Good sense is the best distributed thing in the world, for everyone thinks himself to be so well endowed with it that even those who are the most difficult to please in everything else are not at all wont to desire more of it than they have.[Rene Descartes, Discourse on Method, Cress, tr., Hackett (Indianapolis, IN: 1998), p. 1.]
From Montaigne's "On Presumption":
’Tis commonly said that the justest portion Nature has given us of her favors is that of sense; for there is no one who is not contented with his share: is it not reason? whoever should see beyond that, would see beyond his sight. I think my opinions are good and sound, but who does not think the same of his own?
There are other echoes of the essay, though. For instance, much of the essay is concerned with knowing oneself:
The world looks always opposite; I turn my sight inwards, and there fix and employ it. I have no other business but myself, I am eternally meditating upon myself, considering and tasting myself. Other men’s thoughts are ever wandering abroad, if they will but see it; they are still going forward:—
“No one thinks of descending into himself;”
for my part, I circulate in myself. This capacity of trying the truth, whatever it be, in myself, and this free humor of not over easily subjecting my belief, I owe principally to myself; for the strongest and most general imaginations I have are those that, as a man may say, were born with me; they are natural and entirely my own.
Like Part One of Descartes's Discourse, Montaigne also briefly discusses the emptiness of education:
I willingly fall again into the discourse of the vanity of our education, the end of which is not to render us good and wise, but learned, and she has obtained it. She has not taught us to follow and embrace virtue and prudence, but she has imprinted in us their derivation and etymology; we know how to decline Virtue, if we know not how to love it; if we do not know what prudence is really and in effect, and by experience, we have it however by jargon and heart: we are not content to know the extraction, kindred, and alliances of our neighbors; we desire, moreover, to have them our friends and to establish a correspondence and intelligence with them; but this education of ours has taught us definitions, divisions, and partitions of virtue, as so many surnames and branches of a genealogy, without any further care of establishing any familiarity or intimacy betwixt her and us.
The young Descartes, then, at least as presented in the Discourse, comes through his schooling with a perspective that is broadly Montaignian; having had one of the best educations in Europe, he finds himself with an ironic attitude to it. Becoming a mercenary and tramping about Europe, he continues with his mathematical studies and becomes interested in questions of method, and finds a way out of Montaigne-like skepticism into a way of 'seeking truth in the sciences', through the Montaigne-like approach of descending into himself and trying all truths in himself. The Discourse is a Montaignian history of the mind, with a very un-Montaignian end.
How true this history is of Descartes himself is hard to say; it's certainly the case that Descartes is at least stylizing his account for the purpose of his argument. But as the Discourse is a sort of sales pitch for the method, one could see it as primarily a form of persuasive outreach, trying to reach the Bright Young Things of his age, who, like Bright Young Things of every age, flirt with ideas more on the basis of style than on the basis of substance. What will entice a fashionable skeptic more than to lead them by something presented "as a story or, if you prefer, as a fable" (p. 3) from their fashionably skeptical starting-point to a scientific end? Present it in French and it will certainly get you further than a treatise in technical Latin. Pascal in the Pensées clearly had something like the same idea; one finds the influence of Montaigne as a depiction of a state of mind in Pascal as well, and Pascal's Wager is not a random argument, but one designed for fashionable skeptics who divert themselves with gambling. The rational structure of persuasion is narrative more than argument, bon mot more than jargon, conversation more than analysis; but someone who spends his hours gambling can develop an interest in an argument put in the form of the doctrine of chances. Likewise, we have Descartes appealing to the Montaigne-influenced features of fashionable life and drawing in the reader with his backhanded compliments about his top-notch education. But neither of our literary mathematicians are themselves disciples of Montaigne, although clearly deeply familiar with him; Pascal is appealing to the Montaignian aspect of the French mind in order to lead the reader to Christ -- and Descartes is appealing to the sae aspect of the French mind in order to lead the reader to Descartes.