc\nt ( ! utt] in>tificiality ( ! me?od1 results :i* 9 ! ord9>y ,c>tesian co-ord9ates are ( ! utmo/ complexity1 a v p[]ful argum5t = _! use w\ld be furni%$4 ,b x w\ld be unju/ to ,quat]nions to be cont5t ) su* a /atem5t2 = we are fully entitl$ to say t 9 all cases1 ev5 9 ^? to :i* ! ,c>tesian me?ods seem specially adapt$1 !y give z simple an expression z any o!r me?od2 :ile 9 ! great majority ( cases !y give a va/ly simpl] "o4 ,9 ! common me?ods a judici\s *oice ( co-ord9ates is (t5 ( imm5se importance 9 simplify+ an inve/igation2 9 ,quat]nions !re is usually no *oice1 = 7except :5 !y degrade to m]e scal>s7 !y are 9 g5]al utt]ly indep5d5t ( any p>ticul> directions 9 space1 & select ( !mselves ! mo/ natural ref]5ce l9es = ea* p>ticul> problem4 --,,tait1 ,p4 ,g4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#ahga72 ,nature1 ,vol4 #d1 p4 #bgj4 #agbe4 ,comp>+ a ,quat]nion inve/igation1 no matt] 9 :at dep>tm5t1 ) ! equival5t ,c>tesian "o1 ev5 :5 ! latt] has avail$ itself to ! utmo/ ( ! improvem5ts sugge/$ by ,high] ,algebra1 "o c h>dly help mak+ ! rem>k t !y contra/ ev5 m /rongly ?an ! decimal notation ) ! b9>y scale1 or ) ! old ,greek ari?metic--or ?an ! well-ord]$ subdivisions ( ! metrical sy/em ) ! prepo/]\s no-sy/ems ( ,great ,brita91 a m]e fragm5t ( :i* 79 ! =m ( ,table ( ,weights & ,measures7 =m1 p]haps ! mo/ effective1 if n ! mo/ +5i\s1 ( ! _m in/rum5ts ( torture employ$ 9 our elem5t>y tea*+4--,,tait1 ,p4 ,g4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#ahga72 ,nature1 ,vol4 #d1 p4 #bga4 #agbf4 ,x is true t1 9 ! eyes ( ! pure ma!matician1 ,quat]nions h "o gr& & fatal defect4 ,!y _c be appli$ to space ( n dim5sions1 !y are cont5t$ to deal ) ^? poor ?ree dim5sions 9 :i* m]e mortals are doom$ to dwell1 b :i* _c b\nd ! limitless aspirations ( a ,cayley or a ,sylve/]4 ,f ! physical po9t ( view ?is1 in/ead ( a defect1 is to be reg>d$ z ! greate/ possible recomm5dation4 ,x %[s1 9 fact1 ,quat]nions to be ! special in/rum5t s con/ruct$ = application to ! ,actual z to h ?r[n ov]bo>d ev]y?+ :i* is n absolutely necess>y1 )\t ! slighte/ consid]ation :e!r or no x 0 !reby be+ r5d]$ useless = application to ! ,inconceivable4--,,tait1 ,p4 ,g4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#ahga72 ,nature1 ,vol4 #d1 p4 #bga4 #agbg4 ,!re is an old epigram :i* assigns ! empire ( ! sea to ! ,engli%1 ( ! l& to ! ,fr5*1 & ( ! cl\ds to ! ,g]mans4 ,surely x 0 f ! cl\ds t ! ,g]mans fet*$ & 2 ! ideas :i* ^! symbols h g5]at$ are mu* too important to ! welf>e ( humanity to h come f ! sea or f ! l&4--,,:itehead1 ,a4 ,n4 ,an ,introduction to ,ma!matics 7,new ,york1 #aiaa71 p4 #hf4 #agbh4 ,n[ z to :at p]ta9s to ^! ,surd numb]s 7:i*1 z x 7 by way ( reproa* & calumny1 hav+ no m]it ( _! own are also /yl$ ,irrational1 ,irregul>1 & ,inexplicable7 !y are by _m d5i$ to be numb]s prop]ly speak+1 & are wont to be bani%$ f ari?metic to ano!r ,sci5ce1 7:i* yet is no sci5ce7 viz4 algebra4--,,b>r[1 ,,isaac4 ,ma!matical ,lectures 7,london1 #agcd71 p4 #dd4 #agbi4 ,if x is true z ,:ewell says1 t ! ess5ce ( ! triumphs ( sci5ce & its progress consi/s 9 t x enables u to consid] evid5t & necess>y1 views :i* our ance/ors held to be un9telligible & 7 unable to compreh5d1 !n ! ext5sion ( ! numb] concept to include ! irrational1 & we w at once add1 ! imag9>y1 is ! greate/ =w>d /ep :i* pure ma!matics has ev] tak54--,,hankel1 ,,h]mann4 ,!orie d] ,complex5 ,zahl5 7,leipzig1 #ahfg71 p4 #fj4 #agcj4 ,t ?is subject ( imag9>y magnitudes has hi!rto be5 consid]$ f ! wrong po9t ( view & surr\nd$ by a my/]i\s obscurity1 is to be attribut$ l>gely to an ill-adapt$ notation4 ,if = in/ance1 #a1a1 #a _h be5 call$ direct1 inv]se1 & lat]al units1 in/ead ( positive1 negative1 & imag9>y 7or ev5 impossible7 su* an obscurity w\ld h be5 out ( "q4--,,gauss1 ,c4 ,f4 ,!oria residiorum biquadraticorum1 ,comm5tatio secunda2 ,w]ke1 ,bd4 #b 7,goett+51 #ahfc71 p4 #agg4 #agca4 444 ! imag9>y1 ?is bosom-*ild ( complex my/icism4 --,d,,hr+1 ,,eug54 ,kritis*e ,ges*i*te d] allgeme95 ,pr9cipi5 d] ,me*anik 7,leipzig1 #ahgg71 p4 #eag4 #agcb4 ,judg$ by ! only /&>ds :i* are admissible 9 a pure doctr9e ( numb]s i is imag9>y 9 ! same s5se z ! negative1 ! fraction1 & ! irrational1 b 9 no o!r s5se2 all are alike m]e symbols devis$ = ! sake ( repres5t+ ! results ( op]ations ev5 :5 ^! results are n numb]s 7positive integ]s74--,,f9e1 ,h4 ,b4 ,! ,numb]-,sy/em ( ,algebra 7,bo/on1 #ahij71 p4 #cf4 #agcc4 ,?is symbol -#a is re/rict$ to a precise signification z ! repres5tative ( p]p5dicul>ity 9 quat]nions1 & ?is wond]ful algebra ( space is intimately dep5d5t ^u ! special use ( ! symbol = its symmetry1 elegance1 & p[]4 ,! immortal au?or ( quat]nions has %[n t !re are o!r significations :i* may atta* to ! symbol 9 o!r cases4 ,b ! /ronge/ use ( ! symbol is to be f\nd 9 its magical p[] ( d\bl+ ! actual univ]se1 & plac+ by its side an ideal univ]se1 its exact c\nt]p>t1 ) :i* x c be comp>$ & contra/$1 &1 by means ( curi\sly connect+ fibres1 =m ) x an organic :ole1 f :i* mod]n analysis has develop$ h] surpass+ geometry4--,,peirce1 ,,b5jam94 ,on ! ,uses & ,trans=mations ( ,l9e> ,algebras2 ,am]ican ,j\rnal ( ,ma!matics1 ,vol4 #d 7#ahha71 p4 #baf4 #agcd4 ,! conception ( ! inconceivable imag9>y1 ?is measurem5t ( :at n only does n1 b _c exi/1 is "o ( ! f9e/ a*ievem5ts ( ! human intellect4 ,no "o c d5y t su* imag9+s are inde$ imag9>y4 ,b !y lead to results gr&] ?an any :i* fl[ f ! imag9ation ( ! poet4 ,! imag9>y calculus is "o ( ! ma/]keys to physical sci5ce4 ,^! realms ( ! inconceivable af=d 9 _m places our only mode ( passage to ! doma9s ( positive k4 ,light itself lay 9 d>kness until ?is imag9>y calculus ?rew light ^u light4 ,& 9 all mod]n rese>*es into electricity1 magnetism1 & heat1 & o!r subtile physical inquiries1 ^! are ! mo/ p[]ful in/rum5ts4--,,hill1 ,,?omas4 ,nor? ,am]ican ,review1 ,vol4 #he1 p4 #bce4 #agce4 ,all ! fruitful uses ( imag9>ies1 9 ,geometry1 are ^? :i* beg9 & end ) real quantities1 & use imag9>ies only = ! int]m$iate /eps4 ,n[ 9 all su* cases1 we h a real spatial int]pretation at ! beg9n+ & end ( our argum5t1 :]e alone ! spatial int]pretation is important2 9 ! int]m$iate l9ks1 we are deal+ 9 purely algebraic mann] ) purely algebraic quantities1 & may p]=m any op]ations :i* are algebraically p]missible4 ,if ! quantities ) :i* we end are capable ( spatial int]pretation1 !n1 & only !n1 our results may be reg>d$ z geometrical4 ,to use geometrical language1 9 any o!r case1 is only a conv5i5t help to ! imag9ation4 ,to speak1 = example1 ( projective prop]ties :i* ref] to ! circul> po9ts1 is a m]e memoria te*nica = purely algebraical prop]ties2 ! circul> po9ts are n to be f\nd 9 space1 b only 9 ! auxili>y quantities by :i* geometrical equations are trans=m$4 ,t no contradictions arise f ! geometrical int]pretation ( imag9>ies is n wond]ful2 = !y are int]pret$ solely by ! rules ( ,algebra1 :i* we may admit z valid 9 _! int]pretation to imag9>ies4 ,! p]ception ( space be+ :olly abs5t1 ,algebra rules supreme1 & no inconsi/5cy c arise4--,,russell1 ,,b]tr&4 ,f\ndations ( ,geometry 7,cambridge1 #ahig71 p4 #de4 #agcf4 ,inde$1 if "o und]/&s by algebra ! application ( ari?metic op]ations to composite magnitudes ( all k9ds1 :e!r !y be rational or irrational numb] or space magnitudes1 !n ! le>n$ ,brahm9s ( ,h9do/an are ! true inv5tors ( algebra4--,,hankel1 ,,h]mann4 ,ges*i*te d] ,ma!matik im ,alt]tum und ,mittelalt] 7,leipzig1 #ahgd71 p4 #aie4 #agcg4 ,x is rem>kable to :at ext5t ,indian ma!matics ent]s into ! sci5ce ( our "t4 ,bo? ! =m & ! _s ( ! ari?metic & algebra ( mod]n times are ess5tially ,indian & n ,grecian4--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #ajj4 #agch4 ,!re are _m que/ions 9 ?is sci5ce algebra :i* le>n$ m5 h to ?is "t 9 va9 attempt$ to solve2 & !y h /at$ "s ( ^! que/ions 9 _! writ+s1 to prove t ?is sci5ce conta9s difficulties1 to sil5ce ^? :o pret5d !y f9d no?+ 9 x above _! ability1 to w>n ma!maticians aga9/ und]tak+ to answ] e "q t may be propos$1 & to excite m5 ( g5ius to attempt _! solution4 ,( ^! ,i h select$ sev54 #a4 ,to divide #aj into two p>ts1 su*1 t :5 ea* "p is add$ to its squ>e-root & ! sums multipli$ toge!r1 ! product is equal to ! suppos$ numb]4 #b4 ,:at squ>e is t1 :i* be+ increas$ or dim9i%$ by #aj1 ! sum & rema9d] are bo? squ>e numb]s8 #c4 ,a p]son said he ow$ to ,zaid #aj all b ! squ>e-root ( :at he ow$ to ,amir1 & t he ow$ ,amir #e all b ! squ>e-root ( :at he ow$ ,zaid4 #d4 ,to divide a cube numb] into two cube numb]s4 #e4 ,to divide #aj into two p>ts su*1 t if ea* is divid$ by ! o!r1 & ! two quoti5ts are add$ toge!r1 ! sum is equal to "o ( ! p>ts4 #f4 ,!re are ?ree squ>e numb]s 9 cont9u$ geometric proportion1 su*1 t ! sum ( ! ?ree is a squ>e numb]4 #g4 ,!re is a squ>e1 su*1 t :5 x is increas$ & dim9i%$ by its root & #b1 ! sum & ! diff]5ce are squ>es4--,,khulasat-,,al-,,hisab4 ,algebra2 quot$ 9 ,hutton3 ,a ,philosophical & ,ma!matical ,diction>y 7,london1 #ahae71 ,vol4 #a1 p4 #gj4 #agci4 ,! solution ( su* que/ions z ^! ref]r+ to ! solution ( cubic equations dep5ds on correct judgm5t1 aid$ by ! assi/ance ( ,god4--,,bija ,,ganita4 ,quot$ 9 ,hutton3 ,a ,philosophical & ,ma!matical ,diction>y 7,london1 #ahae71 ,vol4 #a1 p4 #fe4 #agdj4 ,= :at is ! !ory ( det]m9ants8 ,x is an algebra ^u algebra2 a calculus :i* enables u to comb9e & =etell ! results ( algebraical op]ations1 9 ! same way z algebra itself enables u to disp5se ) ! p]=mance ( ! special op]ations ( ari?metic4 ,all analysis mu/ ultimately clo! itself "u ?is =m4--,,sylve/]1 ,j4 ,j4 ,philosophical ,magaz9e1 ,vol4 #a1 7#ahea71 p4 #cjj2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #a1 p4 #bdg4 #agda4 ,fu*s4 ,fa/ m*t' i* nun mod]ne ,algebra /udi]54 ,meph4 ,i* wns*te ni*t eu* irre zu fhr54 ,0 diese ,wiss5s*aft betrifft1 ,es i/ s s*w]1 die le]e ,=m zu meid51 ,und w5n ihr es ni*t re*t begrifft1 ,v]mgt die ,indices ihr kaum zu unt]s*eid54 ,am ,be/5 i/'s1 w5n ihr nur ,e9em traut ,und auf des ,mei/]'s ,=meln baut4 ,im ,ganz5--haltet eu* an die ,symbole4 ,dann geht ihr zu d] ,=s*ung ,wohle ,ins si*re ,rei* d] ,=meln e94 ,fu*s4 ,e9 ,resultat muss beim ,symbole se98 ,meph4 ,s*on gut6 ,nur muss man si* ni*t alzu ng/li* qul54 ,d5n eb51 wo die ,resultate fehl51 ,/ellt e9 ,symbol zur re*t5 ,zeit si* e94 ,symbolis* ls/ si* alles s*reib51 ,ms/ nur im ,allgeme95 bleib54 ,w5n man d] ,glei*ung ,lsung ni*t erkannte1 ,s*reibt man sie als ,det]m9ante4 ,s*reib' 0 du will/1 nur re*ne nie 0 aus4 ,symbole lass5 treffli* si* trakti]51 ,mit e9em ,/ri* i/ alles auszufhr51 ,und mit ,symbol5 kommt man imm] aus4 --,,lasswitz1 ,,kurd4 ,d] ,fau/-,tragdie 7-n7t] ,teil2 ,zeits*rift fr ma!matis*5 und naturwiss5s*aftli*5 ,unt]ri*t1 ,bd4 #ad1 p4 #cag4 ,fu*s4 ,to /udy mod]n algebra ,i'm mo/ p]suad$4 ,meph4 ',twas n my wi% to lead !e a/ray4 ,b z conc]ns ?is sci5ce1 truly ',tis difficult to avoid ! empty =m1 ,& %\ld'/ ?\ lack cle> compreh5sion1 ,sc>cely ! indices ?\'ll "k ap>t4 ',tis safe/ f> to tru/ b "o ,& built ^u y\r ma/]'s =mulas4 ,on ! :ole--cl+ closely to y\r symbols4 ,!n1 = ! weal ( rese>* y may ga9 ,an entrance to ! =mula's sure doma94 ,fu*s4 ,! symbol1 x mu/ lead to "s result8 ,meph4 ,grant$4 ,b nev] worry ab\t results1 ,=1 m9d y1 j :]e ! results are want+ ,a symbol at ! nick ( "t appe>s4 ,to symbolic treatm5t all ?+s yield1 ,provid$ we /ay 9 ! g5]al field4 ,%\ld a solution prove elusive1 ,write ! equation 9 det]m9ant =m4 ,write :at y please1 b nev] calculate4 ,symbols are pati5t & long suff]+1 ,a s+le /roke completes ! :ole affair4 ,symbols = e purpose d suffice4 #agdb4 ,z all roads are said to lead to ,rome1 s ,i f9d1 9 my own case at lea/1 t all algebraic inquiries soon] or lat] end at ! ,capitol ( ,mod]n ,algebra ov] ^: %9+ portal is inscrib$ 8,!ory ( ,inv>iants40--,,sylve/]1 ,j4 ,j4 ,on ,newton's ,rule = ! ,discov]y ( ,imag9>y ,roots2 ,collect$ ,ma!matical ,pap]s1 ,vol4 #b1 p4 #chj4 #agdc4 ,if we consid] ! beauty ( ! !orem ,sylve/]'s ,!orem on ,newton's ,rule = ! ,discov]y ( ,imag9>y ,roots :i* has n[ be5 exp\nd$1 ! int]e/ :i* belongs to ! rule associat$ ) ! great "n ( ,newton1 & ! long lapse ( ye>s dur+ :i* ! reason & ext5t ( t rule rema9$ undiscov]$ by ma!maticians1 among :om ,maclaur91 ,w>+ & ,eul] are explicitly includ$1 we mu/ reg>d ,pr(essor ,sylve/]'s inve/igations made to ! ,!ory ( ,equations 9 mod]n times1 ju/ly to be rank$ ) ^? ( ,f\ri]1 ,/urm & ,cau*y4--,,todhunt]1 ,i4 ,!ory ( ,equations 7,london1 #aijd71 p4 #bej4 #agdd4 ,consid]+ ! rem>kable elegance1 g5]ality1 & simplicity ( ! me?od ,hom]'s ,me?od ( f9d+ ! num]ical values ( ! roots ( an equation1 x is n a little surpris+ t x has n tak5 a m prom95t place 9 curr5t ma!matical textbooks4444 ,z a matt] ( fact1 its _s is purely ari?metical2 & its beauty1 :i* c only be appreciat$ aft] "o has us$ x 9 p>ticul> cases1 is ( t indescribably simple k9d1 :i* di/+ui%es ! use ( position 9 ! decimal notation & ! arrangem5t ( ! simple rules ( ari?metic4 ,x is1 9 %ort1 "o ( ^? ?+s ^: inv5tion 0 ! creation ( a commonplace4 --,,*ry/al1 ,,george4 ,algebra 7,london & ,ed9burgh1 #ahic71 ,vol4 #a1 *ap4 #ae1 sect4 #be4 #agde4 ,to a miss+ memb] ( a family gr\p ( t]ms 9 an algebraical =mula4 ,lone & disc>d$ "o6 divorc$ by fate1 ,f> f ?y wi%$-= fell[s--:i!r art fl[n8 ,:]e l+]e/ ?\ 9 ?y b]eav$ e/ate1 ,l "s lo/ />1 or buri$ meteor /one8 ,?\ m9d/ me mu* ( t presumptu\s "o ,:o lo?1 aught less ?an greate/1 to be great1 ,f ,heav5's imm5sity fell headlong d[n ,to live =lorn1 self-c5tr$1 desolate3 ,or :o1 l ,h]aclid1 h>d exile bore1 ,n[ buoy$ by hope1 n[ /ret*$ on rack ( fe>1 ,till ?ron$ ,a/a1 waft+ to 8 e> ,words ( dim port5t ?r\gh ! ,atlantic ro>1 ,bade him 8! sanctu>y ( ! ,muse rev]e ,& /rew ) flame ! du/ ( ,isis' %ore40 --,,sylve/]1 ,j4 ,j4 ,inaugural ,lecture1 ,ox=d1 #ahhe2 ,nature1 ,vol4 #cc1 p4 #bbh4 #agdf4 ,9 e subject ( inquiry !re are c]ta9 entities1 ! mutual relations ( :i*1 "u v>i\s conditions1 x is desirable to asc]ta94 ,a c]ta9 comb9ation ( ^! entities are submitt$ to c]ta9 processes or are made ! subjects ( c]ta9 op]ations4 ,! !ory ( inv>iants 9 its wide/ sci5tific mean+ det]m9es ^! comb9ations1 elucidates _! prop]ties1 & expresses results :5 possible 9 t]ms ( !m4 ,_m ( ! g5]al pr9ciples ( political sci5ce & economics c be repres5t$ by means ( inv>iantive relations connect+ ! factors :i* ent] z entities into ! special problems4 ,! great pr9ciple ( *emical sci5ce :i* ass]ts t :5 elem5t>y or comp\nd bodies comb9e ) "o ano!r ! total weight ( ! mat]ials is un*ang$1 is ano!r case 9 po9t4 ,aga91 9 physics1 a giv5 mass ( gas "u ! op]ation ( v>y+ pressure & temp]ature has ! well-kn[n inv>iant1 pressure multipli$ by volume & divid$ by absolute temp]ature4444 ,9 ma!matics ! entities "u exam9ation may be ari?metical1 algebraical1 or geometrical2 ! processes to :i* !y are subject$ may be any ( ^? :i* are met ) 9 ma!matical "w4444 ,x is ! pr9ciple :i* is s valuable4 ,x is ! idea ( inv>iance t p]vades today all bran*es ( ma!matics4--,,macmahon1 ,p4 ,a4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#aija72 ,nature1 ,vol4 #fd1 p4 #dha4 #agdg4 ,! !ory ( inv>iants has invad$ ! doma9 ( geometry1 & has almo/ re-creat$ ! analytical !ory2 b x has done m ?an ?is = ! inve/igations ( ,cayley h requir$ a full reconsid]ation ( ! v f\ndations ( geometry4 ,x has ex]cis$ a pr(\nd influ5ce ^u ! !ory ( algebraic equations2 x has made its way into ! !ory ( diff]5tial equations2 & ! g5]alisation ( its ideas is op5+ out new regions ( mo/ advanc$ & pr(\nd functional analysis4 ,& s f> f its c\rse be+ complet$1 its que/ions fully answ]$1 or its int]e/ ext9ct1 !re is no reason to suppose t a t]m c be assign$ to its gr[? & its influ5ce4--,,=sy?1 ,a4 ,r4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#ahig72 ,nature1 ,vol4 #ef1 p4 #cgh4 #agdh4 444 ! doctr9e ( ,inv>iants1 a !ory fill+ ! heav5s l a light-be>+ e!r1 p5etrat+ all ! bran*es ( geometry & analysis1 reveal+ ev]y:]e abid+ configurations 9 ! mid/ ( *ange1 ev]y:]e disclos+ ! et]nal reign ( ! law ( =m4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #bh4 #agdi4 ,x is 9 ! ma!matical doctr9e ( ,inv>iance1 ! realm :]e9 are s\ght & f\nd configurations & types ( be+ t1 amid/ ! swirl & /ress ( c\ntless ho/s ( trans=mations rema9 immutable1 & ! _s dwells 9 contemplation ( ! s]5e & et]nal reign ( ! subtile laws ( ,=m1 x is !re t ,!ology may f9d1 if %e w1 ! cle>e/ conceptions1 ! noble/ symbols1 ! mo/ inspir+ intimations1 ! mo/ illum9at+ illu/rations1 & ! sure/ gu>antees ( ! object ( h] tea*+ & h] que/1 an ,et]nal ,be+1 un*ang+ 9 ! mid/ ( ! univ]sal flux4 --,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #db4 #agej4 ,i ?9k t "y *emi/s desir\s ( rais+ _! sci5ce to its prop] rank w\ld act wisely 9 mak+ !mselves ma/] betimes ( ! !ory ( algebraic =ms4 ,:at me*anics is to physics1 t ,i ?9k is algebraic morphology1 f\nd$ at option on ! !ory ( p>titions or ideal elem5ts1 or bo?1 is de/9$ to be to ! *emi/ry ( ! future 444 inv>iants & isom]ism are si/] !ories4--,,sylve/]1 ,j4 ,j4 ,am]ican ,j\rnal ( ,ma!matics1 ,vol4 #a 7#ahgh71 p4 #abf4 #agea4 ,! great notion ( ,gr\p1 444 ?\gh x _h b>ely m]g$ into consci\sness a hundr$ ye>s ago1 has mean:ile become a concept ( fundam5tal importance & prodigi\s f]tility1 n only af=d+ ! basis ( an impos+ doctr9e--! ,!ory ( ,gr\ps--b !re) s]v+ also z a bond ( union1 a k9d ( connective tissue1 or r z an imm5se c]ebro-sp9al sy/em1 unit+ toge!r a l>ge numb] ( widely dissimil> doctr9es z organs ( a s+le body4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #ab4 #ageb4 ,9 rec5t times ! view becomes m & m preval5t t _m bran*es ( ma!matics are no?+ b ! !ory ( inv>iants ( special gr\ps4--,,lie1 ,,sophus4 ,cont9ui]li*e ,grupp5--,s*eff]s 7,leipzig1 #ahic71 p4 #ffe4 #agec4 ,univ]sal ,algebra has be5 look$ on ) "s suspicion by _m ma!maticians1 z be+ )\t intr9sic ma!matical int]e/ & z be+ comp>atively useless z an eng9e ( inve/igation4444 ,b x may be %[n t ,univ]sal ,algebra has ! same claim to be a s]i\s subject ( ma!matical /udy z any o!r bran* ( ma!matics4--,,:itehead1 ,a4 ,n4 ,univ]sal ,algebra 7,cambridge1 #ahih71 ,preface1 p4 vi4 #aged4 ,function !ory 01 9 effect1 f\nd$ by ,cau*y2 b1 outside 8 own inve/igations1 x at fir/ made sl[ & hesitat+ progress4 ,at ! pres5t day1 its fundam5tal ideas may be said almo/ to gov]n mo/ dep>tm5ts ( ! analysis ( cont9u\s quantity4 ,on _m ( !m1 x has %$ a completely new light2 x has educ$ relations betwe5 !m be=e unkn[n4 ,x may be d\bt$ :e!r any subject is at ! pres5t day s ri*ly end[$ ) v>iety ( me?od & f]tility ( res\rce2 its activity is prodigi\s1 & no less rem>kable ?an its activity is its fre%ness4--,,=sy?1 ,a4 ,r4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#ahig72 ,nature1 ,vol4 #ef1 p4 #cgh4 #agee4 ,let me m5tion "o o!r contribution :i* ?is !ory ,!ory ( functions ( a complex v>iable has made to k ly+ some:at outside our track4 ,dur+ ! rigor\s revision to :i* ! f\ndations ( ! !ory h be5 subject$ 9 its re-e/abli%m5t by ,wei]/rass1 new ideas z reg>ds numb] & cont9uity h be5 introduc$4 ,) him & ) o!rs influ5c$ by him1 !re has !nce sprung a new !ory ( high] ari?metic2 & ) its gr[?1 mu* has concurr5tly be5 effect$ 9 ! elucidation ( ! g5]al notions ( numb] & quantity4444 ,x ?us appe>s to be ! fact t1 z ) ,plato1 or ,desc>tes1 or ,leibnitz1 or ,kant1 ! activity ( pure ma!matics is aga9 l5d+ "s assi/ance to ! bett] compreh5sion ( ^? notions ( "t1 space1 numb]1 quantity1 :i* und]lie a philosophical conception ( ! univ]se4 --,,=sy?1 ,a4 ,r4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#ahig72 ,nature1 ,vol4 #ef1 p4 #cgh4 ,,*apt] ,,xviii ,,geometry #ahja4 ,! sci5ce ( figures is mo/ glori\s & beautiful4 ,b h[ inaptly x has receiv$ ! "n geometry6--,,fris*l9us1 ,n4 ,dialog #a4 #ahjb4 ,plato said t ,god geometrizes cont9ually4--,,plut>*4 ,convivialium disputationum1 lib] #h1 #b4 #ahjc4 4 ,let no "o ignorant ( geometry ent] my door4--,,plato4 ,tzetzes1 ,*iliad1 #h1 #igb4 #ahjd4 ,all ! au?orities agree t he ,plato made a /udy ( geometry or "s exact sci5ce an indisp5sable prelim9>y to t ( philosophy4 ,! inscription ov] ! entrance to 8 s*ool ran 8,let none ignorant ( geometry ent] my door10 & on "o occasion an applicant :o knew no geometry is said to h be5 refus$ admission z a /ud5t4--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #de4 #ahje4 ,=m & size con/itute ! f\ndation ( all se>* = tru?4--,,p>k]1 ,f4 ,w4 ,talks on ,p$agogics 7,new ,york1 #ahid71 p4 #gb4 #ahjf4 ,at pres5t ! sci5ce ( geometry is 9 flat contradiction to ! language :i* geometricians use1 z w h>dly be d5i$ by ^? :o h any acqua9tance ) ! /udy3 = !y speak ( f9d+ ! side ( a squ>e1 & apply+ & add+1 & s on1 z if !y 7 engag$ 9 "s bus9ess1 & z if all _! propositions _h a practical end 9 view3 :]eas 9 reality ! sci5ce is pursu$ :olly = ! sake ( k4 ,c]ta9ly1 he said4 ,!n mu/ n a fur!r admission be made8 ,:at admission8 ,! admission t ?is k at :i* geometry aims is ( ! et]nal1 & n ( ! p]i%+ & transi5t4 ,t may be easily all[$4 ,geometry1 no d\bt1 is ! k ( :at et]nally exi/s4 ,!n1 my noble fri5d1 geometry w draw ! s\l t[>ds tru?1 & create ! m9d ( philosophy1 & raise up t :i* is n[ unhappily all[$ to fall d[n4--,,plato4 ,republic ,j[ett-,davies1 ,bk4 #g1 p4 #ebg4 #ahjg4 ,among !m ! ,greeks geometry 0 held 9 highe/ honor3 no?+ 0 m glori\s ?an ma!matics4 ,b we h limit$ ! usefulness ( ?is art to measur+ & calculat+4 --,,cic]o4 ,tusculanae ,disputationes1 #a1 #b1 #e4 #ahjh4 ,geometria1 ,?r\gh :i* a man ha? ! sleight ,( l5g?1 & br$e1 ( dep?1 ( height4 --,,g[]1 ,,john4 ,confessio ,amantis1 ,bk4 #g4 #ahji4 ,geometrical tru?s are 9 a way asymptotes to physical tru?s1 t is to say1 ! latt] approa* ! =m] indef9itely ne> )\t ev] rea*+ !m exactly4--,d',,alemb]t4 ,quot$ 9 ,rebire3 ,ma?matiques et ,ma?matici5s 7,p>is1 #ahih71 p4 #aj4 #ahaj4 ,geometry exhibits ! mo/ p]fect example ( logical /ratagem4--,,buckle1 ,h4 ,t4 ,hi/ory ( ,civilization 9 ,engl& 7,new ,york1 #ahia71 ,vol4 #b1 p4 #cdb4 #ahaa4 ,x is ! glory ( geometry t f s few pr9ciples1 fet*$ f )\t1 x is able to accompli% s mu*4--,,newton4 ,philosophiae ,naturalis ,pr9cipia ,ma!matica1 ,praefat4 #ahab4 ,geometry is ! application ( /rict logic to ^? prop]ties ( space & figure :i* are self-evid5t1 & :i* !re=e _c be disput$4 ,b ! rigor ( ?is sci5ce is c>ri$ "o /ep fur!r2 = no prop]ty1 h[ev] evid5t x may be1 is all[$ to pass )\t demon/ration1 if t c be giv54 ,! "q is !re=e to demon/rate all geometrical tru?s ) ! smalle/ possible numb] ( assumptions4 --,,de ,,morgan1 ,a4 ,on ! ,/udy & ,difficulties ( ,ma!matics 7,*icago1 #aijb71 p4 #bca4 #ahac4 ,geometry is a true natural sci5ce3--only m simple1 & !re=e m p]fect ?an any o!r4 ,we mu/ n suppose t1 because x admits ! application ( ma!matical analysis1 x is !re=e a purely logical sci5ce1 indep5d5t ( obs]vation4 ,e body /udi$ by geomet]s pres5ts "s primitive ph5om5a :i*1 n be+ discov]able by reason+1 mu/ be due to obs]vation alone4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #a1 *ap4 #c4 #ahad4 ,geometry 9 e proposition speaks a language :i* exp]i5ce nev] d>es to utt]2 & inde$ ( :i* %e b half compreh5ds ! mean+4 ,exp]i5ce sees t ! ass]tions are true1 b %e sees n h[ pr(\nd & absolute is _! tru?4 ,%e unhesitat+ly ass5ts to ! laws :i* geometry deliv]s1 b %e does n pret5d to see ! orig9 ( _! obligation4 ,%e is always ready to ackn[l$ge ! sway ( pure sci5tific pr9ciples z a matt] ( fact1 b %e does n dream ( (f]+ h] op9ion on _! au?ority z a matt] ( "r2 /ill less c %e ju/ly claim to h]self ! s\rce ( t au?ority4--,,:ewell1 ,,william4 ,! ,philosophy ( ! ,inductive ,sci5ces1 ,"p #a1 ,bk4 #a1 *ap4 #f1 sect4 #a 7,london1 #aheh74 #ahae4 ,geometry is ! sci5ce creat$ to give und]/&+ & ma/]y ( ! ext]nal relations ( ?+s2 to make easy ! explanation & description ( su* relations & ! transmission ( ?is ma/]y4--,,hal/$1 ,g4 ,b4 ,proce$+s ( ! ,am]ican ,association = ! ,advancem5t ( ,sci5ce 7#aijd71 p4 #cei4 #ahaf4 ,a ma!matical po9t is ! mo/ indivisible & unique ?+ :i* art c pres5t4--,,donne1 ,,john4 ,lett]s1 #ba4 #ahag4 ,x is c]ta9 t f its complet5ess1 uni=mity & faultlessness1 f its arrangem5t & progressive "*1 & f ! univ]sal adoption ( ! complete/ & be/ l9e ( argum5t1 ,euclid's 8,elem5ts0 /& pre-em95tly at ! head ( all human productions4 ,9 no sci5ce1 9 no dep>tm5t ( k1 has any?+ appe>$ l ?is "w3 = upw>d ( #bjjj ye>s x has comm&$ ! admiration ( mank9d1 & t p]iod has sugge/$ little t[>d its improvem5t4--,,kell&1 ,p4 ,lectures on ! ,pr9ciples ( ,demon/rative ,ma!matics 7,london1 #ahdc71 p4 #ag4 #ahah4 ,9 comp>+ ! p]=mance 9 ,euclid ) t 9 ,ari?metic & ,algebra !re c\ld be no d\bt t ,euclid _h made ! deepe/ & mo/ b5eficial impression3 9 fact x might be ass]t$ t ?is con/itut$ by f> ! mo/ valuable "p ( ! :ole tra9+ to :i* su* p]sons /ud5ts1 ! majority ( :i* 7 n di/+ui%$ = ma!matical ta/e & p[] 7 subject$4--,,todhunt]1 ,i4 ,essay on ,elem5t>y ,geometry2 ,conflict ( ,/udies & o!r ,essays 7,london1 #ahgc71 p4 #afg4 #ahai4 ,9 ,engl& ! geometry /udi$ is t ( ,euclid1 & ,i hope x nev] w be any o!r2 = ?is reason1 t s mu* has be5 writt5 on ,euclid1 & all ! difficulties ( geometry h s uni=mly be5 consid]$ ) ref]5ce to ! =m 9 :i* !y appe> 9 ,euclid1 t ! /udy ( t au?or is a bett] key to a great quantity ( useful read+ ?an any o!r4 --,,de ,,morgan1 ,a4 ,elem5ts ( ,algebra 7,london1 #ahcg71 ,introduction4 #ahbj4 ,?is book ,euclid has be5 = ne>ly tw5ty-two c5turies ! enc\ragem5t & guide ( t sci5tific ?\ght :i* is "o ?+ ) ! progress ( man f a worse to a bett] /ate4 ,! enc\ragem5t2 = x conta9$ a body ( k t 0 really kn[n & c\ld be reli$ on1 & t moreov] 0 gr[+ 9 ext5t & application4 ,= ev5 at ! "t ?is book 0 writt5--%ortly aft] ! f\ndation ( ! ,alex&rian ,museum--,ma!matics 0 no long] ! m]ely ideal sci5ce ( ! ,platonic s*ool1 b _h />t$ on h] c>e] ( conque/ ov] ! :ole _w ( ,ph5om5a4 ,! guide2 = ! aim ( e sci5tific /ud5t ( e subject 0 to br+ 8 k ( t subject into a =m z p]fect z t :i* geometry _h atta9$4 ,f> up on ! great m\nta9 ( ,tru?1 :i* all ! sci5ces hope to scale1 ! =emo/ ( t sacr$ si/]hood 0 se51 beckon+ = ! re/ to foll[ h]4 ,& h5ce %e 0 call$1 9 ! dialect ( ! ,phy?agoreans1 8! purifi] ( ! reasonable s\l40--,,clif=d1 ,w4 ,k4 ,lectures & ,essays 7,london1 #aija71 ,vol4 #a1 p4 #ced4 #ahba4 ,euclid at once ! inspiration & aspiration ( sci5tific ?\ght4--,,clif=d1 ,w4 ,k4 ,lectures & ,essays 7,london1 #aija71 ,vol #a1 p4 #cee4 #ahbb4 ,! 8elem5ts0 ( ! ,great ,alex&rian rema9 = all "t ! fir/1 & "o may v5ture to ass]t1 ! only p]fect model ( logical exactness ( pr9ciples1 & ( rigor\s developm5t ( !orems4 ,if "o w\ld see h[ a sci5ce c be con/ruct$ & develop$ to its m9ute/ details f a v small numb] ( intuitively p]ceiv$ axioms1 po/ulates1 & pla9 def9itions1 by means ( rigor\s1 "o w\ld almo/ say *a/e1 syllogism1 :i* n[h]e makes use ( surreptiti\s or =eign aids1 if "o w\ld see h[ a sci5ce may ?us be con/ruct$ "o mu/ turn to ! elem5ts ( ,euclid4--,,hankel1 ,h4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahhd71 p4 #g4 #ahbc4 ,if we consid] him ,euclid z mean+ to be :at 8 comm5tators h tak5 him to be1 a model ( ! mo/ unscrupul\s =mal rig\r1 we c d5y t he has altoge!r succe$$1 ?