In preparing this book, the author had especially in mind classes in the upper grades of grammar schools, though the work will be found equally well adapted to the needs of any classes of beginners. The ideas which have guided in the treatment of the subject are the following: The study of algebra is a continuation of what the pupil has been doing for years, but it is expected that this new work will result in a knowledge of general truths about numbers, and an increased power of clear thinking. All the differences between this work and that pursued in arithmetic may be traced to the introduction of two new elements, namely, negative numbers and the representation of numbers by letters. The solution of problems is one of the most valuable portions of the work, in that it serves to develop the thought-power of the pupil at the same time that it broadens his knowledge of numbers and their relations. Powers are developed and habits formed only by persistent, long-continued practice. Accordingly, in this book, it is taken for granted that the pupil knows what he may be reasonably expected to have learned from his study of arithmetic; abundant practice is given in the representation of numbers by letters, and great care is taken to make clear the meaning of the minus sign as applied to a single number, together with the modes of operating upon negative numbers; problems are given in every exercise in the book; and, instead of making a statement of what the child is to see in the illustrative example, Questions are asked which shall lead him to second for himself that which he is to learn from the example.