110 NEWTON S PRINCIPIA. 



tageous approximations through series, logarithms, and 

 the arithmetic of sines, would have afforded important faci 

 lities for these inquiries ; because the solution must come 

 always to an integration. Accordingly Euler, D Alem- 

 bert, and Clairaut, availed themselves of that improve 

 ment to investigate the problem, as we have already seen. 

 But soon after their researches had led to the important 

 result formerly described, a great refinement was intro 

 duced into the calculus, which bore directly upon the sub 

 ject of these inquiries ; and this exceedingly facilitated 

 the solution of the problem in its more extended application. 

 We allude to the invention of the Calculation of Varia 

 tions by Euler and Lagrange. 



We have in the introductory part of this Analytical 

 View explained that this calculus enables us to examine 

 the transition of one curve into another in certain circum 

 stances, by showing how those lines may be found which 

 have certain properties in relation to other lines of a 

 different kind, and thus to investigate problems with 

 respect to curves whose nature changes under the inves 

 tigation, because the relation between their co-ordinates 

 is variable, and is indeed the thing sought for. It is 

 evident, therefore, that this calculus has its immediate 

 application to the subject in question. For the effect of 

 the disturbing force is to change at each moment the 

 nature of the path, which, but for that force, would be 

 described; or the inclination of orbits to one another, 

 which, but for such disturbances, would subsist; or the 

 position in space, which, but for the disturbance, these 

 orbits would have. Now, those changes produced by 

 mutual disturbances, really comprise all the effects of the 

 disturbances on the planetary system. Thus, beside the 

 precession of the equinoxes and the motion of the apsides 

 and nodes, which we have just now generally stated, 



