They wrongly believe that the mere subordinate reference of the upper or under position of the lines is the principal point of view. But Lambert's notation is neither a very easy nor a sure way of representation. The notation by triangles adopted by Maass is not so convenient as that by circles.

Gergonne symbolises the relations of circles by simple signs-the identity of two spheres by I, the complete separation by H, the crossing by x, the comprehension of the sphere of the subject in that of the predicate by C, and, lastly, the comprehension of the sphere of the predicate in that of the subject by ɔ. By the use of these signs the representation attains brevity and elegance, but loses immediate intuitiveness.

[Mansel objects to the use of any sensible representations whatsoever. He thinks that to represent the relation of terms in a syllogism by that of figures in a diagram is to lose sight of the distinctive mark of a concept,-that it cannot be presented. The diagrams of Geometry, he says, furnish no precedent, for they illustrate the matter, not the form, of thought. This last statement is scarcely correct.

Hamilton employs, in his Lectures on Logic, the circle notation of Euler, and also a modification of Lambert's linear method. The notation (linear) which he afterwards adopted is very intricate, and while free from the objection that it confounds logical with mathematical extension, does not intuitively represent the logical relations.2

For a history and criticism of various methods of logical notation, cf. Hamilton's Lectures on Logic, i. 256 and ii. 460 ff.] $86. By Conversion follows

I. From the particular affirmative categorical judgment (of the form i): Some S are P, The particular affirmative judgment (also of the form i): At least some P are S.

1 Essai de Dialectique rationnelle in the Annales des Mathématiques, tom. vii. 189-228, 1816-17.

[2 Cf. Lect. on Logic, ii. Appendix, p. 469 ff., and Discussions, pp. 657-661.]

And from the particular conditional judgment:
If A is, в sometimes is,

The particular conditional follows: Sometimes
at least if в is, a is.

The proof results from the comparison of the spheres. The given categorical judgment: Some S are P, when the predicate P belongs only to some S, presupposes two relations of spheres, which are denoted by the Schema:

i, 1.

S

P

i, 2.

Р

But since the possibility is not excluded, that the same predicate P belongs also to other S, the two following relations of the spheres also exist:

i, 3.

SP

i, 4.

S P

These Schemata are to be taken in the same sense as in § 85. Now in i, 1 and i, 3 some P only are S; and in i, 2 and i, 4 all, and therefore at least some P are S. But this is what was to be proved.

In the corresponding hypothetical judgments the relations of the spheres are the same and the result equivalent.

The Conversion of the particular affirmative and of the particular conditional judgment is therefore a Conversio simplex. For both the given judgment, and the

X

judgment arising from the conversion, take the form of the particular affirmative (i).

The Modality of the given judgment and of its converse is the same.

Examples of i, 1 are: Some parallelograms are regular figures;—of i, 2: Some parallelograms are squares ;—of i, 3 : Some parallelograms are divided by a diagonal into two coincident triangles; and of i, 4: Some parallelograms are divided by both diagonals into two coincident triangles. The relation of spheres i, 1 admits of many other modifications. If two spheres are of unequal size, it can happen that most S are P, and relatively very few P are S, or a few S are P, and most P are S. Although the number of S which are P, and of P which are S, is in itself necessarily the same, yet the relation of the sum total of individuals is a different one in each of the two spheres. For example, some, and relatively not a few, planets belonging to our system are heavenly bodies which may be seen by us with the naked eye; but only a very few of the heavenly bodies visible to the naked eye are planets of our system. This conversion is not, therefore, conversio simplex, in the stricter sense that the quantity remains the same in each reference. It is so only in the more general sense, that the judgment remains a particular one, and does not pass over to any other of the four classes of judgments designated by a, e, i, o.

§ 87. By Conversion follows

III. From the universal negative categorical judgment (of the form e): No S is P,

The universal negative judgment (also of the form e): No P is S.

And from the universal negative hypothetical

judgment: If A is, в never is,

The similarly universal negative hypothetical judgment: If в is, a never is.

The validity of these rules may be directly proved by the comparison of spheres. The Schema of the universal negative categorical judgment is the complete separation of the spheres:

i.e. The action or quality which the notion of the predicate P denotes, is to be found in no object which the subject-notion S denotes, and, if it really exists, only in other notions. Hence the judgment: No objects in which the predicate P is found, and which may therefore be denoted by the notion P made substantive, are S. And this is what was to be proved.

The same may be proved indirectly. For if any one P were S, then (according to § 86) some one S would be P; but this is false according to the axiom of Contradiction (§ 77), for it is opposed contradictorily (§ 72) to the given judgment: no S is P. Hence the assertion is false, that any one P is S, and it is true that no P is S; which was to be proved.

The corresponding hypothetical judgment presupposes the analogous relation of spheres:

B

i.e. The case denoted by в is never found where A is present. Whenever в happens, it takes place under other conditions. The case в does not occur together

with the case A; and the case A does not occur with the case B. If в is, A never is; which was to be proved.

The indirect proof may be led here as well as in the universal negative categorical judgment. For if it once happened that when B is, A is also, then (according to § 86) the converse would be true that, once, when a is, B is; this would contradict the given presupposition, that when a is, в never is, and is therefore false. Hence it is false, that when в is, A is once; and the proposition is true: when в is, A never is; which was to be proved.

The converse of the universal negative judgment is therefore not accompanied with any change of quantity, and is throughout simple conversion.

The rule also holds good without exception that Modality remains unchanged in the conversion. If it is apodictically certain that no S is P, the same kind of certainty passes over to the judgment, that no P is S. If that is only probable, or is true only perhaps, and the assertion remains possible, that, perhaps at least some one S is P, then (according to § 86) there is the same possibility for the assertion that, perhaps at least, some one P is S. It does not follow: no P is S; but only: Probably or perhaps, no P is S.

The following are examples of the conversion of the universal affirmative categorical judgment. If the judgment be given as true: No innocent person is unhappy, it follows with equal truth: No unhappy person is innocent. If the proposition be proved: No equilateral triangle is unequiangular, it follows without further mathematical demonstration, by logical conversion: No unequiangular triangle is equilateral (every un