It is well known that the six rotations of an equilateral triangle form a group; that is to say that, if an equilateral triangle ABC occupies a certain triangular space, there are six operations and six only which when performed either singly or in succession will leave the triangle still occupying the same space, but with vertices in different positions. The set of six possible orientations of the movable triangle are drawn below and numbered.

In 1877, the English mathematician J J. Sylvester (1) announced the discovery of a rather complete formal analogy between the mathematical theory of the invariants and covariants of binary quantics and the chemical theory of valency as it then stood.

Desargues’ theorem is concerned with two triangles in perspective for some point O. Let us denote the vertices of the first triangle by , it is then natural to denote the sides by where is the side containing . Now bringing in the second triangle we get sides . Moreover, one pair of corresponding sides is ; it is therefore natural to christen their intersection . The other two meets are ; and, by Desargues' theorem, these three meets are collinear; we therefore christen the axis of perspective .

1. Definition

We define an infinite coprime sequence as an infinite class of positive integers arranged in order of ascending magnitude, in which the highest common divisor of any two members is unity.

2. Note on prime numbers

If an infinite sequence of primes, constructed by some rule which does not invoke a knowledge of the sequence of primes, should ever be found, it will necessarily be an infinite coprime sequence. We thus concern ourselves in this note with the construction of infinite coprime sequences without using a knowledge of the sequence of primes.

What is the effective rate of interest when a loan of £L is paid for by n equal payments of £P at equal intervals of time, as in hire purchase and other forms of time payment?