28 LECTURE IV. 



But when two bodies revolve with equal velocities at different distances, 

 the forces are inversely as the distances ; consequently the forces are, in all 

 cases, directly as the squares of the velocities, and inversely as the dis- 

 tances. 



If two bodies revolve in equal times at different distances, the forces by 

 which they are retained in their orbits are simply as the distances. If one 

 of the stages of the whirling table be placed at twice the distance of the 

 other, it will raise twice as great a weight when the revolutions are per- 

 formed in the same time. 



In general, the forces are as the distances directly, and as the squares of 

 the times of revolution inversely.* Thus the same weight revolving in a 

 double time, at the same distance, will have its effect reduced to one fourth, 

 but at a double distance the effect will again be increased to half of its 

 original magnitude. 



From these principles we may deduce the law which was discovered by 

 Keplerf in the motions of the planetary bodies, but which was first demon- 

 strated by NewtonJ from mechanical considerations. Where the forces 

 vary inversely as the squares of the distances, as in the case of gravitation, 

 the squares of the times of revolution are proportional to the cubes of the 

 distances. Thus if the distance of one body be four times as great as that 

 of another, the cube of 4 being 64, which is the square of 8, the jtime of its 

 revolution will be 8 times as great as that of the first body. It would be 

 easy to show the truth of this proposition experimentally by means of the 

 whirling table, but the proof would be less striking than those of the 

 simpler laws which have already been laid down. 



Hitherto we have supposed the orbit of a revolving body to be a perfect 

 circle ; but it often happens in nature, as for instance in all the planetary 

 motions, that the orbit deviates more or less from a circular form ; and in 

 such cases we may apply another very important law which was also 

 discovered by Kepler ; that the right line joining a revolving body and its 

 centre of attraction, always describes equal areas in equal times, and the 

 velocity of the body is therefore always inversely as the perpendicular 

 drawn from the centre to the tangent. (Plate I. Fig. 14.) 



The demonstration of this law invented by Newton, || was one of the most 

 elegant applications of the geometry of infinites or indivisibles ; a branch of 

 mathematics of which Archimedes laid the foundations, which Cavalleri** 

 and Wallistt greatly advanced, and which Newton^ brought near to per- 

 fection. Its truth may be in some measure shown by an experiment on the 

 revolution of a ball suspended by a long thread, and drawn towards a point 

 immediately under the point of suspension by another thread, which may 



* Principia, I. Prop. 4, Cor. 2. f Harmonice Mundi, lib. v. cap. 3, 8. 



J Principia, I. Prop. 4, Cor. 6; and Prop. 15. 



On the Motion of Mars, 1609, p. 194. || Principia, I. 1. 



** Exercitationes Geometricse, Bonon. 1647. 



ft Arithmetica Innnitorum, Op. fol. Ox. 1699, v. i. p. 365. 



JJ Fluxions, Trans, by Colson, 4to, 1736, Ph. Trans. No. 432. Consult also 

 Taylor, Methodus Incrementorum, 4to, 1715. Maclaurin's Fluxions, 2 vols. 4 to, 

 Lond. 1742. Euler, Calculus Dif. et int. 4 vols. 4to, Pet. 1792. Lacroix, Traitc 

 du Calcul Dif. 3 vols. 4to, Paris. Lagrange, Calcul des Fonctions. 



