QUATEKNIONS AND THE AUSDEHNUNGSLEHRE. 163 



portant 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 quaternionic 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 aA + &B + cC + etc. = eE +/F + etc, 



in which the capitals denote points, and the small letters scalars (or 

 ordinary algebraic quantities), signifies that 



G& + & + C+ etc. = e+f+ etc., 



and also that the centre of gravity of the weights a, b, c, etc., at the 

 points A, B, C, etc., is the same as that of the weights e, f, etc., at 

 the points 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 

 Mobius (Barycentrischer Calcul, 1827). 



We have next the addition of finite straight lines, or quantities 

 located in straight lines (Liniengrossen). The meaning of the 

 equation AB + CD + etc. = EF + GH + 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 Liniengrossen AB and 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, AB is a rotor, AB + CD 

 a motor. In the language of Prof. Ball's Theory of Screws, AB + CD 

 represents either a twist or a wrench. 



