The term measuring number is acquiring currency as a popular synonym for non-negative real number. This is good. Yet the belief persists that the real numbers, by whatever name, must remain fundamentally inaccessible to school children, obscured behind a veil of Dedekind cuts or Cauchy sequences. (Thus Griffiths and Howson, in what is generally a humane book [1], devote several pages to an ingenious, but ultimately condescending, attempt to foreshadow the mysteries of real numbers in terms of points of a line.)

Many communist countries hold internal olympiads in order to identify and thereafter foster special talents. Since 1959 they have held an annual International Mathematical Olympiad (IMO) for all European communist countries (except Albania) and subsequently Mongolia, Cuba and Vietnam. Non-communist countries were first invited in 1967 and France, Sweden, Italy and Great Britain accepted the invitation. Italy has dropped out and Austria, Netherlands and USA now regularly participate. Belgium, Finland and, last year, Greece have also taken part. Each country sends a maximum of eight competitors and two adults.

The Alexandrian Greek mathematician Heron—whom we associate with the formula √{s(s − a)(s − b)(s − c)} for the area of a triangle—got involved in a calculation about a pyramid design which led to the evaluation of √(81 − 144). This occurs in his book Stereometria (C. A.D. 75) and to ‘solve’ it the numbers are turned round thus: √(144 − 81), to give √63 which is taken to be . (Is this a reasonable approximation for √63?) It is not known whether Heron made this transpositional error or whether a copier of his work was responsible. This seems to be the first occasion in which the square root of a negative number was stumbled across—a concept not properly understood for another 1750 years!

“A fair coin is tossed successively until either two heads occur in a row or three tails occur in a row. What is the probability that the sequence ends with two heads?”

Thus the problem was stated on the Cambridge Scholarship paper. My problem on being asked for help was to produce a solution using only sixth form methods, and hopefully a solution which would prove interesting and instructive. Certainly the problem had captured the imagination of my questioners, although the method for its solution was proving to be elusive.

This note describes an example which I have found useful for introducing ideas in linear algebra and group theory. It is simple and yet it is neither trivial nor too abstract.

The reproductions shown here are of p. 49 and Fig. XVI (which refers) from A sure guide to builders by Batty Langley, published in 1729 and dedicated to Thomas Scawen Esq., of Carshalton. The reference is to the construction

“of oval-like figures” and the implication, towards the bottom of the page, is clearly that something very close to “an ellipsis” is what Mr. Langley had in mind. The poor plate engraver was hard put to it to draw an ellipse from the instructions in the text and a careful examination of his effort reveals that his final solution was to draw, very faintly, an ellipse and try to fit tangents to it from the divisions along the sides.