NavList:

May I please ask some questions?

1. I am not a sailor. So, what are the useful ranges of a Bygrave? In  other terms, what are the critical trigonometrical/nautical Regardssituations where the knowledge of angles >89deg or angles <1deg  is important?

Suppose your celestial body is the sun.  If reducing for a meridian crossing (sun on your line of longitude), then the hour angle T is 0 (less than 1 degree).  As a practical matter, the Bygrave is quite compressed under 5 degrees for the LogCosine scale, so reductions when the sun is close to your meridian suffer from accuracy issues.

If reducing when the sun is directly east or west of you, when the hour angle is 90 degrees (sunrise, sunset) similar issues result.

2. Are there practical approximations by polynomials etc. for such extreme cases?

Just use the standard azimuth and altitude equations that can be directly reduced using a calculator.

3. More generally: Are we aiming at historical replicas of the Bygrave, MHR, etc.  or are we seeking at a modern implemetation of the idea? Or both?

Depends upon who you talk to.  Due to cost, we probably won’t be looking at historical replicas. 

4: Similarly: Are we interested in the Bygrave just because it is so interesting - which it is! - or are we looking for the very simplest means to do accurate sight reduction? For the latter, there might be attractive alternatives.

HO229 offers accuracies to 0.1 arc-minutes, about 10-20X the accuracy of the Bygrave.  It is heavier, larger and requires tabular lookup instead of slide rule manipulation. 

Others will have different preferences for tabular reduction or for computer aided reduction.  The beauty of the Bygrave is the small size, small weight and rapidity of reduction

Regards</table