by E. T. Bell
As an example we may verify the associative law for (AB)C and A(BC), which should be equal. First, AB = C; hence (AB)C = CC = I. Again BC = A; hence A(BC) = AA = I. In the same way A(DB) = AI = A; (AD)B = EB = A; thus (AD)B = A(DB).
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The total number of distinct operations in a group is called its order. Here 6 is the order of the group. By inspection of the table we pick out several subgroups, for example, which are of the respective orders 1, 2, 3. This illustrates one of the fundamental theorems proved by Cauchy: the order of any subgroup is a divisor of the order of the group.
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The reader may find it amusing to try to construct groups of orders other than 6. For any given order the number of distinct groups (having different multiplication tables) is finite, but what this number may be for any given order (the general order n) is not known —nor likely to be in our lifetime. So at the very beginning of a theory which on its surface is as simple as dominoes we run into unsolved problems.
Having constructed the “multiplication table” of a group, we forget about its derivation from substitutions (if that happens to be the way the table was made), and regard the table as defining an abstract group; that is, the symbols I, A, B, . . . are given no interpretation beyond that implied by the rule of combination, as in CD = A, DC = E, etc. This abstract point of view is that now current. It was not Cauchy’s, but was introduced by Cayley in 1854. Nor were completely satisfactory sets of postulates for groups stated till the first decade of the twentieth century.
When the operations of a group are interpreted as substitutions, or as the rotations of a rigid body, or in any other department of mathematics to which groups are applicable, the interpretation is called a realization of the abstract group defined by the multiplication table. A given abstract group may have many diverse realizations. This is one of the reasons that groups are of fundamental importance in modern mathematics: one abstract, underlying structure (that summarized in the multiplication table) of one and the same group is the essence of several apparently unrelated theories, and by an intensive study of the properties of the abstract group, a knowledge of the theories in question and their mutual relationships is obtained by one investigation instead of several.
To give but one instance, the set of all rotations which twirl a regular icosahedron (twenty-sided regular solid) about its axes of symmetry, so that after any rotation of the set the volume of the solid occupies the same space as before, forms a group, and this group of rotations, when expressed abstractly, is the same group as that which appears, under permutations of the roots, when we attempt to solve the general equation of the fifth degree. Further, this same group turns up (to anticipate slightly) in the theory of elliptic functions. This suggests that although it is impossible to solve the general quintic algebraically, the equation may be—and in fact is—solvable in terms of the functions mentioned. Finally, all this can be pictured geometrically by describing the rotations of an icosahedron already mentioned. This beautiful unification was the work of Felix Klein (1849–1925) in his book on the icosahedron (1884).
Cauchy was one of the great pioneers in the theory of substitution groups. Since his day an immense amount of work has been done in the subject, and the theory itself has been vastly extended by the accession of infinite groups—groups having an infinity of operations which can be counted off 1, 2, 3, . . . , and further, to groups of continuous motions. In the latter an operation of the group shifts a body into another position by infinitesimal (arbitrarily small) displacements—not like the icosahedral group described above, where the rotations shift the whole body round by a finite amount. This is but one category of infinite groups (the terminology here is not exact, but is sufficient to bring out the point of importance—the distinction between discrete and continuous groups). Just as the theory of finite discrete groups is the structure underlying the theory of algebraic equations, so is the theory of infinite, continuous groups of great service in the theory of differential equations—those of the greatest importance in mathematical physics. So in playing with groups Cauchy was not idling.
To close this description of groups we may indicate how the groups of substitutions discussed by Cauchy have entered the modern theory of atomic structure. A substitution, say (xy), containing precisely two letters in its symbol, is called a transposition. It is easily proved that any substitution is a combination of transpositions. For example,
(abcdef) = (ab)(ac)(ad)(ae)(qf),
from which the rule for writing out any substitution in terms of transpositions is evident.
Now, it is an entirely reasonable hypothesis to assume that the electrons in an atom are identical, that is, one electron is indistinguishable from another. Hence, if in an atom two electrons are interchanged, the atom will remain unchanged. Suppose for simplicity that the atom contains precisely three electrons, say a, b, c. To the group of substitutions on a, b, c (the one whose multiplication table we gave) will correspond all interchanges of electrons leaving the atom invariant—as it was. From this to the spectral lines in the light emitted by an excited gas consisting of atoms may seem a long step, but it has been taken, and one school of experts in quantum mechanics finds a satisfactory background for the elucidation of spectra (and other phenomena associated with atomic structure) in the theory of substitution groups. Cauchy of course foresaw no such applications of the theory which he developed for its own fascinations, nor did he foresee its application to the outstanding riddle of algebraic equations. That triumph was reserved for a boy in his teens whom we shall meet later.
