DEFINITION OF QUATERNIONS BY INDEPENDENT POSTULATES 261 



€1 €2 ^3 ^4 



€1 Oj 62 ^3 ^4 



^2 ^2 " ^i ^4 ^3 



^3 ^3 ^4 ^l ^2 



^4 ^4 ^3 ^2 ^i 



which, apart from symbolism, is the same as the table of § i. 

 § 4. On THE Independence of the Postulates 



If Z) is a domain admitting addition and subtraction, postulates II 

 and III are redundant. 



Aside from this case postulates I-IV are independent, as shown by 

 the following systems : 



(I) Elements o, [±i, o, o, o], [o, ±i, o, o], [o, o, dzi, o], [o, o, o, ±1]. 

 Postulate II holds, since o is an element. 



Postulate III holds, since [ + 1, o, o, o]+[— i, o, o, o] = o; and 

 [o, +i,o,o] + [o, -i,o,o] = o;[o,o, + 1, o] + [o, 0,-1, o]=o; [0,0,0, +1] 

 + [0, o, o, — i]=o. 



Postulate IV holds, since the product of any two elements gives an 

 element. 



Postulate V holds, since no mention is made of the t's. 



Postulate I does not hold, since [i, o, o, o]+[o, i, o, o] is not an 

 element. 



(II) D is the domain of positive integers. 

 Postulate I is true, since addition is possible. 



Postulate III is dependent (conditionally) upon postulate II. There- 

 fore it is not considered here. 



Postulate IV holds for those values of ffj and' &»• which make the pi 

 positive. The values which make the Pi negative are excluded by the 

 statement of postulate IV and the definition of D. 



No mention is made of the t's. Therefore postulate V need not be 

 considered. 



Postulate II does not hold, since zero is not an element. 



(III) Set (II) with zero added. 



