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The Mathematical Principles of Natural Philosophy

Isaac Newton

1846

Exported from Wikisource on July 17, 2023

NEWTON'S PRINCIPIA.

* * *

THE

MATHEMATICAL PRINCIPLES

OF

NATURAL PHILOSOPHY,

BY SIR ISAAC NEWTON;

TRANSLATED INTO ENGLISH BY ANDREW MOTTE.

TO WHICH IS ADDED

NEWTON'S SYSTEM OF THE WORLD;

With a Portrait taken from the Bust in the Royal Observatory at Greenwich.

FIRST AMERICAN EDITION, CAREFULLY REVISED AND CORRECTED,

WITH A LIFE OF THE AUTHOR, BY N. W. CHITTENDEN, M. A., &c.

* * *

NEW-YORK ·

PUBLISHED BY DANIEL ADEE, 45 LIBERTY STREET.

* * *

Entered according to Act of Congress, in the year 1846, by

DANIEL ADEE.

In the Clerk's Office of the Southern District Court of New-York.

(not individually listed)

Dedication 3

Introduction to the American Edition 5

Life of Sir Isaac Newton 9

The Principia.

The Author's Preface 67

Book I.

Definitions 73

Axioms, or Laws of Motion 83

Of the Motion of Bodies

Section I: Of the method of first and last ratios of quantities, by the help whereof we demonstrate the propositions that follow 95

Section II: Of the Invention of Centripetal Forces 103

Section III: Of the motion of bodies in eccentric conic sections 116

Section IV: Of the finding of elliptic, parabolic, and hyperbolic orbits, from the focus given 125

Section V: How the orbits are to be found when neither focus is given 131

Section VI: How the motions are to be found in given orbits 153

Section VII: Concerning the rectilinear ascent and descent of bodies 159

Section VIII: Of the invention of orbits wherein bodies will revolve, being acted upon by any sort of centripetal force 168

Section IX: Of the motion of bodies in movable orbits; and of the motion of the apsides 172

Section X: Of the motion of bodies in given superficies; and of the reciprocal motion of funependulous bodies 182

Section XI: Of the motions of bodies tending to each other with centripetal forces 194

Section XII: Of the attractive forces of sphærical bodies 218

Section XIII: Of the attractive forces of bodies which are not of a sphærical figure 233

Section XIV: Of the motion of very small bodies when agitated by centripetal forces tending to the several parts of any very great body 243

Book II.

Of the Motion of Bodies

Section I: Of the motion of bodies that are resisted in the ratio of the velocity 251

Section II: Of the motion of bodies that are resisted in the duplicate ratio of their velocities 258

Section III: Of the motions of bodies which are resisted partly in the ratio of the velocities, and partly in the duplicate of the same ratio 279

Section IV: Of the circular motion of bodies in resisting mediums 287

Section V: Of the density and compression of fluids; and of hydrostatics 293

Section VI: Of the motion and resistance of funependulous bodies 303

Section VII: Of the motion of fluids and the resistance made to projected bodies 323

Section VIII: Of motion propagated through fluids 356

Section IX: Of the circular motion of fluids 370

Book III.

Book III 383

Rules of Reasoning in Philosophy 384

Phænomena, or Appearances 386

Propositions I-IX (Force of gravity) 390

Propositions X-XXIV (Motions of celestial bodies and the sea) 400

Propositions XXV-XXXIII (Quantity of lunar motions) 419

Propositions XXXVI-XXXVIII (Forces to move the sea) 449

Lemmas I-III, Proposition XXXIX (Precession of equinoxes) 455

Lemmas IV-XI, Propositions XL-XLII (Comets) 460

General Scholium 503

The System of the World. 511

Index to the Principia. 575

This work was published before January 1, 1928, and is in the public domain worldwide because the author died at least 100 years ago.

DEDICATION.

* * *

TO THE

TEACHERS OF THE NORMAL SCHOOL

OF THE STATE OF NEW-YORK.

