WASHINGTON — Riding Air Force One with President Trump back to the nation’s capital on Monday afternoon felt eerily like that grim day in 2005 when President George W. Bush was flying to the White House to confront the catastrophe of Hurricane Katrina.

As the legendary blue-and-white Boeing 747 soared above the clouds, the televisions on board were tuned to wall-to-wall news coverage of a country in crisis below. The ravages of the coronavirus and the market meltdown this week, like the ravages of that storm 15 years ago, played out on the screens for hours — urgently, relentlessly, inexorably, inescapably.

Then and now, there was a palpable sense of a besieged leader confronting a whole new reality, that the world had suddenly changed and so had his presidency. Seen fairly or not as out of touch so high in the sky, Mr. Trump like Mr. Bush before him landed and headed to the White House to take on a challenge he had not fully recognized at first. In each case, the president then headed to the cameras to reassure a country that was not at all convinced.

In Mr. Trump’s case, he addressed the health situation primarily in economic terms, clearly rattled by the largest market drop in more than a decade, as he vowed measures like a payroll tax cut and aid for low-wage workers sidelined because of the virus. “We have a great economy, we have a very strong economy, but this came — this blindsided the world,” he said, including, he acknowledged, himself. “And I think we’ve handled it very, very well.”

The judgments about how well he has handled it have so far fallen largely along the same lines of division about his presidency writ large. In a Quinnipiac College poll, 43 percent approved of how Mr. Trump has managed the coronavirus outbreak while 49 percent disapproved, roughly paralleling his overall approval and disapproval ratings over the past three years.

But crises are moments when presidents can rise above prior troubles or sink deeper into them, as Mr. Bush discovered. A onetime political colossus with a 90 percent approval rating built on his response to the Sept. 11, 2001, attacks, Mr. Bush’s presidency was marred by Katrina, which became a modern metaphor for a mishandled crisis.

Whether Mr. Trump’s approach to the coronavirus outbreak becomes remembered the same way or not, it is too early to say, and they are very different challenges. But the history of presidents grappling with crises is replete with lessons that sometimes go unlearned and examples that go unheeded.

“These are the moments when you’re looking for leadership,” said Doris Kearns Goodwin, the eminent presidential biographer who titled her latest book just that, “Leadership: In Turbulent Times.” Quoting President Franklin D. Roosevelt’s speechwriter, she added: “Robert Sherwood said something like most of the time a president can just be that portrait in a box. But then when something occurs, he has to come out of that box and connect to the people.”

The roster of presidents who could not conquer crisis when it arrived is long and inglorious — James Buchanan, who failed to stop the South from seceding; Grover Cleveland, who failed to stem the Panic of 1893; Herbert Hoover, who failed to overcome the Great Depression; Jimmy Carter, who failed to end the Iranian hostage standoff until his last minutes in office.

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But there are others who rose to the occasion, probably none more than Abraham Lincoln, who found his country falling apart when he took the oath and responded with calm determination and a willingness to work with rivals in a common cause. Roosevelt took office at the depths of the Depression and likewise projected confidence and took decisive action while still leveling with the American people about the scope of the challenge.

While Mr. Trump has tried to reassure the public, saying that the coronavirus was “under control” and would “go away,” he has not sounded to many ears as if he were fully acknowledging the scope of the problem.

Ms. Goodwin said Roosevelt demonstrated the way to balance messages in his inaugural address in 1933. While the speech is often remembered for his line that “the only thing we have to fear is fear itself,” he also made a point of telling Americans that “only a foolish optimist can deny the dark realities of the moment.”

“Instinctively, he is trying to stem the panic,” Ms. Goodwin said. “But what has to be connected to that is the recognition of the problem.”

When World War II threatened, Roosevelt invited Republicans into his cabinet to foster national unity. As the virus spread over the weekend, Mr. Trump declared that he would skip the annual bipartisan St. Patrick’s Day luncheon out of disdain for Speaker Nancy Pelosi, who led his impeachment and ripped up his State of the Union address.

