HISTORY OF MATHEMATICS Volume 24

A HISTOR Y

OF ANALYSIS

Hans Niels Jahnke Editor

American Mathematica l Societ y

London Mathematica l Societ y

https://doi.org/10.1090/hmath/024

Editor ia l B o a r 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

Geschichte de r Analysi s

Originally publishe d i n Germa n b y Spektru m Akademische r Verlag , 199 9

Translated fro m th e Germa n b y th e authors .

2000 Mathematics Subject Classification. Primar y 01A05 , 26-03 , 28-03 , 30-03 , 31-03 , 33-03, 34-03 , 35-03 , 40-03 , 45-03 , 49-03 .

For additiona l informatio n an d update s o n thi s book , visi t

w w w . a m s . o r g / b o o k p a g e s / h m a t h - 2 4

Library o f Congres s Cataloging-in-Publicat io n D a t a

A histor y o f analysis / Han s Niel s Jahnke , editor . p. cm . — (Histor y o f mathematics, ISS N 0899-242 8 ; v. 24)

Includes bibliographica l reference s an d index . ISBN 0-8218-2623- 9 (alk . paper) 1. Mathematica l analysis—History . I . Jahnke, H . N. (Hans Niels) , 1948 - II . Series.

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Contents

Introduction vi i

Chapter 1 . Antiquit y RUDIGER THIEL E 1

1.1. Gree k mathematics ' par t i n th e formatio n o f analysi s 1 1.2. Th e Gree k concep t o f number s an d magnitude s 3 1.3. Problem s o f quadrature . A n example : Th e circl e 1 4 1.4. Archimedes ' contribution s t o infinitesima l mathematic s 2 1 1.5. Th e concep t o f curves i n antiquit y 2 9 1.6. Philosophi c reflection s o n th e infinitesima l 3 2 Bibliography 3 6

Chapter 2 . Precursor s o f Differentiation an d Integratio n JAN VA N MAANE N 4 1

2.1. Scop e an d motivatio n 4 1 2.2. Th e stud y o f curve s i n the 165 9 edition o f Geometria 4 2 2.3. Earl y integration , reflecte d i n th e correspondenc e o f Huygen s an d

Sluse (1658 ) 5 6 2.4. Barro w glimpse s th e "fundamenta l theorem " 6 9 Bibliography 7 1

Chapter 3 . Newton' s Metho d an d Leibniz' s Calculu s NICCOLO GUICCIARDIN I 7 3

3.1. Introductio n 7 3 3.2. Newton' s metho d o f series an d fluxions 7 4 3.3. Leibniz' s differentia l an d integra l calculu s 8 5 3.4. Mathematizin g forc e 9 1 3.5. Newto n versu s Leibni z 9 5 Bibliography 10 2

Chapter 4 . Algebrai c Analysi s i n th e 18t h Centur y HANS NIEL S JAHNK E 10 5

4.1. Concepts , problems , character s 10 5 4.2. Th e exampl e o f the catenar y 10 8 4.3. Taylor' s theore m 11 1 4.4. Th e notio n o f analytica l functio n 11 3 4.5. Calculatin g wit h serie s 11 8 4.6. Limitation s o f the analyti c functio n concep t 12 3 4.7. Lagrange' s algebrai c foundatio n o f analysi s 12 7 4.8. Th e generalit y o f algebr a 13 1

iv CONTENT S

Bibliography 13 2

Chapter 5 . Th e Origin s o f Analyti c Mechanic s i n th e 18t h Centur y MARCO PANZ A 13 7

5.1. Th e principl e o f leas t action : Maupertuis , Eule r an d Lagrang e (1740-1761) 13 8

5.2. Th e analytica l mechanic s 14 7 Bibliography 15 2

Chapter 6 . Th e Foundatio n o f Analysi s i n the 19t h Centur y JESPER LUTZE N 15 5

6.1. Introductio n 15 5 6.2. Th e concep t o f function 15 6 6.3. Cauch y an d th e Cours d'analyse 16 0 6.4. Gauss , Bolzan o an d Abe l 17 3 6.5. Convergenc e o f Fourie r serie s 17 8 6.6. Cauchy' s theore m an d unifor m convergenc e 18 1 6.7. Weierstras s 18 4 6.8. Pathologica l function s an d th e ne w styl e i n analysi s 18 7 6.9. Diffusio n an d acceptanc e o f rigouris t analysi s 18 8 6.10. Breakin g th e rigorou s chain s 19 0 Bibliography 19 1

Chapter 7 . Analysi s an d Physic s i n th e Nineteent h Century : The Cas e o f Boundary-value Problem s T O M ARCHIBAL D 19 7

7.1. Introduction : Mathematica l analysi s i n physic s circ a 180 0 19 7 7.2. Green , Gauss , Dirichlet : Boundary-valu e problem s com e o f ag e 20 2 7.3. Som e late r development s 20 9 7.4. Concludin g remark s 21 1 Bibliography 21 1

Chapter 8 . Comple x Functio n Theory , 1780-190 0 UMBERTO BOTTAZZIN I 21 3

8.1. Introductio n 21 3 8.2. "Th e passag e fro m th e rea l t o th e imaginary " 21 4 8.3. "Comple x function s an d th e integra l theorem " 21 9 8.4. Th e integra l formul a an d th e "calcu l de s limites " 22 5 8.5. Th e emergenc e o f the Frenc h "school " 22 7 8.6. Riemann' s comple x functio n theor y 23 2 8.7. Riemann' s furthe r researc h 23 8 8.8. Th e influenc e o f Riemann' s idea s 24 4 8.9. Weierstrass' s earl y paper s 24 7 8.10. Weierstrass' s Funktionenlehr e 25 0 Bibliography 25 5

Chapter 9 . Theor y o f Measur e an d Integratio n fro m Rieman n t o Lebesgu e THOMAS HOCHKIRCHE N 26 1

9.1. O n th e prehistor y o f Riemann' s integra l 26 1 9.2. Th e Rieman n integra l 26 3

CONTENTS v

9.3. Discussion s 26 6 9.4. Integratio n revisited : Camill e Jorda n 27 5 9.5. Th e developmen t o f the theor y o f measur e 27 8 9.6. Seekin g new directions : Henr i Lebesgu e 28 4 Bibliography 28 9

Chapter 10 . Th e En d o f th e Scienc e o f Quantity : Foundation s o f Analysis , 1860-1910 MORITZ E P P L E 29 1

10.1. Construction s o f rea l number s 29 2 10.2. Th e emergenc e o f se t theor y 30 4 10.3. Th e axiomati c metho d 31 3 Bibliography 32 1

Chapter 11 . Differentia l Equations : A Historica l Overvie w t o circ a 190 0 T O M ARCHIBAL D 32 5

11.1. Introductio n 32 5 11.2. Fro m th e origin s o f calculus t o th e lat e eighteent h centur y 32 6 11.3. Fro m th e Frenc h Revolutio n t o abou t 190 0 33 9 Bibliography 35 1

Chapter 12 . Th e Calculu s o f Variations: A Historica l Surve y CRAIG FRASE R 35 5

12.1. Introductio n 35 5 12.2. Prehistor y 35 6 12.3. Th e Bernoullis , Taylo r an d Eule r 35 7 12.4. Lagrang e 36 1 12.5. Legendr e 36 3 12.6. Jacob i 36 4 12.7. Maye r 36 7 12.8. Erdman n 36 8 12.9. Weierstras s 37 0 12.10. Refinemen t o f Weierstrass' s method s 37 4 12.11. Variationa l method s i n mechanic s 37 7 12.12. Existenc e question s 37 9 Bibliography 380

Chapter 13 . Th e Origin s o f Functiona l Analysi s REINHARD SIEGMUND-SCHULTZ E 38 5

13.1. Introduction : Summar y plu s mathematica l an d historiographica l motivations 38 5

13.2. Th e root s i n th e calculu s o f variation s an d i n th e Italia n Calcolo Funzionale 38 7

13.3. Ascoli' s theorem , th e set-theoreti c impuls e an d Frechet' s analyse generate 38 9

13.4. Th e root s i n th e theor y o f system s o f linea r equation s an d integral equation s 39 1

13.5. Pioneerin g foray s withou t effect : Axio m system s o f Pean o an d Pincherle fo r infinite-dimensiona l vecto r space s 39 4

vi C O N T E N T S

13.6. Th e Hilber t theor y o f integra l equation s an d it s reformulatio n by E . Schmid t 39 4

13.7. Th e faile d attemp t a t a synthesi s b y a n outsider : E . H . Moore' s General Analysis 39 7

13.8. A firs t synthesi s o f Frechet' s analyse generale an d Hilbert -Schmidt's theor y o f integra l equations : Th e theore m o f Ries z and Fische r 39 8

13.9. Furthe r developmen t o f th e theor y o f functionals : Representatio n theorems 40 0

13.10. Ries z an d th e beginnin g o f operato r theor y 40 1 13.11. Banac h an d th e Polis h schoo l 40 2 13.12. Conclusio n 40 3 Bibliography 40 5

Index o f Names 40 9

Subject Inde x 415

Introduction

Analysis a s a n independen t subjec t wa s create d i n th e 17t h centur y durin g the scientifi c revolution . Kepler , Galileo , Descartes , Fermat , Huygens , Newto n and Leibniz , t o mentio n bu t a fe w importan t names , contribute d t o it s genesis . Questions fro m mechanics , optic s an d astronom y playe d a role i n it s earl y days , a s well as problems interna l t o mathematics , suc h a s the calculatio n o f areas , volume s and centre s o f gravit y an d th e analysi s o f involve d curves . Motio n alon g curve d paths unde r th e influenc e o f variabl e force s becam e a n are a o f particula r interes t after Galileo' s stud y o f freel y fallin g bodie s ha d le d t o initia l success . Ou t o f thi s wide variety o f efforts ther e emerge d b y the en d o f the 17t h century , i n the wor k of Newton an d Leibniz , th e ne w mathematica l disciplin e whos e histor y i s the subjec t of the presen t volume . Ver y broadl y stated , th e objec t o f thi s scienc e i s th e stud y of dependencies among variable quantities.

Since that time , no other mathematica l fiel d ha s influenced th e developmen t o f modern scientific thinking as deeply. Th e basic idea of using differential equation s t o gain insight int o the global behaviour o f varying quantities from thei r (infinitesimal ) changes ha s prove d fundamenta l an d fruitfu l fa r beyon d mathematic s an d physic s and ha s shape d ou r overal l scientifi c vie w o f th e world , especiall y ou r notio n o f causality. A t th e en d o f th e 18t h century , i n fact , mos t scientist s ha d com e t o agree tha t processe s i n natur e (an d society ) ar e deterministi c an d obe y law s tha t may b e describe d i n term s o f differentia l equations . Laplace , the n th e maste r o f mathematical physics , suggested tha t a n omniscien t intelligence , enjoying complet e knowledge of these laws and o f the stat e o f the world a t a given point i n time, coul d predict th e furthe r developmen t o f the worl d fo r eve r an d anon . I t wa s th e notio n of a law of nature tha t inspire d th e mathematica l concep t o f function, o f course , but thi s notio n woul d neve r hav e bee n a s influentia l i f mathematica l analysi s ha d not devise d suc h successfu l method s fo r th e stud y o f functiona l dependencies .

