In your Mathematical Gazette for last month you read that “the mathematical theory of Probability is practically dead”—and that from a real authority on the subject whose name is known and honoured wherever statistics are analysed and laws deduced from them.

If z is a function of y having a differential coefficient at a certain point, and y is a function of x having a differential coefficient at the corresponding point, then z is a function of x having a differential coefficient at that point, and this differential coefficient is given by the formula

This theorem, thought not always stated so carefully, will be found in all texts on the calculus; but, so far as I am aware, no English writer, not even the most exact, offers a valid proof.

The Report on “The Teaching of English in England” lays great stress on the important truth that the use of English concerns all teachers, but when it deals with the matter more closely it enumerates teachers of history, languages, geography and science, and says nothing about mathematics (e.g. § 110).

Among the complicated rectilinear patterns employed in Saracenic art there is a group which may be described as “geometrical arabesques,” drawn by a curious method which may be of interest to mathematicians.

There is little doubt that many teachers are seriously bewildered by this Report. It is not an easy document to read and frequently it is difficult to ascertain the precise nature and application of its recommendations. It suffers from attempting to cover too much ground: the reader is confused by the double purpose it tries to serve, (i) instruction on the axiomatic basis of geometry for teachers who have made no special study of the logical foundations of the subject, (ii) pronouncements on practical procedure in teaching and geometrical method.

The main object of the present note is to direct attention to the fact that Lill’s method for solving geometrically the general quadratic equation with real coefficients can be used effectively for the purpose of impressing upon students that even in elementary mathematics some very useful advances have been made recently. In 1867 E. Lui published an article in volume 65 of the Paris Comptes Rendus, from which the following method for solving the equation

In teaching Analytical Geometry very little attention is often given to the interpretation of the form of an equation, although this may often convey a good deal of information; by grouping the terms of an equation in different ways, a variety of useful and interesting results may be obtained with very little labour.

Some suggestions as to the possibilities of this method may be afforded by the cases given below. A considerable number of results are obtained, purely by simple manipulation of the general equation of the conic section in point and line co-ordinates. Except in one instance, no reference has been made to infinite elements; if we include the notions of infinite points and lines, and of the circular points, the power of the method increases very greatly. The only knowledge that has been assumed is that which is quite fundamental in point and line co-ordinates, and the meanings of certain simple typical forms.

I.The first obstacle may be one of sentiment. It is said that in a certain grassy part of the world a man will walk a mile to catch a horse, whereon to ride a quarter of a mile to pay an afternoon call. Similarly, it is not quite respectable to arrive at a mathematical destination, under the gaze of a learned society, at the mere footpace of arithmetic. Even at the expense of considerable time and effort, one should be mounted on the swift steed of symbolic analysis.

The function of the mathematical laboratory can best be understood in the light of the experience of many mathematical teachers. Anyone who has devoted even a comparatively short time to the mathematical training of workers in any branch of applied science must have had the experience that many past students return at frequent intervals for advice and consultation. The difficulties with which they are confronted appear invariably to be associated with the application in more or less concrete form of the mathematical principles and methods which they have learned. I find, as a matter of actual experience, that the complaint, when it is voiced, concerns itself not so much with the width of the mathematical field covered during training as the inability to know when to apply this or that mathematical method, how to attain the so-called solution, and how, when the mathematical solution has been obtained, to derive from it the desired information with reasonable expenditure of labour. It is partly this gap that the mathematical laboratory seeks to fill. An engineer confronted with a problem in heat conduction will not be satisfied should you provide him with an expression for the distribution in temperature in terms of a series of Bessel functions, unless he has at hand the means of immediately calculating to so many decimal places the values of these functions, either from a set of tables or by some direct mechanical process which he can apply. The form of solution provided may be to him more difficult to interpret than the original problem. It is partly because in the past mathematical teachers have been content to regard a problem as solved when, somehow or other, an analytical solution has been obtained, or to regard a problem as insoluble when such a solution cannot be obtained, that the failure exists on the part of the students of that training to utilise the mathematical weapons at their disposal, to modify them according to circumstances, and to interpret their results in actual cases.

Lemma. If p is a prime number, the sum of the products of the numbers 1, 2, 3, ..., p - 1 taken r at a time, where r is any number less than p - 1, is a multiple of p (Lagrange).