In 1916, during 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 with long-range weapons such as firearms).
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.
Lanchester's Square Law
With firearms engaging each other directly with aimed shooting from a distance, they can attack multiple targets and can receive fire from multiple directions. The rate of attrition now depends only on the number of weapons shooting. Lanchester determined that the power of such a force 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.
More precisely, the law specifies the casualties a shooting force will inflict over a period of time, relative to those inflicted by the opposing force. In its basic form, the law is only useful to predict outcomes and casualties by attrition. It does not apply to whole armies, where tactical deployment means not all troops will be engaged all the time. It only works where each unit (soldier, ship, etc.) can kill only one equivalent unit at a time. For this reason, the law does not apply to machine guns, artillery, or nuclear weapons. The law requires an assumption that casualties increase over time: it does not work in situations in which opposing troops kill each other instantly, either by shooting simultaneously or by one side getting off the first shot and inflicting multiple casualties.
Note that Lanchester's Square Law does not apply to technological force, only numerical force; so it requires an N-squared-fold increase in quality to compensate for an N-fold increase in quantity.
Suppose that two armies, Red and Blue are engaging each other in combat. Red is shooting a continuous stream of bullets at Blue. Meanwhile, Blue is shooting a continuous stream of bullets at Red.
Let symbol A represent the number of soldiers in the Red force at the beginning of the battle. Each one has offensive firepower α, which is the number of enemy soldiers it can incapacitate (e.g., kill or injure) per unit time. Likewise, Blue has B soldiers, each with offensive firepower β.
Lanchester’s square law calculates the number of soldiers lost on each side using the following pair of equations. Here, dA/dt represents the rate at which the number of Red soldiers is changing at a particular instant. A negative value indicates the loss of soldiers. Similarly, dB/dt represents the rate of change of the number of Blue soldiers.
- dA/dt = -βB
- dB/dt = -αA
Relation to the Salvo Combat Model
Lanchester’s equations are related to the more recent Salvo combat model equations, with two main differences.
First, Lanchester's original equations form a continuous time model, whereas the basic salvo equations form a discrete time model. In a gun battle, bullets or shells are typically fired in large quantities. Each round has a relatively low chance of hitting its target, and does a relatively small amount of damage. Therefore Lanchester’s equations model gunfire as a stream of firepower that continuously weakens the enemy force over time.
By comparison, cruise missiles typically are fired in relatively small quantities. Each one has a high probability of hitting its target, and carries a relatively powerful warhead. Therefore it makes more sense to model them as a discrete pulse (or salvo) of firepower in a discrete time model.
Second, Lanchester's equations include only offensive firepower, whereas the salvo equations also include defensive firepower. Given their small size and large number, it is not practical to intercept bullets and shells in a gun battle. By comparison, cruise missiles can be intercepted (shot down) by surface-to-air missiles and anti-aircraft guns. So it is important to include such active defenses in a missile combat model.
Lanchester's Law in use
Lanchester's laws have been used to model historical battles for research purposes. Examples include Pickett's Charge of Confederate infantry against Union infantry during the 1863 Battle of Gettysburg, and the 1940 Battle of Britain between the British and German air forces.
- Numbers, Predictions and War, Col T N Dupuy, Macdonald and Jane’s, 1979
- Lanchester F.W., Mathematics in Warfare in The World of Mathematics, Vol. 4 (1956) Ed. Newman, J.R., Simon and Schuster, 2138-2157
- Lanchester Equations and Scoring Systems
- Taylor JG. 1983. Lanchester Models of Warfare, volumes I & II. Operations Research Society of America.
- Armstrong MJ, Sodergren SE, 2015, Refighting Pickett’s Charge: mathematical modeling of the Civil War battlefield, Social Science Quarterly.
- MacKay N, Price C, 2011, Safety in Numbers: Ideas of concentration in Royal Air Force fighter defence from Lanchester to the Battle of Britain, History 96, 304-325.
- Race to the Swift: Thoughts on Twenty-First Century Warfare by Richard E. Simpkin
- "Kicking Butt By the Numbers: Lanchester's Laws", a Designer's Notebook column by Ernest Adams in the Gamasutra webzine
- Lanchester Equations and Scoring Systems, appendix to "Aggregation, Disaggregation, and the 3:1 Rule in Ground Combat" by Paul K. Davis, Rand Corporation publication MR-638-AF/A/OSD
- Lanchester combat models, "Mathematics Today", 2006, Vol 42/5, pages 170-173.
- N-Squared Law: An Examination of one of the Mathematical Theories behind the Dreadnought Battleship by Joseph Czarnecki at Naval Weapons of the World