Men of Mathematics (44 page)

Before entering the Caroline College at the age of fifteen, Gauss had made great headway in the classical languages by private study and help from older friends, thus precipitating a crisis in his career. To his crassly practical father the study of ancient languages was the height of folly. Dorothea Gauss put up a fight for her boy, won, and the Duke subsidized a two-years' course at the Gymnasium. There Gauss' lightning mastery of the classics astonished teachers and students alike.

Gauss himself was strongly attracted to philological studies, but fortunately for science he was presently to find a more compelling attraction in mathematics. On entering college Gauss was already master of the supple Latin in which many of his greatest works are written. It is an ever-to-be-regretted calamity that even the example of Gauss was powerless against the tides of bigoted nationalism which swept over Europe after the French Revolution and the downfall of Napoleon. Instead of the easy Latin which sufficed for Euler and Gauss, and which any student can master in a few weeks, scientific workers must now acquire a reading knowledge of two or three languages in addition to their own. Gauss resisted as long as he could, but even he had to submit when his astronomical friends in Germany pressed him to write some of his astronomical works in German.

Gauss studied at the Caroline College for three years, during which he mastered the more important works of Euler, Lagrange and, above
all, Newton's
Principia.
The highest praise one great man can get is from another in his own class. Gauss never lowered the estimate which as a boy of seventeen he had formed of Newton. Others—Euler, Laplace, Lagrange, Legendre—appear in the flowing Latin of Gauss with the complimentary
clarissimus;
Newton is
summus.

While still at the college Gauss had begun those researches in the higher arithmetic which were to make him immortal. His prodigious powers of calculation now came into play. Going directly to the numbers themselves he experimented with them, discovering by induction recondite general theorems whose proofs were to cost even him an effort. In this way he rediscovered “the gem of arithmetic,”
“theorema aureum,”
which Euler also had come upon inductively, which is known as the law of quadratic reciprocity, and which he was to be the first to prove. (Legendre's attempted proof slurs over a crux.)

The whole investigation originated in a simple question which many beginners in arithmetic ask themselves: How many digits are there in the period of a repeating decimal? To get some light on the problem Gauss calculated the decimal representations of all the fractions
1/n
for
n
= 1 to 1000. He did not find the treasure he was seeking, but something infinitely greater—the law of quadratic reciprocity. As this is quite simply stated we shall describe it, introducing at the same time one of the revolutionary improvements in arithmetical nomenclature and notation which Gauss invented, that of
congruence.
All numbers in what follows are integers (common whole numbers).

If the
difference (a – b
or
b – a)
of two numbers a,
b
is exactly divisible by the number
m,
we say that
a, b
are
congruent
with respect to the modulus
m,
or simply
congruent modulo m
_y
and we symbolize this by writing
a ≡ b
(mod
m).
Thus 100 ≡ 2 (mod 7),
35 ≡
2 (mod 11).

The advantage of this scheme is that it recalls the way we write algebraic equations, traps the somewhat elusive notion of arithmetical divisibility in a compact notation, and suggests that we try to carry over to arithmetic (which is much harder than algebra) some of the manipulations that lead to interesting results in algebra. For example we can “add” equations, and we find that congruences also can be “added,” provided the modulus is the same in all, to give other congruences.

Let
x
denote an unknown number, r and
m
given numbers, of which r is not divisible by
m.
Is there a number
x
such that

x
²
≡ r (mod
m)?

If there is, r is called a
quadratic residue of
m, if not, a
quadratic non-residue of m.

If
r is
a quadratic residue of
m,
then it must be possible to find at least one
x
whose square when divided by
m
leaves the remainder
r
; if r is a quadratic non-residue of
m,
then there is no
x
whose square when divided by
m
leaves the remainder
r.
These are immediate consequences of the preceding definitions.

To illustrate: is 13 a quadratic residue of 17? If so, it must be possible to solve the
congruence

x
²
≡
13 (mod 17)

Trying 1, 2, 3, . . . , we find that
x
* 8, 25, 42, 59, . . . are solutions (8
²
= 64 = 3
×
17 + 13; 25
²
= 625 = 36
×
17 + 13; etc.,) so that IS
is
a quadratic residue of 17. But there is
no
solution of x
²
≡ 5 (mod 17), so 5 is a quadratic non-residue of 17.

It is now natural to ask what are the quadratic residues and non-residues of a given
m?
Namely, given
m
in
x
²
≡ r (mod
m),
what numbers r can appear and what numbers r cannot appear as
x
runs through all the numbers 1, 2, 3, . . .?

Without much difficulty it can be shown that it is sufficient to answer the question when both
r
and
m
are restricted to be primes. So we restate the problem: If
p
is a
given
prime, what primes
q
will make the congruence
x
²
m q
(mod
p)
solvable? This is asking altogether too much in the present state of arithmetic. However, the situation is not utterly hopeless.

There is a beautiful “reciprocity” between
the pair
of congruences

x
²
≡ q
(mod
p), x
²
≡ p
(mod
q),

in which
both
of
p, q
are
primes: both
congruences are
solvable,
or
both
are
unsolvable, unless both
of
p, q
leave the remainder 3 when divided by 4, in which case
one
of the congruences
is
solvable and
the other
is
not.
This is the law of quadratic reciprocity.

It was not easy to prove. In fact it baffled Euler and Legendre. Gauss gave the first proof at the age of nineteen. As this reciprocity is of fundamental importance in the higher arithmetic and in many
advanced parts of algebra, Gauss turned it over and over in his mind for many years, seeking to find its taproot, until in all he had given six distinct proofs, one of which depends upon the straightedge and compass construction of regular polygons.

