Abelian equation, 755
Abelian function, 665
Abelian integral, 653, 663-65, 936, 939-40
Abel’s theorem, 645, 653-55, 935-36
Absolute, 907
Absolute differential calculus, SeeTensor analysis
Absolutely continuous function, 1048
Abstract algebra, 945, 1136-57
Lie algebra, 1153-56
non-associative algebra, 1153-56
see alsoField
Group
Ideal
Ring
Abstraction, 29, 43-44, 171
Academies, 227, 370, 396-97, 401, 404, 621, 1023
Academy of Plato, 27, 42-43, 45, 130, 190
Ad Locos Pianos et Solidos Isagoge,302, 317
Adjoint curve, 935
Aesthetics, 172-73
Affine geometry, 918
Airy’s integral, 1100, 1102-3
Akkadians, 4
Alexandria, 101-3
Algebra: and analysis, 323-24, 361
as analysis, 279-80, 323
Arabic, 191-95
vs, geometry, 49, 136, 197-98, 279-82, 318, 323, 365, 391-92, 834-36
Greek (Alexandrian), 104, 135-44
Hindu, 184-88
Renaissance, 236-37
of throws, 850-51
see also Chaps, 13, 25, 31, 32, 33, 34
Algebra,364
Algebraic geometry, 920, 924-46, 1168-69
Algebraic invariants, 925-32
absolute, 926
complete system, 928-29
covariant, 926
Algebraic numbers, 593-94, 818-26, 980, 996-97
Almagest, 54, 122-25, 133, 159, 191
Analytic continuation, 642-44, 1114
Analytic geometry, SeeCoordinate geometry
Angle, projective definition of, 907, 911
Anharmonic ratio, SeeCross ratio
Apollonian problem, 99
Application of areas, 34, 74-77
Arabs, 181, 190-99, 205
Archimedes’ principle, 165-66
Areas, 1042
calculation of, 343, 348-54, 360
of surfaces, 355, 1020
Arithmetic: Arabic, 191-92
Babylonian, 5-8
Egyptian, 16-18
Greek, (Alexandrian), 131-35
Hindu, 183-86
primitive, 3
Renaissance, 251-59
Arithmetica, 139-43
Arithmetica Universalis, 252, 271-72, 281, 312, 318, 358, 392, 607
Arithmetization of analysis, 947-72, 1025-26
Ars Conjectandi, 273, 442, 451
Ars Magna, 236, 253, 263-65, 267-68
Astrology, 13, 168-69, 179, 196, 202-3, 222
Astronomy, 250, 470-71, 490-97
Arabic, 196
Babylonian, 11-12
Egyptian, 21-22;
Greek, 119, 125-26, 148-49, 151, 154-60
Hindu, 189
see alsoGeocentric theory
Gravitational attraction
Heliocentric
theory
Three-body problem
Asymptotic series, 1097-1109
semiconver-gent, 1097, 1104
see alsoWKBJ method Athens, 37, 101
Atomism, 150, 329
Automorphic functions, 726-30
elliptic modular, 727
Fuchsian, 729
Kleinian, 730
Automorphism, 1142
Axes of a conic, 93-95
Axiom of Archimedes, 81, 991, 1012-13
Axiom of choice, 1002, 1186, 1209
Axiom of reducibility, 1195-96
Axiomatization, 1026-27
of set theory, 1185-87
Axioms, 50, 52, 59-60
of Euclid, 59-60, 1005-7
of Hilbert for geometry, 1010-13
of non-Euclidean geometry, 1015
of number, 988-91
of projective geometry, 1007-10
of set theory, 1185-86
see alsoHubert’s axioms for number
Parallel axiom
Banach space, 1088-91, 1159, 1178
Base, 5-6
Bernoulli numbers, 449, 451
Bessel functions, 480-82, 489, 518-19, 710-11
Bessel inequality, 717, 1083
Beta function, 424
Betti numbers, 1169, 1173 74, 1176-77, 1179-80
Binomial equation, 600, 752-53, 761-64
Binomial theorem, 272-73, 438, 441, 964-65
Binormal, 560-61
Biquadratic reciprocity, 816-18
Biquatemions, 792
Birational transformation, 924, 932
Brachistochrone, 574-75
Branch-cut, 642, 656-57
Branch-point, 641-657
Brianchon’s theorem, 841, 848
Burnside’s problem, 1143
Burali-Forti paradox, 1003, 1182
Byzantine Empire, 180, 191, 205, 216
Calculus, 342-90, 400-35, 615, 798
see alsoMethod of exhaustion
Calculus of extension, 782-85
Calculus of finite differences, 440-41, 452-53
Calculus of variations, 573-91, 685, 739-51, 1076, 1078, 1081
Jacobi condition, 745
Legendre condition, 589-90
Weierstrass conditions, 748
see alsoMinimal surface
Calendar, 12-13, 21-22, 119, 178-79
Cardinal number, 995
Cassinian ovals, 320
