Landmark Writings in Western Mathematics 1640-1940 (eBook)

front cover 1
copyright 6
Table of Contents 7
body 11
0 Introduction 11
Waves in the sea 11
Organisation of the articles 11
Some principal limitations 12
Acknowledgements 18
Bibliography 0 18
1. René Descartes, GÉOMÉTRIE, Latin edition (1649), French edition (1637) 21
Youth, from La Flèche to the REGULAE 22
Descartes in Holland and Stockholm 23
The GÉOMÉTRIE 24
The `construction' of the expressions 24
Compasses, ruler-and-slide, criterion for `continuous motions' 27
The problem of Pappus and an algebraic criterion 28
Tangents 32
Ovals 33
Algebraic equations 33
The `construction' of equations 34
From the GÉOMÉTRIE to the ENUMERATIO 37
`PROLES SINE MATRE CREATA' 38
Bibliography 41
2. John Wallis, Arithmetica infinitorum (1656) 43
Background to the ARITHMETICA INFINITORUM 43
Methods and results in the ARITHMETICA INFINITORUM 45
The motivation to `Wallis's product' 47
Reactions to the ARITHMETICA INFINITORUM 49
Bibliography 2 52
3. Christiaan Huygens, book on the pendulum clock (1673) 53
The three strands of Huygens's research 54
Pendulum clocks 55
The five parts of HOROLOGIUM OSCILLATORIUM 56
On Huygens's mathematical style 61
Unfocused reception 62
Bibliography 3 65
4. Gottfried Wilhelm Leibniz, first three papers on the calculus (1684, 1686, 1693) 66
Leibniz's research on the infinitesimal mathematics 67
The enigmatic first publication 69
The early reception of the differential calculus 72
The first papers on integral calculus 74
The spread of the Leibnizian calculus 75
Bibliography 4 77
5. Isaac Newton, philosophiae naturalis principia mathematica, first edition (1687) 79
Newton's mathematical methods 80
Early studies on the motion of bodies and on planetary motion 83
The PRINCIPIA (1687): definitions and laws 86
The PRINCIPIA (1687): limits 87
The PRINCIPIA (1687): the area law 92
The PRINCIPIA (1687): central forces 93
The PRINCIPIA (1687): Pappus's problem 95
The PRINCIPIA (1687): algebraic non-integrability of ovals 96
The PRINCIPIA (1687): the general inverse problem of central forces 97
The PRINCIPIA (1687): universal gravitation 98
Revisions (1690s), second (1713) and third (1726) editions 98
The impact of the PRINCIPIA 103
Bibliography 5 105
6. Jakob Bernoulli, Ars conjectandi (1713) 108
Background and story of publication 109
Content and structure of the AC 110
Huygens's De rationciniis in ludo aleae with Bernoulli's annotations 112
Combinatorics as the main tool of the art of conjecturing 116
New problems and exercises in the third Part 119
The transition to a calculus of probabilities in the fourth Part 119
The impact of the Ars conjectandi 123
Bibliography 6 123
7. Abraham de Moivre, The doctrine of chances (1718, 1738, 1756) 125
Background and story of the publication 125
The introduction of the Doctrine of chances, and the mathematical requirements for the reader as announced in the preface 127
Generating functions 130
A `new sort of algebra' in the Doctrine of chances 130
The duration of play 132
The approximation of the binomial by the normal distribution 134
Annuities on lives 137
Impact of the DoC 139
Bibliography 7 140
8. George Berkeley, The analyst (1734) 141
Berkeley's life and works 142
The purpose of The analyst 143
The principal arguments 144
Responses to Berkeley 149
Bibliography 8 150
9. Daniel Bernoulli, Hydrodynamica (1738) 151
Daniel Bernoulli's life and work 152
General remarks 152
The Bernoulli equation in the Hydrodynamica 154
Propulsion of ships by jet ejection, and the dynamics of systems with variable mass 156
Some other topics 157
Some general remarks on the style of the Hydrodynamica 159
Early praise and troubles 160
Bibliography 9 161
10. Colin MacLaurin, A treatise of fluxionsA treatise of fluxions (1742) 163
Colin MacLaurin (1698-1746) 163
