Spinoza, Principles of Cartesian Philosophy, Proposition 21, Part 2

by Corry Shores
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[The following is quotation; my summary and commentary is in brackets.]

Baruch Spinoza

The Principles of Cartesian Philosophy

and Metaphysical Thoughts followed by

Lodewijk Meyer Inaugural Dissertation on Matter (1660)

Part II

Proposition 21

If 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 Bs.

[Two bodies are moving at equal speed, but body A is twice as large as B. Thus A is more able to retain its speed in the face of opposing forces, in fact, A has twice as much force to retain its speed. In other words, A has twice the motion of B.]

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.

[We want to understand why it is that if we have two bodies moving at the same speed, with A twice the size of B, then A has twice the capacity to retain its motion. We know that A is twice B's size, so first we will consider A split up in two parts, each of which equal in size to B. We know that because both B's are in motion, they will tend to stay in motion, on account of their having a quantity of force that causes them to persist in their present speed. If we then consider these two B's as joined together, while maintaining the same speed, they will reconstitute as one A. Because this A is made up of two equal parts with equal motion, A's motion will be twice one of its halves, that is to say, twice the motion of one B.]

Not that this follows simple 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.

[Definition 8, briefly:

Local motion is the transfer of one part of matter, or one body, from the vicinity of those bodies that touch it immediately, and are considered as resting, to the vicinity of others.

(46)]

[For A and B to have been set in motion, they needed to be separated from the matter that was once in their vicinity; in other words, they needed to be pushed away from something else. We need to push a body twice as hard as a body half its size to get them both moving at the same speed.]

Spinoza, Baruch. The Principles of Cartesian Philosophy and Metaphysical Thoughts followed by Lodewijk Meyer Inaugural Dissertation on Matter (1660). Transl. Samuel Shirley and Steven Barbone. Indianapolis: Hackett, 1998.