Let A B C be the crooked line of the first semiparabolaster; A D the diameter; D C the base; and let the parallelogram completed be A D C E, whose diagonal is A C. Divide the diameter into two equal parts in F, and draw F H equal and parallel to D C, cutting A C in K, the crooked line in O, and E C in H. Then draw O L parallel to E C, cutting A C in L; and draw L N parallel to the base D C, cutting the crooked line in M, and the strait line E C in N; and produce it on the other side to A D in I. Lastly, through the point M draw P M Q parallel and equal to H C, cutting F H in P; and join C P, A P and A O. I say, the two strait lines A P and P C are equal to the crooked line A B O C. For the line A B O C, being the crooked line of the first semiparabolaster, is generated by the concourse of two motions, one uniform from A to E, the other in the same time accelerated from rest in A to D, so as that the impetus increaseth in proportion perpetually triplicate to that of the increase of the time, or which is all one, the lengths transmitted are in proportion triplicate to that of the times of their transmission; for as the impetus or quicknesses increase, so the lengths transmitted increase also. And because the motion from A to E is uniform, the line A E may serve to represent the time, and consequently the lines, ordinately drawn in the semiparabolaster, will design the parts of time wherein the body, beginning from rest in A, describeth by its motion the crooked line A B O C. And because D C, which represents the greatest acquired impetus, is equal to A E, the same ordinate lines will represent the several augmentations of the impetus increasing from rest in A. Therefore, supposing uniform motion from A to F, in the time F K there will be described, by the concourse of the two uniform motions A F and F K, the line A K uniformly, and K O will be the increase of impetus in the time F K; and by the concourse of the two uniform motions in A F and F O will be described the line A O uniformly. Through the point L draw the strait line L M N parallel to D C, cutting the strait line A D in I, the crooked line A B C in M, and the strait line E C in N; and through the point M the strait line P M Q parallel and equal to H C, cutting D C in Q and F H in P. By the concourse therefore of the two uniform motions in A F and F P in the time F P will be uniformly described the strait line A P; and L M or O P will be the increase of impetus to be added for the time F O. And because the proportion of I N to I L is triplicate to the proportion of I N to I M, the proportion of F H to F O will also be triplicate to the proportion of F H to F P; and the proportional impetus gained in the time F P is P H. So that F H being equal to D C, which designed the whole impetus acquired by the acceleration, there is no more increase of impetus to be computed. Now in the time P H suppose an uniform motion from H to C; and by the two