Leonardo da Vinci: Painter at the Court of Milan (National Gallery)

Leonardo’s masterpieces remain startlingly modern, finds Thomas Calvocoressi.

Leonardo da Vinci: Painter at the Court of Milan
National Gallery, London WC2

There's little that is codified, conspiratorial, overblown or even "blockbuster" about "Leonardo da Vinci: Painter at the Court of Milan", an exhibition that, in truth, befits the latter label more than any other in the National Gallery's recent history. The curator, Luke Syson, has achieved something remarkable: a crowd-pleasing, history-making show that is nevertheless quiet, contemplative and calm, despite its awesome star turns. And these do not disappoint - how could they?

Let's not understate the gallery's extraordinary achievement. After five years in gestation, this exhibition has pulled together nine of Leonardo's 20 or so surviving paintings for the first (and, perhaps, last) time. These are all the works - except one, the mural of The Last Supper, which remains in situ - that he painted during his most prolific period, from 1482 to 1499, as court painter for Ludovico Sforza, ruler of Milan. Each room is dominated by one or two of the masterpieces, surrounded by Leonardo's related sketches, some never shown before.

From the first of his portraits we come to, The Musician, we witness what sets Leonardo apart: a thrilling tension between the strict conventions of courtly and sacred portraiture and his life-giving realism and artistry. In La Belle Ferronnière, although the shape of the woman's face is geometrically perfect, her expression - pensive and slightly suspicious – is arresting in its modernity; she is turned half towards us, as if challenging us to disapprove. The Lady with an Ermine, depicting Sforza's mistress, has a purity of beauty that is meant both to celebrate love and elicit it in the viewer; but she looks away, eluding us with a knowing almost-smile.

It is such nuance and sensitivity to his sitters that gives these paintings their humanity, transcending the relatively stiff and bloodless portraits by followers such as d'Oggiono. In his unfinished painting Saint Jerome and the accompanying anatomical sketches, Leonardo shows how he understands the body literally inside out; the twisted, muscular, lion-loving priest has a naturalistic physicality that elevates the work from blank idealisation but in so doing also lays bare its subject's soul. In these works, Leonardo's skills as mathematician and anatomist and his brilliant imagination collide.

There are many other firsts here: the first time the two almost hallucinatory Virgin of the Rocks paintings have been displayed in the same gallery; the first showing of the recently attributed Christ as Salvator Mundi, with its hazy-faced Jesus and miraculous crystal ball; a highly intricate knot pattern that we are told can be considered the first work of abstract art. What stays with you most, though, are Leonardo's startlingly fresh drawings. From his scribbles of women with their wondrous sense of vitality and movement to his heavily reworked (pentimento), kinetic mother-and-child studies, it is as if - 500 years after his death - he is just around the corner, the ink wet on the page, the breath of genius hanging in the air.