The fountain in the town square had been broken for two years, and when the engineer finally came to repair it, half the town came to watch.
She set the pump and stood back. The water left the nozzle, rose in a long curve, and came down into the pool. She watched it for a while and then took out a notebook.
"Three metres out from the nozzle," she said, "it reaches its highest point, and there it is nine metres above the water. Six metres out, it comes back down to the surface. That is all I need to know about this fountain."
A boy watching asked how she could be so sure about the far side, since she had only measured the near one. She said the curve was symmetrical. Whatever the water does on the way up, it does again in reverse on the way down. The highest point sits exactly midway between the place it leaves the water and the place it returns to it.
Then the mayor asked her to make it taller. She turned the pump up, and the whole curve grew: the top climbed higher and the water landed further out. But the shape did not change, and the highest point stayed exactly halfway along. Every setting of that pump gave a different peak, and the peak was always in the middle.
"That is the useful thing about this curve," she told the mayor, closing the notebook. "You only ever have to find one point on it. The rest of the arch is folded around that point."