Reading the Principia

Cover......Page 1
Half-title......Page 3
Reading the Principia: The Debate on Newton’s Mathematical Methods for Natural Philosophy from 1687 to 1736......Page 5
Copyright......Page 6
Contents......Page 7
1.1.1 Principia as a plural......Page 9
1.1.2 The debate on the mathematical methods of the Principia, 1687-1736......Page 11
1.2 Plan of the work......Page 14
1.3 Conventions: abbreviations, quotations and mathematical notation......Page 20
1.4 Acknowledgments......Page 22
2.2.1 Mathematical background......Page 25
2.2.2 Power series......Page 26
2.2.4 The three rules of Newton\'s analysis......Page 28
2.2.5.1 Definitions......Page 29
2.2.5.3 Algorithm......Page 30
2.2.6.1 Two problems......Page 31
2.2.7 Some general results on quadratures......Page 34
2.3.1 The Ancients versus the Moderns......Page 35
2.3.2 Geometry and Nature: first and ultimate ratios......Page 40
2.4 Conclusion: Newton\'s classicism......Page 45
3.1 Purpose of this chapter......Page 47
3.2 Book 1: an overview......Page 49
3.3 First and ultimate ratios: Section 1, Book 1......Page 51
3.4.1 Synthetic method of first and ultimate ratios applied to the law of areas: Props. 1–2......Page 56
3.4.2 Synthetic method of first and ultimate ratios applied to the geometric representation of central force: Prop. 6......Page 58
3.4.3 Proposition 6 applied to the direct problem: Props. 7-13......Page 59
3.4.4 The inverse problem: Cor. 1 to Props. 11-13......Page 62
3.4.5 Scaling conics: Props. 16 and 17......Page 64
3.5 Quadratures applied to the general inverse problem: Sections 7 and 8, Book 1......Page 66
3.6 The geometric representation of central force (Prop. 6) and infinite series applied to ‘mobile orbits’: Section 9, Book 1......Page 68
3.7 Qualitative geometrical modelling applied to the three-body problem: Section 11, Book 1......Page 73
3.8 Synthetic techniques and analytical quadratures applied to the attraction of extended bodies: Sections 12–13, Book 1......Page 76
3.9 Power series applied to optics: Sections 13-14, Book 1......Page 87
3.11 Geometrical representation of logarithms applied to projectiles in resisting media: Props. 1, 2 and 5, Book 2......Page 89
3.12 Synthetic method of moments and fluxions: Lemma 2, Section 2, Book 2......Page 93
3.13 Application of Lemma 2 to motion in resisting media: Prop. 8, Book 2......Page 94
3.14 Controversial issues......Page 96
3.15 Book 3: some remarks......Page 98
3.16 Conclusion: quadrature avoidance......Page 103
4.1 Purpose of this chapter......Page 107
4.2 Prisca geometria......Page 109
4.3 Newton and geometry in the Principia......Page 112
4.4 Newton and his readers......Page 114
4.5 Newton\'s analytical approach to central force......Page 116
4.6 Newton and analysis and synthesis in the Principia......Page 120
4.7 Conclusion: private discoveries and public style......Page 123
5.1 Purpose of this chapter......Page 126
5.2 Huygens as a mathematician......Page 127
5.3 Huygens and Newton\'s Principia......Page 129
5.4 Proportion theory......Page 133
5.5.1.2 Commentary......Page 137
5.5.2 Huygens\' notes on Proposition 6: Manuscript G, f. 15r......Page 138
5.5.2.1 Text......Page 139
5.5.2.2 Commentary......Page 141
5.6 Conclusion: Huygens\' vs. Newton’s classicism......Page 142
6.2.1 In search of a characteristica universalis......Page 144
6.2.2 Infinite series, 1672-73......Page 145
6.2.3 The geometry of infinitesimals, 1673-74......Page 146
6.2.4 The calculus of infinitesimals, 1675-86......Page 150
6.3 Dynamics......Page 153
6.3.1 First applications of Leibniz\'s calculus to dynamics......Page 155
6.3.2 Comparison of Leibniz\'s and Newton\'s mathematizations of trajectories......Page 160
6.4.1 \'Not equivalent in practice\'......Page 164
6.4.2.1 Actual infinitesimals do not exist......Page 167
6.4.2.2 Infinitesimals as variables......Page 168
6.4.2.3 From infinitesimals to limits......Page 170
6.5 Conclusion: different policies......Page 172
7.1 Purpose of this chapter......Page 177
7.2 Philosophers without Mathematicks......Page 178
7.3 The best geometers and naturalists......Page 185
7.4 Guides to the Principia......Page 195
7.5 Conclusion: private exchanges, public expositions......Page 200
8.1 Purpose of this chapter......Page 203
8.2 The Bernoullian school......Page 204
8.3 \'Une science cabalistique\'......Page 205
8.4 Varignon on central forces and resisted motion......Page 209
8.5 Hermann\'s Phoronomia......Page 213
8.6.1 Johann Bernoulli and the Principia......Page 224
8.6.2.1 The inverse problem of central forces in the Principia: the public solution......Page 225
8.6.2.2 Newton’s unpublished analytical approach to the inverse problem of inverse cube forces......Page 230
8.6.2.3 Bernoulli\'s calculus approach to the inverse problem of central forces......Page 231
8.6.2.4 Bernoulli\'s criticism of Corollary 1......Page 233
8.6.2.5 The analytical approach in the British school: De Moivre, Keill and Cotes......Page 234
8.6.2.6 Keill and Bernoulli......Page 237
8.6.2.7 The acceptance of Corollary 1......Page 238
8.6.2.8 Moral......Page 240
8.6.3.1 Proposition 10: the 1687 version......Page 241
8.6.3.2 An error is announced......Page 244
8.6.3.3 The corrected 1713 version......Page 245
8.6.3.4 The quarrel over Newton\'s understanding of higher-order infinitesimals......Page 248
8.6.3.5 Moral......Page 252
8.7 A remarkable improvement......Page 255
8.8 Conclusion: confronting values......Page 257
9.2 Equivalence......Page 258
9.3.1 Inner validation criteria......Page 261
9.3.2 Policies......Page 263
9.3.3 Research priorities......Page 264
9.4 Postscriptum: a decline of British mathematical science?......Page 265
Appendix......Page 271
References......Page 273
Index......Page 289