\gh we admit t he made ! ne>e/ approa*4 --,,de ,,morgan1 ,a4 ,smi?'s ,diction>y ( ,greek & ,roman ,biography & ,my?ology 7,london1 #aijb72 ,article 8,eucleides40 #ahbd4 ,! ,elem5ts ( ,euclid is z small a "p ( ma!matics z ! ,iliad is ( lit]ature2 or z ! sculpture ( ,phidias is ( ! _w's total art4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #h4 #ahbe4 ,i %\ld rejoice to see 444 ,euclid hon\rably %elv$ or buri$ 8deep] ?an did ev] plummet s\nd0 out ( ! s*oolboys' rea*2 morphology introduc$ into ! elem5ts ( algebra2 projection1 correlation1 & motion accept$ z aids to geometry2 ! m9d ( ! /ud5t quick5$ & elevat$ & 8 fai? awak5$ by e>ly initiation into ! rul+ ideas ( pol>ity1 cont9uity1 inf9ity1 & famili>ization ) ! doctr9es ( ! imag9>y & inconceivable4--,,sylve/]1 ,j4 ,j4 ,a ,plea = ! ,ma!matician2 ,nature1 ,vol4 #a1 p4 #bfa4 #ahbf4 ,! e>ly /udy ( ,euclid made me a hat] ( geometry1 444 & yet1 9 spite ( ?is repugnance1 :i* _h become a second nature 9 me1 :5ev] ,i w5t f> en\gh into any ma!matical "q1 ,i f\nd ,i t\*$1 at la/1 a geometrical bottom4--,,sylve/]1 ,j4 ,j4 ,a ,plea = ! ,ma!matician2 ,nature1 ,vol4 #a1 p4 #bfb4 #ahbg4 ,newton _h s rem>kable a tal5t = ma!matics t ,euclid's ,geometry seem$ to him 8a trifl+ book10 & he wond]$ t any man %\ld h tak5 ! tr\ble to demon/rate propositions1 ! tru? ( :i* 0 s obvi\s to him at ! fir/ glance4 ,b1 on attempt+ to read ! m ab/ruse geometry ( ,desc>tes1 )\t hav+ ma/]$ ! elem5ts ( ! sci5ce1 he 0 baffl$1 & 0 glad to come back aga9 to 8 ,euclid4--,,p>ton1 ,,james4 ,sir ,isaac ,newton4 #ahbh4 ,z to ! ne$ ( improvem5t !re c be no "q :il/ ! reign ( ,euclid cont9ues4 ,my own idea ( a useful c\rse is to beg9 ) ari?metic1 & !n n ,euclid b algebra4 ,next1 n ,euclid1 b practical geometry1 solid z well z plane2 n demon/ration1 b to make acqua9tance4 ,!n n ,euclid1 b elem5t>y vectors1 conjo9$ ) algebra1 & appli$ to geometry4 ,addition fir/2 !n ! scal> product4 ,elem5t>y calculus %\ld g on simultane\sly1 & come into ! vector algebraic geometry aft] a bit4 ,euclid might be an extra c\rse = le>n$ m51 l ,hom]4 ,b ,euclid = *ildr5 is b>b>\s4--,,heaviside1 ,,oliv]4 ,electro-,magnetic ,!ory 7,london1 #ahic71 ,vol4 #a1 p4 #adh4 #ahbi4 ,geometry is no?+ if x be n rigor\s1 & ! :ole educational value ( ! /udy is lo/1 if /rictness ( demon/ration be trifl$ )4 ,! me?ods ( ,euclid are1 by almo/ univ]sal cons5t1 unexceptionable 9 po9t ( rig\r4 --,,smi?1 ,h4 ,j4 ,s4 ,nature1 ,vol4 #h1 p4 #dej4 #ahcj4 ,to seek = pro( ( geometrical propositions by an appeal to obs]vation proves no?+ 9 reality1 except t ! p]son :o has rec\rse to su* gr\nds has no due appreh5sion ( ! nature ( geometrical demon/ration4 ,we h he>d ( p]sons :o conv9ce !mselves by measurem5t t ! geometrical rule respect+ ! squ>es on ! sides ( a "r-angles triangle 0 true3 b ^! 7 p]sons ^: m9ds _h be5 engross$ by practical habits1 & 9 :om speculative developm5t ( ! idea ( space _h be5 /ifl$ by o!r employm5ts4--,,:ewell1 ,,william4 ,! ,philosophy ( ! ,inductive ,sci5ces1 7,london1 #aheh71 ,"p #a1 ,bk4 #b1 *ap4 #a1 sect4 #d4 #ahca4 ,no "o has ev] giv5 s easy & natural a *a9 ( geometrical consequ5ces z ,euclid4 ,!re is a nev]-err+ tru? 9 ! results4--,,de ,,morgan1 ,a4 ,smi?'s ,diction>y ( ,greek & ,roman ,biography & ,my?ology 7,london1 #aijb72 ,article 8,eucleides40 #ahcb4 ,beyond "q1 ,egyptian geometry1 su* z x 01 0 eag]ly /udi$ by ! e>ly ,greek philosoph]s1 & 0 ! g]m f :i* 9 _! h&s grew t magnific5t sci5ce to :i* e ,engli%man is indebt$ = 8 fir/ lessons 9 "r see+ & ?9k+4--,,g[1 ,,james4 ,a ,%ort ,hi/ory ( ,greek ,ma!matics 7,cambridge1 #ahhd71 p4 #aca4 #ahcc4 ,a figure & a /ep onw>d3 ,n a figure & a flor94 --,,motto ,,( ,,! ,,py?agorean ,,bro!rhood4 ,w4 ,b4 ,frankl&3 ,/ory ( ,euclid 7,london1 #aijb71 p4 #cc4 #ahcd4 ,! doctr9e ( proportion1 z laid d[n 9 ! fif? book ( ,euclid1 is1 probably1 /ill unsurpass$ z a ma/]piece ( exact reason+2 al?\gh ! cumbr\sness ( ! =ms ( expression :i* 7 adopt$ 9 ! old geometry has l$ to ! total exclusion ( ?is "p ( ! elem5ts f ! ord9>y c\rse ( geometrical education4 ,a zeal\s def5d] ( ,euclid might add ) tru? t ! gap ?us creat$ 9 ! elem5t>y tea*+ ( ma!matics has nev] be5 adequately suppli$4 --,,smi?1 ,h4 ,j4 ,s4 ,presid5tial ,address ,briti% ,association = ! ,advancem5t ( ,sci5ce 7#ahgc72 ,nature1 ,vol4 #h1 p4 #dea4 #ahce4 ,! ,def9ition 9 ! ,elem5ts1 accord+ to ,clavius1 is ?is3 ,magnitudes are said to be 9 ! same ,reason ratio1 a fir/ to a second1 & a ?ird to a f\r?1 :5 ! ,equimultiples ( ! fir/ & ?ird accord+ to any ,multiplication :atsoev] are bo? toge!r ei!r %ort (1 equal to1 or exce$ ! ,equimultiples ( ! second & f\r?1 if ^? be tak51 :i* answ] "o ano!r4444 ,su* is ,euclid's ,def9ition ( ,proportions2 t sc>e-,cr[ at :i* ! ov] mode/ or slo?ful ,dispositions ( ,m5 are g5]ally affright$3 !y are mode/1 :o di/ru/ _! own ,ability1 z soon z a ,difficulty appe>s1 b !y are slo?ful t w n give "s ,att5tion = ! le>n+ ( ,sci5ces2 z if :ile we are involv$ 9 ,obscurity we c\ld cle> ourselves )\t ,lab\r4 ,bo? ( :i* ,sorts ( ,p]sons are to be admoni%$1 t ! =m] be n disc\rag$1 nor ! latt] refuse a little ,c>e & ,dilig5ce :5 a ,?+ requires "s ,/udy4--,,b>r[1 ,,isaac4 ,ma!matical ,lectures 7,london1 #agcd71 p4 #chh4 #ahcf4 ,( all bran*es ( human k1 !re is none :i*1 l x geometry has sprung a completely arm$ ,m9]va f ! head ( ,jupit]2 none be=e ^: dea?-deal+ ,aegis d\bt & inconsi/5cy h s little d>$ to raise _! eyes4 ,x escapes ! t$i\s & tr\blesome task ( collect+ exp]im5tal facts1 :i* is ! prov9ce ( ! natural sci5ces 9 ! /rict s5se ( ! ^w3 ! sole =m ( its sci5tific me?od is d$uction4 ,conclusion is d$uc$ f conclusion1 & yet no "o ( common s5se d\bts b t ^! geometrical pr9ciples mu/ f9d _! practical application 9 ! real _w ab\t u4 ,l& survey+1 z well z ar*itecture1 ! con/ruction ( ma*9]y no less ?an ma!matical physics1 are cont9ually calculat+ relations ( space ( ! mo/ v>i$ k9ds by geometrical pr9ciples2 !y expect t ! success ( _! con/ructions & exp]im5ts %all agree ) _! calculations2 & no case is kn[n 9 :i* ?is expectation has be5 falsifi$1 provid$ ! calculations 7 made correctly & ) suffici5t data4--,,helmholtz1 ,h4 ,! ,orig9 & ,significance ( ,geometrical ,axioms2 ,popul> ,sci5tific ,lectures ,atk9son1 ,second ,s]ies 7,new ,york1 #ahha71 p4 #bg4 #ahcg4 ,! amaz+ triumphs ( ?is bran* ( ma!matics geometry %[ h[ p[]ful a weapon t =m ( d$uction is :i* proce$s by an artificial rep>ation ( facts1 9 !mselves insep>able4--,,buckle1 ,h4 ,t4 ,hi/ory ( ,civilization 9 ,engl& 7,new ,york1 #ahia71 ,vol4 #b1 p4 #cdc4 #ahch4 ,e !orem 9 geometry is a law ( ext]nal nature1 & might h be5 asc]ta9$ by g5]aliz+ f obs]vation & exp]im5t1 :i* 9 ?is case resolve !mselves into comp>isons & measurem5ts4 ,b x 0 f\nd practicable1 & be+ practicable 0 desirable1 to d$uce ^! tru?s by ratioc9ation f a small numb] ( g5]al laws ( nature1 ! c]ta9ty & univ]sality ( :i* 0 obvi\s to ! mo/ c>eless obs]v]1 & :i* compose ! fir/ pr9ciples & ultimate premises ( ! sci5ce4--,,mill1 ,j4 ,s4 ,sy/em ( ,logic1 ,bk4 #c1 *ap4 #bd1 sect4 #g4 #ahci4 ,all su* reason+s natural philosophy1 *emi/ry1 agriculture1 political economy1 etc4 are1 9 comp>ison ) ma!matics1 v complex2 requir+ s mu* m ?an t does1 beyond ! process ( m]ely d$uc+ ! conclusion logically f ! premises3 s t x is no wond] t ! longe/ ma!matical demon/ration %\ld be mu* m easily con/ruct$ & und]/ood1 ?an a mu* %ort] tra9 ( j reason+ conc]n+ real facts4 ,! =m] has be5 aptly comp>$ to a long & /eep1 b ev5 & regul>1 flight ( /eps1 :i* tries ! brea?1 & ! /r5g?1 & ! p]sev]ance only2 :ile ! latt] resembles a %ort1 b rugg$ & unev51 asc5t up a precipice1 :i* requires a quick eye1 agile limbs1 & a firm /ep2 & 9 :i* we h to tread n[ on ?is side1 n[ on t--ev] consid]+ z we proce$1 :e!r ?is or t projection w af=d room = our foot1 or :e!r "s loose /one may n slide f "u u4 ,!re are probably z _m /eps ( pure reason+ 9 "o ( ! long] ( ,euclid's demon/rations1 z 9 ! :ole ( an argum5tative treatise on "s o!r subject1 occupy+ p]haps a consid]able volume4--,,:ately1 ,r4 ,elem5ts ( ,logic1 ,bk4 #d1 *ap4 #b1 sect4 #e4 #ahdj4 ,geometry t held acqua9tance ) ! />s1 ,& w$d$ s\l to s\l 9 pure/ bond ,( reason1 undi/urb$ by space or "t4 --,,wordswor?4 ,! ,prelude1 ,bk4 #e4 #ahda4 ,! /atem5t t a giv5 individual has receiv$ a s\nd geometrical tra9+ implies t he has segregat$ f ! :ole ( 8 s5se impressions a c]ta9 set ( ^! impressions1 t he has elim9at$ f _! consid]ation all irrelevant impressions 79 o!r words1 acquir$ a subjective comm& ( ^! impressions71 t he has develop$ on ! basis ( ^! impressions an ord]$ & cont9u\s sy/em ( logical d$uction1 & f9ally t he is capable ( express+ ! nature ( ^! impressions & 8 d$uctions !refrom 9 t]ms simple & free f ambiguity4 ,n[ ! slighte/ consid]ation w conv9ce any "o n already conv]sant ) ! idea1 t ! same sequ5ce ( m5tal processes und]lies ! :ole c>e] ( any individual 9 any walk ( life if only he is n conc]n$ entirely ) manual labor2 consequ5tly a full tra9+ 9 ! p]=mance ( su* sequ5ces mu/ be reg>d$ z =m+ an ess5tial "p ( any education wor?y ( ! "n4 ,moreov] ! full appreciation ( su* processes has a high] value ?an is conta9$ 9 ! m5tal tra9+ involv$1 great ?\gh ?is be1 = x induces an appreciation ( intellectual unity & beauty :i* plays = ! m9d t "p :i* ! appreciation ( s*emes ( %ape & color plays = ! arti/ic faculties2 or1 aga91 t "p :i* ! appreciation ( a body ( religi\s doctr9e plays = ! e?ical aspirations4 ,n[ geometry is n ! sole possible basis = inculcat+ ?is appreciation4 ,logic is an alt]native = adults1 provid$ t ! individual is possess$ ( suffici5t wide1 ?\gh r\gh1 exp]i5ce on :i* to base 8 reason+4 ,geometry is1 h[ev]1 highly desirable 9 t ! objective bases are s simple & precise t !y c be grasp$ at an e>ly age1 t ! am\nt ( tra9+ = ! imag9ation is v l>ge1 t ! d$uctive processes are n beyond ! scope ( ord9>y boys1 & f9ally t x af=ds a bett] basis = ex]cise 9 ! art ( simple & exact expression ?an any o!r possible subject ( a s*ool c\rse4 --,,c>son1 ,g4 ,w4 ,l4 ,! ,functions ( ,geometry z a ,subject ( ,education 7,tonbridge1 #aiaj71 p4 #c4 #ahdb4 ,x seems to me t ! ?+ t is want+ 9 ! education ( wom5 is n ! acqua9tance ) any facts1 b accurate & sci5tific habits ( ?\ght1 & ! c\rage to ?9k t true :i* appe>s unlikely4 ,& = supply+ ?is want !re is a special advantage 9 geometry1 namely t x does n require /udy ( a physically labori\s k9d1 b r t rapid intuition :i* wom5 c]ta9ly possess2 s t x is fit to become a sci5tific pursuit = !m4--,,clif=d1 ,w4 ,k4 ,quot$ by ,pollock 9 ,clif=d's ,lectures & ,essays 7,london1 #aija71 ,vol4 #a1 ,introduction1 p4 #dc4 #ahdc4 ,on ! lecture slate ,! circle r\nd$ "u female h&s ,) flawless demon/ration4 --,,t5nyson4 ,! ,pr9cess1 ,,ii1 l4 #dic4 #ahdd4 ,x is pla9 t t "p ( geometry :i* be>s ^u /rategy does conc]n u4 ,= 9 pit*+ camps1 or 9 occupy+ positions1 or 9 clos+ or ext5d+ ! l9es ( an army1 & 9 all ! o!r manuvres ( an army :e!r 9 battle or on ! m>*1 x w make a great diff]5ce to a g5]al1 :e!r he is a geometrician or n4--,,plato4 ,republic1 ,bk4 #g1 p4 #ebf4 #ahde4 ,!n no?+ %\ld be m effectually enact$1 ?an t ! inhabitants ( y\r fair city %\ld le>n geometry4 ,moreov] ! sci5ce has indirect effects1 :i* are n small4 ,( :at k9d are !y8 he said4 ,!re are ! milit>y advantages ( :i* y spoke1 ,i said2 & 9 all dep>tm5ts ( /udy1 z exp]i5ce proves1 any "o :o has /udi$ geometry is inf9itely quick] ( appreh5sion4--,,plato4 ,republic ,j[ett1 ,bk4 #g1 p4 #ebg4 #ahdf4 ,x is d\btful if we h any o!r subject t does s mu* to br+ to ! front ! dang] ( c>elessness1 ( slov5ly reason+1 ( inaccuracy1 & ( =getfulness z ?is sci5ce ( geometry1 :i* has be5 s poli%$ & p]fect$ z ! c5turies h gone on4--,,smi?1 ,d4 ,e4 ,! ,tea*+ ( ,geometry 7,bo/on1 #aiaa71 p4 #ab4 #ahdg4 ,! culture ( ! geometric imag9ation1 t5d+ to produce precision 9 remembrance & inv5tion ( visible =ms w1 !re=e1 t5d directly to increase ! appreciation ( works ( belles-lett]s4--,,hill1 ,,?omas4 ,! ,uses ( ,ma!sis2 ,biblio!ca ,sacra1 ,vol4 #cb1 p4 #ejd4 #ahdh4 ,yet may we n entirely ov]look ,! pleasures ga!r$ f ! rudim5ts ,( geometric sci5ce4 ,?\gh advanc$ ,9 ^! inquiries1 ) regret ,i speak1 ,no f>!r ?an ! ?re%old1 !re ,i f\nd ,bo? elevation & compos$ delight3 ,) ,indian awe & wond]1 ignorance pleas$ ,) its own /ruggles1 did ,i m$itate ,on ! relations ^? ab/ractions be> ,to ,nature's laws4 99 99 99 99 99 ,m frequ5tly f ! same s\rce ,i drew ,a pleasure quiet & pr(\nd1 a s5se ,( p]man5t & univ]sal sway1 ,& p>am\nt belief2 !re1 recogniz$ ,a type1 = f9ite natures1 ( ! "o ,supreme ,exi/5ce1 ! surpass+ life ,:i* to ! b\nd>ies ( space & "t1 ,( melan*oly space & doleful "t1 ,sup]ior & incapable ( *ange1 ,nor t\*$ by welt]+s ( passion--is1 ,& ha? ! "n ( ,god4 ,transc5d5t peace ,& sil5ce did wait ^u ^! ?\ghts ,t 7 a frequ5t com=t to my y\?4 99 99 99 99 99 ,mighty is ! *>m ,( ^? ab/ractions to a m9d beset ,) images & haunt$ by himself1 ,& specially delightful unto me ,0 t cle> syn!sis built up al(t ,s gracefully2 ev5 !n :5 x appe>$ ,n m ?an a m]e play?+1 or a toy ,to s5se embodi$3 n ! ?+ x is ,9 v]ity1 an indep5d5t _w1 ,creat$ out ( pure intellig5ce4 --,,wordswor?4 ,! ,prelude1 ,bk4 #f4 #ahdi4 ',tis told by "o :om /ormy wat]s ?rew1 ,) fell[-suff]]s by ! %ipwreck sp>$1 ,^u a des]t coa/1 t hav+ br\ght ,to l& a s+le volume1 sav$ by *ance1 ,a treatise ( ,geometry1 he wont1 ,al?\gh ( food & clo?+ de/itute1 ,& beyond common wret*$ness depress$1 ,to "p f company1 & take ?is book 7,!n fir/ a self taught pupil 9 its tru?s7 ,to spots remote1 & draw 8 diagrams ,) a long /aff ^u ! s&1 & ?us ,did (t beguile 8 sorr[1 & almo/ ,=get 8 feel+3 --,,wordswor?4 ,! ,prelude1 ,bk4 #f4 #ahej4 ,we /udy art because we receive pleasure f ! great works ( ! ma/]s1 & probably we appreciate !m ! m because we h dabbl$ a little 9 pigm5ts or 9 clay4 ,we d n expect to be compos]s1 or poets1 or sculptors1 b we wi% to appreciate music & lett]s & ! f9e arts1 & to d]ive pleasure f !m & be uplift$ by !m4444 ,s x is ) geometry4 ,we /udy x because we d]ive pleasure f contact ) a great & anci5t body ( le>n+ t has occupi$ ! att5tion ( ma/] m9ds dur+ ! ?\s&s ( ye>s 9 :i* x has be5 p]fect$1 & we are uplift$ by x4 ,to d5y t our pupils d]ive ?is pleasure f ! /udy is to confess ourselves poor tea*]s1 = mo/ pupils d h positive enjoym5t 9 ! pursuit ( geometry1 9 spite ( ! tradition t leads !m to proclaim a g5]al dislike = all /udy4 ,?is enjoym5t is p>tly t ( ! game1--! play+ ( a game t c always be won1 b t _c be won too easily4 ,x is p>tly t ( ! ae/hetic1 ! pleasure ( symmetry ( =m1 ! delight ( fitt+ ?+s toge!r4 ,b probably x lies *iefly 9 ! m5tal uplift t geometry br+s1 ! contact ) absolute tru?1 & ! approa* t "o makes to ! ,inf9ite4 ,we are n q sure ( any "o ?+ 9 biology2 our k ( geology is relatively v slight1 & ! economic laws ( society are unc]ta9 to e "o except "s individual :o attempts to set !m =?2 b be=e ! _w 0 fa%ion$ ! squ>e on ! hypot5use 0 equal to ! sum ( ! squ>es on ! o!r two sides ( a "r triangle1 & x w be s aft] ?is _w is dead2 & ! inhabitant ( ,m>s1 if he exi/s1 probably kn[s its tru? z we "k x4 ,! uplift ( ?is contact ) absolute tru?1 ) tru? et]nal1 gives pleasure to humanity to a great] or less degree1 dep5d+ ^u ! m5tal equipm5t ( ! p>ticul> individual2 b x probably gives an appreciable am\nt ( pleasure to e /ud5t ( geometry :o has a tea*] wor?y ( ! "n4--,,smi?1 ,d4 ,e4 ,! ,tea*+ ( ,geometry 7,bo/on1 #aiaa71 p4 #af4 #ahea4 ,no o!r p]son c judge bett] ( ei!r ! m]its ( a writ] & ! m]its ( 8 works ?an himself2 = none h _h access to a clos] or m delib]ate exam9ation ( !m4 ,x is = ?is reason1 t 9 proportion t ! value ( a "w is intr9sic1 & indep5d5t ( op9ion1 ! less eag]ness w ! au?or feel to conciliate ! suffrages ( ! public4 ,h5ce t inw>d satisfaction1 s pure & s complete1 :i* ! /udy ( geometry yields4 ,! progress :i* an individual makes 9 ?is sci5ce1 ! degree ( em95ce :i* he atta9s 9 x1 all ?is may be measur$ ) ! same rigor\s accuracy z ! me?ods ab\t :i* 8 ?\ghts are employ$4 ,x is only :5 we ent]ta9 "s d\bts ab\t ! ju/ness ( our own /&>d1 t we become anxi\s to relieve ourselves f our unc]ta9ty1 by comp>+ x ) ! /&>d ( ano!r4 ,n[1 9 all matt]s :i* fall "u ! cognizance ( ta/e1 ?is /&>d is necess>ily some:at v>iable2 dep5d+ on a sort ( gross e/imate1 always a little arbitr>y1 ei!r 9 :ole or 9 "p2 & liable to cont9ual alt]ation 9 its dim5sions1 f neglig5ce1 temp]1 or caprice4 ,9 consequ5ce ( ^! circum/ances ,i h no d\bt1 t if m5 liv$ sep>ate f ea* o!r1 & c\ld 9 su* a situation occupy !mselves ab\t any?+ b self-pres]vation1 !y w\ld pref] ! /udy ( ! exact sci5ces to ! cultivation ( ! agreeable arts4 ,x is *iefly on acc\nt ( o!rs1 t a man aims at excell5ce 9 ! latt]1 x is on 8 own acc\nt t he devotes himself to ! =m]4 ,9 a des]t isl&1 accord+ly1 ,i %\ld ?9k t a poet c\ld sc>cely be va92 :]eas a geometrician might /ill enjoy ! pride ( discov]y4--,d',,alemb]t4 ,essai sur les ,g5s ,lettres2 ,melages 7,am/]dam #agfd71 t4 #a1 p4 #ccd4 #aheb4 ,if x 7 requir$ to det]m9e incl9$ planes ( v>y+ incl9ations ( su* l5g?s t a free roll+ body w\ld desc5d on !m 9 equal times1 any "o :o und]/&s ! me*anical laws involv$ w\ld admit t ?is w\ld necessitate sundry prep>ations4 ,b 9 ! circle ! prop] arrangem5t takes place ( its own accord = an inf9ite v>iety ( positions yet ) ! greate/ accuracy 9 ea* individual case4 ,= all *ords :i* meet ! v]tical diamet] :e!r at its highe/ or l[e/ po9t1 & :atev] _! incl9ations1 h ?is 9 common3 t ! free desc5t along !m takes place 9 equal times4 ,i rememb]1 "o bright pupil1 :o1 aft] ,i _h /at$ & demon/rat$ ?is !orem to him1 & he _h caught ! full import ( x1 0 mov$ z by a miracle4 ,&1 inde$1 !re is j cause = a/oni%m5t & wond] :5 "o beholds su* a /range union ( manifold ?+s 9 accordance ) su* fruitful rules 9 s pla9 & simple an object z ! circle4 ,moreov]1 !re is no miracle 9 nature1 :i* because ( its p]vad+ beauty or ord]1 gives great] cause = a/oni%m5t1 unless x be1 = ! reason t its causes are n s cle>ly compreh5d$1 m>vel be+ a daught] ( ignorance4--,,kant4 ,d] e9zig mgli*e ,beweisgrund zu e9] ,demon/ration des ,dase9s ,gottes2 ,w]ke 7,h>t5/e971 ,bd4 #b1 p4 #acg4 #ahec4 ,^! examples tak5 f ! geometry ( ! circle indicate :at a c\ntless numb] ( o!r su* h>monic relations obta9 9 ! prop]ties ( space1 _m ( :i* are manife/$ 9 ! relations ( ! v>i\s classes ( curves 9 high] geometry1 all ( :i*1 besides ex]cis+ ! und]/&+ ?r\gh intellectual insight1 affect ! emotion 9 a simil> or ev5 great] degree ?an ! occasional beauties ( nature4--,,kant4 ,d] e9zig mgli*e ,beweisgrund zu e9] ,demon/ration des ,dase9s ,gottes2 ,w]ke 7,h>t5/e971 ,bd4 #b1 p4 #ach4 #ahed4 ,b nei!r ?irty ye>s1 nor ?irty c5turies1 affect ! cle>ness1 or ! *>m1 ( ,geometrical tru?s4 ,su* a !orem z 8! squ>e ( ! hypot5use ( a "r-angl$ triangle is equal to ! sum ( ! squ>es ( ! sides0 is z dazzl+ly beautiful n[ z x 0 9 ! day :5 ,py?agoras fir/ discov]$ x1 & celebrat$ its adv5t1 x is said1 by sacrific+ a hecatomb ( ox5--a me?od ( do+ honor to ,sci5ce t has always seem$ to me slightly exagg]at$ & uncall$-=4 ,"o c imag9e oneself1 ev5 9 ^! deg5]ate days1 m>k+ ! epo* ( "s brilliant sci5tific discov]y by invit+ a convivial fri5d or two1 to jo9 "o 9 a beef/eak & a bottle ( w9e4 ,b a hecatomb ( ox56 ,x w\ld produce a q inconv5i5t supply ( beef4--,,dodgson1 ,c4 ,l4 ,a ,new ,!ory ( ,p>allels 7,london1 #ahie71 ,introduction1 p4 #af4 #ahee4 ,aft] ,py?agoras discov]$ 8 fundam5tal !orem he sacrific$ a hecatomb ( ox54 ,s9ce t "t all dunces#aj ,o*s5 tremble :5ev] a new tru? is discov]$4--,,bo]ne4 ,quot$ 9 ,moszk[ski3 ,die un/]bli*e ,ki/e 7,b]l91 #aijh71 p4 #ah4 #aj ,9 ! ,g]man v]nacul> a dunce or blockhead is call$ an ox4 #ahef4 ,vom ,py?agories*5 ,lehrsatz4 ,die ,wahrheit1 sie be/eht 9 ,ewigkeit1 ,w5n er/ die blde ,welt ihr ,li*t erkannt3 ,d] ,lehrsatz1 na* ,py?agoras b5annt1 ,gilt heute1 wie er galt 9 se9] ,zeit4 ,e9 ,opf] hat ,py?agoras geweiht ,d5 ,gtt]n1 die d5 ,li*t/rahl ihm ges&t2 ,es ?at5 kund1 ges*la*tet und v]brannt1 ,e9 hund]t ,o*s5 se9e ,dankb>keit4 ,die ,o*s5 seit d5 ,tage1 w5n sie witt]n1 ,dass e9e neue ,wahrheit si* en?lle1 ,erheb5 e9 unm5s*li*es ,gebrlle2 ,py?agoras erfllt sie mit ,entsetz52 ,und ma*tlos1 si* dem ,li*t zu wi$]setz51 ,v]s*iess5 sie die ,aug5 und erzitt]n4 --,,*amisso1 ,,adelb]t ,,von4 ,g$i*te1 #ahce 7,hau%5bus*71 7,b]l91 #ahhi71 p4 #cjb4 ,tru? la/s ?r\gh\t et]nity1 ,:5 once ! /upid _w its light disc]ns3 ,! !orem1 c\pl$ ) ,py?agoras' "n1 ,holds true today1 z't did 9 old5 times4 ,a spl5did sacrifice ,py?agoras br\ght ,! gods1 :o bless$ him ) ?is ray div9e2 ,a great burnt (f]+ ( a hundr$ k9e1 ,proclaim$ af> ! sage's gratitude4 ,n[ s9ce t day1 all cattle blockheads :5 !y sc5t ,new tru? ab\t to see ! light ( day1 ,9 frightful bell[+s manife/ _! dismay2 ,py?agoras fills !m all ) t]ror2 ,& p[]less to %ut out light by error1 ,9 %e] despair !y %ut _! eyes & tremble4 #aheg4 ,to ! "q 8,:i* is ! signally mo/ beautiful ( geometrical tru?s80 ,frankl& replies3 8,"o /> excels ano!r 9 brightness1 b ! v sun w be1 by common cons5t1 a prop]ty ( ! circle ,euclid1 ,book #c1 ,proposition #ca select$ = p>ticul> m5tion by ,dante1 t greate/ ( all expon5ts ( ! beautiful40--,,frankl&1 ,w4 ,b4 ,! ,/ory ( ,euclid 7,london1 #aijb71 p4 #gj4 #aheh4 ,z "o ,:o v]s'd 9 geometric lore1 w\ld fa9 ,measure ! circle2 &1 ?\gh pond]+ long ,& deeply1 t beg9n+1 :i* he ne$s1 ,f9ds n2 e'en su* 0 ,i1 int5t to scan ,! novel wond]1 & trace out ! =m1 ,h[ to ! circle fitt$1 & !re9 ,h[ plac'd3 b ! flight 0 n = my w+2 --,,dante4 ,p>adise ,c>ey ,canto #cc1 l9es #abb-abi4 #ahei4 ,if geometry 7 z mu* oppos$ to our passions & pres5t int]e/s z is e?ics1 we %\ld conte/ x & violate x b little less1 not)/&+ all ! demon/rations ( ,euclid & ( ,ar*im$es1 :i* y w\ld call dreams & believe full ( p>alogisms2 & ,joseph ,scalig]1 ,hobbes1 & o!rs1 :o h writt5 aga9/ ,euclid & ,ar*im$es1 w\ld n f9d !mselves 9 su* a small company z at pres5t4--,,leibnitz4 ,new ,essays conc]n+ ,human ,und]/&+ ,langley1 ,bk4 #a1 *ap4 #b1 sect4 #ab4 #ahfj4 ,i h no fault to f9d ) ^? :o tea* geometry4 ,t sci5ce is ! only "o :i* has n produc$ sects2 x is f\nd$ on analysis & on syn!sis & on ! calculus2 x does n occupy itself ) probable tru?2 moreov] x has ! same me?od 9 e c\ntry4--,,fr$]ick ,,! ,,great4 ,oeuvres 7,deck]71 t4 #g1 p4 #ajj4 #ahfa4 ,!re are1 und\bt$ly1 ! mo/ ample reasons = /at+ bo? ! pr9ciples & !orems ( geometry 9 _! g5]al =m1 4444 ,b1 t an unpractis$ le>n]1 ev5 9 mak+ use ( "o !orem to demon/rate ano!r1 reasons r f p>ticul> to p>ticul> ?an f ! g5]al proposition1 is manife/ f ! difficulty he f9ds 9 apply+ a !orem to a case 9 :i* ! configuration ( ! diagram is extremely unlike t ( ! diagram by :i* ! orig9al !orem 0 demon/rat$4 ,a difficulty :i*1 except 9 cases ( unusual m5tal p[]s1 long practice c alone remove1 & removes *iefly by r5d]+ u famili> ) all ! configurations consi/5t ) ! g5]al conditions ( ! !orem4--,,mill1 ,j4 ,s4 ,sy/em ( ,logic1 ,bk4 #b1 *ap4 #c1 sect4 #c4 #ahfb4 ,! reason :y ,i impute any defect to geometry1 is1 because its orig9al & fundam5tal pr9ciples are d]iv'd m]ely f appe>ances2 & x may p]haps be imag9'd1 t ?is defect mu/ always att5d x1 & keep x f ev] rea*+ a great] exactness 9 ! comp>ison ( objects or ideas1 ?an :at our eye or imag9ation alone is able to atta94 ,i own t ?is defect s f> att5ds x1 z to keep x f ev] aspir+ to a full c]ta9ty4 ,b s9ce ^! fundam5tal pr9ciples dep5d on ! easie/ & lea/ deceitful appe>ances1 !y be/[ on _! consequ5ces a degree ( exactness1 ( :i* ^! consequ5ces are s+ly incapable4--,,hume1 ,d4 ,a ,treatise ( ,human ,nature1 ,"p #c1 sect4 #a4 #ahfc4 ,i h already obs]v$1 t geometry1 or ! art1 by :i* we fix ! proportions ( figures1 ?o' x mu* excels bo? 9 univ]sality & exactness1 ! loose judgm5ts ( ! s5ses & imag9ation2 yet nev] atta9s a p]fect precision & exactness4 ,its fir/ pr9ciples are /ill drawn f ! g5]al appe>ance ( ! objects2 & t appe>ance c nev] af=d u any security1 :5 we exam9e ! prodigi\s m9ut5ess ( :i* nature is susceptible4444 ,!re rema91 !re=e1 algebra & ari?metic z ! only sci5ces1 9 :i* we c c>ry on a *a9 ( reason+ to any degree ( intricacy1 & yet pres]ve a p]fect exactness & c]ta9ty4--,,hume1 ,d4 ,a ,treatise ( ,human ,nature1 ,"p #c1 sect4 #a4 #ahfd4 ,all geometrical reason+ is1 9 ! la/ resort1 circul>3 if we />t by assum+ po9ts1 !y c only be def9$ by ! l9es or planes :i* relate !m2 & if we />t by assum+ l9es or planes1 !y c only be def9$ by ! po9ts ?r\gh :i* !y pass4--,,russell1 ,,b]tr&4 ,f\ndations ( ,geometry 7,cambridge1 #ahig71 p4 #abj4 #ahfe4 ,! description ( "r l9es & circles1 ^u :i* ,geometry is f\nd$1 belongs to ,me*anics4 ,geometry does n tea* u to draw ^! l9es1 b requires !m to be drawn4444 x requires t ! le>n] %\ld fir/ be taught to describe ^! accurately1 be=e he ent]s ^u ,geometry2 !n x %[s h[ by ^! op]ations problems may be solv$4 ,to describe "r l9es & circles are problems1 b n geometrical problems4 ,! solution ( ^! problems is requir$ f ,me*anics2 by ,geometry ! use ( !m1 :5 solv$1 is %[n4444 ,!re=e ,geometry is f\nd$ 9 me*anical practice1 & is no?+ b t "p ( univ]sal ,me*anics :i* accurately proposes & demon/rates ! art ( measur+4 ,b s9ce ! manual arts are *iefly conv]sant 9 ! mov+ ( bodies1 x comes to pass t ,geometry is commonly ref]r$ to _! magnitudes1 & ,me*anics to _! motion4--,,newton4 ,philosophiae ,naturalis ,pr9cipia ,ma!matica1 ,praefat4 #ahff4 ,we mu/1 !n1 admit 444 t !re is an indep5d5t sci5ce ( geometry j z !re is an indep5d5t sci5ce ( physics1 & t ei!r ( ^! may be treat$ by ma!matical me?ods4 ,?us geometry becomes ! simple/ ( ! natural sci5ces1 & its axioms are ( ! nature ( physical laws1 to be te/$ by exp]i5ce & to be reg>d$ z true only )9 ! limits ( error ( obs]vation--,b,,*]1 ,,maxime4 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #b 7#aijd71 p4 #abd4 #ahfg4 ,geometry is n an exp]im5tal sci5ce2 exp]i5ce =ms m]ely ! occasion = our reflect+ ^u ! geometrical ideas :i* pre-exi/ 9 u4 ,b ! occasion is necess>y1 if x did n exi/ we %\ld n reflect1 & if our exp]i5ces 7 diff]5t1 d\btless our reflections w\ld also be diff]5t4 ,space is n a =m ( s5sibility2 x is an in/rum5t :i* s]ves u n to repres5t ?+s to ourselves1 b to reason ^u ?+s4--,,po9c>1 ,h4 ,on ! ,f\ndations ( ,geometry2 ,moni/1 ,vol4 #i 7#ahih-ahii71 p4 #da4 #ahfh4 ,x has be5 said t geometry is an in/rum5t4 ,! comp>ison may be admitt$1 provid$ x is grant$ at ! same "t t ?is in/rum5t1 l ,proteus 9 ! fable1 ought con/antly to *ange its =m4--,,arago4 ,oeuvres1 t4 #b 7#ahed71 p4 #fid4 #ahfi4 ,x is ess5tial t ! treatm5t ( geometry %\ld be rid ( ev]y?+ sup]flu\s1 = ! sup]flu\s is an ob/acle to ! acquisition ( k2 x %\ld select ev]y?+ t embraces ! subject & br+s x to a focus1 = ?is is ( ! highe/ s]vice to sci5ce2 x mu/ h great reg>d bo? to cle>ness & to concis5ess1 = _! opposites tr\ble our und]/&+2 x mu/ aim to g5]alize its !orems1 = ! division ( k into small elem5ts r5d]s x difficult ( compreh5sion4--,,proclus4 ,quot$ 9 ,d4 ,e4 ,smi?3 ,! ,tea*+ ( ,geometry 7,bo/on1 #aiaa71 p4 #ga4 #ahgj4 ,_m are acqua9t$ ) ma!matics1 b ma!sis few "k4 ,= x is "o ?+ to "k a numb] ( propositions & to make "s obvi\s d$uctions f !m1 by accid5t r ?an by any sure me?od ( proc$ure1 ano!r ?+ to "k cle>ly ! nature & "* ( ! sci5ce itself1 to p5etrate into its inmo/ recesses1 & to be in/ruct$ by its univ]sal pr9ciples1 by :i* facility 9 work+ out c\ntless problems & _! pro(s is secur$4 ,= z ! majority ( arti/s1 by copy+ ! same model aga9 & aga91 ga9 c]ta9 te*nical skill 9 pa9t+1 b no o!r k ( ! art ( pa9t+ ?an :at _! eyes sugge/1 s _m1 hav+ read ! books ( ,euclid & o!r geometricians1 are wont to devise1 9 imitation ( !m & to prove "s propositions1 b ! mo/ pr(\nd me?od ( solv+ m difficult demon/rations & problems !y are utt]ly ignorant (4--,,lafaille1 ,j4 ,c4 ,!oremata de ,c5tro ,gravitatis 7,anv]s1 #afcb71 ,praefat4 #ahga4 ,! elem5ts ( plane geometry %\ld prec$e algebra = e reason kn[n to s\nd educational !ory4 ,x is m fundam5tal1 m concrete1 & x deals ) ?+s & _! relations r ?an ) symbols4--,,butl]1 ,n4 ,m4 ,! ,mean+ ( ,education etc4 7,new ,york1 #aije71 p4 #aga4 #ahgb4 ,! reason :y geometry is n s difficult z algebra1 is to be f\nd 9 ! less g5]al nature ( ! symbols employ$4 ,9 algebra a g5]al proposition respect+ numb]s is to be prov$4 ,lett]s are tak5 :i* may repres5t any ( ! numb]s 9 "q1 & ! c\rse ( ! demon/ration1 f> f mak+ use ( a p>ticul> case1 does n ev5 all[ t any reason+1 h[ev] g5]al 9 its nature1 is conclusive1 unless ! symbols are z g5]al z ! argum5ts4444 ,9 geometry on ! contr>y1 at lea/ 9 ! elem5t>y p>ts1 any proposition may be safely demon/rat$ on reason+s on any "o p>ticul> example4444 ,x also af=ds "s facility t ! results ( elem5t>y geometry are 9 _m cases suffici5tly evid5t ( !mselves to ! eye2 = in/ance1 t two sides ( a triangle are great] ?an ! ?ird1 :]eas 9 algebra _m rudim5t>y propositions d]ive no evid5ce f ! s5ses2 = example1 t ab is always divisible )\t a rema9d] by ab4--,,de ,,morgan1 ,a4 ,on ! ,/udy & ,difficulties ( ,ma!matics 7,*icago1 #aijb71 *ap4 #ac4 #ahgc4 ,! pr9cipal *>act]i/ics ( ! anci5t geometry are3-- 7#a7 ,a wond]ful cle>ness & def9it5ess ( its concepts & an almo/ p]fect logical rig\r ( its conclusions4 7#b7 ,a complete want ( g5]al pr9ciples & me?ods4444 ,9 ! demon/ration ( a !orem1 !re 71 = ! anci5t geomet]s1 z _m diff]5t cases requir+ sep>ate pro( z !re 7 diff]5t positions ( ! l9es4 ,! greate/ geomet]s consid]$ x necess>y to treat all possible cases indep5d5tly ( ea* o!r1 & to prove ea* ) equal fulness4 ,to devise me?ods by :i* all ! v>i\s cases c\ld all be dispos$ ( ) "o /roke1 0 beyond ! p[] ( ! anci5ts4--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #fb4 #ahgd4 ,x has be5 obs]v$ t ! anci5t geomet]s made use ( a k9d ( analysis1 :i* !y employ$ 9 ! solution ( problems1 al?