Edby,
I don't even hold that the longitude lines would widen, myself, as I have always been a proponent of the Bi-Polar model. I just see that the analysis on this matter appears to be flawed.
Rowbotham is actually referring to manual ways to find longitude, as was done in his time... The spherical geographical model that is associated with these planar maps is literally a sphere.
I'm not clear what the problem is TBH. Let's say we could pick somewhere suitably far south (and I think New Zealand is a good example).
We then identify a couple of places as far south as possible, a reasonable distance apart with a fairly straight road between them running approximately E-W. Pick two endpoints on this road. Can we find the distance between the two points? Well what would you trust? We can ask Google maps for a route, would you trust the result? Personally I find distances Google maps give me are accurate enough and correspond with reality, you may disagree, I don't know. We could look for road signs along the route with distances (exactly as edby has done in his examples). Would you (within reason) trust them? Again, personally I would tend to. Or perhaps you could suggest another way to determine the distance between the endpoints? Maybe we can find a map you would agree is accurate and estimate distances directly from the map?
Next, can we find the longitude and latitude of the two end points.
Do you agree that in principle we could go anywhere on earth and find through some means or other our latitude and longitude, with or without a suitable map?
A simple way to do this would be to get them from Google maps. A question again of trust. Personally I'd expect to be able to visit a random spot on earth, use "manual methods" (e.g. accurate clocks, a nautical almanac and a sextant) or a GPS device or otherwise to determine position and then find this in close agreement with Google Maps. I'd be very surprised to find a discrepancy of more than a few hundred meters for example.
Do you agree that we can trust Google Maps to give us accurate values for these positions? If not, is there some method you would accept?
If (hypothetically) we get past this point, we have two places a known distance apart then surely it's a simple calculation to determine distance divided by difference in longitude to find the width of a degree of longitude at that particular location.