Analog SFF, July-August 2007

by Dell Magazine Authors

  “And...?” Buridan's voice is heavy with lust. He is a sailor in sight of port, and Heytesbury realizes that he can suspend the matter no further. He opens the satchel and removes two bundles. One is a ream of parchment, quarto, tied between two stiff boards, which he hands to the Paris Master. The other is a smaller bundle tied in a rag. This, he passes to Oresme. Albrecht grumbles. What, no gift for him?

  “Do not fret, my Saxon giant,” the Englishman assures him with a clap to the shoulder. “There is a passage in the Philoponus that will interest you greatly. You recall how Brother Roger wrote, ‘Without experience nothing can be known sufficiently'?"

  “As did Albertus Magnus,” the Saxon replies, defending his namesake. “Remember how he always added to his statements, Fui et vidi experiri. ‘I was there and saw it for myself.’”

  Heytesbury brushes imaginary flies. “Yes, yes. And ‘Pilgrim Pierre,’ and Aquinas, and all the others said alike. But Brother Roger wrote of degrees of experience, and one, which he called the ‘best experience,’ is one in which all the forms affecting the experience have been accounted for by deliberate arrangement."

  By deliberate arrangement? Albrecht purses his lips. “Would not artificial conditions affect the body's natural behavior? It is the natural behavior we wish to understand."

  “Shit!” says Oresme, who is frowning over the pair of spectacles he has found in the bundle. “I'm no old man to need reading glasses."

  Heytesbury turns to him, “Put them on! Put them on!” Then, without sensible pause, turns back to Albrecht and says, “Philoponus thought contrived experiences useful. So did Bacon, who wrote that we learn more through artful vexation of nature than we do through patient observation."

  Albrecht glances at Nicole, who is gawking with wonder out the window of the apartment. “And what does Philoponus say about these ‘best experiences'?” he asks.

  “Yes, what does he say?” asks Buridan, who has been eagerly skimming the pages. He is already making plans to have a bookbinder set them between covers, and to have the university stationers produce several additional copies. His purse can afford the labor. Whether his patience can afford the wait is another matter. Yet the laborious process of copying a single book is one reason why so many become lost to mice and mildew and fire. If only there were some way in which a book could be written once, yet read many times.

  Heytesbury is smug. “Why only that Philoponus, using a contrived experience, determined as your Saxon giant, that, against Aristotle, bodies fall at the same speed, regardless of their weight, and that their speed increases in..."

  “In uniformly difform motion,” says the Saxon. Then, to the startled looks he receives, adds, “It stands obvious, no?"

  “I can see!” Nicole exclaims. But he does not mean that he understands Philoponus, or Heytesbury, or even Albrecht. He has discovered the world beyond his nose-tip. “There is a drover at the corner—” He stands at the window, pointing. “He holds a crook and drives one, three, eight pigs toward the pens. And that lady wears a kirtle in orofrise done up with scenes of hunting, and—Good day to you also, m'lady! And such superb melons!"

  Now he has his master's attention. Buridan asks to see—the new spectacles; not the lady's melons—and Heytesbury explains how Abbot Richard had reasoned that if ‘lentil-shaped’ glasses help a man see close at hand, a concave shape must help him see farther away. “And there are the laws of perspectiva,” he adds, “which Grosseteste used in De iride. Father Abbot drew diagrams of the paths of the rays, after Witelo's methods, as they enter and leave the faces of the lens. The most difficult task was the artisan's. Grinding a concave glass is not as simple as your convex reading lenses."

  There is no help for it. Each must try Nicole's new spectacles, although none else can see more than a blur with them. Albrecht is nettled that Nicole has made himself once more the center of attention.

  * * * *

  It is a rowdy era, as are all those eras when everything flips over. There are brawls high and low. Rhineland barons squabble. Kaiser Ludwig strong-arms Margaret Pocket-Mouth, the Ugly Duchess of Tyrol. The theologians of Paris have declared Pope John a heretic, again; and William of Ockham, safe at the Kaiser's court, wastes his pen on political screeds against him. The French fleet sails into Southampton and fires the town. In the Mediterranean, Genoa and Venice stumble through the final years of their long, drawn-out mutual suicide pact.

