Selected Title s i n This Serie s

23 Kare n Hunge r Parshal l an d Adria n C . Rice , Editors , Mathematic s unbound : Th e evolution o f a n internationa l mathematica l researc h community , 1800-1945 , 200 2 22 Bruc e C . Bernd t an d Rober t A . Rankin , Editors , Ramanujan : Essay s an d surveys , 2001 21 Arman d Borel , Essay s i n the histor y o f Li e groups an d algebrai c groups , 200 1 20 Kolmogoro v i n perspective, 200 0 19 Herman n Grassmann , Extensio n theory , 200 0 18 Jo e Albree , Davi d C . Arney , an d V . Frederic k Rickey , A station favorabl e t o th e pursuits o f science : Primary material s i n the histor y o f mathematics a t th e Unite d State s Military Academy , 200 0 17 Jacque s Hadamar d (Jerem y J . Gra y an d Ab e Shenitzer , Editors) , Non-Euclidea n geometry i n the theor y o f automorphic functions , 199 9 16 P . G . L . Dirichle t (wit h Supplement s b y R . Dedekind) , Lecture s o n numbe r theory , 1999 15 Charle s W . Curtis , Pioneer s o f representation theory : Frobenius , Burnside , Schur , an d Brauer, 199 9 14 Vladimi r Maz'y a an d Tatyan a Shaposhnikova , Jacque s Hadamard , a universa l mathematician, 199 8 13 Lar s Garding , Mathematic s an d mathematicians : Mathematic s i n Swede n befor e 1950 , 1998 12 Walte r Rudin , Th e wa y I remember it , 199 7 11 Jun e Barrow-Green , Poincar e an d th e thre e bod y problem , 199 7 10 Joh n Stillwell , Source s o f hyperbolic geometry , 199 6 9 Bruc e C . Bernd t an d Rober t A . Rankin , Ramanujan : Letter s an d commentary , 199 5 8 Kare n Hunge r Parshal l an d Davi d E . Rowe , Th e emergenc e o f the America n mathematical researc h community , 1876-1900 : J . J . Sylvester , Feli x Klein , an d E . H . Moore , 1994 7 Hen k J . M . Bos , Lecture s i n the histor y o f mathematics, 199 3 6 Smilk a Zdravkovsk a an d Pete r L . Duren , Editors , Golde n year s o f Mosco w mathematics, 199 3 5 Georg e W . Mackey , Th e scop e an d histor y o f commutative an d noncommutativ e harmonic analysis , 199 2 4 Charle s W . M c Arthur, Operation s analysi s i n the U.S . Army Eight h Ai r Forc e i n Worl d War II , 199 0 3 Pete r L . Dure n e t al. , Editors , A century o f mathematics i n America , par t III , 198 9 2 Pete r L . Dure n e t al. , Editors , A century o f mathematics i n America, par t II , 198 9 1 Pete r L . Dure n e t al. , Editors , A century o f mathematics i n America , par t I , 198 8 This page intentionally left blank MATHEMATICS UNBOUND: The Evolution of an International Mathematical Research Community, 1800-1945 This page intentionally left blank https://doi.org/10.1090/hmath/023

HISTORY OF MATHEMATICS Volum e 23

MATHEMATICS UNBOUND: The Evolution of an International Mathematical Research Community, 1800-1945

Karen Hunger Parshall Adrian C. Rice Editors

America n Mathematica l Societ y Londo n Mathematica l Societ y Editorial Boar d American Mathematica l Societ y Londo n Mathematica l Societ y George E . Andrew s Davi d Fowler , Chai r Joseph W . Daube n Jerem y J . Gra y Karen Parshall , Chai r To m Korne r Michael I . Rose n Pete r Neuman n

2000 Mathematics Subject Classification. Primar y 01A55 , 01A60 , 01A70 , 01A72 , 01A73 , 01A74, 01A80 . A complet e listin g o f photograph an d figur e credit s appear s o n pag e xxi .

Library o f Congres s Cataloging-in-Publicatio n Dat a Mathematics unboun d : th e evolutio n o f a n internationa l mathematica l researc h community , 1800-1945 / Kare n Hunge r Parshall , Adria n C . Rice , editors . p. cm . — (Histor y o f mathematics, ISS N 0899-242 8 ; v. 23 ) Includes bibliographica l reference s an d index . ISBN 0-8218-2124- 5 (acid-fre e paper ) 1. Mathematics—Research—History—19t h century . 2 . Mathematics—Research—History — 20th century . 3 . Mathematics—History—19t h century . 4 . Mathematics—History—20t h cen - tury. I . Parshall, Karen Hunger, 1955 - II . Rice, Adrian C . (Adrian Clifford) , 1970 - III . Series.

QA11.2.M28 200 2 510'.72~dc21 2002022744

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisitions Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o reprint-permissionOams.org . © 200 2 by the America n Mathematica l Society . Al l rights reserved . Printed i n the Unite d State s o f America . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . @ Th e pape r use d i n thi s boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . The Londo n Mathematica l Societ y i s incorporated unde r Roya l Charte r and i s registered wit h th e Charit y Commissioners . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 7 06 0 5 0 4 03 0 2 Contents

Acknowledgments xii i

List o f Contributors x v

Photograph an d Figur e Credit s xx i

Chapter 1 . Th e Evolutio n o f a n Internationa l Mathematica l Researc h Community, 1800-1945 : A n Overvie w an d a n Agend a KAREN HUNGE R PARSHAL L AN D ADRIA N C . RIC E 1 Introduction 1 Internationalization, Internationalism , Transnationahsm , Supranationahsm , Multinationalism, Denationalization , .. . : What's i n a Word? 2 The Timeframe : 1800-194 5 5 The Internationalizatio n o f a Mathematica l Researc h Community , 1800-1945: A First Vintag e 6 A Second Vintag e an d Beyon d 1 1 References 1 4

Chapter 2 . Th e En d o f Dominance : Th e Diffusio n o f Frenc h Mathematic s Elsewhere, 1820-187 0 IVOR GRATTAN-GUINNES S 1 7 Multinationalism vs . Internationalism 1 7 French Dominanc e 1 8 Translating th e Frenc h 2 0 Declinl 2 4 Case Study 1 : Real-Variabl e Analysi s 2 5 Case Study 2 : Complex-Variabl e Analysi s 2 7 Case Study 3 : Fro m Energ y Mechanic s to Energetics 2 8 Case Stud y 4 : Celestia l Mechanics , Especiall y Perturbation s 3 0 Case Study 5 : Th e Influenc e o f Gauss 3 2 Concluding Remark s 3 3 References 3 4 Appendix 3 9 Chapter 3 . Spanis h Initiative s t o Brin g Mathematic s i n Spai n int o th e International Mainstrea m ELENA AUSEJ O AN D MARIAN O HORMIGO N 4 5 The International Mainstream : A Problem o f Definition 4 5 The Enlightenmen t 4 6 The Nineteenth Centur y 4 8 The Rol e o f Individuals i n History 5 1 viii CONTENT S

The Firs t Thir d o f the Twentiet h Centur y 5 3 Conclusion 5 7 References 5 8

Chapter 4 . Internationa l Mathematica l Contribution s t o Britis h Scientifi c Journals, 1800-190 0 SLOAN EVAN S DESPEAU X 6 1 Introduction 6 1 Foreign Mathematic s i n Britis h Genera l Scienc e Journal s 6 2 British Specialize d Scienc e Journal s a s a Venu e fo r Foreig n Mathematic s 6 6 Changes i n Foreig n Participatio n throug h th e Nineteent h Centur y 6 9 A Geographica l Profil e o f the Publicatio n Communit y 7 6 Society Involvement an d Persona l Influence : Factor s i n Foreign Participation 7 9 Conclusions 8 3 References 8 3

Chapter 5 . Internationa l Participation i n Liouville's Journal de mathematiques pures et appliquees JESPER LUTZE N 8 9 Introduction 8 9 The Nationa l Enterpris e 8 9 Foreign Contribution s t o Liouville' s Journal 9 1 Countering th e Sens e o f Frenc h Self-sufficienc y 9 3 International Inspiration s fo r Liouville' s Work : Mechanic s 9 5 International Inspiration s fo r Liouville' s Work : Potentia l Theor y 9 7 International Inspiration s fo r Liouville' s Work : Differentia l Geometr y 9 8 Conclusion 10 0 References 10 1

Chapter 6 . Th e Effect s o f War o n France's International Rol e i n Mathematics , 1870-1914 HELENE GISPER T 10 5 Introduction 10 5 The Creatio n o f a Nationa l Mathematica l Society : Nationalis m an d Professionalization 10 6 Structuralization o f the SMF : Establishin g a n Academi c Cente r 11 0 A Non-academi c Periphery : Actor s i n an d Value s o f the AFA S 11 1 Foreign Contribution s t o th e Journal s o f the SM F an d th e AFA S 11 5 Center an d Periphery : Fro m Nationa l Context s t o th e Internationa l Scal e 11 7 Conclusion 11 9 References 12 0

Chapter 7 . Charle s Hermit e an d Germa n Mathematic s i n Franc e THOMAS ARCHIBAL D 12 3 Introduction 12 3 Hermite an d Germa n Mathematica l Value s 12 4 Hermite's Critiqu e o f Radicalism 12 6 The Promotio n o f Franco-German Relation s i n Mathematics 12 8 Bringing Germa n Mathematic s t o Frenc h Students : A Thankles s Tas k 13 1 German Mathematic s an d th e Frenc h Mathematica l Researc h Communit y 13 3 CONTENTS i x

