1. In order to understand the meaning to be attached to Non-Euclidean Geometry and the use of the model measuring tape described below, it is advisable to say a few words upon Euclidean Geometry.

Teachers of elementary Mathematics are generally very glad to find some unfamiliar field into which they can turn their pupils to exercise themselves in elementary processes. The following is a suggestion of a method of approaching certain theorems in connection with the Geometry of the Triangle which, although well known to Mathematicians, are not, as a rule, studied as part of a school course, except perhaps by specialists. The Brocard Points and Circle are generally approached, via the theory of Isogonal Conjugates (see Casey’s Sequel to Euclid, Supplementary Chapter, Section I. etc.). But many of the properties connected with these points can be obtained in a more direct manner, and this suggests some interesting investigations involving nothing but the ordinary processes of Geometry and Trigonometry.

Some teachers may not have noticed that there is a real connection between the parallel postulate and another and more fundamental property of space which it is also more natural to assume, viz. that stated by John Wallis in the seventeenth century:

If particles of positive mass lie on a continuous curve whose concavity is always upwards, their centroid clearly lies above the curve and below the chord joining the extreme particles.

This obvious property gives a simple formula which includes several of the standard inequalities of the Algebra text-books.

Many of us who learned Euclid in our schooldays, feel that something substantial has been lost in the training in precise expression and logical statement that it gave. Even those mathematical teachers who would admit no such thing, must admit, if their experience is anything like my own, that many of their colleagues, particularly science teachers, have deplored the loss of what was at least a standard of reference for logical form.

Whatever theory is finally adopted for the propagation of light, there will always remain the interesting fundamental problem as to how far an object is exactly reproduced by an optical system. In this paper, the enquiry will be considered in its simplest form, and only the simplest types of objects can be considered. Taking the wave theory, let us consider the reproduction of an infinitely distant self-luminous point source, such as a star, through an optical system, such as a telescope. Sir George Airy investigated this case in 1834 ; he shewed what W. Herschel had recorded in 1782, that the image in the focal plane f of the optical system consisted of a central concentration of light, surrounded by light rings of rapidly diminishing intensity (see Fig 2), and gave formulae to shew the falling off in intensity of the light away from the centre of the image. This disc of light, then, is the approximation to the point-image of geometrical optics. The deformation is due to the distortion of the sphericity of the light waves making the image by diffraction at the aperture of the system. As in the propagation of wireless waves, however, diffraction is not the only cause of distortion.

The systematic treatment of these systems by this method has been strangely neglected hitherto by English writers, the only text-book, as far as I am aware, which deals at all with the subject being Salmon’s Conics, sixth edition, 1879, pp. 389, 390, where the states that the properties have been studied by de Jonquières, Chasles, Zeuthen, and Cayley, and that references to the original memoirs will be found in Cayley’s memoir (Phil. Trans., 1867, p. 75). This I have not had the opportunity of consulting. In this article I have followed the Danish mathematician, H. G. Zeuthen, who selected the subject for his Doctor’s Thesis of 1865, which was translated and published in the Nouvelles Annales de Mathématiques in 1866, p. 241, with the title “A New Method of Determining the Characteristics of Systems of Conics.”

Very few teachers ever have the chance of noting the advantage or disadvantage of a change, introduced into their methods of teaching, and of profiting by the same. I desire now to put on record the results of such a change and the method by which they were attained.

The number of systems of teaching dynamics is great, and still the results are deplorable as judged by the knowledge possessed by the average engineer some years after he leaves college. Teachers are either biassed in favour of the Gaussian system or the Perry system, or some other system of their own, and are loath to change because they can train their students to solve the necessary problems, and if a mistake is made by omitting g or some other quantity it is merely a mistake and costs nothing. It is not so with the poor student, however, when he is launched on the world without his teacher to help him.

In a system in which we have given a circle and the relations between it and certain points and lines, certain changes take place in these relations in another system in which a conic takes the place of the circle. To enable these changes to be more easily seen and interpreted, the writer has found it very convenient to use certain modes of expression.

The recent discussion of this subject * cannot fail to have important effects. It must certainly have aroused in many a new conception of the part Similarity should play in Elementary Geometry.

In this article—which embodies an integral part of work on elementary geometry, most of which has recently been published —a presentation of the ideas of Parallelism is given, which is designed to bring the conviction on this subject that has too commonly been lacking, and to show how central its place is in Euclidean Geometry. Together with the discussion of the Plane theory, an indication is given (in 5) of a sufficient elementary treatment of Similarity: “sufficient,” in no disparaging sense, but in the sense of sufficing for every reasonable need at the stage in question.

There is a large and well-defined group of students, in polytechnics and elsewhere, taking a course of mathematics which is very different from that given in schools. Practical Mathematics, as the subject is usually known, is regarded very unfavourably by many mathematicians, and no doubt it has many defects. However, it is possible that some of its critics have failed to realise the difficulties that occur in connection with evening classes. The present article is intended to show that Practical Mathematics meets a real need, and that it may be made really educational. No attempt will be made to conceal the grave weaknesses that often occur in the actual working of the classes. Some suggestions for improvement will be given, and it is hoped that those with experience of the work, either as teachers or inspectors, will give their views on the subject. Ultimately the Mathematical Association may be able to issue a report that will incorporate the most helpful suggestions received.

Construction. As we depart widely from Euclid’s ideas in the proofs of Stage B, it seems to be thought by many people that go-as-you-please methods are good enough for constructions as well. Now, “trial” methods of construction are sometimes useful in more advanced work ; but they are tolerated only when a geometrical method is impossible ; they are means to get a solution, but there is no instruction to be had from such a method. We would therefore urge that a certain severity be still maintained in problems of construction.