Men of Mathematics (45 page)

Speaking for himself Gauss said that he undertook his scientific works only in response to the deepest promptings of his nature, and it was a wholly secondary consideration to him whether they were ever published for the instruction of others. Another statement which Gauss once made to a friend explains both his diary and his slowness in publication. He declared that such an overwhelming horde of new ideas stormed his mind before he was twenty that he could hardly control them and had time to record but a small fraction. The diary contains only the final brief statements of the outcome of elaborate investigations, some of which occupied him for weeks. Contemplating as a youth the close, unbreakable chains of synthetic proofs in which
Archimedes and Newton had tamed their inspirations, Gauss resolved to follow their great example and leave after him only finished works of art, severely perfect, to which nothing could be added and from which nothing could be taken away without disfiguring the whole. The work itself must stand forth, complete, simple, and convincing, with no trace remaining of the labor by which it had been achieved. A cathedral is not a cathedral, he said, till the last scaffolding is down and out of sight. Working with this ideal before him, Gauss preferred to polish one masterpiece several times rather than to publish the broad outlines of many as he might easily have done. His seal, a tree with but few fruits, bore the motto
Pauca sed matura
(Few, but ripe).

The fruits of this striving after perfection were indeed ripe but not always easily digestible. All traces of the steps by which the goal had been attained having been obliterated, it was not easy for the followers of Gauss to rediscover the road he had travelled. Consequently some of his works had to wait for highly gifted interpreters before mathematicians in general could understand them, see their significance for unsolved problems, and go ahead. His own contemporaries begged him to relax his frigid perfection so that mathematics might advance more rapidly, but Gauss never relaxed. Not till long after his death was it known how much of nineteenth-century mathematics Gauss had foreseen and anticipated before the year
1800.
Had he divulged what he knew it is quite possible that mathematics would now be half a century or more ahead of where it is. Abel and Jacobi could have begun where Gauss left off, instead of expending much of their finest effort rediscovering things Gauss knew before they were born, and the creators of non-Euclidean geometry could have turned their genius to other things.

Of himself Gauss said that he was “all mathematician.” This does him an injustice unless it is remembered that “mathematician” in his day included also what would now be termed a mathematical physicist. Indeed his second motto
^II

Thou, nature, art my goddess; to thy laws

My services are bound . . . ,

truly sums up his life of devotion to mathematics and the physical sciences of his time. The “all mathematician” aspect of him is to be
understood only in the sense that he did not scatter his magnificent endowment broadcast over all fields where he might have reaped abundantly, as he blamed Leibniz for doing, but cultivated his greatest gift to perfection.

The three years (October,
1795
-September,
1798)
at the University of Göttingen were the most prolific in Gauss' life. Owing to the generosity of the Duke Ferdinand the young man did not have to worry about finances. He lost himself in his work, making but few friends. One of these, Wolfgang Bolyai, “the rarest spirit I ever knew,” as Gauss described him, was to become a friend for life. The course of this friendship and its importance in the history of non-Euclidean geometry is too long to be told here; Wolfgang's son Johann was to retrace practically the same path that Gauss had followed to the creation of a non-Euclidean geometry, in entire ignorance that his father's old friend had anticipated him. The ideas which had overwhelmed Gauss since his seventeenth year were now caught—partly—and reduced to order. Since
1795
he had been meditating a great work on the theory of numbers. This now took definite shape, and by
1798
the
Disquisitiones Arithmeticae
(Arithmetical Researches) was practically completed.

To acquaint himself with what had already been done in the higher arithmetic and to make sure that he gave due credit to his predecessors, Gauss went to the University of Helmstedt, where there was a good mathematical library, in September,
1798.
There he found that his fame had preceded him. He was cordially welcomed by the librarian and the professor of mathematics, Johann Friedrich Pfaff (
1765–1825),
in whose house he roomed. Gauss and Pfaff became warm friends, although the Pfaff family saw but little of their guest. Pfaff evidently thought it his duty to see that his hard-working young friend took some exercise, for he and Gauss strolled together in the evenings, talking mathematics. As Gauss was not only modest but reticent about his own work, Pfaff probably did not learn as much as he might have. Gauss admired the professor tremendously (he was then the best-known mathematician in Germany), not only for his excellent mathematics, but for his simple, open character. All his life there was but one type of man for whom Gauss felt aversion and contempt, the pretender to knowledge who will not admit his mistakes when he knows he is wrong.

Gauss spent the autumn of
1798
(he was then twenty one) in
Brunswick, with occasional trips to Helmstedt, putting the finishing touches to the
Disquisitiones.
He had hoped for early publication, but the book was held up in the press owing to a Leipzig publisher's difficulties till September, 1801. In gratitude for all that Ferdinand had done for him, Gauss dedicated his book to the Duke—
“Serenissimo Principi ac Domino Carolo Guilielmo Ferdinando.”

