When we write a number in the ordinary way, such as 673, the symbols such as 6, 7, 3 in this example stand for what we call the digits of the number, and a piece of mechanical or electrical equipment which operates with and records the discrete digits of each number is often called a digital calculating machine. In the past twelve years there has been a remarkable development of such machines with two important features. First, they can carry out lengthy and intricate calculations quite automatically once they have been supplied with a specification, in suitable form, of the calculation to be carried out, and secondly, they are very versatile, so that the same machine can be used for many quite different kinds of calculation, for example for evaluating values of a function from its power-series expansion, for solving large systems of linear simultaneous algebraic equations, for finding the characteristic values of matrices, for the step-by-step integration of ordinary differential equations. To express these two features, such machines are sometimes called general-purpose, automatic, digital calculating machines, and my purpose is to give an account of some aspects of such machines, both of the machines themselves and of the way in which they are used. I can only touch on some aspects, since there is not nearly enough time in a single lecture to deal with the subject fully.

It is a far cry from the Conics of Apollonius to the theory of linear spaces and matrix algebra and many men have helped to build the road between. This article is about one of the builders whose work for a time was discarded and whose name was almost forgotten. But his name finds frequent mention now, when the latter aspect of geometry bears so little resemblance to the former, and metrical geometry has given place to projective geometry with its non-metrical homogeneous coordinates, which make it possible to publish a book on coordinate geometry which does not contain a single figure.

We seniors are apt to be told that, in our teaching of analytical geometry, we have been in the habit of giving too much attention to the intricacies of conic sections, and that work on other curves and their tangents and normals would have been more interesting and instructive. There is much force in this criticism, but when we come to examine the alternatives to the conics which are offered to us, they are usually cubic curves, probably given in terms of parameters, such as the “Folium of Descartes” I should like to suggest that some better-known curves, notably the curve of sines and the logarithmic curve, will provide examples even more instructive than the “Folium”, and that these examples have at least the advantage of greater novelty.

1. In a triangle ABC of area Δ, BC = a, CA = b, AB = c are the sides; 0 the centre and R the radius of the circumcircle; ρ1 and ρ2 the radii of the first and second Lemoine circles; σ the radius of the Brocard circle; O9 the centre of the nine-points circle; Ω, Ω′ the Brocard points and ω the Brocard angle; K the Lemoine point; G the centroid; H the orthocentre; ma, mb, mc the medians; ha, hb, hc the altitudes; AL = sa, BS = sb, CT = sc the symmedians corresponding to the sides a, b, c; 2∝ and 2β the axes of the Brocard ellipse of foci Ω, Ω′ and centre V; 2x and 2y the axes of the Lemoine ellipse of foci G, K and centre ϕτ and τ′ the Torricelli segments (the invariant segments which join a vertex of ABC to the vertices of the equilateral triangles constructed on the opposite side, internally and externally to the fundamental triangle); p(X) the power of a point X with respect to the circle O(R).

The object of this paper is to discuss some properties of the Isosceles Tetrahedron, in which each edge is equal to its opposite. The last three theorems are possibly new, the remainder are included for their own interest or for completeness in listing elementary results; in the latter case, proofs are not given.

The mistakes of great men leave long shadows in history Euclid’s proofs in the main are elegant and concise, and apart from style of presentation, with slight alterations form the basis for most modern geometrical texts. Exception must be taken, however, to the earlier introductory propositions. These have been found unsuitable for teaching to children. Some have sought to explain this away by saying that Euclid wrote for mature adults, but that children of 11 or 12 cannot comprehend logical proofs. From other eminent authorities, however, come criticisms of the proofs themselves. Thus Russell says that Euclid’s proposition I, 4, is a tissue of nonsense. Undoubtedly Euclid’s earlier proofs fall far short of perfection, but, perversely enough, the criticisms which have been levelled at them are even more mistaken than are Euclid’s proofs.

A description of the construction and use of the differential analyser has already been given in the Mathematical Gazette by Professor D. R. Hartree (Vol. XXII (1938), page 342); this note arose from the consideration of means of depicting results to problems depending on three dimensions. In using the machine on what are essentially three-dimensional problems (e.g. orbits and trajectories not lying in a plane), results can usually be presented only in tabular form. As a matter of some interest in this field, consideration was given to the possibility of using the machine itself to draw a perspective diagram of such twisted curves in three dimensions. If this were found practicable it was realised that with two such diagrams, representing the space-figure as seen from two points separated by the distance between the eyes, a stereoscope could be used to reproduce a three-dimensional effect.

Every geometrical truth has an individual beauty which delights the mind: but the most fascinating theorems, as an eminent mathematician rightly asserts, are those which have simplicity, depth, and obvious generality I propose to show that nearly all the important circles of the geometry of the triangle—the in-circle, the nine-points circle, the circum-circle, Taylor’s circle, Lemoine’s circle, and many other useful but nameless circles—are each and all particular cases of one fundamental general circle.