"It would seem from Fermat's correspondence with Descartes as if he had thought out the principles of analytical geometry for himself before reading Descartes' Discours, and had realized that from the equation of a curve (or as he calls it the "specific property") all its properties could be deduced. His extant papers on this subject deal however only with the application of infinitesimals to geometry; it seems probable that these papers are a revision of his original manuscripts (which he destroyed) and were written about 1663, but he was certainly in possession of the general idea of his method for finding maxima and minima as early as 1628 or 1629. Kepler had already remarked that the values of a function immediately adjacent to and on either side of a maximum (or minimum) value must be equal. Fermat applied this to a few examples. Thus to find the maximum value of x(a - x) he took a consecutive value of x, namely x - e where e is very small, and put x(a - x) = (x - e) (a - x + e). Simplifying and ultimately putting e = 0 he got x = \frac{1}{2}a. This value of x makes the given expression a maximum. [This] is the principle of Fermat's method, but his analysis is more involved."