MULTIPLE ALGEBKA. 115 



may regard Hamilton's system as equivalent to Grassmann's algebra 

 of vectors. Such practical equivalence is of course consistent with 

 great differences of notation, and of the point of view from which the 

 subject is regarded. 



Perhaps I should add a word in regard to the nature of the problems 

 which require a vector analysis, or the more general form of Grass- 

 mann's point analysis. The distinction of the problems is very marked, 

 and corresponds precisely to the distinction familiar to all analysts 

 between problems which are suitable for Cartesian coordinates, and 

 those which are suitable for the use of tetrahedral, or, in plane 

 geometry, triangular coordinates. Thus, in mechanics, kinematics, 

 astronomy, physics, or crystallography, Grassmann's point analysis 

 will rarely be wanted. One might teach these subjects for years by a 

 vector analysis, and never perhaps feel the need of any of the notions 

 or notations which are peculiar to the point analysis, precisely as in 

 ordinary algebra one might use the Cartesian coordinates in teaching 

 these subjects, without any occasion for the use of tetrahedral coor- 

 dinates. I think of one exception, which, however, confirms the 

 rule. The very important theory of forces acting on a rigid body is 

 much better treated by point analysis than by vector analysis, exactly 

 as in ordinary algebra it is much better treated by tetrahedral coor- 

 dinates than by Cartesian, I mean for the purpose of the elegant 

 development of general propositions. A sufficient theory for the 

 purposes of numerical calculations can easily enough be given by any 

 method, and the most familiar to the student is for such practical 

 purposes of course the best. On the other hand, the projective pro- 

 perties of bodies, the relations of collinearity, and similar subjects, 

 seem to demand the point analysis for their adequate treatment. 



If I have said that the algebra of vectors is contained in the algebra 

 of points, it does not follow that in a certain sense the algebra of 

 points is not deducible from the algebra of vectors. In mathematics, 

 a part often contains the whole. If we represent points by vectors 

 drawn from a common origin, and then develop those relations 

 between such vectors representing points, which are independent of 

 the position of the origin, by this simple process we may obtain a 

 large part, possibly all, of an algebra of points. In this way the 

 vector analysis may be made to serve very conveniently for many of 

 those subjects which I have mentioned as suitable for point analysis. 

 The vector analysis, thus enlarged, is hardly to be distinguished from 

 a point analysis, but the treatment of the subject in this way has 

 somewhat of a makeshift character, as distinguished from the unity 

 and simplicity of the subject when developed directly from the idea of 

 something situated at a point. 



Of those subjects which have no relations to space, the elementary 



