# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

Message:αβγ
Message:abc
 Add Images & Files Posting Code: Name: Email:
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
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.

--~--~---------~--~----~------------~-------~--~----~
To post, email NavList@fer3.com
To unsubscribe, email NavList-unsubscribe@fer3.com
-~----------~----~----~----~------~----~------~--~---

```
Browse Files

Drop Files

### Join NavList

 Name: (please, no nicknames or handles) Email:
 Do you want to receive all group messages by email? Yes No
You can also join by posting. Your first on-topic post automatically makes you a member.

### Posting Code

Enter the email address associated with your NavList messages. Your posting code will be emailed to you immediately.
 Email:

### Email Settings

 Posting Code:

### Custom Index

 Subject: Author: Start date: (yyyymm dd) End date: (yyyymm dd)