Last October, in an internal Pentagon memo leaked to the press, Secretary of Defense Donald Rumsfeld hit on the key question in assessing U.S. progress in the war on Al Qaeda: “Are we capturing, killing or deterring and dissuading more terrorists every day than the madrassas and the radical clerics are recruiting, training and deploying against us?”

Three years after the destruction of the Twin Towers, that question is as vital as ever.

Rumsfeld’s question is key because it recognizes the nature of the enemy: We’re not at war with a state, but with an armed ideology with murderous adherents in more than 60 countries. Responses appropriate to a state-based threat will only rarely be effective against a private, self-organizing, adaptable enemy that can operate without state support or central direction. Indeed, such responses may exacerbate the problem, drawing new recruits to jihad.

Sept. 11, 2001, should have concentrated the mind wonderfully as to the type of enemy we’re fighting. Too often, however, the administration has insisted on “fighting the last war.” Having rightfully removed the one state that was directly related to the terror threat, the Taliban regime in Afghanistan, the administration continued on to Iraq, as if the war against terror was a war against states. But it’s hard to understand how regime change in Iraq aided the war against anti-American terrorism. Iraq appears to have had few, if any, genuine Al Qaeda links and no WMD stockpiles to speak of, much less a plan to pass off weapons of mass destruction to anti-American terrorists.

“Anonymous,” the author of “Imperial Hubris,” a 22-year CIA veteran who ran the Counterterrorist Center’s Bin Laden station from 1996 to 1999, is nobody’s peacenik. But he says that “there is nothing Bin Laden could have hoped for more than the invasion and occupation of Iraq.”

His assessment is echoed by former counterterrorism chief Richard Clarke, who says that the war on Iraq “delivered to Al Qaeda the greatest recruitment propaganda imaginable.”

Are they right? It’s difficult to tell. As Rumsfeld put it in the October memo, “we lack metrics” to know whether the pool of anti-American jihadis is growing or shrinking.

But there are some indications that we are losing that battle of numbers.

On April 1, J. Cofer Black, the State Department’s coordinator for counterterrorism, testified before Congress that there are “growing indications that Al Qaeda’s ideology is spreading well beyond the Middle East, particularly its virulent anti-American rhetoric. This has been picked up by a number of Islamic extremist movements which exist around the globe. This greatly complicates our task in stamping out Al Qaeda, and poses a threat in its own right for the foreseeable future.”

A year after the start of the Iraq war, a Pew Research Center Poll revealed that “large majorities in Jordan (70%) and Morocco (66%) believe suicide bombings carried out against Americans and other Westerners in Iraq are justifiable. Nearly half of those in Pakistan agree (46%).” Sixty-five percent of Pakistanis and 55 percent of Jordanians have a positive view of Bin Laden.

More recently, polls conducted by Zogby International show that the Iraq war has contributed to near-universal hostility toward the United States in the Arab world, with, for example, 98 percent of Egyptians holding negative views toward America. The “radical clerics” that Rumsfeld worries about now have an even more receptive audience.

That’s not to suggest that the war on Al Qaeda should be run as a global popularity contest. Far from it: We need to kill or capture those who mean us harm, and should make no apologies about it. But anti-American sentiment is the lifeblood of jihad. Needlessly increasing it through unnecessary wars in the Middle East nourishes the enemy and swells its ranks.

With the wisdom of hindsight, does the Bush administration fully appreciate this? Perhaps not.

Time magazine has reported that during “a private Aug. 19 conference call with Capitol Hill aides from both parties … senior Pentagon policy official William Luti said there are at least five or six foreign countries with traits that ‘no responsible leader can allow.’ ” There may be more Iraqs in our future.

Meanwhile, in Iraq, the periodic standoffs in Najaf, Sadr City, Fallujah and elsewhere put American servicemen in the untenable position of either having their hands tied in the face of aggression, or responding with overwhelming force, generating civilian casualties and film footage that will surely make its way into jihadist recruitment videos.

In the Defense Department memorandum leaked last October, Secretary Rumsfeld wondered, “Is our current situation such that ‘the harder we work, the behinder we get’?”

Rumsfeld wasn’t talking about Iraq specifically, but his words perfectly describe our current dilemma.