MULTIPLE ALGEBEA. 103 



To show that so simple an expression is really amenable to analytical 

 treatment, I observe that Q may be expressed in terms of any four 

 points (not in the same plane) on the barycentric principle explained 

 above, viz., Q = xA+yB+zC+wD, (25) 



and II may be expressed in terms of combinatorial products of 

 A t B, C, and D, viz., 



U=BxCxD+riCxAxD+DxAxB+ w AxCxB, (26) 



and by these substitutions, by the laws of the combinatorial product 

 to be mentioned hereafter, equation (24) is transformed into 



w<*+xg+yT) + z=0, (27) 



which is identical with the formula of ordinary analysis.* 



I have gone at length into this very simple point in order to 

 illustrate the fact, which I think is a general one, that the modern 

 geometry is not only tending to results which are appropriately 

 expressed in multiple algebra, but that it is actually striving to clothe 

 itself in forms which are remarkably similar to the notations of 

 multiple algebra, only less simple and general and far less amenable 

 to analytical treatment, and therefore, that a certain logical necessity 

 calls for throwing off the yoke under which analytical geometry has 

 so long labored. And lest this should seem to be the utterance of an 

 uninformed enthusiasm, or the echoing of the possibly exaggerated 

 claims of the devotees of a particular branch of mathematical study, 

 I will quote a sentence from Clebsch and one from Clifford, relating 

 to the past and to the future of multiple algebra. The former in 

 his eulogy on Pliicker,t in 1871, speaking of recent advances in 

 geometry, says that " in a certain sense the coordinates of a straight 

 line, and in general a great part of the fundamental conceptions of 

 the newer algebra, are contained in the Ausdehnungslehre of 1844," 

 and Clifford | in the last year of his life, speaking of the Ausdehn- 

 ungslehre, with which he had but recently become acquainted, 

 expresses "his profound admiration of that extraordinary work, and 

 his conviction that its principles will exercise a vast influence upon 

 the future of mathematical science." 



Another subject in which we find a tendency toward the forms 

 and methods of multiple algebra, is the calculus of operations. Our 

 ordinary analysis introduces operators, and the successive operations 

 A and B may be equivalent to the operation C. To express this in 

 an equation we may write 

 BA (x) = C(aj), 



*The letters , T;, , w, here denote the distances of the plane II from the points 

 A, B, C, D, divided by six times the volume of the tetrahedron, A, B, C, D. The 

 letters #, y, z, w, denote the tetrahedral coordinates as above. 



+ GGU. Abhandl., vol. xvi, p. 28. %Amw. Journ. Math., vol. i, p. 350. 



