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." 03-19-34 Mlte Lowdermilk 9-30.30 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 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 1. Addition of algebraic quantities, II. Subtraction of algebraic quantities, III. Multiplication of algebraic quantities, On Simple Equations, involving two or more un- I. 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, |