MATH.-STAT. SIM ISAAC MIBWf OM« NEWTON'S PRINCIPIA. THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY, BY SIR ISAAC NEWTON; TRANSLATED INTO ENGLISH BY ANDREW MOTTE. TO WHICH IS ADDKTV 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 PI. W. CHITTENDEN, M. A., &e. NEW-YORK PUBLISHED BY DANIEL ADEE, 45 LIBERTY STREET. p*- Kntered according to Act of Congress, in the year 1846, by DANIEL ADEE. 3!Ltht Clerk's Office ut'tiie Southern Oisli:ct Court of New-York. TWuey * Lockwoof, Stom 16 Spruce St. N. Y. 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 ora. The days of a drivelling instruction are departing. With us is the opening promise of a better time, wherein genuine man hood 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 hun dred 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 hu man soul — the love of Rule, and the love of Gain ! How shall our liberties continue to be preserved from the graspings of Am bition and the corruptions of Gold ? Not by Bills of Rights 4 DEDICATION. Constitutions, and Statute Books ; but alone by the rightly culti vated hearts and heads of the PEOPLE. They must themselves guard the Ark. It is yours to tit 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 no bleness 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 fa voured 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 Helbrm, ana Popular Teaching, the First American Edition of the PRINCIPIA ol Newton — the greatest wrork 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 gen erally 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 foun tain-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 edu cation 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 d^ad 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, how ever, placed within the reach of all. And in performing this la bour, 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. u Le plus beau monument que l? on puisse clever a la gloire de Newton, c'est une bonne edition de ses ouvrages :" and a monument like unto that we would here V: INTRODUCTION TO 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, un wavering, 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 effec tive to the accomplishment of that end as the study of Geometri cal 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 prefer ence, so strongly marked, is not to be reckoned a mere matter oi 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 completes! mental develop ment. Geometry, the vigorous product of remote time ; blended with the earliest aspirations of Science and the earliest applica tions of Art ; as well in the measures of music as in the move ment of spheres ; as wholly in the structure of the atom as in that of the world; directing MOTION and shaping APPEARANCE; in a wonl, *t the moulding of the created all, is, in comprehensive THE AMERICAN EDITION. Vll 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 ab straction. A more general and thorough cultivation of it should oe 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 enlight ened 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 SIE ISAAC NEWTON. Nec fas est proprius mortal? 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 en deavour, 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 harmo nies, gave to the human heart a new song to the goodness, wis dom, 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 De cember, O. S. 1642 — Christmas day — at Woolsthorpe, in the parish of Colsterworth, in Lincolnshire. His father, John New ton, died at the age of thirty-six, and only a few months after his marriage to Harriet Ayscough, daughter of James Ayscough, oi 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 10 LIFE OF SIR ISAAC NEWTON. so doubtingly hovered, lived ; — lived to a vigorous maturity, to a hale old age ; — lived to become the boast of his country, the won der of his time, and the "ornament of his srjecies." 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, be fore 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 com fortable subsistence. The Woolsthorpe estate together with small one which she possessed at Sewstern, in Leicestershire, yield ed her an income of some eighty pounds ; and upon this limited sum, she had to rely chiefly for the support of herself, and the educa tion 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 Gran- tham, and boarded at the house of Mr. Clark, an apothecary. But even in this excellent seminary, his mental acquisitions con tinued for a while unpromising enough : study apparentlv had no charms for him ; he was very inattentive, and ranked low in the school. One day, however, the boy immediately above our seem ingly 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 itb 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. Gener- LIFE OF SIR ISA VC NEWTON. 11 osity, 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 ol 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 ol 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 ot treadvvheel, 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 h'rst 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-re peated 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 re search^ the patient meditation, the aspiring glance, and the energy 12 LIFE OF SIR ISAAC NEWTON. 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 si lently 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 per sonal 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 mar ried ; 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 widowr, 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 manage ment 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 trust worthy 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 re- LIFE OF SIR ISAAC NEWTON, 13 turned. 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 in tense. His mother, therefore, wisely resolved to give him all the advantages which an education could confer. He was sent back to Grantham 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 upon a new and wider field ; and we find him devoting himself to the pursuit of know ledge with amazing ardour and perseverance. Among other sub jects, 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 unparallelled 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 pre paratory 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 upon them many comments ; and, in these notes on Wallis's work was un doubtedly 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 Intel lectual nimbleness did not belong to him. He saw too far : his, insight was too deep. He dwelt fully, cautiously upon the least subject ; while to the consideration of the greatest, he brought a massive strength joined with a matchless clearness, that, regard less of the merely trivial or unimportant, bore with unerring sa gacity upon the prominences of the subject, and, grappling with its difficulties, rarely failed to surmount them. 14 LIFE OF SIR ISAAC NEWTON His early and fast friend, Dr. Barrow — in compass of inven tion only inferior to Newton — who had been elected Professor of Greek in the University, in 1660, was made Lucasian Profes sor of Mathematics in 1663, and soon afterward delivered his Optical Lectures : the manuscripts of these were revised by New ton, and several oversights corrected, and many important sug gestions made by him ; but they were not published till 1669. In the year 1665, he received the degree of Bachelor of Arts ; and, in 1666, he entered upon those brilliant and imposing dis coveries which have conferred inappreciable benefits upon science, and immortality upon his own name. Newton, himself, states that he was in possession of his Method of Fluxions, " in the year 1666, or before." Infinite quantities had long been a subject of profound investigation ; among the ancients by Archimedes, and Pappus of Alexandria ; among the moderns by Kepler, Cavaleri, Roberval, Fermat and Wallis. With consummate ability Dr. Wallis had improved upon the la- hours of his predecessors : with a higher power, Newton moved forwards from where Wallis stopped. Our author first invented his celebrated BINOMIAL THEOREM. And then, applying this Theorem to the rectification of curves, and to the determination of the surfaces and contents of solids, and the position of their centres of gravity, he discovered the general principle of deducing the areas of curves from the ordinate, by considering the area as a nascent quantity, increasing by continual fluxion in the propor tion of the length of the ordinate, and supposing the abscissa to increase uniformly in proportion to the time. Regarding lines as generated by the motion of points, surfaces by the motion of lines, and solids by the motion of surfaces, and considering that the ordinates, abscissae, &c., of curves thus formed, vary accord ing to a regular law depending on the equation of the curve, he deduced from this equation the velocities with which these quantities are generated, and obtained by the rules of infinite series, the ultimate value required. To the velocities with which every line or quantity is generated, he gave the name of FLUX IONS, and to the lines or quantities themselves, that of FLUENTS. A discovery that successively baffled the acutest and strongest LIFE OF SIR ISAAC NEWTON. 15 intellects : — that, variously modified, has proved of incalculable service in aiding to develope the most abstruse and the highest 'ruths in Mathematics and Astronomy : and that was of itself enough to render any name illustrious in the crowded Annals of Science. At this period, the most distinguished philosophers were direct ing all their energies to the subject of light and the improvement of the refracting telescope. Newton, having applied himself to the grinding of "optic glasses of other figures than spherical," ex perienced the impracticability of executing such lenses ; and con jectured that their defects, and consequently those of refracting telescopes, might arise from some other cause than the imperfect convergency of rays to a single point. He accordingly "procured a triangular glass prism to try therewith the celebrated phenom ena of colours." His experiments, entered upon with zeal, and conducted with that industry, accuracy, and patient thought, for which he was so remarkable, resulted in the grand conclusion, that LIGHT WAS NOT HOMOGENEOUS, BUT CONSISTED OF RAYS, SOME OF WHICH WERE MORE REFRANGIBLE THAN OTHERS. This profound and beautiful discovery opened up a new era in the History of Optics. As bearing, however, directly upon the construc tion of telescopes, he saw that a lens refracting exactly like a prism would necessarily bring the different rays to different foci, at different distances from the glass, confusing and rendering the vision indistinct. Taking for granted that all bodies produced spectra of ^ jtial length, he dismissed all further consideration of the refracting instrument, and took up the principle of reflection. Rays of all colours, he found, were reflected regularly, so that the angle of reflection was equal to the angle of incidence, and hence he concluded that ojitical instruments might be brought to any degree of perfection imaginable, provided reflecting specula of the requisite figure and finish could be obtained. At this stage of his optical researches, he was forced to leave Cambridge on account of the plague which was then desolating England. He retired to Woolsthorpe. The old manor-house, in which he was born, was situated in a beautiful little valley, on the west side of the river Witham ; and here in the quiet home of his boyhood, 2 16 LIFE OF SIR ISAAC NEWTON. he passed his days in serene contemplation, while the stalking pestilence was hurrying its tens of thousands into undistinguisha • ble graves. Towards the close of a pleasant day in the early autumn of 1666, he was seated alone beneath a tree, in his garden, absorbed in meditation. He was a slight young man ; in the twenty-fourth year of his age ; his countenance mild and full of thought. For a century previous, the science of Astronomy had advanced with rapid strides. The human mind had risen from the gloom and bondage of the middle ages, in unparalleled vigour, to unfold the system, to investigate the phenomena, and to establish the laws of the heavenly bodies. Copernicus, Tycho Brahe, Kepler, Galileo, and others had prepared and lighted the way for him who was ta give to their labour its just value, and to their genius its true lustre. At his bidding isolated facts were to take order as parts of one harmonious whole, and sagacious conjectures grow luminous in the certain splendour of demonstrated truth. And this ablest man had come — was here. His mind, familiar with the knowledge of past effort, and its unequalled faculties develop ed in transcendant strength, was now moving on to the very threshold of Its grandest achievement. Step by step the untrod den path was measured, till, at length, the entrance seemed dis closed, and the tireless explorer to stand amid the first opening wonders of the universe. The nature of gravity — that mysterious power which causes all bodies to descend towards the centre of the earth — had, in deed, dawned upon him. And reason busily united link to link of that chain which was yet to be traced joining the least to the vastest, the most remote to the nearest, in one harmonious bond. From the bottoms of the deepest caverns to the summits of the highest mountains, this power suffers no sensible change : may not its action, then, extend to the moon ? Undoubtedly : and furthei reflection convinced him that such a power might be .sufficient for retaining that luminary in her orbit round the earth. But, though this power suffers no sensible variation, in the little change of distance from the earth's centre, at which we may place our- .'«lves, yet, at the distance of the moon, :miy not its force undergo LIFE OF SIR ISAAC NEWTON. 17 more or less diminution ? The conjecture appeared most proba ble : and, in order to estimate what the degree of diminution might be, he considered that if the moon be retained in her orbit by the force of gravity, the primary planets must also be carried round the sun by the like power; and, by comparing the periods of the several planets with their distances from the sun, he found that, if they were held in their courses by any power like gravity, its strength must decrease in the duplicate proportion of the in crease of distance. In forming this conclusion, he supposed the planets to move in perfect circles, concentric to the sun. Now was this the law of the moon's motion ? Was such a force, em anating from the earth and directed to the moon, sufficient, when diminished as the square of the distance, to retain her in her orbit ? To ascertain this master-fact, he compared the space through which heavy bodies fall, in a second of time, at a given distance from the centre of the earth, namely, at its surface, with the space through which the moon falls, as it were, to the earth, in the same time, while revolving in a circular orbit. He was absent from books ; and, therefore, adopted, in computing the earth's diameter, the common estimate of sixty miles to a degree of latitude as then in use among geographers and navigators. The result of his calculations did not, ot course, answer his ex pectations ; hence, he concluded that some other cause, beyond the reach of observation — analogous, perhaps, to the vortices of Des cartes — joined its action to that of the power of gravity upon the rnooil. Though by no means satisfied, he yet abandoned awhile further inquiry, and remained totally silent upon the subject. These rapid marches in the career of discovery, combined with the youth of Newton, seem to evince a penetration the most lively, and an invention the most exuberant. But in him there was a conjunction of influences as extraordinary as fortunate. Study, unbroken, persevering and profound carried on its inform ing and disciplining work upon a genius, natively the greatest, and rendered freest in its movements, and clearest in its vision, through the untrammelling and enlig} tenirig power of religion. And, in this happy concurrence, are to be sought the elements of those amazing abilities, which, grasping, with equal facility, the 18 LIFE OF SIR ISAAC NEWTON. minute and the stupendous, brought these successively to light, and caused science to make them her own. In 1667, Newton was made a Junior Fellow ; and, in the year following, he took his degree of Master of Arts, and was appoint ed to a Senior Fellowship. On his return to Cambridge, in 1668, he resumed his optical labours. Having thought of a delicate method of polishing metal, he proceeded to the construction of his newly projected reflect ing telescope ; a small specimen of which he actually made with his own hands, It was six inches long ; and magnified about forty times ; — a power greater than a refracting instrument of six feet tube could exert with distinctness. Jupiter, with his four satellites, and the horns, or moon-like phases of Venus were plainly visible through it. THIS WAS THE FIRST REFLECTING TELESCOPE EVER EXECUTED AND DIRECTED TO THE HEAVENS. He gave an account of it, in a letter to a friend, dated February 23d, 1668-9 — a letter which is also remarkable for containing the firs' allusion to his discoveries " concerning the nature of light." En couraged by the success of his first experiment, he again executed with his own hands, not long afterward, a second and superior instrument of the same kind. The existence of this having come to the knowledge of the Royal Society of London, in 1671, they requested it of Newton for examination. He accordingly sent it to them, It excited great admiration; it was shown to the king* a drawing and description of it was sent to Paris ; and the tele- scope itself was carefully preserved in the Library of the Society. Newton lived to see his invention in public use, and of eminent service in the cause of science. In the spring of 1669, he wrote to his friend Francis Aston, Esq., then about setting out on his travels, a letter of advice and directions, it was dated May 18th, and is interesting as exhibit ing some of the prominent features in Newton's character. Thus : — " Since in your letter you give me so much liberty of spending my judgment about what may be to your advantage in travelling, 1 shall do it more freely than perhaps otherwise would have been decent, Fir,ct, then, I will lay down some general rules, most of LIFE OF SIR ISAAC NEWTON. 19 which, I bolieA e, you have considered already ; but if any of them be new to you, they may excuse the rest ; if none at all, yet is my punishment more in writing than yours in reading. "When you come into any fresh company. 1. Observe their humours. 2. Suit your own carriage thereto, by which insinua tion you will make their converse more free and open. 3. Let your discourse be more in queries and doubtings than peremptory assertions or disputings, it being the design of travellers to learn, not to teach. Besides, it will persuade your acquaintance that you have the greater esteem of them, and so make them more ready to communicate what they know to you ; whereas nothing sooner occasions disrespect and quarrels than peremptoriness. You will find little or no advantage in seeming wiser or much more ignorant than your company. 4. Seldom discommend any thing though never so bad, or do it but moderately, lest you be unexpectedly forced to an unhandsome retraction. It is safer to commend any thing more than it deserves, than to discommend a thing so much as it deserves; for commendations meet not so often with oppositions, or, at least, are not usually so ill re sented by men that think otherwise, as discommendations ; and you will insinuate into men's favour by nothing sooner than seem ing to approve and commend what they like ; but beware o doing it by comparison. 5. If you be affronted, it is better, in c foreign country, to pass it by in silence, and with a jest, though with some dishonour, than to endeavour revenge ; for, in the first case, your credit's ne'er the worse when you return into England, or come into other company that have not heard of the quarrel. But, in the second case, you may bear the marks of the quarrel while you live, if you outlive it at all. But, if you find yoursell unavoidably engaged, 'tis best, I think, if you can command your passion and language, to keep them pretty evenly at some certain moderate pitch, not much heightening them to exasperate your adversary, or provoke his friends, nor letting them grow overmuch dejected to make him insult. In a word, if you can keep reason above passion, that and watchfulness will be your best defendants. To which purpose you may consider, that, though such excuses is this — He provok't me so much I could not forbear — may pass 20 LIFE OF SIR ISAAC NEWTON. among friends, yet amongst strangers they are insignificant, ina only argue a traveller's weakness. " To these I may add some general heads for inquiries or ob servations, such as at present I can think on. As, 1. To observe the policies, wealth, and state affairs of nations, so far as a soli- fary traveller may conveniently do. 2. Their impositions upon all sorts of people, trades, or commodities, that are remarkable. 3. Their laws and customs, how far they differ from ours. 4. Their trades and arts wherein they excel or come short of us in England. 5. Such fortifications as you shall meet with, their fashion, strength, and advantages for defence, and other such mili tary affairs as are considerable. 6. The power and respect be longing to their degrees of nobility or magistracy. 7. It will not be time misspent to make a catalogue of the names and excellen cies of those men that are most wise, learned, or esteemed in any nation. 8. Observe the mechanism and manner of guiding ships. 9. Observe the products of Nature in several places, especially in mines, with the circumstances of mining and of extracting metals or minerals out of their ore, and of refining them ; and if you meet with any transmutations out of their own species into another (as out of iron into copper, out of any metal into quick silver, out of one salt into another, or into an insipid body, &c.), those, above all, will be worth your noting, being the most lucif- erous, and many times lucriferous experiments, too, in philosophy. 10. The prices of diet and other things. 11. And the staple commodities of places. " These generals (such as at present I could think of), if they will serve for nothing else, yet they may assist you in drawing up a model to regulate your travels by. ' As for particulars, these that follow are all that 1 can now think of, viz. ; whether at Schem- nitium, in Hungary (where there are mines of gold, copper, iron, vitriol, antimony, &c.). they change iron into copper by dissolving »t in a vitriolate water, which they find in cavities of rocks in the mines, and then melting the slimy solution in a stroi ig fire, which in the cooling proves copper. The like is said to be done in other places, which I cannot now remember ; perhaps, too, it may be lone in Italy. For about twenty or thirty years agone there was LIFE OF SIR ISAAC NEWTON, 21 a certain vitriol came from thence (called Roman vitriol), but of a nobler virtue than that which is now called by that name ; which vitriol is not now to be gotten, because, perhaps, they make a greater gain by some such trick as turning iron into copper with it than by selling it. 2. Whether, in Hungary, Sclavonia, Bohemia, near the town Eila, or at the mountains of Bohemia near Silesia, there be rivers whose waters are impregnated with gold ; perhaps, the gold being dissolved by some corrosive water like aqua regis, and the solution carried along with the stream, that runs through the mines. And whether the practice of laying mercury in the rivers, till it be tinged with gold, and then strain ing the mercury through leather, that the gold may stay behind, be a secret yet, or openly practised. 