The dig had uncovered the base of an old arch, and half of the arch itself. The other half had come down some centuries ago and lay in pieces in the sand.
The team's youngest member wanted to know how tall it had been. The archaeologist leading the dig said they could work it out from what was left, and she was confident about it.
She began with the two feet of the arch, both still in place. They stood twelve metres apart. The crown of an arch, she explained, always sits exactly halfway between its feet — six metres in from either side. That much they knew without measuring anything at all.
Then she used the standing half. Three metres in from the left foot, the surviving stonework reached a height of four metres. By symmetry, the fallen half must have reached exactly four metres at three metres in from the right foot. Every point had a partner on the other side, at the same height.
Those three measurements — two feet and one point on the curve — were enough. She fitted the arch's curve to them and read off the height at the crown.
The young man asked whether she was guessing. She said she was not: the shape of an arch is fixed by three points, and they had three. "The fallen half is not lost," she said. "It is mirrored in the half that stayed up."