10 _____ _____ _____ _/ \_ _/ \_/ \_ / \ / / \ \ | x | | y | | m | \_ _/ \_ \_/ _/ \_____/ \_____/ \_____/ 11 _____ _____ _____ _/ \_/ \_/ \_ / / \ / \ \ | x | | y | | m | \_ \_/ \_/ _/ \_____/ \_____/ \_____/ 12 _____ _____ _/___ \_/ \_ / / x \ / \ \ | \___/| | m | \_ y \_/ _/ \_____/ \_____/ Figs. 1 and 4 give 13 _____ _____ _/ \_ _/ ___ \_ / \ / / y \ \ | x | | \___/ | \_ _/ \_ m _/ \_____/ \_____/ From this group (Figs. 5 to 13) we have, by disregarding m, to find the relation of x and y. On examination we find that Figs. 5, 10, 13 express the relation of entire mutual exclusion; that Figs. 6, 11 express partial inclusion and partial exclusion; that Fig. 7 expresses coincidence; that Figs. 8, 12 express entire inclusion of x in y; and that Fig. 9 expresses entire inclusion of y in x. pg182 We thus get five Biliteral Diagrams for x and y, viz. 14 _____ _____ _/ \_ _/ \_ / \ / \ | x | | y | \_ _/ \_ _/ \_____/ \_____/ 15 _____ _____ _/ \_/ \_ / / \ \ | x | | y | \_ \_/ _/ \_____/ \_____/ 16 _____ _/ \_ / \ | xy | \_ _/ \_____/ 17 _____ _/ ___ \_ / / x \ \ | \___/ | \_ y _/ \_____/ 18 _____ _/ ___ \_ / / y \ \ | \___/ | \_ x _/ \_____/ where the only Proposition, represented by them all, is "Some not-y are not-x," i.e. "Some persons, who are not gamblers, are not philosophers"----a result which Euler would hardly have regarded as a _valuable_ one, since he seems to have assumed that a Proposition of this form is _always_ true! (4) _Solution by Venn's Method of Diagrams._ The following Solution has been kindly supplied to me Mr. Venn himself. "The Minor Premiss declares that some of the constituents in my' must be saved: mark these constituents with a cross. _____ _/ + \_ _/___ ___\_ _/|##+#\_/ |\_ / \###/#\ _/ \ | + \|_#_|/ | \_ m \_/ y _/ \_____/ \_____/ The Major declares that all xm must be destroyed; erase it. Then, as some my' is to be saved, it must clearly be my'x'. That is, there must exist my'x'; or eliminating m, y'x'. In common phraseology,