Concluding Remark s 13 5 References 13 6

Chapter 8 . Gost a Mittag-Leffle r an d th e Foundatio n an d Administratio n o f Acta Mathematica JUNE E . BARROW-GREE N 13 9 Introduction 13 9 The Foundin g o f Acta Mathematica 14 0 Poincare, Cantor , Kovalevskaya , an d Acta Mathematica 14 8 Acta Mathematica Volume s 1-2 0 15 5 Conclusion 15 8 References 16 1

Chapter 9 . A n Episod e i n the Evolutio n o f a Mathematical Community : Th e Case o f Cesar e Arzel a a t Bologn a LAURA MARTIN I 16 5 The Bolognes e Contex t 16 5 Galois Theor y i n the Europea n Curriculu m 16 6 Arzela's Source s fo r th e Lecture s 16 9 The Ruffini-Abe l Theore m 17 0 Conclusions 17 5 References 17 5 Appendix 17 8

Chapter 10 . Th e Firs t Internationa l Mathematica l Community : Th e Circolo matematico di Palermo ALDO BRIGAGLI A 17 9 Introduction 17 9 The Circolo matematico di Palermo i n the Sicilia n Cultura l Milie u 17 9 The Circolo matematico di Palermo i n th e Italia n Mathematica l Milie u 18 0 The Rol e o f Giova n Battist a Gucci a i n the Circolo 18 3 The Circolo an d th e Internationalizatio n o f Mathematical Researc h 18 6 A Decad e o f Grea t Development , 1904-191 4 18 7 The Declin e 19 5 Conclusions 19 8 References 19 9

Chapter 11 . Language s fo r Mathematic s an d th e Languag e o f Mathematic s in a Worl d o f Nation s JEREMY J . GRA Y 20 1 Introduction 20 1 A Worl d o f Nations 20 1 National Language s 20 3 International Language s i n Genera l 20 5 International Language s i n Particula r 20 6 Language, Meaning , an d Mathematics : Calculatio n an d Pasigraph y 20 9 Is Mathematics a Language? 21 1 Universal Languag e an d Calculatin g Languag e 21 2 Nineteenth-century Linguistic s 21 4 Language, Meaning , an d Mathematics : Signifie s 21 6 x CONTENT S

Language, Meaning , an d Mathematics : Hilber t an d Husser l 21 8 Language, Meaning , an d Mathematics : Hilber t an d Schrode r 22 1 Conclusion 22 3 References 22 4

Chapter 12 . Th e Emergenc e o f th e Japanes e Mathematica l Communit y i n the Moder n Wester n Style , 1855-194 5 CHIKARA SASAK I 22 9 Japanese Mathematic s fro m Traditiona l t o Moder n 22 9 Chinese Mathematic s an d It s Refor m i n Seventeenth-Centur y Japa n 23 0 Learning Wester n Mathematic s a s Militar y Science , 1855-186 8 23 1 Educational Refor m durin g th e Earl y Meij i Period , 1868-187 7 23 2 The Universit y o f Toky o an d th e Toky o Mathematica l Society , 1877-188 1 23 6 The Germanizatio n o f the Politica l Syste m an d o f Learning, 1881-194 5 23 6 "For th e Nation! " Fujisaw a an d Mathematica l Researc h a t Toky o 23 8 The Kyot o Universit y Schoo l 24 3 The Tohok u Universit y Schoo l 24 3 The Othe r College s an d Universitie s 24 6 Towards Democratizatio n an d Internationalizatio n afte r Worl d Wa r I I 24 6 Conclusion: Genera l Characteristic s o f Mathematica l Studie s i n Moder n Japan 24 7 References 24 9

Chapter 13 . Internationalizin g Mathematic s Eas t an d West : Individual s and Institution s i n th e Emergenc e o f a Moder n Mathematica l Community i n Chin a JOSEPH W . DAUBE N 25 3 Introduction 25 3 Modern Scienc e Emerge s i n China : Th e Self-Strengthenin g Movemen t 25 4 The Beijin g Tongwen Guan (1861-1862 ) 25 6 The Shangha i Tongwen Guan (1863-1864 ) an d th e Jiangnan Arsena l (1865 ) 25 8 The Fuzho u Shipyar d (1866 ) 26 0 Educational Refor m an d th e "Reig n o f On e Hundre d Days " 26 2 Japan a s a n Earl y Rol e Mode l 26 3 England: Supportin g th e Ris e o f Mathematics i n Moder n Chin a 26 4 France: Contributin g t o the Transmissio n o f Modern Mathematic s t o China26 8 Germany: A Mode l fo r Developin g Moder n Mathematic s i n Chin a 27 0 The Unite d State s 27 2 New Institutiona l Model s 27 5 Universities 27 6 The Chines e Mathematica l Societ y 27 7 Conclusion 28 1 References 28 2

Chapter 14 . Chinese-U.S . Mathematica l Relations , 1859-194 9 YIBAO X u 28 7 Introduction 28 7 The Boxe r Indemnit y an d th e Modernizatio n o f Chines e Mathematic s 28 8 Harvard: A n Educationa l Cente r fo r Chines e Mathematician s 29 4 CONTENTS x i

The Institute fo r Advance d Study : A Bridge Betwee n the U.S . and Chin a 29 6 Two Unsuccessfu l Invitation s 30 1 Conclusion 30 4 References 30 5

Chapter 15 . America n Initiative s Towar d Internationalization : Th e Cas e o f Leonard Dickso n DELLA DUMBAUG H FENSTE R 31 1 Introduction 31 1 Dickson i n the Emergen t Perio d o f American Mathematic s 31 3 Dickson's Research : Th e International Exchang e o f Mathematical Idea s 31 8 Dickson an d th e Publicatio n o f Manuscripts an d Book-lengt h Treatise s 32 3 American Mathematics : Unboun d 32 8 References 32 9

Chapter 16 . Th e Effect s o f Naz i Rul e o n the Internationa l Participatio n o f German Mathematicians : A n Overvie w an d Tw o Cas e Studie s REINHARD SIEGMUND-SCHULTZ E 33 5 Introduction 33 5 Internationalization: A Complex o f Factors 33 6 Germany Immediatel y afte r 1933 : The Dogm a o f Antisemitism 33 9 Case Study 1 : Germa n Participatio n i n the Osl o ICM, July 193 6 34 2 Case Stud y 2 : WWI I an d Germa n Internationa l Participation : Haral d Geppert an d Wilhel m Sus s 34 5 Conclusions 34 7 References 34 9 Appendix 35 2

Chapter 17 . War , Refugees , an d th e Creatio n o f a n Internationa l Mathematical Communit y SANFORD L . SEGA L 35 9 Introduction 35 9 Before Worl d Wa r I 36 0 World Wa r I and Afte r 36 2 Hitler's German y an d Mathematica l Refugee s 36 8 The Unite d States : Countr y o f Refuge fo r Mathematician s 37 1 Internationalism unde r America n Leadershi p 37 5 References 37 6

Chapter 18 . Th e Formatio n o f the International Mathematica l Unio n OLLI LEHT O 38 1 Introduction 38 1 Background 38 1 The First Internationa l Congresse s o f Mathematicians 38 4 World Wa r I and it s Aftermath 38 5 The Birt h o f the IM U 38 7 CONTENTS

Opposition t o the Polic y o f Exclusion 38 9 The Dissolutio n o f the IM U 39 1 Conclusion 39 2 Epilogue: The Ne w IMU 39 3 References 39 5

Index 399 Acknowledgments

The present volume evolved from a three-day, international symposium, entitle d "Mathematics Unbound: Th e Evolution o f an International Mathematical Commu - nity, 1800-1945, " an d hel d a t th e Universit y o f Virgini a 27-2 9 May , 199 9 a s par t of the Mathematics Department' s "Emphasi s Year " i n the histor y o f mathematics . The conference , whic h brough t togethe r scholar s fro m eleve n differen t countries , received generou s suppor t fro m th e Nationa l Scienc e Foundatio n i n th e for m o f grant numbe r SBR-981793 3 as well as from the University o f Virginia's Dean o f the College o f Art s an d Sciences , Corcora n Departmen t o f History , an d Departmen t of Mathematics. Thi s financial suppor t notwithstanding , th e conferenc e woul d no t have been a success without the tireless and ever-cheerfu l practica l assistance o f the Mathematics Department' s administrativ e staff , Conni e Abell , Mar y Cline , Juli e Riddleberger, an d Joyc e Stevens . The last twenty-seven months have found us all—authors, referees, and editors— as part o f a truly international , collaborativ e effor t tha t ha s unite d scholar s a t al l levels o f professional development . Th e author s hav e worke d faithfull y an d consci - entiously to produce chapters that responde d meaningfull y t o the critical comment s and suggestion s o f both th e referee s an d th e editors . Fo r thei r part , th e referee s provided thoughtful , detailed , and , abov e all , promp t report s wit h a n ey e towar d thematic tightness . Th e ensuin g dialogu e ha s resulte d i n a serie s o f chapter s tha t have bee n almos t surprisin g i n their interconnectio n an d mutua l complementarity . As editors, w e thank no t onl y the author s an d referee s bu t als o the numerou s people wh o hav e helpe d i n the productio n o f the finished volume . Firs t an d fore - most, Juli e Riddleberge r labore d unde r th e unforgivin g yok e o f LaTeX^s indexin g program, keying the entire index with remarkably good cheer in the face o f technical adversity! A t the America n Mathematica l Society , Chri s Thivierge o f the HMAT H series, Vicki e Ancon a i n the Productio n Department , an d Barbar a Beeto n i n Au - thor Suppor t hav e bee n constan t source s o f information, help , an d guidance . Ou r thanks als o go to Alle n Debus, Ivor Grattan-Guinness, an d Brian Parshall fo r thei r constructive remark s an d observation s o n variou s part s o f th e manuscript . The y seem neve r t o tir e o f request s fro m thei r friend s (o r thei r spous e a s the cas e ma y be) fo r thei r exper t advice . Our wor k o n thi s projec t ha s bee n stimulating , bot h professionall y an d intel - lectually. W e hop e tha t th e end-produc t wil l no t onl y prov e illuminatin g t o ou r readers bu t als o provid e a n impetu s fo r futur e investigation s int o th e fascinatin g historical questio n o f the internationalizatio n o f research-level mathematics .

