In talking about the relations of mathematics and culture or, rather, about the cultural value of mathematics, I think it advisable that I should first say a little about what I suppose the word “culture” to mean. In its general use it contains, I think, two ingredients, as it were. It refers to information and it refers to sensibility An ignorant man, however refined his natural sensibilities may be, is not regarded as cultured. And a man who has amassed an immense knowledge of facts, but is quite unable to distinguish between what is valuable and what is not valuable in his information, is not regarded as cultured. Of these two elements, I am inclined to think that sensibility is the more important. What the old writers on aesthetics used to call taste is, I think, more important than knowledge for the formation of a cultured outlook.

Professor Forder’s elegantly simple method in Note 3068 (Math. Gazette 47 (1963), 237-8) can be extended to prove a more general result. Let k be any positive integer, ξ any irrational number, ar the (r + l)-th quotient in the simple continued fraction for ξ and pr/qr the (r+ l)-th convergent, so that

Most of us, I suppose, can quote a few phrases read or heard, which have greatly affected our outlook on life in general and our own work in particular. Of about half a dozen such, two came vividly into my mind as I recently attended, at the University of London Institute of Education, a course of lectures in which a closer link between the sixth form and the University courses in mathematics was sought. The first goes back to my youth, and comes from Ruskin—“Know what you have to do—and do it!” Ruskin goes on to say that insufficiency of means is far less frequently a source of failure than an inadequate understanding of the thing to be done. The second is more recent, and came as a gentle rebuke from more than one of the nurses at St. Bartholomew’s Hospital, during my stay there after becoming blind, when I apologetically asked for some small non-routine service—“What are we here for?” Surely Ruskin’s precept and the nurses’ question are very relevant to our work.