A Treatise on the Theory of Algebraical Equations (Classic Reprint)

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Excerpt from A Treatise on the Theory of Algebraical Equations The commencement of the publication of a "Treatise on the Theory of Algebraical Expressions, " of which the first number appeared in 1831, under the direction of the "Society for the Diffusion of Useful Knowledge, " having been accidentally interrupted, I was directed by the same Society to compose the present Treatise, keeping in sight the views of the author of the "Algebraical Expressions, " at the same time keeping pace with the advancement made, since the former date, on this subject. The main view of the author of the work quoted, I learned, was to conduct his subject so as insensibly to lead the learner from pure algebraical theories to a knowledge of the principles on which the more advanced branches of analysis depend. To this advice from an excellent analyst I have adhered as well as I was able, but, in consideration of the recent progress of the "Theory of Equations, " I felt it necessary to alter the plan, assuming however the propositions proved in the other work, to which therefore the reader will find several subsequent references. I will now make a short statement of the plan adopted in the present work, premising that no treatise with exclusively the same object has been published of late, as far as I know, either at home or abroad. To collect and methodically digest the scattered elements of this theory, as far as its present advanced state imports, was attended with no inconsiderable difficulties, therefore, though an object of great utility has been, I hope, obtained by the composition of this work, it cannot be expected to be altogether faultless. Before examining algebraical equations theoretically, it appeared necessary to convey a precise idea of the continuous nature of algebraic functions, and to show that their numerical magnitudes may be extended through every quantity from negative to positive infinity, notwithstanding the existence of certain maxima and minima values. This subject is discussed in a series of propositions, the more clearly to impress the reader with the steps of the reasoning. Having attained this object, the ordinary properties of equations relative to the existence, number, limits, and symmetrical relations of the roots, followed as easy consequences, on these deduced properties I have not much dilated, as they have been already ably treated in the work before referred to. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.