MA in.-. v- ;w MA NATU-RA TRANSLAT 'NEWTON'S With 2 : PUBLISH JED NEWTON'S PKINCIPIA. THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY, BY SIR ISAAC NEWTON; TRANSLATED INTO ENGLISH BY ANDREW MOTTE. TO WHICH IS ADDKfV NEWTOFS 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 H. W, CHITTENDEN, M. A., &e. NEW-YORK-PUBLISHED BY DANIEL ADEE, 45 LIBERTY STREET. |* VIA o /vi faltered according to Act of Congress, in the year 1846, b?DANIEL ADEE. Clerk's Office uf'the Southern Oislnct Court of New-York. Twrncy f Lockwooff* Slctw.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 thehorizon of the moral world betoken the grateful dawning of a newora. The days of a drivelling instruction are departing. Withus is the opening promise of a better time, wherein genuine manhood doing its noblest work shall have adequate reward.TEACHER is the highest and most responsible office man can fill.Its dignity is, and will yet be held commensurate with its duty—a duty boundless as man's intellectual capacity, and great as hismoral need—a duty from the performance of which shall emanatean influence not limited to the now and the here, but which surelywill, as time flows into eternity and space into infinity, roll up, ameasureless curse or a measureless blessing, in inconceivableswellings along the infinite curve. It is an office that should beesteemed of even sacred import in this country. Ere long a hundred millions, extending from the Atlantic to the Pacific, fromBaffin's Bay to that of Panama, shall call themselves Americancitizens. What a field for those two master-passions of the human soul—the love of Rule, and the love of Gain! How shallour liberties continue to be preserved from the graspings of Ambition and the corruptions of Gold ? Not by Bills of Rights 4 DEDICATION. Constitutions, and Statute Books ; but alone by the rightly cultivated hearts and heads of the PEOPLE. They must themselvesguard the Ark. It is yours to tit them for the consecratedcharge. Look well to it: for you appear clothed in the majestyof great power ! It is yours to fashion, and to inform , to save,and to perpetuate. You are the Educators of the People : youare the prime Conservators of the public weal. Betray yourtrust, and the sacred fires would go out, and the altars crumbleinto dust: knowledge become lost in tradition, and Christian nobleness a fable! As you, therefore, are multiplied in number,elevated in consideration, increased in means, and fulfill, well andfaithfully, all the requirements of true Teachers, so shall our favoured land lift up her head among the nations of the earth, andcall herself blessed. In conclusion, Gentlemen, to you, as the conspicuous leadersin the vast and honourable labour of Educational Reform, anaPopular Teaching, the First American Edition of the PRINCIPIA oiNewton—the greatest work of the greatest Teacher—is mostrespectfully dedicated. N. W. CHTTTENDEN. INTRODUCTION TO THE AMERICAN EDITION, THAT the PRINCIPIA of Newton should have remained so generally unknown in this country to the present day is a somewhatremarkable fact; because the name of the author, learned withthe very elements of science, is revered at every hearth-stonewhere knowledge and virtue are of chief esteem, while, abroad,in all the high places of the land, the character which that namerecalls is held up as the noblest illustration of what MAN may be,and may do, in the possession and manifestation of pre-eminentintellectual and moral worth ; because the work is celebrated, notonly in the history of one career and one mind, but in the historyof all achievement and human reason itself; because of the spiritof inquiry, which has been aroused, and which, in pursuing itssearchings, is not always satisfied with stopping short of the fountain-head of any given truth ; and, finally, because of the earnestendeavour that has been and is constantly going on, in manysections of the Republic, to elevate the popular standard of education and give to scientific and other efforts a higher and abetter aim. True, the PRINCIPIA has been hitherto inaccessible to popularuse. A few copies in Latin, and occasionally one in English maybe found in some of our larger libraries, or in the possession ofsome ardent disciple of the great Master. But a drad languagein the one case, and an enormous price in both, particularly inthat of the English edition, have thus far opposed very sufficientobstacles to the wide circulation of the work. It is now, however, placed within the reach of all. And in performing this labour, the utmost care has been taken, by collation, revision, andotherwise, to render the First American Edition the most accurateand beautiful in our language. u Le plus beau monument que1' on puisse elever a la gloire de Newton, c'est une bonne editionde ses ouvrages:" and a monument like unto that we would here V: INTRODUCTION TO set up. The PRINCIPIA, above all, glows with the immortality ofa 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 sendforth afar and near, over the troublous waters of life, a pure, unwavering, quenchless light whereby the myriad myriads of barques,richly laden with reason, intelligence and various faculty, areguided through the night and the storm, by the beetling shoreand the hidden rock, the breaker and the shoal, safely into havenscalm and secure. To the teacher and the taught, the scholar and the student, thedevotee of Science and the worshipper of Truth, the PRINCIPIAmust ever continue to be of inestimable value. If to educatemeans, not so much to store the memory with symbols and facts,as to bring