I S A AC N E W T ON THE PRINCIPIA Mathematical Principles of Natural Philosophy 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 UNIVERSITY OF CALIFORNIA PRESS Berkeley Los Angeles London THE PRINCIPIA The publication of this worf^ has been made possible in part by a grant from the National Endowment for the Humanities, an independent federal agency. The publisher also gratefully acknowledges the generous contribution to this boo\ provided by the General Endowment Fund of the Associates of the University of California Press. 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 33/4 in. from top to bottom and 3>1A 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. I S A AC N E W T ON THE PRINCIPIA Mathematical Principles of Natural Philosophy 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 UNIVERSITY OF CALIFORNIA PRESS Berkeley Los Angeles London University of California Press Berkeley and Los Angeles, California University of California Press London, England Copyright © 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 978-0-520-08816-0 (alk. paper).—ISBN 978-0-520-08817-7 (alk. paper) 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 13 14 13 12 11 10 9 The paper used in this publication meets the minimum requirements of ANSI/NISO Z39.48-1992 (R 1997) (Permanence of Paper). @ This translation is dedicated to D. T. WHITESIDE with respect and affection This page intentionally left blank Contents Preface xi A GUIDE TO NEWTON'S PRINCIPIA 1 Contents of the Guide 3 Abbreviations 9 CHAPTER ONE: A Brief History of the Principia 11 CHAPTER TWO: Translating the Principia 26 CHAPTER THREE: Some General Aspects of the Principia 43 CHAPTER FOUR: Some Fundamental Concepts of the Principia 85 CHAPTER FIVE: Axioms, or the Laws of Motion 109 CHAPTER six: The Structure of Book 1 128 CHAPTER SEVEN: The Structure of Book 2 161 CHAPTER EIGHT: The Structure of Book 3 195 CHAPTER NINE: The Concluding General Scholium 274 CHAPTER TEN: How to Read the Principia 293 CHAPTER ELEVEN: Conclusion 369 THE PRINCIPIA (Mathematical Principles of Natural Philosophy) 371 Halley's Ode to Newton 379 Newton's Preface to the First Edition 381 Newton's Preface to the Second Edition 384 Cotes's Preface to the Second Edition 385 Newton's Preface to the Third Edition 400 Definitions 403 Axioms, or the Laws of Motion 416 BOOK 1: THE MOTION OF BODIES 431 BOOK 2: THE MOTION OF BODIES 631 BOOK 3: THE SYSTEM OF THE WORLD 791 General Scholium 939 Index 947