3. There is newly con trived, in Holland, a mill to grind glasses plane withal, and I think polishing them too ; perhaps it will be worth the while to see it. 4. There is in Holland one Borry, who some years since was imprisoned by the Pope, to have extorted from him secrets (as I am told) of great worth, both as to medicine and profit, but he escaped into Holland, where they have granted him a guard. I think he usually goes clothed in green. Pray inquire what you can of him, and whether his ingenuity be any profit to the Dutch. You may inform yourself whether the Dutch have any tricks to keep their ships from being all worm-eaten in their voyages to the Indies. Whether pendulum clocks do any service in finding out the longitude, &c. " I am very weary, and shall not stay to part with a long compliment, only I wish you a good journey, and God be with you." It was not till the month of June, 1669, that our author made known his Method of Fluxions. He then communicated the work which he had composed upon the subject, and entitled, ANALYSIS PER EQUATIONES NUMERO TERMINORUM INFINITAS, to his friend Dr. Barrow. The latter, in a letter dated 20th of the same month, mentioned it to Mr. Collins, and transmitted it to him, on the 31st of July thereafter. Mr. Collins greatly approv> ed of the work ; took a copy of it ; and sent the original back to Dr. Barrow. During the same and the two following years, Mr < LIFE OF SIR ISAAC NEWTON. Collins, by his extensive correspondence, spread the knowledge of this discovery among the mathematicians in England, Scotland, France, Holland and Italy. Dr. Barrow, having resolved to devote himself to Theology, resigned the Lucasian Professorship of Mathematics, in 1669, in favour of Newton, who accordingly received the appointment to the vacant chair. During the years 1669, 1670, and 1671, our author, as such Professor, delivered a course of Optical Lectures. Though these contained his principal discoveries relative to the different re- frangibility of light, yet the discoveries themselves did not be come publicly known, it seems, till he communicated them to the Royal Society, a few weeks after being elected a member there of, in the spring of 1671-2. He now rose rapidly in reputation, and was soon regarded as foremost among the philosophers of the age. His paper on light excited the deepest interest in the Royal Society, who manifested an anxious solicitude to secure the author from the " arrogations of others," and proposed to publish his discourse in the monthly numbers in which the Transactions were given to the world. Newton, gratefully sensible of these expres sions of esteem, willingly accepted of the proposal for publication. He gave them also, at this time, the results of some further ex periments in the decomposition and re-composition of light : — that the same degree of refrangibility always belonged to the same colour, and the same colour to the same degree of refrangibility : that the seven different colours of the spectrum were original, or simple, and that whiteness^ or white light was a compound of all these seven colours. The publication of his new doctrines on light soon called forth violent opposition as to their soundness. Hooke and Huygens — men eminent for ability and learning — were the most conspicuous of the assailants. And though Newton effectually silenced all his adversaries, yet he felt the triumph of little gain in comparison .vith the loss his tranquillity had sustained. He subsequently re- narked in allusion to this controversy — and to one with whom he was destined to have a longer and a bitterer conflict — " I was so persecuted with discussions arising from the publication of m v LIFE OF SIR ISAAC NEWTON. 23 theory ot light, that I blamed my own imprudence for parting with so substantial a blessing as rny quiet to run after a shadow.7' In a communication to Mr. Oldenburg, Secretary of the Royal Society, in 1672, our author stated many valuable suggestions re lative to the construction of REFLECTING MICROSCOPES which he considered even more capable of improvement than telescopes. He also contemplated, about the same time, an edition of Kirick- huysen's Algebra, with notes and additions; partially arranging, as an introduction to the work, a treatise, entitled, A Method of Fluxions ; but he finally abandoned the design. This treatise, however, he resolved, or rather consented, at a late period of his life, to put forth separately ; and the plan would probably have been carried into execution had riot his death intervened. It was translated into English, and published in 1736 by John Colson, Professor of Mathematics in Cambridge. Newton, it is thought, made his discoveries concerning the INFLECTION and DIFFRACTION of light before 1674. The phe nomena of the inflection of light had been first discovered more than ten years before by Grimaldi. And Newton began by re peating one of the experiments of the learned Jesuit — admitting a beam of the sun's light through a small pin hole into a dark chamber : the light diverged from the aperture in the form of a, cone, and the shadows of all bodies placed in this light were larger than might have been expected, and surrounded with three coloured fringes, the nearest being widest, and the most remote the narrowest. Newton, advancing upon this experiment, took exact measures of the diameter of the shadow of a human hair, and of the breadth of the fringes, at different distances behind it, and discovered that these diameters and breadths were not pro portional to the distances at which they were measured. He hence supposed that the rays which passed by the edge of the hair were deflected or turned aside from it, as if by a repulsive force, the nearest rays suffering the greatest, the more remote a less degree of deflection. In explanation of the coloured fringes, he queried : whether the rays which differ in refrangibility do not differ also in flexibility, and whether they are n<»t, by these dif ferent inflections, separated from one another, so as after separa- < LIFE OF SIR ISAAC NEWTON. tion to make the colours in the three fringes above described ? Also, whether the rays, in passing by the edges and sides ol bodies, are not bent several times backwards and forwards with an eel-like motion — the three fringes arising from three such bendings ? His inquiries on this subject were here interrupted and fiever renewed. His Theory of the COLOURS of NATURAL BODIES was commu nicated to the Royal Society, in February, 1675. This is justly regarded as one of the profoundest of his speculations. The fun damental principles of the Theory in brief, are : — That bodies possessing the greatest refractive powers reflect the greatest quantity of light ; and that, at the confines of equally refracting media, there is no reflection. That the minutest particles of al most all natural bodies are in some degree transparent. That between the particles of bodies there are pores, or spaces, either empty or filled with media of a less density than the particles themselves. That these particles, and pores or spaces, have some definite size. Hence he deduced the Transparency, Opacity, and colours of natural bodies. Transparency arises from the particles and their pores being too small to cause reflection at their com mon surfaces — the light all passing through ; Opacity from the opposite cause of the particles and their pores being sufficiently large to reflect the light which is " stopped or stifled7' by the multitude of reflections ; and colours from the particles, accord ing to their several sizes, reflecting rays of one colour and trans mitting those of another — or in other words, the colour that meets the eye is the colour reflected, while all the other rays are transmitted or absorbed. Analogous in origin to the colours of natural bodies, he con sidered the COLOURS OF THIN PLATES. This subject was interest ing and important, and had attracted considerable investigation. He, however, was the first to determine the law of the produc tion of these colours, arid, during the same year made known the results of his researches herein to the Royal Society. His mode of procedure in these experiments was simple and curious. He placed a double convex lens of a large known radius of curvature, the flat surface of a plano-convex object glass. Thus, from UFE OF SIR ISAAC NEWTON. 25 their point of contact at the centre, to the circumference of the lens, he obtained plates of air, or spaces varying from the ex- tremest possible thinness, by slow degrees, to a considerable thick ness. Letting the light fall, every different thickness of this plate of air gave different colours — the point of contact of the lens and glass forming the centre of numerous concentric colored nags. Now the radius of curvature of the lens being known, the thickness of the plate of air, at any given point, or where any par ticular colour appeared, could be exactly determined. Carefully noting, therefore, the order in which the different colours ap peared, he measured, with the nicest accuracy, the different thick* nesses at which the most luminous parts of the rings were pro duced, whether the medium were air, water, or mica — all these substances giving the same colours at different thicknesses ; — the ratio of which he also ascertained. From the phenomena obser ved in these experiments, Newton deduced his Theory of Fits of EASY REFLECTION AND TRANSMISSION of light. It consists in suppos ing that every particle of light, from its first discharge from a lumi nous body, possesses, at equally distant intervals, dispositions to be reflected from, or transmitted through the surfaces of bodies upon which it may fall. For instance, if the rays are in a Fit of Easy Reflection, they are on reaching the surface, repelled, thrown off] or reflected from it ; if, in a Fit of Easy Transmission, they are attracted, drawn in, or transmitted through it. By this Theory of Fits, our author likewise explained the colours of thick plates. He regarded light as consisting of small material particles emitted from shining substances. He thought that these parti cles could be re-combined into solid matter, so that " gross bodies and light, were convertible into one another ;" that the particles of light and the particles of solid bodies acted mutually upon each other ; those of light agitating and heating those of solid bodies, and the latter attracting and repelling the former. Newton was the first to suggest the idea of the POLARIZATION of light. In the paper entitled An Hypothesis Explaining Properties of Light, December, 1675, our author first introduced his opinions re specting Ether — opinions which he afterward abandoned and again 26 LIFE OF SIR S.\AC 1SEWTON. permanently resumed — " A most subtle spirit which pervades" ah bodies, and is expanded through all the heavens. It is electric, and almost, if not quite immeasurably elastic and rare. " By the force and action of which spirit the particles of bodies mutually attract one another, at near distances, and cohere, if contiguous ; and electric bodies operate at greater distances, as well repelling as attracting the neighbouring corpuscles ; and light is emitted, -reflected, refracted, inflected and heats bodies ; and all sensation is excited, and the members of animal bodies move at the com mand of the will, namely, by the vibrations of this spirit, mutu ally propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles." This " spirit" was no anima mundi ; nothing further from the thought of Newton ; but was it not, on his part, a par tial recognition of, or attempt to reach an ultimate material force, or primary element, by means of which, " in the roaring loom of time," this material universe, God's visible garment, may be woven for us ? The Royal Society were greatly interested in the results of some experiments, which our author had, at the same time, com municated to them relative to the excitation of electricity in glass ; and they, after several attempts and further direction from him, succeeded in re-producing the same phenomena. One of the most curious of Newton's minor inquiries related to the connexion between the refractive powers and chemical com position of bodies. He found on comparing the refractive powers and the densities of many different substances, that the former were very nearly proportional to the latter, in the same bodies. Unctuous and sulphureous bodies were noticed as remarkable excep tions — as well as the diamond — their refractive powers being two or three times greater in respect of their densities than in the case of other substances, while, as among themselves, the one was generally proportional to the other. He hence inferred as to the diamond a great degree of combustibility •; — a conjecture which the experiments of modern chemistry have shown to be true. The chemical researches of our author were probably pursued with more or less diligence from the time of his witnessing some LIFE OF .SIR ISAAC NEWTON. 27 ?t' the uractical operations in that science at the Apothecary's at Grantham. DE NATURA ACIDORUM is a short chemical paper, on various topics, and published in Dr. Horsley's Edition of his works. TABULA QUANTITATUM E r GRADUUM COLORIS was in serted iii the Philosophical Transactions ; it contains a compara tive scale of temperature from that of melting ice to that of a small kitchen coal-fire. He regarded fire as a body heated so hot as to emit light copiously ; and flame as a vapour, fume, or ex halation heated so hot as to shine. To elective attraction, by the operation of which the small particles of bodies, as he con ceived, act upon one another, at distances so minute as to escape observation, he ascribed all the various chemical phenomena ot precipitation, combination, solution, and crystallization, and the mechanical phenomena of cohesion and capillary attraction. New ton's chemical views were illustrated and confirmed, in part, at least, in his own life-time. As to the structure of bodies, he was of opinion " that the smallest particles of matter may cohere by the strongest attractions, and compose bigger particles of weaker virtue ; and many of these may cohere and compose bigger par tides whose virtue is still weaker ; and so on for divers succes sions, until the progression end in the biggest particles, on which the operations in chemistry and the colours of natural bodies de pend, and which by adhering, compose bodies of sensible magni tude." There is good reason to suppose that our author was a diligent student of the writings of Jacob Behmen ; and that in conjunction with a relative, Dr. Newton, he was busily engaged, for several months in the earlier part of life, in quest of the philosopher's tincture. " Great Alchymist," however, very imperfectly de scribes the character of Behmen, whose researches into things material and things spiritual, things human and things divine, ai- ford the strongest evidence of a great and original mind. More appropriately here, perhaps, than elsewhere, may be given Newton's account of some curious experiments, made in his own person, on the action of light upon the retina, Locke, who was an intimate friend of our author, wrote to him for his opinion on a certain fact stated in Boyle's Book of Colours. Newton, in 2S LIFE OF SIR ISAAC NEWTON. his reply, dated June 30th, 16(Jl, narrates the following circum stances, which probably took place in the course of his optical researches. Thus : — " The observation you mention in Mr. Boyle's Book of Colours I once tried upon myself with the hazard of my eyes. The manner was this ; I looked a very little while upon the sun in the looking-glass with my right eye, and then turned my eyes into a dark corner of my chamber, arid winked, to observe the impres sion made, and the circles of colours which encompassed it, and how they decayed by degrees, and at last vanished. This I re peated a second and a third time. At the third time, when the phantasm of light and colours about it were almost vanished, in tending my fancy upon them to see their last appearance, I found, to my amazement, that they began to return, and by little and little to become as lively and vivid as when I had newly looked upon the sun. But when I ceased to intend my fancy upon them, they vanished again. After this, I found, that as often as I went into the dark, and intended my mind upon them, as when a man looks earnestly to see anything which is difficult to be seen, I could make the phantasm return without looking any more upon the sun ; and the oftener I made it return, the more easily I could make it return again. And, at length, by repeating this, without looking any more upon the sun, I made such an impression on my eye, that, if I looked upon the clouds, or a book, or any bright object, I saw upon it a round bright spot of light like the sun, and, which is still stranger, though I looked upon the sun with my right eye only, and not with my left, yet my fancy began *o make an impression upon my left eye, as well us upon my right. For if I shut my right eye, or looked upon a book, or the clouds, with my left eye, I could see the spectrum of the sun almost as plain as with my right eye, if I did but intend my fancy a little while upon it ; for at first, if I shut my right eye, and looked with my left, the spectrum of the sun did not appear till I intended my fancy upon it ; but by repeating, this appeared every time more easily. And now, in a few hours' time, I had brought my eyes to such a pass, that I could look upon no blight object with either eye, but I saw the sun before me, so that I durst neither write LIFE OF SIR ISAAC NEWTON. 29 nor read ; but to recover the use of my eyes, shut myself up in my chamber made dark, for three days together, and used all means to divert my imagination from the sun. For if I thought upon him, I presently saw his picture, though I was in the dark. But by keeping in the dark, and employing my mind about other things, I began in three or four days to have some use of my eyes again ; and by forbearing to look upon bright objects, recovered them pretty well, though not so well but that, for some months after, the spectrum of the sun began to return as often as I began to meditate upon the phenomena, even though I lay in bed at mid night with my curtains drawn. But now I have been very well for many years, though I am apt to think, if I durst venture my eyes, I could still make the phantasm return by the power of my fancy. This story I tell you, to let you understand, thaj; in the observation related by Mr. Boyle, the man's fancy probably con curred with the impression made by the sun's light to produce that phantasm of the sun which he constantly saw in bright ob jects. And so your question about the cause of phantasm in volves another about the power of fancy, which I must confess is too hard a knot for me to untie. To place this effect in a constant motion is hard, because the sun ought then to appear perpetually. It seems rather to consist in a disposition of the sensorium to move the imagination strongly, and to be easily moved, both by the imagination and by the light, as often as bright objects are looked upon."J Though Newton had continued silent, yet his thoughts were by no means inactive upon the vast subject of the planetary mo tions. The idea of Universal Gravitation, first caught sight of, so to speak, in the garden at Woolsthorpe, years ago, had gradually expanded upon him. We find him, in a letter to Dr. Hooke, Secretary of the Royal Society, dated in November, 1679, pro posing to verify the motion of the earth by direct experiment, namely, by the observation of the path pursued by a body falling from a considerable height. He had concluded that the path would be spiral ; but Dr. Hooke maintained that it would be an eccentric ellipse iu vacuo, and an ellipti-spiral in a resisting me dium. Our author, aided by this correction of his error, and by 30 LIFE OF SIR ISAAC NEWTON. the discovery that a projectile would move in an elliptical orbil when under the influence of a force varying inversely as the square of the distance, was led to discover " the theorem bj which he afterwards examined the ellipsis ;" and to demonstrate the celebrated proposition that a planet acted upon by an attrac tive force varying inversely as the squares of the distances will describe an elliptical orbit, in one of whose foci the attractive force resides. When he was attending a meeting of the Royal Society, in June 1682, the conversation fell upon the subject of the measure ment of a degree of the meridian, executed by M. Picard, a French Astronomer, in 1679. Newton took a memorandum oi the result ; and afterward, at the earliest opportunity, computed from it the diameter of the earth : furnished with these new data, he resumed his calculation of 1666. As he proceeded therein, he saw that his early expectations were now likely to be realized ; the thick rushing, stupendous results overpowered him ; he be came unable to carry on the process of calculation, and intrusted its completion to one of his friends. The discoverer had, indeed, grasped the master-fact. The law of falling bodies at the earth's surface was at length identified with that which guided the moon in her orbit. And so his GREAT THOUGHT, that had for sixteen years loomed up in dim, gigantic outline, amid the first dawn of a plausible hypothesis, now stood forth, radiant and not less grand, in the mid-day light of demonstrated truth. It were difficult, nay impossible to imagine, even, the influence of a result like this upon a mind like Newton's. It was as if the keystone had been fitted to the glorious arch by which his spirit should ascend to the outskirts of infinite space — spanning the immea surable — weighing the imponderable — computing the incalculable —mapping out the marchings of the planets, and the far-wander ings of the comefs, and catching, bring back to earth some clearer notes of that higher melody which, as a sounding voice, bears perpetual witness to the design and omnipotence of a creating Deity. Newton, extending the law thus obtained, composed a series of about twelve propositions on the motion of the primary planets LIFE OF SIR ISAAC NEWTON. 31 about the sun. These were sent to London, and communicated to the Royal Society about the end of 1683. At or near this pe riod, other philosophers, as Sir Christopher Wren, Dr. Halley, and Dr. Hooke, were engaged in investigating the same subject ; but with no definite or satisfactory results. Dr. Halley, having seen, it is presumed, our author's propositions, went in August, 1684, to Cambridge to consult with him upon the subject. Newton assured him that he had brought the demonstration to perfection. In November, Dr. Halley received a copy of the work ; and, in the following month^ announced . it to the Royal Society, with the author's promise to have it entered upon their Register. Newton, subsequently reminded by the Society of his promise, proceeded in the diligent preparation of the work, and. though suffering an interruption of six weeks, transmitted the manuscript of the first book to London before the end of April. The work was entitled PHILOSOPHI/E NATURALIS PRINCIPIA MATHEMATICA, dedicated to the Royal Society, and presented thereto on the 28th of April, 1685-6. The highest encomiums were passed upon it ; and the council resolved, on the 19th of May, to print it at the expense of the Society, and under the di rection of Dr. Halley. The latter, a few days afterward, com municated these steps to Newton, who, in a reply, dated the 20th of June, holds the following language : — " The proof you sent me I like very well. I designed the whole to consist of three books ; the second was finished last summer, being short, and only wants transcribing, and drawing the cuts fairly. Some new propositions I have since thought on, which I can as well let alone. The third wants the theory of comets. In autumn last, I spent two months in calculation to no purpose for want of a good method, which made me afterward return to the first book, and enlarge it with diverse propositions, some* relating to comets, others to other things found ouf last winter. The third I now design to sup press. Philosophy is such an impertinently litigious lady, that a man had as good be engaged in l«iw-suits as have to do with her. I found it so formerly, and now I can no sooner come near her again, but she gives me warning. The first two books without the third will not so well bear the title of P/iilosophicc Naturalis 3 32 LIFE OF SIR ISAAC NEWTON. Principia Mathematicia ; and thereupon I had altered it to this, De Motu Corporum Libri duo. But after second thought I re tain the former title. It will help the sale of the book, which I ought not to diminish now 'tis yours." This " warning" arose from some pretensions put forth by Dr. Hooke. And though Newton gave a minute and positive refuta tions of such claims, yet, to reconcile all differences, he gener ously added to Prop. IV. Cor. 6, Book I, a Scholium, in which Wren, Hooke and Halley are acknowledged to have indepen dently deduced the law of gravity from the second law of Kepler. The suppression of the third book Dr. Halley could not endure to see. " I must again beg you" says he, " not to let your re sentments run so high as to deprive us of your third book, where in your applications of your mathematical doctrine to the theory of comets, and several curious experiments, which, as I guess by what you write ought to compose it, will undoubtedly render it acceptable to those who will call themselves philosophers without mathematics, which are much the greater number." To these solicitations Newton yielded. There were no "resentments," how ever, as we conceive, in his " design to suppress." He sought peace ; for he loved and valued it above all applause. But, in spite of his efforts for tranquillity's sake, his course of discovery was all along molested by ignorance or presumptuous rivalry. The publication of the great work now went rapidly forwards, The second book was sent to the Society, and presented on the 2d March ; the third, on the 6th April ; and the whole was com pleted and published in the month of May, 1686-7. In the sec ond Lemma of the second book, the fundamental principle of his fiuxionary calculus was, for the first time, given to the world ; but its algorithm or notation did not appear till published in the second volume nf Dr. Wallis's works, in 1693. And thus was ushered into existence The PRINCIPIA — a work to which pre-eminence above all the productions of the human intellect has been awarded — a work that must be esteemed of priceless worth so long as Science has a votary, or a single wor shipper be left to kneel at the altar of Truth. LIFE OF SIR ISAAC NEWTON. 