forth the faculties of the soul and develope them to thefull by healthy nurture and a hardy discipline, then, what so effective to the accomplishment of that end as the study of Geometrical Synthesis ? The Calculus, in some shape or other, is, indeed,necessary to the successful prosecution of researches in the higherbranches of philosophy. But has not the Analytical encroachedupon the Synthetical, and Algorithmic Formulae been employedwhen not requisite, either for the evolution of truth, or even itsapter illustration ? To each method belongs, undoubtedly, anappropriate use. Newton, himself the inventor of Fluxions,censured the handling of Geometrical subjects by Algebraicalcalculations ; and the maturest opinions which he expressed wereadditionally in favour of the Geometrical Method. His preference, so strongly marked, is not to be reckoned a mere matter oltaste ; and his authority should bear with preponderating weightupon the decision of every instructor in adopting what may bedeemed the best plan to insure the completest mental develop-tnent. Geometry, the vigorous product of remote time ; blendedwith the earliest aspirations of Science and the earliest applications of Art; as well in the measures of music as in the movement of spheres ; as wholly in the structure of the atom as in thatof the world; directing MOTION and shaping APPEARANCE; in aword, *t- the moulding of the created all, is, in comprehensive THE AMERICAN EDITION. VU view, the outward form of that Inner Harmony of which and inwhich all things are. Plainly, therefore, this noble study hasother and infinitely higher uses than to increase the power of abstraction. A more general and thorough cultivation of it shouldoe strenuously insisted on. Passing from the pages of Euclid orLegendre, might not the student be led, at the suitable time, tothose of the PRINCIPIA wherein Geometry may be found in varieduse from the familiar to the sublime ? The profoundest and thehappiest results, it is believed, would attend upon this enlargementof our Educational System. Let the PRINCIPIA, then, be gladly welcomed into every Hallwhere a TRUE TEACHER presides. And they who are guided tothe diligent study of this incomparable work, who becomestrengthened by its reason, assured by its evidence, and enlightened by its truths, and who rise into loving communion with thegreat and pure spirit of its author, will go forth from the scenesof their pupilage, and take their places in the world as strong-minded, right-hearted men—such men as the Theory of ourGovernment contemplates and its practical operation absolutelydemands. LIFE OF SIR ISAAC NEWTON. Nee fas est proprius mortali attingere Divos.—HALLEY. FROM the thick darkness of the middle ages man's strugglingspirit emerged as in new birth ; breaking out of the iron controlof that period ; growing strong and confident in the tug and dinof succeeding conflict and revolution, it bounded forwards andupwards with resistless vigour to the investigation of physical andmoral truth; ascending height after height; sweeping afar overthe earth, penetrating afar up into the heavens ; increasing in endeavour, enlarging in endowment; every where boldly, earnestlyout-stretching, till, in the AUTHOR of the PRINCIPIA, one arose,who, grasping the master-key of the universe and treading itscelestial paths, opened up to the human intellect the stupendousrealities of the material world, and, in the unrolling of its harmonies, gave to the human heart a new song to the goodness, wisdom, and majesty of the all-creating, all-sustaining, all-perfectGod. Sir Isaac Newton, in whom the rising intellect seemed to attain,as it were, to its culminating point, was born on the 25th of December, O. S. 1642—Christmas day—at Woolsthorpe, in theparish of Colsterworth, in Lincolnshire. His father, John Newton, died at the age of thirty-six, and only a few months after hismarriage to Harriet Ayscough, daughter of James Ayscough, otRutlandshire. Mrs. Newton, probably wrought upon by theearly loss of her husband, gave premature birth to her only andposthumous child, of which, too, from its extreme diminutiveness,she appeared likely to be soon bereft. Happily, it was otherwisedecreed! 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 ahale old age ;—lived to become the boast of his country, the wonder of his time, and the "ornament of his species." Beyond the grandfather, Robert Newton, the descent of SirIsaac cannot with certainty be traced. Two traditions were heldin the family: one, that they were of Scotch extraction ; theother, that they came originally from Newton, in Lancashire,dwelling, for a time, however, at Westby, county of Lincoln, before the removal to and purchase of Woolsthorpe—about a hundredyears before this memorable birth. The widow Newton was left with the simple means of a comfortable subsistence. The Woolsthorpe estate together withsmall one which she possessed at Sewstern, in Leicestershire, yielded her an income of some eighty pounds; ancl upon this limited sum,she had to rely chiefly for the support of herself, and the education of her child. She continued his nurture for three year&,when, marrying again, she confided the tender charge to the careof her own mother. Great genius is seldom marked by precocious development;and young Isaac, sent, at the usual age, to two day schools atSkillington 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 continued for a while unpromising enough : study apparently had nocharms for him; he was very inattentive, and ranked low in theschool. One day, however, the boy immediately above our seemingly dull student gave him a severe kick in the stomach ; Isaac,deeply affected, but with no outburst of passion, betook himself,with quiet, incessant toil, to his books j he quickly passed abovethe offending classmate ; yet there he stopped not ; the strongspirit was, for once and forever, awakened, and, yielding to itsnoble 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 alternatedwith reflection. " A sober, silent, thinking lad/' yet, the wisestand the kindliest, the indisputable leader of his fallows. Gener- LIFE OF SIR ISAVC NEWTON. 