The following paper is an attempt to explain the action of a class of three-dimensional mechanisms, first from a theoretical standpoint and afterwards to suggest some practical applications. One or two rather more general mechanisms are investigated than have previously been studied and well-known mechanisms are deduced as special cases of these. There is no modern textbook or recent publication dealing with these mechanisms and their properties are not generally known.

Most of us who find ourselves still undergraduates in the third year of war have no great knowledge of Cambridge in peace time. Consequently, only the most superficial comparisons between pre-war and present-day conditions can be offered. It can be said, however, that while the private lives of both senior and junior members of the University have been very much affected by the war, mathematical activities here continue almost as normal.

When asked to give this lecture I was given to understand that this Society would like a change from the discussion of pedagogical methods and school problems, and could do with something that had nothing to do with teaching at all. In other words, I was to talk about stars and not to mention schools. Well, I promise to do that, though I must confess that the fact that members of the Society are teachers was at the back of my mind when choosing the subject. The schoolboy and schoolgirl and the man in the street know that celestial distances are very large; in fact the adjective “astronomical” has been sadly degraded by being applied to such mundane affairs as the figures of the national debt—and a schoolboy of enquiring mind may ask how the astronomers measure such distances. Such an enquiry is often tinged with a certain scepticism—sometimes politely veiled, sometimes not. When you tell a member of the public that a certain nebula is many million light years away and that a light year is 6 billion miles, he is apt to adopt the attitude of mind indicated by the phrase, “You’re telling me”, and to wonder what meaning such a monstrous distance can have: and how does the astronomer know, anyway?

Mathematics existed prior to the mathematician. Before Man calculated, Nature multiplied and divided; her products were based on geometrical forms. Mathematics, in reality, is not a creation of the mathematician. It is a vital spirit, immanent in Nature and Man, and expressed in the forms and processes of Nature and the manifold works of Man. This spirit functions only partly through the mathematician, who limits both its sphere of operation and its method of working. These limitations constitute a serious difficulty for the pupil by cramping the expression of his personality. His love of beauty is ignored, his social sympathy is untouched, his sense of humour lies dormant, his interest in Nature is not aroused, his imagination has little outlet, his artistic, musical and religious tendencies find no contact. On the whole his emotions are starved, his intellectual development is limited and his spiritual life is uninfluenced. These are hardly the conditions for the successful teaching of any subject. Nor does such limited mathematics contribute adequately to the general culture of the pupil. Its limitations are also a serious obstacle to the integration of the school curriculum, one of the most pressing problems.

Of the five regular solids, the octohedron is conspicuous (though perhaps not more so than the cube) for its connections with the others: one of these is new to me and possibly not as well known to others as it deserves to be.