MULTIPLE ALGEBRA. 93 



their terms arranged and transposed, exactly like the ordinary 

 equations of algebra. It follows that the elimination of letters 

 representing points from equations of this kind is performed by the 

 rules of ordinary algebra. This is evidently the beginning of a 

 quadruple algebra, and is identical, as far as it goes, with Grassmann's 

 marvellous geometrical algebra. 



In the same work we find also, for the first time so far as I am 

 aware, the distinction of positive and negative consistently carried 

 out on the designation of segments of lines, of triangles, and of tetra- 

 hedra, viz., that a change in place of two letters, in such expressions 

 as AB, ABC, A BCD, is equivalent to prefixing the negative sign. It 

 is impossible to overestimate the importance of this step, which gives 

 to designations of this kind the generality and precision of algebra. 



Moreover, if A, B, C are three points in the same straight line, and 

 D any point outside of that line, the author observes that we have 



AB + BC+ CA=Q, 

 and also, with D prefixed, 



DAB+DBC+DCA = Q. 



Again, if A, B, C, D are four points in the same plane, and E any 

 point outside of that plane, we have 



ABC- BCD + CD A - DAB = 0, 

 and also, with E prefixed, 



EABC-EBCD+ECDA -EDAB = Q. 



The similarity to multiplication in the derivation of these formulae 

 cannot have escaped the author's notice. Yet he does not seem to 

 have been able to generalize these processes. It was reserved for the 

 genius of Grassmann to see that AB might be regarded as the 

 product of A and B, DAB as the product of D and AB, and EABC 

 as the product of E and ABC. That Mobius could not make this step 

 was evidently due to the fact that he had not the conception of the 

 addition of other multiple quantities than such as may be represented 

 by masses situated at points. Even the addition of vectors (i.e., the 

 fact that the composition of directed lines could be treated as an 

 addition) seems to have been unknown to him at this time, although 

 he subsequently discovered it, and used it in his Mechanik des 

 Himmels, which was published in 1843. This addition of vectors, 

 or geometrical addition, seems to have occurred independently to 

 many persons. 



Seventeen years after the Barycentrischer Calcul, in 1844, the year 

 in which Hamilton's first papers on quaternions appeared in print, 

 Grassmann published his Lineale Ausdehnungslehre, in which he 

 developed the idea and the properties of the external or combinatorial 



