RESULTS AND LIMITS OF SCIENTIFIC METHOD. 451 



no precise explanation can yet be given ? In every such 

 fact there is a possible opening for new discoveries, and it 

 can only be the fault of the investigator's mind if he can 

 look around him and find no scope for the exercise of his 

 faculties. 



The Infinite Incompleteness of the Mathematical 

 Sciences. 



There is one privilege which a certain amount of know- 

 ledge should confer ; it is that of becoming aware of the 

 indefinite weakness of our powers compared with the tasks 

 which they might undertake if stronger. To the poor 

 savage who cannot count twenty, the arithmetical accom- 

 plishments of the ordinary schoolboy are miraculously 

 great in comparison. The schoolboy cannot comprehend 

 the almost infinitely greater powers of the student, who 

 has acquired facility with algebraic processes. The student 

 can but look with feelings of surprise and reverence at 

 the powers of a Newton or Laplace. But the question at 

 once suggests itself, Do the powers of the highest human 

 intellect bear any moderate ratio to the things which are 

 to be understood and calculated \ How many further 

 steps must we take in the rise of mental ability and the 

 extension of mathematical method before we begin to 

 exhaust the knowable \ 



I am inclined to find fault with mathematical writers 

 because they often exult in what they can accomplish, but 

 omit to point out that what they do is but an indefinitely, 

 nay an infinitely, small part of what might be done. They 

 exhibit a general inclination, with few exceptions, not to 

 do so much as mention the existence of problems of an 

 impracticable character. This may be excusable so far as 

 the immediate practical result of their researches is in 

 question, but the custom has the effect of misleading the 



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