1873] GEOMETRICAL REASONING 7 



There are only two courses open to Mr. Mill either to 

 confess that the attempt to square geometry with a 

 preconceived theory has forced him into a grossly erro 

 neous demonstration, or to invent a new formula, viz. 

 that if in two plane figures any number of consecutive 

 sides and angles taken one by one may be made to coin 

 cide, they may also be made to coincide as rigidly con 

 nected wholes. But, then, Mr. Mill must maintain that 

 the man who reads Euc. I. 4 for the first time does not at 

 once conclude the general truth of this formula from the 

 one figure before him, but either brings the formula with 

 him to the proof as a result of previous induction, or 

 requires to pause in the proof, and satisfy himself of 

 the truth of the formula by a comparison of a series of 

 figures. 



It is easily shown, by the same species of analysis as 

 we have adopted here, that wherever a real step is made 

 in geometry we must either use the figure or introduce a 

 new general axiom [not of course in mere converses, as 

 Euc. I. 19, I. 25]. All geometrical construction is in the 

 last resort a means of making clear to the eye complicated 

 relations of figures. 



Now, if we can at once and with certainty conclude 

 from the one case figured in the diagram to the general 

 case if, that is, axioms are proved not by induction, 

 but by intuition, and are necessarily true there is no 

 difficulty about geometrical reasoning ; but if each new 

 axiom is gained by a new induction (and that, on Mr. 

 Mill s showing, an &quot; inductio per enumerationem simpli- 

 cem &quot;), we get a difficulty which Mr. Mill curiously enables 

 us to state in his own words (i. p. 301) &quot; If it were 

 necessary,&quot; in adding a second step to an argument, 

 &quot; to assume some other axiom, the argument would no 

 doubt be weakened.&quot; But, says Mr. Mill, it is the same 

 axiom which is repeated at each step. If this were not 

 so, &quot; the deductions of pure mathematics could hardly 

 fail to be among the most uncertain of argumentative 



