Such is the emotional incontinence of so much American public life it was perhaps inevitable that the subject of Gordon Brown’s dead daughter would be raised — not once, but twice — when he visited the US this week.

President Bush was first out of the traps, at a Camp David press conference where his overfamiliarity with “Gordon, the humorous Scotsman” swiftly descended into rambling about how he was interested in a family which had “suffered unspeakable tragedy”. Mr Brown replied stiffly, thanking his host for a “compassionate” remark.

Then came an NBC Nightly News interview where the Prime Minister was asked more directly how he coped with the death of a newborn daughter and the cystic fibrosis diagnosed in his younger son. “I don’t really talk about it,” said Mr Brown, before muttering something about being a good father.

Neither the President nor NBC meant any harm, explained one of those political strategists that proliferate in Washington. “They probably thought they were doing Brown a favour in getting him on to a personal narrative and life experience.”

The idea is that if you can hear a politician’s pain, you might think he will feel yours. So Mr Bush has admitted being a recovering alcoholic whose “Goodbye Jack Daniels, hello Jesus” moment propelled him into the White House. John McCain rarely goes a full day without mentioning how he was tortured at the hands of the Vietcong.

John Edwards spent much of his 2004 presidential campaign talking about his dead son — and a good portion of this one going on about his wife’s breast cancer. Barack Obama has turned his life story into two books and a political movement.

Yes, birds do it, bees do it, even educated “OEs” (that’s Old Etonians, to those of us who didn’t go there) do it — with David Cameron making more references to his disabled son than might be deemed strictly necessary.

But, as America discovered this week, Mr Brown definitely does not do it. And nor, usually, does Hillary Clinton. These two politicians, hard and serious to the point of being brittle, have much in common. Both are emerging from the shadow of a troubled, even jealous, relationship with more charismatic partners in Tony Blair and Bill Clinton. Both have had to wait an awful long time for a chance to succeed them. And both have sometimes received bad advice that they will not get the top job unless they lighten up and show a more human side.

The difference is that Mr Brown is now Prime Minister and Mrs Clinton is not (yet) President. As such, he appears to have lost some of the self-doubt that once afflicted his sombre style.

It was noticeable that, even as Mr Brown was deftly side-stepping invitations to discuss deeply personal matters in America this week, Mrs Clinton was getting into a lather over a “grossly inappropriate” article in The Washington Post about her cleavage after some recent sartorial diversions from her standard-issue Hillary trouser suit.

No matter how well her angry response plays with women Democratic donors, who swiftly received an e-mail from her campaign urging them to “take a stand” against the pettiness of American political culture (ie, send money now), she should not have risen to it.

While her husband can scarcely open his mouth without natural empathy dribbling from its corners, Mrs Clinton’s efforts to display emotion appear stilted and artificial. Her autobiography, Living History, is one of the most sterile pieces of literature ever written. Look at this from her similarly hygienic website: “The promise of America was very real as Hillary was growing up. She learned that no matter who you are or where you’re from, if you worked hard and played by the rules, you could provide a good life for your family.” Yuck.

Her friends insist that she is a big bag of fun in private but, even if that is true, it does not exactly come across in public. Mr Brown’s aides used to say the same about him and, occasionally, would buy him some leisure wear or issue a list of his top pop tunes on the ground that this sort of thing seemed to work for Mr Blair. It was mutton and lamb: Mr Blair looked comfortable in a Christian sort of way wearing jeans, and he really had been in a rock band; Mr Brown just looked weird.

As Prime Minister, however, he has put a stop to such nonsense. And so far, at least, the voters seem to like it. His strength is his strength.

The same is true of Mrs Clinton, who could learn from Mr Brown’s current popularity. In all three presidential debates this summer against her Democratic rivals, she has come across as the candidate with the experience, toughness, knowledge and intelligence needed to lead America. Her worst moments have come when she tried to laugh, giving off a scratchy clicking noise from her throat.

It may yet prove more difficult for Mrs Clinton, as she tries to become America’s first woman president, to win on qualities traditionally, if wrongly, associated with men. Some suggest there is a dysfunction between her core message of strong leadership and her core support among women. Mr Edwards’s wife, Elizabeth, has already effectively accused Mrs Clinton of behaving too much like a man.

Recent polls suggest otherwise: slowly but surely the seething resentment of those (mostly men) who regard her as some sort of Lady Macbeth figure appears to be diminishing while she has maintained high levels of support among women. Like Mr Brown, she does not need to pastel-shade her image to impress voters.

It could just be that on both sides of the Atlantic, we are ready for more serious leadership. We may not warm very much to Mr Brown or Mrs Clinton and they risk sending us all to sleep with a political style bereft of “narrative”. But at least we will know that those who turn the lights off and tuck us up are grown-ups.

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