Axioms
1. Substance is, by its nature, prior to all its modifications.
2. Things which are different are distinguished either realiter or modaliter.
3. Things which are distinguished realiter either have different attributes, such as Thought and Extension, or are referred to different attributes, as in the case of Understanding and Motion; one of which belongs to Thought, and the other to Extension.
4. Things which have different attributes, as also the things which belong to different attributes, do not have anything the one of the other.
5. That which has not in itself something of another thing, can also not be a cause of the existence of such another thing.
6. It is impossible that that which is a cause of itself should have limited itself.
7. That by which the things are sustained is by its nature prior to such things.
PROPOSITION I
To no substance that exists can one and the same attribute be ascribed that is ascribed to another substance; or (which is the same) in Nature there cannot be two substances, unless they are distinguished realiter.a
Proof If there are two substances, then they are distinct; and consequently (Axiom 2) they are distinguished either realiter or modaliter; not modaliter, for in that case the modes would by their nature be prior to the substance, which is contrary to the first axiom; therefore, realiter; and consequently, what is predicated of the one cannot be predicated of the other, which is what we intended to prove.
PROPOSITION II
One substance cannot be the cause of the existence of another substance.
Proof Such a cause cannot contain in itself anything of such an effect (Prop. 1); because the difference between them is real, and therefore it cannot (Axiom 5) produce it.
PROPOSITION III
Every attribute or substance is by nature infinite, and supremely perfect in its kind.
Proof No substance is produced by another (Prop. 2) and consequently, if it exists, it is either an attribute of God, or it has been its own cause outside God. If the first, then it is necessarily infinite, and supremely perfect in its kind, such as are all other attributes of God. If the second, then it is also necessarily such because (Axiom 6) it could not have limited itself.
PROPOSITION IV
To such an extent does existence pertain by nature to the essence of every substance, that it is impossible to posit in an infinite understanding the Idea of the essence of a substance that does not exist in Nature.
Proof The true essence of an object is something which is realiter different from the Idea of the same object; and this something exists (Axiom 3) either realiter, or is contained in some other thing which exists realiter; from which other thing this essence cannot be distinguished realiter, but only modaliter; such are all the essences of the things which we see, which before they yet existed were already contained in extension, motion, and rest, and when they do exist are not distinguished from extension realiter, but only modaliter. Moreover, it would involve self-contradiction to suppose that the essence of a substance is contained thus in some other thing; because in that case it could not be distinguished from this realiter, contrary to the first proposition; also, it could in that case be produced by the subject which contains it, contrary to the second proposition; and lastly, it could not by its nature be infinite and supremely perfect in its kind, contrary to the third proposition. Therefore, as its essence is not contained in any other thing, it must be a thing that exists through itself.
Corollary Nature is known through itself, and not through any other thing. It consists of infinite attributes every one of them infinite and perfect in its kind; to its essence pertains existence, so that outside it there is no other essence or existence, and it thus coincides exactly with the essence of God who alone is glorious and blessed.
a [B: … in Nature there cannot be posited two substances of one and the same nature.]