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.
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!
“If I have seen less far than others,” Galileo complained in irritation to Aurora, “it is because I was standing on the shoulders of dwarfs.”
She laughed out loud. “Don’t say that to anyone else.”
They flew over and through number theory, theory of equations, probability theory—which was ever so useful, and instantaneously true to experience as well. It was the way of the world, no doubt about it, the way of the world mathematicized; oh how he could have used that! And how broadly it could be applied!