The Laws of Motion

Proposition 19: Motion, regarded in itself, is different from its determination toward a certain direction.

There is no need for a moving body to switch to rest so that it can change direction.

Proof: Suppose that a body A moves in a straight line toward a body B. It is prevented by body B from continuing further.

Therefore (preceding Prop.) A will retain its motion undiminished. It will not switch too rest for even a moment.

However, it continues to move, but is prevented by B from going in its current direction.

Therefore, it will move in the opposite direction (see Chapter 2 of Dioptrics).

Therefore (Ax. 2), direction is different from the essence of motion [momentum]. A moving body that is repelled does not switch to rest in any moment. Q.E.D.

Corollary: Hence it follows that motion is not contrary to motion.

Proposition 20: If a body A collides with a body B and takes it along with it, A will lose as much of its motion as B acquires from A because of its collision with A.

Proof: If you deny this, suppose that B acquires more or less motion from A than A loses.

All this difference must be added to or subtracted from the quantity of motion in the whole of Nature, which is absurd (Prop. 13 Part 2).

Therefore, body B can acquire neither more nor less motion. And so, it will acquire just as much motion as A loses. Q.E.D.

Proposition 21 (See Preceding Diagram)

If a body A is twice as large as B and moves with equal speed, A will also have twice as much motion as B, or twice as much force for retaining a speed equal to B s.

Proof: Suppose that instead of A there are two Bs; that is, by hypothesis, one A divided into two equal parts.

Each B has a force for remaining in the state in which it is (Prop. 14 Part 2), and this force is equal in both Bs (by hypothesis).

If now these two Bs are joined together, their speed remaining the same, they will become one A, whose force and quantity will be equal to two Bs, or twice that of one B. Q.E.D.

Note that this follows simply from the definition of motion. For the greater the moving body, the more the matter that is being separated from other matter.

Therefore there is more separation, that is (Def. 8), more motion. See Note 4 regarding the definition of motion.

Proposition 22 (See Diagram Prop. 20)

If a body A is equal to a body B, and A is moving at twice the speed of B, the force or motion in A will be twice that in B.

Proof: Suppose that B, when it first acqu ired a certain force of motion has acquired four degrees of speed.

If now nothing is added to this, it will continue to move (Prop. 14 Part 2) and persevere in its state.

Suppose that it now acquires an additional force from a further impulse equal to the former.

As a result, it will acquire another four degrees of speed in addition to the previous four degrees, which it will also preserve (same Prop.), that is, it will move twice as fast (i.e., as fast as A), and at the same time it will have twice the force (i.e., a force equal to Ns).

Therefore the motion in A is twice that of B. Q.E.D.

Note that by force in moving bodies we here understand quantity of motion.

This quantity must be greater in equal bodies in proportion to their speed of motion, insofar as by that speed equal bodies become more separated in the same time from immediately contiguous bodies than if they were to move more slowly.

Thus they also have more motion (Def. 8). But in bodies at rest, we understand by force of resistance the quantity of rest. Hence it follows:

Corollary 1: The more slowly bodies move, the more they participate in rest.

For they offer more resistance to more swiftly moving bodies that collide with them and have less force than they, and they also are less separated from immediately contiguous bodies.

Corollary 2: If a body A moves twice as fast as a body B, and B is twice as great as A, there is the same amount of motion in the greater body B as in the smaller body A, and therefore there is also an equal force.

Proof: Let B be twice the size of A, and let A move with twice the speed of B; then let C be half the size of B and move with half the speed of A. Therefore B (Prop. 21 Part 2) will have a motion twice that of C, and A (Prop. 22 Part 2) will have a motion twice that of C.

Therefore (Ax. 1 5) B and A will have equal motion; for the motion of each is twice that of the third body C. Q.E.D.

Corollary 3: Hence it follows that motion is distinct from speed.

For we conceive that, of bodies possessing equal speed, one can have more motion than another (Prop. 21 Part 2), and on the other hand, bodies possessing unequal speed can have equal motion (previous Cor.).

This can also be deduced merely from the definition of motion, for it is nothing but the transfer of one body from the vicinity … , etc.

But here it should be noted that this third corollary is not inconsistent with the first.

We conceive speed in 2 ways:

1. As a body is more or less separated in the same time from immediately contiguous bodies (and to that extent it participates to a greater or lesser degree in motion or rest)

2. As it describes in the same time a longer or shorter line (and to that extent is distinct from motion).

Proposition 23: When the modes of a body are compelled to undergo variation, that variation will always be the least that can be.

Proof: This proposition follows quite clearly from Prop. 14 Part 2.

Proposition 24, RULE 1 (See Diagram Prop. 20)

If two bodies, A and B, are completely equal and move in a straight line toward each other with equal velocity, on colliding with each other they will both be reflected in the opposite direction with no loss of speed.

In this hypothesis it is evident that, in order that the contrariety of these two bodies should be removed, either both must be reflected in the opposite direction or the one must take the other along with it For they are contrary to each other only in respect of their determination, not in respect of motion.

Proof: When A and B collide, they must undergo some variation (Ax. 19).

But because motion is not contrary to motion (Cor. Prop. 19 Part 2), they will not be compelled to lose any of their motion (Ax. 19).

Therefore there will be change only in determination.

But we cannot conceive that only the determination of the one, say B, is changed, unless we suppose that A, by which it would have to be changed, is the stronger (Ax. 20). But this would be contrary to the hypothesis.

There cannot be a change of determination in only the one. Therefore, there will be a change in both, with A’s and B’s changing course in the opposite direction- but not in any other direction (see what is said in Chap. 2 Dioptrics) and preserving their own motion undiminished. Q.E.D.