A

PRACTICAL SYSTEM

OF

ALGEBRA,

DESIGNED FOR THE USE OF

SCHOOLS AND PRIVATE STUDENTS.

BY PETER NICHOLSON,

AND

By J. ROWBOTHAM, F.R.A.S.

"Nil tam difficile est, quod non solertia vincat."

THE SECOND EDITION.

LONDON:

PRINTED FOR BALDWIN AND CRADOCK;

WHITTAKER, TREACHER, AND ARNOT;

J. DUNCAN; AND J. ROWBOTHAM.

1831.

1

PREFACE.

ALGEBRA LGEBRA may justly be considered as one of the most valuable, sublime, and scientific of all human inventions. To thinking and intelligent beings it affords a rich source of gratification; enables them to find out general rules for the solution of all questions that relate to quantity; and lays the foundation of great and noble acquirements. By its general and comprehensive nature, it not only aids the reasoning faculties in the development of Mathematical truths; but it also assists us in arranging our thoughts on other subjects. "Pure Mathema"tics," says Bacon, "do remedy and cure many de"fects in the wit and faculties intellectual; for, if "the wit be dull they sharpen it, if too wandering they fix it, if too inherent in the sense they ab66 stract it."

66

The extent of this science is so great, that it has been called by way of pre-eminence, "The Great "Art;" or, "Universal Mathematics."-It is far superior, in many respects, to both Arithmetic and Geometry, as by it may be solved problems or ques

tions which are quite beyond the reach of either. It would be in vain to attempt to enumerate the various subjects, both practical and theoretical, to which this truly admirable art is applicable. By a train of reasoning founded upon simple mathematical principles, we may often discover, in a few lines, more important truths than can be found in volumes. Algebra is not only useful to the mechanic, the astronomer, the architect, &c. &c., but a knowledge of it ought to be considered as forming a principal part of a good education; for no one can be denominated an accomplished scholar, who is ignorant of its principles.

Considering the many judicious and approved Algebraical works already before the public, the authors would scarcely have ventured to offer any thing new on the subject, had they not felt, from long experience, a want of something more practical, and more in unison with their own ideas, than what is at present to be met with. The chief aim of the writers has been to simplify the subject as much as possible, by endeavouring to give clear and concise definitions and rules;-for want of which, at the commencement of works on Algebra, many persons have been disheartened, and even disgusted with the subject.

It has been the opinion of many, that the works which have been written, are, in many respects, de

ficient both in theory and practice, particularly the latter; and although this treatise commences with the first principles of the science, it nevertheless comprises the demonstrations of the principal rules, and more examples for the scholar's practice than are to be found in any similar publication extant. In addition to a general and easy method of solving equations of the higher orders, it contains the most valuable discoveries that have been made in England, France, Germany, Holland, &c.-The other subjects of importance, to which it may not be improper to call the attention of the reader, are Arithmetical and Geometrical Progression, Binomial Theorem, Series, and easy methods of solving Quadratic and Indeterminate Equations, with their respective demonstrations in notes.

With regard to the correctness of the work, the authors trust it will not be found deficient, each having submitted his part of the manuscript to the other for revisal before it went to press.

The present treatise is, therefore, respectfully submitted to the candid notice of the public, with the

* The greater part of the questions are translated, or taken from the works of foreign Algebraists of established authority, such as Lacroix, L'Abbé Sauri, Don Benito Bails, Euler, Hirsch, Bangma, &c.