ELEMENTS OF ALGEBRA WITH EXERCISES BY GEORGE EGBERT FISHER, M.A., PH.D. AND ISAAC J. SCHWATT, PH.D. ASSISTANT PROFESSORS OF MATHEMATICS IN THE PHILADELPHIA FISHER AND SCHWATT Educ T12 9.01 3 8 8 HARVARD COLLEGE LIBRARY GIFT OF GINN & COMPANY COPYRIGHT, 1899, BY FISHER AND SCHWATT. Norwood Press J. S. Cushing & Co. - Berwick & Smith Norwood Mass. U.S.A. PREFACE. THE unusual character of the recognition which the TEXTBOOK OF ALGEBRA, Part I., has received encourages the authors to believe that a book on the same lines, but in briefer form, will have a still wider field of usefulness. This book retains the distinctive features of the larger volume; but it is in many respects, for younger students, an improvement on the latter. The needs of beginners have been constantly kept in mind. The aim has been to make the transition from ordinary Arithmetic to Algebra natural and easy. No efforts have been spared to present the subject in a simple and clear manner. Yet nothing has been slighted or evaded, and all difficulties have been honestly faced and explained. New terms and ideas have been introduced only when the development of the subject made them necessary. Special attention has been paid to making clear the reason for every step taken. Each principle is first illustrated by particular examples, thus preparing the mind of the student to grasp the meaning of a formal statement of the principle and its proof. Directions for performing the different operations are, as a rule, given after these operations have been illustrated by particular examples. The importance of mental discipline to every student of mathematics has also been fully recognized. On this account great care has been taken to develop the subject in a logical manner. Rigorous, but, as a rule, simple, proofs of all principles have been given. If mathematics is to develop the reasoning power of the student and to teach him to think logically, it is better to omit a proof altogether than to give as a proof logically incorrect statements, thus training the mind of the student in illogical thinking. Concrete illustrations, such as receiving and paying out money, going north and going south, have their proper places, but cannot be said to constitute proofs. If Algebra, like Arithmetic, treats of number, then the laws governing the operations with numbers should be derived from the properties of and the relations between them. The subject-matter in the book has been printed in two sizes of type. The matter in smaller type consists of the formal proofs of principles and of the more difficult portions of each topic treated. The matter given in the larger type is logically complete (except for the proofs of principles), and can be taken up as a first course in the subject. To economize space the exercises have been put in smaller type, and not the explanations and solutions of illustrative examples in the text. It is regarded as more important that the student should have these, which he is to study, most clearly represented rather than the examples which he is to copy and then work from his paper. The attention of teachers is especially invited to the following features of the book: The introductory chapter and the development in Chapter II. of the fundamental operations with algebraic numbers. The use of type-forms in multiplication and division (Chapter VI.) and in factoring (Chapter VIII.). The application of factoring to the solution of equations (Chapters VIII. and XXI.). By the early introduction of this method it has been possible to give problems which lead to quadratic equations before the formal treatment of that topic. The solutions of equations based upon equivalent equations and equivalent systems of equations (Chapter IV., etc.). This method is of extreme importance, even to the beginner. The ordinary way of treating equations is illogical, leads to serious errors, and is therefore also pedagogically wrong. Thus, no one will dispute that if both sides of an equation be multiplied by the same algebraical number an equation is obtained; but whether it is legitimate to assume that the solutions of this equation are the solutions of the given equation is quite another matter. The treatment of irrational equations (Chapter XXIII.). The special suggestions given in the first chapter on problems (Chapter V.), and applied subsequently to assist the student in acquiring facility in translating the verbal language of the problem into the symbolic language of the equation. The discussion of general problems (Chapter XI.) and the interpretation of positive, negative, zero, indeterminate, and infinite solutions of problems (Chapter XII.). The outline of irrational numbers (Chapter XVIII.). The brief introduction to imaginary and complex numbers (Chapter XX.). The exercises are voluminous. The aim has been not only to give examples for sufficient drill in the applications of the principles, but to include also many which tend to develop the thinking power of the student, rather than to develop him in a treadmill way. |