Men of Mathematics (15 page)

The equation
y
³
= x
²
+ 2,
with the restriction that the solution y, x is to be in whole numbers,
is
indeterminate
(because there are more unknowns, namely two,
x
and
y,
than there are equations, namely one, connecting them) and
Diophantine,
after the Greek who was one of the first to insist upon
whole number
solutions of equations or, less stringently, on
rational
(fractional) solutions. There is no difficulty whatever in describing an infinity of solutions
without
the restriction to whole numbers: thus we may give
x any
value we please and then determine
y
by adding 2 to this
x
²
and extracting the cube root of the
result. But the
Diophantine
problem of finding
all
the
whole number
solutions is quite another matter. The solution
y
= 3,
x
= 5 is seen “by inspection”; the difficulty of the problem is to prove that there are
no other
whole numbers
y, x
which will satisfy the equation. Fermat proved that there are none but, as usual, suppressed his proof, and it was not until many years after his death that a proof was found.

This time he was not guessing; the problem is hard; he asserted that he had a proof; a proof was later found. And so for all of his positive assertions with the one exception of the seemingly simple one which he made in his Last Theorem and which mathematicians, struggling for nearly 300 years, have been unable to prove: whenever Fermat asserted that he had
proved
anything, the statement, with the one exception noted, has subsequently been proved. Both his scrupulously honest character and his unrivalled penetration as an arithmetician substantiate the claim made for him by some, but not by all, that he knew what he was talking about when he asserted that he possessed a proof of his theorem.

It was Fermat's custom in reading Bachet's
Diophantus
to record the results of his meditations in brief marginal notes in his copy. The margin was not suited for the writing out of proofs. Thus, in commenting on the eighth problem of the Second Book of Diophantus' Arithmetic, which asks for the solution in rational numbers (fractions or whole numbers) of the equation
x
²
+ y
²
= a
²
,
Fermat comments as follows:

“On the contrary, it is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or, generally, any power above the second into two powers of the same degree: I have discovered a truly marvellous demonstration [of this general theorem] which this margin is too narrow to contain” (Fermat,
Oeuvres,
III,
p. 241
). This is his famous Last Theorem, which he discovered about the year 1637.

To restate this in modern language: Diophantus' problem is to find whole numbers or fractions
x, y, a
such that
x
²
+
y
²
= a
²
;
Fermat asserts that
no
whole numbers or fractions exist such that x
³
+
y
³
= a
³
,
or
x
⁴
+ y
⁴
= a
⁴
, or, generally, such that
x
ⁿ
+ y
ⁿ
=
a
ⁿ
if
n
is a whole number greater than 2.

Diophantus' problem has an infinity of solutions; specimens are
x
= S,
y =
4,
a
= 5;
x =
5,
y
= 12,
a
= 13. Fermat himself gave a proof by his method of infinite descent for the impossibility of x
⁴
+
Y
⁴
=
a
⁴
. Since his day
x
ⁿ
+ y
ⁿ
= a
n
has been proved impossible in whole numbers (or fractions) for a great many numbers
n
(up to all primes* less than
n
= 14000 if none of the numbers
x, y, a
is divisible by
n),
but this is not what is required. A proof disposing of
all
n's greater than 2 is demanded. Fermat said he possessed a “marvellous” proof.

After all that has been said, is it likely that he had deceived himself? It may be left up to the reader. One great arithmetician, Gauss, voted against Fermat. However, the fox who could not get at the grapes declared they were sour. Others have voted for him. Fermat was a mathematician of the first rank, a man of unimpeachable honesty, and an arithmetician without a superior in history.
^III

I
. This statement is sufficiently accurate for the present account. Actually, the values of the variables (coordinates and velocities) which make the function in question
stationary
(neither increasing nor decreasing, roughly) are those required. An
extremum
is stationary; but a
stationary
is not necessarily an extremum.

II
. The reader can easily see that it suffices to dispose of the case where
n
is an odd prime, since, in algebra,
u
^ab
= (
u
^a
)
^b
, where
u, a, b
are any numbers.

III
. In 1908 the late Professor Paul Wolfskehl (German) left 100,000 marks to be awarded to the first person giving a
complete
proof of Fermat's Last Theorem. The inflation after the World War reduced this prize to a fraction of a cent, which is what the mercenary will now get for a proof.

We see . . . that the theory of probabilities is at bottom only common sense reduced to calculation; it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct, often without being able to account for it. . . . It is remarkable that [this] science, which originated in the consideration of games of chance, should have become the most important object of human knowledge.
—P. S. L
APLACE

Y
OUNGER BY TWENTY SEVEN YEARS
than his great contemporary Descartes, Blaise Pascal was born at Clermont, Auvergne, France, on June 19, 1623, and outlived Descartes by twelve years. His father Êtienne Pascal, president of the court of aids at Clermont, was a man of culture and had some claim to intellectual distinction in his own times; his mother, Antoinette Bégone, died when her son was four. Pascal had two beautiful and talented sisters, Gilberte, who became Madame Périer, and Jacqueline, both of whom, the latter especially, played important parts in his life.

Blaise Pascal is best known to the general reader for his two literary classics, the
Pensées
and the
Lettres écrites par Louis de Montalte à un provincial de ses amis
commonly referred to as the “Provincial Letters,” and it is customary to condense his mathematical career to a few paragraphs in the display of his religious prodigies. Here our point of view must necessarily be somewhat oblique, and we shall consider Pascal primarily as a highly gifted mathematician who let his masochistic proclivities for self-torturing and profitless speculations on the sectarian controversies of his day degrade him to what would now be called a religious neurotic.

