PREFACE. Xlll 



moon's apogee when in that position is to the mean hourly 

 motion of the moon as 



l + Jf<7 : 238^. 



The investigation on this point is not entirely satisfactory, 

 and from the alterations made in the MS. Newton evidently 

 felt doubts about the correctness of the coefficient ^ which 

 occurs in this formula. 



From this, however, he deduces quite correctly that the 

 mean annual motion of the apogee resulting would amount 

 to 38 51' 51", whereas the annual motion given in the Astro- 

 nomical Tables is 40 41f . 



The result stated in the scholium to the 1st Edition appears 

 to have been found by a more complete and probably a much 

 more complicated investigation than that contained in the 

 extant MSS. 



The papers also contain a long list of propositions in the 

 Lunar Theory which were evidently intended to be inserted in 

 a second edition, upon which Newton appears to have been 

 engaged in 1694. This list, together with the two lemmas on 

 the motion of the apogee mentioned above, will be found in 

 the Appendix. 



Halley inserted in the Philosophical Transactiom of 1721 a 

 Table of Refractions by Newton, without giving any idea of 

 the method of its formation. 



Kramp, in his Analyse des Refractions, published in 1799, 

 investigates by a new and powerful analytical method the law 

 of atmospheric refraction for rays in the neighbourhood of the 

 horizon. 



On comparing his theoretical results with Newton's Table, 

 he finds a remarkably close agreement, which is enough to show 

 that the Table was also the result of theory, and therefore that 

 Newton must have had some method of his own of solving the 

 difficult problem of horizontal refraction. 



Nothing was known of this method, however, until the pub- 

 lication of the correspondence between Newton and Flamsteed 

 by Mr Baily in 1835. In a letter to Flamsteed, dated De- 

 cember 20th, 1694 1 , Newton tries to explain the foundation of 



1 Baily's Flamsteed p. 145. 



