Newton's Principia for the Common Reader

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ISBN: 9780198526759 | 019852675X
Cover: Paperback
Copyright: 6/12/2003

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Representing a decade's work from one of the world's most distinguished physicists, this major publication is, as far as is known, the first comprehensive analysis of Newton's Principia without recourse to secondary sources. Chandrasekhar analyses some 150 propositions which form a direct chain leading to Newton's formulation of his universal law of gravitation. In each case, Newton's proofs are arranged in a linear sequence of equations and arguments, avoiding the need to unravel the necessarily convoluted style of Newton's connected prose. In almost every case, a modern version of the proofs is given to bring into sharp focus the beauty, clarity, and breathtaking economy of Newton's methods. This book will stimulate great interest and debate among the scientific community, illuminating the brilliance of Newton's work under the steady gaze of Chandrasekhar's rare perception.

Acknowledgements

xxi

Prologue

xxiii

The beginnings and the writing of the Principia

(16)

Introduction

(1)

The plague years

(6)

The year 1679

(1)

The year 1684

(3)

The years 1685--1686: the writing of the Principia

(7)

Basic concepts: Definitions and Axioms

(26)

Introduction

(1)

Basic concepts: Definitions

(5)

Definition I

(1)

Definition II

(1)

Definition III

(1)

Definition IV

(1)

Definition V

(1)

Definition VI

(1)

Definition VII

(1)

Definition VIII

(2)

Basic concepts: the Laws of Motion

(8)

Law I

(1)

Law II

(1)

Law III

(1)

Corollaries I-IV

(3)

Lemma XXIII

(2)

Corollaries V and VI

(1)

The Scholium to the Laws of Motion

(5)

Additional amplifications

(8)

The proportionality of mass and weight and the experiments on the pendulums

(1)

Proposition XXIV, Book II

(2)

Maxwell's reformulation of Newton's Laws of Motion

(4)

The Newtonian principle of relativity

(2)

On the notion of limits and the ratios of evanescent quantities

(14)

Introduction

(1)

Lemma I

(1)

Lemmas II--IV

(3)

Lemmas V--VIII

(3)

Lemmas IX and X

(2)

Lemma XI

(5)

On the motion of particles under centripetal attraction: an introduction to Newton's treatment

(10)

Introduction

(1)

The dynamics of a particle under a general law of centripetal attraction

(3)

The conservation of angular momentum

(1)

The law of areas

(1)

The conservation of energy

(1)

The equation governing r in the orbital plane

(1)

The dynamics of a particle under the inverse-square law of attraction

(3)

The Lenz vector and the Lenz equation

(1)

Kepler's third law

(1)

An alternative derivation of the elliptical orbit

(1)

The accelerations and velocities along a curved orbit

(3)

The law of areas and some relations which follow

(26)

Introduction

(1)

The area theorem

(9)

Proposition I

(2)

Corollaries I--VI

(1)

Proposition II

(1)

Corollaries I and II

(1)

Proposition III

(1)

Corollaries I--IV

(1)

Proposition IV

(1)

Corollaries I--IX

(2)

Proposition V

(1)

Newton's relations for determining the law of centripetal attraction from the orbit

(7)

Proposition VI

(1)

Corollaries I--V

(2)

Proposition VII

(1)

`The same otherwise'

(1)

Corollaries I--III

(2)

Proposition VIII

(1)

Two simple illustrations of the basic relation

(10)

Proposition IX

(2)

`The same otherwise'

(1)

Lemma XII

(1)

Proposition X

(2)

`The same otherwise'

(1)

Corollaries I and II

(4)

The motion of bodies along conic sections

(21)

Introduction

(1)

Proposition XI

(4)

`The same otherwise'

(1)

Proposition XII

(1)

`The same otherwise'

(1)

Proposition XIII: the motion of a body along a parabola

(5)

Lemmas XIII and XIV

(2)

Proposition XIII

100

(2)

Corollaries I and II

102

(1)

Kepler's third law: Propositions XIV and XV

103

(1)

Proposition XIV

103

(1)

Proposition XV

104

(1)

Corollary I

104

(1)

Amplifications: Proposition XVI

104

(10)

Proposition XVI

104

(1)

Corollaries I--IX

105

(2)

