Newton's Principia : $b The mathematical principles of natural philosophy

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 on 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

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

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
[Pg 9]
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 on 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[Pg 10] 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 Woolst

horpe—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 a small one which she possessed at Sewstern, in Leicestershire, yielded her an income of some eighty pounds; and on 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,[Pg 11] 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 qui

te 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 immeasurable 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 on 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[Pg 12] 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. In 1656, Newton's mother was again left a widow, and took up her abode once more at Woolsthorpe. He was now fifteen years of age, and had made great progress in his studies; but she, desirous of his help, and from motives of economy, recalled him from school. Business occupations, however, and the management of the farm, proved utterly distasteful to him. When sent to Grantham Market on Saturdays, he would betake himself to his former lodgings in the apothecary's garret, where some of Mr. Clark's old books employed his thoughts till the aged and trustworthy servant had executed the family commissions and announced the necessity of return: or, at other times, our young philosopher would seat himself under a hedge, by the wayside, and continue his studies till the same faithful personage—proceeding alone to the town and completing the day's business—stopped as he returned.[Pg 13] The more immediate affairs of the farm received no better attention. In fact, his passion for study grew daily more absorbing, and his dislike for every other occupation more intense. His mother, therefore, wisely resolved to give him all the advantages which an education could confer. He was sent back to Grant

ham school, where he remained for some months in busy preparation for his academical studies. At the recommendation of one of his uncles, who had himself studied at Trinity College, Cambridge, Newton proceeded thither, and was duly admitted, on the 5th day of June 1660, in the eighteenth year of his age. The eager student had now entered on a new and wider field; and we find him devoting himself to the pursuit of knowledge with amazing ardour and perseverance. Among other subjects, his attention was soon drawn to that of Judicial Astrology. He exposed the folly of this pseudo-science by erecting a figure with the aid of one or two of the problems of Euclid;—and thus began his study of the Mathematics. His researches into this science were prosecuted with unparalleled vigour and success. Regarding the propositions contained in Euclid as self-evident truths, he passed rapidly over this ancient system—a step which he afterward much regretted—and mastered, without further preparatory study, the Analytical Geometry of Descartes. Wallis's Arithmetic of Infinites, Saunderson's Logic, and the Optics of Kepler, he also studied with great care; writing on them many comments; and, in these notes on Wallis's work was undoubtedly the germ of his fluxionary calculus. His progress was so great that he found himself more profoundly versed than his tutor in many branches of learning. Yet his acquisitions were not gotten with the rapidity of intuition; but they were thoroughly made and firmly secured. Quickness of apprehension, or intellectual nimbleness did not belong to him. He saw too far: his insight was too deep. He dwelt fully, cautiously on the least subject; while to the consideration of the greatest, he brought a massive strength joined with a matchless clearness, that, regardless of the merely trivial or unimportant, bore with unerring sagacity on the prominences of the subject, and, grappling with its difficulties, rarely failed to surmount them. [Pg 14] His early and fast friend, Dr. Barrow—in compass of invention only inferior to Newton—who had been elected Professor of Greek in the University, in 1660, was made Lucasian Professor of Mathematics in 1663, and soon aft