The developmen t o f mathematica l analysi s ha s displaye d uniqu e vitalit y an d momentum. Newto n an d Leibni z wer e thoroughl y consciou s o f th e novelt y an d importance o f thei r creation , ye t on e ca n hardl y imagin e tha t the y coul d hav e anticipated ho w th e scienc e the y invente d woul d develo p i n th e hundre d year s fol -lowing thei r work . Th e sam e migh t b e sai d fo r Eule r an d Cauchy . Th e dept h o f change ca n als o b e assesse d b y considerin g jus t ho w fa r today' s scientifi c thinkin g has distance d itsel f fro m Laplace' s determinism .

The presen t histor y o f analysi s seek s to describ e thi s dramati c developmen t i n all it s dimensions , an d t o d o s o satisfactoril y a discussio n o f genera l trend s woul d be insufficient . Scientifi c progres s i s impelled b y the solutio n o f concrete problems , and s o a reconstructio n o f genera l trend s i n th e histor y o f analysi s clearl y mus t be complemente d b y a n examinatio n o f th e specifi c problems , i n al l thei r variety , which bot h challenge d th e ne w disciplin e an d contribute d t o it s growth .

vii

viii INTRODUCTIO N

For thes e reason s i t seeme d appropriat e t o produc e thi s boo k a s a collectiv e work o f author s wh o ar e prove n historica l expert s i n specifi c fields. Firs t draft s o f the chapters were initially exchanged amon g authors and then discussed an d coordi -nated a t a conferenc e a t th e Universit y o f Essen. Th e final version s o f the chapter s were writte n afte r a ne w roun d o f evaluatio n an d mutua l criticism . Th e resultin g volume manage s t o clarif y th e conceptua l chang e whic h analysi s underwen t i n th e course o f it s development , whil e elucidatin g th e influenc e o f specifi c application s and describin g th e relevanc e o f the biographica l an d philosophica l backgrounds .

The boo k i s aime d a t a broa d audience . Mathematica l example s ar e selecte d and presente d i n suc h a wa y tha t the y ca n b e understoo d b y an y reade r wit h a college background an d a certain opennes s t o mathematica l argumentation . Thos e who would lik e to pursue a topic i n more dept h wil l find a comprehensive referenc e to the sources and the relevant secondar y literature . I f a reliable English translatio n of a sourc e i s available , i t i s quoted an d use d alon g wit h th e source .

The first ten chapters of the book present a chronology o f analysis up to the end of the 19t h century . Chapte r 1 describes those developments o f antiquity whic h th e authors o f the 16t h and 17t h centuries were able to build upon . I n describing work s on infinitesima l analysi s i n th e perio d befor e Newto n an d Leibniz , Chapte r 2 con-centrates o n th e frequentl y underestimate d Cartesia n tradition . Chapte r 3 direct s its attentio n t o th e conceptua l difference s betwee n th e approache s o f Newto n an d Leibniz t o infinitesima l analysi s an d th e relatio n o f these difference s t o mechanics . Chapter 4 analyse s th e chang e fro m a geometrica l t o a n algebrai c conceptio n o f analysis whic h too k plac e i n th e 18t h centur y i n th e work s o f Eule r an d Lagrang e and whic h wa s accompanie d b y th e emergenc e o f the concep t o f function .

The relatio n o f mathematics t o it s applications , mainl y i n physics, i s present i n every chapte r o f thi s narrativ e history . Tw o chapter s focu s directl y o n thi s topic ; however, Chapte r 5 sketches th e genesi s o f analytica l mechanics , whil e Chapte r 7 delineates thos e problem s o f 19th-centur y physics , suc h a s potentia l theory , whic h led t o th e fundamenta l integra l theorem s o f Gauss , Gree n an d Stokes .

The dee p conceptua l chang e i n analysi s brough t abou t i n th e 19t h centur y by Cauch y an d Weierstras s i s analyse d i n Chapte r 6 . Chapte r 8 describe s th e emergence and flowering of the theory of complex functions; th e extensive treatmen t of thi s subjec t i s a specia l featur e o f th e presen t book . Chapte r 9 examine s th e history o f th e concep t o f th e integra l fro m Rieman n t o Lebesgue , th e fascinating , almost paradigmati c developmen t o f a mathematica l concep t i n whic h ever y ste p was motivated b y concrete problems an d intentions . Finally , Chapte r 1 0 deals wit h the foundation s o f analysis i n the secon d hal f o f the 19t h century . Mathematically , it i s about th e emergenc e o f an adequat e theor y o f real number s an d th e genesi s of set theory . Thi s developmen t ha d far-reachin g consequence s fo r mathematic s an d its philosoph y an d culminate d i n the so-calle d foundationa l crisis .

The chronolog y jus t outline d i s complemented b y surve y chapter s o n subject s which could have been integrated int o the narrative onl y a t grea t sacrific e o f clarity. These concer n th e theor y o f differentia l equation s (Chapte r 11) , th e variationa l calculus (Chapte r 12 ) an d functiona l analysi s (Chapte r 13) .

Each chapte r contain s biographie s o f on e o r tw o mathematician s wh o wer e especially influentia l i n th e perio d unde r discussion .

Some explanatio n o f th e reference s i s i n order . I n principle , w e us e th e styl e (author year , page) , fo r example , (Cauch y 1825 , 50). I n the bibliography , however ,

I N T R O D U C T I O N i x

the reader wil l find for (Cauch y 1825 ) both the original publication and it s reprint i n Cauchy's Oeuvres. I n every case the year designates the year of original publication , whereas the page number refer s t o the las t editio n mentione d i n the bibliographica l entry (i n th e abov e example , t o th e Oeuvres).

Some reference s d o sho w tw o years , fo r example , (Eule r 1755/2000 , 53) . Her e 1755 refers t o the yea r o f first publication , wherea s 200 0 is the yea r o f a translatio n of thi s sourc e int o English . I n suc h a case , th e pag e numbe r refer s t o th e pag e in th e translation . Tw o year s ar e als o show n whe n referrin g t o publication s o f academies, a s i n (Eule r 1753/1755 , 234) . Her e 175 3 identifies th e annua l volum e of the academy , whil e 175 5 indicates th e yea r i n whic h i t wa s actuall y publishe d an d the pag e numbe r refer s t o th e Opera . I t wil l b e clea r fro m th e contex t t o whic h o f these tw o case s th e doubl e yea r belongs .

The presen t History of Analysis i s a translatio n int o Englis h fro m th e origina l German boo k publishe d b y Spektru m Akademische r Verla g i n 1999 . Eac h autho r was responsibl e fo r th e translatio n (o r retranslation ) o f hi s chapter . Thi s wa s no proble m fo r th e nativ e speaker s o f English , o f course , whil e th e othe r author s benefited greatl y fro m th e unselfis h an d generou s hel p o f Jerem y Gray , London . Some reviewer s o f the AM S als o mad e significan t bu t anonymou s contribution s t o the polishin g an d simplifyin g o f the Englis h i n the chapter s whic h were not writte n by nativ e speakers . Nicol e Huelsman n investe d a grea t dea l o f skil l an d effor t o n the technica l side , includin g th e productio n o f the indexes . I would lik e to expres s my heartfel t gratitud e t o al l those wh o hav e bee n s o helpful .

Thanks ar e also due to the authors fo r the competence an d caring they investe d in thi s project .

Hans Niels Jahnk e Essen, Decembe r 200 2

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Index o f Name s

Abel, Niel s Henri k (180 2 - 1829) , 173 , 176-178, 181 , 184 , 186 , 189 , 239 , 34 5

d'Alembert, Jean-Baptist e l e Ron d (171 7 -1783), 102 , 105 , 123-125 , 128 , 131, 145, 147 , 148 , 163 , 168 , 169 , 213-215 , 217, 331-333 , 335 , 337 , 34 3

Alexander o f Aphrodisia s (2nd/3r d century), 1 8

Ampere, Andr e Mari e (177 5 - 1836) , 167 , 176, 177 , 189 , 20 8

Anaxagoras (approx . 50 0 - 42 8 BC) , 16 , 3 4 Apollonius o f Perg e (approx . 26 0 - 19 0

BC), 32 , 35 , 42 , 45 , 4 8 Aquinas, Thoma s (122 4 - 1274) , 30 9 Arbogast, Loui s Francoi s Antoin e (175 9 -

1803), 125 , 130 , 16 5 Archimedes (28 7 - 21 2 BC) , 2 , 3 , 9 , 18 ,

20-29, 32 , 35 , 41 , 58, 9 8 Argand, Jea n Rober t (176 8 - 1822) , 216 ,

228 Aristotle (38 4 - 32 2 BC) , 1 , 2 , 7 , 13 , 17 , 18 ,

31, 34 , 35 , 30 9 Arzela, Cesar e (184 7 - 1912) , 273 , 385 , 389 ,

390 Ascoli, Giuli o (184 3 - 1896) , 389 , 39 0 Auzout, Adrie n (162 2 - 1691) , 48 , 4 9

Baire, Ren e Loui s (187 4 - 1932) , 188 , 274 , 399, 40 3

Banach, Stefa n (189 2 - 1945) , 385-387 , 401-404

Barrow, Isaa c (163 0 - 1677) , 1 , 35 , 41, 69-71

Beer, Augus t (182 5 - 1863) , 39 2 Beltrami, Eugeni o (183 5 - 1900) , 245 , 37 8 Bendixson, Iva r Ott o (186 1 - 1935) , 35 1 Berkeley, Georg e (168 5 - 1753) , 127 , 18 8 Bernays, Pau l (188 8 - 1977) , 31 7 Bernoulli, Danie l (170 0 - 1782) , 95 , 102 ,

105, 116 , 119 , 125 , 198 , 343 , 34 5 Bernoulli, Jako b I (165 4 - 1705) , 60 , 69 , 86 ,

105, 106 , 118 , 327-329 , 331 , 355, 357-359

Bernoulli, Johan n I (166 7 - 1748) , 1 , 17 , 34 , 60, 69 , 86 , 92 , 102 , 105-109 , 111-114 , 116-119, 147 , 329 , 330 , 332 , 336 , 337 , 343, 344 , 355 , 358 , 35 9

