Coroll. I. In motion uniformly accelerated, the proportion of the lengths transmitted to that of their times, is compounded of the proportions of their times to their times, and impetus to impetus.
Coroll. II. In motion uniformly accelerated, the lengths transmitted in equal times, taken in continual succession from the beginning of motion, are as the differences of square numbers beginning from unity, namely, as 3, 5, 7, &c. For if in the first time the length transmitted be as 1, in the first and second times the length transmitted will be as 4, which is the square of 2, and in the three first times it will be as 9, which is the square of 3, and in the four first times as 16, and so on. Now the differences of these squares are 3, 5, 7, &c.
Coroll. III. In motion uniformly accelerated from rest, the length transmitted is to another length transmitted uniformly in the same time, but with such impetus as was acquired by the accelerated motion in the last point of that time, as a triangle to a parallelogram, which have their altitude and base common. For seeing the length D E (in [fig. 1]) is passed through with velocity as the triangle A B I, it is necessary that for the passing through of a length which is double to D E, the velocity be as the parallelogram A I; for the parallelogram A I is double to the triangle A B I.
4. In motion, which beginning from rest is so aclerated, that the impetus thereof increases continually in proportion duplicate to the proportion of the times in which it is made, a length transmitted in one time will be to a length transmitted in another time, as the product made by the mean impetus multiplied into the time of one of those motions, to the product of the mean impetus multiplied into the time of the other motion.
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.
Coroll. I. Lengths transmitted with motion so accelerated, that the impetus increase continually in duplicate proportion to that of their times, if the base represent the impetus, are in triplicate proportion of their impetus acquired in the last point of their times. For as the length D E is to the length D P, so is the parallelogram A M to the parallelogram A N, and so the parabola A K I B to the parabola A K F. But the proportion of the parabola A K I B to the parabola A K F is triplicate to the proportion which the base B I has to the base F K. Wherefore also the proportion of D E to D P is triplicate to that of B I to F K.
Coroll. II. Lengths transmitted in equal times succeeding one another from the beginning, by motion so accelerated, that the proportion of the impetus be duplicate to the proportion of the times, are to one another as the differences of cubic numbers beginning at unity, that is as 7, 19, 37, &c. For if in the first time the length transmitted be as 1, the length at the end of the second time will be as 8, at the end of the third time as 27, and at the end of the fourth time as 64, &c.; which are cubic numbers, whose differences are 7, 19, 37, &c.
Coroll. III. In motion so accelerated, as that the length transmitted be always to the length transmitted in duplicate proportion to their times, the length uniformly transmitted in the whole time, and with impetus all the way equal to that which is last acquired, is as a parabola to a parallelogram of the same altitude and base, that is, as 2 to 3. For the parabola A K I B is the impetus increasing in the time A B; and the parallelogram A I is the greatest uniform impetus multiplied into the same time A B. Wherefore the lengths transmitted will be as a parabola to a parallelogram, &c., that is, as 2 to 3.
5. If I should proceed to the explication of such motions as are made by impetus increasing in proportion triplicate, quadruplicate, quintuplicate, &c., to that of their times, it would be a labour infinite and unnecessary. For by the same method by which I have computed such lengths, as are transmitted with impetus increasing in single and duplicate proportion, any man may compute such as are transmitted with impetus increasing in triplicate, quadruplicate, or what other proportion he pleases.
In making which computation he shall find, that where the impetus increase in proportion triplicate to that of the times, there the whole velocity will be designed by the first parabolaster (of which see the next chapter); and the lengths transmitted will be in proportion quadruplicate to that of the times. And in like manner, where the impetus increase in quadruplicate proportion to that of the times, that there the whole velocity will be designed by the second parabolaster, and the lengths transmitted will be in quintuplicate proportion to that of the times; and so on continually.