The Geometry Lesson
by Bill Schweizer © 2007
The frail young boy sat quietly on the park bench then fidgeted nervously for no reason then sat quietly again, his chin balanced on his two slender fists as he looked, now languidly, across the park toward the trees and stone wall in the distance. Finally and reluctantly he sat upright.
For a moment he toyed with a small chamois bag, which he now dangled from his first finger by its drawstring. It contained a student's implements, pencils, nibs, a small flask of ink, a compass, and a measured straightedge. Also within were his personal treasures, some fragments of colored glass, green and blue parrot feathers, a knife, and a pair of discarded spectacles with one lens intact. He opened his copybook and took out from the chamois sack a compass and pencil and a calibrated straightedge.
He was tempted to open his sketchbook, which he held clamped under his arm, but resisted and read the first problem his teacher had dictated earlier that day.
“Divide a straight line into two equal parts without measuring its length.”
Easy enough he thought as he twirled the compass, but the next problem seemed more daunting.
“Divide an angle of unknown degrees in two equal angles.” He knitted his brows together and grimaced at the same time glancing to his left.
In the brief time he had spent in productive procrastination an old gentlemen had taken a seat on his bench and was, himself, sketching intermittently looking up from his sheet of paper to gaze across the park, then scribble violently, then cough.
They had become aware of one another together.
“Hello and good afternoon.” The man spoke first.
He was clearly quite old indeed, a specter from the beginning of the century, dressed in a black long frock coat, very old fashioned, dark trousers and boots. He had a lion's mane of silver hair that a younger man might envy. Balanced against the park bench was a walnut cane with a silver tip the exact color of his hair topped with an ornate handle that seemed to have been shaped in the form of a crouching lizard, also the same silver color, and with red glass eyes that the boy imagined were rubies, and they might have been for all he knew.
The boy answered in greeting, and an exchange of names was requested by the old man
“Antoni,” the boy murmured
The old man in response and quite loud.” Ferdinand.”
“Oh, like the King of Aragon and Castile.”
“You know your history.”
“Oh yes. I love history. I know all the Kings of all the Kingdoms of Spain and the Kings and Queens of England, France, Austria, Sweden…” The boy heard himself boasting and felt embarrassed and stopped.
“Remarkable” the man replied. "You know the history of Catalonia I'm sure. You have studied the Visigoths, the Moors?”
“I think I do. Yes, the Visigoths.” He worried that a quiz would follow, but his worry was needless.
“You and I are most certainly descendants of the Visigoths,” the old man observed with bemused feigned pride.
“Are you here resting from your history lessons?”
“Geometry”
The old man groaned sympathetically.
“Ah geometry. Straight lines and boxes. The bricklayer's science. You are measuring angles and such?"
“Well yes, exactly so.”
“Its always the same even when I was at school. I was a schoolboy once.”
The boy seemed not to listen. “What are you sketching? The park? May I see?”
He hesitated in mock modesty then turned his sketchbook toward the boy. Expecting to see a vista of the park the boy was utterly surprised and astounded by the picture. The man had sketched two ancient warships in intricate detail, galleons with sails billowing and pennants flying, doing battle in an imaginary bay with foaming seas and with a crenellated fortress with countless arched windows in the background. What was more amazing, almost shocking, was the artist's inclusion of the figures of two angels seated laconically on clouds above the battle gazing with mild interest as though they were watching running races at a picnic.
“Oh my goodness,” the exclamation escaped the boy.
“Is the rendering that bad?” his neighbor asked.
“No not bad. No. Wonderful. Marvelous. I just thought you were copying, sketching the park. You kept looking across the field into the distance. I thought you were drawing what you saw. Such detail.”
“That's quite true. That's what I saw in the distance. No artist, even a poor one like myself ever copies exactly. Plato said something like that. Art is a bad copy of a good idea. Or maybe I just said it. I can't be sure. But that is what I saw. Just badly copied. I draw what I see.”
“You saw ships in the park?”
“No. I saw the angels lounging on the clouds. I added the ships.”
What a curious person, the boy thought, but he was intrigued and hoped the conversation would continue, and again, he need not have worried.
