Battles are what the Second World War is remembered for. They are not, mostly, what settled it. It was settled by how many aircraft a country could build in a year, by whether food and fuel could cross an ocean, and — genuinely — by a handful of scientists who were handed operational data and asked what it meant. This page does three of those calculations. One of them, worked out by a physicist looking at convoy records, saved more ships than any battle in the Atlantic.
Drag the timeline. It starts in July 1937, not September 1939, because for China the war was already two years old by the time Germany invaded Poland — where you begin the story depends on where you are standing. Watch the pace: almost nothing for two years, then most of Europe inside fourteen months, then the Soviet Union and the United States within six months of each other, after which the outcome was largely a question of arithmetic. Italy changes colour twice, because it changed sides.
Aircraft built during the war, by country. Germany and Japan were not beaten because their engineers were worse — several of their designs were excellent, and German production kept rising until late 1944 despite being bombed. They were beaten by the size of the other side’s industry. Press the button to add up each side and see the ratio, which is the single most important number on this page.
This is the result that feels wrong until you see the geometry. Ships crossing the Atlantic sailed in convoys, ringed by escorts. Put more ships in the convoy and it sounds like you are simply offering a bigger target. But the ships fill an area while the escorts patrol a perimeter, and area grows faster than perimeter. Double the ships and the fence around them grows by only about forty per cent. Drag the slider and watch the number of escorts needed per ship fall as the convoy grows.
Patrick Blackett, a physicist, was given the Admiralty’s convoy records and found something nobody expected: the number of ships lost per convoy attacked barely depended on how large the convoy was. If a U-boat pack sinks roughly the same number of ships whichever convoy it finds, then putting the same total cargo into fewer, larger convoys means fewer convoys to be found and the same losses each — so the fraction lost falls. Move the slider and watch what happens to a fixed number of ships crossing the ocean as you change only how they are grouped.
Escorts were not the only thing hunting U-boats; aircraft were better at it, because a submarine that has to stay submerged cannot catch a convoy. But an aircraft has to come back. Draw a circle of range around each base and there was a stretch in the middle of the ocean that nothing could reach — and the U-boats knew exactly where it was. Slide the radius of action out and watch it close. The route drawn is the real great circle from Halifax to Liverpool, and the circles are true ranges computed on the sphere, not ellipses drawn on a flat map.
In 1941 Bomber Command believed it was hitting factories. David Bengtson Butt was asked to check, and examined photographs taken automatically as bombs fell. His conclusion: of the crews who reported attacking their target, only about one in three had come within five miles of it. Over the Ruhr, one in ten. Turn that single fact into a scatter of bombs and the consequence becomes obvious — and so does why the policy changed from aiming at factories to aiming at cities, which is a decision this page does not defend, only explain.
Nothing in this section moves, and nothing in it should. Between 60 and 85 million people died in the Second World War — the range is wide because much of the killing happened where records were destroyed or never kept. Most of the dead were civilians. That was new: in the First World War soldiers were the majority.
Six million of those people were Jews, murdered in a deliberate, planned, industrial attempt to destroy the Jewish people of Europe. That is the Holocaust, and it is not a statistic that belongs in a chart with the others — it was not a consequence of fighting, nor of bombing, nor of famine. It was the purpose of a policy. Alongside them the Nazi state murdered Roma and Sinti people, disabled people, Soviet prisoners of war, Polish and other Slavic civilians, gay men, and political and religious opponents. Estimates of civilians killed as a direct result of Nazi ideological policy run from 11 to 17 million people in total.
Of the numbers above this section, one is worth keeping for its own reasons. About 30,000 of Germany’s roughly 40,000 U-boat crewmen did not come back — three in every four. It was very probably the highest loss rate of any substantial fighting force in the war, and it belongs here because the arithmetic in sections 2 and 3 is what killed them. Mathematics applied to a war is still applied to a war.
The mathematics is computed live. Section 3 lays out a real formation and counts it: N ships in a roughly square block have a perimeter of about 4√N ship-widths, so escorts needed per ship falls as 1/√N — doubling a convoy from 32 to 64 ships needs only 1.41 times the escorts, which is 0.71 times as many per ship. Section 4 divides a fixed number of crossings into convoys and multiplies by a loss rate you set. Section 5 walks the real great circle from Halifax to Liverpool in six hundred steps, and at each step takes the least great-circle distance to Gander, Reykjavik or Aldergrove; the range rings are traced round the sphere bearing by bearing rather than drawn as ellipses, which is why they are not circles on the page. The gap closes at a radius of 771 nautical miles. Section 6 converts the Butt figure into a circular error probable by solving 1 − ½^(r²/CEP²) for CEP, then scatters bombs from that distribution: at one crew in three within five miles the CEP is 6.54 miles, and the chance of a given bomb landing within 100 m of the aiming point works out at about one in sixteen thousand.
The historical inputs are quoted and every one of them is a range in the literature. Aircraft production totals differ between sources depending on which trainers and prototypes are counted; the figures here are the commonly cited national totals. Blackett’s finding is presented as what he argued from the records, and the slider lets you disagree with him. The Butt Report itself has been argued over ever since, including by people who thought it too harsh.
The same caution about the map as in the First World War lab. These are modern borders from Natural Earth, which is public domain, and several of the states that fought are gone: the Soviet Union is drawn as Russia, Ukraine, Belarus, the Baltic states, the Caucasus and Central Asia; Yugoslavia as the seven countries that replaced it; Czechoslovakia as Czechia and Slovakia. Colonies and occupied territories are shown by the power that held them, which flattens a great deal — India did not choose to enter the war, and the millions who fought from British, French and Dutch colonies are inside somebody else’s colour here. Countries change colour on the date they entered the fighting, which for the invaded ones is the date they were invaded rather than any decision they took.
Two things this page deliberately does not do. It does not let you move a slider and change how many people died, because that is not a model, it is an obscenity. And it does not pretend that arithmetic explains the war. Production ratios do not explain why Germany invaded the Soviet Union, why the Holocaust happened, or what it was like to be in a convoy or under a bomber. For those you need history, testimony and memoir, not a page of sums. What is here is one narrow, honest slice: the part of the war that was decided by counting.