My contribution to this discussion reminds me of the ill-prepared Divinity student, who, asked in an examination to “Discuss the textual criticism of the Minor Prophets”, began, “Far be it from me to criticise these great and holy men, be it mine rather to give the list of the Kings of Judah and Israel”. For what I have to say has about the same degree of relevance to the topics discussed by the last two speakers as the student’s answer had to the question asked, and my remarks will, moreover, be more in the nature of a sermon than of a mathematical lecture. I hope, however, that they will serve as an illustration of the general subject under discussion, even though they may not bear very directly upon what has already been said.

The eternal question of the distinction to be drawn, in teaching elementary Mechanics to youngsters, between mass and weight has recently thrust itself upon me in discussing with a Science colleague the choice of a textbook on General Science suitable for pupils, aged about 11 or 12 years, beginning the subject. The first year General Science syllabus which he has drawn up excludes as far as possible all quantitative work, my own Mathematics department is left to deal with the fundamental teaching of all work involving units, not only of mere length, area, and such things as invariably occur in elementary mathematical teaching, but also of such others as temperature and our old friends mass, force, weight, density, and so on, which will in time be used in the Mechanics course for which the Mathematics department assumes responsibility and to which the Science department will refer incidentally as it feels the need.

This paper attempts to illustrate the method of representation, which is so powerful a weapon of modern geometry, in terms of elementary Higher Certificate mathematics. The results in their general form are familiar to students of geometry; but the special problem given here has several interesting features of its own.

No originality is claimed for this paper, though I do not know of any other account of the work. It seems, however, worth while to give a brief summary, as it covers a great deal of the ground with which schoolboys should be familiar.

The fundamental idea expressed in this paper is that objects and phenomena are only seen in part, vision being limited to a plane in time about second thick, and showing us things as they are for the instant only but if we wish to picture the Completeness of Things, or Phenomena, we have to include, certainly, time as well as space, we have to retain the image of the movement of the rolling ball, the running machine, the growing plant, the dancing girl, through either the cycle or phase of movement or the life history, as appropriate.