Xll PREFACE. 



he had found by theory for the motion of the moon's apogee, is 

 omitted in the new scholium. 



It is interesting to find among the papers on the Lunar 

 Theory a good many containing Newton's calculations relating 

 to the inequalities which are described in the above scholium. 

 These papers are unfortunately very imperfect, and they have 

 greatly suffered from fire and damp, but there is enough re- 

 maining to give a general idea of Newton's mode of proceeding. 

 The most interesting of these papers relate to the motion of the 

 moon's apogee. Two lemmas are first established which give 

 the motion of the apogee in an elliptic orbit of very small 

 eccentricity due to given small disturbing forces acting, (1) in 

 the direction of the radius vector, and (2) in the direction 

 perpendicular to it. 



These lemmas are carefully written out, as if in preparation 

 for the press, and they were probably at first intended to form 

 part of the Principia. 



Next follows the application of the lemmas to the particular 

 case of the Moon, in which the supposition that the disturbances 

 are represented by changes in the elements of a purely elliptic 

 orbit of small eccentricity would lead to practical inconvenience, 

 and consequently Newton is led to modify that supposition. 

 In the Principia he shows that if the moon's orbit be supposed 

 to have no independent eccentricity, its form will be approxi- 

 mately an oval with the earth in the centre, the smaller axis 

 being in the line of syzygies and the larger in that of quadra- 

 tures, the ratio of these axes being nearly that of 69 to 70. 

 Now when the proper eccentricity of the orbit is taken into 

 account, supposing that eccentricity to be small, Newton 

 assumes that the form of the orbit in which the moon really 

 moves will be related to the form of the oval orbit before 

 mentioned, nearly as an elliptic orbit of small eccentricity with 

 the earth in its focus is related to a circular orbit about the 

 earth in the centre. He then attempts to deduce the horary 

 motion of the moon's apogee for any given position of the 

 apogee with respect to the sun, and his conclusion is that if C 

 denote the cosine of double the angle of elongation of the sun 

 from the moon's apogee, then the mean hourly motion of the 