The TREATISE ON FLUXIONS (1742): foundations 164
Extrema and inflections 171
Limits of series, and the Euler-MacLaurin theorem 173
The method of infinitesimals, and Newton's prime and ultimate ratios 174
Book II 175
Summary remarks 10 176
Impact 177
Bibliography 10 177
11 Jean Le Rond D'Alembert, Traité de dynamique (1743, 1758) 179
Biography of D'Alembert 180
Scientific works 180
Contents of the TRAITÉ DE DYNAMIQUE 181
On the second edition 184
The place of the TREATISE in the work of the author 184
Posterity of the TREATISE 185
Bibliography 11 187
12. Leonhard Euler, book on the calculus of variations (1744) 188
Introduction 12 188
Origins and basic results 189
Foundations of analysis 194
Later developments: Lagrange, Euler and the calculus of variations 197
Bibliography 12 199
13. Leonhard Euler, `Introduction' to analysis (1748) 201
Introduction to the INTRODUCTIO 202
Book I, on analysis 202
Book II on plane and surface geometry 208
On the impact of the INTRODUCTIO 209
Bibliography 13 210
14. Leonhard Euler, treatise on the differential calculus (1755) 211
The second part of Euler's trilogy on mathematical analysis 211
The differential calculus and its foundations 212
Applications of the differential calculus 215
General remarks 14 216
Bibliography 14 217
15. Thomas Bayes, An essay towards solving a problem in the doctrine of chances (1764) 219
Biography 15 220
Bayes's work on chances 221
Price's appendix 223
A posthumous publication 225
Impact and influence of the work 225
Bibliography 15 226
16. Joseph Louis Lagrange, Méchanique analitiqueMéchanique Analitique, Lagrange's|(, first edition (1788) 228
Outline of Lagrange's scientific biography 229
Lagrange's conception of `analytic mechanics' as a science 230
Lagrange's fundamental developments up to the first edition of the Méchanique analitique 231
Contents of the editions 236
Fundamental differences between the Méchanique analitique and the Mécanique analytique and later reviews 238
General assessment and reception of the work 240
Bibliography 16 241
17. Gaspard Monge, Géométrie descriptive, first edition (1795) 245
Introduction 17 246
Gaspard Monge 247
The subject matter of Monge's lectures 248
The principal aims of Monge's course 255
The influence of Monge's lectures 258
Conclusion 17 260
Bibliography 17 260
18. P.S. Laplace, Exposition du système du monde, first edition (1796) Traité de mécanique céleste (1799-1823/1827)
Background 18 263
The Exposition 18 264
The `First Part' of Mécanique céleste, 1799 266
Mécanique céleste, the celestial volume 1 266
Mécanique céleste, the planetary volume 2 270
Mécanique céleste, the numerical Books 6-9 271
Mécanique céleste, the miscellaneous Book 10 272
Immediate influence 273
Mécanique céleste, the miscellaneous volume 5 274
Bibliography 18 276
19. Joseph Louis Lagrange, Théorie des fonctions analytiques, first edition (1797) 278
Introduction 19 278
Algebraic analysis and the function concept 281
Theorems of analysis 283
Methods of approximation 285
Multiplier rule 289
Calculus of variations: sufficiency results 293
Conclusion 19 294
Bibliography 19 295
20. S.F. Lacroix, Traité du calcul différentiel et du calcul intégral, first edition (1797-1800) 297
Mathematical teacher and writer 297
Purposes of the Traité 299
Differential calculus 300
Integral calculus 304
Differences and series 307
The Traité élémentaire and the second edition of the Traité 310
Impact 20 310
Acknowledgement 20 311
Bibliography 20 311
21. Jean-Etienne Montucla, Histoire des mathématiques, second edition (1799-1802) 312
Biography 21 312
Contents of the book 314
Volume 1 (1758 and 1799) 315
Volume 2 (1758 and 1799) 316
Volume III (1802) 317
Volume 4 (1802) 320
Reception of the work 21 321
Bibliography 21 322
22. Carl Friedrich Gauss, Disquisitiones arithmeticae (1801) 323
Introduction 22 324
