Men of Mathematics (62 page)

He tells then how Crelle has been begging him to take up his residence permanently in Berlin. Crelle was already using all his human engineering skill to hoist the Norwegian Abel into a professorship in the University of Berlin. Such was the Germany of 1826. Abel of course was already great, and the sure promise of what he had in him indicated him as the likeliest mathematical successor to Gauss. That he was a foreigner made no difference; Berlin in 1826 wanted the best in mathematics. A century later the best in mathematical physics
was not good enough, and Berlin quite forcibly got rid of Einstein. Thus do we progress. But to return to the sanguine Abel.

“At first I counted on going directly from Berlin to Paris, happy in the promise that Mr.
Crelle
would accompany me. But Mr.
Crelle
was prevented, and I shall have to travel alone. Now I am so constituted that I cannot endure solitude. Alone, I am depressed, I get cantankerous, and I have little inclination for work. So I said to myself it would be much better to go with Mr.
Boeck
to Vienna, and this trip seems to me to be justified by the fact that at Vienna there are men like
Littrow, Burg,
and still others, all indeed excellent mathematicians; add to this that I shall make but this one voyage in my life. Could one find anything but reasonableness in this wish of mine to see something of the life of the South? I could work assiduously enough while travelling. Once in Vienna and leaving there for Paris, it is almost a bee-line via Switzerland. Why shouldn't I see a little of it too? My God! I, even I, have some taste for the beauties of nature, like everybody else. This whole trip would bring me to Paris two months later, that's all. I could quickly catch up the time lost. Don't you think such a trip would do me good?”

*  *  *

So Abel went South, leaving his masterpiece in Cauchy's care to be presented to the Institut. The prolific Cauchy was so busy laying eggs of his own and cackling about them that he had no time to examine the veritable roc's egg which the modest Abel had deposited in the nest. Hachette, a mere pot-washer of a mathematician, presented Abel's
Memoir on a general property of a very extensive class of transcendental functions
to the Paris Academy of Sciences on the tenth of October, 1826. This is the work which Legendre later described in the words of Horace as
“monumentum aere perennius,”
and the five hundred years' work which Hermite said Abel had laid out for future generations of mathematicians. It is one of the crowning achievements of modern mathematics.

What happened to it? Legendre and Cauchy were appointed as referees. Legendre was seventy four, Cauchy thirty nine. The veteran was losing his edge, the captain was in his self-centred prime. Legendre complained (letter to Jacobi, 8 April, 1829) that “we perceived that the memoir was barely legible; it was written in ink almost white, the letters badly formed; it was agreed between us that the author should be asked for a neater copy to be read.” What an alibi! Cauchy took the memoir home, mislaid it, and forgot all about it.

To match this phenomenal feat of forgetfulness we have to imagine an Egyptologist mislaying the Rosetta Stone. Only by a sort of miracle was the memoir unearthed after Abel's death. Jacobi heard of it from Legendre, with whom Abel corresponded after returning to Norway, and in a letter dated 14 March, 1829, Jacobi exclaims, “What a discovery is this of Mr. Abel's! . . . Did anyone ever see the like? But how comes it that this discovery, perhaps the most important mathematical discovery that has been made in our Century, having been communicated to your Academy two years ago, has escaped the attention of your colleagues?” The enquiry reached Norway. To make a long story short, the Norwegian consul at Paris raised a diplomatic row about the missing manuscript and Cauchy dug it up in 1830. Finally it was printed, but not till 1841, in the
Mémoires présentés par divers savants à
l'
Académie royale des sciences de
l'
Institut de France,
vol. 7, pp. 176-264. To crown this epic
in parvo
of crass incompetence, the editor, or the printers, or both between them, succeeded in losing the manuscript before the proof-sheets were read.
^II
The Academy (in 1830) made amends to Abel by awarding him the Grand Prize in Mathematics jointly with Jacobi. Abel, however, was dead.

*  *  *

The opening paragraphs of the memoir indicate its scope.

“The transcendental functions hitherto considered by mathematicians are very few in number. Practically the entire theory of transcendental functions is reduced to that of logarithmic functions, circular and exponential functions, functions which, at bottom, form but a single species. It is only recently that some other functions have begun to be considered. Among the latter, the elliptic transcendents, several of whose remarkable and elegant properties have been developed by Mr. Legendre, hold the first place. The author [Abel] has considered, in the memoir which he has the honor to present to the Academy, a very extended class of functions, namely: all those whose derivatives are expressible by means of algebraic
equations whose coefficients are rational functions of one variable, and he has proved for these functions properties analogous to those of logarithmic and elliptic functions . . . and he has arrived at the following theorem:

“If we have several functions whose derivatives can be roots of
one and the same algebraic equation,
all of whose coefficients are
rational
functions of one variable, we can always express the sum of any number of such functions by an
algebraic
and
logarithmic
function, provided that we establish a certain number of
algebraic
relations between the variables of the functions in question.

“The number of these relations does not depend at all upon the number of functions, but only upon the nature of the particular functions considered . . . .”

*  *  *

The theorem which Abel thus briefly describes is today known as Abel's Theorem. His proof of it has been described as nothing more than “a marvellous exercise in the integral calculus.” As in his algebra, so in his analysis, Abel attained his proof with a superb parsimony. The proof, it may be said without exaggeration, is well within the purview of any seventeen-year-old who has been through a good
first
course in the calculus. There is nothing high-falutin' about the classic simplicity of Abel's own proof. The like cannot be said for some of the nineteenth century expansions and geometrical reworkings of the original proof. Abel's proof is like a statue by Phidias; some of the others resemble a Gothic cathedral smothered in Irish lace, Italian confetti, and French pastry.

