Mathematical Thought from Ancient to Modern Times v1+2+3

Cover......Page 1
Preface to the Three-Volume Paperback Edition of Mathematical Thought......Page 5
Preface......Page 6
Contents......Page 10
Publisher\'s Note to this Three-Volume Paperback Edition......Page 13
1. Where Did Mathematics Begin? (3)......Page 15
2. Political History in Mesopotamia (4)......Page 16
3. The Number Symbols (5)......Page 17
4. Arithmetic Operations (7)......Page 19
5. Babylonian Algebra (8)......Page 20
6. Babylonian Geometry (10)......Page 22
7. The Uses of Mathematics in Babylonia (11)......Page 23
8. Evaluation of Babylonian Mathematics (13)......Page 25
Bibliography (14)......Page 26
1. Background (15)......Page 27
2. The Arithmetic (16)......Page 28
3. Algebra and Geometry (18)......Page 30
4. Egyptian Uses of Mathematics (21)......Page 33
5. Summary (22)......Page 34
Bibliography (23)......Page 35
1. Background (24)......Page 36
2. The General Sources (25)......Page 37
3. The Major Schools of the Classical Period (27)......Page 39
5. The Pythagoreans (28)......Page 40
6. The Eleatic School (34)......Page 46
7. The Sophist School (37)......Page 49
8. The Platonic School (42)......Page 54
9. The School of Eudoxus (48)......Page 60
10. Aristotle and His School (51)......Page 63
Bibliography (54)......Page 66
1. Introduction (56)......Page 68
2. The Background of Euclid\'s Elements (57)......Page 69
3. The Definitions and Axioms of the Elements (58)......Page 70
4. Books I to IV of the Elements (60)......Page 72
5. Book V: The Theory of Proportion (68)......Page 80
6. Book VI: Similar Figures (73)......Page 85
7. Books VII, VIII, and IX: The Theory of Numbers (77)......Page 89
8. Book X: The Classification of Incommensurables (80)......Page 92
9. Books XI, XII, and XIII: Solid Geometry and the Method of Exhaustion......Page 93
10. The Merits and Defects of the Elements (86)......Page 98
11. Other Mathematical Works by Euclid (88)......Page 100
12. The Mathematical Work of Apollonius (89)......Page 101
Bibliography (99)......Page 111
1. The Founding of Alexandria (101)......Page 113
2. The Character of Alexandrian Greek Mathematics (103)......Page 115
3. Areas and Volumes in the Work of Archimedes (105)......Page 117
4. Areas and Volumes in the Work of Heron (116)......Page 128
5. Some Exceptional Curves (117)......Page 129
6. The Creation of Trigonometry (119)......Page 131
7. Late Alexandrian Activity in Geometry (126)......Page 138
Bibliography (130)......Page 142
1. The Symbols and Operations of Greek Arithmetic (131)......Page 143
2. Arithmetic and Algebra as an Independent Development (135)......Page 147
Bibliography (144)......Page 156
1. The Inspiration for Greek Mathematics (145)......Page 157
2. The Beginnings of a Rational View of Nature (146)......Page 158
3. The Development of the Belief in Mathematical Design (147)......Page 159
4. Greek Mathematical Astronomy (154)......Page 166
5. Geography (160)......Page 172
6. Mechanics (162)......Page 174
7. Optics (166)......Page 178
8. Astrology (168)......Page 180
Bibliography (169)......Page 181
1. A Review of the Greek Achievements (171)......Page 183
2. The Limitations of Greek Mathematics (173)......Page 185
3. The Problems Bequeathed by the Greeks (176)......Page 188
4. The Demise of the Greek Civilization (177)......Page 189
Bibliography (182)......Page 194
1. Early Hindu Mathematics (183)......Page 195
2. Hindu Arithmetic and Algebra of the Period A. D. 200–1200 (184)......Page 196
3. Hindu Geometry and Trigonometry of the Period A. D. 200–1200 (188)......Page 200
4. The Arabs (190)......Page 202
5. Arabic Arithmetic and Algebra (191)......Page 203
6. Arabic Geometry and Trigonometry (195)......Page 207
7. Mathematics circa 1300 (197)......Page 209
Bibliography (199)......Page 211