33 The entire work bears the general title of THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY. It consists of three books: the first two, entitled, OF THE MOTION OF BODIES, are occupied with the laws and conditions of motions and forces, and are illus trated with many scholia treating of some of the most general and best established points in philosophy, such as the density and resistance of bodies, spaces void of matter, and the motion of sound and light. From these principles, there is deduced, in the third book, drawn up in as popular a style as possible and entitled, OF THE SYSTEM OF THE WORLD, the constitution of the system of i he world. In regard to this book, the author say^ — " I had, indeed, composed the third Book in a popular method, that it might be read by many ; but afterwards, considering that such as had not suf- ficently entered into the principles could not easily discover the strength of the consequences, nor lay aside the prejudices to which they had been many years accustomed, therefore, to prevent dis putes which might be raised upon such accounts, I chose to reduce the substance of this Book into the form of Propositions (in the mathematical way), which should be read by those only who had first made themselves masters of the principles established in the preceding Books : not that I would advise any one to the previous study of every Proposition of those Books." — "It is enough it one carefully reads the Definitions, the Laws of Motion, and the three first Sections of the first Book. He may then pass on to this Book, and consult such of the remaining Propositions of the first two Books, as the references in this, and his occasions shall re quire." So that " The System of the World" is composed both " in a popular method," and in the form of mathematical Propo sitions. The principle of Universal Gravi' ition, namely, that every particle of matter is attracted by, or gravitates to, every other particle of matter, icith a force inversely proportional to the squares of their distances — is the discovery w? ich characterizes The PRINCIPIA. This principle the author deduced from the mo tion of the moon, and the three laws of Kepler — laws, which Newton, in turn, by his greater law, demonstrated to be true. From the first law of Kepler, namely, the proportionality of LIFE OF SIR ISAAC NEWTON. the areas to t\ie times of their description, our author inferred that the force which retained the planet in its orbit was always directed to the sun ; and from the second, namely, that every planet moves in an ellipse with the sun in one of its foci, he drew the more general inference that the force by which the planet moves round that focus varies inversely as the square of its dis tance therefrom : and he demonstrated that a planet acted upon by such a force could not move in any other curve than a conic section ; showing when the moving body would describe a circu lar, an elliptical, a parabolic, or hyperbolic orbit. He demon strated, too, that this force, or attracting, gravitating power re sided in every, the least particle ; but that, in spherical masses, it operated as if confined to their centres ; so that, one sphere or body will act upon another sphere or body, with a force directly proportional to the quantity of matter, and inversely as the square of the distance between their centres; and that their velocities of mutual approach will be in the inverse ratio of their quantities o*' matter. Thus he grandly outlined the Universal Law. Verify ing its truth by the motions of terrestrial bodies, then by those of the moon and other secondary orbs, he finally embraced, in one mighty generalization, the entire Solar System — all the move ments of all its bodies — planets, satellites and comets — explain ing and harmonizing the many diverse and theretofore inexplica ble phenomena. Guided by the genius of Newton, we see sphere bound to sphere, body to body, particle to particle, atom to mass, the min utest part to the stupendous whole — each to each, each to all, and all to each — in the mysterious bonds of a ceaseless, recipro cal influence. An influence whose workings are shown to be alike present in the globular dew-drop, or oblate-spheroidal earth ; in the falling shower, or vast heaving ocean tides ; in the flying thistle-down, or fixed, ponderous rock ; in the swinging pendulum, or time-measuring sun ; in the varying and unequal moon, or earth's slowly retrograding poles ; in the uncertain meteor, or oiazing comet wheeling swiftly away on its remote, yet determined round. An influence, in fine, that may link system to system through all the star-glowing firmament ; then firmament to iirma- LIFE OF SIR ISAAC NEWTON. 35 merit ; aye, firmament to firmament, again and again, till, con verging home, it may be, to some ineffable centre, where more presently dwells He who inhabiteth immensity, and where infini tudes meet and eternities have their condux, and where around move, in softest, swiftest measure, all the countless hosts that crowd heaven's fathomless deeps. And yet Newton, amid the loveliness and magnitude of Om nipotence, lost not sight of the Almighty One. A secondary, however universal, was not taken for the First Cause. An im pressed force, however diffused and powerful, assumed not the functions of the creating, giving Energy. Material beauties, splendours, and sublimities, however rich in glory, and endless in extent, concealed not the attributes of an intelligent Supreme. From the depths of his own soul, through reason and the WORD, he had risen, a priori, to God : from the heights of Omnipotence, through the design and law of the builded universe, he proved n of their orbits, done also with greater accuracy. In this third edition the resistance of mediums is somewhat more largely handled than before; and new experiments of the resistance of heavy bodies falling in air are added. In the third book, the argument to prove that the moon is retained in its orbit by the force of gravity is enlarged on ; and there are added new observations of Mr. Pound's of the proportion of the diameters of Ju.piter to each other : there are, besides, added Mr. Kirk's observations of the comet in 16SO ; the orbit of that comet com puted in an ellipsis by Dr. Halley ; and the ortit of the comet in computed by Mr. Bradley, OOK I. THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY DEFINITIONS. DEFINITION I. 77w? quantity of matter is the measure of the same, arising from its density and hulk conjutictly. THUS air of a double density, in a double space, is quadruple in quan- ti ty ; in a triple space, sextuple in quantity. The same thing is to be un derstood of snow, and fine dust or powders, that are condensed by compres sion or liquefaction • and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts oi bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body ; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter. DEFINITION II. The quantity of motion is the measure nf tlie same, arising from the velocity and quantity of matter corjunctly. The motion of the whole if the same proportional to the efficacy of the cause that propagates it from the cen tre, through the spaces round about. Thus the magnetic force is greater in one load-stone and less in another according to their sizes and strength of intensity. 76 THE MATHEMATICAL PRINCIPLES DEFINITION VII. The accelerative quantity of a centripetal force is the measure, of tht same, proportional to the velocity which it generates in a given time. Thus the force of the same load-stone is greater at a less distance, and less at a greater : also the force of gravity is greater in valleys, less on tops of exceeding high mountains ; and yet less (as shall hereafter be shown), at greater distances from the body of the earth ; but at equal distan ces, it is the same everywhere ; because (taking away, or allowing for, the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small. DEFINITION VIII. TJie motive quantity of a centripetal force, is the measure of the samt\ proportional to the motion which it generates in a given twip. Thus the weight is greater in a greater body, less in a less body ; and. in the same body, it is greater near to the earth, and less at remoter dis tances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight ; and it is al ways known by the quantity of an equal and contrary force just sufficient to Ifinder the descent of the body. These quantities of forces, we may, for brevity's sake, call by the names of motive, accelerative, and absolute forces ; and, for distinction's sake, con sider them, with respect to the bodies that tend to the centre ; to the places of those bodies ; and to the centre of force towards which they tend ; that is to say, I refer the motive force to the body as an endeavour and propen sity of the whole towards a centre, arising from the propensities of the several parts taken together ; the accelerative force to the place of the body, as a certain power or energy diffused from the centre to all places around to move the bodies that are in them : and the absolute force to the centre, as endued with some cause, without which those motive forces would not be propagated through the spaces round about ; whether that cause be some central body (siuh as is the load-stone, in the centre of the magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical causes and seats. Wherefore the accelerative force will stand in the same relation to the motive, as celerity does to motion. For the quantity of motion arises from the celerity drawn into the quantity of matter : and the motive force arises from the accelerative force drawn into the same quantity of matter. For the sum of the actions of the accelerative force, upon the several ; articles of the body, is the motive force of the whole. Hence it is, that near the OF NATURAL PHILOSOPHY. 77 surface of the earth, where the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body : but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those re gions, where the accelerative gravity is diminished into one half, the weight of a body two or three times less, will be four or six times less. I likewise call attractions and impulses, in the same sense, accelerative, and motive ; and use the words attraction, impulse or propensity of any sort towards a centre, promiscuously, and indifferently, one for another ; considering those forces not physically, but mathematically : wherefore, the reader is not to imagine, that by those words, I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points) ; when at any time I happen to speak of centres as attracting, or as endued with attractive powers. SCHOLIUM. Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be under stood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the re moving of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. I. Absolute, true, and mathematical time, of itself, and from its own na ture flows equably without regard to anything external, and by another name is called duration : relative, apparent, and common time, is some sen sible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ; such as an hour, a day, a month, a year. II. Absolute space, in its own nature, without regard to anything exter nal, remains always similar and immovable. Relative space is some mo vable dimension or measure of the absolute spaces ; which our senses de termine by its position to bodies ; and which is vulgarly taken for immo vable space ; such is the dimension of a subterraneous, an aereal, or celestial space, determined by its position in respect of the earth. Absolute and relative space, are the same in figure and magnitude ; but they do not re main always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains al ways the same, will at one time be one part of the absolute space into which TS THE MATHEMATICAL PRINCIPLES the air passes ; at another time it will be another part of the same, and so. absolutely understood, it will be perpetually mutable. III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space ; not the situation, nor the external surface of the body. For the places of equal solids are always equal ; but their superfices, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts ; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places ; and therefore the place of the whole is the same thing with the sum of the places of the parts, and for that reason, it is internal, and in the whole body. IV. Absolute motion is the translation of a body from one absolute place into another ; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses ; or that part of its cavity which the body fills, and which therefore moves together with the ship : and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body, which relatively rests in the ship, will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space ; partly from the relative motion of the ship on the earth ; and if the body moves also relatively in the ship ; its true motion will arise, partly from the true motion of the earth, in immovable space, and partly from the relative mo tions as well of the ship on the earth, as of the body in the ship ; and from these relative motions will arise the relative motion of the body on the earth. As if that part of the earth, where the ship is, was truly moved toward the east, with a velocity of 10010 parts; while the ship itself, with a fresh gale, and full sails, is carried towards the west, with a velocity ex pressed by 10 of those parts ; but a sailor walks in the ship towards the east, with 1 part of the said velocity ; then the sailor will be moved truly in immovable space towards the east, with a velocity of 10001 parts, and relatively on the earth towards the west, with a velocity of 9 of those parts. Absolute time, in astronomy, is distinguished from relative, by the equa tion or correction of the vulgar time. For the natural days are tr^y un equal, though they are commonly considered as equal, and used for a meas ure of time ; astronomers correct this inequality for their more accurate deducing of the celestial motions. It may be, that there is no such thing as an equable motion, whereby time may H accurately measured. All mo OF NATURAL PHILOSOPHY. 79 tions may be accelerated and retarded; but the true, or equable, progress of absolute time is liable to no change. The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all : and therefore it ought to be distinguished from what are only sensible measures thereof ; and out of which we collect it, by means of the astronomical equation. The necessity of which equation, for deter mining the times of a phamomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter. As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession ; and in space as _to order of situation. It is from their essence or nature that they are places ; and that the primary places of things should be moveable, is absurd. These are therefore the absolute places ; and trans lations out of those places, are the only absolute motions. But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body con sidered as immovable, we define all places ; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And'so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common af fairs ; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred. But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest ; but impossi ble to know, from the position of bodies to one another in our regions whether any of these do keep the same position to that remote body; it follows that absolute rest cannot be determined from the position of bodies in our regions. It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axis of motion ; and the impetus of bodies moving forward, arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, those that are relatively at rest within them, will partake of their motion. Upon which account, the true and absolute motion of a body cannot be Jeter- 8C THE MATHEMATICAL PRINCIPLES mined by the translation of it from those which only seem to rest ; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, beside their translation from near the surrounding ones, partake likewise of their true motions ; and though that translation were not made they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel ; but, if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell. A property, near akin to the preceding, is this, that if a place is moved, whatever is placed therein moves along with it ; and therefore a body, which is moved from a place in motion, partakes also of the motion of its place. Upon which account, all motions, from places in motion, are no other than parts of entire and absolute motions ; and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place ; and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Where fore, entire and absolute motions can be no otherwise determined than by immovable places : and for that reason I did before refer those absolute motions to immovable places, but relative ones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another ; and upon this account must ever remain unmoved ; and do thereby constitute immovable space. The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved : but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies with which the former is compared, that by their giving way, that relation may be changed, in which the re lative rest or motion of this other body did consist. Again, true motion suffers always some change from any force impressed upon the moving body ; but relative motion docs not necessarily undergo any change by such forces. For if the same forces are likewise impressed on those other bodies, with which the comparison is made, that the relative position may be pre served, then that condition will be preserved in which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true suffers some change. Upon which accounts; true motion does by no means consist in such relations. The effects whicli distinguish absolute from relative motion arc, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute cir cular motion., they are greater or less, according t » the quantity of the OF NATURAL PHILOSOPHY. «1 motion. If a vessel, hung: by & }ong cord, is so often turned ubout that the cord is strongly twisted, then filled with water, and held at rest together with the water ; after, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion ; the surface of the water will at first be plain, as before the vessel began to move : but the vessel; by grad ually communicating its motion to the water, will make it begin sensibly ^to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I have experi enced), and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its en deavour to recede from the axis of its motion ; and the true and absolute circular motion of the water, which is here directly contrary to the rela- tivej discovers itself, and may be measured by this endeavour. At first, when the relative motion of the water in the vessel was greatest, it pro duced no endeavour to recede from the axis ; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its en deavour to recede from the axis ; and this endeavour showed the real cir cular motion of the water perpetually increasing, till it had acquired its greatest quantity, wh en the water rested relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect ; but relative motions, in one and the same body, are innumerable, according to the various relations it bears to external bodies, and like other relations, arc altogether destitute of any real effect, any otherwise than they may perhaps par take of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them ; the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes, endeavour to recede from the axis of their motions. Wherefore relative quantities are not the quantities themselves, whose names they bear, but those sensible measures of them (either accurate cr inaccurate), which arc commonly used instead of the measured quantities themselves. And if the meaning of words is to he determined bv their 82 THE MATHEMATICAL PRINCIPLES use, then by the names time, space, place and motion, their measures arv properly to be understood ; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. Upon which account, they do strain the sacred writings, who there interpret those words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quan tities themselves with their relations and vulgar measures. It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent ; be cause the parts of that immovable space, in which those motions are per formed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate : for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions ; partly from the forces, which are the causes and effects of the true motions. For instance, if tAvo globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavour of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decr» ase of the tensicn of 1 le cord, we might infer the increment or decrement of their motions : and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented ; that is, we might discover their hinder- most faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circu lar motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions, we could not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest ; and then, lastly, from the trans lation of the globes among the bodies, we should find the determination oi their motions. But how we are to collect the true motions from their causes, effects, and apparent differences ; and, vice versa, how from the mo tions, either true or apparent, we may come to the knowledge of theii causes and effects, shall be explained more at large in the following tra<;t For to this end it was that I composed it. OF NATURAL PHILOSOPHY. AXIOMS, OR LAWS OF MOTION. LAW I. Hvery body perseveres in its state of rest, or of uniform motion in a ri^ht line, unless it is compelled to change that state by forces impressed thereon. PROJECTILES persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is re tarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve then jDotions both pro gressive and circular for a much longer time. LAW II. The alteration of motion is ever proportional to the motive force imp reus ed ; and is made in the direction of the right line in. which that force is impressed. If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other ; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both. LAW III. To every action there is always opposed an equal reaction : or the mu tual actions of two bodies upon each other are always equal, and di rected to contrary parts. Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. 84 THE MATHEMATICAL PRINCIPLES If a body impinge upon another, and by its force change the motion of (It* other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies ; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally pro portional to the bodies. This law takes place also in attractions, as will be proved in the next scholium. COROLLARY I. A body by two forces conjoined will describe the diagonal of a parallelo gram, in the same time that it wovld describe the sides, by those forces apart. If a body in a given time, by the force M impressed apart in the place A, should with an uniform motion / be carried from A to B ; and by the force N impressed apart in the same place, should be carried from A to c ~\) C ; complete the parallelogram ABCD, and, by both forces acting together, it will in the same time be carried in the diagonal from A to D. For since the force N acts in the direction of the line AC, parallel to BD, this force (by the second law) will not at all alter the velocity generated by the other force M, by which the body is carried towards the line BD. The body therefore will arrive at the line BD in the same time, whether the rorce N be impressed or not ; and therefore at the end of that time it will he found somewhere in the line BD. By the same argument, at the end of the same time it AY ill be found somewhere in the line CD. Therefore it will be found in the point D, where both lines meet. But it will move in ;i right line from A to D, by Law I. COROLLARY II. And hence is explained the composition of any one direct force AD, out of any two oblique forces AC and CD ; and, on the contrary, the re solution of any one direct force AD into two oblique forces AC and CD : which composition and resolution are abundantly confirmed from, mechanics. As if the unequal radii OM and ON drawn from the centre O of any wheel, should sustain the weights A and P by the cords MA and NP ; and the forces of those weights to move the wheel were required. Through the rentre O draw the right line KOL, meeting the cords perpendicularly in A and L; and from the centre O, with OL the greater of the distances OF NATURAL PHILOSOPHY. OK arid OL, describe a circle, meeting the cord MA in D : and drawing OD, make AC paral- "^ lei and DC perpendicular thereto. Now, it being indifferent whether the points K, L, D, of the cords be lixed to the plane of the wheel or not, the weights will have the same effect whether they are suspended from the points K and L, or from D and L. Let the whole force of the weight A be represented by the line AD, and let it be resolved into the forces AC and CD ; of which the force AC, drawing the radius OD directly from the centre, will have no effect to move the wheel : but the other force DC, drawing the radius DO perpendicularly, will have the same effect as if it drew perpendicularly the radius OL equal to OD ; that is, it w ill have the same effect as the weight P, if that weight is to the weight A as the force DC is to the force DA ; that is (because of the sim ilar triangles ADC, DOK), as OK to OD or OL. Therefore the weights A and P, which are reciprocally as the radii OK and OL that lie in the same right line, will be equipollent, and so remain in equilibrio ; which is the well known property of the balance, the lever, and the wheel. If either weight is greater than in this ratio, its force to move the wheel will be so much greater. If the weight p, equal to the weight P, is partly suspended by the cord NJO, partly sustained by the oblique plane pG ; draw p}i, NH, the former perpendicular to the horizon, the latter to the plane pG ; and if the force of the weight p tending downwards is represented by the line /?H, it may be resolved into the forces joN, HN. If there was any plane /?Q, perpendicular to the cord y?N, cutting the other plane pG in a line parallel to the horizon, and the weight p was supported only by those planes pQ, pG, it would press those planes perpendicularly with the forces pN, HN; to wit, the plane joQ, with the force joN, and the plane pG with the force HN. And therefore if the plane pQ was taken away, so thnt the weight might stretch the cord, because the cord, now sustaining the weight, supplies the place of the plane that was removed, it will be strained by the same force joN which pressed upon the plane before. Therefore, the tension of this oblique cord joN will be to that of the other perpendic ular cord PN as jt?N to joH. And therefore if the weight p is to the weight A in a ratio compounded of the reciprocal ratio of the least distances of the cords PN, AM, from the centre of the wheel, and of the direct ratio of pH tojoN, the weights will have the same effect towards moving the wheel, and will therefore sustain each other : as any one may find by experiment. But the weight p pressing upon those two oblique planes, may be con sidered as a wedge between the two internal surfaces of a body split by it; and hence tlif ft IV.P* of th^ v, ^dge and the mallet may be determined; foi 8G THE MATHEMATICAL PRINCIPLES because the force with which the weight p presses the plane pQi is to the force with which the same, whether by its own gravity, or by the blow of a mallet, is impelled in the direction of the line joH towards both the planes, as joN to pH ; and to the force with which it presses the other plane pG, as joN to NH. And thus the force of the screw may be deduced from a like resolution of forces ; it being no other than a wedge impelled with the force of a lever. Therefore the use of this Corollary spreads far and wide, and by that diffusive extent the truth thereof is farther con firmed. For on what has been said depends the whole doctrine of mechan ics variously demonstrated by different authors. For from hence are easily deduced the forces of machines, which are compounded of wheels, pullics, levers, cords, and weights, ascending directly or obliquely, and other mechan ical powers ; as also the force of the tendons to move the bones of animals. COROLLARY III. The (/uaittity of motion, which is collected by taking the sum of the mo tions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action oj bodies among themselves. For action and its opposite re-action are equal, by Law III, and there fore, by Law II, they produce in the motions equal changes towards oppo site parts. Therefore if the motions are directed towards the same parts. whatever is added to the motion of the preceding body will be subducted from the motion of that which follows ; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both ; and therefore the difference of the motions directed towards opposite parts will remain the same. Thus if a spherical body A with two parts of velocity is triple of a spherical body B which follows in the same right line with ten parts of velocity, the motion of A will be to that of B as 6 to 10. Suppose, then, their motions to be of 6 parts and of 10 parts, and the sum will be 16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4, or 5 parts of motion, B will lose as many ; and therefore after reflexion A will proceed With 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the sum remaining always of 16 parts as before. If the body A acquire 9, 10, 11, or 12 parts of motion, and therefore after meeting proceed with 15, 16, 17, or 18 parts, the body B, losing so many parts as A has got, will either proceed with 1 part, having lost 9, or stop and remain at rest, as having lost its whole progressive motion of 10 parts ; or it will go back with 1 part, having not only lost its whole motion, but (if 1 may so say) one part more; or it will go back with 2 parts, because a progressive mo tion of 12 parts is taken off. And so the sums of the Conspiring motions 15 ,1, or 16-1-0, and the differences of the contrary i otions 17 — 1 and OF NATURAL PHILOSOPHY. [S — 2, will always be equal to 16 parts, as they were before tie meeting and reflexion of the bodies. But, the motions being known with whicli the bodies proceed after reflexion, the velocity of either will be also known, by taking the velocity after to the velocity before reflexion, as the motion after is to the motion before. As in the last case, where the motion of tho body A was of 0 parts before reflexion and of IS parts after, and the velocity was of 2 parts before reflexion, the velocity thereof after reflexion will be found to be of 6 parts ; by saying, as the 0 parts of motion before to 18 parts after, so are 2 parts of velocity before reflexion to (5 parts after. But if the bodies are cither not spherical, or, moving in different right lines, impinge obliquely one upon the other, and their mot'ons after re flexion are required, in those cases we are first to determine the position of the plane that touches the concurring bodies in the point of concourse , then the motion of each body (by Corol. II) is to be resolved into two, one perpendicular to that plane, and the other parallel to it. This done, be cause the bodies act upon each other in the direction of a line perpendicu lar to this plane, the parallel motions are to be retained the same after reflexion as before ; and to the perpendicular motions we are to assign equal changes towards the contrary parts ; in such manner that the sum of the conspiring and the difference of the contrary motions may remain the same as before. From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows ; and it would be too tedious to demonstrate every particular that relates to this subject. COROLLARY IV. The common centre of gravity of two or more bodies does not alter its state of 'motion or rest by the actions of the bodies among themselves ; and therefore the common centre of gravity of all bodies acting upon each other (excluding outward actions and impediments) is either at rest, or moves uniformly in a right line. For if two points proceed with an uniform motion in right lines, and their distance be divided in a given ratio, the dividing point will be either at rest, or proceed uniformly in a right line. This is demonstrated here after in Lem. XXIII and its Corol., when the points are moved in the same plane ; and by a like way of arguing, it may be demonstrated when the points are not moved in the same plane. Therefore if any number of Kdies move uniformly in right lines, the common centre of gravity of any two of them is either at rest, or proceeds uniformly in a right line ; because the line which connects the centres of those two bodies so moving is divided at that common centre in a given ratio. In like manner the common centre of those two and that of a third body will be either at rest or moving uni formly in aright line because at that centre the distance 1 etween th«? 88 THE MATHEMATICAL PRINCIPLES common centre of the two bodies, and the centre of this last, is divided in a given ratio. In like manner the common centre of these three, and of a fourth body, is either at rest, or moves uniformly in a right line ; because the distance between the common centre of the three bodies, and the centre of the fourth is there also divided in a given ratio, and so on m itifinitum. Therefore, in a system of bodies where there is neither any mutual action among themselves, nor any foreign force impressed upon them from without, and which consequently move uniformly in right lines, the common centre of gravity of them all is either at rest or moves uniformly forward in a right line. Moreover, in a system of two bodies mutually acting upon each other, since the distances between their centres and the common centre of gravity of both are reciprocally as the bodies, the relative motions of those bodies, whether of approaching to or of receding from that centre, will be equal among themselves. Therefore since the changes which happen to motions are equal and directed to contrary parts, the common centre of those bodies, by their mutual action between themselves, is neither promoted nor re tarded, nor suffers any change as to its state of motion or rest. But in a system of several bodies, because the common centre of gravity of any two acting mutually upon each other suffers no change in its state by that ac tion : and much less the common centre of gravity of the others with which that action does not intervene ; but the distance between those two centres is divided by the common centre of gravity of all the bodies into parts re ciprocally proportional to the total sums of those bodies whose centres they are : and therefore while those two centres retain their state of motion or rest, xhe common centre of all does also retain its state : it is manifest that the common centre of all never suffers any change in the state of its mo tion or rest from the actions of any two bodies between themselves. But in such & system all the actions of the bodies among themselves either hap pen between two bodies, or are composed of actions interchanged between some two bodies ; and therefore they do never produce any alteration in the comrrv n centre of alias to its state of motion or rest. Wherefore tiince that centre, when the bodies do not act mutually one upon another, Oilier is nt rest or moves uniformly forward in some right line, it will, :v\>U7ithst?nding the mutual actions of the bodies among themselves, always jAY-jevere in its state, either of rest, or of proceeding uniformly in a right liiv,, unless it is forced out of this state by the action of some power im- prev^-d from without upon the whole system. And therefore the same law take*1 place in a system consisting of many bodies as in one single body, with wsgard to their persevering in their state of motion or of rest. For the pi \\jressive motion, whether of one single body, or of a whole system of bodies us always to be estimated from the motion of the centre of gravity. COROLLARY V. The motions cf bcdies included in a given space a ~e Ike same among OF NATURAL PHILOSOPHY. 89 themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion. For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are, at first (by sup position), in both cases the same ; and it is from those sums and differences that the collisions and impulses do arise with which the bodies mutually impinge one upon another. Wherefore (by Law II), the effects of those collisions will be equal in both cases ; and therefore the mutual motions of the bodies among themselves in the one case will remain equal to the mutual motions of the bodies among themselves in the other. A clear proof of which we have from the experiment of a ship ; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line. COROLLARY VI. If bodies, any how moved among themselves, are urged in the direct-ton of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had been urged by no such forces. For these forces acting equally (with respect to the quantities of the DO dies to be moved), and in the direction of parallel lines, will (by Law II) move all the bodies equally (as to velocity), and therefore will never pro duce any change in the positions or motions of the bodies among themselves. SCHOLIUM. Hitherto I have laid down such principles as have been received by math ematicians, and are confirmed by abundance of experiments. By the first two Laws and the first two Corollaries, Galileo discovered that the de scent of bodies observed the duplicate ratio of the time, and that the mo tion of projectiles was in the curve of a parabola; experience agreeing with both, unless so far as these motions are a little retarded by the re sistance of the air. When a body is falling, the uniform force of its gravity acting equally, impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities; and in the whole time impresses a whole force, and generates a whole velocity proportional to the time. And the spaces described in proportional times are as the velocities and the times conjunctly ; that is, in a duplicate ratio of the times. And when a body is thrown upwards, its uniform gravity im presses forces and takes off velocities proportional to the times ; and the times of ascending to the greatest heights are as the velocities to be taken off, and those heights are as the velocities and the times conjunetly, or ir, the duplicate ratio of the velocities. And if a body be projected in any direction, the motion arising from its projection jS compounded with the 90 THE MATHEMATICAL PRINCIPLES motion arising from its gravity. As if the body A by its motion of pio- jection alone could describe in a given time the right line AB, and with its motion of falling alone could describe in the same time the altitude AC ; complete the paralello- gram ABDC, and the body by that compounded motion will at the end of the time be found in the place D ; and the curve line AED, which that body describes, will be a parabola, to which the right line AB will be a tangent in A ; and whose ordinate BD will be as the square of the line AB. On the same Laws and Corollaries depend those things which have been demon strated concerning the times of the vibration of pendulums, and are con firmed by the daily experiments of pendulum clocks. By the same, to gether with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huv- gens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication ; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariottc soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have a due regard as well to the resistance of the air bodies. Let the spherical bodies CD F II as to the clastic force of the concurrin A, B be suspended by the parallel and equal strings AC, Bl), from the centres C, D. About these centres, with those intervals, describe the semicircles EAF, GBH, bisected by the radii CA, DB. Bring the body A to any point R of the arc EAF, and (withdrawing the body B) let it go from thence, and after one oscillation suppose it to return to the point V : then RV will be the retardation arising from the resistance of the air. Of this RV let ST be a fourth part, situated in the middle. to wit, so as RS and TV may be equal, and RS may be to ST as 3 to 2 then will ST represent very nearly the retardation during the descent from S to A. Restore the body B to its place: and, supjx sing the body A to be let fall from the point S, the velocity thereof in the place of re flexion A, without sensible error, will be the same as if it had descended m vacit.o from the point T. Upon which account this velocity may be represented by the chord of the arc TA. For it is a proposition well known to geometers, that the velocity of a pendulous body in the loAvest point is as the chord of the arc which it has described in its descent. Aftci OF NATUltAL PHILOSOPHY. 9 I reflexion, suppose the body A comes to the place s, and the body B to the place k. Withdraw the body B, and find the place v, from which if the body A, being let go, should after one oscillation return to the place r, st may be a fourth part of rv. so placed in the middle thereof as to leave is equal to tv, and let the chord of the arc tA represent the velocity which the body A had in the place A immediately after reflexion. For t will be the true and correct place to which the body A should have ascended, if the resistance of the air had been taken off. In the s.ime way we are to correct the place k to which the body B ascends, by finding the place I to which it should have ascended in vacuo. And thus everything may be subjected to experiment, in the same manner as if we were really placed in vacuo. These things being done, we are to take the product (if I may so say) of the body A, by the chord of the arc TA (which represents its velocity), that we may have its motion in the place A immediately before reflexion ; and then by the chord of the arc /A, that we may have its mo tion in the place A immediately after reflexion. And so we are to take the product of the body B by the chord of the arc B/, that we may have the motion of the same immediately after reflexion. And in like manner, when two bodies are let go together from different places, we are to find the motion of each, as well before as after reflexion; and then we may compare the motions between themselves, and collect the effects of the re flexion. Thus trying the thing with pendulums of ten feet, in unequal as well as equal bodies, and making the bodies to concur after a descent through large spaces, as of 8, 12, or 16 feet, I found always, without an error of 3 inches, that when the bodies concurred together directly, equal changes towards the contrary parts were produced in their motions, and, of consequence, that the action and reaction were always equal. As if the body A impinged upon the body B at rest with 9 parts of motion, and losing 7, proceeded after reflexion with 2, the body B was carried back wards with those 7 parts. If the bodies concurred with contrary motions, A with twelve parts of motion, and B with six, then if A receded with J4, B receded with 8 ; to wit, with a deduction of 14 parts of motion on each side. For from the motion of A subducting twelve parts, nothing will remain ; but subducting 2 parts more, a motion will be generated of 2 parts towards the contrary way ; and so, from the motion of the body B of 6 parts, subducting 14 parts, a motion is generated of 8 parts towards the contrary way. But if the bodies were made both to move towards the same way, A, the swifter, with 14 parts of motion, B, the slower, with 5, and after reflexion A went on with 5, B likewise went on with 14 parts ; 9 parts being transferred from A to B. And so in other cases. By the congress and collision of bodies, the quantity of motion, collected from the sum of the motions directed towards the same way, or from the difference, of those that were directed towards contrary ways, was never changed. For the error of an inch or two in measures may be easily ascribed to tht 92 THE MATHEMATICAL PRINCIPLES difficulty of executing everything with accuracy. It was not easy to let go the two pendulums so exactly together that the bodies should impinge one upon the other in the lowermost place AB ; nor to mark the places s, and ky to which the bodies ascended after congress. Nay, and some errors, too, might have happened from the unequal density of the parts of the pen dulous bodies themselves, and from the irregularity of the texture pro ceeding from other causes. But to prevent an objection that may perhaps be alledged against the rule, for the proof of which this experiment was made, as if this rule did suppose that the bodies were either absolutely hard, or at least perfectly elastic (whereas no such bodies are to be found in nature), 1 must add. that the experiments we have been describing, by no means depending upon that quality of hardness, do succeed as well in soft as in hard bodies. For if the rule is to be tried in bodies not perfectly hard, we are only to di minish the reflexion in such a certain proportion as the quantity of the elastic force requires. By the theory of Wren and Huygens, bodies abso lutely hard return one from another with the same velocity with which they meet. But this may be affirmed with more certainty of bodies per fectly elastic. In bodies imperfectly elastic the velocity of the return is to be diminished together with the elastic force ; because that force (except when the parts of bodies are bruised by their congress, or suffer some such extension as happens under the strokes of a hammer) is (as far as I can per ceive) certain and determined, and makes the bodies to return one from the other with a relative velocity, which is in a given ratio to that relative velocity with which they met. This I tried in balls of wool, made up tightly, and strongly compressed. For, first, by letting go the pendulous bodies, and measuring their reflexion, I determined the quantity of their elastic force ; and then, according to this force, estimated the reflexions that ought to happen in other cases of congress. And with this computa tion other experiments made afterwards did accordingly agree ; the balls always receding one from the other with a relative velocity, which was to the relative velocity with which they met as about 5 to 9. Balls of steel returned with almost the same velocity : those of cork with a velocity some-^ thing less ; but in balls of glass the proportion was as about 15 to 16. And thus the third Law, so far as it regards percussions and reflexions, is proved by a theory exactly agreeing with experience. In attractions, I briefly demonstrate the thing after this manner. Sup pose an obstacle is interposed to hinder the congress of any two bodies A. B, mutually attracting one the other : then if either body, as A, is more attracted towards the other body B, than that other body B is towards the first body A, the obstacle will be more strongly urged by the pressure of the body A than by the pressure of the body B, and therefore will not remain in equilibrio : but the stronger pressure will prevail, and will make the system of the two bodies, together with the obstacle, to move directly OF NATURAL PHILOSOPHY. 