President Clinton's decision to send 20,000 U.S. troops to Bosnia to enforce an American-brokered peace settlement contains a simple geopolitical message: The Cold War may be over and the "Soviet threat" may have receded into history, but American blood and treasure remain vital ingredients in the preservation of European security.

"The centrality of American leadership is one of the things that has come home to us during the three years of this administration," said Secretary of State Warren Christopher in an interview this week. "One of the things we concluded in 1995 was that American leadership was necessary to resolve {the Bosnia} problem."

Compare that remark with the phrase used by one of Christopher's predecessors less than four years ago, as the former Yugoslavia began its headlong descent into fratricidal war based on ancient nationalist hatreds, to explain why military intervention by the United States was out of the question. "We do not have a dog in that fight," said James A. Baker III, in a blunt Texan reaction to the bewildering complexity of Balkan politics.

The story of how America overcame a deep reluctance to get militarily involved in a seemingly obscure corner of Europe spans two U.S. administrations and two historical eras. The violent disintegration of the former Yugoslavia coincided with the end of the Cold War, the collapse of communism and the emergence of the United States as the world's sole remaining superpower.

In struggling to find an appropriate response to the Bosnia crisis, U.S. policymakers also were struggling to find a proper place for the United States in the post-Cold War world. The victory over communism produced an initial wave of euphoria in Western capitals, with talk about the construction of a "new world order" and the final triumph of American democratic values. Having borne the burden of the Cold War and the Persian Gulf War, Americans looked forward to the long-promised "peace dividend." Hundreds of thousands of American soldiers began coming home from Europe.

There was an assumption, on both sides of the Atlantic, that the end of Cold War would permit Europeans to resolve their own problems, without constant American tutelage. "The hour of Europe has arrived," proclaimed Luxembourg Foreign Minister Jacques Poos in the middle of 1991, insisting that the Yugoslav crisis was a matter for Europeans themselves to settle.

"The Europeans kept on telling us that this was a problem on their own doorstep, and they would like to take the lead," said Brent Scowcroft, national security adviser to President George Bush. "We were happy to let them do so."

"It was a profound miscalculation," said Assistant Secretary of State Richard C. Holbrooke, who led the American team that helped to negotiate the peace settlement between Serbia, Croatia and Bosnia in Dayton, Ohio. "At the end of the Cold War, we had a brief amnesia attack about our national interests in Europe. We had linked them solely to the Soviet threat. But history came back and bit us, in Yugoslavia."

The policy of treating Yugoslavia as a predominantly "European problem" was finally abandoned only last summer, after the international community suffered a series of humiliations at the hands of the Bosnian Serbs, including the overrunning of two United Nations "safe areas" in eastern Bosnia. Confronted with the nightmare possibility of being obliged to deliver on a commitment to send American troops to Bosnia to rescue the European peacekeepers, the Clinton administration chose instead to fashion a "Pax Americana." Cold War's Thaw

During the Cold War, the need for American military involvement in Europe was obvious to almost everyone. Without the American nuclear umbrella, and 350,000 GI's in West Germany, there would be little to stop Soviet tank divisions pouring into Western Europe. Europe was the most important battleground in the global struggle between the forces of free market democracy and communist dictatorship.

A maverick communist country that broke away from the Soviet bloc in 1948, Yugoslavia found a comfortable niche for itself at the very heart of the East-West divide. The founder of modern Yugoslavia, Marshal Tito, succeeded for more than three decades, until his death in 1980, in playing one bloc off against the other. He used the threat of Soviet intervention to extract informal security guarantees from the United States and to keep his people from internecine quarrels.

Straddling the historic geopolitical fault line between Byzantium and Rome, Yugoslavia was a complex and uneasy mosaic of different religious and ethnic groups, one of several East European countries that came into being after World War I and the 1918 Versailles Conference, following the collapse of the Ottoman and Habsburg empires. Bosnia is a kind of Yugoslavia in miniature, a place where Eastern and Western cultures collide.

It was hardly a coincidence that the strains in the Yugoslav federation began to become unmanageable in 1989, the year of the fall of the Berlin Wall. As the Soviet threat diminished, so too did Yugoslavia's strategic importance. Western governments stopped pouring money into the country. The growing economic crisis deepened the nationalist divisions. Deprived of an external enemy, Yugoslav leaders began a furious search for internal scapegoats.

American politicians who attempted to reason with the squabbling Yugoslav politicians soon threw up their hands in despair. After meeting in Belgrade in July 1991 with the leaders of Serbia, Croatia and Bosnia, Baker complained that it was difficult to get past the 15th century. Scowcroft recalls that Bush would frequently react to news from Yugoslavia with the phrase: "Tell me again what this is all about."

There was a strong argument that the violent breakup of Yugoslavia was a "worst-case scenario" for the rest of the communist world. The way in which Yugoslav communists transformed themselves virtually overnight into nationalists of one kind or another was a harbinger of things to come in the Soviet Union. But the nationalist threat was underestimated by Bush and Baker, whose foreign policy strategy was focused on winning the Gulf War and shoring up the authority of Soviet President Mikhail Gorbachev.