What is the Disquisitiones arithmeticae about? 324
Content and reception of Sections 1-4 327
Content and reception of Section 5 330
Content and reception of Section 7 332
Fame and reactions 22 333
Bibliography 22 334
23. Carl Friedrich Gauss, book on celestial mechanics (1809) 336
Biographical sketch 336
Inception of Gauss's work in orbit determination 337
Gauss's early work on orbit-determination 338
The determination of orbits in TM 341
Themes and topics of Book I 344
Sections 3 and 4 of Book II: the method of least squares perturbations
The impact of TM 347
Bibliography 23 348
24. P.S. Laplace, Théorie analytique des probabilités, first edition (1812) Essai philosophique sur les probabilités, first edition (1814)
The Mont Blanc of mathematical analysis, and its foothills 350
Laplacian probability 351
Publication of the THÉORIE ANALYTIQUE 353
Laplace's THÉORIE and mathematical statistics 353
The ESSAI PHILOSOPHIQUE 357
The legacy 358
Bibliography 24 359
25. A.-L. Cauchy, Cours d'analyse (1821) and Résumé of the calculus (1823) 361
From student to professor 362
The COURS D'ANALYSE: `algebraic analysis' and the theory of limits 363
The COURS D'ANALYSE: continuous functions and infinite series 364
The RÉSUMÉ: a new version of the calculus 366
Reactions at the ECOLE POLYTECHNIQUE: Cauchy's later books 369
The gradual influence of Cauchy's doctrine 370
Bibliography 25 372
26. Joseph Fourier, Théorie analytique de la chaleur (1822) 374
Education and employments 374
The chronology of Fourier's researches 375
Heat diffusion, internal and surface 376
Fourier series and their function 379
Calculating and interpreting the coefficients 380
Non-harmonic series, and the Bessel function 381
The Laplace and Fourier integrals 382
Later work and recognition 382
On the later impact 383
Bibliography 26 384
27. Jean Victor Poncelet, Traité des propriétés projectives des figures, first edition (1822) 386
Poncelet's Traité 386
Jean Victor Poncelet and the other French geometers 392
Michel Chasles 395
Bibliography 27 396
28. A.-L. Cauchy, two memoirs on complex-variable function theory (1825, 1827) 397
Background to the 1814 memoir 398
`Cauchy's theorem' foreshadowed 398
An argument foreshadowing the notion of principal value 401
The first hints of the residue theorem 402
The notes added before publication 405
The background of the 1825 memoir 405
Definite integrals between complex limits 406
The main theorem of the 1825 memoir 407
Taking singularities into account 407
The use of geometrical language 408
Later developments 409
Bibliography 28 410
29. Niels Henrik Abel, paper on the irresolvability of the quintic equation (1826) 411
Introduction: equations in general 411
The problem of the quintic, 1700-1800 416
Niels Henrik Abel 419
Contents of the memoir 420
Reception of the memoir 420
Bibliography 29 421
30. George Green, An essay on the mathematical analysis of electricity and magnetism (1828) 423
Green's little-known entrée 423
Poisson's `simplifying' theorem 424
Green's theorem in Green's book 425
Green's applications to electricity and magnetism 427
Green's later researches 428
The discovery of the book 429
Bibliography 30 431
31. C.G.J. Jacobi, book on elliptic functions (1829) 432
Elliptic integrals 432
Elliptic integrals from Fagnano to Legendre 436
Genesis of the Fundamenta 439
The author 444
Contents of the work 445
Significance of the work 448
Bibliography 31 450
32. Hermann G. Grassmann, Ausdehnungslehre, first edition (1844) 451
A family of mathematicians 452
Dialectics and the theory of tides 452
The new branch of mathematics 454
A muted reception 457
Eventual recognition 458
Bibliography 32 459
33. Karl Georg Christian von Staudt, book on projective geometry (1847) 461
Background and biography 461
Von Staudt's `Geometry of position' 462
On von Staudt's Beiträge zur Geometrie der Lage 465
