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.

On 22 September 1636, Fermat wrote to Roberval, [I, FO.II.XIII, p.71], saying that he had found the quadrature of an infinite number of curves. He named in particular the ‘solid parabola’, y = x3, and said that his method was different from Archimedes‘ quadrature of the parabola. He invited Roberval to share his thoughts on this and other matters.

On 11th October 1636, Roberval replied to Fermat [1. FO.II.XIV, p.75], indicating his method for determining the quadrature of the ‘solid parabola’ and extending it to the quadrature of y = x4 and y = x5.

Problems have been important in the history of mathematics, for the attempts at solution, whether successful or not, have often given results much wider than the original questions. The more obdurate have become notorious. Though circle-squaring and duplicating the cube are obsolete, angle trisection survives. This problem, Fermat’s last theorem and the map-colouring question still appear occasionally in the Problem Bureau’s correspondence.

Cyclic pairs

When you transpose the digits of a two digit number and obtain a multiple of the original, the original may be called a “cyclic pair”. I have before me, a computer printout obtained by Mr F. B. Satterthwaite of 439 East 51st Street, New York 10022, exhibiting all the cyclic pairs for which the multiple of the original obtained by transposing the digits is ⩽ 16. This computer printout exhibits patterns of many kinds. The most striking pattern that Satterthwaite identified was that cyclic pairs do not exist when the base is one less than a prime number, but do exist for every other base ⩾ 5. His claim may be substantiated as follows.

The investigations of Professor Karl Pearson and his coadjutors on the applications of the theory of errors to statistics have greatly increased the importance and extent of the subject.

It is hoped that the following version of one of the methods of arriving at the so-called normal law of error may be of interest from an elementary point of view: affording, as it does, a simple application of the binomial theorem and some interesting graphical work.