Members of the Association will deeply regret to hear of the death of F C. Boon. Readers of the Gazette and members attending the Annual Meetings will not need to be told of the immense work he did for the Association, in his contributions to the Gazette, his part in discussions, the way he literally carried the London Branch as Secretary and as Chairman, his years of strenuous and fruitful service on the Teaching Committee. His stalwart defence of the so-called “duffer”, the essential soundness of his outlook, his fatherly treatment (the boys at Dulwich called him “Daddy Boon”), his patience and kindliness in debate, by these he was known.

In accepting the invitation to address the members of the Association I made one condition, that if I came I should be allowed to speak informally You will, I hope, forgive me if owing to the pressure of meetings, during which I have been trying not to square the circle but to get a “square deal”, I have been unable to prepare a set discourse.

In these days, when payment for goods received is so commonly made in instalments spread over a period, it is not surprising that examination questions on Commercial Arithmetic, or on Arithmetic with a Commercial bias, not infrequently take the form

Some time ago a Hungarian post-graduate, Dr. P. Erdös, studying under Prof. L. J Mordell at Manchester University, enunciated an interesting theorem connected with a triangle, for which he asked that a proof should be given.

I feel somewhat diffident in addressing you on this subject, because I am only too conscious of the fact that my qualifications for doing so are but slight. Most of my audience will have had some experience of teaching algebra to the higher forms in schools, some will have had considerable experience. I have had none. My only knowledge of the subject on which I address you has been gained as an examiner and, a little, as a university don teaching undergraduates who come from your schools. I am as ready as the next man to admit that the examiner does not see everything; to admit, too, that he does not even see all that he thinks he sees. But when all has been said against the examiner—and more than a little has been said of late—the examiner does see something. And so, with these admittedly slight qualifications, I venture to put before you some few points about algebra.

The diagram which it is the purpose of these notes to explain is so simple in idea that it is surprising that it is not well known. Its idea is merely that the shape of a triangle depends only on two data and can therefore be represented in various ways by the coordinates of a point. To do this, suppose the scale of any triangle reduced so that its largest side is unity and let the other sides be x and y in descending order, then the point whose coordinates are x and y will represent the triangle. Moreover x and y are limited by the inequalities 1 ≥ x, x ≥ y, x + y ≥ 1, so the point representing any triangle lies in the quarter of the unit square shown in the diagram. Of this the boundary line x = 1 corresponds to isosceles triangles whose greater sides are equal, the line x = y to isosceles triangles whose smaller sides are equal, and the third line x + y = 1 to nugatory triangles with two angles equal to zero. The circle x2 + y2 = 1 corresponds to right-angled triangles, and triangles are obtuse-angled or acute-angled according as their representative points lie inside or outside this circle.

To find the ratio which gives the greatest chance that through a point, chosen at random within a triangle, six lines may be drawn to divide the triangle in that ratio.

The following paper arose out of an attempt to improve upon the familiar “match puzzles” one sees in the popular press and which are generally too puerile to be interesting. Following some desultory experiments, it developed that the methods of construction postulated are capable of determining all points obtainable with ruler and compass, but no others. The sequence of results below is arranged to establish this conclusion.

Mr. A. W. Siddons (Harrow): As they say at company meetings, I hope that we may take this Report as read, and very good reading it is. I may say at once that I am not here to criticise the Report if by “criticise” we mean “criticise adversely”. There are two points about which I wrote to the July Gazette.

All school teachers of the world who are interested in the development of the teaching of mathematics in general and of the teaching of geometry in particular know the three stages A, B, c which are prescribed for the teachers of geometry in England. The father of this splendid idea is W C. Fletcher, formerly Chief Inspector of the Board of Education in London. He issued in March 1909, as Circular 711, a famous regulation: Teaching of Geometry and Graphic Algebra in Secondary Schools.

In connection with the article on Quadratic Residues in the December number of the Gazette, a neat graphical method, which I first put forward in 1913, may be of interest.

Members of the Association who have read the Report on Secondary Education will be interested to know that the references in the Report to mathematics were considered in January last by the Council and by the Teaching Committee. The following is a copy of the letter sent to the Board of Education in consequence of a resolution of the Teaching Committee. It was signed by the President of the Association as well as by representatives of the Teaching Committee:

“The Teaching Committee of the Mathematical Association have had under consideration the recently published Report on Secondary Education, and they wish to make the following observations.

Members of the Association who read the statement on page 6 of the Mathematical Gazette for February 1939 will remember that the Examinations Sub-Committee is considering first the Algebra syllabuses of the various school certificate examinations.