To the observant and intelligent boy or girl who has ever sailed on the Broads or steered a motor-boat in a busy harbour, the ordinary text-book treatment of Relative Velocity must be unsatisfying. The velocity of a ship (or other moving body) P relative to a ship Q is defined as the velocity of P as it “appears” to an observer on Q. But the exact meaning of “appears” is seldom explained. The fact that in all practical applications, observations are being made from one or other of the moving platforms, not from a stationary airship above them, is generally obscured. Thus the relative velocity of P with respect to Q is presented as something mathematical, obtained by the trick of reversing Q's velocity and adding it vectorially to that of P. Now, anyone who has steered a boat and observed another approaching on a converging course, knows that he is not directly concerned with the “actual” velocities (meaning velocities relative to the earth) of either boat. On the other hand, the velocity relative to him of the other boat is a very real thing. But to obtain it he does not have to go through the mental process of imagining that he is going astern when he knows that he is going ahead. All he does is to look over the side and observe. If the other boat's relative bearing does not change, or if its rate of change is small, he knows that it is time to do something with the tiller.

We propose in this paper to develop equations which give the principal points and lines of the conic formed by the intersection of the general conicoid.

S=ax2+by2+cz2+2fyz+2gzx+2hxy+2ux+2vy+2wz+d=0,

with the plane lx+my+nz+p)=0, in a form convenient for numerical work. We discuss the general case in which special values of the constants do not lead to special cases the slight modifications necessary where such occur become clearly manifest in numerical work. For brevity, the expressions ax+hy+gz+u, etc., will be denoted by the usual symbols X, Y, Z, and for nY - mZ, lZ - nX, mX - lY we will write U, V, W where lU + mV + nW≡0. We can assume without loss of generality that n is not zero.

In teaching algebra, it is obviously desirable to prove the Binomial Theorem for a real (or at any rate for a rational) index without assuming certain theorems on convergence which are used in Euler’s proof. So far as I know, no one has thought it worth while to devise such a proof.

In a previous article I have outlined my conception of the rôle of history in the exposition of mathematical technique. In this article I attempt to provide my bare thesis with a respectable clothing of practicability The treatment of Number and Continuity which follows is a short and very sketchy historical supplement to the technical treatment of the same subjects in, say, the earlier chapters of Hardy’s Pure Mathematics.

FIRST of all I should like to say that I am not responsible for the subject of this paper. Some member of the Association proposed that a discussion should be held on this subject and the Committee that arranges the programme for this meeting asked me to read a paper.

This Association was founded a little over sixty years ago, mainly by schoolmasters, and its main object was to advance mathematical knowledge by improving the teaching of mathematics in schools, so that I am only too glad to welcome any discussion on actual school work, and I hope that what I have to say may provoke a discussion to which many of you will contribute.

After promising to read the paper I sat down to consider what ground it should cover. It seems to me that the subject covers nearly the whole ground of geometry teaching up to the School Certificate stage, so the subject is wide.

It will not be uninteresting to look at the past for a few minutes. I started teaching in the last century, and there was much teaching at that time that had made little advance since the date of the foundation of the A.I.G.T. But for a moment I will go back still earlier and tell you two stories of the middle of the last century

There is a well-known story about the late Archbishop Temple, that he once had to listen to a sermon by a youthful and inexperienced clergyman, and to dine with him afterwards, the young man, by way of making conversation during the meal, ventured to remark, “I think, my lord, that I chose a good text for my sermon” Instantly there came the grim reply, “There was nothing wrong with the text”. It may be that the consequence of my having selected a seductive title which does not possess a very close connection with the actual subject of my address will be that, when we adjourn presently, I may get the impression that my audience has consisted entirely of archbishops.

There is no need to recommend to teachers in this country, familiar with Dobbs’ School Course and with later books of the same tendency, the free use of the ideas of displacement, rotation, and reflection. In the course of a paper read at the Zürich Congress last year, Prof. Thomsen, of Rostock, gave some striking examples of the effective introduction of the language and notation of the theory of groups into work of this kind, and afterwards he agreed willingly to contribute to the Gazette an article on the subject. For applications to their own problems we must refer readers elsewhere, since Prof. Thomsen has paid them the compliment of explaining a point of view that would be quite unsuitable in the school. To some of our readers we must apologise for replacing the original article by a translation, but since there are no familiar symbols or recognisable formulae to facilitate the grasping of a novel theory, we think that a substantial minority will be grateful for this assistance. In making his version, Prof. Neville has taken one or two liberties with the author’s notation.

The remarks I have to offer are made with no little diffidence, but no great reluctance. I am diffident because I feel I can only offer a very humble contribution to your meeting, at the same time, recalling the aims of the Association and the purpose of this meeting, I am glad to bring up for discussion a subject which concerns our teaching in all branches of the subject and at all stages.