Lanchester's Linear Law
For ancient combat, between phalanxes of soldiers with spears, say, one soldier could only ever fight exactly one other soldier at a time. If each soldier kills, and is killed by, exactly one other, then the number of soldiers remaining at the end of the battle is simply the difference between the larger army and the smaller, assuming identical weapons.
The linear law also applies to unaimed fire into an enemy-occupied area. The rate of attrition depends on the density of the available targets in the target area as well as the number of weapons shooting. If two forces, occupying the same land area and using the same weapons, shoot randomly into the same target area, they will both suffer the same rate and number of casualties, until the smaller force is eventually eliminated: the greater probability of any one shot hitting the larger force is balanced by the greater number of shots directed at the smaller force.
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