1. The Beginnings of a European Civilization (200)......Page 212
2. The Materials Available for Learning (201)......Page 213
3. The Role of Mathematics in Early Medieval Europe (202)......Page 214
4. The Stagnation in Mathematics (203)......Page 215
5. The First Revival of the Greek Works (205)......Page 217
6. The Revival of Rationalism and Interest in Nature (206)......Page 218
7. Progress in Mathematics Proper (209)......Page 221
8. Progress in Physical Science (211)......Page 223
9. Summary (213)......Page 225
Bibliography (214)......Page 226
1. Revolutionary Influences in Europe (216)......Page 228
2. The New Intellectual Outlook (218)......Page 230
3. The Spread of Learning (220)......Page 232
4. Humanistic Activity in Mathematics (221)......Page 233
5. The Clamor for the Reform of Science (223)......Page 235
6. The Rise of Empiricism (227)......Page 239
Bibliography (230)......Page 242
1. Perspective (231)......Page 243
2. Geometry Proper (234)......Page 246
3. Algebra (236)......Page 248
4. Trigonometry (237)......Page 249
5. The Major Scientific Progress in the Renaissance (240)......Page 252
6. Remarks on the Renaissance (247)......Page 259
1. Introduction (250)......Page 262
2. The Status of the Number System and Arithmetic (251)......Page 263
3. Symbolism (259)......Page 271
4. The Solution of Third and Fourth Degree Equations (263)......Page 275
5. The Theory of Equations (270)......Page 282
6. The Binomial Theorem and Allied Topics (272)......Page 284
7. The Theory of Numbers (274)......Page 286
8. The Relationship of Algebra to Geometry (278)......Page 290
Bibliography (282)......Page 294
1. The Rebirth of Geometry (285)......Page 297
2. The Problems Raised by the Work on Perspective (286)......Page 298
3. The Work of Desargues (288)......Page 300
4. The Work of Pascal and La Hire (295)......Page 307
5. The Emergence of New Principles (299)......Page 311
Bibliography (301)......Page 313
1. The Motivation for Coordinate Geometry (302)......Page 314
2. The Coordinate Geometry of Fermat (303)......Page 315
3. René Descartes (304)......Page 316
4. Descartes\'s Work in Coordinate Geometry (308)......Page 320
5. Seventeenth-Century Extensions of Coordinate Geometry (317)......Page 329
6. The Importance of Coordinate Geometry (321)......Page 333
Bibliography (324)......Page 336
2. Descartes\'s Concept of Science (325)......Page 337
3. Galileo\'s Approach to Science (327)......Page 339
4. The Function Concept (335)......Page 347
Bibliography (340)......Page 352
1. The Motivation for the Calculus (342)......Page 354
2. Early Seventeenth-Century Work on the Calculus (344)......Page 356
3. The Work of Newton (356)......Page 368
4. The Work of Leibniz (370)......Page 382
5. A Comparison of the Work of Newton and Leibniz (378)......Page 390
6. The Controversy over Priority (380)......Page 392
7. Some Immediate Additions to the Calculus (381)......Page 393
8. The Soundness of the Calculus (383)......Page 395
Bibliography (389)......Page 401
Volume 2 Contents......Page 403
1. The Transformation of Mathematics (391)......Page 407
2. Mathematics and Science (394)......Page 410
3. Communication Among Mathematicians (396)......Page 412
4. The Prospects for the Eighteenth Century (398)......Page 414
Bibliography (399)......Page 415
1. Introduction (400)......Page 416
2. The Function Concept (403)......Page 419
3. The Technique of Integration and Complex Quantities (406)......Page 422
4. Elliptic Integrals (411)......Page 427
5. Further Special Functions (422)......Page 438
6. The Calculus of Functions of Several Variables (425)......Page 441
7. The Attempts to Supply Rigor in the Calculus (426)......Page 442
Bibliography (434)......Page 450
2. Initial Work on Infinite Series (436)......Page 452
3. The Expansion of Functions (440)......Page 456
