5o8 



NATURE 



{March 29, 1888 



Prof. Hertvvig has availed himself of the latest inquiries, 

 we may call attention to two figures of the pineal eye of 

 Chameleon and Hatteria, copied from Prof. Baldwin 

 Spencer's memoir in the Quart, yourn. Micr. Set. of 

 last year. Full justice is done by Prof. Hertwig to 

 Mr. Spencer's researches and their significance. We can 

 cordially recommend this text-book of embryology as 

 presenting a decided advance in scope upon the current 

 German treatises on human embryology, one of its merits 

 being that it embodies, among other good things, the 

 teachings and many of the drawings of our " unvergess- 

 licher " Balfour. E. R. L. 



A TREATISE ON ALGEBRA. 

 A Treatise on Algebra. By C. Smith. (London : Mac- 

 millan, 1888.) 



THIS, the latest text-book on elementary algebra, is 

 intended for the higher classes of schools and for 

 the junior students in the Universities, The title of the 

 book " A Treatise on Algebra," together with the fact that 

 in the preface the book is affirmed to be complete in 

 itself, is likely to convey the impression that the work is 

 more extensive and ambitious in its scope and design than 

 is really the case. In regard to the matter treated of, it 

 covers much the same ground as Todhunter's " Algebra," 

 which it greatly resembles ; it differs from it chiefly in a 

 different arrangement of the parts of the subject, and in 

 the introduction of elementary notions of " elimination " 

 and "determinants." 



As regards rearrangement of the subject-matter, there 

 is one very gratifying novelty : before making any use of 

 infinite series, the author introduces a chapter in which he 

 discusses some of the tests of the convergency of such 

 series. There is no doubt of the soundness of this course, 

 and for this single reason many teachers would be inclined 

 to prefer this book to others of the same nature. 



The principal feature of modern elementary algebraical 

 text-books seems to be that they are written without any 

 reference to the light shed upon the relative importance of 

 different parts of the subject by the progress of algebraical 

 research. A comprehensive survey of the existing know- 

 ledge of the science should induce an author to lead 

 the schoolmaster, and not to follow him. It is not too 

 much to expect that a book like the one under notice 

 should bear some traces of what is taking place in 

 the development of the science to which it seeks to 

 introduce a student. It is perfectly true that certain 

 fundamental notions must necessarily be presented in 

 much the same detail relatively in every book, independ- 

 ently of the date of production; but beyond this an author 

 may easily be too conservative in his ideas to be able to 

 compile a work which shall be of the greatest advantage 

 to a student who intends subsequently to continue his 

 reading at a University or elsewhere. Even from the 

 narrow point of view of an examination it would be advis- 

 able to give some small indications of the directions in 

 which explorations have been recently taking place, for it 

 is well known that problem papers at the Universities 

 and elsewhere frequently contain matter taken from 

 researches quite recently published. The absence of 

 modern ideas in a book gives a teacher but little oppor- 

 tunity of pointing out to promising pupils the roads to the 



frontiers of the science. This is the more to be deplored 

 just now, when a premium is placed at Cambridge upon 

 originality of thought in connection with examinations for 

 Fellowships. 



As an instance of what is meant, it may be observed 

 that the subject of " reversion of series " is omitted alto- 

 gether, although it has of recent years come into great 

 prominence. As a fact, for the last three years one of the 

 chief points of interest in pure mathematics has been 

 Sylvester's theory of reciprocants, which are simply 

 reversion invariants ; that is to say, those functions of the 

 coefficients of a convergent series which remain unaltered 

 after the process of reversion has been carried out. One 

 has a right to expect, for this reason, that a " Treatise on 

 Algebra " published at the present time should make some 

 allusion to the existence of such a process ; in the older 

 text-books, such as Young's " Course of Mathematics," 

 and the "Algebra" published in Chambers' series, the 

 subject received a special heading, whilst in more recent 

 works it appears merely as an example. The present 

 time is not happily chosen for its complete banishment. 



" Scales of notation " give place here to '' systems of 

 numeration " ; this is in accordance with the German 

 " Zahlensysteme," and seems to be a more suitable 

 nomenclature. 



The definitions throughout the book are very carefully 

 given. One or two are open to criticism, as in the case of 

 "cyclical order"; this is defined in reference to a 

 "cyclical change of letters." In modern mathematics 

 this process is termed a "cyclical substitution of the 

 letters," and is one of the fundamental ideas of the ex- 

 tensive " theory of substitutions." There seems to be no 

 good ground for shirking the word " substitution," which 

 fulfils requirements of simplicity and suggestiveness, and 

 is the word with which the student will afterwards become 

 familiar. It seems a pity that in the chapter on per- 

 mutations the opportunity is- not taken to introduce a 

 few of the leading ideas of this theory. 



In defining "symmetrical expressions" the author 

 states that an expression, which remains unaltered by the 

 "cyclical change" is also considered symmetrical; the 

 modern definition of a symmetrical function is that it is 

 such that it remains unaltered when any substitution is 

 impressed upon the letters. The expression {b~c) (c-a) 

 (a-ff), instanced by the author as being also called sym- 

 metrical, is in reality a two-valued (sometimes called an 

 alternating) function, falls under a different (the alternat- 

 ing) " group of substitutions," and is not properly called 

 symmetrical. 



In the chapter on theory of numbers — a particularly 

 clear one — the idea of congruences is happily introduced 

 with Gauss's notation. One would have liked to see also 

 some of the notions of Sylvester's "constructive theory 

 of the partition of numbers," as the ideas are very simple 

 and useful, and moreover algebraically expressible most 

 elegantly. The partition of numbers is rapidly becoming 

 a most important part of the "theory of numbers," a 

 ' fact which must soon be recognized by authors of books 



of the same scope as this one. 

 I Other portions of the book which are well presented 

 are "factors" (including many of the first notions of the 

 '"theory of equations"), "imaginary and complex 

 ' quantities," and " binomial theorem."^ 



