Men of Mathematics (49 page)

The theory of analytic functions of a complex variable was one of the greatest fields of mathematical triumphs in the nineteenth century. Gauss in his letter to Bessel states what amounts to the fundamental theorem in this vast theory, but he hid it away to be rediscovered by Cauchy and later Weierstrass. As this is a landmark in the history of mathematical analysis we shall briefly describe it, omitting all refinements that would be demanded in an exact formulation.

Imagine the complex variable
z
tracing out a closed curve of finite
length without loops or kinks. We have an intuitive notion of what we mean by the “length” of a piece of this curve.

Mark
n
points
P
₁
P
₂
, . . . , P
_n
on the curve so that each of the pieces
P
₁
P
₂
, P
₂
P
₃
, P
₃
P
₄
, . . . , P
_n
P
₁
is not greater than some preassigned finite length
l
. On each of these pieces choose a point, not at either end of the piece; form the value of
f(z)
for the value of
z
corresponding to the point, and multiply this value by the length of the piece in which the point lies. Do the like for
all
the pieces, and add the results. Finally take the limiting value of this sum as the number of pieces is indefinitely increased. This gives the
“line integral”
of
f(z)
for the curve.

When will this line integral be zero? In order that the line integral shall be zero it is sufficient that
f(z)
be
analytic
(uniform and monogenic) at every point
z
on the curve and inside the curve.

Such is the great theorem which Gauss communicated to Bessel in
1811
and which, with another theorem of a similar kind, in the hands of Cauchy who rediscovered it independently, was to yield many of the important results of analysis as corollaries.

*  *  *

Astronomy did not absorb the whole of Gauss' prodigious energies in his middle thirties. The year 1812, which saw Napoleon's Grand Army fighting a desperate rear-guard action across the frozen plains, witnessed the publication of another great work by Gauss, that on the
hyfergeometric series

the dots meaning that the series continues indefinitely according to the law indicated; the next term is

This memoir is another landmark. As has already been noted Gauss was the first of the modern rigorists. In this work he determined the restrictions that must be imposed on the numbers
a, b, c, x
in order that the series shall converge (in the sense explained earlier in this chapter). The series itself was no mere textbook exercise that may be investigated to gain skill in analytical manipulations and then be forgotten. It includes as special cases—obtained by assigning specific values to one or more of
a, b, c, x
—many of the most important series in analysis, for example those by which logarithms, the trigonometric functions, and several of the functions that turn up repeatedly in Newtonian astronomy and mathematical physics are calculated and tabulated; the general binomial theorem also is a special case. By disposing of this series in its general form Gauss slew a multitude at one smash. From this work developed many applications to the differential equations of physics in the nineteenth century.

The choice of such an investigation for a serious effort is characteristic of Gauss. He never published trivialities. When he put out anything it was not only finished in itself but was also so crammed with ideas that his successors were enabled to apply what Gauss had invented to new problems. Although limitations of space forbid discussion of the many instances of this fundamental character of Gauss' contributions to pure mathematics, one cannot be passed over in even the briefest sketch: the work on the law of biquadratic reciprocity. The importance of this was that it gave a new and totally unforeseen direction to the higher arithmetic.

*  *  *

Having disposed of
quadratic
(second degree) reciprocity, it was natural for Gauss to consider the general question of binomial congruences of any degree. If m is a given integer not divisible by the prime
p,
and if
n
is a given positive integer, and if further an integer
x
can be found such that
x
ⁿ
≡ m
(mod
p), m is
called an
n-ic residue
of
p;
when
n
= 4,
m
is a
biquadratic residue
of
p.

The case of
quadratic
binomial congruences
(n
= 2) suggests but little to do when
n
exceeds 2. One of the matters Gauss was to have
included in the discarded eighth section (or possibly, as he told Sophie Germain, in the projected but unachieved second volume) of the
Disquisitiones Arithmeticae
was a discussion of these higher congruences and a search for the corresponding laws of reciprocity, namely the interconnections (as to solvability or non-solvability) of the pair
x
ⁿ
≡
p
(mod
q), x
ⁿ
≡
q
(mod
p
), where
p
,
q
are rational primes. In particular the cases
n
= 3,
n
= 4 were to have been investigated.

