lieves the work will fully answer all the necessary requisitions of an introduction to the higher branches of mathematics. The Logic of Algebra is an object that should not be lost sight of in the study; and in order that the student may be exercised in this, every important principle has been explained and demonstrated. But, at the same time, the explanations have been made simple, and the demonstrations put in such a form (especially in the first part of the work), that they can be easily comprehended by those unaccustomed to the rigid demonstrations of analytical algebra. In the higher departments of mathematics, it is undoubtedly desirable that the formality in stating every proposition, and the course of demonstration required by the precise rules of logic, should be adhered to. But in algebra the case is different. The mind of the student must become gradually habituated to the more abstract modes of thinking and precise methods of reasoning; and, as algebra is commenced in so early a part of the course, a certain degree of familiarity, rather than formality, in the statement of propositions and in their proof, becomes not only excusable, but even necessary. The author, however, has studied precision in the statement of propositions, and endeavoured to make his reasoning explicit. In this way has he endeavoured to make the theory obvious and satisfactory. Believing that a knowledge of the general principles of algebra can be perfected and permanently secured only by frequent and rigid application, the author has endeavoured, throughout the work, to blend theory and practice. For this purpose, a careful selection of problems and exercises has been made from the most approved authors. In the ninth section the author has given a clear and concise view of the theory of Logarithms, and a method of calculating common logarithms, or those in general use, so explicit, and yet so simple, that the student well versed in propor tion and progression may be able to calculate them with ease and facility. As the last three sections treat upon subjects that are seldom called into use by the merely practical algebraist, and yet subjects that are indispensable as an introduction to the higher departments of mathematics, they have been thrown into the form of an Appendix. This work was commenced, and has been carried to its completion, amid the arduous duties incident to the charge of a large and flourishing seminary of learning. Yet labour and care have been bestowed upon every part of it, and that, too, while the author was daily engaged in instructing classes in this interesting and important branch of study; and if, under these circumstances, he has been able to discover the wants of the student, and adapt his work to meet those wants, he will feel amply compensated for his toil. D. W. CLARK. Amenia Seminary, March, 1843. To Reduce a Fraction to its Lowest Terms (139) The Least Common Multiple (140) Reduction of Fractions to a Common Denominator (141) Page Generalization of Algebraic Problems (§ 188-195). Demonstration of Theorems (196-206) Algebraic Demonstration of certain Properties of Numbers (207-213) Problems producing Affected Quadratic Equations Discussion of the General Equation of the Second Degree (338-351) .176 . 184 . 211 Multiplication and Division by Logarithms (404) Involution and Evolution by Logarithms (405) Newton's Method of Approximation (477-480) Resolution of Higher Equations by Trial and Error (481-483) Young's Method of Resolving Higher Equations (484) . Notes 337 343 The ill health of the author while the work was in course of publication has |
