NEWTON, ISAAC. 



NEWTON, ISAAC. 



470 



it is difficult to suppose that a mind such as his, so ardent in the 

 pursuit of truth, could have contented itself with following a few rules 

 of authority without understanding the reasons upon which they were 

 based. But if these considerations be not thought conclusive, we have 

 only to look to the nature of his discoveries during the first six years 

 of his residence at Cambridge, that is, before the completion of his 

 twenty-fourth year, in order to be convinced that he must either have 

 pushed his studies to a very considerable extent before entering the 

 university, or that his subsequent progress was perfectly unparalleled ; 

 for in this period of six years he invented his Binomial Theorem, 

 established the fundamental principles of his doctrine of Fluxions, 

 and demonstrated the law of the force in virtue of which the planets 

 gravitate towards the sun, although, hi consequence of the erroneous 

 measurement of the earth then in use, it was not till afterwards that 

 he was able to show that the same law holds with respect to the moon, 

 and that the force manifested at the earth's surface in the fall of a 

 pebble is identical, as to its nature, with that which pervades the whole 

 planetary system. 



Descartes had already laid open a vast field of research by the 

 successful application of algebra to geometry, and his writings, both 

 mathematical and speculative, were then much read at Cambridge. 

 Alter the perusal of Saundenon's ' Logic ' and the ' Optics ' of Kepler, 

 the attention of Newton was directed to the ' Geometria ' of Descartes, 

 a work which doubtless exercised considerable influence over his sub- 

 sequent pursuit*, by furnishing him with general methods of direct 

 investigation, such as, till the time of Descartes, were totally unknown. 

 Upon the whole however he was much leas indebted to the analytical 

 writings of Descartes than he was to those of his countryman Wallis. 

 He seldom read without making comments upon the text or marginal 

 notes of such parts as appeared to him susceptible of extension or 

 improvement. In this way he completed the perusal of Descartes' s 

 ' Geometry ,' after which he passed on to the 'Arithineticalnfinitorum' 

 of Wallis. In this work the author had suggested a method of obtaining 

 the quadrature of the circle, the practicability of which depended upon 

 an interpolation. Newton set about effecting this, notwithstanding 

 the discouraging declaration of Wallis, that he believed it to be 

 impracticable. The attempt however proved not merely successful, 

 but in the course of his inquiry he was led (1663-64) to a discovery of 

 greater moment, the history of which is given under BINOMIAL 

 THEOREM, in Aura AND Sc. Div. This theorem, combined with the 

 previous labours of Wallis and others, supplied Newton with a method 

 of determining the area and rectification of curves, the surface and 

 content of the solids formed by their revolution, and the position of 

 their centre of gravity ; and by similar means he solved with com- 

 parative ease a number of problems which had hitherto baffled the 

 attempts of mathematicians, or of which solutions had been obtained 

 only under particular circumstances, whereby the real difficulty had 

 been rather evaded than overcome. The almost indefinite application 

 which he continued to make of this method, computing even the 

 numerical values of the funuulie to which his investigations gave rise, 

 as if he regarded the occupation rather as a source of amusement than 

 of labour, may possibly have been suggested by the view, as novel as 

 it was important, which he took of the mode whereby magnitudes of 

 every kind may be conceived to be generated, and by the notion he 

 early entertained of the possibility of deducing the definite Value of 

 a variable magnitude from the velocity of its increase or diminution. 



The nuxionary calculus to which this opinion gave rise was invented 

 by Newton in or before the year 1665. Its history is given in the 

 article FLUXIONS, in ARTS AND Sc. Div. The following year he com- 

 posed bis 'Analysis per Equatioues Numero Termiuorum lufinitas,' a 

 tract which afterwards caused much discussion as to the extent to 

 which it contained the method of fluxions. For some reason, which 

 it ia now difficult to assign, he thought proper to conceal the substance 

 of this tract from the public, and even from his friends. However 

 on the appearance of Mercator's 'Logaritbmoteclmia' in 1668, in which 

 work Newton, having recognised two of the many results to which his 

 binomial theorem had previously conducted him, namely, the develop- 

 ment of log(l-t-x) and the determination of the quadrature of the 

 hyperbola, he communicated the tract above mentioned to his friend 

 Dr, Harrow. This wag no.t till the month of June 1669. Thu 31st of 

 July following, Barrow, with Newton's permission, transmitted the 

 manuscript to Mr. Collins, at the same time acquainting him that it 

 was the production of a young friend of his who possessed a fine genius 

 for such inquiries. Collins took a copy of the manuscript, and returned 

 the original to Dr. Barrow. The copy was afterwards found among 

 Collins' s papers, and attested the year in which the original treatise. 

 had been composed. It was first formally published in 1712, but long 

 previous to that its contents must have been pretty widely diffused 

 through Collins' s correspondence with many of the principal mathe- 

 maticians of the day, both in England and upon the Continent. 



Newton was admitted sub-sizar in 1061, became scholar in 1664, and 

 took his degree of B.A. in 1665. In 1664-65 he was a candidate with 

 Mr. Robert Uvedale for the law-fellowship of Trinity College ; when 

 Barrow, having found the candidates on an equality as regarded attain- 

 ments, conferred the appointment on Mr. Uvedale, he being the elder. 

