Sometimes, this manifests itself in tactics that offer magnificent if unintentional tableaux vivant. I recall a field exercise years ago with the Second Marine Division at Camp Lejeune where, rounding a bend, we found a lieutenant had built a perfect 19th century fortress wall across the road, complete with firing step. The Division Sergeant Major, in whose jeep I was riding, said, "My God, it's the siege of Vicksburg!"

More often, linearity manifests itself in a military service's culture, as a subtle but omnipresent mindset.

It is easy to understand why this is so. Both on land and at sea, tactics became linear right at the beginning of the First Generation in the mid-17th century. In armies, that was when lines of infantrymen two or three deep replaced the square formations of the tercios. In navies, beginning with the British navy in the Dutch Wars, the line ahead replaced the general melee. The two developments were causally related: the line ahead was adopted when generals took command of the British fleet under the Commonwealth of Oliver Cromwell in the mid-17th century.

The First Generation lasted about two centuries, centuries in which the culture of state militaries was formed. Linearity on the battlefield carried over directly into that culture, where it remains today. In Second Generation militaries, such as the American, the tactics too remained largely linear. As late as the First Gulf War in 1991 a battalion commander in the U.S. Second Marine Division was nearly relieved for "breaking the line" when he pulled his unit back to avoid an Iraqi fire sack.

The expectation of linearity lies behind much of the U.S. military's misreading of the current situation in Iraq. If you look at its projections of success, they follow a line. It foresees a linear "building process" where its alliance with some Sunni militias in Anbar province and parts of Baghdad leads to similar alliances elsewhere, with no regression in "pacified" areas. Similarly, it expects the Sunnis to follow their acceptance of U.S. forces with acceptance of the Shiite-dominated government in Baghdad and its army and police.

These lines, which lead to improved security, then mesh with other lines such as economic and political developments that represent the re-emergence of a state in Iraq. It graphs nicely as a series of vectors on a chart, all pointing up. Linearity has marched from Waterloo to PowerPoint.

Unfortunately, Fourth Generation wars, and many other types of war as well, are not linear. Rather, they are chaotic, an unending melee of coming together and splitting apart that leaves an occupier running in place. Seemingly linear progress is matched or exceeded by non-linear regression. The state military perceives the former much more readily than the latter because linearity is what it expects. You find what you seek, whether or not it is there.

The reality in Iraq is that both Sunnis and Shiites are split along many different axes. Factions come together in temporary alliances of convenience, including with the foreign occupiers, only to split apart again and fight former allies. Reality for all parties is local and short term.

To the Iraqis, one alliance, such as with the Americans, does not imply any other alliance, such as with the central government. Arrangements that appear contradictory to us are natural to them. Linear progress toward a set of goals that represent a state is not what they expect. Our linearity and their non-linearity are ships passing in the night.

It will happen from time to time that the chaos shakes out into patterns in which we can see linear progress. But the reality remains chaos, which means the patterns will soon reform into other, quite different shapes. We cannot anticipate what those shapes might be. If we can be quick enough, we may be able to use some of those new shapes, as we have used the unexpected outbreak of fighting between local Sunni militias and al-Qaida. What we must not do, if our orientation is to be accurate, is project these kaleidoscopic pattern shifts in linear terms.

Regrettably, that is what the U.S. military in Iraq is doing now. Given its First Generation heritage, it may not be able to do anything else.

(William S. Lind, expressing his own personal opinion, is director of the Center for Cultural Conservatism at the Free Congress Foundation.)

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