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 equation considered is of the type

and f(δ) and g(δ) are polynomials in δ.

There are three cases to consider These are:

(I) The roots off f(z) = 0 are distinct and no two differ by an integer.

(II) Some of the roots of f(z) = 0 are co-incident.

(III) There exists an integral difference between some of the roots of f(z) = 0.

I will illustrate the kind of problem I am going to discuss by showing how it applies to the case of addition. Suppose we add a series of numbers, each correct to the nearest unit ; this may be tenths, hundredths or any other unit. The maximum error in the sum of n such numbers is ½n units. The argument, almost universally applied, is that as the maximum error is ½n units, the answer is unreliable to that extent. This is bad logic. Let us take an analogy from electricity. When a current is switched on the maximum current is reached only after a time which is infinity. Actually, after a very short time, the difference of the current from the maximum is negligible. Likewise to argue that the sum of n items is unreliable to the extent of ½n units is not merely bad logic, it is completely wide of the truth.

A pilot flying solo is in a somewhat difficult position as regards air navigation. Too much in my opinion has been written about how to navigate an aeroplane for the man with an array of instruments and a comfortable chair and chart table to work at, and not enough for the other poor unfortunate. The pilot navigator (always excepting people like Mollison or Lindbergh, Amy Johnson or Jean Batten) is limited, and has to find his way about mostly by map-reading, but if he does his job properly he should at least find out the wind from the weather people, and use it to calculate his course to steer and his ground speed. These will never be quite right, but will be near enough for him to start off; he can checlr how he is going by his map and make the necessary corrections.

It is the duty of the Teaching Committee to keep under review Reports published by the Association, and accordingly members of this committee were asked last May to answer the following questions :

(a) Do you consider that the Mechanics Report is sufficiently in need of revision or rewriting for a sub-committee to be appointed now for this purpose, or do you consider that any other Report needs revision more urgently?

(b) Which parts of the Mechanics Report do you think are most in need of rewriting?

(c) Are there any topics not included in the Mechanics Report which you consider should be put into a new report?

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.

Though the familiar proofs of the classical theorems on the multiplication of series are intuitively evident, they present, on reflection, many tiresome details, and by their piece-meal character tend to conceal the underlying unity of the results.

In the following account the multiplication theorems are all presented as simple consequences of a central formula in the theory of limits.

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.

1. Let the rod be of length l, line density d, mass M = ld. If I is its moment of inertia about one end, then from the theory of dimensions

where C is dimensionless, i.e. a constant, independent of M and l.