A few years ago I tried making a model of an object from its photograph. Realising that, of course, one could not take direct measurements from the photograph, I had to try to guess the relative dimensions of the object. I ended up, unfortunately, with a model which did not seem to be in the correct proportions. Would it have been possible, I wondered, to find the precise ratios of the different lengths?

This article is addressed to anyone who has ever doodled idly with a fixed compass on a blank sheet of paper. Perhaps he has drawn a circle centre O, picked a point P1 on it, drawn a circle centre P1 intersecting the original circle at P2 and P3, drawn a circle centre P2 producing two new points of intersection, slowly building up an interlacing hexagonal pattern II of points each a distance at least R from each other. Then he has picked some point P* not of II, and drawn a circle centre P*, and almost immediately the pretty pattern has been replaced by a jumble of intersections apparently covering the whole of the paper.

A famous problem in euclidean geometry, associated with the names of J. Steiner (1796-1867) and D. C. Lehmus (1780-1863), is to show that a triangle is isosceles when two internal bisectors are equal. Although the converse is obvious, a direct proof of the statement is far from simple. A large number of demonstrations have been devised, some valid and some spurious, and a considerable literature exists on the subject.

It has often been noted that modular arithmetic provides a rich source of supply of groups. Indeed, a remarkable theorem asserts that any finite commutative group can be found by a sufficiently diligent search through the multiplicative groups and subgroups of modular arithmetic! During the last few years two articles in this area have appeared in Mathematics Teaching. In the first [1], Tim Brand drew attention to the fact that such a multiplicative group can have an identity element other than 1. (For example, 8 is the identity of the group {2,4,8} under multiplication mod 14.) More recently [2] Geoff Saltmarsh described an ingenious way of finding the identity and put forward an interesting conjecture about these groups.

The study of unique factorisation is almost as old as mathematics itself. The so-called Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be factorised into a product of primes in only one way, was probably the first major theorem proved. It is interesting to note that Euclid (c. 300 B.C.) did not give the result in this form. He proved (IX 14) that “if a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it” (see [1]).

If we have a system of m linear equations to solve, it is a great simplification to write them in matrix form

Ax = b,

where A is an m x n matrix of coefficients, b is an m-dimensional vector of constants and x is an n-dimensional vector of unknowns.

The first published description of the geoboard appears to have been given as an incidental illustration to Denis Diderot’s Letter on the blind (1774).

The savants of the eighteenth century were interested in blindness as a problem for speculative philosophy. If the mind is a tabula rasa at birth, how will it develop if the sovereign sense is absent? Diderot held that the blind man would not only form different physical concepts, but different aesthetic and even different moral values. In the end he would inevitably come to a different set of religious beliefs and hence, Diderot argued, the concept of God was relative to our senses.