8 Mr. Venn on the Diagrammatic and Mechanical 



way or other have set before him all those thirty-two com- 

 pounds of which XYZWV is a sample ; that is, he must 

 contemplate the array produced by 160 letters. In compa- 

 rison with most ways of doing that, the sketching out of such 

 a figure is a pleasure, besides being for more expeditious ; for, 

 with a very little practice, any of the diagrams here offered 

 might be drawn in but a minute fraction of the time requisite 

 to write down all the letter-compounds. I can only say for 

 myself that, after having for various purposes worked through 

 hundreds of logical examples, I generally resort to diagrams 

 of this description ; it not only avoids a deal of unpleasant 

 drudgery, but is also a valuable security against error and 

 oversight. The way in which this last advantage is secured 

 will be best seen presently, when we come to inquire how 

 these diagrams are to be used to represent propositions as dis- 

 tinguished from mere terms or classes. 



Beyond five terms it hardly seems as if diagrams offered 

 much substantial help ; but then we do not often have occa- 

 sion to meddle with problems of a purely logical kind which 

 involve such intricacies. If we did have such occasion, viz. 

 to visualize the sixty-four compounds yielded by the six terms 

 X, Y, Z, W, Y, U, the best plan would probably be to take 

 two of the above five-term figures — one for the U part and the 

 other for the not-U part of all the other combinations. This 

 would yield the desired distinctive sixty -four subdivisions, but, 

 of course, it to some extent loses the advantage of the coup 

 oVozil afforded by a single figure. 



We have endeavoured above to employ only symmetrical 

 figures, such as should not merely be an aid to the sense of 

 sight, but should also be to some extent elegant in themselves. 

 But for merely theoretical purposes the rule of formation 

 would be very simple. It would merely be to begin by draw- 

 ing any closed figure, and then proceed to draw others, subject 

 to the one condition that each is to intersect once and once 

 only all the existing subdivisions produced by those which 

 had gone before. Proceeding thus we should naturally select 

 circles as the simplest figures, so long as they would answer 

 our purpose ; that would be, up to three terms inclusive. The 

 two successive modifications, aiming always at simplicity of 

 figure, would then be naturally such as the following (the 

 fifth figure is marked, for clearness, by a dotted line) : — 



