“I remember.”
“I agreed with him for a short time-but it was wishful thinking. I am seeking it in geometry now-or rather I am seeking it where geometry fails.”
“Fails?”
“Come upstairs with me Daniel.”
DANIEL RECOGNIZED THEfirst proof as easily as his own signature. “Objects governed by a centripetal force conserve angular momentum and sweep out equal areas in equal times.”
“You have read my De Motu Corporum in Gyrum?”
“Mr. Halley has made its contents known to the Royal Society,” Daniel said drily.
“Some supporting lemmas come out of this,” Isaac said, pulling another diagram over the first.
“and thence we can move on immediately to”
“That is the great one,” Daniel said. “If the centripetal force is governed by an inverse-square law, then the body moves on an ellipse, or at any rate a conic section.”
“I would say, ‘That heavenly bodies do move on conic sections proves the inverse-square law.’ But we are only speaking of fictions, so far. These proofs only apply to infinitesimal concentrations of mass, which do not exist in the real world. Real heavenly bodies possess geometry-they comprise a vast number of tiny particles arranged in the shape of a sphere. If Universal Gravitation exists, then each of the motes that make up the Earth attracts every other, and attracts the moon as well, and vice versa. And each of the moon’s particles attracts the water in Earth’s oceans to create tides. But how does the spherical geometry of a planet inform its gravity?”
Isaac produced another sheet, much newer-looking than the rest.
Daniel did not recognize this one. At first he thought it was a diagram of an eyeball, like the ones Isaac had made as a student. But Isaac was speaking of planets, not eyes.
A few awkward moments followed.
“Isaac,” Daniel finally said, “ youcan draw a diagram like this one and say, ‘Behold!’ and the proof is finished. I require a bit of explanation.”
“Very well.” Isaac pointed to the circle in the middle of the diagram. “Consider a spherical body-actually an aggregate of countless particles, each of which produces gravitational attraction according to an inverse square law.” Now he reached for the nearest handy object-an inkwell-and set it on the corner of the page, as far away from the “spherical body” as it could go. “What is felt by a satellite, here, on the outside, if the separate attractions of all of those particles are summed and fused into one aggregate force?”
“Far be it from me to tell you how to do physics, Isaac, but this strikes me as an ideal problem for the integral calculus-so why are you solving it geometrically?”
“Why not?”
“Is it because Solomon didn’t have the calculus?”
“The calculus, as some call it, is a harsh method. I prefer to develop my proofs in a more geometric way.”
“Because geometry is ancient, and everything ancient is good.”
“This is idle talk. The result, as anyone can see from contemplating my diagram, is that a spherical body-a planet, moon, or star-having a given quantity of matter, produces a gravitational attraction that is the same as if all of its matter were concentrated into a single geometric point at the center.”
“The same? You mean exactly the same?”
“It is a geometrical proof,” Isaac merely said. “That the particles are spread out into a sphere doesn’t make any difference because the geometry of the sphere is what it is. The gravity is the same.”
Daniel now had to locate a chair; all the blood in his legs seemed to be rushing into his brain.
“If that is true,” he said, “then everything you proved before about point objects-for example that they move along conic section trajectories-”
“Applies without alteration to spherical bodies.”
“To real things. ” Daniel had a queer vision just then of a shattered Temple reconstituting itself: fallen columns rising up from the rubble, and the rubble re-aggregating itself into cherubim and seraphim, a fire sparking on the central altar. “You’ve done it then… created the System of the World.”
“ Godcreated it. I have only found it. Rediscovered what was forgot. Look at this diagram, Daniel. It is all here, it is Truth made manifest, epiphanes. ”
“Now you said before that you were looking for God where Geometry failed.”
“Of course. There is no choice in this,” Isaac said, patting his diagram with a dry hand. “Not even God could have made the world otherwise. The only God here-” Isaac slammed the page hard “-is the God of Spinoza, a God that is everything and therefore nothing.”
“But it seems as if you’ve explained everything.”
“I’ve not explained the inverse square law.”
“You’ve a proof right there saying that if gravity follows an inverse square law, satellites move on conic sections.”
“And Flamsteed says that they do,” Isaac said, yanking the sheaf of notes out of Daniel’s hip pocket. Ignoring the cover letter, he tore the ribbon from the bundle and began to scan the pages. “Therefore gravity does indeed follow an inverse square law. But we may only say so because it is consistent with Flamsteed’s observations. If tonight Flamsteed notices a comet moving in a spiral, it shows that all my work is wrong.”
“You’re saying, why do we need Flamsteed at all?”
“I’m saying that the fact that we do need him proves that God is making choices.”
“Or has made them.”
This caused a sort of queasy sneer to come across Isaac’s face. He closed his eyes and shook his head. “I am not one of those who believes that God made the world and walked away from it, that He has no further choices to make, no ongoing presence in the world. I believe that He is everywhere, making choices all the time.”
“But only because there are certain things you have not explained yet with geometric proofs.”
“As I told you, I seek God where Geometry fails.”
“But perhaps there is an undiscovered proof for the inverse square law. Perhaps it has something to do with vortices in the ?ther.”
“No one has been able to make sense of vortices.”
“Some interaction of microscopic particles, then?”
“Particles traversing the distance from the Sun to Saturn and back, at infinite speed, without being hindered by the ?ther?”
“You’re right, it is impossible to take seriously. What is your hypothesis, Isaac?”
“ Hypothesis non fingo.”
“But that’s not really true. You begin with a hypothesis-I saw several of them scratched in the gravel out there. Then you come up with one of these diagrams. I cannot explain how you do that part, unless God is using you as a conduit. When you are finished, it is no longer a hypothesis but a demonstrated truth.”
“Geometry can never explain gravity.”
“Calculus then?”
“The calculus is just a convenience, a short-hand way of doing geometry.”
“So what is beyond geometry is also beyond calculus.”
“Of course, by definition.”
“The inner workings of gravity, you seem to be saying, are beyond the grasp, or even the reach, of Natural Philosophy. To whom should we appeal, then? Metaphysicians? Theologians? Sorcerers?”
“They are all the same to me,” Isaac said, “and I am one.”
IT WAS AS IFWILLIAMof Orange had searched the world over to find the place most different from Versailles, and had told Eliza to meet him there. At Versailles, everything had been designed and made by men. But here was nothing to see but ocean and sand. Every grain of sand had been put where it was by waves that formed up in the ocean according to occult laws that might have been understood by the Doctor, but not by Eliza.