And as this, where one of the motions is uniformly accelerated, is proved out of the demonstration of [art. 13]; so, when the accelerations are in double proportion to the times, the same may be proved by the demonstration of [art. 14]; and by the same method in all other accelerations, whose proportions to the times are explicable in numbers.

18. If two sides, which contain an angle in any parallelogram, be moved in the same time to the sides opposite to them, one of them with uniform motion, the other with motion uniformly accelerated; that side, which is moved uniformly, will affect as much with its concourse through the whole length transmitted, as it would do if the other motion were also uniform, and the length transmitted by it in the same time were a mean proportional between the whole and the half.

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 dispersed through the whole length A B or C D, yet, so that if it were collected and put together, it would be equal to the same F B or G D. Wherefore, if two sides which contain an angle, &c.; which was to be demonstrated.

19. If two transmitted lengths have to their times any other proportion explicable by number, and the side A B be so divided in E, that A B be to A E in the same proportion which the lengths transmitted have to the times in which they are transmitted, and between A B and A E there be taken a mean proportional A F; it may be shown by the same method, that the side, which is moved with uniform motion, works as much with its concourse through the whole length A B, as it would do if the other motion were also uniform, and the length transmitted in the same time A C were that mean proportional A F.

And thus much concerning motion by concourse.

Vol. 1. Lat. & Eng.
C. XVI.
Fig. 1-11

[Fig 1.][Fig 2.][Fig 3.][Fig 4.][Fig 5.][Fig 6.][Fig 7.][Fig 8.][Fig 9.][Fig 10.][Fig 11.]