$1 234, 235] Lunar Theory 299 



motions of the moon far more accurately than those of 

 D'Alembert, and were even superior in some points to those 

 based on Euler's very much more elaborate second theory ; 

 Clairaut's last tables were seldom in error more than i|', 

 and would hence serve to determine the longitude to 

 within about j. Clairaut's tables were, however, never 

 much used, since Tobias Mayer's as improved by 

 Bradley were found in practice to be a good deal more 

 accurate ; but Mayer borrowed so extensively from observa- 

 tion that his formulae cannot be regarded as true deductions 

 from gravitation in the same sense in which Clairaut's were. 

 Mathematically Euler's second jtheory is the most interest- 

 ing and was of the greatest importance as a basis for later 

 developments. The most modern lunar theory * is in 

 some sense a return to Euler's methods. 



234. Newton's lunar theory may be said to have given a 

 qualitative account of the lunar inequalities known by 

 observation at the time when the Principia was published, 

 and to have indicated others which had not yet been 

 observed. But his attempts to explain these irregularities 

 quantitatively were only partially successful. 



Euler, Clairaut, and D'Alembert threw the lunar theory 

 into an entirely new form by using analytical methods 

 instead of geometrical ; one advantage of this was that by 

 the expenditure of the necessary labour calculations could 

 in general be carried further when required and lead to a 

 higher degree of accuracy. The result of their more 

 elaborate development was that with one exception the 

 inequalities known from observation were explained with a 

 considerable degree of accuracy quantitatively as well as 

 qualitatively ; and thus tables, such as those of Clairaut, 

 based on theory, represented the lunar motions very closely. 

 The one exception was the secular acceleration : we have 

 just seen that Euler failed to explain it ; D'Alembert was 

 equally unsuccessful, and Clairaut does not appear to have 

 considered the question. 



235. The chief inequalities in planetary motion which 

 observation had revealed up to Newton's time were the 

 forward motion of the apses of the earth's orbit and a very 



* That of the distinguished American astronomer Dr. G. W. Hill 

 (chapter xin., 286). 