Charlottesville, Virginia Kare n Hunge r Parshal l 10 August, 2001 Adria n C . Ric e

xiii This page intentionally left blank List o f Contributor s

Thomas Archibal d i s Professo r i n th e Departmen t o f Mathematic s an d Statis - tics a t Acadi a Universit y i n Wolfville , Canada . H e studie d a t th e Universit y o f Waterloo, Yor k University , an d th e Universit y o f Toronto, obtainin g a Ph.D. fro m the las t o f these i n 1987 . Hi s current researc h concern s mathematic s i n Franc e i n the lat e nineteent h century . Othe r interest s includ e the histor y o f the relationshi p between mathematic s an d physics , particularl y i n the nineteent h century , an d th e history o f mathematics i n Canada . Address: Departmen t o f Mathematics and Statistics, Acadia University, Wolfville , NS, BO P 1X0 , Canada . E-mail: [email protected] a

Elena Ausej o earne d he r Ph.D . i n mathematic s a t th e Universit y o f Zaragoz a in 199 1 an d currentl y serve s a s Associat e Professo r o f th e Histor y o f Scienc e a t her alma mater. Th e Secretar y Genera l o f the Organizin g Committe e o f the XI X International Congres s o f the Histor y o f Scienc e (Zaragoza , 1993 ) an d forme r Sec - retary Genera l o f the Spanis h Societ y fo r th e Histor y o f Scienc e an d Technolog y (1986-1993), she is Vice President o f the Spanis h Societ y fo r the History o f Scienc e and Technology . He r researc h focuse s o n th e histor y o f mathematic s an d scienc e (primarily i n Spain ) i n th e nineteent h an d twentiet h centurie s an d o n scientifi c institutionalization. Sh e i s the autho r o f Por la Ciencia y por la Patria: La insti- tutionalization cientifica en Espana en el primer tercio del siglo XX (Sigl o XXI d e Espaha Editore s S.A. , 1993 ) an d ha s edite d (wit h Marian o Hormigon ) th e books , Messengers of Mathematics: European Mathematical Journals 1800-1946 (Sigl o XXI, 1993 ) an d Paradigms and Mathematics (Sigl o XXI, 1996) , among others . Address: Faculta d d e Ciencia s (Matematicas) , Ciuda d Universitaria , E-5000 9 Zaragoza, Spain . E-mail: [email protected] s

June E . Barrow-Gree n i s a Researc h Fello w i n th e Histor y o f Mathematics a t the Ope n University . He r researc h interest s ar e the histor y o f mathematics o f th e nineteenth and twentieth centuries, in particular, dynamica l systems theory. Sh e is the autho r o f Poincare and the Three Body Problem (AMS/LM S HMATH , 1997 ) and i s currently studyin g th e wor k o f Georg e Birkhoff . Sh e i s als o concerned wit h the us e o f databases i n the history o f mathematics an d with the rol e o f the Web i n history o f mathematics education . Address: Facult y o f Mathematics , Th e Ope n University , Milto n Keyne s MK 7 6AA, United Kingdom . E-mail: j [email protected]

XV xvi LIS T O F CONTRIBUTOR S

Aldo Brigagli a earne d hi s degre e i n physic s a t th e Universit y o f Palermo, wher e he i s no w Professo r o f "Matematich e Complementari. " Sinc e th e earl y 1980s , hi s main are a o f researc h ha s bee n th e histor y o f science , particularl y th e histor y o f mathematics i n Sicil y an d th e histor y o f geometr y an d it s interrelation s wit h al - gebra. H e ha s published , wit h Guid o Masotto , / / Circolo matematico di Palermo (Dedalo, 1982 ) and , wit h Cir o Ciliberto , Italian Algebraic Geometry Between the Two World Wars (Circol o matematico d i Palermo, 1995) , as well as articles on the history o f mathematics . Address: Dipartiment o d i Matematica , Universit a d i Palermo , Vi a Archiraf i 34 , 1-90123 Palermo, Italy . E-mail: [email protected] t

Joseph W . Daube n i s Professo r o f Histor y an d Histor y o f Scienc e a t Lehma n College, Cit y Universit y o f Ne w York (CUNY ) an d i s a membe r o f the Ph.D. Pro - gram i n Histor y a t th e Graduat e Cente r o f CUNY . Formerl y edito r o f Historia Mathematica an d a previou s Chai r o f the Internationa l Commissio n o n Histor y o f Mathematics, h e is also the author o f biographies o f Georg Cantor (Harvard , 1979 ) and Abraha m Robinso n (Princeton , 1995) . H e mad e hi s first tri p t o Chin a a s a visiting lecturer fo r fou r month s under the joint auspice s o f the U.S. National Acad - emy o f Sciences and the Chines e Academia Sinic a i n 198 8 and i s currently involve d in a joint ventur e wit h colleague s i n China , Taiwan , an d Singapor e t o produc e a critical Englis h editio n o f th e Ten Classics of Ancient Chinese Mathematics. H e is als o intereste d i n question s o f the transmissio n o f mathematic s betwee n Chin a and othe r countries , both i n antiquity, an d mor e recently , a s modern mathematic s began t o develo p i n Chin a i n the lat e Qin g Dynasty, an d i n the subsequen t cours e of the twentiet h century . Address: Ph.D . Progra m i n History , Th e Graduat e Center , Th e Cit y Universit y of Ne w York, 36 5 Fifth Avenue , Ne w York, Ne w York 10016 , United States . E-mail: [email protected] t

Sloan Evan s Despeau x earne d a B.A . i n mathematic s a t Franci s Mario n Uni - verisity an d a n M.S . i n mathematic s a t Florid a Stat e University . Sh e i s presentl y completing he r Ph.D. i n mathematics a t th e Universit y o f Virginia. He r forthcom - ing thesis i s entitled "Th e Developmen t o f a Publication Community : Nineteenth - century Mathematic s i n British Scientifi c Journals. " Address: Departmen t o f Mathematics, Universit y o f Virginia, Kercho f Hall , P.O . Box 400137, Charlottesville, Virgini a 22904-4137 , United States . E-mail: [email protected] u

Delia Dumbaug h Fenste r i s Assistant Professo r o f Mathematics a t th e Univer - sity o f Richmond . He r researc h focuse s o n th e histor y o f America n mathematics , particularly algebr a an d numbe r theory , i n the lat e nineteenth an d earl y twentiet h centuries. He r curren t wor k consider s th e rol e o f th e Carnegi e Institutio n i n th e development o f American mathematics . Address: Departmen t o f Mathematics an d Compute r Science , University o f Rich- mond, Richmond , Virgini a 23173 , United States . E-mail: [email protected] u

Helene Gisper t teache s th e histor y o f scienc e a t th e Institut universitaire de formation des maitres de Versailles an d a t th e Universit e d'Orsay . A membe r o f LIST O F CONTRIBUTOR S xvi i the Grou p o n th e Histor y an d Diffusio n o f th e Science s (GHDSO) , sh e work s o n the histor y o f mathematics i n France a t th e en d o f the nineteent h centur y a s wel l as on the history o f education an d o n the diffusio n o f the sciences i n France durin g the Third Republic . Address: Institu t universitair e d e formatio n de s maitre s d e Versailles , Group e d'histoire e t d e diffusio n de s science s d'Orsay , Batimen t 407 , Centre universitaire , F-91405 Orsa y Cedex , France . E-mail: [email protected] r

Ivor Grattan-Guinnes s i s Professo r o f th e Histor y o f Mathematic s an d Logi c at Middlese x University , England . H e wa s editor o f the histor y o f science journal, Annals of Science, fro m 197 4 to 1981 . In 1979 , he founded th e journal, History and Philosophy of Logic, and edite d i t unti l 1992 . H e edited th e Companion Encyclo- pedia of the History and Philosophy of the Mathematical Sciences ( 2 vols. , Rout - ledge, 1994 ) and published The Fontana History of the Mathematical Sciences: The Rainbow of Mathematics (Fontana , 1997 ) an d The Search for Mathematical Roots, 1870-1940: Logics, Set Theories, and the Foundations of Mathematics from Cantor through Russell to Godel (Princeton, 2000) . H e i s the Associat e Edito r fo r math - ematicians an d statistician s fo r th e New Dictionary of National Biography, t o b e published i n 2004 . H e i s editin g fo r Elsevie r a collectio n o f essay s o n Landmark Writings in Western Mathematics, 1640-1940, als o to appea r i n 2004 . Address: 43 , St. Leonard' s Road , Bengeo , Herts SG1 4 3JW, Unite d Kingdom . E-mail: [email protected] k

