Recent Titles in This Series

123 M . A * Akivis and B. A. Rosenfeld, Eli e Carton (1869-1951), 199 3 122 Zhan g Guan-Hou, Theor y of entire and meromorphic functions: Deficien t an d asymptotic values and singular directions, 199 3 121 LB . Fesenko and S. V. Vostokov, Loca l fields and their extensions: A constructive approach, 199 3 120 Takeyuk i Hida and Masnyuki Hitsnda, Gaussia n processes, 199 3 119 M . V. Karasev and V. P. Maslov, Nonlinea r Poisson brackets. Geometry and quantization, 199 3 118 Kenkich i Iwasawa, Algebrai c functions, 199 3 117 Bori s Zilber, Uncountabl y categorical theories, 199 3 116 G . M. Fel'dman, Arithmeti c of probability distributions, and characterization problem s on abelian groups, 199 3 115 Nikola i V. Ivanov, Subgroup s of Teichmuller modular groups, 199 2 114 Seiz o ltd, Diffusio n equations , 199 2 113 Michai l Zhitomirskii, Typica l singularities of differential 1-form s and Pfafna n equations, 199 2 112 S . A. Lomov, Introductio n to the general theory of singular perturbations, 199 2 111 Simo n Gindikin, Tub e domains and the Cauchy problem, 199 2 110 B . V. Shabat, Introductio n to complex analysis Part II. Functions of several variables, 1992 109 Isa o Miyadera, Nonlinea r semigroups, 199 2 108 Take o Yokonnma, Tenso r spaces and exterior algebra, 199 2 107 B . M. MakaroY, M. G. Golnzina, A. A. Lodkin, and A. N. Podkorytov, Selecte d problem s in real analysis, 199 2 106 G.-C . Wen, Conforma l mapping s and boundary value problems, 199 2 105 D . R. Yafaev, Mathematica l scattering theory: Genera l theory, 199 2 104 R . L. Dobrushin, R. Kotecky, and S. Shlosman, Wulf f construction: A global shape fro m local interaction, 199 2 103 A . K. Tsikh, Multidimensiona l residue s and their applications, 199 2 102 A . M. II' in, Matchin g of asymptotic expansions of solutions of boundary value problems, 199 2 101 Zhan g Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Don g Zhen-xi, Qualitativ e theory of differential equations , 199 2 100 V . L. Popov, Groups , generators, syzygies, and orbits in invariant theory, 199 2 99 Nori o Shimaknra, Partia l differential operator s of elliptic type, 199 2 98 V . A. Vassiliev, Complement s of discriminants of smooth maps: Topolog y and applications, 199 2 97 Itir o Tamura, Topolog y of foliations: A n introduction, 199 2 96 A . I. Marknshevich, Introductio n to the classical theory o f Abelian functions, 199 2 95 Guangchan g Dong, Nonlinea r partial differential equation s of second order, 199 1 94 Yu . S. Il'yashenko, Finitenes s theorems for limit cycles , 199 1 93 A . T. Fomenko and A. A. Tnzhilin, Element s of the geometry and topology of minimal surfaces i n three-dimensional space, 199 1 92 £ . M. Nikishin and V. N. Sorokin, Rationa l approximations and orthogonality, 199 1 91 Mamor u Mimnra and Hlrosi Toda, Topolog y of Lie groups, I and II, 199 1 90 S . L. Sobolev, Som e applications of functional analysis in mathematical physics, third edition, 199 1 (Continued in the back of this publication) This page intentionally left blank Elle Cartan (1869-1951) ;{V ^

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EU E CARTA N Apri l t, ?S69~May6 » 195 1 10.1090/mmono/123 Translations o f MATHEMATICAL MONOGRAPHS

Volume 12 3

Elie Carta n (1869-1951)

M. A. Akivis B. A. Rosenfel d

American Mathematical Societ y I Providence , Rhode Island 9JIM KAPTA H (1869-1951 )

M. A . AKHBH C B. A . Po3eH(|)ejib,n ;

Translated b y V . V . Goldber g fro m a n origina l Russia n manuscrip t Translation edite d b y Simeo n Ivano v

2000 Mathematics Subject Classification. Primar y 01A70 ; Secondary 01A60 , 01A55 .

ABSTRACT. The scientific biograph y o f one of the greatest mathematicians o f the 20th century, Eli e Cartan (1869-1951) , i s presented, a s well as the development o f Cartan's idea s by mathematician s of the followin g generations .

Photo credits : p . iv-Centre National de la Recherche Scientifique ; pp . 2, 3, 9 , 10 , 17 , 19 , 25, 27 , 28, 29-Henri Cartan ; p . 31-Department o f Geometry, Kaza n University , Tatarstan , Russia

Library o f Congres s Cataloging-in-Publicatio n Dat a Akivis, M . A . (Mak s Aizikovich ) [Elie Kartan (1869-1951) . English ] Elie Carta n (1869-1951)/M . A . Akivis , B . A . Rosenfeld ; [translate d fro m th e Russia n b y V. V . Goldberg ; translatio n edite d b y Simeo n Ivanov] . p. cm.—(Translation s o f mathematical monographs , ISS N 0065-9282 ; v. 123 ) Includes bibliographica l references . ISBN 0-8218-4587- X (acid-free ) 1. Cartan , Elie , 1869-1951 . 2 . Mathematicians-Prance-Biography . 3 . Li e groups . 4. Geometry , Differential . I . Rozenfel'd , B . A . (Bori s Abramovich ) II . Title. III . Series .

QA29.C355A6613 199 3 93-693 2 516.3'76'092-dc20 CI P

Copyright © 199 3 by the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e granted t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Information o n copyin g an d reprintin g ca n b e foun d i n the bac k o f this volume . This volum e wa s typeset usin g ^v^vS-TgX , the America n Mathematica l Society' s T^ X macr o system . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 6 1 5 1 4 1 3 1 2 1 1 Contents

Preface x i Chapter 1 . Th e Life and Work of E . Cartan 1 §1.1. Parents'hom e 1 §1.2. Studen t at a school and a lycee 2 §1.3. Universit y student 4 § 1.4. Docto r of Scienc e 6 §1.5. Professo r 8 §1.6. Academicia n 1 7 §1.7. Th e Cartan famil y 2 4 §1.8. Carta n and the mathematicians o f the world 2 7 Chapter 2. Li e Groups and Algebras 3 3 §2.1. Group s 3 3 §2.2. Li e groups and Lie algebras 3 7 §2.3. Killing' s paper 4 2 §2.4. Cartan' s thesis 4 5 §2.5. Root s of the classical simple Lie groups 4 6 §2.6. Isomorphism s o f complex simpl e Lie groups 5 1 §2.7. Root s of exceptional complex simple Lie groups 5 1 §2.8. Th e Cartan matrices 5 3 §2.9. Th e Weyl groups 5 5 §2.10. Th e Weyl affine group s 6 0 §2.11. Associativ e and alternative algebra s 6 3 §2.12. Cartan' s works on algebras 6 7 §2.13. Linea r representations of simpl e Lie groups 6 9 §2.14. Rea l simple Lie groups 7 3 §2.15. Isomorphism s o f real simple Lie groups 7 8 §2.16. Reductiv e and quasireductive Li e groups 8 2 §2.17. Simpl e Chevalley groups 8 4 §2.18. Quasigroup s and loop s 8 5 viii CONTENT S

Chapter 3 . Projectiv e Space s and Projective Metric s 8 7 §3.1. Rea l spaces 8 7 §3.2. Comple x spaces 9 3 §3.3. Quaternio n space s 9 5 §3.4. Octav e planes 9 6 §3.5. Degenerat e geometries 9 7 §3.6. Equivalen t geometries 10 1 §3.7. Multidimensiona l generalization s o f the Hesse transfe r principle 10 7 §3.8. Fundamenta l element s 10 9 §3.9. Th e duality and triality principles 11 3 §3.10. Space s over algebras with zero divisors 11 6 §3.11. Space s over tensor products of algebras 11 8 §3.12. Degenerat e geometries over algebras 12 1 §3.13. Finit e geometries 12 3 Chapter 4. Li e Pseudogroups and Pfaffia n Equation s 12 5 §4.1. Li e pseudogroups 12 5 §4.2. Th e Kac-Moody algebra s 12 7 §4.3. Pfaffia n equation s 12 9 §4.4. Completel y integrable Pfaffian system s 13 0 §4.5. Pfaffia n system s in involution 13 2 §4.6. Th e algebra of exterior forms 13 4 §4.7. Applicatio n of the theory of systems in involution 13 5 §4.8. Multipl e integrals, integral invariants, and integra l geometry 13 6 §4.9. Differentia l form s an d the Betti numbers 13 9 §4.10. Ne w methods in the theory of partial differential equation s 14 2 Chapter 5 . Th e Method o f Moving Frames and Differentia l Geometry 14 5 §5.1. Movin g trihedra of Frenet and Darboux 14 5 §5.2. Movin g tetrahedra and pentaspheres of Demoulin 14 7 §5.3. Cartan' s moving frames 14 8 §5.4. Th e derivational formula s 15 0 §5.5. Th e structure equations 1 5 2 §5.6. Application s o f the method o f moving frames 15 3 §5.7. Som e geometric examples 15 4 §5.8. Multidimensiona l manifold s i n Euclidean spac e 15 8 §5.9. Minima l manifolds 16 0 §5.10. "Isotropi c surfaces" 16 2 §5.11. Deformatio n an d projective theor y of multidimensiona l manifolds 16 6 CONTENTS i x

§5.12. Invarian t normalization o f manifolds 17 0 §5.13. "Pseudo-conforma l geometr y of hypersurfaces" 17 4 Chapter 6 . Riemannia n Manifolds . Symmetri c Space s 17 7 §6.1. Riemannia n manifold s 17 7 §6.2. Pseudo-Riemannia n manifold s 18 1 §6.3. Paralle l displacement o f vectors 18 1 §6.4. Riemannia n geometr y in an orthogonal frame 18 3 §6.5. Th e problem of embedding a Riemannian manifol d int o a Euclidean spac e 18 4 §6.6. Riemannia n manifold s satisfyin g "th e axiom of plane" 18 5 §6.7. Symmetri c Riemannian space s 18 6 §6.8. Hermitia n space s as symmetric spaces 19 1 §6.9. Element s of symmetry 19 3 §6.10. Th e isotropy groups and orbits 19 6 §6.11. Absolute s of symmetric spaces 19 8 §6.12. Geometr y of the Cartan subgroup s 19 9 §6.13. Th e Cartan submanifolds o f symmetric spaces 20 0 §6.14. Antipoda l manifolds o f symmetric spaces 20 1 §6.15. Orthogona l system s of functions o n symmetric space s 20 2 §6.16. Unitar y representations o f noncompact Lie groups 20 4 §6.17. Th e topology o f symmetric spaces 20 7 §6.18. Homologica l algebra 20 9 Chapter 7 . Generalize d Space s 21 1 §7.1. "Affin e connections " and WeyF s "metric manifolds" 21 1 §7.2. Space s with affin e connectio n 21 2 §7.3. Space s with a Euclidean, isotropic, and metric connection 21 5 §7.4. Affin e connection s in Lie groups and symmetric space s with an affine connectio n 21 6 §7.5. Space s with a projective connectio n 21 9 §7.6. Space s with a conformal connectio n 22 0 §7.7. Space s with a symplectic connection 22 1 §7.8. Th e relativity theory and the unified field theory 22 2 §7.9. Finsle r spaces 22 3 §7.10. Metri c spaces based on the notion o f area 22 5 §7.11. Generalize d space s over algebras 22 6 §7.12. Th e equivalence problem and G-structure s 22 8 §7.13. Multidimensiona l web s 23 1 Conclusion 23 5 Dates of Cartan's Lif e and Activities 23 9 List of Publications o f Elie Cartan 241 x CONTENT S

Appendix A . Rappor t su r les Travaux de M. Cartan, by H. Poincare 26 3 Appendix B. Su r une degenerescence de la geometrie euclidienne , by E. Carta n 27 3 Appendix C . Allocutio n d e M. Elie Cartan 27 5 Appendix D. Th e Influenc e o f France in the Development o f Mathematics 28 1 Bibliography 30 3 Preface

The yea r 198 9 marke d th e 120t h birthda y o f Eli e Carta n (1869-1951) , one o f th e greates t mathematician s o f th e 20t h century , an d 199 1 marke d the 40t h anniversar y o f his death. Th e publication o f this book is timed t o these two dates. Th e book i s written b y two geometers working in two dif - ferent branches of geometry whose foundations wer e created by Cartan. Th e mathematical heritag e o f Cartan i s very wide, and there i s no possibility o f describing all mathematical discoverie s made by him, a t least not in a book of relatively modest size. Becaus e of this, the authors pose fo r themselves a much more modest problem—to describe and evaluate only the most impor- tant o f these discoveries. O f course, the authors are onl y abl e to describe i n detail Cartan' s result s connected wit h thos e branches o f geometr y i n whic h the authors are experts. The book consists of seve n chapters. I n Chapte r 1 the outline o f E . Car - tan's life is given, and in Chapters 2-7 hi s main achievement s are described, namely, in the theory of Lie groups and algebras; in applications of these the- ories to geometry; in the theory of Lie pseudogroups; in the theory of Pfaffia n differential equation s an d it s application t o geometr y b y mean s o f Cartan' s method of moving frames; in the geometry of Riemannian manifolds; and, in particular, in the theory of symmetric spaces created by Cartan; in the theory of spaces of affine connection and other generalized spaces. I n the same chap- ters the main routes of the development of Cartan's ideas by mathematicians of the following generations are given. A t the end of the book a chronology of the main events of E. Cartan's life and a list of his works are presented. Th e references t o Cartan's work s are given by numerals without Cartan' s name , and the other references by first letters of the names of the authors, with nu - merals added for multiple references. Th e appendices contain H. Poincare' s reference o n Cartan' s wor k (1912) ; Cartan's pape r On a degeneracy of Eu- clidean geometry, whic h wa s omitted i n his OEuvre Completes-, hi s speech a t the meetin g i n th e Sorbonn e o n th e occasio n o f hi s 70t h birthda y (1939) ; and hi s lecture, The influence of France in the development of Mathematics (1940). Chapter s 1- 3 an d 6 were written by B. A. Rosenfeld, Chapter s 5 and 7 were written by M. A. Akivis, and Chapter 4 was written b y both authors.

XI PREFACE

The authors express their cordial gratitude to Henri Cartan, a son of E. Car- tan, wh o himself i s one o f the greates t mathematician s o f this century , fo r providing numerous facts for a biography o f his father an d fo r pictures fur - nished by him.

