f ! ,google ,pr9t project47 ,,sci5ce ,,& ,,education ,a ,,s]ies ,,( ,,volumes ,,= ,,! ,,promotion ,,( ,,sci5tific ,,rese>* ,,& ,,educational ,,progress ,,edit$ ,,by ,j4 ,,mcke5 ,,cattell ,,volume ,i--,,! ,,f\ndations ,,( ,,sci5ce ,,"u ,,! ,,same ,,editor%ip ,,sci5ce ,,& ,,education4 ,a s]ies ( volumes = ! promotion ( sci5tific rese>* & educational progress4 ,volume ,i4 ,! ,f\ndations ( ,sci5ce4 ,by ,h4 ,,po9c>4 ,conta9+ ! au?oris$ ,engli% translation by ,george ,bruce ,hal/$ ( ,sci5ce & ,hypo!sis1 ,! ,value ( ,sci5ce1 & ,sci5ce & ,me?od4 ,volume ,,ii4 ,m$ical ,rese>* & ,education4 ,by ,ri*>d ,mills ,pe>ce1 ,william ,h4 ,wel*1 ,w4 ,h4 ,h[ell1 ,frankl9 ,p4 ,mall1 ,lewellys ,f4 ,b>k]1 ,*>les ,s4 ,m9ot1 ,w4 ,b4 ,cannon1 ,w4 ,t4 ,c\ncilman ,!obald ,smi?1 ,g4 ,n4 ,/ew>t1 ,c4 ,m4 ,jackson1 ,e4 ,p4 ,lyon1 ,james ,b4 ,h]rick1 ,john ,m4 ,dodson1 ,c4 ,r4 ,b>de51 ,w4 ,ophuls1 ,s4 ,j4 ,meltz]1 ,james ,ew+1 ,w4 ,w4 ,ke51 ,h5ry ,h4 ,donaldson1 ,*ri/ian ,a4 ,h]t]1 & ,h5ry ,p4 ,b[dit*4 ,volume ,,iii4 ,univ]sity ,control4 ,by ,j4 ,,mcke5 ,,cattell & o!r au?ors4 ,,am]ican ,,m5 ,,( ,,sci5ce4 ,a ,biographical ,directory4 ,,sci5ce4 ,a weekly j\rnal devot$ to ! advancem5t ( sci5ce4 ,! (ficial organ ( ! ,am]ican ,association = ! ,advancem5t ( ,sci5ce4 ,,! ,,popul> ,,sci5ce ,,mon?ly4 ,a mon?ly magaz9e devot$ to ! diffusion ( sci5ce4 ,,! ,,am]ican ,,naturali/4 ,a mon?ly j\rnal devot$ to ! biological sci5ces1 ) special ref]5ce to ! factors ( evolution4 ,,! ,,sci5ce ,,press ,,new ,,york ,,g>rison1 ,n4 ,y4 ,,! ,,f\ndations ,,( ,,sci5ce ,,sci5ce ,,& ,,hypo!sis ,,! ,,value ,,( ,,sci5ce ,,sci5ce ,,& ,,me?od ,,by ,h4 ,,po9c> ,,au?oriz$ ,,translation ,,by ,,george ,,bruce ,,hal/$ ,,) ,a ,,special ,,preface ,,by ,,po9c>1 ,,& ,,an ,,introduction ,,by ,,josiah ,,royce1 ,,h>v>d ,,univ]sity ,,! ,,sci5ce ,,press ,,new ,,york ,,& ,,g>rison1 ,n4 ,y4 #aiac ,copyright1 #aiac ,,by ,,! ,,sci5ce ,,press ,,press ,,( ,,! ,,new ,,era ,,pr9t+ ,,company ,,lanca/]1 ,,pa4 ,,cont5ts ,,page ,h5ri ,po9c> ix ,au?or's ,preface to ! ,translation #c ,,sci5ce ,,& ,,hypo!sis ,introduction by ,royce #i ,introduction #bg ,,"p ,i4 ,numb] & ,magnitude ,,*apt] ,i4--,on ! ,nature ( ,ma!matical ,reason+ #ca ,syllogi/ic ,d$uction #ca ,v]ification & ,pro( #cb ,elem5ts ( ,ari?metic #cc ,reason+ by ,recurr5ce #cg ,induction #dj ,ma!matical ,con/ruction #da ,,*apt] ,,ii4--,ma!matical ,magnitude & ,exp]i5ce #dc ,def9ition ( ,incomm5surables #dd ,! ,physical ,cont9uum #df ,creation ( ! ,ma!matical ,cont9uum #df ,measurable ,magnitude #di ,v>i\s ,rem>ks 7,curves )\t ,tang5ts7 #ej ,! ,physical ,cont9uum ( ,sev]al ,dim5sions #eb ,! ,ma!matical ,cont9uum ( ,sev]al ,dim5sions #ec ,,"p ,,ii4 ,space ,,*apt] ,,iii4--,! ,non-,euclidean ,geometries #ee ,! ,bolyai-,loba*evski ,geometry #ef ,riemann's ,geometry #eg ,! ,surfaces ( ,con/ant ,curvature #eh ,int]pretation ( ,non-,euclidean ,geometries #ei ,! ,implicit ,axioms #fj ,! ,f\r? ,geometry #fb ,lie's ,!orem #fb ,riemann's ,geometries #fc ,on ! ,nature ( ,axioms #fc ,,*apt] ,,iv4--,space & ,geometry #ff ,geometric ,space & ,p]ceptual ,space #ff ,visual ,space #fg ,tactile ,space & ,motor ,space #fh ,*>act]i/ics ( ,p]ceptual ,space #fi ,*ange ( ,/ate & ,*ange ( ,position #gj ,conditions ( ,comp5sation #gb ,solid ,bodies & ,geometry #gb ,law ( ,homog5eity #gd ,! ,non-,euclidean ,_w #ge ,! ,_w ( ,f\r ,dim5sions #gh ,conclusions #gi ,,*apt] ,v4--,exp]i5ce & ,geometry #ha ,geometry & ,a/ronomy #ha ,! ,law ( ,relativity #hc ,be>+ ( ,exp]im5ts #hf ,supplem5t 7,:at is a ,po9t87 #hi ,ance/ral ,exp]i5ce #ia ,,"p ,,iii4 ,=ce ,,*apt] ,,vi4--,! ,classic ,me*anics #ib ,! ,pr9ciple ( ,in]tia #ic ,! ,law ( ,accel]ation #ig ,an?ropomorphic ,me*anics #ajc ,! ,s*ool ( ! ,?read #ajd ,,*apt] ,,vii4--,relative ,motion & ,absolute ,motion #ajg ,! ,pr9ciple ( ,relative ,motion #ajg ,newton's ,argum5t #ajh ,,*apt] ,,viii4--,en]gy & ,!rmodynamics #aae ,en]getics #aae ,!rmodynamics #aai ,g5]al ,conclusions on ,"p ,,iii #abc ,,"p ,,iv4 ,nature ,,*apt] ,,ix4--,hypo!ses 9 ,physics #abg ,! ,rle ( ,exp]im5t & ,g5]alization #abg ,! ,unity ( ,nature #acj ,! ,rle ( ,hypo!sis #acc ,orig9 ( ,ma!matical ,physics #acf ,,*apt] ,x4--,! ,!ories ( ,mod]n ,physics #adj ,mean+ ( ,physical ,!ories #adj ,physics & ,me*anism #add ,pres5t ,/ate ( ! ,sci5ce #adh ,,*apt] ,,xi4--,! ,calculus ( ,probabilities #aee ,classification ( ! ,problems ( ,probability #aeh ,probability 9 ,ma!matics #afa ,probability 9 ! ,physical ,sci5ces #afd ,r\ge et noir #afg ,! ,probability ( ,causes #afi ,! ,!ory ( ,errors #agj ,conclusions #agb ,,*apt] ,,xii4--,optics & ,electricity #agd ,fresnel's ,!ory #agd ,maxwell's ,!ory #age ,! ,me*anical ,explanation ( ,physical ,ph5om5a #agg ,,*apt] ,,xiii4--,electrodynamics #ahd ,ampre's ,!ory #ahd ,clos$ ,curr5ts #ahe ,action ( a ,clos$ ,curr5t on a ,portion ( ,curr5t #ahf ,cont9u\s ,rotations #ahg ,mutual ,action ( ,two ,op5 ,curr5ts #ahi ,induction #aij ,!ory ( ,helmholtz #aia ,difficulties ,rais$ by ^! ,!ories #aic ,maxwell's ,!ory #aic ,r[l&'s ,exp]im5t #aid ,! ,!ory ( ,lor5tz #aif ,,! ,,value ,,( ,,sci5ce ,translator's ,introduction #bja ,does ! ,sci5ti/ ,create ,sci5ce8 #bja ,! ,m9d ,dispell+ ,optical ,illusions #bjb ,euclid n ,necess>y #bjb ,)\t ,hypo!ses1 no ,sci5ce #bjc ,:at ,outcome8 #bjc ,introduction #bje ,,"p ,i4 ,! ,ma!matical ,sci5ces ,,*apt] ,i4--,intuition & ,logic 9 ,ma!matics #baj ,,*apt] ,,ii4--,! ,measure ( ,"t #bbc ,,*apt] ,,iii4--,! ,notion ( ,space #bce ,qualitative ,geometry #bch ,! ,physical ,cont9uum ( ,sev]al ,dim5sions #bdj ,! ,notion ( ,po9t #bdd ,! ,notion ( ,displacem5t #bdg ,visual ,space #beb ,,*apt] ,,iv4--,space & its ,?ree ,dim5sions #bef ,! ,gr\p ( ,displacem5ts #bef ,id5tity ( ,two ,po9ts #bei ,tactile ,space #bfd ,id5tity ( ! ,diff]5t ,spaces #bfh ,space & ,empiricism #bga ,rle ( ! ,semicircul> ,canals #bgf ,,"p ,,ii4 ,! ,physical ,sci5ces ,,*apt] ,v4--,analysis & ,physics #bgi ,,*apt] ,,vi4--,a/ronomy #bhi ,,*apt] ,,vii4--,! ,hi/ory ( ,ma!matical ,physics #big ,! ,physics ( ,c5tral ,=ces #big ,! ,physics ( ! ,pr9ciples #bii ,,*apt] ,,viii4--,! ,pres5t ,crisis 9 ,physics #cjc ,! ,new ,crisis #cjc ,c>not's ,pr9ciple #cjc ,! ,pr9ciple ( ,relativity #cje ,newton's ,pr9ciple #cjh ,lavoisi]'s ,pr9ciple #caj ,may]'s ,pr9ciple #cab ,,*apt] ,,ix4--,! ,future ( ,ma!matical ,physics #cad ,! ,pr9ciples & ,exp]im5t #cad ,! ,rle ( ! ,analy/ #cad ,ab]ration & ,a/ronomy #cae ,electrons & ,spectra #caf ,conv5tions prec$+ ,exp]im5t #cag ,future ,ma!matical ,physics #cai ,,"p ,,iii4 ,! ,objective ,value ( ,sci5ce ,,*apt] ,x4--,is ,sci5ce ,artificial8 #cba ,! ,philosophy ( ,le,roy #cba ,sci5ce1 ,rule ( ,action #cbc ,! ,crude ,fact & ! ,sci5tific ,fact #cbe ,nom9alism & ! ,univ]sal ,inv>iant #ccc ,,*apt] ,,xi4--,sci5ce & ,reality #cdj ,cont+5ce & ,det]m9ism #cdj ,objectivity ( ,sci5ce #cdg ,! ,rotation ( ! ,e>? #cec ,sci5ce = ,its ,own ,sake #ced ,,sci5ce ,,& ,,me?od ,introduction #cei ,,book ,i4 ,sci5ce & ! ,sci5ti/ ,,*apt] ,i4--,! ,*oice ( ,facts #cfb ,,*apt] ,,ii4--,! ,future ( ,ma!matics #cfi ,,*apt] ,,iii4--,ma!matical ,creation #chc ,,*apt] ,,iv4--,*ance #cie ,,book ,,ii4 ,ma!matical ,reason+ ,,*apt] ,i4--,! ,relativity ( ,space #dac ,,*apt] ,,ii4--,ma!matical ,def9itions & ,tea*+ #dcj ,,*apt] ,,iii4--,ma!matics & ,logic #ddh ,,*apt] ,,iv4--,! ,new ,logics #dfj ,,*apt] ,v4--,! ,late/ ,ef=ts ( ! ,logi/icians #dgb ,,book ,,iii4 ,! ,new ,me*anics ,,*apt] ,i4--,me*anics & ,radium #dhf ,,*apt] ,,ii4--,me*anics & ,optics #dif ,,*apt] ,,iii4--,! ,new ,me*anics & ,a/ronomy #eab ,,book ,,iv4 ,a/ronomic ,sci5ce ,,*apt] ,i4--,! ,milky ,way & ! ,!ory ( ,gases #ebc ,,*apt] ,,ii4--,fr5* ,geodesy #ece ,g5]al ,conclusions #edd ,index #edg ,,h5ri ,,po9c> ,,sir ,,george ,,d>w91 wor?y son ( an immortal fa!r1 said1 ref]r+ to :at ,po9c> 0 to him & to 8 "w3 ,he mu/ be reg>d$ z ! presid+ g5ius--or1 %all ,i say1 my patron sa9t8 ,h5ri ,po9c> 0 born ,april #bi1 #ahed1 at ,nancy1 :]e 8 fa!r 0 a physician highly respect$4 ,8 s*ool+ 0 brok5 into by ! w> ( #ahgj-ga1 to get news ( :i* he le>n$ to read ! ,g]man newspap]s4 ,he outclass$ ! o!r boys ( 8 age 9 all subjects & 9 #ahgc pass$ highe/ into ! cole ,polyte*nique1 :]e1 l ,john ,bolyai at ,m>os ,vsrhely1 he foll[$ ! c\rses 9 ma!matics )\t tak+ a note & )\t ! syllabus4 ,he proce$$ 9 #ahge to ! ,s*ool ( ,m9es1 & 0 ,nomm1 ,m>* #bf1 #ahgi4 ,b he won 8 doctorate 9 ! ,univ]sity ( ,p>is1 ,augu/ #a1 #ahgi1 & 0 appo9t$ to tea* 9 ! ,facult des ,sci5ces de ,ca51 ,decemb] #a1 #ahgi1 :5ce he 0 quickly call$ to ! ,univ]sity ( ,p>is1 tea*+ !re f ,octob] #ba1 #ahha1 until 8 dea?1 ,july #ag1 #aiab4 ,s x is an error to say he />t$ z an eng9e]4 ,at ! e>ly age ( ?irty-two he became a memb] ( l',acadmie des ,sci5ces1 &1 ,m>* #e1 #aijh1 0 *os5 ,membre de l',acadmie ,franaise4 ,july #a1 #aiji1 ! numb] ( 8 writ+s 0 #dcf4 ,8 e>lie/ publication 0 9 #ahgh1 & 0 n important4 ,aft]w>d came an essay submitt$ 9 competition = ! ,gr& ,prix (f]$ 9 #ahhj1 b x did n w94 ,sudd5ly !re came a *ange1 a /rik+ fire1 a bur/+ =?1 9 ,febru>y1 #ahha1 & ,po9c> tells u ! v m9ute x happ5$4 ,m\nt+ an omnibus1 at ! mom5t :5 ,i put my foot ^u ! /ep1 ! idea came to me1 )\t any?+ 9 my previ\s ?\ghts seem+ to =e%ad[ x1 t ! trans=mations ,i _h us$ to def9e ! ,fu*sian functions 7 id5tical ) ^? ( non-,euclidean geometry4 ,!reby 0 op5$ a p]spective new & imm5se4 ,moreov]1 ! magic w& ( 8 :ole life-"w _h be5 grasp$1 ! ,aladd9's lamp _h be5 rubb$1 non-,euclidean geometry1 ^: necromancy 0 to op5 up a new !ory ( our univ]se1 ^: brilliant exposition 0 comm5c$ 9 8 book ,sci5ce & ,hypo!sis1 :i* has be5 translat$ into six languages & has already _h a circulation ( ov] #bj1jjj4 ,! non-,euclidean notion is t ( ! possibility ( alt]native laws ( nature1 :i* 9 ! ,introduction to ! lectricit et ,optique1 #aija1 is ?us put3 ,if !re=e a ph5om5on admits ( a complete me*anical explanation1 x w admit ( an inf9ity ( ,o!rs :i* w acc\nt equally well = all ! peculi>ities disclos$ by exp]im5t4 ,! s*eme ( laws ( nature s l>gely due to ,newton is m]ely "o ( an inf9ite numb] ( conceivable rational s*emes = help+ u ma/] & make exp]i5ce2 x is commode1 conv5i5t2 b p]haps ano!r may be va/ly m advantage\s4 ,! old conception ( true has be5 revis$4 ,! fir/ expression ( ! new idea occurs on ! title page ( ,john ,bolyai's m>vel\s ,sci5ce ,absolute ( ,space1 9 ! phrase haud unquam a priori decid5da4 ,) be>+ on ! hi/ory ( ! e>? & moon sy/em & ! orig9 ( d\ble />s1 9 =mulat+ ! geometric crit]ion ( /ability1 ,po9c> prov$ ! exi/5ce ( a previ\sly unkn[n pe>-%ap$ figure1 ) ! possibility t ! progressive de=mation ( ?is figure ) increas+ angul> velocity might result 9 ! break+ up ( ! rotat+ body into two deta*$ masses4 ,( 8 treatise ,les ,m?odes n\velles de la ,m*anique cle/e1 ,sir ,george ,d>w9 says3 ,x is probable t = half a c5tury to come x w be ! m9e f :i* humbl] inve/igators w excavate _! mat]ials4 ,brilliant 0 8 appreciation ( ,po9c> 9 pres5t+ ! gold m$al ( ! ,royal ,a/ronomical ,society4 ,! ?ree o!rs mo/ ak9 9 g5ius are l9k$ ) him by ! ,sylve/] m$al ( ! ,royal ,society1 ! ,loba*evski m$al ( ! ,physico-,ma!matical ,society ( ,kazan1 & ! ,bolyai prize ( ! ,hung>ian ,academy ( ,sci5ces4 ,8 "w mu/ be reckon$ ) ! greate/ ma!matical a*ievem5ts ( mank9d4 ,! k]nel ( ,po9c>'s p[] lies 9 an oracle ,sylve/] (t5 quot$ to me z f ,hesiod3 ,! :ole is less ?an its "p4 ,he p5etrates at once ! div9e simplicity ( ! p]fectly g5]al case1 & !nce desc5ds1 z f ,olympus1 to ! special concrete e>?ly p>ticul>s4 ,a comb9ation ( seem+ly extremely simple analytic & geometric concepts gave necess>y g5]al conclusions ( imm5se scope f :i* sprang a disconc]t+ wild]ness ( possible d$uctions4 ,& s he leaves a noble1 fruitful h]itage4 ,says ,love3 ,8 "r is recogniz$ n[1 & x is n likely t future g5]ations w revise ! judgm5t1 to rank among ! greate/ ma!maticians ( all "t4 ,,george ,,bruce ,,hal/$4 99 99 99 99 99 ,,sci5ce ,,& ,,hypo!sis 99 99 99 99 99 ,,au?or',s ,,preface ,,to ,,! ,,translation ,i am exce$+ly grateful to ,dr4 ,hal/$1 :o has be5 s good z to pres5t my book to ,am]ican read]s 9 a translation1 cle> & fai?ful4 ,e "o kn[s t ?is savant has already tak5 ! tr\ble to translate _m ,european treatises & ?us has p[]fully contribut$ to make ! new cont95t und]/& ! ?\ght ( ! old4 ,"s p love to repeat t ,anglo-,saxons h n ! same way ( ?9k+ z ! ,lat9s or z ! ,g]mans2 t !y h q ano!r way ( und]/&+ ma!matics or ( und]/&+ physics2 t ?is way seems to !m sup]ior to all o!rs2 t !y feel no ne$ ( *ang+ x1 nor ev5 ( kn[+ ! ways ( o!r peoples4 ,9 t !y w\ld beyond "q be wrong1 b ,i d n believe t is true1 or1 at lea/1 t is true no long]4 ,= "s "t ! ,engli% & ,am]icans h be5 devot+ !mselves mu* m ?an =m]ly to ! bett] und]/&+ ( :at is ?\ght & said on ! cont95t ( ,europe4 ,to be sure1 ea* p w pres]ve its *>act]i/ic g5ius1 & x w\ld be a pity if x 7 o!rwise1 suppos+ su* a ?+ possible4 ,if ! ,anglo-,saxons wi%$ to become ,lat9s1 !y w\ld nev] be m ?an bad ,lat9s2 j z ! ,fr5*1 9 seek+ to imitate !m1 c\ld turn out only pretty poor ,anglo-,saxons4 ,& !n ! ,engli% & ,am]icans h made sci5tific conque/s !y alone c\ld h made2 !y w make /ill m ( :i* o!rs w\ld be incapable4 ,x w\ld !re=e be deplorable if !re 7 no long] ,anglo-,saxons4 ,b cont95tals h on _! "p done ?+s an ,engli%man c\ld n h done1 s t !re is no ne$ ei!r = wi%+ all ! _w ,anglo-,saxon4 ,ea* has 8 *>act]i/ic aptitudes1 & ^! aptitudes %\ld be div]se1 else w\ld ! sci5tific conc]t resemble a qu>tet :]e e "o want$ to play ! viol94 ,& yet x is n bad = ! viol9 to "k :at ! violon-cello is play+1 & vice v]sa4 ,?is x is t ! ,engli% & ,am]icans are compreh5d+ m & m2 & f ?is po9t ( view ! translations und]tak5 by ,dr4 ,hal/$ are mo/ opportune & timely4 ,consid] fir/ :at conc]ns ! ma!matical sci5ces4 ,x is frequ5tly said ! ,engli% cultivate !m only 9 view ( _! applications & ev5 t !y despise ^? :o h o!r aims2 t speculations too ab/ract repel !m z savor+ ( metaphysic4 ,! ,engli%1 ev5 9 ma!matics1 are to proce$ always f ! p>ticul> to ! g5]al1 s t !y w\ld nev] h an idea ( ent]+ ma!matics1 z d _m ,g]mans1 by ! gate ( ! !ory ( aggregates4 ,!y are always to hold1 s to speak1 "o foot 9 ! _w ( ! s5ses1 & nev] burn ! bridges keep+ !m 9 communication ) reality4 ,!y ?us are to be incapable ( compreh5d+ or at lea/ ( appreciat+ c]ta9 !ories m int]e/+ ?an utilit>ian1 su* z ! non-,euclidean geometries4 ,accord+ to t1 ! fir/ two p>ts ( ?is book1 on numb] & space1 %\ld seem to !m void ( all sub/ance & w\ld only baffle !m4 ,b t is n true4 ,& fir/ ( all1 are !y su* uncompromis+ reali/s z has be5 said8 ,are !y absolutely refractory1 ,i d n say to metaphysic1 b at lea/ to ev]y?+ metaphysical8 ,recall ! "n ( ,b]keley1 born 9 ,irel& d\btless1 b imm$iately adopt$ by ! ,engli%1 :o m>k$ a natural & necess>y /age 9 ! developm5t ( ,engli% philosophy4 ,is ?is n en\gh to %[ !y are capable ( mak+ asc5sions o!rwise ?an 9 a captive balloon8 ,& to return to ,am]ica1 is n ! ,moni/ publi%$ at ,*icago1 t review :i* ev5 to u seems bold & yet :i* f9ds read]s8 ,& 9 ma!matics8 ,d y ?9k ,am]ican geomet]s are conc]n$ only ab\t applications8 ,f> f x4 ,! "p ( ! sci5ce !y cultivate mo/ devot$ly is ! !ory ( gr\ps ( sub/itutions1 & "u its mo/ ab/ract =m1 ! f>!/ remov$ f ! practical4 ,moreov]1 ,dr4 ,hal/$ gives regul>ly ea* ye> a review ( all productions relative to ! non-,euclidean geometry1 & he has ab\t him a public deeply int]e/$ 9 8 "w4 ,he has initiat$ ?is public into ! ideas ( ,hilb]t1 & he has ev5 writt5 an elem5t>y treatise on ',rational ,geometry1' bas$ on ! pr9ciples ( ! r5[n$ ,g]man savant4 ,to introduce ?is pr9ciple into tea*+ is surely ?is "t to burn all bridges ( reliance ^u s5sory intuition1 & ?is is1 ,i confess1 a boldness :i* seems to me almo/ ra%ness4 ,! ,am]ican public is !re=e mu* bett] prep>$ ?an has be5 ?\ght = inve/igat+ ! orig9 ( ! notion ( space4 ,moreov]1 to analyze ?is concept is n to sacrifice reality to ,i "k n :at phantom4 ,! geometric language is aft] all only a language4 ,space is only a ^w t we h believ$ a ?+4 ,:at is ! orig9 ( ?is ^w & ( o!r words also8 ,:at ?+s d !y hide8 ,to ask ?is is p]missible2 to =bid x w\ld be1 on ! contr>y1 to be a dupe ( words2 x w\ld be to adore a metaphysical idol1 l savage peoples :o pro/rate !mselves be=e a /atue ( wood )\t d>+ to take a look at :at is )94 ,9 ! /udy ( nature1 ! contra/ betwe5 ! ,anglo-,saxon _s & ! ,lat9 _s is /ill great]4 ,! ,lat9s seek 9 g5]al to put _! ?\ght 9 ma!matical =m2 ! ,engli% pref] to express x by a mat]ial repres5tation4 ,bo? d\btless rely only on exp]i5ce = kn[+ ! _w2 :5 !y happ5 to g beyond ?is1 !y consid] _! =ekn[l$ge z only provisional1 & !y ha/5 to ask its def9itive confirmation f nature h]self4 ,b exp]i5ce is n all1 & ! savant is n passive2 he does n wait = ! tru? to come & f9d him1 or = a *ance meet+ to br+ him face to face ) x4 ,he mu/ g to meet x1 & x is = 8 ?9k+ to reveal to him ! way lead+ ?i!r4 ,= t !re is ne$ ( an in/rum5t2 well1 j !re beg9s ! diff]5ce--! in/rum5t ! ,lat9s ord9>ily *oose is n t pref]r$ by ! ,anglo-,saxons4 ,= a ,lat91 tru? c be express$ only by equations2 x mu/ obey laws simple1 logical1 symmetric & fitt$ to satisfy m9ds 9 love ) ma!matical elegance4 ,! ,anglo-,saxon to depict a ph5om5on w fir/ be engross$ 9 mak+ a model1 & he w make x ) common mat]ials1 su* z our crude1 unaid$ s5ses %[ u !m4 ,he also makes a hypo!sis1 he assumes implicitly t nature1 9 h] f9e/ elem5ts1 is ! same z 9 ! complicat$ aggregates :i* alone are )9 ! rea* ( our s5ses4 ,he concludes f ! body to ! atom4 ,bo? !re=e make hypo!ses1 & ?is inde$ is necess>y1 s9ce no sci5ti/ has ev] be5 able to get on )\t !m4 ,! ess5tial ?+ is nev] to make !m unconsci\sly4 ,f ?is po9t ( view aga91 x w\ld be well = ^! two sorts ( physici/s to "k some?+ ( ea* o!r2 9 /udy+ ! "w ( m9ds s unlike _! own1 !y w imm$iately recognize t 9 ?is "w !re has be5 an accumulation ( hypo!ses4 ,d\btless ?is w n suffice to make !m compreh5d t !y on _! "p h made j z _m2 ea* sees ! mote )\t see+ ! beam2 b by _! criticisms !y w w>n _! rivals1 & x may be suppos$ ^! w n fail to r5d] !m ! same s]vice4 ,! ,engli% proc$ure (t5 seems to u crude1 ! analogies !y ?9k !y discov] to u seem at times sup]ficial2 !y are n suffici5tly int]lock$1 n precise en\gh2 !y sometimes p]mit incoh]5ces1 contradictions 9 t]ms1 :i* %ock a geometric _s & :i* ! employm5t ( ! ma!matical me?od w\ld imm$iately h put 9 evid5ce4 ,b mo/ (t5 x is1 on ! o!r h&1 v =tunate t !y h n p]ceiv$ ^! contradictions2 else w\ld !y h reject$ _! model & c\ld n h d$uc$ f x ! brilliant results !y h (t5 made to come out ( x4 ,& !n ^! v contradictions1 :5 !y end by p]ceiv+ !m1 h ! advantage ( %[+ !m ! hypo!tical "* ( _! conceptions1 :]eas ! ma!matical me?od1 by its app>5t rigor & inflexible c\rse1 (t5 inspires 9 u a confid5ce no?+ w>rants1 & prev5ts our look+ ab\t u4 ,f ano!r po9t ( view1 h[ev]1 ! two conceptions are v unlike1 & if all mu/ be said1 !y are v unlike because ( a common fault4 ,! ,engli% wi% to make ! _w out ( :at we see4 ,i mean :at we see ) ! unaid$ eye1 n ! microscope1 nor t /ill m subtile microscope1 ! human head guid$ by sci5tific induction4 ,! ,lat9 wants to make x out ( =mulas1 b ^! =mulas are /ill ! qu9tess5c$ expression ( :at we see4 ,9 a ^w1 bo? w\ld make ! unkn[n out ( ! kn[n1 & _! excuse is t !re is no way ( do+ o!rwise4 ,& yet is ?is legitimate1 if ! unkn[n be ! simple & ! kn[n ! complex8 ,%all we n get ( ! simple a false idea1 if we ?9k x l ! complex1 or worse yet if we /rive to make x out ( elem5ts :i* are !mselves comp\nds8 ,is n ea* great advance accompli%$ precisely ! day "s "o has discov]$ "u ! complex aggregate %[n by our s5ses some?+ f> m simple1 n ev5 resembl+ x--z :5 ,newton replac$ ,kepl]'s ?ree laws by ! s+le law ( gravitation1 :i* 0 some?+ simpl]1 equival5t1 yet unlike8 ,"o is ju/ifi$ 9 ask+ if we are n on ! eve ( j su* a revolution or "o ev5 m important4 ,matt] seems on ! po9t ( los+ its mass1 its solide/ attribute1 & resolv+ itself into electrons4 ,me*anics mu/ !n give place to a broad] conception :i* w expla9 x1 b :i* x w n expla94 ,s x 0 9 va9 ! attempt 0 made 9 ,engl& to con/ruct ! e!r by mat]ial models1 or 9 ,france to apply to x ! laws ( dynamic4 ,! e!r x is1 ! unkn[n1 :i* expla9s matt]1 ! kn[n2 matt] is incapable ( expla9+ ! e!r4 ,,po9c>4 ,,introduction ,,by ,,pr(essor ,,josiah ,,royce ,,h>v>d ,,univ]sity ,! treatise ( a ma/] ne$s no comm5dation ?r\gh ! words ( a m]e le>n]4 ,b1 s9ce my fri5d & =m] fell[ /ud5t1 ! translator ( ?is volume1 has jo9$ ) ano!r ( my colleagues1 ,pr(essor ,cattell1 9 ask+ me to und]take ! task ( call+ ! att5tion ( my fell[ /ud5ts to ! importance & to ! scope ( ,m4 ,po9c>'s volume1 ,i accept ! (fice1 n z "o compet5t to pass judgm5t ^u ! book1 b simply z a le>n]1 desir\s to increase ! numb] ( ^? among/ u :o are already int]e/$ 9 ! type ( rese>*es to :i* ,m4 ,po9c> has s notably contribut$4 ,i ,! bran*es ( inquiry collectively kn[n z ! ,philosophy ( ,sci5ce h und]gone great *anges s9ce ! appe>ance ( ,h]b]t ,sp5c]'s ,fir/ ,pr9ciples1 t volume :i* a l>ge "p ( ! g5]al public 9 ?is c\ntry us$ to reg>d z ! repres5tative comp5d ( all mod]n wisdom relat+ to ! f\ndations ( sci5tific k4 ,! summ>y :i* ,m4 ,po9c> gives1 at ! outset ( 8 own introduction to ! pres5t "w1 :]e he /ates ! view :i* ! 'sup]ficial obs]v]' takes ( sci5tific tru?1 sugge/s1 n inde$ ,sp5c]'s own mo/ *>act]i/ic !ories1 b some?+ ( ! _s 9 :i* _m disciples ( ,sp5c] int]pret+ _! ma/]'s =mulas us$ to conceive ! position :i* sci5ce occupies 9 deal+ ) exp]i5ce4 ,x 0 well kn[n to !m1 inde$1 t exp]i5ce is a con/ant guide1 & an inexhau/ible s\rce bo? ( novel sci5tific results & ( unsolv$ problems2 b ! fundam5tal ,sp5c]ian pr9ciples ( sci5ce1 su* z '! p]si/5ce ( =ce1' ! 'rhy?m ( motion' & ! re/1 7 treat$ by ,sp5c] himself z demon/rably objective1 al?\gh inde$ 'relative' tru?s1 capable ( be+ te/$ once = all by ! 'inconceivability ( ! opposite1' & c]ta9 to hold true = ! :ole 'kn[able' univ]se4 ,?us1 :e!r "o dwelt ^u ! results ( su* a ma!matical proc$ure z t to :i* ,m4 ,po9c> ref]s 9 8 op5+ p>agraphs1 or :e!r1 l ,sp5c] himself1 "o appli$ ! 'fir/ pr9ciples' to regions ( less exact sci5ce1 ?is confid5ce t a c]ta9 or?odoxy reg>d+ ! pr9ciples ( sci5ce 0 e/abli%$ =ev] 0 *>act]i/ic ( ! foll[]s ( ! movem5t 9 "q4 ,exp]i5ce1 light$ up by reason1 seem$ to !m to h pr$et]m9$ = all future "t c]ta9 great !oretical results reg>d+ ! real con/itution ( ! 'kn[able' cosmos4 ,:oev] d\bt$ ?is d\bt$ '! v]dict ( sci5ce4' ,"s ( u well rememb] h[1 :5 ,/allo's ',pr9ciples & ,!ories ( ,mod]n ,physics' fir/ appe>$1 ?is s5se ( sci5tific or?odoxy 0 %ock$ among/ _m ( our ,am]ican read]s & tea*]s ( sci5ce4 ,i myself c recall to m9d "s highly au?oritative reviews ( t "w 9 :i* ! au?or 0 m or less %>ply tak5 to task = 8 ignorant presumption 9 speak+ ) ! fre$om t he !re us$ reg>d+ su* sacr$ possessions ( humanity z ! fundam5tal concepts ( physics4 ,t v book1 h[ev]1 has q lately be5 translat$ into ,g]man z a valuable contribution to "s ( ! mo/ rec5t ef=ts to recon/itute a mod]n 'philosophy ( nature4' ,& :atev] may be o!rwise ?\ght ( ,/allo's critical me?ods1 or ( 8 results1 !re c be no d\bt t1 at ! pres5t mom5t1 if 8 book 7 to appe> = ! fir/ "t1 nobody w\ld attempt to discr$it ! "w m]ely on acc\nt ( its disposition to be agno/ic reg>d+ ! objective reality ( ! concepts ( ! k9etic !ory ( gases1 or on acc\nt ( its call = a logical re>rangem5t ( ! fundam5tal concepts ( ! !ory ( en]gy4 ,we are no long] able s easily to "k h]etics at fir/ sight4 ,= we n[ appe> to /& 9 ?is position3 ,! control ( natural ph5om5a1 :i* ?r\gh ! sci5ces m5 h atta9$1 gr[s daily va/] & m detail$1 & 9 its details m assur$4 ,ph5om5a m5 "k & pr$ict bett] ?an ev]4 ,b reg>d+ ! mo/ g5]al !ories1 & ! mo/ fundam5tal1 ( sci5ce1 !re is no long] any notable sci5tific or?odoxy4 ,?us1 z k gr[s firm] & wid]1 conceptual con/ruction becomes less rigid4 ,! field ( ! !oretical philosophy ( nature--yes1 ! field ( ! logic ( sci5ce--?is :ole region is to-day an op5 "o4 ,:oev] w "w !re mu/ inde$ accept ! v]dict ( exp]i5ce reg>d+ :at happ5s 9 ! natural _w4 ,s f> he is inde$ b\nd4 ,b he may und]take )\t h9drance f m]e tradition ! task ( try+ afre% to r$uce :at happ5s to conceptual unity4 ,! circle-squ>]s & ! inv5tors ( devices = p]petual motion are inde$ /ill z unwelcome 9 sci5tific company z !y 7 9 ! days :5 sci5tific or?odoxy 0 m rigidly def9$2 b t is n because ! f\ndations ( geometry are n[ view$ z completely settl$1 beyond controv]sy1 nor yet because ! 'p]si/5ce ( =ce' has be5 f9ally s def9$ z to make ! 'opposite inconceivable' & ! doctr9e ( en]gy beyond ! rea* ( novel =mulations4 ,no1 ! circle-squ>]s & ! inv5tors ( devices = p]petual motion are to-day discr$it$1 n because ( any unor?odoxy ( _! g5]al philosophy ( nature1 b because _! views reg>d+ special facts & processes /& 9 conflict ) c]ta9 equally special results ( sci5ce :i* !mselves admit ( v v>i\s g5]al !oretical int]pretations4 ,c]ta9 prop]ties ( ! irrational numb] pi are kn[n1 9 suffici5t multitude to ju/ify ! ma!matician 9 decl9+ to li/5 to ! argum5ts ( ! circle-squ>]2 b1 despite great advances1 & despite ! assur$ results ( ,d$ek9d1 ( ,cantor1 ( ,wei]/rass & ( v>i\s o!rs1 ! g5]al !ory ( ! logic ( ! numb]s1 rational & irrational1 /ill pres5ts sev]al important features ( great obscurity2 & ! philosophy ( ! concepts ( geometry yet rema9s1 9 sev]al v notable respects1 unconqu]$ t]ritory1 despite ! "w ( ,hilb]t & ( ,pi]i1 & ( our au?or himself4 ,! ord9>y inv5tors ( ! p]petual motion ma*9es /ill /& 9 conflict ) accept$ g5]alizations2 b nobody kn[s z yet :at ! f9al =m ( ! !ory ( en]gy w be1 nor c any "o say precisely :at place ! ph5om5a ( ! radioactive bodies w occupy 9 t !ory4 ,! al*emi/s w\ld n be welcome work]s 9 mod]n laboratories2 yet "s sorts ( trans=mation & ( evolution ( ! elem5ts are to-day matt]s :i* !ory c f9d x conv5i5t1 ^u occasion1 to treat z m or less exactly def9able possibilities2 :ile "s newly obs]v$ ph5om5a t5d to indicate1 n inde$ t ! anci5t hopes ( ! al*emi/s 7 well f\nd$1 b t ! ultimate con/itution ( matt] is some?+ m flu5t1 less inv>iant1 ?an ! !oretical or?odoxy ( a rec5t p]iod suppos$4 ,aga91 reg>d+ ! f\ndations ( biology1 a !oretical or?odoxy gr[s less possible1 less def9able1 less conceivable 7ev5 z a hope7 ! m k advances4 ,once 'me*anism' & 'vitalism' 7 mutually contradictory !ories reg>d+ ! ultimate con/itution ( liv+ bodies4 ,n[ !y are obvi\sly becom+ m & m 'po9ts ( view1' div]se b n necess>ily conflict+4 ,s f> z y f9d x conv5i5t to limit y\r /udy ( vital processes to ^? ph5om5a :i* di/+ui% liv+ matt] f all o!r natural objects1 y may assume1 9 ! mod]n 'pragmatic' s5se1 ! attitude ( a 'neo-vitali/4' ,s f>1 h[ev]1 z y are able to lay /ress1 ) good results1 ^u ! _m ways 9 :i* ! life processes c be assimilat$ to ^? /udi$ 9 physics & 9 *emi/ry1 y "w z if y 7 a p>tisan ( 'me*anics4' ,9 any case1 y\r special sci5ce prosp]s by reason ( ! empirical discov]ies t y make4 ,& y\r !ories1 :atev] !y are1 mu/ n run c\nt] to any positive empirical results4 ,b o!rwise1 sci5tific or?odoxy no long] pr$et]m9es :at alone x is respectable = y to ?9k ab\t ! nature ( liv+ sub/ance4 ,?is ga9 9 ! fre$om ( !ory1 com+1 z x does1 side by side ) a con/ant increase ( a positive k ( nature1 l5ds itself to v>i\s int]pretations1 & raises v>i\s obvi\s que/ions4 ,,ii ,"o ( ! mo/ natural ( ^! int]pretations1 "o ( ! mo/ obvi\s ( ^! que/ions1 may be readily /at$4 ,is n ! lesson ( all ^! rec5t discussions simply ?is1 t g5]al !ories are simply va91 t a philosophy ( nature is an idle dream1 & t ! results ( sci5ce are coext5sive ) ! range ( actual empirical obs]vation & ( successful pr$iction8 ,if ?is is inde$ ! lesson1 !n ! decl9e ( !oretical or?odoxy 9 sci5ce is--l ! eclipse ( dogma 9 religion--m]ely a fur!r lesson 9 pure positivism1 ano!r pro( t man does be/ :5 he limits himself to ?9k+ ab\t :at c be f\nd 9 human exp]i5ce1 & 9 try+ to plan :at c be done to make human life m controllable & m reasonable4 ,:at we are free to d z we please--is x any long] a s]i\s bus9ess8 ,:at we are free to ?9k z we please--is x ( any fur!r int]e/ to "o :o is 9 se>* ( tru?8 ,if c]ta9 g5]al !ories are m]e conceptual con/ructions1 :i* to-day are1 & to-morr[ are ca/ into ! ov51 :y dignify !m by ! "n ( philosophy8 ,has sci5ce any place = su* !ories8 ,:y be a 'neo-vitali/1' or an 'evolutioni/1' or an 'atomi/1' or an ',en]getik]'8 ,:y n say1 pla9ly3 ,su* & su* ph5om5a1 ?us & ?us describ$1 h be5 obs]v$2 su* & su* exp]i5ces are to be expect$1 s9ce ! hypo!ses by ! t]ms ( :i* we are requir$ to expect !m h be5 v]ifi$ too (t5 to let u reg>d ! agreem5t ) exp]i5ce z due m]ely to *ance2 s mu* !n ) reasonable assurance we "k2 all else is sil5ce--or else is "s matt] to be te/$ by ano!r exp]im5t8 ,:y n limit our philosophy ( sci5ce /rictly to su* a c\nsel ( resignation8 ,:y n sub/itute1 = ! old sci5tific or?odoxy1 simply a confession ( ignorance1 & a resolution to devote ourselves to ! bus9ess ( enl>g+ ! b\nds ( actual empirical k8 ,su* comm5ts ^u ! situation j *>act]iz$ are frequ5tly made4 ,un=tunately1 !y seem n to cont5t ! v age ^: revolt f ! or?odoxy ( traditional !ory1 ^: unc]ta9ty ab\t all !oretical =mulations1 & ^: va/ weal? ( empirical discov]ies & ( rapidly advanc+ special rese>*es1 w\ld seem mo/ to ju/ify ^! v comm5ts4 ,nev] has !re be5 bett] reason ?an !re is to-day to be cont5t1 if rational man c\ld be cont5t1 ) a pure positivism4 ,! spl5did triumphs ( special rese>* 9 ! mo/ v>i\s fields1 ! con/ant increase 9 our practical control ov] nature--^!1 our positive & gr[+ possessions1 /& 9 gl>+ contra/ to ! failure ( ! sci5tific or?odoxy ( a =m] p]iod to fix ! outl9es ( an ultimate cre$ ab\t ! nature ( ! kn[able univ]se4 ,:y n 'take ! ca% & let ! cr$it g'8 ,:y pursue ! elusive !oretical 'unification' any fur!r1 :5 :at we daily get f our sci5ces is an increas+ weal? ( detail$ in=mation & ( practical guidance8 ,z a fact1 h[ev]1 ! kn[n answ] ( our own age to ^! v obvi\s comm5ts is a con/ant multiplication ( new ef=ts t[>ds l>ge & unify+ !ories4 ,if !oretical or?odoxy is no long] cle>ly def9able1 !oretical con/ruction 0 nev] m rife4 ,! hi/ory ( ! doctr9e ( evolution1 ev5 9 its mo/ rec5t phases1 :5 ! !oretical unc]ta9ties reg>d+ ! 'factors ( evolution' are mo/ insi/$ ^u1 is full ( illu/rations ( ?is rem>kable union ( scepticism 9 critical "w ) c\rage reg>d+ ! use ( ! sci5tific imag9ation4 ,! hi/ory ( ^? controv]sies reg>d+ !oretical physics1 "s ( ^: pr9cipal phases ,m4 ,po9c>1 9 8 book1 sket*es ) ! h& ( ! ma/]1 is ano!r illu/ration ( ! consci\sness ( ! "t4 ,m5 h _! fre$om ( ?\ght 9 ^! regions2 & !y feel ! ne$ ( mak+ con/ant & con/ructive use ( ?is fre$om4 ,& ! m5 :o mo/ feel ?is ne$ are by no means 9 ! majority ( cases pr(essional metaphysicians--or /ud5ts :o1 l myself1 h to view all ^! controv]sies among/ ! sci5tific !oreticians f )\t z le>n]s4 ,^! l>ge !oretical con/ructions are due1 on ! contr>y1 9 a great _m cases to special work]s1 :o h be5 driv5 to ! fre$om ( philosophy by ! oppression ( exp]i5ce1 & :o h le>n$ 9 ! conflict ) special problems ! lesson t !y n[ tea* 9 ! =m ( g5]al ideas reg>d+ ! philosophical aspects ( sci5ce4 ,:y1 !n1 does sci5ce actually ne$ g5]al !ories1 despite ! fact t ^! !ories inevitably alt] & pass away8 ,:at is ! s]vice ( a philosophy ( sci5ce1 :5 x is c]ta9 t ! philosophy ( sci5ce :i* is be/ suit$ to ! ne$s ( "o g5]ation mu/ be sup]s$$ by ! advanc+ insight ( ! next g5]ation8 ,:y mu/ t :i* endlessly gr[s1 namely1 man's k ( ! ph5om5al ord] ( nature1 be con/antly unit$ 9 m5's m9ds ) t :i* is c]ta9 to decay1 namely1 ! !oretical =mulation ( special k 9 m or less completely unifi$ sy/ems ( doctr9e8 ,i und]/& our au?or's volume to be 9 ! ma9 an answ] to ?is "q4 ,to be sure1 ! compact & manifold tea*+s :i* ?is text conta9s relate to a great _m diff]5t special issues4 ,a /ud5t int]e/$ 9 ! problems ( ! philosophy ( ma!matics1 or 9 ! !ory ( probabilities1 or 9 ! nature & (fice ( ma!matical physics1 or 9 /ill o!r problems belong+ to ! wide field "h discuss$1 may f9d :at he wants "h & !re 9 ! text1 ev5 9 case ! g5]al issues :i* give ! volume its unity mean little to him1 or ev5 if he diff]s f ! au?or's views reg>d+ ! pr9cipal issues ( ! book4 ,b 9 ! ma91 ?is volume mu/ be reg>d$ z :at its title indicates--a critique ( ! nature & place ( hypo!sis 9 ! "w ( sci5ce & a /udy ( ! logical relations ( !ory & fact4 ,! result ( ! book is a sub/antial ju/ification ( ! sci5tific utility ( !oretical con/ruction--an ab&onm5t ( dogma1 b a v9dication ( ! rights ( ! con/ructive reason4 ,,iii ,! mo/ notable ( ! results ( our au?or's inve/igation ( ! logic ( sci5tific !ories relates1 z ,i und]/& 8 "w1 to a topic :i* ! pres5t /ate ( logical inve/igation1 j summ>iz$1 makes especially important1 b :i* has ?us f> be5 v inadequately treat$ 9 ! text-books ( inductive logic4 ,! useful hypo!ses ( sci5ce are ( two k9ds3 #a4 ,! hypo!ses :i* are valuable precisely because !y are ei!r v]ifiable or else refutable ?r\gh a def9ite appeal to ! te/s furni%$ by exp]i5ce2 & #b4 ,! hypo!ses :i*1 despite ! fact t exp]i5ce sugge/s !m1 are valuable despite1 or ev5 because1 ( ! fact t exp]i5ce c nei!r confirm nor refute !m4 ,! contra/ betwe5 ^! two k9ds ( hypo!ses is a prom95t topic ( our au?or's discussion4 ,hypo!ses ( ! g5]al type :i* ,i h "h plac$ fir/ 9 ord] are ! ones :i* ! text-books ( inductive logic & ^? summ>ies ( sci5tific me?od :i* are cu/om>y 9 ! c\rse ( ! elem5t>y treatises ^u physical sci5ce are already accu/om$ to recognize & to *>act]ize4 ,! value ( su* hypo!ses is inde$ und\bt$4 ,b hypo!ses ( ! type :i* ,i h "h nam$ 9 ! second place are f> less frequ5tly recogniz$ 9 a p]fectly explicit way z useful aids 9 ! "w ( special sci5ce4 ,"o usually ei!r fails to admit _! pres5ce 9 sci5tific "w1 or else rema9s sil5t z to ! reasons ( _! usefulness4 ,our au?or's treatm5t ( ! "w ( sci5ce is !re=e especially m>k$ by ! fact t he explicitly makes prom95t bo? ! exi/5ce & ! sci5tific importance ( hypo!ses ( ?is second type4 ,!y occupy 9 8 discussion a place some:at analog\s to ea* ( ! two di/9ct positions occupi$ by ! 'categories' & ! '=ms ( s5sibility1' on ! "o h&1 & by ! 'regulative pr9ciples ( ! reason1' on ! o!r h&1 9 ! ,kantian !ory ( our k ( nature4 ,t is1 ^! hypo!ses :i* c nei!r be confirm$ nor refut$ by exp]i5ce appe>1 9 ,m4 ,po9c>'s acc\nt1 p>tly 7l ! conception ( 'cont9u\s quantity'7 z devices ( ! und]/&+ :]eby we give conceptual unity & an invisible connect$ness to c]ta9 types ( ph5om5al facts :i* come to u 9 a discrete =m & 9 a confus$ v>iety2 & p>tly 7l ! l>g] organiz+ concepts ( sci5ce7 z pr9ciples reg>d+ ! /ructure ( ! _w 9 its :ol5ess2 i4 e41 z pr9ciples 9 ! light ( :i* we try to int]pret our exp]i5ce1 s z to give to x a totality & an inclusive unity su* z ,euclidean space1 or su* z ! _w ( ! !ory ( en]gy is conceiv$ to possess4 ,?us view$1 ,m4 ,po9c>'s logical !ory ( ?is second class ( hypo!ses und]takes to accompli%1 ) mod]n means & 9 ! light ( to-day's issues1 a "p ( :at ,kant endeavor$ to accompli% 9 8 !ory ( sci5tific k ) ! limit$ means :i* 7 at 8 disposal4 ,^? aspects ( sci5ce :i* are det]m9$ by ! use ( ! hypo!ses ( ?is second k9d appe> 9 our au?or's acc\nt z con/itut+ an ess5tial human way ( view+ nature1 an int]pretation r ?an a portrayal or a pr$iction ( ! objective facts ( nature1 an adju/m5t ( our conceptions ( ?+s to ! int]nal ne$s ( our intellig5ce1 r ?an a grasp+ ( ?+s z !y are 9 !mselves4 ,to be sure1 ,m4 ,po9c>'s view1 9 ?is portion ( 8 "w1 obvi\sly diff]s1 mean:ile1 f t ( ,kant1 z well z ?is agrees1 9 a measure1 ) ! _s ( ! ,kantian epi/emology4 ,i d n mean !re=e to class our au?or z a ,kantian4 ,= ,kant1 ! int]pretations impos$ by ! '=ms ( s5sibility1' & by ! 'categories ( ! und]/&+1' ^u our doctr9e ( nature are rigidly pr$et]m9$ by ! unalt]able '=m' ( our intellectual p[]s4 ,we 'mu/' ?us view facts1 :atev] ! data ( s5se mu/ be4 ,?is1 ( c\rse1 is n ,m4 ,po9c>'s view4 ,a simil>ly rigid pr$et]m9ation also limits ! ,kantian 'ideas ( ! reason' to a c]ta9 set ( pr9ciples ^: guidance ( ! c\rse ( our !oretical inve/igations is inde$ only 'regulative1' b is 'a priori1' & s un*angeable4 ,= ,m4 ,po9c>1 on ! contr>y1 all ?is adju/m5t ( our int]pretations ( exp]i5ce to ! ne$s ( our intellect is some?+ f> less rigid & unalt]able1 & is con/antly subject to ! sugge/ions ( exp]i5ce4 ,we mu/ inde$ int]pret 9 our own way2 b our way is itself only relatively det]m9ate2 x is ess5tially m or less pla/ic2 o!r int]pretations ( exp]i5ce are conceivable4 ,^? t we use are m]ely ! ones f\nd to be mo/ conv5i5t4 ,b ?is conv5i5ce is n absolute necessity4 ,unv]ifiable & irrefutable hypo!ses 9 sci5ce are inde$1 9 g5]al1 indisp5sable aids to ! organization & to ! guidance ( our int]pretation ( exp]i5ce4 ,b x is exp]i5ce itself :i* po9ts out to u :at l9es ( int]pretation w prove mo/ conv5i5t4 ,in/ead ( ,kant's rigid li/ ( a priori '=ms1' we consequ5tly h 9 ,m4 ,po9c>'s acc\nt a set ( conv5tions1 nei!r :olly subjective & arbitr>y1 nor yet impos$ ^u u unambigu\sly by ! ext]nal compulsion ( exp]i5ce4 ,! organization ( sci5ce1 s f> z ?is organization is due to hypo!ses ( ! k9d "h 9 "q1 ?us resembles t ( a con/itutional gov]nm5t--nei!r absolutely necess>y1 nor yet det]m9$ ap>t f ! w ( ! subjects1 nor yet accid5tal--a free1 yet n a caprici\s e/abli%m5t ( good ord]1 9 con=mity ) empirical ne$s4 ,*>act]i/ic rema9s1 h[ev]1 = our au?or1 z1 9 8 decid$ly contra/+ way1 = ,kant1 ! ?\ght t )\t pr9ciples :i* at e /age transc5d precise confirmation ?r\gh su* exp]i5ce z is !n accessible ! organization ( exp]i5ce is impossible4 ,:e!r "o views ^! pr9ciples z conv5tions or z a priori '=ms1' !y may !re=e be describ$ z hypo!ses1 b z hypo!ses t1 :ile ly+ at ! basis ( our actual physical sci5ces1 at once ref] to exp]i5ce & help u 9 deal+ ) exp]i5ce1 & are yet nei!r confirm$ nor refut$ by ! exp]i5ces :i* we possess or :i* we c hope to atta94 ,?ree special in/ances or classes ( in/ances1 accord+ to our au?or's acc\nt1 may be us$ z illu/rations ( ?is g5]al type ( hypo!ses4 ,!y are3 7#a7 ,! hypo!sis ( ! exi/5ce ( cont9u\s ext5sive quanta 9 nature2 7#b7 ,! pr9ciples ( geometry2 7#c7 ,! pr9ciples ( me*anics & ( ! g5]al !ory ( en]gy4 ,9 case ( ea* ( ^! special types ( hypo!ses we are at fir/ dispos$1 ap>t f reflection1 to say t we f9d ! _w to be ?us or ?us1 s t1 = in/ance1 we c confirm ! !sis accord+ to :i* nature conta9s cont9u\s magnitudes2 or c prove or disprove ! physical tru? ( ! po/ulates ( ,euclidean geometry2 or c confirm by def9ite exp]i5ce ! objective validity ( ! pr9ciples ( me*anics4 ,a clos] exam9ation reveals1 accord+ to our au?or1 ! incorrectness ( all su* op9ions4 ,hypo!ses ( ^! v>i\s special types are ne$$2 & _! usefulness c be empirically %[n4 ,!y are 9 t\* ) exp]i5ce2 & t !y are n m]ely arbitr>y conv5tions is also v]ifiable4 ,!y are n a priori necessities2 & we c easily conceive intellig5t be+s ^: exp]i5ce c\ld be be/ int]pret$ )\t us+ ^! hypo!ses4 ,yet ^! hypo!ses are n subject to direct confirmation or refutation by exp]i5ce4 ,!y /& !n 9 %>p contra/ to ! sci5tific hypo!ses ( ! o!r1 & m frequ5tly recogniz$1 type1 i4 e41 to ! hypo!ses :i* c be te/$ by a def9ite appeal to exp]i5ce4 ,to ^! o!r hypo!ses our au?or atta*es1 ( c\rse1 great importance4 ,8 treatm5t ( !m is full ( a liv+ appreciation ( ! significance ( empirical inve/igation4 ,b ! c5tral problem ( ! logic ( sci5ce ?us becomes ! problem ( ! relation betwe5 ! two fundam5tally di/9ct types ( hypo!ses1 i4 e41 betwe5 ^? :i* c n be v]ifi$ or refut$ ?r\gh exp]i5ce1 & ^? :i* c be empirically te/$4 ,,iv ,! detail$ treatm5t :i* ,m4 ,po9c> gives to ! problem ?us def9$ mu/ be le>n$ f 8 text4 ,x is no "p ( my purpose to exp\nd1 to def5d or to trav]se any ( 8 special conclusions reg>d+ ?is matt]4 ,yet ,i c n avoid obs]v+ t1 :ile ,m4 ,po9c> /rictly conf9es 8 illu/rations & 8 expressions ( op9ion to ^? regions ( sci5ce :]e91 z special inve/igator1 he is himself mo/ at home1 ! issues :i* he ?us raises reg>d+ ! logic ( sci5ce are ( ev5 m critical importance & ( m impressive int]e/ :5 "o applies ,m4 ,po9c>'s me?ods to ! /udy ( ! concepts & presuppositions ( ! organic & ( ! hi/orical & social sci5ces1 ?an :5 "o conf9es "o's att5tion1 z our au?or "h does1 to ! physical sci5ces4 ,x belongs to ! prov9ce ( an introduction l ! pres5t to po9t out1 h[ev] briefly & inadequately1 t ! significance ( our au?or's ideas ext5ds f> beyond ! scope to :i* he *ooses to conf9e _! discussion4 ,! hi/orical sci5ces1 & 9 fact all ^? sci5ces su* z geology1 & su* z ! evolution>y sci5ces 9 g5]al1 und]take !oretical con/ructions :i* relate to pa/ "t4 ,hypo!ses relat+ to ! m or less remote pa/ /&1 h[ev]1 9 a position :i* is v int]e/+ f ! po9t ( view ( ! logic ( sci5ce4 ,directly speak+1 no su* hypo!sis is capable ( confirmation or ( refutation1 because we c n return into ! pa/ to v]ify by our own exp]i5ce :at !n happ5$4 ,yet indirectly1 su* hypo!ses may lead to pr$ictions ( com+ exp]i5ce4 ,^! latt] w be subject to control4 ,?us1 ,s*liemann's confid5ce t ! leg5d ( ,troy _h a def9ite hi/orical f\ndation l$ to pr$ictions reg>d+ :at c]ta9 excavations w\ld reveal4 ,9 a s5se some:at diff]5t f t :i* fill$ ,s*liemann's en?usia/ic m9d1 ^! pr$ictions prov$ v]ifiable4 ,! result has be5 a consid]able *ange 9 ! attitude ( hi/orians t[>d ! leg5d ( ,troy4 ,geological inve/igation leads to pr$ictions reg>d+ ! ord] ( ! /rata or ! c\rse ( m9]al ve9s 9 a di/rict1 reg>d+ ! fossils :i* may be discov]$ 9 giv5 =mations1 & s on4 ,^! hypo!ses are subject to ! control ( exp]i5ce4 ,! v>i\s !ories ( evolution>y doctr9e include _m hypo!ses capable ( confirmation & ( refutation by empirical te/s4 ,yet1 despite all su* empirical control1 x /ill rema9s true t :5ev] a sci5ce is ma9ly conc]n$ ) ! remote pa/1 :e!r ?is sci5ce be ar*eology1 or geology1 or an?ropology1 or ,old ,te/am5t hi/ory1 ! pr9cipal !oretical con/ructions always include features :i* no appeal to pres5t or to accessible future exp]i5ce c ev] def9itely te/4 ,h5ce ! suspicion ) :i* /ud5ts ( exp]im5tal sci5ce (t5 reg>d ! !oretical con/ructions ( _! confrres ( ! sci5ces t deal ) ! pa/4 ,! orig9 ( ! races ( m51 ( man himself1 ( life1 ( species1 ( ! planet2 ! hypo!ses ( an?ropologi/s1 ( ar*eologi/s1 ( /ud5ts ( 'high] criticism'--all ^! are matt]s :i* ! m5 ( ! laboratory (t5 reg>d ) a g5]al incr$ulity z belong+ n at all to ! doma9 ( true sci5ce4 ,yet no "o c d\bt ! importance & ! inevitabl5ess ( endeavor+ to apply sci5tific me?od to ^! regions also4 ,sci5ce ne$s !ories reg>d+ ! pa/ hi/ory ( ! _w4 ,& no "o :o looks clos] into ! me?ods ( ^! sci5ces ( pa/ "t c d\bt t v]ifiable & unv]ifiable hypo!ses are 9 all ^! regions inevitably int]wov52 s t1 :ile exp]i5ce is always ! guide1 ! attitude ( ! inve/igator t[>ds exp]i5ce is det]m9$ by int]e/s :i* h to be p>tially due to :at ,i %\ld call t 'int]nal mean+1' t human int]e/ 9 rational !oretical con/ruction :i* inspires ! sci5tific inquiry2 & ! !oretical con/ructions :i* prevail 9 su* sci5ces are nei!r unbias$ reports ( ! actual con/itution ( an ext]nal reality1 nor yet arbitr>y con/ructions ( fancy4 ,^! con/ructions 9 fact resemble 9 a measure ^? :i* ,m4 ,po9c> 9 ?is book has analyz$ 9 ! case ( geometry4 ,!y are con/ructions mold$1 b n pr$et]m9$ 9 _! details1 by exp]i5ce4 ,we report facts2 we let ! facts speak2 b we1 z we inve/igate1 9 ! popul> phrase1 'talk back' to ! facts4 ,we int]pret z well z report4 ,man is n m]ely made = sci5ce1 b sci5ce is made = man4 ,x expresses 8 deepe/ intellectual ne$s1 z well z 8 c>eful obs]vations4 ,x is an ef=t to br+ int]nal mean+s into h>mony ) ext]nal v]ifications4 ,x attempts !re=e to control1 z well z to submit1 to conceive ) rational unity1 z well z to accept data4 ,its arts are ^? direct$ t[>ds self-possession z well z t[>ds an imitation ( ! out] reality :i* we f9d4 ,x seeks !re=e a discipl9$ fre$om ( ?\ght4 ,! discipl9e is z ess5tial z ! fre$om2 b ! latt] has also its place4 ,! !ories ( sci5ce are human1 z well z objective1 int]nally rational1 z well z 7:5 t is possible7 subject to ext]nal te/s4 ,9 a field v diff]5t f t ( ! hi/orical sci5ces1 namely1 9 a sci5ce ( obs]vation & ( exp]im5t1 :i* is at ! same "t an organic sci5ce1 ,i h be5 l$ 9 ! c\rse ( "s /udy ( ! hi/ory ( c]ta9 rese>*es to notice ! exi/5ce ( a !oretical conception :i* has prov$ extremely fruitful 9 guid+ rese>*1 b :i* app>5tly resembles 9 a measure ! type ( hypo!ses ( :i* ,m4 ,po9c> speaks :5 he *>act]izes ! pr9ciples ( me*anics & ( ! !ory ( en]gy4 ,i v5ture to call att5tion "h to ?is conception1 :i* seems to me to illu/rate ,m4 ,po9c>'s view ( ! functions ( hypo!sis 9 sci5tific "w4 ,! mod]n sci5ce ( pa?ology is usually reg>d$ z dat+ f ! e>li] rese>*es ( ,vir*[1 ^: ',cellul> ,pa?ology' 0 ! outcome ( a v c>eful & elaborate induction4 ,vir*[1 himself1 felt a /rong av]sion to m]e speculation4 ,he endeavor$ to keep close to obs]vation1 & to relieve m$ical sci5ce f ! control ( fanta/ic !ories1 su* z ^? ( ! ,naturphilosoph5 _h be54 ,yet ,vir*['s rese>*es 71 z e>ly z #ahdg1 or /ill e>li]1 already "u ! guidance ( a !oretical presupposition :i* he himself /ates z foll[s3 ,we h le>n$ to recognize1 he says1 t diseases are n autonom\s organisms1 t !y are no entities t h ent]$ into ! body1 t !y are no p>asites :i* take root 9 ! body1 b t !y m]ely %[ u ! c\rse ( ! vital processes "u alt]$ conditions 7'dasz sie nur ,ablauf d] ,leb5s]s*e9ung5 unt] v]nd]t5 ,b$+ung5 d>/ell5'74 ,! enorm\s importance ( ?is !oretical presupposition = all ! e>ly successes ( mod]n pa?ological inve/igation is g5]ally recogniz$ by ! exp]ts4 ,i d n d\bt ?is op9ion4 ,x appe>s to be a commonplace ( ! hi/ory ( ?is sci5ce4 ,b 9 ,vir*['s lat] ye>s ?is v presupposition seem$ to "s ( 8 contempor>ies to be call$ 9 "q by ! successes ( rec5t bact]iology4 ,! "q arose :e!r ! !oretical f\ndations ( ,vir*['s pa?ology _h n be5 set aside4 ,& 9 fact ! !ory ( ! p>asitical orig9 ( a va/ numb] ( diseas$ conditions has inde$ come ^u an empirical basis to be g5]ally recogniz$4 ,yet to ! end ( 8 own c>e] ,vir*[ /\tly ma9ta9$ t 9 all its ess5tial significance 8 own fundam5tal pr9ciple rema9$ q unt\*$ by ! new] discov]ies4 ,&1 z a fact1 ?is view c\ld inde$ be ma9ta9$4 ,= if diseases prov$ to be ! consequ5ces ( ! pres5ce ( p>asites1 ! diseases !mselves1 s f> z !y belong$ to ! diseas$ organism1 7 /ill n ! p>asites1 b 71 z be=e1 ! reaction ( ! organism to ! v]nd]te ,b$+ung5 :i* ! pres5ce ( ! p>asites entail$4 ,s ,vir*[ c\ld well insi/4 ,& if ! fam\s pr9ciple 9 "q is only /at$ ) suffici5t g5]ality1 x am\nts simply to say+ t if a disease involves a *ange 9 an organism1 & if ?is *ange is subject to law at all1 !n ! nature ( ! organism & ! reaction ( ! organism to :atev] x is :i* causes ! disease mu/ be und]/ood 9 case ! disease is to be und]/ood4 ,= ?is v reason1 h[ev]1 ,vir*['s !oretical pr9ciple 9 its mo/ g5]al =m c\ld be nei!r confirm$ nor refut$ by exp]i5ce4 ,x w\ld rema9 empirically irrefutable1 s f> z ,i c see1 ev5 if we %\ld le>n t ! devil 0 ! true cause ( all diseases4 ,= ! devil himself w\ld !n simply pr$et]m9e ! v]nd]te ,b$+ung5 to :i* ! diseas$ organism w\ld be react+4 ,let bullets or bact]ia1 poisons or compress$ air1 or ! devil be ! ,b$+ung5 to :i* a diseas$ organism reacts1 ! po/ulate t ,vir*[ /ates 9 ! passage j quot$ w rema9 irrefutable1 if only ?is po/ulate be int]pret$ to meet ! case4 ,= ! pr9ciple 9 "q m]ely says t :atev] entity x may be1 bullet1 or poison1 or devil1 t affects ! organism1 ! disease is n t entity1 b is ! result+ alt]ation 9 ! process ( ! organism4 ,i insi/1 !n1 t ?is pr9ciple ( ,vir*['s is no trial supposition1 no sci5tific hypo!sis 9 ! n>r[] s5se--capable ( be+ submitt$ to precise empirical te/s4 ,x is1 on ! contr>y1 a v preci\s lead+ idea1 a !oretical int]pretation ( ph5om5a1 9 ! light ( :i* obs]vations are to be made--'a regulative pr9ciple' ( rese>*4 ,x is equival5t to a resolution to se>* = ^? detail$ connections :i* l9k ! processes ( disease to ! normal process ( ! organism4 ,su* a se>* und]takes to f9d ! true unity1 :atev] t may prove to be1 :]e9 ! pa?ological & ! normal processes are l9k$4 ,n[ )\t "s su* lead+ idea1 ! cellul> pa?ology itself c\ld nev] h be5 rea*$2 because ! empirical facts 9 "q w\ld nev] h be5 obs]v$4 ,h5ce ?is pr9ciple ( ,vir*['s 0 indisp5sable to ! gr[? ( 8 sci5ce4 ,yet x 0 n a v]ifiable & n a refutable hypo!sis4 ,"o value ( unv]ifiable & irrefutable hypo!ses ( ?is type lies1 !n1 9 ! sort ( empirical inquiries :i* !y initiate1 inspire1 organize & guide4 ,9 ^! inquiries hypo!ses 9 ! n>r[] s5se1 t is1 trial propositions :i* are to be submitt$ to def9ite empirical control1 are inde$ ev]y:]e pres5t4 ,& ! use ( ! o!r sort ( pr9ciples lies :olly 9 _! application to exp]i5ce4 ,yet )\t :at ,i h j propos$ to call ! 'lead+ ideas' ( a sci5ce1 t is1 its pr9ciples ( an unv]ifiable & irrefutable "*1 sugge/$1 b n to be f9ally te/$1 by exp]i5ce1 ! hypo!ses 9 ! n>r[] s5se w\ld lack t guidance :i*1 z ,m4 ,po9c> has %[n1 ! l>g] ideas ( sci5ce give to empirical inve/igation4 ,v ,i h dwelt1 no d\bt1 at too great l5g? ^u "o aspect only ( our au?or's v>i$ & well-balanc$ discussion ( ! problems & concepts ( sci5tific !ory4 ,( ! hypo!ses 9 ! n>r[] s5se & ( ! value ( direct empirical control1 he has also spok5 ) ! au?ority & ! orig9ality :i* belong to 8 position4 ,& 9 deal+ ) ! f\ndations ( ma!matics he has rais$ "o or two que/ions ( great philosophical import into :i* ,i h no "t1 ev5 if ,i _h ! "r1 to ent] "h4 ,9 p>ticul>1 9 speak+ ( ! ess5ce ( ma!matical reason+1 & ( ! difficult problem ( :at makes possible novel results 9 ! field ( pure ma!matics1 ,m4 ,po9c> def5ds a !sis reg>d+ ! (fice ( 'demon/ration by recurr5ce'--a !sis :i* is inde$ disputable1 :i* has be5 disput$ & :i* ,i myself %\ld be dispos$1 s f> z ,i at pres5t und]/& ! matt]1 to modify 9 "s respects1 ev5 9 accept+ ! _s ( our au?or's ass]tion4 ,yet !re c be no d\bt ( ! importance ( ?is !sis1 & ( ! fact t x def9es a *>act]i/ic t is inde$ fundam5tal 9 a wide range ( ma!matical rese>*4 ,! philosophical problems t lie at ! basis ( recurr5t pro(s & processes are1 z ,i h else:]e argu$1 ( ! mo/ fundam5tal importance4 ,^!1 !n1 are a few h9ts relat+ to ! significance ( our au?or's discussion1 & a few reasons = hop+ t our own /ud5ts w pr(it by ! read+ ( ! book z ^? ( o!r nations h already done4 ,( ! p]son & ( ! life-"w ( our au?or a few words are "h1 9 conclusion1 /ill 9 place1 address$1 n to ! /ud5ts ( 8 own sci5ce1 to :om 8 position is well kn[n1 b to ! g5]al read] :o may seek guidance 9 ^! pages4 ,jules ,h5ri ,po9c> 0 born at ,nancy1 9 #ahed1 ! son ( a pr(essor 9 ! ,faculty ( ,m$ic9e at ,nancy4 ,he /udi$ at ! cole ,polyte*nique & at ! cole des ,m9es1 & lat] receiv$ 8 doctorate 9 ma!matics 9 #ahgi4 ,9 #ahhc he began c\rses ( in/ruction 9 ma!matics at ! cole ,polyte*nique2 9 #ahhf receiv$ a pr(essor%ip ( ma!matical physics 9 ! ,faculty ( ,sci5ces at ,p>is2 !n became memb] ( ! ,academy ( ,sci5ces at ,p>is1 9 #ahhg1 & devot$ 8 life to in/ruction & inve/igation 9 ! regions ( pure ma!matics1 ( ma!matical physics & ( cele/ial me*anics4 ,8 li/ ( publi%$ treatises relat+ to v>i\s bran*es ( 8 *os5 sci5ces is long2 & 8 orig9al memoirs h includ$ sev]al mom5t\s inve/igations1 :i* h gone f> to trans=m m ?an "o bran* ( rese>*4 ,8 pres5ce at ! ,int]national ,congress ( ,arts & ,sci5ce 9 ,/4 ,l\is 0 "o ( ! mo/ noticeable features ( t rem>kable ga!r+ ( di/+ui%$ =eign gue/s4 ,9 ,po9c> ! read] meets1 !n1 n "o :o is prim>ily a speculative /ud5t ( g5]al problems = _! own sake1 b an orig9al inve/igator ( ! highe/ rank 9 sev]al di/9ct1 al?\gh int]relat$1 bran*es ( mod]n rese>*4 ,! !ory ( functions--a highly recondite region ( pure ma!matics--owes to him advances ( ! fir/ importance1 = in/ance1 ! def9ition ( a new type ( functions4 ,! 'problem ( ! ?ree bodies1' a fam\s & fundam5tal problem ( cele/ial me*anics1 has receiv$ f 8 /udies a treatm5t ^: significance has be5 recogniz$ by ! highe/ au?orities4 ,8 int]national reputation has be5 confirm$ by ! conf]r+ ( m ?an "o important prize = 8 rese>*es4 ,8 memb]%ip 9 ! mo/ em95t le>n$ societies ( v>i\s nations is widely ext5d$2 8 volumes be>+ ^u v>i\s bran*es ( ma!matics & ( ma!matical physics are us$ by special /ud5ts 9 all p>ts ( ! le>n$ _w2 9 brief1 he is1 z geomet]1 z analy/ & z a !oretical physici/1 a lead] ( 8 age4 ,mean:ile1 z contributor to ! philosophical discussion ( ! bases & me?ods ( sci5ce1 ,m4 ,po9c> has long be5 active4 ,:51 9 #ahic1 ! admirable ,revue de ,mtaphysique et de ,morale began to appe>1 ,m4 ,po9c> 0 soon f\nd among/ ! mo/ satisfactory ( ! contributors to ! "w ( t j\rnal1 ^: (fice x has especially be5 to br+ philosophy & ! v>i\s special sci5ces 7bo? natural & moral7 into a clos] mutual und]/&+4 ,! discussions br\ght toge!r 9 ! pres5t volume are 9 l>ge "p ! outcome ( ,m4 ,po9c>'s contributions to ! ,revue de ,mtaphysique et de ,morale4 ,! read] ( ,m4 ,po9c>'s book is 9 pres5ce1 !n1 ( a great special inve/igator :o is also a philosoph]4 ,,sci5ce ,,& ,,hypo!sis ,,introduction ,= a sup]ficial obs]v]1 sci5tific tru? is beyond ! possibility ( d\bt2 ! logic ( sci5ce is infallible1 & if ! sci5ti/s are sometimes mi/ak51 ?is is only f _! mi/ak+ its rules4 ,! ma!matical v]ities fl[ f a small numb] ( self-evid5t propositions by a *a9 ( impeccable reason+s2 !y impose !mselves n only on u1 b on nature itself4 ,!y fett]1 s to speak1 ! ,creator & only p]mit him to *oose betwe5 "s relatively few solutions4 ,a few exp]im5ts !n w suffice to let u "k :at *oice he has made4 ,f ea* exp]im5t a cr[d ( consequ5ces w foll[ by a s]ies ( ma!matical d$uctions1 & ?us ea* exp]im5t w make kn[n to u a corn] ( ! univ]se4 ,behold :at is = _m p 9 ! _w1 = s*ol>s gett+ _! fir/ notions ( physics1 ! orig9 ( sci5tific c]titude4 ,?is is :at !y suppose to be ! rle ( exp]im5tation & ma!matics4 ,?is same conception1 a hundr$ ye>s ago1 0 held by _m savants :o dream$ ( con/ruct+ ! _w ) z little z possible tak5 f exp]im5t4 ,on a little m reflection x 0 p]ceiv$ h[ great a place hypo!sis occupies2 t ! ma!matician c n d )\t x1 /ill less ! exp]im5t]4 ,& !n x 0 d\bt$ if all ^! con/ructions 7 really solid1 & believ$ t a brea? w\ld ov]?r[ !m4 ,to be skeptical 9 ?is fa%ion is /ill to be sup]ficial4 ,to d\bt ev]y?+ & to believe ev]y?+ are two equally conv5i5t solutions2 ea* saves u f ?9k+4 ,in/ead ( pron\nc+ a summ>y condemnation1 we ought !re=e to exam9e ) c>e ! rle ( hypo!sis2 we %all !n recognize1 n only t x is necess>y1 b t usually x is legitimate4 ,we %all also see t !re are sev]al sorts ( hypo!ses2 t "s are v]ifiable1 & once confirm$ by exp]im5t become fruitful tru?s2 t o!rs1 p[]less to lead u a/ray1 may be useful to u 9 fix+ our ideas2 t o!rs1 f9ally1 are hypo!ses only 9 appe>ance & are r$ucible to disguis$ def9itions or conv5tions4 ,^! la/ are met ) above all 9 ma!matics & ! relat$ sci5ces4 ,!nce precisely x is t ^! sci5ces get _! rigor2 ^! conv5tions are ! "w ( ! free activity ( our m9d1 :i*1 9 ?is doma91 recognizes no ob/acle4 ,"h our m9d c affirm1 s9ce x decrees2 b let u und]/& t :ile ^! decrees are impos$ ^u our sci5ce1 :i*1 )\t !m1 w\ld be impossible1 !y are n impos$ ^u nature4 ,are !y !n arbitr>y8 ,no1 else 7 !y /]ile4 ,exp]im5t leaves u our fre$om ( *oice1 b x guides u by aid+ u to disc]n ! easie/ way4 ,our decrees are !re=e l ^? ( a pr9ce1 absolute b wise1 :o consults 8 c\ncil ( /ate4 ,"s p h be5 /ruck by ?is "* ( free conv5tion recognizable 9 c]ta9 fundam5tal pr9ciples ( ! sci5ces4 ,!y h wi%$ to g5]alize beyond measure1 &1 at ! same "t1 !y h =gott5 t lib]ty is n lic5se4 ,?us !y h rea*$ :at is call$ nom9alism1 & h ask$ !mselves if ! savant is n ! dupe ( 8 own def9itions & if ! _w he ?9ks he discov]s is n simply creat$ by 8 own caprice4#a ,"u ^! conditions sci5ce w\ld be c]ta91 b depriv$ ( significance4 #a ,see ,le ,roy1 ',sci5ce et ,philosophie1' ,revue de ,mtaphysique et de ,morale1 #aija4 ,if ?is 7 s1 sci5ce w\ld be p[]less4 ,n[ e day we see x "w "u our v eyes4 ,t c\ld n be if x taught u no?+ ( reality4 ,/ill1 ! ?+s !mselves are n :at x c rea*1 z ! nave dogmati/s ?9k1 b only ! relations betwe5 ?+s4 ,outside ( ^! relations !re is no kn[able reality4 ,su* is ! conclusion to :i* we %all come1 b = t we mu/ review ! s]ies ( sci5ces f ari?metic & geometry to me*anics & exp]im5tal physics4 ,:at is ! nature ( ma!matical reason+8 ,is is really d$uctive1 z is commonly suppos$8 ,a deep] analysis %[s u t x is n1 t x p>takes 9 a c]ta9 measure ( ! nature ( inductive reason+1 & j because ( ?is is x s fruitful4 ,none ! less does x reta9 its "* ( rigor absolute2 ?is is ! fir/ ?+ t _h to be %[n4 ,kn[+ bett] n[ "o ( ! in/rum5ts :i* ma!matics puts into ! h&s ( ! inve/igator1 we _h to analyze ano!r fundam5tal notion1 t ( ma!matical magnitude4 ,d we f9d x 9 nature1 or d we ourselves introduce x !re8 ,&1 9 ?is latt] case1 d we n risk m>r+ ev]y?+8 ,comp>+ ! r\gh data ( our s5ses ) t extremely complex & subtile concept :i* ma!maticians call magnitude1 we are =c$ to recognize a diff]5ce2 ?is frame into :i* we wi% to =ce ev]y?+ is ( our own con/ruction2 b we h n made x at r&om4 ,we h made x1 s to speak1 by measure & !re=e we c make ! facts fit into x )\t *ang+ :at is ess5tial 9 !m4 ,ano!r frame :i* we impose on ! _w is space4 ,:5ce come ! fir/ pr9ciples ( geometry8 ,are !y impos$ on u by logic8 ,loba*evski has prov$ n1 by creat+ non-,euclidean geometry4 ,is space reveal$ to u by our s5ses8 ,/ill no1 = ! space our s5ses c\ld %[ u diff]s absolutely f t ( ! geomet]4 ,is exp]i5ce ! s\rce ( geometry8 ,a deep] discussion w %[ u x is n4 ,we !re=e conclude t ! fir/ pr9ciples ( geometry are only conv5tions2 b ^! conv5tions are n arbitr>y & if transport$ into ano!r _w 7t ,i call ! non-,euclidean _w & seek to imag9e71 !n we %\ld h be5 l$ to adopt o!rs4 ,9 me*anics we %\ld be l$ to analog\s conclusions1 & %\ld see t ! pr9ciples ( ?is sci5ce1 ?\gh m directly bas$ on exp]im5t1 /ill p>take ( ! conv5tional "* ( ! geometric po/ulates4 ,?us f> nom9alism triumphs2 b n[ we arrive at ! physical sci5ces1 prop]ly s call$4 ,"h ! sc5e *anges2 we meet ano!r sort ( hypo!ses & we see _! f]tility4 ,)\t d\bt1 at fir/ blu%1 ! !ories seem to u fragile1 & ! hi/ory ( sci5ce proves to u h[ ephem]al !y are2 yet !y d n entirely p]i%1 & ( ea* ( !m some?+ rema9s4 ,x is ?is some?+ we mu/ seek to dis5tangle1 s9ce !re & !re alone is ! v]itable reality4 ,! me?od ( ! physical sci5ces re/s on ! induction :i* makes u expect ! repetition ( a ph5om5on :5 ! circum/ances "u :i* x fir/ happ5$ are reproduc$4 ,if all ^! circum/ances c\ld be reproduc$ at once1 ?is pr9ciple c\ld be appli$ )\t fe>2 b t w nev] happ52 "s ( ^! circum/ances w always be lack+4 ,are we absolutely sure !y are unimportant8 ,evid5tly n4 ,t may be probable1 x c n be rigor\sly c]ta94 ,h5ce ! important rle ! notion ( probability plays 9 ! physical sci5ces4 ,! calculus ( probabilities is !re=e n m]ely a recreation or a guide to play]s ( bacc>at1 & we mu/ seek to g deep] ) its f\ndations4 ,"u ?is head ,i h be5 able to give only v incomplete results1 s /rongly does ?is vague in/9ct :i* lets u disc]n probability defy analysis4 ,aft] a /udy ( ! conditions "u :i* ! physici/ works1 ,i h ?\ght prop] to %[ him at "w4 ,= t ,i h tak5 in/ances f ! hi/ory ( optics & ( electricity4 ,we %all see :5ce h sprung ! ideas ( ,fresnel1 ( ,maxwell1 & :at unconsci\s hypo!ses 7 made by ,ampre & ! o!r f\nd]s ( electrodynamics4 ,,"p ,i ,,numb] ,,& ,,magnitude ,,*apt] ,i ,,on ,,! ,,nature ,,( ,,ma!matical ,,reason+ ,i ,! v possibility ( ! sci5ce ( ma!matics seems an insoluble contradiction4 ,if ?is sci5ce is d$uctive only 9 appe>ance1 :5ce does x d]ive t p]fect rigor no "o dreams ( d\bt+8 ,if1 on ! contr>y1 all ! propositions x enunciates c be d$uc$ "o f ano!r by ! rules ( =mal logic1 :y is n ma!matics r$uc$ to an imm5se tautology8 ,! syllogism c tea* u no?+ ess5tially new1 &1 if ev]y?+ is to spr+ f ! pr9ciple ( id5tity1 ev]y?+ %\ld be capable ( be+ r$uc$ to x4 ,%all we !n admit t ! enunciations ( all ^? !orems :i* fill s _m volumes are no?+ b devi\s ways ( say+ ,a is ,a8 ,)\t d\bt1 we c g back to ! axioms1 :i* are at ! s\rce ( all ^! reason+s4 ,if we decide t ^! c n be r$uc$ to ! pr9ciple ( contradiction1 if /ill less we see 9 !m exp]im5tal facts :i* c\ld n p>take ( ma!matical necessity1 we h yet ! res\rce ( class+ !m among syn!tic a priori judgm5ts4 ,?is is n to solve ! difficulty1 b only to baptize x2 & ev5 if ! nature ( syn!tic judgm5ts 7 = u no my/]y1 ! contradiction w\ld n h disappe>$1 x w\ld only h mov$ back2 syllogi/ic reason+ rema9s incapable ( add+ any?+ to ! data giv5 x3 ^! data r$uce !mselves to a few axioms1 & we %\ld f9d no?+ else 9 ! conclusions4 ,no !orem c\ld be new if no new axiom int]v5$ 9 its demon/ration2 reason+ c\ld give u only ! imm$iately evid5t v]ities borr[$ f direct intuition2 x w\ld be only an int]m$i>y p>asite1 & !re=e %\ld we n h good reason to ask :e!r ! :ole syllogi/ic app>atus did n s]ve solely to disguise our borr[+8 ,! contradiction w /rike u ! m if we op5 any book on ma!matics2 on e page ! au?or w ann\nce 8 int5tion ( g5]aliz+ "s proposition already kn[n4 ,does ! ma!matical me?od proce$ f ! p>ticul> to ! g5]al1 &1 if s1 h[ !n c x be call$ d$uctive8 ,if f9ally ! sci5ce ( numb] 7 purely analytic1 or c\ld be analytically d]iv$ f a small numb] ( syn!tic judgm5ts1 x seems t a m9d suffici5tly p[]ful c\ld at a glance p]ceive all its tru?s2 nay m1 we might ev5 hope t "s day "o w\ld inv5t to express !m a language suffici5tly simple to h !m appe> self-evid5t to an ord9>y intellig5ce4 ,if we refuse to admit ^! consequ5ces1 x mu/ be conc$$ t ma!matical reason+ has ( itself a sort ( creative virtue & consequ5tly diff]s f ! syllogism4 ,! diff]5ce mu/ ev5 be pr(\nd4 ,we %all n1 = example1 f9d ! key to ! my/]y 9 ! frequ5t use ( t rule accord+ to :i* "o & ! same uni=m op]ation appli$ to two equal numb]s w give id5tical results4 ,all ^! modes ( reason+1 :e!r or n !y be r$ucible to ! syllogism prop]ly s call$1 reta9 ! analytic "*1 & j because ( t are p[]less4 ,,ii ,! discussion is old2 ,leibnitz tri$ to prove #b & #b make #d2 let u look a mom5t at 8 demon/ration4 ,i w suppose ! numb] #a def9$ & also ! op]ation x #a :i* consi/s 9 add+ unity to a giv5 numb] x4 ,^! def9itions1 :atev] !y be1 d n ent] into ! c\rse ( ! reason+4 ,i def9e !n ! numb]s #b1 #c & #d by ! equalities 7#a7 #a #a #b2 7#b7 #b #a #c2 7#c7 #c #a #d4 ,9 ! same way1 ,i def9e ! op]ation x #b by ! relation3 7#d7 x #b 7x #a7 #a4 ,t presuppos$1 we h #b #a #a #c #a 7,def9ition #b71 #c #a #d 7,def9ition #c71 #b #b 7#b #a7 #a 7,def9ition #d71 :5ce #b #b #d ,q4,e4,d4 ,x c n be d5i$ t ?is reason+ is purely analytic4 ,b ask any ma!matician3 ',t is n a demon/ration prop]ly s call$1' he w say to y3 't is a v]ification4' ,we h conf9$ ourselves to comp>+ two purely conv5tional def9itions & h asc]ta9$ _! id5tity2 we h le>n$ no?+ new4 ,v]ification diff]s f true demon/ration precisely because x is purely analytic & because x is /]ile4 ,x is /]ile because ! conclusion is no?+ b ! premises translat$ into ano!r language4 ,on ! contr>y1 true demon/ration is fruitful because ! conclusion "h is 9 a s5se m g5]al ?an ! premises4 ,! equality #b #b #d is ?us susceptible ( a v]ification only because x is p>ticul>4 ,e p>ticul> enunciation 9 ma!matics c always be v]ifi$ 9 ?is same way4 ,b if ma!matics c\ld be r$uc$ to a s]ies ( su* v]ifications1 x w\ld n be a sci5ce4 ,s a *ess-play]1 = example1 does n create a sci5ce 9 w9n+ a game4 ,!re is no sci5ce ap>t f ! g5]al4 ,x may ev5 be said ! v object ( ! exact sci5ces is to sp>e u ^! direct v]ifications4 ,,iii ,let u1 !re=e1 see ! geomet] at "w & seek to cat* 8 process4 ,! task is n )\t difficulty2 x does n suffice to op5 a "w at r&om & analyze any demon/ration 9 x4 ,we mu/ fir/ exclude geometry1 :]e ! "q is complicat$ by ardu\s problems relative to ! rle ( ! po/ulates1 to ! nature & ! orig9 ( ! notion ( space4 ,= analog\s reasons we c n turn to ! inf9itesimal analysis4 ,we mu/ seek ma!matical ?\ght :]e x has rema9$ pure1 t is1 9 ari?metic4 ,a *oice /ill is necess>y2 9 ! high] p>ts ( ! !ory ( numb]s1 ! primitive ma!matical notions h already und]gone an elaboration s pr(\nd t x becomes difficult to analyze !m4 ,x is1 !re=e1 at ! beg9n+ ( ari?metic t we mu/ expect to f9d ! explanation we seek1 b x happ5s t precisely 9 ! demon/ration ( ! mo/ elem5t>y !orems ! au?ors ( ! classic treatises h %[n ! lea/ precision & rigor4 ,we mu/ n impute ?is to !m z a crime2 !y h yield$ to a necessity2 beg9n]s are n prep>$ = real ma!matical rigor2 !y w\ld see 9 x only useless & irksome subtleties2 x w\ld be a wa/e ( "t to try prematurely to make !m m exact+2 !y mu/ pass ov] rapidly1 b )\t skipp+ /ations1 ! road trav]s$ sl[ly by ! f\nd]s ( ! sci5ce4 ,:y is s long a prep>ation necess>y to become habituat$ to ?is p]fect rigor1 :i*1 x w\ld seem1 %\ld naturally impress itself ^u all good m9ds8 ,?is is a logical & psy*ological problem well wor?y ( /udy4 ,b we %all n take x up2 x is =eign to our purpose2 all ,i wi% to insi/ on is t1 n to fail ( our purpose1 we mu/ reca/ ! demon/rations ( ! mo/ elem5t>y !orems & give !m1 n ! crude =m 9 :i* !y are left1 s z n to h>ass beg9n]s1 b ! =m t w satisfy a skill$ geomet]4 ,,def9ition ,,( ,,addition4--,i suppose already def9$ ! op]ation x #a1 :i* consi/s 9 add+ ! numb] #a to a giv5 numb] x4 ,?is def9ition1 :atev] x be1 does n ent] into our subsequ5t reason+4 ,we n[ h to def9e ! op]ation x a1 :i* consi/s 9 add+ ! numb] a to a giv5 numb] x4 ,suppos+ we h def9$ ! op]ation x 7a - #a71 ! op]ation x a w be def9$ by ! equality 7#a7 x a x 7a - #a7 #a4 ,we %all "k !n :at x a is :5 we "k :at x 7a - #a7 is1 & z ,i h suppos$ t to />t ) we knew :at x #a is1 we c def9e successively & 'by recurr5ce' ! op]ations x #b1 x #c1 etc4 ,?is def9ition des]ves a mom5t's att5tion2 x is ( a p>ticul> nature :i* already di/+ui%es x f ! purely logical def9ition2 ! equality 7#a7 conta9s an inf9ity ( di/9ct def9itions1 ea* hav+ a mean+ only :5 "o kn[s ! prec$+4 ,,prop]ties ,,( ,,addition4--,associativity4--,i say t a 7b c7 7a b7 c4 ,9 fact ! !orem is true = c #a2 x is !n writt5 a 7b #a7 7a b7 #a1 :i*1 ap>t f ! diff]5ce ( notation1 is no?+ b ! equality 7#a71 by :i* ,i h j def9$ addition4 ,suppos+ ! !orem true = c gamma1 ,i say x w be true = c gamma #a4 ,9 fact1 suppos+ 7a b7 gamma a 7b gamma71 x foll[s t 7a b7 gamma #a a 7b gamma7 #a or by def9ition 7#a7 7a b7 7gamma #a7 a 7b gamma #a7 a b 7gamma #a71 :i* %[s1 by a s]ies ( purely analytic d$uctions1 t ! !orem is true = gamma #a4 ,be+ true = c #a1 we ?us see successively t s x is = c #b1 = c #c1 etc4 ,commutativity4--#a ,i say t a #a #a a4 ,! !orem is evid5tly true = a #a2 we c v]ify by purely analytic reason+ t if x is true = a gamma x w be true = a gamma #a2 = !n 7gamma #a7 #a 7#a gamma7 #a #a 7gamma #a72 n[ x is true = a #a1 !re=e x w be true = a #b1 = a #c1 etc41 :i* is express$ by say+ t ! enunciat$ proposition is demon/rat$ by recurr5ce4 #b ,i say t a b b a4 ,! !orem has j be5 demon/rat$ = b #a2 x c be v]ifi$ analytically t if x is true = b beta1 x w be true = b beta #a4 ,! proposition is !re=e e/abli%$ by recurr5ce4 ,,def9ition ,,( ,,multiplication4--,we %all def9e multiplication by ! equalities4 7#a7 a #a a4 7#b7 a b a 7b - #a7 a4 ,l equality 7#a71 equality 7#b7 conta9s an inf9ity ( def9itions2 hav+ def9$ a #a1 x enables u to def9e successively3 a #b1 a #c1 etc4 ,,prop]ties ,,( ,,multiplication4--,di/ributivity4--,i say t 7a b7 c 7a c7 7b c74 ,we v]ify analytically t ! equality is true = c #a2 !n t if ! !orem is true = c gamma1 x w be true = c gamma #a4 ,! proposition is1 !re=e1 demon/rat$ by recurr5ce4 ,commutativity4--#a ,i say t a #a #a a4 ,! !orem is evid5t = a #a4 ,we v]ify analytically t if x is true = a alpha1 x w be true = a alpha #a4 #b ,i say t a b b a4 ,! !orem has j be5 prov5 = b #a4 ,we c\ld v]ify analytically t if x is true = b beta1 x w be true = b beta #a4 ,,iv ,"h ,i /op ?is monoton\s s]ies ( reason+s4 ,b ?is v monotony has ! bett] br\ght out ! proc$ure :i* is uni=m & is met aga9 at ea* /ep4 ,?is proc$ure is ! demon/ration by recurr5ce4 ,we fir/ e/abli% a !orem = n #a2 !n we %[ t if x is true ( n - #a1 x is true ( n1 & !nce conclude t x is true = all ! :ole numb]s4 ,we h j se5 h[ x may be us$ to demon/rate ! rules ( addition & multiplication1 t is to say1 ! rules ( ! algebraic calculus2 ?is calculus is an in/rum5t ( trans=mation1 :i* l5ds itself to _m m diff]+ comb9ations ?an does ! simple syllogism2 b x is /ill an in/rum5t purely analytic1 & incapable ( tea*+ u any?+ new4 ,if ma!matics _h no o!r in/rum5t1 x w\ld !re=e be =?) arre/$ 9 its developm5t2 b x has rec\rse anew to ! same proc$ure1 t is1 to reason+ by recurr5ce1 & x is able to cont9ue its =w>d m>*4 ,if we look closely1 at e /ep we meet aga9 ?is mode ( reason+1 ei!r 9 ! simple =m we h j giv5 x1 or "u a =m m or less modifi$4 ,"h !n we h ! ma!matical reason+ p> excell5ce1 & we mu/ exam9e x m closely4 ,v ,! ess5tial *>act]i/ic ( reason+ by recurr5ce is t x conta9s1 cond5s$1 s to speak1 9 a s+le =mula1 an inf9ity ( syllogisms4 ,t ?is may ! bett] be se51 ,i w /ate "o aft] ano!r ^! syllogisms :i* are1 if y w all[ me ! expression1 arrang$ 9 'cascade4' ,^! are ( c\rse hypo!tical syllogisms4 ,! !orem is true ( ! numb] #a4 ,n[1 if x is true ( #a1 x is true ( #b4 ,!re=e x is true ( #b4 ,n[1 if x is true ( #b1 x is true ( #c4 ,!re=e x is true ( #c1 & s on4 ,we see t ! conclusion ( ea* syllogism s]ves z m9or to ! foll[+4 ,fur!rmore ! majors ( all our syllogisms c be r$uc$ to a s+le =mula4 ,if ! !orem is true ( n - #a1 s x is ( n4 ,we see1 !n1 t 9 reason+ by recurr5ce we conf9e ourselves to /at+ ! m9or ( ! fir/ syllogism1 & ! g5]al =mula :i* conta9s z p>ticul> cases all ! majors4 ,?is nev]-end+ s]ies ( syllogisms is ?us r$uc$ to a phrase ( a few l9es4 ,x is n[ easy to compreh5d :y e p>ticul> consequ5ce ( a !orem c1 z ,i h expla9$ above1 be v]ifi$ by purely analytic proc$ures4 ,if in/ead ( %[+ t our !orem is true ( all numb]s1 we only wi% to %[ x true ( ! numb] #f1 = example1 x w suffice = u to e/abli% ! fir/ #e syllogisms ( our cascade2 #i w\ld be necess>y if we wi%$ to prove ! !orem = ! numb] #aj2 m w\ld be ne$$ = a l>g] numb]2 b1 h[ev] great ?is numb] might be1 we %\ld always end by rea*+ x1 & ! analytic v]ification w\ld be possible4 ,& yet1 h[ev] f> we ?us might g1 we c\ld nev] rise to ! g5]al !orem1 applicable to all numb]s1 :i* alone c be ! object ( sci5ce4 ,to rea* ?is1 an inf9ity ( syllogisms w\ld be necess>y2 x w\ld be necess>y to ov]leap an abyss t ! pati5ce ( ! analy/1 re/rict$ to ! res\rces ( =mal logic alone1 nev] c\ld fill up4 ,i ask$ at ! outset :y "o c\ld n conceive ( a m9d suffici5tly p[]ful to p]ceive at a glance ! :ole body ( ma!matical tru?s4 ,! answ] is n[ easy2 a *ess-play] is able to comb9e f\r moves1 five moves1 9 advance1 b1 h[ev] extraord9>y he may be1 he w nev] prep>e m ?an a f9ite numb] ( !m2 if he applies 8 faculties to ari?metic1 he w n be able to p]ceive its g5]al tru?s by a s+le direct intuition2 to arrive at ! smalle/ !orem he c n disp5se ) ! aid ( reason+ by recurr5ce1 = ?is is an in/rum5t :i* enables u to pass f ! f9ite to ! inf9ite4 ,?is in/rum5t is always useful1 =1 all[+ u to ov]leap at a b\nd z _m /ages z we wi%1 x sp>es u v]ifications1 long1 irksome & monoton\s1 :i* w\ld quickly become impracticable4 ,b x becomes indisp5sable z soon z we aim at ! g5]al !orem1 to :i* analytic v]ification w\ld br+ u cont9ually ne>] )\t ev] enabl+ u to rea* x4 ,9 ?is doma9 ( ari?metic1 we may ?9k ourselves v f> f ! inf9itesimal analysis1 & yet1 z we h j se51 ! idea ( ! ma!matical inf9ite already plays a prepond]ant rle1 & )\t x !re w\ld be no sci5ce1 because !re w\ld be no?+ g5]al4 ,,vi ,! judgm5t on :i* reason+ by recurr5ce re/s c be put "u o!r =ms2 we may say1 = example1 t 9 an inf9ite collection ( diff]5t :ole numb]s !re is always "o :i* is less ?an all ! o!rs4 ,we c easily pass f "o enunciation to ! o!r & ?us get ! illusion ( hav+ demon/rat$ ! legitimacy ( reason+ by recurr5ce4 ,b we %all always be arre/$1 we %all always arrive at an undemon/rable axiom :i* w be 9 reality only ! proposition to be prov$ translat$ into ano!r language4 ,we c n !re=e escape ! conclusion t ! rule ( reason+ by recurr5ce is irr$ucible to ! pr9ciple ( contradiction4 ,nei!r c ?is rule come to u f exp]i5ce2 exp]i5ce c\ld tea* u t ! rule is true = ! fir/ t5 or hundr$ numb]s2 = example1 x c n atta9 to ! indef9ite s]ies ( numb]s1 b only to a portion ( ?is s]ies1 m or less long b always limit$4 ,n[ if x 7 only a "q ( t1 ! pr9ciple ( contradiction w\ld suffice2 x w\ld always all[ ( our develop+ z _m syllogisms z we wi%$2 x is only :5 x is a "q ( includ+ an inf9ity ( !m 9 a s+le =mula1 x is only be=e ! inf9ite t ?is pr9ciple fails1 & !re too1 exp]i5ce becomes p[]less4 ,?is rule1 inaccessible to analytic demon/ration & to exp]i5ce1 is ! v]itable type ( ! syn!tic a priori judgm5t4 ,on ! o!r h&1 we c n ?9k ( see+ 9 x a conv5tion1 z 9 "s ( ! po/ulates ( geometry4 ,:y !n does ?is judgm5t =ce itself ^u u ) an irresi/ible evid5ce8 ,x is because x is only ! affirmation ( ! p[] ( ! m9d :i* kn[s itself capable ( conceiv+ ! indef9ite repetition ( ! same act :5 once ?is act is possible4 ,! m9d has a direct intuition ( ?is p[]1 & exp]i5ce c only give occasion = us+ x & !reby becom+ consci\s ( x4 ,b1 "o w say1 if raw exp]i5ce c n legitimatize reason+ by recurr5ce1 is x s ( exp]im5t aid$ by induction8 ,we see successively t a !orem is true ( ! numb] #a1 ( ! numb] #b1 ( ! numb] #c & s on2 ! law is evid5t1 we say1 & x has ! same w>ranty z e physical law bas$ on obs]vations1 ^: numb] is v great b limit$4 ,"h is1 x mu/ be admitt$1 a /rik+ analogy ) ! usual proc$ures ( induction4 ,b !re is an ess5tial diff]5ce4 ,induction appli$ to ! physical sci5ces is always unc]ta91 because x re/s on ! belief 9 a g5]al ord] ( ! univ]se1 an ord] outside ( u4 ,ma!matical induction1 t is1 demon/ration by recurr5ce1 on ! contr>y1 imposes itself necess>ily because x is only ! affirmation ( a prop]ty ( ! m9d itself4 ,,vii ,ma!maticians1 z ,i h said be=e1 always endeavor to g5]alize ! propositions !y h obta9$1 &1 to seek no o!r example1 we h j prov$ ! equality3 a #a #a a & aft]w>ds us$ x to e/abli% ! equality a b b a :i* is manife/ly m g5]al4 ,ma!matics c1 !re=e1 l ! o!r sci5ces1 proce$ f ! p>ticul> to ! g5]al4 ,?is is a fact :i* w\ld h appe>$ incompreh5sible to u at ! outset ( ?is /udy1 b :i* is no long] my/]i\s to u1 s9ce we h asc]ta9$ ! analogies betwe5 demon/ration by recurr5ce & ord9>y induction4 ,)\t d\bt recurr5t reason+ 9 ma!matics & inductive reason+ 9 physics re/ on diff]5t f\ndations1 b _! m>* is p>allel1 !y advance 9 ! same s5se1 t is to say1 f ! p>ticul> to ! g5]al4 ,let u exam9e ! case a little m closely4 ,to demon/rate ! equality a #b #b a x suffices to twice apply ! rule 7#a7 a #a #a a & write 7#b7 a #b a #a #a #a a #a #a #a a #b a4 ,! equality 7#b7 ?us d$uc$ 9 purely analytic way f ! equality 7#a7 is1 h[ev]1 n simply a p>ticul> ease ( x2 x is some?+ q diff]5t4 ,we c n !re=e ev5 say t 9 ! really analytic & d$uctive "p ( ma!matical reason+ we proce$ f ! g5]al to ! p>ticul> 9 ! ord9>y s5se ( ! ^w4 ,! two memb]s ( ! equality 7#b7 are simply comb9ations m complicat$ ?an ! two memb]s ( ! equality 7#a71 & analysis only s]ves to sep>ate ! elem5ts :i* ent] into ^! comb9ations & to /udy _! relations4 ,ma!maticians proce$ !re=e 'by con/ruction1' !y 'con/ruct' comb9ations m & m complicat$4 ,com+ back !n by ! analysis ( ^! comb9ations1 ( ^! aggregates1 s to speak1 to _! primitive elem5ts1 !y p]ceive ! relations ( ^! elem5ts & f !m d$uce ! relations ( ! aggregates !mselves4 ,?is is a purely analytical proce$+1 b x is n1 h[ev]1 a proce$+ f ! g5]al to ! p>ticul>1 because evid5tly ! aggregates c n be reg>d$ z m p>ticul> ?an _! elem5ts4 ,great importance1 & ju/ly1 has be5 atta*$ to ?is proc$ure ( 'con/ruction1' & "s h tri$ to see 9 x ! necess>y & suffici5t condition = ! progress ( ! exact sci5ces4 ,necess>y1 )\t d\bt2 b suffici5t1 no4 ,= a con/ruction to be useful & n a va9 toil = ! m9d1 t x may s]ve z /epp+-/one to "o wi%+ to m\nt1 x mu/ fir/ ( all possess a sort ( unity enabl+ u to see 9 x some?+ besides ! juxtaposition ( its elem5ts4 ,or1 m exactly1 !re mu/ be "s advantage 9 consid]+ ! con/ruction r ?an its elem5ts !mselves4 ,:at c ?is advantage be8 ,:y reason on a polygon1 = in/ance1 :i* is always decomposable into triangles1 & n on ! elem5t>y triangles8 ,x is because !re are prop]ties app]ta9+ to polygons ( any numb] ( sides & t may be imm$iately appli$ to any p>ticul> polygon4 ,usually1 on ! contr>y1 x is only at ! co/ ( ! mo/ prolong$ ex]tions t !y c\ld be f\nd by /udy+ directly ! relations ( ! elem5t>y triangles4 ,! k ( ! g5]al !orem sp>es u ^! ef=ts4 ,a con/ruction1 !re=e1 becomes int]e/+ only :5 x c be rang$ beside o!r analog\s con/ructions1 =m+ species ( ! same g5us4 ,if ! quadrilat]al is some?+ besides ! juxtaposition ( two triangles1 ?is is because x belongs to ! g5us polygon4 ,moreov]1 "o mu/ be able to demon/rate ! prop]ties ( ! g5us )\t be+ =c$ to e/abli% !m successively = ea* ( ! species4 ,to atta9 t1 we mu/ necess>ily m\nt f ! p>ticul> to ! g5]al1 asc5d+ "o or m /eps4 ,! analytic proc$ure 'by con/ruction' does n oblige u to desc5d1 b x leaves u at ! same level4 ,we c asc5d only by ma!matical induction1 :i* alone c tea* u some?+ new4 ,)\t ! aid ( ?is induction1 diff]5t 9 c]ta9 respects f physical induction1 b q z f]tile1 con/ruction w\ld be p[]less to create sci5ce4 ,obs]ve f9ally t ?is induction is possible only if ! same op]ation c be repeat$ indef9itely4 ,t is :y ! !ory ( *ess c nev] become a sci5ce1 = ! diff]5t moves ( ! same game d n resemble "o ano!r4 ,,*apt] ,,ii ,,ma!matical ,,magnitude ,,& ,,exp]i5ce ,to le>n :at ma!maticians und]/& by a cont9uum1 "o %\ld n inquire ( geometry4 ,! geomet] always seeks to repres5t to himself m or less ! figures he /udies1 b 8 repres5tations are = him only in/rum5ts2 9 mak+ geometry he uses space j z he does *alk2 s too mu* weight %\ld n be atta*$ to non-ess5tials1 (t5 ( no m importance ?an ! :it5ess ( ! *alk4 ,! pure analy/ has n ?is rock to fe>4 ,he has dis5gag$ ! sci5ce ( ma!matics f all =eign elem5ts1 & c answ] our "q3 ',:at exactly is ?is cont9uum ab\t :i* ma!maticians reason8' ,_m analy/s :o reflect on _! art h answ]$ already2 ,monsieur ,tann]y1 = example1 9 8 ,introduction la ?orie des fonctions d'une v>iable4 ,let u />t f ! scale ( :ole numb]s2 betwe5 two consecutive /eps1 int]calate "o or m int]m$i>y /eps1 !n betwe5 ^! new /eps /ill o!rs1 & s on indef9itely4 ,?us we %all h an unlimit$ numb] ( t]ms2 ^! w be ! numb]s call$ fractional1 rational or comm5surable4 ,b ?is is n yet en\gh2 betwe5 ^! t]ms1 :i*1 h[ev]1 are already inf9ite 9 numb]1 x is /ill necess>y to int]calate o!rs call$ irrational or incomm5surable4 ,a rem>k be=e go+ fur!r4 ,! cont9uum s conceiv$ is only a collection ( individuals rang$ 9 a c]ta9 ord]1 inf9ite 9 numb]1 x is true1 b ext]ior to "o ano!r4 ,?is is n ! ord9>y conception1 :]e9 is suppos$ betwe5 ! elem5ts ( ! cont9uum a sort ( intimate bond :i* makes ( !m a :ole1 :]e ! po9t does n exi/ be=e ! l9e1 b ! l9e be=e ! po9t4 ,( ! celebrat$ =mula1 '! cont9uum is unity 9 multiplicity1' only ! multiplicity rema9s1 ! unity has disappe>$4 ,! analy/s are none ! less "r 9 def9+ _! cont9uum z !y d1 = !y always reason on j ?is z soon z !y pique !mselves on _! rigor4 ,b ?is is en\gh to apprise u t ! v]itable ma!matical cont9uum is a v diff]5t ?+ f t ( ! physici/s & t ( ! metaphysicians4 ,x may also be said p]haps t ! ma!maticians :o are cont5t ) ?is def9ition are dupes ( words1 t x is necess>y to say precisely :at ea* ( ^! int]m$i>y /eps is1 to expla9 h[ !y are to be int]calat$ & to demon/rate t x is possible to d x4 ,b t w\ld be wrong2 ! only prop]ty ( ^! /eps :i* is us$ 9 _! reason+s#b is t ( be+ be=e or aft] su* & su* /eps2 !re=e also ?is alone %\ld occur 9 ! def9ition4 #b ,) ^? conta9$ 9 ! special conv5tions :i* s]ve to def9e addition & ( :i* we %all speak lat]4 ,s h[ ! int]m$i>y t]ms %\ld be int]calat$ ne$ n conc]n u2 on ! o!r h&1 no "o w d\bt ! possibility ( ?is op]ation1 unless f =gett+ t possible1 9 ! language ( geomet]s1 simply means free f contradiction4 ,our def9ition1 h[ev]1 is n yet complete1 & ,i return to x aft] ?is ov]-long digression4 ,,def9ition ,,( ,,incomm5surables4--,! ma!maticians ( ! ,b]l9 s*ool1 ,kroneck] 9 p>ticul>1 h devot$ !mselves to con/ruct+ ?is cont9u\s scale ( fractional & irrational numb]s )\t us+ any mat]ial o!r ?an ! :ole numb]4 ,! ma!matical cont9uum w\ld be1 9 ?is view1 a pure creation ( ! m9d1 :]e exp]i5ce w\ld h no "p4 ,! notion ( ! rational numb] seem+ to !m to pres5t no difficulty1 !y h *iefly /riv5 to def9e ! incomm5surable numb]4 ,b be=e produc+ "h _! def9ition1 ,i mu/ make a rem>k to =e/all ! a/oni%m5t x is sure to ar\se 9 read]s unfamili> ) ! cu/oms ( geomet]s4 ,ma!maticians /udy n objects1 b relations betwe5 objects2 ! replacem5t ( ^! objects by o!rs is !re=e indiff]5t to !m1 provid$ ! relations d n *ange4 ,! matt] is = !m unimportant1 ! =m alone int]e/s !m4 ,)\t recall+ ?is1 x w\ld sc>cely be compreh5sible t ,d$ek9d %\ld designate by ! "n incomm5surable numb] a m]e symbol1 t is to say1 some?+ v diff]5t f ! ord9>y idea ( a quantity1 :i* %\ld be measurable & almo/ tangible4 ,let u see n[ :at ,d$ek9d's def9ition is3 ,! comm5surable numb]s c 9 an inf9ity ( ways be p>tition$ into two classes1 su* t any numb] ( ! fir/ class is great] ?an any numb] ( ! second class4 ,x may happ5 t among ! numb]s ( ! fir/ class !re is "o small] ?an all ! o!rs2 if1 = example1 we range 9 ! fir/ class all numb]s great] ?an #b1 & #b itself1 & 9 ! second class all numb]s less ?an #b1 x is cle> t #b w be ! lea/ ( all numb]s ( ! fir/ class4 ,! numb] #b may be *os5 z symbol ( ?is p>tition4 ,x may happ51 on ! contr>y1 t among ! numb]s ( ! second class is "o great] ?an all ! o!rs2 ?is is ! case1 = example1 if ! fir/ class compreh5ds all numb]s great] ?an #b1 & ! second all numb]s less ?an #b1 & #b itself4 ,"h aga9 ! numb] #b may be *os5 z symbol ( ?is p>tition4 ,b x may equally well happ5 t nei!r is !re 9 ! fir/ class a numb] less ?an all ! o!rs1 nor 9 ! second class a numb] great] ?an all ! o!rs4 ,suppose1 = example1 we put 9 ! fir/ class all comm5surable numb]s ^: squ>es are great] ?an #b & 9 ! second all ^: squ>es are less ?an #b4 ,!re is none ^: squ>e is precisely #b4 ,evid5tly !re is n 9 ! fir/ class a numb] less ?an all ! o!rs1 =1 h[ev] ne> ! squ>e ( a numb] may be to #b1 we c always f9d a comm5surable numb] ^: squ>e is /ill clos] to #b4 ,9 ,d$ek9d's view1 ! incomm5surable numb] sqrt7#b7 or 7#b7a_/b is no?+ b ! symbol ( ?is p>ticul> mode ( p>tition ( comm5surable numb]s2 & to ea* mode ( p>tition corresponds ?us a numb]1 comm5surable or n1 :i* s]ves z its symbol4 ,b to be cont5t ) ?is w\ld be to =get too f> ! orig9 ( ^! symbols2 x rema9s to expla9 h[ we h be5 l$ to attribute to !m a sort ( concrete exi/5ce1 &1 besides1 does n ! difficulty beg9 ev5 = ! fractional numb]s !mselves8 ,%\ld we h ! notion ( ^! numb]s if we _h n previ\sly kn[n a matt] t we conceive z inf9itely divisible1 t is to say1 a cont9uum8 ,,! ,,physical ,,cont9uum4--,we ask ourselves !n if ! notion ( ! ma!matical cont9uum is n simply drawn f exp]i5ce4 ,if x 71 ! raw data ( exp]i5ce1 :i* are our s5sations1 w\ld be susceptible ( measurem5t4 ,we might be tempt$ to believe !y really are s1 s9ce 9 ^! latt] days ! attempt has be5 made to measure !m & a law has ev5 be5 =mulat$1 kn[n z ,fe*n]'s law1 accord+ to :i* s5sation is proportional to ! log>i?m ( ! /imulus4 ,b if we exam9e m closely ! exp]im5ts by :i* x has be5 s\ght to e/abli% ?is law1 we %all be l$ to a diametrically opposite conclusion4 ,x has be5 obs]v$1 = example1 t a weight ,a ( #aj grams & a weight ,b ( #aa grams produce id5tical s5sations1 t ! weight ,b is j z indi/+ui%able f a weight ,c ( #ab grams1 b t ! weight ,a is easily di/+ui%$ f ! weight ,c4 ,?us ! raw results ( exp]i5ce may be express$ by ! foll[+ relations3 ,a ,b1 ,b ,c1 ,a ,c1 :i* may be reg>d$ z ! =mula ( ! physical cont9uum4 ,b "h is an intol]able discord ) ! pr9ciple ( contradiction1 & ! ne$ ( /opp+ ?is has compell$ u to inv5t ! ma!matical cont9uum4 ,we are1 !re=e1 =c$ to conclude t ?is notion has be5 creat$ entirely by ! m9d1 b t exp]i5ce has giv5 ! occasion4 ,we c n believe t two quantities equal to a ?ird are n equal to "o ano!r1 & s we are l$ to suppose t ,a is diff]5t f ,b & ,b f ,c1 b t ! imp]fection ( our s5ses has n p]mitt$ ( our di/+ui%+ !m4 ,,creation ,,( ,,! ,,ma!matical ,,cont9uum4--,fir/ ,/age4 ,s f> x w\ld suffice1 9 acc\nt+ = ! facts1 to int]calate betwe5 ,a & ,b a few t]ms1 :i* w\ld rema9 discrete4 ,:at happ5s n[ if we h rec\rse to "s in/rum5t to supplem5t ! feebl5ess ( our s5ses1 if1 = example1 we make use ( a microscope8 ,t]ms su* z ,a & ,b1 be=e indi/+ui%able1 appe> n[ di/9ct2 b betwe5 ,a & ,b1 n[ become di/9ct1 w be int]calat$ a new t]m1 ,d1 t we c di/+ui% nei!r f ,a nor f ,b4 ,despite ! employm5t ( ! mo/ highly p]fect$ me?ods1 ! raw results ( our exp]i5ce w always pres5t ! *>act]i/ics ( ! physical cont9uum ) ! contradiction :i* is inh]5t 9 x4 ,we %all escape x only by incessantly int]calat+ new t]ms betwe5 ! t]ms already di/+ui%$1 & ?is op]ation mu/ be cont9u$ indef9itely4 ,we might conceive ! /opp+ ( ?is op]ation if we c\ld imag9e "s in/rum5t suffici5tly p[]ful to decompose ! physical cont9uum into discrete elem5ts1 z ! telescope resolves ! milky way into />s4 ,b ?is we c n imag9e2 9 fact1 x is ) ! eye we obs]ve ! image magnifi$ by ! microscope1 & consequ5tly ?is image mu/ always reta9 ! *>act]i/ics ( visual s5sation & consequ5tly ^? ( ! physical cont9uum4 ,no?+ di/+ui%es a l5g? obs]v$ directly f ! half ( ?is l5g? d\bl$ by ! microscope4 ,! :ole is homog5e\s ) ! "p2 ?is is a new contradiction1 or r x w\ld be if ! numb] ( t]ms 7 suppos$ f9ite2 9 fact1 x is cle> t ! "p conta9+ few] t]ms ?an ! :ole c\ld n be simil> to ! :ole4 ,! contradiction ceases :5 ! numb] ( t]ms is reg>d$ z inf9ite2 no?+ h9d]s1 = example1 consid]+ ! aggregate ( :ole numb]s z simil> to ! aggregate ( ev5 numb]s1 :i*1 h[ev]1 is only a "p ( x2 &1 9 fact1 to ea* :ole numb] corresponds an ev5 numb]1 its d\ble4 ,b x is n only to escape ?is contradiction conta9$ 9 ! empirical data t ! m9d is l$ to create ! concept ( a cont9uum1 =m$ ( an indef9ite numb] ( t]ms4 ,all happ5s z 9 ! sequ5ce ( :ole numb]s4 ,we h ! faculty ( conceiv+ t a unit c be add$ to a collection ( units2 ?anks to exp]i5ce1 we h occasion to ex]cise ?is faculty & we become consci\s ( x2 b f ?is mom5t we feel t our p[] has no limit & t we c c\nt indef9itely1 ?\gh we h nev] _h to c\nt m ?an a f9ite numb] ( objects4 ,j s1 z soon z we h be5 l$ to int]calate means betwe5 two consecutive t]ms ( a s]ies1 we feel t ?is op]ation c be cont9u$ beyond all limit1 & t !re is1 s to speak1 no intr9sic reason = /opp+4 ,z an abbreviation1 let me call a ma!matical cont9uum ( ! fir/ ord] e aggregate ( t]ms =m$ accord+ to ! same law z ! scale ( comm5surable numb]s4 ,if we aft]w>ds int]calate new /eps accord+ to ! law ( =mation ( incomm5surable numb]s1 we %all obta9 :at we w call a cont9uum ( ! second ord]4 ,second ,/age4--,we h made hi!rto only ! fir/ /ride2 we h expla9$ ! orig9 ( cont9ua ( ! fir/ ord]2 b x is necess>y to see :y ev5 !y are n suffici5t & :y ! incomm5surable numb]s _h to be inv5t$4 ,if we try to imag9e a l9e1 x mu/ h ! *>act]i/ics ( ! physical cont9uum1 t is to say1 we %all n be able to repres5t x except ) a c]ta9 bread?4 ,two l9es !n w appe> to u "u ! =m ( two n>r[ b&s1 &1 if we are cont5t ) ?is r\gh image1 x is evid5t t if ! two l9es cross !y w h a common "p4 ,b ! pure geomet] makes a fur!r ef=t2 )\t entirely r5\nc+ ! aid ( ! s5ses1 he tries to rea* ! concept ( ! l9e )\t bread?1 ( ! po9t )\t ext5sion4 ,?is he c only atta9 to by reg>d+ ! l9e z ! limit t[>d :i* t5ds an ev] n>r[+ b&1 & ! po9t z ! limit t[>d :i* t5ds an ev] less5+ area4 ,& !n1 our two b&s1 h[ev] n>r[ !y may be1 w always h a common area1 ! small] z !y are ! n>r[]1 & ^: limit w be :at ! pure geomet] calls a po9t4 ,?is is :y x is said two l9es :i* cross h a po9t 9 common1 & ?is tru? seems intuitive4 ,b x w\ld imply contradiction if l9es 7 conceiv$ z cont9ua ( ! fir/ ord]1 t is to say1 if on ! l9es trac$ by ! geomet] %\ld be f\nd only po9ts hav+ = coord9ates rational numb]s4 ,! contradiction w\ld be manife/ z soon z "o affirm$1 = example1 ! exi/5ce ( /raights & circles4 ,x is cle>1 9 fact1 t if ! po9ts ^: coord9ates are comm5surable 7 alone reg>d$ z real1 ! circle inscrib$ 9 a squ>e & ! diagonal ( ?is squ>e w\ld n int]sect1 s9ce ! coord9ates ( ! po9t ( int]section are incomm5surable4 ,t w\ld n yet be suffici5t1 because we %\ld get 9 ?is way only c]ta9 incomm5surable numb]s & n all ^? numb]s4 ,b conceive ( a /raight l9e divid$ into two rays4 ,ea* ( ^! rays w appe> to our imag9ation z a b& ( a c]ta9 bread?2 ^! b&s moreov] w encroa* "o on ! o!r1 s9ce !re mu/ be no int]val betwe5 !m4 ,! common "p w appe> to u z a po9t :i* w always rema9 :5 we try to imag9e our b&s n>r[] & n>r[]1 s t we admit z an intuitive tru? t if a /raight is cut into two rays _! common fronti] is a po9t2 we recognize "h ! conception ( ,d$ek9d1 9 :i* an incomm5surable numb] 0 reg>d$ z ! common fronti] ( two classes ( rational numb]s4 ,su* is ! orig9 ( ! cont9uum ( ! second ord]1 :i* is ! ma!matical cont9uum prop]ly s call$4 ,rsum4--,9 recapitulation1 ! m9d has ! faculty ( creat+ symbols1 & x is ?us t x has con/ruct$ ! ma!matical cont9uum1 :i* is only a p>ticul> sy/em ( symbols4 ,its p[] is limit$ only by ! necessity ( avoid+ all contradiction2 b ! m9d only makes use ( ?is faculty if exp]i5ce furni%es x a /imulus !reto4 ,9 ! case consid]$1 ?is /imulus 0 ! notion ( ! physical cont9uum1 drawn f ! r\gh data ( ! s5ses4 ,b ?is notion leads to a s]ies ( contradictions f :i* x is necess>y successively to free ourselves4 ,s we are =c$ to imag9e a m & m complicat$ sy/em ( symbols4 ,t at :i* we /op is n only exempt f int]nal contradiction 7x 0 s already at all ! /ages we h trav]s$71 b nei!r is x 9 contradiction ) v>i\s propositions call$ intuitive1 :i* are d]iv$ f empirical notions m or less elaborat$4 ,,measurable ,,magnitude4--,! magnitudes we h /udi$ hi!rto are n measurable2 we c inde$ say :e!r a giv5 "o ( ^! magnitudes is great] ?an ano!r1 b n :e!r x is twice or ?rice z great4 ,s f>1 ,i h only consid]$ ! ord] 9 :i* our t]ms are rang$4 ,b = mo/ applications t does n suffice4 ,we mu/ le>n to comp>e ! int]val :i* sep>ates any two t]ms4 ,only on ?is condition does ! cont9uum become a measurable magnitude & ! op]ations ( ari?metic applicable4 ,?is c only be done by ! aid ( a new & special conv5tion4 ,we w agree t 9 su* & su* a case ! int]val compris$ betwe5 ! t]ms ,a & ,b is equal to ! int]val :i* sep>ates ,c & ,d4 ,= example1 at ! beg9n+ ( our "w we h set out f ! scale ( ! :ole numb]s & we h suppos$ int]calat$ betwe5 two consecutive /eps n int]m$i>y /eps2 well1 ^! new /eps w be by conv5tion reg>d$ z equidi/ant4 ,?is is a way ( def9+ ! addition ( two magnitudes1 because if ! int]val ,,ab is by def9ition equal to ! int]val ,,cd1 ! int]val ,,ad w be by def9ition ! sum ( ! int]vals ,,ab & ,,ac4 ,?is def9ition is arbitr>y 9 a v l>ge measure4 ,x is n completely s1 h[ev]4 ,x is subject$ to c]ta9 conditions &1 = example1 to ! rules ( commutativity & associativity ( addition4 ,b provid$ ! def9ition *os5 satisfies ^! rules1 ! *oice is indiff]5t1 & x is useless to p>ticul>ize x4 ,,v>i\s ,,rem>ks4--,we c n[ discuss sev]al important que/ions3 #a ,is ! creative p[] ( ! m9d exhau/$ by ! creation ( ! ma!matical cont9uum8 ,no3 ! works ( ,du ,bois-,reymond demon/rate x 9 a /rik+ way4 ,we "k t ma!maticians di/+ui% betwe5 inf9itesimals ( diff]5t ord]s & t ^? ( ! second ord] are inf9itesimal1 n only 9 an absolute way1 b also 9 relation to ^? ( ! fir/ ord]4 ,x is n difficult to imag9e inf9itesimals ( fractional or ev5 ( irrational ord]1 & ?us we f9d aga9 t scale ( ! ma!matical cont9uum :i* has be5 dealt ) 9 ! prec$+ pages4 ,fur!r1 !re are inf9itesimals :i* are inf9itely small 9 relation to ^? ( ! fir/ ord]1 &1 on ! contr>y1 inf9itely great 9 relation to ^? ( ord] #a epsilon1 & t h[ev] small epsilon may be4 ,"h1 !n1 are new t]ms int]calat$ 9 our s]ies1 & if ,i may be p]mitt$ to rev]t to ! phraseology lately employ$ :i* is v conv5i5t ?\gh n consecrat$ by usage1 ,i %all say t ?us has be5 creat$ a sort ( cont9uum ( ! ?ird ord]4 ,x w\ld be easy to g fur!r1 b t w\ld be idle2 "o w\ld only be imag9+ symbols )\t possible application1 & no "o w ?9k ( do+ t4 ,! cont9uum ( ! ?ird ord]1 to :i* ! consid]ation ( ! diff]5t ord]s ( inf9itesimals leads1 is itself n useful en\gh to h won citiz5%ip1 & geomet]s reg>d x only z a m]e curiosity4 ,! m9d uses its creative faculty only :5 exp]i5ce requires x4 #b ,once 9 possession ( ! concept ( ! ma!matical cont9uum1 is "o safe f contradictions analog\s to ^? :i* gave bir? to x8 ,no1 & ,i w give an example4 ,"o mu/ be v wise n to reg>d x z evid5t t e curve has a tang5t2 & 9 fact if we picture ?is curve & a /raight z two n>r[ b&s we c always s dispose !m t !y h a "p 9 common )\t cross+4 ,if we imag9e !n ! bread? ( ^! two b&s to dim9i% indef9itely1 ?is common "p w always subsi/ &1 at ! limit1 s to speak1 ! two l9es w h a po9t 9 common )\t cross+1 t is to say1 !y w be tang5t4 ,! geomet] :o reasons 9 ?is way1 consci\sly or n1 is only do+ :at we h done above to prove two l9es :i* cut h a po9t 9 common1 & 8 intuition might seem j z legitimate4 ,x w\ld deceive him h[ev]4 ,we c demon/rate t !re are curves :i* h no tang5t1 if su* a curve is def9$ z an analytic cont9uum ( ! second ord]4 ,)\t d\bt "s artifice analog\s to ^? we h discuss$ above w\ld h suffic$ to remove ! contradiction2 b1 z ?is is met ) only 9 v exceptional cases1 x has receiv$ no fur!r att5tion4 ,in/ead ( seek+ to reconcile intuition ) analysis1 we h be5 cont5t to sacrifice "o ( ! two1 & z analysis mu/ rema9 impeccable1 we h decid$ aga9/ intuition4 ,,! ,,physical ,,cont9uum ,,( ,,sev]al ,,dim5sions4--,we h discuss$ above ! physical cont9uum z d]iv$ f ! imm$iate data ( our s5ses1 or1 if y wi%1 f ! r\gh results ( ,fe*n]'s exp]im5ts2 ,i h %[n t ^! results are summ$ up 9 ! contradictory =mulas ,a ,b1 ,b ,c1 ,a ,c4 ,let u n[ see h[ ?is notion has be5 g5]aliz$ & h[ f x has come ! concept ( _m-dim5sional cont9ua4 ,consid] any two aggregates ( s5sations4 ,ei!r we c discrim9ate !m "o f ano!r1 or we c n1 j z 9 ,fe*n]'s exp]im5ts a weight ( #aj grams c be di/+ui%$ f a weight ( #ab grams1 b n f a weight ( #aa grams4 ,?is is all t is requir$ to con/ruct ! cont9uum ( sev]al dim5sions4 ,let u call "o ( ^! aggregates ( s5sations an elem5t4 ,t w be some?+ analog\s to ! po9t ( ! ma!maticians2 x w n be altoge!r ! same ?+ h[ev]4 ,we c n say our elem5t is )\t ext5sion1 s9ce we c n di/+ui% x f neighbor+ elem5ts & x is ?us surr\nd$ by a sort ( haze4 ,if ! a/ronomical comp>ison may be all[$1 our 'elem5ts' w\ld be l nebulae1 :]eas ! ma!matical po9ts w\ld be l />s4 ,t be+ grant$1 a sy/em ( elem5ts w =m a cont9uum if we c pass f any "o ( !m to any o!r1 by a s]ies ( consecutive elem5ts su* t ea* is indi/+ui%able f ! prec$+4 ,?is l9e> s]ies is to ! l9e ( ! ma!matician :at an isolat$ elem5t 0 to ! po9t4 ,be=e go+ f>!r1 ,i mu/ expla9 :at is meant by a cut4 ,consid] a cont9uum ,c & remove f x c]ta9 ( its elem5ts :i* = an in/ant we %all reg>d z no long] belong+ to ?is cont9uum4 ,! aggregate ( ! elem5ts s remov$ w be call$ a cut4 ,x may happ5 t1 ?anks to ?is cut1 ,c may be subdivid$ into sev]al di/9ct cont9ua1 ! aggregate ( ! rema9+ elem5ts ceas+ to =m a unique cont9uum4 ,!re w !n be on ,c two elem5ts1 ,a & ,b1 t mu/ be reg>d$ z belong+ to two di/9ct cont9ua1 & ?is w be recogniz$ because x w be impossible to f9d a l9e> s]ies ( consecutive elem5ts ( ,c1 ea* ( ^! elem5ts indi/+ui%able f ! prec$+1 ! fir/ be+ ,a & ! la/ ,b1 )\t "o ( ! elem5ts ( ?is s]ies be+ indi/+ui%able f "o ( ! elem5ts ( ! cut4 ,on ! contr>y1 x may happ5 t ! cut made is insuffici5t to subdivide ! cont9uum ,c4 ,to classify ! physical cont9ua1 we w exam9e precisely :at are ! cuts :i* mu/ be made to subdivide !m4 ,if a physical cont9uum ,c c be subdivid$ by a cut r$uc+ to a f9ite numb] ( elem5ts all di/+ui%able f "o ano!r 7& consequ5tly =m+ nei!r a cont9uum1 nor sev]al cont9ua71 we %all say ,c is a "o-dim5sional cont9uum4 ,if1 on ! contr>y1 ,c c be subdivid$ only by cuts :i* are !mselves cont9ua1 we %all say ,c has sev]al dim5sions4 ,if cuts :i* are cont9ua ( "o dim5sion suffice1 we %all say ,c has two dim5sions2 if cuts ( two dim5sions suffice1 we %all say ,c has ?ree dim5sions1 & s on4 ,?us is def9$ ! notion ( ! physical cont9uum ( sev]al dim5sions1 ?anks to ?is v simple fact t two aggregates ( s5sations are di/+ui%able or indi/+ui%able4 ,,! ,,ma!matical ,,cont9uum ,,( ,,sev]al ,,dim5sions4--,!nce ! notion ( ! ma!matical cont9uum ( n dim5sions has sprung q naturally by a process v l t we discuss$ at ! beg9n+ ( ?is *apt]4 ,a po9t ( su* a cont9uum1 y "k1 appe>s to u z def9$ by a sy/em ( n di/9ct magnitudes call$ its coord9ates4 ,^! magnitudes ne$ n always be measurable2 !re is1 = in/ance1 a bran* ( geometry indep5d5t ( ! measurem5t ( ^! magnitudes1 9 :i* x is only a "q ( kn[+1 = example1 :e!r on a curve ,,abc1 ! po9t ,b is betwe5 ! po9ts ,a & ,c1 & n ( kn[+ :e!r ! arc ,,ab is equal to ! arc ,,bc or twice z great4 ,?is is :at is call$ ,analysis ,situs4 ,?is is a :ole body ( doctr9e :i* has attract$ ! att5tion ( ! greate/ geomet]s & :]e we see fl[ "o f ano!r a s]ies ( rem>kable !orems4 ,:at di/+ui%es ^! !orems f ^? ( ord9>y geometry is t !y are purely qualitative & t !y w\ld rema9 true if ! figures 7 copi$ by a draughtsman s awkw>d z to grossly di/ort ! proportions & replace /raights by /rokes m or less curv$4 ,?r\gh ! wi% to introduce measure next into ! cont9uum j def9$ ?is cont9uum becomes space1 & geometry is born4 ,b ! discussion ( ?is is res]v$ = ,"p ,second4 ,,"p ,,ii ,,space ,,*apt] ,,iii ,,! ,,non-,,euclidean ,,geometries ,e conclusion supposes premises2 ^! premises !mselves ei!r are self-evid5t & ne$ no demon/ration1 or c be e/abli%$ only by rely+ ^u o!r propositions1 & s9ce we c n g back ?us to inf9ity1 e d$uctive sci5ce1 & 9 p>ticul> geometry1 mu/ re/ on a c]ta9 numb] ( undemon/rable axioms4 ,all treatises on geometry beg91 !re=e1 by ! enunciation ( ^! axioms4 ,b among ^! !re is a di/9ction to be made3 ,"s1 = example1 ',?+s :i* are equal to ! same ?+ are equal to "o ano!r1' are n propositions ( geometry1 b propositions ( analysis4 ,i reg>d !m z analytic judgm5ts a priori1 & %all n conc]n myself ) !m4 ,b ,i mu/ lay /ress ^u o!r axioms :i* are peculi> to geometry4 ,mo/ treatises enunciate ?ree ( ^! explicitly3 #a ,?r\gh two po9ts c pass only "o /raight2 #b ,! /raight l9e is ! %orte/ pa? f "o po9t to ano!r2 #c ,?r\gh a giv5 po9t !re is n m ?an "o p>allel to a giv5 /raight4 ,al?\gh g5]ally a pro( ( ! second ( ^! axioms is omitt$1 x w\ld be possible to d$uce x f ! o!r two & f ^?1 mu* m num]\s1 :i* are implicitly admitt$ )\t enunciat+ !m1 z ,i %all expla9 fur!r on4 ,x 0 long s\ght 9 va9 to demon/rate likewise ! ?ird axiom1 kn[n z ,euclid's ,po/ulate4 ,:at va/ ef=t has be5 wa/$ 9 ?is *im]ic hope is truly unimag9able4 ,f9ally1 9 ! fir/ qu>t] ( ! n9ete5? c5tury1 & almo/ at ! same "t1 a ,hung>ian & a ,russian1 ,bolyai & ,loba*evski1 e/abli%$ irrefutably t ?is demon/ration is impossible2 !y h almo/ rid u ( inv5tors ( geometries 'sans po/ulatum'2 s9ce !n ! ,acadmie des ,sci5ces receives only ab\t "o or two new demon/rations a ye>4 ,! "q 0 n exhau/$2 x soon made a great /ride by ! publication ( ,riemann's celebrat$ memoir entitl$3 ,ueb] die ,hypo!s5 wel*e d] ,geometrie zu ,grunde lieg54 ,?is pap] has inspir$ mo/ ( ! rec5t works ( :i* ,i %all speak fur!r on1 & among :i* x is prop] to cite ^? ( ,beltrami & ( ,helmholtz4 ,,! ,,bolyai-,,loba*evski ,,geometry4--,if x 7 possible to d$uce ,euclid's po/ulate f ! o!r axioms1 x is evid5t t 9 d5y+ ! po/ulate & admitt+ ! o!r axioms1 we %\ld be l$ to contradictory consequ5ces2 x w\ld !re=e be impossible to base on su* premises a coh]5t geometry4 ,n[ ?is is precisely :at ,loba*evski did4 ,he assumes at ! />t t3 ,?r\gh a giv5 po9t c be drawn two p>allels to a giv5 /raight4 ,& he reta9s besides all ,euclid's o!r axioms4 ,f ^! hypo!ses he d$uces a s]ies ( !orems among :i* x is impossible to f9d any contradiction1 & he con/ructs a geometry ^: faultless logic is inf]ior 9 no?+ to t ( ! ,euclidean geometry4 ,! !orems are1 ( c\rse1 v diff]5t f ^? to :i* we are accu/om$1 & !y c n fail to be at fir/ a little disconc]t+4 ,?us ! sum ( ! angles ( a triangle is always less ?an two "r angles1 & ! diff]5ce betwe5 ?is sum & two "r angles is proportional to ! surface ( ! triangle4 ,x is impossible to con/ruct a figure simil> to a giv5 figure b ( diff]5t dim5sions4 ,if we divide a circumf]5ce into n equal p>ts1 & draw tang5ts at ! po9ts ( division1 ^! n tang5ts w =m a polygon if ! radius ( ! circle is small en\gh2 b if ?is radius is suffici5tly great !y w n meet4 ,x is useless to multiply ^! examples2 ,loba*evski's propositions h no relation to ^? ( ,euclid1 b !y are n less logically b\nd "o to ano!r4 ,,riemann',s ,,geometry4--,imag9e a _w uniquely peopl$ by be+s ( no ?ickness 7height72 & suppose ^! 'inf9itely flat' animals are all 9 ! same plane & c n get out4 ,admit besides t ?is _w is suffici5tly f> f o!rs to be free f _! influ5ce4 ,:ile we are mak+ hypo!ses1 x co/s u no m to end[ ^! be+s ) reason & believe !m capable ( creat+ a geometry4 ,9 t case1 !y w c]ta9ly attribute to space only two dim5sions4 ,b suppose n[ t ^! imag9>y animals1 :ile rema9+ )\t ?ickness1 h ! =m ( a sph]ical1 & n ( a plane1 figure1 & are all on ! same sph]e )\t p[] to get (f4 ,:at geometry w !y con/ruct8 ,fir/ x is cle> !y w attribute to space only two dim5sions2 :at w play = !m ! rle ( ! /raight l9e w be ! %orte/ pa? f "o po9t to ano!r on ! sph]e1 t is to say1 an arc ( a great circle2 9 a ^w1 _! geometry w be ! sph]ical geometry4 ,:at !y w call space w be ?is sph]e on :i* !y mu/ /ay1 & on :i* happ5 all ! ph5om5a !y c "k4 ,_! space w !re=e be unb\nd$ s9ce on a sph]e "o c always g =w>d )\t ev] be+ /opp$1 & yet x w be f9ite2 "o c nev] f9d ! end ( x1 b "o c make a t\r ( x4 ,well1 ,riemann's geometry is sph]ical geometry ext5d$ to ?ree dim5sions4 ,to con/ruct x1 ! ,g]man ma!matician _h to ?r[ ov]bo>d1 n only ,euclid's po/ulate1 b also ! fir/ axiom3 ,only "o /raight c pass ?r\gh two po9ts4 ,on a sph]e1 ?r\gh two giv5 po9ts we c draw 9 g5]al only "o great circle 7:i*1 z we h j se51 w\ld play ! rle ( ! /raight = our imag9>y be+s72 b !re is an exception3 if ! two giv5 po9ts are diametrically opposite1 an inf9ity ( great circles c be drawn ?r\gh !m4 ,9 ! same way1 9 ,riemann's geometry 7at lea/ 9 "o ( its =ms71 ?r\gh two po9ts w pass 9 g5]al only a s+le /raight2 b !re are exceptional cases :]e ?r\gh two po9ts an inf9ity ( /raights c pass4 ,!re is a sort ( opposition betwe5 ,riemann's geometry & t ( ,loba*evski4 ,?us ! sum ( ! angles ( a triangle is3 ,equal to two "r angles 9 ,euclid's geometry2 ,less ?an two "r angles 9 t ( ,loba*evski2 ,great] ?an two "r angles 9 t ( ,riemann4 ,! numb] ( /raights ?r\gh a giv5 po9t t c be drawn coplan> to a giv5 /raight1 b n[h]e meet+ x1 is equal3 ,to "o 9 ,euclid's geometry2 ,to z]o 9 t ( ,riemann2 ,to inf9ity 9 t ( ,loba*evski4 ,add t ,riemann's space is f9ite1 al?\gh unb\nd$1 9 ! s5se giv5 above to ^! two words4 ,,! ,,surfaces ,,( ,,con/ant ,,curvature4--,"o objection /ill rema9$ possible4 ,! !orems ( ,loba*evski & ( ,riemann pres5t no contradiction2 b h[ev] num]\s ! consequ5ces ^! two geomet]s h drawn f _! hypo!ses1 !y mu/ h /opp$ be=e exhau/+ !m1 s9ce _! numb] w\ld be inf9ite2 :o c say !n t if !y _h pu%$ _! d$uctions f>!r !y w\ld n h ev5tually rea*$ "s contradiction8 ,?is difficulty does n exi/ = ,riemann's geometry1 provid$ x is limit$ to two dim5sions2 9 fact1 z we h se51 two-dim5sional ,riemannian geometry does n diff] f sph]ical geometry1 :i* is only a bran* ( ord9>y geometry1 & consequ5tly is beyond all discussion4 ,beltrami1 9 correlat+ likewise ,loba*evski's two-dim5sional geometry ) a bran* ( ord9>y geometry1 has equally refut$ ! objection s f> z x is conc]n$4 ,"h is h[ he accompli%$ x4 ,consid] any figure on a surface4 ,imag9e ?is figure trac$ on a flexible & inext5sible canvas appli$ ov] ?is surface 9 su* a way t :5 ! canvas is displac$ & de=m$1 ! v>i\s l9es ( ?is figure c *ange _! =m )\t *ang+ _! l5g?4 ,9 g5]al1 ?is flexible & inext5sible figure c n be displac$ )\t leav+ ! surface2 b !re are c]ta9 p>ticul> surfaces = :i* su* a movem5t w\ld be possible2 ^! are ! surfaces ( con/ant curvature4 ,if we resume ! comp>ison made above & imag9e be+s )\t ?ickness liv+ on "o ( ^! surfaces1 !y w reg>d z possible ! motion ( a figure all ( ^: l9es rema9 con/ant 9 l5g?4 ,on ! contr>y1 su* a movem5t w\ld appe> absurd to animals )\t ?ickness liv+ on a surface ( v>iable curvature4 ,^! surfaces ( con/ant curvature are ( two sorts3 ,"s are ( positive curvature1 & c be de=m$ s z to be appli$ ov] a sph]e4 ,! geometry ( ^! surfaces r$uces itself !re=e to ! sph]ical geometry1 :i* is t ( ,riemann4 ,! o!rs are ( negative curvature4 ,beltrami has %[n t ! geometry ( ^! surfaces is none o!r ?an t ( ,loba*evski4 ,! two-dim5sional geometries ( ,riemann & ,loba*evski are ?us correlat$ to ! ,euclidean geometry4 ,,int]pretation ,,( ,,non-,,euclidean ,,geometries4--,s vani%es ! objection s f> z two-dim5sional geometries are conc]n$4 ,x w\ld be easy to ext5d ,beltrami's reason+ to ?ree-dim5sional geometries4 ,! m9ds t space ( f\r dim5sions does n repel w see no difficulty 9 x1 b !y are few4 ,i pref] !re=e to proce$ o!rwise4 ,consid] a c]ta9 plane1 :i* ,i %all call ! fundam5tal plane1 & con/ruct a sort ( diction>y1 by mak+ correspond ea* to ea* a d\ble s]ies ( t]ms writt5 9 two columns1 j z correspond 9 ! ord9>y diction>ies ! words ( two languages ^: significance is ! same3 ,space3 ,portion ( space situat$ above ! fundam5tal plane4 ,plane3 ,sph]e cutt+ ! fundam5tal plane or?ogonally4 ,/raight3 ,circle cutt+ ! fundam5tal plane or?ogonally4 ,sph]e3 ,sph]e4 ,circle3 ,circle4 ,angle3 ,angle4 ,di/ance betwe5 two po9ts3 ,log>i?m ( ! cross ratio ( ^! two po9ts & ! int]sections ( ! fundam5tal plane ) a circle pass+ ?r\gh ^! two po9ts & cutt+ x or?ogonally4 ,etc41 ,etc4 ,n[ take ,loba*evski's !orems & translate !m ) ! aid ( ?is diction>y z we translate a ,g]man text ) ! aid ( a ,g]man-,engli% diction>y4 ,we %all ?us obta9 !orems ( ! ord9>y geometry4 ,= example1 t !orem ( ,loba*evski3 '! sum ( ! angles ( a triangle is less ?an two "r angles' is translat$ ?us3 ,if a curvil9e> triangle has = sides circle-arcs :i* prolong$ w\ld cut or?ogonally ! fundam5tal plane1 ! sum ( ! angles ( ?is curvil9e> triangle w be less ?an two "r angles4 ,?us1 h[ev] f> ! consequ5ces ( ,loba*evski's hypo!ses are pu%$1 !y w nev] lead to a contradiction4 ,9 fact1 if two ( ,loba*evski's !orems 7 contradictory1 x w\ld be ! same ) ! translations ( ^! two !orems1 made by ! aid ( our diction>y1 b ^! translations are !orems ( ord9>y geometry & no "o d\bts t ! ord9>y geometry is free f contradiction4 ,:5ce comes ?is c]ta9ty & is x ju/ifi$8 ,t is a "q ,i c n treat "h because x w\ld require to be enl>g$ ^u1 b :i* is v int]e/+ & ,i ?9k n insoluble4 ,no?+ rema9s !n ( ! objection above =mulat$4 ,?is is n all4 ,loba*evski's geometry1 susceptible ( a concrete int]pretation1 ceases to be a va9 logical ex]cise & is capable ( applications2 ,i h n ! "t to speak "h ( ^! applications1 nor ( ! aid t ,kle9 & ,i h gott5 f !m = ! integration ( l9e> diff]5tial equations4 ,?is int]pretation moreov] is n unique1 & sev]al diction>ies analog\s to ! prec$+ c\ld be con/ruct$1 :i* w\ld enable u by a simple 'translation' to trans=m ,loba*evski's !orems into !orems ( ord9>y geometry4 ,,! ,,implicit ,,axioms4--,are ! axioms explicitly enunciat$ 9 our treatises ! sole f\ndations ( geometry8 ,we may be assur$ ( ! contr>y by notic+ t aft] !y are successively ab&on$ !re are /ill left ov] "s propositions common to ! !ories ( ,euclid1 ,loba*evski & ,riemann4 ,^! propositions mu/ re/ on premises ! geomet]s admit )\t enunciation4 ,x is int]e/+ to try to dis5tangle !m f ! classic demon/rations4 ,/u>t ,mill has claim$ t e def9ition conta9s an axiom1 because 9 def9+ "o affirms implicitly ! exi/5ce ( ! object def9$4 ,?is is go+ mu* too f>2 x is r>e t 9 ma!matics a def9ition is giv5 )\t its be+ foll[$ by ! demon/ration ( ! exi/5ce ( ! object def9$1 & :5 ?is is disp5s$ ) x is g5]ally because ! read] c easily supply x4 ,x mu/ n be =gott5 t ! ^w exi/5ce has n ! same s5se :5 x ref]s to a ma!matical entity & :5 x is a "q ( a mat]ial object4 ,a ma!matical entity exi/s1 provid$ its def9ition implies no contradiction1 ei!r 9 itself1 or ) ! propositions already admitt$4 ,b if ,/u>t ,mill's obs]vation c n be appli$ to all def9itions1 x is none ! less j = "s ( !m4 ,! plane is sometimes def9$ z foll[s3 ,! plane is a surface su* t ! /raight :i* jo9s any two ( its po9ts is :olly on ?is surface4 ,?is def9ition manife/ly hides a new axiom2 x is true we might *ange x1 & t w\ld be pref]able1 b !n we %\ld h to enunciate ! axiom explicitly4 ,o!r def9itions w\ld sugge/ reflections n less important4 ,su*1 = example1 is t ( ! equality ( two figures2 two figures are equal :5 !y c be sup]pos$2 to sup]pose !m "o mu/ be displac$ until x co9cides ) ! o!r2 b h[ %all x be displac$8 ,if we %\ld ask ?is1 no d\bt we %\ld be told t x mu/ be done )\t alt]+ ! %ape & z a rigid solid4 ,! vici\s circle w\ld !n be evid5t4 ,9 fact ?is def9ition def9es no?+2 x w\ld h no mean+ = a be+ liv+ 9 a _w :]e !re 7 only fluids4 ,if x seems cle> to u1 t is because we are us$ to ! prop]ties ( natural solids :i* d n diff] mu* f ^? ( ! ideal solids1 all ( ^: dim5sions are inv>iable4 ,yet1 imp]fect z x may be1 ?is def9ition implies an axiom4 ,! possibility ( ! motion ( a rigid figure is n a self-evid5t tru?1 or at lea/ x is s only 9 ! fa%ion ( ,euclid's po/ulate & n z an analytic judgm5t a priori w\ld be4 ,moreov]1 9 /udy+ ! def9itions & ! demon/rations ( geometry1 we see t "o is oblig$ to admit )\t pro( n only ! possibility ( ?is motion1 b "s ( its prop]ties besides4 ,?is is at once se5 f ! def9ition ( ! /raight l9e4 ,_m defective def9itions h be5 giv51 b ! true "o is t :i* is impli$ 9 all ! demon/rations :]e ! /raight l9e ent]s3 ,x may happ5 t ! motion ( a rigid figure is su* t all ! po9ts ( a l9e belong+ to ?is figure rema9 motionless :ile all ! po9ts situat$ outside ( ?is l9e move4 ,su* a l9e w be call$ a /raight l9e4 ,we h design$ly1 9 ?is enunciation1 sep>at$ ! def9ition f ! axiom x implies4 ,_m demon/rations1 su* z ^? ( ! cases ( ! equality ( triangles1 ( ! possibility ( dropp+ a p]p5dicul> f a po9t to a /raight1 presume propositions :i* are n enunciat$1 = !y require ! admission t x is possible to transport a figure 9 a c]ta9 way 9 space4 ,,! ,,f\r? ,,geometry4--,among ^! implicit axioms1 !re is "o :i* seems to me to m]it "s att5tion1 because :5 x is ab&on$ a f\r? geometry c be con/ruct$ z coh]5t z ^? ( ,euclid1 ,loba*evski & ,riemann4 ,to prove t a p]p5dicul> may always be erect$ at a po9t ,a to a /raight ,,ab1 we consid] a /raight ,,ac movable ar\nd ! po9t ,a & initially co9cid5t ) ! fix$ /raight ,,ab2 & we make x turn ab\t ! po9t ,a until x comes into ! prolongation ( ,,ab4 ,?us two propositions are presuppos$3 ,fir/1 t su* a rotation is possible1 & next t x may be cont9u$ until ! two /raights come into ! prolongation "o ( ! o!r4 ,if ! fir/ po9t is admitt$ & ! second reject$1 we are l$ to a s]ies ( !orems ev5 /rang] ?an ^? ( ,loba*evski & ,riemann1 b equally exempt f contradiction4 ,i %all cite only "o ( ^! !orems & t n ! mo/ s+ul>3 ,a real /raight may be p]p5dicul> to itself4 ,,lie',s ,,!orem4--,! numb] ( axioms implicitly introduc$ 9 ! classic demon/rations is great] ?an necess>y1 & x w\ld be int]e/+ to r$uce x to a m9imum4 ,x may fir/ be ask$ :e!r ?is r$uction is possible1 :e!r ! numb] ( necess>y axioms & t ( imag9able geometries are n inf9ite4 ,a !orem ( ,sophus ,lie dom9ates ?is :ole discussion4 ,x may be ?us enunciat$3 ,suppose ! foll[+ premises are admitt$3 #a ,space has n dim5sions2 #b ,! motion ( a rigid figure is possible2 #c ,x requires p conditions to det]m9e ! position ( ?is figure 9 space4 ,! numb] ( geometries compatible ) ^! premises w be limit$4 ,i may ev5 add t if n is giv51 a sup]ior limit c be assign$ to p4 ,if !re=e ! possibility ( motion is admitt$1 !re c be inv5t$ only a f9ite 7& ev5 a r small7 numb] ( ?ree-dim5sional geometries4 ,,riemann',s ,,geometries4--,yet ?is result seems contradict$ by ,riemann1 = ?is savant con/ructs an inf9ity ( diff]5t geometries1 & t to :i* 8 "n is ord9>ily giv5 is only a p>ticul> case4 ,all dep5ds1 he says1 on h[ ! l5g? ( a curve is def9$4 ,n[1 !re is an inf9ity ( ways ( def9+ ?is l5g?1 & ea* ( !m may be ! />t+ po9t ( a new geometry4 ,t is p]fectly true1 b mo/ ( ^! def9itions are incompatible ) ! motion ( a rigid figure1 :i* 9 ! !orem ( ,lie is suppos$ possible4 ,^! geometries ( ,riemann1 9 _m ways s int]e/+1 c\ld nev] !re=e be o!r ?an purely analytic & w\ld n l5d !mselves to demon/rations analog\s to ^? ( ,euclid4 ,,on ,,! ,,nature ,,( ,,axioms4--,mo/ ma!maticians reg>d ,loba*evski's geometry only z a m]e logical curiosity2 "s ( !m1 h[ev]1 h gone f>!r4 ,s9ce sev]al geometries are possible1 is x c]ta9 ours is ! true "o8 ,exp]i5ce no d\bt tea*es u t ! sum ( ! angles ( a triangle is equal to two "r angles2 b ?is is because ! triangles we deal ) are too little2 ! diff]5ce1 accord+ to ,loba*evski1 is proportional to ! surface ( ! triangle2 w x n p]haps become s5sible :5 we %all op]ate on l>g] triangles or :5 our measurem5ts %all become m precise8 ,! ,euclidean geometry w\ld ?us be only a provisional geometry4 ,to discuss ?is op9ion1 we %\ld fir/ ask ourselves :at is ! nature ( ! geometric axioms4 ,are !y syn!tic a priori judgm5ts1 z ,kant said8 ,!y w\ld !n impose !mselves ^u u ) su* =ce t we c\ld n conceive ! contr>y proposition1 nor build ^u x a !oretic edifice4 ,!re w\ld be no non-,euclidean geometry4 ,to be conv9c$ ( x take a v]itable syn!tic a priori judgm5t1 ! foll[+1 = in/ance1 ( :i* we h se5 ! prepond]ant rle 9 ! fir/ *apt]3 ,if a !orem is true = ! numb] #a1 & if x has be5 prov$ t x is true ( n #a provid$ x is true ( n1 x w be true ( all ! positive :ole numb]s4 ,!n try to escape f t &1 d5y+ ?is proposition1 try to f\nd a false ari?metic analog\s to non-,euclidean geometry--x c n be done2 "o w\ld ev5 be tempt$ at fir/ blu% to reg>d ^! judgm5ts z analytic4 ,moreov]1 resum+ our fiction ( animals )\t ?ickness1 we c h>dly admit t ^! be+s1 if _! m9ds are l ours1 w\ld adopt ! ,euclidean geometry :i* w\ld be contradict$ by all _! exp]i5ce4 ,%\ld we !re=e conclude t ! axioms ( geometry are exp]im5tal v]ities8 ,b we d n exp]im5t on ideal /raights or circles2 x c only be done on mat]ial objects4 ,on :at !n c\ld be bas$ exp]im5ts :i* %\ld s]ve z f\ndation = geometry8 ,! answ] is easy4 ,we h se5 above t we con/antly reason z if ! geometric figures behav$ l solids4 ,:at geometry w\ld borr[ f exp]i5ce w\ld !re=e be ! prop]ties ( ^! bodies4 ,! prop]ties ( light & its rectil9e> propagation h also giv5 rise to "s ( ! propositions ( geometry1 & 9 p>ticul> ^? ( projective geometry1 s t f ?is po9t ( view "o w\ld be tempt$ to say t metric geometry is ! /udy ( solids1 & projective1 t ( light4 ,b a difficulty rema9s1 & x is insurm\ntable4 ,if geometry 7 an exp]im5tal sci5ce1 x w\ld n be an exact sci5ce1 x w\ld be subject to a cont9ual revision4 ,nay1 x w\ld f ?is v day be convict$ ( error1 s9ce we "k t !re is no rigor\sly rigid solid4 ,! axioms ( geometry !re=e are nei!r syn!tic a priori judgm5ts nor exp]im5tal facts4 ,!y are conv5tions2 our *oice among all possible conv5tions is guid$ by exp]im5tal facts2 b x rema9s free & is limit$ only by ! necessity ( avoid+ all contradiction4 ,?us x is t ! po/ulates c rema9 rigor\sly true ev5 ?\gh ! exp]im5tal laws :i* h det]m9$ _! adoption are only approximative4 ,9 o!r words1 ! axioms ( geometry 7,i d n speak ( ^? ( ari?metic7 are m]ely disguis$ def9itions4 ,!n :at are we to ?9k ( t "q3 ,is ! ,euclidean geometry true8 ,x has no mean+4 ,z well ask :e!r ! metric sy/em is true & ! old measures false2 :e!r ,c>tesian coord9ates are true & pol> coord9ates false4 ,"o geometry c n be m true ?an ano!r2 x c only be m conv5i5t4 ,n[1 ,euclidean geometry is1 & w rema91 ! mo/ conv5i5t3 #a ,because x is ! simple/2 & x is s n only 9 consequ5ce ( our m5tal habits1 or ( ,i "k n :at direct intuition t we may h ( ,euclidean space2 x is ! simple/ 9 itself1 j z a polynomial ( ! fir/ degree is simpl] ?an "o ( ! second2 ! =mulas ( sph]ical trigonometry are m complicat$ ?an ^? ( plane trigonometry1 & !y w\ld /ill appe> s to an analy/ ignorant ( _! geometric signification4 #b ,because x accords suffici5tly well ) ! prop]ties ( natural solids1 ^? bodies :i* our h&s & our eyes comp>e & ) :i* we make our in/rum5ts ( measure4 ,,*apt] ,,iv ,,space ,,& ,,geometry ,let u beg9 by a little p>adox4 ,be+s ) m9ds l ours1 & hav+ ! same s5ses z we1 b )\t previ\s education1 w\ld receive f a suitably *os5 ext]nal _w impressions su* t !y w\ld be l$ to con/ruct a geometry o!r ?an t ( ,euclid & to localize ! ph5om5a ( t ext]nal _w 9 a non-,euclidean space1 or ev5 9 a space ( f\r dim5sions4 ,z = u1 ^: education has be5 accompli%$ by our actual _w1 if we 7 sudd5ly transport$ into ?is new _w1 we %\ld h no difficulty 9 ref]r+ its ph5om5a to our ,euclidean space4 ,conv]sely1 if ^! be+s 7 transport$ into our environm5t1 !y w\ld be l$ to relate our ph5om5a to non-,euclidean space4 ,nay m2 ) a little ef=t we likewise c\ld d x4 ,a p]son :o %\ld devote 8 exi/5ce to x might p]haps atta9 to a realization ( ! f\r? dim5sion4 ,,geometric ,,space ,,& ,,p]ceptual ,,space4--,x is (t5 said ! images ( ext]nal objects are localiz$ 9 space1 ev5 t !y c n be =m$ except on ?is condition4 ,x is also said t ?is space1 :i* s]ves ?us z a ready prep>$ frame = our s5sations & our repres5tations1 is id5tical ) t ( ! geomet]s1 ( :i* x possesses all ! prop]ties4 ,to all ! good m9ds :o ?9k ?us1 ! prec$+ /atem5t mu/ h appe>$ q extraord9>y4 ,b let u see :e!r !y are n subject to an illusion t a m pr(\nd analysis w\ld dissipate4 ,:at1 fir/ ( all1 are ! prop]ties ( space1 prop]ly s call$8 ,i mean ( t space :i* is ! object ( geometry & :i* ,i %all call geometric space4 ,! foll[+ are "s ( ! mo/ ess5tial3 #a ,x is cont9u\s2 #b ,x is inf9ite2 #c ,x has ?ree dim5sions2 #d ,x is homog5e\s1 t is to say1 all its po9ts are id5tical "o ) ano!r2 #e ,x is isotropic1 t is to say1 all ! /raights :i* pass ?r\gh ! same po9t are id5tical "o ) ano!r4 ,comp>e x n[ to ! frame ( our repres5tations & our s5sations1 :i* ,i may call p]ceptual space4 ,,visual ,,space4--,consid] fir/ a purely visual impression1 due to an image =m$ on ! bottom ( ! ret9a4 ,a cursory analysis %[s u ?is image z cont9u\s1 b z possess+ only two dim5sions2 ?is already di/+ui%es f geometric space :at we may call pure visual space4 ,besides1 ?is image is enclos$ 9 a limit$ frame4 ,f9ally1 !re is ano!r diff]5ce n less important3 ?is pure visual space is n homog5e\s4 ,all ! po9ts ( ! ret9a1 aside f ! images :i* may !re be =m$1 d n play ! same rle4 ,! yell[ spot c 9 no way be reg>d$ z id5tical ) a po9t on ! bord] ( ! ret9a4 ,9 fact1 n only does ! same object produce !re mu* m vivid impressions1 b 9 e limit$ frame ! po9t occupy+ ! c5t] ( ! frame w nev] appe> z equival5t to a po9t ne> "o ( ! bord]s4 ,no d\bt a m pr(\nd analysis w\ld %[ u t ?is cont9uity ( visual space & its two dim5sions are only an illusion2 x w\ld sep>ate x !re=e /ill m f geometric space1 b we %all n dwell on ?is rem>k4 ,sight1 h[ev]1 enables u to judge ( di/ances & consequ5tly to p]ceive a ?ird dim5sion4 ,b e "o kn[s t ?is p]ception ( ! ?ird dim5sion r$uces itself to ! s5sation ( ! ef=t at accommodation x is necess>y to make1 & to t ( ! conv]g5ce :i* mu/ be giv5 to ! two eyes1 to p]ceive an object di/9ctly4 ,^! are muscul> s5sations altoge!r diff]5t f ! visual s5sations :i* h giv5 u ! notion ( ! fir/ two dim5sions4 ,! ?ird dim5sion !re=e w n appe> to u z play+ ! same rle z ! o!r two4 ,:at may be call$ complete visual space is !re=e n an isotropic space4 ,x has1 x is true1 precisely ?ree dim5sions1 :i* means t ! elem5ts ( our visual s5sations 7^? at lea/ :i* comb9e to =m ! notion ( ext5sion7 w be completely def9$ :5 ?ree ( !m are kn[n2 to use ! language ( ma!matics1 !y w be functions ( ?ree indep5d5t v>iables4 ,b exam9e ! matt] a little m closely4 ,! ?ird dim5sion is reveal$ to u 9 two diff]5t ways3 by ! ef=t ( accommodation & by ! conv]g5ce ( ! eyes4 ,no d\bt ^! two indications are always concordant1 !re is a con/ant relation betwe5 !m1 or1 9 ma!matical t]ms1 ! two v>iables :i* measure ^! two muscul> s5sations d n appe> to u z indep5d5t2 or aga91 to avoid an appeal to ma!matical notions already r ref9$1 we may g back to ! language ( ! prec$+ *apt] & enunciate ! same fact z foll[s3 ,if two s5sations ( conv]g5ce1 ,a & ,b1 are indi/+ui%able1 ! two s5sations ( accommodation1 ,a' & ,b'1 :i* respectively accompany !m1 w be equally indi/+ui%able4 ,b "h we h1 s to speak1 an exp]im5tal fact2 a priori no?