Jeremy J . Gra y i s Senior Lecturer i n Mathematics at the Open University, wher e he i s the Directo r o f the Centr e fo r th e Histor y o f the Mathematica l Sciences . H e is also an Affiliate d Researc h Schola r i n the Department o f History and Philosoph y of Scienc e a t th e Universit y o f Cambridge , England . H e work s o n th e histor y o f mathematics i n the nineteent h an d twentiet h centuries , wit h a particula r interes t in comple x functio n theor y an d geometry , an d als o o n issue s i n th e philosoph y and socia l significanc e o f mathematics. Hi s mos t recen t publicatio n i s The Hilbert Challenge (Oxford , 2000) . Address: Facult y o f Mathematics , Th e Ope n University , Milto n Keyne s MK 7 6AA, Unite d Kingdom . E-mail: [email protected] k

Mariano Hormigo n earne d degree s i n mathematic s befor e takin g hi s Ph.D . i n Philosophy a t th e Universida d Autonom a d e Madrid i n 1982 . H e has been Profes - sor o f the Histor y o f Scienc e i n the Scienc e Facult y o f the Universit y o f Zaragoz a since 1986 . Presiden t o f the Organizin g Committe e o f the XI X Internationa l Con - gress o f the History o f Scienc e (Zaragoza , 1993 ) and Pas t Presiden t o f the Spanis h Society fo r the Histor y o f Scienc e and Technolog y (1984-1993) , h e i s a membe r o f the Academi e international e d'histoir e de s science s (sinc e 1993 ) an d o f th e Inter - national Commissio n o n Histor y o f Mathematic s (sinc e 1997) . H e ha s serve d a s the edito r o f LLULL, th e journal o f the Spanis h Societ y fo r th e Histor y o f Scienc e and Technology , sinc e 1981 . H e ha s edite d (wit h Elen a Ausejo) , Messengers of Mathematics: European Mathematical Journals 1800-1946 (Sigl o XX I d e Espah a xviii LIS T O F CONTRIBUTOR S

Editores S.A. , 1993 ) an d Paradigms and Mathematics (Sigl o XXI , 1996) . Hi s re - search center s o n th e histor y o f mathematic s an d scienc e (primaril y i n Spain ) i n the nineteent h an d twentiet h centurie s an d o n scienc e an d ideology . Address: Faculda d d e Ciencia s (Matematicas) , Ciuda d Universitaria , E-5000 9 Zaragoza, Spain . E-mail: [email protected] s

Olli Lehto received his Ph.D. in mathematics i n 194 9 at the University o f Helsinki, where he served a s Associate Professo r (1956-1961) , Professor (1961-1988) , Recto r (1983-1988), an d Chancello r (1988-1993) . H e wa s Presiden t o f th e Internationa l Congress o f Mathematicians i n 1978 , a Member o f the Executive Committe e o f the International Mathematical Union (1975-1990) , the Union's Secretary (1983-1990) , a Membe r o f the Administrativ e Boar d o f the International Associatio n o f Univer - sities (1985-1995) , and the Association's Vice-Presiden t (1990-1995) . H e is both a member (1962 ) and an honorary membe r (2001 ) o f the Finnish Academ y o f Science and Letters . H e acte d a s Presiden t o f th e Delegatio n o f th e Finnis h Academie s (1979-1999), wa s Chai r o f th e Boar d o f th e Finnis h Cultura l Foundatio n (1991 - 1994), an d th e latter' s Honorar y Presiden t i n 1998 . I n mathematics , hi s specia l field i s comple x analysi s and , afte r retirement , als o th e histor y o f mathematics . He i s the autho r o f Mathematics Without Borders: A History of the International Mathematical Union (Springer-Verlag , 1998) . Address: Departmen t o f Mathematics, University o f Helsinki, P.O. Box 4 (Yliopis- tonkatu 5) , 00014 Helsinki, Finland . E-mail: [email protected] i

Jesper Liitze n obtaine d a Master' s Degre e (1976 ) an d a Ph.D. (1980 ) fro m th e Department o f History o f Science a t Aarhu s University . A Habilitation followe d i n 1990 from Copenhage n University. H e is currently Professo r o f History o f the Exac t Sciences in the Department o f Mathematics, Copenhagen University. Th e author o f Joseph Liouville: Master of Pure and Applied Mathematics (Springer-Verlag , 1990 ) and The Prehistory of the Theory of Distributions (Springer-Verlag , 1980) , h e i s currently workin g o n a book o n Heinric h Hertz' s Principles of Mechanics. Address: Mathematic s Institute , Copenhage n University , Universitetsparke n 5 , 2100 Copenhagen, Denmark . E-mail: [email protected] k

Laura Martini earne d her laurea from the University o f Siena in 1997 with a thesis on "Cesar e Arzela' s Lecture s o n Galoi s Theory , Bologn a 1886-1887 " directe d b y Laura Tot i Rigatelli , Sh e i s presentl y a t wor k o n a Ph.D . i n mathematic s a t th e University o f Virginia. He r mai n researc h interest s li e i n the histor y o f algebr a i n the nineteenth century and in the social development o f mathematics i n nineteenth- century Italy . Address: Departmen t o f Mathematics, Universit y o f Virginia, Kercho f Hall , P.O. Box 400137 , Charlottesville , Virgini a 22904-4137 , United States . E-mail: [email protected] u

Karen Hunge r Parshal l i s Professor o f History an d Mathematic s a t th e Univer - sity o f Virginia, a former edito r o f Historia Mathematica, an d th e curren t Chai r o f the International Commissio n o n History o f Mathematics. He r researc h focuse s o n the histor y o f nineteenth- an d twentieth-centur y mathematic s wit h a special focu s LIST O F CONTRIBUTOR S xi x on development s i n algebra . I n 1994 , he r boo k wit h Davi d Rowe , The Emergence of the American Mathematical Research Community, 1876-1900: J. J. Sylvester, Felix Klein, and E. H. Moore, appeare d i n th e AMS/LM S HMAT H series . Sinc e then, sh e has co-edite d wit h Pau l Theerman , Experiencing Nature: Proceedings of a Conference in Honor of Allen G. Debus (Kluwer , 1997) , an d ha s publishe d th e book, James Joseph Sylvester: Life and Work in Letters (Oxford , 1998) . Address: Department s o f History an d Mathematics , Universit y o f Virginia, Ker - chof Hall , P.O. Bo x 400137, Charlottesville, Virgini a 22904-4137 , United States . E-mail: [email protected] u

Adrian C . Ric e receive d a B.Sc. in mathematics fro m Universit y Colleg e Londo n in 199 2 an d a Ph.D . i n th e histor y o f mathematic s fro m Middlese x Universit y i n 1997 fo r a thesis o n Augustu s D e Morgan an d th e developmen t o f university-leve l mathematics i n nineteenth-centur y London . H e i s Assistan t Professo r o f Mathe - matics at Randolph-Macon Colleg e in Ashland, Virginia , where his research focuse s on the nineteenth- an d earl y twentieth-century Britis h mathematical researc h com - munity wit h especia l referenc e t o the rol e o f the Londo n Mathematica l Society . Address: Departmen t o f Mathematics , Randolph-Maco n College , Ashland , Vir - ginia 23005-5505 , United States . E-mail: [email protected] u

Chikara Sasak i studie d mathematic s a t Tohok u Universit y an d th e histor y o f science a t Princeto n University . H e receive d hi s Ph.D . fro m Princeto n fo r a the - sis o n "Descartes' s Mathematica l Thought. " H e i s no w Professo r o f Histor y an d Philosophy o f Scienc e a t th e Universit y o f Toky o an d a membe r o f the Executiv e Committee o f th e Internationa l Commissio n o n th e Histor y o f Mathematics . H e was awarde d th e Suntor y Priz e fo r Socia l Science s an d Humanitie s fo r th e yea r 1993. Address: Departmen t o f Histor y an d Philosoph y o f Science , Graduat e Schoo l o f Arts an d Sciences , Universit y o f Tokyo , 3-8-1 , Komaba , Meguro-ku , Toky o 153 - 0041, Japan . E-mail: [email protected] p

Sanford L . Sega l i s Professor o f Mathematics a t th e Universit y o f Rochester an d was Chai r o f the departmen t ther e fro m 197 9 to 1987 . H e obtaine d hi s Ph.D . i n Mathematics i n 196 3 from th e Universit y o f Colorado (supervise d b y Sarvadama n Chowla) an d sinc e then ha s pursued researc h i n mathematics, th e history o f math- ematics, and mathematics education. H e has been a visiting lecturer at the Univer - sity o f Nottingham (1972-1973 ) an d a research fello w a t th e Federa l Universit y o f Rio de Janeiro (summe r 1982) . H e received Fulbright grant s i n 1958-195 9 to stud y in Mainz , German y an d i n 1965-196 6 fo r researc h a t th e Universit y o f Vienna . More recently , h e receive d a n Alexande r vo n Humbold t fellowshi p t o German y i n 1988. Wor k durin g thi s las t an d succeedin g trip s t o German y wer e i n preparatio n for a book, Mathematicians under the Nazis, t o be published shortly . H e is also the author o f the book , Nine Introductions in Complex Analysis (Elsevier , 1981) . Address: Departmen t o f Mathematics , Universit y o f Rochester , Rochester , Ne w York 14627 , United States . E-mail: [email protected] u xx LIS T O F CONTRIBUTOR S

