To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology. The relationships of men to God and to the universe have each an accurate, and even a numerical, side, capable of scientific treatment, called pure and applied mathematics. It is more than an accident that theology, in the narrower sense, continually uses mathematical terms and illustrations, some of which we shall consider. Since truth is God, and therefore one, its different departments are related and overlapping; each is capable of illuminating the others, and the lower aspects are continually recognised as symbols and sacraments of the higher.

A new and general proof of theorem [1] is presented, which accommodates directed angles and covers all possible configurations of line and conic. A sign convention for the use of directed angles in this setting is developed, and examples are given showing how this fundamental theorem can be used to establish, much more directly and succinctly, the many tangent/focus properties of conies.

A square matrix is said to be doubly-stochastic if its elements are non-negative and if all row-sums and column-sums are equal to 1. The study of doubly-stochastic matrices was initiated by I. Schur and was subsequently taken up by Hardy, Littlewood, and Polya, who proved the following fundamental proposition [1, Theorem 46].

I Admire the ingenuity of the method by which Prof. Nanson establishes the expansions of sin x and cos x as power series in x. But I must confess that his attempt to ‘simplify’ the ‘accepted accurate’ proofs of these expansions seems to me a fundamentally mistaken one. Nor can I admit that the result is at all satisfactory ‘from the elementary didactic point of view.’ His proof is to my mind essentially an artificial verification, and altogether unnatural, in the sense that it has no place in any natural and logical way of developing the ideas which lead to the result. I speak with diffidence as one who has had far less experience of teaching than Prof. Nanson. But I am convinced that my criticism is a fair one: and I think that many of the proofs usually given, in English text-books, of many of the theorems of ‘Higher Trigonometry,’ are open to fair criticism on similar grounds, when (as is too seldom the case) they are not to be condemned for ‘hopeless complexity’ or utter lack of force.

The aim of this note is simply to encourage dialogue between mathematics and physics teachers. In most school physics and chemistry courses, any mention of quantum theory has been limited to a brief qualitative discussion, or has been restricted to the simple Bohr theory. The new Nuffield Advanced Physics material [1] goes much further than this, with the opportunity to apply some quite simple mathematics, supported by a computer analysis. We shall here give a gross simplification of the end of a long story in the physics course.

The use of the Greek letter π to denote the ratio of the circumference of a circle to its diameter appears to have been popularised by Euler following his adoption of the symbol in 1737 and later in his many popular textbooks. The original use of the symbol for this purpose was in fact by the Welsh mathematician William Jones who used it his book Synopsis Palmariorum Matheseos published in 1706. This book was described in 1824, over a hundred years later as follows ‘This work has long been esteemed as a compendious, but comprehensive summary of Mathematical Science’.

The object of this note is to give a simple geometrical proof of the following well-known