Given Tehran's defiant response to the European and American effort to constrain its nuclear program, it is time for bolder diplomacy out of Washington. U.S. President George W. Bush should take a page from the playbook of Ronald Reagan, who negotiated with an evil Soviet regime--competing in the war of ideas, but addressing the enemy's security concerns through arms-control agreements.

Iran's intransigence is both deeply unfortunate and perfectly predictable. It is unfortunate because Tehran's refusal to suspend its uranium-enrichment operations immediately--as demanded in July by the U.N. Security Council in a legally binding resolution--suggests that Iran is moving more quickly than expected toward a nuclear-weapons capability. Tehran has now turned the nuclear crisis into a test of the whole U.N. Security Council system. And Russia and China's current position, threatening to veto any biting sanctions against Iran, suggests that the Security Council may well fail this crucial test.

Tehran's response is predictable, however, because the offer on the table contains both inadequate economic carrots and barely credible threats of sanctions and military force. The carrots appeared impressive at first glance--in return for a suspension of enrichment we reportedly promised to provide light-water nuclear reactors and to help Iran with civil aviation and telecommunications technology. But we did not offer the one incentive that might possibly work, security guarantees that could reduce Iran's desire for nuclear weapons.

This omission is striking. The Iranian government can't talk openly about their security concerns because that would blow their cover story that the nuclear program is only for energy production. And Washington does not want to discuss such worries because it wants to keep open the possibility of removing the regime by force. "Security assurances are not on the table," U.S. Secretary of State Condoleezza Rice too cleverly argued this spring: "It is a little strange to talk about security guarantees ... I thought the Iranian position was that they weren't developing a nuclear bomb."

This is partly a crisis of our own making, as the Bush administration has practiced the reverse of Teddy Roosevelt's maxim--speaking loudly and carrying a small stick. Think about how Tehran reacted when Bush stated (in his second Inaugural Address), "The rulers of outlaw regimes can know that we still believe as Abraham Lincoln did: 'Those who deny freedoms to others deserve it not for themselves; and, under the rule of a just God, cannot long retain it." Or when Bush dramatically told reporters last April that "all options are on the table," in direct response to a question about whether he was considering a nuclear attack against Iran. Such statements only encourage Iran to develop a nuclear deterrent quickly, before the United States can carry out its perceived aggressive intent. Last month, Iran's National Security Council Secretary Ali Larijani pointedly complained about such rhetoric. "How can a side that wants to topple the regime also attempt to negotiate?"

Given the current vulnerability of U.S. forces in Iraq, the deteriorating situation in Afghanistan, and the lack of Israeli success against Hizbullah, Iranian officials seem confident that they face no immediate threat of a U.S. military assault. But they are clearly worried that Bush just might attack Iran right before he leaves office in January 2009, or that his successor might do so once U.S. forces withdraw from Iraq.

The best way to prevent a nuclear Iran is for Washington to offer the kind of security assurances that might reduce support in Tehran for building a nuclear arsenal. It will be hard to make such assurances credible, but a public U.S. promise to take forcible regime change off the table, and a U.N. Security Council commitment to protect the "political sovereignty" of Iran could help. Involving the Security Council could also pull China and Russia back into the nonproliferation coalition and enhance the U.N.'s legitimacy.

There is very little time left, which means negotiations should begin despite Iran's unfortunate opening position. Tehran's response reportedly indicated a willingness to negotiate all aspects of its nuclear program, so working out an agreement for Iran to limit itself to low-level uranium enrichment might still be possible. This would work only if Tehran accepts full IAEA inspections and a freeze on future centrifuge construction. Will they? The one thing that might cause Tehran to do so, and that would compensate for any loss of face, would be an assurance that the United States will not launch another preventive war, as it did in Iraq, to remove the Iranian regime. If in turn we get a nuclear-free Iran, that's a good deal for the West as well.

North Korea is poised to flight test a ballistic missile that may have intercontinental range -- an action the Bush administration declares would be provocative. Others have called for sanctions if the flight test occurs, the use of U.S. ballistic-missile defenses to intercept the missile in flight or a pre-emptive attack against the missile-launch site. But is this missile test a red line that, if crossed, seriously threatens U.S. security and, hence, warrants strong action? The simple answer is "No."