Categoricalness, 1014-15
Catenary, 382, 472, 579
Cauchy integral formula, 639
Cauchy integral theorem, 599, 636-37, 639-40, 668-69
Cauchy-Lipschilz theorem, 718
Caucliy-Riemann equations, 627-28, 635, 658
Ca\alicri’s theorem, 349
Caylcy-Hamilton theorem, 807-9
Cayley numbers, 792
Chain, 1171-72
Chain, oscillations of, 480-81
Chain rule, 376
Chaldeans, 4
Characteristic equation: of a determinant, 800-801
of a differential equation, 485
of a matrix, 807
of quadratic forms, 800-801
Characteristic function, 706-7, 715-16, 1063, 1068
Characteristic root: of a determinant, 800-1
of a matrix, 807-8
Characteristic triangle, 346, 375, 387
Characteristic value, 480, 673, 706-7, 715-16, 1063, 1068
Characteristics, theory of, 535-38, 700-703
Christianity, 180-81, 200-201, 203-5
Christoffel symbols, 894, 1129
Chiistoffel’s quadrilinear form, 898
μ-ply form, 898-99
Circular points, 845, 854
Cissoid, 118-19, 286, 355
Clebsch-Gordan theorem, 929
Clifford algebra, 792
Commutator, 1142
Complete continuity, 1065
Complete quadrilateral, 127, 292-93
Completeness of axioms, 1015, 1207
Completeness of characteristic functions, 7’7
Complex, 1171
Complex function theory, 626-70, 686-87, 934-35
Complex integers, 817-19
Complex number, 143, 253-54, 407-8, 594-95, 635-36, 687, 775-76, 815-16
geometrical representation, 594-95, 628-32
logarithms of, 408-11, 594
Composition indices, 762
Composition series, 762, 767
Computing machines, 258-59
Conchoid, 117-18, 286
Conformai mapping, 236, 370, 570-71, 666-67, 887
see alsoMap-making
Congruence of lines, 567-69
Congruence of numbers, 813-18
Conic sections, 47-48, 88-99, 286, 299, 303-43, 319, 837-38, 847-49
Conic Sections, 27, 56, 90-99, 172
Connectivity, 660-62, 937
Consistency, 880, 913-17, 1013-14, 1038, 1182, 1186, 1206, 1209
Construction problems, 38-42, 47-48, 118, 195, 234-35, 309-10, 312-4, 753-54, 763-64, 840
Content, 1041-44
Continued fraction, 187, 254-56, 459-60, 1114-16
Continuity, 405, 950-54
uniform, 953
Continuous change, 299
Continuous transformation, 1160-61
Continuous transformation group, 1139-40, 1154-55
Continuum hypothesis, 1002, 1209
Convergence, 460-66
Cauchy, 961-66
strong, 1072-73
weak, 1072-73
see alsoSummability
Coordinate geometry, 302-24, 544-54
higher plane curves, 547-54, 855-58
importance of, 321-24
three-dimensional, 320-21, 545-47
see alsoConic sections
Quadric surfaces
Cours d’analyse algébrique,948, 950, 951-52, 963-64
Covariant, 926
Covariant differentiation, 1127-30
Cramer’s paradox, 553-54, 856
Cramer’s rule, 606
Cremona transformation, 933-34
Cross ratio, 121-22, 127-28, 219-20, 848-49, 851, 907
Cross- cut, 660
Cubic equation, 193-95, 237, 263-67, 269
Cubic reciprocity, 816, 818
Cuneiform, 5
Curl, 781, 785, 789
Curvature, 364, 378, 382-83, 555-56, 559-60
of a manifold, 891-93, 895-96, 1126-27
mean, 884
of surfaces, 563-64, 883-85, 887
Curve: concept of, 174, 312-14, 1017-22, 1162
length of, 343, 348, 354-55, 414-16, 958, 1020
see alsoHilbert curve
Jordan curve
Peano curve
Curve and equation, 303-4, 310-17
see alsoHigher plane curves
Algebraic geometry
Curvilinear coordinates, 687-89, 713-14
Cusp, SeeSingular points of curves Cycle, 1172
Cycloid, 338, 350-53, 355, 367, 469, 472, 556, 575
Cyclotomic equation, SeeBinomial equation
De Analyst per Aequationes Numero Ter-minorum Infinitas, 359, 361, 381, 438
De Revolutionibus Orbium Coelestium, 241
Deferent, 157
Deficiency, 552
Definition: Aristotle’s concept of, 51-52
in Euclid, 58-59, 68-71, 73, 78, 81-82, 1006
Del, 781
DeMorgan’s laws, 1189
Derivative, 344-48, 359-65, 371-78, 384-88, 425-33, 954-56
Desargues’s involution theorem, 292-93
