From one end of Britain to the other Megalithic Man has left us examples of geometric constructions of various designs. Originally set out on the ground by means of stakes, rope and measuring rods, the designs were rendered permanent by large stones placed along the outlines. The erection of the larger of these stones was an enormous task. It is certain that the engineers controlling the operations were working to pre-conceived plans which were set out with care and accuracy. Further, it appears that the unit of length used was the same from Land’s End to John o’ Groats. That this unit was 5·44 ft. I have shown in Ref. 1, and many more sites surveyed since writing that paper have amply confirmed statistically this value. Half of this, 2·72 ft., might be called the Megalithic Yard, but whether this or the Megalithic Fathom (5·44 ft.) was the length of the standard rod carried about the country it is not possible to say The important thing is that in a statistically significant number of cases in any group of circles the major diameter is approximately an integral number of fathoms. The group can be formed according to locality, type of circle, or size; the result is the same. We also find this dimension occurring in the distances of outliers from the construction centre and the distances between circles where there are two close together. The method of analysis used was that given by Broadbent in Ref. 5.

Methods have been described that facilitate the discovery of economical dissections. They are summarized in §§ 1 and 2 that follow.

1. Each of the figures to be dissected is made an element of a strip with parallel sides; 1 and 2 show strips formed from a Latin or tau cross and a pentagon. The first is a P-strip (prototype parallelogram), in which all elements are the same way up. The second is a T-strip (prototype trapezium), in which alternate ones are inverted. Points such as A and B in 1 are called congruent, a term copied from Whittaker and Watson's Modern Analysis, p. 430.

A few years ago there appeared on the commercial market a certain puzzle called “Maestro”. This consists of packing into a 6 x 10 rectangle the twelve shapes depicted in Fig. 1, (where they are shown packed into an annular region bounded by two concentric squares). It is also possible to pack these shapes into other simple geometrical figures. The aesthetic attraction of this problem lies in the fact that the twelve shapes concerned form a complete set in the sense that any other shape consisting of five abutting squares is congruent to one of them.

The general term (see Fig. 1) in Pascal’s Triangle may be defined as:

(1a)

where

n and r are positive integers, where n≥r.

is equivalent to the binomial coefficient, and is equal to n!/[r!(n-r)!] (1b)

We discuss the deduction of real euclidean theorems from complex projective theorems, and the reverse process: when is it permissible, in proving a complex projective theorem, to project two given points into the circular points at infinity and use real euclidean methods? We emphasise the difference between the real and the complex, and the consequences of using real numbers to define euclidean geometry, and complex numbers to define projective geometry.

The difficulty of a good calendar lies in the fact that the earth travels round the sun in 365 days, or to be precise in 365 days 5 hours, 48 mins., 46 secs, which, we should note, is less than 365 days by just over 11 minutes.