On the mathematical side Pascal is perhaps the greatest might-have-been in history. He had the misfortune to precede Newton by only a few years and to be a contemporary of Descartes and Fermat, both more stable men than himself. His most novel work, the creation of the mathematical theory of probability, was shared with
Fermat, who could easily have done it alone. In geometry, for which he is famous as a sort of infant prodigy, the creative idea was supplied by a man—Desargues—of much lesser celebrity.

In his outlook on experimental science Pascal had a far clearer vision than Descartes—from a modern point of view—of the scientific method. But he lacked Descartes' singleness of aim, and although he did some first-rate work, allowed himself to be deflected from what he might have done by his morbid passion for religious subtleties.

It is useless to speculate on what Pascal might have done. Let his life tell what he actually did. Then, if we choose, we can sum him up as a mathematician by saying that he did what was in him and that no man can do more. His life is a running commentary on two of the stories or similes in that New Testament which was his constant companion and unfailing comfort: the parable of the talents, and the remark about new wine bursting old bottles (or skins). If ever a wonderfully gifted man buried his talent, Pascal did; and if ever a medieval mind was cracked and burst asunder by its attempt to hold the new wine of seventeenth-century science, Pascal's was. His great gifts were bestowed upon the wrong person.

At the age of seven Pascal moved from Clermont with his father and sisters to Paris. About this time the father began teaching his son. Pascal was an extremely precocious child. Both he and his sisters appear to have had more than their share of nature's gifts. But poor Blaise inherited (or acquired) a wretched physique along with his brilliant mind, and Jacqueline, the more gifted of his sisters, seems to have been of the same stripe as her brother, for she too fell a victim to morbid religiosity.

At first everything went well enough. Pascal senior, astonished at the ease with which his son absorbed the stock classical education of the day, tried to hold the boy down to a reasonable pace to avoid injuring his health. Mathematics was taboo, on the theory that the young genius might overstrain himself by using his head. His father was an excellent drillmaster but a poor psychologist. His ban on mathematics naturally excited the boy's curiosity. One day when he was about twelve Pascal demanded to know what geometry was about. His father gave him a clear description. This set Pascal off like a hare after his true vocation. Contrary to his own opinion in later life he had been called by God, not to torment the Jesuits, but to be a great
mathematician. But his hearing was defective at the time and he got his orders confused.

What happened when Pascal began the study of geometry has become one of the legends of mathematical precocity. In passing it may be remarked that infant prodigies in mathematics do not invariably blow up as they are sometimes said to do. Precocity in mathematics has often been the first flush of a glorious maturity, in spite of the persistent superstition to the contrary. In Pascal's case early mathematical genius was not extinguished as he grew up but stifled under other interests. The ability to do first-class mathematics persisted, as will be seen from the episode of the cycloid, late into his all too brief life, and if anything is to be blamed for his comparatively early mathematical demise it is probably his stomach. His first spectacular feat was to prove, entirely on his own initiative, and without a hint from any book, that the sum of the angles of a triangle is equal to two right angles. This encouraged him to go ahead at a terrific pace.

Realizing that he had begotten a mathematician, Pascal senior wept with joy and gave his son a copy of Euclid's
Elements.
This was quickly devoured, not as a task, but as play. The boy gave up his games to geometrize. In connection with Pascal's rapid mastery of Euclid, sister Gilberte permits herself an overappreciative fib. It is true that Pascal had found out and proved several of Euclid's propositions for himself before he ever saw the book. But what Gilberte romances about her brilliant young brother is less probable than a throw of a billion aces in succession with one die, for the reason that it is infinitely improbable. Gilberte declared that her brother had rediscovered for himself the first thirty two propositions of Euclid, and that he had found them
in the same order
as that in which Euclid sets them forth. The thirty second proposition is indeed the famous one about the sum of the angles of a triangle which Pascal rediscovered. Now, there may be only one way of doing a thing right, but it seems more likely that there are an infinity of ways of doing it wrong. We know today that Euclid's allegedly rigorous demonstrations, even in the first four of his propositions, are no proofs at all. That Pascal faithfully duplicated all of Euclid's oversights on his own account is an easy story to tell but a hard one to believe. However, we can forgive Gilberte for bragging. Her brother was worth it. At the age of fourteen he was admitted to the weekly scientific discussions,
conducted by Mersenne, out of which the French Academy of Sciences developed.

While young Pascal was fast making a geometer of himself, old Pascal was making a thorough nuisance of
himself
with the authorities on account of his honesty and general uprightness. In particular he disagreed with Cardinal Richelieu over a little matter of imposing taxes. The Cardinal was incensed; the Pascal family went into hiding till the storm blew over. It is said that the beautiful and talented Jacqueline rescued the family and restored her father to the light of the Cardinal's countenance by her brilliant acting, incognito, in a play presented for Richelieu's entertainment. On inquiring the name of the charming young artiste who had captivated his clerical fancy, and being told that she was the daughter of his minor enemy, Richelieu very handsomely forgave the whole family and planted the father in a political job at Rouen. From what is known of that wily old serpent, Cardinal Richelieu, this pleasing tale is probably a fish story. Anyhow, the Pascals once more found a job and security at Rouen. There young Pascal met the tragic dramatist Corneille, who was duly impressed with the boy's genius. At the time Pascal was all mathematician, so probably Corneille did not suspect that his young friend was to become one of the great creators of French prose.