1} osity, modesty, and a love of truth distinguished him then as everafterwards. He did not often join his classmates in play ; but hewould contrive for them various amusements of a scientific kind.Paper kites he introduced; carefully determining their best formand proportions, and the position and number of points wherebyto attach the string. He also invented paper lanterns ; theseserved ordinarily to guide the way to school in winter mornings,but occasionally for quite another purpose ; they were attached tothe tails of kites in a dark night, to the dismay of the country peopledreading portentous comets, and to the immeasureable delight othis companions. To him, however, young as he was, life seemedto have become an earnest thing. When not occupied with hisstudies, his mind would be engrossed with mechanical contrivances;now imitating, now inventing. He became singularly skilful in theuse of his little saws, hatchets, hammers, and other tools. Awindmill was erected near Grantham ; during the operations oithe workmen, he was frequently present; in a short time, he hadcompleted a perfect working model of it, \vhich elicited generaladmiration. Not content, however, with this exact imitation, heconceived the idea of employing, in the place of sails, animal power ,and, adapting the construction of his mill accordingly, he enclosedin it a mouse, called the miller, and which by acting on a sort ottreadwheel, gave motion to the machine. He invented, too, amechanical carriage—having four wheels, and put in motion witha handle worked by the person sitting inside. The measurementof time early drew his attention. He first constructed a waterclock, in proportions somewhat like an old-fashioned house clock.The index of the dial plate was turned by a piece of wood actedupon by dropping water. This instrument, though long used byhimself, and by Mr. Clark's family, did not satisfy his inquiringmind. His thoughts rose to the sun ; and, by careful and oft-repeated observations of the solar movements, he subsequentlyformed many dials. One of these, named Isaac's dial, was theaccurate result of years' labour, and was frequently referred tofor the hour of the day by the country people. May we not discern in these continual efforts—the diligent research; the patient meditation, the aspiring glance, and the energy 12 LIFE OF SIR ISAAC NEWTON. of discovery—the stirring elements of that wondrous spirit,which, clear, calm, and great, moved, in after years, throughdeep onward through deep of Nature's mysteries, unlocking herstrongholds, dispelling darkness, educing order—everywhere silently conquering. Newton had an early and decided taste for drawing. Pictures,taken sometimes from copies, but often from life, and drawn,coloured and framed by himself, ornamented his apartment. Hewas skilled also, in poetical composition, " excelled in makingverses ;" some of these were borne in remembrance and repeated,seventy years afterward, by Mrs. Vincent, for whom, in earlyyouth, as Miss Storey, he formed an ardent attachment. Shewras the sister of a physician resident near Woolsthorpe; butNewton's intimate acquaintance with her began at Grantham.where they were both numbered among the inmates of the samehouse. Two or three years younger than himself, of great personal beauty, and unusual talent, her society afforded him thegreatest pleasure ; and their youthful friendship, it is believed,gradually rose to a higher passion; but inadequacy of fortuneprevented their union. Miss Storey was afterwards twice married ; Newton, never; his esteem for her continued unabatedduring life, accompanied by numerous acts of attention andkindness. In 1656, Newton's mother was again left a widowr, and tookup her abode once more at Woolsthorpe. He was now fifteenyears of age, and had made great progress in his studies ; but she,desirous of his help, and from motives of economy, recalled himfrom school. Business occupations, however, and the management of the farm, proved utterly distasteful to him. When sent toGrantham Market on Saturdays, he would betake himself to hisformer lodgings in the apothecary's garret, where some of Mr.Clark's old books employed his thoughts till the aged and trustworthy servant had executed the family commissions and announcedthe necessity of return : or, at other times, our young philosopherwould seat himself under a hedge, by the wayside, and continuehis studies till the same faithful personage—proceeding alone tothe 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 nobetter attention. In fact, his passion for study grew daily moreabsorbing, and his dislike for every other occupation more intense. His mother, therefore, wisely resolved to give him all theadvantages which an education could confer. He was sent backto Grantham school, where he remained for some months in busypreparation for his academical studies. At the recommendationof 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 widerfield; and we find him devoting himself to the pursuit of knowledge