Around the time of World War I, July 28, 1914 to November 11, 1918, many mathematicians and engineers, including Frederick W. Lanchester, became fascinated by the dynamics of the battlefield.  Various mathematical models were proposed in an effort to explain--and to predict--how military forces interacted on the battlefield.  During World War I these mathematical investigations were mainly academic, although during World War II the United States government actually applied these models to make important decisions about the Battle of Iwo Jima in which the American forces seized control of the Japanese island of Iwo Jima.  Outnumbered and outgunned by the Americans, the Japanese were defeated even before the battle began although the American forces suffered many casualties and injuries.