The Principia: Mathematical Principles of Natural Philosophy. The Authoritative Translation and Guide

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Portrait of Isaac Newton at about the age of sixty, a drawing presented by Newton to David Gregory. For details see the following page.
Portrait of Isaac Newton at about the age of sixty, presented by Newton to David Gregory (1661–1708). This small oval drawing (roughly 3¾ in. from top to bottom and 3¼ in. from left to right) is closely related to the large oval portrait in oils made by Kneller in 1702, which is considered to be the second authentic portrait made of Newton. The kinship between this drawing and the oil painting can be seen in the pose, the expression, and such unmistakable details as the slight cast in the left eye and the button on the shirt. Newton is shown in both this drawing and the painting of 1702 in his academic robe and wearing a luxurious wig, whereas in the previous portrait by Kneller (now in the National Portrait Gallery in London), painted in 1689, two years after the publication of the Principia, Newton is similarly attired but is shown with his own shoulder-length hair.
This drawing was almost certainly made after the painting, since Kneller’s preliminary drawings for his paintings are usually larger than this one and tend to concentrate on the face rather than on the details of the attire of the subject. The fact that this drawing shows every detail of the finished oil painting is thus evidence that it was copied from the finished portrait. Since Gregory died in 1708, the drawing can readily be dated to between 1702 and 1708. In those days miniature portraits were commonly used in the way that we today would use portrait photographs. The small size of the drawing indicates that it was not a copy made in preparation for an engraved portrait but was rather made to be used by Newton as a gift.
The drawing captures Kneller’s powerful representation of Newton, showing him as a person with a forceful personality, poised to conquer new worlds in his recently gained position of power in London. This high level of artistic representation and the quality of the drawing indicate that the artist responsible for it was a person of real talent and skill.
The drawing is mounted in a frame, on the back of which there is a longhand note reading: “This original drawing of Sir Isaac Newton, belonged formerly to Professor Gregory of Oxford; by him it was bequeathed to his youngest son (Sir Isaac’s godson) who was later Secretary of Sion College; & by him left by Will to the Revd. Mr. Mence, who had the Goodness to give it to Dr. Douglas; March 8th 1870.”
David Gregory first made contact with Newton in the early 1690s, and although their relations got off to a bad start, Newton did recommend Gregory for the Savilian Professorship of Astronomy at Oxford, a post which he occupied until his death in 1708. As will be evident to readers of the Guide, Gregory is one of our chief sources of information concerning Newton’s intellectual activities during the 1690s and the early years of the eighteenth century, the period when Newton was engaged in revising and planning a reconstruction of his Principia. Gregory recorded many conversations with Newton in which Newton discussed his proposed revisions of the Principia and other projects and revealed some of his most intimate and fundamental thoughts about science, religion, and philosophy. So far as is known, the note on the back of the portrait is the only record that Newton stood godfather to Gregory’s youngest son.
THE PRINCIPIA Mathematical Principles of Natural Philosophy
The Authoritative Translation by I. Bernard Cohen and Anne Whitman assisted by Julia Budenz
Preceded by A GUIDE TO NEWTON’S PRINCIPIA by I. Bernard Cohen
UNIVERSITY OF CALIFORNIA PRESS
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© 1999 by The Regents of the University of California
Library of Congress Cataloging-in-Publication Data
Newton, Isaac, Sir, 1642–1727. [Principia. English] The Principia: mathematical principles of natural philosophy / Isaac
Newton; a new translation by I. Bernard Cohen and Anne Whitman assisted by Julia Budenz; preceded by a guide to Newton’s Principia by I. Bernard Cohen.
p. cm. Includes bibliographical references and index. ISBN: 978-0-520-29087-7 (cloth : alk. paper) ISBN: 978-0-520-29088-4 (pbk. : alk. paper) ISBN: 978-0-520-96481-5 (ebook) 1. Newton, Isaac, Sir, 1642–1727. Principia. 2. Mechanics—Early works
to 1800. 3. Celestial mechanics—Early works to 1800. I. Cohen, I. Bernard, 1914– . II. Whitman, Anne Miller, 1937–1984. III. Title. QA803.N413 1999 531—dc21
99-10278 CIP
Printed in the United States of America
25 24 23 22 21 20 19 18 17 16 10 9 8 7 6 5 4 3 2 1
The paper used in this publication meets the minimum requirements of ANSI/NISO Z39.48-1992 (R 2002) (Permanence of Paper). ******ebook converter DEMO Watermarks*******
with respect and affection
A GUIDE TO NEWTON’S PRINCIPIA
Contents of the Guide Abbreviations CHAPTER ONE: A Brief History of the Principia CHAPTER TWO: Translating the Principia CHAPTER THREE: Some General Aspects of the Principia CHAPTER FOUR: Some Fundamental Concepts of the
Principia CHAPTER FIVE: Axioms, or the Laws of Motion CHAPTER SIX: The Structure of Book 1 CHAPTER SEVEN: The Structure of Book 2 CHAPTER EIGHT: The Structure of Book 3 CHAPTER NINE: The Concluding General Scholium CHAPTER TEN: How to Read the Principia CHAPTER ELEVEN: Conclusion Notes to the Guide
THE PRINCIPIA (Mathematical Principles of Natural Philosophy)
Contents of the Principia
Halley’s Ode to Newton Newton’s Preface to the First Edition Newton’s Preface to the Second Edition Cotes’s Preface to the Second Edition Newton’s Preface to the Third Edition Definitions Axioms, or the Laws of Motion
BOOK 1: THE MOTION OF BODIES
BOOK 3: THE SYSTEM OF THE WORLD
General Scholium Notes to the Principia
Index
Abbreviations
CHAPTER ONE: A Brief History of the Principia 1.1 The Origins of the Principia 1.2 Steps Leading to the Composition and
Publication of the Principia 1.3 Revisions and Later Editions
CHAPTER TWO: Translating the Principia 2.1 Translations of the Principia: English Versions
by Andrew Motte (1729), Henry Pemberton (1729–?), and Thomas Thorp (1777)
2.2 Motte’s English Translation 2.3 The Motte-Cajori Version; The Need for a New
Translation 2.4 A Problem in Translation: Passive or Active
Voice in Book 1, Secs. 2 and 3; The Sense of “Acta” in Book 1, Prop. 1
2.5 “Continuous” versus “Continual”; The Problem of “Infinite”
CHAPTER THREE: Some General Aspects of the Principia 3.1 The Title of Newton’s Principia; The Influence
of Descartes 3.2 Newton’s Goals: An Unpublished Preface to the
Principia 3.3 Varieties of Newton’s Concepts of Force in the
Principia 3.4 The Reorientation of Newton’s Philosophy of
Nature; Alchemy and the Principia 3.5 The Reality of Forces of Attraction; The
Newtonian Style 3.6 Did Newton Make the Famous Moon Test of the
Theory of Gravity in the 1660s? 3.7 The Continuity in Newton’s Methods of
Studying Orbital Motion: The Three Approximations (Polygonal, Parabolic, Circular)
3.8 Hooke’s Contribution to Newton’s Thinking about Orbital Motion
3.9 Newton’s Curvature Measure of Force: New Findings (by Michael Nauenberg)
3.10 Newton’s Use of “Centrifugal Force” in the Principia
CHAPTER FOUR: Some Fundamental Concepts of the Principia
4.1 Newton’s Definitions 4.2 Quantity of Matter: Mass (Def. 1) 4.3 Is Newton’s Definition of Mass Circular? 4.4 Newton’s Determination of Masses 4.5 Newton’s Concept of Density 4.6 Quantity of Motion: Momentum (Def. 2) 4.7 “Vis Insita”: Inherent Force and the Force of
Inertia (Def. 3) 4.8 Newton’s Acquaintance with “Vis Insita” and
“Inertia”
4.9 Impressed Force; Anticipations of Laws of Motion (Def. 4)
4.10 Centripetal Force and Its Three Measures (Defs. 5, 6–8)
4.11 Time and Space in the Principia; Newton’s Concept of Absolute and Relative Space and Time and of Absolute Motion; The Rotating- Bucket Experiment
CHAPTER FIVE: Axioms, or the Laws of Motion 5.1 Newton’s Laws of Motion or Laws of Nature 5.2 The First Law; Why Both a First Law and a
Second Law? 5.3 The Second Law: Force and Change in Motion 5.4 From Impulsive Forces to Continually Acting
Forces: Book 1, Prop. 1; Is the Principia Written in the Manner of Greek Geometry?
