Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/117

from the origin perpendicularly toward the plane and in length equal to the reciprocal of the distance of the plane from the origin. The equation

$\sigma '''={\frac {\sigma '+\sigma ''}{2}}$

(9)

will have precisely the same meaning as equation (3), and

$\rho '''={\frac {\rho '+\rho ''}{2}}$

(10)

will have precisely the same meaning as equation (7), viz., that the point $\rho '''$ is in the middle between $\rho '$ and $\rho ''.$ That the point $\rho$ lies in the plane $\sigma$ is expressed by equating to unity the product of $\rho$ and $\sigma$ called by Grassmann internal, or by Hamilton called the scalar part of the product taken negatively. By whatever name called, the quantity in question is the product of the lengths of the vectors and the cosine of the included angle. It is of course immaterial what particular sign we use to express this product, as whether we write

$\rho .\sigma =1,$or$S\rho \sigma =-1.$

(11)

I should myself prefer the simplest possible sign for so simple a relation. It may be observed that $\rho$ and $\sigma$ may be expressed as the geometrical sum of their components parallel to a set of perpendicular axes, viz.,

$\rho =xi+yj+zk,$$\sigma =\xi i+\eta j+\zeta k.$

(12)

By substitution of these values, equation (11) becomes by the laws of this kind of multiplication

$x\xi +y\eta +z\zeta =1.$

(13)

My object in going over these elementary matters is to call attention to the very roundabout way in which the ordinary analysis makes out to represent a point or a plane by a single letter, as distinguished from the directness and simplicity of the notations of multiple algebra, and also to the fact that the representations of points and planes by single letters in the ordinary analysis are not, when obtained, as amenable to analytical treatment as are the notations of multiple algebra.

I have compared that form of the ordinary analysis which relates to Cartesian axes with a vector analysis. But the case is essentially the same if we compare the form of ordinary analysis which relates to a fundamental tetrahedron with Grassmann's geometrical analysis, founded on the point as the elementary quantity.

In the method of ordinary analysis a point is represented by four coordinates, of which each represents the distance of the point from a plane of the tetrahedron divided by the distance of the opposite vertex from the same plane. The equation of a plane may be put in the form

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

(14)