THE STUDY OF MATHEMATICS 61 



tions, no barrier to the creative activity embodying in 

 splendid edifices the passionate aspiration after the per- 

 fect from which all great work springs. Remote from 

 human passions, remote even from the pitiful facts of 

 nature, the generations have gradually created an 

 ordered cosmos, where pure thought can dwell as in its 

 natural home, and where one, at least, of our nobler 

 impulses can escape from the dreary exile of the actual 

 world. 



So little, however, have mathematicians aimed at 

 beauty, that hardly anything in their work has had this 

 conscious purpose. Much, owing to irrepressible instincts, 

 which were better than avowed beliefs, has been moulded 

 by an unconscious taste ; but much also has been spoilt 

 by false notions of what was fitting. The characteristic 

 excellence of mathematics is only to be found where the 

 reasoning is rigidly logical : the rules of logic are to 

 mathematics what those of structure are to architecture. 

 In the most beautiful work, a chain of argument is pre- 

 sented in which every link is important on its own 

 account, in which there is an air of ease and lucidity 

 throughout, and the premises achieve more than would 

 have been thought possible, by means which appear 

 natural and inevitable. Literature embodies what is 

 general in particular circumstances whose universal 

 significance shines through their individual dress ; but 

 mathematics endeavours to present whatever is most 

 general in its purity, without any irrelevant trappings. 



How should the teaching of mathematics be conducted 

 so as to communicate to the learner as much as possible 

 of this high ideal ? Here experience must, in a great 

 measure, be our guide ; but some maxims may result 

 from our consideration of the ultimate purpose to be 

 achieved. 



