Lanchester’s Linear Law (for ancient combat)
Assuming identical weapons and morale, in ancient warfare, one soldier could 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.
Lanchester’s Square Law (for modern combat)
In modern combat, in battles of attrition, with modern weapons, the unit can attack multiple targets and can receive fire from multiple sources. The power of such an army is proportional not to the number of units in difference, but to the square of the number of units.
In order to win by attrition, the killing rate of the technology must advance in proportion to the square of the number of opponents it is meant to kill. It takes an N-squared-fold increase in quality to make up for an N-fold increase in quantity.
Image CC US Army