If ever a generous patron deserved the homage of his protégé, Ferdinand deserved that of Gauss. When the young genius was worried ill about his future after leaving Göttingen—he tried unsuccessfully to get pupils—the Duke came to his rescue, paid for the printing of his doctoral dissertation (University of Helmstedt, 1799), and granted him a modest pension which would enable him to continue his scientific work unhampered by poverty.
“Your
kindness,” Gauss says in his dedication, “freed me from all other responsibilities and enabled me to assume this exclusively.”

*  *  *

Before describing the
Disquisitiones
we shall glance at the dissertation for which Gauss was awarded his doctor's degree
in absentia
by the University of Helmstedt in 1799:
Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus revolvi posse
(A New Proof that Every Rational Integral Function of One Variable Can Be Resolved into Real Factors of the First or Second Degree).

There is only one thing wrong with this landmark in algebra. The first two words in the title would imply that Gauss had merely added a
new
proof to others already known. He should have omitted “nova.” His was the
first
proof. (This assertion will be qualified later.) Some before him had published what they supposed were proofs of this theorem—usually called the fundamental theorem of algebra—but none had attained a proof. With his uncompromising demand for logical and mathematical rigor Gauss insisted upon a
proof,
and gave the first. Another, equivalent, statement of the theorem says that every algebraic equation in one unknown has a root, an assertion which beginners often take for granted as being true without having the remotest conception of what it means.

If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics. It is just as doubtful whether the assertion that every algebraic equation has a root
means anything until we say
what sort
of a root the equation has. Vaguely, we feel that a
number
will “satisfy” the equation but that half a pound of butter will not.

Gauss made this feeling precise by proving that all the roots of any algebraic equation are “numbers” of the form
a + bi,
where
a, b
are real numbers (the numbers that correspond to the distances, positive, zero, or negative, measured from a fixed point O on a given straight line, as on the x-axis in Descartes' geometry), and
i
is the square root of −1. The new sort of “number”
a + bi
is called
complex.

Incidentally, Gauss was one of the first to give a coherent account of complex numbers and to interpret them as labelling the points of a plane, as is done today in elementary textbooks on algebra.

The Cartesian coordinates of
P
are
(a, b);
the point
P
is also labelled
a + bi.
Thus to every point of the plane corresponds precisely one complex number; the numbers corresponding to the points on
XOX'
are “real,” those on
YOY'
“pure imaginary” (they are all of the type
ic,
where
c
is a real number).

The word “imaginary” is the great algebraical calamity, but it is too well established for mathematicians to eradicate. It should never have been used. Books on elementary algebra give a simple interpretation of imaginary numbers in terms of rotations. Thus if we interpret the multiplication
i × c,
where
c
is real, as a rotation about O of the segment
Oc
through one right angle,
Oc
is rotated onto
0Y
; another multiplication by z, namely
i × i X
c,
rotates
Oc
through another
right angle, and hence the total effect is to rotate
Oc
through two right angles, so that
+Oc
becomes
—Oc.
As an operation, multiplication by
i
×
i
has the same effect as multiplication by −1; multiplication by
i
has the same effect as a rotation through a right angle, and these interpretations (as we have just seen) are consistent. If we like we may now write
i
×
i
= −1, in operations, or
i
²
= −1; so that the operation of rotation through a right angle is symbolized by

All this of course proves nothing. It is not meant to prove anything.
There is nothing to be proved;
we
assign
to the symbols and operations of algebra
any meanings whatever
that will lead to consistency. Although the
interpretation
by means of rotations
proves
nothing, it may suggest that there is no occasion for anyone to muddle himself into a state of mystic wonderment over nothing about the grossly misnamed “imaginaries.” For further details we must refer to almost any Schoolbook on elementary algebra.

Gauss thought the theorem that every algebraic equation has a root in the sense just explained so important that he gave four distinct proofs, the last when he was seventy years old. Today some would transfer the theorem from algebra (which restricts itself to processes that can be carried through in a finite number of steps) to analysis. Even Gauss
assumed
that the graph of a polynomial is a continuous curve and that if the polynomial is of odd degree the graph must cross the axis at least once. To any beginner in algebra this is obvious. But today it is
not obvious
without proof, and attempts to prove it again lead to the difficulties connected with continuity and the infinite. The roots of so simple an equation as
x
²
−2 = 0 cannot be computed exactly in any finite number of steps. More will be said about this when we come to Kronecker. We proceed now to the
Disquisitiones Arithmeticae.