"There was some excessive optimism," acknowledged Baker's former State Department deputy, Lawrence S. Eagleburger, who had served as U.S. ambassador to Belgrade and played a key role in shaping the Bush administration's Balkan strategy. "It seems to me that the Yugoslav collapse came too soon. If we had three or four years to think about it, we might have been able to deal with it a little better."

With hindsight, Eagleburger now concedes it was a mistake to let the Europeans take the lead in handling the Yugoslav crisis. Without American leadership, the Europeans were unable to reach a common approach. The Germans had political, historic and economic ties to Croatia and Slovenia, which were part of the old Austro-Hungarian Empire Britain and France, by contrast, had historic ties with Serbia. The divisions between Western governments sent a series of confusing signals to Yugoslav leaders.

Eagleburger said that he has asked himself "a thousand times" whether there was anything the Bush administration could have done to prevent the Yugoslav tragedy. He still believes that military intervention was a political impossibility, particularly in view of the effort that the United States had just put into the Gulf War. The only possible way out might have been for the United States to convene a pan-European Congress to agree to a set of rules for the breakup of Yugoslavia.

"That might have worked," said Eagleburger. "Having said that, I still believe that the differences between us were so substantial that it would have been very difficult to have achieved a consensus between the major powers."

Other foreign policy thinkers strongly disagree with this conclusion. "The best time to have intervened with force was during the Bush administration, when you could have saved tens of thousands of lives and achieved a territorial settlement much better than at Dayton," said Warren Zimmermann, who was the U.S. ambassador to Yugoslavia when Baker and Eagleburger were making Bush administration policy. "The serious use of NATO air power at any point in the war would have deterred the Serbs." The Balkan Quagmire After Clinton came to office in January 1993, American policy veered back and forth between a desire to end the killing and human rights atrocities in Bosnia and a horror at getting involved in the Balkan quagmire. Administration officials now concede that Clinton had an uncertain grasp of foreign policy at the beginning of his presidency: It took time for him to become comfortable with the idea of combining diplomacy with force.

"The growing confidence of the president in the international sphere is simply a fact that has developed as he gained experience over time," said Christopher. "The president . . . began to see how much the world looked to the United States for a resolution of the problems."

Clinton helped to undermine Christopher's ill-fated trip to Europe in May 1993 to canvass support for a policy of using airstrikes to assist the Muslim-led Bosnian government. While his secretary of state was in London, Clinton began to have second thoughts about the policy. He told his aides that he had been reading a book called "Balkan Ghosts" by Robert D. Kaplan that suggested that Balkan hatreds were so deep-seated that there was little America could do about them.

More recently, according to a senior White House official, Clinton has been influenced by another book about Bosnia, by the British writer Noel Malcolm, which takes a much more sanguine view of Bosnian history. The Malcolm book makes the point that relations between Bosnia's different ethnic and religious groups were remarkably harmonious for much of the last 600 years.

Asked why it took Clinton nearly three years to make up his mind about Bosnia, aides argue that he had a lot of other foreign policy problems on his plate, such as the fate of democracy in Russia. They also contend that the opportunity to take decisive action in Bosnia only really presented itself this summer, in the form of a string of Croatian victories over the Serbs. This had the effect of shattering the myth of Serb invincibility, and providing an opening for energetic U.S. diplomacy.

"That is rubbish," said Richard N. Perle, a senior Pentagon official under the Reagan administration and longtime supporter of the Bosnian cause. "A real leader doesn't wait for opportunities. He shapes them, and makes them happen. The fact is that we sat on the sidelines for nearly four years."

In the end, the administration's decision to take the lead in Bosnia seemed to depend on politics as much as geopolitics. The fall of the U.N. "safe area" of Srebrenica in early July exposed the emptiness of American promises to protect the Bosnian Muslims and undermined the credibility of NATO, which had used "pinprick airstrikes" in a vain attempt to hold back the Serbs. On Bastille Day, July 14, French President Jacques Chirac taunted Clinton by saying that the post of leader of the free world was now "vacant."

According to administration sources, some White House domestic advisers urged Clinton to disengage completely from Bosnia at this point and work for the withdrawal of the U.N. force. His foreign policy advisers insisted that dispatching American troops to Bosnia to extricate the U.N. peacekeepers involved many more risks than sending them in to enforce a peace settlement. It came down to a choice between two unpleasant alternatives.

Last week, Clinton went on television to urge Americans to "help the nations of Europe end their worst nightmare since World War II." Even so, in the eyes of European commentators, there is still a tentativeness about the American commitment to Europe four years after the end of the Cold War.

"It would be wrong for Europeans and Americans to conclude that Bosnia sets a precedent for the way America will react in future crises," said Christoph Bertram, diplomatic correspondent of Die Zeit and a recent fellow at the Carnegie Endowment for International Peace in Washington. "What we are seeing here is selective involvement, and selective engagement, with the timing related as much to the domestic political agenda as the international one."

In his television address last week, Clinton laid out a doctrine of American leadership that was very different from John F. Kennedy's ringing declaration that "we shall pay any price, bear any burden, meet any hardship . . . to assure the survival and success of liberty." "We cannot stop war for all time, but we can stop some wars," the president said. "We can't do everything, but we must do what we can." CAPTION: Bosnian Serbs mourn at graves of young men killed in fighting near Sarajevo.