The techniques of modern ‘digital’ communication have made familiar the idea that any piece of information, whether a written message, a photograph or even a sound can be transmitted in the form of a number. From this point of view a secret code consists of a finite subset U of the positive integers ℤ+ (the possible messages), a finite subset V of ℤ+ (the possible coded messages) together with a function T:U → V (the encoding function) and a function S:V→U (the decoding function) such that ST:U → U is the identity.

Remark. In fact even this definition fails to cover all possibilities since, for example, we could suppose T ‘multivalued’ with the value T(u) being chosen at random from a set Q(u)⊆Q such that ν∊Q(u) implies S(ν) = u. But we must start somewhere.

As a simple example let us take U = V = {n:0 ≤ n ≤ N − 1} and define T:U→V and S:V→U by the relations T(u) = u + M mod N, S(ν) = ν−M mod N. We consider the problem faced by ‘opponents’ who wish to decipher messages written using this code.

In general we must assume that our opponents know or guess the method of coding that we use. For the sake of illustration we may suppose that their information includes the value of N but not, at least initially, the value of M. Thus if we choose M at random and only use the code once, it is unbreakable, since trial decodes S′r(ν) = ν − r mod N allowing r to run from 0 to N − 1 will give all possible messages without any indication of which to choose.

J.B.S. Haidane was one of those intellectual aristocratic radicals who have enlivened British life and thought for the last two hundred years. He pursued three distinct but simultaneous careers as an experimental physiologist (like his father), as a mathematical geneticist and statistician (he, Fisher and Wright were the first people to give extensive mathematical treatments of Darwinian evolution from the Mendelian viewpoint) and as a scientific journalist. One of the high spots of his stormy public life was his dismissal from Cambridge at the instigation of the ‘Sex Viri’ (‘Six Men’, a kind of disciplinary committee) on moral grounds (he had been a co-respondent in a divorce suit). He fought back and gained reinstatement (and the ‘Six Men’ added one to their number to become the ‘Septem Viri’). (On a more peaceful level he wrote a curious children's book My Friend Mr Leaky which I remember as one of my childhood favourites.) Here is an article which he wrote for Eureka (the Cambridge undergraduate mathematical journal) in 1941. It was entitled ‘The Faking of Genetical Results’.

My father published a number of papers on blood analysis. In the proofs of one of them the following sentence, or something very like it, occurred: ‘Unless the blood is very thoroughly faked, it will be found that duplicate determinations rarely agree.’ Every biochemist will sympathise with this opinion. I may add that the verb ‘to lake’, when applied to blood, means to break up the corpuscles so that it becomes transparent.

In genetical work also, duplicates rarely agree unless they are faked.

We are used to thinking of stars as being so far away as to act as point sources of light. The fact that they appear to us as ‘twinkling’ patches of light is due to atmospheric effects. But, surprising as it may seem to a layman, the nearest stars are sufficiently close that, if it were not for the effects of the atmosphere, a good photograph using a good telescope would show them as tiny discs. Since observations of the nearest stars at six-monthly intervals (i.e. using a diameter of the earth's orbit as a surveyor's base line) enable astronomers to measure the distance of these stars, knowledge of the apparent diameter (i.e. the diameters of the discs on the photographic plate) would then enable us to calculate the true diameters of the nearest stars.

However, the blurring due to atmospheric effects is much greater than the apparent diameter we wish to observe. How can we get round this problem? Soon we will be able to use the ‘big science’ method and spend our way out of trouble by putting our telescope in orbit above the atmosphere. A more elegant (and considerably cheaper) solution has been found by Labeyrie.

Suppose we photograph a point source at time t and suppose that, without atmospheric effects, it would appear at 0 on our photographic plate. Owing to atmospheric effects we obtain a picture whose ‘brightness’ at a point x on the plate is λKt(x) (where λ is the ‘brightness’ of our original point source).

(The contents of the next two chapters are based on the excellent biography by Herivel Joseph Fourier The Man and the Physicist, Oxford 1975.)

Joseph Fourier was born in 1768 in Auxerre, the ninth child of a master tailor. Although the death of his father left him an orphan at the age of ten, his intelligence gained him a free place at the local Benedictine school. At the end of a brilliant school career he applied to enter the artillery only to be informed that such a profession was only open to those of noble blood and was closed to him ‘even if he were a second Newton’.

Fourier began to prepare to enter the Benedictine teaching order but, whatever his plans may have been, the course of his life was violently altered by the outbreak of the French Revolution. ‘As the natural ideas of equality developed, it was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests and to free from this double yoke the long usurped soil of Europe. I readily became enamoured of this cause, in my opinion the greatest and the most beautiful which any nation has undertaken.’

However, the Revolution was soon threatened by problems of its own making. The collapse of royal authority and the effects of revolutionary zeal and political misjudgement created administrative chaos whilst at the same time driving France into war against most of Europe.