Had I been called upon to address the Association a few years earlier on a subject of this nature, my task would have been much more difficult. Model atoms were then becoming almost as numerous as the sands of the sea. The very simplicity of the quantitative phenomena shown by the simplest atoms was an obstacle to progress. For example, the simple spectrum of Hydrogen,— the so-called Balmer series,—consisting of a set of frequencies given by the formula

where n and m take all possible integral values, was somewhat too simple if considered by itself instead of in relation to phenomena of a widely different type. And as a consequence, we had at least five completely different forms of suggested structure of a Hydrogen atom, each of which, by suitable mathematical artifices, could be made to yield such a set of frequencies.

This is a remarkable book in many respects, and well worthy of the attention of those teachers of mechanics who have not yet made its acquaintance at first hand. Though written in 1784, some of its pages are greatly superior to those in many of the text-books now in use in schools, and one is particularly impressed by the stress laid on experimental verification of the various laws of mechanics, and by the extreme care shown in the planning and execution of the experiments proposed. The following quotations from it and the notes thereon are given, because I feel sure that they will prove of use to those who want Mechanics to become a more interesting subject to their pupils and to stand for something more than what is usually covered by the name “Applied Mathematics.” I have added a few parentheses, but only, I hope, where they were necessary.

The determination of the group of partial fractions corresponding to the powers of a linear factor x+b in the denominator of a rational function f (x)/{(x+b)nφ(x)} is one of the comparatively rare algebraic problems whose theoretical solutions are feasible in practice. The numerator of (x+b)n−r is the coefficient of yr in the expansion of f(−b+y)/φ(−b+y), and the calculation of the group of numerators is effected by means of two Horner transformations followed by one division, which can of course be arranged synthetically; no operation need be taken further than the term in yn−1, whatever the degrees of f(x) and φ(x).

In a previous note it has been pointed out as a corollary to the proof of Ptolemy’s Theorem that, if in a cyclic quadrilateral ABCD, AB.CD = BC.DA, and if M be the mid-point of the diagonal BD (2m),

and

so that A and C are each the image in DB of the inverse of the other w.r.t. M, the constant of inversion being m2.

Euclid does not define the inclination to one another, or the mutual perpendicularity of non-intersecting straight lines. (For brevity I shall omit the adjective straight before lines.)

The following definition is justified by the 10th Proposition of Bk. XI. Def. : The inclination of two non-intersecting lines to one another, or the “ angle between them ” is the angle between two intersecting straight lines which are parallel to them respectively.

If it be granted that the study of trigonometry in the secondary school course should not be limited to the numerical solution of triangles but should include some treatment of form as an introduction to higher mathematical analysis, then the importance of the trigonometrical identity must be admitted. A certain amount of valuable training can be obtained from the proper study of the identity if this study be systematic; but there seems little to be gained by working through the customary set of miscellaneous examples inserted at odd intervals in the trigonometrical text-book.

A Study of the questions on this subject set in the recent geometry papers of the London Intermediate Examinations will show that there is considerable danger that what should be the chief aim in the subject may be subordinated to the subsidiary aim of affording practice in evaluating and plotting results.

The chief aim should be, I think, a careful study of those important curves in Physics and in Higher Mathematics which best lend themselves to trigonometrical rather than to algebraic treatment. In the actual plotting of these curves much varied practice in the use of tables will, of course, be obtained; but this should be considered of distinctly minor importance.

Principles. 1. If K be the centre of a conic and P a point not on it, the diameter conjugate to KP is parallel to the polar of P.

2. Conjugate diameters form a pencil in involution. Hence if S=0 and S' =0 be the equations of two known pairs of conjugate diameters, the general equation is S + λS' =0.

The method then is to find the diameters conjugate to KA, KB, KC, where ABC is the triangle of reference, by means of 1, and then to write down the general equation by means of 2.

The solid which is the three-dimensional analogue of the plane quadrilateral appears to have been neglected by geometers, possibly on account of the difficulty of visualising it adequately from a plane figure. A model, however, makes an interesting object and is readily constructed from thin stiff wire and thread. The extraordinary number of collinearities, which are its principal feature, then become at once obvious. For the benefit of any one who wishes to construct such a model I would suggest the desirability of commencing with the external diagonal triangle PQR and the line P′Q′R′ (see Figs. 1 and 2), as otherwise it is difficult to get a compact model.

I fear you may think us rather unpractical people in Hull, for while my colleague, Mr. Forder, will presently speak to you on the subject of infinity, my own topic has not a little of the unreal or imaginary involved in it—but at least, if I become imaginary and Mr. Forder tends to infinity, we shall not, like ‘I’ and ‘J,’ be both at infinity and unreal at once. My own thesis is not a new one, for it is at least as old as Plato. It is a claim for greater attention to the more beautiful aspects of mathematics. What one feels is that most people give to Mathematics and Science very definite and, as I think, very limited spheres of influence in the educational domain. In their view it is the duty of school science work to beget accuracy of observation, skilful manipulation and the power to make general deductions from observed facts—it is an essentially materialistic and utilitarian view. As a matter of fact, Natural Science deals with the profoundest mysteries. So, with them, Mathematics is a science cold and austere—ruthlessly precise and devoid of imagination. And yet, as a matter of fact, it deals with truth and beauty, its forms are those of rhythm and symmetry, and in it imagination has no limits. Now I fear very much that we who teach these subjects are apt to accept without demur these low valuations of mathematics and science as means of general education.

My desire is to emphasize the importance and simplicity of the notion of “differentials” in introducing a student to the calculus. I find that pupils have but little difficulty in realizing the ideas connected with them, and that the notion of a differential coefficient as the ultimate value of the ratio of two connected differentials comes automatically.

Vectors in two dimensions are generally studied in schools; I do not suppose that the study is often pushed into three dimensions. But it is difficult to understand a modern book on electricity without some knowledge of three-dimensional vector notation.

The particular point that I wish to bring before you is that the two-dimensional case is often presented in such a way that its relation to the more general case is obscured.