Now wondering what this implies for the Bedford level!
Suppose the water really is flat, but that the light curves below the straight line joining the bridges. And suppose k = -1, so the curvature of the light is equal to the hypothetical radius of the earth r, 4000km. I think we can solve this by Pythagoras. Let m be the midpoint of about 1km. Finally let h be the apparent height of the midmarker over the line of sight.
h = r - sqrt(r^2 - m^2) = 7cm (or 2.7 inches)
So if the water was flat and there were negative refraction of 1, it would still not be enough to explain what they actually saw.