In the preparation of this textbook, no pains have been spared by the author to obtain from the best sources, such as the later treatises in highest repute, memoirs of scientific bodies, and mathematical journals in English, French, and German, the materials for a book suited to the present state of mathematical science and the wants of teachers and students. The French treatises furnish excellent models of the theory of Algebra, the German of ingenuity and brevity of notation and exposition, the English of practical adaptation and variety of illustration and example; and from these, after a careful comparison of many authors in each language, demonstrations have been selected and introduced verbatim when they seemed incapable of improvement; but when ever the slightest alteration or amalgamation, or the entire remodeling of them, could give additional clearness or elegance, the labor has not been spared. The work will be found to contain all that is important in the higher parts of Algebra, upon which usually separate treatises are thought necessary, as well as the elementary expositions suited to beginners. Every variety of symbol and of example has been introduced. Those articles of this volume which constitute a minimum course of Algebra requisite for the prosecution of the higher branches of mathematics are indicated. A more extended course, such as would ordinarily be advisable, is also pointed out. The rest may very well be reserved for reference, as the student's own discovery of his wants, in the advanced stages of mathematical pursuit, shall call it in requisition.