A numerical illustration will illuminate the statement of the law. First, take
p
= 5,
q
= 13. Since both of 5, 13 leave the remainder 1 on division by 4,
both
of
x
²
≡ 13 (mod 5),
x
²
≡
5 (mod 13) must be
solvable,
or
neither
is solvable. The latter is the case for this pair. For
p
= 13,
q
= 17, both of which leave the remainder 1 on division by 4, we get
x
²
≡
17 (mod 13),
x
²
≡ 13 (mod 17), and
both,
or
neither
again must be solvable. The former is the case here: the first congruence has the solutions
x
= 2, 15, 28, . . .; the second has the solutions
x
= 8, 25, 42, . . . . There remains to be tested only the case when
both of p, q
leave the remainder 3 on division by 4. Take
p
= 11,
q
= 19. Then, according to the
law, precisely one
of
x
²
≡ 19 (mod 11),
x
²
≡ 11 (mod 19) must be solvable. The first congruence has no solution; the second has the solutions 7, 26, 45, . . ..

The mere discovery of such a law was a notable achievement. That it was first proved by a boy of nineteen will suggest to anyone who tries to prove it that Gauss was more than merely competent in mathematics.

When Gauss left the Caroline College in October, 1795 at the age of eighteen to enter the University of Göttingen he was still undecided whether to follow mathematics or philology as his life work. He had already invented (when he was eighteen) the method of “least squares,” which today is indispensable in geodetic surveying, in the reduction of observations and indeed in all work where the “most probable” value of anything that is measured is to be inferred from a large number of measurements. (The most probable value is furnished by making the sum of the squares of the “residuals”—roughly, divergences from assumed exactness—a minimum.) Gauss shares this honor with Legendre who published the method independently in 1806. This work was the beginning of Gauss' interest in the theory of errors of observation. The Gaussian law of normal distribution of errors and its accompanying bell-shaped curve is familiar today to all who handle statistics, from high-minded intelligence testers to unscrupulous market manipulators.

*  *  *

March 30, 1796, marks the turning point in Gauss' career. On that
day, exactly a month before his twentieth year opened, Gauss definitely decided in favor of mathematics. The study of languages was to remain a lifelong hobby, but philology lost Gauss forever on that memorable day in March.

As has already been told in the chapter on Fermat the regular polygon of seventeen sides was the die whose lucky fall induced Gauss to cross his Rubicon. The same day Gauss began to keep his scientific diary
(Notizenjournal).
It is one of the most precious documents in the history of mathematics. The first entry records his great discovery.

The diary came into scientific circulation only in 1898, forty three years after the death of Gauss, when the Royal Society of Göttingen borrowed it from a grandson of Gauss for critical study. It consists of nineteen small octavo pages and contains 146 extremely brief statements of discoveries or results of calculations, the last of which is dated July 9, 1814. A facsimile reproduction was published in 1917 in the tenth volume (part l) of Gauss' collected works, together with an exhaustive analysis of its contents by several expert editors. Not all of Gauss' discoveries in the prolific period from 1796 to 1814 by any means are noted. But many of those that are jotted down suffice to establish Gauss' priority in fields—elliptic functions, for instance—where some of his contemporaries refused to believe he had preceded them. (Recall that Gauss was born in 1777.)

Things were buried for years or decades in this diary that would have made half a dozen great reputations had they been published promptly. Some were never made public during Gauss' lifetime, and he never claimed in anything he himself printed to have anticipated others when they caught up with him. But the record stands. He did anticipate some who doubted the word of his friends. These anticipations were not mere trivialities. Some of them became major fields of nineteenth century mathematics.

A few of the entries indicate that the diary was a strictly private affair of its author's. Thus for July 10, 1796, there is the entry

ETPHKA!
num =
Δ + Δ + Δ.

Translated, this echoes Archimedes' exultant “Eureka!” and states that every positive integer is the sum of three triangular numbers—such a number is one of the sequence 0, 1, 3, 6, 10, 15, . . . where each (after 0) is of the form
½n(n +
l),
n
being any positive integer. Another way of saying the same thing is that every number of the
form
8n
+ 3 is a sum of three odd squares: 3 = 1
²
+ 1
²
+ 1
²
; 11 = 1 + 1 + 3
²
; 19 = 1
²
+ 3
²
+ 3
²
, etc. It is not easy to prove this from scratch.

Less intelligible is the cryptic entry for October 11, 1796, “Vicimus GEGAN.” What dragon had Gauss conquered this time? Or what giant had he overcome on April 8, 1799, when he boxes REV. GALEN up in a neat rectangle? Although the meaning of these is lost forever the remaining 144 are for the most part clear enough. One in particular is of the first importance, as we shall see when we come to Abel and Jacobi: the entry for March 19, 1797, shows that Gauss had already discovered the double periodicity of certain elliptic functions. He was then not quite twenty. Again, a later entry shows that Gauss had recognized the double periodicity in the general case. This discovery of itself, had he published it, would have made him famous. But he never published it.

Why did Gauss hold back the great things he discovered? This is easier to explain than his genius—if we accept his own simple statements, which will be reported presently. A more romantic version is the story told by W. W. R. Ball in his well-known history of mathematics. According to this, Gauss submitted his first masterpiece, the
Disquisitiones Arithmeticae,
to the French Academy of Sciences, only to have it rejected with a sneer. The undeserved humiliation hurt Gauss so deeply that he resolved thenceforth to publish only what anyone would admit was above criticism in both matter and form. There is nothing in this defamatory legend. It was disproved once for all in 1935, when the officers of the French Academy ascertained by an exhaustive search of the permanent records that the
Disquisitiones
was never even submitted to the Academy, much less rejected.