Splay the Scale-Keeper

She makes them work it in both directions, because Advanced students must travel the scale both ways. "The map says the road is four inches long, at a scale of one to a hundred. How long on the ground?" Four hundred inches. "The real bridge is thirty feet; how long do I draw it at one to a hundred?" Point-three of a foot — a hair over a third of an inch. "Multiply to go from paper to ground; divide to go from ground to paper. The scale factor is the bridge between the two worlds, and it works in either direction as long as you keep it the same for every length."

Then she springs the trap the whole lesson is built around. "Enlarge a square garden by a scale factor of three. Each side triples — a two-yard side becomes six. But how much bigger is the garden — the area you must plant?" The children say three. Splay shakes her head. "Watch. The old garden is two by two — four square yards. The new one is six by six — thirty-six. Thirty-six over four is nine. Not three. When every length triples, the area grows by three times three — by the factor squared."

She lets that settle, because it overturns their first instinct. "Length scales by the factor. Area scales by the factor squared. Double every length and the area quadruples; triple them and the area grows nine-fold. It is why a map at one to a hundred packs ten thousand real paces of ground into one paper pace of area — a hundred squared. If you ever scale a shape and its area grows by the plain factor, you have made an error somewhere, because area cannot help but square."

A quiet boy named Reave, who wanted to enlarge a drawing of his father's boat, had made exactly that error — tripled the length, tripled the sailcloth, and run desperately short of cloth. Splay found his working. "Your lengths were perfect," she said. "But sail is an area. Triple the boat and you need nine times the cloth, not three. The shape stayed true — you scaled every length by the same factor, which is the hard part, and you got it right. You only forgot that area lives in two directions at once, so the factor has to act twice."

Reave redid it, nine-fold, and the paper boat came out true. What stayed with him, he told her, was not the correction but the fairness underneath the whole craft: that a shape holds together only if you shrink or grow every part by the same amount — no favourites, no shortcuts.

Splay said that was the maker's oldest discipline. "The moment you scale one length more than another, the shape lies. A faithful map, a true enlargement, an honest model — they all rest on the same even-handedness: one factor, applied to everything." Reave looked at the small true boat beside the large one and felt the specific satisfaction of a copy that had kept its shape.