Propositions 1 to 20 Icon

  1. Substance is by nature prior to its modifications.

  2. Two substances, whose attributes are different, have nothing in common.

  • For each must exist in itself, and be conceived through itself.
  • In other words, the conception of one does not imply the conception of the other.
  1. Things which have nothing in common cannot be one the cause of the other.

Proof: If they have nothing in common, it follows that one cannot be apprehended by means of the other (Axiom 5).

  • Therefore, one cannot be the cause of the other (Axiom 4). Q.E.D.
  1. Two or more distinct things are distinguished one from the other, either:
  • by the difference of the attributes of the substances, or
  • the difference of their modifications.

Proof: Everything which exists, exists either in itself or in something else (Axiom 1).

  • Nothing is granted (by Def.3 and 5.) in addition to the understanding, except substance and its modifications.
  • Therefore, nothing is given besides the understanding.
    • The understanding allows several things to be distinguished from the other, except the substances.
  1. Two or more substances cannot have the same nature or attribute.

Proof: Distinct substances must be distinguished from the other, either:

  • by the difference of their attributes, or
    • In this case, an identical attribute can only exist in one.
  • by the difference of their modifications (Prop. 4).
    • In this case, since substance is naturally prior to its modifications (Prop. 1) then, setting the modifications aside, and considering substance in itself, (Def. 3. and 6), we cannot conceive one substance different from another
    • This means there is only 1 subtance
  1. One substance cannot be produced by another substance.

Proof: It is impossible for 2 substances to have an identical attribute (Prop. 2).

  • Therefore (Prop. 3), one cannot be the cause of the other, neither can one be produced by the other. Q.E.D.

Corollary: It follows that a substance cannot be produced by anything external to itself.

  • This is because, in the universe, nothing is granted other than substances and their modifications. This is from Axiom 1 and Def. 3 and 5.

By the last Proposition, substance cannot be produced by another substance.

  • Therefore, it cannot be produced by anything external to itself. Q.E.D.
    • If substance were produced by an external cause, the knowledge of it would depend on the knowledge of its cause (Axiom 4), and (by Def. 3) it would itself not be substance.
  1. Existence belongs to the nature of substances.

Proof: Substance cannot be produced by anything external (Corollary, Prop 6).

  • Therefore, it must be its own cause.
  • Its essence necessarily involves existence, or existence belongs to its nature.
  1. Every substance is necessarily infinite.

Proof: There can only be 1 substance with an identical attribute.

  • Existence follows from its nature (Prop. 7).

Therefore, its nature involves existence, either as finite or infinite.

  • If it were finite then (by Def. 2) it would then be limited by something else of the same kind.
    • This would mean there would be 2 substances with an identical attribute, which is absurd (Prop. 5).

Therefore it exists as infinite. Q.E.D.

Note 1

  • Finite existence involves a partial negation.
  • Infinite existence is the absolute affirmation of the given nature.

It follows (solely from Prop. 7) that every substance is necessarily infinite.

Note 2

People who think about things loosely, and have not been used to know them by their primary causes will find it difficult to comprehend the demonstration of Prop. 7.

Such persons:

  • make no distinction between the modifications of substances and the substances themselves, and
  • are ignorant of how things are produced.

Hence, they may attribute to substances the beginning which they observe in natural objects.

Those who are ignorant of true causes, make complete confusion. They think that:

  • trees might talk just as well as men
  • men might be formed from stones as well as from seed
  • any form might be changed into any other.

So, also, those who confuse the 2 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.

But if people considered the nature of substance, they would have no doubt about the truth of Prop. 7.

In fact, this proposition would be a universal axiom, and accounted a truism.

For, by substance, would be understood that which is in itself, and is conceived through itself. It 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.

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 conceived.

Whereas the only truth substances can have, external to the intellect, must consist in their existence, because they are conceived through themselves.

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 no 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.

It must, then, necessarily be admitted that the existence of substance as its essence is an eternal truth.

  1. The more reality or being a thing has, the greater the number of its attributes (Def. 4).

  2. Each particular attribute of the one substance must be conceived through itself.

Proof: An attribute is that which the intellect perceives of substance, as constituting its essence (Def. 4), and, therefore, must be conceived through itself (Def. 3). Q.E.D.


