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RICE UNIVERSITY STUDIES

(3)                                   = O.

ax

Now, Fermat does not actually form a derivative by a limiting process,
and he does not express his condition as we have just done; but he applies
an operational procedure which ought to delight an algebraist of today.
He replaces the variable
x by x + E, forms the expansion

P(x + E) = P(χ) + P1(x) ■£ + •■• + Pb(x) ∙ E",

and asserts that the maxima and minima are among those points for
which

(4)                            P1(X) = O,

[5, pp. 183 ff.]; [5, ρ. 382]. The algebraic purity of the procedure is com-
mendable, but there is a price on it. With this derivation of his criterion (4),
Fermat cannot properly prove his assertion, and he knows it. And so he
exorcizes the ghost of (the “algebraist”) Diophantus (ca. 250 a.d.) to stand
mathematical surety for him that his assertion is all right. The editors of the
collected works of Fermat were apparently puzzled by this invocation of the
shades of Diophantus, and in a terse footnote they seem to make Diophantus
into an analyst for the nonce. Moritz Cantor, however, observes perspica-
ciously that to Fermat “even infinitesimal considerations were emanations
of number theoretic conception al izations” [.9, p. 858]. Yet as late as 1934
an editor of a German translation of Fermat’s essay most gratuitously
remarks that “Diophantus employs the word ‘approximation procedure’
(Arithmetica, v. 14 and 17) in a sense different from Fermat’s”
[10, ρ. 44].

Continuity. Leibniz was apparently the first knowingly to associate with
the notion of function the attribute of continuity. This was a meaningful
“first,” and we are going to make some remarks on the meaning of it. It must
be stated however that, in the main, Leibniz reflected on this association
philosophically rather than mathematically, so that working mathematicians
probably did not become aware of this association, directly, that is. In-
directly they may have indeed been influenced by it, but it would be difficult
to establish this. Specifically, it would not be easy to trace back Cauchy’s pre-
occupation with the phenomenon of continuity of functions in working
mathematics to Leibniz’ reflections, over a century before, on continuity
of functions in natural and other philosophy.

We must return to the Greeks for a proper beginning. Greek rationality
was aware of continuity from the first [77], and the Greeks had a standard
word for “continuous”
(synechés), which, literally, can best be rendered by



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