100 MULTIPLE ALGEBRA. 



It is to be noticed that on account of the indeterminateness of the 

 x, y, and z t this method, regarded as an analytical artifice, is identical 

 with that of Lagrange, also that in multiple algebra we should have 

 an equation of precisely the same form as (3) to express the same 

 relation between the planes, but that the equation would be explained 

 to the student in a totally different manner. This we shall see more 

 particularly hereafter. 



It is curious that we have thus a simpler notation for a plane than 

 for a point. This however may be reversed. If we commence with 

 the notion of the coordinates of a plane, (f, q, f, the equation of a 

 point (i.e., the equation between 77, f which will hold for every 

 plane passing through the point) will be 



a+3M + s?=l, (5) 



where x, y, z are the coordinates of the point. Now if we set 



q = xg+yti + zg, (6) 



we may regard the single letter q as representing the point, and use 

 it, in many cases, instead of the coordinates x, y, 0, which indeed it 

 implicitly contains. Thus we may write 



for the three equations 



Here, by an analytical artifice, we come to equations identical in 

 form and meaning with those used by Hamilton, Grassmann, and even 

 by Mobius in 1827. But the explanations of the formulae would 

 differ widely. The methods of the founders of multiple algebra are 

 characterized by a bold simplicity, that of the modern geometry by a 

 somewhat bewildering ingenuity. That p and q represent the same 

 expression (in one case x, y, z, and in the other (-, 77, f being indeter- 

 minate) is a circumstance which may easily become perplexing. I am 

 not quite certain that it would be convenient to use both of these 

 abridged notations at the same time. In fact, if the geometer using 

 these methods were asked to express by an equation in p and q that 

 the point q lies in the plane p, he might find himself somewhat 

 entangled in the meshes of his own ingenuity, and need some new 

 artifice to extricate himself. I do not mean that his genius might 

 not possibly be equal to the occasion, but I do mean very seriously 

 that it is a vicious method which requires any ingenuity or any 

 artifice to express so simple a relation. 



If we use the methods of multiple algebra which are most com- 

 parable to those just described, a point is naturally represented by a 

 vector (p) drawn to it from the origin, a plane by a vector (or) drawn 



