xi.] PHILOSOPHY OF INDUCTIVE INFERENCE. 235 



senses must soon stop, whereas the mental powers of de- 

 ductive reasoning can proceed to an unlimited degree of ap- 

 proximation. Geometrical truths, then, are incapable of 

 verification ; and, if so, they cannot even be learnt by 

 observation. How can I have learnt by observation a pro- 

 position of which I cannot even prove the truth by obser- 

 vation, when I am in possession of it ? All that observa- 

 tion or empirical trial can do is to suggest propositions, of 

 which the truth may afterwards be proved deductively. 



If Viviani's story is to be believed, Galileo endeavoured 

 to satisfy himself about the area of the cycloid by cutting 

 out several large cycloids in pasteboard, and then compar- 

 ing the areas of the curve and the generating circle by 

 weighing them. In every trial the curve seemed to be 

 rather less than three times the circle, so that Galileo, we 

 are told, began to suspect that the ratio was not precisely 

 3 to i. It is quite clear, however, that no process of 

 weighing or measuring could ever prove truths like these, 

 and it remained for Torricelli to show what his master 

 Galileo had only guessed at. 1 



Much has been said about the peculiar certainty of 

 mathematical reasoning, but it is only certainty of deduc- 

 tive reasoning, and equal certainty attaches to all correct 

 logical deduction. If a triangle be right-angled, the 

 square on the hypothenuse will undoubtedly equal the 

 sum of the two squares on the other sides ; but I can 

 never be sure that a triangle is right-angled : so I can be 

 certain that nitric acid will not dissolve gold, provided I 

 know that the substances employed really correspond to 

 those on which I tried the experiment previously. Here 

 is like certainty of inference, and like doubt as to the 

 facts. 



Discrimination of Certainty and Probability. 



We can never recur too often to the truth that our 

 knowledge of the laws and future events of the external 

 world is only probable. The mind itself is quite capable 

 of possessing certain knowledge, and it is well to discri- 

 minate carefully between what we can and cannot know 



1 Life of Galileo, Society for the Diffusion of Useful Knowledge, 

 p. 102. 