5.5 The Third Law: Action and Reaction 5.6 Corollaries and Scholium: The Parallelogram
Rule, Simple Machines, Elastic and Inelastic Impact
5.7 Does the Concept of Energy and Its Conservation Appear in the Principia?
5.8 Methods of Proof versus Method of Discovery; The “New Analysis” and Newton’s Allegations about How the Principia Was Produced; The Use of Fluxions in the Principia
CHAPTER SIX: The Structure of Book 1 6.1 The General Structure of the Principia 6.2 Sec. 1: First and Ultimate Ratios 6.3 Secs. 2–3: The Law of Areas, Circular Motion,
Newton’s Dynamical Measure of Force; The
Centripetal Force in Motion on an Ellipse, on a Hyperbola, and on a Parabola
6.4 The Inverse Problem: The Orbit Produced by an Inverse-Square Centripetal Force
6.5 Secs. 4—5: The Geometry of Conics 6.6 Problems of Elliptical Motion (Sec. 6); Kepler’s
Problem 6.7 The Rectilinear Ascent and Descent of Bodies
(Sec. 7) 6.8 Motion under the Action of Any Centripetal
Forces (Sec. 8); Prop. 41 6.9 Specifying the Initial Velocity in Prop. 41; The
Galileo-Plato Problem 6.10 Moving Orbits (Sec. 9) and Motions on Smooth
Planes and Surfaces (Sec. 10) 6.11 The Mutual Attractions of Bodies (Sec. 11) and
the Newtonian Style 6.12 Sec. 11: The Theory of Perturbations (Prop. 66
and Its Twenty-Two Corollaries) 6.13 Sec. 12 (Some Aspects of the Dynamics of
Spherical Bodies); Sec. 13 (The Attraction of Nonspherical Bodies); Sec. 14 (The Motion of “Minimally Small” Bodies)
CHAPTER SEVEN: The Structure of Book 2 7.1 Some Aspects of Book 2 7.2 Secs. 1, 2, and 3 of Book 2: Motion under
Various Conditions of Resistance 7.3 Problems with Prop. 10 7.4 Problems with the Diagram for the Scholium to
Prop. 10 7.5 Secs. 4 and 5: Definition of a Fluid; Newton on
Boyle’s Law; The Definition of “Simple Pendulum”
7.6 Secs. 6 and 7: The Motion of Pendulums and the Resistance of Fluids to the Motions of Pendulums and Projectiles; A General Scholium (Experiments on Resistance to Motion)
7.7 The Solid of Least Resistance; The Design of Ships; The Efflux of Water
7.8 Sec. 8: Wave Motion and the Motion of Sound 7.9 The Physics of Vortices (Sec. 9, Props. 51—53);
Vortices Shown to Be Inconsistent with Keplerian Planetary Motion
7.10 Another Way of Considering Book 2: Some Achievements of Book 2 (by George E. Smith)
CHAPTER EIGHT: The Structure of Book 3; The System of the World
8.1 The Structure of Book 3 8.2 From Hypotheses to Rules and Phenomena 8.3 Newton’s “Rules for Natural Philosophy” 8.4 Newton’s “Phenomena” 8.5 Newton’s Hyp. 3 8.6 Props. 1–5: The Principles of Motion of Planets
and of Their Satellites; The First Moon Test 8.7 Jupiter’s Perturbation of Saturn as Evidence for
Universal Gravity 8.8 Planetary Perturbations: The Interaction of
Jupiter and Saturn (by George E. Smith) 8.9 Props. 6 and 7: Mass and Weight 8.10 Prop. 8 and Its Corollaries (The Masses and
Densities of the Sun, Jupiter, Saturn, and the Earth)
8.11 Props. 9—20: The Force of Gravity within Planets, the Duration of the Solar System, the Effects of the Inverse-Square Force of Gravity (Further Aspects of Gravity and the System of
the World): Newton’s Hyp. 1 8.12 Prop. 24: Theory of the Tides; The First
Enunciation of the Principle of Interference 8.13 Props. 36–38: The Tidal Force of the Sun and
the Moon; The Mass and Density (and Dimensions) of the Moon
8.14 Props. 22, 25–35: The Motion of the Moon 8.15 Newton and the Problem of the Moon’s Motion
(by George E. Smith) 8.16 The Motion of the Lunar Apsis (by George E.
Smith) 8.17 Prop. 39: The Shape of the Earth and the
Precession of the Equinoxes 8.18 Comets (The Concluding Portion of Book 3)
CHAPTER NINE: The Concluding General Scholium 9.1 The General Scholium: “Hypotheses non fingo” 9.2 “Satis est”: Is It Enough? 9.3 Newton’s “Electric and Elastic” Spirit 9.4 A Gloss on Newton’s “Electric and Elastic”
Spirit: An Electrical Conclusion to the Principia
CHAPTER TEN: How to Read the Principia 10.1 Some Useful Commentaries and Reference
Works; Newton’s Directions for Reading the Principia
10.2 Some Features of Our Translation; A Note on the Diagrams
10.3 Some Technical Terms and Special Translations (including Newton’s Use of “Rectangle” and “Solid”)
10.4 Some Trigonometric Terms (“Sine,” “Cosine,” “Tangent,” “Versed Sine” and “Sagitta,”
“Subtense,” “Subtangent”); “Ratio” versus “Proportion”; “Q.E.D.,” “Q.E.F.,” “Q.E.I.,” and “Q.E.O.”
10.5 Newton’s Way of Expressing Ratios and Proportions
10.6 The Classic Ratios (as in Euclid’s Elements, Book 5)
10.7 Newton’s Proofs; Limits and Quadratures; More on Fluxions in the Principia
10.8 Example No. 1: Book 1, Prop. 6 (Newton’s Dynamical Measure of Force), with Notes on Prop. 1 (A Centrally Directed Force Acting on a Body with Uniform Linear Motion Will Produce Motion according to the Law of Areas)
10.9 Example No. 2: Book 1, Prop. 11 (The Direct Problem: Given an Ellipse, to Find the Force Directed toward a Focus)
10.10 Example No. 3: A Theorem on Ellipses from the Theory of Conics, Needed for Book 1, Props. 10 and 11
10.11 Example No. 4: Book 1, Prop. 32 (How Far a Body Will Fall under the Action of an Inverse- Square Force)
10.12 Example No. 5: Book 1, Prop. 41 (To Find the Orbit in Which a Body Moves When Acted On by a Given Centripetal Force and Then to Find the Orbital Position at Any Specified Time)
10.13 Example No. 6: Book 1, Lem. 29 (An Example of the Calculus or of Fluxions in Geometric Form)
10.14 Example No. 7: Book 3, Prop. 19 (The Shape of the Earth)
10.15 Example No. 8: A Theorem on the Variation of Weight with Latitude (from Book 3, Prop. 20)
10.16 Example No. 9: Book 1, Prop. 66, Corol. 14,
Needed for Book 3, Prop. 37, Corol. 3 10.17 Newton’s Numbers: The “Fudge Factor” 10.18 Newton’s Measures 10.19 A Puzzle in Book 1, Prop. 66, Corol. 14 (by
George E. Smith)
CHAPTER ELEVEN: Conclusion
Notes to the Guide
Contents of the Principia
Halley’s Ode to Newton Newton’s Preface to the First Edition Newton’s Preface to the Second Edition Cotes’s Preface to the Second Edition Newton’s Preface to the Third Edition Definitions Axioms, or the Laws of Motion
Section 1 Lemma 1 Lemma 2 Lemma 3
COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4
Lemma 4 COROLLARY
COROLLARY 2 COROLLARY 3
COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5
Scholium Lemma 11
CASE 1 CASE 2 CASE 3 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5
Scholium Section 2
Proposition 1, Theorem 1 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6
Proposition 2, Theorem 2 CASE 1
Scholium Proposition 3 Theorem 3
Scholium Proposition 4, Theorem 4
COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7 COROLLARY 8 COROLLARY 9
Scholium Proposition 5, Problem 1 Proposition 6, Theorem 5
Proposition 7, Problem 2 COROLLARY 1 COROLLARY 2 COROLLARY 3
Proposition 8, Problem 3 Scholium Proposition 9, Problem 4 Lemma 12 Proposition 10, Problem 5
Scholium Section 3
Proposition 11, Problem 6 Proposition 12, Problem 7 Lemma 13 Lemma 14
COROLLARY 1 COROLLARY 2 COROLLARY 3
Proposition 13, Problem 8 COROLLARY 1 COROLLARY 2
Proposition 14, Theorem 6 COROLLARY
Proposition 16, Theorem 8 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7 COROLLARY 8
Scholium Section 4
