The teaching of Algebra in the earlier stages ought to consist in a gradual generalisation of Arithmetic; in other words, Algebra ought, in the first instance, to be taught as Arithmetica Universalis in the strictest sense. I suppose that the student has gone in this way the length of, say, the solution of problems by means of simple or perhaps even quadratic equations, and that he is more or less familiar with the construction of literal formulae, such, for example, as that for the amount of a sum of money during a given term at simple interest.
Then it becomes necessary, if Algebra is to be anything more than a mere bundle of unconnected rules, to lay down generally the three fundamental laws of the subject, and to proceed deductively - in short, to introduce the idea of Algebraic Form, which is the foundation of all the modern developments of Algebra and the secret of analytical geometry, the most beautiful of all its applications. Such is the course followed from the beginning in this work
As mathematical education stands at present in this country, the first part might be used in the higher classes of our secondary schools and in the lower courses of our colleges and universities. It will be seen on looking through the pages that the only knowledge required outside of Algebra proper is familiarity with the definition of the trigonometrical functions and a knowledge of their fundamental addition-theorem.
The first object I have set before me is to develop Algebra as a science, and thereby to increase its usefulness as an educational discipline. I have also endeavoured so to lay the foundations that nothing shall have to be un-learned and as little as possible added when the student comes to the higher parts of the subject. The neglect of this consideration I have found to be one of the most important of the many defects of the English text-books hitherto in vogue. Where immediate practical application comes in question, I have striven to adapt the matter to that end as far as the main general educational purpose would allow. I have also endeavoured, so far as possible, to give complete information on every subject taken up, or, in default of that, to indicate the proper sources; so that the book should serve the student' both as a manual and as a book of reference. The introduction here and there of historical notes is intended partly to serve the purpose just mentioned, and partly to familiarise the student with the great names of the science, and to open for him a vista beyond the boards of an elementary text-book.
As examples of the special features of this book, I may ask the attention of teachers to chapters iv. and v. With respect to the opening chapter, which the beginner will doubtless find the hardest in the book, I should mention that it was written as a suggestion to the teacher how to connect the general laws of Algebra with the former experience of the pupil. In writing, this chapter I had to remember that I was engaged in writing, not a book on the philosophical nature of the first principles of Algebra, but the first chapter of a book on their consequences. Another peculiarity of the work is the large amount of illustrative matter, which I thought necessary to prevent the vagueness which dims the learner's vision of pure theory; this has swollen the book to dimensions and corresponding price that require some apology. The chapters on the theory of the complex variable and on the equivalence of systems of equations, the free use of graphical illustrations, and the elementary discussion of problems on maxima and minima, although new features in an English text-book, stand so little in need of apology with the scientific public that I offer none.
The order of the matter, the character of the illustrations, and the method of exposition generally, are the result of some ten years' experience as a university teacher. I have adopted now this, now that deviation from accepted English usages solely at the dictation of experience. It was only after my own ideas bad been to a considerable extent thus fixed that I did what possibly I ought to have done sooner, viz., consulted foreign elementary treatises. I then found that wherever there bad been free consideration of the subject the results had been much the same. I thus derived moral support, and obtained numberless hints on matters of detail, the exact sources of which it would be difficult to indicate. I may mention, however, as specimens of the class of treatises referred to, the elementary text-books of Baltzer in German and Collin in French. Among the treatises to which I am indebted in the matter of theory and logic, I should mention the works of De Morgan, Peacock, Lipschitz, and Serret. Many of the exercises have been either taken from my own class examination papers or constructed expressly to illustrate some theoretical point discussed in the text. For the rest I am heavily indebted to the examination papers of the various colleges in Cambridge. I had originally intended to indicate in all cases the sources, but soon I found recurrences which rendered this difficult, if not impossible.
The order in which the matter is arranged will doubt-less seem strange to many teachers, but a little reflection will, I think, convince them that it could easily be justified. There is, however, no necessity that, at a first reading, the order of the chapters should be exactly adhered to. I think that, in a final reading, the order I have given should be followed, as it seems to me to be the natural order into which the subjects fall after they have been fully comprehended in their relation to the fundamental laws of Algebra.
With respect to the very large number of Exercises, I should mention that they have been given for the convenience of the teacher, in order that he might have, year by year, in using the book, a sufficient variety to prevent mere rote-work on the part of his pupils. I should much deprecate the idea that any one pupil is to work all the exercises at the first or at any reading. We do too much of that kind of work in this country.
I have to acknowledge personal obligations to Professor Tait, to Dr Thomas Muir, and to my assistant, Mr R E Allardice, for criticism and suggestions regarding the theoretical part of the work; to these gentlemen and to Messrs Mackay and A Y Fraser for proof reading, and for much assistance in the tedious work of verifying the answers to exercises. In this latter part of the work I am also indebted to my pupil, Mr J Mackenzie, and to my old friend and former tutor, Dr David Rennet of Aberdeen.
Notwithstanding the kind assistance of my friends and the care I have taken myself, there must remain many errors both in the text and in the answers to the exercises notification of which either to my publishers or to myself will be gratefully received.
EDINBURGH, 26th June 1886.
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