\gh !y begrudg$ to po/]ity ! k ( x4--,,desc>tes4 ,rules = ! ,direction ( ! ,m9d2 ,! ,philosophy ( ,desc>tes ,torrey 7,new ,york1 #ahib71 p4 #fh4 #ahge4 ,! anci5ts /udi$ geometry ) ref]5ce to ! bodies "u notice1 or specially3 ! mod]ns /udy x ) ref]5ce to ! ph5om5a to be consid]$1 or g5]ally4 ,! anci5ts extract$ all !y c\ld out ( "o l9e or surface1 be=e pass+ to ano!r2 & ea* inquiry gave little or no assi/ance 9 ! next4 ,! mod]ns1 s9ce ,desc>tes1 employ !mselves on que/ions :i* relate to any figure :atev]4 ,!y ab/ract1 to treat by itself1 e "q relat+ to ! same geometrical ph5om5on1 9 :atev] bodies x may be consid]$4 ,geomet]s c ?us rise to ! /udy ( new geometrical conceptions1 :i*1 appli$ to ! curves inve/igat$ by ! anci5ts1 h br\ght out new prop]ties nev] suspect$ by !m4--,,comte4 ,positive ,philosophy ,m>t9eau1 ,bk4 #a1 *ap4 #c4 #ahgf4 ,x is a/oni%+ t ?is subject projective geometry %\ld be s g5]ally ignor$1 = ma!matics (f]s no?+ m attractive4 ,x possesses ! concret5ess ( ! anci5t geometry )\t ! t$i\s p>ticul>ity1 & ! p[] ( ! analytical geometry )\t ! reckon+1 & by ! beauty ( its ideas & me?ods illu/rates ! e/hetic g5]ality :i* is ! *>m ( high] ma!matics1 b :i* ! elem5t>y ma!matics g5]ally lacks4 ,report ( ! ,committee ( ,t5 on ,second>y ,s*ool ,/udies 7,*icago1 #ahid71 p4 #aaf4 #ahgg4 ,!re exi/ a small numb] ( v simple fundam5tal relations :i* conta9 ! s*eme1 accord+ to :i* ! rema9+ mass ( !orems 9 projective geometry p]mit ( ord]ly & easy developm5t4 ,by a prop] appropriation ( a few fundam5tal relations "o becomes ma/] ( ! :ole subject2 ord] takes ! place ( *aos1 "o beholds h[ all p>ts fit naturally into ea* o!r1 & arrange !mselves s]ially 9 ! mo/ beautiful ord]1 & h[ relat$ p>ts comb9e into well-def9$ gr\ps4 ,9 ?is mann] "o arrives1 z x 71 at ! elem5ts1 :i* nature h]self employs 9 ord] to end[ figures ) numb]less prop]ties ) ! utmo/ economy & simplicity4--,,/e9]1 ,j4 ,w]ke1 ,bd4 #a 7#ahha71 p4 #bcc4 #ahgh4 ,euclid once said to 8 k+ ,ptolemy1 :o1 z is easily und]/ood1 f\nd ! pa9/ak+ /udy ( ! 8,elem5ts0 repellant1 8,!re exi/s no royal road to ma!matics40 ,b we may add3 ,mod]n geometry is a royal road4 ,x has disclos$ 8! organism1 by means ( :i* ! mo/ het]og5e\s ph5om5a 9 ! _w ( space are unit$ "o ) ano!r0 7,/e9]71 & has1 z we may say )\t exagg]ation1 almo/ atta9$ to ! sci5tific ideal4--,,hankel1 ,h4 ,die ,entwickelung d] ,ma!matik 9 d5 letzt5 ,jahrhund]t5 7,tb+51 #ahfi74 #ahgi4 ,! two ma!matically fundam5tal ?+s 9 projective geometry are anh>monic ratio1 & ! quadrilat]al con/ruction4 ,ev]y?+ else foll[s ma!matically f ^! two4 --,,russell1 ,,b]tr&4 ,f\ndations ( ,geometry 7,cambridge1 #ahig71 p4 #abb4 #ahhj4 444 ,projective ,geometry3 a b\ndless doma9 ( c\ntless fields :]e reals & imag9>ies1 f9ites & inf9ites1 ent] on equal t]ms1 :]e ! _s delights 9 ! arti/ic balance & symmetric int]play ( a k9d ( conceptual & logical c\nt]po9t1--an en*ant$ realm :]e ?\ght is d\ble & fl[s ?r\gh\t 9 p>allel /reams4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,arts 7,new ,york1 #aijh71 p4 #b4 #ahha4 ,! anci5ts1 9 ! e>ly days ( ! sci5ce1 made great use ( ! graphic me?od1 ev5 9 ! =m ( con/ruction2 z :5 ,ari/>*us ( ,samos e/imat$ ! di/ance ( ! sun & moon f ! e>? on a triangle con/ruct$ z ne>ly z possible 9 resemblance to ! "r-angl$ triangle =m$ by ! ?ree bodies at ! in/ant :5 ! moon is 9 quadrature1 & :5 !re=e an obs]vation ( ! angle at ! e>? w\ld def9e ! triangle4 ,ar*im$es himself1 ?\gh he 0 ! fir/ to introduce calculat$ det]m9ations into geometry1 frequ5tly us$ ! same means4 ,! introduction ( trigonometry less5$ ! practice2 b did n aboli% x4 ,! ,greeks & ,arabians employ$ x /ill = a great numb] ( inve/igations = :i* we n[ consid] ! use ( ! ,calculus indisp5sable4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #a1 *ap4 #c4 #ahhb4 ,a ma!matical problem may usually be attack$ by :at is t]m$ 9 milit>y p>lance ! me?od ( 8sy/ematic approa*20 t is to say1 its solution may be gradually felt =1 ev5 ?\gh ! successive /eps lead+ to t solution _c be cle>ly =ese54 ,b a ,descriptive ,geometry problem mu/ be se5 ?r\gh & ?r\gh be=e x c be attempt$4 ,! entire scope ( its conditions1 z well z ea* /ep t[>d its solution1 mu/ be grasp$ by ! imag9ation4 ,x mu/ be 8tak5 by assault40 --,,cl>ke1 ,g4 ,s4 ,quot$ 9 ,w4 ,s4 ,hall3 ,descriptive ,geometry 7,new ,york1 #aijb71 *ap4 #a4 #ahhc4 ,! gr& use ( ,descriptive ,geometry is 9 its application to ! indu/rial arts2--its few ab/ract problems1 capable ( inv>iable solution1 relat+ ess5tially to ! contacts & int]sections ( surfaces2 s t all ! geometrical que/ions :i* may arise 9 any ( ! v>i\s arts ( con/ruction1--z /one-cutt+1 c>p5try1 p]spective1 dial+1 =tification1 etc41--c always be treat$ z simple individual cases ( a s+le !ory1 ! solution be+ c]ta9ly obta9able ?r\gh ! p>ticul> circum/ances ( ea* case4 ,?is creation mu/ be v important 9 ! eyes ( philosoph]s :o ?9k t all human a*ievem5t1 ?us f>1 is only a fir/ /ep t[>d a philosophical r5ovation ( ! lab\rs ( mank9d2 t[>ds t precision & logical "* :i* c alone ensure ! future progression ( all arts4444 ,( ,descriptive ,geometry1 x may fur!r be said t x usefully ex]cises ! /ud5t's faculty ( ,imag9ation1--( conceiv+ ( complicat$ geometrical comb9ations 9 space2 & t1 :ile x belongs to ! geometry ( ! anci5ts by ! "* ( its solutions1 x approa*es to ! geometry ( ! mod]ns by ! nature ( ! que/ions :i* compose x4--,,comte1 ,a4 ,positive ,philosophy ,m>t9eau1 ,bk4 #a1 *ap4 #c4 #ahhd4 ,!re is p]haps no?+ :i* s occupies1 z x 71 ! middle position ( ma!matics1 z trigonometry4 --,,h]b>t1 ,j4 ,f4 ,idee e9es ,,abc d] ,ans*auung2 ,w]ke 7,kehrba*7 7,lang5salza1 #ahij71 ,bd4 #a1 p4 #agd4 #ahhe4 ,trigonometry conta9s ! sci5ce ( cont9ually undulat+ magnitude3 mean+ magnitude :i* becomes alt]nately great] & less1 )\t any t]m9ation to succession ( increase & decrease4444 ,all trigonometric functions are n undulat+3 b x may be /at$ t 9 common algebra no?+ b inf9ite s]ies undulate3 9 trigonometry no?+ b inf9ite s]ies d n undulate4--,,de ,,morgan1 ,a4 ,trigonometry & ,d\ble ,algebra 7,london1 #ahdi71 ,bk4 #a1 *ap4 #a4 #ahhf4 ,s9 is odi\s to me1 ev5 ?\gh ,laplace made use ( x2 %\ld x be fe>$ t s9 might become ambigu\s1 :i* w\ld p]haps nev] occur1 or at mo/ v r>ely :5 speak+ ( s9 771 well !n1 let u write 7s971 b n s91 :i* by analogy %\ld signify s97s974--,,gauss4 ,gauss-,s*uma*] ,briefwe*sel1 ,bd4 #c1 p4 #bib2 ,bd4 #d1 p4 #fc4 #ahhg4 ,p]haps to ! /ud5t !re is no "p ( elem5t>y ma!matics s repulsive z is sph]ical trigonometry4--,,tait1 ,p4 ,g4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,quat]nions40 #ahhh4 8,napi]'s ,rule ( circul> p>ts0 is p]haps ! happie/ example ( artificial memory t is kn[n4--,,cajori1 ,f4 ,hi/ory ( ,ma!matics 7,new ,york1 #ahig71 p4 #afe4 #ahhi4 ,! analytical equations1 unkn[n to ! anci5ts1 :i* ,desc>tes fir/ introduc$ into ! /udy ( curves & surfaces1 are n re/rict$ to ! prop]ties ( figures1 & to ^? prop]ties :i* are ! object ( rational me*anics2 !y apply to all ph5om5a 9 g5]al4 ,!re _c be a language m univ]sal & m simple1 m free f errors & obscurities1 t is to say1 bett] adapt$ to express ! inv>iable relations ( nature4--,,f\ri]4 ,?orie ,analytique de la ,*aleur1 ,disc\rs ,prlim9aire4 #ahij4 ,x is impossible n to feel /irr$ at ! ?\ght ( ! emotions ( m5 at c]ta9 hi/oric mom5ts ( adv5ture & discov]y--,columbus :5 he fir/ saw ! ,we/]n %ore1 ,piz>ro :5 he />$ at ! ,pacific ,ocean1 ,frankl9 :5 ! electric sp>k came f ! /r+ ( 8 kite1 ,galileo :5 he fir/ turn$ 8 telescope to ! heav5s4 ,su* mom5ts are also grant$ to /ud5ts 9 ! ab/ract regions ( ?\ght1 & high among !m mu/ be plac$ ! morn+ :5 ,desc>tes lay 9 b$ & inv5t$ ! me?od ( co-ord9ate geometry4--,,:itehead1 ,a4 ,n4 ,an ,introduction to ,ma!matics 7,new ,york1 #aiaa71 p4 #abb4 #ahia4 ,x is (t5 said t an equation conta9s only :at has be5 put into x4 ,x is easy to reply t ! new =m "u :i* ?+s are f\nd (t5 con/itutes by itself an important discov]y4 ,b !re is some?+ m3 analysis1 by ! simple play ( its symbols1 may sugge/ g5]alizations f> beyond ! orig9al limits4--,,pic>d1 ,e4 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #b 7#aije71 p4 #dji4 #ahib4 ,x is n ! ,simplicity ( ! ,equation1 b ! ,eas9ess ( ! ,description1 :i* is to det]m9e ! ,*oice ( our ,l9es = ! ,con/ructions ( ,problems4 ,= ! ,equation t expresses a ,p>abola is m simple ?an t t expresses ! ,circle1 & yet ! ,circle1 by its m simple ,con/ruction1 is admitt$ be=e x4--,,newton4 ,! ,l9e> ,con/ructions ( ,equations2 ,univ]sal ,ari?metic 7,london1 #agfi71 ,vol4 #b1 p4 #dfh4 #ahic4 ,! pursuit ( ma!matics unfolds its =mative p[] completely only ) ! transition f ! elem5t>y subjects to analytical geometry4 ,unque/ionably ! simple/ geometry & algebra already accu/om ! m9d to %>p quantitative ?9k+1 z also to assume z true only axioms & :at has be5 prov54 ,b ! repres5tation ( functions by curves or surfaces reveals a new _w ( concepts & tea*es ! use ( "o ( ! mo/ fruitful me?ods1 :i* ! human m9d ev] employ$ to increase its own effectiv5ess4 ,:at ! discov]y ( ?is me?od by ,vieta & ,desc>tes br\ght to humanity1 t x br+s today to e "o :o is 9 any measure end[$ = su* ?+s3 a life-epo*-mak+ beam ( light ,li*tblick4 ,?is me?od has its roots 9 ! f>!/ dep?s ( human cognition & s has an entirely diff]5t significance1 ?an ! mo/ +5i\s artifice :i* s]ves a special purpose4--,,bois-,,reymond1 ,,emil ,,du4 ,r$51 ,bd4 #a 7,leipzig1 #ahhe71 p4 #bhg4 #ahid4 ,song ( ! ,screw4 ,a mov+ =m or rigid mass1 ,"u :ate'er conditions ,along successive screws mu/ pass ,betwe5 ea* two positions4 ,x turns ar\nd & slides along-- ,?is is ! burd5 ( my song4 ,! pit* ( screw1 if multipli$ ,by angle ( rotation1 ,w give ! di/ance x mu/ glide ,9 motion ( translation4 ,inf9ite pit* means pure translation1 ,& z]o pit* means pure rotation4 ,two motions on two giv5 screws1 ,) amplitudes at pleasure1 ,into a ?ird screw-motion fuse1 ,^: amplitude we measure ,by p>allelogram con/ruction 7,a v obvi\s d$uction74 ,its axis cuts ! nodal l9e ,:i* to bo? screws is normal1 ,& g5]ates a =m div9e1 ,^: "n1 9 language =mal1 ,is 8surface-rul$ ( ?ird degree40 ,cyl9droid is ! "n = me4 ,rotation r\nd a giv5 l9e ,is l a =ce along1 ,if to say c\ple y decl9e1 ,y're cle>ly 9 ! wrong2-- ',tis obvi\s1 ^u reflection1 ,a l9e is n a m]e direction4 ,s c\ples ) translations too ,9 all respects agree2 ,& ?us !re c5tres 9 ! screw ,a wondr\s h>mony ,( ,k9ematics & ( ,/atics1-- ,! sweete/ ?+ 9 ma!matics4 ,! =ces on "o giv5 screw1 ,) motion on a second1 ,9 g5]al "s "w w d1 ,^: magnitude is reckon$ ,by angle1 =ce1 & :at we call ,! coeffici5t virtual4 ,rotation n[ to =ce conv]t1 ,& =ce into rotation2 ,un*ang$ ! "w1 we c ass]t1 ,9 spite ( trans=mation4 ,& if two screws no "w c claim1 ,reciprocal w be _! "n4 ,five numb]s w a screw def9e1 ,a screw+ motion1 six2 ,= f\r w give ! axial l9e1 ,"o m ! pit* w fix2 ,& h5ce we always c contrive ,"o screw reciprocal to five4 ,screws--two1 ?ree1 f\r or five1 comb9$ 7,no "q "h ( six71 ,yield o!r screws :i* are conf9$ ,)9 "o screw complex4 ,?us we obta9 ! cle>e/ notion ,( fre$om & con/ra9t ( motion4 ,9 complex ,,iii1 ?ree sev]al screws ,at e po9t y f9d1 ,or if y "o direction *oose1 ,"o screw is to y\r m9d2 ,& complexes ( ord] ,,iii4 ,_! own reciprocals may be4 ,9 ,,iv1 :]ev] y arrive1 ,y f9d ( screws a cone1 ,on e l9e ( complex ,v4 ,!re is precisely "o2 ,at ea* po9t ( ?is complex ri*1 ,a plane ( screws h giv5 pit*4 ,b "t w\ld fail me to disc\rse ,( ,ord] & ,degree2 ,( ,impulse1 ,en]gy & ,=ce1 ,& ,reciprocity4 ,all ^! & m1 = motions small1 ,h be5 discuss$ by ,dr4 ,ball4 --,,anonym\s4 ,,*apt] ,,xix ,,! ,,calculus ,,& ,,alli$ ,,topics #aija4 ,x may be said t ! conceptions ( diff]5tial quoti5t & integral1 :i* 9 _! orig9 c]ta9ly g back to ,ar*im$es1 7 introduc$ into sci5ce by ! inve/igations ( ,kepl]1 ,desc>tes1 ,cavali]i1 ,f]mat & ,wallis4444 ,! capital discov]y t diff]5tiation & integration are inv]se op]ations belongs to ,newton & ,leibnitz4--,,lie1 ,,sophus4 ,leipzig] ,b]i*te1 #dg 7#ahie71 ,ma?4-phys4 ,classe1 p4 #ec4 #aijb4 ,x appe>s t ,f]mat1 ! true inv5tor ( ! diff]5tial calculus1 consid]$ t calculus z d]iv$ f ! calculus ( f9ite diff]5ces by neglect+ inf9itesimals ( high] ord]s z comp>$ ) ^? ( a l[] ord]4444 ,newton1 ?r\gh 8 me?od ( fluxions1 has s9ce r5d]$ ! calculus m analytical1 he also simplifi$ & g5]aliz$ ! me?od by ! inv5tion ( 8 b9omial !orem4 ,leibnitz has enri*$ ! diff]5tial calculus by a v happy notation4 --,,laplace4 ,ls ,intgrales ,df9ies1 etc42 ,oeuvres1 t4 #ab 7,p>is1 #ahih71 p4 #cei4 #aijc4 ,pr(essor ,peacock's ,algebra1 & ,mr4 ,:ewell's ,doctr9e ( ,limits %\ld be /udi$ by e "o :o desires to compreh5d ! evid5ce ( ma!matical tru?s1 & ! mean+ ( ! obscure processes ( ! calculus2 :ile1 ev5 aft] ma/]+ ^! treatises1 ! /ud5t w h mu* to le>n on ! subject f ,m4 ,comte1 ( ^: admirable "w "o ( ! mo/ admirable portions is t 9 :i* he may truly be said to h creat$ ! philosophy ( ! high] ma!matics4 --,,mill1 ,j4 ,s4 ,sy/em ( ,logic1 ,bk4 #c1 *ap4 #bd1 sect4 #f4 #aijd4 ,if we mu/ conf9e ourselves to "o sy/em ( notation !n !re c be no d\bt t t :i* 0 inv5t$ by ,leibnitz is bett] fitt$ = mo/ ( ! purposes to :i* ! inf9itesimal calculus is appli$ ?an t ( fluxions1 & = "s 7su* z ! calculus ( v>iations7 x is inde$ almo/ ess5tial4--,,ball1 ,w4 ,w4 ,r4 ,hi/ory ( ,ma!matics 7,london1 #aija71 p4 #cga4 #aije4 ,! diff]5ce betwe5 ! me?od ( inf9itesimals & t ( limits 7:5 exclusively adopt$7 is1 t 9 ! latt] x is usual to reta9 evanesc5t quantities ( high] ord]s until ! end ( ! calculation & !n neglect !m4 ,on ! o!r h&1 su* quantities are neglect$ f ! comm5cem5t 9 ! inf9itesimal me?od1 f ! conviction t !y _c affect ! f9al result1 z !y mu/ disappe> :5 we proce$ to ! limit4--,,williamson1 ,b4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,inf9itesimal ,calculus10 sect4 #ad4 #aijf4 ,:5 we h grasp$ ! _s ( ! inf9itesimal me?od1 & h v]ifi$ ! exactness ( its results ei!r by ! geometrical me?od ( prime & ultimate ratios1 or by ! analytical me?od ( d]iv$ functions1 we may employ inf9itely small quantities z a sure & valuable means ( %ort5+ & simplify+ our pro(s4--,,lagrange4 ,m*anique ,analytique1 ,preface2 ,oeuvres1 t4 #b 7,p>is1 #ahhh71 p4 #ad4 #aijg4 ,! ess5tial m]it1 ! sublimity1 ( ! inf9itesimal me?od lies 9 ! fact t x is z easily p]=m$ z ! simple/ me?od ( approximation1 & t x is z accurate z ! results ( an ord9>y calculation4 ,?is advantage w\ld be lo/1 or at lea/ greatly impair$1 if1 "u ! pret5se ( secur+ great] accuracy ?r\gh\t ! :ole process1 we 7 to sub/itute = ! simpl] me?od giv5 by ,leibnitz1 "o less conv5i5t & less 9 h>mony ) ! probable c\rse ( natural ev5ts4444 ,! objections :i* h be5 rais$ aga9/ ! inf9itesimal me?od are bas$ on ! false supposition t ! errors due to neglect+ inf9itely small quantities dur+ ! actual calculation w cont9ue to exi/ 9 ! result ( ! calculation4--,,c>not1 ,l4 ,rflections sur la ,mtaphysique du ,calcul ,inf9itsimal 7,p>is1 #ahac71 p4 #bae4 #aijh4 ,a limit+ ratio is nei!r m nor less difficult to def9e ?an an inf9itely small quantity4--,,c>not1 ,l4 ,rflections sur la ,mtaphysique du ,calcul ,inf9itsimal 7,p>is1 #ahac71 p4 #baj4 #aiji4 ,a limit is a peculi> & fundam5tal conception1 ! use ( :i* 9 prov+ ! propositions ( ,high] ,geometry _c be sup]s$$ by any comb9ation ( o!r hypo!ses & def9itions4 ,! axiom j not$ t :at is true up to ! limit is true at ! limit1 is involv$ 9 ! v conception ( a limit3 & ?is pr9ciple1 ) its consequ5ces1 leads to all ! results :i* =m ! subject ( ! high] ma!matics1 :e!r prov$ by ! consid]ation ( evanesc5t triangles1 by ! processes ( ! ,diff]5tial ,calculus1 or 9 any o!r way4 --,,:ewell1 ,w4 ,! ,philosophy ( ! ,inductive ,sci5ces1 ,"p #a1 bk4 #b1 *ap4 #ab1 sect4 #a1 7,london1 #aheh74 #aiaj4 ,! diff]5tial calculus has all ! exactitude ( o!r algebraic op]ations4--,,laplace4 ,?orie ,analytique des ,probabilits1 ,introduction2 ,oeuvres1 t4 #g 7,p>is1 #ahhf71 p4 #cg4 #aiaa4 ,! me?od ( fluxions is probably "o ( ! greate/1 mo/ subtle1 & sublime discov]ies ( any age3 x op5s a new _w to our view1 & ext5ds our k1 z x 71 to inf9ity2 c>ry+ u beyond ! b\nds t seem$ to h be5 prescrib$ to ! human m9d1 at lea/ inf9itely beyond ^? to :i* ! anci5t geometry 0 conf9$4--,,hutton1 ,,*>les4 ,a ,philosophical & ,ma!matical ,diction>y 7,london1 #ahae71 ,vol4 #a1 p4 #ebe4 #aiab4 ,! /ates & conditions ( matt]1 z !y occur 9 nature1 are 9 a /ate ( p]petual flux1 & ^! qualities may be effectively /udi$ by ! ,newtonian me?od 7,me?odus fluxionem7 :5ev] !y c be ref]r$ to numb] or subject$ to measurem5t 7real or imag9>y74 ,by ! aid ( ,newton's calculus ! mode ( action ( natural *anges f mom5t to mom5t c be portray$ z fai?fully z ^! words repres5t ! ?\ghts at pres5t 9 my m9d4 ,f ?is1 ! law :i* controls ! :ole process c be det]m9$ ) unmi/akable c]ta9ty by pure calculation4--,,mellor1 ,j4 ,w4 ,high] ,ma!matics = ,/ud5ts ( ,*emi/ry & ,physics 7,london1 #aijb71 ,prologue4 #aiac4 ,! calculus is ! greate/ aid we h to ! appreciation ( physical tru? 9 ! broade/ s5se ( ! ^w4 --,,osgood1 ,w4 ,f4 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #ac 7#aijg71 p4 #dfg4 #aiad4 ,inf9itesimal analysis is ! mo/ p[]ful weapon ( ?\ght yet devis$ by ! wit ( man4--,,smi?1 ,w4 ,b4 ,inf9itesimal ,analysis 7,new ,york1 #ahih71 ,preface1 p4 vii4 #aiae4 ,! me?od ( ,fluxions is ! g5]al key by help :]e( ! mod]n ma!maticians unlock ! secrets ( ,geometry1 & consequ5tly ( ,nature4 ,&1 z x is t :i* ha? enabl$ !m s rem>kably to outgo ! anci5ts 9 discov]+ !orems & solv+ problems1 ! ex]cise & application !re( is become ! ma9 if n sole employm5t ( all ^? :o 9 ?is age pass = pr(\nd geomet]s4--,,b]keley1 ,,george4 ,! ,analy/1 sect4 #c4 #aiaf4 ,i h at la/ become fully satisfi$ t ! language & idea ( inf9itesimals %\ld be us$ 9 ! mo/ elem5t>y in/ruction--"u all safegu>ds ( c\rse4--,,de ,,morgan1 ,a4 ,graves' ,life ( ,w4 ,r4 ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #dgi4 #aiag4 ,pupils %\ld be taught h[ to diff]5tiate & h[ to integrate simple algebraic expressions be=e we attempt to tea* !m geometry & ^! o!r complicat$ ?+s4 ,! dreadful fe> ( ! symbols is entirely brok5 d[n 9 ^? cases :]e at ! beg9n+ ! tea*+ ( ! calculus is adopt$4 ,!n aft] ! pupil has ma/]$ ^? symbols y may beg9 geometry or any?+ y please4 ,i w\ld also aboli% out ( ! s*ool t ?+ call$ geometrical conics4 ,!re is a great deal ( sup]/ition ab\t conic sections4 ,! /ud5t %\ld be taught ! symbols ( ! calculus & ! simple/ use ( ^! symbols at ! e>lie/ age1 in/ead ( ^! be+ left ov] until he has gone to ! ,college or ,univ]sity4 --,,?ompson1 ,s4 ,p4 ,p]ry's ,tea*+ ( ,ma!matics 7,london1 #aijb71 p4 #di4 #aiah4 ,e "o v]s$ 9 ! matt] w agree t ev5 ! elem5ts ( a sci5tific /udy ( nature c be und]/ood only by ^? :o h a k ( at lea/ ! elem5ts ( ! diff]5tial & integral calculus1 z well z ( analytical geometry--i4e4 ! s-call$ l[] "p ( ! high] ma!matics4444 ,we %\ld raise ! "q1 :e!r suffici5t "t c\ld n be res]v$ 9 ! curricula ( at lea/ ! sci5ce high s*ools ,realan/alt5 to make room = ^! subjects4444 ,! fir/ consid]ation w\ld be to entirely relieve f ! ma!matical requirem5ts ( ! univ]sity ,ho*s*ule c]ta9 classes ( /ud5ts :o c get along )\t ext5d$ ma!matical k1 or to make ! necess>y ma!matical k accessible to !m 9 a mann] :i*1 = v>i\s reasons1 has n yet be5 adopt$ by ! univ]sity4 ,among su* /ud5ts ,i w\ld c\nt ar*itects1 also ! *emi/s & 9 g5]al ! /ud5ts ( ! s-call$ descriptive natural sci5ces4 ,i am moreov] ( ! op9ion--& ?is has be5 = long a favorite idea ( m9e--1 t x w\ld be v useful to m$ical /ud5ts to acquire su* ma!matical k z is indicat$ by ! above describ$ mode/ limits2 = x seems impossible to und]/& f>-rea*+ physiological inve/igations1 if "o is t]rifi$ z soon z a diff]5tial or integration symbol appe>s4--,,kle91 ,f4 ,jahresb]i*t d] ,deuts*5 ,ma!matik] ,v]e9igung1 ,bd4 #b 7#aijb71 p4 #aca4 #aiai4 ,common integration is only ! memory ( diff]5tiation 444 ! diff]5t artifices by :i* integration is effect$1 are *anges1 n f ! kn[n to ! unkn[n1 b f =ms 9 :i* memory w n s]ve u to ^? 9 :i* x w4--,,de ,,morgan1 ,a4 ,transactions ,cambridge ,philosophical ,society1 ,vol4 #h 7#ahdd71 p4 #ahh4 #aibj4 ,giv5 = "o in/ant an intellig5ce :i* c\ld compreh5d all ! =ces by :i* nature is animat$ & ! respective positions ( ! be+s :i* compose x1 if moreov] ?is intellig5ce 7 va/ en\gh to submit ^! data to analysis1 x w\ld embrace 9 ! same =mula bo? ! movem5ts ( ! l>ge/ bodies 9 ! univ]se & ^? ( ! lighte/ atom3 to x no?+ w\ld be unc]ta91 & ! future z ! pa/ w\ld be pres5t to its eyes4 ,! human m9d (f]s a feeble outl9e ( t intellig5ce1 9 ! p]fection :i* x has giv5 to a/ronomy4 ,its discov]ies 9 me*anics & 9 geometry1 jo9$ to t ( univ]sal gravity1 h enabl$ x to compreh5d 9 ! same analytical expressions ! pa/ & future /ates ( ! _w sy/em4--,,laplace4 ,?orie ,analytique des ,probabilits1 ,introduction2 ,oeuvres1 t4 #g 7,p>is1 #ahhf71 p4 #f4 #aiba4 ,!re is p]haps ! same relation betwe5 ! action ( natural selection dur+ "o g5]ation & ! accumulat$ result ( a hundr$ ?\s& g5]ations1 t !re exi/s betwe5 diff]5tial & integral4 ,h[ seldom are we able to foll[ completely ?is latt] relation al?\gh we subject x to calculation4 ,d we on t acc\nt d\bt ! correctness ( our integrations8--,,bois-,,reymond1 ,,emil ,,du4 ,r$51 ,bd4 #a 7,leipzig1 #ahhe71 p4 #bbh4 #aibb4 ,x seems to be expect$ ( e pilgrim up ! slopes ( ! ma!matical ,p>nassus1 t he w at "s po9t or o!r ( 8 j\rney sit d[n & inv5t a def9ite integral or two t[>ds ! increase ( ! common /ock4--,,sylve/]1 ,j4 ,j4 ,notes to ! ,m$itation on ,poncelet's ,!orem2 ,ma!matical ,pap]s1 ,vol4 #b1 p4 #bad4 #aibc4 ,! exp]im5tal v]ification ( a !ory conc]n+ any natural ph5om5on g5]ally re/s on ! result ( an integration4--,,mellor1 ,j4 ,w4 ,high] ,ma!matics = ,/ud5ts ( ,*emi/ry & ,physics 7,new ,york1 #aijb71 p4 #aej4 #aibd4 ,among all ! ma!matical discipl9es ! !ory ( diff]5tial equations is ! mo/ important4444 ,x furni%es ! explanation ( all ^? elem5t>y manife/ations ( nature :i* involve "t4444--,,lie1 ,,sophus4 ,leipzig] ,b]i*te1 #dg 7#ahie72 ,ma?4-phys4 ,classe1 p4 #bfb4 #aibe4 ,if ! ma!matical expression ( our ideas leads to equations :i* _c be integrat$1 ! work+ hypo!sis w ei!r h to be v]ifi$ "s o!r way1 or else relegat$ to ! great repository ( unv]ifi$ speculations4--,,mellor1 ,j4 ,w4 ,high] ,ma!matics = ,/ud5ts ( ,*emi/ry & ,physics 7,new ,york1 #aijb71 p4 #aeg4 #aibf4 ,x is well kn[n t ! c5tral problem ( ! :ole ( mod]n ma!matics is ! /udy ( ! transc5d5tal functions def9$ by diff]5tial equations4--,,kle91 ,f4 ,lectures on ,ma!matics 7,new ,york1 #aiaa71 p4 #h4 #aibg4 ,e "o kn[s :at a curve is1 until he has /udi$ en\gh ma!matics to become confus$ ?r\gh ! c\ntless numb] ( possible exceptions4444 ,a curve is ! totality ( po9ts1 ^: co-ord9ates are functions ( a p>amet] :i* may be diff]5tiat$ z (t5 z may be requir$4--,,kle91 ,f4 ,elem5t> ,ma!matik vom hh]5 ,/&punkte aus4 7,leipzig4 #aiji7 ,vol4 #b1 p4 #ced4 #aibh4 ,f\ri]'s !orem is n only "o ( ! mo/ beautiful results ( mod]n analysis1 b x may be said to furni% an indisp5sable in/rum5t 9 ! treatm5t ( ne>ly e recondite "q 9 mod]n physics4 ,to m5tion only sonor\s vibrations1 ! propagation ( electric signals along telegraph wires1 & ! conduction ( heat by ! e>?'s cru/1 z subjects 9 _! g5]ality intractable )\t x1 is to give b a feeble idea ( its importance4--,,?omson ,,& ,,tait4 ,elem5ts ( ,natural ,philosophy1 *ap4 #a4 #aibi4 ,! pr9cipal advantage aris+ f ! use ( hyp]bolic functions is t !y br+ to light "s curi\s analogies betwe5 ! integrals ( c]ta9 irrational functions4 --,,by]ly1 ,w4 ,e4 ,integral ,calculus 7,bo/on1 #ahij71 p4 #cj4 #aicj4 ,hyp]bolic functions are extremely useful 9 e bran* ( pure physics & 9 ! applications ( physics :e!r to obs]vational & exp]im5tal sci5ces or to te*nology4 ,?us :5ev] an entity 7su* z light1 velocity1 electricity1 or radio-activity7 is subject to gradual absorption or ext9ction1 ! decay is repres5t$ by "s =m ( hyp]bolic functions4 ,m]cator's projection is likewise comput$ by hyp]bolic functions4 ,:5ev] me*anical /ra9s are reg>d$ great en\gh to be measur$ !y are mo/ simply express$ 9 t]ms ( hyp]bolic functions4 ,h5ce geological de=mations inv>iably lead to su* expressions4444--,,walcott1 ,c4 ,d4 ,smi?sonian ,ma!matical ,tables1 ,hyp]bolic ,functions 7,wa%+ton1 #aiji71 ,adv]tisem5t4 #aica4 ,geometry may sometimes appe> to take ! lead ov] analysis1 b 9 fact prec$es x only z a s]vant goes be=e 8 ma/] to cle> ! pa? & light him on ! way4 ,! int]val betwe5 ! two is z wide z betwe5 empiricism & sci5ce1 z betwe5 ! und]/&+ & ! reason1 or z betwe5 ! f9ite & ! inf9ite4--,,sylve/]1 ,j4 ,j4 ,philosophic ,magaz9e1 ,vol4 #ca 7#ahff71 p4 #eba4 #aicb4 ,nature h]self exhibits to u measurable & obs]vable quantities 9 def9ite ma!matical dep5d5ce2 ! conception ( a function is sugge/$ by all ! processes ( nature :]e we obs]ve natural ph5om5a v>y+ accord+ to di/ance or to "t4 ,ne>ly all ! 8kn[n0 functions h pres5t$ !mselves 9 ! attempt to solve geometrical1 me*anical1 or physical problems4--,,m]z1 ,j4 ,t4 ,a ,hi/ory ( ,european ,?\ght 9 ! ,n9ete5? ,c5tury 7,ed9burgh & ,london1 #aijc71 p4 #fif4 #aicc4 ,t fl[] ( mod]n ma!matical ?\ght--! notion ( a function4--,,mccormack1 ,,?omas ,j4 ,on ! ,nature ( ,sci5tific ,law & ,sci5tific ,explanation1 ,moni/1 ,vol4 #aj 7#ahii-aijj71 p4 #eee4 #aicd4 ,fu*s4 ,i* b9 von all$em s con/]ni]t1 ,als wrde mir e9 ,kreis im ,kopfe quadri]t4 ,meph4 ,na*h] vor ali5 &]n ,sa*5 ,ms/ ihe eu* an die ,funktion5-,!orie ma*54 ,da seht1 dass ihr tiefs9nig fas/1 ,0 si* zu integri]5 ni*t pas/4 ,an ,!orem5 wird's eu* ni*t fehl51 ,ms/ nur die ,v]s*w9dungspunkte zhl51 ,umkehr51 abbild51 auf d] ,eb'ne 'rumfahr5 ,und mit d5 ,!ta-,produkt5 ni*t sp>54 --,,lasswitz1 ,,kurd4 ,d] ,fau/-,tragdie 7-n7t] ,tiel2 ,zeits*rift fr d5 ma?4-natur4 ,unt]ri*t1 ,bd4 #ad 7#ahhc71 p4 #caf4 ,fu*s4 ,y\r words fill me ) an awful dread1 ,seems l a circle 7 squ>$ 9 my head4 ,meph4 ,next 9 ord] y c]ta9ly ought ,on function-!ory be/[ y\r ?\ght1 ,& p5etrate ) contemplation ,:at resi/s y\r attempts at integration4 ,y'll f9d no de>? ( !orems !re-- ,to vani%+-po9ts give prop] c>e-- ,enum]ate1 reciprocate1 ,nor =get to del9eate1 ,trav]se ! plane f end to end1 ,& !ta-functions freely sp5d4 #aice4 ,! /ud5t %\ld avoid f\nd+ results ^u div]g5t s]ies1 z ! "q ( _! legitimacy is disput$ ^u gr\nds to :i* no answ] comm&+ any?+ l g5]al ass5t has yet be5 giv54 ,b !y may be us$ z means ( discov]y1 provid$ t _! results be v]ifi$ by o!r means be=e !y are consid]$ z e/abli%$4--,,de ,,morgan1 ,a4 ,trigonometry & ,d\ble ,algebra 7,london1 #ahdi71 p4 #ee4 #aicf4 ,!re is no?+ n[ :i* ev] gives me any ?\ght or c>e 9 algebra except div]g5t s]ies1 :i* ,i _c foll[ ! ,fr5* 9 reject+4--,,de ,,morgan1 ,a4 ,graves' ,life ( ,w4 ,r4 ,hamilton 7,new ,york1 #ahhb-ahhi71 ,vol4 #c1 p4 #bdi4 #aicg4 ,x is a /range vicissitude ( our sci5ce t ^! div]g5t s]ies :i* e>ly 9 ! c5tury 7 suppos$ to be bani%$ once & = all f rigor\s ma!matics %\ld at its close be knock+ at ! door = readmission4--,,pi]pont1 ,j4 ,congress ( ,arts & ,sci5ces 7,bo/on & ,new ,york1 #aije71 ,vol4 #a1 p4 #dgf4 #aich4 ,z5o 0 conc]n$ ) ?ree problems4444 ,^! are ! problem ( ! inf9itesimal1 ! inf9ite1 & cont9uity4444 ,f him to our own day1 ! f9e/ intellects ( ea* g5]ation 9 turn attack$ ^! problems1 b a*iev$ broadly speak+ no?