AS A MENTAL OPERATION 17 



periodic times are as the cubes of the radii, the centripetal 

 forces of the bodies will be inversely as the squares of 

 the radii (i.e. the distances from the centre); and vice 

 versa. Further, he deduces these laws for all kinds of 

 curves. 



The derivative laws of dynamkal mechanics thus 

 deduced from the laws of motion served Newton in 

 their turn as his major premisses for the deduction of 

 the cause of planetary motions in ellipses round the sun. 

 But first he wanted these facts themselves as his minor 

 premisses. He derives them from Kepler's laws, which 

 he calls Phaenomena. They are three : planets move in 

 ellipses round the sun in one of the foci ; they describe, 

 by the radii drawn, not to the earth, but to the sun, 

 areas proportional to the times of description ; and the 

 squares of their periodic times are as the cubes of their 

 mean distances from the sun. Starting then from 

 mechanical laws in the major, and from astronomical 

 phenomena in the minor, Newton was ready to deduce 

 the real cause of planetary motions in ellipses round 

 the sun ; namely, their centripetal force, or gravitation 

 to the sun. 



His deductions, which are formal analytic syllogisms 

 from fact to cause, are as follows : 



(i) Every body which moves in a curve, and by a 

 radius drawn to a point describes about that point areas 

 proportional to the times, is urged by a centripetal force 

 to that point (Principia, Book I, Prop. 2). 



The planets by the radii, drawn, not to the earth, but 

 to the sun, describe areas proportional to the times of 

 description (Ibid., Book III, Phenomenon 5 : Kepler s 

 Second Law). 



Therefore, the planets are urged by a centripetal force 

 to the sun (Ibid., Book III, Prop. 2). 



