by E. T. Bell
Cauchy’s childhood fell in the bloodiest period of the Revolution. The schools were closed. Having no need of science or culture at the moment, the Commune either left the cultured and men of science to starve or carted them off to the guillotine. To escape the obvious danger Cauchy senior moved his family to his country place in the village of Arcueil. There he sat out the Terror, half starved himself and feeding his wife and infant son largely from what scanty fruits and vegetables he could raise. As a consequence Cauchy grew up delicate and underdeveloped physically. He was nearly twenty before he began to recover from this early malnutrition, and all his life had to watch his health.
This retreat, gradually becoming less strict, lasted nearly eleven years, during which Cauchy senior undertook the education of his children. He wrote his own textbooks, several of them in the fluent verse of which he was master. Verse, he believed, made grammar, history and, above all, morals less repulsive to the juvenile mind. Young Cauchy thus acquired his own uncontrolled fluency in both French and Latin verse which he indulged all his life. His verse abounds in noble sentiments loftily expressed and admirably reflects the piety of his blameless life but is otherwise undistinguished. A large share of the lessons was devoted to narrow religious instruction, in which the mother assisted ably.
Arcueil adjoined the imposing estates of the Marquis Laplace and Count Claude-Louis Berthollet (1748-1822), the distinguished and eccentric chemist who kept his head in the Terror because he knew all about gunpowder. The two were great friends. Their gardens were separated by a common wall with a gate to which each had a key. In spite of the fact that both the mathematician and the chemist were anything but pious, Cauchy senior scraped an acquaintance with his distinguished and well-fed neighbors.
Berthollet never went anywhere. Laplace was more sociable and presently began dropping in at his friend’s cottage, where he was struck by the spectacle of young Cauchy, too feeble physically to be tearing round like a properly nourished boy, poring over his books and papers like a penitent monk and seeming to enjoy it. Before long Laplace discovered that the boy had a phenomenal mathematical talent and advised him to husband his strength. Within a few years Laplace was to be listening apprehensively to Cauchy’s lectures on infinite series, fearing that the bold young man’s discoveries in convergence might have destroyed the whole vast edifice of his own celestial mechanics. “The system of the world” came within a hairsbreadth of going to smash that time; a slightly greater ellipticity of the Earth’s almost circular orbit, and the infinite series on which Laplace had based his calculations would have diverged. Luckily his astronomical intuition had preserved him from disaster, as he discovered on rising with a sigh of infinite relief after a prolonged testing of the convergence of all his series by Cauchy’s methods.
On January 1, 1800, Cauchy senior, who had kept discreetly in touch with Paris, was elected Secretary of the Senate. His office was in the Luxembourg Palace. Young Cauchy shared the office, using a corner as his study. Thus it came about that he frequently saw Lagrange—then Professor at the Polytechnique—who dropped in frequently to discuss business with Secretary Cauchy. Lagrange soon became interested in the boy and, like Laplace, was struck by his mathematical talent. On one occasion when Laplace and several other notables were present, Lagrange pointed to young Cauchy in his corner and said, “You see that little young man? Well! He will supplant all of us in so far as we are mathematicians.”
To Cauchy senior Lagrange gave some sound advice, believing that the delicate boy might burn himself out: “Don’t let him touch a mathematical book till he is seventeen.” Lagrange meant higher mathematics. And on another occasion: “If you don’t hasten to give Augustin a solid literary education his tastes will carry him away; he will be a great mathematician but he won’t know how to write his own language.” The father took this advice from the greatest mathematician of the age to heart and gave his son a sound literary education before turning him loose on advanced mathematics.
After his father had done all he could for him, Cauchy entered the Central School of the Panthéon, at about the age of thirteen. Napoleon had instituted several prizes in the school and a sort of grand sweepstakes prize for all the schools of France in the same class. From the first Cauchy was the star of the school, carrying off the first prizes in Greek, Latin composition, and Latin verse. On leaving the school in 1804 he won the sweepstakes and a special prize in humanities. The same year Cauchy received his first communion, a solemn and beautiful occasion in the life of any Catholic and trebly so to him.
