Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/234

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the centre S, at the interval SC, in the ſubduplicate ratio of AC to AO or SK (by prop. 33.) and this velocity is to the velocity of a body deſcribing the circle OKk in the ſubduplicate ratio of SK to SC (by cor. 6. prop. 4.) and ex æquo, the firſt velocity to the laſt, that is the little line Cc to the arc Kk, 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 Kk, and conſequently AC to SK as AC x Kk to ST x Dd, and thence SK x KL equal to ST x Dd. and equal to , that is, the area KSk equal to the area SDd. Therefore in every moment of time two equal particles, KSk and SDd, of areas are generated which, if their magnitude is diminiſhed and their number increaſed in infinitum, obtain the ratio of equality, and conſenquently (by cor. lem. 4.) the whole areas together generated are always equal. Q. E. D.

Plate 16, Figure 2
Plate 16, Figure 2

Case 2. But if the figure DES (Fig. 2.) is a parabola, we ſhall find as above CD x Cc to ST x Dd as TC to TS, that is, as 2 to 1; and that therefore is equal to . But the velocity of the falling body in C is equal to the velocity with which a circle may be uniformly deſcribed at the interval , (by prop. 34.) And this velocity to the velocity with which a circle may be deſcribed with the radius SK, that is, the little line Cc to the arc Kk is (by cor. 5. prop. 4.) in the ſubduplicate ratio of SK to ; that is, in the ratio of SK to . Wherefore is equal to , and therefore equal to ; that is, the area KSk is equal to the area SDd as above. Q. E. D.