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    Bygrave formula accuracy on 10 inch slide rule
    From: Paul Hirose
    Date: 2009 Jul 06, 15:19 -0700

    My computer simulation of the Bygrave sight reduction formulas, worked
    on a 10 inch slide rule, had altitude accuracy of 1.8' and azimuth
    accuracy of 2.0'. Those are the square roots of the mean squared errors.
    In a run of 500,000 random sight reduction problems, 95% of the 
    solutions were correct within plus or minus 3.7' in altitude, and 95% 
    were within plus or minus 4.2' in azimuth.
    The maximum altitude error seen by the program was 10.4'. Worst cases
    always seem to occur around 40° - 50° altitude. Note that the Bygrave
    solution reads altitude on the tangent scale, which is most compressed
    at 45°.
    The maximum azimuth error seen by the program was 28.0'. Worst cases
    occur at high altitudes.
    I was suspicious of the accuracies reported by the test program. They
    seemed too good, so I worked six random problems (generated by the
    program) by hand on a 10 inch rule. Altitude errors (minutes) were +1.1,
    -5.2, 0.0, +1.1, -2.3, -.6. Azimuth errors were -.8, +.1, +.3, -1.3,
    +1.1, -1.4. These results suggest the program's modeling of slide rule
    errors is realistic.
    To operate the slide rule I wore reading glasses but did not use my 
    hands free magnifying glass, though it would have helped a good deal.
    My program generates each sight reduction problem from a random azimuth
    and altitude, the latter being weighted so the simulated stars tend to
    have constant density everywhere in the sky instead of packing closer
    with increasing altitude. Altitudes less than 5° or greater than 80° are
    A random observer latitude between 0° and 70° is generated in similar
    Each azimuth, altitude, and latitude combination is converted to LHA and
    declination. The sight reduction module converts these values back to
    azimuth and altitude, injecting a random error in each slide rule
    operation, then compares results to the correct values.
    Slide rule error is assumed to be .1% RMS per multiplication or division
    (which involves two settings and one reading). The Bygrave azimuth
    formula requires *three* settings and one reading, so for that
    calculation I increase the error accordingly.
    I have modeled the Bygrave formulas on a standard slide rule, but not
    the Bygrave rule itself. Its error should be in inverse proportion to
    its scale length relative to a 10 inch rule.
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