“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 description 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.