Newton's Principia for the Common Reader

by Chandrasekhar, S.

• ISBN13: 9780198526759
• ISBN10: 019852675X
• Edition: Reprint
• Format: Paperback
• Copyright: 2003-06-12
• Publisher: Clarendon Press

Summary

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.

Table of Contents

Acknowledgements	xxi
Prologue	xxiii
The beginnings and the writing of the Principia	1	(16)
Introduction	1	(1)
The plague years	1	(6)
The year 1679	7	(1)
The year 1684	7	(3)
The years 1685--1686: the writing of the Principia	10	(7)
Basic concepts: Definitions and Axioms	17	(26)
Introduction	17	(1)
Basic concepts: Definitions	17	(5)
Definition I	17	(1)
Definition II	18	(1)
Definition III	18	(1)
Definition IV	19	(1)
Definition V	19	(1)
Definition VI	20	(1)
Definition VII	20	(1)
Definition VIII	20	(2)
Basic concepts: the Laws of Motion	22	(8)
Law I	22	(1)
Law II	23	(1)
Law III	23	(1)
Corollaries I-IV	24	(3)
Lemma XXIII	27	(2)
Corollaries V and VI	29	(1)
The Scholium to the Laws of Motion	30	(5)
Additional amplifications	35	(8)
The proportionality of mass and weight and the experiments on the pendulums	35	(1)
Proposition XXIV, Book II	35	(2)
Maxwell's reformulation of Newton's Laws of Motion	37	(4)
The Newtonian principle of relativity	41	(2)
On the notion of limits and the ratios of evanescent quantities	43	(14)
Introduction	43	(1)
Lemma I	43	(1)
Lemmas II--IV	44	(3)
Lemmas V--VIII	47	(3)
Lemmas IX and X	50	(2)
Lemma XI	52	(5)
On the motion of particles under centripetal attraction: an introduction to Newton's treatment	57	(10)
Introduction	57	(1)
The dynamics of a particle under a general law of centripetal attraction	58	(3)
The conservation of angular momentum	58	(1)
The law of areas	59	(1)
The conservation of energy	60	(1)
The equation governing r in the orbital plane	60	(1)
The dynamics of a particle under the inverse-square law of attraction	61	(3)
The Lenz vector and the Lenz equation	62	(1)
Kepler's third law	63	(1)
An alternative derivation of the elliptical orbit	64	(1)
The accelerations and velocities along a curved orbit	64	(3)
The law of areas and some relations which follow	67	(26)
Introduction	67	(1)
The area theorem	67	(9)
Proposition I	67	(2)
Corollaries I--VI	69	(1)
Proposition II	70	(1)
Corollaries I and II	70	(1)
Proposition III	71	(1)
Corollaries I--IV	71	(1)
Proposition IV	72	(1)
Corollaries I--IX	73	(2)
Proposition V	75	(1)
Newton's relations for determining the law of centripetal attraction from the orbit	76	(7)
Proposition VI	76	(1)
Corollaries I--V	77	(2)
Proposition VII	79	(1)
`The same otherwise'	80	(1)
Corollaries I--III	80	(2)
Proposition VIII	82	(1)
Two simple illustrations of the basic relation	83	(10)
Proposition IX	83	(2)
`The same otherwise'	85	(1)
Lemma XII	86	(1)
Proposition X	86	(2)
`The same otherwise'	88	(1)
Corollaries I and II	89	(4)
The motion of bodies along conic sections	93	(21)
Introduction	93	(1)
Proposition XI	93	(4)
`The same otherwise'	96	(1)
Proposition XII	97	(1)
`The same otherwise'	98	(1)
Proposition XIII: the motion of a body along a parabola	98	(5)
Lemmas XIII and XIV	98	(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