Reinhard Siegmund-Schultz e studie d mathematic s a t th e Universit y o f Halle . Since 1975 , hi s researc h ha s focuse d o n th e histor y o f mathematics , primaril y o n the history o f functional analysi s and the social history o f mathematics i n the Thir d Reich. H e i s author o f Mathematische Berichterstattung in Hitlerdeutschland: Der Niedergang des Jahrbuchs ilber die Fortschritte der Mathematik (1869-1945) (Van - denhoeck & Ruprecht, 1993) , Mathematiker auf der Flucht vor Hitler: Quellen und Studien zur Emigration einer Wissenschaft (Vieweg , 1998) , and Rockefeller and the Internationalization of Mathematics Between the World Wars (Birkhause r Verlag , 2001). H e i s currentl y Professo r fo r Histor y o f Mathematic s a t Agde r Universit y College i n Kristiansand, Norway . Address: Departmen t o f Mathematics, Agde r University College , Serviceboks 422, N-4604 Kristiansand, Norway . E-mail: [email protected] o

Yibao X u studie d th e histor y o f Chines e mathematic s a t th e Institut e fo r th e History o f Science , Inne r Mongoli a Norma l University , Huhhot , China , wher e h e received hi s M.Sc. in July, 1991 . Subsequently , h e worked a s an editor a t the Offic e of Jiangx i Loca l Histor y Editin g Committee , Nanchang , China . I n August , 1995 , he entere d th e Graduat e Schoo l an d Universit y Cente r o f th e Cit y Universit y o f New York (CUNY ) t o study the history o f science and i s now a doctoral candidat e in the Ph.D. Program i n History, CUNY . Address: Ph.D . Progra m i n History , Th e Graduat e Center , Th e Cit y Universit y of Ne w York, 36 5 Fifth Avenue , Ne w York, Ne w York 10016 , United States . E-mail: [email protected] m Photograph an d Figur e Credit s

We gratefull y acknowledg e th e kindnes s o f th e followin g fo r grantin g thes e permissions: Cambridge Universit y Librar y Photograph o f the Ru e de s Nations a t th e Pari s Exhibitio n o f 1900 ; cover an d p. 206 ; Fro m The Paris Exhibition, 1900. Ed . D . Croa l Thomson . Assiste d b y Herbert E . Butle r an d E . G . Halton . London : H . Virtu e an d Co. , 1901 . Facin g page 111 ; By permission o f the Syndic s o f Cambridge Universit y Library . Institut Mittag-Leffle r Photograph o f Gosta Mittag-Leffle r (1846-1927) ; p. 138 ; By permissio n o f the Institut Mittag-Leffler . Professor Re n Nanhen g Photograph o f th e Annua l Meetin g o f th e Chines e Mathematica l Societ y o n the Occasio n o f It s Fiftiet h Anniversary , Fuda n University , Shanghai , Decembe r 1985; p. 278 ; Reproduced b y permissio n o f Professo r Re n Nanhen g o f the Chines e Mathematical Society . Springer-Verlag Sketch o f L. E. Dickson, made during his lecture at the International Congress of Mathematicians, Strasbourg , p . 312; From International Mathematical Congresses: An Illustrated History. B y Donal d J . Albers , Geral d L . Alexanderson , an d Con - stance Reid . Ne w York: Springer-Verlag , 1987 .

The followin g image s ar e i n the publi c domain : Cartoon "Demonstrier e z u Hause!" b y D. Gulbransson; p . 368; from Simplicis- simus (1919) . Thi s cartoon als o appeared i n Paul Forman . "Scientifi c Internation - alism an d th e Weima r Physicists : Th e Ideolog y an d It s Manipulatio n i n German y after Worl d Wa r I. " Isis 6 4 (1973):151-18 0 o n p. 150 . List o f Editoria l Boar d member s o f Acta Mathematica i n 1882 ; p . 141 ; fro m Acta Mathematica 1 (1882) . Paragraph fro m Sectio n 15 0 of Cesare Arzela's lectures on Galois theory; p. 172; from th e Bortolott i Collection , Mathematic s Library , Universit y o f Bologna . Photograph o f Charle s Hermite ; p . 122 ; fro m (Euvres de Charles Hermite. Ed. Emil e Picard . 4 Vols. Paris : Gauthier-Villars , 1905-1917 , 3 : frontispiece . Table o f Contents o f the firs t volum e o f Acta Mathematical p . 145 ; from Acta Mathematica 1 (1882) .

xxi xxii PHOTOGRAP H AN D FIGUR E CREDIT S

Title pag e o f Yonagaw a Shunsan' s Yosan Yoho fro m 1857 ; bac k cove r an d p. 233. Yonagawa's explanatio n o f fou r fundamenta l arithmetica l operation s an d o f proportion i n Yosan Yoho; p. 234 ; appears o n foli o 7r . This page intentionally left blank Index

Abel, Niel s Henrik , 93 , 143 , 170-17 1 Association fo r the Improvement o f Geomet - Academia Sinica , 296 , 300, 302-30 3 rical Teaching , 6 7 Academic des sciences (Paris) , 6 , 90 , 94 , Association francaise pour Vavancement des 125, 129 , 189 , 385-38 6 sciences (AFAS) , 108, 112-117, 183, 187 Accademia dei Lincei, 187 , 190—191 , 19 7 Association o f Collegiat e Alumna e (ACA) , Acta Mathematica, 126 , 134 , 186 , 198 , 202- 317 203, 363-36 4 Aumann, Georg , 338-33 9 and Gost a Mittag-Leffler , 139-16 1 Aydelotte, Frank , 298-30 0 and Geor g Cantor , 150-15 3 and Henr i Poincare , 148-15 0 Babbage, Charles , 6 1 and Sony a Kovalevskaya , 153-15 5 Bachmann, Paul , 16 8 first 2 0 volumes, 155-15 8 Bacon, Francis , 7 , 1 1 founding of , 140-14 8 and a first vintag e o f heat, 7- 8 Adams, Joh n Couch , 72-7 4 Bacon, Roger , 2 , 5 Akizuki Yasuo , 24 3 Bails, Benito , 47-4 8 Albert, A . Adrian , 323 , 37 2 Balz, Erwin , 23 5 American Journal of Mathematics, 141 , 158, Banach, Stefan , 34 4 313 Bansho Shirabe-sho, see also Institut e fo r American Mathematical Monthly, 312-31 3 the Investigatio n o f Barbarian Book s American Mathematica l Societ y (AMS) , 56 , Basile, Ernesto , 17 9 108, 118 , 184 , 188 , 202 , 314 , 317 , 319 , Battaglini, Giuseppe , 16 5 327, 359 , 370, 390 , 393-39 4 Behnke, Heinrich , 340 , 353, 35 4 Bulletin of the, 204 , 29 0 Beijing University , 276-27 7 Transactions of the, 312-31 3 Beltrami, Eugenio , 165 , 18 1 American mathematic s Bertrand, Joseph , 97 , 125 , 13 5 developments in , 311-32 9 Betti, Enrico , 71 , 165-167, 181-18 2 influence o n Chines e mathematics , 272 - Bianchi, Luigi , 166 , 16 9 275, 287-30 5 Bieberbach, Ludwig , 339-340, 345, 352, 355, Analysis, 5 6 365, 391-39 2 complex, 27-28 , 13 2 Birkhoff, Garrett , 348-34 9 functional, 19 5 BirkhofT, Georg e David , 193 , 34 9 numerical, 5 6 Blaschke, Wilhelm , 271-272 , 280 , 292-293 , real, 25-2 7 302, 338-339 , 354 , 370-37 1 Annales de mathematiques pures et appliquees, Bocher, Maxime , 29 5 90-91, 15 8 Bohr, Harald , 33 6 Annali di matematica pura ed applicata, 71 , Bolza, Oskar , 314 , 36 1 158, 181-182 , 28 0 Borchardt, Car l Wilhelm , 129-13 0 Antisemitism, 339-341 , 368-37 2 Borel, Emile , 269 , 345-34 6 Appell, Paul , 134 , 38 9 Bortolotti, Ettore , 166 , 19 4 Arago, Frangois , 9 0 Bossut, Charles , 21 1 Archiv for mathematik og naturvidenskab, Bourbaki, Nicholas , 37 6 159 Bourlet, Carlo , 10 5 Arithmetics o f algebras, 320-32 2 Bouty, Emile , 10 8 Arnold, Matthew , 36 1 Boxer Indemnity , 266-267 , 288-29 1 Artin, Emil , 241-242 , 279 , 355, 36 9 Boxer Rebellio n (o r Uprising), 253 , 263, 288 Arzela, Cesare , 10 , 165-166 , 168-17 2 Brahe, Tycho , 5

