Vol. xliv.] philosophical transactions. 31§ 



He went on to examine the place and motion of the apogee, and theory of the 

 increase and decrease of the eccentricity, as well as the greatest and least eccen- 

 tricities themselves ; all which agreed so well with the tables, about the sun's 

 mean distances, that he dare not venture to make any alteration in them : in- 

 deed the 6th equation does not so well account for the variation of the motion of 

 the apogee, and change of the eccentricity, according to the greater or less dis- 

 tance of the sun from the earth ; and therefore he set himself to compute what 

 change this difFeience of the sun's action on the lunar orbit would introduce in 

 the moon's place in every situation of the sun and lunar orbit ; and found, after 

 many tedious computations, that the sun being in apogee, this change, where 

 greatest, would amount to about 4', and to 4' l6", when the sun is in perigee. 

 In other distances of the sun from the earth, this greatest change is proportional 

 to the difference of the cubes of the mean and present distances ; and in every 

 situation of the moon, and of her orbit, the present is to the greatest equation, 

 nearly as the sine of the excess of the moon's mean anomaly above twice the an- 

 nual argument to radius. It increases the moon's longitude, when the sun is in his 



•fp • } Semicircle, and that excess/ ' ]■ than 180°; and diminishes 



it when otherwise.* 



In fine, he compared the theory of the moon, as to her longitude, with se- 

 veral observations, as well in the octants and semi-octants, as in the syzygies and 

 quadratures, and found such an agreement when the above corrections were 

 made, as seemed rather to be wished than hoped for, considering the many in- 

 equalities with which the sun's action disturbs the motion of the moon, and the 

 defects to which the best observations are liable. 



He compared 100 observed longitudes of the moon with the tables; viz. 25 

 eclipses of the moon, all, except the first, taken from Flamsteed's Historia Coe- 

 lestis, the Philosophical Transactions, and the Memoirs of the Royal Academy 

 of Sciences; the 2 great eclipses of the sun in 1706 and 1715 ; 25 select places 

 of the moon from Flamsteed's Historia Coelestis, and 48 of those longitudes of 

 the moon computed from Flamsteed's Observations by Dr. Halley (as he sup- 

 poses) printed in the first edition of the Historia Coelestis. The difference be- 

 tween the observed and computed places, run from to 4', sometimes in excess, 

 sometimes in defect. 



Several observed latitudes of the moon, which he compared with the tables, 

 show them to be very near the truth, both in the motion of the nodes, and also 

 in the quantity and variation of the inclination. 



* If this equation be increased and diminished in a direct ratio of the moon's horizontal parallax, it 

 will become more exact. And if it were always diminished by a fourth or perhaps a third part, it 

 would agree better with observations. — Orig. 



