Few subjects deserve attention better than the problem of teaching higher mathematics to the average student.

Any man who does work in the world will some day, even though he may be a classical scholar investigating variants in manuscripts, be the worse for not knowing more mathematics ; and if he is not too closely limited by his environment he may even realise this fact. A goodly proportion of the work of the world must be done by quite ordinary folk. Something will be gained if the average student is helped even a little by being enabled to use the calculus and to think of its machinery as an engineer thinks of a drill, and not as a West Coast native thinks of a Ju-ju.

An illustration may be taken from the Astronomer Royal's recent text-book. A star of the third magnitude is very much brighter than a star of the fourth magnitude. But the fourth-magnitude stars are more numerous than the third-magnitude stars, so that in fact we receive more light from the fourth-magnitude than from the third-magnitude stars. To raise all fourth-magnitude stars to the third magnitude would increase our light more than to raise all third-magnitude stars to the second magnitude.

My first duty is to thank you for the honour you have conferred upon me in electing me as your President for the year. Once all mathematicians were astronomers and all astronomers were mathematicians. But now only some astronomers are mathematicians, and I fear that I can only claim to be one to a very limited extent. It was, however, my good fortune to be a member of a college where your ex-President was a lecturer, and to have learned there, from him and others of a very distinguished society, something of the pleasure to be obtained from the study of Mathematics.

Of all the problems which have perplexed teachers of mathematics in this generation, probably none has been more irritating and insistent than the choice of assumptions which must be made in each branch of the science. In geometry, in analysis, in mechanics, one and the same difficulty arises. Are we to prove that any two sides of a triangle are greater than the third? That the limit of the sum of a finite number of functions is equal to the sum of their limits? That the total momentum of two bodies is uninfluenced by their mutual action? And in every such case, on what is the proof to depend? A clear understanding of the answers to such questions, or better still, a clear understanding of principles by which answers may be found, would go far to co-ordinate and simplify elementary teaching; the object of this paper is to state such principles, and indicate their application.

In his interesting article on Newton in the July Gazette Dr. Rouse Ball had space only for a slight reference to Newton’s views on what we should now call the Foundations of the Differential Calculus. The subject “ involves,” as Dr. Rouse Ball has remarked, “ some awkward questions of philosophy which before Weierstrass’s researches were usually slurred over.” Some of the logical difficulties incident to Newton’s method of approach were clearly seen and pointed out by a contemporary, the celebrated Bishop Berkeley, in a tract which drew much attention at the time, and was followed by a considerable controversy.

Another useful application is that the parabola, in the form y = –k.x2, represents the path of a projectile. The boys at this stage have already dealt with equations of the form y= ±(x–a)(x–b), and are able to find the points where the curve is cut by the x or the y-axis.

The current elementary discussion of the Simple Pendulum is unsatisfactory, in that the problem is not reduced with sufficient directness to a case of S.H.M. It has to be artificially prefaced by consideration of a curvilinear motion in which the tangential resolute of the acceleration is proportional in magnitude to the arcual distance from a point of the path; and the discussion of this motion raises new points of difficulty altogether out of proportion to its significance in this connection. The net result is that the student’s appreciation of this first of the important applications of S.H.M. to “small oscillations” is lost, in dismay at finding that, even after he has mastered the essential difficulties of the S.H.M. itself, the first good application of it brings yet another awkward hurdle. And the physicist or engineer may well make this another case for railing impatiently at the devices for “dodging the Calculus” by which elementary theory is so apt to be obscured. Nevertheless, the teacher of Mathematics knows how important it is to postpone such Calculus difficulties as, e.g., to pave the way to the complete primitive of the S.H.M. differential equation and its uses (one of which gives the only clear-cut way of handling the ordinary discussion of the Simple Pendulum). But the postponement must not be obtained at the disproportionate cost of artificial complications which, for the sake of “elementary” treatment, cast a fog round the important features of the argument, without contributing anything of independent value to the student’s store of knowledge.

