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    Re: Bygrave formula accuracy on 10 inch slide rule
    From: Gary LaPook
    Date: 2009 Jul 7, 22:06 -0700

    The rms of 1.8 minutes using the Bygrave method is much better than
    the 4.7' error level you found using the standard sine-cosine formulas
    and confirms what everyone has written about the accuracy of the
    Bygrave. As you point out, the longer scale on a Bygrave would result
    in an even higher level of accuracy. Since you found 1.8 with just a
    ten inch standard slide rule the Byrave, which has a scale 7.85 times
    longer than a ten inch rule, would produce much better accuracy and
    certainly no worse accuracy than the ten inch rule. So the claim of
    one or two minute accuracy with a Bygrave appears to be validated and
    is consistent with my own tests on my recreation of the Bygrave.
    
    gl
    
    On Jul 6, 3:19�pm, Paul Hirose  wrote:
    > 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
    > rejected.
    >
    > A random observer latitude between 0� and 70� is generated in similar
    > fashion.
    >
    > 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.
    >
    > --
    > I filter out messages with attachments or HTML.
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