4. The Manipulation of Series (442)......Page 458
5. Trigonometric Series (454)......Page 470
6. Continued Fractions (459)......Page 475
7. The Problem of Convergence and Divergence (460)......Page 476
Bibliography (466)......Page 482
1. Motivations (468)......Page 484
2. First Order Ordinary Differential Equations (471)......Page 487
3. Singular Solutions (476)......Page 492
4. Second Order Equations and the Riccati Equations (478)......Page 494
5. Higher Order Equations (484)......Page 500
6. The Method of Series (488)......Page 504
7. Systems of Differential Equations (490)......Page 506
8. Summary (499)......Page 515
Bibliography (501)......Page 517
1. Introduction (502)......Page 518
2. The Wave Equation (503)......Page 519
3. Extensions of the Wave Equation (515)......Page 531
4. Potential Theory (522)......Page 538
5. First Order Partial Differential Equations (531)......Page 547
6. Monge and the Theory of Characteristics (536)......Page 552
7. Monge and Nonlinear Second Order Equations (538)......Page 554
8. Systems of First Order Partial Differential Equations (540)......Page 556
9. The Rise of the Mathematical Subject (542)......Page 558
Bibliography (543)......Page 559
2. Basic Analytic Geometry (544)......Page 560
3. Higher Plane Curves (547)......Page 563
4. The Beginnings of Differential Geometry (554)......Page 570
5. Plane Curves (555)......Page 571
6. Space Curves (557)......Page 573
7. The Theory of Surfaces (562)......Page 578
8. The Mapping Problem (570)......Page 586
Bibliography (571)......Page 587
1. The Initial Problems (573)......Page 589
2. The Early Work of Euler (577)......Page 593
3. The Principle of Least Action (579)......Page 595
4. The Methodology of Lagrange (582)......Page 598
5. Lagrange and Least Action (587)......Page 603
6. The Second Variation (589)......Page 605
Bibliography (590)......Page 606
1. Status of the Number System (592)......Page 608
2. The Theory of Equations (597)......Page 613
3. Determinants and Elimination Theory (606)......Page 622
4. The Theory of Numbers (608)......Page 624
Bibliography (612)......Page 628
1. The Rise of Analysis (614)......Page 630
2. The Motivation for the Eighteenth-Century Work (616)......Page 632
3. The Problem of Proof (617)......Page 633
4. The Metaphysical Basis (619)......Page 635
5. The Expansion of Mathematical Activity (621)......Page 637
6. A Glance Ahead (623)......Page 639
Bibliography (625)......Page 641
2. The Beginnings of Complex Function Theory (626)......Page 642
3. The Geometrical Representation of Complex Numbers (628)......Page 644
4. The Foundation of Complex Function Theory (632)......Page 648
5. Weierstrass\'s Approach to Function Theory (642)......Page 658
6. Elliptic Functions (644)......Page 660
7. Hyperelliptic Integrals and Abel\'s Theorem (651)......Page 667
8. Riemann and Multiple-Valued Functions (655)......Page 671
9. Abelian Integrals and Functions (663)......Page 679
10. Conformal Mapping (666)......Page 682
11. The Representation of Functions and Exceptional Values (667)......Page 683
Bibliography (669)......Page 685
2. The Heat Equation and Fourier Series (671)......Page 687
3. Closed Solutions; the Fourier Integral (679)......Page 695
4. The Potential Equation and Green\'s Theorem (681)......Page 697
5. Curvilinear Coordinates (687)......Page 703
6. The Wave Equation and the Reduced Wave Equation (690)......Page 706
7. Systems of Partial Differential Equations (696)......Page 712
8. Existence Theorems (699)......Page 715
Bibliography (707)......Page 723
2. Series Solutions and Special Functions (709)......Page 725
3. Sturm-Liouville Theory (715)......Page 731
4. Existence Theorems (717)......Page 733
5. The Theory of Singularities (721)......Page 737
6. Automorphic Functions (726)......Page 742
7. Hill\'s Work on Periodic Solutions of Linear Equations (730)......Page 746
