congresso dei matematici 1900.pdf

8/14/2019 congresso dei matematici 1900.pdf

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1 9 0 0 . ] PAR IS CONGRESS OF MATH EMATICIANS. 57

T H E I N T E R N A T I O N A L C O NG RE SS O F M A T H E M A T I C I A N S I N P A R I S .A T the Zurich Congress of 1897 it was agreed to hold thene xt congress in P ar i s in 1900 , th e Fre nch M athem at ica lSocie ty being charge d w ith the prep arat io ns . Circulars ha vebeen i s sued a t in te rva l s dur ing the las t e igh teen months ,ca l l ing the a t ten t ion of mathemat ic ians to the a r rangem en ts in progre ss . T he congress was f inally an nou nce d

fo r Augus t 6 th - l l t h , a nd t he ope n ing ge ne ra l m e e t ing wa sheld in the Pala is des Congrs , in the Exhibi t ion grounds, a t9 .30 on the m orn ing of M onday , Au gus t 6 th . M. Po inc arwas e lected President , M. Hermite , who of course was notpresen t , be ing the President d'honneur. Th e execu t iveboa rd was con st i tu ted as follows : v ice-presidents , M M .Cz ube r (V ie nna ) , Ge i se r (Zu r i c h ) , Gorda n (E r l a nge n ) fGre e nh i l l (London) , L inde lo f (He l s ing fo r s ) , L inde m a nn(Munich) , Mi t tag-Lef f le r (S tockholm) , Moore (Chicago ,a b s e n t ) , T i k h o m a n d r i t z k y ( K h a r k o f f ) , V o l t e r r a ( T u r i n ) ,Ze uthen ( Cop enhagen ) ; secre tar ies , MM . Ben dixson(S toc kho lm ) , Ca pe l l i (Na p le s ) , M inkowsk i (Zu r i c h ) ,P t a sz yc k i (S t . Pe t e r sbu rg ) , W hi t e he a d (Ca m br idge , a b se n t ) ; genera l sec re ta ry , M. Duporcq (P ar i s ) . Af te r th eannouncement of the off icers of the sect ions and the namesof the official delegates, and a very few words from thePres iden t , the two addresses o f the day , bo th in French ,we re de live red b y MM . M. Cantor (He ide lberg ) an d V ol te r ra (Tur in) ; each occupied about th ree -quar te r s o f anhou r .M . CANTOR : Sur Vhistoriographie des mathmatiques.

Dur ing the cen tury drawing to i t s c lose the charac te r o fm ath em atic s ha s chang ed ; i t s devotees are no w different ia ted into geometers , analysts , a lgebrais ts , ar i thmet ic ians^as t rono me rs , theore t ica l phys ic i s ts , and h i s to r iographers .Th ese las t m ak e no c la im to adv anc ing th e sc ience i tse lf ;they press nei ther towards the arc t ic pole of the theory offunct ions , no r to w ard s the an tarc t ic pole of a lgebra ; th eyexplore nei ther the s teep surfaces of geometry nor thedep ths of di f ferent ia l equ at ions . Th eir tas k is ra th er todraw up gu ides and char t s , to ind ica te by wha t rou tes ther e su l t s ha ve be e n ob t a ine d , a nd wha t im por t a n t po in t sha ve been passed by w itho ut sufficient explorat ion . T hi swork began wi th the His to ry of Eudemus of Rhodes , B . C ,


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5 8 PAR IS CONGRESS OF MATHE MATICIANS. [N O V . ,300, of which only a f ragment has been preserved, jus t suf-ficient to excite l ively reg ret for th e loss of th e wh ole. D uring the nex t two thousand years the re were many ba ldchronicles of mathematics , but his tor iography as a sc iencebegins wi th Montuc la . No twi th s tand ing the e r rors , unavoidab le a t th at t im e, to be found in the two volume s ofhis H is toire des math m atiqu es (1st edi tion, 1758, 2d edit ion wi th two vo lumes by La lan de , 1799) , M ontuc la e s tencore e t res tera peut- t re toujours un modle que tout histor io gra ph e des sc iences doi t suivre . K s tne r publ ishedfour volumes of his Geschichte der Mathematik in the las tfour ye ars of his l ife, 1796-1800. H e ha s been alte rn ate lyover-praised and deprecia ted ; Gauss referred to him as thebes t poe t among the mathemat ic ians , the bes t mathem at ic ian am ong the poets of his day. H is his tory is noreal h is tory, i t i s ra ther a cata logue raisonne,bu t i t i s n evertheless invaluable , on account of i ts conscient ious analysisof a number of works , which, wi th their authors , would beo therwise abso lu te ly unk no wn to us now. A t about the samed a t e , 1797-1799, there appeared the two volumes of Cos-sa l i ' s Stor ia cr i t ica delF a lgebra , deal ing ex haust ive ly w iththe per iod 1200-1600 ; as regards I ta ly only, i t i s t rue , butth en du r ing th i s pe riod the I ta l i an a lgebra was of impo rtan ce far surpas sing th at of an y other cou ntry . Cos-sal i ' s labors for the e lucidat ion of Leonard of Pisa andCardan are of specia l meri t .Bossut publ ished in 1810 his His toire des mathmatiques ;in th is he gives only rapid aperus of t he gen eral develop m ent , in te res t ing to those th a t kn ow a l read y , use less tothose th a t need to l ea rn . In the p resen t cen tury we h avef irst Chasles , to wh om th e speaker pa id a w arm person a l t r ib u te . In h i s Ape ru h i s to r ique publ i shed in 1837 ,the notes , deal ing with geometry, ca lcula t ion, a lgebra ,mechanics , which a t ta in the dimensions of memoirs , formth e mode l pa r t of the vo lume, the tex t , the ac tu a l i' Aper u, ' 'be ing but a very condensed s ta tement of the his tory ofsy nth et ic geom etry. Th e other his tor ical work of Chasles,the Rapport sur les progrs de la gomtrie of 1870,is ser iously affected by his igno rance of th e Ge rma n lang uage. T he yea rs 1837-1841 saw th e publ icat ion of Li br i ' sH is to ire d es sc iences m ath m atiqu es en I ta l i e , f rom th eear l ies t t imes up to the middle of the 17th century, awo rk which owing to the au tho r ' s adm irab le s ty le sel i t comme un roman, mme dans les par t ies o e l le n 'en es tpa s u n . No t wi th s tand ing Li br i ' s imm ense se rv ices in thestudy of manuscr ipts , h is his tory is v i t ia ted, as a his tor ical


