The last examination of the Cambridge Mathematical Tripos, when the candidates were placed in each class in Order of Merit and not in alphabetical order, was held in 1909. Last year was therefore the fiftieth anniversary of the end of a famous epoch. The University Calendar used annually to reproduce all the lists from 1747, the year when they were first printed, and they made fascinating reading, well known to mathematicians. Of course candidates were examined before that date, but full details have not been preserved. It was earlier called ‘The Senate House Examination’, and not before the start of the Classical Tripos in 1824 did the name gradually change.

I am convinced that many difficulties that arise in the Primary and Secondary schools would never do so if from the very beginning division were taught differently

What I wish to urge is a complete break away from the traditional way of setting out division.

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.