time by constructing, with aid of his method; geometrically any algebraic curve, remained again unnoticed. Need we marvel if Grassmann turned his attention to other subjects,—to Schleiermacher's philosophy, to politics, to philology? Still, articles by him continued to appear in Crelle's Journal, and in 1862 came out the second part of his Ausdehnungslehre. It was intended to show better than the first part the broad scope of the Ausdehnungslehre, by considering not only geometric applications, but by treating also of algebraic functions, infinite series, and the differential and integral calculus. But the second part was no more appreciated than the first. At the age of fifty-three, this wonderful man, with heavy heart, gave up mathematics, and directed his energies to the study of Sanskrit, achieving in philology results which were better appreciated, and which vie in splendour with those in mathematics.
Common to the Ausdehnungslehre and to quaternions are geometric addition, the function of two vectors represented in quaternions by and , and the linear vector functions. The quaternion is peculiar to Hamilton, while with Grassmann we find in addition to the algebra of vectors a geometrical algebra of wide application, and resembling Möbius's Barycentrische Calcul, in which the point is the fundamental element. Grassmann developed the idea of the "external product," the "internal product," and the "open product." The last we now call a matrix. His Ausdehnungslehre has very great extension, having no limitation to any particular number of dimensions. Only in recent years has the wonderful richness of his discoveries begun to be appreciated. A second edition of the Ausdehnungslehre of 1844 was printed in 1877. C. S. Peirce gave a representation of Grassmann's system in the logical notation, and E. W. Hyde of the University of Cincinnati wrote the first text-book on Grassmann's calculus in the English language.
