VOL. XLIX.] PHILOSOPHICAL TRANSACTIONS. 23 



vp being given, Ln is as pn X pi. From the centre t describe the arc rp of a 

 circle perpendicular to the equator lp ; then, in the spherical triangle lrp, the 

 tangent of the angle rlp, (viz. of the inclination of the ecliptic and equator,) to the 

 tangent of the arc rp, i. e. - to --^, as radius 1 to pi the sine of the arc pl; 



hence is pi = '^-^--. Also in the same triangle it is Z> : 1 :: mt : rh the sine 



6 X PM *^ 



of the elliptic arc rl, that is, mt = /? X RH. Further, pn is as pm X mt by 

 lemma 1 ; therefore pn X pi and also Ln is as rh^; that is, the horary preces- 

 sion of the equinoxes, generated by the sun's force, is in the duplicate ratio of 

 the sine of the sun's distance from the equinox. And because the sum of all the 

 RH% in which time the sun completes his period, is half the sum of all the tr'^, 

 therefore the annual precession of the equinoxes is the half of that which the 

 sun, always remaining in the quadratures of the equinoxes, that is in the solstices, 

 could generate in the same time. 



Let therefore the sun be in the solstitial colure, and let lp and lr (fig. 4) be 

 quadrants of a circle, and Lr the measure of the angle Ljbw or p/)N ; hence, in 

 the triangle Lrw, lw or the horary precession of the equinoxes, is in this case to 

 Lr, or to the angle p/;w, as 1 to ^ : but the angle p/jn (drawing fjs perpendicular 

 to the radius tp,) is to double the angle ¥ps, that is to the angle pt/), which is 

 the eartli's horary motion about its axis, as the force which acts according to 

 PN, to the centrifugal force at the equator, that is, by lemma 1, as X -; 



3bc t^ 



(fig. 1) to I ; or, because in this case mt is = b, and sm = c, as — X - to 1 ; 

 and the horary motion of the earth about its axis, is to the horary motion of 

 the sun, as * to i : hence, compounding the ratios, the horary precession of the 



3c t 



equinoxes, is to the horary motion of the sun, as — X - to 1 , and in the same 



ratio is the annual precession to the annual motion of the sun. 



Therefore the annual precession of the equinoxes, on the hypothesis that the 



sun remains all that time immovable in the solstice, will be — X - X300'',and 

 the true annual precession will be the half of this. But as the sun acts not only 

 on the equatorial circle, as is supposed in this proposition, but on the whole 

 matter dispersed over the interior globe of the earth, and the globe itself must 

 participate of the motion generated by this force, therefore the precession is di- 

 minished in the ratio composed of the ratio of 2 to 5 by lemma 2, and of the 

 ratio of 5ab to 2ca by lemma 3 ; therefore the annual precession of the equi- 

 noxes arising from the solar force, at length produces 



^^ X ^ X ? X *- X 360°= ^' X '- X ^ X 360». 



4 « 5 2ac 4 « AC 



Let then the earth's greater diameter be to the less, as 230 to 229, and it 



