$ Mr. Venn on the Diagrammatic and Mechanical 



The leading conception of this scheme is then simple enough; 

 but it involves some consideration in order to decide upon the 

 most effective and symmetrical plan of carrying it out. Up 

 to three terms, indeed, there is but little opening for any 

 difference; and as the departure from the familiar Eulerian 

 plan has to be made from the very first, we will examine 

 these simpler cases somewhat carefully. The diagram 



for two terms, then, is to be thus drawn : — ( x 0/ ^ n ^ ne 



common plan this would represent a. proposition, and is, indeed, 

 very commonly taken as illustrative of the proposition <l Some 

 X is Y."* With us it does not as yet represent a proposi- 

 tion at all, but only the framework into which propositions 

 can be fitted ; that is, it represents only the four combinations 

 indicated by the letter-compounds XY, XY, XY, XY. Xow 

 conceive that we have to reckon also with the presence, 

 and consequently with the absence, of Z. We just draw a 



third circle intersecting the two above, thus, 



and we have the eight compartments or classes which we 

 need. The subdivisions thus produced correspond precisely 

 with the letter-combinations. Quote one of these latter, and 

 the appropriate class-division is ready to meet it ; put a finger 

 on any compartment, and the letter indication is unambiguous. 

 Moreover both schemes, that of letters and that of spaces, 

 agree in being mutually exclusive and collectively exhaustive 

 in respect of all their elements. No one of the elements 

 trespasses upon the ground of any other ; and amongst them 

 they account for all possibilities. Either scheme, therefore, 

 may be taken as a fair representative of the other. 



Beyond three terms circles fail us, since we cannot draw a 

 fourth circle which shall intersect three others in the wav re- 

 quired. But there is no theoretic difficulty in carrying out 

 the scheme indefinitely. Of course any closed figure will do 

 as well as a circle, since all that we demand of it, in order 

 that it shall adequately represent the contents of a class, is 

 that it shall have an inside and an outside, so as to indicate 

 what does and what does not belong to the class. There is 

 nothing to prevent us from going on for ever thus drawing- 

 successive figures, doubling the consequent number of sub- 

 divisions. The only objection is, that since diagrams are pri- 



* It really takes, however, three common propositions to exhaust its 

 significance ; for the figure involves in addition the two statements u Some 

 X is not Y," and " Some Y is not X." 



