150 THE THEORY OF GRAVITATION. 



Third. That the ellipticity of an orbit is the necessary consequence 

 of gravitation directed toward its focus, and reciprocally proportional 

 to the square of the distance, is the converse of Proposition XI of the 

 same book. This proposition has been more simply demonstrated as a 

 consequence of the fiftieth of Book III of the conies of Appolonius. 



I may pause here, since in maintaining that the laws of Kepler are 

 an easy consequence of the system of atoms I have not pretended that 

 their application to complex cases readily follows from the slight knowl- 

 edge of geometry possessed by the ancients. Nevertheless, I may add — 



Fourth. That the Proposition XI of the Principia once attained it 

 does not appear to me difficult to establish the fiftieth, which extends 

 our second consequences to ellipses — that is to say, which proves that 

 in ellipses as well, the squares of the periodic times about an attracting 

 body (placed in one of the foci) are proportional to the cubes of the 

 mean distances. 



XVII. 



Let us now see how the laws of Galileo may be derived from the 

 hypothesis of the impulsion of the atoms. 



The blows of corpuscles, moving with a velocity more rapid than 

 light, upon a body which has fallen three or four seconds, would be sen- 

 sibly of the same strength as the preceding blows had been upon the 

 same body when it had only fallen one or two seconds. 1 Hence the suc- 

 cessive accelerations of the body in equal times must be sensibly equal, 

 and the velocity at any instant must be sensibly proportional to the 

 time elapsed since the beginning of the fall. From this it follows neces- 

 sarily that the spaces traversed since the beginning are sensibly pro- 

 portional to the squares of the total times, 2 and will be sensibly pro- 

 portional to the succeesive odd numbers. 



1 To assign to these corpuscles the velocity of sound even would be sufficient. For 

 the velocity of souud is more than thirty-four times as rapid as that of a body which 

 has fallen one second, or more than seventeen times as great as that of one that has 

 fallen two seconds, etc. Hence with the increasing velocity of the falling body the 

 accelerating impulses impressed by the corpuscles would be more feeble than at the 

 beginning of the fall by one thirty-fourth at the end of one second, by two thirty- 

 fourths at the end of two seconds, etc. This gradual decrease of acceleration would 

 not be perceived in the longest times of fall which are ordinarily measured. How 

 much less therefore would they be perceived if we assume for the corpuscles the 

 velocity of light, which is nine hundred thousand times as great as that of sound. 



2 Demonstration : I divide the two times which are to be compared into an equal 

 number of parts, so small that the body may be conceived as falling with equal 

 rapidity during the whole duration of one of these parts. And I observe that the 

 two bodies which are compared will have, at the beginning of each of the corre- 

 sponding parts of the two times, velocities proportional to the times then elapsed, 

 and consequently to the entire times. Hence the small spaces traversed at these cor- 

 responding instants will be traversed with a velocity proportional to the times com- 

 pared. 



But the elementary spaces fallen through will be proportional not only to the 

 velocities with which they are traversed, but also to the portions of time occupied 

 in traversing them, and consequently to the whole time's. Therefore the small cor- 



