THE STUDY OF MATHEMATICS 65 



old lines, the new work is considered to be appallingly 

 difficult, abstruse, and obscure ; and it must be con- 

 fessed that the discoverer, as is so often the case, has 

 hardly himself emerged from the mists which the light 

 of his intellect is dispelling. But inherently, the new 

 doctrine of the infinite, to all candid and inquiring 

 minds, has facilitated the mastery of higher mathematics ; 

 for hitherto, it has been necessary to learn, by a long 

 process of sophistication, to give assent to arguments 

 which, on first acquaintance, were rightly judged to be 

 confused and erroneous. So far from producing a fear- 

 less belief in reason, a bold rejection of whatever failed 

 to fulfil the strictest requirements of logic, a mathematical 

 training, during the past two centuries, encouraged the 

 belief that many things, which a rigid inquiry would 

 reject as fallacious, must yet be accepted because they 

 work in what the mathematician calls ' practice." By 

 this means, a timid, compromising spirit, or else a sacer- 

 dotal belief in mysteries not intelligible to the profane, 

 has been bred where reason alone should have ruled. All 

 this it is now time to sweep away ; let those who wish to 

 penetrate into the arcana of mathematics be taught at 

 once the true theory in all its logical purity, and in the 

 concatenation established by the very essence of the 

 entities concerned. 



If we are considering mathematics as an end in itself, 

 and not as a technical training for engineers, it is very 

 desirable to preserve the purity and strictness of its 

 reasoning. Accordingly those who have attained a 

 sufficient familiarity with its easier portions should be 

 led backward from propositions to which they have 

 assented as self-evident to more and more fundamental 

 principles from which what had previously appeared as 

 premises can be deduced. They should be taught 



