Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/179

Rh are the advantages to be gained by the use of the quaternion? This question, unlike the preceding, is one into which a personal equation will necessarily enter. Everyone will naturally prefer the methods with which he is most familiar; but I think that it may be safely affirmed that in the majority of cases in this field the advantage derived from the use of the quaternion is either doubtful or very trifling. There remains a residuum of cases in which a substantial advantage is gained by the use of the quatemionic method. Such cases, however, so far as my own observation and experience extend, are very exceptional. If a more extended and careful inquiry should show that they are ten times as numerous as I have found them, they would still be exceptional.

We have now to inquire what we find in the Ausdehnungslehre in the way of a geometrical algebra, that is wanting in quaternions. In addition to an algebra of vectors, the Ausdehnungslehre affords a system of geometrical algebra in which the point is the fundamental element, and which for convenience I shall call Grassmann's algebra of points. In this algebra we have first the addition of points, or quantities located at points, which may be explained as follows. The equation in which the capitals denote points, and the small letters scalars (or ordinary algebraic quantities), signifies that  and also that the centre of gravity of the weights $$a, b, c,$$ etc., at the points $$\text{A, B, C,}$$ etc., is the same as that of the weights $$e, f,$$ etc., at the points $$\text{E, F,}$$ etc. (It will be understood that negative weights are allowed as well as positive.) The equation is thus equivalent to four equations of ordinary algebra. In this Grassmann was anticipated by Möbius (Barycentrischer Calcul, 1827).

We have next the addition of finite straight lines, or quantities located in straight lines (Liniengrössen). The meaning of the equation $$\text{AB} + \text{CD} + \text{etc.} = \text{EF} + \text{GH} + \text{etc.}$$ will perhaps be understood most readily, if we suppose that each member represents a system of forces acting on a rigid body. The equation then signifies that the two systems are equivalent. An equation of this form is therefore equivalent to six ordinary equations. It will be observed that the Liniengrössen $$\text{AB}$$ and $$\text{CD}$$ are not simply vectors; they have not merely length and direction, but they are also located each in a given line, although their position within those lines is immaterial. In Clifford's terminology, $$\text{AB}$$ is a rotor, $$\text{AB} + \text{CD}$$ a motor. In the language of Prof. Ball's Theory of Screws, $$\text{AB} + \text{CD}$$ represents either a twist or a wrench.