Let the straight line A B (in fig. 5) be understood to be moved with uniform motion to C D; and let another movent in the strait line A C be supposed to be moved in the same time to B D, but with motion uniformly accelerated, that is, with such motion, that the proportion of the spaces which are transmitted be always duplicate to that of the times, till the impetus acquired be B D equal to the strait line A C; and let the semiparabola A G D B be described. I say that by the concourse of those two movents, the body will be carried through the semiparabolical crooked line A G D. For let the parallelogram A B D C be completed; and from the point E, taken anywhere in the strait line A B, let E F be drawn parallel to A C and cutting the crooked line in G; and lastly, through the point G let H I be drawn parallel to the strait lines A B and C D. Seeing therefore the proportion of A B to A E is by supposition duplicate to the proportion of E F to E G, that is, of the time A C to the time A H, at the same time when A C is in E F, A B will be in H I; and therefore the moved body will be in the common point G. And so it will always be, in what part soever of A B the point E be taken. Wherefore the moved body will always be found in the parabolical line A G D; which was to be demonstrated. 10. If a body be carried by two movents together, which meet in any given angle, and are moved the one uniformly, the other with impetus increasing from rest, till it be equal to that of the uniform motion, and with such acceleration, that the proportion of the lengths transmitted be every where triplicate to that of the times in which they are transmitted; the line, in which that body is moved, will be the crooked line of the first semiparabolaster of two means, whose base is the impetus last acquired.