According to the introduction, Scharlau and Opolka's book is based on a
course of lectures given by Scharlau. They say, about the history in the
book, "we do not aim at completeness but hope to stir up the interests of
our readers...." There are lists of general references at the end of each
chapter, but little to no documentation of specifics. Mahoney's book is
one of the references at the end of the chapter on Fermat.
I went back and quickly reread their chapter on Fermat. Unless I missed
something, they give no evidence whatsoever for their assertion that the
marginal note was written in 1637. I had naively interpreted their
passage as follows: The date of 1637 is (in some unmentioned way)
authoritatively established. Given that date and the other factors
mentioned, it "seems obvious" that "Fermat quickly became aware of the
incompleteness of the 'proof' of 1637."
Julio having raised the doubt, I also am now curious to know what
justifies that date.
Jack V. Wales, Jr.
5025 Thacher Road
Ojai, CA 93023