61 



upon the mind of the learner, and in this respect, the old system was 

 far superior to the new. Mr. Hawthorne then analyzed two or three 

 examples given by Mr. Brent (including the 5th and 16th propositions of 

 the first book of " Euclid "), and pointed out that the modern system of 

 demonstrating these theorems was not only illogical, but involved mental 

 impossibilities. Professor De Morgan had expressed his opinion that 

 the love of accuracy has declined wherever " Euclid" has been abandoned, 

 and he (Mr. Hawthorne) believed that the abandonment of " Euclid " 

 would be introducing too much of the sensuous into the educational 

 system, and was calculated to produce very serious injury. On this and 

 many other grounds, he was a firm believer in " Euclid " as a text-book. 

 At the same time, many minor improvements might be made in it ; and 

 he would also introduce boys of, say, from ten to thirteen years of age, 

 to the modern system, so as to give them an insight into the practical 

 bearing of geometry. 



Mr. Brent was rather surprised to hear Mr. Hawthorne quote 

 Professor De Morgan in support of his arguments ; as he (Mr. Brent) 

 had made an extract from the Professor's writings in support of his own 

 views. The difference between the last speaker and himself seemed to 

 be, that while the former looked upon geometry as abstract ideas, he 

 regarded it as concrete quantities. 



Mr. E,. Gillies agreed with Mr. Hawthorne, whOj however, had not 

 fully brought out the point that the solution of these problems was 

 purely a mental process, and whenever they brought in the mechanical, 

 they brought in a source of error. Consequently, any demonstration 

 introducing the mechanical, failed in accuracy. Any one who had to 

 apply mathematics to practical purposes, knew very well that there was 

 a very great difierence between the theory and the practical results. It 

 was of the utmost importance, in teaching mathematics, that every 

 possible source of error should be eliminated from the processes. This, 

 however, did not touch the question with which Mr. Brent started, viz., 

 that it woiild be advantageous to teach elementary mathematics by the 

 modern method. It was doubtful whether any benefit was gained by 

 teaching young children mathematics ; and while, by the modern method, 

 a learner might get a sufficient general idea of mathematics for practical 

 purposes, he would not obtain the mental training afibrded by " Euclid." 



The Rev. Mr. Stuart said no doubt Mr. Brent, in teaching mathe- 

 matics to children, found the " sensuous " method convenient ; and he 

 thought that children should not be taught mathematics until they were 

 thirteen years of age. The mere advantage of brevity in demonstration 



