Proposition 8

PROPOSITION 8. THEOREM

63. A body having uniform absolute motion will always be moving, and with the same speed now that it had at an any earlier time, unless an external cause should act on it or have acted on it.

DEMONSTRATION

If the speed with which a body moved was not kept the same always, then the speed might either increase or decrease. Moreover from this cause it might tend towards rest, which, since it is never allowed to do this as a consequence of (22), cannot happen.

Likewise for the body to have emerged into its present state from rest is considered to be absurd.

Besides if this body situated in infinite empty space should be considered in this way, which is that it has speeded up or slowed down, there is nor reason why it should have a greater or lesser speed than what it already has, on account of which it must always be moving with the same speed. Q. E. D.

Corollary

64. Therefore how much we should see the speed of the moving body to have either increased or decreased, we must attribute the change to some external cause.

PROPOSITION 9. THEOREM.

65. The body with a given absolute motion shall progress in a straight line, or the distance that it describes shall be a straight line.

DEMONSTRATION

There is no reason, if this body is considered to be placed in infinite and empty space, why it should depart from moving in a straight line from one region to another.

From which it is to be concluded, that it is from the nature of the body itself that it moves in a straight line.

On account of which also in the world, where indeed with sufficient reason for this principle to cease being applicable, it is nevertheless observed that any body moving progresses along a straight line, unless obviously it is prevented.

Q. E. D

Corollary 1

66. From these two propositions that universal law can be set out : every body provided with motion progresses uniformly in a straight line.

Corollary 2

67. Therefore a body, that was compelled by some external cause to progress in a curve AM (Fig. 7), if, when it arrived at M, these causes suddenly ceased, then it would progress with the speed which it had at M, uniformly following the direction it would have from that time of being freed.

Truly, it is the tangent MT of no element of the curve other than the direction produced at M that is relinquished, on account of which the body at M progresses along the tangent MT with that uniform speed that the body had.

Scholium 1

68. These laws regarding absolute rest and motion have been gathered together under one authority. And this is the law NEWTON thus proposed in the Principia, as he has stated : Every body persists in its own state of being at rest or of moving uniformly in a direction, unless in as much as it is forced to change that state by impressed forces.

Corollary 3

69. Moreover these laws are concerned with the continuation of motion as applied to absolute motion, and not these in relative motion that retain their force. As indeed it can be possible, that a body in a state of relative rest not to be maintained at rest, if also by no external cause it should be agitated (59), thus also the motions bodies had relative to others, are not always to be moved in the same direction uniformly.

Corollary 4

70. Therefore when a body has been disturbed, by no external cause, that may move in some non-uniform relative way, and yet either it is required by consideration to remain at rest or to move in some uniform manner. From this it is to be understood in some manner, how much the relative state differs from the absolute state.

Scholium 2

71. In the foundations of theoretical astronomy, as have been set out by Newton, the sun and the fixed stars are established as being entirely non-affected by external causes, or the effect is so small as to be negligible.

Though however we can neither see the sun to be in uniform motion nor to be progressing in a fixed direction with respect to the earth; , yet it is certain that the sun is either at rest or is moving uniformly in a fixed direction.

It is necessary that irregularities in the motion of the sun should be observed from the position of the earth.

DEFINITION 8.

72. The direction of the motion is to be determined by a straight line, along which the moving body is trying to progress, and according to this the body progresses, unless it is impeded by some external cause.

Corollary

73. Therefore a body having an absolute motion, unless it is affected by other causes, always keeps the same speed and moves in the same direction.

DEFINITION 9

74. The force of inertia is in all bodies is that in situ faculty of the body to maintain its state of rest or of continuing in its present state of motion in a straight line.

[This is another supposed Newtonian idea; by whom inertia was regarded as a force. See, e. g. Cohn’s translation of the Principia, p. 96, § 4.7. However, as Euler points out, it had its origins with Kepler.]

Corollary

75. Though indeed we have demonstrated with sufficient reasons that a body remains in a state of rest or of a continuation of uniform motion in a straight line, yet now we will note that this is not the effecting cause of the phenomenon, for that is situated in the nature of the body itself.

This depends on the nature of the bodies themselves itself and is the reason for the conservation of the state of a body, and it is called the force of inertia.

Scholium

76. KEPLER first formed this notion.

He attributed to it a force, that all bodies have, all of these are resistant to that which is trying to disturb their state; and calling this inertia is better than calling it resistance, agreeing with the idea that there is a perseverance, that we have connected to with our ideas.

But it is easily understood these definitions are not different from what we have already asserted ; for it is the same force of continuing or of remaining at rest, and which offers resistance to hindrances to the state of the motion.

I have used this definition rather than the KEPLER one, since it has not yet been agreed, how bodies with disturbing forces resist changes in their motion.

This force of resistance has its own origin from the facility of remaining at rest or of continuing to move; and therefore should be explained from this basis.

[Curiously enough, some engineering texts on physics still treat the inertial force on the same footing as other forces, so that the sum of the forces on any body is always zero; and likewise with the sources of e.m.f. in circuits.]