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.

There are in the Mathematical Gazette for October 1924, three communications on the above subject.

It will make my remarks on these clearer if I refer to the contents of my paper on the Postulate of Parallels in the Gazette for December 1923, and make an addition thereto. In that paper the demonstrations of Euc. I. 27 and 29 are deduced from Wallis’s Postulate of Similarity, viz.: “To every figure there exists a similar figure of arbitrary magnitude” taken as a substitute for Euclid’s Postulate of Parallels. The demonstration of Euc. I. 27 in this manner was first given (I believe) by Professor Nunn in the Gazette of May 1922. He has not, however, published in full his demonstration of Euc. I. 29. He was so kind as to communicate it to me, but I hope he will some day publish it in full. It is entirely different from the demonstration of Euc. I. 29 which I published in the Gazette of December 1923, which involves three successive applications of the Postulate of Similarity and makes use of Euc. I. 27 as proved by Professor Nunn, but does not include the proof of any other proposition.

There is little doubt that many teachers are seriously bewildered by this Report. It is not an easy document to read and frequently it is difficult to ascertain the precise nature and application of its recommendations. It suffers from attempting to cover too much ground: the reader is confused by the double purpose it tries to serve, (i) instruction on the axiomatic basis of geometry for teachers who have made no special study of the logical foundations of the subject, (ii) pronouncements on practical procedure in teaching and geometrical method.

In a system in which we have given a circle and the relations between it and certain points and lines, certain changes take place in these relations in another system in which a conic takes the place of the circle. To enable these changes to be more easily seen and interpreted, the writer has found it very convenient to use certain modes of expression.

One of the best-known examples of a caustic curve in Geometrical Optics is that obtained by reflection from the arc of a circle of rays emanating from a point on the circumference of the circle. The enveloping curve is known to be a cardioid. It has not, however, I think, been pointed out that the analysis •of this problem leads to a very simple construction from which a cardioid •can be rapidly and accurately drawn.

Now that the value of models in the teaching of Geometry has received emphasis in the Report on the Teaching of Geometry in Schools, a great impetus may be expected in their development. This expectancy combined with the article by Mr. Langley (Math. Gazette, May 1924), has led the writer to think that some account of a device for three-dimensional representation may be of interest to readers. In its present form the model is so arranged that figures for very general straight-line problems can be built up in a very few munites. It has been in active use for about a year and has proved of valuable assistance both in trigonometrical and in pure geometry.