The Theory of Determinants in the Historical Order of DevelopmentPart I. General Determinants up to 1841.
Part II. Special Determinants up to 1841.
Thomas Muir, M.A., LL.D., F.R.S.Superintendent-General of Education in Cape Colony
The way in which the material for a history of the theory of Determinants has been accumulated is quite similar to that which has been observed in the case of other branches of science.
In the middle of the eighteenth century one of the independent discoverers of the fundamental idea, viz., Cramer, was fortunate enough to attract attention to it, and in time it became the common property of mathematicians in France and elsewhere. As it slowly spread it naturally also received accretions and developments, and of the dozen or so of writers who thus handled it in the sixty years that followed Cramer's publication there were of course a few who by a more or less casual reference kept alive the memory of some of their predecessors. It was then taken up by Cauchy, and, thanks to the prestige of his name and to the inherent excellence of his extensive monograph, its position as a theory of importance became more firmly assured. The thirty years that followed Cauchy's memoir resembled the sixty that preceded it, save that the number of contributors was considerably larger. Then another great analyst, Jacobi, the most noteworthy of those contributors, produced in Germany a monograph similar in extent and value to Cauchy's, and the importance of the subject in the eyes of mathematicians became still more enhanced. As a consequence, the single decade following gave rise to quite as many new contributions as the preceding three decades had done, and closed with the appearance of the first separately published elementary treatise on the subject, viz., Spottiswoode's. The preface to this contains the first notable historical sketch of the theory, and includes references to the writings of twelve outstanding mathematicians, beginning with Cramer (1750) and ending with the author's own contemporaries, Cayley, Sylvester and Hermite. In the same year (1850) there also occurred something out of the ordinary, for the correspondence between Leibnitz and the Marquis de I'Hôpital having been published from manuscripts in the Royal Library at Hanover, the striking discovery was made that more than half-a-century before Cramer's time the fundamental idea of determinants had been clear to Leibnitz, and had been expounded with considerable fullness by him in a letter to his friend. So strongly attractive had the subject now become to mathematicians that in the single year succeeding the publication of Spottiswoode's short treatise a greater number of separate contributions to the theory made their appearance than in the whole sixty-year period from Cramer to Cauchy. The wants of students everywhere had to be attended to: a second edition of Spottiswoode was consequently prepared for Crelle's Journal in 1853; a textbook by Brioschi was published at Pavia in 1854; French and German translations of Brioschi in 1856; and an elementary exposition by Bellavitis in 1857. So far as historical material is concerned, the last-mentioned work was of little account; that of Brioschi resembled Spottiswoode's, the number of references, however, being greater. Of quite a different character was the text-book by BALTZER, which was published at Leipzig the year after the German translation of Brioschi had appeared at Berlin, an important part of the new author's plan being to deal methodically with the history of the subject by means of footnotes. On the enunciation of almost every theorem a note with historical references was added at the foot of the page, the result being that in the portion (thirty-four pages) devoted expressly to the pure theory of determinants about as many separate writings are referred to as there are pages. This was a marked advance, and although during the next twenty years the publication of text-books became more frequent - in fact, if we include those of every language and of every scope, we shall find an average of about one per year - Baltzer's dominated the field; enlarged editions of it appeared in 1864, 1870, and 1875, and the historical notes grew correspondingly in number. Of the other text-books only one, Günther's, which was published in 1875, sought to follow the historical line taken by Baltzer and to add to the supply of material. Then in 1876 another new departure took place, this being the year in which the first writings were published which dealt with the history alone, the one being an academic thesis by E J Mellberg printed at Helsingfors, and the other a memoir presented by F J Studnieka to the Bohemian Society of Sciences.
