Let the movent A B (in fig. 4) have strait and uniform motion, and be moved till it come into the place C D; and let another movent A C, having likewise strait and uniform motion, and making with the movent A B any given angle C A B, be understood to be moved in the same time to D B; and let the body be placed in the point of their concourse, A. I say the line which that body describes with its motion is a strait line. For let the parallelogram A B D C be completed, and its diagonal A D be drawn; and in the strait line A B let any point E be taken; and from it let E F be drawn parallel to the strait lines A C and B D, cutting A D in G; and through the point G let H I be drawn parallel to the strait lines A B and C D; and lastly, let the measure of the time be A C. Seeing therefore both the motions are made in the same time, when A B is in C D, the body also will be in C D; and in like manner, when A C is in B D, the body will be in B D. But A B is in C D at the same time when A C is in B D; and therefore the body will be in C D and B D at the same time; wherefore it will be in the common point D. Again, seeing the motion from A C to B D is uniform, that is, the spaces transmitted by it are in proportion to one another as the times in which they are transmitted, when A C is in E F, the proportion of A B to A E will be the same with that of E F to E G, that is, of the time A C to the time A H. Wherefore A B will be in H I in the same time in which A C is in E F, so that the body will at the same time be in E F and H I, and therefore in their common point G. And in the same manner it will be, wheresoever the point E be taken between A and B. Wherefore the body will always be in the diagonal A D; which was to be demonstrated. Coroll. From hence it is manifest, that the body will be carried through the same strait line A D, though the motion be not uniform, provided it have like acceleration; for the proportion of A B to A E will always be the same with that of A C to A H. [Sidenote: If a body be carried by two movents together, one of them being moved with uniform, the other with accelerated motion, and the proportion of their lengths to their times being explicable in numbers, how to find out what line that body describes.] 9. If a body be carried by two movents together, which meet in any given angle, and are moved, the one uniformly, the other with motion uniformly accelerated from rest, that is, that the proportion of their impetus be as that of their times, that is, that the proportion of their lengths be duplicate to that of the lines of their times, till the line of greatest impetus acquired by acceleration be equal to that of the line of time of the uniform motion; the line in which the body is carried will be the crooked line of a semiparabola, whose base is the impetus last acquired, and vertex the point of rest.