An Universal Affirmative may also state a relation between two terms whose denotation is co-extensive. A definition always does this, as _Man is a rational animal_; and this, of course, we cannot represent by two distinct circles, but at best by one with a thick circumference, to suggest that two coincide, thus: [Illustration: FIG. 2.] The Particular Affirmative Proposition may be represented in several ways. In the first place, bearing in mind that 'Some' means 'some at least, it may be all,' an I. proposition may be represented by Figs. 1 and 2; for it is true that _Some horned animals ruminate_, and that _Some men are rational_. Secondly, there is the case in which the 'Some things' of which a predication is made are, in fact, not all; whilst the predicate, though not given as distributed, yet might be so given if we wished to state the whole truth; as if we say _Some men are Chinese_. This case is also represented by Fig. 1, the outside circle representing 'Men,' and the inside one 'Chinese.' Thirdly, the predicate may appertain to some only of the subject, but to a great many other things, as in _Some horned beasts are domestic_; for it is true that some are not, and that certain other kinds of animals are, domestic. This case, therefore, must be illustrated by overlapping circles, thus: [Illustration: FIG. 3.] The Universal Negative is sufficiently represented by a single Fig. (4): two circles mutually exclusive, thus: [Illustration: FIG. 4.] That is, _No horned beasts are carnivorous_. Lastly, the Particular Negative may be represented by any of the Figs. 1, 3, and 4; for it is true that _Some ruminants are not hollow-horned_, that _Some horned animals are not domestic_, and that _Some horned beasts are not carnivorous_. Besides their use in illustrating the denotative force of propositions, these circles may be employed to verify the results of Obversion, Conversion, and the secondary modes of Immediate Inference. Thus the Obverse of A. is clear enough on glancing at Figs. 1 and 2; for if we agree that whatever term's denotation is represented by a given circle, the denotation of the contradictory term shall be represented by the space outside that circle; then if it is true that _All hollow horned animals are ruminants_, it is at the same time true that _No hollow-horned animals are not-ruminants_; since none of the hollow-horned are found outside the palisade that encloses the ruminants. The Obverse of I., E. or O. may be verified in a similar manner. As to the Converse, a Definition is of course susceptible of Simple Conversion, and this is shown by Fig. 2: 'Men are rational animals' and 'Rational animals are men.' But any other A. proposition is presumably convertible only by limitation, and this is shown by Fig. 1; where _All hollow-horned animals are ruminants_, but we can only say that _Some ruminants are hollow-horned_.