Two attributes are conceived as distinct. One is conceived without the help of the other.

Yet we cannot conclude that they constitute 2 entities, or 2 different substances.

The nature of substance is 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.

Each expresses the reality or being of substance.

It is then alright to ascribe several attributes to one substance. This is because it is natural 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.

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.

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 in vain. –>

  1. God, or substance, consisting of infinite attributes, of which each expresses eternal and infinite essentiality, necessarily exists.

Proof 1: If this is denied, think that God does not exist and therefore his essence does not involve existence. But Prop. 7 says this is absurd. Therefore, God necessarily exists.

Proof 2: Everything that exists has a cause or reason.

If that cause does not exist, a cause must also be granted to explain why that other cause did not exist.

  • This cause must either be contained in the nature of the thing in question, or be external to it.

For instance, the reason for the non-existence of a square circle is indicated in its nature.

The existence of substance follows also solely from its nature, inasmuch as its nature involves existence. (See Prop. 7)

But the reason for the existence of a triangle or a circle does not follow from the nature of those shapes, but from the order of universal nature in space.

From the latter it follows, either that a triangle necessarily exists, or that it is impossible that it should exist.

It follows therefrom that a thing necessarily exists, if no cause or reason be granted which prevents its existence.

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.

If such a reason or cause should 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.

For if it were of the same nature, God, by that very fact, would be admitted to exist.

But substance of another nature could have nothing in common with God (by Prop. 2), and therefore would be unable either to cause or to destroy his existence.

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.

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. Therefore, God necessarily exists. Q.E.D.

Proof 3: The potentiality of non-existence is a negation of power, and contrariwise the potentiality of existence is a power, as is obvious.

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. Now we exist either in ourselves, or in something else which necessarily exists (see Axiom. 1 and Prop. 7).

Therefore a being absolutely infinite.

In other words, God (Def. 6) necessarily exists. Q.E.D.


In this last proof, I have purposely shown God’s existence à posteriori, so that the proof might be more easily followed, not because, from the same premises, God’s existence does not follow à priori.

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.

Therefore, a being absolutely infinite, such as God, has from himself an absolutely infinite power of existence.

  • Hence he absolutely exists.

Many people will be unable to see the force of this proof because they are used only to consider those things which flow from external causes.

  • They only see such things which quickly come into existence and quickly disappears.

They think that those things that are not so easily brought into existence as more difficult to conceive and therefore more complicated.

This proverb is correct: “What comes quickly, goes quickly.”

I am only referring to substances which (by Prop. vi.) cannot be produced by any external cause.

  • I am not talking about things which come to exist through causes external to themselves.

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.

On the contrary, 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.

Thus, the perfection of a thing does not annul its existence, but, on the contrary, asserts it. 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.

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.

This, I think, will be evident to every moderately attentive reader.

  1. No attribute of substance can be conceived from which it would follow that substance can be divided.

Proof: The parts into which substance as thus conceived would be divided either will retain the nature of substance, or they will not.

If the former, then (by Prop. 8) each part will necessarily be infinite, and (by Prop. 6) self—caused, and (by Prop. 5) will perforce consist of a different attribute, so that, in that case, several substances could be formed out of one substance, which (by Prop. 6) is absurd.

Moreover, the parts (by Prop. 2) would have nothing in common with their whole, and the whole (by Def. iv. and Prop. 10) could both exist and be conceived without its parts, which everyone will admit to be absurd.

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 Prop. 7) is absurd. –>

  1. Substance absolutely infinite is indivisible.
  1. Besides God, no substance can be granted or conceived.
  1. Whatsoever is, is in God, and without God nothing can be, or be conceived.
  1. From the necessity of the divine nature must follow an infinite number of things in infinite ways. That is, all things which can fall within the sphere of infinite intellect.
  1. God acts solely by the laws of his own nature, and is not constrained by anyone.
  1. God is the indwelling and not the transient cause of all things.
  1. God, and all the attributes of God, are eternal.
  1. God’s existence and are one and the same.
  1. All things which follow from the absolute nature of any attribute of God must always exist and be infinite, or, in other words, are eternal and infinite through the said attribute.


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