I broke away from the kiss, still holding him to me. His mouth was glossy with saliva. He rubbed it dry with the back of his hand, smiling at me again. I looked at him as if I hadn't seen him for years. The sherry color of his brown eyes, his slightly askew nose, his thick lips.

"Are you in trouble?" he asked.

"No. Why?"

"I wasn't expecting you for two months."

"Well, it's your lucky day, then, isn't it? Come on, let's eat before they run out of food."

THE INVERSE CASCADE.

Hope Clearwater buys four parsnips in the market today. She is delighted and very surprised to find them. She asks the trader where they came from. Nigeria, the trader says. Hope doesn't believe her, and questions her skeptically: "Where exactly in Nigeria?" The trader is not pleased to have her word doubted. "Jos," she says, and turns away. Hope remembers that Jos is situated high on a plateau in central Nigeria. All sorts of fruit and vegetables can be grown there because of its cool nights and dry days-even raspberries and strawberries.

The parsnips remind her of a story John told her about an old professor of his. This man had worked for a time on the problem of turbulent diffusion. For his experiments he required a large number of floats that had to be at once highly visible and not affected by wind. For this reason balls-both rubber and Ping-Pong-would be of no use. After experimenting with turnips and potatoes, the professor discovered that the ideal vegetable was the parsnip. Its rough, conic configuration, and the fact that in the water most of the vegetable was below the surface, made it a stable float and indifferent to all but the severest breezes.

What the professor did was to paint several dozen parsnips white and tip sackfuls of them off a bridge over the River Cam. The white parsnips would float downstream, be caught in eddies, cluster and circle in side-swirls or flow downriver in long bobbing strings, photographed by the professor's assistants standing with cameras on both banks of the river at intervals of twenty yards.

The professor (Hope can't remember his name) bad done useful work, John said. The problem was that, despite his imaginative experimental technique, his thinking was too rigid. He believed that turbulence was caused by a cascade of energy from large eddies to small. But John's own work-his breakthrough, as he termed it-had shown that this was only part of the story. In every case of turbulence, in whatever medium, there is also an inverse cascade, a flow of energy from the small eddy back to the large. Hope remembers clearly the day John proved this. He had explained it to her, his voice hoarse with excitement. Disorder, he said, is not simply handed down a chain; some of it is always being handed back again. Once that fact is grasped, a great deal that was baffling about turbulent systems becomes far easier to understand.

When John Clearwater came back from America he was in good form. He had met someone at the conference-a statistician-who had helped him enormously, almost without knowing it. He told Hope about these new avenues that had opened up, new potentials he could now see. Hope laughed with him, with genuine pleasure-with relief perhaps-at this excitement. Once again what he told her-he spent two hours trying to explain it to her-meant little or nothing, but she felt pleased, reassured. That time, after his return from America, calmed her, staunched the thin hemorrhage of anxiety and doubt. All, it seemed, was well again.

He worked as hard as ever, leaving the flat at eight in the morning and not returning, usually, until nine at night, but their hours together seemed to Hope to recapture some of the vivacity and edge of their first months of marriage. Later, when she looked back, she realized that they had merely been going through another phase. (She found she could demarcate the phases in her married life as efficiently as a historian-they seemed as precise as the circles of growth in the trunk of a tree.) In this particular phase a new enthusiasm dominated-the cinema. They went fairly frequently to the cinema and theater, whenever the mood took them, or whenever some triumphant succes d'estime seemed to demand it. But now John wanted to go out every second or third night. And at first it was fun. His absorption was so intense and single-minded, and the pleasure he took in the cinema so manifestly good for his spirits, that it was a privilege, she reckoned, to share in it. But after six weeks of this, amounting to over two dozen visits-some films were returned to two or three times-she began to find that the strain of accompanying him was growing and she started making excuses.

Part of the problem was that he insisted sitting very close to the screen, in the front row preferably, and certainly no farther back than the third, so that his entire field of vision was dominated by the projected image. At first this was oddly exhilarating, and Hope would emerge from the cinema with her head reverberating like a gong from the big booming pictures, breathless and jangled.

His other idiosyncrasy, however, was harder to take and began to irritate her. He was scrupulous about the type of film they saw. Reviews in as many newspapers as possible were studied and collated, and he built up a small library of film reference books in an attempt to ensure that the film they were going to see would fulfill the demands he made of it. She accused him, jokingly at first, of being the only person she had ever met who sought to entertain himself in a wholly prescriptive way. It was the very opposite of random; he wanted to take no risks. "How can you enjoy yourself, how can you have fun, without an element of risk?" He paid no attention. It was not a narrow censorship he indulged in-he was very keen on horror films and violent thrillers-it was simply that he believed, with a fundamentalist's zeal, that a true film, a film that was true to the nature of its own form, had to have a happy ending.

