Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/119

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 $\Pi$ may be expressed in terms of combinatorial products of $A,B,C,$ and $D,$ viz.,

$\Pi =\xi B\times C\times D+\eta C\times A\times D+\zeta D\times A\times B+\omega A\times C\times B,$

(26)

and by these substitutions, by the laws of the combinatorial product to be mentioned hereafter, equation (24) is transformed into

$w\omega +x\xi +y\eta +z\zeta =0,$

(27)

which is identical with the formula of ordinary analysis.^[1]

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 Plücker,^[2] 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^[3] in the last year of his life, speaking of the Ausdehnungslehre, 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

$BA(x)=C(x),$

1. The letters $\xi ,\eta ,\zeta ,\omega ,$ here denote the distances of the plane $\Pi$ from the points $A,B,C,D,$ divided by six times the volume of the tetrahedron, $A,B,C,D.$ The letters $x,y,z,w,$ denote the tetrahedral coordinates as above.
2. Gött. Abhandl., vol. xvi, p. 28.
3. Amer. Journ. Math., vol. i, p. 350.