Lemma 15 Proposition 18, Problem 10 Proposition 19, Problem 11 Proposition 20, Problem 12
CASE 1 CASE 2 CASE 3 CASE 4
Lemma 16 CASE 1 CASE 2 CASE 3
Proposition 21, Problem 13 Scholium
Section 5 Lemma 17
Lemma 18 COROLLARY
Scholium Lemma 19
CASE 1 CASE 2 COROLLARY 1 COROLLARY 2 COROLLARY 3
Lemma 21 Proposition 22, Problem 14
CASE 1 CASE 2
Proposition 24, Problem 16 Lemma 22 Proposition 25, Problem 17 Proposition 26, Problem 18 Lemma 23
COROLLARY
Lemma 25 COROLLARY 1 COROLLARY 2 COROLLARY 3
Proposition 27, Problem 19 Scholium Lemma 26
COROLLARY
COROLLARY
Section 6 Proposition 30, Problem 22
Lemma 28 COROLLARY
Section 7 Proposition 32, Problem 24
CASE 1 CASE 2 CASE 3
Proposition 33, Theorem 9 COROLLARY 1 COROLLARY 2
Proposition 34, Theorem 10 Proposition 35, Theorem 11
CASE 1 CASE 2
Proposition 36, Problem 25 Proposition 37, Problem 26 Proposition 38, Theorem 12
Proposition 39, Problem 27
Proposition 42, Problem 29 Section 9
Proposition 43, Problem 30 Proposition 44, Theorem 14
COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6
Proposition 45, Problem 31 EXAMPLE 1 EXAMPLE 2 EXAMPLE 3 COROLLARY 1 COROLLARY 2
Section 10 Proposition 46, Problem 32 Proposition 47, Theorem 15 Scholium Proposition 48, Theorem 16
Proposition 50, Problem 33 COROLLARY
COROLLARY
Proposition 59, Theorem 22 Proposition 60, Theorem 23 Proposition 61, Theorem 24 Proposition 62, Problem 38 Proposition 63, Problem 39 Proposition 64, Problem 40 Proposition 65, Theorem 25
Proposition 66, Theorem 26 CASE 1 CASE 2 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7 COROLLARY 8 COROLLARY 9 COROLLARY 10 COROLLARY 11 COROLLARY 12 COROLLARY 13 COROLLARY 14 COROLLARY 15 COROLLARY 16 COROLLARY 17 COROLLARY 18 COROLLARY 19 COROLLARY 20 COROLLARY 21 COROLLARY 22
Proposition 69, Theorem 29 COROLLARY 1 COROLLARY 2 COROLLARY 3
Scholium Section 12
Proposition 70, Theorem 30 Proposition 71, Theorem 31 Proposition 72, Theorem 32
Proposition 73, Theorem 33 Scholium Proposition 74, Theorem 34
Proposition 75, Theorem 35 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4
Proposition 76, Theorem 36 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7
Proposition 77, Theorem 37 CASE 1 CASE 2 CASE 3 CASE 4 CASE 5 CASE 6
Scholium Lemma 29 Proposition 79, Theorem 39 Proposition 80, Theorem 40
Proposition 81, Problem 41 EXAMPLE 1 EXAMPLE 2 EXAMPLE 3
Proposition 82, Theorem 41 Proposition 83, Problem 42 Proposition 84, Problem 43 Scholium
Section 13 Proposition 85, Theorem 42 Proposition 86, Theorem 43 Proposition 87, Theorem 44
COROLLARY 1
COROLLARY
Scholium Section 14
Proposition 94, Theorem 48 CASE 1 CASE 2
Proposition 95, Theorem 49 Proposition 96, Theorem 50 Scholium Proposition 97, Problem 47
COROLLARY
CASE 1 CASE 2 COROLLARY
Proposition 3, Problem 1 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4
Proposition 4, Problem 2 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7
Section 2 Proposition 5, Theorem 3
Proposition 6, Theorem 4 ******ebook converter DEMO Watermarks*******
Proposition 7, Theorem 5 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5
Lemma 2 CASE 1 CASE 2 CASE 3 CASE 4 CASE 5 CASE 6 COROLLARY 1 COROLLARY 2 COROLLARY 3
Proposition 9, Theorem 7 CASE 1 CASE 2 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7
COROLLARY 1 COROLLARY 2 EXAMPLE 1 EXAMPLE 2 EXAMPLE 3 EXAMPLE 4
Scholium RULE 1 RULE 2 RULE 3 RULE 4 RULE 5 RULE 6 RULE 7 RULE 8
Proposition 13, Theorem 10 CASE 1 CASE 2 CASE 3 COROLLARY
CASE 1 CASE 2
Lemma 3 Proposition 15, Theorem 12
COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7 COROLLARY 8 COROLLARY 9
Scholium Proposition 17, Problem 4 Proposition 18, Problem 5
Section 5 Definition of a Fluid Proposition 19, Theorem 14
CASE 1 CASE 2 CASE 3 CASE 4 CASE 5 CASE 6 CASE 7
COROLLARY
Proposition 20, Theorem 15 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7 COROLLARY 8 COROLLARY 9
Scholium Proposition 23, Theorem 18 Scholium
COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7
COROLLARY 1
COROLLARY 2 Proposition 28, Theorem 23 Proposition 29, Problem 6
Proposition 31, Theorem 25 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5
General Scholium Section 7
Proposition 33, Theorem 27 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6
Proposition 34, Theorem 28 Scholium Proposition 35, Problem 7
CASE 1 CASE 2 CASE 3
COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7
Scholium Proposition 36, Problem 8
CASE 1 CASE 2 CASE 3 CASE 4 CASE 5 CASE 6 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7 COROLLARY 8 COROLLARY 9 COROLLARY 10
Lemma 4 Proposition 37, Theorem 29
Scholium Lemma 5
Proposition 39, Theorem 31 Scholium Proposition 40, Problem 9 Scholium
EXPERIMENT 1 EXPERIMENT 2 EXPERIMENT 3 EXPERIMENT 4 EXPERIMENT 5 EXPERIMENT 6 EXPERIMENT 7 EXPERIMENT 8 EXPERIMENT 9 EXPERIMENT 10 EXPERIMENT 11 EXPERIMENT 12 EXPERIMENT 13 EXPERIMENT 14
COROLLARY
CASE 1 CASE 2 COROLLARY
Proposition 45, Theorem 36 Proposition 46, Problem 10
Proposition 48, Theorem 38 CASE 1 CASE 2 CASE 3
Proposition 50, Problem 12 Scholium
Section 9 Hypothesis Proposition 51, Theorem 39
COROLLARY 6 Proposition 52, Theorem 40
CASE 1 CASE 2 CASE 3 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7 COROLLARY 8 COROLLARY 9 COROLLARY 10 COROLLARY 11
BOOK 3: THE SYSTEM OF THE WORLD
Rules for the Study of Natural Philosophy Rule 1 Rule 2 Rule 3 Rule 4
Phenomena Phenomenon 1 Phenomenon 2 Phenomenon 3
Phenomenon 4 Phenomenon 5 Phenomenon 6
Propositions Proposition 1, Theorem 1 Proposition 2, Theorem 2 Proposition 3, Theorem 3
COROLLARY
Proposition 8, Theorem 8 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4
Proposition 9, Theorem 9 Proposition 10, Theorem 10 Hypothesis 1
COROLLARY
Scholium Proposition 15, Problem 1 Proposition 16, Problem 2 Proposition 17, Theorem 15 Proposition 18, Theorem 16 Proposition 19, Problem 3 Proposition 20, Problem 4 Proposition 21, Theorem 17 Proposition 22, Theorem 18 Proposition 23, Problem 5 Proposition 24, Theorem 19 Proposition 25, Problem 6 Proposition 26, Problem 7 Proposition 27, Problem 8
Proposition 28, Problem 9 Proposition 29, Problem 10 Proposition 30, Problem 11
Proposition 32, Problem 13 Proposition 33, Problem 14
Proposition 35, Problem 16 Scholium Proposition 36, Problem 17
COROLLARY
Proposition 37, Problem 18 COROLLARY 1 COROLLARY 2 COROLLARY 3 COROLLARY 4 COROLLARY 5 COROLLARY 6 COROLLARY 7 COROLLARY 8 COROLLARY 9 COROLLARY 10
Proposition 38, Problem 19 COROLLARY
Lemma 1 Lemma 2 Lemma 3 Hypothesis 2 Proposition 39, Problem 20 Lemma 4
Lemma 5 CASE 1 CASE 2 COROLLARY
Lemma 6 Lemma 7 Lemma 8
COROLLARY
COROLLARY
EXAMPLE
Proposition 42, Problem 22 OPERATION 1 OPERATION 2 OPERATION 3
General Scholium Notes to the Principia
Preface
ALTHOUGH NEWTON’s PRINCIPIA has been translated into many languages, the last complete translation into English (indeed, the only such complete translation) was produced by Andrew Motte and published in London more than two and a half centuries ago. This translation was printed again and again in the nineteenth century and in the 1930s was modernized and revised as a result of the efforts of Florian Cajori. This latter version, with its partial modernization and partial revision, has become the standard English text of the Principia.