Bernoulli, Nicola s I (168 7 - 1759) , 106 , 109 , 122

Bernoulli, Nicola s I I (169 5 - 1726) , 33 1 Bessel, Friedric h Wilhel m (178 4 - 1846) ,

221, 345 , 39 5 Betti, Enric o (182 3 - 1892) , 234 , 244 , 245 ,

252, 25 4 Bolza, Osca r (185 7 - 1942) , 356 , 371 , 375,

377 Bolzano, Bernar d (178 1 - 1848) , 1 , 161 ,

173-176, 185-187 , 217 , 29 6 Bombelli, Rafae l (152 6 - 1572) , 21 3 Borchardt, Car l Wilhel m (181 7 - 1880) ,

232, 240 , 245 , 36 5 Borel, Emil e (187 1 - 1956) , 186 , 253 , 261,

277, 281-286 , 31 2 Bouquet, Jean-Claud e (181 9 - 1885) , 184 ,

229, 232 , 234 , 244 , 246 , 252 , 339 , 340 , 343, 346 , 347 , 35 0

Brill, Alexande r Wilhel m vo n (184 2 -1935), 23 5

Brioschi, Francesc o (182 4 - 1897) , 24 4 Briot, Charle s August e Alber t (181 7 -

1882), 184 , 229 , 232 , 234 , 244 , 246 , 252, 339 , 340 , 343 , 346, 347 , 35 0

Broden, Thorste n (185 7 - 1931) , 27 1 Brouwer, Luitze n Egber t us Ja n (188 1 -

1966), 308 , 320 , 40 2 Bryson o f Heraclei a (45 0 - 37 0 BC) , 9 , 17 ,

18 Buee, Adrien-Quenti n (174 8 - 1826) , 22 8 Burali-Forti, Cesar e (186 1 - 1931) , 312 , 31 3

Campanus o f Novar a (approx . 121 0 -1296), 1 8

Cantor, Geor g (184 5 - 1918) , 1 , 11 , 12 , 35 , 186, 210 , 278-282 , 292 , 293 , 295, 299-302, 304-313 , 316-320 , 386 , 389 , 390, 39 7

410 INDEX O F NAME S

Caratheodory, Constanti n (187 3 - 1950) , 378

Carnot, Lazar e Nicola s Marguerit e (175 3 -1823), 29 , 16 4

Cartan, Henr i Pau l (bor n 1904) , 25 3 Casorati, Felic e (183 5 - 1890) , 185 , 229 ,

244-246 Cauchy, Augustin-Loui s (178 9 - 1857) , 109 ,

120, 123 , 131 , 155-158 , 160-182 , 184 , 185, 187 , 189 , 191 , 209, 213-236 , 239 , 244, 246 , 248 , 249 , 253 , 255 , 262-264 , 271, 339-342 , 345-349 , 39 2

Cavalieri, Bonaventur a (1598 ? - 1647) , 25 , 27, 28 , 35 , 41 , 56, 58 , 60 , 62 , 66 , 69 , 8 9

Cavendish, Henr y (173 1 - 1810) , 20 2 Chebotarev, Nikola i Grigorievic h (189 4 -

1947), 1 7 Chittenden, Edwar d Wilso n (188 5 - 1977) ,

398 Christina o f Swede n (162 6 - 1689) , 4 2 Christoffel, Elwi n Brun o (182 9 - 1900) , 21 1 Clairaut, Alexi s Claud e (171 3 - 1765) , 95 ,

102, 105 , 332-33 4 Clausius, Rudol f (182 2 - 1888) , 208 , 20 9 Clavius, Christop h (153 7 - 1612) , 10 , 4 2 Clebsch, Rudol f Friedric h Alfre d (183 3 -

1872), 239 , 345 , 367 , 368 , 376 , 37 8 Cohen, Pau l Josep h (bor n 1934) , 31 1 Commandino, Federig o (150 9 - 1575) , 35 6 Condorcet, Mari e Jea n Antoin e Nicola s

Caritat, Marqui s d e (174 3 - 1794) , 333 , 335, 34 4

Conon o f Samo s (befor e 21 2 BC) , 2 1 Coriolis, Gaspar d Gusta v d e (179 2 - 1843) ,

217, 34 7 Cotes, Roge r (168 2 - 1716) , 11 6 Coulomb, Charle s August e d e (173 6 -

1806), 20 1 Crelle, Augus t Leopol d (178 0 - 1855) , 24 9

Darboux, Jean-Gasto n (184 2 - 1917) , 27 , 187, 189 , 266 , 268-273 , 275 , 277 , 286 , 339, 342 , 34 7

Dedekind, Richar d (183 1 - 1916) , 1 , 11 , 12, 155, 233 , 243 , 245 , 263 , 265 , 292 , 293 , 297-299, 302 , 306 , 307 , 310-312 , 314 , 317

Deinostratus (4t h centur y BC) , 3 0 Delaunay, Charle s Eugen e (181 4 - 1872) ,

366 Democritus o f Abder a (approx . 46 0 - 37 0

BC), 25 , 3 3 Descartes, Ren e (159 6 - 1650) , 1 , 29 , 30 ,

35, 41-48 , 50-55 , 75 , 76 , 92 , 100 , 213 , 309, 327 , 33 0

Dini, Uliss e (184 5 - 1918) , 271 , 27 3 Diodes (1s t centur y BC?) , 56-5 8

Dirichlet, Johan n Pete r Gusta v (Lejeune ) (1805 - 1859) , 127 , 157 , 158 , 180 , 181, 186-189, 201 , 202, 208-210 , 232 , 233 , 236, 243 , 261-263 , 266 , 267 , 297 , 339 , 379

du Bois-Reymond , Pau l (183 1 - 1889) , 181 , 187, 189 , 190 , 280 , 281 , 295, 300 , 301, 312, 340 , 348 , 35 5

Eisenstein, Ferdinan d Gotthol d Ma x (182 3 - 1852) , 232 , 23 3

Epicharmus (poet ) (53 0 - 44 0 BC) , 3 Eratosthenes o f Kyren e (approx . 28 4 - 20 0

BC), 21 , 23 Erdmann, G . (2n d hal f o f 19t h century) ,

355, 368-37 0 Euclid (approx . 30 0 BC) , 2-14 , 18-22 , 27 ,

28, 32 , 34 , 156 , 168 , 297 , 29 8 Eudemus o f Rhodo s (4t h centur y BC) , 1 6 Eudoxus o f Knido s (approx . 40 0 - 34 7 BC) ,

6-10, 1 8 Euler, Leonhar d (170 7 - 1783) , 1 , 17 , 20 ,

95, 102 , 105-108 , 111 , 113-119 , 121-128, 130 , 131 , 137 , 138 , 140-147 , 156-158, 160-162 , 164 , 165 , 167 , 169 , 171, 173 , 188 , 190 , 191 , 198 , 213-216 , 218, 233 , 241 , 244, 251 , 267, 291 , 329, 331-338, 343 , 348 , 355-357 , 359-363 , 369, 372 , 376 , 378 , 37 9

Eutocius o f Ascalo n (5th/6t h century) , 5 8

Faraday, Michae l (179 1 - 1867) , 37 3 Fejer, Lipo t (188 0 - 1959) , 19 0 Fermat, Pierr e d e (160 1 - 1665) , 1 , 42 , 47 ,

51-54, 69 , 8 0 Fischer, Erns t (187 5 - 1945) , 385 , 398 , 39 9 Fourier, Jean-Baptiste-Josep h d e (176 8 -

1830), 126 , 156-158 , 164 , 165 , 170 , 177, 179-181 , 191 , 199-201 , 219, 261, 262, 272 , 289 , 343 , 391 , 392

Frechet, Mauric e Ren e (187 8 - 1973) , 385-387, 389-391 , 394, 397-40 4

Fraenkel, Abraha m (189 1 - 1965) , 316 , 31 7 Fredholm, Eri k Iva r (186 6 - 1927) , 385 ,

387, 393 , 395 , 40 0 Frege, Gottlo b (184 8 - 1925) , 292 , 300 ,

302-304, 311 , 312, 314 , 31 8 Frobenius, Geor g Ferdinan d (184 9 - 1917) ,

190, 34 0 Fuchs, Immanue l Lazaru s (183 3 - 1902) ,

242, 345 , 346 , 34 8

Godel, Kur t (190 6 - 1978) , 311 , 319 Galilei, Galile o (156 4 - 1642) , 28 , 32 , 55 ,

92, 109 , 356 , 35 7 Gauss, Car l Friedric h (177 7 - 1855) , 35 ,

173, 174 , 177 , 181 , 198 , 202 , 203 , 205-209, 215 , 216 , 221 , 227, 228 , 232-234, 236 , 241-243 , 251 , 262, 34 5

INDEX O F NAME S 411

Gauss, Car l Friedric h (1777-1855) , 20 5 Gentzen, Gerhar d Kar l Eric h (190 9 - 1945) ,

320 Gergonne, Josep h Dia z (177 1 - 1859) , 21 6 Gibbs, Josia h Willar d (183 9 - 1903) , 20 8 Goldbach, Christia n vo n (169 0 - 1764) , 116 ,

118, 122 , 33 1 Goursat, Eduar d Jean-Baptist e (185 8 -

1936), 229 , 255 , 356 , 371 , 376 Grandi, Guid o (167 1 - 1742) , 12 1 Grassmann, Herman n Giinthe r (180 9 -

1877), 291-293 , 39 4 Green, Georg e (179 3 - 1841) , 198 , 202-210 ,

223, 37 9 Gregorius a St . Vincent o (158 4 - 1667) , 1 8 Gregory, Jame s (163 7 - 1675) , 66 , 69 , 111,

357 Gudermann, Christop h (179 8 - 1852) , 184 ,

185, 247 , 24 8 Guldin, Habaku k Pau l (157 7 - 1643) , 56 ,

62, 6 8

Holder, Ludwi g Ott o (185 9 - 1937) , 190 , 210, 34 7

Hadamard, Jacque s (186 5 - 1963) , 253 , 339, 356, 385-391 , 393 , 397 , 400 , 403 , 40 4

Hahn, Han s (187 9 - 1934) , 386 , 401 , 402 Hamilton, Willia m Rowa n (180 5 - 1865) ,

208, 228 , 291 , 293, 368 , 375 , 377 , 37 8 Hankel, Herman n (183 9 - 1873) , 32 , 130 ,

158, 176 , 187 , 188 , 190 , 208 , 228 , 266-269, 271 , 278-280, 292-295 , 298 , 299, 301 , 303, 307 , 309 , 310 , 314 , 316 , 320

Hansteen, Christoffe r (178 4 - 1873) , 176 , 177

Harnack, Car l Gusta v Axe l (185 1 - 1888) , 190, 280 , 28 1

Hartogs, Friedric h (187 4 - 1943) , 24 0 Hausdorff, Feli x (186 8 - 1942) , 317 , 39 1 Heaviside, Olive r (185 0 - 1925) , 190 , 191,