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By the age of twenty seven (in 1816) Cauchy had raised himself to the front rank of living mathematicians. His only serious rival was the reticent Gauss, twelve years older than himself. Cauchy’s memoir of 1814 on definite integrals with complex-number limits inaugurated his great career as the independent creator and unequalled developer of the theory of functions of a complex variable. For the technical terms we must refer to the chapter on Gauss—who had reached the fundamental theorem in 1811, three years before Cauchy. Cauchy’s luxuriantly detailed memoir on the subject was published only in 1827. The delay was due possibly to the length of the work—about 180 pages. Cauchy thought nothing of hurling massive works of from 80 to 300 pages at the Academy or the Polytechnique to be printed out of their stinted funds.
The following year (1815) Cauchy created a sensation by proving one of the great theorems which Fermat had bequeathed to a baffled posterity: every positive integer is a sum of three “triangles,” four “squares,” five “pentagons,” six “hexagons,” and so on, zero in each case being counted as a number of the kin
d concerned. A “triangle” is one of the numbers 0, 1, 3, 6, 10, 15, 21, . . . got by building up regular (equilateral) triangles out of dots,
“squares” are built up similarly,
where the “bordering” by which one square is obtained from its predecessor is evident. Similarly “pentagons” are regular pentagons built up by dots; and so on for “hexagons” and the rest. This was not easy to prove. In fact it had been too much for Euler, Lagrange, and Legendre. Gauss had early proved the case of “triangles.”
As if to show that he was not limited to first-rate work in pure mathematics Cauchy next captured the Grand Prize offered by the Academy in 1816 for a “theory of the propagation of waves on the surface of a heavy fluid of indefinite depth”—ocean waves are close enough to this type for mathematical treatment. This finally (when printed) ran to more than 300 pages. At the age of twenty seven Cauchy found himself being strongly “rushed” for membership in the Academy of Sciences—a most unusual honor for so young a man. The very first vacancy in the mathematical section would fall to him, he was assured on the quiet. So far as popularity is concerned this was the highwater mark of Cauchy’s career.
In 1816, then, Cauchy was ripe for election to the Academy. But there were no vacancies. Two of the seats however might soon be expected to be empty owing to the age of the incumbents: Monge was seventy, L. M. N. Carnot sixty three. Monge we have already met; Carnot was a precursor of Poncelet. Carnot held his seat in the Academy on account of his researches which restored and extended the synthetic geometry of Pascal and Desargues, and for his heroic attempt to put the calculus on a firm logical foundation. Outside of mathematics Carnot made an enviable name for himself in French history, being the genius who in 1793 organized fourteen armies to defeat the half million troops hurled against France by the united antidemocratic reactionaries of Europe. When Napoleon seized the power for himself in 1796, Carnot was banished for opposing the tyrant: “I am an irreconcilable enemy of all kings,” said Carnot. After the Russian campaign of 1812 Carnot offered his services as a soldier, but with one stipulation. He would fight for France, not for the French Empire of Napoleon.
In the reorganization of the Academy of Sciences during the political upheaval after Napoleon’s glorious “Hundred Days” following his escape from Elba, Carnot and Monge were expelled. Carnot’s successor took his seat without much being said, but when young Cauchy calmly sat down in Monge’s chair the storm broke. The expulsion of Monge was sheer political indecency, and whoever profited by it showed at least that he lacked the finer sensibilities. Cauchy of course was well within his rights and his conscience.
The hippopotamus is said to have a tender heart by those who have eaten that delicacy baked, so a thick skin is not necessarily a reliable index to what is inside a man. Worshipping the Bourbons as he did, and believing the dynasty to be the direct representatives of Heaven sent to govern France—even when Heaven sent an incompetent clown like Charles X—Cauchy was merely doing his loyal duty to Heaven and to France when he slipped into Monge’s chair. That he was sincere and not merely self-seeking will appear from his subsequent devotion to the sanctified Charles.
Honorable and important positions now came thick and fast to the greatest mathematician in France—still well under thirty. Since 1815 (when he was twenty six) Cauchy had been lecturing on analysis at the Polytechnique. He was now made Professor, and before long was appointed also at the Collège de France and the Sorbonne. Everything began coming his way. His mathematical activity was incredible; sometimes two full length papers would be laid before the Academy in the same week. In addition to his own research he drew up innumerable reports on the memoirs of others submitted to the Academy, and found time to emit an almost constant stream of short papers on practically all branches of mathematics, pure and applied. He became better known than Gauss to the mathematicians of Europe. Savants as well as students came to hear his beautifully clear expositions of the new theories he was developing, particularly in analysis and mathematical physics. His auditors included well-known mathematicians from Berlin, Madrid, and St. Petersburg.
In the midst of all this work Cauchy found time to do his courting. His fancy, Aloise de Bure, whom he married in 1818 and with whom he lived for nearly forty years, was the daughter of a cultured old family and, like himself, an ardent Catholic. They had two daughters, who were brought up as Cauchy had been.
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One great work of this period may be noted. Encouraged by Laplace and others, Cauchy in 1821 wrote up for publication the course of lectures on analysis he had been giving at the Polytechnique. This is the work which for long set the standard in rigor. Even today Cauchy’s definitions of limit and continuity, and much of what he wrote on the convergence of infinite series in this course of lectures, will be found in any carefully written book on the calculus. An extract from the preface will show what he had in mind and what he accomplished.
