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From the AUTHOR'S PREFACE. This little book is an attempt to present simply and concisely the elementary principles of the Extensive Analysis as fully developed in the comprehensive treatises of Hermann Grassmann, restricting the treatment however to the geometry of two and three dimensional space. It is designed to set forth, as far as is possible in so brief a work, the remarkable adaptability and effectiveness of the methods used as applied to the various problems and operations of geometry and mechanics. The ideas of direction and position appear to the writer to be as simple and fundamental as that of magnitude, and an algebra which deals directly with all three of these ideas should not be greatly more difficult than the ordinary one, which deals with magnitude only. The result of using such an algebra is an extraordinary gain in the brevity of operations and the expressiveness of formulas and equations. Some of the terms belonging to this general subject are frequently employed in modern text-books on mechanics and physics, even when no use is made of the algebraic systems from which they are derived. It is hoped that this little book may do something to interest students, and to help toward bringing in the time when the methods as well as the ideas of this calculus shall come into general use. Cincinnati, O., December, 1905.

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