208 Heine, Heinric h Eduar d (182 1 - 1881) , 186 ,

210, 272 , 292 , 299-302 , 304-306 , 310 , 365

Helly, Eduar d (188 4 - 1943) , 386 , 39 2 Heraclitus (5t h centur y BC) , 3 3 Herigone, Pierr e (158 0 - 1643) , 5 1 Hermann, Jako b (167 8 - 1733) , 92 , 94 , 137 ,

333 Hermite, Charle s (182 2 - 1901) , 231 , 232,

242, 244 , 254 , 30 7 Hero o f Alexandri a (40 ? - 120?) , 1 5 Hesse, Ludwi g Ott o (181 1 - 1874) , 365-367 ,

376 Heuraet, Hendric k va n (163 4 - 1660?) , 47 ,

49 Hieron o f Syracus e (30 6 - 21 5 BC) , 2 1

Hilbert, Davi d (186 2 - 1943) , 32 , 34 , 156 , 242, 292 , 295 , 302 , 311-314 , 316 , 318-320, 356 , 374-376 , 379 , 380 , 385 , 387, 390 , 392-401 , 40 4

Hill, Georg e Willia m (183 8 - 1914) , 392 , 393

Hindenburg, Car l Friedric h (174 1 - 1808) , 130

Hipparchus (approx . 18 0 - 12 7 BC) , 2 0 Hippasus o f Metapon t (approx . 45 0 BC) , 4 Hippias o f Elli s (approx . 40 0 BC) , 3 0 Hippocrates o f Chio s (approx . 45 0 BC) , 1 6 Holmboe, Bern t Michae l (179 5 - 1850) , 176 ,

177 Hopkins, Willia m (179 3 - 1866) , 20 5 l'Hospital, Guillaum e Francoi s Antoin e

(1661 - 1704) , 41 , 60, 86 , 95 , 105 , 106 , 108

Hudde, Ja n (162 8 - 1704) , 47 , 49-51 , 53 , 54, 75 , 8 0

Hurwitz, Adol f (185 9 - 1919) , 188 , 29 5 Huygens, Christiaa n (162 9 - 1695) , 41 , 42,

47, 49 , 56-58 , 60 , 63-69 , 81 , 85, 92 , 100, 109 , 32 7

Huygens, Constantij n (159 6 - 1687) , 4 2

Ibn al-Haytha m (96 5 - approx . 1038) , 1 6

Jacobi, Car l Gusta v Jaco b (180 4 - 1851) , 232, 239 , 240 , 246 , 247 , 340 , 345 , 347 , 355, 356 , 364-368 , 372 , 375-37 8

Jordan, Camill e Mari e Ennemon d (183 8 -1922), 189 , 242 , 261 , 275-278, 284 , 288, 40 4

Jungius, Joachi m (158 7 - 1657) , 10 9

Kepler, Johanne s (157 1 - 1630) , 41 , 56, 61-63, 66 , 9 2

Killing, Wilhel m Kar l Josep h (184 7 - 1923) , 249

Kirchhoff, Gusta v Rober t (182 4 - 1887) , 191, 208 , 209 , 34 8

Klugel, Geor g Simo n (173 9 - 1812) , 2 Klein, Christia n Feli x (184 9 - 1925) , 233 ,

238, 243 , 246 , 25 5 Kneser, Adol f (186 2 - 1930) , 356 , 371 , 373,

378 Koch, Niel s Fabia n Helg e vo n (187 0 -

1924), 392 , 39 3 Kossak, Erns t (183 9 - 1892) , 29 5 Kovalevskaya, Sofi a (Sonja ) Wasiljewn a

(1850 - 1891) , 343 , 346-34 8 Kramp, Christia n (176 0 - 1826) , 24 9 Kronecker, Leopol d (182 3 - 1891) , 178 , 185 ,

243, 245 , 246 , 292 , 303 , 307 , 31 2 Kummer, Erns t Eduar d (181 0 - 1893) , 227 ,

241, 243 , 291 , 304

Levy, Pau l (188 6 - 1971) , 385 , 389 , 398 , 40 0

412 INDEX O F NAME S

Lacroix, Sylvestr e Francoi s (176 5 - 1843) , 28, 161 , 164 , 169 , 171 , 369

Lagrange, Josep h Loui s (173 6 - 1813) , 102 , 105, 107 , 120 , 125 , 127-131 , 137 , 138 , 145-152, 156 , 157 , 161 , 162 , 164-166 , 169, 171 , 175 , 176 , 188 , 189 , 198 , 207 , 216, 218 , 332-340 , 345 , 355 , 361-365 , 367-370, 37 7

Lame, Gabrie l (179 5 - 1870) , 34 0 Lambert, Johan n Heinric h (172 8 - 1777) ,

105, 12 0 Laplace, Pierr e Simo n (174 9 - 1827) , 105 ,

120, 130 , 131 , 197-202 , 213 , 214 , 216 , 247, 332 , 336 , 34 4

Laugwitz, Detle f (bor n 1932) , 164 , 18 8 Laurent, Pierr e Alphons e (181 3 - 1854) , 22 6 Lebesgue, Henr i Leo n (187 5 - 1941) , 188 ,

261, 271-275 , 277 , 284-289 , 390 , 394 , 398, 39 9

Legendre, Adrien-Mari e (175 2 - 1833) , 105 , 117, 214 , 215 , 217 , 355 , 356 , 363 , 364 , 366, 372 , 37 6

Leibniz, Gottfrie d Wilhel m (164 6 - 1716) , 23, 29 , 34 , 58 , 60 , 69 , 71 , 73-75, 77-81 , 85-92, 94-102 , 105-109 , 111-114 , 116 , 117, 121 , 122 , 164 , 170 , 191 , 309, 325 , 327, 329 , 330 , 334 , 35 8

Levi, Eugeni o Eli a (188 3 - 1917) , 24 0 Liouville, Josep h (180 9 - 1882) , 189 , 205 ,

208, 227 , 229-232 , 240 , 307 , 331, 343-345, 347 , 349 , 35 0

Lipschitz, Rudol f Ott o Sigismun d (183 2 -1903), 298 , 340 , 342 , 346 , 34 7

Locke, Joh n (163 2 - 1704) , 30 9

Meray, Charle s Rober t (183 5 - 1911) , 155 , 300, 30 1

Mach, Erns t (183 8 - 1916) , 35 8 Maclaurin, Coli n (169 8 - 1746) , 92 , 100 ,

105, 107 , 113 , 127 , 12 8 Manfredi, Gabriell o (168 1 - 1761) , 8 6 Maupertuis, Pierr e Loui s Morea u d e (169 8

- 1759) , 138 , 139 , 141-14 4 Maxwell, Jame s Cler k (183 1 - 1879) , 208 ,

373 Mayer, Christia n Gusta v Adol f (183 9 -

1908), 355 , 367 , 368 , 375 , 377 , 37 8 Mersenne, Mari n (158 8 - 1648) , 55 , 5 8 Mittag-Leffler, Magnu s Gusta v (184 6 -

1927), 3 , 247-249 , 252 , 266 , 34 8 Moivre, Abraha m d e (166 7 - 1754) , 111 ,

120 Monge, Gaspar d (174 6 - 1818) , 105 , 107 ,

131, 338 , 339 , 34 5 Montucla, Jea n Etienn e (172 5 - 1799) , 14 2 Moore, Eliaki m Hasting s (186 2 - 1932) ,

316, 385 , 397 , 39 8 Morera, Giacint o (185 6 - 1909) , 22 9

Morgan, Augustu s d e (180 6 - 1871) , 291, 294

Napier, Joh n Lor d o f Merchisto n (155 0 -1617), 3 5

Navier, Claud e Loui s Mari e Henr i (178 5 -1836), 18 9

Neumann, Car l Gottfrie d (183 2 - 1925) , 208, 210 , 350 , 39 2

Neumann, Fran z Erns t (179 8 - 1895) , 208 , 209, 36 5

von Neumann , Joh n (Janos ) (190 3 - 1957) , 317, 386 , 387 , 397 , 40 4

Nevanlinna, Rol f Herman n (189 5 - 1980) , 247, 25 3

Newton, Isaa c (164 3 - 1727) , 32 , 35 , 41, 51, 60, 66 , 69 , 71 , 73-85, 89 , 91-102 , 105-107, 111 , 112 , 116 , 120 , 127 , 152 , 163, 189 , 230 , 325 , 326 , 336 , 357 , 35 8

Nicomedes (2n d centur y BC ) , 3 0 Nobili, Leopold o (178 4 - 1835) , 20 9 Noether, Ma x (Brill ; Noethe r 184 4 - 1921) ,

235

Ohm, Marti n (179 2 - 1872) , 201 , 249 Oldenburg, Henr y (161 8 - 1677) , 7 4 Oresme, Nicol e (132 3 - 1382) , 1 , 35 , 11 8 Osgood, Willia m Fog g (186 4 - 1943) , 273 ,

274, 288 , 37 1 Ostrogradsky, Michae l Wassilewitsc h (180 1

- 1862) , 20 7 Oughtred, Willia m (157 5 - 1660) , 7 5

Pappus o f Alexandri a (approx . 300) , 2 , 29 , 30, 42 , 43 , 47 , 35 6

Parmenides (approx . 52 0 - 45 0 BC) , 7 , 3 3 Pascal, Blais e (162 3 - 1662) , 69 , 8 6 Pasch, Morit z (184 3 - 1930) , 15 6 Peacock, Georg e (179 1 - 1858) , 291 , 294 Peano, Giusepp e (185 8 - 1939) , 156 , 270 ,

275, 292 , 313 , 394 , 39 7 Pell, Joh n (161 1 - 1685) , 2 2 Philolaus (5t h centur y BC) , 5 Philoponus, Johanne s (6t h centur y BC) ,

17, 3 1 Picard, Charle s Emil e (185 6 - 1941) , 252 ,

253, 339 , 349 , 350 , 391 , 402 Pieri, Mari o (I86 0 - 1913) , 15 6 Pincherle, Salvator e (185 3 - 1936) , 254 ,

255, 385 , 391 , 394, 397 , 400 , 40 4 Plato (43 7 - 34 7 BC) , 7 , 10 , 12 , 18 , 33-3 5 Plutarch (4 5 - 120) , 2 1 Poincare, Jules-Henr i (185 4 - 1912) , 172 ,

187, 190 , 243 , 281 , 319, 320 , 339 , 350 , 351, 386 , 392 , 393 , 40 2

Poisson, Simeo n Deni s (178 1 - 1840) , 161 , 171, 177 , 179 , 199 , 201 , 202, 204 , 206 , 207, 214-219 , 238 , 34 5

