CHEZ FERMAT A.D. 1637' Erkka Maula and Eero Kasanen Abstract ...

156 ERKKA MAULA & EERO KASANEN
tian and four dimensions were needed for the new numbers. As
Kline puts it, geometric hindsight can show us that the rotation
and the stretching or contraction of a given vector in physical
space requires four parameters; three angles and a stretching
factor. More formally, Frobenius has proved thst "the only linear
associative algebras with real coefficients (of the primary units),
with a finite number of primary units, a unit element for multiplication,
and obeying the product law are those of the resl
numbers, lhe complex numbers and real quaternions" (cf. Morris
Kline 1972:779,793). Nevertheless, triples' of integers are flexible
in other respects. Although three-dimensional vectors are used
in the vector analysis of modern physics or in mathematics as a
special case of a n-dimensional linear algebra, it seems that the
study of triangular numbers as separate entities has been
sidestepped because of the historical process that led to the
physics-orienled quaternions. Second, interpreted as integerside
triangles, they constituted Fermat's ontology of mathematics.
Their metamorphosis into furca crosses we have barely
alluded to (cf. Fig.3).
There is no need to dwell on the ancestry of mathematical
atomism (at times distinguished from, at times merging together
with, physical atomism). It may be recalled, however, that a very
eloquent prologue was given in Zeno's "Stadion" and "Flying
Arrow". In the subsequent dialogue, atoms or indivisibles were
made use of in many ways by men like Democritus, Plato,
Aristotle, Archimedes, Heron, Oresme, Nicholas of Cusa, Galileo,
Kepler, Pascal, Huygens, Cavalieri, Torricelli and Roberval besides
Fermat. With sorne notable exceptions, like Kronecker,
atomism is rejected in modern analysis; its kingdom is number
theory.
It is worthwhile, however, to get rid of a misunderstanding
as to Fermat's use of the auxiliary variable E. In his method of
the maxima and minima, Fermat's procedure is aImost precisely
the same now employed in the differential calculus, except that
dx is substituted for E. No wonder, therefore, that Fermat's
argument for his method is at times interpreted in terms of the
limit concept (sa that E becomes a variable quantity approaching
zero; cf. for instance Duhamel's (1864) "Mémoire sur la méthode
des maxima et minima de Fermat, et sur les méthodes des tangentes
de Fermat et Descartes"). A much more reasonable interpretation
is,. however, that Fermat let E vanish in the sense of
actually being zero (cf. Tannery 1902:344; Wallner 1904:122-123;
Boyer 1949:154 ff.). We have adopted this interpretation
throughout. For firstly, in so doing we need not ascribe to
Fermat advanced notions not corroborated by his own words.