“What is this task that bores you so severely?”
Had he looked bored? Maybe so.
“Dividing lines and angles.”
“Oh yes I know that. Very useful. But of course you must apply yourself. We need to know these things to build the roads we travel and the boxes in which we live.”
“Boxes?”
“Just my opinion. We all live in boxes. Different sizes, heights, but boxes all the same. The King of Bavaria lives in a box. Decorated beautifully to be sure, but a box nevertheless, and he thinks it is a palace. Why is he King and not I?”
"Are you a teacher?” the boy inquired respectfully.
"Ah no, the furthest thing away. A truant. Isn't this the park where truants congregate?” The boy looked puzzled and the old man laughed. He added some details to the sketch than continued. “I also struggled with bisecting lines and angles.” He paused. “Then I surrendered. Now I bisect four and twenty lines each morning before breakfast.”
The lesson was forgotten now. “I want to go back to the boxes. What did you mean?”
“Please don't let me discourage you. Mathematics, geometry, these are the tools of creation for men. They are tools of great utility without doubt, but without inspiration that comes from God they are worthless. We build, we forge, we construct. We make the round earth level and then erect our boxes. What else can we do?”
More sketching, more details, if that could be possible, then again speaking.
“Have you ever seen a cave? I should hope you have. How old are you, thirteen, fourteen?”
“Thirteen. Yes. I've seen caves” He recalled an outing where they had explored some small caves.
“God made those caves. Are the walls straight? Was the floor level? Was there a rectangular porch? A door with brass hinges?”
“You already know the answer.”
“You see, man has great difficulty in creating anything as simple and useful and durable as what God has already put on earth. He can copy as we talked about. He can divide lines, divide angles, compute the volumes of cubes and pyramids but ultimately what he builds comes out in the form of boxes. An old man's opinion, take it for what it's worth.”
To the thirteen-year-old boy this seemed more complicated than the lesson he had abandoned.
The old man kept on now unconcerned if anyone was listening or not but the boy listened intently thinking that he might hear more about the angels in the park. “So lines and angles and measures are man's tools. On the other hand God loves circles and curves, ellipses and parabolas. God abhors straight lines. In all his creation God had drawn straight lines only twice. The sides of crystals, that's one.”
He paused as tough forgetting his second example.
“And the other?”
“And the other straight line God has drawn is the road from Heaven to Barcelona.”
The boy's eyes widened.
“That road is straight because when angels are sent to earth they come to this world through Barcelona.” He smiled. “Sometimes they come here to Reus. I think the two I saw had just recently arrived. I think they may have come to guide someone back with them.”
“You mean someone is about to die?” The boy was frightened and curious.
“Someone is always about to die somewhere, but to answer your question or perhaps not answer it, why they are here is none of my business.”
The boy looked up to the clouds and saw only clouds.
“You see I'm sure your priest tells you God is perfect therefore he has no beginning and no end. There is no disagreement as to that fact even among those who worship other gods. A circle has no beginning and no end. God's mind, which is beyond our understanding, is like a circle. It moves and thinks and creates around and through the universe in circular patterns, beautiful circular patterns that repeat themselves in such ways that they become circles themselves.”
This somewhat opaque sermon was interrupted when they heard the sound of shouts and arguing.
A commotion had arisen across the gravel path, which distracted them both. And they both jumped up, the elderly man much slower but with equal curiosity. The boy arrived first, the old man hobbling after. There stood a man with a woman berating him severely in loud voice and what they saw and heard summarized the controversy immediately. The man had with him a dog and the dog was harnessed to a small cart like a tumbrel that was filled with vegetables. The dog had three legs and the woman was incensed that its master was treating it with such apparent cruelty.
“Can you believe it Sir?” She spoke to the elderly man who was by now inspecting the animal and stroking its head while he supported its muzzle in the other hand and speaking reassuringly in dog language.
“I see no evidence of cruelty whatever,” he said. “This is obviously a happy and proud animal.” The dog confirmed the old man's assessment wagging his tail side to side furiously and haughtily thrusting his nose high in the air.
“But Sir, this animal is maimed and forced to do such work like a beast of burden. It's, it's inhuman.”
