They continued beyond the silver building for about an hour, he reckoned, passing crowded plazas extending to left and right, some of them open to the black sky, most roofed by ice. As the hour passed, he learned better how to walk in the light downward pull. This strange lightness was suggestive of all kinds of things, including the idea that weight was perhaps proportional to the size of the planet one stood on. Another sign that Europa must be fairly small.
“Where are you taking me?” he said.
“To a person who may be able to answer your questions. Or maybe you would call it a machine.”
“A machine? So none of you know?”
“No no, this person is a kind of … composite. A person quite like you, in fact—a physicist and mathematician, quite famous.”
“Good,” Galileo said. “I want some explanations.”
They came to a lake, and stepped down into a long low boat, like a gondola. When they were settled in its bow, a boatman cast off and they hummed slowly over clear blue water, leaving a wake that ran in a clean curl that was slower than it would have been on the lagoon. Greenish blues pulsed overhead and in waves around them, and Galileo could not tell how deep the lake might be, as the many subtle shades of creamy blue bobbed darker and lighter, but always opaque. Royal blue, sky blue, azure, turquoise, aquamarine—all these bounced against each other in long bands, and it also seemed that waves of cobalt were passing through the other blues, staining them as it pulsed by, as if they boated through the veins of a beating blue heart. The buildings behind the broad fondamenta to their left looked like clean blocks of ice, painted in pastels that held their color manfully even in the omnipresent green-blue glow, contradicting what Galileo thought he knew of color theory. The sight of one curving row of waterfront buildings reminded Galileo strongly of the Grand Canal, and he saw the city was a kind of Venice carved in ice. “Why doesn’t it melt?”
“It’s all cladded. Sheathed in diamond, in fact.”
People promenaded on the fondamenta just as they would have at home. Some of them looked out on the water, but not at Ganymede and Galileo; theirs was only one watercraft among many. All the wakes on the water created a fine curvilinear slow-motion cross-chop. The ice ceiling overhead was thicker in some places than in others, judging by the differences in the green-blue. Pulses most definitely were running through it.
“What are those waves of color running through the roof?” he asked.
“The other moons exert tidal forces against the tug of Jupiter proper. We shine a type of light through the ice to reveal the stresses in it, so we can see these tides’ interactions.”
“How do you keep these canals and lakes liquid?”
“We heat them,” Ganymede said patiently. “In places you will see steam. In other places we will break through a skim of ice as we progress along certain canals.”
“But you don’t know how the water is heated, do you?”
“That is not one of the more difficult accomplishments of our technology believe me.”
Their boat hummed up to a fondamenta made of something like black stone. As they climbed out of the boat, Galileo asked, “Where do you get rock?”
“From meteorites, called here dropstones. One or two big ones will supply enough material for an entire city, as it just supplements the local ice.”
“How many people live in this Venice of yours?”
“This is Rhadamanthys Linea. About a million people.”
“That many! And how many cities like this are there on Europa?”
“Maybe a hundred.”
“A hundred millions!”
“It’s a big moon, as you know.”
Overhead the broad crossing arcs of cobalt and violet pulsed from before them to behind them. Galileo said, “The patterns of light are so complicated, it seems there must be more than four influences.”
“All the Jovian moons pull a bit on the rest.”
“But are there more than four moons?”
“There are about ninety.”
“Ninety?”
“Most are very small. Some are out of the plane of the rest. In any case they all have a pull, no matter how slight, and with the ice overhead charged as the locals have charged it, every change in tug registers piezoelectrically.”
“Why do they charge it that way?”
Ganymede shrugged. “They like the way it looks.”
They were now walking down a broad crowded street flanked by long low buildings. Low carts moved at a running pace, without anything pulling them. Before them a cluster of very tall angular buildings reached right up to the ice ceiling.
“It must be the Tower of Babel,” Galileo said.
“Well, there is a great deal of confusion inside it, to be sure. And people who want it to fall.”
Soon they reached these tall buildings, and outside one they entered a glass antechamber, which then rose on the outside wall so fast that Galileo’s ears popped, surprising him. He always had a small earache in his right ear, and now it throbbed unhappily. So it seemed that in some sense his body was here too. “If I am here, how am I also back in Italy, lost in one of my syncopes?”
“You are here in a complementary potentiality.”
The glass antechamber stopped and a door opened on its inner side. They stepped out on a smooth broad roof terrace the color of malachite, just under the ice ceiling. Ganymede led Galileo to a small group of people congregated against a railing that overlooked the city. From here Galileo could see far down the canal; it developed a mirror surface in just the place a waterblink would have appeared on Earth, about halfway to the horizon. From there on it looked like a silver road through undulant blue buildings. Venice had looked just so on certain moony nights, and again Galileo wondered if he were dreaming.
Ganymede said, “This is Galileo Galilei, the first scientist, here in a proleptic entanglement.”
“Ah yes,” said a tall old woman at the center of the group. “We heard you were coming. Welcome to Rhadamanthys.”
