For example, let A B (in the 8th figure) be a length transmitted in the time A C, with uniform motion; and let A H be another length transmitted in the same time with motion uniformly accelerated, so that the impetus last acquired be G H equal to A C; and in A H let any part A I be taken, and transmitted in part of the time A C with uniform motion; and let another part A B be taken and transmitted in the same part of the time A C with motion uniformly accelerated; I say, that as A H is to A B, so will A B be to A I. Let B D be drawn parallel and equal to H G, and divided in the midst at E, and between B D and B E let a mean proportional be taken as B F; and the strait line A G, by the demonstration of art. 13, shall pass through F. And dividing A H in the midst at I, A B shall be a mean proportional between A H and A I. Again, because A I and A B are described by the same motions, if I K be drawn parallel and equal to B F or A M, and divided in the midst at N, and between I K and I N be taken the mean proportional I L, the strait line A F will, by the demonstration of the same art. 13, pass through L. And dividing A B in the midst at O, the line A I will be a mean proportional between A B and A O. Where A B is divided in I and O, in like manner as A H is divided in B and I; and as A H to A B, so is A B to A I. Which was to be proved. Coroll. Also as A H to A B, so is H B to B I; and so also B I to I O. And as this, where one of the motions is uniformly accelerated, is proved out of the demonstration of art. 13; so, when the accelerations are in double proportion to the times, the same may be proved by the demonstration of art. 14; and by the same method in all other accelerations, whose proportions to the times are explicable in numbers. 18. If two sides, which contain an angle in any parallelogram, be moved in the same time to the sides opposite to them, one of them with uniform motion, the other with motion uniformly accelerated; that side, which is moved uniformly, will affect as much with its concourse through the whole length transmitted, as it would do if the other motion were also uniform, and the length transmitted by it in the same time were a mean proportional between the whole and the half.