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.

To begin at the beginning, Euclid built up a system of geometrical theory two thousand two hundred years ago. What perfect clearness there is in his logical procedure! The logic is not only sound, but evident. No wonder that Spinoza wanted to write a geometry of ethics ; the reason was that he wanted a perfect logical sequence.

[71. vol. v. p. 330] Mr. Shovelton’s proof (vol. vii. p. 153) may be shortened by using central-difference operators. The theorem itself may be made clearer by splitting it up into two:

§ 1. The Anharmonic Ratio expressed in terms of Squared Distances. In Cremona’s notation, the symbol (AB, CD) stands for the quantity

where ABCD are four collinear points, A, B and C, D are Conjugate pairs. It is easy to show that

There is, of course, nothing really new to be said about the elementary theory of the Elliptic Functions sn, cn, dn; but it may be not too late in the day for a didactic note on the existence and one-formedness of these functions to have some usefulness. One reason, and perhaps the chief one, why the older analysis has fallen into disfavour in the treatment of elliptic functions is the obscurity on the subject of one-formedness which prevails in so many presentations of the early parts of the theory. Halphen’s Fonctions Elliptiques is exceptionally valuable in that it neither avoids nor postpones the difficulty; but it is common for analysts who follow the older methods either to pass the matter over without proper emphasis, or to confess weakness by appealing to the ideas of the general theory of functions, thus encouraging a belief that the newer analysis is absolutely necessary for soundness.

The following paper was read at the meeting of the London Branch, which took place on the 8th of March last at the London Day Training College. It contains in outline the substance of the first few chapters of A School Course in Geometry, to be published shortly by Messrs. Longmans, Green & Co. Many details, necessarily omitted in a short preliminary paper, will be found fully treated in the forthcoming volume.

Suppose that a curve Г of constant shape moves in space so as to trace out a surface Σ. The problem has suggested itself to various authors whether it is possible for Г to be always a line of curvature, geodesic, asymptotic line, etc., on Σ. The following elementary method of solving such problems may be of interest.

When, with the other members, I received a request from the Secretary, it seemed only fair to him that I should express my willingness to do whatever might be in my power towards the provision of a subject for discussion ; and therefore, with an earnest wish that he would if possible ignore my suggestions, I proposed the rather well worn subject of this paper. To my dismay he did not ignore the suggestions, and in consequence you will have to listen to the trite observations of one who, sufficiently remote from great centres of education, and in great measure precluded from intimate contact with modern methods, is nevertheless sufficiently interested in them to form theories, or rather opinions, of his own. Those to whom the element of novelty may be absent in the points that I proceed to consider will, I trust, extend a charitable reception on the great moral principle of the Western States that the pianist who does his best is not to be shot.

In the ordinary course of lecturing to students who come stamped with the approval of some of the entrance examination Boards, and their name is legion, I find that, especially when they have had a more or less practical course, very great difficulties arise on account of their weak grasp of mathematical reasoning.