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

148 ERKKA MAULA & EERO KASANEN
two squares is the greatest one because the arithmetical mean of
E' .... 2 and E"'3 has been subtracted. In that case the problem
becomes one of determining the maximum sum, where Fermat
could have used his method of maxima and minima with the
characteristic auxiliary variable E.
It is worth while to conclude this section on porisms with a
philosophieal note. Every mathematician interested in FLT must
have faced the fact that FLT is an extremely isolated proposition.
Now, a proof is ordinarily so conducted - no matter which
method is employed - that either an absurdity, contradiction or
compatibility is established with respect to previously proved
propositions or accepted axioms, although even the accepted
rules of inference might do. As a rule, the ab sur dit y, contradiction
or compatibility can ultimately be referred to an axiome The
porisms, despite their great generality, as in Porism 1, are not
axioms, however. When a contradiction (or absurdity, or compatibility)
is established with respect to a porism, the proof might
be considered less than perfect by our standards. Fermat must
have thought differently, following Pappus who had stated about
Euclid's Porisms that "the se porisms embody a theory subtle,
natural, necessary, and of considerable generality, which is
fascinating to those who can see and produce results" [7,1:431,
in a sentence bracketed by Hultsch who edited the Collection,
but in full agreement with Pappus' words and tenor in the same
context]. At any event, Fermat's use of Lemma 8 (which rests on
Porism 1) and Porisms 2-3, should be considered as an attempt tD
break the isolation of FLT. In this respect it is to be compared
with Yoichi Miyaoka's reliance on the principles of avantgarde
theoretical physics, in his attempt at a proof of FLT (1988).
As to Porism 2, it is far less precisely worded than the
modern tas te demands. That is due to the historical restriction:
Fermat did not use the Umit concept, contrary ta what earlier
interpretations often claim (see below). Finally, in Porism 3,
Fermat may well have tried the geometric mean instead of the
arithmetical mean. But that leads to a blind alley.
Prao! Reconstruction
In our proof reconstruction of Prop. 5 by Fermat's methods we
employ redu~tio ad absurdum and attempt to prove that if (a,b,c)
is a (2k)-potent acute-angled scalene triangle with integer sides
su ch that a > b > c > 0 are relatively prime, p(a,k), p(b,k), p(c,k)
are sides of an acute-angled triangle, p(a,k+1), p(b,k+1), p(c,k+l)
are sides of an obtuse-angled triangle, their geometric means