Page:Carroll - Euclid and His Modern Rivals.djvu/276

238 if thought convenient or necessary, to supply any additional matter. But for my part I think there are grave objections to any large increase in the extent of the course of synthetical geometry which is to be prepared for examination purposes. One great drawback to our present system of mathematical instruction and examination is the monotony which prevails in many parts. When a mathematical subject has been studied so far as to master the essential principles, little more is gained by pursuing these principles into almost endless applications. On this account we may be disposed to regard with slender satisfaction the expenditure of much time on geometrical conic sections; the student seems to gain only new facts, but no fresh ideas or principles. Thus after a moderate course of synthetical geometry such as Euclid supplies, it may be most advantageous for the student to pass on to other subjects like analytical geometry and trigonometry which present him with ideas of another kind, and not mere repetitions of those with which he is already familiar.

It has been said, and apparently with great justice, that examination in elementry geometry under a system of unrestricted text-books will be a very troublesome process; for it is obvious that in different systems the demonstration of a particular proposition may be more or less laborious, and so may be entitled to more or fewer marks. This perplexity is certainly felt by examiners as regards geometrical conic sections; and by teachers also who may be uncertain as to the particular system which the examiners may prefer or favour. It has been asserted that the objection thus raised is imaginary, and that "the manuals of geometry will not differ from one another nearly so widely as the manuals of algebra or chemistry: yet it is not difficult to examine in algebra and chemistry." But I am unable to feel the confidence thus expressed. It seems to me that much more variety may be expected in treatises on geometry than on algebra; certainly if we may judge from the experience of the