93 towards the parts on which B lies ; arid in free spaces, to go forward in infmitiim with a motion perpetually accelerated ; which is absurd and contrary to the first Law. For, by the first Law, the system ought to per severe in its state of rest, or of moving uniformly forward in a right line : and therefore the bodies must equally press the obstacle, and be equally attracted one by the other. I made the experiment on the loadstone and iron. If these, placed apart in proper vessels, are made to float by one another in standing water, neither of them will propel the other ; but, by being equally attracted, they will sustain each other's pressure, and rest at last in an equilibrium. So the gravitation betwixt the earth and its parts is mutual. Let the earth FI be cut by any plane EG into two parts EGF and EGI, and their weights one towards the other will be mutually equal. For if by another plane HK, parallel to the former EG, the greater partFJ EGI is cut into two parts EGKH and HKI. whereof HKI is equal to the part EFG, first cut oft', it is evident that the middle part EGKH, will have no propension by its proper weight towards either side, but will hang as it were, and rest in an equilibrium betwixt both. But the one extreme part HKI will with its whole weight bear upon and press the middle part towards the other extreme part EGF : and therefore the force with which EGI, the sum of the parts HKI and EGKH, tends towards the third part EGF, is equal to the weight of the part HKI, that is, to the weight of the third part EGF. And therefore the weights of the two parts EGI and EGF, one towards the other, are equal, as I was to prove. And in deed if those weights were not equal, the whole earth floating in the non- resisting aether would give way to the greater weight, and, retiring from it, would be carried off in infinitum. And as those bodies are equipollent in the congress and reflexion, whose velocities are reciprocally as their innate forces, so in the use of mechanic instruments those agents are equipollent, and mutually sustain each the contrary pressure of the other, whose velocities, estimated according to the determination of the forces, are reciprocally as the forces. So those weights are of equal force to move the arms of a balance; which during the play of the balance are reciprocally as their velocities upw ards and downwards ; that is, if the ascent or descent is direct, those weights are of equal force, which are reciprocally as the distances of the points at which they are suspended from the axis oi the balance : but if they are turned aside by the interposition of oblique planes, or other ob stacles, and made to ascend or descend obliquely, those bodies will be equipollent, wThich are reciprocally as the heights of their ascent and de scent taken according to the perpendicular ; and that on account of the determination of gravity downwards. 94 THE MATHEMATICAL PRINCIPLES And in like manner in the pully, or in a combination of pullies, the force of a hand drawing the rope directly, which is to the weight, whethel ascending directly or obliquely, as the velocity of the perpendicular ascent of the weight to the velocity of the hand that draws the rope, will sustain the weight. In clocks and such like instruments, made up from a combination of wheels, the contrary forces that promote and impede the motion of the wheels, if they are reciprocally as the velocities of the parts of the wheel on which they are impressed, will mutually sustain the one the other. The force of the screw to press a body is to the force of the hand that turns the handles by which it is moved as the circular velocity of the handle in that part where it is impelled by the hand is to the progressive velocity of the screw towards the pressed body. The forces by which the wedge presses or drives the two parts of the wood it cleaves are to the force of the mallet upon the wedge as the pro- press of the wedge in the direction of the force impressed upon it by the mallet is to the velocity with which the parts of the wood yield to the wedge, in the direction of lines perpendicular to the sides of the wedge. And the like account is to be given of all machines. The power and use of machines consist only in this, that by diminishing the velocity we may augment the force, and the contrary : from whence in all sorts of proper machines, we have the solution of this problem ; 7 move a given weight with a given power, or with a given force to over come any other given resistance. For if machines are so contrived that the velocities of the agent and resistant are reciprocally as their forces, the agent will just sustain the resistant, but with a greater disparity of ve locity will overcome it. So that if the disparity of velocities is so great as to overcome all that resistance which commonly arises either from the attrition of contiguous bodies as they slide by one another, or from the cohesion of continuous bodies that are to be separated, or from the weights of bodies to be raised, the excess of the force remaining, after all those re sistances are overcome, will produce an acceleration of motion proportional thereto, as well in the parts of {he machine as in the resisting body. But to treat of mechanics is not my present business. I was only willing to show by those examples the great extent and certainty of the third Law ot motion. For if we estimate the action of the agent from its force and velocity conjunctly, and likewise the reaction of the impediment conjuncth from the velocities of its several parts, and from the forces of resistance arising from the attrition, cohesion, weight, and acceleration of those parts, the action and reaction 'YL the use of all sorts of machines will b" found always equal to one another. And so far as the action is propagated by the intervening instruments, and at last impressed upon tic resisting body, the ultimate determination of the action will be always contrary to the determination of the reaction. OF NATURAL PHILOSOPHY 95 BOOK I. OF THE MOTION OF BODIES. SECTION I. Of the method of first and last ratios of quantities, by the help wJicreoj we demonstrate the propositions that follow. LEMMA I. Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal. If you deny it, suppose them to be ultimately unequal, and let D be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D ; which is against the supposition, LEMMA II. If in any figure AacE, terminated by the right (f lines A a. AE, and the curve acE, there be in scribed any number of parallelograms Ab, Be, Cd, fyc., comprehended under equal bases AB, BC, CD, ^«c., and the sides, Bb, Cc, Dd, ^c., parallel to one side Aa of the figure ; and the parallelograms aKbl, bLcm, cMdn, §*c., are com pleted. Then if the breadth of those parallelo- \ grams be supposed to be diminisJied, and their X BF C D |; number to be augmented in infinitum : / say, that :he ultimate ratios 'which the inscribed fignre AKbLcMdD, the tin nmscribed figure AalbmcndoE, and en rvilijiear figure AabcdE, will have to one another, are ratios of equality. For the difference of the inscribed and circumscribed figures is the sum of the parallelograms K7, Lw, M//. Do. that is (from the equality of all their bases), the rectangle under one of their bases K6 and the sum of their altitudes Aa, that is, the rectangle ABla. But this rectangle, because M a 96 THE MATHEMATICAL PRINCIPLES [BOOK 1 its breadth AB is supposed diminished • in infinitum, becomes less than any given space. And therefore (by Lem. I) the figures inscribed and circumscribed become ultimately equal one to the other; and much more will the intermediate curvilinear figure be ultimately equal to either* Q.E.D. LEMMA III. The same ultimate ratios are also ratios of equality, when the breadth^ AB, BC, DC, fyc., of the parallelograms are unequal, and are all di minished in infinitum. For suppose AF equal to the greatest breadth, and complete the parallelogram FAaf. This parallelo gram will be greater than the difference of the in scribed and circumscribed figures ; but, because its breadth AF is diminished in infinitum, it will be come less than any given rectangle. Q.E.D. COR. 1. Hence the ultimate sum of those evanes cent parallelograms will in all parts coincide with the curvilinear figure. A BF C D E COR. 2. Much more will the rectilinear figure^comprehendcd under tne chords of the evanescent arcs ab, be, cd, (fee., ultimately coincide with tl.c curvilinear figure. COR. 3. And also the circumscribed rectilinear figure comprehended under the tangents of the same arcs. COR. 4 And therefore these ultimate figures (as to their perimeters acE) are not rectilinear, but curvilinear limi's of rectilinear figures. LEMMA IV. If in two figures AacE, PprT, you inscribe (as before) two ranks of parallelograms, an equal number in each rank, and, when their breadths are diminished in infinitum. the ultimate ratios of the parallelograms in one figure to those in the other, each to each respec tively, are the same; I say, that those two figures AacE, PprT, are to one another in that same ratio. For as the parallelograms in the one are severally to p the parallelograms in the other, so (by composition) is the < sum of all in the one to the sum of all in the other : and so is the one figure to the other; because (by Lem. Ill) the former figure to the former sum, and the latter figure to the latter sum, are both in the ratio of equality. Q.E.D. COR. Hence if two quantities of any kind are any how divided into an equal number of parts, and those A £ SEC. I.] OF NATURAL PHILOSOPHY. 97 parts, when their number is augmented, and their magnitude diminished in infinitum, have a given ratio one to the other, the first to the first, the second to the second, and so on in order, the whole quantities will be one to the other in that same given ratio. For if, in the figures of this Lemma, the parallelograms are taken one to the other in the ratio of the parts, the sum of the parts will always be as the sum of the parallelograms ; and therefore supposing the number of the parallelograms and parts to be aug mented, and their magnitudes diminished in infinitum, those sums will be in the ultimate ratio of the parallelogram in the one figure to the corres pondent parallelogram in the other ; that is (by the supposition), in the ultimate ratio of any part of the one quantity to the correspondent part of the other. LEMMA V. In similar figures, all sorts of homologous sides, whether curvilinear or rectilinear, are proportional ; and the areas are in the duplicate ratio of the homologous sides. LEMMA VI. If any arc ACB, given in position, is snb- _j tended by its chord AB, and in any point A, in the middle of the contiinied curva ture, is touched by a right line AD, pro duced both ways ; then if the points A R and B approach one another and meet, I say, the angle RAT), contained between, the chord and the tangent, will be dimin- ? ished in infinitum, a/id ultimately will vanish. For if that angle does not vanish, the arc ACB will contain with the tangent AD an angle equal to a rectilinear angle ; and therefore the cur vature at the point A will not be continued, which is against the supposi tion. LEMMA VII. The same things being supposed, I say that the ultimate ratio of the arc, chord, and tangent, any one to any other, is the ratio of equality. For while the point B approaches towards the point A, consider always AB and AD as produced to the remote points b and d, and parallel to the secant BD draw bd : and let the arc Acb be always similar to the arc ACB. Then, supposing the points A and B to coincide, the angle dAb will vanish, by the preceding Lemma; and therefore the right lines Ab, Arf (which are always finite), and the intermediate arc Acb, will coincide, and become equal among themselves. Wheref ,re, the right lines AB, AD, 98 THE MATHEMATICAL PRINCIPLES [SEC. I. and the intermediate arc ACB (which are always proportional to the former), will vanish, and ultimately acquire the ratio of equality. Q.E.D. COR. 1. Whence if through B we draw A BP parallel to the tangent, always cutting any right line AF passing through A in F/— i- P, this line BP will be ultimately in the ratio of equality with the evanescent arc ACB ; because, completing the parallelogram APBD, it is always in a ratio of equality with AD. COR. 2. And if through B and A more right lines are drawn, as BE, I5D, AF, AG, cutting the tangent AD and its parallel BP : the ultimate ratio of all the abscissas AD, AE, BF, BG, and of the chord and arc AB, any one to any other, will be the ratio of equality. COR. 3. And therefore in all our reasoning about ultimate ratios, we may freely use any one of those lines for any other. LEMMA VIII. If the right lines AR, BR, with the arc ACB, the chord AB, and the tangent AD, constitute three triangles RAB. RACB, RAD, and the points A and B approach and meet : I say, that the ultimate form oj these evanescent triangles is that of similitude, and their ultimate ratio that of equality. For while the point B approaches towards A the point A, consider always AB, AD, AR, as produced to the remote points b, d, and r, and rbd as drawn parallel to RD, and let the arc Acb be always similar to the arc ACB. Then supposing the points A and B to coincide, the angle bAd will vanish ; and therefore the three triangles rAb, rAcb,rAd ^which are always finite), will coincide, and on that account become both similar and equal. And therefore the triangles RAB. RACB, RAD which are always similar and proportional to these, will ultimately be come both similar and equal among themselves. Q..E.D. COR. And hence in all reasonings about ultimate ratios, we may indif ferently use any one of those triangles for any other. LEMMA IX. If a ngnt line AE. and a curve tine ABC, both given by position, cut each other in a given angle, A ; and to that right line, in another given angle, BD, CE are ordinately applied, meeting the curve in B, C : and the points B and C together approach towards and meet in the point A : / say, that the areas of the triangles ABD, ACE, wilt ultimately be one to the other in the duplicate ratio of the sides. BOOK LI OF NATURAL PHILOSOPHY. For while the points B, C, approach towards the point A, suppose always AD to be produced to the remote points d and . e, so as Ad, Ae may be proportional to AD, AE ; and the ordinates db, ec, to be drawn parallel to the ordinates DB and EC, and meeting AB and AC produced D in b and c. Let the curve A be be similar to the curve A BC, and draw the right line Ag- so as to touch both curves in A, and cut the ordinates DB, EC, db ec, in F, G, J] g. Then, supposing the length Ae to remain the same, let the points B and C meet in the point A ; and the angle cAg vanishing, the curvilinear areas AW, Ace will coincide with the rectilinear areas A/rf, Age ; and therefore (by Lem. V) will be one to the other in the duplicate ratio of the sides Ad, Ae. But the areas ABD, ACE are always proportional to these areas ; and so the sides AD, AE are to these sides. And therefore the areas ABD, ACE are ultimately one to the other in the duplicate ratio of the sides AD, AE. Q.E.D. LEMMA X. The spaces which a bodij describes by any finite force urging it. whether that force is determined and immutable, or is continually augmented or continually diminished, are in the very beginning of the motion one to the other in the duplicate ratio of the times. Let the times be represented by the lines AD, AE, and the velocities generated in those times by the ordinates DB, EC. The spaces described with these velocities will be as the areas ABD, ACE. described by those ordinates, that is, at the very beginning of the motion (by Lem. IX), in the duplicate ratio of the times AD, AE. Q..E.D. COR. 1. And hence one may easily infer, that the errors of bodies des cribing similar parts of similar figures in proportional times, are nearly as the squares of the times in which they are generated ; if so be these errors are generated by any equal forces similarly applied to the bodies, and measured by the distances of the bodies from those places of the sim ilar figures, at which, without the action of those forces, the bodies would have arrived in those proportional times. COR. 2. But the errors that are generated by proportional forces, sim ilarly applied to the bodies at similar parts of the similar figures, are as the forces and the squares of the times conjuiu tly. COR. 3. The same thing is to be understood of any spaces whatsoever described by bodies urged with different forces ; all which, in the very be- g'nning of the motion, are as the forces and the squares of the times conjunctly. 100 THE MATHEMATICAL PRINCIPLES I SEC. 1 COR. 4. And therefore the forces are as the spaces described in the very beginning of the motion directly, and the squares of the times inversely. COR. 5. And the squares of the times are as the spaces described direct ly, und the forces inversely. SCHOLIUM. If in comparing indetermined quantities of different sorts one with another, any one is said to be as any other directly or inversely, the mean ing is, that the former is augmented or diminished in the same ratio with the latter, or with its reciprocal. And if any one is said to be as any other two or more directly or inversely, the meaning is, that the first is aug mented or diminished in the ratio compounded of the ratios in which the others, or the reciprocals of the others, are augmented or diminished. As if A is said to be as B directly, and C directly, and D inversely, the mean ing is, that A is augmented or diminished in the same ratio with B X C X -jj-, that is to say, that A and -£• arc one to the other in a given ratio. LEMMA XL The evanescent subtense of the angle of contact, in all curves which at the point of contact have a finite curvature, is ultimately in the dupli cate rati1) of the subtense of the conterminate arc. CASE 1. Let AB be that arc, AD its tangent, BD the subtense of the angle of contact perpendicular on the tangent, AB the subtense of the arc. Draw BG perpendicular to the subtense AB, and AG to the tan gent AD, meeting in G ; then let the points D, B, and G. approach to the points d, b, and g, and suppose J to be the ultimate intersection of the lines BG, AG, when the points D, B, have come to A. It is evident that the distance GJ may be less than any assignable. But (from the nature of the circles passing through the points A, B, G, A, b, g,) AE2= AG X BD, and A62 = Ag X bd ; and therefore the ratio of AB2 to Ab2 is compounded oi the ratios of AG to Ag, and of Ed to bd. But because GJ may be as sumed of less length than any assignable, the ratio of AG to Ag may be such as to differ from the ratio of equality by less than any assignable difference ; and therefore the ratio of AB2 to Ab2 may be such as to differ from the ratio of BD to bd by less than any assignable difference. There fore, by Lem. I, the ultimate ratio of AB2 to Ab2 is the same with tho ul timate ratio of BD to bd. Q.E.D. CASE 2. Now let BD be inclined to AD in any given an*r1r, and the ultimate ratio of BD to bd will always be the same as before, and there fore the same with the ratio of AB2 to Ab2. Q.E-P BOOK I.] OF NATURAL PHILOSOPHY. 101 CASE 3. And if we suppose the angle D not to be given, but that the right line BD converges to a given point, or is determined by any other condition whatever ; nevertheless the angles D, d, being determined by the same law, will always draw nearer to equality, arid approach nearer to each other than by any assigned difference, and therefore, by Lem. I, will at lust be* equal ; and therefore the lines BD; bd arc in the same ratio to each other as before. Q.E.D. COR. 1. Therefore since the tangents AD, Ad, the arcs AB, Ab, and their sines, BC, be, become ultimately equal to the chords AB, Ab} their squares will ultimately become as the subtenses BD, bd. COR. 2. Their squares are also ultimately as the versed sines of the arcs, bisecting the chords, and converging to a given point. For those versed sines are as the subtenses BD, bd. COR. 3. And therefore the versed sine is in the duplicate ratio of the time in which a body will describe the arc with a given velocity. COR. 4. The rectilinear triangles ADB, Adb are ultimately in the triplicate ratio of the sides AD, Ad, c and in a sesquiplicate ratio of the sides DB, db ; as being in the ratio compounded of the sides AD to DB, and of Ad to db. So also the triangles ABC, Abe are ultimately in the triplicate ratio of the sides BC, be. What I call the sesquiplicate ratio is the subduplicate of the triplicate, as being compounded of the simple and subduplicate ratio. j COR. 5. And because DB, db are ultimately paral- g lei and in the duplicate ratio of the lines AD, Ad, the ultimate curvilinear areas ADB, Adb will be (by the nature of the para bola) two thirds of the rectilinear triangles ADB, Adb and the segments AB, Ab will be one third of the same triangles. And thence those areas and those segments will be in the triplicite ratio as well of the tangents AD, Ad, as of the chords and arcs AB, AB. SCHOLIUM. But we have all along supposed the angle of contact to be neither infi nitely greater nor infinitely less than the angles of contact made by cir cles and their tangents ; that is, that the curvature at the point A is neither infinitely small nor infinitely great, or that the interval AJ is of a finite mag nitude. For DB may be taken as AD3 : in which case no circle can be drawn through the point A, between the tangent AD and the curve AB, and therefore the angle of contact will be infinitely less than those of circles. And by a like reasoning, if DB be made successfully as AD4, AD5, AD8, AD7, etc., we shall have a series of angles of contact, proceeding in itifini- tum, wherein every succeeding term is infinitely less than the pre- 102 THE MATHEMATICAL PRINCIPLES [BOOK 1 ceding. And if DB be made successively as AD2, AD|, AD^, AD], AD| AD7, &c., we shall have another infinite series of angles of contact, the first of which is of the same sort with those of circles, the second infinitely greater, and every succeeding one infinitely greater than the preceding. But between any two of these angles another series of intermediate angles of contact may be interposed, proceeding both ways in infinitum. wherein every succeeding angle shall be infinitely greater or infinitely less than the preceding. As if between the terms AD2 and AD3 there were interposed the series AD'f, ADy, AD49, AD|, AD?, AD|, AD^1, AD^, AD^7, &c. And again, between any two angles of this series, a new series of intermediate angles may be interposed, differing from one another by infinite intervals. Nor is nature confined to any bounds. Those things which have been demonstrated of curve lines, and the euperfices which they comprehend, may be easily applied to the curve su- perfices and contents of solids. These Lemmas are premised to avoid the tediousness of deducing perplexed demonstrations ad absurdnm, according to the method of the ancient geometers. For demonstrations are more contracted by the method of indivisibles : but because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the follow ing propositions to the first and last sums and ratios of nascent and evane scent quantities, that is, to the limits of those sums and ratios ; and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is performed as by the method of indivisibles ; and now those principles being demonstrated, we may use them with more safety. Therefore if hereafter I should happen to consider quantities as made up of particles, or should use little curve lines for right ones, I would not be un- (lerstood to mean indivisibles, but evanescent divisible quantities : not the sums and ratios of determinate parts, but always the limits of sums and ratios ; and that the force of such demonstrations always depends on the method laid down in the foregoing Lemmas. Perhaps it may be objected, that there is no ultimate proportion, of evanescent quantities ; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alledged, that a body arriving at a cer tain place, and there stopping has no ultimate velocity : because the velo city, before the body comes to the place, is not its ultimate velocity ; when it has arrived, is none i ut the answer is easy; for by the ultimate ve locity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives ; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ra tio of evanescent quantities is to Le understood the ratio of the ijuantitiea SEC. II.] OF NATURAL PHILOSOPHY. 