The bulk of this book is a compendious catalogue of qualitative explanations of various physical and astronomical phenomena. These things are not Descartes's best work and have proved to be of limited impact and value. More interesting, in my opinion, is Descartes's attempts to build up physics on the basis of his overall philosophy, and his cogito argument in particular. This perspective is really only sustained in parts I and II of the book, so I shall limit my discussion to those parts.
Descartes's philosophical method is modelled on the method of Euclid's Elements, as is clear from Descartes's preface:
"One must begin by searching for ... first causes, that is, for Principles [which] must be so clear and so evident that the human mind cannot doubt of their truth when it attentively considers them ... And then, one must attempt to deduce from these Principles the knowledge of the things which depend upon them, in such a way that there is nothing in the whole sequence of deductions which one makes from them which is not very manifest." (xvii-xviii)
But Descartes is not content with merely adopting the Euclidean method---he also justifies it. He does this by showing that it survives even the most critical examination possible, namely that announced in the first sentence of the text: "whoever is searching for truth must, once in his life, doubt all things" (I.1).
The Euclidean method is the only philosophical method to survive this critical abyss, by the following chain of reasoning.
First we prove our own existence. "We can indeed easily suppose that there is no God, no heaven, no material bodies; and yet even that we ourselves have no hands, or feet, in short, no body; yet we do not on that account suppose that we, who are thinking such things, are nothing: for it is contradictory for us to believe that that which thinks, at the very time when it is thinking, does not exist. And, accordingly, this knowledge, _I think, therefore I am,_ is the first and most certain to be acquired by and present itself to anyone who is philosophizing in correct order." (I.7)
"The knowledge of remaining things depend on a knowledge of God," because the next things the mind feels certain of are basic mathematical facts, but it cannot trust these judgments unless it knows that its creator is not deceitful. Thus "the mind ... discovers [in itself] certain common notions, and forms various proofs from these; and as long as it is concentrating on these proofs it is entirely convinced that they are true. Thus, for example, the mind has in itself the ideas of numbers and figures, and also has among its common notions, _that if equals are added to equals, the results will be equal,_ and other similar ones; from which it is easily proved that the three angles of a triangle are equal to two right angles, etc." But the mind "does not yet know whether it was perhaps created of such a nature that it errs even in those things which appear most evident to it." Therefore "the mind sees that it rightly doubts such things, and cannot have any certain knowledge until it has come to know the author of its origin." (I.13)
The existence of God is established to Descartes's satisfaction by several dubious arguments, most notably the following. "Just as, for example, the mind is entirely convinced that a triangle has three angles which are equal to two right angles, because it perceives that the fact that its three angles equal two right angles is necessarily contained in the idea of a triangle; so, solely because it perceives that necessary and eternal existence is contained in the idea of a supremely perfect being, the mind must clearly conclude that a supremely perfect being exists." (I.14) And all the more since it is "very well know from [our] natural enlightenment" "that that which is more perfect is not produced by an efficient and total cause which is less perfect; and moreover that there cannot be in us the idea or image of anything, of which there does not exist somewhere (either in us or outside us), some Original, which truly contains all its perfections. And because we in no way find in ourselves those supreme perfections of which we have the idea; from that fact alone we rightly conclude that they exist, or certainly once existed, in something different from us; that is, in God." (I.18)
"It follows from this that all the things which we clearly perceive are true, and that the doubts previously listed are removed" (I.30), since "God is not the cause of errors," owing to his perfection, seeing as "the will to deceive certainly never proceeds from anything other than malice, or fear, or weakness; and, consequently, cannot occur in God." (I.29) "Thus, Mathematical truths must no longer be mistrusted by us, since they are most manifest." (I.30)
In the same way we can be sure that material objects exist, since otherwise "it would be impossible to devise any reason for not thinking Him a deceiver" (II.1). But the argument forces upon us the restriction "that the nature of body does not consist in weight, hardness, color, or other similar properties; but in extension alone" (II.4), since a body can easily be conceived to be deprived of its secondary properties (cf. also II.11), but not its extension.
Physics, therefore, must be based on a theory of extended matter and nothing else. Two key characteristics of Cartesian physics follow quite naturally from this starting point, and are indeed introduced almost immediately: relativity of space (II.13-14) and contact mechanics (II.36-52).
The first is a quite unavoidable corollary of Descartes's starting point, since his perspective does not admit the possibility of space as a concept separate from body. Thus he is compelled to argue that "the names 'place' or 'space' do not signify a thing different from the body which is said to be in the place; but only designate its size, shape and situation among other bodies" (II.13). "So when we say that a thing is in a certain place, we understand only that it is in a certain situation in relation to other things" (II.14).
A second rather straightforward consequence of Descartes's starting point is that contact mechanics is the fundamental phenomena in terms of which all other physics must be construed. And indeed Descartes offers a detailed account of contact mechanics almost at once, in II.36-52.