The Silver Rule

Many years ago, more than I actually care to count, I was blessed with the perfect tenth-grade geometry teacher. I don't think I realized it at the time but looking back from here she was the quintessential model of what God would want a geometry teacher to be.

For starters she looked the part, a strong angular woman in her fifties with silver hair and steel rimmed glasses. She stood straight and tall, always perfectly groomed in tweed skirts and starched high collar blouses. And even her name was perfect. Miss Rule. Miss Mary Rule.

In class she was stern and precise, just what you'd expect. But after school when she was alone correcting papers and I'd come in to ask her about something she'd taught that day, we'd sit facing each other across the corner of her desk and she would smile.

We talked about theorems and proofs and what I wanted to do with my life, and oh, the places she took my active and curious mind were wondrous. She opened me up to whole new ways of thinking and being. I loved her class and am grateful to this day to have been her student.

Like the day in class when she presented a proof contradicting the Pythagorean Theorem. You know the one, that the squares of the two sides of a right triangle are equal to the square of the hypotenuse, a² + b² = c², probably the best known theorem in all geometry.

"I'm going to show you how Pythagoras was wrong," she said in a very stern voice. Rising from her desk she walked to the blackboard and picked up the chalk. "Pay attention."

First she drew a right triangle on the board, labeling the corners A, B, and C, the sides a and b, and the hypotenuse c. Then she marked the midpoints of the sides, D and E, and drew perpendiculars from those points to where they met on the hypotenuse, a point she labeled F. She spoke and wrote on the board at the same time.

There was now a rectangle, B,D,F,E, inscribed within the original triangle, flanked by two smaller triangles, ADF and FEC. "AB equals AD plus FE. BC equals DF plus EC. So AB plus BC equals AD plus DF plus FE plus EC," she said. "Are you with me?"

She paused, turned towards us and looked around the room. The room was silent and I leaned forward just a bit from my third row seat to be a little closer to the action.

She then bisected lines AD, DF, FE, and EC, and drew new perpendiculars to the hypotenuse within the smaller triangles ADF and FEC similar to the ones she had previously done. "This distance, too," she said, following the step pattern that had been created on the blackboard, "is also equal to AB plus BC. Are you still with me?" I sat there mesmerized. Something was happening here that didn't feel right. I just didn't know what it was.

Again she inscribed a rectangle within each of the newly formed small triangles, dividing every line in two. She was facing the board as she spoke, drawing more and more triangles on the black surface, triangles that obviously kept getting ever smaller.

"We can keep doing this," she said, pausing between phrases to give her time to draw a new set of bisected lines and ever smaller triangles on the board, "the lines getting ever smaller... ever closer together... narrower than the width of the chalk - eventually - becoming impossible to draw...".

She turned and rubbed the chalk dust off her hands. "Finally, they become infinitely small, becoming at that point, for all practical purposes, equal to c, the hypotenuse." She looked around the room for an objection, but there was none. "Therefore, a + b = c, not a² + b² = c². So Pythagoras was obviously wrong," She stated this last with a triumphant smile at our dumbfounded expressions and put down the chalk with a flourish.

My mind was racing. No! This couldn't be! Logic had become an enemy. Something was wrong. Something had to be wrong. She had spent all semester beating Pythagoras into our heads. She couldn't be refuting all of that now, just like that! Could she?

It looked like the lines were equal but just because you couldn't see something didn't mean it wasn't there! Right? All those little bisected lines couldn't just have disappeared? Could they? No. No.

Yes!!! Suddenly there it was, in my mind, clear as could be.

"Miss Rule, Miss Rule," my hand was waving wildly even though I wasn't sure what I was going to say. "Maybe you can't draw all those little steps because the chalk is too thick. But they're still there. You just can't see them." My mind was spinning. "And, and - there are so many of them, even if they are so small, they'd still add up to the length of the sides, AB + BC. They're always there. So a² + b² = c² still works. You see?" My chest was pounding. I'd contradicted my favorite teacher in front of the whole class. I was going to get it now for sure.

Her stern face melted into a smile, "Of course. Very good. That's the whole point. As you approach infinity, the laws of logic go out the window. Don't be fooled by trusting only what you can see and measure. Anyway, this was just for fun. Let's get back to some real work."

That afternoon, on the way home from school, I zig-zagged across the empty lot at the corner.

Somehow, as a youth I knew it intuitively. There are limits to what we can see with the external eye. And not seeing something does not mean it isn't there. Within ourselves, as within the depths of a geometry class blackboard, infinitesimal steps reach out in a meaningful, non-measurable direction towards an intuitive reality we know to be true, but cannot measure in material terms.