In their objective sense, the three figures of the syllogism declare that everything rational is manifested as a triple syllogism; that is to say, each one of the members takes in turn the place of the extremes, as well as of the mean which reconciles them. Such, for example, is the case with the three branches of philosophy; the Logical Idea, Nature, and Mind. As we first see them, Nature is the middle term which links the others together. Nature, the totality immediately before us, unfolds itself into the two extremes of the Logical Idea and Mind. But Mind is Mind only when it is mediated through nature. Then, in the second place, Mind, which we know as the principle of individuality, or as the actualising principle, is the mean; and Nature and the Logical Idea are the extremes. It is Mind which cognises the Logical Idea in Nature and which thus raises Nature to its essence. In the third place again the Logical Idea itself becomes the mean: it is the absolute substance both of mind and of nature, the universal and all-pervading principle. These are the members of the Absolute Syllogism. 188.] In the round by which each constituent function assumes successively the place of mean and of the two extremes, their specific difference from each other has been superseded. In this form, where there is no distinction between its constituent elements, the syllogism at first has for its connective link equality, or the external identity of understanding. This is the Quantitative or Mathematical Syllogism: if two things are equal to a third, they are equal to one another. * * * * * Everybody knows that this Quantitative syllogism appears as a mathematical axiom, which like other axioms is said to be a principle that does not admit of proof, and which indeed being self-evident does not require such proof. These mathematical axioms however are really nothing but logical propositions, which, so far as they enunciate definite and particular thoughts, are deducible from the universal and self-characterising thought. To deduce them, is to give their proof. That is true of the Quantitative syllogism, to which mathematics gives the rank of an axiom. It is really the proximate result of the qualitative or immediate syllogism. Finally, the Quantitative syllogism is the syllogism in utter formlessness. The difference between the terms which is required by the notion is suspended. Extraneous circumstances alone can decide what propositions are to be premisses here: and therefore in applying this syllogism we make a pre-supposition of what has been elsewhere proved and established.