with amazing ardour and perseverance. Among other subjects, his attention was soon drawn to that of Judicial AstrologyHe exposed the folly of this pseudo-science by erecting a figurewith the aid of one or two of the problems of Euclid ;—and thusbegan his study of the Mathematics. His researches into thisscience were prosecuted with unparallelled vigour and success.Regarding the propositions contained in Euclid as self-evidenttruths, he passed rapidly over this ancient system—a step whichhe afterward much regretted—and mastered, without further preparatory study, the Analytical Geometry of Descartes. Wallis'sArithmetic of Infinites, Saunderson's Logic, and the Optics ofKepler, he also studied with great care ; writing upon themmany comments; and, in these notes on Wallis's work was undoubtedly the germ of his fluxionary calculus. His progress wasso great that he found himself more profoundly versed than his tutorin many branches of learning. Yet his acquisitions were notgotten with the rapidity of intuition; but they were thoroughlymade and firmly secured. Quickness of apprehension, or Intellectual nimbleness did not belong to him. He saw too far : his,insight was too deep. He dwelt fully, cautiously upon the leastsubject; while to the consideration of the greatest, he brought amassive strength joined with a matchless clearness, that, regardless of the merely trivial or unimportant, bore with unerring sagacity upon the prominences of the subject, and, grappling withits difficulties, rarely failed to surmount them. 14 LIKE OK SIR ISAAC NEWTON His early and last friend, Dr. Barrow—in compass of invention only inferior to Newton—who had been elected Professorof Greek in the University, in 1660, was made Lucasian Professor of Mathematics in 1663, and soon afterward delivered hisOptical Lectures : the manuscripts of these were revised by Newton, and several oversights corrected, and many important suggestions 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 discoveries which have conferred inappreciable benefits upon science,and immortality upon his own name. Newton, himself, states that he was in possession of his Methodof Fluxions, " in the year 1666, or before." Infinite quantitieshad long been a subject of profound investigation; among theancients by Archimedes, and Pappus of Alexandria; among themoderns by Kepler, Cavaleri, Roberval, Fermat and Wallis.With consummate ability Dr. Wallis had improved upon the labours of his predecessors: with a higher power, Newton movedforwards from where Wallis stopped. Our author first inventedhis celebrated BINOMIAL THEOREM. And then, applying thisTheorem to the rectification of curves, and to the determinationof the surfaces and contents of solids, and the position of theircentres of gravity, he discovered the general principle of deducingthe areas of curves from the ordinate, by considering the area asa nascent quantity, increasing by continual fluxion in the proportion of the length of the ordinate, and supposing the abscissato increase uniformly in proportion to the time. Regarding linesas generated by the motion of points, surfaces by the motion oflines, and solids by the motion of surfaces, and considering thatthe ordinates, abscissae, &c., of curves thus formed, vary according to a regular law depending on the equation of the curve,he deduced from this equation the velocities with which thesequantities are generated, and obtained by the rules of infiniteseries, the ultimate value required. To the velocities with whichevery line or quantity is generated, he gave the name of FLUXIONS, and to the lines or quantities themselves, that of FLUENTS.A discovery that successively baffled the acutest and strongest LIFE OK SIR ISAAC NEWTON. 15 intellects :—that, variously modified, has proved of incalculableservice in aiding to develope the most abstruse and the highestTuths in Mathematics and Astronomy: and that was of itselfenough to render any name illustrious in the crowded Annals ofScience. At this period, the most distinguished philosophers were directing all their energies to the subject of light and the improvementof the refracting telescope. Newton, having applied himself tothe grinding of "optic glasses of other figures than spherical," experienced the impracticability of executing such lenses ; and conjectured that their defects, and consequently those of refractingtelescopes, might arise from some other cause than the imperfectconvergency of rays to a single point. He accordingly "procureda triangular glass prism to try therewith the celebrated phenomena of colours." His experiments, entered upon with zeal, and3onducted with that industry, accuracy, and patient thought, forwhich 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 theHistory of Optics. As bearing, however, directly upon the construction of telescopes, he saw that a lens refracting exactly like a prismwould necessarily bring the different rays to different foci, atdifferent distances from the glass, confusing and rendering thevision indistinct. Taking for granted that all bodies producedspectra of ^ jtial length, he dismissed all further consideration ofthe refracting instrument, and took up the principle of reflection.Rays of all colours, he found, were reflected regularly, so that theangle of reflection was equal to the angle of incidence, and hencehe concluded that optical instruments might be brought to anydegree of perfection imaginable, provided reflecting specula ofthe requisite figure and finish could be obtained. At this stageof his optical researches, he was forced t