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 uppoe EG to coincide with the perpendicular AB; and the velocity of the body in any place E will be as a right line whoe power is the curvilinear area $$\scriptstyle \frac 12ABGE$$. Q. E. I.

In EG take EM reciprocally proportional to a right line whoe power is the area $$\scriptstyle \frac 12ABGE$$, and let VLM be a curve line wherein the point M is always placed, and to which the right line AB produced is an aymptote, and the time in which the body is falling decribes 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 mall 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 uch, that a right line whoe power is the area $$\scriptstyle \frac 12ABGE$$, is as the velocity of the decending body, the area it elf will be as the quare 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 $$\scriptstyle \frac 12ABGE$$ as VV + 2VI + II; and by diviion 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 firt ratio's of thoe quantities when jut nacent, the length DF is as the quantity $$\textstyle \frac {2VI}{DE}$$ and therefore alo as half that quantity $$\textstyle \frac {I \times V}{DE}$$. But the time, in which the body in