“13) In a 19th century French experiment by M. M. Biot and Arago a powerful lamp with good reflectors was placed on the summit of Desierto las Palmas in Spain and able to be seen all the way from Camprey on the Island of Iviza. Since the elevation of the two points were identical and the distance between covered nearly 100 miles, if Earth were a ball 25,000 miles in circumference, the light should have been more than 6600 feet, a mile and a quarter, below the line of sight!”
Interestingly, the Free Masons are responsible for the ball Earth conspiracy and yet the flat Earthers cite the work of 19th century scientist and Free Mason, Francois Arago.
Francois Arago was involved in a project with Jean-Baptiste Biot to complete the meridian arc measurements started by JBJ Delambre. I believe this is the study referenced, since it placed him in the mountains of Spain. I cannot find the specifics published on how his research was conducted. I did find this book: The Shadow of Enlightenment: Optical and Political Transparency in France.
The two were trying to measure the length of the ball Earth’s meridian to establish the size of a meter which was defined as one ten millionth of a quarter of the Earth’s meridian. To do this, Arago positioned himself on one of the peaks of Desierto de las Palmas (presumably on Bartolo peak which would afford the greatest view from 2391 ft). Biot then traveled along to other areas trying to obtain the highest elevation he could. They then lit fires, amplified the light with mirrors, and used a Borda repeating circle to measure angles between triangulation points.
On the Ibiza, it is noted they had their greatest challenge because of the distance which they thought might be impossible. Biot took his position on the mountaintop Campvey (I am unsure what the elevation is of this mountain, but the highest elevation on the island is 1558 ft). In fact, they used 8 mirrors to attempt this instead of the usual 2 or 3. After 6 months of attempts, only Biot saw Arago’s light. Arago did not see Biot’s.
After my research, two things stand out. First, Arago did not see Biot’s light and so it is at the very least difficult to accomplish this task and possible that Biot falsified his reporting (there were other tensions ongoing with hostility between France and Spain that could have encouraged him to do so). Second, the two were not at the same height. The question remains therefore, could they see each other on ball Earth?
To do this I constructed this crazy diagram of the Earth with an observe at a height (h), trying to view a light or other object and a different height (x). The angles a and b can be calculated using trigonometry since the radius (R) of Earth is known. The distance between these two locations is listed as 100 miles by the author above and 150 km (93 mi) in the book. So if the combined distance of the two locations is equal to or greater than 93 miles then it is possible to see even on a ball Earth.
Remembering in trigonometry that the cosine of an angle is equal to the adjacent side of a right triangle divided by the hypotenuse you end up with the angle identified by the following equations:
Then once having the angle and remembering that each mile of earth circumference is equal to 0.01446 degrees, it is trivial to calculate the distances.
Using the height of 2391 feet for Arago and 1558 feet for Biot, I calculate that they could see each other from 108 miles apart (within the actual distance).
Lastly, the calculated loss of height equal to 6600 feet is for an observer’s height equal to zero. This is completely different than the actual problem.