Gentlemen:—

A stirring freshness in the air, and ruddy streaks upon the horizon of the moral world betoken the grateful dawning of a new era. The days of a drivelling instruction are departing. With us is the opening promise of a better time, wherein genuine manhood doing its noblest work shall have adequate reward. Teacher is the highest and most responsible office man can fill. Its dignity is, and will yet be held commensurate with its duty—a duty boundless as man's intellectual capacity, and great as his moral need—a duty from the performance of which shall emanate an influence not limited to the now and the here, but which surely will, as time flows into eternity and space into infinity, roll up, a measureless curse or a measureless blessing, in inconceivable swellings along the infinite curve. It is an office that should be esteemed of even sacred import in this country. Ere long a hundred millions, extending from the Atlantic to the Pacific, from Baffin's Bay to that of Panama, shall call themselves American citizens. What a field for those two master-passions of the human soul—the love of Rule, and the love of Gain! How shall our liberties continue to be preserved from the graspings of Ambition and the corruptions of Gold? Not by Bills of Rights Constitutions, and Statute Books; but alone by the rightly cultivated hearts and heads of the People. They must themselves guard the Ark. It is yours to fit them for the consecrated charge. Look well to it: for you appear clothed in the majesty of great power! It is yours to fashion, and to inform, to save, and to perpetuate. You are the Educators of the People: you are the prime Conservators of the public weal. Betray your trust, and the sacred fires would go out, and the altars crumble into dust: knowledge become lost in tradition, and Christian nobleness a fable! As you, therefore, are multiplied in number, elevated in consideration, increased in means, and fulfill, well and faithfully, all the requirements of true Teachers, so shall our favoured land lift up her head among the nations of the earth, and call herself blessed.

In conclusion, Gentlemen, to you, as the conspicuous leaders in the vast and honourable labour of Educational Reform, and Popular Teaching, the First American Edition of the Principia of Newton—the greatest work of the greatest Teacher—is most respectfully dedicated.

N. W. CHITTENDEN.

INTRODUCTION TO THE AMERICAN EDITION.

That the Principia of Newton should have remained so generally unknown in this country to the present day is a somewhat remarkable fact; because the name of the author, learned with the very elements of science, is revered at every hearth-stone where knowledge and virtue are of chief esteem, while, abroad, in all the high places of the land, the character which that name recalls is held up as the noblest illustration of what Man may be, and may do, in the possession and manifestation of pre-eminent intellectual and moral worth; because the work is celebrated, not only in the history of one career and one mind, but in the history of all achievement and human reason itself; because of the spirit of inquiry, which has been aroused, and which, in pursuing its searchings, is not always satisfied with stopping short of the fountain-head of any given truth; and, finally, because of the earnest endeavour that has been and is constantly going on, in many sections of the Republic, to elevate the popular standard of education and give to scientific and other efforts a higher and a better aim.

True, the Principia has been hitherto inaccessible to popular use. A few copies in Latin, and occasionally one in English may be found in some of our larger libraries, or in the possession of some ardent disciple of the great Master. But a dead language in the one case, and an enormous price in both, particularly in that of the English edition, have thus far opposed very sufficient obstacles to the wide circulation of the work. It is now, however, placed within the reach of all. And in performing this labour, the utmost care has been taken, by collation, revision, and otherwise, to render the First American Edition the most accurate and beautiful in our language. "Le plus beau monument que l'on puisse élever à la gloire de Newton, c'est une bonne édition de ses ouvrages:" and a monument like unto that we would here set up. The Principia, above all, glows with the immortality of a transcendant mind. Marble and brass dissolve and pass away; but the true creations of genius endure, in time and beyond time, forever: high upon the adamant of the indestructible, they send forth afar and near, over the troublous waters of life, a pure, unwavering, quenchless light whereby the myriad myriads of barques, richly laden with reason, intelligence and various faculty, are guided through the night and the storm, by the beetling shore and the hidden rock, the breaker and the shoal, safely into havens calm and secure.

