The object of the first conversation (Math. Gazette, March 1928) was to suggest the advantages of using a definition of right angle different from that of Euclid for the study of “spaces” of which the dimensions can be distinguished from each other, a very common type of space. For brevity we will refer to this study as “Dimensional Geometry,” and call the right angles “dimensional right angles.” The second conversation is intended to suggest that “dimensional spaces” facilitate the study of Non-Euclidean Geometry.

One of the charms of elementary geometry is that there is no theorem so well known but that one may hope to discover improvements or at least interesting variations in the proof; and the same thing is true of the constructions.

In order to decide between variations in these it is necessary to have some way of measuring the complexity of a construction. This is done by counting the number of settings of the ruler or compass and the number of lines or arcs drawn.

Every attempt to give an elementary rigorous course of Analysis is beset by several unsurmountable difficulties. Every teacher of the subject will readily agree that the two main obstacles are (i) Dedekind’s theorem, and (ii) the Heine-Borel theorem. Of these the first may be rendered less apparent, for in an elementary treatment there is no great objection to the postulation of the existence of an upper and a lower bound for every bounded linear set of numbers. The second cannot, however, be avoided, for the important theorems of the differential calculus, such as Rolle’s theorem, the Mean-value theorem, Cauchy’s formula, and Taylor’s theorem can only be rigorously proved by appealing to certain properties of continuous functions, which in their turn depend upon the Heine-Borel theorem, or upon some alternative simpler theorem on intervals such as the one which Carslaw uses to replace it.

We are greatly indebted to the correspondents who have supplied so much valuable information regarding the history and development of the finger-method of multiplication, and to our Editor for the summary of them which will be found in the notes in this number of the Gazette.

Introduction. Many students interest themselves in finding a function which is equal to factorial n for integral values of n. There is a certain fascination about this function : it is so different from the transcendental functions of applied mathematics.

Ladies and Gentlemen,—Before this meeting commenced, Professor Lodge asked me if I was doing much examining now, to which I answered “No, none.” Then he asked what qualification I thought I had got for speaking on the best method of examining. I do not want to speak much on the best method, but I think it is time that members of this Association had the opportunity of blowing off a lot of steam, and I hope that many will have something to say about the school certificate examinations. My desire is that many different people should give their views and their criticisms.

Voting, in some form or other, must have been one of the activities of primitive man: and one can easily picture the cave-dweller expounding, perhaps with illustrations on the wall of his dwelling, some of the difficulties of theory and practice quite as earnestly as we shall consider the subject to-day, and, I am inclined to think, with no less success. The announcement of the results of any system was probably no less exciting than the clash of spear on shield that declared the verdict in Athens, or the public vote by a show of hands. Secret voting was only permitted when individual persons were directly concerned. It will be remembered that John Stuart Mill’s judgment is in exact accord with the ancient practice: his illustration of the exceptional case being that of club membership, where the “blackball” of the old-fashioned ballot-box still represents the pebble of ancient Greece. In Rome voting “by classes” was frequently used, until corruption led to the domination of the wealthy; and a form of ballot by the tabella was used about B.C. 120.

When it is said that Descartes was the inventor of analytical geometry, the impression is often given that by Cartesian Geometry is simply meant a system in which the position of a point is determined by means of its coordinates referred to two fixed straight lines, either at right angles or oblique. A little consideration shows us that it is not correct to say that this system was invented by Descartes, as we find that Apollonius, Bk. V, Prop. 52, gives the values of the coordinates of the centre of curvature of a point on a conic referred to two rectangular axes. It will no doubt interest the readers of the Mathematical Gazette to know what is was that Descartes invented, and what led him to it.

The work of writers like Rouse Ball in England, and Cajori and D. E. Smith in America, in revealing the fascinating story of the history of- arithmetic, has shown the subject developing slowly but surely along the corridor of time. This development, which culminates in present-day arithmetic, proceeds not only by increase of subject-matter, but involves suppression and modification both of matter and methods. It is not the haphazard growth of the wild plant, but the steady and systematic development which results from the pruning and shaping of the skilled and sympathetic gardener.

It has been suggested that a sketch, limited to the real plane, of the theory of homogeneous coordinates and points at infinity which is developed for complex space in my Prolegomena to Analytical Geometry, would be of interest to teachers who without wanting to go deeply into the matter are humanly discontented with the crude treatment that makes of infinity “the place where things happen that don’t”.