AUTHOR'S PREFACE
The author of the present volume feels some embarrassment in having already offered to the public a work upon the "Theory of Determinants." The apparently general acceptability of this former work, which has now reached its third edition, doubtless led to his being invited by the editors of Higher Mathematics to prepare for them a chapter upon the same subject. This was done without the least thought of its publication as a separate volume. Now that its issue as such, along with the other chapters, is requested by both the publishers and the editors of Higher Mathematics, it is but just to the author that the above circumstances should be understood lest he be suspected of entertaining an unseemly desire to keep himself before the mathematical public by vain repetition.
The limitations imposed have permitted the addition of only a few articles to the work as originally published; principally those treating of linear substitutions, quantics, invariance, covariance, and functional determinants. Determinants of special forms have not been considered, nor is there' the least reference to the application of determinants to geometry. It is hoped, however, that the work may prove useful to the constantly increasing number of students who, while not wishing to specialize in mathematics, desire to obtain the comprehensive view of its methods and processes essential to the successful pursuit of the exact sciences in general.
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An excerpt from the beginning of the first chapter:
Introduction.
As early as 1693 Leibnitz arrived at some vague notions regarding the functions which we now know as determinants. His researches in this subject, the first account of which is contained in his correspondence with De L'Hospital, resulted simply in the statement of some rather clumsy rules for eliminating the unknowns from systems of linear equations, and exerted no influence whatever upon subsequent investigations in the same direction. It was over half a century later, in 1750, that Gabriel Cramer first formulated an intelligible and general definition of the functions, based upon the recognition of the two classes of permutations, as presently to be set forth.
Though Cramer failed to recognize, even to the same extent as Leibnitz, the importance of the functions thus defined, the development of the subject from this time on has been almost continuous and often rapid. The name " determinant" is due to Gauss, who, with Vandermonde, Lagrange, Cauchy, Jacobi, and others, ranks among the great pioneers in this development.
