342 PROF. W. STANLEY JEVONS ON 



XYZ, 



x\Z, 



XYz, 



xXz, 



XyZ, 



xyZ, 



Xyz, 



xyz, 



and these altogether constitute the universe, of which the 

 number is n. The problem is at once seen to be indeter- 

 minate in reality ; for there are eight classes, of which the 

 number would have to be determined, and there are only 

 six known quantities, namely, u, x, y, z, m, and n^ by which 

 to determine them. Accordingly we find that Mr. De 

 Morgan^s conclusons, though not absolutely erroneous, 

 have little or no meaning. From the premises he infers 

 that (m + n — y) or more X^s are Z^s. Now 



m + 7i-y:=: (XY) + (YZ) -Y 

 = (XYZ)-(^Y-^). 



Thus Mr. De Morgan represents the number of the whole 

 class, XZ, by a quantity indefinitely less than its own 

 part, XYZ. It is quite true that if the second side 

 (XYZ) — (^Y^") of this equation has value, there must be 

 at least this number of X"'s which are Z''s ; but as (xYz) 

 may exceed (XYZ) in any degree, this may give zero or a 

 negative result, while there is really a large number of 

 XZ^s. The true and complete expression for the number 

 of XZ^s is found as follows : — 



(XZ) = (XYZ) + (X2/Z) 



= (XYZ) + (XY^) + (XYZ) -f (.z-YZ) - (Y) + (XyZ) + (^Y^) 



= m + m' -\- n + n' —y + (X?/Z) + (xYz) . 



Among these seven quantities, only m, n, and y are definitely 

 given. The two m' and n are two indefinite quantities, 

 expressing the uncertainty in the number of XY^s and 

 YZ's, while there are two other unknown quantities, the 

 numbers of XyZ's and aiYs's arising in the course of the 

 problem. 



