NEWTON S PRINCIPIA. 113 



unobserved in Halley s time, and it was discussed acutely 

 by Mayer ; but its cause was first discovered by Laplace : 

 it is the sun s action upon the moon, combined with the 

 variation in the orbit of the earth, the eccentricity of 

 which has been diminishing regularly, though by an 

 extremely small quantity (only V. 0000007667 of the 

 greater axis of our orbit); so that the orbit has been 

 slowly approaching more and more to the circular form. 



It is a great proof of the usefulness of the calculus in 

 these investigations, that this great geometrician appears 

 to have discovered the connexion between the earth s 

 diminishing eccentricity and the acceleration of the moon s 

 mean motion, by the careful examination of the mere 

 equation or algebraical expression. For the reciprocal 



of the semi-axis of the moon s orbit -, as influenced by 



the sun s attraction combined with the earth s, is found 

 to be represented by an expression, which, among other 



terms, contains this : ^ in which a* is the 



semi-axis of the earth s orbit, nf the mass of the sun, 

 and e the eccentricity of the earth s orbit. Consequently, 



as e* decreases, increases, the term being negative; 



and therefore a itself decreases as e&quot; decreases ; in other 

 words, the moon s orbit is diminished, and her velocity 

 augmented, in consequence of the earth s eccentricity 

 decreasing. But if the diminution of the greater axis is 

 not admitted as necessarily lessening the orbit, we may 

 recollect the relation between the times and the mean 

 distances, the squares of the former being as the cubes 

 of the latter; and the mean motion is, of course, in 

 versely as the periodic time. However Laplace fur 

 nishes us with a still closer reason, and illustrates the 



