658 



NATURE 



[January 20, 192 1 



Copan to England by Maudslay, such as Altar R 

 and part of the frieze of Temple II., are in the 

 Victoria and Albert Museum. As a matter of fact, 

 all the original sculptures collected by Maudslay 

 were transferred to the British Museum shortly 

 after their temporary deposition at South Ken- 

 sington. 



In conclusion, Dr. Morley's work is scientific 

 and scholarly. As a scientific man and a scholar 

 he aimed at perfection; he has achieved a land- 

 mark. Can higher praise be given ? 



T. A. Joyce. 



The History of Determinants. 



The Theory of Determinants in the Historical 

 Order of Development. By Sir Thomas Muir. 

 Vol. iii., The Period 1861 to 1880. Pp. xxvi + 

 503. (London: Macmillan and Co., Ltd., 

 1920.) Price 355. net. 



THE period covered by this volume is perhaps 

 the most important in the history of the 

 subject. During that time three important 

 branches of pure mathematics attained vast 

 dimensions — invariant-theory, analytical geo- 

 metry, and the general theory of algebraic 

 numbers. In each of these, familiarity with 

 determinants and their manipulation is essential, 

 so a great many students mastered the deter- 

 minant calculus, and applied it to a variety of 

 problems. Incidentally, the properties of deter- 

 minants aroused interest for their own sake ; 

 numerous papers dealing with them were pub- 

 lished, and, above all, several treatises on the 

 subject made their appearance, in which a com- 

 pact notation replaced all the old cumbersome 

 symbols, and practically all the theorems of the 

 determinant calculus proper were expounded in a 

 simple and orderly way. 



What we may call the derivative part of 

 the theory consists mainly of classifying de- 

 terminants of special types. Thus in the 

 present volume we have separate chapters 

 on axisymmetric determinants, symmetric de- 

 terminants, alternants, recurrents, Wronskians, 

 Jacobians, etc. (sixteen chapters or so). Broadly 

 speaking, these types come from two sources — 

 either as the outcome of a particular research, not 

 primarily concerned with determinants (thus con- 

 tinuants arose from the theory of continued frac- 

 tions) ; or else from intrinsic characters belong- 

 ing to the array from which the determinant is 

 formed, as in the case of symmetric determin- 

 ants. Of course, any special type of determinant 

 can be specified per se ; we are thinking rather of 

 the way in which the discussion of particular types 

 NO, 2673, VOL. 106] 



actually originated. A remarkable example is 

 Smith's arithmetical determinant (p. 116), of n 

 rows and columns, the value of which is the pro- 

 duct tf}{i).<f>(2) . . . <^(n), where <i>(m) means the 

 "totient" of m — namely, the number of integers 

 prime to m and not exceeding it. 



In a book such as this, one feature is almost 

 sure to present itself. We shall find some excel- 

 lent work unaccountably neglected, and results of 

 first-rate importance only becoming generally 

 known and appreciated after re-discovery, when 

 their original authors are dead. The cases in 

 this volume which strike the attention are those 

 of Trudi and Reiss. Reiss's work on compound 

 determinants goes back as far as 1867; the 

 analysis of it on pp. 181-90 (in modern nota- 

 tion) shows its importance, and is worth study, 

 because the theory of compound determinants is 

 perhaps the one part of general determinant- 

 calculus not yet fully reduced to its complete and 

 simplest form. 



In many applications the rank and elementary 

 divisors of a determinant (or matrix) are of 

 primary importance. The elementary divisors of 

 an array depend upon the arithmetical or alge- 

 braical character of the field to which the elements 

 of the array belong. Consequently, the deter- 

 mination of them does not properly belong to 

 determinant-theory; on the other hand, the rank 

 of an array is immediately calculable, on the 

 assumption that we can calculate the "value" of 

 any minor determinant, or, at any rate, decide 

 whether it is or is not zero. It must often have 

 been difficult for the author to decide when a 

 theorem in matrices should or should not be con- 

 sidered one relating to determinants. Rank is 

 referred to several times ; apparently theorems 

 about elementary divisors have been omitted. In 

 the case when the elements of an array are ordinary- 

 integers it is clear from Smith's paper on linear 

 indeterminate equations and congruences (1861) 

 that he was then perfectly familiar with the exist- 

 ence and properties of elementary divisors; to 

 that extent he anticipated the theory of Weier- 

 strass, Frobenius, and others. 



Opinions may differ about Sir T. Muir's choice 

 of a subject on which to bestow his labour ; some 

 readers may regret that he did not select a branch 

 of mathematics of a less circumscribed and sub- 

 sidiary kind than determinant-theory undoubtedly 

 is. But all will agree in admiring the ability and im- 

 partiality with which this labour of love has been 

 accomplished, and rejoice to know that the fourth 

 and final volume is nearly complete in manuscript. 

 Histories of other branches of mathematics are 

 badly wanted, and this work is a model of what 

 such histories ought to be. G. B. M. 