Moscow, Russia M. A. Akivis

University Park, PA, U.S.A. B. A. Rosenfeld APPENDIX A

Rapport sur les Travaux de M. Cartan

fait a la Faculte des Sciences de FUniversite d e Pans

PAR R POINCARfi 1

Le role preponderant de la theorie des groupes en mathematiques a ete longtemps insoupgonne; il y a quatre-vingts ans, le nom meme de groupe etait ignore. Ces t GALOI S qui, l e premier, e n a eu une notio n claire , mai s c'es t seulement depuis les travaux de KLEIN et surtout de Lie que Ton a commence a voir qu'il n' y a presque aucun e theori e mathematique o u cett e notion n e tienue une place importante. On avait cependant remarque comment se font presque toujours les progres des mathematiques; c'est par generalisation sans doute, mais cette generalisa- tion n e s'exerce pa s dans un sen s quelconque. O n a pu dir e qu e l a mathe - matique es t Tar t de donner l e meme nom a des choses differentes . L e jour ou on a donne le nom d'addition geometrique a la composition de s vecteurs, on a fait un progres serieux, s i bien que la theorie des vecteurs s e trouvait a moitie faite; on en a fait u n autre du meme genre quand on a donne le nom de multiplication a une certaine operation portant sur les quaternions. I I est inutile de multiplier les exemples, car toutes les mathematiques y passeraient. Par cette similitude de nom, en efFet , on me t e n evidence une similitude d e fait, un e sorte de parallelisme qu i aurai t p u echappe r a l'attention. O n n' a plus ensuite qu'a calquer, pour ainsi dire, la theorie nouvelle sur une theorie ancienne deja connue . II fau t s'entendre , toutefois : i l fau t donne r l e mem e no m a de s chose s differentes, mai s a la conditio n qu e ce s choses soient differente s quan t a la matiere, mais non quant a la forme. A quoi tient c e phenomene mathema - tique si souvent constate? E t d'autre part en quoi consiste cette communaute de forme qui subsiste sous la diversite de la matiere? Ell e tient a ce que toute theorie mathematiqu e est , e n dernier e analyse , Fetud e de s propriete s d'u n groupe d'operations, c'est-a-dire d'un system e forme par certaines operations

lActa Mathematica 38 (1914), 137-145 .

263 264 A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE fondamentales e t pa r toutes les combinaisons qu'o n e n peut faire . Si , dan s une autre theorie, on etudie d'autres operations qui se combinent d'apres les memes lois, on verra naturellement se derouler une suite de theoremes corre- spondant un a un a ceux de la premiere theorie, et les deux theories pourront se developper avec un parallelisme parfait ; il suffir a d'u n artific e d e langage, comme ceux dont nous parlions tout a l'heure, pou r que ce parallelisme de- vienne manifeste e t donne presque Timpression d'une identite complete. O n dit alor s qu e le s deu x groupe s d'operation s son t isomorphe s o u bien qu'il s ont meme structure. Si alors o n depouille l a theorie mathematiqu e d e se qu i n' y apparai t qu e comme u n accident , c'est-a-dir e d e s a matiere , i l n e rester a que Pessentiel , c'est-a-dire la forme; et cette forme, qui constitue pour ainsi dire le squelette solide de la theorie, ce sera la structure du groupe. On distinguera parm i le s groupes possible s quatr e categorie s principales , sans compter certains groupes etranges ou composites qui ne rentrent dans au- cune categorie, ou qui participent des caracteres de deux ou plusieurs d'entre elles. C e sont: I. Le s groupes discontinus et finis, ou groupes de Galois; ce sont ceux qui presiden t a la resolutio n de s equation s algebriques , a la theori e des permutations, etc II. Le s groupes discontinus et infinis; ce sont ceux que Ton rencontre dans l a theorie de s fonctions elliptiques , de s fonction s fuchsienne s etc III. Le s groupes continus et finis ou groupes de LIE proprement dits; ce son t ceu x auxquel s s e rattachen t le s principale s theorie s geome - triques, telle s qu e l a geometri e euclidienne , l a geometri e non - euclidienne, la geometrie projective, etc IV. Le s groupes continus et infinis, beaucou p plus complexes, beau- coup plu s rebelle s au x effort s d u geometrie . li s son t e n connexio n naturelle avec la theorie des equations aux derivees partielles. M. CARTA N a fai t fair e de s progre s important s a no s connaissance s su r ere e e trois d e ce s categories , l a 1 , l a 3 , e t l a 4 . I I s'es t principalemen t place a u poin t d e vu e l e plu s abstrai t d e l a structure , d e l a form e pure , independamment d e la matiere, c'est-a-dire, dan s l'espece, d u nombre e t du choix des variables independantes. Groupes continus et finis

Je commencera i pa r les groupe s continu s e t finis, qu i on t et e introduit s par LI E dan s l a science ; l e savan t norvegie n a fait commaitr e le s principe s fondamentaux d e la theorie, e t il a montre en particulier que la structure de ces groupe s depen d d'u n certai n nombr e d e constante s qu'i l design e pa r la lettre c affecte e d'u n triple indice et entre lesquelles il doit y avoir certaines A RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE 265 relations. D a enseigne egalement commen t on pouvait construir e l e groupe quand on connaissai t ce s constantes . Mai s i l restai t a discuter le s diverse s manieres de satisfaire aux relations qui doivent avoir lieu entre les constantes c; o n pouvait supposer que les divers types de structure seraient extremement nombreux et extremement varies, de sorte que l'enumeration en serait a peu pres impossible. I I ne semble pas eu etre tout a fait ainsi, au moins en ce qui concerne les groupes simples. La distinction entre les groupes simples et les groupes composes est due a GALOIS et elle est essentielle, puisque les groupes composes peuvent toujours etre construit s e n partan t de s groupe s simples . I I est clai r qu e l e premie r probleme a resoudre est la construction des groupes simples. Vers 1890, KILLING a annonce que tous les groupes simples continus et fi- nis rentrent: soit dans quatre grands types generaux deja signales par LIE, soit dans cinq types particuliers don t les ordres sont respectivemen t 14 , 52 , 78, 133, et 248. Cetai t la un resultat d'une tres haute importance; malheureuse- ment toutes les demonstrations etaien t fausses ; il ne restait qu e des apergus denues de toute force probante. II etait reserve a M. CARTAN de transformer ces apergus en demonstrations rigoureuses; il suffit d'avoir lu le memoire de KILLING pour comprendre com- bien cett e tach e etai t difficile . L a method e repos e su r l a consideratio n d e Fequation caracteristique , e t e n particulie r d e l a form e quadratiqu e y/ r{e) qui es t le coefficien t d e o/~ 2 dan s cette equation ; cette consideratio n per - met de reconnaitre si le groupe integrable, ou de trouver son plus grand sous groupe invarian t integrable , o u enfi n d e reconnaitr e s i l e group e es t simpl e ou semisimple. M. CARTAN a donne une maniere de former, dans chaque type, les groupes lineaires simples dont le nombre des variables est aussi petit que possible. Une des plus importantes applications des groupes de LIE est Integration des equation s differentielle s ordinaire s o u partielle s qu i son t inalteree s pa r les transformations d'u n groupe . M . CARTA N a applique cett e method e a u cas des systemes d'equations aux derivees partielles don't Pintegrale generate ne depen d que de constantes arbitraires . Le s operations a faire son t toutes de nature rationnelle ou algebrique.

Groupes discontinus et finis

M. CARTAN a fait faire aussi un progres important a la theorie des groupes de GALOIS , e n les rattachant a celle des nombre s complexes. O n sait qu'o n designe pa r nombres complexes de s expressions algebrique s susceptible s d e subir des operations qui peuvent etre regardees comme des generalisations de Taddition e t d e la multiplication, e t auxquelles on peu t applique r les regies ordinaires du calcul ave c cette differenc e qu e l a multiplication, quoiqu e as- sociative, n'es t pas commutative. L a plus conn u des systeme s d e nombre s 266 A . RAPPORT SUR LES TRAVAUX D E M. CARTAN, B Y H. POINCARE

complexes a reg u l e no m d e quaternion s e t o n e n a fai t de s application s nombreuses en Mecanique e t en Physique Mathematique. Ces nombres complexes ont u n lien intim e ave c les groupes de Lie e t en particulier ave c les groupe s lineaires simplemen t transitifs ; i l y a , a c e su - jet, u n theorem e d e M . POINCAR E don t M . CARTA N a donne un e nouvell e demonstration. L a theorie de s nombres complexe s a ete pousse e plu s loi n par M. M. SCHEFFER S et MOLLIEN qui en ont entrepris la classification e t ont les premiers mis en evidence Pimportance de la distinction entre les systemes a quaternions et les systemes sans quaternions. M. CARTA N es t arriv e a resoudr e completemen t l e probleme , pa r un e heureuse adaptatio n de s methode s qu i lu i avaien t reuss i dan s l'etud e de s groupes de Lie. I I a pris comme point de depart une equation caracteristique qui n'est pas tout a fait la meme que celle qu'on envisage a propos des groupes de Lie, mai s qui se prete a une discussion analogue . M . CARTA N a montre comment on peut construire un systeme quelconque par la combinaison d'un systfeme pseudonu l e t de systemes simple s e t comment le s systemes simple s se reduisent aux quaternions generalises; comment enfi n les systemes dits de la 2 e class e s e deduisen t facilemen t d e ceu x d e l a \ ere classe . I I a etudi e aussi le cas ou les coefficients son t des nombres reels. Ces resultats ne constituent pas, comme on pourrait etre tente de la croire, une simple curiosite mathematique. li s sont au contraire susceptibles dupli- cations nombreuses. E n particulier, ils se rattachent a la theorie des groupes de GALOIS ; i l es t clai r que les lois d e la composition de s substitutions d'u n groupe d e GALOI S son t associatives , san s etr e commutatives ; elle s peuven t done etr e regardee s comm e le s regie s d e l a multiplicatio n d'u n system e d'unites complexes ; et pa r consequent elle s definissen t u n systeme d e nom- bres complexes. O r si on applique a ce systeme le theoreme de M. CARTAN , on retrouve, de la fagon la plus simple et pour ainsi dire d'un trait de plume, les resultats que M. FROBENIU S avait obtenus par une tout autre voie et qui avaient ete regardes a juste titre comme le plus grand progres que la theorie des groupes de GALOI S eut fait depuis longtemps. On peut , pa r cette voie , reconnaitr e quel s son t le s groupe s lineaire s le s plus simples qui sont isomorthes a un groupe de GALOI S donne, ce qui nous conduit a u probleme de l'integration algebriqu e des equations differentielle s lineaires. M . POINCAR £ a eu Poccasion d'applique r les principes de M . CAR - TAN a l'integration algebriqu e d'une equation lineaire.

Groupes continus et infini s

La determination de s groupes continus infini s present e beaucoup plu s de difficultes qu e celle des groupes finis et c'est l a que M . CARTA N a deploye le plus d'originalit e e t d'ingeniosite . I I s'est restrein t d'ailleur s a une certain e classe d e groupe s infinis , l a plu s important e a u poin t d e vu e de s applica - tions, e t celle su r laquelle l'attention d e Lie avait surtou t ete attiree, je veux A. RAPPOR T SUR LES TRAVAUX DE M. CARTAN, B Y H. POINCARE 26 7 parler des groupes dont les transformations finies dependent de fonctions ar- bitrages d'u n o u de pluseurs parametres, ou, plus generalement, d e ceux oil les variables transformees, considerees comme fonctions de s variables prim- itives, constituen t l'integra l genera l d'u n system e d'equation s au x derivee s partielles. M. CARTAN s'est d'ailleurs servi, dans cette etude, de resultats importants qu'il avai t obtenu s dan s de s travau x anterieur s relatif s au x equation s au x derivees partielle s e t au x equations d e PFAFF , travau x don t nou s parleron s plus loin. La theorie d e la structure, telle que LIE l'expose dan s I'etude des groupes finis, n'est pa s susceptible d'etr e immediatemen t generalise e e t etendue au x groupes infinis. M . CARTA N lui substitue done une autre theorie de la struc- ture, equivalente a la premiere en ce qui concerne les groupes finis, mais sus- ceptible d e generalisation. S i / es t un e fonction quelconqu e de s variable s x, e t si les XJ represente d les symboles de Lie, on aura identiquement:

*/+£*,/*>, = ° les a) i etan t des expressions de Pfaff dependant des parametres du groupe et de leurs differentielles . Au lieu de faire jouer le role essentiel aux symboles XJ , comm e le faisait LIE, M . CARTAN l'attribue aux expressions de PFAFF W qu i sont invariantes par les substitutions d u groupe des parametres. Le s relations qui definissen t la structur e s e presenten t alor s sou s un e autr e forme . A u lie u d e relation s lineaires entre les X f e t leurs crochets, nous aurons des relations lineaires entre les covariants bilineaires des co et des combinaisons bilineaires de ces meme expressons . L e coefficients d e ce s relation s son t le s meme s dan s le s deux cas, quoique dans un autre ordre; ce sont les constantes c d e LIE. Sans sortir encore du domaine des groupes finis, M. CARTAN a illustre cette theorie nouvelle en Fappliquant a des exemples concrets, et en particulier au groupe des deplacements d e I'espace; il a montre comment ell e s e rattachait a l a theori e classiqu e d u triedr e mobil e d e M . DARBOU X e t commen t ell e permettait I'etude des invariants differentiels de s surfaces et en particulier de ceux de certaines surfaces imaginaires remarquables. Voyons maintenan t commen t ce s notion s peuven t etr e etendue s au x groupes infinis. L a notion d'isomorphisme holoedriqu e peut etre facilement definie en ce qui concerne les groupes finis, parce que Ton n'a qu'a faire cor- responds un e a une les transformations infinitesimale s de s deu x groupe s a comparer. Nou s n e pouvons plus employer ce profced e lorsqu e les transfor - mations infinitesimale s son t en nombre infini ; M . CARTA N donn e done un e definition differente, quoique equivalente a la premiere dans le cas ou celle-ci a u n sens . U n group e es t l e prolongemen t d'u n autr e quand i l transform e les memes variables que cet autre et de la meme maniere et qu'il transform e 268 A. RAPPORT SUR LES TRAVAUX DE M. CARTAN, BY H. POINCARE en mem e temp s d'autre s variable s auxiliaires . Pa r exemple , l e group e de s deplacements de s points d e l'espac e aur a pour prolongement l e group e de s deplacements des droites ou celui des cercles de l'espace. Deu x groupes sont alors isomorphes quan d deux de leurs prolongements sont semblables. La theoreme fondamental de LIE peut alors etre etendu aux groupes infinis; on montre que tout groupe infini es t isomorphe a u groupe qui laisse invari - antes a la fois certaines fonctions U e t certaines expressions de PFAFF CO e t cb. Le s differentiates totale s des U s'experimen t lineairemen t e n fonction s des co , les covariants bilineaires des co (mais non ceux des cb) s'exprimen t bilineairement e n fonction s de s co et cb. Le s coefficient s d e ces relation s lineaires ou bilineaires jouent l e role des constantes c d e LIE . C e sont de s fonctions de s invariant s U . C e qui caracteris e le s groupe s transitifs , c'es t qu'il n'y a pas d'invariants et par consequent que les coefficients s e reduisent a des constantes. C e qui caracterise les groupes finis, c'est que les expressions n'existent pas. Les coefficients e n question peuvent-ils etr e choisis arbitrairement ? Non , ils son t assujetti s a certaines condition s qu e M . CARTA N determin e e t qu e peuvent etr e regardee s comme l a generalization de s conditions de structur e de LIE. Les trois theoreme s fondamentau x d e LI E s e trouvent don e etendu s au x groupes infinis, d e sorte que M. CARTA N a fait pou r ces groupes ce que LI E avait fait pour les groupes finis. Cette analyse a mis en evidence de s resultats tout a fait surprenants . U n groupe fini est toujours isomorphe a un groupe transitif, pa r exemple a celui qu'on appelle so n groupe parametrique, e t on aurai t p u etre tente de croire qu'il e n etait d e meme pour les groupes infinis , puisqu'a u premie r abord la demonstration ne semblait mettre en ceuvre que la notion generate de groupe. Au contraire, M. CARTAN a montre qu'il existe les groupes infinis qui ne sont isomorphes a aucun groupe transitif . Ce n'est pas tout: u n groupe infini peu t etre meriedriquement isomorph e a lui-meme , u n group e infin i peu t n'admettr e aucu n sou s group e invarian t maximum, etc.,.... L a notion d u prolongemen t norma l perme t ensuit e a M. CARTA N d e determine r tou s le s groupe s isomorphe s a un group e infin i donne. Citon s un resultat particulier. Le s groupes qui ne dependent que de fonctions arbitrage s d'un argument , s'il s sont transitifs, son t isomorphes au groupe general d'une variable. Etant donne u n groupe defini pa r ses equations d e structure, M. CARTA N montre qu'on peut determiner les equations de structure de tous ses-groupes par des proced e puremen t algebrique s e t appliqu e cett e method e a des ca s particuliers tels que celles du groupe general de deux variables ou il retrouve, par une voie nouvelle, quelques sous groupes deja connus et importants par leurs applications. Si Ton se donne deux systemes differentiels et un groupe, on peut se deman- der s'il y a des transformations d u groupe qui transforment u n des systemes A. RAPPOR T SUR LES TRAVAUX DE M. CARTAN, B Y H. POINCARE 26 9

