We six remaining advanced calculus survivors stared out the open second floor windows at the sun-dappled leaves on the oak trees just outside. The 110 year-old building was not air-conditioned, but the tall windows with their low sills allowed a fresh-smelling morning breeze to pulse its way into the classroom. The spring weather was perfect. We were concentrating on an afternoon of cutting our college classes while lying in the grass under those oak trees, letting the warmth soak into us after the long Illinois winter.
Our instructor stood at the front of the room, chalk in hand, going over our homework from the previous class. He wore his usual shapeless black suit and wrinkled white shirt without a tie. It made him look even taller and thinner than he already was. His worn, scuff-streaked shoes echoed his disheveled, gray-streaked hair.
He had taught math virtually forever at our liberal arts college. Everyone described him the same way: “Quirky, but brilliant.” He had PhDs in both mathematics and philosophy from University of Edinburgh in the UK. He was such a brilliant mathematician that we weren’t smart enough to really fathom how brilliant he was.
You could find him nearly every afternoon and well into the evening sitting at the bar in one of the college hangouts in town.He always smiled at us when he saw us come in. His wife had died some years before. People who knew him well said that without her, he’d become even more eccentric.
He stuffed his college office with mathematics books, journals, and papers – on shelves, side tables, chairs and nearly every other free surface. Over-full ashtrays were scattered about on top of the heaps. The blackboard on one wall was always crammed with densely packed rows of equations, none of which could we understand. His big wooden desk was stacked literally three feet high with papers -- stacked so high the pile seemed to defy physical laws of friction and gravity.
Amazingly, though, when we went in to ask about a test score, he’d mumble, with a cigarette dangling out of the corner of his mouth, “Oh, um, ah, yes, your test, um, ah, Mr…. , ah. Yes, your test. From last week … your, ah, test.” And his arm would disappear nearly to the elbow into the stack on his desk and seemingly blindly he’d pull out a paper or test booklet. Magically it would nearly always be the paper he was looking for. Squinting at the paper through the smoke, he’d say things like, “Um, ah, it, ah, appears you passed, sir. Only a C+, I’m afraid, but perhaps I, um, made the test a bit, ah, stout shall we say? Anyway, ah, here … … um.” We’d take the test, say thanks, and turn to go. He’d already have gone back to whatever math problem he was working on.
Today, though, we were sitting at our desks in the classroom, working through the solutions to the homework. He wrote a particularly difficult problem on the blackboard, and turned to us, eyebrows raised in question. The brightest of us piped up, “Dr. H we couldn’t get this at all. We worked on it together last night. No matter where we started, we always came up with an obviously incorrect result. We finally decided the problem must be wrongly stated in the book.” The rest of us nodded in agreement.
He turned and looked with renewed interest at the problem on the board for a moment then he turned toward us again. His pale blue eyes darted around the room, looking over our heads. He said, “Yes, well. The problem is, ah, correct, you see, and it’s quite an, um, interesting one. Can anyone suggest where we might start?”
We threw out ideas that we had tried and that had failed us the previous night. He nodded and gave a crooked-toothed, but sympathetic smile to each suggestion.
Our instructor stood at the front of the room, chalk in hand, going over our homework from the previous class. He wore his usual shapeless black suit and wrinkled white shirt without a tie. It made him look even taller and thinner than he already was. His worn, scuff-streaked shoes echoed his disheveled, gray-streaked hair.
He had taught math virtually forever at our liberal arts college. Everyone described him the same way: “Quirky, but brilliant.” He had PhDs in both mathematics and philosophy from University of Edinburgh in the UK. He was such a brilliant mathematician that we weren’t smart enough to really fathom how brilliant he was.
You could find him nearly every afternoon and well into the evening sitting at the bar in one of the college hangouts in town.He always smiled at us when he saw us come in. His wife had died some years before. People who knew him well said that without her, he’d become even more eccentric.
He stuffed his college office with mathematics books, journals, and papers – on shelves, side tables, chairs and nearly every other free surface. Over-full ashtrays were scattered about on top of the heaps. The blackboard on one wall was always crammed with densely packed rows of equations, none of which could we understand. His big wooden desk was stacked literally three feet high with papers -- stacked so high the pile seemed to defy physical laws of friction and gravity.