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1 9 0 0 . ] PARIS CONGRESS OF MATHEM ATICIANS. 5 9work, by his misplaced patr iot ism ; according to him al lp rogress in mathemat ics i s due to the I ta l i ans , wi th perhapsa few sca t te red Fre nch wr i te r s . W he n he finds an I ta l i an inpossess ion of any ideas or methods, no mat ter whence der ived , he a t once credi ts h im with the ir d iscovery. In an ycase, it is not possible to give any true idea of the history ofm athe m at ics by t rac ing i t in one coun t ry on ly . I f the reis an int ern at i on al sc ience, i t i s m athe m atics ; i t bea rsno s tam p of na t ion al i ty . I n consider ing th e ear l ies t t im es,i t i s impossible to unders tand the course of mathematics inone country without fol lowing i t in others a lso ; to unders tand Greek m athem at ics , we m us t k now someth ing ofEgypt and Babylon ia ; the mathemat ics o f the Arabs cannotbe explained without reference to Egypt , Greece, and India .After the invent ion of pr int ing, so long as Lat in was in use ,mathemat ics had no count ry ; and even when the f ron t ie r swere fa int ly marked by the use of di f ferent languages , theywere speedi ly obl i tera ted for most mathematic ians .Passing rapidly over Gerhardt and Quetele t , wi th a fewwords of recogni t ion, M. Cantor spoke of Nesselmann'sDie Algebra der Griechen, 1842, a u n chef d ' uvr e d igned ' t re mis a ct de 1'Ape ru h i s to r ique de C ha s l es ; o fA rn e t h ' s G esch ich te de r re inen M athem at ik , 1852 , whichwould have been an excel lent book, i f the author had made abe t ter app ort io nm ent of his space to his m ater ia l parts of thew ork fourm il lent de rem arq ues aussi spir i tuel les que profon des ' ' ; of H an ke l ' s pos thu mo us f ragment , 1876 , u u ntorse d 'une te l le beaut qu ' i l eut t pi t i de ne pas le mett re au grand jour ; and of the p r ince Ba ldassa re Boncom-pa gn i 's d is interes ted labors on behalf of his to r iogra phy .I n th i s ske tch he passed over m any au tho rs tous auss im or ts qu e l eur s l ivres ; gard ons-n ous de les ressusci ter ' ' ;and avoided a l l ment ion of l iv ing authors for very obviousrea so ns. H e br ou gh t his ad dre ss to a close by a forecast ofthe mode in which the h i s to ry of more recen t mathemat icsm us t be w r i t t e n . Reg ard ing Lagra nge as the founder ofmodern mathematics , th is gives 1759 as the s tar t ing point ;and from this year on, the different subjects will have to betrea ted in specia l volum es. Th is how ever wil l be insuf-f ic ient ; the development of the l ines of thought that runthr ou gh a l l thes e different bra nch es of m athe m atics m us tbe trace d in one final volu m e, th e H isto ry of Id ea s ; difficultto w ri te , cer ta in ly , bu t indispensab le , for as Jacob i sa id ,1 Mathematics is a sc ience of which i t i s impossible to unde r s t a nd a ny one pa r t w i thou t knowing a l l t he o the r s .


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6 0 P ARI S CONGRES S OF MATHEMATI CIANS , [ N o v . ,V . V O L T E R R A : Trois analystes italiens; Betti, Briosehi, Casorati.The scientif ic existence of I taly as a nation dates from ajou rn ey wh ich Bet t i , Br ioschi , and Casorat i too k tog eth erin the autumn of 1858, with the object of enter ing into rela t ions wi th the fo remos t mathemat ic ians o f France andG erm any . I t i s to th e teach ing, labors , an d devot ion ofthes e thr ee , to their inf luence in the organ izat ion of advanced s tudies , to the f r iendly sc ient i f ic re la t ions that theyins t i tu te d be tween I ta ly and foreign count r ies , th a t theexistence of a school of analysts in I taly is due.The extent of their joint influence, affecting minds ofmany diverse casts , i s largely due to the dif ferences in theirnatural facul t ies , in the c i rcumstances of their l ives , and inthe ir acqu ired tendencies . Br ioschi , ' ' toujours jeu ne par soncarac t re e t tou jours m r pa r son esp r i t / 7 a Lom bard bybir th , was a t f i rs t an engineer ; but a t an ear ly age he acquired a profound knowledge of the c lass ical mathematicalw ork s , an d was called to the chair of m echan ics a t P a via a tth e age of 25. H e founded th e Po lytech nic School a t M ilan,and held the directorship unt i l h is death ; in his capaci tyof Sen ator, he w as ac tive in public affairs ; he found tim eto engage in publ ic works and in engineer ing ; and up tothe las t , as Director of the Annali di Matematica,a nd P re s ident of the Accademia dei Lincei , he was one of the leadersof th e m athe m at ica l mo vem ent in I t a ly . A grea t co n t ra s tto th is activ e l ife i s offered by th e calm existence of B ett i .He was born in a mountain vi l lage in Tuscany ; a t 34 hebecame a professor in the Univers i ty of Pisa , and a t 41Director of the Scuola normale super iore of Pisa , whoseorganizat ion is much l ike that of the cole normale supr ie ur e of Par is ; he took no pa r t in pol i t ica l m ove m ents . H eloved scientif ic researches for their own sake exclusively,wi thout regard to the resu l t s they might p roduce in thescienti f ic world , or to the ir impo rtanc e in teachin g. H edid not care for publ ishing his researches ; and even whenhe d id under take th i s , he was ap t to push i t a s ide , a t t ra cte d by new ideas . Th e know ledge th at h is in te l lectualconception could be realized was all-sufficient for him ; hedid not give himself the t rouble of carrying i t out in deta i l .When once he had obta ined a c lear vis ion of hidden t ruths ,an d had cons t ruc ted in h i s own min d a sys tem in whichthe y proceeded direct ly f rom th e s implest pr inciples , tou t ta i t fa i t pour Bet t i .Casorat i wag born and l ived a t Pavia ; he passed throughthe var ious g rades in the Univers i ty , where a t the t ime of


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1 9 0 0 . ] P A R I S C ON GR ES S O F M A T H E M A T I C IA N S . 6 1his de ath he wa s professor of inf ini tes imal analysis . H el ived and worked a lmost exclusively for his pupi ls ; a l l h isworks bear the s tamp of the pract ical teacher , bent one luc ida t ing some obscur i ty , cor rec t ing some e r ror , expou nding some theo ry. A l l h is w ri t ings we re in a def ini te re lat ion to h i s un ivers i ty t each ing ; in h i s mind the re was nodis t inct ion between the work of the savant and the work ofthe professor.The fundamental d i f ferences in the three can be broughtout most c lear ly by a comparison of their a t t i tude towardsth e the ory of funct ions . T he deve lopm ent of th i s the or yexhib i t s th ree wel l -marked per iods , cor responding to theth re e dis t inc t p hases t h at can be recognized in th e his to ryof any mathematical subject ; these three phases , however ,correspond a lso to three dis t inct modes of regarding quest ion s in analy sis , each of which h as i ts advo cates . In th ef i rs t ins tance, the discovery of facts is a l l - important , andpar t ic u la r theor ies a re e labora ted . Th ere a re no un i formmethods ; every ques t ion i s a t t acked on i t s own mer i t s andm etho ds a re created as occasion ar ises ; th e ideas an d resu l tsdisengage them selves f inally f rom long calcula t ion s . In th etheory of functions this manifests i tself in the heroic period,personified in Eu ler , Jac ob i , Abel ; an d this m an ne r ofapproach ing ques t ions i s na tur a l to B r iosch i , the eng ineerand pract ical man, with his extraordinary gif t for deal ingw i th fo rmidable ca lcu la t ions . H e rem ained fa ith fu l to th eclass ical method, never a t t racted by the second phase ,w hich he even scorned so me wh at . In th is second phase ,ideas replace calcula t ions ; the phi losophic spir i t tak es cont ro l an d d em ands a genera l method inc lud ing th e wholesubjec t in one bod}^ of do ctrin e. Th is desire foun d its fulfilment in the second p eriod of th e the or y of fu nctio ns, inthe w orks of C auchy , W eie rs t rass , and K ieman n, who der iveeverything f rom the very sources of the fundamental concept ions . To thi s per iod belongs Be t t i th e phi losoph er . H isbroa d an d cul t iv ated m ind loved phi losophic system s ; h isTuscan indolence (which is not in te l lectual id leness) causedhim to de l igh t in med i ta t ion ra t he r tha n in mecha nica llabor . Cu riously enou gh, his na m e is associa ted with th etheo ry of W eiers t rass jus t as surely as with th at of E iem an n ;h i s educa t ion had made h im an a lgebra i s t whi le na turemeant him for a physic is t .In the f inal per iod the theor ies f ind their appropr ia te appl icat ions , their most sui table forms ; they are ref ined bycr i t ic ism, and cast in t o a didac t ic mould. Th e nam e ofCas orat i , cr i t ic an d tea che r , i s associa ted w ith this t h i rd