+4444 ,wei]/rass1 ,d$ek9d1 & ,cantor1 444 h completely solv$ !m4 ,_! solutions 444 are s cle> z to leave no long] ! slighte/ d\bt ( difficulty4 ,?is a*ievem5t is probably ! greate/ ( :i* ! age c boa/4444 ,! problem ( ! inf9itesimal 0 solv$ by ,wei]/rass1 ! solution ( ! o!r two 0 begun by ,d$ek9d & def9itely accompli%$ by ,cantor4--,,russell1 ,,b]tr&4 ,int]national ,mon?ly1 ,vol4 #d 7#aija71 p4 #hi4 #aici4 ,x 0 n till ,leibnitz & ,newton1 by ! discov]y ( ! diff]5tial calculus1 _h dispell$ ! anci5t d>kness :i* envelop$ ! conception ( ! inf9ite1 & _h cle>ly e/abli%$ ! conception ( ! cont9u\s & cont9u\s *ange1 t a full & productive application ( ! newly-f\nd me*anical conceptions made any progress4--,,helmholtz1 ,h4 ,aim & ,progress ( ,physical ,sci5ce2 ,popul> ,lectures ,flight 7,new ,york1 #aijj71 p4 #cgb4 #aidj4 ,! idea ( an inf9itesimal involves no contradiction4444 ,z a ma!matician1 ,i pref] ! me?od ( inf9itesimals to t ( limits1 z f> easi] & less infe/$ ) sn>es4--,,pi]ce1 ,c4 ,f4 ,! ,law ( ,m9d2 ,moni/1 ,vol4 #b 7#ahia-ahib71 pp4 #edc1 #ede4 #aida4 ,! *ief objection aga9/ all ab/ract reason+s is d]iv$ f ! ideas ( space & "t2 ideas1 :i*1 9 common life & to a c>eless view1 are v cle> & intelligible1 b :5 !y pass ?r\gh ! scrut9y ( ! pr(\nd sci5ces 7& !y are ! *ief object ( ^! sci5ces7 af=d pr9ciples1 :i* seem full ( obscurity & contradiction4 ,no prie/ly dogmas1 inv5t$ on purpose to tame & subdue ! rebelli\s reason ( mank9d1 ev] %ock$ common s5se m ?an ! doctr9e ( ! inf9ite divisibility ( ext5sion1 ) its consequ5ces2 z !y are pomp\sly display$ by all geometricians & metaphysicians1 ) a k9d ( triumph & exultation4 ,a real quantity1 inf9itely less ?an any f9ite quantity1 conta9+ quantities inf9itely less ?an itself1 & s on 9 inf9itum2 ?is is an edifice s bold & prodigi\s1 t x is too weighty = any pret5d$ demon/ration to support1 because x %ocks ! cle>e/ & mo/ natural pr9ciples ( human reason4 ,b :at r5d]s ! matt] m extraord9>y1 is1 t ^! seem+ly absurd op9ions are support$ by a *a9 ( reason+1 ! cle>e/ & mo/ natural2 nor is x possible = u to all[ ! premises )\t admitt+ ! consequ5ces4 ,no?+ c be m conv9c+ & satisfactory ?an all ! conclusions conc]n+ ! prop]ties ( circles & triangles2 & yet1 :5 ^! are once receiv$1 h[ c we d5y1 t ! angle ( contact betwe5 a circle & its tang5t is inf9itely less ?an any rectil9eal angle1 t z y may increase ! diamet] ( ! circle 9 inf9itum1 ?is angle ( contact becomes /ill less1 ev5 9 inf9itum1 & t ! angle ( contact betwe5 o!r curves & _! tang5ts may be inf9itely less ?an ^? betwe5 any circle & its tang5t1 & s on1 9 inf9itum8 ,! demon/ration ( ^! pr9ciples seems z unexceptionable z t :i* proves ! ?ree angles ( a triangle to be equal to two "r ones1 ?\gh ! latt] op9ion be natural & easy1 & ! =m] big ) contradiction & absurdity4 ,reason "h seems to be ?r[n into a k9d ( amazem5t & susp5se1 :i*1 )\t ! sugge/ion ( any sceptic1 gives h] a diffid5ce ( h]self1 & ( ! gr\nd on :i* %e treads4 ,%e sees a full light1 :i* illum9ates c]ta9 places2 b t light bord]s ^u ! mo/ pr(\nd d>kness4 ,& betwe5 ^! %e is s dazzl$ & conf\nd$1 t %e sc>cely c pron\nce ) c]ta9ty & assurance conc]n+ any "o object4--,,hume1 ,,david4 ,an ,inquiry conc]n+ ,human ,und]/&+1 ,sect4 #ab1 "p #b4 #aidb4 ,he :o c dige/ a second or ?ird fluxion1 a second or ?ird diff]5ce1 ne$ n1 me?9ks1 be squeami% ab\t any po9t 9 ,div9ity4--,,b]keley1 ,g4 ,! ,analy/1 sect4 #g4 #aidc4 ,& :at are ^! fluxions8 ,! velocities ( evanesc5t increm5ts4 ,& :at are ^! same evanesc5t increm5ts8 ,!y are nei!r f9ite quantities1 nor quantities inf9itely small1 nor yet no?+4 ,may we n call !m gho/s ( dep>t$ quantities8--,,b]keley1 ,g4 ,! ,analy/1 sect4 #ce4 #aidd4 ,x is said t ! m9ute/ errors are n to be neglect$ 9 ma!matics2 t ! fluxions are cel]ities1 n proportional to ! f9ite increm5ts1 ?\gh ev] s small2 b only to ! mom5ts or nasc5t increm5ts1 :]e( ! proportion alone1 & n ! magnitude1 is consid]$4 ,& ( ! a=esaid fluxions !re be o!r fluxions1 :i* fluxions ( fluxions are call$ second fluxions4 ,& ! fluxions ( ^! second fluxions are call$ ?ird fluxions3 & s on1 f\r?1 fif?1 six?1 etc41 ad inf9itum4 ,n[1 z our ,s5se is /ra9$ & puzzl$ ) ! p]ception ( objects extremely m9ute1 ev5 s ! ,imag9ation1 :i* faculty d]ives f s5se1 is v mu* /ra9$ & puzzl$ to frame cle> ideas ( ! lea/ p>ticle ( "t1 or ! lea/ increm5t g5]at$ !re93 & mu* m to compreh5d ! mom5ts1 or ^? increm5ts ( ! fl[+ quantities 9 /atus nasc5ti1 9 _! fir/ orig9 or beg9n+ to exi/1 be=e !y become f9ite p>ticles4 ,& x seems /ill m difficult to conceive ! ab/ract$ velocities ( su* nasc5t imp]fect entities4 ,b ! velocities ( ! velocities1 ! second1 ?ird1 f\r?1 & fif? velocities1 etc41 exce$1 if ,i mi/ake n1 all human und]/&+4 ,! fur!r ! m9d analyse? & pursue? ^! fugitive ideas ! m x is lo/ & bewild]$2 ! objects1 at fir/ fleet+ & m9ute1 soon vani%+ out ( sight4 ,c]ta9ly1 9 any s5se1 a second or ?ird fluxion seems an obscure ,my/]y4 ,! incipi5t cel]ity ( an incipi5t cel]ity1 ! nasc5t augm5t ( a nasc5t augm5t1 i4e4 ( a ?+ :i* ha? no magnitude2 take x 9 :at light y please1 ! cle> conception ( x w1 if ,i mi/ake n1 be f\nd impossible2 :e!r x be s or no ,i appeal to ! trial ( e ?9k+ read]4 ,& if a second fluxion be inconceivable1 :at are we to ?9k ( ?ird1 f\r?1 fif? fluxions1 & s on )\t end4--,,b]keley1 ,g4 ,! ,analy/1 sect1 #d4 #aide4 ,! inf9ite divisibility ( f9ite ext5sion1 ?\gh x is n expressly laid d[n ei!r z an axiom or !orem 9 ! elem5ts ( t sci5ce1 yet x is ?r\gh\t ! same ev]y:]e suppos$ & ?\ght to h s insep>able & ess5tial a connection ) ! pr9ciples & demon/rations 9 ,geometry1 t ma!maticians nev] admit x into d\bt1 or make ! lea/ "q ( x4 ,&1 z ?is notion is ! s\rce :5ce d spr+ all ^? amus+ geometrical p>adoxes :i* h su* a direct repugnancy to ! pla9 common s5se ( mank9d1 & are admitt$ ) s mu* reluctance into a m9d n yet debau*$ by le>n+2 s x is ! pr9cipal occasion ( all t nice & extreme subtility :i* r5d]s ! /udy ( ,ma!matics s difficult & t$i\s4--,,b]keley1 ,g4 ,on ! ,pr9ciples ( ,human ,k1 ,sect4 #abc4 #aidf4 ,to avoid misconception1 x %\ld be borne 9 m9d t inf9itesimals are n reg>d$ z be+ actual quantities 9 ! ord9>y acceptation ( ! words1 or z capable ( exact repres5tation4 ,!y are introduc$ = ! purpose ( abridgm5t & simplification ( our reason+s1 & are an ultimate phase ( magnitude :5 x is conceiv$ by ! m9d z capable ( dim9ution bel[ any assign$ quantity1 h[ev] small4444 ,moreov] su* quantities are neglect$1 n1 z ,leibnitz /at$1 because !y are inf9itely small 9 comp>ison ) ^? t are reta9$1 :i* w\ld produce an inf9itely small error1 b because !y mu/ be neglect$ to obta9 a rigor\s result2 s9ce su* result mu/ be def9ite & det]m9ate1 & consequ5tly indep5d5t ( ^! v>iable indef9itely small quantities4 ,x may be add$ t ! precise pr9ciples ( ! inf9itesimal calculus1 l ^? ( any o!r sci5ce1 _c be ?or\ghly appreh5d$ except by ^? :o h already /udi$ ! sci5ce1 & made "s progress 9 ! application ( its pr9ciples4 --,,williamson1 ,b4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,inf9itesimal ,calculus10 ,sect4 #ab1 #ad4 #aidg4 ,we admit1 9 geometry1 n only inf9ite magnitudes1 t is to say1 magnitudes great] ?an any assignable magnitude1 b inf9ite magnitudes inf9itely great]1 ! "o ?an ! o!r4 ,?is a/oni%es our dim5sion ( bra9s1 :i* is only ab\t six in*es long1 five broad1 & six 9 dep?1 9 ! l>ge/ heads4 --,,voltaire4 ,a ,philosophical ,diction>y2 ,article 8,inf9ity40 7,bo/on1 #ahha74 #aidh4 ,inf9ity is ! l& ( ma!matical hocus pocus4 ,!re ,z]o ! magician is k+4 ,:5 ,z]o divides any numb] he *anges x )\t reg>d to its magnitude into ! inf9itely small great81 & inv]sely1 :5 divid$ by any numb] he begets ! inf9itely great small84 ,9 ?is doma9 ! circumf]5ce ( ! circle becomes a /raight l9e1 & !n ! circle c be squ>$4 ,"h all ranks are aboli%$1 = ,z]o r$uces ev]y?+ to ! same level "o way or ano!r4 ,happy is ! k+dom :]e ,z]o rules6--,,c>us1 ,,paul4 ,logical & ,ma!matical ,?\ght2 ,moni/1 ,vol4 #bj 7#aiji-aiaj71 p4 #fi4 #aidi4 ,great fleas h little fleas ^u _! backs to bite 'em1 ,& little fleas h less] fleas1 & s ad inf9itum4 ,& ! great fleas !mselves1 9 turn1 h great] fleas to g on2 ,:ile ^! aga9 h great] /ill1 & great] /ill1 & s on4 --,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #cgg4 #aiej4 ,we h adroitly def9$ ! inf9ite 9 ari?metic by a loveknot1 9 ?is mann] 2 b we possess n !re=e ! cle>] notion ( x4--,,voltaire4 ,a ,philosophical ,diction>y2 ,article 8,inf9ity40 7,bo/on1 #ahha74 #aiea4 ,i prote/ aga9/ ! use ( inf9ite magnitude z some?+ complet$1 :i* 9 ma!matics is nev] p]missible4 ,inf9ity is m]ely a facon de p>l]1 ! real mean+ be+ a limit :i* c]ta9 ratios approa* indef9itely ne>1 :ile o!rs are p]mitt$ to increase )\t re/riction4--,,gauss4 ,brief an ,s*uma*] 7#ahca72 ,w]ke1 ,bd4 #h p4 #baf4 #aieb4 ,9 spite ( ! ess5tial diff]5ce betwe5 ! conceptions ( ! pot5tial & ! actual inf9ite1 ! =m] signify+ a v>iable f9ite magnitude increas+ beyond all f9ite limits1 :ile ! latt] is a fix$1 con/ant quantity ly+ beyond all f9ite magnitudes1 x happ5s only too (t5 t ! "o is mi/ak5 = ! o!r4444 ,ow+ to a ju/ifiable av]sion to su* illegitimate actual inf9ities & ! influ5ce ( ! mod]n epicuric-mat]iali/ic t5d5cy1 a c]ta9 horror inf9iti has gr[n up 9 ext5d$ sci5tific circles1 :i* f9ds its classic expression & support 9 ! lett] ( ,gauss see #aiea1 yet x seems to me t ! consequ5t uncritical rejection ( ! legitimate actual inf9ite is no less] violation ( ! nature ( ?+s1 :i* mu/ be tak5 z !y are4--,,cantor1 ,g4 ,zum ,problem des actual5 ,un5dli*52 ,natur und ,(f5b>ung1 ,bd4 #cb 7#ahhf71 p4 #bbf4 #aiec4 ,! ,inf9ite is (t5 conf\nd$ ) ! ,indef9ite1 b ! two conceptions are diametrically oppos$4 ,in/ead ( be+ a quantity ) unassign$ yet assignable limits1 ! ,inf9ite is n a quantity at all1 s9ce x nei!r admits ( augm5tation nor dim9ution1 hav+ no assignable limits2 x is ! op]ation ( cont9u\sly )draw+ any limits t may h be5 assign$3 ! endless addition ( new quantities to ! old3 ! flux ( cont9uity4 ,! ,inf9ite is no m a quantity ?an ,z]o is a quantity4 ,if ,z]o is ! sign ( a vani%$ quantity1 ! ,inf9ite is a sign ( t cont9uity ( ,exi/5ce :i* has be5 ideally divid$ into discrete p>ts 9 ! affix+ ( limits4 --,,lewes1 ,g4 ,h4 ,problems ( ,life & ,m9d 7,bo/on1 #ahge71 ,vol4 #b1 p4 #chd4 #aied4 ,a great deal ( misund]/&+ is avoid$ if x be rememb]$ t ! t]ms inf9ity1 inf9ite1 z]o1 inf9itesimal mu/ be int]pret$ 9 connexion ) _! context1 & admit a v>iety ( mean+s accord+ to ! way 9 :i* !y are def9$4--,,ma!ws1 ,g4 ,b4 ,!ory ( ,numb]s 7,cambridge1 #ahib71 ,"p #a1 sect4 #ajd4 #aiee4 ,?is fur!r is obs]vable 9 numb]1 t x is t :i* ! m9d makes use ( 9 measur+ all ?+s t by u are measurable1 :i* pr9cipally are expansion & duration2 & our idea ( inf9ity1 ev5 :5 appli$ to ^?1 seems to be no?+ b ! inf9ity ( numb]4 ,= :at else are our ideas ( ,et]nity & ,imm5sity1 b ! repeat$ additions ( c]ta9 ideas ( imag9$ p>ts ( duration & expansion1 ) ! inf9ity ( numb]2 9 :i* we c come to no end ( addition8 --,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #b1 *ap4 #af1 sect4 #h4 #aief4 ,b ( all o!r ideas1 x is numb]1 :i* ,i ?9k furni%es u ) ! cle>e/ & mo/ di/9ct idea ( inf9ity we are capable (4--,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #b1 *ap4 #ag1 sect4 #i4 #aieg4 ,will/ du ins ,un5dli*e s*reit58 ,geh nur im ,endli*5 na* all5 ,seit56 ,will/ du di* am ,ganz5 erquick51 ,s mus/ du das ,ganze im ,kle9/5 erblick54 --,,goe!4 ,gott1 ,gemt und ,welt 7#ahae74 ,w\ld'/ ?\ ! inf9ite essay8 ,! f9ite b trav]se 9 e way4 ,w\ld'/ 9 ! :ole delight ?y he>t8 ,le>n to disc]n ! :ole 9 its m9ute/ "p4 #aieh4 ,i* hufe ungeheure ,zahl51 ,gebrge ,million5 auf1 ,i* setze ,zeit auf ,zeit und ,welt auf ,welt zu ,hauf1 ,und w5n i* von d] graus5 ,hh' ,mit ,s*w9deln wi$] na* dir seh1' ,i/ alle ,ma*t d] ,zahl1 v]mehrt zu taus5dmal51 ,no* ni*t e9 ,!il von dir4 ,i* zieh' sie ab1 und du lieg/ ganz vor mir4 --,,hall]1 ,,albr4 ,,von4 ,quot$ 9 ,hegel3 ,wiss5s*aft d] ,logik1 ,bu* #a1 ,abs*nitt #b1 ,kap4 #b1 ,c1 b4 ,numb]s ^u numb]s pile1 ,m\nta9s millions high1 ,"t on "t & _w on _w amass1 ,!n1 if f ! dreadful hight1 alas6 ,dizzy-bra9$1 ,i turn !e to behold1 ,all ! p[] ( numb]1 increas$ ?\s&fold1 ,n yet may mat* ?y "p4 ,subtract :at ,i w1 :olly :ole ?\ art4 #aiei4 ,a collection ( t]ms is inf9ite :5 x conta9s z p>ts o!r collections :i* h j z _m t]ms 9 x z x has4 ,if y c take away "s ( ! t]ms ( a collection1 )\t dim9i%+ ! numb] ( t]ms1 !n !re is an inf9ite numb] ( t]ms 9 ! collection4--,,russell1 ,,b]tr&4 ,int]national ,mon?ly1 ,vol4 #d 7#aija71 p4 #ic4 #aifj4 ,an assemblage 7ensemble1 collection1 gr\p1 manifold7 ( elem5ts 7?+s1 no matt] :at7 is inf9ite or f9ite accord+ z x has or has n a "p to :i* ! :ole is j equival5t 9 ! s5se t betwe5 ! elem5ts compos+ t "p & ^? compos+ ! :ole !re subsi/s a unique & reciprocal 7"o-to-"o7 correspond5ce4--,,keys]1 ,c4 ,j4 ,! ,axioms ( ,inf9ity2 ,hibb]t ,j\rnal1 ,vol4 #b 7#aijc-aijd71 p4 #eci4 #aifa4 ,:]eas 9 =m] times ! ,inf9ite betray$ its pres5ce n inde$ to ! faculties ( ,logic b only to ! spiritual ,imag9ation & ,s5sibility1 ma!matics has %[n 444 t ! /ructure ( ,transf9ite ,be+ is op5 to exploration by ! organon ( ,?\ght4--,,keys]1 ,c4 ,j4 ,lectures on ,sci5ce1 ,philosophy & ,art 7,new ,york1 #aijh71 p4 #db4 #aifb4 ,! ma!matical !ory ( probability is a sci5ce :i* aims at r$uc+ to calculation1 :]e possible1 ! am\nt ( cr$5ce due to propositions or /atem5ts1 or to ! occurr5ce ( ev5ts1 future or pa/1 m especially z cont+5t or dep5d5t ^u o!r propositions or ev5ts ! probability ( :i* is kn[n4--,,cr(ton1 ,m4 ,w4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article1 8,probability40 #aifc4 ,! !ory ( probabilities is at bottom no?+ b common s5se r$uc$ to calculus2 x enables u to appreciate ) exactness t :i* accurate m9ds feel ) a sort ( in/9ct = :i* (ttimes !y are unable to acc\nt4 ,if we consid] ! analytical me?ods to :i* ?is !ory has giv5 bir?1 ! tru? ( ! pr9ciples on :i* x is bas$1 ! f9e & delicate logic :i* _! employm5t 9 ! solution ( problems requires1 ! public utilities ^: e/abli%m5t re/s ^u x1 ! ext5sion :i* x has receiv$ & :i* x may /ill receive ?r\gh its application to ! mo/ important problems ( natural philosophy & ! moral sci5ces2 if aga9 we obs]ve t1 ev5 9 matt]s :i* _c be submitt$ to ! calculus1 x gives u ! sure/ sugge/ions = ! guidance ( our judgm5ts1 & t x tea*es u to avoid ! illusions :i* (t5 mislead u1 !n we %all see t !re is no sci5ce m wor?y ( our contemplations nor a m useful "o = admission to our sy/em ( public education4--,,laplace4 ,?orie ,analytique des ,probabilitis1 ,introduction2 ,oeuvres1 t4 #g 7,p>is1 #ahhf71 p4 #aec4 #aifd4 ,x is a tru? v c]ta9 t1 :5 x is n 9 our p[] to det]m9e :at is true1 we ought to foll[ :at is mo/ probable4--,,desc>tes4 ,disc\rse on ,me?od1 ,"p #c4 #aife4 ,z demon/ration is ! %[+ ! agreem5t or disagreem5t ( two ideas1 by ! int]v5tion ( "o or m pro(s1 :i* h a con/ant1 immutable1 & visible connexion "o ) ano!r2 s probability is no?+ b ! appe>ance ( su* an agreem5t or disagreem5t1 by ! int]v5tion ( pro(s1 ^: connexion is n con/ant & immutable1 or at lea/ is n p]ceiv$ to be s1 & x is en\gh to induce ! m9d to judge ! proposition to be true or false1 r ?an contr>y4--,,locke1 ,,john4 ,an ,essay conc]n+ ,human ,und]/&+1 ,bk4 #d1 *ap4 #ae1 sect4 #a4 #aiff4 ,! diff]5ce betwe5 necess>y & cont+5t tru?s is inde$ ! same z t betwe5 comm5surable & incomm5surable numb]s4 ,= ! r$uction ( comm5surable numb]s to a common measure is analog\s to ! demon/ration ( necess>y tru?s1 or _! r$uction to su* z are id5tical4 ,b z1 9 ! case ( surd ratios1 ! r$uction involves an inf9ite process1 & yet approa*es a common measure1 s t a def9ite b un5d+ s]ies is obta9$1 s also cont+5t tru?s require an inf9ite analysis1 :i* ,god alone c accompli%4--,,leibnitz4 ,philosophis*e ,s*rift5 ,g]h>dt ,bd4 #g 7,b]l91 #ahij71 p4 #bjj4 #aifg4 ,! !ory 9 "q !ory ( probability af=ds an excell5t illu/ration ( ! application ( ! !ory ( p]mutation & comb9ations :i* is ! fundam5tal "p ( ! algebra ( discrete quantity2 x =ms 9 ! elem5t>y p>ts an excell5t logical ex]cise 9 ! accurate use ( t]ms & 9 ! nice discrim9ation ( %ades ( mean+2 &1 above all1 x ent]s into ! regulation ( "s ( ! mo/ important practical conc]ns ( mod]n life4--,,*ry/al1 ,,george4 ,algebra1 ,vol4 #b 7,ed9burgh1 #ahhi71 *ap4 #cf1 sect4 #a4 #aifh4 ,!re is possibly no bran* ( ma!matics at once s int]e/+1 s bewild]+1 & ( s great practical importance z ! !ory ( probabilities4 ,its hi/ory reveals bo? ! wond]s t c be accompli%$ & ! b\nds t _c be transc5d$ by ma!matical sci5ce4 ,x is ! l9k betwe5 rigid d$uction & ! va/ field ( inductive sci5ce4 ,a complete !ory ( probabilities w\ld be ! complete !ory ( ! =mation ( belief4 ,x is c]ta9ly a pity !n1 t1 to quote ,m4 ,b]tr&1 8"o _c well und]/& ! calculus ( probabilities )\t hav+ read ,laplace's "w10 & t 8"o _c read ,laplace's "w )\t hav+ prep>$ oneself = x by ! mo/ pr(\nd ma!matical /udies40--,,davis1 ,e4 ,w4 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #a 7#ahid-ahie71 p4 #af4 #aifi4 ,! mo/ important que/ions ( life are1 = ! mo/ "p1 really only problems ( probability4 ,/rictly speak+ "o may ev5 say t ne>ly all our k is problematical2 & 9 ! small numb] ( ?+s :i* we are able to "k ) c]ta9ty1 ev5 9 ! ma!matical sci5ces !mselves1 induction & analogy1 ! pr9cipal means = discov]+ tru?1 are bas$ on probabilities1 s t ! entire sy/em ( human k is connect$ ) ?is !ory4--,,laplace4 ,?orie ,analytique des ,probabilitis1 ,introduction2 ,oeuvres1 t4 #g 7,p>is1 #ahhf71 p4 #e4 #aigj4 ,!re is no m rem>kable feature 9 ! ma!matical !ory ( probability ?an ! mann] 9 :i* x has be5 f\nd to h>monize )1 & ju/ify1 ! conclusions to :i* mank9d h be5 l$1 n by reason+1 b by in/9ct & exp]i5ce1 bo? ( ! individual & ( ! race4 ,at ! same "t x has correct$1 ext5d$1 & inve/$ !m ) a def9it5ess & precision ( :i* ^! crude1 ?\gh s\nd1 appreciations ( common s5se 7 till !n devoid4--,,cr(ton1 ,m4 ,w4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,probability40 #aiga4 ,x is rem>kable t a sci5ce probabilities :i* began ) ! consid]ation ( games ( *ance1 %\ld h become ! mo/ important object ( human k4--,,laplace4 ,?orie ,analytique des ,probabilitis1 ,introduction2 ,oeuvres1 t4 #g 7,p>is1 #ahhf71 p4 #aeb4 #aigb4 ,n mu* has be5 add$ to ! subject ( probability s9ce ! close ( ,laplace's c>e]4 ,! hi/ory ( sci5ce records m ?an "o p>allel to ?is abatem5t ( activity4 ,:5 su* a g5ius has dep>t$1 ! field ( 8 lab\rs seems exhau/$ = ! "t1 & little left to be glean$ by 8 successors4 ,x is to be regrett$ t s little rema9s to u ( ! inn] work+s ( su* gift$ m9ds1 & ( ! clue by :i* ea* ( _! discov]ies 0 rea*$4 ,! didactic & syn!tic =m 9 :i* ^! are pres5t$ to ! _w reta9s b fa9t traces ( ! skilful inductions1 ! ke5 & delicate p]ception ( fitness & analogy1 & ! p[] ( imag9ation 444 :i* h d\btless guid$ su* a ma/] z ,laplace or ,newton 9 %ap+ out su* great designs--only ! m9or details ( :i* h rema9$ ov]1 to be suppli$ by ! less cunn+ h& ( comm5tator & disciple4--,,cr(ton1 ,m4 ,w4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,probability40 #aigc4 ,! !ory ( errors may be def9$ z t bran* ( ma!matics :i* is conc]n$1 fir/1 ) ! expression ( ! resultant effect ( "o or m s\rces ( error to :i* comput$ & obs]v$ quantities are subject2 &1 secondly1 ) ! det]m9ation ( ! relation betwe5 ! magnitude ( an error & ! probability ( its occurr5ce4--,,woodw>d1 ,r4 ,s4 ,probability & ,!ory ( ,errors 7,new ,york1 #aijf71 p4 #cj4 #aigd4 ,( all ! applications ( ! doctr9e ( probability none is ( great] utility ?an ! !ory ( errors4 ,9 a/ronomy1 geodesy1 physics1 & *emi/ry1 z 9 e sci5ce :i* atta9s precision 9 measur+1 weigh+1 & comput+1 a k ( ! !ory ( errors is indisp5sable4 ,by ! aid ( ?is !ory ! exact sci5ces h made great progress dur+ ! n9ete5? c5tury1 n only 9 ! actual det]m9ations ( ! con/ants ( nature1 b also 9 ! fixation ( cle> ideas z to ! possibilities ( future conque/s 9 ! same direction4 ,no?+1 = example1 is m satisfactory & in/ructive 9 ! hi/ory ( sci5ce ?an ! success ) :i* ! unique me?od ( lea/ squ>es has be5 appli$ to ! problems pres5t$ by ! e>? & ! o!r memb]s ( ! sol> sy/em4 ,s great1 9 fact1 are ! practical value & !oretical importance ( lea/ squ>es1 t x is frequ5tly mi/ak5 = ! :ole !ory ( errors1 & is sometimes reg>d$ z embody+ ! major "p ( ! doctr9e ( probability itself4--,,woodw>d1 ,r4 ,s4 ,probability & ,!ory ( ,errors 7,new ,york1 #aijf71 pp4 #i-aj4 #aige4 ,direct & inv]se ratios h be5 appli$ by an +5i\s au?or to measure human affections1 & ! moral wor? ( actions4 ,an em95t ,ma!matician attempt$ to asc]ta9 by calculation1 ! ratio 9 :i* ! evid5ce ( facts mu/ decrease 9 ! c\rse ( "t1 & fix$ ! p]iod :5 ! evid5ce ( ! facts on :i* ,*ri/ianity is f\nd$ %all become evanesc5t1 & :5 9 consequ5ce no fai? %all be f\nd on ! e>?4--,,reid1 ,,?omas4 ,essays on ! ,p[]s ( ! ,human ,m9d 7,ed9burgh1 #ahab71 ,vol4 #b1 p4 #djh4 ,,*apt] ,,xx ,,! ,,fundam5tal ,,concepts1 ,,"t ,,& ,,space #bjja4 ,kant's ,doctr9e ( ,"t4 ,i4 ,"t is n an empirical concept d$uc$ f any exp]i5ce1 = nei!r co-exi/5ce nor succession w\ld ent] into our p]ception1 if ! repres5tation ( "t 7 n giv5 a priori4 ,only :5 ?is repres5tation a priori is giv51 c we imag9e t c]ta9 ?+s happ5 at ! same "t 7simultane\sly7 or at diff]5t times 7successively74 ,,ii4 ,"t is a necess>y repres5tation on :i* all intuitions dep5d4 ,we _c take away "t f ph5om5a 9 g5]al1 ?\gh we c well take away ph5om5a out ( "t4 ,9 "t alone is reality ( ph5om5a possible4 ,all ph5om5a may vani%1 b "t itself 7z ! g5]al condition ( _! possibility7 _c be done away )4 ,,iii4 ,on ?is a priori necessity dep5ds also ! possibility ( apodictic pr9ciples ( ! relations ( "t1 or ( axioms ( "t 9 g5]al4 ,"t has "o dim5sion only2 diff]5t times are n simultane\s1 b successive1 :ile diff]5t spaces are nev] successive1 b simultane\s4 ,su* pr9ciples _c be d]iv$ f exp]i5ce1 because exp]i5ce c\ld n imp>t to !m absolute univ]sality nor apodictic c]ta9ty4444 ,,iv4 ,"t is n a discursive1 or :at is call$ a g5]al concept1 b a pure =m ( s5su\s intuition4 ,diff]5t times are p>ts only ( "o & ! same "t4444 ,v4 ,to say t "t is inf9ite means no m ?an t e def9ite quantity ( "t is possible only by limitations ( "o "t :i* =ms ! f\ndation ( all times4 ,! orig9al repres5tation ( "t mu/ !re=e be giv5 z unlimit$4 ,b :5 ! p>ts !mselves & e quantity ( an object c be repres5t$ z det]m9$ by limitation only1 ! :ole repres5tation _c be giv5 by concepts 7= 9 t case ! p>tial repres5tation comes fir/71 b mu/ be f\nd$ on imm$iate intuition4--,,kant1 ,i4 ,critique ( ,pure ,reason ,max ,mll] 7,new ,york1 #aijj71 pp4 #bd-be4 #bjjb4 ,kant's ,doctr9e ( ,space4 ,i4 ,space is n an empirical concept :i* has be5 d]iv$ f ext]nal exp]i5ce4 ,= 9 ord] t c]ta9 s5sations %\ld be ref]r$ to some?+ outside myself1 i4e4 to some?+ 9 a diff]5t "p ( space f t :]e ,i am2 aga91 9 ord] t ,i may be able to repres5t !m z side by side1 t is1 n only z diff]5t1 b z 9 diff]5t places1 ! repres5tation ( space mu/ already be !re4444 ,,ii4 ,space is a necess>y repres5tation a priori1 =m+ ! v f\ndation ( all ext]nal intuitions4 ,x is impossible to imag9e t !re %\ld be no space1 ?\gh "o might v well imag9e t !re %\ld be space )\t objects to fill x4 ,space is !re=e reg>d$ z a condition ( ! possibility ( ph5om5a1 n z a det]m9ation produc$ by !m2 x is a repres5tation a priori :i* necess>ily prec$es all ext]nal ph5om5a4 ,,iii4 ,on ?is necessity ( an a priori repres5tation ( space re/s ! apodictic c]ta9ty ( all geometrical pr9ciples1 & ! possibility ( _! con/ruction a priori4 ,= if ! intuition ( space 7 a concept ga9$ a po/]iori1 borr[$ f g5]al ext]nal exp]i5ce1 ! fir/ pr9ciples ( ma!matical def9ition w\ld be no?+ b p]ceptions4 ,!y w\ld be expos$ to all ! accid5ts ( p]ception1 & !re be+ b "o /raight l9e betwe5 two po9ts w\ld n be a necessity1 b only some?+ taught 9 ea* case by exp]i5ce4 ,:atev] is d]iv$ f exp]i5ce possesses a relative g5]ality only1 bas$ on induction4 ,we %\ld !re=e n be able to say m ?an t1 s f> z hi!rto obs]v$1 no space has yet be5 f\nd hav+ m ?an ?ree dim5sions4 ,,iv4 ,space is n a discursive or s-call$ g5]al concept ( ! relations ( ?+s 9 g5]al1 b a pure intuition4 ,=1 fir/ ( all1 we c imag9e "o space only1 & if we speak ( _m spaces1 we mean p>ts only ( "o & ! same space4 ,nor c ^! p>ts be consid]$ z antec$5t to ! "o & all-embrac+ space &1 z x 71 its compon5t p>ts out ( :i* an aggregate is =m$1 b !y c be ?\ght ( z exi/+ )9 x only4 ,space is ess5tially "o2 its multiplicity1 & !re=e ! g5]al concept ( spaces 9 g5]al1 arises entirely f limitations4 ,h5ce x foll[s t1 ) respect to space1 an intuition a priori1 :i* is n empirical1 mu/ =m ! f\ndation ( all conceptions ( space4444 ,v4 ,space is repres5t$ z an inf9ite giv5 quantity4 ,n[ x is q true t e concept is to be ?\ght z a repres5tation1 :i* is conta9$ 9 an inf9ite numb] ( diff]5t possible repres5tations 7z _! common *>act]i/ic71 & !re=e compreh5ds !m3 b no concept1 z su*1 c be ?\ght z if x conta9$ 9 itself an inf9ite numb] ( repres5tations4 ,nev]!less1 space is s ?\ght 7= all p>ts ( inf9ite space exi/ simultane\sly74 ,consequ5tly1 ! orig9al repres5tation ( space is an intuition a priori1 & n a concept4--,,kant1 ,i4 ,critique ( ,pure ,reason ,max ,mll] 7,new ,york1 #aijj71 pp4 #ah-bj & ,supplem5t #h4 #bjjc4 ,s*op5hau]'s ,pr$icabilia a priori4#aa ,,( ,,"t ,,( ,,space #a4 ,!re is b "o "t1 all #a4 ,!re is b "o space1 diff]5t times are p>ts ( all diff]5t spaces are x4 p>ts ( x4 #b4 ,diff]5t times are n #b4 ,diff]5t spaces are n simultane\s b successive4 successive b simultane\s4 #c4 ,ev]y?+ 9 "t may be #c4 ,ev]y?+ 9 space may be ?\ght ( z non-exi/5t1 ?\ght ( z non-exi/5t1 b n "t4 b n space4 #d4 ,"t has ?ree divisions3 #d4 ,space has ?ree dim5sions3 pa/1 pres5t & future1 height1 bread?1 & :i* =m two directions l5g?4 ) a po9t ( indiff]5ce4 #e4 ,"t is inf9itely #e4 ,space is inf9itely divisible4 divisible4 #f4 ,"t is homog5e\s & a #f4 ,space is homog5e\s & a cont9uum3 i4e4 no "p is cont9uum3 i4e4 no "p diff]5t f ano!r1 nor is diff]5t f ano!r1 sep>at$ by some?+ nor sep>at$ by some?+ :i* is n "t4 :i* is n space4 #g4 ,"t has no beg9n+ nor #g4 ,space has no limits end1 b all beg9n+ & ,grnz51 b all limits end is 9 "t4 are 9 space4 #h4 ,"t makes c\nt+ #h4 ,space makes measurem5t possible4 possible4 #i4 ,rhy?m exi/s only 9 "t4 #i4 ,symmetry exi/s only 9 space4 #aj4 ,! laws ( "t are a #aj4 ,! laws ( space are a priori conceptions4 priori conceptions4 #aa4 ,"t is p]ceptible a #aa4 ,space is imm$iately priori1 b only by a p]ceptible a priori4 means ( a l9e-image4 #ab4 ,"t has no p]man5ce b #ab4 ,space nev] passes b is passes ! mom5t x is p]man5t ?r\gh\t pres5t4 all "t4 #ac4 ,"t nev] re/s4 #ac4 ,space nev] moves4 #ad4 ,ev]y?+ 9 "t has #ad4 ,ev]y?+ 9 space has duration4 position4 #ae4 ,"t has no duration1 b #ae4 ,space has no motion1 b all duration is 9 "t2 all motion is 9 space2 "t is ! p]si/5ce ( space is ! *ange 9 :at is p]man5t 9 position ( t :i* contra/ ) its re/less moves 9 contra/ to its c\rse4 imp]turbable re/4 #af4 ,motion is only possible 9 #af4 ,motion is only possible 9 "t4 space4 #ag4 ,velocity1 ! space be+ #ag4 ,velocity1 ! "t be+ ! same1 is 9 ! inv]se ! same1 is 9 ! direct ratio ( ! "t4 ratio ( ! space4 #ah4 ,"t is n directly #ah4 ,space is measurable directly measurable by means ( ?r\gh itself & itself b only by means ( indirectly ?r\gh motion motion :i* takes place 9 :i* takes place 9 bo? bo? space & "t4444 "t & space4444 #ai4 ,"t is omnipres5t3 ea* #ai4 ,space is et]nal3 ea* "p ( x is ev]y:]e4 "p ( x exi/s always4 #bj4 ,9 "t alone all ?+s #bj4 ,9 space alone all ?