Impact 33 465
Bibliography 33 466
34. Bernhard Riemann, thesis on the theory of functionscomplex function theory|( of a complex variablefunction of a complex variable|( (1851) 468
The theory of functions of a complex variable before Riemann 469
Biography of Riemann 471
The thesis 472
Riemann's publications from 1857 474
The reaction of Weierstrass 475
The positive reception of Riemann's thesis 475
A complex of theories 476
Concluding remark 34 477
Bibliography 34 477
35. William Rowan HamiltonHamilton|(, Lectures on quaternionsLectures|( (1853) 480
From prodigy to sage 480
The origin of quaternions 481
The Lectures 482
Reception and subsequent development 486
Quaternions versus vectors: J.W. Gibbs and E.B. Wilson 487
Bibliography 35 489
36. George Boole, An investigation of the laws of thought on which are founded the mathematical theory of logic and probabilities (1854) 490
A self-made mathematician 490
Boole's initial `analysis' of logic 492
Boole's mature `Investigation' of logic 492
The algebraic methods of deduction and elimination 494
Boole's treatment of probability theory 496
The religious connotation of Boole's logic 496
Boole's gradual influence 497
Bibliography 36 498
37. Johann Peter Gustav Lejeune-Dirichlet, Vorlesungen über Zahlentheorie, first edition (1863) 500
A posthumous textbook 501
The simplification of Gauss's Disquisitiones arithmeticae 502
Analysis and arithmetic 506
Dedekind's supplements X and XI: towards the theory of ideals 507
The influence of the Vorlesungen 508
Bibliography 37 510
38. Bernhard Riemann, posthumous Thesis on the representation of functions by trigonometric series (1867) 511
Background 38 511
Riemann's historical analysis of the integral and trigonometric series 512
Riemann on the integral 517
The problem of the uniqueness of the representation 519
The final article: examples illustrating the diversity and complexity of trigonometric series 521
Bibliography 38 524
39. Bernhard Riemann, posthumous thesis `On the hypotheses which lie at the foundation of geometry' (1867) 526
Bernhard Riemann (1826-1866) 527
The lecture 527
The intellectual context 531
Bolyai and Lobachevsky and the discovery of non-Euclidean geometry 535
Beltrami, Poincaré, and Klein on non-Euclidean geometry 537
The later reception of non-Euclidean geometry 538
Bibliography 39 539
40. William Thomson and Peter Guthrie Tait, Treatise on natural philosophy, first edition (1867) 541
The place of T& T' in Thomson's work
Collaboration with Tait 544
Kinematics 545
Dynamics of energy 548
Extremum principles 550
Abstract dynamics 551
Reception 40 552
Bibliography 40 553
41. Stanley Jevons, The theory of political economy, first edition (1871) 554
A new theory of value 555
The law of exchange and the trading bodies 558
Concluding remarks 561
Bibliography 41 562
42. Felix Klein's Erlangen Program, `Comparative considerations of recent geometrical researches' (1872) 564
On the biography of Klein 565
The Erlangen Program 566
The reception of the Erlangen Program 570
Bibliography 42 572
43. Richard Dedekind, Stetigkeit und irrationale Zahlen (1872) 573
Introduction: the problem of incommensurables 573
The author 575
Dedekind's view of the problem of continuity 576
Dedekind's solution of the problem 577
Reception of the work 580
Bibliography 43 583
44. James Clerk Maxwell, A treatise on electricity and magnetism, first edition (1873) 584
Education and career 585
Mathematical theories of electricity and magnetism in the first half of the 19th century 586
Faraday and Thomson on the notion of field 587
Maxwell and the theoretical reform of electromagnetism 589
The publication, functions and structure of the Treatise 590
Mathematical structures in the Treatise 591
Electrostatics and electrokinetics 595
Magnetism and electromagnetism 596
