Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/116

It is to be noticed that on account of the indeterminateness of the $x,y,$ and $z,$ this method, regarded as an analytical artifice, is identical with that of Lagrange, also that in multiple algebra we should have an equation of precisely the same form as (3) to express the same relation between the planes, but that the equation would be explained to the student in a totally different manner. This we shall see more particularly hereafter.

It is curious that we have thus a simpler notation for a plane than for a point. This however may be reversed. If we commence with the notion of the coordinates of a plane, $\xi ,\eta ,\zeta ,$ the equation of a point (i.e., the equation between $\xi ,\zeta ,\eta$ which will hold for every plane passing through the point) will be

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

(5)

where $x,y,z$ are the coordinates of the point. Now if we set

$q=x\xi +y\eta +z\zeta ,$

(6)

we may regard the single letter $q$ as representing the point, and use it, in many cases, instead of the coordinates $x,y,z,$ which indeed it implicitly contains. Thus we may write

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

(7)

for the three equations

$x'''={\frac {x'+x''}{2}},$⁠$y'''={\frac {y'+y''}{2}},$⁠$z'''={\frac {z'+z''}{2}}\cdot$

(8)

Here, by an analytical artifice, we come to equations identical in form and meaning with those used by Hamilton, Grassmann, and even by Möbius in 1827. But the explanations of the formulæ would differ widely. The methods of the founders of multiple algebra are characterized by a bold simplicity, that of the modern geometry by a somewhat bewildering ingenuity. That $p$ and $q$ represent the same expression (in one case $x,y,z$ and in the other $\xi ,\eta ,\zeta$ being indeterminate) is a circumstance which may easily become perplexing. I am not quite certain that it would be convenient to use both of these abridged notations at the same time. In fact, if the geometer using these methods were asked to express by an equation in $p$ and $q$ that the point $q$ lies in the plane $p,$ he might find himself somewhat entangled in the meshes of his own ingenuity, and need some new artifice to extricate himself. I do not mean that his genius might not possibly be equal to the occasion, but I do mean very seriously that it is a vicious method which requires any ingenuity or any artifice to express so simple a relation.

If we use the methods of multiple algebra which are most comparable to those just described, a point is naturally represented by a vector $(\rho )$ drawn to it from the origin, a plane by a vector $(\sigma )$ drawn