Definitions
1.By that which is self-caused，I mean that of which the essence involves exist-ence，or that of which the nature is only conceivable as existent。
2.A thing is called finite after its kind，when it can be limited by another thing of the same nature。
For instance，a body is called finite because we always conceive another greater body。So，also，a thought is limited by another thought，but a body is not limited by thought，nor a thought by body。
3.By substance，I mean that which is in itself，and is conceived through itself；in other words，that of which a conception can be formed independently of any other con-ception。
4.By attribute，I mean that which the intellect perceives as constituting the es-sence of substance。
5.By mode，I mean the modifications of substance，or that which is in another through which it is also conceived。
6.By God，I mean a being absolutely infinite-that is，a substance consisting of infinite attributes，of which each expresses eternal and infinite essentiality。
Explanation。-I say absolutely infinite，not infinite after its kind；for，of a thing infinite only after its kind，infinite attributes may be denied；but if something is abso-lutely infinite，whatever expresses essence and involves no negation pertains to its es-sence。
7.That thing is called free，which exists solely by the necessity of its own nature，and of which the action is determined by itself alone。On the other hand，that thing is necessary，or rather constrained，which is determined by something external to itself to a fixed and definite method of existence or action。
8.By eternity，I mean existence itself，in so far as it is conceived necessarily to follow solely from the definition of that which is eternal。
Explanation。-Existence of this kind is conceived as an eternal truth，like the es-sence of a thing，and，therefore，cannot be explained by means of continuance or time，though continuance may be conceived without a beginning or end。
Axioms
1.Everything which exists，exists either in itself or in something else。
2.That which cannot be conceived through anything else must be conceived through itself。
3.From a given definite cause an effect necessarily follows；and，on the other hand，if no definite cause be granted，it is impossible that an effect can follow。
4.The knowledge of an effect depends on and involves the knowledge of a cause。
5.Things which have nothing in common cannot be understood，the one by means of the other；the conception of one does not involve the conception of the other。
6.A true idea must agree with its object。
7.If a thing can be conceived as non-existing，its essence does not involve exist-ence。
Propositions
PROP。1.Substance is by nature prior to its modifications。
Proof。-This is clear from Def。3.and 5.
PROP。2.Two substances whose attributes are different have nothing in common。
Proof。-Also evident from Def。3.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。
PROP。3.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，and，therefore，one cannot be the cause of the other。
PROP。4.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。-Everything which exists，exists either in itself or in something else，-that is，outside the understanding there is nothing except substance and its modifications。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，their attrib-utes and modifications。
PROP。5.In nature there cannot be two or more substances of the same nature or attribute。
Proof。-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。If only by the difference of their attributes，it will be granted that there cannot be more than one with an identical attribute。But if by the difference of their modifications-as substance is naturally prior to its modifications，-it follows that set-ting the modifications aside，and considering substance in itself，that is truly，there can-not be conceived one substance different from another，-that is，there cannot be granted several substances，but one substance only。
PROP。6.One substance cannot be produced by another substance。
Proof。-In nature there cannot be two substances of the same attribute，i。e。，which have anything common to them both，and，therefore，one cannot be the cause of another，neither can one be produced by the other。
Corollary。-Hence it follows that a substance cannot be produced by anything ex-ternal to itself。For in nature there is nothing except substances and their modifications。Now substance cannot be produced by another substance，therefore it cannot be pro-duced by anything external itself。This is shown still more readily by the absurdity of the contradictory。For，if substance be produced by an external cause，the knowledge of it would depend on the knowledge of its cause，and it would itself not be substance。
PROP。7.Existence belongs to the nature of substance。
Proof。-Substance cannot be produced by anything external，it must，therefore，be its own cause-that is，its essence necessarily involves existence，or existence be-longs to its nature。
PROP。8.Every substance is necessarily infinite。
Proof。-There can be only one substance with an identical attribute，and existence follows from its nature。Its nature，therefore，involves existence，either as finite or infi-nite。It does not exist as finite，for it would then be limited by something else of the same kind，which would also necessarily exist；and there would be two substances with an identical attribute，which is absurd。It therefore exists as infinite。
PROP。9.The more reality or being a thing has the greater the number of its attrib-utes。
PROP。10.Each particular attribute of the one substance must be conceived through itself。
Proof。-An attribute is that which the intellect perceives of substance，as constitu-ting its essence，and，therefore，must be conceived through itself。
PROP。11.God，or substance，consisting of infinite attributes，of which each ex-presses eternal and infinite essentiality，necessarily exists。
Proof。-If this be denied，conceive，if possible，that God does not exist；then his essence does not involve existence。But this is absurd。Therefore God necessarily exists。
Another proof。-Of everything whatsoever a cause or reason must be assigned，ei-ther 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。This rea-son or cause must either be contained in the nature of the thing，or be external to it。For instance，the reason for the non-existence of a square circle is indicated in its nature，namely，because it would involve a contradiction。On the other hand，the existence of substance follows also solely from its nature，inasmuch as its nature involves existence。
