52 Mr Venn, On geometrical diagrams for the [Dec G, 



culty, but merely state the fact that general satisfaction being felt 

 with the Eulerian plan no serious attempts were made to modify 

 it. Indeed, except on the part of those who wrote and thought 

 under the influence of Boole, directly or indirectly, it was scarcely 

 likely that need should be felt for any more generalized scheme. 



The essential characteristic of the Eulerian plan being that of 

 representing directly and immediately the inclusion and exclusion 

 of classes, it is clear that the employment of circles as distinguished 

 from any other closed figures is a mere accident. Nor have 

 circles in fact always been employed. Thus Ploucquet, — whose 

 system however, as he himself pointed out, in contrast with that 

 of Lambert, is essentially symbolic and not diagrammatic, — has 

 made use of squares. Kant (Logik, I. § 21) and De Morgan [Formal 

 Logic, p. 9) have introduced or suggested both a square and a 

 circle into the same diagram, one standing for subject and the 

 other for predicate ; with the view of distinguishing between these. 

 Mr R. G. Latham (Logic, p. 88) and Mr Leechman (Logic, p. 66) 

 have a square, circle, and triangle all in one figure for the same 

 purpose, presumably, of distinguishing between the three terms in 

 the syllogism. Bolzano again, in one of the examples above ad- 

 duced, has had resort to parallelograms : to which indeed, or to 

 ellipses or to some such figure, it is evident he must have appealed 

 if he wished to set before us the outcome of four class terms. 

 But to regard these as constituting distinct schemes of notation 

 would be merely idle. They all do exactly the same thing, viz. 

 they aim at so arranging two (or more) closed figures that these 

 shall represent the mutual relation of inclusion and exclusion of 

 the various classes denoted by the terms we employ. 



There is one modification of this plan which deserves passing 

 notice both on its own account and because it has been so mis- 

 judged by Hamilton. It is that of Maass. In order to understand 

 it we must recall one essential defect of the customary plan. 

 Representing as this does the final outcome of the class relation, 

 it is clear that every fresh proposition demands a diagram new 

 from the beginning. If we have drawn a scheme for "All A is B" 

 we must abandon it and draw another for "All B is A ". Seeing 

 this, apparently, Maass took two fixed lines enclosing an angle, 

 and regarded the third line which combined with them to form 

 the necessary closed figure, as movable. Hence only one line had 

 to be altered in order to meet the new information contained in 

 such a second proposition. (Logik, p. 294.) 



Thus let AB and AG be the fixed lines ; and the triangle ADE 

 represent the class X, and AFG represent the class Y. If FG 

 remain where it is we have "All X is Y", whilst in order to 

 represent "All X is all F" we have only to conceive it transferred 



