Across a six-mile reach, three equal-height markers should put the middle one about 6.0 feet above the straight line joining the ends. Wallace’s 1870 field diagrams show that pattern. Rowbotham’s earlier low-eye boat sighting tested a different line through air immediately above the water.
The two historical accounts are often treated as one experiment. Their observer heights, targets and comparisons differ, so they need different calculations.
Rowbotham described watching a boat travel six statute miles with his eye roughly eight inches above the canal. In the 1881 account, the flag top was five feet above the water. He said the boat and flag remained visible and interpreted that as a flat-water result.
The 1870 field report placed a telescope and two distant signals at 13 feet 4 inches above the water, spaced three miles apart. The middle signal did not align with the line from the observing endpoint to the far marker. Wallace reported the middle signal roughly 5½ feet from that line in a field sketch.
On a sphere, the same height above water at each station is not the same height above a single, six-mile straight line. Using a mean Earth radius of 6,371.0084 km, the midpoint above the endpoint chord is 6.00 feet. A horizontal tangent at one endpoint gives about 24 feet of drop at the far end; that is a different line and must not be substituted for the six-foot midpoint result.
Wallace’s roughly 5½-foot value came from the recorded optical diagram, not a modern precision survey. A simple effective-radius illustration gives 5.22 feet at an assumed refraction fraction of 0.13, or 4.80 feet at 0.20. Those are sensitivity examples, not measurements of the air over the canal on that day.
For an eight-inch eye, a straight ray just grazing a smooth spherical water surface reaches a geometric horizon about 1.00 mile away. At six miles, the far water lies behind that horizon by a sight-line clearance equivalent to 16.67 feet of target height. A five-foot flag would remain about 11.67 feet below the limiting straight ray in this idealised model.
That calculation does not establish what Rowbotham actually saw. Light close to water can bend through temperature layers, while the 1881 account supplies no measured air profile for his observation. It reports his sighting and interpretation; it does not provide the conditions needed to reconstruct the optical path. Wallace raised the markers and compared three positions, so his test asks a more direct geometric question.
The three-marker field result agrees with a curved water surface and an end-to-end chord. Rowbotham’s reported low-eye visibility remains a historical observation with an unmeasured near-water light path; it cannot by itself establish a flat surface. A modern repeat would record marker heights, water level, sight-line angles and temperature gradients together.
An archived July 2016 post from this site reported a group revisit of the canal that June. Its captured text gives no instrument heights or optical readings, so it cannot replace the dimensions and diagrams in the earlier records.