268 THE PRINCIPLES OF SCIENCE. 



supposed act of induction, that it would succeed in all 

 other cases. Professor de Morgan has recorded a proposed 

 mode of trisecting the angle which could not be dis- 

 criminated by the senses from a true general solution, 

 except when it was applied to very obtuse angles. 5 In 

 all such cases, it has always turned out either that the 

 angle was not trisected at all, or that only certain 

 particular angles could be thus trisected. They were 

 misled by some apparent or special coincidence, and only 

 deductive proof could establish the truth and generality 

 of the result. In this case, deductive proof shows that the 

 problem, as attempted, is impossible, and that angles 

 generally cannot be trisected by common geometrical 

 methods. 



Geometrical Reasoning. 



This view of the matter is strongly supported by the 

 further consideration of geometrical reasoning. No skill 

 and care could ever enable us to verify absolutely any one 

 geometrical pro-position. Rousseau, in his Emile* tells us 

 that we should teach a child geometry by causing him to 

 measure and compare figures by superposition. While a 

 child was yet incapable of general reasoning, this would 

 doubtless be an instructive exercise ; but it never could 

 teach geometry, nor prove the truth of any one proposition. 

 All our figures are rude approximations, and they may 

 happen to seem unequal when they should be equal, 

 and equal when they should be unequal. Moreover, 

 figures may from chance be equal in case after case, and 

 yet there may be no general reason why they should be 

 so. The results of deductive geometrical reasoning are 



' Budget of Paradoxes,' p. 257. 



* I2mo. Amsterdam, 1762, vol. i. p. 401. 