dans l'autr e e t quelle s elle s sont ; o n peu t s e demande r egalemen t s'i l y a dans le groupe des transformations qu i n'altereront pas Tun de ces systemes differentiels e t qu i naturellemen t formeron t u n sous-groupe . L'etud e d e c e sous-groupe a fait egalemen t l'objet d'u n memoir e de M. CARTA N . Enfin M . CARTA N s'es t propose e n c e qui concerne les groupes infinis , l e meme probleme qu'il avait resolu pour les groupes finis, la formation d e tous les groupe s simples . I I a montr e qu'ic i aussi , le s groupes simple s peuven t se ramener a un nombr e restrein t d e types; ceu x qui son t primitif s e t d'o u Ton peu t deduir e tou s le s groupe s transitif s simple s s e repartissen t e n si x grandes classes ; quant au x groupes simples qui ne sont isomorphe s a aucu n groupe transitif, ils peuvent etre deduits des precedents par des procedes des procedes que M. CARTAN nous fait connaitre . Le probleme propose se trouve done entierement resolu .

Equations aux derivees partielles

Le probleme d e 1'integration d'u n system e d'equation s au x derivee s par - tielles a fai t l'obje t d e travau x nombreux . M . CARTA N s'es t plac e pou r l'etudier a u n poin t d e vu e particulier ; i l remplac e l e system e d'equation s aux derivee s partielles par l e systeme correspondan t d'equation s d e PFAFF , e'est-a-dire d'equations au x differentielles totales . Dans l a theori e de s expression s d e PFAFF , i l y a un e notion , introduit e par M . M . FROBENIU S e t DARBOUX , qu i joue u n rol e extrememen t impor - tant, c'es t cell e d u covarian t bilineaire ; nou s avon s dej a v u apparaltr e c e covariant a propos de la theorie des groupes infinis. M . CARTAN en a donne une interpretatio n nouvell e a l'aid e d u calcu l d e GRASSMANN , e t cett e in - terpretation l'a conduit a une generalisation. D e chaque expression de PFAFF , il deduit un e seri e d'expression s differentielle s qu'i l appell e se s derivees; l a derivee premier e es t l a covarian t bilineaire ; l a derive e n e es t n + 1 foi s lineaire. C'es t en cherchant quelle est la premiere de ces derivees qui s'annul e identiquement qu e To n reconnaitr a si , e t jusqu'a que l point , i l es t possi - ble d e reduir e l e nombr e de s variable s independante s su r lesquelle s port e l'expression. Cette consideratio n a permis a M. CARTA N d e retrouver sou s une form e extremement simpl e tous les resultats connus relatifs au probleme de PFAF F et un asse z grand nombre de resultats entierement nouveaux . Comment maintenan t cel a peut-i l servi r a l a resolutio n d'u n system e d'equations d e PFAFF , e t surtout a reconnaitre que l est l e degre d'arbitrair e que comporte Fintegrale generale d'un pareil systeme? C'es t e n se servant de la notion d'involutio n qu e M. CARTA N a resol u cett e question. U n system e est di t e n involutio n si , jusqu'a un e certain e valeu r d e m , pa r tout e mul - tiplicity integrat e a m dimension s pass e une multiplicity integrat e a m + 1 dimensions. M . CARTAN donne une maniere de reconnaitre si un systeme est 270 A . RAPPORT SUR LES TRAVAUX D E M. CARTAN, B Y H. POINCARE

en involution pour les valeurs de m inferieure s a un nombr e donne, et, par la, de savoir combien la solution generate contient de fonctions arbitrages de 1, de 2, .. . , de n variables . On retrouve ainsi sous une forme nouvelle la theorie des caracteristiques de CAUCHY, celle des caracteristiques de MONGE, celle des solutions singulieres, etc.,... ; on retrouve egalement sou s une forme plu s simple tous les resultats de M . RIQUIER . M. CARTAN a applique sa methode a un certain nombre de cas particuliers ou Pintegratio n peu t s e faire pa r des equations differentielle s ordinaires . I I Fa egalement complete e e n s'aidant d e la theorie des groupes qui lui etait s i familiere; i l a ains i reconn u de s ca s o u To n peu t determine r le s invariant s d'un system e de PFAFF , san s en determine r le s caracteristiques, c'est-a-dir e d'une fago n rationnelle , e t d'autres o u les caracteristiques s'obtiennen t san s integration.

Conclusions

On voit que les problemes traites par M. CARTA N sont parmi les plus im- portants, les plus abstraits et les plus generaux don't s'occupent le s Mathema- tiques; ainsi que nous Favons dit, la theorie des groupes est, pour ainsi diew, la Mathematiqu e entiere , depouille e d e s a matier e e t reduit e a un e form e pure. Ce t extrem e degr e d'abstractio n a san s dout e rend u mo n expos e u n peu aride ; pou r fair e apprecie r chacu n de s resultats , i l m'aurai t fall u pou r ainsi dire lui restituer l a matiere dont i l avait ete depouille; mais cette resti- tution peu t s e fair e d e mill e fa?on s differentes ; e t c'es t cett e form e unqu e que To n retrouve ains i sou s une foule d e vetements divers , que constitue l e lien commu n entr e de s theorie s mathematique s qu'o n s'etonn e souven t d e trouver s i voisines. M. CARTA N e n a donn e recemmen t u n exempl e curieux . O n connai t Fimportance e n Physiqu e Mathematique s d e c e qu'o n a appel e l e group e de LORENTZ ; c'es t su r c e group e qu e reposen t no s idee s nouvelle s su r l e principe d e relativit e .et su r Dynamiqu e d e FElectron . D'u n autr e cot e , LAGUERRE a autrefois introdui t e n geometrie un groupe de transformation s qui changen t le s sphere s e n spheres . Ce s de s groupes son t isomorphes , d e sorte que mathematiquement ces deux theories, l'une physique, l'autre geome- trique, ne presentent pa s de difference essentielle . Les rapprochement s d e c e genr e s e presenteron t e n foul e a ceu x qu i etudieront ave c soin le s travaux de LI E e t de M. CARTAN . M . CARTA N n'e n a pourtant signal e qu'un peti t nombre , parce que, courant a u plus presse, i l s'est attach e a la form e seulemen t e t n e s'es t preoccup e qu e raremen t de s diverses matieres dont o n la pouvait revetir . Les resultat s le s plu s important s enonce s pa r M . CARTA N lu i appartien - nent bien e n propre. E n c e qui concerne le s groupes d e LIE , o n n'avai t qu e A. RAPPOR T SU R LE S TRAVAUX D E M CARTAN , B Y H. POINCAR E 27 1 des enonces et pas de demonstration; en ce qui concerne les groupes de GA- LOIS, o n avai t le s theoremes de FROBENIU S qui avaien t et e rigoureusement demontres, mai s par un e method e entieremen t differente ; enfi n e n c e qui concerne les groupes infinis on n'avait rien: pou r ces groupes infinis, Foeuvre de M. CARTAN correspond a ce qu'a ete pour les groupes finis Toeuvre de LIE, celle de KILLING, et celle de CARTAN lui-meme. This page intentionally left blank APPENDIX B

Sur une degenerescence de la geometrie euclidienne PAR

M. ELI E CARTA N

Professeur a la Faculte des Sciences de Paris1

La geometrie dans un plan isotrope differe profondemen t d e la geometrie plane classique ; le s lignes qui jouent dan s u n plan nonisotrop e l e role des circonferences sont, dans un plan isotrope, des paraboles toutes tangentes en un mem e poin t a la droute d e Finfini. S i Ton prend pou r ax e des y un e parallele a la direction isotrop e uniqu e d u plan, l e groupe d e la geometrie euclidienne du plan isotrope est la forme: = x + a, {/ = ex + hy + b, Pare elementaire ds d'un e courbe etant reduit a dx . L a notion ordinaire de courbure disparait , mai s i l s' y substitu e un e pseudocourbure egal e a 4T^ , lorsque la courbe est definie par y = f(x). Le groupe (1 ) est un sous-groupe du plus gran d group e affin e qu i laisse invariant le point a Tinfini dans la direction Oy , a savoir:

/2x fx^kx +a, \y' = cx + hy +b; un autre sous-groupe invariant de ce dernier, a savoir le groupe (3) = kx +a, {/': ex + y +b, peut etre pris comme base d'une geometrie plane a direction isotrop e privi- legiee. Dan s cette geometrie, qui est en un certain sen s une degenerescence de la geometrie euclidienne, on peut definir la longueur d'un vecteur parallele

Assoc. Franc. Avanc. des Sciences, 59* session , Nantes, 1935 , 128-130 .

273 274 B . SU R UN E DEGENERESCENC E D E L A GEOMETRI E EUCLIDIENNE , B Y E . CARTA N a la direction isotrope comme etant la difference de s ordonnees y e t y d e son extremite et de son origine, mais la notion de longueur disparait pour les vecteurs nonisotropes. La geometrie fondee sur le groupe (3) est interessante; on voit tout de suite qu'etant donne e une ligne plane autre qu'une droite, o n peut defini r d'un e maniere intrinseque un element d'arc ds pa r la formule

2 2 2 tA\ J dxd y-dyd x JI ( , , 2 (4) ds = ^ ^ = / (x)dx . Le secon d membr e es t e n effe t l e rappor t d e deu x aires , Fair e d u par - allelogramme construi t su r le s deu x vecteurs {dx , dy) e t {d2x, d 2y), e t Faire du parallelogramme construit sur les deux vecteurs (dx , dy) e t (0 , 1). Cet elemen t d'arc est identiquement nu l quand l a ligne consideree es t un e droite. S i Ton attache a chaque point de la ligne deux vecteurs f e t N, l e premier tangent k la ligne et de composantes ^, ^ , le second parallele a Oy e t de longueur 1 , on a les formules de Frenet generalisees:

(5) -di = T; ~dI = kT+N> 17 = °- Le coefficien t k = ^4/^7 = -5 >; ) es t l a courbure. Les courbes d e ds2 ' ds 2 J (*) courbure nulle sont les paraboles tangentes a la droite de Tinfini a u point a 1'infini sur Oy. La courbure est du reste un invariant pour le groupe general (2). Ce quit donn e un certain intere t a la geometrie precedente, c'est qu'ell e se presente d'elle-meme quand on veut chercher des proprietes geometriques intrinsfequement attachee s k une integrale f F(x, y 9 y , y")dx, o u F es t une fonctio n donne e d e x , y, y = $£, y" = ~4 ; une propriet e es t dit e intrinseque si elle ne depend pas du choix des coordonnees x, y. S i la fonc- tion F sereduita -y/y 77, le plus grand groupe qui laisse invariante Tintegrale est precisement le groupe (3). S i F est de la forme J y + ^S^y ), ou A e t B son t des fonctions d e x,y ? on a un e geometrie qu e joue par rapport a la geometrie de groupe (3) le meme role que la geometrie riemannienne par rapport a la geometrie euclidienne, avec cette difference que Fespace doit etre regarde comme engendre non par des points (x, y) mai s par des elements lineaires {x , y, y); Fespace est un espace d'elements lineaires a connexion affine, assimilable au voisinage de chaque element lineaire a un plan euclidien isotrope de groupe (3). Un autre cas particulier interessant es t celui de Fintegrale / y/y dx qui est liee a la geometrie affine unimodulaire . APPENDIX C