To the teacher and the taught, the scholar and the student, the devotee of Science and the worshipper of Truth, the Principia must ever continue to be of inestimable value. If to educate means, not so much to store the memory with symbols and facts, as to bring forth the faculties of the soul and develope them to the full by healthy nurture and a hardy discipline, then, what so effective to the accomplishment of that end as the study of Geometrical Synthesis? The Calculus, in some shape or other, is, indeed, necessary to the successful prosecution of researches in the higher branches of philosophy. But has not the Analytical encroached upon the Synthetical, and Algorithmic Formulae been employed when not requisite, either for the evolution of truth, or even its apter illustration? To each method belongs, undoubtedly, an appropriate use. Newton, himself the inventor of Fluxions, censured the handling of Geometrical subjects by Algebraical calculations; and the maturest opinions which he expressed were additionally in favour of the Geometrical Method. His preference, so strongly marked, is not to be reckoned a mere matter of taste; and his authority should bear with preponderating weight upon the decision of every instructor in adopting what may be deemed the best plan to insure the completest mental development. Geometry, the vigorous product of remote time; blended with the earliest aspirations of Science and the earliest applications of Art; as well in the measures of music as in the movement of spheres; as wholly in the structure of the atom as in that of the world; directing Motion and shaping Appearance; in a word, at the moulding of the created all, is, in comprehensive view, the outward form of that Inner Harmony of which and in which all things are. Plainly, therefore, this noble study has other and infinitely higher uses than to increase the power of abstraction. A more general and thorough cultivation of it should be strenuously insisted on. Passing from the pages of Euclid or Legendre, might not the student be led, at the suitable time, to those of the Principia wherein Geometry may be found in varied use from the familiar to the sublime? The profoundest and the happiest results, it is believed, would attend upon this enlargement of our Educational System.

Let the Principia, then, be gladly welcomed into every Hall where a true teacher presides. And they who are guided to the diligent study of this incomparable work, who become strengthened by its reason, assured by its evidence, and enlightened by its truths, and who rise into loving communion with the great and pure spirit of its author, will go forth from the scenes of their pupilage, and take their places in the world as strong-minded, right-hearted men—such men as the Theory of our Government contemplates and its practical operation absolutely demands.

LIFE OF

SIR ISAAC NEWTON.

* * *

Nec fas est proprius mortali attingere Divos.—Halley.

* * *

From the thick darkness of the middle ages man's struggling spirit emerged as in new birth; breaking out of the iron control of that period; growing strong and confident in the tug and din of succeeding conflict and revolution, it bounded forwards and upwards with resistless vigour to the investigation of physical and moral truth; ascending height after height; sweeping afar over the earth, penetrating afar up into the heavens; increasing in endeavour, enlarging in endowment; every where boldly, earnestly out-stretching, till, in the Author of the Principia, one arose, who, grasping the master-key of the universe and treading its celestial paths, opened up to the human intellect the stupendous realities of the material world, and, in the unrolling of its harmonies, gave to the human heart a new song to the goodness, wisdom, and majesty of the all-creating, all-sustaining, all-perfect God.

Sir Isaac Newton, in whom the rising intellect seemed to attain, as it were, to its culminating point, was born on the 25th of December, O. S. 1642—Christmas day—at Woolsthorpe, in the parish of Colsterworth, in Lincolnshire. His father, John Newton, died at the age of thirty-six, and only a few months after his marriage to Harriet Ayscough, daughter of James Ayscough, of Rutlandshire. Mrs. Newton, probably wrought upon by the early loss of her husband, gave premature birth to her only and posthumous child, of which, too, from its extreme diminutiveness, she appeared likely to be soon bereft. Happily, it was otherwise decreed! The tiny infant, on whose little lips the breath of life so doubtingly hovered, lived;—lived to a vigorous maturity, to a hale old age;—lived to become the boast of his country, the wonder of his time, and the "ornament of his species."

Beyond the grandfather, Robert Newton, the descent of Sir Isaac cannot with certainty be traced. Two traditions were held in the family: one, that they were of Scotch extraction; the other, that they came originally from Newton, in Lancashire, dwelling, for a time, however, at Westby, county of Lincoln, before the removal to and purchase of Woolsthorpe—about a hundred years before this memorable birth.