The dynamical theory of electrokinetic phenomena, and the general equations of the electromagnetic field 597
The electromagnetic theory of light 600
The Maxwellians 603
The experiments of Hertz and their impact 604
Larmor and the notion of electron 605
Bibliography 44 606
45. J.W. Strutt, Third Baron Rayleigh, The theory of sound, first edition (1877-1878) 608
Rayleigh's early research on sound 608
The publication of The theory of sound 609
The book as compared with its predecessors 610
On Rayleigh's mathematical methods in the book 612
Rayleigh on waves and vibrations 613
Presenting original researches 614
The influence of the book on acoustics and elsewhere 616
Bibliography 45 618
46. Georg Cantor, paper on the `Foundations of a general set theory' (1883) 620
Cantor's way in 621
Early work on trigonometric series: derived sets 621
Cantor's theory of real numbers 622
The descriptive theory of point sets 623
The GRUNDLAGEN: a general theory of sets and transfinite ordinal numbers 624
The Continuum Hypothesis 627
Cantor's nervous breakdowns 627
Transfinite cardinal numbers: the alephs () 628
Cantor and the DEUTSCHE MATHEMATIKER-VEREINIGUNG 629
Transfinite mathematics and Cantor's manic depression 629
Consequences of the GRUNDLAGEN for later mathematics 631
Bibliography 46 631
47. Richard Dedekind (1888) and Giuseppe Peano (1889), booklets on the foundations of arithmetic 633
Dedekind: biography and background 634
Peano: biography and background 636
Dedekind's theory 639
Peano's theory 641
Appraisal and impact 643
Bibliography 47 645
48. Henri Poincaré, memoir on the three-body problem (1890) 647
Introduction 48 647
Origin and significance of the three-body problem 648
Poincaré's work before TBP 649
The publication of TBP 650
The content of TBP 651
Poincaré on celestial mechanics after TBP 655
The reception of TBP 655
The resolution of the three-body problem and some later developments 656
Bibliography 48 658
49. Oliver Heaviside, Electrical papers (1892) 659
General outline of the ELECTRICAL PAPERS 659
The algebra of vectors 662
Heaviside's operational calculus 667
On Heaviside's later work 671
Bibliography 49 672
50. Walter William Rouse Ball, Mathematical recreations and problems of past and present times, first edition (1892) 673
Historical background 674
The late 19th century 676
Walter William Rouse Ball (1850-1925) 678
The publication of Ball's book 679
Examples of new material in the book 679
Concluding remark 50 682
Bibliography 50 682
51. Alexandr Mikhailovich Lyapunov, thesis on the stability of motion (1892) 684
The author 685
The aim and the inspiration of the Dissertation 685
Lyapunov's concept of stability 686
The first method of Lyapunov 688
The second method of Lyapunov 688
The case of autonomous systems 689
The case of periodic systems 690
The influence of Poincaré's work on Lyapunov's Dissertation 691
The early reception of the work of Lyapunov on stability 693
The later development of Lyapunov stability 694
Bibliography 51 694
52. Heinrich Hertz, posthumous book on mechanics (1894) 697
Education and employments 697
Mechanics, a race with death 698
Why mechanics? 699
Images of nature 700
Geometry of systems of points 702
Dynamics 705
Reception and impact 52 707
Bibliography 52 708
53. Heinrich Weber, Lehrbuch der Algebra (1895-1896) 710
Background 53 710
Heinrich Weber's career 711
The Introduction to the Lehrbuch 712
The three volumes 713
Impact 53 718
Bibliography 53 719
54. David Hilbert, report on algebraic number fields (`Zahlbericht') (1897) 720
A report and almost a textbook 720
The preface: number theory and arithmetisation 721
Dedekind versus Kronecker, arithmetic versus algebra 722
Content and structure 724
Later reactions 54 728
Bibliography 54 729
55. David Hilbert, Grundlagen der Geometrie, first edition (1899) 730
A new direction of mathematical thought in 1899 730
