As expected, the Louvre’s new “Leonardo da Vinci” show is mobbed. Timed tickets control the crush inside, but the place is still packed. The strain on the museum’s online reservation site is such that the Louvre is apologizing for the delays, and it’s difficult to get tickets before December.

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In the months leading up to last week’s public opening, the exhibition was engulfed in the din of what Saul Bellow called “event glamour.” Thirteen years in the making, the show was plagued by obstacles and all but suffocated by intrigue and hype.

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(You want to mount a Leonardo exhibition? Knock yourself out. Understand first, though, that you’ll have to beg to borrow works from a Saudi despot, the Queen of England, Bill Gates, and the U.S. and Russian governments; defeat a bunch of Italian nationalists in court; and placate a gang of Leonardo experts as vicious and territorial as stoats.)

The result is nonetheless magnificent. It may be decades before anything like “Leonardo da Vinci” is staged again. So, for all the hype surrounding it, the exhibition offers a precious opportunity to find a bit of focus in the fog, to draw something substantial out of the sfumato. Who was this elusive genius? How did his mind work? What, among the things he left behind, still presses in on us today?

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In Paris, in a state of high anticipatory excitement, I had plenty of time to ponder these questions. Standing with the crowds outside the Louvre’s Pyramid and then, inside the exhibition, craning to see over the curling locks and bald pates of fellow patrons, I remembered a passage Leonardo penned about the dynamics of crowds.

“All the bystanders at an event worthy of note adopt various gestures of admiration,” he wrote in his treatise on painting, the Codex Urbinas. “And if the event is of a devotional kind all the onlookers direct their eyes with various expressions of devotion toward [it].”

With this in mind, I found myself inside the dark, devotional galleries paying special attention to the gestures, expressions and looks of blinking astonishment on the faces of my fellow patrons. All of us together seemed to be sharing unaccountable things, our inner lives linked by invisible filaments.

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Writing in 1869, the English essayist and art critic Walter Pater pictured Leonardo “tracking the sources of expression to their subtlest retreats.” He had it right, I think. For all his exemplary curiosity about the external world, Leonardo’s deepest concern was to express the idea — the fact — of inner life, to stake a claim on the soul’s inherent beauty.

One of the show’s finest moments occurs in the very first gallery. At the center of the room stands a monumental bronze sculpture by Leonardo’s teacher, Verrocchio, showing Saint Thomas poking doubtfully at the wound in the risen Christ’s torso. Both Jesus and Thomas wear flowing garments and are well beyond life-size. They are surrounded by a sweeping arc of Leonardo’s drapery studies (in distemper brushed on primed linen).

A subtler, more involving way to kick off the show is hard to imagine. It’s as if we’ve been deposited in a life drawing class, as students around a model. Like Saint Thomas, we see up close the young Leonardo’s studies, each one capturing in two dimensions a palpable, three-dimensional reality: the very same folds we can see in Verrocchio’s sculpture.

The claim is often made that Leonardo’s mind was too supple, active and curious to be contained by any single art form — even painting. This isn’t true, according to the show’s curators, Vincent Delieuvin and Louis Frank. They want us to let go of the idea of Leonardo as a freakish and fatally distracted polymath — inventor, naturalist, mathematician and so on — and recognize that painting was at the heart of all his endeavors. If they are right, as I’m ready to believe they are, the question is only why he painted so little.

We are almost halfway into the exhibition before we see the first painting by Leonardo — the “Benois Madonna” from St. Petersburg. By the show’s end, we have seen only six others. (The Mona Lisa would have taken the total to eight — out of a known total of about 14 or 15 — but in the interests of crowd control she has remained upstairs in the Louvre’s Salle des Etats.)

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It’s not that there aren’t other great things to see. The “Vitruvian Man,” Leonardo’s drawing of the body’s ideal proportions, is here, a late arrival (after an Italian court order preventing its traveling was overturned). Alas, it will be in the show for only eight weeks. It’s thrilling to be in the presence of one of the world’s most reproduced artworks (NASA even used a version of it as a patch on astronauts’ spacesuits). The curators have placed it, almost insouciantly, at the back of a room full of Leonardo’s scientific sketches and mathematical speculations.