Proposition XVII

107

(3)

Corollaries I--IV

110

(2)

Scholium

112

(1)

A personal reflection

113

(1)

Supplement: on dual laws of centripetal attraction

114

(229)

A recapitulation

114

(5)

A digression

115

(1)

The orbit described is an ellipse

115

(1)

The orbit described is a hyperbola

116

(1)

A body orbiting the conjugate hyperbola with the centre of attraction at S

117

(1)

The self-duality of the inverse-fifth power law of attraction

117

(2)

The mapping of orbits described in the complex plane

119

(3)

The dual laws of centripetal forces

122

(5)

Kepler's equation and its solution

127

(16)

Introduction

127

(1)

Kepler's equation

128

(2)

Proposition XXX

130

(3)

Corollaries I--III

132

(1)

Lemma XXVIII

133

(6)

Proposition XXXI

139

(4)

Scholium

141

(2)

The rectilinear ascent and descent of bodies

143

(22)

Introduction

143

(1)

An ab initio treatment of rectilinear motion

143

(5)

The elliptic case

144

(1)

The hyperbolic case

145

(2)

The parabolic case

147

(1)

The velocity of the body

148

(1)

Proposition XXXII

148

(2)

Proposition XXXIII

150

(4)

A personal reflection

153

(1)

Proposition XXXIV

153

(1)

The reduction of the problem of rectilinear motion to one in circular motion

154

(6)

Proposition XXXV

154

(1)

Case I

155

(1)

Case II

156

(1)

Proposition XXXVI

157

(1)

Proposition XXXVII

158

(2)

A pause

160

(1)

Proposition XXXVIII

160

(1)

Corollaries I and II

161

(1)

The initial-value problem

161

(4)

Proposition XXXIX

161

(2)

Corollaries I--III

163

(2)

The conservation of energy and the initial-value problem

165

(18)

Introduction

165

(1)

The energy integral

165

(3)

Proposition XL

165

(3)

Corollaries I and II

168

(1)

Proposition XLI

168

(4)

Corollaries I and II

172

(1)

The mystery of the missing corollary

172

(2)

Motion under an inverse-cube law of centripetal attraction

174

(6)

Some observations

179

(1)

Proposition XLII

180

(3)

On revolving orbits

183

(18)

Introduction

183

(1)

The theorem of revolving orbits

184

(1)

Proposition XLIII

184

(3)

Proposition XLIV

187

(5)

Corollaries I--VI

189

(3)

Proposition XLV

192

(9)

Examples 1--3

194

(2)

Corollaries I and II

196

(5)

A pause

201

(4)

Introduction

201

(1)

Proposition XLVI

201

(4)

Proposition XLVII

203

(2)

The two-body problem

205

(14)

Introduction

205

(1)

The two-body problem: the general theorems

206

(7)

Proposition LVII

206

(2)

Proposition LVIII

208

(1)

Corollaries I--III

209

(1)

Proposition LIX

210

(1)

Proposition LX

211

(1)

Proposition LXI

212

(1)

Initial-value problems

213

(2)

Proposition LXII

213

(1)

Proposition LXIII

214

(1)

The solution of a many-body problem

215

(4)

Proposition LXIV

215

(4)

The method of the variation of the elements of a Kepler orbit and Newton's lunar theory: an introduction to Propositions LXV--LXIX

219

(16)

Introduction

219

(1)

The basic equations, definitions, and the coordinate system adopted

219

(4)

The variation of the elements

223

(5)

Variation of h

223

(1)

Variation of ι

223

(1)

Variation of Ω

224

(1)

Variation of e

224

(1)

Variation of ω = (ν, e)

225

(1)

Variation of a and n

226

(1)

Variation of Kepler's equation

227

(1)

Summary

228

(1)

Application of the method of the variation of the elements to lunar motion

228

(7)

The disturbing function

229

(1)

The components of F( = (Fr, Fα, Fh))

230

(3)

Application of the variational equations

233

(2)

The three-body problem: the foundations of Newton's lunar theory

235

(34)

Introduction

235

(1)

Proposition LXV

235

(2)

Cases I and II

236

(1)

Corollary III

236

(1)

Proposition LXVI

237

(2)

Cases I and II

237

(2)