+ prev5ts our suppos+ ! contr>y1 & if ! contr>y takes place1 if ^! two muscul> s5sations v>y indep5d5tly ( "o ano!r1 we %all h to take acc\nt ( "o m indep5d5t v>iable1 & 'complete visual space' w appe> to u z a physical cont9uum ( f\r dim5sions4 ,we h "h ev51 ,i w add1 a fact ( ext]nal exp]i5ce4 ,no?+ prev5ts our suppos+ t a be+ ) a m9d l ours1 hav+ ! same s5se organs t we h1 may be plac$ 9 a _w :]e light w\ld only rea* him aft] hav+ trav]s$ reflect+ m$ia ( complicat$ =m4 ,! two indications :i* s]ve u 9 judg+ di/ances w\ld cease to be connect$ by a con/ant relation4 ,a be+ :o %\ld a*ieve 9 su* a _w ! education ( 8 s5ses w\ld no d\bt attribute f\r dim5sions to complete visual space4 ,,tactile ,,space ,,& ,,motor ,,space4--',tactile space' is /ill m complicat$ ?an visual space & f>!r remov$ f geometric space4 ,x is sup]flu\s to repeat = t\* ! discussion ,i h giv5 = sight4 ,b ap>t f ! data ( sight & t\*1 !re are o!r s5sations :i* contribute z mu* & m ?an !y to ! g5esis ( ! notion ( space4 ,^! are kn[n to e "o2 !y accompany all our movem5ts1 & are usually call$ muscul> s5sations4 ,! correspond+ frame con/itutes :at may be call$ motor space4 ,ea* muscle gives rise to a special s5sation capable ( augm5t+ or ( dim9i%+1 s t ! totality ( our muscul> s5sations w dep5d ^u z _m v>iables z we h muscles4 ,f ?is po9t ( view1 motor space w\ld h z _m dim5sions z we h muscles4 ,i "k x w be said t if ! muscul> s5sations contribute to =m ! notion ( space1 x is because we h ! s5se ( ! direction ( ea* movem5t & t x makes an integrant "p ( ! s5sation4 ,if ?is 7 s1 if a muscul> s5sation c\ld n arise except accompani$ by ?is geometric s5se ( direction1 geometric space w\ld inde$ be a =m impos$ ^u our s5sibility4 ,b ,i p]ceive no?+ at all ( ?is :5 ,i analyze my s5sations4 ,:at ,i d see is t ! s5sations :i* correspond to movem5ts 9 ! same direction are connect$ 9 my m9d by a m]e association ( ideas4 ,x is to ?is association t :at we call '! s5se ( direction' is r$ucible4 ,?is feel+ !re=e c n be f\nd 9 a s+le s5sation4 ,?is association is extremely complex1 = ! contraction ( ! same muscle may correspond1 accord+ to ! position ( ! limbs1 to movem5ts ( v diff]5t direction4 ,besides1 x is evid5tly acquir$2 x is1 l all associations ( ideas1 ! result ( a habit2 ?is habit itself results f v num]\s exp]i5ces2 )\t any d\bt1 if ! education ( our s5ses _h be5 accompli%$ 9 a diff]5t environm5t1 :]e we %\ld h be5 subject$ to diff]5t impressions1 contr>y habits w\ld h aris5 & our muscul> s5sations w\ld h be5 associat$ accord+ to o!r laws4 ,,*>act]i/ics ,,( ,,p]ceptual ,,space4--,?us p]ceptual space1 "u its triple =m1 visual1 tactile & motor1 is ess5tially diff]5t f geometric space4 ,x is nei!r homog5e\s1 nor isotropic2 "o c n ev5 say t x has ?ree dim5sions4 ,x is (t5 said t we 'project' into geometric space ! objects ( our ext]nal p]ception2 t we 'localize' !m4 ,has ?is a mean+1 & if s :at8 ,does x mean t we repres5t to ourselves ext]nal objects 9 geometric space8 ,our repres5tations are only ! reproduction ( our s5sations2 !y c !re=e be rang$ only 9 ! same frame z ^!1 t is to say1 9 p]ceptual space4 ,x is z impossible = u to repres5t to ourselves ext]nal bodies 9 geometric space1 z x is = a pa9t] to pa9t on a plane canvas objects ) _! ?ree dim5sions4 ,p]ceptual space is only an image ( geometric space1 an image alt]$ 9 %ape by a sort ( p]spective1 & we c repres5t to ourselves objects only by br++ !m "u ! laws ( ?is p]spective4 ,!re=e we d n repres5t to ourselves ext]nal bodies 9 geometric space1 b we reason on ^! bodies z if !y 7 situat$ 9 geometric space4 ,:5 x is said !n t we 'localize' su* & su* an object at su* & su* a po9t ( space1 :at does x mean8 ,x simply means t we repres5t to ourselves ! movem5ts x w\ld be necess>y to make to rea* t object2 & "o may n say t to repres5t to oneself ^! movem5ts1 x is necess>y to project ! movem5ts !mselves 9 space & t ! notion ( space mu/1 consequ5tly1 pre-exi/4 ,:5 ,i say t we repres5t to ourselves ^! movem5ts1 ,i mean only t we repres5t to ourselves ! muscul> s5sations :i* accompany !m & :i* h no geometric "* :atev]1 :i* consequ5tly d n at all imply ! preexi/5ce ( ! notion ( space4 ,,*ange ,,( ,,/ate ,,& ,,*ange ,,( ,,position4--,b1 x w be said1 if ! idea ( geometric space is n impos$ ^u our m9d1 & if1 on ! o!r h&1 none ( our s5sations c furni% x1 h[ c\ld x h come into exi/5ce8 ,?is is :at we h n[ to exam9e1 & x w take "s "t1 b ,i c summ>ize 9 a few words ! attempt at explanation t ,i am ab\t to develop4 ,none ( our s5sations1 isolat$1 c\ld h conduct$ u to ! idea ( space2 we are l$ to x only 9 /udy+ ! laws1 accord+ to :i* ^! s5sations succe$ ea* o!r4 ,we see fir/ t our impressions are subject to *ange2 b among ! *anges we asc]ta9 we are soon l$ to make a di/9ction4 ,at "o "t we say t ! objects :i* cause ^! impressions h *ang$ /ate1 at ano!r "t t !y h *ang$ position1 t !y h only be5 displac$4 ,:e!r an object *anges its /ate or m]ely its position1 ?is is always translat$ = u 9 ! same mann]3 by a modification 9 an aggregate ( impressions4 ,h[ !n c\ld we h be5 l$ to di/+ui% betwe5 ! two8 ,x is easy to acc\nt =4 ,if !re has only be5 a *ange ( position1 we c re/ore ! primitive aggregate ( impressions by mak+ movem5ts :i* replace u opposite ! mobile object 9 ! same relative situation4 ,we ?us correct ! modification t happ5$ & we ree/abli% ! initial /ate by an inv]se modification4 ,if x is a "q ( sight1 = example1 & if an object *anges its place be=e our eye1 we c 'foll[ x ) ! eye' & ma9ta9 its image on ! same po9t ( ! ret9a by appropriate movem5ts ( ! eyeball4 ,^! movem5ts we are consci\s ( because !y are volunt>y & because !y are accompani$ by muscul> s5sations1 b t does n mean t we repres5t !m to ourselves 9 geometric space4 ,s :at *>act]izes *ange ( position1 :at di/+ui%es x f *ange ( /ate1 is t x c always be correct$ 9 ?is way4 ,x may !re=e happ5 t we pass f ! totality ( impressions ,a to ! totality ,b 9 two diff]5t ways3 #a ,involunt>ily & )\t exp]i5c+ muscul> s5sations2 ?is happ5s :5 x is ! object :i* *anges place2 #b ,volunt>ily & ) muscul> s5sations2 ?is happ5s :5 ! object is motionless1 b we move s t ! object has relative motion ) ref]5ce to u4 ,if ?is be s1 ! passage f ! totality ,a to ! totality ,b is only a *ange ( position4 ,x foll[s f ?is t sight & t\* c\ld n h giv5 u ! notion ( space )\t ! aid ( ! 'muscul> s5se4' ,n only c\ld ?is notion n be d]iv$ f a s+le s5sation or ev5 f a s]ies ( s5sations1 b :at is m1 an immobile be+ c\ld nev] h acquir$ x1 s9ce1 n be+ able to correct by 8 movem5ts ! effects ( ! *anges ( position ( ext]ior objects1 he w\ld h _h no reason :atev] to di/+ui% !m f *anges ( /ate4 ,j z little c\ld he h acquir$ x if 8 motions _h n be5 volunt>y or 7 unaccompani$ by any s5sations4 ,,conditions ,,( ,,comp5sation4--,h[ is a l comp5sation possible1 ( su* sort t two *anges1 o!rwise indep5d5t ( ea* o!r1 reciprocally correct ea* o!r8 ,a m9d already famili> ) geometry w\ld reason z foll[s3 ,evid5tly1 if !re is to be comp5sation1 ! v>i\s p>ts ( ! ext]nal object1 on ! "o h&1 & ! v>i\s s5se organs1 on ! o!r h&1 mu/ be 9 ! same relative position aft] ! d\ble *ange4 ,&1 = t to be ! case1 ! v>i\s p>ts ( ! ext]nal object mu/ likewise h reta9$ 9 ref]5ce to ea* o!r ! same relative position1 & ! same mu/ be true ( ! v>i\s p>ts ( our body 9 reg>d to ea* o!r4 ,9 o!r words1 ! ext]nal object1 9 ! fir/ *ange1 mu/ be displac$ z is a rigid solid1 & s mu/ x be ) ! :ole ( our body 9 ! second *ange :i* corrects ! fir/4 ,"u ^! conditions1 comp5sation may take place4 ,b we :o z yet "k no?+ ( geometry1 s9ce = u ! notion ( space is n yet =m$1 we c n reason ?us1 we c n =esee a priori :e!r comp5sation is possible4 ,b exp]i5ce tea*es u t x sometimes happ5s1 & x is f ?is exp]im5tal fact t we />t to di/+ui% *anges ( /ate f *anges ( position4 ,,solid ,,bodies ,,& ,,geometry4--,among surr\nd+ objects !re are "s :i* frequ5tly und]go displacem5ts susceptible ( be+ ?us correct$ by a correlative movem5t ( our own body2 ^! are ! solid bodies4 ,! o!r objects1 ^: =m is v>iable1 only exceptionally und]go l displacem5ts 7*ange ( position )\t *ange ( =m74 ,:5 a body *anges its place & its %ape1 we c no long]1 by appropriate movem5ts1 br+ back our s5se-organs into ! same relative situation ) reg>d to ?is body2 consequ5tly we c no long] ree/abli% ! primitive totality ( impressions4 ,x is only lat]1 & z a consequ5ce ( new exp]i5ces1 t we le>n h[ to decompose ! bodies ( v>iable =m into small] elem5ts1 su* t ea* is displac$ almo/ 9 accordance ) ! same laws z solid bodies4 ,?us we di/+ui% 'de=mations' f o!r *anges ( /ate2 9 ^! de=mations1 ea* elem5t und]goes a m]e *ange ( position1 :i* c be correct$1 b ! modification und]gone by ! aggregate is m pr(\nd & is no long] susceptible ( correction by a correlative movem5t4 ,su* a notion is already v complex & mu/ h be5 relatively late 9 appe>+2 moreov] x c\ld n h aris5 if ! obs]vation ( solid bodies _h n already taught u to di/+ui% *anges ( position4 ,!re=e1 if !re 7 no solid bodies 9 nature1 !re w\ld be no geometry4 ,ano!r rem>k also des]ves a mom5t's att5tion4 ,suppose a solid body to occupy successively ! positions alpha & beta2 9 its fir/ position1 x w produce on u ! totality ( impressions ,a1 & 9 its second position ! totality ( impressions ,b4 ,let !re be n[ a second solid body1 hav+ qualities entirely diff]5t f ! fir/1 = example1 a diff]5t color4 ,suppose x to pass f ! position alpha1 :]e x gives u ! totality ( impressions ,a'1 to ! position beta1 :]e x gives ! totality ( impressions ,b'4 ,9 g5]al1 ! totality ,a w h no?+ 9 common ) ! totality ,a'1 nor ! totality ,b ) ! totality ,b'4 ,! transition f ! totality ,a to ! totality ,b & t f ! totality ,a' to ! totality ,b' are !re=e two *anges :i* 9 !mselves h 9 g5]al no?+ 9 common4 ,& yet we reg>d ^! two *anges bo? z displacem5ts &1 fur!rmore1 we consid] !m z ! same displacem5t4 ,h[ c t be8 ,x is simply because !y c bo? be correct$ by ! same correlative movem5t ( our body4 ',correlative movem5t' !re=e con/itutes ! sole connection betwe5 two ph5om5a :i* o!rwise we nev] %\ld h dreamt ( lik5+4 ,on ! o!r h&1 our body1 ?anks to ! numb] ( its articulations & muscles1 may make a multitude ( diff]5t movem5ts2 b all are n capable ( 'correct+' a modification ( ext]nal objects2 only ^? w be capable ( x 9 :i* our :ole body1 or at lea/ all ^? ( our s5se-organs :i* come into play1 are displac$ z a :ole1 t is1 )\t _! relative positions v>y+1 or 9 ! fa%ion ( a solid body4 ,to summ>ize3 #a ,we are l$ at fir/ to di/+ui% two categories ( ph5om5a3 ,"s1 involunt>y1 unaccompani$ by muscul> s5sations1 are attribut$ by u to ext]nal objects2 ^! are ext]nal *anges2 ,o!rs1 opposite 9 "* & attribut$ by u to ! movem5ts ( our own body1 are int]nal *anges2 #b ,we notice t c]ta9 *anges ( ea* ( ^! categories may be correct$ by a correlative *ange ( ! o!r category2 #c ,we di/+ui% among ext]nal *anges ^? :i* h ?us a correlative 9 ! o!r category2 ^! we call displacem5ts2 & j s among ! int]nal *anges1 we di/+ui% ^? :i* h a correlative 9 ! fir/ category4 ,?us are def9$1 ?anks to ?is reciprocity1 a p>ticul> class ( ph5om5a :i* we call displacem5ts4 ,! laws ( ^! ph5om5a con/itute ! object ( geometry4 ,,law ,,( ,,homog5eity4--,! fir/ ( ^! laws is ! law ( homog5eity4 ,suppose t1 by an ext]nal *ange alpha1 we pass f ! totality ( impressions ,a to ! totality ,b1 !n t ?is *ange alpha is correct$ by a correlative volunt>y movem5t beta1 s t we are br\ght back to ! totality ,a4 ,suppose n[ t ano!r ext]nal *ange alpha' makes u pass anew f ! totality ,a to ! totality ,b4 ,exp]i5ce tea*es u t ?is *ange alpha' is1 l alpha1 susceptible ( be+ correct$ by a correlative volunt>y movem5t beta' & t ?is movem5t beta' corresponds to ! same muscul> s5sations z ! movem5t beta :i* correct$ alpha4 ,?is fact is usually enunciat$ by say+ t space is homog5e\s & isotropic4 ,x may also be said t a movem5t :i* has once be5 produc$ may be repeat$ a second & a ?ird "t1 & s on1 )\t its prop]ties v>y+4 ,9 ! fir/ *apt]1 :]e we discuss$ ! nature ( ma!matical reason+1 we saw ! importance :i* mu/ be attribut$ to ! possibility ( repeat+ indef9itely ! same op]ation4 ,x is f ?is repetition t ma!matical reason+ gets its p[]2 x is1 !re=e1 ?anks to ! law ( homog5eity1 t x has a hold on ! geometric facts4 ,= complet5ess1 to ! law ( homog5eity %\ld be add$ a multitude ( o!r analog\s laws1 into ! details ( :i* ,i d n wi% to ent]1 b :i* ma!maticians sum up 9 a ^w by say+ t displacem5ts =m 'a gr\p4' ,,! ,,non-,,euclidean ,,_w4--,if geometric space 7 a frame impos$ on ea* ( our repres5tations1 consid]$ individually1 x w\ld be impossible to repres5t to ourselves an image /ripp$ ( ?is frame1 & we c\ld *ange no?+ ( our geometry4 ,b ?is is n ! case2 geometry is only ! rsum ( ! laws accord+ to :i* ^! images succe$ ea* o!r4 ,no?+ !n prev5ts u f imag9+ a s]ies ( repres5tations1 simil> 9 all po9ts to our ord9>y repres5tations1 b succe$+ "o ano!r accord+ to laws diff]5t f ^? to :i* we are accu/om$4 ,we c conceive !n t be+s :o receiv$ _! education 9 an environm5t :]e ^! laws 7 ?us upset might h a geometry v diff]5t f ours4 ,suppose1 = example1 a _w enclos$ 9 a great sph]e & subject to ! foll[+ laws3 ,! temp]ature is n uni=m2 x is greate/ at ! c5t]1 & dim9i%es 9 proportion to ! di/ance f ! c5t]1 to s9k to absolute z]o :5 ! sph]e is rea*$ 9 :i* ?is _w is enclos$4 ,to specify /ill m precisely ! law 9 accordance ) :i* ?is temp]ature v>ies3 ,let ,r be ! radius ( ! limit+ sph]e2 let r be ! di/ance ( ! po9t consid]$ f ! c5t] ( ?is sph]e4 ,! absolute temp]ature %all be proportional to ,r#b - r#b4 ,i %all fur!r suppose t1 9 ?is _w1 all bodies h ! same coeffici5t ( dilatation1 s t ! l5g? ( any rule is proportional to its absolute temp]ature4 ,f9ally1 ,i %all suppose t a body transport$ f "o po9t to ano!r ( diff]5t temp]ature is put imm$iately into !rmal equilibrium ) its new environm5t4 ,no?+ 9 ^! hypo!ses is contradictory or unimag9able4 ,a movable object w !n become small] & small] 9 proportion z x approa*es ! limit-sph]e4 ,note fir/ t1 ?\gh ?is _w is limit$ f ! po9t ( view ( our ord9>y geometry1 x w appe> inf9ite to its inhabitants4 ,9 fact1 :5 ^! try to approa* ! limit-sph]e1 !y cool (f & become small] & small]4 ,!re=e ! /eps !y take are also small] & small]1 s t !y c nev] rea* ! limit+ sph]e4 ,if1 = u1 geometry is only ! /udy ( ! laws accord+ to :i* rigid solids move1 = ^! imag9>y be+s x w be ! /udy ( ! laws ( motion ( solids di/ort$ by ! diff]5ces ( temp]ature j spok5 (4 ,no d\bt1 9 our _w1 natural solids likewise und]go v>iations ( =m & volume due to w>m+ or cool+4 ,b we neglect ^! v>iations 9 lay+ ! f\ndations ( geometry1 because1 besides _! be+ v slight1 !y are irregul> & consequ5tly seem to u accid5tal4 ,9 our hypo!tical _w1 ?is w\ld no long] be ! case1 & ^! v>iations w\ld foll[ regul> & v simple laws4 ,moreov]1 ! v>i\s solid pieces ( :i* ! bodies ( its inhabitants w\ld be compos$ w\ld und]go ! same v>iations ( =m & volume4 ,i w make /ill ano!r hypo!sis2 ,i w suppose light trav]ses m$ia div]sely refractive & su* t ! index ( refraction is inv]sely proportional to ,r#b - r#b4 ,x is easy to see t1 "u ^! conditions1 ! rays ( light w\ld n be rectil9e>1 b circul>4 ,to ju/ify :at prec$es1 x rema9s = me to %[ t c]ta9 *anges 9 ! position ( ext]nal objects c be correct$ by correlative movem5ts ( ! s5ti5t be+s inhabit+ ?is imag9>y _w1 & t 9 su* a way z to re/ore ! primitive aggregate ( impressions exp]i5c$ by ^! s5ti5t be+s4 ,suppose 9 fact t an object is displac$1 und]go+ de=mation1 n z a rigid solid1 b z a solid subject$ to unequal dilatations 9 exact con=mity to ! law ( temp]ature above suppos$4 ,p]mit me = brevity to call su* a movem5t a non-,euclidean displacem5t4 ,if a s5ti5t be+ happ5s to be 9 ! neighborhood1 8 impressions w be modifi$ by ! displacem5t ( ! object1 b he c ree/abli% !m by mov+ 9 a suitable mann]4 ,x suffices if f9ally ! aggregate ( ! object & ! s5ti5t be+1 consid]$ z =m+ a s+le body1 has und]gone "o ( ^? p>ticul> displacem5ts ,i h j call$ non-,euclidean4 ,?is is possible if x be suppos$ t ! limbs ( ^! be+s dilate accord+ to ! same law z ! o!r bodies ( ! _w !y inhabit4 ,al?\gh f ! po9t ( view ( our ord9>y geometry !re is a de=mation ( ! bodies 9 ?is displacem5t & _! v>i\s p>ts are no long] 9 ! same relative position1 nev]!less we %all see t ! impressions ( ! s5ti5t be+ h once m become ! same4 ,9 fact1 ?\gh ! mutual di/ances ( ! v>i\s p>ts may h v>i$1 yet ! p>ts orig9ally 9 contact are aga9 9 contact4 ,!re=e ! tactile impressions h n *ang$4 ,on ! o!r h&1 tak+ into acc\nt ! hypo!sis made above 9 reg>d to ! refraction & ! curvature ( ! rays ( light1 ! visual impressions w also h rema9$ ! same4 ,^! imag9>y be+s w !re=e l ourselves be l$ to classify ! ph5om5a !y witness & to di/+ui% among !m ! '*anges ( position' susceptible ( correction by a correlative volunt>y movem5t4 ,if !y con/ruct a geometry1 x w n be1 z ours is1 ! /udy ( ! movem5ts ( our rigid solids2 x w be ! /udy ( ! *anges ( position :i* !y w ?us h di/+ui%$ & :i* are none o!r ?an ! 'non-,euclidean displacem5ts'2 x w be non-,euclidean geometry4 ,?us be+s l ourselves1 educat$ 9 su* a _w1 w\ld n h ! same geometry z ours4 ,,! ,,_w ,,( ,,f\r ,,dim5sions4--,we c repres5t to ourselves a f\r-dim5sional _w j z well z a non-,euclidean4 ,! s5se ( sight1 ev5 ) a s+le eye1 toge!r ) ! muscul> s5sations relative to ! movem5ts ( ! eyeball1 w\ld suffice to tea* u space ( ?ree dim5sions4 ,! images ( ext]nal objects are pa9t$ on ! ret9a1 :i* is a two-dim5sional canvas2 !y are p]spectives4 ,b1 z eye & objects are movable1 we see 9 succession v>i\s p]spectives ( ! same body1 tak5 f diff]5t po9ts ( view4 ,at ! same "t1 we f9d t ! transition f "o p]spective to ano!r is (t5 accompani$ by muscul> s5sations4 ,if ! transition f ! p]spective ,a to ! p]spective ,b1 & t f ! p]spective ,a' to ! p]spective ,b' are accompani$ by ! same muscul> s5sations1 we lik5 !m "o to ! o!r z op]ations ( ! same nature4 ,/udy+ !n ! laws accord+ to :i* ^! op]ations comb9e1 we recognize t !y =m a gr\p1 :i* has ! same /ructure z t ( ! movem5ts ( rigid solids4 ,n[1 we h se5 t x is f ! prop]ties ( ?is gr\p we h d]iv$ ! notion ( geometric space & t ( ?ree dim5sions4 ,we und]/& ?us h[ ! idea ( a space ( ?ree dim5sions c\ld take bir? f ! pageant ( ^! p]spectives1 ?\gh ea* ( !m is ( only two dim5sions1 s9ce !y foll[ "o ano!r accord+ to c]ta9 laws4 ,well1 j z ! p]spective ( a ?ree-dim5sional figure c be made on a plane1 we c make t ( a f\r-dim5sional figure on a picture ( ?ree 7or ( two7 dim5sions4 ,to a geomet] ?is is only *ild's play4 ,we c ev5 take ( ! same figure sev]al p]spectives f sev]al diff]5t po9ts ( view4 ,we c easily repres5t to ourselves ^! p]spectives1 s9ce !y are ( only ?ree dim5sions4 ,imag9e t ! v>i\s p]spectives ( ! same object succe$ "o ano!r1 & t ! transition f "o to ! o!r is accompani$ by muscul> s5sations4 ,we %all ( c\rse consid] two ( ^! transitions z two op]ations ( ! same nature :5 !y are associat$ ) ! same muscul> s5sations4 ,no?+ !n prev5ts u f imag9+ t ^! op]ations comb9e accord+ to any law we *oose1 = example1 s z to =m a gr\p ) ! same /ructure z t ( ! movem5ts ( a rigid solid ( f\r dim5sions4 ,"h !re is no?+ unpicturable1 & yet ^! s5sations are precisely ^? :i* w\ld be felt by a be+ possess$ ( a two-dim5sional ret9a :o c\ld move 9 space ( f\r dim5sions4 ,9 ?is s5se we may say ! f\r? dim5sion is imag9able4 ,,conclusions4--,we see t exp]i5ce plays an indisp5sable rle 9 ! g5esis ( geometry2 b x w\ld be an error !nce to conclude t geometry is1 ev5 9 "p1 an exp]im5tal sci5ce4 ,if x 7 exp]im5tal1 x w\ld be only approximative & provisional4 ,& :at r\gh approximation6 ,geometry w\ld be only ! /udy ( ! movem5ts ( solids2 b 9 reality x is n occupi$ ) natural solids1 x has = object c]ta9 ideal solids1 absolutely rigid1 :i* are only a simplifi$ & v remote image ( natural solids4 ,! notion ( ^! ideal solids is drawn f all p>ts ( our m9d1 & exp]i5ce is only an occasion :i* induces u to br+ x =? f !m4 ,! object ( geometry is ! /udy ( a p>ticul> 'gr\p'2 b ! g5]al gr\p concept pre-exi/s1 at lea/ pot5tially1 9 our m9ds4 ,x is impos$ on u1 n z =m ( our s5se1 b z =m ( our und]/&+4 ,only1 f among all ! possible gr\ps1 t mu/ be *os5 :i* w be1 s to speak1 ! /&>d to :i* we %all ref] natural ph5om5a4 ,exp]i5ce guides u 9 ?is *oice )\t =c+ x ^u u2 x tells u n :i* is ! true/ geometry1 b :i* is ! mo/ conv5i5t4 ,notice t ,i h be5 able to describe ! fanta/ic worlds above imag9$ )\t ceas+ to employ ! language ( ord9>y geometry4 ,&1 9 fact1 we %\ld n h to *ange x if transport$ ?i!r4 ,be+s educat$ !re w\ld d\btless f9d x m conv5i5t to create a geometry diff]5t f ours1 & bett] adapt$ to _! impressions4 ,z = u1 9 face ( ! same impressions1 x is c]ta9 we %\ld f9d x m conv5i5t n to *ange our habits4 ,,*apt] ,v ,,exp]i5ce ,,& ,,geometry #a4 ,already 9 ! prec$+ pages ,i h sev]al times tri$ to %[ t ! pr9ciples ( geometry are n exp]im5tal facts & t 9 p>ticul> ,euclid's po/ulate c n be prov5 exp]im5tally4 ,h[ev] decisive appe> to me ! reasons already giv51 ,i believe ,i %\ld emphasize ?is po9t because "h a false idea is pr(\ndly root$ 9 _m m9ds4 #b4 ,if we con/ruct a mat]ial circle1 measure its radius & circumf]5ce1 & see if ! ratio ( ^! two l5g?s is equal to pi1 :at %all we h done8 ,we %all h made an exp]im5t on ! prop]ties ( ! matt] ) :i* we con/ruct$ ?is r\nd ?+1 & ( t ( :i* ! measure us$ 0 made4 #c4 ,,geometry ,,& ,,a/ronomy4--,! "q has also be5 put 9 ano!r way4 ,if ,loba*evski's geometry is true1 ! p>allax ( a v di/ant /> w be f9ite2 if ,riemann's is true1 x w be negative4 ,^! are results :i* seem )9 ! rea* ( exp]im5t1 & !re h be5 hopes t a/ronomical obs]vations might enable u to decide betwe5 ! ?ree geometries4 ,b 9 a/ronomy '/raight l9e' means simply 'pa? ( a ray ( light4' ,if !re=e negative p>allaxes 7 f\nd1 or if x 7 demon/rat$ t all p>allaxes are sup]ior to a c]ta9 limit1 two c\rses w\ld be op5 to u2 we might ei!r r5\nce ,euclidean geometry1 or else modify ! laws ( optics & suppose t light does n travel rigor\sly 9 a /raight l9e4 ,x is ne$less to add t all ! _w w\ld reg>d ! latt] solution z ! m advantage\s4 ,! ,euclidean geometry has1 !re=e1 no?+ to fe> f fre% exp]im5ts4 #d4 ,is ! position t5able1 t c]ta9 ph5om5a1 possible 9 ,euclidean space1 w\ld be impossible 9 non-,euclidean space1 s t exp]i5ce1 9 e/abli%+ ^! ph5om5a1 w\ld directly contradict ! non-,euclidean hypo!sis8 ,= my "p ,i ?9k no su* "q c be put4 ,to my m9d x is precisely equival5t to ! foll[+1 ^: absurdity is pat5t to all eyes3 are !re l5g?s expressible 9 met]s & c5timet]s1 b :i* c n be measur$ 9 fa?oms1 feet & in*es1 s t exp]i5ce1 9 asc]ta9+ ! exi/5ce ( ^! l5g?s1 w\ld directly contradict ! hypo!sis t !re are fa?oms divid$ into six feet8 ,exam9e ! "q m closely4 ,i suppose t ! /raight l9e possesses 9 ,euclidean space any two prop]ties :i* ,i %all call ,a & ,b2 t 9 non-,euclidean space x /ill possesses ! prop]ty ,a1 b no long] has ! prop]ty ,b2 f9ally ,i suppose t 9 bo? ,euclidean & non-,euclidean space ! /raight l9e is ! only l9e hav+ ! prop]ty ,a4 ,if ?is 7 s1 exp]i5ce w\ld be capable ( decid+ betwe5 ! hypo!sis ( ,euclid & t ( ,loba*evski4 ,x w\ld be asc]ta9$ t a def9ite concrete object1 accessible to exp]im5t1 = example1 a p5cil ( rays ( light1 possesses ! prop]ty ,a2 we %\ld conclude t x is rectil9e>1 & !n inve/igate :e!r or n x has ! prop]ty ,b4 ,b ?is is n s2 no prop]ty exi/s :i*1 l ?is prop]ty ,a1 c be an absolute crit]ion enabl+ u to recognize ! /raight l9e & to di/+ui% x f e o!r l9e4 ,%all we say1 = in/ance3 ! foll[+ is su* a prop]ty3 ! /raight l9e is a l9e su* t a figure ( :i* ?is l9e =ms a "p c be mov$ )\t ! mutual di/ances ( its po9ts v>y+ & s t all po9ts ( ?is l9e rema9 fix$8 ,?is1 9 fact1 is a prop]ty :i*1 9 ,euclidean or non-,euclidean space1 belongs to ! /raight & belongs only to x4 ,b h[ %all we asc]ta9 exp]im5tally :e!r x belongs to ?is or t concrete object8 ,x w be necess>y to measure di/ances1 & h[ %all "o "k t any concrete magnitude :i* ,i h measur$ ) my mat]ial in/rum5t really repres5ts ! ab/ract di/ance8 ,we h only pu%$ back ! difficulty4 ,9 reality ! prop]ty j enunciat$ is n a prop]ty ( ! /raight l9e alone1 x is a prop]ty ( ! /raight l9e & di/ance4 ,= x to s]ve z absolute crit]ion1 we %\ld h to be able to e/abli% n only t x does n also belong to a l9e o!r ?an ! /raight & to di/ance1 b 9 addition t x does n belong to a l9e o!r ?an ! /raight & to a magnitude o!r ?an di/ance4 ,n[ ?is is n true4 ,x is !re=e impossible to imag9e a concrete exp]im5t :i* c be int]pret$ 9 ! ,euclidean sy/em & n 9 ! ,loba*evskian sy/em1 s t ,i may conclude3 ,no exp]i5ce w ev] be 9 contradiction to ,euclid's po/ulate2 nor1 on ! o!r h&1 w any exp]i5ce ev] contradict ! po/ulate ( ,loba*evski4 #e4 ,b x is n en\gh t ! ,euclidean 7or non-,euclidean7 geometry c nev] be directly contradict$ by exp]i5ce4 ,might x n happ5 t x c accord ) exp]i5ce only by violat+ ! pr9ciple ( suffici5t reason or t ( ! relativity ( space8 ,i w expla9 myself3 consid] any mat]ial sy/em2 we %all h to reg>d1 on ! "o h&1 '! /ate' ( ! v>i\s bodies ( ?is sy/em 7= in/ance1 _! temp]ature1 _! electric pot5tial1 etc471 &1 on ! o!r h&1 _! position 9 space2 & among ! data :i* enable u to def9e ?is position we %all1 moreov]1 di/+ui% ! mutual di/ances ( ^! bodies1 :i* def9e _! relative positions1 f ! conditions :i* def9e ! absolute position ( ! sy/em & its absolute ori5tation 9 space4 ,! laws ( ! ph5om5a :i* w happ5 9 ?is sy/em w dep5d on ! /ate ( ^! bodies & _! mutual di/ances2 b1 because ( ! relativity & passivity ( space1 !y w n dep5d on ! absolute position & ori5tation ( ! sy/em4 ,9 o!r words1 ! /ate ( ! bodies & _! mutual di/ances at any in/ant w dep5d solely on ! /ate ( ^! same bodies & on _! mutual di/ances at ! initial in/ant1 b w n at all dep5d on ! absolute initial position ( ! sy/em or on its absolute initial ori5tation4 ,?is is :at = brevity ,i %all call ! law ( relativity4 ,hi!rto ,i h spok5 z a ,euclidean geomet]4 ,z ,i h said1 an exp]i5ce1 :atev] x be1 admits ( an int]pretation on ! ,euclidean hypo!sis2 b x admits ( "o equally on ! non-,euclidean hypo!sis4 ,well1 we h made a s]ies ( exp]im5ts2 we h int]pret$ !m on ! ,euclidean hypo!sis1 & we h recogniz$ t ^! exp]im5ts ?us int]pret$ d n violate ?is 'law ( relativity4' ,we n[ int]pret !m on ! non-,euclidean hypo!sis3 ?is is always possible2 only ! non-,euclidean di/ances ( our diff]5t bodies 9 ?is new int]pretation w n g5]ally be ! same z ! ,euclidean di/ances 9 ! primitive int]pretation4 ,w our exp]im5ts1 int]pret$ 9 ?is new mann]1 /ill be 9 accord ) our 'law ( relativity'8 ,& if !re 7 n ?is accord1 %\ld we n h also ! "r to say exp]i5ce _h prov5 ! falsity ( ! non-,euclidean geometry8 ,x is easy to see t ?is is an idle fe>2 9 fact1 to apply ! law ( relativity 9 all rigor1 x mu/ be appli$ to ! entire univ]se4 ,= if only a "p ( ?is univ]se 7 consid]$1 & if ! absolute position ( ?is "p happ5$ to v>y1 ! di/ances to ! o!r bodies ( ! univ]se w\ld likewise v>y1 _! influ5ce on ! "p ( ! univ]se consid]$ w\ld consequ5tly augm5t or dim9i%1 :i* might modify ! laws ( ! ph5om5a happ5+ !re4 ,b if our sy/em is ! entire univ]se1 exp]i5ce is p[]less to give in=mation ab\t its absolute position & ori5tation 9 space4 ,all t our in/rum5ts1 h[ev] p]fect$ !y may be1 c tell u w be ! /ate ( ! v>i\s p>ts ( ! univ]se & _! mutual di/ances4 ,s our law ( relativity may be ?us enunciat$3 ,! read+s we %all be able to make on our in/rum5ts at any in/ant w dep5d only on ! read+s we c\ld h made on ^! same in/rum5ts at ! initial in/ant4 ,n[ su* an enunciation is indep5d5t ( e int]pretation ( exp]im5tal facts4 ,if ! law is true 9 ! ,euclidean int]pretation1 x w also be true 9 ! non-,euclidean int]pretation4 ,all[ me "h a %ort digression4 ,i h spok5 above ( ! data :i* def9e ! position ( ! v>i\s bodies ( ! sy/em2 ,i %\ld likewise h spok5 ( ^? :i* def9e _! velocities2 ,i %\ld !n h _h to di/+ui% ! velocities ) :i* ! mutual di/ances ( ! diff]5t bodies v>y2 &1 on ! o!r h&1 ! velocities ( translation & rotation ( ! sy/em1 t is to say1 ! velocities ) :i* its absolute position & ori5tation v>y4 ,to fully satisfy ! m9d1 ! law ( relativity %\ld be expressible ?us3 ,! /ate ( bodies & _! mutual di/ances at any in/ant1 z well z ! velocities ) :i* ^! di/ances v>y at ?is same in/ant1 w dep5d only on ! /ate ( ^? bodies & _! mutual di/ances at ! initial in/ant1 & ! velocities ) :i* ^! di/ances v>y at ?is initial in/ant1 b !y w n dep5d ei!r ^u ! absolute initial position ( ! sy/em1 or ^u its absolute ori5tation1 or ^u ! velocities ) :i* ?is absolute position & ori5tation v>i$ at ! initial in/ant4 ,unhappily ! law ?us enunciat$ is n 9 accord ) exp]im5ts1 at lea/ z !y are ord9>ily int]pret$4 ,suppose a man be transport$ to a planet ^: heav5s 7 always cov]$ ) a ?ick curta9 ( cl\ds1 s t he c\ld nev] see ! o!r />s2 on t planet he w\ld live z if x 7 isolat$ 9 space4 ,yet ?is man c\ld become aw>e t x turn$1 ei!r by measur+ its oblat5ess 7done ord9>ily by ! aid ( a/ronomic obs]vations1 b capable ( be+ done by purely geodetic means71 or by repeat+ ! exp]im5t ( ,f\cault's p5dulum4 ,! absolute rotation ( ?is planet c\ld !re=e be made evid5t4 ,t is a fact :i* %ocks ! philosoph]1 b :i* ! physici/ is compell$ to accept4 ,we "k t f ?is fact ,newton inf]r$ ! exi/5ce ( absolute space2 ,i myself am q unable to adopt ?is view4 ,i %all expla9 :y 9 ,"p ,,iii4 ,= ! mom5t x is n my int5tion to ent] ^u ?is difficulty4 ,!re=e ,i mu/ resign myself1 9 ! enunciation ( ! law ( relativity1 to includ+ velocities ( e k9d among ! data :i* def9e ! /ate ( ! bodies4 ,h[ev] t may be1 ?is difficulty is ! same = ,euclid's geometry z = ,loba*evski's2 ,i !re=e ne$ n tr\ble myself ) x1 & h only m5tion$ x incid5tally4 ,:at is important is ! conclusion3 exp]im5t c n decide betwe5 ,euclid & ,loba*evski4 ,to sum up1 :i*ev] way we look at x1 x is impossible to discov] 9 geometric empiricism a rational mean+4 #f4 ,exp]im5ts only tea* u ! relations ( bodies to "o ano!r2 none ( !m be>s or c be> on ! relations ( bodies ) space1 or on ! mutual relations ( diff]5t p>ts ( space4 ,yes1 y reply1 a s+le exp]im5t is insuffici5t1 because x gives me only a s+le equation ) sev]al unkn[ns2 b :5 ,i %all h made en\gh exp]im5ts ,i %all h equations en\gh to calculate all my unkn[ns4 ,to "k ! height ( ! ma9ma/ does n suffice = calculat+ ! age ( ! capta94 ,:5 y h measur$ e bit ( wood 9 ! %ip y w h _m equations1 b y w "k 8 age no bett]4 ,all y\r measurem5ts be>+ only on y\r bits ( wood c reveal to y no?+ except conc]n+ ^! bits ( wood4 ,j s y\r exp]im5ts1 h[ev] num]\s !y may be1 be>+ only on ! relations ( bodies to "o ano!r1 w reveal to u no?+ ab\t ! mutual relations ( ! v>i\s p>ts ( space4 #g4 ,w y say t if ! exp]im5ts be> on ! bodies1 !y be> at lea/ ^u ! geometric prop]ties ( ! bodies8 ,b1 fir/1 :at d y und]/& by geometric prop]ties ( ! bodies8 ,i assume t x is a "q ( ! relations ( ! bodies ) space2 ^! prop]ties are !re=e inaccessible to exp]im5ts :i* be> only on ! relations ( ! bodies to "o ano!r4 ,?is alone w\ld suffice to %[ t !re c be no "q ( ^! prop]ties4 ,/ill let u beg9 by com+ to an und]/&+ ab\t ! s5se ( ! phrase3 geometric prop]ties ( bodies4 ,:5 ,i say a body is compos$ ( sev]al p>ts1 ,i assume t ,i d n enunciate !re9 a geometric prop]ty1 & ?is w\ld rema9 true ev5 if ,i agre$ to give ! improp] "n ( po9ts to ! smalle/ p>ts ,i consid]4 ,:5 ,i say t su* a "p ( su* a body is 9 contact ) su* a "p ( su* ano!r body1 ,i enunciate a proposition :i* conc]ns ! mutual relations ( ^! two bodies & n _! relations ) space4 ,i suppose y w grant me ^! are n geometric prop]ties2 at lea/ ,i am sure y w grant me ^! prop]ties are indep5d5t ( all k ( metric geometry4 ,?is presuppos$1 ,i imag9e t we h a solid body =m$ ( eight sl5d] iron rods1 ,,oa1 ,,ob1 ,,oc1 ,,od1 ,,oe1 ,,(1 ,,og1 ,,oh1 unit$ at "o ( _! extremities ,o4 ,let u besides h a second solid body1 = example a bit ( wood1 to be m>k$ ) ?ree little flecks ( ink :i* ,i %all call alpha1 beta1 gamma4 ,i fur!r suppose x asc]ta9$ t alphabetagamma may be br\ght into contact ) ,,ago 7,i mean alpha ) ,a1 & at ! same "t beta ) ,g & gamma ) ,o71 !n t we may br+ successively into contact alphabetagamma ) ,,bgo1 ,,cgo1 ,,dgo1 ,,ego1 ,,fgo1 !n ) ,,aho1 ,,bho1 ,,*o1 ,,dho1 ,,eho1 ,,fho1 !n alphagamma successively ) ,,ab1 ,,bc1 ,,cd1 ,,de1 ,,ef1 ,,fa4 ,^! are det]m9ations we may make )\t hav+ 9 advance any notion ab\t =m or ab\t ! metric prop]ties ( space4 ,!y 9 no wise be> on ! 'geometric prop]ties ( bodies4' ,& ^! det]m9ations w n be possible if ! bodies exp]im5t$ ^u move 9 accordance ) a gr\p hav+ ! same /ructure z ! ,loba*evskian gr\p 7,i mean accord+ to ! same laws z solid bodies 9 ,loba*evski's geometry74 ,!y suffice !re=e to prove t ^! bodies move 9 accordance ) ! ,euclidean gr\p1 or at lea/ t !y d n move accord+ to ! ,loba*evskian gr\p4 ,t !y are compatible ) ! ,euclidean gr\p is easy to see4 ,= !y c\ld be made if ! body alphabetagamma 0 a rigid solid ( our ord9>y geometry pres5t+ ! =m ( a "r-angl$ triangle1 & if ! po9ts ,,abcdefgh 7 ! summits ( a polyh$ron =m$ ( two regul> hexagonal pyramids ( our ord9>y geometry1 hav+ = common base ,,abcdef & = apices ! "o ,g & ! o!r ,h4 ,suppose n[ t 9 place ( ! prec$+ det]m9ation x is obs]v$ t z above alphabetagamma c be successively appli$ to ,,ago1 ,,bgo1 ,,cgo1 ,,dgo1 ,,ego1 ,,aho1 ,,bho1 ,,*o1 ,,dho1 ,,eho1 ,,fho1 !n t alphabeta 7& no long] alphagamma7 c be successively appli$ to ,,ab1 ,,bc1 ,,cd1 ,,de1 ,,ef & ,,fa4 ,^! are det]m9ations :i* c\ld be made if non-,euclidean geometry 7 true1 if ! bodies alphabetagamma & ,,oabcdefgh 7 rigid solids1 & if ! fir/ 7 a "r-angl$ triangle & ! second a d\ble regul> hexagonal pyramid ( suitable dim5sions4 ,!re=e ^! new det]m9ations are n possible if ! bodies move accord+ to ! ,euclidean gr\p2 b !y become s if x be suppos$ t ! bodies move accord+ to ! ,loba*evskian gr\p4 ,!y w\ld suffice1 !re=e 7if "o made !m71 to prove t ! bodies 9 "q d n move accord+ to ! ,euclidean gr\p4 ,?us1 )\t mak+ any hypo!sis ab\t =m1 ab\t ! nature ( space1 ab\t ! relations ( bodies to space1 & )\t attribut+ to bodies any geometric prop]ty1 ,i h made obs]vations :i* h enabl$ me to %[ 9 "o case t ! bodies exp]im5t$ ^u move accord+ to a gr\p ^: /ructure is ,euclidean1 9 ! o!r case t !y move accord+ to a gr\p ^: /ructure is ,loba*evskian4 ,& "o may n say t ! fir/ aggregate ( det]m9ations w\ld con/itute an exp]im5t prov+ t space is ,euclidean1 & ! second an exp]im5t prov+ t space is non-,euclidean4 ,9 fact "o c\ld imag9e 7,i say imag9e7 bodies mov+ s z to r5d] possible ! second s]ies ( det]m9ations4 ,& ! pro( is t ! fir/ me*anician met c\ld con/ruct su* bodies if he c>$ to take ! pa9s & make ! outlay4 ,y w n conclude f t1 h[ev]1 t space is non-,euclidean4 ,nay1 s9ce ! ord9>y solid bodies w\ld cont9ue to exi/ :5 ! me*anician _h con/ruct$ ! /range bodies ( :i* ,i h j spok51 x w\ld be necess>y to conclude t space is at ! same "t ,euclidean & non-,euclidean4 ,suppose1 = example1 t we h a great sph]e ( radius ,r & t ! temp]ature decreases f ! c5t] to ! surface ( ?is sph]e accord+ to ! law ( :i* ,i h spok5 9 describ+ ! non-,euclidean _w4 ,we might h bodies ^: expansion w\ld be negligible & :i* w\ld act l ord9>y rigid solids2 &1 on ! o!r h&1 bodies v dilatable & :i* w\ld act l non-,euclidean solids4 ,we might h two d\ble pyramids ,,oabcdefgh & ,o',a',b',c',d',e',f',g',h' & two triangles alphabetagamma & alpha'beta'gamma'4 ,! fir/ d\ble pyramid might be rectil9e> & ! second curvil9e>2 ! triangle alphabetagamma might be made ( inexpansible matt] & ! o!r ( a v dilatable matt]4 ,x w\ld !n be possible to make ! fir/ obs]vations ) ! d\ble pyramid ,,oah & ! triangle alphabetagamma1 & ! second ) ! d\ble pyramid ,o',a',h' & ! triangle alpha'beta'gamma'4 ,& !n exp]im5t w\ld seem to prove fir/ t ! ,euclidean geometry is true & !n t x is false4 ,exp]im5ts !re=e h a be>+1 n on space1 b on bodies4 ,,supplem5t #h4 ,to complete ! matt]1 ,i ought to speak ( a v delicate "q1 :i* w\ld require long developm5t2 ,i %all conf9e myself to summ>iz+ "h :at ,i h exp\nd$ 9 ! ,revue de ,mtaphysique et de ,morale & 9 ,! ,moni/4 ,:5 we say space has ?ree dim5sions1 :at d we mean8 ,we h se5 ! importance ( ^? 'int]nal *anges' reveal$ to u by our muscul> s5sations4 ,!y may s]ve to *>act]ize ! v>i\s attitudes ( our body4 ,take arbitr>ily z orig9 "o ( ^! attitudes ,a4 ,:5 we pass f ?is initial attitude to any o!r attitude ,b1 we feel a s]ies ( muscul> s5sations1 & ?is s]ies ,s w def9e ,b4 ,obs]ve1 h[ev]1 t we %all (t5 reg>d two s]ies ,s & ,s' z def9+ ! same attitude ,b 7s9ce ! initial & f9al attitudes ,a & ,b rema9+ ! same1 ! int]m$i>y attitudes & ! correspond+ s5sations may diff]74 ,h[ !n %all we recognize ! equival5ce ( ^! two s]ies8 ,because !y may s]ve to comp5sate ! same ext]nal *ange1 or m g5]ally because1 :5 x is a "q ( comp5sat+ an ext]nal *ange1 "o ( ! s]ies c be replac$ by ! o!r4 ,among ^! s]ies1 we h di/+ui%$ ^? :i* ( !mselves alone c comp5sate an ext]nal *ange1 & :i* we h call$ 'displacem5ts4' ,z we c n discrim9ate betwe5 two displacem5ts :i* are too close toge!r1 ! totality ( ^! displacem5ts pres5ts ! *>act]i/ics ( a physical cont9uum2 exp]i5ce tea*es u t !y are ^? ( a physical cont9uum ( six dim5sions2 b we d n yet "k h[ _m dim5sions space itself has1 we mu/ fir/ solve ano!r "q4 ,:at is a po9t ( space8 ,ev]ybody ?9ks he kn[s1 b t is an illusion4 ,:at we see :5 we try to repres5t to ourselves a po9t ( space is a black speck on :ite pap]1 a speck ( *alk on a blackbo>d1 always an object4 ,! "q %\ld !re=e be und]/ood z foll[s3 ,:at d ,i mean :5 ,i say ! object ,b is at ! same po9t t ! object ,a occupi$ j n[8 ,or fur!r1 :at crit]ion w enable me to appreh5d ?is8 ,i mean t1 al?\gh ,i h n budg$ 7:i* my muscul> s5se tells me71 my fir/ f+] :i* j n[ t\*$ ! object ,a t\*es at pres5t ! object ,b4 ,i c\ld h us$ o!r crit]ia2 = in/ance ano!r f+] or ! s5se ( sight4 ,b ! fir/ crit]ion is suffici5t2 ,i "k t if x answ]s yes1 all ! o!r crit]ia w give ! same response4 ,i "k x by exp]i5ce1 ,i c n "k x a priori4 ,= ! same reason ,i say t t\* c n be ex]cis$ at a di/ance2 ?is is ano!r way ( enunciat+ ! same exp]im5tal fact4 ,& if1 on ! contr>y1 ,i say t sight acts at a di/ance1 x means t ! crit]ion furni%$ by sight may respond yes :ile ! o!rs reply no4 ,& 9 fact1 ! object1 al?\gh mov$ away1 may =m its image at ! same po9t ( ! ret9a4 ,sight responds yes1 ! object has rema9$ at ! same po9t & t\* answ]s no1 because my f+] :i* j n[ t\*$ ! object1 at pres5t t\*es x no long]4 ,if exp]i5ce _h %[n u t "o f+] may respond no :5 ! o!r says yes1 we %\ld likewise say t t\* acts at a di/ance4 ,9 %ort1 = ea* attitude ( my body1 my fir/ f+] det]m9es a po9t1 & ?is x is1 & ?is alone1 :i* def9es a po9t ( space4 ,to ea* attitude corresponds ?us a po9t2 b x (t5 happ5s t ! same po9t corresponds to sev]al diff]5t attitudes 79 ?is case we say our f+] has n budg$1 b ! re/ ( ! body has mov$74 ,we di/+ui%1 !re=e1 among ! *anges ( attitude ^? :]e ! f+] does n budge4 ,h[ are we l$ !reto8 ,x is because (t5 we notice t 9 ^! *anges ! object :i* is 9 contact ) ! f+] rema9s 9 contact ) x4 ,range1 !re=e1 9 ! same class all ! attitudes obta9able f ea* o!r by "o ( ! *anges we h ?us di/+ui%$4 ,to all ! attitudes ( ! class w correspond ! same po9t ( space4 ,!re=e to ea* class w correspond a po9t & to ea* po9t a class4 ,b "o may say t :at exp]i5ce arrives at is n ! po9t1 x is ?is class ( *anges or1 bett]1 ! correspond+ class ( muscul> s5sations4 ,& :5 we say space has ?ree dim5sions1 we simply mean t ! totality ( ^! classes appe>s to u ) ! *>act]i/ics ( a physical cont9uum ( ?ree dim5sions4 ,"o might be tempt$ to conclude t x is exp]i5ce :i* has taught u h[ _m dim5sions space has4 ,b 9 reality "h also our exp]i5ces h be>+1 n on space1 b on our body & its relations ) ! neighbor+ objects4 ,moreov] !y are excessively crude4 ,9 our m9d pre-exi/$ ! lat5t idea ( a c]ta9 numb] ( gr\ps--^? ^: !ory ,lie has develop$4 ,:i* gr\p %all we *oose1 to make ( x a sort ( /&>d ) :i* to comp>e natural ph5om5a8 ,&1 ?is gr\p *os51 :i* ( its sub-gr\ps %all we take to *>act]ize a po9t ( space8 ,exp]i5ce has guid$ u by %[+ u :i* *oice be/ adapts itself to ! prop]ties ( our body4 ,b its rle is limit$ to t4 ,,ance/ral ,,exp]i5ce ,x has (t5 be5 said t if individual exp]i5ce c\ld n create geometry ! same is n true ( ance/ral exp]i5ce4 ,b :at does t mean8 ,is x meant t we c\ld n exp]im5tally demon/rate ,euclid's po/ulate1 b t our ance/ors h be5 able to d x8 ,n 9 ! lea/4 ,x is meant t by natural selection our m9d has adapt$ itself to ! conditions ( ! ext]nal _w1 t x has adopt$ ! geometry mo/ advantage\s to ! species3 or 9 o!r words ! mo/ conv5i5t4 ,?is is entirely 9 con=mity ) our conclusions2 geometry is n true1 x is advantage\s4 ,,"p ,,iii ,,=ce ,,*apt] ,,vi ,,! ,,classic ,,me*anics ,! ,engli% tea* me*anics z an exp]im5tal sci5ce2 on ! cont95t x is always exp\nd$ z m or less a d$uctive & a priori sci5ce4 ,! ,engli% are "r1 t goes )\t say+2 b h[ c\ld ! o!r me?od h be5 p]si/$ 9 s long8 ,:y h ! cont95tal savants :o h s\ght to get out ( ! ruts ( _! pr$ecessors be5 usually unable to free !mselves completely8 ,on ! o!r h&1 if ! pr9ciples ( me*anics are only ( exp]im5tal orig91 are !y n !re=e only approximate & provisional8 ,might n new exp]im5ts "s day lead u to modify or ev5 to ab&on !m8 ,su* are ! que/ions :i* naturally obtrude !mselves1 & ! difficulty ( solution comes pr9cipally f ! fact t ! treatises on me*anics d n cle>ly di/+ui% betwe5 :at is exp]im5t1 :at is ma!matical reason+1 :at is conv5tion1 :at is hypo!sis4 ,t is n all3 #a ,!re is no absolute space & we c conceive only ( relative motions2 yet usually ! me*anical facts are enunciat$ z if !re 7 an absolute space to :i* to ref] !m4 #b ,!re is no absolute "t2 to say two durations are equal is an ass]tion :i* has by itself no mean+ & :i* c acquire "o only by conv5tion4 #c ,n only h we no direct intuition ( ! equality ( two durations1 b we h n ev5 direct intuition ( ! simultaneity ( two ev5ts occurr+ 9 diff]5t places3 ?is ,i h expla9$ 9 an article entitl$ ,la mesure du temps4#c #c ,revue de ,mtaphysique et de ,morale1 t4 ,,vi41 pp4 #a-ac 7,janu>y1 #ahih74 #d ,f9ally1 our ,euclidean geometry is itself only a sort ( conv5tion ( language2 me*anical facts might be enunciat$ ) ref]5ce to a non-,euclidean space :i* w\ld be a guide less conv5i5t ?an1 b j z legitimate z1 our ord9>y space2 ! enunciation w\ld ?us become mu* m complicat$1 b x w\ld rema9 possible4 ,?us absolute space1 absolute "t1 geometry itself1 are n conditions :i* impose !mselves on me*anics2 all ^! ?+s are no m antec$5t to me*anics ?an ! ,fr5* language is logically antec$5t to ! v]ities "o expresses 9 ,fr5*4 ,we might try to enunciate ! fundam5tal laws ( me*anics 9 a language indep5d5t ( all ^! conv5tions2 we %\ld ?us )\t d\bt get a bett] idea ( :at ^! laws are 9 !mselves2 ?is is :at ,m4 ,&rade has attempt$ to d1 at lea/ 9 "p1 9 8 ,leons de mcanique physique4 ,! enunciation ( ^! laws w\ld become ( c\rse mu* m complicat$1 because all ^! conv5tions h be5 devis$ expressly to abridge & simplify ?is enunciation4 ,z = me1 save 9 :at conc]ns absolute space1 ,i %all ignore all ^! difficulties2 n t ,i fail to appreciate !m1 f> f t2 b we h suffici5tly exam9$ !m 9 ! fir/ two p>ts ( ! book4 ,i %all !re=e admit1 provisionally1 absolute "t & ,euclidean geometry4 ,,! ,,pr9ciple ,,( ,,in]tia4--,a body act$ on by no =ce c only move uni=mly 9 a /raight l9e4 ,is ?is a tru? impos$ a priori ^u ! m9d8 ,if x 7 s1 h[ %\ld ! ,greeks h fail$ to recognize x8 ,h[ c\ld !y h believ$ t motion /ops :5 ! cause :i* gave bir? to x ceases8 ,or aga9 t e body if no?+ prev5ts1 w move 9 a circle1 ! noble/ ( motions8 ,if x is said t ! velocity ( a body c n *ange if !re is no reason = x to *ange1 c\ld x n be ma9ta9$ j z well t ! position ( ?is body c n *ange1 or t ! curvature ( its trajectory c n *ange1 if no ext]nal cause int]v5es to modify !m8 ,is ! pr9ciple ( in]tia1 :i* is n an a priori tru?1 !re=e an exp]im5tal fact8 ,b has any "o ev] exp]im5t$ on bodies )drawn f ! action ( e =ce8 &1 if s1 h[ 0 x kn[n t ^! bodies 7 subject$ to no =ce8 ,! example ord9>ily cit$ is t ( a ball roll+ a v long "t on a m>ble table2 b :y d we say x is subject$ to no =ce8 ,is ?is because x is too remote f all o!r bodies to exp]i5ce any appreciable action f !m8 ,yet x is n f>!r f ! e>? ?an if x 7 ?r[n freely into ! air2 & e "o kn[s t 9 ?is case x w\ld exp]i5ce ! influ5ce ( gravity due to ! attraction ( ! e>?4 ,tea*]s ( me*anics usually pass rapidly ov] ! example ( ! ball2 b !y add t ! pr9ciple ( in]tia is v]ifi$ indirectly by its consequ5ces4 ,!y express !mselves badly2 !y evid5tly mean x is possible to v]ify v>i\s consequ5ces ( a m g5]al pr9ciple1 ( :i* t ( in]tia is only a p>ticul> case4 ,i %all propose = ?is g5]al pr9ciple ! foll[+ enunciation3 ,! accel]ation ( a body dep5ds only ^u ! position ( ?is body & ( ! neighbor+ bodies & ^u _! velocities4 ,ma!maticians w\ld say ! movem5ts ( all ! mat]ial molecules ( ! univ]se dep5d on diff]5tial equations ( ! second ord]4 ,to make x cle> t ?is is really ! natural g5]alization ( ! law ( in]tia1 ,i %all beg y to p]mit me a bit ( fiction4 ,! law ( in]tia1 z ,i h said above1 is n impos$ ^u u a priori2 o!r laws w\ld be q z compatible ) ! pr9ciple ( suffici5t reason4 ,if a body is subject$ to no =ce1 9 lieu ( suppos+ its velocity n to *ange1 x might be suppos$ t x is its position or else its accel]ation :i* is n to *ange4 ,well1 imag9e = an in/ant t "o ( ^! two hypo!tical laws is a law ( nature & replaces our law ( in]tia4 ,:at w\ld be its natural g5]alization8 ,a mom5t's ?\ght w %[ u4 ,9 ! fir/ case1 we mu/ suppose t ! velocity ( a body dep5ds only ^u its position & ^u t ( ! neighbor+ bodies2 9 ! second case t ! *ange ( accel]ation ( a body dep5ds only ^u ! position ( ?is body & ( ! neighbor+ bodies1 ^u _! velocities & ^u _! accel]ations4 ,or to speak ! language ( ma!matics1 ! diff]5tial equations ( motion w\ld be ( ! fir/ ord] 9 ! fir/ case1 & ( ! ?ird ord] 9 ! second case4 ,let u slightly modify our fiction4 ,suppose a _w analog\s to our sol> sy/em1 b :]e1 by a /range *ance1 ! orbits ( all ! planets are )\t ecc5tricity & )\t incl9ation4 ,suppose fur!r t ! masses ( ^! planets are too slight = _! mutual p]turbations to be s5sible4 ,a/ronom]s inhabit+ "o ( ^! planets c\ld n fail to conclude t ! orbit ( a /> c only be circul> & p>allel to a c]ta9 plane2 ! position ( a /> at a giv5 in/ant w\ld !n suffice to det]m9e its velocity & its :ole pa?4 ,! law ( in]tia :i* !y w\ld adopt w\ld be ! fir/ ( ! two hypo!tical laws ,i h m5tion$4 ,imag9e n[ t ?is sy/em is "s day trav]s$ ) great velocity by a body ( va/ mass1 com+ f di/ant con/ellations4 ,all ! orbits w\ld be pr(\ndly di/urb$4 ,/ill our a/ronom]s w\ld n be too greatly a/oni%$2 !y w\ld v well div9e t ?is new /> 0 alone to blame = all ! mis*ief4 ,b1 !y w\ld say1 :5 x is gone1 ord] w ( itself be ree/abli%$2 no d\bt ! di/ances ( ! planets f ! sun w n rev]t to :at !y 7 be=e ! cataclysm1 b :5 ! p]turb+ /> is gone1 ! orbits w aga9 become circul>4 ,x w\ld only be :5 ! di/urb+ body 0 gone & :5 nev]!less ! orbits1 9 lieu ( aga9 becom+ circul>1 became elliptic1 t ^! a/ronom]s w\ld become consci\s ( _! error & ! necessity ( remak+ all _! me*anics4 ,i h dwelt some:at ^u ^! hypo!ses because x seems to me "o c cle>ly compreh5d :at our g5]aliz$ law ( in]tia really is only 9 contra/+ x ) a contr>y hypo!sis4 ,well1 n[1 has ?is g5]aliz$ law ( in]tia be5 v]ifi$ by exp]im5t1 or c x be8 ,:5 ,newton wrote ! ,pr9cipia he q reg>d$ ?is tru? z exp]im5tally acquir$ & demon/rat$4 ,x 0 s 9 8 eyes1 n only ?r\gh ! an?ropomorphism ( :i* we %all speak fur!r on1 b ?r\gh ! "w ( ,galileo4 ,x 0 s ev5 f ,kepl]'s laws !mselves2 9 accordance ) ^! laws1 9 fact1 ! pa? ( a planet is completely det]m9$ by its initial position & initial velocity2 ?is is j :at our g5]aliz$ law ( in]tia requires4 ,= ?is pr9ciple to be only 9 appe>ance true1 = "o to h cause to dread hav+ "s day to replace x by "o ( ! analog\s pr9ciples ,i h j n[ contra/$ ) x1 w\ld be necess>y our hav+ be5 misl$ by "s amaz+ *ance1 l t :i*1 9 ! fiction above develop$1 l$ into error our imag9>y a/ronom]s4 ,su* a hypo!sis is too unlikely to delay ov]4 ,no "o w believe t su* co9cid5ces c happ52 no d\bt ! probability ( two ecc5tricities be+ bo? precisely null1 to )9 errors ( obs]vation1 is n less ?an ! probability ( "o be+ precisely equal to #j4a1 = in/ance1 & ! o!r to #j4b1 to )9 errors ( obs]vation4 ,! probability ( a simple ev5t is n less ?an t ( a complicat$ ev5t2 & yet1 if ! fir/ happ5s1 we %all n cons5t to attribute x to *ance2 we %\ld n believe t nature _h act$ expressly to deceive u4 ,! hypo!sis ( an error ( ?is sort be+ disc>d$1 x may !re=e be admitt$ t 9 s f> z a/ronomy is conc]n$1 our law has be5 v]ifi$ by exp]im5t4 ,b a/ronomy is n ! :ole ( physics4 ,may we n fe> le/ "s day a new exp]im5t %\ld come to falsify ! law 9 "s doma9 ( physics8 ,an exp]im5tal law is always subject to revision2 "o %\ld always expect to see x replac$ by a m precise law4 ,yet no "o s]i\sly ?9ks t ! law we are speak+ ( w ev] be ab&on$ or am5d$4 ,:y8 ,precisely because x c nev] be subject$ to a decisive te/4 ,fir/ ( all1 9 ord] t ?is trial %\ld be complete1 x w\ld be necess>y t aft] a c]ta9 "t all ! bodies 9 ! univ]se %\ld rev]t to _! initial positions ) _! initial velocities4 ,x might !n be se5 :e!r1 />t+ f ?is mom5t1 !y w\ld resume _! orig9al pa?s4 ,b ?is te/ is impossible1 x c be only p>tially appli$1 &1 h[ev] well x is made1 !re w always be "s bodies :i* w n rev]t to _! initial positions2 ?us e d]ogation ( ! law w easily f9d its explanation4 ,?is is n all2 9 a/ronomy we see ! bodies ^: motions we /udy & we usually assume t !y are n subject$ to ! action ( o!r invisible bodies4 ,"u ^! conditions our law mu/ inde$ be ei!r v]ifi$ or n v]ifi$4 ,b x is n ! same 9 physics2 if ! physical ph5om5a are due to motions1 x is to ! motions ( molecules :i* we d n see4 ,if !n ! accel]ation ( "o ( ! bodies we see appe>s to u to dep5d on some?+ else besides ! positions or velocities ( o!r visible bodies or ( invisible molecules ^: exi/5ce we h be5 previ\sly l$ to admit1 no?+ prev5ts our suppos+ t ?is some?+ else is ! position or ! velocity ( o!r molecules ^: pres5ce we h n be=e suspect$4 ,! law w f9d itself safegu>d$4 ,p]mit me to employ ma!matical language a mom5t to express ! same ?\ght "u ano!r =m4 ,suppose we obs]ve n molecules & asc]ta9 t _! #cn coord9ates satisfy a sy/em ( #cn diff]5tial equations ( ! f\r? ord] 7& n ( ! second ord] z ! law ( in]tia w\ld require74 ,we "k t by introduc+ #cn auxili>y v>iables1 a sy/em ( #cn equations ( ! f\r? ord] c be r$uc$ to a sy/em ( #fn equations ( ! second ord]4 ,if !n we suppose ^! #cn auxili>y v>iables repres5t ! coord9ates ( n invisible molecules1 ! result is aga9 9 con=mity ) ! law ( in]tia4 ,to sum up1 ?is law1 v]ifi$ exp]im5tally 9 "s p>ticul> cases1 may unhesitat+ly be ext5d$ to ! mo/ g5]al cases1 s9ce we "k t 9 ^! g5]al cases exp]im5t no long] is able ei!r to confirm or to contradict x4 ,,! ,,law ,,( ,,accel]ation4--,! accel]ation ( a body is equal to ! =ce act+ on x divid$ by its mass4 ,c ?is law be v]ifi$ by exp]im5t8 ,= t x w\ld be necess>y to measure ! ?ree magnitudes :i* figure 9 ! enunciation3 accel]ation1 =ce & mass4 ,i assume t accel]ation c be measur$1 = ,i pass ov] ! difficulty aris+ f ! measurem5t ( "t4 ,b h[ measure =ce1 or mass8 ,we d n ev5 "k :at !y are4 ,:at is mass8 ,accord+ to ,newton1 x is ! product ( ! volume by ! d5sity4 ,accord+ to ,?omson & ,tait1 x w\ld be bett] to say t d5sity is ! quoti5t ( ! mass by ! volume4 ,:at is =ce8 ,x is1 replies ,lagrange1 t :i* moves or t5ds to move a body4 ,x is1 ,kir*h(f w say1 ! product ( ! mass by ! accel]ation4 ,b !n1 :y n say ! mass is ! quoti5t ( ! =ce by ! accel]ation8 ,^! difficulties are inextricable4 ,:5 we say =ce is ! cause ( motion1 we talk metaphysics1 & ?is def9ition1 if "o 7 cont5t ) x1 w\ld be absolutely /]ile4 ,= a def9ition to be ( any use1 x mu/ tea* u to measure =ce2 moreov] t suffices2 x is n at all necess>y t x tea* u :at =ce is 9 itself1 nor :e!r x is ! cause or ! effect ( motion4 ,we mu/ !re=e fir/ def9e ! equality ( two =ces4 ,:5 %all we say two =ces are equal8 ,x is1 we are told1 :51 appli$ to ! same mass1 !y impress ^u x ! same accel]ation1 or :51 oppos$ directly "o to ! o!r1 !y produce equilibrium4 ,?is def9ition is only a %am4 ,a =ce appli$ to a body c n be unc\pl$ to hook x up to ano!r body1 z "o unc\ples a locomotive to atta* x to ano!r tra94 ,x is !re=e impossible to "k :at accel]ation su* a =ce1 appli$ to su* a body1 w\ld impress ^u su* ano!r body1 if x 7 appli$ to x4 ,x is impossible to "k h[ two =ces :i* are n directly oppos$ w\ld act1 if !y 7 directly oppos$4 ,x is ?is def9ition we try to mat]ialize1 s to speak1 :5 we measure a =ce ) a dynamomet]1 or 9 balanc+ x ) a weight4 ,two =ces ,f & ,f'1 :i* = simplicity ,i w suppose v]tical & direct$ upw>d1 are appli$ respectively to two bodies ,c & ,c'2 ,i susp5d ! same heavy body ,p fir/ to ! body ,c1 !n to ! body ,c'2 if equilibrium is produc$ 9 bo? cases1 ,i %all conclude t ! two =ces ,f & ,f' are equal to "o ano!r1 s9ce !y are ea* equal to ! weight ( ! body ,p4 ,b am ,i sure ! body ,p has reta9$ ! same weight :5 ,i h transport$ x f ! fir/ body to ! second8 ,f> f x2 ,i am sure ( ! contr>y2 ,i "k ! int5sity ( gravity v>ies f "o po9t to ano!r1 & t x is /rong]1 = in/ance1 at ! pole ?an at ! equator4 ,no d\bt ! diff]5ce is v slight &1 9 practise1 ,i %all take no acc\nt ( x2 b a prop]ly con/ruct$ def9ition %\ld h ma!matical rigor2 ?is rigor is lack+4 ,:at ,i say ( weight w\ld evid5tly apply to ! =ce ( ! resili5cy ( a dynamomet]1 :i* ! temp]ature & a multitude ( circum/ances may cause to v>y4 ,?is is n all2 we c n say ! weight ( ! body ,p may be appli$ to ! body ,c & directly balance ! =ce ,f4 ,:at is appli$ to ! body ,c is ! action ,a ( ! body ,p on ! body ,c2 ! body ,p is submitt$ on its "p1 on ! "o h&1 to its weight2 on ! o!r h&1 to ! reaction ,r ( ! body ,c on ,p4 ,f9ally1 ! =ce ,f is equal to ! =ce ,a1 s9ce x balances x2 ! =ce ,a is equal to ,r1 9 virtue ( ! pr9ciple ( ! equality ( action & reaction2 la/ly1 ! =ce ,r is equal to ! weight ( ,p1 s9ce x balances x4 ,x is f ^! ?ree equalities we d$uce z consequ5ce ! equality ( ,f & ! weight ( ,p4 ,we are !re=e oblig$ 9 ! def9ition ( ! equality ( ! two =ces to br+ 9 ! pr9ciple ( ! equality ( action & reaction2 on ?is acc\nt1 ?is pr9ciple mu/ no long] be reg>d$ z an exp]im5tal law1 b z a def9ition4 ,= recogniz+ ! equality ( two =ces "h1 we are !n 9 possession ( two rules3 equality ( two =ces :i* balance2 equality ( action & reaction4 ,b1 z we h se5 above1 ^! two rules are insuffici5t2 we are oblig$ to h rec\rse to a ?ird rule & to assume t c]ta9 =ces1 z1 = in/ance1 ! weight ( a body1 are con/ant 9 magnitude & direction4 ,b ?is ?ird rule1 z ,i h said1 is an exp]im5tal law2 x is only approximately true2 x is a bad def9ition4 ,we are !re=e r$uc$ to ,kir*h(f's def9ition2 =ce is equal to ! mass multipli$ by ! accel]ation4 ,?is 'law ( ,newton' 9 its turn ceases to be reg>d$ z an exp]im5tal law1 x is n[ only a def9ition4 ,b ?is def9ition is /ill insuffici5t1 = we d n "k :at mass is4 ,x enables u d\btless to calculate ! relation ( two =ces appli$ to ! same body at diff]5t in/ants2 x tea*es u no?+ ab\t ! relation ( two =ces appli$ to two diff]5t bodies4 ,to complete x1 x is necess>y to g back anew to ,newton's ?ird law 7equality ( action & reaction71 reg>d$ aga91 n z an exp]im5tal law1 b z a def9ition4 ,two bodies ,a & ,b act "o ^u ! o!r2 ! accel]ation ( ,a multipli$ by ! mass ( ,a is equal to ! action ( ,b ^u ,a2 9 ! same way1 ! product ( ! accel]ation ( ,b by its mass is equal to ! reaction ( ,a ^u ,b4 ,z1 by def9ition1 action is equal to reaction1 ! masses ( ,a & ,b are 9 ! inv]se ratio ( _! accel]ations4 ,"h we h ! ratio ( ^! two masses def9$1 & x is = exp]im5t to v]ify t ?is ratio is con/ant4 ,t w\ld be all v well if ! two bodies ,a & ,b alone 7 pres5t & remov$ f ! action ( ! re/ ( ! _w4 ,?is is n at all ! case2 ! accel]ation ( ,a is n due m]ely to ! action ( ,b1 b to t ( a multitude ( o!r bodies ,c1 ,d1444 ,to apply ! prec$+ rule1 x is !re=e necess>y to sep>ate ! accel]ation ( ,a into _m compon5ts1 & disc]n :i* ( ^! compon5ts is due to ! action ( ,b4 ,?is sep>ation w\ld /ill be possible1 if we %\ld assume t ! action ( ,c ^u ,a is simply adjo9$ to t ( ,b ^u ,a1 )\t ! pres5ce ( ! body ,c modify+ ! action ( ,b ^u ,a2 or ! pres5ce ( ,b modify+ ! action ( ,c ^u ,a2 if we %\ld assume1 consequ5tly1 t any two bodies attract ea* o!r1 t _! mutual action is along _! jo9 & dep5ds only ^u _! di/ance ap>t2 if1 9 a ^w1 we assume ! hypo!sis ( c5tral =ces4 ,y "k t to det]m9e ! masses ( ! cele/ial bodies we use a :olly diff]5t pr9ciple4 ,! law ( gravitation tea*es u t ! attraction ( two bodies is proportional to _! masses2 if r is _! di/ance ap>t1 m & m' _! masses1 k a con/ant1 _! attraction w be kmm'_/r#b4 ,:at we are measur+ !n is n mass1 ! ratio ( =ce to accel]ation1 b ! attract+ mass2 x is n ! in]tia ( ! body1 b its attract+ =ce4 ,?is is an indirect proc$ure1 ^: employm5t is n !oretically indisp5sable4 ,x might v well h be5 t attraction 0 inv]sely proportional to ! squ>e ( ! di/ance )\t be+ proportional to ! product ( ! masses1 t x 0 equal to f_/r#b1 b )\t our hav+ f kmm'4 ,if x 7 s1 we c\ld nev]!less1 by obs]vation ( ! relative motions ( ! heav5ly bodies1 measure ! masses ( ^! bodies4 ,b h we ! "r to admit ! hypo!sis ( c5tral =ces8 ,is ?is hypo!sis rigor\sly exact8 ,is x c]ta9 x w nev] be contradict$ by exp]im5t8 ,:o w\ld d>e affirm t8 ,& if we mu/ ab&on ?is hypo!sis1 ! :ole edifice s labori\sly erect$ w crumble4 ,we h no long] ! "r to speak ( ! compon5t ( ! accel]ation ( ,a due to ! action ( ,b4 ,we h no means ( di/+ui%+ x f t due to ! action ( ,c or ( ano!r body4 ,! rule = ! measurem5t ( masses becomes inapplicable4 ,:at rema9s !n ( ! pr9ciple ( ! equality ( action & reaction8 ,if ! hypo!sis ( c5tral =ces is reject$1 ?is pr9ciple %\ld evid5tly be enunciat$ ?us3 ! geometric resultant ( all ! =ces appli$ to ! v>i\s bodies ( a sy/em isolat$ f all ext]nal action w be null4 ,or1 9 o!r words1 ! motion ( ! c5t] ( gravity ( ?is sy/em w be rectil9e> & uni=m4 ,!re x seems we h a means ( def9+ mass2 ! position ( ! c5t] ( gravity evid5tly dep5ds on ! values attribut$ to ! masses2 x w be necess>y to dispose ( ^! values 9 su* a way t ! motion ( ! c5t] ( gravity may be rectil9e> & uni=m2 ?is w always be possible if ,newton's ?ird law is true1 & possible 9 g5]al only 9 a s+le way4 ,b !re exi/s no sy/em isolat$ f all ext]nal action2 all ! p>ts ( ! univ]se are subject m or less to ! action ( all ! o!r p>ts4 ,! law ( ! motion ( ! c5t] ( gravity is rigor\sly true only if appli$ to ! entire univ]se4 ,b !n1 to get f x ! values ( ! masses1 x w\ld be necess>y to obs]ve ! motion ( ! c5t] ( gravity ( ! univ]se4 ,! absurdity ( ?is consequ5ce is manife/2 we "k only relative motions2 ! motion ( ! c5t] ( gravity ( ! univ]se w rema9 = u et]nally unkn[n4 ,!re=e no?+ rema9s & our ef=ts h be5 fruitless2 we are driv5 to ! foll[+ def9ition1 :i* is only an av[al ( p[]lessness3 masses are coeffici5ts x is conv5i5t to introduce into calculations4 ,we c\ld recon/ruct all me*anics by attribut+ diff]5t values to all ! masses4 ,?is new me*anics w\ld n be 9 contradiction ei!r ) exp]i5ce or ) ! g5]al pr9ciples ( dynamics 7pr9ciple ( in]tia1 proportionality ( =ces to masses & to accel]ations1 equality ( action & reaction1 rectil9e> & uni=m motion ( ! c5t] ( gravity1 pr9ciple ( areas74 ,only ! equations ( ?is new me*anics w\ld be less simple4 ,let u und]/& cle>ly3 x w\ld only be ! fir/ t]ms :i* w\ld be less simple1 t is ^? exp]i5ce has already made u acqua9t$ )2 p]haps "o c\ld alt] ! masses by small quantities )\t ! complete equations ga9+ or los+ 9 simplicity4 ,h]tz has rais$ ! "q :e!r ! pr9ciples ( me*anics are rigor\sly true4 ,9 ! op9ion ( _m physici/s1 he says1 x is inconceivable t ! remote/ exp]i5ce %\ld ev] *ange any?+ 9 ! immovable pr9ciples ( me*anics2 & yet1 :at comes f exp]i5ce may always be rectifi$ by exp]i5ce4 ,aft] :at we h j said1 ^! fe>s w appe> gr\ndless4 ,! pr9ciples ( dynamics at fir/ appe>$ to u z exp]im5tal tru?s2 b we h be5 oblig$ to use !m z def9itions4 ,x is by def9ition t =ce is equal to ! product ( mass by accel]ation2 "h1 !n1 is a pr9ciple :i* is h5ce=? beyond ! rea* ( any fur!r exp]im5t4 ,x is 9 ! same way by def9ition t action is equal to reaction4 ,b !n1 x w be said1 ^! unv]ifiable pr9ciples are absolutely devoid ( any significance2 exp]im5t c n contradict !m2 b !y c tea* u no?+ useful2 !n :at is ! use ( /udy+ dynamics8 ,?is ov]-ha/y condemnation w\ld be unju/4 ,!re is n 9 nature any sy/em p]fectly isolat$1 p]fectly remov$ f all ext]nal action2 b !re are sy/ems almo/ isolat$4 ,if su* a sy/em be obs]v$1 "o may /udy n only ! relative motion ( its v>i\s p>ts "o 9 ref]5ce to ano!r1 b also ! motion ( its c5t] ( gravity 9 ref]5ce to ! o!r p>ts ( ! univ]se4 ,we asc]ta9 !n t ! motion ( ?is c5t] ( gravity is almo/ rectil9e> & uni=m1 9 con=mity ) ,newton's ?ird law4 ,t is an exp]im5tal tru?1 b x c n be invalidat$ by exp]i5ce2 9 fact1 :at w\ld a m precise exp]im5t tea* u8 ,x w\ld tea* u t ! law 0 only almo/ true2 b t we knew already4 ,we c n[ und]/& h[ exp]i5ce has be5 able to s]ve z basis = ! pr9ciples ( me*anics & yet w nev] be able to contradict !m4 ,,an?ropomorphic ,,me*anics4--,kir*h(f1 x w be said1 has only act$ 9 ob$i5ce to ! g5]al t5d5cy ( ma!maticians t[>d nom9alism2 f ?is 8 ability z a physici/ has n sav$ him4 ,he want$ a def9ition ( =ce1 & he took = x ! fir/ proposition t pres5t$ itself2 b we ne$ no def9ition ( =ce3 ! idea ( =ce is primitive1 irr$ucible1 indef9able2 we all "k :at x is1 we h a direct intuition ( x4 ,?is direct intuition comes f ! notion ( ef=t1 :i* is famili> to u f infancy4 ,b fir/1 ev5 ?\gh ?is direct intuition made kn[n to u ! real nature ( =ce 9 itself1 x w\ld be insuffici5t z a f\ndation = me*anics2 x w\ld besides be :olly useless4 ,:at is ( importance is n to "k :at =ce is1 b to "k h[ to measure x4 ,:atev] does n tea* u to measure x is z useless to me*anics z is1 = in/ance1 ! subjective notion ( w>m? & cold to ! physici/ :o is /udy+ heat4 ,?is subjective notion c n be translat$ into numb]s1 !re=e x is ( no use2 a sci5ti/ ^: sk9 0 an absolutely bad conductor ( heat & :o1 consequ5tly1 w\ld nev] h felt ei!r s5sations ( cold or s5sations ( w>m?1 c\ld read a !rmomet] j z well z any "o else1 & t w\ld suffice him = con/ruct+ ! :ole !ory ( heat4 ,n[ ?is imm$iate notion ( ef=t is ( no use to u = measur+ =ce2 x is cle>1 = in/ance1 t ,i %\ld feel m fatigue 9 lift+ a weight ( fifty kilos ?an a man accu/om$ to c>ry burd5s4 ,b m ?an t3 ?is notion ( ef=t does n tea* u ! real nature ( =ce2 x r$uces itself f9ally to a remembrance ( muscul> s5sations1 & x w h>dly be ma9ta9$ t ! sun feels a muscul> s5sation :5 x draws ! e>?4 ,all t c !re be s\ght is a symbol1 less precise & less conv5i5t ?an ! arr[s ! geomet]s use1 b j z remote f ! reality4 ,an?ropomorphism has play$ a consid]able hi/oric rle 9 ! g5esis ( me*anics2 p]haps x w /ill at times furni% a symbol :i* w appe> conv5i5t to "s m9ds2 b x c n s]ve z f\ndation = any?+ ( a truly sci5tific or philosophic "*4 ',,! ,,s*ool ,,( ,,! ,,?read4'--,m4 ,&rade1 9 8 ,leons de mcanique physique1 has rejuv5at$ an?ropomorphic me*anics4 ,to ! s*ool ( me*anics to :i* ,kir*h(f belongs1 he opposes t :i* he biz>rely calls ! s*ool ( ! ?read4 ,?is s*ool tries to r$uce ev]y?+ to ! consid]ation ( c]ta9 mat]ial sy/ems ( negligible mass1 envisag$ 9 ! /ate ( t5sion & capable ( transmitt+ consid]able ef=ts to di/ant bodies1 sy/ems ( :i* ! ideal type is ! ?read4 ,a ?read :i* transmits any =ce is slightly elongat$ "u ! action ( ?is =ce2 ! direction ( ! ?read tells u ! direction ( ! =ce1 ^: magnitude is measur$ by ! elongation ( ! ?read4 ,"o may !n conceive an exp]im5t su* z ?is4 ,a body ,a is atta*$ to a ?read2 at ! o!r extremity ( ! ?read any =ce acts :i* v>ies until ! ?read takes an elongation alpha2 ! accel]ation ( ! body ,a is not$2 ,a is deta*$ & ! body ,b atta*$ to ! same ?read2 ! same =ce or ano!r =ce acts anew1 & is made to v>y until ! ?read takes aga9 ! elongation alpha2 ! accel]ation ( ! body ,b is not$4 ,! exp]im5t is !n r5ew$ ) bo? ,a & ,b1 b s t ! ?read takes ! elongation beta4 ,! f\r obs]v$ accel]ations %\ld be proportional4 ,we h ?us an exp]im5tal v]ification ( ! law ( accel]ation above enunciat$4 ,or /ill bett]1 a body is submitt$ to ! simultane\s action ( sev]al id5tical ?reads 9 equal t5sion1 & by exp]im5t x is s\ght :at mu/ be ! ori5tations ( all ^! ?reads t ! body may rema9 9 equilibrium4 ,we h !n an exp]im5tal v]ification ( ! law ( ! composition ( =ces4 ,b1 aft] all1 :at h we done8 ,we h def9$ ! =ce to :i* ! ?read is subject$ by ! de=mation und]gone by ?is ?read1 :i* is reasonable en\gh2 we h fur!r assum$ t if a body is atta*$ to ?is ?read1 ! ef=t transmitt$ to x by ! ?read is equal to ! action ?is body ex]cises on ?is ?read2 aft] all1 we h !re=e us$ ! pr9ciple ( ! equality ( action & reaction1 9 consid]+ x1 n z an exp]im5tal tru?1 b z ! v def9ition ( =ce4 ,?is def9ition is j z conv5tional z ,kir*h(f's1 b f> less g5]al4 ,all =ces are n transmitt$ by ?reads 7besides1 to be able to comp>e !m1 !y w\ld all h to be transmitt$ by id5tical ?reads74 ,ev5 if x %\ld be conc$$ t ! e>? is atta*$ to ! sun by "s invisible ?read1 at lea/ x w\ld be admitt$ t we h no means ( measur+ its elongation4 ,n9e times out ( t51 consequ5tly1 our def9ition w\ld be at fault2 no sort ( s5se c\ld be attribut$ to x1 & x w\ld be necess>y to fall back on ,kir*h(f's4 ,:y !n take ?is dt\r8 ,y admit a c]ta9 def9ition ( =ce :i* has a mean+ only 9 c]ta9 p>ticul> cases4 ,9 ^! cases y v]ify by exp]im5t t x leads to ! law ( accel]ation4 ,on ! /r5g? ( ?is exp]im5t1 y !n take ! law ( accel]ation z a def9ition ( =ce 9 all ! o!r cases4 ,w\ld x n be simpl] to consid] ! law ( accel]ation z a def9ition 9 all cases1 & to reg>d ! exp]im5ts 9 "q1 n z v]ifications ( ?is law1 b z v]ifications ( ! pr9ciple ( reaction1 or z demon/rat+ t ! de=mations ( an ela/ic body dep5d only on ! =ces to :i* ?is body is subject$8 ,& ?is is )\t tak+ into acc\nt t ! conditions "u :i* y\r def9ition c\ld be accept$ are nev] fulfill$ except imp]fectly1 t a ?read is nev] )\t mass1 t x is nev] remov$ f e =ce except ! reaction ( ! bodies atta*$ to its extremities4 ,&rade's ideas are nev]!less v int]e/+2 if !y d n satisfy our logical crav+1 !y make u und]/& bett] ! hi/oric g5esis ( ! fundam5tal ideas ( me*anics4 ,! reflections !y sugge/ %[ u h[ ! human m9d has rais$ itself f a nave an?ropomorphism to ! pres5t conceptions ( sci5ce4 ,we see at ! />t a v p>ticul> & 9 sum r crude exp]im5t2 at ! f9i%1 a law p]fectly g5]al1 p]fectly precise1 ! c]ta9ty ( :i* we reg>d z absolute4 ,?is c]ta9ty we ourselves h be/[$ ^u x volunt>ily1 s to speak1 by look+ ^u x z a conv5tion4 ,are ! law ( accel]ation1 ! rule ( ! composition ( =ces !n only arbitr>y conv5tions8 ,conv5tions1 yes2 arbitr>y1 no2 !y w\ld be if we lo/ sight ( ! exp]im5ts :i* l$ ! creators ( ! sci5ce to adopt !m1 & :i*1 imp]fect z !y may be1 suffice to ju/ify !m4 ,x is well t f "t to "t our att5tion is c>ri$ back to ! exp]im5tal orig9 ( ^! conv5tions4 ,,*apt] ,,vii ,,relative ,,motion ,,& ,,absolute ,,motion ,,! ,,pr9ciple ,,( ,,relative ,,motion4--,! attempt has sometimes be5 made to atta* ! law ( accel]ation to a m g5]al pr9ciple4 ,! motion ( any sy/em mu/ obey ! same laws1 :e!r x be ref]r$ to fix$ axes1 or to movable axes c>ri$ along 9 a rectil9e> & uni=m motion4 ,?is is ! pr9ciple ( relative motion1 :i* =ces itself ^u u = two reasons3 fir/1 ! commone/ exp]i5ce confirms x1 & second1 ! contr>y hypo!sis is s+ul>ly repugnant to ! m9d4 ,assume x !n1 & consid] a body subject$ to a =ce2 ! relative motion ( ?is body1 9 ref]5ce to an obs]v] mov+ ) a uni=m velocity equal to ! initial velocity ( ! body1 mu/ be id5tical to :at its absolute motion w\ld be if x />t$ f re/4 ,we conclude h5ce t its accel]ation c n dep5d ^u its absolute velocity2 ! attempt has ev5 be5 made to d]ive f ?is a demon/ration ( ! law ( accel]ation4 ,!re long 7 traces ( ?is demon/ration 9 ! regulations = ! degree ,b4 s ,sc4 ,x is evid5t t ?is attempt is idle4 ,! ob/acle :i* prev5t$ our demon/rat+ ! law ( accel]ation is t we _h no def9ition ( =ce2 ?is ob/acle subsi/s 9 its entirety1 s9ce ! pr9ciple invok$ has n furni%$ u ! def9ition we lack$4 ,! pr9ciple ( relative motion is none ! less highly int]e/+ & des]ves /udy = its own sake4 ,let u fir/ try to enunciate x 9 a precise mann]4 ,we h said above t ! accel]ations ( ! diff]5t bodies =m+ "p ( an isolat$ sy/em dep5d only on _! relative velocities & positions1 & n on _! absolute velocities & positions1 provid$ ! movable axes to :i* ! relative motion is ref]r$ move uni=mly 9 a /raight l9e4 ,or1 if we pref]1 _! accel]ations dep5d only on ! diff]5ces ( _! velocities & ! diff]5ces ( _! coord9ates1 & n on ! absolute values ( ^! velocities & coord9ates4 ,if ?is pr9ciple is true = relative accel]ations1 or r = diff]5ces ( accel]ation1 9 comb9+ x ) ! law ( reaction we %all !nce d$uce t x is /ill true ( absolute accel]ations4 ,x !n rema9s to be se5 h[ we may demon/rate t ! diff]5ces ( ! accel]ations dep5d only on ! diff]5ces ( ! velocities & ( ! coord9ates1 or1 to speak 9 ma!matical language1 t ^! diff]5ces ( coord9ates satisfy diff]5tial equations ( ! second ord]4 ,c ?is demon/ration be d$uc$ f exp]im5ts or f a priori consid]ations8 ,recall+ :at we h said above1 ! read] c answ] = himself4 ,?us enunciat$1 9 fact1 ! pr9ciple ( relative motion s+ul>ly resembles :at ,i call$ above ! g5]aliz$ pr9ciple ( in]tia2 x is n altoge!r ! same ?+1 s9ce x is a "q ( ! diff]5ces ( coord9ates & n ( ! coord9ates !mselves4 ,! new pr9ciple tea*es u !re=e some?+ m ?an ! old1 b ! same discussion is applicable & w\ld lead to ! same conclusions2 x is unnecess>y to return to x4 ,,newton',s ,,argum5t4--,"h we enc\nt] a v important & ev5 some:at disconc]t+ "q4 ,i h said ! pr9ciple ( relative motion 0 = u n solely a result ( exp]im5t & t a priori e contr>y hypo!sis w\ld be repugnant to ! m9d4 ,b !n1 :y is ! pr9ciple true only if ! motion ( ! movable axes is rectil9e> & uni=m8 ,x seems t x ought to impose itself ^u u ) ! same =ce1 if ?is motion is v>i$1 or at any rate if x r$uces to a uni=m rotation4 ,n[1 9 ^! two cases1 ! pr9ciple is n true4 ,i w n dwell long on ! case :]e ! motion ( ! axes is rectil9e> )\t be+ uni=m2 ! p>adox does n be> a mom5t's exam9ation4 ,if ,i am on bo>d1 & if ! tra91 /rik+ any ob/acle1 /ops sudd5ly1 ,i %all be ?r[n aga9/ ! seat 9 front ( me1 al?\gh ,i h n be5 directly subject$ to any =ce4 ,!re is no?+ my/]i\s 9 t2 if ,i h und]gone ! action ( no ext]nal =ce1 ! tra9 itself has exp]i5c$ an ext]nal impact4 ,!re c be no?+ p>adoxical 9 ! relative motion ( two bodies be+ di/urb$ :5 ! motion ( "o or ! o!r is modifi$ by an ext]nal cause4 ,i w pause long] on ! case ( relative motions ref]r$ to axes :i* rotate uni=mly4 ,if ! heav5s 7 always cov]$ ) cl\ds1 if we _h no means ( obs]v+ ! />s1 we nev]!less might conclude t ! e>? turns r\nd2 we c\ld le>n ?is f its flatt5+ or aga9 by ! ,f\cault p5dulum exp]im5t4 ,& yet1 9 ?is case1 w\ld x h any mean+1 to say ! e>? turns r\nd8 ,if !re is no absolute space1 c "o turn )\t turn+ 9 ref]5ce to some?+ else8 &1 on ! o!r h&1 h[ c\ld we admit ,newton's conclusion & believe 9 absolute space8 ,b x does n suffice to asc]ta9 t all possible solutions are equally repugnant to u2 we mu/ analyze1 9 ea* case1 ! reasons = our repugnance1 s z to make our *oice intellig5tly4 ,! long discussion :i* foll[s w !re=e be excus$4 ,let u resume our fiction3 ?ick cl\ds hide ! />s f m51 :o c n obs]ve !m & are ignorant ev5 ( _! exi/5ce2 h[ %all ^! m5 "k ! e>? turns r\nd8 ,ev5 m ?an our ance/ors1 no d\bt1 !y w reg>d ! gr\nd :i* be>s !m z fix$ & immovable2 !y w await mu* long] ! adv5t ( a ,cop]nicus4 ,b 9 ! end ! ,cop]nicus w\ld come--h[8 ,! /ud5ts ( me*anics 9 ?is _w w\ld n at fir/ be confront$ ) an absolute contradiction4 ,9 ! !ory ( relative motion1 besides real =ces1 two fictiti\s =ces are met :i* are call$ ord9>y & comp\nd c5trifugal =ce4 ,our imag9>y sci5ti/s c\ld !re=e expla9 ev]y?+ by reg>d+ ^! two =ces z real1 & !y w\ld n see !re9 any contradiction ( ! g5]aliz$ pr9ciple ( in]tia1 = ^! =ces w\ld dep5d1 ! "o on ! relative positions ( ! v>i\s p>ts ( ! sy/em1 z real attractions d1 ! o!r on _! relative velocities1 z real frictions d4 ,_m difficulties1 h[ev]1 w\ld soon awak5 _! att5tion2 if !y succe$$ 9 realiz+ an isolat$ sy/em1 ! c5t] ( gravity ( ?is sy/em w\ld n h an almo/ rectil9e> pa?4 ,!y w\ld invoke1 to expla9 ?is fact1 ! c5trifugal =ces :i* !y w\ld reg>d z real1 & :i* !y w\ld attribute no d\bt to ! mutual actions ( ! bodies4 ,only !y w\ld n see ^! =ces become null at great di/ances1 t is to say 9 proportion z ! isolation 0 bett] realiz$2 f> f x2 c5trifugal =ce increases indef9itely ) ! di/ance4 ,?is difficulty w\ld seem to !m already suffici5tly great2 & yet x w\ld n /op !m long2 !y w\ld soon imag9e "s v subtile m$ium1 analog\s to our e!r1 9 :i* all bodies w\ld be imm]s$ & :i* w\ld ex]t a repell5t action ^u !m4 ,b ?is is n all4 ,space is symmetric1 & yet ! laws ( motion w\ld n %[ any symmetry2 !y w\ld h to di/+ui% betwe5 "r & left4 ,x w\ld be se5 = in/ance t cyclones turn always 9 ! same s5se1 :]eas by reason ( symmetry ^! w9ds %\ld turn indiff]5tly 9 "o s5se & 9 ! o!r4 ,if our sci5ti/s by _! labor _h succe$$ 9 r5d]+ _! univ]se p]fectly symmetric1 ?is symmetry w\ld n rema91 ev5 ?\gh !re 0 no app>5t reason :y x %\ld be di/urb$ 9 "o s5se r ?an 9 ! o!r4 ,!y w\ld get !mselves out ( ! difficulty d\btless1 !y w\ld inv5t some?+ :i* w\ld be no m extraord9>y ?an ! glass sph]es ( ,ptolemy1 & s x w\ld g on1 complications accumulat+1 until ! long-expect$ ,cop]nicus sweeps !m all away at a s+le /roke1 say+3 ,x is mu* simpl] to assume ! e>? turns r\nd4 ,& j z our ,cop]nicus said to u3 ,x is m conv5i5t to suppose ! e>? turns r\nd1 s9ce ?us ! laws ( a/ronomy are expressible 9 a mu* simpl] language2 ?is "o w\ld say3 ,x is m conv5i5t to suppose ! e>? turns r\nd1 s9ce ?us ! laws ( me*anics are expressible 9 a mu* simpl] language4 ,?is does n preclude ma9ta9+ t absolute space1 t is to say ! m>k to :i* x w\ld be necess>y to ref] ! e>? to "k :e!r x really moves1 has no objective exi/5ce4 ,h5ce1 ?is affirmation3 '! e>? turns r\nd' has no mean+1 s9ce x c be v]ifi$ by no exp]im5t2 s9ce su* an exp]im5t1 n only c\ld n be ei!r realiz$ or dream$ by ! bolde/ ,jules ,v]ne1 b c n be conceiv$ ( )\t contradiction2 or r ^! two propositions3 '! e>? turns r\nd1' &1 'x is m conv5i5t to suppose ! e>? turns r\nd' h ! same mean+2 !re is no?+ m 9 ! "o ?an 9 ! o!r4 ,p]haps "o w n be cont5t ev5 ) t1 & w f9d x already %ock+ t among all ! hypo!ses1 or r all ! conv5tions we c make on ?is subject1 !re is "o m conv5i5t ?an ! o!rs4 ,b if x has be5 admitt$ )\t difficulty :5 x 0 a "q ( ! laws ( a/ronomy1 :y %\ld x be %ock+ 9 t :i* conc]ns me*anics8 ,we h se5 t ! coord9ates ( bodies are det]m9$ by diff]5tial equations ( ! second ord]1 & t s are ! diff]5ces ( ^! coord9ates4 ,?is is :at we h call$ ! g5]aliz$ pr9ciple ( in]tia & ! pr9ciple ( relative motion4 ,if ! di/ances ( ^! bodies 7 det]m9$ likewise by equations ( ! second ord]1 x seems t ! m9d ought to be entirely satisfi$4 ,9 :at measure does ! m9d get ?is satisfaction & :y is x n cont5t ) x8 ,to acc\nt = ?is1 we _h bett] take a simple example4 ,i suppose a sy/em analog\s to our sol> sy/em1 b :]e "o c n p]ceive fix$ />s =eign to ?is sy/em1 s t a/ronom]s c obs]ve only ! mutual di/ances ( ! planets & ! sun1 & n ! absolute longitudes ( ! planets4 ,if we d$uce directly f ,newton's law ! diff]5tial equations :i* def9e ! v>iation ( ^! di/ances1 ^! equations w n be ( ! second ord]4 ,i mean t if1 besides ,newton's law1 "o knew ! initial values ( ^! di/ances & ( _! d]ivatives ) respect to ! "t1 t w\ld n suffice to det]m9e ! values ( ^! same di/ances at a subsequ5t in/ant4 ,!re w\ld /ill be lack+ "o datum1 & ?is datum might be = in/ance :at a/ronom]s call ! area-con/ant4 ,b "h two diff]5t po9ts ( view may be tak52 we may di/+ui% two sorts ( con/ants4 ,to ! eyes ( ! physici/ ! _w r$uces to a s]ies ( ph5om5a1 dep5d+1 on ! "o h&1 solely ^u ! initial ph5om5a2 on ! o!r h&1 ^u ! laws :i* b9d ! consequ5ts to ! antec$5ts4 ,if !n obs]vation tea*es u t a c]ta9 quantity is a con/ant1 we %all h ! *oice betwe5 two conceptions4 ,ei!r we %all assume t !re is a law requir+ ?is quantity n to v>y1 b t by *ance1 at ! beg9n+ ( ! ages1 x _h1 r ?an ano!r1 ?is value x has be5 =c$ to keep ev] s9ce4 ,?is quantity might !n be call$ an accid5tal con/ant4 ,or else we %all assume1 on ! contr>y1 t !re is a law ( nature :i* imposes ^u ?is quantity su* a value & n su* ano!r4 ,we %all !n h :at we may call an ess5tial con/ant4 ,= example1 9 virtue ( ,newton's laws1 ! duration ( ! revolution ( ! e>? mu/ be con/ant4 ,b if x is #cff sid]eal days & some?+ ov]1 & n #cjj or #djj1 ?is is 9 consequ5ce ( ,i "k n :at initial *ance4 ,?is is an accid5tal con/ant4 ,if1 on ! contr>y1 ! expon5t ( ! di/ance :i* figures 9 ! expression ( ! attractive =ce is equal to -#b & n to -#c1 ?is is n by *ance1 b because ,newton's law requires x4 ,?is is an ess5tial con/ant4 ,i "k n :e!r ?is way ( giv+ *ance its "p is legitimate 9 itself1 & :e!r ?is di/9ction is n some:at artificial2 x is c]ta9 at lea/ t1 s long z nature %all h secrets1 ?is di/9ction w be 9 application extremely arbitr>y & always prec>i\s4 ,z to ! area-con/ant1 we are accu/om$ to reg>d x z accid5tal4 ,is x c]ta9 our imag9>y a/ronom]s w\ld d ! same8 ,if !y c\ld h comp>$ two diff]5t sol> sy/ems1 !y w\ld h ! idea t ?is con/ant may h sev]al diff]5t values2 b my v supposition 9 ! beg9n+ 0 t _! sy/em %\ld appe> z isolat$1 & t !y %\ld obs]ve no /> =eign to x4 ,"u ^! conditions1 !y w\ld see only "o s+le con/ant :i* w\ld h a s+le value absolutely inv>iable2 !y w\ld be l$ )\t any d\bt to reg>d x z an ess5tial con/ant4 ,a ^w 9 pass+ to =e/all an objection3 ! inhabitants ( ?is imag9>y _w c\ld nei!r obs]ve nor def9e ! area-con/ant z we d1 s9ce ! absolute longitudes escape !m2 t w\ld n preclude _! be+ quickly l$ to notice a c]ta9 con/ant :i* w\ld introduce itself naturally into _! equations & :i* w\ld be no?+ b :at we call ! area-con/ant4 ,b !n see :at w\ld happ54 ,if ! area-con/ant is reg>d$ z ess5tial1 z dep5d+ ^u a law ( nature1 to calculate ! di/ances ( ! planets at any in/ant x w suffice to "k ! initial values ( ^! di/ances & ^? ( _! fir/ d]ivatives4 ,f ?is new po9t ( view1 ! di/ances w be det]m9$ by diff]5tial equations ( ! second ord]4 ,yet w\ld ! m9d ( ^! a/ronom]s be completely satisfi$8 ,i d n believe s2 fir/1 !y w\ld soon p]ceive t 9 diff]5tiat+ _! equations & ?us rais+ _! ord]1 ^! equations became mu* simpl]4 ,& above all !y w\ld be /ruck by ! difficulty :i* comes f symmetry4 ,x w\ld be necess>y to assume diff]5t laws1 accord+ z ! aggregate ( ! planets pres5t$ ! figure ( a c]ta9 polyh$ron or ( ! symmetric polyh$ron1 & "o w\ld escape f ?is consequ5ce only by reg>d+ ! area-con/ant z accid5tal4 ,i h tak5 a v special example1 s9ce ,i h suppos$ a/ronom]s :o did n at all consid] t]re/rial me*anics1 & ^: view 0 limit$ to ! sol> sy/em4 ,our univ]se is m ext5d$ ?an !irs1 z we h fix$ />s1 b /ill x too is limit$1 & s we might reason on ! totality ( our univ]se z ! a/ronom]s on _! sol> sy/em4 ,?us we see t f9ally we %\ld be l$ to conclude t ! equations :i* def9e di/ances are ( an ord] sup]ior to ! second4 ,:y %\ld we be %ock$ at t1 :y d we f9d x p]fectly natural = ! s]ies ( ph5om5a to dep5d ^u ! initial values ( ! fir/ d]ivatives ( ^! di/ances1 :ile we hesitate to admit t !y may dep5d on ! initial values ( ! second d]ivatives8 ,?is c only be because ( ! habits ( m9d creat$ 9 u by ! con/ant /udy ( ! g5]aliz$ pr9ciple ( in]tia & its consequ5ces4 ,! values ( ! di/ances at any in/ant dep5d ^u _! initial values1 ^u ^? ( _! fir/ d]ivatives & also ^u some?+ else4 ,:at is ?is some?+ else8 ,if we w n admit t ?is may be simply "o ( ! second d]ivatives1 we h only ! *oice ( hypo!ses4 ,ei!r x may be suppos$1 z is ord9>ily done1 t ?is some?+ else is ! absolute ori5tation ( ! univ]se 9 space1 or ! rapidity ) :i* ?is ori5tation v>ies2 & ?is supposition may be correct2 x is c]ta9ly ! mo/ conv5i5t solution = geometry2 x is n ! mo/ satisfactory = ! philosoph]1 because ?is ori5tation does n exi/4 ,or x may be suppos$ t ?is some?+ else is ! position or ! velocity ( "s invisible body2 ?is has be5 done by c]ta9 p]sons :o h ev5 call$ x ! body alpha1 al?\gh we are doom$ nev] to "k any?+ ( ?is body b its "n4 ,?is is an artifice entirely analog\s to t ( :i* ,i spoke at ! end ( ! p>agraph devot$ to my reflections on ! pr9ciple ( in]tia4 ,b1 aft] all1 ! difficulty is artificial4 ,provid$ ! future indications ( our in/rum5ts c dep5d only on ! indications !y h giv5 u or w\ld h giv5 u =m]ly1 ?is is all t is necess>y4 ,n[ z to ?is we may re/ easy4 ,,*apt] ,,viii ,,en]gy ,,& ,,!rmodynamics ,,en]getics4--,! difficulties inh]5t 9 ! classic me*anics h l$ c]ta9 m9ds to pref] a new sy/em !y call en]getics4 ,en]getics took its rise z an outcome ( ! discov]y ( ! pr9ciple ( ! cons]vation ( en]gy4 ,helmholtz gave x its f9al =m4 ,x beg9s by def9+ two quantities :i* play ! fundam5tal rle 9 ?is !ory4 ,!y are k9etic en]gy1 or vis viva1 & pot5tial en]gy4 ,all ! *anges :i* bodies 9 nature c und]go are regulat$ by two exp]im5tal laws3 #a ,! sum ( k9etic en]gy & pot5tial en]gy is con/ant4 ,?is is ! pr9ciple ( ! cons]vation ( en]gy4 #b ,if a sy/em ( bodies is at ,a at ! "t t#j & at ,b at ! "t t#a1 x always goes f ! fir/ situation to ! second 9 su* a way t ! mean value ( ! diff]5ce betwe5 ! two sorts ( en]gy1 9 ! int]val ( "t :i* sep>ates ! two epo*s t#j & t#a1 may be z small z possible4 ,?is is ,hamilton's pr9ciple1 :i* is "o ( ! =ms ( ! pr9ciple ( lea/ action4 ,! en]getic !ory has ! foll[+ advantages ov] ! classic !ory3 #a ,x is less incomplete2 t is to say1 ,hamilton's pr9ciple & t ( ! cons]vation ( en]gy tea* u m ?an ! fundam5tal pr9ciples ( ! classic !ory1 & exclude c]ta9 motions n realiz$ 9 nature & :i* w\ld be compatible ) ! classic !ory3 #b ,x saves u ! hypo!sis ( atoms1 :i* x 0 almo/ impossible to avoid ) ! classic !ory4 ,b x raises 9 its turn new difficulties3 ,! def9itions ( ! two sorts ( en]gy w\ld raise difficulties almo/ z great z ^? ( =ce & mass 9 ! fir/ sy/em4 ,yet !y may be gott5 ov] m easily1 at lea/ 9 ! simple/ cases4 ,suppose an isolat$ sy/em =m$ ( a c]ta9 numb] ( mat]ial po9ts2 suppose ^! po9ts subject$ to =ces dep5d+ only on _! relative position & _! mutual di/ances1 & indep5d5t ( _! velocities4 ,9 virtue ( ! pr9ciple ( ! cons]vation ( en]gy1 a function ( =ces mu/ exi/4 ,9 ?is simple case ! enunciation ( ! pr9ciple ( ! cons]vation ( en]gy is ( extreme simplicity4 ,a c]ta9 quantity1 accessible to exp]im5t1 mu/ rema9 con/ant4 ,?is quantity is ! sum ( two t]ms2 ! fir/ dep5ds only on ! position ( ! mat]ial po9ts & is indep5d5t ( _! velocities2 ! second is proportional to ! squ>e ( ^! velocities4 ,?is resolution c take place only 9 a s+le way4 ,! fir/ ( ^! t]ms1 :i* ,i %all call ,u1 w be ! pot5tial en]gy2 ! second1 :i* ,i %all call ,t1 w be ! k9etic en]gy4 ,x is true t if ,t ,u is a con/ant1 s is any function ( ,t ,u1 ,phi7,t ,u74 ,b ?is function ,phi7,t ,u7 w n be ! sum ( two t]ms ! "o indep5d5t ( ! velocities1 ! o!r proportional to ! squ>e ( ^! velocities4 ,among ! functions :i* rema9 con/ant !re is only "o :i* enjoys ?is prop]ty1 t is ,t ,u 7or a l9e> function ( ,t ,u1 :i* comes to ! same ?+1 s9ce ?is l9e> function may always be r$uc$ to ,t ,u by *ange ( unit & ( orig974 ,?is !n is :at we %all call en]gy2 ! fir/ t]m we %all call pot5tial en]gy & ! second k9etic en]gy4 ,! def9ition ( ! two sorts ( en]gy c !re=e be c>ri$ ?r\gh )\t any ambiguity4 ,x is ! same ) ! def9ition ( ! masses4 ,k9etic en]gy1 or vis viva1 is express$ v simply by ! aid ( ! masses & ! relative velocities ( all ! mat]ial po9ts ) ref]5ce to "o ( !m4 ,^! relative velocities are accessible to obs]vation1 &1 :5 we "k ! expression ( ! k9etic en]gy z function ( ^! relative velocities1 ! coeffici5ts ( ?is expression w give u ! masses4 ,?us1 9 ?is simple case1 ! fundam5tal ideas may be def9$ )\t difficulty4 ,b ! difficulties reappe> 9 ! m complicat$ cases &1 = in/ance1 if ! =ces1 9 lieu ( dep5d+ only on ! di/ances1 dep5d also on ! velocities4 ,= example1 ,web] supposes ! mutual action ( two electric molecules to dep5d n only on _! di/ance1 b on _! velocity & _! accel]ation4 ,if mat]ial po9ts %\ld attract ea* o!r accord+ to an analog\s law1 ,u w\ld dep5d on ! velocity1 & might conta9 a t]m proportional to ! squ>e ( ! velocity4 ,among ! t]ms proportional to ! squ>es ( ! velocities1 h[ di/+ui% ^? :i* come f ,t or f ,u8 ,consequ5tly1 h[ di/+ui% ! two p>ts ( en]gy8 ,b /ill m2 h[ def9e en]gy itself8 ,we no long] h any reason to take z def9ition ,t ,u r ?an any o!r function ( ,t ,u1 :5 ! prop]ty :i* *>act]iz$ ,t ,u has disappe>$1 t1 namely1 ( be+ ! sum ( two t]ms ( a p>ticul> =m4 ,b ?is is n all2 x is necess>y to take acc\nt1 n only ( me*anical en]gy prop]ly s call$1 b ( ! o!r =ms ( en]gy1 heat1 *emical en]gy1 electric en]gy1 etc4 ,! pr9ciple ( ! cons]vation ( en]gy %\ld be writt53 ,t ,u ,q con/4 :]e ,t w\ld repres5t ! s5sible k9etic en]gy1 ,u ! pot5tial en]gy ( position1 dep5d+ only on ! position ( ! bodies1 ,q ! int]nal molecul> en]gy1 "u ! !rmal1 *emic or electric =m4 ,all w\ld g well if ^! ?ree t]ms 7 absolutely di/9ct1 if ,t 7 proportional to ! squ>e ( ! velocities1 ,u indep5d5t ( ^! velocities & ( ! /ate ( ! bodies1 ,q indep5d5t ( ! velocities & ( ! positions ( ! bodies & dep5d5t only on _! int]nal /ate4 ,! expression = ! en]gy c\ld be resolv$ only 9 "o s+le way into ?ree t]ms ( ?is =m4 ,b ?is is n ! case2 consid] electrifi$ bodies2 ! electro/atic en]gy due to _! mutual action w evid5tly dep5d ^u _! *>ge1 t is to say1 on _! /ate2 b x w equally dep5d ^u _! position4 ,if ^! bodies are 9 motion1 !y w act "o ^u ano!r electrodynamically & ! electrodynamic en]gy w dep5d n only ^u _! /ate & _! position1 b ^u _! velocities4 ,we !re=e no long] h any means ( mak+ ! sep>ation ( ! t]ms :i* %\ld make "p ( ,t1 ( ,u & ( ,q1 & ( sep>at+ ! ?ree p>ts ( en]gy4 ,if 7,t ,u ,q7 is con/ant s is any function phi7,t ,u ,q74 ,if ,t ,u ,q 7 ( ! p>ticul> =m ,i h above consid]$1 no ambiguity w\ld result2 among ! functions phi7,t ,u ,q7 :i* rema9 con/ant1 !re w\ld only be "o ( ?is p>ticul> =m1 & t ,i %\ld conv5e to call en]gy4 ,b z ,i h said1 ?is is n rigor\sly ! case2 among ! functions :i* rema9 con/ant1 !re is none :i* c be put rigor\sly "u ?is p>ticul> =m2 h5ce1 h[ *oose among !m ! "o :i* %\ld be call$ en]gy8 ,we no long] h any?+ to guide u 9 our *oice4 ,!re only rema9s = u "o enunciation ( ! pr9ciple ( ! cons]vation ( en]gy3 ,!re is some?+ :i* rema9s con/ant4 ,"u ?is =m x is 9 its turn out ( ! rea* ( exp]im5t & r$uces to a sort ( tautology4 ,x is cle> t if ! _w is gov]n$ by laws1 !re w be quantities :i* w rema9 con/ant4 ,l ,newton's laws1 &1 = an analog\s reason1 ! pr9ciple ( ! cons]vation ( en]gy1 f\nd$ on exp]im5t1 c\ld no long] be invalidat$ by x4 ,?is discussion %[s t 9 pass+ f ! classic to ! en]getic sy/em progress has be5 made2 b at ! same "t x %[s ?is progress is insuffici5t4 ,ano!r objection seems to me /ill m grave3 ! pr9ciple ( lea/ action is applicable to rev]sible ph5om5a2 b x is n at all satisfactory 9 s f> z irrev]sible ph5om5a are conc]n$2 ! attempt by ,helmholtz to ext5d x to ?is k9d ( ph5om5a did n succe$ & c\ld n succe$2 9 ?is reg>d ev]y?+ rema9s to be done4 ,! v /atem5t ( ! pr9ciple ( lea/ action has some?+ ab\t x repugnant to ! m9d4 ,to g f "o po9t to ano!r1 a mat]ial molecule1 act$ ^u by no =ce1 b requir$ to move on a surface1 w take ! geodesic l9e1 t is to say1 ! %orte/ pa?4 ,?is molecule seems to "k ! po9t :i!r x is to g1 to =esee ! "t x w\ld take to rea* x by su* & su* a r\te1 & !n to *oose ! mo/ suitable pa?4 ,! /atem5t pres5ts ! molecule to u1 s to speak1 z a liv+ & free be+4 ,cle>ly x w\ld be bett] to replace x by an enunciation less objectionable1 & :]e1 z ! philosoph]s w\ld say1 f9al causes w\ld n seem to be sub/itut$ = effici5t causes4 ,,!rmodynamics4#d--,! rle ( ! two fundam5tal pr9ciples ( !rmodynamics 9 all bran*es ( natural philosophy becomes daily m important4 ,ab&on+ ! ambiti\s !ories ( =ty ye>s ago1 :i* 7 encumb]$ by molecul> hypo!ses1 we are try+ to-day to erect ^u !rmodynamics alone ! entire edifice ( ma!matical physics4 ,w ! two pr9ciples ( ,may] & ( ,clausius assure to x f\ndations solid en\gh = x to la/ "s "t8 ,no "o d\bts x2 b :5ce comes ?is confid5ce8 #d ,! foll[+ l9es are a p>tial reproduction ( ! preface ( my book ,!rmodynamique4 ,an em95t physici/ said to me "o day propos ( ! law ( errors3 ,all ! _w believes x firmly1 because ! ma!maticians imag9e t x is a fact ( obs]vation1 & ! obs]v]s t x is a !orem ( ma!matics4 ,x 0 long s = ! pr9ciple ( ! cons]vation ( en]gy4 ,x is no long] s to-day2 no "o is ignorant t ?is is an exp]im5tal fact4 ,b !n :at gives u ! "r to attribute to ! pr9ciple itself m g5]ality & m precision ?an to ! exp]im5ts :i* h s]v$ to demon/rate x8 ,?is is to ask :e!r x is legitimate1 z is done e day1 to g5]alize empirical data1 & ,i %all n h ! presumption to discuss ?is "q1 aft] s _m philosoph]s h va9ly /riv5 to solve x4 ,"o ?+ is c]ta92 if ?is p[] 7 d5i$ u1 sci5ce c\ld n exi/ or1 at lea/1 r$uc$ to a sort ( inv5tory1 to ! asc]ta9+ ( isolat$ facts1 x w\ld h no value = u1 s9ce x c\ld give no satisfaction to our crav+ = ord] & h>mony & s9ce x w\ld be at ! same "t incapable ( =esee+4 ,z ! circum/ances :i* h prec$$ any fact w probably nev] be simultane\sly reproduc$1 a fir/ g5]alization is already necess>y to =esee :e!r ?is fact w be reproduc$ aga9 aft] ! lea/ ( ^! circum/ances %all be *ang$4 ,b e proposition may be g5]aliz$ 9 an inf9ity ( ways4 ,among all ! g5]alizations possible1 we mu/ *oose1 & we c only *oose ! simple/4 ,we are !re=e l$ to act z if a simple law 71 o!r ?+s be+ equal1 m probable ?an a complicat$ law4 ,half a c5tury ago ?is 0 frankly confess$1 & x 0 proclaim$ t nature loves simplicity2 %e has s9ce too (t5 giv5 u ! lie4 ,to-day we no long] confess ?is t5d5cy1 & we reta9 only s mu* ( x z is indisp5sable if sci5ce is n to become impossible4 ,9 =mulat+ a g5]al1 simple & precise law on ! basis ( exp]im5ts relatively few & pres5t+ c]ta9 div]g5ces1 we h !re=e only obey$ a necessity f :i* ! human m9d c n free itself4 ,b !re is some?+ m1 & ?is is :y ,i dwell ^u ! po9t4 ,no "o d\bts t ,may]'s pr9ciple is de/9$ to survive all ! p>ticul> laws f :i* x 0 obta9$1 j z ,newton's law has surviv$ ,kepl]'s laws1 f :i* x sprang1 & :i* are only approximative if acc\nt be tak5 ( p]turbations4 ,:y does ?is pr9ciple occupy ?us a sort ( privileg$ place among all ! physical laws8 ,!re are _m little reasons = x4 ,fir/ ( all x is believ$ t we c\ld n reject x or ev5 d\bt its absolute rigor )\t admitt+ ! possibility ( p]petual motion2 ( c\rse we are on our gu>d at su* a prospect1 & we ?9k ourselves less ra% 9 affirm+ ,may]'s pr9ciple ?an 9 d5y+ x4 ,t is p]haps n :olly accurate2 ! impossibility ( p]petual motion implies ! cons]vation ( en]gy only = rev]sible ph5om5a4 ,! impos+ simplicity ( ,may]'s pr9ciple likewise contributes to /r5g!n our fai?4 ,9 a law d$uc$ imm$iately f exp]im5t1 l ,m>iotte's1 ?is simplicity w\ld r seem to u a reason = di/ru/2 b "h ?is is no long] ! case2 we see elem5ts1 at fir/ sight disp>ate1 arrange !mselves 9 an unexpect$ ord] & =m a h>moni\s :ole2 & we refuse to believe t an un=ese5 h>mony may be a simple effect ( *ance4 ,x seems t our conque/ is ! de>] to u ! m ef=t x has co/ u1 or t we are ! sur] ( hav+ wre/$ h] true secret f nature ! m jeal\sly %e has hidd5 x f u4 ,b ^? are only little reasons2 to e/abli% ,may]'s law z an absolute pr9ciple1 a m pr(\nd discussion is necess>y4 ,b if ?is be attempt$1 x is se5 t ?is absolute pr9ciple is n ev5 easy to /ate4 ,9 ea* p>ticul> case x is cle>ly se5 :at en]gy is & at lea/ a provisional def9ition ( x c be giv52 b x is impossible to f9d a g5]al def9ition = x4 ,if we try to enunciate ! pr9ciple 9 all its g5]ality & apply x to ! univ]se1 we see x vani%1 s to speak1 & no?+ is left b ?is3 ,!re is some?+ :i* rema9s con/ant4 ,b has ev5 ?is any mean+8 ,9 ! det]m9i/ hypo!sis1 ! /ate ( ! univ]se is det]m9$ by an extremely great numb] n ( p>amet]s :i* ,i %all call x#a1 x#b1444 xn4 ,z soon z ! values ( ^! n p>amet]s at any in/ant are kn[n1 _! d]ivatives ) respect to ! "t are likewise kn[n & consequ5tly ! values ( ^! same p>amet]s at a prec$+ or subsequ5t in/ant c be calculat$4 ,9 o!r words1 ^! n p>amet]s satisfy n diff]5tial equations ( ! fir/ ord]4 ,^! equations admit ( n - #a integrals & consequ5tly !re are n - #a functions ( x#a1 x#b1444 xn1 :i* rema9 con/ant4 ,if !n we say !re is some?+ :i* rema9s con/ant1 we only utt] a tautology4 ,we %\ld ev5 be puzzl$ to say :i* among all our integrals %\ld reta9 ! "n ( en]gy4 ,besides1 ,may]'s pr9ciple is n und]/ood 9 ?is s5se :5 x is appli$ to a limit$ sy/em4 ,x is !n assum$ t p ( our p>amet]s v>y indep5d5tly1 s t we only h n - p relations1 g5]ally l9e>1 betwe5 our n p>amet]s & _! d]ivatives4 ,to simplify ! enunciation1 suppose t ! sum ( ! "w ( ! ext]nal =ces is null1 z well z t ( ! quantities ( heat giv5 (f to ! outside4 ,!n ! signification ( our pr9ciple w be3 ,!re is a comb9ation ( ^! n - p relations ^: fir/ memb] is an exact diff]5tial2 & !n ?is diff]5tial vani%+ 9 virtue ( our n - p relations1 its integral is a con/ant & ?is integral is call$ en]gy4 ,b h[ c x be possible t !re are sev]al p>amet]s ^: v>iations are indep5d5t8 ,t c only happ5 "u ! influ5ce ( ext]nal =ces 7al?\gh we h suppos$1 = simplicity1 t ! algebraic sum ( ! effects ( ^! =ces is null74 ,9 fact1 if ! sy/em 7 completely isolat$ f all ext]nal action1 ! values ( our n p>amet]s at a giv5 in/ant w\ld suffice to det]m9e ! /ate ( ! sy/em at any subsequ5t in/ant1 provid$ always we reta9 ! det]m9i/ hypo!sis2 we come back !re=e to ! same difficulty z above4 ,if ! future /ate ( ! sy/em is n entirely det]m9$ by its pres5t /ate1 ?is is because x dep5ds besides ^u ! /ate ( bodies ext]nal to ! sy/em4 ,b !n is x probable t !re exi/ betwe5 ! p>amet]s x1 :i* def9e ! /ate ( ! sy/em1 equations indep5d5t ( ?is /ate ( ! ext]nal bodies8 & if 9 c]ta9 cases we believe we c f9d su*1 is ?is n solely 9 consequ5ce ( our ignorance & because ! influ5ce ( ^! bodies is too slight = our exp]im5t+ to detect x8 ,if ! sy/em is n reg>d$ z completely isolat$1 x is probable t ! rigor\sly exact expression ( its int]nal en]gy w dep5d on ! /ate ( ! ext]nal bodies4 ,aga91 ,i h above suppos$ ! sum ( ! ext]nal "w 0 null1 & if we try to free ourselves f ?is r artificial re/riction1 ! enunciation becomes /ill m difficult4 ,to =mulate ,may]'s pr9ciple 9 an absolute s5se1 x is !re=e necess>y to ext5d x to ! :ole univ]se1 & !n we f9d ourselves face to face ) ! v difficulty we s\ght to avoid4 ,9 conclusion1 us+ ord9>y language1 ! law ( ! cons]vation ( en]gy c h only "o signification1 :i* is t !re is a prop]ty common to all ! possibilities2 b on ! det]m9i/ hypo!sis !re is only a s+le possibility1 & !n ! law has no long] any mean+4 ,on ! indet]m9i/ hypo!sis1 on ! contr>y1 x w\ld h a mean+1 ev5 if x 7 tak5 9 an absolute s5se2 x w\ld appe> z a limitation impos$ ^u fre$om4 ,b ?is ^w rem9ds me t ,i am digress+ & am on ! po9t ( leav+ ! doma9 ( ma!matics & physics4 ,i *eck myself !re=e & w /ress ( all ?is discussion only "o impression1 t ,may]'s law is a =m flexible en\gh = u to put into x almo/ :atev] we wi%4 ,by t ,i d n mean x corresponds to no objective reality1 nor t x r$uces itself to a m]e tautology1 s9ce1 9 ea* p>ticul> case1 & provid$ "o does n try to pu% to ! absolute1 x has a p]fectly cle> mean+4 ,?is flexibility is a reason = believ+ 9 its p]man5ce1 & z1 on ! o!r h&1 x w disappe> only to lose itself 9 a high] h>mony1 we may "w ) confid5ce1 support+ ourselves ^u x1 c]ta9 be=eh& t our labor w n be lo/4 ,almo/ ev]y?+ ,i h j said applies to ! pr9ciple ( ,clausius4 ,:at di/+ui%es x is t x is express$ by an inequality4 ,p]haps x w be said x is ! same ) all physical laws1 s9ce _! precision is always limit$ by errors ( obs]vation4 ,b !y at lea/ claim to be fir/ approximations1 & x is hop$ to replace !m little by little by laws m & m precise4 ,if1 on ! o!r h&1 ! pr9ciple ( ,clausius r$uces to an inequality1 ?is is n caus$ by ! imp]fection ( our means ( obs]vation1 b by ! v nature ( ! "q4 ,,g5]al ,,conclusions ,,on ,,"p ,,?ird ,! pr9ciples ( me*anics1 !n1 pres5t !mselves to u "u two diff]5t aspects4 ,on ! "o h&1 !y are tru?s f\nd$ on exp]im5t & approximately v]ifi$ s f> z conc]ns almo/ isolat$ sy/ems4 ,on ! o!r h&1 !y are po/ulates applicable to ! totality ( ! univ]se & reg>d$ z rigor\sly true4 ,if ^! po/ulates possess a g5]ality & a c]ta9ty :i* are lack+ to ! exp]im5tal v]ities :5ce !y are drawn1 ?is is because !y r$uce 9 ! la/ analysis to a m]e conv5tion :i* we h ! "r to make1 because we are c]ta9 be=eh& t no exp]im5t c ev] contradict x4 ,?is conv5tion1 h[ev]1 is n absolutely arbitr>y2 x does n spr+ f our caprice2 we adopt x because c]ta9 exp]im5ts h %[n u t x w\ld be conv5i5t4 ,?us is expla9$ h[ exp]im5t c\ld make ! pr9ciples ( me*anics1 & yet :y x c n ov]turn !m4 ,comp>e ) geometry3 ,! fundam5tal propositions ( geometry1 z = in/ance ,euclid's po/ulate1 are no?+ m ?an conv5tions1 & x is j z unreasonable to inquire :e!r !y are true or false z to ask :e!r ! metric sy/em is true or false4 ,only1 ^! conv5tions are conv5i5t1 & x is c]ta9 exp]im5ts :i* h taught u t4 ,at fir/ blu%1 ! analogy is complete2 ! rle ( exp]im5t seems ! same4 ,"o w !re=e be tempt$ to say3 ,ei!r me*anics mu/ be reg>d$ z an exp]im5tal sci5ce1 & !n ! same mu/ hold = geometry2 or else1 on ! contr>y1 geometry is a d$uctive sci5ce1 & !n "o may say z mu* ( me*anics4 ,su* a conclusion w\ld be illegitimate4 ,! exp]im5ts :i* h l$ u to adopt z m conv5i5t ! fundam5tal conv5tions ( geometry be> on objects :i* h no?+ 9 common ) ^? geometry /udies2 !y be> on ! prop]ties ( solid bodies1 on ! rectil9e> propagation ( light4 ,!y are exp]im5ts ( me*anics1 exp]im5ts ( optics2 !y c n 9 any way be reg>d$ z exp]im5ts ( geometry4 ,& ev5 ! pr9cipal reason :y our geometry seems conv5i5t to u is t ! diff]5t p>ts ( our body1 our eye1 our limbs1 h ! prop]ties ( solid bodies4 ,on ?is acc\nt1 our fundam5tal exp]im5ts are preem95tly physiological exp]im5ts1 :i* be>1 n on space :i* is ! object ! geomet] mu/ /udy1 b on 8 body1 t is to say1 on ! in/rum5t he mu/ use = ?is /udy4 ,on ! contr>y1 ! fundam5tal conv5tions ( me*anics1 & ! exp]im5ts :i* prove to u t !y are conv5i5t1 be> on exactly ! same objects or on analog\s objects4 ,! conv5tional & g5]al pr9ciples are ! natural & direct g5]alization ( ! exp]im5tal & p>ticul> pr9ciples4 ,let x n be said t ?us ,i trace artificial fronti]s betwe5 ! sci5ces2 t if ,i sep>ate by a b>ri] geometry prop]ly s call$ f ! /udy ( solid bodies1 ,i c\ld j z well erect "o betwe5 exp]im5tal me*anics & ! conv5tional me*anics ( ! g5]al pr9ciples4 ,9 fact1 :o does n see t 9 sep>at+ ^! two sci5ces ,i mutilate !m bo?1 & t :at w rema9 ( conv5tional me*anics :5 x %all be isolat$ w be only a v small ?+ & c 9 no way be comp>$ to t sup]b body ( doctr9e call$ geometry8 ,"o sees n[ :y ! tea*+ ( me*anics %\ld rema9 exp]im5tal4 ,only ?us c x make u compreh5d ! g5esis ( ! sci5ce1 & t is indisp5sable = ! complete und]/&+ ( ! sci5ce itself4 ,besides1 if we /udy me*anics1 x is to apply x2 & we c apply x only if x rema9s objective4 ,n[1 z we h se51 :at ! pr9ciples ga9 9 g5]ality & c]ta9ty !y lose 9 objectivity4 ,x is1 !re=e1 above all ) ! objective side ( ! pr9ciples t we mu/ be famili>iz$ e>ly1 & t c be done only by go+ f ! p>ticul> to ! g5]al1 in/ead ( ! inv]se4 ,! pr9ciples are conv5tions & disguis$ def9itions4 ,yet !y are drawn f exp]im5tal laws2 ^! laws h1 s to speak1 be5 exalt$ into pr9ciples to :i* our m9d attributes an absolute value4 ,"s philosoph]s h g5]aliz$ too f>2 !y believ$ ! pr9ciples 7 ! :ole sci5ce & consequ5tly t ! :ole sci5ce 0 conv5tional4 ,?is p>adoxical doctr9e1 call$ nom9alism1 w n be> exam9ation4 ,h[ c a law become a pr9ciple8 ,x express$ a relation betwe5 two real t]ms ,a & ,b4 ,b x 0 n rigor\sly true1 x 0 only approximate4 ,we introduce arbitr>ily an int]m$i>y t]m ,c m or less fictiti\s1 & ,c is by def9ition t :i* has ) ,a exactly ! relation express$ by ! law4 ,!n our law is sep>at$ into an absolute & rigor\s pr9ciple :i* expresses ! relation ( ,a to ,c & an exp]im5tal law1 approximate & subject to revision1 :i* expresses ! relation ( ,c to ,b4 ,x is cle> t1 h[ev] f> ?is p>tition is pu%$1 "s laws w always be left rema9+4 ,we g to ent] n[ ! doma9 ( laws prop]ly s call$4 ,,"p ,,iv ,,nature ,,*apt] ,,ix ,,hypo!ses ,,9 ,,physics ,,! ,r,,le ,,( ,,exp]im5t ,,& ,,g5]alization4--,exp]im5t is ! sole s\rce ( tru?4 ,x alone c tea* u any?+ new2 x alone c give u c]ta9ty4 ,^! are two po9ts t c n be que/ion$4 ,b !n1 if exp]im5t is ev]y?+1 :at place w rema9 = ma!matical physics8 ,:at has exp]im5tal physics to d ) su* an aid1 "o :i* seems useless & p]haps ev5 dang]\s8 ,& yet ma!matical physics exi/s1 & has done unque/ionable s]vice4 ,we h "h a fact t mu/ be expla9$4 ,! explanation is t m]ely to obs]ve is n en\gh4 ,we mu/ use our obs]vations1 & to d t we mu/ g5]alize4 ,?is is :at m5 always h done2 only z ! memory ( pa/ errors has made !m m & m c>eful1 !y h obs]v$ m & m1 & g5]aliz$ less & less4 ,e age has ridicul$ ! "o be=e x1 & accus$ x ( hav+ g5]aliz$ too quickly & too navely4 ,desc>tes piti$ ! ,ionians2 ,desc>tes1 9 8 turn1 makes u smile4 ,no d\bt our *ildr5 w "s day laugh at u4 ,b c we n !n pass ov] imm$iately to ! goal8 ,is n ?is ! means ( escap+ ! ridicule t we =esee8 ,c we n be cont5t ) j ! b>e exp]im5t8 ,no1 t is impossible2 x w\ld be to mi/ake utt]ly ! true nature ( sci5ce4 ,! sci5ti/ mu/ set 9 ord]4 ,sci5ce is built up ) facts1 z a h\se is ) /ones4 ,b a collection ( facts is no m a sci5ce ?an a heap ( /ones is a h\se4 ,& above all ! sci5ti/ mu/ =esee4 ,c>lyle has some:]e said some?+ l ?is3 ,no?+ b facts are ( importance4 ,john ,lackl& pass$ by "h4 ,"h is some?+ t is admirable4 ,"h is a reality = :i* ,i w\ld give all ! !ories 9 ! _w4 ,c>lyle 0 a fell[ c\ntryman ( ,bacon2 b ,bacon w\ld n h said t4 ,t is ! language ( ! hi/orian4 ,! physici/ w\ld say r3 ,john ,lackl& pass$ by "h2 t makes no diff]5ce to me1 = he nev] w pass ?is way aga94 ,we all "k t !re are good exp]im5ts & poor ones4 ,! latt] w accumulate 9 va92 ?\gh "o may h made a hundr$ or a ?\s&1 a s+le piece ( "w by a true ma/]1 by a ,pa/eur1 = example1 w suffice to tumble !m into oblivion4 ,bacon w\ld h well und]/ood ?is2 x is he :o inv5t$ ! phrase ,exp]im5tum crucis4 ,b ,c>lyle w\ld n h und]/ood x4 ,a fact is a fact4 ,a pupil has read a c]ta9 numb] on 8 !rmomet]2 he has tak5 no precaution2 no matt]1 he has read x1 & if x is only ! fact t c\nts1 "h is a reality ( ! same rank z ! p]egr9ations ( ,k+ ,john ,lackl&4 ,:y is ! fact t ?is pupil has made ?is read+ ( no int]e/1 :ile ! fact t a skill$ physici/ _h made ano!r read+ might be on ! contr>y v important8 ,x is because f ! fir/ read+ we c\ld n inf] any?+4 ,:at !n is a good exp]im5t8 ,x is t :i* in=ms u ( some?+ besides an isolat$ fact2 x is t :i* enables u to =esee1 t is1 t :i* enables u to g5]alize4 ,= )\t g5]alization =ekn[l$ge is impossible4 ,! circum/ances "u :i* "o has work$ w nev] reproduce !mselves all at once4 ,! obs]v$ action !n w nev] recur2 ! only ?+ t c be affirm$ is t "u analog\s circum/ances an analog\s action w be produc$4 ,9 ord] to =esee1 !n1 x is necess>y to invoke at lea/ analogy1 t is to say1 already !n to g5]alize4 ,no matt] h[ timid "o may be1 /ill x is necess>y to int]polate4 ,exp]im5t gives u only a c]ta9 numb] ( isolat$ po9ts4 ,we mu/ unite ^! by a cont9u\s l9e4 ,?is is a v]itable g5]alization4 ,b we d m2 ! curve t we %all trace w pass betwe5 ! obs]v$ po9ts & ne> ^! po9ts2 x w n pass ?r\gh ^! po9ts !mselves4 ,?us "o does n re/rict himself to g5]aliz+ ! exp]im5ts1 b corrects !m2 & ! physici/ :o %\ld try to ab/a9 f ^! corrections & really be cont5t ) ! b>e exp]im5t1 w\ld be =c$ to enunciate "s v /range laws4 ,! b>e facts1 !n1 w\ld n be en\gh = u2 & t is :y we mu/ h sci5ce ord]$1 or r organiz$4 ,x is (t5 said exp]im5ts mu/ be made )\t a preconceiv$ idea4 ,t is impossible4 ,n only w\ld x make all exp]im5t b>r51 b t w\ld be attempt$ :i* c\ld n be done4 ,e "o c>ries 9 8 m9d 8 own conception ( ! _w1 ( :i* he c n s easily rid himself4 ,we mu/1 = in/ance1 use language2 & our language is made up only ( preconceiv$ ideas & c n be o!rwise4 ,only ^! are unconsci\s preconceiv$ ideas1 a ?\s& times m dang]\s ?an ! o!rs4 ,%all we say t if we introduce o!rs1 ( :i* we are fully consci\s1 we %all only aggravate ! evil8 ,i ?9k n4 ,i believe r t !y w s]ve z c\nt]balances to ea* o!r--,i 0 go+ to say z antidotes2 !y w 9 g5]al accord ill ) "o ano!r--!y w come into conflict ) "o ano!r1 & !reby =ce u to reg>d ?+s "u diff]5t aspects4 ,?is is en\gh to emancipate u4 ,he is no long] a slave :o c *oose 8 ma/]4 ,?us1 ?anks to g5]alization1 ea* fact obs]v$ enables u to =esee a great _m o!rs2 only we mu/ n =get t ! fir/ alone is c]ta91 t all o!rs are m]ely probable4 ,no matt] h[ solidly f\nd$ a pr$iction may appe> to u1 we are nev] absolutely sure t exp]im5t w n contradict x1 if we und]take to v]ify x4 ,! probability1 h[ev]1 is (t5 s great t practically we may be cont5t ) x4 ,x is f> bett] to =esee ev5 )\t c]ta9ty ?an n to =esee at all4 ,"o mu/1 !n1 nev] disda9 to make a v]ification :5 opportunity (f]s4 ,b all exp]im5t is long & difficult2 ! work]s are few2 & ! numb] ( facts t we ne$ to =esee is imm5se4 ,comp>$ ) ?is mass ! numb] ( direct v]ifications t we c make w nev] be any?+ b a negligible quantity4 ,( ?is few t we c directly atta91 we mu/ make ! be/ use2 x is v necess>y to get f e exp]im5t ! greate/ possible numb] ( pr$ictions1 & ) ! highe/ possible degree ( probability4 ,! problem is1 s to speak1 to increase ! yield ( ! sci5tific ma*9e4 ,let u comp>e sci5ce to a libr>y t ought to gr[ cont9ually4 ,! libr>ian has at 8 disposal = 8 pur*ases only insuffici5t funds4 ,he ought to make an ef=t n to wa/e !m4 ,x is exp]im5tal physics t is entru/$ ) ! pur*ases4 ,x alone1 !n1 c enri* ! libr>y4 ,z = ma!matical physics1 its task w be to make out ! catalogue4 ,if ! catalogue is well made1 ! libr>y w n be any ri*]1 b ! read] w be help$ to use its ri*es4 ,& ev5 by %[+ ! libr>ian ! gaps 9 8 collections1 x w enable him to make a judici\s use ( 8 funds2 :i* is all ! m important because ^! funds are entirely inadequate4 ,su*1 !n1 is ! rle ( ma!matical physics4 ,x mu/ direct g5]alization 9 su* a mann] z to increase :at ,i j n[ call$ ! yield ( sci5ce4 ,by :at means x c arrive at ?is1 & h[ x c d x )\t dang]1 is :at rema9s = u to inve/igate4 ,,! ,,unity ,,( ,,nature4--,let u notice1 fir/ ( all1 t e g5]alization implies 9 "s measure ! belief 9 ! unity & simplicity ( nature4 ,z to ! unity !re c be no difficulty4 ,if ! diff]5t p>ts ( ! univ]se 7 n l ! memb]s ( "o body1 !y w\ld n act on "o ano!r1 !y w\ld "k no?+ ( "o ano!r2 & we 9 p>ticul> w\ld "k only "o ( ^! p>ts4 ,we d n ask1 !n1 if nature is "o1 b h[ x is "o4 ,z = ! second po9t1 t is n su* an easy matt]4 ,x is n c]ta9 t nature is simple4 ,c we )\t dang] act z if x 78 ,!re 0 a "t :5 ! simplicity ( ,m>iotte's law 0 an argum5t invok$ 9 favor ( its accuracy2 :5 ,fresnel himself1 aft] hav+ said 9 a conv]sation ) ,laplace t nature 0 n conc]n$ ab\t analytical difficulties1 felt himself oblig$ to make explanations1 9 ord] n to /rike too h>d at prevail+ op9ion4 ,to-day ideas h greatly *ang$2 & yet1 ^? :o d n believe t natural laws h to be simple1 are /ill (t5 oblig$ to act z if !y did4 ,!y c\ld n entirely avoid ?is necessity )\t mak+ impossible all g5]alization1 & consequ5tly all sci5ce4 ,x is cle> t any fact c be g5]aliz$ 9 an inf9ity ( ways1 & x is a "q ( *oice4 ,! *oice c be guid$ only by consid]ations ( simplicity4 ,let u take ! mo/ commonplace case1 t ( int]polation4 ,we pass a cont9u\s l9e1 z regul> z possible1 betwe5 ! po9ts giv5 by obs]vation4 ,:y d we avoid po9ts mak+ angles & too abrupt turns8 ,:y d we n make our curve describe ! mo/ caprici\s zig-zags8 ,x is because we "k be=eh&1 or believe we "k1 t ! law to be express$ c n be s complicat$ z all t4 ,we may calculate ! mass ( ,jupit] f ei!r ! movem5ts ( its satellites1 or ! p]turbations ( ! major planets1 or ^? ( ! m9or planets4 ,if we take ! av]ages ( ! det]m9ations obta9$ by ^! ?ree me?ods1 we f9d ?ree numb]s v close toge!r1 b diff]5t4 ,we might int]pret ?is result by suppos+ t ! coeffici5t ( gravitation is n ! same 9 ! ?ree cases4 ,! obs]vations w\ld c]ta9ly be mu* bett] repres5t$4 ,:y d we reject ?is int]pretation8 ,n because x is absurd1 b because x is ne$lessly complicat$4 ,we %all only accept x :5 we are =c$ to1 & t is n yet4 ,to sum up1 ord9>ily e law is held to be simple till ! contr>y is prov$4 ,?is cu/om is impos$ ^u physici/s by ! causes t ,i h j expla9$4 ,b h[ %all we ju/ify x 9 ! pres5ce ( discov]ies t %[ u e day new details t are ri*] & m complex8 ,h[ %all we ev5 reconcile x ) ! belief 9 ! unity ( nature8 ,= if ev]y?+ dep5ds on ev]y?+1 relation%ips :]e s _m div]se factors ent] c no long] be simple4 ,if we /udy ! hi/ory ( sci5ce1 we see happ5 two inv]se ph5om5a1 s to speak4 ,sometimes simplicity hides "u complex appe>ances2 sometimes x is ! simplicity :i* is app>5t1 & :i* disguises extremely complicat$ realities4 ,:at is m complicat$ ?an ! confus$ movem5ts ( ! planets8 ,:at simpl] ?an ,newton's law8 ,"h nature1 mak+ sport1 z ,fresnel said1 ( analytical difficulties1 employs only simple means1 & by comb9+ !m produces ,i "k n :at inextricable tangle4 ,"h x is ! hidd5 simplicity :i* mu/ be discov]$4 ,examples ( ! opposite ab\nd4 ,9 ! k9etic !ory ( gases1 "o deals ) molecules mov+ ) great velocities1 ^: pa?s1 alt]$ by incessant collisions1 h ! mo/ caprici\s =ms & trav]se space 9 e direction4 ,! obs]vable result is ,m>iotte's simple law4 ,e individual fact 0 complicat$4 ,! law ( great numb]s has ree/abli%$ simplicity 9 ! av]age4 ,"h ! simplicity is m]ely app>5t1 & only ! co>s5ess ( our s5ses prev5ts our p]ceiv+ ! complexity4 ,_m ph5om5a obey a law ( proportionality4 ,b :y8 ,because 9 ^! ph5om5a !re is some?+ v small4 ,! simple law obs]v$1 !n1 is only a result ( ! g5]al analytical rule t ! inf9itely small increm5t ( a function is proportional to ! increm5t ( ! v>iable4 ,z 9 reality our increm5ts are n inf9itely small1 b v small1 ! law ( proportionality is only approximate1 & ! simplicity is only app>5t4 ,:at ,i h j said applies to ! rule ( ! sup]position ( small motions1 ! use ( :i* is s fruitful1 & :i* is ! basis ( optics4 ,& ,newton's law itself8 ,its simplicity1 s long undetect$1 is p]haps only app>5t4 ,:o kn[s :e!r x is n due to "s complicat$ me*anism1 to ! impact ( "s subtile matt] animat$ by irregul> movem5ts1 & :e!r x has n become simple only ?r\gh ! action ( av]ages & ( great numb]s8 ,9 any case1 x is difficult n to suppose t ! true law conta9s complem5t>y t]ms1 :i* w\ld become s5sible at small di/ances4 ,if 9 a/ronomy !y are negligible z modify+ ,newton's law1 & if ! law ?us rega9s its simplicity1 x w\ld be only because ( ! imm5sity ( cele/ial di/ances4 ,no d\bt1 if our means ( inve/igation %\ld become m & m p5etrat+1 we %\ld discov] ! simple "u ! complex1 !n ! complex "u ! simple1 !n aga9 ! simple "u ! complex1 & s on1 )\t our be+ able to =esee :at w be ! la/ t]m4 ,we mu/ /op some:]e1 & t sci5ce may be possible we mu/ /op :5 we h f\nd simplicity4 ,?is is ! only gr\nd on :i* we c re> ! edifice ( our g5]alizations4 ,b ?is simplicity be+ only app>5t1 w ! gr\nd be firm en\gh8 ,?is is :at mu/ be inve/igat$4 ,= t purpose1 let u see :at "p is play$ 9 our g5]alizations by ! belief 9 simplicity4 ,we h v]ifi$ a simple law 9 a good _m p>ticul> cases2 we refuse to admit t ?is agreem5t1 s (t5 repeat$1 is simply ! result ( *ance1 & conclude t ! law mu/ be true 9 ! g5]al case4 ,kepl] notices t a planet's positions1 z obs]v$ by ,ty*o1 are all on "o ellipse4 ,nev] = a mom5t does he h ! ?\ght t by a /range play ( *ance ,ty*o nev] obs]v$ ! heav5s except at a mom5t :5 ! real orbit ( ! planet happ5$ to cut ?is ellipse4 ,:at does x matt] !n :e!r ! simplicity be real1 or :e!r x cov]s a complex reality8 ,:e!r x is due to ! influ5ce ( great numb]s1 :i* levels d[n individual diff]5ces1 or to ! greatness or smallness ( c]ta9 quantities1 :i* all[s u to neglect c]ta9 t]ms1 9 no case is x due to *ance4 ,?is simplicity1 real or app>5t1 always has a cause4 ,we c always foll[1 !n1 ! same c\rse ( reason+1 & if a simple law has be5 obs]v$ 9 sev]al p>ticul> cases1 we c legitimately suppose t x w /ill be true 9 analog\s cases4 ,to refuse to d ?is w\ld be to attribute to *ance an inadmissible rle4 ,!re is1 h[ev]1 a diff]5ce4 ,if ! simplicity 7 real & ess5tial1 x w\ld resi/ ! increas+ precision ( our means ( measure4 ,if !n we believe nature to be ess5tially simple1 we mu/1 f a simplicity t is approximate1 inf] a simplicity t is rigor\s4 ,?is is :at 0 done =m]ly2 & ?is is :at we no long] h a "r to d4 ,! simplicity ( ,kepl]'s laws1 = example1 is only app>5t4 ,t does n prev5t _! be+ applicable1 v ne>ly1 to all sy/ems analog\s to ! sol> sy/em2 b x does prev5t _! be+ rigor\sly exact4 ,,! ,r,,le ,,( ,,hypo!sis4--,all g5]alization is a hypo!sis4 ,hypo!sis1 !n1 has a necess>y rle t no "o has ev] conte/$4 ,only1 x ought always1 z soon z possible & z (t5 z possible1 to be subject$ to v]ification4 ,&1 ( c\rse1 if x does n /& ?is te/1 x ought to be ab&on$ )\t res]ve4 ,?is is :at we g5]ally d1 b sometimes ) r an ill humor4 ,well1 ev5 ?is ill humor is n ju/ifi$4 ,! physici/ :o has j r5\nc$ "o ( 8 hypo!ses ought1 on ! contr>y1 to be full ( joy2 = he has f\nd an unexpect$ opportunity = discov]y4 ,8 hypo!sis1 ,i imag9e1 _h n be5 adopt$ )\t consid]ation2 x took acc\nt ( all ! kn[n factors t x seem$ c\ld ent] into ! ph5om5on4 ,if ! te/ does n support x1 x is because !re is some?+ unexpect$ & extraord9>y2 & because !re is go+ to be some?+ f\nd t is unkn[n & new4 ,has ! disc>d$ hypo!sis1 !n1 be5 b>r58 ,f> f t1 x may be said x has r5d]$ m s]vice ?an a true hypo!sis4 ,n only has x be5 ! occasion ( ! decisive exp]im5t1 b1 )\t hav+ made ! hypo!sis1 ! exp]im5t w\ld h be5 made by *ance1 s t no?+ w\ld h be5 d]iv$ f x4 ,"o w\ld h se5 no?+ extraord9>y2 only "o fact ! m w\ld h be5 catalogu$ )\t d$uc+ f x ! lea/ consequ5ce4 ,n[ on :at condition is ! use ( hypo!sis )\t dang]8 ,! firm det]m9ation to submit to exp]im5t is n en\gh2 !re are /ill dang]\s hypo!ses2 fir/1 & above all1 ^? :i* are tacit & unconsci\s4 ,s9ce we make !m )\t kn[+ x1 we are p[]less to ab&on !m4 ,"h aga91 !n1 is a s]vice t ma!matical physics c r5d] u4 ,by ! precision t is *>act]i/ic ( x1 x compels u to =mulate all ! hypo!ses t we %\ld make )\t x1 b unconsci\sly4 ,let u notice besides t x is important n to multiply hypo!ses beyond measure1 & to make !m only "o aft] ! o!r4 ,if we con/ruct a !ory bas$ on a numb] ( hypo!ses1 & if exp]im5t condemns x1 :i* ( our premises is x necess>y to *ange8 ,x w be impossible to "k4 ,& inv]sely1 if ! exp]im5t succe$s1 %all we believe t we h demon/rat$ all ! hypo!ses at once8 ,%all we believe t ) "o s+le equation we h det]m9$ sev]al unkn[ns8 ,we mu/ equally take c>e to di/+ui% betwe5 ! diff]5t k9ds ( hypo!ses4 ,!re are fir/ ^? :i* are p]fectly natural & f :i* "o c sc>cely escape4 ,x is difficult n to suppose t ! influ5ce ( bodies v remote is q negligible1 t small movem5ts foll[ a l9e> law1 t ! effect is a cont9u\s function ( its cause4 ,i w say z mu* ( ! conditions impos$ by symmetry4 ,all ^! hypo!ses =m1 z x 71 ! common basis ( all ! !ories ( ma!matical physics4 ,!y are ! la/ t ought to be ab&on$4 ,!re is a second class ( hypo!ses1 t ,i %all t]m neutral4 ,9 mo/ que/ions ! analy/ assumes at ! beg9n+ ( 8 calculations ei!r t matt] is cont9u\s or1 on ! contr>y1 t x is =m$ ( atoms4 ,he might h made ! opposite assumption )\t *ang+ 8 results4 ,he w\ld only h _h m tr\ble to obta9 !m2 t is all4 ,if1 !n1 exp]im5t confirms 8 conclusions1 w he ?9k t he has demon/rat$1 = in/ance1 ! real exi/5ce ( atoms8 ,9 optical !ories two vectors are introduc$1 ( :i* "o is reg>d$ z a velocity1 ! o!r z a vortex4 ,"h aga9 is a neutral hypo!sis1 s9ce ! same conclusions w\ld h be5 rea*$ by tak+ precisely ! opposite4 ,! success ( ! exp]im5t1 !n1 c n prove t ! fir/ vector is inde$ a velocity2 x c only prove "o ?+1 t x is a vector4 ,?is is ! only hypo!sis t has really be5 introduc$ 9 ! premises4 ,9 ord] to give x t concrete appe>ance :i* ! weakness ( our m9ds requires1 x has be5 necess>y to consid] x ei!r z a velocity or z a vortex1 9 ! same way t x has be5 necess>y to repres5t x by a lett]1 ei!r x or y4 ,! result1 h[ev]1 :atev] x may be1 w n prove t x 0 "r or wrong to reg>d x z a velocity any m ?an x w prove t x 0 "r or wrong to call x x & n y4 ,^! neutral hypo!ses are nev] dang]\s1 if only _! "* is n misund]/ood4 ,!y may be useful1 ei!r z devices = computation1 or to aid our und]/&+ by concrete images1 to fix our ideas z ! say+ is4 ,!re is1 !n1 no occasion to exclude !m4 ,! hypo!ses ( ! ?ird class are ! real g5]alizations4 ,!y are ! ones t exp]im5t mu/ confirm or invalidate4 ,:e!r v]ifi$ or condemn$1 !y w always be fruitful4 ,b = ! reasons t ,i h set =?1 !y w only be fruitful if !y are n too num]\s4 ,,orig9 ,,( ,,ma!matical ,,physics4--,let u p5etrate fur!r1 & /udy m closely ! conditions t h p]mitt$ ! developm5t ( ma!matical physics4 ,we obs]ve at once t ! ef=ts ( sci5ti/s h always aim$ to resolve ! complex ph5om5on directly giv5 by exp]im5t into a v l>ge numb] ( elem5t>y ph5om5a4 ,?is is done 9 ?ree diff]5t ways3 fir/1 9 "t4 ,in/ead ( embrac+ 9 its entirety ! progressive developm5t ( a ph5om5on1 ! aim is simply to connect ea* in/ant ) ! in/ant imm$iately prec$+ x4 ,x is admitt$ t ! actual /ate ( ! _w dep5ds only on ! imm$iate pa/1 )\t be+ directly influ5c$1 s to speak1 by ! memory ( a di/ant pa/4 ,?anks to ?is po/ulate1 in/ead ( /udy+ directly ! :ole succession ( ph5om5a1 x is possible to conf9e ourselves to writ+ its 'diff]5tial equation4' ,= ,kepl]'s laws we sub/itute ,newton's law4 ,next we try to analyze ! ph5om5on 9 space4 ,:at exp]im5t gives u is a confus$ mass ( facts pres5t$ on a /age ( consid]able ext5t4 ,we mu/ try to discov] ! elem5t>y ph5om5on1 :i* w be1 on ! contr>y1 localiz$ 9 a v small region ( space4 ,"s examples w p]haps make my ?\ght bett] und]/ood4 ,if we wi%$ to /udy 9 all its complexity ! di/ribution ( temp]ature 9 a cool+ solid1 we %\ld nev] succe$4 ,ev]y?+ becomes simple if we reflect t "o po9t ( ! solid c n give up its heat directly to a di/ant po9t2 x w give up its heat only to ! po9ts 9 ! imm$iate neighborhood1 & x is by degrees t ! fl[ ( heat c rea* o!r p>ts ( ! solid4 ,! elem5t>y ph5om5on is ! ex*ange ( heat betwe5 two contigu\s po9ts4 ,x is /rictly localiz$1 & is relatively simple1 if we admit1 z is natural1 t x is n influ5c$ by ! temp]ature ( molecules ^: di/ance is s5sible4 ,i b5d a rod4 ,x is go+ to take a v complicat$ =m1 ! direct /udy ( :i* w\ld be impossible4 ,b ,i %all be able1 h[ev]1 to attack x1 if ,i obs]ve t its flexure is a result only ( ! de=mation ( ! v small elem5ts ( ! rod1 & t ! de=mation ( ea* ( ^! elem5ts dep5ds only on ! =ces t are directly appli$ to x1 & n at all on ^? :i* may act on ! o!r elem5ts4 ,9 all ^! examples1 :i* ,i might easily multiply1 we admit t !re is no action at a di/ance1 or at lea/ at a great di/ance4 ,?is is a hypo!sis4 ,x is n always true1 z ! law ( gravitation %[s u4 ,x mu/1 !n1 be submitt$ to v]ification4 ,if x is confirm$1 ev5 approximately1 x is preci\s1 = x w enable u to make ma!matical physics1 at lea/ by successive approximations4 ,if x does n /& ! te/1 we mu/ look = some?