Lanchester's square law

Why does a force twice the size win with far more than half its strength intact?

Two hundred aircraft meet one hundred of equal quality and fight until one side is gone. The intuitive answer is that the larger side finishes with a hundred — each of the smaller side's aircraft trades itself for one of yours. Frederick Lanchester, the English engineer who worked this out in 1916 while thinking about air combat, said the answer is about a hundred and seventy-three.

That is a startling amount of surplus, and it did not come from a better aeroplane or a cleverer pilot. Both sides are identical here except in number. So where did the extra seventy-three come from? Something about the arithmetic of a fight must be doing work that the trading picture leaves out.

Try to say precisely what "trading one for one" assumes. It assumes each combatant is paired with an opponent and the two of them resolve their private duel. Under that picture the larger side really does finish with a hundred, and Lanchester agreed — for that kind of fight. He called it the linear law, and thought it described a spear line where a man can only reach the man in front of him.

Now change one thing. Suppose everyone can shoot at anyone, at range, and choose their target. What breaks?

The pairing breaks. There is no private duel any more. Ask yourself what determines how fast your side loses aircraft in the next minute — and the answer is no longer "how many of us there are", it is "how many of them there are, shooting". Losses are inflicted by the enemy's numbers, not your own.

Hold both sides of that at once, because it is the whole idea. Your rate of loss depends on their number. Their rate of loss depends on yours. And you outnumber them two to one, so they start dying twice as fast as you do.

Now let a minute pass and ask the question again. They have lost more than you. So they now have even fewer shooters, so your losses slow further, while their losses continue at nearly the old rate. Do you see what has happened? The advantage has fed on itself. The ratio between the two forces does not stay at two to one — it widens, and it widens faster the wider it gets. That is the signature of a compounding process, and indeed if you solve the pair of equations properly the numbers move as exponential functions of time rather than in a straight line.

Where does the square come from? From what that compounding conserves. Work through the algebra and the quantity that stays fixed throughout the battle is the difference of the squares of the two forces. So the survivors of the larger side are the square root of four hundred minus one hundred — the square root of three hundred, in hundreds, or about a hundred and seventy-three. Combat power under these conditions goes as numbers squared times individual effectiveness.

That last sentence has a sharp practical edge worth pausing on. If power goes as the square of numbers but only linearly with quality, then doubling your numbers is worth four times as much, while doubling each unit's effectiveness is worth only twice as much. To match a force twice your size you would need each of your units to be four times as good. Concentration is not a preference in this model; it is the dominant term.

Which is why splitting a superior force is so costly here. Fight the enemy's hundred with a hundred, then with your other hundred, and you win two even fights and lose almost everything — the same men, sequenced differently, throw the square away entirely.

An honest caveat, because this is a model and not a law of nature. It assumes both sides are in range of each other, that everyone can find a target, that effectiveness stays constant, and that nobody breaks or runs. Real battles supply terrain, surprise, morale, command failures, and forces that never fully engage. Attempts to fit the equations to historical battles have produced mixed results — sometimes a good fit, often not — and the model is best treated as an argument about why concentration pays, not as a predictor of casualty lists.

The most common misreading is that the square law says a bigger force is more likely to win. It says something more specific: given that it wins, it wins with disproportionately more left over, and that surplus is available for the next fight. The law is about the cost of victory, and the cost is what makes concentration compound across a campaign rather than only within one engagement.

The second is treating the square as universal. Lanchester offered two laws deliberately, and which one applies is a question about the kind of combat. Where fire is aimed and everyone can engage everyone — gunnery, air combat, modern direct fire — the square applies. Where engagements are pairwise, or fire is unaimed into an area, the linear law is the better description, and there numbers buy exactly what you would naively expect. Guerrilla fighting is usually argued to sit closer to the linear end, which is part of why small forces can persist against large ones for years.

• The linear law, and why insurgency and dispersed fighting are argued to live nearer that end.
• Defeat in detail: the operational art of forcing an opponent to fight in halves.
• How the model changes when the two sides differ in quality, and what exchange rate a smaller force would need.

Every term the collection defines is gathered in the glossary.