MULTIPLE ALGEBEA. 99 



electricity, or magnetism, gives us a scalar quantity instead of a vector 

 as the subject of study; and in mechanics generally the use of the 

 force-function substitutes a simple quantity for a complex. This 

 method is in reality not different from that just mentioned, since 

 Lagrange's indeterminate equation expresses, at least in its origin, 

 the variation of the force-function. It is indeed the real beauty of 

 Lagrange's method that it is not so much an analytical artifice, as the 

 natural development of the subject. 



In modern analytical geometry we find methods in use which are 

 exceedingly ingenious, and give forms curiously like those of multiple 

 algebra, but which, at least if logically carried out very far, are exces- 

 sively artificial, and that for the expression of the simplest things. 

 The simplest conceptions of the geometry of three dimensions are 

 points and planes, and the simplest relation between these is that a 

 point lies in a plane. Let us see how these notions have been handled 

 by means of ordinary algebra, and by multiple algebra. It will illus- 

 trate the characteristic difference of the methods, perhaps as well as 

 the reading of an elaborate treatise. 



In multiple algebra a point is designated by a single letter, just as 

 it is in what is called synthetic geometry, and as it generally is by the 

 ordinary analyst when he is not writing equations. But in his 

 equations, instead of a single letter the analyst introduces several 

 letters (coordinates) to represent the point. 



A plane may be represented in multiple algebra as in synthetic 

 geometry by a single letter ; in the ordinary algebra it is sometimes 

 represented by three coordinates, for which it is most convenient to 

 take the reciprocals of the segments cut off by the plane on three axes. 

 But the modern analyst has a more ingenious method of representing 

 the plane. He observes that the equation of the plane may be written 



fx + W + z = l, . (1) 



where rj, f are the reciprocals of the segments, and x, y, z are the 

 coordinates of any point in the plane. Now if we set 



p = (x + W + &, (2) 



this letter will represent an expression which represents the plane. In 

 fact, we may say that p implicitly contains rj, and f, which are the 

 coordinates of the plane. We may therefore speak of the plane p, and 

 for many purposes can introduce the letter p into our equations instead 

 f n> f For example, the equation 



is equivalent to the three equations 



t i / 



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