THEORY OF GEOMETRICAL REASONING 



Mr. Mill s Theory of Geometrical Reasoning Mathematically Tested. 1 



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. I. 5, 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 &quot; against 

 the form of the geometrical syllogism, as exhibited by 

 Mr. Mill, the logician will have no objections to allege &quot; 

 (Mansel s Aldrich, 3rd 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. 



1 Communicated by Professor Tait to the Royal Society of Edinburgh 

 on February i, 1869, and published in its Proceedings, vol. vi. pp. 477- 

 483- 



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