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6 2 PARIS CONGRESS OF MATHEM ATICIANS. [N O V . ,pha se. H is wo rk, Teo r ica del le funzioni di var iabi l i com-plesse , ha s served mo re th an a ny other o ne book to popular izein I t a l y th e fun dam enta l c oncept ions of th e theo ry of funct i ons , for the reason that, while reading i t , diff icult ies disap pe ar. T he influence of this book is not confined to pro fessed an aly sts ; any one a t te m ptin g to t race the development o f mathemat ics in I t a ly dur ing th i s ha l f cen tury wi l lf ind that analysts and pure geometers have inf luenced oneano ther . Fo r ins tance , the ideas of E iem an n a re a t thefoundat ion of many of the works of I ta l ian geometers , andwhile the actual in t roduct ion of these ideas was due toB et t i , i t i s th is book of C aso rat i ' s th at ha s carr ied the meverywhere and a t t rac ted the a t ten t ion of geomete rs .This comparison of the work of these three analysts inthe region that they had in common^gives no idea , however ,of th e ex ten t of th e labo rs an d influence of each one . F orth is i t wou ld be necessary to dwel l on the work of Cas orat iin the theory of di f ferent ia l equat ions , in analyt ical andinf ini tes imal geometry; of Bet t i in mathematical physicsand algebra, he being one of the first to accept the newideas of Galois ; of Brioschi in mechanics, algebra, and geom etr y. T he field in wh ich B etti an d Briosc hi f irst obtaine drenown was in fact that of a lgebra ; their names wil l a lwaysbe associa ted w ith tha t of K ron eck er as second only to I I er -mite in their work on the equat ion of the f i f th degree , anequ at ion whose com plete solut ion was due to an d securedimmor ta l i ty fo r M. Hermi te .

This concluded the business of the f i rs t general meet ing,with the except ion of one or two formal announcements rela t i ng to secretar ia l m at te rs . Th is was the only one of t hem ee t ings to be he ld in t he Exh ib i t ion gro und s ; a l l th eoth ers were held a t th e Sorbonne. Six sect ions were orga nized for the presentation of special papers, to meet on the7th, 8th, 9th, and 10th of August, as follows :Sec tion I , A r i thm et ic and Algebra ; Tuesday , Th urs dayan d Fr ida y morn ing s ; p res iden t , M. H ub er t , sec re ta ry ,M . Ca r t a n .Sec t ion I I , Ana lys i s ; Tuesday and Thursday morn ings ;p res iden t , M. Pa in lev , sec re ta ry , M. Hadamard .Sec tion I I I , Geomet ry ; Tue sday and Th urs da y a f te r noons ; p res iden t , M. Darbo ux , sec re ta ry , M. M ew en -glowski .Sec t ion IV, Mechanics and Mathemat ica l Phys ics ; Tuesday and Th urs da y a f te rnoons ; p res iden t , M. Larm or , sec re ta ry , M. Levi -Civ i t .


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1 9 0 0 . ] PARIS CONGRESS OF MATHEM ATICIANS. 6 3Sec t ion V, Bib l iography and His to ry ; Wednesday morning and a f te rnoon and Fr id ay m orn ing ; p res iden t , P r in ceRoland Bonapar te , sec re ta ry , M. d 'Ocagne .Sec tion V I , Teach ing and Methods ; W ednesda y mo rning and a f te rnoon an d Fr ida y m orn ing ; p res iden t , M.Cantor , sec re ta ry , M. La isan t .Sec t ions V and VI , however , amalgamated and sa t a s onesect ion, thus making the sect ions the same as those a tZurich . I t ha rd ly seems advisable to give a comp lete l is tof the papers, as all will be given in the full official report,

w hic h will app ear sh ortl y ; i t seem s be tter to give soncraaccount of the most in teres t ing.In Sect ion I the most not iceable communicat ion wasth a t of M. H en ri Pad , of Li l le , ' 'Ape ru sur les dveloppements rcents de la thor ie des f ract ions cont inues . 7 T h eobject of th is com m unica t ion was th e discussion of the que st ion as to w ha t is to be u nde rs tood by the developm ent ofa funct ion as a con t inu ed f ract ion, an d th e exa m inat ion ofth e consequen ces of th e answ er obta ined . Fo r th e function


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6 4 PARIS CONGRESS OF MATH EMATICIANS. [N O V . ,M. Pad then ind ica ted the two ways in which these re sul ts can be general ized, the extension they involve in a l lthe appl icat ions of cont inued f ract ions hi ther to made, andthe impor tan t consequences to which they lead , bo th in thet h e o r y of func t ions , wh ere they have a l ready in t roducedth e quest ion of the use of diverge nt pow er-series , an d in th etheory of numbers .In Sect ion I I , the f i rs t paper read on Tuesday morningw a s M . T i k h o m a n d r i t z k y ' s L ' v a n o u i s s e m e n t d e s fo nct i ons 0 de p lus ieu rs va r iab les indpend antes . The funct ionuhIh ofp independent va r iab les un= SIh vanishes

1 wh en som e of th e po ints I #.,y.\ fallat ( , y ); 2 whenthey are on an adjoint curve of the first kind, ?>(#w~2, yn~2)= 0. If w ith W eier s t ras s w e def ine th is funct ion by th ee qua t ion

^-i}.)-/^1-^^, i)/ p \ p *iw h e r e J k , = 2 , 2)

(FE* den ot ing th e inte gra l of th e second species , wh ich be-comes inf ini te when (# . , y.) falls at (aft, 6 ft,))> n^ s proper tyof 0 must be der ived f rom those of the funct ion (2) .Forth is purpose, consider ing in the f i rs t p lace the funct ion

( t f . ) >P H2 n , 3)

( ( * ' , y') de no t inga poin t ve ry near to (ak, bk)) , whe re t hep o i n t s (x', y'), (x(, y() are the inf ini t ies , and ( , y^), (a{,ya.)the zeros, of the principal function of (,yz

