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

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Evolution of the Flat Bygrave
From: Robin Stuart
Date: 2021 Jul 12, 07:45 -0700

Calculations by slide rule fundamentally involve transferring lengths from one logarithmic scale to another. A Bygrave slide rule consists of two concentric cylindrical scales which are popularly described as being an inner (log-)cotangent scale and an outer (log-)cosine scale.

In Gary Lapook's original flat Bygrave https://sites.google.com/site/fredienoonan/other-flight-navigation-information/modern-bygrave-slide-rule transferring lengths between scales was achieved by having the cosine scale printed on transparent material that could be overlaid on the cotangent scale. In order to ensure that any required angle can be accessed without running off the edge of the sheet all values on the scale appear twice. That is to say, an angle marker appearing near the middle of some row in the scale will be repeated on the left hand side of the scale in the next row up.

In 2014 http://fer3.com/arc/m2.aspx/Flat-Bygrave-alternative-configuration-Stuart-mar-2014-g27166 I pointed out that this repetition could be avoided if two 0° markers were provided on the cosine scale. One of these sits at the left hand end of the bottom row and the other is at the right hand end of the next, unprinted, row down. This means that the scale lengths can be doubled for the same area of paper compared to the original design. The procedures for using them is analogous to those of a standard slide rule in which, depending on where the numbers fall, either the left or right hand end of the sliding scale is aligned with a number on the fixed scale.

Postscript code for producing Bygrave scales with full explanations in various configurations can be found here http://fer3.com/arc/m2.aspx/Postscript-code-for-making-Bygrave-Scales-Stuart-jan-2015-g29918. Additional information concerning configuring Ghostscript can be found here http://fer3.com/arc/m2.aspx/flat-Bygrave-Stuart-feb-2017-g38421.

Before his passing in 2015 Hanno Ix experimented with this design and used a beam compass to transfer lengths from one scale to the other http://fer3.com/arc/m2.aspx/Flat-Bygrave-alternative-configuration-Stuart-jul-2014-g28182. These scales were designed to be printed on A3 paper. When using a beam compass it is necessary to keep track of the number of horizontal rows the compass pointers span and preserve this when transferring lengths between scales. In order to assist in this he had me number the rows in both scales. He also had me label the two cosine scale pointers “A” and “B” to aid in explaining the steps required to perform calculations http://fer3.com/arc/m2.aspx/Borrowed-Bygrave-Stuart-jan-2016-g34208. Although Hanno seemed to be satisfied with this approach I always thought it was inconvenient and that there must be an easier way keep track of this information without having to perform any mental subtractions.

I have built a prototype “square compass” that satisfies this requirements. It consists of pointers that slide on 2 arms at right angles to each other. One of the arms is laid parallel to the horizontal scales when setting the pointers and the orientation is maintained when moving to the other scale. The attached document gives an illustrated example of how to carry out calculations with a flat Bygrave and the square compass. Some enhancements have also been made to the scales originally produced for Hanno Ix and these are attached,

Robin Stuart

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