103 not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the ve locity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and pro portions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to use in determining and de monstrating any other thing that is likewise geometrical. It may also be objected, that if the ultimate ratios of evanescent quan tities are given, their ultimate magnitudes will be also given : and so all quantities will consist of indivisibles, which is contrary to what Euclid has demonstrated concerning incommensurables, in the 10th Book of his Elements. But this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge ; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in wfinitum. This thing will appear more evident in quantities infinitely great. If two quantities, whose dif ference is given, be augmented in infin&um, the ultimate ratio of these quantities will be given, to wit, the ratio of equality ; but it does not from thence follow, that the ultimate or greatest quantities themselves, whose ratio that is, will be given. Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be al ways diminished without end. SECTION II. Of the Invention of Centripetal Forces. PROPOSITION I. THEOREM 1. The areas, which revolving bodies describe by radii drawn to an ^mmo- vable centra of force do lie in tJ:e same immovable planes, and are pro- portional to the times in which they are described. For suppose the time to be divided into equal parts, and in the first part of that time let the body by its innate force describe the right line AB In the second part of that time, the same would (by Law I.), if not hindered, proceel directly to c, alo ILJ; the line Be equal to AB ; so that by the radii AS, BS, cS, draw. i to the centre, the equal areas ASB, BSc, would be de- 104 THE MATHEMATICAL PRINCIPLES [BOOK I scribed. But when the body is arrived at B, suppose that a centripetal force acts at once with a great im pulse, and, turning aside the body from the right line Be, compels it afterwards to con tinue its motion along the right line BC. Draw cC parallel to BS meeting BC in C ; and at the end of the second part of the time, the body (by Cor. I. of the Laws) will be found in C, in the same plane with the triangle A SB. Join SC, and, because s SB and Cc are parallel, the triangle SBC will be equal to the triangle SBc, and therefore also to the triangle SAB. By the like argument, if the centripetal force acts successively in C, D, E. &c., and makes the body, in each single particle of time, to describe the right lines CD, DE, EF7 &c., they will all lie in the same plane : and the triangle SCD will be equal to the triangle SBC, and SDE to SCD, and SEF to SDE. And therefore, in equal times, equal areas are described in one immovable plane : and, by composition, any sums SADS, SAFS, of those areas, are one to the other as the times in which they are described. Now let the number of those triangles be augmented, and their breadth diminished in wjinitum ; and (by Cor. 4, Lem. III.) their ultimate perimeter ADF will be a curve line : and therefore the centripetal force, by which the body is perpetually drawn back from the tangent of this curve, will act continually ; and any described areas SADS, SAFS, which are always proportional to the times of de scription, will, in this case also, be proportional to those times. Q.E.D. COR. 1. The velocity of a body attracted towards an immovable centre, in spaces void of resistance, is reciprocally as the perpendicular let fall from that centre on the right line that touches the orbit. For the veloci ties in those places A, B, C, D, E. are as the bases AB, BC, CD, DE, EF. of equal triangles ; and these bases are reciprocally as the perpendiculars let fall upon them. COR. 2. If the chords AB, BC of two arcs, successively described in equal times by the same body, in spaces void of resistance, are completed into a parallelogram ABCV, and the diagonal BV of this parallelogram; in the position which it ultimately acquires when those arcs are diminished in irifinitum, is produced both ways, it will pass through the centre of force. COR. 3. If the chords AB, BC, and DE, EF, cf arcs described in equal SEC. II.] OF NATURAL PHILOSOPHY. 105 times, in spaces void of resistance, are completed into the parallelograms ABCV, DEFZ : the forces in B and E are one to the other in the ulti mate ratio of the diagonals BV, EZ, when those arcs are diminished in infinitum. For the motions BC and EF of the body (by Cor. 1 of the Laws) are compounded of the motions Be, BV, and E/", EZ : but BV and EZ, which are equal to Cc and F/, in the demonstration of this Proposi tion, were generated by the impulses of the centripetal force in B and E; and are therefore proportional to those impulses. COR. 4. The forces by which bodies, in spaces void of resistance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the versed sines of arcs described in equal times ; which versed sines tend to the centre of force, and bisect the chords when those arcs are diminished to infinity. For such versed sines are the halves of the diagonals mentioned in Cor. 3. COR. 5. And therefore those forces are to the force of gravity as the said versed sines to the versed sines perpendicular to the horizon of those para bolic arcs which projectiles describe in the same time. COR. 6. And the same things do all hold good (by Cor. 5 of the Laws), when the planes in which the bodies are moved, together with the centres of force which are placed in those planes, are not at rest, but move uni formly forward in right lines. PROPOSITION II. THEOREM II. Every body that moves in any curve line described in a plane, and by a radius, drawn to a point either immovable, or moving forward with an uniform rectilinear motion, describes about that point areas propor tional to the times, is urged by a centripetal force directed to that point CASE. 1. For every body that moves in a curve line, is (by Law 1) turned aside from its rectilinear course by the action of some force that impels it. And that force by which the body is turned off from its rectilinear course, and is made to describe, in equal times, the equal least triangles SAB, SBC, SCD, &c., about the immovable point S (by Prop. XL. Book 1, Elem. and Law II), acts in the place B, according to the direction of a line par- 1U6 THE MATHEMATICAL PRINCIPLES [BOOK f. allel K cC. that is, in the direction of the line BS. and in the place C, accordii g to the direction of a line parallel to dD, that is, in the direction of the line CS, (fee.; and therefore acts always in the direction of lines tending to the immovable point S. Q.E.I). CASE. 2. And (by Cor. 5 of the Laws) it is indifferent whether the su- perfices in which a body describes a curvilinear figure be quiescent, or moves together with the body, the figure described, and its point S, uniformly forward in right lines. COR. 1. In non-resisting spaces or mediums, if the areas are not propor tional to the times, the forces are not directed to the point in which the radii meet ; but deviate therefrom in. consequently or towards the parts to which the motion is directed, if the description of the areas is accelerated ; but in antecedentia, if retarded. COR. 2. And even in resisting mediums, if the description of the areas is accelerated, the directions of the forces deviate from the point in which the radii meet, towards the parts to which the motion tends. SCHOLIUM. A body may be urged by a centripetal force compounded of several forces ; in which case the meaning of the Proposition is, that the force which results out of all tends to the point S. But if any force acts per petually in the direction of lines perpendicular to the described surface, this force will make the body to deviate from the plane of its motion : but will neither augment nor diminish the quantity of the described surface and is therefore to be neglected in the composition of forces. PROPOSITION III. THEOREM III. Every body, that by a radius drawn to the centre of another body, how soever moved, describes areas about that centre proportional to iJie times, is urged by a force compounded out of the centripetal force Bending fo that other body, and of all the accelerative force by which that other body is impelled. Let L represent the one, and T the other body ; and (by Cor. 0 of the Laws) if both bodies are urged in the direction of parallel lines, by a ne T force equal and contrary to that by which the second body T is tinned, the first body L will go on to describe about the other body T the same areas as before : but the force by which that other body T was urged will be now destroyed by an equal and contrary force; and therefore (by Law I.) that other body T, now left to itself, will either rest, or move uniformly forward in a right line : and the first body L impelled by the difference of the forces, that is, by the force remaining, will go on to describe about the other body T areas proportional to the times. And therefore (by Theor. II.) the difference ;f the forces is directed to the other body T as its centre. Q.E.D SEC. IL] OF NATURAL PHILOSOPHY. 107 Co.*. 1. Hence if the one body L, by a radius drawn to the other body T, describes areas proportional to the times ; and from the whole force, by which the firr.t body L is urged (whether that force is simple, or, according to Cor. 2 of the Laws, compounded out of several forces), we subduct (by the same Cor.) that whole accelerative force by which the other body is urged ; the who_e remaining force by which the first body is urged will tend to the ( ther body T, as its centre. COR. 2. And, if these areas are proportional to the times nearly, the re maining force will tend to the other body T nearly. COR. 3. And vice versa, if the remaining force tends nearly to the other body T, those areas will be nearly proportional to the times. COR. 4. If the body L, by a radius drawn to the other body T, describes areas, which, compared with the times, are very unequal ; and that other body T be either at rest, or moves uniformly forward in a right line : the action of the centripetal force tending to that other body T is either none at all, or it is mixed and compounded with very powerful actions of other forces : and the whole force compounded of them all, if they are many, is directed to another (immovable or moveaJble) centre. The same thing ob tains, when the other body is moved by any motion whatsoever ; provided that centripetal force is taken, wrhich remains after subducting that whole force acting upon that other body T. SCHOLIUM. Because the equable description of areas indicates that a centre is re spected by that force with which the body is most affected, and by which it is drawn back from its rectilinear motion, and retained in its orbit ; why may we not be allowed, in the following discourse, to use the equable de scription of areas as an indication of a centre, about which all circular motion is performed in free spaces ? PROPOSITION IV. THEOREM IV. The centripetal forces of bodies, which by equable 'motions describe differ ent circles, tend to the centres of the same circles ; and are one to tJie other as the squares of t/ie arcs described in equal times applied to the radii of the circles. These forces tend to the centres of the circles (by Prop. II., and Cor. 2, Prop. L), and are one to another as the versed sines of the least arcs de scribed in equal times (by Cor. 4, Prop. I.) ; that is, as the squares of the same arcs applied to the diameters of the circles (by Lem. VII.) ; and there fore since those arcs are as arcs described in any equal times, and the dia- me ers ace as the radii, the forces will be as the squares of any arcs de- scr bed in the same time applied to the radii of the circles. Q.E.D. ^OR. 1. Therefore, since those arcs are as the velocities of the bodies. I OS THE MATHEMATICAL PRINCIPLES [BOOK . the centripetal forces are in a ratio compounded of the duplicate ra'jio of the velocities directly, and of the simple ratio of the radii inversely. COR. 2. And since the periodic times are in a ratio compounded of the ratio of the radii directly, and the ratio of the velocities inversely, the cen tripetal forces, are in a ratio compounded of the ratio of the radii directly, and the duplicate ratio of the periodic times inversely. COR, 3. Whence if the periodic times are equal, and the velocities therefore as the radii, the centripetal forces will be also as the radii ; and tke contrary. COR. 4. If the periodic times and the velocities are both in the subdu- plicate ratio of the radii, the centripetal forces will be equal among them selves ; and the contrary. COR. 5. If the periodic times are as the radii, and therefore the veloci ties equal, the centripetal forces will be reciprocally as the radii ; and the contrary. COR. 6. If the periodic times are in the sesquiplicate ratio of the radii, and therefore the velocities reciprocally in the subduplicate ratio of the radii, the centripetal forces will be in the duplicate ratio of the radii in versely : and the contrary. COR. 7. And universally, if the periodic time is as any power Rn of the radius R, and therefore the velocity reciprocally as the power Rn ] of the radius, the centripetal force will be reciprocally as the power R2n 1 of the radius; and the contrary. COR. 8. The same things all hold concerning the times, the velocities, and forces by which bodies describe the similar parts of any similar figures that have their centres in a similar position with those figures ; as appears by applying the demonstration of the preceding cases to those. And the application is easy, by only substituting the equable description of areas in the place of equable motion, and using the distances of the bodies from the centres instead of the radii. COR. 9. From the same demonstration it likewise follows, that the arc which a body, uniformly revolving in a circle by means of a given centri petal force, describes in any time, is a mean proportional between the diameter of the circle, and the space which the same body falling by the same given force would descend through in the same given time. SCHOLIUM. The case of the 6th Corollary obtains in the celestial bodies (as Sir Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed) ; and therefore in what follows, I intend to treat more at large of those things which relate to centripetal force decreasing in a duplicate ratio of the distances from the centres. Moreover, by means of the preceding Proposition and its Corollaries, we SEC. II.] OF NATURAL PHILOSOPHY. 109 may discover the proportion of a centripetal force to any other known force, such as that of gravity. For if a body by means of its gravity re volves in a circle concentric to the earth, this gravity is the centripetal force of that body. But from the descent of heavy bodies, the time of one entire revolution, as well as the arc described in any given time, is given (by Cor. 9 of this Prop.). And by such propositions, Mr. Huygens, in his excellent book De Horologio Oscillatorio, has compared the force of gravity with the centrifugal forces of revolving bodies. The preceding Proposition may be likewise demonstrated after this manner. In any circle suppose a polygon to be inscribed of any number of sides. And if a body, moved with a given velocity along the sides of the polygon, is reflected from the circle at the several angular points, the force, with which at every reflection it strikes the circle, will be as its velocity : and therefore the sum of the forces, in a given time, will be as that ve locity and the number of reflections conjunctly ; that is (if the species of the polygon be given), as the length described in that given time, and in creased or diminished in the ratio of the same length to the radius of the circle ; that is, as the square of that length applied to the radius ; and therefore the polygon, by having its sides diminished in'inftnitum, coin cides with the circle, as the square of the arc described in a given time ap plied to the radius. This is the centrifugal force, with which the body impels the circle ; and to which the contrary force, wherewith the circle continually repels the body towards the centre, is equal. PROPOSITION V. PROBLEM I. There being given, in any places, the velocity with which a body de scribes a given figure, by means of forces directed to some common centre : to find that centre. Let the three right lines PT, TQV, VR touch the figure described in as many points, P, Q, R, and meet in T and V. On the tan gents erect the perpendiculars PA, QB, RC, reciprocally proportional to the velocities of the body in the points P, Q, R, from which the perpendiculars were raised ; that is, so that PA may be to QB as the velocity in Q to the velocity in P, and QB to RC as the velocity in R to the velocity in Q. Through the ends A, B, C, of the perpendiculars draw AD, DBE, EC, at right angles, meeting in D and E : and the right lines TD, VE produced, will meet in S, the centre re quired. For the perpendiculars let fall from the centre S on the tangents PT. QT. are reciprocally as the velocities of the bodies in the points P and Q 110 THE MATHEMATICAL PRINCIPLES [BOOK 1 (by Cor. 1, Prop. I.), and therefore, by construction, as the perpendiculars AP, BQ, directly ; that is, as the perpendiculars let fall from the point D on the tangents. Whence it is easy to infer that the points S, D, T, are in one right line. And by the like argument the points S, E, V are also in one right line ; and therefore the centre S is in the point where the right lines TD; YE meet. Q.E.D. PROPOSITION VL THEOREM V. In a space void of resistance, if a body revolves in any orbit about an im movable centre, and in the least time describes any arc just then, na scent ; and the versed sine of that arc is supposed to be drawn bisect ing the chord, and produced passing through the centre of force: the centripetal force in the middle of the arc will be as the versed sine di rectly and the square of the time inversely. For the versed sine in a given time is as the force (by Cor. 4, Prop. 1) ; and augmenting the time in any ratio, because the arc will be augmented in the same ratio, the versed sine will be augmented in the duplicate of that ratio (by Cor. 2 and 3, Lem. XL), and therefore is as the force and the square of the time. Subduct on both sides the duplicate ratio of the time, and the force will be as the versed sine directly, arid the square of the time inversely. Q.E.D. And the same thing may also be easily demonstrated by Corol. 4? T,em. X. COR. 1. If a body P revolving about the centre S describes a curve line APQ,, which a right line ZPR touches in any point P ; and from any other point Q, of the curve, QJl is drawn parallel to the distance SP, meeting the tangent in R ; and QT is drawn perpen- (licular to the distance SP ; the centripetal force will be reciprocally as the sp2 x Q/r2 solid - — :— , if the solid be taken of that magnitude which it ulti- mately acquires when the points P and Q, coincide. For Q,R is equal to the versed sine of double the arc QP, whose middle is P : and double the triangle SQP, or SP X Q,T is proportional to the time in which that double arc is described ; and therefore may be used for the exponent of the time. COR. 2. By a like reasoning, the centripetal force is reciprocally as the SY2 X QJP2 solid - 7^5 - ; if SY is a perpendicular from the centre of force on PR the tangent of the orbit. For the rectangles SY X QP and SP X Q,T are equal. SEC. II.] OF NATURAL PHILOSOPHY. Ill COR. 3. If the orbit is cither a circle, or touches or cuts a circle c< ncen- trically, that is, contains with a circle the least angle of contact or sec tion, having the same curvature rnd the same radius of curvature at the point P : and if PV be a chord of this circle, drawn from the body through the centre of force ; the centripetal force will be reciprocally as the solid QP2 SY2 X PV. For PV is - . COR. 4. The same things being supposed, the centripetal force is as the square of the velocity directly, and that chord inversely. For the velocity is reciprocally as the perpendicular SY, by Cor. 1. Prop. I. COR. 5. Hence if any curvilinear figure APQ, is given, and therein a point S is also given, to which a centripetal force is perpetually directed. that law of centripetal force may be found, by which the body P will bcj continually drawn back from a rectilinear course, and. being detained in the perimeter of that figure, will describe the same by a perpetual revolu- SP2 x QT2 tion. That is, we are to find, by computation, either the solid ---- - or the solid SY2 X PV, reciprocally proportional to this force. Example: of this we shall give in the following Problems. PROPOSITION VII. PROBLEM II. Tf a body revolves in the circumference of a circle; it is proposed to finii the law of centripetal force directed to any given, point. Let VQPA be the circumference of the circle ; S the given point to which as to a centre the force tends : P the body mov ing in the circumference ; Q the next place into which it is to move; and PRZ the tangent of the circle at the preceding place. Through the point S draw the v chord PV, and the diameter VA of the circle : join AP, and draw Q,T perpen dicular to SP, which produced, may meet the tangent PR in Z ; and lastly, through the point Q, draw LR parallel to SP, meeting the circle" in L, and the tangent PZ in R. And, because of the similar triangles ZQR, ZTP. VPA, we shall have RP2, that is. QRL to QT2 as AV2 to PV2. And QRlj x PV2 SI3-' therefore '- —TS -- is equal to QT2. Multiply those equals by -'. and the points P and Q, coinciding, for RL write PV ; then we shall have SP-' X PV5 SP2 x QT2 — • And therefore fl»r Cor 1 and 5. Prop. VI.) 112 THE MATHEMATICAL PRINCIPLES [BOOK I, SP2 X PV3 the centripetal force is reciprocally as - — ry^~ — J that is (because AV2 ia given), reciprocally as the square of the distance or altitude SP, and the 3ube of the chord PV conjunctly. Q.E.L The same otherwise. On the tangent PR produced let fall the perpendicular SY ; and (be cause of the similar triangles SYP, VPA), we shall have AV to PV as SP SP X PV SP2 >< PV3 to SY, and therefore -- ^~— - = SY, and - — ^- = SY2 X PV. A V A V And therefore (by Corol. 