+ else analog\s2 = !re are /ill o!r means ( arriv+ at ! elem5t>y ph5om5on4 ,if sev]al bodies act simultane\sly1 x may happ5 t _! actions are indep5d5t & are simply add$ to "o ano!r1 ei!r z vectors or z scal>s4 ,! elem5t>y ph5om5on is !n ! action ( an isolat$ body4 ,or aga91 we h to deal ) small movem5ts1 or m g5]ally ) small v>iations1 :i* obey ! well-kn[n law ( sup]position4 ,! obs]v$ movem5t w !n be decompos$ into simple movem5ts1 = example1 s\nd into its h>monics1 :ite light into its mono*romatic compon5ts4 ,:5 we h discov]$ 9 :at direction x is advisable to look = ! elem5t>y ph5om5on1 by :at means c we rea* x8 ,fir/ ( all1 x w (t5 happ5 t 9 ord] to detect x1 or r to detect ! "p ( x useful to u1 x w n be necess>y to p5etrate ! me*anism2 ! law ( great numb]s w suffice4 ,let u take aga9 ! in/ance ( ! propagation ( heat4 ,e molecule emits rays t[>d e neighbor+ molecule4 ,accord+ to :at law1 we d n ne$ to "k4 ,if we %\ld make any supposition 9 reg>d to ?is1 x w\ld be a neutral hypo!sis & consequ5tly useless & incapable ( v]ification4 ,&1 9 fact1 by ! action ( av]ages & ?anks to ! symmetry ( ! m$ium1 all ! diff]5ces are level$ d[n1 & :atev] hypo!sis may be made1 ! result is always ! same4 ,! same circum/ance is pres5t$ 9 ! !ory ( electricity & 9 t ( capill>ity4 ,! neighbor+ molecules attract & repel "o ano!r4 ,we d n ne$ to "k accord+ to :at law2 x is en\gh = u t ?is attraction is s5sible only at small di/ances1 & t ! molecules are v num]\s1 t ! m$ium is symmetrical1 & we %all only h to let ! law ( great numb]s act4 ,"h aga9 ! simplicity ( ! elem5t>y ph5om5on 0 hidd5 "u ! complexity ( ! resultant obs]vable ph5om5on2 b1 9 its turn1 ?is simplicity 0 only app>5t1 & conceal$ a v complex me*anism4 ,! be/ means ( arriv+ at ! elem5t>y ph5om5on w\ld evid5tly be exp]im5t4 ,we ought by exp]im5tal contrivance to dissociate ! complex %eaf t nature (f]s to our rese>*es1 & to /udy ) c>e ! elem5ts z mu* isolat$ z possible4 ,= example1 natural :ite light w\ld be decompos$ into mono*romatic lights by ! aid ( ! prism1 & into pol>iz$ light by ! aid ( ! pol>iz]4 ,un=tunately t is nei!r always possible nor always suffici5t1 & sometimes ! m9d mu/ out/rip exp]im5t4 ,i %all cite only "o example1 :i* has always /ruck me =cibly4 ,if ,i decompose :ite light1 ,i %all be able to isolate a small "p ( ! spectrum1 b h[ev] small x may be1 x w reta9 a c]ta9 bread?4 ,likewise ! natural lights1 call$ mono*romatic1 give u a v n>r[ l9e1 b n1 h[ev]1 inf9itely n>r[4 ,x might be suppos$ t by /udy+ exp]im5tally ! prop]ties ( ^! natural lights1 by work+ ) f9] & f9] l9es ( ! spectrum1 & by pass+ at la/ to ! limit1 s to speak1 we %\ld succe$ 9 le>n+ ! prop]ties ( a light /rictly mono*romatic4 ,t w\ld n be accurate4 ,suppose t two rays emanate f ! same s\rce1 t we pol>ize !m fir/ 9 two p]p5dicul> planes1 !n br+ !m back to ! same plane ( pol>ization1 & try to make !m int]f]e4 ,if ! light 7 /rictly mono*romatic1 !y w\ld int]f]e4 ,) our lights1 :i* are ne>ly mono*romatic1 !re w be no int]f]5ce1 & t no matt] h[ n>r[ ! l9e4 ,9 ord] to be o!rwise x w\ld h to be sev]al million times z n>r[ z ! f9e/ kn[n l9es4 ,"h1 !n1 ! passage to ! limit w\ld h deceiv$ u4 ,! m9d mu/ out/rip ! exp]im5t1 & if x has done s ) success1 x is because x has all[$ itself to be guid$ by ! in/9ct ( simplicity4 ,! k ( ! elem5t>y fact enables u to put ! problem 9 an equation4 ,no?+ rema9s b to d$uce f ?is by comb9ation ! complex fact t c be obs]v$ & v]ifi$4 ,?is is :at is call$ integration1 & is ! bus9ess ( ! ma!matician4 ,x may be ask$ :y1 9 physical sci5ces1 g5]alization s readily takes ! ma!matical =m4 ,! reason is n[ easy to see4 ,x is n only because we h num]ical laws to express2 x is because ! obs]vable ph5om5on is due to ! sup]position ( a great numb] ( elem5t>y ph5om5a all alike4 ,?us q naturally are introduc$ diff]5tial equations4 ,x is n en\gh t ea* elem5t>y ph5om5on obeys simple laws2 all ^? to be comb9$ mu/ obey ! same law4 ,!n only c ! int]v5tion ( ma!matics be ( use2 ma!matics tea*es u 9 fact to comb9e l ) l4 ,its aim is to le>n ! result ( a comb9ation )\t ne$+ to g ov] ! comb9ation piece by piece4 ,if we h to repeat sev]al times ! same op]ation1 x enables u to avoid ?is repetition by tell+ u 9 advance ! result ( x by a sort ( induction4 ,i h expla9$ ?is above1 9 ! *apt] on ma!matical reason+4 ,b = ?is1 all ! op]ations mu/ be alike4 ,9 ! opposite case1 x w\ld evid5tly be necess>y to resign ourselves to do+ !m 9 reality "o aft] ano!r1 & ma!matics w\ld become useless4 ,x is !n ?anks to ! approximate homog5eity ( ! matt] /udi$ by physici/s t ma!matical physics c\ld be born4 ,9 ! natural sci5ces1 we no long] f9d ^! conditions3 homog5eity1 relative indep5d5ce ( remote p>ts1 simplicity ( ! elem5t>y fact2 & ?is is :y naturali/s are oblig$ to resort to o!r me?ods ( g5]alization4 ,,*apt] ,x ,,! ,,!ories ,,( ,,mod]n ,,physics ,,mean+ ,,( ,,physical ,,!ories4--,! laity are /ruck to see h[ ephem]al sci5tific !ories are4 ,aft] "s ye>s ( prosp]ity1 !y see !m successively ab&on$2 !y see ru9s accumulate ^u ru9s2 !y =esee t ! !ories fa%ionable to-day w %ortly succumb 9 _! turn & h5ce !y conclude t ^! are absolutely idle4 ,?is is :at !y call ! bankruptcy ( sci5ce4 ,_! skepticism is sup]ficial2 !y give no acc\nt to !mselves ( ! aim & ! rle ( sci5tific !ories2 o!rwise !y w\ld compreh5d t ! ru9s may /ill be good = some?+4 ,no !ory seem$ m solid ?an t ( ,fresnel :i* attribut$ light to motions ( ! e!r4 ,yet n[ ,maxwell's is pref]r$4 ,does ?is mean ! "w ( ,fresnel 0 9 va98 ,no1 because ! aim ( ,fresnel 0 n to f9d out :e!r !re is really an e!r1 :e!r x is or is n =m$ ( atoms1 :e!r ^! atoms really move 9 ?is or t s5se2 8 object 0 to =esee optical ph5om5a4 ,n[1 ,fresnel's !ory always p]mits ( ?is1 to-day z well z be=e ,maxwell4 ,! diff]5tial equations are always true2 !y c always be integrat$ by ! same proc$ures & ! results ( ?is integration always reta9 _! value4 ,& let no "o say t ?us we r$uce physical !ories to ! rle ( m]e practical recipes2 ^! equations express relations1 & if ! equations rema9 true x is because ^! relations pres]ve _! reality4 ,!y tea* u1 n[ z !n1 t !re is su* & su* a relation betwe5 "s ?+ & "s o!r ?+2 only ?is some?+ =m]ly we call$ motion2 we n[ call x electric curr5t4 ,b ^! appellations 7 only images sub/itut$ = ! real objects :i* nature w et]nally hide f u4 ,! true relations betwe5 ^! real objects are ! only reality we c atta9 to1 & ! only condition is t ! same relations exi/ betwe5 ^! objects z betwe5 ! images by :i* we are =c$ to replace !m4 ,if ^! relations are kn[n to u1 :at matt] if we deem x conv5i5t to replace "o image by ano!r4 ,t "s p]iodic ph5om5on 7an electric oscillation1 = in/ance7 is really due to ! vibration ( "s atom :i*1 act+ l a p5dulum1 really moves 9 ?is or t s5se1 is nei!r c]ta9 nor int]e/+4 ,b t betwe5 electric oscillation1 ! motion ( ! p5dulum & all p]iodic ph5om5a !re exi/s a close relation%ip :i* corresponds to a pr(\nd reality2 t ?is relation%ip1 ?is similitude1 or r ?is p>allelism ext5ds into details2 t x is a consequ5ce ( m g5]al pr9ciples1 t ( en]gy & t ( lea/ action2 ?is is :at we c affirm2 ?is is ! tru? :i* w always rema9 ! same "u all ! co/umes 9 :i* we may deem x useful to deck x out4 ,num]\s !ories ( disp]sion h be5 propos$2 ! fir/ 0 imp]fect & conta9$ only a small "p ( tru?4 ,aft]w>ds came t ( ,helmholtz2 !n x 0 modifi$ 9 v>i\s ways1 & its au?or himself imag9$ ano!r f\nd$ on ! pr9ciples ( ,maxwell4 ,b1 :at is rem>kable1 all ! sci5ti/s :o came aft] ,helmholtz rea*$ ! same equations1 />t+ f po9ts ( dep>ture 9 appe>ance v widely sep>at$4 ,i w v5ture to say ^! !ories are all true at ! same "t1 n only because !y make u =esee ! same ph5om5a1 b because !y put 9 evid5ce a true relation1 t ( absorption & anomal\s disp]sion4 ,:at is true 9 ! premises ( ^! !ories is :at is common to all ! au?ors2 ?is is ! affirmation ( ?is or t relation betwe5 c]ta9 ?+s :i* "s call by "o "n1 o!rs by ano!r4 ,! k9etic !ory ( gases has giv5 rise to _m objections1 :i* we c\ld h>dly answ] if we pret5d$ to see 9 x ! absolute tru?4 ,b all ^! objections w n preclude its hav+ be5 useful1 & p>ticul>ly s 9 reveal+ to u a relation true & b = x pr(\ndly hidd51 t ( ! gase\s pressure & ! osmotic pressure4 ,9 ?is s5se1 !n1 x may be said to be true4 ,:5 a physici/ f9ds a contradiction betwe5 two !ories equally de> to him1 he sometimes says3 ,we w n bo!r ab\t t1 b hold firmly ! two ends ( ! *a91 ?\gh ! int]m$iate l9ks are hidd5 f u4 ,?is argum5t ( an emb>rass$ !ologian w\ld be ridicul\s if x 7 necess>y to attribute to physical !ories ! s5se ! laity give !m4 ,9 case ( contradiction1 "o ( !m at lea/ mu/ !n be reg>d$ z false4 ,x is no long] ! same if 9 !m be s\ght only :at %\ld be s\ght4 ,may be !y bo? express true relations & ! contradiction is only 9 ! images :]e) we h clo!d ! reality4 ,to ^? :o f9d we re/rict too mu* ! doma9 accessible to ! sci5ti/1 ,i answ]3 ,^! que/ions :i* we int]dict to y & :i* y regret1 are n only insoluble1 !y are illusory & devoid ( mean+4 ,"s philosoph] pret5ds t all physics may be expla9$ by ! mutual impacts ( atoms4 ,if he m]ely means !re are betwe5 physical ph5om5a ! same relations z betwe5 ! mutual impacts ( a great numb] ( balls1 well & good1 t is v]ifiable1 t is p]haps true4 ,b he means some?+ m2 & we ?9k we und]/& x because we ?9k we "k :at impact is 9 itself2 :y8 ,simply because we h (t5 se5 games ( billi>ds4 ,%all we ?9k ,god1 contemplat+ 8 "w1 feels ! same s5sations z we 9 wat*+ a billi>d mat*8 ,if we d n wi% to give ?is biz>re s5se to 8 ass]tion1 if nei!r d we wi% ! re/rict$ s5se ,i h j expla9$1 :i* is good s5se1 !n x has none4 ,hypo!ses ( ?is sort h !re=e only a metaphorical s5se4 ,! sci5ti/ %\ld no m int]dict !m ?an ! poet does metaphors2 b he ought to "k :at !y are wor?4 ,!y may be useful to give a c]ta9 satisfaction to ! m9d1 & !y w n be injuri\s provid$ !y are only indiff]5t hypo!ses4 ,^! consid]ations expla9 to u :y c]ta9 !ories1 suppos$ to be ab&on$ & f9ally condemn$ by exp]im5t1 sudd5ly arise f _! a%es & recomm5ce a new life4 ,x is because !y express$ true relations2 & because !y _h n ceas$ to d s :51 = "o reason or ano!r1 we felt x necess>y to enunciate ! same relations 9 ano!r language4 ,s !y reta9$ a sort ( lat5t life4 ,sc>cely fifte5 ye>s ago 0 !re any?+ m ridicul\s1 m navely antiquat$1 ?an ,c\lomb's fluids8 ,& yet "h !y are reappe>+ "u ! "n ( electrons4 ,:]e9 d ^! p]man5tly electrifi$ molecules diff] f ,c\lomb's electric molecules8 ,x is true t 9 ! electrons ! electricity is support$ by a little1 a v little matt]2 9 o!r words1 !y h a mass 7& yet ?is is n[ conte/$72 b ,c\lomb did n d5y mass to 8 fluids1 or1 if he did1 x 0 only ) reluctance4 ,x w\ld be ra% to affirm t ! belief 9 electrons w n aga9 suff] eclipse2 x 0 none ! less curi\s to note ?is unexpect$ resurrection4 ,b ! mo/ /rik+ example is ,c>not's pr9ciple4 ,c>not set x up />t+ f false hypo!ses2 :5 x 0 se5 t heat is n inde/ructible1 b may be trans=m$ into "w1 8 ideas 7 completely ab&on$2 aft]w>ds ,clausius return$ to !m & made !m f9ally triumph4 ,c>not's !ory1 "u its primitive =m1 express$1 aside f true relations1 o!r inexact relations1 dbris ( antiquat$ ideas2 b ! pres5ce ( ^! latt] did n *ange ! reality ( ! o!rs4 ,clausius _h only to disc>d !m z "o lops (f dead bran*es4 ,! result 0 ! second fundam5tal law ( !rmodynamics4 ,!re 7 always ! same relations2 ?\gh ^! relations no long] subsi/$1 at lea/ 9 appe>ance1 betwe5 ! same objects4 ,?is 0 en\gh = ! pr9ciple to reta9 its value4 ,& ev5 ! reason+s ( ,c>not h n p]i%$ because ( t2 !y 7 appli$ to a mat]ial ta9t$ ) error2 b _! =m 7t is to say1 ! ess5tial7 rema9$ correct4 ,:at ,i h j said illum9ates at ! same "t ! rle ( g5]al pr9ciples su* z ! pr9ciple ( lea/ action1 or t ( ! cons]vation ( en]gy4 ,^! pr9ciples h a v high value2 !y 7 obta9$ 9 seek+ :at !re 0 9 common 9 ! enunciation ( num]\s physical laws2 !y repres5t !re=e1 z x 71 ! qu9tess5ce ( innum]able obs]vations4 ,h[ev]1 f _! v g5]ality a consequ5ce results to :i* ,i h call$ att5tion 9 ,*apt] ,,viii1 namely1 t !y c no long] be v]ifi$4 ,z we c n give a g5]al def9ition ( en]gy1 ! pr9ciple ( ! cons]vation ( en]gy signifies simply t !re is some?+ :i* rema9s con/ant4 ,well1 :atev] be ! new notions t future exp]im5ts %all give u ab\t ! _w1 we are sure 9 advance t !re w be some?+ !re :i* w rema9 con/ant & :i* may be call$ en]gy4 ,is ?is to say t ! pr9ciple has no mean+ & vani%es 9 a tautology8 ,n at all2 x signifies t ! diff]5t ?+s to :i* we give ! "n ( en]gy are connect$ by a true k9%ip2 x affirms a real relation betwe5 !m4 ,b !n if ?is pr9ciple has a mean+1 x may be false2 x may be t we h n ! "r to ext5d indef9itely its applications1 & yet x is c]ta9 be=eh& to be v]ifi$ 9 ! /rict acceptation ( ! t]m2 h[ !n %all we "k :5 x %all h atta9$ all ! ext5sion :i* c legitimately be giv5 x8 ,j simply :5 x %all cease to be useful to u1 t is1 to make u correctly =esee new ph5om5a4 ,we %all be sure 9 su* a case t ! relation affirm$ is no long] real2 = o!rwise x w\ld be fruitful2 exp]im5t1 )\t directly contradict+ a new ext5sion ( ! pr9ciple1 w yet h condemn$ x4 ,,physics ,,& ,,me*anism4--,mo/ !ori/s h a con/ant pr$ilection = explanations borr[$ f me*anics or dynamics4 ,"s w\ld be satisfi$ if !y c\ld expla9 all ph5om5a by motions ( molecules attract+ ea* o!r accord+ to c]ta9 laws4 ,o!rs are m exact+2 !y w\ld suppress attractions at a di/ance2 _! molecules %\ld foll[ rectil9e> pa?s f :i* !y c\ld be made to deviate only by impacts4 ,o!rs aga91 l ,h]tz1 suppress =ces also1 b suppose _! molecules subject$ to geometric atta*m5ts analog\s1 = in/ance1 to ^? ( our l9kages2 !y try ?us to r$uce dynamics to a sort ( k9ematics4 ,9 a ^w1 all w\ld b5d nature into a c]ta9 =m outside ( :i* _! m9d c\ld n feel satisfi$4 ,w nature be suffici5tly flexible = t8 ,we %all exam9e ?is "q 9 ,*apt] ,,xii1 propos ( ,maxwell's !ory4 ,:5ev] ! pr9ciples ( en]gy & ( lea/ action are satisfi$1 we %all see n only t !re is always "o possible me*anical explanation1 b t !re is always an inf9ity ( !m4 ,?anks to a well-kn[n !orem ( ,knig's on l9kages1 x c\ld be %[n t we c1 9 an inf9ity ( ways1 expla9 ev]y?+ by atta*m5ts aft] ! mann] ( ,h]tz1 or also by c5tral =ces4 ,)\t d\bt x c\ld be demon/rat$ j z easily t ev]y?+ c always be expla9$ by simple impacts4 ,= t1 ( c\rse1 we ne$ n be cont5t ) ord9>y matt]1 ) t :i* falls "u our s5ses & ^: motions we obs]ve directly4 ,ei!r we %all suppose t ?is common matt] is =m$ ( atoms ^: int]nal motions elude u1 ! displacem5t ( ! totality alone rema9+ accessible to our s5ses4 ,or else we %all imag9e "s "o ( ^? subtile fluids :i* "u ! "n ( e!r or "u o!r names1 h at all times play$ s great a rle 9 physical !ories4 ,(t5 "o goes fur!r & reg>ds ! e!r z ! sole primitive matt] or ev5 z ! only true matt]4 ,! m mod]ate consid] common matt] z cond5s$ e!r1 :i* is no?+ />tl+2 b o!rs r$uce /ill fur!r its importance & see 9 x no?+ m ?an ! geometric locus ( ! e!r's s+ul>ities4 ,= in/ance1 :at we call matt] is = ,"l ,kelv9 only ! locus ( po9ts :]e ! e!r is animat$ by vortex motions2 = ,riemann1 x 0 ! locus ( po9ts :]e e!r is con/antly de/roy$2 = o!r m rec5t au?ors1 ,wie*]t or ,l>mor1 x is ! locus ( po9ts :]e ! e!r und]goes a sort ( torsion ( a v p>ticul> nature4 ,if ! attempt is made to occupy "o ( ^! po9ts ( view1 ,i ask myself by :at "r %all we ext5d to ! e!r1 "u pretext t ?is is ! true matt]1 me*anical prop]ties obs]v$ 9 ord9>y matt]1 :i* is only false matt]4 ,! anci5t fluids1 caloric1 electricity1 etc41 7 ab&on$ :5 x 0 p]ceiv$ t heat is n inde/ructible4 ,b !y 7 ab&on$ = ano!r reason also4 ,9 mat]ializ+ !m1 _! individuality 01 s to speak1 emphasiz$1 a sort ( abyss 0 op5$ betwe5 !m4 ,?is _h to be fill$ up on ! com+ ( a m vivid feel+ ( ! unity ( nature1 & ! p]ception ( ! intimate relations :i* b9d toge!r all its p>ts4 ,n only did ! old physici/s1 9 multiply+ fluids1 create entities unnecess>ily1 b !y broke real ties4 ,x is n suffici5t = a !ory to affirm no false relations1 x mu/ n hide true relations4 ,& does our e!r really exi/8 ,we "k ! orig9 ( our belief 9 ! e!r4 ,if light rea*es u f a di/ant />1 dur+ sev]al ye>s x 0 no long] on ! /> & n yet on ! e>?2 x mu/ !n be some:]e & su/a9$1 s to speak1 by "s mat]ial support4 ,! same idea may be express$ "u a m ma!matical & m ab/ract =m4 ,:at we asc]ta9 are ! *anges und]gone by mat]ial molecules2 we see1 = in/ance1 t our photographic plate feels ! consequ5ces ( ph5om5a ( :i* ! inc&esc5t mass ( ! /> 0 ! !at] sev]al ye>s be=e4 ,n[1 9 ord9>y me*anics ! /ate ( ! sy/em /udi$ dep5ds only on its /ate at an in/ant imm$iately ant]ior2 !re=e ! sy/em satisfies diff]5tial equations4 ,on ! contr>y1 if we %\ld n believe 9 ! e!r1 ! /ate ( ! mat]ial univ]se w\ld dep5d n only on ! /ate imm$iately prec$+1 b on /ates mu* old]2 ! sy/em w\ld satisfy equations ( f9ite diff]5ces4 ,x is to escape ?is d]ogation ( ! g5]al laws ( me*anics t we h inv5t$ ! e!r4 ,t w\ld /ill only oblige u to fill up1 ) ! e!r1 ! int]planet>y void1 b n to make x p5etrate ! bosom ( ! mat]ial m$ia !mselves4 ,fizeau's exp]im5t goes fur!r4 ,by ! int]f]5ce ( rays :i* h trav]s$ air or wat] 9 motion1 x seems to %[ u two diff]5t m$ia int]p5etrat+ & yet *ang+ place "o ) reg>d to ! o!r4 ,we seem to t\* ! e!r ) ! f+]4 ,yet exp]im5ts may be conceiv$ :i* w\ld make u t\* x /ill m ne>ly4 ,suppose ,newton's pr9ciple1 ( ! equality ( action & reaction1 no long] true if appli$ to matt] alone1 & t we h e/abli%$ x4 ,! geometric sum ( all ! =ces appli$ to all ! mat]ial molecules w\ld no long] be null4 ,x w\ld be necess>y !n1 if we did n wi% to *ange all me*anics1 to introduce ! e!r1 9 ord] t ?is action :i* matt] appe>$ to exp]i5ce %\ld be c\nt]balanc$ by ! reaction ( matt] on some?+4 ,or aga91 suppose we discov] t optical & electrical ph5om5a are influ5c$ by ! motion ( ! e>?4 ,we %\ld be l$ to conclude t ^! ph5om5a might reveal to u n only ! relative motions ( mat]ial bodies1 b :at w\ld seem to be _! absolute motions4 ,aga91 an e!r w\ld be necess>y1 t ^! s-call$ absolute motions %\ld n be _! displacem5ts ) reg>d to a void space1 b _! displacem5ts ) reg>d to some?+ concrete4 ,%all we ev] arrive at t8 ,i h n ?is hope1 ,i %all soon say :y1 & yet x is n s absurd1 s9ce o!rs h _h x4 ,= in/ance1 if ! !ory ( ,lor5tz1 ( :i* ,i %all speak 9 detail fur!r on 9 ,*apt] ,,xiii41 7 true1 ,newton's pr9ciple w\ld n apply to matt] alone1 & ! diff]5ce w\ld n be v f> f be+ accessible to exp]im5t4 ,on ! o!r h&1 _m rese>*es h be5 made on ! influ5ce ( ! e>?'s motion4 ,! results h always be5 negative4 ,b ^! exp]im5ts 7 und]tak5 because ! outcome 0 n sure 9 advance1 &1 inde$1 accord+ to ! rul+ !ories1 ! comp5sation w\ld be only approximate1 & "o might expect to see precise me?ods give positive results4 ,i believe t su* a hope is illusory2 x 0 none ! less int]e/+ to %[ t a success ( ?is sort w\ld op5 to u1 9 "s sort1 a new _w4 ,& n[ ,i mu/ be p]mitt$ a digression2 ,i mu/ expla91 9 fact1 :y ,i d n believe1 despite ,lor5tz1 t m precise obs]vations c ev] put 9 evid5ce any?+ else ?an ! relative displacem5ts ( mat]ial bodies4 ,exp]im5ts h be5 made :i* %\ld h disclos$ ! t]ms ( ! fir/ ord]2 ! results h be5 negative2 c\ld t be by *ance8 ,no "o has assum$ t2 a g5]al explanation has be5 s\ght1 & ,lor5tz has f\nd x2 he has %[n t ! t]ms ( ! fir/ ord] mu/ de/roy ea* o!r1 b n ^? ( ! second4 ,!n m precise exp]im5ts 7 made2 !y also 7 negative2 nei!r c\ld ?is be ! effect ( *ance2 an explanation 0 necess>y2 x 0 f\nd2 !y always are f\nd2 ( hypo!ses !re is nev] lack4 ,b ?is is n en\gh2 :o does n feel t ?is is /ill to leave to *ance too great a rle8 ,w\ld n t also be a *ance1 ?is s+ul> co9cid5ce :i* br\ght x ab\t t a c]ta9 circum/ance %\ld come j 9 ! nick ( "t to de/roy ! t]ms ( ! fir/ ord]1 & t ano!r circum/ance1 :olly diff]5t1 b j z opportune1 %\ld take ^u itself to de/roy ^? ( ! second ord]8 ,no1 x is necess>y to f9d an explanation ! same = ! "o z = ! o!r1 & !n ev]y?+ leads u to ?9k t ?is explanation w hold good equally well = ! t]ms ( high] ord]1 & t ! mutual de/ruction ( ^! t]ms w be rigor\s & absolute4 ,,pres5t ,,/ate ,,( ,,! ,,sci5ce4--,9 ! hi/ory ( ! developm5t ( physics we di/+ui% two inv]se t5d5cies4 ,on ! "o h&1 new bonds are cont9ually be+ discov]$ betwe5 objects :i* _h seem$ de/9$ to rema9 =ev] unconnect$2 scatt]$ facts cease to be /rang]s to "o ano!r2 !y t5d to arrange !mselves 9 an impos+ syn!sis4 ,sci5ce advances t[>d unity & simplicity4 ,on ! o!r h&1 obs]vation reveals to u e day new ph5om5a2 !y mu/ long await _! place & sometimes1 to make "o = !m1 a corn] ( ! edifice mu/ be demoli%$4 ,9 ! kn[n ph5om5a !mselves1 :]e our crude s5ses %[$ u uni=mity1 we p]ceive details f day to day m v>i$2 :at we believ$ simple becomes complex1 & sci5ce appe>s to advance t[>d v>iety & complexity4 ,( ^! two inv]se t5d5cies1 :i* seem to triumph turn ab\t1 :i* w w98 ,if x be ! fir/1 sci5ce is possible2 b no?+ proves ?is a priori1 & x may well be fe>$ t aft] hav+ made va9 ef=ts to b5d nature 9 spite ( h]self to our ideal ( unity1 subm]g$ by ! ev]-ris+ flood ( our new ri*es1 we mu/ r5\nce classify+ !m1 ab&on our ideal1 & r$uce sci5ce to ! regi/ration ( innum]able recipes4 ,to ?is "q we c n reply4 ,all we c d is to obs]ve ! sci5ce ( to-day & comp>e x ) t ( ye/]day4 ,f ?is exam9ation we may d\btless draw "s enc\ragem5t4 ,half a c5tury ago1 hope ran high4 ,! discov]y ( ! cons]vation ( en]gy & ( its trans=mations _h reveal$ to u ! unity ( =ce4 ,?us x %[$ t ! ph5om5a ( heat c\ld be expla9$ by molecul> motions4 ,:at 0 ! nature ( ^! motions 0 n exactly kn[n1 b no "o d\bt$ t x soon w\ld be4 ,= light1 ! task seem$ completely accompli%$4 ,9 :at conc]ns electricity1 ?+s 7 less advanc$4 ,electricity _h j annex$ magnetism4 ,?is 0 a consid]able /ep t[>d unity1 & a decisive /ep4 ,b h[ %\ld electricity 9 its turn ent] into ! g5]al unity1 h[ %\ld x be r$uc$ to ! univ]sal me*anism8 ,( t no "o _h any idea4 ,yet ! possibility ( ?is r$uction 0 d\bt$ by none1 !re 0 fai?4 ,f9ally1 9 :at conc]ns ! molecul> prop]ties ( mat]ial bodies1 ! r$uction seem$ /ill easi]1 b all ! detail rema9$ hazy4 ,9 a ^w1 ! hopes 7 va/ & animat$1 b vague4 ,to-day1 :at d we see8 ,fir/ ( all1 a prime progress1 imm5se progress4 ,! relations ( electricity & light are n[ kn[n2 ! ?ree realms1 ( light1 ( electricity & ( magnetism1 previ\sly sep>at$1 =m n[ b "o2 & ?is annexation seems f9al4 ,?is conque/1 h[ev]1 has co/ u "s sacrifices4 ,! optical ph5om5a subord9ate !mselves z p>ticul> cases "u ! electrical ph5om5a2 s long z !y rema9$ isolat$1 x 0 easy to expla9 !m by motions t 7 suppos$ to be kn[n 9 all _! details1 t 0 a matt] ( c\rse2 b n[ an explanation1 to be acceptable1 mu/ be easily capable ( ext5sion to ! entire electric doma94 ,n[ t is a matt] n )\t difficulties4 ,! mo/ satisfactory !ory we h is t ( ,lor5tz1 :i*1 z we %all see 9 ! la/ *apt]1 expla9s electric curr5ts by ! motions ( little electrifi$ p>ticles2 x is unque/ionably ! "o :i* be/ expla9s ! kn[n facts1 ! "o :i* illum9ates ! greate/ numb] ( true relations1 ! "o ( :i* mo/ traces w be f\nd 9 ! f9al con/ruction4 ,nev]!less1 x /ill has a s]i\s defect1 :i* ,i h indicat$ above2 x is contr>y to ,newton's law ( ! equality ( action & reaction2 or r1 ?is pr9ciple1 9 ! eyes ( ,lor5tz1 w\ld n be applicable to matt] alone2 = x to be true1 x w\ld be necess>y to take acc\nt ( ! action ( ! e!r on matt] & ( ! reaction ( matt] on ! e!r4 ,n[1 f :at we "k at pres5t1 x seems probable t ?+s d n happ5 9 ?is way4 ,h[ev] t may be1 ?anks to ,lor5tz1 ,fizeau's results on ! optics ( mov+ bodies1 ! laws ( normal & anomal\s disp]sion & ( absorption f9d !mselves l9k$ to "o ano!r & to ! o!r prop]ties ( ! e!r by bonds :i* beyond any d\bt w nev] m be brok54 ,see ! facility ) :i* ! new ,zeeman effect has f\nd its place already & has ev5 aid$ 9 classify+ ,f>aday's magnetic rotation :i* _h defi$ ,maxwell's ef=ts2 ?is facility abundantly proves t ! !ory ( ,lor5tz is n an artificial assemblage de/9$ to fall asund]4 ,x w probably h to be modifi$1 b n de/roy$4 ,b ,lor5tz _h no aim beyond t ( embrac+ 9 "o totality all ! optics & electrodynamics ( mov+ bodies2 he nev] pret5d$ to give a me*anical explanation ( !m4 ,l>mor goes fur!r2 reta9+ ! !ory ( ,lor5tz 9 ess5tials1 he grafts ^u x1 s to speak1 ,mac,cullagh's ideas on ! direction ( ! motions ( ! e!r4 ,accord+ to him1 ! velocity ( ! e!r w\ld h ! same direction & ! same magnitude z ! magnetic =ce4 ,h[ev] +5i\s ?is attempt may be1 ! defect ( ! !ory ( ,lor5tz rema9s & is ev5 aggravat$4 ,) ,lor5tz1 we d n "k :at are ! motions ( ! e!r2 ?anks to ?is ignorance1 we may suppose !m su* t1 comp5sat+ ^? ( matt]1 !y ree/abli% ! equality ( action & reaction4 ,) ,l>mor1 we "k ! motions ( ! e!r1 & we c asc]ta9 t ! comp5sation does n take place4 ,if ,l>mor has fail$1 z x seems to me he has1 does t mean t a me*anical explanation is impossible8 ,f> f x3 ,i h said above t :5 a ph5om5on obeys ! two pr9ciples ( en]gy & ( lea/ action1 x admits ( an inf9ity ( me*anical explanations2 s x is1 !re=e1 ) ! optical & electrical ph5om5a4 ,b ?is is n en\gh3 = a me*anical explanation to be good1 x mu/ be simple2 = *oos+ x among all :i* are possible1 !re %\ld be o!r reasons besides ! necessity ( mak+ a *oice4 ,well1 we h n z yet a !ory satisfy+ ?is condition & consequ5tly good = some?+4 ,mu/ we lam5t ?is8 ,t w\ld be to =get :at is ! goal s\ght2 ?is is n me*anism2 ! true1 ! sole aim is unity4 ,we mu/ !re=e set b\nds to our ambition2 let u n try to =mulate a me*anical explanation2 let u be cont5t ) %[+ t we c\ld always f9d "o if we wi%$ to4 ,9 ?is reg>d we h be5 successful2 ! pr9ciple ( ! cons]vation ( en]gy has receiv$ only confirmations2 a second pr9ciple has come to jo9 x1 t ( lea/ action1 put "u ! =m :i* is suitable = physics4 ,x also has always be5 v]ifi$1 at lea/ 9 s f> z conc]ns rev]sible ph5om5a :i* ?us obey ! equations ( ,lagrange1 t is to say1 ! mo/ g5]al laws ( me*anics4 ,irrev]sible ph5om5a are mu* m rebelli\s4 ,yet ^! also are be+ coord9at$1 & t5d to come into unity2 ! light :i* has illum9at$ !m has come to u f ,c>not's pr9ciple4 ,long did !rmodynamics conf9e itself to ! /udy ( ! dilatation ( bodies & _! *anges ( /ate4 ,= "s "t pa/ x has be5 gr[+ bold] & has consid]ably ext5d$ its doma94 ,we owe to x ! !ory ( ! galvanic batt]y & t ( ! !rmoelectric ph5om5a2 !re is n 9 all physics a corn] t x has n explor$1 & x has attack$ *emi/ry itself4 ,ev]y:]e ! same laws reign2 ev]y:]e1 "u ! div]sity ( appe>ances1 is f\nd aga9 ,c>not's pr9ciple2 ev]y:]e also is f\nd t concept s prodigi\sly ab/ract ( entropy1 :i* is z univ]sal z t ( en]gy & seems l x to cov] a reality4 ,radiant heat seem$ de/9$ to escape x2 b rec5tly we h se5 t submit to ! same laws4 ,9 ?is way fre% analogies are reveal$ to u1 :i* may (t5 be foll[$ into detail2 ohmic resi/ance resembles ! viscosity ( liquids2 hy/]esis w\ld resemble r ! friction ( solids4 ,9 all cases1 friction w\ld appe> to be ! type :i* ! mo/ v>i\s irrev]sible ph5om5a copy1 & ?is k9%ip is real & pr(\nd4 ,( ^! ph5om5a a me*anical explanation1 prop]ly s call$1 has also be5 s\ght4 ,!y h>dly l5t !mselves to x4 ,to f9d x1 x 0 necess>y to suppose t ! irrev]sibility is only app>5t1 t ! elem5t>y ph5om5a are rev]sible & obey ! kn[n laws ( dynamics4 ,b ! elem5ts are extremely num]\s & bl5d m & m1 s t to our crude sight all appe>s to t5d t[>d uni=mity1 t is1 ev]y?+ seems to g =w>d 9 ! same s5se )\t hope ( return4 ,! app>5t irrev]sibility is ?us only an effect ( ! law ( great numb]s4 ,b1 only a be+ ) inf9itely subtile s5ses1 l ,maxwell's imag9>y demon1 c\ld dis5tangle ?is inextricable ske9 & turn back ! c\rse ( ! univ]se4 ,?is conception1 :i* atta*es itself to ! k9etic !ory ( gases1 has co/ great ef=ts & has n1 on ! :ole1 be5 fruitful2 b x may become s4 ,?is is n ! place to exam9e :e!r x does n lead to contradictions & :e!r x is 9 con=mity ) ! true nature ( ?+s4 ,we signalize1 h[ev]1 ,m4 ,g\y's orig9al ideas on ! ,br[nian movem5t4 ,accord+ to ?is sci5ti/1 ?is s+ul> motion %\ld escape ,c>not's pr9ciple4 ,! p>ticles :i* x puts 9 sw+ w\ld be small] ?an ! l9ks ( t s compact$ ske92 !y w\ld !re=e be fitt$ to dis5tangle !m & h5ce to make ! _w g backw>d4 ,we %\ld almo/ see ,maxwell's demon at "w4 ,to summ>ize1 ! previ\sly kn[n ph5om5a are bett] & bett] classifi$1 b new ph5om5a come to claim _! place2 mo/ ( ^!1 l ! ,zeeman effect1 h at once f\nd x4 ,b we h ! ca?ode rays1 ! ,x-rays1 ^? ( uranium & ( radium4 ,h]e9 is a :ole _w :i* no "o suspect$4 ,h[ _m unexpect$ gue/s mu/ be /[$ away8 ,no "o c yet =esee ! place !y w occupy4 ,b ,i d n believe !y w de/roy ! g5]al unity2 ,i ?9k !y w r complete x4 ,on ! "o h&1 9 fact1 ! new radiations seem connect$ ) ! ph5om5a ( lum9esc5ce2 n only d !y excite fluoresc5ce1 b !y sometimes take bir? 9 ! same conditions z x4 ,nor are !y )\t k9%ip ) ! causes :i* produce ! electric sp>k "u ! action ( ! ultra-violet light4 ,f9ally1 & above all1 x is believ$ t 9 all ^! ph5om5a are f\nd true ions1 animat$1 x is true1 by velocities incomp>ably great] ?an 9 ! electrolytes4 ,t is all v vague1 b x w all become m precise4 ,phosphoresc5ce1 ! action ( light on ! sp>k1 ^! 7 regions r isolat$ & consequ5tly some:at neglect$ by inve/igators4 ,"o may n[ hope t a new pa? w be con/ruct$ :i* w facilitate _! communications ) ! re/ ( sci5ce4 ,n only d we discov] new ph5om5a1 b 9 ^? we ?\ght we knew1 un=ese5 aspects reveal !mselves4 ,9 ! free e!r1 ! laws reta9 _! maje/ic simplicity2 b matt]1 prop]ly s call$1 seems m & m complex2 all t is said ( x is nev] m ?an approximate1 & at ea* in/ant our =mulas require new t]ms4 ,nev]!less ! frames are n brok52 ! relations t we h recogniz$ betwe5 objects we ?\ght simple /ill subsi/ betwe5 ^! same objects :5 we "k _! complexity1 & x is t alone :i* is ( importance4 ,our equations become1 x is true1 m & m complicat$1 9 ord] to embrace m closely ! complexity ( nature2 b no?+ is *ang$ 9 ! relations :i* p]mit ! d$uc+ ( ^! equations "o f ano!r4 ,9 a ^w1 ! =m ( ^! equations has p]si/$4 ,take1 = example1 ! laws ( reflection3 ,fresnel _h e/abli%$ !m by a simple & s$uctive !ory :i* exp]im5t seem$ to confirm4 ,s9ce !n m precise rese>*es h prov$ t ?is v]ification 0 only approximate2 !y h %[n ev]y:]e traces ( elliptic pol>ization4 ,b1 ?anks to ! help t ! fir/ approximation gave u1 we f\nd =?) ! cause ( ^! anomalies1 :i* is ! pres5ce ( a transition lay]2 & ,fresnel's !ory has subsi/$ 9 its ess5tials4 ,b !re is a reflection we c n help mak+3 ,all ^! relations w\ld h rema9$ unp]ceiv$ if "o _h at fir/ suspect$ ! complexity ( ! objects !y connect4 ,x has long be5 said3 ,if ,ty*o _h _h in/rum5ts t5 times m precise nei!r ,kepl]1 nor ,newton1 nor a/ronomy w\ld ev] h be54 ,x is a mis=tune = a sci5ce to be born too late1 :5 ! means ( obs]vation h become too p]fect4 ,?is is to-day ! case ) physical *emi/ry2 its f\nd]s are emb>rass$ 9 _! g5]al grasp by ?ird & f\r? decimals2 happily !y are m5 ( a robu/ fai?4 ,! bett] "o kn[s ! prop]ties ( matt] ! m "o sees cont9uity reign4 ,s9ce ! labors ( ,&rews & ( van d] ,waals1 we get an idea ( h[ ! passage is made f ! liquid to ! gase\s /ate & t ?is passage is n abrupt4 ,simil>ly1 !re is no gap betwe5 ! liquid & solid /ates1 & 9 ! proce$+s ( a rec5t congress is to be se51 alongside ( a "w on ! rigidity ( liquids1 a memoir on ! fl[ ( solids4 ,by ?is t5d5cy no d\bt simplicity loses2 "s ph5om5on 0 =m]ly repres5t$ by sev]al /raight l9es1 n[ ^! /raights mu/ be jo9$ by curves m or less complicat$4 ,9 comp5sation unity ga9s notably4 ,^? cut-(f categories quiet$ ! m9d1 b !y did n satisfy x4 ,f9ally ! me?ods ( physics h invad$ a new doma91 t ( *emi/ry2 physical *emi/ry is born4 ,x is /ill v "y1 b we already see t x w enable u to connect su* ph5om5a z electrolysis1 osmosis & ! motions ( ions4 ,f ?is rapid exposition1 :at %all we conclude8 ,ev]y?+ consid]$1 we h approa*$ unity2 we h n be5 z quick z 0 hop$ fifty ye>s ago1 we h n always tak5 ! pr$ict$ way2 b1 f9ally1 we h ga9$ ev] s mu* gr\nd4 ,,*apt] ,,xi ,,! ,,calculus ,,( ,,probabilities ,d\btless x w be a/oni%+ to f9d "h ?\ghts ab\t ! calculus ( probabilities4 ,:at has x to d ) ! me?od ( ! physical sci5ces8 ,& yet ! que/ions ,i %all raise )\t solv+ pres5t !mselves naturally to ! philosoph] :o is ?9k+ ab\t physics4 ,s f> is ?is ! case t 9 ! two prec$+ *apt]s ,i h (t5 be5 l$ to use ! words 'probability' & '*ance4' ',pr$ict$ facts1' z ,i h said above1 'c only be probable4' ,h[ev] solidly f\nd$ a pr$iction may seem to u to be1 we are nev] absolutely sure t exp]im5t w n prove x false4 ,b ! probability is (t5 s great t practically we may be satisfi$ ) x4 ,& a little fur!r on ,i h add$3 ,see :at a rle ! belief 9 simplicity plays 9 our g5]alizations4 ,we h v]ifi$ a simple law 9 a great numb] ( p>ticul> cases2 we refuse to admit t ?is co9cid5ce1 s (t5 repeat$1 c be a m]e effect ( *ance4444 ,?us 9 a multitude ( circum/ances ! physici/ is 9 ! same position z ! gambl] :o reckons up 8 *ances4 ,z (t5 z he reasons by induction1 he requires m or less consci\sly ! calculus ( probabilities1 & ?is is :y ,i am oblig$ to introduce a p>5!sis1 & int]rupt our /udy ( me?od 9 ! physical sci5ces 9 ord] to exam9e a little m closely ! value ( ?is calculus1 & :at confid5ce x m]its4 ,! v "n calculus ( probabilities is a p>adox4 ,probability oppos$ to c]ta9ty is :at we d n "k1 & h[ c we calculate :at we d n "k8 ,yet _m em95t savants h occupi$ !mselves ) ?is calculus1 & x c n be d5i$ t sci5ce has drawn !refrom no small advantage4 ,h[ c we expla9 ?is app>5t contradiction8 ,has probability be5 def9$8 ,c x ev5 be def9$8 ,& if x c n1 h[ d>e we reason ab\t x8 ,! def9ition1 x w be said1 is v simple3 ! probability ( an ev5t is ! ratio ( ! numb] ( cases favorable to ?is ev5t to ! total numb] ( possible cases4 ,a simple example w %[ h[ incomplete ?is def9ition is4 ,i ?r[ two dice4 ,:at is ! probability t "o ( ! two at lea/ turns up a six8 ,ea* die c turn up 9 six diff]5t ways2 ! numb] ( possible cases is #f #f #cf2 ! numb] ( favorable cases is #aa2 ! probability is #aa_/cf4 ,t is ! correct solution4 ,b c\ld ,i n j z well say3 ,! po9ts :i* turn up on ! two dice c =m #f #g_/b #ba diff]5t comb9ations8 ,among ^! comb9ations #f are favorable2 ! probability is #f_/ba4 ,n[ :y is ! fir/ me?od ( enum]at+ ! possible cases m legitimate ?an ! second8 ,9 any case x is n our def9ition t tells u4 ,we are !re=e oblig$ to complete ?is def9ition by say+3 '444 to ! total numb] ( possible cases provid$ ^! cases are equally probable4' ,s1 !re=e1 we are r$uc$ to def9+ ! probable by ! probable4 ,h[ c we "k t two possible cases are equally probable8 ,w x be by a conv5tion8 ,if we place at ! beg9n+ ( ea* problem an explicit conv5tion1 well & good4 ,we %all !n h no?+ to d b apply ! rules ( ari?metic & ( algebra1 & we %all complete our calculation )\t our result leav+ room = d\bt4 ,b if we wi% to make ! slighte/ application ( ?is result1 we mu/ prove our conv5tion 0 legitimate1 & we %all f9d ourselves 9 ! pres5ce ( ! v difficulty we ?\ght to escape4 ,w x be said t good s5se suffices to %[ u :at conv5tion %\ld be adopt$8 ,alas6 ,m4 ,b]tr& has amus$ himself by discuss+ ! foll[+ simple problem3 ,:at is ! probability t a *ord ( a circle may be great] ?an ! side ( ! inscrib$ equilat]al triangle8 ,! illu/ri\s geomet] successively adopt$ two conv5tions :i* good s5se seem$ equally to dictate & ) "o he f\nd #a_/b1 ) ! o!r #a_/c4 ,! conclusion :i* seems to foll[ f all ?is is t ! calculus ( probabilities is a useless sci5ce1 & t ! obscure in/9ct :i* we may call good s5se1 & to :i* we are wont to appeal to legitimatize our conv5tions1 mu/ be di/ru/$4 ,b nei!r c we subscribe to ?is conclusion2 we c n d )\t ?is obscure in/9ct4 ,)\t x sci5ce w\ld be impossible1 )\t x we c\ld nei!r discov] a law nor apply x4 ,h we ! "r1 = in/ance1 to enunciate ,newton's law8 ,)\t d\bt1 num]\s obs]vations are 9 accord ) x2 b is n ?is a simple effect ( *ance8 ,besides h[ d we "k :e!r ?is law1 true = s _m c5turies1 w /ill be true next ye>8 ,to ?is objection1 y w f9d no?+ to reply1 except3 ',t is v improbable4' ,b grant ! law4 ,?anks to x1 ,i believe myself able to calculate ! position ( ,jupit] a ye> f n[4 ,h ,i ! "r to believe ?is8 ,:o c tell if a gigantic mass ( enorm\s velocity w n betwe5 n[ & t "t pass ne> ! sol> sy/em1 & produce un=ese5 p]turbations8 ,"h aga9 ! only answ] is3 ',x is v improbable4' ,f ?is po9t ( view1 all ! sci5ces w\ld be only unconsci\s applications ( ! calculus ( probabilities4 ,to condemn ?is calculus w\ld be to condemn ! :ole ( sci5ce4 ,i %all dwell lightly on ! sci5tific problems 9 :i* ! int]v5tion ( ! calculus ( probabilities is m evid5t4 ,9 ! =efront ( ^! is ! problem ( int]polation1 9 :i*1 kn[+ a c]ta9 numb] ( values ( a function1 we seek to div9e ! int]m$iate values4 ,i %all likewise m5tion3 ! celebrat$ !ory ( errors ( obs]vation1 to :i* ,i %all return lat]2 ! k9etic !ory ( gases1 a well-kn[n hypo!sis1 :]e9 ea* gase\s molecule is suppos$ to describe an extremely complicat$ trajectory1 b 9 :i*1 ?r\gh ! effect ( great numb]s1 ! mean ph5om5a1 alone obs]vable1 obey ! simple laws ( ,m>iotte & ,gay-,lussac4 ,all ^! !ories are bas$ on ! laws ( great numb]s1 & ! calculus ( probabilities w\ld evid5tly involve !m 9 its ru94 ,x is true t !y h only a p>ticul> int]e/ & t1 save z f> z int]polation is conc]n$1 ^! are sacrifices to :i* we might readily be resign$4 ,b1 z ,i h said above1 x w\ld n be only ^! p>tial sacrifices t w\ld be 9 "q2 x w\ld be ! legitimacy ( ! :ole ( sci5ce t w\ld be *all5g$4 ,i q see t x might be said3 ,we are ignorant1 & yet we mu/ act4 ,= action1 we h n "t to devote ourselves to an inquiry suffici5t to dispel our ignorance4 ,besides1 su* an inquiry w\ld dem& an inf9ite "t4 ,we mu/ !re=e decide )\t kn[+2 we are oblig$ to d s1 hit or miss1 & we mu/ foll[ rules )\t q believ+ !m4 ,:at ,i "k is n t su* & su* a ?+ is true1 b t ! be/ c\rse = me is to act z if x 7 true4 ,! calculus ( probabilities1 & consequ5tly sci5ce itself1 w\ld !nce=? h m]ely a practical value4 ,un=tunately ! difficulty does n ?us disappe>4 ,a gambl] wants to try a c\p2 he asks my advice4 ,if ,i give x to him1 ,i %all use ! calculus ( probabilities1 b ,i %all n gu>antee success4 ,?is is :at ,i %all call subjective probability4 ,9 ?is case1 we might be cont5t ) ! explanation ( :i* ,i h j giv5 a sket*4 ,b suppose t an obs]v] is pres5t at ! game1 t he notes all its c\ps1 & t ! game goes on a long "t4 ,:5 he makes a summ>y ( 8 book1 he w f9d t ev5ts h tak5 place 9 con=mity ) ! laws ( ! calculus ( probabilities4 ,?is is :at ,i %all call objective probability1 & x is ?is ph5om5on :i* has to be expla9$4 ,!re are num]\s insurance companies :i* apply ! rules ( ! calculus ( probabilities1 & !y di/ribute to _! %>ehold]s divid5ds ^: objective reality c n be conte/$4 ,to invoke our ignorance & ! necessity to act does n suffice to expla9 !m4 ,?us absolute skepticism is n admissible4 ,we may di/ru/1 b we c n condemn en bloc4 ,discussion is necess>y4 ,i4 ,,classification ,,( ,,! ,,problems ,,( ,,probability4--,9 ord] to classify ! problems :i* pres5t !mselves propos ( probabilities1 we may look at !m f _m diff]5t po9ts ( view1 &1 fir/1 f ! po9t ( view ( g5]ality4 ,i h said above t probability is ! ratio ( ! numb] ( favorable cases to ! numb] ( possible cases4 ,:at = want ( a bett] t]m ,i call ! g5]ality w increase ) ! numb] ( possible cases4 ,?is numb] may be f9ite1 z1 = in/ance1 if we take a ?r[ ( ! dice 9 :i* ! numb] ( possible cases is #cf4 ,t is ! fir/ degree ( g5]ality4 ,b if we ask1 = example1 :at is ! probability t a po9t )9 a circle is )9 ! inscrib$ squ>e1 !re are z _m possible cases z !re are po9ts 9 ! circle1 t is to say1 an inf9ity4 ,?is is ! second degree ( g5]ality4 ,g5]ality c be pu%$ fur!r /ill4 ,we may ask ! probability t a function w satisfy a giv5 condition4 ,!re are !n z _m possible cases z "o c imag9e diff]5t functions4 ,?is is ! ?ird degree ( g5]ality1 to :i* we rise1 = in/ance1 :5 we seek to f9d ! mo/ probable law 9 con=mity ) a f9ite numb] ( obs]vations4 ,we may place ourselves at a po9t ( view :olly diff]5t4 ,if we 7 n ignorant1 !re w\ld be no probability1 !re w\ld be room = no?+ b c]ta9ty4 ,b our ignorance c n be absolute1 = !n !re w\ld no long] be any probability at all1 s9ce a little light is necess>y to atta9 ev5 ?is unc]ta9 sci5ce4 ,?us ! problems ( probability may be class$ accord+ to ! great] or less dep? ( ?is ignorance4 ,9 ma!matics ev5 we may set ourselves problems ( probability4 ,:at is ! probability t ! fif? decimal ( a log>i?m tak5 at r&om f a table is a '#i'8 ,!re is no hesitation 9 answ]+ t ?is probability is #a_/aj2 "h we possess all ! data ( ! problem4 ,we c calculate our log>i?m )\t rec\rse to ! table1 b we d n wi% to give ourselves ! tr\ble4 ,?is is ! fir/ degree ( ignorance4 ,9 ! physical sci5ces our ignorance becomes great]4 ,! /ate ( a sy/em at a giv5 in/ant dep5ds on two ?+s3 ,its initial /ate1 & ! law accord+ to :i* t /ate v>ies4 ,if we "k bo? ?is law & ?is initial /ate1 we %all h !n only a ma!matical problem to solve1 & we fall back ^u ! fir/ degree ( ignorance4 ,b x (t5 happ5s t we "k ! law1 & d n "k ! initial /ate4 ,x may be ask$1 = in/ance1 :at is ! pres5t di/ribution ( ! m9or planets8 ,we "k t f all "t !y h obey$ ! laws ( ,kepl]1 b we d n "k :at 0 _! initial di/ribution4 ,9 ! k9etic !ory ( gases1 we assume t ! gase\s molecules foll[ rectil9e> trajectories1 & obey ! laws ( impact ( ela/ic bodies4 ,b1 z we "k no?+ ( _! initial velocities1 we "k no?+ ( _! pres5t velocities4 ,! calculus ( probabilities only enables u to pr$ict ! mean ph5om5a :i* w result f ! comb9ation ( ^! velocities4 ,?is is ! second degree ( ignorance4 ,f9ally x is possible t n only ! initial conditions b ! laws !mselves are unkn[n4 ,we !n rea* ! ?ird degree ( ignorance & 9 g5]al we c no long] affirm any?+ at all z to ! probability ( a ph5om5on4 ,x (t5 happ5s t in/ead ( try+ to guess an ev5t1 by means ( a m or less imp]fect k ( ! law1 ! ev5ts may be kn[n & we want to f9d ! law2 or t in/ead ( d$uc+ effects f causes1 we wi% to d$uce ! causes f ! effects4 ,^! are ! problems call$ probability ( causes1 ! mo/ int]e/+ f ! po9t ( view ( _! sci5tific applications4 ,i play c>t ) a g5tleman ,i "k to be p]fectly hone/4 ,he is ab\t to deal4 ,:at is ! probability ( 8 turn+ up ! k+8 ,x is #a_/h4 ,?is is a problem ( ! probability ( effects4 ,i play ) a g5tleman :om ,i d n "k4 ,he has dealt t5 times1 & he has turn$ up ! k+ six times4 ,:at is ! probability t he is a %>p]8 ,?is is a problem 9 ! probability ( causes4 ,x may be said t ?is is ! ess5tial problem ( ! exp]im5tal me?od4 ,i h obs]v$ n values ( x & ! correspond+ values ( y4 ,i h f\nd t ! ratio ( ! latt] to ! =m] is practically con/ant4 ,!re is ! ev5t1 :at is ! cause8 ,is x probable t !re is a g5]al law accord+ to :i* y w\ld be proportional to x1 & t ! small div]g5cies are due to errors ( obs]vation8 ,?is is a type ( "q t "o is ev] ask+1 & :i* we unconsci\sly solve :5ev] we are engag$ 9 sci5tific "w4 ,i am n[ go+ to pass 9 review ^! diff]5t categories ( problems1 discuss+ 9 succession :at ,i h call$ above subjective & objective probability4 ,,ii4 ,,probability ,,9 ,,ma!matics4--,! impossibility ( squ>+ ! circle has be5 prov$ s9ce #ahhb2 b ev5 be=e t date all geomet]s consid]$ t impossibility z s 'probable1' t ! ,academy ( ,sci5ces reject$ )\t exam9ation ! alas6 too num]\s memoirs on ?is subject1 t "s unhappy madm5 s5t 9 e ye>4 ,0 ! ,academy wrong8 ,evid5tly n1 & x knew well t 9 act+ ?us x did n run ! lea/ risk ( /ifl+ a discov]y ( mom5t4 ,! ,academy c\ld n h prov$ t x 0 "r2 b x knew q well t its in/9ct 0 n mi/ak54 ,if y _h ask$ ! ,academicians1 !y w\ld h answ]$3 ,we h comp>$ ! probability t an unkn[n savant %\ld h f\nd out :at has be5 va9ly s\ght = s long1 ) ! probability t !re is "o madman ! m on ! e>?2 ! second appe>s to u ! great]4 ,^! are v good reasons1 b !re is no?+ ma!matical ab\t !m2 !y are purely psy*ological4 ,& if y _h press$ !m fur!r !y w\ld h add$3 ,:y d y suppose a p>ticul> value ( a transc5d5tal function to be an algebraic numb]2 & if pi 7 a root ( an algebraic equation1 :y d y suppose ?is root to be a p]iod ( ! function s9 #bx1 & n ! same ab\t ! o!r roots ( ?is same equation8 ,to sum up1 !y w\ld h invok$ ! pr9ciple ( suffici5t reason 9 its vague/ =m4 ,b :at c\ld !y d$uce f x8 ,at mo/ a rule ( conduct = ! employm5t ( _! "t1 m usefully sp5t at _! ord9>y "w ?an 9 read+ a lucubration t inspir$ 9 !m a legitimate di/ru/4 ,b :at ,i call above objective probability has no?+ 9 common ) ?is fir/ problem4 ,x is o!rwise ) ! second problem4 ,consid] ! fir/ #aj1jjj log>i?ms t we f9d 9 a table4 ,among ^! #aj1jjj log>i?ms ,i take "o at r&om4 ,:at is ! probability t its ?ird decimal is an ev5 numb]8 ,y w n hesitate to answ] #a_/b2 & 9 fact if y pick out 9 a table ! ?ird decimals ( ^! #aj1jjj numb]s1 y w f9d ne>ly z _m ev5 digits z odd4 ,or if y pref]1 let u write #aj1jjj numb]s correspond+ to our #aj1jjj log>i?ms1 ea* ( ^! numb]s be+ #a if ! ?ird decimal ( ! correspond+ log>i?m is ev51 & -#a if odd4 ,!n take ! mean ( ^! #aj1jjj numb]s4 ,i d n hesitate to say t ! mean ( ^! #aj1jjj numb]s is probably #j1 & if ,i 7 actually to calculate x ,i %\ld v]ify t x is extremely small4 ,b ev5 ?is v]ification is ne$less4 ,i might h rigor\sly prov$ t ?is mean is less ?an #j4jjc4 ,to prove ?is result1 ,i %\ld h _h to make a r long calculation = :i* !re is no room "h1 & = :i* ,i conf9e myself to cit+ an article ,i publi%$ 9 ! ,revue gnrale des ,sci5ces1 ,april #ae1 #ahii4 ,! only po9t to :i* ,i wi% to call att5tion is ! foll[+3 9 ?is calculation1 ,i %\ld h ne$$ only to re/ my case on two facts1 to wit1 t ! fir/ & second d]ivatives ( ! log>i?m rema91 9 ! int]val consid]$1 betwe5 c]ta9 limits4 ,h5ce ?is important consequ5ce t ! prop]ty is true n only ( ! log>i?m1 b ( any cont9u\s function :atev]1 s9ce ! d]ivatives ( e cont9u\s function are limit$4 ,if ,i 0 c]ta9 be=eh& ( ! result1 x is fir/1 because ,i _h (t5 obs]v$ analog\s facts = o!r cont9u\s functions2 & next1 because ,i made 9 my m9d1 9 a m or less unconsci\s & imp]fect mann]1 ! reason+ :i* l$ me to ! prec$+ inequalities1 j z a skill$ calculator be=e f9i%+ 8 multiplication takes into acc\nt :at x %\ld come to approximately4 ,& besides1 s9ce :at ,i call my intuition 0 only an incomplete summ>y ( a piece ( true reason+1 x is cle> :y obs]vation has confirm$ my pr$ictions1 & :y ! objective probability has be5 9 agreem5t ) ! subjective probability4 ,z a ?ird example ,i %all *oose ! foll[+ problem3 ,a numb] u is tak5 at r&om1 & n is a giv5 v l>ge integ]4 ,:at is ! probable value ( s9 nu8 ,?is problem has no mean+ by itself4 ,to give x "o a conv5tion is ne$$4 ,we %all agree t ! probability = ! numb] u to lie betwe5 a & a is equal to phi7a7da2 t x is !re=e proportional to ! inf9itely small int]val da1 & equal to ?is multipli$ by a function phi7a7 dep5d+ only on a4 ,z = ?is function1 ,i *oose x arbitr>ily1 b ,i mu/ assume x to be cont9u\s4 ,! value ( s9 nu rema9+ ! same :5 u increases by #bpi1 ,i may )\t loss ( g5]ality assume t u lies betwe5 #j & #bpi1 & ,i %all ?us be l$ to suppose t phi7a7 is a p]iodic function ^: p]iod is #bpi4 ,! probable value s\ght is readily express$ by a simple integral1 & x is easy to %[ t ?is integral is less ?an #bpi,mk_/nk1 ,mk be+ ! maximum value ( ! k? d]ivative ( phi7u74 ,we see !n t if ! k? d]ivative is f9ite1 our probable value w t5d t[>d #j :5 n increases indef9itely1 & t m rapidly ?an #a_/nk - #a4 ,! probable value ( s9 nu :5 n is v l>ge is !re=e naught4 ,to def9e ?is value ,i requir$ a conv5tion2 b ! result rema9s ! same :atev] t conv5tion may be4 ,i h impos$ ^u myself only slight re/rictions 9 assum+ t ! function phi7a7 is cont9u\s & p]iodic1 & ^! hypo!ses are s natural t we may ask ourselves h[ !y c be escap$4 ,exam9ation ( ! ?ree prec$+ examples1 s diff]5t 9 all respects1 has already giv5 u a glimpse1 on ! "o h&1 ( ! rle ( :at philosoph]s call ! pr9ciple ( suffici5t reason1 &1 on ! o!r h&1 ( ! importance ( ! fact t c]ta9 prop]ties are common to all cont9u\s functions4 ,! /udy ( probability 9 ! physical sci5ces w lead u to ! same result4 ,,iii4 ,,probability ,,9 ,,! ,,physical ,,sci5ces4--,we come n[ to ! problems connect$ ) :at ,i h call$ ! second degree ( ignorance1 ^?1 namely1 9 :i* we "k ! law1 b d n "k ! initial /ate ( ! sy/em4 ,i c\ld multiply examples1 b w take only "o4 ,:at is ! probable pres5t di/ribution ( ! m9or planets on ! zodiac8 ,we "k !y obey ! laws ( ,kepl]4 ,we may ev51 )\t at all *ang+ ! nature ( ! problem1 suppose t _! orbits are all circul>1 & situat$ 9 ! same plane1 & t we "k ?is plane4 ,on ! o!r h&1 we are 9 absolute ignorance z to :at 0 _! initial di/ribution4 ,h[ev]1 we d n hesitate to affirm t _! di/ribution is n[ ne>ly uni=m4 ,:y8 ,let b be ! longitude ( a m9or planet 9 ! initial epo*1 t is to say1 ! epo* z]o4 ,let a be its mean motion4 ,its longitude at ! pres5t epo*1 t is to say at ! epo* t1 w be at b4 ,to say t ! pres5t di/ribution is uni=m is to say t ! mean value ( ! s9es & cos9es ( multiples ( at b is z]o4 ,:y d we ass]t ?is8 ,let u repres5t ea* m9or planet by a po9t 9 a plane1 to wit1 by a po9t ^: coord9ates are precisely a & b4 ,all ^! repres5tative po9ts w be conta9$ 9 a c]ta9 region ( ! plane1 b z !y are v num]\s ?is region w appe> dott$ ) po9ts4 ,we "k no?+ else ab\t ! di/ribution ( ^! po9ts4 ,:at d we d :5 we wi% to apply ! calculus ( probabilities to su* a "q8 ,:at is ! probability t "o or m repres5tative po9ts may be f\nd 9 a c]ta9 portion ( ! plane8 ,9 our ignorance1 we are r$uc$ to mak+ an arbitr>y hypo!sis4 ,to expla9 ! nature ( ?is hypo!sis1 all[ me to use1 9 lieu ( a ma!matical =mula1 a crude b concrete image4 ,let u suppose t ov] ! surface ( our plane has be5 spread an imag9>y sub/ance1 ^: d5sity is v>iable1 b v>ies cont9u\sly4 ,we %all !n agree to say t ! probable numb] ( repres5tative po9ts to be f\nd on a portion ( ! plane is proportional to ! quantity ( fictiti\s matt] f\nd !re4 ,if we h !n two regions ( ! plane ( ! same ext5t1 ! probabilities t a repres5tative po9t ( "o ( our m9or planets is f\nd 9 "o or ! o!r ( ^! regions w be to "o ano!r z ! mean d5sities ( ! fictiti\s matt] 9 ! "o & ! o!r region4 ,"h !n are two di/ributions1 "o real1 9 :i* ! repres5tative po9ts are v num]\s1 v close toge!r1 b discrete l ! molecules ( matt] 9 ! atomic hypo!sis2 ! o!r remote f reality1 9 :i* our repres5tative po9ts are replac$ by cont9u\s fictiti\s matt]4 ,we "k t ! latt] c n be real1 b our ignorance =ces u to adopt x4 ,if aga9 we _h "s idea ( ! real di/ribution ( ! repres5tative po9ts1 we c\ld arrange x s t 9 a region ( "s ext5t ! d5sity ( ?is imag9>y cont9u\s matt] w\ld be ne>ly proportional to ! numb] ( ! repres5tative po9ts1 or1 if y wi%1 to ! numb] ( atoms :i* are conta9$ 9 t region4 ,ev5 t is impossible1 & our ignorance is s great t we are =c$ to *oose arbitr>ily ! function :i* def9es ! d5sity ( our imag9>y matt]4 ,only we %all be =c$ to a hypo!sis f :i* we c h>dly get away1 we %all suppose t ?is function is cont9u\s4 ,t is suffici5t1 z we %all see1 to enable u to rea* a conclusion4 ,:at is at ! in/ant t ! probable di/ribution ( ! m9or planets8 ,or r :at is ! probable value ( ! s9e ( ! longitude at ! in/ant t1 t is to say ( s9 7at b78 ,we made at ! outset an arbitr>y conv5tion1 b if we adopt x1 ?is probable value is entirely def9$4 ,divide ! plane into elem5ts ( surface4 ,consid] ! value ( s9 7at b7 at ! c5t] ( ea* ( ^! elem5ts2 multiply ?is value by ! surface ( ! elem5t1 & by ! correspond+ d5sity ( ! imag9>y matt]4 ,take !n ! sum = all ! elem5ts ( ! plane4 ,?is sum1 by def9ition1 w be ! probable mean value we seek1 :i* w ?us be express$ by a d\ble integral4 ,x may be ?\ght at fir/ t ?is mean value dep5ds on ! *oice ( ! function :i* def9es ! d5sity ( ! imag9>y matt]1 & t1 z ?is function phi is arbitr>y1 we c1 accord+ to ! arbitr>y *oice :i* we make1 obta9 any mean value4 ,?is is n s4 ,a simple calculation %[s t our d\ble integral decreases v rapidly :5 t increases4 ,?us ,i c\ld n q tell :at hypo!sis to make z to ! probability ( ?is or t initial di/ribution2 b :atev] ! hypo!sis made1 ! result w be ! same1 & ?is gets me out ( my difficulty4 ,:atev] be ! function phi1 ! mean value t5ds t[>d z]o z t increases1 & z ! m9or planets h c]ta9ly accompli%$ a v great numb] ( revolutions1 ,i may ass]t t ?is mean value is v small4 ,i may *oose phi z ,i wi%1 save always "o re/riction3 ?is function mu/ be cont9u\s2 &1 9 fact1 f ! po9t ( view ( subjective probability1 ! *oice ( a discont9u\s function w\ld h be5 unreasonable4 ,= in/ance1 :at reason c\ld ,i h = suppos+ t ! initial longitude might be exactly #j1 b t x c\ld n lie betwe5 #j & #a8 ,b ! difficulty reappe>s if we take ! po9t ( view ( objective probability1 if we pass f our imag9>y di/ribution 9 :i* ! fictiti\s matt] 0 suppos$ cont9u\s to ! real di/ribution 9 :i* our repres5tative po9ts =m1 z x 71 discrete atoms4 ,! mean value ( s9 7at b7 w be repres5t$ q simply by 7#a_/n7,sigma s9 7at b71 n be+ ! numb] ( m9or planets4 ,9 lieu ( a d\ble integral ref]r+ to a cont9u\s function1 we %all h a sum ( discrete t]ms4 ,& yet no "o w s]i\sly d\bt t ?is mean value is practically v small4 ,our repres5tative po9ts be+ v close toge!r1 our discrete sum w 9 g5]al diff] v little f an integral4 ,an integral is ! limit t[>d :i* a sum ( t]ms t5ds :5 ! numb] ( ^! t]ms is indef9itely increas$4 ,if ! t]ms are v num]\s1 ! sum w diff] v little f its limit1 t is to say f ! integral1 & :at ,i said ( ?is latt] w /ill be true ( ! sum itself4 ,nev]!less1 !re are exceptions4 ,if1 = in/ance1 = all ! m9or planets1 b pi_/#b - at1 ! longitude = all ! planets at ! "t t w\ld be pi_/#b1 & ! mean value w\ld evid5tly be equal to unity4 ,= ?is to be ! case1 x w\ld be necess>y t at ! epo* #j1 ! m9or planets mu/ h all be5 ly+ on a spiral ( peculi> =m1 ) its spires v close toge!r4 ,e "o w admit t su* an initial di/ribution is extremely improbable 7&1 ev5 suppos+ x realiz$1 ! di/ribution w\ld n be uni=m at ! pres5t "t1 = example1 on ,janu>y #a1 #aiac1 b x w\ld become s a few ye>s lat]74 ,:y !n d we ?9k ?is initial di/ribution improbable8 ,?is mu/ be expla9$1 because if we _h no reason = reject+ z improbable ?is absurd hypo!sis ev]y?+ w\ld break d[n1 & we c\ld no long] make any affirmation ab\t ! probability ( ?is or t pres5t di/ribution4 ,once m we %all invoke ! pr9ciple ( suffici5t reason to :i* we mu/ always recur4 ,we might admit t at ! beg9n+ ! planets 7 di/ribut$ almo/ 9 a /raight l9e4 ,we might admit t !y 7 irregul>ly di/ribut$4 ,b x seems to u t !re is no suffici5t reason = ! unkn[n cause t gave !m bir? to h act$ along a curve s regul> & yet s complicat$1 :i* w\ld appe> to h be5 expressly *os5 s t ! pres5t di/ribution w\ld n be uni=m4 ,,iv4 ,,r\ge ,,et ,,noir4--,! que/ions rais$ by games ( *ance1 su* z r\lette1 are1 fundam5tally1 entirely analog\s to ^? we h j treat$4 ,= example1 a :eel is p>tition$ into a great numb] ( equal subdivisions1 alt]nately r$ & black4 ,a ne$le is :irl$ ) =ce1 & aft] hav+ made a great numb] ( revolutions1 x /ops be=e "o ( ^! subdivisions4 ,! probability t ?is division is r$ is evid5tly #a_/b4 ,! ne$le describes an angle !ta1 includ+ sev]al complete revolutions4 ,i d n "k :at is ! probability t ! ne$le may be :irl$ ) a =ce su* t ?is angle %\ld lie betwe5 !ta & !tad!ta2 b ,i c make a conv5tion4 ,i c suppose t ?is probability is phi7!ta7d!ta4 ,z = ! function phi7!ta71 ,i c *oose x 9 an entirely arbitr>y mann]4 ,!re is no?+ t c guide me 9 my *oice1 b ,i am naturally l$ to suppose ?is function cont9u\s4 ,let epsilon be ! l5g? 7measur$ on ! circumf]5ce ( radius #a7 ( ea* r$ & black subdivision4 ,we h to calculate ! integral ( phi7!ta7d!ta1 ext5d+ x1 on ! "o h&1 to all ! r$ divisions &1 on ! o!r h&1 to all ! black divisions1 & to comp>e ! results4 ,consid] an int]val #b;epsilon1 compris+ a r$ division & a black division :i* foll[s x4 ,let ,m & m be ! greate/ & lea/ values ( ! function phi7!ta7 9 ?is int]val4 ,! integral ext5d$ to ! r$ divisions w be small] ?an ,sigma,mepsilon2 ! integral ext5d$ to ! black divisions w be great] ?an ,sigmamepsilon2 ! diff]5ce w !re=e be less ?an ,sigma7,m - m7epsilon4 ,b1 if ! function !ta is suppos$ cont9u\s2 if1 besides1 ! int]val epsilon is v small ) respect to ! total angle describ$ by ! ne$le1 ! diff]5ce ,m - m w be v small4 ,! diff]5ce ( ! two integrals w !re=e be v small1 & ! probability w be v ne>ly #a_/b4 ,we see t )\t kn[+ any?+ ( ! function !ta1 ,i mu/ act z if ! probability 7 #a_/b4 ,we und]/&1 on ! o!r h&1 :y1 if1 plac+ myself at ! objective po9t ( view1 ,i obs]ve a c]ta9 numb] ( c\ps1 obs]vation w give me ab\t z _m black c\ps z r$4 ,all play]s "k ?is objective law2 b x leads !m into a rem>kable error1 :i* has be5 (t5 expos$1 b into :i* !y always fall aga94 ,:5 ! r$ has won1 = in/ance1 six times runn+1 !y bet on ! black1 ?9k+ !y are play+ a safe game2 because1 say !y1 x is v r>e t r$ w9s sev5 times runn+4 ,9 reality _! probability ( w9n+ rema9s #a_/b4 ,obs]vation %[s1 x is true1 t s]ies ( sev5 consecutive r$s are v r>e1 b s]ies ( six r$s foll[$ by a black are j z r>e4 ,!y h notic$ ! r>ity ( ! s]ies ( sev5 r$s2 if !y h n rem>k$ ! r>ity ( six r$s & a black1 x is only because su* s]ies /rike ! att5tion less4 ,v4 ,,! ,,probability ,,( ,,causes4--,we n[ come to ! problems ( ! probability ( causes1 ! mo/ important f ! po9t ( view ( sci5tific applications4 ,two />s1 = in/ance1 are v close toge!r on ! cele/ial sph]e4 ,is ?is app>5t contiguity a m]e effect ( *ance8 ,are ^! />s1 al?\gh on almo/ ! same visual ray1 situat$ at v diff]5t di/ances f ! e>?1 & consequ5tly v f> f "o ano!r8 ,or1 p]haps1 does ! app>5t correspond to a real contiguity8 ,?is is a problem on ! probability ( causes4 ,i recall fir/ t at ! outset ( all problems ( ! probability ( effects t h hi!rto occupi$ u1 we h always _h to make a conv5tion1 m or less ju/ifi$4 ,& if 9 mo/ cases ! result 01 9 a c]ta9 measure1 indep5d5t ( ?is conv5tion1 ?is 0 only because ( c]ta9 hypo!ses :i* p]mitt$ u to reject a priori discont9u\s functions1 = example1 or c]ta9 absurd conv5tions4 ,we %all f9d some?+ analog\s :5 we deal ) ! probability ( causes4 ,an effect may be produc$ by ! cause ,a or by ! cause ,b4 ,! effect has j be5 obs]v$4 ,we ask ! probability t x is due to ! cause ,a4 ,?is is an a po/]iori probability ( cause4 ,b ,i c\ld n calculate x1 if a conv5tion m or less ju/ifi$ did n tell me 9 advance :at is ! a priori probability = ! cause ,a to come into play2 ,i mean ! probability ( ?is ev5t = "s "o :o _h n obs]v$ ! effect4 ,! bett] to expla9 myself ,i g back to ! example ( ! game ( c>t m5tion$ above4 ,my adv]s>y deals = ! fir/ "t & he turns up a k+4 ,:at is ! probability t he is a %>p]8 ,! =mulas ord9>ily taught give #h_/i1 a result evid5tly r surpris+4 ,if we look at x clos]1 we see t ! calculation is made z if1 be=e sitt+ d[n at ! table1 ,i _h consid]$ t !re 0 "o *ance 9 two t my adv]s>y 0 n hone/4 ,an absurd hypo!sis1 because 9 t case ,i %\ld h c]ta9ly n play$ ) him1 & ?is expla9s ! absurdity ( ! conclusion4 ,! conv5tion ab\t ! a priori probability 0 unju/ifi$1 & t is :y ! calculation ( ! a po/]iori probability l$ me to an inadmissible result4 ,we see ! importance ( ?is prelim9>y conv5tion4 ,i %all ev5 add t if none 7 made1 ! problem ( ! a po/]iori probability w\ld h no mean+4 ,x mu/ always be made ei!r explicitly or tacitly4 ,pass to an example ( a m sci5tific "*4 ,i wi% to det]m9e an exp]im5tal law4 ,?is law1 :5 ,i "k x1 c be repres5t$ by a curve4 ,i make a c]ta9 numb] ( isolat$ obs]vations2 ea* ( ^! w be repres5t$ by a po9t4 ,:5 ,i h obta9$ ^! diff]5t po9ts1 ,i draw a curve betwe5 !m1 /riv+ to pass z ne> to !m z possible & yet pres]ve = my curve a regul> =m1 )\t angul> po9ts1 or inflections too acc5tuat$1 or brusque v>iation ( ! radius ( curvature4 ,?is curve w repres5t = me ! probable law1 & ,i assume n only t x w tell me ! values ( ! function int]m$iate betwe5 ^? :i* h be5 obs]v$1 b also t x w give me ! obs]v$ values !mselves m exactly ?an direct obs]vation4 ,?is is :y ,i make x pass ne> ! po9ts1 & n ?r\gh ! po9ts !mselves4 ,"h is a problem 9 ! probability ( causes4 ,! effects are ! measurem5ts ,i h record$2 !y dep5d on a comb9ation ( two causes3 ! true law ( ! ph5om5on & ! errors ( obs]vation4 ,kn[+ ! effects1 we h to seek ! probability t ! ph5om5on obeys ?is law or t1 & t ! obs]vations h be5 affect$ by ?is or t error4 ,! mo/ probable law !n corresponds to ! curve trac$1 & ! mo/ probable error ( an obs]vation is repres5t$ by ! di/ance ( ! correspond+ po9t f ?is curve4 ,b ! problem w\ld h no mean+ if1 be=e any obs]vation1 ,i _h n fa%ion$ an a priori idea ( ! probability ( ?is or t law1 & ( ! *ances ( error to :i* ,i am expos$4 ,if my in/rum5ts are good 7& t ,i knew be=e mak+ ! obs]vations71 ,i %all n p]mit my curve to dep>t mu* f ! po9ts :i* repres5t ! r\gh measurem5ts4 ,if !y are bad1 ,i may g a little fur!r away f !m 9 ord] to obta9 a less s9u\s curve2 ,i %all sacrifice m to regul>ity4 ,:y !n is x t ,i seek to trace a curve )\t s9uosities8 ,x is because ,i consid] a priori a law repres5t$ by a cont9u\s function 7or by a function ^: d]ivatives ( high ord] are small71 z m probable ?an a law n satisfy+ ^! conditions4 ,)\t ?is belief1 ! problem ( :i* we speak w\ld h no mean+2 int]polation w\ld be impossible2 no law c\ld be d$uc$ f a f9ite numb] ( obs]vations2 sci5ce w\ld n exi/4 ,fifty ye>s ago physici/s consid]$1 o!r ?+s be+ equal1 a simple law z m probable ?an a complicat$ law4 ,!y ev5 invok$ ?is pr9ciple 9 favor ( ,m>iotte's law z aga9/ ! exp]im5ts ( ,regnault4 ,to-day !y h repudiat$ ?is belief2 & yet1 h[ _m times are !y compell$ to act z ?\gh !y /ill held x6 ,h[ev] t may be1 :at rema9s ( ?is t5d5cy is ! belief 9 cont9uity1 & we h j se5 t if ?is belief 7 to disappe> 9 its turn1 exp]im5tal sci5ce w\ld become impossible4 ,,vi4 ,,! ,,!ory ,,( ,,errors4--,we are ?us l$ to speak ( ! !ory ( errors1 :i* is directly connect$ ) ! problem ( ! probability ( causes4 ,"h aga9 we f9d effects1 to wit1 a c]ta9 numb] ( discordant obs]vations1 & we seek to div9e ! causes1 :i* are1 on ! "o h&1 ! real value ( ! quantity to be measur$2 on ! o!r h&1 ! error made 9 ea* isolat$ obs]vation4 ,x is necess>y to calculate :at is a po/]iori ! probable magnitude ( ea* error1 & consequ5tly ! probable value ( ! quantity to be measur$4 ,b z ,i h j expla9$1 we %\ld n "k h[ to und]take ?is calculation if we did n admit a priori1 t is to say1 be=e all obs]vation1 a law ( probability ( errors4 ,is !re a law ( errors8 ,! law ( errors admitt$ by all calculators is ,gauss's law1 :i* is repres5t$ by a c]ta9 transc5d5tal curve kn[n "u ! "n ( '! bell4' ,b fir/ x is prop] to recall ! classic di/9ction betwe5 sy/ematic & accid5tal errors4 ,if we measure a l5g? ) too long a met]1 we %all always f9d too small a numb]1 & x w be ( no use to measure sev]al times2 ?is is a sy/ematic error4 ,if we measure ) an accurate met]1 we may1 h[ev]1 make a mi/ake2 b we g wrong1 n[ too mu*1 n[ too little1 & :5 we take ! mean ( a great numb] ( measurem5ts1 ! error w t5d to gr[ small4 ,^! are accid5tal errors4 ,x is evid5t f ! fir/ t sy/ematic errors c n satisfy ,gauss's law2 b d ! accid5tal errors satisfy x8 ,a great numb] ( demon/rations h be5 attempt$2 almo/ all are crude p>alogisms4 ,nev]!less1 we may demon/rate ,gauss's law by />t+ f ! foll[+ hypo!ses3 ! error committ$ is ! result ( a great numb] ( p>tial & indep5d5t errors2 ea* ( ! p>tial errors is v little & besides1 obeys any law ( probability1 provid$ t ! probability ( a positive error is ! same z t ( an equal negative error4 ,x is evid5t t ^! conditions w be (t5 b n always fulfill$1 & we may res]ve ! "n ( accid5tal = errors :i* satisfy !m4 ,we see t ! me?od ( lea/ squ>es is n legitimate 9 e case2 9 g5]al ! physici/s are m di/ru/ful ( x ?an ! a/ronom]s4 ,?is is1 no d\bt1 because ! latt]1 besides ! sy/ematic errors to :i* !y & ! physici/s are subject alike1 h to control ) an extremely important s\rce ( error :i* is :olly accid5tal2 ,i mean atmosph]ic undulations4 ,s x is v curi\s to he> a physici/ discuss ) an a/ronom] ab\t a me?od ( obs]vation4 ,! physici/1 p]suad$ t "o good measurem5t is wor? m ?an _m bad ones1 is be=e all conc]n$ ) elim9at+ by d9t ( precautions ! lea/ sy/ematic errors1 & ! a/ronom] says to him3 ',b ?us y c obs]ve only a small numb] ( />s2 ! accid5tal errors w n disappe>4' ,:at %\ld we conclude8 ,mu/ we cont9ue to use ! me?od ( lea/ squ>es8 ,we mu/ di/+ui%4 ,we h elim9at$ all ! sy/ematic errors we c\ld suspect2 we "k well !re are /ill o!rs1 b we c n detect !m2 yet x is necess>y to make up our m9d & adopt a def9itive value :i* w be reg>d$ z ! probable value2 & = t x is evid5t ! be/ ?+ to d is to apply ,gauss's me?od4 ,we h only appli$ a practical rule ref]r+ to subjective probability4 ,!re is no?+ m to be said4 ,b we wi% to g f>!r & affirm t n only is ! probable value s mu*1 b t ! probable error 9 ! result is s mu*4 ,?is is absolutely illegitimate2 x w\ld be true only if we 7 sure t all ! sy/ematic errors 7 elim9at$1 & ( t we "k absolutely no?+4 ,we h two s]ies ( obs]vations2 by apply+ ! rule ( lea/ squ>es1 we f9d t ! probable error 9 ! fir/ s]ies is twice z small z 9 ! second4 ,! second s]ies may1 h[ev]1 be bett] ?an ! fir/1 because ! fir/ p]haps is affect$ by a l>ge sy/ematic error4 ,all we c say is t ! fir/ s]ies is probably bett] ?an ! second1 s9ce its accid5tal error is small]1 & we h no reason to affirm t ! sy/ematic error is great] = "o ( ! s]ies ?an = ! o!r1 our ignorance on ?is po9t be+ absolute4 ,,vii4 ,,conclusions4--,9 ! l9es :i* prec$e1 ,i h set _m problems )\t solv+ any ( !m4 ,yet ,i d n regret hav+ writt5 !m1 because !y w p]haps invite ! read] to reflect on ^! delicate que/ions4 ,h[ev] t may be1 !re are c]ta9 po9ts :i* seem well e/abli%$4 ,to und]take any calculation ( probability1 & ev5 = t calculation to h any mean+1 x is necess>y to admit1 z po9t ( dep>ture1 a hypo!sis or conv5tion :i* has always some?+ arbitr>y ab\t x4 ,9 ! *oice ( ?is conv5tion1 we c be guid$ only by ! pr9ciple ( suffici5t reason4 ,un=tunately ?is pr9ciple is v vague & v ela/ic1 & 9 ! cursory exam9ation we h j made1 we h se5 x take _m diff]5t =ms4 ,! =m "u :i* we h met x mo/ (t5 is ! belief 9 cont9uity1 a belief :i* x w\ld be difficult to ju/ify by apodeictic reason+1 b )\t :i* all sci5ce w\ld be impossible4 ,f9ally ! problems to :i* ! calculus ( probabilities may be appli$ ) pr(it are ^? 9 :i* ! result is indep5d5t ( ! hypo!sis made at ! outset1 provid$ only t ?is hypo!sis satisfies ! condition ( cont9uity4 ,,*apt] ,,xii ,,optics ,,& ,,electricity ,,fresnel',s ,,!ory4--,! be/ example#e t c be *os5 ( physics 9 ! mak+ is ! !ory ( light & its relations to ! !ory ( electricity4 ,?anks to ,fresnel1 optics is ! be/ develop$ "p ( physics2 ! s-call$ wave-!ory =ms a :ole truly satisfy+ to ! m9d4 ,we mu/ n1 h[ev]1 ask ( x :at x c n give u4 #e ,?is *apt] is a p>tial reproduction ( ! prefaces ( two ( my works3 ,?orie ma?matique de la lumire 7,p>is1 ,naud1 #ahhi71 & lectricit et optique 7,p>is1 ,naud1 #aija74 ,! object ( ma!matical !ories is n to reveal to u ! true nature ( ?+s2 ?is w\ld be an unreasonable pret5sion4 ,_! sole aim is to coord9ate ! physical laws :i* exp]im5t reveals to u1 b :i*1 )\t ! help ( ma!matics1 we %\ld n be able ev5 to /ate4 ,x matt]s little :e!r ! e!r really exi/s2 t is ! affair ( metaphysicians4 ,! ess5tial ?+ = u is t ev]y?+ happ5s z if x exi/$1 & t ?is hypo!sis is conv5i5t = ! explanation ( ph5om5a4 ,aft] all1 h we any o!r reason to believe 9 ! exi/5ce ( mat]ial objects8 ,t1 too1 is only a conv5i5t hypo!sis2 only ?is w nev] cease to be s1 :]eas1 no d\bt1 "s day ! e!r w be ?r[n aside z useless4 ,b ev5 at t day1 ! laws ( optics & ! equations :i* translate !m analytically w rema9 true1 at lea/ z a fir/ approximation4 ,x w always be useful1 !n1 to /udy a doctr9e t unites all ^! equations4 ,! undulatory !ory re/s on a molecul> hypo!sis4 ,= ^? :o ?9k !y h ?us discov]$ ! cause "u ! law1 ?is is an advantage4 ,= ! o!rs x is a reason = di/ru/4 ,b ?is di/ru/ seems to me z little ju/ifi$ z ! illusion ( ! =m]4 ,^! hypo!ses play only a second>y "p4 ,!y might be sacrific$4 ,!y usually are n1 because !n ! explanation w\ld lose 9 cle>ness2 b t is ! only reason4 ,9 fact1 if we look$ clos] we %\ld see t only two ?+s are borr[$ f ! molecul> hypo!ses3 ! pr9ciple ( ! cons]vation ( en]gy & ! l9e> =m ( ! equations1 :i* is ! g5]al law ( small movem5ts1 z ( all small v>iations4 ,?is expla9s :y mo/ ( ,fresnel's conclusions rema9 un*ang$ :5 we adopt ! electromagnetic !ory ( light4 ,,maxwell',s ,,!ory4--,maxwell1 we "k1 connect$ by a close bond two p>ts ( physics until !n entirely =eign to "o ano!r1 optics & electricity4 ,by bl5d+ ?us 9 a va/] :ole1 9 a high] h>mony1 ! optics ( ,fresnel has n ceas$ to be alive4 ,its v>i\s p>ts subsi/1 & _! mutual relations are /ill ! same4 ,only ! language we us$ to express !m has *ang$2 &1 on ! o!r h&1 ,maxwell has reveal$ to u o!r relations1 be=e unsuspect$1 betwe5 ! diff]5t p>ts ( optics & ! doma9 ( electricity4 ,:5 a ,fr5* read] fir/ op5s ,maxwell's book1 a feel+ ( uneas9ess & (t5 ev5 ( mi/ru/ m+les at fir/ ) 8 admiration4 ,only aft] a prolong$ acqua9tance & at ! co/ ( _m ef=ts does ?is feel+ disappe>4 ,!re are ev5 "s em95t m9ds t nev] lose x4 ,:y are ! ,engli% sci5ti/'s ideas ) su* difficulty acclimatiz$ among u8 ,x is1 no d\bt1 because ! education receiv$ by ! majority ( enlight5$ ,fr5*m5 pr$isposes !m to appreciate precision & logic above e o!r quality4 ,! old !ories ( ma!matical physics gave u 9 ?is respect complete satisfaction4 ,all our ma/]s1 f ,laplace to ,cau*y1 h proce$$ 9 ! same way4 ,/>t+ f cle>ly /at$ hypo!ses1 !y d$uc$ all _! consequ5ces ) ma!matical rigor1 & !n comp>$ !m ) exp]im5t4 ,x seem$ _! aim to give e bran* ( physics ! same precision z cele/ial me*anics4 ,a m9d accu/om$ to admire su* models is h>d to suit ) a !ory4 ,n only w x n tol]ate ! lea/ appe>ance ( contradiction1 b x w dem& t ! v>i\s p>ts be logically connect$ ) "o ano!r1 & t ! numb] ( di/9ct hypo!ses be r$uc$ to m9imum4 ,?is is n all2 x w h /ill o!r dem&s1 :i* seem to me less reasonable4 ,beh9d ! matt] :i* our s5ses c rea*1 & :i* exp]im5t tells u (1 x w desire to see ano!r1 & 9 its eyes ! only real1 matt]1 :i* w h only purely geometric prop]ties1 & ^: atoms w be no?+ b ma!matical po9ts1 subject to ! laws ( dynamics alone4 ,& yet ^! atoms1 invisible & )\t color1 x w seek by an unconsci\s contradiction to repres5t to itself & consequ5tly to id5tify z closely z possible ) common matt]4 ,!n only w x be fully satisfi$ & imag9e t x has p5etrat$ ! secret ( ! univ]se4 ,if ?is satisfaction is deceitful1 x is none ! less difficult to r5\nce4 ,?us1 on op5+ ,maxwell1 a ,fr5*man expects to f9d a !oretical :ole z logical & precise z ! physical optics bas$ on ! hypo!sis ( ! e!r2 he ?us prep>es = himself a disappo9tm5t :i* ,i %\ld l to sp>e ! read] by in=m+ him imm$iately ( :at he mu/ look = 9 ,maxwell1 & :at he c n f9d !re4 ,maxwell does n give a me*anical explanation ( electricity & magnetism2 he conf9es himself to demon/rat+ t su* an explanation is possible4 ,he %[s also t optical ph5om5a are only a special case ( electromagnetic ph5om5a4 ,f e !ory ( electricity1 "o c !re=e d$uce imm$iately a !ory ( light4 ,! conv]se un=tunately is n true2 f a complete explanation ( light1 x is n always easy to d]ive a complete explanation ( electric ph5om5a4 ,?is is n easy1 9 p>ticul>1 if we wi% to />t f ,fresnel's !ory4 ,d\btless x w\ld n be impossible2 b nev]!less we mu/ ask :e!r we are n go+ to be =c$ to r5\nce admirable results t we ?\ght def9itely acquir$4 ,t seems a /ep backw>d2 & _m good m9ds are n will+ to submit to x4 ,:5 ! read] %all h cons5t$ to limit 8 hopes1 he w /ill enc\nt] o!r difficulties4 ,! ,engli% sci5ti/ does n try to con/ruct a s+le edifice1 f9al & well ord]$2 he seems r to erect a great numb] ( provisional & indep5d5t con/ructions1 betwe5 :i* communication is difficult & sometimes impossible4 ,take z example ! *apt] 9 :i* he expla9s electro/atic attractions by pressures & t5sions 9 ! dielectric m$ium4 ,?is *apt] might be omitt$ )\t mak+ !reby ! re/ ( ! book less cle> or complete2 &1 on ! o!r h&1 x conta9s a !ory complete 9 itself :i* "o c\ld und]/& )\t hav+ read a s+le l9e t prec$es or foll[s4 ,b x is n only indep5d5t ( ! re/ ( ! "w2 x is difficult to reconcile ) ! fundam5tal ideas ( ! book4 ,maxwell does n ev5 attempt ?is reconciliation2 he m]ely says3 ,i h n be5 able to make ! next /ep1 namely1 to acc\nt by me*anical consid]ations = ^! /resses 9 ! dielectric4 ,?is example w suffice to make my ?\ght und]/ood2 ,i c\ld cite _m o!rs4 ,?us :o w\ld suspect1 9 read+ ! pages devot$ to magnetic rot>y pol>ization1 t !re is an id5tity betwe5 optical & magnetic ph5om5a8 ,"o mu/ n !n flatt] himself t he c avoid all contradiction2 to t x is necess>y to be resign$4 ,9 fact1 two contradictory !ories1 provid$ "o does n m+le !m1 & if "o does n seek 9 !m ! basis ( ?+s1 may bo? be useful in/rum5ts ( rese>*2 & p]haps ! read+ ( ,maxwell w\ld be less sugge/ive if he _h n op5$ up to u s _m new & div]g5t pa?s4 ,! fundam5tal idea1 h[ev]1 is ?us a little obscur$4 ,s f> is ?is ! case t 9 ! majority ( popul>iz$ v]sions x is ! only po9t completely left aside4 ,i feel1 !n1 t ! bett] to make its importance /& out1 ,i ought to expla9 9 :at ?is fundam5tal idea consi/s4 ,b = t a %ort digression is necess>y4 ,,! ,,me*anical ,,explanation ,,( ,,physical ,,ph5om5a4--,!re is 9 e physical ph5om5on a c]ta9 numb] ( p>amet]s :i* exp]im5t rea*es directly & all[s u to measure4 ,i %all call ^! ! p>amet]s q4 ,obs]vation !n tea*es u ! laws ( ! v>iations ( ^! p>amet]s2 & ^! laws c g5]ally be put 9 ! =m ( diff]5tial equations1 :i* connect ! p>amet]s q ) ! "t4 ,:at is x necess>y to d to give a me*anical int]pretation ( su* a ph5om5on8 ,"o w try to expla9 x ei!r by ! motions ( ord9>y matt]1 or by ^? ( "o or m hypo!tical fluids4 ,^! fluids w be consid]$ z =m$ ( a v great numb] ( isolat$ molecules m4 ,:5 %all we say1 !n1 t we h a complete me*anical explanation ( ! ph5om5on8 ,x w be1 on ! "o h&1 :5 we "k ! diff]5tial equations satisfi$ by ! coord9ates ( ^! hypo!tical molecules m1 equations :i*1 moreov]1 mu/ con=m to ! pr9ciples ( dynamics2 &1 on ! o!r h&1 :5 we "k ! relations t def9e ! coord9ates ( ! molecules m z functions ( ! p>amet]s q accessible to exp]im5t4 ,^! equations1 z ,i h said1 mu/ con=m to ! pr9ciples ( dynamics1 &1 9 p>ticul>1 to ! pr9ciple ( ! cons]vation ( en]gy & ! pr9ciple ( lea/ action4 ,! fir/ ( ^! two pr9ciples tea*es u t ! total en]gy is con/ant & t ?is en]gy is divid$ into two p>ts3 #a ,! k9etic en]gy1 or vis viva1 :i* dep5ds on ! masses ( ! hypo!tical molecules m1 & _! velocities1 & :i* ,i %all call ,t4 #b ,! pot5tial en]gy1 :i* dep5ds only on ! coord9ates ( ^! molecules & :i* ,i %all call ,u4 ,x is ! sum ( ! two en]gies ,t & ,u :i* is con/ant4 ,:at n[ does ! pr9ciple ( lea/ action tell u8 ,x tells u t to pass f ! initial position occupi$ at ! in/ant t#j to ! f9al position occupi$ at ! in/ant t#a1 ! sy/em mu/ take su* a pa? t1 9 ! int]val ( "t t elapses betwe5 ! two in/ants t#j & t#a1 ! av]age value ( '! action' 7t is to say1 ( ! diff]5ce betwe5 ! two en]gies ,t & ,u7 %all be z small z possible4 ,if ! two functions ,t & ,u are kn[n1 ?is pr9ciple suffices to det]m9e ! equations ( motion4 ,among all ! possible ways ( pass+ f "o position to ano!r1 !re is evid5tly "o = :i* ! av]age value ( ! action is less ?an = any o!r4 ,!re is1 moreov]1 only "o2 & x results f ?is t ! pr9ciple ( lea/ action suffices to det]m9e ! pa? foll[$ & consequ5tly ! equations ( motion4 ,?us we obta9 :at are call$ ! equations ( ,lagrange4 ,9 ^! equations1 ! indep5d5t v>iables are ! coord9ates ( ! hypo!tical molecules m2 b ,i n[ suppose t "o takes z v>iables ! p>amet]s q directly accessible to exp]im5t4 ,! two p>ts ( ! en]gy mu/ !n be express$ z functions ( ! p>amet]s q & ( _! d]ivatives4 ,!y w evid5tly appe> "u ?is =m to ! exp]im5t]4 ,! latt] w naturally try to def9e ! pot5tial & ! k9etic en]gy by ! aid ( quantities t he c directly obs]ve4#f #f ,we add t ,u w dep5d only on ! p>amet]s q1 t ,t w dep5d on ! p>amet]s q & _! d]ivatives ) respect to ! "t & w be a homog5e\s polynomial ( ! second degree ) respect to ^! d]ivatives4 ,t grant$1 ! sy/em w always g f "o position to ano!r by a pa? su* t ! av]age action %all be a m9imum4 ,x matt]s little t ,t & ,u are n[ express$ by ! aid ( ! p>amet]s q & _! d]ivatives2 x matt]s little t x is also by means ( ^! p>amet]s t we def9e ! initial & f9al positions2 ! pr9ciple ( lea/ action rema9s always true4 ,n[ "h aga91 ( all ! pa?s t lead f "o position to ano!r1 !re is "o = :i* ! av]age action is a m9imum1 & !re is only "o4 ,! pr9ciple ( lea/ action suffices1 !n1 to det]m9e ! diff]5tial equations :i* def9e ! v>iations ( ! p>amet]s q4 ,! equations ?us obta9$ are ano!r =m ( ! equations ( ,lagrange4 ,to =m ^! equations we ne$ to "k nei!r ! relations t connect ! p>amet]s q ) ! coord9ates ( ! hypo!tical molecules1 nor ! masses ( ^! molecules1 nor ! expression ( ,u z a function ( ! coord9ates ( ^! molecules4 ,all we ne$ to "k is ! expression ( ,u z a function ( ! p>amet]s1 & t ( ,t z a function ( ! p>amet]s q & _! d]ivatives1 t is1 ! expressions ( ! k9etic & ( ! pot5tial en]gy z functions ( ! exp]im5tal data4 ,!n we %all h "o ( two ?+s3 ei!r = a suitable *oice ( ! functions ,t & ,u1 ! equations ( ,lagrange1 con/ruct$ z we h j said1 w be id5tical ) ! diff]5tial equations d$uc$ f exp]im5ts2 or else !re w exi/ no functions ,t & ,u1 = :i* ?is agreem5t takes place4 ,9 ! latt] case x is cle> t no me*anical explanation is possible4 ,! necess>y condition = a me*anical explanation to be possible is !re=e t we c *oose ! functions ,t & ,u 9 su* a way z to satisfy ! pr9ciple ( lea/ action1 :i* involves t ( ! cons]vation ( en]gy4 ,?is condition1 moreov]1 is suffici5t4 ,suppose1 9 fact1 t we h f\nd a function ,u ( ! p>amet]s q1 :i* repres5ts "o ( ! p>ts ( ! en]gy2 t ano!r "p ( ! en]gy1 :i* we %all repres5t by ,t1 is a function ( ! p>amet]s q & _! d]ivatives1 & t x is a homog5e\s polynomial ( ! second degree ) respect to ^! d]ivatives2 & f9ally t ! equations ( ,lagrange1 =m$ by means ( ^! two functions1 ,t & ,u1 con=m to ! data ( ! exp]im5t4 ,:at is necess>y 9 ord] to d$uce f ?is a me*anical explanation8 ,x is necess>y t ,u c be reg>d$ z ! pot5tial en]gy ( a sy/em & ,t z ! vis viva ( ! same sy/em4 ,!re is no difficulty z to ,u1 b c ,t be reg>d$ z ! vis viva ( a mat]ial sy/em8 ,x is easy to %[ t ?is is always possible1 & ev5 9 an inf9ity ( ways4 ,i w conf9e myself to ref]r+ = m details to ! preface ( my "w1 'lectricit et optique4' ,?us if ! pr9ciple ( lea/ action c n be satisfi$1 no me*anical explanation is possible2 if x c be satisfi$1 !re is n only "o1 b an inf9ity1 :5ce x foll[s t z soon z !re is "o !re is an inf9ity ( o!rs4 ,"o m obs]vation4 ,among ! quantities t exp]im5t gives u directly1 we %all reg>d "s z functions ( ! coord9ates ( our hypo!tical molecules2 ^! are our p>amet]s q4 ,we %all look ^u ! o!rs z dep5d5t n only on ! coord9ates1 b on ! velocities1 or1 :at comes to ! same ?+1 on ! d]ivatives ( ! p>amet]s q1 or z comb9ations ( ^! p>amet]s & _! d]ivatives4 ,& !n a "q pres5ts itself3 among all ^! quantities measur$ exp]im5tally1 :i* %all we *oose to repres5t ! p>amet]s q8 ,:i* %all we pref] to reg>d z ! d]ivatives ( ^! p>amet]s8 ,?is *oice rema9s arbitr>y to a v l>ge ext5t2 b1 = a me*anical explanation to be possible1 x suffices if we c make ! *oice 9 su* a way z to accord ) ! pr9ciple ( lea/ action4 ,& !n ,maxwell ask$ himself :e!r he c\ld make ?is *oice & t ( ! two en]gies ,t & ,u1 9 su* a way t ! electrical ph5om5a w\ld satisfy ?is pr9ciple4 ,exp]im5t %[s u t ! en]gy ( an electromagnetic field is decompos$ into two p>ts1 ! electro/atic en]gy & ! electrodynamic en]gy4 ,maxwell obs]v$ t if we reg>d ! fir/ z repres5t+ ! pot5tial en]gy ,u1 ! second z repres5t+ ! k9etic en]gy ,t2 if1 moreov]1 ! electro/atic *>ges ( ! conductors are consid]$ z p>amet]s q & ! int5sities ( ! curr5ts z ! d]ivatives ( o!r p>amet]s q2 "u ^! conditions1 ,i say1 ,maxwell obs]v$ t ! electric ph5om5a satisfy ! pr9ciple ( lea/ action4 ,!nce=? he 0 c]ta9 ( ! possibility ( a me*anical explanation4 ,if he _h expla9$ ?is idea at ! beg9n+ ( 8 book in/ead ( relegat+ x to an obscure "p ( ! second volume1 x w\ld n h escap$ ! majority ( read]s4 ,if1 !n1 a ph5om5on admits ( a complete me*anical explanation1 x w admit ( an inf9ity ( o!rs1 t w r5d] an acc\nt equally well ( all ! p>ticul>s reveal$ by exp]im5t4 ,& ?is is confirm$ by ! hi/ory ( e bran* ( physics2 9 optics1 = in/ance1 ,fresnel believ$ vibration to be p]p5dicul> to ! plane ( pol>ization2 ,neumann reg>d$ x z p>allel to ?is plane4 ,an 'exp]im5tum crucis' has long be5 s\ght :i* w\ld enable u to decide betwe5 ^! two !ories1 b x has n be5 f\nd4 ,9 ! same way1 )\t leav+ ! doma9 ( electricity1 we may asc]ta9 t ! !ory ( two fluids & t ( ! s+le fluid bo? acc\nt 9 a fa%ion equally satisfactory = all ! obs]v$ laws ( electro/atics4 ,all ^! facts are easily explicable1 ?anks to ! prop]ties ( ! equations ( ,lagrange :i* ,i h j recall$4 ,x is easy n[ to compreh5d :at is ,maxwell's fundam5tal idea4 ,to demon/rate ! possibility ( a me*anical explanation ( electricity1 we ne$ n preoccupy ourselves ) f9d+ ?is explanation itself2 x suffices u to "k ! expression ( ! two functions ,t & ,u1 :i* are ! two p>ts ( en]gy1 to =m ) ^! two functions ! equations ( ,lagrange & !n to comp>e ^! equations ) ! exp]im5tal laws4 ,among all ^! possible explanations1 h[ make a *oice = :i* ! aid ( exp]im5t fails u8 ,a day w come p]haps :5 physici/s w n int]e/ !mselves 9 ^! que/ions1 inaccessible to positive me?ods1 & w ab&on !m to ! metaphysicians4 ,?is day has n yet arriv$2 man does n resign himself s easily to be =ev] ignorant ( ! f\ndation ( ?+s4 ,our *oice c !re=e be fur!r guid$ only by consid]ations :]e ! "p ( p]sonal appreciation is v great2 !re are1 h[ev]1 solutions t all ! _w w reject because ( _! :imsicality1 & o!rs t all ! _w w pref] because ( _! simplicity4 ,9 :at conc]ns electricity & magnetism1 ,maxwell ab/a9s f mak+ any *oice4 ,x is n t he sy/ematically disda9s all t is unatta9able by positive me?ods2 ! "t he has devot$ to ! k9etic !ory ( gases suffici5tly proves t4 ,i w add t if1 9 8 great "w1 he develops no complete explanation1 he _h previ\sly attempt$ to give "o 9 an article 9 ! ,philosophical ,magaz9e4 ,! /rang5ess & ! complexity ( ! hypo!ses he _h be5 oblig$ to make _h l$ him aft]w>ds to give ?is up4 ,! same _s is f\nd ?r\gh\t ! :ole "w4 ,:at is ess5tial1 t is to say :at mu/ rema9 common to all !ories1 is made prom95t2 all t w\ld only be suitable to a p>ticul> !ory is ne>ly always pass$ ov] 9 sil5ce4 ,?us ! read] f9ds himself 9 ! pres5ce ( a =m almo/ devoid ( matt]1 :i* he is at fir/ tempt$ to take = a fugitive %ad[ n to be grasp$4 ,b ! ef=ts to :i* he is ?us condemn$ =ce him to ?9k & he ends by compreh5d+ :at 0 (t5 r artificial 9 ! !oretic con/ructs he _h previ\sly only wond]$ at4 ,,*apt] ,,xiii ,,electrodynamics ,! hi/ory ( electrodynamics is p>ticul>ly in/ructive f our po9t ( view4 ,ampre entitl$ 8 immortal "w1 ',?orie des phnomnes lectrodynamiques1 uniquem5t fonde sur l'expri5ce4' ,he !re=e imag9$ t he _h made no hypo!sis1 b he _h made !m1 z we %all soon see2 only he made !m )\t be+ consci\s ( x4 ,8 successors1 on ! o!r h&1 p]ceiv$ !m1 s9ce _! att5tion 0 attract$ by ! weak po9ts 9 ,ampre's solution4 ,!y made new hypo!ses1 ( :i* ?is "t !y 7 fully consci\s2 b h[ _m times x 0 necess>y to *ange !m be=e arriv+ at ! classic sy/em ( to-day :i* is p]haps n yet f9al2 ?is we %all see4 ,i4 ,,amp]e',s ,,!ory4--,:5 ,ampre /udi$ exp]im5tally ! mutual actions ( curr5ts1 he op]at$ & he only c\ld op]ate ) clos$ curr5ts4 ,x 0 n t he d5i$ ! possibility ( op5 curr5ts4 ,if two conductors are *>g$ ) positive & negative electricity & br\ght into communication by a wire1 a curr5t is e/abli%$ go+ f "o to ! o!r1 :i* cont9ues until ! two pot5tials are equal4 ,accord+ to ! ideas ( ,ampre's "t ?is 0 an op5 curr5t2 ! curr5t 0 kn[n to g f ! fir/ conductor to ! second1 x 0 n se5 to return f ! second to ! fir/4 ,s ,ampre consid]$ z op5 curr5ts ( ?is nature1 = example1 ! curr5ts ( dis*>ge ( cond5s]s2 b he c\ld n make !m ! objects ( 8 exp]im5ts because _! duration is too %ort4 ,ano!r sort ( op5 curr5t may also be imag9$4 ,i suppose two conductors1 ,a & ,b1 connect$ by a wire ,,amb4 ,small conduct+ masses 9 motion fir/ come 9 contact ) ! conductor ,b1 take f x an electric *>ge1 leave contact ) ,b & move along ! pa? ,,bna1 &1 transport+ ) !m _! *>ge1 come into contact ) ,a & give to x _! *>ge1 :i* returns !n to ,b along ! wire ,,amb4 ,n[ !re we h 9 a s5se a clos$ circuit1 s9ce ! electricity describes ! clos$ circuit ,,bnamb2 b ! two p>ts ( ?is curr5t are v diff]5t4 ,9 ! wire ,,amb1 ! electricity is displac$ ?r\gh a fix$ conductor1 l a voltaic curr5t1 ov]com+ an ohmic resi/ance & develop+ heat2 we say t x is displac$ by conduction4 ,9 ! "p ,,bna1 ! electricity is c>ri$ by a mov+ conductor2 x is said to be displac$ by convection4 ,if !n ! curr5t ( convection is consid]$ z altoge!r analog\s to ! curr5t ( conduction1 ! circuit ,,bnamb is clos$2 if1 on ! contr>y1 ! convection curr5t is n 'a true curr5t' &1 = example1 does n act on ! magnet1 !re rema9s only ! conduction curr5t ,,amb1 :i* is op54 ,= example1 if we connect by a wire ! two poles ( a ,holtz ma*9e1 ! *>g$ rotat+ disc transf]s ! electricity by convection f "o pole to ! o!r1 & x returns to ! fir/ pole by conduction ?r\gh ! wire4 ,b curr5ts ( ?is sort are v difficult to produce ) appreciable int5sity4 ,) ! means at ,ampre's disposal1 we may say t ?is 0 impossible4 ,to sum up1 ,ampre c\ld conceive ( ! exi/5ce ( two k9ds ( op5 curr5ts1 b he c\ld op]ate on nei!r because !y 7 n /rong en\gh or because _! duration 0 too %ort4 ,exp]im5t !re=e c\ld only %[ him ! action ( a clos$ curr5t on a clos$ curr5t1 or1 m accurately1 ! action ( a clos$ curr5t on a portion ( a curr5t1 because a curr5t c be made to describe a clos$ circuit compos$ ( a mov+ "p & a fix$ "p4 ,x is possible !n to /udy ! displacem5ts ( ! mov+ "p "u ! action ( ano!r clos$ curr5t4 ,on ! o!r h&1 ,ampre _h no means ( /udy+ ! action ( an op5 curr5t1 ei!r on a clos$ curr5t or ano!r op5 curr5t4 #a4 ,! ,case ( ,clos$ ,curr5ts4--,9 ! case ( ! mutual action ( two clos$ curr5ts1 exp]im5t reveal$ to ,ampre rem>kably simple laws4 ,i recall rapidly "h ^? :i* w be useful to u 9 ! sequel3 #a ,if ! int5sity ( ! curr5ts is kept con/ant1 & if ! two circuits1 aft] hav+ und]gone any de=mations & displacem5ts :atsoev]1 return f9ally to _! initial positions1 ! total "w ( ! electrodynamic actions w be null4 ,9 o!r words1 !re is an electrodynamic pot5tial ( ! two circuits1 proportional to ! product ( ! int5sities1 & dep5d+ on ! =m & relative position ( ! circuits2 ! "w ( ! electrodynamic actions is equal to ! v>iation ( ?is pot5tial4 #b ,! action ( a clos$ sol5oid is null4 #c ,! action ( a circuit ,c on ano!r voltaic circuit ,c' dep5ds only on ! 'magnetic field' develop$ by ?is circuit4 ,at ea* po9t 9 space we c 9 fact def9e 9 magnitude & direction a c]ta9 =ce call$ magnetic =ce1 :i* enjoys ! foll[+ prop]ties3 7a7 ,! =ce ex]cis$ by ,c on a magnetic pole is appli$ to t pole & is equal to ! magnetic =ce multipli$ by ! magnetic mass ( t pole2 7b7 ,a v %ort magnetic ne$le t5ds to take ! direction ( ! magnetic =ce1 & ! c\ple to :i* x t5ds to r$uce is proportional to ! magnetic =ce1 ! magnetic mom5t ( ! ne$le & ! s9e ( ! dip ( ! ne$le2 7c7 ,if ! circuit ,c is displac$1 ! "w ( ! electrodynamic action ex]cis$ by ,c on ,c' w be equal to ! increm5t ( ! 'fl[ ( magnetic =ce' :i* passes ?r\gh ! circuit4 #b4 ,action ( a ,clos$ ,curr5t on a ,portion ( ,curr5t4--,ampre n hav+ be5 able to produce an op5 curr5t1 prop]ly s call$1 _h only "o way ( /udy+ ! action ( a clos$ curr5t on a portion ( curr5t4 ,?is 0 by op]at+ on a circuit ,c compos$ ( two p>ts1 ! "o fix$1 ! o!r movable4 ,! movable "p 01 = in/ance1 a movable wire alphabeta ^: extremities alpha & beta c\ld slide along a fix$ wire4 ,9 "o ( ! positions ( ! movable wire1 ! end alpha re/$ on ! ,a ( ! fix$ wire & ! extremity beta on ! po9t ,b ( ! fix$ wire4 ,! curr5t circulat$ f alpha to beta1 t is to say1 f ,a to ,b along ! movable wire1 & !n x return$ f ,b to ,a along ! fix$ wire4 ,?is curr5t 0 !re=e clos$4 ,9 a second position1 ! movable wire hav+ slipp$1 ! extremity alpha re/$ on ano!r po9t ,a' ( ! fix$ wire1 & ! extremity beta on ano!r po9t ,b' ( ! fix$ wire4 ,! curr5t circulat$ !n f alpha to beta1 t is to say f ,a' to ,b' along ! movable wire1 & x aft]w>ds return$ f ,b' to ,b1 !n f ,b to ,a1 !n f9ally f ,a to ,a'1 always foll[+ ! fix$ wire4 ,! curr5t 0 !re=e also clos$4 ,if a l curr5t is subject$ to ! action ( a clos$ curr5t ,c1 ! movable "p w be displac$ j z if x 7 act$ ^u by a =ce4 ,ampre assumes t ! app>5t =ce to :i* ?is movable "p ,,ab seems ?us subject$1 repres5t+ ! action ( ! ,c on ! portion alphabeta ( ! curr5t1 is ! same z if alphabeta 7 trav]s$ by an op5 curr5t1 /opp+ at alpha & beta1 9 place ( be+ trav]s$ by a clos$ curr5t :i* aft] arriv+ at beta returns to alpha ?r\gh ! fix$ "p ( ! circuit4 ,?is hypo!sis seems natural en\gh1 & ,ampre made x unconsci\sly2 nev]!less x is n necess>y1 s9ce we %all see fur!r on t ,helmholtz reject$ x4 ,h[ev] t may be1 x p]mitt$ ,ampre1 ?\gh he _h nev] be5 able to produce an op5 curr5t1 to enunciate ! laws ( ! action ( a clos$ curr5t on an op5 curr5t1 or ev5 on an elem5t ( curr5t4 ,! laws are simple3 #a ,! =ce :i* acts on an elem5t ( curr5t is appli$ to ?is elem5t2 x is normal to ! elem5t & to ! magnetic =ce1 & proportional to ! compon5t ( ?is magnetic =ce :i* is normal to ! elem5t4 #b ,! action ( a clos$ sol5oid on an elem5t ( curr5t is null4 ,b ! electrodynamic pot5tial has disappe>$1 t is to say t1 :5 a clos$ curr5t & an op5 curr5t1 ^: int5sities h be5 ma9ta9$ con/ant1 return to _! initial positions1 ! total "w is n null4 #c4 ,cont9u\s ,rotations4--,among electrodynamic exp]im5ts1 ! mo/ rem>kable are ^? 9 :i* cont9u\s rotations are produc$ & :i* are sometimes call$ unipol> induction exp]im5ts4 ,a magnet may turn ab\t its axis2 a curr5t passes fir/ ?r\gh a fix$ wire1 ent]s ! magnet by ! pole ,n1 = example1 passes ?r\gh half ! magnet1 em]ges by a slid+ contact & re5t]s ! fix$ wire4 ,! magnet !n beg9s to rotate cont9u\sly )\t be+ able ev] to atta9 equilibrium2 ?is is ,f>aday's exp]im5t4 ,h[ is x possible8 ,if x 7 a "q ( two circuits ( inv>iable =m1 ! "o ,c fix$1 ! o!r ,c' movable ab\t an axis1 ?is latt] c\ld nev] take on cont9u\s rotation2 9 fact !re is an electrodynamic pot5tial2 !re mu/ !re=e be necess>ily a position ( equilibrium :5 ?is pot5tial is a maximum4 ,cont9u\s rotations are !re=e possible only :5 ! circuit ,c' is compos$ ( two p>ts3 "o fix$1 ! o!r movable ab\t an axis1 z is ! case 9 ,f>aday's exp]im5t4 ,"h aga9 x is conv5i5t to draw a di/9ction4 ,! passage f ! fix$ to ! movable "p1 or inv]sely1 may take place ei!r by simple contact 7! same po9t ( ! movable "p rema9+ con/antly 9 contact ) ! same po9t ( ! fix$ "p71 or by a slid+ contact 7! same po9t ( ! movable "p com+ successively 9 contact ) div]se po9ts ( ! fix$ "p74 ,x is only 9 ! second case t !re c be cont9u\s rotation4 ,?is is :at !n happ5s3 ,! sy/em t5ds to take a position ( equilibrium2 b1 :5 at ! po9t ( rea*+ t position1 ! slid+ contact puts ! movable "p 9 communication ) a new po9t ( ! fix$ "p2 x *anges ! connections1 x *anges !re=e ! conditions ( equilibrium1 s t ! position ( equilibrium flee+1 s to say1 be=e ! sy/em :i* seeks to atta9 x1 rotation may take place indef9itely4 ,ampre assumes t ! action ( ! circuit on ! movable "p ( ,c' is ! same z if ! fix$ "p ( ,c' did n exi/1 & !re=e z if ! curr5t pass+ ?r\gh ! movable "p 7 op54 ,he concludes !re=e t ! action ( a clos$ on an op5 curr5t1 or inv]sely t ( an op5 curr5t on a clos$ curr5t1 may give rise to a cont9u\s rotation4 ,b ?is conclusion dep5ds on ! hypo!sis ,i h enunciat$ & :i*1 z ,i said above1 is n admitt$ by ,helmholtz4 #d4 ,mutual ,action ( ,two ,op5 ,curr5ts4--,9 :at conc]ns ! mutual actions ( two op5 curr5ts1 & 9 p>ticul> t ( two elem5ts ( curr5t1 all exp]im5t breaks d[n4 ,ampre has rec\rse to hypo!sis4 ,he supposes3 #a ,t ! mutual action ( two elem5ts r$uces to a =ce act+ along _! jo92 #b ,t ! action ( two clos$ curr5ts is ! resultant ( ! mutual actions ( _! div]se elem5ts1 :i* are besides ! same z if ^! elem5ts 7 isolat$4 ,:at is rem>kable is t "h aga9 ,ampre makes ^! hypo!ses unconsci\sly4 ,h[ev] t may be1 ^! two hypo!ses1 toge!r ) ! exp]im5ts on clos$ curr5ts1 suffice to det]m9e completely ! law ( ! mutual action ( two elem5ts4 ,b !n mo/ ( ! simple laws we h met 9 ! case ( clos$ curr5ts are no long] true4 ,9 ! fir/ place1 !re is no electrodynamic pot5tial2 nor 0 !re any1 z we h se51 9 ! case ( a clos$ curr5t act+ on an op5 curr5t4 ,next !re is1 prop]ly speak+1 no magnetic =ce4 ,&1 9 fact1 we h giv5 above ?ree diff]5t def9itions ( ?is =ce3 #a ,by ! action on a magnetic pole2 #b ,by ! director c\ple :i* ori5tates ! magnetic ne$le2 #c ,by ! action on an elem5t ( curr5t4 ,b 9 ! case :i* n[ occupies u1 n only ^! ?ree def9itions are no long] 9 h>mony1 b ea* has lo/ its mean+1 & 9 fact3 #a ,a magnetic pole is no long] act$ ^u simply by a s+le =ce appli$ to ?is pole4 ,we h se5 9 fact t ! =ce due to ! action ( an elem5t ( curr5t on a pole is n appli$ to ! pole1 b to ! elem5t2 x may moreov] be replac$ by a =ce appli$ to ! pole & by a c\ple2 #b ,! c\ple :i* acts on ! magnetic ne$le is no long] a simple director c\ple1 = its mom5t ) respect to ! axis ( ! ne$le is n null4 ,x breaks up into a director c\ple1 prop]ly s call$1 & a supplem5t>y c\ple :i* t5ds to produce ! cont9u\s rotation ( :i* we h above spok52 #c ,f9ally ! =ce act+ on an elem5t ( curr5t is n normal to ?is elem5t4 ,9 o!r words1 ! unity ( ! magnetic =ce has disappe>$4 ,let u see 9 :at ?is unity consi/s4 ,two sy/ems :i* ex]cise ! same action on a magnetic pole w ex]t also ! same action on an indef9itely small magnetic ne$le1 or on an elem5t ( curr5t plac$ at ! same po9t ( space z ?is pole4 ,well1 ?is is true if ^! two sy/ems conta9 only clos$ curr5ts2 ?is w\ld no long] be true if ^! two sy/ems conta9$ op5 curr5ts4 ,x suffices to rem>k1 = in/ance1 t1 if a magnetic pole is plac$ at ,a & an elem5t at ,b1 ! direction ( ! elem5t be+ along ! prolongation ( ! sect ,,ab1 ?is elem5t :i* w ex]cise no action on ?is pole w1 on ! o!r h&1 ex]cise an action ei!r on a magnetic ne$le plac$ at ! po9t ,a1 or on an elem5t ( curr5t plac$ at ! po9t ,a4 #e4 ,induction4--,we "k t ! discov]y ( electrodynamic induction soon foll[$ ! immortal "w ( ,ampre4 ,z long z x is only a "q ( clos$ curr5ts !re is no difficulty1 & ,helmholtz has ev5 rem>k$ t ! pr9ciple ( ! cons]vation ( en]gy is suffici5t = d$uc+ ! laws ( induction f ! electrodynamic laws ( ,ampre4 ,b always on "o condition1 z ,b]tr& has well %[n2 t we make besides a c]ta9 numb] ( hypo!ses4 ,! same pr9ciple aga9 p]mits ?is d$uction 9 ! case ( op5 curr5ts1 al?\gh ( c\rse we c n submit ! result to ! te/ ( exp]im5t1 s9ce we c n produce su* curr5ts4 ,if we try to apply ?is mode ( analysis to ,ampre's !ory ( op5 curr5ts1 we rea* results calculat$ to surprise u4 ,9 ! fir/ place1 induction c n be d$uc$ f ! v>iation ( ! magnetic field by ! =mula well kn[n to savants & practicians1 &1 9 fact1 z we h said1 prop]ly speak+ !re is no long] a magnetic field4 ,b1 fur!r1 if a circuit ,c is subject$ to ! induction ( a v>iable voltaic sy/em ,s1 if ?is sy/em ,s be displac$ & de=m$ 9 any way :atev]1 s t ! int5sity ( ! curr5ts ( ?is sy/em v>ies accord+ to any law :atev]1 b t aft] ^! v>iations ! sy/em f9ally returns to its initial situation1 x seems natural to suppose t ! mean electromotive =ce induc$ 9 ! circuit ,c is null4 ,?is is true if ! circuit ,c is clos$ & if ! sy/em ,s conta9s only clos$ curr5ts4 ,?is w\ld no long] be true1 if "o accepts ! !ory ( ,ampre1 if !re 7 op5 curr5ts4 ,s t n only induction w no long] be ! v>iation ( ! fl[ ( magnetic =ce1 9 any ( ! usual s5ses ( ! ^w1 b x c n be repres5t$ by ! v>iation ( any?+ :atev]4 ,,ii4 ,,!ory ,,( ,,helmholtz4--,i h dwelt ^u ! consequ5ces ( ,ampre's !ory1 & ( 8 me?od ( expla9+ op5 curr5ts4 ,x is difficult to ov]look ! p>adoxical & artificial "* ( ! propositions to :i* we are ?us l$4 ,"o c n help ?9k+ 't c n be s4' ,we und]/& !re=e :y ,helmholtz 0 l$ to seek some?+ else4 ,helmholtz rejects ,ampre's fundam5tal hypo!sis1 to wit1 t ! mutual action ( two elem5ts ( curr5t r$uces to a =ce along _! jo94 ,he assumes t an elem5t ( curr5t is n subject$ to a s+le =ce1 b to a =ce & a c\ple4 ,x is j ?is :i* gave rise to ! celebrat$ polemic betwe5 ,b]tr& & ,helmholtz4 ,helmholtz replaces ,ampre's hypo!sis by ! foll[+3 two elem5ts always admit ( an electrodynamic pot5tial dep5d+ solely on _! position & ori5tation2 & ! "w ( ! =ces t !y ex]cise1 "o on ! o!r1 is equal to ! v>iation ( ?is pot5tial4 ,?us ,helmholtz c no m d )\t hypo!sis ?an ,ampre2 b at lea/ he does n make "o )\t explicitly ann\nc+ x4 ,9 ! case ( clos$ curr5ts1 :i* are alone accessible to exp]im5t1 ! two !ories agree4 ,9 all o!r cases !y diff]4 ,9 ! fir/ place1 contr>y to :at ,ampre suppos$1 ! =ce :i* seems to act on ! movable portion ( a clos$ curr5t is n ! same z w\ld act ^u ?is movable portion if x 7 isolat$ & con/itut$ an op5 curr5t4 ,let u return to ! circuit ,c'1 ( :i* we spoke above1 & :i* 0 =m$ ( a movable wire alphabeta slid+ on a fix$ wire4 ,9 ! only exp]im5t t c be made1 ! movable portion alphabeta is n isolat$1 b is "p ( a clos$ circuit4 ,:5 x passes f ,,ab to ,a',b'1 ! total electrodynamic pot5tial v>ies = two reasons3 #a ,x und]goes a fir/ increase because ! pot5tial ( ,a',b' ) respect to ! circuit ,c is n ! same z t ( ,,ab2 #b ,x takes a second increm5t because x mu/ be increas$ by ! pot5tials ( ! elem5ts ,,aa'1 ,,bb' ) respect to ,c4 ,x is ?is d\ble increm5t :i* repres5ts ! "w ( ! =ce to :i* ! portion ,,ab seems subject$4 ,if1 on ! contr>y1 alphabeta 7 isolat$1 ! pot5tial w\ld und]go only ! fir/ increase1 & ?is fir/ increm5t alone w\ld measure ! "w ( ! =ce :i* acts on ,,ab4 ,9 ! second place1 !re c\ld be no cont9u\s rotation )\t slid+ contact1 &1 9 fact1 t1 z we h se5 propos ( clos$ curr5ts1 is an imm$iate consequ5ce ( ! exi/5ce ( an electrodynamic pot5tial4 ,9 ,f>aday's exp]im5t1 if ! magnet is fix$ & if ! "p ( ! curr5t ext]ior to ! magnet runs along a movable wire1 t movable "p may und]go a cont9u\s rotation4 ,b ?is does n mean to say t if ! contacts ( ! wire ) ! magnet 7 suppress$1 & an op5 curr5t 7 to run along ! wire1 ! wire w\ld /ill take a movem5t ( cont9u\s rotation4 ,i h j said 9 fact t an isolat$ elem5t is n act$ ^u 9 ! same way z a movable elem5t mak+ "p ( a clos$ circuit4 ,ano!r diff]5ce3 ,! action ( a clos$ sol5oid on a clos$ curr5t is null accord+ to exp]im5t & accord+ to ! two !ories4 ,its action on an op5 curr5t w\ld be null accord+ to ,ampre2 x w\ld n be null accord+ to ,helmholtz4 ,f ?is foll[s an important consequ5ce4 ,we h giv5 above ?ree def9itions ( magnetic =ce4 ,! ?ird has no mean+ "h s9ce an elem5t ( curr5t is no long] act$ ^u by a s+le =ce4 ,no m has ! fir/ any mean+4 ,:at1 9 fact1 is a magnetic pole8 ,x is ! extremity ( an indef9ite l9e> magnet4 ,?is magnet may be replac$ by an indef9ite sol5oid4 ,= ! def9ition ( magnetic =ce to h any mean+1 x w\ld be necess>y t ! action ex]cis$ by an op5 curr5t on an indef9ite sol5oid %\ld dep5d only on ! position ( ! extremity ( ?is sol5oid1 t is to say1 t ! action on a clos$ sol5oid %\ld be null4 ,n[ we h j se5 t su* is n ! case4 ,on ! o!r h&1 no?+ prev5ts our adopt+ ! second def9ition1 :i* is f\nd$ on ! measurem5t ( ! director c\ple :i* t5ds to ori5tate ! magnetic ne$le4 ,b if x is adopt$1 nei!r ! effects ( induction nor ! electrodynamic effects w dep5d solely on ! di/ribution ( ! l9es ( =ce 9 ?is magnetic field4 ,,iii4 ,,difficulties ,,rais$ ,,by ,,^! ,,!ories4--,! !ory ( ,helmholtz is 9 advance ( t ( ,ampre2 x is necess>y1 h[ev]1 t all ! difficulties %\ld be smoo!d away4 ,9 ! "o z 9 ! o!r1 ! phrase 'magnetic field' has no mean+1 or1 if we give x "o1 by a m or less artificial conv5tion1 ! ord9>y laws s famili> to all electricians no long] apply2 ?us ! electromotive =ce induc$ 9 a wire is no long] measur$ by ! numb] ( l9es ( =ce met by ?is wire4 ,& our repugnance does n come alone f ! difficulty ( r5\nc+ invet]ate habits ( language & ( ?\ght4 ,!re is some?+ m4 ,if we d n believe 9 action at a di/ance1 electrodynamic ph5om5a mu/ be expla9$ by a modification ( ! m$ium4 ,x is precisely ?is modification t we call 'magnetic field4' ,& !n ! electrodynamic effects mu/ dep5d only on ?is field4 ,all ^! difficulties arise f ! hypo!sis ( op5 curr5ts4 ,,iv4 ,,maxwell',s ,,!ory4--,su* 7 ! difficulties rais$ by ! dom9ant !ories :5 ,maxwell appe>$1 :o ) a /roke ( ! p5 made !m all vani%4 ,to 8 m9d1 9 fact1 all curr5ts are clos$ curr5ts4 ,maxwell assumes t if 9 a dielectric ! electric field happ5s to v>y1 ?is dielectric becomes ! seat ( a p>ticul> ph5om5on1 act+ on ! galvanomet] l a curr5t1 & :i* he calls curr5t ( displacem5t4 ,if !n two conductors be>+ contr>y *>ges are put 9 communication by a wire1 9 ?is wire dur+ ! dis*>ge !re is an op5 curr5t ( conduction2 b !re are produc$ at ! same "t 9 ! surr\nd+ dielectric1 curr5ts ( displacem5t :i* close ?is curr5t ( conduction4 ,we "k t ,maxwell's !ory leads to ! explanation ( optical ph5om5a1 :i* w\ld be due to extremely rapid electrical oscillations4 ,at t epo* su* a conception 0 only a bold hypo!sis1 :i* c\ld be support$ by no exp]im5t4 ,at ! end ( tw5ty ye>s1 ,maxwell's ideas receiv$ ! confirmation ( exp]im5t4 ,h]tz succe$$ 9 produc+ sy/ems ( electric oscillations :i* reproduce all ! prop]ties ( light1 & only diff] f x by ! l5g? ( _! wave2 t is to say z violet diff]s f r$4 ,9 "s measure he made ! syn!sis ( light4 ,x might be said t ,h]tz has n demon/rat$ directly ,maxwell's fundam5tal idea1 ! action ( ! curr5t ( displacem5t on ! galvanomet]4 ,?is is true 9 a s5se4 ,:at he has %[n 9 sum is t electromagnetic induction is n propagat$ in/antane\sly z 0 suppos$2 b ) ! spe$ ( light4 ,b to suppose !re is no curr5t ( displacem5t1 & induction is propagat$ ) ! spe$ ( light2 or to suppose t ! curr5ts ( displacem5t produce effects ( induction1 & t ! induction is propagat$ in/antane\sly1 comes to ! same ?+4 ,?is c n be se5 at ! fir/ glance1 b x is prov$ by an analysis ( :i* ,i mu/ n ?9k ( giv+ ev5 a summ>y "h4 ,v4 ,,r[l&',s ,,exp]im5t4--,b z ,i h said above1 !re are two k9ds ( op5 conduction curr5ts4 ,!re are fir/ ! curr5ts ( dis*>ge ( a cond5s] or ( any conductor :atev]4 ,!re are also ! cases 9 :i* electric dis*>ges describe a clos$ cont\r1 be+ displac$ by conduction 9 "o "p ( ! circuit & by convection 9 ! o!r "p4 ,= op5 curr5ts ( ! fir/ sort1 ! "q might be consid]$ z solv$2 !y 7 clos$ by ! curr5ts ( displacem5t4 ,= op5 curr5ts ( ! second sort1 ! solution appe>$ /ill m simple4 ,x seem$ t if ! curr5t 7 clos$1 x c\ld only be by ! curr5t ( convection itself4 ,= t x suffic$ to assume t a 'convection curr5t1' t is to say a *>g$ conductor 9 motion1 c\ld act on ! galvanomet]4 ,b exp]im5tal confirmation 0 lack+4 ,x appe>$ difficult 9 fact to obta9 a suffici5t int5sity ev5 by augm5t+ z mu* z possible ! *>ge & ! velocity ( ! conductors4 ,x 0 ,r[l&1 an extremely skillful exp]im5t]1 :o fir/ triumph$ ov] ^! difficulties4 ,a disc receiv$ a /rong electro/atic *>ge & a v great spe$ ( rotation4 ,an a/atic magnetic sy/em plac$ beside ! disc und]w5t deviations4 ,! exp]im5t 0 made twice by ,r[l&1 once 9 ,b]l91 once 9 ,baltimore4 ,x 0 aft]w>ds repeat$ by ,him/$t4 ,^! physici/s ev5 ann\nc$ t !y _h succe$$ 9 mak+ quantitative measurem5ts4 ,9 fact1 = tw5ty ye>s ,r[l&'s law 0 admitt$ )\t objection by all physici/s4 ,besides ev]y?+ seem$ to confirm x4 ,! sp>k c]ta9ly does produce a magnetic effect4 ,n[ does x n seem probable t ! dis*>ge by sp>k is due to p>ticles tak5 f "o ( ! electrodes & transf]r$ to ! o!r electrode ) _! *>ge8 ,is n ! v spectrum ( ! sp>k1 9 :i* we recognize ! l9es ( ! metal ( ! electrode1 a pro( ( x8 ,! sp>k w\ld !n be a v]itable curr5t ( convection4 ,on ! o!r h&1 x is also admitt$ t 9 an electrolyte ! electricity is c>ri$ by ! ions 9 motion4 ,! curr5t 9 an electrolyte w\ld !re=e be also a curr5t ( convection2 n[1 x acts on ! magnetic ne$le4 ,! same = ca?ode rays4 ,crookes attribut$ ^! rays to a v subtile matt] *>g$ ) electricity & mov+ ) a v great velocity4 ,he reg>d$ !m1 9 o!r words1 z curr5ts ( convection4 ,n[ ^! ca?ode rays are deviat$ by ! magnet4 ,9 virtue ( ! pr9ciple ( action & reaction1 !y %\ld 9 turn deviate ! magnetic ne$le4 ,x is true t ,h]tz believ$ he _h demon/rat$ t ! ca?ode rays d n c>ry electricity1 & t !y d n act on ! magnetic ne$le4 ,b ,h]tz 0 mi/ak54 ,fir/ ( all1 ,p]r9 succe$$ 9 collect+ ! electricity c>ri$ by ^! rays1 electricity ( :i* ,h]tz d5i$ ! exi/5ce2 ! ,g]man sci5ti/ appe>s to h be5 deceiv$ by effects due to ! action ( ,x-rays1 :i* 7 n yet discov]$4 ,aft]w>ds1 & q rec5tly1 ! action ( ! ca?ode rays on ! magnetic ne$le has be5 put 9 evid5ce4 ,?us all ^! ph5om5a reg>d$ z curr5ts ( convection1 sp>ks1 electrolytic curr5ts1 ca?ode rays1 act 9 ! same mann] on ! galvanomet] & 9 con=mity ) ,r[l&'s law4 ,,vi4 ,,!ory ,,( ,,lor5tz4--,we soon w5t f>!r4 ,accord+ to ! !ory ( ,lor5tz1 curr5ts ( conduction !mselves w\ld be true curr5ts ( convection4 ,electricity w\ld rema9 insep>ably connect$ ) c]ta9 mat]ial p>ticles call$ electrons4 ,! circulation ( ^! electrons ?r\gh bodies w\ld produce voltaic curr5ts4 ,& :at w\ld di/+ui% conductors f insulators w\ld be t ! "o c\ld be trav]s$ by ^! electrons :ile ! o!rs w\ld arre/ _! movem5ts4 ,! !ory ( ,lor5tz is v attractive4 ,x gives a v simple explanation ( c]ta9 ph5om5a :i* ! e>li] !ories1 ev5 ,maxwell's 9 its primitive =m1 c\ld n expla9 9 a satisfactory way2 = example1 ! ab]ration ( light1 ! p>tial c>ry+ away ( lum9\s waves1 magnetic pol>ization & ! ,zeeman effect4 ,"s objections /ill rema9$4 ,! ph5om5a ( an electric sy/em seem$ to dep5d on ! absolute velocity ( translation ( ! c5t] ( gravity ( ?is sy/em1 :i* is contr>y to ! idea we h ( ! relativity ( space4 ,support$ by ,m4 ,crmieu1 ,m4 ,lippmann has pres5t$ ?is objection 9 a /rik+ =m4 ,imag9e two *>g$ conductors ) ! same velocity ( translation2 !y are relatively at re/4 ,h[ev]1 ea* ( !m be+ equival5t to a curr5t ( convection1 !y ought to attract "o ano!r1 & by measur+ ?is attraction we c\ld measure _! absolute velocity4 ,no6 repli$ ! p>tisans ( ,lor5tz4 ,:at we c\ld measure 9 t way is n _! absolute velocity1 b _! relative velocity ) respect to ! e!r1 s t ! pr9ciple ( relativity is safe4 ,:atev] !re may be 9 ^! latt] objections1 ! edifice ( electrodynamics1 at lea/ 9 its broad l9es1 seem$ def9itively con/ruct$4 ,ev]y?+ 0 pres5t$ "u ! mo/ satisfactory aspect4 ,! !ories ( ,ampre & ( ,helmholtz1 made = op5 curr5ts :i* no long] exi/$1 seem$ to h no long] any?+ b a purely hi/oric int]e/1 & ! inextricable complications to :i* ^! !ories l$ 7 almo/ =gott54 ,?is quiesc5ce has be5 rec5tly di/urb$ by ! exp]im5ts ( ,m4 ,crmieu1 :i* = a mom5t seem$ to contradict ! result previ\sly obta9$ by ,r[l&4 ,b fre% rese>*es h n confirm$ !m1 & ! !ory ( ,lor5tz has victori\sly /ood ! te/4 ,! hi/ory ( ^! v>iations w be none ! less in/ructive2 x w tea* u to :at pitfalls ! sci5ti/ is expos$1 & h[ he may hope to escape !m4 99 99 99 99 99 ,,! ,,value ,,( ,,sci5ce 99 99 99 99 99 ,,translator',s ,,introduction #a4 ,does ! ,sci5ti/ create ,sci5ce8--,pr(essor ,rados ( ,budape/ 9 8 report to ! ,hung>ian ,academy ( ,sci5ce on ! aw>d to ,po9c> ( ! ,bolyai prize ( t5 ?\s& cr[ns1 speak+ ( him z unque/ionably ! mo/ p[]ful inve/igator 9 ! doma9 ( ma!matics & ma!matical physics1 *>act]iz$ him z ! intuitive g5ius draw+ ! inspiration = 8 wide-rea*+ rese>*es f ! exhau/less f\nta9 ( geometric & physical intuition1 yet work+ ?is inspiration out 9 detail ) m>vel\s logical ke5ness4 ,) 8 brilliant creative g5ius 0 comb9$ ! capacity = %>p & successful g5]alization1 pu%+ f> out ! b\nd>ies ( ?\ght 9 ! mo/ widely diff]5t doma9s1 s t 8 works mu/ be rank$ ) ! greate/ ma!matical a*ievem5ts ( all "t4 ,f9ally1 says ,rados1 p]mit me to make especial m5tion ( 8 int5sely int]e/+ book1 ',! ,value ( ,sci5ce1' 9 :i* he 9 a way has laid d[n ! sci5ti/'s cre$4 ,n[ :at is ?is cre$8 ,s5se may act z /imulus1 z sugge/ive1 yet n to awak5 a dormant depiction1 or to educe ! conception ( an ar*etypal =m1 b r to /rike ! h\r = creation1 to summon to "w a sculptor capable ( smoo?+ a ,v5us ( ,milo out ( ! =mless clay4 ,k is n a gift ( b>e exp]i5ce1 nor ev5 made solely out ( exp]i5ce4 ,! creative activity ( m9d is 9 ma!matics p>ticul>ly cle>4 ,! axioms ( geometry are conv5tions1 disguis$ def9itions or unprovable hypo!ses precreat$ by auto-active animal & human m9ds4 ,b]tr& ,russell says ( projective geometry3 ,x takes no?+ f exp]i5ce1 & has1 l ari?metic1 a creature ( ! pure intellect = its object4 ,x deals ) an object ^: prop]ties are logically d$uc$ f its def9ition1 n empirically discov]$ f data4 ,!n does ! sci5ti/ create sci5ce8 ,?is is a "q ,po9c> "h dissects ) a ma/] h&4 ,! physiologic-psy*ologic inve/igation ( ! space problem mu/ give ! mean+ ( ! words geometric fact1 geometric reality4 ,po9c> "h subjects to ! mo/ successful analysis ev] made ! tridim5sionality ( our space4 #b4 ,! ,m9d ,dispell+ ,optical ,illusions4--,actual p]ception ( spatial prop]ties is accompani$ by movem5ts correspond+ to its "*4 ,9 ! case ( optical illusions1 ) ! s-call$ false p]ceptions eye-movem5ts are closely relat$4 ,b ?\gh ! p]ceiv$ object & its environm5t rema9 con/ant1 ! suffici5tly p[]ful m9d c1 z we say1 dispel ^! illusions1 ! p]ception itself be+ creatively *ang$4 ,photo-graphs tak5 at int]vals dur+ ! pres5ce ( ^! optical illusions1 dur+ ! *ange1 p]haps gradual & unconsci\s1 9 ! p]ception1 & aft] ^! illusions h1 z ! phrase is1 f9ally disappe>$1 %[ q cle>ly t *anges 9 eye-movem5ts correspond+ to ^? int]nally creat$ 9 p]ception itself successively occur4 ,:at is call$ accuracy ( movem5t is creat$ by :at is call$ correctness ( p]ception4 ,! high] creation 9 ! p]ception is ! det]m9+ cause ( an improvem5t1 a precision 9 ! motion4 ,?us we see correct p]ception 9 ! individual help+ to make t c]ebral organization & accurate motor adju/m5t on :i* its possibility & p]man5ce seem 9 s f> to dep5d4 ,s-call$ correct p]ception is connect$ ) a long-cont9u$ process ( p]ceptual education motiv$ & initiat$ f )94 ,h[ ?is may take place is "h illu/rat$ at l5g? by our au?or4 #c4 ,euclid n ,necess>y4--,geometry is a con/ruction ( ! intellect1 9 application n c]ta9 b conv5i5t4 ,z ,s*ill] says1 :5 we see ^! facts z cle>ly z ! developm5t ( metageometry has compell$ u to see !m1 we mu/ surely confess t ! ,kantian acc\nt ( space is hopelessly & demon/rably antiquat$4 ,z ,royce says 9 ',kant's ,doctr9e ( ! ,basis ( ,ma!matics1' ,t v use ( intuition :i* ,kant reg>d$ z geometrically ideal1 ! mod]n geomet] reg>ds z sci5tifically defective1 because surreptiti\s4 ,no ma!matical exactness )\t explicit pro( f assum$ pr9ciples--su* is ! motto ( ! mod]n geomet]4 ,b suppose ! reason+ ( ,euclid purifi$ ( ?is comp>atively surreptiti\s appeal to intuition4 ,suppose t ! pr9ciples ( geometry are made q explicit at ! outset ( ! treatise1 z ,pi]i & ,hilb]t or ,pr(essor ,hal/$ or ,dr4 ,vebl5 makes 8 pr9ciples explicit 9 8 rec5t treatm5t ( geometry4 ,!n1 inde$1 geometry becomes = ! mod]n ma!matician a purely rational sci5ce4 ,b v few /ud5ts ( ! logic ( ma!matics at ! pres5t "t c see any w>rant 9 ! analysis ( geometrical tru? = reg>d+ j ! ,euclidean sy/em ( pr9ciples z possess+ any discov]able necessity4 ,yet ! environm5tal & p]haps h]$it>y premiums on ,euclid /ill make ev5 ! sci5ti/ ?9k ,euclid mo/ conv5i5t4 #d4 ,)\t ,hypo!ses1 no ,sci5ce4--,nobody ev] obs]v$ an equidi/antial1 b also nobody ev] obs]v$ a /raight l9e4 ,em]son's ,uriel ,gave 8 s5tim5t div9e ,aga9/ ! be+ ( a l9e4 ,l9e 9 ,nature is n f\nd4 ,cle>ly n1 be+ an eject f man's m9d4 ,:at is call$ 'a k ( facts' is usually m]ely a subjective realization t ! old hypo!ses are /ill suffici5tly ela/ic to s]ve 9 "s doma92 t is1 ) a suffici5cy ( consci\s or unconsci\s omissions & doctor+s & fudg+s m or less wilful4 ,9 ! pres5t book we see ! v f\ndation rocks ( sci5ce1 ! cons]vation ( en]gy & ! inde/ructibility ( matt]1 beat+ aga9/ ! b>s ( _! cages1 seem+ly anxi\s to take w+ away into ! empyrean1 to *ase ! once div9e p>allel po/ulate brok5 loose f ,euclid & ,kant4 #e4 ,:at ,outcome8--,:at n[ is ! def9ite1 ! p]man5t outcome8 ,:at new islets raise _! frond$ palms 9 air )9 ?\ght's musical doma98 ,ov] :at age-gray b>ri]s rise ! fragrant floods ( ?is new spr+-tide1 r$ol5t ( ! wolf-haunt$ =e/ ( ,transylvania1 ( f> ,erdly's plung+ riv]1 ,m>os ! bitt]1 or broad "m ,volga at ,kazan8 ,:at victory h]ald$ ! great rocket = :i* "y ,loba*evski1 ! wid['s son1 0 ca/ into prison8 ,:at sev]+ ( age-old m5tal fett]s symboliz$ "y ,bolyai's cutt+-(f ) 8 ,damascus blade ! spikes driv5 into 8 door-po/1 & /rew+ ov] ! sod ! ?irte5 ,au/rian cavalry (fic]s8 ,?is book by ! greate/ ma!matician ( our "t gives weightie/ & mo/ *>m+ answ]4 ,,george ,,bruce ,,hal/$4 ,,introduction ,! se>* = tru? %\ld be ! goal ( our activities2 x is ! sole end wor?y ( !m4 ,d\btless we %\ld fir/ b5d our ef=ts to assuage human suff]+1 b :y8 ,n to suff] is a negative ideal m surely atta9$ by ! annihilation ( ! _w4 ,if we wi% m & m to free man f mat]ial c>es1 x is t he may be able to employ ! lib]ty obta9$ 9 ! /udy & contemplation ( tru?4 ,b sometimes tru? fright5s u4 ,& 9 fact we "k t x is sometimes deceptive1 t x is a phantom nev] %[+ itself = a mom5t except to ceaselessly flee1 t x mu/ be pursu$ fur!r & ev] fur!r )\t ev] be+ atta9$4 ,yet to "w "o mu/ /op1 z "s ,greek1 ,ari/otle or ano!r1 has said4 ,we also "k h[ cruel ! tru? (t5 is1 & we wond] :e!r illusion is n m consol+1 yea1 ev5 m brac+1 = illusion x is :i* gives confid5ce4 ,:5 x %all h vani%$1 w hope rema9 & %all we h ! c\rage to a*ieve8 ,?us w\ld n ! horse h>ness$ to 8 treadmill refuse to g1 7 8 eyes n b&ag$8 ,& !n to seek tru? x is necess>y to be indep5d5t1 :olly indep5d5t4 ,if1 on ! contr>y1 we wi% to act1 to be /rong1 we %\ld be unit$4 ,?is is :y _m ( u fe> tru?2 we consid] x a cause ( weakness4 ,yet tru? %\ld n be fe>$1 = x alone is beautiful4 ,:5 ,i speak "h ( tru?1 assur$ly ,i ref] fir/ to sci5tific tru?2 b ,i also mean moral tru?1 ( :i* :at we call ju/ice is only "o aspect4 ,x may seem t ,i am misus+ words1 t ,i comb9e ?us "u ! same "n two ?+s hav+ no?+ 9 common2 t sci5tific tru?1 :i* is demon/rat$1 c 9 no way be lik5$ to moral tru?1 :i* is felt4 ,& yet ,i c n sep>ate !m1 & :osoev] loves ! "o c n help lov+ ! o!r4 ,to f9d ! "o1 z well z to f9d ! o!r1 x is necess>y to free ! s\l completely f prejudice & f passion2 x is necess>y to atta9 absolute s9c]ity4 ,^! two sorts ( tru? :5 discov]$ give ! same joy2 ea* :5 p]ceiv$ beams ) ! same spl5dor1 s t we mu/ see x or close our eyes4 ,la/ly1 bo? attract u & flee f u2 !y are nev] fix$3 :5 we ?9k to h rea*$ !m1 we f9d t we h /ill to advance1 & he :o pursues !m is condemn$ nev] to "k repose4 ,x mu/ be add$ t ^? :o fe> ! "o w also fe> ! o!r2 = !y are ! ones :o 9 ev]y?+ are conc]n$ above all ) consequ5ces4 ,9 a ^w1 ,i lik5 ! two tru?s1 because ! same reasons make u love !m & because ! same reasons make u fe> !m4 ,if we ought n to fe> moral tru?1 /ill less %\ld we dread sci5tific tru?4 ,9 ! fir/ place x c n conflict ) e?ics4 ,e?ics & sci5ce h _! own doma9s1 :i* t\* b d n int]p5etrate4 ,! "o %[s u to :at goal we %\ld aspire1 ! o!r1 giv5 ! goal1 tea*es u h[ to atta9 x4 ,s !y c nev] conflict s9ce !y c nev] meet4 ,!re c no m be immoral sci5ce ?an !re c be sci5tific morals4 ,b if sci5ce is fe>$1 x is above all because x c n give u happ9ess4 ,( c\rse x c n4 ,we may ev5 ask :e!r ! bea/ does n suff] less ?an man4 ,b c we regret t e>?ly p>adise :]e man brute-l 0 really immortal 9 kn[+ n t he mu/ die8 ,:5 we h ta/$ ! apple1 no suff]+ c make u =get its savor4 ,we always come back to x4 ,c\ld x be o!rwise8 ,z well ask if "o :o has se5 & is bl9d w n long = ! light4 ,man1 !n1 c n be happy ?r\gh sci5ce1 b to-day he c mu* less be happy )\t x4 ,b if tru? be ! sole aim wor? pursu+1 may we hope to atta9 x8 ,x may well be d\bt$4 ,read]s ( my little book ',sci5ce & ,hypo!sis' already "k :at ,i ?9k ab\t ! "q4 ,! tru? we are p]mitt$ to glimpse is n altoge!r :at mo/ m5 call by t "n4 ,does ?is mean t our mo/ legitimate1 mo/ imp]ative aspiration is at ! same "t ! mo/ va98 ,or c we1 despite all1 approa* tru? on "s side8 ,?is x is :i* mu/ be inve/igat$4 ,9 ! fir/ place1 :at in/rum5t h we at our disposal = ?is conque/8 ,is n human intellig5ce1 m specifically ! intellig5ce ( ! sci5ti/1 susceptible ( inf9ite v>iation8 ,volumes c\ld be writt5 )\t exhau/+ ?is subject2 ,i1 9 a few brief pages1 h only t\*$ x lightly4 ,t ! geomet]'s m9d is n l ! physici/'s or ! naturali/'s1 all ! _w w\ld agree2 b ma!maticians !mselves d n resemble ea* o!r2 "s recognize only implacable logic1 o!rs appeal to intuition & see 9 x ! only s\rce ( discov]y4 ,& ?is w\ld be a reason = di/ru/4 ,to m9ds s unlike c ! ma!matical !orems !mselves appe> 9 ! same light8 ,tru? :i* is n ! same = all1 is x tru?8 ,b look+ at ?+s m closely1 we see h[ ^! v diff]5t work]s collaborate 9 a common task :i* c\ld n be a*iev$ )\t _! coop]ation4 ,& t already reassures u4 ,next mu/ be exam9$ ! frames 9 :i* nature seems enclos$ & :i* are call$ "t & space4 ,9 ',sci5ce & ,hypo!sis' ,i h already %[n h[ relative _! value is2 x is n nature :i* imposes !m ^u u1 x is we :o impose !m ^u nature because we f9d !m conv5i5t4 ,b ,i h spok5 ( sc>cely m ?an space1 & p>ticul>ly quantitative space1 s to say1 t is ( ! ma!matical relations ^: aggregate con/itutes geometry4 ,i %\ld h %[n t x is ! same ) "t z ) space & /ill ! same ) 'qualitative space'2 9 p>ticul>1 ,i %\ld h inve/igat$ :y we attribute ?ree dim5sions to space4 ,i may be p>don$ !n = tak+ up aga9 ^! important que/ions4 ,is ma!matical analysis1 !n1 ^: pr9cipal object is ! /udy ( ^! empty frames1 only a va9 play ( ! m9d8 ,x c give to ! physici/ only a conv5i5t language2 is ?is n a m$iocre s]vice1 :i*1 /rictly speak+1 c\ld be done )\t2 & ev5 is x n to be fe>$ t ?is artificial language may be a veil int]pos$ betwe5 reality & ! eye ( ! physici/8 ,f> f x2 )\t ?is language 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* is1 we %all see1 ! only true objective reality4 ,! be/ expression ( ?is h>mony is law4 ,law is "o ( ! mo/ rec5t conque/s ( ! human m9d2 !re /ill are p :o live 9 ! pres5ce ( a p]petual miracle & are n a/oni%$ at x4 ,on ! contr>y1 we x is :o %\ld be a/oni%$ at nature's regul>ity4 ,m5 dem& ( _! gods to prove _! exi/5ce by miracles2 b ! et]nal m>vel is t !re are n miracles )\t cease4 ,! _w is div9e because x is a h>mony4 ,if x 7 rul$ by caprice1 :at c\ld prove to u x 0 n rul$ by *ance8 ,?is conque/ ( law we owe to a/ronomy1 & j ?is makes ! gr&eur ( ! sci5ce r ?an ! mat]ial gr&eur ( ! objects x consid]s4 ,x 0 altoge!r natural1 !n1 t cele/ial me*anics %\ld be ! fir/ model ( ma!matical physics2 b s9ce !n ?is sci5ce has develop$2 x is /ill develop+1 ev5 rapidly develop+4 ,& x is already necess>y to modify 9 c]ta9 po9ts ! s*eme f :i* ,i drew two *apt]s ( ',sci5ce & ,hypo!sis4' ,9 an address at ! ,/4 ,l\is exposition1 ,i s\ght to survey ! road travel$2 ! result ( ?is inve/igation ! read] %all see f>!r on4 ,! progress ( sci5ce has seem$ to imp]il ! be/ e/abli%$ pr9ciples1 ^? ev5 :i* 7 reg>d$ z fundam5tal4 ,yet no?+ %[s !y w n be sav$2 & if ?is comes ab\t only imp]fectly1 !y w /ill subsi/ ev5 ?\gh !y are modifi$4 ,! advance ( sci5ce is n comp>able to ! *anges ( a city1 :]e old edifices are pitilessly torn d[n to give place to new1 b to ! cont9u\s evolution ( zoologic types :i* develop ceaselessly & end by becom+ unrecognizable to ! common sight1 b :]e an exp]t eye f9ds always traces ( ! prior "w ( ! c5turies pa/4 ,"o mu/ n ?9k !n t ! old-fa%ion$ !ories h be5 /]ile & va94 ,7 we to /op !re1 we %\ld f9d 9 ^! pages "s reasons = confid5ce 9 ! value ( sci5ce1 b _m m = di/ru/+ x2 an impression ( d\bt w\ld rema92 x is ne$ful n[ to set ?+s to rights4 ,"s p h exagg]at$ ! rle ( conv5tion 9 sci5ce2 !y h ev5 gone s f> z to say t law1 t sci5tific fact itself1 0 creat$ by ! sci5ti/4 ,?is is go+ mu* too f> 9 ! direction ( nom9alism4 ,no1 sci5tific laws are n artificial creations2 we h no reason to reg>d !m z accid5tal1 ?