366 THE MATHEMATICAL PRINCIPLES [BOOK 1 a right line whose square is equal to the area 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 in 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 square is equal to the area ABGE; is as the velocity of the descend ing body, the area itself 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 VY, and the area ABGE as YY + 2VI -f II; and by division, the area DFGE as 2 VI -f LI, and therefore ^ will be as -- ^r that is. if we take the first ratios of those quantities when just nascent, the 2YI length DF is as the quantity -