214 



NEWTON 



obtained as many portions of the whole area as there 

 were terms, and, by their addition, obtained the total. 

 But the far more extensive, and, in some respects, 

 unlimited applications that Newton made of this rule, 

 depended on a general principle, which he had made 

 out, and which consisted in determining, from the 

 manner in which quantities gradually increase, what 

 are the values to which they ultimately arrive. To 

 effect this, Newton regards them, not as aggregates 

 of small homogeneous parts, but as the results of 

 continued motion, lines being considered as described 

 by the movement of points, surfaces by that of lines, 

 solids by that of surfaces, and angles by the rotation 

 of their sides. Again, considering that the quantities 

 so formed are greater or smaller, in equal times, 

 according as the velocity with which they are 

 developed is more or less rapid, he endeavours to 

 determine their ultimate values from the expression 

 for these velocities, which he calls fluxions, naming 

 the quantities themselves fluents. In fact, when any 

 given curve, surface, or solid, is generated in this 

 manner, the different elements which either compose 

 or belong to it, such as the ordinates, the abscissas, 

 the lengths of the arcs, the solid contents, the in- 

 clinations of the tangent planes, and of the tangents, 

 all vary, differently and unequally indeed, but never- 

 theless according to a regular law, depending on the 

 equation of the curve, surface, or solid under considera- 

 tion. Hence Newton was able to deduce from this equa- 

 tion the fluxions of all these elements, in terms of any 

 one of the variables, and of the fluxion of this varia- 

 ble, considered as indeterminate ; then, by expanding 

 into series, he transformed the expression, so obtained, 

 into finite or infinite series of monomial terms, to 

 which Wallis's rule became applicable ; thus*, by 

 applying it successively to each, and taking the sum 

 of the results, he obtained the ultimate value, that is, 

 the fluent of the element, which he had been con- 

 sidering. In this consists the method of fluxions, of 

 which Newton, from that time, laid the foundation, 

 and which, eleven years later, Leibnitz again dis- 

 covered, and presented to the world in a different 

 form, that of the differential calculus. Newton 

 made these important discoveries before completing 

 his twenty-third year, and collected them in a manu- 

 script, entitled Analysis per jEquationes Numero 

 Terminorum infinitas, but did not communicate them 

 to any one. 



About this time (1665), being obliged to quit 

 Cambridge on account of the plague, he retired to 

 Woolsthorpe, and now turned his attention more 

 closely to subjects of natural philosophy. As he was 

 one day sitting under an apple-tn-e. the fall of an 

 apple led him to reflect on the nature of that remark- 

 able principle which urges all bodies towards the 

 centre of the earth. " Why," he asked himself, 

 " may not this power extend to the moon? and, if so, 

 what more would be necessary to retain her in her 

 orbit about the earth?" He considered that if the 

 moon was retained about the earth by terrestrial 

 gravity, the planets, which move round the sun, 

 ought, similarly, to be retained in their orbs by their 

 gravity towards that body. Setting out with the law 

 of Kepler, that the squares of the times of revolution 

 of the different planets are proportional to the cubes 

 of their distances from the sun, Newton found, by 

 calculation, that the force of solar gravity decreases 

 proportionally to the square of the distance ; and 

 having thus determined the law of the gravity of the 

 planets towards the sun. he endeavoured to apply it 

 to the moon ; that is, to determine the velocity of her 

 motion round the earth by means of her distance, 

 as settled by astronomers, and of the intensity of 

 gravity, as shown by the fell of bodies at the earth's 

 surface. To make this calculation, it is necessary to 



know exactly the distance from the surface to the 

 centre of the earth, expressed in parts of the same 

 measure that is used in marking the spaces described, 

 in a given time, by Jailing bodies at the earth's sur- 

 face ; for their velocity is the first term of compari- 

 son that determines the intensity of gravity at this 

 distance from the centre, which we apply afterwards 

 at the moon's distance, by diminishing it proportion* 

 ably to the square of the distance. It tiien remains 

 only to be seen if gravity, when thus diminished, has 

 precisely the degree of energy necessary to counter- 

 act the centrifugal force of the moon, caused by her 

 observed motion in her orbit. Unfortunately, at that 

 time, there existed no correct measure of the earth's 

 dimensions. (See Degrees, Measurement of.) New- 

 ton was obliged to employ the imperfect measures 

 then in use, and found that they gave for the force 

 which retains the moon in her orbit, a value greater 

 by one-sixth than that which results from her observed 

 circular velocity. This small difference seemed, to 

 his cautious mind, a strong proof against his bold 

 conjecture. He imagined that some unknown cause 

 modified, in the case of the moon, the general law of 

 gravity indicated by the motion of the planets. Yet 

 he did not abandon his leading notion, but deter- 

 mined to wait till study and reflection should reveal 

 to him this supposed unknown cause. In 1666, he 

 returned to Cambridge, was chosen fellow of his 

 college (Trinity college) in 1667, and, the next year, 

 was admitted A. M.; but he did not disclose his 

 secrets even to his instructer, doctor Barrow. In 

 1668, however, Mercator published his Logarithmo- 

 technia, in which he had obtained the area of the 

 hyperbola referred to its asymptotes, by expanding its 

 ordinate into an infinite series, which was the main se- 

 cret of Newton's method. Barrow showed this work 

 to Newton, who immediately gave him his own trea- 

 tise (the Analysis, &c ), but did not yet publish it. 



In the course of 1666, his attention had been ac- 

 cidentally drawn to the phenomena of the refraction 

 of light through prisms. His experiments led him to 

 conclude that light, as it emanates from the radiating 

 bodies, is not a simple and homogeneous substance, 

 but that it is composed of a number of rays, endowed 

 with unequal refrangibility, and possessing different 

 colouring properties. More than two years elapsed 

 before he returned to his researches on this subject ; 

 but, in 1669, being appointed professor of mathe- 

 matics, and preparing to lecture on optics, he en- 

 deavoured to mature his first results, and composed 

 a complete treatise, in which the fundamental pro- 

 perties of light were unfolded, established, and 

 arranged by means of experiments aione, without 

 any mixture of hypothesis a novelty at that time 

 almost as surprising as these properties themselves. 

 Thus it appears, that the three great discoveries 

 which form the glory of his life, the method of 

 Fluxions, the Theory of Universal Gravitation, and 

 the Decomposition of Light, were conceived before 

 the completion of his twenty-fourth year. 



In 1672, Newton was chosen a fellow of the royal 

 society, to which he communicated a description of a 

 new arrangement for reflecting-telescopes, which 

 rendered them more convenient by diminishing their 

 length without weakening their magnifying powers, 

 and, soon after, the first part of his labours on the 

 analysis of light. When the first feelings of surprise 

 and admiration, excited by this noble work, had sub- 

 sided, the society appointed three members to study 

 it fully, and report upon it. Hooke, a man of exten- 

 sive acquirements and an original turn of thought, 

 but of excessive desire of renown, being one of the 

 members, undertook to draw up the report. Instead 

 of discussing the new facts, as presented by the ex- 

 periments of Newton, he examined them merely in 



