ciple, or employ a process, with the rationale of which he is not already acquainted. The reference by Articles will always enable him to trace any subject back to its first principles. The limits of a preface will not permit a statement of the peculiarities of the work, nor is it necessary, as those who are interested to know will examine it for themselves. It is, however, proper to remark, that Quadratic Equations have received more than usual attention. The same may be said of Radicals, of the Binomial Theorem, and of Logarithms, all of which are so useful in other branches of Mathematics. On some subjects it was necessary to be brief, to bring the work within suitable limits. For example, what is here given of the Theory of Equations, is to be regarded merely as an outline of the more practical and interesting parts of the subject, which alone is sufficient for a distinct treatise, as may be seen by reference to the works of Young or Hymers in English, or of DeFourcy or Reynaud in French. Some topics and exercises, deemed both useful and interesting, will be found here, not hitherto presented to the notice of students. But these, as well as the general manner of treating the subject, are submitted, with deference, to the intelligent educational public, to whom the author is already greatly indebted for the favor with which his previous works have been received. WOODWARD COLLEGE, May, 1852. In this NEW ELECTROTYPE EDITION, the whole volume has been subjected to a careful and thorough revision. The examples, where they were thought to be needlessly multiplied, have been reduced; the rules and demonstrations abridged, and other methods of proof, in a few instances, substituted. It is confidently believed that these modifications, while they do not impair the integrity or change the essential features of the book, will materially enhance its value, and secure the approbation of all intelligent teachers. March, 1866. CONTENTS. MULTIPLICATION-Preliminary principle. Rule of Coefficients-of Exponents Rule of the Signs-General Rule Multiplication by Detached Coefficients. Remarks on Algebraic Multiplication DIVISION-Rule of Signs-Coefficients-Exponents . Division of a Monomial by a Monomial. II. THEOREMS, FACTORING, ETC. ALGEBRAIC THEOREMS-Square of the sum of two quantities. Square of the difference of two quantities Product of Sum and Difference Transfer of factors in a fraction. 87-95 114-119 121 122 123 125-126 Miscellaneous Propositions in Fractions. Theorems in Fractions-Miscellaneous exercises. Elimination by Substitution-Comparison-Addition, etc. V.-SUPPLEMENT TO SIMPLE EQUATIONS. Generalization-Formation of Rules-Examples Negative Solutions-Discussion of Problems-Couriers VI.-FORMATION OF POWERS-EXTRACTION OF ROOTS- RADICALS-INEQUALITIES. Square Root of Monomials-Of Polynomials CUBE ROOT of Numbers-Approximate Cube Roots Cube Root of Monomials-Of Polynomials. Fourth Root-Sixth Root-Nth Root, etc. Signs of the Roots-Nth Root of Monomials 173-179 Multiplication and Division in Fractional Exponents 212-213 Varieties of Trinomials-Form of Fourth Degree Simultaneous Quadratic Equations Pure Equations-Affected Equations VIII.-RATIO-PROPORTION-PROGRESSIONS. RATIO-Kinds-Antecedent and Consequent Ratio Multiplication and Division of Ratio of Equality-Of greater and less Inequality Ratio Compound-Duplicate-Triplicate Product of Means equal to Product of Extremes . Proportion from two Equal Products Product of the Extremes equal to the Square of the Mean Proportion from equality of Antecedents and Consequents Like Powers or Roots of Proportionals are in Proportion Products of Proportionals are in Proportion ARTICLES. 254-256 ARITHMETICAL PROGRESSION-Increasing and Decreasing. Last Term-Rule-Sum of Series-Rule-Table . . To insert m Arithmetical Means between two numbers, Ex. 16 GEOMETRICAL PROGRESSION-Increasing and Decreasing Last Term-How to find it-Sum of Series-Rule Sum of Decreasing Infinite Geometrical Series To insert m Geometrical Means between two numbers, Ex. 19 Circulating Decimals-To find the value of . |