This is quintessential Descartes, and a concise, eloquent and candid expression of the main themes of his philosophy.
In my review I wish to stress a particular aspect of Descartes's method which is neglected in most commentaries, including that of the present translator, namely the fact that it is directly modelled on the axiomatic method of Greek mathematics, and Euclid's Elements in particular. Descartes makes it quite clear that his intention is to widen the scope of the mathematical method to philosophy in general:
"I was most keen on mathematics, because of its certainty and the incontrovertibility of its proofs; but I did not yet see its true use. Believing as I did that its only application was to the mechanical arts, I was astonished that nothing more exalted had been built on such sure and solid foundations." (9 = AT 7)
Indeed, Descartes's definitive statement of his method is such an apt description of the Elements that it could easily have been written by Euclid himself as a preface to this work. Here I quote it in its entirety and point out the obvious parallels with Euclid.
"The first [principle of my method] was never to accept anything as true that I did not incontrovertibly know to be so; that is to say, carefully to avoid both prejudice and premature conclusions; and to include nothing in my judgements other than that which presented itself to my mind so clearly and distinctly, that I would have no occasion to doubt it." (17 = AT 18) This is of course a perfect description of the way Euclid bases his entire work on a few evident postulates and common notions.
"The second was to divide all the difficulties under examination into as many parts as possible, and as many as was required to solve them in the best way." (17 = AT 18) Just as, e.g., Euclid's proof of the Pythagorean theorem relies on some 26 previous propositions, and so on for all other theorems.
"The third was to conduct my thoughts in a given order, beginning with the simplest and most easily understood objects, and gradually ascending, as it were step by step, to the knowledge of the most complex; and positing an order even on those which do not have a natural order or precedence." (17 = AT 18) Again it is hard to imagine how any work could fit this description more perfectly than Euclid's Elements. The last point in particular is something of a peculiarity of mathematics. In mathematics, when faced with two equivalent statements, one picks arbitrarily which to prove first and which to derive as a corollary, and this has nothing to do with any kind of causal hierarchy between them.
"The last was to undertake such complete enumerations and such general surveys that I would be sure to have left nothing out." (17 = AT 19) Cf., for example, Euclid's exhaustive and systematic treatments of "geometric algebra" in Book II and beyond, irrational magnitudes in Book X, and regular polyhedra in Book XIII.
Descartes immediately goes on the emphasise again that his method is modelled on mathematics:
"The long chains of reasonings, every one simple and easy, which geometers habitually employ to reach their most difficult proofs had given me cause to suppose that all those things which fall within the domain of human understanding follow on from each other in the same way, and that as long as one stops oneself taking anything to be true that is not true and sticks to the right order so as to deduce one thing from another, there can be nothing so remote that one cannot eventually reach it, nor so hidden that one cannot discover it. And I had little difficulty in determining those with which it was necessary to begin, for I already knew that I had to begin with the simplest and the easiest to understand; and considering that of all those who had up to now sought truth in the sphere of human knowledge, only mathematicians have been able to discover any proofs, that is, any certain and incontrovertible arguments, I did not doubt that I should begin as they had done." (17-18 = AT 19; cf. 16-19 generally)