Desargues’s theorem, 291, 845-46
Descartes’s rule of signs, 270
Descriptive geometry, 235
Determinant, 606-8, 795-804
elementary divisors of, 803
infinite, 732
invariant factors, 803
similar, 802
see alsoMatrices
Developable surface, 564-68
polar, 566
Dialogue on the Great World Systems, 327-28
Dialogues Concerning Two New Sciences, 229, 328, 332-33, 338, 348, 468, 993
Diameter of a conic, 93, 98-99, 294-95
Differential, 372-78, 385, 429-30, 433, 615, 954
Differential geometry, 554-71, 881-902, 1130-35
Differential invariants, 889-902, 1122-23, 1127-28, 1130
Dimension, 1021, 1161-62, 1177
Dirichlet principle, 659-60, 684-86, 704-5, 939
Dirichlet problem, 685, 703-5
Dirichlet series, 830
Discourse on Method, 226-27, 304-5
Discrete vs, continuous, 34-35, 52-53, 175-76
Disquisitiones Arithmeticae, 752, 813-15, 826-28, 870
Distance, projective definition of, 908, 910-11
Divergence, 781, 785, 789, 1129
Divergence theorem, 790
Divergent series, 466, 973-77, 1096-1120
see alsoAsymptotic series
Infinite series
Division algebra, 1151
Domain of rationality, SeeField
Double Points, SeeSingular points of curves
Duality, topological, 1174, 1179
Duality theorem, 845-46, 848-49
Dupin indicatrix, 569
e, 258, 439, 459-60, 593, 980-81
Eastern Roman Empire, 180, 191, 205, 216
Edge of regression, 566-67
Eigcnfunction, SeeCharacteristic function
Eigenvalue, SeeCharacteristic value
Elastica, 413, 558
Elasticity, 468-69, 485, 697-98, 739
Eleatic school, 27, 34-37
Elements of Euclid, 25, 27, 33, 37, 52, 56-88, 172, 220, 393, 507, 866, 1005-7
Elimination, SeeResultant
Ellipoidal harmonics, 712-13
Elliptic functions, 644-51
addition theorem of, 648
Empiricism, 227-30, 330
Encyclopédie, 432-33, 465, 510, 595, 597, 616
Entire functions, 667-68
Envelope, 378, 556-57
Epicycle, 157
Epicycloid, 235, 367
Erlanger Programm, 917-20, 1139, 1159
Essential singularity, 641
Euclidean algorithm, 78-79, 187
Euclidean geometry, 861-63, 1005-22
Alexandrian Greek, 104, 107-19, 126-29
Arabic, 195
Babylonian, 10-11
Egyp-tion, 19-20
Classical Greek, 24-55, 58-99
Hindu, 188-89
nineteenth century, 837-40
solid, 47, 81-3
spherical, 89, 119-21
Euler-Maclaurin summation formula, 452
Euler-Poincaré formula, 1174
Euler’s constant, 450, 892
Eulcr’s differential equation, 578, 584, 589
Euler’s differential geometry theorem, 563
Euler’s topology theorem, 1163-64
Evolute, 98, 555-56
Existence, 599
in algebra, 597-98
in Euclidean geometry, 52, 60, 174, 1010-11
proof of, 52, 176-77
see alsoOrdinary differential equation
Partial differential equation Exponents, 260-61
Fagnano’s theorem, 415-16
Fermat’s “theorem”, 276-77, 609, 818-20
Field, 755, 757, 821-25, 931, 1146-50
adjunction, 821, 1148
characteristic of, 1148
extension, 821, 1148
finite, 114-950
Galois theory of, 1149
non-commutative, 1151
p-adic, 1146-47
Fixed point theorem, 1177-79
Fluent, 361
Fluxion, 361
Folium of Descartes, 549
Formalism, 1203-8
Foundations, 979-1022
algebra, 176, 282, 772-75, 980
analysis, 383-89, 426-34, 947-77
arithmetic, 176, 596-97, 775-76, 950, 972
geometry, 1005-22
of mathematics, 1182-1210
Fourier integral, 679-81
Fourier series, 456-59, 513-14, 674-78, 966-72, 1040, 1046-47, 1119
see alsoTrigonometric series
Fourier transform, 681, 1052-53, 1074
integral, 679-81
Fredholm alternative theorem, 1060, 1067-69, 1090-91
Fresnel integrals, 1100
Function concept, 338-40, 403-6, 505-7, 677-79, 949-54
Functional, 1077-81
differential of, 1080
semi-continuity of, 1079
Functional analysis, 1076-95, 1159
Functions of bounded variation, 971, 1048
Functions of real variables, 1040-50
Fundamental group, 1174-75
Fundamental theorem: of algebra, 595, 597-98
