Why on earth did the Prime Minister think it a good idea to follow his barnstorming speech at the UN with an appearance on the Late Show with David Letterman? Perhaps he genuinely saw it, as he said, as a chance to present Britain. Perhaps he thought his face recognition in the US left something to be desired. Perhaps he wanted to show that anything Boris Johnson does, he can do better, or perhaps he just thought of it as another of life’s little challenges. Each explanation has a measure of plausibility; each is thoroughly bad.

The late-night American talk show is a genre we do not really have in Britain. It is not the BBC Jimmy Young Show of old, where Margaret Thatcher proved such a hit, but nor is it anything like the sober and periodically pugilistic Newsnight. It is neither old-style Parkinson nor new-style Jonathan Ross; elements of showbiz, satire and politics come together. The entertainment is in the mix – and in the dominance of the host.

Anyone with the faintest ambition to subject himself to the Letterman experience should take careful note of the studio arrangement before they do. David Letterman appears genial, even tame, compared with the Paxmans of this world. His humour is of the unsubtle, American kind; if it turns negative, it wounds rather than kills. He can goad. He can expose. He can switch gears from nice to nasty in a trice. But the spectrum of his manner is secondary to where he sits. His desk and chair are substantially higher than those assigned to his guest – whether that guest is a B-list film star or President of the United States. And that is the trap.

There are very few ways in which a guest can be anything like equal to the host. One is if that guest is the US President, as it was last week. Any President, even George W. Bush, has to be comfortable with a degree of showmanship to be elected. Add the aura and gravitas conferred by the highest office in the land, and he is pretty much on a par.

Almost the only other way to compete with Letterman is by force of personality – but be warned. Mayor Boris, for all his loveable British bombast, was judged a success on Letterman. But he only just held his own. And if even Boris cannot make the US talk show his own, think how hard it must be for everyone else.

David Cameron has a very British geniality, but he is not in that league. A cut-glass English accent may still open American doors, but it is not enough, by itself, to allow a British politician to cut a figure on the US stage. You can count on the fingers of one hand the British politicians who have gone down well in the US in recent decades. Thatcher, of course: distinctive, forthright, no mincing of words. John Prescott and the late Mo Mowlam were rare successes. Peter Mandelson never hacked it; despite his lack of languages, he was better liked in Europe. Americans demand character, with a capital C.

And that should have been one warning to Cameron. He looks and sounds like a prime minister, but you would not describe force of personality as necessarily his strongest suit.

It should never have come to this, though. There are some things that are simply not compatible with the dignity of office and, for a serving British prime minister, going on an American talk show – any talk show – is one. The minor Royals disporting themselves on the game show It’s a Knockout comes to mind. If Cameron wants to say something to a US audience, he has other means at his disposal – summit press conferences, one-on-one interviews – that carry nothing like the risk. A British prime minister does not have to – indeed should not – try to recast himself as a celebrity, in the US or anywhere else. The whole package is demeaning.

It is always tempting for UK politicians, seduced by talk of the “special relationship”, to believe that the English language gives them a hotline to domestic opinion across the Atlantic. But all references to the “special relationship” should be hedged about by the other cliché, about two nations divided by a common language. Our modes of expression, physical gestures and humour are all very different. The risks of misunderstanding are legion.

In the event, Cameron kept his dignity, just about. His mistakes were relatively minor; Magna Carta a slip. He came across as an agreeable, if forgettable, guy. He could have done a lot worse. It would just have been better if he had not done it at all.

At which point, it would be easy to say: lighten up; don’t take it so seriously. But Cameron is head of our Government and Letterman has enough Hollywood starlets clamouring for airtime without needing to accommodate a British prime minister. Which side, one wonders, initiated the approach, did the begging?

In 1995, quite soon after becoming French president, Jacques Chirac went on The Larry King Show. He saw an opportunity to set a new diplomatic tone. He knew the US better than most French politicians, having worked there as a student. In the US, the response was curious to lukewarm – and did not last. In France, though, there was an outcry. Chirac was lambasted from right and left for compromising his office. The talk-show format was one crime; but his decision to speak English was even worse. Chastened, Chirac never repeated his mistake. And neither should David Cameron.