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TREATISE

ON THE

ELEMENTS OF ALGEBRA.

BY

JAMES BRYCE, JUN. M.A. F.G.S.

MASTER OF THE MATHEMATICAL DEPARTMENT IN THE

HIGH SCHOOL OF GLASGOW.

SECOND EDITION,

WITH VARIOUS ADDITIONS AND IMPROVEMENTS.

LONDON: LONGMAN, BROWN, GREEN, AND LONGMANS.

EDINBURGH: ADAM AND CHARLES BLACK.
DUBLIN: CUMMING AND FERGUSON.

1846.

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PREFACE.

When this work was first announced as preparing for publication, a near relative of the Author was to have been associated with him in its execution ; but circumstances having occurred which withdrew that gentleman from mathematical pursuits, the only portions of the Work composed by him, are the Definitions, the explanatory parts of Addition and Subtraction, and the Note at the end, containing a defence of the principle on which the Definitions are founded; and controverting the views of Dr. Peacock, and other eminent mathematicians, respecting the supposed distinction between arithmetical and symbolical Algebra.

Great pains have been taken to express the Definitions and Rules in the most simple and accurate manner, so that their meaning should be at once apparent, and that the student should, from the first, be accustomed to the most rigid mathematical exactness. As the theory of every operation is fully given, and a copious collection of examples subjoined to each Rule, this Treatise combines more theory and practice than any hitherto published: and it has therefore been necessary, in order that the Work might sell at a moderate price, to omit several subjects usually embraced in books on Algebra. It was the Author's intention to treat these subjects separately in a supplementary volume; but want of leisure has hitherto prevented him from executing this part of his plan.

The intelligent teacher will perceive that it is of great importance to introduce Equations, and questions producing Equations, at an early period of the student's progress. The examples of Simple Equations are so arranged, that a few of them may be used after the student is acquainted with Addition and Subtraction ; a few more after he has learned Division ;—the rest will require a knowledge of Fractions and Surds. In a regular treatise, it is necessary to bring together all that relates to the same subject, in whatever order the subjects themselves may be taken up; but, in the practice of teaching, it is not desirable to adhere strictly to this arrangement.

Many important improvements have been introduced into this edition. The chapter on the theory of numbers has been omitted, and that on the solution of simultaneous Equations considerably shortened, in order to make room for several new subjects, such as Detached Coefficients, Imaginary Quantities, Piling of Balls, &c. The chapters on Proportion and Progression, Indeterminate Coefficients, and Logarithms, have been much enlarged, and a great many other alterations introduced throughout the Work, which cannot be particularly specified, but which, with those already mentioned, will, it is confidently hoped, render it more acceptable to students of mathematics, and fully adapt it to the present advanced state of analytical science.

The Author cannot conclude without stating, that he is indebted for advice and assistance on various subjects contained in this Work to the gentleman already mentioned, and to his friend and former pupil, Mr. E. T. R. Moncrieff, of Trinity College, Dublin.

HIGH SCHOOL OF GLASGOW, SEPT. 26, 1846.

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