8. Nonlinear Differential Equations: The Qualitative Theory (732)......Page 748
Bibliography (738)......Page 754
2. Mathematical Physics and the Calculus of Variations (739)......Page 755
3. Mathematical Extensions of the Calculus of Variations Proper (745)......Page 761
4. Related Problems in the Calculus of Variations (749)......Page 765
Bibliography (750)......Page 766
2. Binomial Equations (752)......Page 768
3. Abefs Work on the Solution of Equations by Radicals (754)......Page 770
4. Galois\'s Theory of Solvability (755)......Page 771
5. The Geometric Construction Problems (763)......Page 779
6. The Theory of Substitution Groups (764)......Page 780
Bibliography (770)......Page 786
1. The Foundation of Algebra on Permanence of Form (772)......Page 788
2. The Search for a Three-Dimensional \"Complex Number\" (776)......Page 792
3. The Nature of Quaternions (779)......Page 795
4. Grassmann\' s Calculus of Extension (782)......Page 798
5. From Quaternions to Vectors (785)......Page 801
6. Linear Associative Algebras (791)......Page 807
Bibliography (794)......Page 810
2. Some New Uses of Determinants (795)......Page 811
3. Determinants and Quadratic Forms (799)......Page 815
4. Matrices (804)......Page 820
Bibliography (812)......Page 828
2. The Theory of Congruences (813)......Page 832
3. Algebraic Numbers (818)......Page 837
4. The Ideals of Dedekind (822)......Page 841
5. The Theory of Forms (826)......Page 845
6. Analytic Number Theory (829)......Page 848
Bibliography (832)......Page 851
1. The Renewal of Interest in Geometry (834)......Page 853
2. Synthetic Euclidean Geometry (837)......Page 856
3. The Revival of Synthetic Projective Geometry (840)......Page 859
4. Algebraic Projective Geometry (852)......Page 871
5. Higher Plane Curves and Surfaces (855)......Page 874
Bibliography (859)......Page 878
2. The Status of Euclidean Geometry About 1800 (861)......Page 880
3. The Research on the Parallel Axiom (863)......Page 882
4. Foreshadowings of Non-Euclidean Geometry (867)......Page 886
5. The Creation of Non-Euclidean Geometry (870)......Page 888
6. The Technical Content of Non-Euclidian Geometry (874)......Page 893
7. The Claims of Lobatchevsky and Bolyai to Priority (877)......Page 896
8. The Implications of Non-Euclidean Geometry (879)......Page 898
Bibliography (881)......Page 900
2. Gauss\'s Differential Geometry (882)......Page 901
3. Riemann\'s Approach to Geometry (889)......Page 908
4. The Successors of Riemann (896)......Page 915
5. Invariants of Differential Forms (899)......Page 918
Bibliography (902)......Page 921
2. Surfaces as Models of Non-Euclidean Geometry (904)......Page 923
3. Projective and Metric Geometry (906)......Page 925
4. Models and the Consistency Problem (913)......Page 932
5. Geometry from the Transformation Viewpoint (917)......Page 936
6. The Reality of Non-Euclidean Geometry (921)......Page 940
Bibliography (923)......Page 942
1. Background (924)......Page 943
2. The Theory of Algebraic Invariants (925)......Page 944
3. The Concept of Birational Transformations (932)......Page 951
4. The Function-Theoretic Approach to Algebraic Geometry (934)......Page 953
5. The Uniformization Problem (937)......Page 956
6. The Algebraic-Geometric Approach (939)......Page 958
7. The Arithmetic Approach (942)......Page 961
8. The Algebraic Geometry of Surfaces (943)......Page 962
Bibliography (946)......Page 965
1. Introduction (947)......Page 966
2. Functions and Their Properties (949)......Page 968
3. The Derivative (954)......Page 973
4. The Integral (956)......Page 975
5. Infinite Series (961)......Page 980
6. Fourier Series (966)......Page 985
7. The Status of Analysis (972)......Page 991
Bibliography (977)......Page 996
1. Introduction (979)......Page 998