PJ*, v top)- 7 ^ 7 Z r~ (4>r\ z> y x yy )


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1 9 0 0 . ] PARIS CONGRESS OF MATHEMA TICIANS. 6 5we see, by the secon d form of th is function , th a t in th etw o cases one of th e inf ini ties of th e funct ion in th e nu m era tor being absorbed by one of i ts arbi t rary zeros (1 (xpJ yp)by (, y ; 2 (xp, yp) by (x 'p_v 2/^ -0) , the other wil l beabsorbed by one of i ts non-arbi t rary zeros ( % VaA. Hencein th e two cases, a t th e l imit , for (V, y') = (aA, bk), one of theinte gra ls in (3 ) w ill becom e infinite l ike for x= ak ; akt h u s 0 wil l vanish.More general interest was taken in M. Mit tag-Lefi ier 'spap ers , wh ich fol lowed. (iSu r fonct ion a na lyt iq ue et express ion an a ly t iq ue . U ne appl ica t ion de la thor ie des sriesn-fois infinies. l Sur u ne extension de la sr ie de T ay lo r .I n these M . Mit tag-Leff ler reported on his recen t resear che s* in the theo ry of func t ions . Le t f(a), f (a),

i ' ( ) , de termine an e lement P(x |a) = 2 Jn() {x


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6 6 P ARI S CONGRESS OF MATHEMATI CI ANS. [ N o v . ,which the coefficients cw>m are given ini t ia l ly and do not depend on a or onf(a), ' (a) , .The express ion

G n(x | a)

A = A 2 = A ^ o V ' - V V /leads to a l imi t ing express ion ^ 6 rn(# I a) with the following proper t ies : I t i s uniformly convergent for every regionin te r io r to the s ta r A} but never uniformly convergent fora region containing a ver tex of A. W i t h i n A i t r epresen tst h e b r a n c hfA(x) of f(x).I t is perfectly possible that w1im00 Gn(# |a) may convergeouts ide J . ; th e s tar A is no t a sta r of con verge nce forn l^O n(x\a). M. M i t tag-Lem er has shown th a t i t i s pos s ible to replace ^i 1 ^ GJjx |a) by another express ion forwhich J . i s a s tar of convergence. *J G n(x\a) was obtained from an w-fold series in x by m a k ing t he m a x im umvalues of Xv X2, , Xn proceed s im ultaneo usly to th e l im itoo. If th e passage to the l im it is performed in an oth erAl A2 \n w2 n* n2 ^w ay, viz., by ta k in g 2 2 2 in place of 2 2 * 2Al=0 A2=0 An=0 A l= 0 A2 = 0 \n=0a nd t he n m a k ing ^n, n_ly , ^ te n d successively, in the order named, to inf ini ty , the express ion

h A2 K (x a \A i + ' + A *flL(*l a) = 2 2 - 2 v J ^ ^ Wl T )A = 0 A2= 0 A = 0 \ /(where the c 'a are given numerical constants of which

1 1 / 1 \ A l+ A2 1 / 1 \ ^1+A2+\S

while for values of n > 3 the y ar e a lgebraic i r ra t i on als)yie lds th 3 desired resul t ; 1oom #w (x \ a) has the s ta r A as a starof convergence, and represents fA(x) wi th in A. W r i t i n gn= 1, th e series is seen to be simply T ay lo r 's se ries ; ingeneral i t i s an extension of Taylor 's ser ies .I n th e course of his rem ark s M. Mit tag-Lefner referred torecent researches of M. Borel ; this led to a discussion in


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1 9 0 0 . ] PARIS CONGRESS OF MATHEM ATICIANS. 67wh ich M M. Bore l , Ha da m ard , and P a in lev took par t , onthe na ture o f the connec t ion be tween ii analyt ic express ionin a complex var iab le a? an d ana lyt ic funct ion in # .

In Sec t ion I I I papers were p resen ted by MM. Love t t ,On contact t ransformat ions between the essent ia l e lementsof space ; M acfarlane , A pp lica tion s of space an alys is toc u rv i l ine a r c oo rd in a t e s ; S t r i ngha m , Or thogona l t r a ns form at ions in e l l ip t ic or in hyperbo l ic spac e ; Am odeo,an d othe rs . I n Sect ion I V ve ry few pap ers were read ;one appointed meet ing of the sect ion was not held , andsome of the papers in tended for the sect ion were presentedat the joint s i t t ing of Sect ions Y and VI, which was t ransformed m om enta r i ly in to a s i t t in g of Section I V , to he arM M . H a da m a rd : R e l a t i o ns e n t r e l es c a ra c t r i st ique srel les e t les caractr is t iques imaginaires pour les quat ionsd if f ren tiel les p lus ieurs va r iab les i nd p en da nt es ; andV ol te r ra : Com ment on passe de Vquation de Poisson caractr is t ique imaginaire une quat ion semblable carac t r i s t ique re l le .The communica t ions made in Sec t ions V and VI , whi lenot necessar i ly the most valuable mathematical ly , were yetof the most general in teres t , and lend themselves best toan y genera l repor t . Th e s i t t ing was opened by M. H uber t ' s address , in German, on the future problems ofm athe m at ics . Th e l ines a long which we m ay expec t thedevelopment of any science which is progressing in a cont inu ou s m ann er can be de tec ted by an exam ina t ion of theprob lem s to wh ich a t te nt io n is specia l ly paid . Am ongthes e m ost im po rtanc e is to be a t tache d to those th at aresharply defined and stand out well ; such, for example, as

th e prob lem of thre e bodies. I n th e earlier stages of a nysc ience , p rob lems presen t themse lves na tura l ly th rough exper ie nce , as is exemplified in m athe m atics by th e dupl icat ion of the cube and the quadrature of the c i rc le , and a t ala ter date by the quest ions ar is ing with reference to inf ini tes imal analysis and the theory of the potent ia l ; but asthe science progresses, i t is the logical faculty of the intellect that imposes on us problems such as are found in thetheor ies of pr ime numbers , e l l ip t ic funct ions , e tc .As to our a im wi th regard to any prob lem, the re mus t bea definite result of some kind, i t cannot be laid aside untilwe have obta ined e i ther a sa t isfactory solut ion or a r igorousdem on stra t ion of th e impossibi l i ty of a solut ion. T hem athe m atica l r igo r th at is essent ia l in the t rea tm en t of a


