Let A B (in the 8th figure) be a length, transmitted with uniform motion in the time A C; and let it be required to find another length, which shall be transmitted in the same time with motion uniformly accelerated, so that the line of the impetus last acquired be equal to the strait line A C. Let the parallelogram A B D C be completed; and let B D be divided in the middle at E; and between B E and B D let B F be a mean proportional; and let A F be drawn and produced till it meet with C D produced in G; and lastly, let the parallelogram A C G H be completed. I say, A H is the length required. For as duplicate proportion is to single proportion, so let A H be to A I, that is, let A I be the half of A H; and let I K be drawn parallel to the strait line A C, and cutting the diagonal A D in K, and the strait line A G in L. Seeing therefore A I is the half of A H, I L will also be the half of B D, that is, equal to B E; and I K equal to B F; for B D, that is, G H, B F, and B E, that is, I L, being continual proportionals, A H, A B and A I will also be continual proportionals. But as A B is to A I, that is, as A H is to A B, so is B D to I K, and so also is G H, that is, B D to B F; and therefore B F and I K are equal. Now the proportion of A H to A I is duplicate to the proportion of A B to A I, that is, to that of B D to I K, or of G H to I K. Wherefore the point K will be in a parabola, whose diameter is A H, and base G H, which G H is equal to A C. The body therefore proceeding from rest in A, with motion uniformly accelerated in the time A C, when it has passed through the length A H, will acquire the impetus G H equal to the time A C, that is, such impetus, as that with it the body will pass through the length A C in the time A C. Wherefore any length being given, &c., which was propounded to be done. 14. Any length being given, which in a given time is transmitted with uniform motion, to find out what length shall be transmitted in the same time with motion so accelerated, that the lengths transmitted be continually in triplicate proportion to that of their times, and the line of the impetus last of all acquired be equal to the line of time given. Let the given length A B (in the 9th figure) be transmitted with uniform motion in the time A C; and let it be required to find what length shall be transmitted in the same time with motion so accelerated, that the lengths transmitted be continually in triplicate proportion to that of their times, and the impetus last acquired be equal to the time given. Let the parallelogram A B D C be completed; and let B D be so divided in E, that B E be a third part of the whole B D; and let B F be a mean proportional between B D and B E; and let A F be drawn and produced till it meet the strait line C D in G; and lastly, let the parallelogram A C G H be completed. I say, A H is the length required.