ioo MATHEMATICS 



differences of opinion amongst teachers as to the 

 proportions in which these two elements should be 

 combined. There has been some expression of 

 opinion in favour of again stereotyping the subject 

 by means of the imposition of a single authorized 

 text book; but although the carrying out of such 

 a proposal would undoubtedly be a convenience 

 for the examiner, it is to be deprecated in the 

 interest of the freedom of the teacher. Even if 

 the accuracy of the results of examination were 

 thereby increased, it would be a very insufficient 

 reason for fettering the teacher; for one of the 

 most serious evils from which our educational 

 system has suffered has been the subordination of 

 the freedom of the teaching to the exigencies of 

 examinations. Although we now possess schemes 

 of Geometry in which, by the placing of all the 

 intuitional elements at the beginning, in the form 

 of postulates and definitions, the subsequent 

 treatment is made purely deductive, it will be 

 recognized by all who are acquainted with the 

 nature, number, and complication of these postu- 

 lates and definitions, that such treatment of the 

 subject is quite unsuitable for use except by 

 specialists in an advanced stage of their course. 

 That Euclid's Elements of Geometry embodied 

 a purely logical treatment of the subject, of unim- 

 peachable rigour, is far from being true. At 

 numerous points in the course of his reasoning, 

 Euclid has recourse to unproved and unstated 



