Three main purposes have controlled the author in the preparation of this work :- First, to provide, in a single volume, of convenient size, an elementary treatise upon Algebra, sufficiently simple for the use of beginners in the science - sufficiently full on the subjects embraced to render their rediscussion in a subsequent volume unnecessary, and to afford a range of topics comprehensive enough to meet the wants of our common and Public High School, and for a good preparation to enter any of our Colleges or Universities. Second, to train the pupil to methods of reasoning, rather than in mere methods of operating. Third, to present the whole in such a form as to make the book a convenient one to use in the class room.