else of A, or of C, there is nothing to hinder a syllogism, it will not however appertain to B from the assumptions. Nor when C is predicated of something else, and that of another, and this last of a third, if none of these belong to B, neither thus will there be a syllogism with reference to B, since in short we say that there never will be a syllogism of one thing in respect of another unless a certain middle is assumed, which refers in some way to each extreme in predication. For a syllogism is simply from premises, but that which pertains to this in relation to that, is from premises belonging to this in relation to that, but it is impossible to assume a premise relating to B, if we neither affirm nor deny any thing of it, or again of A in relation to B, if we assume nothing common, but affirm or deny certain peculiarities of each. Hence a certain middle of both must be taken, which unites the predications, if there shall be a syllogism of one in relation to the other; now if it is necessary to assume something common to both, this happens in a three-fold manner, (since we either predicate A of C, and C of B, or C of both or both of C,) but these are the before-mentioned figures—it is evident that every syllogism is necessarily produced by some one of these figures, for there is the same reasoning, if A be connected with B, even through many media, for the figure in many media will be the same.
Wherefore that all ostensive syllogisms are perfected by the above-named figures is clear, also that those per impossibile (are so perfected) will appear from these, for all syllogisms concluding per impossibile collect the false, but they prove by hypothesis the original proposition, when contradiction being admitted some impossibility results, as for instance that the diameter of a square is incommensurate with the side, because, a common measure being given, the odd would be equal to the even.
