During the last third of the eighteenth century Watt embarked on a total redesign of the Newcomen steam engine turning it from a special purpose machine to an economical general purpose power source. This involved the solution not just of one but many technical problems each one requiring either a new idea (like the separate condenser) or the astute adaptation of an old one (like the centrifugal governor which we shall briefly return to in Chapter 72).

One of the problems that had to be overcome was that of turning the rectilinear motion of a piston into the circular motion of a wheel and vice versa. Spurred on by his partner Boulton, Watt patented five different solutions. The one we shall discuss is called Watt's parallelogram. Watt is reported to have been prouder of this invention than of any other.

In its simplest form it consists of three bars AB, BC and CD linked as shown in Figure 42.1 with A and D fixed but D and C free to move. The ‘tracer point’ Q is on BC. A little experiment will show the reader that as C rotates round D, Q performs quite a complicated motion. However, for many purposes it is sufficient that the path of Q is approximately linear for rotations of CD through a small angle θ(∣θ∣ < 20° say).

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).