PRE FACE. The first edition of my Algebra was received with unox. pected favor. Almost immediately after its publication, it was adopted as a text-book in half a dozen colleges, besides numerous academies and schools; and the most flattering testimonials were received from every part of the country. I have thus been stimulated to increased exertions to render it less unworthy of public favor. Every line of it has been subjected to a thorough revision. The work has been read by two successive classes in the University, and wherever improvement seemed practicable, alterations have been freely made. I have also availed myself of the suggestions of sev. eral professors in other colleges. This edition will accordingly be found to differ considerably from the preceding. Alterations, more or less important, have been made on nearly every page. Among these may be mentioned the addition of Continued Fractions, the Extraction of the Roots of Numbers, Elimination by means of the Greatest Common Divisor, and a large collection of Miscellaneous Examples. It is believed that this treatise contains as much of Algebra as can be profitably read in the time allotted to this study in most of our colleges, and that those subjects have been selected which are most important in a course of mathematical study. These materials I have endeavored to combine, so as to form a consistent treatise. I have aimed to cultivate in the mind of the student a habit of generalization, and to lead nim to reduce every principle to its most general form. At the same time, I have been solicitous not to discourage the young beginner, who frequently finds it much more difficult to comprehend a general than a particular proposition. Accordingly, many of the Problems have been twice stated. I first give a simple numerical problem, and then repeat the same problem in a more general form. I have labored to develop, in a clear and intelligible manner, the most important properties of equations, and have bestowed great pains upon the selection of examples to illustrate these properties. Throughout the work I have endeavored to render the most important principles so prominent as to arrest attention, and I have reduced them, as far as practicable, to the form of concise and simple rules. It is believed that, in respect of difficulty, this treatise need not discourage any youth of fifteen years of age who possesses average abilities, while it is designed to form close habits of reasoning, and cultivate a truly philosophical spirit in more mature minds. CONTENT S. DEFINITIONS.—Difference between Algebra and Arithmetic. Case of Monomials.-Rule for the Exponents ............. Maltiplication by detached Coefficients .................................... To reduce a mixed Quantity to the Form of a Fraction .. To reduce Fractions to a common Denominator................ Addition and Subtraction of Fractions ................................ Definitions.--Axioms employed ............ Equations solved by Subtraction and Addition... Equations solved by Division and Multiplication ..... EQUATIONS WITH TWO OR MORE UNKNOWN QUANTITIES. Elimination by Substitution........ By Comparison.-By Addition and Subtraction ...... Equations containing three unknown Quantities .... Equations containing m unknown Quantities......... TVOLUTION AND RADICAL QUANTITIES. To extract a Root of a Monomial.................. Sign of the Root.Square Root of a Trinomial ....... Irrational Quantities.-Fractional Exponents .......... To reduce Surds to their most simple Forms ............... To reduce a Rational Quantity to the Form of a Surd.. To reduce Surds to a common Index ............. Addition and Subtraction of Sard Quantities ........ Multiplication and Division of Surd Quantities ........ Involution and Evolution of Surd Quantities ........ To find Multipliers which shall render Surds Rational To reduce a Sard Fraction to a Rational Numerator or Denominator ...... .... 131 |