Though old she was still straight, and stood a head taller than Galileo. Pendulant silver earrings emerged directly from her ear holes and then curved and seemed to dive into her neck. He bowed to her briefly, looked to his guide, muttered, “And where is the mathematician?”
Ganymede indicated the old woman. “This is she. Aurora.”
Galileo tried to conceal his surprise. “I thought you said it was a machine,” he said to cover himself.
“That’s partly true,” the willowy crone said. “I am interfaced to various artifactual entities.”
Galileo kept a straight face, although the idea struck him as monstrous, like jamming one of his military compasses through an ear into one’s brain. And in fact there were those earrings.
“Come with me,” Aurora said, taking him by the arm and moving him down the altana railing a short distance. Low creaks and hums that seemed to come from the ceiling kept them from being able to hear the other conversations on the terrace.
“It’s a pleasure to meet you,” the ancient woman said politely. She had a voice like Ganymede’s, hoarse and croaky, and her Latin had the same odd accent. “You are often called the first scientist.”
“That would be an honor, but I was not the first.”
“I agree with you. But you were the first mathematical experimentalist.”
“Was I?”
“So it seems from what we read in history, and see in the entanglements. One must always make assumptions, of course. And the past is always changing. But as far as we can tell, you tried only to assert what you could demonstrate and describe mathematically. This is science. Wasn’t it you who said that? That the world is written in mathematics?”
“I like that,” Galileo admitted. “If it’s true.”
“It’s partly true.” Although she looked troubled. “Reality is mathematical, as long as you understand that uncertainty and contingency can be mathematically described, without them becoming any more certain.”
“Teach me,” Galileo said. “Teach me how you breathe here, and what these tides of color are, and—teach me everything. I want to know everything! Teach me everything you have learned since my time.”
She smiled, pleased by his effrontery. “That would take a while.”
“I don’t care!”
She glanced at him curiously. “It would take years, even for one of your intelligence.”
“Can’t you do it quickly? Give me the short version?”
“The short version doesn’t give you real understanding. It’s only a matter of metaphors, images that don’t really convey the situation. The mathematics is what you want, and that took a great number of people many centuries to develop. Now no one learns more than a small percentage of what there is, and even that takes many years.”
“Maybe not for me!”
“Even for you.”
Galileo shook his head. “I don’t want to take years. I don’t have years.”
Aurora seemed to consult the patterns of intersecting waves in their low ice sky. She said, “There is a drug complex we can give you that would enable you to learn faster. A synaptic velocinestic, it is called, made of a particular mixture of brain chemicals. With the help of it, one can accomplish a certain forcing. Networks bloom in the brain extremely rapidly. It’s useful in certain situations.”
“An alchemical preparation?”
“Yes, if you like.”
“Is it safe?” Thinking of the half-crazed alchemists he had met, pursuing something like witchcraft in their foul workshops, poisoned by their own hand.
“Yes, we think so. It is mildly carcinogenic, but it won’t kill you. Although some people have felt distressed afterward, I’ve heard. But I have taken it, and felt no such thing.”
This, from a machine mind. Galileo could not stop a snort from emerging, though he curbed his tongue. After considering it briefly, he said, “Give me this preparation of yours. And then who will teach me the mathematics? You?”
She gave him an amused look. “One of our machines.”
“Another machine?”
“It’s a standard curriculum, designed for use with the velocinestic. It will be faster than I could be, and clearer too. I will oversee the process.”
“Do it then. I want to know!”
Her people gave him a tight-fitting helmet, made of a mesh of metals in a dense weave. They insisted he sit down, and got him settled into what looked like a small throne tilted onto its back.
Recumbent in it, he stared at the ice ceiling. It was pulsing rapidly in dense interference patterns, waves from three directions tossing off brief glints of sapphire iridescence. These triple peaks formed their own moving pattern, like sunlight on windblown water. Even if there had been just the four big moons, the Galilean moons (such a good name), their tugging would of course create a very complex pattern. He had been so sure that the tides on Earth were the result of the ocean sloshing around in its basins of stone, shifting as the Earth both rotated and flew around the sun, creating differential speeds. But here they said it was not true. In that case, what caused tides? The tug of celestial bodies—but that was astrology all over again. And yet they seemed to be saying it was so. Was astrology right, then, with its celestial influences and its action at a distance, action without any mechanical forces applied? He hated such nonexplaining explanations!
And yet here they were. He looked at Aurora’s assistants, hovering over the bank of machines against the wall. He hoped the treatment would work, that it would not kill or derange him.
They slipped their preparation into his blood using a hollow needle that they inserted painlessly into his skin—an ugly little experience. He held his breath as they did this, and when he finally exhaled and inhaled, the world ballooned. He saw immediately that he was thinking several trains of thought at once, and they all meshed in a contrapuntal fugue that his father would have very much enjoyed hearing, if it were music, which in a sense it seemed to be: a polyphonic singing of his ideas, each strand taking its part in the larger music. To a certain extent his thinking had always felt that way, with any number of accompaniments running under the aria of the voice of thought. Now these descants were choral, and loud, while at the same time architectonically fitted to the melody. He could think six or ten thoughts at once, and at the same time think about his thinking, and contemplate the whole score.