"Don't be so ridiculous," Hope said, when he first put this to her.

"No, honestly...a film that doesn't have a happy ending is-" He paused. "-is misunderstanding the basis of all cinema."

"OK. OK. How many counterexamples do you want me to give you? Two dozen? Three dozen?"

"No. Don't you see?" He was enjoying himself. "Put it this way. The essence of all art is positive. At root. So in the one great popular art form, the popular art form, this motive has to be even more powerful."

Hope wondered then if he were teasing her. But his expression-candid, intense-belied this. "Nonsense," she said. "Rubbish. What more can I say?"

But he wasn't joking.

And so it went on. And so John only saw those films that did not, as he saw it, demean or betray cinema's true purpose. Hope came to realize fairly swiftly that going to see these films was in a real sense therapeutic for him. They functioned as a kind of drug, and she began to see how his close-up, all-enveloping, dream-fulfilling cinema buoyed him up and kept him floating. Those few weeks of ease she had experienced after his return from the conference began to be eroded once again by the slow drip, drip of worry.

Hope looked up at John's taut, stretched face as he came. She saw his brow crumple, his cheeks concave, and heard a grunting deep in his throat. Then he exhaled and smiled and lowered his head until their noses met. He settled his weight on his elbows as Hope touched his wiry hair. He fitted his head into the angle of her neck and shoulder and exhaled again, his breath warm and moist against her skin. Inside her she felt the small shiftings and slippings as his penis detumesced. She sensed a complementary swelling of love for him in her throat as she dragged her fingers over his head, down across the thick hair that grew on his neck, trailing them lightly across the flaky blur of big freckles on his shoulder blades, making him shiver.

Catching the thin, sour smell of fresh sweat from his armpits, she slipped her hand into his armpit, feeling the hairs slick and clotted between her fingers. She kissed his neck, pressing her nose into his neck, smelling his own particular scent, his spoor. She remembered thinking once, before she married, what kind of man she wanted to live with, and had run through the various types that seemed most commonly on offer-the caring ones, the bastards, the strong ones, the moneyed, the humorists, the saints-and had decided that what she wanted was not a model or an archetype, but somebody quite different. A man. A person. Different from her.

Hope held and smelled this real person that she had found. Then she slipped her fingers into her mouth and tasted his salt sweat. She reached down his spine to touch the small, flat button of a mole that grew four inches above the cleft in his buttocks and reveled selfishly in the quiddity of this individual who was hers, whom she possessed.... Intimacy made her melancholy and exhilarated. She turned her head and kissed him on the mouth, forcing his teeth apart with a blunt, strong tongue and then sucking his own tongue into her mouth, tasting his saliva.

She pushed him over onto his back and felt his flaccid penis slide wetly from her.

"Ah. Sheets," he said.

"I love you, John," she said. "And don't you forget it."

"I won't. But I've forgotten the tissues."

"Then the deal's off."

It was a Sunday morning. He brought her a mug of tea and then went out to buy newspapers and bread. She shouted at him to put some music on the record player before he left. He couldn't have heard her because the door closed and there was only silence.

She rolled over and sat on the edge of the bed looking down at her lap and thighs, thinking dully that she was putting on weight. She cupped her soft stomach with both hands-she was. She sighed, and then, absentmindedly, with the backs of her fingers, gently stroked her pubic hair-unusually thick, she thought, a brash, dense triangle-and thought about John, and the cinema, their first wedding anniversary, which was approaching, the holiday they were going to take, and how it would be.

She stood up, walked through to the sitting room and crouched in front of the record player. A thick plank of sun lay across the dining table illuminating the wreckage of their evening meal, the dregs of wine in the opaque, smeary glasses, the congealed scraps on the uncleared plates.

She put on a record and stood up, humming along. And then, somehow, her mood, a phrase in the music, the sun on the table made the moment magically thicken and hold. For an instant she forgot where she was, her gaze unfocused and she seemed to see John, in her mind's eye, hurrying back to the flat. She saw the sunny street, the shiny cars, the comical way he was trying to read the newspapers as he walked, his arms full of groceries. The shadows the buildings cast were striped obliquely across the street, light and shade. John walked through gloom and glare toward her.

The odd trance passed. She shivered, naked, in the sitting room. She ran back to their bed and slid between just warm sheets.