Allocution de M. Elie Cartan

A la fin d e cett e emouvant e ceremonie , apre s tou s le s eloge s don t vou s m'avez combl e et que j'ai conscience de n'avoir qu'imparfaitement merites , permettez que ma pensee se reporte vers ceux qui ne sont plus et qui auraient ete s i fiers de le s entendre. J e pens e a mo n per e e t a m a mere , humble s paysans qui pendant leu r longue vie ont donne a leurs enfants Fexempl e du travail joyeusement accompli et des charges vaiUamment acceptees. Cest a u bruit de Fenclume resonnant chaque matin des Faube que mon enfance a ete bercee, et je vois encore ma mere actionnant le metier du canut, aux instants que lui laissaient libres les soins de ses enfants e t les soucis du menage, En mem e temp s qu' a me s parent s je pens e a me s premier s maitres , le s instituteurs de FEcole primaire de mon village de Dolomieu, M. Collomb, et surtout M. Dupuis; ils donnaient a plus de deux cents gargons un enseigne - ment precis dont j'appreciai plus tard la valeur. Je suis oblige d'avouer—et je n'en a i pas honte—que j'etais un excellen t eleve ; j'etais capabl e d'enumere r sans hesitation les sous-prefectures de n'importe quel departement, et aucune subtilite de s regie s du particip e pass e n e m'echappait. U n jour u n delegu e cantonal qu i s'appelai t Antoni n Dubos t e t qu i devai t plu s tar d deveni r u n des plus hauts personnages de FEtat vint inspecter Fecole; cette visite orienta toute m a vie. I I fut decid e que je me presenterais a u concour s de s bourses des lycees; M. Dupuis dirigea ma preparation avec un devouement affectueu x que je n'oublierai jamais. Tou t cela me valut un beau voyage a Grenoble, ou je subis sans trop d'emoi de s epreuves pas trop redoutables. J e fus reg u bril- lamment, ce qui remplit M. Dupuis de fierte et grace a Fappui de M. Dubost, qui s'interessa pendan t toute s a vie avec une affection tout e paternelle a ma carriere et a mes succes, je fu s gratifi e d'un e bourse complete au Colleg e de Vienne. A Fage de dix ans je quittai done joyeux le foyer paternel, sans me douter que bie n pe u d e jours m e suffiraien t pou r regrette r c e qu e je perdais . I I fallut m'adapte r a la vie d'internat qu e je devais mener pendant plu s de dix ans. Apre s cinq an s d e colleg e pendant lesquel s je du s mettr e le s bouchee s doubles, m a bours e fu t transfere e a u Lyce e d e Grenobl e o u j'achevai me s etudes classiques par la rhetorique et la philosophic, puis au Lycee Janson-de- Sailly, qui etait dans toute la fraicheur d e sa premiere jeunesse, rayonnant du

275 276 C. ALLOCUTIO N D E M . ELI E CARTAN succes que venait d'obtenir Le Dantec re$u premier a l'Ecole Normals J'eu s a Janso n de s professeurs remarquables , Salomo n Bloc h e n mathematique s elementaires A , et en mathematiques speciale s Emile Lacour dont tu a s su, mon cher Tresse, sans Pavoir connu comme professeur, depeindre la noblesse de caractere. C'est dans cette classe que j'eus comme camarade, avec Eugene Perreau qui devait entrer avec moi a l'Ecole Normale, Jean Perrin, plus jeune que nous , e t qui devai t deveni r une de s plus grandes gloire s de l a scienc e frangaise. Cest ave c emotion , mo n che r Tresse , qu e je t'a i entend u evoque r no s annees d'Ecole Normale. Je ne suis pas sur que le recul du temps n'ait pas em- belli le souvenir que tu as garde de moi et du role que j'aurais joue aupres de mes camarades. C e que je me rappelle, c'est en effet une camaraderie frater - nelle et une collaboration qui s'est montree surtout assez etroite dans Pannee de preparation a Fagregation . J e vois encore le s seances ou l e soir, reuni s dans une salle quelconque, nous ecoutions Tun de nous exposer la legon qu'il devait faire le lendemain. L a les critiques etaient libres et franches e t com- bien profitables. J e me rappelle particulierement un e legon sur Pintersection des quadriques qu i nous frappa pou r la maniere elegant e et neuv e dont l a question etait congue; Pauteur de cette legon etait Arthur Tresse. Tu as parle tout a l'heure, mo n cher ami, de Padmiration qu e nous pro- duisaient les cours de M. Emile Picard, qui excellait a nous ouvrir de vastes perspectives dans un domaine encore nouveau pour nous . A l'Ecole mem e c'est Jule s Tannery qui exerga sur nous la plus profonde influence ; pa r une sorte de transposition mysterieus e due a Fensemble de toute sa personne, a son regard peut-etre, le respect de la rigueur dont il nous montrait la necessite en mathematiques devenait une vertu morale, la franchise, l a loyaute le re- spect de soi-meme. Comm e on T a dit deja, Tannery etait notre conscience: c'est pourquoi nous Faimions, c'est pourquoi nous avons voue a sa memoire un culte fidele. Nous admirions auss i 1'eleganc e de certaine s conference s d e Kcenigs , la clarte de l'enseignement d e Goursat. A la Sorbonne c'etait l a limpidite des cours de Mecanique rationnelle d'Appell, 1'eleganc e incomparable de s cours de Darboux. Le s legons qui nous produisaient 1'impressio n l a plus profonde peut-etre etaien t celle s d'Hermite, don t l e visag e e t le s yeux d'un e beaut e admirable s'illuminaien t comm e s'i l contemplai t a u sei n d e l a Divinit e c e monde eternel des nombres et des formes don t nou s parlait tout a l'heureu M. Picard. Tannery, Goursat, Appell, Darboux, Picard, Hermite, que de grands noms s'offraient a Padmiration d e notre jeunesse. J e n'ai pa s parle du gean t de s Mathematiques, Henri Poincare, dont les legons passaient bien au-dessus de nos tetes ; i l n'es t aucun e branch e de s mathematique s moderne s qu i n'ai t subi so n empreinte , e t vou s comprendre z qu e je gard e a s a memoir e un e particuliere reconnaissance puisque le dernier travail de sa vie si brusquement interrompue a ete un rapport su r mon ceuvr e scientifique. D e cette illustre C. ALLOCUTIO N D E M . ELI E CARTAN 277 pleiade de grands mathematiciens, vous seul, mon cher Maitre, nous restez; nous admiron s toujour s votr e jeuness e e t je m e felicit e qu e mo n ag e m e donne encore le privilege d'entendre retrace r m a carriere scientifiqu e pa r l e maitre admire qui, il y a un demi-siecle, m'initiait a l'Analyse mathematique, presentait me s premiere s note s a FAcademi e e t etai t l e rapporteu r d e mo n jury de these. Apres ma these dont le sujet, tu Fas peut-etre oublie, mon cher Tresse, me fut signal e par toi a ton retour d e Leipzig ou tu avai s ete Felev e de Sophu s Lie, je fus nomme maitre de conferences a Montpellier. J e garde le meilleur souvenir des quinze ans que j'ai passes en province, a Montpellier d'abord, a Lyon, et a Nancy ensuite. C e furent de s annees de meditation dans le calme, et tou t c e qu e j'ai fai t plu s tar d es t conten u e n germ e dan s me s travau x murement medites de cette periode. C'es t a Nancy que je commenga i a m e familiariser avec les vastes auditoires. J'avai s a y enseigner le s elements de l'Analyse aux eleves de FInstitut electrotechnique et de Mecanique appliquee. Institut encor e jeune, mai s dej a prosper e sou s la direction d e Fhomm e a u devouement admirabl e qu'etai t Vogt . Ce t enseignement m'interessai t beau - coup e t j'eus la satisfactio n d e senti r tou t d e suit e l e contac t s'etabli r ave c les eleves. J e me trouvai ainsi prepar e a Fenseignement de s mathematique s generates qui devait m'etre confi e un peu plus tard a la Sorbonne . Cest u n enseignement analogu e que je donne a FEcole de Physique e t de Chimie depui s vingt-neu f ans . Dan s l a mesur e o u je merit e le s eloge s af - fectueux qu e votr e amiti e m' a prodigues , mo n che r Langevin , je sui s tre s heureux d'avoi r p u vou s aide r a realise r l e dessei n qu i vou s tien t a coeur , celui de faire de FEcole technique que vous dirigez un veritable etablissement d'enseignement superieu r en assurant au x eleves une culture theorique forte- ment organisee . L a tache , l a encore , m' a et e rendu e facil e pa r l e couran t de sympathi e qu i n' a cess e d'uni r l e matre e t le s eleves , toujour s attentif s et desireu x d'acqueri r le s connaissance s don t il s reconnaissen t eux-meme s Futilite pour leur carriere future. C e n'est pas sans un vif regre t que je quit- terai bientot, cett e Ecol e a laquelle me rattachent tan t d e liens ; mon depar t ne pourra affaibli r le s sentiments d'admiration qu e j'eprouve pour l e savant et Fhomme qui la dirige. Tu a s retrace tout a Fheure, mon cher Maurain, en termes qui m'ont par - ticulierement touche , venant d e Fami, d u doye n affectueusemen t vener e d e tous se s collegues, ma carriere d e professeur a la Sorbonne. Cel a a toujour s ete pour moi une grande joie que d'enseigner; je me suis toujours interess e a ce que j'enseignais: c'es t une condition necessaire et peut-etre suffisante pou r interesser ceu x qui vous ecoutent. S i ma prochaine mis e a la retraite n e me vieillit pas premeturement, i l me sera agreable de donner de temps en temps quelques serie s de legons sur de s sujet s qu e je n'a i pa s encore e u Foccasio n d'enseigner. C'est a FEcole Normale que s'est exercee une grande partie de ma carriere de professeur ; pendan t quelqu e quatorz e an s j'y a i e u tou t mo n service . I I 278 C. ALLOCUTIO N D E M. ELI E CARTAN

est vrai qu e j'y comprends le s annees de guerre, pendant lesquelle s je vou s ai accueill i k plusieur s reprises , mo n che r Julia , lorsqu e gran d bless e vou s veniez vous reposer dans notre vieille Ecole des operations successives qu'on etait oblig e de vous fair e subi r a u Va l d e Grace . I I est difficil e d'imagine r un auditoir e plu s interessant qu e celu i PEcol e Normale; devan t lu i o n peu t aborder tou s le s probleme s e t j'en a i abord e u n certai n nombre . J'a i et e heureux d'entendr e d e vous, mon che r Bruhat , e t d e vous, mo n che r Julia , l'opinion qu'on t bie n voul u garde r de moi me s eleves . C e sont maintenan t des maitres; un grand nombre enseignent dans les Facultes. L'un d'eux, celui que ses camarades de Janson envoyaien t passe r leurs colles chez Cartan, es t Tun des plus jeunes membres de PAcademie des Sciences. Nous, leurs aines, nous avons la grande joie de voir sortir de PEcole Nor- male des generations successive s de brillants mathematiciens ; nou s somme s assures ains i qu'ell e n'abdiqu e pa s l e role d e pepiniere de s mathematique s qu'elle joue depuis longtemps et qui inspira autrefois a Sophus Lie Pidee de lui dedier so n gran d trait e su r l a theorie de s groupes. E t puisque , pa r un e pensee touchante, le fils de Sophus Lie a voulu marquer ce Jubile par Penvoi du buste de son pere, ne serait-il pas naturel que la place de ce buste soit a la bibliotheque des Sciences de PEcole Normale? I I rappellerait aux promotions successives a la fois l e grand mathematicien norvegie n e t les normaliens qu i ont ete ses eleves a Leipzig et ont illustr e PEcole, les Vessiot, le s Tresse, les Drach. Mon che r Bruhat , vou s ave z parl e e n terme s qu i m e son t alle s a u cceu r de la dynastie normalienne des Cartan. M e permettrez-vous d'adjoindre au x deux noms d'Henri Carta n et d'Helene Carta n le s noms de deux autres nor- maliens qu i m'on t et e tre s chers ? L e premier es t celui d e mo n beau-frer e Antoine Bianconi, caciqu e litteraire d e l a promotion d e 1903 , dont l a mor t sur le champ de bataille interrompit Poeuvr e philosophique qu'i l meditait e t qui promettait d'etre importante . L e second est celui de ma plus jeune sceu r Anna Cartan, dont le succes au concours d'entree a Sevres m'avait rempli de joyeuse fierte; elev e elle aussi d e Jules Tannery, don t ell e ne pouvait parle r sans emotion, elle a termine prematurement s a brillante carriere comme pro- fesseur au Lycee annexe de Sevres. I I m'est doux de penser qu'elle est un peu presente ici, en voyant au milieu de nous la compagne de promotion a qui la liait une tendre affection, m a chere amie Madame la Directrice de PEcole de Sevres. Mon che r Julia , c'es t ave c empressemen t qu e je m e sui s associ e a votr e projet d e fonde r pou r le s jeunes mathematicien s u n cercl e d'etudes , votr e seminaire, o u ce s jeune s gens , travaillan t e n collaboration , exposeraien t chaque annee une questio n important e d e Mathematiques. Vou s nous ave z dit a ce propos que les jeunes sentent; sans peut-etre trop s e Pavouer, l e be- soin d e s'appuye r su r leur s aines . E n entendan t tou t a Pheure Dieudonne , nous avons compris combien vous aviez raison. C. ALLOCUTIO N D E M . ELI E CARTAN 279

Mon cher Dieudonne, les paroles que vous m'avez adressees me touchent au del a d e toute expression. Elle s montrent qu e vou s ave z Fenthousiasm e de la jeunesse, vertu que je vous souhaite de conserver toute votre vie. Ce t enthousiasme ne vous a-t-il pas fait depasser la mesure? J'aurai s certes mau- vaise grace a vous contredire, mais je sui s assez age pour savoi r ne pas tirer de vos eloges un orgueil deplace, sachant tres bien que si j'ai les qualites que vous m'attribuez, il m'en manque un certain nombre d'autres qui m'auraient permis de rendr e plus d e service s a Fenseignemen t e t a la science ; elle s n e sont san s doute pa s dans m a nature , mai s je n'a i peut-etr e pa s eu asse z d e ferme volonte pour les acquerir. Mon cher Demoulin, nou s sommes lie s par une vieille amitie e t de nom- breux souvenir s communs ; nou s avons ecout e ensembl e le s maitres don t je rappelais le s noms tout a Fheure . J e sui s tres sensibl e au x felicitations qu e vous m'apportez a u nom des savants etrangers. J e remercie particulieremen t tous ceux d'entre eux , et je les vois ici nombreux, qu i ont tenu a assister e n personne a cette ceremonie. Leu r presence m'est precieuse et 1'empressemen t avec leque l de s savant s d e nombreuse s nation s etrangere s on t bie n voul u s'associer a mon Jubile m'a vivement touche. Dan s le monde trouble ou nous vivons, il est indispensable que la collaboration internationale, au moins dans le domaine scientifique , soi t maintenue malgr e tous les obstacles. En meme temp s qu'au x delegue s etrangers , j'adresse mes remerciement s aux amis, aux collegues, aux elfeve s qui ont bien voulu repondre a Fappel du Comite jubilaire. J e remercie les membres de ce Comite qui ont accept e d e donner leu r concours a l'organisation d e cette fete, e t surtou t mo n collegu e et am i Darmoi s qui , ave c Faide d e mon elev e Ehresmann, a pris su r lu i l a part la plus lourde de cette organisation . Plusieurs des orateurs precedents, et j'en suis particulierement touche, ont tenu a associe r l e nom d e l a compagn e d e m a vi e a cett e commemoratio n de ma carriere scientifique . Depui s plus d e trente-six an s elle est la flamme ardente qui anime le foyer familial. No s enfants nous ont reserve de grandes joies; l a douleu r n e nou s a pa s et e epargnee . Nou s n'oublieron s jamai s 1'empressement ave c lequel l e Comite a tenu a faire sienn e la pieuse pense e de rendre presente ici, grace au grand artiste qu'est M. Charles Munch, Fame de Fenfant dispar u dont toi, mon cher Tresse, vous, mon cher Julia, et vous, mon che r Dieudonne, ave z s u evoquer la memoire e n termes s i emouvants. La ceremonie de ce matin, ou vous avez tenu a ne pas dissocier Fhomme d u professeur e t du savant, nous a donne a ma femme e t a moi les plus grandes joies qui puissent encore nous etre reservees. This page intentionally left blank APPENDIX D

