03< 



NA TURE 



[Oct. 4, 1888 



of their respective vertices, and some simple trigono- 

 metrical formulas. On p. 173, in this chapter, we 

 notice a curious misprint : in each of three successive 

 formulae (the usual expressions for the sine, cosine, and 

 tangent of half an angle of a triangle in terms of its 

 sides) a capital V takes the place of the sign of the 

 square root. 



The opening paragraph of the first book tells us of the 

 origin of determinants, citing as evidence of their inven- 

 tion by Leibnitz his celebrated letter to L'Hopital, dated 

 April 28, 1693. Their re-discovery by Cramer in 1750, 

 and the rule (for the solution of a system of linear equa- 

 tions) which still bears his name, are next mentioned ; 

 but authors of a more modern date are summarily 

 dismissed with the following brief notice : — 



" Desde el tiempo de Cramer la teoria de las determin- 

 antes ha hecho notables progresos debido a los trabajos 

 de Vandermonde, Laplace, Gauss, Cauchy, Jacobi, 

 Sylvester, Muir, Baltzer y otros, no habiendo rama de las 

 matematicas en que no haya sido aplicada con ventaja." 



We are not, however, left entirely in the dark as to the 

 contributions to the theory made by these writers ; for 

 some theorems are called by the names of their respective 

 authors, and a large number of others have these names 

 indicated in brackets. For instance, the proposition 

 which concludes the third chapter in the first book is 

 thus enunciated : — 



" Descomponer una determinante de orden n 6s[mo en 

 una suma de productos formados cada uno de una 

 determinante de orden ^Csimo y de una determinante de 

 orden [« -^Jcsimo / Laplace } ." 



This is immediately preceded by — 



"Teorema de Cauchy. — Si se elige una fila y una 

 columna de una determinante cualquiera, el elemento 

 comiin de ellas multiplicado por el respectivo comple- 

 ment algebraico, mas la suma de productos obtenidos 

 multiplicando el producto de un elemento de la fila y de 

 la columna por su respectivo complemento algebraico, es 

 equivalente a la determinante dada." 



The way in which these two propositions are treated in 

 the present work will serve to exemplify the methods 

 employed by its compilers for imparting knowledge to 

 their readers. The proof of Laplace's theorem given by 

 Scott, in § 5, chap. iii. of his " Determinants,'' is clearer 

 than any other we are acquainted with ; but it depends 

 on some of the properties of alternate numbers. It is 

 true that these properties are of the simplest kind, but 

 then the notion of alternate numbers is a highly abstract 

 one, quite as much so as the idea of a four-dimensional 

 space. In order, therefore, to convey a clear conception 

 of Laplace's theorem to students of average capacity, 

 our authors have turned it into a problem, and, by con- 

 sidering what Prof. Sylvester calls a simple diagrammatic 

 case, have shown how this problem can be solved, thereby 

 bringing the theorem within the grasp of those whose 

 minds are as yet unprepared to revel luxuriously in such 

 abstractions as the alternate numbers. 



On the other hand, the proof of Cauchy's theorem 

 and the illustrative example appended to it have been 

 reproduced, with only some slight vertal alterations, from 

 § 62 of Muir's " Determinants," where the theorem in 

 question is presented in a form eminently adapted for 

 elementary instruction* 



The first book ends with a rule for the division of 

 determinants, which may be briefly stated thus : To 

 divide |« ln | by \b\ x \ assume the quotient to be |.t- ln | and 

 equate each element of the determinant formed by 

 multiplying \x ln \ and \fr ln \ to the corresponding element 

 of KJ. 



The values of the elements x u , x n , .... x nn , of the 

 assumed quotient |.v ln | will then be determined by solving 

 a system of equations of the form 



*ii x in + K *m +....+ K x im = a nn . 



The article containing this rule should be expunged 

 from all future editions of the work. Its practical inutility 

 becomes apparent when we remember that, on solving 

 the system of equations to which it leads, each x is found 

 in the form of the quotient of two determinants ; so that 

 we have to perform many divisions instead of one. Those 

 who are practically engaged in the work of mathematical 

 tuition in the University of Buenos Ayres will doubtless 

 be able to suggest other improvements, and if these 

 suggestions are attended to, students in that University 

 will possess in the second edition of the " Teoria 

 Elemental " an introduction to the theory of determinants 

 written in their own language and suited to their require- 

 ments. 



In some respects we do not desire to see any im- 

 provement. The appearance of the book is as attractive 

 as good paper, wide margins, and a bold clear type can 

 make it. The authors have chosen for their motto the 

 appropriate quotation from Sylvester : " For what is the 

 theory of determinants ? It is an algebra upon algebra ; 

 a calculus which enables us to combine and foretell the 

 results of algebraical operations, in the same way as 

 algebra enables us to dispense with the performance of 

 the special operations of arithmetic." The table of 

 contents is a model of completeness, and gives the 

 enunciations of the theorems in full instead of merely 

 indicating the pages and articles in which they occur. 

 The volume ends with a selected list of treatises on 

 determinants " que pueden servir de texto y que son 

 dignas de especial mencion." This will be of use to 

 students who only want to be told what authors they 

 should read, for the names mentioned are few and well 

 chosen ; while those whose object is to improve their 

 acquaintance with the bibliography of determinants may 

 fully satisfy their desire by consulting the two papers by 

 Muir in the Quarterly Journal of Mathematics (one of 

 them published in 1881, the other in 1886) to which 

 reference is made. 



Responding to the invitation — " agradecenamos las 

 indicaciones que se nos hicieran sobre omisiones 6 errores 

 que no hubieramos advertido " — we call attention to a slight 

 misprint in this reference, in which the word '" Quarterly " 

 has been mis-spelt " Quaterly." With the exception of 

 those previously mentioned, no other erratum has come 

 under our notice. 



OUR BOOK SHELF. 



The Geological History of Plants. By Sir J. W. Dawson, 

 C.M.G., LL.D., F.R.S., &c. 8vo, pp. 290. With Illus- 

 trations. " International Scientific Series." (London : 

 Kegan Paul, Trench, and Co., 1888.) 

 THIS book gives, in a connected form, a summary of the 

 development of the vegetable kingdom in geological time. 