399 400 INDEX

Breal, Michel , 205 , 21 6 Circolo matematico di Palermo, 9 , 17 9 Bridgman, Elija h Coleman , 28 7 and internationalizatio n o f mathematica l Brioschi, Francesco , 71 , 165, 181-18 2 research, 186-18 7 British Associatio n fo r th e Advancemen t o f decline of , 195-19 8 Science (BAAS) , 64-65 , 82, 116 , 20 4 Giovan Gucci a and , 183-18 5 British mathematic s Italian mathematica l contex t of , 180-18 2 developments in , 61-8 3 Rendiconti del, 140 , 186 , 191-194 , 19 8 influence o n Chines e mathematics , 264 - Sicilian contex t of , 179-18 0 268 Class field theory , 241-24 2 influence o n Japanes e mathematics , 23 6 Condillac, Etienn e Bonnot , Abb e de , 21 1 Brouwer, L . E . J. , 216-217 , 340 , 36 5 Copley Medal , 65 , 8 2 Bulletin de Ferrusac, 24 , 9 0 Courant, Richard , 197 , 270 , 340 , 368 , 374 - Bulletin des sciences mathematiques et as- 376 tronomiques, 105 , 108 , 130 , 134 , 147 , Couturat, Louis , 206-20 7 189 Crelle, Augus t Leopold , 22 , 90, 15 8 Cremona, Luigi , 67 , 165 , 181-18 2 Cambridge and Dublin Mathematical Jour- Cruson, Johan n Philip , 23-2 4 nal, 68 , 71, 10 1 Curie, Marie , 155 , 36 4 Cambridge Mathematical Journal, 67-6 8 Czech Societ y o f Mathematicians, 11 9 Cambridge Philosophica l Society , 6 4 Transactions of the, 15 8 d'Aiquebelle, Paul , 26 1 Cantor, Georg , 116 , 359, 382-38 4 Darboux, Gaston , 105 , 134 , 182 , 385-38 6 and Acta Mathematical 150-15 3 Darwin, Si r George , 38 5 and se t theory , 150—15 3 De Franchis , Michele , 196-19 7 Capelli, Alfredo , 16 9 de l a Valle e Poussin , Charles , 196 , 38 6 Carandinos, Ioannis , 2 3 De Morgan , Augustus , 2 7 Caratheodory, Constantin , 353 , 35 4 Dedekind, Richard , 16 7 Carnegie Institution o f Washington, 319-320 , Dehn, Max , 36 9 326-328 Delaunay, Charles-Eugene , 31 , 72-73 Carnegie, Andrew , 31 8 Denationalization, 2- 4 Carnot, Lazare , 2 9 Deng Encong , 29 0 Cartan, Elie , 280 , 29 2 Deutsche Kongrefizentrale, 35 2 Casorati, Felice , 18 1 Deutsche Mathematik, 34 0 Casorati-WeierstraB Theorem , 13 9 Deutsche Mathematiker-Vereinigung (DMV) , Cauchy, Augustin-Louis , 25-28 , 32 , 9 4 108, 118 , 183-184 , 188 , 202, 33 7 Ceng Jiongzhi , 27 0 Dickson, Leonar d Eugene , 10 , 78-79 , 311 - Ceng Yuanrong , 290-29 1 329, 364 , 372-37 3 Cerruti, Valentino , 19 1 Dirichlet, Pete r Legeune , 36 0 Chao Yuanren , 294-29 5 Donati, Luigi , 16 5 Chasles, Michel , 82 , 97, 106-107 , 114 , 18 4 Du Bois-Reymond , Paul , 123-124 , 12 6 Chebyshev, Pafnuti , 15 4 Du Pasquier , Loui s Gustave , 321-32 3 Chelini, Domenico , 16 5 Dutch mathematics , influenc e o n Japanes e Chen Jiangong , 245 , 26 4 mathematics, 230-23 2 Chen Yu-why , 30 1 Chern Shiing-shen , 272 , 280 , 292 , 298-301 , Echegaray y Eizaguirre , Jose , 51-5 3 303-305 Ecole normale superieure (ENS) , 109-110 , Chevalley, Claude , 37 6 112 China Foundatio n fo r th e Promotio n o f Ed - Ecole polytechnique, 18 , 26, 28, 49, 108-110 , ucation an d Culture , 291-29 4 112, 114 , 118 , 13 1 Chinese Mathematical Society , 277-280, 294, Edinburgh Mathematical Societ y (EMS), 66- 296, 301 , 303 67 Journal of the, 277-280 , 29 1 Proceedings of the, 15 8 Chinese mathematics, developments in, 253- Educational Times, 6 8 281, 287-30 5 Einstein, Albert , 36 3 Chinese Students ' Allianc e o f America, 29 0 Encke, Johan n Franz , 3 1 Chow Wei-liang , 30 1 Energetics, 29-3 0 Christoffel, Elwi n Bruno , 23 8 Enneper, Alfred , 7 2 Chung Kai-lai , 29 2 Enriques, Federigo , 21 8 INDEX 401

Enseignement mathematique, 105 , 119 , 186 , Geppert, Harald , 345-347 , 353 , 35 4 188-189, 19 8 Gerbert d'Aurilla c (Pop e Sylveste r II) , 1 , 5 Enzyklopddie der mathematischen Wissen- Gergonne, Josep h Diaz , 9 0 schaften, 340 , 362 , 38 2 German mathematic s Esperanto, 207-20 9 developments in , 335-34 9 Euler, Leonhard , 2 , 6 , 3 0 diffusion of , 93 , 97, 100 , 133-13 5 dominance of , 361-36 2 Faa d e Bruno , Francesco , 7 2 influence o n American mathematics , 313 - Fan Ky , 30 1 318 Fehr, Henri , 36 7 influence o n Chines e mathematics , 270 - Feng Zuxun , 29 4 272 Fermat's Las t Theorem , 9 4 influence o n Frenc h mathematics , 32-33 , Ferrel, William , 73-7 4 93, 95-100 , 123-13 6 Ferrusac, Andre-Etienne-Just-Pascal-Joseph - influence o n Japanes e mathematics , 236 - Francois d'Audebart , Baro n de , 2 4 246 Fibonacci, Leonardo , 1-2 , 5 translation of , 93 , 134 , 15 0 Fields Medal , 195 , 242-24 3 Giquel, Prosper-Marie , 26 1 Fields, Joh n Charles , 364-365 , 389-39 0 Glaisher, Jame s Whitbrea d Lee , 67-6 8 Filopanti, Quirico , 16 5 Godeaux, Lucien , 27 9 Finite field theory , 314-315 , 318-31 9 Godel, Kurt , 22 3 Forsyth, Andre w Russell , 2 8 Gottingen University , 132 , 167 , 197 , 24 1 Foundations o f mathematics, 218-22 3 Gottingen Seven , 12 3 Franco-Prussian War , 24-25 , 93 , 105 , 107 - Graustein, William , 323-32 4 109, 112-113 , 124 , 126 , 131 , 141 , 183 , Green, George , 9 7 202, 360-36 1 Gregory, Duncan , 6 7 Franklin, Benjamin , 6 5 Group theory , 78-79 , 314-315 , 32 4 Frattini, Giovanni , 16 9 Guang Xu , Emperor , 26 3 Frege, Gottlob , 206-207 , 210 , 219 , 22 1 Guccia, Giova n Battista , 10 , 183-185 , 187 - French mathematic s 191, 193-195 , 19 8 decline of , 24-2 5 Gugino, Edoardo , 19 8 developments in , 17-34 , 89-91 , 93-101 , 105-120, 123-13 6 Hadamard, Jacques , 269-270 , 279 , 30 2 dominance of , 17-20 , 89-90 , 93 , 360-36 1 Hale, Georg e Ellery , 38 9 influence o n Chines e mathematics , 268 - Halmos, Paul , 32 4 270 Hamilton, Willia m Rowan , 9 5 influence o n Japanes e mathematics , 23 2 Hansen, Pete r Andreas , 72-73 , 7 5 influence o n Spanis h mathematics , 46-5 1 Hardy, Godfre y Harold , 139 , 161 , 278-279 , translation of , 20-24 , 39-44 , 13 1 363, 389-39 0 Friedrichs, Kurt , 34 0 Harvard University , 294-29 6 Fryer, John , 256 , 259-260, 265 , 273-27 4 Hasse, Helmut , 339-341 , 346 , 353 , 35 4 Fu Zhongsun , 26 7 Hayashi Tsuruichi , 243-24 5 Fujisawa Rikitaro , 238-240 , 243 , 24 7 He Yunhung, 29 0 Fujiwara Matsusaburo , 243-24 5 Hecke, Erich , 35 5 Fukuzawa Yukichi , 235-236 , 23 8 Hermite, Charles , 10 , 68-71 , 82 , 105 , 123 , Fuzhou Shipyard , 260-262 , 26 8 142-144, 150 , 18 6 career of , 125-12 6 Galilei, Galile o law o f reciprocity, 6 8 Discourses on Two New Sciences, 2 political view s of , 126-12 7 Galois theory , 166—16 9 Herschel, John , 6 1 Garcia d e Galdeano , Zoel , 52-5 3 Hilbert, David , 196 , 204 , 218-223, 241, 363, Garcia Sa n Pedro , Fernando , 4 9 365-366, 39 2 Gauss, Car l Friedrich , 31-33 , 93 , 97, 99-10 0 23 problems of , 204 , 219, 38 5 Disquisitiones arithmetics, 3 2 Hironaka Heisuke , 24 3 Disquisitiones generates circa superficies Hirst, Thoma s Archer , 67 , 80-82, 14 7 curvas, 10 0 Hopf, Eberhard , 354 , 37 0 Geometry Hopf, Heinz , 297 , 33 8 algebraic, 19 5 Hoppe, Reinhold , 71-7 2 differential, 33 , 98-10 0 Hoiiel, Guillaume-Jules , 134 , 182 , 20 4 projective, 5 6 Hu Kunsheng , 277-278 , 29 1 402 INDEX

Hu Mingfu , 294-29 5 of mathematics, supranationa l aspect s of , Hua Hengfang , 25 9 10-12 Hua Luogeng , 267 , 278-280 , 291-293 , 301, Invariant theory , 6 8 303-304 Italian mathematics , development s in , 165 — Humboldt, Wilhel m von , 8 175, 179-19 8 Hurewicz, Witold , 302 , 30 4 Ito Hirobumi , 237-23 8 Hurwitz, Adolf , 321 , 360 Ivory, James , 8 0 Huygens, Christiaan , 2 , 6 Iwakura diplomati c mission , 23 7