The Influence of France in the Development of Mathematics1

Like any science, mathematics is a common, international possession; it is the commonwealth that belongs to all developed nations, the commonwealth to which ever y nation contributes according to its abilities. I t would be un- acceptable if any well-regarded mathematician would decline to pay awed re- spect to the great foreign minds of the past: Galilei from Italy, Newton from England, Eule r from Switzerland , Abe l fro m Norway , Leibniz , Gauss , an d Riemann from Germany , to mention but the most significant . The y opened new route s i n differen t fields of th e scienc e that , withou t them , woul d no t be wha t i t i s today . However , I hope t o mak e you realiz e tha t the Frenc h mathematicians mad e on e of th e mos t noteworth y contribution s t o th e de- velopment of mathematics , and that, whe n it comes to the number of great mathematical minds, France does not take second place to any other nation. I am honored and pleased to be given this opportunity to talk about this par- ticular subject in front of a friendly audienc e and in a country tied with my own by many common memories. In mathematics, as in any other science, there are two kinds of scientists: those who open royal avenues by coming up with ne w ideas, usually simpl e ones but nevertheless ones that have not occurred to anyone else; and those who, on the vast land cleared by the first, till their own gardens, often picking tasty fruits, an d sometimes collectin g magnificen t harvests . Whe n it come s to the development o f an y science, th e latte r ar e not simpl y significan t bu t rather indispensable; however , i t i s clea r that the name s o f th e forme r ar e those that are remembered and honored. Thos e ar e the people abou t whom I speak today. Joseph Bertran d tells us that, a t a Fontainebleau receptio n for the Dutch ambassador, King Henri IV took pleasure in recalling great Frenchmen who, by thei r achievement s i n literatur e an d art , exceede d thei r foreig n rivals . "Those I myself admire, " said the Dutchman , b y training a mathematician

This talk was presented by Elie Cartan in the French Institute m Belgrade, Yugoslavia, on February 27 , 1940 . Th e tal k wa s translate d fro m Frenc h int o Serbia n b y Milora d B . Protic , published i n 194 0 i n th e Yugoslavia n journa l Saturn and i n 194 1 a s a separat e boo k wit h the introductio n writte n b y Mihail o Petrovi c (se e [190]) . Fo r this Appendi x th e lectur e wa s translated from Serbian into English by Dr. Jelen a B . Gill, who also wrote all footnotes. 281 282 D . TH E INFLUENC E O F FRANC E I N TH E DEVELOPMEN T O F MATHEMATIC S

whose field was geometry, "bu t I mus t notic e that, s o far , Franc e faile d t o produce an y mathematicians." "Romanu s s e trompe!" crie d Henr i I V and , having at once turned to one of the servants, asked that M . de la Bigottier e be brought in . Th e first great Frenc h mathematician , M . de l a Bigottiere — whose real name was Francois Viete (1540-1603)—was the founder o f mod- ern algebra. H e was the first to realize that the procedure for solvin g special numeric equations would be simplified i f the operational symbolism—whos e beginnings can be traced back to the ancient times—was applied to letters as well; also, he deserves most o f the credit for th e systematic development o f that idea, and he predicted its unbounded expansion . A t the end o f the six - teenth century, when Galilei and an advanced geometry school brought fam e to Italy, it was Frangois Viete who secured fo r Franc e a distinguished plac e in the process of foundin g moder n mathematics . I should tel l you that, fo r quite some time, Viete was in contact with one of your first mathematicians, Marin Getaldic (1566-1626), who was born in Dubrovnik and who, in Paris, in the year 1600 , published one of Viete's last works. For France, the seventeenth centur y was particularly glorious . I n the his- tory of mathematics, mechanics, and physics , three names fro m thi s perio d especially stand out: Descartes , Pascal, and Fermat . A philosopher , mathematician , an d physicist , Ren e Descarte s (1596 - 1650) i s frequentl y considere d th e originato r o f a ne w er a i n th e histor y of th e human mind . A s a physicist , h e witnesse d a defea t o f hi s attempt s to explai n th e world; however, hi s idea that al l physica l phenomena ca n b e expressed i n terms o f spac e and motio n ha s retained it s attractiveness unti l the present day, because the founder o f the general theory of relativity him - self believed that i t may be possible to interpre t physic s by using geometri c terms (it was nothing but the past development o f mathematics that enable d Einstein t o carr y hi s idea s furthe r tha n Descarte s coul d have) . Eve n i f w e deny him credit fo r the creation o f analytical geometry (1637) , we must no t undermine his role in mathematics. I t is known that Gree k geometers freel y used numbers and computations in their thinking, but fo r them the numbers had no t ye t completel y los t th e geometri c characte r the y ha d i n hellenisti c science; as the words "square" and "cube " stand fo r bot h th e number s an d the geometric forms i t is clear that the common speec h o f today stil l show s traces o f thi s doubl e use . Descarte s wa s th e first t o us e abstrac t number s systematically to represent geometric forms and to convert geometric reason- ing into computations. I n tha t wa y he create d a n extraordinaril y powerfu l tool. T o him w e must ascribe the growth of geometry that stemmed primar - ily fro m analytica l an d differentia l geometry ; h e enriche d th e latte r wit h a general method fo r finding tangents o f algebraicall y define d curves . Thank s to analytical geometry, mathematicians not only succeeded i n understandin g a space of any number of dimensions but also learned to think geometricall y in such a space. I t is possible to say that it is in fact analytical geometry that taught mathematician s to fee l comfortabl e in , fo r example , a spheric three- D. TH E INFLUENCE O F FRANCE IN THE DEVELOPMEN T O F MATHEMATICS 28 3 dimensional space , i.e., the one that, onl y recently , physicist s starte d usin g to explain physical phenomena. Al l of this represents, although remote, nev- ertheless unquestionabl e consequence s o f Descartes's idea s an d results. I n algebra, it is to him that we owe the rule about the signs. I n pure geometry, he should be credited with a theorem that, having been independently discov- ered by Euler, no w bears Euler's name. A result of analysis situs , a science unknown at the time, this theorem establishes the relationships between the number o f vertices, edges , an d sides o f a conve x polyhedron . Finally , i n mechanics, Descartes' s principl e o f conservation o f linear momentu m pro - vides an illustration of the intuition that required nothing more than a proper refinement to bring about on e of the basic principles of classic mechanics. Even i n his early youth, Blais e Pasca l (1623-1662) , a somewhat strang e but extraordinary genius, exhibited an unusual talent for geometry by writing, at the age of sixteen, Traite sur les sections coniques, a treatise about curve s that are most frequently studie d as flat conic section and play an important role in Kepler's planetary laws . Pasca l use d the results of his contemporar y Gerald Desargues, who was one of the most significant Frenc h geometers and who, alongsid e Pascal , wa s a forefather o f projective geometry . B y taking, in a way similar to Desargues's, th e perspective a s a starting point , Pasca l succeeded i n reducing all properties of conic section s to a property tha t he called "L 'hexagramme mystique": i f a hexagon is inscribed int o a cone, the three points at which pairs of opposite sides cross each other always lie on a straight line. Eve n by this resul t Pasca l demonstrate d th e creative power of an eminent geometer . As soon as Pascal the forefather o f projective geometry established himself, Pascal th e founder o f mathematical probabilit y too k th e stage. Whe n his friend Chevalie r de Mere asked him a couple of questions concerning a game of chance, Pascal answered the m by reducing all possible outcomes to those most basic . Pierr e de Fermat, o n the other hand , cam e up with th e same answer but in a completely differen t way . Th e evolution of the principles of mathematical probabilit y i s well illustrated i n the letters exchanged between Pascal and Fermat. Th e scope of this new research did not escape Pascal: "By connecting the exactness o f a mathematical approac h wit h th e uncertainty of chance, " he was known t o say, "the new science can rightly be given an astounding name—Geometr y o f Chance." Fro m th e famous bettin g proof , it is known to what exten t his research and thinking were influence d b y his interest i n this ne w geometry. I t is also know n tha t thi s geometr y playe d an instrumental rol e in the development o f modern science , in which entir e portions of physics are nothing but chapters of mathematical probability, and many of the laws of physics are nothing but laws of chance. Pierre de Fermat (1601-1665) , whom we mentioned earlier , is one of the greatest mathematical geniuses . He became a counselor of the parliament at

The old nam e fo r topology . 284 D . TH E INFLUENCE O F FRANCE IN TH E DEVELOPMEN T O F MATHEMATIC S the ag e of thirt y an d held that position unti l hi s death . Althoug h his voca- tion did not predestine him for mathematical fame , he made sur e to devote enough tim e to his favorit e avocation . Ferma t i s especiall y famou s fo r hi s research in arithmetic and number theory. O n the margins of a copy of Dio- phantus's work about undefined equation s (whic h was published in 161 2 by Bache de Meriziac, the author of Problemes plaisants et delectables) he wrote a number of important theorems without proofs; it is a matter of common be- lief that he was in possession of their proofs. Th e most famous among those theorems is the one frequently calle d Fermat's Last Theorem—according to which the sum of the nth degrees of two integers cannot equal the nth degree of a third integer for an y integer n tha t is greater than two. Thi s theorem inspired a wealth o f result s whos e authors , i n spit e o f havin g a t thei r dis- posal modern algebraic results that had been unknown to Fermat, have never been able either to prove or disprove it. I t has been believed for a long time that, even if the theorem is wrong in general, it might in fact be wrong only for some values for n ; however , i t is by no means known if th e number of the value s fo r whic h i t i s wron g is finite o r infinite . Throug h th e researc h prompted b y thi s singl e theorem—conducte d i n nearl y al l mathematicall y developed theories—Ferma t influence d th e growt h o f numbe r theory . Hi s contemporaries readil y recognize d hi s extraordinar y skill s i n tha t field. I n one of hi s letters, Pasca l wrot e that hi s ow n result s i n numbe r theory wer e surpassed by Fermat's and that his was but to admire them. The first half of the seventeenth century was an era of strong advancement of integra l and differential calculus . Wit h respect to integral calculus (deter- mining areas and volumes, finding centers of gravity), it is enough to mention Cavalieri3 and de Roberval4. A s Fermat's own research, however, went quite far in this field, we ar e indebted to hi m fo r th e classica l integratio n proce - dures. O n the other hand, once while trying to fight a tremendous toothache by solving roulette problems, Pascal accidentall y discovered a procedure fo r obtaining integral s of highe r powers of trigonometri c functions. Th e names of those whom we have been talking about are found i n differential calculu s as well (the tangent problem). B y his method ude maximis et minimis", Fer- mat introduced the notion of an infinitesimally smal l number. Lagrang e and Laplace considered Fermat to be the actual founder of infinitesimal calculus , while Emil e Picard 5 believe d Pascal' s work s abou t roulett e to represen t th e beginnings of integral calculus. Originally , Leibniz scribbled his formulae of infinitesimal calculu s on a copy of one of Pascal's manuscripts, which, as he himself put it, had suddenly showed him the way. It woul d b e unfai r t o conclud e th e accoun t o f thes e grea t mind s with - out mentionin g that , a t the ag e of twenty-eight , Pasca l constructe d th e first

3Bonaventura Cavalieri (1598-1647). 4Gilles Personne Roberval (1602-1675). 5Charles Emile Picard (1856-1941). D. TH E INFLUENCE O F FRANCE IN TH E DEVELOPMEN T O F MATHEMATIC S 28 5

arithmetic machine , capabl e o f addin g an d subtracting . Du e t o hi s wor k Traite de Vequilibre des liqueurs, Pascal ca n b e considered—togethe r wit h Archimedes—one o f th e founder s o f hydrostatics ; thi s i s wh y i t come s a s no surpris e tha t th e barre l h e use d t o chec k wha t i s toda y know n a s Pas- cal's Principle is displayed next to his death mask in the little chapel erected in th e churchyar d o f Por t Royal . Finall y le t m e mentio n th e experiment s concerning atmospheri c pressure , which , i t i s suspected , h e conducte d un - der the influenc e o f Mersenne 6, th e sou l o f a smal l grou p of philosophers , mathematicians, an d physicists that, before th e creation of th e Academy of Sciences in 1666 , represented the first small but lively academy. Those were fortunate times when one and the same man could be accom- plished i n philosophy , mathematics , an d physics , an d whe n a philosophe r such as Malebranche7 coul d have the extraordinary feeling that colors might be related to the number of vibrations of which light is composed! II The secon d hal f o f th e seventeent h an d the beginnin g o f th e eighteent h century were dominated by Christian Huygens (1629-1695) fro m the Nether- lands, Isaac Newton (1642-1727) fro m England, and Gottfried Wilhelm von Leibniz (1646-1716 ) fro m Germany . I t should be enough t o mention tha t the last two are credited with the discovery or, rather, the systematization of infinitesimal calculus , whil e th e first is famou s fo r his work s i n differentia l geometry, rational and applied mechanics, and especially his works concern- ing the theory of light (i n which he originated and developed an undulatory theory as opposed to Newton's particle theory). I n this period, a remarkable scientific revolutio n wa s triggered b y Newton's proo f tha t star s and object s on Eart h move accordin g to th e sam e law s of mechanics , namely , tha t on e and the same law, the law of gravitation, explains the motion of planets, the moon, an d comets as well a s the existenc e of Earth' s gravity, hig h an d low tide, and so on. I t was Newton's genius that created an entirely new science— celestial mechanics. Bu t even if the earliest beginnings of this science did take place in England , i t was France tha t provided a particularly fertil e soi l fo r its futur e development . T o realiz e this, i t i s enoug h to recal l th e name s of those whose works contributed the most to its growth: Clairaut , d'Alembert , Euler, Lagrange, Laplace, Gauss, Cauchy, Poisson, Le Verrier, Tisserand, and finally an d especially—Henri Poincare . I pause for a moment on the first of them, Clairaut. Th e second in a family of twenty-on e children , wit h a fathe r wh o wa s a teache r o f mathematics , Alexis Claud e Clairau t (1713-1765 ) demonstrate d talent s simila r t o thos e of Pascal ; however , unlik e Pascal , hi s first works i n n o wa y reveale d th e significance o f thos e tha t followed . H e sen t hi s first announcement t o th e Academy of Science s before reachin g the ag e of thirteen , an d addresse d a n