Motte’s version is often almost as opaque to the modern reader as Newton’s Latin original, since Motte used such older and unfamiliar expressions as “subsesquialterate” ratio. Additionally, there are statements in which the terms are no longer immediately comprehensible today, such as book 3, prop. 8, corol. 3, in which Motte writes that “the densities of dissimilar spheres are as those weights applied to the diameters of the spheres,” a statement unaltered in the Motte-Cajori version. Of course, a little thought reveals that Newton was writing about the densities of nonhomogeneous spheres and was concluding with a reference to the weights divided by the diameters. The Motte-Cajori version, as explained in §2.3 of the Guide to the present translation, is also not satisfactory because it too is frequently difficult to read and, what is more important, does not always present an authentic rendition of Newton’s original. The discovery of certain extraordinary examples in which scholars have
been misled in this regard was a chief factor in our decision to produce the present translation.
When we completed our Latin edition, somewhat awed by the prospect of undertaking a wholly new translation, we thought of producing a new edition of Motte’s English version, with notes that either would give the reader a modern equivalent of difficult passages in Motte’s English prose or would contain some aids to help the reader with certain archaic mathematical expressions. That is, since Motte’s text had been a chief means of disseminating Newton’s science for over two centuries, we considered treating it as an important historical document in its own right. Such a plan was announced in our Latin edition, and we even prepared a special interleaved copy of the facsimile of the 1729 edition to serve as our working text.*
After the Latin edition appeared, however, many colleagues and some reviewers of that edition insisted that it was now our obligation to produce a completely new translation of the Principia, rather than confine our attentions to Motte’s older pioneering work. We were at first reluctant to accept this assignment, not only because of the difficulty and enormous labor involved, but also because of our awareness that we ourselves would thereby become responsible for interpretations of Newton’s thought for a long period of time.
Goaded by our colleagues and friendly critics, Anne Whitman and I finally agreed to produce a wholly new version of the Principia. We were fortunate in obtaining a grant from the National Science Foundation to support our efforts. Many scholars offered good advice, chief among them our good friends D. T. Whiteside and R. S. Westfall. In particular, Whiteside stressed for us that we should pay no attention to any existing translation, not even consulting any other version on occasions when we might be puzzled, until after our own assignment had been fully completed. Anyone who has had to translate a technical text will appreciate the importance of this advice, since it is all too easy to be influenced by other translations, even ******ebook converter DEMO Watermarks*******
to the extent of unconsciously repeating their errors. Accordingly, during the first two or three rounds of translation and revision, we recorded puzzling or doubtful passages, and passages for which we hoped to produce a final version that would be less awkward than our preliminary efforts, reserving for some later time a possible comparison of our version with others. It should be noted that in the final two rounds of our revision, while checking some difficult passages and comparing some of our renditions with others, the most useful works for such purpose were Whiteside’s own translation of an early draft of what corresponds to most of book 1 of the Principia and the French translation made in the mid-eighteenth century by the marquise du Châtelet. On some difficult points, we also profited from the exegeses and explanations in the Le Seur and Jacquier Latin edition and the Krylov Russian edition. While neither Anne Whitman nor I could read Russian, we did have the good fortune to have two former students, Richard Kotz and Dennis Brezina, able and willing to translate a number of Krylov’s notes for us.
The translation presented here is a rendition of the third and final edition of Newton’s Principia into present-day English with two major aims: to make Newton’s text understandable to today’s reader and yet to preserve Newton’s form of mathematical expression. We have thus resisted any temptation to rewrite Newton’s text by introducing equations where he expressed himself in words. We have, however, generally transmuted such expressions as “subsesquiplicate ratio” into more simply understandable terms. These matters are explained at length in §§10.3–10.5 of the Guide.
After we had completed our translation and had checked it against Newton’s Latin original several times, we compared our version with Motte’s and found many of our phrases to be almost identical, except for Motte’s antique mathematical expressions. This was especially the case in the mathematical portions of books 1 and 2 and the early part of book 3. After all, there are not many ways of saying ******ebook converter DEMO Watermarks*******
that a quantity A is proportional to another quantity B. Taking into account that Motte’s phrasing represents the prose of Newton’s own day (his translation was published in 1729) and that in various forms his rendition has been the standard for the English-reading world for almost three centuries, we decided that we would maintain some continuity with this tradition by making our phrasing conform to some degree to Motte’s. This comparison of texts did show, however, that Motte had often taken liberties with Newton’s text and had even expanded Newton’s expressions by adding his own explanations—a result that confirmed the soundness of the advice that we not look at Motte’s translation until after we had completed our own text.
This translation was undertaken in order to provide a readable text for students of Newton’s thought who are unable to penetrate the barrier of Newton’s Latin. Following the advice of scholarly friends and counselors, we have not overloaded the translation with extensive notes and comments of the sort intended for specialists, rather allowing the text to speak for itself. Much of the kind of editorial comment and explanation that would normally appear in such notes may be found in the Guide. Similarly, information concerning certain important changes in the text from edition to edition is given in the Guide, as well as in occasional textual notes. The table of contents for the Guide, found on pages 3–7, will direct the reader to specific sections of the Principia, or even to particular propositions.
The Guide to the present translation is intended to be just that—a kind of extended road map through the sometimes labyrinthine pathways of the Principia. Some propositions, methods, and concepts are analyzed at length, and in some instances critical details of Newton’s argument are presented and some indications are given of the alterations produced by Newton from one edition to the next. Sometimes reference is made to secondary works where particular topics are discussed, but no attempt has ******ebook converter DEMO Watermarks*******
been made to indicate the vast range of scholarly information relating to this or that point. That is, I have tended to cite, in the text and in the footnote references, primarily those works that either have been of special importance for my understanding of some particular point or some sections of the Principia or that may be of help to the reader who wishes to gain a more extensive knowledge of some topic. As a result, I have not had occasion in the text to make public acknowledgment of all the works that have been important influences on my own thinking about the Principia and about the Newtonian problems associated with that work. In this rubric I would include, among others, the important articles by J. E. McGuire, the extremely valuable monographs on many significant aspects of Newton’s science and philosophic background by Maurizio Mamiani (which have not been fully appreciated by the scholarly world because they are written in Italian), the two histories of mechanics by Réne Dugas and the antecedent documentary history by Léon Jouguet, the analysis of Newton’s concepts and methods by Pierre Duhem and Ernst Mach, and monographic studies by Michel Blay, G. Bathélemy, Pierre Costabel, and A. Rupert Hall, and by François de Gandt.
I also fear that in the Guide I may not have sufficiently stressed how greatly my understanding of the Principia has profited from the researches of D. T. Whiteside and Curtis Wilson and from the earlier commentaries of David Gregory, of Thomas Le Seur and François Jacquier, and of Alexis Clairaut. The reader will find, as I have done, that R. S Westfall’s Never at Rest not only provides an admirable guide to the chronology of Newton’s life and the development of his thought in general, but also analyzes the whole range of Newton’s science and presents almost every aspect of the Principia in historical perspective.
All students of the Principia find a guiding beacon in D. T. Whiteside’s essays and his texts and commentaries in his edition of Newton’s Mathematical Papers, esp. vols. 6 and 8 (cited on p. 9 below). On Newton’s astronomy, the concise ******ebook converter DEMO Watermarks*******
analysis by Curtis Wilson (cited on p. 10 below) has been of enormous value. Many of the texts quoted in the Guide have been translated into English. It did not seem necessary to mention this fact again and again.
From the very start of this endeavor, Anne Whitman and I were continuously aware of the awesome responsibility that was placed on our shoulders, having in mind all too well the ways in which even scholars of the highest distinction have been misled by inaccuracies and real faults in the current twentieth-century English version. We recognized that no translator or editor could boast of having perfectly understood Newton’s text and of having found the proper meaning of every proof and construction. We have ever been aware that a translation of a work as difficult as Newton’s Principia will certainly contain some serious blunders or errors of interpretation. We were not so vain that we were always sure that we fully understood every level of Newton’s meaning. We took comfort in noting that even Halley, who probably read the original Principia as carefully as anyone could, did not always fully understand the mathematical significance of Newton’s text. We therefore, in close paraphrase of Newton’s own preface to the first edition, earnestly ask that everything be read with an open mind and that the defects in a subject so difficult may be not so much reprehended as kindly corrected and improved by the endeavors of our readers.
I. B. C.
Some Acknowledgments
Anne Whitman died in 1984, when our complete text was all but ready for publication, being our fourth (and in some cases fifth and even sixth) version. It was her wish, as well as mine, that this translation be dedicated to the scholar whose knowledge of almost every aspect of Newton’s mathematics, science, and life is unmatched in our time
and whose own contributions to knowledge have raised the level of Newtonian scholarship to new heights.
We are fortunate that Julia Budenz has been able to help us with various aspects of producing our translation and especially in the final stages of preparing this work for publication.
It has been a continual joy to work with the University of California Press. I am especially grateful to Elizabeth Knoll, for her thoughtfulness with regard to every aspect of converting our work into a printed book, and to Rose Vekony, for the care and wisdom she has exercised in seeing this complex work through production, completing the assignment so skillfully begun by Rebecca Frazier. I have profited greatly from the many wise suggestions made by Nicholas Goodhue, whose command of Latin has made notable improvements in both the Guide and the translation. One of the fortunate aspects of having the translation published by the University of California Press is that we have been able to use the diagrams (some with corrections) of the older version.