Proposition LXVI (continued): Corollaries I--VI

239

(8)

The perturbing function

239

(2)

The centripetal attraction

241

(1)

The perturbed orbit

242

(1)

The variation of the `constant of areas'

243

(1)

The determination of x

243

(1)

Corollaries I--VI

244

(3)

Proposition LXVI (continued): Corollaries VII and VIII---the rotation of the line of apsides

247

(3)

Corollary VII

247

(3)

Corollary VIII

250

(1)

Proposition LXVI (continued): Corollaries IX--XVII

250

(9)

Corollary IX: the variation of the eccentricity

250

(2)

Corollary X: the variation of the inclination

252

(2)

Corollary XI: the variation of the direction of the ascending node (Ω)

254

(1)

Corollary XII

255

(1)

Further elaborations: Corollaries XIII--XVII

256

(1)

Corollary XIII

256

(1)

Corollary XIV

257

(1)

Corollary XV

257

(1)

Corollary XVI

257

(1)

Corollary XVII

257

(1)

A personal reflection

258

(1)

Proposition LXVI (continued): Corollaries XVIII--XXII

259

(6)

Corollary XVIII

260

(1)

Corollary XIX

260

(2)

Corollary XX

262

(1)

Corollary XXI

263

(1)

Corollary XXII

263

(2)

Propositions LXVII--LXIX

265

(4)

Corollary

265

(1)

Proposition LXIX and Corollaries I and II

265

(2)

Corollary III

267

(1)

Scholium

267

(2)

`The superb theorems'

269

(34)

Introduction

269

(1)

Scholium

269

(1)

Propositions LXX--LXXII

269

(6)

Proposition LXX

269

(1)

Proposition LXXI

270

(3)

A personal reflection

273

(1)

Proposition LXXII

273

(2)

Corollary III

275

(1)

Propositions LXXIII--LXXV

275

(5)

Proposition LXXIII

275

(1)

Proposition LXXIV

276

(2)

Corollary III

278

(1)

Proposition LXXV

278

(1)

Corollaries I--IV

279

(1)

Proposition LXXVI

280

(3)

Corollaries I--IX

281

(2)

Propositions LXXVII and LXXVIII

283

(4)

Proposition LXXVII

283

(1)

Cases 1--6

283

(3)

Proposition LXXVIII

286

(1)

Corollary

286

(1)

Scholium

286

(1)

Lemma XXIX and Propositions LXXIX--LXXXI

287

(6)

Examples 1--3

291

(2)

Proposition LXXXII and the discovery of the method of inversion and the principle of images

293

(5)

Proposition LXXXII

294

(2)

On the discovery of the method of images

296

(1)

Method of Inversion

297

(1)

Propositions LXXXIII and LXXXIV

298

(4)

Proposition LXXXIII

298

(1)

Proposition LXXXIV

299

(1)

Scholium

299

(3)

Some personal reflections

302

(1)

Attraction by non-spherical bodies

303

(20)

Introduction

303

(1)

How we may discriminate between different laws of centripetal attraction

303

(3)

Proposition LXXXV

304

(1)

Proposition LXXXVI

305

(1)

The scaling law

306

(1)

Proposition LXXXVII and Corollary I

306

(1)

Corollary II

307

(1)

Propositions LXXXVIII and LXXXIX

307

(2)

Proposition LXXXVIII

307

(1)

Corollary

308

(1)

Proposition LXXXIX

308

(1)

Corollary

309

(1)

The attraction by circular discs and round solids at points along their axes

309

(4)

Proposition XC

309

(1)

Corollaries I--III

310

(1)

Proposition XCI

311

(1)

Corollary I

312

(1)

Corollaries II and III of Proposition XCI and Proposition XCII

313

(4)

Corollary II

313

(3)

Corollary III

316

(1)

Proposition XCII

317

(1)

Proposition XCIII

317

(6)

Cases 1 and 2

318

(1)

Corollaries I--III

319

(1)

Scholium

320

(3)

A digression into Opticks

323

(20)

Introduction

323

(1)

Propositions XCIV--XCVI

323

(4)

Proposition XCIV

324

(2)

Proposition XCV

326

(1)

Proposition XCVI

326

(1)

The Scholium

327

(2)

The ovals of Descartes

329

(5)