+s are successive4 are simultane\s4 #ba4 ,"t makes possible ! #ba4 ,space makes possible ! *ange ( accid5ts4 endurance ( sub/ance4 #bb4 ,ea* "p ( "t conta9s #bb4 ,no "p ( space conta9s all sub/ance4 ! same sub/ance z ano!r4 #bc4 ,"t is ! pr9cipium #bc4 ,space is ! pr9cipium individuationis4 individuationis4 #bd4 ,! n[ is )\t #bd4 ,! po9t is )\t duration4 ext5sion4 #be4 ,"t ( itself is empty & #be4 ,space is ( itself empty indet]m9ate4 & indet]m9ate4 #bf4 ,ea* mom5t is condition$ #bf4 ,! relation ( ea* by ! "o :i* prec$es b\nd>y 9 space to e x1 & only s f> z ?is o!r is det]m9$ by its "o has ceas$ to exi/4 relation to any "o4 7,pr9ciple ( suffici5t 7,pr9ciple ( suffici5t reason ( be+ 9 "t47 reason ( be+ 9 space47 #bg4 ,"t makes ,ari?metic #bg4 ,space makes ,geometry possible4 possible4 #bh4 ,! simple elem5t ( #bh4 ,! elem5t ( ,geometry ,ari?metic is unity4 is ! po9t4 --,,s*op5hau]1 ,a4 ,die ,welt als ,vor/ellung und ,wille2 ,w]ke 7,frau5/dt7 7,leipzig1 #ahgg71 ,bd4 #b1 p4 #ee4 #aa ,s*op5hau]'s table conta9s a ?ird column head$ 8( matt]0 :i* has "h be5 omitt$4 #bjjd4 ,! cle> possession ( ! ,idea ( ,space is ! fir/ requisite = all geometrical reason+2 & ?is cle>ness ( idea may be te/$ by exam9+ :e!r ! axioms (f] !mselves to ! m9d z evid5t4--,,:ewell1 ,,william4 ,! ,philosophy ( ! ,inductive ,sci5ces1 ,"p #a1 ,bk4 #b1 *ap4 #d1 sect4 #d 7,london1 #aheh74 #bjje4 ,geometrical axioms are nei!r syn!tic a priori conclusions nor exp]im5tal facts4 ,!y are conv5tions3 our *oice1 among/ all possible conv5tions1 is guid$ by exp]im5tal facts2 b x rema9s free1 & is only limit$ by ! necessity ( avoid+ all contradiction4444 ,9 o!r words1 axioms ( geometry are only def9itions 9 disguise4 ,t be+ s :at ought "o to ?9k ( ?is "q3 ,is ! ,euclidean ,geometry true8 ,! "q is nons5se4 ,"o might z well ask :e!r ! metric sy/em is true & ! old measures false2 :e!r ,c>tesian co-ord9ates are true & pol> co-ord9ates false4 --,,po9c>1 ,h4 ,non-,euclidean ,geometry2 ,nature1 ,vol #de 7#ahia-ahib71 p4 #djg4 #bjjf4 ,i d 9 no wise %>e ?is view t ! axioms are arbitr>y propositions :i* we assume :olly at w1 & t 9 l mann] ! fundam5tal conceptions are 9 ! end only arbitr>y symbols ) :i* we op]ate b consid] x ! dea? ( all sci5ce3 9 my judgm5t ! axioms ( geometry are n arbitr>y1 b reasonable propositions :i* g5]ally h ! orig9 9 space intuition & ^: sep>ate cont5t & sequ5ce is controll$ by reasons ( exp$i5cy4--,,kle91 ,f4 ,elem5t>ma!matik vom hh]5 ,/&punkte aus 7,leipzig1 #aiji71 ,bd4 #b1 p4 #chd4 #bjjg4 ,euclid's ,po/ulate #e ,! ,p>allel ,axiom4 ,t1 if a /raight l9e fall+ on two /raight l9es make ! int]ior angles on ! same side less ?an two "r angles1 ! two /raight l9es1 if produc$ indef9itely1 meet on t side on :i* are ! angles less ?an ! two "r angles4--,,euclid4 ,! ,?irte5 ,books ( ,euclid's ,elem5ts ,t4 ,l4 ,hea? ,vol4 #a 7,cambridge1 #aijh71 p4 #bjb4 #bjjh4 ,x mu/ be admitt$ t ,euclid's ,p>allel ,axiom is unsatisfactory z ! basis ( a !ory ( p>allel /raight l9es4 ,x _c be reg>d$ z ei!r simple or self-evid5t1 & x !re=e falls %ort ( ! ess5tial *>act]i/ics ( an axiom4444--,,hall1 ,h4 ,s4 & ,,/ev5s1 ,f4 ,h4 ,euclid's ,elem5ts 7,london1 #ahib71 p4 #ee4 #bjji4 ,we may /ill well decl>e ! p>allel axiom ! simple/ assumption :i* p]mits u to repres5t spatial relations1 & s x w be true g5]ally1 t concepts & axioms are n imm$iate facts ( intuition1 b r ! idealizations ( ^! facts *os5 = reasons ( exp$i5cy4--,,kle91 ,f4 ,elem5t>ma!matik vom1 hh]5 ,/&punkte aus 7,leipzig1 #aiji71 ,bd4 #b1 p4 #chb4 #bjaj4 ,! *>act]i/ic features ( our space are n necessities ( ?\ght1 & ! tru? ( ,euclid's axioms1 9 s f> z !y specially diff]5tiate our space f o!r conceivable spaces1 mu/ be e/abli%$ by exp]i5ce & by exp]i5ce only4--,,ball1 ,r4 ,s4 ,encyclop$ia ,britannica1 #i? ,edition2 ,article 8,measurem5t40 #bjaa4 ,ma!matical & physiological rese>*es h %[n t ! space ( exp]i5ce is simply an actual case ( _m conceivable cases1 ab\t ^: peculi> prop]ties exp]i5ce alone c in/ruct u4--,,ma*1 ,,ern/4 ,popul> ,sci5tific ,lectures 7,*icago1 #aiaj71 p4 #bje4 #bjab4 ,! famili> def9ition3 ,an axiom is a self-evid5t tru?1 means if x means any?+1 t ! proposition :i* we call an axiom has be5 approv$ by u 9 ! light ( our exp]i5ce & intuition4 ,9 ?is s5se ma!matics has no axioms1 = ma!matics is a =mal subject ov] :i* =mal & n mat]ial implication reigns4--,,wilson1 ,e4 ,b4 ,bullet9 ,am]ican ,ma!matical ,society1 ,vol4 #b 7#aijd-aije71 p4 #ha4 #bjac4 ,! pro( ( self-evid5t propositions may seem1 to ! un9itiat$1 a some:at frivol\s occupation4 ,to ?is we might reply t x is (t5 by no means self-evid5t t "o obvi\s proposition foll[s f ano!r obvi\s proposition2 s t we are really discov]+ new tru?s :5 we prove :at is evid5t by a me?od :i* is n evid5t4 ,b a m int]e/+ retort is1 t s9ce p h tri$ to prove obvi\s propositions1 !y h f\nd t _m ( !m are false4 ,self-evid5ce is (t5 a m]e w-o'-!-wisp1 :i* is sure to lead u a/ray if we take x z our guide4--,,russell1 ,,b]tr&4 ,rec5t ,"w on ! ,pr9ciples ( ,ma!matics2 ,int]national ,mon?ly1 ,vol4 #d 7#aija71 p4 #hf4 #bjad4 ,! problem ( ,euclid's ,p>allel ,axiom is n[ at a p> ) ! squ>+ ( ! circle & ! trisection ( an angle by means ( rul] & compass4 ,s f> z ! ma!matical public is conc]n$1 ! fam\s problem ( ! p>allel is settl$ = all "t4--,,"y1 ,,john ,,wesley4 ,fundam5tal ,concepts ( ,algebra & ,geometry 7,new ,york1 #aiaa71 p4 #cb4 #bjae4 ,if ! ,euclidean assumptions are true1 ! con/itution ( ^? p>ts ( space :i* are at an inf9ite di/ance f u1 8geometry ^u ! plane at inf9ity10 is j z well kn[n z ! geometry ( any portion ( ?is room4 ,9 ?is inf9ite & ?or\ghly well-kn[n space ! ,univ]se is situat$ dur+ at lea/ "s portion ( an inf9ite & ?or\ghly well-kn[n "t4 ,s t "h we h real k ( some?+ at lea/ t conc]ns ! ,cosmos2 some?+ t is true ?r\gh\t ! ,imm5sities & ! ,et]nities4 ,t some?+ ,lobat*ewsky & 8 successors h tak5 away4 ,! geomet] ( to-day kn[s no?+ ab\t ! nature ( ! actually exi/+ space at an inf9ite di/ance2 he kn[s no?+ ab\t ! prop]ties ( ?is pres5t space 9 a pa/ or future et]nity4 ,he kn[s1 inde$1 t ! laws assum$ by ,euclid are true ) an accuracy t no direct exp]im5t c approa*1 n only 9 ?is place :]e we are1 b 9 places at a di/ance f u t no a/ronom] has conceiv$2 b he kn[s ?is z ( ,"h & ,n[2 beyond ?is range is a ,!re & ,!n ( :i* he kn[s no?+ at pres5t1 b may ultimately come to "k m4--,,clif=d1 ,w4 ,k4 ,lectures & ,essays 7,new ,york1 #aija71 ,vol4 #a1 pp4 #ceh-cei4 #bjaf4 ,! tru? is t o!r sy/ems ( geometry are possible1 yet aft] all1 ^! o!r sy/ems are n spaces b o!r me?ods ( space measurem5ts4 ,!re is "o space only1 ?\gh we may conceive ( _m diff]5t manifolds1 :i* are contrivances or ideal con/ructions inv5t$ = ! purpose ( det]m9+ space4--,,c>us1 ,,paul4 ,sci5ce1 ,vol4 #ah 7#aijc71 p4 #ajf4 #bjag4 ,z ,i h =m]ly /at$ t f ! philosophic side ,non-,euclidean ,geometry has z yet n frequ5tly met ) full und]/&+1 s ,i mu/ n[ emphasize t x is univ]sally recogniz$ 9 ! sci5ce ( ma!matics2 inde$1 = _m purposes1 z = in/ance 9 ! mod]n !ory ( functions1 x is us$ z an extremely conv5i5t means = ! visual repres5tation ( highly complicat$ ari?metical relations4 --,,kle91 ,f4 ,elem5t>ma!matik vom hh]5 ,/&punkte aus 7,leipzig1 #aiji71 ,bd4 #b1 p4 #cgg4 #bjah4 ,ev]y?+ 9 physical sci5ce1 f ! law ( gravitation to ! build+ ( bridges1 f ! spectroscope to ! art ( navigation1 w\ld be pr(\ndly modifi$ by any consid]able inaccuracy 9 ! hypo!sis t our actual space is ,euclidean4 ,! obs]v$ tru? ( physical sci5ce1 !re=e1 con/itutes ov]:elm+ empirical evid5ce t ?is hypo!sis is v approximately correct1 ev5 if n rigidly true4 --,,russell1 ,,b]tr&4 ,f\ndations ( ,geometry 7,cambridge1 #ahig71 p4 #f4 #bjai4 ,! mo/ sugge/ive & notable a*ievem5t ( ! la/ c5tury is ! discov]y ( ,non-,euclidean geometry4--,,hilb]t1 ,d4 ,quot$ by ,g4 ,d4 ,fit* 9 ,mann+'s 8,! ,f\r? ,dim5sion ,simply ,expla9$10 7,new ,york1 #aiaj71 p4 #eh4 #bjbj4 ,non-,euclidean geometry--primate among ! emancipators ( ! human intellect4444--,,keys]1 ,c4 ,j4 ,! ,f\ndations ( ,ma!matics2 ,sci5ce ,hi/ory ( ! ,univ]se1 ,vol4 #h 7,new ,york1 #aiji71 p4 #aib4 #bjba4 ,e high s*ool tea*] ,gymnasial-lehr] mu/ ( necessity "k some?+ ab\t non-euclidean geometry1 because x is "o ( ! few bran*es ( ma!matics :i*1 by means ( c]ta9 cat*-phrases1 has become kn[n 9 wid] circles1 & conc]n+ :i* any tea*] is consequ5tly liable to be ask$ at any "t4 ,9 physics !re are _m su* matt]s--almo/ e new discov]y is ( ?is k9d--:i*1 ?r\gh c]ta9 cat*-words h become topics ( common conv]sation1 & ab\t :i* !re=e e tea*] mu/ ( c\rse be in=m$4 ,?9k ( a tea*] ( physics :o kn[s no?+ ( ,ro5tg5 rays or ( radium2 no bett] impression w\ld be made by a ma!matician :o is unable to give in=mation conc]n+ non-euclidean geometry4 --,,kle91 ,f4 ,elem5t>ma!matik vom hh]5 ,/&punkte aus 7,leipzig1 #aiji71 ,bd4 #b1 p4 #cgh4 #bjbb4 ,:at ,vesalius 0 to ,gal51 :at ,cop]nicus 0 to ,ptolemy1 t 0 ,lobat*ewsky to ,euclid4 ,!re is1 inde$1 a some:at in/ructive p>allel betwe5 ! la/ two cases4 ,cop]nicus & ,lobat*ewsky 7 bo? ( ,slavic orig94 ,ea* ( !m has br\ght ab\t a revolution 9 sci5tific ideas s great t x c only be comp>$ ) t wr\ght by ! o!r4 ,& ! reason ( ! transc5d5t importance ( ^! two *anges is t !y are *anges 9 ! conception ( ! ,cosmos4444 ,& 9 virtue ( ^! two revolutions ! idea ( ! ,univ]se1 ! ,macrocosm1 ! ,all1 z subject ( human k1 & !re=e ( human int]e/1 has fall5 to pieces4--,,clif=d1 ,w4 ,k4 ,lectures & ,essays 7,new ,york1 #aija71 ,vol4 #a1 pp4 #cef1 #ceh4 #bjbc4 ,i am exce$+ly sorry t ,i h fail$ to avail myself ( our =m] great] proximity to le>n m ( y\r "w on ! f\ndations ( geometry2 x surely w\ld h sav$ me mu* useless ef=t & giv5 me m peace1 ?an "o ( my disposition c enjoy s long z s mu* is left to consid] 9 a matt] ( ?is k9d4 ,i h myself made mu* progress 9 ?is matt] 7?\gh my o!r het]og5e\s occupations h left me b little "t = ?is purpose72 ?\gh ! c\rse :i* ,i h pursu$ does n lead z mu* to ! desir$ end1 :i* y assure me y h rea*$1 z to ! que/ion+ ( ! tru? ( geometry4 ,x is true t ,i h f\nd mu* :i* _m w\ld accept z pro(1 b :i* 9 my e/imation proves no?+1 = in/ance1 if x c\ld be %[n t a rectil9e> triangle is possible1 ^: area is great] ?an t ( any giv5 surface1 !n ,i c\ld rigor\sly e/abli% ! :ole ( geometry4 ,n[ mo/ p1 no d\bt1 w\ld grant ?is z an axiom1 b n ,i2 x is conceivable t1 h[ev] di/ant ap>t ! v]tices ( ! triangle might be *os51 its area might yet always be bel[ a c]ta9 limit4 ,i h f\nd sev]al o!r su* !orems1 b none ( !m satisfies me4--,,gauss4 ,lett] to ,bolyai 7#agii72 ,w]ke1 ,bd4 #h 7,gtt+51 #aijj71 p4 #aei4 #bjbd4 ,on ! supposition t ,euclidean geometry is n valid1 x is easy to %[ t simil> figures d n exi/2 9 t case ! angles ( an equilat]al triangle v>y ) ! side 9 :i* ,i see no absurdity at all4 ,! angle is a function ( ! side & ! sides are functions ( ! angle1 a function :i*1 ( c\rse1 at ! same "t involves a con/ant l5g?4 ,x seems some:at ( a p>adox to say t a con/ant l5g? c\ld be giv5 a priori z x 71 b 9 ?is aga9 ,i see no?+ inconsi/5t4 ,inde$1 x w\ld be desirable t ,euclidean geometry 7 n valid1 = !n we %\ld possess a g5]al a priori /&>d ( measure4--,,gauss4 ,lett] to ,g]l+ 7#ahaf72 ,w]ke1 ,bd4 #h 7,gtt+51 #aijj71 p4 #afi4 #bjbe4 ,i am conv9c$ m & m t ! necess>y tru? ( our geometry _c be demon/rat$1 at lea/ n by ! human intellect to ! human und]/&+4 ,p]haps 9 ano!r _w we may ga9 o!r insights into ! nature ( space :i* at pres5t are unatta9able to u4 ,until !n we mu/ consid] geometry z ( equal rank n ) ari?metic1 :i* is purely a priori1 b ) me*anics4--,,gauss4 ,lett] to ,olb]s 7#ahag72 ,w]ke1 ,bd4 #h 7,gtt+51 #aijj71 p4 #agg4 #bjbf4 ,!re is no d\bt t x c be rigor\sly e/abli%$ t ! sum ( ! angles ( a rectil9e> triangle _c exce$ #ahj4 ,b x is o!rwise ) ! /atem5t t ! sum ( ! angles _c be less ?an #ahj2 ?is is ! real ,gordian knot1 ! rocks :i* cause ! wreck ( all4444 ,i h be5 occupi$ ) ! problem ov] ?irty ye>s & ,i d\bt if anyone has giv5 x m s]i\s att5tion1 ?\gh ,i h nev] publi%$ any?+ conc]n+ x4 ,! assumption t ! angle sum is less ?an #ahj leads to a peculi> geometry1 entirely diff]5t f ! ,euclidean1 b ?r\gh\t consi/5t ) itself4 ,i h develop$ ?is geometry to my own satisfaction s t ,i c solve e problem t arises 9 x ) ! exception ( ! det]m9ation ( a c]ta9 con/ant :i* _c be det]m9$ a priori4 ,! l>g] "o assumes ?is con/ant ! m ne>ly "o approa*es ! ,euclidean geometry1 an inf9itely l>ge value makes ! two co9cide4 ,! !orems ( ?is geometry seem 9 "p p>adoxical1 & to ! unpractic$ absurd2 b on a clos] & calm reflection x is f\nd t 9 !mselves !y conta9 no?+ impossible4444 ,all my ef=ts to discov] "s contradiction1 "s inconsi/5cy 9 ?is ,non-,euclidean geometry h be5 fruitless1 ! "o ?+ 9 x t seems contr>y to reason is t space w\ld h to conta9 a def9itely det]m9ate 7?\gh to u unkn[n7 l9e> magnitude4 ,h[ev]1 x seems to me t not)/&+ ! mean+less ^w-wisdom ( ! metaphysicians we "k really too little1 or no?+1 conc]n+ ! true nature ( space to conf\nd :at appe>s unnatural ) ! absolutely impossible4 ,%\ld ,non-,euclidean geometry be true1 & ?is con/ant be> "s relation to magnitudes :i* come )9 ! doma9 ( t]re/rial or cele/ial measurem5t1 x c\ld be det]m9$ a po/]iori4 --,,gauss4 ,lett] to ,taur9us 7#ahbd72 ,w]ke1 ,bd4 #h 7,gtt+51 #aijj71 p4 #ahg4 #bjbg4 ,!re is also ano!r subject1 :i* ) me is ne>ly =ty ye>s old1 to :i* ,i h aga9 giv5 "s ?\ght dur+ leisure h\rs1 ,i mean ! f\ndations ( geometry4444 ,"h1 too1 ,i h consolidat$ _m ?+s1 & my conviction has1 if possible become m firm t geometry _c be completely e/abli%$ on a priori gr\nds4 ,9 ! mean "t ,i %all probably n = a long "t yet put my v ext5d$ inve/igations conc]n+ ?is matt] 9 %ape = publication1 possibly n :ile ,i live1 = ,i fe> ! cry ( ! ,botians :i* w\ld arise %\ld ,i express my :ole view on ?is matt]4--,x is curi\s too1 t besides ! kn[n gap 9 ,euclid's geometry1 to fill :i* all ef=ts till n[ h be5 9 va91 & :i* w nev] be fill$1 !re exi/s ano!r defect1 :i* to my k no "o ?us f> has criticis$ & :i* 7?\gh possible7 x is by no means easy to remove4 ,?is is ! def9ition ( a plane z a surface :i* :olly conta9s ! l9e jo9+ any two po9ts4 ,?is def9ition conta9s m ?an is necess>y to ! det]m9ation ( ! surface1 & tacitly involves a !orem :i* dem&s pro(4--,,gauss4 ,lett] to ,bessel 7#ahbi72 ,w]ke1 ,bd4 #h 7,gtt+51 #aijj71 p4 #bjj4 #bjbh4 ,i w add t ,i h rec5tly receiv$ f ,hung>y a little pap] on ,non-,euclidean geometry1 9 :i* ,i r$iscov] all my own ideas & results work$ out ) great elegance1 4444 ,! writ] is a v "y ,au/rian (fic]1 ! son ( "o ( my e>ly fri5ds1 ) :om ,i (t5 discuss$ ! subject 9 #agih1 al?\gh my ideas 7 at t "t f> remov$ f ! developm5t & maturity :i* !y h receiv$ ?r\gh ! orig9al reflections ( ?is "y man4 ,i consid] ! "y geomet] v4 ,bolyai a g5ius ( ! fir/ rank4--,,gauss4 ,lett] to ,g]l+ 7#ahcb72 ,w]ke1 ,bd4 #h 7,gtt+51 #aijj71 p4 #bba4 #bjbi4 ,?9k ( ! image ( ! _w 9 a convex mirror4444 ,a well-made convex mirror ( mod]ate ap]ture repres5ts ! objects 9 front ( x z app>5tly solid & 9 fix$ positions beh9d its surface4 ,b ! images ( ! di/ant horizon & ( ! sun 9 ! sky lie beh9d ! mirror at a limit$ di/ance1 equal to its focal l5g?4 ,betwe5 ^! & ! surface ( ! mirror are f\nd ! images ( all ! o!r objects be=e x1 b ! images are dim9i%$ & flatt5$ 9 proportion to ! di/ance ( _! objects f ! mirror4444 ,yet e /raight l9e or plane 9 ! out] _w is repres5t$ by a /raight 8 l9e or plane 9 ! image4 ,! image ( a man measur+ ) a rule a /raight l9e f ! mirror1 w\ld contract m & m ! f>!r he w5t1 b ) 8 %runk5 rule ! man 9 ! image w\ld c\nt out exactly ! same numb] ( c5timet]s z ! real man4 ,&1 9 g5]al1 all geometrical measurem5ts ( l9es & angles made ) regul>ly v>y+ images ( real in/rum5ts w\ld yield exactly ! same results z 9 ! out] _w1 all l9es ( sight 9 ! mirror w\ld be repres5t$ by /raight l9es ( sight 9 ! mirror4 ,9 %ort1 ,i d n see h[ m5 9 ! mirror are to discov] t _! bodies are n rigid solids & _! exp]i5ces good examples ( ! correctness ( ,euclidean axioms4 ,b if !y c\ld look out ^u our _w z we look into !irs )\t ov]/epp+ ! b\nd>y1 !y mu/ decl>e x to be a picture 9 a sph]ical mirror1 & w\ld speak ( u j z we speak ( !m2 & if two inhabitants ( ! diff]5t worlds c\ld communicate ) "o ano!r1 nei!r1 z f> z ,i c see1 w\ld be able to conv9ce ! o!r t he _h ! true1 ! o!r ! di/ort$1 relation4 ,inde$ ,i _c see t su* a "q w\ld h any mean+ at all1 s long z me*anical consid]ations are n mix$ up ) x4--,,helmholtz1 ,h4 ,on ! ,orig9 & ,significance ( ,geometrical ,axioms2 ,popul> ,sci5tific ,lectures1 second s]ies 7,new ,york1 #ahha71 pp4 #eg-ei4 #bjcj4 ,t space conceiv$ ( z a locus ( po9ts has b ?ree dim5sions ne$s no argum5t f ! ma!matical po9t ( view2 b j z little c we f ?is po9t ( view prev5t ! ass]tion t space has really f\r or an inf9ite numb] ( dim5sions ?\gh we p]ceive only ?ree4 ,! !ory ( multiply-ext5d$ manifolds1 :i* ent]s m & m into ! =egr\nd ( ma!matical rese>*1 is f its v nature p]fectly indep5d5t ( su* an ass]tion4 ,b ! =m ( expression1 :i* ?is !ory employs1 has inde$ gr[n out ( ?is conception4 ,in/ead ( ref]r+ to ! individuals ( a manifold1 we speak ( ! po9ts ( a high] space1 etc4 ,9 itself ?is =m ( expression has _m advantages1 9 t x facilitates compreh5sion by call+ up geometrical intuition4 ,b x has ?is disadvantage1 t 9 ext5d$ circles1 inve/igations conc]n+ manifolds ( any numb] ( dim5sions are consid]$ s+ul> alongside ! above-m5tion$ conception ( space4 ,?is view is )\t ! lea/ f\ndation4 ,! inve/igations 9 "q w\ld inde$ f9d imm$iate geometric applications if ! conception 7 valid b its value & purpose1 be+ indep5d5t ( ?is conception1 re/s ^u its ess5tial ma!matical cont5t4--,,kle91 ,f4 ,ma!matis*e ,annal51 ,bd4 #dc 7#ahic71 p4 #ie4 #bjca4 ,we are l$ naturally to ext5d ! language ( geometry to ! case ( any numb] ( v>iables1 /ill us+ ! ^w po9t to designate any sy/em ( values ( n v>iables 7! cord9ates ( ! po9t71 ! ^w space 7( n dim5sions7 to designate ! totality ( all ^! po9ts or sy/ems ( values1 curves or surface to designate ! spread compos$ ( po9ts ^: cord9ates are giv5 functions 7) ! prop] re/rictions7 ( "o or two p>amet]s 7! /raight l9e or plane1 :5 !y are l9e> fractional functions ) ! same d5om9ator71 etc4 ,su* an ext5sion has come to be a necessity 9 a l>ge numb] ( inve/igations1 9 ord] z well to give !m ! greate/ g5]ality z to pres]ve 9 !m ! intuitive "* ( geometry4 ,b x has be5 not$ t 9 su* use ( geometric language we are no long] con/ruct+ truly a geometry1 = ! =ms t we h be5 consid]+ are ess5tially analytic1 & t1 = example1 ! g5]al projective geometry con/ruct$ 9 ?is way is 9 sub/ance no?+ m ?an ! algebra ( l9e> trans=mations4 --,,segre1 ,,corradi4 ,rivi/a di ,matematica1 ,vol4 ,i 7#ahia71 p4 #ei4 ,j4 ,w4 ,"y4 #bjcb4 ,^? :o c1 9 common algebra1 f9d a squ>e root ( #a1 w be at no loss to f9d a f\r? dim5sion 9 space 9 :i* ,,abc may become ,,abcd3 or1 if !y _c f9d x1 !y h b to imag9e x1 & call x an impossible dim5sion1 subject to all ! laws ( ! ?ree we f9d possible4 ,& j z #a 9 common algebra1 gives all its significant comb9ations true1 s w\ld x be ) any numb] ( dim5sions ( space :i* ! speculator might *oose to call into impossible exi/5ce4 --,,de ,,morgan1 ,a4 ,trigonometry & ,d\ble ,algebra 7,london1 #ahdi71 ,"p #b1 *ap4 #c4 #bjcc4 ,! doctr9e ( non-,euclidean spaces & ( hyp]spaces 9 g5]al possesses ! highe/ intellectual int]e/1 & x requires a f>-sight$ man to =etell t x c nev] h any practical importance4--,,smi?1 ,w4 ,b4 ,introductory ,mod]n ,geometry 7,new ,york1 #ahic71 p4 #bgd4 #bjcd4 ,accord+ to 8 frequ5tly express$ view1 ,gauss consid]$ ! ?ree dim5sions ( space z specific peculi>ities ( ! human s\l2 p1 :i* are unable to compreh5d ?is1 he designat$ 9 8 humor\s mood by ! "n ,botians4 ,we c\ld imag9e ourselves1 he said1 z be+s :i* are consci\s ( b two dim5sions2 high] be+s might look at u 9 a l mann]1 & cont9u+ jok+ly1 he said t he _h laid aside c]ta9 problems :i*1 :5 9 a high] /ate ( be+1 he hop$ to inve/igate geometrically4--,,s>torius1 ,w4 ,v4 ,,walt]%aus54 ,gauss zum ,g$*tniss 7,leipzig1 #ahef71 p4 #ha4 #bjce4 ,!re is _m a rational logos1 & ! ma!matician has high delight 9 ! contemplation ( 9consi/5t sy/ems ( consi/5t relation%ips4 ,!re are1 = example1 a ,euclidean geometry & m ?an "o species ( non-,euclidean4 ,z !ories ( a giv5 space1 ^! are n compatible4 ,if our univ]se be1 z ,plato ?\ght1 & nature-sci5ce takes = grant$1 a space-condition$1 geometris$ affair1 "o ( ^! geometries may be1 none ( !m may be1 n all ( !m c be1 valid 9 x4 ,b 9 ! va/] _w ( ?\ght1 all ( !m are valid1 !re !y co-exi/1 & int]lace among !mselves & o!rs1 z diff]+ compon5t /ra9s ( a high]1 /rictly sup]natural1 hyp]cosmic1 h>mony4--,,keys]1 ,c4 ,j4 ,! ,univ]se & ,beyond2 ,hibb]t ,j\rnal1 ,vol4 #c 7#aijd-aije71 p4 #cac4 #bjcf4 ,! introduction into geometrical "w ( conceptions su* z ! inf9ite1 ! imag9>y1 & ! relations ( hyp]space1 none ( :i* c be directly imag9$1 has a psy*ological significance well wor?y ( exam9ation4 ,x gives a deep insight into ! res\rces & work+ ( ! human m9d4 ,we arrive at ! bord]l& ( ma!matics & psy*ology4 --,,m]z1 ,j4 ,t4 ,hi/ory ( ,european ,?\ght 9 ! ,n9ete5? ,c5tury 7,ed9burgh & ,london1 #aijc71 p4 #gaf4 #bjcg4 ,among ! spl5did g5]alizations effect$ by mod]n ma!matics1 !re is none m brilliant or m inspir+ or m fruitful1 & none m comm5surate ) ! limitless imm5sity ( be+ itself1 ?an t :i* produc$ ! great concept designat$ 444 hyp]space or multidim5sional space4 --,,keys]1 ,c4 ,j4 ,ma!matical ,emancipations2 ,moni/1 ,vol4 #af 7#aijf71 p4 #fe4 #bjch4 ,! great g5]alization ( hyp]space has made x possible to enri*1 quick5 & beautify analysis ) ! t]se1 s5su\s1 arti/ic1 /imulat+ language ( geometry4 ,on ! o!r h&1 ! hyp]spaces are 9 !mselves immeasurably int]e/+ & inexhau/ibly ri* fields ( rese>*4 ,n only does ! geometrician f9d light 9 !m = ! illum9ation ( o!rwise d>k & undiscov]$ prop]ties ( ord9>y spaces ( intuition1 b he also discov]s !re wondr\s /ructures q unkn[n to ord9>y space4444 ,x is by creation ( hyp]spaces t ! rational _s secures release f limitation4 ,9 !m x lives ev] joy\sly1 su/a9$ by an unfail+ s5se ( inf9ite fre$om4--,,keys]1 ,c4 ,j4 ,ma!matical ,emancipations2 ,moni/1 ,vol4 #af 7#aijf71 p4 #hc4 #bjci4 ,ma!maticians :o busy !mselves a great deal ) ! =mal !ory ( f\r-dim5sional space1 seem to acquire a capacity = imag9+ ?is =m z easily z ! ?ree-dim5sional =m ) :i* we are all famili>4--,,o/wald1 ,w4 ,natural ,philosophy ,seltz]1 7,new ,york1 #aiaj71 p4 #gg4 #bjdj4 ,fu*s4 ,0 soll i* nun ab] d5n /udi]58 ,meph4 ,ihr knnt es mit analytis*] ,geometrie probi]54 ,da wird d] ,raum eu* wohl dressi]t1 ,9 ,coord9at5 e+es*nrt1 ,dass ihr ni*t etwa auf gut ,glck ,von d] ,figur gew9nt e9 ,/ck4 ,dann lehret man eu* man*5 ,tag1 ,dass1 0 ihr son/ auf e95 ,s*lag ,con/rui]tet im ,raume frei1 ,e9e ,glei*ung dazu ntig sei4 ,zw> w> dem ,m5s*5 zu se9] ,erbauung ,die dreidim5sionale ,raumans*auung1 ,dass er sieht1 0 um ihn passi]t1 ,und die ,figur5 si* con/rui]t-- ,d] ,analytik] tritt h]e9 ,und bewei/1 das knnte au* &]s se94 ,glei*ung51 die auf dem ,papi]e /ehn1 ,die ms/' man au* knn5 im ,raume sehn2 ,und knnte man's ni*t con/rui]51 ,da ms/e man's &]s def9i]54 ,d5n 0 man =mt na* ,zahl5gesetz5 ,ms/' uns au* geometris* erletz54 ,drum 9 d5 un5dli* f]n5 beid5 ,imag9r5 ,punkt5 mss5 si* s*neid5 ,alle ,kreise fe9 sub]li*1 ,au* ,p>allel51 die treff5 si*1 ,und im ,raume kann man daneb5 ,all]lei ,krmmungsmasse erleb54 ,die ,=meln s9d alle wahr und s*n1 ,w>um soll5 sie ni*t zu deut5 gehn8 ,da preis5's die ,s*l] all] ,ort51 ,dass das ,g]ade i/ krumm geword54 ,ni*t-,euklidis* n5nt's die ,geometrie1 ,spott$ ihr] selb/1 und weiss ni*t wie4 ,fu*s4 ,kann eu* ni*t eb5 ganz v]/ehn4 ,meph4 ,das soll d5 ,philosoph5 au* s gehn4 ,do* w5n ihr l]nt alles r$uci]5 ,und gehrig trans=mi]51 ,bis die ,=meln d5 ,s9n v]li]51 ,dann v]/eht ihr ma!mati% zu spekuli]54 --,,lasswitz1 ,,kurd4 ,d] ,fau/-,tragdie 7-n7t] ,teil2 ,zeits*rift fr d5 ma?-naturw4 ,unt]ri*t1 ,bd4 #ad 7#ahhh71 p4 #caf4 ,fu*s4 ,to :at /udy !n %\ld ,i myself apply8 ,meph4 ,beg9 ) analytical geometry4 ,!re all space is prop]ly tra9$1 ,by cord9ates well re/ra9$1 ,t no "o by "s lucky assay ,c>ry "s "p ( ! figure away4 ,next ?\'ll be taught to realize1 ,con/ructions won't help !e to geometrize1 ,& ! result ( a free con/ruction ,requires an equation = prop] d$uction4 ,?ree-dim5sional space relation ,exi/s = human edification1 ,t he may see :at ab\t him transpires1 ,& con/ruct su* figures z he requires4 ,ent]s ! analy/4 ,=?) y see ,t all ?is might o!rwise be4 ,equations1 writt5 ) p5cil or p51 ,mu/ be visible 9 space1 & :5 ,difficulties 9 con/ruction arise1 ,we ne$ only def9e x o!rwise4 ,=1 :at is =m$ aft] laws ari?metic ,mu/ also yield "s delight geometric4 ,!re=e we mu/ n object ,t all circles int]sect ,9 ! circul> po9ts at inf9ity4 ,& all p>allels1 !y decl>e1 ,if produc$ mu/ meet some:]e4 ,s 9 space1 x c't be d5i$1 ,any old curvature may abide4 ,! =mulas are all f9e & true1 ,!n :y %\ld !y n h a mean+ too8 ,pupils ev]y:]e praise _! fate ,t t n[ is crook$ :i* once 0 /raight4 ,non-,euclidean1 9 f9e d]ision1 ,is :at x's call$ by ! geometrician4 ,fu*s4 ,i d n fully foll[ !e4 ,meph4 ,no bett] does philosophy4 ,to ma/] ma!matical speculation1 ,c>efully le>n to r$uce y\r equation ,by an adequate trans=mation ,till ! =mulas are devoid ( int]pretation4 ,,*apt] ,,xxi ,,p>adoxes ,,& ,,curiosities #baja4 ,! pseudoma? is a p]son :o h&les ma!matics z a monkey h&les ! razor4 ,! creature tri$ to %ave himself z he _h se5 8 ma/] d2 b1 n hav+ any notion ( ! angle at :i* ! razor 0 to be held1 he cut 8 own ?roat4 ,he nev] tri$ x a second "t1 poor animal6 b ! pseudoma? keeps on 9 8 "w1 proclaims himself clean %av$1 & all ! re/ ( ! _w hairy4 ,! graphoma? is a p]son :o1 hav+ no ma!matics1 attempts to describe a ma!matician4 ,noveli/s p]=m 9 ?is way3 ev5 ,walt] ,scott n[ & !n burns 8 f+]s4 ,8 dream+ calculator1 ,davy ,ramsay1 swe>s 8by ! bones ( ! immortal ,napi]40 ,scott ?\ght t ! philoma?s wor%ipp$ relics3 s !y d 9 "o s5se4--,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #dgc4 #bajb4 ,pro( requires a p]son :o c give & a p]son :o c receive4444 ,a bl9d man said1 ,z to ! ,sun1 ,i'll take my ,bible oa? !re's none2 ,= if !re _h be5 "o to %[ ,!y w\ld h %[n x long ago4 ,h[ came he su* a goose to be8 ,did he n "k he c\ldn't see8 ,n he4 --,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #bfb4 #bajc4 ,ma!matical rese>*1 ) all its weal? ( hidd5 treasure1 is all too apt to yield no?+ to our rese>*3 = x is haunt$ by c]ta9 ignes fatui--delusive phantoms1 t float be=e u1 & seem s fair1 & are all b 9 our grasp1 s ne>ly t x nev] seems to ne$ m ?an "o /ep fur!r1 & ! prize %all be ours6 ,alas = him :o has be5 turn$ aside f real rese>* by "o ( ^! spectres--:o has f\nd a music 9 its mock+ laught]--& :o wa/es 8 life & en]gy 9 ! desp]ate *ase6--,,dodgson1 ,c4 ,l4 ,a new ,!ory ( ,p>allels 7,london1 #ahie71 ,introduction4 #bajd4 ,z lightn+ cle>s ! air ( impalpable vap\rs1 s an incisive p>adox frees ! human intellig5ce f ! le?>gic influ5ce ( lat5t & unsuspect$ assumptions4 ,p>adox is ! slay] ( ,prejudice4--,,sylve/]1 ,j4 ,j4 ,on a ,lady's ,fan etc4 ,collect$ ,ma!matical ,pap]s1 ,vol4 #c1 p4 #cf4 #baje4 ,:5 a p>adox] p>ades capital lett]s & diagrams :i* are z good z ,newton's to all :o "k no?+ ab\t x1 "s p]sons wond] :y sci5ce does n rise & triturate ! :ole ?+4 ,?is is :y3 all :o are fit to read ! refutation are satisfi$ already1 & c1 if !y please1 detect ! p>adox] = !mselves4 ,^? :o are n fit to d ?is w\ld n "k ! diff]5ce betwe5 ! true answ] & ! new capitals & diagrams on :i* ! delight$ p>adox] w\ld decl>e t he _h crumbl$ ! philosoph]s1 & n !y him4 --,,de ,,morgan1 ,a4 ,a ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #dhd4 #bajf4 ,demon/rative reason nev] raises ! cry ( ,*ur* 9 ,dang]6 & x _c h any ,diction>y ( h]esies except a ,budget ( ,p>adoxes4 ,mi/ak5 claimants are left to ,"t & 8 ext+ui%]1 ) ! approbation ( all non-claimants3 !re is no ne$ ( a succession ( exposures4 ,"t gets ?r\gh ! job 9 8 own workmanlike mann]4--,,de ,,morgan1 ,a4 ,a ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #dhe4 #bajg4 ,d',israeli speaks ( ! 