Superphysics
Propositions 7-11

# Rules of Motion

by Spinoza

## Proposition 7: No body moves into the place of another body unless at the same time that other body moves into the place of another body.

Proof: (See Diagram of Next Proposition) If body A moves into the place of a body B, then the space of body B will now contain A and B.

The space will now contain twice the amount of corporeal substance, which is absurd (Prop. 4 Part 2).

Therefore no body moves into the place of another*, … etc. Q.E.D.

*Superphysics Note: We explain this with our Law of Conservation of Idea

## Proposition 8: When a body moves into the place of another body, at the same moment of time the place quitted by it is occupied by another body immediately contiguous to it.

Proof: If a body B moves toward 0, bodies A and C at the same moment of time will either move toward each other and touch each other, or they will not.

If they move toward each other and touch each other, what we have proposed is granted.

But, by hypothesis, this is not B. Therefore it is another body, which at JJ the same moment of time moves into B’s place. And because it moves into B’s place at the same moment of time, it can be none other than that which is immediately contiguous, according to Scholium Prop. 6 Part

1. For there we demonstrated that there can be no motion from one place to another such that it does not require a time other than which there is always a shorter time.

From this it follows that the space of body B cannot be occupied at the same moment of time by another body that would have to move through some space before it moved into B’s space. Therefore only a body immediately contiguous to B moves into its place at the same moment of time. Q.E.D.

Scholium: Because the parts of matter are in reality distinct from one another (Art. 61 Principia Part I), one can exist without another (Cor. Prop. 7 Part I). and they do not depend on one another. So all those fictions about Sympathy and Antipathy must be rejected as false. Furthermore, because the cause of an effect must always be positive (Ax. 8 Part I). it must never be said that a body moves to avoid there being a vacuum. It moves only through the impulse of another body.

Corollary: In every motion, a complete circle of bodies moves at the same time.

Proof: At the time when body I moves into the place of body 2, body 2 must move into the place of another body, say, body 3, and so on (Prop. 7 Part 2).

Again, at the same moment of time as body 1 moves into the place of body 2, the place quitted by body I must be occupied by another body (Prop. 8 Part 2), let us say body 8 or another body immediately contiguous to body I. Because this occurs only through the impulse of another body (Schol. to this Prop.). which is here supposed to be body I, all these moving bodies cannot be in the same straight line (Ax. 21) but (Def. 9) form a complete circle. Q.E.D.

## Proposition 9: If a circular tube ABC is full of water and is four times as wide at A as at B, then at the time that the water (or any other fluid body) at A begins to move toward B, the water at B will move at four times that speed.

Proof: When all the water at A moves toward B, the same amount of water must at the same time move into its place from C, which is immediately contiguous to A (Prop. 8 Part 2). And from B the same amount of water will have to move into the place ofC (same Prop.). Therefore (Ax. 14) it will move at 4 times that speed. Q.E.D.

What we say about the circular tube must also apply to all unequal spaces through which bodies moving at the same time are compelled to pass; for the proof will be the same in the other cases.

Lemma: If 2 semicircles A and B are described about the same center, the space between their circumferences �B �OD will be everywhere the same. But if two semicircles C and D are described about different centers, the space between their circumferences will be everywhere unequal. The proof is evident merely from the definition of a circle.

## Proposition 10: The fluid body that moves through the tube ABC (of Prop. 9) receives an indefinite number of degrees of speed.

Proof: The space between A and B is everywhere unequal (previous Lemma).

Therefore (Prop. 9 Part 2) the speed at which the fluid body passes through the tube ABC will be unequal at all points. Furthermore, because we conceive in thought an indefinite number of spaces ever smaller and smaller between A and B (Prop. 5 Part 2), we shall also conceive its inequalities of speed, which are at all points, as indefinite. Therefore (Prop. 9 Part 2) the degrees of speed will be indefinite in number. Q.E.D.

## Proposition 11: The matter that flows through the tube ABC (of Prop. 9) is divided into an indefinite number of particles.

Proof: The matter that flows through the tube ABC acquires at the same time an indefinite number of degrees of speed (Prop. 10 Part 2). Therefore (Ax. 16) it has an indefinite number of parts into which it is in reality divided. Q.E.D.

Scholium: So far we have been dealing with the nature of motion.

Its cause is twofold:

1. The primary or general cause, which is the cause of all the motions in the world,

2. The particular cause, whereby it comes about that individual parts of matter acquire motions that they did not have before.

As to the general cause, because we must admit nothing (Prop. 14 Part I and Schol. Prop. 15 Part 1)3 but what we clearly and distinctly perceive, and because we clearly and distinctly understand no other cause than God, the creator of matter, it is obvious that no other general cause but God must be admitted.