I gladly acknowledge and record some truly extraordinary acts of scholarly friendship. Three colleagues—George E. Smith, Richard S. Westfall, and Curtis Wilson—not only gave my Guide a careful reading, sending me detailed commentaries for its improvement; these three colleagues also checked our translation and sent me many pages of detailed criticisms and useful suggestions for its improvement. I am particularly indebted to George Smith of Tufts University for having allowed me to make use of his as yet unpublished Companion to Newton’s “Principia,” a detailed analysis of the Principia proposition by proposition. Smith used the text of our translation in his seminar on the Principia at Tufts during the academic year 1993/94 and again during 1997/98. Our final version has profited greatly from the suggestions of the students, who were required to study the actual text of the Principia from beginning to end. I am happy to be able to include in the Guide a general ******ebook converter DEMO Watermarks*******
presentation he was written (in his dual capacity as a philosopher of science and a specialist in fluid mechanics) on the contents of book 2 and also two longish notes, one on planetary perturbations, the other on the motion of the lunar apsis. I have also included a note by Prof. Michael Nauenberg of the University of California, Santa Cruz, on his current research into the origins of some of Newton’s methods.
I am grateful to the University Library, Cambridge, for permission to quote extracts and translations of various Newton MSS. I gladly record here my deep gratitude to the staff and officers of the UL for their generosity, courtesy, kindness, and helpfulness over many years.
I am especially grateful to Robert S. Pirie for permission to reproduce the miniature portrait which serves as frontispiece to this work. The following illustrations are reproduced, with permission, from books in the Grace K. Babson Collection of the Works of Sir Isaac Newton, Burndy Library, Dibner Institute for the History of Science and Technology: the title pages of the first and second editions of the Principia; the half title, title page, and dedication of the third edition; and the diagrams for book 2, prop. 10, in the Jacquier and Le Seur edition of the Principia.
I gratefully record the continued and generous support of this project by the National Science Foundation, which also supported the prior production of our Latin edition with variant readings. Without such aid this translation and Guide would never have come into being.
Finally, I would like to thank the Alfred P. Sloan Foundation for a grant that made it possible to add an index to the second printing.
ADDENDUM
I am particularly grateful to four colleagues who helped me read the proofs. Bruce Brackenridge checked the ******ebook converter DEMO Watermarks*******
proofs of the Guide and shared with me many of his insights into the methods used by Newton in the Principia, while George Smith worked through the proofs of each section of the Guide and also helped me check the translation. Michael Nauenberg and William Harper helped me find errors in the Guide. A student, Luis Campos, gave me the benefit of his skill at proofreading.
I also gladly acknowledge the importance of correspondence with Mary Ann Rossi which helped me to clarify certain grammatical puzzles. Edmund J. Kelly kindly sent me the fruit of his detailed textual study of the Motte- Cajori version of the Principia.
Readers’ attention is called to three collections of studies that are either in process of publication or appeared too late to be used in preparing the Guide: Planetary Astronomy from the Renaissance to the Rise of Astrophysics, Part B: The Eighteenth and Nineteenth Centuries, ed. René Taton and Curtis Wilson (Cambridge: Cambridge University Press, 1995); Isaac Newton’s Natural Philosophy, ed. Jed Buchwald and I. B. Cohen (Cambridge: MIT Press, forthcoming); and The Foundations of Newtonian Scholarship: Proceedings of the 1997 Symposium at the Royal Society, ed. R. Dalitz and M. Nauenberg (Singapore: World Scientific, forthcoming). Some of the chapters in these collections, notably those by Michael Nauenberg, either suggest revisions of the interpretations set forth in the Guide or offer alternative interpretations. Other contributions of Nauenberg are cited in the notes to the Guide.
A GUIDE TO NEWTON’S PRINCIPIA
By I. Bernard Cohen with contributions by Michael Nauenberg (§3.9) and George E. Smith (§§7.10, 8.8, 8.15, 8.16, and 10.19)
Abbreviations
Throughout the notes of the Guide, ten works are referred to constantly under the following standardized abbreviations:
Anal. View Henry Lord Brougham and E. J. Routh, Analytical View of Sir Isaac Newton’s “Principia” (London: Longman, Brown, Green, and Longmans, 1855; reprint, with an introd. by I. B. Cohen, New York and London: Johnson Reprint Corp., 1972).
Background John Herivel, The Background to Newton’s “Principia”: A Study of Newton’s Dynamical Researches in the Years 1664–84 (Oxford: Clarendon Press, 1965).
Corresp. The Royal Society’s edition of The Correspondence of Isaac Newton, 7 vols., ed. H. W. Turnbull (vols. 1–3), J. F. Scott (vol. 4), A. Rupert Hall and Laura Tilling (vols. 5–7) (Cambridge: Cambridge University Press, 1959–1977).
Essay W. W. Rouse Ball, An Essay on Newton’s “Principia” (London: Macmillan and Co., 1893; reprint, with an introd. by I. B. Cohen, New York and London: Johnson Reprint Corp., 1972).
Introduction I. B. Cohen, Introduction to Newton’s “Principia” (Cambridge, Mass.: Harvard University Press; Cambridge: Cambridge University Press, 1971).
Math. Papers The Mathematical Papers of Isaac Newton, 8 vols., ed. D. T. Whiteside (Cambridge: Cambridge University Press, 1967– 1981).
Never at Rest R. S. Westfall, Never at Rest: A Biography of Isaac Newton (Cambridge, London, New York: Cambridge University Press, 1980).
“Newt. Achievement”
Curtis Wilson, “The Newtonian Achievement in Astronomy,” in Planetary Astronomy from the Renaissance to the Rise of Astrophysics, Part A: Tycho Brahe to Newton, ed. René Taton and Curtis Wilson, vol. 2A of The General History of Astronomy (Cambridge and New York: Cambridge University Press, 1989), pp. 233–274.
Newt. Revolution I. B. Cohen, The Newtonian Revolution (Cambridge, London, New York: Cambridge University Press, 1980).
Unpubl. Sci. Papers Unpublished Scientific Papers of Isaac Newton, ed. A. Rupert Hall and Marie Boas Hall (Cambridge: Cambridge University Press, 1962).
A useful bibliographical guide to Newton’s writings and the vast secondary literature concerning Newton’s life, thought, and influence is Peter Wallis and Ruth Wallis, Newton and Newtoniana, 1672–1975: A Bibliography (Folkestone: Dawson, 1977). A generally helpful reference work is Derek Gjertsen, The Newton Handbook (London and New York:
Routledge and Kegan Paul, 1986).
The abbreviation “ULC” in such references as “ULC MS Add. 3968” refers to the University Library of Cambridge University.
A Brief History of the Principia
1.1 The Origins of the Principia
Isaac Newton’s Principia was published in 1687. The full title is Philosophiae Naturalis Principia Mathematica, or Mathematical Principles of Natural Philosophy. A revised edition appeared in 1713, followed by a third edition in 1726, just one year before the author’s death in 1727. The subject of this work, to use the name assigned by Newton in the first preface, is “rational mechanics.” Later on, Leibniz introduced the name “dynamics.” Although Newton objected to this name,1 “dynamics” provides an appropriate designation of the subject matter of the Principia, since “force” is a primary concept of that work. Indeed, the Principia can quite properly be described as a study of a variety of forces and the different kinds of motions they produce. Newton’s eventual goal, achieved in the third of the three “books” of which the Principia is composed, was to apply the results of the prior study to the system of the world, to the motions of the heavenly bodies. This subject is generally known today by the name used a century or so later by Laplace, “celestial mechanics.”
The history of how the Principia came into being has been told and retold.2 In the summer of 1684, the astronomer Edmond Halley visited Newton in order to find out
whether he could solve a problem that had baffled Christopher Wren, Robert Hooke, and himself: to find the planetary orbit that would be produced by an inverse- square central force. Newton knew the answer to be an ellipse.3 He had solved the problem of elliptical orbits earlier, apparently in the period 1679–1680 during the course of an exchange of letters with Hooke. When Halley heard Newton’s reply, he urged him to write up his results. With Halley’s prodding and encouragement, Newton produced a short tract which exists in several versions and will be referred to as De Motu4 (On Motion), the common beginning of all the titles Newton gave to the several versions. Once started, Newton could not restrain the creative force of his genius, and the end product was the Principia. In his progress from the early versions of De Motu to the Principia, Newton’s conception of what could be achieved by an empirically based mathematical science had become enlarged by several orders of magnitude.