About this time, while engaged in writing my own so-called "Treatise on the Theory of Determinants," I had occasion to look into the question of the authorship and history of the various theorems, and I was reluctantly forced to the conclusion that much inaccurate statement prevailed - in regard to such matters and that the whole subject was worthy of serious investigation. A resolution was accordingly taken to set about collecting the titles of all the writings which had appeared on the theory up to the end of 1880. The task was not an easy one, as will readily be understood by those who know how scanty and defective are the bibliographical aids at the disposal of mathematicians, and how often the titles given by investigators to their memoirs are imperfect and even misleading in regard to' the nature of the contents. The outcome of the search was published in 1881 in the October number of the Quarterly Journal of Mathematics (vol. xviii. pp. 110-149) under the title of "A List of Writings on Determinants." It contained 589 entries arranged in chronological order. Some three or four years afterwards, when there had been time to test the completeness of the earlier portion of the list, the writings included in it were taken up in historical succession and suitable abstracts or reviews of them made for publication in the Proceedings of the Royal Society of Edinburgh; the first contribution of this kind was presented to the Society in the beginning of the year 1886. At the same time there was -being prepared an additional list of writings containing omitted titles, 84 in number, belonging to the period of the first list, and 176 titles belonging to the further period 1881-1885. This second list appeared in 1886 in the June number of the Quarterly Journal of Mathematics (vol. xxi. pp. 299-320). In 1890 a collection was made of the contributions, just mentioned, which had up to that date been printed in the Edinburgh Proceedings, and with the consent of the Society was published separately. Unfortunately in that year all this train of work had to be laid aside on account of the pressure of official duties, and ten years elapsed before it could be resumed. It was thus not until March 1900 that a second series of analytic abstracts began to appear in the Edinburgh Proceedings, and that the preparation of a third list of writings was methodically undertaken. The period to be covered by this list was the fifteen years 1886-1900; and as the number of writers interested in the subject had in these years continued to increase, and as closer examination of the literature of the previous periods had led to new finds, the resulting compilation was more extensive than the first two put together. It was presented to the South African Association for the Advancement of Science at its inaugural meeting in April 1903 and was published in the Report; it is also to be found in the Quarterly Journal of Mathematics for December 1904 and February 1905 (vol. xxxvi. pp. 171-267). The number of titles in the three lists is about 1740; they furnish, it is hoped, an almost complete guide to the literature of the theory of determinants from the earliest times to the close of the nineteenth century.
From these later labours it became manifest that it was undesirable in the way of separate publication to issue merely another volume as a continuation of, and similar to, that of the year 1900. The better course clearly was to reproduce the material of that volume along with the intercalations necessitated in it by the existence of subsequently discovered papers, and to follow this up in such a way as to give finally within the compass of a reasonably sized volume a full history of the subject in all its branches up to about the middle of the nineteenth century. This is what is here attempted.
The plan followed is not to give one connected history of determinants as a whole, but to give separately the history of each of the sections into which the subject has been divided, viz., to deal with determinants in general, and thereafter in order with the various special forms. This will not only tend to smoothness in the narrative by doing away with the necessity of frequent harkings back, but it will also be of material importance to investigators who may wish to find out what has already been done in advancing any particular department of the subject. To this end, also, each new result as it appears will be numbered in Roman figures; and if the same result be obtained in a different way, or be generalised, by a subsequent worker, it will be marked among the contributions of the latter with the same Roman figures, followed by an Arabic numeral. Thus the theorem regarding the effect of the transposition of two rows of a determinant will be found under Vandermonde, marked with the number xi., and the information intended thus to be conveyed is that in the order of discovery the said theorem was the eleventh noteworthy result obtained: while the mark xi. 2, which occurs under Laplace, is meant to show that the theorem was not then heard of for the first time, but that Laplace contributed something additional to our knowledge of it. In this way any reader who will take the trouble to look up the sequence xi., xi. 2, xi. 3, etc., may be certain, it is hoped, of obtaining the full history of the theorem in question.
The early foreshadowings of a new domain of science, and tentative gropings at a theory of it, are so difficult for the historian to represent without either conveying too much or too little, that the only satisfactory way of dealing with a subject in its earliest stages seems to be to reproduce the exact words of the authors where essential parts of the theory are concerned. This I have resolved to do, although to some it may have the effect of rendering. the account at the commencement somewhat dry and forbidding.