LECTURES ON THE LUNAR THEORY. [LECT. 



we may reduce this case to the former by rewriting 

 x = -(A 1 + B 1 )cos (11^ + 0,) + - (A 1 



Probably we have here the reason why circular motions and epicycles 

 were first employed. 



Tycho Brahe (1546 1601) discovered the existence of another inequality 

 in the Moon's Longitude quite different from the Elliptic Inequality and 

 the Evection. He found it bore reference to the position of the Sun with 

 regard to the Moon; so that when the Sun and the Moon were in 

 conjunction or opposition or quadratures the position of the Moon was 

 quite well represented by the existing theory, but from conjunction to the 

 quadrature following, her position was more advanced than the place as- 

 signed to it, reaching a maximum of some 35' about half-way; and in the 

 second quadrant it was just as much behind. This inequality he called 

 the Variation ; it was the first that Newton accounted for theoretically, 

 and if we were to suppose the Moon and Sun to move, except for mutual 

 disturbance, in pure circles in the same plane, it is the only one that 

 would present itself. 



The next significant step was made by Horrox (1619 1641) who repre- 

 sented the Evection geometrically by motion in a variable ellipse, and gave 

 very approximately the law of variation of the eccentricity and the motion 

 of the apse. He supposed the focus of the orbit to move in an epicycle 

 about its mean place. 



Newton's Principia did not profess to be and was not intended for 

 a complete exposition of the Lunar Theory. It was fragmentary ; its object 

 was to shew that the more prominent irregularities admitted of explanation 

 on his newly discovered theory of universal gravitation. He explained the 

 Variation completely, and traced its effects in Radius Vector as well as 

 in Longitude; and he also saw clearly that the change of eccentricity and 

 motion of the apse that constitute the Evection could be explained on his 

 principles, but he did not give the investigation in the Principia, even 

 to the extent to which he had actually carried it. The approximations 

 are more difficult in this case than in that of the Variation, and require 

 to be carried further in order to furnish results of the same accuracy as 

 had already been obtained by Horrox from observation. He was more 



