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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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```
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