Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/181

Rh anyone who considers what generalized multiplication in connection with additive relations has done in other fields, as in quaternions, or in the theory of matrices, or in the algebra of logic. For a single example, if we multiply the equation by $$\text{PQ}$$ ($$\text{P}$$ and $$\text{Q}$$ being any two points), we have  which will be recognised as expressing an important theorem of statics.

The field in which Grassmann's algebra of points, as distinguished from his algebra of vectors, finds its especial application and utility is nearly coincident with that in which, when we use the methods of ordinary algebra, tetrahedral or anharmonic coordinates are more appropriate than rectilinear. In fact, Grassmann's algebra of points may be regarded as the application of the methods of multiple algebra to the notions connected with tetrahedral coordinates, just as his or Hamilton's algebra of vectors may be regarded as the application of the methods of multiple algebra to the notions connected with rectilinear coordinates. These methods, however, enrich the field to which they are applied with new notiona Thus the notion of the coordinates of a line in space, subsequently introduced by Plücker, was first given in the Ausdehnungslehre of 1844. It should also be observed that the utility of a multiple algebra when it takes the place of an ordinary algebra of four coordinates, is very much greater than when it takes the place of three coordinates, for the same reason that a multiple algebra taking the place of three coordinates is very much more useful than one taking the place of two. Grassmann's algebra of points will always command the admiration of geometers and analysts, and furnishes an instrument of marvellous power to the former, and in its general form, as applicable to space of any number of dimensions, to tiie latter. To the physicist an algebra of points is by no means so indispensable an instrument as an algebra of vectors.

Grassmann's algebra of vectors, which we have described as coincident with a part of Hamilton's system, is not really anything separate from his algebra of points, but constitutes a part of it, the vector arising when one point is subtracted from another. Yet it constitutes a whole, complete in itself, and we may separate it from the larger system to facilitate comparison with the methods of Hamilton.

We have, then, as geometrical algebras published in 1844, an algebra of vectors common to Hamilton and Grassmann, augmented on Hamilton's side by the quaternion, and on Grassmann's by his algebra of points. This statement should be made with the