of arithmetic, 79-80, 817, 819, 824
of the calculus, 373-74, 956-57
Galois theory, 752-64
Galoisian equation, 763
Gamma function, 423-24
Generality, 394
Generalized coordinates, 588-89
Generators of a group, 1141, 1143
Genus, 654-55, 661, 936-40, 1166-67
geometrical, 944
numerical, 944-45
Geocentric theory, 154-60, 241
Geodesic, 562-63, 575, 577, 886, 891
Geodesy, 116-17
Geography, 119, 160-62
Geometrical algebra, 62-67, 76-77, 108, 195
Geometry vs, analysis, 614-16
Geometry of numbers, 829
Geometry from the transformation viewpoint, 917-20
Gnomon, 30-31
Coldbach’s “theorem”, 616
Gradient, 781, 786, 789, 901-2
Gravitational attraction, 343, 357-58, 367-68, 469, 490-94, 522-30
Greek mathematics summarized, 171-77
Greek science, 49-50, 103, 145-69
Green’s function, 683-85, 692, 1069
Green’s theorem, 683, 692-96
Gregory-Newton formula, 440-41
Group, 757-58
Abelian, 767
abstract, 769, 1137-46
alternating, 762
composite, 766, 1142
continuous transformation, 1139-40, 1154-55
discontinuous, 726-30
of an equation, 758
index, 759
linear transformation, 768-69, monotlromy, 723
order of, 759
permutation, seesubstitution
primitive, 765, 1142
simple, 766, 1142
solvable, 762, 1142-43
substitution, 758, 764-69
symmetric, 762
in topology, 1180
transitive, 765, 1142
Group character, 1145
Group representation, 768, 1143-45
Hahn-Banach theorem, 1090
Hamilton-Jacobi equation, 744
Hamilton’s equations of motion, 742-43
Harmonic function, 685
Harmonic oscillator, 482
Harmonic set of points, 96, 128, 293, 300, 851
Hauptvermutungof Poincaré, 1176
Heine-Borei theorem, 953-54
Heliocentric theory, 241-47, 327-28
Helmholtz equation, SeePartial differential equation
Hermite function, 714
Hessian, 858, 917-28
Hieratic, 15-16
Hieroglyphic, 15
Higher degree equation, 270, 599-606, 752-63
see alsoAbelian equation
Binomial equation
Galois theory
Higher plane curves, 547-54
projective approach, 855-58
Hilbert cube, 1161
Hilbert curve, 1018-19
Hilbert space, 1074, 1082, 1088, 1092-95, 1159
Hubert’s axioms for number, 990-92
Hubert’s basis theorem, 930-31, 1158
Hubert’s invariant integral, 749
Hubert’s Nullstellensalz, 943
Hilbert-Schmidt theorem, 1064, 1068
Hindus, 183-90
Holomorphic, 642
Homeomorphism, 1158
Homogeneous coordinates, 853-54
Homologous figures, 842
Homology, 1180-81
Homomorphism, 767, 1139
Homotopy, SeeFundamented group
Horn angle, 67-68
Humanist movement, 221-23, 236
Huygens’s principle, 691, 694
Hydrodynamics, 368, 520, 540-42, 686, 686-97
hydrostatics, 165-66, 211
Hyperbolic functions, 404
Hyperelliptic integrals, 651-53
Hypernumbers, 782-85
see alsoLinear associative algebra
Quaternion Hypocycloid, 367
Ideal, 822-23, 1150-51, 1153
Ideal numbers, 819-20
Imaginary elements in geometry, 843-45
Impredicative definition, 1184
In Artem Analyticam Isagoge,261
Incommensurable ratio, SeeIrrational number
Independence of axioms, 1014
Indeterminate equations, 140-43, 187-88
Index of a curve, 736-37
Indivisibles, 349-51
Infinite series, 360-61, 410, 436-66, 465, 480-82, 488-89
convergence and divergence, 460-66, 961-66
Euler transformation of, 452-53
harmonic, 443-44, 449-50
hypergeometric, 489
uniform convergence of, 964-66
see alsoDivergent series
Fourier series
Ordinary differential equation
Special functions
Trigonometric series
Infinitesimal, 69, 361, 385, 388, 429
see alsoDifferential
Infinity, 53, 69, 175-77
Inflection points, 549, 551-52, 556, 587, 857
Institutiones Calculi Differentialis, 402, 430, 463
Institutiones Calculi Integralis, 402, 489, 542
Integral, 360-62, 372-80, 956-61
Lebesque-Stieltjes, 1050
Riemann, 9585g
Stieltjes, 1041
Integral equations, 1052-74, 1086, 1089-91
