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 follow- ing: 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. Al