The Motion of Bodies

Proposition 32: If a body B is surrounded by particles in motion which impel it with equal force in all directions, it will remain unmoved in the same place as long as no other cause occurs.

Proof: This proposition is self-evident.

If it were to move in any direction through the impulse of particles coming from one direction, the particles that move it would be impelling it with greater force than other particles that at the same time are impelling it in the opposite direction, with no effect (Ax. 20).9

This would be contrary to the hypothesis.

Proposition 33: The Body B can be moved in any direction by any additional force, however small.

Proof: All bodies immediately contiguous to B are in motion (by hypothesis). B (Prop. 32) remains unmoved. As soon as they touch B they will be reflected in another direction while retaining their original motion (Prop. 28 Part 2).

Thus body B is all the time automatically being left by immediately contiguous bodies.

And so, whatever magnitude is assigned to B, no action is required to separate it from immediately contiguous bodies (Note 4 of Def. 8).

So any external force striking against it, however small, is bound to be greater than the force that B possesses for remaining in the same place.

This is because B possesses no force for adhering to its immediately contiguous bodies.

When added to the impulse of those particles that together with it are impelling B by external force in the same direction, it is also bound to be greater than the force of other particles impelling B in the opposite direction (for, disregarding this external force, the one force was supposed to be equal to the other).

Therefore (Ax. 20), body B will be moved in any direction by this external force, however small it be imagined. Q.E.D.

Proposition 34: The Body B cannot move more quickly than it is impelled by the external force, even though the particles by which it is surrounded are in much swifter motion.

Proof: The particles that, together with the external force, are impelling B in the same direction are faster than the external force can move B.

Yet (by hypothesis) they have no more force than the bodies that are repelling B in the opposite direction.

And so, they will use up all the power of their determination merely in resisting these, without imparting any speed to B (Prop. 32 Part 2).

Therefore, because no other circumstances or causes are supposed, B will not receive any amount of speed from any cause other than the external force.

Therefore (Part 1, Ax. 8) it cannot move more quickly than it is impelled by the external force. Q.E.D.

Proposition 35: When body B is thus moved by an external impulse, it receives the greatest part of its motion from the bodies by which it is constantly surrounded, and not from the external force.

Proof: Even though body B is imagined to be very large, it must be moved by even the smallest impulse (Prop. 33 Part 2).

Let B be 4 times as large as the external body by whose force it is impelled.

Both must move at the same speed (preceding Prop.). There will be 4 times as much motion in B as in the external body by which it is impelled (Prop. 21 Part 2).

Therefore (Ax. 8 Part I), it does not have the principal part of its motion from the external cause.

Apart from this cause, no causes are supposed other than the bodies by which it is constantly surrounded (for B is supposed to be not moving of itself).

Then it is only from the bodies by which it is surrounded (Ax. 7 Part I) that it receives the principal part of its motion, and not from the external cause. Q.E.D.

Here, we cannot say, as previously, that the motion of particles coming from one direction is required in order to resist the motion of particles coming from the opposite direction.

For bodies moving toward each other with equal motion (as these are supposed) are contrary only by determination,1O and not by motion (Cor. Prop. 19 Part 2).

And so in resisting one another they use up only their determination, and not their motion.

Therefore body B can receive no determination, and consequently (Cor. Prop. 27 Part 2) no speed insofar as that is distinct from motion-from adjacent bodies.

But it can receive motion; indeed, when the extra force is added, it must necessarily be moved by them, as we have demonstrated in this proposition and as can be clearly seen from the manner of the proof of Proposition 33.10 See Prop. 24 Part 2, where d IS demonstrated that two bodies, In reslstmg one another, expend their determination, not their mohon