A queen, her lover and John Locke

Two centuries ago, an affair sparked a Danish Enlightenment. Now a film takes up the amazing story.

The opening scene, where a beautiful woman strokes a horse, and then walks by a river in a rain-soaked landscape, is like one of those paintings where you see a figure, and a face, and a certain kind of light, and glimpse a world. Within moments, she's writing a letter. Within a few more moments, she's back in a sunlit field, in a different country and a different time. In moments, it is clear that this beautiful woman, in this rain-soaked landscape, has an extraordinary story to tell.

The woman is Caroline Mathilde (played by Alicia Vikander), the sister of the British King George III and the central character of Nikolaj Arcel's award-winning new film, A Royal Affair. In that sunlit field, she's 15 and on the brink of an arranged marriage to the new king of Denmark, who's just 17.

"If you can get the King to visit your bedchamber on the first evening," her mother tells her, "you'll be perceived as a great success." But when she finally meets Christian VII (Mikkel Boe Folsgaard) who's hiding behind a tree, and she watches him stare out of the carriage window, and then greet his dog as if it was the only living creature he cared about, it's clear that life at the Danish court is going to be a fair bit tougher than life at home.

"Why," says Christian to his new court physician, "did I have to end up with that boring cow?... Make her fun. I want a fun queen!" Christian's idea of "fun" is a trip to the theatre or brothel. He's as bored by the trappings of office as he is by his beautiful wife. But Johann Struensee, his very handsome doctor (Mads Mikkelsen, Best Actor at Cannes for his role in The Hunt), doesn't find Caroline Mathilde boring at all. When he examines her, in his study at the palace, sparks fly.

On a walk in the rain, they talk about Rousseau, and Voltaire, and Locke. Struensee talks about personal freedom, and he practises what he preaches. After a dance at a court ball so electric with erotic tension it has the cinema audience practically swooning, they kiss. Soon, they're spending nights in each other's arms. Soon, they're spending days discussing reforms – mass inoculation, outlawing torture, new rights for peasants – they hope to persuade the King to persuade the privy council to introduce. For a while, it looks as though they can do no wrong.

"For a while," as Caroline Mathilde says in the film, "it felt like we could do something – bring about change." And for a while, at least in the film, they did. After Struensee persuaded the king to abolish the privy council, they launched a radical programme of reform. And then, it all started to go wrong. There was a palace coup. There was a gory death.

The astonishing thing about this astonishing film is that everything that happens in it is true. Caroline Mathilde did, at 15, marry a mad young king she'd never met. She did have an affair with his charismatic physician, Johann Struensee, who had so much influence on his royal patient that he became a kind of king himself. And they did develop a programme, inspired by the key writers of the Enlightenment, of mass reform. For a few short years, 200 years ago, the country now known as one of the world leaders in social democracy became, as the film says, "a pioneering country across Europe".

Stella Tillyard, the historian and author of a wonderful book about the period also called A Royal Affair, which the makers of the film claim is a coincidence, stumbled on it pretty much by accident. "As soon as I did," she told me, "I realised it was the really, really interesting story amongst George III's siblings. I also knew that the British and American markets were not going to bear, at that point, a story just about Denmark."

What a difference a Killing makes. And a Borgen. And a Bridge. Now you probably only have to say the word "Denmark" to have publishers and film studios begging you to take their cash. But whatever they threw at this film, it was worth it. A Royal Affair is beautiful, and poetic, and funny, and sad. It's also, according to Tillyard, amazingly accurate.

"As an ordinary film-goer, you'd think 'was all that true?', but the thing is the king was more mad, and the queen was more angry, and Struensee was more radical than it's possible to show."

There are some details in the book that are even stranger than the ones in the film. Caroline Mathilde's maids, for example, put white powder on the floor round her bed, in order to capture evidence of Struensee's visits. And Christian's dog really did become a privy counsellor.

But this film isn't just a fascinating story that's grippingly told. It's also an important reminder of what Tillyard calls in her book "the boldest attempt ever mounted to shift human consciousness from a god-centred to a man-centred view both of the world and of the life that could be lived in it." It is, in other words, an important reminder of the values of the Enlightenment, and of how swiftly they can pass.

"I think," said Tillyard, "one of the reasons they could make this film now is because of the lurch to the right. One of the things that liberals now want to say is, 'we had a radical past, the Enlightenment, and we really must hold on to it, because all over Europe now we're in danger of losing those gains'."

'A Royal Affair' is out now