For the line A B O C being a parabolical line, is generated by the concourse of two motions, one uniform from A to E, the other in the same time uniformly accelerated from rest in A to D. And because the motion from A to E is uniform, A E may represent the times of both those motions from the beginning to the end. Let therefore A E be the time; and consequently the lines ordinately applied in the semiparabola will design the parts of time wherein the body, that describeth the line A B O C, is in every point of the same; so that as at the end of the time A E or D C it is in C, so at the end of the time F O it will be in O. And because the velocity in A D is increased uniformly, that is, in the same proportion with the time, the same lines ordinately applied in the semiparabola will design also the continual augmentation of the impetus, till it be at the greatest, designed by the base D C. Therefore supposing uniform motion in the line A F, in the time F K the body in A by the concourse of the two uniform motions in A F and F K will be moved uniformly in the line A K; and K O will be the increase of the impetus or swiftness gained in the time F K; and the line A O will be uniformly described by the concourse of the two uniform motions in A F and F O in the time F O. From O draw O L parallel to E C, cutting A C in L; and draw L N parallel to D C, cutting E C in N, and the parabolical line in M; and produce it on the other side to A D in I; and I N, I M and I L will be, by the construction of a parabola, in continual proportion, and equal to the three lines F H, F P and F O; and a strait line parallel to E C passing through M will fall on P; and therefore O P will be the increase of impetus gained in the time F O or I L. Lastly, produce P M to C D in Q; and Q C or M N or P H will be the increase of impetus proportional to the time F P or I M or D Q. Suppose now uniform motion from H to C in the time P H. Seeing therefore in the time F P with uniform motion and the impetus increased in proportion to the times, is described the straight line A P; and in the rest of the time and impetus, namely, P H, is described the line C P uniformly; it followeth that the whole line A P C is described with the whole impetus, and in the same time wherewith is described the parabolical line A B C; and therefore the line A P C, made of the two strait lines A P and P C, is equal to the parabolical line A B C; which was to be proved. [Sidenote: To find a strait line equal to the crooked line of the first semiparabolaster or to the crooked line of any other of the deficient figures of the table of art. 3 of the preceding chapter.] 2. To find a strait line equal to the crooked line of the first semiparabolaster. [Sidenote: [Discussion of Figure 18.2]]