In tackling a problem, we needed to show that a certain 3 × 3 determinant is non-zero. This determinant could be expressed in terms of the symmetric, homogeneous function N = Σx.Σx2 + 6xyz - 2Σx3 of three variables, where, for example Σx2 = x2 + y2 + z2. We first looked into several classical books, to see if there was a relevant inequality. There was not, nor do they contain anything other than ad hoc techniques for dealing with the inequalities they give. In this paper we give a general method for dealing with a certain class of such inequalities; it might be instructive for undergraduates, partly to exercise them in applying a moderately complicated algorithm, and partly to show an application of 2- and 3- variable Calculus.

James R. Newman observed that “the most painful thing about mathematics is how far away you are from being able to use it after you have learned it” ([1], Vol. 3, p. 1978). Perhaps there is some sort of natural justice, then, in the fact that one of the most pleasant things about mathematics is that you don’t need to know very much about the subject in order to enjoy the puzzles, pastimes and toys that the study of mathematics has produced.

This article gives instructions for a new way to construct the mathematician’s toy which is called a “rotating ring”. I publish it in hopes that it may give you some pleasure and also provide a means by which you can entice your students to study the more serious aspects of solid geometry.

The history of mathematics is an abundant field for making surprising discoveries. Some luck is necessary, but whereas the literary tradition has been extensively explored by scholars, finding a nugget in the history of mathematics is not so difficult as turning up a new fragment by Jane Austen, or a lost piece by Mozart. In my case I struck lucky in the theory of matrices.

As is well known, the theory of matrices arose during the 1850s – a useful benchmark is the publication of Arthur Cayley's celebrated memoir of 1858, A memoir on the theory of matrices, a paper which generations of mathematicians have taken as signalling the beginning of matrix theory. Even today the opening paragraphs serve as a useful introduction to the subject.

One of the pleasant aspects of research into mathematical history is the way in which existing material, passed over in the conventional account, may allow us not only to establish the bare, if unexpected, fact of a priority but, more importantly, to assess the significance of major currents in mathematical thought with greater precision. Briggs’ partial discovery, in anticipation of Newton, of the general binomial expansion,

is an excellent case in point.

In the preceding discussion of the Galileo-Newton theory, force has been introduced as measured by mass-acceleration, in accordance with the generally accepted procedure. The existence of laws, connecting force with the conditions accompanying it, has been referred to as a fundamental fact which makes the theory possible; though, as a matter of logical order, all details as to such laws have been put into a secondary position. The recognition of the fact that acceleration was the thing, which could be connected by laws with the conditions under which a body moved, has been mentioned as Galileo’s great achievement. This was historically the first and most important, and was probably the most difficult, step to be taken. There is no record of any suggestion of it before Galileo’s time, and no theory of motion of any value had been constructed on any other basis. Perhaps the nearest approach to such a theory was the study of the celestial motions on the basis of circular motion; and this was not capable of progressing beyond the empirical stage.

Information Theory was developed to provide electrical engineers with a calculus for comparing telecommunications systems as transmitters of “information”—an ambiguous word used here in a technical sense to be defined. Since 1948 the theory has been rapidly developed and widely applied, but the results of its application to telecommunication engineering have been disappointing in some respects. Although it has been applied indiscriminately to all kinds of communication systems including language, it has nevertheless stimulated new ways of thinking about the storage and transmission of every kind of “information” in the most general sense of that word. It has attracted the attention of workers in fields as diverse as neurology and librarianship, statistical mechanics and psychology, cryptography and the study of social insects. Its terminology and concepts have provided stimulating analogies and have helped to link studies as apparently unrelated as thermodynamics and semantics. It is evidently a branch of applied mathematics of wide general interest.