It follows from what has been said that every concept, just because it is abstract and incapable of presentation in perception, and is therefore not a completely determined idea, has what is called extension or sphere, even in the case in which only one real object exists that corresponds to it. Now we always find that the sphere of one concept has something in common with the sphere of other concepts. That is to say, part of what is thought under one concept is the same as what is thought under other concepts; and conversely, part of what is thought under these concepts is the same as what is thought under the first; although, if they are really different concepts, each of them, or at least one of them, contains something which the other does not contain; this is the relation in which every subject stands to its predicate. The recognition of this relation is called judgment. The representation of these spheres by means of figures in space, is an exceedingly happy idea. It first occurred to Gottfried Plouquet, who used squares for the purpose. Lambert, although later than him, used only lines, which he placed under each other. Euler carried out the idea completely with circles. Upon what this complete analogy between the relations of concepts, and those of figures in space, ultimately rests, I am unable to say. It is, however, a very fortunate circumstance for logic that all the relations of concepts, according to their possibility, _i.e._, _a priori_, may be made plain in perception by the use of such figures, in the following way:— (1.) The spheres of two concepts coincide: for example the concept of necessity and the concept of following from given grounds, in the same way the concepts of _Ruminantia_ and _Bisulca_ (ruminating and cloven-hoofed animals), also those of vertebrate and red-blooded animals (although there might be some doubt about this on account of the annelida): they are convertible concepts. Such concepts are represented by a single circle which stands for either of them. (2.) The sphere of one concept includes that of the other. [Illustration: Category "horse" within category "animal".] (3.) A sphere includes two or more spheres which exclude each other and fill it. [Illustration: Circle divided into thirds "right", "acute", and "obtuse".] (4.) Two spheres include each a part of the other. [Illustration: Two overlapping circles, one "flower" and one "red".] (5.) Two spheres lie in a third, but do not fill it. [Illustration: A large circle, "matter", within which are two other circles, "water" and "earth".] This last case applies to all concepts whose spheres have nothing immediately in common, for there is always a third sphere, often a much wider one, which includes both.