For the next ten months he studied mathematics intensively with a good tutor, and in 1805 at the age of sixteen passed second into the Polytechnique. There his experiences were not altogether happy among the ribald young skeptics who hazed him unmercifully for making a public exhibition of his religious observances. But Cauchy kept his temper and even tried to convert some of his scorners.
From the Polytechnique Cauchy passed to the civil engineering school (Ponts et Chaussées) in 1807. Although only eighteen he easily beat young men of twenty who had been two years in the school, and was early marked for special service. On completing his training in March, 1810, Cauchy was at once given an important commission. His ability and bold originality had singled him out as a man for whom red tape should be cut, even at the risk of lopping off some older man’s head in the process. Whatever else may be said of Napoleon, he took ability wherever he found it.
In March, 1810, when Cauchy left Paris, “light of baggage, but full of hope,” for Cherbourg on his first commission, the battle of Waterloo (June 18, 1815) was still over five years in the future, and Napoleon still confidently expected to take England by the neck and rub its nose in its own fragrant sod. Before an invasion could be launched an enormous fleet was necessary, and this had yet to be built. Harbors and fortifications to defend the shipyards from the seagoing British were the first detail to be disposed of in the glamorous dream. Cherbourg for many reasons was the logical point to begin all these grandiose operations which were to hasten “the day of glory” the French had been yelling about ever since the fall of the Bastille. Hence the gifted young Cauchy’s assignment to Cherbourg to become a great military engineer.
In his light baggage Cauchy carried only four books, the Mécanique céleste of Laplace, the Traité des fonctions analytiques of Lagrange, Thomas à Kempis’ Imitation of Christ, and a copy of Virgil’s works—an unusual assortment for an ambitious young military engineer. Lagrange’s treatise was to be the very book which caused its author’s prophecy that “this young man will supplant all of us” to come true first, as it inspired Cauchy to seek some theory of functions free from the glaring defects of Lagrange’s.
The third on the list was to occasion Cauchy some distress, for with it and his aggressive piety he rather got on the nerves of his practical associates who were anxious to get on with their job of killing. But Cauchy soon showed them by turning the other cheek that he had at least read the book. “You’ll soon get over all that,” they assured him. To which Cauchy replied by sweetly asking them to point out what was wrong in his conduct and he would gladly correct it. What answer this drew has not survived.
Rumors that her darling boy was fast becoming an infidel or worse reached the ears of his anxious mother. In a letter long enough and full enough of pious sentiments to calm all the mothers who ever sent their sons to the front or anywhere near it Cauchy reassured her, and she was happy once more. The conclusion of the letter shows that the holy Cauchy was quite capable of holding his own against his tormentors, who had hinted he was slightly cracked.
“It is therefore ridiculous to suppose that religion can turn anybody’s head, and if all the insane were sent to insane asylums, more philosophers than Christians would be found there.” Is this a slip on Cauchy’s part, or did he really mean that no Christians are philosophers? He signs off with a flash from the other side of his head: “But enough of this—it is more profitable for me to work at certain Memoirs o
n Mathematics.” Precisely; but every time he saw a windmill waving its gigantic arms against the sky he was off again full tilt.
Cauchy stayed approximately three years at Cherbourg. Outside of his heavy duties his time was well spent. In a letter of July S, 1811, he describes his crowded life. “I get up at four and am busy from morning to night. My ordinary work is augmented this month by the arrival of the Spanish prisoners. We had only eight days’ warning, and during those eight days we had to build barracks and prepare camp beds for 1200 men. . . . At last our prisoners are lodged and covered—since the last two days. They have camp beds, straw, food, and count themselves very fortunate. . . . Work doesn’t tire me; on the contrary it strengthens me and I am in perfect health.”
