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    Re: Comments on Bygrave/MHR1 calculations
    From: Gary LaPook
    Date: 2016 Aug 8, 01:44 -0700

    Both the Bygrave and the MHR-1 have scales going all the way to 180 degrees so the Bygrave produces azimuths angles that are always referenced to the opposite pole. The MHR-1 could do it the same way but the instructions have you limiting the azimuth angle to 90 degrees so they must be related to either pole and there are rules provided to accomplish this. My recreations of the Bygrave have scales going only to 90 degrees to avoid clutter so my azimuth angles are also referenced to each pole and I provided rule for this. 

    Bygrave provides a way for calculating the altitude when the declination is outside the limits of the sllide rule scales. Conspicuously missing, however, is a method for finding the azimuth is such cases. Nor does the MHR-1 provide such a method. I developed a way to find the azimuth in these special cases. 

    ----------------------------------------------------------------------------------

    So your MHR-1 doesn't provide for cases where the declination is less than 
    20', the lowest mark on the cotangent scale. Bygrave says to use the same 
    procedure in this case as in the case where the azimuth is near 90?, simply 
    interchange the declination and the latitude and then compute the altitude 
    and this works fine and you get accurate altitudes. But the azimuth that you 
    derive in this process is not the correct azimuth and is thrown away and not 
    used for plotting the LOP just as in the case of azimuths near 90?. Bygrave 
    gives no instruction for computing the azimuth in this case. It cannot be 
    computed in the normal way since the first thing you need to to is find 
    declination on the cotangent scale and values less than 20' are not on the 
    Bygrave or MHR-1. This is an important special case since the sun's 
    declination is in this range for several days around each equinox. I 
    developed an approximation that works well giving azimuths within one degree 
    of he correct value and usually much closer. I simply skip the first step, he 
    derivation of "y" (Bygrave's terminology), "W" (my terminology). I simply set 
    "y" equal to declination and then proceed normally.

    http://fer3.com/arc/m2.aspx/New-compact-backup-CELNAV-system-RENAMED-Accuracyof-Bygrave-Slide-LaPook-apr-2009-g7947

    I also found a way to produce an exact azimuth for this special case:

    http://fer3.com/arc/m2.aspx/Computing-azimuth-with-Bygrave-special-cases-LaPook-feb-2010-g11826

    http://fer3.com/arc/m2.aspx/New-compact-backup-CELNAV-system-RENAMED-Accuracyof-Bygrave-Slide-LaPook-apr-2009-g7911

    http://fer3.com/arc/m2.aspx/New-compact-backup-CELNAV-system-RENAMED-Accuracyof-Bygrave-Slide-Morris-apr-2009-g7917

    http://fer3.com/arc/m2.aspx/All-Bygrave-slide-rule-posts-LaPook-jan-2014-g26194

    http://fer3.com/arc/img/106329.bygrave%20manual.pdf

    This link gets you an index to all the posts containing the word "Bygrave" 

    http://www.fer3.com/arc/sort2.aspx?y=1996&y2=2016&subject=bygrave

    And this picks up some additional posts:

    http://fer3.com/arc/m2.aspx/All-Bygrave-slide-rule-posts-LaPook-jan-2014-g26195

       
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