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6 8 PARIS CONGRESS OF MATHEM ATICIANS. [N O V . ,problem does no t requ ire complicated d em onstra t ions ; i trequires only that the resul t be obta ined by a f ini te numberof logical steps from a finite number of hypotheses furnishedby the p rob lem itself; in seeking this r igor we may findsimplic i ty . T he prop er t re atm en t of any problem depen dson 1 a complete system of axioms, by means of whichthe conceptions are defined, 2 a system of symbols appropr ia te to the concept ions with which the problem deals ;thus a demonstra t ion by means of geometr ical symbols is asleg i t imate as an a r i thmet ica l one , p rov ided tha t the ax iomson wh ich i t i s based are perfectly unde rs tood. Th e m ereformulation of these axioms is in some cases i tself the problem, as for ins ta nce in ar i th m et ic an d physics . Am ongthe ten problems that M. Huber t specif ied in par t icular asfi t ted to advance mathematics, No. 2 is that of f inding someone system of independent compat ible axioms governingand def ining ar i thmetical concept ions , and No. 3 is thesame quest ion for the calculus of probabi l i t ies , ra t ional mechan ics , an d physics . Oth er problem s are to prove th at eiirZist r a nsc e nde n t a l whe n z is an a lgebraic i r ra t ion al ; and th atthe solut ion of the general equat ion of the 7th degree canno t b e obta ined by m ean s of a f inite nu m be r of op eratio nsinvolv ing on ly two parame te rs . In geom et ry , the re la t ivesi tuat ion of the c i rcui ts that a plane curve of ass igned ordercan possess , wi th the corresponding quest ion as regardssurfaces , demands invest igat ion ; in the theory of funct ionsthere is the quest ion of the express ion of two var iables ,connec ted by any ana ly t ic re la t ion wha tever , a s un i formfunct ions of a s ingle parameter zfor Poincar ' s theorem(Bulletin de la Socit mathm atique de France, volume 11(1 88 3) ) is subject to some l imita t ions . Th ese are bu t afew of the problems that M. Hilber t ment ioned, and thesewere a selection from a much longer l ist for which he referred to an ar t ic le abo ut to app ear in th e Nachrichten derKgl. Gesellschaft der W issenschaften zu G ttingen. 1900. I nthe course of a ra ther desul tory discussion that fol lowed thereading of th is paper , the c la im was made, though apparen t ly wi thout adequa te g rounds , tha t more had been done asreg ards th e equat ion of the 7th degree (by some G erm anw ri t er ) tha n th e au tho r of t he paper was wil l ing to a l low.A more prec i se ob jec t ion was taken to M. Hi lber t ' s r e marks on the ax ioms of a r i thmet ic by M. Peano , whoclaimed that such a system as that specif ied as desirablehas a l ready been es tabl ished by his compatr iots MM. Bural i -For t i , Padoa, Pier i , in memoirs referred to on pp. 3-5 ofno . 1, volume 7 of the Rivista di Matematica.


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1 9 0 0 . ] PARIS CONGRESS OF MATHEM ATICIANS. 69M. Hubert was followed by M. Fujisawa, the official delega te f rom Japan , who gave , in Engl i sh , a ve ry in te res t ingaccount of the mathematics of the older Japanese school .I t is difficult to follow the course of Japanese mathematics ;the re a re some two thousand manuscr ip t vo lumes s t i l l to bet ranscr ibed ; in these m uch v a luab le work i s mixed u p wi thw ha t is pu rely e lem enta ry and even t r iv ia l . Th e diff icultyof arr iving a t any c lear idea is great ly increased by the facttha t pub l ica t ion of resu l t s was no t cus tomary ; they werepreserv ed to a gre at ex ten t only by oral t ransm ission. So

far as th e books ha ve been deciphered a nd col la ted, one facts tands out with ever- increasing c learness , and that is thats ide by s ide with one less important school of Japanesem athe m at ics the re ex is t s ano ther ea r l ie r sys tem of m athem at ics of a pecu l iar kin d, which h ad i ts or igin in Ja pa n,and was developed there ent i re ly f ree f rom any externalinfluences.The mathematics of the f i rs t k ind, probably der ived f romthe Chinese a t a very ear ly date , d isplays a not iceable lackof r igo r ; for in sta nc e, V lO is accepte d as th e va lu e of TT.As to content ; in a lgebra , the solut ion of s imple equat ionsand the formulae for the sum of an ar i thmetical or geometr ical progress ion w ere kno w n ; in geometry, the r ight-angled t r iangle with s ides proport ional to 3 , 4 , 5 was used,w ith some propo si t ions reg ard ing regu lar polygons ; mag icsquares were discussed, even so far as those containing thef i rs t 400 nu m ber s . Bam boo rods were used for purpose s ofcalcu la t ion ; these we re placed o ne above a no the r to indicate addi t ion, s ide by s ide to indicate mult ipl icat ion,d iagona l ly to denote sub t rac t ion .The o ther pa r t o f Japanese mathemat ics , tha t ind igenousto th e cou nt ry , is of mo re imp or tance and in te res t . I t appe ars th a t the m athem at ic ian s of th i s school ma de use oflocal value in express ing numbers , invented zero for themselves , an d u sed the c i rc le as th e symbol for zero. Th eywere fami l ia r wi th imaginar ies and complex numbers ; andwe re such a dep ts a t ca lcu la tion th a t they found the va lueof TV co rrect to 49 places of decimals . M. Fu j isaw a exp la ined tha t the knowledge of th i s pa r t o f Japanese mathemat ics so far obta ined is very f ragmentary, the unexploredpar t offers an a t t ract ive f ie ld of research for Japanese whom ay care to devote them selves to i t . I t i s a m at t er ofpure ly h i s to r ica l in te res t , a s the p resen t t each ing of mathemat ics in Japan is in no sense founded on i t ; for , verywise ly as he th inks , the Japanese educa t iona l au thor i t i e smade an ent i re ly f resh s tar t , sweeping away al l t race of th iso lder educa t iona l sys tem.


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7 0 PAR IS CONGRESS OF MATHEM ATICIANS. [N O V . ,The president of the sect ion, M. Cantor , then spoke ofthe dif f icul t ies he encountered, when wri t ing his Geschichteder Mathematik , in f inding out anything about the ear l ierJap an es e m athe m atics . W he n he did f inal ly he ar of awo rk of reference i t tu rn ed out to be w ri t te n in Ja pa ne se .With reference to the ear l ies t use of zero, he expressed theopin ion tha t i t was p robably due to the Babylon ians , about1700 B. C.Another paper of in teres t in these sect ions was that of M.Pa doa (Rom e ) on F r ida y m orn ing : Un nouv eau sys tme

i r rduc t ib le de pos tu la t s pour l ' a lgbre . N am ing the object , entier ( in teger ) , two undef ined der iva t ives , successifa ndsymtrique,are considered. Th e seven pos tula tes are1. If a is an integer , then suc. a is an integer .2 . If a is an integer , then sym. a. is an integer .3 . If a is an integer, then sym. (sym. a) = a.4 . If a is an integer , then sym. \sac . [sym. (sue. a) ] }= a .5 . There ex is t s an in teger x such tha t sym. x x.6 . There do not exis t two dif ferent in tegers x, y, suchtha t sym . x = x, a nd sym . y= y.7 . If a class u of objects satisfies the conditions( i ) i t conta ins some one integer ,( i i) if i t contains an integer x i t conta ins a lso sue. x,(i i i) if i t contains suc. x i t conta ins a lso x,then every integer belongs to the c lass u.These postula tes def ine an a lgebraic f ie ld , whose natureis at once seen to agree with that of the natural f ield, suc. xbe ing in te rpre te d as 1 + x, and sym. x as x, M. Pad oadid not get beyond this definit ion, possibly because he hadentered so minutely into the deta i ls of the proof of the independence of the seven pos tu la tes tha t he had exhaus tedhis a l lowance of t ime.A grea t pa r t o f the Fr iday morn ing s i t t ing of these twosect ions was devoted to the discussion of a resolut ion, proposed by M. Leau, in favor of the adoption of some specialart if icial lan gu ag e a s th e vehicle for all scientif ic com m unica t ions . Th oug h no par t i cu la r l anguage was named in ther e so lu ti on , i t wa s m a d e c le a r t ha t E s p e ra n to wa s t helangu age in tende d . I t s advoca tes , MM, Leau , Padoa , Boc-cardi , Laisant , and others , d isc la imed any wish to subst i tu te i t for natural languages , but urged i ts adopt ion as thevehicle for in ternat ional in tercourse ; th is view they upheldwi th g rea t ea rnes tness o n behal f of h um an i ty , a s M.L ais an t pu t i t . T he opposi te view was upheld with equ alearn estne ss , i f less vehem ence, by MM . Schroeder , Va ssi-