On top of all this good work pour la gloire de la belle France Cauchy found time for research. As early as December, 1810, he had begun “to go over again all the branches of Mathematics, beginning with Arithmetic and finishing with Astronomy, clearing up obscurities, applying [my own methods] to the simplification of proofs and the discovery of new propositions.” And still on top of this the amazing young man found time to instruct others who begged for lessons so that they might rise in their profession, and he even assisted the mayor of Cherbourg by conducting school examinations. In this way he learned to teach. He still had time for hobbies.
The Moscow fiasco of 1812, war against Prussia and Austria, and the thorough drubbing he got at the battle of Leipzig in October, 1813, all distracted Napoleon’s attention from the dream of invading England, and the works at Cherbourg languished. Cauchy returned to Paris in 1813, worn out by overwork. He was then only twenty four, but he had already attracted the attention of the leading mathematicians of France by his brilliant researches, particularly the memoir on polyhedra and that on symmetric functions. As the nature of both is easily understood, and each offers suggestions of the very first importance to the mathematics of today, we shall briefly describe them.
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The first is of only minor interest in itself. What is significant regarding it today is the extraordinary acuteness of the criticism which Malus levelled at it. By a curious historical coincidence Malus was exactly one hundred years ahead of his times in objecting to Cauchy’s reasoning in the precise manner in which he did. The Academy had proposed as its prize problem “To perfect in some essential point the theory of polyhedra,” and Lagrange had suggested this as a promising research for young Cauchy to undertake. In February, 1811, Cauchy submitted his first memoir on the theory of polyhedra. This answered negatively a question asked by Poinsot (1777-1859): is it possible that regular polyhedra other than those having 4, 6, 8, 12, or 20 faces exist? In the second part of this memoir Cauchy extended the formula of Euler, given in the school books on solid geometry, connecting the number of edges (E), faces (F), and vertices (V) of a polyhedron, E + 2 = F + V.
This work was printed. Legendre thought highly of it and encouraged Cauchy to continue, which Cauchy did in a second memoir (January, 1812). Legendre and Malus (1775-1812) were the referees. Legendre was enthusiastic and predicted great things for the young author. But Malus was more reserved.
Êtienne-Louis Malus was not a professional mathematician but an ex-officer of engineers in Napoleon’s campaigns in Germany and Egypt, who made himself famous by his accidental discovery of the polarization of light by reflexion. So possibly his objections struck young Cauchy as just the sort of captious criticisms to be expected from an obstinate physicist. In proving his most important theorems Cauchy had used the “indirect method” familiar to all beginners in geometry. It was to this method of proof that Malus objected.
In proving a proposition by the indirect method, a contradiction is deduced from the assumed falsity of the proposition; whence it follows, in Aristotelian logic, that the proposition is true. Cauchy could not meet the objection by supplying direct proofs, and Malus gave in—still unconvinced that Cauchy had proved anything. When we come to the conclusion of the whole story (in the last chapter) we shall see the same objection being raised in other connections by the intuitionists. If Malus failed to make Cauchy see the point in 1812, Malus was avenged by Brouwer in 1912 and thereafter when Brouwer succeeded in making some of Cauchy’s successors in mathematical analysis at least see that there is a point to be seen. Aristotelian logic, as Malus was trying to tell Cauchy, is not always a safe method of reasoning in mathematics.
Passing to the theory of substitutions, begun systematically by Cauchy, and elaborated by him in a long series of papers in the middle 1840’s, which developed into the theory of finite groups, we shall presently illustrate the underlying notions by a simple example. First, however, the leading properties of a group of operations may be described informally.
Operations will be denoted by capital letters, A, B, C, D, . . .; and the performance of two operations in succession, say A first, B second, will be indicated by juxtaposition thus, AB. Note that BA, by what has just been said, means that B is performed first, A second; so that AB and BA are not necessarily the same operation. For example, if A is the operation “add 10 to a given number,” and B is the operation “divide a given number by 10,” AB applied to x gives while BA gives or and the resulting fractions are unequal; hence AB and BA are distinct.
