Mr Mill's Theory of Geometrical Reasoning Mathematically Tested

Abstract

An amusing and instructive example of the way in which logicians are accustomed to dogmatise upon the theory of sciences that they do not understand, is afforded by Mr Mill's explanation of the nature of geometrical reasoning.Those who remember that Mr Mill assures Dr Whewell that he has conscientiously studied geometry (Logic, 7th ed. I. 270), will probably find some difficulty in believing that the demonstration of Euc. I5, which Mr Mill offers as an illustration of the justice of his theory of geometrical reasoning, depends on the axiom, that triangles, having two sides equal each to each, are equal in all respects. Such, nevertheless, is the case; and when one sees this absurdity pass unmodified from edition to edition of Mr Mill's Logic, and when even Mansel, Mr Mill's watchful enemy, tells us that “against the form of the geometrical syllogism, as exhibited by Mr Mill, the logician will have no objections to allege” (Mansel's Aldrich, 3d ed., p. 255), one cannot but think that logic would make more progress if logicians would give a little more attention to the processes they profess to explain.