Prop. 1. All attribute must be conceived through itself, and through itself alone.
Dem. An attribute is that which the intellect perceives to constitute the essence of substance (Def. 4), and hence must be conceived through itself (Def. 3), which was the first point; and through itself alone; for if an attribute could be conceived through something else, since it constitutes the essence of substance, a substance could be also thus conceived, which is absurd (Def. 3). Therefore, etc.
Prop. 2. No attribute can possess a mode.
Dem. A mode, if it exist, is an accident of substance or (since attribute constitutes the essence of substance) of attribute, or that which is in something else, by whose means it must be conceived (Def. 5). But an attribute must be conceived through itself alone (Prop. 1), and hence cannot possess a mode. Q. E. D.
Prop. 3. Nothing exists but attributes and modes.
Dem. For all things can be conceived only through themselves or through something else. But that which is conceived through itself is an attribute (Defs. 4 and 3, or Prop. 1), and that which is conceived through something else is a mode. Therefore, etc.
Cor. 1. Nothing exists but attributes. If not, suppose something else exists, which (by preceding Prop.) must be a mode. But since a mode (Def. 5) cannot be in itself alone, it must be in something else (Ax. 1), which can only be an attribute, since nothing else exists. But this is absurd, since (Prop. 2) no attribute can possess a mode. Therefore, etc.
Cor. 2. Things can be distinguished only by their attributes. For nothing else exists.
Cor. 3. Two or more attributes of the same nature cannot be given. For since things (preceding Cor.) can be distinguished only by their attributes, if there were two of the same nature they could not be distinguished — i. e., would be one and the same (Ax. 4), Therefore, etc.
Prop. 4. Every attribute is infinite.
Dem. For if finite it must be limited by an attribute (Cor. 1, Prop. 3) of the same nature (Def. 2), which is impossible (Cor. 3, Prop. 3). Therefore, etc.
Cor. There are no finite things. For nothing exists but attributes (Cor. 1, Prop. 3), and they are all infinite.
