In October 1943, the oceanographer Arthur Doodson, head of the Liverpool Tidal Institute, got a letter from his friend Commander Ian Farquharson. For months, Doodson and the machine he’d perfected had provided the British Navy with key tidal information to launch their attacks on Nazi-occupied France, but this time Farquharson’s letter was far more cryptic: It listed no specific latitude or longitude, and it referred to a new location he called “Position Z.”

“The place is nameless and the constants inferred,” he wrote to Doodson. “There is in fact, very little data for it, I am gambling on the inferred shallow water constants giving something like the right answer.”

Position Z was the northern coast of Normandy, where Farquharson had surreptitiously been sending two-manned submarines to take water measurements in the hopes they’d help predict the tides at potential landing sites for D-Day. Since it is the 74th anniversary of D-Day and France is still France, we know that Doodson and his machine were able to calculate “something like the right answer” using Farquharson’s measurements. Doing so, however, was an extremely difficult feat.

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Troops approach the beach at Normandy, thanks to the work of Arthur Doodson and his tide-calculating machine

What Doodson had perfected was a tide-predicting machine, an extremely complicated invention that, as it was developed over the years, took into account an immense amount of data on tides. Fortunately, Doodson was an internationally respected genius when it came to doing this tidal math. Accounts from the Biographical Memoirs of Fellows of the Royal Society describe his ability to do all this calculating in his head. For the tidal calculations at “Position Z,” however, even the smallest error would have resulted in the allied forces plunging headlong into a series of underwater booby traps laid out by German forces. And so, Doodson created two machines to mimic what he did in his mind.

The two machines he kept in separate rooms in his Liverpool observatory included the simpler Kelvin 10-component tide-predicting machine and the 1.2-ton Roberts Lége Universal model, which summed up all of the components using over 30 different pulleys and wheels.

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A tide-predicting machine on display in Japan. This machine was modeled after the one used in Liverpool to predict the tides at Normandy.

According to the United States National Oceanic and Atmospheric Administration, there are three basic tidal patterns Doodson had to keep track of, all of which are controlled by the movement of celestial bodies: “In general, most areas have two high tides and two low tides each day. When the two highs and the two lows are about the same height, the pattern is called a semi-daily or semidiurnal tide. If the high and low tides differ in height, the pattern is called a mixed semidiurnal tide. Some areas … have only one high and one low tide each day. This is called a diurnal tide.”

Doodson knew that tracking this pattern of high tides and low tides over time would produce peaks and valleys — in other words, a sine wave. Adding them all together would create a rough picture of how each one of these different waves combine to create the overall tide of a specific place. Rough estimates, however, would not cut it for World War II.

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Different constituents are combined to create an estimate of a tide. Each constituent is caused by the gravitational pull of a different planet or star. Charles Darwin's son, George, was one of the first to figure this out, because figuring out things is what the Darwin family does.  

In a 1926 Nature article published years before the war, Doodson explained how his machine worked: “A predicting machine sums the number of harmonic variations, transmitted vertically to pulleys … round which passes a wire or chain which is fixed at one end, and carries a recording pen at the free end.”

Soon after he received that critical letter in 1943, Doodson calibrated the machine with measurements provided by Farquharson. Then, after some four hours winding and prodding, it spat out a rough estimate of the tides at Position Z, deciding the fate of Nazi-occupied France.

What makes this feat even more remarkable is that Doodson was working with limited data — he was initially only given eleven measurements — so he had to do a certain amount of adjusting and recalculating. But, this kind of thing was a simple task for Doodson, who claimed in the same 1926 Nature article: “The simplicity of the harmonic methods of analysis and prediction is very fascinating.”

Eventually, Doodson’s calculations made predicting the tides possible, though deciding what to do with that information became a point of contention between the British Navy and Army. While the Navy needed shallower water to reveal traps in the sand, the Army wanted to maximize the number of boats it could have in the sea at high tide. Wesleyan University professor of Earth science Suzanne O’Connell, Ph.D., explained the final decision in Slate in 2014:

An Army-Navy compromise was struck: The invasion would begin one to three hours after low tide. The necessary tide and moon conditions in 1944 were on June 5, 6, and 7.

Nevertheless, without Doodson’s calculations, the opportunity for such a compromise might not even have existed. And had the troops missed the crucial window they needed to storm Normandy’s shores, our world might look a lot different. Now, 74 years later, we can all be grateful for those two enormous tide calculators — and the one guy who could do some very impressive mental math.