“Madam, I'm lame myself. This dog and I are brothers.” He leaned forward on the cane and then pulled up his trouser cuff. Stuck in the top of his boot where a leg should have been was a wooden post not much thicker than his cane and the same color. To no one in particular he explained. “Souvenir of the French occupation. Lost it when I was fifteen. Haven't missed it much.”
Now he spoke again, directly to the woman. “What would you do with me? Put me out to pasture? Shut me away? I would be happier lame with useful work to do and the sun shining on my face than cooped up in an infirmary.”
He added. “Or a kennel.”
The dog's owner stepped forward to explain himself. The dog had been run down by a horse cart and, although the leg had been severed, the dog had survived the loss of limb. The owner had, at first, devised a wheel support for the dog on which he balanced the stub of the rear leg. It became immediately apparent to the inventor that the contraption was unnecessary, but the dog enjoyed it as something that pleased the master. He seemed to like to pull it as he followed his master along. The wheeled support evolved into a cart and then ultimately the vegetable cart.
The boy looked more closely at the cart. He was fascinated by its design and fabrication and by the decoration the man had included. The detail was as intricate as the sketch of the pirate ships he had just seen. How remarkable that this little park was filled with such art. A museum of oddities. One needed just to know where to look. The sides of the cart were painted lovingly with colorful pictures of the same dog but depicted in fine full health with four legs. Smaller colorful pictures of birds, butterflies and flowers in crimson, gold, and cobalt blue adorned the wheels and rails of the cart. The ornamentation on the wagon was endless, wrapping around to the rear of the cart and continuing on the other side. While the comparison seemed absurd, the boy thought of pictures he had seen of Michelangelo's frescoes on the ceiling of the Sistine Chapel, and as he looked further the comparison seemed more apt.
The officious woman pretended to be placated, and the ingenious grocery monger and his dog went their way, the dog setting a brisk pace despite his missing hind quarter, the master with all his limbs struggling to keep up.
The artist-veteran and the boy returned to the bench and the old man continued his lecture pontificating broadly as though no interruption had occurred.
“Let us get your lesson out of the way so that we can continue our discussion of more important things like scribbling and sketching and naval battles.”
He took the measured straight edge, drew a long line, and then with the foot of the compass at each end of the line drew two circles, each large enough to cross the other. He connected the points where the circles intersected with the straight edge and scratched another straight line with a flourish. "There you are.”
Setting down the book, he looked from side to side at the bench then reached out with his cane and, reversing it, deftly scythed a dandelion with the lizard handle, which he scooped into his hand,
“Now lets get into higher mathematics. Look at this flower. It is unique. Similar to millions like it but none quite identical. Basically a circle but a circle made of curved petals also similar but ultimately unique.” He began to count the petals from the center outward. “Now count with me. One and one make two. Two and one make three, three and two five, five and three eight and so on. If you continue adding the sums of the two previous numbers you get a clue, a clue, not a solution, as to how this flower was imagined. Isn't it more interesting to imagine how God dreamed that flower than to cut lines in half.”
The boy gave no argument. He had none.
The old man tied the flower around a coat button seemingly inspired by the grandiose ornamentation of the dogcart.
“Do you know what is the most perfect structure?”
The boy ventured his best guess. “The Alhambra?”
“Well, I'm sorry. It was a poor question. Certainly, there's no correct answer. I was thinking of an egg. Perfectly curved, perfectly balanced, perfectly simple and, to the chicken inside, perfectly habitable. Well, that's my opinion anyway. If I were the architect for the King of Bavaria I would build him an egg. Then I would paint it with dogs and butterflies” He laughed raucously and the boy joined adding, “And with pirate ships.”
The would-be teacher resumed the podium.
“Now let's bisect an angle." He took the boy's compass, scribed three successive curves and drew a line that precisely divided an angle of unknown degrees into two angles half as big and exactly equal.
“Do you know what remains of the seven wonders of the ancient world?”
The boy knew this answer. “The Great Pyramids in Egypt.”