In thinking about this test, one must not lose sight of two paramount goals: rolling back North Korea's nuclear weapons program and the eventual peaceful reunification of North and South Korea. Ballistic missiles constitute a serious threat to the U.S. homeland only when armed with nuclear warheads and they are only one delivery means for such weapons. In this sense, ballistic missiles are of secondary concern. By most estimates, North Korea has sufficient nuclear material for a few nuclear explosive devices, but whether they can design a nuclear weapon that satisfies the size, weight and delivery constraints associated with intercontinental-range ballistic missiles is far from obvious.

If North Korea tests a three-stage version of the Taepodong-2 missile, it will likely attempt to put a satellite into orbit, just as it did in 1998 when it failed to place a satellite into orbit with the smaller Taepodong-1 missile. North Korea has a sovereign right to launch satellites, or to test ballistic missiles for that matter. International protocol requires launch notification and restrictions on air and marine traffic for reasons of range safety -- steps North Korea failed to take in 1998 -- but no international agreement bars this test. True, North Korea agreed to a unilateral moratorium on ballistic missile flight tests in 1999, pending further talks with the United States regarding North Korea's missile program, but the Bush administration refused to join these talks. North Korea leader Kim Jong Il reaffirmed this flight test moratorium in the 2002 Pyongyang Declaration signed with Japan, but this document is not a legally binding commitment.

If successful, this flight test would demonstrate that North Korea can produce rockets large enough to carry payloads intercontinental distances. However, this does not translate into an immediate threat because North Korea has not demonstrated that it can build a nuclear warhead that is small enough to fit on top of a Taepodong-2 missile and that can survive re-entry into the atmosphere after flying intercontinental distances.

Given vastly superior U.S. conventional and nuclear forces, deterrence should dissuade North Korea from ever using such missiles, except for saber rattling, or worse selling nuclear weapons or nuclear material abroad (this is a serious red line). Kim Jong Il may be a ruthless totalitarian leader, with little regard for the welfare of his people, but he is not suicidal.

More important, these missiles would be highly vulnerable to pre-emptive attack in the midst of a crisis, which is when pre-emption makes sense, because these missiles are large and easy to detect, they are not mobile, and they take many hours, if not days, to erect in a vertical position and fuel -- precisely the activity that generated this concern.

On the other hand, U.S. sanctions against North Korea in the wake of a test flight could backfire. They would likely cause rifts with other friendly parties to the Six Party talks aimed at eliminating North Korea's nuclear weapons, especially China. U.S. national missile defenses may not be within range, depending on the flight trajectory, to intercept this flight test. If this unproven U.S. missile defense were to fail and North Korea's flight test succeed, the Bush administration would be embarrassed, and Kim Jong Il triumphant. And, pre-emptive attack against the test facility would be a unilateral act of war at a time when U.S. unilateralism has hurt more than helped U.S. vital interests. South Korea would adamantly oppose such adventurism because Seoul is vulnerable to retribution, being within artillery range of the Demilitarized Zone.

So, what should the United States do on the eve of this flight test? Nothing, beyond expressing its dismay that North Korea appears to favor conflict over cooperation.

A Taepodong-2 flight test allows the United States to learn more about this missile than North Korea, given the concentration of technical intelligence assets in the area, which would help resolve the question of whether this missile, in fact, constitutes a serious threat to the U.S. homeland. In addition, such a test would isolate North Korea further and reinvigorate the Six Party Talks by encouraging South Korea, China, Japan, Russia and the United States to overcome their differences and create a united front to persuade North Korea to renounce its nuclear weapon program, which is the real threat.

Stepping back, U.S. leaders should see that North Korea is a mouse and the United States the elephant. Contrary to popular mythology, elephants are not, and should not be, afraid of mice.

North Korean technicians are reportedly in the final stages of fueling a long-range ballistic missile that some experts estimate can deliver a deadly payload to the United States. The last time North Korea tested such a missile, in 1998, it sent a shock wave around the world, but especially to the United States and Japan, both of which North Korea regards as archenemies. They recognized immediately that a missile of this type makes no sense as a weapon unless it is intended for delivery of a nuclear warhead.

A year later North Korea agreed to a moratorium on further launches, which it upheld -- until now. But there is a critical difference between now and 1998. Today North Korea openly boasts of its nuclear deterrent, has obtained six to eight bombs' worth of plutonium since 2003 and is plunging ahead to make more in its Yongbyon reactor. The six-party talks aimed at containing North Korea's weapons of mass destruction have collapsed.