Proclus (41 2 - 485) , 18 , 32 , 3 5

INDEX O F NAME S 413

Protagoras (48 0 - 42 1 BC) , 3 5 Prym, Friedric h Emi l (184 1 - 1915) , 23 3 Ptolemy, Claudiu s (approx . 8 5 - 165) , 1 ,

20, 21 , 28 Puiseux, Victo r Alexandr e (182 0 - 1883) ,

224, 230 , 231 , 235, 236 , 24 4 Pythagoras o f Samo s (approx . 58 0 - 50 0

BC), 15 , 16 , 316 , 32 0

Riccati, Jacop o Francesco , Gra f (167 6 -1754), 86 , 33 1

Richard, Jule s Antoin e (186 2 - 1956) , 312 , 313

Richelot, Friedric h Juliu s (180 8 - 1875) , 36 5 Riemann, Bernhar d Geor g Friedric h (182 6 -

1866), 26 , 157 , 172 , 187-190 , 205 , 208-210, 213 , 214 , 230 , 232-247 , 250 , 253, 255 , 261 , 263-275, 277 , 278 , 280 , 284, 286 , 288 , 291 , 292, 297 , 305 , 308 , 346, 350 , 37 9

Riesz, Frederi c (188 0 - 1956) , 385 , 387 , 394 , 398, 399 , 401 , 402

Roberval, Gille s Personn e d e (160 2 - 1675) , 42, 55 , 5 6

Robinson, Abraha m (191 8 - 1974) , 164 , 19 1 Rosenhain, Johan n Geor g (181 6 - 1887) ,

365 Russell, Bertran d Arthu r Willia m (187 2 -

1970), 33 , 303 , 312 , 313 , 318, 31 9

Schauder, Julius z Pawe l (189 9 - 1943) , 40 3 Schmidt, Erhar d (187 6 - 1959) , 385 , 386 ,

393-395, 399 , 40 0 Schmieden, Cur t (190 5 - 1992) , 16 4 Schoenflies, Arthu r Morit z (185 3 - 1928) ,

283, 284 , 385 , 39 7 van Schooten , Fran s (161 5 - 1660) , 41-43 ,

45-55, 60 , 7 5 van Schooten , Frans , Sr . (1581 ? - 1645) , 4 6 van Schooten , Piete r (163 4 - 1679) , 4 7 Schopenhauer, Arthu r (178 8 - 1860) , 2 8 Schumacher, Heinric h Christia n (178 0 -

1850), 17 3 Schwartz, Lauren t (bor n 1915) , 19 1 Schwarz, Herman n Amandu s (184 3 - 1921) ,

176, 186 , 210 , 211 , 238, 243 , 250 , 345 , 350, 39 5

Seidel, Philip p Ludwi g vo n (182 1 - 1896) , 181, 182 , 184 , 237 , 36 5

Simplicius (approx . 50 0 - 549) , 1 9 Simpson, Thoma s (171 0 - 1761) , 9 5 Skolem, Thoral f Alber t (187 7 - 1904) , 316 ,

317 Sluse, Ren e - Francoi s d e (162 2 - 1685) , 41,

56-66, 68 , 6 9 Smith, Henr y Joh n Stanle y (182 6 - 1883) ,

190, 269 , 278-280 , 39 8 Spinoza, Baruc h (163 2 - 1677) , 30 9 Sporus (2n d thir d o f th e 2n d century) , 3 1

Steiner, Jaco b (179 6 - 1863) , 18 9 Steinhaus, Hug o Dyoniz y (188 7 - 1972) , 40 3 Stevin, Simo n (154 8 - 1620) , 4 , 1 0 Stieltjes, Thoma s Jea n (185 6 - 1894) , 190 ,

401 Stirling, Jame s (169 2 - 1770) , 105 , 113 ,

116, 11 8 Stokes, Georg e Gabrie l (181 9 - 1903) ,

181-184, 198 , 202 , 207 , 20 8 Stolz, Ott o (184 2 - 1905) , 8 , 12 , 28 1 Sturm, Jaque s Charle s Francoi s (180 3 -

1855), 189 , 343 , 344 , 35 0

Tait, Pete r Guthri e (183 1 - 1901) , 20 8 Taylor, Broo k (168 5 - 1731) , 105 , 111-113 ,

123, 330 , 331 , 334, 343 , 357-35 9 Theatetus (approx . 41 4 - 36 9 BC) , 1 4 Themistius (31 7 - 388) , 1 7 Thomae, Johanne s Kar l (184 0 - 1921) , 292 ,

301, 302 , 304 , 310 , 314 , 316 , 31 8 Thomson, Si r Willia m (Lor d Kelvi n o f

Largs) (182 4 - 1907) , 202 , 205 , 208 , 209, 37 9

Todhunter, Isaa c (182 0 - 1884) , 368-37 1 Torricelli, Evangelist a (160 8 - 1647) , 55 ,

58-60, 6 6 Truel, Henri-Dominiqu e (19t h century) , 21 6

Valerio, Luc a (155 2 - 1618) , 2 8 Varignon, Pierr e (165 4 - 1722) , 92 , 102 ,

121, 137 , 13 8 Veronese, Giusepp e (185 4 - 1917) , 31 2 Viete, Frangoi s (154 0 - 1603) , 1 , 2 , 20 , 31,

75 Voltaire, Frangois-Mari e (1694-1778) , 7 4 Volterra, Vit o (186 0 - 1940) , 190 , 271 , 280,

385-389, 391-393 , 400 , 403 , 40 4

Wallis, Joh n (161 6 - 1703) , 56 , 66-69 , 75 , 76, 8 1

Weber, Heinric h (185 4 - 1913) , 29 7 Weber, Wilhel m Eduar d (180 4 - 1891) , 206 ,

233 Weierstrass, Kar l Theodo r Wilhel m (181 5 -

1897), 155 , 156 , 184-191 , 209, 210 , 214, 228 , 232 , 237 , 238 , 240 , 241 , 243, 245-255, 292 , 293 , 295-299 , 301-304 , 340, 343 , 346-349 , 356 , 370-376 , 378 , 379, 39 4

Wessel, Caspa r (174 5 - 1818) , 21 6 Weyl, Herman n (188 5 - 1955) , 32 , 236 , 246 ,

247, 312 , 319 , 320 , 39 7 Whewell, Willia m (179 4 - 1866) , 37 7 Whitehead, Alfre d Nort h (186 1 - 1947) ,

318, 31 9 Wijnquist (2n d hal f o f 18t h century) , 1 7 Witt, Joha n d e (162 5 - 1672) , 47 , 4 9 Wolff, Christia n Freiher r vo n (167 9 - 1754) ,

2, 12 1

414 INDEX O F

Zeno o f Ele a (approx . 49 5 - 43 0 BC) , 1 , 33, 34

Zenodorous (2n d centur y BC?) , 35 6 Zermelo, Erns t Friedric h Ferdinan d (187 1 -

1953), 309 , 316-318 , 356 , 37 5

Subject Inde x

s algorithm, 355 , 36 1 function, 19 1

e — 8- formalism, 186 , 37 7 Ecole Polytechnique , 160 , 167 , 189 , 226 ,

232, 262 , 277 , 34 0

academy, 32 , 106 , 125 , 145 , 147 , 185 , 187 , 207, 214 , 216 , 223 , 224 , 226-228 , 230-232, 236 , 238 , 243 , 244 , 247 , 249 , 250, 333 , 348 , 359 , 398 , 39 9

acceleration, 41 , 150 , 15 1 accumulation point , 252 , 296 , 305 , 306 , 39 0 Achilles an d th e tortoise , 3 3 algorithm, 13 , 14 , 42-44 , 52 , 71 , 73, 78 , 80 ,

81, 87 , 89 , 91 , 96-101, 213 , 36 1 alternating method , 210 , 23 8 analyse generate , 385 , 389 , 390 , 398 , 400 ,

403, 40 4 analysis

algebraic, 128 , 130 , 160 , 36 2 complex, 107 , 209 , 214 , 232 , 255 , 281,

340, 37 9 foundation of , 127 , 130 , 155 , 160 , 173 ,

174, 176 , 185 , 188 , 198 , 245 , 250 , 296, 301 , 303, 304 , 306 , 311 , 314, 319-321, 340 , 36 1

higher vs . lower , 11 3 in contras t t o synthesis , 1 , 2 nonstandard, 12 , 29 , 35 , 164 , 191 , 312 real, 217 , 232 , 305 , 310 , 320 , 341 , 356

analytical mechanics , 137 , 152 , 232 , 37 8 Analytical Society , 18 9 area la w (Kepler) , 93 , 9 5 area o f integration , 275 , 28 8 arithmetic, 2 , 5 , 6 , 10-12 , 49 , 56 , 66 , 67 ,

69, 156 , 185 , 188 , 207 , 226 , 228 , 241, 255, 292 , 294 , 297-304 , 310 , 311 , 314, 316, 318-32 1

arithmetization, 156 , 25 5 astronomy, 21 , 29 , 41 , 60, 105 , 120 , 190 ,

197, 205 , 33 6 attraction o f ellipsoids , 198 , 205 , 206 , 20 9 Ausdehnungslehre, 39 4

axiom Archimedean, 3 , 8 , 9 , 11 , 12 , 18 , 31 5 axioms o f se t theory , 311 , 316, 317 , 32 1 bridging principle , 30 6 cutting postulate , 30 2 of calculation , 31 5 of choice , 309 , 317 , 40 1 of combination , 31 5 of completeness , 314 , 31 5 of continuity , 31 5 of order , 31 5

axiomatics, 22 , 318 , 38 5

boundary condition , 123 , 200 , 203 , 330 , 337, 339 , 34 3

boundary-value problem , 197 , 198 , 202 , 204, 205 , 207-210 , 337 , 339 , 39 2

boundedness, 203 , 204 , 208 , 210 , 224 , 254 , 265, 266 , 271 , 273-275 , 287 , 288 , 340-342, 347 , 386 , 402 , 40 3

brachistochrones, 105 , 106 , 33 0 branch point , 217 , 225 , 227 , 229 , 230 ,

234-236, 239 , 241 , 242, 246 , 34 6

calcolo funzionale , 385 , 387 , 39 4 calcul de s limites , 225 , 226 , 249 , 34 0 calculus, 27 , 28 , 33 , 41 , 47, 56 , 69 , 74 , 85 ,

86, 95 , 101 , 102 , 105-109 , 111-114 , 120, 127 , 128 , 130 , 137 , 149 , 156 , 160 , 161, 163 , 168 , 170 , 171 , 173 , 191 , 197, 214, 222 , 272 , 280 , 288 , 325-327 , 330 , 334, 348 , 36 1

calculus o f operators , 13 0 catenary, 29 , 105 , 108 , 109 , 111 , 142 , 33 0 caustics, 10 5 celestial mechanics , 41 , 160 , 197 , 336 , 39 2 center o f gravity , 1 , 22 , 24 , 28 , 41 , 62, 68 ,

