168 QUATEENIONS AND THE AUSDEHNUNGSLEHRE. 



"mode of multiplying couples" But I cannot find anything like 

 Grassmann's external or internal multiplication in this memoir, which 

 is concerned, as the title pretty clearly indicates, with the theory of 

 the complex quantities of ordinary algebra. 



It is difficult to understand the statements respecting the Ausdehn- 

 ungslehre, which seem to imply that Grassmann's two kinds of 

 multiplication were subject to some kind of limitation to a plane. 

 The external product is not limited in the first Ausdehnungslehre even 

 to three dimensions. The internal, which is a comparatively simple 

 matter, is mentioned in the first Ausdehnungslehre only in the preface, 

 where it is defined, and placed beside the external product as relating 

 to directed lines. There is not the least suggestion of any difference 

 in the products in respect to the generality of their application to 

 vectors. 



The misunderstanding seems to have arisen from the following 

 sentence in Grassmann's preface : " And in general, in the consideration 

 of angles in space, difficulties present themselves, for the complete 

 (allseitig) solution of which I have not yet had sufficient leisure." 

 It is not surprising that Grassmann should have required more time 

 for the development of some parts of his system, when we consider 

 that Hamilton, on his discovery of quaternions, estimated the time 

 which he should wish to devote to them at ten or fifteen years (see 

 his letter to Prof. Tait in the North British Review for September 

 1866), and actually took several years to prepare for the press as 

 many pages as Grassmann had printed in 1844. But any speculation 

 as to the questions which Grassmann may have had principally in 

 mind in the sentence quoted, and the particular nature of the diffi- 

 culties which he found in them, however interesting from other points 

 of view, seems a very precarious foundation for a comparison of the 

 systems of Hamilton and Grassmann as published in the years 

 1843-44. Such a comparison should be based on the positive evidence 

 of doctrines and methods actually published. 



Such a comparison I have endeavoured to make, or rather to 

 indicate the basis on which it may be made, so far as systems of 

 geometrical algebra are concerned. As a contribution to analysis in 

 general, I suppose that there is no question that Grassmann's system 

 is of indefinitely greater extension, having no limitation to any 

 particular number of dimensions. 



