He fires a woman out of a cannon, makes her disappear and then reappear inside a box that is inside a chest that has been hanging in plain sight of the audience since the show began.

He then transforms the woman into a Bengal tiger.

How does he do it?

"Very carefully," says Harry Blackstone. That's his general approach to the complicated business of being a magician -- particularly one named Blackstone.

There are very few barnstorming magicians these days. One reason is that people don't say "gee whiz" as easily as they did in a pre-TV age. In the high-tech '80s, it's harder to draw gasps from a crowd.

That's why the current Blackstone production -- which opens a two-week run at the Warner tonight -- includes an elephant, a camel, a donkey, a hawk, a rabbit and some pink doves (as well as the tiger", a woman who is cut in half by a buzz saw, more than 6,000 flowers that sprout everywhere in an "enchanted garden" sequence, and a light bulb that stays lighted while it is unscrewed from a lamp and floats out into the audience.

"I'm not up there to demonstrate apparatus," says Blackstone, 45. "I want to give them a sense of spectacle -- one critic called it 'magic a la Ziegfeld," and that's about right. The audiences have been educated now. They're used to television, which brings them the best talent availble and shows it in close-up.You can't expect to go on the road with anything less than the best."

Harry Blackstone Jr. should know: He has spent his life since early infancy in show business, most of it on the road -- as a singer, an actor, a magician, a TV producer and, by accident, stage-manager of Ricahrd Nixon's inaugural in 1969

His father was The Great Blackstone, whose legendary traveling show flourished in the '20s and '30s and who once performed for Calvin Coolidge int the White House.

Harry Jr. made his stage debut with that show before he was 6 months old and learned the business inside out. Yet after his father retired the act, shortly before his death in 1965, the son hesitated for a long time before deciding to bring the name back before the public.

"Of course, I'd be a fool to change my name," he says, "but for a long time I hestitated because I felt that one legend in a family was enough. Besides the enormous expense of taking a show like this on the road, I didn't want to do it until I felt I could do justice to the name. It takes a great sense of confidence . . . or foolhardiness."

And it takes a lot of money. To be precise: "750,000 for the special equipment, scenery and costumes; $175,000 for the aniamls (which are inspected once a month by two government agencies); and $61,700 per week in salaries, insurance and transportation costs.

"We travel with about 55 tons of luggage," says Blackstone, "plus the elephant, who weighs 3.25 tons. When I consider the effect of that floating light bulb in terms of its weight, I soemtimes wonder whether I'm carrying a lot of unnecessary luggage.

"We have three 45-foot semis to carry the show. A lot of stuff gets left in the trunk."

In the family tradition Blackstone's wife, Gay, travels with the show. She appears myteriously in illusions (as his mother did) and is shot from a cannon, turned into a tiger and assaulted with a buzz saw.For the Washington engagement, she will be returning from a leave of absence during which she gave birth to 11-week-old Bellamie, Blackstone's fourth daughter. The older sisters -- Cynthia 16, Adrienne, 13, and Tracey, 11 -- live with their grandmother at the family home in Redlands, Calif.

His son Harry Blackstone III, has already begun a show-business career (he was one of the backup voices on Sammy Davis Jr.'s recording of "Candy Man"), but is currently heading an oil-exploration team in Guatemala -- "Doing whatever it is 22-year-old American boys do in Guatemala," his father says ominously. But at the moment, he is wondering more about his elephant than his son.

During an advance visit to scout facilities at the Warner, Blackstone was told that the stage might not be able to bear the weight of Misty, the elephant. Blackstone seemed not at all perturbed. "The elephant will decide," he said. "She always test a stage before she wil put her weight on it. Last year, we played in 119 cities and she never went through a floor."

Before going into magic on this grand scale about seven year ago, Blackstone had tried his hand in many other branches of show business. After returning from military service in Korea in 1956-57, he was part of a vocal trio called Tom, Dick and Harry that did the 4 a.m.-to-noon shift at a bar at Lake Tahoe. The Tom and Dick in that trio were named smothers, and when they moved on to greater things, Blackstone went along as a producer and director of their shows.

"Around Christmas of 1968 Tommy told me, "I need a tap-dancing magician,' and I promised to have one in my office the next day. When he came in and asked me, 'Where's the magician?' I told him, 'I'm here.'