2. Algebraic and Transcendental Numbers (980)......Page 999
3. The Theory of Irrational Numbers (982)......Page 1001
4. The Theory of Rational Numbers (987)......Page 1006
5. Other Approaches to the Real Number System (990)......Page 1009
6. The Concept of an Infinite Set (992)......Page 1011
7. The Foundation of the Theory of Sets (994)......Page 1013
8. Transfinite Cardinals and Ordinals (998)......Page 1017
9. The Status of Set Theory by 1900 (1002)......Page 1021
Bibliography (1004)......Page 1023
1. The Defects in Euclid (1005)......Page 1024
2. Contributions to the Foundations of Projective Geometry (1007)......Page 1026
3. The Foundations of Euclidean Geometry (1010)......Page 1029
4. Some Related Foundational Work (1015)......Page 1034
5. Some Open Questions (1017)......Page 1036
Bibliography (1022)......Page 1041
1. The Chief Features of the Nineteenth-Century Developments (1023)......Page 1042
2. The Axiomatic Movement (1026)......Page 1045
3. Mathematics as Man\'s Creation (1028)......Page 1047
4. The Loss of Truth (1032)......Page 1051
5. Mathematics as the Study of Arbitrary Structures (1036)......Page 1055
6. The Problem of Consistency (1038)......Page 1057
Bibliography (1039)......Page 1058
1. The Origins (1040)......Page 1059
3. Early Work on Content and Measure (1041)......Page 1060
4. The Lebesgue Integral (1044)......Page 1063
Bibliography (1050)......Page 1069
1. Introduction (1052)......Page 1071
2. The Beginning of a General Theory (1056)......Page 1075
3. The Work of Hilbert (1060)......Page 1079
4. The Immediate Successors of Hilbert (1070)......Page 1089
5. Extensions of the Theory (1073)......Page 1092
Bibliography (1075)......Page 1094
1. The Nature of Functional Analysis (1076)......Page 1095
2. The Theory of Functionals (1077)......Page 1096
3. Linear Functional Analysis (1081)......Page 1100
4. The Axiomatization of Hilbert Space (1091)......Page 1110
Bibliography (1095)......Page 1114
1. Introduction (1096)......Page 1115
2. The Informal Uses of Divergent Series (1098)......Page 1117
3. The Formal Theory of Asymptotic Series (1103)......Page 1122
4. Summability (1109)......Page 1128
Bibliography (1120)......Page 1139
1. The Origins of Tensor Analysis (1122)......Page 1141
2. The Notion of a Tensor (1123)......Page 1142
3. Covariant Differentiation (1127)......Page 1146
4. Parallel Displacement (1130)......Page 1149
5. Generalizations of Riemannian Geometry (1133)......Page 1152
Bibliography (1135)......Page 1154
1. The Nineteenth-Century Background (1136)......Page 1155
2. Abstract Group Theory (1137)......Page 1156
3. The Abstract Theory of Fields (1146)......Page 1165
4. Rings (1150)......Page 1169
5. Non-Associative Algebras (1153)......Page 1172
6. The Range of Abstract Algebra (1156)......Page 1175
Bibliography (1157)......Page 1176
1. The Nature of Topology (1158)......Page 1177
2. Point Set Topology (1159)......Page 1178
3. The Beginnings of Combinational Topology (1163)......Page 1182
4. The Combinational Work of Poincaré (1170)......Page 1189
5. Combinatorial Invariants (1176)......Page 1195
6. Fixed Point Theorems (1177)......Page 1196
7. Generalizations and Extensions (1179)......Page 1198
Bibliography (1181)......Page 1200
1. Introduction (1182)......Page 1201
2. The Paradoxes of Set Theory (1183)......Page 1202
3. The Axiomatization of Set Theory (1185)......Page 1204
4. The Rise of Mathematical Logic (1187)......Page 1206
5. The Logistic School (1192)......Page 1211
6. The Intuitionist School (1197)......Page 1216
7. The Formalist School (1203)......Page 1222
8. Some Recent Developments (1208)......Page 1227
List of Abbreviations (i)......Page 1231
B......Page 1233
C......Page 1234
F......Page 1235
H......Page 1236
K......Page 1237
M......Page 1238
P......Page 1239
S......Page 1240
X......Page 1241
Z......Page 1242