The Nobel Prizewinning physicist Patrick Blackett 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 report was on convoy size, marshalling statistics in support of large convoys as a means to reduce merchant ship sinkings.
Here, however, I want to go over the other crucial element of that report, on the value of very long-range “VLR” aircraft, mostly US 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. Swartz and Zee have books on how to do it, and it’s also the basis of my namesake (but not my relative) David MacKay’s classic “Sustainable Energy, without the hot air”.
Back-of-the-envelope physics is the art of estimation. It doesn’t need algebra, just basic arithmetic. (When the numbers are very large or very small it’s called “Fermi estimation”, and you need to be comfortable with large powers of 10.) There are lots of useful ideas to have in your toolkit for doing it, and Blackett’s calculation exemplifies one: 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, the population of the UK, 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 lifetime of a Liberator? Over Germany it might be 20 sorties. Over the Atlantic, facing fewer dangers, and including accidents on take-off or landing, it’s about 45 sorties.
Then he compared convoys with and without air cover. In 43 days of convoys without cover, 75 ships were sunk, while in 38 days with cover, 24 ships were sunk. So air cover is worth 75/43 – 24/38 = 1.1 ships per day of cover.
Next, the 38 days of cover required 147 sorties, or 3.9 sorties per day, or 3.9/1.1 = 3.5 sorties per ship saved. So a Liberator saves 45/3.5=13 ships.
Further, a Liberator sinks U boats. Those 147 sorties included 43 attacks, and typically about 8% of attacks resulted in a sinking, so a Liberator should sink about 45 x 43/147 x 0.08 = 1 U-boat. Then he needed a rule of thumb, which was that a U-boat sunk is worth about 3 ships saved (source unknown).
So a Liberator is worth about 13+3 = 16 ships saved. That’s a hell of a lot of war effort, and a lot more (although this calculation isn’t in the report) than the damage likely to be caused to the German war effort by 45 sorties’ worth of bombing Germany.
It’s a neat calculation, which you can find in AU (43) 40 in CAB 86/3 at Kew, a paper for the UK Cabinet Anti-U-boat Warfare Committee meeting of 5th February 1943. But at the time it immediately got bogged down in politics, disparaged as “slide rule strategy” – Blackett noted that Whitehall had “an allergy to arithmetic”. But the Naval and Air Staffs digested it and (in AU(43)68) renewed the call for more aircraft – my sense is that, although the staffs were clearly “words” people rather than “numbers” people, they accepted his reasoning.
An entertaining counter-blast came from Arthur “Bomber” Harris, in AU (43) 96, and a more tendentious and fallacious argument you’d be hard put to find nowadays outside…. well, I won’t say where one encounters them these days; I’m sure you have your own examples. It commits most of the sins found in my diagnostic bible for such rubbish, Robert Thouless’ “Straight and Crooked Thinking”. 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 are still closing the air gap, still providing cover where otherwise there is none, Blackett’s calculation stands – and the number of U-boats overall has little to do with it, since they find it hard to operate where there is air cover, whether alone or in numbers. And 16 merchant ships saved per aircraft, even over the life of a Liberator, surely multiplies up by the necessary number of Liberators (several squadrons) to give a non-negligible number.
There are a couple of lessons here. Blackett has a quantified model. When someone puts forward a model, anyone can critique it. Harris doesn’t – so his argument is slippery, rhetoric rather than logic. (Indeed where does one start arguing with him? No part of his argument is built on any quantified foundations at all, 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. There are 200 tanks on a merchant ship in convoy to Russia. How much effort will it take to destroy it? 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 1000 tons to build a U-boat that will sink 3 ships and 600 tanks, might it be better to use the steel to build your own tanks instead? Well, 1000 tons will build only about 50 Pz.IV tanks instead, so probably not. A factor larger than 10 should be convincing enough.
Such estimates are great fun, can be crucial to effective quantitative argument and critical thinking in many real-world applications, and are an art that can be learned. They were what Fields Medal-winning mathematician Tim Gowers had in mind for teaching mathematics to non-mathematicians in UK schools, and I had the good fortune to be involved in the project from its inception through various working groups (ACME, MEI, …). It’s now embodied in the OCR Critical Mathematics qualification (now Core Maths A). Does it put across the interest, importance and excitement? Here’s a sample exam paper. Make your own mind up.
Weighing the Fog of War 