It cannot be said too often that each portion of a mathematical curriculum must be chosen with some definite purposes in mind, and that these must be the controlling influence in the scheme of development. Such purposes may be roughly divided into two classes : the first includes all desires to illustrate or elucidate general ideas and principles, and the second includes all desires to relate one part of the subject to another, or to some other domain of thought. It is not here implied that possible fulfilment of some such purposes enforces the inclusion of any particular branch of mathematics ; its difficulty may be too great for the students in question, and in any case only a limited amount can be included in a given time-table. But it is very definitely intended that no branch shall be included merely because it happens to be sufficiently easy or to tickle the fancy. There is ample real and human interest to be found without sacrifice of purpose ; and while undue difficulty is to be deprecated, it must always be remembered that definite achievement resulting from continued effort is of the very essence of the learning of mathematics.

The majority of the boys who enter the Technical Day School of the Borough Polytechnic Institute come from elementary schools, a few only come from secondary schools. The general age of entry is about 13 years, and the boys are about up to the level of the VIIth standard of a good Council School.

As far as we are able, we select for admission boys who have a mechanical or constructive bias. All the boys who enter are expected to take the full three-years' course of work. At the end of that time they go into engineering workshops.

If we wish to develop a race of geometers, I think we should begin by encouraging the habit of speculation. This we do nowadays by using squared paper for plotting graphs. Here, however, we place ourselves very much in the hands of the printer : and although ideas of geometrical shape are enlarged and systematised, the right angle plays too great a part in whatever knowledge of curves is acquired ; indeed a knowledge of curves is not the professed object in plotting graphs so much as a visualised representation of corresponding changes in related physical quantities.

In teaching elementary geometry there lingers an aversion from the use of dividers in pricking out equal distances : for the transference of distances is not recognised in the postulates of Euclid, and Euclid's influence remains.

Later on a study of the conics, carried on for a prolonged period, and the application of the theorems of Pascal and Brianchon, must exercise their fascination. Ideas about infinity become clarified : the use of parallels is seen to be necessary, to exercise the right accorded by Euclid's first postulate to join any given point to any other, even if one be at infinity. But the mind inevitably tends to become obsessed with the second-degree idea. Mere straight-edge methods are found to be so potent in the construction of conics that metrical properties are neglected. Primary notions about equal distances interfere with the whole scheme of a projective geometry confined to the conics ; and the bisection of a given finite straight line becomes a search for the harmonic conjugate of the point at infinity in its direction, with regard to its two ends.

Some time ago, when attempting to generalise certain methods of solving differential equations by operators, the writer came across the following mode of deducing Fourier’s Theorem symbolically.

The method is so simple that it seems improbable that it should be unknown. The nearest approach to it known to the author, however, occurs in Oliver Heaviside’s Electromagnetic Theory, Vol. II. §§ 275 et seq., but the investigation there proceeds on somewhat different lines, and the interpretation of the operators used requires to be carefully defined.

It does not seem to be sufficiently realised that most of the theory usually classed under “Centre of Gravity” is purely geometrical in cliaracter ; and, on the other hand, such impossible terms as “centre of gravity of an area” still offend the eye. This paper suggests a presentation of the subject free from such defects ; but its main object is to establish a sounder transition from the treatment of discrete point-systems, which is the essential groundwork of the theory, to the much further reaching treatment of continuous point-systems. The fact that this transition remains in an unsatisfactory state is an instance of the reluctance to face the necessity for explicit recognition of certain fundamental ideas of the Infinitesimal Calculus, in elementary Mathematics. It is not long since the proper conception of tangent was admitted as suitable for an elementary course of Geometry. Still more recently the more abstruse conception of arc-length has been faced. From what follows it may appear that the conception of centroid awaits its turn.

The inhabitants of the three kingdoms which constitute Great Britain did not come into contact with the civilisation of the ancient East. Nor, as far as we know, did they have the advantage of direct relations with the Greeks—that race, beloved of Nature, which succeeded in placing all the sciences and all the fundamental arts on so secure a basis as to defy for two thousand years any attempt at radical innovation.