There is ground for a possible misunderstanding in Abel's opening paragraph. Abel no doubt was merely being kindly courteous to an old man who had patronized him—in the bad sense—on first acquaintance, but who, nevertheless, had spent most of his long working life on an important problem without seeing what it was all about. It is not true that Legendre had discussed the elliptic
functions,
as Abel's words might imply; what Legendre spent most of his life over was elliptic
integrals,
which are as different from
elliptic functions
as a horse is from the cart it pulls, and therein precisely is the crux and the germ of one of Abel's greatest contributions to mathematics. The matter is quite simple to anyone who has had a school course in trigonometry; to obviate tedious explanations of elementary matters this much will be assumed in what follows presently.

For those who have forgotten all about trigonometry, however, the
essence, the
methodology,
of Abel's epochal advance can be analogized thus. We alluded to the cart and the horse. The frowsy proverb about putting the cart before the horse describes what Legendre did; Abel saw that if the cart was to move forward the horse should precede it. To take another instance: Francis Galton, in his statistical studies of the relation between poverty and chronic drunkenness, was led, by his impartial mind, to a reconsideration of all the self-righteous platitudes by which indignant moralists and economic crusaders with an axe to grind evaluate such social phenomena. Instead of assuming that people are depraved
because
they drink to excess, Galton
inverted
this hypothesis and assumed temporarily that people drink to excess
because
they have inherited no moral guts from their ancestors, in short,
because
they
are
depraved. Brushing aside all the vaporous moralizings of the reformers, Galton took a firm grip on a
scientific,
unemotional,
workable
hypothesis to which he could apply the impartial machinery of mathematics. His work has not yet registered socially. For the moment we need note only that Galton, like Abel,
inverted
his problem—turned it upside-down and inside-out, back-end-to and foremost-end-backward. Like Hiawatha and his fabulous mittens, Galton put the skinside inside and the inside outside.

All this is far from being obvious or a triviality. It is one of the most powerful methods of mathematical discovery (or invention) ever devised, and Abel was the first human being to use it consciously as an engine of research. “You must always invert,” as Jacobi said when asked the secret of his mathematical discoveries. He was recalling what Abel and he had done. If the solution of a problem becomes hopelessly involved, try turning the problem backwards, put the quaesita for the data and vice versa. Thus if we find Cardan's character incomprehensible when we think of him as
a
son of his father, shift the emphasis,
invert
it, and see what we get when we analyse Cardan's father as
the
begetter and endower of his son. Instead of studying “inheritance” concentrate on “endowing.” To return to those who remember some trigonometry.

Suppose mathematicians had been so blind as not to see that sin
x,
cos
x
and the other
direct
trigonometric functions are simpler to use, in the addition formulas and elsewhere, than the
inverse
functions sin
^-1
x,
cos
^-1
x.
Recall the formula sin
(x
+ y) in terms of sines and cosines of
x
and
y,
and contrast it with the formula for sin
^-1
(x + y)
in terms of
x
and
y.
Is not the former incomparably simpler, more elegant, more “natural” than the latter? Now, in the integral calculus, the
inverse
trigonometric functions present themselves naturally as definite integrals of simple algebraic irrationalities (second degree); such integrals appear when we seek to find the length of an arc of a circle by means of the integral calculus. Suppose the
inverse
trigonometric functions had
first
presented themselves this way. Would it not have been “more natural” to consider the
inverses
of these functions, that is, the familiar trigonometric functions themselves as the
given
functions to be studied and analyzed? Undoubtedly; but in shoals of more advanced problems, the simplest of which is that of finding the length of the arc of an
ellipse
by the integral calculus, the awkward
inverse
“elliptic” (not “circular,” as for the arc of a circle) functions presented themselves
first.
It took Abel to see that
these
functions should be “inverted” and studied, precisely as in the case of sin x, cos
x
instead of sin
^-1
x, cos
^-1
x.
Simple, was it not? Yet Legendre, a great mathematician, spent more than
forty years
over his “elliptic integrals” (the awkward “inverse functions” of his problem) without ever once suspecting that he should
invert.
^III
This extremely simple, uncommonsensical way of looking at an apparently simple but profoundly recondite problem was one of the greatest mathematical advances of the nineteenth century.

All this however was but the beginning, although a sufficiently tremendous beginning—like Kipling's dawn coming up like thunder—of what Abel did in his magnificent theorem and in his work on elliptic functions. The trigonometric or circular functions have a single real period, thus sin
(x
+
2π)
= sin
x,
etc. Abel discovered that his new functions provided by the inversion of an elliptic integral have precisely
two
periods, whose ratio is imaginary. After that, Abel's followers in this direction—Jacobi, Rosenhain, Weierstrass, Riemann, and many more—mined deeply into Abel's great theorem and by carrying on and extending his ideas discovered functions of
n
variables having
2n
periods. Abel himself carried the exploitation of his discoveries far. His successors have applied all this work to geometry, mechanics, parts of mathematical physics, and other tracts of mathematics,
solving important problems which, without this work initiated by Abel, would have been unsolvable.

*  *  *

While in Paris Abel consulted good physicians for what he thought was merely a persistent cold. He was told that he had tuberculosis of the lungs. He refused to believe it, wiped the mud of Paris off his boots, and returned to Berlin for a short visit. His funds were running low; about seven dollars was the extent of his fortune. An urgent letter brought a loan from Holmboë after some delay. It must not be supposed that Abel was a chronic borrower on no prospects. He had good reason for believing that he should have a paying job when he got home. Moreover, money was still owed to him. On Holmboë's loan of about sixty dollars Abel existed and researched from March till May, 1827. Then, all his resources exhausted, he turned homeward and arrived in Kristiania completely destitute.