xi.] PHILOSOPHY OF INDUCTIVE INFERENCE. 233 



in semicircles are right angles ; but no number of instances, 

 and no possible accuracy of measurement would really 

 establish the truth of that general law. Availing ourselves 

 of the suggestion furnished by the figures, we can only 

 investigate deductively the consequences which flow from 

 the definition of a circle, until we discover among them the 

 property of containing right angles. Persons have thought 

 that they had discovered a method of trisecting angles by 

 plane geometrical construction, because a certain complex 

 arrangement of lines and circles had appeared to trisect an 

 angle in every case tried by them, and they inferred, by a 

 supposed act of induction, that it would succeed in all 

 other cases. De Morgan has recorded a proposed mode of 

 trisecting the angle which could not be discriminated by 

 the senses from a true general solution, except when it was 

 applied to very obtuse angles. 1 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. The trisectors were misled by some apparent or 

 special coincidence, and only deductive proof could es- 

 tablish the truth and generality of the result. In this par- 

 ticular case, deductive proof shows that the problem 

 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 proposition. Rousseau, in his Umile, 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 



1 Budget of Paradoxes, p. 257. 



