the use of “Euclid” as an elementary 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 difficult, 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 in Continental schools no such difficulty has been experienced. With regard to the first objection, there is no reason, why in a modern textbook, 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

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. R. Gillies agreed with Mr. Hawthorne, who, 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 difference 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 would 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 afforded by “Euclid.”

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