3 and 5, Prop. VI), the centripetal force is recip- SP2 X PV3 rocally as - ~~ry¥~~~ I *na* *s (because AV is given), reciprocally as SP" X PV3. Q.E.I. Con. 1. Hence if the given point S, to which the centripetal force al ways tends, is placed in the circumference of the circle, as at V, the cen tripetal force will be reciprocally as the quadrato-cube (or fifth power) of the altitude SP. COR. 2. The force by which the body P in the circle APTV revolves about the centre of force S is to the force by which the same body P may re volve in the same circle, and in the same periodic time, about any other centre of force R, as RP2 X SP to the cube of the right line SG, which, from the first centre of force S is drawn parallel to the distance PR of the body from the second centre of force R, meeting the tangent PG of the orbit in G. For by the construction of this Proposition, the former force is to the latter as RP2 X PT3 to SP2 X PV3; that is, as SP3 X PV3 SP X RP2 to -- p™ — ; or (because of the similar triangles PSG, TPV) to SGS. COR. 3. The force by which the body P in any orbit revolves about the centre of force S, is to the force by which the same body may revolve in the same orbit, and the same periodic time, about any other centre of force R. as the solid SP X RP2, contained under the distance of the body from the first centre of force S, and the square of its distance from the sec ond centre of force R, to the cube of the right line SG, drawn from the first centre of the force S, parallel to the distance RP of the body from fch*3 second centre of force R, meeting the tangent PG of the orbit in G. For the force in this orbit at any point P is the same as in a circle of the same curvature. SJSG. IL] OF NATURAL PHILOSOPHY. 113 PROPOSITION VIII. PROBLEM III. If a body mi ues in the semi-circuwferencePQA: it is proposed to find the law of the centripetal force tending to a point S, so remote, that all the lines PS. RS drawn thereto, may be taken for parallels. From C, the centre of the semi-circle, let the semi-diameter CA he drawn, cutting the parallels at right angles in M and N, and join CP. Because of the similar triangles CPM, PZT, and RZQ, we shall have CP2 to PM2 as PR2 to QT2; and, from the na ture of the circle, PR2 is equal to the rect angle QR X RN + QN, or, the points P, Q coinciding, to the rectangle QR x 2PM. Therefore CP2 is to PM2 as QR X 2PM to QT2; and QT2 2PM3 QT2 X SP2 2PM3 X SP2 therefore (by QR Corol. 8PM3 X SP2 , and QR And 1 and 5, Prop. VI.), the centripetal force is reciprocally as 2SP2 . that is (neglecting the given ratio -ppr)> reciprocally as PM3. Q.E.L And the same thing is likewise easily inferred from the preceding Pro position. SCHOLIUM. And by a like reasoning, a body will be moved in an ellipsis, or even ia an hyperbola, or parabola, by a centripetal force which is reciprocally ae the cube of the ordinate directed to an infinitely remote centre of force. PROPOSITION IX. PROBLEM IV. If a body revolves in a spiral PQS, cutting all the radii SP, SQ, fyc., in a given angle; it is proposed to find thelaio of the centripetal force tending to tJie centre of that spiral. Suppose the inde finitely small angle AY PSQ to be given ; be cause, then, all the angles are given, the figure SPRQT will , _ be given in specie. v QT Q,T2 Therefore the ratio -7^- is also given, and „„ is as QT, that is (be lot IX QK cause the figure is given in specie), as SP. But if the angle PSQ is any way changed, the right line QR, subtending the angle of contact QPU tU THE MATHEMATICAL PRINCIPLES [BOOK J (by Lemma XI) will be changed in the duplicate ratio of PR or QT QT2 Therefore the ratio ~TVD~ remains the same as before, that is, as SP. And QT2 x SP2 — -^ is as SP3, and therefore (by Corol. 1 and 5, Prop. YI) the centripetal force is reciprocally as the cube of the distance SP. Q.E.I. The same otherwise. The perpendicular SY let fall upon the tangent, and the chord PY of the circle concentrically cutting the spiral, are in given ratios to the height SP ; and therefore SP3 is as SY2 X PY, that is (by Corol. 3 and 5, Prop. YI) reciprocally as the centripetal force. LEMMA XII. All parallelograms circumscribed about any conjugate diameters of a given ellipsis or hyperbola are equal among themselves. This is demonstrated by the writers on the conic sections. PROPOSITION X. PROBLEM Y. If a body revolves in an ellipsis ; it is proposed to find the law of thi centripetal force tending to the centre of the ellipsis. Suppose CA, CB to be semi-axes of the ellipsis; GP, DK, con jugate diameters ; PF, Q,T perpendiculars to those diameters; Qvan ^rdinate to the diame ter GP ; and if the parallelogram QvPR be completed, then (by the properties of the jonic sections) the rec- langle PvG will be to Qv2 as PC2 to CD2; and (because of the similar triangles Q^T, PCF), Qi>2 to QT2 as PC2 to PF2 ; and, by com position, the ratio of PvG to QT2 is compounded of the ratio of PC2 1< QT2 CD2, and of the ratio of PC2 to PF2, that is, vG to -p— as PC; to_92L^_P_]^_. Put QR for Pr, and (by Lem. XII) BC X CA for CD K PF ; also (the points P and Q coinciding) 2PC for rG; and multiply- SEC. II.] OF NATURAL PHILOSOPHY. 115 QT2 x PC2 ing the extremes and means together, we shall have rfo~ equal to 2BC2 X CA2 — pp — — . Therefore (by Cor. 5, Prop. VI), the centripetal force is 2BC2 X CA2 reciprocally as — — ry~ ; that is (because 2I3C2 X CA2 is given), re ciprocally as-r^v; that is, directly as the distance PC. QEI. I O TJie same otherwise. [n the right line PG on the other side of the point T, take the point u so that Tu may be equal to TV ; then take uV, such as shall be to v G as DC2 to PC2. And because Qr9 is to PvG as DC'2 to PC2 (by the conic sections), we shall have Qv2 •-= Pi' X «V. Add the rectangle n.Pv to both sides, and the square of the chord of the arc PQ, will be equal to the rect angle VPv ; and therefore a circle which touches the conic section in P, and passes through the point Q,, will pass also through the point V. Now let the points P and Q, meet, and the ratio of nV to rG, which is the same with the ratio of DC2 to PC2, will become the ratio of PV to PG, or PV 2DC2 to 2PC : and therefore PY will be equal to „„ — . And therefore the force by which the body P revolves in the ellipsis will be reciprocally as 2 DC2 — ry— X PF2 (by Cor. 3, Prop. VI) ; that is (because 2DC2 X PF2 is I O given) directly as PC. Q.E.I. COR. 1. And therefore the force is as the distance of the body from the centre of the ellipsis ; and, vice versa, if the force is as the distance, the body will move in an ellipsis whose centre coincides with the centre of force, or perhaps in a circle into which the ellipsis may degenerate. COR. 2. And the periodic times of the revolutions made in all ellipses whatsoever about the same centre will be equal. For those times in sim ilar ellipses will be equal (by Corol. 3 and S, Prop. IV) ; but in ellipses that have their greater axis common, they are one to another as the whole areas of the ellipses directly, and the parts of the areas described in the same time inversely: that is, as the lesser axes directly, and the velocities of the bodies in their principal vertices inversely ; :hat is, as those lesser axes dirtily, and the ordinates to the same point %f the common axes in versely ; and therefore (because of the equality of the direct and inverse ratios) in the ratio of equality. SCHOLIUM. If the ellipsis, by having its centre removed to an infinite distance, de generates into a parabola, the body will move in tin's parabola ; and the 116 THE MATHEMATICAL PRINCIPLES [BOOK I force, now tending to a centre infinitely remote, will become equable. Which is Galileo s theorem. And if the parabolic section of the cone (by changing the inclination of the cutting plane to the cone) degenerates into an hyperbola, the body will move in the perimeter of this hyperbola, hav ing its centripetal force changed into a centrifugal force. And in like manner as in the circle, or in the ellipsis, if the forces are directed to the centre of the figure placed in the abscissa, those forces by increasing or di minishing the ordinates in any given ratio, or even by changing the angle of the inclination of the ordinates to the abscissa, are always augmented or diminished in the ratio of the distances from the centre ; provided the periodic times remain equal ; so also in all figures whatsoever, if the ordi- nates are augmented or diminished in any given ratio, or their inclination is any way changed, the periodic time remaining the same, the forces di rected to any centre placed in the abscissa are in the several ordinatee augmented or diminished in the ratio of the distances from the centre SECTION III. Of the motion of bodies in eccentric conic sections. PROPOSITION XL PROBLEM VI. If a body revolves in an ellipsis ; it is required to find the law of the centripetal force tending to the focus of the ellipsis. Let S be the focus of the ellipsis. Draw SP cutting the diame ter DK of the ellipsis in E, and the ordinate Qv in x ; and com plete the parallelogram d.rPR, It is evident that EP is equal to the greater semi-axis AC : for drawing HI froln the other focus H of the ellipsis parallel to EC, because CS, CH are equal, ES, El will be also equal ; so that EP is the half sum of PS, PI, that is (because of the parallels HI, PR, and the equal angles IPR, HPZ), of PS, PH, which taken together are equal to the whole axis 2AC. Draw QT perpendicu lar to SP, and putting L for the princi al latus rectum of the ellipsis (or for III. OF NATURAL PHILOSOPHY. 117 L X ^R t0 L X Py aS ^R t0 PV that 1S> US PE or AC to PC ; and L X Pv to GvP as L to Gy ; and GvP to Qi>2 as to CD- ; and by (Corol. 2, Lem. VII) the points Q, and P coinciding, Qv* is to Q,r- in the ratio of equality ; and Q,.r2 or Qv2 is to Q,T2 as EP2 to PF2, that is, as CA2 to PF2, or (by Lem. XII) as CD'2 to CB2. And com pounding all those ratios together, we shall have L X QR to Q,T2 as AC X L X PC2 X CD2, or 2CB2 X PC2 X CD2 to PC X Gv X CD2 X CB2, or as 2PC to Gv. But the points Q and P coinciding, 2PC and Gr are equal. And therefore the quantities L X QR and Q,T2, proportional SP2 to these, will be also equal. Let those equals be drawn into-p^B"? and L SP2 X QT2 X SP2 will become equal to -- ^p — — . And therefore (by Corol. 1 and 5, Prop. VI) the centripetal force is reciprocally as L X SP2, that is, re ciprocally in the duplicate ratio of the distance SP. Q.E.I. The same otherwise. Since the force tending to the centre of the ellipsis, by which the body P may revolve in that ellipsis, is (by Corol. 1, Prop. X.) as the distance CP of the body from the centre C of the ellipsis ; let CE be drawn paral lel to the tangent PR of the ellipsis : and the force by which the same body P may revolve about any other point S of the ellipsis, if CE and PS in- PE3 tersect in E, will be as ^T3, (by Cor. 3, Prop. VII.) ; that is, if the point S is the focus of the ellipsis, and therefore PE be given as SP2 recipro cally. Q.E.I. With the same brevity with which we reduced the fifth Problem to the parabola, and hyperbola, we might do the like here : but because of the dignity of the Problem and its use in what follows, I shall confirm the other cases by particular demonstrations. PROPOSITION XII. PROBLEM VII. Suppose a body to move in an hyperbola ; it is required to find lite law of the centripetal force tending to the focus of that figure. Let CA, CB be the semi-axes of the hyperbola ; PG, KD other con jugate diameters ; PF a perpendicular to the diameter KD ; and Qv an ordinate to the diameter GP. Draw SP cutting the diameter DK in E, and the ordinate Qv in x, and complete the parallelogram QRP.r. It is evident that EP is equal to the semi-transverse axis AC ; for drawing HE, from the other focus H of the hyperbola, parallel to EC, because CS, TH are equal, ES El will be also equal ; so that EP is the half difference J1S THE MATHEMATICAL PRINCIPLES [Book I .of PS, PI; that is (be cause of the parallels IH, PR, and the equal angles IPR, HPZ), of PS, PH, the difference of which is equal to the whole axis 2AC. Draw Q,T perpen dicular to SP; and put ting L for the principal latus rectum of the hy perbola (that is, for 2BC2\ .... -Tp- ) 7 we shall have L X QR to L X Pv as QR to Pv, or Px to Pv, that is (because of the similar tri angles Pxv, PEC), as PE to PC, or AC to PC. And L X Pv will be to Gv X Pv as L to Gv; and (by the properties of the conic sections) the rec tangle G?'P is to Q,v2 as PC2 to CD2 ; and by (Cor. 2, Lem. VII.), Qv2 to Qa* the points Q and P coinciding, becomes a ratio of equality ; and Q,.r2 or Qv2 is to Q,T2 as EP2 to PF2, that is, as CA2 to PF2, or (by Lem. XII.) as CD2 to CB2 : and, compounding all those ratios together, we shall have L X Q,R to Q,T2 as AC X L X PC2 X CD2, or 2CB2 X PC2 X CD2 to PC X Gv X CD2 X CB2, or as 2PC to Gv. But the points P and Q, coinciding. 2PC and Gv are equal. And therefore the quantities L X Q,R arid Q.T2, propor tional to them, will be also equal. Let those equals be drawn into SP2 sp2 x o/r2 ^, and we shall have L X SP2 equal to ^^ . And therefore (by Cor. 1. and 5, Prop. VI.) the centripetal force is reciprocally as L X SP'2. 'hat is, reciprocally in the duplicate ratio of the distance SP. Q,.E.I. TJie same otherwise. Find out the force tending from the centre C of the hype rbola. This will be proportional to the distance CP. But from thence (by Cor. 3, Prop. PE3 VII.) the force tending to the focus S will be as -^-^ th; (t is, because PE is given reciprocally as SP-. Q,.E.I. SEC. III.] OF NATURAL PHILOSOPHY. 119 And the same way may it be demonstrated, that the body having its cen tripetal changed into a centrifugal force, will move in the conjugate hy perbola. LEMMA XIII. The latus rectum of a parabola belonging to any vertex is quadruple the distance of that vertex from the focus of thejigurc. This is demonstrated by the writers on the conic sections. LEMMA XIV. Tlie perpendicular, let fall from the focus of a parabola on its tangent, is a mean proportional between the. distances of the focus from the poini of contact, and from the principal vertex of the figure. For, let AP be the parabola, S its focus, A its principal vertex, P the point of contact, PO an ordinate to the principal diameter. PM the tangent meeting the principal diameter in M. and SN the perpendicular from the fo-~ M A s o cus on the tangent : join AN, and because of the equal lines MS and SP, MN and NP, MA and AC, the right lines AN, OP, will be parallel ; and thence the triangle SAN will be right-angled at A, and similar to the equal triangles SNM, SNP j therefore PS is to SN as SN to SA. Q.E.D. COR. 1. PS2 is to SN2 as PS to SA. COR. 2. And because SA is given, SN- will be as PS. COR. 3. And the concourse of any tangent PM, with the right line SN. drawn from the focus per] endicular on the tangent, falls in the right line AN that touches the parabola in the principal vertex. PROPOSITION XIII. PROBLEM VIII. If a body moves in the perimeter of a parabola ; it is required to find the. law of the centripetal force tending to the focus of that figure. Retaining the construction of the preceding Lemma, let P be the body in the perimeter of the parabola ; and from the place Q,, into which it is next to succeed, draw QH parallel IS!. and Q,T perpendicular to SP, as also Qv parallel to the tan gent, and mating the diame ter PG in v, and the distance — 120 THE MATHEMATICAL PRINCIPLES [BOOK I. SP in x. Now. because of the similar triangles Pxv, SPM, and of the equal sides SP, SM of the one, the sides Px or Q,R and Pv of the other will be also equal. But (by the conic sections) the square of the ordinate Q,y is equal to the rectangle under the latus rectum and the segment Pv of the diameter ; that is (by Lem. XIII.), to the rectangle 4PS X Pv, or 4PS X Q,R ; and the points P and Q, coinciding, the ratio of Qv to Q,.r (by Cor. 2, Lem. VII.,) becomes a ratio of equality. And therefore Q,#2, in this case, becomes equal to the rectangle 4PS X Q,R. But (because of the similar triangles Q#T, SPN), Q^'2 is to QT2 as PS2 to SN2, that is (by Cor. 1, Lem. XIV.), as PS to SA ; that is, as 4PS X QR to 4SA x QR, and therefore (by Prop. IX. Lib. V., Elem.) QT* and 4SA X QR are SP2 SP2 X QT2 equal. Multiply these equals by ^-^-, and ^5 — -will become equal to SP2 X 4SA : and therefore (by Cor. 1 and 5, Prop. VL), the centripetal force is reciprocally as SP2 X 4S A ; that is, because 4SA is given, recipro cally in the duplicate ratio of the distance SP. Q.E.I. COR. 1. From the three last Propositions it follows, that if any body P goes from the place P with any velocity in the direction of any right line PR, and at the same time is urged by the action of a centripetal force that is reciprocally proportional to the square of the distance of the places from the centre, the body will move in one of the conic sections, having its fo cus in the centre of force ; and the contrary. For the focus, the point of contact, and the position of the tangent, being given, a conic section may be described, which at that point shall have a given curvature. But the curvature is given from the centripetal force and velocity of the body be ing given ; and two orbits, mutually touching one the other, cannot be de scribed by the same centripetal force and the same velocity. COR. 2. If the velocity with which the body goes from its place P is such, that in any infinitely small moment of time the lineola PR may be thereby describe I: and the centripetal force such as in the same time to move the same body through the space QR ; the body will move in one of QT2. the conic sections, whose principal latus rectum is the quantity Tjfr in its ultimate state, when thelineoke PR, QR are diminished in infinitum. In these Corollaries I consider the circle as an ellipsis ; and I except the case where the body descends to the centre in a right line. PROPOSITION XIV. THEOREM VI. Tf several bodies revolve about one common centre, and the centripetal force is reciprocally in tlie duplicate ratio of the distance of places from the centre ; I say, that the principal latera recta of tfieir orbits are in the duplicate ratio of the areas, which the bodies by radii drawn to the centre describe it\ the same time. SEC. HI. OF NATURAL PHILOSO1 HY. For (by Cor 2, Prop. XIII) the latus rectum QT*. L is equal to the quantity-^-in its ultimate state when the points P and Q, coincide. But the lineola QR in a given time is as the gen erating centripetal force ; that is (by supposi- QT2 tion), reciprocally as SP2. And therefore-^-^ is as Q.T2 X SP2 ; that is, the latus rectum L is in the duplicate ratio of the area QT X SP. Q.E.D. C?OR. Hence the whole area of the ellipsis, and the rectangle under the axes, which is proportional to it, is in the ratio compounded of the subdu- plicate ratio of the latus rectum, and the ratio of the periodic time. For the whole area is as the area QT X SP, described in a given time, mul tiplied by the periodic time. PROPOSITION XV. THEOREM VII. The same things being supposed, J say, that the periodic times in ellip ses are in the sesquiplicate ratio of their greater axes. For the lesser axis is a mean proportional between the greater axis and the latus rectum ; and, therefore, the rectangle under the axes is in the ratio compounded of the subduplicate ratio of the latus rectum and the sesquiplicate ratio of the greater axis. But this rectangle (by Cor. o. Prop. XIV) is in a ratio compounded of the subduplicate ratio of the latus rectum, and the ratio of the periodic time. Subduct from both sides the subduplicate ratio of the latus rectum, and there will remain the ses quiplicate ratio of the greater axis, equal to the ratio of the periodic time. Q.E.D. COR. Therefore the periodic times in ellipses are the same as in circles whose diameters are equal to the greater axes of the ellipses. PROPOSITION XVI. THEOREM VIII. The same things being supposed, and right lines being drawn to the bodies that shall touch the orbits, and perpendiculars being let fall on those tangents from the common focus ; I say, that the velocities oj the bodies are in a ratio compounded of the ratio of the perpendiculars inversely, and the, subduplicate ratio of the principal later a recta direct!]). From the focus S draw SY perpendicular to the tangent PR, and the velocity of the body P will be reciprocally in the subduplicate ratio of the SY2 quantity -y— . For that velocity is as the infinitely small arc PQ de- 122 THE MATHEMATICAL PRINCIPLES [BOOK I. scribed in a given moment of time, that is (by Lem. "VII), as the tangent PR ; that is (because of the proportionals PR to Q,T, and SP to SP X Q,T SY), as ~y — — ; or as SY reciprocally, and SP X Q,T directly ; but SP X QT is as the area described in the given time, that is (by Prop. XIV), in the subduplicate ratio of the latus rectum. Q.E.D. COR. 1. The principal latera recta are in a ratio compounded of the duplicate ratio of the perpendiculars and the duplicate ratio of the ve locities. COR. 2. The velocities of bodies, in their greatest and least distances from the common focus, are in the ratio compounded of the ratio of the distan ces inversely, and the subduplicate ratio of the principal latera recta di rectly. For those perpendiculars are now the distances. COR. 3. Arid therefore the velocity in a conic section, at its greatest or least distance from the focus, is to the velocity in a circle, at the same dis tance from the centre, in the subduplicate ratio of the principal latus rec tum to the double of that distance. COR. 4. The velocities of the bodies revolving in ellipses, at their mean distances from the common focus, are the same as those of bodies revolving in circles, at the same distances ; that is (by Cor. 6. Prop. IV), recipro cally in the subduplicate ratio of the distances. For the perpendiculars are now the lesser semi-axes, and these are as mean proportionals between the distances and the latera recta. Let this ratio inversely be compounded with the subduplicate ratio of the latera recta directly, and we shall have the subduplicate ratio of the distance inversely. COR. 5. In the same figure, or even in different figures, whose principal latera recta are equal, the velocity of a body is reciprocally as the perpen dicular let fall from the focus on the tangent. COR. 6. In a parabola, the velocity is reciprocally in the subduplicate ratio.of the distance of the body from the focus of the figure; it is more variable in the ellipsis, and less in the hyperbola, than according to this ratio. For (by Cor. 2, Lem. XIV) the perpendicular let fall from the focus on the tangent of a parabola is in the subduplicate ratio of the dis tance. In the hyperbola the perpendicular is less variable ; in the ellipsis more. COR. 7. In a parabola, the velocity of a body at any distance from the focus is to the velocity of a body revolving in a circle, at the same distance from the centre, in the subduplicate ratio of the number 2 to 1 ; in the ellipsis it is less, and in the hyperbola greater, than according to this ratio, For (by Cor. 2 of this Prop.) the velocitv at the vertex of a parabola is ir SEC. III.] OF NATURAL PHILOSOPHY. 