NEWTON S PRTNCIPIA. Ill 



the alteration in the form of the curve includes also the 

 change of its eccentricity, and the acceleration or retar 

 dation of the motion itself. Hence, we have at once 

 proved that the determination of those effects which arise 

 from disturbing forces, is in a peculiar manner the pro 

 vince of this new and refined analysis, the Calculus of 

 Variations. Therefore, beside the facilities afforded by 

 the improvement of older methods of investigation, the 

 addition of this new instrument to our means of solving 

 the problem has established an entirely novel method, 

 and opened an almost unknown field of inquiry, from 

 which the original author of all these discoveries was 

 necessarily shut out. Instead, therefore, of minutely going 

 over the steps of his solution, as applied to the celestial 

 motions, we shall show the course which he pursued by 

 demonstrating its fundamental principles ; but we shall 

 begin by stating concisely the results of the more recent 

 investigations as affecting the science of physical astro 

 nomy, and shall reserve the fuller discussion of this subject 

 for the account of Laplace s work. 



In considering the motions of the planets and their 

 satellites round the sun, we may first regard him as from 

 his magnitude and distance so little affected by their 

 attractions, that his motion is trifling, and cannot sen 

 sibly affect that of the other bodies ; so that he may be 

 viewed as at rest ; and then the smaller bodies will both 

 move round one another, and round the larger and more 

 distant body as if he were fixed. But not only will the 

 movement of these bodies be thus affected by their mu 

 tual actions ; they will also be affected in their motions round 

 one another by the action of the third body, the sun ; and 

 this action will disturb and alter their relative motions, as 

 regards both their velocities, the forms, and the positions 

 of their orbits. Thus the position of the moon s path 