If the effects of two operations X, T are the same, X and T are said to be equal (or equivalent), and this is expressed by writing X = Y.
The next fundamental notion is that of associativity. If for every triple of operations, say U, V, W is any triple, in the set, (UV)W = U(VW), the set is said to satisfy the associative law. By (UV)W is meant that UV is performed first, then, on the result, W is performed; by U(VW) is meant that U is performed first, then, on the result of this VW is performed.
The last fundamental notion is that of an identical operation, or an identity: an operation I which leaves unchanged whatever it operates on is called an identity.
With these notions we can now state the simple postulates which define a group of operations.
A set of operations I, A, B, C, . . . , X, T, . . . is said to form a group if the postulates (1) – (4) are satisfied.
(1) There is a rule of combination applicable to any pair X, Y of operationsI in the set such that the result, denoted by XY, of combining X, Y, in this order, according to the rule of combination, is a uniquely determined operation in the set.
(2) For any three operations X, Y, Z in the set, the rule in (1) is associative, namely (XY)Z = X(YZ).
(3) There is a unique identity I in the set, such that, for every operation X in the set, IX = XI = X.
(4) If X is any operation in the set, there is in the set a unique operation, say X′, such that XX′ = I (it can be easily proved that X′X = i also).
These postulates contain redundancies deducible from other statements in (1) - (4), but in the form given the postulates are easier to grasp. To illustrate a group we shall take a very simple example relating to permutations (arrangements) of letters. This may seem trivial, but such permutation or substitution groups were found to be the long-sought clue to the algebraic solvability of equations.
There are precisely 6 orders in which the 3 letters a, b, c can be written, namely abc, acb, bca, bac, cab, cba. Take any one of these, say the first abc, as the initial order. By what permutations of the letters can we pass from this to the remaining 5 arrangements? To pass from abc to acb it is sufficient to interchange, or permute, b and c. To indicate the operation of permuting b and c, we write (be), which is read, “b into c, and c into b.” From abc to bca we pass by a into b, b into c, and c into a, which is written (abc). The order abc itself is obtained from abc by no change, namely a into a, b into b, c into c, which is the identity substitution and is denoted by I. Proceeding similarly with all 6 orders
abc, acb, bca, bac, cab, cba,
we get the corresponding substitutions,
I, (bc), (abc), (ab), (acb), (ac).
The “rule of combination” in the postulates is here as follows. Take any t
wo of the substitutions, say (be) and (acb), and consider the effect of these applied successively in the order stated, namely (be) first and (acb) second: (be) carries b into c, then (acb) carries c into b. Thus b is left as it was. Take the next letter, c, in (be): by (be), c is carried into b, which, by (acb) is carried into a; thus c is carried into a. Continuing, we see what a is now carried into: (be) leaves a as it was, but (acb) carries a into c. Finally then the total effect of (be) followed by (acb) is seen to be (ca), which we indicate by writing (be)(acb) = (ca) = (ac).
In the same way it is easily verified that
(acb) (abc) = (abc) (acb) = I;
(abc)(ac) = (ab); (bc)(ac) = (acb),
and so on for all possible pairs. Thus postulate (1) is satisfied for these 6 substitutions, and it can be checked that (2), (3), (4) are also satisfied.
All this is summed up in the “multiplication table” of the group, which we shall write out, denoting the substitutions by the letters under them (to save space),
I, (be), (abc), (ab), (acb), (ac)
I, A, B, C, D, E.
In reading the table any letter, say C, is taken from the left-hand column, and any letter, say D, from the top row, and the entry, here A, where the corresponding row and column intersect is the result of CD. Thus CD = A, DC = E, EA = B, and so on.