There is a workshop in the old part of town where the owner has never owned a protractor, and every corner he has made in thirty years is true.
His method is older than his tools. When a new frame has to sit at a right angle to the wall, he measures three metres along one edge and marks it. He measures four metres along the other edge and marks that. Then he measures the diagonal between the two marks.
If that diagonal is exactly five metres, the corner is true. If it comes out at 5.2 metres, the corner is open — out of square — and the frame will rock for the rest of its life. If it comes out short, at 4.8, the corner is too tight and the door will bind every winter.
Three, four, five is the smallest set of whole numbers that does this, and it works at any scale: 30 centimetres and 40 centimetres give a 50-centimetre diagonal; six metres and eight give ten. The apprentice is taught to use the biggest triangle the room allows, because a small error in a short measurement becomes a large error across a long wall.
Two things are worth noticing. The carpenter never measures the angle itself — he measures three lengths and lets them prove the angle. And he does it before the nails go in, not after.
The sign above his bench says only this: Check, then cut.