As first conceived, the Principia consisted of two “books” and bore the simple title De Motu Corporum (On the Motion of Bodies).5 This manuscript begins, as does the Principia, with a series of Definitions and Laws of Motion, followed by a book 1 whose subject matter more or less corresponds to book 1 of the Principia.6 The subject matter of book 2 of this early draft is much the same as that of book 3 of the Principia. In revising this text for the Principia, Newton limited book 1 to the subject of forces and motion in free spaces, that is, in spaces devoid of any resistance. Book 2 of the Principia contains an expanded version of the analysis of motion in resisting mediums, plus discussions of pendulums,7 of wave motion, and of the physics of vortices. In the Principia, the system of the world became the subject of what is there book 3, incorporating much that had been in the older book 2 but generally recast in a new form. As Newton explained in the final Principia, while introducing book 3, he had originally presented this subject in a popular manner, but then decided to recast it in a more mathematical form so that it would not be read ******ebook converter DEMO Watermarks*******
by anyone who had not first mastered the principles of rational mechanics. Even so, whole paragraphs of the new book 3 were copied word for word from the old book 2.8
1.2 Steps Leading to the Composition and Publication of the Principia
The history of the development of Newton’s ideas concerning mechanics, more specifically dynamics, has been explored by many scholars and is still the subject of active research and study.9 The details of the early development of Newton’s ideas about force and motion, however interesting in their own right, are not directly related to the present assignment, which is to provide a reader’s guide to the Principia. Nevertheless, some aspects of this prehistory should be of interest to every prospective reader of the Principia. In the scholium to book 1, prop. 4, Newton refers to his independent discovery (in
the 1660s) of the rule for the force in uniform circular
motion (at speed v along a circle of radius r), a discovery usually attributed to Christiaan Huygens, who formally announced it to the world in his Horologium Oscillatorium of 1673.10 It requires only the minimum skill in algebraic
manipulation to combine the rule with Kepler’s third
law in order to determine that in a system of bodies in
uniform circular motion the force is proportional to or
is inversely proportional to the square of the distance. Of course, this computation does not of itself specify anything about the nature of the force, whether it is a centripetal or a centrifugal force or whether it is a force in the sense of the later Newtonian dynamics or merely a Cartesian “conatus,” or endeavor. In a manuscript note Newton later ******ebook converter DEMO Watermarks*******
claimed that at an early date, in the 1660s, he had actually
applied the rule to the moon’s motion, much as he does
later on in book 3, prop. 4, of the Principia, in order to confirm his idea of the force of “gravity extending to the Moon.”11 In this way he could counter Hooke’s allegation that he had learned the concept of an inverse-square force of gravity from Hooke.
A careful reading of the documents in question shows that sometime in the 1660s, Newton made a series of computations, one of which was aimed at proving that what was later known as the outward or centrifugal force arising from the earth’s rotation is less than the earth’s gravity, as it must be for the Copernican system to be possible. He then computed a series of forces. Cartesian vortical endeavors are not the kind of forces that, in the Principia, are exerted by the sun on the planets to keep them in a curved path or the similar force exerted by the earth on the moon. At this time, and for some years to come, Newton was deeply enmeshed in the Cartesian doctrine of vortices. He had no concept of a “force of gravity” acting on the moon in anything like the later sense of the dynamics of the Principia. These Cartesian “endeavors” (Newton used Descartes’s own technical term, “conatus”) are the magnitude of the planets’ endeavors to fly out of their orbits. Newton concludes that since the cubes of the distances of the planets from the sun are “reciprocally as the squared numbers of their revolutions in a given time,” their “conatus to recede from the Sun will be reciprocally as the squares of their distances from the Sun.”12
Newton also made computations to show that the endeavor or “conatus” of receding from the earth’s surface (caused by the earth’s daily rotation) is 12½ times greater than the orbital endeavor of the moon to recede from the earth. He concludes that the force of receding at the earth’s surface is “4000 and more times greater than the ******ebook converter DEMO Watermarks*******
endeavor of the Moon to recede from the Earth.” In other words, “Newton had discovered an interesting
mathematical correlation within the solar vortex,”13 but he plainly had not as yet invented the radically new concept of a centripetal dynamical force, an attraction that draws the planets toward the sun and the moon toward the earth.14 There was no “twenty years’ delay” (from the mid- 1660s to the mid-1680s) in Newton’s publication of the theory of universal gravity, as was alleged by Florian Cajori.15
In 1679/80, Hooke initiated an exchange of correspondence with Newton on scientific topics. In the course of this epistolary interchange, Hooke suggested to Newton a “hypothesis” of his own devising which would account for curved orbital motion by a combination of two motions: an inertial or uniform linear component along the tangent to the curve and a motion of falling inward toward a center. Newton told Hooke that he had never heard of this “hypothesis.”16 In the course of their letters, Hooke urged Newton to explore the consequences of his hypothesis, advancing the opinion or guess that in combination with the supposition of an inverse-square law of solar-planetary force, it would lead to the true planetary motions.17 Hooke also wrote that the inverse-square law would lead to a rule for orbital speed being inversely proportional to the distance of a planet from the sun.18
Stimulated by Hooke, Newton apparently then proved that the solar-planetary force is as the inverse square of the distance, a first step toward the eventual Principia.
We cannot be absolutely certain of exactly how Newton proceeded to solve the problem of motion in elliptical orbits, but most scholars agree that he more or less followed the path set forth in the tract De Motu which he wrote after Halley’s visit a few years later in 1684.19
Essentially, this is the path from props. 1 and 2 of book 1 to prop. 4, through prop. 6, to props. 10 and 11. Being secretive by nature, Newton didn’t tell Hooke of his
achievement. In any event, he would hardly have announced so major a discovery to a jealous professional rival, nor in a private letter. What may seem astonishing, in retrospect, is not that Newton did not reveal his discovery to Hooke, but that Newton was not at once galvanized into expanding his discovery into the eventual Principia.
Several aspects of the Hooke-Newton exchange deserve to be noted. First, Hooke was unable to solve the problem that arose from his guess or his intuition; he simply did not have sufficient skill in mathematics to be able to find the orbit produced by an inverse-square force. A few years later, Wren and Halley were equally baffled by this problem. Newton’s solution was, as Westfall has noted, to invert the problem, to assume the path to be an ellipse and find the force rather than “investigating the path in an inverse-square force field.”20 Second, there is no certainty that the tract De Motu actually represents the line of Newton’s thought after corresponding with Hooke; Westfall, for one, has argued that a better candidate would be an essay in English which Newton sent later to John Locke, a position he maintains in his biography of Newton.21 A third point is that Newton was quite frank in admitting (in private memoranda) that the correspondence with Hooke provided the occasion for his investigations of orbital motion that eventually led to the Principia.22 Fourth, as we shall see in §3.4 below, the encounter with Hooke was associated with a radical reorientation of Newton’s philosophy of nature that is indissolubly linked with the Principia. Fifth, despite Newton’s success in proving that an elliptical orbit implies an inverse-square force, he was not at that time stimulated—as he would be some four years later—to move ahead and to create modern rational mechanics. Sixth, Newton’s solution of the problems of planetary force depended on both his own new concept of a dynamical measure of force (as in book 1, prop. 6) and his recognition of the importance of Kepler’s law of areas.23 A final point to be made is that most scholarly ******ebook converter DEMO Watermarks*******
analyses of Newton’s thoughts during this crucial period concentrate on conceptual formulations and analytical solutions, whereas we know that both Hooke24 and Newton made important use of graphical methods, a point rightly stressed by Curtis Wilson.25 Newton, in fact, in an early letter to Hooke, wrote of Hooke’s “acute Letter having put me upon considering . . . the species of this curve,” saying he might go on to “add something about its description quam proxime,”26 or by graphic methods. The final proposition in the Principia (book 3, prop. 42) declared its subject (in the first edition) to be: “To correct a comet’s trajectory found graphically.” In the second and third editions, the text of the demonstration was not appreciably altered, but the statement of the proposition now reads: “To correct a comet’s trajectory that has been found.”
When Newton wrote up his results for Halley (in the tract De Motu) and proved (in the equivalent of book 1, prop. 11) that an elliptical orbit implies an inverse-square central force, he included in his text the joyous conclusion: “Therefore the major planets revolve in ellipses having a focus in the center of the sun; and the radii to the sun describe areas proportional to the times, exactly [“omnino”] as Kepler supposed.”27 But after some reflection, Newton recognized that he had been considering a rather artificial situation in which a body moves about a mathematical center of force. In nature, bodies move about other bodies, not about mathematical points. When he began to consider such a two-body system, he came to recognize that in this case each body must act on the other. If this is true for one such pair of bodies, as for the sun-earth system, then it must be so in all such systems. In this way he concluded that the sun (like all the planets) is a body on which the force acts and also a body that gives rise to the force. It follows at once that each planet must exert a perturbing force on every other planet in the solar system. The consequence must be, as Newton recognized almost at once, that “the displacement of the sun from the center of gravity” may ******ebook converter DEMO Watermarks*******
have the effect that “the centripetal force does not always tend to” an “immobile center” and that “the planets neither revolve exactly in ellipses nor revolve twice in the same orbit.” In other words, “Each time a planet revolves it traces a fresh orbit, as happens also with the motion of the moon, and each orbit is dependent upon the combined motions of all the planets, not to mention their actions upon each other.”28 This led him to the melancholy conclusion: “Unless I am much mistaken, it would exceed the force of human wit to consider so many causes of motion at the same time, and to define the motions by exact laws which would allow of any easy calculation.”
