Chapter 1i

Proposition 12

Euler
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PROPOSITION 12. PROBLEM

  1. The body A (Fig. 10) moves in some manner along the line AL with absolute motion. The body B moves along the line BM.

The relative motion of body A is required with respect to body B.

SOLUTION

The arcs AL and BM are cut from the curves, which are traversed in equal times. Therefore the body A will be found at L, when B reaches M.

But since the relative motion of the body A with respect to B is desired, the body B is taken as at rest.

Whereby that known motion is transferred at B from M along the line MB, and the body L reaches N, drawn parallel to LN and equal to MB.

Therefore the curve, on which the point N is found in this way, will be the path for the body A with the relative motion relative described.

By this relative motion the arc AN is traversed in the same time, by which the arcs AL and BM are completed. From which the relative speed at N is also to become known. Q. E. I.

Corollary 1

  1. Therefore in this manner the relative motion of any body can be determined, with respect to the extent of the motion of some other moving body. [p. 36]

Corollary 2

  1. It is also understood from the solution, how from the given curves AN and AL together with the motions along these, it is possible to find a curve BM and the motion along that curve. And the curve AL is defined from the curves BM and AN.

Corollary 3

  1. The curve BM , on account of the arbitrary choice of the point B, can be taken in an endless number of other positions. However since the arc BM, which is described in the same time as AL and AN, always subtends BM equal and parallel to the line LN, which will itself always be similar and equal and parallel to itself, and the motion along that line will always be the same as along BM. Scholium.

  2. These are the motions, which are concerned with the comparison of absolute and relative motion that I have judged worthy of reporting on.

Moreover there is another way in which it is customary to described relative motion, as the motion along AN is called the motion of the body A, which actually is moving along the line AL, and with such for the motion observed from the body B moving along BM. Truly this motion will be observed with the body B placed in relative rest and the point B as considered at rest. Thus the relative motion of the stars with respect to the earth agrees with this motion, that we living on the earth consider as at rest.

The earth from B has moved forwards to M and the star from A to L, we see that star from M following into the region ML and in the distance ML. Since indeed we are not able to apprehend motions from the place B any more, but still now we consider ourselves to be at B, then we will see the star [p. 37] from B and not at L, but at N, clearly in the same place and the same distance. Therefore the right line BN will be equal and parallel to the line ML, as we have found by considering our method.

GENERAL SCHOLIUM

  1. These laws of motion, which a body observes that is left to itself, either at rest or in continued motion, are seen particularly for these indefinitely small bodies, which can be consider as points.

For in bodies of finite magnitudes, of which the individual parts have their own motions, the body will exert itself to observe these laws, but which will not always be possible to happen on account of the state of the body.

The body therefore will follow a motion which is composed from the individual exertions of the parts of the body, and this hitherto on account of the insufficiency of the principles has not been possible to be defined, but this discussion is to be differed to the following.

The different kinds of bodies will therefore supply the needs for the primary division of our work.

  1. Very small bodies as points.

  2. Finite bodies which are rigid and are not allowed to change shape

  3. Flexible bodies.

  4. Those which allow extension and contraction.

  5. The solution of the motion of many bodies, that are impeded by others, that their own motion may be completed as they exert themselves.

  6. The motion of fluids

For these bodies we will not only see, how the remainder of the motion is to be continued [p. 38]; but in addition we will inquire, how these are affected by the external causes or forces.

  1. The large scale variation of the whole body can be inferred, whether it is free or not.

I define a non-free state as when bodies are impeded, by which they are unable to progress in that direction, and which they try to overcome.

The motion of pendular bodies is of this kind which, since the are unable to descent directly, as they try, and so make oscillations.

For the free state is to be understood : when bodies are progressing and which come upon no impediments to their motion anywhere, not only on account of their own force, or from disturbing forces pulling on them. Therefore it appears, from the things Mechanics will have as its agenda , and that there are many which have not even been touched upon.

For besides the motion of points, which have been dealt with hitherto, nevertheless there are so few that it will be necessary to derive nearly all from first principles.

I begin therefore with the motion of free points with any kinds of disturbing forces, because these left to themselves will follow the motion shown in this chapter.

Hence on account of this I have resolved for the First Volume to be concerned with the motion of free points, and the following Volume truly set up to explore the motion of points which are not in free motion ; in both of which, and which will occur, as with these already dealt with, so likewise for these that follow, I shall derive the motions from first principles using the analytical method.

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