 In 10B7 he became junior fellow, took the degree of ALA., and became 

 senior fellow in 1668. lie succeeded Dr. Barrow as Lucasiau professor 

 of mathematics in 1669. 



The raging of the plas^ie in 1665-66, induced Newton to quit 

 Cambridge and retire to Woolsthorpe. Here it was that he beuan to 

 reflect more particularly upon the nature of the force by which bodies 

 at the earth's surface are drawn towards its centre, and to conjecture 

 that the same force might possibly extend to the moon, and there be 

 of sufficient intensity to counteract the centrifugal force of that 

 satellite, and thereby retain it in its orbit about the earth. To com- 

 pare this hypothesis with observation, it was necessary to determine 

 the law according to which the intensity of such a force would vary 

 with the distance from the earth's centre ; for although no sensible 

 variation can be detected within the narrow limits of direct observation, 

 namely, the summit of the highest mountains or the bottom of the 

 deepest mines, it was reasonable to presume that some variation would 

 be appreciable at the distance of the moon, and in such case only could 

 the force be just sufficient to counteract the centrifugal tendency of 

 the revolving satellite. To a mind so habituated to generalise, it was 

 a natural extension of his hypothesis to suppose that the same kind 

 of force which incessantly deflects the moon from a rectilinear path, 

 might likewise act upon the planets so as to retain them in their 

 orbits about the sun. Now, the assumption of an attractive forco 

 emanating from the sun was at this time far from being a novelty, and 

 it had even been asserted by Bouillaud that, if such a force really 

 existed, its intensity would vary inversely as the square of the distance 

 from the attracting body ; but neither Bouillaud nor those who enter- 

 tained similar opinions had given any proof, either empirically or 

 otherwise, of what they had asserted ; and certainly none appear to 

 have attempted to establish that the forces which retain the planets 

 in their orbs were identical, as to their nature, with that which draws 

 a stone, when let fall, to the surface of the earth. Newton showed 

 that the law of the inverse square of the distance is that which really 

 exists in nature ; and further, that this law was a necessary conse- 

 quence of the analogy already discovered by Kepler between the 

 periodic times and the mean distances of the planets. The following 

 will convey a notion of the line of reasoning by which Newton arrived 

 at this result. The intensity of the force, whatever may be its nature, 

 which counteracts the centrifugal force of a planet, is proportional to 

 the versed sine of the arc described in a given time; so that, if the 

 time be small, the force will be proportional to the square of the arc 

 divided by the planet's mean distance, or to the square of the linear 

 velocity by the distance. If therefore for the velocity we substitute 

 the ratio of the mean distance to the periodic time, which is pro- 

 portional to it, we shall find that the force varies as the distance by 

 the square of the periodic time, that is, by Kepler's law, as the distance 

 by the cube of the distance, that is, inversely as the square of the 

 distance. Having thus established the law whereby the planets 

 gravitate towards the sun, he proceeded to examine whether the same 

 law regulated the gravitation of the moon towards the earth. . At this 

 point it is that Newton's reasoning first rests upon conjecture, namely, 

 that the force manifested at the earth's surface in the fall of a stone, is 

 identical with that which is constantly deflecting the moon towards 

 the centre of the earth ; and that the law of its variation was the 

 same as that which he had determined fur the planets. If such were 

 the case, the distance fallen through by the moon in one second of 

 time ought to bear the same ratio to the distance fallen through by a 

 body at the surface of the earth in one second, which the square of 

 the earth's radius bears to the square of the moon's mean distance. 

 The length of the earth's radius, which entered as a necessary element 

 in the verification of his conjecture, was at that time very imperfectly 

 known (a degree of latitude being estimated at only 60 miles, instead 

 of 694 miles, its more correct length) ; the consequence of which was, 

 that the result of his calculation indicated a force at the distance of 

 the moon greater, by nearly one-sixth, than that deduced from direct 

 observation. This difference, which many would have considered 

 sufficiently small to establish the correctness of the hypothesis, was 

 regarded by Newton rather as a direct refutation of its truth. He 

 therefore laid aside further consideration of the subject, suspecting, 

 says Whiston ('Memoirs' of himself), that sorue unknown cause, 

 perhaps similar to the vortices of Descartes, modified, in the case of 

 the moon, the law which he had satisfactorily established with regard 

 to the planets. 



In 1666, the plague having subsided, ho returned to Cambridge, 

 but without mentioning to any of his friends the interesting inquiry 

 which, during his absence, had occupied so much of his attention. In 

 this way the discarded hypothesis lay dormant for sixteen years. 

 In 1682, when attending a meeting of the Koyal Society, he casually 

 heard of the measurement of au arc of the meridian which had been 

 executed by Picard three years before. Having taking a note of the 

 result, and thence deduced the length of the earth's radius, he resumed 

 his former calculation ; but hi the course of the work, observing that 

 the conclusion he had formerly anticipated was about to be realised, 

 his ardour is said to have brought on a state otexcitemeut and nervous 

 irritability which precluded hia further progress, so that the comple- 

 tion of the calculation was confided to a friend. The following year 

 he transmitted to London a few propositions on the motion of bodies 

 acted on by centripetal forces, which were shortly after communicated 

 to the Koyal Society, and constitute the leading propositions of the 

 'Priucipia.' The manuscript of this work, entitled ' Philosophise 

 Naturalia Principia Mathematica,' was presented to the Koyal Society 