“I have sought to give to the methods [of analysis] all the rigor which is demanded in geometry, in such a way as never to refer to reasons drawn from the generality of algebra. [As it would be put today, the formalism of algebra.] Reasons of this kind, although commonly enough admitted, above all in the passage from convergent to divergent series, and from real quantities to imaginary, cannot be considered, it seems to me, as anything more than inductions which occasionally suggest the truth, but which agree but little with the boasted exactitude of mathematics. We must also observe that they tend to cause an indefinite validity to be attributed to algebraical formulas,II while, in reality, the majority of these formulas subsist only under certain conditions, and for certain values of the quantities which they contain. By determining these conditions and values, and by fixing precisely the meaning of the notations I make use of, I shall dispel all uncertainty.”
Cauchy’s productivity was so prodigious that he had to found a sort of journal of his own, the Exercises de Mathématiques (1826-30), continued in a second series as Exercises d’Analyse Mathématique et de Physique, for the publication of his expository and original work in pure and applied mathematics. These works were eagerly bought and studied, and did much to reform mathematical taste before 1860.
One aspect of Cauchy’s terrific activity is rather amusing. In 1835 the Academy of Sciences began publishing its weekly bulletin (the Comptes rendus). Here was a virgin dumping ground for Cauchy, and he began swamping the new publication with notes and lengthy memoirs—sometimes more than one a week. Dismayed at the rapidly mounting bill for printing, the Academy passed a rule, in force today, prohibiting the publication of an article over four pages long. This cramped Cauchy’s luxuriant style, and his longer memoirs, including a great one of 300 pages on the theory of numbers, were published elsewhere.
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Happily married and as prolific in his research as a spawning salmon, Cauchy was ripe for the jester when the revolution of 1830 unseated his beloved Charles. Fate never enjoyed a heartier laugh than it did when it motioned Cauchy to rise from Monge’s chair in the Academy and follow his anointed King into exile. Cauchy could not refuse; he had sworn a solemn oath of allegiance to Charles, and to Cauchy an oath was an oath, even if sworn to a deaf donkey. To his credit, Cauchy, at the age of forty, gave up all his positions and went into voluntary exile.
He was not sorry to go. The bloodied streets of Paris had turned his sensitive stomach. He firmly believed that good King Charles was in no way responsible for the gory mess.
Leaving his family in Paris, but not resigning his seat in the Academy, Cauchy went first to Switzerland, where he sought distraction in scientific conferences and research. He never asked the slightest favor from Charles and did not even know that the exiled king was aware of his voluntary sacrifice for a matter of principle. Shortly a more enlightened Charles, Charles Albert, King of Sardinia, heard that the renowned Cauchy was out of a job and made one for him as Professor of Mathematical Physics at Turin. Cauchy was perfectly happy. He quickly learned Italian and deliv
ered his lectures in that language.
Presently overwork and excitement made him ill, and to his regret (as he wrote to his wife) he was forced to abandon evening work for a time. A vacation in Italy, with a visit to the Pope for good measure, completely restored him, and he returned to Turin, eagerly anticipating a long life devoted to teaching and research. But presently the obtuse Charles × butted into the retiring mathematician’s life like a brainless goat and, in seeking to reward his loyal follower, did him a singular disservice. In 1833 Cauchy was entrusted with the education of Charles’ heir, the thirteen-year-old Duke of Bordeaux. The job of male nurse and elementary tutor was the last thing on earth that Cauchy desired. Nevertheless he dutifully reported to Charles at Prague and took up the cross of loyalty. The following year he was joined by his family.
The education of the heir to the Bourbons proved no sinecure. From early morning to late evening, with barely time out for meals, Cauchy was pestered by the royal brat. Not only the elementary lessons of an ordinary school course had to be hammered somehow or another into the pampered boy, but Cauchy was detailed to see that his charge did not fall down and skin his knees on his gambols in the park. Needless to say the major part of Cauchy’s instruction consisted in intimate talks on the peculiar brand of moral philosophy to which he was addicted; so perhaps it is as well that France finally decided not to take the Bourbons back to its heart, but to leave them and their innumerable descendants as prizes to be raffled off to the daughters of millionaires in the international marriage bureau.
In spite of almost constant attendance on his pupil Cauchy somehow managed to keep his mathematics going, dashing into his private quarters at odd moments to jot down a formula or scribble a hasty paragraph. The most impressive work of this period was the long memoir on the dispersion of light, in which Cauchy attempted to explain the phenomenon of dispersion (the separation of white light into colors owing to different refrangibilities of the colored lights composing the white) on the hypothesis that light is caused by the vibrations of an elastic solid. This work is of great interest in the history of physics, as it exemplified the tendency of the nineteenth century to try to account for physical phenomena in terms of mechanical models instead of merely constructing an abstract, mathematical theory to correlate observations. This was a departure from the prevailing practice of Newton and his successors—although there had been attempts to “explain” gravitation mechanically.
