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OF

ALGEBRA,

THEORETICAL AND PRACTICAL,

FOR THE

Use of Schools and Private Students:

CONTAINING THE

FUNDAMENTAL RULES, FRACTIONS, INVOLUTION AND EVOLUTION, SURDS, EQUATIONS
OF ALL DEGREES, PROGRESSIONS, SERIES, LOGARITHMS AND THEIR APPLICATIONS,
PROPERTIES OF NUMBERS, CONTINUED FRACTIONS AND THEIR USES, THE INDETER-
MINATE OR DIOPHANTINE ANALYSIS, PROBABILITIES, LIFE ANNUITIES, ETC.

WITH

NUMEROUS EXERCISES UNDER EACH HEAD, AND A LARGE COLLECTION OF

MISCELLANEOUS QUESTIONS.

BY ALEXANDER INGRAM,

Author of a Concise System of Mathematics, Elements of Arithmetic, &c.

AND

JAMES TROTTER,

Of the Scottish Naval and Military Academy, Author of a Manual of Logarithms,
Key to Ingram's Mathematics, &c.

BODL

[graphic]

EDINBURGH:

OLIVER & BOYD, TWEEDDALE COUR

SIMPKIN, MARSHALL, & CO., LONDON.

MDCCCXLIV.

[Price Four Shillings bound.]

Printed by Oliver & Boyd,

Tweeddale Court, High Street, Edinburgh,

PREFACE.

THE groundwork of the following Treatise belongs to the late Mr INGRAM, than whom few men knew better what was suited to the capacities of the young. The public have already pronounced a favourable opinion on the Algebra contained in his Concise System of Mathematics; and in extending it to its present size every endeavour has been used to adhere as much as possible to the original plan. The chief object of that work was to combine theory and practice in such a manner that the principle of every rule should be thoroughly understood before proceeding to solve the exercises under the rule; and it is hoped that, at least in the higher departments, this has been fully attained. Usefulness and simplicity have been considered of far more importance than originality, and therefore free use has been made of the labours of our older authors for many of the exercises. With these, however, will be found interspersed many new and interesting questions; while care has been taken that every part of the work should be so clearly and simply expressed, that the most ordinary capacity can scarcely fail to comprehend it.

Although among the quadratic equations of the original work there were given several questions involving two unknown quantities, yet, as no specific rules were laid down for the solution of such equations, it has been thought expedient to treat these in a regular and systematic manner. The portion of the volume devoted to the properties of numbers may probably by some students be considered too lengthy for an elementary treatise; but by those who are fond of theoretical speculations, and alive to the intrinsic value of the subject, no such fault will be found. Continued fractions and their various applications have been

treated at considerable length, though not to a greater extent than their importance seemed to demand. The chief feature, however, to which the Publishers would draw attention, is the very full and complete exposition of the Indeterminate or Diophantine Analysis. Of all the branches of Algebra this is the most difficult and abstruse :-the various substitutions necessary to effect the solutions of the problems, the ingenuity and dexterity requisite in making the original assumptions, and the caution and judgment indispensable in evolving the final result, tend more to sharpen and invigorate the intellectual powers than the study of any other subject in the whole course of human investigation.

Great attention having been devoted to the selection of the exercises, it is confidently hoped that they will be found appropriate, and also of sufficient number and variety to initiate the student in the various rules; while the very large collection of miscellaneous questions at the end of the volume will afford a vast fund of profit and amusement to those who may undertake their solution, and will at the same time test their knowledge of the principles previously laid down.

Although every care has been taken to ensure accuracy, yet it is possible that errors may be found. These, however, when noticed, will be corrected; and in a second edition such alterations and additions will be made as may suggest themselves in teaching the work.

EDINBURGH, August 12, 1844.

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