i8 73 ] GEOMETRICAL REASONING 5 



Mr. Mill s example, as we have said, is Euclid I. 5, 

 which he undertakes to deduce from the original deductive 

 foundations. We have first (p. 241) 

 some preliminary remarks, which 

 afford a remarkably happy instance 

 of the way in which Mr. Mill is ac 

 customed to keep himself safe from 

 all opponents, by alternately sup 

 porting each of two contrary views 

 of a subject. &quot; First,&quot; says he, 

 speaking of the angles ABE, CBE ; 

 ACD, BCD, &quot;it could be perceived 

 intuitively that their differences were 

 the angles at the base.&quot; If this intuition is really a step 

 in the proof, then since intuition is just actual looking at 

 the figure, what becomes of the doctrine that the figure 

 is not essential, or of the still more fundamental doctrine, 

 that no general truth can flow from a single intuition ? ] 

 In this, however, Mr. Mill only falls foul of himself. A 

 more serious matter is, that when he sets about his 

 regular demonstration, he falls foul of the truths of 

 geometry. 



Having shown that AD = AE, Mr. Mill proceeds thus : 

 &quot; Both these pairs of straight lines &quot; [AC, AB : AD, AE] 

 &quot; have the property of equality ; which is a mark that, 

 if applied to each other, they will coincide. Coinciding 

 altogether means coinciding in every part, and of course 

 at their extremities, D, E, and B, C.&quot; Now, &quot; straight 

 lines, having their extremities coincident, coincide. BE 

 and CD have been brought within this formula by the 

 preceding induction ; they will, therefore, coincide.&quot; [!] 

 If Mr. Mill generalises this conclusion, I think he will 



1 This is no mere slip on Mr. Mill s part. To show that the angles at 

 the base are the differences of the angles in question, without appealing 

 to the figure, we must have a new axiom [proved, of course, by induc 

 tion I], viz. that if a side of a triangle be produced to any point, the line 

 joining that point with the opposite angle falls wholly without the 

 triangle. 



