364 



MEMOIRS OF THE AMERICAN ACADEMY. ' 



same way as those given abov 



In 



ds, all such algebras are complications 



and modifications of the algebra of (156). It is very likely that this is true of all 

 algebras whatever. The algebra of (156), which is of such a fundamental character 

 in reference to pure algebra and our logical notation, has been shown by Professor 



Peirce to be the algebra of Hamilton's quaterni 



In fact, if we put 



1 



i+l. 



i 



n 



pji 



(Vi 



aH + ab J)/-f (y/1 



a l b 



ah S)k 



vtt 



b*Jl 



J 



•r 



by/1 



<? J i -f ( 



ac 



«Vi 



#Vi 



c 2 



(n/i 



a 2 c-{-a\/l 



#yi 



- *V)j 



ac 



« 2 y/l 



#yi 



*+(va 



a 2 c-\-a^l 



b 2 s/l 



(?)j)k+b \/l 



<?JL 



h' 



UM-\-{yJ\ 



a 2 sj\ 



b 2 c-\-a\jl—c 2 -\-{a\Jl 



b 2 c 



flVl 



*)->)/ 



(fl 



«Vi 



Pe + a)/l 



<? 



(as/1 



b 2 c 



a 2 s/l 



<?)J)k 



bcJl 



where a,b,c, are scalars, then !/,/,# are the four fundamental factors of quaternions, 



the multiplication-table of which is as follows : 



1 



t 



3 



k f 



I 



a 



J 



•/ 



k' 



It 



part of my present purpose to consider the bearing upon the philosophy 



of space of this occurrence, in pure logic, of the algebra which expresses 



all the 



properties of space ; but it is proper to point out that one method of 



this notation would b 



transform the 



given 



Hamilton's quaternions (after representing them 



as 



. expressions 

 parated into 



the form of 

 entary rela- 



and then to make use of geometrical reasoning. The following formula) 



assist this process. Take the quaterni 



9 



+ yj + zk + wl 



where i W and * are scalars. The conditions of g being a scalar, vector, etc. (th 



