THE STUDY OF MATHEMATICS 69 



the terrible sense of impotence, of weakness, of exile amid 

 hostile powers, which is too apt to result from acknow- 

 ledging the ail-but omnipotence of alien forces. To 

 reconcile us, by the exhibition of its awful beauty, to the 

 reign of Fate which is merely the literary personifica- 

 tion of these forces is the task of tragedy. But mathe- 

 matics takes us still further from what is human, into the 

 region of absolute necessity, to which not only the actual 

 world, but every possible world, must conform ; and 

 even here it builds a habitation, or rather finds a habita- 

 tion eternally standing, where our ideals are fully satisfied 

 and our best hopes are not thwarted. It is only when we 

 thoroughly understand the entire independence of our- 

 selves, which belongs to this world that reason finds, that 

 we can adequately realise the profound importance of its 

 beauty. 



Not only is mathematics independent of us and our 

 thoughts, but in another sense we and the whole universe 

 of existing things are independent of mathematics. The 

 apprehension of this purely ideal character is indispens- 

 able, if we are to understand rightly the place of 

 mathematics as one among the arts. It was formerly sup- 

 posed that pure reason could decide, in some respects, as 

 to the nature of the actual world : geometry, at least, was 

 thought to deal with the space in which we live. But we 

 now know that pure mathematics can never pronounce 

 upon questions of actual existence : the world of reason, 

 in a sense, controls the world of fact, but it is not at any 

 point creative of fact, and in the application of its results 

 to the world in time and space, its certainty and precision 

 are lost among approximations and working hypotheses. 

 The objects considered by mathematicians have, in the 

 past, been mainly of a kind suggested by phenomena ; 

 but from such restrictions the abstract imagination 



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