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1 9 0 0 . ] P ARI S CONGRES S OF MATHEMATI CIANS . 7 1lief, Maggi , and others , chief ly on the ground that any suchlanguage is ent i re ly unnecessary ; as M. Maggi remarked,m athem at ics a l ready has a un iversa l l anguage , the languageof formulae. I n th e en d th e sugg estion of M. Vassil ief w asadopted, that the Congress should place i tse l f on record asopposed to any unnecessary divers i ty in the languages employed, that is , pract ical ly , to the use of any language forsc ien t i f i c purposes o ther than Engl i sh , French , German,an d I ta l ia n, tho ug h these lang uage s were n ot specified inthe reso lu t ion adopted .

On Sa turday , Augus t 11 th , the conc lud ing genera l mee t ing was held a t 9 a . m. T he f i rs t business was to dete rm ineth e t im e and place for th e nex t m eet ing. A t Zurich, Pro fessor Klein , on behalf of the German Mathematical Socie ty ,had expressed their great desi re that the thi rd congressshould be held in G erm any ; an d a def inite in vi ta t ion toth is effect was now laid before the Co ngress, an d un an imo usly accepted. T he place of mee t ing wil l probab ly beBaden-Baden ; the date decided upon is 1904, and the t imeis to be e i the r a t th e begin ning or th e end of the sum m erva c a t i on . 'N o o ther bus iness was t ransac ted , and the twogeneral addresses appointed for the day were then del iveredby MM . Mit tag-LefHer an d Poin car . Im m edi ate ly af terthe conclusion of the President 's address , he dismissed theCongress wi th the words a La sance est leve, le congrses t c los .

M. Mit tag-Leff ler ' s address was ent i t led li Une page de lav i e de W e ie r s t r a s s ; in this he considered in some detailW e ie r s t r a s s ' s a t t i t ude t owa rds som e o f t he m a the m a t i c a lideas of his t ime, i l lus t ra t ing by copious extracts f rom hiscorrespondence; unfor tunate ly i t i s not possible to give anyad eq ua te acco unt of i t . M. Poincar 7s can be given m orefully.I I . P O I N C A R E : DU rle de l'intuition et de la logique en math-matiques.I t i s obvious that there are two ent i re ly dif ferent typesof mind among mathemat ic ians , mani fes t ing themse lves intwo d i s t inc t m ethods of t r ea t in g m athem at ica l ques t ions .Those of the first type are dominated by logic ; those of thesecond are guided by int i t iou . Th ey ma y be cal led analysts and geometers , though i t i s not real ly a quest ion ofth e subject w i th which the y deal ; the ana lyst rem ains anana lys t even when work ing a t geomet ry , and the geomete r


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7 2 PAR IS CONGRESS OF MATH EMA TICIANS. [1ST0V.?em ploy ing himself on pu re an aly sis is st i l l a geom eter. ISToris the dis t inct ion a mere mat ter of educat ion ; a man is borna mathemat ic ian , he does no t become one ; and e i ther he isbo rn an an aly st or he is born a geom eter . T he tw o ty pesof mind are equally necessary for the progress of the science ; each has accom plished grea t th in gs th at would ha vebeen impossible to the other .At f i rs t s ight the ancients seem to have a l l been intui t ion-al is ts , bu t th is impression disapp ears on c loser s tud y. E ucl id , for ins tance, was a logic ian, even though every s toneof his edif ice is du e to int ui t i on . T he na tu ra l te nde ncieshav e no t changed , on ly the i r man i fes ta tion . Th ere hasbeen an evolut ion, due to the increasing recogni t ion of thefact that in tui t ion cannot give r igor , nor even cer ta inty ; aproof that re l ies on concrete images may be very decept ive .I t was soon real ized th at r igor can no t be expected in th edemonstra t ions unless i t i s to be found in the def ini t ions ;so long as th e objects of reaso ning were given s imply bythe bodi ly senses or the imaginat ion, there was no preciseidea on wh ich reaso ning could be based. T hu s th e effortsof th e logic ians we re conc entra ted on th e def init ions , oneresu l t of wh ich i s th a t m athem at ics has become a r i thm e-t ized.The quest ion ar ises , i s th is evolut ion endedhave we a tlas t a t ta ined to absolute r igor , or do we deceive ourselvesas our fa thers did ? Phi loso phers te l l us th at i t i s impossibleto e l iminate in tui t ion a l together f rom our reasonings, for noscience can spr ing f rom p ure logic a lone. To design ate . th iso ther essen t ia l , we hav e no na m e b u t in tu i t ion ; bu t th i scovers m an y different ideas . T he re is (1) the appea l tothe bodi ly senses and to imaginat ion ; (2) general izat ion byinduc t ion ; (3 ) the in tu i t ion of pure number ; on th i s l a s ta ve r i t ab le mathemat ica l method i s based , whi le f rom thef irst two no ce r ta in ty can be der ive d. T he analy sis of th epre sen t d ay con structs i ts dem ons tra t ion s sole ly f rom syllog isms and th i s in tu i t ion of pure number ; we may say tha ta t las t absolute r igor is a t ta ined.T he phi losophers now object th at w ha t h as b een gaine din r igor has been lost in ac tua l i ty ; th e approach tow ard th elogical ideal has been secured by cut t ing the t ies with reali ty . Fo r the sake of the dem ons t ra t ion a m athem at ica ldefinit ion is substi tuted for the object, and i t st i l l remainsto prove that the concrete real i ty answers to the def ini t ion.But as th i s i s an exper imenta l t ru th , i t i s no t the bus inessof m ath em atics to es tabl ish i t . I t i s a grea t step forward tohav e separa ted these two th in gs ; never the less the re i s some-


