There are no vacuums nor atoms

Proposition 1: Hardness, weight, and the other sensible qualities may be separated from a body. But the nature of the body will still remain unimpaired

Proof: In the case of hardness, such as of this stone, our sense-perception merely indicates to us that hard bodies resist the movement of our hands (Ax. 3).

Therefore (Part 1, Prop. 14) hardness is also only just this.

If the rock is reduced to the finest powder, its parts will readily give way (Ax. 12). Yet, it will not lose the nature of body (Ax. 5). Q.E.D.

In the case of weight and the other sensible qualities, the proof proceeds in the same way.

Proposition 2: The nature of body or matter consists only in extension [metaphysical space]

Proof: The nature of body is not lost as a result of the loss of sensible qualities (Part 2, Prop. 1).

Therefore, these do not constitute its essence (Ax. 2).

Therefore, nothing is left but extension [metaphysical space] and its affections (Ax. 7).

So if extension is taken away, nothing will remain pertaining to the nature of body. It will be completely annulled.

Therefore, (Ax. 2) the nature of body is only in extension. Q.E.D.

Corollary: Space and body do not differ in reality.

Proof: The following are the same in reality:

• body and extension [metaphysical space] (previous Prop.)
• space and extension (Def. 6).

Therefore (Ax. 1 5) space and body do not differ in reality. Q.E.D.

Scholium: We say that God is everywhere. But it does not mean that God is extended [metaphysical space] (i.e. [previous Prop.], that God is corporeal).

His ubiquity [metaphysical everywhere] refers only to:

• his power
• his concurrence to preserve all things.

In this way, God’s ubiquity does not refer to body or extension any more than to angels and human souls.

When we say that his power is everywhere, we do not exclude his essence. Where his power is, there too is his essence (Cor. Prop. 17 Part I). We intend to exclude only bodily nature.

God is everywhere not by a corporeal power but by his divine power or essence, which serves alike to preserve extension [metaphysical space] and thinking things [abstract minds] (Part 1, Prop. 17).

The latter he certainly could not have preserved if his essence were corporeal.

Proposition 3: There is no vacuum

Proof: A vacuum is extension [metaphysical space] without corporeal substance [metaphysical aether] (Def. 3). It is (Prop. 2 Part 2) body without body, which is absurd.

For a fuller explanation, read Articles 17 and 18 Part 2 of the Principia. It says that:

• nothingness has no properties
• bodies with nothing in between necessarily touch one another

Proposition 4: One part of a body does not occupy more space at one time than at another. Conversely, the same space does not contain more body at one time than at another

Proof: Space and body do not differ in reality (Cor. Prop. 2 Part 2).

Therefore when we say that a space is not greater at one time than at another (Ax. 1 3), we are also saying thata body cannot be greater (i.e., occupy more space) at one time than at another, which was our first point.

Space and body do not differ in reality.

It follows that when we say that body cannot occupy more space at one time than at another, we are also saying that the same space cannot contain more body at one time than at another. Q.E.D.

Corollary: Bodies that occupy equal space, say, gold and air, have the same amount of matter or corporeal substance.

Proof: Corporeal substance [metaphysical aether] consists only in extension [metaphysical space] (Prop. 2 Part 2), not in:

• hardness (e.g., of gold)
• softness (e.g. , of air)
• any of the sensible qualities (Prop. I Part 2)

By hypothesis, there is the same amount of space or (Def. 6) extension [metaphysical space] in the one as in the other. Therefore, there will also be the same amount of corporeal substance. Q.E.D.

Proposition 5: There are no atoms

Proof: Atoms are parts of matter that are, by their own nature, indivisible (Def. 3).

But the nature of matter consists in extension (Prop. 2 Part 2), which is divisible, however small it be (Ax. 9 and Def. 7).

Therefore, however small a part of matter may be, it is by its own nature divisible. That is, there are no atoms, or parts of matter that are by their own nature indivisible. Q.E.D.

Scholium: The question of atoms has always been a difficult and complicated one.

Some assert that there must be atoms, arguing from the impossibility of an infinite being greater than another infinite.

If 2 quantities-say A and 2A (double of A)-are infinitely divisible, they can also be divided in actuality into an infinite number of parts by the power of God, who understands their infinitely many parts with a single intuition.

Therefore, because one infinite cannot be greater than another infinite, quantity A will be equal to its double, which is absurd.

They ask:

• whether half an infinite number is also infinite
• whether it is even or odd
• etc

To all this, Descartes replied that we must not reject what comes within the scope of our intellect, and is therefore clearly and distinctly conceived, because of other things that exceed our intellect or grasp, and that are therefore only perceived very inadequately by us.

The infinite and its properties exceed the finite human intellect.

And so it would be foolish to reject as false, or to doubt, what we clearly and distinctly conceive concerning space, on the grounds that we do not comprehend the infinite.

This is why Descartes considers as indefinite those things in which we can see no boundaries, such as the extension of the world, the divisibility of the parts of matter, etc. Read Art. 26 Part I of the Principia.