360 



MEMOIRS OF THE AMERICAN ACADEMY. 



which is (it + l) n + l where tl is the number of correlates which the conjugative has. 

 At present, I shall consider only the simple relatives. 



The existence of an elementary relation supposes the existence of mutually exclu- 

 sive pairs of classes. The first members of those pairs have something in common 

 which discriminates them from the second members, and may therefore be united in 

 one class, while the second members are united into a second class. Thus pupil is 

 not an elementary relative unless there Is an absolute distinction between those who 

 teach and those who are taught. We have, therefore, two general absolute terms 

 which are mutually exclusive, " body of teachers in a school," and " body of pupils in 

 a school." These terms are general because it remains undetermined what school is 

 referred to. I shall call the two mutually exclusive absolute terms which any system 

 of elementary relatives supposes, the universal extremes of that system. There are 

 certain characters in respect to the possession of which both members of any one of 

 the pairs between which there is a certain elementary relation agree. Thus, the 

 body of teachers and the body of pupils in any school agree in respect to the country 

 and age in which they live, etc., etc. Such characters I term scalar characters for 

 the system of elementary relatives to which they are so related ; and the relatives 

 written with a comma which signify the possession of such characters, I term scalars 

 for the system. Thus, supposing French teachers have only French pupils and vice 

 versa, the relative 



will be a scalar for the system " colleague : teacher : pupil : schoolmate." Tf 

 mentary relative for which s, is a scalar, 



(154.) 



s,r = rs 



Let c, t,p, s, denote the four elementary relatives of any system; such as colleague, 

 teacher, pupil, schoolmate; and let a,,b,,c, ,d,, be scalars for this system. Then 

 any relative which is capable of expression in the form 



a,c -|- b,£ + c,p + d,s . 



* 



I shall call a logical quaternion. Let such relatives be denoted by q, q\ q", etc. It is 

 plain, then, from what has been said, that any relative may be regarded as resolvable 

 into a logical sum of logical quaternions. 



The multiplication of elementary relatives of the same system follows a very simple 

 law. For if u and v be the two universal extremes of the system c,t,p,s, we may 



write 



c = u:u t = u:v p = v:u s = v:v 



