Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/115

electricity, or magnetism, gives us a scalar quantity instead of a vector as the subject of study; and in mechanics generally the use of the force-function substitutes a simple quantity for a complex. This method is in reality not different from that just mentioned, since Lagrange's indeterminate equation expresses, at least in its origin, the variation of the force-function. It is indeed the real beauty of Lagrange's method that it is not so much an analytical artifice, as the natural development of the subject.

In modern analytical geometry we find methods in use which are exceedingly ingenious, and give forms curiously like those of multiple algebra, but which, at least if logically carried out very far, are excessively artificial, and that for the expression of the simplest things. The simplest conceptions of the geometry of three dimensions are points and planes, and the simplest relation between these is that a point lies in a plane. Let us see how these notions have been handled by means of ordinary algebra, and by multiple algebra. It will illustrate the characteristic difference of the methods, perhaps as well as the reading of an elaborate treatise.

In multiple algebra a point is designated by a single letter, just as it is in what is called synthetic geometry, and as it generally is by the ordinary analyst when he is not writing equations. But in his equations, instead of a single letter the analyst introduces several letters (coordinates) to represent the point.

A plane may be represented in multiple algebra as in synthetic geometry by a single letter; in the ordinary algebra it is sometimes represented by three coordinates, for which it is most convenient to take the reciprocals of the segments cut off by the plane on three axes. But the modern analyst has a more ingenious method of representing the plane. He observes that the equation of the plane may be written

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

(1)

where $\xi ,\eta ,\zeta$ are the reciprocals of the segments, and $x,y,z$ are the coordinates of any point in the plane. Now if we set

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

(2)

this letter will represent an expression which represents the plane. In fact, we may say that $p$ implicitly contains $\xi ,\eta ,$ and $\zeta ,$ which are the coordinates of the plane. We may therefore speak of the plane $p,$ and for many purposes can introduce the letter $p$ into our equations instead of $\xi ,\eta ,\zeta .$ For example, the equation

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

(3)

is equivalent to the three equations

$\xi '''={\frac {\xi '+\xi ''}{2}},$⁠$\eta '''={\frac {\eta '+\eta ''}{2}},$⁠$\zeta '''={\frac {\zeta '+\zeta ''}{2}}\cdot$

(4)