Fredholm, 1055, 1058-60
Volterra, 1055-58
Interpolation, 422-23, 440-41, 454-56
Intersections of curves, 553, 856-58
Introductie Arithmetica, 136-38
Introducilo in Analysin Infinitorum, 392, 402, 404-5, 430, 506, 554, 558
Intuitionism, 1197-1203
Invariance, 299-300, 920, 925-32
Inversion, 932-33
Involute, 555
Involution, 128, 292-93
Ionia, 25, 146
Ionian school, 27-28
Irrational number, 8, 18, 32-33, 48-49
72-73, 80-81, 104, 134-35, 143, 149, 173, 176, 185-86, 191-92, 197, 209, 251-52, 593-94
definition, 982-87
Irreducible polynomials, 755
Isochrone, 471, 556
Isoperimetric figures, 126, 576-77, 838-39
Isoperimetric theorem, 838-39
Jacobian, 927-28
Jordan canonical form, 810
Jordan curve, 1017
Jordan curve theorem, 1017, 1179
Jordan-Holder theorem, 762, 767
Journals, 397, 624-25
Kepler’s laws, 244-45, 367
Klein bottle, 1168
Koenigsberg bridge problem, 1163-64
Kummer surface, 859
La Géométrie, 270-71, 281, 305, 307-17, 345
Lagrange’s equations of motion, 588-89, 740
Laméfunctions, 712-13
Laplace coefficients, See Legendre polynomials
Laplace transform, 1052
Laplace’s equation, See Potential theory
Laplacian, 785-86, 900-2, 1129-30
Latent roots, See Characteristic equation
Latitude of forms, 210-11
Laurent expansion, 641
Law of excluded middle, 1202
Law of inertia, 799-800
Lebesgue integral, 1044-50, 1071
Lebesgue-Stieltjes integral, 1050
Legendre polynomial, 526-31, 711-12
associated, 531
Lemniscate, 320, 416-20, 549
Length, projective definition of, 908, 910-11
Les Méthodes nouvelles de la mécanique céleste, 735, 1105, 1170
L’Hospital’s rule, 383
Library of Alexandria, 102, 180
Lie algebra, See Abstract algebra
Line coordinates, 855
Line curve, 848
Line, structure of, 52-53
Linear algebraic equations, 606, 803-4
Linear associative algebra, 791-94, 1151-53
Linear transformations, 768-69, 917-20
Liouville’s theorem, 667
Logarithm function, 354, 403-4
Logarithms, 256-58
Logic, 52-53
see also Mathematical logic
Logicism, 1192-97
Logistica, 131
Logistica numerosa, 261-62
Logistica speciosa, 261-62
Lunes of Hippocrates, 41-42
Lyceum of Aristotle, 27
Maclaurin’s theorem, 442
Magnitudes of Eudoxus, 48-49, 68-73
see also Irrational number
Map problem, 1166
Map-making, 161-62, 235-36, 286, 564-65, 570-71
Mastery of nature, 226, 307
Mathematical design of nature, 153, 213-14, 218-19, 326, 328-29
Mathematical induction, 272
Mathematical logic, 281, 1187-92
Mathematical Principles of Natural Philosophy, 334-35, 358, 364-69, 380, 395, 440, 470, 492-93, 497, 573
Mathematical societies, 625
Mathematics and reality, 392-94, 879-80, 1028-31
Mathematics and science, 325-35, 394-96, 616-17
see also Methodology, of science
Mathieu functions, 713-14
Matrices, 804-12
congruent, 811
elementary divisors of, 809
equivalent, 809
Hermitian, 808
infinite, 811
invariant factors of, 809
inverse, 807
minimal polynomial of, 808
orthogonal, 809-10
rank of, 809
similar, 810
trace of, 808
transpose, 807
see also Characteristic equation
Characteristic root
Maxima and minima, 97-98, 343, 347-48, 838-39
Maxwell’s equations, 698-99
Mean value theorem, 464, 955
Means of numbers, 32
Measure, 1044-46
Mécanique analytique, 493, 541, 615, 949, 1029
Mécanique céleste, 495, 498-99, 530, 542, 972
Mechanics, 129, 162-66, 211-12, 286, 616
center of gravity, 129, 164, 211, 235, 343, 348
motion, 153, 162-63, 211-12, 335-38, 469
see also Astronomy
Pendulum motion
Projectile motion
Medicine, 169, 196, 203
Menelaus’s theorem, 121
Meromorphic functions, 642, 668
Metamathematics, 1205-6
Method of exhaustion, 50, 83-85, 108, 112-15, 177, 343
Method of infinite descent, 275
Methodology: in algebra, 268-70
in geometry, 286, 300, 302, 308, 322
