Splay the Scale-Keeper

Her given name was Furlong, the same as the town, though she had been called Splay since she was ten, when she copied her father's great wall-map onto a card small enough for a shepherd's pocket, tripling nothing and shrinking everything by the same careful factor, and the shepherd came back a season later to say he had never once been lost.

She understood, young, what a scale factor is. It is a single number that multiplies every length. Shrink by a factor and you get a map; enlarge by a factor and you get a plan a builder can work from. And the one iron rule, the one her father repeated until it was part of her, was that the factor must touch everything equally. You cannot scale the length and forget the width. You cannot enlarge the door and leave the window. A shape stays true only when every length is multiplied by the same factor; the moment you play favorites, the shape distorts and the map lies.

A cartographer from the RatioRealm academy came to Furlong for a map of the northern passes and found Splay, grown now, teaching an apprentice to read a scale-bar — how many real miles hid in one inch of paper.

"You are working with scale factor," the cartographer said. "Scale drawings and maps. Our students grasp the arithmetic but forget that the factor must be the same for every measurement, and their shapes come out crooked. You have the iron rule."

Splay had never wanted to leave the valley. But she liked the thought of children learning to hold big things faithfully in small hands. She has taught scale for seven years.

In her classroom Splay keeps a small drawing of a house and a scale-bar beneath it: 1 finger-width = 2 strides. She begins the same way each year. "This drawing is small," she says, "but it is true — every length was shrunk by the same factor. The door is one finger-width on the paper; how tall is the real door?" The children read the scale-bar: one finger-width is two strides, so the door is two strides tall. "And the wall, three finger-widths on the paper?" Six strides. "The chimney, four?" Eight. The children see that reading a scaled drawing is just multiplying every paper-length by the same factor. Then she shows them the danger: she draws a second house, but she cheats — the door scaled by two, the roof scaled by three — and the class laughs, because the poor house comes out lopsided, its roof sliding off. "That," says Splay, "is what happens the moment the factor stops being the same. The shape only stays true if every length gets the same multiplier."