The memoir of 1825 breaks new ground with all the boldness of the great pioneers. After many false starts which led to intolerable complexity Gauss discovered the “natural” way to the heart of his problem. The
rational
integers 1, 2, 3, . . . are
not
those appropriate to the statement of the law of
biquadratic
reciprocity, as they are for
quadratic;
a totally new species of
integers
must be invented. These are called the
Gaussian complex integers
and are all those complex numbers of the form
a + bi
in which
a, b
are
rational integers
and
i
denotes

To state the law of biquadratic reciprocity an exhaustive preliminary discussion of the laws of arithmetical divisibility for such
complex integers
is necessary. Gauss gave this, thereby inaugurating the theory of algebraic numbers—that which he probably had in mind when he gave his estimate of Fermat's Last Theorem. For
cubic
reciprocity
(n
= 3) he also found the right way in a similar manner. His work on this was found in his posthumous papers.

The significance of this great advance will become clearer when we follow the careers of Kummer and Dedekind. For the moment it is sufficient to say that Gauss' favorite disciple, Eisenstein, disposed of cubic reciprocity. He further discovered an astonishing connection between the law of biquadratic reciprocity and certain parts of the theory of elliptic functions, in which Gauss had travelled far but had refrained disclosing what he found.

Gaussian complex
integers
are of course a subclass of
all
complex
numbers,
and it might be thought that the
algebraic
theory of
all
the numbers would yield the
arithmetical
theory of the included
integers
as a trivial detail. Such is by no means the case. Compared to the arithmetical theory the algebraic is childishly easy. Perhaps a reason why this should be so is suggested by the
rational numbers
(numbers of the form
a/b,
where
a, b
are rational integers). We can
always
divide one rational number by another and get
another
rational number:
a/b
divided by
c/d
yields the rational number
ad/bc.
But a rational
integer
divided by another rational integer is not always another rational integer: 7 divided by 8 gives ⅞ Hence if we must restrict ourselves to
integers,
the case of interest for the theory of numbers, we have tied our hands and hobbled our feet before we start. This is one of the reasons why the higher arithmetic is harder than algebra, higher or elementary.

*  *  *

Equally significant advances in geometry and the applications of mathematics to geodesy, the Newtonian theory of attraction, and electromagnetism were also to be made by Gauss. How was it possible for one man to accomplish this colossal mass of work of the highest order? With characteristic modesty Gauss declared that “If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.” Possibly. Gauss' explanation recalls Newton's. Asked how he had made discoveries in astronomy surpassing those of all his predecessors, Newton replied, “By always thinking about them.” This may have been plain to Newton; it is not to ordinary mortals.

Part of the riddle of Gauss is answered by his
involuntary
preoccupation with mathematical ideas—which itself of course demands explanation. As a young man Gauss would be “seized” by mathematics. Conversing with friends he would suddenly go silent, overwhelmed by thoughts beyond his control, and stand staring rigidly oblivious of his surroundings. Later he controlled his thoughts—or they lost their control over him—and he consciously directed all his energies to the solution of a difficulty till he succeeded. A problem once grasped was never released till he had conquered it, although several might be in the foreground of his attention simultaneously.

In one such instance (referring to the
Disquisitiones,
page
636)
he relates how for four years scarcely a week passed that he did not spend some time trying to settle whether a certain sign should be plus or minus. The solution finally came of itself in a flash. But to imagine that it would have blazed out of itself like a new star without the “wasted” hours is to miss the point entirely. Often after spending days or weeks fruitlessly over some research Gauss would find on resuming work after a sleepless night that the obscurity had vanished and the whole solution shone clear in his mind. The capacity for intense and prolonged concentration was part of his secret.