We don’t know exactly how Newton reached this conclusion, but a major factor may have been the recognition of the need to take account of the third law, that to every action there must be an equal and opposite reaction. Yet, in the texts of De Motu, the third law does not appear explicitly among the “laws” or “hypotheses.” We do have good evidence, however, that Newton was aware of the third law long before writing De Motu.29 In any event, the recognition that there must be interplanetary perturbations was clearly an essential step on the road to universal gravity and the Principia.30
In reviewing this pre-Principia development of Newton’s dynamics, we should take note that by and large, Newton has been considering exclusively the motion of a particle, of unit mass. Indeed, a careful reading of the Principia will show that even though mass is the subject of the first definition at the beginning of the Principia, mass is not a primary variable in Newton’s mode of developing his dynamics in book 1. In fact, most of book 1 deals exclusively with particles. Physical bodies with significant dimensions or shapes do not appear until sec. 12, “The attractive forces of spherical bodies.”
Newton’s concept of mass is one of the most original concepts of the Principia. Newton began thinking about mass some years before Halley’s visit. Yet, in a series of definitions which he wrote out some time after De Motu and ******ebook converter DEMO Watermarks*******
before composing the Principia, mass does not appear as a primary entry. We do not have documents that allow us to trace the development of Newton’s concept of mass with any precision. We know, however, that two events must have been important, even though we cannot tell whether they initiated Newton’s thinking about mass or reinforced ideas that were being developed by Newton. One of these was the report of the Richer expedition, with evidence that indicated that weight is a variable quantity, depending on the terrestrial latitude. Hence weight is a “local” property and cannot be used as a universal measure of a body’s quantity of matter. Another was Newton’s study of the comet of 1680. After he recognized that the comet turned around the sun and after he concluded that the sun’s action on the comet cannot be magnetic, he came to believe that Jupiter must also exert an influence on the comet. Clearly, this influence must derive from the matter in Jupiter, Jupiter acting on the comet just as it does on its satellites.
Once Newton had concluded that planets are centers of force because of their matter or mass, he sought some kind of empirical confirmation of so bold a concept. Since Jupiter is by far the most massive of all the planets, it was obvious that evidence of a planetary force would be most manifest in relation to the action of Jupiter on a neighboring planet. In happened that in 1684/85 the orbital motions of Jupiter and Saturn were bringing these two planets to conjunction. If Newton’s conclusion were correct, then the interactions of these two giant planets should show the observable effects of an interplanetary force. Newton wrote to the astronomer John Flamsteed at the Royal Observatory at Greenwich for information on this point. Flamsteed reported that Saturn’s orbital speed in the vicinity of Jupiter did not exactly follow the expected path, but he could not detect the kind of effect or perturbation that Newton had predicted.31 As we shall see, the effect predicted by Newton does occur, but its magnitude is so tiny that Flamsteed could never have ******ebook converter DEMO Watermarks*******
observed it. Newton needed other evidence to establish the validity of his force of universal gravity.
Newton’s discovery of interplanetary forces as a special instance of universal gravity enables us to specify two primary goals of the Principia. The first is to show the conditions under which Kepler’s laws of planetary motion are exactly or accurately true; the second is to explore how these laws must be modified in the world of observed nature by perturbations, to show the effects of perturbations on the motions of planets and their moons.32
It is well known that after the Principia was presented to the Royal Society, Hooke claimed that he should be given credit for having suggested to Newton the idea of universal gravity. We have seen that Hooke did suggest to Newton that the sun exerts an inverse-square force on the planets, but Newton insisted that he didn’t need Hooke to suggest to him that there is an inverse-square relation. Furthermore, Newton said that this was but one of Hooke’s guesses. Newton again and again asserted that Hooke didn’t know enough mathematics to substantiate his guess, and he was right. As the mathematical astronomer Alexis Clairaut said of Hooke’s claim, a generation later, it serves “to show what a distance there is between a truth that is glimpsed and a truth that is demonstrated.”33
In explaining his position with respect to Hooke’s guess, Newton compared his own work with that of Hooke and Kepler. Newton evidently believed that he himself had “as great a right” to the inverse-square law “as to the ellipsis.” For, just “as Kepler knew the orb to be not circular but oval, and guessed it to be elliptical, so Mr. Hooke, without knowing what I have found out since his letters to me,”34
knew only “that the proportion was duplicata quam proxime at great distances from the centre,” and “guessed it to be so accurately, and guessed amiss in extending that proportion down to the very centre.” But, unlike Hooke, “Kepler guessed right at the ellipsis,” so that “Mr. Hooke
found less of the proportion than Kepler of the ellipsis.”35
Newton believed that he himself deserved credit for the law of elliptical orbits, as well as the law of the inverse square, on the grounds that he had proved both in their generality.36 In the Principia (e.g., in the “Phenomena” in book 3), Newton gave Kepler credit only for the third or harmonic law. At the time that Newton was writing his Principia, there were alternatives to the area law that were in use in making tables of planetary motion. Newton proposed using the eclipses of Jupiter’s satellites (and later of those of Saturn) to show that this law holds to a high degree. But the law of elliptical orbits was of a different sort because there was no observational evidence that would distinguish between an ellipse and other ovals. Thus there may have been very different reasons for not giving Kepler credit for these two laws.
At one point during the exchange of letters with Halley on Hooke’s claims to recognition, Newton—in a fit of pique —threatened to withdraw book 3 altogether.37 We do not know how serious this threat was, but Halley was able to explain matters and to calm Newton’s rage. Halley deserves much praise for his services as midwife to Newton’s brainchild. Not only was he responsible for goading Newton into writing up his preliminary results; he encouraged Newton to produce the Principia. At an early stage of composition of the Principia, as I discovered while preparing the Latin edition with variant readings, Halley even helped Newton by making suggestive comments on an early draft of book 1, the manuscript of which no longer exists.38
Although publication of the Principia was sponsored by the Royal Society, there were no funds available for the costs of printing, and so Halley had to assume those expenses.39
Additionally, he edited the book for the printer, saw to the making of the woodcuts of the diagrams, and read the proofs. He wrote a flattering ode to Newton that introduces the Principia in all three editions,40 and he also
wrote a book review that was published in the Royal Society’s Philosophical Transactions.41
1.3 Revisions and Later Editions
Within a decade of publication of the Principia, Newton was busy with a number of radical revisions, including an extensive restructuring of the opening sections.42 He planned to remove secs. 4 and 5, which are purely geometrical and not necessary to the rest of the text, and to publish them separately.43 He also developed plans to include a mathematical supplement on his methods of the calculus, his treatise De Quadratura. Many of the proposed revisions and restructurings of the 1690s are recorded in Newton’s manuscripts; others were reported in some detail by David Gregory.44 When Newton began to produce a second edition, however, with the aid of Roger Cotes, the revisions were of a quite different sort. Some of the major or most interesting alterations are given in the notes to the present translation. The rest are to be found in the apparatus criticus of our Latin edition of the Principia with variant readings.45
There were a number of truly major emendations that appeared in the second edition, some of which involved a complete replacement of the original text. One of these was the wholly new proof of book 2, prop. 10, a last- minute alteration in response to a criticism made by Johann Bernoulli.46 Another occurred in book 2, sec. 7, on the motion of fluids and the resistance encountered by projectiles, where most of the propositions and their proofs are entirely different in the second edition from those of the first edition. That is, the whole set of props. 34–40 of the first edition were cast out and replaced.47
This complete revision of sec. 7 made it more appropriate to remove to the end of sec. 6 the General Scholium on pendulum experiments which originally had been at the end of sec. 7. This was a more thorough revision of the ******ebook converter DEMO Watermarks*******
text than occurred in any other part of the Principia. Another significant novelty of the second edition was the
introduction of a conclusion to the great work, the celebrated General Scholium that appears at the conclusion of book 3. The original edition ended rather abruptly with a discussion of the orbits of comets, a topic making up about a third of book 3. Newton had at first essayed a conclusion, but later changed his mind. His intentions were revealed in 1962 by A. Rupert Hall and Marie Boas Hall, who published the original drafts. In these texts, Newton shows that he intended to conclude the Principia with a discussion of the forces between the particles of matter, but then thought better of introducing so controversial a topic. While preparing the second edition, Newton thought once again of an essay on “the attraction of the small particles of bodies,” but on “second thought” he chose “rather to add but one short Paragraph about that part of Philosophy.”48 The conclusion he finally produced is the celebrated General Scholium, with its oft- quoted slogan “Hypotheses non fingo.” This General Scholium ends with a paragraph about a “spirit” which has certain physical properties, but whose laws have not as yet been determined by experiment. Again thanks to the researches of A. Rupert Hall and Marie Boas Hall, we now know that while composing this paragraph, Newton was thinking about the new phenomena of electricity.49
Another change that occurs in the second edition is in the beginning of book 3. In the first edition, book 3 opened with a preliminary set of Hypotheses.50 Perhaps in reply to the criticism in the Journal des Sçavans,51 Newton now renamed the “hypotheses” and divided them into several classes. Some became Regulae Philosophandi, or “Rules for Natural Philosophy,” with a new rule (no. 3). Others became “Phenomena,” with new numerical data. Yet another was transferred to a later place in book 3, where it became “hypothesis 1.”
