The first twenty-two chapters are identical with the author's "Elementary Algebra," whose general plan and scope are stated in its preface as follows: "The author has aimed to make this treatment of elementary algebra simple and practical, without, however, sacrificing scientific accuracy and thoroughness. "Particular care has been bestowed upon those chapters which in the customary courses offer the greatest difficulties to the beginner, especially problems and factoring. The presentation of problems as given in Chapter V will be found to be quite a departure from the customary way of treating the subject, and it is hoped that this treatment will materially diminish the difficulty of this topic for young students. "In factoring, instead of the usual multiplicity of cases, comparatively few methods are given, but these few are treated thoroughly. The cross-product method for factoring quadratic trinomials has been simplified by considering the common monomial factors (§ 116, 4); and in this form the method seemed to be preferable to the other prevailing methods. The criticism that the cross-product method is based upon guessing has no value, since all other devices are equally based upon guessing; in fact, these methods have to be empirical until quadratic equations furnish a scientific means of factoring. "Applications taken from geometry, physics, and commercial life are numerous, but care has been taken not to introduce illustrations so complex as to require the expenditure of time for the teaching of physics or geometry. In cases, however, in which a physical or geometric formula produced an example equally good as the putting together of symbols at random, the formula has been used, as in numerical substitution, proportion, literal equations, etc. "The book is designed to meet the requirements for admission to our best universities and colleges, in particular the requirements of the College Entrance Examination Board. This made it necessary to introduce the theory of proportions and graphical methods into the first year's work, an innovation. which seems to mark a distinct gain from the pedagogical point of view. "By studying proportions during the first year's work, the student will be able to utilize this knowledge where it is most needed, viz. in geometry; while in the usual course proportions are studied a long time after their principal application. "Graphical methods have not only a great practical value, but they unquestionably furnish a very good antidote against 'the tendency of school algebra to degenerate into a mechanical application of memorized rules.' This topic has been represented in a simple, elementary way, and it is hoped that some of the modes of representation given will be considered improvements upon the prevailing methods, e.g. the finding of roots to several decimal places (§ 305) and the solution of quadratic equations (§ 330). The entire work in graphical methods has been so arranged that teachers who wish a shorter course may omit these chapters." The author desires to acknowledge his indebtedness to Messrs. William P. Manguse and B. A. Heydrick for the careful reading of the proofs and for many valuable suggestions. NEW YORK, October, 1905. ARTHUR SCHULTZE. |