22 PHILOSOPHICAL TRANSACTIONS. [aNNO 1755. 



any point of the circumference is as its distance from that diameter, the motion 

 of the whole circumference will be expressed by 4cb- : hence, if instead of the 

 circumference there be substituted a very thin ring, the motion of the ring will 

 be to the motion of the globe whose semidiameter is cb, as 4cb^ to -^J^^^ x cb^; 

 that is, in the ratio composed of the ratio of the matter in the ring to the 

 matter in the globe, and of the ratio of double the square of the diameter to 

 triple the square of the quadrantal arc of a circle ; as Newton demonstrated. 

 And thus, if the less semidiameter of the earth be to the greater as 229 to 230, 

 and the whole matter diffused over the interior globe of the earth he conceived 

 to coalesce, as Newton supposes, into a uniform ring surrounding the equator, 

 the motion of the ring will be to that of the interior globe, as 459O to 485223; 

 and the notion of the ring to that of the whole earth, as 4590 to 489813. 



It may be observed here that this proportion of the motions, viz. which is 

 derived from the hypothesis, that the whole matter in the interior globe of the 

 earth coalesces into a superior ring about the equator, is a very little erroneous : 

 for it appears that the single particles of matter, in their own places, will not 

 take each the same motion from the rotation of the earth, as when they are sup- 

 posed to adhere together as in that hypothesis. But, in the investigation of the 

 mean precession of the equinoxes, Newton omits that difference of the motions, 

 as being very small, and of no consequence. But now, by the late discoveries 

 in astronomy, when we must search out more accurately the proportion of the 

 forces of the sun and moon, and of each of their effects, it seems necessary to 

 have regard to that difference, and therefore to this lemma we must annex the 

 following proposition. 



■ 'Prop. 1. Prob. To Investigate the Mean Precession of the Equinoxes gene- 

 i^ted by the Force of the Sun. Let spq (fig. 3) represent the earth's equator, 

 ARL the ecliptic, tl the line of the intersection of the planes of the equator and 

 ecliptic, and pm a perjjendicular from the point p of the equator on the plane 

 aT supposed perpendicular to the ecliptic. Taking any very small arc p/> of the 

 equator, let pn be double the space wliich a body can run through perpendi- 

 cular to the equator, impelled by the force defined in lemma 1st, in the time in 

 which the point p revolving with the equator describes the arc p/); bv which 

 means, after that particle of time the plane of the equator being changed into 

 the position ssjm, and now cutting the ecliptic in n, then the arc lw will be the 

 recess of the intersection of the equator and ecliptic, or the precession of the 

 equinoxes. On ^pn demit the perpendicular Lr, and on tl the perpendicular 

 pi; then since the lines pn, Lr are as the sines of the arcs p/;, pl, it will be p/): 

 PN :: PI : Lr; and writing b for the sine, and c for the cosine of the inclination 

 <rf the ecliptic and equator, to radius 1, in the rightangled triangle i.rn it will be 



P N X P I . 



b: 1 :: Lr : lw, therefore p& X b: pn :: pi : l??, and lw = -r : thcreforej the arc 



