A meeting of this branch was held on February 20th at the Friars’ School, Bangor—the President, Dr. Bryan, in the chair. Mr. R. W. Jones, Headmaster of Glanadda Elementary School, opened a discussion on the teaching of the foundations of arithmetic to children of 6 or 8 years old. After referring to the difficulty of forming any ideas as to a child’s conception of number, and mentioning instances of the vague answers given by a child who is asked to estimate some large number of objects, Mr. Jones proceeded to show how the four fundamental laws of arithmetic could be gradually and insensibly taught to children by means of experiments involving simple operations in weighing and measuring. He emphasised the necessity for using familiar illustrations in expounding the four rules and brought out two points as worthy of special notice: (1) that in teaching the elements to children it is absolutely necessary to confine one’s attention to small numbers; (2) that the time spent in teaching the multiplication tables to children of 6 or 8 years was not well spent, since for small numbers at any rate, they could be learnt insensibly, and the higher tables could be learnt with much greater facility in the child’s third year at school. Most of those present took part in the interesting discussion which followed, generally expressing their agreement with Mr. Jones’s views.

The Inaugural Meeting of the above Society was held at Taunton School, Southampton, on Friday, October 22nd, by kind invitation of the Headmaster, Mr. S. J. Gubb, when seventeen members were present.

The summer meeting of the North Wales branch was held on 22nd May at the County School, Beaumaris, where the headmaster, Mr. Madoc Jones, entertained the members present. In opening a discussion on the teaching of geometry, Mr. Botting, mathematical tutor at the Normal College, Bangor, referred to the recent circular of the Board of Education, which recommends three stages, each occupying one year : (1) children to be grounded in fundamental ideas of space, lines, angles, etc., without actual de-finitions, and accustomed to instruments of measurement ; (2) the acquisition of a number of fundamental propositions, but not in a formal way, e.g. I. 13-15 in Euclid, and some propositions about parallel lines, the equality of triangles, etc.; (3) rigid deductive proofs to be begun, all fresh propositions being treated as original work like riders. He also said that propositions should not be taken singly, but in groups together, and that the power of working riders is the only real test of advance in geometry

In an article on models of functions in the October (1908) number of the Gazette it was suggested that some knowledge of one-to-one correspondence of operands ought to be acquired by means of models before going on to the correspondence of operands to functions, and of functions to functions. In particular, it is easy and of importance to make models of some of the roots of one. It is needless to say that the order of subjects actually adopted in schools is widely different from that here suggested. It is usual to begin with the addition and multiplication of numbers, that is, with a particular kind of correspondence not of operands but of functions. Even the words function and correspondence are not met with until a comparatively late stage, and the fact is ignored that the numbers 2, 3, 4, etc., are functions, though not one-to-one functions.

Several societies have during the past few years considered the teaching of mathematics and of science ; since, however, they have confined their attention to the subjects separately, and in many instances to particular types of school, the Joint Committee decided not to rely upon data previously collected, but to send a circular to all the schools mentioned in the School-marters Year Book, asking for details of the conditions under which the subjects are at present taught. This has been done and replies received from about 300 schools representing all types of non-primary education: a summary of the replies will be found in Appendix B, and the Committee wish to express their thanks to those gentlemen who, by so fully answering the questions asked, have made it possible to present a report dealing with a larger number of schools than has been hitherto possible.

The Committee wish to make several recommendations ; they however recognise with pleasure that the returns prove not only that a considerable advance has been made during the past ten years in the teaching of mathematics and science, but that in many cases a definite attempt has been made to correlate this teaching.

Describe a circle ABC (Figs. 1 and 2). Draw two chords AB, BC intersecting within it. Describe eight circles (a, a′ b, b′, etc.), each of which touches ABC and the chords AD, BC; construct a triangle by joining any three of the points of intersections of ABC and the two chords; then the chords of contact and the line of centres of two circles taken in suitable pairs out of the eight circles are concurrent, the point of concurrence being equidistant from the three sides of the triangle.

A Meeting of the North Wales branch was held on November 20th at the Girls’ County School, Bangor. The proposed rules for local branches were read, and their consideration, as far as this branch is concerned, was postponed until after the general meeting of the Association.

In last year’s address I called attention to the need of some symbol to denote unknown digits in approximate numerical calculations. The present note is commuunicated in the hopes that the matter may be submitted either to the Council or to a special committee of the Association in order that some definite recommendation may be adopted.

The last section of Arithmetica Infinitorum is devoted to the ingenious argument by which Wallis suceeded in reaching the first modern expression of π as an infinite series.

We begin with an argument which, although in Wallis’s hands it proved abortive as an attempt to determine π, is yet useful for the comprehension of his more successful procedure, and has (as we shall see) great historical importance.