Two armies of equal skill meet; each side’s losses are proportional to how many the ENEMY still has firing. The brutal arithmetic that falls out: fighting power grows with the SQUARE of numbers, so a force twice the size wins with four times the margin. Concentration of force, proven. Slide the attacker’s numbers and watch the square bite.
Lanchester’s aimed-fire model: dA/dt = −βB, dB/dt = −αA. Dividing gives αA·dA = βB·dB, so αA² − βB² is CONSERVED through the battle — the square law. With equal quality (α=β), the winner’s survivors are exactly √(A₀² − B₀²). The instrument integrates the real ODEs. A fail-loud self-check runs 5 vs 3 and throws unless the stronger side ends with √(25−9)=4 survivors and the weaker is wiped — why splitting the enemy or concentrating your own is decisive, not merely helpful.
The SQUARE law is for aimed fire where all units engage; ancient melee (only the front rank fights) follows Lanchester’s LINEAR law (advantage ∝ N, not N²). Real combat adds morale, terrain, logistics; the ODE invariant is exact (Lanchester 1916).