6Marin Mersenne (1588-1648). 7Nicolas de Malebranche (1638-1715). 286 D . TH E INFLUENC E O F FRANCE I N TH E DEVELOPMEN T O F MATHEMATIC S article abou t line s wit h doubl e curvature s a t th e ag e o f sixteen . H e wa s eighteen when, against the existing rules, the king named him a member of the Academy of Sciences, the Division of Mechanics. I shall refrain fro m tellin g you about his research in the field of pure mathematics in general and abou t the part connected with solving differential equation s in particular—the latter of which should be well known to all who studied differential equations—an d focus instead on those results that made him famous. Newto n and Huygen s came up no t onl y wit h a theoretical proo f that , instea d o f bein g a perfec t sphere, th e eart h i s a spher e flattene d a t th e poles , bu t als o wit h a wa y t o calculate the measure o f flatness. However , whe n i n 1701 , at th e Pyrenees, Cassini8 determined the degree of arc of the Paris meridian, their conclusions came to be questioned. Afte r debate s tha t wer e occasionall y confusin g bu t always lively , i n 173 6 th e Academ y o f Science s decide d t o launch , unde r the guidance of de Maupertuis9, an expedition that woul d travel to Laplan d to determin e the degre e of the Laplan d meridia n arc . Workin g under ver y hard conditions , whic h wer e further complicate d b y sno w and pola r night , the team—which included Clairaut as well—came up with a numerical value that wa s remarkabl y large r tha n th e on e Cassin i ha d obtaine d i n France , hence provin g beyon d an y doub t tha t th e eart h i s indee d flattene d a t th e poles. Understandably , d e Maupertui s wo n laurel s fo r th e succes s o f th e expedition: wit h his head wrapped i n a bear skin , his hand pressin g agains t a globe , h e pose d fo r a portrait. Bu t Clairau t continue d t o thin k abou t a possible cause of the earth's polar flatness and tried theoretically to determine the shape that a fluid planet would assum e under the influenc e o f Newton' s attraction. Th e results of his research were published i n 174 3 in La Theorie de la Figure de la Terre, the book that d'Alembert characterize d as a classical account o f everythin g tha t ha d bee n don e b y tha t time , th e accoun t tha t marked an important date in the history of celestial mechanics. I n addition , Clairaut explaine d th e motio n o f the moo n an d i n s o doing contributed t o Newton's lunar theory. H e summarized hi s results from thi s field in Theorie de la Lune, a book publishe d i n 1732 , to which , tw o year s later, h e adde d numerical tables , which, as Fontaine had put it , made it possible to find out "every step that the moon makes in the sky". A few years later, by predicting the next return o f Halley' s comet, Clairaut reache d popula r recognitio n an d fame. Afte r explainin g that the perturbations cause d by Saturn would dela y the retur n o f Halley' s come t fo r abou t on e hundred day s and th e influenc e of Jupiter woul d dela y i t fo r a n additiona l fiv e hundre d an d eightee n days , he predicted that its next passage through the perihelion would occur around April 13 , 1759 , but cautione d that , du e t o numerou s othe r factor s tha t h e had to neglect , this date might be of f by up to one month—indeed, Halley' s comet passed through the perihelion on March 13 , 1759. Almost one century

Jacques Cassini (1677-1756). Pierre Louis Moreau de Maupertuis (1698-1759). D. TH E INFLUENCE O F FRANC E I N TH E DEVELOPMEN T O F MATHEMATIC S 28 7 later, by determining the position o f an until-then-unknow n plane t that ha d been th e mai n caus e o f th e disturbanc e o f Uranus , Frenc h astronome r L e Verrier10 attaine d nearly the same glory. Ill The secon d hal f o f th e eighteent h centur y wa s dominated b y Eule r an d Lagrange and, in a somewhat lesser degree, by d'Alembert . Leonhard Eule r (1707-1783) , "th e princ e o f mathematicians" , wa s bor n in Base l and spen t part o f his life i n St . Petersbur g and Berlin . Hi s geniu s glowed i n al l area s o f mathematics , an d hi s wor k ha s ha d significan t an d lasting influence . I wil l alway s remembe r th e deligh t I experience d whil e reading his Introduction to the infinitesimal analysis, the book that was given to m e a s a n awar d a t th e en d o f m y final yea r o f gymnasium : i t opene d a whol e ne w worl d i n fron t o f me , preparin g m e to understan d bette r th e lectures I would attend at the Sorbonne and in l'Ecole Normale. Jean L e Ron d D'Alember t (1717-1783 ) lef t hi s trace i n man y differen t areas of mathematics . A well-known algebrai c theorem tha t bears his nam e asserts that th e tota l numbe r o f solution s (rea l an d complex ) o f a rationa l equation equal s th e highes t degre e o f th e variable . Althoug h d'Alembert' s proof o f thi s resul t wa s wrong , i t shoul d b e mentione d tha t Euler' s proof , based on completel y differen t principles , was not without flaws . Onl y whe n the famous mathematician Gaus s entered the mathematical scen e was a cor- rect proo f found , an d onl y wit h Cauchy' s appearanc e wa s a rea l an d ver y simple justificatio n o f thi s theore m established . I n analysi s I shal l men - tion onl y th e first correc t formulation—whic h cam e fro m d'Alembert—o f a partia l differentia l equatio n describin g vibrations o f strings . An d finally, it i s well worth mentionin g that , i n mechanics , d'Alember t cam e u p wit h a principle—nowadays known as d'Alembert's principle—which paved the way for Lagrange' s analytical mechanics . Joseph Louis Lagrange (1736-1813) was born in Torino, in a French fam - ily; although , lik e Euler , h e spen t a fe w year s i n Berlin, i n 178 7 h e mad e his permanen t hom e i n Paris , entitlin g Franc e t o conside r hi m on e o f he r very ow n mos t celebrate d minds . H e i s trul y on e o f th e mos t significan t mathematicians o f all times. H e worked i n all fields of mathematics. I n th e theory o f numbers h e proved Fermat's theore m fo r th e power four. I n alge- bra, through developin g a unique method fo r solvin g a polynomial equatio n by reducing it to an equation o f a lowe r degree, he cleared a path fo r Abel , Gauss, and Galois ; in addition, h e demonstrated tha t polynomia l equation s of the fift h degre e cannot be solved in the way used for solvin g those of the third an d fourt h degree . I n analysis , he gave the method fo r solvin g partia l differential equation s of the first order and came up with the notion of a sin- gular solution. I n function theory , he attempted but did not quite succeed in establishing a rigorous foundation fo r infinitesima l calculus , the area whos e

Urbam Jea n Joseph L e Verner (1811-1877) . 288 D . TH E INFLUENC E O F FRANCE IN THE DEVELOPMEN T O F MATHEMATIC S principles had not yet been developed with desired exactness but whose con- sequences were nevertheless trusted. However , i n spit e o f this lac k o f ful l success, his metho d o f considerin g function s i n a n abstrac t way , indepen - dent of their geometric or mechanical meaning, had remarkable influence in preparing the terrain fo r the modern theory o f functions. Lagrange' s talent for generalizin g became truly obviou s in his works concerning the calculus of variations. The calculus of variations was developed durin g the eighteenth century , through the works of Bernoulli and Euler, both from Switzerland . It s roots are i n som e problem s o f geometr y an d mechanics , th e simples t o f whic h might be the problem o f determining the shortest pat h betwee n two points on the same surface; here, the unknown quantity is not a number but, much more complexly, a line consisting of infinitely man y points. D e Maupertuis was the one who, by his Principle o f Leas t Action, reduced the problem of determining a trajector y o f a particl e i n a give n forc e field to a proble m of maxima and minima, giving special importance to this kind o f calculus. It shoul d no t b e forgotten, however , tha t b y that tim e Fermat ha d alread y reduced the laws of optics to a similar principle, according to which the path chosen by light is the shortest in terms of time. B y applying the infinitesimal variation o n a n unknow n lin e an d b y showin g ho w that variatio n ca n b e calculated, Lagrang e introduce d a genera l metho d int o a theor y i n whic h nearly every problem required a special procedure in order to be solved. I shal l omit Lagrange' s wor k in celestia l mechanic s and , instead , devot e more time to his most significant work, M&anique Analytique (1788). Galilei, Descartes, Huygens, Leibniz, Newton, and d'Alember t graduall y develope d all of the grand principles of modern mechanics. Bu t the problem of deter- mining the trajectory o f a syste m governe d b y given force s wa s frequentl y complicated by the necessity to take into account unknown relations between the forces. Wit h ingenious intuition, i n the cas e without frictio n Lagrang e completely removed the difficulty and gave a general procedure for determin- ing equations that woul d giv e the trajectory i n question : t o achiev e this it is enough to determine the active force o f tha t syste m a s well as the work of that forc e fo r a n infinitel y smal l movemen t o f th e system . Asid e fro m practical importance, this wonderful creatio n has remarkable philosophica l importance becaus e i t completel y illuminate s everythin g tha t is , fro m th e point o f view of mechanica l properties, important i n a syste m of particles. In this respect, Lagrange's genius is equal to that of Descartes, the creator of analytical geometry. The so-called Lagrange's equations in Mechanique Analytique represented an analytical model for various mechanical explanations of certain physical theories. Fro m that poin t o f vie w this work has great philosophical signifi - cance; but, although it is the most important work of the nineteenth century, it created the impression that everything can be explained by the principles of mechanics—an impression as erroneous as Descartes's belief that everything D. TH E INFLUENC E O F FRANCE IN THE DEVELOPMEN T O F MATHEMATIC S 28 9

can be explained in terms of geometry—which i s the reason that, today, it is completely abandoned. Nevertheless , it illustrates the ability of mathematic s to provide physicists with the tools they require to carry out their theories. Some of the extraordinary mind s were inclined to see danger in the man- ufacturing o f structures (similar to the one created by Lagrange) that offere d insights into infinit e array s o f phenomena ; the y feare d tha t suc h structure s might cause a loss of connection with reality. Fo r instance, the great geome- ter Poncelet , know n fo r hi s work s in mechanics , avoide d usin g Lagrange' s method and, instead, preferred followin g to the last detail the influences an d interactions of various forces in order to determine, step by step, their actual works. The same type of skepticism prevented Poncelet from using analytical geometry and prompted him, instead , to examine directly relations between various geometri c figures by applyin g principle s o f classi c geometry. Wit h respect to acceptin g the latest results , there are indeed tw o kinds o f minds , both equally important for the development of science and both found amon g great French mathematicians .

IV

Visible as early as the end of the eighteenth century, the French superiority in mathematics became especially clear during the French Revolution and at the beginning of the nineteenth century . Amon g the great names of that er a one must include Monge, Laplace, and Legendre . Pierre Simon de Laplace (1749-1827) owe d his reputation to his research in celestia l mechanics , summarize d i n hi s charmin g treatis e Exposition du Systeme du Monde. Th e peculia r resul t statin g tha t eve n th e finest detail s of almos t al l celestia l phenomen a ca n b e explaine d evolve d int o scientifi c determinism, accordin g to which, in order to be able to determine position s and velocities of cosmic particles at a given time, it i s enough to know their positions and velocities at any other time, provided i t is known, in addition , which principle s regulat e the forces—modele d afte r th e force s o f Newton' s gravitation—that th e particles are governed by. Fo r a long time mathemati - cal physics developed according to this result; only recently, electromagnetism and atomic physics succeeded in proving it to be wrong. Still , this result had strong influenc e o n th e developmen t o f science . A very significan t treatise , Theorie Analytique des Probabilities (1812), is another one for which we are grateful to Laplace; the most important part of this work deals with the appli- cation of the notion of probability in the theory of least squares, the possibil- ity of which had been indicated by Legendre. Whil e studying the inclinatio n of a n ellipsoid , Laplac e introduce d spherica l function s b y means o f whic h one can expres s any function dependen t o n a point o n a sphere. W e should not forget Laplace' s famous equation which is satisfied b y Newton's potential function; thi s equation i s of extraordinary importanc e i n many problems of analysis, geometry, mechanics, and physics. 290 D . TH E INFLUENC E O F FRANC E IN TH E DEVELOPMEN T O F MATHEMATIC S Adrien Marie Legendre (1752-1833) is responsible for the rejuvenation of number theory, previously successfully treated by Euler. Althoug h Euler was the first one to publish the reciprocity law in arithmetic, Legendre explained it clearly and partially proved it; the law is named after Legendre . Gauss was the third mathematician discover this same law, but the first one to construct a correc t and complet e proof. Legendre' s significan t wor k o f severa l years, Sur les Integrates Elliptique, a tract in two volumes, was published i n 182 5 and 1826 . There he presented a complete study of integrals involving square roots of fourth-degree polynomials and developed different form s that can be given to them. Althoug h with this work Legendre became a forefather o f the marvelous theory o f elliptica l functions, h e let Jacobi and Abe l take credit for its founding. Finally , let us mention his Elements de Geometrie (1794), a work which had numerous editions and which, in schools of the Anglo-Saxon countries, soon replaced Euclid's theory; in the history of the non-Euclidean geometries, this work had definitive importance. Gaspard Monge (1746-1818) was one of the best French geometers. There are tw o reason s why . First , b y foundin g moder n projectiv e geometry , h e joined the long process of development of perspective, the theory whose prin- ciples had been known to Italian renaissance painters, which Desargues and Pascal applied to the theory of conic sections, and which, following the pre- vious two, the French geometer de la Hire11 expanded to the theory of poles and polars of a circle. Mong e systematized projective geometry and enriched it wit h construction s o n surface s tha t ar e no t flat. O n th e other hand , b y his treatise Applications de VAnalyse a la Geometrie h e gav e a substantial boost to differential geometry , the field that wa s separated fro m Descartes' s analytical geometr y by Euler's and Meusnier' s significan t work s concerning the properties of surfaces; it is Monge to whom we are indebted for the no- tion o f measure of curvature , a s well as for it s application i n stereometry ; it wa s his idea to characterize a vast famil y o f surface s b y obtaining them as a solution se t o f a singl e partia l differentia l equation . H e manage d t o integrate the equatio n o f minima l surfaces , surface s whic h hav e been an d still are an object o f important research , and which had been obtained first in Plateau's experiments. Mong e presented hi s theories durin g his lectures at PEcol e Normale—the schoo l founde d i n 179 5 as a convent—a s wel l as at TEcol e Polytechnique (a t whic h Lagrange and Laplac e taught a s well). I am pleased to have a chance to mention Dupin 1 , for he was one of the nu- merous students wit h who m Mong e worked; Dupi n i s known fo r hi s work Developpement de Geometrie, i n which he introduced th e notions o f conju - gated tangent s an d indicatri x a t a poin t o f a surface ; also , Dupi n ca n b e considered a creator of a new branch of geometry.

