62 MYSTICISM AND LOGIC 



One of the chief ends served by mathematics, when 

 rightly taught, is to awaken the learner's belief in reason, 

 his confidence in the truth of what has been demon- 

 strated, and in the value of demonstration. This purpose 

 is not served by existing instruction ; but it is easy to 

 see ways in which it might be served. At present, in 

 what concerns arithmetic, the boy or girl is given a set 

 of rules, which present themselves as neither true nor 

 false, but as merely the will of the teacher, the way in 

 which, for some unfathomable reason, the teacher prefers 

 to have the game played. To some degree, in a study of 

 such definite practical utility, this is no doubt unavoid- 

 able ; but as soon as possible, the reasons of rules should 

 be set forth by whatever means most readily appeal to 

 the childish mind. In geometry, instead of the tedious 

 apparatus of fallacious proofs for obvious truisms which 

 constitutes the beginning of Euclid, the learner should 

 be allowed at first to assume the truth of everything 

 obvious, and should be instructed in the demonstrations 

 of theorems which are at once startling and easily verifi- 

 able by actual drawing, such as those in which it is shown 

 that three or more lines meet in a point. In this way 

 belief is generated ; it is seen that reasoning may lead 

 to startling conclusions, which nevertheless the facts will 

 verify ; and thus the instinctive distrust of whatever is 

 abstract or rational is gradually overcome. Where 

 theorems are difficult, they should be first taught as 

 exercises in geometrical drawing, until the figure has 

 become thoroughly familiar ; it will then be an agreeable 

 advance to be taught the logical connections of the 

 various lines or circles that occur. It is desirable also 

 that the figure illustrating a theorem should be drawn in 

 all possible cases and shapes, that so the abstract relations 

 with which geometry is concerned may of themselves 