of science, 223, 325-35
Methodus Fluxionum et Serierum Infini-tarum, 361-62, 364, 381, 439, 470
Meusnier’s theorem, 564
Miletus, 25, 27
Minimal surface, 539, 57g, 585-86, 750
Möbius band, 1165-66
Modular system, 825
Module, 931, 943
Moment problem, 1072, 1117
Morley’s theorem, 839-40
Multiple integrals, 425-26, 798, 1048-49
Multiple points of curves, See Singular points of curves
Multiple-valued complex functions, 641-42, 655-62
Museum of Alexandria, 102
Music, 148, 478-79, 481, 515, 521-22, 693-94
see also Vibrating-string problem
Napier’s rule, 240
Navier-Stokes equations, 696-97
Navigation, 119, 250, 286, 336-37, 470-71
N-dimensional geometry, 782, 890, 1028-31
Negative number, 143, 185, 192, 252-53, 592-93
Neumann problem, 685
Newton’s laws of motion, 366-67, 490
Newton’s parallelogram, 439-40, 552
Newton-Raphson method, 381
Nine-point circle, 837
Non-Archimedian geometry, 1016
Non-Euclidean geometry, 729, 861-81, 947, 1014, 1025
applicability, 872-73, 877, 921-22
axioms for, 1015
consistency, 880, 913-17
hyperbolic, 905-6
implications, 879-80
models, 888, 905-6, 912-17
priority of creation, 877-79
single and double elliptic, 904, 906, 912-13
see also Riemannian geometry
Non-Riemannian geometries, 1133-35
Normal, 560-61
Number, 29-30
amicable, 31, 278, 610
hexagonal, 31-32
pentagonal, 31
perfect, 31, 78, 137, 278, 610
polygonal, 137, 277-78, 829
prime, see Prime number
square, 30
triangular, 29-30, 828
see also Complex number, Irrational nummer, Negative number
Theory of numbers
Operator, 1076-77, 1082, 1085-89, 1094
Her-mitian, 1092-93
Opticks, 358
Optics, 88, 166-68, 196, 212-13, 285-86, 307, 314-15, 357, 579-81, 740
Ordinal number, 1000-1
Ordinary differential equation, 468-500, 578, 709-38
adjoint, 487
Bernoulli’s 474
Bessel’s, 488-89, 519
Clairaut’s 476-77
exact, 476
existence theorems, 717-21, 1178-79
first order, 451, 471-78
Fuchsian, 721-22, 724-26
higher order, 484-87
hypergeometric, 489, 712, 723
Lamé’s, 721-22
Legendre’s, 529, 711
linear, 485-87, 730-32
Mathieu’s, 713-14
method of series, 488-89, 709-12
nonlinear, 483-84, 732-38
periodic solutions, 713-14, 730-32
Riccati, 483-84
second order, 478-84
singular solutions of, 476-78
systems of, 490-92, 735, 742
variation of parameters, 497-99
Weber’s, 714
see also Asymptotic series, Auto-morphic functions
Qualitative theory of ordinary differential equations
Sturm-Liouville theory
Summability
Orthogonal system of functions, 716, 1066
Orthogonal trajectories, 474-75
Osculating circle, 556
Osculating plane, 559, 561
Ostrogradsky’s theorem, See Divergence theorem
Oval of Descartes, 315-16
P-adic fields, 1146-47
Pappus’s theorem, 128, 297-98
Pappus-Guldin theorem, 129
Papyri, 16, 20, 25, 132
Parabolic cylinder functions, 714
Paradoxes of set theory, 1182-85
Parallel axiom, 60, 177, 852, 863-67, 916, 1012, 1014
Parallel displacement, 1130-33
Parseval inequality, 1093
Parscval’s theorem, 716-17, 971, 1047
Partial derivative, 425
Partial differential equation, 362, 502-43, 567-68, 671-707
classification, 700-701
existence theorems, 685, 699-707, 1178-79
first order, 532-35
Hamilton-Jacobi, 744
heat equation, 672-75, 679, 687-89
Helmholtz, 693-96, 1056
nonlinear, 536-40
Poisson ’s, 682, 684-85
potential, 524-529, 659, 681-87, 703-5
reduced wave equation, 693-94
separation of variables, 516-17, 673-74
systems of, 540-42, 696-99
total, 532
wave equation, 502-22, 690-94
Pascal triangle, 272-73
Pascal’s theorem, 297-98, 848
Pasch’s axiom, 1011-12
Peano curve, 1018, 1161
Pell’s equation, 278, 610-11
Pendulum motion, 337, 469, 471-72, 47g, 556
Periodicity modules, 641, 662
Permanence of form, 773-75
