In the Gazette, No. 248 (February, 1938), pp. 50-57, Mr. W J Dobbs gave a very interesting and complete descriptive account of the 18 equilateral “Morley” triangles of a given triangle. The article is especially memorable for its diagrams (1-8) and the present reader is asked to have them at hand for reference.

Although some years have elapsed since the first appearance of Professor Halsted’s translations of Lobatchewsky’s Theory of Parallels and John Bolyai’s Science Absolute of Space, they are not so widely known and studied as they deserve to be. There is nothing in the mathematics of these small volumes of a kind to deter any one from reading them; they are short and elementary treatises, disclosing a system of geometry quite distinct from Euclid, by rejecting the axiom concerning parallels. The Science Absolute of Space is a most interesting exposition of one branch of non-Euclidean geometry; but it is unfortunately extremely condensed, and occasionally obscure. It would have been a great advantage if explanatory notes had been added to the English translation.

The following problems concern a rigid four-legged table whose feet may be considered as coplanar points, on a floor to be treated as rigid, conveniently smooth and large, but not flat.

1. (a) If the feet form a square, show that there is a position where all feet are on the floor.

2. (b) As (a), but with isosceles trapezium in place of square.

Of course (b) includes (a), but the diminished symmetry calls for a subtler argument.

1. (c) If the feet do not form a cyclic quadrilateral, show that there may be no position with all feet on the floor.

Some three weeks into the academic year I presented, off the cuff, the following morsel to engage some lower-sixth formers who had polished off some routine derivatives all too quickly:

Sketch the curve whose equation is given by x3 + y3 + 3xy – 1 = 0 , and find the gradient at the point (–1 , –1).

Since this was near the end of a double period I thought no more of it. At the end of the next double period one student remarked that he had been having difficulty in sketching it. “It seems to be a straight line”, he said. I rashly suggested that he could well be mistaken, but promised to look at it. If my faith in the student had been greater I could have saved myself some labour searching for simple solutions of the cubic. I had noticed the solution (½ , ½) by the way. I began to have delusions of grandeur, imagining myself to be the natural successor of Diophantus, as I found the solutions (5,–4) and (–4,5).

The first published description of the geoboard appears to have been given as an incidental illustration to Denis Diderot’s Letter on the blind (1774).

The savants of the eighteenth century were interested in blindness as a problem for speculative philosophy. If the mind is a tabula rasa at birth, how will it develop if the sovereign sense is absent? Diderot held that the blind man would not only form different physical concepts, but different aesthetic and even different moral values. In the end he would inevitably come to a different set of religious beliefs and hence, Diderot argued, the concept of God was relative to our senses.

In Sir Thomas Heath’s entrancing Manual of Greek Mathematics it is stated on p. 52 that the Babylonians discovered “a most perfect or musical proportion” between two numbers, which Pythagoras introduced into Greece. This is the ratio a ½(a + b):: 2ab/(a + b) b. The two middle terms are the arithmetic and harmonic means of the extreme terms. Not knowing of this when considering the fundamental principle of the logarithm, it seemed to me that the use of these two means should assist in evaluating the natural logarithm of a ratio in which the numerator and denominator are the first and last terms. In this way I found the surprisingly exact but approximate expression for the numerical value of such a logarithm which appeared in Nature of 14th March, 1931.