\gh x be impossible to prove !y are n4 ,does ! h>mony ! human intellig5ce ?9ks x discov]s 9 nature exi/ outside ( ?is intellig5ce8 ,no1 beyond d\bt a reality completely indep5d5t ( ! m9d :i* conceives x1 sees or feels x1 is an impossibility4 ,a _w z ext]ior z t1 ev5 if x exi/$1 w\ld = u be =ev] inaccessible4 ,b :at we call objective reality is1 9 ! la/ analysis1 :at is common to _m ?9k+ be+s1 & c\ld be common to all2 ?is common "p1 we %all see1 c only be ! h>mony express$ by ma!matical laws4 ,x is ?is h>mony !n :i* 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 ,,"p ,i ,,! ,,ma!matical ,,sci5ces ,,*apt] ,i ,,intuition ,,& ,,logic ,,9 ,,ma!matics ,i ,x is impossible to /udy ! works ( ! great ma!maticians1 or ev5 ^? ( ! less]1 )\t notic+ & di/+ui%+ two opposite t5d5cies1 or r two entirely diff]5t k9ds ( m9ds4 ,! "o sort are above all preoccupi$ ) logic2 to read _! works1 "o is tempt$ to believe !y h advanc$ only /ep by /ep1 aft] ! mann] ( a ,vauban :o pu%es on 8 tr5*es aga9/ ! place besieg$1 leav+ no?+ to *ance4 ,! o!r sort are guid$ by intuition & at ! fir/ /roke make quick b sometimes prec>i\s conque/s1 l bold cavalrym5 ( ! advance gu>d4 ,! me?od is n impos$ by ! matt] treat$4 ,?\gh "o (t5 says ( ! fir/ t !y are analy/s & calls ! o!rs geomet]s1 t does n prev5t ! "o sort f rema9+ analy/s ev5 :5 !y "w at geometry1 :ile ! o!rs are /ill geomet]s ev5 :5 !y occupy !mselves ) pure analysis4 ,x is ! v nature ( _! m9d :i* makes !m logicians or intuitionali/s1 & !y c n lay x aside :5 !y approa* a new subject4 ,nor is x education :i* has develop$ 9 !m "o ( ! two t5d5cies & /ifl$ ! o!r4 ,! ma!matician is born1 n made1 & x seems he is born a geomet] or an analy/4 ,i %\ld l to cite examples & !re are surely pl5ty2 b to acc5tuate ! contra/ ,i %all beg9 ) an extreme example1 tak+ ! lib]ty ( seek+ x 9 two liv+ ma!maticians4 ,m4 ,mray wants to prove t a b9omial equation always has a root1 or1 9 ord9>y words1 t an angle may always be subdivid$4 ,if !re is any tru? t we ?9k we "k by direct intuition1 x is ?is4 ,:o c\ld d\bt t an angle may always be divid$ into any numb] ( equal p>ts8 ,m4 ,mray does n look at x t way2 9 8 eyes ?is proposition is n at all evid5t & to prove x he ne$s sev]al pages4 ,on ! o!r h&1 look at ,pr(essor ,kle93 he is /udy+ "o ( ! mo/ ab/ract que/ions ( ! !ory ( functions3 to det]m9e :e!r on a giv5 ,riemann surface !re always exi/s a function admitt+ ( giv5 s+ul>ities4 ,:at does ! celebrat$ ,g]man geomet] d8 ,he replaces 8 ,riemann surface by a metallic surface ^: electric conductivity v>ies accord+ to c]ta9 laws4 ,he connects two ( its po9ts ) ! two poles ( a batt]y4 ,! curr5t1 says he1 mu/ pass1 & ! di/ribution ( ?is curr5t on ! surface w def9e a function ^: s+ul>ities w be precisely ^? call$ = by ! enunciation4 ,d\btless ,pr(essor ,kle9 well kn[s he has giv5 "h only a sket*2 nev]!less he has n hesitat$ to publi% x2 & he w\ld probably believe he f9ds 9 x1 if n a rigor\s demon/ration1 at lea/ a k9d ( moral c]ta9ty4 ,a logician w\ld h reject$ ) horror su* a conception1 or r he w\ld n h _h to reject x1 because 9 8 m9d x w\ld nev] h orig9at$4 ,aga91 p]mit me to comp>e two m51 ! honor ( ,fr5* sci5ce1 :o h rec5tly be5 tak5 f u1 b :o bo? ent]$ long ago into immortality4 ,i speak ( ,m4 ,b]tr& & ,m4 ,h]mite4 ,!y 7 s*ol>s ( ! same s*ool at ! same "t2 !y _h ! same education1 7 "u ! same influ5ces2 & yet :at a diff]5ce6 ,n only does x blaze =? 9 _! writ+s2 x is 9 _! tea*+1 9 _! way ( speak+1 9 _! v look4 ,9 ! memory ( all _! pupils ^! two faces are /amp$ 9 dea?less l9es2 = all :o h _h ! pleasure ( foll[+ _! tea*+1 ?is remembrance is /ill fre%2 x is easy = u to evoke x4 ,:ile speak+1 ,m4 ,b]tr& is always 9 motion2 n[ he seems 9 combat ) "s outside enemy1 n[ he outl9es ) a ge/ure ( ! h& ! figures he /udies4 ,pla9ly he sees & he is eag] to pa9t1 ?is is :y he calls ge/ure to 8 aid4 ,) ,m4 ,h]mite1 x is j ! opposite2 8 eyes seem to %un contact ) ! _w2 x is n )\t1 x is )9 he seeks ! vision ( tru?4 ,among ! ,g]man geomet]s ( ?is c5tury1 two names above all are illu/ri\s1 ^? ( ! two sci5ti/s :o f\nd$ ! g5]al !ory ( functions1 ,wei]/rass & ,riemann4 ,wei]/rass leads ev]y?+ back to ! consid]ation ( s]ies & _! analytic trans=mations2 to express x bett]1 he r$uces analysis to a sort ( prolongation ( ari?metic2 y may turn ?r\gh all 8 books )\t f9d+ a figure4 ,riemann1 on ! contr>y1 at once calls geometry to 8 aid2 ea* ( 8 conceptions is an image t no "o c =get1 once he has caught its mean+4 ,m rec5tly1 ,lie 0 an intuitionali/2 ?is might h be5 d\bt$ 9 read+ 8 books1 no "o c\ld d\bt x aft] talk+ ) him2 y saw at once t he ?\ght 9 pictures4 ,madame ,kovalevski 0 a logician4 ,among our /ud5ts we notice ! same diff]5ces2 "s pref] to treat _! problems 'by analysis1' o!rs 'by geometry4' ,! fir/ are incapable ( 'see+ 9 space1' ! o!rs are quickly tir$ ( long calculations & become p]plex$4 ,! two sorts ( m9ds are equally necess>y = ! progress ( sci5ce2 bo? ! logicians & ! intuitionali/s h a*iev$ great ?+s t o!rs c\ld n h done4 ,:o w\ld v5ture to say :e!r he pref]r$ t ,wei]/rass _h nev] writt5 or t !re _h nev] be5 a ,riemann8 ,analysis & syn!sis h !n bo? _! legitimate rles4 ,b x is int]e/+ to /udy m closely 9 ! hi/ory ( sci5ce ! "p :i* belongs to ea*4 ,,ii ,/range6 ,if we read ov] ! works ( ! anci5ts we are tempt$ to class !m all among ! intuitionali/s4 ,& yet nature is always ! same2 x is h>dly probable t x has begun 9 ?is c5tury to create m9ds devot$ to logic4 ,if we c\ld put ourselves into ! fl[ ( ideas :i* reign$ 9 _! "t1 we %\ld recognize t _m ( ! old geomet]s 7 9 t5d5cy analy/s4 ,euclid1 = example1 erect$ a sci5tific /ructure :]e9 8 contempor>ies c\ld f9d no fault4 ,9 ?is va/ con/ruction1 ( :i* ea* piece h[ev] is due to intuition1 we may /ill to-day1 )\t mu* ef=t1 recognize ! "w ( a logician4 ,x is n m9ds t h *ang$1 x is ideas2 ! intuitional m9ds h rema9$ ! same2 b _! read]s h requir$ ( !m great] concessions4 ,:at is ! cause ( ?is evolution8 ,x is n h>d to f9d4 ,intuition c n give u rigor1 nor ev5 c]ta9ty2 ?is has be5 recogniz$ m & m4 ,let u cite "s examples4 ,we "k !re exi/ cont9u\s functions lack+ d]ivatives4 ,no?+ is m %ock+ to intuition ?an ?is proposition :i* is impos$ ^u u by logic4 ,our fa!rs w\ld n h fail$ to say3 ,x is evid5t t e cont9u\s function has a d]ivative1 s9ce e curve has a tang5t4 ,h[ c intuition deceive u on ?is po9t8 ,x is because :5 we seek to imag9e a curve we c n repres5t x to ourselves )\t wid?2 j s1 :5 we repres5t to ourselves a /raight l9e1 we see x "u ! =m ( a rectil9e> b& ( a c]ta9 bread?4 ,we well "k ^! l9es h no wid?2 we try to imag9e !m n>r[] & n>r[] & ?us to approa* ! limit2 s we d 9 a c]ta9 measure1 b we %all nev] atta9 ?is limit4 ,& !n x is cle> we c always picture ^! two n>r[ b&s1 "o /raight1 "o curv$1 9 a position su* t !y encroa* slightly "o ^u ! o!r )\t cross+4 ,we %all ?us be l$1 unless w>n$ by a rigor\s analysis1 to conclude t a curve always has a tang5t4 ,i %all take z second example ,diri*let's pr9ciple on :i* re/ s _m !orems ( ma!matical physics2 to-day we e/abli% x by reason+ v rigor\s b v long2 h]et(ore1 on ! contr>y1 we 7 cont5t ) a v summ>y pro(4 ,a c]ta9 integral dep5d+ on an arbitr>y function c nev] vani%4 ,h5ce x is conclud$ t x mu/ h a m9imum4 ,! flaw 9 ?is reason+ /rikes u imm$iately1 s9ce we use ! ab/ract t]m function & are famili> ) all ! s+ul>ities functions c pres5t :5 ! ^w is und]/ood 9 ! mo/ g5]al s5se4 ,b x w\ld n be ! same _h we us$ concrete images1 _h we1 = example1 consid]$ ?is function z an electric pot5tial2 x w\ld h be5 ?\ght legitimate to affirm t electro/atic equilibrium c be atta9$4 ,yet p]haps a physical comp>ison w\ld h awak5$ "s vague di/ru/4 ,b if c>e _h be5 tak5 to translate ! reason+ into ! language ( geometry1 int]m$iate betwe5 t ( analysis & t ( physics1 d\btless ?is di/ru/ w\ld n h be5 produc$1 & p]haps "o might ?us1 ev5 to-day1 /ill deceive _m read]s n =ew>n$4 ,intuition1 !re=e1 does n give u c]ta9ty4 ,?is is :y ! evolution _h to happ52 let u n[ see h[ x happ5$4 ,x 0 n sl[ 9 be+ notic$ t rigor c\ld n be introduc$ 9 ! reason+ unless fir/ made to ent] into ! def9itions4 ,= ! mo/ "p ! objects treat$ ( by ma!maticians 7 long ill def9$2 !y 7 suppos$ to be kn[n because repres5t$ by means ( ! s5ses or ! imag9ation2 b "o _h only a crude image ( !m & n a precise idea on :i* reason+ c\ld take hold4 ,x 0 !re fir/ t ! logicians _h to direct _! ef=ts4 ,s1 9 ! case ( incomm5surable numb]s4 ,! vague idea ( cont9uity1 :i* we owe to intuition1 resolv$ itself into a complicat$ sy/em ( inequalities ref]r+ to :ole numb]s4 ,by t means ! difficulties aris+ f pass+ to ! limit1 or f ! consid]ation ( inf9itesimals1 are f9ally remov$4 ,to-day 9 analysis only :ole numb]s are left or sy/ems1 f9ite or inf9ite1 ( :ole numb]s b\nd toge!r by a net ( equality or inequality relations4 ,ma!matics1 z !y say1 is ari?metiz$4 ,,iii ,a fir/ "q pres5ts itself4 ,is ?is evolution end$8 ,h we f9ally atta9$ absolute rigor8 ,at ea* /age ( ! evolution our fa!rs also ?\ght !y _h rea*$ x4 ,if !y deceiv$ !mselves1 d we n likewise *eat ourselves8 ,we believe t 9 our reason+s we no long] appeal to intuition2 ! philosoph]s w tell u ?is is an illusion4 ,pure logic c\ld nev] lead u to any?+ b tautologies2 x c\ld create no?+ new2 n f x alone c any sci5ce issue4 ,9 "o s5se ^! philosoph]s are "r2 to make ari?metic1 z to make geometry1 or to make any sci5ce1 some?+ else ?an pure logic is necess>y4 ,to designate ?is some?+ else we h no ^w o!r ?an intuition4 ,b h[ _m diff]5t ideas are hidd5 "u ?is same ^w8 ,comp>e ^! f\r axioms3 7#a7 ,two quantities equal to a ?ird are equal to "o ano!r2 7#b7 if a !orem is true ( ! numb] #a & if we prove t x is true ( n #a if true = n1 !n w x be true ( all :ole numb]s2 7#c7 if on a /raight ! po9t ,c is betwe5 ,a & ,b & ! po9t ,d betwe5 ,a & ,c1 !n ! po9t ,d w be betwe5 ,a & ,b2 7#d7 ?r\gh a giv5 po9t !re is n m ?an "o p>allel to a giv5 /raight4 ,all f\r are attribut$ to intuition1 & yet ! fir/ is ! enunciation ( "o ( ! rules ( =mal logic2 ! second is a real syn!tic a priori judgm5t1 x is ! f\ndation ( rigor\s ma!matical induction2 ! ?ird is an appeal to ! imag9ation2 ! f\r? is a disguis$ def9ition4 ,intuition is n necess>ily f\nd$ on ! evid5ce ( ! s5ses2 ! s5ses w\ld soon become p[]less2 = example1 we c n repres5t to ourselves a *iliagon1 & yet we reason by intuition on polygons 9 g5]al1 :i* include ! *iliagon z a p>ticul> case4 ,y "k :at ,poncelet und]/ood by ! pr9ciple ( cont9uity4 ,:at is true ( a real quantity1 said ,poncelet1 %\ld be true ( an imag9>y quantity2 :at is true ( ! hyp]bola ^: asymptotes are real1 %\ld !n be true ( ! ellipse ^: asymptotes are imag9>y4 ,poncelet 0 "o ( ! mo/ intuitive m9ds ( ?is c5tury2 he 0 passionately1 almo/ o/5tati\sly1 s2 he reg>d$ ! pr9ciple ( cont9uity z "o ( 8 bolde/ conceptions1 & yet ?is pr9ciple did n re/ on ! evid5ce ( ! s5ses4 ,to assimilate ! hyp]bola to ! ellipse 0 r to contradict ?is evid5ce4 ,x 0 only a sort ( precoci\s & in/9ctive g5]alization :i*1 moreov]1 ,i h no desire to def5d4 ,we h !n _m k9ds ( intuition2 fir/1 ! appeal to ! s5ses & ! imag9ation2 next1 g5]alization by induction1 copi$1 s to speak1 f ! proc$ures ( ! exp]im5tal sci5ces2 f9ally1 we h ! intuition ( pure numb]1 :5ce arose ! second ( ! axioms j enunciat$1 :i* is able to create ! real ma!matical reason+4 ,i h %[n above by examples t ! fir/ two c n give u c]ta9ty2 b :o w s]i\sly d\bt ! ?ird1 :o w d\bt ari?metic8 ,n[ 9 ! analysis ( to-day1 :5 "o c>es to take ! tr\ble to be rigor\s1 !re c be no?+ b syllogisms or appeals to ?is intuition ( pure numb]1 ! only intuition :i* c n deceive u4 ,x may be said t to-day absolute rigor is atta9$4 ,,iv ,! philosoph]s make /ill ano!r objection3 ,:at y ga9 9 rigor1 !y say1 y lose 9 objectivity4 ,y c rise t[>d y\r logical ideal only by cutt+ ! bonds :i* atta* y to reality4 ,y\r sci5ce is infallible1 b x c only rema9 s by imprison+ itself 9 an ivory t[] & r5\nc+ all relation ) ! ext]nal _w4 ,f ?is seclusion x mu/ g out :5 x w\ld attempt ! slighte/ application4 ,= example1 ,i seek to %[ t "s prop]ty p]ta9s to "s object ^: concept seems to me at fir/ indef9able1 because x is intuitive4 ,at fir/ ,i fail or mu/ cont5t myself ) approximate pro(s2 f9ally ,i decide to give to my object a precise def9ition1 & ?is enables me to e/abli% ?is prop]ty 9 an irreproa*able mann]4 ,& !n1 say ! philosoph]s1 x /ill rema9s to %[ t ! object :i* corresponds to ?is def9ition is inde$ ! same made kn[n to y by intuition2 or else t "s real & concrete object ^: con=mity ) y\r intuitive idea y believe y imm$iately recognize corresponds to y\r new def9ition4 ,only !n c\ld y affirm t x has ! prop]ty 9 "q4 ,y h only displac$ ! difficulty4 ,t is n exactly s2 ! difficulty has n be5 displac$1 x has be5 divid$4 ,! proposition to be e/abli%$ 0 9 reality compos$ ( two diff]5t tru?s1 at fir/ n di/+ui%$4 ,! fir/ 0 a ma!matical tru?1 & x is n[ rigor\sly e/abli%$4 ,! second 0 an exp]im5tal v]ity4 ,exp]i5ce alone c tea* u t "s real & concrete object corresponds or does n correspond to "s ab/ract def9ition4 ,?is second v]ity is n ma!matically demon/rat$1 b nei!r c x be1 no m ?an c ! empirical laws ( ! physical & natural sci5ces4 ,x w\ld be unreasonable to ask m4 ,well1 is x n a great advance to h di/+ui%$ :at long 0 wrongly confus$8 ,does ?is mean t no?+ is left ( ?is objection ( ! philosoph]s8 ,t ,i d n int5d to say2 9 becom+ rigor\s1 ma!matical sci5ce takes a "* s artificial z to /rike e "o2 x =gets its hi/orical orig9s2 we see h[ ! que/ions c be answ]$1 we no long] see h[ & :y !y are put4 ,?is %[s u t logic is n en\gh2 t ! sci5ce ( demon/ration is n all sci5ce & t intuition mu/ reta9 its rle z complem5t1 ,i 0 ab\t to say z c\nt]poise or z antidote ( logic4 ,i h already _h occasion to insi/ on ! place intuition %\ld hold 9 ! tea*+ ( ! ma!matical sci5ces4 ,)\t x "y m9ds c\ld n make a beg9n+ 9 ! und]/&+ ( ma!matics2 !y c\ld n le>n to love x & w\ld see 9 x only a va9 logoma*y2 above all1 )\t intuition !y w\ld nev] become capable ( apply+ ma!matics4 ,b n[ ,i wi% be=e all to speak ( ! rle ( intuition 9 sci5ce itself4 ,if x is useful to ! /ud5t x is /ill m s to ! creative sci5ti/4 ,v ,we seek reality1 b :at is reality8 ,! physiologi/s tell u t organisms are =m$ ( cells2 ! *emi/s add t cells !mselves are =m$ ( atoms4 ,does ?is mean t ^! atoms or ^! cells con/itute reality1 or r ! sole reality8 ,! way 9 :i* ^! cells are arrang$ & f :i* results ! unity ( ! individual1 is n x also a reality mu* m int]e/+ ?an t ( ! isolat$ elem5ts1 & %\ld a naturali/ :o _h nev] /udi$ ! elephant except by means ( ! microscope ?9k himself suffici5tly acqua9t$ ) t animal8 ,well1 !re is some?+ analog\s to ?is 9 ma!matics4 ,! logician cuts up1 s to speak1 ea* demon/ration into a v great numb] ( elem5t>y op]ations2 :5 we h exam9$ ^! op]ations "o aft] ! o!r & asc]ta9$ t ea* is correct1 are we to ?9k we h grasp$ ! real mean+ ( ! demon/ration8 ,%all we h und]/ood x ev5 :51 by an ef=t ( memory1 we h become able to repeat ?is pro( by reproduc+ all ^! elem5t>y op]ations 9 j ! ord] 9 :i* ! inv5tor _h arrang$ !m8 ,evid5tly n2 we %all n yet possess ! entire reality2 t ,i "k n :at1 :i* makes ! unity ( ! demon/ration1 w completely elude u4 ,pure analysis puts at our disposal a multitude ( proc$ures ^: infallibility x gu>antees2 x op5s to u a ?\s& diff]5t ways on :i* we c emb>k 9 all confid5ce2 we are assur$ ( meet+ !re no ob/acles2 b ( all ^! ways1 :i* w lead u mo/ promptly to our goal8 ,:o %all tell u :i* to *oose8 ,we ne$ a faculty :i* makes u see ! end f af>1 & intuition is ?is faculty4 ,x is necess>y to ! explor] = *oos+ 8 r\te2 x is n less s to ! "o foll[+ 8 trail :o wants to "k :y he *ose x4 ,if y are pres5t at a game ( *ess1 x w n suffice1 = ! und]/&+ ( ! game1 to "k ! rules = mov+ ! pieces4 ,t w only enable y to recognize t ea* move has be5 made con=mably to ^! rules1 & ?is k w truly h v little value4 ,yet ?is is :at ! read] ( a book on ma!matics w\ld d if he 7 a logician only4 ,to und]/& ! game is :olly ano!r matt]2 x is to "k :y ! play] moves ?is piece r ?an t o!r :i* he c\ld h mov$ )\t break+ ! rules ( ! game4 ,x is to p]ceive ! inw>d reason :i* makes ( ?is s]ies ( successive moves a sort ( organiz$ :ole4 ,?is faculty is /ill m necess>y = ! play] himself1 t is1 = ! inv5tor4 ,let u drop ?is comp>ison & return to ma!matics4 ,= example1 see :at has happ5$ to ! idea ( cont9u\s function4 ,at ! outset ?is 0 only a s5sible image1 = example1 t ( a cont9u\s m>k trac$ by ! *alk on a blackbo>d4 ,!n x became little by little m ref9$2 ere long x 0 us$ to con/ruct a complicat$ sy/em ( inequalities1 :i* reproduc$1 s to speak1 all ! l9es ( ! orig9al image2 ?is con/ruction f9i%$1 ! c5t]+ ( ! ar*1 s to say1 0 remov$1 t crude repres5tation :i* _h tempor>ily s]v$ z support & :i* 0 aft]w>d useless 0 reject$2 !re rema9$ only ! con/ruction itself1 irreproa*able 9 ! eyes ( ! logician4 ,& yet if ! primitive image _h totally disappe>$ f our recollection1 h[ c\ld we div9e by :at caprice all ^! inequalities 7 erect$ 9 ?is fa%ion "o ^u ano!r8 ,p]haps y ?9k ,i use too _m comp>isons2 yet p>don /ill ano!r4 ,y h d\btless se5 ^? delicate assemblages ( silici\s ne$les :i* =m ! skeleton ( c]ta9 sponges4 ,:5 ! organic matt] has disappe>$1 !re rema9s only a frail & elegant lace-"w4 ,true1 no?+ is !re except silica1 b :at is int]e/+ is ! =m ?is silica has tak51 & we c\ld n und]/& x if we did n "k ! liv+ sponge :i* has giv5 x precisely ?is =m4 ,?us x is t ! old intuitive notions ( our fa!rs1 ev5 :5 we h ab&on$ !m1 /ill impr9t _! =m ^u ! logical con/ructions we h put 9 _! place4 ,?is view ( ! aggregate is necess>y = ! inv5tor2 x is equally necess>y = :oev] wi%es really to compreh5d ! inv5tor4 ,c logic give x to u8 ,no2 ! "n ma!maticians give x w\ld suffice to prove ?is4 ,9 ma!matics logic is call$ analysis & analysis means division1 dissection4 ,x c h1 !re=e1 no tool o!r ?an ! scalpel & ! microscope4 ,?us logic & intuition h ea* _! necess>y rle4 ,ea* is indisp5sable4 ,logic1 :i* alone c give c]ta9ty1 is ! in/rum5t ( demon/ration2 intuition is ! in/rum5t ( inv5tion4 ,,vi ,b at ! mom5t ( =mulat+ ?is conclusion ,i am seiz$ ) scruples4 ,at ! outset ,i di/+ui%$ two k9ds ( ma!matical m9ds1 ! "o sort logicians & analy/s1 ! o!rs intuitionali/s & geomet]s4 ,well1 ! analy/s also h be5 inv5tors4 ,! names ,i h j cit$ make my insi/5ce on ?is unnecess>y4 ,"h is a contradiction1 at lea/ app>5tly1 :i* ne$s explanation4 ,& fir/1 d y ?9k ^! logicians h always proce$$ f ! g5]al to ! p>ticul>1 z ! rules ( =mal logic w\ld seem to require ( !m8 ,n ?us c\ld !y h ext5d$ ! b\nd>ies ( sci5ce2 sci5tific conque/ is to be made only by g5]alization4 ,9 "o ( ! *apt]s ( ',sci5ce & ,hypo!sis1' ,i h _h occasion to /udy ! nature ( ma!matical reason+1 & ,i h %[n h[ ?is reason+1 )\t ceas+ to be absolutely rigor\s1 c\ld lift u f ! p>ticul> to ! g5]al by a proc$ure ,i h call$ ma!matical induction4 ,x is by ?is proc$ure t ! analy/s h made sci5ce progress1 & if we exam9e ! detail itself ( _! demon/rations1 we %all f9d x !re at ea* in/ant beside ! classic syllogism ( ,ari/otle4 ,we1 !re=e1 see already t ! analy/s are n simply mak]s ( syllogisms aft] ! fa%ion ( ! s*ola/ics4 ,besides1 d y ?9k !y h always m>*$ /ep by /ep ) no vision ( ! goal !y wi%$ to atta98 ,!y mu/ h div9$ ! way lead+ ?i!r1 & = t !y ne$$ a guide4 ,?is guide is1 fir/1 analogy4 ,= example1 "o ( ! me?ods ( demon/ration de> to analy/s is t f\nd$ on ! employm5t ( dom9ant functions4 ,we "k x has already s]v$ to solve a multitude ( problems2 9 :at consi/s !n ! rle ( ! inv5tor :o wi%es to apply x to a new problem8 ,at ! outset he mu/ recognize ! analogy ( ?is "q ) ^? :i* h already be5 solv$ by ?is me?od2 !n he mu/ p]ceive 9 :at way ?is new "q diff]s f ! o!rs1 & !nce d$uce ! modifications necess>y to apply to ! me?od4 ,b h[ does "o p]ceive ^! analogies & ^! diff]5ces8 ,9 ! example j cit$ !y are almo/ always evid5t1 b ,i c\ld h f\nd o!rs :]e !y w\ld h be5 mu* m deeply hidd52 (t5 a v uncommon p5etration is necess>y = _! discov]y4 ,! analy/s1 n to let ^! hidd5 analogies escape !m1 t is1 9 ord] to be inv5tors1 mu/1 )\t ! aid ( ! s5ses & imag9ation1 h a direct s5se ( :at con/itutes ! unity ( a piece ( reason+1 ( :at makes1 s to speak1 its s\l & inmo/ life4 ,:5 "o talk$ ) ,m4 ,h]mite1 he nev] evok$ a s5su\s image1 & yet y soon p]ceiv$ t ! mo/ ab/ract entities 7 = him l liv+ be+s4 ,he did n see !m1 b he p]ceiv$ t !y are n an artificial assemblage & t !y h "s pr9ciple ( int]nal unity4 ,b1 "o w say1 t /ill is intuition4 ,%all we conclude t ! di/9ction made at ! outset 0 only app>5t1 t !re is only "o sort ( m9d & t all ! ma!maticians are intuitionali/s1 at lea/ ^? :o are capable ( inv5t+8 ,no1 our di/9ction corresponds to some?+ real4 ,i h said above t !re are _m k9ds ( intuition4 ,i h said h[ mu* ! intuition ( pure numb]1 :5ce comes rigor\s ma!matical induction1 diff]s f s5sible intuition to :i* ! imag9ation1 prop]ly s call$1 is ! pr9cipal contributor4 ,is ! abyss :i* sep>ates !m less pr(\nd ?an x at fir/ appe>$8 ,c\ld we recognize ) a little att5tion t ?is pure intuition itself c\ld n d )\t ! aid ( ! s5ses8 ,?is is ! affair ( ! psy*ologi/ & ! metaphysician & ,i %all n discuss ! "q4 ,b ! ?+'s be+ d\btful is en\gh to ju/ify me 9 recogniz+ & affirm+ an ess5tial diff]5ce betwe5 ! two k9ds ( intuition2 !y h n ! same object & seem to call into play two diff]5t faculties ( our s\l2 "o w\ld ?9k ( two se>*-lights direct$ ^u two worlds /rang]s to "o ano!r4 ,x is ! intuition ( pure numb]1 t ( pure logical =ms1 :i* illum9es & directs ^? we h call$ analy/s4 ,?is x is :i* enables !m n alone to demon/rate1 b also to inv5t4 ,by x !y p]ceive at a glance ! g5]al plan ( a logical edifice1 & t too )\t ! s5ses appe>+ to int]v5e4 ,9 reject+ ! aid ( ! imag9ation1 :i*1 z we h se51 is n always infallible1 !y c advance )\t fe> ( deceiv+ !mselves4 ,happy1 !re=e1 are ^? :o c d )\t ?is aid6 ,we mu/ admire !m2 b h[ r>e !y are6 ,among ! analy/s !re w !n be inv5tors1 b !y w be few4 ,! majority ( u1 if we wi%$ to see af> by pure intuition alone1 w\ld soon feel ourselves seiz$ ) v]tigo4 ,our weakness has ne$ ( a /aff m solid1 &1 despite ! exceptions ( :i* we h j spok51 x is none ! less true t s5sible intuition is 9 ma!matics ! mo/ usual in/rum5t ( inv5tion4 ,apropos ( ^! reflections1 a "q comes up t ,i h n ! "t ei!r to solve or ev5 to enunciate ) ! developm5ts x w\ld admit (4 ,is !re room = a new di/9ction1 = di/+ui%+ among ! analy/s ^? :o above all use pure intuition & ^? :o are fir/ ( all preoccupi$ ) =mal logic8 ,m4 ,h]mite1 = example1 :om ,i h j cit$1 c n be class$ among ! geomet]s :o make use ( ! s5sible intuition2 b nei!r is he a logician1 prop]ly s call$4 ,he does n conceal 8 av]sion to purely d$uctive proc$ures :i* />t f ! g5]al & end 9 ! p>ticul>4 ,,*apt] ,,ii ,,! ,,measure ,,( ,,"t ,i ,s long z we d n g outside ! doma9 ( consci\sness1 ! notion ( "t is relatively cle>4 ,n only d we di/+ui% )\t difficulty pres5t s5sation f ! remembrance ( pa/ s5sations or ! anticipation ( future s5sations1 b we "k p]fectly well :at we mean :5 we say t ( two consci\s ph5om5a :i* we rememb]1 "o 0 ant]ior to ! o!r2 or t1 ( two =ese5 consci\s ph5om5a1 "o w be ant]ior to ! o!r4 ,:5 we say t two consci\s facts are simultane\s1 we mean t !y pr(\ndly int]p5etrate1 s t analysis c n sep>ate !m )\t mutilat+ !m4 ,! ord] 9 :i* we arrange consci\s ph5om5a does n admit ( any arbitr>9ess4 ,x is impos$ ^u u & ( x we c *ange no?+4 ,i h only a s+le obs]vation to add4 ,= an aggregate ( s5sations to h become a remembrance capable ( classification 9 "t1 x mu/ h ceas$ to be actual1 we mu/ h lo/ ! s5se ( its inf9ite complexity1 o!rwise x w\ld h rema9$ pres5t4 ,x mu/1 s to speak1 h cry/alliz$ ar\nd a c5t] ( associations ( ideas :i* w be a sort ( label4 ,x is only :5 !y ?us h lo/ all life t we c classify our memories 9 "t z a botani/ arranges dri$ fl[]s 9 8 h]b>ium4 ,b ^! labels c only be f9ite 9 numb]4 ,on t score1 psy*ologic "t %\ld be discont9u\s4 ,:5ce comes ! feel+ t betwe5 any two in/ants !re are o!rs8 ,we arrange our recollections 9 "t1 b we "k t !re rema9 empty comp>tm5ts4 ,h[ c\ld t be1 if "t 7 n a =m pre-exi/5t 9 our m9ds8 ,h[ c\ld we "k !re 7 empty comp>tm5ts1 if ^! comp>tm5ts 7 reveal$ to u only by _! cont5t8 ,,ii ,b t is n all2 into ?is =m we wi% to put n only ! ph5om5a ( our own consci\sness1 b ^? ( :i* o!r consci\snesses are ! !at]4 ,b m1 we wi% to put !re physical facts1 ^! ,i "k n :at ) :i* we p space & :i* no consci\sness sees directly4 ,?is is necess>y because )\t x sci5ce c\ld n exi/4 ,9 a ^w1 psy*ologic "t is giv5 to u & mu/ ne$s create sci5tific & physical "t4 ,!re ! difficulty beg9s1 or r ! difficulties1 = !re are two4 ,?9k ( two consci\snesses1 :i* are l two worlds imp5etrable "o to ! o!r4 ,by :at "r d we /rive to put !m into ! same mold1 to measure !m by ! same /&>d8 ,is x n z if "o /rove to measure l5g? ) a gram or weight ) a met]8 ,& besides1 :y d we speak ( measur+8 ,we "k p]haps t "s fact is ant]ior to "s o!r1 b n by h[ mu* x is ant]ior4 ,!re=e two difficulties3 7#a7 ,c we trans=m psy*ologic "t1 :i* is qualitative1 into a quantitative "t8 7#b7 ,c we r$uce to "o & ! same measure facts :i* transpire 9 diff]5t worlds8 ,,iii ,! fir/ difficulty has long be5 notic$2 x has be5 ! subject ( long discussions & "o may say ! "q is settl$4 ,we h n a direct intuition ( ! equality ( two int]vals ( "t4 ,! p]sons :o believe !y possess ?is intuition are dupes ( an illusion4 ,:5 ,i say1 f noon to "o ! same "t passes z f two to ?ree1 :at mean+ has ?is affirmation8 ,! lea/ reflection %[s t by itself x has none at all4 ,x w only h t :i* ,i *oose to give x1 by a def9ition :i* w c]ta9ly possess a c]ta9 degree ( arbitr>9ess4 ,psy*ologi/s c\ld h done )\t ?is def9ition2 physici/s & a/ronom]s c\ld n2 let u see h[ !y h manag$4 ,to measure "t !y use ! p5dulum & !y suppose by def9ition t all ! beats ( ?is p5dulum are ( equal duration4 ,b ?is is only a fir/ approximation2 ! temp]ature1 ! resi/ance ( ! air1 ! b>ometric pressure1 make ! pace ( ! p5dulum v>y4 ,if we c\ld escape ^! s\rces ( error1 we %\ld obta9 a mu* clos] approximation1 b x w\ld /ill be only an approximation4 ,new causes1 hi!rto neglect$1 electric1 magnetic or o!rs1 w\ld introduce m9ute p]turbations4 ,9 fact1 ! be/ *ronomet]s mu/ be correct$ f "t to "t1 & ! corrections are made by ! aid ( a/ronomic obs]vations2 arrangem5ts are made s t ! sid]eal clock m>ks ! same h\r :5 ! same /> passes ! m]idian4 ,9 o!r words1 x is ! sid]eal day1 t is1 ! duration ( ! rotation ( ! e>?1 :i* is ! con/ant unit ( "t4 ,x is suppos$1 by a new def9ition sub/itut$ = t bas$ on ! beats ( ! p5dulum1 t two complete rotations ( ! e>? ab\t its axis h ! same duration4 ,h[ev]1 ! a/ronom]s are /ill n cont5t ) ?is def9ition4 ,_m ( !m ?9k t ! tides act z a *eck on our globe1 & t ! rotation ( ! e>? is becom+ sl[] & sl[]4 ,?us w\ld be expla9$ ! app>5t accel]ation ( ! motion ( ! moon1 :i* w\ld seem to be go+ m rapidly ?an !ory p]mits because our wat*1 :i* is ! e>?1 is go+ sl[4 ,,iv ,all ?is is unimportant1 "o w say2 d\btless our in/rum5ts ( measurem5t are imp]fect1 b x suffices t we c conceive a p]fect in/rum5t4 ,?is ideal c n be rea*$1 b x is en\gh to h conceiv$ x & s to h put rigor into ! def9ition ( ! unit ( "t4 ,! tr\ble is t !re is no rigor 9 ! def9ition4 ,:5 we use ! p5dulum to measure "t1 :at po/ulate d we implicitly admit8 ,x is t ! duration ( two id5tical ph5om5a is ! same2 or1 if y pref]1 t ! same causes take ! same "t to produce ! same effects4 ,& at fir/ blu%1 ?is is a good def9ition ( ! equality ( two durations4 ,b take c>e4 ,is x impossible t exp]im5t may "s day contradict our po/ulate8 ,let me expla9 myself4 ,i suppose t at a c]ta9 place 9 ! _w ! ph5om5on alpha happ5s1 caus+ z consequ5ce at ! end ( a c]ta9 "t ! effect alpha'4 ,at ano!r place 9 ! _w v f> away f ! fir/1 happ5s ! ph5om5on beta1 :i* causes z consequ5ce ! effect beta'4 ,! ph5om5a alpha & beta are simultane\s1 z are also ! effects alpha' & beta'4 ,lat]1 ! ph5om5on alpha is reproduc$ "u approximately ! same conditions z be=e1 & simultane\sly ! ph5om5on beta is also reproduc$ at a v di/ant place 9 ! _w & almo/ "u ! same circum/ances4 ,! effects alpha' & beta' also take place4 ,let u suppose t ! effect alpha' happ5s p]ceptibly be=e ! effect beta'4 ,if exp]i5ce made u witness su* a sight1 our po/ulate w\ld be contradict$4 ,= exp]i5ce w\ld tell u t ! fir/ duration alphaalpha' is equal to ! fir/ duration betabeta' & t ! second duration alphaalpha' is less ?an ! second duration betabeta'4 ,on ! o!r h&1 our po/ulate w\ld require t ! two durations alphaalpha' %\ld be equal to ea* o!r1 z likewise ! two durations betabeta'4 ,! equality & ! inequality d$uc$ f exp]i5ce w\ld be incompatible ) ! two equalities d$uc$ f ! po/ulate4 ,n[ c we affirm t ! hypo!ses ,i h j made are absurd8 ,!y are 9 no wise contr>y to ! pr9ciple ( contradiction4 ,d\btless !y c\ld n happ5 )\t ! pr9ciple ( suffici5t reason seem+ violat$4 ,b to ju/ify a def9ition s fundam5tal ,i %\ld pref] "s o!r gu>antee4 ,v ,b t is n all4 ,9 physical reality "o cause does n produce a giv5 effect1 b a multitude ( di/9ct causes contribute to produce x1 )\t our hav+ any means ( discrim9at+ ! "p ( ea* ( !m4 ,physici/s seek to make ?is di/9ction2 b !y make x only approximately1 &1 h[ev] !y progress1 !y nev] w make x except approximately4 ,x is approximately true t ! motion ( ! p5dulum is due solely to ! e>?'s attraction2 b 9 all rigor e attraction1 ev5 ( ,sirius1 acts on ! p5dulum4 ,"u ^! conditions1 x is cle> t ! causes :i* h produc$ a c]ta9 effect w nev] be reproduc$ except approximately4 ,!n we %\ld modify our po/ulate & our def9ition4 ,in/ead ( say+3 ',! same causes take ! same "t to produce ! same effects1' we %\ld say3 ',causes almo/ id5tical take almo/ ! same "t to produce almo/ ! same effects4' ,our def9ition !re=e is no long] any?+ b approximate4 ,besides1 z ,m4 ,cal9on v ju/ly rem>ks 9 a rec5t memoir3#g ,"o ( ! circum/ances ( any ph5om5on is ! velocity ( ! e>?'s rotation2 if ?is velocity ( rotation v>ies1 x con/itutes 9 ! reproduction ( ?is ph5om5on a circum/ance :i* no long] rema9s ! same4 ,b to suppose ?is velocity ( rotation con/ant is to suppose t we "k h[ to measure "t4 #g ,etude sur les div]ses gr&eurs1 ,p>is1 ,gau?i]-,vill>s1 #ahig4 ,our def9ition is !re=e n yet satisfactory2 x is c]ta9ly n t :i* ! a/ronom]s ( :om ,i spoke above implicitly adopt1 :5 !y affirm t ! t]re/rial rotation is sl[+ d[n4 ,:at mean+ accord+ to !m has ?is affirmation8 ,we c only und]/& x by analyz+ ! pro(s !y give ( _! proposition4 ,!y say fir/ t ! friction ( ! tides produc+ heat mu/ de/roy vis viva4 ,!y invoke !re=e ! pr9ciple ( vis viva1 or ( ! cons]vation ( en]gy4 ,!y say next t ! secul> accel]ation ( ! moon1 calculat$ accord+ to ,newton's law1 w\ld be less ?an t d$uc$ f obs]vations unless ! correction relative to ! sl[+ d[n ( ! t]re/rial rotation 7 made4 ,!y invoke !re=e ,newton's law4 ,9 o!r words1 !y def9e duration 9 ! foll[+ way3 "t %\ld be s def9$ t ,newton's law & t ( vis viva may be v]ifi$4 ,newton's law is an exp]im5tal tru?2 z su* x is only approximate1 :i* %[s t we /ill h only a def9ition by approximation4 ,if n[ x be suppos$ t ano!r way ( measur+ "t is adopt$1 ! exp]im5ts on :i* ,newton's law is f\nd$ w\ld none ! less h ! same mean+4 ,only ! enunciation ( ! law w\ld be diff]5t1 because x w\ld be translat$ into ano!r language2 x w\ld evid5tly be mu* less simple4 ,s t ! def9ition implicitly adopt$ by ! a/ronom]s may be summ$ up ?us3 ,"t %\ld be s def9$ t ! equations ( me*anics may be z simple z possible4 ,9 o!r words1 !re is n "o way ( measur+ "t m true ?an ano!r2 t :i* is g5]ally adopt$ is only m conv5i5t4 ,( two wat*es1 we h no "r to say t ! "o goes true1 ! o!r wrong2 we c only say t x is advantage\s to con=m to ! indications ( ! fir/4 ,! difficulty :i* has j occupi$ u has be51 z ,i h said1 (t5 po9t$ out2 among ! mo/ rec5t works 9 :i* x is consid]$1 ,i may m5tion1 besides ,m4 ,cal9on's little book1 ! treatise on me*anics ( ,&rade4 ,,vi ,! second difficulty has up to ! pres5t attract$ mu* less att5tion2 yet x is altoge!r analog\s to ! prec$+2 & ev51 logically1 ,i %\ld h spok5 ( x fir/4 ,two psy*ological ph5om5a happ5 9 two diff]5t consci\snesses2 :5 ,i say !y are simultane\s1 :at d ,i mean8 ,:5 ,i say t a physical ph5om5on1 :i* happ5s outside ( e consci\sness1 is be=e or aft] a psy*ological ph5om5on1 :at d ,i mean8 ,9 #aegb1 ,ty*o ,brahe notic$ 9 ! heav5s a new />4 ,an imm5se conflagration _h happ5$ 9 "s f> di/ant heav5ly body2 b x _h happ5$ long be=e2 at lea/ two hundr$ ye>s 7 necess>y = ! light f t /> to rea* our e>?4 ,?is conflagration !re=e happ5$ be=e ! discov]y ( ,am]ica4 ,well1 :5 ,i say t2 :51 consid]+ ?is gigantic ph5om5on1 :i* p]haps _h no witness1 s9ce ! satellites ( t /> 7 p]haps un9habit$1 ,i say ?is ph5om5on is ant]ior to ! =mation ( ! visual image ( ! isle ( ,espaola 9 ! consci\sness ( ,*ri/oph] ,columbus1 :at d ,i mean8 ,a little reflection is suffici5t to und]/& t all ^! affirmations h by !mselves no mean+4 ,!y c h "o only z ! outcome ( a conv5tion4 ,,vii ,we %\ld fir/ ask ourselves h[ "o c\ld h _h ! idea ( putt+ into ! same frame s _m worlds imp5etrable to "o ano!r4 ,we %\ld l to repres5t to ourselves ! ext]nal univ]se1 & only by s do+ c\ld we feel t we und]/ood x4 ,we "k we nev] c atta9 ?is repres5tation3 our weakness is too great4 ,b at lea/ we desire ! ability to conceive an inf9ite intellig5ce = :i* ?is repres5tation c\ld be possible1 a sort ( great consci\sness :i* %\ld see all1 & :i* %\ld classify all 9 its "t1 z we classify1 9 our "t1 ! little we see4 ,?is hypo!sis is inde$ crude & incomplete1 because ?is supreme intellig5ce w\ld be only a demigod2 inf9ite 9 "o s5se1 x w\ld be limit$ 9 ano!r1 s9ce x w\ld h only an imp]fect recollection ( ! pa/2 & x c\ld h no o!r1 s9ce o!rwise all recollections w\ld be equally pres5t to x & = x !re w\ld be no "t4 ,& yet :5 we speak ( "t1 = all :i* happ5s outside ( u1 d we n unconsci\sly adopt ?is hypo!sis2 d we n put ourselves 9 ! place ( ?is imp]fect god2 & d n ev5 ! a!i/s put !mselves 9 ! place :]e god w\ld be if he exi/$8 ,:at ,i h j said %[s u1 p]haps1 :y we h tri$ to put all physical ph5om5a into ! same frame4 ,b t c n pass = a def9ition ( simultaneity1 s9ce ?is hypo!tical intellig5ce1 ev5 if x exi/$1 w\ld be = u imp5etrable4 ,x is !re=e necess>y to seek some?+ else4 ,,viii ,! ord9>y def9itions :i* are prop] = psy*ologic "t w\ld suffice u no m4 ,two simultane\s psy*ologic facts are s closely b\nd toge!r t analysis c n sep>ate )\t mutilat+ !m4 ,is x ! same ) two physical facts8 ,is n my pres5t ne>] my pa/ ( ye/]day ?an ! pres5t ( ,sirius8 ,x has also be5 said t two facts %\ld be reg>d$ z simultane\s :5 ! ord] ( _! succession may be inv]t$ at w4 ,x is evid5t t ?is def9ition w\ld n suit two physical facts :i* happ5 f> f "o ano!r1 & t1 9 :at conc]ns !m1 we no long] ev5 und]/& :at ?is rev]sibility w\ld be2 besides1 succession itself mu/ fir/ be def9$4 ,,ix ,let u !n seek to give an acc\nt ( :at is und]/ood by simultaneity or antec$5ce1 & = ?is let u analyze "s examples4 ,i write a lett]2 x is aft]w>d read by ! fri5d to :om ,i h address$ x4 ,!re are two facts :i* h _h = _! !at] two diff]5t consci\snesses4 ,9 writ+ ?is lett] ,i h _h ! visual image ( x1 & my fri5d has _h 9 8 turn ?is same visual image 9 read+ ! lett]4 ,?\gh ^! two facts happ5 9 imp5etrable worlds1 ,i d n hesitate to reg>d ! fir/ z ant]ior to ! second1 because ,i believe x is its cause4 ,i he> ?und]1 & ,i conclude !re has be5 an electric dis*>ge2 ,i d n hesitate to consid] ! physical ph5om5on z ant]ior to ! auditory image p]ceiv$ 9 my consci\sness1 because ,i believe x is its cause4 ,behold !n ! rule we foll[1 & ! only "o we c foll[3 :5 a ph5om5on appe>s to u z ! cause ( ano!r1 we reg>d x z ant]ior4 ,x is !re=e by cause t we def9e "t2 b mo/ (t51 :5 two facts appe> to u b\nd by a con/ant relation1 h[ d we recognize :i* is ! cause & :i* ! effect8 ,we assume t ! ant]ior fact1 ! antec$5t1 is ! cause ( ! o!r1 ( ! consequ5t4 ,x is !n by "t t we def9e cause4 ,h[ save ourselves f ?is petitio pr9cipii8 ,we say n[ po/ hoc1 ergo propt] hoc2 n[ propt] hoc1 ergo po/ hoc2 %all we escape f ?is vici\s circle8 ,x ,let u see1 n h[ we succe$ 9 escap+1 = we d n completely succe$1 b h[ we try to escape4 ,i execute a volunt>y act ,a & ,i feel aft]w>d a s5sation ,d1 :i* ,i reg>d z a consequ5ce ( ! act ,a2 on ! o!r h&1 = :atev] reason1 ,i inf] t ?is consequ5ce is n imm$iate1 b t outside my consci\sness two facts ,b & ,c1 :i* ,i h n witness$1 h happ5$1 & 9 su* a way t ,b is ! effect ( ,a1 t ,c is ! effect ( ,b1 & ,d ( ,c4 ,b :y8 ,if ,i ?9k ,i h reason to reg>d ! f\r facts ,a1 ,b1 ,c1 ,d1 z b\nd to "o ano!r by a causal connection1 :y range !m 9 ! causal ord] ,a ,b ,c ,d1 & at ! same "t 9 ! *ronologic ord] ,a ,b ,c ,d1 r ?an 9 any o!r ord]8 ,i cle>ly see t 9 ! act ,a ,i h ! feel+ ( hav+ be5 active1 :ile 9 und]go+ ! s5sation ,d ,i h t ( hav+ be5 passive4 ,?is is :y ,i reg>d ,a z ! initial cause & ,d z ! ultimate effect2 ?is is :y ,i put ,a at ! beg9n+ ( ! *a9 & ,d at ! end2 b :y put ,b be=e ,c r ?an ,c be=e ,b8 ,if ?is "q is put1 ! reply ord9>ily is3 we "k t x is ,b :i* is ! cause ( ,c because we always see ,b happ5 be=e ,c4 ,^! two ph5om5a1 :5 witness$1 happ5 9 a c]ta9 ord]2 :5 analog\s ph5om5a happ5 )\t witness1 !re is no reason to inv]t ?is ord]4 ,d\btless1 b take c>e2 we nev] "k directly ! physical ph5om5a ,b & ,c4 ,:at we "k are s5sations ,b' & ,c' produc$ respectively by ,b & ,c4 ,our consci\sness tells u imm$iately t ,b' prec$es ,c' & we suppose t ,b & ,c succe$ "o ano!r 9 ! same ord]4 ,?is rule appe>s 9 fact v natural1 & yet we are (t5 l$ to dep>t f x4 ,we he> ! s\nd ( ! ?und] only "s seconds aft] ! electric dis*>ge ( ! cl\d4 ,( two fla%es ( lightn+1 ! "o di/ant1 ! o!r ne>1 c n ! fir/ be ant]ior to ! second1 ev5 ?\gh ! s\nd ( ! second comes to u be=e t ( ! fir/8 ,,xi ,ano!r difficulty2 h we really ! "r to speak ( ! cause ( a ph5om5on8 ,if all ! p>ts ( ! univ]se are int]*a9$ 9 a c]ta9 measure1 any "o ph5om5on w n be ! effect ( a s+le cause1 b ! resultant ( causes inf9itely num]\s2 x is1 "o (t5 says1 ! consequ5ce ( ! /ate ( ! univ]se a mom5t be=e4 ,h[ enunciate rules applicable to circum/ances s complex8 ,& yet x is only ?us t ^! rules c be g5]al & rigor\s4 ,n to lose ourselves 9 ?is inf9ite complexity1 let u make a simpl] hypo!sis4 ,consid] ?ree />s1 = example1 ! sun1 ,jupit] & ,saturn2 b1 = great] simplicity1 reg>d !m z r$uc$ to mat]ial po9ts & isolat$ f ! re/ ( ! _w4 ,! positions & ! velocities ( ?ree bodies at a giv5 in/ant suffice to det]m9e _! positions & velocities at ! foll[+ in/ant1 & consequ5tly at any in/ant4 ,_! positions at ! in/ant t det]m9e _! positions at ! in/ant t h z well z _! positions at ! in/ant t - h4 ,ev5 m2 ! position ( ,jupit] at ! in/ant t1 toge!r ) t ( ,saturn at ! in/ant t a1 det]m9es ! position ( ,jupit] at any in/ant & t ( ,saturn at any in/ant4 ,! aggregate ( positions occupi$ by ,jupit] at ! in/ant t e & ,saturn at ! in/ant t a e is b\nd to ! aggregate ( positions occupi$ by ,jupit] at ! in/ant t & ,saturn at ! in/ant t a1 by laws z precise z t ( ,newton1 ?\gh m complicat$4 ,!n :y n reg>d "o ( ^! aggregates z ! cause ( ! o!r1 :i* w\ld lead to consid]+ z simultane\s ! in/ant t ( ,jupit] & ! in/ant t a ( ,saturn8 ,9 answ] !re c only be reasons1 v /rong1 x is true1 ( conv5i5ce & simplicity4 ,,xii ,b let u pass to examples less artificial2 to und]/& ! def9ition implicitly suppos$ by ! savants1 let u wat* !m at "w & look = ! rules by :i* !y inve/igate simultaneity4 ,i w take two simple examples1 ! measurem5t ( ! velocity ( light & ! det]m9ation ( longitude4 ,:5 an a/ronom] tells me t "s /ell> ph5om5on1 :i* 8 telescope reveals to him at ?is mom5t1 happ5$1 nev]!less1 fifty ye>s ago1 ,i seek 8 mean+1 & to t end ,i %all ask him fir/ h[ he kn[s x1 t is1 h[ he has measur$ ! velocity ( light4 ,he has begun by suppos+ t light has a con/ant velocity1 & 9 p>ticul> t its velocity is ! same 9 all directions4 ,t is a po/ulate )\t :i* no measurem5t ( ?is velocity c\ld be attempt$4 ,?is po/ulate c\ld nev] be v]ifi$ directly by exp]im5t2 x might be contradict$ by x if ! results ( diff]5t measurem5ts 7 n concordant4 ,we %\ld ?9k ourselves =tunate t ?is contradiction has n happ5$ & t ! slight discordances :i* may happ5 c be readily expla9$4 ,! po/ulate1 at all ev5ts1 resembl+ ! pr9ciple ( suffici5t reason1 has be5 accept$ by ev]ybody2 :at ,i wi% to emphasize is t x furni%es u ) a new rule = ! inve/igation ( simultaneity1 entirely diff]5t f t :i* we h enunciat$ above4 ,?is po/ulate assum$1 let u see h[ ! velocity ( light has be5 measur$4 ,y "k t ,roem] us$ eclipses ( ! satellites ( ,jupit]1 & s\ght h[ mu* ! ev5t fell beh9d its pr$iction4 ,b h[ is ?is pr$iction made8 ,x is by ! aid ( a/ronomic laws2 = in/ance ,newton's law4 ,c\ld n ! obs]v$ facts be j z well expla9$ if we attribut$ to ! velocity ( light a little diff]5t value f t adopt$1 & suppos$ ,newton's law only approximate8 ,only ?is w\ld lead to replac+ ,newton's law by ano!r m complicat$4 ,s = ! velocity ( light a value is adopt$1 su* t ! a/ronomic laws compatible ) ?is value may be z simple z possible4 ,:5 navigators or geograph]s det]m9e a longitude1 !y h to solve j ! problem we are discuss+2 !y mu/1 )\t be+ at ,p>is1 calculate ,p>is "t4 ,h[ d !y accompli% x8 ,!y c>ry a *ronomet] set = ,p>is4 ,! qualitative problem ( simultaneity is made to dep5d ^u ! quantitative problem ( ! measurem5t ( "t4 ,i ne$ n take up ! difficulties relative to ?is latt] problem1 s9ce above ,i h emphasiz$ !m at l5g?4 ,or else !y obs]ve an a/ronomic ph5om5on1 su* z an eclipse ( ! moon1 & !y suppose t ?is ph5om5on is p]ceiv$ simultane\sly f all po9ts ( ! e>?4 ,t is n altoge!r true1 s9ce ! propagation ( light is n in/antane\s2 if absolute exactitude 7 desir$1 !re w\ld be a correction to make accord+ to a complicat$ rule4 ,or else f9ally !y use ! telegraph4 ,x is cle> fir/ t ! reception ( ! signal at ,b]l91 = in/ance1 is aft] ! s5d+ ( ?is same signal f ,p>is4 ,?is is ! rule ( cause & effect analyz$ above4 ,b h[ mu* aft]8 ,9 g5]al1 ! duration ( ! transmission is neglect$ & ! two ev5ts are reg>d$ z simultane\s4 ,b1 to be rigor\s1 a little correction w\ld /ill h to be made by a complicat$ calculation2 9 practise x is n made1 because x w\ld be well )9 ! errors ( obs]vation2 its !oretic necessity is none ! less f our po9t ( view1 :i* is t ( a rigor\s def9ition4 ,f ?is discussion1 ,i wi% to emphasize two ?+s3 7#a7 ,! rules appli$ are exce$+ly v>i\s4 7#b7 ,x is difficult to sep>ate ! qualitative problem ( simultaneity f ! quantitative problem ( ! measurem5t ( "t2 no matt] :e!r a *ronomet] is us$1 or :e!r acc\nt mu/ be tak5 ( a velocity ( transmission1 z t ( light1 because su* a velocity c\ld n be measur$ )\t measur+ a "t4 ,,xiii ,to conclude3 ,we h n a direct intuition ( simultaneity1 nor ( ! equality ( two durations4 ,if we ?9k we h ?is intuition1 ?is is an illusion4 ,we replace x by ! aid ( c]ta9 rules :i* we apply almo/ always )\t tak+ c\nt ( !m4 ,b :at is ! nature ( ^! rules8 ,no g5]al rule1 no rigor\s rule2 a multitude ( little rules applicable to ea* p>ticul> case4 ,^! rules are n impos$ ^u u & we might amuse ourselves 9 inv5t+ o!rs2 b !y c\ld n be ca/ aside )\t greatly complicat+ ! enunciation ( ! laws ( physics1 me*anics & a/ronomy4 ,we !re=e *oose ^! rules1 n because !y are true1 b because !y are ! mo/ conv5i5t1 & we may recapitulate !m z foll[s3 ,! simultaneity ( two ev5ts1 or ! ord] ( _! succession1 ! equality ( two durations1 are to be s def9$ t ! enunciation ( ! natural laws may be z simple z possible4 ,9 o!r words1 all ^! rules1 all ^! def9itions are only ! fruit ( an unconsci\s opportunism4 ,,*apt] ,,iii ,,! ,,notion ,,( ,,space #a4 ,introduction ,9 ! articles ,i h h]et(ore devot$ to space ,i h above all emphasiz$ ! problems rais$ by non-,euclidean geometry1 :ile leav+ almo/ completely aside o!r que/ions m difficult ( approa*1 su* z ^? :i* p]ta9 to ! numb] ( dim5sions4 ,all ! geometries ,i consid]$ _h ?us a common basis1 t tridim5sional cont9uum :i* 0 ! same = all & :i* diff]5tiat$ itself only by ! figures "o drew 9 x or :5 "o aspir$ to measure x4 ,9 ?is cont9uum1 primitively amorph\s1 we may imag9e a network ( l9es & surfaces1 we may !n conv5e to reg>d ! me%es ( ?is net z equal to "o ano!r1 & x is only aft] ?is conv5tion t ?is cont9uum1 become measurable1 becomes ,euclidean or non-,euclidean space4 ,f ?is amorph\s cont9uum c !re=e arise indiff]5tly "o or ! o!r ( ! two spaces1 j z on a blank %eet ( pap] may be trac$ indiff]5tly a /raight or a circle4 ,9 space we "k rectil9e> triangles ! sum ( ^: angles is equal to two "r angles2 b equally we "k curvil9e> triangles ! sum ( ^: angles is less ?an two "r angles4 ,! exi/5ce ( ! "o sort is n m d\btful ?an t ( ! o!r4 ,to give ! "n ( /raights to ! sides ( ! fir/ is to adopt ,euclidean geometry2 to give ! "n ( /raights to ! sides ( ! latt] is to adopt ! non-,euclidean geometry4 ,s t to ask :at geometry x is prop] to adopt is to ask1 to :at l9e is x prop] to give ! "n /raight8 ,x is evid5t t exp]im5t c n settle su* a "q2 "o w\ld n ask1 = in/ance1 exp]im5t to decide :e!r ,i %\ld call ,,ab or ,,cd a /raight4 ,on ! o!r h&1 nei!r c ,i say t ,i h n ! "r to give ! "n ( /raights to ! sides ( non-,euclidean triangles because !y are n 9 con=mity ) ! et]nal idea ( /raight :i* ,i h by intuition4 ,i grant1 inde$1 t ,i h ! intuitive idea ( ! side ( ! ,euclidean triangle1 b ,i h equally ! intuitive idea ( ! side ( ! non-,euclidean triangle4 ,:y %\ld ,i h ! "r to apply ! "n ( /raight to ! fir/ ( ^! ideas & n to ! second8 ,:]e9 does ?is syllable =m an integrant "p ( ?is intuitive idea8 ,evid5tly :5 we say t ! ,euclidean /raight is a true /raight & t ! non-,euclidean /raight is n a true /raight1 we simply mean t ! fir/ intuitive idea corresponds to a m notewor?y object ?an ! second4 ,b h[ d we decide t ?is object is m notewor?y8 ,?is "q ,i h inve/igat$ 9 ',sci5ce & ,hypo!sis4' ,x is "h t we saw exp]i5ce come 94 ,if ! ,euclidean /raight is m notewor?y ?an ! non-,euclidean /raight1 x is s *iefly because x diff]s little f c]ta9 notewor?y natural objects f :i* ! non-,euclidean /raight diff]s greatly4 ,b1 x w be said1 ! def9ition ( ! non-,euclidean /raight is artificial2 if we = a mom5t adopt x1 we %all see t two circles ( diff]5t radius bo? receive ! "n ( non-,euclidean /raights1 :ile ( two circles ( ! same radius "o c satisfy ! def9ition )\t ! o!r be+ able to satisfy x1 & !n if we transport "o ( ^! s-call$ /raights )\t de=m+ x1 x w cease to be a /raight4 ,b by :at "r d we consid] z equal ^! two figures :i* ! ,euclidean geomet]s call two circles ) ! same radius8 ,x is because by transport+ "o ( !m )\t de=m+ x we c make x co9cide ) ! o!r4 ,& :y d we say ?is transportation is effect$ )\t de=mation8 ,x is impossible to give a good reason = x4 ,among all ! motions conceivable1 !re are "s ( :i* ! ,euclidean geomet]s say t !y are n accompani$ by de=mation2 b !re are o!rs ( :i* ! non-,euclidean geomet]s w\ld say t !y are n accompani$ by de=mation4 ,9 ! fir/1 call$ ,euclidean motions1 ! ,euclidean /raights rema9 ,euclidean /raights & ! non-,euclidean /raights d n rema9 non-,euclidean /raights2 9 ! motions ( ! second sort1 or non-,euclidean motions1 ! non-,euclidean /raights rema9 non-,euclidean /raights & ! ,euclidean /raights d n rema9 ,euclidean /raights4 ,x has1 !re=e1 n be5 demon/rat$ t x 0 unreasonable to call /raights ! sides ( non-,euclidean triangles2 x has only be5 %[n t t w\ld be unreasonable if "o cont9u$ to call ! ,euclidean motions motions )\t de=mation2 b x has at ! same "t be5 %[n t x w\ld be j z unreasonable to call /raights ! sides ( ,euclidean triangles if ! non-,euclidean motions 7 call$ motions )\t de=mation4 ,n[ :5 we say t ! ,euclidean motions are ! true motions )\t de=mation1 :at d we mean8 ,we simply mean t !y are m notewor?y ?an ! o!rs4 ,& :y are !y m notewor?y8 ,x is because c]ta9 notewor?y natural bodies1 ! solid bodies1 und]go motions almo/ simil>4 ,& !n :5 we ask3 ,c "o imag9e non-,euclidean space8 ,t means3 ,c we imag9e a _w :]e !re w\ld be notewor?y natural objects affect+ almo/ ! =m ( non-,euclidean /raights1 & notewor?y natural bodies frequ5tly und]go+ motions almo/ simil> to ! non-,euclidean motions8 ,i h %[n 9 ',sci5ce & ,hypo!sis' t to ?is "q we mu/ answ] yes4 ,x has (t5 be5 obs]v$ t if all ! bodies 9 ! univ]se 7 dilat$ simultane\sly & 9 ! same proportion1 we %\ld h no means ( p]ceiv+ x1 s9ce all our measur+ in/rum5ts w\ld gr[ at ! same "t z ! objects !mselves :i* !y s]ve to measure4 ,! _w1 aft] ?is dilatation1 w\ld cont9ue on its c\rse )\t any?+ appris+ u ( s consid]able an ev5t4 ,9 o!r words1 two worlds simil> to "o ano!r 7und]/&+ ! ^w similitude 9 ! s5se ( ,euclid1 ,book ,,vi47 w\ld be absolutely indi/+ui%able4 ,b m2 worlds w be indi/+ui%able n only if !y are equal or simil>1 t is1 if we c pass f "o to ! o!r by *ang+ ! axes ( coord9ates1 or by *ang+ ! scale to :i* l5g?s are ref]r$2 b !y w /ill be indi/+ui%able if we c pass f "o to ! o!r by any 'po9t-trans=mation' :atev]4 ,i w expla9 my mean+4 ,i suppose t to ea* po9t ( "o corresponds "o po9t ( ! o!r & only "o1 & inv]sely2 & besides t ! coord9ates ( a po9t are cont9u\s functions1 o!rwise altoge!r arbitr>y1 ( ! correspond+ po9t4 ,i suppose besides t to ea* object ( ! fir/ _w corresponds 9 ! second an object ( ! same nature plac$ precisely at ! correspond+ po9t4 ,i suppose f9ally t ?is correspond5ce fulfill$ at ! initial in/ant is ma9ta9$ indef9itely4 ,we %\ld h no means ( di/+ui%+ ^! two worlds "o f ! o!r4 ,! relativity ( space is n ord9>ily und]/ood 9 s broad a s5se2 x is ?us1 h[ev]1 t x w\ld be prop] to und]/& x4 ,if "o ( ^! univ]ses is our ,euclidean _w1 :at its inhabitants w call /raight w be our ,euclidean /raight2 b :at ! inhabitants ( ! second _w w call /raight w be a curve :i* w h ! same prop]ties 9 relation to ! _w !y inhabit & 9 relation to ! motions t !y w call motions )\t de=mation4 ,_! geometry w1 !re=e1 be ,euclidean geometry1 b _! /raight w n be our ,euclidean /raight4 ,x w be its trans=m by ! po9t-trans=mation :i* c>ries ov] f our _w to !irs4 ,! /raights ( ^! m5 w n be our /raights1 b !y w h among !mselves ! same relations z our /raights to "o ano!r4 ,x is 9 ?is s5se ,i say _! geometry w be ours4 ,if !n we wi% aft] all to proclaim t !y deceive !mselves1 t _! /raight is n ! true /raight1 if we /ill are unwill+ to admit t su* an affirmation has no mean+1 at lea/ we mu/ confess t ^! p h no means :atev] ( recogniz+ _! error4 #b4 ,qualitative ,geometry ,all t is relatively easy to und]/&1 & ,i h already s (t5 repeat$ x t ,i ?9k x ne$less to expatiate fur!r on ! matt]4 ,euclidean space is n a =m impos$ ^u our s5sibility1 s9ce we c imag9e non-,euclidean space2 b ! two spaces1 ,euclidean & non-,euclidean1 h a common basis1 t amorph\s cont9uum ( :i* ,i spoke 9 ! beg9n+4 ,f ?is cont9uum we c get ei!r ,euclidean space or ,loba*evskian space1 j z we c1 by trac+ ^u x a prop] graduation1 trans=m an ungraduat$ !rmomet] into a ,fahr5heit or a ,raumur !rmomet]4 ,& !n comes a "q3 ,is n ?is amorph\s cont9uum1 t our analysis has all[$ to survive1 a =m impos$ ^u our s5sibility8 ,if s1 we %\ld h enl>g$ ! prison 9 :i* ?is s5sibility is conf9$1 b x w\ld always be a prison4 ,?is cont9uum has a c]ta9 numb] ( prop]ties1 exempt f all idea ( measurem5t4 ,! /udy ( ^! prop]ties is ! object ( a sci5ce :i* has be5 cultivat$ by _m great geomet]s & 9 p>ticul> by ,riemann & ,betti & :i* has receiv$ ! "n ( analysis situs4 ,9 ?is sci5ce ab/raction is made ( e quantitative idea &1 = example1 if we asc]ta9 t on a l9e ! po9t ,b is betwe5 ! po9ts ,a & ,c1 we %all be cont5t ) ?is asc]ta9m5t & %all n tr\ble to "k :e!r ! l9e ,,abc is /raight or curv$1 nor :e!r ! l5g? ,,ab is equal to ! l5g? ,,bc1 or :e!r x is twice z great4 ,! !orems ( analysis situs h1 !re=e1 ?is peculi>ity1 t !y w\ld rema9 true if ! figures 7 copi$ by an inexp]t draftsman :o %\ld grossly *ange all ! proportions & replace ! /raights by l9es m or less s9u\s4 ,9 ma!matical t]ms1 !y are n alt]$ by any 'po9t-trans=mation' :atsoev]4 ,x has (t5 be5 said t metric geometry 0 quantitative1 :ile projective geometry 0 purely qualitative4 ,t is n altoge!r true4 ,! /raight is /ill di/+ui%$ f o!r l9es by prop]ties :i* rema9 quantitative 9 "s respects4 ,! real qualitative geometry is1 !re=e1 analysis situs4 ,! same que/ions :i* came up apropos ( ! tru?s ( ,euclidean geometry1 come up anew apropos ( ! !orems ( analysis situs4 ,are !y obta9able by d$uctive reason+8 ,are !y disguis$ conv5tions8 ,are !y exp]im5tal v]ities8 ,are !y ! *>act]i/ics ( a =m impos$ ei!r ^u our s5sibility or ^u our und]/&+8 ,i wi% simply to obs]ve t ! la/ two solutions exclude ea* o!r4 ,we c n admit at ! same "t t x is impossible to imag9e space ( f\r dim5sions & t exp]i5ce proves to u t space has ?ree dim5sions4 ,! exp]im5t] puts to nature a "q3 ,is x ?is or t8 & he c n put x )\t imag9+ ! two t]ms ( ! alt]native4 ,if x 7 impossible to imag9e "o ( ^! t]ms1 x w\ld be futile & besides impossible to consult exp]i5ce4 ,!re is no ne$ ( obs]vation to "k t ! h& ( a wat* is n m>k+ ! h\r #ae on ! dial1 because we "k be=eh& t !re are only #ab1 & we c\ld n look at ! m>k #ae to see if ! h& is !re1 because ?is m>k does n exi/4 ,note likewise t 9 analysis situs ! empirici/s are disemb>rass$ ( "o ( ! grave/ objections t c be level$ aga9/ !m1 ( t :i* r5d]s absolutely va9 9 advance all _! ef=ts to apply _! !sis to ! v]ities ( ,euclidean geometry4 ,^! v]ities are rigor\s & all exp]im5tation c only be approximate4 ,9 analysis situs approximate exp]im5ts may suffice to give a rigor\s !orem &1 = in/ance1 if x is se5 t space c n h ei!r two or less ?an two dim5sions1 nor f\r or m ?an f\r1 we are c]ta9 t x has exactly ?ree1 s9ce x c\ld n h two & a half or ?ree & a half4 ,( all ! !orems ( analysis situs1 ! mo/ important is t :i* is express$ 9 say+ t space has ?ree dim5sions4 ,?is x is t we are ab\t to consid]1 & we %all put ! "q 9 ^! t]ms3 ,:5 we say t space has ?ree dim5sions1 :at d we mean8 #c4 ,! ,physical ,cont9uum ( ,sev]al ,dim5sions ,i h expla9$ 9 ',sci5ce & ,hypo!sis' :5ce we d]ive ! notion ( physical cont9uity & h[ t ( ma!matical cont9uity has aris5 f x4 ,x happ5s t we are capable ( di/+ui%+ two impressions "o f ! o!r1 :ile ea* is indi/+ui%able f a ?ird4 ,?us we c readily di/+ui% a weight ( #ab grams f a weight ( #aj grams1 :ile a weight ( #aa grams c\ld be di/+ui%$ f nei!r ! "o nor ! o!r4 ,su* a /atem5t1 translat$ into symbols1 may be writt53 ,a ,b1 ,b ,c1 ,a ,c4 ,?is w\ld be ! =mula ( ! physical cont9uum1 z crude exp]i5ce gives x to u1 :5ce arises an intol]able contradiction t has be5 obviat$ by ! introduction ( ! ma!matical cont9uum4 ,?is is a scale ( :i* ! /eps 7comm5surable or incomm5surable numb]s7 are inf9ite 9 numb] b are ext]ior to "o ano!r1 in/ead ( encroa*+ on "o ano!r z d ! elem5ts ( ! physical cont9uum1 9 con=mity ) ! prec$+ =mula4 ,! physical cont9uum is1 s to speak1 a nebula n resolv$2 ! mo/ p]fect in/rum5ts c\ld n atta9 to its resolution4 ,d\btless if we measur$ ! weights ) a good balance in/ead ( judg+ !m by ! h&1 we c\ld di/+ui% ! weight ( #aa grams f ^? ( #aj & #ab grams1 & our =mula w\ld become3 ,a ,b1 ,b ,c1 ,a ,c4 ,b we %\ld always f9d betwe5 ,a & ,b & betwe5 ,b & ,c new elem5ts ,d & ,e1 su* t ,a ,d1 ,d ,b1 ,a ,b2 ,b ,e1 ,e ,c1 ,b ,c1 & ! difficulty w\ld only h rec$$ & ! nebula w\ld always rema9 unresolv$2 ! m9d alone c resolve x & ! ma!matical cont9uum x is :i* is ! nebula resolv$ into />s4 ,yet up to ?is po9t we h n introduc$ ! notion ( ! numb] ( dim5sions4 ,:at is meant :5 we say t a ma!matical cont9uum or t a physical cont9uum has two or ?ree dim5sions8 ,fir/ we mu/ introduce ! notion ( cut1 /udy+ fir/ physical cont9ua4 ,we h se5 :at *>act]izes ! physical cont9uum4 ,ea* ( ! elem5ts ( ?is cont9uum consi/s ( a manifold ( impressions2 & x may happ5 ei!r t an elem5t c n be discrim9at$ f ano!r elem5t ( ! same cont9uum1 if ?is new elem5t corresponds to a manifold ( impressions n suffici5tly diff]5t1 or1 on ! contr>y1 t ! discrim9ation is possible2 f9ally x may happ5 t two elem5ts indi/+ui%able f a ?ird may1 nev]!less1 be di/+ui%$ "o f ! o!r4 ,t po/ulat$1 if ,a & ,b are two di/+ui%able elem5ts ( a cont9uum ,c1 a s]ies ( elem5ts may be f\nd1 ,e#a1 ,e#b1 4441 ,en1 all belong+ to ?is same cont9uum ,c & su* t ea* ( !m is indi/+ui%able f ! prec$+1 t ,e#a is indi/+ui%able f ,a1 & ,en indi/+ui%able f ,b4 ,!re=e we c g f ,a to ,b by a cont9u\s r\te & )\t quitt+ ,c4 ,if ?is condition is fulfill$ = any two elem5ts ,a & ,b ( ! cont9uum ,c1 we may say t ?is cont9uum ,c is all 9 "o piece4 ,n[ let u di/+ui% c]ta9 ( ! elem5ts ( ,c :i* may ei!r be all di/+ui%able f "o ano!r1 or !mselves =m "o or sev]al cont9ua4 ,! assemblage ( ! elem5ts ?us *os5 arbitr>ily among all ^? ( ,c w =m :at ,i %all call ! cut or ! cuts4 ,take on ,c any two elem5ts ,a & ,b4 ,ei!r we c also f9d a s]ies ( elem5ts ,e#a1 ,e#b1 4441 ,en1 su*3 7#a7 t !y all belong to ,c2 7#b7 t ea* ( !m is indi/+ui%able f ! foll[+1 ,e#a indi/+ui%able f ,a & ,en f ,b2 7#c7 & besides t none ( ! elem5ts ,e is indi/+ui%able f any elem5t ( ! cut4 ,or else1 on ! contr>y1 9 ea* ( ! s]ies ,e#a1 ,e#b1 4441 ,en satisfy+ ! fir/ two conditions1 !re w be an elem5t ,e indi/+ui%able f "o ( ! elem5ts ( ! cut4 ,9 ! fir/ case we c g f ,a to ,b by a cont9u\s r\te )\t quitt+ ,c & )\t meet+ ! cuts2 9 ! second case t is impossible4 ,if !n = any two elem5ts ,a & ,b ( ! cont9uum ,c1 x is always ! fir/ case :i* pres5ts itself1 we %all say t ,c rema9s all 9 "o piece despite ! cuts4 ,?us1 if we *oose ! cuts 9 a c]ta9 way1 o!rwise arbitr>y1 x may happ5 ei!r t ! cont9uum rema9s all 9 "o piece or t x does n rema9 all 9 "o piece2 9 ?is latt] hypo!sis we %all !n say t x is divid$ by ! cuts4 ,x w be notic$ t all ^! def9itions are con/ruct$ 9 sett+ out solely f ?is v simple fact1 t two manifolds ( impressions sometimes c be discrim9at$1 sometimes c n be4 ,t po/ulat$1 if1 to divide a cont9uum1 x suffices to consid] z cuts a c]ta9 numb] ( elem5ts all di/+ui%able f "o ano!r1 we say t ?is cont9uum is ( "o dim5sion2 if1 on ! contr>y1 to divide a cont9uum1 x is necess>y to consid] z cuts a sy/em ( elem5ts !mselves =m+ "o or sev]al cont9ua1 we %all say t ?is cont9uum is ( sev]al dim5sions4 ,if to divide a cont9uum ,c1 cuts =m+ "o or sev]al cont9ua ( "o dim5sion suffice1 we %all say t ,c is a cont9uum ( two dim5sions2 if cuts suffice :i* =m "o or sev]al cont9ua ( two dim5sions at mo/1 we %all say t ,c is a cont9uum ( ?ree dim5sions2 & s on4 ,to ju/ify ?is def9ition x is prop] to see :e!r x is 9 ?is way t geomet]s introduce ! notion ( ?ree dim5sions at ! beg9n+ ( _! works4 ,n[1 :at d we see8 ,usually !y beg9 by def9+ surfaces z ! b\nd>ies ( solids or pieces ( space1 l9es z ! b\nd>ies ( surfaces1 po9ts z ! b\nd>ies ( l9es1 & !y affirm t ! same proc$ure c n be pu%$ fur!r4 ,?is is j ! idea giv5 above3 to divide space1 cuts t are call$ surfaces are necess>y2 to divide surfaces1 cuts t are call$ l9es are necess>y2 to divide l9es1 cuts t are call$ po9ts are necess>y2 we c g no fur!r1 ! po9t c n be divid$1 s ! po9t is n a cont9uum4 ,!n l9es :i* c be divid$ by cuts :i* are n cont9ua w be cont9ua ( "o dim5sion2 surfaces :i* c be divid$ by cont9u\s cuts ( "o dim5sion w be cont9ua ( two dim5sions2 f9ally1 space :i* c be divid$ by cont9u\s cuts ( two dim5sions w be a cont9uum ( ?ree dim5sions4 ,?us ! def9ition ,i h j giv5 does n diff] ess5tially f ! usual def9itions2 ,i h only endeavor$ to give x a =m applicable n to ! ma!matical cont9uum1 b to ! physical cont9uum1 :i* alone is susceptible ( repres5tation1 & yet to reta9 all its precision4 ,moreov]1 we see t ?is def9ition applies n alone to space2 t 9 all :i* falls "u our s5ses we f9d ! *>act]i/ics ( ! physical cont9uum1 :i* w\ld all[ ( ! same classification2 t x w\ld be easy to f9d !re examples ( cont9ua ( f\r1 ( five1 dim5sions1 9 ! s5se ( ! prec$+ def9ition2 su* examples occur ( !mselves to ! m9d4 ,i %\ld expla9 f9ally1 if ,i _h ! "t1 t ?is sci5ce1 ( :i* ,i spoke above & to :i* ,riemann gave ! "n ( analysis situs1 tea*es u to make di/9ctions among cont9ua ( ! same numb] ( dim5sions & t ! classification ( ^! cont9ua re/s also on ! consid]ation ( cuts4 ,f ?is notion has aris5 t ( ! ma!matical cont9uum ( sev]al dim5sions 9 ! same way t ! physical cont9uum ( "o dim5sion eng5d]$ ! ma!matical cont9uum ( "o dim5sion4 ,! =mula ,a ,c1 ,a ,b1 ,b ,c1 :i* summ$ up ! data ( crude exp]i5ce1 impli$ an intol]able contradiction4 ,to get free f x1 x 0 necess>y to introduce a new notion :ile /ill respect+ ! ess5tial *>act]i/ics ( ! physical cont9uum ( sev]al dim5sions4 ,! ma!matical cont9uum ( "o dim5sion admitt$ ( a scale ^: divisions1 inf9ite 9 numb]1 correspond$ to ! diff]5t values1 comm5surable or n1 ( "o same magnitude4 ,to h ! ma!matical cont9uum ( n dim5sions1 x w suffice to take n l scales ^: divisions correspond to diff]5t values ( n indep5d5t magnitudes call$ coord9ates4 ,we ?us %all h an image ( ! physical cont9uum ( n dim5sions1 & ?is image w be z fai?ful z x c be aft] ! det]m9ation n to all[ ! contradiction ( :i* ,i spoke above4 #d4 ,! ,notion ( ,po9t ,x seems n[ t ! "q we put to ourselves at ! />t is answ]$4 ,:5 we say t space has ?ree dim5sions1 x w be said1 we mean t ! manifold ( po9ts ( space satisfies ! def9ition we h j giv5 ( ! physical cont9uum ( ?ree dim5sions4 ,to be cont5t ) t w\ld be to suppose t we "k :at is ! manifold ( po9ts ( space1 or ev5 "o po9t ( space4 ,n[ t is n z simple z "o might ?9k4 ,e "o believes he kn[s :at a po9t is1 & x is j because we "k x too well t we ?9k !re is no ne$ ( def9+ x4 ,surely we c n be requir$ to "k h[ to def9e x1 because 9 go+ back f def9ition to def9ition a "t mu/ come :5 we mu/ /op4 ,b at :at mom5t %\ld we /op8 ,we %all /op fir/ :5 we rea* an object :i* falls "u our s5ses or t we c repres5t to ourselves2 def9ition !n w become useless2 we d n def9e ! %eep to a *ild2 we say to him3 ,see ! %eep4 ,s1 !n1 we %\ld ask ourselves if x is possible to repres5t to ourselves a po9t ( space4 ,^? :o answ] yes d n reflect t !y repres5t to !mselves 9 reality a :ite spot made ) ! *alk on a blackbo>d or a black spot made ) a p5 on :ite pap]1 & t !y c repres5t to !mselves only an object or r ! impressions t ?is object made on _! s5ses4 ,:5 !y try to repres5t to !mselves a po9t1 !y repres5t ! impressions t v little objects made !m feel4 ,x is ne$less to add t two diff]5t objects1 ?\gh bo? v little1 may produce extremely diff]5t impressions1 b ,i %all n dwell on ?is difficulty1 :i* w\ld /ill require "s discussion4 ,b x is n a "q ( t2 x does n suffice to repres5t "o po9t1 x is necess>y to repres5t a c]ta9 po9t & to h ! means ( di/+ui%+ x f an o!r po9t4 ,& 9 fact1 t we may be able to apply to a cont9uum ! rule ,i h above exp\nd$ & by :i* "o may recognize ! numb] ( its dim5sions1 we mu/ rely ^u ! fact t two elem5ts ( ?is cont9uum sometimes c & sometimes c n be di/+ui%$4 ,x is necess>y !re=e t we %\ld 9 c]ta9 cases "k h[ to repres5t to ourselves a specific elem5t & to di/+ui% x f an o!r elem5t4 ,! "q is to "k :e!r ! po9t t ,i repres5t$ to myself an h\r ago is ! same z ?is t ,i n[ repres5t to myself1 or :e!r x is a diff]5t po9t4 ,9 o!r words1 h[ d we "k :e!r ! po9t occupi$ by ! object ,a at ! in/ant alpha is ! same z ! po9t occupi$ by ! object ,b at ! in/ant beta1 or /ill bett]1 :at ?is means8 ,i am seat$ 9 my room2 an object is plac$ on my table2 dur+ a second ,i d n move1 no "o t\*es ! object4 ,i am tempt$ to say t ! po9t ,a :i* ?is object occupi$ at ! beg9n+ ( ?is second is id5tical ) ! po9t ,b :i* x occupies at its end4 ,n at all2 f ! po9t ,a to ! po9t ,b is #cj kilomet]s1 because ! object has be5 c>ri$ along 9 ! motion ( ! e>?4 ,we c n "k :e!r an object1 be x l>ge or small1 has n *ang$ its absolute position 9 space1 & n only c we n affirm x1 b ?is affirmation has no mean+ & 9 any case c n correspond to any repres5tation4 ,b !n we may ask ourselves if ! relative position ( an object ) reg>d to o!r objects has *ang$ or n1 & fir/ :e!r ! relative position ( ?is object ) reg>d to our body has *ang$4 ,if ! impressions ?is object makes ^u u h n *ang$1 we %all be incl9$ to judge t nei!r has ?is relative position *ang$2 if !y h *ang$1 we %all judge t ?is object has *ang$ ei!r 9 /ate or 9 relative position4 ,x rema9s to decide :i* ( ! two4 ,i h expla9$ 9 ',sci5ce & ,hypo!sis' h[ we h be5 l$ to di/+ui% ! *anges ( position4 ,moreov]1 ,i %all return to t fur!r on4 ,we come to "k1 !re=e1 :e!r ! relative position ( an object ) reg>d to our body has or has n rema9$ ! same4 ,if n[ we see t two objects h reta9$ _! relative position ) reg>d to our body1 we conclude t ! relative position ( ^! two objects ) reg>d to "o ano!r has n *ang$2 b we rea* ?is conclusion only by indirect reason+4 ,! only ?+ t we "k directly is ! relative position ( ! objects ) reg>d to our body4 ,a =tiori x is only by indirect reason+ t we ?9k we "k 7&1 moreov]1 ?is belief is delusive7 :e!r ! absolute position ( ! object has *ang$4 ,9 a ^w1 ! sy/em ( coord9ate axes to :i* we naturally ref] all ext]ior objects is a sy/em ( axes inv>iably b\nd to our body1 & c>ri$ ar\nd ) u4 ,x is impossible to repres5t to oneself absolute space2 :5 ,i try to repres5t to myself simultane\sly objects & myself 9 motion 9 absolute space1 9 reality ,i repres5t to myself my own self motionless & see+ move ar\nd me diff]5t objects & a man t is ext]ior to me1 b t ,i conv5e to call me4 ,w ! difficulty be solv$ if we agree to ref] ev]y?+ to ^! axes b\nd to our body8 ,%all we "k !n :at is a po9t ?us def9$ by its relative position ) reg>d to ourselves8 ,_m p]sons w answ] yes & w say t !y 'localize' ext]ior objects4 ,:at does ?is mean8 ,to localize an object simply means to repres5t to oneself ! movem5ts t w\ld be necess>y to rea* x4 ,i w expla9 myself4 ,x is n a "q ( repres5t+ ! movem5ts !mselves 9 space1 b solely ( repres5t+ to oneself ! muscul> s5sations :i* accompany ^! movem5ts & :i* d n presuppose ! preexi/5ce ( ! notion ( space4 ,if we suppose two diff]5t objects :i* successively occupy ! same relative position ) reg>d to ourselves1 ! impressions t ^! two objects make ^u u w be v diff]5t2 if we localize !m at ! same po9t1 ?is is simply because x is necess>y to make ! same movem5ts to rea* !m2 ap>t f t1 "o c n j see :at !y c\ld h 9 common4 ,b1 giv5 an object1 we c conceive _m diff]5t s]ies ( movem5ts :i* equally enable u to rea* x4 ,if !n we repres5t to ourselves a po9t by repres5t+ to ourselves ! s]ies ( muscul> s5sations :i* accompany ! movem5ts :i* enable u to rea* ?is po9t1 !re w be _m ways entirely diff]5t ( repres5t+ to oneself ! same po9t4 ,if "o is n satisfi$ ) ?is solution1 b wi%es1 = in/ance1 to br+ 9 ! visual s5sations along ) ! muscul> s5sations1 !re w be "o or two m ways ( repres5t+ to oneself ?is same po9t & ! difficulty w only be increas$4 ,9 any case ! foll[+ "q comes up3 ,:y d we ?9k t all ^! repres5tations s diff]5t f "o ano!r /ill repres5t ! same po9t8 ,ano!r rem>k3 ,i h j said t x is to our own body t we naturally ref] ext]ior objects2 t we c>ry ab\t ev]y:]e ) u a sy/em ( axes to :i* we ref] all ! po9ts ( space & t ?is sy/em ( axes seems to be inv>iably b\nd to our body4 ,x %\ld be notic$ t rigor\sly we c\ld n speak ( axes inv>iably b\nd to ! body unless ! diff]5t p>ts ( ?is body 7 !mselves inv>iably b\nd to "o ano!r4 ,z ?is is n ! case1 we ought1 be=e ref]r+ ext]ior objects to ^! fictiti\s axes1 to suppose our body br\ght back to ! initial attitude4 #e4 ,! ,notion ( ,displacem5t ,i h %[n 9 ',sci5ce & ,hypo!sis' ! prepond]ant rle play$ by ! movem5ts ( our body 9 ! g5esis ( ! notion ( space4 ,= a be+ completely immovable !re w\ld be nei!r space nor geometry2 9 va9 w\ld ext]ior objects be displac$ ab\t him1 ! v>iations :i* ^! displacem5ts w\ld make 9 8 impressions w\ld n be attribut$ by ?is be+ to *anges ( position1 b to simple *anges ( /ate2 ?is be+ w\ld h no means ( di/+ui%+ ^! two sorts ( *anges1 & ?is di/9ction1 fundam5tal = u1 w\ld h no mean+ = him4 ,! movem5ts t we impress ^u our memb]s h z effect ! v>y+ ( ! impressions produc$ on our s5ses by ext]nal objects2 o!r causes may likewise make !m v>y2 b we are l$ to di/+ui% ! *anges produc$ by our own motions & we easily discrim9ate !m = two reasons3 7#a7 because !y are volunt>y2 7#b7 because !y are accompani$ by muscul> s5sations4 ,s we naturally divide ! *anges t our impressions may und]go into two categories to :i* p]haps ,i h giv5 an inappropriate designation3 7#a7 ! int]nal *anges1 :i* are volunt>y & accompani$ by muscul> s5sations2 7#b7 ! ext]nal *anges1 hav+ ! opposite *>act]i/ics4 ,we !n obs]ve t among ! ext]nal *anges are "s :i* c be correct$1 ?anks to an int]nal *ange :i* br+s ev]y?+ back to ! primitive /ate2 o!rs c n be correct$ 9 ?is way 7x is ?us t1 :5 an ext]ior object is displac$1 we may !n by *ang+ our own position replace ourselves z reg>ds ?is object 9 ! same relative position z be=e1 s z to ree/abli% ! orig9al aggregate ( impressions2 if ?is object 0 n displac$1 b *ang$ its /ate1 t is impossible74 ,!nce comes a new di/9ction among ext]nal *anges3 ^? :i* may be s correct$ we call *anges ( position2 & ! o!rs1 *anges ( /ate4 ,?9k1 = example1 ( a sph]e ) "o hemisph]e blue & ! o!r r$2 x fir/ pres5ts to u ! blue hemisph]e1 !n x s revolves z to pres5t ! r$ hemisph]e4 ,n[ ?9k ( a sph]ical vase conta9+ a blue liquid :i* becomes r$ 9 consequ5ce ( a *emical reaction4 ,9 bo? cases ! s5sation ( r$ has replac$ t ( blue2 our s5ses h exp]i5c$ ! same impressions :i* h succe$$ ea* o!r 9 ! same ord]1 & yet ^! two *anges are reg>d$ by u z v diff]5t2 ! fir/ is a displacem5t1 ! second a *ange ( /ate4 ,:y8 ,because 9 ! fir/ case x is suffici5t = me to g ar\nd ! sph]e to place myself opposite ! blue hemisph]e & ree/abli% ! orig9al blue s5sation4 ,/ill m2 if ! two hemisph]es1 9 place ( be+ r$ & blue1 _h be5 yell[ & gre51 h[ %\ld ,i h int]pret$ ! revolution ( ! sph]e8 ,be=e1 ! r$ succe$$ ! blue1 n[ ! gre5 succe$s ! yell[2 & yet ,i say t ! two sph]es h und]gone ! same revolution1 t ea* has turn$ ab\t its axis2 yet ,i c n say t ! gre5 is to yell[ z ! r$ is to blue2 h[ !n am ,i l$ to decide t ! two sph]es h und]gone ! same displacem5t8 ,evid5tly because1 9 "o case z 9 ! o!r1 ,i am able to ree/abli% ! orig9al s5sation by go+ ar\nd ! sph]e1 by mak+ ! same movem5ts1 & ,i "k t ,i h made ! same movem5ts because ,i h felt ! same muscul> s5sations2 to "k x1 ,i d n ne$1 !re=e1 to "k geometry 9 advance & to repres5t to myself ! movem5ts ( my body 9 geometric space4 ,ano!r example3 ,an object is displac$ be=e my eye2 its image 0 fir/ =m$ at ! c5t] ( ! ret9a2 !n x is =m$ at ! bord]2 ! old s5sation 0 c>ri$ to me by a n]ve fib] end+ at ! c5t] ( ! ret9a2 ! new s5sation is c>ri$ to me by ano!r n]ve fib] />t+ f ! bord] ( ! ret9a2 ^! two s5sations are qualitatively diff]5t2 o!rwise1 h[ c\ld ,i di/+ui% !m8 ,:y !n am ,i l$ to decide t ^! two s5sations1 qualitatively diff]5t1 repres5t ! same image1 :i* has be5 displac$8 ,x is because ,i c foll[ ! object ) ! eye & by a displacem5t ( ! eye1 volunt>y & accompani$ by muscul> s5sations1 br+ back ! image to ! c5t] ( ! ret9a & ree/abli% ! primitive s5sation4 ,i suppose t ! image ( a r$ object has gone f ! c5t] ,a to ! bord] ,b ( ! ret9a1 !n t ! image ( a blue object goes 9 its turn f ! c5t] ,a to ! bord] ,b ( ! ret9a2 ,i %all decide t ^! two objects h und]gone ! same displacem5t4 ,:y8 ,because 9 bo? cases ,i %all h be5 able to ree/abli% ! primitive s5sation1 & t to d x ,i %all h _h to execute ! same movem5t ( ! eye1 & ,i %all "k t my eye has execut$ ! same movem5t because ,i %all h felt ! same muscul> s5sations4 ,if ,i c\ld n move my eye1 %\ld ,i h any reason to suppose t ! s5sation ( r$ at ! c5t] ( ! ret9a is to ! s5sation ( r$ at ! bord] ( ! ret9a z t ( blue at ! c5t] is to t ( blue at ! bord]8 ,i %\ld only h f\r s5sations qualitatively diff]5t1 & if ,i 7 ask$ if !y are connect$ by ! proportion ,i h j /at$1 ! "q w\ld seem to me ridicul\s1 j z if ,i 7 ask$ if !re is an analog\s proportion betwe5 an auditory s5sation1 a tactile s5sation & an olfactory s5sation4 ,let u n[ consid] ! int]nal *anges1 t is1 ^? :i* are produc$ by ! volunt>y movem5ts ( our body & :i* are accompani$ by muscul> *anges4 ,!y give rise to ! two foll[+ obs]vations1 analog\s to ^? we h j made on ! subject ( ext]nal *anges4 #a4 ,i may suppose t my body has mov$ f "o po9t to ano!r1 b t ! same attitude is reta9$2 all ! p>ts ( ! body h !re=e reta9$ or resum$ ! same relative situation1 al?\gh _! absolute situation 9 space may h v>i$4 ,i may suppose t n only has ! position ( my body *ang$1 b t its attitude is no long] ! same1 t1 = in/ance1 my arms :i* be=e 7 fold$ are n[ /ret*$ out4 ,i %\ld !re=e di/+ui% ! simple *anges ( position )\t *ange ( attitude1 & ! *anges ( attitude4 ,bo? w\ld appe> to me "u =m ( muscul> s5sations4 ,h[ !n am ,i l$ to di/+ui% !m8 ,x is t ! fir/ may s]ve to correct an ext]nal *ange1 & t ! o!rs c n1 or at lea/ c only give an imp]fect correction4 ,?is fact ,i proce$ to expla9 z ,i w\ld expla9 x to "s "o :o already knew geometry1 b x ne$ n !nce be conclud$ t x is necess>y already to "k geometry to make ?is di/9ction2 be=e kn[+ geometry ,i asc]ta9 ! fact 7exp]im5tally1 s to speak71 )\t be+ able to expla9 x4 ,b m]ely to make ! di/9ction betwe5 ! two k9ds ( *ange1 ,i d n ne$ to expla9 ! fact1 x suffices me to asc]ta9 x4 ,h[ev] t may be1 ! explanation is easy4 ,suppose t an ext]ior object is displac$2 if we wi% ! diff]5t p>ts ( our body to resume ) reg>d to ?is object _! initial relative position1 x is necess>y t ^! diff]5t p>ts %\ld h resum$ likewise _! initial relative position ) reg>d to "o ano!r4 ,only ! int]nal *anges :i* satisfy ?is latt] condition w be capable ( correct+ ! ext]nal *ange produc$ by ! displacem5t ( t object4 ,if1 !re=e1 ! relative position ( my eye ) reg>d to my f+] has *ang$1 ,i %all /ill be able to replace ! eye 9 its initial relative situation ) reg>d to ! object & ree/abli% ?us ! primitive visual s5sations1 b !n ! relative position ( ! f+] ) reg>d to ! object w h *ang$ & ! tactile s5sations w n be ree/abli%$4 #b4 ,we asc]ta9 likewise t ! same ext]nal *ange may be correct$ by two int]nal *anges correspond+ to diff]5t muscul> s5sations4 ,"h aga9 ,i c asc]ta9 ?is )\t kn[+ geometry2 & ,i h no ne$ ( any?+ else2 b ,i proce$ to give ! explanation ( ! fact1 employ+ geometrical language4 ,to g f ! position ,a to ! position ,b ,i may take sev]al r\tes4 ,to ! fir/ ( ^! r\tes w correspond a s]ies ,s ( muscul> s5sations2 to a second r\te w correspond ano!r s]ies ,s''1 ( muscul> s5sations :i* g5]ally w be completely diff]5t1 s9ce o!r muscles w be us$4 ,h[ am ,i l$ to reg>d ^! two s]ies ,s & ,s'' z correspond+ to ! same displacem5t ,,ab8 ,x is because ^! two s]ies are capable ( correct+ ! same ext]nal *ange4 ,ap>t f t1 !y h no?+ 9 common4 ,let u n[ consid] two ext]nal *anges3 alpha & beta1 :i* %all be1 = in/ance1 ! rotation ( a sph]e half blue1 half r$1 & t ( a sph]e half yell[1 half gre52 ^! two *anges h no?+ 9 common1 s9ce ! "o is = u ! pass+ ( blue into r$ & ! o!r ! pass+ ( yell[ into gre54 ,consid]1 on ! o!r h&1 two s]ies ( int]nal *anges ,s & ,s''2 l ! o!rs1 !y w h no?+ 9 common4 ,& yet ,i say t alpha & beta correspond to ! same displacem5t1 & t ,s & ,s'' correspond also to ! same displacem5t4 ,:y8 ,simply because ,s c correct alpha z well z beta & because alpha c be correct$ by ,s'' z well z by ,s4 ,& !n a "q sugge/s itself3 ,if ,i h asc]ta9$ t ,s corrects alpha & beta & t ,s'' corrects alpha1 am ,i c]ta9 t ,s'' likewise corrects beta8 ,exp]im5t alone c tea* u :e!r ?is law is v]ifi$4 ,if x 7 n v]ifi$1 at lea/ approximately1 !re w\ld be no geometry1 !re w\ld be no space1 because we %\ld h no m int]e/ 9 classify+ ! int]nal & ext]nal *anges z ,i h j done1 &1 = in/ance1 9 di/+ui%+ *anges ( /ate f *anges ( position4 ,x is int]e/+ to see :at has be5 ! rle ( exp]i5ce 9 all ?is4 ,x has %[n me t a c]ta9 law is approximately v]ifi$4 ,x has n told me h[ space is1 & t x satisfies ! condition 9 "q4 ,i knew1 9 fact1 be=e all exp]i5ce1 t space satisfi$ ?is condition or t x w\ld n be2 nor h ,i any "r to say t exp]i5ce told me t geometry is possible2 ,i v well see t geometry is possible1 s9ce x does n imply contradiction2 exp]i5ce only tells me t geometry is useful4 #f4 ,visual ,space ,al?\gh motor impressions h _h1 z ,i h j expla9$1 an altoge!r prepond]ant influ5ce 9 ! g5esis ( ! notion ( space1 :i* nev] w\ld h tak5 bir? )\t !m1 x w n be )\t int]e/ to exam9e also ! rle ( visual impressions & to inve/igate h[ _m dim5sions 'visual space' has1 & = t purpose to apply to ^! impressions ! def9ition ( #c4 ,a fir/ difficulty pres5ts itself3 consid] a r$ color s5sation affect+ a c]ta9 po9t ( ! ret9a2 & on ! o!r h& a blue color s5sation affect+ ! same po9t ( ! ret9a4 ,x is necess>y t we h "s means ( recogniz+ t ^! two s5sations1 qualitatively diff]5t1 h some?+ 9 common4 ,n[1 accord+ to ! consid]ations exp\nd$ 9 ! prec$+ p>agraph1 we h be5 able to recognize ?is only by ! movem5ts ( ! eye & ! obs]vations to :i* !y h giv5 rise4 ,if ! eye 7 immovable1 or if we 7 unconsci\s ( its movem5ts1 we %\ld n h be5 able to recognize t ^! two s5sations1 ( diff]5t quality1 _h some?+ 9 common2 we %\ld n h be5 able to dis5gage f !m :at gives !m a geometric "*4 ,! visual s5sations1 )\t ! muscul> s5sations1 w\ld h no?+ geometric1 s t x may be said !re is no pure visual space4 ,to d away ) ?is difficulty1 consid] only s5sations ( ! same nature1 r$ s5sations1 = in/ance1 diff]+ "o f ano!r only z reg>ds ! po9t ( ! ret9a t !y affect4 ,x is cle> t ,i h no reason = mak+ su* an arbitr>y *oice among all ! possible visual s5sations1 = ! purpose ( unit+ 9 ! same class all ! s5sations ( ! same color1 :atev] may be ! po9t ( ! ret9a affect$4 ,i %\ld nev] h dreamt ( x1 _h ,i n be=e le>n$1 by ! means we h j se51 to di/+ui% *anges ( /ate f *anges ( position1 t is1 if my eye 7 immovable4 ,two s5sations ( ! same color affect+ two diff]5t p>ts ( ! ret9a w\ld h appe>$ to me z qualitatively di/9ct1 j z two s5sations ( diff]5t color4 ,9 re/rict+ myself to r$ s5sations1 ,i !re=e impose ^u myself an artificial limitation & ,i neglect sy/ematically "o :ole side ( ! "q2 b x is only by ?is artifice t ,i am able to analyze visual space )\t m+l+ any motor s5sation4 ,imag9e a l9e trac$ on ! ret9a & divid+ 9 two its surface2 & set ap>t ! r$ s5sations affect+ a po9t ( ?is l9e1 or ^? diff]+ f !m too little to be di/+ui%$ f !m4 ,! aggregate ( ^! s5sations w =m a sort ( cut t ,i %all call ,c1 & x is cle> t ?is cut suffices to divide ! manifold ( possible r$ s5sations1 & t if ,i take two r$ s5sations affect+ two po9ts situat$ on "o side & ! o!r ( ! l9e1 ,i c n pass f "o ( ^! s5sations to ! o!r 9 a cont9u\s way )\t pass+ at a c]ta9 mom5t ?r\gh a s5sation belong+ to ! cut4 ,if1 !re=e1 ! cut has n dim5sions1 ! total manifold ( my r$ s5sations1 or if y wi%1 ! :ole visual space1 w h n #a4 ,n[1 ,i di/+ui% ! r$ s5sations affect+ a po9t ( ! cut ,c4 ,! assemblage ( ^! s5sations w =m a new cut ,c'4 ,x is cle> t ?is w divide ! cut ,c1 always giv+ to ! ^w divide ! same mean+4 ,if1 !re=e1 ! cut ,c' has n dim5sions1 ! cut ,c w h n #a & ! :ole ( visual space n #b4 ,if all ! r$ s5sations affect+ ! same po9t ( ! ret9a 7 reg>d$ z id5tical1 ! cut ,c' r$uc+ to a s+le elem5t w\ld h #j dim5sions1 & visual space w\ld h #b4 ,& yet mo/ (t5 x is said t ! eye gives u ! s5se ( a ?ird dim5sion1 & enables u 9 a c]ta9 measure to recognize ! di/ance ( objects4 ,:5 we seek to analyze ?is feel+1 we asc]ta9 t x r$uces ei!r to ! consci\sness ( ! conv]g5ce ( ! eyes1 or to t ( ! ef=t ( accommodation :i* ! cili>y muscle makes to focus ! image4 ,two r$ s5sations affect+ ! same po9t ( ! ret9a w !re=e be reg>d$ z id5tical only if !y are accompani$ by ! same s5sation ( conv]g5ce & also by ! same s5sation ( ef=t ( accommodation or at lea/ by s5sations ( conv]g5ce & accommodation s slightly diff]5t z to be indi/+ui%able4 ,on ?is acc\nt ! cut ,c' is itself a cont9uum & ! cut ,c has m ?an "o dim5sion4 ,b x happ5s precisely t exp]i5ce tea*es u t :5 two visual s5sations are accompani$ by ! same s5sation ( conv]g5ce1 !y are likewise accompani$ by ! same s5sation ( accommodation4 ,if !n we =m a new cut ,c'' ) all ^? ( ! s5sations ( ! cut ,c'1 :i* are accompani$ by a c]ta9 s5sation ( conv]g5ce1 9 accordance ) ! prec$+ law !y w all be indi/+ui%able & may be reg>d$ z id5tical4 ,!re=e ,c'' w n be a cont9uum & w h #j dim5sion2 & z ,c'' divides ,c' x w !nce result t ,c' has "o1 ,c two & ! :ole visual space ?ree dim5sions4 ,b w\ld x be ! same if exp]i5ce _h taught u ! contr>y & if a c]ta9 s5sation ( conv]g5ce 7 n always accompani$ by ! same s5sation ( accommodation8 ,9 ?is case two s5sations affect+ ! same po9t ( ! ret9a & accompani$ by ! same s5se ( conv]g5ce1 two s5sations :i* consequ5tly w\ld bo? app]ta9 to ! cut ,c''1 c\ld nev]!less be di/+ui%$ s9ce !y w\ld be accompani$ by two diff]5t s5sations ( accommodation4 ,!re=e ,c'' w\ld be 9 its turn a cont9uum & w\ld h "o dim5sion 7at lea/72 !n ,c' w\ld h two1 ,c ?ree & ! :ole visual space w\ld h f\r dim5sions4 ,w x !n be said t x is exp]i5ce :i* tea*es u t space has ?ree dim5sions1 s9ce x is 9 sett+ out f an exp]im5tal law t we h come to attribute ?ree to x8 ,b we h !re9 p]=m$1 s to speak1 only an exp]im5t 9 physiology2 & z also x w\ld suffice to fit ov] ! eyes glasses ( suitable con/ruction to put an end to ! accord betwe5 ! feel+s ( conv]g5ce & ( accommodation1 are we to say t putt+ on spectacles is en\gh to make space h f\r dim5sions & t ! optician :o con/ruct$ !m has giv5 "o m dim5sion to space8 ,evid5tly n2 all we c say is t exp]i5ce has taught u t x is conv5i5t to attribute ?ree dim5sions to space4 ,b visual space is only "o "p ( space1 & 9 ev5 ! notion ( ?is space !re is some?+ artificial1 z ,i h expla9$ at ! beg9n+4 ,! real space is motor space & ?is x is t we %all exam9e 9 ! foll[+ *apt]4 ,,*apt] ,,iv ,,space ,,& ,,its ,,?ree ,,dim5sions #a4 ,! ,gr\p ( ,displacem5ts ,let u sum up briefly ! results obta9$4 ,we propos$ to inve/igate :at 0 meant 9 say+ t space has ?ree dim5sions & we h ask$ fir/ :at is a physical cont9uum & :5 x may be said to h n dim5sions4 ,if we consid] diff]5t sy/ems ( impressions & comp>e !m ) "o ano!r1 we (t5 recognize t two ( ^! sy/ems ( impressions are indi/+ui%able 7:i* is ord9>ily express$ 9 say+ t !y are too close to "o ano!r1 & t our s5ses are too crude1 = u to di/+ui% !m7 & we asc]ta9 besides t two ( ^! sy/ems c sometimes be discrim9at$ f "o ano!r ?\gh indi/+ui%able f a ?ird sy/em4 ,9 t case we say ! manifold ( ^! sy/ems ( impressions =ms a physical cont9uum ,c4 ,& ea* ( ^! sy/ems is call$ an elem5t ( ! cont9uum ,c4 ,h[ _m dim5sions has ?is cont9uum8 ,take fir/ two elem5ts ,a & ,b ( ,c1 & suppose !re exi/s a s]ies ,sigma ( elem5ts1 all belong+ to ! cont9uum ,c1 ( su* a sort t ,a & ,b are ! two extreme t]ms ( ?is s]ies & t ea* t]m ( ! s]ies is indi/+ui%able f ! prec$+4 ,if su* a s]ies ,sigma c be f\nd1 we say t ,a & ,b are jo9$ to "o ano!r2 & if any two elem5ts ( ,c are jo9$ to "o ano!r1 we say t ,c is all ( "o piece4 ,n[ take on ! cont9uum ,c a c]ta9 numb] ( elem5ts 9 a way altoge!r arbitr>y4 ,! aggregate ( ^! elem5ts w be call$ a cut4 ,among ! v>i\s s]ies ,sigma :i* jo9 ,a to ,b1 we %all di/+ui% ^? ( :i* an elem5t is indi/+ui%able f "o ( ! elem5ts ( ! cut 7we %all say t ^! are !y :i* cut ! cut7 & ^? ( :i* all ! elem5ts are di/+ui%able f all ^? ( ! cut4 ,if all ! s]ies ,sigma :i* jo9 ,a to ,b cut ! cut1 we %all say t ,a & ,b are sep>at$ by ! cut1 & t ! cut divides ,c4 ,if we c n f9d on ,c two elem5ts :i* are sep>at$ by ! cut1 we %all say t ! cut does n divide ,c4 ,^! def9itions laid d[n1 if ! cont9uum ,c c be divid$ by cuts :i* d n !mselves =m a cont9uum1 ?is cont9uum ,c has only "o dim5sion2 9 ! contr>y case x has sev]al4 ,if a cut =m+ a cont9uum ( #a dim5sion suffices to divide ,c1 ,c w h #b dim5sions2 if a cut =m+ a cont9uum ( #b dim5sions suffices1 ,c w h #c dim5sions1 etc4 ,?anks to ^! def9itions1 we c always recognize h[ _m dim5sions any physical cont9uum has4 ,x only rema9s to f9d a physical cont9uum :i* is1 s to speak1 equival5t to space1 ( su* a sort t to e po9t ( space corresponds an elem5t ( ?is cont9uum1 & t to po9ts ( space v ne> "o ano!r correspond indi/+ui%able elem5ts4 ,space w h !n z _m dim5sions z ?is cont9uum4 ,! int]m$iation ( ?is physical cont9uum1 capable ( repres5tation1 is indisp5sable2 because we c n repres5t space to ourselves1 & t = a multitude ( reasons4 ,space is a ma!matical cont9uum1 x is inf9ite1 & we c repres5t to ourselves only physical cont9ua & f9ite objects4 ,! diff]5t elem5ts ( space1 :i* we call po9ts1 are all alike1 &1 to apply our def9ition1 x is necess>y t we "k h[ to di/+ui% ! elem5ts f "o ano!r1 at lea/ if !y are n too close4 ,f9ally absolute space is nons5se1 & x is necess>y = u to beg9 by ref]r+ space to a sy/em ( axes inv>iably b\nd to our body 7:i* we mu/ always suppose put back 9 ! initial attitude74 ,!n ,i h s\ght to =m ) our visual s5sations a physical cont9uum equival5t to space2 t c]ta9ly is easy & ?is example is p>ticul>ly appropriate = ! discussion ( ! numb] ( dim5sions2 ?is discussion has enabl$ u to see 9 :at measure x is all[able to say t 'visual space' has ?ree dim5sions4 ,only ?is solution is incomplete & artificial4 ,i h expla9$ :y1 & x is n on visual space b on motor space t x is necess>y to br+ our ef=ts to be>4 ,i h !n recall$ :at is ! orig9 ( ! di/9ction we make betwe5 *anges ( position & *anges ( /ate4 ,among ! *anges :i* occur 9 our impressions1 we di/+ui%1 fir/ ! int]nal *anges1 volunt>y & accompani$ by muscul> s5sations1 & ! ext]nal *anges1 hav+ opposite *>act]i/ics4 ,we asc]ta9 t x may happ5 t an ext]nal *ange may be correct$ by an int]nal *ange :i* ree/abli%es ! primitive s5sations4 ,! ext]nal *anges1 capable ( be+ correct$ by an int]nal *ange are call$ *anges ( position1 ^? n capable ( x are call$ *anges ( /ate4 ,! int]nal *anges capable ( correct+ an ext]nal *ange are call$ displacem5ts ( ! :ole body2 ! o!rs are call$ *anges ( attitude4 ,n[ let alpha & beta be two ext]nal *anges1 alpha' & beta' two int]nal *anges4 ,suppose t a may be correct$ ei!r by alpha' or by beta'1 & t alpha' c correct ei!r alpha or beta2 exp]i5ce tells u !n t beta' c likewise correct beta4 ,9 ?is case we say t alpha & beta correspond to ! same displacem5t & also t alpha' & beta' correspond to ! same displacem5t4 ,t po/ulat$1 we c imag9e a physical cont9uum :i* we %all call ! cont9uum or gr\p ( displacem5ts & :i* we %all def9e 9 ! foll[+ mann]4 ,! elem5ts ( ?is cont9uum %all be ! int]nal *anges capable ( correct+ an ext]nal *ange4 ,two ( ^! int]nal *anges alpha' & beta' %all be reg>d$ z indi/+ui%able3 7#a7 if !y are s naturally1 t is1 if !y are too close to "o ano!r2 7#b7 if alpha' is capable ( correct+ ! same ext]nal *ange z a ?ird int]nal *ange naturally indi/+ui%able f beta'4 ,9 ?is second case1 !y w be1 s to speak1 indi/+ui%able by conv5tion1 ,i mean by agree+ to disreg>d circum/ances :i* might di/+ui% !m4 ,our cont9uum is n[ entirely def9$1 s9ce we "k its elem5ts & h fix$ "u :at conditions !y may be reg>d$ z indi/+ui%able4 ,we ?us h all t is necess>y to apply our def9ition & det]m9e h[ _m dim5sions ?is cont9uum has4 ,we %all recognize t x has six4 ,! cont9uum ( displacem5ts is1 !re=e1 n equival5t to space1 s9ce ! numb] ( dim5sions is n ! same2 x is only relat$ to space4 ,n[ h[ d we "k t ?is cont9uum ( displacem5ts has six dim5sions8 ,we "k x by exp]i5ce4 ,x w\ld be easy to describe ! exp]im5ts by :i* we c\ld arrive at ?is result4 ,x w\ld be se5 t 9 ?is cont9uum cuts c be made :i* divide x & :i* are cont9ua2 t ^! cuts !mselves c be divid$ by o!r cuts ( ! second ord] :i* yet are cont9ua1 & t ?is w\ld /op only aft] cuts ( ! six? ord] :i* w\ld no long] be cont9ua4 ,f our def9itions t w\ld mean t ! gr\p ( displacem5ts has six dim5sions4 ,t w\ld be easy1 ,i h said1 b t w\ld be r long2 & w\ld x n be a little sup]ficial8 ,?is gr\p ( displacem5ts1 we h se51 is relat$ to space1 & space c\ld be d$uc$ f x1 b x is n equival5t to space1 s9ce x has n ! same numb] ( dim5sions2 & :5 we %all h %[n h[ ! notion ( ?is cont9uum c be =m$ & h[ t ( space may be d$uc$ f x1 x might always be ask$ :y space ( ?ree dim5sions is mu* m famili> to u ?an ?is cont9uum ( six dim5sions1 & consequ5tly d\bt$ :e!r x 0 by ?is det\r t ! notion ( space 0 =m$ 9 ! human m9d4 #b4 ,id5tity ( ,two ,po9ts ,:at is a po9t8 ,h[ d we "k :e!r two po9ts ( space are id5tical or diff]5t8 ,or1 9 o!r words1 :5 ,i say3 ,! object ,a occupi$ at ! in/ant alpha ! po9t :i* ! object ,b occupies at ! in/ant beta1 :at does t mean8 ,su* is ! problem we set ourselves 9 ! prec$+ *apt]1 #d4 ,z ,i h expla9$ x1 x is n a "q ( comp>+ ! positions ( ! objects ,a & ,b 9 absolute space2 ! "q !n w\ld manife/ly h no mean+4 ,x is a "q ( comp>+ ! positions ( ^! two objects ) reg>d to axes inv>iably b\nd to my body1 suppos+ always ?is body replac$ 9 ! same attitude4 ,i suppose t betwe5 ! in/ants alpha & beta ,i h mov$ nei!r my body nor my eye1 z ,i "k f my muscul> s5se4 ,nor h ,i mov$ ei!r my head1 my arm or my h&4 ,i asc]ta9 t at ! in/ant alpha impressions t ,i attribut$ to ! object ,a 7 transmitt$ to me1 "s by "o ( ! fib]s ( my optic n]ve1 ! o!rs by "o ( ! s5sitive tactile n]ves ( my f+]2 ,i asc]ta9 t at ! in/ant beta o!r impressions :i* ,i attribute to ! object ,b are transmitt$ to me1 "s by ?is same fib] ( ! optic n]ve1 ! o!rs by ?is same tactile n]ve4 ,"h ,i mu/ pause = an explanation2 h[ am ,i told t ?is impression :i* ,i attribute to ,a1 & t :i* ,i attribute to ,b1 impressions :i* are qualitatively diff]5t1 are transmitt$ to me by ! same n]ve8 ,mu/ we suppose1 to take = example ! visual s5sations1 t ,a produces two simultane\s s5sations1 a s5sation purely lum9\s a & a color$ s5sation a'1 t ,b produces 9 ! same way simultane\sly a lum9\s s5sation b & a color$ s5sation b'1 t if ^! diff]5t s5sations are transmitt$ to me by ! same ret9al fib]1 a is id5tical ) b1 b t 9 g5]al ! color$ s5sations a' & b' produc$ by diff]5t bodies are diff]5t8 ,9 t case x w\ld be ! id5tity ( ! s5sation a :i* accompanies a' ) ! s5sation b :i* accompanies b'1 :i* w\ld tell t all ^! s5sations are transmitt$ to me by ! same fib]4 ,h[ev] x may be ) ?is hypo!sis & al?\gh ,i am l$ to pref] to x o!rs consid]ably m complicat$1 x is c]ta9 t we are told 9 "s way t !re is some?+ 9 common betwe5 ^! s5sations a a' & b b'1 )\t :i* we %\ld h no means ( recogniz+ t ! object ,b has tak5 ! place ( ! object ,a4 ,!re=e ,i d n fur!r insi/ & ,i recall ! hypo!sis ,i h j made3 ,i suppose t ,i h asc]ta9$ t ! impressions :i* ,i attribute to ,b are transmitt$ to me at ! in/ant beta by ! same fib]s1 optic z well z tactile1 :i*1 at ! in/ant alpha1 _h transmitt$ to me ! impressions t ,i attribut$ to ,a4 ,if x is s1 we %all n hesitate to decl>e t ! po9t occupi$ by ,b at ! in/ant beta is id5tical ) ! po9t occupi$ by ,a at ! in/ant alpha4 ,i h j enunciat$ two conditions = ^! po9ts be+ id5tical2 "o is relative to sight1 ! o!r to t\*4 ,let u consid] !m sep>ately4 ,! fir/ is necess>y1 b is n suffici5t4 ,! second is at once necess>y & suffici5t4 ,a p]son kn[+ geometry c\ld easily expla9 ?is 9 ! foll[+ mann]3 ,let ,o be ! po9t ( ! ret9a :]e is =m$ at ! in/ant alpha ! image ( ! body ,a2 let ,m be ! po9t ( space occupi$ at ! in/ant alpha by ?is body ,a2 let ,m' be ! po9t ( space occupi$ at ! in/ant beta by ! body ,b4 ,= ?is body ,b to =m its image 9 ,o1 x is n necess>y t ! po9ts ,m & ,m' co9cide2 s9ce vision acts at a di/ance1 x suffices = ! ?ree po9ts ,o ,m ,m' to be 9 a /raight l9e4 ,?is condition t ! two objects =m _! image on ,o is !re=e necess>y1 b n suffici5t = ! po9ts ,m & ,m' to co9cide4 ,let n[ ,p be ! po9t occupi$ by my f+] & :]e x rema9s1 s9ce x does n budge4 ,z t\* does n act at a di/ance1 if ! body ,a t\*es my f+] at ! in/ant alpha1 x is because ,m & ,p co9cide2 if ,b t\*es my f+] at ! in/ant beta1 x is because ,m' & ,p co9cide4 ,!re=e ,m & ,m' co9cide4 ,?us ?is condition t if ,a t\*es my f+] at ! in/ant alpha1 ,b t\*es x at ! in/ant beta1 is at once necess>y & suffici5t = ,m & ,m' to co9cide4 ,b we :o1 z yet1 d n "k geometry c n reason ?us2 all t we c d is to asc]ta9 exp]im5tally t ! fir/ condition relative to sight may be fulfill$ )\t ! second1 :i* is relative to t\*1 b t ! second c n be fulfill$ )\t ! fir/4 ,suppose exp]i5ce _h taught u ! contr>y1 z might well be2 ?is hypo!sis conta9s no?+ absurd4 ,suppose1 !re=e1 t we _h asc]ta9$ exp]im5tally t ! condition relative to t\* may be fulfill$ )\t t ( sight be+ fulfill$ & t1 on ! contr>y1 t ( sight c n be fulfill$ )\t t ( t\* be+ also4 ,x is cle> t if ?is 7 s we %\ld conclude t x is t\* :i* may be ex]cis$ at a di/ance1 & t sight does n op]ate at a di/ance4 ,b ?is is n all2 up to ?is "t ,i h suppos$ t to det]m9e ! place ( an object ,i h made use only ( my eye & a s+le f+]2 b ,i c\ld j z well h employ$ o!r means1 = example1 all my o!r f+]s4 ,i suppose t my fir/ f+] receives at ! in/ant alpha a tactile impression :i* ,i attribute to ! object ,a4 ,i make a s]ies ( movem5ts1 correspond+ to a s]ies ,s ( muscul> s5sations4 ,aft] ^! movem5ts1 at ! in/ant alpha'1 my second f+] receives a tactile impression t ,i attribute likewise to ,a4 ,aft]w>d1 at ! in/ant beta1 )\t my hav+ budg$1 z my muscul> s5se tells me1 ?is same second f+] transmits to me anew a tactile impression :i* ,i attribute ?is "t to ! object ,b2 ,i !n make a s]ies ( movem5ts1 correspond+ to a s]ies ,s' ( muscul> s5sations4 ,i "k t ?is s]ies ,s' is ! inv]se ( ! s]ies ,s & corresponds to contr>y movem5ts4 ,i "k ?is because _m previ\s exp]i5ces h %[n me t if ,i made successively ! two s]ies ( movem5ts correspond+ to ,s & to ,s'1 ! primitive impressions w\ld be ree/abli%$1 9 o!r words1 t ! two s]ies mutually comp5sate4 ,t settl$1 %\ld ,i expect t at ! in/ant beta'1 :5 ! second s]ies ( movem5ts is end$1 my fir/ f+] w\ld feel a tactile impression attributable to ! object ,b8 ,to answ] ?is "q1 ^? already kn[+ geometry w\ld reason z foll[s3 ,!re are *ances t ! object ,a has n budg$1 betwe5 ! in/ants alpha & alpha'1 nor ! object ,b betwe5 ! in/ants beta & beta'2 assume ?is4 ,at ! in/ant alpha1 ! object ,a occupi$ a c]ta9 po9t ,m ( space4 ,n[ at ?is in/ant x t\*$ my fir/ f+]1 & z t\* does n op]ate at a di/ance1 my fir/ f+] 0 likewise at ! po9t ,m4 ,i aft]w>d made ! s]ies ,s ( movem5ts & at ! end ( ?is s]ies1 at ! in/ant alpha'1 ,i asc]ta9$ t ! object ,a t\*$ my second f+]4 ,i !nce conclude t ?is second f+] 0 !n at ,m1 t is1 t ! movem5ts ,s _h ! result ( br++ ! second f+] to ! place ( ! fir/4 ,at ! in/ant beta ! object ,b has come 9 contact ) my second f+]3 z ,i h n budg$1 ?is second f+] has rema9$ at ,m2 !re=e ! object ,b has come to ,m2 by hypo!sis x does n budge up to ! in/ant beta'4 ,b betwe5 ! in/ants beta & beta' ,i h made ! movem5ts ,s'2 z ^! movem5ts are ! inv]se ( ! movem5ts ,s1 !y mu/ h = effect br++ ! fir/ f+] 9 ! place ( ! second4 ,at ! in/ant beta' ?is fir/ f+] w1 !re=e1 be at ,m2 & z ! object ,b is likewise at ,m1 ?is object ,b w t\* my fir/ f+]4 ,to ! "q put1 ! answ] %\ld !re=e be yes4 ,we :o d n yet "k geometry c n reason ?us2 b we asc]ta9 t ?is anticipation is ord9>ily realiz$2 & we c always expla9 ! exceptions by say+ t ! object ,a has mov$ betwe5 ! in/ants alpha & alpha'1 or ! object ,b betwe5 ! in/ants beta & beta'4 ,b c\ld n exp]i5ce h giv5 a contr>y result8 ,w\ld ?is contr>y result h be5 absurd 9 itself8 ,evid5tly n4 ,:at %\ld we h done !n if exp]i5ce _h giv5 ?is contr>y result8 ,w\ld all geometry ?us h become impossible8 ,n ! lea/ 9 ! _w4 ,we %\ld h cont5t$ ourselves ) conclud+ t t\* c op]ate at a di/ance4 ,:5 ,i say1 t\* does n op]ate at a di/ance1 b sight op]ates at a di/ance1 ?is ass]tion has only "o mean+1 :i* is z foll[s3 ,to recognize :e!r ,b occupies at ! in/ant beta ! po9t occupi$ by ,a at ! in/ant alpha1 ,i c use a multitude ( diff]5t crit]ia4 ,9 "o my eye int]v5es1 9 ano!r my fir/ f+]1 9 ano!r my second f+]1 etc4 ,well1 x is suffici5t = ! crit]ion relative to "o ( my f+]s to be satisfi$ 9 ord] t all ! o!rs %\ld be satisfi$1 b x is n suffici5t t ! crit]ion relative to ! eye %\ld be4 ,?is is ! s5se ( my ass]tion4 ,i cont5t myself ) affirm+ an exp]im5tal fact :i* is ord9>ily v]ifi$4 ,at ! end ( ! prec$+ *apt] we analyz$ visual space2 we saw t to eng5d] ?is space x is necess>y to br+ 9 ! ret9al s5sations1 ! s5sation ( conv]g5ce & ! s5sation ( accommodation2 t if ^! la/ two 7 n always 9 accord1 visual space w\ld h f\r dim5sions 9 place ( ?ree2 we also saw t if we br\ght 9 only ! ret9al s5sations1 we %\ld obta9 'simple visual space1' ( only two dim5sions4 ,on ! o!r h&1 consid] tactile space1 limit+ ourselves to ! s5sations ( a s+le f+]1 t is 9 sum to ! assemblage ( positions ?is f+] c occupy4 ,?is tactile space t we %all analyze 9 ! foll[+ section & :i* consequ5tly ,i ask p]mission n to consid] fur!r = ! mom5t1 ?is tactile space1 ,i say1 has ?ree dim5sions4 ,:y has space prop]ly s call$ z _m dim5sions z tactile space & m ?an simple visual space8 ,x is because t\* does n op]ate at a di/ance1 :ile vision does op]ate at a di/ance4 ,^! two ass]tions h ! same mean+ & we h j se5 :at ?is is4 ,n[ ,i return to a po9t ov] :i* ,i pass$ rapidly 9 ord] n to int]rupt ! discussion4 ,h[ d we "k t ! impressions made on our ret9a by ,a at ! in/ant alpha & ,b at ! in/ant beta are transmitt$ by ! same ret9al fib]1 al?\gh ^! impressions are qualitatively diff]5t8 ,i h sugge/$ a simple hypo!sis1 :ile add+ t o!r hypo!ses1 decid$ly m complex1 w\ld seem to me m probably true4 ,"h !n are ^! hypo!ses1 ( :i* ,i h already said a ^w4 ,h[ d we "k t ! impressions produc$ by ! r$ object ,a at ! in/ant alpha1 & by ! blue object ,b at ! in/ant beta1 if ^! two objects h be5 imag$ on ! same po9t ( ! ret9a1 h some?+ 9 common8 ,! simple hypo!sis above made may be reject$ & we may suppose t ^! two impressions1 qualitatively diff]5t1 are transmitt$ by two diff]5t ?\gh contigu\s n]v\s fib]s4 ,:at means h ,i !n ( kn[+ t ^! fib]s are contigu\s8 ,x is probable t we %\ld h none1 if ! eye 7 immovable4 ,x is ! movem5ts ( ! eye :i* h told u t !re is ! same relation betwe5 ! s5sation ( blue at ! po9t ,a & ! s5sation ( blue at ! po9t ,b ( ! ret9a z betwe5 ! s5sation ( r$ at ! po9t ,a & ! s5sation ( r$ at ! po9t ,b4 ,!y h %[n u1 9 fact1 t ! same movem5ts1 correspond+ to ! same muscul> s5sations1 c>ry u f ! fir/ to ! second1 or f ! ?ird to ! f\r?4 ,i d n emphasize ^! consid]ations1 :i* belong1 z "o sees1 to ! "q ( local signs rais$ by ,lotze4 #c4 ,tactile ,space ,?us ,i "k h[ to recognize ! id5tity ( two po9ts1 ! po9t occupi$ by ,a at ! in/ant alpha & ! po9t occupi$ by ,b at ! in/ant beta1 b only on "o condition1 namely1 t ,i h n budg$ betwe5 ! in/ants alpha & beta4 ,t does n suffice = our object4 ,suppose1 !re=e1 t ,i h mov$ 9 any mann] 9 ! int]val betwe5 ^! two in/ants1 h[ %all ,i "k :e!r ! po9t occupi$ by ,a at ! in/ant alpha is id5tical ) ! po9t occupi$ by ,b at ! in/ant beta8 ,i suppose t at ! in/ant alpha1 ! object ,a 0 9 contact ) my fir/ f+] & t 9 ! same way1 at ! in/ant beta1 ! object ,b t\*es ?is fir/ f+]2 b at ! same "t my muscul> s5se has told me t 9 ! int]val my body has mov$4 ,i h consid]$ above two s]ies ( muscul> s5sations ,s & ,s'1 & ,i h said x sometimes happ5s t we are l$ to consid] two su* s]ies ,s & ,s' z inv]se "o ( ! o!r1 because we h (t5 obs]v$ t :5 ^! two s]ies succe$ "o ano!r our primitive impressions are ree/abli%$4 ,if !n my muscul> s5se tells me t ,i h mov$ betwe5 ! two in/ants alpha & beta1 b s z to feel successively ! two s]ies ( muscul> s5sations ,s & ,s' t ,i consid] inv]ses1 ,i %all /ill conclude1 j z if ,i _h n budg$1 t ! po9ts occupi$ by ,a at ! in/ant alpha & by ,b at ! in/ant beta are id5tical1 if ,i asc]ta9 t my fir/ f+] t\*es ,a at ! in/ant alpha1 & ,b at ! in/ant beta4 ,?is solution is n yet completely satisfactory1 z "o w see4 ,let u see1 9 fact1 h[ _m dim5sions x w\ld make u attribute to space4 ,i wi% to comp>e ! two po9ts occupi$ by ,a & ,b at ! in/ants alpha & beta1 or 7:at am\nts to ! same ?+ s9ce ,i suppose t my f+] t\*es ,a at ! in/ant alpha & ,b at ! in/ant beta7 ,i wi% to comp>e ! two po9ts occupi$ by my f+] at ! two in/ants alpha & beta4 ,! sole means ,i use = ?is comp>ison is ! s]ies ,sigma ( muscul> s5sations :i* h accompani$ ! movem5ts ( my body betwe5 ^! two in/ants4 ,! diff]5t imag9able s]ies ,sigma =m evid5tly a physical cont9uum ( :i* ! numb] ( dim5sions is v great4 ,let u agree1 z ,i h done1 n to consid] z di/9ct ! two s]ies ,sigma & ,sigma ,s ,s'1 :5 ,s & ,s' are inv]ses "o ( ! o!r 9 ! s5se above giv5 to ?is ^w2 9 spite ( ?is agreem5t1 ! aggregate ( di/9ct s]ies ,sigma w /ill =m a physical cont9uum & ! numb] ( dim5sions w be less b /ill v great4 ,to ea* ( ^! s]ies ,sigma corresponds a po9t ( space2 to two s]ies ,sigma & ,sigma' ?us correspond two po9ts ,m & ,m'4 ,! means we h hi!rto us$ enable u to recognize t ,m & ,m' are n di/9ct 9 two cases3 7#a7 if ,sigma is id5tical ) ,sigma'2 7#b7 if ,sigma' ,sigma ,s ,s'1 ,s & ,s' be+ inv]ses "o ( ! o!r4 ,if 9 all ! o!r cases we %\ld reg>d ,m & ,m' z di/9ct1 ! manifold ( po9ts w\ld h z _m dim5sions z ! aggregate ( di/9ct s]ies ,sigma1 t is1 mu* m ?an ?ree4 ,= ^? :o already "k geometry1 ! foll[+ explanation w\ld be easily compreh5sible4 ,among ! imag9able s]ies ( muscul> s5sations1 !re are ^? :i* correspond to s]ies ( movem5ts :]e ! f+] does n budge4 ,i say t if "o does n consid] z di/9ct ! s]ies ,sigma & ,sigma sigma1 :]e ! s]ies sigma corresponds to movem5ts :]e ! f+] does n budge1 ! aggregate ( s]ies w con/itute a cont9uum ( ?ree dim5sions1 b t if "o reg>ds z di/9ct two s]ies ,sigma & ,sigma' unless ,sigma' ,sigma ,s ,s'1 ,s & ,s' be+ inv]ses1 ! aggregate ( s]ies w con/itute a cont9uum ( m ?an ?ree dim5sions4 ,9 fact1 let !re be 9 space a surface ,a1 on ?is surface a l9e ,b1 on ?is l9e a po9t ,m4 ,let ,c#j be ! aggregate ( all s]ies ,sigma4 ,let ,c#a be ! aggregate ( all ! s]ies ,sigma1 su* t at ! end ( correspond+ movem5ts ! f+] is f\nd ^u ! surface ,a1 & ,c#b or ,c#c ! aggregate ( s]ies ,sigma su* t at ! end ! f+] is f\nd on ,b1 or at ,m4 ,x is cle>1 fir/ t ,c#a w con/itute a cut :i* w divide ,c#j1 t ,c#b w be a cut :i* w divide ,c#a1 & ,c#c a cut :i* w divide ,c#b4 ,!nce x results1 9 accordance ) our def9itions1 t if ,c#c is a cont9uum ( n dim5sions1 ,c#j w be a physical cont9uum ( n #c dim5sions4 ,!re=e1 let ,sigma & ,sigma' ,sigma sigma be two s]ies =m+ "p ( ,c#c2 = bo?1 at ! end ( ! movem5ts1 ! f+] is f\nd at ,m2 !nce results t at ! beg9n+ & at ! end ( ! s]ies sigma ! f+] is at ! same po9t ,m4 ,?is s]ies sigma is !re=e "o ( ^? :i* correspond to movem5ts :]e ! f+] does n budge4 ,if ,sigma & ,sigma sigma are n reg>d$ z di/9ct1 all ! s]ies ( ,c#c bl5d into "o2 !re=e ,c#c w h #j dim5sion1 & ,c#j w h #c1 z ,i wi%$ to prove4 ,if1 on ! contr>y1 ,i d n reg>d ,sigma & ,sigma sigma z bl5d+ 7unless sigma ,s ,s'1 ,s & ,s' be+ inv]ses71 x is cle> t ,c#c w conta9 a great numb] ( s]ies ( di/9ct s5sations2 because1 )\t ! f+] budg+1 ! body may take a multitude ( diff]5t attitudes4 ,!n ,c#c w =m a cont9uum & ,c#j w h m ?an ?ree dim5sions1 & ?is also ,i wi%$ to prove4 ,we :o d n yet "k geometry c n reason 9 ?is way2 we c only v]ify4 ,b !n a "q arises2 h[1 be=e kn[+ geometry1 h we be5 l$ to di/+ui% f ! o!rs ^! s]ies sigma :]e ! f+] does n budge8 ,x is1 9 fact1 only aft] hav+ made ?is di/9ction t we c\ld be l$ to reg>d ,sigma & ,sigma sigma z id5tical1 & x is on ?is condition alone1 z we h j se51 t we c arrive at space ( ?ree dim5sions4 ,we are l$ to di/+ui% ! s]ies sigma1 because x (t5 happ5s t :5 we h execut$ ! movem5ts :i* correspond to ^! s]ies sigma ( muscul> s5sations1 ! tactile s5sations :i* are transmitt$ to u by ! n]ve ( ! f+] t we h call$ ! fir/ f+]1 p]si/ & are n alt]$ by ^! movem5ts4 ,exp]i5ce alone tells u t & x alone c\ld tell u4 ,if we h di/+ui%$ ! s]ies ( muscul> s5sations ,s ,s' =m$ by ! union ( two inv]se s]ies1 x is because !y pres]ve ! totality ( our impressions2 if n[ we di/+ui% ! s]ies sigma1 x is because !y pres]ve c]ta9 ( our impressions4 7,:5 ,i say t a s]ies ( muscul> s5sations ,s 'pres]ves' "o ( our impressions ,a1 ,i mean t we asc]ta9 t if we feel ! impression ,a1 !n ! muscul> s5sations ,s1 we /ill feel ! impression ,a aft] ^! s5sations ,s47 ,i h said above x (t5 happ5s t ! s]ies sigma d n alt] ! tactile impressions felt by our fir/ f+]2 ,i said (t51 ,i did n say always4 ,?is x is t we express 9 our ord9>y language by say+ t ! tactile impressions w\ld n be alt]$ if ! f+] has n mov$1 on ! condition t nei!r has ! object ,a1 :i* 0 9 contact ) ?is f+]1 mov$4 ,be=e kn[+ geometry1 we c\ld n give ?is explanation2 all we c\ld d is to asc]ta9 t ! impression (t5 p]si/s1 b n always4 ,b t ! impression (t5 cont9ues is en\gh to make ! s]ies sigma appe> rem>kable to u1 to lead u to put 9 ! same class ! s]ies ,sigma & ,sigma sigma1 & h5ce n reg>d !m z di/9ct4 ,"u ^! conditions we h se5 t !y w eng5d] a physical cont9uum ( ?ree dim5sions4 ,behold !n a space ( ?ree dim5sions eng5d]$ by my fir/ f+]4 ,ea* ( my f+]s w create "o l x4 ,x rema9s to consid] h[ we are l$ to reg>d !m z id5tical ) visual space1 z id5tical ) geometric space4 ,b "o reflection be=e go+ fur!r2 accord+ to ! =ego+1 we "k ! po9ts ( space1 or m g5]ally ! f9al situation ( our body1 only by ! s]ies ( muscul> s5sations reveal+ to u ! movem5ts :i* h c>ri$ u f a c]ta9 initial situation to ?is f9al situation4 ,b x is cle> t ?is f9al situation w dep5d1 on ! "o h&1 ^u ^! movem5ts &1 on ! o!r h&1 ^u ! initial situation f :i* we set out4 ,n[ ^! movem5ts are reveal$ to u by our muscul> s5sations2 b no?+ tells u ! initial situation2 no?+ c di/+ui% x = u f all ! o!r possible situations4 ,?is puts well 9 evid5ce ! ess5tial relativity ( space4 #d4 ,id5tity ( ! ,diff]5t ,spaces ,we are !re=e l$ to comp>e ! two cont9ua ,c & ,c' eng5d]$1 = in/ance1 "o by my fir/ f+] ,d1 ! o!r by my second f+] ,d'4 ,^! two physical cont9ua bo? h ?ree dim5sions4 ,to ea* elem5t ( ! cont9uum ,c1 or1 if y pref]1 to ea* po9t ( ! fir/ tactile space1 corresponds a s]ies ( muscul> s5sations ,sigma1 :i* c>ry me f a c]ta9 initial situation to a c]ta9 f9al situation4#h ,moreov]1 ! same po9t ( ?is fir/ space w correspond to ,sigma & ,sigma sigma1 if sigma is a s]ies ( :i* we "k t x does n make ! f+] ,d move4 #h ,9 place ( say+ t we ref] space to axes rigidly b\nd to our body1 p]haps x w\ld be bett] to say1 9 con=mity to :at prec$es1 t we ref] x to axes rigidly b\nd to ! initial situation ( our body4 ,simil>ly to ea* elem5t ( ! cont9uum ,c'1 or to ea* po9t ( ! second tactile space1 corresponds a s]ies ( s5sations ,sigma'1 & ! same po9t w correspond to ,sigma' & to ,sigma' sigma'1 if sigma' is a s]ies :i* does n make ! f+] ,d' move4 ,:at makes u di/+ui% ! v>i\s s]ies designat$ sigma f ^? call$ sigma' is t ! fir/ d n alt] ! tactile impressions felt by ! f+] ,d & ! second pres]ve ^? ! f+] ,d' feels4 ,n[ see :at we asc]ta93 9 ! beg9n+ my f+] ,d' feels a s5sation ,a'2 ,i make movem5ts :i* produce muscul> s5sations ,s2 my f+] ,d feels ! impression ,a2 ,i make movem5ts :i* produce a s]ies ( s5sations sigma2 my f+] ,d cont9ues to feel ! impression ,a1 s9ce ?is is ! *>act]i/ic prop]ty ( ! s]ies sigma2 ,i !n make movem5ts :i* produce ! s]ies ,s' ( muscul> s5sations1 inv]se to ,s 9 ! s5se above giv5 to ?is ^w4 ,i asc]ta9 !n t my f+] ,d' feels anew ! impression ,a'4 7,x is ( c\rse und]/ood t ,s has be5 suitably *os547 ,?is means t ! s]ies ,s sigma ,s'1 pres]v+ ! tactile impressions ( ! f+] ,d'1 is "o ( ! s]ies ,i h call$ sigma'4 ,inv]sely1 if "o takes any s]ies sigma'1 ,s' sigma' ,s w be "o ( ! s]ies t we call sigma'4 ,?us if ,s is suitably *os51 ,s sigma ,s' w be a s]ies sigma'1 & by mak+ sigma v>y 9 all possible ways1 we %all obta9 all ! possible s]ies sigma'4 ,n yet kn[+ geometry1 we limit ourselves to v]ify+ all t1 b "h is h[ ^? :o "k geometry w\ld expla9 ! fact4 ,9 ! beg9n+ my f+] ,d' is at ! po9t ,m1 9 contact ) ! object a1 :i* makes x feel ! impression ,a'4 ,i make ! movem5ts correspond+ to ! s]ies ,s2 ,i h said t ?is s]ies %\ld be suitably *os51 ,i %\ld s make ?is *oice t ^! movem5ts c>ry ! f+] ,d to ! po9t orig9ally occupi$ by ! f+] ,d'1 t is1 to ! po9t ,m2 ?is f+] ,d w ?us be 9 contact ) ! object a1 :i* w make x feel ! impression ,a4 ,i !n make ! movem5ts correspond+ to ! s]ies sigma2 9 ^! movem5ts1 by hypo!sis1 ! position ( ! f+] ,d does n *ange1 ?is f+] !re=e rema9s 9 contact ) ! object a & cont9ues to feel ! impression ,a4 ,f9ally ,i make ! movem5ts correspond+ to ! s]ies ,s'4 ,z ,s' is inv]se to ,s1 ^! movem5ts c>ry ! f+] ,d' to ! po9t previ\sly occupi$ by ! f+] ,d1 t is1 to ! po9t ,m4 ,if1 z may be suppos$1 ! object a has n budg$1 ?is f+] ,d' w be 9 contact ) ?is object & w feel anew ! impression ,a'4444 ,q4,e4,d4 ,let u see ! consequ5ces4 ,i consid] a s]ies ( muscul> s5sations ,sigma4 ,to ?is s]ies w correspond a po9t ,m ( ! fir/ tactile space4 ,n[ take aga9 ! two s]ies ,s & ,s'1 inv]ses ( "o ano!r1 ( :i* we h j spok54 ,to ! s]ies ,s ,sigma ,s' w correspond a po9t ,n ( ! second tactile space1 s9ce to any s]ies ( muscul> s5sations corresponds1 z we h said1 a po9t1 :e!r 9 ! fir/ space or 9 ! second4 ,i am go+ to consid] ! two po9ts ,n & ,m1 ?us def9$1 z correspond+4 ,:at au?orizes me s to d8 ,= ?is correspond5ce to be admissible1 x is necess>y t if two po9ts ,m & ,m'1 correspond+ 9 ! fir/ space to two s]ies ,sigma & ,sigma'1 are id5tical1 s also are ! two correspond+ po9ts ( ! second space ,n & ,n'1 t is1 ! two po9ts :i* correspond to ! two s]ies ,s ,sigma ,s' & ,s ,sigma' ,s'4 ,n[ we %all see t ?is condition is fulfill$4 ,fir/ a rem>k4 ,z ,s & ,s' are inv]ses ( "o ano!r1 we %all h ,s ,s' #j1 & consequ5tly ,s ,s' ,sigma ,sigma ,s ,s' ,sigma1 or aga9 ,sigma ,s ,s' ,sigma' ,sigma ,sigma'2 b x does n foll[ t we h ,s ,sigma ,s' ,sigma2 because1 ?\gh we h us$ ! addition sign to repres5t ! succession ( our s5sations1 x is cle> t ! ord] ( ?is succession is n indiff]5t3 we c n1 !re=e1 z 9 ord9>y addition1 inv]t ! ord] ( ! t]ms2 to use abridg$ language1 our op]ations are associative1 b n commutative4 ,t fix$1 9 ord] t ,sigma & ,sigma' %\ld correspond to ! same po9t ,m ,m' ( ! fir/ space1 x is necess>y & suffici5t = u to h ,sigma' ,sigma sigma4 ,we %all !n h3 ,s ,sigma' ,s' ,s ,sigma sigma ,s' ,s ,sigma ,s' ,s sigma ,s'4 ,b we h j asc]ta9$ t ,s sigma ,s' 0 "o ( ! s]ies sigma'4 ,we %all !re=e h3 ,s ,sigma' ,s' ,s ,sigma ,s' sigma'1 :i* means t ! s]ies ,s ,sigma' ,s' & ,s ,sigma ,s' correspond to ! same po9t ,n ,n' ( ! second space4 ,q4,e4,d4 ,our two spaces !re=e correspond po9t = po9t2 !y c be 'trans=m$' "o into ! o!r2 !y are isomorphic4 ,h[ are we l$ to conclude !nce t !y are id5tical8 ,consid] ! two s]ies sigma & ,s sigma ,s' sigma'4 ,i h said t (t51 b n always1 ! s]ies sigma pres]ves ! tactile impression ,a felt by ! f+] ,d2 & simil>ly x (t5 happ5s1 b n always1 t ! s]ies sigma' pres]ves ! tactile impression ,a' felt by ! f+] ,d'4 ,n[ ,i asc]ta9 t x happ5s v (t5 7t is1 mu* m (t5 ?an :at ,i h j call$ '(t5'7 t :5 ! s]ies sigma has pres]v$ ! impression ,a ( ! f+] ,d1 ! s]ies sigma' pres]ves at ! same "t ! impression ,a' ( ! f+] ,d'2 &1 inv]sely1 t if ! fir/ impression is alt]$1 ! second is likewise4 ,t happ5s v (t51 b n always4 ,we int]pret ?is exp]im5tal fact by say+ t ! unkn[n object a :i* gives ! impression ,a to ! f+] ,d is id5tical ) ! unkn[n object a' :i* gives ! impression ,a' to ! f+] ,d'4 ,& 9 fact :5 ! fir/ object moves1 :i* ! disappe>ance ( ! impression ,a tells u1 ! second likewise moves1 s9ce ! impression ,a' disappe>s likewise4 ,:5 ! fir/ object rema9s motionless1 ! second rema9s motionless4 ,if ^! two objects are id5tical1 z ! fir/ is at ! po9t ,m ( ! fir/ space & ! second at ! po9t ,n ( ! second space1 ^! two po9ts are id5tical4 ,?is is h[ we are l$ to reg>d ^! two spaces z id5tical2 or bett]1 ?is is :at we mean :5 we say t !y are id5tical4 ,:at we h j said ( ! id5tity ( ! two tactile spaces makes unnecess>y our discuss+ ! "q ( ! id5tity ( tactile space & visual space1 :i* c\ld be treat$ 9 ! same way4 #e4 ,space & ,empiricism ,x seems t ,i am ab\t to be l$ to conclusions 9 con=mity ) empiri/ic ideas4 ,i h1 9 fact1 s\ght to put 9 evid5ce ! rle ( exp]i5ce & to analyze ! exp]im5tal facts :i* int]v5e 9 ! g5esis ( space ( ?ree dim5sions4 ,b :atev] may be ! importance ( ^! facts1 !re is "o ?+ we mu/ n =get & to :i* besides ,i h m ?an once call$ att5tion4 ,^! exp]im5tal facts are (t5 v]ifi$ b n always4 ,t evid5tly does n mean t space has (t5 ?ree dim5sions1 b n always4 ,i "k well t x is easy to save oneself & t1 if ! facts d n v]ify1 x w be easily expla9$ by say+ t ! ext]ior objects h mov$4 ,if exp]i5ce succe$s1 we say t x tea*es u ab\t space2 if x does n succe$1 we hie to ext]ior objects :i* we accuse ( hav+ mov$2 9 o!r words1 if x does n succe$1 x is giv5 a fillip4 ,^! fillips are legitimate2 ,i d n refuse to admit !m2 b !y suffice to tell u t ! prop]ties ( space are n exp]im5tal tru?s1 prop]ly s call$4 ,if we _h wi%$ to v]ify o!r laws1 we c\ld h succe$$ also1 by giv+ o!r analog\s fillips4 ,%\ld we n always h be5 able to ju/ify ^! fillips by ! same reasons8 ,"o c\ld at mo/ h said to u3 ',y\r fillips are d\btless legitimate1 b y abuse !m2 :y move ! ext]ior objects s (t58' ,to sum up1 exp]i5ce does n prove to u t space has ?ree dim5sions2 x only proves to u t x is conv5i5t to attribute ?ree to x1 because ?us ! numb] ( fillips is r$uc$ to a m9imum4 ,i w add t exp]i5ce br+s u into contact only ) repres5tative space1 :i* is a physical cont9uum1 nev] ) geometric space1 :i* is a ma!matical cont9uum4 ,at ! v mo/ x w\ld appe> to tell u t x is conv5i5t to give to geometric space ?ree dim5sions1 s t x may h z _m z repres5tative space4 ,! empiric "q may be put "u ano!r =m4 ,is x impossible to conceive physical ph5om5a1 ! me*anical ph5om5a1 = example1 o!rwise ?an 9 space ( ?ree dim5sions8 ,we %\ld ?us h an objective exp]im5tal pro(1 s to speak1 indep5d5t ( our physiology1 ( our modes ( repres5tation4 ,b x is n s2 ,i %all n "h discuss ! "q completely1 ,i %all conf9e myself to recall+ ! /rik+ example giv5 u by ! me*anics ( ,h]tz4 ,y "k t ! great physici/ did n believe 9 ! exi/5ce ( =ces1 prop]ly s call$2 he suppos$ t visible mat]ial po9ts are subject$ to c]ta9 invisible bonds :i* jo9 !m to o!r invisible po9ts & t x is ! effect ( ^! invisible bonds t we attribute to =ces4 ,b t is only a "p ( 8 ideas4 ,suppose a sy/em =m$ ( n mat]ial po9ts1 visible or n2 t w give 9 all #cn coord9ates2 let u reg>d !m z ! coord9ates ( a s+le po9t 9 space ( #cn dim5sions4 ,?is s+le po9t w\ld be con/ra9$ to rema9 ^u a surface 7( any numb] ( dim5sions #cn7 9 virtue ( ! bonds ( :i* we h j spok52 to g on ?is surface f "o po9t to ano!r1 x w\ld always take ! %orte/ way2 ?is w\ld be ! s+le pr9ciple :i* w\ld sum up all me*anics4 ,:atev] %\ld be ?\ght ( ?is hypo!sis1 :e!r we be allur$ by its simplicity1 or repell$ by its artificial "*1 ! simple fact t ,h]tz 0 able to conceive x1 & to reg>d x z m conv5i5t ?an our habitual hypo!ses1 suffices to prove t our ord9>y ideas1 &1 9 p>ticul>1 ! ?ree dim5sions ( space1 are 9 no wise impos$ ^u me*anics ) an inv9cible =ce4 #f4 ,m9d & ,space ,exp]i5ce1 !re=e1 has play$ only a s+le rle1 x has s]v$ z occasion4 ,b ?is rle 0 none ! less v important2 & ,i h ?\ght x necess>y to give x prom95ce4 ,?is rle w\ld h be5 useless if !re exi/$ an a priori =m impos+ itself ^u our s5sitivity1 & :i* 0 space ( ?ree dim5sions4 ,does ?is =m exi/1 or1 if y *oose1 c we repres5t to ourselves space ( m ?an ?ree dim5sions8 ,& fir/ :at does ?is "q mean8 ,9 ! true s5se ( ! ^w1 x is cle> t we c n repres5t to ourselves space ( f\r1 nor space ( ?ree1 dim5sions2 we c n fir/ repres5t !m to ourselves empty1 & no m c we repres5t to ourselves an object ei!r 9 space ( f\r1 or 9 space ( ?ree1 dim5sions3 7#a7 ,because ^! spaces are bo? inf9ite & we c n repres5t to ourselves a figure 9 space1 t is1 ! "p 9 ! :ole1 )\t repres5t+ ! :ole1 & t is impossible1 because x is inf9ite2 7#b7 because ^! spaces are bo? ma!matical cont9ua1 & we c repres5t to ourselves only ! physical cont9uum2 7#c7 because ^! spaces are bo? homog5e\s1 & ! frames 9 :i* we enclose our s5sations1 be+ limit$1 c n be homog5e\s4 ,?us ! "q put c only be und]/ood 9 "o way2 is x possible to imag9e t1 ! results ( ! exp]i5ces relat$ above hav+ be5 diff]5t1 we might h be5 l$ to attribute to space m ?an ?ree dim5sions2 to imag9e1 = in/ance1 t ! s5sation ( accommodation might n be con/antly 9 accord ) ! s5sation ( conv]g5ce ( ! eyes2 or inde$ t ! exp]i5ces ( :i* we h spok5 9 #b1 & ( :i* we express ! result by say+ 't t\* does n op]ate at a di/ance1' might h l$ u to an inv]se conclusion4 ,& !n yes evid5tly t is possible2 f ! mom5t "o imag9es an exp]i5ce1 "o imag9es j !reby ! two contr>y results x may give4 ,t is possible1 b t is difficult1 because we h to ov]come a multitude ( associations ( ideas1 :i* are ! fruit ( a long p]sonal exp]i5ce & ( ! /ill long] exp]i5ce ( ! race4 ,is x ^! associations 7or at lea/ ^? ( !m t we h inh]it$ f our ance/ors71 :i* con/itute ?is a priori =m ( :i* x is said t we h pure intuition8 ,!n ,i d n see :y "o %\ld decl>e x refractory to analysis & %\ld d5y me ! "r ( inve/igat+ its orig94 ,:5 x is said t our s5sations are 'ext5d$' only "o ?+ c be meant1 t is t !y are always associat$ ) ! idea ( c]ta9 muscul> s5sations1 correspond+ to ! movem5ts :i* enable u to rea* ! object :i* causes !m1 :i* enable u1 9 o!r words1 to def5d ourselves aga9/ x4 ,& x is j because ?is association is useful = ! def5se ( ! organism1 t x is s old 9 ! hi/ory ( ! species & t x seems to u inde/ructible4 ,nev]!less1 x is only an association & we c conceive t x may be brok52 s t we may n say t s5sation c n ent] consci\sness )\t ent]+ 9 space1 b t 9 fact x does n ent] consci\sness )\t ent]+ 9 space1 :i* means1 )\t be+ entangl$ 9 ?is association4 ,no m c ,i und]/& "o's say+ t ! idea ( "t is logically subsequ5t to space1 s9ce we c repres5t x to ourselves only "u ! =m ( a /raight l9e2 z well say t "t is logically subsequ5t to ! cultivation ( ! prairies1 s9ce x is usually repres5t$ arm$ ) a scy!4 ,t "o c n repres5t to himself simultane\sly ! diff]5t p>ts ( "t1 goes )\t say+1 s9ce ! ess5tial "* ( ^! p>ts is precisely n to be simultane\s4 ,t does n mean t we h n ! intuition ( "t4 ,s f> z t goes1 no m %\ld we h t ( space1 because nei!r c we repres5t x1 9 ! prop] s5se ( ! ^w1 = ! reasons ,i h m5tion$4 ,:at we repres5t to ourselves "u ! "n ( /raight is a crude image :i* z ill resembles ! geometric /raight z x does "t itself4 ,:y has x be5 said t e attempt to give a f\r? dim5sion to space always c>ries ?is "o back to "o ( ! o!r ?ree8 ,x is easy to und]/&4 ,consid] our muscul> s5sations & ! 's]ies' !y may =m4 ,9 consequ5ce ( num]\s exp]i5ces1 ! ideas ( ^! s]ies are associat$ toge!r 9 a v complex wo(1 our s]ies are class$4 ,all[ me1 = conv5i5ce ( language1 to express my ?\ght 9 a way altoge!r crude & ev5 inexact by say+ t our s]ies ( muscul> s5sations are class$ 9 ?ree classes correspond+ to ! ?ree dim5sions ( space4 ,( c\rse ?is classification is mu* m complicat$ ?an t1 b t w suffice to make my reason+ und]/ood4 ,if ,i wi% to imag9e a f\r? dim5sion1 ,i %all suppose ano!r s]ies ( muscul> s5sations1 mak+ "p ( a f\r? class4 ,b z all my muscul> s5sations h already be5 class$ 9 "o ( ! ?ree pre-exi/5t classes1 ,i c only repres5t to myself a s]ies belong+ to "o ( ^! ?ree classes1 s t my f\r? dim5sion is c>ri$ back to "o ( ! o!r ?ree4 ,:at does t prove8 ,?is3 t x w\ld h be5 necess>y fir/ to de/roy ! old classification & replace x by a new "o 9 :i* ! s]ies ( muscul> s5sations %\ld h be5 di/ribut$ into f\r classes4 ,! difficulty w\ld h disappe>$4 ,x is pres5t$ sometimes "u a m /rik+ =m4 ,suppose ,i am enclos$ 9 a *amb] betwe5 ! six impassable b\nd>ies =m$ by ! f\r walls1 ! floor & ! ceil+2 x w be impossible = me to get out & to imag9e my gett+ out4 ,p>don1 c y n imag9e t ! door op5s1 or t two ( ^! walls sep>ate8 ,b ( c\rse1 y answ]1 "o mu/ suppose t ^! walls rema9 immovable4 ,yes1 b x is evid5t t ,i h ! "r to move2 & !n ! walls t we suppose absolutely at re/ w be 9 motion ) reg>d to me4 ,yes1 b su* a relative motion c n be arbitr>y2 :5 objects are at re/1 _! relative motion ) reg>d to any axes is t ( a rigid solid2 n[1 ! app>5t motions t y imag9e are n 9 con=mity ) ! laws ( motion ( a rigid solid4 ,yes1 b x is exp]i5ce :i* has taught u ! laws ( motion ( a rigid solid2 no?+ w\ld prev5t our imag9+ !m diff]5t4 ,to sum up1 = me to imag9e t ,i get out ( my prison1 ,i h only to imag9e t ! walls seem to op51 :5 ,i move4 ,i believe1 !re=e1 t if by space is und]/ood a ma!matical cont9uum ( ?ree dim5sions1 7 x o!rwise amorph\s1 x is ! m9d :i* con/ructs x1 b x does n con/ruct x out ( no?+2 x ne$s mat]ials & models4 ,^! mat]ials1 l ^! models1 preexi/ )9 x4 ,b !re is n a s+le model :i* is impos$ ^u x2 x has *oice2 x may *oose1 = in/ance1 betwe5 space ( f\r & space ( ?ree dim5sions4 ,:at !n is ! rle ( exp]i5ce8 ,x gives ! indications foll[+ :i* ! *oice is made4 ,ano!r ?