Nov 



lO, 



1887] 



NATURE 



27 



A TEXT-BOOK OF ALGEBRA. 

 A Text-book of Algebra. By W. Steadman Aldis, M.A 



(Oxford : Clarendon Press, 1887.) 

 'T^HIS work is, as we are told in the preface, " the out- 

 A come of lectures delivered in the College of Physical 

 Science at Newcastle-upon-Tyne." It discusses, more 

 fully than is usual in books on algebra, the fundamental 

 principles of the science, and its aim is to be of service 

 to the independent student who has not the advantage 

 of "access to large libraries, or intercourse with other 

 mathematical scholars." The object of the author, as 

 might be expected from his eminence as a mathemati- 

 cian and his experience as a teacher, is, in our judgment, 

 likely to be successfully attained in the use of his work. 

 The book is hardly adapted for those students whose 

 object it is to attain such skill and facility in algebraical 

 work as is necessary to face an examination paper in 

 algebra, set by the University examiner of the present 

 day. The examples, though sufficient for illustrating the 

 pnnciples, are not numerous enough for the purpose of 

 developing such skill, nor selected with that special 

 object ; and such aids to the attainment of exactness as 

 the various tentative methods of finding the factors of 

 algebraical expressions of different forms, and other aids 

 to insight into their constitution, are only incidentally 

 alluded to. Still, even this class of students will find it 

 a book worthy of reference, when they are revising the 

 fundamental principles on which the science is based, 

 and realizing that all its operations are reducible to a few 

 fundamental laws. 



The book is divided into four sections, treating respect- 

 ively of the fundamental laws and the algebraical opera- 

 tions founded thereon, of equations, of series, and of 

 arithmetical applications. 



In the first chapter, headed " Arithmetical Notions," the 

 arithmetical basis of algebra is laid down in a careful 

 discussion of the laws of the simple operations of arith- 

 metic ; the commutative laws of addition and multiplica- 

 tion, and the distributive and associative laws of multi- 

 plication, being shown to result, both for integral and 

 fractional numbers, from our fundamental conceptions of 

 number. In this chapter particular numbers only are 

 used, and the expression of the results by the use of letters 

 denotmg any numbers whatever is relegated to the follow- 

 ing chapter on "Algebraical Notation." This would seem 

 to indicate that Prof. Aldis agrees with a commonly 

 accepted notion, that algebra begins with the introduction 

 of letters to denote unspecified numbers. We hold, on 

 the contrary, that, in arithmetic, letters may, and ou'ght 

 to, be freely used to express the unknown quantities of a 

 question, or to sum up in general terms properties of 

 numbers or rules which have been established for solving 

 particular problems ; and that only when a result has been 

 obtained by means of organized algebraical operations 

 instead of by ordinary reasoning, has algebra, properly 

 so called, been employed. 'ft-/ 



In the second chapter the general results of the first 

 are summed up in a series of formulae, numbered (i) to 

 (21), to which are afterwards added others, numbered (22) 

 to (25), expressing the laws of indices. The extension of 

 the use of the signs -f and - to indicate opposite affec- 



tions of the quantiti-es denoted by the letters to which 

 they are prefixed is carefully explained ; and it is shown 

 by the Illustration of "steps" that still wider interpreta- 

 tions may be given to the symbols and formula. Upon 

 this foundation the subsequent chapters dealing with the 

 elementary operations on algebraical expressions are 

 based, explicit reference being made to one or other of 

 the formute by its number to justify each step in the 

 establishment of the various processes. This method of 

 procedure is sound and logical in itself, yet we fear that 

 the effect of referring to so many apparently independent 

 formulae must be confusing to the student, and likely to give 

 him incorrect ideas as to the number of independent laws 

 to which all algebraical operations are reducible. This 

 might have been avoided by a preliminary discussion of 

 the formula, showing that with the understanding that 

 the letters may denote either positive or negative quanti- 

 ties they are reducible to some five or six fundamental 

 laws, to which, rather than to the particular exemplifica- 

 tions of the laws in these formula, it would have been 

 better in the sequel to refer. Thus the formulae numbered 

 (U, (2), (3) are all included in the commutative law of 

 addition or aggregation-that in an aggregate of positive 

 and negative terms the order of aggregation is indifferent : 

 so, too, (3), (4), (5), (6) are summed up in the " Rule of 

 Signs "—that the addition of a positive aggregate of terms 

 is equivalent to the addition of each term with its 

 actual sign, and that of a negative aggregate is equivalent 

 to the addition of each term with its sign reversed, and 

 similarly for other groups of the formulte. 



The discussion of the highest common factor, lowest 

 common multiple, and fractions, is followed by a chapter 

 on fractional and negative indices, at the outset of which 

 the question of incommensurables is discussed, and it 

 is shown by apt illustrations that the literal symbols of 

 algebra may represent incommensurable as well as com- 

 mensurable quantities, since the same laws hold good for 

 the former as have been established for the latter. We 

 should have expected that, as a natural sequel to this 

 chapter, logarithms and their properties and uses would 

 have been discussed, but we find no mention even of the 

 word till we come, much later on in the book, to the 

 Exponential Series. There is no logical necessity for 

 postponing the discussion of the nature and properties of 

 logarithms till we can show how their values can be 

 practically calculated, while the enormous practical im- 

 portance of an acquaintance with their theory and use is 

 a good reason for its introduction at the earliest possible 

 stage. 



A chapter on surds and impossible quantities concludes 

 the first section. In this it is shown that the impossible 

 quantity of ordinary algebra is only relatively impossible, 

 since it becomes interpretable as an " operational 

 quantity " when the letter to which it is attached is taken 

 to denote a length in a definite direction— a view which is 

 further illustrated by the discussion of the cube roots of 

 unity as " operational quantities." This, though not a 

 full account of the matter, is satisfactory so far as it 

 extends, and sufficient for the student at this stage. 



The specially distinctive features of Prof. Aldis's work 

 are contained in this first section. We trust we have 

 made it plain that we think it well worthy of the study of 



