270 THE PRINCIPLES OF SCIENCE. 



nearer than i part in 540. We might imagine measure 

 ments so accurately executed as to give us eight or ten 

 places correctly. But the power of the hands and senses 

 must soon stop, whereas the mental powers of deductive 

 reasoning can proceed to an unlimited degree of approxi 

 mation. Geometrical truths, then, are incapable of verifi 

 cation ; and, if so, they cannot even be learnt by observa 

 tion. How can I have learnt by observation a proposition 

 of which I cannot even prove the truth by observation, 

 when I am in possession of it \ All that observation or 

 empirical trial can do is to suggest propositions, of which 

 the truth may afterwards be proved deductively. By 

 drawing a number of right-angled triangles on paper, 

 with squares upon their sides, and cutting out and 

 weighing these squares very accurately, I might have 

 reason to suspect the existence of the relation of equality 

 proved in Euclid s 47th Proposition ; but no process of 

 weighing or measuring could ever prove it, nor could it 

 ever assure me that the like degree of approximation 

 would exist in untried cases. 



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. 