8six follies ( sci5ce10--! quadrature1 ! duplication1 ! p]petual motion1 ! philosoph]'s /one1 magic1 & a/rology4 ,he might z well h add$ ! trisection1 to make ! my/ic numb] sev52 b _h he done s1 he w\ld /ill h be5 v l5i5t2 only sev5 follies 9 all sci5ce1 f ma!matics to *emi/ry6 ,sci5ce might h said to su* a judge--z convicts us$ to say :o got sev5 ye>s1 expect+ x = life1 8,?ank y1 my ,"l1 & may y sit !re until !y are ov]10--may ! ,curiosities ( ,lit]ature outlive ! ,follies ( ,sci5ce6--,,de ,,morgan1 ,a4 ,a ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #ga4 #bajh4 ,montucla says1 speak+ ( ,france1 t he f9ds ?ree notions preval5t among cyclomet]s3 #a4 ,t !re is a l>ge rew>d (f]$ = success2 #b4 ,t ! longitude problem dep5ds on t success2 #c4 ,t ! solution is ! great end & object ( geometry4 ,! same ?ree notions are equally preval5t among ! same class 9 ,engl&4 ,no rew>d has ev] be5 (f]$ by ! gov]nm5t ( ei!r c\ntry4 ,! longitude problem 9 no way dep5ds ^u p]fect solution2 exi/+ approximations are suffici5t to a po9t ( accuracy f> beyond :at c be want$4 ,& geometry1 cont5t ) :at exi/s1 has long pass$ on to o!r matt]s4 ,sometimes a cyclomet] p]suades a skipp] :o has made l& 9 ! wrong place t ! a/ronom]s are at fault1 = us+ a wrong measure ( ! circle2 & ! skipp] ?9ks x a v com=table solution6 ,& ?is is ! utmo/ t ! problem has to d ) longitude4--,,de ,,morgan1 ,a4 ,a ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #if4 #baji4 ,gregory ,/4 ,v9c5t is ! greate/ ( circle-squ>]s1 & 8 inve/igations l$ him into _m tru?s3 he f\nd ! prop]ty ( ! arc ( ! hyp]bola :i* l$ to ,napi]'s log>i?ms be+ call$ hyp]bolic4 ,montucla says ( him1 ) sly tru?1 t no "o ev] squ>$ ! circle ) s mu* g5ius1 or1 except+ 8 pr9cipal object1 ) s mu* success4 --,,de ,,morgan1 ,a4 ,a ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #gj4 #baaj4 ,:5 ,i rea*$ geometry1 & became acqua9t$ ) ! proposition ! pro( ( :i* has be5 s\ght = c5turies1 ,i felt irresi/ibly impell$ to try my p[]s at its discov]y4 ,y w consid] me fooli% if ,i confess t ,i am /ill e>ne/ly ( ! op9ion to h succe$$ 9 my attempt4--,,bolzano1 ,,b]n>d4 ,selb/biographie 7,wi51 #ahge71 p4 #ai4 #baaa4 ,! ,!ory ( ,p>allels4 ,x is kn[n t to complete ! !ory x is only necess>y to demon/rate ! foll[+ proposition1 :i* ,euclid assum$ z an axiom3 ,prop4 ,if ! sum ( ! int]ior angles ,,ecf & ,,dbc :i* two /raight l9es ,,ec & ,,db make ) a ?ird l9e ,,cp is less ?an two "r angles1 ! l9es1 if suffici5tly produc$1 w int]sect4 ,illu/ration3 ,a geometrical draw+ ( p>allel l9es & int]sect+ l9es to accompany pro(4 ,pro(4 ,con/ruct ,,pca equal to ! supplem5t ,,pbd ( ,,cbd1 & ,,ecf1 ,,fcg1 etc4 ea* equal to ,,ace1 s t ,,acf #b4;,,ace1 ,,acg #c4;,,ace1 etc4 ,!n h[ev] small ! angle ,,ace may be1 !re exi/s "s numb] n su* t n4,,ace ,,a* w be equal to or great] ?an ,,acp4 ,aga91 take ,,bi1 ,,il1 etc4 ea* equal to ,,cb1 & draw ,,ik1 ,,lm1 etc4 p>allel to ,,bd1 !n ! figures ,,acbd1 ,,dbik1 ,,kilm1 etc4 are congru5t1 & ,,acik #b4;,,abcd1 ,,aclm #c4;,,acbd1 etc4 ,take ,,acno n4,,acbd1 n hav+ ! same value z 9 ! expression ,,a* n4,,ace1 !n ,,acno is c]ta9ly less ?an ,,acp1 s9ce ,,acno mu/ be increas$ by ,,onp to be equal to ,,acp4 ,x foll[s t ,,acno is also less ?an ,,a*1 & by tak+ ! n? "p ( ea* ( ^!1 t ,,acbd is less ?an ,,ace4 ,b if ,,ace is great] ?an ,,acbd1 ,,ce & ,,bd mu/ int]sect1 = o!rwise ,,ace w\ld be a "p ( ,,acbd4 ,j\rnal fr ,ma!matik1 ,bd4 #b 7#ahcd71 p4 #aih4 #baab4 ,are y sure t x is impossible to trisect ! angle by ,euclid8 ,i h n to lam5t a s+le h\r ?r[n away on ! attempt1 b fancy t x is r a tact1 a feel+1 ?an a pro(1 :i* makes u ?9k t ! ?+ _c be done4 ,b w\ld ,gauss's inscription ( ! regul> polygon ( sev5te5 sides h seem$1 a c5tury ago1 mu* less an impossible ?+1 by l9e & circle8--,,hamilton1 ,w4 ,r4 ,lett] to ,de ,morgan 7#aheb74 #baac4 ,"o ( ! mo/ curi\s ( ^! cases geometrical p>adox]s 0 t ( a /ud5t1 ,i am n sure b a graduate1 ( ! ,univ]sity ( ,virg9ia1 :o claim$ t geomet]s 7 9 error 9 assum+ t a l9e _h no ?ickness4 ,he publi%$ a s*ool geometry bas$ on 8 views1 :i* receiv$ ! endorsem5t ( a well-kn[n ,new ,york s*ool (ficial &1 on ! basis ( ?is1 0 actually endors$1 or came v ne> be+ endors$1 z a text-book 9 ! public s*ools ( ,new ,york4 --,,newcomb1 ,,simon4 ,! ,rem9isc5ces ( an ,a/ronom] 7,bo/on & ,new ,york1 #aijc71 p4 #chh4 #baad4 ,:at di/+ui%es ! /raight l9e & circle m ?an any?+ else1 & prop]ly sep>ates !m = ! purpose ( elem5t>y geometry8 ,_! self-simil>ity4 ,e in* ( a /raight l9e co9cides ) e o!r in*1 & (f a circle ) e o!r (f ! same circle4 ,:]e1 !n1 did ,euclid fail8 ,9 n introduc+ ! ?ird curve1 :i* has ! same prop]ty--! screw4 ,! "r l9e1 ! circle1 ! screw--! repres5tations ( translation1 rotation1 & ! two comb9$--ought to h be5 ! in/rum5ts ( geometry4 ,) a screw we %\ld nev] h he>d ( ! impossibility ( trisect+ an angle1 squ>+ ! circle1 etc4--,,de ,,morgan1 ,a4 ,quot$ 9 ,graves' ,life ( ,sir ,w4 ,r4 ,hamilton1 ,vol4 #c 7,new ,york1 #ahhi71 p4 #cdb4 #baae4 ,mad ,ma!sis alone 0 unconf9$1 ,too mad = m]e mat]ial *a9s to b9d1 ,n[ to pure space lifts h] ec/atic />e1 ,n[1 runn+ r\nd ! circle1 f9ds x squ>e4 --,,pope1 ,,alex&]4 ,! ,dunciad1 ,bk4 #d1 l9es #ca-cd4 #baaf4 ,or is't a t>t idea1 to procure ,an edge1 & keep ! practic s\l 9 ure1 ,l t de> ,*ymic du/1 or puzzl+ quadrature8 --,,qu>les1 ,,philip4 ,quot$ by ,de ,morgan3 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #dcf4 #baag4 ,quale 'l geometra *e tutto s' affige ,p] misur> lo c]*io1 e non ritruova1 ,p5s&o qual pr9cipio ond' egli indige4 --,,dante4 ,p>adise1 canto #cc1 l9es #abb-abe4 ,z do? ! exp]t geomet] appe> ,:o seeks to squ>e ! circle1 & ^: skill ,f9ds n ! law ) :i* 8 c\rse to /e]4#ab ,quot$ 9 ,frankl&'s ,/ory ( ,euclid 7,london1 #aijb71 p4 #aja4 #ab ,= ano!r r5dition ( ^! same l9es see #aheh4 #baah4 ,9 ,ma!maticks he 0 great] ,?an ,ty*o ,brahe1 or ,erra ,pat]3 ,= he1 by ,geometrick scale1 ,c\ld take ! size ( ,pots ( ,ale2 ,resolve by ,signs & ,tang5ts /reight1 ,if ,bread or ,butt] want$ weight2 ,& wisely tell :at h\r o' ?' day ,! ,clock do? /rike1 by ,algebra4 --,,butl]1 ,,samuel4 ,hudibras1 ,"p #a1 canto #a1 l9es #aai-abf4 #baai4 ,i h (t5 be5 surpris$ t ,ma!matics1 ! qu9tess5ce ( tru?1 %\ld h f\nd admir]s s few & s languid4 ,frequ5t consid]ations & m9ute scrut9y h at l5g? unravell$ ! cause2 viz4 t ?\gh ,reason is fea/$1 ,imag9ation is />v$2 :il/ ,reason is luxuriat+ 9 its prop] ,p>adise1 ,imag9ation is we>ily travell+ on a dre>y des]t4--,,col]idge1 ,,samuel4 ,a ,ma!matical ,problem4 #babj4 ,at la/ we ent]$ ! palace1 & proce$$ into ! *amb] ( pres5ce :]e ,i saw ! k+ seat$ on 8 ?rone1 att5d$ on ea* side by p]sons ( prime quality4 ,be=e ! ?rone1 0 a l>ge table fill$ ) globes & sph]es1 & ma!matical in/rum5ts ( all k9ds4 ,8 maje/y took n ! lea/ notice ( u1 al?\gh our entrance 0 n )\t suffici5t noise1 by ! conc\rse ( all p]sons belong+ to ! c\rt4 ,b he 0 !n deep 9 a problem1 & we att5d$ an h\r1 be=e he c\ld solve x4 ,!re /ood by him1 on ea* side1 a "y page ) flaps 9 _! h&s1 & :5 !y saw he 0 at leisure1 "o ( !m g5tly /ruck 8 m\?1 & ! o!r 8 "r e>2 at :i* he />t$ l "o awak$ on ! sudd51 & look+ t[>d me & ! company ,i 0 91 recollect$ ! occasion ( our com+1 :]e( he _h be5 in=m$ be=e4 ,he spake "s words1 :]eupon imm$iately a "y man ) a flap came to my side1 & flapt me g5tly on ! "r e>1 b ,i made signs1 z well z ,i c\ld1 t ,i _h no occasion = su* an in/rum5t2 :i*1 z ,i aft]w>ds f\nd1 gave 8 maje/y1 & ! :ole c\rt1 a v mean op9ion ( my und]/&+4 ,! k+1 z f> z ,i c\ld conjecture1 ask$ me sev]al que/ions1 & ,i address$ myself to him 9 all ! languages ,i _h4 ,:5 x 0 f\nd1 t ,i c\ld nei!r und]/& nor be und]/ood1 ,i 0 conduct$ by 8 ord] to an ap>tm5t 9 8 palace1 7?is pr9ce be+ di/+ui%$ above all 8 pr$ecessors1 = 8 hospitality to /rang]s7 :]e two s]vants 7 appo9t$ to att5d me4 ,my d9n] 0 br\ght1 & f\r p]sons ( quality1 did me ! hon\r to d9e ) me4 ,we _h two c\rses ( ?ree di%es ea*4 ,9 ! fir/ c\rse1 !re 0 a %\ld] ( mutton cut into an equilat]al triangle1 a piece ( beef into a rhomboides1 & a pudd+ into a cycloid4 ,! second c\rse1 01 two ducks truss$ up 9 ! =m ( fiddles2 sausages & pudd+s1 resembl+ flutes & haut-boys1 & a brea/ ( veal 9 ! %ape ( a h>p4 ,! s]vants cut our bread into cones1 cyl9d]s1 p>allelograms1 & sev]al o!r ma!matical figures4--,,swift1 ,,jona?an4 ,gulliv]'s ,travels2 ,a ,voyage to ,laputa2 ,*ap4 #b4 #baba4 ,^? to :om ! k+ _h entru/$ me1 obs]v+ h[ ill ,i 0 clad1 ord]$ a taylor to come next morn+1 & take measure = a suit ( cloa?s4 ,?is op]ator did 8 (fice aft] a diff]5t mann]1 f ^? ( 8 trade 9 ,europe4 ,he fir/ took my altitude by a quadrant1 & !n1 ) rule & compasses1 describ$ ! dim5sions & outl9es ( my :ole body1 all :i* he ent]$ ^u pap]2 & 9 six days1 br\ght my cloa?s v ill made1 & q out ( %ape1 by happ5+ to mi/ake a figure 9 ! calculation4 ,b my com=t 01 t ,i obs]v$ su* accid5ts v frequ5t1 & little reg>d$4 --,,swift1 ,,jona?an4 ,gulliv]'s ,travels2 ,a ,voyage to ,laputa1 ,*ap4 #b4 #babb4 ,! k ,i _h 9 ma!matics1 gave me great assi/ance 9 acquir+ _! phraseology1 :i* dep5d$ mu* ^u t sci5ce1 & music2 & 9 ! latt] ,i 0 n unskill$4 ,_! ideas are p]petually conv]sant 9 l9es & figures4 ,if !y w\ld1 = example1 praise ! beauty ( a woman1 or any o!r animal1 !y describe x by rhombs1 circles1 p>allelograms1 ellipses1 & o!r geometrical t]ms1 or by words ( art drawn f music1 ne$less "h to repeat4 ,i obs]v$ 9 ! k+'s kit*5 all sorts ( ma!matical & musical in/rum5ts1 aft] ! figures ( :i*1 !y cut up ! jo9ts t 7 s]v$ to 8 maje/y's table4--,,swift1 ,,jona?an4 ,gulliv]'s ,travels2 ,a ,voyage to ,laputa1 ,*ap4 #b4 #babc4 ,i 0 at ! ma!matical s*ool1 :]e ! ma/] taught 8 pupils1 aft] a me?od1 sc>ce imag9able to u 9 ,europe4 ,! propositions1 & demon/rations1 7 fairly writt5 on a ?9 waf]1 ) ink compos$ ( a cephalic t9cture4 ,?is1 ! /ud5t 0 to swall[ ^u a fa/+ /oma*1 & = ?ree days foll[+1 eat no?+ b bread & wat]4 ,z ! waf] dige/$1 ! t9cture m\nt$ to 8 bra91 be>+ ! proposition along ) x4 ,b ! success has n hi!rto be5 answ]able1 p>tly by "s error 9 ! quantum or composition1 & p>tly by ! p]v]s5ess ( lads2 to :om ?is bolus is s nause\s1 t !y g5]ally /eal aside1 & dis*>ge x upw>ds1 be=e x c op]ate2 nei!r h !y be5 yet p]suad$ to use s long an ab/95ce z ! prescription requires4--,,swift1 ,,jona?an4 ,gulliv]'s ,travels2 ,a ,voyage to ,laputa1 ,*ap4 #e4 #babd4 ,x is wor? obs]v+ t "s ( ^? :o disp>age "s bran* ( /udy 9 :i* !y are defici5t1 w (t5 affect m contempt = x ?an !y really feel4 ,& n unfrequ5tly !y w take pa9s to h x ?\ght t !y are !mselves well v]s$ 9 x1 or t !y easily might be1 if !y ?\ght x wor? :ile2--9 %ort1 t x is n f hang+ too high t ! grapes are call$ s\r4 ,?us1 ,swift1 9 ! p]son ( ,gulliv]1 repres5ts himself1 :ile d]id+ ! extravagant passion = ,ma!matics among ! ,laputians1 z be+ a good ma!matician4 ,yet he betrays 8 utt] ignorance1 by speak+ 8( a pudd+ 9 ! =m ( a cycloid30 evid5tly tak+ ! cycloid = a figure1 in/ead ( a l9e4 ,?is may help to expla9 ! difficulty he is said to h _h 9 obta9+ 8 ,degree4--,,:ately1 ,r4 ,annotations to ,bacon's ,essays1 ,essay ,l4 #babe4 ,x is natural to ?9k t an ab/ract sci5ce _c be ( mu* importance 9 ! affairs ( human life1 because x has omitt$ f its consid]ation ev]y?+ ( real int]e/4 ,x w be rememb]$ t ,swift1 9 8 description ( ,gulliv]'s voyage to ,laputa1 is ( two m9ds on ?is po9t4 ,he describes ! ma!maticians ( t c\ntry z silly & useless dream]s1 ^: att5tion has to be awak5$ by flapp]s4 ,also1 ! ma!matical tailor measures 8 height by a quadrant1 & d$uces 8 o!r dim5sions by a rule & compasses1 produc+ a suit ( v ill-fitt+ clo!s4 ,on ! o!r h&1 ! ma!maticians ( ,laputa1 by _! m>vell\s inv5tion ( ! magnetic isl& float+ 9 ! air1 rul$ ! c\ntry & ma9ta9$ _! asc5d5cy ov] _! subjects4 ,swift1 inde$1 liv$ at a "t peculi>ly unsuit$ = gibes at contempor>y ma!maticians4 ,newton's ,pr9cipia _h j be5 writt51 "o ( ! great =ces :i* h trans=m$ ! mod]n _w4 ,swift might j z well h laugh$ at an e>?quake4 --,,:itehead1 ,a4 ,n4 ,an ,introduction to ,ma!matics 7,new ,york1 #aiaa71 p4 #aj4 #babf4 ,illu/ration3 ,a geometrical draw+ includ+ squ>e & f\r triangles to demon/rate a graphical pro( ( ! !orem ( ,py?agoras z describ$ 9 ! poem4 ,"h ,i am z y may see a b - ab ,:5 two ,triangles on me /& ,squ>e ( hypo!ne is plann'd ,b if ,i /& on !m in/ead1 ,! squ>es ( bo? ! sides are read4 --,,airy1 ,g4 ,b4 ,quot$ 9 ,graves' ,life ( ,sir ,w4 ,r4 ,hamilton1 ,vol4 #c 7,new ,york1 #ahhi71 p4 #ejb4 #babg4 #c4ada #eib #fec #ehi #gic #bch #dfb #fdc #chc #bgi 444 #c #a #d #a #e #i ,n[ ,i1 ev5 ,i1 w\ld celebrate #b #f #e #c #e ,9 rhymes inapt1 ! great #h #i #g #i ,immortal ,syracusan1 rival$ nev]more1 #c #b #c #h #d ,:o 9 8 wondr\s lore1 #f #b #f ,pass$ on be=e1 #d #c #c #h #c #b #g #i ,left m5 8 guidance h[ to circles m5surate4 --,,orr1 ,a4 ,c4 ,lit]>y ,dige/1 ,vol4 #cb 7#aijf71 p4 #hd4 #babh4 ,i take f a biographical diction>y ! fir/ five names ( poets1 ) _! ages at dea?4 ,!y are ,aag>d1 di$ at #dh4 ,abeille1 8 8 #gf4 ,abulola1 8 8 #hd4 ,abun[as1 8 8 #dh4 ,accords1 8 8 #de4 ,^! five ages h ! foll[+ *>act]s 9 common3-- #a4 ,! diff]5ce ( ! two digits compos+ ! numb] divid$ by ?ree1 leaves a rema9d] ( "o4 #b4 ,! fir/ digit rais$ to ! p[] indicat$ by ! second1 & !n divid$ by ?ree1 leaves a rema9d] ( "o4 #c4 ,! sum ( ! prime factors ( ea* age1 includ+ "o z a prime factor1 is divisible by ?ree4--,,peirce1 ,c4 ,s4 ,a ,!ory ( ,probable ,inf]5ce2 ,/udies 9 ,logic 7,bo/on1 #ahhc71 p4 #afc4 #babi4 ,9 view ( ! fact t ! (f]$ prize = ! solution ( ! problem ( ,f]mat's ,great] ,!orem is ab\t #be1jjj & t lack ( m>g9al space 9 8 copy ( ,diophantus 0 ! reason giv5 by ,f]mat = n communicat+ 8 pro(1 "o might be tempt$ to wi% t "o c\ld s5d cr$it = a dime back ?r\gh ! ages to ,f]mat & ?us secure ?is covet$ prize1 if x actually exi/$4 ,?is might1 h[ev]1 result m s]i\sly ?an "o w\ld at fir/ suppose2 = if ,f]mat _h b\ght on cr$it a dime's wor? ( pap] ev5 dur+ ! ye> ( 8 dea?1 #affe1 & if ?is bill _h be5 draw+ comp\nd int]e/ at ! rate ( six p] c5t1 s9ce t "t1 ! bill w\ld n[ am\nt to m ?an sev5 times z mu* z ! prize4 --,,mill]1 ,g4 ,a4 ,"s ,?\ghts on ,mod]n ,ma!matical ,rese>*2 ,sci5ce1 ,vol4 #ce 7#aiab71 p4 #hha4 #bacj4 ,if ! ,indians hadn't sp5t ! #bd4 ,9 #afbf ,pet] ,m9uit1 fir/ gov]nor ( ,new ,ne!rl&1 pur*as$ ,manhattan ,isl& f ! ,indians = ab\t #bd4 ,! rate ( int]e/ on money is high] 9 new c\ntries1 & gradually decreases z weal? accumulates4 ,)9 ! pres5t g5]ation ! legal rate 9 ! /ate has fall5 f #g to #f4 ,assume = simplicity a uni=m rate ( #g f #afbf to ! pres5t1 & suppose t ! ,indians _h put _! #bd at int]e/ at t rate 7bank+ facilities 9 ,new ,york be+ always tak5 = grant$67 & _h add$ ! int]e/ to ! pr9cipal ye>ly4 ,:at w\ld be ! am\nt n[1 aft] #bhj ye>s8 #bd 7#a4jg7bhj m ?an #d1jdb1jjj1jjj4 ,! late/ tax assessm5t available at ! "t ( writ+ gives ! realty = ! bor\gh ( ,manhattan z #c1hbj1ged4aha4 ,?is is e/imat$ to be #gh ( ! actual value1 mak+ ! actual value a little m ?an #d1hih1djj1jjj4 ,! am\nt ( ! ,indians' money w\ld !re=e be m ?an ! pres5t assess$ valuation b less ?an ! actual valuation4 --,,:ite1 ,w4 ,f4 ,a ,scrap-book ( ,elem5t>y ,ma!matics 7,*icago1 #aijh71 pp4 #dg-dh4 #baca4 ,see ,my/]y to ,ma!matics fly6--,,pope1 ,,alex&]4 ,! ,dunciad1 ,bk4 #d1 l9e #fdg4 #bacb4 ,! ,py?agoreans & ,platoni/s 7 c>ri$ fur!r by ?is love ( simplicity4 ,py?agoras1 by 8 skill 9 ma!matics1 discov]$ t !re c be no m ?an five regul> solid figures1 t]m9at$ by plane surfaces :i* are all simil> & equal2 to wit1 ! tetrah$ron1 ! cube1 ! octah$ron1 ! dodecah$ron1 & ! eicosih$ron4 ,z nature works 9 ! mo/ simple & regul> way1 he ?\ght t all elem5t>y bodies mu/ h "o or o!r ( ^? regul> figures2 & t ! discov]y ( ! prop]ties & relations ( ! regul> solids mu/ be a key to op5 ! my/]ies ( nature4 ,?is notion ( ! ,py?agoreans & ,platoni/s has und\bt$ly great beauty & simplicity4 ,accord+ly x prevail$1 at lea/ to ! "t ( ,euclid4 ,he 0 a ,platonic philosoph]1 & is said to h wrote all ! books ( 8 ,elem5ts1 9 ord] to discov] ! prop]ties & relations ( ! five regul> solids4 ,! anci5t tradition ( ! int5tion ( ,euclid 9 writ+ 8 elem5ts1 is c\nt5anc$ by ! "w itself4 ,= ! la/ book ( ! elem5ts treats ( ! regul> solids1 & all ! prec$+ are subs]vi5t to ! la/4--,,reid1 ,,?omas4 ,essays on ! ,p[]s ( ! ,human ,m9d 7,ed9burgh1 #ahab71 ,vol4 #b1 p4 #djj4 #bacc4 ,9 ! ,timu ( ,plato x is ass]t$ t ! p>ticles ( ! v>i\s elem5ts h ! =ms ( ^! ! regul> solids4 ,fire has ! ,pyramid2 ,e>? has ! ,cube2 ,wat] ! ,octah$ron2 ,air ! ,icosah$ron2 & ! ,dodecah$ron is ! plan ( ! ,univ]se itself4 ,x 0 natural t :5 ,plato _h le>nt t o!r ma!matical prop]ties _h a be>+ ^u ! con/itution ( ! ,univ]se1 he %\ld suppose t ! s+ul> prop]ty ( space1 :i* ! exi/5ce ( ?is limit$ & v>i$ class ( solids impli$1 %\ld h "s correspond+ prop]ty 9 ! ,univ]se1 :i* exi/s 9 space4 --,,:ewell1 ,w4 ,hi/ory ( ! ,inductive ,sci5ces1 #crd ,edition1 ,additions to ,bk4 #b4 #bacd4 ,! orbit ( ! e>? is a circle3 r\nd ! sph]e to :i* ?is circle belongs1 describe a dodecah$ron2 ! sph]e includ+ ?is w give ! orbit ( ,m>s4 ,r\nd ,m>s describe a tetrah$ron2 ! circle includ+ ?is w be ! orbit ( ,jupit]4 ,describe a cube r\nd ,jupit]'s orbit2 ! circle includ+ ?is w be ! orbit ( ,saturn4 ,n[ inscribe 9 ! e>?'s orbit an icosah$ron2 ! circle inscrib$ 9 x w be ! orbit ( ,v5us4 ,inscribe an octah$ron 9 ! orbit ( ,v5us2 ! circle inscrib$ 9 x w be ,m]cury's orbit4 ,?is is ! reason ( ! numb] ( ! planets4--,,kepl]4 ,my/]ium ,cosmographicum ,:ewell4 #bace4 ,x w n be ?\ght surpris+ t ,plato expect$ t ,a/ronomy1 :5 fur!r advanc$1 w\ld be able to r5d] an acc\nt ( _m ?+s = :i* %e has n acc\nt$ ev5 to ?is day4 ,?us1 9 ! passage 9 ! sev5? ,book ( ! ,republic1 he says t ! philosoph] requires a reason = ! proportion ( ! day to ! mon?1 & ! mon? to ! ye>1 deep] & m sub/antial ?an m]e obs]vation c give4 ,yet ,a/ronomy has n yet %[n u any reason :y ! proportion ( ! times ( ! e>?'s rotation on its axis1 ! moon's revolution r\nd ! e>?1 & ! e>?'s revolution r\nd ! sun1 might n h be5 made by ! ,creator q diff]5t f :at !y are4 ,b 9 ask+ ,ma!matical ,a/ronomy = reasons :i* %e _c give1 ,plato 0 only do+ :at a great a/ronomical discov]]1 ,kepl]1 did at a lat] p]iod4 ,"o ( ! que/ions :i* ,kepl] especially wi%$ to h answ]$ 01 :y !re are five planets1 & :y at su* p>ticul> di/ances f ! sun8 ,& x is /ill m curi\s t he ?\ght he _h f\nd ! reason ( ^! ?+s1 9 ! relation ( ^? five regul> solids :i* ,plato 0 desir\s ( introduc+ into ! philosophy ( ! univ]se4444 ,kepl] reg>ds ! law :i* ?us det]m9es ! numb] & magnitude ( ! planet>y orbits by means ( ! five regul> solids z a discov]y no less rem>kable & c]ta9 ?an ! ,?ree ,laws :i* give 8 "n its imp]i%able place 9 ! hi/ory ( a/ronomy4--,,:ewell1 ,w4 ,hi/ory ( ! ,inductive ,sci5ces1 #crd ,edition1 ,additions to ,bk4 #c4 #bacf4 ,py?agorean philosoph]s 444 ma9ta9$ t ( two combatants1 he w\ld conqu]1 ! sum ( ! numb]s express$ by ! *>act]s ( ^: names exce$$ ! sum ( ^? express$ by ! o!r4 ,x 0 ^u ?is pr9ciple t !y expla9$ ! relative pr[ess & fate ( ! h]oes 9 ,hom]1 1 & 1 ! sum ( ! numb]s 9 ^: names are #hfa1 #abbe1 & #abgf respectively4--,,peacock1 ,,george4 ,encyclop$ia ( ,pure ,ma!matics 7,london1 #ahdg72 ,article 8,ari?metic10 sect4 #ch4 #bacg4 ,r\nd numb]s are always false4--,,johnson1 ,,samuel4 ,johnsoniana2 ,apo!gms1 ,s5tim5t1 etc4 #bach4 ,num]o deus imp>e gaudet ,god 9 numb] odd rejoices4 --,,virgil4 ,eclogue1 #h1 #gg4 #baci4 ,:y is x t we ent]ta9 ! belief t = e purpose odd numb]s are ! mo/ effectual8--,,pl9y4 ,natural ,hi/ory1 ,bk4 #bh1 *ap4 #e4 #badj4 8,!n "h goes ano!r10 says he1 8to make sure1 ,=e !re's luck 9 odd numb]s10 says ,rory ,o',moore4 --,,lov]1 ,s4 ,rory ,o',moore4 #bada4 ,?is is ! ?ird "t2 ,i hope1 good luck lies 9 odd numb]s4444 ,!y say1 !re is div9ity 9 odd numb]s1 ei!r 9 nativity1 *ance1 or dea?4--,,%akespe>e4 ,! ,m]ry ,wives ( ,w9dsor1 ,act #e1 sc5e #a4 #badb4 ,to add to gold5 numb]s1 gold5 numb]s4--,,deck]1 ,,?omas4 ,pati5t ,grissell1 ,act #a1 sc5e #a4 #badc4 ,i've read t ?+s inanimate h mov$1 ,&1 z ) liv+ s\ls1 h be5 in=m'd1 ,by magic numb]s & p]suasive s\nd4 --,,congreve1 ,,ri*>d4 ,! ,morn+ ,bride1 ,act #a1 sc5e #a4 #badd4 444 ! ,yancos on ! ,amazon1 ^: "n = ?ree is ,poett>r>or9co>oac1 ( a l5g? suffici5tly =midable to ju/ify ! rem>k ( ,la ,condam9e3 ,heureusem5t p\r ceux qui ont faire avec eux1 leur ,ari?metique ne va pas plus lo94--,,peacock1 ,,george4 ,encyclop$ia ( ,pure ,ma!matics 7,london1 #ahdg72 ,article 8,ari?metic10 sect4 #cb4 #bade4 ,!re are ?ree pr9cipal s9s1 av>ice1 luxury1 & pride2 ?ree sorts ( satisfaction = s91 fa/+1 almsgiv+1 & pray]2 ?ree p]sons (f5d$ by s91 ,god1 ! s9n] himself1 & 8 neighb\r2 ?ree witnesses 9 heav51 ,pat]1 v]bum1 & spiritus sanctus2 ?ree degrees ( p5it5ce1 contrition1 confession1 & satisfaction1 :i* ,dante has repres5t$ z ! ?ree /eps ( ! ladd] t lead to purgatory1 ! fir/ m>ble1 ! second black & rugg$ /one1 & ! ?ird r$ porphyry4 ,!re are ?ree sacr$ ord]s 9 ! *ur* militant1 sub-diaconati1 diaconiti1 & presbyt]ati2 !re are ?ree p>ts1 n )\t my/]y1 ( ! mo/ sacr$ body made by ! prie/ 9 ! mass2 & ?ree times he says ,agnus ,dei1 & ?ree times1 ,sanctus2 & if we well consid] all ! dev\t acts ( ,*ri/ian wor%ip1 !y are f\nd 9 a t]n>y comb9ation2 if we wi% rightly to p>take ( ! holy communion1 we mu/ ?ree times express our contrition1 ,dom9e non sum dignus2 b :o c say m ( ! t]n>y numb] 9 a %ort] compass1 ?an :at ! prophet says1 tu signaculum sanctae tr9itatis4 ,!re are ?ree ,furies 9 ! inf]nal regions2 ?ree ,fates1 ,atropos1 ,la*esis1 & ,clo?o4 ,!re are ?ree !ological virtues3 ,fides1 spes1 & *>itas4 ,tria sunt p]icula mundi3 ,equum curr]e2 navig>e1 et sub tyranno viv]e4 ,!re are ?ree enemies ( ! s\l3 ! ,devil1 ! _w1 & ! fle%4 ,!re are ?ree ?+s :i* are ( no e/eem3 ! /r5g? ( a port]1 ! advice ( a poor man1 & ! beauty ( a beautiful woman4 ,!re are ?ree v[s ( ! ,m9orite ,fri>s3 pov]ty1 ob$i5ce1 & *a/ity4 ,!re are ?ree t]ms 9 a cont9u$ proportion4 ,!re are ?ree ways 9 :i* we may commit s93 corde1 ore1 ope4 ,?ree pr9cipal ?+s 9 ,p>adise3 glory1 ri*es1 & ju/ice4 ,!re are ?ree ?+s :i* are especially displeas+ to ,god3 an av>ici\s ri* man1 a pr\d poor man1 & a luxuri\s old man4 ,& all ?+s1 9 %ort1 are f\nd$ 9 ?ree2 t is1 9 numb]1 9 weight1 & 9 measure4 --,,pacioli1 ,au?or ( ! fir/ pr9t$ treatise on ari?metic4 ,quot$ 9 ,encyclop$ia ( ,pure ,ma!matics 7,london1 #ahdg72 ,article 8,ari?metic10 sect4 #ij4 #badf4 ,ah6 :y1 ye ,gods1 %\ld two & two make f\r8 --,,pope1 ,,alex&]4 ,! ,dunciad1 ,bk4 #b1 l9e #bhe4 #badg4 ,by him :o /ampt ,! ,f\r ^u ! m9d1-- ,! ,f\r1 ! f\nt ( nature's endless /ream4 --,ascrib$ to ,,py?agoras4 ,quot$ 9 ,:ewell's ,hi/ory ( ! ,inductive ,sci5ces1 ,bk4 #d1 *ap4 #c4 #badh4 ,along ! skiey ar* ! goddess trode1 ,& s\ght ,h>monia's augu/ abode2 ,! univ]sal plan1 ! my/ic ,f\r1 ,def9es ! figure ( ! palace floor4 ,solid & squ>e ! anci5t fabric /&s1 ,rais$ by ! labors ( unnumb]$ h&s4 --,,nonnus4 ,dionysiac1 #da1 #bge-bhj4 ,:ewell4 #badi4 ,! numb] sev5ty-sev5 figures ! abolition ( all s9s by baptism4444 ,! numb] t5 signifies ju/ice & beatitude1 result+ f ! creature1 :i* makes sev5 ) ! ,tr9ity1 :i* is ?ree3 !re=e x is t ,god's comm&m5ts are t5 9 numb]4 ,! numb] elev5 d5otes s91 because x transgresses t54444 ,?is numb] sev5ty-sev5 is ! product ( elev51 figur+ s91 multipli$ by sev51 & n by t51 = sev5 is ! numb] ( ! creature4 ,?ree repres5ts ! s\l1 :i* is 9 "s sort an image ( ,div9ity2 & f\r repres5ts ! body1 on acc\nt ( its f\r qualities4444 --,,/4 ,,augu/9e4 ,s]mon #da1 art4 #bc4 #baej4 ,heliodorus says t ! ,nile is no?+ else ?an ! ye>1 f\nd+ 8 op9ion on ! fact t ! numb]s express$ by ! lett]s 1 ,nile1 are 9 ,greek ari?metic1 #ej2 #e2 #aj2 #cj2 #gj2 #bjj2 & ^! figures make up toge!r #cfe1 ! numb] ( days 9 ! ye>4 ,littell's ,liv+ ,age1 ,vol4 #aag1 p4 #chj4 #baea4 ,9 treat+ #fff1 ,bungus ,petri ,bungi ,b]gomatis ,num]orum my/]ia1 ,b]gamo1 #aeia a good ,ca?olic1 c\ld n complim5t ! ,pope ) x1 b he fixes x on ,m>t9 ,lu!r ) a little =c+4 ,if f ,a to ,i repres5t #a-i1 f ,k to ,s #aj-ij1 & f ,t to ,z #ajj-ejj1 we see-- ,m ,a ,r ,t ,i ,n ,l ,u ,t ,e ,r ,a #cj #a #hj #ajj #i #dj #bj #bjj #ajj #e #hj #a :i* gives #fff4 ,aga9 9 ,hebrew1 ,lult] ,hebraiz$ =m ( ,lu!r does ! same3-- #bjj #djj #cj #f #cj --,,de ,,morgan1 ,a4 ,budget ( ,p>adoxes 7,london1 #ahgb71 p4 #cg4 #baeb4 ,/ifel1 ! mo/ acute & orig9al ( ! e>ly ma!maticians ( ,g]many1 444 relates 444 t :il/ a monk at ,essl+5 9 #aebj1 & :5 infect$ by ! writ+s ( ,lu!r1 he 0 read+ 9 ! libr>y ( 8 conv5t ! #ac? ,*apt] ( ,revelations1 x /ruck 8 m9d t ! ,bea/ mu/ signify ! ,pope1 ,leo ,x42 ,he !n proce$$ 9 pi\s hope to make ! calculation ( ! sum ( ! num]al lett]s 9 ,leo decimus1 :i* he f\nd to be ,m1 ,d1 ,c1 ,l1 ,v1 ,i2 ! sum :i* ^! =m$ 0 too great by ,m1 & too little by ,x2 b he be?\ght him aga91 t he has se5 ! "n writt5 ,leo ,x41 & t !re 7 t5 lett]s 9 ,leo decimus1 f ei!r ( :i* he c\ld obta9 ! defici5t numb]1 & by int]pret+ ! ,m to mean my/]ium1 he f\nd ! numb] requir$1 a discov]y :i* gave him su* unspeakable com=t1 t he believ$ t 8 int]pretation mu/ h be5 an imm$iate inspiration ( ,god4 --,,peacock1 ,,george4 ,encyclop$ia ( ,pure ,ma!matics 7,london1 #ahdg72 ,article 8,ari?metic10 sect4 #hi4 #baec4 ,p]haps ! be/ anagram ev] made is t by ,dr4 ,burney on ,horatio ,nelson1 s happily trans=m$ into ! ,lat9 s5t5ce s tru?ful ( ! great admiral1 ,honor e/ a ,nilo4 ,read+ ?is1 "o is almo/ p]suad$ t ! hit conta9$ 9 x has a mean+ provid$ by provid5ce or fate4 ,?is is also amus+ly illu/rat$ 9 ! case ( ! ,fr5*man ,&r ,pujom1 :o1 us+ j z i1 f\nd 9 8 "n ! anagram1 ,p5du ,riom4 ,riom be+ ! seat ( ju/ice = ! prov9ce ( ,auv]gne1 ! poor fell[1 impell$ by a sort ( infatuation1 actually committ$ a capital (f5ce 9 t prov9ce1 & 0 hang$ at ,riom1 t ! anagram might be fulfill$4 ,new ,am]ican ,cyclop$ia1 ,vol4 #a2 ,article 8,anagram40 #baed4 ,! mo/ rem>kable pseudonym ( transpos$ names adopt$ by au?ors is ! "n ( 8,voltaire10 :i* ! celebrat$ philosoph] assum$ in/ead ( 8 family "n1 8,franois ,m>ie ,ar\et10 & :i* is n[ g5]ally all[$ to be an anagram ( 8,ar\et1 l4 j410 t is1 ,ar\et ! y\ng]4 ,encyclop$ia ,britannica1 #aa? ,edition2 ,article 8,anagram40 #baee4 ,p]haps ! mo/ beautiful anagram t has ev] be5 compos$ is by ,jablonsky1 a =m] rector ( ! s*ool at ,lissa4 ,! occasion 0 ! foll[+3 ,:5 :ile a "y man k+ ,/anislaus ( ,pol& return$ f a j\rney1 ! :ole h\se ( ,lesc9sky assembl$ to welcome ! family heir4 ,on ?is occasion ,jablonsky arrang$ = a s*ool program1 ! clos+ numb] ( :i* consi/$ ( a ballet by ?irte5 pupils imp]sonat+ y\?ful h]oes4 ,ea* ( !m c>ri$ a %ield on :i* appe>$ 9 gold "o ( ! lett]s ( ! words ,domus ,lesc9ia4 ,at ! end ( ! fir/ dance ! *ildr5 7 s arrang$ t ! lett]s on _! %ields spell$ ! words ,domus ,lesc9ia4 ,at ! end ( ! second dance !y read3 ades incolumis 7s\nd ?\ art "h74 ,aft] ! ?ird3 omnis es lucida 7:olly brilliant art ?\72 aft] ! f\r?3 lucida sis om5 7bright be ! om574 ,!n3 mane sidus loci 7rema9 our c\ntry's />72 & aga93 sis columna ,dei 7be a column ( ,god72 & f9ally3 ,i6 sc&e solium 7,proce$1 asc5d ! ?rone74 ,?is la/ 0 ! m beautiful s9ce x prov$ a true prophecy4 ,ev5 m artificial are ! anagrams :i* trans=m "o v]se into ano!r4 ,?us an ,italian s*ol> beheld 9 a dream ! l9e f ,horace3 ,grata sup]v5iet1 quae non sp]abitur1 hora4 ,?is a fri5d *ang$ to ! anagram3 ,e/ v5tura ,rhos9a p>ataque nub]e pigro4 ,?is induc$ ! s*ol>1 ?\gh an old man1 to m>ry an unkn[n lady by ! "n ( ,ros9a4--,,heis1 ,,edu>d4 ,algebrais*e ,aufgab5 7,kln1 #ahih71 p4 #cca4 #baef4 ,! foll[+ v]ses read ! same :e!r read =w>d or backw>d3-- ,aspice6 nam r>o mittit timor arma1 nec ipsa ,si se m5te reget1 non teg]et ,nemesis2#ac also1 ,sator ,arepo t5et op]a rotas4 --,,heis1 ,,edu>d4 ,algebrais*e ,aufgab5 7,kln1 #ahih71 p4 #cbh4 #ac ,! beg9n+ ( a poem :i* ,johannes a ,lasco wrote on ! c\nt ,k>l von ,sd]manl&4 #baeg4 ,!re is a c]ta9 spiral ( a peculi> =m on :i* a po9t may h be5 approa*+ = c5turies ! c5t]1 & h ne>ly rea*$ x1 be=e we discov] t its rate ( approa* is accel]at$4 ,! fir/ ?\ght ( ! obs]v]1 on see+ ! accel]ation1 w\ld be to say t x w\ld rea* ! c5t] soon] ?an he _h be=e suppos$4 ,b z ! po9t comes ne> ! c5t] x sudd5ly1 al?\gh /ill mov+ "u ! same simple law z f ! beg9n+1 makes a v %ort turn ^u its pa? & flies (f rapidly almo/ 9 a /raight l9e1 out to an inf9ite di/ance4 ,?is illu/rates t app>5t brea* ( cont9uity :i* we sometimes f9d 9 a natural law2 t app>5tly sudd5 *ange ( "* :i* we sometimes see 9 man4--,,hill1 ,,?omas4 ,uses ( ,ma!sis2 ,biblio!ca ,sacra1 ,vol4 #cb1 p4 #eba4 #baeh4 ,"o ( ! mo/ rem>kable ( ,babbage's illu/rations ( miracles has nev] _h ! consid]ation 9 ! popul> m9d :i* x des]ves2 ! illu/ration drawn f ! exi/5ce ( isolat$ po9ts fulfill+ ! equation ( a curve4444 ,!re are def9itions ( curves :i* describe n only ! positions ( e po9t 9 a c]ta9 curve1 b also ( "o or m p]fectly isolat$ po9ts2 & if we %\ld attempt to get by induction ! def9ition1 f ! obs]vation ( ! po9ts on ! curve1 we might fail altoge!r to include ^! isolat$ po9ts2 :i*1 nev]!less1 al?