+3 :5ce does space get its quantitative "*8 ,x comes f ! rle :i* ! s]ies ( muscul> s5sations play 9 its g5esis4 ,^! are s]ies :i* may repeat !mselves1 & x is f _! repetition t numb] comes2 x is because !y c repeat !mselves indef9itely t space is inf9ite4 ,& f9ally we h se51 at ! end ( section #c1 t x is also because ( ?is t space is relative4 ,s x is repetition :i* has giv5 to space its ess5tial *>act]i/ics2 n[1 repetition supposes "t2 ?is is en\gh to tell t "t is logically ant]ior to space4 #g4 ,rle ( ! ,semicircul> ,canals ,i h n hi!rto spok5 ( ! rle ( c]ta9 organs to :i* ! physiologi/s attribute ) reason a capital importance1 ,i mean ! semicircul> canals4 ,num]\s exp]im5ts h suffici5tly %[n t ^! canals are necess>y to our s5se ( ori5tation2 b ! physiologi/s are n entirely 9 accord2 two oppos+ !ories h be5 propos$1 t ( ,ma*-,delage & t ( ,m4 de ,cyon4 ,m4 de ,cyon is a physiologi/ :o has made 8 "n illu/ri\s by important discov]ies on ! inn]vation ( ! he>t2 ,i c n1 h[ev]1 agree ) 8 ideas on ! "q be=e u4 ,n be+ a physiologi/1 ,i hesitate to criticize ! exp]im5ts he has direct$ aga9/ ! adv]se !ory ( ,ma*-,delage2 x seems to me1 h[ev]1 t !y are n conv9c+1 because 9 _m ( !m ! total pressure 0 made to v>y 9 "o ( ! canals1 :ile1 physiologically1 :at v>ies is ! diff]5ce betwe5 ! pressures on ! two extremities ( ! canal2 9 o!rs ! organs 7 subject$ to pr(\nd lesions1 :i* mu/ alt] _! functions4 ,besides1 ?is is n important2 ! exp]im5ts1 if !y 7 irreproa*able1 might be conv9c+ aga9/ ! old !ory4 ,!y w\ld n be conv9c+ = ! new !ory4 ,9 fact1 if ,i h rightly und]/ood ! !ory1 my expla9+ x w be en\gh = "o to und]/& t x is impossible to conceive ( an exp]im5t confirm+ x4 ,! ?ree pairs ( canals w\ld h z sole function to tell u t space has ?ree dim5sions4 ,japanese mice h only two pairs ( canals2 !y believe1 x w\ld seem1 t space has only two dim5sions1 & !y manife/ ?is op9ion 9 ! /range/ way2 !y put !mselves 9 a circle1 &1 s ord]$1 !y sp9 rapidly ar\nd4 ,! lampreys1 hav+ only "o pair ( canals1 believe t space has only "o dim5sion1 b _! manife/ations are less turbul5t4 ,x is evid5t t su* a !ory is inadmissible4 ,! s5se-organs are design$ to tell u ( *anges :i* happ5 9 ! ext]ior _w4 ,we c\ld n und]/& :y ! ,creator %\ld h giv5 u organs de/9$ to cry )\t cease3 ,rememb] t space has ?ree dim5sions1 s9ce ! numb] ( ^! ?ree dim5sions is n subject to *ange4 ,we mu/1 !re=e1 come back to ! !ory ( ,ma*-,delage4 ,:at ! n]ves ( ! canals c tell u is ! diff]5ce ( pressure on ! two extremities ( ! same canal1 & !reby3 7#a7 ! direction ( ! v]tical ) reg>d to ?ree axes rigidly b\nd to ! head2 7#b7 ! ?ree compon5ts ( ! accel]ation ( translation ( ! c5t] ( gravity ( ! head2 7#c7 ! c5trifugal =ces develop$ by ! rotation ( ! head2 7#d7 ! accel]ation ( ! motion ( rotation ( ! head4 ,x foll[s f ! exp]im5ts ( ,m4 ,delage t x is ?is la/ indication :i* is mu* ! mo/ important2 d\btless because ! n]ves are less s5sible to ! diff]5ce ( pressure itself ?an to ! brusque v>iations ( ?is diff]5ce4 ,! fir/ ?ree indications may ?us be neglect$4 ,kn[+ ! accel]ation ( ! motion ( rotation ( ! head at ea* in/ant1 we d$uce f x1 by an unconsci\s integration1 ! f9al ori5tation ( ! head1 ref]r$ to a c]ta9 initial ori5tation tak5 z orig94 ,! circul> canals contribute1 !re=e1 to in=m u ( ! movem5ts t we h execut$1 & t on ! same gr\nd z ! muscul> s5sations4 ,:51 !re=e1 above we speak ( ! s]ies ,s or ( ! s]ies ,sigma1 we %\ld say1 n t ^! 7 s]ies ( muscul> s5sations alone1 b t !y 7 s]ies at ! same "t ( muscul> s5sations & ( s5sations due to ! semicircul> canals4 ,ap>t f ?is addition1 we %\ld h no?+ to *ange 9 :at prec$es4 ,9 ! s]ies ,s & ,sigma1 ^! s5sations ( ! semicircul> canals evid5tly hold a v important place4 ,yet alone !y w\ld n suffice1 because !y c tell u only ( ! movem5ts ( ! head2 !y tell u no?+ ( ! relative movem5ts ( ! body or ( ! memb]s 9 reg>d to ! head4 ,& m1 x seems t !y tell u only ( ! rotations ( ! head & n ( ! translations x may und]go4 ,,"p ,,ii ,,! ,,physical ,,sci5ces ,,*apt] ,v ,,analysis ,,& ,,physics ,i ,y h d\btless (t5 be5 ask$ ( :at good is ma!matics & :e!r ^! delicate con/ructions entirely m9d-made are n artificial & born ( our caprice4 ,among ^? :o put ?is "q ,i %\ld make a di/9ction2 practical p ask ( u only ! means ( money-mak+4 ,^! m]it no reply2 r w\ld x be prop] to ask ( !m :at is ! good ( accumulat+ s mu* weal? & :e!r1 to get "t to acquire x1 we are to neglect art & sci5ce1 :i* alone give u s\ls capable ( enjoy+ x1 '& = life's sake to sacrifice all reasons = liv+4' ,besides1 a sci5ce made solely 9 view ( applications is impossible2 tru?s are fecund only if b\nd toge!r4 ,if we devote ourselves solely to ^? tru?s :5ce we expect an imm$iate result1 ! int]m$i>y l9ks are want+ & !re w no long] be a *a94 ,! m5 mo/ disda9ful ( !ory get f x1 )\t suspect+ x1 _! daily bread2 depriv$ ( ?is food1 progress w\ld quickly cease1 & we %\ld soon congeal into ! immobility ( old ,*9a4 ,b en\gh ( uncompromis+ practicians6 ,besides ^!1 !re are ^? :o are only int]e/$ 9 nature & :o ask u if we c enable !m to "k x bett]4 ,to answ] ^!1 we h only to %[ !m ! two monum5ts already r\gh-hewn1 ,cele/ial ,me*anics & ,ma!matical ,physics4 ,!y w\ld d\btless conc$e t ^! /ructures are well wor? ! tr\ble !y h co/ u4 ,b ?is is n en\gh4 ,ma!matics has a triple aim4 ,x mu/ furni% an in/rum5t = ! /udy ( nature4 ,b t is n all3 x has a philosophic aim &1 ,i d>e ma9ta91 an e/hetic aim4 ,x mu/ aid ! philosoph] to fa?om ! notions ( numb]1 ( space1 ( "t4 ,& above all1 its adepts f9d !re9 delights analog\s to ^? giv5 by pa9t+ & music4 ,!y admire ! delicate h>mony ( numb]s & =ms2 !y m>vel :5 a new discov]y op5s to !m an unexpect$ p]spective2 & has n ! joy !y ?us feel ! e/hetic "*1 ev5 ?\gh ! s5ses take no "p !re98 ,only a privileg$ few are call$ to enjoy x fully1 x is true1 b is n ?is ! case = all ! noble/ arts8 ,?is is :y ,i d n hesitate to say t ma!matics des]ves to be cultivat$ = its own sake1 & ! !ories inapplicable to physics z well z ! o!rs4 ,ev5 if ! physical aim & ! e/hetic aim 7 n unit$1 we ought n to sacrifice ei!r4 ,b m3 ^! two aims are insep>able & ! be/ means ( atta9+ "o is to aim at ! o!r1 or at lea/ nev] to lose sight ( x4 ,?is is :at ,i am ab\t to try to demon/rate 9 sett+ =? ! nature ( ! relations betwe5 ! pure sci5ce & its applications4 ,! ma!matician %\ld n be = ! physici/ a m]e purveyor ( =mulas2 !re %\ld be betwe5 !m a m intimate collaboration4 ,ma!matical physics & pure analysis are n m]ely adjac5t p[]s1 ma9ta9+ good neighborly relations2 !y mutually int]p5etrate & _! _s is ! same4 ,?is w be bett] und]/ood :5 ,i h %[n :at physics gets f ma!matics & :at ma!matics1 9 return1 borr[s f physics4 ,,ii ,! physici/ c n ask ( ! analy/ to reveal to him a new tru?2 ! latt] c\ld at mo/ only aid him to =esee x4 ,x is a long "t s9ce "o /ill dreamt ( =e/all+ exp]im5t1 or ( con/ruct+ ! entire _w on c]ta9 premature hypo!ses4 ,s9ce all ^? con/ructions 9 :i* "o yet took a nave delight x is an age1 to-day only _! ru9s rema94 ,all laws are !re=e d$uc$ f exp]im5t2 b to enunciate !m1 a special language is ne$ful2 ord9>y language is too poor1 x is besides too vague1 to express relations s delicate1 s ri*1 & s precise4 ,?is !re=e is "o reason :y ! physici/ c n d )\t ma!matics2 x furni%es him ! only language he c speak4 ,& a well-made language is no indiff]5t ?+2 n to g beyond physics1 ! unkn[n man :o inv5t$ ! ^w heat devot$ _m g5]ations to error4 ,heat has be5 treat$ z a sub/ance1 simply because x 0 designat$ by a sub/antive1 & x has be5 ?\ght inde/ructible4 ,on ! o!r h&1 he :o inv5t$ ! ^w electricity _h ! unm]it$ good =tune to implicitly end[ physics ) a new law1 t ( ! cons]vation ( electricity1 :i*1 by a pure *ance1 has be5 f\nd exact1 at lea/ until n[4 ,well1 to cont9ue ! simile1 ! writ]s :o embelli% a language1 :o treat x z an object ( art1 make ( x at ! same "t a m supple in/rum5t1 m apt = r5d]+ %ades ( ?\ght4 ,we und]/&1 !n1 h[ ! analy/1 :o pursues a purely e/hetic aim1 helps create1 j by t1 a language m fit to satisfy ! physici/4 ,b ?is is n all3 law spr+s f exp]im5t1 b n imm$iately4 ,exp]im5t is individual1 ! law d$uc$ f x is g5]al2 exp]im5t is only approximate1 ! law is precise1 or at lea/ pret5ds to be4 ,exp]im5t is made "u conditions always complex1 ! enunciation ( ! law elim9ates ^! complications4 ,?is is :at is call$ 'correct+ ! sy/ematic errors4' ,9 a ^w1 to get ! law f exp]im5t1 x is necess>y to g5]alize2 ?is is a necessity impos$ ^u ! mo/ circumspect obs]v]4 ,b h[ g5]alize8 ,e p>ticul> tru? may evid5tly be ext5d$ 9 an inf9ity ( ways4 ,among ^! ?\s& r\tes op5+ be=e u1 x is necess>y to make a *oice1 at lea/ provisional2 9 ?is *oice1 :at %all guide u8 ,x c only be analogy4 ,b h[ vague is ?is ^w6 ,primitive man knew only crude analogies1 ^? :i* /rike ! s5ses1 ^? ( colors or ( s\nds4 ,he nev] w\ld h dreamt ( lik5+ light to radiant heat4 ,:at has taught u to "k ! true1 pr(\nd analogies1 ^? ! eyes d n see b reason div9es8 ,x is ! ma!matical _s1 :i* disda9s matt] to cl+ only to pure =m4 ,?is x is :i* has taught u to give ! same "n to ?+s diff]+ only 9 mat]ial1 to call by ! same "n1 = in/ance1 ! multiplication ( quat]nions & t ( :ole numb]s4 ,if quat]nions1 ( :i* ,i h j spok51 _h n be5 s promptly utiliz$ by ! ,engli% physici/s1 _m p]sons w\ld d\btless see 9 !m only a useless fancy1 & yet1 9 tea*+ u to lik5 :at appe>ances sep>ate1 !y w\ld h already r5d]$ u m apt to p5etrate ! secrets ( nature4 ,su* are ! s]vices ! physici/ %\ld expect ( analysis2 b = ?is sci5ce to be able to r5d] !m1 x mu/ be cultivat$ 9 ! broade/ fa%ion )\t imm$iate expectation ( utility--! ma!matician mu/ h work$ z arti/4 ,:at we ask ( him is to help u to see1 to disc]n our way 9 ! labyr9? :i* op5s be=e u4 ,n[1 he sees be/ :o /&s highe/4 ,examples ab\nd1 & ,i limit myself to ! mo/ /rik+4 ,! fir/ w %[ u h[ to *ange ! language suffices to reveal g5]alizations n be=e suspect$4 ,:5 ,newton's law has be5 sub/itut$ = ,kepl]'s we /ill "k only elliptic motion4 ,n[1 9 s f> z conc]ns ?is motion1 ! two laws diff] only 9 =m2 we pass f "o to ! o!r by a simple diff]5tiation4 ,& yet f ,newton's law may be d$uc$ by an imm$iate g5]alization all ! effects ( p]turbations & ! :ole ( cele/ial me*anics4 ,if1 on ! o!r h&1 ,kepl]'s enunciation _h be5 reta9$1 no "o w\ld ev] h reg>d$ ! orbits ( ! p]turb$ planets1 ^? complicat$ curves ( :i* no "o has ev] writt5 ! equation1 z ! natural g5]alizations ( ! ellipse4 ,! progress ( obs]vations w\ld only h s]v$ to create belief 9 *aos4 ,! second example is equally des]v+ ( consid]ation4 ,:5 ,maxwell began 8 "w1 ! laws ( electro-dynamics admitt$ up to 8 "t acc\nt$ = all ! kn[n facts4 ,x 0 n a new exp]im5t :i* came to invalidate !m4 ,b 9 look+ at !m "u a new bias1 ,maxwell saw t ! equations became m symmetrical :5 a t]m 0 add$1 & besides1 ?is t]m 0 too small to produce effects appreciable ) ! old me?ods4 ,y "k t ,maxwell's a priori views await$ = tw5ty ye>s an exp]im5tal confirmation2 or1 if y pref]1 ,maxwell 0 tw5ty ye>s ahead ( exp]im5t4 ,h[ 0 ?is triumph obta9$8 ,x 0 because ,maxwell 0 pr(\ndly /eep$ 9 ! s5se ( ma!matical symmetry2 w\ld he h be5 s1 if o!rs be=e him _h n /udi$ ?is symmetry = its own beauty8 ,x 0 because ,maxwell 0 accu/om$ to '?9k 9 vectors1' & yet x 0 ?r\gh ! !ory ( imag9>ies 7neomonics7 t vectors 7 introduc$ into analysis4 ,& ^? :o inv5t$ imag9>ies h>dly suspect$ ! advantage :i* w\ld be obta9$ f !m = ! /udy ( ! real _w1 ( ?is ! "n giv5 !m is pro( suffici5t4 ,9 a ^w1 ,maxwell 0 p]haps n an able analy/1 b ?is ability w\ld h be5 = him only a useless & bo!rsome baggage4 ,on ! o!r h&1 he _h 9 ! highe/ degree ! intimate s5se ( ma!matical analogies4 ,!re=e x is t he made good ma!matical physics4 ,maxwell's example tea*es u /ill ano!r ?+4 ,h[ %\ld ! equations ( ma!matical physics be treat$8 ,%\ld we simply d$uce all ! consequ5ces & reg>d !m z intangible realities8 ,f> f x2 :at !y %\ld tea* u above all is :at c & :at %\ld be *ang$4 ,x is ?us t we get f !m some?+ useful4 ,! ?ird example goes to %[ u h[ we may p]ceive ma!matical analogies betwe5 ph5om5a :i* h physically no relation ei!r app>5t or real1 s t ! laws ( "o ( ^! ph5om5a aid u to div9e ^? ( ! o!r4 ,! v same equation1 t ( ,laplace1 is met 9 ! !ory ( ,newtonian attraction1 9 t ( ! motion ( liquids1 9 t ( ! electric pot5tial1 9 t ( magnetism1 9 t ( ! propagation ( heat & 9 /ill _m o!rs4 ,:at is ! result8 ,^! !ories seem images copi$ "o f ! o!r2 !y are mutually illum9at+1 borr[+ _! language f ea* o!r2 ask electricians if !y d n felicitate !mselves on hav+ inv5t$ ! phrase fl[ ( =ce1 sugge/$ by hydrodynamics & ! !ory ( heat4 ,?us ma!matical analogies n only may make u =esee physical analogies1 b besides d n cease to be useful :5 ^! latt] fail4 ,to sum up1 ! aim ( ma!matical physics is n only to facilitate = ! physici/ ! num]ical calculation ( c]ta9 con/ants or ! integration ( c]ta9 diff]5tial equations4 ,x is besides1 x is above all1 to reveal to him ! hidd5 h>mony ( ?+s 9 mak+ him see !m 9 a new way4 ,( all ! p>ts ( analysis1 ! mo/ elevat$1 ! pure/1 s to speak1 w be ! mo/ fruitful 9 ! h&s ( ^? :o "k h[ to use !m4 ,,iii ,let u n[ see :at analysis owes to physics4 ,x w\ld be necess>y to h completely =gott5 ! hi/ory ( sci5ce n to rememb] t ! desire to und]/& nature has _h on ! developm5t ( ma!matics ! mo/ con/ant & happie/ influ5ce4 ,9 ! fir/ place ! physici/ sets u problems ^: solution he expects ( u4 ,b 9 propos+ !m to u1 he has l>gely paid u 9 advance = ! s]vice we %all r5d] him1 if we solve !m4 ,if ,i may be all[$ to cont9ue my comp>ison ) ! f9e arts1 ! pure ma!matician :o %\ld =get ! exi/5ce ( ! ext]ior _w w\ld be l a pa9t] :o knew h[ to h>moni\sly comb9e colors & =ms1 b :o lack$ models4 ,8 creative p[] w\ld soon be exhau/$4 ,! comb9ations :i* numb]s & symbols may =m are an inf9ite multitude4 ,9 ?is multitude h[ %all we *oose ^? :i* are wor?y to fix our att5tion8 ,%all we let ourselves be guid$ solely by our caprice8 ,?is caprice1 :i* itself w\ld besides soon tire1 w\ld d\btless c>ry u v f> ap>t & we %\ld quickly cease to und]/& ea* o!r4 ,b ?is is only ! small] side ( ! "q4 ,physics w d\btless prev5t our /ray+1 b x w also pres]ve u f a dang] mu* m =midable2 x w prev5t our ceaselessly go+ ar\nd 9 ! same circle4 ,hi/ory proves t physics has n only =c$ u to *oose among problems :i* came 9 a cr[d2 x has impos$ ^u u su* z we %\ld )\t x nev] h dream$ (4 ,h[ev] v>i$ may be ! imag9ation ( man1 nature is /ill a ?\s& times ri*]4 ,to foll[ h] we mu/ take ways we h neglect$1 & ^! pa?s lead u (t5 to summits :5ce we discov] new c\ntries4 ,:at c\ld be m useful6 ,x is ) ma!matical symbols z ) physical realities2 x is 9 comp>+ ! diff]5t aspects ( ?+s t we are able to compreh5d _! inn] h>mony1 :i* alone is beautiful & consequ5tly wor?y ( our ef=ts4 ,! fir/ example ,i %all cite is s old we are tempt$ to =get x2 x is nev]!less ! mo/ important ( all4 ,! sole natural object ( ma!matical ?\ght is ! :ole numb]4 ,x is ! ext]nal _w :i* has impos$ ! cont9uum ^u u1 :i* we d\btless h inv5t$1 b :i* x has =c$ u to inv5t4 ,)\t x !re w\ld be no inf9itesimal analysis2 all ma!matical sci5ce w\ld r$uce itself to ari?metic or to ! !ory ( sub/itutions4 ,on ! contr>y1 we h devot$ to ! /udy ( ! cont9uum almo/ all our "t & all our /r5g?4 ,:o w regret x2 :o w ?9k t ?is "t & ?is /r5g? h be5 wa/$8 ,analysis unfolds be=e u inf9ite p]spectives t ari?metic nev] suspects2 x %[s u at a glance a maje/ic assemblage ^: array is simple & symmetric2 on ! contr>y1 9 ! !ory ( numb]s1 :]e reigns ! un=ese51 ! view is1 s to speak1 arre/$ at e /ep4 ,d\btless x w be said t outside ( ! :ole numb] !re is no rigor1 & consequ5tly no ma!matical tru?2 t ! :ole numb] hides ev]y:]e1 & t we mu/ /rive to r5d] transp>5t ! scre5s :i* cloak x1 ev5 if to d s we mu/ resign ourselves to int]m9able repetitions4 ,let u n be su* puri/s & let u be grateful to ! cont9uum1 :i*1 if all spr+s f ! :ole numb]1 0 alone capable ( mak+ s mu* proce$ !refrom4 ,ne$ ,i also recall t ,m4 ,h]mite obta9$ a surpris+ advantage f ! introduction ( cont9u\s v>iables into ! !ory ( numb]s8 ,?us ! :ole numb]'s own doma9 is itself invad$1 & ?is invasion has e/abli%$ ord] :]e disord] reign$4 ,see :at we owe to ! cont9uum & consequ5tly to physical nature4 ,f\ri]'s s]ies is a preci\s in/rum5t ( :i* analysis makes cont9ual use1 x is by ?is means t x has be5 able to repres5t discont9u\s functions2 ,f\ri] inv5t$ x to solve a problem ( physics relative to ! propagation ( heat4 ,if ?is problem _h n come up naturally1 we %\ld nev] h d>$ to give discont9uity its rights2 we %\ld /ill long h reg>d$ cont9u\s functions z ! only true functions4 ,! notion ( function has be5 !reby consid]ably ext5d$ & has receiv$ f "s logician-analy/s an un=ese5 developm5t4 ,^! analy/s h ?us adv5tur$ into regions :]e reigns ! pure/ ab/raction & h gone z f> away z possible f ! real _w4 ,yet x is a problem ( physics :i* has furni%$ !m ! occasion4 ,aft] ,f\ri]'s s]ies1 o!r analog\s s]ies h ent]$ ! doma9 ( analysis2 !y h ent]$ by ! same door2 !y h be5 imag9$ 9 view ( applications4 ,! !ory ( p>tial diff]5tial equations ( ! second ord] has an analog\s hi/ory4 ,x has be5 develop$ *iefly by & = physics4 ,b x may take _m =ms1 because su* an equation does n suffice to det]m9e ! unkn[n function1 x is necess>y to adjo9 to x complem5t>y conditions :i* are call$ conditions at ! limits2 :5ce _m diff]5t problems4 ,if ! analy/s _h ab&on$ !mselves to _! natural t5d5cies1 !y w\ld nev] h kn[n b "o1 t :i* ,madame ,kovalevski has treat$ 9 h] celebrat$ memoir4 ,b !re are a multitude ( o!rs :i* !y w\ld h ignor$4 ,ea* ( ! !ories ( physics1 t ( electricity1 t ( heat1 pres5ts u ^! equations "u a new aspect4 ,x may1 !re=e1 be said t )\t ^! !ories we %\ld n "k p>tial diff]5tial equations4 ,x is ne$less to multiply examples4 ,i h giv5 en\gh to be able to conclude3 :5 physici/s ask ( u ! solution ( a problem1 x is n a duty-s]vice !y impose ^u u1 x is on ! contr>y we :o owe !m ?anks4 ,,iv ,b ?is is n all2 physics n only gives u ! occasion to solve problems2 x aids u to f9d ! means !reto1 & t 9 two ways4 ,x makes u =esee ! solution2 x sugge/s argum5ts to u4 ,i h spok5 above ( ,laplace's equation :i* is met 9 a multitude ( div]se physical !ories4 ,x is f\nd aga9 9 geometry1 9 ! !ory ( con=mal repres5tation & 9 pure analysis1 9 t ( imag9>ies4 ,9 ?is way1 9 ! /udy ( functions ( complex v>iables1 ! analy/1 alongside ( ! geometric image1 :i* is 8 usual in/rum5t1 f9ds _m physical images :i* he may make use ( ) ! same success4 ,?anks to ^! images1 he c see at a glance :at pure d$uction w\ld %[ him only successively4 ,he masses ?us ! sep>ate elem5ts ( ! solution1 & by a sort ( intuition div9es be=e be+ able to demon/rate4 ,to div9e be=e demon/rat+6 ,ne$ ,i recall t ?us h be5 made all ! important discov]ies8 ,h[ _m are ! tru?s t physical analogies p]mit u to pres5t & t we are n 9 condition to e/abli% by rigor\s reason+6 ,= example1 ma!matical physics introduces a great numb] ( developm5ts 9 s]ies4 ,no "o d\bts t ^! developm5ts conv]ge2 b ! ma!matical c]titude is lack+4 ,^! are s _m conque/s assur$ = ! inve/igators :o %all come aft] u4 ,on ! o!r h&1 physics furni%es u n alone solutions2 x furni%es u besides1 9 a c]ta9 measure1 argum5ts4 ,x w suffice to recall h[ ,felix ,kle91 9 a "q relative to ,riemann surfaces1 has _h rec\rse to ! prop]ties ( electric curr5ts4 ,x is true1 ! argum5ts ( ?is species are n rigor\s1 9 ! s5se ! analy/ atta*es to ?is ^w4 ,& "h a "q arises3 ,h[ c a demon/ration n suffici5tly rigor\s = ! analy/ suffice = ! physici/8 ,x seems !re c n be two rigors1 t rigor is or is n1 & t1 :]e x is n !re c n be d$uction4 ,?is app>5t p>adox w be bett] und]/ood by recall+ "u :at conditions numb] is appli$ to natural ph5om5a4 ,:5ce come 9 g5]al ! difficulties enc\nt]$ 9 seek+ rigor8 ,we /rike !m almo/ always 9 seek+ to e/abli% t "s quantity t5ds to "s limit1 or t "s function is cont9u\s1 or t x has a d]ivative4 ,n[ ! numb]s ! physici/ measures by exp]im5t are nev] kn[n except approximately2 & besides1 any function always diff]s z little z y *oose f a discont9u\s function1 & at ! same "t x diff]s z little z y *oose f a cont9u\s function4 ,! physici/ may1 !re=e1 at w suppose t ! function /udi$ is cont9u\s1 or t x is discont9u\s2 t x has or has n a d]ivative2 & may d s )\t fe> ( ev] be+ contradict$1 ei!r by pres5t exp]i5ce or by any future exp]im5t4 ,we see t ) su* lib]ty he makes sport ( difficulties :i* /op ! analy/4 ,he may always reason z if all ! functions :i* occur 9 8 calculations 7 entire polynomials4 ,?us ! sket* :i* suffices = physics is n ! d$uction :i* analysis requires4 ,x does n foll[ !nce t "o c n aid 9 f9d+ ! o!r4 ,s _m physical sket*es h already be5 trans=m$ into rigor\s demon/rations t to-day ?is trans=mation is easy4 ,!re w\ld be pl5ty ( examples did ,i n fe> 9 cit+ !m to tire ! read]4 ,i hope ,i h said en\gh to %[ t pure analysis & ma!matical physics may s]ve "o ano!r )\t mak+ any sacrifice "o to ! o!r1 & t ea* ( ^! two sci5ces %\ld rejoice 9 all :i* elevates its associate4 ,,*apt] ,,vi ,,a/ronomy ,gov]nm5ts & p>liam5ts mu/ f9d t a/ronomy is "o ( ! sci5ces :i* co/ mo/ de>3 ! lea/ in/rum5t co/s hundr$s ( ?\s&s ( doll>s1 ! lea/ obs]vatory co/s millions2 ea* eclipse c>ries ) x supplem5t>y appropriations4 ,& all t = />s :i* are s f> away1 :i* are complete /rang]s to our electoral conte/s1 & 9 all probability w nev] take any "p 9 !m4 ,x mu/ be t our politicians h reta9$ a remnant ( idealism1 a vague in/9ct = :at is gr&2 truly1 ,i ?9k !y h be5 calumniat$2 !y %\ld be enc\rag$ & %[n t ?is in/9ct does n deceive !m1 t !y are n dupes ( t idealism4 ,we might inde$ speak to !m ( navigation1 ( :i* no "o c und]e/imate ! importance1 & :i* has ne$ ( a/ronomy4 ,b ?is w\ld be to take ! "q by its small] side4 ,a/ronomy is useful because x raises u above ourselves2 x is useful because x is gr&2 t is :at we %\ld say4 ,x %[s u h[ small is man's body1 h[ great 8 m9d1 s9ce 8 intellig5ce c embrace ! :ole ( ?is dazzl+ imm5sity1 :]e 8 body is only an obscure po9t1 & enjoy its sil5t h>mony4 ,?us we atta9 ! consci\sness ( our p[]1 & ?is is some?+ :i* c n co/ too de>1 s9ce ?is consci\sness makes u mighti]4 ,b :at ,i %\ld wi% be=e all to %[ is1 to :at po9t a/ronomy has facilitat$ ! "w ( ! o!r sci5ces1 m directly useful1 s9ce x has giv5 u a s\l capable ( compreh5d+ nature4 ,?9k h[ dim9i%$ humanity w\ld be if1 "u heav5s con/antly ov]cl\d$1 z ,jupit]'s mu/ be1 x _h =ev] rema9$ ignorant ( ! />s4 ,d y ?9k t 9 su* a _w we %\ld be :at we are8 ,i "k well t "u ?is somb] vault we %\ld h be5 depriv$ ( ! light ( ! sun1 necess>y to organisms l ^? :i* inhabit ! e>?4 ,b if y please1 we %all assume t ^! cl\ds are phosphoresc5t & emit a s(t & con/ant light4 ,s9ce we are mak+ hypo!ses1 ano!r w co/ no m4 ,well6 ,i repeat my "q3 ,d y ?9k t 9 su* a _w we %\ld be :at we are8 ,! />s s5d u n only t visible & gross light :i* /rikes our bodily eyes1 b f !m also comes to u a light f> m subtle1 :i* illum9ates our m9ds & ^: effects ,i %all try to %[ y4 ,y "k :at man 0 on ! e>? "s ?\s&s ( ye>s ago1 & :at he is to-day4 ,isolat$ amid/ a nature :]e ev]y?+ 0 a my/]y to him1 t]rifi$ at ea* unexpect$ manife/ation ( incompreh5sible =ces1 he 0 incapable ( see+ 9 ! conduct ( ! univ]se any?+ b caprice2 he attribut$ all ph5om5a to ! action ( a multitude ( little g5ii1 fanta/ic & exact+1 & to act on ! _w he s\ght to conciliate !m by means analog\s to ^? employ$ to ga9 ! good graces ( a m9i/] or a deputy4 ,ev5 8 failures did n enlight5 him1 any m ?an to-day a begg> refus$ is disc\rag$ to ! po9t ( ceas+ to beg4 ,to-day we no long] beg ( nature2 we comm& h]1 because we h discov]$ c]ta9 ( h] secrets & %all discov] o!rs ea* day4 ,we comm& h] 9 ! "n ( laws %e c n *all5ge1 because !y are h]s2 ^! laws we d n madly ask h] to *ange1 we are ! fir/ to submit to !m4 ,nature c only be gov]n$ by obey+ h]4 ,:at a *ange mu/ our s\ls h und]gone to pass f ! "o /ate to ! o!r6 ,does any "o believe t1 )\t ! lessons ( ! />s1 "u ! heav5s p]petually ov]cl\d$ t ,i h j suppos$1 !y w\ld h *ang$ s quickly8 ,w\ld ! metamorphosis h be5 possible1 or at lea/ w\ld x n h be5 mu* sl[]8 ,& fir/ ( all1 a/ronomy x is :i* taught t !re are laws4 ,! ,*aldeans1 :o 7 ! fir/ to obs]ve ! heav5s ) "s att5tion1 saw t ?is multitude ( lum9\s po9ts is n a confus$ cr[d w&]+ at r&om1 b r a discipl9$ army4 ,d\btless ! rules ( ?is discipl9e escap$ !m1 b ! h>moni\s spectacle ( ! />ry night suffic$ to give !m ! impression ( regul>ity1 & t 0 9 itself already a great ?+4 ,besides1 ^! rules 7 disc]n$ by ,hipp>*us1 ,ptolemy1 ,cop]nicus1 ,kepl]1 "o aft] ano!r1 & f9ally1 x is ne$less to recall t ,newton x 0 :o enunciat$ ! olde/1 ! mo/ precise1 ! mo/ simple1 ! mo/ g5]al ( all natural laws4 ,& !n1 taught by ?is example1 we h se5 our little t]re/rial _w bett] &1 "u ! app>5t disord]1 !re also we h f\nd aga9 ! h>mony t ! /udy ( ! heav5s _h reveal$ to u4 ,x also is regul>1 x also obeys immutable laws1 b !y are m complicat$1 9 app>5t conflict "o ) ano!r1 & an eye untra9$ by o!r sights w\ld h se5 !re only *aos & ! reign ( *ance or caprice4 ,if we _h n kn[n ! />s1 "s bold spirits might p]haps h s\ght to =esee physical ph5om5a2 b _! failures w\ld h be5 frequ5t1 & !y w\ld h excit$ only ! d]ision ( ! vulg>2 d we n see1 t ev5 9 our day ! meteorologi/s sometimes deceive !mselves1 & t c]ta9 p]sons are incl9$ to laugh at !m4 ,h[ (t5 w\ld ! physici/s1 di%e>t5$ by s _m *ecks1 h fall5 into disc\ragem5t1 if !y _h n _h1 to su/a9 _! confid5ce1 ! brilliant example ( ! success ( ! a/ronom]s6 ,?is success %[$ !m t nature obeys laws2 x only rema9$ to "k :at laws2 = t !y only ne$$ pati5ce1 & !y _h ! "r to dem& t ! sceptics %\ld give !m cr$it4 ,?is is n all3 a/ronomy has n only taught u t !re are laws1 b t f ^! laws !re is no escape1 t ) !m !re is no possible compromise4 ,h[ mu* "t %\ld we h ne$$ to compreh5d t fact1 if we _h kn[n only ! t]re/rial _w1 :]e ea* elem5tal =ce w\ld always seem to u 9 conflict ) o!r =ces8 ,a/ronomy has taught u t ! laws are inf9itely precise1 & t if ^? we enunciate are approximative1 x is because we d n "k !m well4 ,ari/otle1 ! mo/ sci5tific m9d ( antiquity1 /ill accord$ a "p to accid5t1 to *ance1 & seem$ to ?9k t ! laws ( nature1 at lea/ "h bel[1 det]m9e only ! l>ge features ( ph5om5a4 ,h[ mu* has ! ev]-increas+ precision ( a/ronomical pr$ictions contribut$ to correct su* an error1 :i* w\ld h r5d]$ nature un9telligible6 ,b are ^! laws n local1 v>y+ 9 diff]5t places1 l ^? :i* m5 make2 does n t :i* is tru? 9 "o corn] ( ! univ]se1 on our globe1 = in/ance1 or 9 our little sol> sy/em1 become error a little f>!r away8 ,& !n c\ld x n be ask$ :e!r laws dep5d+ on space d n also dep5d ^u "t1 :e!r !y are n simple habitudes1 transitory1 !re=e1 & ephem]al8 ,aga9 x is a/ronomy t answ]s ?is "q4 ,consid] ! d\ble />s2 all describe conics2 ?us1 z f> z ! telescope c>ries1 x does n rea* ! limits ( ! doma9 :i* obeys ,newton's law4 ,ev5 ! simplicity ( ?is law is a lesson = u2 h[ _m complicat$ ph5om5a are conta9$ 9 ! two l9es ( its enunciation2 p]sons :o d n und]/& cele/ial me*anics may =m "s idea ( x at lea/ f ! size ( ! treatises devot$ to ?is sci5ce2 & !n x may be hop$ t ! complication ( physical ph5om5a likewise hides f u "s simple cause /ill unkn[n4 ,x is !re=e a/ronomy :i* has %[n u :at are ! g5]al *>act]i/ics ( natural laws2 b among ^! *>act]i/ics !re is "o1 ! mo/ subtle & ! mo/ important ( all1 :i* ,i %all ask leave to /ress4 ,h[ 0 ! ord] ( ! univ]se und]/ood by ! anci5ts2 = in/ance1 by ,py?agoras1 ,plato or ,ari/otle8 ,x 0 ei!r an immutable type fix$ once = all1 or an ideal to :i* ! _w s\ght to approa*4 ,kepl] himself /ill ?\ght ?us :51 = in/ance1 he s\ght :e!r ! di/ances ( ! planets f ! sun _h n "s relation to ! five regul> polyh$rons4 ,?is idea conta9$ no?+ absurd1 b x 0 /]ile1 s9ce nature is n s made4 ,newton has %[n u t a law is only a necess>y relation betwe5 ! pres5t /ate ( ! _w & its imm$iately subsequ5t /ate4 ,all ! o!r laws s9ce discov]$ are no?+ else2 !y are 9 sum1 diff]5tial equations2 b x is a/ronomy :i* furni%$ ! fir/ model = !m1 )\t :i* we %\ld d\btless long h err$4 ,a/ronomy has also taught u to set at naught appe>ances4 ,! day ,cop]nicus prov$ t :at 0 ?\ght ! mo/ /able 0 9 motion1 t :at 0 ?\ght mov+ 0 fix$1 he %[$ u h[ deceptive c\ld be ! infantile reason+s :i* spr+ directly f ! imm$iate data ( our s5ses4 ,true1 8 ideas did n easily triumph1 b s9ce ?is triumph !re is no long] a prejudice s invet]ate t we c n %ake x (f4 ,h[ c we e/imate ! value ( ! new weapon ?us won8 ,! anci5ts ?\ght ev]y?+ 0 made = man1 & ?is illusion mu/ be v t5aci\s1 s9ce x mu/ ev] be combat$4 ,yet x is necess>y to dive/ oneself ( x2 or else "o w be only an et]nal myope1 incapable ( see+ ! tru?4 ,to compreh5d nature "o mu/ be able to get out ( self1 s to speak1 & to contemplate h] f _m diff]5t po9ts ( view2 o!rwise we nev] %all "k m ?an "o side4 ,n[1 to get out ( self is :at he :o ref]s ev]y?+ to himself c n d4 ,:o deliv]$ u f ?is illusion8 ,x 0 ^? :o %[$ u t ! e>? is only "o ( ! smalle/ planets ( ! sol> sy/em1 & t ! sol> sy/em itself is only an imp]ceptible po9t 9 ! inf9ite spaces ( ! /ell> univ]se4 ,at ! same "t a/ronomy taught u n to be afraid ( big numb]s4 ,?is 0 ne$ful1 n only = kn[+ ! heav5s1 b to "k ! e>? itself2 & 0 n s easy z x seems to u to-day4 ,let u try to g back & picture to ourselves :at a ,greek w\ld h ?\ght if told t r$ light vibrates f\r hundr$ millions ( millions ( times p] second4 ,)\t any d\bt1 su* an ass]tion w\ld h appe>$ to him pure madness1 & he nev] w\ld h l[]$ himself to te/ x4 ,to-day a hypo!sis w no long] appe> absurd to u because x obliges u to imag9e objects mu* l>g] or small] ?an ^? our s5ses are capable ( %[+ u1 & we no long] compreh5d ^? scruples :i* arre/$ our pr$ecessors & prev5t$ !m f discov]+ c]ta9 tru?s simply because !y 7 afraid ( !m4 ,b :y8 ,x is because we h se5 ! heav5s enl>g+ & enl>g+ )\t cease2 because we "k t ! sun is #aej millions ( kilomet]s f ! e>? & t ! di/ances ( ! ne>e/ />s are hundr$s ( ?\s&s ( times great] yet4 ,habituat$ to ! contemplation ( ! inf9itely great1 we h become apt to compreh5d ! inf9itely small4 ,?anks to ! education x has receiv$1 our imag9ation1 l ! eagle's eye t ! sun does n dazzle1 c look tru? 9 ! face4 ,0 ,i wrong 9 say+ t x is a/ronomy :i* has made u a s\l capable ( compreh5d+ nature2 t "u heav5s always ov]ca/ & />less1 ! e>? itself w\ld h be5 = u et]nally un9telligible2 t we %\ld !re h se5 only caprice & disord]2 & t1 n kn[+ ! _w1 we %\ld nev] h be5 able to subdue x8 ,:at sci5ce c\ld h be5 m useful8 ,& 9 ?us speak+ ,i put myself at ! po9t ( view ( ^? :o only value practical applications4 ,c]ta9ly1 ?is po9t ( view is n m9e2 z = me1 on ! contr>y1 if ,i admire ! conque/s ( indu/ry1 x is above all because if !y free u f mat]ial c>es1 !y w "o day give to all ! leisure to contemplate nature4 ,i d n say3 ,sci5ce is useful1 because x tea*es u to con/ruct ma*9es4 ,i say3 ,ma*9es are useful1 because 9 work+ = u1 !y w "s day leave u m "t to make sci5ce4 ,b f9ally x is wor? rem>k+ t betwe5 ! two po9ts ( view !re is no antagonism1 & t man hav+ pursu$ a dis9t]e/$ aim1 all else has be5 add$ unto him4 ,augu/e ,comte has said some:]e1 t x w\ld be idle to seek to "k ! composition ( ! sun1 s9ce ?is k w\ld be ( no use to sociology4 ,h[ c\ld he be s %ort-sight$8 ,h we n j se5 t x is by a/ronomy t1 to speak 8 language1 humanity has pass$ f ! !ological to ! positive /ate8 ,he f\nd an explanation = t because x _h happ5$4 ,b h[ has he n und]/ood t :at rema9$ to d 0 n less consid]able & w\ld be n less pr(itable8 ,physical a/ronomy1 :i* he seems to condemn1 has already begun to be> fruit1 & x w give u mu* m1 = x only dates f ye/]day4 ,fir/ 0 discov]$ ! nature ( ! sun1 :at ! f\nd] ( positivism wi%$ to d5y u1 & !re bodies 7 f\nd :i* exi/ on ! e>?1 b _h "h rema9$ undiscov]$2 = example1 helium1 t gas almo/ z light z hydrog54 ,t already contradict$ ,comte4 ,b to ! spectroscope we owe a lesson preci\s 9 a q diff]5t way2 9 ! mo/ di/ant />s1 x %[s u ! same sub/ances4 ,x might h be5 ask$ :e!r ! t]re/rial elem5ts 7 n due to "s *ance :i* _h br\ght toge!r m t5u\s atoms to con/ruct ( !m ! m complex edifice t ! *emi/s call atom2 :e!r1 9 o!r regions ( ! univ]se1 o!r =tuit\s meet+s _h n eng5d]$ edifices entirely diff]5t4 ,n[ we "k t ?is is n s1 t ! laws ( our *emi/ry are ! g5]al laws ( nature1 & t !y owe no?+ to ! *ance :i* caus$ u to be born on ! e>?4 ,b1 x w be said1 a/ronomy has giv5 to ! o!r sci5ces all x c give !m1 & n[ t ! heav5s h procur$ = u ! in/rum5ts :i* enable u to /udy t]re/rial nature1 !y c\ld )\t dang] veil !mselves =ev]4 ,aft] :at we h j said1 is !re /ill ne$ to answ] ?is objection8 ,"o c\ld h reason$ ! same 9 ,ptolemy's "t2 !n also m5 ?\ght !y knew ev]y?+1 & !y /ill _h almo/ ev]y?+ to le>n4 ,! />s are maje/ic laboratories1 gigantic crucibles1 su* z no *emi/ c\ld dream4 ,!re reign temp]atures impossible = u to realize4 ,_! only defect is be+ a little f> away2 b ! telescope w soon br+ !m ne> to u1 & !n we %all see h[ matt] acts !re4 ,:at good =tune = ! physici/ & ! *emi/6 ,matt] w !re exhibit itself to u "u a ?\s& diff]5t /ates1 f ^? r>efi$ gases :i* seem to =m ! nebul & :i* are lum9\s ) ,i "k n :at glimm]+ ( my/]i\s orig91 ev5 to ! inc&esc5t />s & to ! planets s ne> & yet s diff]5t4 ,p]*ance ev51 ! />s w "s day tea* u some?+ ab\t life2 t seems an ins5sate dream & ,i d n at all see h[ x c be realiz$2 b1 a hundr$ ye>s ago1 w\ld n ! *emi/ry ( ! />s h also appe>$ a mad dream8 ,b limit+ our views to horizons less di/ant1 !re /ill w rema9 to u promises less cont+5t & yet suffici5tly s$uctive4 ,if ! pa/ has giv5 u mu*1 we may re/ assur$ t ! future w give u /ill m4 ,9 sum1 x is incr$ible h[ useful belief 9 a/rology has be5 to humanity4 ,if ,kepl] & ,ty*o ,brahe made a liv+1 x 0 because !y sold to nave k+s pr$ictions f\nd$ on ! conjunctions ( ! />s4 ,if ^! pr9ces _h n be5 s cr$ul\s1 we %\ld p]haps /ill believe t nature obeys caprice1 & we %\ld /ill wall[ 9 ignorance4 ,,*apt] ,,vii ,,! ,,hi/ory ,,( ,,ma!matical ,,physics ,! ,pa/ & ! ,future ( ,physics4--,:at is ! pres5t /ate ( ma!matical physics8 ,:at are ! problems x is l$ to set itself8 ,:at is its future8 ,is its ori5tation ab\t to be modifi$8 ,t5 ye>s h5ce w ! aim & ! me?ods ( ?is sci5ce appe> to our imm$iate successors 9 ! same light z to ourselves2 or1 on ! contr>y1 are we ab\t to witness a pr(\nd trans=mation8 ,su* are ! que/ions we are =c$ to raise 9 ent]+ to-day ^u our inve/igation4 ,if x is easy to prop\nd !m3 to answ] is difficult4 ,if we felt tempt$ to risk a pr$iction1 we %\ld easily resi/ ?is temptation1 by ?9k+ ( all ! /upidities ! mo/ em95t savants ( a hundr$ ye>s ago w\ld h utt]$1 if "s "o _h ask$ !m :at ! sci5ce ( ! n9ete5? c5tury w\ld be4 ,!y w\ld h ?\ght !mselves bold 9 _! pr$ictions1 & aft] ! ev5t1 h[ v timid we %\ld h f\nd !m4 ,d n1 !re=e1 expect ( me any prophecy4 ,b if1 l all prud5t physicians1 ,i %un giv+ a prognosis1 yet ,i c n disp5se ) a little diagno/ic2 well1 yes1 !re are indications ( a s]i\s crisis1 z if we might expect an approa*+ trans=mation4 ,/ill1 be n too anxi\s3 we are sure ! pati5t w n die ( x1 & we may ev5 hope t ?is crisis w be salut>y1 = ! hi/ory ( ! pa/ seems to gu>antee u ?is4 ,?is crisis1 9 fact1 is n ! fir/1 & to und]/& x1 x is important to recall ^? :i* h prec$$4 ,p>don !n a brief hi/orical sket*4 ,! ,physics ( ,c5tral ,=ces4--,ma!matical physics1 z we "k1 0 born ( cele/ial me*anics1 :i* gave bir? to x at ! end ( ! eighte5? c5tury1 at ! mom5t :5 x itself atta9$ its complete developm5t4 ,dur+ its fir/ ye>s especially1 ! infant /rik+ly resembl$ its "m4 ,! a/ronomic univ]se is =m$ ( masses1 v great1 no d\bt1 b sep>at$ by int]vals s imm5se t !y appe> to u only z mat]ial po9ts4 ,^! po9ts attract ea* o!r inv]sely z ! squ>e ( ! di/ance1 & ?is attraction is ! sole =ce :i* influ5ces _! movem5ts4 ,b if our s5ses 7 suffici5tly ke5 to %[ u all ! details ( ! bodies :i* ! physici/ /udies1 ! spectacle ?us disclos$ w\ld sc>cely diff] f ! "o ! a/ronom] contemplates4 ,!re also we %\ld see mat]ial po9ts1 sep>at$ f "o ano!r by int]vals1 enorm\s 9 comp>ison ) _! dim5sions1 & describ+ orbits accord+ to regul> laws4 ,^! inf9itesimal />s are ! atoms4 ,l ! />s prop]1 !y attract or repel ea* o!r1 & ?is attraction or ?is repulsion1 foll[+ ! /raight l9e :i* jo9s !m1 dep5ds only on ! di/ance4 ,! law accord+ to :i* ?is =ce v>ies z function ( ! di/ance is p]haps n ! law ( ,newton1 b x is an analog\s law2 9 place ( ! expon5t -#b1 we h probably a diff]5t expon5t1 & x is f ?is *ange ( expon5t t arises all ! div]sity ( physical ph5om5a1 ! v>iety ( qualities & ( s5sations1 all ! _w1 color$ & sonor\s1 :i* surr\nds u2 9 a ^w1 all nature4 ,su* is ! primitive conception 9 all its purity4 ,x only rema9s to seek 9 ! diff]5t cases :at value %\ld be giv5 to ?is expon5t 9 ord] to expla9 all ! facts4 ,x is on ?is model t ,laplace1 = example1 con/ruct$ 8 beautiful !ory ( capill>ity2 he reg>ds x only z a p>ticul> case ( attraction1 or1 z he says1 ( univ]sal gravitation1 & no "o is a/oni%$ to f9d x 9 ! middle ( "o ( ! five volumes ( ! ',mcanique cle/e4' ,m rec5tly ,briot believes he p5etrat$ ! f9al secret ( optics 9 demon/rat+ t ! atoms ( e!r attract ea* o!r 9 ! inv]se ratio ( ! six? p[] ( ! di/ance2 & ,maxwell himself1 does he n say some:]e t ! atoms ( gases repel ea* o!r 9 ! inv]se ratio ( ! fif? p[] ( ! di/ance8 ,we h ! expon5t -#f1 or -#e1 9 place ( ! expon5t -#b1 b x is always an expon5t4 ,among ! !ories ( ?is epo*1 "o alone is an exception1 t ( ,f\ri]2 9 x are inde$ atoms act+ at a di/ance "o ^u ! o!r2 !y mutually transmit heat1 b !y d n attract1 !y nev] budge4 ,f ?is po9t ( view1 ,f\ri]'s !ory mu/ h appe>$ to ! eyes ( 8 contempor>ies1 to ^? ( ,f\ri] himself1 z imp]fect & provisional4 ,?is conception 0 n )\t gr&eur2 x 0 s$uctive1 & _m among u h n f9ally r5\nc$ x2 !y "k t "o w atta9 ! ultimate elem5ts ( ?+s only by pati5tly dis5tangl+ ! complicat$ ske9 t our s5ses give u2 t x is necess>y to advance /ep by /ep1 neglect+ no int]m$i>y2 t our fa!rs 7 wrong 9 wi%+ to skip /ations2 b !y believe t :5 "o %all h arriv$ at ^! ultimate elem5ts1 !re aga9 w be f\nd ! maje/ic simplicity ( cele/ial me*anics4 ,nei!r has ?is conception be5 useless2 x has r5d]$ u an ine/imable s]vice1 s9ce x has contribut$ to make precise ! fundam5tal notion ( ! physical law4 ,i w expla9 myself2 h[ did ! anci5ts und]/& law8 ,x 0 = !m an int]nal h>mony1 /atic1 s to say1 & immutable2 or else x 0 l a model t nature tri$ to imitate4 ,= u a law is some?+ q diff]5t2 x is a con/ant relation betwe5 ! ph5om5on ( to-day & t ( to-morr[2 9 a ^w1 x is a diff]5tial equation4 ,behold ! ideal =m ( physical law2 well1 x is ,newton's law :i* fir/ clo!d x =?4 ,if !n "o has acclimat$ ?is =m 9 physics1 x is precisely by copy+ z f> z possible ?is law ( ,newton1 t is by imitat+ cele/ial me*anics4 ,?is is1 moreov]1 ! idea ,i h tri$ to br+ out 9 ,*apt] ,,vi4 ,! ,physics ( ! ,pr9ciples4--,nev]!less1 a day arriv$ :5 ! conception ( c5tral =ces no long] appe>$ suffici5t1 & ?is is ! fir/ ( ^? crises ( :i* ,i j n[ spoke4 ,:at 0 done !n8 ,! attempt to p5etrate into ! detail ( ! /ructure ( ! univ]se1 to isolate ! pieces ( ?is va/ me*anism1 to analyze "o by "o ! =ces :i* put !m 9 motion1 0 ab&on$1 & we 7 cont5t to take z guides c]ta9 g5]al pr9ciples ! express object ( :i* is to sp>e u ?is m9ute /udy4 ,h[ s8 ,suppose we h be=e u any ma*9e2 ! initial :eel "w & ! f9al :eel "w alone are visible1 b ! transmission1 ! int]m$i>y ma*9]y by :i* ! movem5t is communicat$ f "o to ! o!r1 is hidd5 9 ! int]ior & escapes our view2 we d n "k :e!r ! communication is made by ge>+ or by belts1 by connect+-rods or by o!r contrivances4 ,d we say t x is impossible = u to und]/& any?+ ab\t ?is ma*9e s long z we are n p]mitt$ to take x to pieces8 ,y "k well we d n1 & t ! pr9ciple ( ! cons]vation ( en]gy suffices to det]m9e = u ! mo/ int]e/+ po9t4 ,we easily asc]ta9 t ! f9al :eel turns t5 times less quickly ?an ! initial :eel1 s9ce ^! two :eels are visible2 we are able !nce to conclude t a c\ple appli$ to ! "o w be balanc$ by a c\ple t5 times great] appli$ to ! o!r4 ,= t !re is no ne$ to p5etrate ! me*anism ( ?is equilibrium & to "k h[ ! =ces comp5sate ea* o!r 9 ! int]ior ( ! ma*9e2 x suffices to be assur$ t ?is comp5sation c n fail to occur4 ,well1 9 reg>d to ! univ]se1 ! pr9ciple ( ! cons]vation ( en]gy is able to r5d] u ! same s]vice4 ,! univ]se is also a ma*9e1 mu* m complicat$ ?an all ^? ( indu/ry1 ( :i* almo/ all ! p>ts are pr(\ndly hidd5 f u2 b 9 obs]v+ ! motion ( ^? t we c see1 we are able1 by ! aid ( ?is pr9ciple1 to draw conclusions :i* rema9 true :atev] may be ! details ( ! invisible me*anism :i* animates !m4 ,! pr9ciple ( ! cons]vation ( en]gy1 or ,may]'s pr9ciple1 is c]ta9ly ! mo/ important1 b x is n ! only "o2 !re are o!rs f :i* we c d]ive ! same advantage4 ,^! are3 ,c>not's pr9ciple1 or ! pr9ciple ( ! degradation ( en]gy4 ,newton's pr9ciple1 or ! pr9ciple ( ! equality ( action & reaction4 ,! pr9ciple ( relativity1 accord+ to :i* ! laws ( physical ph5om5a mu/ be ! same = a /ation>y obs]v] z = an obs]v] c>ri$ along 9 a uni=m motion ( translation2 s t we h n & c n h any means ( disc]n+ :e!r or n we are c>ri$ along 9 su* a motion4 ,! pr9ciple ( ! cons]vation ( mass1 or ,lavoisi]'s pr9ciple4 ,i w add ! pr9ciple ( lea/ action4 ,! application ( ^! five or six g5]al pr9ciples to ! diff]5t physical ph5om5a is suffici5t = our le>n+ ( !m all t we c\ld reasonably hope to "k ( !m4 ,! mo/ rem>kable example ( ?is new ma!matical physics is1 beyond "q1 ,maxwell's electromagnetic !ory ( light4 ,we "k no?+ z to :at ! e!r is1 h[ its molecules are dispos$1 :e!r !y attract or repel ea* o!r2 b we "k t ?is m$ium transmits at ! same "t ! optical p]turbations & ! electrical p]turbations2 we "k t ?is transmission mu/ take place 9 con=mity ) ! g5]al pr9ciples ( me*anics1 & t suffices u = ! e/abli%m5t ( ! equations ( ! electromagnetic field4 ,^! pr9ciples are results ( exp]im5ts boldly g5]aliz$2 b !y seem to d]ive f _! v g5]ality a high degree ( c]ta9ty4 ,9 fact1 ! m g5]al !y are1 ! m frequ5t are ! opportunities to *eck !m1 & ! v]ifications multiply+1 tak+ ! mo/ v>i$1 ! mo/ unexpect$ =ms1 end by no long] leav+ place = d\bt4 ,utility ( ! ,old ,physics4--,su* is ! second phase ( ! hi/ory ( ma!matical physics & we h n yet em]g$ f x4 ,%all we say t ! fir/ has be5 useless8 t dur+ fifty ye>s sci5ce w5t ! wrong way1 & t !re is no?+ left b to =get s _m accumulat$ ef=ts t a vici\s conception condemn$ 9 advance to failure8 ,n ! lea/ 9 ! _w4 ,d y ?9k ! second phase c\ld h come into exi/5ce )\t ! fir/8 ,! hypo!sis ( c5tral =ces conta9$ all ! pr9ciples2 x involv$ !m z necess>y consequ5ces2 x involv$ bo? ! cons]vation ( en]gy & t ( masses1 & ! equality ( action & reaction1 & ! law ( lea/ action1 :i* appe>$1 x is true1 n z exp]im5tal tru?s1 b z !orems2 ! enunciation ( :i* _h at ! same "t some?+ m precise & less g5]al ?an "u _! pres5t =m4 ,x is ! ma!matical physics ( our fa!rs :i* has famili>iz$ u little by little ) ^! v>i\s pr9ciples2 :i* has habituat$ u to recognize !m "u ! diff]5t ve/m5ts 9 :i* !y disguise !mselves4 ,!y h be5 comp>$ ) ! data ( exp]i5ce1 x has be5 se5 h[ x 0 necess>y to modify _! enunciation to adapt !m to ^! data2 !reby !y h be5 ext5d$ & consolidat$4 ,?us !y came to be reg>d$ z exp]im5tal tru?s2 ! conception ( c5tral =ces became !n a useless support1 or r an emb>rassm5t1 s9ce x made ! pr9ciples p>take ( its hypo!tical "*4 ,! frames !n h n brok51 because !y are ela/ic2 b !y h enl>g$2 our fa!rs1 :o e/abli%$ !m1 did n labor 9 va91 & we recognize 9 ! sci5ce ( to-day ! g5]al traits ( ! sket* :i* !y trac$4 ,,*apt] ,,viii ,,! ,,pres5t ,,crisis ,,( ,,ma!matical ,,physics ,! ,new ,crisis4--,are we n[ ab\t to ent] ^u a ?ird p]iod8 ,are we on ! eve ( a second crisis8 ,^! pr9ciples on :i* we h built all1 are !y ab\t to crumble away 9 _! turn8 ,?is has be5 = "s "t a p]t95t "q4 ,:5 ,i speak ?us1 y no d\bt ?9k ( radium1 t gr& revolutioni/ ( ! pres5t "t1 & 9 fact ,i %all come back to x pres5tly2 b !re is some?+ else4 ,x is n alone ! cons]vation ( en]gy :i* is 9 "q2 all ! o!r pr9ciples are equally 9 dang]1 z we %all see 9 pass+ !m successively 9 review4 ,c>not's ,pr9ciple4--,let u comm5ce ) ! pr9ciple ( ,c>not4 ,?is is ! only "o :i* does n pres5t itself z an imm$iate consequ5ce ( ! hypo!sis ( c5tral =ces2 m ?an t1 x seems1 if n to directly contradict t hypo!sis1 at lea/ n to be reconcil$ ) x )\t a c]ta9 ef=t4 ,if physical ph5om5a 7 due exclusively to ! movem5ts ( atoms ^: mutual attraction dep5d$ only on ! di/ance1 x seems t all ^! ph5om5a %\ld be rev]sible2 if all ! initial velocities 7 rev]s$1 ^! atoms1 always subject$ to ! same =ces1 ought to g ov] _! trajectories 9 ! contr>y s5se1 j z ! e>? w\ld describe 9 ! retrograde s5se ?is same elliptic orbit :i* x describes 9 ! direct s5se1 if ! initial conditions ( its motion _h be5 rev]s$4 ,on ?is acc\nt1 if a physical ph5om5on is possible1 ! inv]se ph5om5on %\ld be equally s1 & "o %\ld be able to reasc5d ! c\rse ( "t4 ,n[1 x is n s 9 nature1 & ?is is precisely :at ! pr9ciple ( ,c>not tea*es u2 heat c pass f ! w>m body to ! cold body2 x is impossible aft]w>d to make x take ! inv]se r\te & to ree/abli% diff]5ces ( temp]ature :i* h be5 effac$4 ,motion c be :olly dissipat$ & trans=m$ into heat by friction2 ! contr>y trans=mation c nev] be made except p>tially4 ,we h /riv5 to reconcile ?is app>5t contradiction4 ,if ! _w t5ds t[>d uni=mity1 ?is is n because its ultimate p>ts1 at fir/ unlike1 t5d to become less & less diff]5t2 x is because1 %ift+ at r&om1 !y end by bl5d+4 ,= an eye :i* %\ld di/+ui% all ! elem5ts1 ! v>iety w\ld rema9 always z great2 ea* gra9 ( ?is du/ pres]ves its orig9ality & does n model itself on its neighbors2 b z ! bl5d becomes m & m intimate1 our gross s5ses p]ceive only ! uni=mity4 ,?is is :y1 = example1 temp]atures t5d to a level1 )\t ! possibility ( go+ backw>ds4 ,a drop ( w9e falls into a glass ( wat]2 :atev] may be ! law ( ! int]nal motion ( ! liquid1 we %all soon see x color$ ( a uni=m rosy t9t1 & h[ev] mu* f ?is mom5t "o may %ake x aft]w>ds1 ! w9e & ! wat] d n seem capable ( aga9 sep>at+4 ,"h we h ! type ( ! irrev]sible physical ph5om5on3 to hide a gra9 ( b>ley 9 a heap ( :eat1 ?is is easy2 aft]w>ds to f9d x aga9 & get x out1 ?is is practically impossible4 ,all ?is ,maxwell & ,boltzmann h expla9$2 b ! "o :o has se5 x mo/ cle>ly1 9 a book too little read because x is a little difficult to read1 is ,gibbs1 9 8 ,elem5t>y ,pr9ciples ( ,/ati/ical ,me*anics4' ,= ^? :o take ?is po9t ( view1 ,c>not's pr9ciple is only an imp]fect pr9ciple1 a sort ( concession to ! infirmity ( our s5ses2 x is because our eyes are too gross t we d n di/+ui% ! elem5ts ( ! bl5d2 x is because our h&s are too gross t we c n =ce !m to sep>ate2 ! imag9>y demon ( ,maxwell1 :o is able to sort ! molecules "o by "o1 c\ld well con/ra9 ! _w to return backw>d4 ,c x return ( itself8 ,t is n impossible2 t is only inf9itely improbable4 ,! *ances are t we %\ld wait a long "t = ! conc\rse ( circum/ances :i* w\ld p]mit a retrogradation2 b soon] or lat] !y w occur1 aft] ye>s ^: numb] x w\ld take millions ( figures to write4 ,^! res]vations1 h[ev]1 all rema9$ !oretic2 !y 7 n v disquiet+1 & ,c>not's pr9ciple reta9$ all its practical value4 ,b "h ! sc5e *anges4 ,! biologi/1 arm$ ) 8 microscope1 long ago notic$ 9 8 prep>ations irregul> movem5ts ( little p>ticles 9 susp5sion2 ?is is ! ,br[nian movem5t4 ,he fir/ ?\ght ?is 0 a vital ph5om5on1 b soon he saw t ! inanimate bodies danc$ ) no less ardor ?an ! o!rs2 !n he turn$ ! matt] ov] to ! physici/s4 ,unhappily1 ! physici/s rema9$ long un9t]e/$ 9 ?is "q2 "o conc5trates ! light to illum9ate ! microscopic prep>ation1 ?\ght !y2 ) light goes heat2 !nce inequalities ( temp]ature & 9 ! liquid int]ior curr5ts :i* produce ! movem5ts ref]r$ to4 ,x occurr$ to ,m4 ,g\y to look m closely1 & he saw1 or ?\ght he saw1 t ?is explanation is unt5able1 t ! movem5ts become brisk] z ! p>ticles are small]1 b t !y are n influ5c$ by ! mode ( illum9ation4 ,if !n ^! movem5ts nev] cease1 or r are reborn )\t cease1 )\t borr[+ any?+ f an ext]nal s\rce ( en]gy1 :at ought we to believe8 ,to be sure1 we %\ld n on ?is acc\nt r5\nce our belief 9 ! cons]vation ( en]gy1 b we see "u our eyes n[ motion trans=m$ into heat by friction1 n[ inv]sely heat *ang$ into motion1 & t )\t loss s9ce ! movem5t la/s =ev]4 ,?is is ! contr>y ( ,c>not's pr9ciple4 ,if ?is be s1 to see ! _w return backw>d1 we no long] h ne$ ( ! inf9itely ke5 eye ( ,maxwell's demon2 our microscope suffices4 ,bodies too l>ge1 ^?1 = example1 :i* are a t5? ( a millimet]1 are hit f all sides by mov+ atoms1 b !y d n budge1 because ^! %ocks are v num]\s & ! law ( *ance makes !m comp5sate ea* o!r2 b ! small] p>ticles receive too few %ocks = ?is comp5sation to take place ) c]ta9ty & are incessantly knock$ ab\t4 ,& behold already "o ( our pr9ciples 9 p]il4 ,! ,pr9ciple ( ,relativity4--,let u pass to ! pr9ciple ( relativity2 ?is n only is confirm$ by daily exp]i5ce1 n only is x a necess>y consequ5ce ( ! hypo!sis ( c5tral =ces1 b x is irresi/ibly impos$ ^u our good s5se1 & yet x also is assail$4 ,consid] two electrifi$ bodies2 ?\gh !y seem to u at re/1 !y are bo? c>ri$ along by ! motion ( ! e>?2 an electric *>ge 9 motion1 ,r[l& has taught u1 is equival5t to a curr5t2 ^! two *>g$ bodies are1 !re=e1 equival5t to two p>allel curr5ts ( ! same s5se & ^! two curr5ts %\ld attract ea* o!r4 ,9 measur+ ?is attraction1 we %all measure ! velocity ( ! e>?2 n its velocity 9 relation to ! sun or ! fix$ />s1 b its absolute velocity4 ,i well "k :at w be said3 ,x is n its absolute velocity t is measur$1 x is its velocity 9 relation to ! e!r4 ,h[ unsatisfactory t is6 ,is x n evid5t t f ! pr9ciple s und]/ood we c\ld no long] inf] any?+8 ,x c\ld no long] tell u any?+ j because x w\ld no long] fe> any contradiction4 ,if we succe$ 9 measur+ any?+1 we %all always be free to say t ?is is n ! absolute velocity1 & if x is n ! velocity 9 relation to ! e!r1 x might always be ! velocity 9 relation to "s new unkn[n fluid ) :i* we might fill space4 ,inde$1 exp]im5t has tak5 ^u itself to ru9 ?is int]pretation ( ! pr9ciple ( relativity2 all attempts to measure ! velocity ( ! e>? 9 relation to ! e!r h l$ to negative results4 ,?is "t exp]im5tal physics has be5 m fai?ful to ! pr9ciple ?an ma!matical physics2 ! !ori/s1 to put 9 accord _! o!r g5]al views1 w\ld n h sp>$ x2 b exp]im5t has be5 /ubborn 9 confirm+ x4 ,! means h be5 v>i$2 f9ally ,mi*elson pu%$ precision to its la/ limits2 no?+ came ( x4 ,x is precisely to expla9 ?is ob/9acy t ! ma!maticians are =c$ to-day to employ all _! +5uity4 ,_! task 0 n easy1 & if ,lor5tz has got ?r\gh x1 x is only by accumulat+ hypo!ses4 ,! mo/ +5i\s idea 0 t ( local "t4 ,imag9e two obs]v]s :o wi% to adju/ _! timepieces by optical signals2 !y ex*ange signals1 b z !y "k t ! transmission ( light is n in/antane\s1 !y are c>eful to cross !m4 ,:5 /ation ,b p]ceives ! signal f /ation ,a1 its clock %\ld n m>k ! same h\r z t ( /ation ,a at ! mom5t ( s5d+ ! signal1 b ?is h\r augm5t$ by a con/ant repres5t+ ! duration ( ! transmission4 ,suppose1 = example1 t /ation ,a s5ds its signal :5 its clock m>ks ! h\r ,o1 & t /ation ,b p]ceives x :5 its clock m>ks ! h\r t4 ,! clocks are adju/$ if ! sl[ness equal to t repres5ts ! duration ( ! transmission1 & to v]ify x1 /ation ,b s5ds 9 its turn a signal :5 its clock m>ks ,o2 !n /ation ,a %\ld p]ceive x :5 its clock m>ks t4 ,! timepieces are !n adju/$4 ,& 9 fact !y m>k ! same h\r at ! same physical in/ant1 b on ! "o condition1 t ! two /ations are fix$4 ,o!rwise ! duration ( ! transmission w n be ! same 9 ! two s5ses1 s9ce ! /ation ,a1 = example1 moves =w>d to meet ! optical p]turbation emanat+ f ,b1 :]eas ! /ation ,b flees be=e ! p]turbation emanat+ f ,a4 ,! wat*es adju/$ 9 t way w n m>k1 !re=e1 ! true "t2 !y w m>k :at may be call$ ! local "t1 s t "o ( !m w be sl[ ( ! o!r4 ,x matt]s little1 s9ce we h no means ( p]ceiv+ x4 ,all ! ph5om5a :i* happ5 at ,a1 = example1 w be late1 b all w be equally s1 & ! obs]v] w n p]ceive x1 s9ce 8 wat* is sl[2 s1 z ! pr9ciple ( relativity requires1 he w h no means ( kn[+ :e!r he is at re/ or 9 absolute motion4 ,unhappily1 t does n suffice1 & complem5t>y hypo!ses are necess>y2 x is necess>y to admit t bodies 9 motion und]go a uni=m contraction 9 ! s5se ( ! motion4 ,"o ( ! diamet]s ( ! e>?1 = example1 is %runk by "o two-hundr$-million? 9 consequ5ce ( our planet's motion1 :ile ! o!r diamet] reta9s its normal l5g?4 ,?us ! la/ little diff]5ces are comp5sat$4 ,& !n1 !re is /ill ! hypo!sis ab\t =ces4 ,=ces1 :atev] be _! orig91 gravity z well z ela/icity1 w\ld be r$uc$ 9 a c]ta9 proportion 9 a _w animat$ by a uni=m translation2 or1 r1 ?is w\ld happ5 = ! compon5ts p]p5dicul> to ! translation2 ! compon5ts p>allel w\ld n *ange4 ,resume1 !n1 our example ( two electrifi$ bodies2 ^! bodies repel ea* o!r1 b at ! same "t if all is c>ri$ along 9 a uni=m translation1 !y are equival5t to two p>allel curr5ts ( ! same s5se :i* attract ea* o!r4 ,?is electrodynamic attraction dim9i%es1 !re=e1 ! electro/atic repulsion1 & ! total repulsion is feebl] ?an if ! two bodies 7 at re/4 ,b s9ce to measure ?is repulsion we mu/ balance x by ano!r =ce1 & all ^! o!r =ces are r$uc$ 9 ! same proportion1 we p]ceive no?+4 ,?us all seems arrang$1 b are all ! d\bts dissipat$8 ,:at w\ld happ5 if "o c\ld communicate by non-lum9\s signals ^: velocity ( propagation diff]$ f t ( light8 ,if1 aft] hav+ adju/$ ! wat*es by ! optical proc$ure1 we wi%$ to v]ify ! adju/m5t by ! aid ( ^! new signals1 we %\ld obs]ve discrepancies :i* w\ld r5d] evid5t ! common translation ( ! two /ations4 ,& are su* signals inconceivable1 if we admit ) ,laplace t univ]sal gravitation is transmitt$ a million times m rapidly ?an light8 ,?us1 ! pr9ciple ( relativity has be5 valiantly def5d$ 9 ^! latt] times1 b ! v en]gy ( ! def5se proves h[ s]i\s 0 ! attack4 ,newton's ,pr9ciple4--,let u speak n[ ( ! pr9ciple ( ,newton1 on ! equality ( action & reaction4 ,?is is intimately b\nd up ) ! prec$+1 & x seems inde$ t ! fall ( ! "o w\ld involve t ( ! o!r4 ,?us we mu/ n be a/oni%$ to f9d "h ! same difficulties4 ,electrical ph5om5a1 accord+ to ! !ory ( ,lor5tz1 are due to ! displacem5ts ( little *>g$ p>ticles1 call$ electrons1 imm]s$ 9 ! m$ium we call e!r4 ,! movem5ts ( ^! electrons produce p]turbations 9 ! neighbor+ e!r2 ^! p]turbations propagate !mselves 9 e direction ) ! velocity ( light1 & 9 turn o!r electrons1 orig9ally at re/1 are made to vibrate :5 ! p]turbation rea*es ! p>ts ( ! e!r :i* t\* !m4 ,! electrons1 !re=e1 act on "o ano!r1 b ?is action is n direct1 x is accompli%$ ?r\gh ! e!r z int]m$i>y4 ,"u ^! conditions c !re be comp5sation betwe5 action & reaction1 at lea/ = an obs]v] :o %\ld take acc\nt only ( ! movem5ts ( matt]1 t is1 ( ! electrons1 & :o %\ld be ignorant ( ^? ( ! e!r t he c\ld n see8 ,evid5tly n4 ,ev5 if ! comp5sation %\ld be exact1 x c\ld n be simultane\s4 ,! p]turbation is propagat$ ) a f9ite velocity2 x1 !re=e1 rea*es ! second electron only :5 ! fir/ has long ago ent]$ ^u its re/4 ,?is second electron1 !re=e1 w und]go1 aft] a delay1 ! action ( ! fir/1 b w c]ta9ly n at t mom5t react ^u x1 s9ce ar\nd ?is fir/ electron no?+ any long] budges4 ,! analysis ( ! facts p]mits u to be /ill m precise4 ,imag9e1 = example1 a ,h]tzian oscillator1 l ^? us$ 9 wireless telegraphy2 x s5ds out en]gy 9 e direction2 b we c provide x ) a p>abolic mirror1 z ,h]tz did ) 8 smalle/ oscillators1 s z to s5d all ! en]gy produc$ 9 a s+le direction4 ,:at happ5s !n accord+ to ! !ory8 ,! app>atus recoils1 z if x 7 a cannon & ! project$ en]gy a ball2 & t is contr>y to ! pr9ciple ( ,newton1 s9ce our projectile "h has no mass1 x is n matt]1 x is en]gy4 ,! case is /ill ! same1 moreov]1 ) a beacon light provid$ ) a reflector1 s9ce light is no?+ b a p]turbation ( ! electromagnetic field4 ,?is beacon light %\ld recoil z if ! light x s5ds out 7 a projectile4 ,:at is ! =ce t %\ld produce ?is recoil8 ,x is :at is call$ ! ,maxwell-,b>?oli pressure4 ,x is v m9ute1 & x has be5 difficult to put x 9 evid5ce ev5 ) ! mo/ s5sitive radiomet]s2 b x suffices t x exi/s4 ,if all ! en]gy issu+ f our oscillator falls on a receiv]1 ?is w act z if x _h receiv$ a me*anical %ock1 :i* w repres5t 9 a s5se ! comp5sation ( ! oscillator's recoil2 ! reaction w be equal to ! action1 b x w n be simultane\s2 ! receiv] w move on1 b n at ! mom5t :5 ! oscillator recoils4 ,if ! en]gy propagates itself indef9itely )\t enc\nt]+ a receiv]1 ! comp5sation w nev] occur4 ,%all we say t ! space :i* sep>ates ! oscillator f ! receiv] & :i* ! p]turbation mu/ pass ov] 9 go+ f ! "o to ! o!r is n void1 t x is full n only ( e!r1 b ( air1 or ev5 9 ! int]planet>y spaces ( "s fluid subtile b /ill pond]able2 t ?is matt] und]goes ! %ock l ! receiv] at ! mom5t :5 ! en]gy rea*es x1 & recoils 9 its turn :5 ! p]turbation quits x8 ,t w\ld save ,newton's pr9ciple1 b t is n true4 ,if en]gy 9 its diffusion rema9$ always atta*$ to "s mat]ial sub/ratum1 !n matt] 9 motion w\ld c>ry along light ) x1 & ,fizeau has demon/rat$ t x does no?+ ( ! sort1 at lea/ = air4 ,mi*elson & ,morley h s9ce confirm$ ?is4 ,x might be suppos$ also t ! movem5ts ( matt] prop] are exactly comp5sat$ by ^? ( ! e!r2 b t w\ld lead u to ! same reflections z be=e n[4 ,! pr9ciple s und]/ood w expla9 ev]y?+1 s9ce1 :atev] might be ! visible movem5ts1 we always c\ld imag9e hypo!tical movem5ts :i* comp5sate !m4 ,b if x is able to expla9 ev]y?+1 ?is is because x does n enable u to =esee any?+2 x does n enable u to decide betwe5 ! diff]5t possible hypo!ses1 s9ce x expla9s ev]y?+ be=eh&4 ,x !re=e becomes useless4 ,& !n ! suppositions t x w\ld be necess>y to make on ! movem5ts ( ! e!r are n v satisfactory4 ,if ! electric *>ges d\ble1 x w\ld be natural to imag9e t ! velocities ( ! div]se atoms ( e!r d\ble also2 b1 = ! comp5sation1 x w\ld be necess>y t ! mean velocity ( ! e!r quadruple4 ,?is is :y ,i h long ?\ght t ^! consequ5ces ( !ory1 contr>y to ,newton's pr9ciple1 w\ld end "s day by be+ ab&on$1 & yet ! rec5t exp]im5ts on ! movem5ts ( ! electrons issu+ f radium seem r to confirm !m4 ,lavoisi]'s ,pr9ciple4--,i arrive at ! pr9ciple ( ,lavoisi] on ! cons]vation ( mass4 ,c]ta9ly1 ?is is "o n to be t\*$ )\t unsettl+ all me*anics4 ,& n[ c]ta9 p]sons ?9k t x seems true to u only because 9 me*anics m]ely mod]ate velocities are consid]$1 b t x w\ld cease to be true = bodies animat$ by velocities comp>able to t ( light4 ,n[ ^! velocities are believ$ at pres5t to h be5 realiz$2 ! ca?ode rays & ^? ( radium may be =m$ ( v m9ute p>ticles or ( electrons :i* are displac$ ) velocities small] no d\bt ?an t ( light1 b :i* might be its "o t5? or "o ?ird4 ,^! rays c be deflect$1 :e!r by an electric field1 or by a magnetic field1 & we are able1 by comp>+ ^! deflections1 to measure at ! same "t ! velocity ( ! electrons & _! mass 7or r ! relation ( _! mass to _! *>ge74 ,b :5 x 0 se5 t ^! velocities approa*$ t ( light1 x 0 decid$ t a correction 0 necess>y4 ,^! molecules1 be+ electrifi$1 c n be displac$ )\t agitat+ ! e!r2 to put !m 9 motion x is necess>y to ov]come a d\ble in]tia1 t ( ! molecule itself & t ( ! e!r4 ,! total or app>5t mass t "o measures is compos$1 !re=e1 ( two p>ts3 ! real or me*anical mass ( ! molecule & ! electrodynamic mass repres5t+ ! in]tia ( ! e!r4 ,! calculations ( ,abraham & ! exp]im5ts ( ,kaufmann h !n %[n t ! me*anical mass1 prop]ly s call$1 is null1 & t ! mass ( ! electrons1 or1 at lea/1 ( ! negative electrons1 is ( exclusively electrodynamic orig94 ,?is is :at =ces u to *ange ! def9ition ( mass2 we c n any long] di/+ui% me*anical mass & electrodynamic mass1 s9ce !n ! fir/ w\ld vani%2 !re is no mass o!r ?an electrodynamic in]tia4 ,b 9 ?is case ! mass c no long] be con/ant2 x augm5ts ) ! velocity1 & x ev5 dep5ds on ! direction1 & a body animat$ by a notable velocity w n oppose ! same in]tia to ! =ces :i* t5d to deflect x f its r\te1 z to ^? :i* t5d to accel]ate or to ret>d its progress4 ,!re is /ill a res\rce2 ! ultimate elem5ts ( bodies are electrons1 "s *>g$ negatively1 ! o!rs *>g$ positively4 ,! negative electrons h no mass1 ?is is und]/ood2 b ! positive electrons1 f ! little we "k ( !m1 seem mu* great]4 ,p]haps !y h1 besides _! electrodynamic mass1 a true me*anical mass4 ,! real mass ( a body w\ld1 !n1 be ! sum ( ! me*anical masses ( its positive electrons1 ! negative electrons n c\nt+2 mass s def9$ might /ill be con/ant4 ,alas6 ?is res\rce also evades u4 ,recall :at we h said ( ! pr9ciple ( relativity & ( ! ef=ts made to save x4 ,& x is n m]ely a pr9ciple :i* x is a "q ( sav+1 x is ! indubitable results ( ! exp]im5ts ( ,mi*elson4 ,well1 z 0 above se51 ,lor5tz1 to acc\nt = ^! results1 0 oblig$ to suppose t all =ces1 :atev] _! orig91 7 r$uc$ 9 ! same proportion 9 a m$ium animat$ by a uni=m translation2 ?is is n suffici5t2 x is n en\gh t ?is take place = ! real =ces1 x mu/ also be ! same = ! =ces ( in]tia2 x is !re=e necess>y1 he says1 t ! masses ( all ! p>ticles be influ5c$ by a translation to ! same degree z ! electromagnetic masses ( ! electrons4 ,s ! me*anical masses mu/ v>y 9 accordance ) ! same laws z ! electrodynamic masses2 !y c n1 !re=e1 be con/ant4 ,ne$ ,i po9t out t ! fall ( ,lavoisi]'s pr9ciple involves t ( ,newton's8 ,?is latt] signifies t ! c5t] ( gravity ( an isolat$ sy/em moves 9 a /raight l9e2 b if !re is no long] a con/ant mass1 !re is no long] a c5t] ( gravity1 we no long] "k ev5 :at ?is is4 ,?is is :y ,i said above t ! exp]im5ts on ! ca?ode rays appe>$ to ju/ify ! d\bts ( ,lor5tz conc]n+ ,newton's pr9ciple4 ,f all ^! results1 if !y 7 confirm$1 w\ld arise an entirely new me*anics1 :i* w\ld be1 above all1 *>act]iz$ by ?is fact1 t no velocity c\ld surpass t ( light1#i any m ?an any temp]ature c fall bel[ absolute z]o4 #i ,because bodies w\ld oppose an increas+ in]tia to ! causes :i* w\ld t5d to accel]ate _! motion2 & ?is in]tia w\ld become inf9ite :5 "o approa*$ ! velocity ( light4 ,no m = an obs]v]1 c>ri$ along himself 9 a translation he does n suspect1 c\ld any app>5t velocity surpass t ( light2 & ?is w\ld be !n a contradiction1 if we did n recall t ?is obs]v] w\ld n use ! same clocks z a fix$ obs]v]1 b1 inde$1 clocks m>k+ 'local "t4' ,"h we are !n fac+ a "q ,i cont5t myself ) /at+4 ,if !re is no long] any mass1 :at becomes ( ,newton's law8 ,mass has two aspects3 x is at ! same "t a coeffici5t ( in]tia & an attract+ mass ent]+ z factor into ,newtonian attraction4 ,if ! coeffici5t ( in]tia is n con/ant1 c ! attract+ mass be8 ,t is ! "q4 ,may]'s ,pr9ciple4--,at lea/1 ! pr9ciple ( ! cons]vation ( en]gy yet rema9$ to u1 & ?is seem$ m solid4 ,%all ,i recall to y h[ x 0 9 its turn ?r[n into discr$it8 ,?is ev5t has made m noise ?an ! prec$+1 & x is 9 all ! memoirs4 ,f ! fir/ words ( ,becqu]el1 &1 above all1 :5 ! ,curies _h discov]$ radium1 x 0 se5 t e radioactive body 0 an inexhau/ible s\rce ( radiation4 ,its activity seem$ to subsi/ )\t alt]ation ?r\gh\t ! mon?s & ! ye>s4 ,?is 0 9 itself a /ra9 on ! pr9ciples2 ^! radiations 7 9 fact en]gy1 & f ! same morsel ( radium ?is issu$ & =ev] issu$4 ,b ^! quantities ( en]gy 7 too slight to be measur$2 at lea/ t 0 ! belief & we 7 n mu* disquiet$4 ,! sc5e *ang$ :5 ,curie be?\ght himself to put radium 9 a calorimet]2 x 0 !n se5 t ! quantity ( heat incessantly creat$ 0 v notable4 ,! explanations propos$ 7 num]\s2 b 9 su* case we c n say1 ! m ! bett]4 ,9 s f> z no "o ( !m has prevail$ ov] ! o!rs1 we c n be sure !re is a good "o among !m4 ,s9ce "s "t1 h[ev]1 "o ( ^! explanations seems to be gett+ ! upp] h& & we may reasonably hope t we hold ! key to ! my/]y4 ,sir ,w4 ,ramsay has /riv5 to %[ t radium is 9 process ( trans=mation1 t x conta9s a /ore ( en]gy enorm\s b n inexhau/ible4 ,! trans=mation ( radium !n w\ld produce a million times m heat ?an all kn[n trans=mations2 radium w\ld we> itself out 9 #a1bej ye>s2 ?is is q %ort1 & y see t we are at lea/ c]ta9 to h ?is po9t settl$ "s hundr$s ( ye>s f n[4 ,:ile wait+1 our d\bts rema94 ,,*apt] ,,ix ,,! ,,future ,,( ,,ma!matical ,,physics ,! ,pr9ciples & ,exp]im5t4--,9 ! mid/ ( s mu* ru91 :at rema9s /&+8 ,! pr9ciple ( lea/ action is hi!rto intact1 & ,l>mor appe>s to believe t x w long survive ! o!rs2 9 reality1 x is /ill m vague & m g5]al4 ,9 pres5ce ( ?is g5]al collapse ( ! pr9ciples1 :at attitude w ma!matical physics take8 ,& fir/1 be=e too mu* excitem5t1 x is prop] to ask if all t is really true4 ,all ^! d]ogations to ! pr9ciples are enc\nt]$ only among inf9itesimals2 ! microscope is necess>y to see ! ,br[nian movem5t2 electrons are v light2 radium is v r>e1 & "o nev] has m ?an "s milligrams ( x at a "t4 ,&1 !n1 x may be ask$ :e!r1 besides ! inf9itesimal se51 !re 0 n ano!r inf9itesimal unse5 c\nt]poise to ! fir/4 ,s !re is an int]locutory "q1 &1 z x seems1 only exp]im5t c solve x4 ,we %all1 !re=e1 only h to h& ov] ! matt] to ! exp]im5t]s1 &1 :ile wait+ = !m to f9ally decide ! debate1 n to preoccupy ourselves ) ^! disquiet+ problems1 & to tranquilly cont9ue our "w z if ! pr9ciples 7 /ill unconte/$4 ,c]tes1 we h mu* to d )\t leav+ ! doma9 :]e !y may be appli$ 9 all security2 we h en\gh to employ our activity dur+ ?is p]iod ( d\bts4 ,! ,rle ( ! ,analy/4--,& z to ^! d\bts1 is x inde$ true t we c d no?+ to disemb>rass sci5ce ( !m8 ,x mu/ inde$ be said1 x is n alone exp]im5tal physics t has giv5 bir? to !m2 ma!matical physics has well contribut$4 ,x is ! exp]im5t]s :o h se5 radium ?r[ out en]gy1 b x is ! !ori/s :o h put 9 evid5ce all ! difficulties rais$ by ! propagation ( light across a m$ium 9 motion2 b = ^! x is probable we %\ld n h become consci\s ( !m4 ,well1 !n1 if !y h done _! be/ to put u into ?is emb>rassm5t1 x is prop] also t !y help u to get out ( x4 ,!y mu/ subject to critical exam9ation all ^! new views ,i h j outl9$ be=e y1 & ab&on ! pr9ciples only aft] hav+ made a loyal ef=t to save !m4 ,:at c !y d 9 ?is s5se8 ,t is :at ,i w try to expla94 ,x is a "q be=e all ( endeavor+ to obta9 a m satisfactory !ory ( ! electrodynamics ( bodies 9 motion4 ,x is !re especially1 z ,i h suffici5tly %[n above1 t difficulties accumulate4 ,x is useless to heap up hypo!ses1 we c n satisfy all ! pr9ciples at once2 s f>1 "o has succe$$ 9 safegu>d+ "s only on condition ( sacrific+ ! o!rs2 b all hope ( obta9+ bett] results is n yet lo/4 ,let u take1 !n1 ! !ory ( ,lor5tz1 turn x 9 all s5ses1 modify x little by little1 & p]haps ev]y?+ w arrange itself4 ,?us 9 place ( suppos+ t bodies 9 motion und]go a contraction 9 ! s5se ( ! motion1 & t ?is contraction is ! same :atev] be ! nature ( ^! bodies & ! =ces to :i* !y are o!rwise subject$1 c\ld we n make a m simple & natural hypo!sis8 ,we might imag9e1 = example1 t x is ! e!r :i* is modifi$ :5 x is 9 relative motion 9 ref]5ce to ! mat]ial m$ium :i* p5etrates x1 t1 :5 x is ?us modifi$1 x no long] transmits p]turbations ) ! same velocity 9 e direction4 ,x might transmit m rapidly ^? :i* are propagat$ p>allel to ! motion ( ! m$ium1 :e!r 9 ! same s5se or 9 ! opposite s5se1 & less rapidly ^? :i* are propagat$ p]p5dicul>ly4 ,! wave surfaces w\ld no long] be sph]es1 b ellipsoids1 & we c\ld disp5se ) t extraord9>y contraction ( all bodies4 ,i cite ?is only z an example1 s9ce ! modifications t might be essay$ w\ld be evid5tly susceptible ( inf9ite v>iation4 ,ab]ration & ,a/ronomy4--,x is possible also t a/ronomy may "s day furni% u data on ?is po9t2 %e x 0 9 ! ma9 :o rais$ ! "q 9 mak+ u acqua9t$ ) ! ph5om5on ( ! ab]ration ( light4 ,if we make crudely ! !ory ( ab]ration1 we rea* a v curi\s result4 ,! app>5t positions ( ! />s diff] f _! real positions because ( ! e>?'s motion1 & z ?is motion is v>iable1 ^! app>5t positions v>y4 ,! real position we c n asc]ta91 b we c obs]ve ! v>iations ( ! app>5t position4 ,! obs]vations ( ! ab]ration %[ u1 !re=e1 n ! e>?'s motion1 b ! v>iations ( ?is motion2 !y c n1 !re=e1 give u in=mation ab\t ! absolute motion ( ! e>?4 ,at lea/ ?is is true 9 fir/ approximation1 b ! case w\ld be no long] ! same if we c\ld appreciate ! ?\s&?s ( a second4 ,!n x w\ld be se5 t ! amplitude ( ! oscillation dep5ds n alone on ! v>iation ( ! motion1 a v>iation :i* is well kn[n1 s9ce x is ! motion ( our globe on its elliptic orbit1 b on ! mean value ( ?is motion1 s t ! con/ant ( ab]ration w\ld n be q ! same = all ! />s1 & ! diff]5ces w\ld tell u ! absolute motion ( ! e>? 9 space4 ,?is1 !n1 w\ld be1 "u ano!r =m1 ! ru9 ( ! pr9ciple ( relativity4 ,we are f>1 x is true1 f appreciat+ ! ?\s&? ( a second1 b1 aft] all1 say "s1 ! e>?'s total absolute velocity is p]haps mu* great] ?an its relative velocity ) respect to ! sun4 ,if1 = example1 x 7 #cjj kilomet]s p] second 9 place ( #cj1 ?is w\ld suffice to make ! ph5om5on obs]vable4 ,i believe t 9 reason+ ?us "o admits a too simple !ory ( ab]ration4 ,mi*elson has %[n u1 ,i h told y1 t ! physical proc$ures are p[]less to put 9 evid5ce absolute motion2 ,i am p]suad$ t ! same w be true ( ! a/ronomic proc$ures1 h[ev] f> precision be c>ri$4 ,h[ev] t may be1 ! data a/ronomy w furni% u 9 ?is reg>d w "s day be preci\s to ! physici/4 ,mean:ile1 ,i believe t ! !ori/s1 recall+ ! exp]i5ce ( ,mi*elson1 may anticipate a negative result1 & t !y w\ld accompli% a useful "w 9 con/ruct+ a !ory ( ab]ration :i* w\ld expla9 ?is 9 advance4 ,electrons & ,spectra4--,?is dynamics ( electrons c be approa*$ f _m sides1 b among ! ways lead+ ?i!r is "o :i* has be5 some:at neglect$1 & yet ?is is "o ( ^? :i* promise u ! mo/ surprises4 ,x is movem5ts ( electrons :i* produce ! l9es ( ! emission spectra2 ?is is prov$ by ! ,zeeman effect2 9 an inc&esc5t body :at vibrates is s5sitive to ! magnet1 !re=e electrifi$4 ,?is is a v important fir/ po9t1 b no "o has gone f>!r4 ,:y are ! l9es ( ! spectrum di/ribut$ 9 accordance ) a regul> law8 ,^! laws h be5 /udi$ by ! exp]im5t]s 9 _! lea/ details2 !y are v precise & comp>atively simple4 ,a fir/ /udy ( ^! di/ributions recalls ! h>monics enc\nt]$ 9 ac\/ics2 b ! diff]5ce is great4 ,n only are ! numb]s ( vibrations n ! successive multiples ( a s+le numb]1 b we d n ev5 f9d any?+ analog\s to ! roots ( ^? transc5d5tal equations to :i* we are l$ by s _m problems ( ma!matical physics3 t ( ! vibrations ( an ela/ic body ( any =m1 t ( ! ,h]tzian oscillations 9 a g5]ator ( any =m1 ! problem ( ,f\ri] = ! cool+ ( a solid body4 ,! laws are simpl]1 b !y are ( :olly o!r nature1 & to cite only "o ( ^! diff]5ces1 = ! h>monics ( high ord]1 ! numb] ( vibrations t5ds t[>d a f9ite limit1 in/ead ( increas+ indef9itely4 ,t has n yet be5 acc\nt$ =1 & ,i believe t !re we h "o ( ! mo/ important secrets ( nature4 ,a ,japanese physici/1 ,m4 ,nagaoka1 has rec5tly propos$ an explanation2 accord+ to him1 atoms are compos$ ( a l>ge positive electron surr\nd$ by a r+ =m$ ( a great numb] ( v small negative electrons4 ,su* is ! planet ,saturn ) its r+s4 ,?is is a v int]e/+ attempt1 b n yet :olly satisfactory2 ?is attempt %\ld be r5ew$4 ,we w p5etrate1 s to speak1 into ! inmo/ recess ( matt]4 ,& f ! p>ticul> po9t ( view :i* we to-day occupy1 :5 we "k :y ! vibrations ( inc&esc5t bodies diff] ?us f ord9>y ela/ic vibrations1 :y ! electrons d n behave l ! matt] :i* is famili> to u1 we %all bett] compreh5d ! dynamics ( electrons & x w be p]haps m easy = u to reconcile x ) ! pr9ciples4 ,conv5tions ,prec$+ ,exp]im5t4--,suppose1 n[1 t all ^! ef=ts fail1 &1 aft] all1 ,i d n believe !y w1 :at mu/ be done8 ,w x be necess>y to seek to m5d ! brok5 pr9ciples by giv+ :at we ,fr5* call a c\p de p\ce8 ,t evid5tly is always possible1 & ,i retract no?+ ( :at ,i h said above4 ,h y n writt51 y might say if y wi%$ to seek a qu>rel ) me--h y n writt5 t ! pr9ciples1 ?\gh ( exp]im5tal orig91 are n[ unassailable by exp]im5t because !y h become conv5tions8 ,& n[ y h j told u t ! mo/ rec5t conque/s ( exp]im5t put ^! pr9ciples 9 dang]4 ,well1 =m]ly ,i 0 "r & to-day ,i am n wrong4 ,=m]ly ,i 0 "r1 & :at is n[ happ5+ is a new pro( ( x4 ,take1 = example1 ! calorimetric exp]im5t ( ,curie on radium4 ,is x possible to reconcile x ) ! pr9ciple ( ! cons]vation ( en]gy8 ,?is has be5 attempt$ 9 _m ways4 ,b !re is among !m "o ,i %\ld l y to notice2 ?is is n ! explanation :i* t5ds to-day to prevail1 b x is "o ( ^? :i* h be5 propos$4 ,x has be5 conjectur$ t radium 0 only an int]m$i>y1 t x only /or$ radiations ( unkn[n nature :i* fla%$ ?r\gh space 9 e direction1 trav]s+ all bodies1 save radium1 )\t be+ alt]$ by ?is passage & )\t ex]cis+ any action ^u !m4 ,radium alone took f !m a little ( _! en]gy & aft]w>d gave x out to u 9 v>i\s =ms4 ,:at an advantage\s explanation1 & h[ conv5i5t6 ,fir/1 x is unv]ifiable & ?us irrefutable4 ,!n aga9 x w s]ve to acc\nt = any d]ogation :atev] to ,may]'s pr9ciple2 x answ]s 9 advance n only ! objection ( ,curie1 b all ! objections t future exp]im5t]s might accumulate4 ,?is new & unkn[n en]gy w\ld s]ve = ev]y?+4 ,?is is j :at ,i said1 & !re) we are %[n t our pr9ciple is unassailable by exp]im5t4 ,b !n1 :at h we ga9$ by ?is /roke8 ,! pr9ciple is intact1 b !nce=? ( :at use is x8 ,x enabl$ u to =esee t 9 su* or su* circum/ance we c\ld c\nt on su* a total quantity ( en]gy2 x limit$ u2 b n[ t ?is indef9ite provision ( new en]gy is plac$ at our disposal1 we are no long] limit$ by any?+2 &1 z ,i h writt5 9 ',sci5ce & ,hypo!sis1' if a pr9ciple ceases to be fecund1 exp]im5t )\t contradict+ x directly w nev]!less h condemn$ x4 ,future ,ma!matical ,physics4--,?is1 !re=e1 is n :at w\ld h to be done2 x w\ld be necess>y to rebuild anew4 ,if we 7 r$uc$ to ?is necessity2 we c\ld moreov] console ourselves4 ,x w\ld n be necess>y !nce to conclude t sci5ce c weave only a ,p5elope's web1 t x c raise only ephem]al /ructures1 :i* x is soon =c$ to demoli% f top to bottom ) its own h&s4 ,z ,i h said1 we h already pass$ ?r\gh a l crisis4 ,i h %[n y t 9 ! second ma!matical physics1 t ( ! pr9ciples1 we f9d traces ( ! fir/1 t ( c5tral =ces2 x w be j ! same if we mu/ "k a ?ird4 ,j s ) ! animal t exuviates1 t breaks its too n>r[ c>apace & makes itself a fre% "o2 "u ! new envelope "o w recognize ! ess5tial traits ( ! organism :i* h p]si/$4 ,we c n =esee 9 :at way we are ab\t to exp&2 p]haps x is ! k9etic !ory ( gases :i* is ab\t to und]go developm5t & s]ve z model to ! o!rs4 ,!n ! facts :i* fir/ appe>$ to u z simple !reaft] w\ld be m]ely resultants ( a v great numb] ( elem5t>y facts :i* only ! laws ( *ance w\ld make coop]ate = a common end4 ,physical law w\ld !n assume an entirely new aspect2 x w\ld no long] be solely a diff]5tial equation1 x w\ld take ! "* ( a /ati/ical law4 ,p]haps1 too1 we %all h to con/ruct an entirely new me*anics t we only succe$ 9 cat*+ a glimpse (1 :]e1 in]tia increas+ ) ! velocity1 ! velocity ( light w\ld become an impassable limit4 ,! ord9>y me*anics1 m simple1 w\ld rema9 a fir/ approximation1 s9ce x w\ld be true = velocities n too great1 s t ! old dynamics w\ld /ill be f\nd "u ! new4 ,we %\ld n h to regret hav+ believ$ 9 ! pr9ciples1 & ev51 s9ce velocities too great = ! old =mulas w\ld always be only exceptional1 ! sure/ way 9 practise w\ld be /ill to act z if we cont9u$ to believe 9 !m4 ,!y are s useful1 x w\ld be necess>y to keep a place = !m4 ,to det]m9e to exclude !m altoge!r w\ld be to deprive oneself ( a preci\s weapon4 ,i ha/5 to say 9 conclusion t we are n yet !re1 & z yet no?+ proves t ! pr9ciples w n come =? f out ! fray victori\s & intact4#aj #aj ,^! consid]ations on ma!matical physics are borr[$ f my ,/4 ,l\is address4 ,,"p ,,iii ,,! ,,objective ,,value ,,( ,,sci5ce ,,*apt] ,x ,,is ,,sci5ce ,,artificial8 #a4 ,! ,philosophy ( ,m4 ,le,roy ,!re are _m reasons = be+ sceptics2 %\ld we pu% ?is scepticism to ! v end or /op on ! way8 ,to g to ! end is ! mo/ tempt+ solution1 ! easie/ & t :i* _m h adopt$1 despair+ ( sav+ any?+ f ! %ipwreck4 ,among ! writ+s inspir$ by ?is t5d5cy x is prop] to place 9 ! fir/ rank ^? ( ,m4 ,le,roy4 ,?is ?9k] is n only a philosoph] & a writ] ( ! greate/ m]it1 b he has acquir$ a deep k ( ! exact & physical sci5ces1 & ev5 has %[n r>e p[]s ( ma!matical inv5tion4 ,let u recapitulate 9 a few words 8 doctr9e1 :i* has giv5 rise to num]\s discussions4 ,sci5ce consi/s only ( conv5tions1 & to ?is circum/ance solely does x owe its app>5t c]titude2 ! facts ( sci5ce &1 a =tiori1 its laws are ! artificial "w ( ! sci5ti/2 sci5ce !re=e c tea* u no?+ ( ! tru?2 x c only s]ve u z rule ( action4 ,"h we recognize ! philosophic !ory kn[n "u ! "n ( nom9alism2 all is n false 9 ?is !ory2 its legitimate doma9 mu/ be left x1 b out ( ?is x %\ld n be all[$ to g4 ,?is is n all2 ,m4 ,le,roy's doctr9e is n only nom9ali/ic2 x has besides ano!r *>act]i/ic :i* x d\btless owes to ,m4 ,b]gson1 x is anti-intellectuali/ic4 ,accord+ to ,m4 ,le,roy1 ! intellect de=ms all x t\*es1 & t is /ill m true ( its necess>y in/rum5t 'disc\rse4' ,!re is reality only 9 our fugitive & *ang+ impressions1 & ev5 ?is reality1 :5 t\*$1 vani%es4 ,& yet ,m4 ,le,roy is n a sceptic2 if he reg>ds ! intellect z incurably p[]less1 x is only to give m scope to o!r s\rces ( k1 to ! he>t1 = in/ance1 to s5tim5t1 to in/9ct or to fai?4 ,h[ev] great my e/eem = ,m4 ,le,roy's tal5t1 :atev] ! +5uity ( ?is !sis1 ,i c n :olly accept x4 ,c]tes1 ,i am 9 accord on _m po9ts ) ,m4 ,le,roy1 & he has ev5 cit$1 9 support ( 8 view1 v>i\s passages ( my writ+s :i* ,i am by no means dispos$ to reject4 ,i ?9k myself only ! m b\nd to expla9 :y ,i c n g ) him all ! way4 ,m4 ,le,roy (t5 compla9s ( be+ accus$ ( scepticism4 ,he c\ld n help be+1 ?\gh ?is accusation is probably unju/4 ,are n appe>ances aga9/ him8 ,nom9ali/ 9 doctr9e1 b reali/ at he>t1 he seems to escape absolute nom9alism only by a desp]ate act ( fai?4 ,! fact is t anti-intellectuali/ic philosophy 9 reject+ analysis & 'disc\rse1' j by t condemns itself to be+ intransmissible2 x is a philosophy ess5tially int]nal1 or1 at ! v lea/1 only its negations c be transmitt$2 :at wond] !n t = an ext]nal obs]v] x takes ! %ape ( scepticism8 ,!re9 lies ! weak po9t ( ?is philosophy2 if x /rives to rema9 fai?ful to itself1 its en]gy is sp5t 9 a negation & a cry ( en?usiasm4 ,ea* au?or may repeat ?is negation & ?is cry1 may v>y _! =m1 b )\t add+ any?+4 ,&1 yet1 w\ld x n be m logical 9 rema9+ sil5t8 ,see1 y h writt5 long articles2 = t1 x 0 necess>y to use words4 ,& !re9 h y n be5 mu* m 'discursive' & consequ5tly mu* f>!r f life & tru? ?an ! animal :o simply lives )\t philosophiz+8 ,w\ld n ?is animal be ! true philosoph]8 ,h[ev]1 because no pa9t] has made a p]fect portrait1 %\ld we conclude t ! be/ pa9t+ is n to pa9t8 ,:5 a zoologi/ dissects an animal1 c]ta9ly he 'alt]s x4' ,yes1 9 dissect+ x1 he condemns himself to nev] "k all ( x2 b 9 n dissect+ x1 he w\ld condemn himself to nev] "k any?+ ( x & consequ5tly to nev] see any?+ ( x4 ,c]tes1 9 man are o!r =ces besides 8 intellect2 no "o has ev] be5 mad en\gh to d5y t4 ,! fir/ com] makes ^! bl9d =ces act or lets !m act2 ! philosoph] mu/ speak ( !m2 to speak ( !m1 he mu/ "k ( !m ! little t c be kn[n1 he %\ld !re=e see !m act4 ,h[8 ,) :at eyes1 if n ) 8 intellect8 ,he>t1 in/9ct1 may guide x1 b n r5d] x useless2 !y may direct ! look1 b n replace ! eye4 ,x may be grant$ t ! he>t is ! workman1 & ! intellect only ! in/rum5t4 ,yet is x an in/rum5t n to be done )\t1 if n = action1 at lea/ = philosophiz+8 ,!re=e a philosoph] really anti-intellectuali/ic is impossible4 ,p]haps we %all h to decl>e = ! supremacy ( action2 always x is our intellect :i* w ?us conclude2 9 all[+ prec$5ce to action x w ?us reta9 ! sup]iority ( ! ?9k+ re$4 ,?is also is a supremacy n to be disda9$4 ,p>don ^! brief reflections & p>don also _! brevity1 sc>cely skimm+ ! "q4 ,! process ( intellectualism is n ! subject ,i wi% to treat3 ,i wi% to speak ( sci5ce1 & ab\t x !re is no d\bt2 by def9ition1 s to speak1 x w be intellectuali/ic or x w n be at all4 ,precisely ! "q is1 :e!r x w be4 #b4 ,sci5ce1 ,rule ( ,action ,= ,m4 ,le,roy1 sci5ce is only a rule ( action4 ,we are p[]less to "k any?+ & yet we are laun*$1 we mu/ act1 & at all haz>ds we h e/abli%$ rules4 ,x is ! aggregate ( ^! rules t is call$ sci5ce4 ,x is ?us t m51 desir\s ( div]sion1 h in/itut$ rules ( play1 l ^? ( tric-trac = in/ance1 :i*1 bett] ?an sci5ce itself1 c\ld rely ^u ! pro( by univ]sal cons5t4 ,x is ?us likewise t1 unable to *oose1 b =c$ to *oose1 we toss up a co91 head or tail to w94 ,! rule ( tric-trac is inde$ a rule ( action l sci5ce1 b does any "o ?9k ! comp>ison j & n see ! diff]5ce8 ,! rules ( ! game are arbitr>y conv5tions & ! contr>y conv5tion might h be5 adopt$1 :i* w\ld h be5 none ! less good4 ,on ! contr>y1 sci5ce is a rule ( action :i* is successful1 g5]ally at lea/1 & ,i add1 :ile ! contr>y rule w\ld n h succe$$4 ,if ,i say1 to make hydrog5 cause an acid to act on z9c1 ,i =mulate a rule :i* succe$s2 ,i c\ld h said1 make di/ill$ wat] act on gold2 t also w\ld h be5 a rule1 only x w\ld n h succe$$4 ,if1 !re=e1 sci5tific 'recipes' h a value1 z rule ( action1 x is because we "k !y succe$1 g5]ally at lea/4 ,b to "k ?is is to "k some?+ & !n :y tell u we c "k no?+8 ,sci5ce =esees1 & x is because x =esees t x c be useful & s]ve z rule ( action4 ,i well "k t its previsions are (t5 contradict$ by ! ev5t2 t %[s t sci5ce is imp]fect1 & if ,i add t x w always rema9 s1 ,i am c]ta9 t ?is is a prevision :i*1 at lea/1 w nev] be contradict$4 ,always ! sci5ti/ is less (t5 mi/ak5 ?an a prophet :o %\ld pr$ict at r&om4 ,besides ! progress ?\gh sl[ is cont9u\s1 s t sci5ti/s1 ?\gh m & m bold1 are less & less misl$4 ,?is is little1 b x is en\gh4 ,i well "k t ,m4 ,le,roy has some:]e said t sci5ce 0 mi/ak5 (t5] ?an "o ?\ght1 t comets sometimes play$ tricks on a/ronom]s1 t sci5ti/s1 :o app>5tly are m51 did n will+ly speak ( _! failures1 & t1 if !y %\ld speak ( !m1 !y w\ld h to c\nt m defeats ?an victories4 ,t day1 ,m4 ,le,roy evid5tly ov]rea*$ himself4 ,if sci5ce did n succe$1 x c\ld n s]ve z rule ( action2 :5ce w\ld x get its value8 ,because x is 'liv$1' t is1 because we love x & believe 9 x8 ,! al*emi/s _h recipes = mak+ gold1 !y lov$ !m & _h fai? 9 !m1 & yet our recipes are ! good ones1 al?\gh our fai? be less lively1 because !y succe$4 ,!re is no escape f ?is dilemma2 ei!r sci5ce does n enable u to =esee1 & !n x is valueless z rule ( action2 or else x enables u to =esee1 9 a fa%ion m or less imp]fect1 & !n x is n )\t value z means ( k4 ,x %\ld n ev5 be said t action is ! goal ( sci5ce2 %\ld we condemn /udies ( ! /> ,sirius1 "u pretext t we %all probably nev] ex]cise any influ5ce on t />8 ,to my eyes1 on ! contr>y1 x is ! k :i* is ! end1 & ! action :i* is ! means4 ,if ,i felicitate myself on ! indu/rial developm5t1 x is n alone because x furni%es a facile argum5t to ! advocates ( sci5ce2 x is above all because x gives to ! sci5ti/ fai? 9 himself & also because x (f]s him an imm5se field ( exp]i5ce :]e he cla%es aga9/ =ces too colossal to be tamp]$ )4 ,)\t ?is balla/1 :o kn[s :e!r he w\ld n quit solid gr\nd1 s$uc$ by ! mirage ( "s s*ola/ic novelty1 or :e!r he w\ld n despair1 believ+ he _h fa%ion$ only a dream8 #c4 ,! ,crude ,fact & ! ,sci5tific ,fact ,:at 0 mo/ p>adoxical 9 ,m4 ,le,roy's !sis 0 t affirmation t ! sci5ti/ creates ! fact2 ?is 0 at ! same "t its ess5tial po9t & x is "o ( ^? :i* h be5 mo/ discuss$4 ,p]haps1 says he 7,i well believe t ?is 0 a concession71 x is n ! sci5ti/ t creates ! fact 9 ! r\gh2 x is at lea/ he :o creates ! sci5tific fact4 ,?is di/9ction betwe5 ! fact 9 ! r\gh & ! sci5tific fact does n by itself appe> to me illegitimate4 ,b ,i compla9 fir/ t ! b\nd>y has n be5 trac$ ei!r exactly or precisely2 & !n t ! au?or has seem$ to suppose t ! crude fact1 n be+ sci5tific1 is outside ( sci5ce4 ,f9ally1 ,i c n admit t ! sci5ti/ creates )\t re/ra9t ! sci5tific fact1 s9ce x is ! crude fact :i* imposes x ^u him4 ,! examples giv5 by ,m4 ,le,roy h greatly a/oni%$ me4 ,! fir/ is tak5 f ! notion ( atom4 ,! atom *os5 z example ( fact6 ,i av[ t ?is *oice has s disconc]t$ me t ,i pref] to say no?+ ab\t x4 ,i h evid5tly misund]/ood ! au?or's ?\ght & ,i c\ld n fruitfully discuss x4 ,! second case tak5 z example is t ( an eclipse :]e ! crude ph5om5on is a play ( light & %ad[1 b :]e ! a/ronom] c n int]v5e )\t introduc+ two =eign elem5ts1 to wit1 a clock & ,newton's law4 ,f9ally1 ,m4 ,le,roy cites ! rotation ( ! e>?2 x has be5 answ]$3 b ?is is n a fact1 & he has repli$3 x 0 "o = ,galileo1 :o affirm$ x1 z = ! inquisitor1 :o d5i$ x4 ,x always rema9s t ?is is n a fact 9 ! same s5se z ^? j spok5 ( & t to give !m ! same "n is to expose "o's self to _m confusions4 ,"h !n are f\r degrees3 #a4 ,x gr[s d>k1 says ! cl[n4 #b4 ,! eclipse happ5$ at n9e o'clock1 says ! a/ronom]4 #c4 ,! eclipse happ5$ at ! "t d$ucible f ! tables con/ruct$ accord+ to ,newton's law1 says he aga94 #d4 ,t results f ! e>?'s turn+ ar\nd ! sun1 says ,galileo f9ally4 ,:]e !n is ! b\nd>y betwe5 ! fact 9 ! r\gh & ! sci5tific fact8 ,to read ,m4 ,le,roy "o w\ld believe t x is betwe5 ! fir/ & ! second /age1 b :o does n see t !re is a great] di/ance f ! second to ! ?ird1 & /ill m f ! ?ird to ! f\r?4 ,all[ me to cite two examples :i* p]haps w enlight5 u a little4 ,i obs]ve ! deviation ( a galvanomet] by ! aid ( a movable mirror :i* projects a lum9\s image or spot on a divid$ scale4 ,! crude fact is ?is3 ,i see ! spot displace itself on ! scale1 & ! sci5tific fact is ?is3 a curr5t passes 9 ! circuit4 ,or aga93 :5 ,i make an exp]im5t ,i %\ld subject ! result to c]ta9 corrections1 because ,i "k ,i mu/ h made errors4 ,^! errors are ( two k9ds1 "s are accid5tal & ^! ,i %all correct by tak+ ! mean2 ! o!rs are sy/ematic & ,i %all be able to correct ^? only by a ?or\gh /udy ( _! causes4 ,! fir/ result obta9$ is !n ! fact 9 ! r\gh1 :ile ! sci5tific fact is ! f9al result aft] ! f9i%$ corrections4 ,reflect+ on ?is latt] example1 we are l$ to subdivide our second /age1 & 9 place ( say+3 #b4 ,! eclipse happ5$ at n9e o'clock1 we %all say3 #b;a4 ,! eclipse happ5$ :5 my clock po9t$ to n9e1 & #b;b4 ,my clock be+ t5 m9utes sl[1 ! eclipse happ5$ at t5 m9utes pa/ n9e4 ,& ?is is n all3 ! fir/ /age also %\ld be subdivid$1 & n betwe5 ^! two subdivisions w be ! lea/ di/ance2 x is necess>y to di/+ui% betwe5 ! impression ( obscurity felt by "o witness+ an eclipse1 & ! affirmation3 ,x gr[s d>k1 :i* ?is impression extorts f him4 ,9 a s5se x is ! fir/ :i* is ! only true fact 9 ! r\gh1 & ! second is already a sort ( sci5tific fact4 ,n[ !n our scale has six /ages1 & ev5 ?\gh !re is no reason = halt+ at ?is figure1 !re we %all /op4 ,:at /rikes me at ! />t is ?is4 ,at ! fir/ ( our six /ages1 ! fact1 /ill completely 9 ! r\gh1 is1 s to speak1 individual1 x is completely di/9ct f all o!r possible facts4 ,f ! second /age1 already x is no long] ! same4 ,! enunciation ( ! fact w\ld suit an inf9ity ( o!r facts4 ,s soon z language int]v5es1 ,i h at my comm& only a f9ite numb] ( t]ms to express ! %ades1 9 numb] inf9ite1 t my impressions might cov]4 ,:5 ,i say3 ,x gr[s d>k1 t well expresses ! impressions ,i feel 9 be+ pres5t at an eclipse2 b ev5 9 obscurity a multitude ( %ades c\ld be imag9$1 & if1 in/ead ( t actually realiz$1 _h happ5$ a slightly diff]5t %ade1 yet ,i %\ld /ill h enunciat$ ?is o!r fact by say+3 ,x gr[s d>k4 ,second rem>k3 ev5 at ! second /age1 ! enunciation ( a fact c only be true or false4 ,?is is n s ( any proposition2 if ?is proposition is ! enunciation ( a conv5tion1 x c n be said t ?is enunciation is true1 9 ! prop] s5se ( ! ^w1 s9ce x c\ld n be true ap>t f me & is true only because ,i wi% x to be4 ,:51 = in/ance1 ,i say ! unit = l5g? is ! met]1 ?is is a decree t ,i promulgate1 x is n some?+ asc]ta9$ :i* =ces itself ^u me4 ,x is ! same1 z ,i ?9k ,i h else:]e %[n1 :5 x is a "q1 = example1 ( ,euclid's po/ulate4 ,:5 ,i am ask$3 ,is x gr[+ d>k8 ,i always "k :e!r ,i ought to reply yes or no4 ,al?\gh an inf9ity ( possible facts may be susceptible ( ?is same enunciation1 x gr[s d>k1 ,i %all always "k :e!r ! fact realiz$ belongs or does n belong among ^? :i* answ] to ?is enunciation4 ,facts are class$ 9 categories1 & if ,i am ask$ :e!r ! fact t ,i asc]ta9 belongs or does n belong 9 su* a category1 ,i %all n hesitate4 ,d\btless ?is classification is suffici5tly arbitr>y to leave a l>ge "p to man's fre$om or caprice4 ,9 a ^w1 ?is classification is a conv5tion4 ,?is conv5tion be+ giv51 if ,i am ask$3 ,is su* a fact true8 ,i %all always "k :at to answ]1 & my reply w be impos$ ^u me by ! witness ( my s5ses4 ,if !re=e1 dur+ an eclipse1 x is ask$3 ,is x gr[+ d>k8 all ! _w w answ] yes4 ,d\btless ^? speak+ a language :]e bright 0 call$ d>k1 & d>k bright1 w\ld answ] no4 ,b ( :at importance is t8 ,9 ! same way1 9 ma!matics1 :5 ,i h laid d[n ! def9itions1 & ! po/ulates :i* are conv5tions1 a !orem h5ce=? c only be true or false4 ,b to answ] ! "q3 ,is ?is !orem true8 x is no long] to ! witness ( my s5ses t ,i %all h rec\rse1 b to reason+4 ,a /atem5t ( fact is always v]ifiable1 & = ! v]ification we h rec\rse ei!r to ! witness ( our s5ses1 or to ! memory ( ?is witness4 ,?is is prop]ly :at *>act]izes a fact4 ,if y put ! "q to me3 ,is su* a fact true8 ,i %all beg9 by ask+ y1 if !re is occasion1 to /ate precisely ! conv5tions1 by ask+ y1 9 o!r words1 :at language y h spok52 !n once settl$ on ?is po9t1 ,i %all int]rogate my s5ses & %all answ] yes or no4 ,b x w be my s5ses t w h made answ]1 x w n be y :5 y say to me3 ,i h spok5 to y 9 ,engli% or 9 ,fr5*4 ,is !re some?+ to *ange 9 all t :5 we pass to ! foll[+ /ages8 ,:5 ,i obs]ve a galvanomet]1 z ,i h j said1 if ,i ask an ignorant visitor3 ,is ! curr5t pass+8 he looks at ! wire to try to see some?+ pass2 b if ,i put ! same "q to my assi/ant :o und]/&s my language1 he w "k ,i mean3 ,does ! spot move8 & he w look at ! scale4 ,:at diff]5ce is !re !n betwe5 ! /atem5t ( a fact 9 ! r\gh & ! /atem5t ( a sci5tific fact8 ,! same diff]5ce z betwe5 ! /atem5t ( ! same crude fact 9 ,fr5* & 9 ,g]man4 ,! sci5tific /atem5t is ! translation ( ! crude /atem5t into a language :i* is di/+ui%$ above all f ! common ,g]man or ,fr5*1 because x is spok5 by a v mu* small] numb] ( p4 ,yet let u n g too fa/4 ,to measure a curr5t ,i may use a v great numb] ( types ( galvanomet]s or besides an electrodynamomet]4 ,& !n :5 ,i %all say !re is runn+ 9 ?is circuit a curr5t ( s _m amp]es1 t w mean3 if ,i adapt to ?is circuit su* a galvanomet] ,i %all see ! spot come to ! division a2 b t w mean equally3 if ,i adapt to ?is circuit su* an electrodynamomet]1 ,i %all see ! spot g to ! division b4 ,& t w mean /ill _m o!r ?+s1 because ! curr5t c manife/ itself n only by me*anical effects1 b by effects *emical1 !rmal1 lum9\s1 etc4 ,"h !n is "o same /atem5t :i* suits a v great numb] ( facts absolutely diff]5t4 ,:y8 ,x is because ,i assume a law accord+ to :i*1 :5ev] su* a me*anical effect %all happ51 su* a *emical effect w happ5 also4 ,previ\s exp]im5ts1 v num]\s1 h nev] %[n ?is law to fail1 & !n ,i h und]/ood t ,i c\ld express by ! same /atem5t two facts s inv>iably b\nd "o to ! o!r4 ,:5 ,i am ask$3 ,is ! curr5t pass+8 ,i c und]/& t t means3 ,w su* a me*anical effect happ58 ,b ,i c und]/& also3 ,w su* a *emical effect happ58 ,i %all !n v]ify ei!r ! exi/5ce ( ! me*anical effect1 or t ( ! *emical effect2 t w be indiff]5t1 s9ce 9 bo? cases ! answ] mu/ be ! same4 ,& if ! law %\ld "o day be f\nd false8 ,if x 0 p]ceiv$ t ! concordance ( ! two effects1 me*anical & *emical1 is n con/ant8 ,t day x w\ld be necess>y to *ange ! sci5tific language to free x f a grave ambiguity4 ,& aft] t8 ,is x ?\ght t ord9>y language by aid ( :i* are express$ ! facts ( daily life is exempt f ambiguity8 ,%all we !nce conclude t ! facts ( daily life are ! "w ( ! gramm>ians8 ,y ask me3 ,is !re a curr5t8 ,i try :e!r ! me*anical effect exi/s1 ,i asc]ta9 x & ,i answ]3 ,yes1 !re is a curr5t4 ,y und]/& at once t t means t ! me*anical effect exi/s1 & t ! *emical effect1 t ,i h n inve/igat$1 exi/s likewise4 ,imag9e n[1 suppos+ an impossibility1 ! law we believe true1 n to be1 & ! *emical effect n to exi/4 ,"u ?is hypo!sis !re w be two di/9ct facts1 ! "o directly obs]v$ & :i* is true1 ! o!r inf]r$ & :i* is false4 ,x may /rictly be said t we h creat$ ! second4 ,s t error is ! "p ( man's p]sonal collaboration 9 ! creation ( ! sci5tific fact4 ,b if we c say t ! fact 9 "q is false1 is ?is n j because x is n a free & arbitr>y creation ( our m9d1 a disguis$ conv5tion1 9 :i* case x w\ld be nei!r true nor false4 ,& 9 fact x 0 v]ifiable2 ,i _h n made ! v]ification1 b ,i c\ld h made x4 ,if ,i answ]$ amiss1 x 0 because ,i *ose to reply too quickly1 )\t hav+ ask$ nature1 :o alone knew ! secret4 ,:51 aft] an exp]im5t1 ,i correct ! accid5tal & sy/ematic errors to br+ out ! sci5tific fact1 ! case is ! same2 ! sci5tific fact w nev] be any?+ b ! crude fact translat$ into ano!r language4 ,:5 ,i %all say3 ,x is su* an h\r1 t w be a %ort way ( say+3 ,!re is su* a relation betwe5 ! h\r indicat$ by my clock1 & ! h\r x m>k$ at ! mom5t ( ! pass+ ( su* a /> & su* ano!r /> across ! m]idian4 ,& ?is conv5tion ( language once adopt$1 :5 ,i %all be ask$3 ,is x su* an h\r8 x w n dep5d ^u me to answ] yes or no4 ,let u pass to ! /age be=e ! la/3 ! eclipse happ5$ at ! h\r giv5 by ! tables d$uc$ f ,newton's laws4 ,?is is /ill a conv5tion ( language :i* is p]fectly cle> = ^? :o "k cele/ial me*anics or simply = ^? :o h ! tables calculat$ by ! a/ronom]s4 ,i am ask$3 ,did ! eclipse happ5 at ! h\r pr$ict$8 ,i look 9 ! nautical almanac1 ,i see t ! eclipse 0 ann\nc$ = n9e o'clock & ,i und]/& t ! "q means3 ,did ! eclipse happ5 at n9e o'clock8 ,!re /ill we h no?+ to *ange 9 our conclusions4 ,! sci5tific fact is only ! crude fact translat$ into a conv5i5t language4 ,x is true t at ! la/ /age ?+s *ange4 ,does ! e>? rotate8 ,is ?is a v]ifiable fact8 ,c\ld ,galileo & ! ,gr& ,inquisitor1 to settle ! matt]1 appeal to ! witness ( _! s5ses8 ,on ! contr>y1 !y 7 9 accord ab\t ! appe>ances1 & :atev] _h be5 ! accumulat$ exp]i5ces1 !y w\ld h rema9$ 9 accord ) reg>d to ! appe>ances )\t ev] agree+ on _! int]pretation4 ,x is j on t acc\nt t !y 7 oblig$ to h rec\rse to proc$ures ( discussion s unsci5tific4 ,?is is :y ,i ?9k !y did n disagree ab\t a fact3 we h n ! "r to give ! same "n to ! rotation ( ! e>?1 :i* 0 ! object ( _! discussion1 & to ! facts crude or sci5tific we h hi!rto pass$ 9 review4 ,aft] :at prec$es1 x seems sup]flu\s to inve/igate :e!r ! fact 9 ! r\gh is outside ( sci5ce1 because !re c nei!r be sci5ce )\t sci5tific fact1 nor sci5tific fact )\t fact 9 ! r\gh1 s9ce ! fir/ is only ! translation ( ! second4 ,& !n1 has "o ! "r to say t ! sci5ti/ creates ! sci5tific fact8 ,fir/ ( all1 he does n create x f no?+1 s9ce he makes x ) ! fact 9 ! r\gh4 ,consequ5tly he does n make x freely & z he *ooses4 ,h[ev] able ! work] may be1 8 fre$om is always limit$ by ! prop]ties ( ! raw mat]ial on :i* he works4 ,aft] all1 :at d y mean :5 y speak ( ?is free creation ( ! sci5tific fact & :5 y take z example ! a/ronom] :o int]v5es actively 9 ! ph5om5on ( ! eclipse by br++ 8 clock8 ,d y mean3 ,! eclipse happ5$ at n9e o'clock2 b if ! a/ronom] _h wi%$ x to happ5 at t51 t dep5d$ only on him1 he _h only to advance 8 clock an h\r8 ,b ! a/ronom]1 9 p]petrat+ t bad joke1 w\ld evid5tly h be5 guilty ( an equivocation4 ,:5 he tells me3 ,! eclipse happ5$ at n9e1 ,i und]/& t n9e is ! h\r d$uc$ f ! crude indication ( ! p5dulum by ! usual s]ies ( corrections4 ,if he has giv5 me solely t crude indication1 or if he has made corrections contr>y to ! habitual rules1 he has *ang$ ! language agre$ ^u )\t =ew>n+ me4 ,if1 on ! contr>y1 he took c>e to =ew>n me1 ,i h no?+ to compla9 (1 b !n x is always ! same fact express$ 9 ano!r language4 ,9 sum1 all ! sci5ti/ creates 9 a fact is ! language 9 :i* he enunciates x4 ,if he pr$icts a fact1 he w employ ?is language1 & = all ^? :o c speak & und]/& x1 8 pr$iction is free f ambiguity4 ,moreov]1 ?is pr$iction once made1 x evid5tly does n dep5d ^u him :e!r x is fulfill$ or n4 ,:at !n rema9s ( ,m4 ,le,roy's !sis8 ,?is rema9s3 ! sci5ti/ int]v5es actively 9 *oos+ ! facts wor? obs]v+4 ,an isolat$ fact has by itself no int]e/2 x becomes int]e/+ if "o has reason to ?9k t x may aid 9 ! pr$iction ( o!r facts2 or bett]1 if1 hav+ be5 pr$ict$1 its v]ification is ! confirmation ( a law4 ,:o %all *oose ! facts :i*1 correspond+ to ^! conditions1 are wor?y ! fre$om ( ! city 9 sci5ce8 ,?is is ! free activity ( ! sci5ti/4 ,& t is n all4 ,i h said t ! sci5tific fact is ! translation ( a crude fact into a c]ta9 language2 ,i %\ld add t e sci5tific fact is =m$ ( _m crude facts4 ,?is is suffici5tly %[n by ! examples cit$ above4 ,= in/ance1 = ! h\r ( ! eclipse my clock m>k$ ! h\r alpha at ! in/ant ( ! eclipse2 x m>k$ ! h\r beta at ! mom5t ( ! la/ transit ( ! m]idian ( a c]ta9 /> t we take z orig9 ( "r asc5sions2 x m>k$ ! h\r gamma at ! mom5t ( ! prec$+ transit ( ?is same />4 ,!re are ?ree di/9ct facts 7/ill x w be notic$ t ea* ( !m results itself f two simultane\s facts 9 ! r\gh2 b let u pass ?is ov]74 ,9 place ( t ,i say3 ,! eclipse happ5$ at ! h\r #bd 7alpha-beta7_/7beta-gamma71 & ! ?ree facts are comb9$ 9 a s+le sci5tific fact4 ,i h conclud$ t ! ?ree read+s1 alpha1 beta1 gamma made on my clock at ?ree diff]5t mom5ts lack$ int]e/ & t ! only ?+ int]e/+ 0 ! comb9ation 7alpha-beta7_/7beta-gamma7 ( ! ?ree4 ,9 ?is conclusion is f\nd ! free activity ( my m9d4 ,b ,i h ?us us$ up my p[]2 ,i c n make ?is comb9ation 7alpha-beta7_/7beta-gamma7 h su* a value & n su* ano!r1 s9ce ,i c n influ5ce ei!r ! value ( alpha1 or t ( beta1 or t ( gamma1 :i* are impos$ ^u me z crude facts4 ,9 sum1 facts are facts1 & if x happ5s t !y satisfy a pr$iction1 ?is is n an effect ( our free activity4 ,!re is no precise fronti] betwe5 ! fact 9 ! r\gh & ! sci5tific fact2 x c only be said t su* an enunciation ( fact is m crude or1 on ! contr>y1 m sci5tific ?an su* ano!r4 #d4 ',nom9alism' & '! ,univ]sal ,inv>iant' ,if f facts we pass to laws1 x is cle> t ! "p ( ! free activity ( ! sci5ti/ w become mu* great]4 ,b did n ,m4 ,le,roy make x /ill too great8 ,?is is :at we are ab\t to exam9e4 ,recall fir/ ! examples he has giv54 ,:5 ,i say3 ,phosphorus melts at #dd1 ,i ?9k ,i am enunciat+ a law2 9 reality x is j ! def9ition ( phosphorus2 if "o %\ld discov] a body :i*1 possess+ o!rwise all ! prop]ties ( phosphorus1 did n melt at #dd1 we %\ld give x ano!r "n1 t is all1 & ! law w\ld rema9 true4 ,j s :5 ,i say3 ,heavy bodies fall+ freely pass ov] spaces proportional to ! squ>es ( ! times1 ,i only give ! def9ition ( free fall4 ,:5ev] ! condition %all n be fulfill$1 ,i %all say t ! fall is n free1 s t ! law w nev] be wrong4 ,x is cle> t if laws 7 r$uc$ to t1 !y c\ld n s]ve 9 pr$iction2 !n !y w\ld be good = no?+1 ei!r z means ( k or z pr9ciple ( action4 ,:5 ,i say3 ,phosphorus melts at #dd1 ,i mean by t3 ,all bodies possess+ su* or su* a prop]ty 7to wit1 all ! prop]ties ( phosphorus1 save fus+-po9t7 fuse at #dd4 ,s und]/ood1 my proposition is inde$ a law1 & ?is law may be useful to me1 because if ,i meet a body possess+ ^! prop]ties ,i %all be able to pr$ict t x w fuse at #dd4 ,d\btless ! law may be f\nd to be false4 ,!n we %all read 9 ! treatises on *emi/ry3 ,!re are two bodies :i* *emi/s long conf\nd$ "u ! "n ( phosphorus2 ^! two bodies diff] only by _! po9ts ( fusion4 ,t w\ld evid5tly n be ! fir/ "t = *emi/s to atta9 to ! sep>ation ( two bodies !y 7 at fir/ n able to di/+ui%2 su*1 = example1 are neodymium & praseodymium1 long conf\nd$ "u ! "n ( didymium4 ,i d n ?9k ! *emi/s mu* fe> t a l mis*ance w ev] happ5 to phosphorus4 ,& if1 to suppose ! impossible1 x %\ld happ51 ! two bodies w\ld probably n h id5tically ! same d5sity1 id5tically ! same specific heat1 etc41 s t aft] hav+ det]m9$ ) c>e ! d5sity1 = in/ance1 "o c\ld /ill =esee ! fusion po9t4 ,x is1 moreov]1 unimportant2 x suffices to rem>k t !re is a law1 & t ?is law1 true or false1 does n r$uce to a tautology4 ,w x be said t if we d n "k on ! e>? a body :i* does n fuse at #dd :ile hav+ all ! o!r prop]ties ( phosphorus1 we c n "k :e!r x does n exi/ on o!r planets8 ,d\btless t may be ma9ta9$1 & x w\ld !n be inf]r$ t ! law 9 "q1 :i* may s]ve z a rule ( action to u :o inhabit ! e>?1 has yet no g5]al value f ! po9t ( view ( k1 & owes its int]e/ only to ! *ance :i* has plac$ u on ?is globe4 ,?is is possible1 b1 if x 7 s1 ! law w\ld be valueless1 n because x r$uc$ to a conv5tion1 b because x w\ld be false4 ,! same is true 9 :at conc]ns ! fall ( bodies4 ,x w\ld d me no good to h giv5 ! "n ( free fall to falls :i* happ5 9 con=mity ) ,galileo's law1 if ,i did n "k t else:]e1 9 su* circum/ances1 ! fall w be probably free or approximately free4 ,t !n is a law :i* may be true or false1 b :i* does n r$uce to a conv5tion4 ,suppose ! a/ronom]s discov] t ! />s d n exactly obey ,newton's law4 ,!y w h ! *oice betwe5 two attitudes2 !y may say t gravitation does n v>y exactly z ! inv]se ( ! squ>e ( ! di/ance1 or else !y may say t gravitation is n ! only =ce :i* acts on ! />s & t !re is 9 addition a diff]5t sort ( =ce4 ,9 ! second case1 ,newton's law w be consid]$ z ! def9ition ( gravitation4 ,?is w be ! nom9ali/ attitude4 ,! *oice betwe5 ! two attitudes is free1 & is made f consid]ations ( conv5i5ce1 ?\gh ^! consid]ations are mo/ (t5 s /rong t !re rema9s practically little ( ?is fre$om4 ,we c break up ?is proposition3 7#a7 ,! />s obey ,newton's law1 into two o!rs2 7#b7 gravitation obeys ,newton's law2 7#c7 gravitation is ! only =ce act+ on ! />s4 ,9 ?is case proposition 7#b7 is no long] any?+ b a def9ition & is beyond ! te/ ( exp]im5t2 b !n x w be on proposition 7#c7 t ?is *eck c be ex]cis$4 ,?is is inde$ necess>y1 s9ce ! result+ proposition 7#a7 pr$icts v]ifiable facts 9 ! r\gh4 ,x is ?anks to ^! artifices t by an unconsci\s nom9alism ! sci5ti/s h elevat$ above ! laws :at !y call pr9ciples4 ,:5 a law has receiv$ a suffici5t confirmation f exp]im5t1 we may adopt two attitudes3 ei!r we may leave ?is law 9 ! fray2 x w !n rema9 subject$ to an incessant revision1 :i* )\t any d\bt w end by demon/rat+ t x is only approximative4 ,or else we may elevate x into a pr9ciple by adopt+ conv5tions su* t ! proposition may be c]ta9ly true4 ,= t ! proc$ure is always ! same4 ,! primitive law enunciat$ a relation betwe5 two facts 9 ! r\gh1 ,a & ,b2 betwe5 ^! two crude facts is introduc$ an ab/ract int]m$i>y ,c1 m or less fictiti\s 7su* 0 9 ! prec$+ example ! impalpable entity1 gravitation74 ,& !n we h a relation betwe5 ,a & ,c t we may suppose rigor\s & :i* is ! pr9ciple2 & ano!r betwe5 ,c & ,b :i* rema9s a law subject to revision4 ,! pr9ciple1 h5ce=? cry/alliz$1 s to speak1 is no long] subject to ! te/ ( exp]im5t4 ,x is n true or false1 x is conv5i5t4 ,great advantages h (t5 be5 f\nd 9 proce$+ 9 t way1 b x is cle> t if all ! laws _h be5 trans=m$ into pr9ciples no?+ w\ld be left ( sci5ce4 ,e law may be brok5 up into a pr9ciple & a law1 b !reby x is v cle> t1 h[ev] f> ?is p>tition be pu%$1 !re w always rema9 laws4 ,nom9alism has !re=e limits1 & ?is is :at "o might fail to recognize if "o took to ! v lett] ,m4 ,le,roy's ass]tions4 ,a rapid review ( ! sci5ces w make u compreh5d bett] :at are ^! limits4 ,! nom9ali/ attitude is ju/ifi$ only :5 x is conv5i5t2 :5 is x s8 ,exp]im5t tea*es u relations betwe5 bodies2 ?is is ! fact 9 ! r\gh2 ^! relations are extremely complicat$4 ,in/ead ( envisag+ directly ! relation ( ! body ,a & ! body ,b1 we introduce betwe5 !m an int]m$i>y1 :i* is space1 & we envisage ?ree di/9ct relations3 t ( ! body ,a ) ! figure ,a' ( space1 t ( ! body ,b ) ! figure ,b' ( space1 t ( ! two figures ,a' & ,b' to ea* o!r4 ,:y is ?is det\r advantage\s8 ,because ! relation ( ,a & ,b 0 complicat$1 b diff]$ little f t ( ,a' & ,b'1 :i* is simple2 s t ?is complicat$ relation may be replac$ by ! simple relation betwe5 ,a' & ,b' & by two o!r relations :i* tell u t ! diff]5ces betwe5 ,a & ,a'1 on ! "o h&1 betwe5 ,b & ,b'1 on ! o!r h&1 are v small4 ,= example1 if ,a & ,b are two natural solid bodies :i* are displac$ ) slight de=mation1 we envisage two movable rigid figures ,a' & ,b'4 ,! laws ( ! relative displacem5t ( ^! figures ,a' & ,b' w be v simple2 !y w be ^? ( geometry4 ,& we %all aft]w>d add t ! body ,a1 :i* always diff]s v little f ,a'1 dilates f ! effect ( heat & b5ds f ! effect ( ela/icity4 ,^! dilatations & flexions1 j because !y are v small1 w be = our m9d relatively easy to /udy4 ,j imag9e to :at complexities ( language x w\ld h be5 necess>y to be resign$ if we _h wi%$ to compreh5d 9 ! same enunciation ! displacem5t ( ! solid1 its dilatation & its flexure8 ,! relation betwe5 ,a & ,b 0 a r\gh law1 & 0 brok5 up2 we n[ h two laws :i* express ! relations ( ,a & ,a'1 ( ,b & ,b'1 & a pr9ciple :i* expresses t ( ,a' ) ,b'4 ,x is ! aggregate ( ^! pr9ciples t is call$ geometry4 ,two o!r rem>ks4 ,we h a relation betwe5 two bodies ,a & ,b1 :i* we h replac$ by a relation betwe5 two figures ,a' & ,b'2 b ?is same relation betwe5 ! same two figures ,a' & ,b' c\ld j z well h replac$ advantage\sly a relation betwe5 two o!r bodies ,a'' & ,b''1 entirely diff]5t f ,a & ,b4 ,& t 9 _m ways4 ,if ! pr9ciples ( geometry _h n be5 inv5t$1 aft] hav+ /udi$ ! relation ( ,a & ,b1 x w\ld be necess>y to beg9 aga9 ab ovo ! /udy ( ! relation ( ,a'' & ,b''4 ,t is :y geometry is s preci\s4 ,a geometrical relation c advantage\sly replace a relation :i*1 consid]$ 9 ! r\gh /ate1 %\ld be reg>d$ z me*anical1 x c replace ano!r :i* %\ld be reg>d$ z optical1 etc4 ,yet let no "o say3 ,b t proves geometry an exp]im5tal sci5ce2 9 sep>at+ its pr9ciples f laws :5ce !y h be5 drawn1 y artificially sep>ate x itself f ! sci5ces :i* h giv5 bir? to x4 ,! o!r sci5ces h likewise pr9ciples1 b t does n preclude our hav+ to call !m exp]im5tal4 ,x mu/ be recogniz$ t x w\ld h be5 difficult n to make ?is sep>ation t is pret5d$ to be artificial4 ,we "k ! rle t ! k9ematics ( solid bodies has play$ 9 ! g5esis ( geometry2 %\ld x !n be said t geometry is only a bran* ( exp]im5tal k9ematics8 ,b ! laws ( ! rectil9e> propagation ( light h also contribut$ to ! =mation ( its pr9ciples4 ,mu/ geometry be reg>d$ bo? z a bran* ( k9ematics & z a bran* ( optics8 ,i recall besides t our ,euclidean space :i* is ! prop] object ( geometry has be5 *os51 = reasons ( conv5i5ce1 f among a c]ta9 numb] ( types :i* preexi/ 9 our m9d & :i* are call$ gr\ps4 ,if we pass to me*anics1 we /ill see great pr9ciples ^: orig9 is analog\s1 &1 z _! 'radius ( action1' s to speak1 is small]1 !re is no long] reason to sep>ate !m f me*anics prop] & to reg>d ?is sci5ce z d$uctive4 ,9 physics1 f9ally1 ! rle ( ! pr9ciples is /ill m dim9i%$4 ,& 9 fact !y are only introduc$ :5 x is ( advantage4 ,n[ !y are advantage\s precisely because !y are few1 s9ce ea* ( !m v ne>ly replaces a great numb] ( laws4 ,!re=e x is n ( int]e/ to multiply !m4 ,besides an outcome is necess>y1 & = t x is ne$ful to end by leav+ ab/raction to take hold ( reality4 ,su* are ! limits ( nom9alism1 & !y are n>r[4 ,m4 ,le,roy has insi/$1 h[ev]1 & he has put ! "q "u ano!r =m4 ,s9ce ! enunciation ( our laws may v>y ) ! conv5tions t we adopt1 s9ce ^! conv5tions may modify ev5 ! natural relations ( ^! laws1 is !re 9 ! manifold ( ^! laws some?+ indep5d5t ( ^! conv5tions & :i* may1 s to speak1 play ! rle ( univ]sal inv>iant8 ,= in/ance1 ! fiction has be5 introduc$ ( be+s :o1 hav+ be5 educat$ 9 a _w diff]5t f ours1 w\ld h be5 l$ to create a non-,euclidean geometry4 ,if ^! be+s 7 aft]w>d sudd5ly transport$ into our _w1 !y w\ld obs]ve ! same laws z we1 b !y w\ld enunciate !m 9 an entirely diff]5t way4 ,9 tru? !re w\ld /ill be some?+ 9 common betwe5 ! two enunciations1 b ?is is because ^! be+s d n yet diff] en\gh f u4 ,be+s /ill m /range may be imag9$1 & ! "p common to ! two sy/ems ( enunciations w %r9k m & m4 ,w x ?us %r9k 9 conv]g5ce t[>d z]o1 or w !re rema9 an irr$ucible residue :i* w !n be ! univ]sal inv>iant s\ght8 ,! "q calls = precise /atem5t4 ,is x desir$ t ?is common "p ( ! enunciations be expressible 9 words8 ,x is cle>1 !n1 t !re are n words common to all languages1 & we c n pret5d to con/ruct ,i "k n :at univ]sal inv>iant :i* %\ld be und]/ood bo? by u & by ! fictiti\s non-,euclidean geomet]s ( :om ,i h j spok52 no m ?an we c con/ruct a phrase :i* c be und]/ood bo? by ,g]mans :o d n und]/& ,fr5* & by ,fr5* :o d n und]/& ,g]man4 ,b we h fix$ rules :i* p]mit u to translate ! ,fr5* enunciations into ,g]man1 & inv]sely4 ,x is = t t gramm>s & diction>ies h be5 made4 ,!re are also fix$ rules = translat+ ! ,euclidean language into ! non-,euclidean language1 or1 if !re are n1 !y c\ld be made4 ,& ev5 if !re 7 nei!r int]pret] nor diction>y1 if ! ,g]mans & ! ,fr5*1 aft] hav+ liv$ c5turies 9 sep>ate worlds1 f\nd !mselves all at once 9 contact1 d y ?9k !re w\ld be no?+ 9 common betwe5 ! sci5ce ( ! ,g]man books & t ( ! ,fr5* books8 ,! ,fr5* & ! ,g]mans w\ld c]ta9ly end by und]/&+ ea* o!r1 z ! ,am]ican ,indians end$ by und]/&+ ! language ( _! conqu]ors aft] ! arrival ( ! ,spani%4 ,b1 x w be said1 d\btless ! ,fr5* w\ld be capable ( und]/&+ ! ,g]mans ev5 )\t hav+ le>n$ ,g]man1 b ?is is because !re rema9s betwe5 ! ,fr5* & ! ,g]mans some?+ 9 common1 s9ce bo? are m54 ,we %\ld /ill atta9 to an und]/&+ ) our hypo!tical non-,euclideans1 ?\gh !y be n m51 because !y w\ld /ill reta9 some?+ human4 ,b 9 any case a m9imum ( humanity is necess>y4 ,?is is possible1 b ,i %all obs]ve fir/ t ?is little humanness :i* w\ld rema9 9 ! non-,euclideans w\ld suffice n only to make possible ! translation ( a little ( _! language1 b to make possible ! translation ( all _! language4 ,n[1 t !re mu/ be a m9imum is :at ,i conc$e2 suppose !re exi/s ,i "k n :at fluid :i* p5etrates betwe5 ! molecules ( our matt]1 )\t hav+ any action on x & )\t be+ subject to any action com+ f x4 ,suppose be+s s5sible to ! influ5ce ( ?is fluid & ins5sible to t ( our matt]4 ,x is cle> t ! sci5ce ( ^! be+s w\ld diff] absolutely f ours & t x w\ld be idle to seek an 'inv>iant' common to ^! two sci5ces4 ,or aga91 if ^! be+s reject$ our logic & did n admit1 = in/ance1 ! pr9ciple ( contradiction4 ,b truly ,i ?9k x )\t int]e/ to exam9e su* hypo!ses4 ,& !n1 if we d n pu% :imsicality s f>1 if we introduce only fictiti\s be+s hav+ s5ses analog\s to ours & s5sible to ! same impressions1 & moreov] admitt+ ! pr9ciples ( our logic1 we %all !n be able to conclude t _! language1 h[ev] diff]5t f ours x may be1 w\ld always be capable ( translation4 ,n[ ! possibility ( translation implies ! exi/5ce ( an inv>iant4 ,to translate is precisely to dis5gage ?is inv>iant4 ,?us1 to deciph] a cryptogram is to seek :at 9 ?is docum5t rema9s inv>iant1 :5 ! lett]s are p]mut$4 ,:at n[ is ! nature ( ?is inv>iant x is easy to und]/&1 & a ^w w suffice u4 ,! inv>iant laws are ! relations betwe5 ! crude facts1 :ile ! relations betwe5 ! 'sci5tific facts' rema9 always dep5d5t on c]ta9 conv5tions4 ,,*apt] ,,xi ,,sci5ce ,,& ,,reality #e4 ,cont+5ce & ,det]m9ism ,i d n int5d to treat "h ! "q ( ! cont+5ce ( ! laws ( nature1 :i* is evid5tly insoluble1 & on :i* s mu* has already be5 writt54 ,i only wi% to call att5tion to :at diff]5t mean+s h be5 giv5 to ?is ^w1 cont+5ce1 & h[ advantage\s x w\ld be to di/+ui% !m4 ,if we look at any p>ticul> law1 we may be c]ta9 9 advance t x c only be approximate4 ,x is1 9 fact1 d$uc$ f exp]im5tal v]ifications1 & ^! v]ifications 7 & c\ld be only approximate4 ,we %\ld always expect t m precise measurem5ts w oblige u to add new t]ms to our =mulas2 ?is is :at has happ5$1 = in/ance1 9 ! case ( ,m>iotte's law4 ,moreov] ! /atem5t ( any law is necess>ily incomplete4 ,?is enunciation %\ld comprise ! enum]ation ( all ! antec$5ts 9 virtue ( :i* a giv5 consequ5t c happ54 ,i %\ld fir/ describe all ! conditions ( ! exp]im5t to be made & ! law w\ld !n be /at$3 ,if all ! conditions are fulfill$1 ! ph5om5on w happ54 ,b we %all be sure ( n hav+ =gott5 any ( ^! conditions only :5 we %all h describ$ ! /ate ( ! entire univ]se at ! in/ant t2 all ! p>ts ( ?is univ]se may1 9 fact1 ex]cise an influ5ce m or less great on ! ph5om5on :i* mu/ happ5 at ! in/ant t dt4 ,n[ x is cle> t su* a description c\ld n be f\nd 9 ! enunciation ( ! law2 besides1 if x 7 made1 ! law w\ld become incapable ( application2 if "o requir$ s _m conditions1 !re w\ld be v little *ance ( _! ev] be+ all realiz$ at any mom5t4 ,!n z "o c nev] be c]ta9 ( n hav+ =gott5 "s ess5tial condition1 x c n be said3 ,if su* & su* conditions are realiz$1 su* a ph5om5on w occur2 x c only be said3 ,if su* & su* conditions are realiz$1 x is probable t su* a ph5om5on w occur1 v ne>ly4 ,take ! law ( gravitation1 :i* is ! lea/ imp]fect ( all kn[n laws4 ,x enables u to =esee ! motions ( ! planets4 ,:5 ,i use x1 = in/ance1 to calculate ! orbit ( ,saturn1 ,i neglect ! action ( ! />s1 & 9 do+ s ,i am c]ta9 ( n deceiv+ myself1 because ,i "k t ^! />s are too f> away = _! action to be s5sible4 ,i ann\nce1 !n1 ) a quasi-c]titude t ! coord9ates ( ,saturn at su* an h\r w be compris$ betwe5 su* & su* limits4 ,yet is t c]titude absolute8 ,c\ld !re n exi/ 9 ! univ]se "s gigantic mass1 mu* great] ?an t ( all ! kn[n />s & ^: action c\ld make itself felt at great di/ances8 ,t mass might be animat$ by a colossal velocity1 & aft] hav+ circulat$ f all "t at su* di/ances t its influ5ce _h rema9$ hi!rto ins5sible to u1 x might come all at once to pass ne> u4 ,surely x w\ld produce 9 our sol> sy/em enorm\s p]turbations t we c\ld n h =ese54 ,all t c be said is t su* an ev5t is :olly improbable1 & !n1 in/ead ( say+3 ,saturn w be ne> su* a po9t ( ! heav5s1 we mu/ limit ourselves to say+3 ,saturn w probably be ne> su* a po9t ( ! heav5s4 ,al?\gh ?is probability may be practically equival5t to c]ta9ty1 x is only a probability4 ,= all ^! reasons1 no p>ticul> law w ev] be m ?an approximate & probable4 ,sci5ti/s h nev] fail$ to recognize ?is tru?2 only !y believe1 "r or wrong1 t e law may be replac$ by ano!r clos] & m probable1 t ?is new law w itself be only provisional1 b t ! same movem5t c cont9ue indef9itely1 s t sci5ce 9 progress+ w possess laws m & m probable1 t ! approximation w end by diff]+ z little z y *oose f exactitude & ! probability f c]titude4 ,if ! sci5ti/s :o ?9k ?us are "r1 /ill c\ld x be said t ! laws ( nature are cont+5t1 ev5 ?\gh ea* law1 tak5 9 p>ticul>1 may be qualifi$ z cont+5t8 ,or mu/ "o require1 be=e conclud+ ! cont+5ce ( ! natural laws1 t ?is progress h an end1 t ! sci5ti/ f9i% "s day by be+ arre/$ 9 8 se>* = a clos] & clos] approximation1 & t1 beyond a c]ta9 limit1 he !reaft] meet 9 nature only caprice8 ,9 ! conception ( :i* ,i h j spok5 7& :i* ,i %all call ! sci5tific conception71 e law is only a /atem5t imp]fect & provisional1 b x mu/ "o day be replac$ by ano!r1 a sup]ior law1 ( :i* x is only a crude image4 ,no place !re=e rema9s = ! int]v5tion ( a free w4 ,x seems to me t ! k9etic !ory ( gases w furni% u a /rik+ example4 ,y "k t 9 ?is !ory all ! prop]ties ( gases are expla9$ by a simple hypo!sis2 x is suppos$ t all ! gase\s molecules move 9 e direction ) great velocities & t !y foll[ rectil9eal pa?s :i* are di/urb$ only :5 "o molecule passes v ne> ! sides ( ! vessel or ano!r molecule4 ,! effects our crude s5ses enable u to obs]ve are ! mean effects1 & 9 ^! means1 ! great deviations comp5sate1 or at lea/ x is v improbable t !y d n comp5sate2 s t ! obs]vable ph5om5a foll[ simple laws su* z t ( ,m>iotte or ( ,gay-,lussac4 ,b ?is comp5sation ( deviations is only probable4 ,! molecules incessantly *ange place & 9 ^! cont9ual displacem5ts ! figures !y =m pass successively ?r\gh all possible comb9ations4 ,s+ly ^! comb9ations are v num]\s2 almo/ all are 9 con=mity ) ,m>iotte's law1 only a few deviate f x4 ,^! also w happ51 only x w\ld be necess>y to wait a long "t = !m4 ,if a gas 7 obs]v$ dur+ a suffici5tly long "t x w\ld c]ta9ly be f9ally se5 to deviate1 = a v %ort "t1 f ,m>iotte's law4 ,h[ long w\ld x be necess>y to wait8 ,if x 7 desir$ to calculate ! probable numb] ( ye>s1 x w\ld be f\nd t ?is numb] is s great t to write only ! numb] ( places ( figures employ$ w\ld /ill require half a score places ( figures4 ,no matt]2 en\gh t x may be done4 ,i d n c>e to discuss "h ! value ( ?is !ory4 ,x is evid5t t if x be adopt$1 ,m>iotte's law w !reaft] appe> only z cont+5t1 s9ce a day w come :5 x w n be true4 ,& yet1 ?9k y ! p>tisans ( ! k9etic !ory are adv]s>ies ( det]m9ism8 ,f> f x2 !y are ! mo/ ultra ( me*ani/s4 ,_! molecules foll[ rigid pa?s1 f :i* !y dep>t only "u ! influ5ce ( =ces :i* v>y ) ! di/ance1 foll[+ a p]fectly det]m9ate law4 ,!re rema9s 9 _! sy/em n ! smalle/ place ei!r = fre$om1 or = an evolution>y factor1 prop]ly s-call$1 or = any?+ :atev] t c\ld be call$ cont+5ce4 ,i add1 to avoid mi/ake1 t nei!r is !re any evolution ( ,m>iotte's law itself2 x ceases to be true aft] ,i "k n h[ _m c5turies2 b at ! end ( a fraction ( a second x aga9 becomes true & t = an incalculable numb] ( c5turies4 ,& s9ce ,i h pron\nc$ ! ^w evolution1 let u cle> away ano!r mi/ake4 ,x is (t5 said3 ,:o kn[s :e!r ! laws d n evolve & :e!r we %all n "o day discov] t !y 7 n at ! ,c>bonif]\s epo* :at !y are to-day8 ,:at are we to und]/& by t8 ,:at we ?9k we "k ab\t ! pa/ /ate ( our globe1 we d$uce f its pres5t /ate4 ,& h[ is ?is d$uction made8 ,x is by means ( laws suppos$ kn[n4 ,! law1 be+ a relation betwe5 ! antec$5t & ! consequ5t1 enables u equally well to d$uce ! consequ5t f ! antec$5t1 t is1 to =esee ! future1 & to d$uce ! antec$5t f ! consequ5t1 t is1 to conclude f ! pres5t to ! pa/4 ,! a/ronom] :o kn[s ! pres5t situation ( ! />s c f x d$uce _! future situation by ,newton's law1 & ?is is :at he does :5 he con/ructs ephem]ides2 & he c equally d$uce f x _! pa/ situation4 ,! calculations he ?us c make c n tea* him t ,newton's law w cease to be true 9 ! future1 s9ce ?is law is precisely 8 po9t ( dep>ture2 n m c !y tell him x 0 n true 9 ! pa/4 ,/ill1 9 :at conc]ns ! future1 8 ephem]ides c "o day be te/$ & our desc5dants w p]haps recognize t !y 7 false4 ,b 9 :at conc]ns ! pa/1 ! geologic pa/ :i* _h no witnesses1 ! results ( 8 calculation1 l ^? ( all speculations :]e we seek to d$uce ! pa/ f ! pres5t1 escape by _! v nature e species ( te/4 ,s t if ! laws ( nature 7 n ! same 9 ! ,c>bonif]\s age z at ! pres5t epo*1 we %all nev] be able to "k x1 s9ce we c "k no?+ ( ?is age1 only :at we d$uce f ! hypo!sis ( ! p]man5ce ( ^! laws4 ,p]haps x w be said t ?is hypo!sis might lead to contradictory results & t we %all be oblig$ to ab&on x4 ,?us1 9 :at conc]ns ! orig9 ( life1 we may conclude t !re h always be5 liv+ be+s1 s9ce ! pres5t _w %[s u always life spr++ f life2 & we may also conclude t !re h n always be51 s9ce ! application ( ! exi/5t laws ( physics to ! pres5t /ate ( our globe tea*es u t !re 0 a "t :5 ?is globe 0 s w>m t life on x 0 impossible4 ,b contradictions ( ?is sort c always be remov$ 9 two ways2 x may be suppos$ t ! actual laws ( nature are n exactly :at we h assum$2 or else x may be suppos$ t ! laws ( nature actually are :at we h assum$1 b t x has n always be5 s4 ,x is evid5t t ! actual laws w nev] be suffici5tly well kn[n = u n to be able to adopt ! fir/ ( ^! two solutions & = u to be con/ra9$ to inf] ! evolution ( natural laws4 ,on ! o!r h&1 suppose su* an evolution2 assume1 if y wi%1 t humanity la/s suffici5tly long = ?is evolution to h witnesses4 ,! same antec$5t %all produce1 = in/ance1 diff]5t consequ5ts at ! ,c>bonif]\s epo* & at ! ,quat]n>y4 ,t evid5tly means t ! antec$5ts are closely alike2 if all ! circum/ances 7 id5tical1 ! ,c>bonif]\s epo* w\ld be indi/+ui%able f ! ,quat]n>y4 ,evid5tly ?is is n :at is suppos$4 ,:at rema9s is t su* antec$5t1 accompani$ by su* accessory circum/ance1 produces su* consequ5t2 & t ! same antec$5t1 accompani$ by su* o!r accessory circum/ance1 produces su* o!r consequ5t4 ,"t does n ent] into ! affair4 ,! law1 su* z ill-in=m$ sci5ce w\ld h /at$ x1 & :i* w\ld h affirm$ t ?is antec$5t always produces ?is consequ5t1 )\t tak+ acc\nt ( ! accessory circum/ances1 ?is law1 :i* 0 only approximate & probable1 mu/ be replac$ by ano!r law m approximate & m probable1 :i* br+s 9 ^! accessory circum/ances4 ,we always come back1 !re=e1 to t same process :i* we h analyz$ above1 & if humanity %\ld discov] some?+ ( ?is sort1 x w\ld n say t x is ! laws :i* h evolut$1 b ! circum/ances :i* h *ang$4 ,"h1 !re=e1 are sev]al diff]5t s5ses ( ! ^w cont+5ce4 ,m4 ,le,roy reta9s !m all & he does n suffici5tly di/+ui% !m1 b he introduces a new "o4 ,exp]im5tal laws are only approximate1 & if "s appe> to u z exact1 x is because we h artificially trans=m$ !m into :at ,i h above call$ a pr9ciple4 ,we h made ?is trans=mation freely1 & z ! caprice :i* has det]m9$ u to make x is some?