TO MYSTICISM AND LOGIC 



should be wholly free. A reciprocal liberty must thus be 

 accorded : reason cannot dictate to the world of facts, 

 but the facts cannot restrict reason's privilege of dealing 

 with whatever objects its love of beauty may cause to 

 seem worthy of consideration. Here, as elsewhere, we 

 build up our own ideals out of the fragments to be found 

 in the world ; and in the end it is hard to say whether 

 the result is a creation or a discovery. 



It is very desirable, in instruction, not merely to per- 

 suade the student of the accuracy of important theorems, 

 but to persuade him in the way which itself has, of all 

 possible ways, the most beauty. The true interest of a 

 demonstration is not, as traditional modes of exposition 

 suggest, concentrated wholly in the result ; where this 

 does occur, it must be viewed as a defect, to be remedied, 

 if possible, by so generalising the steps of the proof that 

 each becomes important in and for itself. An argument 

 which serves only to prove a conclusion is like a story 

 subordinated to some moral which it is meant to teach : 

 for aesthetic perfection no part of the whole should be 

 merely a means. A certain practical spirit, a desire for 

 rapid progress, for conquest of new realms, is responsible 

 for the undue emphasis upon results which prevails in 

 mathematical instruction. The better way is to propose 

 some theme for consideration ^in geometry, a figure 

 having important properties ; in analysis, a function of 

 w^hich the study is illuminating, and so on. Whenever 

 proofs depend upon some only of the marks by which we 

 define the object to be studied, these marks should be 

 isolated and investigated on their own account. For it 

 is a defect, in an argument, to employ more premisses 

 than the conclusion demands : what mathematicians call 

 elegance results from employing only the essential prin- 

 ciples in virtue of which the thesis is true. It is a merit in 