  And in Paris, it is morning and a gang of young townies have spotted Nicole Oresme returning to Buridan's quarters in the University. Town has hated gown ever since the Pope freed the universities of local laws and exactions, and this bird is too easy a prey to pass up. Nicole might have seen them sooner, save that he is gawking everywhere in fascination of his new spectacles.

  And what a spectacle he makes of it! Stopping, peering, laughing in delight. The laughter strikes the laborers as being at their expense. Another sneer from the haughty scholars at the common workman. Once again, the young Norman has made himself the center of attention, though he awakens only slowly to the honor.

  Nicole finally notices the train of journeymen and apprentices he has acquired, sees their roughened, horny hands, hears their sniggering laughter. Perhaps there is no more harm in them than mere mockery, but the young scholar suddenly feels very small and very alone, and so he bolts suddenly toward the safety of the university.

  It is the very worst thing he could have done. He is a flushed bird in flight! His gown flaps like wings. Even his cries for help sound remarkably avian. His pursuers are falcons launched.

  Norman sandals slap cobblestones down narrow lanes. He overturns laundry baskets, thrusts aside screeching harpies. A stone hurtles past him—and he thinks, madly, of the prior day's discussion of bodies in motion. A second stone resets his academic priorities. Six-to-one is not fair odds, but he doubts his pursuers would care. He turns another corner...

  ...and Albrecht of Saxony is suddenly there, with his long, grave Saxon face and clumsy demeanor. This fails to dampen the townies’ humor. Two scholars? It is still three-to-one!

  Save that one is a farm boy and has grown up wrestling with calves and other livestock. He may be long and thin, but every thumb-length is tough as rope. Besides, he has a club—a billet snatched up from the construction site, and he knows its use. Rural Saxony has not schooled him in meekness. A swing breaks a pursuer's forearm, drawing a howl; a stab blows the wind from the brisket of another. The townies grumble and draw back. But others have come in response to their shouts.

  Albrecht directs a fighting retreat, but the university is too far and the crowd now too many. Stones begin to fly again and what had begun as a near-amiable thrashing may soon end in riot and murder. Albrecht and Nicole back up a narrow alley, instinctively warding their flanks.

  Then the militia are about them: a dozen halberdiers in the livery of the university corporation.

  An unworldly scholar or two is one thing; grim-faced men who know how to kill is quite another. The mob breaks up sullenly. One reckless youth hurls a final stone—and is felled by the butt end of a poleaxe. That is the end of it. A few shouted imprecations follow—"staircase wit"—but words are nothing compared to sticks or stones. Albrecht throws his billet-club to the ground. His fingers tremble, but he does not permit Nicole to notice.

  The two scholars take stock while the militia escorts them into the university precincts, where university law prevails. A few bruises. A cut on Albrecht's cheek. And Nicole's proud new eyeglasses broken.

  “But the glass is intact,” Buridan comforts him when he inspects the wreckage later. He seems more concerned for the marvelous invention than for his two students. He had given them but a glance of amusement, and cautioned them against brawling, “unless the numbers be more in your favor.” He is more concerned that the militia left the grounds to effect the rescue, something he will now need to square with the Provost of the City. Heytesbury arrives from his rooms, attracted by the commotion and, informed of the circumstance
s, recounts tales of mighty combats in Oxford town. Hundreds of scholars massed against a like number of townies and armed with tight-packed balls of snow and ice.

  “The ice is the worst,” he gravely assures them.

  Nicole thinks stones worse than ice, and knows a little pride that he has endured such combat. When he tells the kitchen wench later, the size of the mob has swollen and the billet-club is in his own hands. Three downed at a blow! She pretends to believe him.

  “You were correct,” Buridan tells his senior student after Nicole has parted to rest from his ordeal. “It is obvious."

  Albrecht blinks. “Obvious that...?” he says, creating an expectant silence for his master to fill.

  “That falling bodies exhibit uniformly difform motion. The velocity increases with each increment of distance fallen."

  Heytesbury purses his lips. “Obvious to you, perhaps, John.... “He also wonders why it has taken the Paris Master a full day to determine the obvious.

