The gyroscope is one of the least reliable devices known to man. I don't know why you would present this as evidence of a round earth other than trying to drown us in nonsense and dazzle us with barely relevant links.
The gyroscope rotates for the same reason the Foucault pendulum rotates.
Sure, if you really want to believe that gyroscopes are unreliable, I can't help you much. I see no evidence for this. I would invite you to read the following link:
http://en.wikipedia.org/wiki/GyrotheodoliteIt basically says that surveyors use gyroscopes to determine the location of north by supplying resisting torque to procession until the gyroscope settles in one direction without processional torque, parallel to the axis of the earth's rotation. These are often used in tunneling, notably in the chunnel.
If you really want to believe that stars/other celestial bodies produce torque on gyroscopes, I'd like to see some evidence.
But I really want to talk about the sagnac effect, which is the basis of of the last three papers, where each one uses a different type of wave to measure rotation. (lasers, atomic waves, waves in a superfluid)
I will focus on fibre optic gyroscopes/ring laser gyroscopes, as these are simple, obtainable(FOG's are), and easy to understand. Basically, a laser is fired through a half-silvered mirror. Half the light goes one way around a large coiled loop, and half the light goes the other way. They are then recombined and interfered. Due to rotation, after the light goes through in either direction, the wavelength of light will be slightly shifted for each direction. This will establish a beat frequency on the detector which is proportional to the angular velocity of the ring. The same equation holds for a ring laser as a fiber optic gyroscope (they are very similar):
δf beat frequency
A enclosed area
P perimeter (beam path length)
λ optical wavelength
n normal vector to A
Ω rotation vector
Source:
http://www.fs.wettzell.de/LKREISEL/G/LaserGyros.htmln (dot)
Ω = Ωsin(Ɵ), where Ɵ is shown as below with
Ω being on the z axis, and
n being the demonstrated vector:
There is a demonstrable way to determine Ɵ from knowing δf, A, P, λ, and Ω (the magnitude of the angular momentum vector of the earth).
And Ɵ happens to more or less be the latitude, with a small correction factor due to the earth not being a perfect sphere. There is similarly a correction term for east west.
Source:
http://www.fs.wettzell.de/LKREISEL/G/LaserGyros.htmlIf the earth were flat, even if the flat earth were rotating, Ɵ would not be dependent on latitude, it would be always π/2, so
n (dot)
Ω would always be Ω, which is not the case.
Note that this is a very accurate way of measuring rotation rates and has nothing to do with the principle of a regular spinning gyroscope, yet it too demonstrates that the earth rotates, and that latitude is the dominant factor in determining the angle of the "up unit vector" to of the earth's axis of rotation.