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THE LATEST IMPROVEMENTS.
ADAPTED TO THE USE OP SCHOOLS AND COLLEGES.
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CHARLES W. HACKLEY, S.T.D.,
FBOFESSOB OF MATHEMATICS AND ASTBONOMT IN COLUMBIA COLLEGE, NEW TOBX.
HARPER & BROTHERS, PUBLISHERS.
83 CLIFF STREET, NEW YORK.
Entered, according to Act of Congress, in the year 184G. By Harper & Brothers, In the Clerk's Office of the Southern District of New York. PREFACE.
In the preparation of the following work no pains have been spared to obtain from the best sources, such as the later treatises in highest repute, memoirs of scientific bodies, and mathematical journals in English, French, and German, the materials for a book suited to the present state of mathematical science and the wants of teachers and students.
The work contains much that has never before appeared in an English dress, and almost every part will be found to present some new feature. No attempt, however, has been made at originality, unless for the benefit of the student, and in the belief that the existing expositions or processes were inferior. The object has simply been, by any and all means, to make the best book, without aiming so much at individual reputation as at the author's own convenience and that of others, devoted, like himself, to the noble task of guiding the youthful votaries of science.
The French treatises furnish excellent models of the theory of Algebra, the German of ingenuity and brevity of notation and exposition, the English of practical adaptation and variety of illustration and example; and from these, after a careful comparison of many authors in each language, demonstrations have been selected and introduced verbatim when they seemed incapable of improvement; but whenever the slightest alteration or amalgamation, or the entire remodeling of them, could give additional clearness or elegance, the lima? labor has not been spared.
The work will be found to contain all that is important in the higher parts of Algebra, upon which usually separate treatises are thought necessary, as well as the elementary expositions suited to beginners. Every variety of symbol and of example has been introduced.
On page 11, those articles of this volume are indicated which constitute a minimum course of Algebra requisite for the prosecution of the higher branches of mathematics. A more extended course, such as would ordinarily be advisable, is also pointed out. The rest may very well be reserved for reference, as the student's own discovery of