This is one story for a book I am writing, titled "Twelve Like You". It will be written for twelve year olds, telling the stories of other twelve year olds who have overcome hardships and achieved fame for themselves. Hypatia (born 350-370? died 415), for example, was a pioneer mathematician; as a woman in an era where women were relegated to domestic roles, she broke the mold.
* * *
“Hypatia, you can do this. My bright daughter can do this.”
“I don’t think I can.”
“You will. And you will do it better than I can.”
“You must be joking, father. I am not a mathematician. You are. You are the greatest in all of Alexandria. That means, in all of Egypt! ‘Theon, the Great Professor,’ they call you. I have heard your students say that no one is smarter than you.”
“If that is so, Hypatia, you are my daughter, and you should grow up to be even smarter than I.”
Hypatia had heard this before and rolled her eyes, but her father did not see her. He was smiling to himself when he noticed how she held the unrolled papyrus under her elbows against the table-top. It reminded him of how she could not – or would not – hold it that way just a few years ago. Instead, she used to let the roll wind up, snapping as it went, with her laughing and smiling each time it did. Now she was twelve, tall, with his dark, bright eyes, her mother’s beauty, and quick at almost everything. He was proud of how well she could write. She had practiced long hours until she learned how to use the goose’s quill to make clear, even letters and numbers. It was still a game for her—making each letter the same size, making each number look perfect. His writing had never been so clear.
Hypatia liked to draw, too. On this sheet of papyrus, she had drawn many different shapes—circles, squares, triangles. She tried to make them look like solid figures. For the circles, she drew a curved line across the middle. For the squares, she added sides and a top, being careful to not let the lines cross or look too big or too small. To the triangles, she added a smaller triangle off to one side, sharing one of its three sides with the larger one. And when she added shadows, as though the light shone on the figures from one side, they looked even more solid.
One shape delighted her more than any of the others. A cone. She drew a cone over and over again, each one slightly larger than the one before it, until she came to the edge of the papyrus.
“A cone is like a circle, father. Actually, many circles, each one smaller than the one below, all stacked on top of each other. And it is like a triangle, too. A triangle made up by a pole in the center of its base and a line that stretches from the top of the pole to the edge of the circle. When the circle spins, it makes triangles. Countless triangles, side by side, until it comes back aroung to the starting point. But a cone is different from both a circle and a triangle, because it is a solid. Circles and triangles are flat on the paper. My cones look like they are standing up, just like my cubes, my spheres, my pyramids.”
“Yes, Hypatia. You make me proud, how you can draw and see and think about things. Now make me a real cone out of this sheet of papyrus.”
“I’ll try.”
Hypatia rolled the papyrus into a cylinder. Then she made one end of the cylinder tighter until it came to a point. Doing this made the other end spread out. She tore off a curved segment of the larger end, so it would stand flat on the table. But as soon as she took her hand away from it, her shape spread out again into a flat piece of papyrus.
“You need some glue or wax. Take this candle,” her father said.
Hypatia made the cone again and sealed the seam with hot wax where the edges met. Now it stood on the table.
“So, here’s my question again, Hypatia. Tell me how we can measure the amount of grain we can put in such a cone. We can tell how much grain will fill your cone. But what do we need to know to be able to find out how much grain we can put into a cone of any size? Of any height or circumference at its base? I want you to make a rule that we can follow for any cone. The women in the market place would love to know the answer, so they can know how much grain or beans or how much of any other item they will get when what they buy fills a big cone. Or a medium or small cone. The merchants will sing your praises. Whoever solves this problem will be praised in all of Alexandria. In all the world, I believe.”
“Yes, father, I understand the problem. But I don’t know the answer,” Hypatia said, rolling her cone on its side and watching it come back around to its starting point.
“There must be a rule, a clear way, true in all cases, to solve this problem,” Theon said. “I will tell you now, Hypatia, that I don’t know the answer either. But I think you will be able to figure it out someday. Think about it when you go to sleep tonight.”
“I will.”
“It’s time for your bed now. We can talk again about it in the morning.”
