product, a conception which is perhaps to be regarded as the greatest monument of the author's genius. This volume was to have been followed by another, of the nature of which some intimation was given in the preface and in the work itself. We are especially told that the internal product,[1] which for vectors is identical except in sign with the scalar part of Hamilton's product (just as Grassmann's external product of two vectors is practically identical with the vector part of Hamilton's product), and the open product,[2] which in the language of to-day would be called a matrix, were to be treated in the second volume. But both the internal product of vectors and the open product are clearly defined, and their fundamental properties indicated, in this first volume.
This remarkable work remained unnoticed for more than twenty years, a fact which was doubtless due in part to the very abstract and philosophical manner in which the subject was presented. In consequence of this neglect the author changed his plan, and instead of a supplementary volume published in 1862 a single volume entitled Ausdehnungslehre, in which were treated, in an entirely different style, the same topics as in the first volume, as well as those which he had reserved for the second.
Deferring for the moment the discussion of these topics in order to follow the course of events, we find in the year following the first Ausdehnungslehre a remarkable memoir of Saint-Venant[3], in which are clearly described the addition both of vectors and of oriented areas, the differentiation of these with respect to a scalar quantity, and a multiplication of two vectors and of a vector and an oriented area. These multiplications, called by the author geometrical, are entirely identical with Grassmann's external multiplication of the same quantities.
It is a striking fact in the history of the subject, that the short period of less than two years was marked by the appearance of well-developed and valuable systems of multiple algebra by British, German, and French authors, working apparently entirely independently of one another. No system of multiple algebra had appeared before, so far as I know, except such as were confined to additive processes with multiplication by scalars, or related to the ordinary double algebra of imaginaiy quantities. But the appearance of a single one of these systems would have been sufficient to mark an epoch, perhaps the most important epoch in the history of the subject.
In 1863 and 1854, Cauchy published several memoirs on what he called clefs algébriques.[4] These were units subject generally to
