For let A B (in fig. 2) represent a time, in whose first instant A let the impetus be as the point A; but as the time proceeds, so let the impetus increase continually in duplicate proportion to that of the times, till in the last point of time B the impetus acquired be B I; then taking the point F anywhere in the time A B, let the impetus F K acquired in the time A F be ordinately applied to that point F. Seeing therefore the proportion of F K to B I is supposed to be duplicate to that of A F to A B, the proportion of A F to A B will be subduplicate to that of F K to B I; and that of A B to A F will be (by chap. XIII. art. 16) duplicate to that of B I to F K; and consequently the point K will be in a parabolical line, whose diameter is A B and base B I; and for the same reason, to what point soever of the time A B the impetus acquired in that time be ordinately applied, the strait line designing that impetus will be in the same parabolical line A K I. Wherefore the mean impetus multiplied into the whole time A B will be the parabola A K I B, equal to the parallelogram A M, which parallelogram has for one side the line of time A B and for the other the line of the impetus A L, which is two-thirds of the impetus B I; for every parabola is equal to two-thirds of that parallelogram with which it has its altitude and base common. Wherefore the whole velocity in A B will be the parallelogram A M, as being made by the multiplication of the impetus A L into the time A B. And in like manner, if F N be taken, which is two-thirds of the impetus F K, and the parallelogram F O be completed, F O will be the whole velocity in the time A F, as being made by the uniform impetus A O or F N multiplied into the time A F. Let now the length transmitted in the time A B and with the velocity A M be the strait line D E; and lastly, let the length transmitted in the time A F with the velocity A N be D P; I say that as A M is to A N, or as the parabola A K I B to the parabola A K F, so is D E to D P. For as A M is to F L, that is, as A B is to A F, so let D E be to D G. Now the proportion of A M to A N is compounded of the proportions of A M to F L, and of F L to A N. But as A M to F L, so by construction is D E to D G; and as F L is to A N (seeing the time in both is the same, namely, A F), so is the length D G to the length D P; for lengths transmitted in the same time are to one another as their velocities are. Wherefore by ordinate proportion, as A M is to A N, that is, as the mean impetus A L multiplied into its time A B, is to the mean impetus A O multiplied into A F, so is D E to D P; which was to be proved.