The Motion of Liquid Bodies

Proposition 36: If a body (e.g., our hand) can move in any direction with equal motion without resistance then the bodies in the space where it moves would move with equal speed and direction with that body (hand)

Proof: Any space through which a body can move is bound to be full of bodies (Prop. 3 Part 2).

Point 1: The space through which our hand can thus move is filled with bodies which will move in the way I have described.

For if you deny this, let them be supposed to be at rest, or to move in a different way.

If they are at rest, they will necessarily resist the motion of the hand. This will transfer the hand’s motion to them them (Prop. 14 Part 2). This will make them move together with the hand in the same direction and speed (Prop. 20 Part 2).

But in the hypothesis they are supposed not to resist. Therefore these bodies are in motion.

Point 2: They must be moving in all directions.

If you deny this, suppose that there is some direction in which they are not moving, say from A toward B.

Therefore if the hand is moving from A towards B, then it will necessarily meet moving bodies (by the first part of this proof), bodies with a determination different from that of the hand.

Therefore, they will resist it (Prop. 14 Part 2) until they move along with the hand in the same direction (Prop. 24 and Schol. Prop. 27 Part 2).

But, by hypothesis, they do not resist the hand. Therefore they will be moving in all directions.

Point 3: These bodies will be moving in all directions equaling one another in force of speed.

If they were not moving with equal force of speed, suppose that those that are moving from A towards B are not moving with as much force of speed as those that are moving from A towards C.

Assume the hand can move with equal motion in all directions without resistance.

Therefore, if the hand were to move from A towards B with the same speed with which bodies are moving from A towards C, the bodies moving from A towards B will resist the hand (Prop. 14 Part 2) until they move with a force of speed equal to that of the hand (Prop. 31 Part 2).

But this is contrary to the hypothesis. Therefore they will move with equal force of speed in all directions.

**Point 4: If the bodies are not moving with the same force of speed as the hand, then the hand will either move more slowly, with less force of speed, or more quickly, with greater force of speed, than the bodies. **

If the former, the hand will resist the bodies that are following it in the same direction (Prop. 31 Part 2).

If the latter, the bodies that the hand is following and with which it is moving in the same direction will resist it (same Prop.).

Each of these is contrary to the hypothesis. Therefore, because the hand can move neither more slowly nor more quickly than the bodies, it will move with the same force of speed as the bodies. Q.E.D.

Why do I say ‘with equal force of speed’ and not simply ‘with equal speed’? The answer is in Scholium Cor. Prop. 27 Part 2.

If you then ask whether the hand, while moving (e.g. , from A toward B), does not resist bodies that are moving at the same time with equal force from B toward A, read Prop. 33 Part 2.

It explains that their force is balanced by the force of the bodies moving together with the hand at the same time from A toward B (for, by the third part of this Prop., these two forces are equal).

Proposition 37: If a body A can be moved in any direction by any force, it must necessarily be surrounded by bodies that are moving at the same speed as one another.

Proof: Body A must be surrounded on all sides by bodies (Prop. 6 Part 2) that are moving equally in all directions.

For if they were at rest, body A could not be moved in any direction by any force, however small. It could only be moved by a force that could at least be able to move along with itself the bodies immediately contiguous to A (Ax. 20 Part 2).

Again, if the bodies by which A is surrounded were moving with greater force in one direction than in another-say, with greater force from B toward C than from C toward B-then because it is surrounded on all sides by moving bodies (as we have just now demonstrated), the bodies moving from B towards C would necessarily take A along with them in the same direction (by what we have demonstrated in Prop. 33).

So it is not any force, however small, that will suffice to move A toward B.

It must be exactly such as would counterbalance the excess of motion of the bodies coming from B toward C (Ax. 20).

Therefore they must be moving with equal force in all directions. Q.E.D.

Scholium: This is the case with fluid bodies.

It follows that fluid bodies are those that are divided into many tiny particles moving with equal force in all directions.

Although those particles cannot be seen by any eye, even a lynx’s, one must not deny what we have now clearly demonstrated.

For from our previously stated Props. 10 and 11, a minuteness of nature such as cannot be determined or attained by any thought, not to say the senses, is sufficiently proved.

Furthermore, bodies resist other bodies merely by their rest.

We perceive of hardness as the parts of hard bodies resisting the motion of our hands.

Read Arts. 54, 55, 56 Part 2 of the Principia.

End of Part 2