+ em95tly cont+5t1 we h communicat$ ?is cont+5ce to ! law itself4 ,x is 9 ?is s5se t we h ! "r to say t det]m9ism supposes fre$om1 s9ce x is freely t we become det]m9i/s4 ,p]haps x w be f\nd t ?is is to give l>ge scope to nom9alism & t ! introduction ( ?is new s5se ( ! ^w cont+5ce w n help mu* to solve all ^? que/ions :i* naturally arise & ( :i* we h j be5 speak+4 ,i d n at all wi% to inve/igate "h ! f\ndations ( ! pr9ciple ( induction2 ,i "k v well t ,i %\ld n succe$2 x is z difficult to ju/ify ?is pr9ciple z to get on )\t x4 ,i only wi% to %[ h[ sci5ti/s apply x & are =c$ to apply x4 ,:5 ! same antec$5t recurs1 ! same consequ5t mu/ likewise recur2 su* is ! ord9>y /atem5t4 ,b r$uc$ to ^! t]ms ?is pr9ciple c\ld be ( no use4 ,= "o to be able to say t ! same antec$5t recurr$1 x w\ld be necess>y = ! circum/ances all to be reproduc$1 s9ce no "o is absolutely indiff]5t1 & = !m to be exactly reproduc$4 ,&1 z t w nev] happ51 ! pr9ciple c h no application4 ,we %\ld !re=e modify ! enunciation & say3 ,if an antec$5t ,a has once produc$ a consequ5t ,b1 an antec$5t ,a'1 slightly diff]5t f ,a1 w produce a consequ5t ,b'1 slightly diff]5t f ,b4 ,b h[ %all we recognize t ! antec$5ts ,a & ,a' are 'slightly diff]5t'8 ,if "s "o ( ! circum/ances c be express$ by a numb]1 & ?is numb] has 9 ! two cases values v ne> toge!r1 ! s5se ( ! phrase 'slightly diff]5t' is relatively cle>2 ! pr9ciple !n signifies t ! consequ5t is a cont9u\s function ( ! antec$5t4 ,& z a practical rule1 we rea* ?is conclusion t we h ! "r to int]polate4 ,?is is 9 fact :at sci5ti/s d e day1 & )\t int]polation all sci5ce w\ld be impossible4 ,yet obs]ve "o ?+4 ,! law s\ght may be repres5t$ by a curve4 ,exp]im5t has taught u c]ta9 po9ts ( ?is curve4 ,9 virtue ( ! pr9ciple we h j /at$1 we believe ^! po9ts may be connect$ by a cont9u\s graph4 ,we trace ?is graph ) ! eye4 ,new exp]im5ts w furni% u new po9ts ( ! curve4 ,if ^! po9ts are outside ( ! graph trac$ 9 advance1 we %all h to modify our curve1 b n to ab&on our pr9ciple4 ,?r\gh any po9ts1 h[ev] num]\s !y may be1 a cont9u\s curve may always be pass$4 ,d\btless1 if ?is curve is too caprici\s1 we %all be %ock$ 7& we %all ev5 suspect errors ( exp]im5t71 b ! pr9ciple w n be directly put at fault4 ,fur!rmore1 among ! circum/ances ( a ph5om5on1 !re are "s t we reg>d z negligible1 & we %all consid] ,a & ,a' z slightly diff]5t if !y diff] only by ^! accessory circum/ances4 ,= in/ance1 ,i h asc]ta9$ t hydrog5 unites ) oxyg5 "u ! influ5ce ( ! electric sp>k1 & ,i am c]ta9 t ^! two gases w unite anew1 al?\gh ! longitude ( ,jupit] may h *ang$ consid]ably 9 ! int]val4 ,we assume1 = in/ance1 t ! /ate ( di/ant bodies c h no s5sible influ5ce on t]re/rial ph5om5a1 & t seems 9 fact requisite1 b !re are cases :]e ! *oice ( ^! practically indiff]5t circum/ances admits ( m arbitr>9ess or1 if y *oose1 requires m tact4 ,"o m rem>k3 ,! pr9ciple ( induction w\ld be inapplicable if !re did n exi/ 9 nature a great quantity ( bodies l "o ano!r1 or almo/ alike1 & if we c\ld n inf]1 = in/ance1 f "o bit ( phosphorus to ano!r bit ( phosphorus4 ,if we reflect on ^! consid]ations1 ! problem ( det]m9ism & ( cont+5ce w appe> to u 9 a new light4 ,suppose we 7 able to embrace ! s]ies ( all ph5om5a ( ! univ]se 9 ! :ole sequ5ce ( "t4 ,we c\ld envisage :at might be call$ ! sequ5ces2 ,i mean relations betwe5 antec$5t & consequ5t4 ,i d n wi% to speak ( con/ant relations or laws1 ,i envisage sep>ately 7individually1 s to speak7 ! diff]5t sequ5ces realiz$4 ,we %\ld !n recognize t among ^! sequ5ces !re are no two altoge!r alike4 ,b1 if ! pr9ciple ( induction1 z we h j /at$ x1 is true1 !re w be ^? almo/ alike & t c be class$ alongside "o ano!r4 ,9 o!r words1 x is possible to make a classification ( sequ5ces4 ,x is to ! possibility & ! legitimacy ( su* a classification t det]m9ism1 9 ! end1 r$uces4 ,?is is all t ! prec$+ analysis leaves ( x4 ,p]haps "u ?is mode/ =m x w seem less appall+ to ! morali/4 ,x w d\btless be said t ?is is to come back by a det\r to ,m4 ,le,roy's conclusion :i* a mom5t ago we seem$ to reject3 we are det]m9i/s volunt>ily4 ,& 9 fact all classification supposes ! active int]v5tion ( ! classifi]4 ,i agree t ?is may be ma9ta9$1 b x seems to me t ?is det\r w n h be5 useless & w h contribut$ to enlight5 u a little4 #f4 ,objectivity ( ,sci5ce ,i arrive at ! "q set by ! title ( ?is article3 ,:at is ! objective value ( sci5ce8 ,& fir/ :at %\ld we und]/& by objectivity8 ,:at gu>antees ! objectivity ( ! _w 9 :i* we live is t ?is _w is common to u ) o!r ?9k+ be+s4 ,?r\gh ! communications t we h ) o!r m51 we receive f !m ready-made reason+s2 we "k t ^! reason+s d n come f u & at ! same "t we recognize 9 !m ! "w ( reasonable be+s l ourselves4 ,& z ^! reason+s appe> to fit ! _w ( our s5sations1 we ?9k we may inf] t ^! reasonable be+s h se5 ! same ?+ z we2 ?us x is we "k we h n be5 dream+4 ,su*1 !re=e1 is ! fir/ condition ( objectivity2 :at is objective mu/ be common to _m m9ds & consequ5tly transmissible f "o to ! o!r1 & z ?is transmission c only come ab\t by t 'disc\rse' :i* inspires s mu* di/ru/ 9 ,m4 ,le,roy1 we are ev5 =c$ to conclude3 no disc\rse1 no objectivity4 ,! s5sations ( o!rs w be = u a _w et]nally clos$4 ,we h no means ( v]ify+ t ! s5sation ,i call r$ is ! same z t :i* my neighbor calls r$4 ,suppose t a *]ry & a r$ poppy produce on me ! s5sation ,a & on him ! s5sation ,b & t1 on ! contr>y1 a leaf produces on me ! s5sation ,b & on him ! s5sation ,a4 ,x is cle> we %all nev] "k any?+ ab\t x2 s9ce ,i %all call r$ ! s5sation ,a & gre5 ! s5sation ,b1 :ile he w call ! fir/ gre5 & ! second r$4 ,9 comp5sation1 :at we %all be able to asc]ta9 is t1 = him z = me1 ! *]ry & ! r$ poppy produce ! same s5sation1 s9ce he gives ! same "n to ! s5sations he feels & ,i d ! same4 ,s5sations are !re=e intransmissible1 or r all t is pure quality 9 !m is intransmissible & =ev] imp5etrable4 ,b x is n ! same ) relations betwe5 ^! s5sations4 ,f ?is po9t ( view1 all t is objective is devoid ( all quality & is only pure relation4 ,c]tes1 ,i %all n g s f> z to say t objectivity is only pure quantity 7?is w\ld be to p>ticul>ize too f> ! nature ( ! relations 9 "q71 b we und]/& h[ "s "o c\ld h be5 c>ri$ away into say+ t ! _w is only a diff]5tial equation4 ,) due res]ve reg>d+ ?is p>adoxical proposition1 we mu/ nev]!less admit t no?+ is objective :i* is n transmissible1 & consequ5tly t ! relations betwe5 ! s5sations c alone h an objective value4 ,p]haps x w be said t ! e/hetic emotion1 :i* is common to all mank9d1 is pro( t ! qualities ( our s5sations are also ! same = all m5 & h5ce are objective4 ,b if we ?9k ab\t ?is1 we %all see t ! pro( is n complete2 :at is prov$ is t ?is emotion is ar\s$ 9 ,john z 9 ,james by ! s5sations to :i* ,james & ,john give ! same "n or by ! correspond+ comb9ations ( ^! s5sations2 ei!r because ?is emotion is associat$ 9 ,john ) ! s5sation ,a1 :i* ,john calls r$1 :ile p>allelly x is associat$ 9 ,james ) ! s5sation ,b1 :i* ,james calls r$2 or bett] because ?is emotion is ar\s$1 n by ! qualities !mselves ( ! s5sations1 b by ! h>moni\s comb9ation ( _! relations ( :i* we und]go ! unconsci\s impression4 ,su* a s5sation is beautiful1 n because x possesses su* a quality1 b because x occupies su* a place 9 ! wo( ( our associations ( ideas1 s t x c n be excit$ )\t putt+ 9 motion ! 'receiv]' :i* is at ! o!r end ( ! ?read & :i* corresponds to ! arti/ic emotion4 ,:e!r we take ! moral1 ! e/hetic or ! sci5tific po9t ( view1 x is always ! same ?+4 ,no?+ is objective except :at is id5tical = all2 n[ we c only speak ( su* an id5tity if a comp>ison is possible1 & c be translat$ into a 'money ( ex*ange' capable ( transmission f "o m9d to ano!r4 ,no?+1 !re=e1 w h objective value except :at is transmissible by 'disc\rse1' t is1 intelligible4 ,b ?is is only "o side ( ! "q4 ,an absolutely disord]$ aggregate c\ld n h objective value s9ce x w\ld be un9telligible1 b no m c a well-ord]$ assemblage h x1 if x does n correspond to s5sations really exp]i5c$4 ,x seems to me sup]flu\s to recall ?is condition1 & ,i %\ld n h dream$ ( x1 if x _h n lately be5 ma9ta9$ t physics is n an exp]im5tal sci5ce4 ,al?\gh ?is op9ion has no *ance ( be+ adopt$ ei!r by physici/s or by philosoph]s1 x is well to be w>n$ s z n to let oneself slip ov] ! declivity :i* w\ld lead ?i!r4 ,two conditions are !re=e to be fulfill$1 & if ! fir/ sep>ates reality#aa f ! dream1 ! second di/+ui%es x f ! romance4 #aa ,i "h use ! ^w real z a synonym ( objective2 ,i ?us con=m to common usage2 p]haps ,i am wrong1 our dreams are real1 b !y are n objective4 ,n[ :at is sci5ce8 ,i h expla9$ 9 ! prec$+ article1 x is be=e all a classification1 a mann] ( br++ toge!r facts :i* appe>ances sep>ate1 ?\gh !y 7 b\nd toge!r by "s natural & hidd5 k9%ip4 ,sci5ce1 9 o!r words1 is a sy/em ( relations4 ,n[ we h j said1 x is 9 ! relations alone t objectivity mu/ be s\ght2 x w\ld be va9 to seek x 9 be+s consid]$ z isolat$ f "o ano!r4 ,to say t sci5ce c n h objective value s9ce x tea*es u only relations1 ?is is to reason backw>d1 s9ce1 precisely1 x is relations alone :i* c be reg>d$ z objective4 ,ext]nal objects1 = in/ance1 = :i* ! ^w object 0 inv5t$1 are really objects & n fleet+ & fugitive appe>ances1 because !y are n only gr\ps ( s5sations1 b gr\ps cem5t$ by a con/ant bond4 ,x is ?is bond1 & ?is bond alone1 :i* is ! object 9 itself1 & ?is bond is a relation4 ,!re=e1 :5 we ask :at is ! objective value ( sci5ce1 t does n mean3 ,does sci5ce tea* u ! true nature ( ?+s8 b x means3 ,does x tea* u ! true relations ( ?+s8 ,to ! fir/ "q1 no "o w\ld hesitate to reply1 no2 b ,i ?9k we may g f>!r2 n only sci5ce c n tea* u ! nature ( ?+s2 b no?+ is capable ( tea*+ x to u1 & if any god knew x1 he c\ld n f9d words to express x4 ,n only c we n div9e ! response1 b if x 7 giv5 to u we c\ld und]/& no?+ ( x2 ,i ask myself ev5 :e!r we really und]/& ! "q4 ,:51 !re=e1 a sci5tific !ory pret5ds to tea* u :at heat is1 or :at is electricity1 or life1 x is condemn$ be=eh&2 all x c give u is only a crude image4 ,x is1 !re=e1 provisional & crumbl+4 ,! fir/ "q be+ out ( reason1 ! second rema9s4 ,c sci5ce tea* u ! true relations ( ?+s8 ,:at x jo9s toge!r %\ld t be put asund]1 :at x puts asund] %\ld t be jo9$ toge!r8 ,to und]/& ! mean+ ( ?is new "q1 x is ne$ful to ref] to :at 0 said above on ! conditions ( objectivity4 ,h ^! relations an objective value8 ,t means3 ,are ^! relations ! same = all8 ,w !y /ill be ! same = ^? :o %all come aft] u8 ,x is cle> t !y are n ! same = ! sci5ti/ & ! ignorant p]son4 ,b t is unimportant1 because if ! ignorant p]son does n see !m all at once1 ! sci5ti/ may succe$ 9 mak+ him see !m by a s]ies ( exp]im5ts & reason+s4 ,! ?+ ess5tial is t !re are po9ts on :i* all ^? acqua9t$ ) ! exp]im5ts made c rea* accord4 ,! "q is to "k :e!r ?is accord w be durable & :e!r x w p]si/ = our successors4 ,x may be ask$ :e!r ! unions t ! sci5ce ( to-day makes w be confirm$ by ! sci5ce ( to-morr[4 ,to affirm t x w be s we c n invoke any a priori reason2 b ?is is a "q ( fact1 & sci5ce has already liv$ long en\gh = u to be able to f9d out by ask+ its hi/ory :e!r ! edifices x builds /& ! te/ ( "t1 or :e!r !y are only ephem]al con/ructions4 ,n[ :at d we see8 ,at ! fir/ blu%1 x seems to u t ! !ories la/ only a day & t ru9s ^u ru9s accumulate4 ,to-day ! !ories are born1 to-morr[ !y are ! fa%ion1 ! day aft] to-morr[ !y are classic1 ! f\r? day !y are sup]annuat$1 & ! fif? !y are =gott54 ,b if we look m closely1 we see t :at ?us succumb are ! !ories prop]ly s call$1 ^? :i* pret5d to tea* u :at ?+s are4 ,b !re is 9 !m some?+ :i* usually survives4 ,if "o ( !m taught u a true relation1 ?is relation is def9itively acquir$1 & x w be f\nd aga9 "u a new disguise 9 ! o!r !ories :i* w successively come to reign 9 place ( ! old4 ,take only a s+le example3 ,! !ory ( ! undulations ( ! e!r taught u t light is a motion2 to-day fa%ion favors ! electromagnetic !ory :i* tea*es u t light is a curr5t4 ,we d n consid] :e!r we c\ld reconcile !m & say t light is a curr5t1 & t ?is curr5t is a motion4 ,z x is probable 9 any case t ?is motion w\ld n be id5tical ) t :i* ! p>tisans ( ! old !ory presume1 we might ?9k ourselves ju/ifi$ 9 say+ t ?is old !ory is de?ron$4 ,& yet some?+ ( x rema9s1 s9ce betwe5 ! hypo!tical curr5ts :i* ,maxwell supposes !re are ! same relations z betwe5 ! hypo!tical motions t ,fresnel suppos$4 ,!re is1 !re=e1 some?+ :i* rema9s ov] & ?is some?+ is ! ess5tial4 ,?is x is :i* expla9s h[ we see ! pres5t physici/s pass )\t any emb>rassm5t f ! language ( ,fresnel to t ( ,maxwell4 ,d\btless _m connections t 7 believ$ well e/abli%$ h be5 ab&on$1 b ! greate/ numb] rema9 & x w\ld seem mu/ rema94 ,& = ^!1 !n1 :at is ! measure ( _! objectivity8 ,well1 x is precisely ! same z = our belief 9 ext]nal objects4 ,^! latt] are real 9 ?is1 t ! s5sations !y make u feel appe> to u z unit$ to ea* o!r by ,i "k n :at inde/ructible cem5t & n by ! haz>d ( a day4 ,9 ! same way sci5ce reveals to u betwe5 ph5om5a o!r bonds f9] b n less solid2 ^! are ?reads s sl5d] t !y long rema9$ unp]ceiv$1 b once notic$ !re rema9s no way ( n see+ !m2 !y are !re=e n less real ?an ^? :i* give _! reality to ext]nal objects2 small matt] t !y are m rec5tly kn[n1 s9ce nei!r c p]i% be=e ! o!r4 ,x may be said1 = in/ance1 t ! e!r is no less real ?an any ext]nal body2 to say ?is body exi/s is to say !re is betwe5 ! color ( ?is body1 its ta/e1 its smell1 an intimate bond1 solid & p]si/5t2 to say ! e!r exi/s is to say !re is a natural k9%ip betwe5 all ! optical ph5om5a1 & nei!r ( ! two propositions has less value ?an ! o!r4 ,& ! sci5tific syn!ses h 9 a s5se ev5 m reality ?an ^? ( ! ord9>y s5ses1 s9ce !y embrace m t]ms & t5d to absorb 9 !m ! p>tial syn!ses4 ,x w be said t sci5ce is only a classification & t a classification c n be true1 b conv5i5t4 ,b x is true t x is conv5i5t1 x is true t x is s n only = me1 b = all m52 x is true t x w rema9 conv5i5t = our desc5dants2 x is true f9ally t ?is c n be by *ance4 ,9 sum1 ! sole objective reality consi/s 9 ! relations ( ?+s :5ce results ! univ]sal h>mony4 ,d\btless ^! relations1 ?is h>mony1 c\ld n be conceiv$ outside ( a m9d :i* conceives !m4 ,b !y are nev]!less objective because !y are1 w become1 or w rema91 common to all ?9k+ be+s4 ,?is w p]mit u to rev]t to ! "q ( ! rotation ( ! e>? :i* w give u at ! same "t a *ance to make cle> :at prec$es by an example4 #g4 ,! ,rotation ( ! ,e>? 444 ,!re=e1 h ,i said 9 ,sci5ce & ,hypo!sis1 ?is affirmation1 ! e>? turns r\nd1 has no mean+ 444 or r ^! two propositions1 ! e>? turns r\nd1 &1 x is m conv5i5t to suppose t ! e>? turns r\nd1 h "o & ! same mean+4 ,^! words h giv5 rise to ! /range/ int]pretations4 ,"s h ?\ght !y saw 9 !m ! rehabilitation ( ,ptolemy's sy/em1 & p]haps ! ju/ification ( ,galileo's condemnation4 ,^? :o _h read att5tively ! :ole volume c\ld n1 h[ev]1 delude !mselves4 ,?is tru?1 ! e>? turns r\nd1 0 put on ! same foot+ z ,euclid's po/ulate1 = example4 ,0 t to reject x8 ,b bett]2 9 ! same language x may v well be said3 ,^! two propositions1 ! ext]nal _w exi/s1 or1 x is m conv5i5t to suppose t x exi/s1 h "o & ! same mean+4 ,s ! hypo!sis ( ! rotation ( ! e>? w\ld h ! same degree ( c]titude z ! v exi/5ce ( ext]nal objects4 ,b aft] :at we h j expla9$ 9 ! f\r? "p1 we may g f>!r4 ,a physical !ory1 we h said1 is by s mu* ! m true z x puts 9 evid5ce m true relations4 ,9 ! light ( ?is new pr9ciple1 let u exam9e ! "q :i* occupies u4 ,no1 !re is no absolute space2 ^! two contradictory propositions3 ',! e>? turns r\nd' & ',! e>? does n turn r\nd' are1 !re=e1 nei!r ( !m m true ?an ! o!r4 ,to affirm "o :ile d5y+ ! o!r1 9 ! k9ematic s5se1 w\ld be to admit ! exi/5ce ( absolute space4 ,b if ! "o reveals true relations t ! o!r hides f u1 we c nev]!less reg>d x z physically m true ?an ! o!r1 s9ce x has a ri*] cont5t4 ,n[ 9 ?is reg>d no d\bt is possible4 ,behold ! app>5t diurnal motion ( ! />s1 & ! diurnal motion ( ! o!r heav5ly bodies1 & besides1 ! flatt5+ ( ! e>?1 ! rotation ( ,f\cault's p5dulum1 ! gyration ( cyclones1 ! trade-w9ds1 :at n else8 ,= ! ,ptolemai/ all ^! ph5om5a h no bond betwe5 !m2 = ! ,cop]nican !y are produc$ by ! "o same cause4 ,9 say+1 ! e>? turns r\nd1 ,i affirm t all ^! ph5om5a h an intimate relation1 & t is true1 & t rema9s true1 al?\gh !re is n & c n be absolute space4 ,s mu* = ! rotation ( ! e>? ^u itself2 :at %all we say ( its revolution ar\nd ! sun8 ,"h aga91 we h ?ree ph5om5a :i* = ! ,ptolemai/ are absolutely indep5d5t & :i* = ! ,cop]nican are ref]r$ back to ! same orig92 !y are ! app>5t displacem5ts ( ! planets on ! cele/ial sph]e1 ! ab]ration ( ! fix$ />s1 ! p>allax ( ^! same />s4 ,is x by *ance t all ! planets admit an inequality ^: p]iod is a ye>1 & t ?is p]iod is precisely equal to t ( ab]ration1 precisely equal besides to t ( p>allax8 ,to adopt ,ptolemy's sy/em is to answ]1 yes2 to adopt t ( ,cop]nicus is to answ]1 no2 ?is is to affirm t !re is a bond betwe5 ! ?ree ph5om5a1 & t also is true1 al?\gh !re is no absolute space4 ,9 ,ptolemy's sy/em1 ! motions ( ! heav5ly bodies c n be expla9$ by ! action ( c5tral =ces1 cele/ial me*anics is impossible4 ,! intimate relations t cele/ial me*anics reveals to u betwe5 all ! cele/ial ph5om5a are true relations2 to affirm ! immobility ( ! e>? w\ld be to d5y ^! relations1 t w\ld be to fool ourselves4 ,! tru? = :i* ,galileo suff]$ rema9s1 !re=e1 ! tru?1 al?\gh x has n altoge!r ! same mean+ z = ! vulg>1 & its true mean+ is mu* m subtle1 m pr(\nd & m ri*4 #h4 ,sci5ce = ,its ,own ,sake ,n aga9/ ,m4 ,le,roy d ,i wi% to def5d sci5ce = its own sake2 maybe ?is is :at he condemns1 b ?is is :at he cultivates1 s9ce he loves & seeks tru? & c\ld n live )\t x4 ,b ,i h "s ?\ghts to express4 ,we c n "k all facts & x is necess>y to *oose ^? :i* are wor?y ( be+ kn[n4 ,accord+ to ,tol/oi1 sci5ti/s make ?is *oice at r&om1 in/ead ( mak+ x1 :i* w\ld be reasonable1 ) a view to practical applications4 ,on ! contr>y1 sci5ti/s ?9k t c]ta9 facts are m int]e/+ ?an o!rs1 because !y complete an unf9i%$ h>mony1 or because !y make "o =esee a great numb] ( o!r facts4 ,if !y are wrong1 if ?is hi]>*y ( facts t !y implicitly po/ulate is only an idle illusion1 !re c\ld be no sci5ce = its own sake1 & consequ5tly !re c\ld be no sci5ce4 ,z = me1 ,i believe !y are "r1 &1 = example1 ,i h %[n above :at is ! high value ( a/ronomical facts1 n because !y are capable ( practical applications1 b because !y are ! mo/ in/ructive ( all4 ,x is only ?r\gh sci5ce & art t civilization is ( value4 ,"s h wond]$ at ! =mula3 sci5ce = its own sake2 & yet x is z good z life = its own sake1 if life is only mis]y2 & ev5 z happ9ess = its own sake1 if we d n believe t all pleasures are ( ! same quality1 if we d n wi% to admit t ! goal ( civilization is to furni% alcohol to p :o love to dr9k4 ,e act %\ld h an aim4 ,we mu/ suff]1 we mu/ "w1 we mu/ pay = our place at ! game1 b ?is is = see+'s sake2 or at ! v lea/ t o!rs may "o day see4 ,all t is n ?\ght is pure no?+ness2 s9ce we c ?9k only ?\ghts & all ! words we use to speak ( ?+s c express only ?\ghts1 to say !re is some?+ o!r ?an ?\ght1 is !re=e an affirmation :i* c h no mean+4 ,& yet--/range contradiction = ^? :o believe 9 "t--geologic hi/ory %[s u t life is only a %ort episode betwe5 two et]nities ( dea?1 & t1 ev5 9 ?is episode1 consci\s ?\ght has la/$ & w la/ only a mom5t4 ,?\ght is only a gleam 9 ! mid/ ( a long night4 ,b x is ?is gleam :i* is ev]y?+4 99 99 99 99 99 ,,sci5ce ,,& ,,me?od 99 99 99 99 99 ,,introduction ,i br+ toge!r "h diff]5t /udies relat+ m or less directly to que/ions ( sci5tific me?odology4 ,! sci5tific me?od consi/s 9 obs]v+ & exp]im5t+2 if ! sci5ti/ _h at 8 disposal inf9ite "t1 x w\ld only be necess>y to say to him3 ',look & notice well'2 b1 z !re is n "t to see ev]y?+1 & z x is bett] n to see ?an to see wrongly1 x is necess>y = him to make *oice4 ,! fir/ "q1 !re=e1 is h[ he %\ld make ?is *oice4 ,?is "q pres5ts itself z well to ! physici/ z to ! hi/orian2 x pres5ts itself equally to ! ma!matician1 & ! pr9ciples :i* %\ld guide ea* are n )\t analogy4 ,! sci5ti/ con=ms to !m in/9ctively1 & "o c1 reflect+ on ^! pr9ciples1 =etell ! future ( ma!matics4 ,we %all und]/& !m bett] yet if we obs]ve ! sci5ti/ at "w1 & fir/ ( all x is necess>y to "k ! psy*ologic me*anism ( inv5tion &1 9 p>ticul>1 t ( ma!matical creation4 ,obs]vation ( ! processes ( ! "w ( ! ma!matician is p>ticul>ly in/ructive = ! psy*ologi/4 ,9 all ! sci5ces ( obs]vation acc\nt mu/ be tak5 ( ! errors due to ! imp]fections ( our s5ses & our in/rum5ts4 ,luckily1 we may assume t1 "u c]ta9 conditions1 ^! errors are 9 "p self-comp5sat+1 s z to disappe> 9 ! av]age2 ?is comp5sation is due to *ance4 ,b :at is *ance8 ,?is idea is difficult to ju/ify or ev5 to def9e2 & yet :at ,i h j said ab\t ! errors ( obs]vation1 %[s t ! sci5ti/ c n neglect x4 ,x !re=e is necess>y to give a def9ition z precise z possible ( ?is concept1 s indisp5sable yet s illusive4 ,^! are g5]alities applicable 9 sum to all ! sci5ces2 & = example ! me*anism ( ma!matical inv5tion does n diff] s5sibly f ! me*anism ( inv5tion 9 g5]al4 ,lat] ,i attack que/ions relat+ m p>ticul>ly to c]ta9 special sci5ces & fir/ to pure ma!matics4 ,9 ! *apt]s devot$ to ^!1 ,i h to treat subjects a little m ab/ract4 ,i h fir/ to speak ( ! notion ( space2 e "o kn[s space is relative1 or r e "o says s1 b _m ?9k /ill z if !y believ$ x absolute2 x suffices to reflect a little h[ev] to p]ceive to :at contradictions !y are expos$4 ,! que/ions ( tea*+ h _! importance1 fir/ 9 !mselves1 !n because reflect+ on ! be/ way to make new ideas p5etrate virg9 m9ds is at ! same "t reflect+ on h[ ^! notions 7 acquir$ by our ance/ors1 & consequ5tly on _! true orig91 t is to say1 9 reality on _! true nature4 ,:y d *ildr5 usually und]/& no?+ ( ! def9itions :i* satisfy sci5ti/s8 ,:y is x necess>y to give !m o!rs8 ,?is is ! "q ,i set myself 9 ! succe$+ *apt] & ^: solution %\ld1 ,i ?9k1 sugge/ useful reflections to ! philosoph]s occupi$ ) ! logic ( ! sci5ces4 ,on ! o!r h&1 _m geomet]s believe we c r$uce ma!matics to ! rules ( =mal logic4 ,unhe>d-( ef=ts h be5 made to d ?is2 to accompli% x1 "s h n hesitat$1 = example1 to rev]se ! hi/oric ord] ( ! g5esis ( our conceptions & to try to expla9 ! f9ite by ! inf9ite4 ,i believe ,i h succe$$ 9 %[+1 = all ^? :o attack ! problem unprejudic$1 t "h !re is a fallaci\s illusion4 ,i hope ! read] w und]/& ! importance ( ! "q & p>don me ! aridity ( ! pages devot$ to x4 ,! conclud+ *apt]s relative to me*anics & a/ronomy w be easi] to read4 ,me*anics seems on ! po9t ( und]go+ a complete revolution4 ,ideas :i* appe>$ be/ e/abli%$ are assail$ by bold innovators4 ,c]ta9ly x w\ld be premature to decide 9 _! favor at once simply because !y are innovators4 ,b x is ( int]e/ to make kn[n _! doctr9es1 & ?is is :at ,i h tri$ to d4 ,z f> z possible ,i h foll[$ ! hi/oric ord]2 = ! new ideas w\ld seem too a/oni%+ unless we saw h[ !y arose4 ,a/ronomy (f]s u maje/ic spectacles & raises gigantic problems4 ,we c n dream ( apply+ to !m directly ! exp]im5tal me?od2 our laboratories are too small4 ,b analogy ) ph5om5a ^! laboratories p]mit u to atta9 may nev]!less guide ! a/ronom]4 ,! ,milky ,way1 = example1 is an assemblage ( suns ^: movem5ts seem at fir/ caprici\s4 ,b may n ?is assemblage be comp>$ to t ( ! molecules ( a gas1 ^: prop]ties ! k9etic !ory ( gases has made kn[n to u8 ,x is ?us by a r\ndab\t way t ! me?od ( ! physici/ may come to ! aid ( ! a/ronom]4 ,f9ally ,i h endeavor$ to give 9 a few l9es ! hi/ory ( ! developm5t ( ,fr5* geodesy2 ,i h %[n ?r\gh :at p]sev]+ ef=ts1 & (t5 :at dang]s1 ! geodesi/s h procur$ = u ! k we h ( ! figure ( ! e>?4 ,is ?is !n a "q ( me?od8 ,yes1 )\t d\bt1 ?is hi/ory tea*es u 9 fact by :at precautions x is necess>y to surr\nd a s]i\s sci5tific op]ation & h[ mu* "t & pa9s x co/s to conqu] "o new decimal4 ,,book ,i ,,sci5ce ,,& ,,! ,,sci5ti/ ,,*apt] ,i ,,! ,,*oice ,,( ,,facts ,tol/oi some:]e expla9s :y 'sci5ce = its own sake' is 9 8 eyes an absurd conception4 ,we c n "k all facts1 s9ce _! numb] is practically inf9ite4 ,x is necess>y to *oose2 !n we may let ?is *oice dep5d on ! pure caprice ( our curiosity2 w\ld x n be bett] to let ourselves be guid$ by utility1 by our practical & above all by our moral ne$s2 h we no?+ bett] to d ?an to c\nt ! numb] ( lady-bugs on our planet8 ,x is cle> ! ^w utility has n = him ! s5se m5 ( affairs give x1 & foll[+ !m mo/ ( our contempor>ies4 ,little c>es he = indu/rial applications1 = ! m>vels ( electricity or ( automobilism1 :i* he reg>ds r z ob/acles to moral progress2 utility = him is solely :at c make man bett]4 ,= my "p1 x ne$ sc>ce be said1 ,i c\ld nev] be cont5t ) ei!r ! "o or ! o!r ideal2 ,i want nei!r t plutocracy grasp+ & mean1 nor t democracy goody & m$iocre1 occupi$ solely 9 turn+ ! o!r *eek1 :]e w\ld dwell sages )\t curiosity1 :o1 %unn+ excess1 w\ld n die ( disease1 b w\ld surely die ( ennui4 ,b t is a matt] ( ta/e & is n :at ,i wi% to discuss4 ,! "q nev]!less rema9s & %\ld fix our att5tion2 if our *oice c only be det]m9$ by caprice or by imm$iate utility1 !re c be no sci5ce = its own sake1 & consequ5tly no sci5ce4 ,b is t true8 ,t a *oice mu/ be made is inconte/able2 :atev] be our activity1 facts g quick] ?an we1 & we c n cat* !m2 :ile ! sci5ti/ discov]s "o fact1 !re happ5 milli>ds ( milli>ds 9 a cubic millimet] ( 8 body4 ,to wi% to comprise nature 9 sci5ce w\ld be to want to put ! :ole into ! "p4 ,b sci5ti/s believe !re is a hi]>*y ( facts & t among !m may be made a judici\s *oice4 ,!y are "r1 s9ce o!rwise !re w\ld be no sci5ce1 yet sci5ce exi/s4 ,"o ne$ only op5 ! eyes to see t ! conque/s ( indu/ry :i* h enri*$ s _m practical m5 w\ld nev] h se5 ! light1 if ^! practical m5 alone _h exi/$ & if !y _h n be5 prec$$ by unselfi% devotees :o di$ poor1 :o nev] ?\ght ( utility1 & yet _h a guide f> o!r ?an caprice4 ,z ,ma* says1 ^! devotees h sp>$ _! successors ! tr\ble ( ?9k+4 ,^? :o might h work$ solely 9 view ( an imm$iate application w\ld h left no?+ beh9d !m1 &1 9 face ( a new ne$1 all mu/ h be5 begun ov] aga94 ,n[ mo/ m5 d n love to ?9k1 & ?is is p]haps =tunate :5 in/9ct guides !m1 = mo/ (t51 :5 !y pursue an aim :i* is imm$iate & ev] ! same1 in/9ct guides !m bett] ?an reason w\ld guide a pure intellig5ce4 ,b in/9ct is r\t9e1 & if ?\ght did n fecundate x1 x w\ld no m progress 9 man ?an 9 ! bee or ant4 ,x is ne$ful !n to ?9k = ^? :o love n ?9k+1 &1 z !y are num]\s1 x is ne$ful t ea* ( our ?\ghts be z (t5 useful z possible1 & ?is is :y a law w be ! m preci\s ! m g5]al x is4 ,?is %[s u h[ we %\ld *oose3 ! mo/ int]e/+ facts are ^? :i* may s]ve _m times2 ^! are ! facts :i* h a *ance ( com+ up aga94 ,we h be5 s =tunate z to be born 9 a _w :]e !re are su*4 ,suppose t in/ead ( #fj *emical elem5ts !re 7 #fj milli>ds ( !m1 t !y 7 n "s common1 ! o!rs r>e1 b t !y 7 uni=mly di/ribut$4 ,!n1 e "t we pick$ up a new pebble !re w\ld be great probability ( its be+ =m$ ( "s unkn[n sub/ance2 all t we knew ( o!r pebbles w\ld be wor?less = x2 be=e ea* new object we %\ld be z ! new-born babe2 l x we c\ld only obey our caprices or our ne$s4 ,biologi/s w\ld be j z mu* at a loss if !re 7 only individuals & no species & if h]$ity did n make sons l _! fa!rs4 ,9 su* a _w !re w\ld be no sci5ce2 p]haps ?\ght & ev5 life w\ld be impossible1 s9ce evolution c\ld n !re develop ! pres]vational in/9cts4 ,happily x is n s2 l all good =tune to :i* we are accu/om$1 ?is is n appreciat$ at its true wor?4 ,:i* !n are ! facts likely to reappe>8 ,!y are fir/ ! simple facts4 ,x is cle> t 9 a complex fact a ?\s& circum/ances are unit$ by *ance1 & t only a *ance /ill mu* less probable c\ld reunite !m anew4 ,b are !re any simple facts8 ,& if !re are1 h[ recognize !m8 ,:at assurance is !re t a ?+ we ?9k simple does n hide a dreadful complexity8 ,all we c say is t we ought to pref] ! facts :i* seem simple to ^? :]e our crude eye disc]ns unlike elem5ts4 ,& !n "o ( two ?+s3 ei!r ?is simplicity is real1 or else ! elem5ts are s intimately m+l$ z n to be di/+ui%able4 ,9 ! fir/ case !re is *ance ( our meet+ anew ?is same simple fact1 ei!r 9 all its purity or ent]+ itself z elem5t 9 a complex manifold4 ,9 ! second case ?is intimate mixture has likewise m *ances ( recurr+ ?an a het]og5e\s assemblage2 *ance kn[s h[ to mix1 x kn[s n h[ to dis5tangle1 & to make ) multiple elem5ts a well-ord]$ edifice 9 :i* some?+ is di/+ui%able1 x mu/ be made expressly4 ,! facts :i* appe> simple1 ev5 if !y are n s1 w !re=e be m easily reviv$ by *ance4 ,?is x is :i* ju/ifies ! me?od in/9ctively adopt$ by ! sci5ti/1 & :at ju/ifies x /ill bett]1 p]haps1 is t (t-recurr+ facts appe> to u simple1 precisely because we are us$ to !m4 ,b :]e is ! simple fact8 ,sci5ti/s h be5 seek+ x 9 ! two extremes1 9 ! inf9itely great & 9 ! inf9itely small4 ,! a/ronom] has f\nd x because ! di/ances ( ! />s are imm5se1 s great t ea* ( !m appe>s b z a po9t1 s great t ! qualitative diff]5ces are effac$1 & because a po9t is simpl] ?an a body :i* has =m & qualities4 ,! physici/ on ! o!r h& has s\ght ! elem5t>y ph5om5on 9 fictively cutt+ up bodies into inf9itesimal cubes1 because ! conditions ( ! problem1 :i* und]go sl[ & cont9u\s v>iation 9 pass+ f "o po9t ( ! body to ano!r1 may be reg>d$ z con/ant 9 ! int]ior ( ea* ( ^! little cubes4 ,9 ! same way ! biologi/ has be5 in/9ctively l$ to reg>d ! cell z m int]e/+ ?an ! :ole animal1 & ! outcome has %[n 8 wisdom1 s9ce cells belong+ to organisms ! mo/ diff]5t are m alike1 = ! "o :o c recognize _! resemblances1 ?an are ^! organisms !mselves4 ,! sociologi/ is m emb>rass$2 ! elem5ts1 :i* = him are m51 are too unlike1 too v>iable1 too caprici\s1 9 a ^w1 too complex2 besides1 hi/ory nev] beg9s ov] aga94 ,h[ !n *oose ! int]e/+ fact1 :i* is t :i* beg9s aga98 ,me?od is precisely ! *oice ( facts2 x is ne$ful !n to be occupi$ fir/ ) creat+ a me?od1 & _m h be5 imag9$1 s9ce none imposes itself1 s t sociology is ! sci5ce :i* has ! mo/ me?ods & ! fewe/ results4 ,!re=e x is by ! regul> facts t x is prop] to beg92 b aft] ! rule is well e/abli%$1 aft] x is beyond all d\bt1 ! facts 9 full con=mity ) x are erelong )\t int]e/ s9ce !y no long] tea* u any?+ new4 ,x is !n ! exception :i* becomes important4 ,we cease to seek resemblances2 we devote ourselves above all to ! diff]5ces1 & among ! diff]5ces are *os5 fir/ ! mo/ acc5tuat$1 n only because !y are ! mo/ /rik+1 b because !y w be ! mo/ in/ructive4 ,a simple example w make my ?\ght pla9]3 ,suppose "o wi%es to det]m9e a curve by obs]v+ "s ( its po9ts4 ,! practician :o conc]ns himself only ) imm$iate utility w\ld obs]ve only ! po9ts he might ne$ = "s special object4 ,^! po9ts w\ld be badly di/ribut$ on ! curve2 !y w\ld be cr[d$ 9 c]ta9 regions1 r>e 9 o!rs1 s t x w\ld be impossible to jo9 !m by a cont9u\s l9e1 & !y w\ld be unavailable = o!r applications4 ,! sci5ti/ w proce$ diff]5tly2 z he wi%es to /udy ! curve = itself1 he w di/ribute regul>ly ! po9ts to be obs]v$1 & :5 en\gh are kn[n he w jo9 !m by a regul> l9e & !n he w h ! entire curve4 ,b = t h[ does he proce$8 ,if he has det]m9$ an extreme po9t ( ! curve1 he does n /ay ne> ?is extremity1 b goes fir/ to ! o!r end2 aft] ! two extremities ! mo/ in/ructive po9t w be ! mid-po9t1 & s on4 ,s :5 a rule is e/abli%$ we %\ld fir/ seek ! cases :]e ?is rule has ! greate/ *ance ( fail+4 ,!nce1 among o!r reasons1 come ! int]e/ ( a/ronomic facts1 & ! int]e/ ( ! geologic pa/2 by go+ v f> away 9 space or v f> away 9 "t1 we may f9d our usual rules entirely ov]turn$1 & ^! gr& ov]turn+s aid u ! bett] to see or ! bett] to und]/& ! little *anges :i* may happ5 ne>] to u1 9 ! little corn] ( ! _w :]e we are call$ to live & act4 ,we %all bett] "k ?is corn] = hav+ travel$ 9 di/ant c\ntries ) :i* we h no?+ to d4 ,b :at we ought to aim at is less ! asc]ta9m5t ( resemblances & diff]5ces ?an ! recognition ( lik5esses hidd5 "u app>5t div]g5ces4 ,p>ticul> rules seem at fir/ discordant1 b look+ m closely we see 9 g5]al t !y resemble ea* o!r2 diff]5t z to matt]1 !y are alike z to =m1 z to ! ord] ( _! p>ts4 ,:5 we look at !m ) ?is bias1 we %all see !m enl>ge & t5d to embrace ev]y?+4 ,& ?is x is :i* makes ! value ( c]ta9 facts :i* come to complete an assemblage & to %[ t x is ! fai?ful image ( o!r kn[n assemblages4 ,i w n fur!r insi/1 b ^! few words suffice to %[ t ! sci5ti/ does n *oose at r&om ! facts he obs]ves4 ,he does n1 z ,tol/oi says1 c\nt ! lady-bugs1 because1 h[ev] int]e/+ lady-bugs may be1 _! numb] is subject to caprici\s v>iations4 ,he seeks to cond5se mu* exp]i5ce & mu* ?\ght into a sl5d] volume2 & t is :y a little book on physics conta9s s _m pa/ exp]i5ces & a ?\s& times z _m possible exp]i5ces ^: result is kn[n be=eh&4 ,b we h z yet look$ at only "o side ( ! "q4 ,! sci5ti/ does n /udy nature because x is useful2 he /udies x because he delights 9 x1 & he delights 9 x because x is beautiful4 ,if nature 7 n beautiful1 x w\ld n be wor? kn[+1 & if nature 7 n wor? kn[+1 life w\ld n be wor? liv+4 ,( c\rse ,i d n "h speak ( t beauty :i* /rikes ! s5ses1 ! beauty ( qualities & ( appe>ances2 n t ,i und]value su* beauty1 f> f x1 b x has no?+ to d ) sci5ce2 ,i mean t pr(\nd] beauty :i* comes f ! h>moni\s ord] ( ! p>ts & :i* a pure intellig5ce c grasp4 ,?is x is :i* gives body1 a /ructure s to speak1 to ! iridesc5t appe>ances :i* flatt] our s5ses1 & )\t ?is support ! beauty ( ^! fugitive dreams w\ld be only imp]fect1 because x w\ld be vague & always fleet+4 ,on ! contr>y1 intellectual beauty is suffici5t unto itself1 & x is = its sake1 m p]haps ?an = ! future good ( humanity1 t ! sci5ti/ devotes himself to long & difficult labors4 ,x is1 !re=e1 ! que/ ( ?is especial beauty1 ! s5se ( ! h>mony ( ! cosmos1 :i* makes u *oose ! facts mo/ fitt+ to contribute to ?is h>mony1 j z ! arti/ *ooses f among ! features ( 8 model ^? :i* p]fect ! picture & give x "* & life4 ,& we ne$ n fe> t ?is in/9ctive & unav[$ prepossession w turn ! sci5ti/ aside f ! se>* = ! true4 ,"o may dream a h>moni\s _w1 b h[ f> ! real _w w leave x beh9d6 ,! greate/ arti/s t ev] liv$1 ! ,greeks1 made _! heav5s2 h[ %abby x is beside ! true heav5s1 ours6 ,& x is because simplicity1 because gr&eur1 is beautiful1 t we pref]ably seek simple facts1 sublime facts1 t we delight n[ to foll[ ! maje/ic c\rse ( ! />s1 n[ to exam9e ) ! microscope t prodigi\s littl5ess :i* is also a gr&eur1 n[ to seek 9 geologic "t ! traces ( a pa/ :i* attracts because x is f> away4 ,we see too t ! long+ = ! beautiful leads u to ! same *oice z ! long+ = ! useful4 ,& s x is t ?is economy ( ?\ght1 ?is economy ( ef=t1 :i* is1 accord+ to ,ma*1 ! con/ant t5d5cy ( sci5ce1 is at ! same "t a s\rce ( beauty & a practical advantage4 ,! edifices t we admire are ^? :]e ! ar*itect has kn[n h[ to proportion ! means to ! end1 :]e ! columns seem to c>ry gaily1 )\t ef=t1 ! weight plac$ ^u !m1 l ! graci\s c>yatids ( ! ,ere*!um4 ,:5ce comes ?is concordance8 ,is x simply t ! ?+s :i* seem to u beautiful are ^? :i* be/ adapt !mselves to our intellig5ce1 & t consequ5tly !y are at ! same "t ! implem5t ?is intellig5ce kn[s be/ h[ to use8 ,or is !re "h a play ( evolution & natural selection8 ,h ! peoples ^: ideal mo/ con=m$ to _! highe/ int]e/ ext]m9at$ ! o!rs & tak5 _! place8 ,all pursu$ _! ideals )\t ref]5ce to consequ5ces1 b :ile ?is que/ l$ "s to de/ruction1 to o!rs x gave empire4 ,"o is tempt$ to believe x4 ,if ! ,greeks triumph$ ov] ! b>b>ians & if ,europe1 heir ( ,greek ?\ght1 dom9ates ! _w1 x is because ! savages lov$ l\d colors & ! clamor\s tones ( ! drum :i* occupi$ only _! s5ses1 :ile ! ,greeks lov$ ! intellectual beauty :i* hides b5ea? s5su\s beauty1 & ?is intellectual beauty x is :i* makes intellig5ce sure & /rong4 ,d\btless su* a triumph w\ld horrify ,tol/oi1 & he w\ld n l to ackn[l$ge t x might be truly useful4 ,b ?is dis9t]e/$ que/ ( ! true = its own beauty is sane also & able to make man bett]4 ,i well "k t !re are mi/akes1 t ! ?9k] does n always draw !nce ! s]5ity he %\ld f9d !re91 & ev5 t !re are sci5ti/s ( bad "*4 ,mu/ we1 !re=e1 ab&on sci5ce & /udy only morals8 ,:at6 ,d y ?9k ! morali/s !mselves are irreproa*able :5 !y come d[n f _! p$e/al8 ,,*apt] ,,ii ,,! ,,future ,,( ,,ma!matics ,to =esee ! future ( ma!matics1 ! true me?od is to /udy its hi/ory & its pres5t /ate4 ,is ?is n = u ma!maticians 9 a way a pr(essional proc$ure8 ,we are accu/om$ to extrapolate1 :i* is a means ( d$uc+ ! future f ! pa/ & pres5t1 & z we well "k :at ?is am\nts to1 we run no risk ( deceiv+ ourselves ab\t ! range ( ! results x gives u4 ,we h _h hi!rto prophets ( evil4 ,!y bli!ly reit]ate t all problems capable ( solution h already be5 solv$1 & t no?+ is left b glean+4 ,happily ! case ( ! pa/ reassures u4 ,(t5 x 0 ?\ght all problems 7 solv$ or at lea/ an inv5tory 0 made ( all admitt+ solution4 ,& !n ! s5se ( ! ^w solution enl>g$1 ! insoluble problems became ! mo/ int]e/+ ( all1 & o!rs un=ese5 pres5t$ !mselves4 ,= ! ,greeks a good solution 0 "o employ+ only rul] & compasses2 !n x became "o obta9$ by ! extraction ( roots1 !n "o us+ only algebraic or log>i?mic functions4 ,! pessimi/s ?us f\nd !mselves always outflank$1 always =c$ to retreat1 s t at pres5t ,i ?9k !re are no m4 ,my int5tion1 !re=e1 is n to combat !m1 z !y are dead2 we well "k t ma!matics w cont9ue to develop1 b ! "q is h[1 9 :at direction8 ,y w answ]1 '9 e direction1' & t is p>tly true2 b if x 7 :olly true x w\ld be a little appall+4 ,our ri*es w\ld soon become encumb]+ & _! accumulation w\ld produce a m$ley z imp5etrable z ! unkn[n true 0 = ! ignorant4 ,! hi/orian1 ! physici/1 ev51 mu/ make a *oice among facts2 ! head ( ! sci5ti/1 :i* is only a corn] ( ! univ]se1 c\ld nev] conta9 ! univ]se entire2 s t among ! innum]able facts nature (f]s1 "s w be pass$ by1 o!rs reta9$4 ,j s1 a =tiori1 9 ma!matics2 no m c ! geomet] hold fa/ pell-mell all ! facts pres5t+ !mselves to him2 all ! m because he x is1 almo/ ,i _h said 8 caprice1 t creates ^! facts4 ,he con/ructs a :olly new comb9ation by putt+ toge!r its elem5ts2 nature does n 9 g5]al give x to him ready made4 ,d\btless x sometimes happ5s t ! ma!matician und]takes a problem to satisfy a ne$ 9 physics2 t ! physici/ or eng9e] asks him to calculate a numb] = a c]ta9 application4 ,%all x be said t we geomet]s %\ld limit ourselves to await+ ord]s1 &1 9 place ( cultivat+ our sci5ce = our own delectation1 try only to accommodate ourselves to ! wants ( our patrons8 ,if ma!matics has no o!r object besides aid+ ^? :o /udy nature1 x is f ^! we %\ld await ord]s4 ,is ?is way ( look+ at x legitimate8 ,c]ta9ly n2 if we _h n cultivat$ ! exact sci5ces = !mselves1 we %\ld n h creat$ ma!matics ! in/rum5t1 & ! day ! call came f ! physici/ we %\ld h be5 helpless4 ,nor d ! physici/s wait to /udy a ph5om5on until "s urg5t ne$ ( mat]ial life has made x a necessity = !m2 & !y are "r4 ,if ! sci5ti/s ( ! eighte5? c5tury _h neglect$ electricity z be+ 9 _! eyes only a curiosity )\t practical int]e/1 we %\ld h _h 9 ! tw5tie? c5tury nei!r telegraphy1 nor electro-*emi/ry1 nor electro-te*nics4 ,! physici/s1 compell$ to *oose1 are !re=e n guid$ 9 _! *oice solely by utility4 ,h[ !n d !y *oose betwe5 ! facts ( nature8 ,we h expla9$ x 9 ! prec$+ *apt]3 ! facts :i* int]e/ !m are ^? capable ( lead+ to ! discov]y ( a law1 & s !y are analog\s to _m o!r facts :i* d n seem to u isolat$1 b closely gr\p$ ) o!rs4 ,! isolat$ fact attracts all eyes1 ^? ( ! layman z well z ( ! sci5ti/4 ,b :at ! g5u9e physici/ alone kn[s h[ to see1 is ! bond :i* unites _m facts ^: analogy is pr(\nd b hidd54 ,! /ory ( ,newton's apple is probably n true1 b x is symbolic2 let u speak ( x !n z if x 7 true4 ,well !n1 we mu/ believe t be=e ,newton pl5ty ( m5 _h se5 apples fall2 n "o knew h[ to conclude any?+ !refrom4 ,facts w\ld be /]ile 7 !re n m9ds capable ( *oos+ among !m1 disc]n+ ^? beh9d :i* some?+ 0 hidd51 & ( recogniz+ :at is hid+1 m9ds :i* "u ! crude fact p]ceive ! s\l ( ! fact4 ,we f9d j ! same ?+ 9 ma!matics4 ,f ! v>i$ elem5ts at our disposal we c get millions ( diff]5t comb9ations2 b "o ( ^! comb9ations1 9 s f> z x is isolat$1 is absolutely void ( value4 ,(t5 we h tak5 great pa9s to con/ruct x1 b x s]ves no purpose1 if n p]haps to furni% a task 9 second>y education4 ,q o!rwise w x be :5 ?is comb9ation %all f9d place 9 a class ( analog\s comb9ations & we %all h notic$ ?is analogy4 ,we are no long] 9 ! pres5ce ( a fact1 b ( a law4 ,& ^u t day ! real discov]] w n be ! workman :o %all h pati5tly built up c]ta9 ( ^! comb9ations2 x w be he :o br+s to light _! k9%ip4 ,! fir/ w h se5 m]ely ! crude fact1 only ! o!r w h p]ceiv$ ! s\l ( ! fact4 ,(t5 to fix ?is k9%ip x suffices him to make a new ^w1 & ?is ^w is creative4 ,! hi/ory ( sci5ce furni%es u a cr[d ( examples famili> to all4 ,! celebrat$ ,vi5na philosoph] ,ma* has said t ! rle ( sci5ce is to produce economy ( ?\ght1 j z ma*9es produce economy ( ef=t4 ,& t is v true4 ,! savage reckons on 8 f+]s or by heap+ pebbles4 ,9 tea*+ *ildr5 ! multiplication table we sp>e !m lat] innum]able pebble bun*+s4 ,"s "o has already f\nd out1 ) pebbles or o!rwise1 t #f times #g is #db & has _h ! idea ( not+ ! result1 & s we ne$ n d x ov] aga94 ,he did n wa/e 8 "t ev5 if he reckon$ = pleasure3 8 op]ation took him only two m9utes2 x w\ld h tak5 9 all two milli>ds if a milli>d m5 _h _h to d x ov] aft] him4 ,! importance ( a fact !n is measur$ by its yield1 t is to say1 by ! am\nt ( ?\ght x p]mits u to sp>e4 ,9 physics ! facts ( great yield are ^? ent]+ into a v g5]al law1 s9ce f x !y enable u to =esee a great numb] ( o!rs1 & j s x is 9 ma!matics4 ,suppose ,i h und]tak5 a complicat$ calculation & labori\sly rea*$ a result3 ,i %all n be comp5sat$ = my tr\ble if !reby ,i h n become capable ( =esee+ ! results ( o!r analog\s calculations & guid+ !m ) a c]ta9ty t avoids ! grop+s to :i* "o mu/ be resign$ 9 a fir/ attempt4 ,on ! o!r h&1 ,i %all n h wa/$ my "t if ^! grop+s !mselves h end$ by reveal+ to me ! pr(\nd analogy ( ! problem j treat$ ) a mu* m ext5d$ class ( o!r problems2 if !y h %[n me at once ! resemblances & diff]5ces ( ^!1 if 9 a ^w !y h made me p]ceive ! possibility ( a g5]alization4 ,!n x is n a new result ,i h won1 x is a new p[]4 ,! simple example t comes fir/ to m9d is t ( an algebraic =mula :i* gives u ! solution ( a type ( num]ic problems :5 f9ally we replace ! lett]s by numb]s4 ,?anks to x1 a s+le algebraic calculation saves u ! pa9s ( ceaselessly beg9n+ ov] aga9 new num]ic calculations4 ,b ?is is only a crude example2 we all "k !re are analogies inexpressible by a =mula & all ! m preci\s4 ,a new result is ( value1 if at all1 :5 9 unify+ elem5ts long kn[n b hi!rto sep>ate & seem+ /rang]s "o to ano!r x sudd5ly introduces ord] :]e app>5tly disord] reign$4 ,x !n p]mits u to see at a glance ea* ( ^! elem5ts & its place 9 ! assemblage4 ,?is new fact is n m]ely preci\s by itself1 b x alone gives value to all ! old facts x comb9es4 ,our m9d is weak z are ! s5ses2 x w\ld lose itself 9 ! _w's complexity 7 ?is complexity n h>moni\s2 l a ne>-sight$ p]son1 x w\ld see only ! details & w\ld be =c$ to =get ea* ( ^! details be=e exam9+ ! foll[+1 s9ce x w\ld be incapable ( embrac+ all4 ,! only facts wor?y our att5tion are ^? :i* introduce ord] into ?is complexity & s make x accessible4 ,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 results2 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 ,aga9 ! comp>ison ( ! ,ere*!um comes to my m9d1 b ,i mu/ n use x too (t54 ,x is = ! same reason t1 :5 a r long calculation has l$ to "s simple & /rik+ result1 we are n satisfi$ until we h %[n t we %\ld h be5 able to =esee1 if n ?is entire result1 at lea/ its mo/ *>act]i/ic traits4 ,:y8 ,:at prev5ts our be+ cont5t ) a calculation :i* has told u1 x seems1 all we wi%$ to "k8 ,x is because1 9 analog\s cases1 ! long calculation might n aga9 avail1 & t ?is is n s ab\t ! reason+ (t5 half intuitive :i* w\ld h enabl$ u to =esee4 ,?is reason+ be+ %ort1 we see at a s+le glance all its p>ts1 s t we imm$iately p]ceive :at mu/ be *ang$ to adapt x to all ! problems ( ! same nature :i* c occur4 ,& !n x enables u to =esee if ! solution ( ^! problems w be simple1 x %[s u at lea/ if ! calculation is wor? und]tak+4 ,:at we h j said suffices to %[ h[ va9 x w\ld be to seek to replace by any me*anical proc$ure ! free initiative ( ! ma!matician4 ,to obta9 a result ( real value1 x is n en\gh to gr9d out calculations1 or to h a ma*9e to put ?+s 9 ord]2 x is n ord] alone1 x is unexpect$ ord]1 :i* is wor? :ile4 ,! ma*9e may gnaw on ! crude fact1 ! s\l ( ! fact w always escape x4 ,s9ce ! middle ( ! la/ c5tury1 ma!maticians are m & m desir\s ( atta9+ absolute rigor2 !y are "r1 & ?is t5d5cy w be m & m acc5tuat$4 ,9 ma!matics rigor is n ev]y?+1 b )\t x !re is no?+4 ,a demon/ration :i* is n rigor\s is no?+ness4 ,i ?9k no "o w conte/ ?is tru?4 ,b if x 7 tak5 too lit]ally1 we %\ld be l$ to conclude t be=e #ahbj1 = example1 !re 0 no ma!matics2 ?is w\ld be manife/ly excessive2 ! geomet]s ( t "t und]/ood volunt>ily :at we expla9 by prolix disc\rse4 ,?is does n mean t !y did n see x at all2 b !y pass$ ov] x too rapidly1 & to see x well w\ld h necessitat$ tak+ ! pa9s to say x4 ,b is x always ne$ful to say x s _m times8 ,^? :o 7 ! fir/ to emphasize exactness be=e all else h giv5 u argum5ts t we may try to imitate2 b if ! demon/rations ( ! future are to be built on ?is model1 ma!matical treatises w be v long2 & if ,i fe> ! l5g!n+s1 x is n solely because ,i deprecate encumb]+ libr>ies1 b because ,i fe> t 9 be+ l5g!n$ out1 our demon/rations may lose t appe>ance ( h>mony ^: usefulness ,i h j expla9$4 ,! economy ( ?\ght is :at we %\ld aim at1 s x is n en\gh to supply models = imitation4 ,x is ne$ful = ^? aft] u to be able to disp5se ) ^! models &1 9 place ( repeat+ an argum5t already made1 summ>ize x 9 a few words4 ,& ?is has already be5 atta9$ at times4 ,= in/ance1 !re 0 a type ( reason+ f\nd ev]y:]e1 & ev]y:]e alike4 ,!y 7 p]fectly exact b long4 ,!n all at once ! phrase 'uni=mity ( conv]g5ce' 0 hit ^u & ?is phrase made ^? argum5ts ne$less2 we 7 no long] call$ ^u to repeat !m1 s9ce !y c\ld be und]/ood4 ,^? :o conqu] difficulties !n d u a d\ble s]vice3 fir/ !y tea* u to d z !y at ne$1 b above all !y enable u z (t5 z possible to avoid do+ z !y1 yet )\t sacrifice ( exactness4 ,we h j se5 by "o example ! importance ( words 9 ma!matics1 b _m o!rs c\ld be cit$4 ,x is h>d to believe h[ mu* a well-*os5 ^w c economize ?\ght1 z ,ma* says4 ,p]haps ,i h already said some:]e t ma!matics is ! art ( giv+ ! same "n to diff]5t ?+s4 ,x is prop] t ^! ?+s1 diff]+ 9 matt]1 be alike 9 =m1 t !y may1 s to speak1 run 9 ! same mold4 ,:5 ! language has be5 well *os51 we are a/oni%$ to see t all ! pro(s made = a c]ta9 object apply imm$iately to _m new objects2 !re is no?+ to *ange1 n ev5 ! words1 s9ce ! names h become ! same4 ,a well-*os5 ^w usually suffices to d away ) ! exceptions f :i* ! rules /at$ 9 ! old way suff]2 ?is is :y we h creat$ negative quantities1 imag9>ies1 po9ts at inf9ity1 & :at n4 ,& exceptions1 we mu/ n =get1 are p]nici\s because !y hide ! laws4 ,well1 ?is is "o ( ! *>act]i/ics by :i* we recognize ! facts :i* yield great results4 ,!y are ^? :i* all[ ( ^! happy innovations ( language4 ,! crude fact !n is (t5 ( no great int]e/2 we may po9t x out _m times )\t hav+ r5d]$ great s]vice to sci5ce4 ,x takes value only :5 a wis] ?9k] p]ceives ! relation = :i* x /&s1 & symbolizes x by a ^w4 ,moreov] ! physici/s d j ! same4 ,!y h inv5t$ ! ^w 'en]gy1' & ?is ^w has be5 prodigi\sly fruitful1 because x also made ! law by elim9at+ ! exceptions1 s9ce x gave ! same "n to ?+s diff]+ 9 matt] & l 9 =m4 ,among words t h _h ! mo/ =tunate influ5ce ,i w\ld select 'gr\p' & 'inv>iant4' ,!y h made u see ! ess5ce ( _m ma!matical reason+s2 !y h %[n u 9 h[ _m cases ! old ma!maticians consid]$ gr\ps )\t kn[+ x1 & h[1 believ+ !mselves f> f "o ano!r1 !y sudd5ly f\nd !mselves ne> )\t kn[+ :y4 ,to-day we %\ld say t !y _h dealt ) isomorphic gr\ps4 ,we n[ "k t 9 a gr\p ! matt] is ( little int]e/1 ! =m alone c\nts1 & t :5 we "k a gr\p we ?us "k all ! isomorphic gr\ps2 & ?anks to ^! words 'gr\p' & 'isomorphism1' :i* cond5se 9 a few syllables ?is subtile rule & quickly make x famili> to all m9ds1 ! transition is imm$iate & c be done ) e economy ( ?\ght ef=t4 ,! idea ( gr\p besides atta*es to t ( trans=mation4 ,:y d we put su* a value on ! inv5tion ( a new trans=mation8 ,because f a s+le !orem x enables u to get t5 or tw5ty2 x has ! same value z a z]o adjo9$ to ! "r ( a :ole numb]4 ,?is !n x is :i* has hi!rto det]m9$ ! direction ( ma!matical advance1 & j z c]ta9ly w det]m9e x 9 ! future4 ,b to ?is end ! nature ( ! problems :i* come up contributes equally4 ,we c n =get :at mu/ be our aim4 ,9 my op9ion ?is aim is d\ble4 ,our sci5ce bord]s ^u bo? philosophy & physics1 & we "w = our two neighbors2 s we h always se5 & %all /ill see ma!maticians advanc+ 9 two opposite directions4 ,on ! "o h&1 ma!matical sci5ce mu/ reflect ^u itself1 & t is useful s9ce reflect+ on itself is reflect+ on ! human m9d :i* has creat$ x1 all ! m because x is ! v "o ( its creations = :i* x has borr[$ lea/ f )\t4 ,?is is :y c]ta9 ma!matical speculations are useful1 su* z ^? devot$ to ! /udy ( ! po/ulates1 ( unusual geometries1 ( peculi> functions4 ,! m ^! speculations div]ge f ord9>y conceptions1 & consequ5tly f nature & applications1 ! bett] !y %[ u :at ! human m9d c create :5 x frees itself m & m f ! tyranny ( ! ext]nal _w1 ! bett] !re=e !y let u "k x 9 itself4 ,b x is t[>d ! o!r side1 ! side ( nature1 t we mu/ direct ! bulk ( our army4 ,!re we meet ! physici/ or ! eng9e]1 :o says to u3 ,please integrate ?is diff]5tial equation = me2 ,i might ne$ x 9 a week 9 view ( a con/ruction :i* %\ld be f9i%$ by t "t4 ,?is equation1 we answ]1 does n come "u "o ( ! integrable types2 y "k !re are n _m4 ,yes1 ,i "k2 b !n :at good are y8 ,usually to und]/& ea* o!r is en\gh2 ! eng9e] 9 reality does n ne$ ! integral 9 f9ite t]ms2 he ne$s to "k ! g5]al look ( ! integral function1 or he simply wants a c]ta9 numb] :i* c\ld readily be d$uc$ f ?is integral if x 7 kn[n4 ,usually x is n kn[n1 b ! numb] c be calculat$ )\t x if we "k exactly :at numb] ! eng9e] ne$s & ) :at approximation4 ,=m]ly an equation 0 consid]$ solv$ only :5 its solution _h be5 express$ by aid ( a f9ite numb] ( kn[n functions2 b t is possible sc>cely once 9 a hundr$ times4 ,:at we always c d1 or r :at we %\ld always seek to d1 is to solve ! problem qualitatively s to speak2 t is to say1 seek to "k ! g5]al =m ( ! curve :i* repres5ts ! unkn[n function4 ,x rema9s to f9d ! quantitative solution ( ! problem2 b if ! unkn[n c n be det]m9$ by a f9ite calculation1 x may always be repres5t$ by a conv]g5t inf9ite s]ies :i* enables u to calculate x4 ,c t be reg>d$ z a true solution8 ,we are told t ,newton s5t ,leibnitz an anagram almo/ l ?is3 aaaaabbbeeeeij1 etc4 ,leibnitz naturally und]/ood no?+ at all ( x2 b we1 :o h ! key1 "k t ?is anagram meant1 translat$ into mod]n t]ms3 ,i c integrate all diff]5tial equations2 & we are tempt$ to say t ,newton _h ei!r great luck or /range delusions4 ,he m]ely wi%$ to say he c\ld =m 7by ! me?od ( indet]m9ate coeffici5ts7 a s]ies ( p[]s =mally satisfy+ ! propos$ equation4 ,su* a solution w\ld n satisfy u to-day1 & = two reasons3 because ! conv]g5ce is too sl[ & because ! t]ms foll[ ea* o!r )\t obey+ any law4 ,on ! contr>y1 ! s]ies ,!ta seems to u to leave no?+ to be desir$1 fir/ because x conv]ges v quickly 7?is is = ! practical man :o wi%es to get at a numb] z quickly z possible7 & next because we see at a glance ! law ( ! t]ms 7?is is to satisfy ! e/hetic ne$ ( ! !ori/74 ,b !n !re are no long] solv$ problems & o!rs :i* are n2 !re are only problems m or less solv$1 accord+ z !y are solv$ by a s]ies conv]g+ m or less rapidly1 or rul$ by a law m or less h>moni\s4 ,x (t5 happ5s h[ev] t an imp]fect solution guides u t[>d a bett] "o4 ,sometimes ! s]ies conv]ges s sl[ly t ! computation is impracticable & we h only succe$$ 9 prov+ ! possibility ( ! problem4 ,& !n ! eng9e] f9ds ?is a mock]y1 & ju/ly1 s9ce x w n aid him to complete 8 con/ruction by ! date fix$4 ,he little c>es to "k if x w b5efit eng9e]s ( ! tw5ty-second c5tury4 ,b z = u1 we ?9k diff]5tly & we are sometimes happi] to h sp>$ our gr&*ildr5 a day's "w ?an to h sav$ our contempor>ies an h\r4 ,sometimes by grop+1 empirically1 s to speak1 we rea* a =mula suffici5tly conv]g5t4 ,:at m d y want8 says ! eng9e]4 ,& yet1 9 spite ( all1 we are n satisfi$2 we %\ld h lik$ to =esee t conv]g5ce4 ,:y8 ,because if we _h kn[n h[ to =esee x once1 we w\ld "k h[ to =esee x ano!r "t4 ,we h succe$$2 t is a small matt] 9 our eyes if we c n validly expect to d s aga94 ,9 proportion z sci5ce develops1 its total compreh5sion becomes m difficult2 !n we seek to cut x 9 pieces & to be satisfi$ ) "o ( ^! pieces3 9 a ^w1 to specialize4 ,if we w5t on 9 ?is way1 x w\ld be a griev\s ob/acle to ! progress ( sci5ce4 ,z we h said1 x is by unexpect$ union betwe5 its div]se p>ts t x progresses4 ,to specialize too mu* w\ld be to =bid ^! draw+s toge!r4 ,x is to be hop$ t congresses l ^? ( ,heidelb]g & ,rome1 by putt+ u 9 t\* ) "o ano!r1 w op5 = u vi/as ov] neighbor+ doma9s & oblige u to comp>e !m ) our own1 to range some:at abroad f our own little village2 ?us !y w be ! be/ rem$y = ! dang] j m5tion$4 ,b ,i h l+]$ too long ov] g5]alities2 x is "t to ent] into detail4 ,let u pass 9 review ! v>i\s special sci5ces :i* comb9$ make ma!matics2 let u see :at ea* has accompli%$1 :i!r x t5ds & :at we may hope f x4 ,if ! prec$+ views are correct1 we %\ld see t ! greate/ advances 9 ! pa/ h happ5$ :5 two ( ^! sci5ces h unit$1 :5 we h become consci\s ( ! simil>ity ( _! =m1 despite ! diff]5ce ( _! matt]1 :5 !y h s model$ !mselves ^u ea* o!r t ea* c\ld pr(it by ! o!r's conque/s4 ,we %\ld at ! same "t =esee 9 comb9ations ( ! same sort ! progress ( ! future4 ,,ari?metic ,progress 9 ari?metic has be5 mu* sl[] ?an 9 algebra & analysis1 & x is easy to see :y4 ,! feel+ ( cont9uity is a preci\s guide :i* ! ari?metician lacks2 ea* :ole numb] is sep>at$ f ! o!rs--x has1 s to speak1 its own individuality4 ,ea* ( !m is a sort ( exception & ?is is :y g5]al !orems are r>] 9 ! !ory ( numb]s2 ?is is also :y ^? :i* exi/ are m hidd5 & long] elude ! se>*]s4 ,if ari?metic is beh9d algebra & analysis1 ! be/ ?+ = x to d is to seek to model itself ^u ^! sci5ces s z to pr(it by _! advance4 ,! ari?metician ought !re=e to take z guide ! analogies ) algebra4 ,^! analogies are num]\s & if1 9 _m cases1 !y h n yet be5 /udi$ suffici5tly closely to become utilizable1 !y at lea/ h long be5 =ese51 & ev5 ! language ( ! two sci5ces %[s !y h be5 recogniz$4 ,?us we speak ( transc5d5t numb]s & ?us we acc\nt = ! future classification ( ^! numb]s already hav+ z model ! classification ( transc5d5t functions1 & /ill we d n z yet v well see h[ to pass f "o classification to ! o!r2 b _h x be5 se51 x w\ld already h be5 accompli%$ & w\ld no long] be ! "w ( ! future4 ,! fir/ example t comes to my m9d is ! !ory ( congru5ces1 :]e is f\nd a p]fect p>allelism to ! !ory ( algebraic equations4 ,surely we %all succe$ 9 complet+ ?is p>allelism1 :i* mu/ hold = in/ance betwe5 ! !ory ( algebraic curves & t ( congru5ces ) two v>iables4 ,& :5 ! problems relative to congru5ces ) sev]al v>iables %all be solv$1 ?is w be a fir/ /ep t[>d ! solution ( _m que/ions ( indet]m9ate analysis4 ,,algebra ,! !ory ( algebraic equations w /ill long hold ! att5tion ( geomet]s2 num]\s & v diff]5t are ! sides :5ce x may be attack$4 ,we ne$ n ?9k algebra is end$ because x gives u rules to =m all possible comb9ations2 x rema9s to f9d ! int]e/+ comb9ations1 ^? :i* satisfy su* & su* a condition4 ,?us w be =m$ a sort ( indet]m9ate analysis :]e ! unkn[ns w no long] be :ole numb]s1 b polynomials4 ,?is "t x is algebra :i* w model itself ^u ari?metic1 foll[+ ! analogy ( ! :ole numb] to ! integral polynomial ) any coeffici5ts or to ! integral polynomial ) integral coeffici5ts4 ,,geometry ,x looks z if geometry c\ld conta9 no?+ :i* is n already includ$ 9 algebra or analysis2 t geometric facts are only algebraic or analytic facts express$ 9 ano!r language4 ,x might !n be ?\ght t aft] our review !re w\ld rema9 no?+ m = u to say relat+ specially to geometry4 ,?is w\ld be to fail to recognize ! importance ( well-con/ruct$ language1 n to compreh5d :at is add$ to ! ?+s !mselves by ! me?od ( express+ ^! ?+s & consequ5tly ( gr\p+ !m4 ,fir/ ! geometric consid]ations lead u to set ourselves new problems2 ^! may be1 if y *oose1 analytic problems1 b su* z we nev] w\ld h set ourselves 9 connection ) analysis4 ,analysis pr(its by !m h[ev]1 z x pr(its by ^? x has to solve to satisfy ! ne$s ( physics4 ,a great advantage ( geometry lies 9 ! fact t 9 x ! s5ses c come to ! aid ( ?\ght1 & help f9d ! pa? to foll[1 & _m m9ds pref] to put ! problems ( analysis into geometric =m4 ,unhappily our s5ses c n c>ry u v f>1 & !y des]t u :5 we wi% to so> beyond ! classical ?ree dim5sions4 ,does ?is mean t1 beyond ! re/rict$ doma9 :]e9 !y seem to wi% to imprison u1 we %\ld rely only on pure analysis & t all geometry ( m ?an ?ree dim5sions is va9 & objectless8 ,! greate/ ma/]s ( a prec$+ g5]ation w\ld h answ]$ 'yes'2 to-day we are s famili>iz$ ) ?is notion t we c speak ( x1 ev5 9 a univ]sity c\rse1 )\t ar\s+ too mu* a/oni%m5t4 ,b :at good is x8 ,t is easy to see3 ,fir/ x gives u a v conv5i5t t]m9ology1 :i* expresses concisely :at ! ord9>y analytic language w\ld say 9 prolix phrases4 ,moreov]1 ?is language makes u call l ?+s by ! same "n & emphasize analogies x w nev] aga9 let u =get4 ,x enables u !re=e /ill to f9d our way 9 ?is space :i* is too big = u & :i* we c n see1 always recall+ visible space1 :i* is only an imp]fect image ( x d\btless1 b :i* is nev]!less an image4 ,"h aga91 z 9 all ! prec$+ examples1 x is analogy ) ! simple :i* enables u to compreh5d ! complex4 ,?is geometry ( m ?an ?ree dim5sions is n a simple analytic geometry2 x is n purely quantitative1 b qualitative also1 & x is 9 ?is respect above all t x becomes int]e/+4 ,!re is a sci5ce call$ analysis situs & :i* has = its object ! /udy ( ! positional relations ( ! diff]5t elem5ts ( a figure1 ap>t f _! sizes4 ,?is geometry is purely qualitative2 its !orems w\ld rema9 true if ! figures1 in/ead ( be+ exact1 7 r\ghly imitat$ by a *ild4 ,we may also make an analysis situs ( m ?an ?ree dim5sions4 ,! importance ( analysis situs is enorm\s & c n be too mu* emphasiz$2 ! advantage obta9$ f x by ,riemann1 "o ( its *ief creators1 w\ld suffice to prove ?is4 ,we mu/ a*ieve its complete con/ruction 9 ! high] spaces2 !n we %all h an in/rum5t :i* w enable u really to see 9 hyp]space & supplem5t our s5ses4 ,! problems ( analysis situs w\ld p]haps n h sugge/$ !mselves if ! analytic language alone _h be5 spok52 or r1 ,i am mi/ak51 !y w\ld h occurr$ surely1 s9ce _! solution is ess5tial to a cr[d ( que/ions 9 analysis1 b !y w\ld h come s+ly1 "o aft] ano!r1 & )\t our be+ able to p]ceive _! common bond4 ,,cantorism ,i h spok5 above ( our ne$ to g back cont9ually to ! fir/ pr9ciples ( our sci5ce1 & ( ! advantage ( ?is = ! /udy ( ! human m9d4 ,?is ne$ has inspir$ two endeavors :i* h tak5 a v prom95t place 9 ! mo/ rec5t annals ( ma!matics4 ,! fir/ is ,cantorism1 :i* has r5d]$ our sci5ce su* conspicu\s s]vice4 ,cantor introduc$ into sci5ce a new way ( consid]+ ma!matical inf9ity4 ,"o ( ! *>act]i/ic traits ( ,cantorism is t 9 place ( go+ up to ! g5]al by build+ up con/ructions m & m complicat$ & def9+ by con/ruction1 x />ts f ! g5us supremum & def9es only1 z ! s*ola/ics w\ld h said1 p] g5us proximum et diff]5tiam specificam4 ,!nce comes ! horror x has sometimes inspir$ 9 c]ta9 m9ds1 = in/ance 9 ,h]mite1 ^: favorite idea 0 to comp>e ! ma!matical to ! natural sci5ces4 ,) mo/ ( u ^! prejudices h be5 dissipat$1 b x has come to pass t we h enc\nt]$ c]ta9 p>adoxes1 c]ta9 app>5t contradictions t w\ld h delight$ ,z5o1 ! ,eleatic & ! s*ool ( ,meg>a4 ,& !n ea* mu/ seek ! rem$y4 ,= my "p1 ,i ?9k1 & ,i am n ! only "o1 t ! important ?+ is nev] to introduce entities n completely def9able 9 a f9ite numb] ( words4 ,:atev] be ! cure adopt$1 we may promise ourselves ! joy ( ! doctor call$ 9 to foll[ a beautiful pa?ologic case4 ,,! ,,inve/igation ,,( ,,! ,,po/ulates ,on ! o!r h&1 ef=ts h be5 made to enum]ate ! axioms & po/ulates1 m or less hidd51 :i* s]ve z f\ndation to ! diff]5t !ories ( ma!matics4 ,pr(essor ,hilb]t has obta9$ ! mo/ brilliant results4 ,x seems at fir/ t ?is doma9 w\ld be v re/rict$ & !re w\ld be no?+ m to d :5 ! inv5tory %\ld be end$1 :i* c\ld n take long4 ,b :5 we %all h enum]at$ all1 !re w be _m ways ( classify+ all2 a good libr>ian always f9ds some?+ to d1 & ea* new classification w be in/ructive = ! philosoph]4 ,"h ,i end ?is review :i* ,i c\ld n dream ( mak+ complete4 ,i ?9k ^! examples w suffice to %[ by :at me*anism ! ma!matical sci5ces h made _! progress 9 ! pa/ & 9 :at direction !y mu/ advance 9 ! future4 ,,*apt] ,,iii ,,ma!matical ,,creation ,! g5esis ( ma!matical creation is a problem :i* %\ld int5sely int]e/ ! psy*ologi/4 ,x is ! activity 9 :i* ! human m9d seems to take lea/ f ! outside _w1 9 :i* x acts or seems to act only ( itself & on itself1 s t 9 /udy+ ! proc$ure ( geometric ?\ght we may hope to rea* :at is mo/ ess5tial 9 man's m9d4 ,?is has long be5 appreciat$1 & "s "t back ! j\rnal call$ ,l'enseignem5t ma?matique1 edit$ by ,laisant & ,fehr1 began an inve/igation ( ! m5tal habits & me?ods ( "w ( diff]5t ma!maticians4 ,i _h f9i%$ ! ma9 outl9es ( ?is article :5 ! results ( t inquiry 7 publi%$1 s ,i h h>dly be5 able to utilize !m & %all conf9e myself to say+ t ! majority ( witnesses confirm my conclusions2 ,i d n say all1 = :5 ! appeal is to univ]sal suffrage unanimity is n to be hop$4 ,a fir/ fact %\ld surprise u1 or r w\ld surprise u if we 7 n s us$ to x4 ,h[ does x happ5 !re are p :o d n und]/& ma!matics8 ,if ma!matics invokes only ! rules ( logic1 su* z are accept$ by all normal m9ds2 if its evid5ce is bas$ on pr9ciples common to all m51 & t none c\ld d5y )\t be+ mad1 h[ does x come ab\t t s _m p]sons are "h refractory8 ,t n e "o c inv5t is n[ise my/]i\s4 ,t n e "o c reta9 a demon/ration once le>n$ may also pass4 ,b t n e "o c und]/& ma!matical reason+ :5 expla9$ appe>s v surpris+ :5 we ?9k ( x4 ,& yet ^? :o c foll[ ?is reason+ only ) difficulty are 9 ! majority3 t is und5iable1 & w surely n be ga9said by ! exp]i5ce ( second>y-s*ool tea*]s4 ,& fur!r3 h[ is error possible 9 ma!matics8 ,a sane m9d %\ld n be guilty ( a logical fallacy1 & yet !re are v f9e m9ds :o d n trip 9 brief reason+ su* z occurs 9 ! ord9>y do+s ( life1 & :o are incapable ( foll[+ or repeat+ )\t error ! ma!matical demon/rations :i* are long]1 b :i* aft] all are only an accumulation ( brief reason+s :olly analog\s to ^? !y make s easily4 ,ne$ we add t ma!maticians !mselves are n infallible8 ,! answ] seems to me evid5t4 ,imag9e a long s]ies ( syllogisms1 & t ! conclusions ( ! fir/ s]ve z premises ( ! foll[+3 we %all be able to cat* ea* ( ^! syllogisms1 & x is n 9 pass+ f premises to conclusion t we are 9 dang] ( deceiv+ ourselves4 ,b betwe5 ! mom5t 9 :i* we fir/ meet a proposition z conclusion ( "o syllogism1 & t 9 :i* we re5c\nt] x z premise ( ano!r syllogism occasionally "s "t w elapse1 sev]al l9ks ( ! *a9 w h unroll$2 s x may happ5 t we h =gott5 x1 or worse1 t we h =gott5 its mean+4 ,s x may happ5 t we replace x by a slightly diff]5t proposition1 or t1 :ile reta9+ ! same enunciation1 we attribute to x a slightly diff]5t mean+1 & ?us x is t we are expos$ to error4 ,(t5 ! ma!matician uses a rule4 ,naturally he beg9s by demon/rat+ ?is rule2 & at ! "t :5 ?is pro( is fre% 9 8 memory he und]/&s p]fectly its mean+ & its be>+1 & he is 9 no dang] ( *ang+ x4 ,b subsequ5tly he tru/s 8 memory & aft]w>d only applies x 9 a me*anical way2 & !n if 8 memory fails him1 he may apply x all wrong4 ,?us x is1 to take a simple example1 t we sometimes make slips 9 calculation because we h =gott5 our multiplication table4 ,accord+ to ?is1 ! special aptitude = ma!matics w\ld be due only to a v sure memory or to a prodigi\s =ce ( att5tion4 ,x w\ld be a p[] l t ( ! :i/-play] :o rememb]s ! c>ds play$2 or1 to g up a /ep1 l t ( ! *ess-play] :o c visualize a great numb] ( comb9ations & hold !m 9 8 memory4 ,e good ma!matician ought to be a good *ess-play]1 & inv]sely2 likewise he %\ld be a good comput]4 ,( c\rse t sometimes happ5s2 ?us ,gauss 0 at ! same "t a geomet] ( g5ius & a v precoci\s & accurate comput]4 ,b !re are exceptions2 or r ,i err2 ,i c n call !m exceptions )\t ! exceptions be+ m ?an ! rule4 ,gauss x is1 on ! contr>y1 :o 0 an exception4 ,z = myself1 ,i mu/ confess1 ,i am absolutely incapable ev5 ( add+ )\t mi/akes4 ,9 ! same way ,i %\ld be b a poor *ess-play]2 ,i w\ld p]ceive t by a c]ta9 play ,i %\ld expose myself to a c]ta9 dang]2 ,i w\ld pass 9 review sev]al o!r plays1 reject+ !m = o!r reasons1 & !n f9ally ,i %\ld make ! move fir/ exam9$1 hav+ meantime =gott5 ! dang] ,i _h =ese54 ,9 a ^w1 my memory is n bad1 b x w\ld be insuffici5t to make me a good *ess-play]4 ,:y !n does x n fail me 9 a difficult piece ( ma!matical reason+ :]e mo/ *ess-play]s w\ld lose !mselves8 ,evid5tly because x is guid$ by ! g5]al m>* ( ! reason+4 ,a ma!matical demon/ration is n a simple juxtaposition ( syllogisms1 x is syllogisms plac$ 9 a c]ta9 ord]1 & ! ord] 9 :i* ^! elem5ts are plac$ is mu* m important ?an ! elem5ts !mselves4 ,if ,i h ! feel+1 ! intuition1 s to speak1 ( ?is ord]1 s z to p]ceive at a glance ! reason+ z a :ole1 ,i ne$ no long] fe> le/ ,i =get "o ( ! elem5ts1 = ea* ( !m w take its allott$ place 9 ! array1 & t )\t any ef=t ( memory on my "p4 ,x seems to me !n1 9 repeat+ a reason+ le>n$1 t ,i c\ld h inv5t$ x4 ,?is is (t5 only an illusion2 b ev5 !n1 ev5 if ,i am n s gift$ z to create x by myself1 ,i myself re-inv5t x 9 s f> z ,i repeat x4 ,we "k t ?is feel+1 ?is intuition ( ma!matical ord]1 t makes u div9e hidd5 h>monies & relations1 c n be possess$ by e "o4 ,"s w n h ei!r ?is delicate feel+ s difficult to def9e1 or a /r5g? ( memory & att5tion beyond ! ord9>y1 & !n !y w be absolutely incapable ( und]/&+ high] ma!matics4 ,su* are ! majority4 ,o!rs w h ?is feel+ only 9 a slight degree1 b !y w be gift$ ) an uncommon memory & a great p[] ( att5tion4 ,!y w le>n by he>t ! details "o aft] ano!r2 !y c und]/& ma!matics & sometimes make applications1 b !y _c create4 ,o!rs1 f9ally1 w possess 9 a less or great] degree ! special intuition ref]r$ to1 & !n n only c !y und]/& ma!matics ev5 if _! memory is no?+ extraord9>y1 b !y may become creators & try to inv5t ) m or less success accord+ z ?is intuition is m or less develop$ 9 !m4 ,9 fact1 :at is ma!matical creation8 ,x does n consi/ 9 mak+ new comb9ations ) ma!matical entities already kn[n4 ,any "o c\ld d t1 b ! comb9ations s made w\ld be inf9ite 9 numb] & mo/ ( !m absolutely )\t int]e/4 ,to create consi/s precisely 9 n mak+ useless comb9ations & 9 mak+ ^? :i* are useful & :i* are only a small m9ority4 ,inv5tion is disc]nm5t1 *oice4 ,h[ to make ?is *oice ,i h be=e expla9$2 ! ma!matical facts wor?y ( be+ /udi$ are ^? :i*1 by _! analogy ) o!r facts1 are capable ( lead+ u to ! k ( a ma!matical law j z exp]im5tal facts lead u to ! k ( a physical law4 ,!y are ^? :i* reveal to u unsuspect$ k9%ip betwe5 o!r facts1 long kn[n1 b wrongly believ$ to be /rang]s to "o ano!r4 ,among *os5 comb9ations ! mo/ f]tile w (t5 be ^? =m$ ( elem5ts drawn f doma9s :i* are f> ap>t4 ,n t ,i mean z suffic+ = inv5tion ! br++ toge!r ( objects z disp>ate z possible2 mo/ comb9ations s =m$ w\ld be entirely /]ile4 ,b c]ta9 among !m1 v r>e1 are ! mo/ fruitful ( all4 ,to inv5t1 ,i h said1 is to *oose2 b ! ^w is p]haps n :olly exact4 ,x makes "o ?9k ( a pur*as] be=e :om are display$ a l>ge numb] ( samples1 & :o exam9es !m1 "o aft] ! o!r1 to make a *oice4 ,"h ! samples w\ld be s num]\s t a :ole lifetime w\ld n suffice to exam9e !m4 ,?is is n ! actual /ate ( ?+s4 ,! /]ile comb9ations d n ev5 pres5t !mselves to ! m9d ( ! inv5tor4 ,nev] 9 ! field ( 8 consci\sness d comb9ations appe> t are n really useful1 except "s t he rejects b :i* h to "s ext5t ! *>act]i/ics ( useful comb9ations4 ,all goes on z if ! inv5tor 7 an exam9] = ! second degree :o w\ld only h to "q ! c&idates :o _h pass$ a previ\s exam9ation4 ,b :at ,i h hi!rto said is :at may be obs]v$ or inf]r$ 9 read+ ! writ+s ( ! geomet]s1 read+ reflectively4 ,x is "t to p5etrate deep] & to see :at goes on 9 ! v s\l ( ! ma!matician4 ,= ?is1 ,i believe1 ,i c d be/ by recall+ memories ( my own4 ,b ,i %all limit myself to tell+ h[ ,i wrote my fir/ memoir on ,fu*sian functions4 ,i beg ! read]'s p>don2 ,i am ab\t to use "s te*nical expressions1 b !y ne$ n fright5 him1 = he is n oblig$ to und]/& !m4 ,i %all say1 = example1 t ,i h f\nd ! demon/ration ( su* a !orem "u su* circum/ances4 ,?is !orem w h a b>b>\s "n1 unfamili> to _m1 b t is unimportant2 :at is ( int]e/ = ! psy*ologi/ is n ! !orem b ! circum/ances4 ,= fifte5 days ,i /rove to prove t !re c\ld n be any functions l ^? ,i h s9ce call$ ,fu*sian functions4 ,i 0 !n v ignorant2 e day ,i seat$ myself at my "w table1 /ay$ an h\r or two1 tri$ a great numb] ( comb9ations & rea*$ no results4 ,"o ev5+1 contr>y to my cu/om1 ,i drank black c(fee & c\ld n sleep4 ,ideas rose 9 cr[ds2 ,i felt !m collide until pairs int]lock$1 s to speak1 mak+ a /able comb9ation4 ,by ! next morn+ ,i _h e/abli%$ ! exi/5ce ( a class ( ,fu*sian functions1 ^? :i* come f ! hyp]geometric s]ies2 ,i _h only to write out ! results1 :i* took b a few h\rs4 ,!n ,i want$ to repres5t ^! functions by ! quoti5t ( two s]ies2 ?is idea 0 p]fectly consci\s & delib]ate1 ! analogy ) elliptic functions guid$ me4 ,i ask$ myself :at prop]ties ^! s]ies mu/ h if !y exi/$1 & ,i succe$$ )\t difficulty 9 =m+ ! s]ies ,i h call$ !ta-,fu*sian4 ,j at ?is "t ,i left ,ca51 :]e ,i 0 !n liv+1 to g on a geologic excursion "u ! auspices ( ! s*ool ( m9es4 ,! *anges ( travel made me =get my ma!matical "w4 ,hav+ rea*$ ,c\tances1 we ent]$ an omnibus to g "s place or o!r4 ,at ! mom5t :5 ,i put my foot on ! /ep ! idea came to me1 )\t any?+ 9 my =m] ?\ghts seem+ to h pav$ ! way = x1 t ! trans=mations ,i _h us$ to def9e ! ,fu*sian functions 7 id5tical ) ^? ( non-,euclidean geometry4 ,i did n v]ify ! idea2 ,i %\ld n h _h "t1 z1 ^u tak+ my seat 9 ! omnibus1 ,i w5t on ) a conv]sation already comm5c$1 b ,i felt a p]fect c]ta9ty4 ,on my return to ,ca51 = consci5ce' sake ,i v]ifi$ ! result at my leisure4 ,!n ,i turn$ my att5tion to ! /udy ( "s ari?metical que/ions app>5tly )\t mu* success & )\t a suspicion ( any connection ) my prec$+ rese>*es4 ,disgu/$ ) my failure1 ,i w5t to sp5d a few days at ! seaside1 & ?\ght ( some?+ else4 ,"o morn+1 walk+ on ! bluff1 ! idea came to me1 ) j ! same *>act]i/ics ( brevity1 sudd5ness & imm$iate c]ta9ty1 t ! ari?metic trans=mations ( indet]m9ate t]n>y quadratic =ms 7 id5tical ) ^? ( non-,euclidean geometry4 ,return$ to ,ca51 ,i m$itat$ on ?is result & d$uc$ ! consequ5ces4 ,! example ( quadratic =ms %[$ me t !re 7 ,fu*sian gr\ps o!r ?an ^? correspond+ to ! hyp]geometric s]ies2 ,i saw t ,i c\ld apply to !m ! !ory ( !ta-,fu*sian s]ies & t consequ5tly !re exi/$ ,fu*sian functions o!r ?an ^? f ! hyp]geometric s]ies1 ! ones ,i !n knew4 ,naturally ,i set myself to =m all ^! functions4 ,i made a sy/ematic attack ^u !m & c>ri$ all ! outworks1 "o aft] ano!r4 ,!re 0 "o h[ev] t /ill held out1 ^: fall w\ld involve t ( ! :ole place4 ,b all my ef=ts only s]v$ at fir/ ! bett] to %[ me ! difficulty1 :i* inde$ 0 some?+4 ,all ?is "w 0 p]fectly consci\s4 ,!reupon ,i left = ,mont-,valri51 :]e ,i 0 to g ?r\gh my milit>y s]vice2 s ,i 0 v diff]5tly occupi$4 ,"o day1 go+ along ! /reet1 ! solution ( ! difficulty :i* _h /opp$ me sudd5ly appe>$ to me4 ,i did n try to g deep into x imm$iately1 & only aft] my s]vice did ,i aga9 take up ! "q4 ,i _h all ! elem5ts & _h only to arrange !m & put !m toge!r4 ,s ,i wrote out my f9al memoir at a s+le /roke & )\t difficulty4 ,i %all limit myself to ?is s+le example2 x is useless to multiply !m4 ,9 reg>d to my o!r rese>*es ,i w\ld h to say analog\s ?+s1 & ! obs]vations ( o!r ma!maticians giv5 9 ,l'enseignem5t ma?matique w\ld only confirm !m4 ,mo/ /rik+ at fir/ is ?is appe>ance ( sudd5 illum9ation1 a manife/ sign ( long1 unconsci\s prior "w4 ,! rle ( ?is unconsci\s "w 9 ma!matical inv5tion appe>s to me inconte/able1 & traces ( x w\ld be f\nd 9 o!r cases :]e x is less evid5t4 ,(t5 :5 "o works at a h>d "q1 no?+ good is accompli%$ at ! fir/ attack4 ,!n "o takes a re/1 long] or %ort]1 & sits d[n anew to ! "w4 ,dur+ ! fir/ half-h\r1 z be=e1 no?+ is f\nd1 & !n all ( a sudd5 ! decisive idea pres5ts itself to ! m9d4 ,x might be said t ! consci\s "w has be5 m fruitful because x has be5 int]rupt$ & ! re/ has giv5 back to ! m9d its =ce & fre%ness4 ,b x is m probable t ?is re/ has be5 fill$ out ) unconsci\s "w & t ! result ( ?is "w has aft]w>d reveal$ itself to ! geomet] j z 9 ! cases ,i h cit$2 only ! revelation1 in/ead ( com+ dur+ a walk or a j\rney1 has happ5$ dur+ a p]iod ( consci\s "w1 b indep5d5tly ( ?is "w :i* plays at mo/ a rle ( excitant1 z if x 7 ! goad /imulat+ ! results already rea*$ dur+ re/1 b rema9+ unconsci\s1 to assume ! consci\s =m4 ,!re is ano!r rem>k to be made ab\t ! conditions ( ?is unconsci\s "w3 x is possible1 & ( a c]ta9ty x is only fruitful1 if x is on ! "o h& prec$$ & on ! o!r h& foll[$ by a p]iod ( consci\s "w4 ,^! sudd5 inspirations 7& ! examples already cit$ suffici5tly prove ?is7 nev] happ5 except aft] "s days ( volunt>y ef=t :i* has appe>$ absolutely fruitless & :5ce no?+ good seems to h come1 :]e ! way tak5 seems totally a/ray4 ,^! ef=ts !n h n be5 z /]ile z "o ?9ks2 !y h set ago+ ! unconsci\s ma*9e & )\t !m x w\ld n h mov$ & w\ld h produc$ no?+4 ,! ne$ = ! second p]iod ( consci\s "w1 aft] ! inspiration1 is /ill easi] to und]/&4 ,x is necess>y to put 9 %ape ! results ( ?is inspiration1 to d$uce f !m ! imm$iate consequ5ces1 to arrange !m1 to ^w ! demon/rations1 b above all is v]ification necess>y4 ,i h spok5 ( ! feel+ ( absolute c]titude accompany+ ! inspiration2 9 ! cases cit$ ?is feel+ 0 no deceiv]1 nor is x usually4 ,b d n ?9k ?is a rule )\t exception2 (t5 ?is feel+ deceives u )\t be+ any ! less vivid1 & we only f9d x out :5 we seek to put on foot ! demon/ration4 ,i h especially notic$ ?is fact 9 reg>d to ideas com+ to me 9 ! morn+ or ev5+ 9 b$ :ile 9 a semi-hypnagogic /ate4 ,su* are ! realities2 n[ = ! ?\ghts !y =ce ^u u4 ,! unconsci\s1 or1 z we say1 ! sublim9al self plays an important rle 9 ma!matical creation2 ?is foll[s f :at we h said4 ,b usually ! sublim9al self is consid]$ z purely automatic4 ,n[ we h se5 t ma!matical "w is n simply me*anical1 t x c\ld n be done by a ma*9e1 h[ev] p]fect4 ,x is n m]ely a "q ( apply+ rules1 ( mak+ ! mo/ comb9ations possible accord+ to c]ta9 fix$ laws4 ,! comb9ations s obta9$ w\ld be exce$+ly num]\s1 useless & cumb]some4 ,! true "w ( ! inv5tor consi/s 9 *oos+ among ^! comb9ations s z to elim9ate ! useless ones or r to avoid ! tr\ble ( mak+ !m1 & ! rules :i* mu/ guide ?is *oice are extremely f9e & delicate4 ,x is almo/ impossible to /ate !m precisely2 !y are felt r ?an =mulat$4 ,"u ^! conditions1 h[ imag9e a sieve capable ( apply+ !m me*anically8 ,a fir/ hypo!sis n[ pres5ts itself3 ! sublim9al self is 9 no way inf]ior to ! consci\s self2 x is n purely automatic2 x is capable ( disc]nm5t2 x has tact1 delicacy2 x kn[s h[ to *oose1 to div9e4 ,:at d ,i say8 ,x kn[s bett] h[ to div9e ?an ! consci\s self1 s9ce x succe$s :]e t has fail$4 ,9 a ^w1 is n ! sublim9al self sup]ior to ! consci\s self8 ,y recognize ! full importance ( ?is "q4 ,b\tr\x 9 a rec5t lecture has %[n h[ x came up on a v diff]5t occasion1 & :at consequ5ces w\ld foll[ an affirmative answ]4 7,see also1 by ! same au?or1 ,sci5ce et ,religion1 pp4 #cac ff47 ,is ?is affirmative answ] =c$ ^u u by ! facts ,i h j giv58 ,i confess t1 = my "p1 ,i %\ld hate to accept x4 ,reexam9e ! facts !n & see if !y are n compatible ) ano!r explanation4 ,x is c]ta9 t ! comb9ations :i* pres5t !mselves to ! m9d 9 a sort ( sudd5 illum9ation1 aft] an unconsci\s work+ some:at prolong$1 are g5]ally useful & f]tile comb9ations1 :i* seem ! result ( a fir/ impression4 ,does x foll[ t ! sublim9al self1 hav+ div9$ by a delicate intuition t ^! comb9ations w\ld be useful1 has =m$ only ^!1 or has x r =m$ _m o!rs :i* 7 lack+ 9 int]e/ & h rema9$ unconsci\s8 ,9 ?is second way ( look+ at x1 all ! comb9ations w\ld be =m$ 9 consequ5ce ( ! automatism ( ! sublim9al self1 b only ! int]e/+ ones w\ld break into ! doma9 ( consci\sness4 ,& ?is is /ill v my/]i\s4 ,:at is ! cause t1 among ! ?\s& products ( our unconsci\s activity1 "s are call$ to pass ! ?re%old1 :ile o!rs rema9 bel[8 ,is x a simple *ance :i* conf]s ?is privilege8 ,evid5tly n2 among all ! /imuli ( our s5ses1 = example1 only ! mo/ int5se fix our att5tion1 unless x has be5 drawn to !m by o!r causes4 ,m g5]ally ! privileg$ unconsci\s ph5om5a1 ^? susceptible ( becom+ consci\s1 are ^? :i*1 directly or indirectly1 affect mo/ pr(\ndly our emotional s5sibility4 ,x may be surpris+ to see emotional s5sibility invok$ propos ( ma!matical demon/rations :i*1 x w\ld seem1 c int]e/ only ! intellect4 ,?is w\ld be to =get ! feel+ ( ma!matical beauty1 ( ! h>mony ( numb]s & =ms1 ( geometric elegance4 ,?is is a true e/hetic feel+ t all real ma!maticians "k1 & surely x belongs to emotional s5sibility4 ,n[1 :at are ! ma!matic entities to :i* we attribute ?is "* ( beauty & elegance1 & :i* are capable ( develop+ 9 u a sort ( e/hetic emotion8 ,!y are ^? ^: elem5ts are h>moni\sly dispos$ s t ! m9d )\t ef=t c embrace _! totality :ile realiz+ ! details4 ,?is h>mony is at once a satisfaction ( our e/hetic ne$s & an aid to ! m9d1 su/a9+ & guid+2 ,& at ! same "t1 9 putt+ "u our eyes a well-ord]$ :ole1 x makes u =esee a ma!matical law4 ,n[1 z we h said above1 ! only ma!matical facts wor?y ( fix+ our att5tion & capable ( be+ useful are ^? :i* c tea* u a ma!matical law4 ,s t we rea* ! foll[+ conclusion3 ,! useful comb9ations are precisely ! mo/ beautiful1 ,i mean ^? be/ able to *>m ?is special s5sibility t all ma!maticians "k1 b ( :i* ! pr(ane are s ignorant z (t5 to be tempt$ to smile at x4 ,:at happ5s !n8 ,among ! great numb]s ( comb9ations bl9dly =m$ by ! sublim9al self1 almo/ all are )\t int]e/ & )\t utility2 b j = t reason !y are also )\t effect ^u ! e/hetic s5sibility4 ,consci\sness w nev] "k !m2 only c]ta9 ones are h>moni\s1 &1 consequ5tly1 at once useful & beautiful4 ,!y w be capable ( t\*+ ?is special s5sibility ( ! geomet] ( :i* ,i h j spok51 & :i*1 once ar\s$1 w call our att5tion to !m1 & ?us give !m occasion to become consci\s4 ,?is is only a hypo!sis1 & yet "h is an obs]vation :i* may confirm x3 :5 a sudd5 illum9ation seizes ^u ! m9d ( ! ma!matician1 x usually happ5s t x does n deceive him1 b x also sometimes happ5s1 z ,i h said1 t x does n /& ! te/ ( v]ification2 well1 we almo/ always notice t ?is false idea1 _h x be5 true1 w\ld h gratifi$ our natural feel+ = ma!matical elegance4 ,?us x is ?is special e/hetic s5sibility :i* plays ! rle ( ! delicate sieve ( :i* ,i spoke1 & t suffici5tly expla9s :y ! "o lack+ x w nev] be a real creator4 ,yet all ! difficulties h n disappe>$4 ,! consci\s self is n>r[ly limit$1 & z = ! sublim9al self we "k n its limitations1 & ?is is :y we are n too reluctant 9 suppos+ t x has be5 able 9 a %ort "t to make m diff]5t comb9ations ?an ! :ole life ( a consci\s be+ c\ld encompass4 ,yet ^! limitations exi/4 ,is x likely t x is able to =m all ! possible comb9ations1 ^: numb] w\ld fright5 ! imag9ation8 ,nev]!less t w\ld seem necess>y1 because if x produces only a small "p ( ^! comb9ations1 & if x makes !m at r&om1 !re w\ld be small *ance t ! good1 ! "o we %\ld *oose1 w\ld be f\nd among !m4 ,p]haps we ought to seek ! explanation 9 t prelim9>y p]iod ( consci\s "w :i* always prec$es all fruitful unconsci\s labor4 ,p]mit me a r\gh comp>ison4 ,figure ! future elem5ts ( our comb9ations z some?+ l ! hook$ atoms ( ,epicurus4 ,dur+ ! complete repose ( ! m9d1 ^! atoms are motionless1 !y are1 s to speak1 hook$ to ! wall2 s ?is complete re/ may be indef9itely prolong$ )\t ! atoms meet+1 & consequ5tly )\t any comb9ation betwe5 !m4 ,on ! o!r h&1 dur+ a p]iod ( app>5t re/ & unconsci\s "w1 c]ta9 ( !m are deta*$ f ! wall & put 9 motion4 ,!y fla% 9 e direction ?r\gh ! space 7,i 0 ab\t to say ! room7 :]e !y are enclos$1 z w\ld1 = example1 a sw>m ( gnats or1 if y pref] a m le>n$ comp>ison1 l ! molecules ( gas 9 ! k9ematic !ory ( gases4 ,!n _! mutual impacts may produce new comb9ations4 ,:at is ! rle ( ! prelim9>y consci\s "w8 ,x is evid5tly to mobilize c]ta9 ( ^! atoms1 to unhook !m f ! wall & put !m 9 sw+4 ,we ?9k we h done no good1 because we h mov$ ^! elem5ts a ?\s& diff]5t ways 9 seek+ to assemble !m1 & h f\nd no satisfactory aggregate4 ,b1 aft] ?is %ak+ up impos$ ^u !m by our w1 ^! atoms d n return to _! primitive re/4 ,!y freely cont9ue _! dance4 ,n[1 our w did n *oose !m at r&om2 x pursu$ a p]fectly det]m9$ aim4 ,! mobiliz$ atoms are !re=e n any atoms :atsoev]2 !y are ^? f :i* we might reasonably expect ! desir$ solution4 ,!n ! mobiliz$ atoms und]go impacts :i* make !m ent] into comb9ations among !mselves or ) o!r atoms at re/ :i* !y /ruck aga9/ 9 _! c\rse4 ,aga9 ,i beg p>don1 my comp>ison is v r\gh1 b ,i sc>cely "k h[ o!rwise to make my ?\ght und]/ood4 ,h[ev] x may be1 ! only comb9ations t h a *ance ( =m+ are ^? :]e at lea/ "o ( ! elem5ts is "o ( ^? atoms freely *os5 by our w4 ,n[1 x is evid5tly among ^! t is f\nd :at ,i call$ ! good comb9ation4 ,p]haps ?is is a way ( less5+ ! p>adoxical 9 ! orig9al hypo!sis4 ,ano!r obs]vation4 ,x nev] happ5s t ! unconsci\s "w gives u ! result ( a some:at long calculation all made1 :]e we h only to apply fix$ rules4 ,we might ?9k ! :olly automatic sublim9al self p>ticul>ly apt = ?is sort ( "w1 :i* is 9 a way exclusively me*anical4 ,x seems t ?9k+ 9 ! ev5+ ^u ! factors ( a multiplication we might hope to f9d ! product ready made ^u our awak5+1 or aga9 t an algebraic calculation1 = example a v]ification1 w\ld be made unconsci\sly4 ,no?+ ( ! sort1 z obs]vation proves4 ,all "o may hope f ^! inspirations1 fruits ( unconsci\s "w1 is a po9t ( dep>ture = su* calculations4 ,z = ! calculations !mselves1 !y mu/ be made 9 ! second p]iod ( consci\s "w1 t :i* foll[s ! inspiration1 t 9 :i* "o v]ifies ! results ( ?is inspiration & d$uces _! consequ5ces4 ,! rules ( ^! calculations are /rict & complicat$4 ,!y require discipl9e1 att5tion1 w1 & !re=e consci\sness4 ,9 ! sublim9al self1 on ! contr>y1 reigns :at ,i %\ld call lib]ty1 if we might give ?is "n to ! simple abs5ce ( discipl9e & to ! disord] born ( *ance4 ,only1 ?is disord] itself p]mits unexpect$ comb9ations4 ,i %all make a la/ rem>k3 :5 above ,i made c]ta9 p]sonal obs]vations1 ,i spoke ( a night ( excitem5t :5 ,i work$ 9 spite ( myself4 ,su* cases are frequ5t1 & x is n necess>y t ! abnormal c]ebral activity be caus$ by a physical excitant z 9 t ,i m5tion$4 ,x seems1 9 su* cases1 t "o is pres5t at 8 own unconsci\s "w1 made p>tially p]ceptible to ! ov]-excit$ consci\sness1 yet )\t hav+ *ang$ its nature4 ,!n we vaguely compreh5d :at di/+ui%es ! two me*anisms or1 if y wi%1 ! work+ me?ods ( ! two egos4 ,& ! psy*ologic obs]vations ,i h be5 able ?us to make seem to me to confirm 9 _! g5]al outl9es ! views ,i h giv54 ,surely !y h ne$ ( x1 = !y are & rema9 9 spite ( all v hypo!tical3 ! int]e/ ( ! que/ions is s great t ,i d n rep5t ( hav+ submitt$ !m to ! read]4 ,,*apt] ,,iv ,,*ance ,i ,h[ d>e we speak ( ! laws ( *ance8 ,is n *ance ! anti!sis ( all law8 ,s says ,b]tr& at ! beg9n+ ( 8 ,calcul des probabiltits4 ,probability is oppos$ to c]titude2 s x is :at we d n "k & consequ5tly x seems :at we c\ld n calculate4 ,"h is at lea/ app>5tly a contradiction1 & ab\t x mu* has already be5 writt54 ,& fir/1 :at is *ance8 ,! anci5ts di/+ui%$ betwe5 ph5om5a seem+ly obey+ h>moni\s laws1 e/abli%$ once = all1 & ^? :i* !y attribut$ to *ance2 ^! 7 ! ones unpr$ictable because rebelli\s to all law4 ,9 ea* doma9 ! precise laws did n decide ev]y?+1 !y only drew limits betwe5 :i* *ance might act4 ,9 ?is conception ! ^w *ance _h a precise & objective mean+2 :at 0 *ance = "o 0 also *ance = ano!r & ev5 = ! gods4 ,b ?is conception is n ours to-day4 ,we h become absolute det]m9i/s1 & ev5 ^? :o want to res]ve ! rights ( human free w let det]m9ism reign undivid$ly 9 ! inorganic _w at lea/4 ,e ph5om5on1 h[ev] m9ute1 has a cause2 & a m9d inf9itely p[]ful1 inf9itely well-in=m$ ab\t ! laws ( nature1 c\ld h =ese5 x f ! beg9n+ ( ! c5turies4 ,if su* a m9d exi/$1 we c\ld n play ) x at any game ( *ance2 we %\ld always lose4 ,9 fact = x ! ^w *ance w\ld n h any mean+1 or r !re w\ld be no *ance4 ,x is because ( our weakness & our ignorance t ! ^w has a mean+ = u4 ,&1 ev5 )\t go+ beyond our feeble humanity1 :at is *ance = ! ignorant is n *ance = ! sci5ti/4 ,*ance is only ! measure ( our ignorance4 ,=tuit\s ph5om5a are1 by def9ition1 ^? ^: laws we d n "k4 ,b is ?is def9ition altoge!r satisfactory8 ,:5 ! fir/ ,*aldean %eph]ds foll[$ ) _! eyes ! movem5ts ( ! />s1 !y knew n z yet ! laws ( a/ronomy2 w\ld !y h dream$ ( say+ t ! />s move at r&om8 ,if a mod]n physici/ /udies a new ph5om5on1 & if he discov]s its law ,tuesday1 w\ld he h said ,monday t ?is ph5om5on 0 =tuit\s8 ,moreov]1 d we n (t5 invoke :at ,b]tr& calls ! laws ( *ance1 to pr$ict a ph5om5on8 ,= example1 9 ! k9etic !ory ( gases we obta9 ! kn[n laws ( ,m>iotte & ( ,gay-,lussac by means ( ! hypo!sis t ! velocities ( ! molecules ( gas v>y irregul>ly1 t is to say at r&om4 ,all physici/s w agree t ! obs]vable laws w\ld be mu* less simple if ! velocities 7 rul$ by any simple elem5t>y law :atsoev]1 if ! molecules 71 z we say1 organiz$1 if !y 7 subject to "s discipl9e4 ,x is due to *ance1 t is to say1 to our ignorance1 t we c draw our conclusions2 & !n if ! ^w *ance is simply synonym\s ) ignorance :at does t mean8 ,mu/ we !re=e translate z foll[s8 ,y ask me to pr$ict = y ! ph5om5a ab\t to happ54 ,if1 unluckily1 ,i knew ! laws ( ^! ph5om5a ,i c\ld make ! pr$iction only by inextricable calculations & w\ld h to r5\nce attempt+ to answ] y2 b z ,i h ! good =tune n to "k !m1 ,i w answ] y at once4 ,& :at is mo/ surpris+1 my answ] w be "r4 ,s x mu/ well be t *ance is some?+ o!r ?an ! "n we give our ignorance1 t among ph5om5a ^: causes are unkn[n to u we mu/ di/+ui% =tuit\s ph5om5a ab\t :i* ! calculus ( probabilities w provisionally give in=mation1 f ^? :i* are n =tuit\s & ( :i* we c say no?+ s long z we %all n h det]m9$ ! laws gov]n+ !m4 ,= ! =tuit\s ph5om5a !mselves1 x is cle> t ! in=mation giv5 u by ! calculus ( probabilities w n cease to be true ^u ! day :5 ^! ph5om5a %all be bett] kn[n4 ,! director ( a life insurance company does n "k :5 ea* ( ! insur$ w die1 b he relies ^u ! calculus ( probabilities & on ! law ( great numb]s1 & he is n deceiv$1 s9ce he di/ributes divid5ds to 8 /ockhold]s4 ,^! divid5ds w\ld n vani% if a v p5etrat+ & v indiscreet physician %\ld1 aft] ! policies 7 sign$1 reveal to ! director ! life *ances ( ! insur$4 ,?is doctor w\ld dissipate ! ignorance ( ! director1 b he w\ld h no influ5ce on ! divid5ds1 :i* evid5tly are n an outcome ( ?is ignorance4 ,,ii ,to f9d a bett] def9ition ( *ance we mu/ exam9e "s ( ! facts :i* we agree to reg>d z =tuit\s1 & to :i* ! calculus ( probabilities seems to apply2 we !n %all inve/igate :at are _! common *>act]i/ics4 ,! fir/ example we select is t ( un/able equilibrium2 if a cone re/s ^u its apex1 we "k well t x w fall1 b we d n "k t[>d :at side2 x seems to u *ance alone w decide4 ,if ! cone 7 p]fectly symmetric1 if its axis 7 p]fectly v]tical1 if x 7 act$ ^u by no =ce o!r ?an gravity1 x w\ld n fall at all4 ,b ! lea/ defect 9 symmetry w make x lean slightly t[>d "o side or ! o!r1 & if x leans1 h[ev] little1 x w fall altoge!r t[>d t side4 ,ev5 if ! symmetry 7 p]fect1 a v slight tremor1 a brea? ( air c\ld make x incl9e "s seconds ( arc2 ?is w be en\gh to det]m9e its fall & ev5 ! s5se ( its fall :i* w be t ( ! initial incl9ation4 ,a v slight cause1 :i* escapes u1 det]m9es a consid]able effect :i* we c n help see+1 & !n we say ?is effect is due to *ance4 ,if we c\ld "k exactly ! laws ( nature & ! situation ( ! univ]se at ! initial in/ant1 we %\ld be able to pr$ict exactly ! situation ( ?is same univ]se at a subsequ5t in/ant4 ,b ev5 :5 ! natural laws %\ld h no fur!r secret = u1 we c\ld "k ! initial situation only approximately4 ,if t p]mits u to =esee ! subsequ5t situation ) ! same degree ( approximation1 ?is is all we require1 we say ! ph5om5on has be5 pr$ict$1 t x is rul$ by laws4 ,b ?is is n always ! case2 x may happ5 t slight diff]5ces 9 ! initial conditions produce v great diff]5ces 9 ! f9al ph5om5a2 a slight error 9 ! =m] w\ld make an enorm\s error 9 ! latt]4 ,pr$iction becomes impossible & we h ! =tuit\s ph5om5on4 ,our second example w be v analog\s to ! fir/ & we %all take x f meteorology4 ,:y h ! meteorologi/s su* difficulty 9 pr$ict+ ! wea!r ) any c]ta9ty8 ,:y d ! ra9s1 ! tempe/s !mselves seem to u to come by *ance1 s t _m p]sons f9d x q natural to pray = ra9 or %9e1 :5 !y w\ld ?9k x ridicul\s to pray = an eclipse8 ,we see t great p]turbations g5]ally happ5 9 regions :]e ! atmosph]e is 9 un/able equilibrium4 ,! meteorologi/s are aw>e t ?is equilibrium is un/able1 t a cyclone is aris+ some:]e2 b :]e !y c n tell2 "o-t5? ( a degree m or less at any po9t1 & ! cyclone bur/s "h & n !re1 & spreads its ravages ov] c\ntries x w\ld h sp>$4 ,?is we c\ld h =ese5 if we _h kn[n t t5? ( a degree1 b ! obs]vations 7 nei!r suffici5tly close nor suffici5tly precise1 & = ?is reason all seems due to ! ag5cy ( *ance4 ,"h aga9 we f9d ! same contra/ betwe5 a v slight cause1 unappreciable to ! obs]v]1 & important effects1 :i* are sometimes trem5d\s disa/]s4 ,let u pass to ano!r example1 ! di/ribution ( ! m9or planets on ! zodiac4 ,_! initial longitudes may h be5 any longitudes :atev]2 b _! mean motions 7 diff]5t & !y h revolv$ = s long a "t t we may say !y are n[ di/ribut$ at r&om along ! zodiac4 ,v slight initial diff]5ces betwe5 _! di/ances f ! sun1 or1 :at comes to ! same ?+1 betwe5 _! mean motions1 h end$ by giv+ enorm\s diff]5ces betwe5 _! pres5t longitudes4 ,an excess ( ! ?\s&? ( a second 9 ! daily mean motion w give 9 fact a second 9 ?ree ye>s1 a degree 9 t5 ?\s& ye>s1 an entire circumf]5ce 9 ?ree or f\r million ye>s1 & :at is t to ! "t :i* has pass$ s9ce ! m9or planets deta*$ !mselves f ! nebula ( ,laplace8 ,aga9 !re=e we see a slight cause & a great effect2 or bett]1 slight diff]5ces 9 ! cause & great diff]5ces 9 ! effect4 ,! game ( r\lette does n take u z f> z might seem f ! prec$+ example4 ,assume a ne$le to be turn$ on a pivot ov] a dial divid$ into a hundr$ sectors alt]nately r$ & black4 ,if x /ops on a r$ sector ,i w92 if n1 ,i lose4 ,evid5tly all dep5ds ^u ! initial impulse ,i give ! ne$le4 ,! ne$le w make1 suppose1 t5 or tw5ty turns1 b x w /op soon] or n s soon1 accord+ z ,i %all h pu%$ x m or less /rongly4 ,x suffices t ! impulse v>y only by a ?\s&? or a two ?\s&? to make ! ne$le /op ov] a black sector or ov] ! foll[+ r$ "o4 ,^! are diff]5ces ! muscul> s5se c n di/+ui% & :i* elude ev5 ! mo/ delicate in/rum5ts4 ,s x is impossible = me to =esee :at ! ne$le ,i h />t$ w d1 & ?is is :y my he>t ?robs & ,i hope ev]y?+ f luck4 ,! diff]5ce 9 ! cause is imp]ceptible1 & ! diff]5ce 9 ! effect is = me ( ! highe/ importance1 s9ce x means my :ole /ake4 ,,iii ,p]mit me1 9 ?is connection1 a ?\ght some:at =eign to my subject4 ,"s ye>s ago a philosoph] said t ! future is det]m9$ by ! pa/1 b n ! pa/ by ! future2 or1 9 o!r words1 f k ( ! pres5t we c\ld d$uce ! future1 b n ! pa/2 because1 said he1 a cause c h only "o effect1 :ile ! same effect might be produc$ by sev]al diff]5t causes4 ,x is cle> no sci5ti/ c subscribe to ?is conclusion4 ,! laws ( nature b9d ! antec$5t to ! consequ5t 9 su* a way t ! antec$5t is z well det]m9$ by ! consequ5t z ! consequ5t by ! antec$5t4 ,b :5ce came ! error ( ?is philosoph]8 ,we "k t 9 virtue ( ,c>not's pr9ciple physical ph5om5a are irrev]sible & ! _w t5ds t[>d uni=mity4 ,:5 two bodies ( diff]5t temp]ature come 9 contact1 ! w>m] gives up heat to ! cold]2 s we may =esee t ! temp]ature w equalize4 ,b once equal1 if ask$ ab\t ! ant]ior /ate1 :at c we answ]8 ,we might say t "o 0 w>m & ! o!r cold1 b n be able to div9e :i* =m]ly 0 ! w>m]4 ,& yet 9 reality ! temp]atures w nev] rea* p]fect equality4 ,! diff]5ce ( ! temp]atures only t5ds asymptotically t[>d z]o4 ,!re comes a mom5t :5 our !rmomet]s are p[]less to make x kn[n4 ,b if we _h !rmomet]s a ?\s& times1 a hundr$ ?\s& times z s5sitive1 we %\ld recognize t !re /ill is a slight diff]5ce1 & t "o ( ! bodies rema9s a little w>m] ?an ! o!r1 & s we c\ld say ?is x is :i* =m]ly 0 mu* ! w>m]4 ,s !n !re are1 contr>y to :at we f\nd 9 ! =m] examples1 great diff]5ces 9 cause & slight diff]5ces 9 effect4 ,flamm>ion once imag9$ an obs]v] go+ away f ! e>? ) a velocity great] ?an t ( light2 = him "t w\ld h *ang$ sign4 ,hi/ory w\ld be turn$ ab\t1 & ,wat]loo w\ld prec$e ,au/]litz4 ,well1 = ?is obs]v]1 effects & causes w\ld be inv]t$2 un/able equilibrium w\ld no long] be ! exception4 ,because ( ! univ]sal irrev]sibility1 all w\ld seem to him to come out ( a sort ( *aos 9 un/able equilibrium4 ,all nature w\ld appe> to him deliv]$ ov] to *ance4 ,,iv ,n[ = o!r examples :]e we %all see some:at diff]5t *>act]i/ics4 ,take fir/ ! k9etic !ory ( gases4 ,h[ %\ld we picture a receptacle fill$ ) gas8 ,innum]able molecules1 mov+ at high spe$s1 fla% ?r\gh ?is receptacle 9 e direction4 ,at e in/ant !y /rike aga9/ its walls or ea* o!r1 & ^! collisions happ5 "u ! mo/ div]se conditions4 ,:at above all impresses u "h is n ! littl5ess ( ! causes1 b _! complexity1 & yet ! =m] elem5t is /ill f\nd "h & plays an important rle4 ,if a molecule deviat$ "r or left f its trajectory1 by a v small quantity1 comp>able to ! radius ( action ( ! gase\s molecules1 x w\ld avoid a collision or su/a9 x "u diff]5t conditions1 & t w\ld v>y ! direction ( its velocity aft] ! impact1 p]haps by n9ety degrees or by a hundr$ & eighty degrees4 ,& ?is is n all2 we h j se5 t x is necess>y to deflect ! molecule be=e ! cla% by only an inf9itesimal1 to produce its deviation aft] ! collision by a f9ite quantity4 ,if !n ! molecule und]goes two successive %ocks1 x w suffice to deflect x be=e ! fir/ by an inf9itesimal ( ! second ord]1 = x to deviate aft] ! fir/ enc\nt] by an inf9itesimal ( ! fir/ ord]1 & aft] ! second hit1 by a f9ite quantity4 ,& ! molecule w n und]go m]ely two %ocks2 x w und]go a v great numb] p] second4 ,s t if ! fir/ %ock has multipli$ ! deviation by a v l>ge numb] ,a1 aft] n %ocks x w be multipli$ by ,an4 ,x w !re=e become v great n m]ely because ,a is l>ge1 t is to say because little causes produce big effects1 b because ! expon5t n is l>ge1 t is to say because ! %ocks are v num]\s & ! causes v complex4 ,take a second example4 ,:y d ! drops ( ra9 9 a %[] seem to be di/ribut$ at r&om8 ,?is is aga9 because ( ! complexity ( ! causes :i* det]m9e _! =mation4 ,ions are di/ribut$ 9 ! atmosph]e4 ,= a long :ile !y h be5 subject$ to air-curr5ts con/antly *ang+1 !y h be5 caught 9 v small :irlw9ds1 s t _! f9al di/ribution has no long] any relation to _! initial di/ribution4 ,sudd5ly ! temp]ature falls1 vapor cond5ses1 & ea* ( ^! ions becomes ! c5t] ( a drop ( ra94 ,to "k :at w be ! di/ribution ( ^! drops & h[ _m w fall on ea* pav+-/one1 x w\ld n be suffici5t to "k ! initial situation ( ! ions1 x w\ld be necess>y to compute ! effect ( a ?\s& little caprici\s air-curr5ts4 ,& aga9 x is ! same if we put gra9s ( p[d] 9 susp5sion 9 wat]4 ,! vase is pl\gh$ by curr5ts ^: law we "k n1 we only "k x is v complicat$4 ,at ! end ( a c]ta9 "t ! gra9s w be di/ribut$ at r&om1 t is to say uni=mly1 9 ! vase2 & ?is is due precisely to ! complexity ( ^! curr5ts4 ,if !y obey$ "s simple law1 if = example ! vase revolv$ & ! curr5ts circulat$ ar\nd ! axis ( ! vase1 describ+ circles1 x w\ld no long] be ! same1 s9ce ea* gra9 w\ld reta9 its initial altitude & its initial di/ance f ! axis4 ,we %\ld rea* ! same result 9 consid]+ ! mix+ ( two liquids or ( two f9e-gra9$ p[d]s4 ,& to take a gross] example1 ?is is also :at happ5s :5 we %uffle play+-c>ds4 ,at ea* /roke ! c>ds und]go a p]mutation 7analog\s to t /udi$ 9 ! !ory ( sub/itutions74 ,:at w happ58 ,! probability ( a p>ticul> p]mutation 7= example1 t br++ to ! n? place ! c>d occupy+ ! phi7n7? place be=e ! p]mutation7 dep5ds ^u ! play]'s habits4 ,b if ?is play] %uffles ! c>ds long en\gh1 !re w be a great numb] ( successive p]mutations1 & ! result+ f9al ord] w no long] be gov]n$ by aught b *ance2 ,i mean to say t all possible ord]s w be equally probable4 ,x is to ! great numb] ( successive p]mutations1 t is to say to ! complexity ( ! ph5om5on1 t ?is result is due4 ,a f9al ^w ab\t ! !ory ( errors4 ,"h x is t ! causes are complex & multiple4 ,to h[ _m sn>es is n ! obs]v] expos$1 ev5 ) ! be/ in/rum5t6 ,he %\ld apply himself to f9d+ out ! l>ge/ & avoid+ !m4 ,^! are ! ones giv+ bir? to sy/ematic errors4 ,b :5 he has elim9at$ ^?1 admitt+ t he succe$s1 !re rema9 _m small ones :i*1 _! effects accumulat+1 may become dang]\s4 ,!nce come ! accid5tal errors2 & we attribute !m to *ance because _! causes are too complicat$ & too num]\s4 ,"h aga9 we h only little causes1 b ea* ( !m w\ld produce only a slight effect2 x is by _! union & _! numb] t _! effects become =midable4 ,v ,we may take /ill a ?ird po9t ( view1 less important ?an ! fir/ two & ^u :i* ,i %all lay less /ress4 ,:5 we seek to =esee an ev5t & exam9e its antec$5ts1 we /rive to se>* into ! ant]ior situation4 ,?is c\ld n be done = all p>ts ( ! univ]se & we are cont5t to "k :at is pass+ 9 ! neighborhood ( ! po9t :]e ! ev5t %\ld occur1 or :at w\ld appe> to h "s relation to x4 ,an exam9ation c n be complete & we mu/ "k h[ to *oose4 ,b x may happ5 t we h pass$ by circum/ances :i* at fir/ sight seem$ completely =eign to ! =ese5 happ5+1 to :i* "o w\ld nev] h dream$ ( attribut+ any influ5ce & :i* nev]!less1 contr>y to all anticipation1 come to play an important rle4 ,a man passes 9 ! /reet go+ to 8 bus9ess2 "s "o kn[+ ! bus9ess c\ld h told :y he />t$ at su* a "t & w5t by su* a /reet4 ,on ! ro( works a til]4 ,! contractor employ+ him c\ld 9 a c]ta9 measure =esee :at he w\ld d4 ,b ! pass]-by sc>cely ?9ks ( ! til]1 nor ! til] ( him2 !y seem to belong to two worlds completely =eign to "o ano!r4 ,& yet ! til] drops a tile :i* kills ! man1 & we d n hesitate to say ?is is *ance4 ,our weakness =bids our consid]+ ! entire univ]se & makes u cut x up into slices4 ,we try to d ?is z little artificially z possible4 ,& yet x happ5s f "t to "t t two ( ^! slices react ^u ea* o!r4 ,! effects ( ?is mutual action !n seem to u to be due to *ance4 ,is ?is a ?ird way ( conceiv+ *ance8 ,n always2 9 fact mo/ (t5 we are c>ri$ back to ! fir/ or ! second4 ,:5ev] two worlds usually =eign to "o ano!r come ?us to react ^u ea* o!r1 ! laws ( ?is reaction mu/ be v complex4 ,on ! o!r h&1 a v slight *ange 9 ! initial conditions ( ^! two worlds w\ld h be5 suffici5t = ! reaction n to h happ5$4 ,h[ little 0 ne$$ = ! man to pass a second lat] or ! til] to drop 8 tile a second soon]4 ,,vi ,all we h said /ill does n expla9 :y *ance obeys laws4 ,does ! fact t ! causes are slight or complex suffice = our =esee+1 if n _! effects 9 ea* case1 at lea/ :at _! effects w be1 on ! av]age8 ,to answ] ?is "q we _h bett] take up aga9 "s ( ! examples already cit$4 ,i %all beg9 ) t ( ! r\lette4 ,i h said t ! po9t :]e ! ne$le w /op dep5ds ^u ! initial pu% giv5 x4 ,:at is ! probability ( ?is pu% hav+ ?is or t value8 ,i "k no?+ ab\t x1 b x is difficult = me n to suppose t ?is probability is repres5t$ by a cont9u\s analytic function4 ,! probability t ! pu% is compris$ betwe5 alpha & alpha epsilon w !n be s5sibly equal to ! probability ( its be+ compris$ betwe5 alpha epsilon & alpha #b;epsilon1 provid$ epsilon be v small4 ,?is is a prop]ty common to all analytic functions4 ,m9ute v>iations ( ! function are proportional to m9ute v>iations ( ! v>iable4 ,b we h assum$ t an exce$+ly slight v>iation ( ! pu% suffices to *ange ! color ( ! sector ov] :i* ! ne$le f9ally /ops4 ,f alpha to alphaepsilon x is r$1 f alphaepsilon to alpha#b;epsilon x is black2 ! probability ( ea* r$ sector is !re=e ! same z ( ! foll[+ black1 & consequ5tly ! total probability ( r$ equals ! total probability ( black4 ,! datum ( ! "q is ! analytic function repres5t+ ! probability ( a p>ticul> initial pu%4 ,b ! !orem rema9s true :atev] be ?is datum1 s9ce x dep5ds ^u a prop]ty common to all analytic functions4 ,f ?is x foll[s f9ally t we no long] ne$ ! datum4 ,:at we h j said = ! case ( ! r\lette applies also to ! example ( ! m9or planets4 ,! zodiac may be reg>d$ z an imm5se r\lette on :i* h be5 toss$ _m little balls ) diff]5t initial impulses v>y+ accord+ to "s law4 ,_! pres5t di/ribution is uni=m & indep5d5t ( ?is law1 = ! same reason z 9 ! prec$+ case4 ,?us we see :y ph5om5a obey ! laws ( *ance :5 slight diff]5ces 9 ! causes suffice to br+ on great diff]5ces 9 ! effects4 ,! probabilities ( ^! slight diff]5ces may !n be reg>d$ z proportional to ^! diff]5ces !mselves1 j because ^! diff]5ces are m9ute1 & ! inf9itesimal increm5ts ( a cont9u\s function are proportional to ^? ( ! v>iable4 ,take an entirely diff]5t example1 :]e int]v5es especially ! complexity ( ! causes4 ,suppose a play] %uffles a pack ( c>ds4 ,at ea* %uffle he *anges ! ord] ( ! c>ds1 & he may *ange !m 9 _m ways4 ,to simplify ! exposition1 consid] only ?ree c>ds4 ,! c>ds :i* be=e ! %uffle occupi$ respectively ! places #abc1 may aft] ! %uffle occupy ! places #abc1 #bca1 #cab1 #cba1 #acb1 #bac4 ,ea* ( ^! six hypo!ses is possible & !y h respectively = probabilities3 p#a1 p#b1 p#c1 p#d1 p#e1 p#f4 ,! sum ( ^! six numb]s equals #a2 b ?is is all we "k ( !m2 ^! six probabilities dep5d naturally ^u ! habits ( ! play] :i* we d n "k4 ,at ! second %uffle & ! foll[+1 ?is w recomm5ce1 & "u ! same conditions2 ,i mean t p#d = example repres5ts always ! probability t ! ?ree c>ds :i* occupi$ aft] ! n? %uffle & be=e ! n #a? ! places #abc1 occupy ! places #cba aft] ! n #a? %uffle4 ,& ?is rema9s true :atev] be ! numb] n1 s9ce ! habits ( ! play] & 8 way ( %uffl+ rema9 ! same4 ,b if ! numb] ( %uffles is v great1 ! c>ds :i* be=e ! fir/ %uffle occupi$ ! places #abc may1 aft] ! la/ %uffle1 occupy ! places #abc1 #bca1 #cab1 #cba1 #acb1 #bac & ! probability ( ^! six hypo!ses w be s5sibly ! same & equal to #a_/f2 & ?is w be true :atev] be ! numb]s p#a 444 p#f :i* we d n "k4 ,! great numb] ( %uffles1 t is to say ! complexity ( ! causes1 has produc$ uni=mity4 ,?is w\ld apply )\t *ange if !re 7 m ?an ?ree c>ds1 b ev5 ) ?ree c>ds ! demon/ration w\ld be complicat$2 let x suffice to give x = only two c>ds4 ,!n we h only two possibilities #ab1 #ba ) ! probabilities p#a & p#b #a - p#a4 ,suppose n %uffles & suppose ,i w9 "o franc if ! c>ds are f9ally 9 ! initial ord] & lose "o if !y are f9ally inv]t$4 ,!n1 my ma!matical expectation w be 7p#a - p#b7n4 ,! diff]5ce p#a - p#b is c]ta9ly less ?an #a2 s t if n is v great my expectation w be z]o2 we ne$ n le>n p#a & p#b to be aw>e t ! game is equitable4 ,!re w\ld always be an exception if "o ( ! numb]s p#a & p#b 0 equal to #a & ! o!r naught4 ,!n x w\ld n apply because our initial hypo!ses w\ld be too simple4 ,:at we h j se5 applies n only to ! mix+ ( c>ds1 b to all mix+s1 to ^? ( p[d]s & ( liquids2 & ev5 to ^? ( ! molecules ( gases 9 ! k9etic !ory ( gases4 ,to return to ?is !ory1 suppose = a mom5t a gas ^: molecules c n mutually cla%1 b may be deviat$ by hitt+ ! insides ( ! vase :]e9 ! gas is conf9$4 ,if ! =m ( ! vase is suffici5tly complex ! di/ribution ( ! molecules & t ( ! velocities w n be long 9 becom+ uni=m4 ,b ?is w n be s if ! vase is sph]ical or if x has ! %ape ( a cuboid4 ,:y8 ,because 9 ! fir/ case ! di/ance f ! c5t] to any trajectory w rema9 con/ant2 9 ! second case ?is w be ! absolute value ( ! angle ( ea* trajectory ) ! faces ( ! cuboid4 ,s we see :at %\ld be und]/ood by conditions too simple2 !y are ^? :i* cons]ve some?+1 :i* leave an inv>iant rema9+4 ,are ! diff]5tial equations ( ! problem too simple = u to apply ! laws ( *ance8 ,?is "q w\ld seem at fir/ view to lack precise mean+2 n[ we "k :at x means4 ,!y are too simple if !y cons]ve some?+1 if !y admit a uni=m integral4 ,if some?+ 9 ! initial conditions rema9s un*ang$1 x is cle> ! f9al situation c no long] be indep5d5t ( ! initial situation4 ,we come f9ally to ! !ory ( errors4 ,we "k n to :at are due ! accid5tal errors1 & precisely because we d n "k1 we are aw>e !y obey ! law ( ,gauss4 ,su* is ! p>adox4 ,! explanation is ne>ly ! same z 9 ! prec$+ cases4 ,we ne$ "k only "o ?+3 t ! errors are v num]\s1 t !y are v slight1 t ea* may be z well negative z positive4 ,:at is ! curve ( probability ( ea* ( !m8 ,we d n "k2 we only suppose x is symmetric4 ,we prove !n t ! resultant error w foll[ ,gauss's law1 & ?is result+ law is indep5d5t ( ! p>ticul> laws :i* we d n "k4 ,"h aga9 ! simplicity ( ! result is born ( ! v complexity ( ! data4 ,,vii ,b we are n ?r\gh ) p>adoxes4 ,i h j recall$ ! figm5t ( ,flamm>ion1 t ( ! man go+ quick] ?an light1 = :om "t *anges sign4 ,i said t = him all ph5om5a w\ld seem due to *ance4 ,t is true f a c]ta9 po9t ( view1 & yet all ^! ph5om5a at a giv5 mom5t w\ld n be di/ribut$ 9 con=mity ) ! laws ( *ance1 s9ce ! di/ribution w\ld be ! same z = u1 :o1 see+ !m unfold h>moni\sly & )\t com+ out ( a primal *aos1 d n reg>d !m z rul$ by *ance4 ,:at does t mean8 ,= ,lum51 ,flamm>ion's man1 slight causes seem to produce great effects2 :y d n ?+s g on z = u :5 we ?9k we see gr& effects due to little causes8 ,w\ld n ! same reason+ be applicable 9 8 case8 ,let u return to ! argum5t4 ,:5 slight diff]5ces 9 ! causes produce va/ diff]5ces 9 ! effects1 :y are ^! effects di/ribut$ accord+ to ! laws ( *ance8 ,suppose a diff]5ce ( a millimet] 9 ! cause produces a diff]5ce ( a kilomet] 9 ! effect4 ,if ,i w9 9 case ! effect corresponds to a kilomet] be>+ an ev5 numb]1 my probability ( w9n+ w be #a_/b4 ,:y8 ,because to make t1 ! cause mu/ correspond to a millimet] ) an ev5 numb]4 ,n[1 accord+ to all appe>ance1 ! probability ( ! cause v>y+ betwe5 c]ta9 limits w be proportional to ! di/ance ap>t ( ^! limits1 provid$ ?is di/ance be v small4 ,if ?is hypo!sis 7 n admitt$ !re w\ld no long] be any way ( repres5t+ ! probability by a cont9u\s function4 ,:at n[ w happ5 :5 great causes produce small effects8 ,?is is ! case :]e we %\ld n attribute ! ph5om5on to *ance & :]e on ! contr>y ,lum5 w\ld attribute x to *ance4 ,to a diff]5ce ( a kilomet] 9 ! cause w\ld correspond a diff]5ce ( a millimet] 9 ! effect4 ,w\ld ! probability ( ! cause be+ compris$ betwe5 two limits n kilomet]s ap>t /ill be proportional to n8 ,we h no reason to suppose s1 s9ce ?is di/ance1 n kilomet]s1 is great4 ,b ! probability t ! effect lies betwe5 two limits n millimet]s ap>t w be precisely ! same1 s x w n be proportional to n1 ev5 ?\gh ?is di/ance1 n millimet]s1 be small4 ,!re is no way !re=e ( repres5t+ ! law ( probability ( effects by a cont9u\s curve4 ,?is curve1 und]/&1 may rema9 cont9u\s 9 ! analytic s5se ( ! ^w2 to inf9itesimal v>iations ( ! abscissa w correspond inf9itesimal v>iations ( ! ord9ate4 ,b practically x w n be cont9u\s1 s9ce v small v>iations ( ! ord9ate w\ld n correspond to v small v>iations ( ! abscissa4 ,x w\ld become impossible to trace ! curve ) an ord9>y p5cil2 t is :at ,i mean4 ,s :at mu/ we conclude8 ,lum5 has no "r to say t ! probability ( ! cause 78 cause1 our effect7 %\ld be repres5t$ necess>ily by a cont9u\s function4 ,b !n :y h we ?is "r8 ,x is because ?is /ate ( un/able equilibrium :i* we h be5 call+ initial is itself only ! f9al outcome ( a long previ\s hi/ory4 ,9 ! c\rse ( ?is hi/ory complex causes h work$ a great :ile3 !y h contribut$ to produce ! mixture ( elem5ts & !y h t5d$ to make ev]y?+ uni=m at lea/ )9 a small region2 !y h r\nd$ (f ! corn]s1 smoo!d d[n ! hills & fill$ up ! valleys4 ,h[ev] caprici\s & irregul> may h be5 ! primitive curve giv5 ov] to !m1 !y h work$ s mu* t[>d mak+ x regul> t f9ally !y deliv] ov] to u a cont9u\s curve4 ,& ?is is :y we may 9 all confid5ce assume its cont9uity4 ,lum5 w\ld n h ! same reasons = su* a conclusion4 ,= him complex causes w\ld n seem ag5ts ( equalization & regul>ity1 b on ! contr>y w\ld create only inequality & diff]5tiation4 ,he w\ld see a _w m & m v>i$ come =? f a sort ( primitive *aos4 ,! *anges he c\ld obs]ve w\ld be = him un=ese5 & impossible to =esee4 ,!y w\ld seem to him due to "s caprice or ano!r2 b ?is caprice w\ld be q diff]5t f our *ance1 s9ce x w\ld be oppos$ to all law1 :ile our *ance /ill has its laws4 ,all ^! po9ts call = l5g?y explications1 :i* p]haps w\ld aid 9 ! bett] compreh5sion ( ! irrev]sibility ( ! univ]se4 ,,viii ,we h s\ght to def9e *ance1 & n[ x is prop] to put a "q4 ,has *ance ?us def9$1 9 s f> z ?is is possible1 objectivity8 ,x may be que/ion$4 ,i h spok5 ( v slight or v complex causes4 ,b :at is v little = "o may be v big = ano!r1 & :at seems v complex to "o may seem simple to ano!r4 ,9 "p ,i h already answ]$ by say+ precisely 9 :at cases diff]5tial equations become too simple = ! laws ( *ance to rema9 applicable4 ,b x is fitt+ to exam9e ! matt] a little m closely1 because we may take /ill o!r po9ts ( view4 ,:at means ! phrase 'v slight'8 ,to und]/& x we ne$ only g back to :at has already be5 said4 ,a diff]5ce is v slight1 an int]val is v small1 :5 )9 ! limits ( ?is int]val ! probability rema9s s5sibly con/ant4 ,& :y may ?is probability be reg>d$ z con/ant )9 a small int]val8 ,x is because we assume t ! law ( probability is repres5t$ by a cont9u\s curve1 cont9u\s n only 9 ! analytic s5se1 b practically cont9u\s1 z already expla9$4 ,?is means t x n only pres5ts no absolute hiatus1 b t x has nei!r sali5ts nor re5trants too acute or too acc5tuat$4 ,& :at gives u ! "r to make ?is hypo!sis8 ,we h already said x is because1 s9ce ! beg9n+ ( ! ages1 !re h always be5 complex causes ceaselessly act+ 9 ! same way & mak+ ! _w t5d t[>d uni=mity )\t ev] be+ able to turn back4 ,^! are ! causes :i* little by little h flatt5$ ! sali5ts & fill$ up ! re5trants1 & ?is is :y our probability curves n[ %[ only g5tle undulations4 ,9 milli>ds ( milli>ds ( ages ano!r /ep w h be5 made t[>d uni=mity1 & ^! undulations w be t5 times z g5tle2 ! radius ( mean curvature ( our curve w h become t5 times z great4 ,& !n su* a l5g? z seems to u to-day n v small1 s9ce on our curve an arc ( ?is l5g? c n be reg>d$ z rectil9eal1 %\ld on ! contr>y at t epo* be call$ v little1 s9ce ! curvature w h become t5 times less & an arc ( ?is l5g? may be s5sibly id5tifi$ ) a sect4 ,?us ! phrase 'v slight' rema9s relative2 b x is n relative to su* or su* a man1 x is relative to ! actual /ate ( ! _w4 ,x w *ange its mean+ :5 ! _w %all h become m uni=m1 :5 all ?+s %all h bl5d$ /ill m4 ,b !n d\btless m5 c no long] live & mu/ give place to o!r be+s--%\ld ,i say f> small] or f> l>g]8 ,s t our crit]ion1 rema9+ true = all m51 reta9s an objective s5se4 ,& on ! o!r h& :at means ! phrase 'v complex'8 ,i h already giv5 "o solution1 b !re are o!rs4 ,complex causes we h said produce a bl5d m & m intimate1 b aft] h[ long a "t w ?is bl5d satisfy u8 ,:5 w x h accumulat$ suffici5t complexity8 ,:5 %all we h suffici5tly %uffl$ ! c>ds8 ,if we mix two p[d]s1 "o blue1 ! o!r :ite1 !re comes a mom5t :5 ! t9t ( ! mixture seems to u uni=m because ( ! feebl5ess ( our s5ses2 x w be uni=m = ! presbyte1 =c$ to gaze f af>1 be=e x w be s = ! myope4 ,& :5 x has become uni=m = all eyes1 we /ill c\ld pu% back ! limit by ! use ( in/rum5ts4 ,!re is no *ance = any man ev] to disc]n ! inf9ite v>iety :i*1 if ! k9etic !ory is true1 hides "u ! uni=m appe>ance ( a gas4 ,& yet if we accept ,g\y's ideas on ! ,br[nian movem5t1 does n ! microscope seem on ! po9t ( %[+ u some?+ analog\s8 ,?is new crit]ion is !re=e relative l ! fir/2 & if x reta9s an objective "*1 x is because all m5 h approximately ! same s5ses1 ! p[] ( _! in/rum5ts is limit$1 & besides !y use !m only exceptionally4 ,,ix ,x is j ! same 9 ! moral sci5ces & p>ticul>ly 9 hi/ory4 ,! hi/orian is oblig$ to make a *oice among ! ev5ts ( ! epo* he /udies2 he rec\nts only ^? :i* seem to him ! mo/ important4 ,he !re=e cont5ts himself ) relat+ ! mo/ mom5t\s ev5ts ( ! sixte5? c5tury1 = example1 z likewise ! mo/ rem>kable facts ( ! sev5te5? c5tury4 ,if ! fir/ suffice to expla9 ! second1 we say ^! con=m to ! laws ( hi/ory4 ,b if a great ev5t ( ! sev5te5? c5tury %\ld h = cause a small fact ( ! sixte5? c5tury :i* no hi/ory reports1 :i* all ! _w has neglect$1 !n we say ?is ev5t is due to *ance4 ,?is ^w has !re=e ! same s5se z 9 ! physical sci5ces2 x means t slight causes h produc$ great effects4 ,! greate/ bit ( *ance is ! bir? ( a great man4 ,x is only by *ance t meet+ ( two g]m9al cells1 ( diff]5t sex1 conta9+ precisely1 ea* on its side1 ! my/]i\s elem5ts ^: mutual reaction mu/ produce ! g5ius4 ,"o w agree t ^! elem5ts mu/ be r>e & t _! meet+ is /ill m r>e4 ,h[ slight a ?+ x w\ld h requir$ to deflect f its r\te ! c>ry+ sp]matozoon4 ,x w\ld h suffic$ to deflect x a t5? ( a millimet] & ,napoleon w\ld n h be5 born & ! de/9ies ( a cont95t w\ld h be5 *ang$4 ,no example c bett] make u und]/& ! v]itable *>act]i/ics ( *ance4 ,"o m ^w ab\t ! p>adoxes br\ght out by ! application ( ! calculus ( probabilities to ! moral sci5ces4 ,x has be5 prov5 t no ,*amb] ( ,deputies w ev] fail to conta9 a memb] ( ! opposition1 or at lea/ su* an ev5t w\ld be s improbable t we might )\t fe> wag] ! contr>y1 & bet a million aga9/ a s\4 ,condorcet has /riv5 to calculate h[ _m jurors x w\ld require to make a judicial error practically impossible4 ,if we _h us$ ! results ( ?is calculation1 we %\ld c]ta9ly h be5 expos$ to ! same disappo9tm5ts z 9 bett+1 on ! fai? ( ! calculus1 t ! opposition w\ld nev] be )\t a repres5tative4 ,! laws ( *ance d n apply to ^! que/ions4 ,if ju/ice be n always met$ out to accord ) ! be/ reasons1 x uses less ?an we ?9k ! me?od ( ,bridoye4 ,?is is p]haps to be regrett$1 = !n ! sy/em ( ,condorcet w\ld %ield u f judicial errors4 ,:at is ! mean+ ( ?is8 ,we are tempt$ to attribute facts ( ?is nature to *ance because _! causes are obscure2 b ?is is n true *ance4 ,! causes are unkn[n to u1 x is true1 & !y are ev5 complex2 b !y are n suffici5tly s1 s9ce !y cons]ve some?+4 ,we h se5 t ?is x is :i* di/+ui%es causes 'too simple4' ,:5 m5 are br\ght toge!r !y no long] decide at r&om & indep5d5tly "o ( ano!r2 !y influ5ce "o ano!r4 ,multiplex causes come into action4 ,!y worry m51 dragg+ !m to "r or left1 b "o ?+ !re is !y c n de/roy1 ?is is _! ,panurge flock-(-%eep habits4 ,& ?is is an inv>iant4 ,x ,difficulties are inde$ involv$ 9 ! application ( ! calculus ( probabilities to ! exact sci5ces4 ,:y are ! decimals ( a table ( log>i?ms1 :y are ^? ( ! numb] pi di/ribut$ 9 accordance ) ! laws ( *ance8 ,else:]e ,i h already /udi$ ! "q 9 s f> z x conc]ns log>i?ms1 & !re x is easy4 ,x is cle> t a slight diff]5ce ( argum5t w give a slight diff]5ce ( log>i?m1 b a great diff]5ce 9 ! six? decimal ( ! log>i?m4 ,always we f9d aga9 ! same crit]ion4 ,b z = ! numb] pi1 t pres5ts m difficulties1 & ,i h at ! mom5t no?+ wor? :ile to say4 ,!re w\ld be _m o!r que/ions to resolve1 _h ,i wi%$ to attack !m be=e solv+ t :i* ,i m specially set myself4 ,:5 we rea* a simple result1 :5 we f9d = example a r\nd numb]1 we say t su* a result c n be due to *ance1 & we seek1 = its explanation1 a non-=tuit\s cause4 ,& 9 fact !re is only a v slight probability t among #aj1jjj numb]s *ance w give a r\nd numb]2 = example1 ! numb] #aj1jjj4 ,?is has only "o *ance 9 #aj1jjj4 ,b !re is only "o *ance 9 #aj1jjj = ! occurr5ce ( any o!r "o numb]2 & yet ?is result w n a/oni% u1 nor w x be h>d = u to attribute x to *ance2 & t simply because x w be less /rik+4 ,is ?is a simple illusion ( ours1 or are !re cases :]e ?is way ( ?9k+ is legitimate8 ,we mu/ hope s1 else 7 all sci5ce impossible4 ,:5 we wi% to *eck a hypo!sis1 :at d we d8 ,we c n v]ify all its consequ5ces1 s9ce !y w\ld be inf9ite 9 numb]2 we cont5t ourselves ) v]ify+ c]ta9 ones & if we succe$ we decl>e ! hypo!sis confirm$1 because s mu* success c\ld n be due to *ance4 ,& ?is is always at bottom ! same reason+4 ,i c n completely ju/ify x "h1 s9ce x w\ld take too mu* "t2 b ,i may at lea/ say t we f9d ourselves confront$ by two hypo!ses1 ei!r a simple cause or t aggregate ( complex causes we call *ance4 ,we f9d x natural to suppose t ! fir/ %\ld produce a simple result1 & !n1 if we f9d t simple result1 ! r\nd numb] = example1 x seems m likely to u to be attributable to ! simple cause :i* mu/ give x almo/ c]ta9ly1 ?an to *ance :i* c\ld only give x once 9 #aj1jjj times4 ,x w n be ! same if we f9d a result :i* is n simple2 *ance1 x is true1 w n give ?is m ?an once 9 #aj1jjj times2 b nei!r has ! simple cause any m *ance ( produc+ x4 ,,book ,,ii ,,ma!matical ,,reason+ ,,*apt] ,i ,,! ,,relativity ,,( ,,space ,i ,x is impossible to repres5t to oneself empty space2 all our ef=ts to imag9e a pure space1 :5ce %\ld be exclud$ ! *ang+ images ( mat]ial objects1 c result only 9 a repres5tation :]e vividly color$ surfaces1 = example1 are replac$ by l9es ( fa9t coloration1 & we c n g to ! v end 9 ?is way )\t all vani%+ & t]m9at+ 9 no?+ness4 ,!nce comes ! irr$ucible relativity ( space4 ,:oev] speaks ( absolute space uses a mean+less phrase4 ,?is is a tru? long proclaim$ by all :o h reflect$ ^u ! matt]1 b :i* we are too (t5 l$ to =get4 ,i am at a det]m9ate po9t 9 ,p>is1 place du ,pan?on = in/ance1 & ,i say3 ,i %all come back "h to-morr[4 ,if ,i be ask$3 ,d y mean y w return to ! same po9t ( space1 ,i %all be tempt$ to answ]3 yes2 & yet ,i %all be wrong1 s9ce by to-morr[ ! e>? w h j\rney$ h5ce1 c>ry+ ) x ! place du ,pan?on1 :i* w h travel$ ov] m ?an two million kilomet]s4 ,& if ,i tri$ to speak m precisely1 ,i %\ld ga9 no?+1 s9ce our globe has run ov] ^! two million kilomet]s 9 its motion ) relation to ! sun1 :ile ! sun 9 its turn is displac$ ) ref]5ce to ! ,milky ,way1 :ile ! ,milky ,way itself is d\btless 9 motion )\t our be+ able to p]ceive its velocity4 ,s t we are completely ignorant1 & always %all be1 ( h[ mu* ! place du ,pan?on is displac$ 9 a day4 ,9 sum1 ,i meant to say3 ,to-morr[ ,i %all see aga9 ! dome & ! p$im5t ( ! ,pan?on1 & if !re 7 no ,pan?on my phrase w\ld be mean+less & space w\ld vani%4 ,?is is "o ( ! mo/ commonplace =ms ( ! pr9ciple ( ! relativity ( space2 b !re is ano!r1 ^u :i* ,delbeuf has p>ticul>ly insi/$4 ,suppose t 9 ! night all ! dim5sions ( ! univ]se become a ?\s& times great]3 ! _w w h rema9$ simil> to itself1 giv+ to ! ^w similitude ! same mean+ z 9 ,euclid1 ,book ,,vi4 ,only :at 0 a met] long w measure !nce=? a kilomet]1 :at 0 a millimet] long w become a met]4 ,! b$ :]eon ,i lie & my body itself w be enl>g$ 9 ! same proportion4 ,:5 ,i awake to-morr[ morn+1 :at s5sation %all ,i feel 9 pres5ce ( su* an a/\nd+ trans=mation8 ,well1 ,i %all p]ceive no?+ at all4 ,! mo/ precise measurem5ts w be incapable ( reveal+ to me any?+ ( ?is imm5se convulsion1 s9ce ! measures ,i use w h v>i$ precisely 9 ! same proportion z ! objects ,i seek to measure4 ,9 reality1 ?is convulsion exi/s only = ^? :o reason z if space 7 absolute4 ,if ,i = a mom5t h reason$ z !y d1 x is ! bett] to br+ out t _! way ( see+ implies contradiction4 ,9 fact x w\ld be bett] to say t1 space be+ relative1 no?+ at all has happ5$1 :i* is :y we h p]ceiv$ no?+4 ,has "o ! "r1 !re=e1 to say he kn[s ! di/ance betwe5 two po9ts8 ,no1 s9ce ?is di/ance c\ld und]go enorm\s v>iations )\t our be+ able to p]ceive !m1 provid$ ! o!r di/ances h v>i$ 9 ! same proportion4 ,we h j se5 t :5 ,i say3 ,i %all be "h to-morr[1 ?is does n mean3 ,to-morr[ ,i %all be at ! same po9t ( space :]e ,i am to-day1 b r3 ,to-morr[ ,i %all be at ! same di/ance f ! ,pan?on z to-day4 ,& we see t ?is /atem5t is no long] suffici5t & t ,i %\ld say3 ,to-morr[ & to-day my di/ance f ! ,pan?on w be equal to ! same numb] ( times ! height ( my body4 ,b ?is is n all2 ,i h suppos$ ! dim5sions ( ! _w to v>y1 b t at lea/ ! _w rema9$ always simil> to itself4 ,we might g mu* fur!r1 & "o ( ! mo/ a/oni%+ !ories ( mod]n physics furni%es u ! occasion4 ,accord+ to ,lor5tz & ,fitzg]ald1 all ! bodies borne along 9 ! motion ( ! e>? und]go a de=mation4 ,?is de=mation is1 9 reality1 v slight1 s9ce all dim5sions p>allel to ! movem5t ( ! e>? dim9i% by a hundr$ million?1 :ile ! dim5sions p]p5dicul> to ?is movem5t are un*ang$4 ,b x matt]s little t x is slight1 t x exi/s suffices = ! conclusion ,i am ab\t to draw4 ,& besides1 ,i h said x 0 slight1 b 9 reality ,i "k no?+ ab\t x2 ,i h myself be5 victim ( ! t5aci\s illusion :i* makes u believe we conceive an absolute space2 ,i h ?\ght ( ! motion ( ! e>? 9 its elliptic orbit ar\nd ! sun1 & ,i h all[$ ?irty kilomet]s z its velocity4 ,b its real velocity 7,i mean1 ?is "t1 n its absolute velocity1 :i* is mean+less1 b its velocity ) relation to ! e!r71 ,i d n "k t1 & h no means ( kn[+ x3 x is p]haps1 #aj1 #ajj times great]1 & !n ! de=mation w be #ajj1 #aj1jjj times m4 ,c we %[ ?is de=mation8 ,evid5tly n2 "h is a cube ) edge "o met]2 9 consequ5ce ( ! e>?'s displacem5t x is de=m$1 "o ( its edges1 t p>allel to ! motion1 becomes small]1 ! o!rs d n *ange4 ,if ,i wi% to assure myself ( x by aid ( a met] measure1 ,i %all measure fir/ "o ( ! edges p]p5dicul> to ! motion & %all f9d t my /&>d met] fits ?is edge exactly2 & 9 fact nei!r ( ^! two l5g?s is *ang$1 s9ce bo? are p]p5dicul> to ! motion4 ,!n ,i wi% to measure ! o!r edge1 t p>allel to ! motion2 to d ?is ,i displace my met] & turn x s z to apply x to ! edge4 ,b ! met]1 hav+ *ang$ ori5tation & become p>allel to ! motion1 has und]gone1 9 its turn1 ! de=mation1 s t ?