Sir Transverse

The trouble was that the long sides were parallel. And every time a family wanted to divide a field — among siblings after a parent died, or among neighbours after a boundary dispute, or among co-owners after a marriage — they wanted to do it fairly.

Fairness, in fields, has a precise meaning. It means: every owner gets a strip of the same proportional width along the parallel sides.

If you have a field 300 paces long and three siblings, you want three strips each 100 paces wide. If you have a field 240 paces long and four siblings, you want four strips each 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 you want to cut diagonal strips across the field — to give each sibling a piece that touches the road, say, or a piece that touches the stream. And then the math becomes interesting.

Sir Transverse, who was thin and stork-legged and had been called Sir since he was six (it was a family nickname; no one knew why), discovered when he was nineteen — his second year as a surveyor — that if you cut a field with a diagonal line — a transversal — across two parallel boundary lines, and then cut another diagonal line parallel to the first one, the strip between the two diagonals had a width that was always proportional to the strip's distance from the parallel boundaries.

This is the intercept theorem. It is one of the oldest results in geometry. 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 the time he was twenty-three, he could walk into a field, look at the disputing parties, ask a single question — "How do you want the strips to touch the road?" — and within ten minutes lay out fair strips with nothing but his measuring-staff and a knotted cord. He never measured the field as a whole. He never calculated areas. He simply cut parallel lines with parallel transversals and let the proportions take care of themselves.

The disputing parties always left satisfied. This made Sir Transverse, by the time he was thirty, the most-requested surveyor in three provinces.

He spent thirty years doing this work. He divided, by his own careful count, one thousand four hundred and 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 colleagues at the land-registry office said he had the soul of a ratio.

When the GeometryForge academy was looking for someone to teach proportional reasoning — specifically the intercept theorem and the broader theory of transversals cutting parallel lines — the academy master had heard about Sir Transverse from his nephew, who had been one of the disputing parties in a particularly stubborn family-field division. The nephew said: "He is the only person I have ever met who treats fair division as a geometric theorem instead of an argument."

The academy master wrote Sir Transverse a letter. Sir Transverse, who was forty-nine and beginning to think his knees would not survive another wet season of field-walking, accepted.

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 the way he learned it — by walking. He lays a long strip of cloth on the floor (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. Walk across the cloth, holding your string taut. Choose any angle you like, but stay straight."

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

He then asks the students to walk a second transversal — parallel to the first — and the class measures the strip between them. The strip is always proportional. Always. Whatever angle the students chose, the ratio of the strip's width to the distance from the parallel boundaries holds.