ISAAC NEWTON 89 



the known paths of the moons and planets, which are directed 

 from all sides in each case towards a fixed point. Newton 

 thus arrives at a thorough discussion of such 'centripetal' 

 forces, concerning which he states a large number of theo- 

 rems. He here covers in part the same ground as Huygens 

 with his 'centrifugal' force;^ for centripetal and centrifugal 

 forces are equal to one another, but opposite in direction, 

 in the case of a circular path. Newton then investigated the 

 form of the path, and the motion in it, in the case of various 

 laws of centripetal force, both with and without frictional 

 force, which latter may again be proportional to the first or 

 second power of the velocity; here a great development of 

 geometry took place. ^ It appeared that motion according to 

 all three laws of Kepler, only occurs when the centripetal 

 force is directed towards a focus of the elliptical path (more 

 generally a path in the form of a conic section), and when it 

 acts according to the inverse square of the distance, and is 

 proportional to the mass, all frictional forces being absent. 

 According to the third law of motion, the force between sun 

 and planet, earth and moon must be mutual, and it follows 

 from this, that both must be in motion about their common 

 centre of gravity, and also that the masses of both enter 

 equally into the magnitude of the force. The law of the actual 

 force acting between sun and planets, gravitation, was thus 

 grasped. Since the four moons of Jupiter and the moons of 

 Saturn (in Newton's time five had already been found and 

 observed) act according to the same laws, as Newton shows 

 in detail, the law of gravitation is also proved for these bodies, 

 and finally also for the comets, since, as Newton, with the 

 important assistance of his pupil Halley, shows in the case of 

 several of them, they move in conic sections with the sun as 



* Newton recognises Huygens' previous work by a special note 

 (Principia, lib. 1, sec. 2, prop. 4, scolium). 



2 Newton also studied for the first time, in a special essay, lines of the 

 third order; altogether, he made very great extensions of the analytical 

 geometry founded by Descartes. 



