The methods usually adopted for demonstrating the Lorentz Transformation are algebraic. Algebra is a powerful means of arriving at results, but does not in itself explain those results. In the present matter, the average reader is left with a sense of puzzledom, baffled by the paradoxes evolved by the algebra ; the results are demonstrated, but he is unsatisfied. The following paper is an attempt to treat the matter from first principles, avoiding algebra and getting down to the plain common sense. The results are obtained quite simply ; paradox is avoided. Some observations are hazarded at the end on the questions of simultaneity, “before” and “after,” in which the view taken differs somewhat radically from that ordinarily adopted.

In 1569 Gerhardt Mercator (1512-1594) constructed the chart now called Mercator’s Map. On it the meridians and parallels are sets of parallel straight lines at right angles, a course on the Earth’s surface always cutting the meridians at the same angle is a straight line cutting the meridians on the map at the same angle, and the shape of any very small area on the map is the same as the shape of the corresponding small area upon the Earth. This chart can be made by gradually increasing the distances between the degrees of latitude in advancing from the equator towards the poles. But though Mercator did, in fact, construct such a map, it is not known on what principle he proceeded, and it is conjectured that he obtained his division of the meridians by observing on a globe on which rhumb-lines had been drawn the intersection of these lines with the different meridians.

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.

Charles Godfrey was born in October, 1873. At an early age he went to King Edward’s School, Birmingham, where he was head of the school, a position which depended entirely on a boy’s ability in classics, and he spent at least five years in the top mathematical division under Rawdon Levett. The top mathematical division in those days consisted of a dozen boys, and generally contained three future major scholars of Trinity. Godfrey went up to Trinity as a major scholar in 1892, and was fourth wrangler in a brilliant year (1895), with Bromwich, Grace and E. T. Whittaker above him; in the following year he was placed in the first division of the first class of Part II. of the Mathematical Tripos; later on he won the Isaac Newton Studentship.

The uses of vectors are many and varied. The value of the vector notation and of vector methods of calculation is becoming recognised in mechanics and physics, but few teachers use them to any extent in teaching pure mathematics, although they appear to be equally useful in this sphere.

I have found vector methods especially useful for students who require a “practical” knowledge; they are well adapted to numerical work, and often provide a welcome change from the drudgery of memorising formulae, and calculating results by substitution of numerical values of the symbols—a dull task, lacking in educational value, which forms such a large part of the work of the technical student. Vector solutions are usually short and simple, and proceed from first principles ; they should thus be preferred on logical grounds.

1. It is a commonplace of Biological Science that each individual, in his physical development, reproduces in an abridged form the successive stages through which the race has passed in its remote history. Educational experience suggests that a similar process occurs in mental development, so that, generally speaking, ideas have to be introduced in the order of their historical occurrence, if unnecessary resistance is not to be encountered.

As soon as the Calculus had descended from its academic heights and had been persuaded to frequent the schoolroom, there was bound to be trouble about e.

The desirability of affording facilities, to mathematical teachers and others, for post-graduate studies in mathematics during the summer vacation, was commented on by the writer in an article in the March, issue of this Gazette. It was there proposed that a “Summer School” should be organized for this purpose by teachers themselves, acting through the Mathematical Association, and after the publication of the article the suggestion was more formally communicated to the Association through the Manchester branch. It proved to be unlikely, however, that the matter could be taken up by the representative body, whose members met rather infrequently, and who were already fully occupied in other business of the Association, in sufficient time to make arrangements for such a School during the present year. The proposal was therefore brought before the Extra-Mural Committee of the University of Manchester, who agreed to the experiment being made under their auspices, and voted a grant for organizing and advertising the project. The Board of Education were next approached, and they agreed to recognize the School and to make a limited grant to cover a possible deficit on the cost of working it. Provisional arrangements were proceeded with meanwhile, and in April and the early part of May circulars announcing the School and inviting students to register were distributed to the heads of the mathematical departments of nearly every secondary, grammar and high school, and many technical schools and colleges, throughout England and Wales. The School was also advertised in various educational journals. The response was satisfactory, thirty-two students being registered, whereas twenty was the number decided upon as the minimum required if the School was to be held.

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.

A striking article under this title appeared in Nature of 10th May, 1924. The anonymous writer deplored the present position of mathematical physics in England, and suggested that the mathematical course at Cambridge was now much inferior to what it was in the days of Kelvin, Stokes and Maxwell, in that the physical aspect had been almost crowded out, leaving pure mathematics predominant. As a result, it was declared, there was only one man out of all those who had taken the Mathematical Tripos in the last forty years who had any claim to be recognised for his researches in mathematical physics apart from astronomy and relativity, in which it was admitted that excellent work had been done. The remainder of the article was a strong plea for the inclusion of physics in every mathematical course.