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Published online by Cambridge University Press: 03 November 2016
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.
Lecture given to the Invariant Society, University of Oxford, in March 1958.
page 185 note 1 If F is not closed, we replace it by its closure.
page 185 note 2 The existence of such a circle follows immediately by the properties of continuous functions.
page 185 note 3 We use the fact that the distance between two closed disjoint sets is positive.
page 185 note 4 By the angle of a circular arc we mean the angle it subtends at the centre of the circle.
page 187 note 1 A system of figures is said to be non-overlapping if no two figures have a point in common.
page 187 note 2 It is assumed that the squares we consider are closed: this convention does not affect the nature of the problem.
page 188 note 1 If S1 contains several squares of maximal size, we select any one of them.
page 190 note 1 A figure is said to be convex if, whenever the points A and B belong to it, so does the entire segment AB.
