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Camfroge: PRINTED BY C. J. CLAY, M.A.

AT THE UNIVERSITY PRESS.

PREFACE TO THE SECOND EDITION.

This work contains all the propositions which are usually included in elementary treatises on algebra, and a large number of examples for exercise.

My chief object has been to render the work easily intelligible. Students should be encouraged to examine carefully the language of the book they are using, so that they may ascertain its meaning or be able to point out exactly where their difficulties arise. The language, therefore, ought to be simple and precise; and it is essential that apparent conciseness should not be gained at the expense of clearness.

In attempting, however, to render the work easily intelligible, I trust I have neither impaired the accuracy of the demonstrations nor contracted the limits of the subject; on the contrary, I think it will be found that in both these respects I have advanced beyond the line traced out by previous elementary writers.

The present treatise is divided into a large number of chapters, each chapter being, as far as possible, complete in itself. Thus the student is not perplexed by attempting to master too much at once; and if he should not succeed in fully comprehending any chapter, he will not be precluded from going on to the next, reserving the difficulties for future consideration: the latter point is of especial importance to those students who are without the aid of a teacher.

The order of succession of the several chapters is to some extent arbitrary, because the position which any one of them should occupy must depend partly upon its difficulty and partly upon its importance. But, since each chapter is nearly independent, it will be in the power of the teacher to abandon the order laid down in the book and to adopt another at his discretion.

The examples have been selected with a view to illustrate every part of the subject, and, as the number of them is about sixteen hundred and fifty, I trust they will supply ample exercise for the student. Complicated and difficult problems have been excluded, because they consume time and energy

be spent more profitably on other branches of mathematics. Each set of examples has been carefully arranged, commencing with some which are very simple and proceeding gradually to others

which may

which are less obvious; those sets which are entitled Miscellaneous Examples, together with a few in each of the other sets, may be omitted by the student who is reading the subject for the first time. The answers to the examples, with hints for the solution of some in which assistance may be needed, are given at the end of the book.

I will now give some account of the sources from which the present treatise has been derived.

Dr Wood's Algebra has been so long published that it has become public property, and it is so well known to teachers that an elementary writer would naturally desire to make use of it to some extent. The first edition of that work appeared in 1795, and the tenth in 1835; the tenth edition was the last issued in Dr Wood's life-time. The chapters on Surds, Ratio, and Proportion, in my Algebra are almost entirely taken from Dr Wood's Algebra. I have also frequently used Dr Wood's examples either in my text or in my collections of examples. Moreover, in the st: ent of rules in the elementary part of my book I have often followed Dr Wood, as, for example, in the Rule for Long Division; the statement of such rules must be almost identical in all works on Algebra. I should have been glad to have had the advantage of Dr Wood's authority to a greater extent, but the requirements of the present state of mathematical instruction rendered this impossible. The tenth edition of Dr Wood's Algebra contains little more than half the matter of the present work, and half of it is devoted to subjects which are now usually studied in distinct treatises, namely, Arithmetic, the Theory of Equations, the application of Algebra to Geometry, and portions of the Summation of Series; the larger part of the remainder, from its brevity and incompleteness, is now unsuitable to the wants of students. Thus, on the whole, a very small number of pages comprises all that I have been able to retain of Dr Wood's Algebra.

For additional matter I have chiefly had recourse to the Treatise on Arithmetic and Algebra in the Library of Useful Knowledge, and the works of Bourdon, Lefebure de Fourcy, and Mayer and Choquet; I have also studied with great advantage the Algebra of Professor De Morgan and other works of the same author which bear upon the subject of Algebra.

I have also occasionally consulted the edition of Wood's Algebra published by Mr Lund in 1841, Hind's Algebra, 1841, Colenso's Algebra, 1849, and Goodwin's Elementary Course of Mathematics, 1853. In the composition of my book I took extreme care to avoid trespassing upon the works of these recent English authors. My rule was not to insert a proposition in the few cases where any doubt existed as to the right to do so, unless I found it in two or more of these authors; if I found it in so many places I concluded that it might be considered common property, and I inserted it in my own language and style.

Although I have not hesitated to use the materials which were available in preceding authors, yet much of the present work is peculiar to it; and I believe it will be found that my Algebra contains more that is new to elementary works, and more that is original, than any of the popular English works of similar plan. Originality however in an elementary work is rarely an advantage; and in publishing the first edition of my Algebra I felt some apprehension that I had deviated too far from the ordinary methods. I have had great satisfaction in receiving from eminent teachers favourable opinions of the work generally and also of those parts which are peculiar to it.

Several years have elapsed since I resolved to publish an Algebra and began to arrange the materials. Thus all the important chapters in the present work have been written and rewritten, and repeatedly revised by myself and my friends. With respect to some parts, which were original at the time when they first occurred to me, I have been anticipated in publication; this applies, for example, to Arts. 520, 611, and 677. I mention this, not as attaching any importance to such points, but merely because otherwise it might appear that I had been indebted for them to preceding authors. My manuscripts on these articles were in use among my pupils before the date in which, so far as I know, these articles were printed; indeed it was not until after my first edition was published that I saw the latter two articles in print elsewhere. Some portions of the present work were written long before I had any intention of publication; the chapter on the Multinomial Theorem, for example, was drawn up about fifteen years ago for the use of a fellow-student.

The task of preparing an elementary treatise is far from easy, and I must therefore request the indulgence of teachers and students for any defects which they may discover either in my plan, or in the mode of executing it. I have to return my thanks to many able mathematicians who have favoured me with suggestions, which have been of great service to me in preparing the Second Edition ; and I trust I shall still continue to receive similar valuable remarks.

I. TODHUNTER.
St John's COLLEGE,

February, 1860.

THE THIRD EDITION has been carefully revised; and some additions have been made to the text, to the examples, and to the answers and the hints given at the end of the book. A treatise on the Theory of Equations has been drawn up by the author, to form a sequel to the Algebra; and the student is referred to that treatise as a suitable continuation of the present work.

ST JOHN'S COLLEGE,

June, 1862.

SOME additions have been made to the text in the fourth edition and some improvements have been effected; these changes are chiefly due to the suggestions of persons engaged in teaching the subject: and the writer returns his thanks to those who have thus assisted him with the results of their experience.

March, 1866.

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