For as triplicate proportion is to single proportion, so let A H be to another line, A I, that is, make A I a third part of the whole A H; and let I K be drawn parallel to the strait line A C, cutting the diagonal A D in K, and the strait line A G in L; then, as A B is to A I, so let A I be to another, A N; and from the point N let N Q be drawn parallel to A C, cutting A G, A D, and F K produced in P, M, and O; and last of all, let F O and L M be drawn, which will be equal and parallel to the strait lines B N and I N. By this construction, the lengths transmitted A H, A B, A I, and A N, will be continual proportionals; and, in like manner, the times G H, B F, I L and N P, that is, N Q, N O, N M and N P, will be continual proportionals, and in the same proportion with A H, A B, A I and A N. Wherefore the proportion of A H, A B, A I and A N. Wherefore the proportion of A H to A N is the same with that of B D, that is, of N Q to N P; and the proportion of N Q to N P triplicate to that of N Q to N O, that is, triplicate to that of B D to I K; wherefore also the length A H is to the length A N in triplicate proportion to that of the time B D, to the time I K; and therefore the crooked line of the first three-sided figure of two means whose diameter is A H, and base G H equal to A C, shall pass through the point O; and consequently, A H shall be transmitted in the time A C, and shall have its last acquired impetus G H equal to A C, and the proportions of the lengths acquired in any of the times triplicate to the proportions of the times themselves. Wherefore A H is the length required to be found out. By the same method, if a length be given which is transmitted with uniform motion in any given time, another length may be found out which shall be transmitted in the same time with motion so accelerated, that the lengths transmitted shall be to the times in which they are transmitted, in proportion quadruplicate, quintuplicate, and so on infinitely. For if B D be divided in E, so that B D be to B E as 4 to 1; and there be taken between B D and B E a mean proportional F B; and as A H is to A B, so A B be made to a third, and again so that third to a fourth, and that fourth to a fifth, A N, so that the proportion of A H to A N be quadruplicate to that of A H to A B, and the parallelogram N B F O be completed, the crooked line of the first three-sided figure of three means will pass through the point O; and consequently, the body moved will acquire the impetus G H equal to A C in the time A C. And so of the rest. 15. Also, if the proportion of the lengths transmitted be to that of their times, as any number to any number, the same method serves for the finding out of the length transmitted with such impetus, and in such time.