24 PHILOSOPHICAL TRANSACTIONS. [anNO 1755. 



wilY ioe — = r^; and, the inclination of the ecliptic being 23° 28' 30', the 

 nnual precession of the equinoxes produced by the solar force will be 10.583. 

 But if the ratio 178 to 177 be that of the earth's diameters, as it has been de- 

 rived from late observations, then will — = ,^r-„, and the annual precession of 



AC W8 ^ 



the equinox 13^.675. 



If the communication of motion between the interior globe of the earth and 

 the exterior matter be according to the Newtonian hypothesis, as explained in 

 the corol. to lemma 3, and the earth's greater diameter be to the less as 230 to 

 229, then the annual precession of the equinoxes by the sun's force will be 

 - X - X r^^f?-, X 360« = 9'M24 = 9" 7'" 20"-. And if the inclination 

 of the ecliptic be supposed 23^^°, that precession will become 9" 7'" 20"'^ as 

 Newton found it. q. e. i. 



CoROL. 1 . Put now, with Bradley, the whole mean annual precession equal 

 to 50".3; then from it deduct 10 '.583, and there will remain 39 ".7 17 for the 

 mean annual precession arising from the force of the moon, and then the lunar 

 force will be to the solar as 3.753 to 1, on the hypothesis that the ratio of the 

 earth's diameters is that of 230 to 229; but if the ratio be that of 178 to 177, 

 the earth being uniformly dense, from 50".3 taking 13".675, the annuaJ preces- 

 sion generated by the lunar force will be 36 ".625, and the lunar force to the solar 

 as 2.678 to 1. 



Corol 2. Take now in the ecliptic the arc r^, fig. 3, described by the sun in 

 a very small time, as an hour, and draw qh parallel to rh; then because of what 

 is said in the proposition, the horary precession of the equinoxes, the sun being 

 in any place r, is to the mean horary precession, as rh^ to -I-tr^, or, since rh : 

 TR :: uh : R9, as rh X hA to 4- tr X R^, the true precession will be to the mean 

 precession, while the sun describes the arc lr, as the space lrh to the sector 

 LTR, and their difference to the mean precession, as the triangle trh to the 

 sector ltr: therefore, lr being = 45°, that is, in the octants of the equinoxes 

 with the sun, this difference or equation, which then becomes the greatest, 

 writing d for the circumference of the circle whose radius is 1, is =: 

 19 •^^- or — rH— ; hence arises this theorem: " The motion of the sun is to the 



2d 2a 



motion of the equinoxes generated by the sun's force, as radius is to the sine of 

 double the greatest equation of the equinoxes." Whence in the former case the 

 greatest equation produces 51 ''', in the latter l"5"'. In other places, that equa 

 tion is to the greatest equation, as the sine of double the sun's distance from the 

 nearest equinox or solstice, to radius, as is evident: which is to be added to the 

 mean motion while the sun passes from the solstices to the equinoxes, but sub- 

 tracted while he passes from the equinoxes to the solstices. 



