REFORM OF TEACHING 99 



unique position of holding a practical monopoly as 

 a scientific text book, has given an opportunity 

 for carrying out experiments in methods of teaching 

 the very important subject of Geometry. The 

 view that the study of Geometry, in its earliest 

 stages, should be of a practical kind, appealing 

 to sensuous intuition and observation, has met 

 with very general acceptance. There is however 

 much difference of opinion as to the relations 

 between the practical and the abstract treatment 

 of the subject in the later stages. By the advocates 

 of the use of Euclid or of some other text book on 

 more or less similar lines great stress is laid upon 

 the importance of the study of Geometry as affording 

 a training in logical thinking of the deductive 

 kind. Those who lay most weight upon the prac- 

 tical side of the subject regard a detailed knowledge 

 of spatial relations as the chief object to be attained. 

 The greater part of the actual teaching of the sub- 

 ject is very properly based upon a compromise be- 

 tween the two extreme views of the objects to be 

 attained. There is evidence that the purely prac- 

 tical treatment of the subject has been carried to 

 an unfortunate extreme by some teachers ; and 

 this has naturally led in some quarters to a strong 

 reaction in favour of more rigorous methods. That 

 the most effective methods of teaching the subject 

 must provide for both the logical and the intuitional 

 treatment of spatial relations is quite clear; 

 although there will no doubt continue to be 



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