The Law of Conservation of Momentum and Mass

Proposition 12: God is the principal cause of motion.

Proof: See the immediately preceding Scholium.

Proposition 13: God still preserves his concurrence the same quantity of motion and rest that he originally gave to matter.

Proof: God is the cause of motion and rest (Prop. 12 Part 2).

He continues to preserve them by that same power by which he created them (Ax. 10 Part 1 ). The quantity also remains the same as when he first created them (Cor. Prop. 20 Part 1 ). Q.E.D.

Scholium 1: Theology says that God does many things at his own good pleasure to display his power to men.

Those things that depend merely on his good pleasure are known only by divine revelation,

To prevent philosophy from being confused with theology, they are not to be admitted in philosophy, where enquiry is restricted to what reason tells us.

Scholium 2: Motion is nothing but a mode of moving matter.

But it nevertheless has a fixed and determinate quantity.

Read Art. 36 Part 2 of the Principia.

Proposition 14: Each single thing, insofar as it is simple and undivided and is considered only in itself, always perseveres in the same state, as far as in it lies.*

Many take this proposition as an axiom, but we shall demonstrate it.

Proof: Everything is in a certain state only by the concurrence of God (Prop. 12 Part 1).

God is constant in his works (Cor. Prop. 20 Part 1 ).

If we pay no attention to any external causes (i.e., particular causes) but consider the thing only in itself, we must affirm that it always perseveres in the state in which it is. Q.E.D.

Corollary: A body that is once in motion always continues to move unless it is checked by external causes.

Proof: This is obvious from the preceding proposition. But to correct prejudice concerning motion, read Arts. 37 and 38 Part 2 of the Principia.

Proposition 15: Every body in motion tends of itself to continue to move in a straight line, not in a curved line.*

This proposition could well be considered as an axiom, but 1 shall demonstrate it.

Proof: Motion has only God for its cause (Prop. 12 Part 2).

It never has of itself any force to exist (Ax. 10 Part 1 ).

At every moment, it continues to be created by God. This is demonstrated in connection with the Axiom just cited.

We attend only to the nature of the motion. But we can never attribute to it a timespan that can be conceived as greater than another timespan.

But if it is said that it pertains to the nature of a moving body to describe by its movement a curve, we should be attributing to the nature of motion a longer timespan than when it is supposed to be in the nature of a moving body to tend to continue to move in a straight line (Ax. 1 7).

We cannot attribute such duration to the nature of motion, then neither can we posit that it is of the nature of a moving body to continue to move in a curve.

It must continue to move only in a straight line. Q.E.D.

Scholium: Perhaps many will think that this proof is equally effective in showing that it does not pertain to the nature of motion to describe a straight line as in showing that it does not pertain to the nature of motion to describe a curved line.

This is because there cannot be posited a straight line other than which there is no shorter, whether straight or curved, nor any curved line other than which there is no shorter curve.

However, I nevertheless hold that the proof proceeds correctly, because it concludes what was required to be proved solely from the universal essence of lines, that is, their essential specific difference, and not from the length of individual lines, that is, their accidental specific difference.

Motion is the transfer of one body from one location to another.

So unless we conceive this transfer in its simplest form-that is, as proceeding in a straight line-we are attaching to motion something not contained in its essence or definition, and so not pertaining to its nature.

Corollary: It follows that every body that moves in a curve is continuously deviating from the line that it should move in if it were alone.

This is through the force of an external cause (Prop. 14 Part 2).