138 circle, 1 , 3 , 8 , 14-17 , 19 , 20 , 22-24 , 27 ,

29-32, 35 , 44 , 45 , 47 , 50 , 55-58 , 60-64 , 67-69, 111 , 115 , 116 , 189 , 356 , 357 , 36 7

circular system , 23 0 cissoid, 29 , 41 , 56-58, 60-69 , 8 1 Combinatorial School , 13 0

416 SUBJECT INDE X

commensurable, 7 , 10 , 13 , 14 , 2 3 compactness, 155 , 186 , 240 , 386 , 39 0 conchoid, 48 , 50 , 52-5 4 condensation o f singularities , 187 , 267 , 27 1 condition o f equilibrium , 137-139 , 14 8 conformal mapping , 205 , 210 , 237 , 238 ,

243, 35 0 conic sections , 22 , 24 , 29 , 32 , 45 , 47 , 48 , 5 9 conjugate points , 355 , 366-368 , 37 6 consistency, 309 , 314 , 317-31 9 constant o f integration , 140 , 331 , 334-336 ,

349 constructive (i n opposit e t o

nonconstructive), 283 , 286 , 295 , 29 6 content

inner, 276 , 28 3 Jordan, 28 5 outer, 276 , 281 , 28 3

continuation analytic, 233 , 239 , 240 , 242 , 243 , 246 ,

249, 254 , 255 , 34 6 continued fractions , 5 , 13 , 113 , 115 , 13 0 continuity

axiom of , 10 , 1 8 completely continuous , 397 , 401 , 402 in Cauchy' s sense , 161 , 164-168 , 172 ,

179, 218 , 221 , 228, 26 2 in Euler' s sense , 164-16 6 pointwise, 166 , 174 , 18 6 uniform, 155 , 166 , 167 , 171 , 186 , 29 9

continuum hypothesis , 311 , 318 convergence, 121 , 13 0

arbitrarily slow , 18 2 Moore-Smith, 39 8 of a Fourie r series , 11 , 157 , 17 8 of a sequence , 12 , 33 , 29 1 of a series , 76 , 118 , 121 , 167 , 244 , 248 ,

253, 272 , 28 2 pointwise, 177 , 18 6 uniform, 168 , 181 , 182 , 184-186 , 248 ,

255, 269 , 272 , 273 , 280 , 39 5 convergence test , 121 , 167 , 168 , 218 , 299 ,

300 coordinate axes , 44 , 198 , 27 6 cosmology, 4 2 Cours d'analyse , 156 , 157 , 160 , 162-164 ,

166, 167 , 170 , 172 , 174 , 175 , 177 , 189 , 216-219, 221 , 223 , 225 , 227-229 , 262 , 277, 28 4

criterion Cauchy, 167 , 176 , 299 , 30 0 of integrability , 214 , 264 , 266 , 27 8

cross section , 58 , 59 , 10 5 curvature, 79 , 33 1 curve

algebraic, 51 , 70, 230 , 240 , 35 0 characteristic, 338 , 33 9 classification, 2 9

family o f curves , 67 , 68 , 109 , 373 , 37 8 length o f a , 14 , 41 , 110 , 36 8 mechanical, 29 , 30 , 32 6 mixed, 12 4 of highe r order , 46 , 4 8 transcendental, 46 , 5 5

cycloid, 55 , 56 , 67 , 91 , 123 , 357 , 358 , 37 0

Dedekind cuts , 11 , 293, 297 , 298 , 31 4 definition

axiomatic, 282 , 292 , 316 , 320 , 386 , 40 4 impredicative, 31 9 predicative, 31 9

derivative of a function , 90 , 107 , 108 , 128 , 129 , 159 ,

169, 176 , 191 , 198 , 199 , 203 , 208, 209, 214 , 223 , 225-227 , 229 , 235 , 237, 255 , 271 , 272, 275 , 288 , 325 , 326, 329 , 332 , 337 , 338 , 340 , 342 , 343, 347 , 349 , 350 , 355 , 359 , 360 , 363, 364 , 368 , 369 , 375 , 38 8

of a set , 306 , 308 , 30 9 Schwarzian, 24 3

diagonal argument , 30 7 difference

infinitely small , 14 9 differentiability

complex, 169 , 22 6 differential

complete (total) , 109 , 130 , 203 , 33 7 of a root , 9 0 of highe r order , 89 , 90 , 108 , 328 , 32 9 partial, 20 0

differential equatio n first-order, 339 , 340 , 34 9 hypergeometric, 24 1 linear, 116 , 222 , 240 , 242 , 338 , 345 , 346 ,

349, 36 6 ordinary, 109 , 200 , 331 , 335-340, 342 ,

343, 345 , 347 , 37 8 partial, 102 , 123 , 125 , 127 , 130 , 165 ,

198-200, 204 , 209 , 232 , 233 , 333, 336-340, 343 , 347-350 , 375 , 377-379, 39 1

singular solution , 33 0 total, 34 5

differentiation term-by-term, 176 , 186 , 25 5

Dirichlet kernel , 18 0 Dirichlet monster , 263 , 28 8 distribution, 179 , 191 , 203, 205-20 7

theory o f distributions , 191 , 351, 403 double poin t o f intersection , 44 , 4 5 dual, 40 1 duplication o f th e cube , 5 8 dynamics, 91 , 95, 146 , 147 , 149 , 345 , 364 ,

377

eigenvalue, 343 , 344 , 39 5

SUBJECT INDE X 417

electrostatics, 197 , 201 , 20 3 ellipse, 32 , 44-50 , 55 , 14 1 energy

kinetic, 140 , 14 4 potential, 145 , 148 , 19 8

envelope, 109 , 334 , 335 , 338 , 33 9 equality, 5 , 9 , 10 , 12 , 13 , 123 , 244 , 295 , 33 7

algebraic, 12 3 equation

Hamilton-Jacobi, 37 8 heat equation , 20 0 indicial equation , 33 2 Laplace's, 198-201 , 219, 238 , 37 9 of condition , 147 , 149 , 332 , 33 3 of motion , 144 , 148 , 149 , 151 , 209, 377 ,

378 symbolic, 21 7

equilibrium of a flui d mass , 14 3 of a lever , 14 2 of a syste m o f bodies , 138 , 143 , 147 , 14 8

Erdmann corne r conditions , 36 9 Eudoxan theor y o f proportions , 1 , 5-7 , 11 ,

12 exhaustion, 18 , 19 , 26 , 27 , 41 , 58 , 97 , 98 ,

101, 12 7 existence, 5 , 7 , 12 , 18 , 19 , 27 , 31 , 132 , 140 ,

155, 161 , 166 , 168 , 171 , 172 , 175 , 186 , 198, 202 , 204 , 206-210 , 221 , 223 , 226 , 229, 233 , 236 , 238-240 , 242 , 245 , 252 , 254, 293 , 294 , 296 , 299 , 301 , 303 , 307 , 309, 314 , 316-318 , 336 , 339-342 , 346-349, 351 , 356, 379 , 380 , 390 , 394 , 395, 398 , 399 , 401 , 40 2

expansion binomial, 112 , 115 , 12 2 power series , 107 , 114 , 115 , 128-130 , 226 ,

229, 244 , 346 , 34 7 expression

algebraic, 14 , 42 , 48 , 52 , 16 2 analytic, 20 , 107 , 114 , 116 , 121 , 122 , 124 ,

127, 128 , 156-158 , 161 , 162 , 164 , 172, 180 , 188 , 233 , 241 , 252 , 261 , 369, 38 7

symbolic, 119 , 217 , 223 , 22 7 extrema

strong extrem a i n opposit e t o wea k extrema (i n variationa l analysis) , 370

extremal, 356 , 371-376 , 37 8 broken, 35 5

extreme value , 22 , 41 , 49, 51-54 , 79 , 113 , 138-140, 207 , 235 , 236 , 253 , 271 , 342 , 344, 356 , 361 , 363 , 365 , 38 9

final cause , 13 9 fluent, 78 , 79 , 81-84 , 95 , 96 , 100 , 101 , 326,

330

fluxion, 66 , 74 , 78-80 , 82 , 84 , 85 , 92 , 93 , 95-97, 100 , 101 , 107 , 112 , 113 , 120 , 127, 128 , 326 , 33 0

higher-order, 84 , 8 5 force

accelerative, 138 , 143 , 15 0 central, 138 , 140-148 , 37 4 external, 14 6 internal, 14 6 live, 37 7

Fourier coefficient , 170 , 201 , 263 , 396 , 398 , 399

Fourier series , 11 , 155 , 157 , 158 , 164 , 178-181, 184 , 187 , 188 , 190 , 261 , 305, 306, 39 9

Fredholm alternative , 393 , 40 2 Fredholm resolvents , 39 5 function

P - , 241-24 3 abelian, 185 , 234 , 238 , 239 , 245 , 246 , 249 ,

250, 253 , 254 , 34 6 algebraic, 114 , 224 , 229-231 , 234 ,

236-239, 246 , 250 , 34 4 analytic, 113 , 123 , 124 , 128 , 185 , 210 ,

229, 232 , 235 , 249 , 250 , 252-255 , 295, 296 , 342 , 346 , 34 8

arbitrary, 120 , 123 , 125-130 , 203 , 261-263, 280 , 289 , 338 , 34 3

continuous, 27 , 124 , 156-159 , 161 , 165 , 166, 168-170 , 172 , 173 , 175-178 , 181, 182 , 184 , 186 , 187 , 207 , 217 , 218, 221 , 223, 230 , 235-237 , 240 , 254, 262-264 , 270 , 271 , 274 , 299 , 369, 389-392 , 395-397 , 39 9

cosine, 11 6 doubly periodic , 227 , 231 , 232 , 239 , 24 0 eigenfunction, 343 , 344 , 39 5 elliptic, 155 , 184 , 185 , 224 , 229 , 231 , 232 ,

238, 239 , 241 , 247 , 250 , 253 , 254 , 340, 36 4

entire, 251-25 3 excess function , 370 , 37 2 exponential function , 115 , 116 , 34 4 gamma-function, 174 , 241 , 25 1 Green's, 198 , 202-205 , 209 , 21 0 harmonic, 214 , 235 , 236 , 23 8 holomorphic, 221 , 225 , 226 , 231 , 253 , 25 5 imaginary, 215 , 218-222 , 22 8 implicit, 76 , 157 , 37 0 irrational, 8 0 line function , 388-39 0 linearly discontinuou s (Hankel) , 267 , 28 0 meromorphic, 239 , 240 , 244 , 251-25 3 modular, 247 , 25 3 monodromic, 227 , 229-23 1 monogenic, 229 , 231 , 253 , 25 4 multi-valued, 114 , 218 , 224 , 229 , 234 , 23 6

