Why Numbers Win Squared

Lanchester’s laws of combat, tested by simulation

Frederick Lanchester was an English engineer who built one of Britain’s first cars and worked out the theory of how wings generate lift. In 1916, with the First World War in its third year, he turned the same habit of mind on the fighting itself. His book Aircraft in Warfare contains a pair of differential equations that say how much an extra thousand soldiers are worth. The answer depends entirely on how the soldiers fight, and in one of the two cases the answer is startling: the strength of an army grows with the square of its size.

A Russian officer, M. Osipov, published much the same equations in 1915, and the pair are usually called Lanchester’s laws anyway. This post does not just derive them. It builds a small battle simulator, runs a hundred fighters against seventy in two different ways, and checks what comes out against the algebra. It is also the first piece of a small wargame: the combat rule that decides what happens when two armies meet.

Two ways to fight

Imagine two armies, A and B, with the same kind of soldier on each side. Everything depends on who can hurt whom.

In aimed fire, such as rifles or artillery, every soldier can shoot at any enemy in range. The more shooters an army has, the…