“A good question this time and the correct answer given. And what are they but piles of rubble. All the other structures, monuments, statues that once were the other wonders of the ancient world are all rubble now. The pyramids remain because they started out as a pile of rubble. Designed by geometry teachers drawing straight lines again and bisecting angles, day in-day out, under the Egyptian sun. There's no sun like the sun of Catalonia do you not agree? They tried to reach the sky, the Egyptians, but the ultimate height of their monuments was predetermined by the size of the base. Arches, that's what are needed to climb, to rise, to ascend. Do you see what I'm saying? You and I, sitting on this bench, looking at flowers in the same sunlight that warmed the Visigoths and the Moors, and talking about eggs and admiring dogcarts, have surpassed, at least in thought and ambition, the men who built the pyramids. Let's congratulate one another.” He gave the boy a military salute and the boy returned it.
“May I see your drawing of the ships again?”
“Most certainly. Feel free to criticize, only save your criticism until I'm gone.”
The old man opened the leather sketchbook again, and the boy marveled at the circles and curves and arcs that made the ships, the flags, the clouds and the angels and nowhere in the picture could he detect a straight line.
The boy was perennially sickly and frail, “rheumatic disease” the Doctors called it, and a spell of weakness prevented his return to the park for many weeks.
When he was once again able to visit the park again there was no artist to be found, no dog and chariot, and no help with his lessons. The same was true when next he returned and the next time after that and even after. At a moment when he finally had almost forgotten that afternoon and the unexpected help with his geometric constructions he found the bench again empty but leaning precariously against it was the lizard-handled cane. Jeopardizing his recovery he sat on the bench and waited until dark for the owner to return and when he did not appear the boy took possession of the cane for safekeeping.
He returned to the park with the cane a dozen times intending to return it to its owner and hoping to see again the man's fanciful drawings and listen to his strange theories, but the opportunity never repeated and no one could help.
He inquired after the old gentlemen to the gatekeeper and was told that there had been a lame old gentlemen noticed in the park from time to time who had stopped coming at all some time ago. “Quite a relic, that one. An old soldier of the very old days. Probably passed on.”
And that was all.
* * *
The old man woke that morning rested and content. He did not feel old. Quite the contrary. There was valuable work, pleasant work, that needed his attention, and then there was the Catalonian sun to warm his face.
Another day to imagine. Another day to observe, to sketch perhaps. Another day to please God. And the City of Barcelona to please him with its infinite variety to captivate him as it had done infallibly since he arrived at the age of seventeen.
As he looked into the wardrobe for a shirt his eyes rested on a dusty object on an upper shelf, which he recognized as his mathematical copybook from childhood. Such a thing to still possess. He thought to get it for a moment and then relented. His eyes found something else of interest, an antique he had long forgotten.
Leaning in a corner was an ancient cane he had found so long ago it seemed another lifetime, ancient now and ancient even the moment he had first seen it, with a silver tip still gleaming and the ruby eyes of its lizard handle still bright and ferocious. In that instant, a recollection crystallized in his mind of a warm afternoon in the park in Reus, the old man and the painted cart came rushing back and he imagined now how pleasant it might be to spend this sunny afternoon conversing with the man about art, building, mathematics, geometry, dogs and, of course, lizards.
On a whim he took the cane with him as a walking stick as he set about on his habitual walk in the city. He walked the route he had walked a thousand times on other sunny afternoons toward the Church of St. Philipp Neri, intending to pray before proceeding on to his studio. He would certainly have noticed the tram coming toward him if he had not for a moment imagined that he saw reclining on two puffy clouds hovering in the afternoon sky, two angels looking lazily down on him and on the onrushing tram.
Next he heard in his mind an almost imperceptible sound that he imagined might be like the sound of a dandelion being plucked in a park in spring, and in an explosion of colored light like the sizzling fireworks that were launched from the shore in summer, the street, the buildings around him, the people and the City itself dimmed and then disappeared. When the blaze of light subsided he saw, stretching from the center of Barcelona to the outskirts of the City, a road leading away through cool green canopies of tropical plants and palm, cascading with brilliantly colored flowers of yellow, blue and red and purple leading into the brilliant ineffable distance, a beautiful road, level and inviting, and straight as the sides of a crystal.
The End