In the campaign he was perhaps the greatest Geiger counter of the age. When compassion was called for, he was compassionate. When inspiration was needed, he was inspiring. When determination was required, he was determined. He had a perfect ear and perfect pitch.

Now Barack Obama’s struggling. In response to the most important but most overlooked recent survey question, one posed by the Wall Street Journal/NBC News poll, almost a third of Americans said they “do not really relate” to him. Not so bad, you might say, given that 47 percent of the country didn’t vote for him. But this is why the 30 percent do-not-relate figure is staggering: The figure was only 8 percent when the president was inaugurated.

It’s not for want of trying. Maybe it’s because he’s trying too hard.

He is, after all, the same man who launched a thousand swoons, who set the press to reverie, who had the magic eraser that swept away all of the attributes on Hillary Rodham Clinton’s resume and made the nation hunger for his touch. America had never seen anything like it, and the fact that he was black and had grown up in Hawaii and Indonesia just added to the sweetness.

Not long ago, America felt good about Barack Obama and, what is probably most important, Americans felt good about feeling good about him. But now only half the country believes he has “strong leadership qualities,” which would be pretty decent except for the fact that 70 percent felt that way when he was inaugurated. He’s lost eight percentage points in that category just since the beginning of the year.

In an earlier age, we might not have cared if the people did “not really relate” to the president. That’s because not all the people mattered, politically speaking. Only men who owned property could vote — until Vermont loosened things up in 1791 and allowed ballot access to all men. Women were given the vote in 1920. But it wasn’t until the last several decades, when all the exclusions and phony poll taxes and literacy tests were swept away, that we approached universal adult suffrage.

For about two-thirds of the life of the country, the chances of an American hearing or seeing the president, and thus having a plausible chance to decide whether he or she could “really relate” to the chief executive, were infinitesimal. That is why two early 20th-century presidents who likely could not have related to 90 percent of the country were able to get elected to the White House (William Howard Taft and Woodrow Wilson).

The five presidents who on the surface might seem to have been best able to relate to the public present a mixed record.

Ulysses S. Grant, the people’s general from Galena, Ill., and Warren G. Harding, the hard-drinking and hard-loving son of Blooming Grove, Ohio, are regarded as among the worst presidents, though Grant may not deserve the description.

Andrew Jackson, who introduced populism to American politics long before the word was part of American politics, and Harry Truman, the artillery captain from Independence, Mo., have substantial presidential reputations, both well-earned.

The fifth, Abraham Lincoln, stands in a category of his own.

Lincoln, after all, was the man who said God must have loved the common people because he made so many of them.

Then again, the homely Lincoln likely could not have been elected in the television age. His image was seen through the Matthew Brady photographs, of course, but he almost certainly would not have had a winning TV style.

Now, of course, there is an appealing anti-style that we call Lincolnesque, but the 16th president couldn’t have drawn upon that. (Winston Churchill found throughout most of his life that being Churchillian was a curse, not a blessing. Only later, when the polymath descended from the first Duke of Marlborough turned out to be right about Adolf Hitler, did we embrace the style known as Churchillian.)

Even so, style is important in a president, and Americans react to it in unpredictable ways.

Two Democratic aristocrats of the 20th century, Harvard men both, had it. Franklin Delano Roosevelt and John F. Kennedy were reared in luxury and privilege, yet they possessed the gift of relating to the American people. Herbert Hoover, an orphan reared in remote, snowy West Branch, Iowa, did not have that gift despite growing up in more common circumstances.

The mystery of the American presidency is why and how different occupants of the White House relate to Americans — and how they defy expectations and experience.

No political scientist’s algorithm can explain why, for example, the erudition of Kennedy seemed appealing while the apparent elitism of Obama seems alienating. There is no explanation why the patrician patois of FDR seemed right for an age of deprivation while the gin-and-tonic and tennis-whites style of George H.W. Bush seemed discordant even before the economy turned sour at the end of the 41st president’s term.

Campaigns are supposed to be dress rehearsals for the presidency but often they are indicative of almost nothing. Prime example: Jimmy Carter.

Carter ran for the White House against the backdrop of Richard Nixon’s “imperial presidency,” and his I-won’t-lie-to-you rhetoric and his carry-my-own-suitcase style appealed to a country tired of perfidy and privilege. Once in the White House, he was seen not as sympathetic but as stiff, not as warm but as aloof, not as an efficient engineer but as a bumbling bumpkin.

Then there was the campaign of Barack Obama, four years out of the Illinois state senate. Theatrically and rhetorically it was almost perfect. Technically it was innovative. Politically it was sure-footed.

But nothing about it prepared Obama for the presidency and nothing about it prepared the American people for Obama’s presidency. On the contrary.

Not long ago Russian President Dmitri A. Medvedev recalled a telephone call with Obama that lasted an hour and 45 minutes. “The ear starts getting stiff,” Medvedev said.

For Obama, whose party faces difficult midterm congressional elections only four months from now and who faces a re-election battle only 28 months away, stiff ears could be a serious liability.