368 HISTOEY OF MECHANICS. 



One consequence of the synthetical form adopted by Newton in the 

 Principia, was, that his successors had the problem of the solar sys- 

 tem to begin entirely anew. Those who would not do this, made no 

 progress, as was long the case with the English. Clairaut says, that he 

 tried for a long time to make some use of Newton's labors ; but that, 

 at last, he resolved to take up the subject in an independent manner. 

 This, accordingly, he did, using analysis throughout, and following 

 methods not much different from those still employed. "We do not 

 now speak of the comparison of this theory with observation, except to 

 remark, that both by the agreements and by the discrepancies of this 

 comparison, Clairaut and other writers were perpetually driven on to 

 carry forwards the calculation to a greater and greater degree of ac- 

 curacy. 



One of the most important of the cases in which this happened, was 

 that of the movement of the Apogee of the Moon ; and in this case, a 

 mode of approximating to the truth, which had been depended on as 

 nearly exact, was, after having caused great perplexity, found by Clairaut 

 and Euler to give only half the truth. This same Problem of Three 

 Bodies was the occasion of a memoir of Clairaut, which gained the 

 prize of the Academy of St. Petersburg in 1751 ; and, finally, of his 

 Theorie de la Lune, published in 1765. D'Alembert labored at the 

 same time on the same problem ; and the value of their methods, and 

 the merit of the inventors, unhappily became a subject of controversy 

 between those two great mathematicians. Euler also, in 1753, pub- 

 lished 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 cer- 

 tain algebraical equations, which express the mechanical conditions of 

 the motion. The operation, by which the result is to be obtained, in- 

 volves the process of integration ; which, in this instance, cannot be 

 performed in an immediate and definite manner ; since the quantitie> 

 thus to be operated on depend upon the moon's position, and thus re- 

 quire us to know the very thing which we have to determine by the 

 operation. The result must be got at, therefore, by successive approx- 

 imations : we must first find a quantity near the truth ; and then, by 

 iii' 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 relations of position between the sun 



