Picture President Teddy Roosevelt tipping his hat at its threshold in 1907 and sending the entire American Navy, all 16 battleships, around the world.
By then, the Spanish American War had made the United States an empire, and Roosevelt talked about speaking softly but carrying a big stick.
"Theodore Roosevelt, in a way, marks the beginning of the aspirations for an American century, not realized at the time," says Fareed Zakaria, managing editor of Foreign Affairs magazine. He contends it was far from evident in Roosevelt's day that this would be the American century.
In fact, Zakaria believes that more than three-quarters of the century had elapsed before it was really clear. Even after the United States won two world wars, it took the end of the Cold War, when the Berlin Wall came down and the Soviet Union collapsed.
"The Cold War was a struggle to determine which side history was on, the United States or the Soviet Union, and it was a struggle that really went to the heart of what it meant to be American," says Zakaria.
With time running out on the 20th century, CBS News Sunday Morning asked four professional observers of American life to comment on it. Two of them have chosen to become American citizens, and two were born in the United States.
Fareed Zakaria, already mentioned above, was born in India. Harold Evans, a journalist for half of the 20th century and a publisher, is from England. Laurie Anderson grew up in the Chicago suburbs, but for years has been associated with the avant-garde art scene in New York. And New Yorker David O. Russell has written as well as directed three movies, all of them social commentary of sorts.
You might wonder what the American century has come to if you saw Three Kings, David Russell's recent film about the Gulf War.
"These soldiers were sent on a mission of freedom, which in some senses was how the century started, with Theodore Roosevelt saying, 'Let's have an empire,...and let's be the beacon of light,'" says Russell."And yet, I think, in some ways it was just a mission of securing oil."
Russell can't resist the contradiction.
"I saw the whole war as almost like a snapshot of American culture at the end of the century in many respects," he says.
The culture can be viewed as being as much about what Americans have as who Americans believe they are.
"I came to the United States on the steamship Franconia in 1956 sailing through the mist, past the Statue of Liberty,...and I became infatuated immediately with the country, with its amazing prosperity," says Evan.
Evans is about to retire as vice chairman and editor of the New York Daily News and of U.S. News and World Report. He is also the author of a best-selling book called The American Century.
"I come from wartime England, where we listened to Franklin Roosevelt and Winston Churchill and were living basically on starvation rations," says Evans. "And to open a refrigerator and see the kind of tectonic piles of T-bone steaks and the gallons of oranges, I thought America was preparing for doomsday....The prosperity was incredible."
"I think it really is a very old thread that goes through our culture of people who came over here....We're searchers," says Anderson.
A performance artist, Anderson says her appearances outside the United States have given her insight into the American character, not to mention the American century. Her current show in Paris is based on Herman Melville's classic Moby Dick.
"In Melville's case, he built a big whale, and it was the symbol of everything you wanted and looked for....It means everything...terror; it means beauty; it means nature," says Anderson.
"It means a lot of things. It's what you want," she says. "In other words, longing,...the American dream or whatever we call it."
"The American appetite for prestige,...for fame,...there's a sort of Pac-Man part of us that doesn't stop," Russell explains. "And that's those three guys in Three Kings. They just want to get stuff."
American stuff, for these observers, represents what's good and bad about how the American century is turning out.
"Somebody said...the good things in America go crazy," says Russell. "So that is the American way,...and now we're exporting it to other countries."
But look what happened in Seattle during the World Trade Organization talks. Some of the rioters didn't like the idea that the rest of the world has figured out how to make and sell all that stuff cheaper than Americans can. Exporting the American way, and American jobs with it, can be a double-edged sword.
"Capitalism and democracy, which are the two great forces at work today and will be in the 21st century, are enormously energetic forces," says Zakaria. "They're unsentimental about the world as it exists. They are essentially forces for change, for very rapid change, and that is frightening to some people."
And change is driven by technology.
"I open my computer and it goes bong, Laurie's hard drive; that's what it's called," says Anderson. "I'm thinking, 'My mind is being sucked into this box; it's such a trick because things don't really fit in that box.'"
As an artist, Anderson is constantly confronting this fact of American life: assessing who Americans are as opposed to who Americans always were.
"The American century has to do with technology, and the biggest challenge, and it's a really rucial one, is to not to slip into our machines....That we have to control them," says Anderson.
Control was what Henry Luce, founder of Time magazine, had in mind when he coined the phrase, "the American century," in 1941. With the world at war, he wanted the United States to seize the century - to accept moral as well as political and economic leadership.
"Moralism is a very strong concept, but moralism can be made into a dirty word," says Evans. "I prefer to think of it as a kind of idealism, too, the ideal of creating a democracy where people will be equal, will have dignity, will get an education."
Although not always achieved, it was this ideal that convinced Evans to become an American citizen and to raise his children in the United States. Writing his book, he concluded the best of the American century was the GI bill.
"Every time a GI, an ex-GI, went up those steps to get his diploma, a light bulb went off in everybody's head. 'Look, this is what democracy is about,'" Evans continues.
Evans, who covered civil rights as a journalist in the '50s and '60s, deplores America's record on race, however.
"What was done to the American black was the worst thing in the American century," he says.
The other observers agree. But in spite of such ugly mars on the gloss of U.S. national self-image, all of them are comfortable holding up a mirror and seeing goodness.
"It's a pretty stable, generous empire as screwed up as it may be and as destructive as it may be," says Russell. "So with all the consumptive crap that we're propagating, I think we also propagate the notion of cultural tolerance, in many respects, and racial tolerance and individual rights."
So here at the end of the star-spangled century, with Lady Liberty lighting the way from the 20th to the 21st, one can wonder where she will lead.
"I would be prepared to say it will be another American century," says Evans.
"I kept thinking that somehow this was going to be the Chinese century," says Russell.
Says Anderson: "I have no idea,...and I'm not going to be here in 100 years, so I'll never know,...but I love a mystery."
"It's difficult to come up with any answer other than America...again," says Zakaria.
At 34, Zakaria is part of the new America that will inherit the new century. It isn't so very different than the beginning of the old one, as the United States is now what it was then: an immigrant nation constantly reinventing itself.
"The real place to get a sense of the new America is Silicon Valley, where a third of the new companies started up are started by Indian or Chinese-born Americans," Zakaria explains. "That will be, in some ways, the dominant, energetic, trend-setting part of society."
So just as CBS News Sunday Morning began this story in New York City at the dawn of what came to be the Ameican century, it ends in Los Angeles, as the sun slides into the Pacific Ocean. Any suggestion of symbolism is intentional; anything else is pure guess!