VOL. I.VII.3 PHILOSOPHICAL TRANSACTIONS. 413 



2. Let TA represent the moon's mean distance from the earth. Take tv, such 

 that TA may be to tv, in the x-" ^ 



duplicate proportion of the pe- 1^^'^' I i i ' i B 



riodic montli to the sidereal ^ D E -^l G- T <y 



year. Take tg, such that vt ^»*.-.->^ 



may be to tg in the proportion of the moon's accelerating attraction to the earth, 

 to the sun's mean disturbance of that attraction. Take te such that te may be 

 to TA, as TA to TG. Take eo, such that the rectangle eoa may be equal to 3tk 

 X TA. On the centre t, with the interval ta, describe a circle. Draw ox per- 

 pendicular to AB, meeting the circle in x. Take ad = at. The proportion of 

 TA to TV being given, and ta being given, tv is given. But the proportion of 

 TV to TG is given. Therefore tg is given, and the proportion of tg to ta is 

 given. TG : TA = ta: te. Therefore the proportion of ta to te is given. There- 

 fore TE is given. Therefore 3te x ta is given. Therefore eo X oa is given. 

 And ea (= TE — ta) is given. Therefore ao is given. But ab (= 2at) is 

 given. Therefore ob is given. Therefore ao X oB is given, ao X ob = ox* 

 (by the circle). Therefore ox^, and consequently o^ is given. But db (= 3at) 

 is given. Therefore the proportion of db to ox is given. And the proportion 

 of DB to ox, is that of the mean distance of the sun, to the mean distance of 

 the moon. 



This is in brief the method of my computation. The computation is as follows: 

 The periodic month is to the anomalistic month, as 57600 to SSOQl. 

 Therefore, in fig. 1, ta^ : tp* = 57600^ : 58001^ = 3317760000: 3374564281 



33^760000 = 1-017 1212748963135=864197330^864197530, &c. 

 Hence, by extracting the cube root, I find ta: tp = 1 : 1.003674827053. 

 Therefore put TA= 1. Then tp= 1.005674827033; and ap = 0.005674827053, 



^^"^' 5^7TTrP= 0-002797722 = xm. 



The square of the periodic month is to the square of the sidereal year, as 1 to 

 178.725. 



Therefore TA : tv = I : 178.725, fig. 2. 

 But TV : TG = 1 : 0.002797722. 

 Therefore ta: tg = 1 : 178.725 X 0.002797722 = 1 : 0.50002286445. 

 TA : TG = TB : TA. Therefore te : ta = 1 : 0.50002286445. 

 Therefore put te = 1 . 



Then ta = 0.50002286445 

 And ea = 0.49997713535 

 And 3te X ta = 1 .50006859335 = EOA. 

 Hence AO (= /te x ta ^ea* — Iea) = 1.0000365 8292. 

 But AB = 2ta = 1.00004572890. 



