Page:Newton's Principia (1846).djvu/290

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284
[Book II.
the mathematical principles

II of this Book) the moment KL of AK will be equal to or , and the moment KLON of the area AbNK will be equal to or .

Case 1. Now if the body ascends, and the gravity be as AB² + BD², BET being a circle, the line AC, which is proportional to the gravity, will be , and DP² or AP² + 2BAP + AB² + BD² will be AK Z + AC Z or CK Z; and therefore the area DTV will be to the area DPQ as DT² or DB² to CK Z.

Case 2. If the body ascends, and the gravity be as AB² - BD², the line AC will be , and DT² will be to DP² as DF² or DB² to BP² - BD² or AP² + 2BAP + AB² - BD², that is, to AK Z +

Principia1846-283.png

AC Z or CK Z. And therefore the area DTV will be to the area DPQ as DB² to CK 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 ; the area DTV will be to the area DPQ, as DB² to CK Z: as above.

Since, therefore, these areas are always in this ratio, if for the area