If a body be carried on with uniform motion by two movents together, which meet in an angle, the line by which it passes will be a strait line, subtending the complement of that angle to 2 right angles.
8. If a body be carried by two movents together, which move with strait and uniform motion, and concur in any given angle, the line by which that body passes will be a strait line.
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.
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.
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.
Let the strait line A B (in the [6th figure]) be moved uniformly to C D; and let another movent A C be moved at the same time to B D with motion so accelerated, that the proportion of the lengths transmitted be everywhere triplicate to the proportion of their times; and let the impetus acquired in the end of that motion be B D, equal to the strait line A C; and lastly, let A G D be the crooked line of the first semiparabolaster of two means. I say, that by the concourse of the two movents together, the body will be always in that 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 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, triplicate 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; and by consequent, the body will always be in the crooked line A G D; which was to be demonstrated.
11. By the same method it may be shown, what line it is that is made by the motion of a body carried by the concourse of any two movents, which are moved one of them uniformly, the other with acceleration, but in such proportions of spaces and times as are explicable by numbers, as duplicate, triplicate, &c., or such as may be designed by any broken number whatsoever. For which this is the rule. Let the two numbers of the length and time be added together; and let their sum be the denominator of a fraction, whose numerator must be the number of the length. Seek this fraction in the table of the third [article] of the XVIIth chapter; and the line sought will be that, which denominates the three-sided figure noted on the left hand; and the kind of it will be that, which is numbered above over the fraction. For example, let there be a concourse of two movents, whereof one is moved uniformly, the other with motion so accelerated, that the spaces are to the times as 5 to 3. Let a fraction be made whose denominator is the sum of 5 and 3, and the numerator 5, namely the fraction 5⁄8. Seek in the table, and you will find 5⁄8 to be the third in that row, which belongs to the three-sided figure of four means. Wherefore the line of motion made by the concourse of two such movents, as are last of all described, will be the crooked line of the third parabolaster of four means.