  “But it is clear from the theory of the impetus,” Buridan declares. “What causes a body to fall? Some say that a body's substantial form causes it to fall; but that begs the question. I say it is the body's gravity, its weight. But consider now that a body's weight is constant..."

  “And yet it clearly moves faster and faster as it falls,” Albrecht adds. “So gravitas cannot be the cause of the difform motion, since an unchanging thing cannot cause a changing thing."

  Heytesbury scratches his head. “Proximity to its natural place? The longer the body falls, the closer it is to its place; and so, as a lover rushes as he nears his beloved, it moves faster."

  “Unconvincing,” said Buridan, dryly. “What else might it be?"

  Albrecht tugs on his chin. “Rarefaction?” he suggests. “A body moving through air becomes warm through friction, and warmer air is more rarified and so presents less resistance to the falling body."

  Buridan shakes his head. “But no. I will tell you. In the beginning, gravitas alone moves the body and it moves slowly. But in moving, the body acquires an impetus. This impetus together with its original gravitas now moves it. We may call this ‘accidental heaviness,’ to distinguish it from the body's ‘substantial heaviness.’ The motion thereby becomes faster; and by the amount it is faster, so the impetus becomes more intense, adding still more accidental heaviness."

  Heytesbury is rendered momentarily mute. Then he hollers, “Oswy!” and before he can articulate his desire, his long-suffering servant has appeared and placed a palimpsest on the table before him, proffering a quill. “Ink!” cries the Englishman, a request fulfilled by Albrecht, who, standing by the window, is closest to Buridan's desk. “This parchment is already marked up,” Heytesbury complains. “Lend me your razor, John. I need to scrape it off."

  Buridan hands him the razor, remarking that it had once belonged to his teacher, before he went off to the Kaiser's court. “A countryman of yours."

  Heytesbury blinks, studies the instrument, purses his lips. “Ockham's razor? He certainly knew how to clear a page. Hah!” For the next few moments, Heytesbury makes notations on the sheet. “I must see if there be a way to express your theory in the arithmetic of fractions. Bradwardine has a pleasing notion which he styles ‘instantaneous velocity.’”

  * * * *

  Buridan had sent Oresme to rest from his ordeal, but he is not in the master's bedchamber when Albrecht comes to fetch him. The Saxon sits upon a stool and considers the possibilities then, shaking his head, he departs for the servants’ quarters, where he finds the younger man swyving the serving wench, Lizette. “The master desires to see us,” he announces while the two scramble for their clothing. Nicole gives him a dark look and Albrecht shrugs. “He told you to lie down."

  “He didn't say ‘alone.’”

  Albrecht grunts and glances at the young woman, who clutches her cover-slut to her. He smiles politely while Nicole pulls up his hose.

  “What is it?” Nicole asks as he hops down the hall in the Saxon's wake, tugging on his shoe.

  “The Master desires us to contrive an experience."

  * * * *

  “I have paid the stationer to copy that section of the Philoponus which deals with contrived experiences,” Buridan explains when they have forgathered in the instruction room. “He should have rough copies for you tomorrow. I desire you master that section and contrive an experience, in imitation of Philoponus, to proof whether our Albrecht has correctly described falling bodies."

  Heytesbury, sitting to the side at a writing desk, scribbling on parchment with quill and straightedge, speaks without looking up, “Meanwhile, I will employ the compounding of fractions to express all this in mathematical form."

  Albrecht says, “I see no reason why the world should be reducible to mere mathematics. In the sensible world, there are no infinite lines, no dimensionless points, no perfect spheres tangent to perfect planes."

  Heytesbury turns and lifts his reading spectacles from his nose. “My dear boy...” He is but three years Albrecht's senior, but he has determined and incepted and is a Fellow of Oxford. “My dear boy,” he says, “Light is the first form that came to primary matter at creation. The entire world thus results from the propagation of luminous species; and, as light propagates rectilinearly as a succession of waves, we can describe it using rays and reflections according to geometrical laws. Hence, to understand the geometry of space and time is to understand space and time, what!"

  Albrecht is stubborn. “You cannot mean that even the bricks of this building are a form of light!"