Idiom Neutral , 20 8 Jacobi, Car l Gusta v Jacob , 94-97 , 125 , 36 1 Ido, 20 9 Jahrbuch iiber die Fortschritte der Mathe- Inoue Kowashi , 237-23 8 matik, 345 , 38 2 Institute fo r Advance d Stud y (IAS) , 296 - Japanese mathematic s 301 developments in , 229-24 9 Institute fo r th e Investigatio n o f Barbaria n influence o n Chines e mathematics , 263 - Books (Bansho Shirabe-sho), 231 , 235 264 Intermediate des mathemattciens, 38 3 Jiang Lifu , 295-296 , 30 1 International Commissio n o n Mathematica l Jiang Zehan , 291 , 295-298, 302-30 3 Education (ICME) , 342 , 36 7 Jiangnan Arsenal , 258-260 , 26 5 International Commissio n o n Mathematica l Jordan, Camille , 93 , 16 8 Instruction (ICMI) , 105 , 119 , 39 4 Traite des substitutions et des equations International Commissio n o n th e Teachin g algebriques, 16 8

of Mathematics, 385 , 38 8 Journal de mathematiques pures et appliquees 1 International Conference s 81, 141 , 143, 18 9 German participatio n in , 356-35 7 Journal fur die reine und angewandte Math- International Congres s fo r Mechanics , 34 1 ematik, 22 , 89-92 , 96 , 100 , 129 , 141 , International Congres s o f Mathematician s 143-144, 150 , 152 , 16 8 (ICM), 118-119, 139-140, 201, 320, 335, Julia, Gaston , 345-34 6 348, 36 7 Bologna (1928) , 366 , 391-39 2 Kakeya Soichi , 24 4 Cambridge (1912) , 195 , 38 5 Kakutani Shizuo , 24 4 Kyoto (1990) , 24 8 Kamke, Erich , 355 , 369-37 0 Oslo (1936) , 342-344 , 370 , 39 3 Kant, Immanuel , 21 1 Paris (1900) , 204 , 384-38 5 Kawai Jittaro , 24 3 Rome (1908) , 187 , 190-191 , 38 5 Ke Zhao , 28 0 Strasbourg(1920), 241, 320, 322, 364, 387- Kempner, Aubrey , 32 3 388 Kepler, Johannes , 2 , 5 Toronto (1924) , 322-323, 389-39 0 Kikuchi Dairoku , 235-236 , 239-240 , 24 7 Zurich (1897) , 187 , 202 , 204 , 359 , 38 4 Klein, Felix , 140 , 148 , 204 , 241 , 313-314 , Zurich (1932) , 242 , 269 , 39 2 324, 360-363 , 367 , 383, 38 5 International Congres s o f Philosophy , 206 - Kline, Joh n Robert , 39 4 207 Kneser, Hellmuth , 36 5 International Counci l o f Scientifi c Union s Knopp, Konrad , 270-27 1 (ICSU), 39 4 Kodaira Kunihiko , 24 2 International Mathematical Union (IMU) , 9, Koenigs, Gabriel , 364 , 365, 388-39 2 345, 348 , 364 , 366-367 , 381 , 386-388 , Kondo Yoitsu , 24 5 390-394 Kovalevskaya, Sonya , 11 , 364 International Researc h Counci l (IRC) , 386 , and Acta Mathematical 153-15 5 388, 390-391 , 393-39 4 Kronecker, Leopold , 71 , 130-131, 144 , 151- International science , 12-1 3 152, 168 , 23 8 Internationalism, 2 , 6, 12-13 , 179-180 , 201- Kubota Tadahiko , 24 4 203, 205-206 , 337 , 348 , 360-367 , 375 - Kummer, Erns t Eduard , 33 , 94 376 Kyoto Imperia l University , 24 3 scientific, 3 , 12 , 140-148 , 159 , 381-38 2 Internationalization, 2 , 4 , 6 Lacroix, Sylvestr e Francois , 21 , 13 2 of mathematics , 6-7 , 11 , 186-195 , 336 - Ladd-Franklin, Christine , 31 7 339, 384-385 , 393-39 4 Lagrange, Josep h Louis , 2 , 6 , 28-3 1 of mathematics , ope n researc h area s in , Mechanique analitique, 28-2 9 11-14, 25-33,117-12 0 Laisant, Charles-Ange , 105 , 187 , 38 3 INDEX 403

Lamb, Horace , 7 6 Mechanics, 95-9 7 Lambert, Johann , 21 1 Carnotian, 2 9 Lame functions , 9 8 celestial, 30-3 2 Lame, Gabriel , 94 , 15 5 energy, 28-3 0 Landau, Edmund , 185 , 190 , 197 fluid, 71-7 2 Languages Lagrangian, 28-2 9 mathematical, 212-21 4 relativist, 5 6 symbolic, 209-21 1 Meiji period , 229 , 233-23 8 universal, 206-20 9 Meray, Charles , 20 7 Laplace, Pierr e Simon , 31 , 95 Messenger of Mathematics, 67-68 , 75—7 7 Traite de mecanique celeste, 21 , 95 Miller, Georg e A. , 78 , 37 3 Latino sin e Flexione , 20 8 Min Sihe , 30 1 Least squares , metho d of , 3 1 Minkowski, Hermann , 8 2 Lefschetz, Solomon , 29 6 Mittag-Leffler, Gosta , 9-10 , 123-124 , 126 , Leibniz, Gottfrie d Wilhelm , 211 , 213-21 4 130, 132 , 186 , 190 , 194-195 , 363-364 , Levi-Civita, Tullio , 194 , 34 1 385, 387-38 9 Li Shanlan , 256-257 , 265 , 270, 27 2 and Acta Mathematica, 139-16 1 Lie, Sophus , 140-142 , 147-148 , 16 7 Mittag-Leffler Theorem , 13 9 Lietzmann, Walter , 342-344 , 35 4 Mohr, Ernst , 37 4 Linguistics, 214-21 6 Monge, Gaspard , 48-4 9 Liouville, Joseph , 10 , 33 , 89-94 , 100-101 , Moore, Eliakim Hastings, 290, 297-298, 313- 158, 20 4 315, 317-319 , 324-325 , 327-328 , 361 - differential geometr y of , 98-10 0 362 mechanics of , 95—9 7 Moore, Rober t Lee , 325-326, 373-37 4 potential theor y of , 97-9 8 Mori Shigefumi , 24 3 Littlewood, Joh n Edensor , 278-279 , 36 3 Morse, Marston , 296-297 , 393-39 4 Liu Jin-nian , 291 , 295 Moscow Mathematica l Society , 183 , 35 9 Locarno Pact , 365 , 39 0 Miiller, Max , 21 5 Logic, 19 5 Multinationalism, 17 , 33-34, 20 1 algebraic, 209-21 1 and -ization , 2 London Mathematica l Societ y (LMS) , 66 - Murray, David , 23 5 67, 82 , 106-107, 117 , 183 , 188, 198 , 359 Journal of the, 28 0 Nagasaki Naval Training Institute (Nagasaki Proceedings of the, 82 , 106-107 , 158 , 184 , Kaigun Denshu-sho), 23 1 204 Nagel, Ernest , 21 8 Loomis, Elias , 28 8 Nakagawa Senkichi , 242-24 3 Louis XIV , 6 Nationalism, 6 , 89-91, 100-101 , 106-109, 115, Lunar theory , 72-7 5 359-362, 36 7 Nationalization, 5 , 202-20 4 Ma Zuju , 26 4 of mathematics, 8 , 105-115, 180-182, 232- Mac Lane , Saunders , 32 4 247, 254-263 , 275-280, 313-31 8 Maclaurin, Colin , 9 5 Nature, 146-14 7 Malmsten, Car l Johan , 142-14 3 Netto, Eugen , 168-17 0 Manchester Literar y an d Philosophica l So - Lehrbuch der Algebra, 24 0 ciety, 6 4 Neugebauer, Otto , 340 , 36 9 Mannoury, Gerrit , 216-21 7 Neumann, Bernhard , 36 9 Maschke, Heinrich , 314 , 36 1 Neumann, Franz , 36 1 Mateer, Calvi n Wilson , 272-274 , 287-28 8 Newson, Ma y Winston , 31 7 Matematicheskii Sbomik. 15 8 Neyman, Jerzy . 37 4 Mathematical Associatio n o f America (MAA) , Noether, Emmy , 246 , 270 , 279 , 37 3 118 Noether, Max , 19 0 Mathematical Laborator y an d Semina r Novi, Giovanni , 16 7 (LSM), 46 , 53-54 , 56 , 5 8 Number theory , 319-32 0 Mathematical Reviews, 34 1 Mathematical Societ y o f Japan, 247-24 8 Odriozola, Jos e de , 4 9 Mathematische Annalen, 150-151 , 168 , 24 1 Ogura Kinnosuke , 24 5 Mathematisches Forschungsinstitut, Oberwol- Oka Kiyoshi , 24 3 fach, 346-347 , 35 3 Okada Yoshitomo , 24 4 Matsunaga Yoshisuke , 23 0 Opium Wars , 231 , 253-255, 287, 28 9 404 INDEX

