Southern District of New-York, ss. BE IT REMEMBERED, That on the first day of July in the furty. eigbth year of the Independence of the United States of America, JAMES RYAN, of the said District, has depositell in this office ibe title of a Book, the right wiereof be claims as Authet, in the words following, lo wit: « An Elementary Treatise on Alg ora, theoretical and practical, adapted to the Instruction of Youth in Schools and Colleges. By James Ryan, Author of a Key to Bunnycastle's Algebra. To wbich is added, an Appendix, containing an Algebraic Method of demonstrating the Propositions in the fifth book of Euclid's Elements, according to the text and arrangement in Simson's edition, by Robert Adrain, LL.D. FAP.S. F.A.A.S., &r and Professor of Mathematics and Natural Philosophy, in Columbia College New-York.” In conformity to the Act of Congress of the United States, entitled “ An Act for 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 proprietors of such copies, during the times therein mentioned, and extending the benefits thereof to ibe arts of designing. engraving, and etching historical, and other prints." JAMES DILL, 9-3:30 22611 ADVERTISEMENT. As UTILITY is the great object aimed at in. this Publication, I have spared no pains to make a careful selection of materials, from the mosi approved sources, which may tend to elucidate, in a full and clear manner, the Elements of Algebra, both in theory and practice. Those authors of whose labours I have principally availed myself, are Euler, Clai , Newton, Simpson, Emerson, Wood, Bonnycastle, Bridge, and Bland. To Bland's Algebraical Problems, (a work compiled for the use of Students in one of the first Universities in Europe), I am chiefly indebted for the problems in Simple, Pure, and Quadratic Equations. han By permission of the learned Dr. Adrain, I have added, as an Appendix, his method of demonstrating algebraically_ the propositions in the fifth book of Euclid's Ele. ments. CONTENTS. Page On the Addition, Subtraction, Multiplication and Division 1. Addition of algebraic quantities, II. Subtraction of algebraic quantities, III. Multiplication of algebraic quantities, IV. Division of algebraic quantities, 1. Theory of algebraic fractions, . II. Method of finding the greatest common divisor III. Method of finding the least common multiple IV. Reduction of algebraic fractions, V. Addition and subtraction of algebraic fractions, 129 VI. Multiplication and division of algebraic frac- On Simple Equations involving only one unknown I. Reduction of simple equations, II. Resolution of simple equations, involving only On Simple Equations, involving two or more un- 1. Elimination of unknown quantities from any II. Resolution of simple equations, involving two Examples in which the preceding rules are ap- plied, in the solution of simple equations, involving two unknown quantities, CHAPTER V. |