Men of Mathematics (89 page)

Since Hermite's time many numbers (and classes of numbers) have been proved transcendental. What is likely to remain a high-water mark on the shores of this dark sea for some time may be noted in passing. In 1934 the young Russian mathematician Alexis Gelfond proved that
all
numbers of the type
a
^b
,
where
a
is neither 0 nor 1 and
b
is
any irrational algebraic number,
are transcendental. This disposes of the seventh of David Hilbert's list of twenty three outstanding mathematical problems which he called to the attention of mathematicians
at the Paris International Congress in 1900. Note that “irrational” is
necessary
in the statement of Gelfond's theorem (if
b = n/m,
where
n, m
are rational integers, then
a
^b
,
where
a
is any algebraic number, is a root of
x
^m
—a
ⁿ
= 0, and it can be shown that this equation is equivalent to one in which all the coefficients are rational integers.

Hermite's unexpected victory over the obstinate
e
inspired mathematicians to hope that
π
would presently be subdued in a similar manner. For himself, however, Hermite had had enough of a good thing. “I shall risk nothing,” he wrote to Borchardt, “on an attempt to prove the transcendence of the number
π
. If others undertake this enterprise, no one will be happier than I at their success, but believe me, my dear friend, it will not fail to cost them some efforts.” Nine years later (in 1882) Ferdinand Lindemann of the University of Munich, using methods very similar to those which had sufficed Hermite to dispose of
e,
proved that
π
is transcendental, thus settling forever the problem of “squaring the circle.” From what Lindemann proved it follows that it is impossible with straightedge and compass alone to construct a square whose area is equal to that of any given circle—a problem which had tormented generations of mathematicians since before the time of Euclid.

As cranks are still tormented by the problem, it may be in order to state concisely how Lindemann's proof settles the matter. He proved that
π
is
not
an
algebraic
number. But any
geometrical
problem that
is
solvable by the aid of straightedge and compass alone, when
restated
in its equivalent
algebraic
form, leads to one or more algebraic equations with rational integer coefficients which can be solved by successive extractions of
square roots.
As
π
satisfies no such equation, the circle cannot be “squared” with the implements named. If other mechanical apparatus is permitted, it is easy to square the circle. To all but mild lunatics the problem has been completely dead for over half a century. Nor is there any merit at the present time in computing
Π
to a large number of decimal places—more accuracy in this respect is already available than is ever likely to be of use to the human race if it survives for a billion to the billionth power years. Instead of trying to do the impossible, mystics may like to contemplate the following useful relation between
e, π,
−1 and
till it becomes as plain to them as Buddha's navel is to a blind Hindu swami,

Anyone who can perceive this mystery intuitively will not need to square the circle.

Since Lindemann settled
π
the one outstanding unsolved problem that attracts amateurs is Fermat's “Last Theorem.” Here an amateur with real genius undoubtedly has a chance. Lest this be taken as an invitation to all and sundry to swamp the editors of mathematical journals with attempted proofs, we recall what happened to Lindemann when he boldly tackled the famous theorem. If this does not suggest that more than ordinary talent will be required to settle Fermat, nothing can. In 1901 Lindemann published a memoir of seventeen pages purporting to contain the long-sought proof. The vitiating error being pointed out, Lindemann, undaunted, spent the best part of the next seven years in attempting to patch the unpatchable, and in 1907 published sixty three pages of alleged proof which were rendered nonsensical by a slip in reasoning near the very beginning.

Great as were Hermite's contributions to the technical side of mathematics, his steadfast adherence to the ideal that science is beyond nations and above the power of creeds to dominate or to stultify was perhaps an even more significant gift to civilization in the long view of things as they now appear to a harassed humanity. We can only look back on his serene beauty of spirit with a poignant regret that its like is nowhere to be found in the world of science today. Even when the arrogant Prussians were humiliating Paris in the Franco-Prussian war, Hermite, patriot though he was, kept his head, and he saw clearly that the mathematics of “the enemy” was mathematics and nothing else. Today, even when a man of science does take the civilized point of view, he is not impersonal about his supposed broadmindedness, but aggressive, as befits a man on the defensive. To Hermite it was so obvious that knowledge and wisdom are not the prerogatives of any sect, any creed, or any nation that he never bothered to put his instinctive sanity into words. In respect of what Hermite knew by instinct our generation is two centuries behind him. He died, loved the world over, on January 14, 1901.

I
. For example, as in the simple quadratic
x
²
—a
= 0: the roots are
and
the “many-valuedness” of the radical involved, here a square root, or irrationality of the
second
degree, appears in the double sign, ±, when we say briefly that the
two
roots are
The formula giving the
three
roots of cubic equations involves the three-valued irrationality
which has the
three
values

II
. Strictly,
ae
^x
,
where
a
does not depend upon
x,
is the most general, but the “multiplicative constant”
a
is trivial here.

All results of the profoundest mathematical investigation must ultimately be expressible in the simple form of properties of the integers.
—L
EOPOLD
K
RONECKER

P
ROFESSIONAL MATHEMATICIANS
who could properly be called business men are extremely rare. The one who most closely approximates to this ideal is Kronecker (1823-1891), who did so well for himself by the time he was thirty that thereafter he was enabled to devote his superb talents to mathematics in considerably greater comfort than most mathematicians can afford.

The obverse of Kronecker's career is to be found—according to a tradition familiar to American mathematicians—in the exploits of John Pierpont Morgan, founder of the banking house of Morgan and Company. If there is anything in this tradition, Morgan as a student in Germany showed such extraordinary mathematical ability that his professors tried to induce him to follow mathematics as his life work and even offered him a university position in Germany which would have sent him off to a flying start. Morgan declined and dedicated his gifts to finance, with results familiar to all. Speculators (in academic studies, not Wall Street) may amuse themselves by reconstructing world history on the hypothesis that Morgan had stuck to mathematics.