# The Works of Charles Hermite

Charles Hermite died in January 1901. His works were published by Gauthier-Villars, Paris, under the auspices of the Académie des Sciences. They were edited by Émile Picard and appeared in four volumes: Volume 1 in 1906; Volume 2 in 1908; Volume 3 in 1912; and Volume 4 in 1917. Reviews of the four volumes appeared in:

Volume 1. The Mathematical Gazette 3 (55) (1906), 270-271;
Volume 2. The Mathematical Gazette 4 (76) (1908), 399;
Volume 3. The Mathematical Gazette 7 (106) (1913), 160;
Volume 4. The Mathematical Gazette 9 (139) (1919), 331-332.

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:

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