\gh ! edge be n a met] long1 x w fit exactly1 ,i %all f9d out no?+4 ,y ask !n ( :at use is ! hypo!sis ( ,lor5tz & ( ,fitzg]ald if no exp]im5t c p]mit ( its v]ification8 ,x is my exposition t has be5 incomplete2 ,i h spok5 only ( measurem5ts t c be made ) a met]2 b we c also measure a l5g? by ! "t x takes light to trav]se x1 on condition we suppose ! velocity ( light con/ant & indep5d5t ( direction4 ,lor5tz c\ld h acc\nt$ = ! facts by suppos+ ! velocity ( light great] 9 ! direction ( ! e>?'s motion ?an 9 ! p]p5dicul> direction4 ,he pref]r$ to suppose t ! velocity is ! same 9 ^! diff]5t directions b t ! bodies are small] 9 ! "o ?an 9 ! o!r4 ,if ! wave surfaces ( light _h und]gone ! same de=mations z ! mat]ial bodies we %\ld nev] h p]ceiv$ ! ,lor5tz-,fitzg]ald de=mation4 ,9 ei!r case1 x is n a "q ( absolute magnitude1 b ( ! measure ( ?is magnitude by means ( "s in/rum5t2 ?is in/rum5t may be a met]1 or ! pa? trav]s$ by light2 x is only ! relation ( ! magnitude to ! in/rum5t t we measure2 & if ?is relation is alt]$1 we h no way ( kn[+ :e!r x is ! magnitude or ! in/rum5t :i* has *ang$4 ,b :at ,i wi% to br+ out is1 t 9 ?is de=mation ! _w has n rema9$ simil> to itself2 squ>es h become rectangles1 circles ellipses1 sph]es ellipsoids4 ,& yet we h no way ( kn[+ :e!r ?is de=mation be real4 ,evid5tly "o c\ld g mu* fur!r3 9 place ( ! ,lor5tz-,fitzg]ald de=mation1 ^: laws are p>ticul>ly simple1 we c\ld imag9e any de=mation :atsoev]4 ,bodies c\ld be de=m$ accord+ to any laws1 z complicat$ z we might wi%1 we nev] %\ld notice x provid$ all bodies )\t exception 7 de=m$ accord+ to ! same laws4 ,9 say+1 all bodies )\t exception1 ,i include ( c\rse our own body & ! light rays emanat+ f diff]5t objects4 ,if we look at ! _w 9 "o ( ^? mirrors ( complicat$ %ape :i* de=m objects 9 a biz>re way1 ! mutual relations ( ! diff]5t p>ts ( ?is _w w\ld n be alt]$2 if1 9 fact two real objects t\*1 _! images likewise seem to t\*4 ,( c\rse :5 we look 9 su* a mirror we see inde$ ! de=mation1 b ?is is because ! real _w subsi/s alongside ( its de=m$ image2 & !n ev5 7 ?is real _w hidd5 f u1 some?+ !re is c\ld n be hidd51 ourself2 we c\ld n cease to see1 or at lea/ to feel1 our body & our limbs :i* h n be5 de=m$ & :i* cont9ue to s]ve u z in/rum5ts ( measure4 ,b if we imag9e our body itself de=m$ 9 ! same way z if se5 9 ! mirror1 ^! in/rum5ts ( measure 9 _! turn w fail u & ! de=mation w no long] be asc]ta9able4 ,consid] 9 ! same way two worlds images ( "o ano!r2 to ea* object ,p ( ! _w ,a corresponds 9 ! _w ,b an object ,p'1 its image2 ! coord9ates ( ?is image ,p' are det]m9ate functions ( ^? ( ! object ,p2 moreov] ^! functions may be any :atsoev]2 ,i only suppose !m *os5 once = all4 ,betwe5 ! position ( ,p & t ( ,p' !re is a con/ant relation2 :at ?is relation is1 matt]s n2 en\gh t x be con/ant4 ,well1 ^! two worlds w be indi/+ui%able "o f ! o!r4 ,i mean ! fir/ w be = its inhabitants :at ! second is = its4 ,& s x w be z long z ! two worlds rema9 /rang]s to ea* o!r4 ,suppose we liv$ 9 _w ,a1 we %all h con/ruct$ our sci5ce & 9 p>ticul> our geometry2 dur+ ?is "t ! inhabitants ( _w ,b w h con/ruct$ a sci5ce1 & z _! _w is ! image ( ours1 _! geometry w also be ! image ( ours or1 bett]1 x w be ! same4 ,b if = u "s day a w9d[ is op5$ ^u _w ,b1 h[ we %all pity !m3 ,poor ?+s1 we %all say1 !y ?9k !y h made a geometry1 b :at !y call s is only a grotesque image ( ours2 _! /raights are all twi/$1 _! circles are hump$1 _! sph]es h caprici\s inequalities4 ,& we %all nev] suspect !y say ! same ( u1 & "o nev] w "k :o is "r4 ,we see 9 h[ broad a s5se %\ld be und]/ood ! relativity ( space2 space is 9 reality amorph\s & ! ?+s :i* are !re9 alone give x a =m4 ,:at !n %\ld be ?\ght ( t direct intuition we %\ld h ( ! /raight or ( di/ance8 ,s little h we intuition ( di/ance 9 itself t 9 ! night1 z we h said1 a di/ance might become a ?\s& times great] )\t our be+ able to p]ceive x1 if all o!r di/ances _h und]gone ! same alt]ation4 ,& ev5 9 a night ! _w ,b might be sub/itut$ = ! _w ,a )\t our hav+ any way ( kn[+ x1 & !n ! /raight l9es ( ye/]day w\ld h ceas$ to be /raight & we %\ld nev] notice4 ,"o "p ( space is n by itself & 9 ! absolute s5se ( ! ^w equal to ano!r "p ( space2 because if s x is = u1 x w\ld n be = ! dwell]s 9 _w ,b2 & ^! h j z mu* "r to reject our op9ion z we to condemn !irs4 ,i h else:]e %[n :at are ! consequ5ces ( ^! facts f ! viewpo9t ( ! idea we %\ld =m ( non-,euclidean geometry & o!r analog\s geometries2 to t ,i d n c>e to return2 & to-day ,i %all take a some:at diff]5t po9t ( view4 ,,ii ,if ?is intuition ( di/ance1 ( direction1 ( ! /raight l9e1 if ?is direct intuition ( space 9 a ^w does n exi/1 :5ce comes our belief t we h x8 ,if ?is is only an illusion1 :y is ?is illusion s t5aci\s8 ,x is prop] to exam9e into ?is4 ,we h said !re is no direct intuition ( size & we c only arrive at ! relation ( ?is magnitude to our in/rum5ts ( measure4 ,we %\ld !re=e n h be5 able to con/ruct space if we _h n _h an in/rum5t to measure x2 well1 ?is in/rum5t to :i* we relate ev]y?+1 :i* we use in/9ctively1 x is our own body4 ,x is 9 relation to our body t we place ext]ior objects1 & ! only spatial relations ( ^! objects t we c repres5t are _! relations to our body4 ,x is our body :i* s]ves u1 s to speak1 z sy/em ( axes ( coord9ates4 ,= example1 at an in/ant alpha1 ! pres5ce ( ! object ,a is reveal$ to me by ! s5se ( sight2 at ano!r in/ant1 beta1 ! pres5ce ( ano!r object1 ,b1 is reveal$ to me by ano!r s5se1 t ( he>+ or ( t\*1 = in/ance4 ,i judge t ?is object ,b occupies ! same place z ! object ,a4 ,:at does t mean8 ,fir/ t does n signify t ^! two objects occupy1 at two diff]5t mom5ts1 ! same po9t ( an absolute space1 :i* ev5 if x exi/$ w\ld escape our cognition1 s9ce1 betwe5 ! in/ants alpha & beta1 ! sol> sy/em has mov$ & we c n "k its displacem5t4 ,t means ^! two objects occupy ! same relative position ) ref]5ce to our body4 ,b ev5 ?is1 :at does x mean8 ,! impressions t h come to u f ^! objects h foll[$ pa?s absolutely diff]5t1 ! optic n]ve = ! object ,a1 ! ac\/ic n]ve = ! object ,b4 ,!y h no?+ 9 common f ! qualitative po9t ( view4 ,! repres5tations we are able to make ( ^! two objects are absolutely het]og5e\s1 irr$ucible "o to ! o!r4 ,only ,i "k t to rea* ! object ,a ,i h j to ext5d ! "r arm 9 a c]ta9 way2 ev5 :5 ,i ab/a9 f do+ x1 ,i repres5t to myself ! muscul> s5sations & o!r analog\s s5sations :i* w\ld accompany ?is ext5sion1 & ?is repres5tation is associat$ ) t ( ! object ,a4 ,n[1 ,i likewise "k ,i c rea* ! object ,b by ext5d+ my "r arm 9 ! same mann]1 an ext5sion accompani$ by ! same tra9 ( muscul> s5sations4 ,& :5 ,i say ^! two objects occupy ! same place1 ,i mean no?+ m4 ,i also "k ,i c\ld h rea*$ ! object ,a by ano!r appropriate motion ( ! left arm & ,i repres5t to myself ! muscul> s5sations :i* w\ld h accompani$ ?is movem5t2 & by ?is same motion ( ! left arm1 accompani$ by ! same s5sations1 ,i likewise c\ld h rea*$ ! object ,b4 ,& t is v important1 s9ce ?us ,i c def5d myself aga9/ dang]s m5ac+ me f ! object ,a or ! object ,b4 ,) ea* ( ! bl[s we c be hit1 nature has associat$ "o or m p>ries :i* p]mit ( our gu>d+ ourselves4 ,! same p>ry may respond to sev]al /rokes2 & s x is1 = in/ance1 t ! same motion ( ! "r arm w\ld h all[$ u to gu>d at ! in/ant alpha aga9/ ! object ,a & at ! in/ant beta aga9/ ! object ,b4 ,j s1 ! same /roke c be p>ri$ 9 sev]al ways1 & we h said1 = in/ance1 ! object ,a c\ld be rea*$ indiff]5tly ei!r by a c]ta9 movem5t ( ! "r arm or by a c]ta9 movem5t ( ! left arm4 ,all ^! p>ries h no?+ 9 common except w>d+ (f ! same bl[1 & ?is x is1 & no?+ else1 :i* is meant :5 we say !y are movem5ts t]m9at+ at ! same po9t ( space4 ,j s1 ^! objects1 ( :i* we say !y occupy ! same po9t ( space1 h no?+ 9 common1 except t ! same p>ry gu>ds aga9/ !m4 ,or1 if y *oose1 imag9e innum]able telegraph wires1 "s c5tripetal1 o!rs c5trifugal4 ,! c5tripetal wires w>n u ( accid5ts happ5+ )\t2 ! c5trifugal wires c>ry ! rep>ation4 ,connections are s e/abli%$ t :5 a c5tripetal wire is trav]s$ by a curr5t ?is acts on a relay & s />ts a curr5t 9 "o ( ! c5trifugal wires1 & ?+s are s arrang$ t sev]al c5tripetal wires may act on ! same c5trifugal wire if ! same rem$y suits sev]al ills1 & t a c5tripetal wire may agitate diff]5t c5trifugal wires1 ei!r simultane\sly or 9 lieu "o ( ! o!r :5 ! same ill may be cur$ by sev]al rem$ies4 ,x is ?is complex sy/em ( associations1 x is ?is table ( di/ribution1 s to speak1 :i* is all our geometry or1 if y wi%1 all 9 our geometry t is in/9ctive4 ,:at we call our intuition ( ! /raight l9e or ( di/ance is ! consci\sness we h ( ^! associations & ( _! imp]i\s "*4 ,& x is easy to und]/& :5ce comes ?is imp]i\s "* itself4 ,an association w seem to u by s mu* ! m inde/ructible z x is m anci5t4 ,b ^! associations are n1 = ! mo/ "p1 conque/s ( ! individual1 s9ce _! trace is se5 9 ! new-born babe3 !y are conque/s ( ! race4 ,natural selection _h to br+ ab\t ^! conque/s by s mu* ! m quickly z !y 7 ! m necess>y4 ,on ?is acc\nt1 ^? ( :i* we speak mu/ h be5 ( ! e>lie/ 9 date1 s9ce )\t !m ! def5se ( ! organism w\ld h be5 impossible4 ,f ! "t :5 ! cellules 7 no long] m]ely juxtapos$1 b 7 call$ ^u to give mutual aid1 x 0 ne$ful t a me*anism organize analog\s to :at we h describ$1 s t ?is aid miss n its way1 b =e/all ! p]il4 ,:5 a frog is decapitat$1 & a drop ( acid is plac$ on a po9t ( its sk91 x seeks to wipe (f ! acid ) ! ne>e/ foot1 &1 if ?is foot be amputat$1 x sweeps x (f ) ! foot ( ! opposite side4 ,!re we h ! d\ble p>ry ( :i* ,i h j spok51 all[+ ! combat+ ( an ill by a second rem$y1 if ! fir/ fails4 ,& x is ?is multiplicity ( p>ries1 & ! result+ coord9ation1 :i* is space4 ,we see to :at dep?s ( ! unconsci\s we mu/ desc5d to f9d ! fir/ traces ( ^! spatial associations1 s9ce only ! inf]ior p>ts ( ! n]v\s sy/em are involv$4 ,:y be a/oni%$ !n at ! resi/ance we oppose to e attempt made to dissociate :at s long has be5 associat$8 ,n[1 x is j ?is resi/ance t we call ! evid5ce = ! geometric tru?s2 ?is evid5ce is no?+ b ! repugnance we feel t[>d break+ ) v old habits :i* h always prov$ good4 ,,iii ,! space s creat$ is only a little space ext5d+ no f>!r ?an my arm c rea*2 ! int]v5tion ( ! memory is necess>y to pu% back its limits4 ,!re are po9ts :i* w rema9 out ( my rea*1 :atev] ef=t ,i make to /ret* =? my h&2 if ,i 7 fa/5$ to ! gr\nd l a hydra polyp1 = in/ance1 :i* c only ext5d its t5tacles1 all ^! po9ts w\ld be outside ( space1 s9ce ! s5sations we c\ld exp]i5ce f ! action ( bodies !re situat$1 w\ld be associat$ ) ! idea ( no movem5t all[+ u to rea* !m1 ( no appropriate p>ry4 ,^! s5sations w\ld n seem to u to h any spatial "* & we %\ld n seek to localize !m4 ,b we are n fix$ to ! gr\nd l ! l[] animals2 we c1 if ! enemy be too f> away1 advance t[>d him fir/ & ext5d ! h& :5 we are suffici5tly ne>4 ,?is is /ill a p>ry1 b a p>ry at long range4 ,on ! o!r h&1 x is a complex p>ry1 & into ! repres5tation we make ( x ent] ! repres5tation ( ! muscul> s5sations caus$ by ! movem5ts ( ! legs1 t ( ! muscul> s5sations caus$ by ! f9al movem5t ( ! arm1 t ( ! s5sations ( ! semicircul> canals1 etc4 ,we mu/1 besides1 repres5t to ourselves1 n a complex ( simultane\s s5sations1 b a complex ( successive s5sations1 foll[+ ea* o!r 9 a det]m9ate ord]1 & ?is is :y ,i h j said ! int]v5tion ( memory 0 necess>y4 ,notice moreov] t1 to rea* ! same po9t1 ,i may approa* ne>] ! m>k to be atta9$1 s z to h to /ret* my arm less4 ,:at m8 ,x is n "o1 x is a ?\s& p>ries ,i c oppose to ! same dang]4 ,all ^! p>ries are made ( s5sations :i* may h no?+ 9 common & yet we reg>d !m z def9+ ! same po9t ( space1 s9ce !y may respond to ! same dang] & are all associat$ ) ! notion ( ?is dang]4 ,x is ! pot5tiality ( w>d+ (f ! same /roke :i* makes ! unity ( ^! diff]5t p>ries1 z x is ! possibility ( be+ p>ri$ 9 ! same way :i* makes ! unity ( ! /rokes s diff]5t 9 k9d1 :i* may m5ace u f ! same po9t ( space4 ,x is ?is d\ble unity :i* makes ! individuality ( ea* po9t ( space1 &1 9 ! notion ( po9t1 !re is no?+ else4 ,! space be=e consid]$1 :i* might be call$ re/rict$ space1 0 ref]r$ to coord9ate axes b\nd to my body2 ^! axes 7 fix$1 s9ce my body did n move & only my memb]s 7 displac$4 ,:at are ! axes to :i* we naturally ref] ! ext5d$ space8 t is to say ! new space j def9$4 ,we def9e a po9t by ! sequ5ce ( movem5ts to be made to rea* x1 />t+ f a c]ta9 initial position ( ! body4 ,! axes are !re=e fix$ to ?is initial position ( ! body4 ,b ! position ,i call initial may be arbitr>ily *os5 among all ! positions my body has successively occupi$2 if ! memory m or less unconsci\s ( ^! successive positions is necess>y = ! g5esis ( ! notion ( space1 ?is memory may g back m or less f> into ! pa/4 ,!nce results 9 ! def9ition itself ( space a c]ta9 indet]m9ation1 & x is precisely ?is indet]m9ation :i* con/itutes its relativity4 ,!re is no absolute space1 !re is only space relative to a c]ta9 initial position ( ! body4 ,= a consci\s be+ fix$ to ! gr\nd l ! l[] animals1 & consequ5tly kn[+ only re/rict$ space1 space w\ld /ill be relative 7s9ce x w\ld h ref]5ce to 8 body71 b ?is be+ w\ld n be consci\s ( ?is relativity1 because ! axes ( ref]5ce = ?is re/rict$ space w\ld be un*ang+6 ,d\btless ! rock to :i* ?is be+ w\ld be fett]$ w\ld n be motionless1 s9ce x w\ld be c>ri$ along 9 ! movem5t ( our planet2 = u consequ5tly ^! axes w\ld *ange at ea* in/ant2 b = him !y w\ld be *angeless4 ,we h ! faculty ( ref]r+ our ext5d$ space n[ to ! position ,a ( our body1 consid]$ z initial1 aga9 to ! position ,b1 :i* x _h "s mom5ts aft]w>d1 & :i* we are free to reg>d 9 its turn z initial2 we make !re=e at ea* in/ant unconsci\s trans=mations ( coord9ates4 ,?is faculty w\ld be lack+ 9 our imag9>y be+1 & f n hav+ travel$1 he w\ld ?9k space absolute4 ,at e in/ant1 8 sy/em ( axes w\ld be impos$ ^u him2 ?is sy/em w\ld h to *ange greatly 9 reality1 b = him x w\ld be always ! same1 s9ce x w\ld be always ! only sy/em4 ,q o!rwise is x ) u1 :o at ea* in/ant h _m sy/ems betwe5 :i* we may *oose at w1 on condition ( go+ back by memory m or less f> into ! pa/4 ,?is is n all2 re/rict$ space w\ld n be homog5e\s2 ! diff]5t po9ts ( ?is space c\ld n be reg>d$ z equival5t1 s9ce "s c\ld be rea*$ only at ! co/ ( ! greate/ ef=ts1 :ile o!rs c\ld be easily atta9$4 ,on ! contr>y1 our ext5d$ space seems to u homog5e\s1 & we say all its po9ts are equival5t4 ,:at does t mean8 ,if we />t f a c]ta9 place ,a1 we c1 f ?is position1 make c]ta9 movem5ts1 ,m1 *>act]iz$ by a c]ta9 complex ( muscul> s5sations4 ,b1 />t+ f ano!r position1 ,b1 we make movem5ts ,m' *>act]iz$ by ! same muscul> s5sations4 ,let a1 !n1 be ! situation ( a c]ta9 po9t ( ! body1 ! end ( ! index f+] ( ! "r h& = example1 9 ! initial position ,a1 & b ! situation ( ?is same index :51 />t+ f ?is position ,a1 we h made ! motions ,m4 ,aft]w>ds1 let a' be ! situation ( ?is index 9 ! position ,b1 & b' its situation :51 />t+ f ! position ,b1 we h made ! motions ,m'4 ,well1 ,i am accu/om$ to say t ! po9ts ( space a & b are relat$ to ea* o!r j z ! po9ts a' & b'1 & ?is simply means t ! two s]ies ( movem5ts ,m & ,m' are accompani$ by ! same muscul> s5sations4 ,& z ,i am consci\s t1 9 pass+ f ! position ,a to ! position ,b1 my body has rema9$ capable ( ! same movem5ts1 ,i "k !re is a po9t ( space relat$ to ! po9t a' j z any po9t b is to ! po9t a1 s t ! two po9ts a & a' are equival5t4 ,?is is :at is call$ ! homog5eity ( space4 ,&1 at ! same "t1 ?is is :y space is relative1 s9ce its prop]ties rema9 ! same :e!r x be ref]r$ to ! axes ,a or to ! axes ,b4 ,s t ! relativity ( space & its homog5eity are "o sole & same ?+4 ,n[1 if ,i wi% to pass to ! great space1 :i* no long] s]ves only = me1 b :]e ,i may lodge ! univ]se1 ,i get !re by an act ( imag9ation4 ,i imag9e h[ a giant w\ld feel :o c\ld rea* ! planets 9 a few /eps2 or1 if y *oose1 :at ,i myself %\ld feel 9 pres5ce ( a m9iature _w :]e ^! planets 7 replac$ by little balls1 :ile on "o ( ^! little balls mov$ a liliputian ,i %\ld call myself4 ,b ?is act ( imag9ation w\ld be impossible = me _h ,i n previ\sly con/ruct$ my re/rict$ space & my ext5d$ space = my own use4 ,,iv ,:y n[ h all ^! spaces ?ree dim5sions8 ,g back to ! table ( di/ribution ( :i* we h spok54 ,we h on ! "o side ! li/ ( ! diff]5t possible dang]s2 designate !m by ,a#a1 ,a#b1 etc42 &1 on ! o!r side1 ! li/ ( ! diff]5t rem$ies :i* ,i %all call 9 ! same way ,b#a1 ,b#b1 etc4 ,we h !n connections betwe5 ! contact /uds or pu% buttons ( ! fir/ li/ & ^? ( ! second1 s t :51 = in/ance1 ! ann\nc] ( dang] ,a#c functions1 x w put or may put 9 action ! relay correspond+ to ! p>ry ,b#d4 ,z ,i h spok5 above ( c5tripetal or c5trifugal wires1 ,i fe> le/ "o see 9 all ?is1 n a simple comp>ison1 b a description ( ! n]v\s sy/em4 ,su* is n my ?\ght1 & t = sev]al reasons3 fir/ ,i %\ld n p]mit myself to put =? an op9ion on ! /ructure ( ! n]v\s sy/em :i* ,i d n "k1 :ile ^? :o h /udi$ x speak only circumspectly2 aga9 because1 despite my incompet5ce1 ,i well "k ?is s*eme w\ld be too simpli/ic2 & f9ally because on my li/ ( p>ries1 "s w\ld figure v complex1 :i* might ev51 9 ! case ( ext5d$ space1 z we h se5 above1 consi/ ( _m /eps foll[$ by a movem5t ( ! arm4 ,x is n a "q !n ( physical connection betwe5 two real conductors b ( psy*ologic association betwe5 two s]ies ( s5sations4 ,if ,a#a & ,a#b = in/ance are bo? associat$ ) ! p>ry ,b#a1 & if ,a#a is likewise associat$ ) ! p>ry ,b#b1 x w g5]ally happ5 t ,a#b & ,b#b w also !mselves be associat$4 ,if ?is fundam5tal law 7 n g5]ally true1 !re w\ld exi/ only an imm5se confusion & !re w\ld be no?+ resembl+ a conception ( space or a geometry4 ,h[ 9 fact h we def9$ a po9t ( space4 ,we h done x 9 two ways3 x is on ! "o h& ! aggregate ( ann\nc]s ,a 9 connection ) ! same p>ry ,b2 x is on ! o!r h& ! aggregate ( p>ries ,b 9 connection ) ! same ann\nc] ,a4 ,if our law 0 n true1 we %\ld say ,a#a & ,a#b correspond to ! same po9t s9ce !y are bo? 9 connection ) ,b#a2 b we %\ld likewise say !y d n correspond to ! same po9t1 s9ce ,a#a w\ld be 9 connection ) ,b#b & ! same w\ld n be true ( ,a#b4 ,?is w\ld be a contradiction4 ,b1 f ano!r side1 if ! law 7 rigor\sly & always true1 space w\ld be v diff]5t f :at x is4 ,we %\ld h categories /rongly contra/$ betwe5 :i* w\ld be portion$ out on ! "o h& ! ann\nc]s ,a1 on ! o!r h& ! p>ries ,b2 ^! categories w\ld be excessively num]\s1 b !y w\ld be entirely sep>at$ "o f ano!r4 ,space w\ld be compos$ ( po9ts v num]\s1 b discrete2 x w\ld be discont9u\s4 ,!re w\ld be no reason = rang+ ^! po9ts 9 "o ord] r ?an ano!r1 nor consequ5tly = attribut+ to space ?ree dim5sions4 ,b x is n s2 p]mit me to resume = a mom5t ! language ( ^? :o already "k geometry2 ?is is q prop] s9ce ?is is ! language be/ und]/ood by ^? ,i wi% to make und]/& me4 ,:5 ,i desire to p>ry ! /roke1 ,i seek to atta9 ! po9t :5ce comes ?is bl[1 b x suffices t ,i approa* q ne>4 ,!n ! p>ry ,b#a may answ] = ,a#a & = ,a#b1 if ! po9t :i* corresponds to ,b#a is suffici5tly ne> bo? to t correspond+ to ,a#a & to t correspond+ to ,a#b4 ,b x may happ5 t ! po9t correspond+ to ano!r p>ry ,b#b may be suffici5tly ne> to ! po9t correspond+ to ,a#a & n suffici5tly ne> ! po9t correspond+ to ,a#b2 s t ! p>ry ,b#b may answ] = ,a#a )\t answ]+ = ,a#b4 ,= "o :o does n yet "k geometry1 ?is translates itself simply by a d]ogation ( ! law /at$ above4 ,& !n ?+s w happ5 ?us3 ,two p>ries ,b#a & ,b#b w be associat$ ) ! same w>n+ ,a#a & ) a l>ge numb] ( w>n+s :i* we %all range 9 ! same category z ,a#a & :i* we %all make correspond to ! same po9t ( space4 ,b we may f9d w>n+s ,a#b :i* w be associat$ ) ,b#b )\t be+ associat$ ) ,b#a1 & :i* 9 comp5sation w be associat$ ) ,b#c1 :i* ,b#c 0 n associat$ ) ,a#a1 & s =?1 s t we may write ! s]ies ,b#a1 ,a#a1 ,b#b1 ,a#b1 ,b#c1 ,a#c1 ,b#d1 ,a#d1 :]e ea* t]m is associat$ ) ! foll[+ & ! prec$+1 b n ) ! t]ms sev]al places away4 ,ne$less to add t ea* ( ! t]ms ( ^! s]ies is n isolat$1 b =ms "p ( a v num]\s category ( o!r w>n+s or ( o!r p>ries :i* h ! same connections z x1 & :i* may be reg>d$ z belong+ to ! same po9t ( space4 ,! fundam5tal law1 ?\gh admitt+ ( exceptions1 rema9s !re=e almo/ always true4 ,only1 9 consequ5ce ( ^! exceptions1 ^! categories1 9 place ( be+ entirely sep>at$1 encroa* p>tially "o ^u ano!r & mutually p5etrate 9 a c]ta9 measure1 s t space becomes cont9u\s4 ,on ! o!r h&1 ! ord] 9 :i* ^! categories are to be rang$ is no long] arbitr>y1 & if we ref] to ! prec$+ s]ies1 we see x is necess>y to put ,b#b betwe5 ,a#a & ,a#b & consequ5tly betwe5 ,b#a & ,b#c & t we c\ld n = in/ance put x betwe5 ,b#c & ,b#d4 ,!re is !re=e an ord] 9 :i* are naturally arrang$ our categories :i* correspond to ! po9ts ( space1 & exp]i5ce tea*es u t ?is ord] pres5ts itself "u ! =m ( a table ( triple entry1 & ?is is :y space has ?ree dim5sions4 ,v ,s ! *>act]i/ic prop]ty ( space1 t ( hav+ ?ree dim5sions1 is only a prop]ty ( our table ( di/ribution1 an int]nal prop]ty ( ! human intellig5ce1 s to speak4 ,x w\ld suffice to de/roy c]ta9 ( ^! connections1 t is to say ( ! associations ( ideas to give a diff]5t table ( di/ribution1 & t might be en\gh = space to acquire a f\r? dim5sion4 ,"s p]sons w be a/oni%$ at su* a result4 ,! ext]nal _w1 !y w ?9k1 %\ld c\nt = some?+4 ,if ! numb] ( dim5sions comes f ! way we are made1 !re might be ?9k+ be+s liv+ 9 our _w1 b :o might be made diff]5tly f u & :o w\ld believe space has m or less ?an ?ree dim5sions4 ,has n ,m4 de ,cyon said t ! ,japanese mice1 hav+ only two pair ( semicircul> canals1 believe t space is two-dim5sional8 ,& !n ?is ?9k+ be+1 if he is capable ( con/ruct+ a physics1 w\ld he n make a physics ( two or ( f\r dim5sions1 & :i* 9 a s5se w\ld /ill be ! same z ours1 s9ce x w\ld be ! description ( ! same _w 9 ano!r language8 ,x seems 9 fact t x w\ld be possible to translate our physics into ! language ( geometry ( f\r dim5sions2 to attempt ?is translation w\ld be to take great pa9s = little pr(it1 & ,i %all conf9e myself to cit+ ! me*anics ( ,h]tz :]e we h some?+ analog\s4 ,h[ev]1 x seems t ! translation w\ld always be less simple ?an ! text1 & t x w\ld always h ! air ( a translation1 t ! language ( ?ree dim5sions seems ! bett] fitt$ to ! description ( our _w1 al?\gh ?is description c be rigor\sly made 9 ano!r idiom4 ,besides1 our table ( di/ribution 0 n made at r&om4 ,!re is connection betwe5 ! w>n+ ,a#a & ! p>ry ,b#a1 ?is is an int]nal prop]ty ( our intellig5ce2 b :y ?is connection8 ,x is because ! p>ry ,b#a af=ds means effectively to gu>d aga9/ ! dang] ,a#a2 & ?is is a fact ext]ior to u1 ?is is a prop]ty ( ! ext]ior _w4 ,our table ( di/ribution is !re=e only ! translation ( an aggregate ( ext]ior facts2 if x has ?ree dim5sions1 ?is is because x has adapt$ itself to a _w hav+ c]ta9 prop]ties2 & ! *ief ( ^! prop]ties is t !re exi/ natural solids ^: displacem5ts foll[ s5sibly ! laws we call laws ( motion ( rigid solids4 ,if !re=e ! language ( ?ree dim5sions is t :i* p]mits u mo/ easily to describe our _w1 we %\ld n be a/oni%$2 ?is language is copi$ f our table ( di/ribution2 & x is 9 ord] to be able to live 9 ?is _w t ?is table has be5 e/abli%$4 ,i h said we c\ld conceive1 liv+ 9 our _w1 ?9k+ be+s ^: table ( di/ribution w\ld be f\r-dim5sional & :o consequ5tly w\ld ?9k 9 hyp]space4 ,x is n c]ta9 h[ev] t su* be+s1 admitt+ !y 7 born !re1 c\ld live !re & def5d !mselves aga9/ ! ?\s& dang]s by :i* !y w\ld !re be assail$4 ,,vi ,a few rem>ks to end )4 ,!re is a /rik+ contra/ betwe5 ! r\ghness ( ?is primitive geometry1 r$ucible to :at ,i call a table ( di/ribution1 & ! inf9ite precision ( ! geomet]s' geometry4 ,& yet ?is is born ( t2 b n ( t alone2 x mu/ be made fecund by ! faculty we h ( con/ruct+ ma!matical concepts1 su* z t ( gr\p1 = in/ance2 x 0 ne$ful to seek among ! pure concepts t :i* be/ adapts itself to ?is r\gh space ^: g5esis ,i h s\ght to expla9 & :i* is common to u & ! high] animals4 ,! evid5ce = c]ta9 geometric po/ulates1 we h said1 is only our repugnance to r5\nc+ v old habits4 ,b ^! po/ulates are inf9itely precise1 :ile ^! habits h some?+ ab\t !m ess5tially pliant4 ,:5 we wi% to ?9k1 we ne$ po/ulates inf9itely precise1 s9ce ?is is ! only way to avoid contradiction2 b among all ! possible sy/ems ( po/ulates1 !re are "s we dislike to *oose because !y are n suffici5tly 9 accord ) our habits2 h[ev] pliant1 h[ev] ela/ic !y may be1 ^! h a limit ( ela/icity4 ,we see t if geometry is n an exp]im5tal sci5ce1 x is a sci5ce born apropos ( exp]i5ce2 t we h creat$ ! space x /udies1 b adapt+ x to ! _w :]e9 we live4 ,we h select$ ! mo/ conv5i5t space1 b exp]i5ce has guid$ our *oice2 z ?is *oice has be5 unconsci\s1 we ?9k x has be5 impos$ ^u u2 "s say exp]i5ce imposes x1 o!rs t we are born ) our space ready made2 we see f ! prec$+ consid]ations1 :at 9 ^! two op9ions is ! "p ( tru?1 :at ( error4 ,9 ?is progressive education ^: outcome has be5 ! con/ruction ( space1 x is v difficult to det]m9e :at is ! "p ( ! individual1 :at ! "p ( ! race4 ,h[ f> c\ld "o ( u1 transport$ f bir? to an entirely diff]5t _w1 :]e 7 dom9ant1 = in/ance1 bodies mov+ 9 con=mity to ! laws ( motion ( non-,euclidean solids1 r5\nce ! ance/ral space to build a space completely new8 ,! "p ( ! race seems inde$ prepond]ant2 yet if to x we owe r\gh space1 ! s(t space ,i h spok5 (1 ! space ( ! high] animals1 is x n to ! unconsci\s exp]i5ce ( ! individual we owe ! inf9itely precise space ( ! geomet]8 ,?is is a "q n easy to solve4 ,yet we cite a fact %[+ t ! space our ance/ors h bequea!d u /ill reta9s a c]ta9 pla/icity4 ,"s hunt]s le>n to %oot fi% "u wat]1 ?\gh ! image ( ^! fi% be turn$ up by refraction4 ,besides !y d x in/9ctively3 !y !re=e h le>n$ to modify _! old in/9ct ( direction2 or1 if y *oose1 to sub/itute = ! association ,a#a1 ,b#a1 ano!r association ,a#a1 ,b#b1 because exp]i5ce %[$ !m ! fir/ w\ld n "w4 ,,*apt] ,,ii ,,ma!matical ,,def9itions ,,& ,,tea*+ #a4 ,i %\ld speak "h ( g5]al def9itions 9 ma!matics2 at lea/ t is ! title1 b x w be impossible to conf9e myself to ! subject z /rictly z ! rule ( unity ( action w\ld require2 ,i %all n be able to treat x )\t t\*+ ^u a few o!r relat$ que/ions1 & if ?us ,i am =c$ f "t to "t to walk on ! bord]+ fl[]-b$s on ! "r or left1 ,i pray y be> ) me4 ,:at is a good def9ition8 ,= ! philosoph] or ! sci5ti/ x is a def9ition :i* applies to all ! objects def9$1 & only ^?2 x is ! "o satisfy+ ! rules ( logic4 ,b 9 tea*+ x is n t2 a good def9ition is "o und]/ood by ! s*ol>s4 ,h[ does x happ5 t s _m refuse to und]/& ma!matics8 ,is t n some?+ ( a p>adox8 ,lo & behold6 a sci5ce appeal+ only to ! fundam5tal pr9ciples ( logic1 to ! pr9ciple ( contradiction1 = in/ance1 to t :i* is ! skeleton1 s to speak1 ( our intellig5ce1 to t ( :i* we c n dive/ ourselves )\t ceas+ to ?9k1 & !re are p :o f9d x obscure6 & !y are ev5 9 ! majority6 ,t !y are incapable ( inv5t+ may pass1 b t !y d n und]/& ! demon/rations %[n !m1 t !y rema9 bl9d :5 we %[ !m a light :i* seems to u fla%+ pure flame1 ?is x is :i* is altoge!r prodigi\s4 ,& yet !re is no ne$ ( a wide exp]i5ce ) exam9ations to "k t ^! bl9d m5 are 9 no wise exceptional be+s4 ,?is is a problem n easy to solve1 b :i* %\ld engage ! att5tion ( all ^? wi%+ to devote !mselves to tea*+4 ,:at is x1 to und]/&8 ,has ?is ^w ! same mean+ = all ! _w8 ,to und]/& ! demon/ration ( a !orem1 is t to exam9e successively ea* ( ! syllogisms compos+ x & to asc]ta9 its correctness1 its con=mity to ! rules ( ! game8 ,likewise1 to und]/& a def9ition1 is ?is m]ely to recognize t "o already kn[s ! mean+ ( all ! t]ms employ$ & to asc]ta9 t x implies no contradiction8 ,= "s1 yes2 :5 !y h done ?is1 !y w say3 ,i und]/&4 ,= ! majority1 no4 ,almo/ all are mu* m exact+2 !y wi% to "k n m]ely :e!r all ! syllogisms ( a demon/ration are correct1 b :y !y l9k toge!r 9 ?is ord] r ?an ano!r4 ,9 s f> z to !m !y seem eng5d]$ by caprice & n by an intellig5ce always consci\s ( ! end to be atta9$1 !y d n believe !y und]/&4 ,d\btless !y are n !mselves j consci\s ( :at !y crave & !y c\ld n =mulate _! desire1 b if !y d n get satisfaction1 !y vaguely feel t some?+ is lack+4 ,!n :at happ5s8 ,9 ! beg9n+ !y /ill p]ceive ! pro(s "o puts "u _! eyes2 b z ^! are connect$ only by too sl5d] a ?read to ^? :i* prec$e & ^? :i* foll[1 !y pass )\t leav+ any trace 9 _! head2 !y are soon =gott52 a mom5t bright1 !y quickly vani% 9 night et]nal4 ,:5 !y are f>!r on1 !y w no long] see ev5 ?is ephem]al light1 s9ce ! !orems lean "o ^u ano!r & ^? !y w\ld ne$ are =gott52 ?us x is !y become incapable ( und]/&+ ma!matics4 ,?is is n always ! fault ( _! tea*]2 (t5 _! m9d1 :i* ne$s to p]ceive ! guid+ ?read1 is too lazy to seek & f9d x4 ,b to come to _! aid1 we fir/ mu/ "k j :at h9d]s !m4 ,o!rs w always ask ( :at use is x2 !y w n h und]/ood if !y d n f9d ab\t !m1 9 practise or 9 nature1 ! ju/ification ( su* & su* a ma!matical concept4 ,"u ea* ^w !y wi% to put a s5sible image2 ! def9ition mu/ evoke ?is image1 s t at ea* /age ( ! demon/ration !y may see x trans=m & evolve4 ,only ^u ?is condition d !y compreh5d & reta94 ,(t5 ^! deceive !mselves2 !y d n li/5 to ! reason+1 !y look at ! figures2 !y ?9k !y h und]/ood & !y h only se54 #b4 ,h[ _m diff]5t t5d5cies6 ,mu/ we combat !m8 ,mu/ we use !m8 ,& if we wi% to combat !m1 :i* %\ld be favor$8 ,mu/ we %[ ^? cont5t ) ! pure logic t !y h se5 only "o side ( ! matt]8 ,or ne$ we say to ^? n s *eaply satisfi$ t :at !y dem& is n necess>y8 ,9 o!r words1 %\ld we con/ra9 ! "y p to *ange ! nature ( _! m9ds8 ,su* an attempt w\ld be va92 we d n possess ! philosoph]'s /one :i* w\ld enable u to transmute "o into ano!r ! metals confid$ to u2 all we c d is to "w ) !m1 adapt+ ourselves to _! prop]ties4 ,_m *ildr5 are incapable ( becom+ ma!maticians1 to :om h[ev] x is necess>y to tea* ma!matics2 & ! ma!maticians !mselves are n all ca/ 9 ! same mold4 ,to read _! works suffices to di/+ui% among !m two sorts ( m9ds1 ! logicians l ,wei]/rass = example1 ! intuitives l ,riemann4 ,!re is ! same diff]5ce among our /ud5ts4 ,! "o sort pref] to treat _! problems 'by analysis' z !y say1 ! o!rs 'by geometry4' ,x is useless to seek to *ange any?+ ( t1 & besides w\ld x be desirable8 ,x is well t !re are logicians & t !re are intuitives2 :o w\ld d>e say :e!r he pref]r$ t ,wei]/rass _h nev] writt5 or t !re nev] _h be5 a ,riemann8 ,we mu/ !re=e resign ourselves to ! div]sity ( m9ds1 or bett] we mu/ rejoice 9 x4 #c4 ,s9ce ! ^w und]/& has _m mean+s1 ! def9itions :i* w be be/ und]/ood by "s w n be be/ suit$ to o!rs4 ,we h ^? :i* seek to produce an image1 & ^? :]e we conf9e ourselves to comb9+ empty =ms1 p]fectly intelligible1 b purely intelligible1 :i* ab/raction has depriv$ ( all matt]4 ,i "k n :e!r x be necess>y to cite examples4 ,let u cite !m1 anyh[1 & fir/ ! def9ition ( fractions w furni% u an extreme case4 ,9 ! prim>y s*ools1 to def9e a fraction1 "o cuts up an apple or a pie2 x is cut up m5tally ( c\rse & n 9 reality1 because ,i d n suppose ! budget ( ! prim>y in/ruction all[s ( su* prodigality4 ,at ! ,normal ,s*ool1 on ! o!r h&1 or at ! college1 x is said3 a fraction is ! comb9ation ( two :ole numb]s sep>at$ by a horizontal b>2 we def9e by conv5tions ! op]ations to :i* ^! symbols may be submitt$2 x is prov$ t ! rules ( ^! op]ations are ! same z 9 calculat+ ) :ole numb]s1 & we asc]ta9 f9ally t multiply+ ! fraction1 accord+ to ^! rules1 by ! d5om9ator gives ! num]ator4 ,?is is all v well because we are address+ "y p long famili>iz$ ) ! notion ( fractions ?r\gh hav+ cut up apples or o!r objects1 & ^: m9d1 matur$ by a h>d ma!matical education1 has come little by little to desire a purely logical def9ition4 ,b ! dbutant to :om "o %\ld try to give x1 h[ dumf\nd$6 ,su* also are ! def9itions f\nd 9 a book ju/ly admir$ & greatly honor$1 ! ,f\ndations ( ,geometry by ,hilb]t4 ,see 9 fact h[ he beg9s3 ,we ?9k ?ree sy/ems ( ,,?+s :i* we %all call po9ts1 /raights & planes4 ,:at are ^! '?+s'8 ,we "k n1 nor ne$ we "k2 x w\ld ev5 be a pity to seek to "k2 all we h ! "r to "k ( !m is :at ! assumptions tell u2 ?is = example3 ,two di/9ct po9ts always det]m9e a /raight1 :i* is foll[$ by ?is rem>k3 9 place ( det]m9e1 we may say ! two po9ts are on ! /raight1 or ! /raight goes ?r\gh ^! two po9ts or jo9s ! two po9ts4 ,?us 'to be on a /raight' is simply def9$ z synonym\s ) 'det]m9e a /raight4' ,behold a book ( :i* ,i ?9k mu* good1 b :i* ,i %\ld n recomm5d to a s*ool boy4 ,yet ,i c\ld d s )\t fe>1 he w\ld n read mu* ( x4 ,i h tak5 extreme examples & no tea*] w\ld dream ( go+ t f>4 ,b ev5 /opp+ %ort ( su* models1 does he n already expose himself to ! same dang]8 ,suppose we are 9 a class2 ! pr(essor dictates3 ! circle is ! locus ( po9ts ( ! plane equidi/ant f an int]ior po9t call$ ! c5t]4 ,! good s*ol> writes ?is phrase 9 8 note-book2 ! bad s*ol> draws faces2 b nei!r und]/&s2 !n ! pr(essor takes ! *alk & draws a circle on ! bo>d4 ,ah6 ?9k ! s*ol>s1 :y did he n say at once3 a circle is a r+1 we %\ld h und]/ood4 ,d\btless ! pr(essor is "r4 ,! s*ol>s' def9ition w\ld h be5 ( no avail1 s9ce x c\ld s]ve = no demon/ration1 s9ce besides x w\ld n give !m ! salut>y habit ( analyz+ _! conceptions4 ,b "o %\ld %[ !m t !y d n compreh5d :at !y ?9k !y "k1 lead !m to be consci\s ( ! r\ghness ( _! primitive conception1 & ( !mselves to wi% x purifi$ & made precise4 #d4 ,i %all return to ^! examples2 ,i only wi%$ to %[ y ! two oppos$ conceptions2 !y are 9 viol5t contra/4 ,?is contra/ ! hi/ory ( sci5ce expla9s4 ,if we read a book writt5 fifty ye>s ago1 mo/ ( ! reason+ we f9d !re seems lack+ 9 rigor4 ,!n x 0 assum$ a cont9u\s function c *ange sign only by vani%+2 to-day we prove x4 ,x 0 assum$ ! ord9>y rules ( calculation are applicable to incomm5surable numb]s2 to-day we prove x4 ,_m o!r ?+s 7 assum$ :i* sometimes 7 false4 ,we tru/$ to intuition2 b intuition c n give rigor1 nor ev5 c]ta9ty2 we see ?is m & m4 ,x tells u = in/ance t e curve has a tang5t1 t is to say t e cont9u\s function has a d]ivative1 & t is false4 ,& z we s\ght c]ta9ty1 we _h to make less & less ! "p ( intuition4 ,:at has made necess>y ?is evolution8 ,we h n be5 sl[ to p]ceive t rigor c\ld n be e/abli%$ 9 ! reason+s1 if x 7 n fir/ put into ! def9itions4 ,! objects occupy+ ma!maticians 7 long ill def9$2 we ?\ght we knew !m because we repres5t$ !m ) ! s5ses or ! imag9ation2 b we _h ( !m only a r\gh image & n a precise concept ^u :i* reason+ c\ld take hold4 ,x is !re t ! logicians w\ld h done well to direct _! ef=ts4 ,s = ! incomm5surable numb]1 ! vague idea ( cont9uity1 :i* we owe to intuition1 has resolv$ itself into a complicat$ sy/em ( inequalities be>+ on :ole numb]s4 ,?us h f9ally vani%$ all ^? difficulties :i* fright5$ our fa!rs :5 !y reflect$ ^u ! f\ndations ( ! inf9itesimal calculus4 ,to-day only :ole numb]s are left 9 analysis1 or sy/ems f9ite or inf9ite ( :ole numb]s1 b\nd by a plexus ( equalities & inequalities4 ,ma!matics we say is ari?metiz$4 #e4 ,b d y ?9k ma!matics has atta9$ absolute rigor )\t mak+ any sacrifice8 ,n at all2 :at x has ga9$ 9 rigor x has lo/ 9 objectivity4 ,x is by sep>at+ itself f reality t x has acquir$ ?is p]fect purity4 ,we may freely run ov] its :ole doma91 =m]ly bri/l+ ) ob/acles1 b ^! ob/acles h n disappe>$4 ,!y h only be5 mov$ to ! fronti]1 & x w\ld be necess>y to vanqui% !m anew if we wi%$ to break ov] ?is fronti] to ent] ! realm ( ! practical4 ,we _h a vague notion1 =m$ ( incongru\s elem5ts1 "s a priori1 o!rs com+ f exp]i5ces m or less dige/$2 we ?\ght we knew1 by intuition1 its pr9cipal prop]ties4 ,to-day we reject ! empiric elem5ts1 reta9+ only ! a priori2 "o ( ! prop]ties s]ves z def9ition & all ! o!rs are d$uc$ f x by rigor\s reason+4 ,?is is all v well1 b x rema9s to be prov$ t ?is prop]ty1 :i* has become a def9ition1 p]ta9s to ! real objects :i* exp]i5ce _h made kn[n to u & :5ce we drew our vague intuitive notion4 ,to prove t1 x w\ld be necess>y to appeal to exp]i5ce1 or to make an ef=t ( intuition1 & if we c\ld n prove x1 our !orems w\ld be p]fectly rigor\s1 b p]fectly useless4 ,logic sometimes makes mon/]s4 ,s9ce half a c5tury we h se5 arise a cr[d ( biz>re functions :i* seem to try to resemble z little z possible ! hone/ functions :i* s]ve "s purpose4 ,no long] cont9uity1 or p]haps cont9uity1 b no d]ivatives1 etc4 ,nay m1 f ! logical po9t ( view1 x is ^! /range functions :i* are ! mo/ g5]al1 ^? "o meets )\t seek+ no long] appe> except z p>ticul> case4 ,!re rema9s = !m only a v small corn]4 ,h]et(ore :5 a new function 0 inv5t$1 x 0 = "s practical end2 to-day !y are inv5t$ expressly to put at fault ! reason+s ( our fa!rs1 & "o nev] w get f !m any?+ m ?an t4 ,if logic 7 ! sole guide ( ! tea*]1 x w\ld be necess>y to beg9 ) ! mo/ g5]al functions1 t is to say ) ! mo/ biz>re4 ,x is ! beg9n] t w\ld h to be set grappl+ ) ?is t]atologic museum4 ,if y d n d x1 ! logicians might say1 y w a*ieve rigor only by /ages4 #f4 ,yes1 p]haps1 b we c n make s *eap ( reality1 & ,i mean n only ! reality ( ! s5sible _w1 :i* h[ev] has its wor?1 s9ce x is to combat aga9/ x t n9e t5?s ( y\r /ud5ts ask ( y weapons4 ,!re is a reality m subtile1 :i* makes ! v life ( ! ma!matical be+s1 & :i* is q o!r ?an logic4 ,our body is =m$ ( cells1 & ! cells ( atoms2 are ^! cells & ^! atoms !n all ! reality ( ! human body8 ,! way ^! cells are arrang$1 :5ce results ! unity ( ! individual1 is x n also a reality & mu* m int]e/+8 ,a naturali/ :o nev] _h /udi$ ! elephant except 9 ! microscope1 w\ld he ?9k he knew ! animal adequately8 ,x is ! same 9 ma!matics4 ,:5 ! logician %all h brok5 up ea* demon/ration into a multitude ( elem5t>y op]ations1 all correct1 he /ill w n possess ! :ole reality2 ?is ,i "k n :at :i* makes ! unity ( ! demon/ration w completely escape him4 ,9 ! edifices built up by our ma/]s1 ( :at use to admire ! "w ( ! mason if we c n compreh5d ! plan ( ! ar*itect8 ,n[ pure logic c n give u ?is appreciation ( ! total effect2 ?is we mu/ ask ( intuition4 ,take = in/ance ! idea ( cont9u\s function4 ,?is is at fir/ only a s5sible image1 a m>k trac$ by ! *alk on ! blackbo>d4 ,little by little x is ref9$2 we use x to con/ruct a complicat$ sy/em ( inequalities1 :i* reproduces all ! features ( ! primitive image2 :5 all is done1 we h remov$ ! c5t]+1 z aft] ! con/ruction ( an ar*2 ?is r\gh repres5tation1 support !nce=? useless1 has disappe>$ & !re rema9s only ! edifice itself1 irreproa*able 9 ! eyes ( ! logician4 ,& yet1 if ! pr(essor did n recall ! primitive image1 if he did n re/ore mom5t>ily ! c5t]+1 h[ c\ld ! /ud5t div9e by :at caprice all ^! inequalities h be5 scaffold$ 9 ?is fa%ion "o ^u ano!r8 ,! def9ition w\ld be logically correct1 b x w\ld n %[ him ! v]itable reality4 #g4 ,s back we mu/ return2 d\btless x is h>d = a ma/] to tea* :at does n entirely satisfy him2 b ! satisfaction ( ! ma/] is n ! unique object ( tea*+2 we %\ld fir/ give att5tion to :at ! m9d ( ! pupil is & to :at we wi% x to become4 ,zoologi/s ma9ta9 t ! embryonic developm5t ( an animal recapitulates 9 brief ! :ole hi/ory ( its ance/ors ?r\gh\t geologic "t4 ,x seems x is ! same 9 ! developm5t ( m9ds4 ,! tea*] %\ld make ! *ild g ov] ! pa? 8 fa!rs trod2 m rapidly1 b )\t skipp+ /ations4 ,= ?is reason1 ! hi/ory ( sci5ce %\ld be our fir/ guide4 ,our fa!rs ?\ght !y knew :at a fraction 01 or cont9uity1 or ! area ( a curv$ surface2 we h f\nd !y did n "k x4 ,j s our s*ol>s ?9k !y "k x :5 !y beg9 ! s]i\s /udy ( ma!matics4 ,if )\t w>n+ ,i tell !m3 ,no1 y d n "k x2 :at y ?9k y und]/&1 y d n und]/&2 ,i mu/ prove to y :at seems to y evid5t1 & if 9 ! demon/ration ,i support myself ^u premises :i* to !m seem less evid5t ?an ! conclusion1 :at %all ! un=tunates ?9k8 ,!y w ?9k t ! sci5ce ( ma!matics is only an arbitr>y mass ( useless subtilities2 ei!r !y w be disgu/$ ) x1 or !y w play x z a game & w rea* a /ate ( m9d l t ( ! ,greek sophi/s4 ,lat]1 on ! contr>y1 :5 ! m9d ( ! s*ol>1 famili>iz$ ) ma!matical reason+1 has be5 matur$ by ?is long frequ5tation1 ! d\bts w arise ( !mselves & !n y\r demon/ration w be welcome4 ,x w awak5 new d\bts1 & ! que/ions w arise successively to ! *ild1 z !y arose successively to our fa!rs1 until p]fect rigor alone c satisfy him4 ,to d\bt ev]y?+ does n suffice1 "o mu/ "k :y he d\bts4 #h4 ,! pr9cipal aim ( ma!matical tea*+ is to develop c]ta9 faculties ( ! m9d1 & among !m intuition is n ! lea/ preci\s4 ,x is ?r\gh x t ! ma!matical _w rema9s 9 contact ) ! real _w1 & if pure ma!matics c\ld d )\t x1 x w\ld always be necess>y to h rec\rse to x to fill up ! *asm :i* sep>ates ! symbol f reality4 ,! practician w always h ne$ ( x1 & = "o pure geomet] !re %\ld be a hundr$ practicians4 ,! eng9e] %\ld receive a complete ma!matical education1 b = :at %\ld x s]ve him8 ,to see ! diff]5t aspects ( ?+s & see !m quickly2 he has no "t to hunt mice4 ,x is necess>y t1 9 ! complex physical objects pres5t$ to him1 he %\ld promptly recognize ! po9t :]e ! ma!matical tools we h put 9 8 h&s c take hold4 ,h[ c\ld he d x if we %\ld leave betwe5 in/rum5ts & objects ! deep *asm holl[$ out by ! logicians8 #i4 ,besides ! eng9e]s1 o!r s*ol>s1 less num]\s1 are 9 _! turn to become tea*]s2 !y !re=e mu/ g to ! v bottom2 a k deep & rigor\s ( ! fir/ pr9ciples is = !m be=e all indisp5sable4 ,b ?is is no reason n to cultivate 9 !m intuition2 = !y w\ld get a false idea ( ! sci5ce if !y nev] look$ at x except f a s+le side1 & besides !y c\ld n develop 9 _! /ud5ts a quality !y did n !mselves possess4 ,= ! pure geomet] himself1 ?is faculty is necess>y2 x is by logic "o demon/rates1 by intuition "o inv5ts4 ,to "k h[ to criticize is good1 to "k h[ to create is bett]4 ,y "k h[ to recognize if a comb9ation is correct2 :at a pr$icam5t if y h n ! art ( *oos+ among all ! possible comb9ations4 ,logic tells u t on su* & su* a way we are sure n to meet any ob/acle2 x does n say :i* way leads to ! end4 ,= t x is necess>y to see ! end f af>1 & ! faculty :i* tea*es u to see is intuition4 ,)\t x ! geomet] w\ld be l a writ] :o %\ld be v]s$ 9 gramm> b _h no ideas4 ,n[ h[ c\ld ?is faculty develop if1 z soon z x %[$ itself1 we *ase x away & proscribe x1 if we le>n to set x at naught be=e kn[+ ! good ( x4 ,& "h p]mit a p>5!sis to insi/ ^u ! importance ( writt5 ex]cises4 ,writt5 compositions are p]haps n suffici5tly emphasiz$ 9 c]ta9 exam9ations1 at ! polyte*nic s*ool1 = in/ance4 ,i am told !y w\ld close ! door aga9/ v good s*ol>s :o h ma/]$ ! c\rse1 ?or\ghly und]/&+ x1 & :o nev]!less are incapable ( mak+ ! slighte/ application4 ,i h j said ! ^w und]/& has sev]al mean+s3 su* /ud5ts only und]/& 9 ! fir/ way1 & we h se5 t suffices nei!r to make an eng9e] nor a geomet]4 ,well1 s9ce *oice mu/ be made1 ,i pref] ^? :o und]/& completely4 #aj4 ,b is ! art ( s\nd reason+ n also a preci\s ?+1 :i* ! pr(essor ( ma!matics ought be=e all to cultivate8 ,i take good c>e n to =get t4 ,x %\ld occupy our att5tion & f ! v beg9n+4 ,i %\ld be di/ress$ to see geometry deg5]ate into ,i "k n :at ta*ymetry ( l[ grade & ,i by no means subscribe to ! extreme doctr9es ( c]ta9 ,g]man ,ob]lehr]4 ,b !re are occasions en\gh to ex]cise ! s*ol>s 9 correct reason+ 9 ! p>ts ( ma!matics :]e ! inconv5i5ces ,i h po9t$ out d n pres5t !mselves4 ,!re are long *a9s ( !orems :]e absolute logic has reign$ f ! v fir/ &1 s to speak1 q naturally1 :]e ! fir/ geomet]s h giv5 u models we %\ld con/antly imitate & admire4 ,x is 9 ! exposition ( fir/ pr9ciples t x is necess>y to avoid too mu* subtility2 !re x w\ld be mo/ disc\rag+ & moreov] useless4 ,we c n prove ev]y?+ & we c n def9e ev]y?+2 & x w always be necess>y to borr[ f intuition2 :at does x matt] :e!r x be done a little soon] or a little lat]1 provid$ t 9 us+ correctly premises x has furni%$ u1 we le>n to reason s\ndly4 #aa4 ,is x possible to fulfill s _m oppos+ conditions8 ,is ?is possible 9 p>ticul> :5 x is a "q ( giv+ a def9ition8 ,h[ f9d a concise /atem5t satisfy+ at once ! uncompromis+ rules ( logic1 our desire to grasp ! place ( ! new notion 9 ! totality ( ! sci5ce1 our ne$ ( ?9k+ ) images8 ,usually x w n be f\nd1 & ?is is :y x is n en\gh to /ate a def9ition2 x mu/ be prep>$ = & ju/ifi$4 ,:at does t mean8 ,y "k x has (t5 be5 said3 e def9ition implies an assumption1 s9ce x affirms ! exi/5ce ( ! object def9$4 ,! def9ition !n w n be ju/ifi$1 f ! purely logical po9t ( view1 until "o %all h prov$ t x involves no contradiction1 nei!r 9 ! t]ms1 nor ) ! v]ities previ\sly admitt$4 ,b ?is is n en\gh2 ! def9ition is /at$ to u z a conv5tion2 b mo/ m9ds w revolt if we wi% to impose x ^u !m z an arbitr>y conv5tion4 ,!y w be satisfi$ only :5 y h answ]$ num]\s que/ions4 ,usually ma!matical def9itions1 z ,m4 ,li>d has %[n1 are v]itable con/ructions built up :olly ( m simple notions4 ,b :y assemble ^! elem5ts 9 ?is way :5 a ?\s& o!r comb9ations 7 possible8 ,is x by caprice8 ,if n1 :y _h ?is comb9ation m "r to exi/ ?an all ! o!rs8 ,to :at ne$ does x respond8 ,h[ 0 x =ese5 t x w\ld play an important rle 9 ! developm5t ( ! sci5ce1 t x w\ld abridge our reason+s & our calculations8 ,is !re 9 nature "s famili> object :i* is s to speak ! r\gh & vague image ( x8 ,?is is n all2 if y answ] all ^! que/ions 9 a satisfactory mann]1 we %all see inde$ t ! new-born _h ! "r to be baptiz$2 b nei!r is ! *oice ( a "n arbitr>y2 x is ne$ful to expla9 by :at analogies "o has be5 guid$ & t if analog\s names h be5 giv5 to diff]5t ?+s1 ^! ?+s at lea/ diff] only 9 mat]ial & are alli$ 9 =m2 t _! prop]ties are analog\s & s to say p>allel4 ,at ?is co/ we may satisfy all incl9ations4 ,if ! /atem5t is correct en\gh to please ! logician1 ! ju/ification w satisfy ! intuitive4 ,b !re is /ill a bett] proc$ure2 :]ev] possible1 ! ju/ification %\ld prec$e ! /atem5t & prep>e = x2 "o %\ld be l$ on to ! g5]al /atem5t by ! /udy ( "s p>ticul> examples4 ,/ill ano!r ?+3 ea* ( ! p>ts ( ! /atem5t ( a def9ition has z aim to di/+ui% ! ?+ to be def9$ f a class ( o!r neighbor+ objects4 ,! def9ition w be und]/ood only :5 y h %[n1 n m]ely ! object def9$1 b ! neighbor+ objects f :i* x is prop] to di/+ui% x1 :5 y h giv5 a grasp ( ! diff]5ce & :5 y h add$ explicitly3 ?is is :y 9 /at+ ! def9ition ,i h said ?is or t4 ,b x is "t to leave g5]alities & exam9e h[ ! some:at ab/ract pr9ciples ,i h exp\nd$ may be appli$ 9 ari?metic1 geometry1 analysis & me*anics4 ,,ari?metic #ab4 ,! :ole numb] is n to be def9$2 9 return1 "o ord9>ily def9es ! op]ations ^u :ole numb]s2 ,i believe ! s*ol>s le>n ^! def9itions by he>t & atta* no mean+ to !m4 ,= t !re are two reasons3 fir/ !y are made to le>n !m too soon1 :5 _! m9d z yet feels no ne$ ( !m2 !n ^! def9itions are n satisfactory f ! logical po9t ( view4 ,a good def9ition = addition is n to be f\nd j simply because we mu/ /op & c n def9e ev]y?+4 ,x is n def9+ addition to say x consi/s 9 add+4 ,all t c be done is to />t f a c]ta9 numb] ( concrete examples & say3 ! op]ation we h p]=m$ is call$ addition4 ,= subtraction x is q o!rwise2 x may be logically def9$ z ! op]ation inv]se to addition2 b %\ld we beg9 9 t way8 ,"h also />t ) examples1 %[ on ^! examples ! reciprocity ( ! two op]ations2 ?us ! def9ition w be prep>$ = & ju/ifi$4 ,j s aga9 = multiplication2 take a p>ticul> problem2 %[ t x may be solv$ by add+ sev]al equal numb]s2 !n %[ t we rea* ! result m quickly by a multiplication1 an op]ation ! s*ol>s already "k h[ to d by r\t9e & out ( t ! logical def9ition w issue naturally4 ,division is def9$ z ! op]ation inv]se to multiplication2 b beg9 by an example tak5 f ! famili> notion ( p>tition & %[ on ?is example t multiplication reproduces ! divid5d4 ,!re /ill rema9 ! op]ations on fractions4 ,! only difficulty is = multiplication4 ,x is be/ to exp\nd fir/ ! !ory ( proportion2 f x alone c come a logical def9ition2 b to make acceptable ! def9itions met at ! beg9n+ ( ?is !ory1 x is necess>y to prep>e = !m by num]\s examples tak5 f classic problems ( ! rule ( ?ree1 tak+ pa9s to introduce fractional data4 ,nei!r %\ld we fe> to famili>ize ! s*ol>s ) ! notion ( proportion by geometric images1 ei!r by appeal+ to :at !y rememb] if !y h already /udi$ geometry1 or 9 hav+ rec\rse to direct intuition1 if !y h n /udi$ x1 :i* besides w prep>e !m to /udy x4 ,f9ally ,i %all add t aft] def9+ multiplication ( fractions1 x is ne$ful to ju/ify ?is def9ition by %[+ t x is commutative1 associative & di/ributive1 & call+ to ! att5tion ( ! auditors t ?is is e/abli%$ to ju/ify ! def9ition4 ,"o sees :at a rle geometric images play 9 all ?is2 & ?is rle is ju/ifi$ by ! philosophy & ! hi/ory ( ! sci5ce4 ,if ari?metic _h rema9$ free f all admixture ( geometry1 x w\ld h kn[n only ! :ole numb]2 x is to adapt itself to ! ne$s ( geometry t x inv5t$ any?+ else4 ,,geometry ,9 geometry we meet =?) ! notion ( ! /raight l9e4 ,c ! /raight l9e be def9$8 ,! well-kn[n def9ition1 ! %orte/ pa? f "o po9t to ano!r1 sc>cely satisfies me4 ,i %\ld />t simply ) ! rul] & %[ at fir/ to ! s*ol> h[ "o may v]ify a rul] by turn+2 ?is v]ification is ! true def9ition ( ! /raight l9e2 ! /raight l9e is an axis ( rotation4 ,next he %\ld be %[n h[ to v]ify ! rul] by slid+ & he w\ld h "o ( ! mo/ important prop]ties ( ! /raight l9e4 ,z to ?is o!r prop]ty ( be+ ! %orte/ pa? f "o po9t to ano!r1 x is a !orem :i* c be demon/rat$ apodictically1 b ! demon/ration is too delicate to f9d a place 9 second>y tea*+4 ,x w be wor? m to %[ t a rul] previ\sly v]ifi$ fits on a /ret*$ ?read4 ,9 pres5ce ( difficulties l ^! "o ne$ n dread to multiply assumptions1 ju/ify+ !m by r\gh exp]im5ts4 ,x is ne$ful to grant ^! assumptions1 & if "o admits a few m ( !m ?an is /rictly necess>y1 ! evil is n v great2 ! ess5tial ?+ is to le>n to reason s\ndly on ! assumptions admitt$4 ,uncle ,s>cey1 :o lov$ to repeat1 (t5 said t at ! !at] ! spectator accepts will+ly all ! po/ulates impos$ ^u him at ! beg9n+1 b ! curta9 once rais$1 he becomes uncompromis+ on ! logic4 ,well1 x is j ! same 9 ma!matics4 ,= ! circle1 we may />t ) ! compasses2 ! s*ol>s w recognize at ! fir/ glance ! curve trac$2 !n make !m obs]ve t ! di/ance ( ! two po9ts ( ! in/rum5t rema9s con/ant1 t "o ( ^! po9ts is fix$ & ! o!r movable1 & s we %all be l$ naturally to ! logical def9ition4 ,! def9ition ( ! plane implies an axiom & ?is ne$ n be hidd54 ,take a draw+ bo>d & %[ t a mov+ rul] may be kept con/antly 9 complete contact ) ?is plane & yet reta9 ?ree degrees ( fre$om4 ,comp>e ) ! cyl9d] & ! cone1 surfaces on :i* an appli$ /raight reta9s only two degrees ( fre$om2 next take ?ree draw+ bo>ds2 %[ fir/ t !y w glide :ile rema9+ appli$ to "o ano!r & ?is ) ?ree degrees ( fre$om2 & f9ally to di/+ui% ! plane f ! sph]e1 %[ t two ( ^! bo>ds :i* fit a ?ird w fit ea* o!r4 ,p]haps y are surpris$ at ?is incessant employm5t ( mov+ ?+s2 ?is is n a r\gh artifice2 x is mu* m philosophic ?an "o w\ld at fir/ ?9k4 ,:at is geometry = ! philosoph]8 ,x is ! /udy ( a gr\p4 ,& :at gr\p8 ,t ( ! motions ( solid bodies4 ,h[ def9e ?is gr\p !n )\t mov+ "s solids8 ,%\ld we reta9 ! classic def9ition ( p>allels & say p>allels are two coplan> /raights :i* d n meet1 h[ev] f> !y be prolong$8 ,no1 s9ce ?is def9ition is negative1 s9ce x is unv]ifiable by exp]im5t1 & consequ5tly c n be reg>d$ z an imm$iate datum ( intuition4 ,no1 above all because x is :olly /range to ! notion ( gr\p1 to ! consid]ation ( ! motion ( solid bodies :i* is1 z ,i h said1 ! true s\rce ( geometry4 ,w\ld x n be bett] to def9e fir/ ! rectil9e> translation ( an inv>iable figure1 z a motion :]e9 all ! po9ts ( ?is figure h rectil9e> trajectories2 to %[ t su* a translation is possible by mak+ a squ>e glide on a rul]8 ,f ?is exp]im5tal asc]ta9m5t1 set up z an assumption1 x w\ld be easy to d]ive ! notion ( p>allel & ,euclid's po/ulate itself4 ,,me*anics ,i ne$ n return to ! def9ition ( velocity1 or accel]ation1 or o!r k9ematic notions2 !y may be advantage\sly connect$ ) t ( ! d]ivative4 ,i %all insi/1 on ! o!r h&1 ^u ! dynamic notions ( =ce & mass4 ,i am /ruck by "o ?+3 h[ v f> ! "y p :o h receiv$ a high-s*ool education are f apply+ to ! real _w ! me*anical laws !y h be5 taught4 ,x is n only t !y are incapable ( x2 !y d n ev5 ?9k ( x4 ,= !m ! _w ( sci5ce & ! _w ( reality are sep>at$ by an imp]vi\s p>tition wall4 ,if we try to analyze ! /ate ( m9d ( our s*ol>s1 ?is w a/oni% u less4 ,:at is = !m ! real def9ition ( =ce8 ,n t :i* !y recite1 b t :i*1 cr\*+ 9 a nook ( _! m9d1 f !re directs x :olly4 ,"h is ! def9ition3 =ces are arr[s ) :i* "o makes p>allelograms4 ,^! arr[s are imag9>y ?+s :i* h no?+ to d ) any?+ exi/+ 9 nature4 ,?is w\ld n happ5 if !y _h be5 %[n =ces 9 reality be=e repres5t+ !m by arr[s4 ,h[ %all we def9e =ce8 ,i ?9k ,i h else:]e suffici5tly %[n !re is no good logical def9ition4 ,!re is ! an?ropomorphic def9ition1 ! s5sation ( muscul> ef=t2 ?is is really too r\gh & no?+ useful c be drawn f x4 ,"h is h[ we %\ld g3 fir/1 to make kn[n ! g5us =ce1 we mu/ %[ "o aft] ! o!r all ! species ( ?is g5us2 !y are v num]\s & v diff]5t2 !re is ! pressure ( fluids on ! insides ( ! vases :]e9 !y are conta9$2 ! t5sion ( ?reads2 ! ela/icity ( a spr+2 ! gravity work+ on all ! molecules ( a body2 friction2 ! normal mutual action & reaction ( two solids 9 contact4 ,?is is only a qualitative def9ition2 x is necess>y to le>n to measure =ce4 ,= t beg9 by %[+ t "o =ce may be replac$ by ano!r )\t de/roy+ equilibrium2 we may f9d ! fir/ example ( ?is sub/itution 9 ! balance & ,borda's d\ble weigh+4 ,!n %[ t a weight may be replac$1 n only by ano!r weight1 b by =ce ( a diff]5t nature2 = in/ance1 ,prony's brake p]mits replac+ weight by friction4 ,f all ?is arises ! notion ( ! equival5ce ( two =ces4 ,! direction ( a =ce mu/ be def9$4 ,if a =ce ,f is equival5t to ano!r =ce ,f' appli$ to ! body consid]$ by means ( a /ret*$ /r+1 s t ,f may be replac$ by ,f' )\t affect+ ! equilibrium1 !n ! po9t ( atta*m5t ( ! /r+ w be by def9ition ! po9t ( application ( ! =ce ,f'1 & t ( ! equival5t =ce ,f2 ! direction ( ! /r+ w be ! direction ( ! =ce ,f' & t ( ! equival5t =ce ,f4 ,f t1 pass to ! comp>ison ( ! magnitude ( =ces4 ,if a =ce c replace two o!rs ) ! same direction1 x equals _! sum2 %[ = example t a weight ( #bj grams may replace two #aj-;gram weights4 ,is ?is en\gh8 ,n yet4 ,we n[ "k h[ to comp>e ! int5sity ( two =ces :i* h ! same direction & same po9t ( application2 we mu/ le>n to d x :5 ! directions are diff]5t4 ,= t1 imag9e a /r+ /ret*$ by a weight & pass+ ov] a pulley2 we %all say t ! t5sor ( ! two legs ( ! /r+ is ! same & equal to ! t5sion weight4 ,?is def9ition ( ours enables u to comp>e ! t5sions ( ! two pieces ( our /r+1 &1 us+ ! prec$+ def9itions1 to comp>e any two =ces hav+ ! same direction z ^! two pieces4 ,x %\ld be ju/ifi$ by %[+ t ! t5sion ( ! la/ piece ( ! /r+ rema9s ! same = ! same t5sor weight1 :atev] be ! numb] & ! disposition ( ! reflect+ pulleys4 ,x has /ill to be complet$ by %[+ ?is is only true if ! pulleys are frictionless4 ,once ma/] ( ^! def9itions1 x is to be %[n t ! po9t ( application1 ! direction & ! int5sity suffice to det]m9e a =ce2 t two =ces = :i* ^! ?ree elem5ts are ! same are always equival5t & may always be replac$ by "o ano!r1 :e!r 9 equilibrium or 9 movem5t1 & ?is :atev] be ! o!r =ces act+4 ,x mu/ be %[n t two concurr5t =ces may always be replac$ by a unique resultant2 & t ?is resultant rema9s ! same1 :e!r ! body be at re/ or 9 motion & :atev] be ! o!r =ces appli$ to x4 ,f9ally x mu/ be %[n t =ces ?us def9$ satisfy ! pr9ciple ( ! equality ( action & reaction4 ,exp]im5t x is1 & exp]im5t alone1 :i* c tea* u all t4 ,x w suffice to cite c]ta9 common exp]im5ts1 :i* ! s*ol>s make daily )\t suspect+ x1 & to p]=m be=e !m a few exp]im5ts1 simple & well *os54 ,x is aft] hav+ pass$ ?r\gh all ^! me&]s t "o may repres5t =ces by arr[s1 & ,i %\ld ev5 wi% t 9 ! developm5t ( ! reason+s return 7 made f "t to "t f ! symbol to ! reality4 ,= in/ance x w\ld n be difficult to illu/rate ! p>allelogram ( =ces by aid ( an app>atus =m$ ( ?ree /r+s1 pass+ ov] pulleys1 /ret*$ by weights & 9 equilibrium :ile pull+ on ! same po9t4 ,kn[+ =ce1 x is easy to def9e mass2 ?is "t ! def9ition %\ld be borr[$ f dynamics2 !re is no way ( do+ o!rwise1 s9ce ! end to be atta9$ is to give und]/&+ ( ! di/9ction betwe5 mass & weight4 ,"h aga91 ! def9ition %\ld be l$ up to by exp]im5ts2 !re is 9 fact a ma*9e :i* seems made expressly to %[ :at mass is1 ,atwood's ma*9e2 recall also ! laws ( ! fall ( bodies1 t ! accel]ation ( gravity is ! same = heavy z = light bodies1 & t x v>ies ) ! latitude1 etc4 ,n[1 if y tell me t all ! me?ods ,i extol h long be5 appli$ 9 ! s*ools1 ,i %all rejoice ov] x m ?an be surpris$ at x4 ,i "k t on ! :ole our ma!matical tea*+ is good4 ,i d n wi% x ov]turn$2 t w\ld ev5 di/ress me4 ,i only desire bett]m5ts sl[ly progressive4 ,?is tea*+ %\ld n be subject$ to brusque oscillations "u ! caprici\s bla/ ( ephem]al fads4 ,9 su* tempe/s its high educative value w\ld soon f\nd]4 ,a good & s\nd logic %\ld cont9ue to be its basis4 ,! def9ition by example is always necess>y1 b x %\ld prep>e ! way = ! logical def9ition1 x %\ld n replace x2 x %\ld at lea/ make ?is wi%$ =1 9 ! cases :]e ! true logical def9ition c be advantage\sly giv5 only 9 advanc$ tea*+4 ,und]/& t :at ,i h "h said does n imply giv+ up :at ,i h writt5 else:]e4 ,i h (t5 _h occasion to criticize c]ta9 def9itions ,i extol to-day4 ,^! criticisms hold good completely4 ,^! def9itions c only be provisory4 ,b x is by way ( !m t we mu/ pass4 ,,*apt] ,,iii ,,ma!matics ,,& ,,logic ,,introduction ,c ma!matics be r$uc$ to logic )\t hav+ to appeal to pr9ciples peculi> to ma!matics8 ,!re is a :ole s*ool1 ab\nd+ 9 ardor & full ( fai?1 /riv+ to prove x4 ,!y h _! own special language1 :i* is )\t words1 us+ only signs4 ,?is language is und]/ood only by ! initiates1 s t common]s are dispos$ to b[ to ! tr5*ant affirmations ( ! adepts4 ,x is p]haps n unpr(itable to exam9e ^! affirmations some:at closely1 to see if !y ju/ify ! p]emptory tone ) :i* !y are pres5t$4 ,b to make cle> ! nature ( ! "q x is necess>y to ent] ^u c]ta9 hi/orical details & 9 p>ticul> to recall ! "* ( ! works ( ,cantor4 ,s9ce long ago ! notion ( inf9ity _h be5 introduc$ into ma!matics2 b ?is inf9ite 0 :at philosoph]s call a becom+4 ,! ma!matical inf9ite 0 only a quantity capable ( increas+ beyond all limit3 x 0 a v>iable quantity ( :i* x c\ld n be said t x _h pass$ all limits1 b only t x c\ld pass !m4 ,cantor has und]tak5 to introduce into ma!matics an actual inf9ite1 t is to say a quantity :i* n only is capable ( pass+ all limits1 b :i* is reg>d$ z hav+ already pass$ !m4 ,he has set himself que/ions l ^!3 ,are !re m po9ts 9 space ?an :ole numb]s8 ,are !re m po9ts 9 space ?an po9ts 9 a plane8 etc4 ,& !n ! numb] ( :ole numb]s1 t ( ! po9ts ( space1 etc41 con/itutes :at he calls a transf9ite c>d9al numb]1 t is to say a c>d9al numb] great] ?an all ! ord9>y c>d9al numb]s4 ,& he has occupi$ himself 9 comp>+ ^! transf9ite c>d9al numb]s4 ,9 arrang+ 9 a prop] ord] ! elem5ts ( an aggregate conta9+ an inf9ity ( !m1 he has also imag9$ :at he calls transf9ite ord9al numb]s ^u :i* ,i %all n dwell4 ,_m ma!maticians foll[$ 8 lead & set a s]ies ( que/ions ( ! sort4 ,!y s famili>iz$ !mselves ) transf9ite numb]s t !y h come to make ! !ory ( f9ite numb]s dep5d ^u t ( ,cantor's c>d9al numb]s4 ,9 _! eyes1 to tea* ari?metic 9 a way truly logical1 "o %\ld beg9 by e/abli%+ ! g5]al prop]ties ( transf9ite c>d9al numb]s1 !n di/+ui% among !m a v small class1 t ( ! ord9>y :ole numb]s4 ,?anks to ?is dt\r1 "o might succe$ 9 prov+ all ! propositions relative to ?is little class 7t is to say all our ari?metic & our algebra7 )\t us+ any pr9ciple =eign to logic4 ,?is me?od is evid5tly contr>y to all sane psy*ology2 x is c]ta9ly n 9 ?is way t ! human m9d proce$$ 9 con/ruct+ ma!matics2 s its au?ors d n dream1 ,i ?9k1 ( introduc+ x into second>y tea*+4 ,b is x at lea/ logic1 or1 bett]1 is x correct8 ,x may be d\bt$4 ,! geomet]s :o h employ$ x are h[ev] v num]\s4 ,!y h accumulat$ =mulas & !y h ?\ght to free !mselves f :at 0 n pure logic by writ+ memoirs :]e ! =mulas no long] alt]nate ) explanatory disc\rse z 9 ! books ( ord9>y ma!matics1 b :]e ?is disc\rse has completely disappe>$4 ,un=tunately !y h rea*$ contradictory results1 :at are call$ ! cantorian ant9omies1 to :i* we %all h occasion to return4 ,^! contradictions h n disc\rag$ !m & !y h tri$ to modify _! rules s z to make ^? disappe> :i* _h already %[n !mselves1 )\t be+ sure1 = all t1 t new ones w\ld n manife/ !mselves4 ,x is "t to adm9i/] ju/ice on ^! exagg]ations4 ,i d n hope to conv9ce !m2 = !y h liv$ too long 9 ?is atmosph]e4 ,besides1 :5 "o ( _! demon/rations has be5 refut$1 we are sure to see x resurrect$ ) insignificant alt]ations1 & "s ( !m h already ris5 sev]al times f _! a%es4 ,su* long ago 0 ! ,l]nan hydra ) its fam\s heads :i* always grew aga94 ,h]cules got ?r\gh1 s9ce 8 hydra _h only n9e heads1 or elev52 b "h !re are too _m1 "s 9 ,engl&1 "s 9 ,g]many1 9 ,italy1 9 ,france1 & he w\ld h to give up ! /ruggle4 ,s ,i appeal only to m5 ( good judgm5t unprejudic$4 ,i ,9 ^! latt] ye>s num]\s works h be5 publi%$ on pure ma!matics & ! philosophy ( ma!matics1 try+ to sep>ate & isolate ! logical elem5ts ( ma!matical reason+4 ,^! works h be5 analyz$ & exp\nd$ v cle>ly by ,m4 ,c\turat 9 a book entitl$3 ,! ,pr9ciples ( ,ma!matics4 ,= ,m4 ,c\turat1 ! new works1 & 9 p>ticul> ^? ( ,russell & ,peano1 h f9ally settl$ ! controv]sy1 s long p5d+ betwe5 ,leibnitz & ,kant4 ,!y h %[n t !re are no syn!tic judgm5ts a priori 7,kant's phrase to designate judgm5ts :i* c nei!r be demon/rat$ analytically1 nor r$uc$ to id5tities1 nor e/abli%$ exp]im5tally71 !y h %[n t ma!matics is entirely r$ucible to logic & t intuition "h plays no rle4 ,?is is :at ,m4 ,c\turat has set =? 9 ! "w j cit$2 ?is he says /ill m explicitly 9 8 ,kant jubilee disc\rse1 s t ,i he>d my neighbor :isp]3 ,i well see ?is is ! c5t5>y ( ,kant's dea?4 ,c we subscribe to ?is conclusive condemnation8 ,i ?9k n1 & ,i %all try to %[ :y4 ,,ii ,:at /rikes u fir/ 9 ! new ma!matics is its purely =mal "*3 ,we ?9k1 says ,hilb]t1 ?ree sorts ( ?+s1 :i* we %all call po9ts1 /raights & planes4 ,we conv5e t a /raight %all be det]m9$ by two po9ts1 & t 9 place ( say+ ?is /raight is det]m9$ by ^! two po9ts1 we may say x passes ?r\gh ^! two po9ts1 or t ^! two po9ts are situat$ on ?is /raight4 ,:at ^! ?+s are1 n only we d n "k1 b we %\ld n seek to "k4 ,we h no ne$ to1 & "o :o nev] _h se5 ei!r po9t or /raight or plane c\ld geometrize z well z we4 ,t ! phrase to pass ?r\gh1 or ! phrase to be situat$ ^u may ar\se 9 u no image1 ! fir/ is simply a synonym ( to be det]m9$ & ! second ( to det]m9e4 ,?us1 be x und]/ood1 to demon/rate a !orem1 x is nei!r necess>y nor ev5 advantage\s to "k :at x means4 ,! geomet] might be replac$ by ! logic piano imag9$ by ,/anley ,jevons2 or1 if y *oose1 a ma*9e might be imag9$ :]e ! assumptions 7 put 9 at "o end1 :ile ! !orems came out at ! o!r1 l ! leg5d>y ,*icago ma*9e :]e ! pigs g 9 alive & come out trans=m$ into hams & sausages4 ,no m ?an ^! ma*9es ne$ ! ma!matician "k :at he does4 ,i d n make ?is =mal "* ( 8 geometry a reproa* to ,hilb]t4 ,?is is ! way he %\ld g1 giv5 ! problem he set himself4 ,he wi%$ to r$uce to a m9imum ! numb] ( ! fundam5tal assumptions ( geometry & completely enum]ate !m2 n[1 9 reason+s :]e our m9d rema9s active1 9 ^? :]e intuition /ill plays a "p1 9 liv+ reason+s1 s to speak1 x is difficult n to introduce an assumption or a po/ulate :i* passes unp]ceiv$4 ,x is !re=e only aft] hav+ c>ri$ back all ! geometric reason+s to a =m purely me*anical t he c\ld be sure ( hav+ accompli%$ 8 design & f9i%$ 8 "w4 ,:at ,hilb]t did = geometry1 o!rs h tri$ to d = ari?metic & analysis4 ,ev5 if !y _h entirely succe$$1 w\ld ! ,kantians be f9ally condemn$ to sil5ce8 ,p]haps n1 = 9 r$uc+ ma!matical ?\ght to an empty =m1 x is c]ta9ly mutilat$4 ,ev5 admitt+ x 7 e/abli%$ t all ! !orems c\ld be d$uc$ by proc$ures purely analytic1 by simple logical comb9ations ( a f9ite numb] ( assumptions1 & t ^! assumptions are only conv5tions2 ! philosoph] w\ld /ill h ! "r to inve/igate ! orig9s ( ^! conv5tions1 to see :y !y h be5 judg$ pref]able to ! contr>y conv5tions4 ,& !n ! logical correctness ( ! reason+s lead+ f ! assumptions to ! !orems is n ! only ?+ :i* %\ld occupy u4 ,! rules ( p]fect logic1 are !y ! :ole ( ma!matics8 ,z well say ! :ole art ( play+ *ess r$uces to ! rules ( ! moves ( ! pieces4 ,among all ! con/ructs :i* c be built up ( ! mat]ials furni%$ by logic1 *oice mu/ be made2 ! true geomet] makes ?is *oice judici\sly because he is guid$ by a sure in/9ct1 or by "s vague consci\sness ( ,i "k n :at m pr(\nd & m hidd5 geometry1 :i* alone gives value to ! edifice con/ruct$4 ,to seek ! orig9 ( ?is in/9ct1 to /udy ! laws ( ?is deep geometry1 felt1 n /at$1 w\ld also be a f9e employm5t = ! philosoph]s :o d n want logic to be all4 ,b x is n at ?is po9t ( view ,i wi% to put myself1 x is n ?us ,i wi% to consid] ! "q4 ,! in/9ct m5tion$ is necess>y = ! inv5tor1 b x w\ld seem at fir/ we might d )\t x 9 /udy+ ! sci5ce once creat$4 ,well1 :at ,i wi% to inve/igate is if x be true t1 ! pr9ciples ( logic once admitt$1 "o c1 ,i d n say discov]1 b demon/rate1 all ! ma!matical v]ities )\t mak+ a new appeal to intuition4 ,,iii ,i once said no to ?is "q3#ab %\ld our reply be modifi$ by ! rec5t works8 ,my say+ no 0 because ! pr9ciple ( complete induction seem$ to me at once necess>y to ! ma!matician & irr$ucible to logic4 ,! /atem5t ( ?is pr9ciple is3 ,if a prop]ty be true ( ! numb] #a1 & if we e/abli% t x is true ( n #a provid$ x be ( n1 x w be true ( all ! :ole numb]s4 ,!re9 ,i see ! ma!matical reason+ p> excell5ce4 ,i did n mean to say1 z has be5 suppos$1 t all ma!matical reason+s c be r$uc$ to an application ( ?is pr9ciple4 ,exam9+ ^! reason+s closely1 we !re %\ld see appli$ _m o!r analog\s pr9ciples1 pres5t+ ! same ess5tial *>act]i/ics4 ,9 ?is category ( pr9ciples1 t ( complete induction is only ! simple/ ( all & ?is is :y ,i h *os5 x z type4 #ab ,see ,sci5ce & ,hypo!sis1 *apt] ,i4 ,! curr5t "n1 pr9ciple ( complete induction1 is n ju/ifi$4 ,?is mode ( reason+ is none ! less a true ma!matical induction :i* diff]s f ord9>y induction only by its c]titude4 ,,iv ,,def9itions ,,& ,,assumptions ,! exi/5ce ( su* pr9ciples is a difficulty = ! uncompromis+ logicians2 h[ d !y pret5d to get out ( x8 ,! pr9ciple ( complete induction1 !y say1 is n an assumption prop]ly s call$ or a syn!tic judgm5t a priori2 x is j simply ! def9ition ( :ole numb]4 ,x is !re=e a simple conv5tion4 ,to discuss ?is way ( look+ at x1 we mu/ exam9e a little closely ! relations betwe5 def9itions & assumptions4 ,let u g back fir/ to an article by ,m4 ,c\turat on ma!matical def9itions :i* appe>$ 9 l',enseignem5t ma?matique1 a magaz9e publi%$ by ,gau?i]-,vill>s & by ,georg at ,g5eva4 ,we %all see !re a di/9ction betwe5 ! direct def9ition & ! def9ition by po/ulates4 ,! def9ition by po/ulates1 says ,m4 ,c\turat1 applies n to a s+le notion1 b to a sy/em ( notions2 x consi/s 9 enum]at+ ! fundam5tal relations :i* unite !m & :i* enable u to demon/rate all _! o!r prop]ties2 ^! relations are po/ulates4 ,if previ\sly h be5 def9$ all ^! notions b "o1 !n ?is la/ w be by def9ition ! ?+ :i* v]ifies ^! po/ulates4 ,?us c]ta9 indemon/rable assumptions ( ma!matics w\ld be only disguis$ def9itions4 ,?is po9t ( view is (t5 legitimate2 & ,i h myself admitt$ x 9 reg>d = in/ance to ,euclid's po/ulate4 ,! o!r assumptions ( geometry d n suffice to completely def9e di/ance2 ! di/ance !n w be1 by def9ition1 among all ! magnitudes :i* satisfy ^! o!r assumptions1 t :i* is su* z to make ,euclid's po/ulate true4 ,well ! logicians suppose true = ! pr9ciple ( complete induction :at ,i admit = ,euclid's po/ulate2 !y want to see 9 x only a disguis$ def9ition4 ,b to give !m ?is "r1 two conditions mu/ be fulfill$4 ,/u>t ,mill says e def9ition implies an assumption1 t by :i* ! exi/5ce ( ! def9$ object is affirm$4 ,accord+ to t1 x w\ld no long] be ! assumption :i* might be a disguis$ def9ition1 x w\ld on ! contr>y be ! def9ition :i* w\ld be a disguis$ assumption4 ,/u>t ,mill meant ! ^w exi/5ce 9 a mat]ial & empirical s5se2 he meant to say t 9 def9+ ! circle we affirm !re are r\nd ?+s 9 nature4 ,"u ?is =m1 8 op9ion is inadmissible4 ,ma!matics is indep5d5t ( ! exi/5ce ( mat]ial objects2 9 ma!matics ! ^w exi/ c h only "o mean+1 x means free f contradiction4 ,?us rectifi$1 ,/u>t ,mill's ?\ght becomes exact2 9 def9+ a ?+1 we affirm t ! def9ition implies no contradiction4 ,if !re=e we h a sy/em ( po/ulates1 & if we c demon/rate t ^! po/ulates imply no contradiction1 we %all h ! "r to consid] !m z repres5t+ ! def9ition ( "o ( ! notions ent]+ !re94 ,if we c n demon/rate t1 x mu/ be admitt$ )\t pro(1 & t !n w be an assumption2 s t1 seek+ ! def9ition "u ! po/ulate1 we %\ld f9d ! assumption "u ! def9ition4 ,usually1 to %[ t a def9ition implies no contradiction1 we proce$ by example1 we try to make an example ( a ?+ satisfy+ ! def9ition4 ,take ! case ( a def9ition by po/ulates2 we wi% to def9e a notion ,a1 & we say t1 by def9ition1 an ,a is any?+ = :i* c]ta9 po/ulates are true4 ,if we c prove directly t all ^! po/ulates are true ( a c]ta9 object ,b1 ! def9ition w be ju/ifi$2 ! object ,b w be an example ( an ,a4 ,we %all be c]ta9 t ! po/ulates are n contradictory1 s9ce !re are cases :]e !y are all true at ! same "t4 ,b su* a direct demon/ration by example is n always possible4 ,to e/abli% t ! po/ulates imply no contradiction1 x is !n necess>y to consid] all ! propositions d$ucible f ^! po/ulates consid]$ z premises1 & to %[ t1 among ^! propositions1 no two are contradictory4 ,if ^! propositions are f9ite 9 numb]1 a direct v]ification is possible4 ,?is case is infrequ5t & un9t]e/+4 ,if ^! propositions are inf9ite 9 numb]1 ?is direct v]ification c no long] be made2 rec\rse mu/ be _h to proc$ures :]e 9 g5]al x is necess>y to invoke j ?is pr9ciple ( complete induction :i* is precisely ! ?+ to be prov$4 ,?is is an explanation ( "o ( ! conditions ! logicians %\ld satisfy1 & fur!r on we %all see !y h n done x4 ,v ,!re is a second4 ,:5 we give a def9ition1 x is to use x4 ,we !re=e %all f9d 9 ! sequel ( ! exposition ! ^w def9$2 h we ! "r to affirm1 ( ! ?+ repres5t$ by ?is ^w1 ! po/ulate :i* has s]v$ = def9ition8 ,yes1 evid5tly1 if ! ^w has reta9$ its mean+1 if we d n attribute to x implicitly a diff]5t mean+4 ,n[ ?is is :at sometimes happ5s & x is usually difficult to p]ceive x2 x is ne$ful to see h[ ?is ^w comes into our disc\rse1 & if ! gate by :i* x has ent]$ does n imply 9 reality a def9ition o!r ?an t /at$4 ,?is difficulty pres5ts itself 9 all ! applications ( ma!matics4 ,! ma!matical notion has be5 giv5 a def9ition v ref9$ & v rigor\s2 & = ! pure ma!matician all d\bt has disappe>$2 b if "o wi%es to apply x to ! physical sci5ces = in/ance1 x is no long] a "q ( ?is pure notion1 b ( a concrete object :i* is (t5 only a r\gh image ( x4 ,to say t ?is object satisfies1 at lea/ approximately1 ! def9ition1 is to /ate a new tru?1 :i* exp]i5ce alone c put beyond d\bt1 & :i* no long] has ! "* ( a conv5tional po/ulate4 ,b )\t go+ beyond pure ma!matics1 we also meet ! same difficulty4 ,y give a subtile def9ition ( numb]s2 !n1 once ?is def9ition giv51 y ?9k no m ( x2 because1 9 reality1 x is n x :i* has taught y :at numb] is2 y long ago knew t1 & :5 ! ^w numb] fur!r on is f\nd "u y\r p51 y give x ! same s5se z ! fir/ com]4 ,to "k :at is ?is mean+ & :e!r x is ! same 9 ?is phrase or t1 x is ne$ful to see h[ y h be5 l$ to speak ( numb] & to introduce ?is ^w into ^! two phrases4 ,i %all n = ! mom5t dilate ^u ?is po9t1 because we %all h occasion to return to x4 ,?us consid] a ^w ( :i* we h giv5 explicitly a def9ition ,a2 aft]w>ds 9 ! disc\rse we make a use ( x :i* implicitly supposes ano!r def9ition ,b4 ,x is possible t ^! two def9itions designate ! same ?+4 ,b t ?is is s is a new tru? :i* mu/ ei!r be demon/rat$ or admitt$ z an indep5d5t assumption4 ,we %all see f>!r on t ! logicians h n fulfill$ ! second condition any bett] ?an ! fir/4 ,,vi ,! def9itions ( numb] are v num]\s & v diff]5t2 ,i =ego ! enum]ation ev5 ( ! names ( _! au?ors4 ,we %\ld n be a/oni%$ t !re are s _m4 ,if "o among !m 0 satisfactory1 no new "o w\ld be giv54 ,if ea* new philosoph] occupy+ himself ) ?is "q has ?\ght he mu/ inv5t ano!r "o1 ?is 0 because he 0 n satisfi$ ) ^? ( 8 pr$ecessors1 & he 0 n satisfi$ ) !m because he ?\ght he saw a petitio pr9cipii4 ,i h always felt1 9 read+ ! writ+s devot$ to ?is problem1 a pr(\nd feel+ ( discom=t2 ,i 0 always expect+ to run aga9/ a petitio pr9cipii1 & :5 ,i did n imm$iately p]ceive x1 ,i fe>$ ,i _h ov]look$ x4 ,?is is because x is impossible to give a def9ition )\t us+ a s5t5ce1 & difficult to make a s5t5ce )\t us+ a numb] ^w1 or at lea/ ! ^w sev]al1 or at lea/ a ^w 9 ! plural4 ,& !n ! declivity is slipp]y & at ea* in/ant !re is risk ( a fall into petitio pr9cipii4 ,i %all devote my att5tion 9 :at foll[s only to ^? ( ^! def9itions :]e ! petitio pr9cipii is mo/ ably conceal$4 ,,vii ,,pasigraphy ,! symbolic language creat$ by ,peano plays a v gr& rle 9 ^! new rese>*es4 ,x is capable ( r5d]+ "s s]vice1 b ,i ?9k ,m4 ,c\turat atta*es to x an exagg]at$ importance :i* mu/ a/oni% ,peano himself4 ,! ess5tial elem5t ( ?is language is c]ta9 algebraic signs :i* repres5t ! diff]5t conjunctions3 if1 &1 or1 !re=e4 ,t ^! signs may be conv5i5t is possible2 b t !y are de/9$ to revolutionize all philosophy is a diff]5t matt]4 ,x is difficult to admit t ! ^w if acquires1 :5 writt5 ,c1 a virtue x _h n :5 writt5 if4 ,?is inv5tion ( ,peano 0 fir/ call$ pasigraphy1 t is to say ! art ( writ+ a treatise on ma!matics )\t us+ a s+le ^w ( ord9>y language4 ,?is "n def9$ its range v exactly4 ,lat]1 x 0 rais$ to a m em95t dignity by conf]r+ on x ! title ( logi/ic4 ,?is ^w is1 x appe>s1 employ$ at ! ,milit>y ,academy1 to designate ! art ( ! qu>t]ma/] ( cavalry1 ! art ( m>*+ & canton+ troops2 b "h no confusion ne$ be fe>$1 & x is at once se5 t ?is new "n implies ! design ( revolutioniz+ logic4 ,we may see ! new me?od at "w 9 a ma!matical memoir by ,burali-,=ti1 entitl$3 ,una ,que/ione sui num]i transf9iti1 ins]t$ 9 ,volume ,,xi ( ! ,r5diconti del circolo matematico di ,pal]mo4 ,i beg9 by say+ ?is memoir is v int]e/+1 & my tak+ x "h z example is precisely because x is ! mo/ important ( all ^? writt5 9 ! new language4 ,besides1 ! un9itiat$ may read x1 ?anks to an ,italian int]l9e> translation4 ,:at con/itutes ! importance ( ?is memoir is t x has giv5 ! fir/ example ( ^? ant9omies met 9 ! /udy ( transf9ite numb]s & mak+ s9ce "s ye>s ! despair ( ma!maticians4 ,! aim1 says ,burali-,=ti1 ( ?is note is to %[ !re may be two transf9ite numb]s 7ord9als71 a & b1 su* t a is nei!r equal to1 great] ?an1 nor less ?an b4 ,to reassure ! read]1 to compreh5d ! consid]ations :i* foll[1 he has no ne$ ( kn[+ :at a transf9ite ord9al numb] is4 ,n[1 ,cantor _h precisely prov$ t betwe5 two transf9ite numb]s z betwe5 two f9ite1 !re c be no o!r relation ?an equality or inequality 9 "o s5se or ! o!r4 ,b x is n ( ! sub/ance ( ?is memoir t ,i wi% to speak "h2 t w\ld c>ry me mu* too f> f my subject2 ,i only wi% to consid] ! =m1 & j to ask if ?is =m makes x ga9 mu* 9 rigor & :e!r x ?us comp5sates = ! ef=ts x imposes ^u ! writ] & ! read]4 ,fir/ we see ,burali-,=ti def9e ! numb] #a z foll[s3 #a iota,t',ko7n7u1 h7epsilon7uepsilon,un71 a def9ition em95tly fitt$ to give an idea ( ! numb] #a to p]sons :o _h nev] he>d speak ( x4 ,i und]/& ,peanian too ill to d>e risk a critique1 b /ill ,i fe> ?is def9ition conta9s a petitio pr9cipii1 consid]+ t ,i see ! figure #a 9 ! fir/ memb] & ,un 9 lett]s 9 ! second4 ,h[ev] t may be1 ,burali-,=ti />ts f ?is def9ition &1 aft] a %ort calculation1 rea*es ! equation3 7#bg7 #a;epsilon,no1 :i* tells u t ,"o is a numb]4 ,& s9ce we are on ^! def9itions ( ! fir/ numb]s1 we recall t ,m4 ,c\turat has also def9$ #j & #a4 ,:at is z]o8 ,x is ! numb] ( elem5ts ( ! null class4 ,& :at is ! null class8 ,x is t conta9+ no elem5t4 ,to def9e z]o by null1 & null by no1 is really to abuse ! weal? ( language2 s ,m4 ,c\turat has introduc$ an improvem5t 9 8 def9ition1 by writ+3 #j iota,lambda3phix ,lambdainv]t$ c,lambda 7xepsilonphix71 :i* means3 z]o is ! numb] ( ?+s satisfy+ a condition nev] satisfi$4 ,b z nev] means 9 no case ,i d n see t ! progress is great4 ,i ha/5 to add t ! def9ition ,m4 ,c\turat gives ( ! numb] #a is m satisfactory4 ,"o1 says he 9 sub/ance1 is ! numb] ( elem5ts 9 a class 9 :i* any two elem5ts are id5tical4 ,x is m satisfactory1 ,i h said1 9 ?is s5se t to def9e #a1 he does n use ! ^w "o2 9 comp5sation1 he uses ! ^w two4 ,b ,i fe>1 if ask$ :at is two1 ,m4 ,c\turat w\ld h to use ! ^w "o4 ,,viii ,b to return to ! memoir ( ,burali-,=ti2 ,i h said 8 conclusions are 9 direct opposition to ^? ( ,cantor4 ,n[1 "o day ,m4 ,hadam>d came to see me & ! talk fell ^u ?is ant9omy4 ,burali-,=ti's reason+1 ,i said1 does x n seem to y irreproa*able8 ,no1 & on ! contr>y ,i f9d no?+ to object to 9 t ( ,cantor4 ,besides1 ,burali-,=ti _h no "r to speak ( ! aggregate ( all ! ord9al numb]s4 ,p>don1 he _h ! "r1 s9ce he c\ld always put ,omega ,t'7,no1epsilon74 ,i %\ld l to "k :o 0 to prev5t him1 & c x be said a ?+ does n exi/1 :5 we h call$ x ,omega8 ,x 0 9 va91 ,i c\ld n conv9ce him 7:i* besides w\ld h be5 sad1 s9ce he 0 "r74 ,0 x m]ely because ,i d n speak ! ,peanian ) en\gh eloqu5ce8 ,p]haps2 b betwe5 ourselves ,i d n ?9k s4 ,?us1 despite all ?is pasigraphic app>atus1 ! "q 0 n solv$4 ,:at does t prove8 ,9 s f> z x is a "q only ( prov+ "o a numb]1 pasigraphy suffices1 b if a difficulty pres5ts itself1 if !re is an ant9omy to solve1 pasigraphy becomes impot5t4 ,,*apt] ,,iv ,,! ,,new ,,logics ,i ,! ,russell ,logic ,to ju/ify its pret5sions1 logic _h to *ange4 ,we h se5 new logics arise ( :i* ! mo/ int]e/+ is t ( ,russell4 ,x seems he has no?+ new to write ab\t =mal logic1 z if ,ari/otle !re _h t\*$ bottom4 ,b ! doma9 ,russell attributes to logic is inf9itely m ext5d$ ?an t ( ! classic logic1 & he has put =? on ! subject views :i* are orig9al & at times well w>rant$4 ,fir/1 ,russell subord9ates ! logic ( classes to t ( propositions1 :ile ! logic ( ,ari/otle 0 above all ! logic ( classes & took z its po9t ( dep>ture ! relation ( subject to pr$icate4 ,! classic syllogism1 ,socrates is a man1 etc41 gives place to ! hypo!tical syllogism3 ,if ,a is true1 ,b is true2 n[ if ,b is true1 ,c is true1 etc4 ,& ?is is1 ,i ?9k1 a mo/ happy idea1 because ! classic syllogism is easy to c>ry back to ! hypo!tical syllogism1 :ile ! inv]se trans=mation is n )\t difficulty4 ,& !n ?is is n all4 ,russell's logic ( propositions is ! /udy ( ! laws ( comb9ation ( ! conjunctions if1 &1 or1 & ! negation n4 ,9 add+ "h two o!r conjunctions1 & & or1 ,russell op5s to logic a new field4 ,! symbols &1 or foll[ ! same laws z ! two signs & 1 t is to say ! commutative associative & di/ributive laws4 ,?us & repres5ts logical multiplication1 :ile or repres5ts logical addition4 ,?is also is v int]e/+4 ,russell rea*es ! conclusion t any false proposition implies all o!r propositions true or false4 ,m4 ,c\turat says ?is conclusion w at fir/ seem p>adoxical4 ,x is suffici5t h[ev] to h correct$ a bad !sis 9 ma!matics to recognize h[ "r ,russell is4 ,! c&idate (t5 is at great pa9s to get ! fir/ false equation2 b t once obta9$1 x is only sport !n = him to accumulate ! mo/ surpris+ results1 "s ( :i* ev5 may be true4 ,,ii ,we see h[ mu* ri*] ! new logic is ?an ! classic logic2 ! symbols are multipli$ & all[ ( v>i$ comb9ations :i* are no long] limit$ 9 numb]4 ,has "o ! "r to give ?is ext5sion to ! mean+ ( ! ^w logic8 ,x w\ld be useless to exam9e ?is "q & to seek ) ,russell a m]e qu>rel ab\t words4 ,grant him :at he dem&s2 b be n a/oni%$ if c]ta9 v]ities decl>$ irr$ucible to logic 9 ! old s5se ( ! ^w f9d !mselves n[ r$ucible to logic 9 ! new s5se--some?+ v diff]5t4 ,a great numb] ( new notions h be5 introduc$1 & ^! are n simply comb9ations ( ! old4 ,russell kn[s ?is1 & n only at ! beg9n+ ( ! fir/ *apt]1 ',! ,logic ( ,propositions1' b at ! beg9n+ ( ! second & ?ird1 ',! ,logic ( ,classes' & ',! ,logic ( ,relations1' he introduces new words t he decl>es indef9able4 ,& ?is is n all2 he likewise introduces pr9ciples he decl>es indemon/rable4 ,b ^! indemon/rable pr9ciples are appeals to intuition1 syn!tic judgm5ts a priori4 ,we reg>d !m z intuitive :5 we meet !m m or less explicitly enunciat$ 9 ma!matical treatises2 h !y *ang$ "* because ! mean+ ( ! ^w logic has be5 enl>g$ & we n[ f9d !m 9 a book entitl$ ,treatise on ,logic8 ,!y h n *ang$ nature2 !y h only *ang$ place4 ,,iii ,c\ld ^! pr9ciples be consid]$ z disguis$ def9itions8 ,x w\ld !n be necess>y to h "s way ( prov+ t !y imply no contradiction4 ,x w\ld be necess>y to e/abli% t1 h[ev] f> "o foll[$ ! s]ies ( d$uctions1 he w\ld nev] be expos$ to contradict+ himself4 ,we might attempt to reason z foll[s3 ,we c v]ify t ! op]ations ( ! new logic appli$ to premises exempt f contradiction c only give consequ5ces equally exempt f contradiction4 ,if !re=e aft] n op]ations we h n met contradiction1 we %all n enc\nt] x aft] n #a4 ,?us x is impossible t !re %\ld be a mom5t :5 contradiction beg9s1 :i* %[s we %all nev] meet x4 ,h we ! "r to reason 9 ?is way8 ,no1 = ?is w\ld be to make use ( complete induction2 & rememb]1 we d n yet "k ! pr9ciple ( complete induction4 ,we !re=e h n ! "r to reg>d ^! assumptions z disguis$ def9itions & only "o res\rce rema9s = u1 to admit a new act ( intuition = ea* ( !m4 ,moreov] ,i believe ?is is inde$ ! ?\ght ( ,russell & ,m4 ,c\turat4 ,?us ea* ( ! n9e indef9able notions & ( ! tw5ty indemon/rable propositions 7,i believe if x 7 ,i t did ! c\nt+1 ,i %\ld h f\nd "s m7 :i* are ! f\ndation ( ! new logic1 logic 9 ! broad s5se1 presupposes a new & indep5d5t act ( our intuition & 7:y n say x87 a v]itable syn!tic judgm5t a priori4 ,on ?is po9t all seem agre$1 b :at ,russell claims1 & :at seems to me d\btful1 is t aft] ^! appeals to intuition1 t w be ! end ( x2 we ne$ make no o!rs & c build all ma!matics )\t ! int]v5tion ( any new elem5t4 ,,iv ,m4 ,c\turat (t5 repeats t ?is new logic is altoge!r indep5d5t ( ! idea ( numb]4 ,i %all n amuse myself by c\nt+ h[ _m num]al adjectives 8 exposition conta9s1 bo? c>d9al & ord9al1 or indef9ite adjectives su* z sev]al4 ,we may cite1 h[ev]1 "s examples3 ,! logical product ( two or m propositions is44442 ,all propositions are capable only ( two values1 true & false2 ,! relative product ( two relations is a relation2 ,a relation exi/s betwe5 two t]ms1 etc41 etc4 ,sometimes ?is inconv5i5ce w\ld n be unavoidable1 b sometimes also x is ess5tial4 ,a relation is incompreh5sible )\t two t]ms2 x is impossible to h ! intuition ( ! relation1 )\t hav+ at ! same "t t ( its two t]ms1 & )\t notic+ !y are two1 because1 if ! relation is to be conceivable1 x is necess>y t !re be two & only two4 ,v ,ari?metic ,i rea* :at ,m4 ,c\turat calls ! ord9al !ory :i* is ! f\ndation ( ari?metic prop]ly s call$4 ,m4 ,c\turat beg9s by /at+ ,peano's five assumptions1 :i* are indep5d5t1 z has be5 prov$ by ,peano & ,padoa4 #a4 ,z]o is an integ]4 #b4 ,z]o is n ! successor ( any integ]4 #c4 ,! successor ( an integ] is an integ]4 ,to ?is x w\ld be prop] to add1 ,e integ] has a successor4 #d4 ,two integ]s are equal if _! successors are4 ,! fif? assumption is ! pr9ciple ( complete induction4 ,m4 ,c\turat consid]s ^! assumptions z disguis$ def9itions2 !y con/itute ! def9ition by po/ulates ( z]o1 ( successor1 & ( integ]4 ,b we h se5 t = a def9ition by po/ulates to be acceptable we mu/ be able to prove t x implies no contradiction4 ,is ?is ! case "h8 ,n at all4 ,! demon/ration c n be made by example4 ,we c n take a "p ( ! integ]s1 = in/ance ! fir/ ?ree1 & prove !y satisfy ! def9ition4 ,if ,i take ! s]ies #j1 #a1 #b1 ,i see x fulfils ! assumptions #a1 #b1 #d & #e2 b to satisfy assumption #c x /ill is necess>y t #c be an integ]1 & consequ5tly t ! s]ies #j1 #a1 #b1 #c1 fulfil ! assumptions2 we might prove t x satisfies assumptions #a1 #b1 #d1 #e1 b assumption #c requires besides t #d be an integ] & t ! s]ies #j1 #a1 #b1 #c1 #d fulfil ! assumptions1 & s on4 ,x is !re=e impossible to demon/rate ! assumptions = c]ta9 integ]s )\t prov+ !m = all2 we mu/ give up pro( by example4 ,x is necess>y !n to take all ! consequ5ces ( our assumptions & see if !y conta9 no contradiction4 ,if ^! consequ5ces 7 f9ite 9 numb]1 ?is w\ld be easy2 b !y are inf9ite 9 numb]2 !y are ! :ole ( ma!matics1 or at lea/ all ari?metic4 ,:at !n is to be done8 ,p]haps /rictly we c\ld repeat ! reason+ ( numb] ,,iii4 ,b z we h said1 ?is reason+ is complete induction1 & x is precisely ! pr9ciple ( complete induction ^: ju/ification w\ld be ! po9t 9 "q4 ,,vi ,! ,logic ( ,hilb]t ,i come n[ to ! capital "w ( ,hilb]t :i* he communicat$ to ! ,congress ( ,ma!maticians at ,heidelb]g1 & ( :i* a ,fr5* translation by ,m4 ,pi]re ,b\tr\x appe>$ 9 l',enseignem5t ma?matique1 :ile an ,engli% translation due to ,hal/$ appe>$ 9 ,! ,moni/4#ac ,9 ?is "w1 :i* conta9s pr(\nd ?\ghts1 ! au?or's aim is analog\s to t ( ,russell1 b on _m po9ts he div]ges f 8 pr$ecessor4 #ac ',! ,f\ndations ( ,logic & ,ari?metic1' ,moni/1 ,,xv41 #cch-ceb4 ,b1 he says 7,moni/1 p4 #cdj71 on att5tive consid]ation we become aw>e t 9 ! usual exposition ( ! laws ( logic c]ta9 fundam5tal concepts ( ari?metic are already employ$2 = example1 ! concept ( ! aggregate1 9 "p also ! concept ( numb]4 ,we fall ?us into a vici\s circle & !re=e to avoid p>adoxes a p>tly simultane\s developm5t ( ! laws ( logic & ari?metic is requisite4 ,we h se5 above t :at ,hilb]t says ( ! pr9ciples ( logic 9 ! usual exposition applies likewise to ! logic ( ,russell4 ,s = ,russell logic is prior to ari?metic2 = ,hilb]t !y are 'simultane\s4' ,we %all f9d fur!r on o!r diff]5ces /ill great]1 b we %all po9t !m out z we come to !m4 ,i pref] to foll[ /ep by /ep ! developm5t ( ,hilb]t's ?\ght1 quot+ textually ! mo/ important passages4 ,let u take z ! basis ( our consid]ation fir/ ( all a ?\ght-?+ #a 7"o7 7p4 #cda74 ,notice t 9 s do+ we 9 no wise imply ! notion ( numb]1 because x is und]/ood t #a is "h only a symbol & t we d n at all seek to "k its mean+4 ,! tak+ ( ?is ?+ toge!r ) itself respectively two1 ?ree or m times4444 ,ah6 ?is "t x is no long] ! same2 if we introduce ! words 'two1' '?ree1' & above all 'm1' 'sev]al1' we introduce ! notion ( numb]2 & !n ! def9ition ( f9ite :ole numb] :i* we %all pres5tly f9d1 w come too late4 ,our au?or 0 too circumspect n to p]ceive ?is begg+ ( ! "q4 ,s at ! end ( 8 "w he tries to proce$ to a truly pat*+-up process4 ,hilb]t !n introduces two simple objects #a & 1 & consid]s all ! comb9ations ( ^! two objects1 all ! comb9ations ( _! comb9ations1 etc4 ,x goes )\t say+ t we mu/ =get ! ord9>y mean+ ( ^! two signs & n attribute any to !m4 ,aft]w>ds he sep>ates ^! comb9ations into two classes1 ! class ( ! exi/5t & ! class ( ! non-exi/5t1 & till fur!r ord]s ?is sep>ation is entirely arbitr>y4 ,e affirmative /atem5t tells u t a c]ta9 comb9ation belongs to ! class ( ! exi/5t2 e negative /atem5t tells u t a c]ta9 comb9ation belongs to ! class ( ! non-exi/5t4 ,,vii ,note n[ a diff]5ce ( ! highe/ importance4 ,= ,russell any object :atsoev]1 :i* he designates by x1 is an object absolutely undet]m9$ & ab\t :i* he supposes no?+2 = ,hilb]t x is "o ( ! comb9ations =m$ ) ! symbols #a & 2 he c\ld n conceive ( ! introduction ( any?+ o!r ?an comb9ations ( objects already def9$4 ,moreov] ,hilb]t =mulates 8 ?\ght 9 ! neate/ way1 & ,i ?9k ,i mu/ reproduce 9 ext5so 8 /atem5t 7p4 #cdh73 ,9 ! assumptions ! arbitr>ies 7z equival5t = ! concept 'e' & 'all' 9 ! cu/om>y logic7 repres5t only ^? ?\ght-?+s & _! comb9ations ) "o ano!r1 :i* at ?is /age are laid d[n z fundam5tal or are to be newly def9$4 ,!re=e 9 ! d$uction ( inf]5ces f ! assumptions1 ! arbitr>ies1 :i* occur 9 ! assumptions1 c be replac$ only by su* ?\ght-?+s & _! comb9ations4 ,also we mu/ duly rememb]1 t ?r\gh ! sup]-addition & mak+ fundam5tal ( a new ?\ght-?+ ! prec$+ assumptions und]go an enl>gem5t ( _! validity1 & :]e necess>y1 are to be subject$ to a *ange 9 con=mity ) ! s5se4 ,! contra/ ) ,russell's view-po9t is complete4 ,= ?is philosoph] we may sub/itute = x n only objects already kn[n1 b any?+4 ,russell is fai?ful to 8 po9t ( view1 :i* is t ( compreh5sion4 ,he />ts f ! g5]al idea ( be+1 & enri*es x m & m :ile re/rict+ x1 by add+ new qualities4 ,hilb]t on ! contr>y recognizes z possible be+s only comb9ations ( objects already kn[n2 s t 7look+ at only "o side ( 8 ?\ght7 we might say he takes ! view-po9t ( ext5sion4 ,,viii ,let u cont9ue ) ! exposition ( ,hilb]t's ideas4 ,he introduces two assumptions :i* he /ates 9 8 symbolic language b :i* signify1 9 ! language ( ! un9itiat$1 t e quality is equal to itself & t e op]ation p]=m$ ^u two id5tical quantities gives id5tical results4 ,s /at$1 !y are evid5t1 b ?us to pres5t !m w\ld be to misrepres5t ,hilb]t's ?\ght4 ,= him ma!matics has to comb9e only pure symbols1 & a true ma!matician %\ld reason ^u !m )\t preconceptions z to _! mean+4 ,s 8 assumptions are n = him :at !y are = ! common p4 ,he consid]s !m z repres5t+ ! def9ition by po/ulates ( ! symbol 77 h]et(ore void ( all signification4 ,b to ju/ify ?is def9ition we mu/ %[ t ^! two assumptions lead to no contradiction4 ,= ?is ,hilb]t us$ ! reason+ ( our numb] ,,iii1 )\t appe>+ to p]ceive t he is us+ complete induction4 ,,ix ,! end ( ,hilb]t's memoir is altoge!r enigmatic & ,i %all n lay /ress ^u x4 ,contradictions accumulate2 we feel t ! au?or is dimly consci\s ( ! petitio pr9cipii he has committ$1 & t he seeks va9ly to pat* up ! holes 9 8 argum5t4 ,:at does ?is mean8 ,at ! po9t ( prov+ t ! def9ition ( ! :ole numb] by ! assumption ( complete induction implies no contradiction1 ,hilb]t )draws z ,russell & ,c\turat )drew1 because ! difficulty is too great4 ,x ,geometry ,geometry1 says ,m4 ,c\turat1 is a va/ body ( doctr9e :]e9 ! pr9ciple ( complete induction does n ent]4 ,t is true 9 a c]ta9 measure2 we c n say x is entirely abs5t1 b x ent]s v slightly4 ,if we ref] to ! ,rational ,geometry ( ,dr4 ,hal/$ 7,new ,york1 ,john ,wiley & ,sons1 #aijd7 built up 9 accordance ) ! pr9ciples ( ,hilb]t1 we see ! pr9ciple ( induction ent] = ! fir/ "t on page #aad 7unless ,i h made an ov]sight1 :i* is q possible74#ad #ad ,second ed41 #aijg1 p4 #hf2 ,fr5* ed41 #aiaa1 p4 #ig4 ,g4 ,b4 ,h4 ,s geometry1 :i* only a few ye>s ago seem$ ! doma9 :]e ! reign ( intuition 0 unconte/$1 is to-day ! realm :]e ! logicians seem to triumph4 ,no?+ c\ld bett] measure ! importance ( ! geometric works ( ,hilb]t & ! pr(\nd impress !y h left on our conceptions4 ,b be n deceiv$4 ,:at is aft] all ! fundam5tal !orem ( geometry8 ,x is t ! assumptions ( geometry imply no contradiction1 & ?is we c n prove )\t ! pr9ciple ( induction4 ,h[ does ,hilb]t demon/rate ?is ess5tial po9t8 ,by lean+ ^u analysis & ?r\gh x ^u ari?metic & ?r\gh x ^u ! pr9ciple ( induction4 ,& if ev] "o inv5ts ano!r demon/ration1 x w /ill be necess>y to lean ^u ?is pr9ciple1 s9ce ! possible consequ5ces ( ! assumptions1 ( :i* x is necess>y to %[ t !y are n contradictory1 are inf9ite 9 numb]4 ,,xi ,conclusion ,our conclusion /raightway is t ! pr9ciple ( induction c n be reg>d$ z ! disguis$ def9ition ( ! entire _w4 ,"h are ?ree tru?s3 7#a7 ,! pr9ciple ( complete induction2 7#b7 ,euclid's po/ulate2 7#c7 ! physical law accord+ to :i* phosphorus melts at #dd 7cit$ by ,m4 ,le ,roy74 ,^! are said to be ?ree disguis$ def9itions3 ! fir/1 t ( ! :ole numb]2 ! second1 t ( ! /raight l9e2 ! ?ird1 t ( phosphorus4 ,i grant x = ! second2 ,i d n admit x = ! o!r two4 ,i mu/ expla9 ! reason = ?is app>5t inconsi/5cy4 ,fir/1 we h se5 t a def9ition is acceptable only on condition t x implies no contradiction4 ,we h %[n likewise t = ! fir/ def9ition ?is demon/ration is impossible2 on ! o!r h&1 we h j recall$ t = ! second ,hilb]t has giv5 a complete pro(4 ,z to ! ?ird1 evid5tly x implies no contradiction4 ,does ?is mean t ! def9ition gu>antees1 z x %\ld1 ! exi/5ce ( ! object def9$8 ,we are "h no long] 9 ! ma!matical sci5ces1 b 9 ! physical1 & ! ^w exi/5ce has no long] ! same mean+4 ,x no long] signifies abs5ce ( contradiction2 x means objective exi/5ce4 ,y already see a fir/ reason = ! di/9ction ,i made betwe5 ! ?ree cases2 !re is a second4 ,9 ! applications we h to make ( ^! ?ree concepts1 d !y pres5t !mselves to u z def9$ by ^! ?ree po/ulates8 ,! possible applications ( ! pr9ciple ( induction are innum]able2 take1 = example1 "o ( ^? we h exp\nd$ above1 & :]e x is s\ght to prove t an aggregate ( assumptions c lead to no contradiction4 ,= ?is we consid] "o ( ! s]ies ( syllogisms we may g on ) 9 />t+ f ^! assumptions z premises4 ,:5 we h f9i%$ ! n? syllogism1 we see we c make /ill ano!r & ?is is ! n #a?4 ,?us ! numb] n s]ves to c\nt a s]ies ( successive op]ations2 x is a numb] obta9able by successive additions4 ,?is !re=e is a numb] f :i* we may g back to unity by successive subtractions4 ,evid5tly we c\ld n d ?is if we _h n n - #a1 s9ce !n by subtraction we %\ld always obta9 aga9 ! same numb]4 ,s ! way we h be5 l$ to consid] ?is numb] n implies a def9ition ( ! f9ite :ole numb] & ?is def9ition is ! foll[+3 ,a f9ite :ole numb] is t :i* c be obta9$ by successive additions2 x is su* t n is n equal to n - #a4 ,t grant$1 :at d we d8 ,we %[ t if !re has be5 no contradiction up to ! n? syllogism1 no m w !re be up to ! n #a?1 & we conclude !re nev] w be4 ,y say3 ,i h ! "r to draw ?is conclusion1 s9ce ! :ole numb]s are by def9ition ^? = :i* a l reason+ is legitimate4 ,b t implies ano!r def9ition ( ! :ole numb]1 :i* is z foll[s3 ,a :ole numb] is t on :i* we may reason by recurr5ce4 ,9 ! p>ticul> case x is t ( :i* we may say t1 if ! abs5ce ( contradiction up to ! "t ( a syllogism ( :i* ! numb] is an integ] c>ries ) x ! abs5ce ( contradiction up to ! "t ( ! syllogism ^: numb] is ! foll[+ integ]1 we ne$ fe> no contradiction = any ( ! syllogisms ^: numb] is an integ]4 ,! two def9itions are n id5tical2 !y are d\btless equival5t1 b only 9 virtue ( a syn!tic judgm5t a priori2 we c n pass f "o to ! o!r by a purely logical proc$ure4 ,consequ5tly we h no "r to adopt ! second1 aft] hav+ introduc$ ! :ole numb] by a way t presupposes ! fir/4 ,on ! o!r h&1 :at happ5s ) reg>d to ! /raight l9e8 ,i h already expla9$ ?is s (t5 t ,i hesitate to repeat x aga91 & %all conf9e myself to a brief recapitulation ( my ?\ght4 ,we h n1 z 9 ! prec$+ case1 two equival5t def9itions logically irr$ucible "o to ! o!r4 ,we h only "o expressible 9 words4 ,w x be said !re is ano!r :i* we feel )\t be+ able to ^w x1 s9ce we h ! intuition ( ! /raight l9e or s9ce we repres5t to ourselves ! /raight l9e8 ,fir/ ( all1 we c n repres5t x to ourselves 9 geometric space1 b only 9 repres5tative space1 & !n we c repres5t to ourselves j z well ! objects :i* possess ! o!r prop]ties ( ! /raight l9e1 save t ( satisfy+ ,euclid's po/ulate4 ,^! objects are '! non-,euclidean /raights1' :i* f a c]ta9 po9t ( view are n entities void ( s5se1 b circles 7true circles ( true space7 or?ogonal to a c]ta9 sph]e4 ,if1 among ^! objects equally capable ( repres5tation1 x is ! fir/ 7! ,euclidean /raights7 :i* we call /raights1 & n ! latt] 7! non-,euclidean /raights71 ?is is prop]ly by def9ition4 ,& arriv+ f9ally at ! ?ird example1 ! def9ition ( phosphorus1 we see ! true def9ition w\ld be3 ,phosphorus is ! bit ( matt] ,i see 9 yond] flask4 ,,xii ,& s9ce ,i am on ?is subject1 /ill ano!r ^w4 ,( ! phosphorus example ,i said3 ,?is proposition is a real v]ifiable physical law1 because x means t all bodies hav+ all ! o!r prop]ties ( phosphorus1 save its po9t ( fusion1 melt l x at #dd4 ,& x 0 answ]$3 ,no1 ?is law is n v]ifiable1 because if x 7 %[n t two bodies resembl+ phosphorus melt "o at #dd & ! o!r at #ej1 x might always be said t d\btless1 besides ! po9t ( fusion1 !re is "s o!r unkn[n prop]ty by :i* !y diff]4 ,t 0 n q :at ,i meant to say4 ,i %\ld h writt51 ,all bodies possess+ su* & su* prop]ties f9ite 9 numb] 7to wit1 ! prop]ties ( phosphorus /at$ 9 ! books on *emi/ry1 ! fusion-po9t except$7 melt at #dd4 ,& ! bett] to make evid5t ! diff]5ce betwe5 ! case ( ! /raight & t ( phosphorus1 "o m rem>k4 ,! /raight has 9 nature _m images m or less imp]fect1 ( :i* ! *ief are ! light rays & ! rotation axis ( ! solid4 ,suppose we f9d ! ray ( light does n satisfy ,euclid's po/ulate 7= example by %[+ t a /> has a negative p>allax71 :at %all we d8 ,%all we conclude t ! /raight be+ by def9ition ! trajectory ( light does n satisfy ! po/ulate1 or1 on ! o!r h&1 t ! /raight by def9ition satisfy+ ! po/ulate1 ! ray ( light is n /raight8 ,assur$ly we are free to adopt ! "o or ! o!r def9ition & consequ5tly ! "o or ! o!r conclusion2 b to adopt ! fir/ w\ld be /upid1 because ! ray ( light probably satisfies only imp]fectly n m]ely ,euclid's po/ulate1 b ! o!r prop]ties ( ! /raight l9e1 s t if x deviates f ! ,euclidean /raight1 x deviates no less f ! rotation axis ( solids :i* is ano!r imp]fect image ( ! /raight l9e2 :ile f9ally x is d\btless subject to *ange1 s t su* a l9e :i* ye/]day 0 /raight w cease to be /raight to-morr[ if "s physical circum/ance has *ang$4 ,suppose n[ we f9d t phosphorus does n melt at #dd1 b at #dc4i4 ,%all we conclude t phosphorus be+ by def9ition t :i* melts at #dd1 ?is body t we did call phosphorus is n true phosphorus1 or1 on ! o!r h&1 t phosphor\s melts at #dc4i8 ,"h aga9 we are free to adopt ! "o or ! o!r def9ition & consequ5tly ! "o or ! o!r conclusion2 b to adopt ! fir/ w\ld be /upid because we c n be *ang+ ! "n ( a sub/ance e "t we det]m9e a new decimal ( its fusion-po9t4 ,,xiii ,to sum up1 ,russell & ,hilb]t h ea* made a vigor\s ef=t2 !y h ea* writt5 a "w full ( orig9al views1 pr(\nd & (t5 well w>rant$4 ,^! two works give u mu* to ?9k ab\t & we h mu* to le>n f !m4 ,among _! results1 "s1 _m ev51 are solid & de/9$ to live4 ,b to say t !y h f9ally settl$ ! debate betwe5 ,kant & ,leibnitz & ru9$ ! ,kantian !ory ( ma!matics is evid5tly incorrect4 ,i d n "k :e!r !y really believ$ !y _h done x1 b if !y believ$ s1 !y deceiv$ !mselves4 ,,*apt] ,v ,,! ,,late/ ,,ef=ts ,,( ,,! ,,logi/icians ,i ,! logicians h attempt$ to answ] ! prec$+ consid]ations4 ,= t1 a trans=mation ( logi/ic 0 necess>y1 & ,russell 9 p>ticul> has modifi$ on c]ta9 po9ts 8 orig9al views4 ,)\t ent]+ into ! details ( ! debate1 ,i %\ld l to return to ! two que/ions to my m9d mo/ important3 ,h ! rules ( logi/ic demon/rat$ _! fruitfulness & infallibility8 ,is x true !y af=d means ( prov+ ! pr9ciple ( complete induction )\t any appeal to intuition8 ,,ii ,! ,infallibility ( ,logi/ic ,on ! "q ( f]tility1 x seems ,m4 ,c\turat has nave illusions4 ,logi/ic1 accord+ to him1 l5ds inv5tion '/ilts & w+s1' & on ! next page3 ,t5 ye>s ago1 ,peano publi%$ ! fir/ edition ( 8 ,=mulaire4 ,h[ is t1 t5 ye>s ( w+s & n to h fl[n6 ,i h ! highe/ e/eem = ,peano1 :o has done v pretty ?+s 7= in/ance 8 'space-fill+ curve1' a phrase n[ disc>d$72 b aft] all he has n gone fur!r nor high] nor quick] ?an ! majority ( w+less ma!maticians1 & w\ld h done j z well ) 8 legs4 ,on ! contr>y ,i see 9 logi/ic only %ackles = ! inv5tor4 ,x is no aid to concis5ess--f> f x1 & if tw5ty-sev5 equations 7 necess>y to e/abli% t #a is a numb]1 h[ _m w\ld be ne$$ to prove a real !orem8 ,if we di/+ui%1 ) ,:itehead1 ! individual x1 ! class ( :i* ! only memb] is x & :i* %all be call$ iotax1 !n ! class ( :i* ! only memb] is ! class ( :i* ! only memb] is x & :i* %all be call$ mux1 d y ?9k ^! di/9ctions1 useful z !y may be1 g f> to quick5 our pace8 ,logi/ic =ces u to say all t is ord9>ily left to be und]/ood2 x makes u advance /ep by /ep2 ?is is p]haps sur] b n quick]4 ,x is n w+s y logi/icians give u1 b lead+-/r+s4 ,& !n we h ! "r to require t ^! lead+-/r+s prev5t our fall+4 ,?is w be _! only excuse4 ,:5 a bond does n be> mu* int]e/1 x %\ld at lea/ be an inve/m5t = a fa!r ( a family4 ,%\ld y\r rules be foll[$ bl9dly8 ,yes1 else only intuition c\ld enable u to di/+ui% among !m2 b !n !y mu/ be infallible2 = only 9 an infallible au?ority c "o h a bl9d confid5ce4 ,?is1 !re=e1 is = y a necessity4 ,infallible y %all be1 or n at all4 ,y h no "r to say to u3 ,x is true we make mi/akes1 b s d y4 ,= u to blund] is a mis=tune1 a v great mis=tune2 = y x is dea?4 ,nor may y ask3 ,does ! infallibility ( ari?metic prev5t errors 9 addition8 ,! rules ( calculation are infallible1 & yet we see ^? blund] :o d n apply ^! rules2 b 9 *eck+ _! calculation x is at once se5 :]e !y w5t wrong4 ,"h x is n at all ! case2 ! logicians h appli$ _! rules1 & !y h fall5 into contradiction2 & s true is ?is1 t !y are prep>+ to *ange ^! rules & to sacrifice ! notion ( class4 ,:y *ange !m if !y 7 infallible8 ,we are n oblig$1 y say1 to solve hic et nunc all possible problems4 ,oh1 we d n ask s mu* ( y4 ,if1 9 face ( a problem1 y w\ld give no solution1 we %\ld h no?+ to say2 b on ! contr>y y give u two ( !m & ^? contradictory1 & consequ5tly at lea/ "o false2 ?is x is :i* is failure4 ,russell seeks to reconcile ^! contradictions1 :i* c only be done1 accord+ to him1 by re/rict+ or ev5 sacrific+ ! notion ( class4 ,& ,m4 ,c\turat1 discov]+ ! success ( 8 attempt1 adds3 ,if ! logicians succe$ :]e o!rs h fail$1 ,m4 ,po9c> w rememb] ?is phrase1 & give ! honor ( ! solution to logi/ic4 ,b no6 ,logi/ic exi/s1 x has its code :i* has already _h f\r editions2 or r ?is code is logi/ic itself4 ,is ,mr4 ,russell prep>+ to %[ t "o at lea/ ( ! two contradictory reason+s has transgress$ ! code8 ,n at all2 he is prep>+ to *ange ^! laws & to abrogate a c]ta9 numb] ( !m4 ,if he succe$s1 ,i %all give ! honor ( x to ,russell's intuition & n to ! ,peanian logi/ic :i* he w h de/roy$4 ,,iii ,! ,lib]ty ( ,contradiction ,i made two pr9cipal objections to ! def9ition ( :ole numb] adopt$ 9 logi/ic4 ,:at says ,m4 ,c\turat to ! fir/ ( ^! objections8 ,:at does ! ^w exi/ mean 9 ma!matics8 ,x means1 ,i said1 to be free f contradiction4 ,?is ,m4 ,c\turat conte/s4 ,logical exi/5ce1 says he1 is q ano!r ?+ f ! abs5ce ( contradiction4 ,x consi/s 9 ! fact t a class is n empty4 ,to say3 a's exi/1 is1 by def9ition1 to affirm t ! class a is n null4 ,& d\btless to affirm t ! class a is n null1 is1 by def9ition1 to affirm t a's exi/4 ,b "o ( ! two affirmations is z d5ud$ ( mean+ z ! o!r1 if !y d n bo? signify1 ei!r t "o may see or t\* a's :i* is ! mean+ physici/s or naturali/s give !m1 or t "o may conceive an a )\t be+ drawn into contradictions1 :i* is ! mean+ giv5 !m by logicians & ma!maticians4 ,= ,m4 ,c\turat1 x is n non-contradiction t proves exi/5ce1 b x is exi/5ce t proves non-contradiction4 ,to e/abli% ! exi/5ce ( a class1 x is necess>y !re=e to e/abli%1 by an example1 t !re is an individual belong+ to ?is class3 ,b1 x w be said1 h[ is ! exi/5ce ( ?is individual prov$8 ,mu/ n ?is exi/5ce be e/abli%$1 9 ord] t ! exi/5ce ( ! class ( :i* x is a "p may be d$uc$8 ,well1 no2 h[ev] p>adoxical may appe> ! ass]tion1 we nev] demon/rate ! exi/5ce ( an individual4 ,individuals1 j because !y are individuals1 are always consid]$ z exi/5t4444 ,we nev] h to express t an individual exi/s1 absolutely speak+1 b only t x exi/s 9 a class4 ,m4 ,c\turat f9ds 8 own ass]tion p>adoxical1 & he w c]ta9ly n be ! only "o4 ,yet x mu/ h a mean+4 ,x d\btless means t ! exi/5ce ( an individual1 alone 9 ! _w1 & ( :i* no?+ is affirm$1 c n involve contradiction2 9 s f> z x is all alone x evid5tly w n emb>rass any "o4 ,well1 s let x be2 we %all admit ! exi/5ce ( ! individual1 'absolutely speak+1' b no?+ m4 ,x rema9s to prove ! exi/5ce ( ! individual '9 a class1' & = t x w always be necess>y to prove t ! affirmation1 ,su* an individual belongs to su* a class1 is nei!r contradictory 9 itself1 nor to ! o!r po/ulates adopt$4 ,x is !n1 cont9ues ,m4 ,c\turat1 arbitr>y & mislead+ to ma9ta9 t a def9ition is valid only if we fir/ prove x is n contradictory4 ,"o c\ld n claim 9 pr\d] & m en]getic t]ms ! lib]ty ( contradiction4 ,9 any case1 ! onus prob&i re/s ^u ^? :o believe t ^! pr9ciples are contradictory4 ,po/ulates are presum$ to be compatible until ! contr>y is prov$1 j z ! accus$ p]son is presum$ innoc5t4 ,ne$less to add t ,i d n ass5t to ?is claim4 ,b1 y say1 ! demon/ration y require ( u is impossible1 & y c n ask u to jump ov] ! moon4 ,p>don me2 t is impossible = y1 b n = u1 :o admit ! pr9ciple ( induction z a syn!tic judgm5t a priori4 ,& t w\ld be necess>y = y1 z = u4 ,to demon/rate t a sy/em ( po/ulates implies no contradiction1 x is necess>y to apply ! pr9ciple ( complete induction2 ?is mode ( reason+ n only has no?+ 'biz>re' ab\t x1 b x is ! only correct "o4 ,x is n 'unlikely' t x has ev] be5 employ$2 & x is n h>d to f9d 'examples & prec$5ts' ( x4 ,i h cit$ two su* in/ances borr[$ f ,hilb]t's article4 ,he is n ! only "o to h us$ x1 & ^? :o h n done s h be5 wrong4 ,:at ,i h blam$ ,hilb]t = is n 8 hav+ rec\rse to x 7a born ma!matician su* z he c\ld n fail to see a demon/ration 0 necess>y & ?is ! only "o possible71 b 8 hav+ rec\rse )\t recogniz+ ! reason+ by recurr5ce4 ,,iv ,! ,second ,objection ,i po9t$ out a second error ( logi/ic 9 ,hilb]t's article4 ,to-day ,hilb]t is excommunicat$ & ,m4 ,c\turat no long] reg>ds him z ( ! logi/ic cult2 s he asks if ,i h f\nd ! same fault among ! or?odox4 ,no1 ,i h n se5 x 9 ! pages ,i h read2 ,i "k n :e!r ,i %\ld f9d x 9 ! ?ree hundr$ pages !y h writt5 :i* ,i h no desire to read4 ,only1 !y mu/ commit x ! day !y wi% to make any application ( ma!matics4 ,?is sci5ce has n z sole object ! et]nal contemplation ( its own navel2 x has to d ) nature & "s day x w t\* x4 ,!n x w be necess>y to %ake (f purely v]bal def9itions & to /op pay+ oneself ) words4 ,to g back to ! example ( ,hilb]t3 always ! po9t at issue is reason+ by recurr5ce & ! "q ( kn[+ :e!r a sy/em ( po/ulates is n contradictory4 ,m4 ,c\turat w d\btless say t !n ?is does n t\* him1 b x p]haps w int]e/ ^? :o d n claim1 z he does1 ! lib]ty ( contradiction4 ,we wi% to e/abli%1 z above1 t we %all nev] enc\nt] contradiction aft] any numb] ( d$uctions :atev]1 provid$ ?is numb] be f9ite4 ,= t1 x is necess>y to apply ! pr9ciple ( induction4 ,%\ld we "h und]/& by f9ite numb] e numb] to :i* by def9ition ! pr9ciple ( induction applies8 ,evid5tly n1 else we %\ld be l$ to mo/ emb>rass+ consequ5ces4 ,to h ! "r to lay d[n a sy/em ( po/ulates1 we mu/ be sure !y are n contradictory4 ,?is is a tru? admitt$ by mo/ sci5ti/s2 ,i %\ld h writt5 by all be=e read+ ,m4 ,c\turat's la/ article4 ,b :at does ?is signify8 ,does x mean t we mu/ be sure ( n meet+ contradiction aft] a f9ite numb] ( propositions1 ! f9ite numb] be+ by def9ition t :i* has all prop]ties ( recurr5t nature1 s t if "o ( ^! prop]ties fails--if1 = in/ance1 we come ^u a contradiction--we %all agree to say t ! numb] 9 "q is n f9ite8 ,9 o!r words1 d we mean t we mu/ be sure n to meet contradictions1 on condition ( agree+ to /op j :5 we are ab\t to enc\nt] "o8 ,to /ate su* a proposition is en\gh to condemn x4 ,s1 ,hilb]t's reason+ n only assumes ! pr9ciple ( induction1 b x supposes t ?is pr9ciple is giv5 u n z a simple def9ition1 b z a syn!tic judgm5t a priori4 ,to sum up3 ,a demon/ration is necess>y4 ,! only demon/ration possible is ! pro( by recurr5ce4 ,?is is legitimate only if we admit ! pr9ciple ( induction & if we reg>d x n z a def9ition b z a syn!tic judgm5t4 ,v ,! ,cantor ,ant9omies ,n[ to exam9e ,russell's new memoir4 ,?is memoir 0 writt5 ) ! view to conqu] ! difficulties rais$ by ^? ,cantor ant9omies to :i* frequ5t allusion has already be5 made4 ,cantor ?\ght he c\ld con/ruct a sci5ce ( ! inf9ite2 o!rs w5t on 9 ! way he op5$1 b !y soon ran f\l ( /range contradictions4 ,^! ant9omies are already num]\s1 b ! mo/ celebrat$ are3 #a4 ,! ,burali-,=ti ant9omy2 #b4 ,! ,z]melo-,knig ant9omy2 #c4 ,! ,ri*>d ant9omy4 ,cantor prov$ t ! ord9al numb]s 7! "q is ( transf9ite ord9al numb]s1 a new notion introduc$ by him7 c be rang$ 9 a l9e> s]ies2 t is to say t ( two unequal ord9als "o is always less ?an ! o!r4 ,burali-,=ti proves ! contr>y2 & 9 fact he says 9 sub/ance t if "o c\ld range all ! ord9als 9 a l9e> s]ies1 ?is s]ies w\ld def9e an ord9al great] ?an all ! o!rs2 we c\ld aft]w>ds adjo9 #a & w\ld obta9 aga9 an ord9al :i* w\ld be /ill great]1 & ?is is contradictory4 ,we %all return lat] to ! ,z]melo-,knig ant9omy :i* is ( a slightly diff]5t nature4 ,! ,ri*>d ant9omy#ae is z foll[s3 ,consid] all ! decimal numb]s def9able by a f9ite numb] ( words2 ^! decimal numb]s =m an aggregate ,e1 & x is easy to see t ?is aggregate is c\ntable1 t is to say we c numb] ! diff]5t decimal numb]s ( ?is assemblage f #a to inf9ity4 ,suppose ! numb]+ effect$1 & def9e a numb] ,n z foll[s3 ,if ! n? decimal ( ! n? numb] ( ! assemblage ,e is #j1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i ! n? decimal ( ,n %all be3 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #a1 #a #ae ,revue gnrale des sci5ces1 ,june #cj1 #aije4 ,z we see1 ,n is n equal to ! n? numb] ( ,e1 & z n is arbitr>y1 ,n does n app]ta9 to ,e & yet ,n %\ld belong to ?is assemblage s9ce we h def9$ x ) a f9ite numb] ( words4 ,we %all lat] see t ,m4 ,ri*>d has himself giv5 ) mu* sagacity ! explanation ( 8 p>adox & t ?is ext5ds1 mutatis mut&is1 to ! o!r l p>adoxes4 ,aga91 ,russell cites ano!r q amus+ p>adox3 ,:at is ! lea/ :ole numb] :i* c n be def9$ by a phrase compos$ ( less ?an a hundr$ ,engli% words8 ,?is numb] exi/s2 & 9 fact ! numb]s capable ( be+ def9$ by a l phrase are evid5tly f9ite 9 numb] s9ce ! words ( ! ,engli% language are n inf9ite 9 numb]4 ,!re=e among !m w be "o less ?an all ! o!rs4 ,&1 on ! o!r h&1 ?is numb] does n exi/1 because its def9ition implies contradiction4 ,?is numb]1 9 fact1 is def9$ by ! phrase 9 italics :i* is compos$ ( less ?an a hundr$ ,engli% words2 & by def9ition ?is numb] %\ld n be capable ( def9ition by a l phrase4 ,,vi ,zigzag ,!ory & ,no-class ,!ory ,:at is ,mr4 ,russell's attitude 9 pres5ce ( ^! contradictions8 ,aft] hav+ analyz$ ^? ( :i* we h j spok51 & cit$ /ill o!rs1 aft] hav+ giv5 !m a =m recall+ ,epim5ides1 he does n hesitate to conclude3 ,a propositional function ( "o v>iable does n always det]m9e a class4 ,a propositional function 7t is to say a def9ition7 does n always det]m9e a class4 ,a 'propositional function' or 'norm' may be 'non-pr$icative4' ,& ?is does n mean t ^! non-pr$icative propositions det]m9e an empty class1 a null class2 ?is does n mean t !re is no value ( x satisfy+ ! def9ition & capable ( be+ "o ( ! elem5ts ( ! class4 ,! elem5ts exi/1 b !y h no "r to unite 9 a syndicate to =m a class4 ,b ?is is only ! beg9n+ & x is ne$ful to "k h[ to recognize :e!r a def9ition is or is n pr$icative4 ,to solve ?is problem ,russell hesitates betwe5 ?ree !ories :i* he calls ,a4 ,! zigzag !ory2 ,b4 ,! !ory ( limitation ( size2 ,c4 ,! no-class !ory4 ,accord+ to ! zigzag !ory def9itions 7propositional functions7 det]m9e a class :5 !y are v simple & cease to d s only :5 !y are complicat$ & obscure4 ,:o1 n[1 is to decide :e!r a def9ition may be reg>d$ z simple en\gh to be acceptable8 ,to ?is "q !re is no answ]1 if x be n ! loyal av[al ( a complete inability3 ,! rules :i* enable u to recognize :e!r ^! def9itions are pr$icative w\ld be extremely complicat$ & c n comm5d !mselves by any plausible reason4 ,?is is a fault :i* might be rem$i$ by great] +5uity or by us+ di/9ctions n yet po9t$ out4 ,b hi!rto 9 seek+ ^! rules1 ,i h n be5 able to f9d any o!r direct+ pr9ciple ?an ! abs5ce ( contradiction4 ,?is !ory !re=e rema9s v obscure2 9 ?is night a s+le light--! ^w zigzag4 ,:at ,russell calls ! 'zigzag9ess' is d\btless ! p>ticul> *>act]i/ic :i* di/+ui%es ! argum5t ( ,epim5ides4 ,accord+ to ! !ory ( limitation ( size1 a class w\ld cease to h ! "r to exi/ if x 7 too ext5d$4 ,p]haps x might be inf9ite1 b x %\ld n be too mu* s4 ,b we always meet aga9 ! same difficulty2 at :at precise mom5t does x beg9 to be too mu* s8 ,( c\rse ?is difficulty is n solv$ & ,russell passes on to ! ?ird !ory4 ,9 ! no-classes !ory x is =bidd5 to speak ! ^w 'class' & ?is ^w mu/ be replac$ by v>i\s p]iphrases4 ,:at a *ange = logi/ic :i* talks only ( classes & classes ( classes6 ,x becomes necess>y to remake ! :ole ( logi/ic4 ,imag9e h[ a page ( logi/ic w\ld look ^u suppress+ all ! propositions :]e x is a "q ( class4 ,!re w\ld only be "s scatt]$ survivors 9 ! mid/ ( a blank page4 ,app>5t r>i nantes 9 gurgite va/o4 ,be t z x may1 we see h[ ,russell hesitates & ! modifications to :i* he submits ! fundam5tal pr9ciples he has hi!rto adopt$4 ,crit]ia are ne$$ to decide :e!r a def9ition is too complex or too ext5d$1 & ^! crit]ia c only be ju/ifi$ by an appeal to intuition4 ,x is t[>d ! no-classes !ory t ,russell f9ally incl9es4 ,be t z x may1 logi/ic is to be remade & x is n cle> h[ mu* ( x c be sav$4 ,ne$less to add t ,cantorism & logi/ic are alone "u consid]ation2 real ma!matics1 t :i* is good = some?+1 may cont9ue to develop 9 accordance ) its own pr9ciples )\t bo!r+ ab\t ! /orms :i* rage outside x1 & g on /ep by /ep ) its usual conque/s :i* are f9al & :i* x nev] has to ab&on4 ,,vii ,! ,true ,solution ,:at *oice ought we to make among ^! diff]5t !ories8 ,x seems to me t ! solution is conta9$ 9 a lett] ( ,m4 ,ri*>d ( :i* ,i h spok5 above1 to be f\nd 9 ! ,revue gnrale des sci5ces ( ,june #cj1 #aije4 ,aft] hav+ set =? ! ant9omy we h call$ ,ri*>d's ant9omy1 he gives its explanation4 ,recall :at has already be5 said ( ?is ant9omy4 ,e is ! aggregate ( all ! numb]s def9able by a f9ite numb] ( words1 )\t introduc+ ! notion ( ! aggregate ,e itself4 ,else ! def9ition ( ,e w\ld conta9 a vici\s circle2 we mu/ n def9e ,e by ! aggregate ,e itself4 ,n[ we h def9$ ,n ) a f9ite numb] ( words1 x is true1 b ) ! aid ( ! notion ( ! aggregate ,e4 ,& ?is is :y ,n is n "p ( ,e4 ,9 ! example select$ by ,m4 ,ri*>d1 ! conclusion pres5ts itself ) complete evid5ce & ! evid5ce w appe> /ill /rong] on consult+ ! text ( ! lett] itself4 ,b ! same explanation holds good = ! o!r ant9omies1 z is easily v]ifi$4 ,?us ! def9itions :i* %\ld be reg>d$ z n pr$icative are ^? :i* conta9 a vici\s circle4 ,& ! prec$+ examples suffici5tly %[ :at ,i mean by t4 ,is x ?is :i* ,russell calls ! 'zigzag9ess'8 ,i put ! "q )\t answ]+ x4 ,,viii ,! ,demon/rations ( ! ,pr9ciple ( ,induction ,let u n[ exam9e ! pret5d$ demon/rations ( ! pr9ciple ( induction & 9 p>ticul> ^? ( ,:itehead & ( ,burali-,=ti4 ,we %all speak ( ,:itehead's fir/1 & take advantage ( c]ta9 new t]ms happily introduc$ by ,russell 9 8 rec5t memoir4 ,call recurr5t class e class conta9+ z]o1 & conta9+ n #a if x conta9s n4 ,call inductive numb] e numb] :i* is a "p ( all ! recurr5t classes4 ,^u :at condition w ?is latt] def9ition1 :i* plays an ess5tial rle 9 ,:itehead's pro(1 be 'pr$icative' & consequ5tly acceptable8 ,9 accordance ) :at has be5 said1 x is necess>y to und]/& by all ! recurr5t classes1 all ^? 9 ^: def9ition ! notion ( inductive numb] does n ent]4 ,else we fall aga9 ^u ! vici\s circle :i* has eng5d]$ ! ant9omies4 ,n[ ,:itehead has n tak5 ?is precaution4 ,:itehead's reason+ is !re=e fallaci\s2 x is ! same :i* l$ to ! ant9omies4 ,x 0 illegitimate :5 x gave false results2 x rema9s illegitimate :5 by *ance x leads to a true result4 ,a def9ition conta9+ a vici\s circle def9es no?+4 ,x is ( no use to say1 we are sure1 :atev] mean+ we may give to our def9ition1 z]o at lea/ belongs to ! class ( inductive numb]s2 x is n a "q ( kn[+ :e!r ?is class is void1 b :e!r x c be rigor\sly delim9at$4 ,a 'non-pr$icative' class is n an empty class1 x is a class ^: b\nd>y is undet]m9$4 ,ne$less to add t ?is p>ticul> objection leaves 9 =ce ! g5]al objections applicable to all ! demon/rations4 ,,ix ,burali-,=ti has giv5 ano!r demon/ration4#af ,b he is oblig$ to assume two po/ulates3 ,fir/1 !re always exi/s at lea/ "o inf9ite class4 ,! second is ?us express$3 uepsilon,k7,k - iota,lambda7 inv]t$ c u v'u4 ,! fir/ po/ulate is n m evid5t ?an ! pr9ciple to be prov$4 ,! second n only is n evid5t1 b x is false1 z ,:itehead has %[n2 z moreov] any recruit w\ld see at ! fir/ glance1 if ! axiom _h be5 /at$ 9 intelligible language1 s9ce x means t ! numb] ( comb9ations :i* c be =m$ ) sev]al objects is less ?an ! numb] ( ^! objects4 #af ,9 8 article ',le classi f9ite1' ,atti di ,tor9o1 ,vol4 ,,xxxii4 ,x ,z]melo's ,assumption ,a fam\s demon/ration by ,z]melo re/s ^u ! foll[+ assumption3 ,9 any aggregate 7or ! same 9 ea* aggregate ( an assemblage ( aggregates7 we c always *oose at r&om an elem5t 7ev5 if ?is assemblage ( aggregates %\ld conta9 an inf9ity ( aggregates74 ,?is assumption _h be5 appli$ a ?\s& times )\t be+ /at$1 b1 once /at$1 x ar\s$ d\bts4 ,"s ma!maticians1 = in/ance ,m4 ,borel1 resolutely reject x2 o!rs admire x4 ,let u see :at1 accord+ to 8 la/ article1 ,russell ?9ks ( x4 ,he does n speak out1 b 8 reflections are v sugge/ive4 ,& fir/ a picturesque example3 ,suppose we h z _m pairs ( %oes z !re are :ole numb]s1 & s t we c numb] ! pairs f "o to inf9ity1 h[ _m %oes %all we h8 ,w ! numb] ( %oes be equal to ! numb] ( pairs8 ,yes1 if 9 ea* pair ! "r %oe is di/+ui%able f ! left2 x w 9 fact suffice to give ! numb] #bn - #a to ! "r %oe ( ! n? pair1 & ! numb] #bn to ! left %oe ( ! n? pair4 ,no1 if ! "r %oe is j l ! left1 because a simil> op]ation w\ld become impossible--unless we admit ,z]melo's assumption1 s9ce !n we c\ld *oose at r&om 9 ea* pair ! %oe to be reg>d$ z ! "r4 ,,xi ,conclusions ,a demon/ration truly f\nd$ ^u ! pr9ciples ( analytic logic w be compos$ ( a s]ies ( propositions4 ,"s1 s]v+ z premises1 w be id5tities or def9itions2 ! o!rs w be d$uc$ f ! premises /ep by /ep4 ,b ?\gh ! bond betwe5 ea* proposition & ! foll[+ is imm$iately evid5t1 x w n at fir/ sight appe> h[ we get f ! fir/ to ! la/1 :i* we may be tempt$ to reg>d z a new tru?4 ,b if we replace successively ! diff]5t expressions !re9 by _! def9ition & if ?is op]ation be c>ri$ z f> z possible1 !re w f9ally rema9 only id5tities1 s t all w r$uce to an imm5se tautology4 ,logic !re=e rema9s /]ile unless made fruitful by intuition4 ,?is ,i wrote long ago2 logi/ic pr(esses ! contr>y & ?9ks x has prov$ x by actually prov+ new tru?s4 ,by :at me*anism8 ,:y 9 apply+ to _! reason+s ! proc$ure j describ$--namely1 replac+ ! t]ms def9$ by _! def9itions--d we n see !m dissolve into id5tities l ord9>y reason+s8 ,x is because ?is proc$ure is n applicable to !m4 ,& :y8 ,because _! def9itions are n pr$icative & pres5t ?is sort ( hidd5 vici\s circle :i* ,i h po9t$ out above2 non-pr$icative def9itions c n be sub/itut$ = ! t]ms def9$4 ,"u ^! conditions logi/ic is n /]ile1 x eng5d]s ant9omies4 ,x is ! belief 9 ! exi/5ce ( ! actual inf9ite :i* has giv5 bir? to ^? non-pr$icative def9itions4 ,let me expla94 ,9 ^! def9itions ! ^w 'all' figures1 z is se5 9 ! examples cit$ above4 ,! ^w 'all' has a v precise mean+ :5 x is a "q ( a f9ite numb] ( objects2 to h ano!r "o1 :5 ! objects are inf9ite 9 numb]1 w\ld require !re be+ an actual 7giv5 complete7 inf9ity4 ,o!rwise all ^! objects c\ld n be conceiv$ z po/ulat$ ant]iorly to _! def9ition1 & !n if ! def9ition ( a notion ,n dep5ds ^u all ! objects ,a1 x may be infect$ ) a vici\s circle1 if among ! objects ,a are "s indef9able )\t ! int]v5tion ( ! notion ,n itself4 ,! rules ( =mal logic express simply ! prop]ties ( all possible classifications4 ,b = !m to be applicable x is necess>y t ^! classifications be immutable & t we h no ne$ to modify !m 9 ! c\rse ( ! reason+4 ,if we h to classify only a f9ite numb] ( objects1 x is easy to keep our classifications )\t *ange4 ,if ! objects are indef9ite 9 numb]1 t is to say if "o is con/antly expos$ to see+ new & un=ese5 objects arise1 x may happ5 t ! appe>ance ( a new object may require ! classification to be modifi$1 & ?us x is we are expos$ to ant9omies4 ,!re is no actual 7giv5 complete7 inf9ity4 ,! ,cantorians h =gott5 ?is1 & !y h fall5 into contradiction4 ,x is true t ,cantorism has be5 ( s]vice1 b ?is 0 :5 appli$ to a real problem ^: t]ms 7 precisely def9$1 & !n we c\ld advance )\t fe>4 ,logi/ic also =got x1 l ! ,cantorians1 & enc\nt]$ ! same difficulties4 ,b ! "q is to "k :e!r !y w5t ?is way by accid5t or :e!r x 0 a necessity = !m4 ,= me1 ! "q is n d\btful2 belief 9 an actual inf9ity is ess5tial 9 ! ,russell logic4 ,x is j ?is :i* di/+ui%es x f ! ,hilb]t logic4 ,hilb]t takes ! view-po9t ( ext5sion1 precisely 9 ord] to avoid ! ,cantorian ant9omies4 ,russell takes ! view-po9t ( compreh5sion4 ,consequ5tly = him ! g5us is ant]ior to ! species1 & ! summum g5us is ant]ior to all4 ,t w\ld n be inconv5i5t if ! summum g5us 0 f9ite2 b if x is inf9ite1 x is necess>y to po/ulate ! inf9ite1 t is to say to reg>d ! inf9ite z actual 7giv5 complete74 ,& we h n only inf9ite classes2 :5 we pass f ! g5us to ! species 9 re/rict+ ! concept by new conditions1 ^! conditions are /ill inf9ite 9 numb]4 ,because !y express g5]ally t ! envisag$ object pres5ts su* or su* a relation ) all ! objects ( an inf9ite class4 ,b t is anci5t hi/ory4 ,russell has p]ceiv$ ! p]il & takes c\nsel4 ,he is ab\t to *ange ev]y?