Let us begin with a lesson I gave to IC (aged 12) three weeks ago. We are trying to construct a safe method of encoding words. We begin with a simple substitution alphabet in which the 26 letters are paired off with the numbers from 1 to 26.

Teaching a class of 33 girls any subject whatsoever would always be interesting, but when the subject is mathematics and when the exact nature of the course is left to the teacher, things become exciting! This was the case during the last term at a Girls’ Grammar School in Leicester where I was given a free hand to tackle any aspect of mathematics with the form 5Sc. Their average age was 15 (exactly: we had laboriously evaluated it the previous term when considering mean values!) and their ability was very high. Although they were still one year from O-level, I thought this to be an excellent opportunity for trying some “Modern Mathematics.”

A recent conference, held at Southampton early in September, marked the end of the first stage, and of the first twelve months’ work, of this Project. Its objectives are now well-known, namely, the experimental introduction of entirely new mathematical syllabuses in certain schools, together with the setting of appropriate G.C.E. papers, and the writing of text-books and teachers’ guides. But it may be of some interest to fill in some further details of the work.

Several lines of thought led up to the present experiment in cooperation which has just started in Abingdon. I would like to start by trying to summarize them.

We are apt, in the grammar schools, to criticise the universities. We may say that their influence on our syllabus is bad. We may think that the teaching they give to our prize pupils is poor, the supervision unsuitable, the courses wrongly devised. These things we say and think and I am not concerned here with the rights and wrongs; but do we realize that these are precisely the things that primary teachers are saying and thinking about us?

In the concluding chapter of “On Teaching Mathematics” (1), we said (a) Mathematics is a subject which must be taught well at an early stage of a child’s schooling, and (b) the youngest children should not be taught by those insensitive to the subject. How much do we in the grammar schools know of the primary teacher and of what is being taught today in the primary schools?

The method of writing , … as recurring decimal fractions by cyclic permutation of the six figures 142857 is well known, but it is perhaps not so well known that this is but a tiny piece of a general pattern of behaviour of all fractions with denominators which are prime to the radix.

Suppose E is a vector space over the field of complex numbers with a complex valued scalar product ( , ), with the properties and (x, x) ≠ 0 when x ≠ 0, defined over it.

If a1, …, an and b1, …, bn are two sets of real numbers, then (a1b1 + … + anbb)2 ≤ (a12 + … + an2)(b12 + … + bn2), the equality holding if and only if either 3 a real number λ such that ar — λbr (r = 1, …, n) with some br ≠ 0 or b1 = … = bn = 0. Write a = (a1, …, an) and b = (b1, …, bn). Suppose a ≠ 0 and b ≠ 0, for otherwise the inequality is trivial. Write с = a — λb.

Choose λ so as to make b, c=0, i.e take

the equality holding if and only if e=0, i.e if and only if a=λb,

2. Suppose x1...,xn and y1 are two sets of linearly independent vectors in the space E defined on the previous note.

Mobius‘ function has the important property of inverting functional relations [1] of the type and giving in turn (Also relations (0.1) and (0.2) follow respectively from (0.11) and (0.21))

I establish in this paper that no other arithmetical function μ*(n) can perform the above inversion. In other words, I prove that Inversion is the characteristic property of Mobius‘ μ-function. Theorem 1 below asserts that if for one pair of functions g(n) and f(n) (subject to a special condition) relations (0.1) and (0.11) are true with an arithmetical function μ*(n) (possibly depending on g and f), then μ*(n) coincides with the Mobius‘-function. Theorem 2 below gives a similar result in connection with functions G(x) and F(x) defined for all real x ≥ 1.