Construction. As we depart widely from Euclid’s ideas in the proofs of Stage B, it seems to be thought by many people that go-as-you-please methods are good enough for constructions as well. Now, “trial” methods of construction are sometimes useful in more advanced work ; but they are tolerated only when a geometrical method is impossible ; they are means to get a solution, but there is no instruction to be had from such a method. We would therefore urge that a certain severity be still maintained in problems of construction.

Most mathematical students are familiar with Pascal’s statement, made in a communication to a learned society in Paris in 1653, that he had a number of mathematical treatises, chiefly geometrical, ready for publication, amongst them being the following : Conicorum opus completum, et conica Apollonii et alia innumera unica fere propositione amplectens ; quod quidem nondum sex decimum aetatis annum assecutus excogitavi ; et deinde in ordinem congessi.

As teachers of elementary mathematics I suppose we are all familiar with the Circular 711 of the Board of Education. That circular is now out of print, and its place has been taken by Circular 851, which, I think, is not so well known. The purpose of the latter circular is to state the matter afresh, to explain it more fully, and to justify the attitude adopted. The main practical suggestion of these two circulars has been so largely adopted in present-day teaching in this country that I propose at the outset to base my remarks largely upon it.

One of the best-known examples of a caustic curve in Geometrical Optics is that obtained by reflection from the arc of a circle of rays emanating from a point on the circumference of the circle. The enveloping curve is known to be a cardioid. It has not, however, I think, been pointed out that the analysis •of this problem leads to a very simple construction from which a cardioid •can be rapidly and accurately drawn.

For many years it has been my custom to take the junior students, that is to say the lowest class, in Mathematics for one hour a week: and a part of the session has been given to what may be called the simpler treatment of the principles of Arithmetic. No attempt has been made to touch the more difficult parts of this subject: no effort has been made to base, for example, the idea of fractions upon a purely abstract foundation.

I Think that everyone here will agree with me that we, who try to teach mathematics, have a heavily loaded curriculum and we are apt to become the slaves of public examinations. According to the age of our pupils, Intermediate or Matriculation hangs like the sword of Damocles above us, and we dare not be led far astray from the mark-gaining part of our work.

Probably the most interesting example of this mixture is that given by Mr. R. Steele in the Introduction (p. xvii) to The Earliest Arithmetics in English, where he quotes from a British Museum MS. (Harl. 4350) a set of dates of successive births

These form an early example, for Britain, of the transition.

The following remarks are based on the assumption, which I believe to be a true one, that there is a desire for more advanced mathematical knowledge among many people in this country, such as those engaged in teaching mathematics in secondary schools, technological institutions, and colleges, as well as others not professionally associated with mathematics. At present the means of satisfying this desire, at least as regards organised courses of post-graduate study available for persons employed in teaching or otherwise, are almost entirely lacking in England. The most that is available, and that only in a few university centres, consists of occasional lectures given under the auspices of universities and such bodies as mathematical societies and branches of the Mathematical Association. Able and interesting as these lectures may be, the knowledge to be acquired from them must, in the nature of things, be somewhat disconnected and superficial. Their value lies in the introductory glimpse which they give of unfamiliar branches of study, or in their rapid survey of a wide field of knowledge. They are useful complements to more systematic study, but are not a substitute for it.