Nowadays every Fifth Form pupil uses logarithm tables. A discussion of the construction of the difference columns, with graphical illustration, would prepare him for the methods to be described below, of an iterative process for the approximate solution of any equation. Incidentally it would help him to realise more fully than he usually does to what degree of accuracy his logarithmic work is reliable. The method of obtaining logarithms by using powers of 1·1, as described in the Algebra Beport, uses the same principle of proportional parts, and, properly treated, helps to establish the general principle that, when the differences for unit increments in the entries used are constant, results obtained by proportional interpolation are accurate to the same number of figures as the entries.

Although I am to speak “from the University point of view” I wish to consider not only how the small number of boys who will go to a University may best be prepared for their work there, but also the general problem of the teaching of Algebra in the upper forms of schools. This problem is connected with a larger question, whether the balance of subjects is right in present-day school teaching in mathematics, and perhaps you will permit me to start with a short digression on this point.

Mr. J. A. Edgar: On beginning the study of astronomy it is imperative for the student to gain familiarity with the brightest stars and with the classical constellations and their relative positions. The stars apparently all shift their positions westward during the night, and there is further a seasonal drift due to the annual revolution of the earth about the sun, so that a permanent acquaintance with the principal groups must depend on an intimate knowledge of their relative positions. In imparting this knowledge it is tedious and inefficient to point out the groups to the student, and published star maps are usually confusing, as they present too many configurations at once. Much can be gained by the student constructing his own maps. In this way the star chart grows, gradually extending familiarity with the sky. The first chart to construct is that showing the circumpolar group of stars. For this purpose polar graph paper must be used, and coordinates analogous to latitude and longitude are adopted.

Population trends are determined by fertility and mortality. The crudest method of measuring fertility and mortality consists in relating the number of births and of deaths to the total population. The birth and death rates thus obtained show the proportion by which a population increases through births and decreases through deaths. But since these rates are calculated without regard to the sex and age composition of the population they do not afford an adequate gauge for the measurement of fertility and mortality The disturbing influence of the sex and age composition is eliminated by computing fertility tables and life tables.

In the Rules of this Association, there is a declaration that its object shall be

Having regard to such a policy in action, having regard also to the professional duties of what probably is the majority of the members, I must appeal for a measure of your indulgence when my address, in retiring from office as your President, is devoted to some reflections concerning the present range of elementary Applied Mathematics.

If a tetrahedron with its opposite edges perpendicular is inscribed in a conicoid, its orthocentre (the intersection of the perpendiculars from the vertices to the opposite faces) will lie on the conicoid if the asymptotic cone of the conicoid has three perpendicular generators.

When I was younger this was a very well-known subject, and actually was called Science Subject No. 1 in the days of the Board of Education’s South Kensington Examination. Engineers (mechanical and civil) and craftsmen were taught it as a fundamental subject, and even teachers in training were expected to know something about it. The rise of Practical Mathematics put it into the background—wrongly I think—but there are signs that a healthy interest is being taken in this section again.

In 1934, the Board of Education asked the Mathematical Association to submit its views on the mathematical aspects of the organisation and interrelation of schools. In its report, the sub-committee appointed for this purpose, of which I was a member, made it clear that at present the mathematics up to School Certificate standard in schools of secondary education was inadequate in its scope and its direction for all but those pupils who will continue to read mathematics in a university course. If we remember that probably only some 10% of the population of our public and secondary schools passes on to any form of university career, and if we agree with the opinion of the sub-committee, we have before us a problem vast in its importance and extent and one which should be tackled without delay

In 1816 there appeared a translation by Dr. John Taylor of a treatise on Hindu Arithmetic and Geometry, published at the expense of the Literary Society of Bombay The author of this treatise was Bhascara Acharya (1114-c. 1185) and the treatise itself is of some importance in the history of Hindu mathematics. A discussion of this treatise, which is called the Lilawati, may be found by those interested in Rouse Ball’s History of Mathematics, pp. 150-154. It is however with the appendix to Taylor’s translation that we are here concerned. Little is known of Taylor himself except that he was born in Edinburgh, took his M.D. degree there in 1804 and died in Persia in 1821. He is believed to have published other translations from the Sanscrit besides the Lilawati. However, the appendix of this translation is entitled “A Short Account of the Present Mode of Teaching Arithmetic in Hindu Schools” and it may be of some interest to give a brief description of its contents.

I want to begin with an explanation as to my position in coming here to address you. I am not an expert upon the history of mathematics. I have come in order to tell you something of my experiences in trying to teach a little of the history of mathematics to sundry university students. This had its origin in the determination of Professor Milne, even while he was teaching at Clifton, to propagate a knowledge of the history of mathematics amongst students who, as a rule, know absolutely nothing of it. He continued on that crusade in a true missionary spirit at the University of Leeds.

The subject of discussion this morning is “The bearing of higher geometry on the school course”. The difficulties of geometry are well-known, though not all of you will entirely endorse the opinion expressed about 350 years ago by a certain Hubert Languet, who wrote to Sir Philip Sidney as follows: “I know not whether it is wise to apply your mind to geometry, though it is a noble study and well worthy of a fine understanding, but . . . you are not overcheerful by nature, and it is a study which will make you still more grave; and as it requires the strongest application of the mind, it is likely to wear out the powers of the intellect, and very much to impair the health.”

Let O, A, B, C be any four points not necessarily in one plane. Let H and K divide OA and BC proportionally, while M and N divide OB and AC proportionally. Then we shall prove that HK and MN meet at T such that OA, MN, BC are proportionally divided at H, T, K, while OB, HK, AC are proportionally divided at M, T, N, the geometrical relations being conveniently, and diagrammatically, expressed thus.

N.B.—In what follows we consider the conic as the projection of the circle and assume only its descriptive (including cross-ratio) properties as known.

1. A circle, centre g1 is cut by a secant line at ww′. Any line cpp′ through c, the pole of ww′, cuts the circle at p and again at p′. The lines wp and w′p′ intersect at d. The locus of d for the variable secant line through c is a hyperbola touching the given circle at ww′ (If w′p and wp′ meet at d′, the locus of d′ is the same hyperbola.)

In opening this discussion on the influence of the university on the school course in geometry I have two dangers to avoid. Many of the things which it would seem natural to say are true and a few of them may be important, but when I wrote them down and took a second look at them, I saw that they were not concerned in any specific way with geometry; we sometimes forget that there is a Section of the British Association which exists to rescue the rest of us from the generalities and platitudes to which teachers are only too prone. The other danger is of entering so far into particulars as to compose a syllabus round which we might be expected to quarrel instead of talking about principles on which we can hope to agree; here I am fortunate in the subjects chosen by the speakers who are to follow me, for where what I have to say is most in need of illustration, their expositions will provide you with examples worked out at the appropriate length.