I'm guessing you have seen the Ed Chamberlain long scale rule archive maintained by Rod Lovett: https://osgalleries.org/longscale/gallerycolour.cgi. It has lots of examples of rules with circular, spiral and helical scales. The Appoullot Logz rules are particularly interesting. They have a three revolution spiral scale, but the spacing between spirals is not regular - the outer spiral is especially irregular. Also, the sine and tan scales are printed back to back on a two revolution spiral scale, but the scale has a break where it 'jumps over' the three revolution spiral scale. This makes the rule quite confusing to use.
You mention how the single cycle scale on the circumference of the Atlas is used to determine the first 3 or so digits of the result of a multiplication or division, which is needed to identify the spiral on which the higher resolution result is read. A different approach is used by the Ross Precision Computer, which has a 25 revolution spiral scale. The Ross computer has a linear slide rule on an arm that extends across all the spirals. This linear rule is used to calculate the lower resolution result, which places the cursor of the linear rule against the spiral on which to read the higher resolution result.
This works because, as you describe, the values on the spiral scale vary proportionally to the angle 𝜃. A consequence of this is that for any radius drawn from the center of the rule, the values read on successive spirals that intersect with the radius are logarithmically distributed (e.g., as you show in the picture of a radius connecting the end points of each revolution on the Atlas scale). Since the scales of the linear slide rule on the Ross computer are also logarithmically distributed, the result on the linear rule appears against the spiral on which the high resolution result is to be read.
Thanks, Eamonn. Yes, the long scale archive is a wonderful resource. Thanks for bringing that up. It is very much worth a look for everyone. In general, it is very interesting to see the many ideas and subsequent trials that were made to find more accuracy on a manageable sized device.
Kurt, yes, it took a lot of work to manually lay out the scales. This JoS article has a letter from Richard Gilson, son of Clair Gilson, who founded the Gilson slide rule company. Richard mentions how he spent an entire summer figuring out the positions of the lines on each of the scales to five figure accuracy for the Gilson Binary slide rule: https://osgalleries.org/journal/displayarticle.cgi?match=2.2/V2.2P8.pdf
Nicely put together Mike.
I'm guessing you have seen the Ed Chamberlain long scale rule archive maintained by Rod Lovett: https://osgalleries.org/longscale/gallerycolour.cgi. It has lots of examples of rules with circular, spiral and helical scales. The Appoullot Logz rules are particularly interesting. They have a three revolution spiral scale, but the spacing between spirals is not regular - the outer spiral is especially irregular. Also, the sine and tan scales are printed back to back on a two revolution spiral scale, but the scale has a break where it 'jumps over' the three revolution spiral scale. This makes the rule quite confusing to use.
You mention how the single cycle scale on the circumference of the Atlas is used to determine the first 3 or so digits of the result of a multiplication or division, which is needed to identify the spiral on which the higher resolution result is read. A different approach is used by the Ross Precision Computer, which has a 25 revolution spiral scale. The Ross computer has a linear slide rule on an arm that extends across all the spirals. This linear rule is used to calculate the lower resolution result, which places the cursor of the linear rule against the spiral on which to read the higher resolution result.
This works because, as you describe, the values on the spiral scale vary proportionally to the angle 𝜃. A consequence of this is that for any radius drawn from the center of the rule, the values read on successive spirals that intersect with the radius are logarithmically distributed (e.g., as you show in the picture of a radius connecting the end points of each revolution on the Atlas scale). Since the scales of the linear slide rule on the Ross computer are also logarithmically distributed, the result on the linear rule appears against the spiral on which the high resolution result is to be read.
Looking forward to the article on helix rules.
Thanks, Eamonn. Yes, the long scale archive is a wonderful resource. Thanks for bringing that up. It is very much worth a look for everyone. In general, it is very interesting to see the many ideas and subsequent trials that were made to find more accuracy on a manageable sized device.
Very clear and understandable. But your calculus is a lot better than mine. 😆
It's very interesting that this was all calculated so long ago to layout the scales.
Kurt, yes, it took a lot of work to manually lay out the scales. This JoS article has a letter from Richard Gilson, son of Clair Gilson, who founded the Gilson slide rule company. Richard mentions how he spent an entire summer figuring out the positions of the lines on each of the scales to five figure accuracy for the Gilson Binary slide rule: https://osgalleries.org/journal/displayarticle.cgi?match=2.2/V2.2P8.pdf
Thank you. I read the article. Amazing. What a story.