Lanchester’s laws are concerned with balancing military
strength of various units.
If one is dealing with ancient combat (i.e., pre-gunpowder)
Lanchester Linear Law applies, the common sense
“relative strength is proportional to number of combatants”.
(e.g., if the unit Alfa has twice as many men as unit Bravo,
unit Alfa is twice as strong)
It applies because one man is engaging in combat with
only one hostile man. If each pair of combatants kills
each other, the number of men remaining after the battle
is the larger army minus the smaller army.
But post gunpowder, Lanchester Square Law applies.
The relative strength is proportional to the *square*
of the number of combatants.
If unit Alfa has three times as many men, it is 3^2 = 9
times as powerful.
This is because with gunpowder combatants can engage
more than one hostile and come under attack from
more than one hostile.
Unit Alfa is concentrating three times as much firepower
on unit Bravo compared to Bravo’s firepower. And as
important, unit Bravo’s firepower is being diluted over
three times as many targets.
The number of units remaining would be
R = sqrt( a^2 – b^2)
If Alfa’s rifles are twice as efficient as Bravo’s,
if the two units are of equal size, Alfa will win.
But the Square law makes it easy to overcome the
efficiency with mere numbers. If Bravo has three
times as many units, they will win even with
Alfa’s advantage in weapons.
Specifically, if Bravo is three times bigger, it
has a strength nine times that of Alfa. Alfa’s
weapons reduces that strength to “only” 4.5,
so Bravo still destroys Alfa.
As Dr. Paulos put it, it takes an N-squared-fold increase in
quality to make up for an N-fold increase in quantity.
That’s a tall order.
Lanchester Laws do not take into account many other
important factors, but they can come in handy when setting
up the cost and production rate of different unit types.