Into the New Millennium: Trailblazing Tales From Analog Science Fiction and Fact, 2000 - 2010 (14 page)

And nearly drops them both. The image now appears large and close, as if the building has toppled in through his very window! Startled, he checks with his own eyes and is gratified to see the structure still in its proper place. The image, then, is mere illusion. He sets the glasses aside and reminds himself to take them to the optician for repair.

Albrecht and Nicole desire to measure the time taken by a falling body to traverse successive distances and, having no notion how this might be accomplished, have decided to consult one who works with time for a living. "A mere artisan," grumbles Nicole as they search out the street of the clockmakers.

"And we have need of his art," Albrecht answers cheerfully. He asks of a man passing in the other direction the name of the most skilled horologist and receives in return the name of Fernand of Quoeux, "the third shop on the right."

Albrecht thanks him, and he and Nicole resume a previous argument.

"If the body rolls down a
ramp
, as Philoponus used," Albrecht says, "then the motion is constrained and not natural. It does not fall free, hence no
gravitas in decendendo
. Why can't we just drop two balls of different weight from the balcony?"

"Because," Nicole tells him again, "the Englishman desires to
measure
the distance fallen and the time taken, and we cannot do that with free fall. It passes too quickly. We must retard the bodies' motions by rolling them down inclines. The Master is confident that the relationship will remain apparent even if the actual velocities are less. Have you never read
On the theory of weights
? Jordanus wrote it when he was old, and corrected all the mistakes in his earlier book. He solved the problem of motion on a ramp, something which the ancients never did. We need only eliminate resistance from the material of the ramp. Grease, perhaps."

Grease . . . He is still wondering why that sufflator moved backward. It seems to him a more interesting puzzle than Albrecht's. Could the method of contrived experience determine
that
question? First, he must repeat the experience to learn whether a common course of nature obtains. Then . . . Then, what?

Albrecht is puzzled by his companion's unwonted silence; but, grateful for the respite, he does not interrupt the young man's thoughts until they have arrived at the clockmaker's shop.

Fernand of Quoeux owns a broad face with thick, pendulous lips and a basset hound's liquid eyes. His hair is white and sparse, save on his lip, where it flourishes. He is engaged in close discussion with Georges the carpenter, who owns a shop on the next street. Nicole is delighted to discover in them two fellow Normans, and they fall into a discussion in which "pockets" replace "sacks" and "hardelles" replace "filles" and they agree that the weather is "muggy" and not, as the French would say, "humid." Albrecht finally interrupts the reunion and explains to the clockmaker what they want and why.

The carpenter has stayed to listen. "Time and distance of a falling body?" he laughs. "Of what use is that to know?"

Nicole and Albrecht are struck dumb. They had never thought of using it for anything. Finally, Nicole says, "So you know how much time you have to get out from under." He does not add "fool" as an address, but the carpenter hears it anyway. He swells like a banty cock, but the clockmaker places a hand on his shoulder to quiet him. Should an argument breaks out, he might lose the work!

"If you would build this inclined plane they want, Georges," he tells his friend, "the payment would be a practical result."

"Pfui! A simple task, mere apprentice work. Not worth my time. How big would you want it?"

Nicole has been thinking about this very matter. "Fifteen or twenty shoes tall. And both the slope and the height of the incline must be adjustable." Albrecht lifts an eyebrow, but says nothing.

Carpenter and clockmaker look at each other, adding pounds and pennies with an arithmetic skill that would shame the Calculators of Merton. Finally, Fernand asks, with a skeptical study of their robes—for he has been cozened by scholars before—how they would pay for the labor. When they tell them that my sir the Rector will pay for all, their eyes light up and the price is adjusted upward by a compounding of fractions.

Days pass while Buridan awaits the clock, and the ramp grows in the courtyard behind his office. The carpenter and his boys hammer away before an ever-shifting audience of curious scholars, who propose an ever shifting cloud of speculation over its intended use.

One day, while Heytesbury, Albrecht, and Nicole wait in the Rector's office for Buridan's appearance, the Englishman explains Grosseteste's view of
uniformly
difform motion. He paces as is his wont—a peripatetic scholar, indeed!—and waves his arms the while. Nicole whispers to Albrecht that the best way to silence the Englishman would be to hold his arms to his sides; and he is prepared to compare the arm motions to those of a bellows, which also forces air out of an orifice, when Heytesbury mentions the Mean Speed Theorem: "Whether a latitude of velocity commences from some finite degree," he explains, "every latitude, so long as it terminates at some finite degree, and so long as latitudes are gained or lost uniformly during some assigned period of time, will traverse a distance exactly equal to what it would traverse in an equal period of time if it were moved uniformly at its mean degree of velocity."

Albrecht is rendered mute as he tries to pierce through the manifest grammar to the occulted mathematics. "So, a body in uniformly difform motion," he ventures, "would the same distance cover as it would have covered had it possessed simple uniform motion at the mean speed." Heytesbury nods happily, and Albrecht wonders why this Grosseteste could not have stated it in that manner.

Nicole has been sketching on a scrap of paper. He represents the passage of time with longitude, drawing a line from left to right. The successive latitudes of velocity acquired, he represents by perpendicular lines drawn upward. If increments of velocity are attained uniformly, each line's extension is successively longer by the same amount. He sees that the lines approximate a right triangle. But if the passage of time is a continuum, as even the ancients recognized, he may replace his procession of sticks with the simple triangular figure. The height of the triangle signifies the final form of velocity attained.

He draws a horizontal line whose latitude is half the height of the triangle.

And gasps.

The Englishman is correct in every detail! A few theorems of Euclid and the conclusion follows!

The others come to his side and study what he has drawn. Nicole stammers something about "equal triangles" and "similar sides" but the gist of it is that the area of the triangle ABC produced by uniformly difform motion has precisely the same area as the rectangle ABGF produced by simple uniform motion.

"Then the
area
enclosed by the figure somehow signifies the distance traversed," Heytesbury cries. He stands suddenly erect and turns his head to look off into the distance. "If distance is to time as area is to length . . . Hah! Area is the doubling of length, so distance must be proportional to the double of time! No, by His wounds!" he swears and flies to Buridan's desk, where he seizes the quill and scratches away on palimpsest. "The area of the triangle is half the length of the base doubled by the height, so distance must be proportional to
half
the doubling of time!" He turns in the chair and stares wide-eyed at the two students. "It remains only to discover the constant of proportionality!"

Nicole's mouth drops open. He is mesmerized by an image of geometry and arithmetic blending into a harmonious whole. A unified mathematics! "We could say ‘square' the ratio instead of ‘double' it," he ventures, applying geometric language to arithmetic.

"Indeed we could," Heytesbury agrees. "But what would we say if we were to compound the ratio
in quarto
? Or reduce it by three-fourths? But . . ."

But whatever thoughts engage his mind are forgotten when Buridan storms into the room. Under his arm, he bears a new copy of the Philoponus; while under his considerable nose, he wears drooping, down-turned lips. As the lips, too, are considerable, the overall effect is one of grave displeasure. "What is it, John?" Heytesbury asks in concern.

The Philoponus thuds to the desk; the Rector, to his seat. "That little Greek catamite," he exclaims, "discovered the impetus! He stole my idea!"

"He's dead, John," Heytesbury informs him. "Long ago."

"I know," The Rector sighs, "but it was a humbling experience, once I realized what Philoponus meant by the ‘carrier.' You could have told me!"

Heytesbury spreads his hands. "I didn't realize. I'm a logician, not a physicist."

"He wrote that his ‘
carrier
' was proportional to the weight of the body, so that a bullet, a cork, and a feather, thrown with the same force, will travel different distances. Once I read that . . . Holy blue! Do you know what most disturbs me?"

"What?"

Buridan raps the book with his knuckles. "That the book was so long lost! Think, William! How far might the philosophy of nature have come had we known of the impetus since Cremona's day?" He shrugs from his elbows in a gesture so Gallic that Albrecht nearly laughs aloud.

"But Master," says Nicole, "surely the Philoponus might have been read by many, but never understood until a mind sufficiently supple considered it!"

Buridan's laugh is froglike. "Well, William, you see that my students learn the most important lesson of all!"

"Permit me to inspect your lips," Albrecht tells Nicole later as they walk the streets to their quarters.

"What? Why? Are they soiled?" The Norman wipes them with a kerchief.

"No," says Albrecht. "I thought they might have turned brown from kissing our Master's arse."

Nicole shoves the laughing Saxon, with no more effect than if he had shoved a tree. Rather it is the Norman who staggers backward a few steps.

"What make you of the Englishman's notion?" asks Albrecht. "That the traversed distance is proportional to half the weight of the body and the doubling—I mean, the ‘squaring'—of the elapsed time." He cocks his head, his gaze on some unseen world. "If the body be uniform and the space a void. But would it be true in a plenum and for a heterogeneous body? Suppose we drop two bodies in water? One may fall more slowly than the other depending on its weight . . . A stone and a cork . . ."

Stone and cork? Nicole grabs him suddenly by the sleeve. "Wait!" And the two come to a halt in the crowded street, earning curses from carters and housewives who find their way suddenly blocked. "No, not the weight . . . Think latitude of forms! A
uniformly
difform distribution of forms must result from a single agent. Apply heat to one end of a rod, and the heat in the rod will be difform—the near end hotter, the far end cooler, just as light dims difformly from its source!"

"And . . ." asks the senior, who has grown conscious of the milling stream of humanity parting angrily around them. He gently presses his companion to the side of the street.

"And . . ." Nicole stammers, "and . . . We distinguish the total amount of heat in a body from the intensity of the heat, no? If two bodies of different size contain the same amount of heat, we say the
intensity
of the heat, the ‘temperature,' is greater in the smaller body. So if two bodies of different size contain the same
quantity
of gravity,
gravitas secundum numerositatem
, then there must be an intensity of latitude,
gravitas secundum speciem
, a
specific
gravity, that differs between them. What if motion is due to the difference in the gravitas speciem between the body and the medium? Your cork . . . A body that falls through air may float on water."

Albrecht grunts. "No, I think the Englishman has right. The answer lies in the difference between rest weight and moving weight. Master Buridan says that, once impressed upon it, a body's impetus is permanent until dissipated by resistance. What if the impetus is naturally in a body, at least in potency, and is only
actualized
by the moving agent? Then weight itself is but the result of a quantity of prime matter and its natural impetus to motion."

"How can a body resting on the ground have motion?" Nicole asks skeptically. So saying, he imparts an impetus to a stone on the dusty lane and it skitters off to strike a mule on the hoof. The mule balks and the muleteer curses them.

"
Potential
motion," Albrecht tells him as they run off. "It
would
have a velocity, toward the center of the world, if the ground weren't holding it back."

The idea is so absurd that Nicole cannot stop laughing until they reach the corner where they go their separate ways. It is only after they have parted, that the Norman recalls his master's answers to Aristotle's objections to the motion of the earth. Suppose,
secundum imaginationem
, that the earth turns with a diurnal motion from west to east. Why do we not feel a consistent easterly wind,
versus
a northerly wind? Why do people not continually stumble as they try to keep their footing on a ground in motion? Why does a stone thrown upward not fall a league to the west if the earth moves eastwardly below it?

Buridan's answer had been that if the air and the people are also moving, the appearances would be saved. The stone, of course, would be pushed to the east by the moving air. Only the Objection of the Arrow had stumped him. An arrow loosed upward cuts through the air and is not borne by it.

But now Nicole sees that Buridan has not pressed his argument far enough! Consider the arrow at rest. If the earth turns,
the arrow is already moving toward the east!
It has a horizontal impetus imparted by the earth, as well as its own natural downward motion. He stares at a loose stone resting in the dusty lane.
It could be moving
, he thinks,
toward the east at many leagues every hour. Like the air, like the people, like the great Ocean Sea. Like me
. It is a heady thought, and it nearly makes him as dizzy as Aristotle once supposed a turning earth ought, until he realizes that he has
assumed
the earth's diurnal motion, not demonstrated it. He has merely completed his master's demonstration that the appearances would be saved whether the earth turned or the heavens.