98 MULTIPLE ALGEBKA. 



the latter happen to be the fundamental ideas, and those which ought 

 to direct the whole course of thought. 



I have taken a very simple illustration, perhaps the very first 

 theorem which meets the student after those immediately connected 

 with the introductory definitions, both because the simplest illustration 

 is really the best, and because I am here most at home. But the 

 principles of multiple algebra seem to me to shed a flood of light into 

 every corner of the subjects usually treated under the title of deter- 

 minants, the subject gaining as much in breadth from the new notions 

 as in simplicity from the new notations; and in the more intricate 

 subjects of invariants, co variants, etc., I believe that the principles of 

 multiple algebra are ready to perform an equal service. Certainly 

 they make many things seem very simple to me which I should 

 otherwise find difficult of comprehension. 



Let us turn to geometry. 



If we were asked to characterize in a single word our modern 

 geometry, we would perhaps say that it is a geometry of position. 

 Now position is essentially a multiple quantity, or, if you prefer, is 

 naturally represented in algebra by a multiple quantity. And the 

 growth in this century of the so-called synthetic as opposed to 

 analytical geometry seems due to the fact that by the ordinary 

 analysis geometers could not easily express, except in a cumbersome 

 and unnatural manner, the sort of relations in which they were par- 

 ticularly interested. With the introduction of the notations of multiple 

 algebra, this difficulty falls away, and with it the opposition between 

 synthetic and analytical geometry. 



It is, however, interesting and very instructive to observe how the 

 ingenuity of mathematicians has often triumphed over the limitations 

 of ordinary algebra. A conspicuous example and one of the simplest 

 is seen in the Mecanique Analytique, where the author, by the use 

 of what are sometimes called indeterminate equations, is able to write 

 in one equation the equivalent of an indefinite number. Thus the 

 equation 



by the indeterminateness of the values of dx, dy, dz, is made equiva- 

 lent to the three equations 



Z = 0, F=0, Z=0. 

 It is instructive to compare this with 



which is the form that Hamilton or Grassmann would have used. 

 The use of this analytical artifice, if such it can be called, runs all 

 through the work and is fairly characteristic of it. 



Again, the introduction of the potential in the theory of gravity, or 



