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 <

cfuhb-ac...@earthlink.net> 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.