\gh /&+ alone1 z miracles to ! obs]v] ( ! c\rse ( ! po9ts 9 ! curve1 are nev]!less rigor\sly includ$ 9 ! law ( ! curve4--,,hill1 ,,?omas4 ,uses ( ,ma!sis2 ,biblio!ca ,sacra1 ,vol4 #cb1 p4 #eaf4 #baei4 ,pure ma!matics is ! magician's real w&4--,,novalis4 ,s*rift51 ,zweit] ,teil 7,b]l91 #aija71 p4 #bbc4 #bafj4 ,miracles1 consid]$ z ant9atural facts1 are ama!matical1 b !re are no miracles 9 ?is s5se1 & ^? s call$ may be compreh5d$ by means ( ma!matics1 = to ma!matics no?+ is miracul\s4--,,novalis4 ,s*rift51 ,zweit] ,teil 7,b]l91 #aiaa71 p4 #bbb4 ,,index ,black-fac$ numb]s ref] to au?ors ,abbreviations3--m4 ma!matics1 ma?4 ma!matical1 ma?'n4 ma!matician4 ,abbott1 #ajja4 ,ab/ract me?od1 ,developm5t (1 #gbi4 ,ab/ract nature ( m41 ,reason =1 #fch4 ,ab/ract reason+1 ,objection to1 #aida4 ,ab/ractness1 ma?41 ,comp>$ ) logical1 #acjd4 ,adams1 ,h5ry1 ,m4 & hi/ory1 #aeii4 ,ma?'ns practice fre$om1 #bjh1 #hje4 ,adams1 ,john1 ,me?od 9 m41 #bbf4 ,a5eid1 ,eul]'s k (1 #hei4 ,aes*ylus4 ,on numb]1 #afjf4 ,aim 9 tea*+ m41 #eja-ejh1 #eag1 #hdd4 ,airy1 ,py?agorean !orem1 #babf4 ,ak5side1 #aecb4 ,alex&]1 #ija1 #ijb4 ,algebra1 ,*apt] ,,xvii4 ,def9itions (1 #aaj1 #agad1 #agae4 ,problems 91 #cbj1 #ecj1 #agch4 ,( use to gr[n m51 #dbe4 ,& geometry1 #ebe-ebg1 #afaj1 #agjg4 ,advantages (1 #agja1 #agjc1 #agje4 ,laws (1 #agjh-agaj4 ,z an art1 #agaa4 ,review (1 #agac4 ,designations (1 #agag4 ,orig9 (1 #agcf4 ,burlesque on mod]n1 #agda4 ,hume on1 #ahfc4 ,algebraic notation1 value (1 #abac1 #abad4 ,algebraic treatises1 ,h[ to read1 #fja4 ,amusem5ts 9 m41 #ijd1 #ije4 ,anagrams1 ,on ,de ,morgan1 #idg4 ,on ,domus ,lesc9ia1 #baee4 ,on ,flam/e$1 #ifh4 ,on ,macaulay1 #iif4 ,on ,nelson1 #baec4 ,on ,newton1 #ajbh4 ,on ,voltaire1 #baed4 ,analysis1 ,invigorates ! faculty ( resolution1 #daf4 ,relation ( geometry to1 #aica4 ,analytical geometry1 #ahhi1 #ahij1 #ahic4 ,me?od (1 #caj4 ,importance (1 #idi4 ,burlesque on1 #bjdj4 ,anci5t geometry1 ,*>act]i/ics (1 #gab1 #gad4 ,comp>$ ) mod]n1 #agaa-agaf4 ,me?od (1 #adbe1 #ahgc-ahge4 ,anci5ts1 ,m4 among !1 #cba4 ,anecdotes1 ,*apt]s1 ,,ix1 ,x4 ,ang]1 ,m4 de/roys pr$isposition to1 #deh4 ,angl+ l m41 #gci4 ,anglo-,danes1 ,aptitude = m41 #hcf4 ,anglo-,saxons1 ,aptitude = m41 #hcg4 ,newton z repres5tative (1 #ajad4 ,anonym\s1 ,song ( ! screw1 #ahid4 ,appolonius1 #gab1 #gad4 ,approximate m41 ,:y n suffici5t1 #aeah4 ,aptitude = m41 #eji1 #eaj1 #ebj1 #hcf-hch1 #igf1 #afag4 ,arabic notation1 #afad4 ,arago1 ,m4 ! enemy ( sci5tific romances1 #bfg4 ,eul]1 8analysis inc>nate10 #ifa4 ,eul] z a comput]1 #ifb4 ,on ,kepl]'s discov]y1 #ihb4 ,newton's ef=ts sup]human1 #ajjf4 ,on probabilities1 #aeia4 ,geometry z an in/rum5t1 #ahfh4 ,arbu?not1 ,m4 frees f prejudice1 cr$ulity & sup]/ition1 #ddi4 ,m4 ! fri5d ( religion1 #deh4 ,m4 comp>$ to music1 #aaab4 ,on ma?1 reason+1 #aejc4 ,ar*im$es1 ,8 ma*9es1 #ijc1 #ijd4 ,e/imate ( ma?1 appliances1 #ijd-ijf1 #ijh4 ,wordswor? on1 #ijf4 ,s*ill] on1 #ijg4 ,& eng9e]+1 #ijh4 ,dea? (1 #iji4 ,8 tomb1 #iaj4 ,comp>$ ) ,newton1 #iaa4 ,"* ( 8 "w1 #iab1 #iac4 ,appli$ m41 #acab4 ,ar*itecture & m41 #bgf4 ,ar*ytas1 #ijd4 ,& ,plato1 #adbg4 ,ari/ippus ! ,cyr5aic1 #hde4 ,ari/otle1 #iad4 ,on relation ( m4 to e/hetics1 #cah4 ,ari?metic1 ,*apt] ,,xvi4 ,def9itions (1 #ajf1 #aaj1 #afaa1 #afab1 #agad4 ,em]son on advantage ( /udy (1 #djh4 ,problems 91 #ebh4 ,a ma/]-key1 #aega4 ,bas$ on concept ( "t1 #afac4 ,me?od ( tea*+1 #afah4 ,purpose ( tea*+1 #ded1 #afbd4 ,z logic1 #afbd1 #afbe4 ,! que5 ( m41 #afdb4 ,high]1 #agee4 ,hume on1 #ahfc4 ,ari?metical !orems1 #afci4 ,art1 ,m4 z a f9e1 ,*apt] ,,xi ,arts1 ,m4 & !1 #aefh-aegj1 #aegc4 ,a/ronomy & m41 #aeed1 #aeei1 #aefb-aefg4 8,auge et imp]a410 #fca4 ,au?ority 9 sci5ce1 #aebh4 ,axioms1 #eah1 #bjae4 ,9 geometry1 #ahab1 #bjjd1 #bjjf4 ,def4 9 disguise1 #bjje4 ,euclid's1 #bjjg-bjaj1 #bjad4 ,nature (1 #bjab4 ,pro(s (1 #bjac4 ,& ! idea ( space1 #bjjd4 ,babbage1 #ibc4 ,bacon1 ,"l1 ,classification ( m41 #ajf4 ,m4 makes m5 subtile1 #bdh4 ,view ( m41 #caf1 #iae1 #iaf4 ,m4 held 9 high e/eem by anci5ts1 #cba4 ,on ! g5]aliz+ p[] ( m41 #cbg4 ,on ! value ( ma?1 /udies1 #daj4 ,m4 develops conc5tration ( m9d1 #daa4 ,m4 cures di/raction ( m9d1 #dab4 ,m4 ess5tial to /udy ( nature1 #dcf4 ,8 view ( m41 #iae1 #iaf4 ,8 k ( m41 #iag1 #iah4 ,m4 & logic1 #acaj4 ,gr[? ( m41 #aeaa4 ,bacon1 ,rog]1 ,neglect ( m4 works injury to all sci5ce1 #caj4 ,on ! value ( m41 #aedg4 ,ba91 ,importance ( m4 9 education1 #ddb4 ,on ! *>m ( ! /udy ( m41 #dec4 ,m4 & sci5ce tea*+1 #ebb4 ,tea*+ ( ari?metic1 #afah4 ,ball1 ,r4 ,s41 #bjaj4 ,ball1 ,w4 ,w4 ,r41 ,on ,babbage1 #ibc4 ,on ,demoivre's dea?1 #idd4 ,de ,morgan & ! actu>y1 #ide4 ,gauss z a/ronom]1 #iga4 ,laplace's 8,x is easy to see40 #ihf4 ,lagrange1 ,laplace & ,gauss contra/$1 #iic4 ,newton's int]e/ 9 *emi/ry & !ology1 #ajae4 ,on ,newton's me?od ( "w1 #ajbf4 ,on ,newton's discov]y ( ! calculus1 #ajbg4 ,gauss's e/imate ( ,newton1 #ajbi4 ,m4 & philosophy1 #adag4 ,advance 9 physics1 #aecj4 ,plato on geometry1 #ahjd4 ,notation ( ! calculus1 #aijd4 ,b>nett1 ,m4 ! type ( p]fect reason+1 #cjg4 ,b>r[1 ,on ! me?od ( m41 #bac1 #bbg4 ,eulogy ( m41 #ccj4 ,m4 z a discipl9e ( ! m9d1 #djb4 ,m4 & eloqu5ce1 #hcj4 ,philosophy & m41 #adcj4 ,uses ( m41 #aegb4 ,on surd numb]s1 #agbh4 ,euclid's def9ition ( proportion1 #ahce4 ,beattie1 #adca4 ,beauty ( m41 #dec1 #hbd1 #abjh4 ,consi/s 9 simplicity1 #bdb1 #cae4 ,sylve/] on1 #aaja4 ,russell on1 #aajd4 ,"y on1 #aaaj4 ,kumm] on1 #aaaa4 ,:ite on1 #aaai4 ,& tru?1 #aaad4 ,boltzmann on1 #aaaf4 ,beltrami1 ,on read+ ( ! ma/]s1 #fad4 ,b]keley1 ,on geometry z logic1 #dbh4 ,on ma?4 symbols1 #abad4 ,on fluxions1 #aiae1 #aidb-aidd4 ,on inf9ite divisibility1 #aide4 ,b]n\lli1 ,daniel1 #iai4 ,b]n\lli1 ,james1 ,leg5d = 8 tomb1 #ibj1 #ibb4 ,computation ( sum ( t5? p[]s ( numb]s1 #iba4 ,discussion ( log>i?mic spiral1 #ibb4 ,b]!lot1 ,m4 inspires respect = tru?1 #dch4 ,bija ,ganita1 ,solution ( problems1 #agci4 ,bill+sley1 ,m4 beautifies ! m9d1 #cai4 ,b9>y ari?metic1 #iia4 ,biology & m41 #aegi-aeha4 ,biot1 ,laplace's 8,x is easy to see10 #ihf4 ,b*]1 ,m4 lik5$ to pa9t+1 #aajc4 ,int]relation ( m4 & logic1 #acac4 ,geometry z a natural sci5ce1 #ahff4 ,bo]ne1 ,on ,py?agoras1 #ahee4 ,bois-,reymond1 ,on ! analytic me?od1 #ahic4 ,natural selection & ! calculus1 #aiba4 ,boltzmann1 ,on beauty 9 m41 #aaaf4 ,bolyai1 ,janos1 ,duel ) (fic]s1 #ibd4 ,univ]sal language1 #ibe4 ,sci5ce absolute ( space1 #ibf4 ,bolyai1 ,wolfgang1 #ibg4 ,on ,gauss1 #igb4 ,bolzano1 #ibh4 ,cur$ by ,euclid1 #ibi4 ,p>allel axiom1 #baaj4 ,book-keep+1 ,importance ( ! art (1 #aega4 ,boole1 ,m4 ,e4 #gai4 ,boole's ,laws ( ,?\ght1 #acah4 ,borda-,dem\l9s1 ,philosophy & m41 #adje4 ,boswell1 #iha4 ,b[dit*1 ,on ,laplace's 8,?us x pla9ly appe>s10 #ihe4 ,boyle1 ,usefulness ( m4 to physics1 #dcg4 ,m4 & sci5ce1 #aeac1 #aecc4 ,ignorance ( m41 #aegg4 ,m4 & physiology1 #aehb4 ,w+s ( m41 #afbf4 ,advantages ( algebra1 #agjc4 ,brahmagupta1 ,e/imate ( m41 #cbj4 ,brew/]1 ,on ,eul]'s k ( ! ,a5eid1 #iei4 ,on ,eul] z a comput]1 #ifc4 ,on ,newton's fame1 #ajjb4 ,br\gham1 #abjb4 ,buckle1 ,on geometry1 #ahaj1 #ahcg4 ,burke1 ,on ! value ( m41 #ddg4 ,burkh>dt1 ,on discov]y 9 m41 #fah4 ,on univ]sal symbolism1 #abba4 ,butl]1 ,n4 ,m41 ,m4 demon/rates ! supremacy ( ! human reason1 #cji4 ,m4 ! mo/ a/\nd+ intellectual creation1 #gjg4 ,geometry be=e algebra1 #ahga4 ,butl]1 ,samuel1 #baah4 ,by]ly1 ,on hyp]bolic functions1 #aibi4 ,cajori1 ,on ! value ( ! hi/ory ( m41 #fae4 ,on ,bolyai1 #ibg4 ,cayley's view ( ,euclid1 #icf4 ,on ! ext5t ( ,eul]'s "w1 #ifj4 ,on ,eul]'s ma?4 p[]1 #ifd4 ,on ! ,d>m/aett] prize1 #ifg4 ,on ,sylve/]'s fir/ class at ,johns ,hopk9s1 #ajca4 ,on music & m4 among ! ,py?agoreans1 #aacj4 ,on ! greate/ a*ievem5t ( ! ,h9doos1 #afae4 ,on mod]n calculation1 #afad4 ,on review 9 ari?metic1 #agac4 ,on ,indian m41 #agcg4 ,on ! *>act]i/ics ( anci5t geometry1 #ahgc4 ,on ,napi]'s rule1 #ahhh4 ,calculat+ ma*9es1 #afda4 ,calculation1 ,importance (1 #fjb4 ,n ! sole object ( m41 #bfh4 ,calculus1 ,*apt] ,,xix4 ,f\ndation ( #bec4 ,z a me?od1 #cji4 ,may be taught at an e>ly age1 #eai1 #aiag1 #aiah4 ,cambridge m41 #hcf1 #abaj4 ,cantor1 ,on fre$om 9 m41 #bje1 #bjg4 ,on ! "* ( ,gauss's writ+1 #ige4 ,z5o's problem1 #aich4 ,on ! inf9ite1 #aieb4 ,c>lisle life tables1 #idf4 ,c>not1 ,on limit+ ratios1 #aijh4 ,on ! inf9itesimal me?od1 #aijg4 ,c>son1 ,value ( geometrical tra9+1 #ahda4 ,c>tesian me?od1 #ahhi1 #ahij4 ,c>us1 ,e/imate ( m41 #cbf4 ,m4 reveals sup]natural ,god1 #dfj4 ,numb] & nature1 #afjc4 ,z]o & inf9ity1 #aidh4 ,non-euclidean geometry1 #bjaf4 ,ca!dral1 8,petrifi$ ma!matics10 #aaaj4 ,causation 9 m41 #bea1 #bed4 ,cayley1 ,advantage ( mod]n geometry ov] anci5t1 #gaa4 ,on ! imag9>y1 #gbb4 ,sylve/] on1 #icj4 ,noe!r on1 #ica4 ,8 /yle1 #icb4 ,=sy? on1 #icb-icd4 ,8 me?od1 #icc4 ,comp>$ ) ,eul]1 #icd4 ,h]mite on1 #ice4 ,8 view ( ,euclid1 #icf4 ,8 e/imate ( quat]nions1 #icg4 ,m4 & philosophy1 #adbj4 ,c]ta9ty ( m41 #bbb1 #addj-addb1 #afbh1 #ahfc4 ,*amisso1 ,py?agorean !orem1 #ahef4 ,*ancellor1 ,m4 develops obs]vation1 imag9ation & reason1 #dcc4 ,*apman1 ,diff]5t aspects ( m41 #bfe4 ,*>act]i/ics ( m41 #bbe1 #bbi1 #bdg1 #bfc4 ,*>act]i/ics ( mod]n m41 #gbj1 #gbd-gbi4 ,*>m 9 m41 #aaae1 #afdj1 #ahdh4 ,*asles1 ,advantage ( mod]n geometry ov] anci5t1 #gab4 ,*ecks 9 m41 #bcj4 ,*emi/ry & m41 #aebj1 #aefj1 #aefa1 #agej4 ,*ess1 ,m4 l1 #hdj4 ,*ry/al1 ,def9ition ( m41 #aac4 ,def9ition ( quantity1 #aae4 ,on problem solv+1 #eca4 ,on mod]n text-books1 #ecc4 ,h[ to read m41 #fjg4 ,8 algebra1 #fce4 ,on ,b]n\lli's numb]s1 #iba4 ,on ma?4 v]sus logical ab/ractness1 #acjd4 ,rules ( algebra1 #agaj4 ,on univ]sal ari?metic1 #agag4 ,on ,horn]'s me?od1 #agdd4 ,on probabilities1 #aifg4 ,cic]o1 ,decad5ce ( geometry among ,romans1 #ahjg4 ,circle1 ,prop]ties (1 #aheb1 #aheg4 ,circle-squ>]s1 #bajh1 #baji4 ,cl>ke1 ,descriptive geometry1 #ahhb4 ,classic problems1 ,hilb]t on1 #fbg4 ,clebs*1 ,on ma?4 rese>*1 #fdd4 ,clif=d1 ,on direct usefulness ( ma?4 results1 #feb4 ,correspond5ce ! c5tral idea ( mod]n m41 #gbf4 ,8 vision1 #ich4 ,8 me?od1 #ici4 ,8 k ( languages1 #idj4 ,8 physical /r5g?1 #ida4 ,on ,helmholtz1 #igi4 ,on m4 & m9]alogy1 #aeeh4 ,on algebra & good ,engli%1 #agab4 ,euclid ! enc\ragem5t & guide ( sci5tific ?\ght1 #ahbj4 ,euclid ! inspiration & aspiration ( sci5tific ?\ght1 #ahba4 ,on geometry = girls1 #ahdb4 ,on ,euclid's axioms1 #bjae4 ,on non-,euclidean geometry1 #bjbb ,colburn1 #ifg4 ,col]idge1 ,on problems 9 m41 #ecd4 ,proposition1 g5tle maid1 #adai4 ,m4 ! qu9tess5ce ( tru?1 #bjai4 ,colton1 ,on ! effect ( ma?4 tra9+1 #dag4 ,comm5surable numb]s1 #aiff4 ,comm]ce & m41 #aega4 ,committee ( ,t51 ,on figures 9 geometry1 #ebd4 ,on projective geometry1 #ahgf4 ,common s5se1 ,m4 ! e!realization (1 #cab4 ,computation1 ,n m41 #eae4 ,& m41 #haj4 ,n conc]n$ ) significance ( numb]s1 #afda4 ,comte1 ,on ! object ( m41 #ajc4 ,on ! bus9ess ( concrete m41 #ajd4 ,m4 ! indisp5sable basis ( all education1 #ccd4 ,mill on1 #idb4 ,hamilton on1 #idc4 ,m4 & logic1 #acjh1 #acad1 #acbe4 ,on ,kant's view ( m41 #adcg4 ,e/imate ( m41 #aejd4 ,m4 ess5tial to sci5tific education1 #aeje4 ,m4 & natural philosophy1 #aejf4 ,m4 & physics1 #aece1 #aeea4 ,m4 & sci5ce1 #aecf4 ,m4 & biology1 #aegh1 #aehj1 #aeha4 ,m4 & social sci5ce1 #aehg4 ,e inquiry r$ucible to a "q ( numb]1 #afjb4 ,def9ition ( algebra & ari?metic1 #agad4 ,geometry a natural sci5ce1 #ahac4 ,anci5t & mod]n me?ods1 #ahge4 ,on ! graphic me?od1 #ahha4 ,on descriptive geometry1 #ahhc4 ,mill's e/imate (1 #aijc4 ,congreve1 #badc4 ,congru5ce1 ,symbol (1 #afdf4 ,conic sections1 #feh1 #ffj1 #aeda1 #aedb4 ,conjecture1 ,m4 free f1 #bcd4 ,cont+5t tru?s1 #aiff4 ,controv]sies 9 m41 #bae1 #bdc1 #ahei4 ,correlation 9 m41 #ebe-ebg1 #agjg1 #agaj4 ,correspond5ce1 ,concept (1 #gbe1 #gbf4 ,c\lomb1 #aeaf4 ,c\nt+1 ,e problem c be solv$ by1 #afja4 ,c\rnot1 ,on ! object ( m41 #bfh4 ,on algebraic notation1 #abac4 ,advantage ( ma?1 notation1 #abbj4 ,craig1 ,on ! orig9 ( a new sci5ce1 #fdf4 ,cr$ulity1 ,m4 frees m9d f1 #dej4 ,cremona1 ,on ,engli% text-books1 #fji4 ,cr(ton1 ,on value ( probabilities1 #aeij4 ,on probabilities1 #aieb1 #aigj1aigb4 ,cromwell1 ,on m4 & public s]vice1 #cbh4 ,curiosities1 ,*apt] ,,xxi4 ,curtius1 ,m4 & philosophy1 #adji4 ,curve1 ,def9ition (1 #aibg4 ,cyclomet]s1 ,notions (1 #bajh4 ,cyclotomy dep5ds on numb] !ory1 #afdg4 ,d',alemb]t1 ,on rigor 9 m41 #ecf4 ,geometry z logic1 #acaa4 ,algebra is g5]\s1 #agjb4 ,geometrical v]sus physical tru?s1 #ahji4 ,/&>ds 9 m41 #ahea4 ,dante1 #aheh1 #baag4 ,d>m/aett] prize1 #babi4 ,davis1 ,on ,sylve/]'s me?od1 #ajce4 ,m4 & sci5ce1 #aeaj4 ,on probability1 #aifh4 ,decimal fractions1 #abag1 #afad4 ,deck]1 #badb4 ,d$ek9d1 ,z5o's ,problem1 #aich4 ,d$uction1 ,:y necess>y1 #bai4 ,m4 bas$ on1 #bbd4 ,& ,intuition1 #adac4 ,dee1 ,on ! nature ( m41 #bfa4 ,def9itions ( m41 ,*apt] ,i4 ,also #bjje4 ,democritus1 #cba4 ,demoivre1 ,8 dea?1 #idd4 ,demon/rations1 ,locke on1 #bcf4 ,outside ( m41 #acab4 ,9 m41 #adbc4 ,de ,morgan1 ,imag9ation 9 m41 #beh4 ,m4 z an ex]cise 9 reason+1 #dcj4 ,on difficulties 9 m41 #eba4 ,on correlation 9 m41 #ebe4 ,on extempore lectures1 #edj4 ,on read+ algebraic works1 #fja4 ,on num]ical calculations1 #fjb4 ,on practice problems1 #fjc4 ,on ! value ( ! hi/ory ( m41 #fae1 #faf4 ,on ma?'ns41 #hab4 ,on ,bacon's k ( m41 #iah4 ,& ! actu>y1 #ide4 ,on life tables1 #idf4 ,anagrams' on 8 "n1 #idg4 ,on translations ( ,euclid1 #iec4 ,euclid's elem5ts comp>$ ) ,newton's ,pr9cipia1 #ied4 ,eul] & ,did]ot1 #iff4 ,lagrange & ! p>allel axiom1 #ihd4 ,anagram on ,macaulay's "n1 #iif4 ,anagrams on ,newton's "n1 #ajbh4 ,on ma?1 notation1 #abaf4 ,antagonism ( m4 & logic1 #acae4 ,on ,g]man metaphysics1 #adaf4 ,on m4 & sci5ce1 #aecg4 ,on m4 & physics1 #aech4 ,on ! advantages ( algebra1 #agja4 ,on algebra z an art1 #agaa4 ,on d\ble algebra & quat]nions1 #agbj4 ,on assumptions 9 geometry1 #ahab4 ,on ,euclid 9 s*ools1 #ahai4 ,euclid n faultless1 #ahbc4 ,on ,euclid's rigor1 #ahca4 ,geometry be=e algebra1 #ahgb4 ,on trigonometry1 #ahhe4 ,on ! calculus 9 elem5t>y in/ruction1 #aiaf4 ,on integration1 #aiai4 ,on div]g5t s]ies1 #aice1 #aicf4 ,ad inf9itum1 #aidi4 ,on ! f\r? dim5sion1 #bjcb4 ,pseudoma? & graphoma?1 #baja4 ,on pro(1 #bajb4 ,on p>adox]s1 #baje4 ,budget ( p>adoxes1 #bajf4 ,on ,d',israeli's six follies ( sci5ce1 #bajg4 ,on notions ( cyclomet]s1 #bajh4 ,on ,/4 ,v9c5t1 #baji4 ,:]e ,euclid fail$1 #baad4 ,on ! numb] ( ! bea/1 #baea4 ,desc>tes1 ,on ! use ( ! t]m m41 #ajb4 ,on intuition & d$uction1 #bai1 #adac4 ,ma?'ns alone arrive at pro(s1 #hag4 ,! mo/ completely ma?4 type ( m9d1 #idh4 ,hankel on1 #idi4 ,mill on1 #iej4 ,hankel on1 #adjd4 ,on m4 & philosophy1 #adbe1 #adcd4 ,e/imate ( m41 #adbf4 ,unpopul>ity (1 #aeja4 ,on ! c]ta9ty ( m41 #afbh4 ,on ! me?od ( ! anci5ts1 #ahgd4 ,on probable tru?1 #aifd4 ,descriptive geometry1 #ahhb1 #ahhc4 ,dessoir1 ,m4 & m$ic9e1 #aehe4 ,det]m9ants1 #agdj1 #agda4 ,did]ot & ,eul]1 #iff4 ,diff]5tial calculus1 ,*apt] ,,xix4 ,& sci5tific physics1 #aedi4 ,diff]5tial equations1 #aedi-aeeb1 #aibd1 #aibf4 ,difficulties 9 m41 #bdj1 #eba1 #fje-fjg1 #fcd1 #gcd1 #gce4 ,dillmann1 ,m4 a royal sci5ce1 #bjd4 ,on m4 z a high s*ool subject1 #dja4 ,anci5t & mod]n geometry comp>$1 #gae4 ,on ignorance (1 #hjg4 ,on m4 z a language1 #abjd4 ,numb] regulates all ?+s1 #aeje4 ,diri*let1 ,on ma?1 discov]y1 #fbe4 ,z a /ud5t ( ,gauss1 #igg4 ,discov]y 9 m41 #fag-fbb1 #fbe4 ,d-ism v]sus dot-age1 #ibc4 ,disquisitiones ,ari?meticae1 #ige1 #igg1 #afcg1 #afch4 ,d',israeli1 #bjjg4 ,div]g5t s]ies1 #aice-aicg4 8,divide et imp]a10 #fca4 ,div9e "* ( m41 #cbe1 #cbi4 8,div9ez avant de demontr]10 #fcj4 ,division ( labor 9 m41 #fca1 #fcb4 ,dodgson1 ,on ! *>m (1 #cjb4 ,py?agorean !orem1 #ahed4 ,ignes fatui 9 m41 #bajc4 ,dolbe>1 ,on exp]im5t 9 ma?4 rese>*1 #fac4 ,domus ,lesc9ia1 ,anagram on1 #baee4 ,donne1 #ahaf4 ,dot-age v]sus d-ism1 #ibc4 ,durfee1 ,on ,sylve/]'s =getfulness1 #ajch4 ,dutton1 ,on ! e?ical value ( m41 #ddf4 8,eadem mutata resurgo40 #ibj1 #ibb4 ,e*ols1 ,on ! e?ical value ( m41 #dee4 ,economics & m41 #aeic1 #aeid4 ,ed9burgh ,review1 ,m4 & a/ronomy1 #aefe1 #aeff4 ,education1 ,place ( m4 91 #ccd1 #djh4 ,/udy ( ari?metic bett] ?an rhetoric1 #djh4 ,m4 z an in/rum5t 91 #dac1 #dad4 ,m4 9 prim>y1 #dca4 ,m4 z a common s*ool subject1 #dcb4 ,ba9 on m4 91 #ddb4 ,calculus 9 elem5t>y1 #aiaf1 #aiag4 ,electricity1 ,m4 & ! !ory (1 #aeed4 ,elegance 9 m41 #fdj1 #gbh4 ,ellis1 ,on precocity 9 m41 #hce4 ,on aptitude ( ,anglo-,danes = m41 #hcf4 ,on ,newton's g5ius1 #ajad4 ,em]son1 ,on ,newton & ,laplace1 #ajjc4 ,on poetry & m41 #aabd4 ,end[m5t ( ma?'ns1 #hah4 ,en?usiasm1 #hja4 ,equality1 ,grassmann's def9ition (1 #aje4 ,equations1 #ajd1 #ebf1 #ahia1 #ahib4 ,errors1 ,!ory (1 #aigc1 #aigd4 ,e/hetic elem5t 9 m41 #dec-dee1 #fdj1 #aajb1 #aaje1 #aheb1 #ahec4 ,e/hetic tact1 #fbb4 ,e/hetic value ( m41 #ahdh1 #ahej4 ,e/hetics1 ,relation ( m4 to1 #cah1 #cai1 #dci4 ,e/imates ( m41 ,*apt] ,,iii4 ,see also #acag1 #acbd1 #acbe1 #adbg1 #aejd1 #aejh4 ,e?ical value ( m41 #djb1 #dch1 #ddf1 #ddi1 #dee-deg4 ,euclid1 ,bolzano cur$ by1 #ibi4 ,& ,ptolemy1 #iea1 #ahgh4 ,& ! /ud5t1 #ieb4 ,euclid's ,elem5ts1 ,translations (1 #iec4 ,comp>$ ) ! ,pr9cipia1 #ied4 ,greatness (1 #iee4 ,greate/ ( human productions1 #ahag4 ,p]=mance 91 #ahah4 ,9 ,engli% s*ools1 #ahai4 ,enc\ragem5t & guide1 #ahbj4 ,inspiration & aspiration1 #ahba4 ,! only p]fect model1 #ahbb4 ,n altoge!r faultless1 #ahbc4 ,only a small "p ( m41 #ahbd4 ,n fitt$ = boys1 #ahbe4 ,e>ly /udy (1 #ahbf4 ,newton &1 #ahbg4 ,its place1 #ahbh4 ,unexceptional 9 rigor1 #ahbi4 ,orig9 (1 #ahca4 ,doctr9e ( proportion1 #ahcd4 ,def9ition ( proportion1 #ahce4 ,/eps 9 demon/ration1 #ahci4 ,p>allel axiom1 #bjjg4 ,euclidean geometry1 #gaa1 #gac1 #gae4 ,eudoxus1 #ijd4 ,eul]1 ! myriad-m9d$1 #bee4 ,p5cil outruns intellig5ce1 #fbf4 ,on !oretical inve/igations1 #feg4 ,m]it ( 8 "w1 #ief4 ,! creator ( mod]n ma?4 ?\ght1 #ieg4 ,8 g5]al k1 #ieh4 ,8 k ( ! ,a5eid1 #iei4 ,ext5t ( 8 "w1 #ifj4 8,analysis inc>nate10 #ifa4 ,z a comput]1 #ifb1 #ifc4 ,8 ma?4 p[]1 #ifd4 ,8 ,t5tam5 novae !orae musicae1 #ife4 ,& ,did]ot1 #iff4 ,error 9 ,f]mat's law ( prime numb]s1 #ifg4 ,eureka1 #iaa1 #iag4 ,eurip$es1 #aefh4 ,ev]ett1 ,e/imate ( m41 #cbe4 ,value ( ma?4 tra9+1 #ddc4 ,!oretical inve/igations1 #fef4 ,ari?metic a ma/]-key1 #aega4 ,on m4 & law1 #aeih4 ,exactness1 ,see precision4 ,exam9ations1 #djg4 ,examples1 #dbb4 ,exp]im5t 9 m41 #fab1 #fac1 #aecj1 #aeca4 ,ext5t ( m41 #gcg1 #gch4 ,fairbairn1 #ebh4 ,fallacies1 #faj4 ,f>aday1 ,m4 & physics1 #aeed4 ,f]mat1 #bee1 #ifg1 #aijb4 ,f]mat's !orem1 #babi4 ,figures1 ,committee ( ,t5 on1 #ebd4 ,democritus view (1 #cba4 ,battalions (1 #afca4 ,f9e1 ,def9ition ( numb]1 #afaj4 ,on ! imag9>y1 #agcb4 ,f9e ,art1 ,m4 z a1 ,*apt] ,,xi4 ,fi%]1 ,m4 & economics1 #aeid4 ,fiske1 ,imag9ation 9 m41 #bef4 ,advantage ( m4 z logic1 #acbd4 ,fit*1 ,def9ition ( m41 #abe4 ,m4 9 education1 #dbi4 ,purpose ( tea*+ ari?metic1 #afbd1 #afbe4 ,fizi1 ,orig9 ( ! ,liliwati1 #iie4 ,flam/e$1 ,anagram on1 #ifh4 ,fluxions1 #aiaa1 #aiae1 #aidb-aidd4 ,font5elle1 ,b]n\lli's tomb1 #ibj4 ,=mulas1 ,comp>$ to focus ( a l5s1 #aeae4 ,=sy?1 ,on direct usefulness ( ma?4 results1 #fed4 ,on !oretical inve/igations1 #ffd4 ,progress ( m4 #gjd4 ,on ,cayley1 #icb-icd4 ,on m4 & physics1 #aeci4 ,on m4 & applications1 #aedj4 ,on inv>iants1 #agdg4 ,on function !ory1 #aged1 #agee4 ,fo/]1 ,on m4 & physics1 #aeaf1 #aebb4 ,on exp]im5t 9 m41 #aeca4 ,f\ndations ( m41 #gag4 ,f\r1 ,! numb]1 #badg1 #badh4 ,f\ri]1 ,ma?1 analysis co-ext5sive ) nature1 #bah4 ,on ma?4 rese>*1 #fab4 ,hamilton on1 #ifi4 ,on m4 & physics1 #aeeb1 #aeec4 ,on ! advantage ( ! ,c>tesian me?od1 #ahhi4 ,f\ri]'s !orem1 #aibh4 ,f\r? dim5sion1 #bjcb1 #bjci4 ,frankl&1 ,a41 ,m4 & *emi/ry1 #aefj4 ,frankl&1 ,w4 ,b41 ,motto ( ,py?agorean bro!rhood1 #ahcc4 ,! mo/ beautiful tru? 9 geometry1 #aheg4 ,frankl91 ,b41 ,e/imate ( m41 #cbb4 ,on ! value ( ! /udy ( m41 #cbc4 ,on ! excell5ce ( m41 #cbd4 ,on m4 z a logical ex]cise1 #acjc4 ,frankl91 ,f41 ,on ,sylve/]'s weakness1 #ajcc4 ,fr$]ick ! ,great1 ,on geometry1 #ahfj4 ,fre$om 9 m41 #bje-bjh1 #hje4 ,fr5* m41 #abaj4 ,fresnel1 #ffb4 ,fris*l9us1 #ahja4 ,froebel1 ,m4 a m$iator betwe5 man & nature1 #bfb4 ,function !ory1 #gji1 #agcb1 #aged1 #agee4 ,functional expon5t1 #abaj4 ,functionality1 ,! c5tral idea ( mod]n m41 #bed4 ,correlat$ to life1 #bgb4 ,functions1 #aicb1 #aicc4 ,concept n us$ by ,sylve/]1 #ajcd4 ,fundam5tal concepts1 ,*apt] ,,xx4 ,fuss1 ,on ,eul]'s ,t5tam5 novae !orae musicae1 #ife4 ,galileo1 ,on au?ority 9 sci5ce1 #aebh4 ,galton1 #hch4 ,gauss1 ,8 motto1 #fdi4 ,m]e ma?'ns1 #hbj4 ,& ,newton comp>$1 #hbg4 ,8 p[]1 #ifd4 ,8 favorite pursuits1 #igj4 ,! fir/ ( !oretical a/ronom]s1 #iga4 ,! greate/ ( ari?meticians1 #iga4 ,! ma?4 giant1 #igb4 ,greatness (1 #igc4 ,lectures to ?ree /ud5ts1 #igd4 ,8 /yle & me?od1 #ihc4 ,8 e/imate ( ,newton1 #ajbi4 ,on ! advantage ( new calculi1 #abae4 ,m4 & exp]im5t1 #aeca4 ,8 ,disquisitiones ,ari?meticae1 #afci1 #afdj4 ,m4 ! que5 ( ! sci5ces1 #afdb4 ,on numb] !ory1 #afdd4 ,on imag9>ies1 #agcj4 ,on ! notation s91 #ahhf4 ,on inf9ite magnitude1 #aiej4 ,on non-euclidean geometry1 #bjbc-bjbh4 ,on ! nature ( space1 #bjcd4 ,g5]alization 9 m41 #bde1 #bdf1 #beb1 #bec1 #cbg1 #gbh4 ,g5ius1 #hai4 ,geometrical inve/igations1 #fdb1 #fdc4 ,geometrical tra9+1 ,value (1 #ahda1 #ahdb1 #ahdd-ahdf4 ,geometry1 ,*apt] ,,xviii4 ,bacon's def9ition (1 #ajf4 ,sylve/]'s def9ition (1 #aaj4 ,value to mank9d1 #ccb1 #ddi4 ,& patriotism1 #ccb4 ,an excell5t logic1 #dbh4 ,plato's view (1 #dbi4 ,! f\nta9 ( all ?\ght1 #dea4 ,& algebra1 #ebe-ebg4 ,lack ( concret5ess1 #gaj4 ,advantage ( mod]n ov] anci5t1 #gaa1 #gab4 ,& music1 #ife4 ,& ari?metic1 #afjd4 ,is figur$ algebra1 #agjf4 ,"n inapt1 #ahja4 ,& exp]i5ce1 #ahad4 ,hal/$'s def9ition (1 #ahae4 ,& obs]vation1 #ahcj4 ,controv]sy 91 #ahei4 ,a me*anical sci5ce1 #ahfe4 ,a natural sci5ce1 #ahff4 ,n an exp]im5tal sci5ce1 #ahfg4 ,%\ld come be=e algebra1 #agfg1 #ahga1 #ahgb4 ,& analysis1 #aica4 ,g]ma91 ,algebra is writt5 geometry1 #agjf4 ,gilman1 ,enli/ a great ma?'n1 #hjh4 ,glai%]1 ,on ! 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( m41 #bji1 #baa1 #gjc4 ,hall1 ,g4 ,s41 ,m4 ! ideal & norm ( all c>eful ?9k+1 #cjd4 ,hall & ,/ev5s1 ,on ! p>allel axiom1 #bjjh4 ,hall]1 ,on ! inf9ite1 #aieh4 ,halley1 ,on ,c>tesian geometry1 #gaf4 ,hal/$1 ,on ,bolyai1 #ibd-ibf4 ,on ,sylve/]1 #ajcj1 #ajci4 ,& ,sylve/]1 #ajca1 #ajcb4 ,on m4 z logic1 #acje4 ,def9ition ( geometry1 #ahae4 ,hamilton1 ,sir ,william1 ,8 ignorance ( m41 #igh4 ,hamilton1 ,w4 ,r41 ,importance ( 8 quat]nions1 #ccc4 ,e/imate ( ,comte's ability1 #idc4 ,to ! memory ( ,f\ri]1 #ifi4 ,discov]y 9 light1 #aeeh4 ,on algebra z ! sci5ce ( "t1 #agae1 #agaf4 ,on quat]nions1 #agah4 ,on trisection ( an angle1 #baab4 ,hankel1 ,def9ition ( m41 #aad4 ,on fre$om 9 m41 #bjf4 ,on ! p]man5cy ( ma?4 k1 #baf4 ,on aim 9 m41 #ejh4 ,on isolat$ !orems1 #fba4 ,on tact 9 m41 #fbb4 ,on geometry1 #gad4 ,anci5t & mod]n m4 comp>$1 #gah1 #gbj4 ,v>iability ! c5tral idea 9 mod]n m41 #gbj4 ,*>act]i/ics ( mod]n m41 #gbh4 ,on ,desc>tes1 #idi4 ,on ,eul]'s "w1 #ief4 ,on philosophy & m41 #adjd4 ,on ! orig9 ( m41 #adab4 ,on irrationals & imag9>ies1 #agbi4 ,on ! orig9 ( algebra1 #agcf4 ,euclid ! only p]fect model1 #ahbb4 ,mod]n geometry a royal road1 #ahgh4 ,h>mony1 #cbf1 #abjh4 ,h>ris1 ,m4 gives comm& ov] nature1 #dcd4 ,ha?away1 ,on ,sylve/]1 #ajcf4 ,heat1 ,m4 & ! !ory (1 #aeeb1 #aeec4 ,hea?1 ,"* ( ,ar*im$es' "w1 #iac4 ,heaviside1 ,! place ( ,euclid1 #ahbh4 ,hebrew & ,lat9 races1 ,aptitude = m41 #hch4 ,hegel1 #adag4 ,heiss1 ,fam\s anagrams1 #bjee4 ,rev]sible v]ses1 #bjef4 ,helmholtz1 ,m4 ! pure/ =m ( logical activity1 #bca4 ,m4 requires p]sev]ance & great caution1 #bdj4 ,m4 %\ld take m important place 9 education1 #dda4 ,clif=d on1 #igi4 ,m4 ! pure/ logic1 #acjb4 ,m4 & applications1 #adde4 ,on geometry1 #ahcf4 ,on ! importance ( ! calculus1 #aici4 ,a non-euclidean _w1 #bjbi4 ,h]b>t1 ,def9ition ( m41 #aag4 ,m4 ! pr$om9ant sci5ce1 #bji4 ,on ! me?od ( m41 #bab1 #aegf4 ,m4 ! prie/ess ( def9it5ess & cle>ness1 #bag4 ,on ! importance ( *ecks1 #bcj4 ,on imag9ation 9 m1 #beg4 ,m4 & inv5tion1 #djf4 ,m4 ! *ief subject = common s*ools1 #dcb4 ,on aptitude = m41 #eji4 ,on ! tea*+ ( m41 #eaf4 ,m4 ! greate/ bless+1 #adja4 ,m4 & philosophy1 #adjh4 ,if philosoph]s und]/ood m41 #adae4 ,m4 indisp5sable to sci5ce1 #aejb4 ,m4 & psy*ology1 #aehc1 #afhd4 ,on trigonometry1 #ahhd4 ,h]mite1 ,on ,cayley1 #ice4 ,h]s*el1 ,m4 & a/ronomy1 #aefd4 ,on probabilities1 #aeib4 ,hi]o1 #ijc1 #ijd4 ,high] m41 ,mellor's def9ition (1 #ajh4 ,hilb]t1 ,on ! nature ( m41 #bff4 ,on rigor 9 m41 #ecg4 ,on ! importance ( problems1 #fbd1 #fbh4 ,on ! solvability ( problems1 #fbg4 ,problems %\ld be difficult1 #fbi4 ,on ! ab/ract "* ( m41 #fch4 ,on ari?metical symbols1 #afbg4 ,on non-euclidean geometry1 #bjai4 ,hill1 ,a>on1 ,on ,newton1 #ajji4 ,hill1 ,?omas1 ,on ! _s ( ma!sis1 #bgd4 ,m4 expresses ?