including Leonhard Euler and Joseph-Louis Lagrange. The latter pub-
lished a famous paper, Ref lections on the Resolution of Algebraic Equations,
in which he stated that he planned to return to the solution of the quintic,
which he obviously hoped to solve by radicals.
Paolo Ruffini was the first mathematician to suggest that the quintic
could not be solved by radicals, and he offered a proof of it in General
Theory of Equations in Which It Is Shown That the Algebraic Solution of the
General Equation of Degree Greater Than Four Is Impossible. In it, he states,
“The algebraic solution of general equations of degree greater than four is
always impossible. Behold a very important theorem which I believe I am
able to assert (if I do not err): to present the proof of it is the main reason
for publishing this volume. The immortal Lagrange, with his sublime
ref lections, has provided the basis of my proof.”^9
Unfortunately, that introduction turned out to be prescient—there was
a gap in his proof. However, not only had Ruffini glimpsed the truth, he
had realized that the path to the solution led through an analysis of what
happened to equations when the roots of a polynomial were permuted.
Even though he did not formalize the idea of a permutation group, he
proved many of the initial basic results in the theory.
Ruffini was yet another mathematician to be dogged by bad luck. He
never really received credit for his work—at least in his lifetime. The only
top mathematician to give him the respect he deserved was Augustin-
Louis Cauchy, but when his paper was examined by leading French and
English mathematicians, the reviews were neutral (the English) to unfa-
vorable (the French). Ruffini was never notified that his proof contained a
gap—had a leading mathematician done so, he would have had a shot at
patching the proof. Usually, the person most familiar with a f lawed proof
has the best chance of fixing it—but Ruffini was never given the chance.
Groups in General—Permutation Groups in Particular
One of the most important accomplishments of mathematics is that it has
shown that apparently dissimilar structures possess many important
common attributes. These attributes can be codified into a set of axioms,
and conclusions derived for all structures that satisfy those axioms. One
of the most important such objects is called a group.
To motivate the definition of a group, consider the set of all nonzero real
numbers. The product of any two nonzero real numbers x and y is a
nonzero real number xy; this product satisfies the associative law:
x(yz)(xy)z. The number 1 has the property that for any nonzero real
number, 1xx 1 x. Finally, each nonzero real number has a multiplica-
90 How Math Explains the World