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1 9 0 0 . ] P A R I S C ON GR ES S O F M A T H E M A T I C IA N S . 7 3th in g in th e phi losophic object ion. I n becom ing r igorous,mathematics has assumed a cer ta in character of ar t i f ic ia l i ty ;if i t is clear how questions can be resolved, i t is no longerclear how a nd w hy the y ar ise . W e seek for real i ty ; bu tthis does not res ide in the separate s teps of the demonstrat ion ; i t mus t be sought ra ther in the someth ing tha t makesfor un i ty . T he microscopic exa m inat io n of an e leph antgives no idea of the animal i tself ; the fairy-like structure ofsil icious needles which is all that is left of certain spongescann ot be unde rs tood w i thout re fe rence to the l iv ing spongeby which this form was imposed on the s i l ic ious par t ic les .Logic by i tse l f cannot give the view of the whole which isind ispensab le a l ike to the inven tor and to h im who des i resrea l ly to un de rs tan d the inven to r . Logic , which a loneg ives ce r ta in ty , i s s imply the ins t rument o f demons t ra t ion ;the ins t rument o f d i scovery i s in tu i t ion .Bu t ana lys t s a l so a re inve n tors ; hence th ey cannot a l w ays be proceeding f rom th e gen eral to th e par t ic ula r , asthe rules of formal logic demand, for scientif ic conquests arem ade on ly by genera l iza t ion . Th ere is however a pe r fect ly r igorous process , that of mathematical induct ion, bywhich i t i s possible to pass f rom the par t icular to the general .* F or th e prof itable use of th is , to recognize th e ana logies who se presence m ake s i t appl icable , th e an aly st m us thave the direct feel ing for the uni ty of an argument , for i t ssoul and spir i t ; for him the most abstract ent i t ies must bel iv ing be ings . W h at i s th i s bu t in tu i t io n ? Th is howeverdoes not inval idate the dis t inct ion a l ready drawn, for i t i san int ui t ion ent i r e ly different in n at ur e f rom t he sensiblein tu i t io n founded in im agina t ion a lone , even though psychologis ts may f inal ly pronounce i t a lso to have a sensualfoun dat ion . I t i s th e intu i t ion of pu re logical form, whic htoge ther wi th the in tu i t ion of pure number makes no t on lydemonstra t ion, but a lso discovery, possible to the analyst .Thus a m ong the a na ly s t s i nve n to r s do e x i s t , bu t no t m a ny ;i t r emains t rue tha t the mos t usua l ins t rument o f inven t ionin mathemat ics i s sens ib le in tu i t ion .

On the evening preceding the formal opening of the Congress an inform al reu nio n of the mem bers , abo ut half ofwhom were prese nt , w as held a t th e Caf V ol ta i re . OnTuesday af ternoon, af ter the r is ing of the sect ions , themembers were en te r ta ined a t the Ecole normale supr ieure .At noon on Sunday , Augus t l th , a ve ry success fu l banque t* P o i n c a r , S u r l a n a t u r e d u r a i s o n n e m e n t m a t h m a t i q u e , Bvue demtaphysique et de morale, vol . 2 (1894) , pp . 371-384 .


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7 4 PARIS CONGRESS OF MATHEM ATICIANS. [N O V . ,was he ld a t the Sa l le de l 'Athne-Sa in t -Germain , when , inthe absence of M. Poincar , M. Darboux presided veryplea sant ly . To asts to those presen t , to th e hosts , to theabsent , to M. Darboux, and to the next Congress , were proposed by MM . Darb oux , G e ise r , J . Ta nn ery , S tephanos ,a n d Vassilief. Inv i ta t io ns to recep t ions he ld by the Pres iden t o f the Republ ic and by Pr ince Roland Bonapar te wereaccepted by a number of the members .A very large a t tendance had been expected, on account ofth e add i t ion al a t t ra ct io ns offered by the Ex hib i t ion ; andthe answers to the circulars f irst sent out went far to justifythis ant ic ipat ion, for up to las t December about 1 ,000 mathem at ic ia ns had s ignified their in ten t ion of being presen t , w i th680 m em bers of the ir famil ies . T he m em bership fee wa sfixed at 30 francs, with an additional 5 francs for everym em ber of th e family . As a ma t te r of fact, th e tota l a t t end anc e can har d ly have exceeded 250 in a l l . Th ereseems very l i t t le doubt that a large proport ion were keptaw ay by d is tas te of th e crowds t ha t were supposed to bevis i t ing the Exhibi t ion, and by the rumored dif f icul ty inobtaining accommodat ion, a di f f icul ty that seems to haveexis ted mainly in the c i rculars of the var ious agencies ; butthe great heat of July cer ta inly decided many to s tay awaywho would o therwise have been presen t .T he cou ntr ies repres ented w ere as fol lows : Fra nc e, 90 ;Ge rm any , 25 : U ni ted S ta tes , 17; I t a ly , 15 ; Be lg ium, 13 ; Russ ia , 9 ; A us tr ia , 8 ; Sw itzer land, 8 ; En gla nd , 7 ; Sweden, 7;Denmark, 4; Hol land, 3 ; Spain, 3 ; Roumania , 3 ; Servia , 2 ;Por tugal , 2 ; South America , 4 ; wi th s ingle representa t ivesf rom Turkey , Greece , Norway , Canada , Japan , Mexico .This l is t i s only approximate , as no revised l is t of memberswas i s sued . Am ong the members f rom the Un i ted S ta teswere Professors Allardice , E. W. Brown, Dickson, Ely,Ha ge n , Ha l s t e d , Ha nc oc k , Ha rkne s s , Ke ppe l , Love t t ,Macfar lane, Pel l , Scot t , Str ingham, Webster .Whi le the four l anguages , Engl i sh , French , German, andI ta l ian , were admi t ted on equa l t e rms , by the cons t i tu t ionof t he congresses , th e great prepond eranc e of Fr en ch wasnot iceab le . A t Zurich, th is prepo nde rance existed inf r iend ly in te rcourse , b u t Fre nch and Germ an were abo utequal ly used in the communicat ions , whereas in Par is a l lthe general addresses , and most of the sect ional papers ,we re in Fr en ch , possibly ou t of com pliment to our hosts .Probably a t Baden-Baden French and German wi l l be usedin about equal proport ions as a t Zur ich.The d i s t r ibu t ion of the au thors o f communica t ions among


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1 9 0 0 . ] PARTS CONGRESS OF MATHEM ATICIANS. 7 7cases be beneficial to the author, obliging him to select , subord ina te , and grou p his deta i ls . I t i s to lerably cer ta in th a tif the a uth or re gard s a l l deta i ls as equal ly im po rtan t , h isaud i to rs wi l l r egard a l l a s equa l ly un impor tan t .One thing very forcibly impressed on the l is tener is thatth e pres enta t io n of papers is usu al ly shocking ly bad. Pr esumably the reader desires to be heard and unders tood ; tocompass these ends, ins tead of speaking to the audience, hereads his paper to himself in a monotone that is sometimeshur r ied , som et imes hes i ta t ing , and f requent ly bored . H edoes no t even tak e pa ins to p ronoun ce h i s own langua geclear ly , but s lurs or exaggerates i ts character is t ics , so thathe is of ten bo th tediou s an d inco mp rehensible . The se fai lings are not conf ined to any one nat ional i ty ; on the wholethe I ta l i ans , wi th the i r c lea r and sp i r i t ed enunc ia t ion , comenea rest to being f ree f rom th em . I t would be invidiou s an dim per t inen t to m ent ion nam es ; the spec ia l s inners s i t inbo th high an d low places . B ut i t i s perh aps pard on able torefer to M. Mit tag-Lef l ler ' s presenta t ion of his paper toSect ion IE as showing in ho w ad m irable an d eng aging as tyle th e th ing can be done . I t is not given to everyo ne todo i t w i th thi s char m ; bu t th ere is n o excuse for an ynormally const i tu ted human being, suff ic ient ly versed inmathematics , fa i l ing to interes t a sui table audience for areasonable t ime in tha t which in te res t s himself, a lways provide d th at i t be of suff ic ient nove l ty e i the r in m at t er or inmode of t reatment to jus t i fy him in present ing i t a t a l l .At the Zurich Congress cer ta in mat ters were energet ical lydiscussed in Sect ion V ; extensive support was then givento reso lu t ions in favor o f cons t i tu t ing permanent commiss ions charged to consider 1 general repor ts on the progressof math em at ics , 2 m at te r s o f b ib l iography and te rm inol ogy* 3 * n e possibi l i ty of giving some permanent characterto the Congress , by means of a centra l bureau or otherwise .Th ou gh these reso lu t ions were no t vo ted upon d i rec t ly , i tbeing fe l t that they required more del iberate discussion,ye t a t th e conclud ing general m eet ing the m em bers of th eZurich bureau were appointed a commission to consider theque st ions th at seemed of m ost im porta nce , an d to furnishthe Mathemat ica l Soc ie ty o f France wi th such in format ionon these points as might be useful in preparat ion for theCo ngress of 1900. A t th e joi nt si t t in g of Sections V an dY I in Pa r is M. Dick ste in asked a quest ion on behalf of th emembers , namely , was no t the Congress to hear any th ingof th e de libe ratio ns of th is com m ission ? ISTo satisfactoryansw er was fo rthcoming ; M. L a isan t rep l ied s imply th a t


