We may first shew how Lambert’s lines may be used in such a manner as to be precisely analogous to Euler’s circles. 164 Thus, the four fundamental propositions may be represented as follows:—
These diagrams occupy less space than Euler’s circles. But they seem also to be less intuitively clear and less suggestive. The different cases too are less markedly distinct from one another. It is probable that one would in consequence be more liable to error in employing them.
The different cases may, however, be combined by the use of dotted lines so as to yield but a single diagram for each proposition much more satisfactorily than in Euler’s scheme. Thus, All S is P may be represented by the diagram
where the dotted line indicates that we are uncertain as to whether there is or is not any P which is not S. We obviously get two cases according as we strike out the dots or fill them in, and these are the two cases previously shewn to be compatible with an A proposition.
Again, Some S is P may be represented by the diagram
and here we get the four cases previously given for an I 165 proposition by (a) filling in the dots to the left and striking out those to the right, (b) filling in all the dots, (c) striking them all out, (d) filling in those to the right and striking out those to the left.
Two complete schemes of diagrams may be constructed on this plan, in one of which no part of any S line, and in the other no part of any P line, is dotted.[171] These two schemes are given below to the left and right respectively of the propositional forms themselves.