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 the whole of corporeal Nature。From the latter it must follow，either that a triangle necessarily exists，or that it is impossible that it should exist。So much is self-evident。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 there were such a reason，or cause，it would have to be either in God s very nature or outside it，that is，in 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 noth-ing in common with God，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 necessari-ly exists。
PROP。12.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 each part will necessarily be infinite，and self-caused，and will perforce consist of a different attrib-ute，so that，in that case，several substances could be formed out of one substance，which is absurd。Moreover，the parts would have nothing in common with their whole，and the whole could both exist and be conceived without its parts，which is absurd，as no one will be able to doubt。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 e-qual parts，it would lose the nature of substance，and would cease to exist，which is ab-surd。
PROP。13.Substance absolutely infinite is indivisible。
Proof。-If it could be divided，the parts into which it was divided would either re-tain the nature of absolutely infinite substance，or they would not。If the former，we should have several substances of the same nature，which is absurd。If the latter，then substance absolutely infinite could cease to exist，which is also absurd。
Corollary。-It follows that no substance，and consequently no corporeal substance，in so far as it is substance，is divisible。
PROP。14.Except God，no substance can be or be conceived。
Proof。-As God is a being absolutely infinite，of whom no attribute that expresses the essence of substance can be denied，and he necessarily exists，if any substance be-sides God were granted it would have to be explained by some attribute of God，and thus two substances with the same attribute would exist，which is absurd。Therefore，besides God no substance can be granted，or consequently，be conceived。If it could be con-ceived，it would necessarily have to be conceived as existent；but this is absurd。There-fore，besides God no substance can be granted or conceived。
Corollary 1.-Clearly，therefore：1.God is one，that is only one substance can be granted in the universe，and that substance is absolutely infinite。
Corollary 2.-It follows：2.That extension and thought are either attributes of God or modifications of the attributes of God。
PROP。15.Whatsoever is，is in God，and without God nothing can be，or be con-ceived。
Proof。-Besides God，no substance is granted or can be conceived，that is nothing which is in itself and is conceived through itself。But modes can neither be，nor be con-ceived without substance；wherefore they can only be in the divine nature，and can only through it be conceived。But except for substances and modes there is nothing，there-fore，without God nothing can be，or be conceived。
PROP。16.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。
Proof。-This Proposition will be clear to everyone，who remembers that from the given definition of any thing the intellect infers several Properties，which really necessa-rily follow therefrom（that is，from the actual essence of the thing）；and that it infers more Properties the more the definition of the thing expresses reality，that is，the more reality the essence of the defined thing involves。Now，as the divine nature has absolute-ly infinite attributes，of which each expresses infinite essence after its kind，it follows that from the necessity of its nature an infinite number of things（that is，everything which can fall within the sphere of an infinite intellect）must necessarily follow。
Corollary 1.-Hence it follows，that God is the efficient cause of all that can fall within the sphere of an infinite intellect。
Corollary 2.-It also follows that God is a cause through himself and not an acci-dental cause。
Corollary 3.-It follows，thirdly，that God is the absolutely first cause。
PROP。17.God acts solely by the laws of his own nature，and is not constrained by any one。
Proof。-We have just shown，that from the necessity of the divine nature alone，or（what is the same thing）from the laws of his nature alone，absolutely infinite things fol-low；and we proved，that without God nothing can be，nor be conceived；but that all things are in God。So there can be nothing outside him by which he is determined or compelled to act。Therefore，God acts from the laws of his nature alone，and is com-pelled by no one。
Corollary 1.-It follows：I。That there can be no cause which，either extrinsically or intrinsically，besides the perfection of his own nature，moves God to act。
Corollary 2.-It follows：2.That God is the sole free cause。For God alone exists by the sole necessity of his nature，and acts by the sole necessity of his nature，where-fore God is the sole free cause。
PROP。18.God is the indwelling and not the transient cause of all things。
Proof。-All things which are，are in God，and must be conceived through God，therefore God is the cause of things which are in him。This is the first。Further，besides God there can be no substance，that is nothing in itself external to God。This is the sec-ond。God，therefore，is the indwelling and not the transient cause of all things。
PROP。19.God，and all the attributes of God，are eternal。
Proof。-God is substance，which necessarily exists，that is existence appertains to its nature，or（what is the same thing）follows from its definition；therefore，God is e-ternal。Further，by the attributes of God we must understand that which expresses the essence of the divine substance，in other words，that which appertains to substance；that，I say，should be involved in the attributes of substance。Now eternity appertains to the nature of substance；therefore，eternity must appertain to each of the attributes，and thus all are eternal。
Note。-This Proposition is also evident from the manner in which I demonstrated the existence of God；it is evident，I repeat，from that proof，that the existence of God，like his essence，is an eternal truth Further，I have proved the eternity of God，in an-other manner，which I need not here repeat。