SEC. ni: OF NATURAL PHILOSOPHY. For suppose a body C in the smallest moment of time describes in fal ling the infinitely little line Cc. while another body K, uniformly revolv ing about the centre S in the circle OK/r, describes the arc KA:. Erect the perpendiculars CD, cd, meeting the figure DES in D, d. Join SD, Sf/. SK. SA*; and draw Del meeting the axis AS in T, and thereon let fall the perpendicular SY. CASE 1. If the figure DES is a circle, or a rectangular hyperbola, bisect its transverse diameter AS in O, and SO will be half the latus rectum. And because TC is to TD as Cc to Dd, and TD to TS as CD to S Y ; ex aquo TC will be to TS as CD X Cc to SY X Dd. But (by Cor. 1, Prop. XXXIII) TC is to TS as AC to AO; to wit, if in the coalescence of the points D, d, the ultimate ratios of the lines are taken. Wherefore AC is to AO or SK as CD X Cc to S Y X Vd, Farther, the velocity of the descending body in C IF, to the velocity of a body describing a circle about the centre S, at the interval SC, in the subduplicate ratio of AC to AO or SK (by Pi-op. XXXIII) ; and this velocity is to the velocity of a body describing the circle OKA: in the subduplicate ratio of SK to SC (by Cor. 6, Prop IV) ; and, ex aqnnj the first velocity to the last, that is, the little line Cc to the arc K/r, in the subduplicate ratio of AC to SC, that is, in the ratio of AC to CD. Wherefore CD X Cc is equal to AC X KA*, and consequently AC to SK as AC X KA: to SY X IW. and thence SK X KA: equal to SY X Drf, and iSK X KA: equal to £SY X DC/, that is, the area KSA* equal to the area SDrf. Therefore in every moment of time two equal particles, KSA" and SDrf, of areas are generated, which, if their magnitude is diminished, and their number increased in iiifinif't-w, obtain the ratio cf equality, and consequently (by Cor. Lem. IV), the whole areas together generated are always equal. Q..E.D. CASE 2. But if the figure DES is a parabola, we shall find, as above. CD X Cc to SY X Df/ as TC to TS, that is, as 2 to 1 ; and that therefore