Permutation, See Substitution
Permutations and combinations, 273
Persia, 4, 10
Perspective, 231-34, 286-87
Pi, 10-11, 19, 134-35, 251, 255, 353, 439, 448, 593, 980-82
Picard’s theorems, 668
Platonic school, 42-48
Plücker formulas, 857
Poincaré conjecture, 1175-76
Poincaré-Bendixson theorem, 737
Poincaré’s last theorem, 1178
Point at infinity, 290
Polar coordinates, 319
Pole and polar, 96-97, 294, 298-99, 845
Polyhedra, regular, 47, 85-86
Positional notation, 5-7, 185
Potential theory, 522-29, 659, 681-87, 1055-56
equation, 524-29, 659, 682-87, 703-5
function, 524, 682-86
Power series, 643-44
see also Taylor’s theorem
Precession of the equinoxes, 158, 369
Primary and secondary qualities, 326, 329
Prime number, 78, 277, 609, 830-32
see also Theory of numbers
Prime number theorem Prime number theorem, 830-32
Principia Mathematica, 1193
Principle of continuity, 385-87, 841, 843-45
Principle of duality, 845-46, 848-49, 855
Principle of Least Action, 581-82, 587-89, 620, 739-45
Principle of Least Time, 315, 580-81
Principle of Stationary Phase, 1099
Printing, 217
Probability, 273
Projectile motion, 286, 476, 479
Projection, 232, 287
Projective geometry, 233, 285-301, 834-59, 1007-10
algebraic, 852-59
and metric geometry, 904-23
relation to Euclidean geometry, 850-52, 909, 1033
relation to non-Euclidean geometry, 909-12, 1033
see also Algebraic invariants
Projective plane, 290, 1168
Proof, 14, 20-22, 34, 44-46, 50, 144, 171, 198-99, 282, 383-89, 393-94, 426-34, 617-19, 1024-26
indirect method of, 33, 44-45
Proportion, 32, 137-38, 237
Eudoxian theory of, 68
Pseudosphere, 893, 905
Ptolemy dynasty, 102-3
Puiseux’s theorem, 552-53
Pure and applied mathematics, 1036-38
Pythagorean number philosophy, 219
Pythagorean theorem, 10, 20, 33, 63-64, 184
Pythagorean triples, 10, 31-32, 34
Pythagoreans, 27-34, 49, 147-50
Quadratic equation, 8-9, 19, 186-87, 192-93
solved geometrically, 76-77
Quadratic form, 799-80
reduction to standard form, 799, 801-2
infinite, 1063-66
see also Law of inertia
Quadratic reciprocity, 611-12, 813-15, 817
Quadratrix, 39-40, 48
Quadrature, 42
Quadric surface, 108-10, 168, 545-46, 848
Quadrivium, 146, 149-50, 201-2
Qualitative theory of ordinary differential equations, 732-38, 1170
Quantics, 928
Quantitative versus qualitative knowledge, 333-34
Quartic equations, 267-70
Quaternion, 779-82, 791, 1025
Quintic equation, 763
Rate of change, instantaneous, 344, 360
Reduction of singularities, 941-42
Reformation, 218
Relativity, 894, 1130-31
Religious motivation, 219-20, 359
Residue, 638, 640
Resolvent equation, 604, 760
Resultant, 606-8, 797-98
Revival of Greek works, 205-7, 216-17
Ricci tensor, 1127
Ricci ’s lemma, 1129
Riemann four index symbol, 894, 1125
Riemann hypothesis, 831
Riemann mapping theorem, 666
Riemann problem, 724, 726, 1069
Riemann surface, 656-62, 934-35, 937
Riemann zeta function, 831
Riemannian geometry, 889-99, 1126-27, 1131-33
applicability, 893
Ricmann-Lebesgue lemma, 1046-47
Riemann-Roch theorem, 665, 940
Riesz representation theorem, 1085
Riesz-Fischer theorem, 1072, 1084
Rigor, 947-77, 1025, 1209
see also Proof
Ring, 821-22, 931, 1150-53
Romans, 106, 178-80
Rule of false position, 18-19
Ruled surfaces, 567, 859
Scholastics, 207
Schwarz-Christoffel mapping, 666-67, 687
Schwarz’s inequality, 1083, 1093
Section, 232, 287
Seleucid period, 4, 102
Semicubical parabola, 98, 550
Serret-Frenet formulas, 561-62
Set, 970, 994-1003, 1159
closed, 995-1078, 1160
derived, 970, 1078
enumerable, 995-97
first species, 970
infinite, 992-1003
limit point of, 970
open, 995, 1160
perfect, 995
power of, 995
well-ordered, 1001
see also Space, abstract
