Men of Mathematics (48 page)

If Gauss was simple and thrifty the French invaders of Germany in
1807
were simpler and thriftier. To govern Germany according to their ideas the victors of Auerstedt and Jena fined the losers for more than the traffic would bear. As professor and astronomer at Göttingen Gauss was rated by the extortionists to be good for an involuntary contribution of
2,000
francs to the Napoleonic war chest. This exorbitant sum was quite beyond Gauss' ability to pay.

Presently Gauss got a letter from his astronomical friend Olbers enclosing the amount of the fine and expressing indignation that a scholar should be subjected to such petty extortion. Thanking his generous friend for his sympathy, Gauss declined the money and sent it back at once to the donor.

Not all the French were as thrifty as Napoleon. Shortly after returning Olbers' money Gauss received a friendly little note from Laplace telling him that the famous French mathematician had paid the
2,000
-franc fine for the greatest mathematician in the world and had considered it an honor to be able to lift this unmerited burden from his friend's shoulders. As Laplace had paid the fine in Paris, Gauss was unable to return him the money. Nevertheless he declined to accept Laplace's help. An unexpected (and unsolicited) windfall was presently to enable him to repay Laplace with interest at the current market rate. Word must have got about that Gauss disdained charity. The next attempt to help him succeeded. An admirer in Frankfurt sent
1,000
guilders anonymously. As Gauss could not trace the sender he was forced to accept the gift.

The death of his friend Ferdinand, the wretched state of Germany under French looting, financial straits, and the loss of his first wife all did their part toward upsetting Gauss' health and making his life miserable in his early thirties. Nor did a constitutional predisposition to hypochondria, aggravated by incessant overwork, help matters. His unhappiness was never shared with his friends, to whom he is always the serene correspondent, but is confided—only once—to a private mathematical manuscript. After his appointment to the directorship at Göttingen in
1807
Gauss returned occasionally for three years to
one of the great things noted in his diary. In a manuscript on elliptic functions purely scientific matters are suddenly interrupted by the finely pencilled words “Death were dearer to me than such a life.” His work became his drug.

The years 1811-12 (Gauss was thirty four in 1811) were brighter. With a wife again to care for his young children Gauss began to have some peace. Then, almost exactly a year after his second marriage, the great comet of 1811, first observed by Gauss deep in the evening twilight of August 22, blazed up unannounced. Here was a worthy foe to test the weapons Gauss had invented to subjugate the minor planets.

His weapons proved adequate. While the superstitious peoples of Europe, following the blazing spectacle with awestruck eyes as the comet unlimbered its flaming scimitar in its approach to the Sun, saw in the fiery blade a sharp warning from Heaven that the King of Kings was wroth with Napoleon and weary of the ruthless tyrant, Gauss had the satisfaction of seeing the comet follow the path he had quickly calculated for it to the last decimal. The following year the credulous also saw their own prediction verified in the burning of Moscow and the destruction of Napoleon's Grand Army on the icy plains of Russia.

This is one of those rare instances where the popular explanation fits the facts and leads to more important consequences than the scientific. Napoleon himself had a basely credulous mind—he relied on “hunches,” reconciled his wholesale slaughters with a childlike faith in a beneficent, inscrutable Providence, and believed himself a Man of Destiny. It is not impossible that the celestial spectacle of a harmless comet flaunting its gorgeous tail across the sky left its impress on the subconscious mind of a man like Napoleon and fuddled his judgment. The almost superstitious reverence of such a man for mathematics and mathematicians is no great credit to either, although it has been frequently cited as one of the main justifications for both.

Beyond a rather crass appreciation of the value of mathematics in military affairs, where its utility is obvious even to a blind idiot, Napoleon had no conception of what mathematics as practised by masters like his contemporaries, Lagrange, Laplace, and Gauss, is all about. A quick student of trivial, elementary mathematics at school, Napoleon turned to other things too early to certify his promise and, mathematically, never grew up. Although it seems incredible that a man of Napoleon's demonstrated capacity could so grossly underestimate
the difficulties of matters beyond his comprehension as to patronize Laplace, it is a fact that he had the ludicrous audacity to assure the author of the
Mécanique céleste
that he would read the book the
first free month
he could find. Newton and Gauss might have been equal to the task; Napoleon no doubt could have turned the pages in his month without greatly tiring himself.

It is a satisfaction to record that Gauss was too proud to prostitute mathematics to Napoleon the Great by appealing to the Emperor's vanity and begging him in the name of his notorious respect for all things mathematical to remit the 2,000-franc fine, as some of Gauss' mistaken friends urged him to do. Napoleon would probably have been flattered to exercise his clemency. But Gauss could not forget Ferdinand's death, and he felt that both he and the mathematics he worshipped were better off without the condescension of a Napoleon.