There are also, among other wondrous things, an early landscape sketch; a marvelously animated study for the Madonna With the Fruit Bowl; one of Leonardo’s beloved grotesque heads; two studies of yelling men for the lost painting “The Battle of Anghiari”; and a stupendous late drawing of a deluge.

But this show’s explicit thesis is that, for Leonardo, what really counted was painting. It’s well known that he carried a chip on his shoulder about his illegitimate birth, which cost him the chance to get a formal education. In a culture that had similarly failed to “legitimate” painting, he also resented the special pleading he had to do on the medium’s behalf. Lacking a recognized niche in the hierarchy of the arts, painting “was left without advocates,” he lamented.

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So why did he leave so few paintings behind? Was it indecisiveness? Distraction? A species of perfectionism that was always threatening to paralyze him?

Scientific analyses, using infrared reflectology (full-scale images pepper the exhibition), show how heavily worked his paintings were. Oil paint was still relatively new in Italy, and Leonardo, who had seen what the great Flemish and Venetian artists — Jan van Eyck, Antonello da Messina and Giovanni Bellini — had done with it, loved the stuff. He was constantly revising compositions. And, as he came into his maturity and developed his famous “sfumato” technique, he brushed on paint with increasing freedom.

Evidence of his searching, intuitive approach is at odds with the image of a man who was too caught up in his own intellectual speculations to bother getting his hands dirty. But, of course, there are different kinds of fastidiousness. What stymied Leonardo more than anything, perhaps, was also the source of his originality: Simply put, he was allergic to cliche. Inherited subjects and traditional approaches failed to satisfy him. He was always looking for ways to make it new.

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He was hardly alone in this. Restlessness in relation to tradition was a hallmark of the Renaissance. But the quality Leonardo brought to his restlessness was deeply peculiar. His centrality to the Renaissance is indisputable, but in front of much of his work — the “Saint John the Baptist,” the Mona Lisa, and both versions of the “Virgin of the Rocks” — you feel a strange eclipse obscuring the period’s bright daylight.

His paintings in particular can produce a feeling analogous to what roboticists call “the uncanny valley” — that charged, disturbing gap between the actually human and a convincing robotic stand-in. You could say this tension undergirds all great portraits. (No painting is a real person, after all.) And yet, in Leonardo the quality is different. The double whammy of naturalistic intensity and abstracted idealization does weird things to his subjects — and to their smiles in particular. Stare at the results — a painting like the “Benois Madonna,” for instance, or the late “Saint John the Baptist” — and you begin to fear that a bunch of internal wiring is about to start fizzing and snarling.

Apart from the Louvre’s “The Virgin of the Rocks,” Leonardo’s greatest paintings were all portraits of women: the Mona Lisa, “La Belle Ferronnière” and “Lady With an Ermine.” The best of the three, the fabulously erotic “Lady With an Ermine,” has not traveled from its home in Krakow, Poland. So, with the Mona Lisa in her upstairs confinement, “La Belle Ferronnière” is left to be the show’s singular star.

God, she is beautiful. Emerging like a weirdly solid specter from behind a stone parapet, her stillness is accentuated by the way her arms hang decorously down at her side, by the sharp clarity of her costume and by Leonardo’s subtle idealizing of her features.

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But Leonardo’s signature obsession was movement. And in “La Belle Ferronnière,” despite her physical stillness, this feeling for flux, though subtle, is on full display. Her face is animated not only by the fluttering play of shadow but by the movement of her eyes. Because she is turning to her left, one eye is slightly closer and therefore larger. And as her pupils turn into the shadowy sfumato at the corners of their almond-shaped apertures, Leonardo exposes the whites. The larger of the two is tinged pink. With a slightly queasy shock, you register its full roundedness as it turns in its socket.

The greatness of Leonardo’s portraits of women can’t, finally, be explained away. There is something more enigmatic going on. Pater wrote that his women “seem to be subject to exceptional conditions, to feel powers at work in the common air unfelt by others,” powers they passed on to us “in a chain of secret influences.”

Pater may have been a creature of 19th-century spiritualism and aestheticism, his sensibility as alien to the Renaissance as to our own era of metrics and algorithms. But I like his take on Leonardo because it grapples openly with an idea that was of fundamental importance to the artist himself (an idea that warrants reviving today): that there are chains of hidden influence and a secret rapport between all things, living and dead.