No history of the Christian Church in the 19th century can ignore the tremendous figure of John Henry Newman (1801-1890), striding across the scene from Calvinism through the Oxford Movement to the Cardinal’s hat. Most biographers of John make some brief reference to his brother, Francis William (1805-1897), poking a little fun at his journey through various branches of nonconformity to an austere agnosticism: his eccentric pilgrimage is contrasted with the contrary progress of his brother by William Robbins in The Newman Brothers (London, 1966), and is very sympathetically studied in one section of Basil Willey’s More Nineteenth Century Studies (London, 1956). But in neither of these books is any reference made to his considerable mathematical ability. G. H. Hardy, in his Divergent Series, asserts (p. 22) that Newman was for a time Professor of Mathematics at University College, London; this seems to be a mistake. Mr. T. L. Wren very kindly looked into this matter, and told me that while Newman was Professor of Latin from 1846 to 1863, there is no evidence that he held the chair of mathematics, though of course it is possible that he had occasionally given lectures on mathematics.

On the eastern fringe of Bodmin Moor close to the natural rock formation known as the Cheesewring lies a large granite slab with a mathematical figure chiselled on its surface. Close by is another rock on which is carved, “D. Gumb, 1735”. This is all that remains of a rock shelter inhabited by Daniel Gumb, an eighteenth-century recluse. The rock shelter was demolished during quarrying operations and the inscribed stones which now lie a few yards south of the Cheesewring Quarry are about 100 yards south-west of their original site.

In the present state of flux of mathematical syllabuses and teaching methods pupils are often exposed to two distinct approaches to the solution of two simultaneous equations in two variables.

The debate about how best to introduce complex numbers is no doubt perennial. (See for instance [1] for a brief survey of some of the possibilities.) One flaw which many presentations have is that they appear somewhat arbitrary. For example one can define complex numbers as polynomial residues to the modulus y2 +1: this definition ought to provoke the question as to why this particular modulus should be chosen.

Glancing through George Gamow’s popular excursion into modern biology Mr. Tompkins Inside Himself, written with Martynas Yeas, my eye fell on an amusing account of parity. In it a matador faces a bull in a Möbius ring. He would find, should he run right round the Möbius ring by-passing the animal, that he would face the bull again, but the wrong way up, doing, as it were, a hand-stand. The matador, the authors point out, changes his parity. And standing on one’s head is no way to fight a bull.

Pantactic squares were invented and studied by Astle [1]. From a review by Holzer [2] we learned that the number of distinct pantactic squares remained unknown; Astle himself only mentions that the number of distinguishable pantactic squares runs into hundreds. Therefore we turned the problem over to the computer. By a simple programme in FORTRAN, run on our CDC 3600/3200 computer, we found altogether 800 distinguishable pantactic squares.

In Classroom Notes No. 178 (Math. Gazette, 1968, pp. 376-380), a problem is set and formally solved on a body falling through a tube in the interior of the Earth. One can condone taking the ideal conditions of a frictionless tube and even overlook the fact that no such tube could withstand the stresses present in the Earth below some 30 kilometres depth. But when, as in this Note, it is further assumed (without even reference to any assumption) that g is proportional to r, the calculations, whatever mathematical interest they might have, cannot be regarded as having any relation whatever to the planet Earth.

Recently we bought for our small daughter a “Merit” toy, called “Design Doodler” [1], which consists of a single open chain of 8 flat lozenges, linked end to end, identical in size and shape (see Figure 1), the overall length of each being twice the breadth. The linkages are such that the lozenges can move in their own plane so that, when two ends are so linked, any of the five edges of one can be made coincident with any of the five equivalent edges of the other. No other incidence is possible, and lozenges cannot overlap.

A recent and fascinating application of statistics has been that of analysing the distribution of words in a written document in an attempt to determine the authorship. This article reviews some of the methods used and proposes a simplified test that could be performed using a desk calculator, although access to a digital computer would reduce the work required. An example of the test applied to a manuscript in the British Museum (Stowe 269) is given in detail to illustrate the method.

At the Cambridge Conference, 1969, I proposed the following problem to several members:

Find a binary operation on the reals which is associative but not commutative.