If QL can upgrade its artillery in this way, its rate of hitting QN's artillery (if QN has an equal number of pieces) will no longer be 6 percent per day, but 54 percent (9 x 6 percent). But because we're still assuming that QN has three times as many pieces of artillery, QL destroys only 18 percent (1/3 x 54 percent) of QN's artillery each day.
Moreover, the rate at which QN's artillery takes out QL artillery pieces remains the same as calculated above at 18 percent per day.
The bottom line: it takes a ninefold increase in QL's quality to make up for a threefold increase in QN's quantity. In general, it takes an N-squared-fold increase in quality to make up for an N-fold increase in quantity.
Now to the war where, in some limited respects, the Republican Guard and irregulars are QN and the American and British forces are QL. If the units under discussion have planes, cruise missiles and the like, there is no comparison and Lanchester's Law is not relevant. With tanks and artillery, Lanchester's Law does come into play, and American qualitative superiority again easily wins the day.
It's only when we get down to the level of individual soldiers with rifles in house-to-house fighting that the balance becomes unclear.
It's here that Lanchester's Law suggests that American soldiers' smaller degree of superiority may not always make up for a potential Iraqi numerical advantage (unless weapons more destructive than rifles are unleashed); it takes a big qualitative advantage to overcome a small quantitative one.
Of course, this analysis is necessarily very simplistic and ignores many other factors. Much depends, for example, on whether or not Iraqis are committed to the fight, something that's more likely if they're engaging an essentially all-American army rather than one broadly composed and under the aegis of the United Nations.
Whatever the duration of the war in Iraq, Lanchester's Law reminds us why urban guerilla warfare appeals to those with a weak conventional military. It is a worrisome fact that only at this level is there less scope for technological superiority.
Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy, and the forthcoming A Mathematician Plays the Market, which will be published in the spring. His Who’s Counting? column on ABCNEWS.com appears the first weekend of every month.