Chapter 1i

Proposition 10

Euler
6 min read
Table of Contents

PROPOSITION 11. PROBLEM

  1. A body can move uniformly with absolute motion along the line AL (Fig. 8), and another body likewise moves uniformly too along the line AM. The size of the relative motion of the other body progressing along AM is sought in relation to the body progressing along AL.

SOLUTION

Let the speed of the body progressing along AL be a; and the speed of the other motion along AM be b : and these bodies set out at the same time from the point A. It is evident that, if the two distances AL and AM are taken in the ratio of the speeds a and b, [p. 31] both bodies arrive at the same time at L and M.

Therefore draw the lines ML with the line AM making the angle AML, the sine of which is to the sine of the angle ALM, that AL makes, as AL to AM, i. e. as a to b, L will designate the place at which the body progressing along AL at the same instant, at which the other will be present at M. Since truly the relative motion of any other body with respect to this body can be sought, this body actually moving along AM can be considered as being at rest at A.

Therefore with the point M known from the translation of A to arrive at L, through N draw AN parallel and equal to ML, from A. In a similar manner when the body moving on AM arrives at the nearby place m, the other is found at l, and ml is parallel to ML, since Mm : Ll = b : a = AM : AL.

With the point m in a similar manner translated from A by taking An = ml there comes about l in n, and n is on the same line AN. From these it follows that the body in absolute motion along AL is to be moving in relative motion along the line AN. Moreover the relative speed will be to the absolute speed as Nn to Ll, or as ML to AL.

Which ratio when constant on account of the kind of the given triangle ALM, a body in absolute motion along AL is in uniform relative motion on progressing along the line AN. For with the position of the line AN found by taking the angle LAN so large, in order that the sine of this to the sine of the angle NAM is as b to a. Hence the absolute speed along AL will be as the relative speed along AN as the sine of the angle MAN to the sine of the angle LAM. Q. E. I.

Corollary 1

  1. Therefore a body progressing in absolute uniform motion along a line will also be progressing uniformly along [another] line in relative motion, if the manner of the body, from which the relative motion is indicated, should be progressing uniformly along some line. And that is what we have assumed in the preceding demonstration (77).

Corollary 2

  1. Moreover the construction of the line AN and of the relative speed has been most easily found by being set up in this manner.

By assuming, as we have done, that AL and AM are in the ratio a to b and draw ML, then AN is drawn parallel to this line ML from A, and in this manner the relative motion will be described. Indeed the relative speed will be to the absolute speed as ML to AL

Corollary 3

  1. Likewise it prevails from the same reasoning , that if with AL not the absolute motion, but travels relative and has the same relation to AM . Then truly another relative speed of the body will be produced by the body traveling along AL with respect to the size of the speed of the body traveling along AM.

Corollary 4

  1. It is therefore apparent, how the absolute motion can be changed into relative motions in an endless number of ways, which will always be uniform and made in along straight lines, but only if the absolute motion and the movements of the bodies from which the relative motions arose, were of this kind.

Scholium

  1. In the solution we have taken both bodies setting out from the same point A : but the solution is just as successful, if both bodies in the beginning start out from different points that were put at A and B (Fig. 9).

For the body A progresses uniformly with an absolute motion along the right line AL, and indeed the other B similarly along the right line BM, thus so that the speeds are in the ratio a to b. AL and BM are taken in the same ratio a to b, and both bodies arrive at the same time at L and M.

But since the relative motion of the body A with respect to that of B is required, body B must be considered to be at rest at B. Therefore, for this reason, body B is moved from M to B, as body A arrives from L at N, by drawing BN parallel and equal to ML: I say that the point N is on the line passing through A, thus as the body A is moved relatively along the line AN, in a uniform motion. For draw NL and it will be equal and parallel to BM. With the appearance of triangle ANL from this construction : whereby NL to AL will have the given ratio ; hence, since NL = BM, the ratio AL to BM is the given ratio, which hence, if both were taken once in the ratio a to b, will always be in the same ratio. From which it is apparent that the point N lies on the line AN and the relative speed along AN is to the absolute speed along AL as AN to AL, i. e. in the given ratio. Therefore the relative motion is made along the right line AN and it is uniform.

Corollary 5

  1. If therefore the absolute motion of the body A (Fig. 9) is given along the right line AL, and the uniform relative motion of this is given along AN [p. 34] with whatever speed. then the motion of the body B can be found that arises with respect to the relative motion of the body A.

With the 2 distances AL and AN assumed that are traversed in the same given time, a line BM is drawn through any point B parallel to the line NL: and this will determine the path traversed by a given body B, and the speed of this will be to the absolute speed of the body A along AL, as NL is to AL. Truly the body B will be at the point B at the same time that body A is at A.

Corollary 6

  1. Therefore there are innumerable motions of the body B, since the point B could be assumed as we pleased, from which the same relative motion of the body A came about. But the speed of the body B was always the same and the direction of this followed the parallel line NL.

Corollary 7

  1. Also the uniform absolute motion stretching out along a line is able to be changed into any relative and likewise uniform motion made along a straight line. Indeed the line AN to be drawn can be chosen arbitrarily, and any speed assigned to that line. Indeed this same uniform motion is given to B, and the line of progression of the body B, from which here the relative motion arises.

Corollary 8

  1. In which case that relative motion alone without any external force will be able to continue. Indeed the absolute motions along AL and BM, since they have been made uniform along right lines, [p. 35] by themselves are to be continued. Indeed for as long as this motion lasts, so long also should the relative motion along AN also continue.

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