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On my original Bygrave re-creation both scales are 1.5 inches in 
diameter, within six-thousandths, the thickness of two layers of plastic 
sealing sheets since the cosine scale is fashioned inside these plastic 
sheets. On my HR-1 replica the cotangent scale is 1.61 inches in 
diameter and the cosine scale is 1.75 inches in diameter, a ratio of 1 
to 1.08 and the cosine scale is only .07 inches above the cotangent 
scale which minimizes any parallax problem.


I posted before at:

http://www.fer3.com/arc/m2.aspx?i=106329&y=200809

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British captain Leonard Bygrave invented his celestial navigation 
computer in 1921. It consists of three concentric tubes forming a 
cylindrical slide rule designed for the one purpose of calculating 
azimuth  (Az)  and altitude (Hc) of celestial bodies. The inner tube is 
covered with a spiral log cotangent scale (the patent documents and 
other descriptions identify this as a  tangent scale but it is, in fact, 
a cotangent scale), the second tube has a spiral log cosine scale and 
the third tube carries two pointers used to line up the two trig scales. 
In normal slide rules you can align the scales next to each other but 
with spiral scales this is not possible so the need for the two 
pointers. The advantage is that a spiral scale can be made much longer 
than  a normal ten inch slide rule allowing for much greater accuracy 
and precision. The cotangent scale on the Bygrave slide rule covers the 
range of 0º 20' to 89º 45' by spiraling 44 times around a two and a half 
inch diameter tube making this scale  28.8 feet long! The copy I 
constructed only covers the range of 0º 55' through 89º 15' by spiraling 
37 times around a one and a half inch diameter tube making this scale 
14.5 feet long. Each spiral represents a .1 change in the value of the 
log cotangent. For example, the log cotangent of 20' is 2.2352 and the 
value directly above the 20' mark and up one spiral is 25' 10" which has 
a log cotangent of 2.1352, exactly .1 less than the value one spiral 
down. Going up 44 turns to the top of the scale and directly above the 
20' mark has a log cotangent of -2.1648 exactly 4.4 less than the log 
cotangent of 20' and marked as 89º 36.5'.

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On my version the cotangent scale goes down only to a value of 55' while 
the original Bygrave goes down to 20'. Because each spiral comprises a 
change of log cotangent of .1 it takes 4 complete spirals to cover the 
range from 55' down to 20'. It would take 5 more spirals to extend down 
to 10'. It would take 10 additional spirals to extend the scale down to 
1'. The same thing happens at the other end of the scale. To extend the 
cotangent scale from 89° 15' to 89° 59' would take an additional 12 spirals.
The log cotangent of 1' is 3.536 and the  log cotangent of 89° 59' is 
-3.536 a range of 7.07. Since each spiral covers a change of .1, to make 
a Bygrave that covers this range would require almost 71 spirals!. The 
cotangent scale covering this range on a 1.61 inch diameter tube would 
be 29.9 feet long! Similar things happen with the cosine scale as you 
approach 90°.

The 19 turn cosine scale on my Bygrave re-creation is 7.5 feet long. On 
my HR-1 copy the cotangent scale is 15.6 feet and the cosine scale is 
8.7 feet.

gl



douglas.denny@btopenworld.com wrote:
>
> Original postings:-
> ------------
> Thanks Gary.
> This may lay to rest Douglas' concern regards the relative diameters over
> which the scales lay. Or perhaps not!
> Best Regards
> Brad
> ------------
> Yes I have compared and it is accurate as my other ones and the flat 
> model.
> gl
>
> -------------------
> Brad Morris wrote:
> Hi Gary
> Have you compared numerical output of the re-creation vs the flat
> Bygrave? Is there any degradation in the solution?
> Best Regards
> Brad
>
> =================================
>
> I still have reservations, but am glad that Gary confirms the accuracy 
> appears good despite the different diameters of the tubes.
>
> Gary says in an earlier posting:
>
> "Making the cosine scale presented a bit of a problem. Since the cosine
> tube has an O.D of 1.750 and the cotangent scale was only 1.520 inches
> in diameter it was necessary to print the cosine scale at a larger scale
> than the cotangent scale which was not a problem using Acrobat. However
> when I tried out the scales a problem presented itself. My printer made
> the cosine scale wider but also taller so that the pitch of the scales
> no longer matched. Although it worked this way it caused ambiguity since
> the cursor would sometimes end up between spirals on the cotangent scale
> and either answer (higher or lower than the end of the cursor) could
> have been the correct one. I contacted Dave Walden (the original source
> for the scales I have been using) and he was able to modify the vertical
> and horizontal ratios of the cosine scale so that, when printed out wide
> enough to fit around the tube, the vertical spiral pitch matched the
> cotangent scale. (Thanks again Dave.)".
> -----------
>
> So the diameter difference does indeed noticeably affect pitch of the 
> scales which had to be adjusted, but Gary says he adjusted the scale 
> factor too:-
>
> " it was necessary to print the cosine scale at a larger scale
> than the cotangent scale "
>
> Which leaves me wondering still ...?
>
> I think the answer is the difference in scale length when adjusted 
> like this when helically wound onto the tube is probably quite small 
> compared to the full length of the scales if fully extended. (Imagine 
> a single long straight slide rule). If you have a difference in the 
> scale length of say an inch in a length of 20 feet that is only 0.4% 
> error. It is probably not noticeable.
>
> I should think for exactitude, the inner scale will have to be 
> lengthened by a small factor for consideration of the smaller diameter 
> if printed smaller(and the pitch adjusted accordingly too) to keep the 
> extended lengths the same between the two scales, inner and outer.
>
> --------
> For the fun of it:
> I counted 19 helical turns on the LogCosine scale of Gary's Bygrave SR 
> and 37 turns on the LogCoTan scale in his pictures.
> He quotes diameters of 1.5" for the LogCotan and 1.625" for the 
> LogCosine.
> This gives 176.9" (nearly 15 feet) total length for the LCoTan scale 
> and 97" (8 feet) for the LCosine scale.
>
> Now the LCosine scale is expected to be half the length of the LCoTan 
> scale because of the maths of it, so there seems to be something wrong 
> here in these figures. Perhaps I do not have the right number of 
> helical turns. I would have felt more comfortable if they had worked 
> out at 16 feet and 8 feet or very nearly so for the total lengths 
> involved.
> Over to you .......
>
> Douglas Denny.
> Chichester. England.
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