Hypatia was not tired, even though she had had a busy day. She had been up since before dawn. Like every other day, she had gone outside with her father into the courtyard when he led the boys from the neighborhood in their morning exercises. Hypatia stood in the back and did them, too. She was the only girl in the group. If Theon had not been there at first, the boys would have chased her away. Now they only teased her a little, especially since she was taller and stronger than most of them.
When her father led the boys away to their school and went on himself to the university, Hypatia went back inside and ate breakfast with her mother. Then she did her chores. Every day she walked with their servant to the well and helped bring home fresh water. Then she helped in their garden. Afterwards, she helped her mother and the servant wash and mend the family’s clothes. Only then would she be permitted to begin her studies by herself. Girls in Alexandria during this time were not allowed to go to school. But Hypatia wanted to learn whatever boys learned. Her father wanted her to become a perfect human being—-she laughed each time she heard him say so. Still, she loved him and believed she should try to be as good at everything she did in order to bring him honor: philosophy, mathematics, literature, athletics – whatever she could do, she would do it for him. And Theon did what he could to guide her.
Hypatia knew that someone must have studied and written about cones. She asked her father that night to help her find it in Alexandria’s wonderful library, the most marvelous collection of all of Greek manuscripts in the world: plays, poems, philosophy, sciences and mathematics, everything that was written by such great authors as Aristotle, Plato, Aeschylus, Euripides, Hippocrates, Apollonius, and many, many more.
“I will bring home a work by Apollonius. He wrote about cones. I will have a scribe copy it out for you to study.”
“Who is Apollonius?” Hypatia asked.
“A Greek. He lived several hundred years ago. But he is the only one I know of who wrote on cones.”
A week later, when Hypatia began reading Apollonius’ work, she was fascinated. She soon saw that mathematics had a beauty of its own, just as beautiful as a poem or a painting, because of its harmony: the way each number or group of numbers was understandable in relationship to other numbers; the way numbers could be used to “describe” things – one cube, for example, could be twice as big as its two halves, or four times as big as its quarters, with ways to measure exactly how big it would be based on the length of its sides. She saw, too, that Appolonius had not solved the problem of the volume of cones, but that he gave her clues to how it could be solved.
“If I slice my cone into sections,” she thought, “I will see that each part of the cone will have an ellipse where it is cut. And if I cut it exactly at right angles to its center, the two pieces will each become a circle where they are cut! There must be a rule of some sort that will allow me to figure out what this means.”
Hypatia eventually wrote a commentary on Apollonius’ work, in which she solved the problem. She saw that there were rules or formulas to imagine the segments of sliced cones and how much volume each cone would have, if one knew, first, the area of the circle that formed its base—-which equaled the radius, if multiplied by itself, and then the sum multiplied by pi-—and how tall it was. This gave her the volume of a cylinder. Now what was the relationship between a cylinder and a cone of the same diameter and height? Her father was right: she could do this!
And she did. By then, she was herself teaching in the university. She grew famous for her lectures which attracted students from as far away as Rome and Greece. She was appointed the official philosopher of the city of Alexandria, as famous as any woman of her time in her part of the world. Hypatia became one of the founders of modern mathematics. It would be a long time still before women would begin to be respected for the power of their minds, but Hypatia helped to point them toward equality by using her excellent mind for the benefit of all.
What a wonderful story! If only we could get children to read stories like this in school, it would help them understand the value of mathematics.
I have my own story. Tim and Tom, my younger twin brothers, came to visit me when they were 11. I took them to my office where we had four calculator computers. One of them, I believe it was a Wang, was quite advanced… It could be programmed To turn out form letters like a word processor. I had to go do something, so I showed Tom how to calculate the area of rings on the calculator. I left them alone and returned later, finding out that Tom was enjoying the exercise.
Tom got his bachelor's degree in statistics and computer science, and now manages major projects doing software avionics for major corporations for his consulting firm. His daughter has a degree in environmental engineering and is currently getting certified to teach math and science in schools.
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