Lacroix and the Calculus (Science Networks. Historical Studies)

Contents......Page 7
Acknowledgments......Page 13
0. Introduction......Page 15
0.1 Lacroix and his Traité in the literature......Page 16
0.2.1 Comparisons......Page 17
0.2.2 Structure......Page 18
0.2.3 Notations......Page 19
1.1 Youth and early career (1765-1793)......Page 20
1.2 The most productive years (1793-1806)......Page 27
1.3 Second editions and prestige (1806-1820)......Page 31
1.4 Declining years (1820-1843)......Page 32
2.1 The project of the Traité......Page 34
2.2 The bibliography......Page 36
2.3 Volume I: differential calculus (1797)......Page 38
2.4 Volume II: integral calculus (1798)......Page 45
2.5 Volume III: differences and series (1800)......Page 53
2.6.1 One or two German partial translations......Page 62
2.6.2 The Greek partial and unpublished translation......Page 63
3.1.1 Infinitesimals......Page 65
3.1.2 Limits......Page 69
3.1.3 Carnot on the compensation of errors......Page 71
3.1.4 Power series......Page 72
3.2.1 Dating the Introduction and first two chapters of volume I......Page 80
3.2.2 Functions of one variable......Page 82
3.2.3 Functions of two or more variables......Page 85
3.2.4 Differentiation of equations......Page 87
3.2.5 Particular cases where the Taylor series does not apply......Page 92
3.2.6 Foundations for algebraic analysis......Page 93
3.2.7 Alternative principles for the differential calculus......Page 98
3.2.8 A criticism of Lagrange......Page 99
4. Analytic and differential geometry......Page 101
4.1.1 From "the application of algebra to geometry" to "analytic geometry"......Page 102
4.1.2 Lacroix and analytic geometry......Page 115
4.2.1 Differential geometry of plane curves......Page 125
4.2.2 Differential geometry of surfaces and "curves of double curvature"......Page 138
5.1.1 Conceptions of the integral......Page 152
5.1.2 Constants of integration, particular integrals, and definite integrals......Page 156
5.1.3 Series integration and approximate integration......Page 160
5.2.2 Euler's "generalmethod"......Page 170
5.2.3 "On the nature of integrals, and on the constants that must be added to them"......Page 176
5.2.4 Approximation of solutions of differential equations......Page 183
6.1 Types of solutions of differential equations in the 18th century......Page 189
6.1.1 Terminological complications......Page 190
6.1.2 Singular solutions......Page 191
6.1.3 Geometrical connections......Page 201
6.1.4 The formation of differential equations and their complete and general integrals......Page 228
6.2.1 Differential equations in two variables and their particular solutions......Page 236
6.2.2 Complete and general integrals and particular solutions of partial differential equations......Page 244
6.2.3 Geometrical connections......Page 253
6.2.4 Total differential equations not satisfying the conditions of integrability......Page 261
7.1 Indices......Page 267
7.1.1 Indices from Leibniz to Laplace......Page 268
7.1.2 Indices in Lacroix's Traité......Page 273
7.2.1 The peculiar equivalent to singular integrals in finite difference equations......Page 277
7.2.2 Biot's work and Lacroix's account......Page 281
7.3.1 "Equations in finite and infinitely small differences"......Page 287
7.3.2 Biot's work and Lacroix's account......Page 289
8.1 The Traité élémentaire de calcul… and the Cours élémentaire de mathématiques......Page 293
8.2 Analysis in the early years of the École Polytechnique......Page 299
8.3 Lacroix in the École Polytechnique......Page 304
8.4 From the large Traité to the Traité élémentaire......Page 305
8.5 The principles of the calculus......Page 308
8.6 Analytic and differential geometry......Page 312
8.7.1 Conceptions of the integral and approximate integration of explicit functions......Page 314
8.7.2 Approximate integration of differential equations......Page 317
8.8.1 Formation of differential equations and their types of solution......Page 318
8.8.2 Connections between differential equations and geometry......Page 320
8.9.3 Mixed difference equations......Page 321
8.10.1 The Portuguese translation (Rio de Janeiro, 1812-1814)......Page 322
8.10.2 The English translation (Cambridge, 1816)......Page 323
8.10.3 The German translations (Berlin, 1817; 1830-1831)......Page 326
8.10.4 The Polish translation (Vilnius, 1824)......Page 327
8.10.5 The Italian translation (Florence, 1829)......Page 328
8.10.6 The Greek translation (unpublished; Corfu, 1820's)......Page 329
9.1 Overview of the second edition......Page 330
9.2 The principles of the calculus......Page 334
9.3.1 Analytic geometry......Page 336
9.3.2 Differential geometry......Page 338
9.4.1 Approximate integration of explicit functions and conceptions of the integral......Page 339
9.4.2 Approximate integration of differential equations......Page 341
9.5.1 Differential equations in two variables and their particular solutions......Page 342
9.5.2 Partial differential equations and their particular solutions......Page 344
9.5.3 Geometrical connections......Page 345
9.5.4 Continuity of arbitrary functions......Page 347
9.5.5 Total differential equations not satisfying the conditions of integrability......Page 348
9.6.3 Mixed difference equations......Page 349
10.1.1 Originalities in Lacroix's Traité......Page 351
10.1.2 Misattributions of originality......Page 352
10.2 Impact......Page 353
10.3.1 Lagrange vs Monge; algebra vs geometry......Page 355
10.3.2 Encyclopedism and encyclopédisme......Page 356
10.4 Some further questions......Page 359
A.1 "Memoire sur le Calcul intégral aux différences partielles", 1785......Page 361
A.1.1 Report by Condorcet and Monge......Page 382
A.2 "Memoire sur les surfaces developpables et les equations aux differences ordinaires a trois variables", 1790......Page 384
B. Lacroix's historical appraisal of his own Traité......Page 405
C.1 Lacroix's lectures on differential and integral calculus at the École Polytechnique in 1799-1800......Page 410
C.2.1 Lacroix's views on the syllabus of analysis in 1800......Page 416
C.2.2 The approved programme of analysis for 1800-1801......Page 419
C.3.1 The official programme......Page 424
C.3.2 Summaries of lectures......Page 427
Jacques-Antoine-Joseph Cousin......Page 430
Pietro Paoli......Page 431
Bibliography......Page 433
B......Page 466
D......Page 467
F......Page 468
L......Page 469
M......Page 472
W......Page 473