Presidents in their first terms worry about how the public views them.

Presidents in their second terms worry about how history views them.

That’s why an intriguing poll finding (and that sure is a phrase a poll-weary nation will regard as contradictory) may offer hints to the nature of the second term of the George W. Bush presidency. This poll, released last week, rates Bush in the bottom third of presidents of the last three-quarters of a century.

Now, let’s not get carried away here with celebrations of this poll on the left or denunciations of it on the right. Presidential greatness is almost always espied from afar, and as proof I offer up the reminder that the greatest president, Abraham Lincoln, didn’t even get 40 percent of the vote in 1860 and then ran for re-election in 1864 in a rent nation divided not into Red and Blue but, more tragically, into Blue and Gray.

One more warning before we proceed further. This poll was conducted by Zogby America, which you may remember as the firm convinced that Sen. John F. Kerry of Massachusetts would be choosing his Cabinet just about now. Nobody’s perfect.

Even so, there is real value in poll findings indicating broadly what the public values in a president — and whose presidency the public values. The winner here is Franklin Delano Roosevelt, and the important thing about this (thoroughly unsurprising) result isn’t that FDR was the apostle of the modern welfare state, nor that he fought a great depression and a great war, nor even that he was elected four times. The important thing is that he determined, in the American mind and memory, what a presidency should be.

Historians have fought about what exactly that is, but for the moment let’s say that the FDR model calls for a president who is in the center of the battle, who is a voice for American values and who is a man of action — decisive but above all pragmatic. Flexible, too, for we sometimes forget that in some matters the 32nd president didn’t care what he did as long as it worked.

The next three presidents in the public poll offer us (and the current president) some other guideposts to greatness.

In second place is John F. Kennedy, whose presidency lasted but a thousand days, but whose legend endures even today, which is why the Sotheby’s auction of JFK artifacts, including otherwise worthless wicker baskets and highball glasses, will cause a sensation in February. Kennedy presided over tension-filled days, to be sure, but his value in the presidential pantheon is for his full-throated expression of American idealism. Kennedy’s words have a timelessness to them. They likely will stir and inspire us as much in 2061 as they did in 1961.

Reagan, who ranks third, offers guideposts, too: Be strong, know your own mind, don’t let your agenda be cluttered by peripheral concerns. One more, which I hope you will let me borrow from another great 20th-century Republican president: Speak softly, but carry a big sense of humor … and humility.

Next is Truman. He nearly was defeated for re-election, and a nation that was mild about Harry more than a half-century ago is wild about him today. That tells us something: The crises of the moment pass, but character endures. Truman knew that in 1948 even if the press did not.

Now, in retrospect, we recognize that Truman and Reagan share many attributes, including their instinctive revulsion toward communism, their sense of thrift and their uncanny, and exceedingly lucky, ability to be underestimated by people who thought they were a lot smarter than the president.

The current president shares that. But you can bet that he won’t want to be remembered as the guy sandwiched between two men (his father and Gerald R. Ford) who didn’t manage to win re-election, and behind a third one-term president (Jimmy Carter) whose accomplishments as a former president haven’t fully expunged the memory of his frustrations as president. To escape that fate, he must look at the presidents who rank above them and aim for history, not merely for headlines. Thus President Bush is embarked on his final campaign.

The innards of this poll are instructive about the world in which the president will conduct that campaign. Democrats rank Bill Clinton as their third-greatest president and George W. Bush as the worst, even worse than the one they used to hate the most, Richard M. Nixon. Republicans rank the president third — behind Reagan and FDR. Conservatives place the president in a tie with Truman for fourth, behind Reagan, FDR and (here’s the proof of my idealism argument) Kennedy.

The first prominent presidential poll came in 1948, when Arthur J. Schlesinger Sr. asked historians for their assessments. They ranked Lincoln, George Washington, Franklin Roosevelt, Woodrow Wilson and Andrew Jackson as “great presidents.” He conducted a second poll in 1962 (Jackson fell to “near great”) and his son, Arthur Schlesinger Jr., conducted a similar survey in 1996 (when only Lincoln, Washington and FDR emerged as “great” and Thomas Jefferson skipped ahead into fourth place).

In the professional historians’ 1996 rankings, Kennedy finished 12th, earning him a “high average” ranking and placing him behind James K. Polk, Dwight D. Eisenhower and John Adams. Reagan ranked only as “average,” behind even William Howard Taft and Rutherford B. Hayes — an assessment that prompted conservatives to question the objectivity of historians and a ranking that, in the wake of Reagan’s death, would almost certainly be revised today.

Times change, and presidencies change with them. Even presidencies that have been long completed change, at least when we look back at them. All that is a good reason for presidents not to watch the polls, lest their ranking in the poll that really matters — the one that measures a presidency for history– be affected. Eyes straight ahead, to be sure, but a glimpse in the rearview mirror never hurts.