Men of Mathematics (60 page)

Abel at the time was about fifteen. Up till now he had shown no marked talent for anything except taking his troubles with a sense of humor. Under the kindly, enlightened Holmboë's teaching Abel suddenly discovered what he was. At sixteen he began reading privately and thoroughly digesting the great works of his predecessors, including some of those of Newton, Euler, and Lagrange. Thereafter real mathematics was not only his serious occupation but his fascinating delight. Asked some years later how he had managed to forge ahead so rapidly to the front rank he replied, “By studying the masters, not their pupils”—a prescription some popular writers of textbooks might do well to mention in their prefaces as an antidote to the poisonous mediocrity of their uninspired pedagogics.

Holmboë and Abel soon became close friends. Although the teacher was himself no creative mathematician he knew and appreciated the masterpieces of mathematics, and under his eager suggestions Abel was soon mastering the toughest of the classics, including the
Disquisitiones Arithmeticae
of Gauss.

Today it is a commonplace that many fine things the old masters thought they had proved were not really proved at all. Particularly is this true of some of Euler's work on infinite series and some of Lagrange's on analysis. Abel's keen mind was one of the first to detect the gaps in his predecessors' reasoning, and he resolved to devote a fair share of his lifework to caulking the cracks and making the
reasoning watertight. One of his classics in this direction is the first
proof
of the
general
binomial theorem, special cases of which had been stated by Newton and Euler. It is not easy to give a sound proof in the general case, so perhaps it is not astonishing to find alleged proofs still displayed in the schoolbooks as if Abel had never lived. This proof however was only a detail in Abel's vaster program of cleaning up the theory and application of infinite series.

Abel's father died in
1820
at the early age of forty eight. At the time Abel was eighteen. The care of his mother and six children fell on his shoulders. Confident of himself Abel assumed his sudden responsibilities cheerfully. Abel was a genial and optimistic soul. With no more than strict justice he foresaw himself as an honored and moderately prosperous mathematician in a university chair. Then he could provide for the lot of them in reasonable security. In the meantime he took private pupils and did what he could. In passing it may be noted that Abel was a very successful teacher. Had he been footloose poverty would never have bothered him. He could have earned enough for his own modest needs, somehow or other, at any time. But with seven on his back he had no chance. He never complained, but took it all in his stride as part of the day's work and kept at his mathematical researches in every spare moment.

Convinced that he had one of the greatest mathematicians of all time on his hands, Holmboë did what he could by getting subsidies for the young man and digging down generously into his own none too deep pocket. But the country was poor to the point of starvation and not nearly enough could be done. In those years of privation and incessant work Abel immortalized himself and sowed the seeds of the disease which was to kill him before he had half done his work.

*  *  *

Abel's first ambitious venture was an attack on the general equation of the fifth degree (the “quintic”). All of his great predecessors in algebra had exhausted their efforts to produce a solution, without success. We can easily imagine Abel's exultation when he mistakenly imagined he had succeeded. Through Holmboë the supposed solution was sent to the most learned mathematical scholar of the time in Denmark who, fortunately for Abel, asked for further particulars without committing himself to an opinion on the correctness of the solution. Abel in the meantime had found the flaw in his reasoning. The supposed solution was of course no solution at all. This failure gave
him a most salutary jolt; it jarred him onto the right track and caused him to doubt whether an algebraic solution was possible. He
proved the impossibility.
At the time he was about nineteen. But he had been anticipated, at least in part, in the whole project.

As this question of the general quintic played a rôle in algebra similar to that of a crucial experiment to decide the fate of an entire scientific theory, it is worth a moment's attention. We shall quote presently a few things Abel himself says.

The nature of the problem is easily described. In early school algebra we learn to solve the
general
equations of the
first
and
second
degrees in the unknown
x,
say

ax
+
b
= 0,
ax
²
+
bx
+
c
= 0,

and a little later those of the
third and fourth
degrees, say

ax
³
+
bx
²
+
cx
+
d
= 0,
ax
⁴
+
bx
³
+
cx
²
+
dx
+
e =
0.

That is, we produce
finite
(closed) formulas for each of these
general
equations of the first four degrees, expressing the unknown
x
in terms of the given coefficients
a, b, c, d, e.
A solution such as any one of these four which can be obtained by only a
finite number of additions, multiplications, subtractions, divisions, and extractions of roots,
all these operations being performed on the given coefficients, is called
algebraic.
The important qualification in this definition of an
algebraic solution
is “finite”; there is no difficulty in describing solutions for
any
algebraic equation which contain no extractions of roots at all, but which do imply an
infinity
of the other operations named.

After this success with algebraic equations of the first four degrees, algebraists struggled for nearly three centuries to produce a similar
algebraic solution
for the general quintic

ax
⁵
+
bx
⁴
+
cx
³
+
dx
²
+
ex
+
f
= 0.

They failed. It is here that Abel enters.

The following extracts are given partly to show how a great inventive mathematician thought and partly for their intrinsic interest. They are from Abel's memoir
On the algebraic resolution of equations.