Oppenheimer, J . Robert , 300-30 1 definition o f fo r mathematics , 8 Oscar II , Kin g o f Swede n an d Norway , 143 - Revista matemdtica hispano-americana, 46 , 144, 14 9 53 Osgood, Willia m Fogg , 203 , 293 , 302 , 360 - Reye, Theodor , 23 8 361 Richardson, Rolan d G . D. , 336 , 343 , 367 Riemann, Bernhard , 28 , 3 3 Painleve, Paul , 269 , 365, 389-39 0 Riesz, Marcel , 36 3 Parseval's formula , 23-2 4 Rong Hong , 26 8 Pasch, Moritz , 218 , 22 0 Royal Astronomica l Society , 66 , 72-7 5 Pasteur, Louis , 107 , 13 0 Royal Iris h Academy , 6 4 Peano, Giuseppe , 208 , 210-21 1 Royal Societ y o f Edinburgh, 6 4 Peirce, Charle s Sanders , 210 , 216 , 221-22 2 Royal Societ y o f London , 62-65 , 82 , 385 - Perturbation theory , 30-3 1 386, 38 9 Petersen, Julius , 16 8 Philosophical Transactions of the, 62-64 , Philosophical Magazine, 65 , 78, 8 0 158 Picard, Emile , 133-134 , 196 , 345 , 364-365 , Ruffini, Paolo , 170-171 , 19 4 384-386, 388-389 , 39 1 Ruffini-Abel Theorem , 170-17 4 Pieri, Mario , 218-21 9 Russell, Bertrand , 221-223 , 266-267 , 281 , Pincherle, Salvatore , 165 , 391-39 2 295, 302 , 36 3 Pirandello, Luigi , 17 9 Russian Academ y o f Sciences , 15 4 Plana, Giovanni , 7 2 Poincare, Henri, 134 , 142-143, 149-150, 157, Scandinavian mathematics, developments in , 186, 192-195 , 36 0 140-148 and Acta Mathematica, 148-15 0 Schmidt, Erhard , 342 , 355, 365-36 6 and Fuchsia n functions , 14 8 Schnuse, Christia n Heinrich , 2 2 and th e three-bod y problem , 14 9 Scholz, Heinrich , 35 4 Poinsot, Louis , 80-8 1 School fo r the Yokosuka Navy Shipyard ( Yo- Poisson, Simeon-Denis , 8 0 sosuka Kaigun Zosenjo Kosha), 23 2 Polish mathematics , internationalizatio n of , Schroder, Ernst , 204 , 209-210 , 221-22 2 344 Schuster, Si r Arthur , 38 6 Politics, effect s o n mathematics, 9 , 195-198 , Schwarz, Herman n Amandus , 13 5 335-349, 359-376 , 385-39 3 Schwerdtfeger, Hans , 36 9 Polya, George , 37 4 Science Societ y o f China, 275-27 6 Pontecoulant, Phillip e Gustave Doulcet, Comt e Scriba, Julius , 23 5 de, 73-7 5 Scuola normale superiore, 169 , 18 1 Potential theory , 30 , 97-98, 15 5 Seki Takakazu , 23 0 Prager, Wilhelm , 37 4 Serret, Josep h Alfred , 167 , 171-17 2 Prandtl, Ludwig , 341 , 354 Cours d'algebre superieure, 167-168 , 24 0 Professionalization o f mathematics, 8-9, 48- Set theory , 150-15 3 51, 53-57,106-11 5 Severi, Francesco , 195 , 37 0 Progreso matemdtico, 5 0 Shanghai Polytechni c Institute , 26 5 Puissant, Louis , 2 3 Shen Youcheng , 279 , 29 5 Qin Fen , 277 , 29 4 Shi Xianglin , 29 2 Qinghua University , 277 , 291-29 2 Shoda Kenjiro , 24 6 Quarterly Journal of Pure and Applied Math- Shu Shien-siu , 293 , 30 1 ematics, 68-72 , 79 , 158 , 18 6 Siegel, Car l Ludwig , 355 , 37 0 Sino-Japanese War , 255 , 259 , 262-264 , 277 , Rademacher, Hans , 369 , 37 5 279-280, 293-294 , 30 2 Ramon y Cajal , Santiago , 5 4 Smith, Henr y Joh n Stephen , 8 2 Ramsauer, Carl , 34 7 Societe frangaise de physique, 10 8 Rasch, Johannes , 35 3 Societe frangaise de statistiques de Paris, Rees, Mina , 37 4 118 Reidemeister, Kurt , 35 5 Societe mathematique de France (SMF) , 25, Ren Hongjun , 27 5 56, 107-117 , 182-183 , 188 , 18 9 Resal, Henri , 9 3 Bulletin de la, 15 8 Research communit y Sono Masazo , 24 3 characterization o f international mathemat - Spanish Civi l Engineerin g Corps , 4 7 ical, 1 Spanish Civi l War , 4 6 INDEX 405

Spanish Mathematica l Societ y (SME) , 46 , von Dyck , Walther , 362 , 38 3 52-56, 5 8 von Humboldt , Wilhelm , 21 5 Spanish mathematics , development s in , 45- 58 Walfisz, Arnold , 33 9 Sperner, Emanuel , 272 , 292-293, 30 2 Wang Hsien-chung , 29 2 Stone, Marshall , 39 4 Wang Renfu , 290 , 29 4 Sturm, Rudolf , 81-8 2 Wantzel, Pierr e Laurent , 17 0 Su Buchin , 245 , 28 0 War, effect s o n mathematics, 106-115 , 123- Sun Guangyuan , 29 1 127, 133-135 , 195-198 , 246-249 , 254 - Sun Yat-Sen , 26 8 256, 263-264 , 345-347 , 362-367 , 385 - Supranationalism an d -ization , 2 , 3 391 Suss, Wilhelm , 346-347 , 353 , 354 Wasan (Japanes e mathematics), 229-230, 236, Sylow, Ludvig , 16 7 245 Sylvester, Jame s Joseph , 31 3 Wedderburn, Josep h Henr y Maclagan , 318 - 319 Takagi Teiji , 10 , 240-242, 24 7 Wei Ciluan , 27 0 Tannaka Tadao , 24 5 Weierstrafi, Karl , 26 , 28 , 130-132 , 136 , 144 , Tannaka's dualit y theorem , 24 5 146, 155 , 23 8 Tatsumi Hajime , 23 2 Weil, Andre , 139 , 37 6 Teichmiiller, Oswald , 34 0 Wen Yuqing , 29 0 Thomson, Willia m (Lor d Kelvin) , 68 , 97 , Weyl, Hermann , 302-304 , 36 9 99-101 Whitehead, Alfre d North , 221-22 3 Thullen, Peter , 36 9 Wiener, Norbert , 278 , 293, 30 2 Tidsskrift for Mathematik, 15 9 Wong Yue-kei , 297-29 8 Tilloch, Alexander , 6 5 World Wa r I , 196 , 198 , 336-338 , 359 , 362 - Todhunter, Isaac , 23 6 363, 367 , 375, 385-386, 39 3 Tohoku Imperia l University , 243-24 5 World Wa r II , 198 , 248 , 277 , 281 , 296, 304 , Tohoku Mathematical Journal, 24 4 345-347, 360 , 374-376, 39 3 Tokyo Mathematica l Society , 183 , 236, 24 7 World's Columbia n Expositio n (1893) , 362 , Journal of the {Tokyo Sugaku Zasshi), 23 6 383 Tokyo Schoo l o f Physics, 24 6 Wu Daren , 267 , 280 , 29 2 Tonelli, Alberto , 19 1 Wundt, Wilhelm , 21 2 Tongwen Guan (Colleg e o f Languages ) Wylie, Alexander , 259 , 264-265, 27 4 Beijing, 254 , 256-259 , 26 5 Shanghai, 258-25 9 Xi Gan , 25 7 Torroja Caballe , Eduardo , 52-5 3 Xiao Wencan, 267-26 8 Toyama Masakazu , 23 5 Xiong Qinglai , 268-269 , 29 2 Transnational universalism , 5 Xu Shou , 25 9 Transnationalism an d -ization , 2 , 3 Yanagawa Shunsan , 231-23 2 Uchida Itsumi , 23 0 Yang Wuzhi , 29 1 Ulm, Helmut , 37 0 Yokosuka Kaigun Zosenjo Kosha, see also Unione matematica italiana, 183 , 19 4 School fo r the Yokosuk a Nav y Shipyar d Universalism, 1 2 Yoshiye Takuzi , 24 2 University Colleg e London , 23 6 Young, Grac e Chisholm , 36 6 University o f Bologna , 165-166 , 16 9 Young, Willia m Henry , 366 , 39 2 University o f Chicago , 313-31 9 Yung Wing, 28 9 University o f Tokyo, 236 , 238-24 3 Zamenhof, Loui s Lazarus , 207-20 8 Vahlen, Theodor , 339-340 , 35 5 Zeitschrift fur Mathematik und Physik, 10 1 van de r Waerden , Baertel , 35 5 Zeng Jiongzhi , 27 9 Vasil'ev, Aleksandr , 20 7 Zentralblatt fur Mathematik, 338 , 340-341 , Veblen, Oswald , 298-300 , 367 , 372-37 3 345 Veronese, Giuseppe , 18 2 Zhang Shixun , 30 1 Victoria, Lad y Welby , 21 6 Zhang Zhidong , 26 3 Volapiik, 207-20 8 Zhang Zhongsui , 30 1 Volterra, Vito , 155 , 180 , 188-189 , 191 , 197, Zheng, Zhifan , 29 1 370 Zhou Shaolin , 28 0 von Caemmerer , Hanna , 36 9 Zhu Gongjin , 27 0 406 INDEX

Zhuang Qitai , 292-29 3 Zorn, Max , 36 9 Zorraquin, Mariano , 48-4 9