94 MULTIPLE ALGEBRA. 



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* 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^ 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 con- 

 sequence 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 J, 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 inde- 

 pendently 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 imaginary 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 1853 and 1854, Cauchy published several memoirs on what 

 he called clefs alge'briques. These were units subject generally to 



* See the preface. t See 172. 



Comptes Rendus, vol. xxi, p. 620. Comptes Rendus, vols. xxxvi, ff. 



