by E. T. Bell
When Hermite invented such forms he was interested in finding what numbers are represented by the forms. Over seventy years later it was found that the algebra of Hermitian forms is indispensable in mathematical physics, particularly in the modern quantum theory. Hermite had no idea that his pure mathematics would prove valuable in science long after his death-—indeed, like Archimedes, he never seemed to care much for the scientific applications of mathematics. But the fact that Hermite’s work has given physics a useful tool is perhaps another argument favoring the side that believes mathematicians best justify their abstract existence when left to their own inscrutable devices.
Leaving aside Hermite’s splendid discoveries in the theory of algebraic invariants as too technical for discussion here, we shall pass on in a moment to two of his most spectacular achievements in other fields. The high esteem in which Hermite’s work in invariants was held by his contemporaries may however be indicated by Sylvester’s characteristic remark that “Cayley, Hermite, and I constitute an In-variantive Trinity.” Who was who in this astounding trinity Sylvester omitted to state; but perhaps this oversight is immaterial, as each member of such a trefoil would be capable of transforming himself into himself or into either of his coinvariantive beings.
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The two fields in which Hermite found what are perhaps the most striking individual results in all his beautiful work are those of the general equation of the fifth degree and transcendental numbers. The nature of what he found in the first is clearly indicated in the introduction to his short note Sur la rèsolution de lèquation du cinquième degré (On the Solution of the [general] Equation of the Fifth Degree; published in the Comptes rendus de l‘Académie des Sciences for 1858, when Hermite was thirty six).
“It is known that the general equation of the fifth degree can be reduced, by a substitution [on the unknown x] whose coefficients are determined without using any irrationalities other than square roots or cube roots, to the form
x5—x—a = 0.
[That is, if we can solve this equation for x, then we can solve the general equation of the fifth degree.]
“This remarkable result, due to the English mathematician Jerrard, is the most important step that has been taken in the algebraic theory of equations of the fifth degree since Abel proved that a solution by radicals is impossible. This impossibility shows in fact the necessity for introducing some new analytic element [some new kind of function] in seeking the solution, and, on this account, it seems natural to take as an auxiliary the roots of the very simple equation we have just mentioned. Nevertheless, in order to legitimize its use rigorously as an essential element in the solution of the general equation, it remains to see if this simplicity of form actually permits us to arrive at some idea of the nature of its roots, to grasp what is peculiar and essential in the mode of existence of these quantities, of which nothing is known beyond the fact that they are not expressible by radicals.
“Now it is very remarkable that Jerrard’s equation lends itself with the greatest ease to this research, and is, in the sense which we shall explain, susceptible of an actual analytic solution. For we may indeed conceive the question of the algebraic solution of equations from a point of view different from that which for long has been indicated by the solution of equations of the first four degrees, and to which we are especially committed.
“Instead of expressing the closely interconnected system of roots, considered as functions of the coefficients, by a formula involving many-valued radicals,I we may seek to obtain the roots expressed separately by as many distinct uniform [one-valued] functions of auxiliary variables, as in the case of the third degree. In this case, where the equation
x3—3x + 2a = 0
is under discussion, it suffices, as we know, to represent the coefficient a by the sine of an angle, say A, in order that the roots be isolated as the following well-determined functions
[Hermite is here recalling the familiar “trigonometric solution” of the cubic usually discussed in the second course of school algebra. The “auxiliary variable” is A; the “uniform functions” are here sines.]
“Now it is an entirely similar fact which we have to exhibit concerning the equation
x5—x—a = 0.
Only, instead of sines or cosines, it is the elliptic functions which it is necessary to introduce. . . .”
In short order Hermite then proceeds to solve the general equation of the fifth degree, using for the purpose elliptic functions (strictly, elliptic modular functions, but the distinction is of no importance here). It is almost impossible to convey to a nonmathematician the spectacular brilliance of such a feat; to give a very inadequate simile, Hermite found the famous “lost chord” when no mortal had the slightest suspicion that such an elusive thing existed anywhere in time and space. Needless to say his totally unforeseen success created a sensation in the mathematical world. Better, it inaugurated a new department of algebra and analysis in which the grand problem is to discover and investigate those functions in terms of which the general equation of the nth degree can be solved explicitly in finite form. The best result so far obtained is that of Hermite’s pupil, Poincaré (in the 1880’s), who created the functions giving the required solution. These turned out to be a “natural” generalization of the elliptic functions. The characteristic of those functions that was generalized was periodicity. Further details would take us too far afield here, but if there is space we shall recur to this point when we reach Poincaré.
Hermite’s other sensational isolated result was that which established the transcendence (explained in a moment) of the number denoted in mathematical analysis by the letter e, namely
where 1! means 1, 2! = 1 × 2, 3! = 1 × 2 × 3, 4! = 1 × 2 × 3 × 4, and so on; this number is the “base” of the so-called “natural” system of logarithms, and is approximately 2.718281828. . . . It has been said that it is impossible to conceive of a universe in which e and π (the ratio of the circumference of a circle to its diameter) are lacking. However that may be (as a matter of fact it is false), it is a fact that e turns up everywhere in current mathematics, pure and applied. Why this should be so, at least so far as applied mathematics is concerned, may be inferred from the following fact: ex, considered as a function of x, is the only function of x whose rate of change with respect to x is equal to the function itself—that is, ex is the only function which is equal to its derivative.II
The concept of “transcendence” is extremely simple, also extremely important. Any root of an algebraic equation whose coefficients are rational integers is called an algebraic number. Thus 2.78 are algebraic numbers, because they are roots of the respective algebraic equations x2 + 1 = 0, 50x −139 = 0, in which the coefficients (1, 1 for the first, 50, −139 for the second) are rational integers. A “number” which is not algebraic is called transcendental. Otherwise expressed, a transcendental number is one which satisfies no algebraic equation with rational integer coefficients.
