The growing importance of the elements of group theory in sixth form teaching has been recognised by a number of recent publications, but a very simple presentation of the theory of groups by means of free variable axioms which was introduced by Lorenzen more than twenty-five years ago, and which is particularly well suited for elementary teaching, does not appear to be as well known as it deserves to be.

Lorenzen defines a group in terms of division x/y, instead of multiplication, a device which obviates the need to postulate the existence of a neutral element and an inverse.

The two oldest representations for the number π are infinite product expansions. The first,

is due to Vieta in 1592. The second is Wallis's product dating from 1655:

A vivid memory of mine is of being shown that certain 2 × 2 matrices were ‘really’ complex numbers. The correspondence between matrices of the form

and complex numbers

seemed magical and the additional remark that

was an unexpected bonus. The correspondence has been examined in various articles in the Gazette [1, 2]. That it is an isomorphism is seen by comparing

with

in the case of addition, and, in the case of multiplication, by comparing

with

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.