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the use of " Euclid " as an elementaiy text-book are, first, that it is an 

 excellent training to logical reasoning, which is as valuable as geometrical 

 knowledge, and thus two ends are accomplished at once ; and secondly, 

 that without it, it would be ditScult, if not impossible, to examine in 

 geometry, as no single text-book would be adopted as a standard. 

 Mr. Brent points out that the latter objection is easily disposed of, for 

 ill Continental schools no such difficulty has been experienced. With 

 regard to the first objection, there is no reason why in a modern text- 

 book, the theorems should not be demonstrated by proofs as rigorous as 

 those in "Euclid." As to the logical training afforded by "Euclid," 

 Mr. Brent believes it will be found inferior to that acquired by the use 

 of a good modern text-book. The true test of logical training is the 

 power of producing original work ; and hence the author advocates the 

 use of a small, rather than a large manual of geometry, but one containing 

 a great number of well-arranged exercises. Again, one valuable discipline 

 of logic is to train the mind to scientific order, and thus we should 

 naturally classify geometrical truths according to their subject matter 

 and relation to each other, putting in one division theorems relating to 

 triangles, in another those treating of circles, and so on. This is not 

 done in " Euclid," but could easily be done in a well-arranged geometry. 

 There are also many other objections to " Euclid " as a system of logical 

 reasoning. His treatment of parallels is a well known instance, for it 

 rests on an axiom which is not axiomatic. Others of his axioms are not 

 axioms at all, but definitions or theorems, as " The whole is greater than 

 its part." Having pointed out the objections to the use of " Euclid " as 

 an elementary text-book, Mr. Brent proceeded to give a detailed explanation 

 of the modern method of teaching geometry, and afterwards illustrated 

 his remarks by diagrams on the black board. 



Mr. Hawthorne said Mr. Brent had been kind enough to lend him 

 the manuscript on the previous evening, and he would endeavour to 

 reply to some of the charges of want of logic which had been brought 

 against " Euclid," although those charges rebounded with tremendous 

 force against the modern system. He did not say that "Euclid" was 

 perfectly logical; but Mr. Brent, in almost every instance he had chosen, 

 was far more illogical. They knew the old saying, " There is no royal 

 road to learning," but the 19th century seemed remarkable for royal 

 roads to learning. It must be remembered, in considering this question, 

 that while generally the object of study was to gain a particular 

 end, the main objects in studying geometry were not results, but 

 processes. They must ascertain the effect of the steps and processes 



