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409 



THURSDAY, SEPTEMBER i, 1887. 



HIGHER ALGEBRA. 

 Higher Algebra: a Sequel to Elementary Algebra for 

 Schools. By H. S. Hall, M.A., and S. R. Knight, B.A. 

 (London: Macmillan, 1887.) 



ANYONE who imagined from the shortened title of 

 " Higher Algebra," which appears on the back of 

 this volume, that the work extended to that higher region 

 of algebra to which Salmon's well-known " Lessons " are 

 " introductory," would be surprised to find that it contains 

 little beyond what may fairly be regarded as " elementary 

 algebra." Indeed it appears to us that much that is con- 

 tained in the earlier chapters would have found its place 

 more appropriately in the same authors' " Elementary 

 Algebra for Schools," using their own device of an 

 asterisk to indicate those articles which might, in the 

 case of the ordinary school-boy, be omitted or reserved for 

 a second reading ; and thus the awkwardness of breaking 

 up such subjects as ratio, proportion, and progressions 

 into separate parts, by an arbitrary division into lower 

 and higher, would have been avoided. 



Apart from this defect of plan, the work before us has 

 great merits as a text-book adapted to the wants of the 

 ordinary student of algebra and to the exigencies of 

 examinations. It is a development and improvement upon 

 ** Todhunter," as "Todhunter" was a development and 

 improvement upon " Wood." The main framework is 

 the same : many of the proofs of algebraical theorems 

 have been replaced by better proofs, and new matter has 

 been introduced. Still it remains essentially an artificial 

 framework and has no claim to be regarded as an organic 

 growth from a few central principles, with a correspond- 

 ing natural relation and affiliation of parts. Thus we 

 find the fundamental laws of algebra for the first time 

 gathered together and discussed in the thirty-fourth 

 chapter (p. 429) of this volume, a chapter of " Miscel- 

 laneous Theorems and Examples " for which apparently 

 no fitting place could be found in the framework. 

 It also includes such a fundamental theorem as that 

 known as the " remainder theorem " — that f{a) is the 

 remainder when the rational integral function f{x) is 

 divided by 4: — a — some of its applications, as well as 

 some discussion of symmetrical expressions and identi- 

 ties. 



An elementary algebra, written by a master of modern 

 algebraical science in the light of the higher views of the 

 essential nature of algebra which modern investigations 

 have established, and yet with such simplicity that it may 

 be put into the hands of the school-boy, is a desideratum 

 the advent of which is perhaps foreshadowed, though not 

 fully realized, in respect of simplicity at any rate, in 

 Prof. Chrystal's recent work. It would be obviously 

 unfair to criticize the present work from this point of 

 view : our remaining remarks on it, therefore, will be 

 confined to some matters of detail in the order of the 

 chapters of the book itself. 



Perhaps the strongest part of the book is the ex- 

 amples, both those which are worked out, and those 

 which are added to each chapter as exercises for the 

 pupil. As far as we have been able to examine them, 

 Vol. XXXVI. — No. 931. 



they are sufficiently numerous, well chosen, and in- 

 structive, and also well arranged in each exercise in the 

 order of their difficulty. We are surprised to find in the 

 chapter on " Miscellaneous Equations" that there is no 

 hint or caution given that the root obtained may not 

 satisfy the original equation unless the sign of one or 

 more of the radicals involved in it is changed. In fact, 

 in the example worked out on p. 99, the root x=(ia gives 

 by substitution in the equation 2a — da = \a ! We hold 

 that in all such cases the student should be required to 

 show with what signs of the radicals in the equation each 

 solution is consistent, and what combinations of their 

 signs are impossible ; otherwise more than half the 

 instructiveness of the example is lost. 



The chapters on ratio and proportion need no remark ; 

 but that on variation, as in most books of algebra, is in 

 our opinion unsatisfactory, from the fact that the attention 

 of the student is not called to the distinction between a 

 magnitude and its numerical measure. If A stand for 

 the distance and B for the time, when the speed is uni- 

 form, " A varies as B " is a statement true of the magni- 

 tudes themselves independent of any particular mode of 

 measuring them ; but when from this is deduced the 

 equation A = ;«B, either A and B must be regarded as 

 numerical measures of the distance and time with refer- 

 ence to some particular units, in which case m will have a 

 value depending on the units selected ; or else m is a 

 multiplier which, besides altering the numerical value of 

 B into that of A, converts a time into a distance, an ex- 

 tension of the notion of multiplication which, if admitted, 

 ought to be very carefully noted and explained. 



After chapters on progressions, we come to one on 

 scales of notation, though there is no reason, apart from 

 the traditional place it has occupied in books on algebra, 

 why such simple questions as are discussed in it, which, 

 if arithmetic were rationally taught, would have been 

 treated in connexion with the theory of decimal numera- 

 tion and notation, should be regarded as forming a 

 chapter of " Higher Algebra." The algebraical formulae 

 which encumber this chapter should only be introduced 

 as summing up what has been previously proved in par- 

 ticular instances by direct reasoning from first principles, 

 not in order to prove the propositions themselves. 



It would have been well if the chapter on the theory 

 of quadratic equations had been made one on that of 

 quadratic expressions. By not doing this the oppor- 

 tunity is lost of exemplifying the notion of continuity in 

 the changes of such expressions with the change of the 

 variable both in magnitude and sign and their maximum 

 and minimum values, as well as the introduction of the 

 graph (as Prof. Chrystal has done), to illustrate these 

 changes. 



The authors state in their preface that the part of 

 algebra which is concerned with permutations and com- 

 binations " is made far more intelligible to the beginner 

 by a system of common-sense reasoning from first 

 principles than by the proofs usually found in algebraical 

 text-books," a proposition with which we heartily agree, 

 only that we see no reason why it should be confined to 

 this particular part of algebra. 



When we turn, however, to the chapter on permuta- 

 tions and combinations, except that there is a greater 

 variety of proofs, we fail to find any further appeal to 



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