On the first morning of the job, the roofer and the surveyor gave the architect two different numbers for the same roof, and both of them were right.
The roofer measures from the beam that runs along the top of the wall. Standing on it, he looks up along the slope and reads thirty-five degrees. That is the number his trade has always used, because it tells him how hard the climb will be.
The surveyor measures from the post that stands at the ridge. Looking down along the same slope, her instrument reads fifty-five degrees. That is the number her office needs, because it tells her how much of the roof will show on a plan drawn from above.
The architect wrote both in her notebook and drew a small triangle beside them. The corner where the post meets the beam is a right angle, and the three corners of any triangle come to one hundred and eighty degrees. The two remaining angles must therefore add to ninety. Thirty-five and fifty-five. Two names, one roof.
There is a second consequence, and it is the one the calculator cares about. The roofer's ratio of the height of the roof to the length of the slope is exactly the surveyor's ratio of the same height to the same slope — the two of them are describing one pair of sides from opposite corners. What one calls the sine, the other calls the cosine, and the number underneath is identical.
"So we never have to convert," the roofer said.
"We never did," said the surveyor. "We just assumed we disagreed."