NEWTON S PRINCIPIA. 129 



tances. Let S attract M with a force inversely as the 

 square of the distance ; call the mean distance = 1 ; the 



mean force will be -^ = 1. Let the distance from S, 

 successively taken by M in moving round E, or its true 

 distance, be S M ; thence the force at M is ^ . , 2 . Take 



S 

 S L = ^r and drawing L N parallel to M E, the 



forces at M are L N + Q (Q being a quantity that 

 varies as -..--. and S N. Now L N : ME:: 



S L : S M; and L N = = -. Therefore 



the force acting upon M towards E is as ME + ^^ ; con- 



sequently it will increase the attraction of E, but it will 

 not be inversely as the square of the distance ; and there 

 fore M will not describe an ellipse round E, and the force 

 N S does not tend towards E, nor does the force resulting 

 from compounding L N, or ME, or L N + M E, with 

 N S, tend to E. So that the areas will not be proportional 

 to the times. Therefore, also, this deviation from the ellip 

 tical form and from the proportional description of the areas 

 will be the greater, as the distances L N and ]S T S are smaller. 



Again, let S attract E with a force as cr-; 



o hi 2 



if this were equal to S N, it would, by combining with 

 SN, that is, with the attraction of S on M, produce 

 no alteration in the relative motion of M and E. There 

 fore, that alteration is only caused by the difference 



S 

 between SN and Q-^; wherefore the nearer SN is 



to the proportion of ^, that is (because of the 