Shape of the earth, 469-70, 522
Sieve of Eratosthenes, 138
Simplex, 1171
Singular points of curves, 549-52, 935, 941-42
conjugate point, 550
cusp, 549-50
double point, 549
multiple point, 549
node, 549
Singularities of differential equations, 721-26, 733, 737
Slide rule, 258
Sophists, 27, 37-42
Sound, 482-83, 520-22, 691
Space: abstract, 1078-80, 1159-63
compact, 1078, 1160
complete, 1161
connected, 1160
extremal, 1078
function, 1078-80, 1082-84, 1178
interior point of, 1078
limit point of, 1160
metric, 1079, 1160-61
metrizable, 1161
neighborhood, 1079, 1160-61
normal, 1087-89
perfect, 1078
separable, 1160
see also Banach space
Hilbert space
Set
Space curves, 235, 557-62, 566, 941
Special functions, 422-26, 709, 715
Specialization, 1024
Spherical circle, 845, 854
Spherical functions, 529-30, 711-12
Spherical harmonics, See Spherical functions
Stability of the solar system, 730, 735-36
Stereometry, 235
Sieltjes integral, 1041, 1085
Stirling series, 453, 1097
Stokes line, 1103, 1108
Stokes’theorem, 790
Straight line, infiniteness of, 863
Sturm-Liouville theory, 715-17
Subgroup, 758
invariant, see normal
normal, 761
self-conjugate, see Normal
Substitution, 602-3, 757-58
Sumcrians, 4
Summability, 464, 1098, 1109-20
Abel, 1111
Borei, 1117-19
Cesàro, 1113
Frobenius, 1112, 1119
Holder, 1112-13
Stieltjes, 1115-16
Summation convention, 1127
Superposition, 87
Sylow’s theorem, 768, 1141
Symbolic logic, See Mathematical logic
Symbolism, 10, 139-40, 143-44, 186, 192, 259-63, 340, 378
Symmetric functions, 600, 602
Tangent, 342-46
Tauberian theorem, 1119
Tautochrone, See Isochrone
Taylor’s theorem, 442, 464-65, 964
Tensor analysis, 784, 1122-23
Théorie analytique de la chaleur, 672, 961, 966
Théorie des fonctions analytiques, 406, 430-32, 948, 964, 1029
Theory of algebraic equations, 270-72, 381, 597-606
Theory of forms, 826-29
Theory of numbers, 10, 29-32, 53, 77-80, 136-43, 274-78, 608-12, 813-22, 925
analytic, 829-32
see also Biquadratic reciprocity
Cubic reciprocity
Pell’s equation
Prime number
Prime number theorem
Quadratic reciprocity
Theory of forms
Theory of perturbations, 494-98
Theory of surfaces, 562-69, 859
cubic, 859
differential geometric, 882-88
isometric, 886
quartic, 859
see also Algebraic geometry
Kummer surface
Quad-ric surface
Theory of types, 1195
Theta functions, 649-50
Three-body problem, 369, 492-93, 496-97, 731-33, 735
Topology, 920, 1158-81
combinatorial, 1159, 1163-81
point Set, 1159-63
Torsion: of a complex, 1173-74, 1176-77, 1180
of a curve, 559-62
Tractatus de Quadratura Curvarum, 361-62
Tractrix, 382, 473-74, 905
Transcendental number, 593-94, 980-82
Transfinite number, 992-1003
Transformation of coordinates, 426, 546-47
Translation of books, 206, 220
Triangle inequality, 1079, 1083
Trigonometric series, 454-59, 508-14, 968-70
see also Fourier series.
Trigonometry: Arabic, ’195-96
Greek, 119-26
Hindu, 189
plane, 119-26, 189, 237-40
Renaissance, 237-40
spherical, 119-26, 237-40
Trivium, 202
Truth, 45, 50, 151, 219, 305-6, 326, 330, 619-21, 879-80, 890, 893, 980, 1031-36
Undecidable propositions, 1202
Undefined terms, 51, 988, 1008, 1010
Unicursal curve, 552
Uniformization of curves, 937-39
Universities, 201, 209, 214, 220, 397-98, 621-22
Vector analysis, 776-79, 785-91
Velaria problem, 748
Velocity potential, 525, 686
Vibrating membrane, 518-19, 689, 713
Vibrating-string problem, 478-79, 503-18
Volumes, calculation of, 343, 348
Waring’s theorem, 609
Weierstrass factorization theorem, 667-68
Weierstrass’s theorem, 966
WKBJ method, 1102, 1108
Word problem, 1141, 1143
Zeno’s paradoxes, 35-37, 992
Zero, 6, 132, 185
Zonal harmonics, See Legendre polynomial