Nicolas Poussin intimidates me. This 17th-century French artist, who spent most of his life in Rome, is so profoundly serious it can feel like you’ll never be quite grownup enough to get him. Quail before his solemn depictions of the Seven Sacraments. Melt under the severe gaze of his Self-Portrait in the Louvre. His greatest champion in Britain was the art historian and Soviet spy Anthony Blunt, who presumably found something delicious in possessing a secret knowledge of the Poussin code, the one I’ve never been privy to. It was also possessed by the snobbish writer Anthony Powell, whose novel sequence shares its title with Poussin’s painting A Dance to the Music of Time. Now the mystery is blown wide open. The National Gallery has cracked art’s most elitist code. Its liberating new exhibition unleashes a Poussin who is human, passionate and high on ancient history. This it achieves with a razor-sharp focus on his first 10 years living in Rome and feasting on its pleasures. In the 1620s, this wide-eyed young artist from Normandy arrived among the rivalrous painters, sex-worker models and art-loving cardinals of the city. You feel Poussin’s excitement the moment you enter this exhibition. The eternal city was a cocktail of raw sensuality and classical remains. Right ahead of you in the first room, a towering Grecian urn – the Gaeta Vase, lent by Naples’ National Archaeological Museum – is carved with marble nymphs and maenads banging cymbals and dancing wildly, but frozen in time, ethereal in their cavortings. Two paintings flank it, in which the young Poussin gives these marble figures flesh. His 1625-6 Bacchus and Ariadne, from the Prado, captures his thrill when he was brand new to Rome. Bodies gyrate in a fleshy tumult. At the edge of the drunken crowd, a young woman bares her breasts as she sways. You sense his nerves, a northerner new to this city where drawing nude models is normal. He lets a shadow fall on her bosom, as if scared to go too far. Five years later, when he started The Realm of Flora, lent by the state museums of Dresden, these inhibitions were gone. Under the sway of Flora, goddess of flowers, nude people are relaxing in a garden. As a fountain trickles crystalline water, smooth slender bodies – male, female and ambiguous – are calmly displayed in a mood of cool sensuality. The painting has a silvery lightness, as if infused with summer. Poussin the hedonist – what a surprise. This exhibition cuts through centuries of dusty art snobbery to reveal that in his first decades in Rome, there was no one more alive to its temptations than this thoughtful Frenchman. He reviled Caravaggio, who had died in 1616, as having been born “to destroy painting”. Yet in his sophisticated celebrations of sex and wine, Poussin comes across as Caravaggio’s lewder French cousin. Not a single drawing by Caravaggio survives, and he probably painted his models as they posed in front of him. Poussin, however, drew obsessively. These works, many on loan from the Royal Collection, are this show’s crown jewels. I imagine Blunt, as surveyor of the Queen’s pictures, handled every single one. Using pen and ink, Poussin creates sharp yet lusciously shaded designs that you can watch developing from wild carnal moments to convoluted orgies. A naked satyr balances on a bag of wine, a follower of Bacchus dances in solitary delight. For Poussin, the pagan ancient world of Greece and Rome is a lost Arcadia of amoral delight. Before the coming of Christianity, before the knowledge of sin, the gods and their votaries danced, drank and copulated without rule or reason. That’s how he pictures it, anyway – and his imagination is informed by archaeology. He was the first painter who ever looked at ancient art closely, with scientific eyes. This exhibition includes some of the majestic antiquities upon which he modelled his dream dances. You can look from the stupendous Borghese Vase to his nearby painting The Triumph of Pan, and marvel at the precise way he depicts an ancient vase lying upturned on the ground. The reason the vase is on its side is that, under the wicked gaze of a red-faced idol of the pastoral deity Pan, the party is kicking off. A nymph lets a satyr embrace her as she sits on a goat, baring her leg, throwing her body back: eerily shiny, humanoid masks lie abandoned on the ground; and, in one of Poussin’s forays beyond heterosexuality, a muscular youth puts his arms gently round a drunken male figure. Your eyes are led from one sensual detail to another, in exquisitely structured mayhem. The exhibition includes modern reconstructions of the little wax figures Poussin used to help him plan these intricate rites. But who on earth, apart from himself, was up for such elaborate re-creations of pagan orgies? Cardinal Richelieu, that’s who. The powerful cleric who exerted sway over French politics also found time to commission this and two other “triumphs” of ecstatic abandon from Poussin. They are arrayed here like one frieze of flesh. In The Triumph of Bacchus, a woman balances on a centaur’s back as he rears his horse body in the air. It’s a menage a trois, for a female satyr rears up beside him, looking at her rival with a tolerant grin. In The Triumph of Silenus, an androgynous figure lies drunk while a goat-legged female satyr goes off with her lover to find a quieter spot. What a shame it has to end. The final painting is billed as the exhibition’s highlight but it is a bit of a downer. We see a more sombre Poussin, woken from his fantasy by time and death. A Dance to the Music of Time, lent by the Wallace Collection, depicts four people dancing in a circle, their backs to one another. Their round seems eternal, yet this Arcadian celebration is shadowed by mortality. A child blows bubbles, a stark symbol of life’s brevity, as Old Father Time plays the dance tune. Carry on a bit further into the landscape and you will find a tomb. After all the joyous revelling in Rome’s grand carnality, Poussin became a grave painter of death in the landscape and a historian of the Christian creed. Maybe he just got older. Perhaps he woke up one morning in his studio surrounded by naked wax puppets and felt ashamed. Yet he had already created some of art’s most sustained and luxurious orgies, as this exhibition lavishly proves.