*The first part of this post appeared on the blog in November 2016. The second part was supposed to come out within a week or two, as soon as I found a little more on the post-war use of analog tide-predicting machines. Unfortunately, the search for “a little more” ended up taking way more time than expected and turning up nothing within easy reach. I’d skip the apology if it wasn’t for the fact that the proper references to Anna Carlsson-Hyslop‘s research (discussed in part one) were buried in the source note at the end of this post. Sorry.*

Tide-predicting machines, the first of which appeared in the late nineteenth century, were an elegant mechanical solution to a complex mathematical problem. Used mostly to produce information useful to commercial shipping, during the two world wars they also played an important role in the planning of amphibious operations like the Normandy landings.

That contribution is interesting enough to give them a spot in the history of war, but the basic design of the British machines – using multiple gears to create an analog approximation of a mathematical function – also has an oblique connection to one of the most important technical achievements of the war: the mechanization of cryptanalysis.

Alan Turing is justifiably famous for his role in the breaking of the German Enigma cipher, and particularly for his contribution to designing electro-mechanical computing tools that transformed the process. (Even if popular versions of the story do a terrible disservice to the story by erasing everyone except Turing from the picture. *Imitation Game*, I’m looking at you.) Less well known are some of Turing’s pre-war flirtations with the mechanization of mathematical problem-solving. Andrew Hodges’ biography describes two projects which Turing took on, at least briefly. The first, during his time at Princeton in 1937, was to use electromagnetic relays for binary multiplication to create an extremely large number that could be used as the key for a cipher. This was, as Hodges puts it, a “surprisingly feeble” idea for a cipher but a practical success as far as constructing the relays was concerned.

The second project was an attempt to disprove the Riemann hypothesis about the distribution of prime numbers by calculating the Riemann zeta-function (for the hypothesis and the zeta-function, read Hodges or Wikipedia. There’s no chance of me describing it properly) by showing that not all instances where the function reached zero lay on a single line, as the hypothesis stated. An Oxford mathematician had already calculated the first 104 zeroes using punched-care machines to implement one approximation of the function. Since the zeta-function was the sum of circular functions of different frequencies, just like Thomson’s harmonic analysis of the tides, Turing realized it could be calculated using the same method. Or, more precisely, the machine could rule out enough values that only a few would have to be calculated by hand.

With a grant of £40 from the Royal Society, Turing and Donald MacPhail designed a machine that, like the tide calculators, used meshed gear wheels to approximate the thirty frequencies involved. The blueprint was completed by 17 July 1939 and the grinding of the wheels was underway when the war broke out at Turing joined the Government Code and Cypher School at Bletchley Park.

Nothing in the work that Turing did at Bletchley connected directly to the zeta-function machine, but, as Hodges notes, it was unusual for a mathematician like Turing to have any interest in using machines to tackle abstract problems of this sort. Clearly, though, Turing had been mulling the question of how machines could be applied to pure mathematics long before he became involved in the specific cryptanalytic problems that were tackled at Bletchley.

Of course, the secrecy surrounding code-breaking meant that no hint of the connection, or any of Turing’s wartime work, would have leaked out to those operating the tide-predicting machines in Liverpool or elsewhere. The end of the war meant a return to usual practice, but their strategic importance remained.

Probably the last analog machine to be constructed was a thirty-four constituent machine built in 1952–5 for East Germany (and now in the collection of the German Maritime Museum in Bremen). The Soviet Union had ordered a Kelvin-type machine for forty constituents from Légé and Co. in 1941 that was delivered to the State Oceanographic Institute in Moscow in 1946, on the eve of the Cold War. Bernard Zetler, an oceanographer who worked on tide prediction at the Scripps Institution of Oceanography in San Diego, recalls that he was unable to visit the machine in 1971 because it or its location was classified. The Soviet tide tables certainly were.

The American Tide Predicting Machine No. 2 remained in use until 1966, but played no role in the American amphibious landing at Inchon during the Korean War. The wide tidal range at Inchon meant that the landing needed good tidal information, but rather than making new calculations the existing American and Japanese tide tables were supplemented by first-hand observation by Navy Lieutenant Eugene F. Clark, whose unit reconnoitered the area for two weeks preceding the landings.

When analog machines like Tide Predicting Machine No. 2 were retired, they were replaced by digital computers whose architecture originated in other wartime projects like the ENIAC computer, which had been built to calculate ballistics tables for US artillery. The world’s navies have not relinquished their interest in tools to predict the tides. Their use, though, has never matched the high drama of prediction during the Second World War.

*Source Note*: The D-Day predictions are discussed many places on the internet, but almost all the accounts trace back to an article oceanographer Bruce Parker published in *Physics Today*, adapted from his 2010 book *The Power of the Sea*. Where Parker disagrees with the inventory of machines commissioned by the National Oceanography Centre, Liverpool (itself a descendant of the Liverpool Tidal Institute), I’ve followed Parker. Details on the work of Arthur Doodson and the Liverpool Tidal Institute come from Anna Carlsson-Hyslop‘s work: the articles “Human Computing Practices and Patronage: Antiaircraft Ballistics and Tidal Calculations in First World War Britain,” *Information & Culture: A Journal of History* 50:1 (2015) and “Patronage and Practice in British Oceanography: The Mixed Patronage of Storm Surge Science at the Liverpool Tidal Institute, 1919–1959,” *Historical Studies in the Natural* *Sciences* 46:3 (2016), and her dissertation for the University of Manchester (accessible through the NERC Open Repository). The scientist suspected of Nazi sympathies was Harald Sverdrup, a Norwegian national who worked with Walter Munk on wave prediction methods used in several amphibious landings. Turing’s experiments calculating the Riemann zeta-function appear in Andrew Hodges, *Alan Turing: The Engima* (1983; my edition the 2014 Vintage movie tie-in).