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7 8 PAR IS CONGRESS OF MATH EMATICIANS. [N O V . ,th e M athe m atica l Socie ty of Fr an ce h ad been so tak en u pwi th mate r ia l p repara t ions fo r the Congress tha t i t had no tbeen able to ente r upon an y of these m at te rs . H e took theoppor tun i ty , however , o f d i rec t ing a t ten t ion to the Annuairedes Mathmaticiens, projected by Carr and Naud, as carryingin to effect one suggestion m ad e at Zu rich. T he qu estionthen dropped , bu t i t was fe l t th at th is lef t ma t ter s in a ve ryunsa t isfactory s ta te . I t i s to be hoped th at a di fferent repor t wi l l be given in 1904, that the members of the Badencongress wil l not s imply hear papers and meet f r iends, butha ve a chanc e to consider these m at te rs of in te rna t ion alcon cern. Some questio ns of bus iness arise in every science ;they tend to se t t le themselves by a kind of tenta t ive process, a survival of the f i t tes t , or ra ther by general tac i t consent . T his is of ten th e best process ; any a t tem pt a t forc ingan express ion of general agreement may resul t in checkingdevelopment by encouraging too ear ly a crysta l l iza t ion.But the re a re some mat te r s , depending on concer ted ac t ion ,that are r ipe for decis ion, and that cannot prof i tably be se t t led by any one nation for i tself ; matters in which for wantof a gen eral agre em ent labor m ay be wa sted. Such m at te rsm ay na tura l ly be dec ided by an in te rn a t ion a l congress ,whose decisions will simply have the force of general conse nt. One suc h qu estio n is th at of a classification of m at hem atical sc iences . A t leas t two wel l -know n systems are inus e , an d there m ay be oth ers . M ult ipl icat ion of systems ofclass i f icat ion, l ike the mult ipl icat ion of universal languages ,pract ical ly destroys the good of any and a l l ; the congressou ght to pron oun ce in favor of some one. As to the prep ara t ion of specia l repor ts , i t seems doubtful whether th is wi l lbe don e best by th e congress a t prese nt . In course of t im ei t may assume academic funct ions and responsibi l i t ies , buti t will be necessary for i t to prove i ts continuity before i tcan w ith an y prop r ie ty expect to c ontrol m athe m atica l effor ts . En co ura ge m en t an d recogni t ion wo uld seem to bei ts ap pro pr ia te province a t prese nt in these respects . T heque st ion as to how thi s c on t inu i ty is to be obta ined is ara th er ser ious one, an d deserv ing of careful discussion. Acentra l bureau with var ious funct ions was suggested whenth e m at te r was un de r d i scuss ion a t Zur ich ; bu t the re a reobjection s to thi s. If th e organize rs of each congress willmake i t a point of honor to act on the recommendat ions ofthe preceding congress , taking into considerat ion the resolu t ions passed an d do ing w ha t can be done towards ca r ry ingthem into effect , possibly sufficient continuity may be atta ined without the red tape that would coi l i t se l f about any


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1 9 0 0 . ] MEETING OF THE AMERICAN ASSOCIATION. 7 9pe rm an en t bureau . Some ques t ions a re be t te r le ft un dec ided . In te rna t ion a l agreem ent i s no t wan ted on a ll po in t s ;in te rna t iona l r iva l ry and emula t ion s t i l l have the i r pa r t top lay , he lped by the in te rn a t ion a l f r iendsh ips th a t a re p romoted by such ga ther ings as these in te rna t iona l congresses .In conc lus ion , I mus t express my thanks to many ofthose whose names appear in th is repor t for the ass is tanceI have received f rom them in i ts preparat ion.C H A R L O T T E A N G A S S C O T T .B E Y N M A W R C O L L E G E ,October,1900.

T H E F O R T Y - N I N T H A N N U A L M E E T I N G O F T H EA M E R I C A N A S S O C I A T I O N F O R T H E A D V A N C E M E N T O F S C IE N C E .T H E for ty -n in th an nu a l me e t ing of the Am er ican Asso

cia t ion for the Advancement of Science, which was held a tColumbia Univers i ty June 23-30, was f rom the point of viewof scientif ic work one of the most successful in the historyof th e A ssocia tion. Sixteen affi liated so cieties m et w ith th eAssoc ia t ion and con t r ibu ted grea t ly to the impor tance andinte res t of th e m eet ing. Tw o of these , th e Am erican M athema t ica l Soc ie ty and the As t ronom ica l and As t rophy s ica lSociety, held joint sessions with Section A.The next meet ing of the Associa t ion wil l be held in Denver dur ing the las t week of August , 1901, under thepres iden cy of Professor Mino t of th e H ar v ar d MedicalSchool . Professor Ja m es M cM ahon, Cornel l U nive rs i ty ,w ill be vice -pres iden t of Section A, an d Professor H . C.Lorcl , Ohio Sta te U nive rs i ty , wi l l be secretary . Fo r ty- on eout of the tota l number of for ty-nine annual meet ings of theAssoc ia t ion ha ve been h e ld d ur ing t he m onth of Au gus t ,whi le the recen t New York mee t ing was the f i r s t tha t washe ld in Ju n e . Th e nex t me e t ing wi l l be fa r the r wes t th anhi t he r to , bu t i t seemed to b e th e gen eral opinion t h at th iswas desirable in order to extend the inf luence of the Assoc ia t ion . P i t t sb urg was recomm ended as the p lace of mee t ing in 1902.

T he mee t ings of the sec t ion of m athem at ics and as t ron om yw ere well at t en de d. T h e officers of th is section w ere : vice-p r e s ide n t , Asa ph Ha l l , J r .; secre ta ry , W. M. S t rong ; coun-

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