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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 which may be spent more profitably on other branches of mathematics. Each set of examples has been carefully arranged, commencing with very simple exercises and proceeding gradually to those which are less obvious; those 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.

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 statement 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

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