Other presidents effectively took on dire if less daunting crises. Theodore Roosevelt marshaled the full resources of the government, including the Army and Navy, after the devastating San Francisco earthquake of 1906. Lyndon B. Johnson rushed to New Orleans immediately after Hurricane Betsy, visited a darkened shelter, illuminated his face with a flashlight and called out: “I am your president. I am here to make sure you have the help you need.”

When the stock market plunged in 1987 by more than 22 percent on what became known as Black Monday, still the largest single-day drop in history, Ronald Reagan’s team scrambled to head off a new depression. Howard H. Baker Jr., the White House chief of staff, put in a desperate call to Alan Greenspan, the chairman of the Federal Reserve, who picked up the phone to hear only silence and then a soft, “Help!”

He did. Reagan pledged to work with Congress to tackle the federal deficit while Mr. Greenspan pumped more money into the system and the New York Stock Exchange imposed new limits on automated trading, all of which reassured the markets. The fact that virtually no one remembers Black Monday today is a sign of the success of the response.

“You have to have a president who’s competent, who has confidence in government and who has the skills to make it work,” said Max J. Skidmore, the author of “Presidents, Pandemics and Politics.”

Pandemics, however, are a unique challenge to presidents, as Mr. Trump has discovered, where pure political muscle can go only so far. Mr. Trump too has suggested stimulus to reassure the markets and the Federal Reserve has cut rates. But more money can only do so much to reassure a public worried about shaking hands.

The most intriguing analog to Mr. Trump’s coronavirus challenge could be Woodrow Wilson, who was in charge when the Spanish flu pandemic killed tens of millions of people around the world from 1918 to 1920. While Mr. Trump’s critics have complained that he has not taken the crisis seriously enough, Wilson was far more detached. Consumed with mobilizing the American army during World War I, he made no public effort to lead the country through the outbreak.

“The lesson learned is not to do what Wilson did. Wilson had one focus, and that was the war,” said John M. Barry, the author of “The Great Influenza,” the definitive history of the pandemic. “Wilson never issued a statement of any kind on the outbreak. He never said a word publicly.”

A. Scott Berg, a Wilson biographer, said the country was to some extent lulled into a false sense of security because the first wave of the pandemic was not that fatal. By the time the second wave hit, the president was obsessed with training soldiers and shipping them to France. “That was our primary mission and our secondary mission and our tertiary mission,” Mr. Berg said. “Everything else came after that.”

Other presidents have sought to learn the lessons from Wilson’s experience. When an American died in early 1976 from what some feared was a new strain of the Spanish flu, Gerald R. Ford responded aggressively, pushing Congress to authorize a vast immunization program and vowing to vaccinate every man, woman and child in the country.

Ultimately, the government immunized about 45 million Americans, about a quarter of the population at the time, before it became clear it was not a new strain of the flu and the whole expensive effort had been unnecessary. About 450 cases of the paralyzing Guillain-Barré syndrome were attributed to the vaccine. Mr. Ford was pilloried.

“Ford got a bad rap,” Mr. Skidmore said. “He did the correct thing and the courageous thing.”

For his part, Mr. Bush never recovered from the Katrina debacle, which even more than the Iraq war undercut his reputation for competence. But he learned from it. When the economy crashed in 2008, Mr. Bush was slow to recognize the extent of the threat at first. But once he did, he responded with overwhelming force.

Casting aside his own ideological convictions, he embraced an unprecedented bailout program to keep the banking industry from collapse and pushed it through Congress over the resistance of his fellow Republicans. Recognizing that his own credibility was damaged, he put forth someone he thought the public and lawmakers would trust more, Treasury Secretary Henry M. Paulson, and he reached out to Ms. Pelosi and other Democrats.

It came at a cost to him. The bailout was deeply unpopular. But he decided the hit was worth it to save the system, taking his own lessons from the history of presidential crises. “If we’re really looking at another Great Depression,” he told aides, “you can be damn sure I’m going to be Roosevelt, not Hoover.”

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