proven to have solutions by radicals, but one can say with some assurance
that mathematics is a lot more interesting a subject because the quintic
does not have such a solution.
Mathematics is a language used to describe a variety of phenomena—
but a language needs words. Some of the most important words in the
mat hemat ical lang uage are func t ions. Func t ions, such as powers or roots,
can be combined in two basic ways—algebraically (using addition, sub-
traction, multiplication, and division), and compositionally (one after the
other, like successive shuff les—one can square a number and then take
its cube root). The insolvability of the quintic amounts to a declaration
that the vocabulary of functions that can be constructed with powers and
roots is inadequate to describe the solutions of a certain equation. This
naturally stimulated a search for other functions that could be used to
describe these solutions.
Where do functions come from? Often they arise from need. The trigo-
nometric functions are used for expressing quantities determined by an-
gles, as well as in describing periodic phenomena, and the exponential
and logarithmic functions are used for describing growth and decay proc-
esses. Many functions arise as solutions to important equations (usually
differential equations) that occur in science and engineering. For exam-
ple, Bessel functions (named after the nineteenth-century mathemati-
cian and physicist William Bessel, who was the first to calculate the
distance to a star) occur as solutions to the problem of how a membrane
such as a drum vibrates when it is struck, or how heat is conducted in a
cylindrical bar.
In 1872, the German mathematician Felix Klein was able to find a gen-
eral solution for the quintic in terms of hypergeometric functions, a class
of functions that occur as a solution to the hypergeometric differential
equation.^12 In 1911, Arthur Coble solved the sextic, the general polyno-
mial of degree six, in terms of Kampé de Fériet functions—a class of
functions of which I had never heard and I doubt that 99 percent of living
mathematicians have, either. The trend appears bleak—it looks as if the
general solution to polynomials of ever-higher degree, if such solutions
can be found, will be in terms of ever-more obscure classes of functions.
Functions are indeed like words: their utility depends largely on the fre-
quency with which they are used, and functions (or words) that are so
specialized that only a few know them have limited value.
The solving of equations is central not only to mathematics, but to the
sciences and engineering. Mathematicians may be interested to know
that the solution to a particular equation exists, but to build something it
is necessary to know what that solution is—and to know it to three, five,
The Hope Diamond of Mathematics 97