In a recent issue of the Gazette, S. Parameśwaran introduced S·P numbers. A positive integer n is an S·P number if it equals the sum s of its digits multiplied by the product p of its digits, i.e. n = sp. Subsequently, H. J. Godwin showed that in any number base the number of S·P numbers is finite, and A. F. Beardon and K. Robin McLean enumerated all S·P numbers in base 10 (namely, 1, 135, and 144). We generalise Beardon’s technique from base 10 to an arbitrary base to determine all S·P numbers in bases 2 through 12. Contrary to the claims of Godwin, we find 4-digit and 5-digit S·P numbers in base 11.

The study of Geometrical Optics has not attracted much interest or attention for some years now. at all events until recently, save only among a quite small company of specialists, and, as but one particular result of this, the divorce between optical theory and optical practice has been lamentable, as, for example, in that problem of so great practical importance, the design of the symmetrical optical system. For the one part, the subject has been regarded as isolated and of little theoretical interest, and scarcely susceptible of wide and profound generalisations, then, too, and this more especially from the point of view of university teaching, geometrical optics has seemed to stand apart from the path to a general mathematical education and training, and, in consequence, in several universities the study of it has been almost abandoned, save only in its most elementary aspects.

I am unable to answer Ian Cook’s query (Gazette, note 65.25) concerning the original discoverer of the construction which he quotes from Nicholson’s Carpenter’s new guide, but readers may be interested in learning more about Peter Nicholson, who is variously described in his own books and in biographies as: architect, carpenter, surveyor, practical builder and private teacher of the mathematics.

Of all mathematical problems, those concerned with probability seem to contain ambiguities more often than any others. In the majority of cases the reason is that too little thought has been given to the wording of the question, but occasionally the question sounds very sensible and it is only when a real attempt is made to solve the problem that difficulties in interpretation appear. I believe that this is mainly due to the loose way that the word ‘probability’ is used.