Sir Transverse

Families often needed to divide fields. Maybe parents had died. Then brothers and sisters shared. Neighbors might argue over a line. Sometimes friends owned land together. They always wanted a fair share.

Fairness in fields meant one thing. It meant: each owner got a strip. The strips had to be the same proportional width. This was measured along the parallel sides.

If a field was 300 paces long, and three kids shared it, they wanted three strips. Each strip was 100 paces wide. If a field was 240 paces long, and four kids shared it, they wanted four strips. Each strip was 60 paces wide. Simple.

But fields are not always nicely lined up north-to-south. Sometimes the parallel sides are not even the longest sides. Sometimes they cut diagonal strips. Maybe a strip needed to touch the road. Or reach the stream. Each sibling wanted a piece like that. And then the math becomes interesting.

Sir Transverse was thin. He had long, stork-like legs. Everyone called him Sir. He had been Sir since he was six. It was a family nickname. No one knew why.

When he was nineteen, he found something out. It was his second year as a surveyor. He saw that if you cut a field with a diagonal line. He called this a transversal. It went across two parallel boundary lines. Then he cut another diagonal line. This second line was parallel to the first. The strip between these two diagonals always had a special width. This width was proportional to how far the strip was from the main parallel lines.

This was the intercept theorem. It's a very old math rule. Sir Transverse, of course, did not know it had a name. He simply observed it. He measured it on every field he surveyed for the next four years. It held. Every time.

By age twenty-three, he was a master. He walked into any field. He saw the people arguing. He asked one question: "How do you want the strips to touch the road?" In ten minutes, he laid out fair strips. He used only his measuring-staff. And a knotted cord.

He never measured the whole field. He never figured out its total area. He didn't need to. He just cut parallel lines. He used parallel transversals. The proportions worked themselves out.

The people arguing always left happy. This made Sir Transverse, by age thirty, famous. Everyone wanted him. He was the busiest surveyor in three provinces.

He spent thirty years doing this work. He divided many fields. He counted them carefully. One thousand, four hundred, sixty-two fields. He never had a complaint. He never had a re-survey. He never, even once, made a strip a half-pace too wide or too narrow.

His friends at the land office had a saying about him. They joked that he had the soul of a ratio. It was a funny thing to say, but they meant it.

The GeometryForge academy needed a teacher. They wanted someone to teach proportional reasoning. This meant the intercept theorem. And how transversals cut parallel lines. The academy master heard about Sir Transverse. His nephew told him. The nephew had been in a tough family field fight. Sir Transverse had solved it.

The nephew said, "He is amazing!" He added, "He makes fair division a math rule. Not just an argument."

The academy master wrote Sir Transverse a letter. Sir Transverse was forty-nine. His knees were getting old. He worried about walking in the rain. He accepted the job.

He brought his measuring-staff and a coil of knotted cord. He still has both. He keeps them in the corner of his classroom.

He teaches the intercept theorem his own way. He teaches it by walking. He lays a long cloth strip on the floor. It has two parallel lines marked with chalk. He picks two students. He gives them each a length of red string.

He says, "You are a transversal." He tells them, "Walk across the cloth. Hold your string tight. Choose any angle, but stay straight."

The students walk. The red strings cross the parallel lines at angles. Sir Transverse watches.

Then he asks them to walk a second transversal. This one must be parallel to the first. The class measures the strip between them. The strip is always proportional. Always.