Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/241

 falling decribes the ver mall line DE is as that line directly and the velocity V inverely, and the force will be as the increment I of the velocity directly and the time inverely, and therefore if we take the firt ratio's when thoe quantities are jut nacent as $$\textstyle \frac {I \times V}{DE}$$, that is as the length DF. Therefore a force proportional to DF or EG will caue the body to decend with a velocity that is as the right line whoe power is the area ABGE. Q. E. D.

Moreover ince the time, in which a very mall line DE of a given length may be decribed, is as the velocity inverely, and therefore alo inverely as a right line whoe quare is equal to the area ABFD; and ince the line DL, and by conequence the nacent area DLME, will be as the ame right line inverely: the time will be as the area DLME, and the um of all the times will be as the um of all the area's; that is (by cor. lem. 4.) the whole time in which the line AE is decribed will be as the whole area ATVME. Q. E. D,

Let P be the place from whence a body ought to fall, o as that when urged by any known uniform centripetal force (uch as gravity is vulgarly uppoed to be) it may acquire in the place D a velocity, equal to the velocity which another body, falling by any force whatever, hath acquired in that place D. In the perpendicular DF let there be taken DR, which may be to DF as that uniform force to the other force in the place D. Compleat the rectangle PDRQ, and cut off the area ABFD equal