3Q i^EcruRE IV. 



In general, the forces are a& tlie 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 iucceased to, half of its ori- 

 ginal magnitude. 



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

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

 strated by Newton 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 revohition are proportional to the cubes of the dis- 

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

 another, the cube of 4 being G4, which is the scjuai^e o-f 8, the time^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 simples 

 laws which have already been laid down. 



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

 circle; but it ol ten 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 disco- 

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

 of attraction, always describes equal aieas in Cfjual- times, and the velocity of 

 the body is therefore always inversely as the peri>endicular 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 

 Wallis greatly advanced, and which Newton brought near to perfection. Its 

 truth may be in some measure shown by an experiment on tlie revolution of 

 a ball suspended Ijy a long thread, and drawn towards a point immediately 

 under the point of suspension by another thread, which may either be held 

 in the hand, or have a weight attached to it. The ball being made to re- 

 volve, its motion becomes evidently more rapid when it is drawn by the ho- 



