284 THE MATHEMATICAL PRINCIPLES [BOOK II 2APQ, + 2B A X PU II of this Book) the moment KL of AK will be equal to 2BPQ or Z -, and the moment KLON of the area A£NK will be equal to 2BPQ X LO BPQ, X BD3 ~~Z~~ >r 2Z X OK x~AB" CASE 1. Now if the body ascends, and the gravity be as AB2 + BD9 BET being a circle, the line AC, which is proportional to the gravity AW2 i RF)2 will be - ~ -- , and DP2 or AP2 + 2BAP + AB2 + BD2 will be AK XZ + AC X Z or CK X Z : and therefore the area DTV will be to the area DPQ, as DT2 or I)B2 to CK X Z. CASE 2. If the body ascends, and the gravity be as AB2 — BD2, the A r>2 _ HI)2 line AC will be " -- ^ --- , and DT2 will be to DP2 as DF2 or DB2 Z to BP2— BD2 or AP2 + 2BAP + AB2 — BD2, that is, to AK X Z + H AC X Z or CK X Z. And therefore the area DTV will be to the area DPQ as DB2 to CK X Z. CASE 3. And by the same reasoning, if the body descends, and therefore the gravity is as BD2 — AB2, and the line AC becomes equal to or)2 AB2 5T— r ; the area DTV will be to the area DPQ as DB2 to CK X Z Z : as above. Since, therefore, these areas are always in this ratio, if for the area