Now, given any “number” constructed according to some definite law, it is a meaningful question to ask whether it is algebraic or transcendental. Consider, for example, the following simply defined number,
in which the exponents 2, 6, 24, 120, . . . are the successive “factorials,” namely 2 = 1 × 2, 6 = 1 × 2 × 3, 24 = 1 × 2 × 3 × 4, 120 = 1 × 2 × 3 × 4 × 5, . . . , and the indicated series continues “to infinity” according to the same law as that for the terms given. The next term is the sum of the first three terms is .1 + .01 + .000001, or .110001, and it can be proved that the series does actually define some definite number which is less than .12. Is this number a root of any algebraic equation with rational integer coefficients? The answer is no, although to prove this without having been shown how to go about it is a severe test of high mathematical ability. On the other hand, the number defined by the infinite series
is algebraic; it is the root of 99900 x— 1 = 0 (as may be verified by the reader who remembers how to sum an infinite convergent geomet
rical progression).
The first to prove that certain numbers are transcendental was Joseph Liouville (the same man who encouraged Hermite to write to Jacobi) who, in 1844, discovered a very extensive class of transcendental numbers, of which all those of the form
where n is a real number greater than 1 (the example given above corresponds to n = 10), are among the simplest. But it is probably a much more difficult problem to prove that a particular suspect, like e or π, is or is not transcendental than it is to invent a whole infinite class of transcendentals: the inventive mathematician dictates—to a certain extent—the working conditions, while the suspected number is entire master of the situation, and it is the mathematician in this case, not the suspect, who takes orders which he only dimly understands. So when Hermite proved in 1873 that e (defined a short way back) is transcendental, the mathematical world was not only delighted but astonished at the marvellous ingenuity of the proof.
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 ab, 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 ab, where a is any algebraic number, is a root of xm—an = 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.
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I. For example, as in the simple quadratic x2—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, aex, where a does not depend upon x, is the most general, but the “multiplicative constant” a is trivial here.
CHAPTER TWENTY FIVE
The Doubter
KRONECKER
All results of the profoundest mathematical investigation must ultimately be expressible in the simple form of properties of the integers.—LEOPOLD KRONECKER
PROFESSIONAL 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.
What might have happened to Germany had Kronecker not abandoned finance for mathematics also offers a wide field for speculation. His business abilities were of a high order; he was an ardent patriot with an uncanny insight into European diplomacy and a shrewd cynicism—his admirers called it realism—regarding the unexpressed sentiments cherished by the great Powers for one another.
At first a liberal like so many intellectual young Jews, Kronecker quickly became a rock-ribbed conservative when he saw which side his own abundant bread was buttered on—after hi
s financial exploits, and proclaimed himself a loyal supporter of that callous old truth-doctor Bismarck. The famous episode of the Ems telegram which, according to some, was the electric spark that touched off the Franco-Prussian war in 1870, had Kronecker’s warm approval, and his grasp of the situation was so firm that before the battle of Weissenburg, when even the military geniuses of Germany were doubtful as to the outcome of their bold challenging of France, Kronecker confidently predicted the success of the entire campaign and was proved right in detail. At the time, and indeed all his life, he was on cordial terms with the leading French mathematicians, and he was clear-headed enough not to let his political opinions cloud his just perception of his scientific rivals’ merits. It is perhaps as well that so realistic a man as Kronecker cast his lot with mathematics.
Leopold Kronecker’s life was easy from the day of his birth. The son of prosperous Jewish parents, he was born on December 7, 1823, at Liegnitz, Prussia. By an unaccountable oversight Kronecker’s official biographers (Heinrich Weber and Adolf Kneser) omit all mention of Leopold’s mother, although he probably had one, and concentrate on the father, who owned a flourishing mercantile business. The father was a well educated man with an unquenchable thirst for philosophy which he passed on to Leopold. There was another son, Hugo, seventeen years younger than Leopold, who became a distinguished physiologist and professor at Berne. Leopold’s early education under a private tutor was supervised by the father; Hugo’s upbringing later became the loving duty of Leopold.
In the second stage of his education at the preparatory school for the Gymnasium Leopold was strongly influenced by the co-rector Werner, a man with philosophical and theological leanings, who later taught Kronecker when he entered the Gymnasium. Among other things Kronecker imbibed from Werner was a liberal draught of Christian theology, for which he acquired a lifelong enthusiasm. With what looks like his usual caution, Kronecker did not embrace the Christian faith till practically on his deathbed when, having seen that it did his six children no noticeable mischief, he permitted himself to be converted from Judaism to evangelical Christianity in his sixty eighth year.