\ghts ( ,god1 #bge4 ,value ( m41 #ccb4 ,e/imate ( ,newton's "w1 #ccc4 ,ma?'ns difficult to judge1 #hda4 ,ma?'ns indiff]5t to ord9>y int]e/s ( life1 #hdb4 ,a geomet] mu/ be tri$ by 8 pe]s1 #hdc4 ,on ,b]n\lli's spiral1 #ibb4 ,on ma!sis & poetry1 #aabe4 ,on poesy & m41 #aabf4 ,on m4 z a language1 #abji4 ,ma?1 language untranslatable1 #abaj4 ,on quat]nions1 #agai4 ,on ! imag9>y1 #agcd4 ,on geometry & lit]ature1 #ahdg4 ,m4 & miracles1 #baeg1 #baeh4 ,h9doos1 ,gr&e/ a*ievem5t (1 #afae4 ,hi/ory & m41 #aeii4 ,hi/ory ( m41 #fae1 #faf1 #fbe1 #fce4 ,hobson1 ,def9ition ( m41 #aah4 ,on ! nature ( m41 #beb4 ,functionality ! c5tral idea ( m41 #bfd4 ,on !oretical inve/igations1 #ffc4 ,on ! gr[? ( m41 #gjc4 ,a great ma?'n a great arti/1 #aaji4 ,on m4 & sci5ce1 #aejh4 ,h(fman1 ,sci5ce & poetry n antagoni/ic1 #aabb4 ,holzmll]1 ,on ! tea*+ ( m41 #eah4 ,hook]1 #adcb4 ,hopk9son1 ,m4 a mill1 #bci4 ,horn]'s me?od1 #agdd4 ,h[ison1 ,def9ition ( m41 #acd1 #ace4 ,def9ition ( ari?metic1 #afab4 ,hudson1 ,on ! tea*+ ( m41 #fab4 ,hughes1 ,on sci5ce = its own sake1 #aedf4 ,humboldt1 ,m4 & a/ronomy1 #aefg4 ,hume1 ,on ! advantage ( ma?1 sci5ce1 #adch4 ,on geometry1 #ahfb4 ,on c]ta9ty 9 m41 #ahfc4 ,objection to ab/ract reason+1 #aida4 ,humor 9 m41 #eci4 ,hutton1 ,on ,b]n\lli1 #iai4 ,on ,eul]'s k1 #ieh4 ,on ! me?od ( fluxions1 #aiaa4 ,huxley1 ,negative qualities ( m41 #bej4 ,hyp]-space1 #bjcj1 #bjca1 #bjcc1 #bjcf-bjch4 ,hyp]bolic functions1 #aibi1 #aicj4 ,ignes fatui 9 m41 #bajc4 ,ignorabimus1 ,none 9 m41 #fbg4 ,ignorance ( m41 #caj1 #cca1 #hjg1 #aecg1 #aegg4 ,imag9>ies1 #gbb1 #agbi-agce4 ,imag9ation 9 m41 #bdf1 #bea1 #bec1 #bef-beh1 #dcc1 #ahhc4 ,improvem5t ( elem5t>y m41 #fag4 ,incomm5surable numb]s1 cont+5t tru?s l1 #aiff4 ,indian m41 #agcf1 #agcg4 ,induction 9 m41 #bbj-bbc1 #bdd4 ,& analogy1 #gbd4 ,inf9ite collection1 ,def9ition (1 #aiei1 #aifj4 ,inf9ite divisibility1 #aide4 ,inf9itesimal analysis1 #aiad4 ,inf9itesimals1 #aije-aijg1 #aidj1 #aidf1 #aied4 ,inf9itum1 ,ad1 #aidi4 ,inf9ity & inf9ite magnitude1 #gbc1 #ibh1 #aidg1 #aidh1 #aiej-aieh4 ,integ]s1 ,kroneck] on1 #afcd1 #afce4 ,integral numb]s1 ,m9k[sky on1 #afcf4 ,integrals1 ,inv5tion (1 #aibb4 ,integration1 #aiai-aiba1 #aibc1 #aibe4 ,int]national ,commission on m41 #eja1 #ejb1 #ich4 ,intuition & d$uction1 #adac4 ,inv>iance1 ,correlat$ to life1 #bgb4 ,mac,mahon on1 #agdf4 ,keys] on1 #agdi4 ,inv>iants1 ,*angeless 9 ! mid/ ( *ange1 #bgf4 ,importance ( concept (1 #gbg4 ,sylve/] on1 #agdb4 ,=sy? on1 #agdg4 ,keys] on1 #agdh4 ,lie on1 #ageb4 ,inv5tion 9 m41 #bea1 #bfj4 ,inv]se process1 #abjg4 ,inve/igations1 ,see rese>*4 ,irrationals1 #agbi4 ,isolat$ !orems 9 m41 #fbj1 #fba4 8,x is easy to see10 #ihe1 #ihf1 #ajde4 ,jacobi1 ,8 tal5t = philology1 #ihj4 ,aphorism1 #afce4 ,die 8,ewige ,zahl10 #afdc4 ,jeff]son1 ,on m4 & law1 #aeig4 ,johnson1 ,8 rec\rse to m41 #iha4 ,aptitude = numb]s1 #afag4 ,on r\nd numb]s1 #bacg4 ,j\rnals & transactions1 #fce4 ,j[ett1 ,m4 z an in/rum5t 9 education1 #dac4 ,judgm5t1 ,m4 requires1 #hbc4 ,jupit]'s eclipses1 #aedd4 ,ju/itia1 ,! goddess1 #hbd4 ,juv5al1 ,nemo ma!maticus etc41 #hca4 ,kant1 ,on ! a priori nature ( m41 #acj4 ,m4 foll[s ! safe way ( sci5ce1 #bja4 ,on ! orig9 ( sci5tific m41 #bja4 ,on m4 9 prim>y education1 #dca4 ,m4 ! emb>rassm5t ( metaphysics1 #adjb4 ,8 view ( m41 #adcf1 #adcg4 ,on ! diff]5ce betwe5 m4 & philosophy1 #adcf4 ,on m4 & sci5ce1 #aejh4 ,e/hetic elem5ts 9 m41 #aheb1 #ahec4 ,doctr9e ( "t1 #bjja4 ,doctr9e ( space1 #bjjc4 ,k>p9sky1 ,m4 & effici5cy1 #afgc4 ,kasn]1 8,div9ez avant de demontr]10 #fcj4 ,on mod]n geometry1 #gaj4 ,kell&1 ,on ,euclid's elem5ts1 #ahag4 ,kelv91 ,"l1 ,see ,william ,?omson4 ,kepl]1 ,8 me?od1 #ihb4 ,planet>y orbits & ! regul> solids1 #bacd4 ,keys]1 ,def9ition ( m41 #acb4 ,?ree *>act]i/ics ( m41 #bbe4 ,on ! me?od ( m41 #bdd4 ,on ratioc9ation1 #bdf4 ,m4 n deta*$ f life1 #bgc4 ,on ! _s ( ma!sis1 #bgf4 ,computation n m41 #eae4 ,ma?1 output ( pres5t day1 #gjb4 ,mod]n !ory ( functions1 #gji4 ,m4 & j\rnalism1 #gca4 ,difficulty ( m41 #gce4 ,m4 appeals to :ole m9d1 #hae4 ,end[m5t ( ma?'ns1 #hah4 ,ma?'ns 9 public s]vice1 #hbc4 ,! aim ( ! ma?'n1 #hdd4 ,on ,bolzano1 #ibi4 ,on ,lie1 #iib4 ,on symbolic logic1 #acba4 ,on ! emancipation ( logic1 #acbb4 ,on ! ,pr9cipia ,ma!matica1 #acbf4 ,on inv>iants1 #agbh4 ,on inv>iance1 #agbi4 ,on ! notion ( gr\p1 #agea4 ,on ! elem5ts ( ,euclid1 #ahbd4 ,on protective geometry1 #ahhj4 ,def9ition ( inf9ite assemblage1 #aifj4 ,on ! inf9ite1 #aifa4 ,on non-euclidean geometry1 #bjce4 ,on hyp]-space1 #bjcg1 #bjch4 ,khulasat-al-,hisab1 ,problems1 #agch4 ,kipl+1 #afcc4 ,kir*h(f1 ,arti/ic nature ( 8 works1 #aaaf4 ,kle91 ,def9ition ( m41 #abc4 ,m4 a v]satile sci5ce1 #bfd4 ,aim 9 tea*+1 #ejg1 #eag4 ,analy/s v]sus syn!si/s1 #fea4 ,on !ory & practice1 #ffa4 ,ma?1 aptitudes ( v>i\s races1 #hch4 ,lie's f9al aim1 #iic4 ,lie's g5ius1 #iid4 ,on m4 & sci5ce1 #aebj4 ,fam\s aphorisms1 #afce4 ,calculat+ ma*9es1 #afda4 ,calculus = high s*ools1 #aiah4 ,on diff]5tial equations1 #aibf4 ,def9ition ( a curve1 #aibg4 ,on axioms ( geometry1 #bjjf4 ,on ! p>allel axiom1 #bjji4 ,on non-euclidean geometry1 #bjag1 #bjba4 ,on hyp]-space1 #bjcj4 ,kroneck]1 ,on ! greatness ( ,gauss1 #igc4 ,god made integ]s etc41 #afcd4 ,kumm]1 ,on ,diri*let1 #igg4 ,on beauty 9 m41 #aaaa4 ,la,faille1 ,ma!sis few "k1 #ahgj4 ,lagrange1 ,on correlation ( algebra & geometry1 #ebg4 ,8 /yle & me?od1 #ihc4 ,& ! p>allel axiom1 #ihd4 ,on ,newton1 #ajaa4 ,w+s ( m41 #afjd4 ,union ( algebra & geometry1 #agjg4 ,on ! inf9itesimal me?od1 #aijf4 ,lal&e1 ,m4 9 ,fr5* army1 #cad4 ,langley1 ,m4 9 ,prussia1 #eac4 ,lampe1 ,on division ( labor 9 m41 #fcb4 ,on ,wei]/rass1 #ajdi4 ,wei]/rass & ,sylve/]1 #ajej4 ,qualities common to ma?'ns & arti/s1 #aaac4 ,*>m ( m41 #aaae4 ,gold5 age ( art & m4 co9cid5t1 #aacd4 ,language1 ,*apt] ,,xii4 ,see also #caa1 #dai1 #ddc1 #aebc1 #ahjd1 #ahhi4 ,laplace1 ,on in/ruction 9 m41 #bbj4 ,8 /yle & me?od1 #ihc4 8,?us x pla9ly appe>s10 #ihe1 #ihf4 ,em]son on1 #ajjc4 ,on ,leibnitz1 #iia4 ,on ! language ( analysis1 #abbb4 ,on m4 & nature1 #aebe4 ,on ! orig9 ( ! calculus1 #aijb4 ,on ! exactitude ( ! diff]5tial calculus1 #aiaj4 ,! univ]se 9 a s+le =mula1 #aibj4 ,on probability1 #aifc1 #aifi1 #aiga4 ,laputa1 ,ma?'ns (1 #babj-babb1 ,ma?4 s*ool (1 #babc4 ,lasswitz1 ,on mod]n algebra1 #agda4 ,on function !ory1 #aicd4 ,on non-euclidean geometry1 #bjdj4 ,lat9 squ>es1 #beb4 ,latta1 ,on ,leibnitz's logical calculus1 #acag4 ,law & m41 #aeig1 #aeih4 ,laws ( ?\ght1 #gai1 #acah4 ,lead]%ip1 ,m4 z tra9+ =1 #cag4 ,lecture1 ,prep>ation (1 #edj4 ,lefevre1 ,m4 hateful to weak m9ds1 #gcc4 ,logic & m41 #acji4 ,leibnitz1 ,on difficulties 9 m41 #bda4 ,8 greatness1 #ihg4 ,8 influ5ce1 #ihh4 ,! nature ( 8 "w1 #ihi4 ,8 ma?4 t5d5cies1 #iij4 ,8 b9>y ari?metic1 #iia4 ,on ,newton1 #ajaj4 ,on demon/rations outside ( m41 #acab4 ,ars *>act]i/ica1 #acaf4 ,8 logical calculus1 #acag4 ,union ( philosophical & m4 productivity1 #adjd4 ,m4 & philosophy1 #adce4 ,on ! c]ta9ty ( ma?4 k1 #addb4 ,on controv]sy 9 geometry1 #ahei4 ,8 diff]5tial calculus1 #aijb4 ,8 notation ( ! calculus1 #aijd4 ,on necess>y & cont+5t tru?1 #aiff4 ,lev]ri]1 ,discov]y ( ,neptune1 #aeei4 ,lewes1 ,on ! inf9ite1 #aiec4 ,lie1 ,on c5tral conceptions 9 mod]n m41 #gbg4 ,end[m5t ( ma?'ns1 #hah4 ,! comp>ative anatomi/1 #iib4 ,aim ( 8 "w1 #iic4 ,8 g5ius1 #iid4 ,on gr\ps1 #ageb4 ,on ! orig9 ( ! calculus1 #aija4 ,on diff]5tial equations1 #aibd4 ,liliwati1 ,orig9 (1 #iie4 ,limitations ( ma?4 sci5ce1 #adcg4 ,limits1 ,me?od (1 #aije1 #aijh1 #aiji1 #aidj4 ,l9deman1 ,on m4 & sci5ce1 #aebc4 ,li\ville1 #hbb4 ,lobat*ewsky1 #bjbb4 ,locke1 ,on ! me?od ( m41 #bad1 #bce4 ,on pro(s & demon/rations1 #bcf4 ,on ! unpopul>ity ( m41 #bga4 ,on m4 z a logical ex]cise1 #dbc1 #dbd4 ,m4 cures presumption1 #dbe4 ,ma?1 reason+ ( univ]sal application1 #dbf4 ,on read+ ( classic au?ors1 #fjd4 ,on ,ari/otle1 #iad4 ,on m4 & philosophy1 #adcc4 ,on m4 & moral sci5ce1 #adci1 #addj4 ,on ! c]ta9ty ( ma?4 k1 #addj1 #adda4 ,on unity1 #afjg4 ,on numb]1 #afjh4 ,on demon/rations 9 numb]s1 #afcj4 ,on ! advantages ( algebra1 #agje4 ,on inf9ity1 #aiee1 #aieg4 ,on probability1 #aife4 ,log>i?mic spiral1 #ibb4 ,log>i?mic tables1 #fjb4 ,log>i?ms1 #aebf1 #afad1 #afaf4 ,logic & m41 ,*apt] ,,xiii4 ,see also #dbc-dcj1 #ddb4 ,logical calculus1 #acaf1 #acag4 ,longevity ( ma?'ns1 #hci4 ,lovelace1 ,:y are wise few etc41 #afbi4 ,lov]1 #badj4 ,macaulay1 ,plato & ,bacon1 #caf4 ,on ,ar*im$es1 #ije4 ,bacon's view ( m41 #iae1 #iaf4 ,anagram on 8 "n1 #iif4 ,plato & ,ar*ytas1 #adbg4 ,on ! p[] ( m41 #aebg4 ,macf>lane1 ,on ,tait1 ,maxwell1 ,?omson1 #ajdb4 ,on ,tait & ,hamilton's quat]nions1 #ajdd4 ,ma*1 ,on ?\ght-economy 9 m41 #bjc4 ,m4 seems possess$ ( intellig5ce1 #fbf4 ,on aim ( rese>*1 #fdg4 ,on m4 & c\nt+1 #afja4 ,on ! space ( exp]i5ce1 #bjaa4 ,mac,mahon1 ,lat9 squ>es1 #beb4 ,on ,sylve/]'s b5d ( m9d1 #fde4 ,on ,sylve/]'s /yle1 #ajdj4 ,on ! idea ( inv>iance1 #agdf4 ,magnitude1 ,grassmann's def9ition1 #aje4 ,magnus1 ,on ! aim 9 tea*+ m41 #eje4 ,manhattan ,isl&1 ,co/ (1 #bacj4 ,m>cellus1 ,e/imate ( ,ar*im$es1 #iji4 ,mas*ke1 ,man above me?od1 #fej4 ,ma/]s1 ,on ! read+ ( !1 #fad4 ,ma!matic1 ,sylve/] on use ( t]m1 #aja4 ,bacon's use ( t]m1 #ajf4 ,ma!matical faculty1 ,frequ5cy (1 #hcb4 ,ma!matical mill1 ,!1 #bci1 #ahia4 ,ma!matical productions1 #fdh1 #fdi4 ,ma!matical !ory1 ,:5 complete1 #fcf1 #fcg4 ,ma!matical tra9+1 #ddc1 #ddd4 ,maxims ( ma?'ns1 #fcj1 #fca1 #fdi4 ,n a comput]1 #abaa4 ,intellectual habits ( ma?'ns1 #adbh4 ,! place ( !1 #aebi4 ,*>act]i/ics ( ! m9d ( a1 #aecd4 ,ma!matician1 ,!1 ,*apt] ,,viii4 ,ma!matics1 ,def9itions (1 ,*apt] ,i4 ,objects (1 ,*apt] ,i4 ,nature (1 ,*apt] ,,ii4 ,e/imates (1 ,*apt] ,,iii4 ,value (1 ,*apt] ,,iv4 ,tea*+ (1 ,*apt] ,v4 ,/udy (1 ,*apt] ,,vi4 ,rese>* 91 ,*apt] ,,vi4 ,mod]n1 ,*apt] ,,vii4 ,z a f9e art1 ,*apt] ,,xi4 ,z a language1 ,*apt] ,,xii4 ,also #dde1 #ahad4 ,& logic1 ,*apt] ,,xiii4 ,& philosophy1 ,*apt] ,,xiv4 ,& sci5ce1 ,*apt] ,,xv4 ,& applications1 ,*apt] ,,xv4 ,k mo/ 91 #bad4 ,suppl4 brevity ( life1 #bah4 ,! range (1 #bfi4 ,comp>$ to ,fr5* language1 #caa4 ,! c>e ( great m51 #cbb4 ,& pr(essional education1 #dbi4 ,& sci5ce tea*+1 #ebb4 ,! que5 ( ! sci5ces1 #ige4 ,advantage ov] philosophy1 #adcf1 #adch4 ,z an in/rum5t1 #aejf4 ,= its own sake1 #aedj1 #aeda1 #aede1 #aedf4 ,! w+s (1 #afjd4 ,ma!sis1 #bgd1 #bgf1 #ahgj1 #bjae4 ,ma!ws1 ,on ,disqu4 ,ari?4 #afch4 ,on numb] !ory1 #afci4 ,! symbol 1 #afdf4 ,on ,cyclotomy1 #afdg4 ,laws ( algebra1 #agji4 ,on inf9ite1 z]o1 inf9itesimal1 #aied4 ,maxims ( great ma?'ns1 #fcj1 #fca1 #fdi4 ,maxwell1 #ajdc1 #aaaf4 ,mc,cormack1 ,on ! unpopul>ity ( m41 #bgj4 ,on function1 #aicc4 ,m*anique cle/e1 #ihe1 #ihf4 ,m$ic9e1 ,m4 & ! /udy (1 #aehe1 #aiah4 ,mellor1 ,def9ition ( high] m41 #ajh4 ,conclusions involv$ 9 premises1 #bch4 ,on m4 & sci5ce1 #aefa4 ,on ! calculus1 #aiab4 ,on integration1 #aibc1 #aibe4 ,memory 9 m41 #bec4 ,m5*mus1 #ija4 ,m]e ma?'ns1 #hbj1 #hba4 ,m]z1 ,on ! trans=m+ p[] ( m41 #cjc4 ,on ! dom9ant ideas 9 m41 #gbe4 ,on extreme views 9 m41 #hbg4 ,on ,leibnitz's "w1 #ihi4 ,on ! ma?4 t5d5cy ( ,leibnitz1 #iij4 ,on m4 z a l5s1 #aeae4 ,m4 ext5ds k1 #aebd4 ,disquisitiones ,ari?meticae1 #afcg4 ,on functions1 #aicb4 ,on hyp]-space1 #bjcf4 ,metaphysics1 ,m4 ! only true1 #cje4 ,meteorology & m41 #aeeg4 ,me?od ( m4 #bab-bae1 #bbf1 #bbg1 #bcj1 #bce1 #bdd1 #hjf1 #aegf4 ,metric sy/em1 #agbe4 ,milit>y tra9+1 ,m4 91 #cad1 #dah1 #aegd4 ,mill1 ,on induction 9 m41 #bba1 #bbb4 ,on g5]alization 9 m41 #bde4 ,on ma?4 /udies1 #dji4 ,on m4 9 a sci5tific education1 #ddd4 ,ma?'ns h>d to conv9ce1 #haa4 ,ma?'ns require g5ius1 #hai4 ,on ,comte1 #idb4 ,on ,desc>tes1 #idb1 #idh4 ,on ,sir ,william ,hamilton's ignorance ( m41 #igh4 ,on ,leibnitz1 #ihg4 ,on m4 & philosophy1 #adba4 ,on m4 z tra9+ = philosoph]s1 #adbb4 ,m4 indisp5sable to sci5ce1 #aeai4 ,m4 & social sci5ce1 #aeie4 ,on ! nature ( geometry1 #ahch4 ,on geometrical me?od1 #ahfa4 ,on ! calculus1 #aijc4 ,mill]1 ,on ! ,d>m/aett] prize1 #babi4 ,miln]1 ,geometry & poetry1 #aaah4 ,m9*91 ,on ,engli% text-books1 #eci4 ,m9]alogy & m41 #aeeh4 ,m9k[ski1 ,on integral numb]s1 #afcf4 ,miracles & m41 #baeg1 #baeh1 #bafj4 ,mix$ m41 ,bacon's def9ition (1 #ajf4 ,:ewell's def9ition (1 #ajg4 ,mod]n algebra1 #ajca1 #ajcb1 #afch1 #agda4 ,mod]n geometry1 #agaj-agac1 #gae1 #gaf1 #ahgh4 ,mod]n m41 ,*apt] ,,vii4 ,moebius1 ,ma?'ns con/itute a favorite class1 #hji4 ,m4 a f9e art1 #aajg4 ,moral sci5ce & m41 #adch-addj4 ,moral value ( m41 ,see e?ical value4 ,mottoes1 ,( ma?'ns1 #fcj1 #fca1 #fdi4 ,( ,py?agoreans1 #ahcc4 ,murray1 ,def9ition ( m41 #aaf4 ,music & m41 #aja1 #bgf1 #ife1 #aajg1 #aaab1 #aaaf1 #aabg1 #aabh1 #aacj-aacc1 #aace1 #aacf4 ,my]s1 ,on m4 z a s*ool subject1 #djc4 ,on pleasure 9 m41 #ded4 ,on ! e?ical value ( m41 #deg4 ,on ! value ( ari?metic1 #afbb4 ,my/icism & numb]s1 #bacf-bada1 #badc4 ,napi]'s rule1 #ahhh4 ,napoleon1 ,m4 & ! welf>e ( ! /ate1 #cac4 ,8 int]e/ 9 m41 #cad1 #ajja4 ,natural sci5ce & m41 ,*apt] ,,xv4 ,also #bdd1 #ddd1 #dde1 #eja4 ,natural selection1 #aiba4 ,nature ( m41 ,*apt] ,,ii4 ,see also #hae1 #abae1 #acjh1 #adbf1 #aebe1 #afbh4 ,nature1 ,/udy (1 #dcc-dcf1 #ead1 #eaf1 #fab4 ,navigation & m41 #aedc1 #aedd4 ,nelson1 ,anagram on1 #baec4 ,neptune1 ,discov]y (1 #aeed1 #aeei4 ,newcomb1 ,on geometrical p>adox]s1 #baac4 ,newton1 ,importance ( 8 "w1 #ccc4 ,on correlation 9 m41 #ebf4 ,on problems 9 algebra1 #ecj4 ,& ,gauss comp>$1 #hbg4 ,8 fame1 #ajjb4 ,em]son on1 #ajjc4 ,:ewell on1 #ajjd1 #ajje4 ,arago on1 #ajjf4 ,pope on1 #ajjg4 ,s\!y on1 #ajjh4 ,hill on1 #ajji4 ,leibnitz on1 #ajaj4 ,lagrange on1 #ajaa4 ,no monum5t to1 #ajab4 ,wilson on1 #ajab1 #ajac4 ,8 g5ius1 #ajad4 ,8 int]e/ 9 *emi/ry & !ology1 #ajae4 ,& al*emy1 #ajaf1 #ajag4 ,8 fir/ exp]im5t1 #ajah4 ,z a lectur]1 #ajai4 ,z an acc\ntant1 #ajbj4 ,8 memor&um-book1 #ajba4 ,8 abs5t-m9d$ness1 #ajbb4 ,e/imate ( himself1 #ajbc-ajbe4 ,8 me?od ( "w1 #ajbf4 ,discov]y ( ! calculus1 #ajbg4 ,anagrams on1 #ajbh4 ,gauss's e/imate (1 #ajbi4 ,on geometry1 #ahaa4 ,comp>$ ) ,euclid1 #ahbg4 ,geometry a me*anical sci5ce1 #ahfe4 ,te/ ( simplicity1 #ahib4 ,me?od ( fluxions1 #aijb4 ,newton's rule1 #agdc4 ,nile1 ,orig9 ( "n1 #baej4 ,noe!r1 ,on ,cayley1 #ica4 ,on ,sylve/]1 #ajcd1 #ajda4 ,non-euclidean geometry1 #acbb1 #bjaf-bjbi1 #bjcc1 #bjce1 #bjdj4 ,nonnus1 ,on ! my/ic f\r1 #badh4 ,nor?rup1 ,on ,"l ,kelv91 #ajdh4 ,notation1 ,importance (1 #fcd1 #abbb1 #afdf4 ,value ( algebraic1 #abac1 #abad4 ,crit]ion ( good1 #abaf4 ,on ,arabic1 #abag1 #afad4 ,advantage ( ma?41 #abbj4 ,see also symbolism4 ,notions1 ,c>d9al ( m41 #aaj4 ,indef9able1 #abai4 ,novalis1 ,def9ition ( pure m41 #aab4 ,m4 ! life supreme1 #cbi4 ,)\t en?usiasm no m41 #hja4 ,me?od is ! ess5ce ( m41 #hjf4 ,ma?'ns n good comput]s1 #haj4 ,music & algebra1 #aabh4 ,philosophy & m41 #adjf4 ,m4 & sci5ce1 #aejg1 #aebf4 ,m4 & hi/oric sci5ce1 #aeii4 ,m4 & magic1 #baei4 ,m4 & miracles1 #bafj4 ,numb]1 ,e inquiry r$ucible to a "q (1 #afjb4 ,& nature1 #afjc4 ,regulates all ?+s1 #afje4 ,aes*ylus on1 #afjf4 ,def9ition (1 #afji1 #afaj4 ,& sup]/ition1 #afcb4 ,di/9ctness (1 #agjg4 ,( ! bea/1 #baea1 #baeb4 ,numb]-!ory1 ,! que5 ( m41 #ige4 ,nature (1 #afci4 ,gauss on1 #afdd4 ,smi? on1 #afde4 ,notation 91 #afdf4 ,aid to geometry1 #afdg4 ,my/]y 91 #afdh4 ,numb]-"w1 ,purpose (1 #afbc4 ,numb]s1 ,py?agoras' view (1 #cba4 ,mighty are1 #aefh4 ,aptitude =1 #afag4 ,demon/rations 91 #afcj4 ,prime1 #afdh4 ,necess>y tru?s l1 #aiff4 ,r\nd1 #bacg4 ,odd1 #bach-bada4 ,gold51 #badb4 ,magic1 #badc4 ,obscurity 9 m4 & philosophy1 #adjg4 ,obs]vation 9 m41 #bea-bec1 #bee1 #dcc1 #ahcj4 ,obvi\sness 9 m41 #ihe1 #ihf1 #ajde4 ,olney1 ,on ! nature ( m41 #bec4 ,oratory & m41 #hbi1 #hcj4 ,ord] & arrangem5t1 #gbe4 ,orig9 ( m41 #adab4 ,orr1 ,memory v]se = 1 #babg4 ,osgood1 ,on ! calculus1 #aiac4 ,o/wald1 ,on f\r-dim5sional space1 #bjci4 4 ,9 actu>ial =mula1 #ide4 ,memory v]se =1 #babg4 ,pacioli1 ,on ! numb] ?ree1 #bade4 ,pa9t+ & m41 #aajc1 #aajg4 ,papp]itz1 ,on ! object ( pure m41 #aaa4 ,p>adoxes1 ,*apt] ,,xxi4 ,p>allel axiom1 ,pro( (1 #ihd1 #baaj1 #baaa4 ,see also non-euclidean geometry4 ,p>k]1 ,def9ition ( ari?metic1 #afaa4 ,numb] born 9 sup]/ition1 #afcb4 ,on geometry1 #ahje4 ,p>ton1 ,on ,newton1 #aiag-aiai1 #ajba1 #ajbb1 #ahbg4 ,pascal1 ,logic & m41 #acjf4 ,peacock1 ,on ! my/icism ( ,greek philosoph]s1 #bacf4 ,! ,yankos ^w = ?ree1 #badd4 ,! numb] ( ! bea/1 #baeb4 ,pe>son1 ,m4 & natural selection1 #hcd4 ,peirce1 ,b5jam91 ,def9ition ( m41 #abj4 ,m4 z an arbit]1 #baj4 ,logic dep5d5t on m41 #acja4 ,on ! symbol -#a1 #agcc4 ,peirce1 ,c4 ,s4 ,def9ition ( m41 #acc4 ,on accid5tal relations1 #babh4 ,p]ry1 ,on ! tea*+ ( m41 #eaj1 #eaa1 #fai1 #hcg4 ,p]sons & anecdotes1 ,*apt]s ,,ix & ,x4 ,philosophy & m41 ,*apt] ,,xiv4 ,also #ccb1 #dja1 #dad1 #ddd1 #dde1 #deb4 ,physics & m41 #abi1 #dcg1 #aeaf1 #aecj1 #aece1 #aech1 #aeci1 #aedh1 #aedi1 #aeej1 #aeee1 #aeef4 ,physiology & m41 #aegh1 #aeha1 #aehb4 ,pic>d1 ,on ! use ( equations1 #ahia4 ,pi]ce1 ,on inf9itesimals1 #aidj4 ,pi]pont1 ,gold5 age ( m41 #gja4 ,on ! progress ( m41 #gjh4 ,*>act]i/ics ( mod]n m41 #gag4 ,on v>iability1 #gba4 ,on div]g5t s]ies1 #aicg4 ,plato1 ,8 view ( m41 #caf1 #dbi4 ,m4 a /udy suitable = freem51 #cag4 ,8 conic sections1 #ccb4 ,& ,ar*im$es1 #ijd4 ,union ( ma?4 & philosophical productivity1 #adjd4 ,diagonal ( squ>e1 #adaa4 ,& ,ar*ytas1 #adbg4 ,m4 & ! arts1 #aefg4 ,on ! value ( m41 #aegd4 ,on ari?metic1 #afbj1 #afba4 ,god geometrizes1 #afce1 #afcf4 #agjb4 ,on geometry1 #dbi1 #ahjc1 #ahjd1 #ahjf1 #ahdd1 #ahde4 ,pleasure1 ,elem5t ( 9 m41 #afbb1 #afbi1 #ahdh1 #ahej1 #ahea4 ,pl9y1 #bjci4 ,plus & m9us signs1 #agbg4 ,plut>*1 ,on ,ar*im$es1 #ijc1 #ijd1 #ijh-iaj1 #iab4 ,god geometrizes1 #ahjb4 ,poe1 #dag4 ,poetry & m41 ,wei]/rass on1 #hjb4 ,pr+%eim on1 #aajh4 ,wordswor? on1 #aaag4 ,miln] on1 #aaah4 ,workman on1 #aabj4 ,pollock on1 #aaba4 ,h(fman on1 #aabb4 ,?oreau on1 #aabc4 ,em]son on1 #aabd4 ,hill on1 #aabe1 #aabf4 ,%akespe>e on1 #aabg4 ,po9c>1 ,on elegance 9 m41 #fdj4 ,m4 has a triple end1 #aajb4 ,m4 z a language1 #abjh4 ,geometry n an exp]im5tal sci5ce1 #ahfg4 ,on geometrical axioms1 #bjje4 ,po9t1 #ahaf4 ,political sci5ce1 ,m4 &1 #abja1 #acbd4 ,politics1 ,ma?'ns &1 #had4 ,pollock1 ,on ,clif=d1 #ich-ida1 #aaba4 ,pope1 #ijg1 #bjae1 #bjca1 #bjdf4 ,precision 9 m41 #bbh1 #fci1 #gbh4 ,precocity 9 m41 #hce4 ,pr$icabilia a priori1 #bjjc4 ,press1 ,m4 ignor$ by daily1 #gca1 #gcb4 ,price1 ,*>act]i/ics ( m41 #bdg4 ,on m4 & physics1 #aeej4 ,prime numb]s1 ,sylve/] on1 #afdh4 ,pr9cipia ,ma!matica1 #acbf4 ,pr+%eim1 ,m4 ! sci5ce ( ! self-evid5t1 #bcb4 ,m4 %\ld be /udi$ = its own sake1 #dci4 ,on ! indirect value ( m41 #ddh4 ,on rigor 9 m41 #ece4 ,on m4 & j\rnalism1 #gcb4 ,on ma?'ns 9 public s]vice1 #hbd4 ,ma?'n some:at ( a poet1 #aajh4 ,on music & m41 #aacb4 ,on ! language ( m41 #abaa4 ,on m4 & physics1 #aedh4 ,probabilities1 #ddb1 #hbc1 #aehi1 #aeij-aeib1 #aifb-aigb1 #aige4 ,problem solv+1 #eca1 #ecb4 ,problems1 ,9 m41 #ebc1 #ecd4 ,9 ari?metic1 #ebh4 ,9 algebra1 #ecj4 ,%\ld be simple1 #fjc4 ,9 ,cambridge texts1 #fjh4 ,on solution (1 #faa4 ,on importance (1 #fbd1 #fbh4 ,:at con/itutes good1 #fbi4 ,aid to rese>*1 #fdd4 ,( mod]n m41 #aibf4 ,proclus1 ,ptolemy & ,euclid1 #iea4 ,on *>act]i/ics ( geometry1 #ahfi4 ,progress 9 m41 #bji1 #baa1 #bab1 #baf1 #bah1 #gjb-gje1 #gjh4 ,projective geometry1 #ahgf1 #ahgg1 #ahgi1 #ahhj4 ,proportion1 ,euclid's doctr9e (1 #ahcd4 ,euclid's def9ition (1 #ahce4 ,proposition1 #abai1 #adai4 ,prussia1 ,m4 91 #eac4 ,pseudoma?1 ,def9$1 #baja4 ,psy*ology & m41 #aegf1 #aehc1 #aehd4 ,ptolemy & ,euclid1 #iea4 ,public s]vice1 ,m4 &1 #hbc1 #hbd1 #acjc1 #aegd4 ,public speak+1 ,m4 &1 #dbj1 #hbi1 #hcj4 ,publications1 ,ma?4 ( pres5t day1 #gjb1 #gjc4 ,pure ,m41 ,bacon's def9ition (1 #ajf4 ,:ewell's def9ition (1 #ajg4 ,on ! object (1 #aaa1 #abi4 ,novalis' conception (1 #aab4 ,hobson's def9ition (1 #aah4 ,russell's def9ition (1 #abg1 #abh4 ,pursuit ( m41 #hdb4 ,py?agoras1 ,numb] ! nature ( ?+s1 #cba4 ,union ( ma?1 & philosophical productivity1 #adjd4 ,! numb] f\r1 #badg4 ,py?agorean bro!rhood1 ,motto (1 #ahcc4 ,py?agorean !orem1 #ahed-ahef1 #bjbf4 ,py?agoreans1 ,music & ,m41 #aacj4 ,quadrature1 ,see ,squ>+ ( ! circle4 ,quantity1 ,*ry/al's def9ition (1 #aae4 ,qu>les1 ,on quadrature1 #baaf4 ,quat]nions1 #ccc1 #hda1 #icg1 #ajdd1 #abaj1 #agah-agbf4 ,quetelet1 ,gr[? ( m41 #aead4 ,railway-mak+1 #aegj4 ,read+ ( m41 #fja1 #fjd-fjf4 ,reason1 ,m4 mo/ solid fabric ( human1 #cjh4 ,m4 demon/rates supremacy ( human1 #cji4 ,reason+1 ,m4 a type ( p]fect1 #cjg4 ,m4 z an ex]cise 91 #dbc-dbg1 #dbi1 #dcj1 #aejc4 ,recorde1 ,value ( ari?metic1 #afai4 ,regiomontanus1 #aedc4 ,regul> solids1 #bacb-bace4 ,reid1 ,m4 frees f sophi/ry1 #bae4 ,conjecture has no place 9 m41 #bcd4 ,m4 ! mo/ solid fabric1 #cjh4 ,on ,euclid's elem5ts1 #iee4 ,m4 manife/s :at is impossible #adad4 ,on m4 & philosophy1 #adbc4 ,probability & ,*ri/ianity1 #aige4 ,on ,py?agoras & ! regul> solids1 #bacb4 ,reidt1 ,m1 z an ex]cise 9 language1 #dai4 ,on ! e?ical value ( m41 #def4 ,on aim 9 ma?4 in/ruction1 #ejf4 ,religion & m41 #bgd-bgf1 #dei1 #dfj1 #ajac4 ,rese>* 9 m41 ,*apt] ,,vi4 ,rev]sible v]ses1 #baef4 ,reye1 ,advantages ( mod]n ov] anci5t geometry1 #gad4 ,rhetoric & m41 #aeii4 ,riemann1 ,on m4 & physics1 #aedi4 ,rigor 9 m41 #ece-ech4 ,rosanes1 ,on ! unpopul>ity ( m41 #gcj4 ,royal road1 #bja1 #ija1 #iea1 #aggd4 ,royal sci5ce1 ,m4 a1 #bjd4 ,rudio1 ,on ,eul]1 #ieg4 ,m4 & great arti/s1 #aaje4 ,on m4 & navigation1 #aedc4 ,ru%1 ,m4 cures pr$isposition to ang]1 #deh4 ,russell1 ,def9ition ( m41 #abg1 #abh4 ,on n9ete5? c5tury m41 #gje4 ,*ief triumph ( mod]n m41 #gjf4 ,on ! inf9ite1 #gbc4 ,on beauty 9 m41 #aajd4 ,on ! value ( symbols1 #abai4 ,on ,boole's ,laws ( ,?\ght1 #acah4 ,pr9cipia ,ma!matica1 #acbf4 ,on geometry & philosophy1 #adaj4 ,def9ition ( numb]1 #afji4 ,fruitful uses ( imag9>ies1 #agce4 ,geometrical reason+ circul>1 #ahfd4 ,on projective geometry1 #ahgi4 ,z5o's problems1 #aich4 ,def9ition ( inf9ite collection1 #aiei4 ,on pro(s ( axioms1 #bjac4 ,on non-euclidean geometry1 #bjah4 ,saf=d1 ,on aptitude = m41 #ebj4 ,on m4 & sci5ce1 #aeji4 ,sage1 ,battalions ( figures1 #afca4 ,s>torius1 ,gauss on ! nature ( space1 #bjcd4 ,scepticism1 #deb1 #haa4 ,s*ellba*1 ,e/imate ( m41 #cjf4 ,on tru?1 #aaad4 ,s*ill]1 ,ar*im$es & ! y\?1 #ijg4 ,s*op5hau]1 ,ari?metic re/s on ! concept ( "t1 #afac4 ,pr$icabilia a priori1 #bjjc4 ,s*rd]1 ,m4 z a bran* ( logic1 #acbc4 ,s*ub]t1 ,?ree *>act]i/ics ( m41 #bbi4 ,on controv]sies 9 m41 #bdc4 ,*>act]i/ics ( m41 #bfc4 ,m4 an exclusive sci5ce1 #gcd4 ,sci5ce & m41 ,*apt] ,,xv4 ,m4 an indisp5sible tool (1 #cji4 ,neglect ( m4 works injury to1 #caj4 ,craig on orig9 ( new1 #fdf4 ,greek view (1 #adbi4 ,six follies (1 #bajg4 ,see also #dcc1 #dcf1 #dcg1 #dfa1 #gbe4 ,sci5tific education1 ,ma?4 tra9+ indisp5sable basis (1 #ddd4 ,screw1 ,! song ( !1 #ahid4 ,z an in/rum5t 9 geometry1 #baad4 ,s$gwick1 ,quat]nion ( maladies1 #agbc4 ,segre1 ,on rese>* 9 m41 #fai4 ,:at k9d ( inve/igations are important1 #fda4 ,on ! wor?lessness ( c]ta9 inve/igations1 #fdb1 #fdc4 ,on hyp]-space1 #bjca4 ,s5eca1 ,alex&] & geometry1 #ijb4 ,sev5ty-sev51 ,! numb]1 #badi4 ,%akespe>e1 #aabg1 #aabi1 #bada4 ,%aw1 ,j4 ,b41 ,m4 l game ( *ess1 #hdj4 ,%aw1 ,w4 ,h41 ,m4 & pr(essional life1 #aeif4 ,%]man1 ,m4 & rhetoric1 #aeii4 ,smi?1 ,adam1 #acbd4 ,smi?1 ,d4 ,e41 ,on problem solv+1 #ecb4 ,value ( geometrical tra9+1 #ahdf4 ,reason = /udy+ geometry1 #ahej4 ,smi?1 ,h4 ,j4 ,s41 ,:5 a ma?4 !ory is complet$1 #fcg4 ,on ! gr[? ( m41 #aeba4 ,on m4 & sci5ce1 #aedb4 ,on m4 & physics1 #aeef4 ,on m4 & meteorology1 #aeeg4 ,on numb] !ory1 #afde4 ,rigor 9 ,euclid1 #ahbi4 ,on ,euclid's doctr9e ( proportion1 #ahcd4 ,smi?1 ,w4 ,b41 ,def9ition ( m41 #aba4 ,on inf9itesimal analysis1 #aiad4 ,on non-euclidean & hyp]spaces1 #bjcc4 ,simon1 ,on beauty & tru?1 #aaad4 ,simplicity 9 m41 #cae1 #ebf4 ,s91 ,on ! notation (1 #ahhf4 ,six hundr$ sixty-six1 ,! numb]1 #baea1 #baeb4 ,social sci5ce & m41 #abja1 #aehf1 #aehg4 ,social s]vice1 ,m4 z an aid to1 #cac1 #cad1 #cbh4 ,social value ( m41 #def1 #aehh4 ,solitude & m41 #ahdi1 #ahea4 ,sophi/ry1 ,m4 free f1 #bae4 ,s\nd1 ,m4 & ! !ory (1 #aeea4 ,s\!y1 ,on ,newton1 #ajjh4 ,space1 ,( exp]i5ce1 #bjaa4 ,kant's doctr9e (1 #bjjc4 ,s*op5hau]'s pr$icabilia1 #bjjd4 ,:ewell1 ,on ! idea (1 #bjjd4 ,non-euclidean1 #bjae1 #bjaf1 #bjah4 ,hyp]-1 #bjcj1 #bjca1 #bjcc1 #bjcf-bjch4 ,sp$d+1 ,on ,bacon's k ( m41 #iag4 ,spe]1 ,on m4 & nature-/udy1 #ead4 ,sp5ce1 ,on ,newton1 #ajaf1 #ajbj4 ,sp5c]1 ,on m4 9 ! arts1 #aegj4 ,sph]ical trigonometry1 #ahhg4 ,spira mirabilis1 #ibb4 ,spottiswoode1 ,on ! k+dom ( m41 #bfi4 ,squ>+ ! circle1 #aecg1 #aheh1 #aicd1 #aidh1 #baae-baag4 ,/4 ,augu/9e1 ,! numb] sev5ty sev51 #badi4 ,/4 ,v9c5t1 ,z a circle-squ>]1 #baji4 ,/e9]1 ,on projective geometry1 #ahgg4 ,/ew>t1 ,m4 & facts1 #bcg4 ,on beauty 9 m41 #bdb4 ,:at we mo/ admire 9 m41 #cae4 ,m4 = its own sake1 #ddj4 ,m4 ! noble/ in/ance ( =ce ( ! human m9d1 #deb4 ,ma?'ns & applause1 #haf4 ,m]e ma?'ns1 #hba4 ,%ortcom+s ( ma?'ns1 #hbh4 ,on ! influ5ce ( ,leibnitz1 #ihh4 ,reason supreme1 #adbd4 ,m4 & philosophy comp>$1 #adbh4 ,m4 & natural philosophy1 #aeee4 ,/ifel1 ,! numb] ( ! bea/1 #baeb4 ,/obu1 ,alex&] & ,m5*mus1 #ija4 ,euclid & ! /ud5t1 #ieb4 ,/udy ( m41 ,*apt] ,,vi4 ,sub/itution1 ,concept (1 #gbg4 ,sup]/ition1 ,m4 frees m9d f1 #dej4 ,numb] 0 born 91 #afcb4 ,surd numb]s1 #agbh4 ,surprises1 ,m4 ri* 91 #bjb4 ,swift1 ,on m4 & politics1 #had4 ,! ma?'ns ( ,laputa1 #babj-babb4 ,! ma?4 s*ool ( ,laputa1 #babc4 ,8 ignorance ( m41 #babd1 #babe4 ,sylve/]1 ,on ! use ( ! t]ms ma!matic & ma!matics1 #aja4 ,ord] & arrangem5t ! basic ideas ( m41 #aji1 #aaj4 ,def9ition ( algebra1 #aaj4 ,def9ition ( ari?metic1 #aaj4 ,def9ition ( geometry1 #aaj4 ,on ! object ( pure m41 #abi4 ,m4 requires h>moni\s action ( all ! faculties1 #bjb4 ,answ] to ,huxley1 #bea4 ,on ! nature ( m41 #bea4 ,on obs]vation 9 m41 #bee4 ,inv5tion 9 m41 #bfj4 ,m4 entitl$ to human reg>d1 #cja4 ,on ! e?ical value ( m41 #ddi4 ,on isolat$ !orems1 #fbj4 8,auge et imp]a40 #fca4 ,8 b5t ( m9d1 #fde4 ,apology = imp]fections1 #fdh4 ,on !oretical inve/igations1 #feh4 ,*>act]i/ics ( mod]n m41 #gbd4 ,inve/$ m4 ) halo ( glory1 #gdj4 ,m4 & eloqu5ce1 #hbi4 ,on longevity ( ma?'ns1 #hci4 ,on ,cayley1 #icj4 ,8 view ( ,euclid1 #icf4 ,jacobi's tal5t = philology1 #ihj4 ,8 eloqu5ce1 #ajcj4 ,rese>*es 9 quantics1 #ajcb4 ,8 weakness1 #ajcc1 #ajcf1 #ajcg4 ,"o-sid$ "* ( 8 "w1 #ajcd4 ,8 me?od1 #ajce1 #ajcf1 #ajda4 ,8 =getfulness1 #ajcg1 #ajch4 ,relations ) /ud5ts1 #ajci4 ,8 /yle1 #ajdj1 #ajda4 ,8 *>act]i/ics1 #ajda4 ,8 en?usiasm1 #ajda4 ,! ma?4 ,adam1 #ajdb4 ,& ,wei]/rass1 #ajej4 ,on div9e beauty & ord] 9 m41 #aaja4 ,m4 among ! f9e arts1 #aajf4 ,on music & m41 #aaca4 ,m4 ! qu9tess5ce ( language1 #abje4 ,m4 ! language ( ! univ]se1 #abjf4 ,on prime numb]s1 #afdh4 ,on det]m9ants1 #agdj4 ,on inv>iants1 #agdb4 ,contribution to !ory ( equations1 #agdc4 ,to a miss+ memb] etc41 #agde4 ,inv>iants & isom]ism1 #agej4 ,8 dislike = ,euclid1 #ahbf4 ,on ! inv5tion ( integrals1 #aibb4 ,on geometry & analysis1 #aica4 ,on p>adoxes1 #bajd4 ,symbolic language1 ,m4 z a1 #abjg1 #abab4 ,use (1 #aegc4 ,symbolic logic1 #acaf-acba4 ,symbolism1 ,on ! nature ( ma?41 #abaj4 ,difficulty ( ma?41 #abah4 ,univ]sal impossible1 #abba4 ,see also notation4 ,symbols1 ,burlesque on1 #agda4 ,symbols1 ,m4 leads to ma/]y (1 #dba4 ,value ( ma?41 #abji1 #abab1 #abai4 ,ess5tial to demon/ration1 #acaf4 ,ari?metical1 #afbg4 ,tact 9 m41 #fbb1 #fbc4 ,tait1 ,on ! unpopul>ity ( m41 #gdj4 ,& ,?omson1 #ajdc4 ,& ,hamilton1 #ajdd4 ,on quat]nions1 #agbd-agbf4 ,on sph]ical trigonometry1 #ahhg4 ,tal5t1 ,ma?'ns m5 (1 #hbe4 ,tea*+ ( m41 ,*apt] ,v4 ,t5nyson1 #ahdc4 ,teutonic race1 ,aptitude = m41 #hch4 ,text-books1 ,*ry/al on1 #ecc4 ,m9*9 on1 #eci4 ,cremona on ,engli%1 #fji4 ,glai%] on ne$ (1 #fce4 ,?ales1 #bja4 ,!oretical inve/igations1 #feb-ffd4 ,!ory & practice1 #ffa4 ,?ompson1 ,sylvanus1 ,"l ,kelv9's def9ition ( a ma?'n1 #hbb4 ,cayley's e/imate ( quat]nions1 #icg4 ,?omson's 8,x is obvi\s t10 #ajde4 ,anecdote ( ,"l ,kelv91 #ajdf1 #ajdg4 ,on ! calculus = beg9n]s1 #aiag4 ,?omson1 ,sir ,william1 ,m4 ! only true metaphysics1 #cje4 ,m4 n repulsive to common s5se1 #cab4 ,:at is a ma?'n8 #hbb4 ,& ,tait1 #ajdc4 8,x is obvi\s t10 #ajde4 ,anecdotes conc]n+1 #ajdf1 #ajdg1 #ajdh4 ,on m4 & a/ronomy1 #aefb4 ,on quat]nions1 #agba1 #agbb4 ,?omson & ,tait1 #ajdc4 ,on ,f\ri]'s !orem1 #aibh4 ,?oreau1 ,on poetry & m41 #aabc4 ,?\ght-economy 9 m41 #bjc1 #abji1 #agjd4 ,?ree1 ,! ,yankos ^w =1 #badd4 ,pacioli on ! numb]1 #bade4 ,"t1 ,ari?metic re/s on notion ( #afac4 ,z a concept 9 algebra1 #agae1 #agaf1 #agag4 ,kant's doctr9e (1 #bjja4 ,s*op5hau]'s pr$icabilia1 #bjjc4 ,todhunt]1 ,on m4 z a univ]sity subject1 #dje4 ,on m4 z a te/ ( p]=mance1 #djh4 ,on m4 z an in/rum5t 9 education1 #dad4 ,m4 requires volunt>y ex]tion1 #dae4 ,on ex]cises1 #dbb4 ,on problems1 #ebc1 #fjh4 ,h[ to read m41 #fje1 #fjf4 ,on discov]y 9 elem5t>y m41 #fag4 ,on ,sylve/]'s !orem1 #agdc4 ,on p]=mance 9 ,euclid1 #ahah4 ,trans=mation1 ,concept (1 #gbg4 ,trigonometry1 #ahha1 #ahhd-ahhi4 ,tril9e> co-ord9ates1 #faa4 ,trisection ( angle1 #baab4 ,tru?1 & m41 #cjf4 ,ma?'ns mu/ p]ceive beauty (1 #hjc4 ,& beauty1 #aaad4 ,tzetzes1 ,plato on geom41 #ahjc4 ,unity1 ,locke on ! idea (1 #afjg4 ,univ]sal algebra1 #agec4 ,univ]sal ari?metic1 #agag4 ,univ]sal language1 #ibe4 ,unpopul>ity ( m41 #bgj1 #bga1 #gcj-gcf1 #gch1 #gdj1 #aeja1 #afbh4 ,usefulness1 ,z a pr9ciple 9 rese>*1 #feb-fee1 #fei1 #ffd4 ,uses ( m41 ,see value ( m4 ,value ( m41 ,*apt] ,,iv4 ,see also #ccj1 #ccc1 #adad1 #adbb1 #aeje1 #aejf1 #aeab1 #aebc1 #aebf1 #aebg1 #aecc1 #aeda1 #aedb1 #aedc1 #aedg-aegf1 #afai-afbf1 #ahda1 #ahdd-ahea4 ,v>iability1 ,! c5tral idea ( mod]n m41 #gbj1 #gba4 ,v5n1 ,on m4 z a symbolic language1 #abjg4 ,m4 ! only gate1 #aeag4 ,viola1 ,on ! use ( fallacies1 #faj4 ,virgil1 #bach4 ,voltaire1 ,ar*im$es m imag9ative ?an ,hom]1 #bei4 ,m4 ! /aff ( ! bl9d1 #dfa4 ,on direct usefulness ( results1 #fec4 ,on inf9ite magnitudes1 #aidg4 ,on ! symbol1 #aiej4 ,anagram on1 #baed4 ,walcott1 ,on hyp]bolic functions1 #aicj4 ,walk]1 ,on problems 9 ari?metic1 #ebh4 ,on ! tea*+ ( geometry1 #ebi4 ,wallace1 ,on ! frequ5cy ( ! ma?4 faculty1 #hcb4 ,on m4 & natural selection1 #hcc1 #hcd4 ,p>allel gr[? ( m4 & music1 #aace4 ,walton1 ,angl+ l m41 #gci4 ,web]1 ,on m4 & physics1 #aedi4 ,web/]1 ,e/imate ( m41 #cca4 ,wei]/rass1 ,ma?'ns are poets1 #hjb4 ,anecdote conc]n+1 #ajdi4 ,& ,sylve/]1 #ajej4 ,problem ( inf9itesimals1 #aich4 ,weismann1 ,on ! orig9 ( ! ma?4 faculty1 #aacf4 ,wells1 ,on m4 z a _w language1 #abja4 ,:ately1 ,on m4 z an ex]cise1 #dbg4 ,on m4 & navigation1 #aedd4 ,on geometrical demon/rations1 #ahci4 ,on ,swift's ignorance ( m41 #babd4 ,:e?am1 ,on symbolic logic1 #acai4 ,:ewell1 ,on mix$ & pure ma?41 #ajg4 ,m4 n an inductive sci5ce1 #bbc4 ,nature ( m41 #bbd4 ,value ( geometry1 #dde4 ,on !oretical inve/igations1 #ffj1 #ffb4 ,ma?'ns m5 ( tal5t1 #hbe4 ,fame ( ma?'ns1 #hbf4 ,on ,newton's greatness1 #ajjd4 ,on ,newton's !ory1 #ajje4 ,on ,newton's humility1 #ajbe4 ,on symbols1 #abab4 ,on philosophy & m41 #adbi4 ,on m4 & sci5ce1 #aecd4 ,quotation f ,r4 ,bacon1 #aedg4 ,on m4 & applications1 #aeda4 ,geometry & exp]i5ce1 #ahad4 ,geometry n an inductive sci5ce1 #ahcj4 ,on limits1 #aiji4 ,on ! idea ( space1 #bjjd4 ,on ,plato & ! regul> solids1 #bacc1 #bace4 ,:ite1 ,h4 ,s41 ,on ! gr[? 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