Near the start of Lawvere and Schanuel's Conceptual Mathematics they ask:
The part of Galileo’s work which we discussed is really concerned with only a small portion of space, say the immediate neighbourhood of the tower of Pisa. Since the ground might be uneven, what could be meant by saying that two points are at the same level? Try to describe an experiment for deciding whether two nearby points are at the same level, without using ‘height’ (distance from an imaginary plane of reference.) Try to use the most elementary tools possible.
I can think of putting a spirit level between the two points, or putting a smooth plank between them and seeing if a ball at one point will roll to the other or vice versa. Or putting an altimeter (i.e. a barometer) at both points and seeing if the air pressure is the same. But from the last sentence, I think the authors had something else in mind. Does anyone know what the "correct" answer is?
Edit: The best answer I can think of is to hang a plumb line at each point, and see if they form a right angle with the line joining the points.