+1 &1 :at is easily und]/ood1 he is prep>+ n only to introduce new pr9ciples :i* %all all[ ( op]ations =m]ly =bidd51 b he is prep>+ to =bid op]ations he =m]ly ?\ght legitimate4 ,n cont5t to adore :at he burn$1 he is ab\t to burn :at he ador$1 :i* is m s]i\s4 ,he does n add a new w+ to ! build+1 he saps its f\ndation4 ,! old logi/ic is dead1 s mu* s t already ! zigzag !ory & ! no-classes !ory are disput+ ov] ! succession4 ,to judge ( ! new1 we %all await its com+4 ,,book ,,iii ,,! ,,new ,,me*anics ,,*apt] ,i ,,me*anics ,,& ,,radium ,i ,introduction ,! g5]al pr9ciples ( ,dynamics1 :i* h1 s9ce ,newton1 s]v$ z f\ndation = physical sci5ce1 & :i* appe>$ immovable1 are !y on ! po9t ( be+ ab&on$ or at lea/ pr(\ndly modifi$8 ,?is is :at _m p h be5 ask+ !mselves = "s ye>s4 ,accord+ to !m1 ! discov]y ( radium has ov]turn$ ! sci5tific dogmas we believ$ ! mo/ solid3 on ! "o h&1 ! impossibility ( ! transmutation ( metals2 on ! o!r h&1 ! fundam5tal po/ulates ( me*anics4 ,p]haps "o is too ha/y 9 consid]+ ^! novelties z f9ally e/abli%$1 & break+ our idols ( ye/]day2 p]haps x w\ld be prop]1 be=e tak+ sides1 to await exp]im5ts m num]\s & m conv9c+4 ,none ! less is x necess>y1 f to-day1 to "k ! new doctr9es & ! argum5ts1 already v weighty1 ^u :i* !y re/4 ,9 few words let u fir/ recall 9 :at ^? pr9ciples consi/3 ,a4 ,! motion ( a mat]ial po9t isolat$ & ap>t f all ext]ior =ce is /raight & uni=m2 ?is is ! pr9ciple ( in]tia3 )\t =ce no accel]ation2 ,b4 ,! accel]ation ( a mov+ po9t has ! same direction z ! resultant ( all ! =ces to :i* x is subject$2 x is equal to ! quoti5t ( ?is resultant by a coeffici5t call$ mass ( ! mov+ po9t4 ,! mass ( a mov+ po9t1 s def9$1 is a con/ant2 x does n dep5d ^u ! velocity acquir$ by ?is po9t2 x is ! same :e!r ! =ce1 be+ p>allel to ?is velocity1 t5ds only to accel]ate or to ret>d ! motion ( ! po9t1 or :e!r1 on ! contr>y1 be+ p]p5dicul> to ?is velocity1 x t5ds to make ?is motion deviate t[>d ! "r1 or ! left1 t is to say to curve ! trajectory2 ,c4 ,all ! =ces affect+ a mat]ial po9t come f ! action ( o!r mat]ial po9ts2 !y dep5d only ^u ! relative positions & velocities ( ^! diff]5t mat]ial po9ts4 ,comb9+ ! two pr9ciples ,b & ,c1 we rea* ! pr9ciple ( relative motion1 9 virtue ( :i* ! laws ( ! motion ( a sy/em are ! same :e!r we ref] ?is sy/em to fix$ axes1 or to mov+ axes animat$ by a /raight & uni=m motion ( translation1 s t x is impossible to di/+ui% absolute motion f a relative motion ) ref]5ce to su* mov+ axes2 ,d4 ,if a mat]ial po9t ,a acts ^u ano!r mat]ial po9t ,b1 ! body ,b reacts ^u ,a1 & ^! two actions are two equal & directly opposite =ces4 ,?is is ! pr9ciple ( ! equality ( action & reaction1 or1 m briefly1 ! pr9ciple ( reaction4 ,a/ronomic obs]vations & ! mo/ ord9>y physical ph5om5a seem to h giv5 ( ^! pr9ciples a confirmation complete1 con/ant & v precise4 ,?is is true1 x is n[ said1 b x is because we h nev] op]at$ ) any b v small velocities2 ,m]cury1 = example1 ! fa/e/ ( ! planets1 goes sc>cely #ajj kilomet]s a second4 ,w\ld ?is planet act ! same if x w5t a ?\s& times fa/]8 ,we see !re is yet no ne$ to worry2 :atev] may be ! progress ( automobilism1 x w be long be=e we mu/ give up apply+ to our ma*9es ! classic pr9ciples ( dynamics4 ,h[ !n h we come to make actual spe$s a ?\s& times great] ?an t ( ,m]cury1 equal1 = in/ance1 to a t5? or a ?ird ( ! velocity ( light1 or approa*+ /ill m closely to t velocity8 ,x is by aid ( ! ca?ode rays & ! rays f radium4 ,we "k t radium emits ?ree k9ds ( rays1 designat$ by ! ?ree ,greek lett]s alpha1 beta1 gamma2 9 :at foll[s1 unless ! contr>y be expressly /at$1 x w always be a "q ( ! beta rays1 :i* are analog\s to ! ca?ode rays4 ,aft] ! discov]y ( ! ca?ode rays two !ories appe>$4 ,crookes attribut$ ! ph5om5a to a v]itable molecul> bomb>dm5t2 ,h]tz1 to special undulations ( ! e!r4 ,?is 0 a r5ewal ( ! debate :i* divid$ physici/s a c5tury ago ab\t light2 ,crookes took up ! emission !ory1 ab&on$ = light2 ,h]tz held to ! undulatory !ory4 ,! facts seem to decide 9 favor ( ,crookes4 ,x has be5 recogniz$1 9 ! fir/ place1 t ! ca?ode rays c>ry ) !m a negative electric *>ge2 !y are deviat$ by a magnetic field & by an electric field2 & ^! deviations are precisely su* z ^! same fields w\ld produce ^u projectiles animat$ by a v high velocity & /rongly *>g$ ) electricity4 ,^! two deviations dep5d ^u two quantities3 "o ! velocity1 ! o!r ! relation ( ! electric *>ge ( ! projectile to its mass2 we _c "k ! absolute value ( ?is mass1 nor t ( ! *>ge1 b only _! relation2 9 fact1 x is cle> t if we d\ble at ! same "t ! *>ge & ! mass1 )\t *ang+ ! velocity1 we %all d\ble ! =ce :i* t5ds to deviate ! projectile1 b1 z its mass is also d\bl$1 ! accel]ation & deviation obs]vable w n be *ang$4 ,! obs]vation ( ! two deviations w give u !re=e two equations to det]m9e ^! two unkn[ns4 ,we f9d a velocity ( f #aj1jjj to #cj1jjj kilomet]s a second2 z to ! ratio ( ! *>ge to ! mass1 x is v great4 ,we may comp>e x to ! correspond+ ratio 9 reg>d to ! hydrog5 ion 9 electrolysis2 we !n f9d t a ca?odic projectile c>ries ab\t a ?\s& times m electricity ?an an equal mass ( hydrog5 w\ld c>ry 9 an electrolyte4 ,to confirm ^! views1 we ne$ a direct measurem5t ( ?is velocity to comp>e ) ! velocity s calculat$4 ,old exp]im5ts ( ,j4 ,j4 ,?omson _h giv5 results m ?an a hundr$ times too small2 b !y 7 expos$ to c]ta9 causes ( error4 ,! "q 0 tak5 up aga9 by ,wie*]t 9 an arrangem5t :]e ! ,h]tzian oscillations 7 utiliz$2 results 7 f\nd agree+ ) ! !ory1 at lea/ z to ord] ( magnitude2 x w\ld be ( great int]e/ to repeat ^! exp]im5ts4 ,h[ev] t may be1 ! !ory ( undulations appe>s p[]less to acc\nt = ?is complex ( facts4 ,! same calculations made ) ref]5ce to ! beta rays ( radium h giv5 velocities /ill great]3 #ajj1jjj or #bjj1jjj kilomet]s or m yet4 ,^! velocities greatly surpass all ^? we "k4 ,x is true t light has long be5 kn[n to g #cjj1jjj kilomet]s a second2 b x is n a c>ry+ ( matt]1 :ile1 if we adopt ! emission !ory = ! ca?ode rays1 !re w\ld be mat]ial molecules really impell$ at ! velocities 9 "q1 & x is prop] to inve/igate :e!r ! ord9>y laws ( me*anics are /ill applicable to !m4 ,,ii ,mass ,longitud9al & ,mass ,transv]sal ,we "k t electric curr5ts produce ! ph5om5a ( induction1 9 p>ticul> self-induction4 ,:5 a curr5t increases1 !re develops an electromotive =ce ( self-induction :i* t5ds to oppose ! curr5t2 on ! contr>y1 :5 ! curr5t decreases1 ! electromotive =ce ( self-induction t5ds to ma9ta9 ! curr5t4 ,! self-induction !re=e opposes e v>iation ( ! int5sity ( ! curr5t1 j z 9 me*anics ! in]tia ( a body opposes e v>iation ( its velocity4 ,self-induction is a v]itable in]tia4 ,ev]y?+ happ5s z if ! curr5t c\ld n e/abli% itself )\t putt+ 9 motion ! surr\nd+ e!r & z if ! in]tia ( ?is e!r t5d$1 9 consequ5ce1 to keep con/ant ! int5sity ( ?is curr5t4 ,x w\ld be requisite to ov]come ?is in]tia to e/abli% ! curr5t1 x w\ld be necess>y to ov]come x aga9 to make ! curr5t cease4 ,a ca?ode ray1 :i* is a ra9 ( projectiles *>g$ ) negative electricity1 may be lik5$ to a curr5t2 d\btless ?is curr5t diff]s1 at fir/ sight at lea/1 f ! curr5ts ( ord9>y conduction1 :]e ! matt] does n move & :]e ! electricity circulates ?r\gh ! matt]4 ,?is is a curr5t ( convection1 :]e ! electricity1 atta*$ to a mat]ial vehicle1 is c>ri$ along by ! motion ( ?is vehicle4 ,b ,r[l& has prov$ t curr5ts ( convection produce ! same magnetic effects z curr5ts ( conduction2 !y %\ld produce also ! same effects ( induction4 ,fir/1 if ?is 7 n s1 ! pr9ciple ( ! cons]vation ( en]gy w\ld be violat$2 besides1 ,crmieu & ,p5d] h employ$ a me?od putt+ 9 evid5ce directly ^! effects ( induction4 ,if ! velocity ( a ca?ode corpuscle v>ies1 ! int5sity ( ! correspond+ curr5t w likewise v>y2 & !re w develop effects ( self-induction :i* w t5d to oppose ?is v>iation4 ,^! corpuscles %\ld !re=e possess a d\ble in]tia3 fir/ _! own prop] in]tia1 & !n ! app>5t in]tia1 due to self-induction1 :i* produces ! same effects4 ,!y w !re=e h a total app>5t mass1 compos$ ( _! real mass & ( a fictiti\s mass ( electromagnetic orig94 ,calculation %[s t ?is fictiti\s mass v>ies ) ! velocity1 & t ! =ce ( in]tia ( self-induction is n ! same :5 ! velocity ( ! projectile accel]ates or slack5s1 or :5 x is deviat$2 !re=e s x is ) ! =ce ( ! total app>5t in]tia4 ,! total app>5t mass is !re=e n ! same :5 ! real =ce appli$ to ! corpuscle is p>allel to its velocity & t5ds to accel]ate ! motion z :5 x is p]p5dicul> to ?is velocity & t5ds to make ! direction v>y4 ,x is necess>y !re=e to di/+ui% ! total longitud9al mass f ! total transv]sal mass4 ,^! two total masses dep5d1 moreov]1 ^u ! velocity4 ,?is foll[s f ! !oretical "w ( ,abraham4 ,9 ! measurem5ts ( :i* we speak 9 ! prec$+ section1 :at is x we det]m9e 9 measur+ ! two deviations8 ,x is ! velocity on ! "o h&1 & on ! o!r h& ! ratio ( ! *>ge to ! total transv]sal mass4 ,h[1 "u ^! conditions1 c we make out 9 ?is total mass ! "p ( ! real mass & t ( ! fictiti\s electromagnetic mass8 ,if we _h only ! ca?ode rays prop]ly s call$1 x c\ld n be dream$ (2 b happily we h ! rays ( radium :i*1 z we h se51 are notably swift]4 ,^! rays are n all id5tical & d n behave 9 ! same way "u ! action ( an electric field & a magnetic field4 ,x is f\nd t ! electric deviation is a function ( ! magnetic deviation1 & we are able1 by receiv+ on a s5sitive plate radium rays :i* h be5 subject$ to ! action ( ! two fields1 to photograph ! curve :i* repres5ts ! relation betwe5 ^! two deviations4 ,?is is :at ,kaufmann has done1 d$uc+ f x ! relation betwe5 ! velocity & ! ratio ( ! *>ge to ! total app>5t mass1 a ratio we %all call epsilon4 ,"o might suppose !re are sev]al species ( rays1 ea* *>act]iz$ by a fix$ velocity1 by a fix$ *>ge & by a fix$ mass4 ,b ?is hypo!sis is improbable2 :y1 9 fact1 w\ld all ! corpuscles ( ! same mass take always ! same velocity8 ,x is m natural to suppose t ! *>ge z well z ! real mass are ! same = all ! projectiles1 & t ^! diff] only by _! velocity4 ,if ! ratio epsilon is a function ( ! velocity1 ?is is n because ! real mass v>ies ) ?is velocity2 b1 s9ce ! fictiti\s electromagnetic mass dep5ds ^u ?is velocity1 ! total app>5t mass1 alone obs]vable1 mu/ dep5d ^u x1 ?\gh ! real mass does n dep5d ^u x & may be con/ant4 ,! calculations ( ,abraham let u "k ! law accord+ to :i* ! fictiti\s mass v>ies z a function ( ! velocity2 ,kaufmann's exp]im5t lets u "k ! law ( v>iation ( ! total mass4 ,! comp>ison ( ^! two laws w enable u !re=e to det]m9e ! ratio ( ! real mass to ! total mass4 ,su* is ! me?od ,kaufmann us$ to det]m9e ?is ratio4 ,! result is highly surpris+3 ! real mass is naught4 ,?is has l$ to conceptions :olly unexpect$4 ,:at _h only be5 prov$ = ca?ode corpuscles 0 ext5d$ to all bodies4 ,:at we call mass w\ld be only semblance2 all in]tia w\ld be ( electromagnetic orig94 ,b !n mass w\ld no long] be con/ant1 x w\ld augm5t ) ! velocity2 s5sibly con/ant = velocities up to #a1jjj kilomet]s a second1 x !n w\ld increase & w\ld become inf9ite = ! velocity ( light4 ,! transv]sal mass w\ld no long] be equal to ! longitud9al3 !y w\ld only be ne>ly equal if ! velocity is n too great4 ,! pr9ciple ,b ( me*anics w\ld no long] be true4 ,,iii ,! ,canal ,rays ,at ! po9t :]e we n[ are1 ?is conclusion might seem premature4 ,c "o apply to all matt] :at has be5 prov$ only = su* light corpuscles1 :i* are a m]e emanation ( matt] & p]haps n true matt]8 ,b be=e ent]+ ^u ?is "q1 a ^w mu/ be said ( ano!r sort ( rays4 ,i ref] to ! canal rays1 ! ,kanal/rahl5 ( ,gold/e94 ,! ca?ode1 toge!r ) ! ca?ode rays *>g$ ) negative electricity1 emits canal rays *>g$ ) positive electricity4 ,9 g5]al1 ^! canal rays n be+ repell$ by ! ca?ode1 are conf9$ to ! imm$iate neighborhood ( ?is ca?ode1 :]e !y con/itute ! *amois cu%ion1' n v easy to p]ceive2 b1 if ! ca?ode is pi]c$ ) holes & if x almo/ completely blocks up ! tube1 ! canal rays spread back ( ! ca?ode1 9 ! direction opposite to t ( ! ca?ode rays1 & x becomes possible to /udy !m4 ,x is ?us t x has be5 possible to %[ _! positive *>ge & to %[ t ! magnetic & electric deviations /ill exi/1 z = ! ca?ode rays1 b are mu* feebl]4 ,radium likewise emits rays analog\s to ! canal rays1 & relatively v absorbable1 call$ alpha rays4 ,we c1 z = ! ca?ode rays1 measure ! two deviations & !nce d$uce ! velocity & ! ratio epsilon4 ,! results are less con/ant ?an = ! ca?ode rays1 b ! velocity is less1 z well z ! ratio epsilon2 ! positive corpuscles are less *>g$ ?an ! negative2 or if1 :i* is m natural1 we suppose ! *>ges equal & ( opposite sign1 ! positive corpuscles are mu* ! l>g]4 ,^! corpuscles1 *>g$ ! ones positively1 ! o!rs negatively1 h be5 call$ electrons4 ,,iv ,! ,!ory ( ,lor5tz ,b ! electrons d n m]ely %[ u _! exi/5ce 9 ^! rays :]e !y are end[$ ) enorm\s velocities4 ,we %all see !m 9 v diff]5t rles1 & x is !y t acc\nt = ! pr9cipal ph5om5a ( optics & electricity4 ,! brilliant syn!sis ab\t to be notic$ is due to ,lor5tz4 ,matt] is =m$ solely ( electrons c>ry+ enorm\s *>ges1 &1 if x seems to u neutral1 ?is is because ! *>ges ( opposite sign ( ^! electrons comp5sate ea* o!r4 ,we may imag9e1 = example1 a sort ( sol> sy/em =m$ ( a great positive electron1 ar\nd :i* gravitate num]\s little planets1 ! negative electrons1 attract$ by ! electricity ( opposite "n :i* *>ges ! c5tral electron4 ,! negative *>ges ( ^! planets w\ld balance ! positive *>ge ( ?is sun1 s t ! algebraic sum ( all ^! *>ges w\ld be naught4 ,all ^! electrons swim 9 ! e!r4 ,! e!r is ev]y:]e id5tically ! same1 & p]turbations 9 x are propagat$ accord+ to ! same laws z light or ! ,h]tzian oscillations 9 vacuo4 ,!re is no?+ b electrons & e!r4 ,:5 a lum9\s wave ent]s a "p ( ! e!r :]e electrons are num]\s1 ^! electrons are put 9 motion "u ! influ5ce ( ! p]turbation ( ! e!r1 & !y !n react ^u ! e!r4 ,s w\ld be expla9$ refraction1 disp]sion1 d\ble refraction & absorption4 ,j s1 if = any cause an electron be put 9 motion1 x w\ld tr\ble ! e!r ar\nd x & w\ld give rise to lum9\s waves1 & ?is w\ld expla9 ! emission ( light by inc&esc5t bodies4 ,9 c]ta9 bodies1 ! metals = example1 we %\ld h fix$ electrons1 betwe5 :i* w\ld circulate mov+ electrons enjoy+ p]fect lib]ty1 save t ( go+ out f ! metallic body & break+ ! surface :i* sep>ates x f ! ext]ior void or f ! air1 or f any o!r non-metallic body4 ,^! movable electrons behave !n1 )9 ! metallic body1 z d1 accord+ to ! k9etic !ory ( gases1 ! molecules ( a gas )9 ! vase :]e ?is gas is conf9$4 ,b1 "u ! influ5ce ( a diff]5ce ( pot5tial1 ! negative movable electrons w\ld t5d to g all to "o side1 & ! positive movable electrons to ! o!r4 ,?is is :at w\ld produce electric curr5ts1 & ?is is :y ^! bodies w\ld be conductors4 ,on ! o!r h&1 ! velocities ( our electrons w\ld be ! great] ! high] ! temp]ature1 if we accept ! assimilation ) ! k9etic !ory ( gases4 ,:5 "o ( ^! movable electrons enc\nt]s ! surface ( ! metallic body1 ^: b\nd>y x c n pass1 x is reflect$ l a billi>d ball :i* has hit ! cu%ion1 & its velocity und]goes a sudd5 *ange ( direction4 ,b :5 an electron *anges direction1 z we %all see fur!r on1 x becomes ! s\rce ( a lum9\s wave1 & ?is is :y hot metals are inc&esc5t4 ,9 o!r bodies1 ! dielectrics & ! transp>5t bodies1 ! movable electrons enjoy mu* less fre$om4 ,!y rema9 z if atta*$ to fix$ electrons :i* attract !m4 ,! f>!r !y g away f !m ! great] becomes ?is attraction & t5ds to pull !m back4 ,!y !re=e c make only small excursions2 !y c no long] circulate1 b only oscillate ab\t _! mean position4 ,?is is :y ^! bodies w\ld n be conductors2 moreov] !y w\ld mo/ (t5 be transp>5t1 & !y w\ld be refractive1 s9ce ! lum9\s vibrations w\ld be communicat$ to ! movable electrons1 susceptible ( oscillation1 & !nce a p]turbation w\ld result4 ,i c n "h give ! details ( ! calculations2 ,i conf9e myself to say+ t ?is !ory acc\nts = all ! kn[n facts1 & has pr$ict$ new ones1 su* z ! ,zeeman effect4 ,v ,me*anical ,consequ5ces ,we n[ may face two hypo!ses3 #a ,! positive electrons h a real mass1 mu* great] ?an _! fictiti\s electromagnetic mass2 ! negative electrons alone lack real mass4 ,we might ev5 suppose t ap>t f electrons ( ! two signs1 !re are neutral atoms :i* h only _! real mass4 ,9 ?is case1 me*anics is n affect$2 !re is no ne$ ( t\*+ its laws2 ! real mass is con/ant2 simply1 motions are d]ang$ by ! effects ( self-induction1 z has always be5 kn[n2 moreov]1 ^! p]turbations are almo/ negligible1 except = ! negative electrons :i*1 n hav+ real mass1 are n true matt]4 #b ,b !re is ano!r po9t ( view2 we may suppose !re are no neutral atoms1 & ! positive electrons lack real mass j z ! negative electrons4 ,b !n1 real mass vani%+1 ei!r ! ^w mass w no long] h any mean+1 or else x mu/ designate ! fictiti\s electromagnetic mass2 9 ?is case1 mass w no long] be con/ant1 ! transv]sal mass w no long] be equal to ! longitud9al1 ! pr9ciples ( me*anics w be ov]?r[n4 ,fir/ a ^w ( explanation4 ,we h said t1 = ! same *>ge1 ! total mass ( a positive electron is mu* great] ?an t ( a negative4 ,& !n x is natural to ?9k t ?is diff]5ce is expla9$ by ! positive electron hav+1 besides its fictiti\s mass1 a consid]able real mass2 :i* takes u back to ! fir/ hypo!sis4 ,b we may j z well suppose t ! real mass is null = ^! z = ! o!rs1 b t ! fictiti\s mass ( ! positive electron is mu* ! great] s9ce ?is electron is mu* ! small]4 ,i say advis$ly3 mu* ! small]4 ,&1 9 fact1 9 ?is hypo!sis in]tia is exclusively electromagnetic 9 orig92 x r$uces itself to ! in]tia ( ! e!r2 ! electrons are no long] any?+ by !mselves2 !y are solely holes 9 ! e!r & ar\nd :i* ! e!r moves2 ! small] ^! holes are1 ! m w !re be ( e!r1 ! great]1 consequ5tly1 w be ! in]tia ( ! e!r4 ,h[ %all we decide betwe5 ^! two hypo!ses8 ,by op]at+ ^u ! canal rays z ,kaufmann did ^u ! beta rays8 ,?is is impossible2 ! velocity ( ^! rays is mu* too slight4 ,%\ld ea* !re=e decide accord+ to 8 temp]am5t1 ! cons]vatives go+ to "o side & ! lov]s ( ! new to ! o!r8 ,p]haps1 b1 to fully und]/& ! argum5ts ( ! innovators1 o!r consid]ations mu/ come 94 ,,*apt] ,,ii ,,me*anics ,,& ,,optics ,i ,ab]ration ,y "k 9 :at ! ph5om5on ( ab]ration1 discov]$ by ,bradley1 consi/s4 ,! light issu+ f a /> takes a c]ta9 "t to g ?r\gh a telescope2 dur+ ?is "t1 ! telescope1 c>ri$ along by ! motion ( ! e>?1 is displac$4 ,if !re=e ! telescope 7 po9t$ 9 ! true direction ( ! />1 ! image w\ld be =m$ at ! po9t occupi$ by ! cross+ ( ! ?reads ( ! network :5 ! light has rea*$ ! objective2 & ?is cross+ w\ld no long] be at ?is same po9t :5 ! light rea*$ ! plane ( ! network4 ,we w\ld !re=e be l$ to mis-po9t ! telescope to br+ ! image ^u ! cross+ ( ! ?reads4 ,!nce results t ! a/ronom] w n po9t ! telescope 9 ! direction ( ! absolute velocity ( ! light1 t is to say t[>d ! true position ( ! />1 b j 9 ! direction ( ! relative velocity ( ! light ) ref]5ce to ! e>?1 t is to say t[>d :at is call$ ! app>5t position ( ! />4 ,! velocity ( light is kn[n2 we might !re=e suppose t we h ! means ( calculat+ ! absolute velocity ( ! e>?4 7,i %all soon expla9 my use "h ( ! ^w absolute47 ,no?+ ( ! sort2 we inde$ "k ! app>5t position ( ! /> we obs]ve2 b we d n "k its true position2 we "k ! velocity ( ! light only 9 magnitude & n 9 direction4 ,if !re=e ! absolute velocity ( ! e>? 7 /raight & uni=m1 we %\ld nev] h suspect$ ! ph5om5on ( ab]ration2 b x is v>iable2 x is compos$ ( two p>ts3 ! velocity ( ! sol> sy/em1 :i* is /raight & uni=m2 ! velocity ( ! e>? ) ref]5ce to ! sun1 :i* is v>iable4 ,if ! velocity ( ! sol> sy/em1 t is to say if ! con/ant "p exi/$ alone1 ! obs]v$ direction w\ld be inv>iable4 ,?is position t "o w\ld ?us obs]ve is call$ ! mean app>5t position ( ! />4 ,tak+ acc\nt n[ at ! same "t ( ! two p>ts ( ! velocity ( ! e>?1 we %all h ! actual app>5t position1 :i* describes a little ellipse ar\nd ! mean app>5t position1 & x is ?is ellipse t we obs]ve4 ,neglect+ v small quantities1 we %all see t ! dim5sions ( ?is ellipse dep5d only ^u ! ratio ( ! velocity ( ! e>? ) ref]5ce to ! sun to ! velocity ( light1 s t ! relative velocity ( ! e>? ) reg>d to ! sun has alone come 94 ,b wait6 ,?is result is n exact1 x is only approximate2 let u pu% ! approximation a little f>!r4 ,! dim5sions ( ! ellipse w dep5d !n ^u ! absolute velocity ( ! e>?4 ,let u comp>e ! major axes ( ! ellipse = ! diff]5t />s3 we %all h1 !oretically at lea/1 ! means ( det]m9+ ?is absolute velocity4 ,t w\ld be p]haps less %ock+ ?an x at fir/ seems2 x is a "q1 9 fact1 n ( ! velocity ) ref]5ce to an absolute void1 b ( ! velocity ) reg>d to ! e!r1 :i* is tak5 by def9ition z be+ absolutely at re/4 ,besides1 ?is me?od is purely !oretical4 ,9 fact1 ! ab]ration is v small2 ! possible v>iations ( ! ellipse ( ab]ration are mu* small] yet1 &1 if we consid] ! ab]ration z ( ! fir/ ord]1 !y %\ld !re=e be reg>d$ z ( ! second ord]3 ab\t a million? ( a second2 !y are absolutely inappreciable = our in/rum5ts4 ,we %all f9ally see1 fur!r on1 :y ! prec$+ !ory %\ld be reject$1 & :y we c\ld n det]m9e ?is absolute velocity ev5 if our in/rum5ts 7 t5 ?\s& times m precise6 ,"o might imag9e "s o!r means1 & 9 fact1 s "o has4 ,! velocity ( light is n ! same 9 wat] z 9 air2 c\ld we n comp>e ! two app>5t positions ( a /> se5 ?r\gh a telescope fir/ full ( air1 !n full ( wat]8 ,! results h be5 negative2 ! app>5t laws ( reflection & refraction are n alt]$ by ! motion ( ! e>?4 ,?is ph5om5on is capable ( two explanations3 #a ,x might be suppos$ t ! e!r is n at re/1 b t x is c>ri$ along by ! body 9 motion4 ,x w\ld !n n be a/oni%+ t ! ph5om5a ( refraction are n alt]$ by ! motion ( ! e>?1 s9ce all1 prisms1 telescopes & e!r1 are c>ri$ along toge!r 9 ! same translation4 ,z to ! ab]ration itself1 x w\ld be expla9$ by a sort ( refraction happ5+ at ! surface ( sep>ation ( ! e!r at re/ 9 ! int]/ell> spaces & ! e!r c>ri$ along by ! motion ( ! e>?4 ,x is ^u ?is hypo!sis 7bodily c>ry+ along ( ! e!r7 t is f\nd$ ! !ory ( ,h]tz on ! electrodynamics ( mov+ bodies4 #b ,fresnel1 on ! contr>y1 supposes t ! e!r is at absolute re/ 9 ! void1 at re/ almo/ absolute 9 ! air1 :atev] be ! velocity ( ?is air1 & t x is p>tially c>ri$ along by refractive m$ia4 ,lor5tz has giv5 to ?is !ory a m satisfactory =m4 ,= him1 ! e!r is at re/1 only ! electrons are 9 motion2 9 ! void1 :]e x is only a "q ( ! e!r1 9 ! air1 :]e ?is is almo/ ! case1 ! c>ry+ along is null or almo/ null2 9 refractive m$ia1 :]e p]turbation is produc$ at ! same "t by vibrations ( ! e!r & ^? ( electrons put 9 sw+ by ! agitation ( ! e!r1 ! undulations are p>tially c>ri$ along4 ,to decide betwe5 ! two hypo!ses1 we h ,fizeau's exp]im5t1 comp>+ by measurem5ts ( ! fr+es ( int]f]5ce1 ! velocity ( light 9 air at re/ or 9 motion4 ,^! exp]im5ts h confirm$ ,fresnel's hypo!sis ( p>tial c>ry+ along4 ,!y h be5 repeat$ ) ! same result by ,mi*elson4 ,! !ory ( ,h]tz mu/ !re=e be reject$4 ,,ii ,! ,pr9ciple ( ,relativity ,b if ! e!r is n c>ri$ along by ! motion ( ! e>?1 is x possible to %[1 by means ( optical ph5om5a1 ! absolute velocity ( ! e>?1 or r its velocity ) respect to ! unmov+ e!r8 ,exp]im5t has answ]$ negatively1 & yet ! exp]im5tal proc$ures h be5 v>i$ 9 all possible ways4 ,:atev] be ! means employ$ !re w nev] be disclos$ any?+ b relative velocities2 ,i mean ! velocities ( c]ta9 mat]ial bodies ) ref]5ce to o!r mat]ial bodies4 ,9 fact1 if ! s\rce ( light & ! app>atus ( obs]vation are on ! e>? & p>ticipate 9 its motion1 ! exp]im5tal results h always be5 ! same1 :atev] be ! ori5tation ( ! app>atus ) ref]5ce to ! orbital motion ( ! e>?4 ,if a/ronomic ab]ration happ5s1 x is because ! s\rce1 a />1 is 9 motion ) ref]5ce to ! obs]v]4 ,! hypo!ses s f> made p]fectly acc\nt = ?is g5]al result1 if we neglect v small quantities ( ! ord] ( ! squ>e ( ! ab]ration4 ,! explanation re/s ^u ! notion ( local "t1 introduc$ by ,lor5tz1 :i* ,i %all try to make cle>4 ,suppose two obs]v]s1 plac$ "o at ,a1 ! o!r at ,b1 & wi%+ to set _! wat*es by means ( optical signals4 ,!y agree t ,b %all s5d a signal to ,a :5 8 wat* m>ks an h\r det]m9$ ^u1 & ,a is to put 8 wat* to t h\r ! mom5t he sees ! signal4 ,if ?is alone 7 done1 !re w\ld be a sy/ematic error1 because z ! light takes a c]ta9 "t t to g f ,b to ,a1 ,a's wat* w\ld be beh9d ,b's ! "t t4 ,?is error is easily correct$4 ,x suffices to cross ! signals4 ,a 9 turn mu/ signal ,b1 &1 aft] ?is new adju/m5t1 ,b's wat* w be beh9d ,a's ! "t t4 ,!n x w be suffici5t to take ! ari?metic mean ( ! two adju/m5ts4 ,b ?is way ( do+ supposes t light takes ! same "t to g f ,a to ,b z to return f ,b to ,a4 ,t is true if ! obs]v]s are motionless2 x is no long] s if !y are c>ri$ along 9 a common translation1 s9ce !n ,a1 = example1 w g to meet ! light com+ f ,b1 :ile ,b w flee be=e ! light com+ f ,a4 ,if !re=e ! obs]v]s are borne along 9 a common translation & if !y d n suspect x1 _! adju/m5t w be defective2 _! wat*es w n indicate ! same "t2 ea* w %[ ! local "t belong+ to ! po9t :]e x is4 ,! two obs]v]s w h no way ( p]ceiv+ ?is1 if ! unmov+ e!r c transmit to !m only lum9\s signals all ( ! same velocity1 & if ! o!r signals !y might s5d are transmitt$ by m$ia c>ri$ along ) !m 9 _! translation4 ,! ph5om5on ea* obs]ves w be too soon or too late2 x w\ld be se5 at ! same in/ant only if ! translation did n exi/2 b z x w be obs]v$ ) a wat* t is wrong1 ?is w n be p]ceiv$ & ! appe>ances w n be alt]$4 ,x results f ?is t ! comp5sation is easy to expla9 s long z we neglect ! squ>e ( ! ab]ration1 & = a long "t ! exp]im5ts 7 n suffici5tly precise to w>rant tak+ acc\nt ( x4 ,b ! day came :5 ,mi*elson imag9$ a mu* m delicate proc$ure3 he made rays int]f]e :i* _h trav]s$ diff]5t c\rses1 aft] be+ reflect$ by mirrors2 ea* ( ! pa?s approximat+ a met] & ! fr+es ( int]f]5ce p]mitt+ ! recognition ( a fraction ( a ?\s&? ( a millimet]1 ! squ>e ( ! ab]ration c\ld no long] be neglect$1 & yet ! results 7 /ill negative4 ,!re=e ! !ory requir$ to be complet$1 & x has be5 by ! ,lor5tz-,fitzg]ald hypo!sis4 ,^! two physici/s suppose t all bodies c>ri$ along 9 a translation und]go a contraction 9 ! s5se ( ?is translation1 :ile _! dim5sions p]p5dicul> to ?is translation rema9 un*ang$4 ,?is contraction is ! same = all bodies2 moreov]1 x is v slight1 ab\t "o two-hundr$-million? = a velocity su* z t ( ! e>?4 ,fur!rmore our measur+ in/rum5ts c\ld n disclose x1 ev5 if !y 7 mu* m precise2 our measur+ rods 9 fact und]go ! same contraction z ! objects to be measur$4 ,if ! met] exactly fits :5 appli$ to a body1 if we po9t ! body & consequ5tly ! met] 9 ! s5se ( ! motion ( ! e>?1 x w n cease to exactly fit 9 ano!r ori5tation1 & t al?\gh ! body & ! met] h *ang$ 9 l5g? z well z ori5tation1 & precisely because ! *ange is ! same = "o z = ! o!r4 ,b x is q diff]5t if we measure a l5g?1 n n[ ) a met]1 b by ! "t tak5 by light to pass along x1 & ?is is j :at ,mi*elson has done4 ,a body1 sph]ical :5 at re/1 w take ?us ! =m ( a flatt5$ ellipsoid ( revolution :5 9 motion2 b ! obs]v] w always ?9k x sph]ical1 s9ce he himself has und]gone an analog\s de=mation1 z also all ! objects s]v+ z po9ts ( ref]5ce4 ,on ! contr>y1 ! surfaces ( ! waves ( light1 rema9+ rigor\sly sph]ical1 w seem to him elongat$ ellipsoids4 ,:at happ5s !n8 ,suppose an obs]v] & a s\rce ( light c>ri$ along toge!r 9 ! translation3 ! wave surfaces emanat+ f ! s\rce w be sph]es hav+ z c5t]s ! successive positions ( ! s\rce2 ! di/ance f ?is c5t] to ! actual position ( ! s\rce w be proportional to ! "t elaps$ aft] ! emission1 t is to say to ! radius ( ! sph]e4 ,all ^! sph]es are !re=e homo!tic "o to ! o!r1 ) relation to ! actual position ,s ( ! s\rce4 ,b1 = our obs]v]1 because ( ! contraction1 all ^! sph]es w seem elongat$ ellipsoids1 & all ^! ellipsoids w moreov] be homo!tic1 ) ref]5ce to ! po9t ,s2 ! exc5tricity ( all ^! ellipsoids is ! same & dep5ds solely ^u ! velocity ( ! e>?4 ,we %all s select ! law ( contraction t ! po9t ,s may be at ! focus ( ! m]idian section ( ! ellipsoid4 ,?is "t ! comp5sation is rigor\s1 & ?is x is :i* expla9s ,mi*elson's exp]im5t4 ,i h said above t1 accord+ to ! ord9>y !ories1 obs]vations ( ! a/ronomic ab]ration w\ld give u ! absolute velocity ( ! e>?1 if our in/rum5ts 7 a ?\s& times m precise4 ,i mu/ modify ?is /atem5t4 ,yes1 ! obs]v$ angles w\ld be modifi$ by ! effect ( ?is absolute velocity1 b ! graduat$ circles we use to measure ! angles w\ld be de=m$ by ! translation3 !y w\ld become ellipses2 !nce w\ld result an error 9 reg>d to ! angle measur$1 & ?is second error w\ld exactly comp5sate ! fir/4 ,?is ,lor5tz-,fitzg]ald hypo!sis seems at fir/ v extraord9>y2 all we c say = ! mom5t1 9 its favor1 is t x is only ! imm$iate translation ( ,mi*elson's exp]im5tal result1 if we def9e l5g?s by ! "t tak5 by light to run along !m4 ,h[ev] t may be1 x is impossible to escape ! impression t ! pr9ciple ( relativity is a g5]al law ( nature1 t "o w nev] be able by any imag9able means to %[ any b relative velocities1 & ,i mean by t n only ! velocities ( bodies ) ref]5ce to ! e!r1 b ! velocities ( bodies ) reg>d to "o ano!r4 ,too _m diff]5t exp]im5ts h giv5 concordant results = u n to feel tempt$ to attribute to ?is pr9ciple ( relativity a value comp>able to t1 = example1 ( ! pr9ciple ( equival5ce4 ,9 any case1 x is prop] to see to :at consequ5ces ?is way ( look+ at ?+s w\ld lead u & !n to submit ^! consequ5ces to ! control ( exp]im5t4 ,,iii ,! ,pr9ciple ( ,reaction ,let u see :at ! pr9ciple ( ! equality ( action & reaction becomes 9 ! !ory ( ,lor5tz4 ,consid] an electron ,a :i* = any cause beg9s to move2 x produces a p]turbation 9 ! e!r2 at ! end ( a c]ta9 "t1 ?is p]turbation rea*es ano!r electron ,b1 :i* w be di/urb$ f its position ( equilibrium4 ,9 ^! conditions !re c n be equality betwe5 action & reaction1 at lea/ if we d n consid] ! e!r1 b only ! electrons1 :i* alone are obs]vable1 s9ce our matt] is made ( electrons4 ,9 fact x is ! electron ,a :i* has di/urb$ ! electron ,b2 ev5 9 case ! electron ,b %\ld react ^u ,a1 ?is reaction c\ld be equal to ! action1 b 9 no case simultane\s1 s9ce ! electron ,b c beg9 to move only aft] a c]ta9 "t1 necess>y = ! propagation4 ,submitt+ ! problem to a m exact calculation1 we rea* ! foll[+ result3 ,suppose a ,h]tz dis*>g] plac$ at ! focus ( a p>abolic mirror to :i* x is me*anically atta*$2 ?is dis*>g] emits electromagnetic waves1 & ! mirror reflects all ^! waves 9 ! same direction2 ! dis*>g] !re=e w radiate en]gy 9 a det]m9ate direction4 ,well1 ! calculation %[s t ! dis*>g] recoils l a cannon :i* has %ot out a projectile4 ,9 ! case ( ! cannon1 ! recoil is ! natural result ( ! equality ( action & reaction4 ,! cannon recoils because ! projectile ^u :i* x has act$ reacts ^u x4 ,b "h x is no long] ! same4 ,:at has be5 s5t out is no long] a mat]ial projectile3 x is en]gy1 & en]gy has no mass3 x has no c\nt]p>t4 ,&1 9 place ( a dis*>g]1 we c\ld h consid]$ j simply a lamp ) a reflector conc5trat+ its rays 9 a s+le direction4 ,x is true t1 if ! en]gy s5t out f ! dis*>g] or f ! lamp meets a mat]ial object1 ?is object receives a me*anical pu% z if x _h be5 hit by a real projectile1 & ?is pu% w be equal to ! recoil ( ! dis*>g] & ( ! lamp1 if no en]gy has be5 lo/ on ! way & if ! object absorbs ! :ole ( ! en]gy4 ,!re=e "o is tempt$ to say t !re /ill is comp5sation betwe5 ! action & ! reaction4 ,b ?is comp5sation1 ev5 %\ld x be complete1 is always belat$4 ,x nev] happ5s if ! light1 aft] leav+ its s\rce1 w&]s ?r\gh int]/ell> spaces )\t ev] meet+ a mat]ial body2 x is incomplete1 if ! body x /rikes is n p]fectly absorb5t4 ,are ^! me*anical actions too small to be measur$1 or are !y accessible to exp]im5t8 ,^! actions are no?+ o!r ?an ^? due to ! ,maxwell-,b>?oli pressures2 ,maxwell _h pr$ict$ ^! pressures f calculations relative to electro/atics & magnetism2 ,b>?oli rea*$ ! same result by !rmodynamic consid]ations4 ,?is is h[ ! tails ( comets are expla9$4 ,little p>ticles deta* !mselves f ! nucleus ( ! comet2 !y are /ruck by ! light ( ! sun1 :i* pu%es !m back z w\ld a ra9 ( projectiles com+ f ! sun4 ,! mass ( ^! p>ticles is s little t ?is repulsion sweeps x away aga9/ ! ,newtonian attraction2 s 9 mov+ away f ! sun !y =m ! tails4 ,! direct exp]im5tal v]ification 0 n easy to obta94 ,! fir/ endeavor l$ to ! con/ruction ( ! radiomet]4 ,b ?is in/rum5t turns backw>d1 9 ! s5se opposite to ! !oretic s5se1 & ! explanation ( its rotation1 s9ce discov]$1 is :olly diff]5t4 ,at la/ success came1 by mak+ ! vacuum m complete1 on ! "o h&1 & on ! o!r by n black5+ "o ( ! faces ( ! paddles & direct+ a p5cil ( lum9\s rays ^u "o ( ! faces4 ,! radiometric effects & ! o!r di/urb+ causes are elim9at$ by a s]ies ( pa9s-tak+ precautions1 & "o obta9s a deviation :i* is v m9ute1 b :i* is1 x w\ld seem1 9 con=mity ) ! !ory4 ,! same effects ( ! ,maxwell-,b>?oli pressure are =eca/ likewise by ! !ory ( ,h]tz ( :i* we h be=e spok51 & by t ( ,lor5tz4 ,b !re is a diff]5ce4 ,suppose t ! en]gy1 "u ! =m ( light1 = example1 proce$s f a lum9\s s\rce to any body ?r\gh a transp>5t m$ium4 ,! ,maxwell-,b>?oli pressure w act1 n alone ^u ! s\rce at ! dep>ture1 & on ! body lit up at ! arrival1 b ^u ! matt] ( ! transp>5t m$ium :i* x trav]ses4 ,at ! mom5t :5 ! lum9\s wave rea*es a new region ( ?is m$ium1 ?is pressure w pu% =w>d ! matt] !re di/ribut$ & w put x back :5 ! wave leaves ?is region4 ,s t ! recoil ( ! s\rce has = c\nt]p>t ! =w>d movem5t ( ! transp>5t matt] :i* is 9 contact ) ?is s\rce2 a little lat]1 ! recoil ( ?is same matt] has = c\nt]p>t ! =w>d movem5t ( ! transp>5t matt] :i* lies a little fur!r on1 & s on4 ,only1 is ! comp5sation p]fect8 ,is ! action ( ! ,maxwell-,b>?oli pressure ^u ! matt] ( ! transp>5t m$ium equal to its reaction ^u ! s\rce1 & t :atev] be ?is matt]8 ,or is ?is action by s mu* ! less z ! m$ium is less refractive & m r>efi$1 becom+ null 9 ! void8 ,if we admit ! !ory ( ,h]tz1 :o reg>ds matt] z me*anically b\nd to ! e!r1 s t ! e!r may be entirely c>ri$ along by matt]1 x w\ld be necess>y to answ] yes to ! fir/ "q & no to ! second4 ,!re w\ld !n be p]fect comp5sation1 z requir$ by ! pr9ciple ( ! equality ( action & reaction1 ev5 9 ! lea/ refractive m$ia1 ev5 9 ! air1 ev5 9 ! int]planet>y void1 :]e x w\ld suffice to suppose a residue ( matt]1 h[ev] subtile4 ,if on ! contr>y we admit ! !ory ( ,lor5tz1 ! comp5sation1 always imp]fect1 is ins5sible 9 ! air & becomes null 9 ! void4 ,b we h se5 above t ,fizeau's exp]im5t does n p]mit ( our reta9+ ! !ory ( ,h]tz2 x is necess>y !re=e to adopt ! !ory ( ,lor5tz1 & consequ5tly to r5\nce ! pr9ciple ( reaction4 ,,iv ,consequ5ces ( ! ,pr9ciple ( ,relativity ,we h se5 above ! reasons :i* impel u to reg>d ! pr9ciple ( relativity z a g5]al law ( nature4 ,let u see to :at consequ5ces ?is pr9ciple w\ld lead1 %\ld x be reg>d$ z f9ally demon/rat$4 ,fir/1 x obliges u to g5]alize ! hypo!sis ( ,lor5tz & ,fitzg]ald on ! contraction ( all bodies 9 ! s5se ( ! translation4 ,9 p>ticul>1 we mu/ ext5d ?is hypo!sis to ! electrons !mselves4 ,abraham consid]$ ^! electrons z sph]ical & inde=mable2 x w be necess>y = u to admit t ^! electrons1 sph]ical :5 9 repose1 und]go ! ,lor5tz contraction :5 9 motion & take !n ! =m ( flatt5$ ellipsoids4 ,?is de=mation ( ! electrons w influ5ce _! me*anical prop]ties4 ,9 fact ,i h said t ! displacem5t ( ^! *>g$ electrons is a v]itable curr5t ( convection & t _! app>5t in]tia is due to ! self-induction ( ?is curr5t3 exclusively z conc]ns ! negative electrons2 exclusively or n1 we d n yet "k1 = ! positive electrons4 ,well1 ! de=mation ( ! electrons1 a de=mation :i* dep5ds ^u _! velocity1 w modify ! di/ribution ( ! electricity ^u _! surface1 consequ5tly ! int5sity ( ! convection curr5t !y produce1 consequ5tly ! laws accord+ to :i* ! self-induction ( ?is curr5t w v>y z a function ( ! velocity4 ,at ?is price1 ! comp5sation w be p]fect & w con=m to ! requirem5ts ( ! pr9ciple ( relativity1 b only ^u two conditions3 #a ,t ! positive electrons h no real mass1 b only a fictiti\s electromagnetic mass2 or at lea/ t _! real mass1 if x exi/s1 is n con/ant & v>ies ) ! velocity accord+ to ! same laws z _! fictiti\s mass2 #b ,t all =ces are ( electromagnetic orig91 or at lea/ t !y v>y ) ! velocity accord+ to ! same laws z ! =ces ( electromagnetic orig94 ,x /ill is ,lor5tz :o has made ?is rem>kable syn!sis2 /op a mom5t & see :at foll[s !refrom4 ,fir/1 !re is no m matt]1 s9ce ! positive electrons no long] h real mass1 or at lea/ no con/ant real mass4 ,! pres5t pr9ciples ( our me*anics1 f\nd$ ^u ! con/ancy ( mass1 mu/ !re=e be modifi$4 ,aga91 an electromagnetic explanation mu/ be s\ght ( all ! kn[n =ces1 9 p>ticul> ( gravitation1 or at lea/ ! law ( gravitation mu/ be s modifi$ t ?is =ce is alt]$ by velocity 9 ! same way z ! electromagnetic =ces4 ,we %all return to ?is po9t4 ,all t appe>s1 at fir/ sight1 a little artificial4 ,9 p>ticul>1 ?is de=mation ( electrons seems q hypo!tical4 ,b ! ?+ may be pres5t$ o!rwise1 s z to avoid putt+ ?is hypo!sis ( de=mation at ! f\ndation ( ! reason+4 ,consid] ! electrons z mat]ial po9ts & ask h[ _! mass %\ld v>y z function ( ! velocity n to contrav5e ! pr9ciple ( relativity4 ,or1 /ill bett]1 ask :at %\ld be _! accel]ation "u ! influ5ce ( an electric or magnetic field1 t ?is pr9ciple be n violat$ & t we come back to ! ord9>y laws :5 we suppose ! velocity v slight4 ,we %all f9d t ! v>iations ( ?is mass1 or ( ^! accel]ations1 mu/ be z if ! electron und]w5t ! ,lor5tz de=mation4 ,v ,kaufmann's ,exp]im5t ,we h be=e u1 !n1 two !ories3 "o :]e ! electrons are inde=mable1 ?is is t ( ,abraham2 ! o!r :]e !y und]go ! ,lor5tz de=mation4 ,9 bo? cases1 _! mass increases ) ! velocity1 becom+ inf9ite :5 ?is velocity becomes equal to t ( light2 b ! law ( ! v>iation is n ! same4 ,! me?od employ$ by ,kaufmann to br+ to light ! law ( v>iation ( ! mass seems !re=e to give u an exp]im5tal means ( decid+ betwe5 ! two !ories4 ,unhappily1 8 fir/ exp]im5ts 7 n suffici5tly precise = t2 s he decid$ to repeat !m ) m precautions1 & measur+ ) great c>e ! int5sity ( ! fields4 ,"u _! new =m !y are 9 favor ( ! !ory ( ,abraham4 ,!n ! pr9ciple ( relativity w\ld n h ! rigor\s value we 7 tempt$ to attribute to x2 !re w\ld no long] be reason = believ+ ! positive electrons d5ud$ ( real mass l ! negative electrons4 ,h[ev]1 be=e def9itely adopt+ ?is conclusion1 a little reflection is necess>y4 ,! "q is ( su* importance t x is to be wi%$ ,kaufmann's exp]im5t 7 repeat$ by ano!r exp]im5t]4#ag ,unhappily1 ?is exp]im5t is v delicate & c\ld be c>ri$ out successfully only by a physici/ ( ! same ability z ,kaufmann4 ,all precautions h be5 prop]ly tak5 & we h>dly see :at objection c\ld be made4 #ag ,at ! mom5t ( go+ to press we le>n t ,m4 ,bu*]] has repeat$ ! exp]im5t1 tak+ new precautions1 & t he has obta9$1 contr>y to ,kaufmann1 results confirm+ ! views ( ,lor5tz4 ,!re is "o po9t h[ev] to :i* ,i wi% to draw att5tion3 t is to ! measurem5t ( ! electro/atic field1 a measurem5t ^u :i* all dep5ds4 ,?is field 0 produc$ betwe5 ! two armatures ( a cond5s]2 &1 betwe5 ^! armatures1 !re 0 to be made an extremely p]fect vacuum1 9 ord] to obta9 a complete isolation4 ,!n ! diff]5ce ( pot5tial ( ! two armatures 0 measur$1 & ! field obta9$ by divid+ ?is diff]5ce by ! di/ance ap>t ( ! armatures4 ,t supposes ! field uni=m2 is ?is c]ta98 ,might !re n be an abrupt fall ( pot5tial 9 ! neighborhood ( "o ( ! armatures1 ( ! negative armature1 = example8 ,!re may be a diff]5ce ( pot5tial at ! meet+ ( ! metal & ! vacuum1 & x may be t ?is diff]5ce is n ! same on ! positive side & on ! negative side2 :at w\ld lead me to ?9k s is ! electric valve effects betwe5 m]cury & vacuum4 ,h[ev] slight ! probability t x is s1 x seems t x %\ld be consid]$4 ,,vi ,! ,pr9ciple ( ,in]tia ,9 ! new dynamics1 ! pr9ciple ( in]tia is /ill true1 t is to say t an isolat$ electron w h a /raight & uni=m motion4 ,at lea/ ?is is g5]ally assum$2 h[ev]1 ,l9demann has made objections to ?is view2 ,i d n wi% to take "p 9 ?is discussion1 :i* ,i c n "h exp\nd because ( its too difficult "*4 ,9 any case1 slight modifications to ! !ory w\ld suffice to %elt] x f ,l9demann's objections4 ,we "k t a body subm]g$ 9 a fluid exp]i5ces1 :5 9 motion1 consid]able resi/ance1 b ?is is because our fluids are visc\s2 9 an ideal fluid1 p]fectly free f viscosity1 ! body w\ld /ir up beh9d x a liquid hill1 a sort ( wake2 ^u dep>ture1 a great ef=t w\ld be necess>y to put x 9 motion1 s9ce x w\ld be necess>y to move n only ! body itself1 b ! liquid ( its wake4 ,b1 ! motion once acquir$1 x w\ld p]petuate itself )\t resi/ance1 s9ce ! body1 9 advanc+1 w\ld simply c>ry ) x ! p]turbation ( ! liquid1 )\t ! total vis viva ( ! liquid augm5t+4 ,ev]y?+ w\ld happ5 !re=e z if its in]tia 0 augm5t$4 ,an electron advanc+ 9 ! e!r w\ld behave 9 ! same way3 ar\nd x1 ! e!r w\ld be /irr$ up1 b ?is p]turbation w\ld accompany ! body 9 its motion2 s t1 = an obs]v] c>ri$ along ) ! electron1 ! electric & magnetic fields accompany+ ?is electron w\ld appe> inv>iable1 & w\ld *ange only if ! velocity ( ! electron v>i$4 ,an ef=t w\ld !re=e be necess>y to put ! electron 9 motion1 s9ce x w\ld be necess>y to create ! en]gy ( ^! fields2 on ! contr>y1 once ! movem5t acquir$1 no ef=t w\ld be necess>y to ma9ta9 x1 s9ce ! creat$ en]gy w\ld only h to g along beh9d ! electron z a wake4 ,?is en]gy1 !re=e1 c\ld only augm5t ! in]tia ( ! electron1 z ! agitation ( ! liquid augm5ts t ( ! body subm]g$ 9 a p]fect fluid4 ,& anyh[1 ! negative electrons at lea/ h no o!r in]tia except t4 ,9 ! hypo!sis ( ,lor5tz1 ! vis viva1 :i* is only ! en]gy ( ! e!r1 is n proportional to v#b4 ,d\btless if v is v slight1 ! vis viva is s5sibly proportional to v#b1 ! quantity ( motion s5sibly proportional to v1 ! two masses s5sibly con/ant & equal to ea* o!r4 ,b :5 ! velocity t5ds t[>d ! velocity ( light1 ! vis viva1 ! quantity ( motion & ! two masses increase beyond all limit4 ,9 ! hypo!sis ( ,abraham1 ! expressions are a little m complicat$2 b :at we h j said rema9s true 9 ess5tials4 ,s ! mass1 ! quantity ( motion1 ! vis viva become inf9ite :5 ! velocity is equal to t ( light4 ,!nce results t no body c atta9 9 any way a velocity beyond t ( light4 ,& 9 fact1 9 proportion z its velocity increases1 its mass increases1 s t its in]tia opposes to any new increase ( velocity a great] & great] ob/acle4 ,a "q !n sugge/s itself3 let u admit ! pr9ciple ( relativity2 an obs]v] 9 motion w\ld n h any means ( p]ceiv+ 8 own motion4 ,if !re=e no body 9 its absolute motion c exce$ ! velocity ( light1 b may approa* x z ne>ly z y *oose1 x %\ld be ! same conc]n+ its relative motion ) ref]5ce to our obs]v]4 ,& !n we might be tempt$ to reason z foll[s3 ,! obs]v] may atta9 a velocity ( #bjj1jjj kilomet]s2 ! body 9 its relative motion ) ref]5ce to ! obs]v] may atta9 ! same velocity2 its absolute velocity w !n be #djj1jjj kilomet]s1 :i* is impossible1 s9ce ?is is beyond ! velocity ( light4 ,?is is only a seem+1 :i* vani%es :5 acc\nt is tak5 ( h[ ,lor5tz evaluates local "t4 ,,vii ,! ,wave ( ,accel]ation ,:5 an electron is 9 motion1 x produces a p]turbation 9 ! e!r surr\nd+ x2 if its motion is /raight & uni=m1 ?is p]turbation r$uces to ! wake ( :i* we h spok5 9 ! prec$+ section4 ,b x is no long] ! same1 if ! motion be curvil9e> or v>i$4 ,! p]turbation may !n be reg>d$ z ! sup]position ( two o!rs1 to :i* ,langev9 has giv5 ! names wave ( velocity & wave ( accel]ation4 ,! wave ( velocity is only ! wave :i* happ5s 9 uni=m motion4 ,z to ! wave ( accel]ation1 ?is is a p]turbation altoge!r analog\s to light waves1 :i* />ts f ! electron at ! in/ant :5 x und]goes an accel]ation1 & :i* is !n propagat$ by successive sph]ical waves ) ! velocity ( light4 ,:5ce foll[s3 9 a /raight & uni=m motion1 ! en]gy is :olly cons]v$2 b1 :5 !re is an accel]ation1 !re is loss ( en]gy1 :i* is dissipat$ "u ! =m ( lum9\s waves & goes out to inf9ity across ! e!r4 ,h[ev]1 ! effects ( ?is wave ( accel]ation1 9 p>ticul> ! correspond+ loss ( en]gy1 are 9 mo/ cases negligible1 t is to say n only 9 ord9>y me*anics & 9 ! motions ( ! heav5ly bodies1 b ev5 9 ! radium rays1 :]e ! velocity is v great )\t ! accel]ation be+ s4 ,we may !n conf9e ourselves to apply+ ! laws ( me*anics1 putt+ ! =ce equal to ! product ( accel]ation by mass1 ?is mass1 h[ev]1 v>y+ ) ! velocity accord+ to ! laws expla9$ above4 ,we !n say ! motion is quasi-/ation>y4 ,x w\ld n be ! same 9 all cases :]e ! accel]ation is great1 ( :i* ! *ief are ! foll[+3 #a ,9 inc&esc5t gases c]ta9 electrons take an oscillatory motion ( v high frequ5cy2 ! displacem5ts are v small1 ! velocities are f9ite1 & ! accel]ations v great2 en]gy is !n communicat$ to ! e!r1 & ?is is :y ^! gases radiate light ( ! same p]iod z ! oscillations ( ! electron2 #b ,inv]sely1 :5 a gas receives light1 ^! same electrons are put 9 sw+ ) /rong accel]ations & !y absorb light2 #c ,9 ! ,h]tz dis*>g]1 ! electrons :i* circulate 9 ! metallic mass und]go1 at ! in/ant ( ! dis*>ge1 an abrupt accel]ation & take !n an oscillatory motion ( high frequ5cy4 ,!nce results t a "p ( ! en]gy radiates "u ! =m ( ,h]tzian waves2 #d ,9 an inc&esc5t metal1 ! electrons enclos$ 9 ?is metal are impell$ ) great velocity2 ^u rea*+ ! surface ( ! metal1 :i* !y c n get ?r\gh1 !y are reflect$ & ?us und]go a consid]able accel]ation4 ,?is is :y ! metal emits light4 ,! details ( ! laws ( ! emission ( light by d>k bodies are p]fectly expla9$ by ?is hypo!sis2 #e ,f9ally :5 ! ca?ode rays /rike ! antica?ode1 ! negative electrons1 con/itut+ ^! rays1 :i* are impell$ ) v great velocity1 are abruptly arre/$4 ,because ( ! accel]ation !y ?us und]go1 !y produce undulations 9 ! e!r4 ,?is1 accord+ to c]ta9 physici/s1 is ! orig9 ( ! ,rntg5 rays1 :i* w\ld only be light rays ( v %ort wave-l5g?4 ,,*apt] ,,iii ,,! ,,new ,,me*anics ,,& ,,a/ronomy ,i ,gravitation ,mass may be def9$ 9 two ways3 #a ,by ! quoti5t ( ! =ce by ! accel]ation2 ?is is ! true def9ition ( ! mass1 :i* measures ! in]tia ( ! body4 #b ,by ! attraction ! body ex]cises ^u an ext]ior body1 9 virtue ( ,newton's law4 ,we %\ld !re=e di/+ui% ! mass coeffici5t ( in]tia & ! mass coeffici5t ( attraction4 ,accord+ to ,newton's law1 !re is rigor\s proportionality betwe5 ^! two coeffici5ts4 ,b t is demon/rat$ only = velocities to :i* ! g5]al pr9ciples ( dynamics are applicable4 ,n[1 we h se5 t ! mass coeffici5t ( in]tia increases ) ! velocity2 %\ld we conclude t ! mass coeffici5t ( attraction increases likewise ) ! velocity & rema9s proportional to ! coeffici5t ( in]tia1 or1 on ! contr>y1 t ?is coeffici5t ( attraction rema9s con/ant8 ,?is is a "q we h no means ( decid+4 ,on ! o!r h&1 if ! coeffici5t ( attraction dep5ds ^u ! velocity1 s9ce ! velocities ( two bodies :i* mutually attract are n 9 g5]al ! same1 h[ w ?is coeffici5t dep5d ^u ^! two velocities8 ,^u ?is subject we c only make hypo!ses1 b we are naturally l$ to inve/igate :i* ( ^! hypo!ses w\ld be compatible ) ! pr9ciple ( relativity4 ,!re are a great numb] ( !m2 ! only "o ( :i* ,i %all "h speak is t ( ,lor5tz1 :i* ,i %all briefly exp\nd4 ,consid] fir/ electrons at re/4 ,two electrons ( ! same sign repel ea* o!r & two electrons ( contr>y sign attract ea* o!r2 9 ! ord9>y !ory1 _! mutual actions are proportional to _! electric *>ges2 if !re=e we h f\r electrons1 two positive ,a & ,a'1 & two negative ,b & ,b'1 ! *>ges ( ^! f\r be+ ! same 9 absolute value1 ! repulsion ( ,a = ,a' w be1 at ! same di/ance1 equal to ! repulsion ( ,b = ,b' & equal also to ! attraction ( ,a = ,b'1 or ( ,a' = ,b4 ,if !re=e ,a & ,b are v ne> ea* o!r1 z also ,a' & ,b'1 & we exam9e ! action ( ! sy/em ,a ,b ^u ! sy/em ,a' ,b'1 we %all h two repulsions & two attractions :i* w exactly comp5sate ea* o!r & ! result+ action w be null4 ,n[1 mat]ial molecules %\ld j be reg>d$ z species ( sol> sy/ems :]e circulate ! electrons1 "s positive1 "s negative1 & 9 su* a way t ! algebraic sum ( all ! *>ges is null4 ,a mat]ial molecule is !re=e :olly analog\s to ! sy/em ,a ,b ( :i* we h spok51 s t ! total electric action ( two molecules "o ^u ! o!r %\ld be null4 ,b exp]im5t %[s u t ^! molecules attract ea* o!r 9 consequ5ce ( ,newtonian gravitation2 & !n we may make two hypo!ses3 we may suppose gravitation has no relation to ! electro/atic attractions1 t x is due to a cause entirely diff]5t1 & is simply some?+ additional2 or else we may suppose ! attractions are n proportional to ! *>ges & t ! attraction ex]cis$ by a *>ge #a ^u a *>ge -#a is great] ?an ! mutual repulsion ( two #a *>ges1 or two -#a *>ges4 ,9 o!r words1 ! electric field produc$ by ! positive electrons & t :i* ! negative electrons produce might be sup]pos$ & yet rema9 di/9ct4 ,! positive electrons w\ld be m s5sitive to ! field produc$ by ! negative electrons ?an to ! field produc$ by ! positive electrons2 ! contr>y w\ld be ! case = ! negative electrons4 ,x is cle> t ?is hypo!sis some:at complicates electro/atics1 b t x br+s back into x gravitation4 ,?is 01 9 sum1 ,frankl9's hypo!sis4 ,:at happ5s n[ if ! electrons are 9 motion8 ,! positive electrons w cause a p]turbation 9 ! e!r & produce !re an electric & magnetic field4 ,! same w be ! case = ! negative electrons4 ,! electrons1 positive z well z negative1 und]go !n a me*anical impulsion by ! action ( ^! diff]5t fields4 ,9 ! ord9>y !ory1 ! electromagnetic field1 due to ! motion ( ! positive electrons1 ex]cises1 ^u two electrons ( contr>y sign & ( ! same absolute *>ge1 equal actions ) contr>y sign4 ,we may !n )\t inconv5i5ce n di/+ui% ! field due to ! motion ( ! positive electrons & ! field due to ! motion ( ! negative electrons & consid] only ! algebraic sum ( ^! two fields1 t is to say ! result+ field4 ,9 ! new !ory1 on ! contr>y1 ! action ^u ! positive electrons ( ! electromagnetic field due to ! positive electrons foll[s ! ord9>y laws2 x is ! same ) ! action ^u ! negative electrons ( ! field due to ! negative electrons4 ,let u n[ consid] ! action ( ! field due to ! positive electrons ^u ! negative electrons 7or inv]sely72 x w /ill foll[ ! same laws1 b ) a diff]5t coeffici5t4 ,ea* electron is m s5sitive to ! field creat$ by ! electrons ( contr>y "n ?an to ! field creat$ by ! electrons ( ! same "n4 ,su* is ! hypo!sis ( ,lor5tz1 :i* r$uces to ,frankl9's hypo!sis = slight velocities2 x w !re=e expla91 = ^! small velocities1 ,newton's law4 ,moreov]1 z gravitation goes back to =ces ( electrodynamic orig91 ! g5]al !ory ( ,lor5tz w apply1 & consequ5tly ! pr9ciple ( relativity w n be violat$4 ,we see t ,newton's law is no long] applicable to great velocities & t x mu/ be modifi$1 = bodies 9 motion1 precisely 9 ! same way z ! laws ( electro/atics = electricity 9 motion4 ,we "k t electromagnetic p]turbations spread ) ! velocity ( light4 ,we may !re=e be tempt$ to reject ! prec$+ !ory ^u rememb]+ t gravitation spreads1 accord+ to ! calculations ( ,laplace1 at lea/ t5 million times m quickly ?an light1 & t consequ5tly x c n be ( electromagnetic orig94 ,! result ( ,laplace is well kn[n1 b "o is g5]ally ignorant ( its signification4 ,laplace suppos$ t1 if ! propagation ( gravitation is n in/antane\s1 its velocity ( spread comb9es ) t ( ! body attract$1 z happ5s = light 9 ! ph5om5on ( a/ronomic ab]ration1 s t ! effective =ce is n direct$ along ! /raight jo9+ ! two bodies1 b makes ) ?is /raight a small angle4 ,?is is a v special hypo!sis1 n well ju/ifi$1 &1 9 any case1 entirely diff]5t f t ( ,lor5tz4 ,laplace's result proves no?+ aga9/ ! !ory ( ,lor5tz4 ,,ii ,comp>ison ) ,a/ronomic ,obs]vations ,c ! prec$+ !ories be reconcil$ ) a/ronomic obs]vations8 ,fir/ ( all1 if we adopt !m1 ! en]gy ( ! planet>y motions w be con/antly dissipat$ by ! effect ( ! wave ( accel]ation4 ,f ?is w\ld result t ! mean motions ( ! />s w\ld con/antly accel]ate1 z if ^! />s 7 mov+ 9 a resi/ant m$ium4 ,b ?is effect is exce$+ly slight1 f> too mu* s to be disc]n$ by ! mo/ precise obs]vations4 ,! accel]ation ( ! heav5ly bodies is relatively slight1 s t ! effects ( ! wave ( accel]ation are negligible & ! motion may be reg>d$ z quasi /ation>y4 ,x is true t ! effects ( ! wave ( accel]ation con/antly accumulate1 b ?is accumulation itself is s sl[ t ?\s&s ( ye>s ( obs]vation w\ld be necess>y = x to become s5sible4 ,let u !re=e make ! calculation consid]+ ! motion z quasi-/ation>y1 & t "u ! ?ree foll[+ hypo!ses3 ,a4 ,admit ! hypo!sis ( ,abraham 7electrons inde=mable7 & reta9 ,newton's law 9 its usual =m2 ,b4 ,admit ! hypo!sis ( ,lor5tz ab\t ! de=mation ( electrons & reta9 ! usual ,newton's law2 ,c4 ,admit ! hypo!sis ( ,lor5tz ab\t electrons & modify ,newton's law z we h done 9 ! prec$+ p>agraph1 s z to r5d] x compatible ) ! pr9ciple ( relativity4 ,x is 9 ! motion ( ,m]cury t ! effect w be mo/ s5sible1 s9ce ?is planet has ! greate/ velocity4 ,tiss]& =m]ly made an analog\s calculation1 admitt+ ,web]'s law2 ,i recall t ,web] _h s\ght to expla9 at ! same "t ! electro/atic & electrodynamic ph5om5a 9 suppos+ t electrons 7^: "n 0 n yet inv5t$7 ex]cise1 "o ^u ano!r1 attractions & repulsions direct$ along ! /raight jo9+ !m1 & dep5d+ n only ^u _! di/ances1 b ^u ! fir/ & second d]ivatives ( ^! di/ances1 consequ5tly ^u _! velocities & _! accel]ations4 ,?is law ( ,web]1 diff]5t en\gh f ^? :i* to-day t5d to prevail1 none ! less pres5ts a c]ta9 analogy ) !m4 ,tiss]& f\nd t1 if ! ,newtonian attraction con=m$ to ,web]'s law !re result$1 = ,m]cury's p]ihelion1 secul> v>iation ( #ad1 ( ! same s5se z t :i* has be5 obs]v$ & c\ld n be expla9$1 b small]1 s9ce ?is is #ch4 ,let u recur to ! hypo!ses ,a1 ,b & ,c1 & /udy fir/ ! motion ( a planet attract$ by a fix$ c5t]4 ,! hypo!ses ,b & ,c are no long] di/+ui%$1 s9ce1 if ! attract+ po9t is fix$1 ! field x produces is a purely electro/atic field1 :]e ! attraction v>ies inv]sely z ! squ>e ( ! di/ance1 9 con=mity ) ,c\lomb's electro/atic law1 id5tical ) t ( ,newton4 ,! vis viva equation holds good1 tak+ = vis viva ! new def9ition2 9 ! same way1 ! equation ( areas is replac$ by ano!r equival5t to x2 ! mom5t ( ! quantity ( motion is a con/ant1 b ! quantity ( motion mu/ be def9$ z 9 ! new dynamics4 ,! only s5sible effect w be a secul> motion ( ! p]ihelion4 ,) ! !ory ( ,lor5tz1 we %all f9d1 = ?is motion1 half ( :at ,web]'s law w\ld give2 ) ! !ory ( ,abraham1 two fif?s4 ,if n[ we suppose two mov+ bodies gravitat+ ar\nd _! common c5t] ( gravity1 ! effects are v little diff]5t1 ?\gh ! calculations may be a little m complicat$4 ,! motion ( ,m]cury's p]ihelion w\ld !re=e be #g 9 ! !ory ( ,lor5tz & #e4f 9 t ( ,abraham4 ,! effect moreov] is proportional to 7n#c77;a#b71 :]e n is ! />'s mean motion & a ! radius ( its orbit4 ,= ! planets1 9 virtue ( ,kepl]'s law1 ! effect v>ies !n inv]sely z sqrt7a#e72 x is !re=e ins5sible1 save = ,m]cury4 ,x is likewise ins5sible = ! moon ?\gh n is great1 because a is extremely small2 9 sum1 x is five times less = ,v5us1 & six hundr$ times less = ! moon ?an = ,m]cury4 ,we may add t z to ,v5us & ! e>?1 ! motion ( ! p]ihelion 7= ! same angul> velocity ( ?is motion7 w\ld be mu* m difficult to disc]n by a/ronomic obs]vations1 because ! exc5tricity ( _! orbits is mu* less ?an = ,m]cury4 ,to sum up1 ! only s5sible effect ^u a/ronomic obs]vations w\ld be a motion ( ,m]cury's p]ihelion1 9 ! same s5se z t :i* has be5 obs]v$ )\t be+ expla9$1 b notably slight]4 ,t c n be reg>d$ z an argum5t 9 favor ( ! new dynamics1 s9ce x w always be necess>y to seek ano!r explanation = ! great] "p ( ,m]cury's anomaly2 b /ill less c x be reg>d$ z an argum5t aga9/ x4 ,,iii ,! ,!ory ( ,lesage ,x is int]e/+ to comp>e ^! consid]ations ) a !ory long s9ce propos$ to expla9 univ]sal gravitation4 ,suppose t1 9 ! int]planet>y spaces1 circulate 9 e direction1 ) high velocities1 v t5u\s corpuscles4 ,a body isolat$ 9 space w n be affect$1 app>5tly1 by ! impacts ( ^! corpuscles1 s9ce ^! impacts are equally di/ribut$ 9 all directions4 ,b if two bodies ,a & ,b are pres5t1 ! body ,b w play ! rle ( scre5 & w int]cept "p ( ! corpuscles :i*1 )\t x1 w\ld h /ruck ,a4 ,!n1 ! impacts receiv$ by ,a 9 ! direction opposite t f ,b w no long] h a c\nt]p>t1 or w n[ be only p>tially comp5sat$1 & ?is w pu% ,a t[>d ,b4 ,su* is ! !ory ( ,lesage2 & we %all discuss x1 tak+ fir/ ! view-po9t ( ord9>y me*anics4 ,fir/1 h[ %\ld ! impacts po/ulat$ by ?is !ory take place2 is x accord+ to ! laws ( p]fectly ela/ic bodies1 or accord+ to ^? ( bodies devoid ( ela/icity1 or accord+ to an int]m$iate law8 ,! corpuscles ( ,lesage c n act z p]fectly ela/ic bodies2 o!rwise ! effect w\ld be null1 s9ce ! corpuscles int]cept$ by ! body ,b w\ld be replac$ by o!rs :i* w\ld h reb\nd$ f ,b1 & calculation proves t ! comp5sation w\ld be p]fect4 ,x is necess>y !n t ! impact make ! corpuscles lose en]gy1 & ?is en]gy %\ld appe> "u ! =m ( heat4 ,b h[ mu* heat w\ld ?us be produc$8 ,note t attraction passes ?r\gh bodies2 x is necess>y !re=e to repres5t to ourselves ! e>?1 = example1 n z a solid scre51 b z =m$ ( a v great numb] ( v small sph]ical molecules1 :i* play individually ! rle ( little scre5s1 b betwe5 :i* ! corpuscles ( ,lesage may freely circulate4 ,s1 n only ! e>? is n a solid scre51 b x is n ev5 a cull5d]1 s9ce ! voids occupy mu* m space ?an ! pl5ums4 ,to realize ?is1 recall t ,laplace has demon/rat$ t attraction1 9 trav]s+ ! e>?1 is weak5$ at mo/ by "o t5-million? "p1 & 8 pro( is p]fectly satisfactory3 9 fact1 if attraction 7 absorb$ by ! body x trav]ses1 x w\ld no long] be proportional to ! masses2 x w\ld be relatively weak] = great bodies ?an = small1 s9ce x w\ld h a great] ?ickness to trav]se4 ,! attraction ( ! sun = ! e>? w\ld !re=e be relatively weak] ?an t ( ! sun = ! moon1 & !nce w\ld result1 9 ! motion ( ! moon1 a v s5sible inequality4 ,we %\ld !re=e conclude1 if we adopt ! !ory ( ,lesage1 t ! total surface ( ! sph]ical molecules :i* compose ! e>? is at mo/ ! t5-million? "p ( ! total surface ( ! e>?4 ,d>w9 has prov$ t ! !ory ( ,lesage only leads exactly to ,newton's law :5 we po/ulate p>ticles entirely devoid ( ela/icity4 ,! attraction ex]t$ by ! e>? on a mass #a at a di/ance #a w !n be proportional1 at ! same "t1 to ! total surface ,s ( ! sph]ical molecules compos+ x1 to ! velocity v ( ! corpuscles1 to ! squ>e root ( ! d5sity rho ( ! m$ium =m$ by ! corpuscles4 ,! heat produc$ w be proportional to ,s1 to ! d5sity rho1 & to ! cube ( ! velocity v4 ,b x is necess>y to take acc\nt ( ! resi/ance exp]i5c$ by a body mov+ 9 su* a m$ium2 x c n move1 9 fact1 )\t go+ aga9/ c]ta9 impacts1 9 flee+1 on ! contr>y1 be=e ^? com+ 9 ! opposite direction1 s t ! comp5sation realiz$ 9 ! /ate ( re/ c no long] subsi/4 ,! calculat$ resi/ance is proportional to ,s1 to rho & to v2 n[1 we "k t ! heav5ly bodies move z if !y exp]i5c$ no resi/ance1 & ! precision ( obs]vations p]mits u to fix a limit to ! resi/ance ( ! m$ium4 ,?is resi/ance v>y+ z ,srhov1 :ile ! attraction v>ies z ,ssqrt7rhov71 we see t ! ratio ( ! resi/ance to ! squ>e ( ! attraction is inv]sely z ! product ,sv4 ,we h !re=e a l[] limit ( ! product ,sv4 ,we h already an upp] limit ( ,s 7by ! absorption ( attraction by ! body x trav]ses72 we h !re=e a l[] limit ( ! velocity v1 :i* mu/ be at lea/ #bdajag times t ( light4 ,f ?is we are able to d$uce rho & ! quantity ( heat produc$2 ?is quantity w\ld suffice to raise ! temp]ature #ajbf degrees a second2 ! e>? w\ld receive 9 a giv5 "t #ajbj times m heat ?an ! sun emits 9 ! same "t2 ,i am n speak+ ( ! heat ! sun s5ds to ! e>?1 b ( t x radiates 9 all directions4 ,x is evid5t ! e>? c\ld n long /& su* a rgime4 ,we %\ld n be l$ to results less fanta/ic if1 contr>y to ,d>w9's views1 we end[$ ! corpuscles ( ,lesage ) an ela/icity imp]fect )\t be+ null4 ,9 tru?1 ! vis viva ( ^! corpuscles w\ld n be entirely conv]t$ into heat1 b ! attraction produc$ w\ld likewise be less1 s t x w\ld be only ! "p ( ?is vis viva conv]t$ into heat1 :i* w\ld contribute to produce ! attraction & t w\ld come to ! same ?+2 a judici\s employm5t ( ! !orem ( ! viriel w\ld enable u to acc\nt = ?is4 ,! !ory ( ,lesage may be trans=m$2 suppress ! corpuscles & imag9e ! e!r ov]run 9 all s5ses by lum9\s waves com+ f all po9ts ( space4 ,:5 a mat]ial object receives a lum9\s wave1 ?is wave ex]cises ^u x a me*anical action due to ! ,maxwell-,b>?oli pressure1 j z if x _h receiv$ ! impact ( a mat]ial projectile4 ,! waves 9 "q c\ld !re=e play ! rle ( ! corpuscles ( ,lesage4 ,?is is :at is suppos$1 = example1 by ,m4 ,tommas9a4 ,! difficulties are n remov$ = all t2 ! velocity ( propagation c be only t ( light1 & we are ?us l$1 = ! resi/ance ( ! m$ium1 to an inadmissible figure4 ,besides1 if ! light is all reflect$1 ! effect is null1 j z 9 ! hypo!sis ( ! p]fectly ela/ic corpuscles4 ,t !re %\ld be attraction1 x is necess>y t ! light be p>tially absorb$2 b !n !re is production ( heat4 ,! calculations d n diff] ess5tially f ^? made 9 ! ord9>y !ory ( ,lesage1 & ! result reta9s ! same fanta/ic "*4 ,on ! o!r h&1 attraction is n absorb$ by ! body x trav]ses1 or h>dly at all2 x is n s ) ! light we "k4 ,light :i* w\ld produce ! ,newtonian attraction w\ld h to be consid]ably diff]5t f ord9>y light & be1 = example1 ( v %ort wave l5g?4 ,?is does n c\nt t1 if our eyes 7 s5sible ( ?is light1 ! :ole heav5s %\ld appe> to u mu* m brilliant ?an ! sun1 s t ! sun w\ld seem to u to /& out 9 black1 o!rwise ! sun w\ld repel u in/ead ( attract+ u4 ,= all ^! reasons1 light :i* w\ld p]mit ( ! explanation ( attraction w\ld be mu* m l ,rntg5 rays ?an l ord9>y light4 ,& besides1 ! ,x-rays w\ld n suffice2 h[ev] p5etrat+ !y may seem to u1 !y c\ld n pass ?r\gh ! :ole e>?2 x w\ld be necess>y !re=e to imag9e ,x'-rays mu* m p5etrat+ ?an ! ord9>y ,x-rays4 ,moreov] a "p ( ! en]gy ( ^! ,x'-rays w\ld h to be de/roy$1 o!rwise !re w\ld be no attraction4 ,if y d n wi% x trans=m$ into heat1 :i* w\ld lead to an enorm\s heat production1 y mu/ suppose x radiat$ 9 e direction "u ! =m ( second>y rays1 :i* might be call$ ,x'' & :i* w\ld h to be mu* m p5etrat+ /ill ?an ! ,x'-rays1 o!rwise !y w\ld 9 _! turn d]ange ! ph5om5a ( attraction4 ,su* are ! complicat$ hypo!ses to :i* we are l$ :5 we try to give life to ! !ory ( ,lesage4 ,b all we h said presupposes ! ord9>y laws ( me*anics4 ,w ?+s g bett] if we admit ! new dynamics8 ,& fir/1 c we cons]ve ! pr9ciples ( relativity8 ,let u give at fir/ to ! !ory ( ,lesage its primitive =m1 & suppose space pl\gh$ by mat]ial corpuscles2 if ^! corpuscles 7 p]fectly ela/ic1 ! laws ( _! impact w\ld con=m to ?is pr9ciple ( relativity1 b we "k t !n _! effect w\ld be null4 ,we mu/ !re=e suppose ^! corpuscles are n ela/ic1 & !n x is difficult to imag9e a law ( impact compatible ) ! pr9ciple ( relativity4 ,besides1 we %\ld /ill f9d a production ( consid]able heat1 & yet a v s5sible resi/ance ( ! m$ium4 ,if we suppress ^! corpuscles & rev]t to ! hypo!sis ( ! ,maxwell-,b>?oli pressure1 ! difficulties w n be less4 ,?is is :at ,lor5tz himself has attempt$ 9 8 ,memoir to ! ,am/]dam ,academy ( ,sci5ces ( ,april #be1 #aijj4 ,consid] a sy/em ( electrons imm]s$ 9 an e!r p]meat$ 9 e s5se by lum9\s waves2 "o ( ^! electrons1 /ruck by "o ( ^! waves1 beg9s to vibrate2 its vibration w be syn*ron\s ) t ( light2 b x may h a diff]5ce ( phase1 if ! electron absorbs a "p ( ! incid5t en]gy4 ,9 fact1 if x absorbs en]gy1 ?is is because ! vibration ( ! e!r impels ! electron2 ! electron mu/ !re=e be sl[] ?an ! e!r4 ,an electron 9 motion is analog\s to a convection curr5t2 !re=e e magnetic field1 9 p>ticul> t due to ! lum9\s p]turbation itself1 mu/ ex]t a me*anical action ^u ?is electron4 ,?is action is v slight2 moreov]1 x *anges sign 9 ! curr5t ( ! p]iod2 nev]!less1 ! mean action is n null if !re is a diff]5ce ( phase betwe5 ! vibrations ( ! electron & ^? ( ! e!r4 ,! mean action is proportional to ?is diff]5ce1 consequ5tly to ! en]gy absorb$ by ! electron4 ,i c n "h ent] into ! detail ( ! calculations2 suffice x to say only t ! f9al result is an attraction ( any two electrons1 v>y+ inv]sely z ! squ>e ( ! di/ance & proportional to ! en]gy absorb$ by ! two electrons4 ,!re=e !re c n be attraction )\t absorption ( light &1 consequ5tly1 )\t production ( heat1 & ?is x is :i* det]m9$ ,lor5tz to ab&on ?is !ory1 :i*1 at bottom1 does n diff] f t ( ,lesage-,maxwell-,b>?oli4 ,he w\ld h be5 mu* m dismay$ /ill if he _h pu%$ ! calculation to ! end4 ,he w\ld h f\nd t ! temp]ature ( ! e>? w\ld h to increase #ajab degrees a second4 ,,iv ,conclusions ,i h /riv5 to give 9 few words an idea z complete z possible ( ^! new doctr9es2 ,i h s\ght to expla9 h[ !y took bir?2 o!rwise ! read] w\ld h _h gr\nd to be fright5$ by _! boldness4 ,! new !ories are n yet demon/rat$2 f> f x2 only !y re/ ^u an aggregate ( probabilities suffici5tly weighty = u n to h ! "r to treat !m ) disreg>d4 ,new exp]im5ts w d\btless tea* u :at we %\ld f9ally ?9k ( !m4 ,! knotty po9t ( ! "q lies 9 ,kaufmann's exp]im5t & ^? t may be und]tak5 to v]ify x4 ,9 conclusion1 p]mit me a ^w ( w>n+4 ,suppose t1 aft] "s ye>s1 ^! !ories und]go new te/s & triumph2 !n our second>y education w incur a great dang]2 c]ta9 pr(essors w d\btless wi% to make a place = ! new !ories4 ,novelties are s attractive1 & x is s h>d n to seem highly advanc$6 ,at lea/ !re w be ! wi% to op5 vi/as to ! pupils &1 be=e tea*+ !m ! ord9>y me*anics1 to let !m "k x has _h its day & 0 at be/ good en\gh = t old dolt ,laplace4 ,& !n !y w n =m ! habit ( ! ord9>y me*anics4 ,is x well to let !m "k ?is is only approximative8 ,yes2 b lat]1 :5 x has p5etrat$ to _! v m>r[1 :5 !y %all h tak5 ! b5t ( ?9k+ only ?r\gh x1 :5 !re %all no long] be risk ( _! unle>n+ x1 !n "o may1 )\t inconv5i5ce1 %[ !m its limits4 ,x is ) ! ord9>y me*anics t !y mu/ live2 ?is alone w !y ev] h to apply4 ,:atev] be ! progress ( automobilism1 our vehicles w nev] atta9 spe$s :]e x is n true4 ,! o!r is only a luxury1 & we %\ld ?9k ( ! luxury only :5 !re is no long] any risk ( h>m+ ! necess>y4 ,,book ,,iv ,,a/ronomic ,,sci5ce ,,*apt] ,i ,,! ,,milky ,,way ,,& ,,! ,,!ory ,,( ,,gases ,! consid]ations to be "h develop$ h sc>cely z yet drawn ! att5tion ( a/ronom]s2 !re is h>dly any?+ to cite except an +5i\s idea ( ,"l ,kelv9's1 :i* has op5$ a new field ( rese>*1 b /ill waits to be foll[$ out4 ,nor h ,i orig9al results to imp>t1 & all ,i c d is to give an idea ( ! problems pres5t$1 b :i* no "o hi!rto has und]tak5 to solve4 ,e "o kn[s h[ a l>ge numb] ( mod]n physici/s repres5t ! con/itution ( gases2 gases are =m$ ( an innum]able multitude ( molecules :i*1 at high spe$s1 cross & crisscross 9 e direction4 ,^! molecules probably act at a di/ance "o ^u ano!r1 b ?is action decreases v rapidly ) di/ance1 s t _! trajectories rema9 s5sibly /raight2 !y cease to be s only :5 two molecules happ5 to pass v ne> to ea* o!r2 9 ?is case1 _! mutual attraction or repulsion makes !m deviate to "r or left4 ,?is is :at is sometimes call$ an impact2 b ! ^w impact is n to be und]/ood 9 its usual s5se2 x is n necess>y t ! two molecules come into contact1 x suffices t !y approa* suffici5tly ne> ea* o!r = _! mutual attractions to become s5sible4 ,! laws ( ! deviation !y und]go are ! same z = a v]itable impact4 ,x seems at fir/ t ! disord]ly impacts ( ?is innum]able du/ c eng5d] only an inextricable *aos be=e :i* analysis mu/ recoil4 ,b ! law ( great numb]s1 t supreme law ( *ance1 comes to our aid2 9 pres5ce ( a semi-disord]1 we mu/ despair1 b 9 extreme disord]1 ?is /ati/ical law ree/abli%es a sort ( mean ord] :]e ! m9d c recov]4 ,x is ! /udy ( ?is mean ord] :i* con/itutes ! k9etic !ory ( gases2 x %[s u t ! velocities ( ! molecules are equally di/ribut$ among all ! directions1 t ! rapidity ( ^! velocities v>ies f "o molecule to ano!r1 b t ev5 ?is v>iation is subject to a law call$ ,maxwell's law4 ,?is law tells u h[ _m ( ! molecules move ) su* & su* a velocity4 ,z soon z ! gas dep>ts f ?is law1 ! mutual impacts ( ! molecules1 9 modify+ ! rapidity & direction ( _! velocities1 t5d to br+ x promptly back4 ,physici/s h /riv51 n )\t success1 to expla9 9 ?is way ! exp]im5tal prop]ties ( gases2 = example ,m>iotte's law4 ,consid] n[ ! milky way2 !re also we see an innum]able du/2 only ! gra9s ( ?is du/ are n atoms1 !y are />s2 ^! gra9s move also ) high velocities2 !y act at a di/ance "o ^u ano!r1 b ?is action is s slight at great di/ance t _! trajectories are /raight2 & yet1 f "t to "t1 two ( !m may approa* ne> en\gh to be deviat$ f _! pa?1 l a comet :i* has pass$ too ne> ,jupit]4 ,9 a ^w1 to ! eyes ( a giant = :om our suns w\ld be z = u our atoms1 ! milky way w\ld seem only a bubble ( gas4 ,su* 0 ,"l ,kelv9's lead+ idea4 ,:at may be drawn f ?is comp>ison8 ,9 h[ f> is x exact8 ,?is is :at we are to inve/igate toge!r2 b be=e rea*+ a def9ite conclusion1 & )\t wi%+ to prejudge x1 we =esee t ! k9etic !ory ( gases w be = ! a/ronom] a model he %\ld n foll[ bl9dly1 b f :i* he may advantage\sly draw inspiration4 ,up to ! pres5t1 cele/ial me*anics has attack$ only ! sol> sy/em or c]ta9 sy/ems ( d\ble />s4 ,be=e ! assemblage pres5t$ by ! milky way1 or ! agglom]ation ( />s1 or ! resolvable nebulae x recoils1 because x sees !re9 only *aos4 ,b ! milky way is n m complicat$ ?an a gas2 ! /ati/ical me?ods f\nd$ ^u ! calculus ( probabilities applicable to a gas are also applicable to x4 ,be=e all1 x is important to grasp ! resemblance ( ! two cases1 & _! diff]5ce4 ,"l ,kelv9 has /riv5 to det]m9e 9 ?is mann] ! dim5sions ( ! milky way2 = t we are r$uc$ to c\nt+ ! />s visible 9 our telescopes2 b we are n sure t beh9d ! />s we see1 !re are n o!rs we d n see2 s t :at we %\ld measure 9 ?is way w\ld n be ! size ( ! milky way1 x w\ld be ! range ( our in/rum5ts4 ,! new !ory comes to (f] u o!r res\rces4 ,9 fact1 we "k ! motions ( ! />s ne>e/ u1 & we c =m an idea ( ! rapidity & direction ( _! velocities4 ,if ! ideas above set =? are exact1 ^! velocities %\ld foll[ ,maxwell's law1 & _! mean value w tell u1 s to speak1 t :i* corresponds to ! temp]ature ( our fictiti\s gas4 ,b ?is temp]ature dep5ds itself ^u ! dim5sions ( our gas bubble4 ,9 fact1 h[ w a gase\s mass let loose 9 ! void act1 if its elem5ts attract "o ano!r accord+ to ,newton's law8 ,x w take a sph]ical =m2 moreov]1 because ( gravitation1 ! d5sity w be great] at ! c5t]1 ! pressure also w increase f ! surface to ! c5t] because ( ! weight ( ! out] p>ts drawn t[>d ! c5t]2 f9ally1 ! temp]ature w increase t[>d ! c5t]3 ! temp]ature & ! pressure be+ connect$ by ! law call$ adiabatic1 z happ5s 9 ! successive lay]s ( our atmosph]e4 ,at ! surface itself1 ! pressure w be null1 & x w be ! same ) ! absolute temp]ature1 t is to say ) ! velocity ( ! molecules4 ,a "q comes "h3 ,i h spok5 ( ! adiabatic law1 b ?is law is n ! same = all gases1 s9ce x dep5ds ^u ! ratio ( _! two specific heats2 = ! air & l gases1 ?is ratio is #a4db2 b is x to air t x is prop] to lik5 ! milky way8 ,evid5tly n1 x %\ld be reg>d$ z a mono-atomic gas1 l m]cury vapor1 l argon1 l helium1 t is to say t ! ratio ( ! specific heats %\ld be tak5 equal to #a4ff4 ,&1 9 fact1 "o ( our molecules w\ld be = example ! sol> sy/em2 b ! planets are v small p]sonages1 ! sun alone c\nts1 s t our molecule is inde$ mono-atomic4 ,& ev5 if we take a d\ble />1 x is probable t ! action ( a /range /> :i* might approa* x w\ld become suffici5tly s5sible to deviate ! motion ( g5]al translation ( ! sy/em mu* be=e be+ able to tr\ble ! relative orbits ( ! two compon5ts2 ! d\ble />1 9 a ^w1 w\ld act l an indivisible atom4 ,h[ev] t may be1 ! pressure1 & consequ5tly ! temp]ature1 at ! c5t] ( ! gase\s sph]e w\ld be by s mu* ! great] z ! sph]e 0 l>g] s9ce ! pressure increases by ! weight ( all ! sup]pos$ lay]s4 ,we may suppose t we are ne>ly at ! c5t] ( ! milky way1 & by obs]v+ ! mean prop] velocity ( ! />s1 we %all "k t :i* corresponds to ! c5tral temp]ature ( our gase\s sph]e & we %all det]m9e its radius4 ,we may get an idea ( ! result by ! foll[+ consid]ations3 make a simpl] hypo!sis3 ! milky way is sph]ical1 & 9 x ! masses are di/ribut$ 9 a homog5e\s mann]2 !nce results t ! />s 9 x describe ellipses hav+ ! same c5t]4 ,if we suppose ! velocity becomes no?+ at ! surface1 we may calculate ?is velocity at ! c5t] by ! equation ( vis viva4 ,?us we f9d t ?is velocity is proportional to ! radius ( ! sph]e & to ! squ>e root ( its d5sity4 ,if ! mass ( ?is sph]e 0 t ( ! sun & its radius t ( ! t]re/rial orbit1 ?is velocity w\ld be 7x is easy to see7 t ( ! e>? 9 its orbit4 ,b 9 ! case we h suppos$1 ! mass ( ! sun %\ld be di/ribut$ 9 a sph]e ( radius #a1jjj1jjj times great]1 ?is radius be+ ! di/ance ( ! ne>e/ />s2 ! d5sity is !re=e #ajah times less2 n[1 ! velocities are ( ! same ord]1 !re=e x is necess>y t ! radius be #aji times great]1 be #a1jjj times ! di/ance ( ! ne>e/ />s1 :i* w\ld give ab\t a ?\s& millions ( />s 9 ! milky way4 ,b y w say ^! hypo!sis diff] greatly f ! reality2 fir/1 ! milky way is n sph]ical & we %all soon return to ?is po9t1 & aga9 ! k9etic !ory ( gases is n compatible ) ! hypo!sis ( a homog5e\s sph]e4 ,b 9 mak+ ! exact calculation accord+ to ?is !ory1 we %\ld f9d a diff]5t result1 d\btless1 b ( ! same ord] ( magnitude2 n[ 9 su* a problem ! data are s unc]ta9 t ! ord] ( magnitude is ! sole end to be aim$ at4 ,& "h a fir/ rem>k pres5ts itself2 ,"l ,kelv9's result1 :i* ,i h obta9$ aga9 by an approximative calculation1 agrees s5sibly ) ! evaluations ! obs]v]s h made ) _! telescopes2 s t we mu/ conclude we are v ne> to pi]c+ ?r\gh ! milky way4 ,b t enables u to answ] ano!r "q4 ,!re are ! />s we see because !y %9e2 b may !re n be d>k />s circulat+ 9 ! int]/ell> spaces ^: exi/5ce might long rema9 unkn[n8 ,b !n1 :at ,"l ,kelv9's me?od w\ld give u w\ld be ! total numb] ( />s1 includ+ ! d>k />s2 z 8 figure is comp>able to t ! telescope gives1 ?is means !re is no d>k matt]1 or at lea/ n s mu* z ( %9+ matt]4 ,be=e go+ fur!r1 we mu/ look at ! problem f ano!r angle4 ,is ! milky way ?us con/itut$ truly ! image ( a gas prop]ly s call$8 ,y "k ,crookes has introduc$ ! notion ( a f\r? /ate ( matt]1 :]e gases hav+ become too r>efi$ are no long] true gases & become :at he calls radiant matt]4 ,consid]+ ! slight d5sity ( ! milky way1 is x ! image ( gase\s matt] or ( radiant matt]8 ,! consid]ation ( :at is call$ ! free pa? w furni% u ! answ]4 ,! trajectory ( a gase\s molecule may be reg>d$ z =m$ ( /raight segm5ts unit$ by v small arcs correspond+ to ! successive impacts4 ,! l5g? ( ea* ( ^! segm5ts is :at is call$ ! free pa?2 ( c\rse ?is l5g? is n ! same = all ! segm5ts & = all ! molecules2 b we may take a mean2 ?is is :at is call$ ! mean pa?4 ,?is is ! great] ! less ! d5sity ( ! gas4 ,! matt] w be radiant if ! mean pa? is great] ?an ! dim5sions ( ! receptacle :]e9 ! gas is enclos$1 s t a molecule has a *ance to g across ! :ole receptacle )\t und]go+ an impact2 if ! contr>y be ! case1 x is gase\s4 ,f ?is x foll[s t ! same fluid may be radiant 9 a little receptacle & gase\s 9 a big "o2 ?is p]haps is :y1 9 a ,crookes tube1 x is necess>y to make ! vacuum by s mu* ! m complete z ! tube is l>g]4 ,h[ is x !n = ! milky way8 ,?is is a mass ( gas ( :i* ! d5sity is v slight1 b ^: dim5sions are v great2 has a /> *ances ( trav]s+ x )\t und]go+ an impact1 t is to say )\t pass+ suffici5tly ne> ano!r /> to be s5sibly deviat$ f its r\te6 ,:at d we mean by suffici5tly ne>8 ,t is p]=ce a little arbitr>y2 take x z ! di/ance f ! sun to ,neptune1 :i* w\ld repres5t a deviation ( a doz5 degrees2 suppose !re=e ea* ( our />s surr\nd$ by a protective sph]e ( ?is radius2 c\ld a /raight pass betwe5 ^! sph]es8 ,at ! mean di/ance ( ! />s ( ! milky way1 ! radius ( ^! sph]es w be se5 "u an angle ( ab\t a t5? ( a second2 & we h a ?\s& millions ( />s4 ,put ^u ! cele/ial sph]e a ?\s& million little circles ( a t5? ( a second radius4 ,are ! *ances t ^! circles w cov] a great numb] ( times ! cele/ial sph]e8 ,f> f x2 !y w cov] only its sixte5 ?\s&? "p4 ,s ! milky way is n ! image ( gase\s matt]1 b ( ,crookes' radiant matt]4 ,nev]!less1 z our =ego+ conclusions are happily n at all precise1 we ne$ n s5sibly modify !m4 ,b !re is ano!r difficulty3 ! milky way is n sph]ical1 & we h reason$ hi!rto z if x 71 s9ce ?is is ! =m ( equilibrium a gas isolat$ 9 space w\ld take4 ,to make am5ds1 agglom]ations ( />s exi/ ^: =m is globul> & to :i* w\ld bett] apply :at we h hi!rto said4 ,h]s*el has already endeavor$ to expla9 _! rem>kable appe>ances4 ,he suppos$ ! />s ( ! aggregates uni=mly di/ribut$1 s t an assemblage is a homog5e\s sph]e2 ea* /> w\ld !n describe an ellipse & all ^! orbits w\ld be pass$ ov] 9 ! same "t1 s t at ! end ( a p]iod ! aggregate w\ld take aga9 its primitive configuration & ?is configuration w\ld be /able4 ,unluckily1 ! aggregates d n appe> to be homog5e\s2 we see a cond5sation at ! c5t]1 we %\ld obs]ve x ev5 7 ! sph]e homog5e\s1 s9ce x is ?ick] at ! c5t]2 b x w\ld n be s acc5tuat$4 ,we may !re=e r comp>e an aggregate to a gas 9 adiabatic equilibrium1 :i* takes ! sph]ical =m because ?is is ! figure ( equilibrium ( a gase\s mass4 ,b1 y w say1 ^! aggregates are mu* small] ?an ! milky way1 ( :i* !y ev5 9 probability make "p1 & ev5 ?\gh !y be m d5se1 !y w r pres5t some?+ analog\s to radiant matt]2 n[1 gases atta9 _! adiabatic equilibrium only ?r\gh innum]able impacts ( ! molecules4 ,t might p]haps be adju/$4 ,suppose ! />s ( ! aggregate h j en\gh en]gy = _! velocity to become null :5 !y rea* ! surface2 !n !y may trav]se ! aggregate )\t impact1 b arriv$ at ! surface !y w g back & w trav]se x anew2 aft] a great numb] ( cross+s1 !y w at la/ be deviat$ by an impact2 "u ^! conditions1 we %\ld /ill h a matt] :i* might be reg>d$ z gase\s2 if p]*ance !re _h be5 9 ! aggregate />s ^: velocity 0 great]1 !y h long gone away out ( x1 !y h left x nev] to return4 ,= all ^! reasons1 x w\ld be int]e/+ to exam9e ! kn[n aggregates1 to seek to acc\nt = ! law ( ! d5sities1 & to see if x is ! adiabatic law ( gases4 ,b to return to ! milky way2 x is n sph]ical & w\ld r be repres5t$ z a flatt5$ disc4 ,x is cle> !n t a mass />t+ )\t velocity f ! surface w rea* ! c5t] ) diff]5t velocities1 accord+ z x />ts f ! surface 9 ! neighborhood ( ! middle ( ! disc or j on ! bord] ( ! disc2 ! velocity w\ld be notably great] 9 ! latt] case4 ,n[1 up to ! pres5t1 we h suppos$ t ! prop] velocities ( ! />s1 ^? we obs]ve1 mu/ be comp>able to ^? :i* l masses w\ld atta92 ?is involves a c]ta9 difficulty4 ,we h giv5 above a value = ! dim5sions ( ! milky way1 & we h d$uc$ x f ! obs]v$ prop] velocities :i* are ( ! same ord] ( magnitude z t ( ! e>? 9 its orbit2 b :i* is ! dim5sion we h ?us measur$8 ,is x ! ?ickness8 ,is x ! radius ( ! disc8 ,x is d\btless some?+ int]m$iate2 b :at c we say !n ( ! ?ickness itself1 or ( ! radius ( ! disc8 ,data are lack+ to make ! calculation2 ,i %all conf9e myself to giv+ a glimpse ( ! possibility ( bas+ an evaluation at lea/ approximate ^u a deep] discussion ( ! prop] motions4 ,& !n we f9d ourselves fac+ two hypo!ses3 ei!r ! />s ( ! milky way are impell$ by velocities = ! mo/ "p p>allel to ! galactic plane1 b o!rwise di/ribut$ uni=mly 9 all directions p>allel to ?is plane4 ,if ?is be s1 obs]vation ( ! prop] motions %\ld %[ a prepond]ance ( compon5ts p>allel to ! milky way2 ?is is to be det]m9$1 because ,i d n "k :e!r a sy/ematic discussion has ev] be5 made f ?is view-po9t4 ,on ! o!r h&1 su* an equilibrium c\ld only be provisory1 s9ce because ( impacts ! molecules1 ,i mean ! />s1 w\ld 9 ! long run acquire notable velocities 9 ! s5se p]p5dicul> to ! milky way & w\ld end by sw]v+ f its plane1 s t ! sy/em w\ld t5d t[>d ! sph]ical =m1 ! only figure ( equilibrium ( an isolat$ gase\s mass4 ,or else ! :ole sy/em is impell$ by a common rotation1 & = t reason is flatt5$ l ! e>?1 l ,jupit]1 l all bodies t twirl4 ,only1 z ! flatt5+ is consid]able1 ! rotation mu/ be rapid2 rapid d\btless1 b x mu/ be und]/ood 9 :at s5se ?is ^w is us$4 ,! d5sity ( ! milky way is #ajbc times less ?an t ( ! sun2 a velocity ( rotation sqrt7#ajbe7 times less ?an t ( ! sun1 = x w\ld1 !re=e1 be ! equival5t s f> z conc]ns flatt5+2 a velocity #ajab times sl[] ?an t ( ! e>?1 say a ?irtie? ( a second ( arc 9 a c5tury1 w\ld be a v rapid rotation1 almo/ too rapid = /able equilibrium to be possible4 ,9 ?is hypo!sis1 ! obs]vable prop] motions w\ld appe> to u uni=mly di/ribut$1 & !re w\ld no long] be a prepond]ance ( compon5ts p>allel to ! galactic plane4 ,!y w tell u no?+ ab\t ! rotation itself1 s9ce we belong to ! turn+ sy/em4 ,if ! spiral nebul are o!r milky ways1 =eign to ours1 !y are n borne along 9 ?is rotation1 & we might /udy _! prop] motions4 ,x is true !y are v f> away2 if a nebula has ! dim5sions ( ! milky way & if its app>5t radius is = example #bj1 its di/ance is #aj1jjj times ! radius ( ! milky way4 ,b t makes no diff]5ce1 s9ce x is n ab\t ! translation ( our sy/em t we ask in=mation f !m1 b ab\t its rotation4 ,! fix$ />s1 by _! app>5t motion1 reveal to u ! diurnal rotation ( ! e>?1 ?\gh _! di/ance is imm5se4 ,unluckily1 ! possible rotation ( ! milky way1 h[ev] rapid x may be relatively1 is v sl[ view$ absolutely1 & besides ! po9t+s on nebul c n be v precise2 !re=e ?\s&s ( ye>s ( obs]vations w\ld be necess>y to le>n any?+4 ,h[ev] t may be1 9 ?is second hypo!sis1 ! figure ( ! milky way w\ld be a figure ( f9al equilibrium4 ,i %all n fur!r discuss ! relative value ( ^! two hypo!ses s9ce !re is a ?ird :i* is p]haps m probable4 ,we "k t among ! irresolvable nebul1 sev]al k9ds may be di/+ui%$3 ! irregul> nebul l t ( ,orion1 ! planet>y & annul> nebul1 ! spiral nebul4 ,! spectra ( ! fir/ two families h be5 det]m9$1 !y are discont9u\s2 ^! nebul are !re=e n =m$ ( />s2 besides1 _! di/ribution on ! heav5s seems to dep5d ^u ! milky way2 :e!r !y h a t5d5cy to g away f x1 or on ! contr>y to approa* x1 !y make !re=e a "p ( ! sy/em4 ,on ! o!r h&1 ! spiral nebul are g5]ally consid]$ z indep5d5t ( ! milky way2 x is suppos$ t !y1 l x1 are =m$ ( a multitude ( />s1 t !y are1 9 a ^w1 o!r milky ways v f> away f ours4 ,! rec5t inve/igations ( ,/raton(f t5d to make u reg>d ! milky way itself z a spiral nebula1 & ?is is ! ?ird hypo!sis ( :i* ,i wi% to speak4 ,h[ c we expla9 ! v s+ul> appe>ances pres5t$ by ! spiral nebul1 :i* are too regul> & too con/ant to be due to *ance8 ,fir/ ( all1 to take a look at "o ( ^! repres5tations is en\gh to see t ! mass is 9 rotation2 we may ev5 see :at ! s5se ( ! rotation is2 all ! spiral radii are curv$ 9 ! same s5se2 x is evid5t t ! mov+ w+ lags beh9d ! pivot & t fixes ! s5se ( ! rotation4 ,b ?is is n all2 x is evid5t t ^! nebul c n be lik5$ to a gas at re/1 nor ev5 to a gas 9 relative equilibrium "u ! sway ( a uni=m rotation2 !y are to be comp>$ to a gas 9 p]man5t motion 9 :i* int]nal curr5ts prevail4 ,suppose1 = example1 t ! rotation ( ! c5tral nucleus is rapid 7y "k :at ,i mean by ?is ^w71 too rapid = /able equilibrium2 !n at ! equator ! c5trifugal =ce w drive x away ov] ! attraction1 & ! />s w t5d to break away at ! equator & w =m div]g5t curr5ts2 b 9 go+ away1 z _! mom5t ( rotation rema9s con/ant1 :ile ! radius vector augm5ts1 _! angul> velocity w dim9i%1 & ?is is :y ! mov+ w+ seems to lag back4 ,f ?is po9t ( view1 !re w\ld n be a real p]man5t motion1 ! c5tral nucleus w\ld con/antly lose matt] :i* w\ld g out ( x nev] to return1 & w\ld dra9 away progressively4 ,b we may modify ! hypo!sis4 ,9 proportion z x goes away1 ! /> loses its velocity & ends by /opp+2 at ?is mom5t attraction rega9s possession ( x & leads x back t[>d ! nucleus2 s !re w be c5tripetal curr5ts4 ,we mu/ suppose ! c5tripetal curr5ts are ! fir/ rank & ! c5trifugal curr5ts ! second rank1 if we adopt ! comp>ison ) a troop 9 battle execut+ a *ange ( front2 &1 9 fact1 x is necess>y t ! composite c5trifugal =ce be comp5sat$ by ! attraction ex]cis$ by ! c5tral lay]s ( ! sw>m ^u ! extreme lay]s4 ,besides1 at ! end ( a c]ta9 "t a p]man5t rgime e/abli%es itself2 ! sw>m be+ curv$1 ! attraction ex]cis$ ^u ! pivot by ! mov+ w+ t5ds to sl[ up ! pivot & t ( ! pivot ^u ! mov+ w+ t5ds to accel]ate ! advance ( ?is w+ :i* no long] augm5ts its lag1 s t f9ally all ! radii end by turn+ ) a uni=m velocity4 ,we may /ill suppose t ! rotation ( ! nucleus is quick] ?an t ( ! radii4 ,a "q rema9s2 :y d ^! c5tripetal & c5trifugal sw>ms t5d to conc5trate !mselves 9 radii in/ead ( dissem9at+ !mselves a little ev]y:]e8 ,:y d ^! rays di/ribute !mselves regul>ly8 ,if ! sw>ms conc5trate !mselves1 x is because ( ! attraction ex]cis$ by ! already exi/+ sw>ms ^u ! />s :i* g out f ! nucleus 9 _! neighborhood4 ,aft] an inequality is produc$1 x t5ds to acc5tuate itself 9 ?is way4 ,:y d ! rays di/ribute !mselves regul>ly8 ,t is less obvi\s4 ,suppose !re is no rotation1 t all ! />s are 9 two planes at "r angles1 9 su* a way t _! di/ribution is symmetric ) reg>d to ^! two planes4 ,by symmetry1 !re w\ld be no reason = _! go+ out ( ^! planes1 nor = ! symmetry *ang+4 ,?is configuration w\ld give u !re=e equilibrium1 b ?is w\ld be an un/able equilibrium4 ,if on ! contr>y1 !re is rotation1 we %all f9d an analog\s configuration ( equilibrium ) f\r curv$ rays1 equal to ea* o!r & int]sect+ at #ij1 & if ! rotation is suffici5tly rapid1 ?is equilibrium is /able4 ,i am n 9 position to make ?is m precise3 en\gh if y see t ^! spiral =ms may p]haps "s day be expla9$ by only ! law ( gravitation & /ati/ical consid]ation recall+ ^? ( ! !ory ( gases4 ,:at has be5 said ( int]nal curr5ts %[s x is ( int]e/ to discuss sy/ematically ! aggregate ( prop] motions2 ?is may be done 9 a hundr$ ye>s1 :5 ! second edition is issu$ ( ! *>t ( ! heav5s & comp>$ ) ! fir/1 t we n[ are mak+4 ,b1 9 conclusion1 ,i wi% to call y\r att5tion to a "q1 t ( ! age ( ! milky way or ! nebul4 ,if :at we ?9k we see is confirm$1 we c get an idea ( x4 ,t sort ( /ati/ical equilibrium ( :i* gases give u ! model is e/abli%$ only 9 consequ5ce ( a great numb] ( impacts4 ,if ^! impacts are r>e1 x c come ab\t only aft] a v long "t2 if really ! milky way 7or at lea/ ! agglom]ations :i* are conta9$ 9 x71 if ! nebul h atta9$ ?is equilibrium1 ?is means !y are v old1 & we %all h an inf]ior limit ( _! age4 ,likewise we %\ld h ( x a sup]ior limit2 ?is equilibrium is n f9al & c n la/ always4 ,our spiral nebul w\ld be comp>able to gases impell$ by p]man5t motions2 b gases 9 motion are visc\s & _! velocities end by we>+ out4 ,:at "h corresponds to ! viscosity 7& :i* dep5ds ^u ! *ances ( impact ( ! molecules7 is excessively slight1 s t ! pres5t rgime may p]si/ dur+ an extremely long "t1 yet n =ev]1 s t our milky ways c n live et]nally nor become inf9itely old4 ,& ?is is n all4 ,consid] our atmosph]e3 at ! surface mu/ reign a temp]ature inf9itely small & ! velocity ( ! molecules !re is ne> z]o4 ,b ?is is a "q only ( ! mean velocity2 z a consequ5ce ( impacts1 "o ( ^! molecules may acquire 7r>ely1 x is true7 an enorm\s velocity1 & !n x w ru% out ( ! atmosph]e1 & once out1 x w nev] return2 !re=e our atmosph]e dra9s (f ?us ) extreme sl[ness4 ,! milky way also f "t to "t loses a /> by ! same me*anism1 & t likewise limits its duration4 ,well1 x is c]ta9 t if we compute 9 ?is mann] ! age ( ! milky way1 we %all get enorm\s figures4 ,b "h a difficulty pres5ts itself4 ,c]ta9 physici/s1 rely+ ^u o!r consid]ations1 reckon t suns c h only an ephem]al exi/5ce1 ab\t fifty million ye>s2 our m9imum w\ld be mu* great] ?an t4 ,mu/ we believe t ! evolution ( ! milky way began :5 ! matt] 0 /ill d>k8 ,b h[ h ! />s compos+ x rea*$ all at ! same "t adult age1 an age s briefly to endure8 ,or mu/ !y rea* !re all successively1 & are ^? we see only a feeble m9ority comp>$ ) ^? ext+ui%$ or :i* %all "o day light up8 ,b h[ reconcile t ) :at we h said above on ! abs5ce ( a notewor?y proportion ( d>k matt]8 ,%\ld we ab&on "o ( ! two hypo!ses1 & :i*8 ,i conf9e myself to po9t+ out ! difficulty )\t pret5d+ to solve x2 ,i %all end !re=e ) a big int]rogation po9t4 ,h[ev]1 x is int]e/+ to set problems1 ev5 :5 _! solution seems v f> away4 ,,*apt] ,,ii ,,fr5* ,,geodesy ,e "o und]/&s our int]e/ 9 kn[+ ! =m & dim5sions ( our e>?2 b "s p]sons w p]haps be surpris$ at ! exactitude s\ght aft]4 ,is ?is a useless luxury8 ,:at good are ! ef=ts s exp5d$ by ! geodesi/8 ,%\ld ?is "q be put to a congressman1 ,i suppose he w\ld say3 ,i am l$ to believe t geodesy is "o ( ! mo/ useful ( ! sci5ces2 because x is "o ( ^? co/+ u mo/ de>4 ,i %all try to give y an answ] a little m precise4 ,! great works ( art1 ^? ( peace z well z ^? ( w>1 are n to be und]tak5 )\t long /udies :i* save mu* grop+1 miscalculation & useless exp5se4 ,^! /udies c only be bas$ ^u a good map4 ,b a map w be only a valueless phantasy if con/ruct$ )\t bas+ x ^u a solid framework4 ,z well make /& a human body m9us ! skeleton4 ,n[1 ?is framework is giv5 u by geodesic measurem5ts2 s1 )\t geodesy1 no good map2 )\t a good map1 no great public works4 ,^! reasons w d\btless suffice to ju/ify mu* exp5se2 b ^! are argum5ts = practical m54 ,x is n ^u ^! t x is prop] to insi/ "h2 !re are o!rs high] &1 ev]y?+ consid]$1 m important4 ,s we %all put ! "q o!rwise2 c geodesy aid u ! bett] to "k nature8 ,does x make u und]/& its unity & h>mony8 ,9 reality an isolat$ fact is ( slight value1 & ! conque/s ( sci5ce are preci\s only if !y prep>e = new conque/s4 ,if !re=e a little hump 7 discov]$ on ! t]re/rial ellipsoid1 ?is discov]y w\ld be by itself ( no great int]e/4 ,on ! o!r h&1 x w\ld become preci\s if1 9 seek+ ! cause ( ?is hump1 we hop$ to p5etrate new secrets4 ,well1 :51 9 ! eighte5? c5tury1 ,maup]tuis & ,la ,condam9e brav$ su* opposite climates1 x 0 n solely to le>n ! %ape ( our planet1 x 0 a "q ( ! :ole _w-sy/em4 ,if ! e>? 0 flatt5$1 ,newton triumph$ & ) him ! doctr9e ( gravitation & ! :ole mod]n cele/ial me*anics4 ,& to-day1 a c5tury & a half aft] ! victory ( ! ,newtonians1 ?9k y geodesy has no?+ m to tea* u8 ,we "k n :at is )9 our globe4 ,! %afts ( m9es & bor+s h let u "k a lay] ( #a or #b kilomet]s ?ickness1 t is to say1 ! million? "p ( ! total mass2 b :at is b5ea?8 ,( all ! extraord9>y j\rneys dream$ by ,jules ,v]ne1 p]haps t to ! c5t] ( ! e>? took u to regions lea/ explor$4 ,b ^! deep-ly+ rocks we c n rea*1 ex]cise f af> _! attraction :i* op]ates ^u ! p5dulum & de=ms ! t]re/rial sph]oid4 ,geodesy c !re=e weigh !m f af>1 s to speak1 & tell u ( _! di/ribution4 ,?us w x make u really see ^? my/]i\s regions :i* ,jules ,v]ne only %[$ u 9 imag9ation4 ,?is is n an empty illusion4 ,m4 ,faye1 comp>+ all ! measurem5ts1 has rea*$ a result well calculat$ to surprise u4 ,"u ! oceans1 9 ! dep?s1 are rocks ( v great d5sity2 "u ! cont95ts1 on ! contr>y1 are empty spaces4 ,new obs]vations w modify p]haps ! details ( ^! conclusions4 ,9 any case1 our v5]at$ dean has %[n u :]e to se>* & :at ! geodesi/ may tea* ! geologi/1 desir\s ( kn[+ ! int]ior con/itution ( ! e>?1 & ev5 ! ?9k] wi%+ to speculate ^u ! pa/ & ! orig9 ( ?is planet4 ,& n[1 :y h ,i entitl$ ?is *apt] ,fr5* ,geodesy8 ,x is because1 9 ea* c\ntry1 ?is sci5ce has tak51 m ?an all o!rs1 p]haps1 a national "*4 ,x is easy to see :y4 ,!re mu/ be rivalry4 ,! sci5tific rivalries are always c\rte\s1 or at lea/ almo/ always2 9 any case1 !y are necess>y1 because !y are always fruitful4 ,well1 9 ^? ent]prises :i* require su* long ef=ts & s _m collaborators1 ! individual is effac$1 9 spite ( himself1 ( c\rse2 no "o has ! "r to say3 ?is is my "w4 ,!re=e x is n betwe5 m51 b betwe5 nations t rivalries g on4 ,s we are l$ to seek :at has be5 ! "p ( ,france4 ,h] "p ,i believe we are "r to be pr\d (4 ,at ! beg9n+ ( ! eighte5? c5tury1 long discussions arose betwe5 ! ,newtonians :o believ$ ! e>? flatt5$1 z ! !ory ( gravitation requires1 & ,cass9i1 :o1 deceiv$ by inexact measurem5ts1 believ$ our globe elongat$4 ,only direct obs]vation c\ld settle ! "q4 ,x 0 our ,academy ( ,sci5ces t und]took ?is task1 gigantic = ! epo*4 ,:ile ,maup]tuis & ,clairaut measur$ a degree ( m]idian "u ! pol> circle1 ,b\gu] & ,la ,condam9e w5t t[>d ! ,&es ,m\nta9s1 9 regions !n "u ,spa9 :i* to-day are ! ,republic ( ,ecuador4 ,our envoys 7 expos$ to great h>d%ips4 ,travel+ 0 n z easy z at pres5t4 ,truly1 ! c\ntry :]e ,maup]tuis op]at$ 0 n a des]t & he ev5 enjoy$1 x is said1 among ! ,lapl&]s ^? sweet satisfactions ( ! he>t t real arctic voyag]s nev] "k4 ,x 0 almo/ ! region :]e1 9 our days1 com=table /eam]s c>ry1 ea* summ]1 ho/s ( t\ri/s & "y ,engli% p4 ,b 9 ^? days ,cook's ag5cy did n exi/ & ,maup]tuis really believ$ he _h made a pol> exp$ition4 ,p]haps he 0 n altoge!r wrong4 ,! ,russians & ! ,sw$es c>ry out to-day analog\s measurem5ts at ,spitzb]g51 9 a c\ntry :]e !re is real ice-cap4 ,b !y h q o!r res\rces1 & ! diff]5ce ( "t makes up = t ( latitude4 ,! "n ( ,maup]tuis has rea*$ u mu* scrat*$ by ! claws ( ,doctor ,akakia2 ! sci5ti/ _h ! mis=tune to displease ,voltaire1 :o 0 !n ! k+ ( m9d4 ,he 0 fir/ prais$ beyond measure2 b ! flatt]ies ( k+s are z mu* to be dread$ z _! displeasure1 because ! days aft] are t]rible4 ,voltaire himself knew some?+ ( ?is4 ,voltaire call$ ,maup]tuis1 my amiable ma/] 9 ?9k+1 m>quis ( ! pol> circle1 de> flatt5] out ( ! _w & ,cass9i1 & ev51 flatt]y supreme1 ,sir ,isaac ,maup]tuis2 he wrote him3 ,only ! k+ ( ,prussia d ,i put on a level ) y2 he only lacks be+ a geomet]4 ,b soon ! sc5e *anges1 he no long] speaks ( deify+ him1 z 9 days ( yore ! ,argonauts1 or ( call+ d[n f ,olympus ! c\ncil ( ! gods to contemplate 8 works1 b ( *a9+ him up 9 a madh\se4 ,he speaks no long] ( 8 sublime m9d1 b ( 8 despotic pride1 plat$ ) v little sci5ce & mu* absurdity4 ,i c>e n to relate ^! comico-h]oic combats2 b p]mit me "s reflections on two ( ,voltaire's v]ses4 ,9 8 ',disc\rse on ,mod]ation' 7no "q ( mod]ation 9 praise & criticism71 ! poet has writt53 ,y h confirm$ 9 regions dre> ,:at ,newton disc]n$ )\t go+ abroad4 ,^! two v]ses 7:i* replace ! hyp]bolic praises ( ! fir/ p]iod7 are v unju/1 & d\btless ,voltaire 0 too enlight5$ n to "k x4 ,!n1 only ^? discov]ies 7 e/eem$ :i* c\ld be made )\t leav+ "o's h\se4 ,to-day1 x w\ld r be !ory t "o w\ld make light (4 ,?is is to misund]/& ! aim ( sci5ce4 ,is nature gov]n$ by caprice1 or does h>mony rule !re8 ,t is ! "q4 ,x is :5 x discloses to u ?is h>mony t sci5ce is beautiful & s wor?y to be cultivat$4 ,b :5ce c come to u ?is revelation1 if n f ! accord ( a !ory ) exp]im5t8 ,to seek :e!r ?is accord exi/s or if x fails1 ?is !re=e is our aim4 ,consequ5tly ^! two t]ms1 :i* we mu/ comp>e1 are z indisp5sable ! "o z ! o!r4 ,to neglect "o = ! o!r w\ld be nons5se4 ,isolat$1 !ory w\ld be empty1 exp]im5t w\ld be bl9d2 ea* w\ld be useless & )\t int]e/4 ,maup]tuis !re=e des]ves 8 %>e ( glory4 ,truly1 x w n equal t ( ,newton1 :o _h receiv$ ! sp>k div9e2 nor ev5 t ( 8 collaborator ,clairaut4 ,yet x is n to be despis$1 because 8 "w 0 necess>y1 & if ,france1 out/ripp$ by ,engl& 9 ! sev5te5? c5tury1 has s well tak5 h] rev5ge 9 ! c5tury foll[+1 x is n alone to ! g5ius ( ,clairauts1 d',alemb]ts1 ,laplaces t %e owes x2 x is also to ! long pati5ce ( ! ,maup]tuis & ! ,la ,condam9es4 ,we rea* :at may be call$ ! second h]oic p]iod ( geodesy4 ,france is torn )94 ,all ,europe is arm$ aga9/ h]2 x w\ld seem t ^! gigantic combats might absorb all h] =ces4 ,f> f x2 %e /ill has !m = ! s]vice ( sci5ce4 ,! m5 ( t "t recoil$ be=e no ent]prise1 !y 7 m5 ( fai?4 ,delambre & ,m*a9 7 commission$ to measure an arc go+ f ,dunk]que to ,b>celona4 ,?is "t !re 0 no go+ to ,lapl& or to ,p]u2 ! ho/ile squadrons _h clos$ to u ! ways ?i!r4 ,b1 ?\gh ! exp$itions are less di/ant1 ! epo* is s tr\bl$ t ! ob/acles1 ! p]ils ev51 are j z great4 ,9 ,france1 ,delambre _h to fight aga9/ ! ill-w ( suspici\s municipalities4 ,"o kn[s t ! /eeples1 :i* are visible f s f>1 & c be aim$ at ) precision1 (t5 s]ve z signal po9ts to geodesi/s4 ,b 9 ! region ,delambre trav]s$ !re 7 no long] any /eeples4 ,a c]ta9 proconsul _h pass$ !re1 & boa/$ ( knock+ d[n all ! /eeples ris+ pr\dly above ! humble abode ( ! sans-culottes4 ,pyramids !n 7 built ( planks & cov]$ ) :ite clo? to make !m m visible4 ,t 0 q ano!r ?+3 ) :ite clo?6 ,:at 0 ?is ra% p]son :o1 ^u our heights s rec5tly set free1 d>$ to raise ! hateful /&>d ( ! c\nt]-revolution8 ,x 0 necess>y to bord] ! :ite clo? ) blue & r$ b&s4 ,m*a9 op]at$ 9 ,spa92 ! difficulties 7 o!r2 b !y 7 n less4 ,! ,spani% peasants 7 ho/ile4 ,!re /eeples 7 n lack+3 b to in/all oneself 9 !m ) my/]i\s & p]haps diabolic in/rum5ts1 0 x n sacrilege8 ,! revolutioni/s 7 allies ( ,spa91 b allies smell+ a little ( ! /ake4 ,)\t cease1 writes ,m*a91 !y ?reat5 to but*] u4 ,=tunately1 ?anks to ! exhortations ( ! prie/s1 to ! pa/oral lett]s ( ! bi%ops1 ^! f]oci\s ,spani>ds cont5t$ !mselves ) ?reat5+4 ,"s ye>s aft] ,m*a9 made a second exp$ition into ,spa93 he propos$ to prolong ! m]idian f ,b>celona to ! ,bale>ics4 ,?is 0 ! fir/ "t x _h be5 attempt$ to make ! triangulations ov]pass a l>ge arm ( ! sea by obs]v+ signals in/all$ ^u "s high m\nta9 ( a f>-away isle4 ,! ent]prise 0 well conceiv$ & well prep>$2 x fail$ h[ev]4 ,! ,fr5* sci5ti/ enc\nt]$ all sorts ( difficulties ( :i* he compla9s bitt]ly 9 8 correspond5ce4 ,hell1 he writes1 p]haps ) "s exagg]ation--hell & all ! sc\rges x vomits ^u ! e>?1 tempe/s1 w>1 ! plague & black intrigues are !re=e un*a9$ aga9/ me6 ,! fact is t he enc\nt]$ among 8 collaborators m ( pr\d ob/9acy ?an ( good w & t a ?\s& accid5ts ret>d$ 8 "w4 ,! plague 0 no?+1 ! fe> ( ! plague 0 mu* m r$\btable2 all ^! isles 7 on _! gu>d aga9/ ! neighbor+ isles & fe>$ le/ !y %\ld receive ! sc\rge f !m4 ,m*a9 obta9$ p]mission to disemb>k only aft] long weeks ^u ! condition ( cov]+ all 8 pap]s ) v9eg>2 ?is 0 ! antisepsis ( t "t4 ,disgu/$ & sick1 he _h j ask$ to be recall$1 :5 he di$4 ,arago & ,biot x 0 :o _h ! honor ( tak+ up ! unf9i%$ "w & c>ry+ x on to completion4 ,?anks to ! support ( ! ,spani% gov]nm5t1 to ! protection ( sev]al bi%ops &1 above all1 to t ( a fam\s brig& *ief1 ! op]ations w5t rapidly =w>d4 ,!y 7 successfully complet$1 & ,biot _h return$ to ,france :5 ! /orm bur/4 ,x 0 ! mom5t :5 all ,spa9 took up arms to def5d h] indep5d5ce aga9/ ,france4 ,:y did ?is /rang] climb ! m\nta9s to make signals8 ,x 0 evid5tly to call ! ,fr5* army4 ,arago 0 able to escape ! populace only by becom+ a prison]4 ,9 8 prison1 8 only di/raction 0 read+ 9 ! ,spani% pap]s ! acc\nt ( 8 own execution4 ,! pap]s ( t "t sometimes gave out news prematurely4 ,he _h at lea/ ! consolation ( le>n+ t he di$ ) c\rage & l a ,*ri/ian4 ,ev5 ! prison 0 no long] safe2 he _h to escape & rea* ,algi]s4 ,!re1 he emb>k$ = ,m>seilles on an ,alg]ian vessel4 ,?is %ip 0 captur$ by a ,spani% corsair1 & behold ,arago c>ri$ back to ,spa9 & dragg$ f dungeon to dungeon1 9 ! mid/ ( v]m9 & 9 ! mo/ %ock+ wret*$ness4 ,if x _h only be5 a "q ( 8 subjects & 8 gue/s1 ! dey w\ld h said no?+4 ,b !re 7 on bo>d two lions1 a pres5t f ! ,african sov]eign to ,napoleon4 ,! dey ?reat5$ w>4 ,! vessel & ! prison]s 7 releas$4 ,! port %\ld h be5 prop]ly rea*$1 s9ce !y _h on bo>d an a/ronom]2 b ! a/ronom] 0 seasick1 & ! ,alg]ian seam51 :o wi%$ to make ,m>seilles1 came out at ,b\gie4 ,!nce ,arago w5t to ,algi]s1 trav]s+ ,kabylia on foot 9 ! mid/ ( a ?\s& p]ils4 ,he 0 long deta9$ 9 ,africa & ?reat5$ ) ! convict prison4 ,f9ally he 0 able to get back to ,france2 8 obs]vations1 :i* he _h pres]v$ & safegu>d$ "u 8 %irt1 &1 :at is /ill m rem>kable1 8 in/rum5ts _h trav]s$ unhurt ^! t]rible adv5tures4 ,up to ?is po9t1 n only did ,france hold ! =emo/ place1 b %e occupi$ ! /age almo/ alone4 ,9 ! ye>s :i* foll[1 %e has n be5 inactive & our /aff-(fice map is a model4 ,h[ev]1 ! new me?ods ( obs]vation & calculation h come to u above all f ,g]many & ,engl&4 ,x is only 9 ! la/ =ty ye>s t ,france has rega9$ h] rank4 ,%e owes x to a sci5tific (fic]1 ,g5]al ,p]ri]1 :o has successfully execut$ an ent]prise truly audaci\s1 ! junction ( ,spa9 & ,africa4 ,/ations 7 in/all$ on f\r peaks ^u ! two sides ( ! ,m$it]ranean4 ,= long mon?s !y await$ a calm & limpid atmosph]e4 ,at la/ 0 se5 ! little ?read ( light :i* _h trav]s$ #cjj kilomet]s ov] ! sea4 ,! und]tak+ _h succe$$4 ,to-day h be5 conceiv$ projects /ill m bold4 ,f a m\nta9 ne> ,nice w be s5t signals to ,corsica1 n n[ = geodesic det]m9ations1 b to measure ! velocity ( light4 ,! di/ance is only #bjj kilomet]s2 b ! ray ( light is to make ! j\rney !re & return1 aft] reflection by a mirror in/all$ 9 ,corsica4 ,& x %\ld n w&] on ! way1 = x mu/ return exactly to ! po9t ( dep>ture4 ,ev] s9ce1 ! activity ( ,fr5* geodesy has nev] slack5$4 ,we h no m su* a/oni%+ adv5tures to tell2 b ! sci5tific "w accompli%$ is imm5se4 ,! t]ritory ( ,france beyond ! sea1 l t ( ! "m c\ntry1 is cov]$ by triangles measur$ ) precision4 ,we h become m & m exact+ & :at our fa!rs admir$ does n satisfy u to-day4 ,b 9 proportion z we seek m exactitude1 ! difficulties greatly increase2 we are surr\nd$ by sn>es & mu/ be on our gu>d aga9/ a ?\s& unsuspect$ causes ( error4 ,x is ne$ful1 !re=e1 to create in/rum5ts m & m faultless4 ,"h aga9 ,france has n let h]self be di/anc$4 ,our appliances = ! measurem5t ( bases & angles leave no?+ to desire1 &1 ,i may also m5tion ! p5dulum ( ,colonel ,def=ges1 :i* enables u to det]m9e gravity ) a precision hi!rto unkn[n4 ,! future ( ,fr5* geodesy is at pres5t 9 ! h&s ( ! ,geographic ,s]vice ( ! army1 successively direct$ by ,g5]al ,bassot & ,g5]al ,b]?aut4 ,we c n suffici5tly congratulate ourselves ^u x4 ,= success 9 geodesy1 sci5tific aptitudes are n en\gh2 x is necess>y to be capable ( /&+ long fatigues 9 all sorts ( climates2 ! *ief mu/ be able to w9 ob$i5ce f 8 collaborators & to make ob$i5t 8 native auxili>ies4 ,^! are milit>y qualities4 ,besides1 "o kn[s t1 9 our army1 sci5ce has always m>*$ %\ld] to %\ld] ) c\rage4 ,i add t a milit>y organization assures ! indisp5sable unity ( action4 ,x w\ld be m difficult to reconcile ! rival pret5sions ( sci5ti/s jeal\s ( _! indep5d5ce1 solicit\s ( :at !y call _! fame1 & :o yet mu/ "w 9 conc]t1 ?\gh sep>at$ by great di/ances4 ,among ! geodesi/s ( =m] times !re 7 (t5 discussions1 ( :i* "s ar\s$ long e*oes4 ,! ,academy long res\nd$ ) ! qu>rel ( ,b\gu] & ,la ,condam9e4 ,i d n mean to say t soldi]s are exempt f passion1 b discipl9e imposes sil5ce ^u a too s5sitive self-e/eem4 ,sev]al =eign gov]nm5ts h call$ ^u our (fic]s to organize _! geodesic s]vice3 ?is is pro( t ! sci5tific influ5ce ( ,france abroad has n decl9$4 ,our hydrographic eng9e]s contribute also to ! common a*ievem5t a glori\s cont+5t4 ,! survey ( our coa/s1 ( our colonies1 ! /udy ( ! tides1 (f] !m a va/ doma9 ( rese>*4 ,f9ally ,i may m5tion ! g5]al level+ ( ,france :i* is c>ri$ out by ! +5i\s & precise me?ods ( ,m4 ,lallem&4 ,) su* m5 we are sure ( ! future4 ,moreov]1 "w = !m w n be lack+2 our colonial empire op5s = !m imm5se expanses illy explor$4 ,t is n all3 ! ,int]national ,geodetic ,association has recogniz$ ! necessity ( a new measurem5t ( ! arc ( ,quito1 det]m9$ 9 days ( yore by ,la ,condam9e4 ,x is ,france t has be5 *>g$ ) ?is op]ation2 %e _h e "r to x1 s9ce our ance/ors _h made1 s to speak1 ! sci5tific conque/ ( ! ,cordill]as4 ,besides1 ^! rights h n be5 conte/$ & our gov]nm5t has und]tak5 to ex]cise !m4 ,capta9s ,maura9 & ,lacombe complet$ a fir/ reconnaissance1 & ! rapidity ) :i* !y accompli%$ _! mission1 cross+ ! r\ghe/ regions & climb+ ! mo/ precipit\s summits1 is wor?y ( all praise4 ,x won ! admiration ( ,g5]al ,alf>o1 ,presid5t ( ! ,republic ( ,ecuador1 :o call$ !m 'los hombres de hi]ro1' ! m5 ( iron4 ,! f9al commission !n set out "u ! comm& ( ,lieut5ant-,colonel 7!n ,major7 ,b\rgeois4 ,! results obta9$ h ju/ifi$ ! hopes ent]ta9$4 ,b our (fic]s h enc\nt]$ un=ese5 difficulties due to ! climate4 ,m ?an once1 "o ( !m has be5 =c$ to rema9 sev]al mon?s at an altitude ( #d1jjj met]s1 9 ! cl\ds & ! sn[1 )\t see+ any?+ ( ! signals he _h to aim at & :i* refus$ to unmask !mselves4 ,b ?anks to _! p]sev]ance & c\rage1 !re result$ f ?is only a delay & an increase ( exp5se1 )\t ! exactitude ( ! measurem5ts suff]+ !refrom4 ,,g5]al ,,conclusions ,:at ,i h s\ght to expla9 9 ! prec$+ pages is h[ ! sci5ti/ %\ld guide himself 9 *oos+ among ! innum]able facts (f]$ to 8 curiosity1 s9ce inde$ ! natural limitations ( 8 m9d compel him to make a *oice1 ev5 ?\gh a *oice be always a sacrifice4 ,i h exp\nd$ x fir/ by g5]al consid]ations1 recall+ on ! "o h& ! nature ( ! problem to be solv$ & on ! o!r h& seek+ to bett] compreh5d t ( ! human m9d1 :i* is ! pr9cipal in/rum5t ( ! solution4 ,i !n h expla9$ x by examples2 ,i h n multipli$ !m indef9itely2 ,i also h _h to make a *oice1 & ,i h *os5 naturally ! que/ions ,i _h /udi$ mo/4 ,o!rs w\ld d\btless h made a diff]5t *oice2 b :at diff]5ce1 because ,i believe !y w\ld h rea*$ ! same conclusions4 ,!re is a hi]>*y ( facts2 "s h no rea*2 !y tea* u no?+ b !mselves4 ,! sci5ti/ :o has asc]ta9$ !m has le>n$ no?+ b a fact1 & has n become m capable ( =esee+ new facts4 ,su* facts1 x seems1 come once1 b are n de/9$ to reappe>4 ,!re are1 on ! o!r h&1 facts ( great yield2 ea* ( !m tea*es u a new law4 ,& s9ce a *oice mu/ be made1 x is to ^! t ! sci5ti/ %\ld devote himself4 ,d\btless ?is classification is relative & dep5ds ^u ! weakness ( our m9d4 ,! facts ( slight outcome are ! complex facts1 ^u :i* v>i\s circum/ances may ex]cise a s5sible influ5ce1 circum/ances too num]\s & too div]se = u to disc]n !m all4 ,b ,i %\ld r say t ^! are ! facts we ?9k complex1 s9ce ! intricacy ( ^! circum/ances surpasses ! range ( our m9d4 ,d\btless a m9d va/] & f9] ?an ours w\ld ?9k diff]5tly ( !m4 ,b :at matt]2 we c n use t sup]ior m9d1 b only our own4 ,! facts ( great outcome are ^? we ?9k simple2 may be !y really are s1 because !y are influ5c$ only by a small numb] ( well-def9$ circum/ances1 may be !y take on an appe>ance ( simplicity because ! v>i\s circum/ances ^u :i* !y dep5d obey ! laws ( *ance & s come to mutually comp5sate4 ,& ?is is :at happ5s mo/ (t54 ,& s we h be5 oblig$ to exam9e some:at m closely :at *ance is4 ,facts :]e ! laws ( *ance apply become easy ( access to ! sci5ti/ :o w\ld be disc\rag$ be=e ! extraord9>y complication ( ! problems :]e ^! laws are n applicable4 ,we h se5 t ^! consid]ations apply n only to ! physical sci5ces1 b to ! ma!matical sci5ces4 ,! me?od ( demon/ration is n ! same = ! physici/ & ! ma!matician4 ,b ! me?ods ( inv5tion are v mu* alike4 ,9 bo? cases !y consi/ 9 pass+ up f ! fact to ! law1 & 9 f9d+ ! facts capable ( lead+ to a law4 ,to br+ out ?is po9t1 ,i h %[n ! m9d ( ! ma!matician at "w1 & "u ?ree =ms3 ! m9d ( ! ma!matical inv5tor & creator2 t ( ! unconsci\s geomet] :o among our f> di/ant ance/ors1 or 9 ! mi/y ye>s ( our infancy1 has con/ruct$ = u our in/9ctive notion ( space2 t ( ! adolesc5t to :om ! tea*]s ( second>y education unveil ! fir/ pr9ciples ( ! sci5ce1 seek+ to give und]/&+ ( ! fundam5tal def9itions4 ,ev]y:]e we h se5 ! rle ( intuition & ( ! _s ( g5]alization )\t :i* ^! ?ree /ages ( ma!maticians1 if ,i may s express myself1 w\ld be r$uc$ to an equal impot5ce4 ,& 9 ! demon/ration itself1 ! logic is n all2 ! true ma!matical reason+ is a v]itable induction1 diff]5t 9 _m reg>ds f ! induction ( physics1 b proce$+ l x f ! p>ticul> to ! g5]al4 ,all ! ef=ts t h be5 made to rev]se ?is ord] & to c>ry back ma!matical induction to ! rules ( logic h ev5tuat$ only 9 failures1 illy conceal$ by ! employm5t ( a language inaccessible to ! un9itiat$4 ,! examples ,i h tak5 f ! physical sci5ces h %[n u v diff]5t cases ( facts ( great outcome4 ,an exp]im5t ( ,kaufmann on radium rays revolutionizes at ! same "t me*anics1 optics & a/ronomy4 ,:y8 ,because 9 proportion z ^! sci5ces h develop$1 we h ! bett] recogniz$ ! bonds unit+ !m1 & !n we h p]ceiv$ a species ( g5]al design ( ! *>t ( univ]sal sci5ce4 ,!re are facts common to sev]al sci5ces1 :i* seem ! common s\rce ( /reams div]g+ 9 all directions & :i* are comp>able to t knoll ( ,sa9t ,go?>d :5ce spr+ wat]s :i* f]tilize f\r diff]5t valleys4 ,& !n we c make *oice ( facts ) m disc]nm5t ?an our pr$ecessors :o reg>d$ ^! valleys z di/9ct & sep>at$ by impassable b>ri]s4 ,x is always simple facts :i* mu/ be *os51 b among ^! simple facts we mu/ pref] ^? :i* are situat$ ^u ^! sorts ( knolls ( ,sa9t ,go?>d ( :i* ,i h j spok54 ,& :5 sci5ces h no direct bond1 !y /ill mutually ?r[ light ^u "o ano!r by analogy4 ,:5 we /udi$ ! laws obey$ by gases we knew we _h attack$ a fact ( great outcome2 & yet ?is outcome 0 /ill e/imat$ b5ea? its value1 s9ce gases are1 f a c]ta9 po9t ( view1 ! image ( ! milky way1 & ^? facts :i* seem$ ( int]e/ only = ! physici/1 ere long op5$ new vi/as to a/ronomy q unexpect$4 ,& f9ally :5 ! geodesi/ sees x is necess>y to move 8 telescope "s seconds to see a signal he has set up ) great pa9s1 ?is is a v small fact2 b ?is is a fact ( great outcome1 n only because ?is reveals to him ! exi/5ce ( a small protub]ance ^u ! t]re/rial globe1 t little hump w\ld be by itself ( no great int]e/1 b because ?is protub]ance gives him in=mation ab\t ! di/ribution ( matt] 9 ! int]ior ( ! globe1 & ?r\gh t ab\t ! pa/ ( our planet1 ab\t its future1 ab\t ! laws ( its developm5t4 99 99 99 99 99 ,,index ab]ration ( light1 #cae1 #dif ,abraham1 #caa1 #dij-a1 #eje-g1 #eji1 #eae-f absolute motion1 #ajg ori5tation1 #hc space1 #he1 #ic1 #bdf1 #beg1 #cec accel]ation1 #id1 #ih1 #dhf1 #eji accid5tal con/ant1 #aab errors1 #aga1 #djb accommodation ( ! eyes1 #fg-h action at a di/ance1 #acg addition1 #cd aim ( ma!matics1 #bhj al*emi/s1 #aa ,alf>o1 #edc algebra1 #cgi analogy1 #bbj analysis1 #bah-i1 #bgi analysis situs1 #ec1 #bci1 #cha analy/1 #baj1 #bba ance/ral exp]i5ce1 #ia ,&rade1 #ic1 #ajd1 #bbh ,&rews1 #aec angle sum ( triangle1 #eh ,anglo-,saxons1 #c ant9omies1 #ddi1 #deg1 #dgg ,arago1 #edj-a ,ari/otle1 #bje1 #bib1 #dfj ari?metic1 #cd1 #cgi1 #dda1 #dfc associativity1 #ce assumptions1 #dea1 #dec a/ronomy1 #ha1 #bhi1 #cae1 #eab ,atwood1 #ddf axiom1 #fj1 #fc1 #fe1 #bae ,bacon1 #abh ,b>?oli1 #ejc ,bassot1 #edb beauty1 #cdi1 #cfh ,becqu]el1 #cab ,beltrami1 #ef1 #eh ,b]gson1 #cba ,b]keley1 #d ,b]?aut1 #edb ,b]tr&1 #aef1 #aij1 #baa1 #cie ,betti1 #bci ,biot1 #edj bodies1 solid1 #gb ,boltzmann1 #cjd ,bolyai1 #ef1 #bja1 #bjc ,borel1 #dhb ,b\gu]1 #ecg1 #edb ,b\rgeois1 #edc ,b\tr\x1 #cij1 #dfd ,bradley1 #dif ,briot1 #bih ,br[nian movem5t1 #aeb1 #daj ,bu*]]1 #ejg ,burali-,=ti1 #deg-i1 #dgg1 #dha-b ,ca51 #chg-h ,cal9on1 #bbh canal rays1 #dia-b canals1 semicircul>1 #bgf ,cantor1 #aa1 #ddh-i1 #deg1 #dei1 #dgg ,cantorism1 #cha1 #chb1 #dhj1 #dhd capill>ity1 #bih ,c>lyle1 #abh ,c>not's pr9ciple1 #adc1 #aea1 #cjj1 #cjc-e1 #cii ,cass9i1 #ecg ca?ode rays1 #dhg-ib cells1 #bag c5t] ( gravity1 #ajc c5tral =ces1 #big ,*aldeans1 #bij *ance1 #cie1 #djh *ange ( position1 #gj /ate1 #gj *emi/ry ( ! />s1 #bie circle-squ>]s1 #aa ,clairaut1 #ecg-h ,clausius1 #aai1 #abc1 #adc color s5sation1 #beb ,columbus1 #bbh commutativity1 #ce-f comp5sation1 #gb complete induction1 #dj ,comte1 #bid ,condorcet1 #daa cont+5ce1 #cdj cont9uity1 #agc cont9uum1 #dc amorph\s1 #bch ma!matical1 #df physical1 #df1 #bdj tridim5sional1 #bdj conv5tion1 #ej1 #ic1 #ajf1 #abe1 #agc1 #bjh1 #cag1 #ddj1 #dea conv]g5ce1 #fg-h coord9ates1 #bdd ,cop]nicus1 #aji1 #bia1 #ced ,c\lomb1 #adc1 #eaf ,c\turat1 #dej1 #dec1 #def1 #dfj1 #dfb-c1 #dfg1 #dgb-f creation1 ma!matical1 #chc cre$1 #a ,crmieu1 #afh-i1 #dij crisis1 #cjc ,crookes1 #aie1 #dhh1 #ebg-h crude fact1 #cbf1 #ccj ,curie1 #cab-c1 #cah curr5t1 #ahf curvature1 #eh-i curve1 #bac1 #cdf curves )\t tang5ts1 #ea cut1 #eb1 #bef cyclones1 #cec d',alemb]t1 #ech ,d>w91 #eah-i ,de ,cyon1 #bgf1 #dbg ,d$ek9d1 #dd-e ,def=ges1 #edb def9itions1 #dcj1 #dec de=mation1 #gc1 #dae ,delage1 #bgg ,delambre1 #eci ,delbeuf1 #dad ,desc>tes1 #abg det]m9ism1 #abc1 #cdj diction>y1 #ei didymium1 #ccc dilatation1 #gf dim5sions1 #ec1 #fh1 #gh1 #bda1 #bef1 #dbf direction1 #fi ,diri*let1 #bac disp]sion1 #ada displacem5t1 #gc1 #gg1 #bdg1 #bef di/ance1 #ei1 #bib di/ributivity1 #cf ,du ,bois-,reymond1 #ej e>?1 rotation (1 #cbf1 #cec eclipse1 #cbf electricity1 #agd electrifi$ bodies1 #aag electrodynamic attraction1 #cjh induction1 #ahh mass1 #caa electrodynamics1 #ahd1 #bhb electromagnetic !ory ( light1 #cja electrons1 #caf1 #dib-d1 #eje-h1 #eaj1 #eab-d elephant1 #bag1 #dcf ellipse1 #bae ,em]son1 #bjc empiricism1 #hf1 #bga ,epim5ides1 #dgh-i equation ( ,laplace1 #bhc ,erdly1 #bjc errors1 accid5tal1 #aga1 #djb law (1 #aai sy/ematic1 #aga1 #djb !ory (1 #djb1 #djf e!r1 #ade1 #cea e?ics1 #bje ,euclid1 #fb1 #hf1 #bjb-c1 #bac ,euclidean geometry1 #fe1 #bce-f1 #ccg ,euclid's po/ulate1 #hc1 #ia1 #abd1 #cec1 #ddc1 #dec1 #dfh1 #dgj-a exp]i5ce1 #ij-a exp]im5t1 #abg1 #cag1 #ccf1 #ddf fact1 crude1 #cbf1 #ccj 9 ! r\gh1 #cbg sci5tific1 #cbf facts1 #cfb1 #cga ,fahr5heit1 #bch ,f>aday1 #aej1 #aib ,faye1 #ecf ,fe*n]1 #df1 #eb ,fehr1 #chc f9ite1 #eg ,fitzg]ald1 #dae-f1 #ejj-a1 #eje ,fizeau1 #adf1 #adi1 #cji1 #dih1 #ejd ,flamm>ion1 #djj1 #djf-g flatt5+ ( ! e>?1 #cec =ce1 #gb1 #ih1 #ddd direction (1 #dde -fl[1 #bhd =ces1 c5tral1 #big equival5ce (1 #dde magnitude (1 #dde ,f\cault's p5dulum1 #he1 #aji1 #cec f\r dim5sions1 #gh ,f\ri]1 #bih-i ,f\ri]'s problem1 #cag s]ies1 #bhf ,frankl91 #eac-d ,fresnel1 #acb1 #adj1 #aec1 #agd1 #agf1 #aha1 #cea1 #dih ,fu*sian1 #chg-h function1 #bac cont9u\s1 #bah1 #bhh ,galileo1 #ig1 #cca1 #cec-d gase\s pressure1 #ada gases1 !ory (1 #djj1 #dje1 #ebc ,gauss1 #chd-e1 #djf ,gay-,lussac1 #aeg g5]alize1 #cdb geodesy1 #ece geomet]1 #hc1 #baj1 #dch geometric space1 #ff geometry1 #gb1 #ha1 #abe1 #bjg1 #chj1 #dbh1 #ddb1 #dfg ,euclidean1 #fe1 #ic f\r?1 #fb non-,euclidean1 #ee projective1 #bja qualitative1 #bch rational1 #e1 #dfg ,riemann's1 #eg sph]ic1 #ei ,gibbs1 #cjd ,gold/e91 #dib ,g\y1 #aeb1 #cje1 #daj gravitation1 #eab ,greeks1 #ic1 #cfh ,hadam>d1 #dei ,hal/$1 #c1 #bjc1 #dfd1 #dfg ,hamilton1 #aae helium1 #bid ,helmholtz1 #ef1 #aae1 #aah1 #ada1 #aij1 #aif ,h]cules1 #ddi ,h]mite1 #baa1 #bbj1 #bbb1 #bhe ,h]s*el1 #ebh ,h]tz1 #ajb1 #ade1 #aid-e1 #dbg1 #dhh1 #dih1 #ejb1 #ejd1 #eaj ,h]tzian oscillator1 #cji1 #cag ,hilb]t1 #e1 #aa1 #bjc1 #dcc1 #dej-a1 #dfd-h1 #dga1 #dge-g1 #dhd ,him/$t1 #aie ,hipp>*us1 #bia homog5eity1 #gd1 #dbc homog5e\s1 #fg hydrodynamics1 #bhd hyp]bola1 #bae hypo!ses1 #f1 #ae1 #abg1 #acc hy/]esis1 #aea id5tity ( spaces1 #bfh ( two po9ts1 #bei illusions1 optical1 #bjb incomm5surable numb]s1 #dd induction1 complete1 #dj1 #deb-c1 #dfg-h electromagnetic1 #ahh ma!matical1 #dj1 #bbj pr9ciple (1 #dha in]tia1 #ic1 #dhf1 #dhi1 #ejg inf9ite1 #ddh inf9itesimals1 #ej inquisitor1 #cca integration1 #aci int]polation1 #aca intuition1 #baj1 #bac1 #bae inv>iant1 #ccc ,ionians1 #abg ions1 #aeb irrational numb]1 #dd irrev]sible ph5om5a1 #aea isotropic1 #fg ,japanese mice1 #bgg1 #dbg ,jevons1 #dea ,john ,lackl&1 #abh ,jules ,v]ne1 #aaa1 #ecf ,jupit]1 #aca1 #aeg1 #bca1 #bhi ,kant1 #af1 #fd1 #bjb-c1 #dej-a1 #dga ,kauffman1 #caa1 #dij-a1 #die1 #ejf-g1 #ebb1 #ede ,kazan1 #bjc ,kelv91 #ade1 #ebc-d1 #ebf-g ,kepl]1 #abj1 #acc1 #aec1 #bhb1 #bia-b ,kepl]'s laws1 #acf1 #eaf k9ematics1 #ccg k9etic en]gy1 #aaf !ory ( gases1 #ada ,kir*h(f1 #ih-i1 #ajc-e ,kle91 #fj1 #baa1 #bhg k1 #bja ,knig1 #add1 #dgg ,kovalevski1 #bab1 #bhf ,kroneck]1 #dd ,lacombe1 #edc ,la ,condam9e1 #ece1 #ecg-h1 #edb-c ,lagrange1 #ih1 #aea1 #agi ,laisant1 #chc ,lallam&1 #edc ,langev91 #eji ,laplace1 #bih1 #cih1 #ead-e1 #eah1 #ebb1 #ech ,laplace's equation1 #bhc1 #bhg ,l>mor1 #ade1 #aej ,lavoisi]'s pr9ciple1 #cja1 #caj1 #cab law1 #bjg1 #bia1 #cie ,leibnitz1 #cb1 #dej1 #dga ,le ,roy1 #bh1 #cba-f1 #ccb1 #cce1 #ccg1 #cdg-h1 #ced1 #dfh ,lesage1 #eag-ba ,li>d1 #ddj ,lie1 #fb-c1 #bab light s5sations1 #beb !ory (1 #cea velocity (1 #bcb1 #cab ,l9demann1 #ejh l9e1 #bjc1 #bdc l9kages1 #add ,lippmann1 #aif ,loba*evski1 #bi1 #ef1 #fj1 #fb1 #hc1 #hf1 #bjc ,loba*evski's space1 #bci local "t1 #cjf-g1 #dii logic1 #bad1 #dce1 #ddh1 #dfj-b1 #dfd logi/ic1 #deg1 #dgb-d logi/icians1 #dgb ,lor5tz1 #adg1 #adi1 #aif-g1 #cjf1 #cjh1 #caa1 #cae1 #dae-f1 #dib1 #dih-ejb1 #ejd-i1 #eab1 #ead-f1 #eba ,lotze1 #bfd luck1 #cii ,lum51 #djg-h ,mac,cullagh1 #aej ,ma*1 #cge ,ma*-,delage1 #bgf magnetism1 #adi magnitude1 #di ,m>iotte's law1 #abj1 #acb1 #aeg1 #cdb1 #ebd ,m>os1 #bjc mass1 #ih1 #cab1 #ddf1 #dhf1 #dhi1 #did1 #eae ma!matical analysis1 #bah cont9uum1 #df creation1 #chc induction1 #dj1 #bbj physics1 #acf1 #big1 #cai ma!matics1 #cfi1 #ddh matt]1 #dib ,maup]tuis1 #ece1 #ecg-h ,maura91 #edc ,maxwell1 #adj1 #aeb1 #age1 #agg1 #aha1 #aic1 #bhb-c1 #bih1 #cja1 #cjd-e1 #cea1 #ejc1 #ebd-e ,maxwell-,b>?oli1 #cji1 #ejc-d1 #eai1 #eba ,may]1 #aai1 #abc1 #cjj1 #cab1 #cah measurem5t1 #di ,m*a91 #eci-dj me*anical explanation1 #agg mass1 #cab me*anics1 #ib1 #ddd1 #dhf1 #dif1 #eab an?ropomorphic1 #ajc cele/ial1 #bgi /ati/ical1 #cjd ,mray1 #baa metaphysician1 #bba meteorology1 #cih mice1 #bgg ,mi*elson1 #cjf1 #cji1 #caa1 #caf1 #dih1 #ejj-a milky way1 #ebc-cj ,mill1 ,/u>t1 #fj-a1 #dec-d ,moni/1 #d1 #hi1 #dfd moons ( ,jupit]1 #bcc ,morley1 #cji motion ( liquids1 #bhc ( moon1 #bh ( planets1 #cda relative1 #ajg1 #dhg )\t de=mation1 #bcf multiplication1 #cf muscul> s5sations1 #fi ,nagaoka1 #cag nature1 #abg navigation1 #bhi neodymium1 #ccc neomonics1 #bhc ,neumann1 #aha ,newton1 #he1 #if1 #ih1 #aji1 #aec1 #bia1 #cgj1 #dhf1 #eaf1 #ecf1 #ech ,newton's argum5t1 #ajh1 #ccd1 #cdc law1 #aaa1 #aah1 #acb1 #acf1 #adi1 #aeg1 #bcc1 #bhb1 #bib1 #eab1 #ead-e1 #eah1 #ebe pr9ciple1 #adf1 #cjj1 #cjh-i1 #cab no-class !ory1 #dgh nom9alism1 #bh1 #abe1 #cba1 #ccc1 #cce non-,euclidean geometry1 #ee1 #ei1 #chh language1 #abg space1 #ee1 #bce1 #bcg /raight1 #bcf1 #dgj _w1 #ge numb]1 #ca big1 #hh imag9>y1 #bhc incomm5surable1 #dd transf9ite1 #ddh-i :ole1 #dd1 #dfi objectivity1 #bji1 #cdg1 #cdi1 #djh optical illusions1 #bjb optics1 #agd1 #dif orbit ( ,saturn1 #cda ord]1 #che ori5tation1 #hc osmotic1 #ada ,padoa1 #dfc ,pan?on1 #dad p>allax1 #dgj p>allels1 #ef1 #ddc ,p>is "t1 #bcc p>ry1 #dai-bb1 #dbg p>tition1 #de pasigraphy1 #def-g ,pa/eur1 #abh ,peano1 #dej1 #def-i1 #dfc1 #dgb ,p5d]1 #dij p5dulum1 #bbd ,p]ri]1 #eda ,p]r91 #aie phosphorus1 #ccc1 #dfh1 #dgj-a physical cont9uum1 #df physics1 #abg1 #adj1 #add1 #bgi1 #big physics ( c5tral =ces1 #big ( ! pr9ciples1 #bii ,pi]i1 #aa1 #bjc ,plato1 #bib ,po9c>1 #dgc po9t1 #hi1 #bdd ,poncelet1 #bae po/ulates1 #chb pot5tial en]gy1 #aaf praseodymium1 #ccc pr9ciple1 #abe1 #bii ,c>not's1 #adc1 #aea1 #cjj1 #cjc-e1 #cii ,clausius'1 #aai1 #abc1 #adc ,hamilton's1 #aae ,lavoisi]'s1 #cjj1 #caj ,may]'s1 #aai1 #aba1 #abc1 #cjj1 #cab1 #cah ,newton's1 #adf1 #cjj1 #cjh-i1 #cab ( action & reaction1 #cjj1 #dhg1 #ejb ( cons]vation ( en]gy1 #cjj ( degradation ( en]gy1 #cjj ( in]tia1 #ic1 #dhf1 #ejg ( lea/ action1 #aah1 #cjj ( relativity1 #cjj1 #cje1 #dih1 #eje ,prony1 #dde psy*ologi/1 #chc ,ptolemy1 #aaj1 #bia1 #cec-d ,py?agoras1 #bib quadrature ( ! circle1 #afa qualitative geometry1 #bch space1 #bjg "t1 #bbd quat]nions1 #bhb radiomet]1 #ejc radium1 #cab1 #cah1 #dhf-g ,rados1 #bja ,ramsay1 #cac rational geometry1 #e1 #dfg reaction1 #ejb reality1 #bag1 #cdj1 #cdi ,raumur1 #bch recurr5ce1 #cg ,regnault1 #agj relativity1 #hc1 #cje1 #dag1 #dbc1 #dih1 #eje ,ri*>d1 #dgg-h1 #dhj-a ,riemann1 #ef1 #fb1 #ade1 #bab1 #bci1 #bdc1 #cha1 #dcb surface1 #baa1 #bhg ,roem]1 #bcc ,rntg51 #eaa1 #ebj rotation ( e>?1 #bbe1 #cca1 #cec r\lette1 #djc ,r[l&1 #aid-g1 #cje1 #dhi ,royce1 #bjb ,russell1 #bja1 #dej1 #dfj-b1 #dfd-g1 #dga-d1 #dgg-hb1 #dhd-e ,/4 ,l\is exposition1 #bjh1 #cbj ,s>cey1 #ddb ,saturn1 #bca1 #cag ,s*ill]1 #bjb ,s*liemann1 #ai sci5ce1 #bje1 #cba1 #cbc1 #cdj1 #ced ,sci5ce & ,hypo!sis1 #bje-g1 #bbj1 #bdj1 #bdf-g1 #cai1 #cec1 #deb semicircul> canals1 #bgf s]ies1 developm5t 91 #bhg ,f\ri]'s1 #bhf ,sirius1 #bbf1 #bbi solid bodies1 #gb space1 #ee1 #ff1 #hi1 #bce1 #bef absolute1 #he1 #ic amorph\s1 #dag ,bolyai1 #ef ,euclidean1 #fe geometric1 #ff ,loba*evski's1 #bci motor1 #fi non-,euclidean1 #ee1 #bce1 #bcg ( f\r dim5sions1 #gh p]ceptual1 #ff1 #fi tactile1 #fh1 #bfd visual1 #fg1 #beb spectra1 #caf spectroscope1 #bid ,sp5c]1 #i sponge1 #bai ,/allo1 #aj />s1 #bib /ati/ical me*anics1 #cjd /raight1 #fb1 #hb1 #bcf1 #dcc1 #dej1 #dgj ,/raton(f1 #eca surfaces1 #eh sy/ematic errors1 #aga tactile space1 #fh1 #bfd ,tait1 #ih tang5t1 #ea ,tann]y1 #dc tea*+1 #dcj1 #dcg !rmodynamics1 #aae1 #aai ,?omson1 #ih1 #dhh ?read1 #ajd "t1 #bbc equality1 #bbe local1 #cjf1 #cjg measure (1 #bbc-d ,tiss]&1 #eae-f ,tol/oi1 #ced1 #cfb1 #cfh ,tommas9a1 #eai ,transylvania1 #bjc triangle1 #eh angle sum (1 #eh tru?1 #bje ,ty*o ,brahe1 #acc1 #aec1 #bbh unity ( nature1 #acj univ]sal inv>iant1 #ccc ,uriel1 #bjc van d] ,waals1 #aec ,vauban1 #baj ,vebl51 #bjc velocity ( light1 #bcb1 #cab ,v5us ( ,milo1 #bja v]ification1 #cc ,vir*[1 #ba visual impressions1 #beb space1 #fg1 #beb ,volga1 #bjc ,voltaire1 #ecg-h ,web]1 #aag1 #eae-f ,wei]/rass1 #aa1 #bab1 #dcb ,:itehead1 #dgb1 #dha-b :ole numb]s1 #dd ,wie*]t1 #ade1 #dhh x-rays1 #aeb1 #eaa1 #ebj ,zeeman effect1 #aeb1 #aif1 #cag1 #did ,z5o1 #chb ,z]melo1 #dgg1 #dhb-c zigzag !ory1 #dgh zodiac1 #cih1 #djd 99 99 99 99 99 ,transcrib]'s ,note3 ,! ,greek alphabets are repres5t$ )9 squ>e brackets4 ,= example1 alpha /&s = fir/ ,greek alphabet alpha4 ,squ>e root ( a numb] is repres5t$ us+ ! symbol 7sqrt74 ,t is to say1 sqrt7#be7 /&s = squ>e root ( #be4 ,! sup]script is %[n ) c>at 77 symbol1 example1 #aje /&s = #e? p[] ( #aj4 ,simil>ly1 subscript is repres5t$ by und]score 77 symbol4 ,= in/ance1 n#c /&s = lett] n ) subscript #c4