THE STUDY OF MATHEMATICS 71 



Euclid that he advances as far as he is able to go without 

 employing the axiom of parallels not, as is often said, 

 because this axiom is inherently objectionable, but 

 because, in mathematics, every new axiom diminishes 

 the generality of the resulting theorems, and the greatest 

 possible generality is before all things to be sought. 



Of the effects of mathematics outside its own sphere 

 more has been written than on the subject of its own 

 proper ideal. The effect upon philosophy has, in the 

 past, been most notable, but most varied ; in the seven- 

 teenth century, idealism and rationalism, in the eigh- 

 teenth, materialism and sensationalism, seemed equally 

 its offspring. Of the effect which it is hkely to have in 

 the future it would be very rash to say much ; but in 

 one respect a good result appears probable. Against 

 that kind of scepticism which abandons the pursuit of 

 ideals because the road is arduous and the goal not cer- 

 tainly attainable, mathematics, within its own sphere, is 

 a complete answer. Too often it is said that there is no 

 absolute truth, but only opinion and private judgment ; 

 that each of us is conditioned, in his view of the world, 

 by his own peculiarities, his own taste and bias ; that 

 there is no external kingdom of truth to which, by patience 

 and discipline, we may at last obtain admittance, but only 

 truth for me, for you, for every separate person. By this 

 habit of mind one of the chief ends of human effort is 

 denied, and the supreme virtue of candour, of fearless 

 acknowledgment of what is, disappears from our moral 

 vision. Of such scepticism mathematics is a perpetual 

 reproof ; for its edifice of truths stands unshakable and 

 inexpugnable to all the weapons of doubting cynicism. 



The effects of mathematics upon practical life, though 

 they should not be regarded as the motive of our studies, 

 may be used to answer a doubt to which the solitary 