123 this ratio, and (by Cor. 6 of this Prop, and Prop. IV) the same proportion holds in all distances. And hence, also, in a parabola, the velocity is everywhere equal to the velocity of a body revolving in a circle at half the distance ; in the ellipsis it is less, and in the hyperbola greater. COR. S. The velocity of a body revolving in any conic section is to the velocity of a body revolving in a circle, at the distance of half the princi pal latus rectum of the section, as that distance to the perpendicular let fall from the focus on the tangent of the section. This appears from Cor. 5. COR. 9. Wherefore since (by Cor. 6, Prop. IV), the velocity of a body revolving in this circle is to the velocity of another body revolving in any other circle reciprocally in the subduplicate ratio of the distances; there fore, ex czqiiO) the velocity of a body revolving *in a conic section will be to the velocity of a body revolving* in a circle at the same distance as a mean proportional between that common distance, and half the principal latus rectum of the section, to the perpendicular let fall from the common focus upon the tangent of the section. PROPOSITION XVII. PROBLEM IX. Supposing the centripetal force to be reciprocally proportional to the squares of the distances of places from the centre, and that the abso lute quantity of that force is known ; it is required to determine t/te line which a body will describe that is let go from a given place with a given velocity in the direction of a given right line. Let the centripetal force tending to the point S be such as will make the body p revolve in any given orbit pq ; and suppose the velocity of this body in the place p is known. Then from the place P suppose the body P to be let go with a given ve locity in the direction of the line PR ; but by virtue of a centripetal force to be immediately turned aside from that right line into the conic section PQ,. This, the right line PR will therefore touch in P. Suppose likewise that the right line pr touches the orbit pq in p ; and if from S you suppose perpendiculars let fall on those tangents, the principal latus rectum of the conic section (by Cor. 1, Prop. XVI) will be to the principal latus rectum of that orbit in a ratio compounded of the duplicate ratio of the perpendiculars, and the duplicate ratio of the velocities ; arid is therefore given. Let this latus rectum be L ; the focus S of the conic L24 THE MATHEMATICAL PRINCIPLES [BOOK I section is also given. Let the angle RPH be the complement of the angle RPS to two right ; and the line PH, in which the other focus II is placed, is given by position. Let fall SK perpendicular on PH, and erect the conjugate semi-axis BC ; this done, we shall have SP2— 2KPH + PH2 = SH2 = 4CH2 = 4BH2 — 4BC2 = SP + PH2— L X SiM SP2 + 2SPH + PH2 — L x SP + PH. Add on both sides 2KPH — SP2— PH2 + L X SP + PH, and we shall have L X SP + PH = 2SPH f 2KPH, or SP + PH to PH, as 2SP + 2KP to L. Whence PH is given both in length and position. That is, if the velocity of the body in P is such that the latus rectum L is less than 2SP + 2KP, PH will lie on the same side of the tangent PR writh the line SP ; and therefore the figure will be an ellipsis, which from the given foci S, H, and the principal axis SP + PH, is given also. But if the velocity of the body is so great, that the latus rectum L becomes equal to 2SP + 2KP, the length PH will be infinite ; and therefore, the figure will be a parabola, which has its axis SH parallel to the line PK, and is thence given. But if the body goes from its place P with a yet greater velocity, the length PH is to be taken on the other side the tangent ; and so the tangent pas sing between the foci, the figure will be an hyperbola having its principal axis equal to the difference of the lines SP and PH, and thence is given. Por if the body, in these cases, revolves in a conic section so found, it is demonstrated in Prop. XI, XII, and XIII, that the centripetal force will be reciprocally as the square of the distance of the body from the centre of force S ; and therefore we have rightly determined the line PQ,, which a body let go from a given place P with a given velocity, and in the di rection of the right line PR given by position, would describe with such a force. Q.E.F. COR. 1. Hence in every conic section, from the principal vertex D, the latus rectum L, and the focus S given, the other focus H is given, by taking DH to DS as the latus rectum to the difference between the latus rectum and 4US. For the proportion, SP + PH to PH as 2SP + 2KP to L, becomes, in the case of this Corollary, DS + DH to DH as 4DS to L, and by division DS to DH as 4DS — L to L. COR. 2. Whence if the velocity of a body in the principal vertex D ig given, the orbit may be readily found ; to wit, by taking its latus rectum to twice the distance DS, in the duplicate ratio of this given velocity to the velocity of a body revolving in a circle at the distance DS (by Cor. 3, Prop. XVI.), and then taking DH to DS as the latus rectum to the difference between the latus rectum and 4DS. COR. 3. Hence also if a body move in any conic section, and is forced out of its orbit by any impulse, you may discover the orbit in which it will afterwards pursue its Bourse. For bv compounding the proper motion oi SEC. IV.] OF NATURAL PHILOSOPHY. 125 the body with that motion, which the impulse alone would generate, you will have the motion with which the body will go off from a given place of impulse in the direction of a right line given in position. COR. 4. And if that body is continually disturbed by the action of some foreign force, we may nearly know its course, by collecting the changes which that force introduces in some points, and estimating the continual changes it will undergo in the intermediate places, from the analogy that appears in the progress of the series. SCHOLIUM. If a body P, by means of a centripetal force tending to any given point R, move in the perimeter of any given conic sec tion whose centre is C ; and the law of the centripetal force is required : draw CG parallel to the radius RP, and meet ing the tangent PG of the orbit in G ; and the force required (by Cor. 1, and CG3 Schol. Prop. X., and Cor. 3, Prop. VII.) will be as - SECTION IV. Of the finding of elliptic, parabolic, and hyperbolic orbits, from ttu. focus given. LEMMA XV. If from the two foci S, II, of any ellipsis or hyberbola, we draw to any third point V the right lines SV, H V, whereof one HV is equal to the principal axis of the figure, thai is, to the axis in which the foci are situated, the other, SV, is bisected in T by t/ie perpendicular TR let fall upon it ; that perpendicular TR will somewhere touch the conic section : and, vice versa, if it does touch it, HV will be equal to the principal axis of the figure. For, let the perpendicular TR cut the right line HV, produced, if need be, in R ; and join SR. Be cause TS, TV are equal, therefore the right lines SR, VR, as well as the angles TRS, TRV, will be also equal. Whence the point R will be in the conic section, and the perpen dicular TR will touch the same ; and the contrary. Q.E.D. 126 THE MATHEMATICAL PBINCIP, -ES [BOOK 1 PROPOSITION XVIII. PROBLEM X. From a focus and the principal axes given, to describe elliptic and hy perbolic trajectories, which shall pass through given points, and touch right lines given by position. Let S be the common focus of the figures ; AB A 33 the length of the principal axis of any trajectory ; r p T~* P a point through which the trajectory should \ /R pass ; and TR a right line which it should touch. / \ About the centre P, with the interval AB — SP, £\ S ~~yf if the orbit is an ellipsis, or AB {- SP, if the y> G ^ orbit is an hyperbola, describe the circle HG. On the tangent TR let fall the perpendicular ST, and produce the same to V, so that TV may be equal to ST; and about V as a centre with the interval AB describe the circle FH. In this manner, whether two points P, p, are given, or two tangents TR, tr, or a point P and a tangent TR, we are to describe two circles. Let H be their common intersection, and from the foci S, H, with the given axis describe the trajectory : I say, the thing is done. For (be cause PH -f- SP in the ellipsis, and PH — SP in the hyperbola, is equal to the axis) the described trajectory will pass through the point P, and (by the preceding Lemma) will touch the right line TR. And by the same argument it will either pass through the two points P, p, or touch the two right lines TR, tr. Q.E.F. PROPOSITION XIX. PROBLEM XI. About a given focus, to describe a parabolic trajectory, which shall pass through given points, and touch right lines given by position. Let S be the focus, P a point, and TR a tangent of the trajectory to be described. About P as a centre, with the interval PS, describe the circle FG. From the focus let fall ST perpendicular on the tangent, and produce the same to V, so as TV may be equal to ST. After the same manner another circle fg is to be de scribed, if another point p is given ; or another point v is to be found, if another tangent tr is given; then draw the right line IF, which shall touch the two circles YG,fg, if two points P, p are given ; or pass through the two points V, v, if two tangents TR, tr, are given : or touch the circle FG, and pass through the point V, if the point P and the tangent TR are given. On FI let fall the perpendicular SI, and bisect the same in K ; and with the axis SK and principal vertex K describe a parabola : I say the thing is done. For this parabola (because SK is equal to IK, and SP to FP) will pass through the point P ; and /KS SEC. IV.] OF NATURAL PHILOSOPHY. 127 (by Cor. 3, Lem. XIV) because ST is equal to TV. and STR a light an gle, it will touch the right line TR. Q.E.F. PROPOSITION XX. PROBLEM XII. About a given focus to describe any trajectory given in specie which shah pass through given points, and touch right lines given by position. CASE 1. About the focus S it is re- uired to describe a trajectory ABC, pass ing through two points B, C. Because the trajectory is given in specie, the ratio of the principal axis to the distance of the foci GAS H will be given. In that ratio take KB to BS, and LC to CS. About the centres B, C, with the intervals BK, CL, describe two circles ; and on the right line KL, that touches the same in K and L, let fall the perpendicu lar SG ; which cut in A and a, so that GA may be to AS, and Ga to aS, as KB to BS ; and with the axis A.«, and vertices A, a, describe a trajectory : I say the thing is done. For let H be the other focus of the described figure, and seeing GA is to AS as Ga to aS, then by division we shall have Ga — GA, or Aa to «S — AS, or SH in the same ratio; and therefore in the ratio which the principal axis of the figure to be described has to the distance of its foci ; and therefore the described figure is of the same species with the figure which was to be described. And since KB to BS, and LC to CS, are in the same ratio, this figure will pass through tht- points B, C, as is manifest from the conic sections. CASE 2. About the focus S it is required to describe a trajectory which shall somewhere touch two right lines TR, tr. From the focus on those tangents let fall the perpendiculars ST, St, which produce to V, v, so that TV, tv may be equal to TS, tS. Bisect Vv in O, and j erect the indefinite perpendicular OH, and cut I. the right line VS infinitely produced in K and V k, so that VK be to KS, and VA* to A~S, as the principal axis of the tra jectory to be described is to the distance of its foci. On the diameter K/J describe a circle cutting OH in H ; and with the foci S, H, and principal axis equal to VH, describe a trajectory : I say, the thing is done. For bisecting Kk in X, and joining HX, HS, HV, Hv, because VK is to KS as VA- to A*S ; and by composition, as VK -f- V/c to KS + kS ; and by division, as VA* — VK to kS — KS, that is, as 2VX to 2KX, and 2KX to 2SX, and therefore as VX to HX and HX to SX, the triangles VXH, HXS will be similar ; therefore VH will be to SH as VX to XH ; and therefore as VK to KS. Wherefore VH, the principal axis of the described trajectory, has the same ratio to SH, the distance of the foci, as 12S THE MATHEMATICAL PRINCIPLES [BOOK 1. K S the principal axis of the trajectory which was to be described has to the distance of its foci ; and is therefore of the same species. Arid seeing VH, vH are equal to the principal axis, and VS, vS are perpendicularly bisected by the right lines TR, tr, it is evident (by Lem. XV) that those right lines touch the described trajectory. Q,.E.F. CASE. 3. About the focus S it is required to describe a trajectory, which shall touch a right line TR in a given Point R. On the right line TR Jet fall the perpendicular ST, which produce to V, so that TV may be equal to ST ; join VR, and cut the right line VS indefinitely produced in K and k, so. that VK may be to SK, and V& to SAr, as the principal axis of the ellipsis to be described to the distance of its foci ; and on the diameter KA: describing a circle, cut the H right line VR produced in H ; then with the foci S, H, and principal axis equal to R VH, describe a trajectory : I say, the thing .---' is done. For VH is to SH as VK to SK, V" "1 and therefore as the principal axis of the trajectory which was to be de scribed to the distance of its foci (as appears from what we have demon strated in Case 2) ; and therefore the described trajectory is of the same species with that which was to be described ; but that the right line TR, by which the angle VRS is bisected, touches the trajectory in the point R, is certain from the properties of the conic sections. Q.E.F. CASE 4. About the focus S it is r required to describe a trajectory APB that shall touch a right line TR, and pass through any given point P without the tangent, and shall be similar to the figure apb, described with the principal axis ab, and foci s, h. On the tangent TR let fall the perpendicular ST, which / ..''''',.---"'" produce to V, so that TV may be equal to ST ; and making the an gles hsq, shq, equal to the angles VSP, SVP, about q as a centre, and with an interval which shall be to ab as SP to VS, describe a circle cut ting the figure apb in p : join sp, and draw SH such that it may be to sh as SP is to sp, and may make the angle PSH equal to the angle psh, and the angle VSH equal to the angle pyq. Then with the foci S, H, and B principal axis AB, equal to the distance VH, describe a conic section : I say, the thing is done ; for if sv is drawn so that it shall be to SEC. IV.] OF NATURAL PHILOSOPHY. 129 sp as sh is to sq, and shall make the angle vsp equal to the angle hsq, and the angle vsh equal to the angle psq, the triangles svh, spq, will be similar, and therefore vh will be to pq as sh is to sq ; that is (because of the simi lar triangles VSP, hsq), as VS is to SP? or as ab to pq. Wherefore vh and ab are equal. But, because of the similar triangles VSH, vsh, VH is to SH as vh to sh ; that is, the axis of the conic section now described is to the distance of its foci as the axis ab to the distance of the foci sh ; and therefore the figure now described is similar to the figure aph. But, because the triangle PSH is similar to the triangle psh, this figure passes through the point P ; and because VH is equal to its axis, and VS is per pendicularly bisected by the rght line TR, the said figure touches the right line TR. Q.E.F. LEMMA XVI. From three given points to draw to afonrth point that is not given three right lines whose differences shall be either given, or none at all. CASE 1. Let the given points be A, B, C, and Z the fourth point which we are to find ; because of the given difference of the lines AZ, BZ, the locus of the point Z will be an hyperbola whose foci are A and B, and whose princi pal axis is the given difference. Let that axis be MN. Taking PM to MA as MN is to AB, erect PR perpendicular to AB, and let fall ZR perpendicular to PR ; then from the nature of the hyperbola, ZR will be to AZ as MN is to AB. And by the like argument, the locus of the point Z will be another hyperbola, whose foci are A, C, and whose principal axis is the difference between AZ and CZ ; and QS a perpendicular on AC may be drawn, to which (QS) if from any point Z of this hyperbola a perpendicular ZS is let fall (this ZS), shall be to AZ as the difference between AZ and CZ is to AC. Wherefore the ratios of ZR and ZS to AZ are given, and consequently the ratio of ZR to ZS one to the other ; and therefore if the right lines RP, SQ, meet in T, and TZ and TA are drawn, the figure TRZS will be given in specie, and the right line TZ, in which the point Z is somewhere placed, will be given in position. There will be given also the right line TA, and the angle ATZ ; and because the ratios of AZ and TZ to ZS are given, their ratio to each other is given also ; and thence will be given likewise the triangle ATZ, whose vertex is the point Z. Q.E.I. CASE 2. If two of the three lines, for example AZ and BZ, are equal, draw the right line TZ so as to bisect the right line AB ; then find the triangle ATZ as above. Q.E.I. 130 THE MATHEMATICAL PRINCIPLES [BOOK I. CASE 3. If all the three are equal, the point Z will be placed in the centre of a circle that passes through the points A, B, C. Q.E.I. This problematic Lemma is likewise solved in Apollonius's Book oi Tactions restored by Vieta. PROPOSITION XXL PROBLEM XIII. About a given focus to describe a trajectory that shall pass through given points and touch right Hues given by position. Let the focus S, the point P, and the tangent TR be given, and suppose that the other focus H is to be found. On the tangent let fall the perpendicular ST, which produce to Y, so that TY may be equal to ST, and YH will be equal to the principal axis. Join SP, HP, and SP will be the difference between HP and the principal axis. After this manner, if more tangents TR are given, or more points P. we shall always determine as many lines YH, or PH, drawn from the said points Y or P, to the focus H, which either shall be equal to the axes, or differ from the axes by given lengths SP ; and therefore which shall either be equal among themselves, or shall have given differences ; from whence (by the preceding Lemma). that other focus H is given. But having the foci and the length of the axis (which is either YH, or, if the trajectory be an ellipsis, PH -f SP ; or PH — SP, if it be an hyperbola), the trajectory is given. Q.E.I. SCHOLIUM. When the trajectory is an hyperbola, I do not comprehend its conjugate hyperbola under the name of tins trajectory. For a body going on with a continued motion can never pass out of one hyperbola into its conjugate hyperbola. The case when three points are given is more readily solved thus. Let B, C, I), be the given points. Join BC, CD, and produce them to E, F, so as EB may be to EC as SB to SC ; and FC to FD as SC to SD. On EF drawn and pro duced let fall the perpendiculars SG, BH, and in GS produced indefinitely E take GA to AS, and Ga to aS, as HB is to BS ; then A will be the vertex, and Aa the principal axis of the tra jectory ; which, according as GA is greater than, equal to, or less than SEC. V.] OF NATURAL PHILOSOPHY. 131 AS. will be either an ellipsis, a parabola, or an hyperbola ; the point a in the first case falling on the same side of the line GP as the point A ; in the second, going oft* to an infinite distance ; in the third, falling on the other side of the line GP. For if on GF the perpendiculars CI, DK are let fall, TC will be to HB as EC to EB ; that is, as SO to SB ; and by permutation, 1C to SC as HB to SB, or as GA to SA. And, by the like argument, we may prove that KD is to SD in the same ratio. Where fore the points B, C, D lie in a conic section described about the focus S, in such manner that all the right lines drawn from the focus S to the several points of the section, and the perpendiculars let fall from the same points on the right line GF, are in that given ratio. That excellent geometer M. De la Hire has solved this Problem much after the same way, in his Conies, Prop. XXV., Lib. VIII. SECTION V. How the orbits are to be found when neither focus is given. LEMMA XVII. If from any point P of a given conic section, to the four produced sides AB, CD, AC, DB, of any trapezium ABDC inscribed in that section, as many right lines PQ, PR, PS, PT are drawn in given ang7ei, each line to each side ; the rectangle PQ, X PR of those on the opposite sides AB, CD, will be to the rectangle PS X PT of those on tie other two opposite sides AC, BD, in a given ratio. CASE 1. Let us suppose, first, that the lines drawn to one pair of opposite sides are parallel to either of I ^^ p ;T the other sides ; as PQ and PR to the side AC, and s| PS and PT to the side AB. And farther, that one pair of the opposite sides, as AC and BD, are parallel betwixt themselves; then the right line which bisects^ IQ I3 those parallel sides will be one of the diameters of the 1L conic section, and will likewise bisect RQ. Let O be the point in which RQ is bisected, and PO will be an ordinate to that diameter. Produce PO to K, so that OK may be equal to PO, and OK will be an ordinate on the other side of that diameter. Since, therefore, the points A, B; P and K are placed in the conic section, and PK cuts AB in a given angle, the rectangle PQK (by Prop. XVII., XIX., XXI. and XXI1L, Book III., of Apollonius's Conies) will be to the rectangle AQB in a given ratio. But QK and PR are equal, as being the differences of the equal lines OK, OP, and OQ, OR ; whence the rectangles PQK and PQ X PR are equal ; and therefore the rectangle PQ X PR is to the rectangle A^ B, that Is, to the rectangle PS X PT in a given ratio. Q.E.D 132 THE MATHEMATICAL PRINCIPLES [BOOK I CASE 2. Let us next suppose that the oppo site sides AC and BD of the trapezium are not parallel. Draw Be/ parallel to AC, and meeting as well the right line ST in /, as the conic section in d. Join Cd cutting PQ in r, and draw DM parallel to PQ, cutting Cd in M, and AB in N. Then (because of the similar triangles BTt, DBN), Et or PQ is to Tt as DN to NB. And ^^ Q N so Rr is to AQ or PS as DM to AN. Wherefore, by multiplying the antece- dents by the antecedents, and the consequents by the consequents, as the rectangle PQ X Rr is to the rectangle PS X Tt, so will the rectangle N i)M be to the rectangle ANB ; and (by Case 1) so is the rectangle PQ X Pr to the rectangle PS X Pt : and by division, so is the rectangle PQ X PR to the rectangle PS X PT. Q.E.D. CASE 3. Let us suppose, lastly, the four lines ?Q, PR, PS, PT, not to be parallel to the sides AC, AB, but any way inclined to them. In their place draw Pq, Pr, parallel to AC ; and Ps, Pt parallel to AB ; and because the angles of the triangles PQD ; and there fore the point n falls on the point b, Moreover the angle GPQ, which is half the angle GPD at the centre, is equal to the angle GaD at the circumference \ and the angle GQP, which is half the angle GQD at the centre, is equal to the complement to two right angles of the angle GbD at the circum ference, and therefore equal to the angle Gba. Upon which account the triangles GPQ, Gab, are similar, and Ga is to ab as GP to PQ. ; that is (by construction), as Ga to AB. Wherefore ab and AB are equal; and consequently the triangles abc, ABC, which we have now proved to be similar, are also equal. And therefore since the angles I), E, F, of the triangle DEF do respectively touch the sides ab, ar, be of the triangle afjc/ the figure AECdef may be completed similar and equal to the figure afrcDEFj and by completing it the Problem will be solved. Q.E.F. COR. Hence a right line may be drawn whose parts given in length may be intercepted between three right lines given by position. Suppose the triangle DEF, by the access of its point D to the side EF, arid by having the sides DE, DF placed i>t directum to be changed into a right line whose given part DE is to be interposed between the right lines AB; AC given by position; and its given part DF is to be interposed between the right lines AB; BC, given by position; then, by applying the preceding construction to this case, the Problem will be solved. THE MATHEMATICAL PRINCIPLES [BOOK 1. PROPOSITION XXVIII. PROBLEM XX. To describe a trajectory giren both in kind and magnitude, given parts of which shall be interposed between three right lines given by position. Suppose a trajectory is to be described that may be similar and equal to the curve line DEF, -and may be cut by three right lines AB, AC, BC, given by position, into parts DE and EF, similar and equal to the given parts of this curve line. Draw the right lines DE, EF, DF: and place the angles D, E, F, of this triangle DEF, so as to touch those right lines given by position (by Lem. XXVI). Then about the triangle describe the trajectory, similar and equal to the curve DEF. Q.E.F. LEMMA XXVII. To describe a trapezium given in kind, the angles whereof may be », placed, in respect of four right lines given by position, that are neither all paralhl among themselves, nor converge to one common point, ////// the several angles may touch the several lines. Let the four right lines ABC, AD, BD, CE, be given by position ; the first cutting the second in A, the third in B, and the fourth in C • and suppose a trapezium fghi is to be described that may be similar to the trapezium FCHI, and whose angle /, equal to the given angle F, may touch the right line ABC ; and (lie other angles g, h, i, equal to the other given angles, G, H, I, may touch the other lines AD, BD, CE, re spectively. Join FH, and upon FG. FH, FI describe J% as many segments of circles FSG, FTH, FVI, the first of which FSG may be capable of an angle equal to the angle BAD ; the second FTH capable of an angle equal to the angle CBD ; and the third FVI of an angle equal to the angle ACE. Bnrf>, the segments are to be described towards those sides of the lines FG, FH, FI, that the circular order of the letters FSGF may be the same as of the letters BADB, and that the letters FTHF may turn .ibout in the same order as the letters CBDC and the letters FVIF in the game order as the letters ACE A. Complete the segments into entire

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