Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/109

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 tetrahedra, viz., that a change in place of two letters, in such expressions as $AB,ABC,ABCD,$ 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

$AB+BC+CA=0,$

and also, with $D$ prefixed,

$DAB+DBC+DCA=0.$

Again, if $A,B,C,D$ are four points in the same plane, and $E$ any point outside of that plane, we have

$ABC-BCD+CDA-DAB=0,$

and also, with $E$ prefixed,

$EABC-EBCD+ECDA-EDAB=0.$

The similarity to multiplication in the derivation of these formulæ 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 Möbius 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 HHimmels, 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