Michelangelo, Leonardo, Raphael review – bitchy Da Vinci conquers the army of gnarly nudes

'Art is a serious subject," say posters put up by the Royal Academy in London to champion art in schools. But is the Royal Academy itself serious? Its main galleries are now full of vacant paintings by Michael Craig-Martin , RA, while three of the greatest artists who ever lived are crammed into a couple of rooms round the back.

Michelangelo, Leonardo, Raphael: Florence, c 1504 is based on the rivalrous encounter between Leonardo da Vinci and Michelangelo when both were commissioned to paint battle scenes by the Florentine republic. First, Leonardo was tasked to paint a mural of The Battle of Anghiari, in which Florence had defeated Milanese mercenaries. As Leonardo planned it, Michelangelo was commissioned to paint another battle on the same wall.

This exhibition could have been, should have been, a mighty epic. In the golden age of Lorenzo de' Medici's Florence these two geniuses emerged. They were different ages and had opportunities beyond Florence, so they didn't meet until "c 1504", as the exhibition subtitle has it, when both were back in town.

It was a changed place. A revolution inspired by the preacher Savonarola chucked out the Medici and established a popular republic. Savonarola supported a new democratic assembly and spurred the building of the Great Council Hall. Then he was executed. One of his critics, Niccolò Machiavelli , future author of The Prince, became a powerful political figure.

It was probably Machiavelli who came up with a cunning plan to excite the citizens by getting both Leonardo and Michelangelo to paint histories of Florentine battles in the hall. Then it kicked off. Michelangelo insulted Leonardo by goading him about his failure to finish his bronze horse in Milan. At a meeting to decide the location for Michelangelo's new, nude statue of the biblical hero David, Leonardo suggested the back of the Loggia della Signoria . And its genitals should be covered. Ooh, you bitch, Leonardo.

The Royal Academy's exhibition tells practically none of this. It makes little attempt to bring "Florence c 1504" to life. There's nothing about the hall where the standoff took place: I wanted wall-filling photos, digital projections and hi-tech sculptural and architectural simulacra.

No. This is an academic show. A drawing by Raphael of David from behind has to stand in for Michelangelo's great statue. There is too much Raphael. The show's insistence on treating him as a third contestant in the Renaissance Turner prize is nonsense.

So up to its halfway point this is a bore. Then the lad from Vinci steps into the ring. The fight is on and it's not even a contest. Leonardo devastates Michelangelo. The second part brings together many of their preparatory drawings, plus copies of their battle-scene works. It leaves you struggling to give Michelangelo attention.

For without its political background, Michelangelo's Battle of Cascina is baffling. In a 16th-century copy, naked men are clambering out of a river, rushing to get dressed. In a soft, sensitive sketch, Michelangelo maps out this scene with much warmer beauty.

Is he just indulging his passion for male bodies? One of the models in his drawings turns provocatively, another adopts a boxer-like stance. Michelangelo dwells on back muscles with such gnarly power you feel you are looking at landscapes.

But he is not only pleasing himself – he is satisfying Machiavelli. The story of how a Florentine army was taking a break to swim in the Arno at Cascina when the Pisans attacked, and the soldiers rapidly armed to secure a victory for Florence, is told in Renaissance history books. It was relevant in 1504 because Florence was again at war with Pisa. It was going badly. Machiavelli believed this was because of the bad habit of hiring mercenaries. A republic should have a citizen army always ready – like Michelangelo's energetic nudes.

Michelangelo was painting propaganda. David, too, was installed in front of the civic palace as a symbol of republican readiness. The political ideas behind them go to the heart of republican theory. The Battle of Cascina and David express the belief in citizen soldiers bearing arms that later inspired the US second amendment. But Michelangelo had not seen war and his designs look false beside the blast of Leonardo's The Battle of Anghiari .

Just before this, Leonardo served as military engineer to Cesare Borgia, psychopathic son of Pope Alexander VI. You can witness the soulless, inhuman snarls of the killers he had encountered in a copy of The Battle of Anghiari by Rubens . It's an interpretation rather than a replica but Leonardo's drawings confirm its scene of an old warrior howling from his leathery cruel face as he prepares to chop off an enemy's hand.

This frenzied hate makes Leonardo's preparatory drawings in red chalk and brown ink throb like beating hearts that have just been torn out of an enemy body to eat. On one sheet horses rear, their mouths churning, a furious human face mirrors their madness and a snarling lion is thrown in for good measure. In another, a rearing horse shakes its body so violently Leonardo draws it in a blur of positions that anticipates Eadweard Muybridge and futurism.

If he looks forward, he also looks back. He sees in The Battle of Anghiari a basic, primal human capacity to transform from civilised creature to wild animal. A drawing that reduces a horse to a rush of red chalk lines thrusting through space resembles cave art.

At the same time, Leonardo worked on inventions. On one sheet war sketches are mixed up with cogwheels. His interests, including an attempt to fly, distracted him and the Florentine republic got furious. Michelangelo was called away by Pope Julius II. The Medici retook Florence, desecrated the Great Council Hall and probably destroyed The Battle of Anghiari.

This is a flawed exhibition but you will never forget Leonardo 's vision of war. Fill your nightmares with apocalypse c 1504.