There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.

In the eighteenth century The Ladies’ Diary regularly included mathematical problems. A complete collection of these problems from 1704 to 1760 appears in a volume entitled The Diarian Repository; or Mathematical Register published in 1774 [1]:

“... a collection of more than four hundred and seventy Problems, relating to such various parts of the Mathematics as are here collected from the Diaries, together with their Solutions fully investigated will, we doubt not, prove of considerable service to the young Adventurer in this useful branch of learning...”

In a course on Greek Mathematics for our Classical Sixth some years ago, I noticed a connection between Theon’s and Hero’s sequences of rational approximations to √2 (the “side-and-diameter” numbers). A footnote in Heath [1] refers to a proof published in German in 1886, but I recently came across a generalisation which may be of some interest.

The popular definition of a “straight line” as “the shortest distance between two points” is one of the few pieces of mathematical jargon in general circulation in the English language. This is unfortunate, for it is not a good definition, and one may well wonder who originated it. Euclid is not to blame, for he defined a straight line as “that which lies evenly between its points”. (This is indeed cryptic, and Sir Thomas Heath, in a lengthy analysis of the original Greek, had to confess that “the language is...hopelessly obscure”.)