4IO 



NA TURE 



[Sept. I, 1887 



" common-sense reasoning from first principles " than in 

 other text-books. In fact, in some of the proofs the 

 crucial pointof the proposition, instead of being elaborated, 

 is so condensed as to make it very difficult to understand, 

 though it is certainly put in a form which may be easily 

 carried into an examination to the perplexity of the 

 examiner, who may well be in doubt whether the examinee 

 who reproduces the words really sees the point of the 

 proof. We hold that the true way of appealing to 

 " common sense " is to take particular cases first, and 

 when these are grasped, the general proof becomes easy. 

 Thus, to find the number of permutations of 4 things 

 {a, b, c, d) taken 3 together, it is plain that the arrange- 

 ment 



c d d b b c 



repeated for each of the 4 letters in the top line will give 

 all possible permutations, and that the number is there- 

 fore 4X3x2, and fm-ther that the principle of arrange- 

 ment may be extended to any number of things. This is 

 the essence of the proof given on page 116. It may be 

 said that such exemplification is in the province of the 

 teacher rather than in that of the text-book, but we fear 

 there are many teachers who fail to make things clear in 

 this way to their pupils. 



The proof, or rather proofs, of the binomial theorem 

 for positive integral indices are distinct improvements on 

 the cumbrous proof given in Todhunter, the theorem 

 being shown, as it should be, to be a direct consequence 

 of the multiplication of « binomial factors. Euler's proof 

 for any index is carefully stated, and its crucial point 

 emphasized by a preliminary discussion. 



Following the binomial theorem comes a chapter on 

 logarithms, which in our opinion would have better 

 followed the chapter on indices in the " Elementary 

 Algebra," as that on interest and annuities might have 

 followed those on progressions. The exponential and 

 logarithmic series would then have followed naturally as 

 a development of the binomial theorem. 



The authors have given a chapter on the converge.icy 

 and divergency of series, in which this important subject 

 is treated with unusual care. We may perhaps demur to 

 the sweeping character of the statement (p. 249) that 

 " the use of divergent series in matheiiiatical reasoning 

 leads to erroneous results," but the student cannot be too 

 early or too emphatically warned that a result obtained 

 by the use of divergent series should be verified by other 

 means. 



The chapters which follow treat of intermediate co- 

 efficients, partial fractions, recurring series, continued 

 fractions, indeterminate equations of the first degree, re- 

 curring continued fractions, and indeterminate equations 

 of the second degree, summation of series, the usual ele- 

 mentary theorems of the theory of numbers, the general 

 theory of continued fractions, and probability. All these 

 subjects appear to us to be judiciously and adequately 

 treated, though we should have been glad to see a little 

 more of "common-sense reasoning from first prin- 

 ciples " in the elementary chapter on continued fractions, 

 by which it might easily and with advantage have been 

 made to take its place among the chapters of the " Ele- 



mentary Algebra." In the chapter on summation of 

 series, the authors, as they tell us in the preface, have 

 laid much stress on the "method of differences." As 

 they have gone so far, we think it is a pity that they 

 have not introduced the notation and the elementary 

 propositions of the calculus of differences, which seem 

 to us very naturally to fall within the limits of algebra. 

 In any case, in their use of the symbol 2 they should 

 not have deviated from its proper meaning by making 

 2«, for instance, include the term 7t instead of denoting 

 by it the series ending with n-i. 



Here the ordinary treatises on algebra end. Our 

 authors have, however, very wisely added a chapter on 

 determinants, containing a satisfactory and sufficient 

 discussion of determinants of the second and third orders, 

 with a useful series of examples of their application, and 

 an indication of the general properties of determinants 

 of any order. The study of this chapter will enable the 

 student to read, without difficulty, treatises on analytical 

 geometry, and afford a good introduction to special works 

 on determinants in general. 



Following this is the chapter on miscellaneous theorems 

 and examples, of which we have before spoken, contain- 

 ing a short discussion of the fundamental laws of algebra, 

 then the " remainder theorem," and synthetic division, 

 symmetrical and alternating functions, and elimination. 



While the end of ordinary algebra and its various direct 

 applications is undoubtedly a suitable place for a re- 

 discussion of its fundamental laws, as preliminary to the 

 interpretations of double algebra and to the various 

 higher algebras with different fundamental laws, it is 

 strange that our authors have not found the desirability, 

 indeed the necessity, of introducing the other subjects 

 of this chapter, with the exception perhaps of elimina- 

 tion, at a much earlier stage, and as part of a regular 

 sequence in the development of algebraic operations. 



The book concludes with a chapter containing the 

 elementary parts of the theory of equations — on the 

 whole judiciously selected. We note it, however, as a 

 defect in this, as in all other treatises we have met with, 

 that Horner's process for approximating to the roots of 

 numerical equations is barely mentioned. We hold that 

 the simplicity and generality of this process is such that 

 it ought to be taught, as a rule (without proof), for finding 

 the roots of numbers, in all treatises on arithmetic, to the 

 exclusion of the cumbrous, uninstructive, and utterly use- 

 less method of finding cube roots only, which is usually 

 given ; while the proof of the process, which may be 

 made quite easy and intelligible, and its general applica- 

 tion to numerical equations, ought to occupy a prominent 

 and early position in every treatise on algebra. Every- 

 one who has made himself expert in the use of Horner's 

 method will, we are sure, agree with us that it gives 

 a power in discussing an algebraical expression with 

 numerical coefficients, which can be obtained in no other 

 way. R. B. H. 



OUR BOOK SHELF. 



Outlines of Quantitative Analysis. By A. H. Sexton. 

 (Charles Griffin and Co., 1887.) 



It is perhaps as great an evil to err on the side of trying 

 too much as to do too little where more might be done. 

 In this book, intended, as the author tells us, to be put 



