26 



The computation of the mean motion of the lunar apsids is by 

 far the most important point in Avhich the Newtonian Theory of the 

 Moon, as given in the Principia, appears defective. Newton him- 

 self, in the latter editions of the Principia, seems to have aban- 

 doned the attempt to reconcile or rattier deduce from Theory the 

 motion given by Observation. 



In the fii-st edition he had made the attempt, after stating his re- 

 sults (Scholium, p. 462.) ; he adds, " compulationes autem, ut niinis 

 " perplexas & approximationibus impeditas, neque satis accuratas, 

 " apponere non lubet." 



It may be presumed that he found his method on examination 

 inaccurate, otherwise it cannot be doubted he would have no- 

 ticed it in the subsequent editions, and given, if not the method 

 in detail, the results. 



Machin appears the fii-st after Newton who attempted this 

 problem ; the inadequacy of his solution, and of those of some 

 others of the same nature, will be noticed further on. 



Clairaut, in 1748, had the honour of giving the first exact 

 solution according to the principles of the Newtonian Theory of 

 Gravity, after he had, in the Memoirs of the Royal Academy 

 of Sciences at Paris, announced that the Newtonian law was in- 

 exact, inasmuch that the mean motion of the lunar apsids deduced 

 from that law did not agree with observation. 



Clairaut's result was confirmed by Euler, D'Alemberl and 

 Mayer, and subsequently by other mathematicians. Their re- 

 searches, however, being directed more towards a general theory 

 of the lunar motions, than towards the particular question of the 

 mean motion of the apsids, are so complicated, that the exact thread 

 of reasoning respecting this motion cannot without difficulty be 



