substance follows also solely from its nature, inasmuch as its nature
involves existence. (See [vii] ) 

(11:8) But the reason for the existence of a triangle or a circle does not 
follow from the nature of those figures, but from the order of universal
nature in extension.  (9) From the latter it must follow, either that a
triangle necessarily exists, or that it is impossible that it should exist.
(11:10) So much is self-evident.  (11) It follows therefrom that a thing
necessarily exists, if no cause or reason be granted which prevents it
existence.

(11:12) If, then, no cause or reason can be given, which prevents the
existence of God, or which destroys his existence, we must certainly
conclude that he necessarily does exist.  (13) If such a reason or cause
hould be given, it must either be drawn from the very nature of God, or
be external to him, that is, drawn from another substance of another
nature.  (11:14) For if it were of the same nature, God, by that very fact,
would be admitted to exist.  (15) But substance of another nature could
have nothing in common with God (by [ii] ), and therefore would be unable
either to cause or to destroy his existence.

(11:16) As, then, a reason or cause which would annul the divine existence
cannot be drawn from anything external to the divine nature, such cause
must, perforce, if God does not exist, be drawn from God's own nature,
which would involve a contradiction.  (17) To make such an affirmation
about a being absolutely infinite and supremely perfect, is absurd;
therefore, neither in the nature of God; nor externally to his nature,
can a cause or reason be assigned which would annul his existence.
(11:18) Therefore, God necessarily exists.  Q.E.D.
 
Another proof.- (11:19) The potentiality of non-existence is a negation
of power, and contrariwise the potentiality of existence is a power, as
is obvious.  (20) If, then, that which necessarily exists is nothing but
finite beings, such finite beings are more powerful than a being
absolutely infinite, which is obviously absurd; therefore, either nothing
exists, or else a being absolutely infinite necessarily exists also.
(11:21) Now we exist either in ourselves, or in something else which
necessarily exists (see [A.i] and [vii] )  (22) Therefore a being
absolutely infinite, in other words, God ([D.vi] ), necessarily exists. 
Q.E.D.
 
Note. - (11:23) In this last proof, I have purposely shown God's existence
a posteriori, so that the proof might be more easily followed, not because,
from the same premises, God's existence does not follow a priori.
(11:24) For, as the potentiality of existence is a power, it follows that, 
in proportion as reality increases in the nature of a thing, so also will
it increase its strength for existence.  (25) Therefore a being absolutely
infinite, such as God, has from himself an absolutely infinite power of
existence, and hence he does absolutely exist.  (26) Perhaps there will
be many who will be unable to see the force of this proof, inasmuch as
they are accustomed only to consider those things which flow from external
causes.  (27) Of such things, they see that those which quickly come to
pass, that is, quickly come into existence, quickly also disappear;
whereas they regard as more difficult of accomplishment, that is, not so
easily brought into existence, those things which they conceive as more
complicated.

(11:28) However, to do away with this misconception, I need not here show
the measure of truth in the proverb, "What comes quickly, goes quickly,"
nor discuss whether, from the point of view of universal nature, all
things are equally easy, or otherwise: I need only remark, that I am not
here speaking of things, which come to pass through causes external to
themselves, but only of substances which (by [vi] ) cannot be produced by
any external cause.  (29) Things which are produced by external causes,
whether they consist of many parts or few, owe whatsoever perfection or
reality they possess solely to the efficacy of their external cause, and
therefore their existence arises solely from the perfection of their
external cause, not from their own.  (11:30) Contrariwise, whatsoever
perfection is possessed by substance is due to no external cause;
wherefore the existence of substance must arise solely from its own
nature, which is nothing else but its essence.  (31) Thus, the perfection
of a thing does not annul its existence, but, on the contrary, asserts it.  
(11:32) Imperfection, on the other hand, does annul it; therefore we
cannot be more certain of the existence of anything, than of the existence
of a being absolutely infinite or perfect--that is, of God.  (33) For
inasmuch as his essence excludes all imperfection, and involves absolute
perfection, all cause for doubt concerning his existence is done away,
and the utmost certainty on the question is given.  (11:34) This, I think,
will be evident to every moderately attentive reader.

PROP. [XII]  No attribute of substance can be conceived from which
             it would follow that substance can be divided.

Proof.- (12:1) The parts into which substance as thus conceived would be 
divided, either will retain the nature of substance, or they will not.
(2) If the former, then (by [viii] ) each part will necessarily be
infinite, and (by [vi] ) self-caused, and (by [v] ) will perforce
consist of a different attribute, so that, in that case, several substances
could be formed out of one substance, which (by [vi] ) is absurd.
(12:3) Moreover, the parts (by [ii] ) would have nothing in common
with their whole, and the whole (by [D.iv]  and [x] ) could both
exist and be conceived without its parts, which everyone will admit to
be absurd.  (4) If we adopt the second alternative, namely, that the
parts will not retain the nature of substance, then, if the whole
substance were divided into equal parts, it would lose the nature of
substance, and would cease to exist, which (by [vii] ) is absurd.

PROP. [XIII]  Substance absolutely infinite is indivisible.

Proof.- (13:1) If it could be divided, the parts into which it was divided