The widow Newton was left with the simple means of a comfortable subsistence. The Woolsthorpe estate together with small one which she possessed at Sewstern, in Leicestershire, yielded her an income of some eighty pounds; and upon this limited sum, she had to rely chiefly for the support of herself, and the education of her child. She continued his nurture for three years, when, marrying again, she confided the tender charge to the care of her own mother.

Great genius is seldom marked by precocious development; and young Isaac, sent, at the usual age, to two day schools at Skillington and Stoke, exhibited no unusual traits of character. In his twelfth year, he was placed at the public school at Grantham, and boarded at the house of Mr. Clark, an apothecary. But even in this excellent seminary, his mental acquisitions continued for a while unpromising enough: study apparently had no charms for him; he was very inattentive, and ranked low in the school. One day, however, the boy immediately above our seemingly dull student gave him a severe kick in the stomach; Isaac, deeply affected, but with no outburst of passion, betook himself, with quiet, incessant toil, to his books; he quickly passed above the offending classmate; yet there he stopped not; the strong spirit was, for once and forever, awakened, and, yielding to its noble impulse, he speedily took up his position at the head of all.

His peculiar character began now rapidly to unfold itself. Close application grew to be habitual. Observation alternated with reflection. "A sober, silent, thinking lad," yet, the wisest and the kindliest, the indisputable leader of his fellows. Generosity, modesty, and a love of truth distinguished him then as ever afterwards. He did not often join his classmates in play; but he would contrive for them various amusements of a scientific kind. Paper kites he introduced; carefully determining their best form and proportions, and the position and number of points whereby to attach the string. He also invented paper lanterns; these served ordinarily to guide the way to school in winter mornings, but occasionally for quite another purpose; they were attached to the tails of kites in a dark night, to the dismay of the country people dreading portentous comets, and to the immeasureable delight of his companions. To him, however, young as he was, life seemed to have become an earnest thing. When not occupied with his studies, his mind would be engrossed with mechanical contrivances; now imitating, now inventing. He became singularly skilful in the use of his little saws, hatchets, hammers, and other tools. A windmill was erected near Grantham; during the operations of the workmen, he was frequently present; in a short time, he had completed a perfect working model of it, which elicited general admiration. Not content, however, with this exact imitation, he conceived the idea of employing, in the place of sails, animal power, and, adapting the construction of his mill accordingly, he enclosed in it a mouse, called the miller, and which by acting on a sort of treadwheel, gave motion to the machine. He invented, too, a mechanical carriage—having four wheels, and put in motion with a handle worked by the person sitting inside. The measurement of time early drew his attention. He first constructed a water clock, in proportions somewhat like an old-fashioned house clock. The index of the dial plate was turned by a piece of wood acted upon by dropping water. This instrument, though long used by himself, and by Mr. Clark's family, did not satisfy his inquiring mind. His thoughts rose to the sun; and, by careful and oft-repeated observations of the solar movements, he subsequently formed many dials. One of these, named Isaac's dial, was the accurate result of years' labour, and was frequently referred to for the hour of the day by the country people.

May we not discern in these continual efforts—the diligent research, the patient meditation, the aspiring glance, and the energy of discovery—the stirring elements of that wondrous spirit, which, clear, calm, and great, moved, in after years, through deep onward through deep of Nature's mysteries, unlocking her strongholds, dispelling darkness, educing order—everywhere silently conquering.

Newton had an early and decided taste for drawing. Pictures, taken sometimes from copies, but often from life, and drawn, coloured and framed by himself, ornamented his apartment. He was skilled also, in poetical composition, "excelled in making verses;" some of these were borne in remembrance and repeated, seventy years afterward, by Mrs. Vincent, for whom, in early youth, as Miss Storey, he formed an ardent attachment. She was the sister of a physician resident near Woolsthorpe; but Newton's intimate acquaintance with her began at Grantham, where they were both numbered among the inmates of the same house. Two or three years younger than himself, of great personal beauty, and unusual talent, her society afforded him the greatest pleasure; and their youthful friendship, it is believed, gradually rose to a higher passion; but inadequacy of fortune prevented their union. Miss Storey was afterwards twice married; Newton, never; his esteem for her continued unabated during life, accompanied by numerous acts of attention and kindness.