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

 the centre S, at the interval SC, in the ubduplicate ratio of AC to AO or SK (by prop. 33.) and this velocity is to the velocity of a body decribing the circle OKk in the ubduplicate ratio of SK to SC (by cor. 6. prop. 4.) and ex æquo, the firt velocity to the lat, that is the little line Cc to the arc Kk, in the ubduplicate 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 conequently AC to SK as AC x Kk to ST x Dd, and thence SK x KL equal to ST x Dd. and $$\scriptstyle \frac 12SK \times Kk$$ equal to $$\scriptstyle \frac 12SY \times Dd$$, 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 diminihed and their number increaed in infinitum, obtain the ratio of equality, and conenquently (by cor. lem. 4.) the whole areas together generated are always equal. Q. E. D.

But if the figure DES (Fig. 2.) is a parabola, we hall find as above CD x Cc to ST x Dd as TC to TS, that is, as 2 to 1; and that therefore $$\scriptstyle \frac 14CD \times Cc$$ is equal to $$\scriptstyle \frac 12ST \times Dd$$. But the velocity of the falling body in C is equal to the velocity with which a circle may be uniformly decribed at the interval $$\scriptstyle \frac 12SC$$, (by prop. 34.) And this velocity to the velocity with which a circle may be decribed with the radius SK, that is, the little line Cc to the arc Kk is (by cor. 5. prop. 4.) in the ubduplicate ratio of SK to $$\scriptstyle \frac 12CD$$; that is, in the ratio of SK to $$\scriptstyle \frac 12CD$$. Wherefore $$\scriptstyle \frac 12SK \times Kk$$ is equal to $$\scriptstyle \frac 14CD \times Cc$$, and therefore equal to $$\scriptstyle \frac 12SY \times Dd$$; that is, the area KSk is equal to the area SDd as above. Q. E. D.