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Ruminations on Bygrave and Hohenrechenschieber scale markings
From: Robin Stuart
Date: 2013 Feb 10, 08:49 -0800

In the course of developing and enhancing my Postscript code to produce Bygrave scales http://fer3.com/arc/m2.aspx/Bygrave-Stuart-apr-2011-g16126 I used Ronald van Riet’s “Position Line Slide Rules: Bygrave and Höhenrechenscheiber” and in particular his descriptions of the scales as a key resource.

In the document on page 18 for the HR1 it says

Over the full length of the scales a base line is present as is a line parallel to it at a distance half the length of the major markings, which appear at each full degree. In addition, the following markings are present:

Log cosine scale:
From 3d on 3/4 length line at each half degree (30’)
From 10d on 2/3 length line at each 10’
From 20d to 45d 1/2 length line at each 5’
From 45d to 60d 1/3 length line at each 2’
From 60d on 1/2 length line at each 5’ and 1/3 length line at each 1’

This is identical to the description for the Bygrave on page 8 but is not consistent with what is visible in Figure 23. There one sees

The line parallel to the baseline is at 2/3 the length of the major markings rather than half.

From 3d to 10d the degree marks are 2/3 line length except 0d, 5d and 10d
From 10d on the degree marks are major marking line lengths
From 3d to 10d half degree marks are 1/3 line length
From 10d to ??d half degree marks are 2/3 line length
From 10d to ??d 10’ marks are 1/3 line length
etc.

There is also a feature that is can be seen in both the Bygrave in Figures 9 and 10 and the HR1 that has not been explicitly noted. Beyond some point, unlabeled half degree marks are indicated by having a dot on them. In the HR1 they are fairly large and connected to the scale mark. On the Bygrave they much smaller and do not touch the scale mark. They are present on both the cosine and (co)tangent scales.

If I had to summarize the principle used to lay out of the HR1 log cosine scale it would be: There are 3 possible line lengths; 1, 2/3 and 1/3 of the major mark. Beyond 10d, major marks are used at all round degrees with the marks at labeled degrees being wider than those elsewhere. The 2/3 line length is used to denote major subdivisions a degree (which depends on the position along the scale) and the 1/3 for all others.

The MHR1 in Figure 32 seems to follow a similar pattern except that the lines at full degrees extend all the way up to the baseline of the scale above.

It is not clear that expressing the scale marking lengths as (approximate) fractions of the major markings is the most appropriate way of recording this information. I suspect that it might be better to measure the marking lengths in the absolute units in which they were designed. For the Bygrave that would presumably be 1/32nd’s of an inch and for the HR1 some fraction of a mm. In the case of the HR1 the scale baseline is relatively broad and it would need to be specified whether lengths are measured from its bottom or top. A high resolution photograph of the scales with a precision rule laid against them would be a very useful resource.

I hope that these observations might be helpful in preparing the next release of Ronald’s valuable paper.

Robin Stuart

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