32 



(the Ausdehnungslehre of Grassmann) which almost 

 simultaneously germinated on the continent, nor with 

 ideas more recently developed in America (Pierce's 

 Linear Associative Algebras) ; yet it must always hold 

 its position as an original discovery, and as a repre- 

 sentative of one of the two great groups of generalised 

 algebras, (viz., those the squares of whose units are 

 respectively negative unity and zero) the common origin 

 of which must still be marked on our intellectual map as 

 an unknown region. Well do I recollect how in its early 

 days we used to handle the method as a magician's page 

 might try to wield his master's wand, trembling as it were 

 between hope and fear, and hardly knowing whether to 

 trust our own results until they had been submitted to 

 the present and ever ready counsel of Sir W. R. Hamilton 

 himself. 



To fix our ideas, consider the measurement of a line, or 

 the reckoning of time, or the performance of any mathe- 

 matical operation. A line may be measured in one direc- 

 tion or in the opposite ; time may be reckoned forward or 

 backward ; an operation may be performed or be reversed, 

 it may be done or may be undone ; and if having once 

 reversed any of these processes we reverse it a second time, 

 we shall find that we have come back to the original 

 direction of measurement or reckoning, or to the original 

 kind of operation. 



Suppose, however, that at some stage of a calculation our 

 formulae indicate an alteration in the mode of measure- 

 ment such that if the alteration be repeated, a condition 

 of things, not the same as, but the reverse of the original, 

 will be produced. Or suppose that, at a certain stage, 

 our transformations indicate that time is to be reckoned 

 in some manner different from future or past, but still 

 in a way having definite algebraical connexion with time 



