NavList:

David C you wrote: I cannot get my head around a line that seems to be neither one scale nor the other.

Most professional navigators were ‘trained’, which meant that they simply did what they were told initially and probably didn’t think too deeply about ‘the reasons why’ until they had a lot of practical experience under their belt, which they could use to stabilise their thoughts.  You’re attempting to teach yourself, which must be good for the brain but also very confusing and frustrating at times, so it’s probably not a good idea to work on it too close to bedtime.

Try this.  First, put your present attempts at understanding aside for a second.  Second, accept for a moment that the Earth is a perfect sphere.  It jolly nearly is.  I once read the difference, 6378:6357, is the order of a postage stamp stuck on a football, but that could just be a such a good story it’s not worth disproving.  Third imagine a line on the Earth from the observer to the point on the Earth directly under the star, the sub stellar point (SSP).  This line will be a part of a great circle.  Its direction measured from the observer is the star’s azimuth, 045° say. The length of an entire great circle around the Earth is 360x60 minutes of arc measured from the Earth’s centre.  It’s also 360x60nm.  If Ho is 10’ say greater than Hc, then the observer must be on a line of position 10’ of arc along the great circle from the assumed position towards the SSP.  1’ of arc along the great circle is also 1nm, so the LOP is also 10nm along the great circle towards the SSP.  This is true whatever the star’s azimuth from the assumed position turns out to be. 

The meridians are also great circles; therefore, we can set our dividers to 10nm by measuring 10’ of arc up a meridian line.  We need to be careful with charts which are not constant scale e.g Mercator’s charts.  In this case we must ensure we set our dividers from a part of a meridian close to the assumed position.  Hope this helps.  DaveP