I would be shocked if anyone could measure sqrt(2) to half a part per million using ropes. You'd need to be accurate to a half mm per km of rope, which is well

I would be shocked if anyone could measure sqrt(2) to half a part per million using ropes. You'd need to be accurate to a half mm per km of rope, which is well beyond what you need for practical surveying. You also need to keep the rope essentially perfectly straight, which means doing the experiment on almost perfectly flat ground. You'd need to keep the rope at a controlled tension so it doesn't stretch. You'd be hosed by temperature and humidity changes while walking those kilometers. The rope would also need to be marked or measured to < 1mm accuracy when you're done walking. You'd need to make sure the right angle is square to < 1ppm as well, though maybe there's a clever procedure that corrects this. It's not even easy to do this with laser surveying equipment.