## The Works of Charles Hermite

Volume 1. We give below versions of these reviews. |

**Volume 1.**

The appearance of this volume is very welcome, and its successors will be eagerly awaited. The editor, assisted by the late Professor Xavier Stouff (1861-1903), has corrected various misprints, 'sometimes at the cost of long calculations'; this, and the addition of some brief notes, will save the reader much trouble, and occasional risk of error (e.g. on p. 163 there is a correction of a casual mistake in one of Hermite's letters to Carl Jacobi). This volume contains, amongst other things, the memoirs on quadratic forms, on homogeneous binary forms, and on the theory of the transformation of Abelian functions. Apparently the arrangement is chronological, but although the source of each paper is given, the date is not always stated, and this is somewhat of a blemish in an edition of this kind. Possibly a dated list is reserved for the final volume. By way of preface the editor has reprinted the admirable lecture on the scientific work of Hermite which he delivered at the Sorbonne in 1901. The frontispiece is a portrait of Hermite at the age of about 25, very youthful, somewhat roguish, but with solid brow and brooding eyes. In size the book is convenient, being a large octavo, and it is as needless to commend the printing as the competency of the editor.

**Volume 2.**

As in the first volume, Émile Picard has arranged the contents in chronological order. The memoirs date from 1858 to 1872. In 1856 Hermite had been admitted to the Institute, and six years later a chair was created for him at the Ecole Normale. As we all know he was afterwards appointed through the influence of Louis Pasteur (1822-1895) to the École Polytechnique, and eventually, in 1869, succeeded Jean-Marie Duhamel as Professor of Analysis at the Sorbonne. The memoirs in the present volume are the outcome of his labours during the years when perhaps his fertility was greatest. Fifteen years before, he had written to Carl Jacobi, a letter of a few pages, which placed him, then barely twenty years of age, among the finest analysts of Europe. The letter, it may be added, dealt with a question in connection with hyper-elliptical functions. His special interests for the next twenty years lay in the domain of pure number and, in spite of the inconveniences attending physical frailty, his power of invention was now at its highest. He took his place with James Joseph Sylvester and Arthur Cayley in the creation of the theory of algebraical forms, he wrote his great memoir on the equation of the fifth degree, and he discovered the properties of the modular function, and the nature of modular equations, with their application to the theory of elliptic functions. Henry Bourget (1864-1921) has borne the lion's share in the editing of the memoir on the equation of the fifth degree for the present volume, having fully revised and checked the whole of the calculations. Many other questions of absorbing interest are dealt with in the present volume, and are striking enough if merely as illustrating the width of Hermite's attainment and the elegance of his methods. We hope that as soon as the last of these volumes is published, this worthy monument to one of the greatest names on the roll of French mathematicians will be crowned by at least a selection from his correspondence. For in the letters he wrote and received we would see adequately reflected the mathematical life of Europe almost from the times of Gauss, Cauchy, Jacobi and Dirichlet to that fatal day when stern Death laid his icy finger on one of the best and purest of men. We had almost forgotten to add that an additional interest attaches to Vol. iv., in that it contains an excellent portrait of Hermite at the age of fifty or thereabouts.

**Volume 3.**

The third of the four volumes of the collected works of Hermite contains the Memoirs published between 1872 and 1880. It is of unusual interest, in that it opens with a paper of 34 pages,Sur l'Extension du Theorème de Sturm à un système d'equations simultanées. This dates back to Hermite's younger days, and was only recently discovered among the papers of Joseph Liouville. And it contains the great paper, later to be published in book form,Sur quelques Applications des Fonctions Elliptiques, covering in this volume some 150 pages. Here also is included the famous memoir in which the transcendental nature of e was established. Hermite narrowly missed proving that π is also transcendental. The calculations have been reworked by Henry Bourget (1864-1921), and an error has been discovered. The values now stand:

Ferdinand von Lindemann's proof for π appeared in 1882, nine years after the publication of the proof for e in thee=^{58019}/_{21344};e^{2}=^{157712}/_{21344}correct to the ten millionth.

Comptes Rezndus. Both proofs were complicated. It remained for Adolf Hurwitz and Paul Gordan to publish in the early nineties an elementary demonstration for each. Our younger readers may be interested to hear that solutions of the first three questions on p. 378 of Professor Ernest Hobson'sTrigonometry(new edition) are to be found in this volume in the Memoir -Intégration des Fonctions Transcendantes. But they must look out for misprints and omissions in addition to those given in the table of errata at the end of this volume ... - all noticed in a cursory glance at this beautiful paper. They will find little difficulty in following the author - which cannot be said of other memoirs from the same hand. Camille Jordan used to say that he once heard Gabriel Lamé remark of Hermite's memoirs on the theory of modular functions that in reading them:on a la chair de poule.["It gives one goosebumps."] There is a fine portrait of Hermite at 65 or thereabouts, showing off the fine head to advantage, but marred by a somewhat grim and forbidding aspect, quite alien to the genial and almost gentle look which is characteristic of other portraits we have seen.

