<begin quote>
It has often been said that attempts to prove the
fundamental theorem began with d'Alembert [1746],
and that the first satisfactory proof was given by
Gauss [1799]. This opinion should not be accepted
without question, as the source of it is Gauss
himself. ... The opinion as to which of two
incomplete proofs is more convincing can of
course change with time, and I believe that
Gauss [1799] might be judged differently today.
We can now fill the gaps in d'Alembert [1746]
by appeal to standard methods and theorems,
whereas there is still no easy way to fill the
gap in Gauss [1799]. ... As for the concept of
continuity, neither Gauss nor d'Alembert
understood it very well. ... The first to
appreciate the importance of continuity
for the fundamental theory of algebra was
Bolzano [1817], who proved the continuity
of polynomial functions and attempted a
proof of the intermediate value theorem.
The latter proof was unsatisfactory because
Bolzano had no clear concept of real number
on which to base it, but it did point in
the right direction. When a definition
of real number emerged in the 1870's
(e.g., with Dedekind cuts; Section 4.2),
Weierstrass [1874] rigorously established
the basic properties of continuous functions,
such as the intermediate value theorem and
extreme value theorem. This completed
not only the second proof of Gauss but
also the proof of d'Alembert ...
<end quote>
I'm not sure what you should teach in
a history course, but in any other math
course I would use d'Alembert's proof,
not Gauss's. It is much shorter and
simpler, if you are approaching things
with modern terminology, modern rigor,
modern tools, etc.
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Eric Schechter 615-322-6651 615-662-4442
http://www.math.vanderbilt.edu/~schectex/
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