Newton’s Apple Tree ~ A short story inspired by the limit,
the number zero, Newton, and a favorite Calculus lesson.

Isaac wandered distractedly through the town, paying no
attention to the people he jostled. His mind was walking around a problem, scrabbling
for purchase on its slippery slope. He made his way to the farm and found his
favorite apple tree. He sat down and leaned against the tree.

“What can I do?” he asked the branches overhead. “I know
that the slope of the tangent exists in theory–I’ve drawn it on paper–but does
it exist organically? Does it have a representation in the natural world?”

In response, the tree dropped an apple on Isaac’s head.
BONK!

“Ouch!” yelled Isaac. “Why did you do that?”

“Pick up that apple, you dolt,” said the tree. “Notice its
curved surface? Now rest that stick in your hand against it.”

Isaac did as the tree commanded.

“At how many points does the stick touch the surface of the
apple?” asked the tree.

Isaac looked more closely.

“At only one!” he cried. “This is stupendous! I wonder why I
did not see this before? Many thanks, tree! I have to leave now!”

Isaac hastened to his study where he spent the next
fortnight making calculations. He worked in a fever, like one possessed,
checking and rechecking his figures. His family grew worried about him and
wondered at the agonizing moans emanating from his room.

Isaac’s father was on
the verge of breaking down his door when Isaac emerged from his study. His
parents gasped. Was this pale, disheveled ghost of a man their son?

Isaac stared blankly at his family and his surroundings a
few minutes before stumbling blindly out the door. He headed back to the apple
tree.

“It’s no use,” Isaac groaned. “I shall never find the
fluxion that proves the tangent line slope, even though I know it exists. It
was almost mine; I had the function, the fluent, and the difference quotient
all set up. I was all ready to merge the two secant points into one. So
close…so close…”

BONK! Another apple landed on Isaac’s head.

“OUCH! I say; that was uncalled for! Here I am in the depths
of misery, pouring out my soul, and all you can do is toss apples at me!”

“QUIET!” barked the tree, “Or I will unload an entire bushel
on your head! That’s better. Now, you have your function, your fluent, and your
difference quotient.”

“Yes.”

“Well, then, go ahead and find the limit as

*delta-s*approaches zero.”
“But I can’t!” Isaac wailed. “That would mean division by
zero. I can’t do that!”

“Why not?”

“Are you wanting in the upper story?” said Isaac. “In the
first place, zero is a dangerous number. It does not behave respectably like
the other numbers. No reputable mathematician would ever attempt to divide by
zero–it’s just not done in polite society. If I tried that, I would be worst
than a laughing stock; I would be shunned.”

“What do you care what other people think?” asked the tree.
“You have always considered yourself a mathematical rogue, haven’t you?”

“It’s not merely that,” said Isaac. “If I were to actually
divide by zero successfully, the rational world would collapse. There would be
riots in the streets, dogs with cats, incompetent rulers on the throne–er,
never mind that last one, it’s true anyway. The point is, if I prove division
by zero, then I could prove anything, whether or not it is real. You see my problem?”

“What I see is a person not willing to take a little risk,”
said the tree. “How do you know the limit does not exist unless you actually
prove or disprove it? I say, throw off your shackles of caution and bonds of
convention! Have faith in your difference quotient and believe in the limit. So
you destroy the world–so what? Are you going to let a little thing like that
stop you?”

Isaac slowly rose and faced the tree.

“You are right,” he said. “I am a mathematician. They think
I’m mad anyway, whether or not it’s true. I will do it. I will find that limit
if it is the last thing I do.”

Isaac gently touched the tree.

“Good bye, old friend. We may never see each other again.”

A week passed with the tree wondering what became of Isaac.
Did he or did he not find the slope of the tangent line? Suddenly Isaac came
bounding down the lane. The exultant look on his face said it all.

“I did it!” he said, trying to catch his breath. “I successfully
found the limit!”

“Congratulations!” said the tree. “I see the world did not
end.”

“No, it did not end–it exploded! Oh, tree, the vision I had
of new vistas to be explored. This has opened the door to another world of
mathematics! Why, I foresee engines flying in the air, wagons moving along
without animal power, strange and unusual food of the gods that heals boils,
the pox, and the plague, buildings towering over the city, and…” BONK!

“OUCH! This is really too much! What is the reason for this
apple?

“I thought you might be hungry.”

Isaac picked up the apple. Now that he thought of it, he

__was__hungry.
“Thanks,” he said.

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