Fourteen editions in 100 years 731
Seeing the master in his workshop: Hilbert's manuscripts 732
Foundations of projective geometry: Hilbert's exercise-book (1879) and later 733
Geometry as a system of axioms (1894) 734
A vacation-course for teachers: the kernel of the Festschrift (1898) 736
Lectures and an elaboration on Euclidean geometry (1898-1899) 737
The algebraisation of geometry: the first edition in June 1899 738
The further development of Hilbert's Grundlagen der Geometrie 738
A survey of the intersection theorems and the most important results 741
Reactions and conclusion 55 742
Bibliography 55 742
56. Karl Pearson, paper on the chi square goodness of fit test (1900) 744
Education and employment 744
Chronology and curve fitting 745
The mathematical derivation 748
Some later developments 749
Concluding remarks 56 749
Bibliography 56 750
57. David Hilbert, paper on `Mathematical problems' (1901) 752
Introduction 57 753
The problems 57 753
Concluding remarks 57 762
Bibliography 57 763
58. Lord Kelvin, Baltimore lectures on mathematical physics ((1884), 1904) 768
Biography 58 768
Mathematical field theory 769
The ether as an elastic solid 770
The physical foundation of the Baltimore lectures 770
The contents of the Baltimore lectures 771
Conclusion 776
Bibliography 58 776
59. Henri Lebesgue and René Baire, three books on mathematical analysis (1904-1906) 777
Integrals and functions in the 19th century 778
The authors 59 784
Lebesgue's Leçons sur l'intégration et la recherche des fonctions primitives (1904) 785
Reception of Lebesgue's book 788
Baire's Leçons sur les fonctions discontinues (1905) 789
Reception of Baire's book 792
Lebesgue's Leçons sur les séries trigonométriques (1906) 793
Reception of Lebesgue's second book 795
Joint effect of these three works 796
Bibliography 59 796
60. H.A. Lorentz, Lectures on electron theory, first edition (1909) 798
Biography 60 798
Lorentz's contributions to electromagnetism 799
The lectures 60 801
A note on his impact 803
Bibliography 60 803
61. A.N. Whitehead and Bertrand Russell, Principia mathematica, first edition (1910-1913) 804
The reductionist heritage 804
Collaboration, and fallow years 806
The writing and content of PM 807
Reactions by Russell and his British followers 810
The reception of PM in the United States 812
German-speaking contributions 812
Logic(ism) after Gödel 813
Bibliography 61 814
62 Federigo Enriques and Oscar Chisini, Lectures on `the geometrical theory of equations and algebraic functions' (1915-1934) 815
The authors 62 815
The lectures 62 817
The methods and the content 817
The impact 62 820
Bibliography 62 821
63 Albert Einstein, review paper on general relativity theory (1916) 822
The special theory of relativity 823
The equivalence hypothesis 824
The metric tensor 825
Einstein's collaboration with Marcel Grossmann 826
Coming close to the solution, or so it seems 827
The Entwurf theory 828
The 1914 review article on the Entwurf theory 830
The demise of the Entwurf and the breakthrough to general covariance 831
The 1916 review paper 835
Early reception of the final version of general relativity 838
Going on and beyond general relativity 839
Bibliography 63 841
64. D'Arcy Wentworth Thompson, On growth and form, first edition (1917) 843
Biography 64 844
Structure of the argument 844
Thompson's key methodological example 846
How should such a work be approached today? 847
Evaluation 64 850
Acknowledgement 64 851
Bibliography 64 851
65. Leonard Dickson, History of the theory of numbers (1919-1923) 853
Brief biography of the author 853
Why Dickson may have written his HISTORY OF THE THEORY OF NUMBERS 854
The style and content of Dickson's HISTORY OF THE THEORY OF NUMBERS 855
One salient omission 859
Reception of Dickson's HISTORY OF THE THEORY OF NUMBERS 860
Bibliography 65 862