Newton also made a slight modification in the scholium
following lem. 2 (book 2, sec. 2), in reference to Leibniz’s method of the calculus. He had originally written that Leibniz’s method “hardly differed from mine except in the forms of words and notations.” In the second edition Newton altered this statement by adding that there was another difference between the two methods, namely, in “the generation of quantities.” This scholium and its successive alterations attracted attention because of the controversy over priority in the invention of the calculus. In the third edition, Newton eliminated any direct reference to Leibniz.
Critical readers of the Principia paid close attention to the alteration in the scholium following book 3, prop. 35. In the second edition, the original short text was replaced by a long discussion of Newton’s attempts to apply the theory of gravity to some inequalities of the moon’s motion.52
Much of the text of this scholium had been published separately by David Gregory.53
Many of Newton’s plans for the actual revisions of the first edition, in order to produce a second edition, were entered in two personal copies of the Principia. One of these was specially bound and interleaved. Once the second edition had been published, Newton again prepared an interleaved copy and kept track of proposed alterations or emendations in his interleaved copy and in an annotated copy. These four special copies of the Principia have been preserved among Newton’s books, and their contents have been noted in our Latin edition with variant readings.54
Soon after the appearance of the second edition, Newton began planning for yet another revision. The preface which he wrote for this planned edition of the late 1710s is of great interest in that it tells us in Newton’s own words about some of the features of the Principia he believed to be most significant. It is printed below in §3.2. Newton at this time once again planned to have a treatise on the calculus published together with the Principia. In the end he abandoned this effort. Later on, when he was in his eighties, he finally decided to produce a new edition. He ******ebook converter DEMO Watermarks*******
chose as editor Dr. Henry Pemberton, a medical doctor and authority on pharmacy and an amateur mathematician.
The revisions in the third edition were not quite as extensive as those in the second edition.55 A new rule 4 was added on the subject of induction, and there were other alterations, some of which may be found in the notes to the present translation. An important change was made in the “Leibniz Scholium” in book 2, sec. 2. The old scholium was replaced by a wholly different one. Newton now boldly asserted his own claims to be the primary inventor of the calculus, referring to some correspondence to prove the point.56 Even though Leibniz had been dead for almost a decade, Newton still pursued his rival with dogged obstinacy. Another innovation in the third edition appeared in book 3, where Newton inserted (following prop. 33) two propositions by John Machin, astronomy professor at Gresham College, whose academic title would later lead to the invention of a fictitious scientist in the Motte-Cajori edition.57
By the time of the third edition, Newton seems to have abandoned his earlier attempts to explain the action of gravity by reference to electrical phenomena and had come rather to hope that an explanation might be found in the actions of an “aethereal medium” of varying density.58
In his personal copy of the Principia, in which he recorded his proposed emendations and revisions, he at first had entered an addition to specify that the “spirit” to which he had referred in the final paragraph of the General Scholium was “electric and elastic.”59 Later on, he apparently decided that since he no longer believed in the supreme importance of the electrical theory, he would cancel the whole paragraph. Accordingly, he drew a line through the text, indicating that this paragraph should be omitted. It is one of the oddities of history that Andrew Motte should have learned of Newton’s planned insertion of the modifier “electricus et elasticus” but not of Newton’s proposed elimination of the paragraph. Without ******ebook converter DEMO Watermarks*******
comment, Motte entered “electric and elastic” into his English version of 1729. These words were in due course preserved in the Motte-Cajori version and have been quoted in the English-speaking world ever since.
Translating the Principia
2.1 Translations of the Principia: English Versions by Andrew Motte (1729), Henry Pemberton (1729—?), and Thomas Thorp (1777)
Newton’s Principia has been translated in full, in large part, or in close paraphrase into many languages, including Chinese, Dutch, English, French, German, Italian, Japanese, Mongolian, Portuguese, Romanian, Russian, Spanish, and Swedish.1 The whole treatise has been more or less continuously available in an English version, from 1729 to the present, in a translation made by Andrew Motte and modernized more than a half-century ago. In addition, a new version of book 1 by Thomas Thorp was published in 1777.2
Motte’s translation of 1729 was reprinted in London in 1803 “carefully revised and corrected by W. Davis” and with “a short comment on, and defence of, the Principia, by W. Emerson.”3 This edition was reissued in London in 1819. Motte’s text also served as the basis for the “first American edition, carefully revised and corrected, with a life of the author, by N. W. Chittenden,” apparently published in New York in 1848, with a copyright date of 1846. There were a number of further New York printings.4
We do not know much about Andrew Motte. He was the ******ebook converter DEMO Watermarks*******
brother of Benjamin Motte, a London printer who was the publisher of Andrew’s English version.5 Andrew was the author of a modest work entitled A Treatise of the Mechanical Powers, wherein the Laws of Motion and the Properties of those Powers are explained and demonstrated in an easy and familiar Method, published by his brother in London in 1727. This book, which informs us that Andrew had been a lecturer at Gresham College, displays an acquaintance with the basics of Newtonian physics, providing a useful and correct discussion of the three laws of motion of Newton’s Principia. Andrew Motte was also a skilled draftsman and engraver. Each of the two volumes of his translation contains an allegorical frontispiece which he had drawn and engraved (“A. Motte invenit & fecit”), and a vignette engraved by him introduced each of the three books.
The new translation of book 1 published in 1777 was the work of Robert Thorp. A second edition was published in 1802.6 Thorp had excelled in mathematics as an undergraduate at Cambridge, where he was graduated B.A. in 1758 as “Senior Wrangler.” He taught Newtonian science as a Fellow of Peterhouse, receiving his M.A. in 1761. In 1765 he was one of three annotators of a Latin volume of excerpta from the Principia. He later followed an ecclesiastical career, eventually becoming archdeacon of Durham.
Thorp’s translation was accompanied by a commentary that is perhaps more to be valued than his translation. His goal was to make Newton’s “sublime discoveries” and the “reasonings by which they are established” available to a reading public with a knowledge only of elementary mathematics. He boasted that he had made explicit “every intermediate step omitted by the author.” A sampling of his commentary would seem to indicate that this latter boast was an exaggeration. I have not attempted to check the accuracy of this version, but a single example, discussed in detail in §2.3 below, shows so gross a distortion of Newton’s straight-forward Latin text that one would hardly consider it a reliable rendition of the ******ebook converter DEMO Watermarks*******
Principia.7 Soon after Newton died, Henry Pemberton made a series
of public announcements that he had all but ready for the press an English translation of the Principia, together with an explanatory commentary.8 Neither of these works was ever published, and the manuscripts of both have disappeared.
2.2 Motte’s English Translation
The Motte version was the result of heroic enterprise, the full measure of which can be fully appreciated only by someone who has gone through the same effort. And yet Motte’s text presents great hurdles to today’s reader, for whom the style will seem archaic and at times very hard to follow. Motte uses such unfamiliar language for ratios as “subduplicate,” “sesquiplicate,” and “semialterate.” He also sometimes, but not always, translates into English certain words of Latin or Greek-Latin origin that Newton introduced into his text but that had not yet become standard. An example of this double usage is “phaenomena,” occurring in Motte’s text almost side by side with the English translation “appearances of things.” Another term which Motte sometimes translates and sometimes keeps in Latin is “inertia,” which—like “phaenomena”—appears to have been not yet common either in treatises on mechanics or in ordinary English. Motte introduces “inertia” in def. 3, referring first to “the inactivity of the Mass,” then to “the inactivity of matter,” and finally to “Vis Inertiae, or Force of Inactivity.”
Motte’s translation has additional problems. Some phrases and passages are based on the second edition and do not therefore represent the final state of the third and ultimate authorized edition.9 Motte introduced phrases that are not in Newton’s text at all. Some of these intrusive phrases consist of explanatory insertions to help the reader through an otherwise difficult passage. In the
final sentence of the concluding General Scholium there is an insertion derived from Newton’s proposed manuscript emendation.10
Not only do many of Motte’s technical mathematical terms seem unintelligible today, but his language abounds in archaic phrases that seem awkward and difficult to understand. Motte’s style of punctuation adds yet an additional hurdle for the would-be reader. (On the relation betwe

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The Principia: Mathematical Principles of Natural Philosophy. The Authoritative Translation and Guide