In a recent paper [1], Victor Bryant shows how the number of axioms required to define a vector space can be reduced to seven (in addition to closure requirements). The main result of his article is that commutativity of addition can be deduced from the other axioms. In the present article we show how to reduce this number to six. For certain underlying fields one or more of these axioms can be deduced from the others. However, the six axioms are in general independent; we invite interested readers to show this by constructing their own counter-examples, which the editor of the Gazette will be pleased to receive.

Had I been called upon to address the Association a few years earlier on a subject of this nature, my task would have been much more difficult. Model atoms were then becoming almost as numerous as the sands of the sea. The very simplicity of the quantitative phenomena shown by the simplest atoms was an obstacle to progress. For example, the simple spectrum of Hydrogen,— the so-called Balmer series,—consisting of a set of frequencies given by the formula

where n and m take all possible integral values, was somewhat too simple if considered by itself instead of in relation to phenomena of a widely different type. And as a consequence, we had at least five completely different forms of suggested structure of a Hydrogen atom, each of which, by suitable mathematical artifices, could be made to yield such a set of frequencies.

The problem of finding rational integral numbers which could be made the sides of right triangles, is said by Proclus to have been solved by Pythagoras, by a method which is equivalent to the use of the formula,

where m is any odd integer.

Heath (1) quotes Proclus’ statement as follows:

“The method starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity, and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side the hypotenuse.”

The subject matter of my talk is geometry in the sense that I shall deal with such entities as lines, circles, and squares. But the results I aim at bear little resemblance to the theorems of traditional geometry, which describe properties of given configurations. Our present object is quite different; we shall begin by specifying not configurations but processes (for example processes of selection), and then seek to determine how economically or how effectively these processes can be carried out. Questions of this nature are quantitative and the solution, if it is known, is generally expressed by a number. It is for this reason that I speak of 'arithmetical' geometry. The problems of the type I shall discuss are usually simple to state and require very little in the way of mathematical equipment for their understanding. However, their solution often presents formidable difficulties and is in many cases as yet far from complete. Let us now consider in some detail a few of the characteristic problems in this field.

Can we produce a ratio ψ to rival the golden ratio ϕ? In this article I shall construct a variant of the classical golden rectangle by choosing a different geometrical property. The rectangle described has more advanced mathematical properties than its golden relation, though it is not extravagant. In this sense it could be described as a supergolden rectangle.