You peg the short side: three units of string. The long side: four. The diagonal: five. When the five sits without argument, the corner is square. It is not magic. It is a fact the Greeks mailed forward through a great many other people who also did not want walls to wander.

Gareth watches, arms folded, then unfolds them despite himself.

You draw the pointed rib next. Two centers, two arcs, a peak that can be taller than a round arch on the same span. Maud, who can lecture with a piece of charcoal, says it once:

“Round arch: half a wheel. Height is chained to width. It shoves out like a man spreading his elbows in a crowd. Pointed: the elbows come in. More of the weight goes down the piers. Then you can dare windows—if the flyers take the rest.”

The drawing on the plaster is the vault’s twin, lying down. Templates will be taken from it: thin boards cut to the curve, carried to the bankers, used until the stone agrees.

The south buttresses wait. Canon Odo waits with money-thoughts, which are also stone-thoughts if he is honest.

You write the numbers in the plaster with a stick: 3 along the west of the bay, 4 along the south, 5 across. When the 5 is tight, the corner does not argue. Pointed peaks go up. Iseult will someday pour color into the spaces this math buys. Gareth mutters that he can square a bay too. He can. He still wants a half-wheel on piers that were not poured for a half-wheel.