THE THEORY OF GRAVITATION. 149 



Certainly they had sufficient patience and sagacity to succeed in 

 finding such a method, since they had had enough of these qualities to 

 discover and advance in considerable degree the admirable doctrine of 

 incommensurables, and of exhaustions, although these were not ordi- 

 narily used except in the consideration of the five regular bodies, and 

 were specially derived, it is said, to examine certain very hazardous and 

 even fantastic conjectures of the Pythagoreans and Platonists. 



XV. 



Practically, if one omits from the theory of central forces those curi- 

 ous propositions and generalizations which can only be regarded as its 

 luxuries, as well as the delicate evaluations which are required only for 

 the perfecting of astronomical tables, all the rest may be demonstrated 

 sufficiently for the uses of the physicist by the aid of lemmas less exact 

 and universal than those of the calculus. 



This has indeed been pointed out in some degree by several geome- 

 ters, but it may be realized still further if the reader will undertake by 

 the same or analogous means of simplification to attack other proposi- 

 tions than those already so treated. 



But the probability that the ancients would have been able to accom- 

 plish such demonstrations is still less necessary to the plan which I 

 have proposed to myself, as stated at the beginning of this essay, than 

 the probability that they would have discovered the simple relations 

 mentioned in the thirteenth paragraph. Consequently the reader may, 

 if he prefers, ignore the last three paragraphs and give attention only 

 to matters which I have expressly engaged to establish. 



XVI. 



I declared that the laws of Kepler were necessary consequences of 

 the doctrine that gravitation results from the impulsion of atoms mov- 

 ing in every direction, since Kepler's laws follow directly from those of 

 Newton. I ought, however, to show, for the benefit of readers less 

 versed in the matter, where it may be found proved that the first-men- 

 tioned laws are the natural consequences of the second. 



First. That the law of areas proportional to times is a necessary con- 

 sequence of gravitation, always directed toward a single point, is dem- 

 onstrated by elementary geometry in the first proposition of Newton's 

 Principia. 



Second. That the law of squares of periodic times proportional to the 

 cubes of the distances, for bodies appearing to describe circles, must 

 necessarily follow from a gravitation inversely proportional to the square 

 of the distance constitutes the second part of the sixth corollary to 

 Proposition IV of the same work, and may be demonstrated by ele- 

 mentary methods also for regular polygons, which represent more nearly 

 than exact circles the orbits traversed by bodies diverted slightly from 

 their paths by intermittent collisions. 



