1. The present paper is an attempt to sketch a mode of procedure for teaching Limits and Convergence to beginners. It is felt by many teachers that at present

(1) the subject is introduced too academically;

(2) that its Arithmetical foundation is not brought into sufficient prominence;

(3) that therefore pupils leave school with only a very hazy notion of what it is all about;

While it is clearly impossible for the average high school teacher of mathematics to become familiar with all the modern branches of this subject, it is desirable that he should not be totally ignorant of any extensive branch. The views of a number of eminent mathematicians often furnish one of the simplest as well as one of the most reliable introductions to the nature and the scope of a difficult subject. The following list of quotations has been prepared for the purpose of providing such an introduction for one subject. The list could easily be extended, but the variety and the standing of the mathematicians quoted are perhaps sufficient to inspire confidence. The qnotations are arranged chronologica1ly, beginning with 1892. several of them were published in the American Mathematical Monthly, volume x. 1903, p. 87

In order to make this brief paper as practical as possible, I shall plunge into the subject with little introduction.

I wish to consider the question from three points of view.

1. (1) What is the ideal course?

2. (2) What is the best practicable course?

3. (3) What effect the best practicable course should have on the teaching of the subject in our secondary schools, and also—since the two are so closely connected—in our elementary schools.

I wish to call attention to the value, for some purposes, ot the notation for the ratio ; and for the cross-ratio .

For instance: in Menelaus’ theorem for the property of a transversal meeting the sides of a triangle ABC in the points P, Q, R, the first mentioned notation makes the property shine out very clearly The equation in the form is

,

which conspicuously separates the points on the transversal from the angular points of the triangle.

The few remarks that I propose to make on this subject will hardly deserve the title of a “paper.” You are doubtless aware of the existence of the International Commission on Mathematical Education. The Commission owes its origin to the distinguished American mathematician, Professor D. E. Smith. At the International Mathematical Congress at Rome three years ago Professor Smith proposed that a “commission” be formed to enquire into questions of teaching, this Commission to report to the next meeting of the Congress, at Cambridge, on August 22-28, 1912. There have been intermediate meetings on the teaching part of the subject, and one meeting was held last September at Milan. It was not a general meeting, being attended mainly by official delegates from different countries. I had the honour of being present there, and it may perhaps interest you if I describe very briefly the matters that were under discussion.

WHEN your chairman, Mr. Jackson, asked me to supply a title for the Presidential Address, I could think of no name short enough and suitable for these random remarks, à bâtons rompus for half an hour.

Professor Turner gave the Mathematical Association an account of some new methods in calculation, found successful for Halley’s Comet; and Ray Lankester held forth in the afternoon on his perennial subject of the place of Greek in education, about which Turner has given you his say on a previous occasion.

I must avow myself at the outset in complete disagreement with their opinion, as in Greek we come to the bedrock of all our knowledge in history, literature, science, and art. Greek is the lighthouse to guide us out of barbarism to the human ideal.

The last half-century has seen a great and significant change in the popular estimation of mathematics. Formerly the subject was regarded as utterly unpractical, and therefore useless in the narrow sense of this term, though it was recognised as providing a training, unique in its character, in logical thought and accurate expression. Now it is regarded, and correctly regarded, as having enormous practical importance in science and engineering. Most, if not all, of those discoveries and inventions which are so profoundly modifying civic and national life have found their origin, or development, or both, in the labours of mathematicians, and this fact is widely known. The mathematician is no longer regarded as a dreamer of dreams; he is classed with the doctor, the engineer, the chemist and all those whose specialised labours have had immense import for the human race.

I believe I am the only surviving original-member of that A.I.G.T., to which the President bas referred (The Association for the Improvement of Geometrical Teaching). This Association arose out of a book which I wrote at Dr. Temple’s (the late Archbishop of Canterbury’s) suggestion, after the Commission of Inquiry into Secondary Education (the Endowed Schools Commission), that some new method, simpler than Euclid, should be brought into our geometrical teaching. It was very desirable that there should not be competition between private individual writers of text-books competing for public approval and public use, and so we formed the Association in order to unify the teaching on the new lines.

FEW branches of elementary mathematics have escaped reform in recent years; algebra, trigonometry, elementary geometry, statics, and the calculus have all been transformed to suit the practical needs of the time. But amid all this change, there have been few attempts to alter the methods employed in introducing students to the geometry of the conic. Dr. Filon, in his most interesting treatise on Projective Geometry published in 1909, points out that students learn the same facts about the conic practically three times over, (1) analytically, (2) through a course of what is called “geometrical conies” based on the focus-directrix definition, and (3) protectively ; and his book indicates a method of co-ordinating and uniting the last two systems of approach. Although it is undoubtedly the general custom in this country to start in the first place from the focus definition, it is worth while considering afresh whether this plan is really the most advantageous.

It is interesting to note that historically this was not the starting point of the early geometers. It is true that the focus-directrix property was known to Pappus, but so far from being regarded as fundamental, it was actually lost sight of altogether, until attention was called to it by Newton. The geometrical investigations of Apollonius were based on the conception of the conic as the plane section of a cone, the advantages of which plan may be enumerated roughly as follows:

I propose in this paper to deal with the development during the last twenty or thirty years of the dominating ideas of the reform of mathematical teaching in Germany, and must beg for your indulgence if the theme is one which is already familiar.

In the Mathematical Gazette of December last there appeared a translation of the so-called Meraner Lehrplan of 1905, which, with the appended notes, formed a kind of Declaration of Eights of the Reform Party. It is my intention to give some account of the movement which led to its publication, to indicate briefly the substance of these appended notes, and to give some details of the way in which the general ideas have been introduced into the curricula.