Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/240

always erected, proportional to the centripetal force in that place tending to the centre C; and let BFG be a curve line, the locus of the point G. And in the beginning of the motion ſuppoſe EG to coincide with the perpendicular AB; and the velocity of the body in any place E will be as a right line whoſe power is the curvilinear area $\scriptstyle {\frac {1}{2}}ABGE$. Q. E. I.

In EG take EM reciprocally proportional to a right line whoſe power is the area $\scriptstyle {\frac {1}{2}}ABGE$, and let VLM be a curve line wherein the point M is always placed, and to which the right line AB produced is an aſymptote, and the time in which the body is falling deſcribes the line AE, will be as the curvilinear area ABTVME. Q. E. I.

For in the right line AE let there be taken the very ſmall line DE of a given length, and let DLF be the place of the line EMG, when the body was in D; and if the centripetal force be ſuch, that a right line whoſe power is the area $\scriptstyle {\frac {1}{2}}ABGE$, is as the velocity of the deſcending body, the area it ſelf will be as the ſquare of that velocity; that is, if for the velocities in D and E we write V and V + I, the area ABFD will be as VV, and the area $\scriptstyle {\frac {1}{2}}ABGE$ as VV + 2VI + II; and by diviſion the area DFGE as 2VI + II and therefore $\textstyle {\frac {DFGE}{DE}}$ will be as $\textstyle {\frac {2VI+II}{DE}}$, that is, if we take the firſt ratio's of thoſe quantities when juſt naſcent, the length DF is as the quantity $\textstyle {\frac {2VI}{DE}}$ and therefore alſo as half that quantity $\textstyle {\frac {I\times V}{DE}}$. But the time, in which the body in