"For that show, I produced the entire orchestra and chorus, Tom and Dick, with all their instruments including a double bass, out of a 36-inch cabinet. It was like the tiny car in the circus that dozens of clowns come out of."

That show was the step that got him back into magic on a big scale, after more than 30 years in which he had pracaticed it mostly for friends or small audiences. "Tom Smothers cosigned my $50,000 loan to finance my first show at Lake Tahoe in 1968," Blackstone recalls. "That year, I did 16 weeks at Caesar's Palace and the Flamingo. Since then, I've financed Tommy in a few things."

Before the return to magic, he also had worked as a disc jockey, the host of a children's television show, a radio announcer on a station owned by Lyndon Johnson in Texas, and the voice-over on a variety of television commericals. He was made the stage manager of the 1969 Nixon inaugural at the last minute, because he was in Washington and had a good reputation, and the original manager was ill in California.

"It was total confusion," he says. "I guess they always are. I remember at the last minute we had to get a chaise longue for Bob Hope to use in his act, and we had to bring it to the theater in one of the inaugural limousines.

"Then, in the darkness backstage, I saw someone lying on it and I told him, 'Get off -- that's for Bob Hope.' And when he stood up, I saw it was Frank Sinatra. A few minutes later, I came up to Sinatra again, looked at my notes on the cue, and told him, 'You're on in eight bars.' He gave me a look that could kill and said, 'Theyre playing "My Way." I know the music."'

But for all his varied experience, magic has always been an inescapable heritage -- beginning when he was less than a year old at the Oriental Theatre in Chicago, "My Mother was part of my father's act, and my father had hired a governess to watch me while she was onstage," he recalls. "But one day the governess apparently didn't know about the early show.

"Rather than leave me in the dressing room alone to tear up the costumes and get into the makeup, my mother took me on with her when she appeared mysteriously in one of the illusions. My father said that when he saw my eyes gleaming at the applause, he knew I would have a career in show business."

He left show business to go to high school at the Southern Arizona School ("for incorrigibles," he says. "That wasn't in the name, but it worked out that way.")

Among the private shcool's 64 students, he was probably the least rich and famous member. His roommate was Lance Reventlow (son of Barbara Hutton), his classmates scions or fortunes linked to Revlon, Chase Manhattan and Montgomery Ward.

"I began doing magic and music to create some identity for myself," he says. "I was surrounded by rich kids with enormous social status who looked onme as 'one of those actor children,' and I had to assert my own uniqueness." He asserted it so well that he began working in shows produced at the University of Arizona.

He brought his magic with him when he went to Swarthmore, and there it got him into trouble. A professor interested in parapsychology was hiring students to participate in his experiments. Blackstone went for the money and decided to give the professor his money's worth.

"I started to do magic gimmicks for him," he says. "i would read closed books across the room and do a perfect run reading those special cards they have for these experiments. I blew the old man away. He didn't know I was putting him on and he was ready to stop everythin and write an article about me -- maybe a book.

"Then the dean blew the whistle, and I was given a short leave of absence. That's when the government got me for an all-expenses-paid tour of Korea. I did some magic for the army while I was there."

Blackstone won't say that he doesn't believe in parapsychology or spiritism.

But he will cheerfully admit that his illusions are exactly that: "I go to great lengths to entertain rather than to prove the veracity of what I'm doing. That's why I use the title of magician rather than faith healer or politician.

"I have never seen anything done by psychics and spiritualists that could not be duplicated by trickery."

His own brand of magic, he says, "is called 'Grand Illusion,' and it's a specific kind of theater. I am really an actor who specializes in the role of a magician, and like any actor I have to believe in it while I'm doing it."

The recognition of magic as entertainment rather than something mysterious and sinister dates back only about two centuries, according to Blackstone. "As late as the 1790s," he says, "a woman was burned at the stake in Germany for doing a very simple trick with a handkerchief -- pulling it part and putting it back together again."

Two centuries later, magic carries another punishment -- constant travel. And Blackstone froesees at least a decade on the road before he can settle down and "do magic in one town all the time."

Meanwhile the man who calls himself "the Liberace of magic -- more glitz per square inch than anyone else," is content to make an elephant disappear and cut up his own wife with a buzz-saw.

"Now that I'm bringing the buzz saw to Washington," he says, "maybe Jimmy Carter will want me to help him on his budget."