USMAN SHOUKRY'S LEMMA Muhammad ibn Musa al-Kwarizmi was an Arab mathematician from Khiva, now part of Uzbekistan in the USSR. He lived in the first part of the ninth century A.D. and is remarkable in that he not only gave us the word algebra (from the title of one of his books-Calculation by Restoration and Reduction-al-jabr means "Restoration"), but also, more interestingly, from his name-al-Khwarizmi-is derived the word algorithm. An algorithm is a mechanical procedure for solving a problem in a finite number of steps, a procedure that requires no ingenuity.

Algorithms are much beloved mathematical tools. Computers operate on algorithms. They imply a world of certainty, of rotas and routine, of continuous process. The great celestial machine, programmed and preordained.

However, algorithmic procedures are of little use for phenomena that are irregular and discontinuous. Fairly self-evident, you would have thought, but how often have we tried to solve the problems in our life algorithmically? It doesn't work. I should know.

There is another appellation in the world of mathematics that comes faintly tinged with contempt. A Lemma. A lemma is a proposition that is so simple that it cannot even be called a theorem. I appreciate lemmae-or lemmas, maybe-they seem to have more bearing on my world. "You can't make an omelet without breaking eggs"..."More baste, less speed"...

Usman gave me a lemma once.

We were in bed, it was dark and we had made love. The roof fan buzzed above our heads and the room was cool. I could hear only the steady beat of the fan and the noise of the crickets outside. I turned to him and kissed him.

"Ah, Hope," he said-I couldn't see his smile in the dark, but I could hear it in his voice-"I think you're falling in love with me."

"Think what you like," I said, "but you're wrong."

"You're a difficult person, Hope. Very difficult."

"Well, I am feeling happy," I said. "I'll give you that. You make me happy."

Then he said something in Arabic.

"What's that?"

"It's a saying. What we always say. A warning: 'Never be too happy.'"

Never be too happy. Usman Shoukry's lemma.

Sometimes I wonder if a lemma is closer to an axiom. Axioms are statements that are assumed to be true, that require no formal proof: 2 + 2 = 4. "A line is a length without breadth." Life is full of lemmae, I know. There must be some axioms.

Usman said he would be on the beach that afternoon if I wanted to meet up with him after my provisioning trip. As it happened I was finished by half past three and a hotel taxi took me down to the bathing beach. I saw Usman's car, parked alongside a few others in the shade of a palm grove, and let the taxi go.

The palm trees here were very tall and old; their tensed, curved, gray trunks looked too slim to hold themselves erect, let alone bear the weight of their shaggy crowns and burden of green coconuts. The ground beneath them was grassless and hard, almost as if it had been rolled and swept. This had been an exclusive beach once and all along the shoreline were the remains of wooden beach houses and cabanas. Most had rotted away over the last few years, or had been dismantled for their timber and tar paper roofs. Locals had settled here and a ribbon of shanties, made from the recycled cabanas, lurked in the scrub behind the littoral's tree line. With them had come rubbish dumps and livestock of all kinds. Goats and hens scrounged amongst the palm trees, stray dogs loped along the sand, sniffing curiously at whatever the waves had brought ashore.

One or two of the beach houses were still in good repair. The general manager of the bauxite mines had one, and a few Lebanese and Syrian merchants had clubbed together to keep others functioning. But whatever their efforts, the mood of this stretch of shoreline was inescapably sad, a morose memory of former glories.

I saw Usman standing waist-high in the sea, his torso canted into the green and foamy breakers that rolled powerfully in, smashing and buffeting his body. With particularly large waves he would dive beneath them, hurling himself into their sheer, tight throats just before they crested, and emerge, spitting and delighted, on the other side.

"Usman!" I called and he waved back at me. I sat down on his mat, took off my shoes and lit a cigarette. Behind me, four men played volleyball outside one of the refurbished beach huts. They were brown-Lebanese, I guessed-wore very small swimming trunks and played with histrionic abandon, making unnecessary dives for very gettable balls.

Usman came out of the sea, shaking his head like a dog. He had put on more weight since my last visit and there was a soft overhang of flesh at the waistband of his swimming trunks. He sat down beside me and with delicate, wet fingers helped himself to one of my cigarettes.

"Going to swim?" he said.

"I'm frightened of the undertow, you know that."

"Ah, Hope. That sounds like an epitaph to me-'Hope Clearwater, she was frightened of the undertow.'"

Usman was Egyptian and in his early forties, I guessed. He wouldn't tell me his exact age.

"You're getting fat," I said.

"You're getting too thin."

He spoke very good English, but with quite a heavy accent. He had a strong face which would have looked better if he were less heavy. All his features-nose, eyebrows, lips, chin-appeared to have extra emphasis. His brown torso was quite hairless. His nipples were small and neat, like a boy's.

