An article by Mr. G. H. Hardy in the March number of The Mathematical Gazette (coupled with certain other articles in recent numbers) raises several points which can hardly be allowed to pass without further discussion. It is proposed to discuss these points under three different heads, of which the first two are treated here while the third will find space in a later number of the Gazette.

During the last few years the methods of teaching Mathematics in schools have been subjected to much adverse criticism: this has induced teachers to apply themselves to the task of setting their house in order. It is now generally deemed advisable to give greater prominence to practical applications, to make more use of accurate drawing as a method of solution of problems and for purposes of verification, to curtail numerical calculations by dispensing with figures of doubtful or unnecessary accuracy, and to lay more stress on the employment of graphical methods.

It appears from the recently published life of Archbishop Temple that he is supposed to have found a proof of the proposition that any map can be coloured with four colours, no two adjacent segments of it being coloured alike. The writer had heard of this before the publication of the Life, but had not succeeded in finding what the proof was, and was under the impression that it had never been published, especially as it was not mentioned in Mr. Rouse Ball’s book. As stated in the Life the supposed solution was published in the Journal of Education, June 1, 1889. It turns out to be what one has learned from experience (and the writer owns that he has spent much time on the subject) to expect from any theory which looks like a short solution of this extremely elusive problem.

It has been my experience, and probably that of most teachers, that even advanced students who ought to know better cannot differentiate the simplest algebraic and trigonometric functions without indulging in an extravagant luxuriance of infinite series and making a number of illegitimate assumptions.

The difficulties which arise in teaching the elementary power series appear to form part of a larger question, namely, the proper course of mathematical education after a sufficient knowledge of Elementary Geometry, Algebra and Trigonometry has been acquired. The question of Geometry is hardly germane to the present discussion, and I will only mention it so far as to say that I believe it is the hope of many teachers of mathematics that boys of 15 or 16 may in the future learn far more of Projective and Descriptive Geometry than is the case at present.