MULTIPLE ALGEBKA. 101 



from the origin perpendicularly toward the plane and in length equal 

 to the reciprocal of the distance of the plane from the origin. The 



equation / , // 



,-=2L+f (9) 



will have precisely the same meaning as equation (3), and 



(10) 



will have precisely the same meaning as equation (7), viz., that the 

 point p" is in the middle between p and p". That the point p lies in 

 the plane <r is expressed by equating to unity the product of p and a- 

 called by Grassmann internal, or by Hamilton called the scalar part 

 of the product taken negatively. By whatever name called, the 

 quantity in question is the product of the lengths of the vectors and 

 the cosine of the included angle. It is of course immaterial what 

 particular sign we use to express this product, as whether we write 



/9.o- = l, or Spar= 1. (11) 



I should myself prefer the simplest possible sign for so simple a 

 relation. It may be observed that p and <r may be expressed as the 

 geometrical sum of their components parallel to a set of perpendicular 

 axes, viz., 



p = xi + yj+zk, (T = gi+W + k. (12) 



By substitution of these values, equation (11) becomes by the laws of 

 this kind of multiplication 



0+y*+sf=i. (13) 



My object in going over these elementary matters is to call attention 

 to the very roundabout way in which the ordinary analysis makes 

 out to represent a point or a plane by a single letter, as distinguished 

 from the directness and simplicity of the notations of multiple algebra, 

 and also to the fact that the representations of points and planes by 

 single letters in the ordinary analysis are not, when obtained, as 

 amenable to analytical treatment as are the notations of multiple 

 algebra. 



I have compared that form of the ordinary analysis which relates 

 to Cartesian axes with a vector analysis. But the case is essentially 

 the same if we compare the form of ordinary analysis which relates 

 to a fundamental tetrahedron with Grassmann's geometrical analysis, 

 founded on the point as the elementary quantity. 



In the method of ordinary analysis a point is represented by four 

 coordinates, of which each represents the distance of the point from 

 a plane of the tetrahedron divided by the distance of the opposite 

 vertex from the same plane. The equation of a plane may be put 

 in the form 



Q, (14) 