Amazingly, though, when we went in to ask about a test score, he’d mumble, with a cigarette dangling out of the corner of his mouth, “Oh, um, ah, yes, your test, um, ah, Mr…. , ah. Yes, your test. From last week … your, ah, test.” And his arm would disappear nearly to the elbow into the stack on his desk and seemingly blindly he’d pull out a paper or test booklet. Magically it would nearly always be the paper he was looking for. Squinting at the paper through the smoke, he’d say things like, “Um, ah, it, ah, appears you passed, sir. Only a C+, I’m afraid, but perhaps I, um, made the test a bit, ah, stout shall we say? Anyway, ah, here … … um.” We’d take the test, say thanks, and turn to go. He’d already have gone back to whatever math problem he was working on.
Today, though, we were sitting at our desks in the classroom, working through the solutions to the homework. He wrote a particularly difficult problem on the blackboard, and turned to us, eyebrows raised in question. The brightest of us piped up, “Dr. H we couldn’t get this at all. We worked on it together last night. No matter where we started, we always came up with an obviously incorrect result. We finally decided the problem must be wrongly stated in the book.” The rest of us nodded in agreement.
He turned and looked with renewed interest at the problem on the board for a moment then he turned toward us again. His pale blue eyes darted around the room, looking over our heads. He said, “Yes, well. The problem is, ah, correct, you see, and it’s quite an, um, interesting one. Can anyone suggest where we might start?”
We threw out ideas that we had tried and that had failed us the previous night. He nodded and gave a crooked-toothed, but sympathetic smile to each suggestion.
“Yes, er, ah, um, that’s a logical starting place, but it is quite difficult, um, complicated from there.”
“Um, yes, I can see where you might start with that, but I’m not sure that will work.”
“Ah, no, you should see quite quickly that starting point will, um, immediately lead to a solution that converges to, ah, infinity.”
When we ran out of ideas, he said, “Well, yes, you followed many of the trails that most, um, undergraduate mathematicians might try, but perhaps, ah, let me show you how one of the more simple, uh, more simple solutions might be derived.”
He faced the board, circled part of the problem with the chalk and said, “Now if we convert this to a Fourier series …”, and he began writing rows of equations on the board. “And then if we differentiate this first in x …” and more rows of symbols appeared. “And then in y … “
He began writing this derivation, but then stopped. He stared at the board, chalk dangling in his right hand at his side, his left hand pointing to various lines of mathematics on the board. “Hmm, ah, no, ah, hmmm … hmm. No that isn’t quite it, is it? Do you see? Hmm. Ah, that’s not right. Hmm. Ah! Sorry.”
He grabbed a large eraser and wiped several of the most recent lines from the board. “Ah, from here, no, ah, first, um, we need to use this … “and he circled a bit from higher up the sequence “… and, um, substituting …” and began furiously writing more on the board. I gave up trying to copy into my notes what he was writing.
The bell rang in the hallway signaling end of class. None of us moved. The room was consumed by awed and complete silence. We felt ourselves quietly fade from existence.
He rapidly backed up a step then stood stock still except for his lips moving and his left arm conducting a pattern back and forth across what he’d just written. He ran his right hand through his hair adding a streak of chalky white with the gray.
“Ah, oh, of course. First, of course, it’s obvious that first I must …” With that he again erased several of the latest lines of work and slammed the chalk against the board writing a series of new equations. He mumbled as he worked, “So then, of course, this means …" more symbols … “so that THEN …” more and more symbols on the second board now. Chalk dust drifted around him in the sunbeams. As he attacked the board, broken pieces of chalk flew in the air and left trails down his suit as they fell.
“BUT, BUT … then, I see … I see now." More and more symbols. He stopped yet again. He stepped back, stared, and moved back another step. “Blast! What?” His eyes roamed the equations -- left arm pointing one way, right arm with the chalk pointing another. Sweat stood out on his forehead. “Where? Hmmm. Blast. Hmmm. Where?”
Clack, clack, CLACK went the chalk in staccato sequence.
“ … SO ! AH, SO! And THEN … SO, I have THIS AS THE SOLUTION!”
