Chapter 1h

Proposition 10

Euler
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Table of Contents

PROPOSITION 10. THEOREM

  1. When the space [our inertial frame of reference] from which the relative motion is determined, is either absolutely at rest or moving uniformly in a direction, then the given laws prevail for a state of relative rest or of relative uniform motion.

DEMONSTRATION

If the space is from absolute rest that the relative motion is decided, then the proposition by itself is clear. For in this case the state of rest and relative motions are in agreement with the absolute motions, and thus any body also will be in a state of relative perpetual rest or uniformly moving in a fixed direction (10).

For if indeed that space itself [our inertial frame of reference] should be moved uniformly in the same direction as these bodies, which are relatively at rest, then they will have the same absolute motion that the space itself has. Whereby with each thus progressing uniformly in the same direction by their own nature, thus they are able to continue : and in this case the law is observed (66).

Truly a body, that is relatively moving uniformly in a direction, it also, if it has uniform rectilinear motion, proceeds absolutely with uniform rectilinear motion, as becomes apparent from the following proposition, where it is made transparent. Hence the relative motion here is in agreement with this law and therefore it can continue without an external force. Q. E. D.

[It appears that Euler has abandoned the idea of an inertial force as a working hypothesis in favour of inertial frames, which he calls ‘spaces’, and the rectilinear motion of a body has the same nature in all such spaces; he still maintains, however, that there is a reference inertial frame, which is the one defining absolute rest or motion, determined by the fixed stars. The model suggested by Newton and this one are of course in agreement with each other.]

Corollary 1

  1. Therefore a body not affected by some external cause, that is relatively either at rest, or moving uniformly in some direction, from the evidence the motion of this will be as that indicated, either absolutely at rest or absolutely to be moving uniformly in some direction.

[Thus, motion in any inertial frame can be compared to the absolute motion of the reference frame of the fixed stars.]

Corollary 2

  1. Likewise such relative motion will be preserved, each in its own state, for ever.

For it is not only the body itself that moves, and the body is moved absolutely uniformly in some direction, but also that space itself, relative to which the body is considered, which progresses according to the same law. On account of which each will be continued on its own, and the relative motion will itself continue in this state without any approaching external cause.

Corollary 3

  1. Since the idea we have concerning all motion, is that it is relative (7), and these laws also are not sufficient to recognise how much of the motion the absolute motion any body shall be.

For when we see a body progressing in uniform motion without being affected by an external cause, then we cannot conclude more than to say that this body is either in a state of absolute rest or to be moving with an absolute uniform motion in a given direction.

Truly how much of this motion is absolute cannot be defined, and neither the direction that it has.

[Thus, there is no unique reference frame in which the motion of the body can be defined; and this is especially the case if a special background reference frame of the fixed stars has to be included.]

Corollary 4

  1. Which therefore shall be deduced from the nature of the bodies, that remain in their own state of rest or of uniform motion [p. 30], not only will they pertain to absolute rest and motion, but also to their relative state of rest or motion, and in which space the body, from its own motion is considered to move uniformly in given direction.

Scholium

  1. I am not very concerned with absolute motion, since the relative motion itself is continued by the laws.

Therefore this relative motion we will more often change into other motions of the same kind, as the laws expounded are still observed : if it is allowed, we will contemplate the body in relation to that of another body progressing uniformly in this direction too.

From which reason the body will not cease to move uniformly along a straight line, and that with innumerable different ways that this can be done, the most convenient of these can be selected.

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