Englishman Frederick W. Lanchester (1868-1946) was a major contributor to the foundation of automotive and aeronautical engineering. He also published works on radio, acoustics, warfare, and even relativity. His equations of combat form the basis of the science of operations research. (These equations have been used to formulate business strategy in recent times.) He was the first to describe the aeronautics of lift and drag. His automobile inventions include the gas engine starter, rack-and-pinion steering, disk brakes, four-wheel drive, and fuel injection.

In his historic 1916 paper "Mathematics in Warfare," Lanchester presents two simple differential equations relating force attrition to the number of forces or weapons in opposition and to their effectiveness (see sidebar "Lanchester's Equations"). The equations' solutions show that the effectiveness of a force is directly proportional to the effectiveness of its weapons and to the *square* of its numbers. The following table illustrates how Lanchester's equations would apply in a classic artillery duel:

**The Lanchester Exchange: Artillery Duel**

The table shows 200 weapons arrayed against 100 weapons with equal kill probabilities (Pk) of 10 percent. In the first round, Orange kills 20 Blues, and Blue kills 10 Oranges--leaving 190 Oranges to kill 19 of the remaining 80 Blues, while the Blues kill 8 Oranges in the second round. At the end of the sixth round, all the Blues are gone and 168 Oranges (84 percent) remain.

Note that each side engages only the remaining live targets. If neither side can tell when it has killed a target, as in some artillery duels, both sides must continue to shoot at all the targets, thereby wasting part of their efforts. Lanchester analyzed this problem also and showed that the impact of numbers is a linear not square law.

Bob Everett, former president of The MITRE Corp., noted that that was reasonable, because in the square-law example, "you get one power from the number of weapons shooting at the other side and the other power from the reduced number of targets you have to shoot at."

The advantage of telling dead from live targets is one of the reasons that artillery forces use spotters and counter-battery radar and that air forces use bomb-damage assessment after air attacks.

(Please see, " " for a graphic representation of the difference in attrition.)

Of course, wars aren't fought in accordance with mathematical equations, and there are many other important factors, including leadership, discipline, morale, training, and health. Nevertheless, analysis of battles between conventional forces over the years has supported the thrust of Lanchester's Law: numbers do make a huge difference.

In the 1985 book *Race to the Swift,* British military analyst Richard E. Simpkin notes that "for a conflict between two large, sophisticated mechanized forces, one did not go far wrong with a '1.5 law'--a halfway house between Lanchester's two cases."

Lanchester's paper appears in Volume 4 of mathematician James Newman's delightful *The World of Mathematics* collection. (Incidentally, to give you some feeling of what Newman thinks of our profession, he writes in his commentary on Lanchester, "His writings on these matters, apart from high professional competence, exhibit such striking independence of judgment and boldness of conception that it is surprising to learn he was an engineer." Oh, well.)

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