66. Paul UrysohnUrysohn, Paul|( and Karl MengerMenger, Karl|(, papers on dimension theorydimension theory|( (1923-1926) 864
Ancestry 66 864
Two physicists 866
Paul Urysohn, mathematician 866
Karl Menger, mathematician 870
The joint contribution 871
The impact of Menger and Urysohn 872
Bibliography 66 874
67. R.A. Fisher, Statistical methods for research workers, first edition (1925) 876
The author 67 877
Writing STATISTICAL METHODS 879
Content of the first edition 881
Subsequent editions and books 885
Impact of the book 888
Epilogue 67 889
Bibliography 67 889
68 George David Birkhoff,Dynamical systems (1927) 891
Introduction 68 891
Celestial mechanics: the historical background 892
The contents of Birkhoff's book 893
Birkhoff's main ambition 895
Birkhoff's lecture course and its context 896
On the impact and renaissance of the book 897
Bibliography 68 899
69 P.A.M. Dirac (1930) and J. von Neumann (1932), books on quantum mechanics 902
The discovery of quantum mechanics 902
Background of Dirac's PRINCIPLES 903
The PRINCIPLES OF QUANTUM MECHANICS 906
Later editions of PRINCIPLES 909
Weyl, Hilbert, von Neumann and the mathematical foundations of quantum mechanics 912
The MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS 916
Bibliography 69 919
70 B.L. van der Waerden, Moderne Algebra,first edition (1930-1931) 921
Biographical notes 70 921
Aims and contents of the book: foundations 922
The consolidation of field theory and group theory: Galois theory 925
Ideals and hypercomplex systems: further basic algebraic theory 927
Some special topics in algebra 929
Reception and historical impact of the book 931
Further revision of the book 70 933
Bibliography 70 935
71. Kurt Gödel, paper on the incompleteness theorems (1931) 937
Gödel's life and work 938
Hilbert's program, completeness, and incompleteness 940
An outline of Gödel's results 941
Importance and impact of the incompleteness theorems 943
Bibliography 71 945
72 Walter Andrew Shewhart, Economic control of quality of manufactured product (1931) 946
The industrial problem of the statistical control of quality 946
An approach via statistical physics 947
The dialogue between mathematics and industrial practice 949
Presentation of the book 950
Allowable variability in quality 952
Reception of the book 72 954
Bibliography 72 954
73. Vito Volterra, book on mathematical biology (1931) 956
The origins of Volterra's interest in biomathematics 956
The premises and genesis of the book: research in the 1920s 957
The writing and contents of the book 959
The place of the book among Volterra's works 962
The book's reception and its influence on biomathematical research 962
Bibliography 73 963
74. S. Bochner, lectures on Fourier integrals (1932) 965
The development of the theory of the Fourier integral 965
New kinds of integrals 968
The author 74 972
Bochner's book 74 973
The aftermath: abstract harmonic analysis 977
Bibliography 74 979
75. A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (1933) 980
Introduction 75 980
The background of the Grundbegriffe 981
Axioms for finitary probability theory 983
The application of probability 984
Infinite fields of probability 985
The impact of the Grundbegriffe 987
Bibliography 75 989
76 H. Seifert and W. Threlfall (1934) and P.S. Alexandroff and H. Hopf (1935), books on topology 990
Algebraic topology prior to 1934 990
Seifert and Threlfall, Lehrbuch der Topologie (1934) 992
Alexandroff and Hopf, Topologie (1935) 995
Reception of the two books 997
Bibliography 76 999
77 David Hilbert and Paul Bernays, Grundlagen der Mathematik, first edition (1934, 1939) 1001
Background 77 1001
Naïve proof theory 1002
The first volume 1006
The second volume 1010
Philosophical and mathematical issues 1014
Concluding remarks 1018
Note 77 1018
Bibliography 77 1018
List of Authors 1020
Index 1024