The style of Spinoza Ethics, demonstrated in geometrical order is a tribute to Euclids Elements as the subtitle itself points out.
The Elements is written in using Definitions, Axioms, Propositions & Corollaries.
Definitions explain the meaning of a term, as in the first definition of Ethics which defines the term causa sui, or cause of itself (in the modernised edition by Jonathan Bennet):
In calling something ‘cause of itself’ I mean that its essence involves existence, i.e. that its nature can’t be conceived except as existing.
compare to this to the first definition of Elements which defines a point:
A point is that which has no part
Now, one might say a point is natural & intuitive; it is something that is visible, or 'clear and distinct' to the eye. It takes a moments thought to see that 'it has no parts'. But, actually this covers up a long debate going by Zeno and the atomists which defined an 'atom as having no part'. An atom is metaphysical: it has no basis in reality. Similarly, a point is metaphysical in the same sense. But we say that it is 'clear and distinct' because of the force of the argument, and the clear connection with in the one hand with matter, and the other with geometry. By Euclids time, one supposes the definition of a point was accepted by geometers.
Similarly, by Spinozas time, the definition of 'cause of itself' was accepted by philosophers & theologians; it already summarises a long & distinguished tradition. (I suspect he is referring to Averroes rejection of Avicenna who places essence as prior to existence, he instead follows Aristotle in claiming that for primary substances essence & existence are equivalent. That this trajectory had further momentum is shown by Sartres slogan for Existentialism- existence precedes essence).
A definition, when widely accepted becomes a tautology. It becomes trite, since both sides of the definition becomes terms accepted in themselves; so when they are put together like this it seems nothing new has been said or added. Looking at the definition of a point today, one easily forgets the early & turbulent discussions of the atomists when this definition was forged. For example Plato supported infinite divisibility in the Timaeus against the atomists.
Spinoza uses Descartes terminology 'clear and distinct'. In Descartes Fourth Meditation, he states everything we can percieve 'clearly and distinctly' is true. This comes from his theory of perception. I, for example, right now 'clearly and distinctly' see a mug of coffee next to me, and I cannot deny that. It is true. Descarte is arguing if an idea has a similar conidition of being seen as 'clearly and distinctly' then it is has the same quality of being true.
Spinoza does nothing so 'silly' as proving God. He clearly understands this is something that is of the order of being unquestionable and/or unprovable. His treatise after all is called Ethics, and he is grounding a conception of ethics in a particular style of reasoning on his ontology, as he clearly states in the subtitle: demonstrated in geometrical order.
In the first eight definitions he defines the following terms:
- cause of itself
- finite in its own kind
He follows this with seven axioms, including this one - which is of interest when comparing his system to Kant:
A true idea must agree with its object
It is the inversion of this axiom that Kant starts his Critique of Reason, calling it his 'Copernican Revolution' in recognition of its revolutionary epistemology. That is an object must conform to the idea.
In any axiomatic system, one can easily 'destroy' the system by denying the rigor of a definition, or some part of it; or by denying the truth of an axiom. After all, the whole of Euclids Elements is technically invalidated if one denies there is such a thing as a dimensionless point (and I should point out there modern theories of geometries that do just that - some geometries having no points whatsoever). But this move is a strange one if one wants to understand the importance of the Elements.
Further, axiomatisation is sometimes thought as the beginning of exact knowledge, which in a sense it is; but also it consolidates in an efficient & consistent manner accepted knowledge.
So, Spinoza isn't proving Gods existence as he in fact takes it as a given, he is merely placing his understanding of the nature of God, his ontology & ethics on an efficient basis on concepts generated in the Greek tradition - substance, essence, God ; and a definition is not the same as a tautology, though they appear to have similarities (one does not normally, at least in mathematics, think of definitions and axioms as tautologies, but when we consider two propositions that can be shown to be equivalent on the basis of those definitions and axioms)
Finally, which is something incidental to your question, but I can't resist mentioning, is that a great deal of Spinozas understanding of God is much closer to mainstream Islamic rather than Jewish Or Christian Theology. Since he was from the Portugeuse Jewish diaspora after the collapse of the Islamic Amoravid kingdoms in Iberia, it seems only natural to suspect some influence in that direction.