It is well known that the St. Andrew’s cross (X) occurs as the symbol for multiplication in W. Oughtred’s Clavis Mathematicae, 1631, and also (in the form of the letter X) in an anonymous Appendix which appeared in E. Wright’s 1618 edition of John Napier’s Descriptio. This Appendix is very probably from the pen of Oughtred. The question has arisen, is this the earliest use of X to designate multiplication? It has been answered in the negative—incorrectly so, we think, as we shall endeavour to show.

Much has been said about the danger of dividing the various branches of pure mathematics into “water-tight compartments,” and books have been written with a view to breaking down from the very beginning the barriers between the different subjects.

I feel, however, that to the average boy who has not yet reached Matriculation standard, the division of Mathematics into Arithmetic, Algebra and Geometry is very real and most helpful. A straight-forward question inmensuration, the study of the identity (a+b)2=a 2+2ab+b 2, and the theorem that the square on a line is equal to the sum of the squares on its two segments together with twice the rectangle contained by them, for example, will and should appear as separate questions—the first being linked with other similar exercises in Arithmetic, the second depending on algebraical multiplication and being a study in form rather than in area, and the third turning on a geometrical construction. Only after each has been treated in the appropriate way should we show the pupil the cross-connections existing in the three problems.

There can be few teachers who, when taking part in a discussion on the relative merits of two methods of teaching a given topic, have not wished that it were possible to produce in evidence a clear, accurate and unbiassed account by the child himself of the way in which the particular method under discussion affects him. As it is, we are forced to rely on the experience and insight of the teacher—experience, which is at best very limited, and insight which, if it is often true and sympathetic, can never fully penetrate into the child mind, and is moreover scarcely ever free from a prejudice in favour of one method or the other.

I desire to suggest to those of your readers, who are giving instruction in the Elements of Geometry, that they should try an experiment when they arrive at the stage in which it is necessary to introduce the Postulate of Parallels.

In accepting the honour of being your President, I was told that the office carried only a single duty—that of delivering an address at a then distant date: even the selection of the topic could be deferred for a time. So there lay before me a primrose path of dalliance. I had some hesitation about the selection of a topic. One decision, however, was immediate; my determination was instantly framed to avoid any discussion of the place of mathematics in education, an unending subject, of which perhaps you have heard only too much. It would have been possible to select some special topic, such as functionality, in its historical development; and you would have had a mere mathematical lecture. I have preferred to choose a general subject: my text seems highly technical in phrase, but I shall not keep very closely to it; my purpose rather is to fall back upon my own experience over a wide range of teaching and to exhibit for your consideration some reflexions based on that experience. Moreover, technical as its title is, my address will contain no mathematical symbols. Though my work as a teacher and writer is largely associated with differential equations—probably in the course of a year I receive (with a request for solution) a greater number of insoluble equations than any one here—the reason for its selection is not to be found in any specially personal relation. The reason is that the particular processes affect many subjects in mathematics, and consequently allow general comparisons to be made, as well as particular inferences to be drawn: they constitute a method as powerful in operation and effective in results as any within our domain.

The strength of analytical methods is generality. It is not quite easy to bring out effectively the essential generality of elementary Analytical Geometry, In that connection the following points appear to be of fundamental importance:

1. (i) If P, Q, R are collinear points, the ratio PR : PQ is positive or negative, according as the directions * PQ, PR are the same or opposite.

(ii) (1) Cartesian coordinates of the representative point P of a given plane are determined by reference to chosen axial lines xOx′, yOy′ in the plane, and the question of sign may be put thus, using (i) : If A, B are the points at unit distance from O on the half-lines Ox, Oy, respectively, and if the parallels through P to the axial lines intersect the x-axis at M and the y-axis at N, the coordinates x, y of P are given by x = OM: OA, y = ON: OB. They are essentially one-valued functions of the position of P.

The very great importance to Elementary Euclidean Geometry of the figure intermediate between the Straight Line and the Triangle appears to have been missed: viz. of the plane figure formed by two intersecting straight lines, each unbounded both ways. For this type of figure we use the name “Complete Angle.”

In the address which he had the honour of giving at the Annual Meeting of the Association the object of the writer was twofold: (1) to urge Secondary School teachers to study non-Euclidean Geometry for the purpose of improving their own scholarship; and (2) to sketch a method of approach to the subject which seems both interesting in itself and much easier than that usually adopted. In preparing this paper for the Gazptte he has thought it better to omit the former and to develop the latter a little more fully than time allowed at the meeting; the title has been modified accordingly.

Taking ABC as the triangle of reference, with the usual notation for its sides and angles, let l = s − a, m = s− b, n = s − c. Let I, I1, I2, I3 be the centres of the inscribed and escribed circles DEF, D1E1F1, D2E2F2, D3E3F3.

Through A, B, C draw parallels to the opposite sides forming the triangle A1B1C1, so that AAl, BB1, CC1 are the medians of the triangle ABC, and there are then ten triangles like ABB1, ACB1, etc., of the same area.