length of the nautical nile

[dClbydalpha]

Radius vs radius of curvature

I think the problem is in using the term radius.
I'll try to do this verbally rather than mathematically, so please excuse any clumsy use of words.

The term radius of the earth is used to refer to the distance to the centre (or more correctly the focus) of the ellipse.

In order to carry out the calculation of the length or arc, you need to know the radius of curvature. This is the radius of an equivalent circle that would give the same curvature at that point. As the earth is flatter at the poles, so the radius of curvature is greater and therefore the length of arc required to subtend a given angle.

To try and visualise this imagine squashing a ball. Underneath your hand the surface is nearly flat, which implies a large radius of curvature (a flat plate having an infinite radius of curvature). At the edges the curvature is greatest, giving a small radius of curvature. Push enough and the shape almost becomes a running track with a tight circle at either end and a flat upper and lower surface.

Imagine looking at a cross section through a line of longitude and trying to draw a circle that touches the earth at the poles, this would have to be larger than the earth as the earth would start to curve away more towards the equator. Similarly any circle that touches the equator, would be smaller than the earth as it would curve faster inside the earth as the earth started to flatten. This leads to the effect you have quoted.

In reality it is not that simple, as the Earth is not simply a squashed sphere, but I hope that helps you visualise the answer to your question.

Edit: Just to be clear the international nautical mile is defined as exactly 1,852m. What I think is under discussion is now called a sea-mile, which varies with latitude. The value 1,852m is the closest integer to the mean value of a sea-mile.