Should the United States allow a country openly hostile to it and armed with nuclear weapons to perfect an intercontinental ballistic missile capable of delivering nuclear weapons to U.S. soil? We believe not. The Bush administration has unwisely ballyhooed the doctrine of "preemption," which all previous presidents have sustained as an option rather than a dogma. It has applied the doctrine to Iraq, where the intelligence pointed to a threat from weapons of mass destruction that was much smaller than the risk North Korea poses. (The actual threat from Saddam Hussein was, we now know, even smaller than believed at the time of the invasion.) But intervening before mortal threats to U.S. security can develop is surely a prudent policy.

Therefore, if North Korea persists in its launch preparations, the United States should immediately make clear its intention to strike and destroy the North Korean Taepodong missile before it can be launched. This could be accomplished, for example, by a cruise missile launched from a submarine carrying a high-explosive warhead. The blast would be similar to the one that killed terrorist leader Abu Musab al-Zarqawi in Iraq. But the effect on the Taepodong would be devastating. The multi-story, thin-skinned missile filled with high-energy fuel is itself explosive -- the U.S. airstrike would puncture the missile and probably cause it to explode. The carefully engineered test bed for North Korea's nascent nuclear missile force would be destroyed, and its attempt to retrogress to Cold War threats thwarted. There would be no damage to North Korea outside the immediate vicinity of the missile gantry.

The U.S. military has announced that it has placed some of the new missile defense interceptors deployed in Alaska and California on alert. In theory, the antiballistic missile system might succeed in smashing into the Taepodong payload as it hurtled through space after the missile booster burned out. But waiting until North Korea's ICBM is launched to interdict it is risky. First, by the time the payload was intercepted, North Korean engineers would already have obtained much of the precious flight test data they are seeking, which they could use to make a whole arsenal of missiles, hiding and protecting them from more U.S. strikes in the maze of tunnels they have dug throughout their mountainous country. Second, the U.S. defensive interceptor could reach the target only if it was flying on a test trajectory that took it into the range of the U.S. defense. Third, the U.S. system is unproven against North Korean missiles and has had an uneven record in its flight tests. A failed attempt at interception could undermine whatever deterrent value our missile defense may have.

We should not conceal our determination to strike the Taepodong if North Korea refuses to drain the fuel out and take it back to the warehouse. When they learn of it, our South Korean allies will surely not support this ultimatum -- indeed they will vigorously oppose it. The United States should accordingly make clear to the North that the South will play no role in the attack, which can be carried out entirely with U.S. forces and without use of South Korean territory. South Korea has worked hard to counter North Korea's 50-year menacing of its own country, through both military defense and negotiations, and the United States has stood with the South throughout. South Koreans should understand that U.S. territory is now also being threatened, and we must respond. Japan is likely to welcome the action but will also not lend open support or assistance. China and Russia will be shocked that North Korea's recklessness and the failure of the six-party talks have brought things to such a pass, but they will not defend North Korea.

In addition to warning our allies and partners of our determination to take out the Taepodong before it can be launched, we should warn the North Koreans. There is nothing they could do with such warning to defend the bulky, vulnerable missile on its launch pad, but they could evacuate personnel who might otherwise be harmed. The United States should emphasize that the strike, if mounted, would not be an attack on the entire country, or even its military, but only on the missile that North Korea pledged not to launch -- one designed to carry nuclear weapons. We should sharply warn North Korea against further escalation.

North Korea could respond to U.S. resolve by taking the drastic step of threatening all-out war on the Korean Peninsula. But it is unlikely to act on that threat. Why attack South Korea, which has been working to improve North-South relations (sometimes at odds with the United States) and which was openly opposing the U.S. action? An invasion of South Korea would bring about the certain end of Kim Jong Il's regime within a few bloody weeks of war, as surely he knows. Though war is unlikely, it would be prudent for the United States to enhance deterrence by introducing U.S. air and naval forces into the region at the same time it made its threat to strike the Taepodong. If North Korea opted for such a suicidal course, these extra forces would make its defeat swifter and less costly in lives -- American, South Korean and North Korean.

This is a hard measure for President Bush to take. It undoubtedly carries risk. But the risk of continuing inaction in the face of North Korea's race to threaten this country would be greater. Creative diplomacy might have avoided the need to choose between these two unattractive alternatives. Indeed, in earlier years the two of us were directly involved in negotiations with North Korea, coupled with military planning, to prevent just such an outcome. We believe diplomacy might have precluded the current situation. But diplomacy has failed, and we cannot sit by and let this deadly threat mature. A successful Taepodong launch, unopposed by the United States, its intended victim, would only embolden North Korea even further. The result would be more nuclear warheads atop more and more missiles.

Ashton B. Carter was assistant secretary of defense under President Bill Clinton and William J. Perry was secretary of defense. The writers, who conducted the North Korea policy review while in government, are now professors at Harvard and Stanford, respectively.

In his State of the Union address in January, President Bush stressed the importance of improving math education. He proposed to "train 70,000 high school teachers to lead advanced placement courses in math and science, bring 30,000 math and science professionals to teach in classrooms, and give early help to students who struggle with math."

But where will these teachers come from? And will the training of teachers be sufficient to increase the number of students choosing math and science careers? And why does all this matter?

Because mathematics is the foundation of the natural sciences. It is no coincidence that Isaac Newton, the man who formulated the law of gravitational attraction that revolutionized our understanding of the universe, was also the man who popularized the calculus. And the natural sciences, however pure, are what give us airplanes, cable TV and the Internet.

In the 2003 Program for International Student Assessment, a test that measures math literacy, American 15-year-olds performed worse than their peers in 23 countries, as well as those in Hong Kong. It's not hard to see why. According to the National Council of Teachers of Mathematics, 40 percent of the nation's middle school math teachers do not have the equivalent of an undergraduate minor in math. The average starting salary of a teacher is only $30,000, whereas the average starting salary for a recent college graduate in computer science or engineering is $50,000.

Short of following the British, who have proposed paying experienced math teachers more than $100,000, with a guaranteed minimum of $70,000, where will we find a way to attract the thousands of teachers George Bush wants?

New York State initiated an innovative program to bring teachers from Jamaica for two or four years to teach in New York schools. Jamaica, a developing nation where one U.S. dollar equals 65 Jamaican dollars, is nonetheless a stable, English-speaking nation with an unbroken democratic tradition; it stands poised to beat the United States in establishing the world's first Institute for Mathematical Methods in Counterterrorism. When teachers for the New York program were recruited on the campus of the University of the West Indies, recruiters found more experienced math and science teachers than they ever dreamed they would.

But you can have all the teachers in the world and still not inspire kids to learn math. My friend Autumn e-mailed me about her nephew, Joshua: "He's upset because he's asked several of the math teachers why math is important or what are certain formulas used for -- there has to be a use, correct?"

Autumn told her nephew about my work in counterterrorism and for the television crime drama "Numb3rs." Autumn reported, "He's told his math teachers about you as well, and about the show 'Numb3rs.' He's informing them that through something called lattice theory you are managing to fight terrorists -- all with math."

Mathematics is art, and should be appreciated for its beauty, not simply for its utility. But we cannot expect 11 year-olds to cherish totally order-disconnected topological spaces as much as professional mathematicians do.

As I first proposed in January 2005, television shows like "Numb3rs" (or "Medium") -- where the main characters are mathematicians -- could work with the National Council of Teachers of Mathematics to show kids how math is really used; the council and Texas Instruments are now working together to use "Numb3rs" to promote math literacy in schools.

Another way to inspire kids is to relate mathematics to something they see every day. In order to excite students and draw funding to his school, school superintendent Ronald Ross of Roosevelt, N.Y., has begun looking into the idea of creating a curriculum involving math and counterterrorism. What kinds of topics would students learn?

The opening line of the Oscar-winning movie "A Beautiful Mind" is "Mathematicians won the war." During World War II, the mathematics underlying cryptography played an important role in military planning. Winston Churchill admired Alan Turing, the mathematician who had mastered the German codes, recognizing him as the man who had perhaps made the single greatest individual contribution to defeating Hitler.

At Los Alamos, the lab that built the atomic bomb, Cliff Joslyn uses lattice theory to mine data drawn from thousands of reports of terrorist-related activity to discover patterns and relationships that were previously in shadow.

Lattice theoretical methods developed at MIT tell us the probability that we have disabled a terrorist cell, based on how many men we have taken out and what rank they hold in the organization. Lauren McGough, a Massachusetts high school student, tested the accuracy of this model by getting her classmates to pretend they were terrorists, passing orders down a fictitious chain of command, essentially confirming what the theory predicts.

High school students could learn algebra, trigonometry, calculus and logic while also learning concrete applications involving homeland security. No longer would students yawn and ask, "What is math good for?" Beauty could defeat both terror and boredom.

