imply the conception of the other.

PROP. [III]  Things which have nothing in common cannot be 
             one the cause of the other. 

Proof.- (3:1) If they have nothing in common, it follows that one cannot
be apprehended by means of the other ([A.v] ), and, therefore, one cannot be
the cause of the other ([A.iv] ).  Q.E.D.

PROP. [IV]  Two or more distinct things are distinguished one
            from the other either by the difference of the
            attributes of the substances, or by the difference
            of their modifications.

Proof.- (4:1) Everything which exists, exists either in itself or in 
something else ([A.i] ), that is (by [D.iii] and [D.v] ), nothing is granted
in addition to the understanding, except substance and its modifications.  
(2) Nothing is, therefore, given besides the understanding, by which several 
things may be distinguished one from the other, except the substances, or, 
in other words (see [A.iv] ), their attributes and modifications.  Q.E.D.

PROP. [V]  There cannot exist in the universe two or more
           substances having the same nature or attribute. 

Proof.- (5:1) If several distinct substances be granted, they must be 
distinguished one from the other, either by the difference of their
attributes, or by the difference of their modifications ([iv] ).  (2) If
only by the difference of their attributes, it will be granted that
there cannot be more than one with an identical attribute.  (3) If by
the difference of their modifications, as substance is naturally prior
to its modifications ([i] ), it follows that setting the modifications
aside, and considering substance in itself, that is truly; ([D.iii] and
[D.vi] }, there cannot be conceived one substance different from another,
that is (by [iv] ), there cannot be granted several substances, but
one substance only.  Q.E.D.

PROP. [VI]  One substance cannot be produced by another substance.

Proof.- (6:1) It is impossible that there should be in the universe two 
substances with an identical attribute, i.e. which have anything common
to them both ([ii] ), and, therefore ([iii] ), one cannot be the cause of
another, neither can one be produced by the other.  Q.E.D.

Corollary.- (6:2) Hence it follows that a substance cannot be produced by 
anything external to itself.  (3) For in the universe nothing is granted,
save substances and their modifications (as appears from [A.i] and [D.iii]
and [D.v] ).  (4) Now (by [v] ) substance cannot be produced by another
substance, therefore it cannot be produced by anything external itself.
Q.E.D.

(6:5) This is shown still more readily by the absurdity of the 
contradictory.  (6) For, if substance be produced by an external cause,
the knowledge of it would depend on the knowledge of its cause ([A.iv] ),
and (by [D.iii] ) it would itself not be substance.

PROP. [VII]  Existence belongs to the nature of substance. 

Proof.- (7:1) Substance cannot be produced by anything external (Corollary,
Prop. [vi] ), it must, therefore, be its own cause, that is, its essence
necessarily involves existence, or existence belongs to its nature. 

PROP. [VIII]  Every substance is necessarily infinite. 

Proof.- (8:1) There can be only one substance with an identical attribute, 
and existence follows from its nature ([vii] ); its nature, therefore,
involves existence, either as finite or infinite.  (2) It does not exist
as finite, for (by [D.ii] ) it would then be limited by something else of
the same kind, which would also necessarily exist ([vii] ); and there
would be two substances with an identical attribute, which is absurd ([v] ).
 
(3) It therefore exists as infinite.  Q.E.D.

Note [N.I]- (8:4) As finite existence involves a partial negation, and
infinite existence is the absolute affirmation of the given nature,
it follows (solely from [vii] ) that every substance is necessarily
infinite. 

Note [N.II]- (8:5) No doubt it will be difficult for those who think about 
things loosely, and have not been accustomed to know them by their primary
causes, to comprehend the demonstrations of [vii] : for such persons
make no distinction between the modifications of substances and the
substances themselves, and are ignorant of the manner in which things are
produced; hence they attribute to substances the beginning which they
observe in natural objects.  (8:6) Those who are ignorant of true causes,
make complete confusion, think that trees might talk just as well as men, 
that men might be formed from stones as well as from seed; and imagine
that any form might be changed into any other.  (7) So, also, those who
confuse the two natures, divine and human, readily attribute human passions
to the deity, especially so long as they do not know how passions originate
in the mind.  (8:8) But, if people would consider the nature of substance,
they would have no doubt about the truth of [vii] .  (9) In fact, this 
proposition would be a universal axiom, and accounted a truism.  (10) For,
by substance, would be understood that which is in itself, and is conceived
through itself, that is, something of which the conception requires not the
conception of anything else; whereas modifications exist in something
external to themselves, and a conception of them is formed by means of a
conception of the thing in which they exist.  (8:11) Therefore, we may have
true ideas of non-existent modifications; for, although they may have no
actual existence apart from the conceiving intellect, yet their essence is
so involved in something external to themselves that they may through it be