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PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY,
CINCINNATI COLLEGE.

CINCINNATI:

PUBLISHED BY E. MORGAN & CO.

1845.

Entered according to Act of Congress, in the year 1845,

By O. M. MITCHEL,

In the Clerk's office of the District Court of Ohio.

Stereotyped by J. A. James,
Cincinnati.

Dudbey

5-6-46 54373

PREFACE.

In presenting to the public a new text-book upon Algebra, no pretension is made to originality in the matter of this work. It is believed that the manner of arranging the subject is new and possessed of great advantages. Most of the popular works now in use, treating of the subject of Algebra, are translations from the French, such as Lecroix's, Bourdon's, etc. These works are, in the original, extended and elegant treatises upon this important branch of mathematics, intended rather for the professor than for the student of Algebra. The sole object of the author of this treatise, has been to prepare a work specifically adapted to schools and colleges, a work in which the demonstrations should be full, clear, and definite, releasing the student from the necessity of any reliance except upon himself and his text-book, in his preparation for the black-board; while, at the same time, the professor is relieved from the onerous duty of performing continually the part of a tutor, in the recitation-room. The plan of requiring of the student a full and systematic demonstration of the great rules involved in each subject within the scope of Algebra, used with such great success in the United States Military Academy, is becoming more and more appreciated every day throughout the country. The students at West Point have the advantage of being assisted by those in more advanced classes, in their preparation for demonstrations, required in the class-room. In introducing this system for the first time, the duty required of the teacher is so great, with the common textbooks in use, that few ever have resolution sufficient to accomplish perfectly the drilling of a single class; yet all acknowledge the great advantages of the system, could it be introduced without too great an expense. The work now presented, is intended to meet this difficulty. The entire course is so divided, as to make twenty-one distinct subjects or demonstrations, each a unit in itself; and, all combined, forming as full a treatise upon Algebra, as is believed to be necessary for the prosecution of a full course of mathematics.

At the close of each demonstration, the student will find the work which will be required to be placed upon the black-board, and reference made to the articles in which a full explanation of the work is given.

If he should desire to test his ability to demonstrate the subject, let him turn to the work used in the demonstration, and see if he can explain this work clearly. Should he find himself unable to explain any part of the work, let him turn to the article in which the explanation is given, and a careful study of the subject as there developed, will, it is believed, clear up any difficulties which may present themselves. In case the student finds himself competent to the explanation of the work, let him next try to perform this work upon his slate without the assistance of his book. Should he fail in this, a reference to his textbook will be in general sufficient, without a call upon his instructor for assistance. He is thus taught to depend upon himself; and, indeed, with this work in his hands, the teacher may generally require him to perform his lessons without assistance.

These facts I am able to state from experience, having taught a class from the sheets as they issued from the press.

In preparing a class for examination, the arrangement of the present work furnishes great facilities. Each subject is a unit, upon which the student may perfect himself, and is ever ready to answer the inquiry, how much he knows of his course, and how much he does not know.

Should the system adopted in this work be thoroughly carried out, the very best results cannot fail to follow. An accurate method of study, a clear and systematic manner of expression, habitual self-reliance, and what is still better, a habit of preparation for each recitation, until failure becomes disgrace, cannot fail to be the necessary consequences.

If in this humble effort I shall have succeeded in so presenting this important subject, as to smooth the pathway of the student, and at the same time to relieve the teacher from a part of his onerous duties, I shall be amply compensated for a labor performed amid a multitude of cares and engagements.

CINCINNATI COLLEGE, March, 1845.

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