A fly settled on his leg and he watched it for a while, letting it taste the salt water, before he waved it away. There was a milky haze covering the sun and a breeze off the ocean. I felt warm but not too hot. I lay back on his mat and shut my eyes, listening to the rumble and hiss of the breakers. Grosso Arvore, my chimpanzees and Mallabar seemed very far away.

"I should have brought my swimsuit," I said. "Not to swim. To get brown."

"No, no. Stay white. I like you white. All the European women here are too brown. Be different."

"I hate being so white."

"OK. Get brown, I don't care that much."

I laughed at him. He made me laugh, Usman, but I couldn't really say why. I sensed him lying down on the mat beside me. We were silent for a while. Then I felt his fingers gently touch my face. Then they were in my hair, brushing it back from my forehead.

"Stay white, Hope," he whispered dramatically in my ear. "Stay white for your brown man."

I laughed at him again. "No."

I felt dulled by the warmth and the smoothing motion of his fingers on my head.

"Hey. What's this?" Both sets of fingers were in my hair now, parting the strands to expose my skull. I kept my eyes closed.

"My port-wine mark."

"What do you call it?"

I explained. I had a port-wine mark, a sizable spill, a ragged two inches across, above my left ear, a dark prelate's purple. My hair was so thick you had to search hard to spot it. No pictures exist of me as a bald baby. My parents waited until my hair had fully grown in before they put me in front of a camera.

"In Egypt this means very good luck."

"In England it means good luck too. It's bad luck if it's full on your face."

He looked resigned. "I just said it to make you feel good."

"Thank you." I paused. "Actually it does make me feel good. I often wonder what I would've been like if it had been on my cheek." I squirmed round and rested on an elbow, looking at him. "You wouldn't be lying here for a start."

This time he laughed at me. "Yes. You're probably correct."

"See. It brings me good luck."

I lay back again. A clamorous argument was going on amongst the volleyball players.

"Do you want to go to that Lebanese restaurant tonight?" he said. "I shouldn't be back too late." He sat up. "I have to go now."

"Where?"

"I'm flying."

"A mission?"

"No. I've got to test the wiring. You know, two days ago, I was on reconnaisance. I pressed the camera button and my fuel tanks dropped off."

Usman was a pilot in the federal air force. A mercenary pilot, I should say, not to put too fine a point on it. All the Mig 15s at the airport were flown by foreigners, on hire to the government. Apart from Usman there were two British, three Rhodesians, an American, two Pakistanis and a South African. Their number varied. All had signed contracts and theoretically they were instructors. They were issued uniforms, but did not have to wear them. No discipline was imposed on them. There was a fairly rapid turnover: people who had simply had enough, or casualties. In the year since Usman had been there only one pilot had died while on a mission. Six others had died as a result of mechanical or navigational failures and subsequent crashes. "Your ground crew," Usman said phlegmatically, "is your greatest threat."

I had met Usman on the first provisioning run I had made from Grosso Arvore. I had arrived at the hotel earlier than expected and, hot and thirsty, had gone into the bar for a beer. The barroom was long and thin and was lined with simulated leather. The chairs and tables were modishly Scandinavian, the chairs organic looking, a warped kidney shape with splayed iron legs. The tables were like large paving stones, inlaid with shards of broken colored glass. It was very gloomy, and, because of the simulated leather walls, warm. The two ceiling fans were always switched to full blast. The blurred, whizzing propellers produced a stiff breeze that blew your hair about. I had never been in a bar like it and I grew oddly fond of its singular atmosphere.

When I went in that first afternoon the place was empty. Then I saw someone kneeling at the far end apparently searching for something on the floor. He looked up as I came in. He was wearing khaki trousers and a Hawaiian shirt which, for some reason, made me assume he was the barman.

"Good afternoon," he said. "I'm trying to catch a frog."

I waited while he did this. Then he brought it over to show me: a small, livid, lime-green tree frog, its throat pulsing uncontrollably.

"I'll have a beer," I said. "As soon as you're ready."

He pushed the frog out through some louvred glass windows at one end of the bar before going behind it and pouring me a glass of beer.

"How much?" I said.

"The house will pay."

Then he engaged me in conversation, in the time-honored bar-keep-to-client manner: "Where are you from?" "How long are you staying?" Fairly soon I began to suspect he might be a manager-he seemed far too forward and intelligent to be running a cocktail bar at the Airport Hotel. By the time he asked me to have dinner with him that night I realized I'd been had.

"You thought I was the barman," he said with some glee. "Admit it. I got you." He was very pleased with his subterfuge.

"Not for one second," I said. "I knew it as soon as you opened the bottle," I improvised. I pointed to the bent bottletop lying on the bar. "No barman in Africa would've left that there. He'd pocket it."

"Oh." He looked disappointed. "You sure about that?"