418 SUBJECT INDE X

nowhere differentiable , 176 , 187 , 189 , 246, 250 , 25 4

of a comple x variable , 206 , 213 , 221 , 229, 232, 233 , 237 , 244 , 253 , 34 7

pointwise discontinuou s (Hankel) , 26 9 polynomial, 25 1 potential function , 198 , 199 , 201 , 202,

204, 206 , 207 , 21 0 prime, 25 1 primitive, 78 , 17 1 rational, 114 , 217 , 218 , 222 , 241 , 248,

250, 251 , 253, 344 , 34 6 sine, 116 , 118 , 20 1 single-valued, 11 4 special, 115 , 17 2 transcendental, 76 , 91 , 101 , 113-118 ,

130, 218 , 238-240 , 250 , 25 1 trigonometric, 116 , 125 , 238 , 34 4

functional, 191 , 385, 387 , 390 , 391 , 398, 400-404

functional analysis , 344 , 385-391 , 394 , 396-399, 402-40 4

functional derivative , 38 9 functional equation , 161 , 172 , 21 8 fundamental formul a (Hilbert) , 39 5

general analysi s (Moore) , 385 , 397 , 39 8 generality o f algebra , 131 , 161 , 16 4 generalization, 47 , 49 , 122 , 127 , 131 , 141,

142, 144 , 181 , 207, 334 , 390 , 393 , 394 , 397, 39 9

genus o f a surface , 239 , 24 0 geometrical construction , 1 , 11 , 14 , 15 , 27 ,

42, 48 , 111 , 12 3 gravitation, 197 , 238 , 33 6 Gymnasium, 185 , 247-24 9

Hamilton-Jacobi theory , 340 , 378 , 37 9 harmonic oscillators , 11 6 heat-conduction, heat , 197 , 199-201 , 209 ,

238, 244 , 261 , 262, 34 3 Hilbert's problems , 242 , 31 1 Hudde's rule , 5 5 hyperbola, 32 , 48 , 58 , 59 , 111 , 11 5

incommensurable, 4-7 , 9 , 10 , 12-14 , 2 3 indivisible, 25 , 34 , 35 , 56 , 58-60 , 6 2 infimum, 175 , 265 , 269 , 28 5 infinite

finite/infinite, 32 , 386 , 389 , 391 , 393, 39 8 infinite determinants , 392 , 39 3 infinite product , 113 , 115 , 119 , 130 , 25 1 infinite syste m o f linea r equations , 386 ,

392, 39 5 infinitely large , 115 , 117 , 26 4

inflection point , 53 , 108 , 11 3 initial condition , 219 , 330 , 341 , 349 integrability, 187 , 190 , 263-266 , 268-271 ,

273, 274 , 277-28 0

Lebesgue, 274 , 288 , 399 , 40 1 Riemann, 271 , 273-275, 277 , 28 8

integral / sign , 58 , 89 , 100 , 111 , 17 0 Cauchy, 170 , 181 , 263 concept o f integratio n o f Archimedes , 26 ,

27 definite, 83 , 84 , 160 , 165 , 170 , 171 , 214,

215, 227 , 263 , 275, 280 , 288 , 33 9 improper, 60 , 116 , 214 , 21 5 invariant, 374-37 6 Jordan, 276 , 277 , 28 8 Lebesgue, 261 , 271, 284, 287 , 288 , 397 ,

399 singular, 215 , 223 , 332, 33 5

integral calculus , 329 , 33 0 integral equation , 385 , 386 , 390-395 , 397 ,

398, 400 , 401 , 403 integral formul a

Cauchy, 219 , 223 , 225, 227 , 230 , 244 , 34 2 integrating factor , 332-335 , 337 , 338 , 34 0 integration

approximate, 34 0 complex, 215 , 219 , 223 , 23 5 in finite terms , 336 , 343 , 34 5 multidimensional, 274-27 6 partial, 88 , 90 , 91 , 112 , 204 , 220 , 36 2

interpolation, 6 7 Gregory-Newton interpolatio n formula ,

85 intuition, 127 , 132 , 156 , 175 , 188 , 190 , 238 ,

269, 294 , 295 , 302 , 303 , 308 , 310 , 312 , 320, 386 , 40 0

intuitionism, 32 0 inverse tangen t problems , 32 7 isochrone, 327 , 33 0 isoperimetrical problem , 114 , 33 1

Jacobi identity , 34 5 Jacobi inversio n problem , 239 , 240 , 24 7

Kepler equation , 12 0 Kepler's fruit , 6 1

Lagrange remainder , 13 0 law o f refraction , 13 9 Leibniz's school , 105 , 108 , 114 , 32 7 length o f a n arc , 28 , 73 , 80 , 32 9 lens, 41 , 44 libration o f th e moon , 147 , 14 9 lignes d'arret , 23 0 limit, 11 , 17-19 , 31 , 32, 52 , 65 , 66 , 74 ,

82-84, 92-94 , 96-99 , 107 , 117 , 121 , 125, 128 , 130 , 131 , 159 , 160 , 162-164 , 166, 167 , 169 , 172 , 174 , 175 , 178 , 180 , 183, 186 , 190 , 217 , 218 , 221 , 249, 262 , 264-266, 269-277 , 280 , 287 , 288 , 291, 296, 301 , 306, 307 , 310 , 314 , 330 , 341,

SUBJECT INDE X 419

342, 349 , 356 , 366 , 367 , 380 , 389-391 , 393, 39 8

linear algebra , 386 , 39 5 linearity, 386 , 387 , 40 3 Lipschitz condition , 346 , 34 7 locally compact , 40 2 locally convex , 40 3 logarithm, 76 , 111 , 114-118 , 213 , 29 4 logarithmic indicator , 23 1 Lunes o f Hippocrates , 16 , 1 7

magnetism, 177 , 199 , 202 , 206 , 209 , 23 8 manifold, 240 , 247 , 280 , 283 , 291 , 301, 302,

304, 306 , 308 , 311 , 365 measurability

Borel, 285 , 28 6 Jordan, 285 , 28 6 Lebesgue, 285-28 7

measure inner, 285 , 28 6 Lebesgue, 28 5 outer, 278 , 28 5

measure theory , 261 , 267-269, 275 , 277 , 278, 281 , 284

measurement o f a circle , 20 , 21 , 28 method

arithmetical, 6 9 axiomatic, 313 , 318 , 320 , 39 4 geometrical, 69 , 128 , 24 6 kinematic, 5 5 mechanical (Archimedes) , 3 , 22 , 23 , 25,

35 of majorants , 225 , 340 , 347-34 9 of successiv e approximation , 349 , 39 2 of undetermine d coefficients , 146-149 ,

239, 33 0 philosophical, 4 2 separation o f variables , 199 , 20 0 series expansion , 114 , 116 , 122 , 199 , 20 1 variation o f parameters , 33 6

minimal surface , 24 3 motion

of a syste m o f bodies , 137 , 141 , 142, 145-147, 14 9

virtual, 14 8 motion o f th e moon , 39 2 multiplier

function, 36 8 Lagrange multiplier , 149 , 15 1 rule, 368 , 37 5

neighbourhood, 44 , 52 , 159 , 166 , 168 , 179 , 182, 183 , 246 , 248 , 249 , 342 , 35 7

new concepts , 147 , 155 , 161 , 171 , 188 , 228 , 234, 238 , 261 , 271, 297, 302 , 308 , 30 9

Newton's polygon , 23 0 Newtonian principle s o f mechanics , 13 8 nonconstructive, 295 , 319 , 349 , 39 4 norm, 264 , 339 , 400-40 2

normal, 41 , 44, 46-52 , 54 , 55 , 131 , 204, 23 5 notation (i n Newton' s an d Leibniz' s

calculi), 78-80 , 84 , 89-91 , 100 , 10 1 notion o f function , 107 , 108 , 113 , 123 , 124 ,

127, 132 , 155-158 , 162 , 188 , 218 , 228 , 245, 255 , 261 , 262, 266 , 26 7

number Bernoulli numbers , 11 9 complex, 116 , 117 , 122 , 156 , 213 , 216 ,

221, 227 , 228 , 244 , 254 , 29 3 imaginary, 214 , 21 7 integer, 13 , 118 , 25 4 irrational, 9 , 263 , 294 , 32 0 natural, 4-11 , 35, 117 , 129 , 156 , 175 , 185 ,

218, 235 , 244 , 279 , 292 , 294 , 317 , 31 9 ratio o f numbers , 5 , 6 , 9 , 1 2 rational, 9 , 11 , 12 , 122 , 176 , 218 , 219 ,

254, 292-294 , 312 , 31 4 real, 7 , 8 , 11-13 , 18 , 155 , 156 , 164 , 168 ,

172, 176 , 185 , 221 , 228, 254 , 265 , 270, 292-295 , 311 , 313, 314 , 316 , 318-320, 390 , 39 8

transcendental, 15 6

omnes lineae , 5 8 ontology, ontological , 33 , 107 , 213 , 217 ,

314, 316-318 , 32 1 operator

bounded, 40 4 differential, 189 , 35 1 integral, 395 , 40 0 linear, 394 , 403 , 40 4 symmetrical, 39 7 unbounded, 397 , 40 4

operator theory , 189 , 385 , 393 , 400 , 40 1 optics, 44 , 105 , 16 0 order o f connectivity , 23 4 orthogonality, 344 , 393 , 395 , 399 , 40 2

Pappus's problem , 42 , 43 , 4 7 parabola, 2 , 21 , 25-27, 32 , 48 , 66 , 67 , 109 ,

141 paradox

Burali-Forti, 312 , 31 3 Richard, 312 , 31 3 Russell, 312 , 313 , 31 8

partition, 160 , 170 , 171 , 262-266, 270 , 274-277, 286 , 287 , 34 1

periodicity, 125 , 224 , 24 2 periodicity moduli , 22 4 physics, 4 , 17 , 21 , 22, 35 , 41 , 60, 105 , 106 ,

124, 125 , 127 , 155 , 160 , 185 , 190 , 202 , 205, 211 , 233, 234 , 239 , 250 , 281 , 296, 325, 336 , 373 , 377 , 389 , 394 , 403 , 40 4

pi, 15 , 20 , 22 , 3 1 polar coordinates , 94 , 36 1 pole, 50 , 222 , 225 , 229-231 , 235-237 , 239 ,

