In the problem proposed by A. G. Ioachimescu in 1895, it is asked to be shown that the sequence , defined by , for each , is convergent and its limit lies between -2 and -1.

There have been given many generalisations and other results regarding Ioachimescu’s problem in the literature (see, for example, [2], [3], [4, Theorem 1, parts (a) and (b)], [5, problem P2, parts (i) and (ii)],).

I WOULD like first to thank the Association for the honour they have done me in inviting me to lecture at this meeting. It is, however, an honour which carries with it a certain danger. When a mathematician comes to lecture on a subject in which he has specialised, he is always liable to suffer from the delusion that what has gradually become plain and straightforward to him will at once appear plain and straightforward to others. It may be that I am suffering from this delusion in thinking that I can present the main idea of the geometry of numbers in one lecture.

There has been a long history in the approximation of ellipses by circular arcs in order to simplify their construction and manipulation. Such approximation was of use for a wide variety of applications, in fields such as mathematics (generating figures), astronomy (analysing orbits), art (marking out large oval frames for ceiling painting), architecture (building masonry arches, floor plans, etc), and, more recently, the conversion of fonts from a general conic specification to circular arcs. Documented evidence goes as far back as the Italian Renaissance when various schemes were published by the architect Sebastiano Serlio in the sixteenth century. More contentiously, it has been argued that fifteen centuries previously the Romans used such approximations when designing and building their amphitheatres.

The values of the Radii of Gyration have been worked out for some solid polyhedra, polyhedra with faces only, (shell or box-like), polyhedra with edges only (skeleton), polyhedra with vertices only. The results are given in the tables.

1. Definition

We define an infinite coprime sequence as an infinite class of positive integers arranged in order of ascending magnitude, in which the highest common divisor of any two members is unity.

2. Note on prime numbers

If an infinite sequence of primes, constructed by some rule which does not invoke a knowledge of the sequence of primes, should ever be found, it will necessarily be an infinite coprime sequence. We thus concern ourselves in this note with the construction of infinite coprime sequences without using a knowledge of the sequence of primes.

A linkage is an assembly of coupled bodies or links whose freedom of movement is restricted, after the fixture of one link, by the constraint imposed by their couplings. Frameworks and mechanisms are particular types of linkage. In this article the freedom on a plane or in space of linkages of an unlimited complexity is investigated by a general method whose application can readily be extended to space of four or more dimensions. The results obtained permit the analysis of any given linkage by mere substitution in the appropriate formula; but although the results are primarily intended to assist the analysis of existing linkages, it is shown how they can be used to a limited extent in the synthesis of new linkages to meet given requirements.