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INTRODUCTION

TO

Ꭺ Ꮮ Ꮐ Ꭼ Ᏼ Ꭱ Ꭺ,

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.

FOURTH 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.

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PUBLISHED BY EVERT DUYCKINCK, AND COLLINS & HANNAY,

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Southern District of New-York, ss.

BE IT REMEMBERED, That on the 28th day of December, in the fortysixth 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 Mathematics in the Royal Military Academy, Woolwich. Fourth 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.

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

Ovid."

In conformity to the Act of Congress of the United States, entitled "An Act or the encouragement of Learning, by securing 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 encouragement of Learning, by securing the copies of Maps, Charts, and Books, to the Authors and Proprie. tors of such copies during the times therein mentioned, and extending the benefits thereof to the arts of Designing, Engraving, and Etching Historica! and other Prints."

JAMES DILL,

Clerk of the Southern District of New-York.

W. E. DEAN, PRINTER.

WON ZOJUN RG

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 first 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 truth and received 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.

That the grounds, or fundamental parts, of every sch ence, are dull and unentertaining, is a complaint universally made, and a truth not to be denied ; but then, what is obtained with difficulty is usually remembered with ease; and what is purchased with pain is often possessed with pleasure. The seeds of knowledge are sown in every soil, but it is by proper culture alone that they are cherished and brought to maturity. A few years of early and assiduous application never fails to procure us the reward of our industry: and who is there, who knows the pleasures and advantages which the sciences afford, that would think his time, in this case, misspent, or his labours useless : riches and honours are the gifts of fortune, casually bestowed, or hereditarily received, and are frequently abused by their possessors; but the superiority of wisdom and knowledge is a pre-eminence of merit, which originates with the man, and is the noblest of all distinctions.

Nature, bountiful and wise in all things, has provided us with an infinite variety of scenes, both for our instruction and entertainment; and, like a kind and indulgent parent, admits all her children to an equal participation of her blessings. But, as the modes, situations, and circumstances of life are various, so accident, habit, and education, have each their predominating influence, and give to every mind its particular bias. Where examples of excellence are wanting, the attempts to attain it are but few; but eminence excites attention and produces imitation. To raise the curiosity, and to awaken the listless and dormant powers of younger minds, we have only to point out to them a valuable acquisition, and the means of obtaining it; the active principles are immediately put into motion, and the certainty of the conquest is ensured from a determination to conquer.

But, of all the sciences which serve to call forth this spirit of enterprize and inquiry, there are none more eminently useful than Mathematics. By an early attachment to these elegant and sublime studies, we acquire a habit of reasoning, and an elevation of thought, which fixes the mind, and prepares it for every other pursuit. From a

few simple axioms, and evident principles, we proceed gradually to the most general propositions, and remote analogies; deducing one truth from another, in a chain of argument well connected and logically pursued; which brings us at last, in the most satisfactory manner, to the conclusion, and serves as a general direction in all our inquiries after truth.

And it is not only in this respect that mathematical learning is so highly valuable; it is likewise equally estimable for its practical utility. Almost all the works of art and devices of man have a dependence upon its principles, and are indebted to it for their origin and perfection. The cultivation of these admirable sciences is, therefore, a thing of the utmost importance, and ought to be considered as a principal part of every liberal and well-regulated plan of education. They are the guide of our youth, the perfection of our reason, and the foundation of every great and noble undertaking.

From these considerations, I have been induced to compose an introductory course of mathematical science; and from the kind encouragement which I have hitherto received, am not without hopes of a continuance of the same candour and approbation. Considerable practice as a teacher, and a long attention to the difficulties and obstructions which retard the progress of learners in general, have enabled me to accommodate myself the more easily to their capacities and understandings, And as an earnest de

sire of promoting and diffusing ase knowledge is the chief motive for this undertaking, so no pains or attention shall be wanting to make it as complex and perfect as possible.

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The subject of the present performance is ALGEBRA : which is one of the most important and useful branches of those sciences, and may be justly ecundered as the key to all the rest. Geometry delights us the simplicity of its principles and the elegance of the do metic is confined to its object, aud pribut Algebra or the Analytic Art, is ger, sive, and may be applied with succes

irations; Arithe its application;

and comprehenin all cases where

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