In the Replies to the Second Objections to the Meditations on First Philosophy, Descartes gives a synthetic rendering of the argument of the Meditations (which proceeds analytically). It's done in a right geometrical style, with definitions, postulates, and axioms. The definitions are the obvious things you would expect (the first three definitions are for thought, idea, and objective reality of an idea). The postulates and axioms are somewhat more interesting, I think.
In a footnote to the CSM edition, the editors say about Descartes's use of the term postulate, "Descartes is here playing on words, since what follows is not a set of postulates in the Euclidian sense, but a number of informal requests" (CSM 114). Likewise, Bennett in his paraphrase of the Second Replies (PDF) has an explanatory note in which he gives this purported difference in sense as a reason for thinking that Descartes is not taking the geometrical exposition very seriously. However, I think Descartes is being a bit more deliberate than either of these comments suggest, and is not being as equivocal as they require. Euclid's postulates are put in request format, too; they are things you are asked to do in order to draw your constructions correctly. What is more, looking at the postulates we see that they lay out quite clearly Descartes's entire method. The only postulates that are explicitly used in the proofs that follow are Postulate 2 (reflecting on one's own mind until one can recognize that its existence is certain and, indeed, more certain than the existence of one's body) and Postulate 5 (reflecting on the nature of supremely perfect being as including necessary existence). Most of the postulates are things you are asked to do in order to recognize the axioms as self-evident or things you are asked to do in order to understand what clear and distinct perception are. And, of course, self-evidence and clear and distinct perception are closely linked for Descartes. It's reasonable, I think, to regard Descartes's postulates as guidelines for philosophical meditation, the end result of which is grasping things as clearly and distinctly true, or, as we might also say, as self-evident. This intellectual grasp of the evident is to Descartes's metaphysics what straightedge-and-compass construction is to Euclid's geometry. Metaphysics is more distant from sensible matters than geometry is, so the thing in metaphysics that corresponds to construction in geometry is proportionately more purely abstract and intellectual. It's interesting to compare Descartes's exposition in this respect with the other famous Cartesian synthetic discussion, Spinoza's Ethics. Spinoza gives eight postulates, and they are very different from anything we find in Descartes. All of them have to do with the human body, for instance; in a sense we can say that for Spinoza the human body is to metaphysics what the diagram is to geometry. It is impossible to imagine Descartes thinking in this way; all of his postulates are mental, laying out particular activities of reflection that one must engage in if one is to understand Descartes's arguments. [Added Later: Of course, Spinoza really uses the postulates for ethics, or applying metaphysics to questions of human happiness, rather than metaphysics as such, so this has to be factored into how we understand the difference.]
The postulates also emphasize, I think, that the synthetic approach is not a substitute for the analytic approach: in both cases exactly the same meditation is necessary. This fits with what Descartes himself says about it; he gives the synthetic format not as an alternative but as an "exposition" for those who will have difficulty with keeping track of the overall structure of the Meditations even when they concentrate properly so as to understand its parts; and, says Descartes, "I am convinced that it is the Meditations which will yield by far the greater benefit" (CSM 113). The main point of the synthetic exposition, I take it, is simply to show that all the parts flow in a fairly straightforward way from the act of philosophical meditation, and that, therefore, the whole Meditations hangs together coherently despite its conversational tone and apparent digressions. For Descartes, however, the postulates are the real pith of his entire approach; and that is because postulates by their nature define a method, and Descartes's whole focus is on presenting a method. In discussing the difference between analysis and synthesis, Descartes insists that only analysis is really suitable to discovery, and that it is "the best and truest method of instruction" (CSM 111). He also says that synthesis is harder in metaphysics because "there is nothing which causes so much effort as making our perception of the primary notions clear an ddistinct" (CSM 111), thus showing that the key issue is clear and distinct perception -- which is found entirely in the postulates.
Following the postulates very carefully, we are supposed to be able to grasp the self-evidence of the axioms. Descartes gives ten. Taking out the explanations to leave the bare axiom, and doing some occasional minor re-writing for clarity, those ten axioms of metaphysics are (CSM 116-117):
(1) For every existing thing, it is possible to ask what is the cause of its existence.
(2) No less a cause is required to preserve something than is required to create it in the first place.
(3) What does not exist cannot be the cause of the existence of anything, nor can it be the cause of the existence of any actual perfection of anything.
(4) Whatever reality or perfection there is in a thing is found either formally or eminently in its first and adequate cause.
(5) The objective reality of our ideas needs a cause which contains this reality not merely objectively but formally or eminently.
(6) A substance has more reality than an accident or a mode, and an infinite substance has more reality than a finite substance.
(7) The will of a thinking thing is drawn voluntarily and freely, "but nevertheless inevitably," towards a clearly known good.
(8) Whatever can bring about a greater or more difficult thing can also bring about a lesser thing.
(9) It is a greater thing to create or preserve a substance than to create or preserve the attributes or properties of that substance.
(10)Existence is contained in the idea or concept of every single thing.
Descartes is quite clear that not all of these axioms are strictly necessary; he is giving an exposition, and therefore is listing some things that "should have been introduced as theorems rather than as axioms, had I wished to be more precise" (CSM 116). Axiom 5 is explicitly presented as following from Axiom 4. Given things that Descartes says in the Meditations, I suspect that Axiom 4 in turn is supposed to follow from Axiom 3, and that Axioms 8 and 9 probably also have the same root source. In other words, I suspect that 4, 5, 8, and 9 are all things that Descartes thinks would, in a more detailed exposition, be derived from the combination of Axiom 3 with various definitions.
These axioms are combined with various definitions in order to get four propositions, which I put here along with the axioms (explicitly) used in the proofs provided for them.
I. The existence of God can be known merely by considering his nature: 10.
II. The existence of God can be demonstrated a posteriori merely from the fact that we have an idea of God within us: 3, 5, 6.
III. The existence of God can be demonstrated from the fact that we who possess the idea of God exist: 1, 2, 4, 7, 8, 9.
IV. There is a real distinction between the mind and the body: no axioms explicitly used (the proposition is derived by combining definitions with a corollary of III and Postulate 2).
The upshot of this, I think, is again that the postulates are where the action is: the axioms themselves do relatively little work, and, indeed, seem mostly to just contribute to Descartes's goal of explaining the overall structure of the Meditations. The axioms really are things that you'll inevitably accept if you follow the postulates closely and carefully, and thus show how the conclusions Descartes reaches follow almost directly from his method, properly practiced. This fits with the claims by Descartes that we noted above, and is another notable contrast to Spinoza.