nPhiIlipe d e la Hire (1640-1718). 12Francois Pierre Charles Dupin (1784-1873) . D. TH E INFLUENCE O F FRANCE IN TH E DEVELOPMEN T O F MATHEMATIC S 29 1

V

The mos t remarkabl e name s in Franc e durin g th e first hal f o f th e nine - teenth century were those of Fourier, Cauchy, Poncelet, and Galois. Althoug h quite different fro m eac h other, they all cleared ne w paths in science. Jean Baptiste Joseph Fourie r (1768-1830 ) ca n be considered the founde r of mathematical physics . I shall neglect his important result s in algebra and instead tell you about Theorie Mathematique de la Chaleur, the work he did not publish until 182 2 but which must have been in his thoughts since at least 1807. With this work Fourier opened up a new field in mathematical analysis. "Unknown to the ancient geometers, and for the first time used by Descartes for researchin g curve d line s an d surfaces, " Fourie r says , "analytica l equa - tions are by no means limited to these general phenomena. Sinc e mathemat- ical analysis determines the most diverse relations and measures time, space, forces and temperature, it is safe to say that it is as wide and rich as Nature itself. It always follows the same paths and gives the same interpretations, in that way certifying about the unity, simplicity and stability of the Universe." It shoul d no t b e forgotte n that , accordin g t o Fourier , th e riches t sourc e o f all mathematical discoveries lies in the study of nature. As , for instance, the mathematical theory of heat had a significant influence on the development of pure mathematics, we may say that Fourier's viewpoint was correct. Create d by Fourie r t o hel p him integrat e frequentl y encountere d partia l differentia l equations, the theory of trigonometric series prompted incredibl y many arti- cles, all of which were trying to establish a rigorous foundation fo r this theory as well as to complete and further develo p it. Th e basic problem that needed to be solved was determining which functions ca n be represented in the for m of a Fourier series. A s even many of Fourier's own examples were peculiar, it di d no t tak e muc h t o mak e th e mathematician s trul y puzzled , i n a wa y in which a musician would be puzzled upon discoverin g that, b y combining finite or infinit e number s o f pur e sound s an d thei r variou s multiple s (har - monics), it is possible to create any disconnected sequenc e of sounds. Thes e unusual result s force d mathematician s t o chec k onc e mor e an d specif y th e notion of a function an d to start thinking, bit by bit, about the foundation s of their own science. Thi s is what brought about unbelievabl e consequence s which have not yet fully presented themselves. Grou p theory—a field which so frequently failed mathematicians and which caused many paradoxes that, I am afraid, hav e not yet been successfully resolved—wa s one of the branches of mathematic s tha t eventuall y evolve d fro m thes e efforts ; anothe r branc h that had its origins in the same efforts i s the theory o f function s o f one real variable, a creation o f Frenc h mathematic s fro m th e en d o f th e nineteent h and the beginning of the twentieth century . Augustin Cauch y (1789-1857) , a n extraordinaril y fruitfu l theorist , wa s successful i n al l area s o f mathematics : numbe r theory , geometry , analysi s and celestial mechanics. Unlik e Euler, he did not explore series without first 292 D . TH E INFLUENC E O F FRANC E IN TH E DEVELOPMEN T OF MATHEMATIC S finding ou t whether they made sense, that is, whether they were convergent; in that way, we may say, he opened up the era of exactness. Cauch y discov- ered the general rule—later found independentl y by J. Hadamard13 —whic h explains ho w to determine thos e values o f th e variable fo r whic h a powe r series is convergent. Creatio n o f a theory o f function s wit h a comple x (o r imaginary) variable is another of Cauchy's great accomplishments. Fo r more than thre e centuries , imaginar y quantitie s wer e a scanda l i n mathematics . They were encountered fo r the first time in the sixteenth century, by Italian algebraists, i n the formul a fo r th e root s o f a third-degree equatio n i n th e paradoxical cas e when al l o f th e root s ar e real . But , onc e researcher s go t adjusted to these new quantities and learned how to use them, it was easy to determine important result s concerning real numbers, some of which could not have been obtained in any other way. Sometim e toward the end of the eighteenth century, the Swiss mathematician Argan d explained the secret of imaginary quantities by finding their importance in the possibility of express- ing a vector in a plane whenever one needed to give not only the length of the vector but its orientation as well. Whe n Cauchy started representing a point in the plane by just one imaginary (or , better, complex) quantity instead of two real coordinates, he got the idea of a function wit h a complex variable, a functio n whic h woul d assig n on e poin t i n th e plan e t o anothe r poin t i n the plane. I n this way Cauchy created a whole new world. Th e elements of that worl d are perfectly organized : jus t a s Cuvier14 wa s able to reconstruc t a creatur e fro m th e antediluvia l er a fro m jus t on e piec e o f it s skeleton , a mathematician becam e abl e to reconstruc t on e of Cauchy' s functions, pro - vided he knew its values at every point of the arc, no matter how small the arc might be. The perfect order in this world, its marvelous harmony, and—with the exclusion o f number theory—a lon g sequence o f theorems determinin g properties o f functions an d their numerou s applications, al l leave the mos t magnificent impression . As Cauchy created the right conditions for more discoveries than he could have possibly anticipated, the significance of his opus should be measured by the length of the sequence of works concerning functions o f a complex vari- able. A single theorem from this sequence, whose beauty is in its simplicity, was nearly enough to immortalize the name of Liouville15. Anothe r theorem on the same subject—named afte r Emil e Picard, perhaps the greatest among the living mathematicians—opened vast and until-then hidden horizons, and created a stream of articles that has not yet ceased. By using a viewpoint different fro m Cauchy's , the German mathematician Weierstrass also developed a theory of functions o f a complex variable. Fo r a long time it had been believed that the viewpoint one chose was irrelevant,

Jacques Salomon Hadamar d (1865-1963). 14Georges Cuvier (1769-1832), a French naturalist. 15Joseph Liouville (1809-1882). D. TH E INFLUENC E O F FRANC E I N THE DEVELOPMEN T O F MATHEMATIC S 29 3 but Borel 16 demonstrated—i n on e o f hi s mos t charmin g results—tha t thi s was not true , an d that Cauchy' s viewpoint penetrate s deepe r int o the hear t of th e matter . Bore l indee d too k ou t o f th e plan e s o muc h tha t n o circle , regardless o f ho w small , wa s lef t intact , an d ye t insid e o f wha t remaine d he managed to construct a function that , although satisfyin g al l of Cauchy's requirements, did not satisfy Weierstrass' s definition, that conditions the ex- istence of a function o f a complex variable by the existence of an intact por- tion of the plane. B y starting his celebrated collectio n o f monographs abou t the theory o f functions—the collectio n whose past and present contributor s include mathematicians fro m al l countries—Borel himsel f contribute d a lot to the development o f functions o f a complex variable. With Jea n Victo r Poncele t (1788-1867 ) w e enter th e er a o f pur e geom - etry. Poncele t i s considere d th e founde r o f projectiv e geometry , th e field whose subjec t i s studying those propertie s o f object s tha t d o not chang e i n projections. H e is the on e who discovere d the ne w and ver y usefu l notio n of transformations b y means of reciprocal polars, the transformations whic h make it possible to derive one flat figure from another, with a provision that, peculiarly, the sides of the new figure correspond to the vertices of the old one, and vice versa. Frequently , a transformation o f this type makes it possible to explore the properties of some figure by reducing them to the easier-to-explore properties o f another . Somewha t later , Gergonne 17 use d thi s to deriv e th e duality principle, a principle very important i n projective geometry. Finally , Poncelet wa s the one who discovered th e continuity principle , accordin g t o which if a figure had a certain property, it will retain the same property even after bein g deformed, provide d tha t the ratios between it s various element s were taken into account. B y many simple examples Cauchy proved that thi s principle, as formulated b y Poncelet, was wrong; however, i f formulated i n a slightly different an d much more precise way, this principle is in fact correct . Being very helpful, thi s principle is frequently used . I n geometry, Poncelet' s influence was remarkable: i n Germany, Steiner and Staudt ow e the existence of thei r work s t o Poncelet ; i n France , Chasle s , th e first membe r o f th e department o f highe r geometr y a t th e Sorbonne , wa s the mos t outstandin g representative o f moder n pur e geometry . T o Chasle s w e ar e indebte d fo r the important historical monument L 'Apergu historique sur le Developpement de la Geometrie, whic h le d to the correctio n o f a certain numbe r o f wron g opinions. Before endin g our discussion o f Poncelet, I note that h e played an impor - tant rol e in developing applied mechanics , which h e taught fo r a long time, first in Metz and then at the Sorbonne .

16Emile Bore l (1871-1956). 17Joseph Diez Gergonn e (1771-1859). 18Michel Chasle s (1793-1880). 294 D . TH E INFLUENC E OF FRANCE IN TH E DEVELOPMEN T O F MATHEMATIC S

Evariste Galoi s (1811-1832 ) i s on e o f th e mos t unusua l figures i n th e history o f science . Havin g twic e faile d th e entranc e exa m a t l'Ecol e Poly - technique, i n 183 1 h e wa s accepte d t o TEcol e Normale , onl y t o leav e i t a year later . Takin g a n activ e part i n politic s earne d hi m severa l month s i n prison; not quite twenty-one years old, he was killed in a duel triggered by an insignificant quarrel. H e had presented his mathematical discoveries in equa- tion theory to the Academy of Sciences in two different announcements , but both of them were later lost; fortunately, h e had also published them in sev - eral small articles in Bulletin de Ferussac in 183 0 and also talked about them to his friend Chevalier in a letter written shortly before his death. Som e other results, discovered among his papers, were published i n 1846 , in Liouville' s magazine. The significance of his work can be explained quickly. Tartaglia , Cardano, and Ferrari, Italian algebraists of the sixteenth century, used the second an d third root s t o solv e equation s o f th e thir d an d fourt h degree ; however , al l efforts to solve equations of higher degrees in the same way were in vain. B y showing that some classes of equations can indeed be solved in that same way, Lagrange, Abel, and Gauss contributed a great deal to this problem. Abe l first showed, in 1826 , that a general equation of the fifth degree cannot be solved by means of radicals. I n that way it became clear that the problem, with which mathematicians ha d wrestle d sinc e the sixteenth century , had no t been wel l formulated. Th e glor y for solvin g it belong s to Galois , fo r h e showe d tha t each equation determines a certain number of permutations o f its roots, the permutations forming a so-called group; although applied to the roots, these permutations do not disrup t thei r rationa l interaction s (th e meaning o f th e term "rationa l interactions " need s a n additiona l explanation) . Th e natur e of that grou p determines th e basic properties o f the equation, whethe r i t i s possible to find its roots or not, and, in a general case, the nature of auxiliary equations whos e solvin g would resul t i n solvin g th e origina l equation . B y starting from his own idea, Galois easily found the results of his predecessors and successfull y incorporate d the m int o his own result. The theory of substitution groups, i.e., groups of permutations of a certain number of objects, which was founded by Cauchy, demonstrated its full value through Galois' s works. Galoi s improved it s important aspect s and demon - strated ho w basi c wa s the rol e o f ordinar y groups . Moreover , h e enriche d number theory by introducing ne w classes of imaginar y quantitie s (Galois' s imaginary numbers), each of which was tied to a power of a prime number ; Galois's name is frequently encountere d no t onl y in the theory of equation s but als o in modern algebra. Th e letters he sent to his friend Chevalie r mak e it clea r that i n analysi s he had a s many importan t result s a s in algebr a an d that his works on Abel integrals were twenty-five year s ahead o f those of the famous German mathematician Riemann. Althoug h it makes me sad to think how much science lost by Galois's early death, I must also say that, as Emile Picard onc e put it , "Whe n confronte d wit h suc h a shor t an d turbulen t life , D. TH E INFLUENCE O F FRANC E IN THE DEVELOPMEN T O F MATHEMATICS 29 5 one's respect for the extraordinary min d whic h left so deep trace in science gets even greater." It was Galois's theory that made it possible to explain the miracle whic h allowed imaginar y quantitie s to appear i n the formula fo r solving a third- degree equation wit h rea l roots; indeed, i t became possible to show that, if an equation has all roots real and if it can be solved by means of radicals, then it can be solved by means of square roots only. B y using the same theory, it can also be shown that som e of the ancient problems—such a s the problem of doublin g a cube or the problem o f trisecting an angle—cannot be solved with a ruler and a compass. B y his significant wor k Traite des Substitutions, Jordan19 erecte d a monument in honor of Galois. Being both simpl e an d profound, Galois' s mai n ide a permitte d applica - tions in areas other than algebrai c equations. Emil e Picard and Ernest Ves- siot, for example, considered it highly important in integration of linear dif- ferential equations . I t is noteworthy tha t Drac h an d Vessiot attempte d to extend Galois's theory to solving the most general differential equation s but encountered difficultie s tha t coul d b e overcome onl y i f the original theor y were altered or if, at least, some of its magnificent simplicit y were sacrificed . The development o f science after Galoi s demonstrated the growth of the importance o f group s i n th e mos t divers e branche s o f mathematic s and physics. Norwegia n mathematicia n Sophu s Lie , the founder o f the theory of group s of transformations, introduce d the m int o analysi s and geometry. A great admirer of Galois, he dedicated his momentous opus about groups of transformations (i n 1889) to l'Ecole Normale Superieure . Indeed , the most significant results concerning developing, refining, extending, and finding new applications of Galois's theory were made in France. Poincar e claimed tha t the notion of group had already existed in the spirit of geometry; the axiom that two geometric figures are equal to each other if each of them is equal to a third is in fact identical to the statement that there is a group that regulate s geometry, mor e precisel y a family o f procedures b y which on e figure turns into anothe r tha t i s equal to the first. I t is extraordinarily importan t tha t group theor y i s capable of giving us all concrete, connecte d meaning s tha t can be given to the expression "equal figures"; as it was shown in 1872 by the great German mathematicia n Feli x Klein, exactl y this implies the existence of infinitely man y geometries, each ruled by a special group, as well as by the fact that each geometry can be investigated independently, without resortin g to elementary geometry . Thi s framework encompasse s projectiv e geometry , the field in which tw o figures are considered equa l i f one of them ca n be obtained from th e other by a sequence of projections. VI Since Galois' s deat h on e century ha s passed. Durin g that perio d mathe - matics has developed remarkably ; innumerabl e volume s hav e bee n written ,