Proposition XCVII

329

(1)

Corollary I

330

(1)

A Comment

331

(1)

Corollary II

331

(1)

Proposition XCVIII

332

(2)

The concluding Scholium of Book I

334

(9)

Scholium

334

(1)

Appendix I. An analytic solution for the ovals of Descartes

334

(4)

Appendix II. Maxwell on the ovals of Descartes

338

(2)

Postscript

340

(3)

Introduction to Newton's System of the World (Book III)

343

(194)

Prolegomenon

345

(8)

Rules of reasoning in philosophy

345

(1)

Phenomena

346

(2)

Propositions

348

(5)

The universal law of gravitation

353

(28)

Introduction

353

(2)

Propositions and rules (to which references are made)

353

(2)

Propositions I--III

355

(2)

Proposition IV and the Moon test

357

(4)

Scholium

360

(1)

The emergence of the law of gravitation

361

(1)

Proposition V

361

(1)

Scholium

362

(1)

Proposition VI: the confirmation of the equality of the inertial and the gravitational masses by astronomical data

362

(8)

Proposition VI

362

(7)

Corollaries I--V

369

(1)

Proposition VII: the universal law of gravitation

370

(1)

Corollaries I and II

370

(1)

Propositions VIII and IX: the implications of the `superb theorems'

371

(4)

Proposition VIII

371

(1)

Corollaries I--IV

372

(2)

Data

374

(1)

Mass

374

(1)

Mean density

374

(1)

Surface gravity

374

(1)

Corollary IV

374

(1)

Proposition IX

375

(1)

Propositions X--XIV

375

(6)

Proposition X

375

(1)

Hypothesis I

376

(1)

Proposition XI

376

(1)

Proposition XII

377

(1)

Corollary

378

(1)

Proposition XIII

378

(1)

Proposition XIV

379

(1)

Scholium

379

(2)

The figure of the Earth and of the planets

381

(18)

Introduction

381

(1)

Proposition XVIII and the historical background

381

(3)

Proposition XIX: the method of the canals

384

(10)

Newton's method of the canals

384

(2)

What Newton withheld

386

(3)

Newton's determination of g(ob)pole/g(ob)eq and ε/m

389

(3)

Application to the figure of Jupiter

392

(2)

The variation of gravity over an oblate spheroid

394

(5)

Proposition XX

394

(2)

A personal reflection

396

(3)

On the theory of tides

399

(20)

Introduction

399

(2)

A recapitulation

401

(2)

The tidal force of a distant body acting on the boundary of a spherical body

401

(2)

Proposition XXIV: an annotated version

403

(8)

Propositions XXV, XXXVI, and XXXVII

411

(4)

Proposition XXV

411

(1)

Proposition XXXVI

412

(1)

Corollary

413

(1)

Proposition XXXVII

414

(1)

Appendix: the equilibrium theory of the tides

415

(4)

The lunar theory

419

(36)

Introduction

419

(1)

Propositions XVII and XXI

420

(1)

Proposition XVII

420

(1)

Proposition XXI

420

(1)

Propositions XXII and XXIII

421

(2)

Proposition XXII

421

(2)

Proposition XXIII

423

(1)

Proposition XXVI

423

(2)

Proposition XXVII

425

(1)

Proposition XXVIII

425

(1)

Proposition XXIX

426

(4)

The variation in the ascending node, Ω: Proposition XXX

430

(4)

Proposition XXX

430

(3)

Corollary I

433

(1)

Corollary II

434

(1)

Proposition XXXI

434

(7)

Newton's procedure

436

(3)

Newton's transformation of the equation dΩ/dt for a Kepler ellipse

439

(2)

Propositions XXXII and XXXIII

441

(2)

Proposition XXXII

441

(2)

Proposition XXXIII

443

(1)

The variation of the inclination

443

(5)

Proposition XXXIV

444

(1)

Corollaries I--III

445

(1)

Corollary IV

446

(1)

Proposition XXXV

446

(2)

Scholium

448

(7)

The annual equation

449

(1)

The motion of the apogee and the `Portsmouth equation'

450

(5)

The precession of the equinoxes

455

(22)

Introduction

455

(1)

On the precession of the equinoxes: a current treatment

456

(10)

Euler's equations

458

(1)

