Recently I have been experimenting on the physical properties, dimensions and constants of hens’ eggs. During this work I have had necessarily to look up the formulae for areas and volumes of ellipsoids and to invent approximate formulae for use where the correct formulae entails the use of mathematical tables. I now give them to other students.

The two most common measures of central tendency and dispersion in statistics are the mean and standard deviation on the one hand, and the median and absolute deviation on the other. For most purposes the former measure is preferred for two very good reasons; the first is that the squares of quantities are easier to handle analytically than their moduli; and secondly for all the common symmetrical distributions, such as the normal, uniform and binomial distributions, for which the mean and median coincide, if a sample is taken to estimate the central value, then the mean of that sample has a smaller variance than the median, and is therefore relatively more efficient as an estimator of the central value of the parent population.

There have been many attempts to understand the portrait of the mathematician Luca Pacioli which now hangs in the Museo di Capodimonte, Naples (Plate 1), and which will be familiar to users of the Pacioli 2000 accountancy software. However none of these attempts by art historians engages with the mathematics of the picture, which the artist has gone to such trouble to detail. While to a connoisseur the painting might be thought wooden and peripheral to the history of art, I believe that the artist has recorded the most important meeting of the Renaissance.

Newton was the greatest mathematician of the seventeenth century. Today, almost three centuries afterwards, we are just beginning to realize the full extent and variety of his achievement. Much of his mathematical work has never been published (though it ranges far through the fields of projective geometry and general point-correspondences to number theory and an exhaustive treatment of interpolation by finite differences) and is now, with few exceptions, to be found only in little known manuscripts in the Cambridge University Library. But Newton is remembered above all, and following his own wish, for his creation of the fluxional calculus and the theory of infinite series, two strands of mathematical technique which he bound inseparably together in his “analytick” method.

Problem. A net is made of string round a unit sphere, so that the sphere cannot slip out of it. How long need the string be?

This problem was offered by me as a “Contest Problem” to undergraduates of Cambridge last Lent Term and a correct solution was given by Mr. R. Schwarzenberger of Trinity College. I am publishing my own solution, as his solution is considerably more complicated.

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.

In the preceding note Mr de Visme writes y’(x) for the “density” of primes in the neighbourhood of x and proves heuristically that

What follows is a condensed statement and classification of various proofs of the above well-known theorem.

For two positive quantities, a and b, the proof may be put as follows: