Analog SFF, July-August 2007

by Dell Magazine Authors

  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.

  He stoops to pluck a stone from the lane and, hefting it, tries to feel its eastward motion. But he cannot, any more than a man on a ship can feel the vessel's forward motion by placing his hand to the deck. He and the stone are both moving with the same speed and in the same direction. He sighs, deciding that there is no way to demonstrate the proposition by the senses alone. If he were affixed to the orb of the stars and looked down upon the earth, the earth would appear to turn; even as he, affixed to the earth and looking up, sees the heavens turn.

  Perhaps he could make an experience by “artful vexation of nature.” An experientia facta est, to coin a phrase. A “fact."

  Suddenly exuberant, Nicole hurls the stone high in the air. It comes down, not many leagues westward, but atop the awning of Schmuel the silversmith directly beside him. Schmuel rises from his bench cursing in his outlandish tongue and Nicole laughs and scampers off.

  * * * *

  Fernand delivers the water clock to the university precincts the next day. The apprentices wrestle it from the cart to the courtyard that the Rector has chosen for the contrived experience. The carpenters’ hammers compete with clockmakers’ shouts as they position the device beside the ramp. Curious scholars have formed a circle around the group, laughing and pointing, until the Rector wonders aloud whether the regent masters are waiting to start the lectiones ordinaria. Half the regents are themselves in the gawking crowd, but scholars and masters quickly disperse.

  Heytesbury arrives and views the large basin with awe. “Why, ‘tis a tunne-dish, i'sooth!” he cries in his own outlandish tongue. “It must hold four hogs’ heads!"

  No one knows what he means and, when it is translated, one of the workmen bristles. “This ain't no slaughter-bucket, begging your pardon, sir! Hogs ain't in it.” But the Englishman explains that a hog's head is a measure of volume used in his country and this both placates the men and amuses them with the queer notions of foreigners.

  “Explain it to me again,” Buridan asks his guest. As a physicist, he is unaccustomed to instruments and making measurements. “The weight of t
he water is a surrogate for the passage of time?"

  “Yes, yes!” Heytesbury exclaims. “That is why master Fernand made the basin so great. Is that not right, my good man?” he cries to the horologist, who has learned to ignore the whirlwind. “As the water in a basin diminishes,” he continues to his host, “so too does the weight pushing the water through the orifice; but with so large a reservoir of the water, the press does not sensibly diminish before it can be replenished. Hence, the water will issue forth with a uniform motion, and in equal increments of time we will obtain equal increments of water. As the rolling balls attain each of the distances marked on the inclined plane, we will accumulate the water in a flask, determine its weight, and so approximate the time. Hah! It is really quite pretty!"

  Buridan nods. “But yes, I follow your reasoning, but it seems ... distanced from the direct experience."

  “No more than a rule distances the carpenter from the length of his wood."

  Fernand shows them the five faucettes he has fixed to the basin and instructs them in their use. “First, turn the master faucet,” he tells them. “That will start the flow of water down all five channels and begin filling all five flasks. When your ball reaches the first mark, close the first faucet; at the second mark, the second faucet; and so on.” He instructs them soberly. Privately, he thinks them mad.

  The Englishman tilts his head back. “And what are those contrivances perched atop your basin? Metal birds, what?"

  Fernand stands taller. “But I am a master clockmaker, my sir, not a base metalworker. There must be bells and whistles to mark the times! It would shame me to give any less. As the water drains from the basin, the water in—you see that tall thin column? Yes. The water in that column will drop more rapidly, and a float dropping with it will, at the most minute intervals, cause the chirping of the bird.” He doubts that anyone other than scholars would ever need such minute times, but the challenge has given him a curious satisfaction. And, who knows? Were he to offer such a feature on his clocks...