No sharper contrast between the mathematician and the military genius can be found than that afforded by their respective attitudes to a broken enemy. We have seen how Napoleon treated Ferdinand. When Napoleon fell Gauss did not exult. Calmly and with a detached interest he read everything he could find about Napoleon's life and did his best to understand the workings of a mind like Napoleon's. The effort even gave him considerable amusement. Gauss had a keen sense of humor, and the blunt realism which he had inherited from his hardworking peasant ancestors also made it easy for him to smile at heroics.

*  *  *

The year 1811 might have been a landmark in mathematics comparable to 1801—the year in which the
Disquisitiones Arithmeticae
appeared—had Gauss made public a discovery he confided to Bessel. Having thoroughly understood complex numbers and their geometrical representation as points on the plane of analytic geometry, Gauss proposed himself the problem of investigating what are today called
analytic functions
of such numbers.

The complex number
x + iy,
where
i
denotes
represents the point
(x, y).
For brevity
x + iy
will be denoted by the single letter
z.
As
x, y
independently take on real values in any prescribed continuous manner, the point
z
wanders about over the plane, obviously not at random but in a manner determined by that in which
x, y
assume their values. Any expression containing
z,
such as z
²
, or
1/z
, etc.,
which takes on a
single
definite value when a value is assigned to
z,
is called a
uniform function
of
z.
We shall denote such a function by
f(z).
Thus if
f(z)
is the particular function
z
²
,
so that here
f(z) = (x
+ iy)
²
= x
²
+ 2
ixy
+
i
²
y
²
, =
x
²
–
y
²
+ 2
ixy
(because i
²
= −1), it is clear that when any value is assigned to
z,
namely to
x
+
iy
, for example
x
= 2,
y
= 3, so that
z
= 2 + 3
i
, precisely one value of this
f(z)
is thereby determined; here, for
z
= 2 + 3
i
we get
z
²
= −5 + 12
i
.

Not all uniform functions
f(z)
are studied in the theory of functions of a complex variable; the
monogenic
functions are singled out for exhaustive discussion. The reason for this will be stated after we have described what “monogenic” means.

Let
z
move to another position, say to
z′.
The function
f(z)
takes on another value,
f(z′),
obtained by substituting
z′
for
z.
The
difference f(z′)—f(z)
of the new and old values of the function is now divided by the difference of the new and old values of the variable, thus
[f(z′) +(zy]/(z′ – z),
and, precisely as is done in calculating the slope of a graph to find the derivative of the function the graph represents, we here let
z′
approach
z
indefinitely, so that
f(z′)
approaches
f(z)
simultaneously. But here a remarkable new phenomenon appears.

There is not here a unique way in which
z′
can move into coincidence with
z,
for
z′
may wander about all over the plane of complex
numbers by any of an infinity of different paths before coming into coincidence with
z.
We should not expect the limiting value of
[f(z′)—f(z)]/(z′—z)
when
z′
coincides with
z
to be
the same
for
all
of these paths, and in general it is
not.
But
iff(z)
is such that the limiting value just described
is
the same for
all
paths by which
z′
moves into coincidence with
z,
then
f(z)
is said to be monogenic at
z
(or at the point representing
z). Uniformity
(previously described) and
monogenicity
are distinguishing features of
analytic
functions of a complex variable.

Some idea of the importance of analytic functions can be inferred from the fact that vast tracts of the theories of fluid motion (also of mathematical electricity and representation by maps which do not distort angles) are naturally handled by the theory of
analytic
functions of a complex variable. Suppose such a function
f(z)
is separated into its “real” part (that which does not contain the “imaginary unit”
i)
and its “imaginary” part, say
f(z) = U
+
iV.
For the special analytic function
z
²
we have U =
x
²
—y
²
, V
=
2xy.
Imagine a film of fluid streaming over a plane. If the motion of the fluid is without vortices, a stream line of the motion is obtainable from
some
analytic function f(
^z
) by plotting the curve
U = a,
in which
a
is any real number, and likewise the equipotential lines are obtainable from
V = b (b
any real number). Letting
a, b
range, we thus get a complete picture of the motion for as large an area as we wish. For a given situation, say that of a fluid streaming round an obstacle, the hard part of the problem is to find what analytic function to choose, and the whole matter has been gone at largely backwards: the simple analytic functions have been investigated and the physical problems which they fit have been sought. Curiously enough, many of these artificially prepared problems have proven of the greatest service in aerodynamics and other practical applications of the theory of fluid motion.