QUATERNIONS AND THE AUSDEHNUNGSLEHRE. 165 



anyone who considers what generalized multiplication in connection 

 with additive relations has done in other fields, as in quaternions, or 

 in the theory of matrices, or in the algebra of logic. For a single 

 example, if we multiply the equation 



AB + CD+etc. = EF + GH + etc. 

 by PQ (P and Q being any two points), we have 



ABPQ + CDPQ + etc. = EFPQ + GHPQ + etc., 



which will be recognised as expressing an important theorem of 

 statics. 



The field in which Grassmann's algebra of points, as distinguished 

 from his algebra of vectors, finds its especial application and utility 

 is nearly coincident with that in which, when we use the methods 

 of ordinary algebra, tetrahedral or anharmonic coordinates are more 

 appropriate than rectilinear. In fact, Grassmann's algebra of points 

 may be regarded as the application of the methods of multiple algebra 

 to the notions connected with tetrahedral coordinates, just as his or 

 Hamilton's algebra of vectors may be regarded as the application of 

 the methods of multiple algebra to the notions connected with recti- 

 linear coordinates. These methods, however, enrich the field to 

 which they are applied with new notions. Thus the notion of the 

 coordinates of a line in space, subsequently introduced by Plticker, 

 was first given in the Ausdehnungslehre of 1844. It should also be 

 observed that the utility of a multiple algebra when it takes the place 

 of an ordinary algebra of four coordinates, is very much greater 

 than when it takes the place of three coordinates, for the same reason 

 that a multiple algebra taking the place of three coordinates is very 

 much more useful than one taking the place of two. Grassmann's 

 algebra of points will always command the admiration of geometers 

 and analysts, and furnishes an instrument of marvellous power to the 

 former, and in its general form, as applicable to space of any number 

 of dimensions, to the latter. To the physicist an algebra of points is 

 by no means so indispensable an instrument as an algebra of vectors. 



Grassmann's algebra of vectors, which we have described as co- 

 incident with a part of Hamilton's system, is not really anything 

 separate from his algebra of points, but constitutes a part of it, the 

 vector arising when one point is subtracted from another. Yet it 

 constitutes a whole, complete in itself, and we may separate it from 

 the larger system to facilitate comparison with the methods of 

 Hamilton. 



We have, then, as geometrical algebras published in 1844, an 

 algebra of vectors common to Hamilton and Grassmann, augmented 

 on Hamilton's side by the quaternion, and on Grassmann's by 

 his algebra of points. This statement should be made with the 



