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XXXII. The Logical Spectrum. 
By A. Macrariane, I.A., D.Sc., RS E.* 
ie my work on the Algebra of Logic I investigated the 
meaning of Boole’s analysis by the aid of diagrams. In 
this paper I propose to make a further use of the same method. 
When the class of things considered (that is, the universe) 
is subdivided by not more than three qualities, a modified use 
of Huler’s circles is sufficient. Let a square represent the 
universe of objects considered, and let a circle separate those 
having a particular quality. Diagram 1 represents the two 
classes formed by the presence and absence of the mark a ; 
and diagram 2 represents each of the four classes ab, ab’, a/b, 
Diagram 1. Diagram 2. 
a’b’ formed by the presence or absence of each of the two 
marks a and 6. Here the dash is used to denote non, as a’ for 
non-a. It is a contraction for the complementary expression 
1—a. The diagram 38 is quite general, because it represents 
each of the eight classes abc, abe’, ab/c, ab’, a’be, abe’, a’b’c, 
a'b’c’ formed by the three marks a, 6, and c. But if we 
Diagram 3. Diagram 4. 
attempt by means of four circles to represent each one of the 
sixteen possible classes formed by four marks a, b, c, d, we 
shall find that it is impossible. For example, in diagram 4 
* Communicated by the Author. 
