Pioneer |
Journal where reappraisal appeared |
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Pierre Fermat | Perspectives on Science | 13e |

Pierre Fermat | Foundations of Science | 18d |

Pierre Fermat | Journal of Humanistic Mathematics | 20e, Sections 3.2-3.5 |

The method of *adequality* was the method with which Fermat
approached the problems of the calculus. We analyze its source in the
*parisotes* of Diophantus.
Adequality is a crucial step in Fermat's method of finding maxima,
minima, tangents, and solving other problems that a modern
mathematician would solve using infinitesimal calculus. The method is
presented in a series of short articles in Fermat's collected works
(Tannery).

We show that at least some of the manifestations of adequality amount
to variational techniques exploiting a small, or infinitesimal,
variation *E*.
Fermat's treatment of geometric and physical applications suggests
that an aspect of approximation is inherent in adequality, as well as
an aspect of smallness on the part of *E*, as in the
familiar difference *f(A+E)-f(A)*. We question the relevance
to understanding Fermat of 19th century dictionary definitions
of *parisotes* and adaequare, cited by Breger, and take issue
with his interpretation of adequality, including his novel reading of
Diophantus, and his hypothesis concerning alleged tampering with
Fermat's texts by Carcavy. We argue that Fermat relied on Bachet's
reading of Diophantus.

Diophantus coined the term *parisotes* for mathematical
purposes and used it to refer to the way in which 1321/711 is
approximately equal to 11/6. Bachet performed a semantic calque in
passing from parisoō to adaequo.
We note the similar role of, respectively, adequality and the
Transcendental Law of Homogeneity in the work of, respectively, Fermat
and Leibniz on the problem of maxima and minima.

The first half of the 17th century was a time of intellectual ferment
when wars of natural philosophy were echoes of religious wars, as we
illustrate by a case study of an apparently innocuous mathematical
technique called *adequality* pioneered by the honorable
*judge* Pierre de Fermat, its relation to indivisibles, as well
as to other hocus-pocus. André Weil noted that simple
applications of adequality involving polynomials can be treated purely
algebraically but more general problems like the cycloid curve cannot
be so treated and require additional tools - leading the
*mathematician* Fermat potentially into troubled waters.
Breger attacks Tannery for tampering with Fermat's manuscript but it
is Breger who tampers with Fermat's procedure by moving all terms to
the left-hand side so as to accord better with Breger's own
interpretation emphasizing the *double root* idea. We provide
modern proxies for Fermat's procedures in terms of relations of
infinite proximity as well as the standard part principle. For an
explanation of the technical terms see
Introduction to infinitesimal analysis without the
axiom of choice

See also

Fermat's opponents
(MO)

Lalouvère's
activities as censor (MO)

More on infinitesimals

Stevin

Leibniz

Euler

Cauchy

Riemann

Cantor

Skolem

Infinitesimal topics

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