It is proper and desirable, at most of our meetings, for speakers to talk on mathematical topics to which they may themselves have made significant contributions. In general at such meetings we shall all understand the broad outline, but the details are likely to be obscure to most of us. The presidential address, I have always thought, is in rather a different category. The speaker should ensure that his audience, though originally lacking deep knowledge of the chosen topic, should go away with the impression that they not only understand every word but could also reproduce the arguments, say to a class of sixth-form mathematicians. Since our association is mainly concerned with the refreshment of school mathematics and mathematicians, I hope to talk about a part of numerical mathematics which could be taught in school, which uses “standard” mathematics, and which springs from standard mathematics. I start with a rather lengthy introduction on the general principles of error analysis.

If equilibrium is about to be broken by rotation about the center of the disc, i.e. slipping around both pegs, the forces are as shown.

Text-book miscalculations are regrettable, but they do no more harm than causing the student or the teacher to spend some extra time checking his answer. Errors in principle or in argument on the other hand, whether they be on the part of an author or a teacher, are more serious, especially when they are as plausible as the one which it is the purpose of this article to expose.

After the first part of this article had been written, it was suggested to the writer that indeterminate forces and the initial conditions might have something to do with the discrepancy between the two different answers obtained to Example 2. Initially the disc might not just be resting on the pegs; it might have been rammed down.

According to orthodox teaching texts the tangential component of the reaction between two surfaces in contact is μ times the normal component when slipping is just about to take place. The tangential component cannot exceed this value, so it is asserted, and is equal to this value when slipping is occurring. The coefficient μ is a property of the surfaces in contact and is a constant, independent of the normal component. This may be a basis for some mathematical exercises, but typical texts even go so far as to say that these statements are true within the limits of experimental error (see for instance Quadling and Ramsey, Elementary Mechanics Vol. I, Bell, 1963, p. 58). Yet no experiments on this are normally performed or discussed in schools. One could argue that since the coefficient varies according to the nature of the surfaces it is so particular as to be not worth measuring, but this would equally justify not doing any exercises on the idea.

The following two theorems are extensions of the Cayley-Hamilton theorem. The first gives the explicit expression for AN as a polynomial in A, where A is an arbitrary square matrix. The second expands the first result to the case of a matrix series.

It is remarkable that Cayley, who gave (1895) a purely algebraic treatment of the addition formulae for the Jacobian elliptic functions, never pointed out their essential identity with the ordinary formulae of spherical trigonometry