At the Louvre, my husband and I ducked briefly into the crowds around the “Venus de Milo” and skipped the “Mona Lisa” altogether in favor of Leonardo da Vinci’s “Virgin of the Rocks.” And after braving the masses around “The Winged Victory of Samothrace,” which is easier to see than some of its equally famous counterparts thanks to its position on a grand staircase, we fled back to the marvelous — and shamefully under-attended– Islamic Art wing. The Louvre’s audience-control issues aren’t unique to that institution; at the Rijksmuseum in Amsterdam, you have to work your way up to Rembrandt’s “The Night Watch” through the crowds of people clustering around in the hopes of snapping a selfie with the monumental work, which at least gives your eyes a chance to linger on all of Rembrandt’s magnificent faces.

Though I’m frustrated by trying to look at art through a flurry of lofted cellphones*, I can’t exactly blame anyone for the impulse to snap a picture of a great work. We live in a moment when an experience might as well not have happened unless it’s documented and broadcast as widely as possible. But it is completely exhausting and discouraging that museums haven’t found a way to balance the demands of museum-goers who want to take snapshots of art and people who want to actually look at it with their eyes instead of through a lens.

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The addition of cellphones and digital cameras may have changed what crowds do when they get up close to famous works of art, but the crowds themselves, and the difficulties museums face in dealing with them, aren’t exactly new issues.

In 1907, the Louvre put the “Mona Lisa” under glass after another picture was vandalized, a decision that “has caused dismay to art-lovers, who declare that the effect of the picture is ruined by the false lights and reflections,” the New York Times reported. The painting was stolen in 1911, and after it was recovered in 1913, it was displayed in Italy on its way home to France. “So great was the pressure of the crowds around” the Brera Gallery in Milan “that about 200 police officials and carabineers had difficulty in preserving order.”

When the Khrushchevs visited France in 1960, Madame Khrushchev behaved a lot like modern tourists at the Louvre: “The museum tour left little time for contemplation,” write one correspondent. “She stopped for only a few of the major pieces — such as Venus de Milo, Mona Lisa and Bonaparte’s crown.” And when the French government finally allowed the painting to be displayed in America in 1963, “The president’s mother, Mrs. Joseph P. Kennedy, could not squeeze through the mob, so she found a seat in a side gallery.” More than a million people visited the “Mona Lisa” when it went on display at the Metropolitan Museum of Art in New York later that year.

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Sending works like the “Mona Lisa” on tour might help disperse the crowds at the Louvre by letting more viewers see the piece in their home countries. But Louvre officials have said that the painting won’t be traveling; it’s simply too fragile to weather overseas trips.

So what’s to be done?

What about dedicated spaces for important works? The Sackler Wing at the Metropolitan Museum of Art in New York, which houses the Temple of Dendur, is one of the most magnificent exhibition spaces anywhere. But it works because of the temple’s scale; even if the “Mona Lisa” was exhibited in a bigger room, or even if the “Venus de Milo” had a more dedicated gallery, the pieces aren’t monumental; you’d still want to get fairly close to them to get the full effect.

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That leaves carefully timed tickets as the most realistic way to thin out crowds around the most popular exhibitions. That adds logistical complexity,and potentially extra costs to a visit to a museum like the Louvre, where entrance fees already start at 15 euros (about $17). Giving audiences a limited amount of time with a piece, whether they want to contemplate it or simply document that they were there, might not seem fair. But the current massive crowds aren’t serving anyone, whether your goal is to grab an Instagram-worthy shot in a sea of other cameras or an intimate communion with a lady with an enigmatic smile.