In 1877, when he was 43, the French impressionist Edgar Degas began stopping by the studio of the 33-year-old American Mary Cassatt and offering her a point or two that might embolden her painting.

Their relationship, a close one for a decade, is one of the best known in art history. What is not known is whether the relationship blossomed beyond a few pedagogical pointers into a secret romantic interlude. Each destroyed the letters from the other. Most historians doubt an affair. The artists seemed too strait-laced for that.

No romance is revealed, but a good deal of artistic influence is explored in an unusual exhibition, simply titled “Degas/Cassatt,” that recently opened at the National Gallery of Art and will close Oct. 5. Comprising 70 paintings and prints, a third from the National Gallery and the rest from other museums, the show, which attempts to demonstrate the influence of each on the other, will go nowhere else.

When Degas first came upon a painting by Cassatt, he reportedly said, “There is someone who feels as I do.” Three years later, he called on her and asked her to submit paintings for the next salon of the impressionists. He also told her in a remark that probably sounded more polite in the 19th century than it does now, “No woman has a right to draw like that.”

Although of different nationalities, Degas and Cassatt had much in common. Both came from wealthy families, and neither ever married. But their personalities were far different. Degas loved to dominate dinner parties with his wit and insults. The poet Paul Valéry said Degas “scattered wit, gaiety, terror” at these dinners. “He mimicked and struck home, lavish with his maxims and his taunts, his tirades and his tales.” Cassatt, who was born in Pittsburgh and first studied art in Paris at 19, settled there when she was 30. She was not volatile like Degas but possessed, as one male friend put it, “an electric vitality.” And she proved as fashionable as the women she depicted.

There was some similarity in their work. Unlike most impressionists, both seldom painted a landscape. Many contemporaries regarded them as teacher and pupil. But this is disputed by Kimberly A. Jones of the National Gallery, the curator of the exhibition.

“Cassatt was no mere student of Degas,” she writes in the catalog, “but a peer who displayed a confidence and facility on a par with his own.” Degas, according to this view, helped out a younger artist from time to time and was in turn helped by the younger artist.

The influence of Degas is easiest to document. In 1903, Cassatt discussed her 1878 painting, “Little Girl in a Blue Armchair,” in a letter to a French dealer. She wrote that the model was “a child of friends of M. Degas” and that “he found it to be good and advised me on the background, he even worked on the background.”

The painting is a longtime prize of the National Gallery’s permanent collection, and the show gave Ann Hoenigswald, the gallery’s senior conservator, the opportunity to examine the painting with an infrared camera to find out what Cassatt was talking about.

The evidence persuaded her that Cassatt had originally painted a straight line separating the wall in the back from the floor. The straight line was parallel to the top of the canvas, and the result was boring. This was changed, presumably at Degas’ suggestion, by painting a new room corner into what had been a single wall. The new arrangement produced a pair of diagonals along the two walls coming out of the new corner, creating a feeling of movement in the center of the painting. Hoenigswald also discovered that a piece of the background was painted in a technique that was a trademark of Degas but not Cassatt.

In turn, Jones pointed out an obvious influence of Cassatt on Degas. She experimented with metallic paint and used it in such paintings as “Woman Standing Holding a Fan.” Degas, who had not done so before, began applying metallic paint to some of his work, including the unflattering “Portrait After a Costume Ball (Portrait of Mme. Dietz-Monnin).”

Both artists are well known for paintings and prints of scenes at the Paris Opera. Their works seem like part of a deliberate dialogue: While Cassatt painted scenes of fashionable women in the audience, Degas concentrated on ballet dancers onstage and in rehearsal backstage.

As Jones explains, “As a woman of her class, she couldn’t go backstage.” It would have been regarded as unseemly. As a result, she could not compete with Degas’ delightful scenes of ballet masters rehearsing young dancers. And he was too fascinated by life backstage to compete with Cassatt on the fashions of the audience.

There is no question that they were collaborators in the art of printmaking. In 1879, Degas invited Cassatt to join him and several other artists in creating a periodical that would offer original prints to customers. It would be called Le Jour et La Nuit (Day and Night). Degas was more experienced at printmaking, but Cassatt soon caught up with him.

For the first issue, Degas prepared a complex etching called “Mary Cassatt at the Louvre: The Etruscan Gallery.” It showed the back of the well-dressed Cassatt, wearing a sumptuous hat and leaning on a folded umbrella for balance, as she studied a canvas. Her sister Lydia, reading a guidebook, sat nearby.

Cassatt’s contribution was an etching with a familiar subject, a woman with a fan sitting in the opera house. This print, called “In the Opera Box (No. 3),” omits the detail of her paintings on the same subject and offers a kind of fleeting, impressionist view of the woman caught in the lights of the opera house.

Le Jour et La Nuit folded even before its first issue. Degas evidently decided that printmaking demanded excessive labor for limited reward.

Degas used Cassatt as a model several times and painted an oil portrait as well. Cassatt never painted him.

After the final salon of impressionist painting in 1886, the relationship waned. This was the era when Cassatt began painting her canvases of mothers and children, such as “Child Picking a Fruit.”

The Dreyfus affair, which began in 1894, would have challenged their relationship in any case. In that year, Captain Alfred Dreyfus, the only Jew on the General Staff of the French army, was falsely accused of betraying France by passing military secrets to Germany and sentenced to life imprisonment.

Joining most French intellectuals, the American Cassatt believed Dreyfus innocent. Degas, on the other hand, became vehemently anti-Dreyfus and anti-Semitic. There is a report of at least one open quarrel between Degas and Cassatt over Dreyfus.

As evidence mounted that army officers had forged the evidence against Dreyfus, Degas insisted that the honor of the French army was more important than the fate of the Jew rotting on Devil’s Island off the coast of South America. The government exonerated Dreyfus in 1906, but Degas never accepted this turn in the affair.

The Dreyfus affair, however, did not end their relationship. Cassatt helped her American relatives and friends, especially H.O. and Louisine Havemeyer, amass collections of Degas’ works, often by buying from Degas directly.

Sometimes the sale was out of Degas’ hands. When the Havemeyers purchased a Degas painting from an anonymous collector in 1912 for almost $100,000, it set a world record for the cost of a painting by a living artist. Degas, who received nothing from the sale, said he felt “like a racehorse who has won the Grand Prix and receives only his ration of oats.”

Degas’ health deteriorated so badly a few years later that Cassatt persuaded his niece, a nurse, to live with him permanently. Degas died in 1917 at age 83. Cassatt wrote a friend, “His death is a great deliverance, but I am sad... He was my oldest friend here, and the last great artist of the 19th century — I see no one to replace him.” Cassatt died in 1926, also at age 83.