I'm having some trouble understanding this:

On the Earth's surface, your lattitude is Φ, and your longitude is Θ. Assuming the Earth is a perfect sphere, your posisiton on the Earth's surface is given by: (R is radius of Earth)

P(Θ,Φ) = ( Rcos(Φ)cos(Θ), Rcos(Φ)sin(Θ), Rsin(Φ) )

so, ∂P/∂Φ = < -Rsin(Φ)cos(Θ), -Rsin(Φ)sin(Θ), Rcos(Φ) >

so, |∂P/∂Φ| = R (I won't write the working)

Now, ∂P/∂Θ = < -Rcos(Φ)sin(Θ), Rcos(Φ)cos(Θ), 0 >

so, |∂P/∂Θ| = Rcos(Φ)

Now what I don't get is if P is your position on the surface of the Earth (or any uniform sphere), then why should the rate of change of your position with respect to your latitude..

|∂P/∂Φ| = R

... be any different to the rate of change of position with respect to your longitude?..

|∂P/∂Θ| = Rcos(Φ)

I must be missing something obvious here, because it doesn't make sense that the rate of change of position on the Earths surface should be different when going 'over' the sphere, to going 'around' the sphere....not if the sphere is a perfect sphere.

Cheers