Page:Carroll - Euclid and His Modern Rivals.djvu/282

244 never been laid aside; only a few months since one of our most influential tutors stated that he was accustomed to give a proposition out of the fifth book of Euclid to some candidates for emoluments, and he considered it a very satisfactory constituent of the whole process of testing them.

I should exceedingly regret the omission of the fifth book of Euclid, which I hold to be one of the most important parts of the training supplied by Elementary Geometry. I do not consider it necessary for beginners to go through the entire book; but the leading propositions might be mastered, and the student led to see how they can be developed if necessary. I may refer here to some valuable remarks which have been made on the subject by the writer of a Syllabus of Elementary Geometry… who himself I believe counts with the reformers. He sums up thus "… any easy and unsatisfactory short cuts (and I have sometimes seen an inclination for such) should be scouted, as a simple deception of inexperienced students."

However, I must remark that I see with great satisfaction the following Resolution which was adopted at a recent meeting of the Association for the Improvement of Geometrical Teaching: "That no treatise on Geometry can be regarded as complete without a rigorous treatment of ratio and proportion either by Euclid's method, or by some equally rigorous method of limits." It would be injudicious to lay much stress on resolutions carried by a majority of votes; but at least we have a striking contradiction to the confident statement that Euclid's theory of proportion is dead. We shall very likely see here, what has been noticed before, that a course may be proposed which differs widely from Euclid's, and then, under the guidance of superior knowledge and experience, the wanderers are brought back to the old path. Legendre's return to Euclid's treatment of parallels is a conspicuous example; see the valuable paper by Professor Kelland on Superposition in the Edinburgh Transactions, Vol..

I cannot avoid noticing one objection which has been urged