As regal Katja Schumann rode her Arabian stallion through a series of paper hoops last June 17, she was intently timing not only her Big Apple Circus act but also her contractions. Throughout her routine that night in Bridgeport, Conn., the 37-year-old Danish stunt rider, glittering with spangles, smiled and cracked her whip as if the performance were her only care in the world. Then, at intermission, Schumann turned to Paul Binder, 45, the founder, artistic director and ringmaster of the circus, who is also her husband of nearly three years. "Do you mind if I miss the finale?" Schumann asked casually. "I have other things to do." The usually unflappable Binder paled, gasping, "Why? Are we having a baby?"

For the next hour, Binder, dapper in his red tails, raced back and forth between the ring and the couple's trailer, where Schumann listened to Mozart and practiced her deep breathing for the planned home birth. The midwives they had summoned, however, were late. So at 10:24 p.m., two hours after taking off her costume and makeup, Katja delivered 7-lb., 4-oz. Max Abraham Binder into her husband's waiting hands for the grand finale of the only show that must truly go on.

"It was matter-of-fact, but not without feeling," recalls Schumann, a determined, no-nonsense type. "I was fully prepared to do the matinee the next day, but Paul said no." He only said it once, though. The night after giving birth, Schumann was back jumping through hoops.

Max's debut was an example of the felicitous, frantic teamwork Schumann and Binder have been practicing since Katja joined the troupe six years ago. Compared with most American circuses, the Big Apple is very little indeed, but its scale, modeled after the classic small circuses of Europe, makes each performance highly demanding. While American circuses pour forth scores of gaudy clowns doing pratfalls, the Big Apple features four sardonic jesters who play the crowd much like nightclub performers. There is one ring and only one act at a time, and the atmosphere is distinctly down-home. Ring-ling Bros, and Barnum & Bailey may draw up to 19,000 people per show; Big Apple plays to only 1,500. "Our vision is intimacy," says Binder, who finds his more famous rival big, brassy and utterly unrefined. "Our acts are fine pieces of work built specifically for this circus, and we look for artists with natural gaiety and joy."

Despite the limitations imposed by its size, the Big Apple under Binder's guiding hand has grown from a hand-to-mouth outfit in a makeshift tent to a Manhattan tradition with a passionate following. Now using a heated tent donated by Donald Trump, the circus, celebrating its 10th season, makes eagerly anticipated annual visits to Washington, Boston, Philadelphia and Pittsburgh, and has become a major fixture in New York at Christmastime.

Katja's exquisitely choreographed act is a highlight of the show, and she is world renowned as a horse trainer and sidesaddle rider. "Horses are beautiful, difficult and gratifying, and certain horses I like more than certain people," she says. Schumann has been riding in circuses since she was 10, when she appeared as a ballerina on horseback in Copenhagen's Circus Schumann, which was founded by her great-great-great-grandfather.

Despite the craftsmanship of their operation, Schumann and Binder lead a private life that has much in common with the scruffy lot of circus people everywhere. Since 1984 they have shared two cramped trailers; the present one, located a few steps from their tent at Lincoln Center, has two tiny bedrooms and well-worn furniture, and their children, Max, 6 months, and Katherine Rose, 2½, usually play in a tiny space between their trailer and the next. "I hate living in trailers," Binder confesses. "On some level, every day, I yearn for a house with a garden."

Binder was not raised to the nomadic life. The son of a Brooklyn wine salesman, he grew up loving Broadway shows. "I never went to the circus," he says. "It seemed distant and smelly and seedy to me." After graduating from Dartmouth in 1963, he worked briefly as a TV stage manager for Julia Child's The French Chef, then earned an M.B.A. from Columbia in 1965. At the time, he was planning to marry Vivian Bachrach, a Bennington graduate, and ordinary commerce "seemed a good idea." They wed in 1967 and had a son, Adam, now 20, but the marriage broke up in 1970, and Binder came to realize he cared more about show business than the white-collar kind. He became a talent booker for Merv Griffin, then spruced up some comedy routines he had been fooling with for years and joined the San Francisco Mime Troupe.

Three years later Binder and a friend packed off to Europe with $125 between them to work as street performers, passing the hat from London to Istanbul. "It was a picaresque journey," Binder recalls. "I was looking for my home by going away from it." When the two were invited to join the Nouveau Cirque de Paris, Paul finally found his calling. "I said, 'This feels like home,' " he remembers. "Then I realized America did not have a small, beautiful, little show like the ones in Paris."

Binder returned to New York in 1976 and began shopping around a proposal for a one-ring circus. From the outset he knew it would have to be nonprofit. "If you work at our scale—meaning intimacy with the audience—you can only sell a certain number of seats," he explains. His earnestness impressed corporate donors, and the next year "Mr. Paul," as he is known in the ring, was presiding over 10 acts in Battery Park. In 1981 he lured Schumann from Denmark to be a guest artist. "I had seen her on TV on the Royal Swedish Circus Show when I was 16," he says, "and I had known her name for years."

Schumann, who grew up in a circus wagon in Copenhagen and on tour with her father, the renowned maître ecuyer (master horseman) Max Schumann, has never known anything but circus life. By 12, she was a popular attraction in her own right, attending public school half the year and being tutored on the road by her mother, a circus wife. By the time Binder got in touch with her, she had already won the Prix de la Dame du Cirque and the Gold Medal at the 1976 Circus World Championships.

Schumann's "royal manner," as Binder calls it, immediately attracted him. "I fell in love the minute I saw her," he says. "But I didn't allow myself to feel it. It wouldn't have been appropriate. We lived in a tiny community, something like a small town." Their relationship changed in the summer of 1984, when he visited her in Copenhagen. "The more I could be with him, the happier I was," says Katja. They married in 1985, four months before the birth of Katherine. "I wore something flowered," remembers Schumann. "In Denmark, if you wear white, you're a picture of innocence. I wasn't."

Binder and Schumann have now settled into a demanding routine. Katja exercises her horses each morning and performs one or two shows a day. In between she tends to the kids, who will be tutored along with the other circus children. Paul has delegated much of the management of the circus to aides, but he still roams the ring looking for misplaced wires and supervises all the acts. Their trailer is always filled with shoptalk, and because of their devotion to craft there are sparks. "I can get pretty difficult," Katja says. "That's in my work, which requires so much precision. When I hire someone, I tell them, 'You have to know you're going to get yelled at.' "

She also gets yelled at back. "We're tough on each other and on the people around us because we set such high standards," says Binder. "I can be strident and arrogant."

Binder and Schumann sometimes work six months without a day off, and almost as soon as they have settled into a city it is time to hit the road again—with 13 tractor-trailer loads of props and equipment and two or three vans full of animals. Currently Binder is considering two locations for a permanent Big Apple Circus building in Manhattan. If that materializes, life would be easier, but ease is not the object of the path that Binder and Schumann have chosen. "We pay attention to fantasy and beauty," says Schumann, "because they are so important." In fact it is that attitude that sets them apart. "I think I'm luckier than most people—they are stuck in life with nowhere to go. I couldn't live in the real world," Schumann says frankly. "And I wouldn't want to.