Proposition 6-7 Inertia

PROPOSITION 6. PROBLEM.

51. From the given scale of the times AT (Fig. 6) to find and construct the scale of the speed AN.

SOLUTION

As before, put AM = s, MN = c and MT = t, it is required from the given equation between s and t to find the equation between s and c. Truly, this is easily effected from the above rule found : t = .. For by differentiation it becomes dt = dsc and c = …

[For the ratio .. Therefore the normal TO to the curve is drawn at T, and it will be ds = MO dt MT/MO is the tangent of the angle MOT the normal makes to the axis AO; this is the complement of the angle the gradient of the curve AT makes to the same axis, which is also dt/ds; from which the result follows.]

Therefore as MO is to MT, thus a certain line is made to equal unity, which indicates the seconds for the fourth proportional, which is equal to MN. Therefore from M the interval is taken MQ = 1, and QN is drawn parallel to the normal TO, will be the point N on the scale of the speeds sought. Q. E. I.

EXAMPLE 1

52. Let the scale of the times be a right line inclined at some angle to AM; then t = ms .. and dt = mds. Therefore c = ds is produced. The scale of the speed is therefore a right line parallel to AM, and the body is carried forwards uniformly.

EXAMPLE 2

53. The times are as some power of the distances described, or t = s m and thus ..

From which c = ms1m−1 = m1 s . Whereby, if the curve AT were the .. parabola of APOLLONIUS, i. e. t = s 2 , then m = 12 and c = 2s 2 . From which it is apparent in this case that the scale of the speeds is of the same kind of parabola also.

Corollary

54. If the equation is given between c and t, in the same way the distance traversed s and both the scales of the speeds and of the times can be found. For because c = ds , then ds = cdt et s = cdt …

Scholium

55. Here it is reminded for these, that as far as scales of speeds and times have been described, not only are they seen to apply to absolute motion, but also to pertain to relative motion.

For the nature of this motion has not yet been considered, neither has anything been assumed, that is a special property of absolute motion. Now indeed we will produce some propositions, that are peculiar to absolute motion, and from which in a certain way, the difference between absolute motion and relative motion will be made apparent.

PROPOSITION 7 THEOREM

56. A body remains in a state of absolute rest, unless it is disturbed to move by some external cause.

DEMONSTRATION

We consider this body to exist in infinite and empty space.

There is no reason why the body should be made to move from one place to another rather than stay at rest.

Consequently, because of the lack of reason why it should move, it must remain at rest forever.

Nor indeed does this reason ever change in the universe.

Although it is possible to object that in the universe, there is sufficient reason that it might fall in one place or another.

It cannot be believed that in that empty infinite space, the failure of sufficient reason for a single motion to occur [by this mechanism], is the cause of the body remaining in a single place.

This is because there is no doubt that the nature of the body itself is the cause of this phenomenon.

The failure to move due to any insufficient reason, cannot give the true and essential cause of the event.

But rather, it:

• rigorously demonstrates the true cause
• shows that the hidden nature of the thing is the true essential cause.

This cause does not cease, from the failure of that other insufficient cause to move [the first explanation that Euler rejects].

Thus, the demonstration of ARCHIMEDES illustrates the principle with the equilibrium of two balances each similar to the other, that the truth of the matter can be shown not only in empty space, but also in the world.

[Here two bodies cancel out each other’s turning effects in the earth’s gravity.]

Moreover, each is given a natural reason for this equilibrium, and which is located on the earth also.

In empty space, a body should therefore be able to remain in a state of rest, since it is in the nature of the body for the reason given.

On that account for a body on the earth too, that once it is at rest, except by some other cause acting on it, it can be considered to remain at rest. Q. E. D.

Corollary 1

57. This is why every body shall remain in a state of rest, except that by some external cause it is disturbed to move.

Corollary 2

58. As the fundamentals of this demonstration has been found from the nature of absolute rest, it is hence in error to extend this law to relative rest.

Scholium

59. This law does not extend to relative rest.

We see bodies on ships at relative rest, but if the ship is suddenly shaken violently, the bodies do not remain in a state of rest, but also are shaken and move from their own positions.

Even if before they were at rest, nothing approached to cause this motion.

[The modern physicist would take great exception to these assertions, as the bodies try to maintain their previous state.]

Corollary 3

60. Similarly, a body once at rest must continue to be at rest, except in the circumstance of being affected by some external cause.

This body, in an state of absolute rest, was previously always in a state of rest if it were left to itself.

Since there is no reason, whereby the state of its present position should have come about from one region or another, thus it is also concluded that the body was always present before in its present situation.

Corollary 4

61. Therefore the body, since once it is at rest, if no external cause had neither acted on it or taken it away, then not only henceforth will it always remain at rest in the same place, but also previously it had to be for ever in a state of rest in that place.

Corollary 5

62. Thus, a body once set in a state of absolute motion is never able left to itself, to come to a state of absolute rest.

For if finally it should come to rest, as before it is necessary that it was always at rest, which is contrary to hypothesis.

[One might regard this proposition and others of the same nature as Euler’s musings. The idea of separating absolute and relative motion seems now to be rather naive, but as always, one must suspend judgment until the whole story unfolds in the future propositions. The ideas of absolute time and space were of course set out by Newton; but none of these agree completely with modern physics, which regards all inertial frames as equivalent.]