Suppose now the figure B E D C to be described by the increasing of the point B to the magnitude C D. Seeing therefore the proportion of B C to B F is triplicate to that of C D to F E, the proportion of F E to C D will, by conversion, as I shall presently demonstrate, be triplicate to that B F to B C. Wherefore if the strait line B C be taken for the measure of the time in which the point B is moved, the figure E K B F will represent the sum of all the increasing velocities in the time B F; and the figure D E B C will in like manner represent the sum of all the increasing velocities in the time B C. Seeing therefore the proportion of the figure E K B F to the figure D E B C is compounded of the proportions of altitude to altitude, and base to base; and seeing the proportion of F E to C D is triplicate to that of B F to B C; the proportion of the figure E K B F to the figure D E B C will be quadruplicate to that of B F to B C; that is, the proportion of the sum of the velocities in the time B F, to the sum of the velocities in the time B C, will be quadruplicate to the proportion of B F to B C. Wherefore if a body be moved from B with velocity so increasing, that the velocity acquired in the time B F be to the velocity acquired in the time B C in triplicate proportion to that of the times themselves B F to B C, and the body be carried to F in the time B F; the same body in the time B C will be carried through a line equal to the fifth proportional from B F in the continual proportion of B F to B C. And by the same manner of working, we may determine what spaces are transmitted by velocities increasing according to any other proportions. It remains that I demonstrate the proportion of F E to C D to be triplicate to that of B F to B C. Seeing therefore the proportion of C D, that is, of F G to F E is subtriplicate to that of B C to B F; the proportion of F G to F E will also be subtriplicate to that of F G to F H. Wherefore the proportion of F G to F H is triplicate to that of F G, that is, of C D to F E. But in four continual proportionals, of which the least is the first, the proportion of the first to the fourth, (by the 16th article of chapter XIII.), is subtriplicate to the proportion of the third to the same fourth. Wherefore the proportion of F H to G F is subtriplicate to that of F E to C D; and therefore the proportion of F E to C D is triplicate to that of F H to F G, that is, of B F to B C; which was to be proved.