**Volume 4.**

With this volume of some ninety papers, published between 1880 and 1901, come to a close the pious labours of Émile Picard in erecting a fit monument to Hermite. The four volumes, taken with the Letters of Hermite and Thomas Stieltjes published some years ago, and Professor Paul Mansion's (1844-1919) biographical sketch and bibliography, will form an almost complete record of the long and busy life of the famous mathematician. Portraits have been reproduced for all the previous volumes, and in this we have a reproduction of the beautiful medal due to Jules-Clément Chaplain (1839-1909), struck on the occasion of Hermite's seventieth anniversary. If we remember that the claims of his official duties during the period covered by these papers were of the most exacting character, it is not surprising to find that by far the majority of them are short. A very considerable number of them deal with elliptic functions, the subject on which he had laid down the lines for his future researches at the early age of twenty-eight. In such papers and memoirs as these, he found relaxation amid the manifold preoccupations in which his later years were largely absorbed. It may not be untimely to remind the reader of the especial importance attached by Hermite to abstraction as an invaluable quality in the equipment of those who may be called upon to serve their country in the field. He regarded rigorous mental discipline as the essential preparation for military duties, and maintained that mathematics, more than any other subject of study, cultivated the power of abstraction which is so indispensable to the leader in framing his course of action amid the obscurity and tumult of a battle. And he saw the influence of such studies extending even further in its power to mould the deepest and most sacred conceptions of the human mind. For instance, in a speech on the occasion of his Jubilee, he classed together the two great schools - the École Normale and the École Polytechnique - as two branches of the same family, inseparably united by the common sense of justice and duty - a sense which, in some way which we cannot fathom, is intimately connected with the science of mathematics, and seems to pass from the intellect to the consciousness, upon which it is imposed with all the force of the truths of Geometry. We find scattered throughout this volume the various obituary notices and addresses which he was so often called upon to prepare from his official position and as the doyen of French mathematicians, and from these it would be easy to fill a number of the Gazette with purple patches. It speaks volumes for the man that there are few if any of these from which the human touch is missing, connected as he was with so many of the mathematicians of his day by ties of long friendship. With masterly skill he traces in turn the main features of the life-work of each, its scope, and its place among contemporary contributions to mathematical science. Then he says, of a Halphen: "But as we stand by his grave and speak of his work we remember the colleague, the friend we have lost. His simplicity and modesty were on a par with his genius; he was kind and affectionate, he was untiring in his devotion to his duty. While quite a young officer he was attached to the Army of the North, was made Captain and decorated on the field of battle, at Pont-Noyelles. He was present at the battle of Bapaume. This great mathematician was a soldier. To him we tender the supreme homage of our admiration for his work, our sorrow for the loss we have sustained, and our expression of the affectionate memories which we shall cherish to our latest breath." Or again, of a Kronecker: "But the tongue of praise is stilled in the presence of the sorrows of Science and the emotion caused throughout the world of Mathematics by the cruel loss of this great mathematician. To that sorrowful regret, to those recollections of a laborious life crowned by discoveries of such value, I now add those of a friendship which for thirty years has been the honour of my scientific career, a friendship the like of which it will never again be mine to enjoy." Or of our own Cayley: "I had some share in several of the investigations undertaken by Cayley: the same problems brought us together at the beginning of our career, and I shall never forget his kindness, his great simplicity of character, his absolute devotion to our Science. To that remembrance, which is very dear to me, I add my sorrowful regret, and the homage that I offer to his memory." And of Weierstrass, he said: "The life of our illustrious Colleague has been entirely consecrated to the Science which he served with an absolute devotion. It has been a long life, and full of honours; but as we stand at the brink of the grave which is about to close over him, the sole thought that fills our minds is that of his genius and that universal respect which is due from all men to nobility of character. Weierstrass was a good and upright man; be his guerdon our last tribute of respect to his memory! That memory will live as long as there are men who are eager in the quest for Truth, as long as men can be found to devote their labours to fresh efforts for the advancement of Analysis, and for the progress of the Science of Calculation." Finally, let us quote from his short tribute to his friend Brioschi: "I have shared in the labours of Brioschi: and often our investigations have been directed to a common end. I have followed his noble career, so notable for his unceasing toil and for the great services he has rendered to his country. None can feel more than I the loss of that great mathematician, of that man of stainless honour. The memory of our friendship, and of an intimacy which dates back to our early youth, will ever remain to me as one of the dearest and most cherished possessions of my life." The reader must not fail to read the two addresses delivered respectively at the opening of the new Sorbonne, and at the Académie des Sciences in 1889 and 1890, containing as they do a number of exquisitely conceived pen-pictures of the mathematical and other worthies whom it became the occasion to honour. The volume closes with the affecting speech made by Hermite as the recipient of the Chapelin medal on the occasion of his Jubilee in 1892, and with facsimiles of two pages from a letter to Jules Tannery.

JOC/EFR November 2014

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