SEQUEL TO THE GENERALIZATION. 103 



unhappily became a subject of controversy between 

 those two great mathematicians. Euler also, in 

 1 753, published a Theory of the Moon, which was, 

 perhaps, more useful than either of the others, 

 since it was afterwards the basis of Mayer's method, 

 and of his Tables. It is difficult to give the general 

 reader any distinct notion of these solutions. We 

 may observe, that the quantities which determine 

 the moon's position, are to be determined by 

 means of certain algebraical equations, which ex- 

 press the mechanical conditions of the motion. The 

 operation, by which the result is to be obtained, 

 involves the process of integration ; which, in this 

 instance, cannot be performed in an immediate and 

 definite manner; since the quantities thus to be 

 operated on depend upon the moon's position, and 

 thus require us to know the very thing which we 

 have to determine by the operation. The result 

 must be got at, therefore, by successive approxi- 

 mations: we must first find a quantity near the 

 truth; and then, by the help of this, one nearer 

 still ; and so on ; and, in this manner, the moon's 

 place will be given by a converging series of terms. 

 The form of these terms depends upon the rela- 

 tions of position between the sun and moon, their 

 apogees, the moon's nodes, and other quantities; 

 and by the variety of combinations of which these 

 admit, the terms become very numerous and com- 

 plex. The magnitude of the terms depends also 

 upon various circumstances; as the relative force 



