Let the side A B of the parallelogram A B D C, (in the 11th figure) be understood to be moved with uniform motion till it be coincident with C D; and let the time of that motion be A C or B D. Also in the same time let the side A C be understood to be moved with motion uniformly accelerated, till it be coincident with B D; then dividing A B in the middle in E, let A F be made a mean proportional between A B and A E; and drawing F G parallel to A C, let the side A C be understood to be moved in the same time A C with uniform motion till it be coincident with F G. I say, the whole A B confers as much to the velocity of the body placed in A, when the motion of A C is uniformly accelerated till it comes to B D, as the part A F confers to the same, when the side A C is moved uniformly and in the same time to F G. For seeing A F is a mean proportional between the whole A B and its half A E, B D will (by the 13th article) be the last impetus acquired by A C, with motion uniformly accelerated till it come to the same B D; and consequently, the strait line F B will be the excess, by which the length, transmitted by A C with motion uniformly accelerated, will exceed the length transmitted by the same A C in the same time with uniform motion, and with impetus every where equal to B D. Wherefore, if the whole A B be moved uniformly to C D in the same time in which A C is moved uniformly to F G, the part F B, seeing it concurs not at all with the motion of the side A C which is supposed to be moved only to F G, will confer nothing to its motion. Again, supposing the side A C to be moved to B D with motion uniformly accelerated, the side A B with its uniform motion to C D will less put forwards the body when it is accelerated in all the parallels, than when it is not at all accelerated; and by how much the greater the acceleration is, by so much the less it will put it forwards, as is shown in the 12th article. When therefore A C is in F G with accelerated motion, the body will not be in the side C D at the point G, but at the point D; so that G D will be the excess, by which the length transmitted with accelerated motion to B D exceeds the length transmitted with uniform motion to F G; so that the body by its acceleration avoids the action of the part A F, and comes to the side C D in the time A C, and makes the length C D, which is equal to the length A B. Wherefore uniform motion from A B to C D in the time A C, works no more in the whole length A B upon the body uniformly accelerated from A C to B D, than if A C were moved in the same time with uniform motion to F G; the difference consisting only in this, that when A B works upon the body uniformly moved from A C to F G, that, by which the accelerated motion exceeds the uniform motion, is altogether in F B or G D; but when the same A B works upon the body accelerated, that, by which the accelerated motion exceeds the uniform motion, is