With his final flourish, he crushed the chalk in both hands against the last line he’d squeezed in at the very bottom of the board. Both pieces of chalk broke in two and flew in the air past his triumphant shoulders. With a delighted grin, he literally hopped back from the board and put is foot into the small waste basket near the wall. Losing his balance, he sat down hard on the low window sill, and, still grinning at the blackboard, began a slow motion fall out the open window.
He faced the board, circled part of the problem with the chalk and said, “Now if we convert this to a Fourier series …”, and he began writing rows of equations on the board. “And then if we differentiate this first in x …” and more rows of symbols appeared. “And then in y … “
He began writing this derivation, but then stopped. He stared at the board, chalk dangling in his right hand at his side, his left hand pointing to various lines of mathematics on the board. “Hmm, ah, no, ah, hmmm … hmm. No that isn’t quite it, is it? Do you see? Hmm. Ah, that’s not right. Hmm. Ah! Sorry.”
He grabbed a large eraser and wiped several of the most recent lines from the board. “Ah, from here, no, ah, first, um, we need to use this … “and he circled a bit from higher up the sequence “… and, um, substituting …” and began furiously writing more on the board. I gave up trying to copy into my notes what he was writing.
He filled the entire first blackboard with equations and moved to the second board on his right nearer the window. “Now, if THAT is true,” he said, waving vaguely at something on the first board, “then, of course, then it must also be true that …” and he rapidly wrote across the top of the fresh board. “Um, ah, so then … “, and he wrote yet more, but again stopped with his hand poised over the board.
“Um … hmmm … ah … sorry … … sorry … … … sorry, uh, hmmm. This, er, can’t be correct either. Ah, oh, dear."The bell rang in the hallway signaling end of class. None of us moved. The room was consumed by awed and complete silence. We felt ourselves quietly fade from existence.
He rapidly backed up a step then stood stock still except for his lips moving and his left arm conducting a pattern back and forth across what he’d just written. He ran his right hand through his hair adding a streak of chalky white with the gray.
“Ah, oh, of course. First, of course, it’s obvious that first I must …” With that he again erased several of the latest lines of work and slammed the chalk against the board writing a series of new equations. He mumbled as he worked, “So then, of course, this means …" more symbols … “so that THEN …” more and more symbols on the second board now. Chalk dust drifted around him in the sunbeams. As he attacked the board, broken pieces of chalk flew in the air and left trails down his suit as they fell.
“BUT, BUT … then, I see … I see now." More and more symbols. He stopped yet again. He stepped back, stared, and moved back another step. “Blast! What?” His eyes roamed the equations -- left arm pointing one way, right arm with the chalk pointing another. Sweat stood out on his forehead. “Where? Hmmm. Blast. Hmmm. Where?”
He charged the board, grabbing another piece of chalk from the tray with his left hand. “But if that’s true …” smashing the chalk in his left hand against the board, “… then THAT cannot be true …” slashing through that portion of work with the chalk in his right.
His head swiveled back and forth. “BUT! Oh, of course, but …” And chalk marks again flew across the board. He wrote first with his left hand then his right. He shouted, “And then …” Clack, clack, CLACK went the chalk in staccato sequence.
“ … SO ! AH, SO! And THEN … SO, I have THIS AS THE SOLUTION!”
With his final flourish, he crushed the chalk in both hands against the last line he’d squeezed in at the very bottom of the board. Both pieces of chalk broke in two and flew in the air past his triumphant shoulders. With a delighted grin, he literally hopped back from the board and put is foot into the small waste basket near the wall. Losing his balance, he sat down hard on the low window sill, and, still grinning at the blackboard, began a slow motion fall out the open window.
Our trance of the last several minutes evaporated. I grabbed at his thighs and another guy caught him by his outstretched right hand. He regained his feet and didn’t seem to notice anything amiss. Still smiling he said, “So THAT’S the correct solution, class. Simple, really, except for that one bit. And um, well, simple. Ah, … class dismissed.”
Not knowing what else to do, we grabbed books and papers and rushed for the door.
Not knowing what else to do, we grabbed books and papers and rushed for the door.
We bought our lunches and took them outside to eat in the shade. We talked about the meaning of ‘simple’ and whether being brilliant was all it was cracked up to be. We couldn’t help but glance at the open window above us. Another math class was in session.
