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INTRODUCTION may

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A L G E B R A.

WITH

NOTES AND OBSERVATIONS ;

DESIGNED FOR THE

USE OF SCHOOLS AND PLACES OF PUBLIC EDUCATION,

TO WHICH IS ADDED

AN APPENDIX,

ON THE

APPLICATION OF ALGEBRA TO GEOMETRY.

BY JOHN BONNYCASTLE,
Professor of Mathematics in the Royal Military Academy, Woolwich.

THIRD NEW-YORK, FROM THE LAST LONDON EDITION,

REVISED, CORRECTED, AND ENLARGED, WITH A VARIETY OF EXAMPLES,

AND MANY OTHER USEFUL ADDITIONS,

BY JAMES RYAN,
Author of " An Elementary Treatise on Algebra, Theoretica

and Practical," &c.

Ingenuas didicisse fideliter artes
Emollit mores, nec sinit esse feros. Qvid.

NEW-YORK:
PUBLISHED BY EVERT DUYCKINCK, 68 WATER-STREET,

IP, E, Dean, Printer, 36 Stone Street,

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Southern District of New York, ss.

BE IT REMEMBERED, That on the twenty-eightb day of December, in the forty-sixth year of the Independence of the United States of America, George Long, of the said district, hath deposited in this office the title of a book, the right whereof he claims as proprietor, in the words following, to wit:

" An Introduction to Algebra, with Notes and Observations; designed for the Use of Schools and places of Public Education. To which is added an Appendix, on the Application of Algebra to Geometry. By John Bonnycastle, Professor of Ma. thematics in the Royal Military Academy, Woolwich. Third New-York from the Last London Edition. Revised, corrected, and enlarged, with a variety of Examples, and many other useful Additions, by James Ryan, Author of an Elementary Treatise on Algebra, Theoretical and Practical, &c.

-Ingennas didicisse fideliter artes

Emollit mores, nec sinit esse feros. Coid.
In conformity to the act of Congress of the United States, entitled, “An act for
the encouragement of learning, by securio; the copies of maps, charts, and books, to
the authors and proprietors of such copies, during the time therein mentioned.".

And
also to an act, entitled, “ An act, supplementary to an act, entitled, An act for the
encauragement of learning, by securing the the copies of maps, charts, and books, to
the authors and proprietors of such copies, during the times therein mentioned, and
extending the benefits thereof to the arts of designing, engraving, and etching histo-
rimi and other prints."!

JAMES DILL,
Clerk of the Southren District of New York.

PREFACE.

THE powers of the mind, like those of the body, are increased by frequent exertion ; application and industry supply the place of genius and invention; and even the creative faculty itself may be strengthened and improved by use and perseverance. Uncultivated nature is uniformly rude and imbecile, it being by imitation alone that we at first acquire knowledge, and the means of extending its bounds. A just and perfect acquaintance with the simple elements of science, is a necessary step towards our future progress and advancement; and this, assisted by laborious investigation and habitual inquiry, will constantly lead to eminence and perfection.

Books of rudiments, therefore, concisely written, well digested, and methodically arranged, are treasures of inestimable value ; and too many attempts cannot be made to render them perfect and complete. When the tirst principles of any art or science are firmly fixed and rooted in the mind, their application soon becomes easy, pleasant, and obvious; the understanding is delighted and enlarged ; we conceive clearly, reason distinctly, and form just and satisfactory conclusions. But, on the contrary, when the mind, instead of reposing on the stability of trath and re. ceived principles, is wandering in doubt and uncertainty, our ideas will necessarily be confused and obscure; and every step we take must be attended with fresh difficulties and endless perplexity.

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