Lanchester's laws are mathematical formulae for calculating the strength of military forces.
In 1916, during the height of World War I, Frederick Lanchester devised a series of differential equations to demonstrate the power relationships between opposing forces. Among these are what is known as Lanchester's Linear Law (for ancient combat) and Lanchester's Square Law (for modern combat). In ancient combat, between phalanxes of men with spears, say, one man could only ever fight exactly one other man at a time. If each man kills, and is killed by, exactly one other, then the number of men remaining at the end of the battle is simply the difference between the larger army and the smaller, as you might expect (assuming identical weapons).
In modern combat, however, with artillery pieces firing at each other from a distance, the guns can attack multiple targets and can receive fire from multiple directions. Lanchester determined that the power of such an army is proportional not to the number of units it has, but to the square of the number of units. This is known as Lanchester's Square Law. It relies on the fact that when either side has more units it means that they have also more surface-area that opponent can hit and also more firepower. So when number of units is for an example doubled compared to the enemy then they have 80% chance(or 4:1) of winning.
Note that Lanchester's Square Law does not apply to technological force, only numerical force; so it takes an N-squared-fold increase in quality to make up for an N-fold increase in quantity.
