Axioms and Lemmas

1. To nothing there belong no properties.

2. Whatever can be taken away from a thing without impairing its integrity does not constitute the thing’s essence. But that whose removal destroys a thing constitutes its essence.

3. In the case of hardness, our sense indicates to us nothing else, and we clearly and distinctly understand of it nothing else, than that the parts of hard bodies resist the movement of our hands.

4. If two bodies approach each other, or move away from each other, they will not thereby occupy more or less space.

5. A part of matter, whether it gives way or resists, does not thereby lose the nature of body.

6. Motion, rest, shape, and the like cannot be conceived without extension [metaphysical space].

7. Apart from its sensible qualities, nothing remains in body but extension [metaphysical space] and its affections enumerated in Part 1 of Principia.

8. Any one space or extension cannot be greater at one time than at another.

9. All extension [metaphysical space] can be divided, at least in thought.

No one who has leamed even the elements of mathematics doubts the truth of this axiom.

The space between a tangent and a circle can always be divided by an infinite number of larger circles. The same point is also made obvious by the asymptotes of the hyperbola.

10. No one can conceive the boundaries of an extension or space without at the same time conceiving other spaces beyond those boundaries, immediately following on that space.

11. If matter is manifold, and one piece is not in immediate contact with another, each piece is necessarily comprehended within boundaries beyond which there is no matter.

12. The most minute bodies readily give way to the movement of our hands.

13. One space does not penetrate another space, nor is it greater at one time than at another.

14. If a hollow pipe A is of the same length as C, and C is twice as wide as A, and if a liquid passes through pipe A at twice the speed at which a liquid passes through pipe C, the same amount of matter will .. pass through both pipes in the same space _¥ ) of time. And if in the same time the same amount of matter passes through pipe A as through C, the former will move at twice the speed.

15. Things that agree with a third thing agree with one another; and things that are double a third thing are equal to one another.

16. Matter that moves in diverse ways has at least as many parts, divided in actuality, as there are different degrees of speed to be observed in itat the same time.

17. The shortest line between two points is a straight line.

18. If body A moving from C towards B is repelled by an opposite impulse, it will move along the same line toward C.

19. When bodies having opposite motions collide with each other, they are both- or at least one ofthem-compelled to undergo some change.

20. A change in any thing proceeds from a stronger force.

21. When body 1 moves toward body 2 and pushes it, if as a result of this impulse body 8 moves toward body 1, then bodies 1, 2, 3, etc., cannot be in a straight line. All 8 bodies form a complete circle. See Def. 9.

Lemmas

Lemma 1: Where there is extension or space, there is necessarily the aether.

Proof: Extension or space (Ax. I) cannot be pure nothing. It is therefore something, but not God (Prop. 16 Part I).

Therefore it is a thing that needs only God’s concurrence to exist (Prop. 12 Part I), that is (Def. 2 Part 2), the aether. Q.E.D.

Lemma 2: Compression and decompression are clearly and distinctly conceived by us, although we do not grant that bodies occupy more space in decompression than in compression.

Proof: can be clearly and distinctly conceived from the mere fact that parts of a body may move away from, or toward, one another.

Therefore (Ax. 4) they will not occupy either more or less space.

For if the parts of a body-say, a sponge-by moving toward one another expel the bodies with which its interstices are filled, this in itself will make that body more dense, and its parts will not thereby occupy less space than before (Ax. 4).

If again the parts move away from one another and the gaps are filled by other bodies, there will be rarefaction, but the parts will not occupy more space.

This, which we clearly perceive with the aid of our senses in the case of a sponge, we can conceive with the unaided intellect in the case of all bodies, although their interstices completely escape human sense-perception.

Therefore, rarefaction and condensation are clearly and distinctly conceived by us, etc. Q.E.D.

I wrote these Lemmas first, so that the intellect may rid itself of prejudices concerning space, rarefaction, etc., and better understand what follows.