Well, here are my 2 cents on this topic.
Obviously you cannot use any physical line as it would have mass and the center would dip into the water with the appearance of a convex surface whether or not there actually is a curvature to that surface. We all agree that we need a mass-less way of measuring between 2 objects of significant distance on the water.
Lasers were suggested, but again you have the problem of calibration and aiming the laser consistently at the sensor.
This may sound a little old school, but line of sight works just as well as a laser beam but approaches this from the other angle.
Setting up the experiment is simple enough, and would be best to do from 2 islands that are ideally spaced to eliminate the discrepancies caused by the rise and fall of a ship on the waves... But a ship works too... you just have to keep the wave effect in mind that will change the height at which observations are made.
Take two ships of significant and (ideally uniform) size that they can be easily seen from a vast distance.
From the surface of the water and up one ship, hang brightly luminous lanterns at every 5-10 feet vertically.
Set watches on the second ship (or better yet from a tower on the shore of an island) such that eye level is at every 5-10 feet above the surface of the water. (telescopes and spyglasses and other magnifying devices are advisable for more accurate viewing at much greater distances)
Send the first boat away from the watch-point at a constant rate, and log for each watch point at what times you loose sight of each successive lower beacon of light until the top light is no longer visible.
For FET, all lights should disappear into the vanishing point while approaching the horizon uniformly and at approximately the same time. They would appear to merge together first and there should not be a difference due to the different heights of the various viewpoints.
For RET, eventually the ship will rise to the top of the horizon and then appear to sink below the horizon on the other side, the lower viewpoints will lose sight of the lower beacons of light first and the higher viewpoints will lose sight of the same lights at a later time-frame. As with FET, the lights will appear to approach each other, but with the aid of magnifying devices, you should see the incremental dropping of the lights from view sequentially from bottom to top and at different time intervals based on your different eye levels from your watch points.