NATURE 



481 



THURSDAY, MARCH 24, 1892. 



THE HISTORY OF DETERMINANTS. 

 The Theory of Determinants in the Historical Order of 



its Development. Part I. Determinants in General. 



By Thomas Muir, M.A., LL.D., F.R.S.E. (London : 



Macmillan and Co., 1890.) 

 nPHE theory of determinants is in that borderland 

 -I which separates the "pass" from the "honour" 

 student of pure mathematics. In elementary text-books 

 the subject is rarely more than introduced for the purpose 

 of representing some result of geometry or analysis in a 

 convenient, beautiful, and suggestive form. The essen- 

 tial properties of a determinant are not set forth, but the 

 student is perhaps referred for further information to one 

 or other of the two excellent treatises which are already 

 at our disposal in the English language, viz. those of 

 Mr. Muir and of Mr. R. F. Scott. The value of the idea 

 thus given to a student of the shape and convenient use 

 of a great analytical implement is beyond all question. 

 His imagination and curiosity are alike excited, and the 

 "trick" possibly prevents his passage through life under 

 the delusion that all mathematics are comprised within 

 the covers of the school-books. The honour student, as 

 a matter of course, reads some work on the subject, and 

 is as surely enchanted. He cannot fail to recognize the 

 power and beauty of the notation. He observes that the 

 object of his study is constructive in its nature. He 

 becomes convinced that pure mathematics is one of the 

 fine arts, an J just as a beautiful picture gives pleasure to 

 one who understands painting, just as a fine piece of 

 sculpture delights one who understands modelling, so he 

 sees unfolded to his intellectual eye an exquisite example 

 of constructive art, which his previous mathematical read- 

 ing has fitted him to understand and appreciate, and to 

 regard as a beautiful object of contemplation. The theory 

 of determinants is one of the most artistic subdivisions 

 of mathematical science, and accordingly has never 

 wanted enthusiastic admirers. It is most gratifying to 

 find such an authority as Mr. Muir devoting his leisure 

 to its historical development. Any mathematician taking 

 up this volume would anticipate a treat, and he would 

 not be disappointed. In this first instalment the reader 

 is taken from Leibnitz (1693) to Cauchy (1841). Mr. Muir 

 assigns a chief place of honour to Vandermonde (1771), 

 who, in his " M^moire sur I'EIimination" (Hist, de I'Acad. 

 Roy. des Sciences), denoted a function formed from the 

 coefficients of a set of linear equations by a symbolism 

 which is at once recognized as a condensed form of the 

 determinant matrix of the present day. He was the first 

 to give a connected exposition of the theory, and to give 

 the true fundamental properties of the new functions. 

 His notation, moreover, was exceedingly good, and much 

 superior to that adopted by some subsequent writers who 

 overlooked or neglected his important work. 



Vandermonde has also recently received justice, long 

 denied him, in other branches of analysis, and there is 

 now no doubt that the value and originality of his work 

 entitle him to a place in the ranks of the mathematical 

 pioneers of his time. Up to the close of the eighteenth 

 century the most noteworthy additions were made by 

 NO. I 169, VOL. 45] 



Laplace, Lagrange, and Bezont. We find that Lagrange 

 knew that the discriminant of a binary quantic of the 

 second order is an invariant of the linear transformation. 

 He did not, however, express either the discriminant or 

 the determinant of transformation in a determinant form. 

 The author critically examines the claim of Laplace to be 

 the discoverer of the expansion theorem. He finds that 

 although the case in which as many as possible of the 

 factors of the terms of the expansion are of the second 

 degree had already been given by Vandermonde, and 

 Laplace himself did not give a statement of the rule 

 suited for general application, the claim can in the main 

 be upheld. Hindenburg (1784) and Rothe (1800) took 

 up the subject in Germany, and between them constructed 

 an elementary theory of permutations. Rothe, it is in- 

 teresting to observe, employed a graphical method which 

 will remind the reader of Prof. Sylvester's modern con- 

 structive theory of partitions. Gauss (i 801) followed, and 

 then we come to the important papers of Binet (181 1) 

 and Cauchy (181 2). These memoirs establish the multi- 

 plication theorem in its full generalization. The method 

 adopted by Binet may be described as that of symmetric 

 functions, which he uses freely. He employs identities 

 of the type 



-SMb'c" - ^aS.b'S.c ■\- riabc - S.al.bc - 'Zb:ica - ^.e-Zab, 

 having reference to several systems of quantities. He 

 was not, however, sufficiently acquainted with the theory 

 of such functions ; and was unable to supply rigid proofs 

 of the theorems in determinants which, from his point 

 of view, rested upon these identities. Nowadays, the 

 identity in question will be recognized as the expression 

 of an "elementary" symmetric function (single unitary, 

 and having three parts in its partition) by means of 

 symmetric functions each of which is expressible sym- 

 bolically by a partition composed of one tripartite part. 

 The law of the coefficients, undivulged by Binet, is now 

 perfectly well known. It is, in fact, an easy generaliza- 

 tion of the law by which, in the case of a single system 

 of quantities, the elementary symmetric functions are 

 expressed as functions of the sums of powers of the 

 quantities. Cauchy at the same date (1812) introduced 

 the idea of " fonctions symetriques alternees," which led 

 him to a new symbolic definition of a determinant and 

 to many valuable results. Mr. Muir devotes several 

 pages to an examination of Cauchy's title to share with 

 Binet the credit of the generalized multiplication theorem. 

 He gives his decision against Cauchy, and probably the 

 reader who closely follows the argument will find himself 

 in accord with the historian. Notwithstanding this con- 

 clusion, Cauchy's memoir is excellent in quality and 

 abundant in quantity ; he "opened up a whole avenue of 

 fresh investigation, and one cannot but assign to him the 

 place of highest honour among all the workers from 1693 

 to 1812." 



A retrospect is given of the period 1693-1 81 2 accom- 

 panied by an interesting tabular record. As a means of 

 reference the work appears to be absolutely perfect. 

 Each new result as it appears is marked in Roman 

 figures ; and if the same result be obtained differently^ 

 or be generalized by a subsequent investigator, the same 

 Roman figure is employed, followed by an Arabic 

 numeral. It is found that to this point nearly every 

 important advance is due to French mathematicians. 



