A :PIIQ:BI;EMS OF MODERN SCIENCE 



an investigation is to be classed, though always 

 roughly, as Science or not. 



It is not a practical problem to select the 

 branches of Mathematics to which I should refer 

 as constituting its scientific side in accordance 

 with the scheme of these lectures. The so-called 

 ' useless ' Mathematics merely means, at any point 

 of time, those portions of the logical development 

 of the subject which, for one of two reasons either 

 because they deal with mental conceptions not yet 

 adopted by the more practical physicist or engineer, 

 or because, while dealing with physical phenomena, 

 they are concerned with deductions which have 

 not been tested by practical experience in a 

 laboratory have not hitherto played any part in 

 the elucidation of anything which makes a direct 

 appeal to one of our senses. Such are the parts 

 of the subject which are usually referred to as an 

 Art, and they include a very large number of the 

 most progressive developments of the subject at 

 the present day, and of the most interesting and 

 fundamental problems which mathematicians are 

 endeavouring to solve. 



But we have to face the fact that many of the 

 most important branches of Pure Mathematics, 

 worked out in the first instance on some purely 

 aesthetic ground for their own interest in them- 

 selves, without any ulterior motive such as their 

 application to anything concrete, have often, per- 

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