Around 1940, Alan Turing sat down to write a manual. New recruits were arriving at Bletchley Park with mathematical training and no idea how the German cipher machines worked or what was being done about them, and somebody had to explain it.
The result runs to about 150 typed pages with handwritten corrections and additions throughout. His colleagues called it the Prof’s Book, because Turing was known as Prof. It remained classified for fifty six years, and it is the most direct surviving account of how the Enigma was actually broken, written by the person who did most of the breaking.
What It Is
The formal title is Treatise on the Enigma. It is not a memoir, a report to superiors, or a piece of theory written for publication. It is a working teaching document produced inside a secret organisation for people who needed to become useful quickly.
That purpose shapes everything about it. The tone is informal and occasionally funny. Turing explains things twice when he thinks they are difficult, uses concrete examples with actual letters and rotor settings, and admits when a method is tedious or when he is unsure whether a technique will keep working.
The physical document carries the marks of a manual in use. Sections are amended in pencil, tables are corrected by hand, and the typescript has the slightly improvised quality of something produced under pressure.
It was declassified and transferred to the National Archives at Kew in 1996, and released online by GCHQ in 2012.
The Machine It Describes
The opening chapters explain the Enigma from first principles, and they remain one of the clearest accounts available.
An Enigma operator presses a key, current flows through a plugboard, then through three rotating wheels chosen from a set, each wiring the twenty six letters to twenty six others, then through a reflector and back out through the rotors and plugboard by a different path, lighting a lamp with the enciphered letter. Every keystroke advances the rightmost rotor, so the substitution changes with every letter.
The number of possible configurations is often quoted at around 159 quintillion. Brute force is not available, and was not available to any machine built in the twentieth century.
Turing spends considerable time on the machine’s one structural weakness, which is the reflector. Because the current passes out and back, no letter can ever encipher to itself. That single fact is what makes everything else possible.
The Crib
The central technique described in the treatise depends on guessing part of the plaintext. German operators were creatures of habit: weather reports began predictably, messages contained standard formulae, and some stations sent nearly identical traffic every morning.
Given a guessed fragment, the no letter enciphers to itself rule lets you slide the guess along the intercepted message and eliminate every position where a letter would have to map to itself. What survives is a small number of candidate alignments, each of which imposes a set of logical constraints on the wiring: if this rotor position turns A into T, then it must also turn T into A, and so on around a chain.
Those constraints form loops, and a contradiction anywhere in a loop rules out the whole configuration. This is the logic the Bombe implemented in hardware, testing candidate settings at speed and stopping when no contradiction had been found.
The treatise sets out this reasoning step by step, which is why it is a better guide to the actual work than any dramatisation.
Banburismus
The most original material in the document concerns Naval Enigma, which used a larger set of rotors and more careful operating procedures, and which resisted the methods that worked against army and air force traffic.
Turing’s answer was a statistical procedure he called Banburismus, named after Banbury, where the long perforated sheets it used were printed.
The idea is to compare two intercepted messages and count how often the same letter appears in the same position in both. If the two were enciphered at the same rotor setting, the count will be slightly higher than chance, because the underlying German plaintext has the statistical structure of a real language. If they were not, the count matches random expectation.
The excess is small, so a single comparison proves nothing. What Turing developed was a way of accumulating many weak pieces of evidence into a confident conclusion.
He defined a unit for the weight of evidence, which he called the ban, with a tenth of a ban as the deciban, chosen because it is roughly the smallest change in confidence a person can perceive. Each observation contributes a quantity of bans for or against a hypothesis, and the contributions add up. When the total passes a threshold, the hypothesis is accepted.
This is sequential Bayesian inference, expressed in logarithmic units of evidence. Turing worked it out in 1940 for the practical purpose of eliminating rotor orders before committing scarce Bombe time to them. The Banburismus procedure could cut the number of possible wheel orders from 336 to a handful, which was the difference between reading naval traffic and not.
The theoretical importance is considerable. Turing’s units are directly analogous to the measures at the heart of Shannon’s information theory, published in 1948, and the sequential analysis framework was developed independently and openly by Abraham Wald in the same period. None of Turing’s work could be cited, discussed, or claimed. His assistant I.J. Good published the ideas after the war, and the wartime origin remained secret for decades.
What the Treatise Does Not Claim
One of the more valuable features of the document is its unheroic framing. Turing is describing a set of tools used by a large organisation, and he says so.
The foundations were Polish. Marian Rejewski, Jerzy Różycki and Henryk Zygalski of the Biuro Szyfrów had reconstructed the internal wiring of the military Enigma mathematically in 1932, using permutation theory, and had built both the perforated sheet method and an electromechanical device they called the bomba. In July 1939, five weeks before the invasion, they handed everything to British and French intelligence at a meeting near Warsaw.
The British Bombe is not a copy of the Polish machine and works on a different principle, because the German procedural change that the Polish method depended on had been withdrawn. But Bletchley started from a reconstructed machine and a proven body of technique rather than from nothing, and the treatise treats this as ordinary background rather than a revelation.
Equally, the operational work described in it was carried out by thousands of people, most of them women, running the Bombes, managing the indexes, and processing traffic. Their contribution is visible throughout the document in the procedures it specifies.
Why the Original Matters
There is a large literature about Bletchley Park, and most of it is narrative: the personalities, the secrecy, the impact on the war. The treatise is a different kind of object. It is the technical content, written at the time, for people who needed to do the work.
Reading it, you can follow the reasoning as it was actually applied, including the parts that are laborious and the judgement calls that no summary preserves. It also documents a piece of intellectual history that was invisible for half a century: a mathematician inventing applied Bayesian statistics under wartime conditions, in a document nobody outside the building could read.
Our edition of The Prof’s Book presents the treatise with Turing’s handwritten annotations preserved alongside a remastered typescript, so that the corrections and marginal additions remain visible as part of the document rather than being tidied away.
Where It Sits in His Work
The treatise falls in the middle of Turing’s career and connects the two ends of it.
The 1936 paper on computable numbers had introduced the abstract machine that defines what computation is, entirely as a piece of mathematical logic with no thought of building anything. The Bombe was a physical device that mechanised a specific inferential process, and the experience of designing machinery to do reasoning fed directly into what came afterwards.
After the war Turing worked on stored program computer design at the National Physical Laboratory, and in 1950 published the paper asking whether machines can think. The line from the treatise to that question is short: five years of building machines that weighed evidence and drew conclusions faster than people could.
The full story of the codebreaking effort and the machine that carried it out are worth reading alongside it. The treatise is what those accounts are describing.
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