When he was nineteen years old, Gauss supplied a straightedge-and-
compass construction for the regular heptadecagon—the polygon with
seventeen sides. Moreover, his construction technique showed that poly-
gons with 2 2N
- 1 sides were regular (numbers of this form are known as
Fermat primes,^13 as they were first studied by the French mathemati-
cian Pierre de Fermat, of Fermat’s last theorem fame). More than two
thousand years had elapsed since anyone had shown constructions of
regular polygons other than the constructions known to the ancient
Greeks.
A list of Gauss’s accomplishments would take substantial time and
space—suffice it to say that his career fulfilled its early promise. He is
recognized today as one of the two or three greatest mathematicians of all
time—and this does not even include his noteworthy accomplishments
in the fields of physics and astronomy.
Pierre Wantzel: The Unknown Prodigy
I’m not a mathematical historian, and at the time of the writing of this
book, the name of Pierre Wantzel was unfamiliar to me, and I suspect it
would be equally unfamiliar to many of today’s mathematicians. Wantzel
was born in 1814, the son of a professor of applied mathematics. Like
Gauss, his talent for mathematics manifested itself at an early age—
where Gauss was correcting errors in his father’s accounts, Wantzel was
handling difficult surveying problems when he was only nine years old.
After a brilliant academic career in both high school and college, Wantzel
entered engineering school. However, feeling that he would experience
greater success teaching mathematics than doing engineering, he be-
came a lecturer in analysis at the École Polytechnique—at the same time
that he was a professor of applied mechanics at another college, while also
teaching courses in physics and mathematics at other Parisian universi-
ties.
Gauss had stated that the problems of doubling the cube and trisecting
the angle could not be solved by straightedge-and-compass construction,
but he had not supplied proofs of these assertions. This was standard op-
erating procedure for Gauss in many problems, but it sometimes left his
colleagues in a dilemma as to whether they should work on a particular
problem, only to find that Gauss had previously solved it. Wantzel, how-
ever, was the first to publish proofs of Gauss’s assertions—finally laying
to rest these two problems. Wantzel had also simplified the proof of the
Abel-Ruffini theorem on the roots of polynomials, and used this to show
that an angle was constructable if and only if its sine and cosine were
Impossible Constructions 73