There remained a main melody, or a path through a maze—a maze that was like the delta of the Po. He seemed to look down on it as he sang it. A great number of channels were weaving down a slightly tilted plain. Each channel was a mathematical specialty—some of them shallow and disappearing into the sand, but most making their loop and reconnecting to other flows. A few were the kind of deep channels that ships would use. Upstream they coalesced until there were fewer, scattered streams. Fewer tributaries rather than more, leading up in different directions to sources, often at springs. Water out of the rock.
This was, he saw, an image of mathematics in time. Or maybe it was all time, or humanity in time; but it was the mathematics that sprang out at him.
The fewer channels upstream, in the distant past, well before his time, were where Aurora’s tutorial now led him. Then he was flying over the time stream, or in it, sometimes returning upstream to view a contemporaneous discipline. Mainly he had a general sense of flying downstream, over or occasionally inside some eternal landscape, the nature of which could not be discerned. He inhabited an image he had heard some time before, of history as a river, in which people were water, eroding the banks and depositing soil elsewhere downstream, so that the banks slowly changed and the river ran otherwise than it had, without the water ever noticing the changed courses of the braiding stream.
He tried to turn all the mathematics into geometry, so that he could see it and thereby grasp it. It often worked. It was definitely true what Aurora had said about the preparation. He grasped things he saw the moment he saw them; aspects even leaped out to him in advance as implications, shooting out before him like arrows. He was both in and out at once, back and forth, up and down, ranging widely, flying in stoops and gyres, and always looking forward with an eagle’s eye. The voice of the machine tutor was Aurora’s own hoarse voice, and Aurora herself flew beside him or in him, and sometimes she spoke too in her odd Latin, so that it seemed there were two of her talking. Sometimes Galileo asked questions and all three of them spoke at once and yet he could follow all three lines of thought, which merged in his mind into music, into a trio for lute and two squawky fagatto.
He was shown glimpses of people and places, but always the main thrust of the tutorial was mathematical. He recognized Euclid and Pythagoras, and for a short but incredibly packed moment he was actually with his hero Archimedes, still crucial to the story, hurrah! The Greek’s entire life bloomed in him at once, an island or bubble in the flow of the stream, and for a moment he knew it completely—and thought he saw Ganymede too standing there, and the burning mirror—also the Roman soldier at the terrible end—
Startled, for this was not like the rest of the lesson, he jerked up in his flight, feeling like a crow frightened out of a tree. Then he recognized Regiomontanus, and all that that brilliant man had rescued from the Greeks by way of the Arab texts, and was distracted that way. Then on to Harriot with his algebraic symbols, which Galileo had known would be useful the very first time Castelli showed them to him. Then Copernicus and his system, and Kepler and his polyhedraic formula for planetary distance, which Galileo had not thought was correct, and indeed it was not.
His own sense that all things moved naturally in circles was also shattered, however, as he was introduced to inertia—but that idea had always been on the tip of his tongue, indeed he had said it in slightly different words, as he cried out when he saw it. And then to the law of gravity—Newton’s equation for it caused him to soar up, startled; such a simple deep thing! He had seen the evidence for the laws of both inertia and gravity, he had used them in his parabolic des
cription of falling bodies, but he had not understood what he had used, and now he floated above them, abashed, glowing before their utter simplicity. The force of gravity was simply an inverse power law, easy as kiss your hand, and resulting in obvious solutions to things like Kepler’s orbits, which Kepler had only groped his way to after years of observation and analysis.
So planetary orbits were naturally ellipses, with the sun occupying the major focus, and the other gravitational pulls together locating the minor focus. Of course! Too bad he had never read far enough in Kepler’s crazy tomes to get to these observations; it might have alerted him to the absence of circularity in the heavens—though he might have concluded they were just circles distorted by something he didn’t see. Certainly any idea one had in mind altered what one could see. And yet still, despite his ideas against it, here was attraction and influence at a distance again, without a mechanical force or cause! It was a mystery. It could not be the whole story, could it?
He was not aware he had asked this aloud, but heard Aurora reply: “This is the question that keeps coming up, as you will see. You are by no means the first or the last to dislike what one of us called spooky action at a distance.”
“Well, of course. Who could like that?”
“And yet as you will also come to see, such action is simply everywhere. You will find that there are serious problems with any simple concept of distance. Eventually distance becomes as problematic as time.”
“I don’t understand.”
But already she and her machine voice had flown off to analytic geometry, and then to a form of analyzing motion called the calculus, which was just what he had always needed and never had. And it seemed to have appeared just after his time, worked out by people young when he was old: an irritating Frenchman called Descartes, a German named Leibniz, and the English maniac Newton again, who to Galileo’s chagrin had distilled Galileo’s dynamics in just the way Galileo had struggled to do all his life. So simple when you saw it!