“One of the most interesting problems of algebra is that of the algebraic solution of equations. Thus we find that nearly all mathematicians of distinguished rank have treated this subject. We arrive without difficulty at the general expression of the roots of equations
of the first four degrees. A uniform method for solving these equations was discovered and it was believed to be applicable to an equation of any degree; but in spite of all the efforts of Lagrange and other distinguished mathematicians the proposed end was not reached. That led to the presumption that the solution of general equations was impossible algebraically; but this is what could not be decided, since the method followed could lead to decisive conclusions only in the case where the equations were solvable. In effect they proposed to solve equations without knowing whether it was possible. In this way one might indeed arrive at a solution, although that was by no means certain; but if by ill luck the solution was impossible, one might seek it for an eternity, without finding it. To arrive infallibly at something in this matter, we must therefore follow another road. We can give the problem such a form that it shall always be possible to solve it, as we can always do with any problem.
^I
Instead of asking for a relation of which it is not known whether it exists or not, we must ask whether such a relation is indeed possible. . . . When a problem is posed in this way, the very statement contains the germ of the solution and indicates what road must be taken; and I believe there will be few instances where we shall fail to arrive at propositions of more or less importance, even when the complication of the calculations precludes a complete answer to the problem.”

He goes on to say that this, the true scientific method to be followed, has been but little used owing to the extreme complication of the calculations (algebraic) which it entails; “but,” he adds, “in many instances this complication is only apparent and vanishes after the first attack.” He continues:

“I have treated several branches of analysis in this manner, and although I have often set myself problems beyond my powers, I have nevertheless arrived at a large number of general results which throw a strong light on the nature of those quantities whose elucidation is the object of mathematics. On another occasion I shall give the results at which I have arrived in these researches and the procedure which has led me to them. In the present memoir I shall treat the problem of the algebraic solution of equations in all its generality.”

Presently he states two general inter-related problems which he proposes to discuss:

“1. To find all the equations of any given degree which are solvable algebraically.

2. To determine whether a given equation is or is not solvable algebraically.”

At bottom, he says, these two problems are the same, and although he does not claim a
complete
solution, he does
indicate
an infallible method
(des moyens sûrs)
for disposing of them fully.

Abel's irrepressible inventiveness hurried him on to vaster problems before he had time to return to these; their complete solution—the explicit statement of necessary and sufficient conditions that an algebraic equation be solvable algebraically—was to be reserved for Galois. When this memoir of Abel's was published in 1828, Galois was a boy of sixteen, already well started on his career of fundamental discovery. Galois later came to know and admire the work of Abel; it is probable that Abel never heard the name of Galois, although when Abel visited Paris he and his brilliant successor could have been only a few miles apart. But for the stupidity of Galois' teachers and the loftiness of some of Abel's mathematical “superiors,” it is quite possible that he and Abel might have met.

Epoch-making as Abel's work in algebra was, it is overshadowed by his creation of a new branch of analysis. This, as Legendre said, is Abel's “time-outlasting monument.” If the story of his life adds nothing to the splendor of his accomplishment it at least suggests what the world lost when he died. It is a somewhat discouraging tale. Only Abel's unconquerable cheerfulness and unyielding courage under the stress of poverty and lack of encouragement from the mathematical princes of his day lighten the story. He did however find one generous friend in addition to Holmboë.

*  *  *

In June, 1822, when Abel was nineteen, he completed his required work at the University of Kristiania. Holmboë had done everything possible to relieve the young man's poverty, convincing his colleagues that they too should subscribe to make it possible for Abel to continue his mathematical researches. They were immensely proud of him but they were also poor themselves. Abel quickly outgrew Scandinavia. He longed to visit France, then the mathematical queen of the world, where he could meet his great peers (he was in a class far above some
of them, but he did not know it). He dreamed also of touring Germany and meeting Gauss, the undisputed prince of them all.

Abel's mathematical and astronomical friends persuaded the University to appeal to the Norwegian Government to subsidize the young man for a grand mathematical tour of Europe. To impress the authorities with his worthiness, Abel submitted an extensive memoir which, from its title, was probably connected with the fields of his greatest fame. He himself thought highly enough of it to believe its publication by the University would bring Norway honor, and Abel's opinion of his own work, never more than just, was probably as good as anyone's. Unfortunately the University was having a severe financial struggle of its own, and the memoir was finally lost. After undue deliberation the Government compromised—does any Government ever do anything else?—and instead of doing the only sensible thing, namely sending Abel at once to France and Germany, granted him a subsidy to continue his university studies at Kristiania in order that he might brush up his French and German. That is exactly the sort of decision he might have expected from any body of officials conspicuous for their good hearts and common sense. Common sense however has no business dictating to genius.

Abel dallied a year and a half at Kristiania, not wasting his time, but dutifully keeping his part of the contract by wrestling (not too successfully) with German, getting a fair start on French, and working incessantly at his mathematics. With his incurable optimism he had also got himself engaged to a young woman—Crelly Kemp. At last, on August 27, 1825, when Abel was twenty three, his friends overcame the last objection of the Government, and a royal decree granted him sufficient funds for a year's travel and study in France and Germany. They did not give him much, but the fact that they gave him anything at all in the straitened financial condition of the country says more for the state of civilization in Norway in 1825 than could a whole encyclopaedia of the arts and trades. Abel was grateful. It took him about a month to straighten out his dependents before leaving. But thirteen months before this, innocently believing that all mathematicians were as generous-minded as himself, he had burned one of his ladders before ever setting foot on it.

Out of his own pocket—God only knows how—Abel had paid for the printing of his memoir in which the impossibility of solving the general equation of the fifth degree algebraically is proved. It was a
pretty poor job of printing but the best backward Norway could manage. This, Abel naively believed, was to be his scientific passport to the great mathematicians of the Continent. Gauss in particular, he hoped, would recognize the signal merits of the work and grant him more than a formal interview. He could not know that “the prince of mathematicians” sometimes exhibited anything but a princely generosity to young mathematicians struggling for just recognition.