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"You should not drink so much, perhaps," said Kepler mildly.

"Pah. Something has burst in my gut: look at that! He pointed with grim pride to a basin of bloodied urine on the floor by Kepler's feet. "Last night at Rosenberg 's my bladder was full for three hours, I could not leave the table for fear of seeming gross. You know what these occasions are. "

"No," said Kepler, "I do not."

Tycho scowled, and took a swig of ale. He looked at Kepler keenly for a moment. "Be careful of my family, they will try to hinder you. Watch Tengnagel, he is a fool, but ambitious. Protect my poor dwarf. " He paused. "Remember me, and all I have done for you. Do not let me seem to have lived in vain.

Kepler ascended laughing to his room. All he has done for me! Barbara was there before him, poking among his things. He edged around her to the table and plunged into his papers, mumbling.

"How is he?" she said.

"Eh? Who?"

"Who!"

"O, it's nothing. Too much wine."

She was silent for a moment, standing behind him with her arms folded, nursing enormities. "How can you," she said at last, "how can you be so… so… "

He turned to stare at her. "What. "

"Have you thought, have you, what will become of us when he dies?"

"Good God, woman! He was dining with his fine friends, and drank too much as always, and was too lazy to leave his chair to pee, and injured his bladder. He will be over it by tomorrow. Permit me to know enough of doctoring to recognise mortal illness when-"

"You recognise nothing!" shrieking a fine spray of spit in his face. "Are you alive at all, with your stars and your precious theories and your laws of this and that and and and…" Fat tears sprang from her eyes, her voice broke, and she fled the room.

Tycho failed rapidly. Within the week Kepler was summoned again to his chamber. It was crowded with family and pupils and court emissaries, poised and silent like a gathering in the gloom on the fringes of a dream. Tycho was enthroned in lamplight upon his high bed. The flesh hung in folds on his shrunken face, his eyes were vague. He held Kepler's hand. "Remember me. Do not let me seem to have lived in vain." Kepler could think of nothing to say, and grinned uncontrollably, nodding, nodding. Mistress Christine plucked at the stuff of her gown, looking about her dazedly as if trying to remember something. The dwarf, blotched with tears, made to scramble on to the bed but someone held him back. Kepler noticed for the first time that Elizabeth Brahe was pregnant. Tengnagel skulked at her shoulder. There was a commotion outside the door of the chamber and Felix burst in, spitting Italian over his shoulder at someone outside. He strode to the bed and, thrusting Kepler away, took the Dane's hand in his own. But the Dane was dead.

He was buried, after an utraquist service, in the Teynkirche in Prague. The house on the Hradcany had an air of pained surprise, as if a wing had suddenly and silently collapsed. One morning it was discovered that the Italian had departed, taking Jeppe with him, no one knew to where. Kepler considered going too; but where would he go? And then a message came from the palace informing him that he had been appointed to succeed the Dane as imperial mathematician.

* * *

Everyone said the Emperor Rudolph was harmless, if a little mad, yet when the moment had come at last for Kepler to meet him for the first time, a spasm of fright had crushed the astronomer's heart in its hot fist. That was ten months before the Dane's death. Kepler by then had been in Bohemia nearly a year, but Tycho's grand manner was impervious to hints. He only shrugged and began to hum when Kepler ventured that it was a long time to have held off from this introduction. "His majesty is… difficult."

They trundled up the Hradcany and turned in between the high walls leading to the gate. Everywhere about them lay the economy of snow: a great white and only the black ruts of the road, the no-colour wall. The sky was the colour of a hare's pelt. Their horse stumbled on packed ice, and a scolopendrine beggar scuttled forward and opened his mouth at them through the carriage window in speechless imprecation. On the wooden bridge before the gate they skated ponderously to a halt. The horse stamped and snorted, blowing cones of steam out of flared nostrils. Kepler put his head out at the window. The air was sharp as needles. The gateman, a fat fellow in furs, waddled forth from his box and spoke to the driver, then waved them on. Tycho flung him a coin.

"Ah, "said the Dane, "ah, I detest this country. "He fussed at the sheepskin wrap about his knees. They were in the palace gardens now. Black trees glided slowly past, bare limbs thrown up as if in stark astonishment at the cold. "Why did I ever leave Denmark?"

"Because.

"Well?" staring balefully, daring him. Kepler sighed.

"I do not know. Tell me."

Tycho transferred his gaze to the smoky air outside. "We Brahes have ever been ill-used by royals. My uncle Jörgen Brahe saved King Frederick from drowning in the Sund at Copenhagen, and died himself in the attempt, did you know that?" He did. It was an oft-told tale. The Dane was working himself up into a fine fit of indignation. "And yet that young brat Christian was bold enough to banish me from my island sanctuary, my fabulous Uraniborg, granted to me by royal charter when he was still a snot-nosed mewler on his nurse's knee-did you know that?"  he did, he did, and more. Tycho had ruled on Hveen like a despotic Turk, until even the mild King Christian could no longer countenance it. "Ah, Kepler, the perfidy of princes!" and glared at the palace advancing to meet them through the icy light of afternoon.

They were left to wait outside the chamber of the presence. There were others there before them, dim depressed figures given to sighing, and a crossing and recrossing of legs. It was bitterly cold, and Kepler's feet were numb. His apprehension had yielded before a grey weight of boredom when the groom of the chamber, an immaculately costumed bland little man, approached swiftly and whispered to the Dane, and already there was a hot constriction in Kepler's breast, as if his lungs, getting wind a fraction before he did of the advent at last of the longed-for and dreaded moment, had snatched a quick gulp of air to cushion the shock. He needed to urinate. I think I must go and-will you excuse-?

"Do you know, " said the Emperor, "do you know what one of our mathematici has told us: that if the digits of any double number be transposed, and the result of the transposition be subtracted from the original, or vice versa of course, depending on which is the greater value, then the remainder in all instances shall be divisible by nine. Is this not a wonderful operation? By nine, always." He was a short plump matronly man with melancholy eyes. A large chin nestled like a pigeon in a bit of soft beard. His manner was a blend of eagerness and weary detachment. "But doubtless you, sir, a mathematician yourself, will think it nothing remarkable that numbers should behave in what to us is a strange and marvellous fashion?"

Kepler was busy transposing and subtracting in his head. Was this perhaps a test to which all paying court for the first time were subjected? The Emperor, slack-jawed and softly panting, watched him with an unnerving avidity. He felt as if he were being slowly and ruminatively devoured. "A mathematician, I am that, your majesty, yes," smiling tentatively. "Nevertheless I admit that I cannot say what is the explanation of this phenomenon…" He was discussing mathematics with the ruler of the Holy Roman Empire, the anointed of God and bearer of the crown of Charlemagne. "Perhaps your majesty himself can offer a solution?"

Rudolph shook his head. For a moment he mused in silence, a forefinger palping his lower lip. Then he sighed.

"There is a magic in numbers," he said, "which is beyond rational explanation. You are aware of this, no doubt, in your own work? May be, even, you put to use sometimes this magic?"