Euler's angles

459

(1)

The equations governing precession and nutation

459

(6)

The solar contribution to the precession

465

(1)

The lunar contribution to the precession and the lunisolar precession

465

(1)

Moment of momentum, moment of inertia, and circulation

466

(6)

Lemma I

467

(2)

Lemma II

469

(1)

Lemma III

470

(1)

Hypothesis II

471

(1)

Proposition XXXIX: to find the precession of the equinoxes

472

(3)

A personal reflection

475

(2)

On comets

477

(60)

Introduction

477

(1)

Lemma IV and Proposition XL

478

(3)

Lemma IV

478

(2)

Proposition XL

480

(1)

Corollaries I--IV

480

(1)

Lemma V: Newton's theory of interpolation

481

(19)

Newton's treatment in Methodus Differentialis

482

(1)

Proposition 1

483

(1)

Proposition 2

484

(2)

A modern version of Proposition 2

486

(2)

A modern version of Proposition 3

488

(7)

Propositions 3--6

495

(1)

Proposition 3: Cases 1 & 2

495

(2)

Propositions 4--6

497

(1)

Lemma V

498

(2)

Lemmas VI--XI

500

(14)

Lemma VI

500

(1)

Lemma VII

501

(1)

Lemma VIII

502

(3)

Scholium

505

(3)

Lemma IX

508

(1)

Lemma X and `Lambert's theorem'

509

(1)

Lemma X

509

(2)

`Lambert's theorem'

511

(2)

Lemma XI

513

(1)

Propositions XLI and XLII

514

(21)

Proposition XLI

514

(1)

Recapitulation

515

(1)

The formulation of the problem

516

(1)

Newton's method of solution in the orbital plane

517

(3)

Newton's formulation of the solution

520

(10)

Proposition XLII

530

(3)

Appendix

533

(2)

The general Scholium

535

(2)

Miscellanea

537

(58)

The effect of air-drag on the descent of bodies

539

(16)

Newton's problem and its solution

539

(2)

Lemma III and Proposition XV

541

(8)

Lemma III

541

(2)

Proposition XV

543

(3)

Corollary I

546

(1)

Corollary II

546

(1)

Corollary III

546

(1)

Corollary IV

546

(1)

Corollary V

547

(1)

Corollary VI

547

(1)

Corollary VII

547

(2)

Corollaries VIII and IX

549

(1)

Proposition XVI--XVIII

549

(2)

Proposition XVI

549

(1)

Corollary I

550

(1)

Corollary II

550

(1)

Corollary III

550

(1)

Scholium

550

(1)

Proposition XVII

550

(1)

Proposition XVIII

551

(1)

An alternative method of solution of Newton's problem

551

(4)

The solid of least resistance

555

(16)

Introduction

555

(1)

Proposition XXXIV

556

(2)

The Scholium, I: the frustum of minimum resistance

558

(1)

The Scholium, II: the solid of least resistance

559

(8)

Newton's manner of solution

562

(5)

Amplifications

567

(4)

Newton's kernel-function

567

(1)

The Euler-Lagrange equations

567

(1)

The Legendre test

568

(1)

The Weierstrass test

569

(2)

The problem of the brachistochrone

571

(8)

Introduction

571

(2)

Newton's anonymous solution

573

(2)

The solution derived from its extremal property

575

(4)

Newton's manner of solution

577

(2)

The velocity of sound and of long waves in canals

579

(16)

Introduction

579

(1)

Propositions XLI--XLIII: Newton's conception of wave propagation

580

(1)

Propositions XLIV--XLVI: the propagation of long waves in canals

581

(5)

Proposition XLIV

581

(1)

Corollary I

582

(1)

Corollary II

582

(1)

Corollary III

583

(1)

Proposition XLV

583

(1)

Proposition XLVI

583

(1)

Corollary I

584

(1)

Corollary II

584

(1)

The theory of long waves in canals

584

(2)

A standard treatment of the velocity of sound

586

(1)

Propositions XLVII--L

587

(4)

Proposition XLVII

588

(1)

Proposition XLVIII

589

(1)

Proposition XLIX

589

(1)

Proposition L

590

(1)

Scholium

591

(4)

Epilogue

595

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Newton's Principia for the Common Reader

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