But can you make us laugh? (Charlie Neibergall/Associated Press)

Kings had it easy. All they had to do was cure scrofula with their touch.

The presidency makes more unreasonable demands. It’s not simply the hours of jet travel, the endless strings of babies being handed to you to soothe with your presence, the insistence that you eat whatever bizarre local delicacy is on offer — and like it. Candidates for the nation’s highest office have to do the equivalent of leaping through a ring of fire with a live opossum clutched in their arms in order to earn the right to be the person taken most seriously in any given room.

But one of the strangest associated demands of this position situated at the awkward summit of politics and policy and pomp is the demand for jokes.

Jokes are one of those features that did not originally come standard with good presidents. The closest George Washington came to making jokes was when he stood near Ben Franklin during assemblies. But then the Lincoln model came out, with its facility for self-deprecating quips, and suddenly everyone wanted that feature.

Abraham Lincoln was famously able to laugh at himself — “If I had two faces, would I be wearing this one?” — and even liked to tell jokes about outhouses.

He responded once to someone who asked how he enjoyed the office by saying, “You have heard the story, haven’t you, about the man as he was ridden out of town on a rail, tarred and feathered, somebody asked him how he liked it and his reply was, if it were not for the honor of the thing, he would much rather walk.”

This, in fact, is the type of humor most vitally associated with the presidency.

Everyone is talking about you. The late-night comics, the Twitter comics, “The Daily Show,” “The Colbert Report,” “Saturday Night Live,” the denizens of open mics across the country.

And it helps if you can speak self-deprecatingly about yourself.

President Obama is generally skilled in that area. Last night he showed up on “The Tonight Show With Jay Leno” and quipped that Donald Trump’s beef with him dates back to their days of playing soccer together in Kenya as children.

The late-night wars continue on the campaign trail.

Both candidates have been hitting the late-show circuit. Since taking office, Obama has appeared on David Letterman’s show twice, “The Tonight Show” thrice, “The Daily Show With Jon Stewart” twice, and Jimmy Fallon once. Over that same time, Mitt Romney has been on Leno twice and on Letterman four times, once presenting a list. At one point he tried to complain about Letterman’s bias, but the war quickly fizzled.

Washington Funny and Actually Funny are often two separate breeds. The president’s former speechwriter Jon Lovett is an actually funny person. That is why “former” is in front of his title. He is off creating a sitcom now.

One of the ways to win people over is to get them to laugh with you, not at you. One of the ways to turn people against your opponent is to drive them to laugh at him. Laughter drives home a point, sets people at ease, brings you down to earth.

The presidency, as a modern institution, makes many seemingly contradictory demands on its occupant. You must be funny but not silly. Serious but not grave. Experienced but not decrepit. The cynosure of all eyes but not a celebrity.

The trouble with Obama and Romney, both of whom have all the natural, unforced charm and spontaneous humor of the guy who comes to your high school to talk about drugs, is that they both are perfectly able to deliver jokes. But conceive of them and produce them on their own? That’s the danger zone. Romney, left to his own devices, will start telling you there are no plates like chrome for the Hollandaise. President Obama isn’t that hokey, but a lot of it depends on being in the room at the time. You’d never rank him with Lincoln as a great generator of quotable quips.

Not that there is anything wrong with that, of course. “We can’t all be heroes,” Will Rogers quipped, “because someone has to sit on the curb and clap as they go by.”

Still, the humor bar is another of the odd hurdles the president has to be able to leap. Can’t interact convincingly with a baby? What are you doing running for the nation’s highest office?

Watching Obama on Leno, I was struck not by the thought that his appearance was unpresidential. It’s that the presidency itself is deeply weird. What a strange job this is. You can’t be all things to all people all the time, but maybe you can be some things to all people and all things to some people. And see how this monologue goes.

As the stand-up comedian Mitch Hedberg quipped, “When you’re in Hollywood and you’re a comedian, everybody wants you to do things besides comedy. They say, ‘Okay, you’re a stand-up comedian — can you act? Can you write? Write us a script?’ . . . It’s as though if I were a cook and I worked my [patoot] off to become a good cook, they said, ‘All right, you’re a cook — can you farm?’ ”

Make that stand-up comedy and the presidency.

“Well,” voters say, “you can deliver a punch line. Know anything about foreign policy?”