244, 246 , 250-252 , 265 , 34 6

420 SUBJECT INDE X

polynomial, 41 , 44, 46 , 49 , 108 , 114 , 119 , 129, 130 , 175 , 217 , 227 , 230 , 253 , 346 , 349, 351 , 39 0

potential theory , 155 , 189 , 208-210 , 223 , 232, 233 , 336 , 337 , 379 , 386 , 39 2

prime an d ultimat e ratios , 82-84 , 93 , 9 5 principal valu e (o f a n integral) , 172 , 22 2 principal-axes transformation , 39 5 principle

Dirichlet, 186 , 189 , 190 , 209 , 210 , 233 , 238, 245 , 246 , 250 , 379 , 38 0

Liouville, 227 , 23 2 of economy , 13 9 of leas t action , 138 , 142 , 144 , 145 , 147 ,

367 of maximini s o r minimi s (i n mechanics) ,

145 of virtua l velocities , 138 , 147-149 , 15 1

principle o f continuity , 11 9 probability, 106 , 121 , 281, 282, 284 , 389 ,

400

quadrature of th e circle , 14 , 15 , 6 0 of th e cissoid , 41 , 66, 68 , 6 9 of th e highe r parabolas , 6 6 of th e parabola , 2 , 21 , 25, 27 , 6 6 solvable by , 33 1

quantity complex, 29 6 higher-order infinitesimal , 89 , 22 0 infinitesimal small , 12 , 18 , 35 , 77 , 79 , 80 ,

82, 84 , 86 , 88 , 89 , 92 , 95-98 , 107 , 108, 112 , 113 , 115 , 118 , 127 , 131, 142, 143 , 158 , 159 , 163 , 164 , 166 , 168-170, 175 , 186 , 191 , 222, 325 , 357-360

real, 227 , 291 , 296 rule o f cancellatio n o f higher-orde r

infinitesimal, 80 , 90 , 9 8 variable, 1 , 114 , 157 , 158 , 162 , 16 4

quantum mechanics , 40 4 quaternion, 202 , 208 , 29 3

reciprocal subtraction , 5 , 6 , 12-1 4 recursive, 120 , 294 , 34 9 reductio a d absurdu m (proo f b y doubl e

contradiction), 19 , 27 , 28 , 66 , 127 , 216 , 217

residue, 179 , 219 , 222 , 224 , 227 , 228 , 231, 232, 24 4

rest (th e la w of) , 138 , 139 , 141 , 143-14 5 Riemann maximu m principle , 23 5 Riemann surface , 230 , 236 , 237 , 240 , 245 ,

247 Riemann's conjecture , 243 , 24 4 rigour, 23 , 61 , 155 , 156 , 158 , 160 , 164 ,

176-178, 188-190 , 201 , 207, 217 , 246 , 247, 250 , 255 , 263 , 26 9

scalar product , 386 , 39 9 science o f quantity , 107 , 291 , 293-295, 303 ,

304, 308 , 310-312 , 32 1 separability, 331 , 386 sequence

Cauchy (o r fundamental) , 167 , 174 , 175 , 184, 299 , 300 , 305 , 31 4

of functions , 182 , 248 , 252 , 273-275 , 288 , 291

of natura l numbers , 24 4 of powers , 11 6 of prim e numbers , 24 4 of quantities , 15 9 of rationals , 299 , 300 , 30 5

series, 10 1 arcsin, 81 , 82 sin, 8 2 alternating, 12 1 binomial, 74-76 , 8 1 cosine, 17 9 geometric, 26 , 121 , 122 , 168 , 176 , 179 ,

218, 225 , 34 2 hypergeometric, 174 , 241-24 3 imaginary, 218 , 21 9 infinite, 76 , 86 , 91 , 100 , 174 , 176 , 177 ,

184, 218 , 222 , 239 , 253 , 343, 344, 39 3 lacunary, 246 , 25 4 Lagrange's, 12 0 Laurent, 22 6 log, 76 , 21 9 Neumann's, 386 , 39 2 numerical, 121 , 13 1 of reciproca l squar e numbers , 11 8 power, 75 , 76 , 82 , 84 , 85 , 101 , 107 , 108 ,

112-115, 119-121 , 124 , 128-131 , 150, 165 , 169 , 173 , 185 , 218-220 , 225, 226 , 229 , 230 , 239 , 246 , 248-252, 255 , 346-34 9

reversion of , 12 0 Taylor, 84 , 112 , 113 , 129 , 130 , 169 , 187 ,

225, 248 , 33 0 trigonometric, 11 , 32, 102 , 125 , 131 , 174 ,

178, 186 , 187 , 218 , 239 , 262 , 263 , 280, 304-30 6

set set theory , 31 7 Borel, 282 , 283 , 28 6 Cantor, 279 , 31 1 cardinality, 309 , 31 1 connected, 22 1 dense, 176 , 181 , 187 , 266 , 268 , 271 , 282,

311 infinite, 35 , 185 , 295-297 , 299 , 304 , 305 ,

307, 310-313 , 31 7 nowhere dense , 190 , 269 , 278-280 , 40 3 of conten t zero , 278 , 280 , 28 1 of kin d n , 278 , 28 0

SUBJECT INDE X 421

of measur e zero , 268 , 278 , 280 , 283 , 286 , 288

perfect, 279 , 310 , 31 1 set theory , 252 , 267 , 275 , 277 , 292 , 304 ,

306, 310-313 , 316-318 , 320 , 321, 385, 386 , 390 , 391 , 397, 400 , 40 1

well-defined, 309 , 31 6 well-ordered, 309 , 31 3

single-valued, 114 , 157 , 217 , 218 , 221 , 227, 228, 230 , 231 , 239, 240 , 249-25 4

singularity, 130 , 187 , 188 , 190 , 199 , 204 , 210, 230 , 231 , 233, 235 , 239 , 244 , 250 , 251, 267 , 282 , 335 , 346 , 351 , 36 9

essential singularity , 246 , 250-252 , 34 6 inessential singularity , 25 1

solid o f revolution , 62 , 68 , 21 3 space

Banach, 40 2 Euclidean, 247 , 38 6 function, 351 , 386, 389 , 390 , 393 , 395 ,

399, 401 , 404 Hilbert, 385 , 394 , 397 , 399 , 401 , 404 metric, 390 , 391 , 402, 40 3 normed, 385 , 401 , 402 of continuou s functions , 39 6 of squar e summabl e numbe r series , 39 6 topological, 398 , 40 4 vector, 385 , 394 , 402 , 40 3

spectral theory , 387 , 397 , 40 4 stability, 35 1 Sturm-Liouville theory , 172 , 339 , 34 3 sum

Darboux, 27 , 27 0 lower, 27 , 27 7 upper, 27 , 27 7

superposition, 12 5 supremum, 175 , 176 , 186 , 265 , 269 , 31 9 system o f bodies , 137 , 138 , 142-151 , 19 8

tangent, 1 , 8 , 22 , 32 , 35 , 41 , 49-52, 55-57 , 60-62, 69-71 , 73 , 75 , 78 , 79 , 83 , 85-89 , 108, 110 , 113 , 115 , 327 , 328 , 330 , 334 , 338, 341 , 351, 35 7

technical hig h school , 15 5 theorem

Abel's theorem , 23 9 accumulation-point theorem , 39 0 approximation theorem , 188 , 340 , 390 ,

391 binomial, 172 , 173 , 177 , 178 , 216 , 218 ,

219 Bolzano-Weierstrass theorem , 188 , 255 ,

296 Casorati-Weierstrass-Sokhotskii theorem ,

252 Cauchy-Hadamard theorem , 21 8 existence theore m (Cauchy) , 17 2

existence theore m (Cauchy , Lipschitz) , 340, 34 2

factorization theorem , 25 2 fixed-point theore m (Banach) , 402 , 40 3 fixed-point theore m (Schauder) , 40 3 fundamental theore m (o f calculus) , 69 ,

73, 76 , 78 , 79 , 90 , 165 , 171 , 208, 27 1 fundamental theore m o f algebra , 172 ,

174, 205 , 213 , 215-217 , 23 1 Heine-Borel theorem , 186 , 18 8 Hilbert-Schmidt theorem , 395 , 397 , 39 8 integral theore m (Cauchy) , 219-224 ,

226-229, 244 , 24 8 integral theorem s (Gauss , Green ,

Stokes), 198 , 202 , 206 , 20 7 intermediate valu e theorem , 168 , 170 ,

174-176 Laurent theorem , 24 8 polynomial, 13 0 preparation theore m (Weierstrass) , 25 3 representation theore m (Frechet , Riesz) ,

401 representation theore m (Weierstrass) ,

252 Riemann mappin g theorem , 237 , 35 0 Riemann-Roch theorem , 24 0 Taylor's theorem , 85 , 101 , 111 , 112 , 17 7

theory o f content , 261 , 269, 275 , 280 , 28 3 theory o f integration , 187 , 214 , 216 , 232 ,

261, 269 , 274 , 275 , 280 , 286 , 312 , 40 1 theory o f types , 31 9 topological degree , 40 2 torus, 6 1 transmutation theore m (Leibniz's) , 87 , 90 ,

91 triangle

characteristic, 69 , 86-89 , 10 1 harmonic, 8 6

trisection o f a n angle , 30 , 4 2

uniformization, 231 , 235 uniqueness, 122 , 157-160 , 166 , 173 , 174 ,

178, 186 , 206 , 207 , 217 , 223 , 226 , 229 , 233, 236 , 264 , 270 , 271 , 275, 298 , 305 , 306, 309 , 314 , 315 , 343 , 346 , 350 , 373-375, 378 , 40 0

university, 46 , 106 , 185 . 202 , 205 , 232 , 247 . 249, 267 , 291 , 300, 304 , 347 , 348 , 364 , 365, 370 , 39 4

variable-endpoint formula , 369 , 370 , 372 , 373

variation calculus o f variations , 102 , 106 , 114 , 146 ,

220, 254 , 340 , 355-359 , 361 , 364, 365, 368 , 370 , 371 , 373, 374 , 376-379, 386-388 , 39 0

first, 363 , 36 5 second, 355 , 363-368 , 370 , 371 , 37 8

422 SUBJECT INDE X

variational equation , 146 , 361 , 368, 37 8 variational integral , 360 , 363-365 , 367 ,

372-374, 37 6 variational operator , 36 2 variational principl e (i n mechanics) , 137 ,

149, 233 , 24 0 velocity, 41 , 55, 128 , 32 8

virtual, 137 , 138 , 147-149 , 15 1 vibrating string , 5 , 6 , 123-126 , 261 , 331,

337, 34 3 vis viva , 138 , 140 , 144-146 , 19 8 volume, 1 , 14 , 18 , 21-25 , 27 , 41 , 47, 54 , 56 ,

58, 60-63 , 69 , 206 , 35 6 volume o f revolution , 60 , 61 , 63, 35 7

winding number , 23 1

zero, 49 , 118 , 119 , 231 , 232, 243 , 244 , 251, 252, 271 , 344

double, 4 9