In numerous instances, after deducing one or more principles, I have introduced selections of easy examples to be worked at sight. These are intended to give opportunity for the application of the principles under which they appear, and to cultivate in the studént a quick perception of letter, exponent, sign, and factor. An unusually large number of examples for written work are distributed throughout the book. These have been selected with special reference to the class of pupils for whom the work is intended. They are arranged for two readings. At the first reading it is recommended that all miscellaneous examples, which are generally more difficult than the others, shall be omitted. These, in connection with a review of the definitions and principles, will form a good second reading. Long, pointless examples, requiring much time and labor in their solution, have been generally avoided. The rather extensive treatment of factoring, and the preparation provided for it by the introduction of a partial treatment of involution, a treatise on composition, and one on exact division, it is believed will be commended by teachers generally. No one can expect to make much progress in the study of algebra who is not somewhat of an adept in factoring. The early introduction of the equation, and the frequent return to it, are features so well adapted to practical work that comment upon their merits is unnecessary. The simplicity of the treatment of generalization and specialization, negative solutions, inequalities, binomial surds, and limiting ratios, is a sufficient excuse for their introduction into an elementary treatise on algebra. These subjects may, however, be omitted where a shorter course is desirable, without doing violence to the logic of other parts. In conclusion, I desire to express my deep obligations to my wife, Annie M. Sensenig, whose experience as teacher has been nearly coextensive with mine, and from whom I have received many practical helps and encouragements in the preparation of this work. I am also greatly indebted to Prof. A. J. Rickoff, of New York, for a careful examination of the manuscript before publication, and for many practical hints obtained through his criticisms. DAVID M. SENSENIG. NORMAL SCHOOL, WEST CHESTER, PA.,} June 1, 1888. Concrete examples involving literal quantities Positive and negative quantities . Definitions of quantities Symbols of aggregation-Definitions and principles Simplification of parenthetical expressions. Squaring of binomials-Principles and applications Cubing of binomials-Principles and applications Composition-Definitions and general principles Special principles and applications Cross-multiplication-Principle and application. Factoring the sum or difference of equal odd powers Factoring trinomials composed of binomial factors having a Factoring trinomials composed of any binomial factors Miscellaneous examples in factoring Highest common divisor-Definitions and principles Highest common divisor of monomials Highest common divisor of polynomials The lowest conımon multiple-Definitions Lowest common multiple of monomials Lowest common multiple of polynomials Cancellation-Definitions and principles. Multiplication and division by cancellation. |