“What Blackett did was to ask: What can one aircraft, one Liberator, achieve in the Battle of the Atlantic?”

By Niall MacKay

IN OUR GROUP at the University of York, we’ve been looking at various ways to use modern computer power to investigate counterfactuals in military history. Our forthcoming book, Quantifying Counterfactual Military History from Taylor & Francis, explores this in depth. But the basic idea of trying to quantify what might have happened in a war is surely as old as war itself. The oldest case I’ve personally examined is Thomas Digges’ 1579 Stratioticos. But here I’d like to tell you about a more recent example that uses no more than simple arithmetic and is from World War Two. It involves a physicist, the Battle of the Atlantic and the legendary Consolidated B-24 Liberator four-engine bomber.

Patrick Blackett wasn’t only a Nobel Prize-winning physicist, he was also one of the pioneers of operational research, heading a new Directorate of OR for the Royal Navy during World War Two.

Perhaps his most famous wartime report dealt with convoy size and marshalling statistics in support of large convoys as a means to reduce merchant ship sinkings. Within the report was a crucial element covering the value of very long-range “VLR” aircraft — mostly American-made B-24 Liberators — in providing air cover for convoys. For me, his analysis is Blackett-the-physicist shining through, because it’s exemplary “back of the envelope” physics.

Back-of-the-envelope physics is the art of estimation. It doesn’t need algebra, just basic arithmetic. There are lots of useful ideas to have in your toolkit for doing it, and Blackett’s calculation exemplifies one: the importance of choosing the right denominator.

For example, if you’re comparing government costs or services, often a useful denominator is to compare these per individual. As my father-in-law (like me, a physicist) says, if more people could divide by 67 million (which is the population of the U.K.), they’d find it easier to understand the economic news.

What Blackett did was to make the individual aircraft his denominator, and to ask: What can one aircraft, one Liberator, achieve in the Battle of the Atlantic?

Well, to begin with, what’s the average lifespan of a Liberator in wartime? Over Germany it might be 20 sorties. Over the Atlantic, where planes faced fewer dangers in the form of flak batteries and enemy interceptors, it’s about 45 sorties.

Then Blackett compared convoys with and without air cover. He found that over 43 days, convoys without cover had 75 ships sunk. That 1.74 ships lost per day. By contrast over 38 days of convoy operations with air cover, only 24 ships were sunk, or 0.63 ships lost per day. So, air cover is worth 1.74 minus 0.63, which equals approximately 1.1 ships saved per day thanks to air cover from B-24 bombers.

Next, those 38 days of air cover required 147 sorties by B-24s, or 3.9 sorties per day. So, if we divide the number of sorties per day, which is 3.9, by the number of ships saved per day, which is 1.1, we can find the number of sorties needed to save one ship (3.9/1.1 = 3.5). This means that it takes 3.5 Liberator sorties to save one ship. Therefore, over its average lifespan of 45 sorties, a B-24 will save approximately 13 ships (because 45/3.5=13).

What’s more, Liberators attacked and sank U-boats. Blackett crunched the numbers and found that of those 147 sorties over 38 days of convoy air cover, there were 43 attacks on enemy submarines. Typically, only about eight per cent of attacks resulted in a sinking.

So, the number of U-boats a single Liberator should sink could be calculated: 45 (a plane’s total number of sorties) x 43/147 (the number of attacks per sortie) x 0.08 (the number of U-boats sunk per attack) = 1 U-boat.

Next, Blackett needed a rule of thumb for how many ships were saved per U-boat destroyed. His estimate was that one enemy submarine sunk was worth about three ships saved (I don’t know his source for this, though, it’ll almost certainly be in a document at the U.K. National Archives at Kew).

So, a single B-24 Liberator is worth about 13+3 = 16 ships saved. That represents a hell of a lot for the Allied war effort, and although this calculation isn’t in the report, it’s certainly more than the damage likely to be caused to the Axis powers from 45 Liberator sorties’ worth of bombing over Germany, even considering RAF Bomber Command‘s early overestimates of the efficacy of strategic bombing.

It’s a neat calculation, which you can find in a paper at Kew for the U.K. Cabinet Anti-U-boat Warfare Committee meeting of 5 February 1943 – designated AU (43) 40 in CAB 86/3. But at the time it immediately became bogged down in politics and was disparaged as “slide rule strategy.” Blackett noted that Whitehall had “an allergy to arithmetic.” But the Naval and Air Staffs digested it and in a subsequent paper (in AU (43) 68 at Kew) renewed the call for more aircraft. My sense is that although the staffs were clearly “word” people rather than “numbers” people, they accepted his reasoning.

An entertaining counter-blast came from Arthur “Bomber” Harris in a report (designated AU (43) 96), and a more tendentious and fallacious argument you’d be hard put to find. Much of it is mere assertion. For example: “In view of the very large number of U-boats … the proportion of [U-boat] successes which would be eliminated by accepting the … proposals seems to me to be so small as to be negligible.”

Setting aside that there are no grounds adduced for its seeming thus to him, Harris doesn’t really have a denominator in mind. As long as additional aircraft were still closing the air gap and providing cover where otherwise there would be none, Blackett’s calculation stands. The number of U-boats overall has little to do with it, since they found it hard to operate where there was air cover. And 16 merchant ships saved per aircraft, even over the life of a Liberator, surely multiplies up by the necessary number of B-24s (several squadrons) to give a non-negligible number.

There are a couple of lessons here. Blackett has a quantified model. When someone puts one forward, anyone can critique it. Harris doesn’t, so his argument is slippery rhetoric rather than logic. Indeed, where does one begin to argue with him? No part of his case is built on any quantified foundations, yet both the bombing campaign and the Battle of the Atlantic were wars of attrition to be won or lost on their quantitative effects. But above all, the correct choice of denominator can make the calculation clear. Here, a Liberator clearly achieves more for the war effort over the Atlantic air gap than over Germany.

From the German point of view, imagine making the tank the denominator. Imagine there are 200 tanks on a merchant ship in a convoy to Russia. How much effort will it take to destroy them? Clearly it is easier to torpedo the ship than to have to destroy 200 tanks in battle. And a different or additional denominator can provide a shift in perspective. For example, if steel is the constraint, and it takes 1,000 tons of it to build a U-boat that will sink three ships carrying a total of 600 tanks, might it be better to use the steel to build your own tanks instead? Well, 1,000 tons will build only about 50 Panzer Mk.IV tanks instead, so probably not. A factor larger than 10 should be convincing enough.

Such estimates are great fun and can be crucial to effective quantitative argument and critical thinking in many real-world applications. Even today, attrition is a factor in the war in Ukraine. Attrition is always a numbers game, and government ministers may well have to make difficult decisions about procurement or use. It would be nice to know that they can do numerical estimates when they need to.

Niall MacKay is the co-author of Quantifying Counterfactual Military History. He is a mathematician and theoretical physicist at the University of York. He has interests in military history, operations research and combat modelling. He received his PhD in Theoretical and Mathematical Physics from Durham University.