Whatever you may think of the State of the Union address, when it comes to supporting math education, we should all see pi to pi. President Bush is correct when he says that mathematics education in America must improve if the United States is to stay economically competitive, but the stakes are much higher than that. During the Cold War, the United States would not have tolerated a military gap between itself and its adversaries. Yet today, with 61 percent of all U.S. doctorates in math going to foreigners (15 percent to Chinese), we readily accept a "math gap."

Dollar for dollar, the best defense against our adversaries' weapons of mass destruction may be our allies in the Americas, armed with weapons of math instruction.

Improving math education is not merely a smart idea. It is a matter of national security. Algebra is one revolutionary Islamic concept we cannot afford to neglect or ignore.

Before attacking Iraq in March 2003, the United States told the world that Saddam Hussein had reconstituted his nuclear weapons program in defiance of the United Nations. That claim, used to justify the war, was based on assessments provided by the United States intelligence community. But as everyone now knows, those assessments were wrong. So Jeffrey T. Richelson's history of American nuclear intelligence, including our attempts to learn about Iraq's nuclear program, could hardly be more timely.

In "Spying on the Bomb," Richelson, the author of several books on American intelligence, has brought together a huge amount of information about Washington's efforts to track the nuclear weapons projects of other countries. He examines the nuclear projects of Nazi Germany, the Soviet Union, China, France, Israel, India, South Africa, Taiwan, Libya, Pakistan, Iran and North Korea, as well as Iraq. Through interviews and declassified documents as well as secondary works, he sets out briefly what we currently know about those projects and compares that with assessments of the time.

This may sound like heavy going, but Richelson writes with admirable clarity. And along the way he has fascinating stories to tell: about plans to assassinate the German physicist Werner Heisenberg during World War II; about discussions in the Kennedy and Johnson administrations on the possibility of attacking Chinese nuclear installations; about Indian measures to evade the gaze of American reconnaissance satellites; and about the bureaucratic infighting over the estimates on Iraq.

The United States has put an enormous effort into gathering information about the nuclear projects of other countries. After World War II it equipped aircraft with special filters to pick up radioactive debris from nuclear tests for isotopic analysis. It created a network of stations around the world to register the seismic effects of nuclear explosions. Most important, in 1960 it began to launch reconnaissance satellites that could take detailed photographs of nuclear sites in the Soviet Union and China. Richelson occasionally speculates about the role of communications intercepts and of spies, but these appear from his account to have been much less important than the other methods of collecting information.

Through these means the United States has gathered a vast quantity of data, sometimes to surprising effect. Intelligence played a crucial role in the cold war, for instance, by reducing uncertainty about Soviet nuclear forces. Alongside such successes, however, there have been failures. One notable example concerned the first Soviet test, which took place in August 1949, much sooner than the C.I.A. had predicted. Another was the failure to detect Indian preparations for tests in May 1998, even though at an earlier time the United States, with the help of satellite intelligence, had managed to learn about preparations the Indians were making and to head off their tests.

But the most serious failure of all was in Iraq in 2003, because in no other case did the intelligence assessments serve as justification for the use of military force. The information needed for avoiding political surprise is one thing. That needed for preventive war is quite another, if only because of the consequences of making a mistake.

Beyond making the uncontroversial recommendation that "aggressive and inventive intelligence collection and analysis" should continue, Richelson draws no general conclusions. That is a pity, because his rich material points to issues that cry out for further analysis. He suggests in one or two cases that failures sprang from the mind-set of the intelligence community, but he does not elaborate on this point. He has little to say about relations between policy makers and the intelligence community, even though the quality of intelligence and the use made of it depend heavily on that relationship.

His focus is no less narrow in his discussion of foreign nuclear projects. He concentrates on the programs themselves, paying very little attention to their political context. Does that reflect a technological bias in nuclear intelligence? Would, for example, the prewar assessment of Iraqi nuclear capabilities have been more accurate if it had paid more attention to the broader political and economic circumstances of Hussein's regime?

The task of intelligence has become more complex than it was during the cold war. A single dominant nuclear opponent has now been replaced by a number of nuclear states, along with states and stateless terrorists that are aiming to get their hands on nuclear weapons. Meanwhile, the technology needed for producing nuclear weapons has become easier to acquire.

Many critics believe the recent performance of the intelligence community shows it has not responded adequately to this new situation. Richelson does not have much to say on this question; nor does he discuss the likely impact of the current reforms, initiated in response to the Iraq war, on the quality of intelligence. His reticence may imply that he does not think reform is necessary. Still, it is disappointing that he does not draw on his historical survey to discuss whether new approaches are needed for dealing with nuclear threats, and, if so, what those new approaches might be.