19CamilIe Jordan (1838-1922) . 296 D . TH E INFLUENC E O F FRANCE IN TH E DEVELOPMEN T O F MATHEMATIC S some of which, I must say , take undeserved spac e in libraries. Som e of the theories, just formulate d a t the time o f Galois , have sinc e been profoundl y explored, an d some of them have penetrated other areas of mathematics ; in a word, as mathematics, like other sciences, has been constantly and dramat- ically changing , i t became difficul t fo r a mathematician, n o matte r wh o h e might be, to have true insight into its current state. Ther e are fewer and fewer minds capabl e o f makin g significan t discoverie s i n eithe r pur e o r applie d mathematics. I t is rar e to encounter a genius similar to that of the French- man Andr e Ampere (1775-1836), wh o wa s also a physicist, th e founder o f electrodynamics, an d a remarkable mathematician (h e and Monge share the credit for creatin g the theory of partia l differentia l equation s of th e secon d order). The Frenchman Gabriel Lame (1795-1870) was an analyst, geometer, and the founde r o f elasticit y theory , whil e th e Frenchma n Simeo n Poisso n (1781-1840) i s famous fo r his works in analysis and mathematical physics ; Augustin Fresnel (1788-1827) , th e creator of physica l optics—whos e work s had finally ensured, a t least until the appearanc e of quantu m physics, a tri- umph of the modular theory of light—can be considered a mathematician as well. Instead of giving you a long, and likely tedious, list of names, let us focus on just a few of the greatest contemporary French mathematicians, those who were my professors and to whom I am honored and happy to have a chance to pay respect. Soon afte r bein g admitte d t o l'Ecol e Polytechnique , Charle s Hermit e (1822-1901) wrot e t o th e well-know n professo r Jacobi—who , alon g wit h Abel, was one of the founders of the theory of elliptical functions—and sen t him a n articl e abou t classifyin g Abel' s transcendenta l functions , th e func = tions related to integration of the most general algebraic differentials. Jacobi , who wa s once , unde r simila r circumstances , kindl y receive d b y Legendre , congratulated the young Hermite on his marvelous results. Tha t was only the beginning of regular correspondence between these two great mathematicians. It was Jacobi to whom, at the age of twenty-four, Hermit e sent his discover- ies in advanced algebra , th e discoveries tha t ultimately secure d him a place among the most prominent geometers. Buildin g on the most famous Gauss's results, he confidently approached the algebraic theory of shapes in their most general form and introduced continuous variables into number theory, a field characterized by discontinuity. Th e fact that he was the one who introduced quadratic forms wit h indefinit e conjugat e terms , toda y know n a s Hermite's forms, i s th e reaso n tha t hi s nam e i s on e o f th e mos t frequentl y foun d i n works from quantum physics. I n 1873, Hermite became famous by discover- ing the transcendentality of e , th e base of Neper's logarithm (the existence of transcendentals, the numbers that satisfy n o algebraic equation whose coeffi - cients are rational numbers, had first been demonstrated by Joseph Liouville). As Hermite's resul t made a strong impression, som e expecte d hi m t o prove transcendentality o f n , and thus, consequently, t o destroy foreve r the hope D. TH E INFLUENCE OF FRANCE IN THE DEVELOPMEN T O F MATHEMATICS 29 7 that a circle can be squared wit h a ruler and a compass; however , havin g found inspiration in Hermite's method and having devised a way to modify it properly, Ferdinand Lindemann, a German mathematician, cam e up with a proof instead, securing the honor for himself Hermite always left a profound impressio n on his listeners. "N o one will ever forget the sermon-like sound of Hermite's lectures," said the well-known mathematician Painleve 20, "or the feeling of beauty and revelation that one had to experience while listening to him talk about a marvelous discovery or something that wa s still waiting to be discovered. Hi s word had the ability to ope n vast horizons of science; it conveyed affectio n an d respect for high ideals." Ever y time I had a chance to listen to Hermite, I had before m e an image of quiet and pure joy cause d by contemplations abou t mathematics , joy similar to the one that Beethoven must have felt while feeling his music inside of himself. Gaston Darbou x (1847-1917 ) wa s an analyst an d geometer at the same time. Althoug h he was the initiator of some results in analysis, I shall not talk about that part of his work because it was his work in geometry that brought him recognition. H e surely was not one of the geometers who avoided tar - nishing the beauty of geometry by flattering analysis, and neither was he one of the analysts inclined to reduce geometry to calculations without any con- cern for or interest in their geometric meanings. I n this respect he followed in Monge's footsteps, connecting fine and well-developed geometric intuition with skille d application s o f analysis. Al l of his method s are extraordinar- ily elegan t an d perfectly suite d fo r the subject unde r investigation . Whil e teaching in the department o f higher geometr y at the Sorbonne, wher e he succeeded Michel Chasles, he frequently an d with reverence spoke about the theory of triple orthogonal systems, with pleasure stressing the importance of Lame's works; not less frequently h e spoke about the theory of deformations of planes, the theory which originated in Gauss's Disquisitiones circa Super- ficies Curvas and which, eve n before Darboux , wa s a subject o f significant works of French mathematicians, among whom Ossian Bonnet certainly de- serves a mention. Finally , Darboux demonstrated the usefulness o f a system of local coordinates, i.e., coordinates connected wit h the investigated figure rather than independent o f it. Thank s to the theory of groups, Elie Cartan further developed this approach and adapted it to the most diverse spaces cre- ated as a consequence of general relativity theory. Darbou x had tremendous influence on the development of geometry; of his numerous students and fol- lowers, I shall mention only the well-known Roumanian geometer Tzitzeica, one of the founder s o f the Mathematica l Review s of the Balka n Union , a man whose recent death is still mourned in the world of science. Classi c in its field, Darboux's work Theorie des Surfaces is a splendid monument erected in honor of both analysis and geometry.

'Paul Painleve (1863-1933) . 298 D . TH E INFLUENCE O F FRANC E I N TH E DEVELOPMEN T O F MATHEMATIC S

A story ha s it that, whe n a youn g German mathematicia n expresse d hi s puzzlement over Lagrange's refusal to recognize Gauss as the greatest German geometer, Lagrange told him, "No, he cannot be the greatest German geome- ter for h e i s the greatest Europea n geometer! " I n the sam e spirit on e coul d say that Henr i Poincar e (1854-1912 ) wa s no t onl y a grea t mathematicia n but mathematics itself . I t is impossible to find a branch o f mathematics— a branch o f physic s even—in whic h h e did no t leav e a trace o r which h e did not rejuvenate o r from whic h he did not infer a completely ne w field. Afte r creating Fuchsian functions 21, h e used uniform function s wit h the same pa- rameter to express the coordinates of a point on an algebraic surface, and in that way obtained the result which, before him, was known only for some spe- cial classes of surfaces. H e solved the uniformization proble m i n a way that, at the time, was quite brave. H e was a forerunner o f the theory o f function s with severa l comple x variables . Also , h e create d th e theor y o f differentia l equations in a real field; due to that theory, h e was then abl e to restore th e methods o f celestia l mechanics , t o stud y periodi c solution s o f problem s o f this field, and to investigate stabilit y problems. I n analysis situs, the part o f geometry interested only in those properties of objects that are not affected b y continuous transformations , Poincar e authore d severa l treatise s that woul d become the starting point fo r nearl y all later results in that field. At the Sor- bonne, b y lecturing on al l areas of mathematica l physics , he influence d th e 22 ideas triggered b y Michelson' s experimen t . Wit h hi s earl y death , scienc e lost on e o f it s mos t prominen t leaders . Translate d t o man y languages , hi s scientific-philosophical work s La Science et VHypothese and La Valeur de la Science are well known to the entire world. I n som e ways—one o f which i s well illustrated b y Poincare's words , "Thought i s only a flash in the middl e of a long night, but the flash that means everything"—Poincare ca n be com- pared with Pascal. I t will take a l

It was Poincare himself who named them this way after the German mathematician Lazarus Fuchs; nowadays, these functions ar e called automorphic. 22 Also known as the Michelson-Morley experiment . D. TH E INFLUENCE O F FRANCE IN TH E DEVELOPMEN T O F MATHEMATIC S 29 9 branch o f geometr y especiall y wel l develope d i n Italy . I t i s tru e tha t th e viewpoint o f Italia n geometer s fro m th e pas t centur y wa s more clearl y de - fined than that of Emile Picard, but, as Emile Borel put it, algebraic geometry would be certainly crippled without Picard's contributions . VII The glory of French mathematics created by the greatest results of Hermite, Darboux, Poincare , an d Picar d ha s no t darkened . Indeed , th e flame i s a s strong as it has ever been. A s the time is short, to justify thi s statement I am forced to limit mysel f to just a few names. Gabriel Koenigs was a fine geometer; the elegance of some of his works can be compare d wit h tha t o f Darboux . B y creating ne w transcendentals, Pau l Painleve solve d a proble m tha t eve n t o Poincar e seeme d unapproachable ; Poincare characterized Painleve' s result s in analysis by saying : "Mathemat - ics is a well-ordered continen t whos e countries are united; the work o f Pau l Painleve i s a magnificen t islan d i n a n ocean. " Bu t thi s judgment i s some- what incomplet e because Painleve—who, fo r a long time, taught mechanic s at l'Ecole Polytechnique—also remarkabl y advance d mechanics ; besides, his theoretical researc h prompte d developmen t o f aviatio n i n suc h a measur e that on e may sa y that, thanks to Painleve, aviation i s an exclusivel y Frenc h creation. The results of Jacques Hadamard were numerous and significant: i n arith- metic, he worked on the Riemann's function relate d to the complicated prob- lem o f distributio n o f prim e numbers ; i n geometry , h e researche d geodesi c lines wit h opposit e curvatures ; i n analysis , h e publishe d work s abou t par - tial differentia l equation s i n mathematica l physics. Also , h e gav e a stron g stimulus t o th e calculu s o f variation s an d functiona l analysis , th e ne w sci - ence founded b y the Italian mathematician Volterra . Finally , his seminar at College de France, where all foreign mathematician s wishe d to present thei r latest results, influenced internationa l collaboration i n mathematics. A s he is still young, I may sa y with certainty that his work is far fro m finished. The researc h o f function s wit h comple x variable s ha s alway s bee n ver y successful i n France . Her e I mentio n Emi l Borel ; th e short-live d analys t Fatou; Pau l Montel , famou s fo r hi s theor y concernin g familie s o f norma l functions; Gasto n Julia , know n fo r hi s work s abou t elevatio n o f rationa l functions; an d s o forth . The theory of functions wit h real variables is of almost exclusivel y French origin. Se t u p b y Camill e Jordan's Traite d'Analyse (which , lik e Emil e Pi - card's treatise of the same name, had international influence), founded b y the works of Emile Borel, Henri Lebesgue (wh o defined measur e of a set), Rene Baire (who introduced integrals which today bear his name), and Denjoy (th e creator of the totalization theory) , it introduced unexpecte d harmon y int o a field that ha d bee n neglecte d fo r a lon g time , testimon y t o th e darin g an d talent o f its creators. 300 D . TH E INFLUENC E OF FRANC E I N TH E DEVELOPMEN T O F MATHEMATIC S I cannot but mention Mauric e Frechef s theory of abstract spaces , Bouli- gand's infinitesimal geometry, and Elie Cartan's works in analysis and geom- etry, the last of which I am not qualified to judge. Institut Henri Poincare is born from ne w French enthusiasm fo r research in th e field of mathematica l physics . Emil e Borel , the sou l o f probabilit y theory, started a praise-deserving series of publications in this field, similar to the one in function theory—a series in which Frechet, Paul Levy, and Georges Darmois presented thei r excellen t results . Th e Department o f Theoretica l Physics is headed by Louis de Broglie, the creator of wave mechanics, who restored atomic physics and reconciled the undulatory and corpuscular theory of light. I should not forget to mention the Institut of Mechanics, headed by Henri Villat, known for his results in hydrodynamics, who is also editor of the internationally known collection Memorial des Sciences Mathematiques and editor-in-chief o f Journal de Mathematiques Pures et Appliquees, a journal which, nearl y a century ago , wa s started b y Liouvill e an d whic h fo r quit e some time was edited by Camille Jordan. The account o f French mathematical activit y would be incomplete with- out a mention of l'Ecole Polytechnique and PEcole Normale. Fo r more than a century , grea t Frenc h mathematician s hav e owe d their educatio n t o on e of the two institutions; in the last half-century tha t marvelous role belonged almost exclusivel y to FEcol e Normale, which , eve n a good fifty years ago, Sophus Lie considered a nursery of French mathematics. Youn g talents from many countries hav e been comin g here to ge t the sam e education a s their French colleagues. That is why it is difficult no t to consider Georges Tzitze- ica, whom I already mentioned, to be a French mathematician. Fo r the same reason, I a m incline d t o includ e amon g Frenc h mathematician s m y goo d friend Mihail o Petrovic, a doye n o f Yugosla v mathematics , wh o is widely recognized for his great originality in inventing the spectral method in arith- metic, algebra, an d analysis, and als o fo r creatin g genera l phenomenology , the field which systematically examines the problems of existence of analyt- ical mold s that coul d be used to present simultaneousl y severa l apparentl y different physica l theories . I hop e that yo u wil l no t objec t i f I credi t hi s results to the accomplishments which mathematics owes to France. Thanks to PEcol e Normale, youn g mathematicians ar e read y t o replac e the olde r ones . On e might sa y that i t i s too earl y t o mentio n names , but some o f the m ar e nevertheles s alread y wel l known . I shal l mentio n onl y Jacques Herbrand, whos e works, mercilessly interrupted b y his early death, were announcing a great mathematician, perhaps similar to Evariste Galois. Ladies and gentlemen, it is time for me to finish this talk, for I have already used a great deal of your kind attention. I n conclusion, I would like to make just one remark of general nature. More than any other science, mathematics develops through a sequence of consecutive abstractions. A desire to avoid mistakes forces mathematician s to find and isolate the essence of the problems and entities considered. Car - D. TH E INFLUENCE O F FRANCE IN TH E DEVELOPMEN T O F MATHEMATICS 30 1 ried to an extreme, this procedure justifies the well-known joke according to which a mathematician i s a scientist wh o neither know s what he is talking about or whether whatever he is indeed talking about exists or not. Frenc h mathematicians, however, never enjoyed distancing themselves from reality ; they do know that, although needed, logic is by no means crucial. I n math- ematical activity, like in any other type o f human activity, on e should find a balance of values: there is no doubt that it is important to think correctly, but i t is even more important to formulate th e righ t problems . I n that re- spect, one can freely sa y that French mathematicians not only always knew what they wer e talking about, but als o had the righ t intuition t o select the most fundamenta l problems , those whose solutions produced th e stronges t influence o n the overall development of science. This page intentionally left blank Bibliography

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