But this was not what he wanted and so he began to avoid the funerals after that, and stayed in the quiet stone-garden of Bedlam.

Hooke was there, too, for Gresham’s College had become too crowded with scheming fops. Bedlam was years away from being done. The masons hadn’t even begun work on the wings. But the middle part was built, and on top of it was a round turret with windows on all sides, where Hooke liked to retreat and work, because it was lonely and the light was excellent. Daniel for his part stayed down below, and only went out into the city to meet with Leibniz.

DOCTORGOTTFRIEDWILHELMLEIBNIZpicked up the coffee-pot and tipped it into his cup for the third time, and for the third time nothing came out of it. It had been empty for half an hour. He made a little sigh of regret, and then reluctantly stood up. “I beg your pardon, but I begin a long journey tomorrow. First the Channel crossing-then, between Calais and Paris, we shall have to dodge French regiments, straggling home, abject, starving, and deranged.”

Daniel insisted on paying the bill, and then followed the Doctor out the door. They began strolling in the direction of the inn where Leibniz had been staying. They were not far from the ‘Change. Paving-stones and charred firebrands still littered the unpaved streets.

“Not much divine harmony in evidence, here in London,” Daniel said. “I can only hang my head in shame, as an Englishman.”

“If you and France had conquered the Dutch Republic, you would have much more to be ashamed of,” Leibniz returned.

“When, God willing, you get back to Paris, you can say that your mission was a success: there is no war.”

“It was a failure,” Leibniz said, “we did not prevent the war.”

“When you came to London, Doctor, you said that your philosophick endeavours were nothing more than a cover for diplomacy. But I suspect that it was the other way round.”

“My philosophick endeavours were a failure, too,” Leibniz said.

“You have gained one adherent…”

“Yes. Oldenburg pesters me every day to complete the Arithmetickal Engine.”

“Make that two adherents, then, Doctor.”

Leibniz actually stopped in his tracks and turned to examine Daniel’s face, to see if he was jesting. “I am honored, sir,” he said, “but I would prefer to think of you not as an adherent but as a friend. ”

“Then the honor is all mine.”

They linked arms and walked in silence for a while.

“Paris!” Leibniz said, as if it were the only thing that could get him through the next few days. “When I get back to the Bibliotheque du Roi, I will turn all of my efforts to mathematics.”

“You don’t want to complete the Arithmetickal Engine?”

It was the first time Daniel had ever seen the Doctor show annoyance. “I am a philosopher, not a watchmaker. The philosophickal problems associated with the Arithmetickal Engine have already been solved… I have found my way out of that labyrinth.”

“That reminds me of something you said on your first day in London, Doctor. You mentioned that the question of free will versus predestination is one of the two great labyrinths into which the mind is drawn. What, pray tell, is the other?”

“The other is the composition of the continuum, or: what is space? Euclid assures us that we can divide any distance in half, and then subdivide each of them into smaller halves, and so on, ad infinitum. Easy to say, but difficult to understand…”

“It is more difficult for metaphysicians than for mathematicians, I think,” Daniel said. “As in so many other fields, modern mathematics has given us tools to work with things that are infinitely small, or infinitely large.”

“Perhaps I am too much of a metaphysician, then,” Leibniz said. “I take it, sir, that you are referring to the techniques of infinite sequences and series?”

“Just so, Doctor. But as usual, you are overly modest. You have already demonstrated, before the Royal Society, that you know as much of those techniques as any man alive.”

“But to me, they do not resolve our confusion, so much as give us a way to think about how confused we are. For example-”

Leibniz gravitated toward a sputtering lamp dangling from the overhanging corner of a building. The City of London’s new program to light the streets at night had suffered from the fact that the country was out of money. But in this riotous part of town, where (in the view of Sir Roger L’Estrange, anyway) any shadow might hide a conspiracy of Dissidents, it had been judged worthwhile to spend a bit of whale-oil on street-lamps.

Leibniz fetched a stick from a pile of debris that had been a goldsmith’s shop a week earlier, and stepped into the circle of brown light cast on the dirt by the lamp, and scratched out the first few terms of a series:

“If you sum this series, it will slowly converge on pi. So we have a way to approach the value of pi-to reach toward it, but never to grasp it… much as the human mind can approach divine things, and gain an imperfect knowledge of them, but never look God in the face.”

“It is not necessarily true that infinite series must be some sort of concession to the unknowable, Doctor… they can clarify, too! My friend Isaac Newton has done wizardly things with them. He has learned to approximate any curve as an infinite series.”

Daniel took the stick from Leibniz, then swept out a curve in the dirt. “Far from detracting from his knowledge, this has extended his grasp, by giving him a way to calculate the tangent to a curve at any point.” He carved a straight line above the curve, grazing it at one point.

A black coach rattled up the street, its four horses driven onwards by the coachman’s whip, but veering nervously around piles of debris. Daniel and Leibniz backed into a doorway to let it pass; its wheels exploded a puddle and turned Leibniz’s glyphs and Daniel’s curves into a system of strange canals, and eventually washed them away.

“Would that some of our work last longer than that, ” Daniel said ruefully. Leibniz laughed-for a moment-then walked silently for a hundred yards or so.

“I thought Newton only did Alchemy,” Leibniz said.

“From time to time, Oldenburg or Comstock or I cajole him into writing out some of his mathematical work.”

“Perhaps I need more cajoling,” Leibniz said.

“Huygens can cajole you, when you get back.”

Leibniz shrugged violently, as if Huygens were sitting astride his neck, and needed to be got rid of. “He has tutored me well, to this point. But if all he can do is give me problems that have already been solved by some Englishman, it must mean that he knows no more mathematics than I do.”

“And Oldenburg is cajoling you-but to do the wrong thing.”

“I shall endeavour to have an Arithmetickal Engine built in Paris, to satisfy Oldenburg,” Leibniz sighed. “It is a worthy project, but for now it is a project for a mechanic.”

They came into the light of another street-lamp. Daniel took advantage of it to look at his companion’s face, and gauge his mood. Leibniz looked a good deal more resolute than he had beneath the previous street-lamp. “It is childish of me to expect older men to tell me what to do,” the Doctor said. “No one told me to think about free will versus predestination. I plunged into the middle of the labyrinth, and became thoroughly lost, and then had no choice but to think my way out of it.”

“The second labyrinth awaits you,” Daniel reminded him.

“Yes… it is time for me to plunge into it. Henceforth, that is my only purpose. The next time you see me, Daniel, I will be a mathematician second to none.”

From any other Continental lawyer these words would have been laughably arrogant; but they had come from the mouth of the monster.