BE IT REMEMBERED, That on the first day of July in the fortyeighth year of the Independence of the United States of America, JAMES RYAN, of the said District, has deposited in this office the title of a Book, the right whereof he claims as Author, in the words following, to wit: "An Elementary Treatise on Algebra, theoretical and practical, adapted to the Instruction of Youth in Schools and Colleges. By James Ryan, Author of a Key to Bonnycastle's Algebra. To which 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. F.A.P.S. F.A.A.S., &c. 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 the arts of designing, engraving, and etching historical, and other prints." JAMES DILL, Clerk of the Southern District of New-York ADVERTISEMENT. Wechert - 8 August, 19640 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 most 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, Clairaut, Lacroix, Garnier, Bezout, Lagrange, 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. 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. New-York, July 1, 1824. JAMES RYAN. CONTENTS. Page IV. Reduction of algebraic fractions, V. Addition and subtraction of algebraic fractions, VI. Multiplication and division of algebraic frac- 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, |