I checked the equations and calculated results on this chart/table and I agree with them.
The definition of the geometry of the problem is not quite right, but it does not make that much difference until you get out to further and further distances. But you will notice that the distance that they define as the drop, their variable X, if you think of it like a skyscrapper that you can't see because of the curvature, the further away it is the more it is leaning towards the observer. That is not realistic. The line should be defined not perpendicular to the line of sight, but instead perpendicular to the surface of the curved earth.
If the problem were set up with "skyscrapers" not leaning but instead standing straight, the equation would be
X = -R + (R^2 + L^2)^0.5
X is the drop, or the height of the skyscraper that you can just barely see
R is the Radius of the earth
L is the distance from the observer to the top of the skyscraper that you can just barely see.
The equation as written is valid as long as the units used are the same for all 3 variables, like miles, for example.
In words the equation says X equals minus R plus the square root of the quantity R squared plus L squared.
Now it just so happens that the equation that you often hear referred to as the drop X being 8 inches times the miles L squared is an approximation using the equation of a parabola.
X = 8*L^2 where L is in miles and X is in inches.
I have actually calculated the results for all these different equations on a spreadsheet for comparison. The simple parabola equation is close. At L = 1000 miles, the error in X is about 1.5%. At L = 2000 miles, the error increases to about 6%. And beyond that the error really sky rockets.
Interestingly the parabola equation is actually a little bit better of a fit to the more correct rigorous equation that I have shown above as
X = -R + (R^2 + L^2)^0.5
Here's an online earth curve calculator that is also useful.
https://dizzib.github.io/earth/curve-calc/?d0=8&h0=4&unit=imperial
Sincerely,
Mark
T. Mark Hightower