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The Ausdehnungslehre of 1862 is Grassmann's most mature presentation of his extension theory and captured the full sweep of his mathematical achievements. Compared with Grassmann's first book, Lineale Ausdehnungslehre, this book contains new material, including a detailed development of the inner product and its relation to the concept of angle, the theory of functions from the point of view of extension theory, and Grassmann's contribution to the Pfaff problem. In many ways, this book is the version of Grassmann's system most accessible to contemporary readers.