conceived.  (12) Whereas the only truth substances can have, external to
the intellect, must consist in their existence, because they are conceived
through themselves.  (8:13) Therefore, for a person to say that he has a
clear and distinct, that is, a true idea of a substance, but that he is not
sure whether such substance exists, would be the same as if he said that he
had a true idea, but was not sure whether or not it was false (a little
consideration will make this plain); or if anyone affirmed that substance
is created, it would be the same as saying that a false idea was true, in
short, the height of absurdity.  (8:14) It must, then, necessarily be 
admitted that the existence of substance as its essence is an eternal
truth.  (8:15) And we can hence conclude by another process of reasoning--
that there is but one such substance.  (16) I think that this may
profitably be done at once; and, in order to proceed regularly with the
demonstration, we must premise:-- 

(8:17) 1. The true definition of a thing neither involves nor
          expresses anything beyond the nature of the thing
          defined. From this it follows that--
(8:18) 2. No definition implies or expresses a certain number of
          individuals, inasmuch as it expresses nothing beyond the
          nature of the thing defined.  (18a) For instance, the
          definition of a triangle expresses nothing beyond the actual
          nature of a triangle: it does not imply any fixed number of
          triangles.
(8:19) 3. There is necessarily for each individual existent thing a
          cause why it should exist.
(8:20) 4. This cause of existence must either be contained in the
          nature and definition of the thing defined, or must be
          postulated apart from such definition.

(8:21) It therefore follows that, if a given number of individual things
exist in nature, there must be some cause for the existence of exactly
that number, neither more nor less.  (22) For example, if twenty men exist
in the universe (for simplicity's sake, I will suppose them existing
simultaneously, and to have had no predecessors), and we want to account
for the existence of these twenty men, it will not be enough to show the
cause of human existence in general; we must also show why there are
exactly twenty men, neither more nor less: for a cause must be assigned
for the existence of each individual.  (8:23) Now this cause cannot be
contained in the actual nature of man, for the true definition of man does
not involve any consideration of the number twenty.  (8:24) Consequently,
the cause for the existence of these twenty men, and, consequently, of each 
of them, must necessarily be sought externally to each individual.
(8:25) Hence we may lay down the absolute rule, that everything which
may consist of several individuals must have an external cause.  (26) And,
as it has been shown already that existence appertains to the nature of
substance, existence must necessarily be included in its definition; and
from its definition alone existence must be deducible.  (8:27) But from
its definition (as we have shown, Notes ii., iii.), we cannot infer the
existence of several substances; therefore it follows that there is only
one substance of the same nature.  Q.E.D. 

PROP. [IX]  The more reality or being a thing has the greater the 
            number of its attributes ([D.iv] ).

PROP. [X]  Each particular attribute of the one substance must be 
           conceived through itself.
 
Proof.- (10:1) An attribute is that which the intellect perceives of 
substance, as constituting its essence ([D.iv] ), and, therefore, must
be conceived through itself ([D.iii] ).  Q.E.D. 

Note.- (10:2) It is thus evident that, though two attributes are, in
fact, conceived as distinct, that is, one without the help of the other,
yet we cannot, therefore, conclude that they constitute two entities,
or two different substances.  (3) For it is the nature of substance that 
each of its attributes is conceived through itself, inasmuch as all the
attributes it has have always existed simultaneously in it, and none could
be produced by any other; but each expresses the reality or being of
substance.  (10:4) It is, then, far from an absurdity to ascribe several
attributes to one substance: for nothing in nature is more clear than
that each and every entity must be conceived under some attribute, and
that its reality or being is in proportion to the number of its attributes
expressing necessity or eternity and infinity.  (5) Consequently it is
abundantly clear, that an absolutely infinite being must necessarily be
defined as consisting in infinite attributes each of which expresses a
certain eternal and infinite essence. 

(10:6) If anyone now ask, by what sign shall he be able to distinguish
different substances, let him read the following propositions, which show
that there is but one substance in the universe, and that it is absolutely
infinite, wherefore such a sign would be sought for in vain.

Prop. [XI]  God, or substance, consisting of infinite attributes, of
            which each expresses eternal and infinite essentiality,
            necessarily exists.
 
Proof.- (11:1) If this be denied, conceive, if possible, that God does
not exist: then his essence does not involve existence.  (2) But this
(by [vii] ) is absurd.  (3) Therefore God necessarily exists. 

Another proof.- (11:4) Of everything whatsoever a cause or reason must b
assigned, either for its existence, or for its non-existence, e.g., if a
triangle exist, a reason or cause must be granted for its existence; if,
on the contrary, it does not exist, a cause must also be granted, which
prevents it from existing, or annuls its existence.  (5) This reason or
cause must either be contained in the nature of the thing in question,
or be external to it.  (6) For instance, the reason for the non-existence
of a square circle is indicated in its nature, namely, because it would
involve a contradiction.  (11:7) On the other hand, the existence of