  Buridan intervenes. “Grosseteste's metaphysics need not concern us. That the world is a consequence of geometry strikes us here at Paris as unduly Pythagorean. Abstractions like rays and numbers are constructs of the human mind and cannot be the efficient causes of sensible facts. No, Master William, it is experience of the senses, not mathematics, that will proof the proposition."

  Nicole, listening in unwonted silence, wonders whether the Oxonian and Parisian schools might be united to the benefit of both and the glory of God.

  * * * *

  Several days pass while each engages his particular task.

  Albrecht and Nicole wrangle with the text. Buridan tries to describe the propter quid of freely falling heavy bodies. Heytesbury wrestles with compounded fractions, trying to capture the insights of the physicists in a net of numbers.

  * * * *

  If the impetus continually adds increments of gravitas to a falling body, the Englishman reasons, then the body's weight while descending is greater than its weight at rest. Jordanus of Nemours had distinguished between gravitas secundum situm, or “positional gravity,” and gravitas in descendendo, or “free-falling gravity.” But that bodies in motion become weightier the faster they move is contrary to experience. Or is it? Who can weigh a body while in motion? To rest on the balance beam, its motion must be arrested, so that if one learns its weight, its speed remains unknown. Whereas, to observe the speed, the body cannot be weighed....

  Further, if resting weight and falling weight are distinct, as Jordanus wrote, there must be some as-yet occult form underlying a body's manifest weight, whether falling or no. Something that informed weight without being weight ... Interesting. He scribbles a marginal gloss on the page.

  Now, motion is the successive accumulation of the form of distance, and difform motion is the successive accumulation of the form of velocity. But uniformly difform motion means that equal increments of velocity are obtained at each interval, so the incremental impetus must be proportional to the same quantity at each interval. But impetus is proportional to the weight and speed of the body. So, if moving weight is continually increasing, it must be the rest weight that increases the velocity.

  But how short is the duration of each interval in which velocity is acquired? Time is a continuum, not a succession of discrete moments, so there is no natural and necessary duration to an interval. Then let the intervals become of shorter and shorter duration. But then in n
o time, no distance would be covered, and so no velocity would result! Hah!

  Perhaps if he considers the shortening of intervals as he had the rarefaction of air between approaching planes ... He can then define Bradwardine's “instantaneous velocity” as an extrinsic limit.... The intervals become shorter ad infinitum, but there is no “last” duration ... Hah! A very pretty problem.

  A pretty problem which at times causes him to throw his quill across the room in frustration and to apply his razor to the sheet of palimpsest. If he works the problem much longer, that sheet will be scraped to translucency.

  And meanwhile, Albrecht and Nicole pore over two scratch copies of the Philoponus chapter that they have dubbed On the vexation of nature and which they have obtained from the stationer. They compare the two copies word for word, correct the spellings and disagreements, rush back to the stationer to consult the original (which the clerks are now rendering in full), identify some likely errors that Cremona himself seems to have made, debate whether a ratio has been carelessly inverted, and generally wrangle over the text. No conclusion should be drawn from a text unless it is faithful to the master's copy.

  * * * *

  And meanwhile, Buridan drafts his ideas on uniformly difform motion, incorporating Albrecht's thesis and Heytesbury's sometimes-peculiar suggestions. Let moments approach intervals of no duration? Absurd! Velocity is the ratio of the distance traversed to the time spent, and a ratio over zero is infinite, which would imply that finite motion is infinitely fast at each moment, a foolishness. Heytesbury replies that a duration of zero is extrinsic and, as the distance covered also decreases, the ratio remains always finite. After this, he trails off into confusion, or Buridan fails to follow the trail, or both.

  * * * *

  One evening, Buridan notices Oresme's broken eyeglasses still lying on the corner of his desk and chides himself for having forgotten their repair. Idly, he holds the new-fangled glass at arm's length to inspect the damage; and, inasmuch as he is wearing his reading spectacles at the time, he is astonished to see a blurred image with the seeming of great distance. Yes, a tiny ymago of the building across the street. On a whim, he reverses the two, holding his own lens at arm's length and peering through Nicole's strange new concave glass.