A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Brad Morris
Date: 2013 Jan 31, 06:33 -0500
The 'rule' of 0.1 log cotan per spiral is NOT a rule, rather it is an artifact only of the manufacture of the HR1. It need not hold for other devices, nor if we were to extend the existing scales of the HR1 using the given resolution. It just happens to be that way for the HR1 and Bygrave. It is not a critical feature, just an arbitrary outcome.
Cylindrical slide rules were invented to overcome the extraordinary lengths required of special linear rules. The Otis King and Stanley Fuller rules are both cylindrical, yet could be expressed linearly, if only one is to deal with the length.
The HR2 has larger diameter cylinders, and as Ronald has pointed out, scale markings at a different orientation from the HR2. Ignoring the trig identity issue, Tube I still expresses log(abs(cos(arg))) and Tube II still expresses log(abs(tan(arg))) in the proper relationship to each other. The important point is that there is no need to hold an arbitrary 0.1 log cotan per spiral.
As Wolfgang has no doubt determined, the scales must be manipulated to fit the tubes in manufacture. Yet they are also constrained in that the scales must align. Think of any linear slide rule. Align the indices. Surprise, all the graduations line up (as they must). The same behavior must be held here as well. With concentric tubes, the easiest way to do that is to have the graduations align radially from the center axis of the concentric tubes.
As a thought experiment, double the diameters of the concentric cylinders, yet keep the resolution of the scales the same. Since the circumference of a circle is Pi*D, doubling the diameter doubles the circumference. So twice the circumference halves the turns per log(cotan). Note that the slide rule still functions, its just bigger.
Try the same thought experiment with original diameters yet the scale resolution is smaller by a factor of two. So twice as much information in the same linear displacement, albeit on a cylinder. The slide rule still functions, there just aren't as many turns to realize the same range of inputs. A bit tougher to align, yet still ... it works.
An analogy: I've noticed that every car that has driven by in the past hour is blue. Therefore, all cars are blue.
I've noticed that on the Bygrave, there is 0.1 log cotan per turn. Therefore every cylindrical rule for solving navigation triangles must follow that rule.
BOY! now I am really confused! I took a close look at figure 29 and the cotan scale of the HR2 does not follow the pattern of the HR1 and Bygrave in that the values on the scale do NOT change by 0.1 log cotan per spiral. I am at a loss to explain how the scales on the HR2 can work.
You can see this for yourself. Take the log cotan of 81 degrees which is -0.80 so 7 spirals up (to the left) the log cotan should -1.50 which is 88° 11' but instead we find (in figure 29) 89° 19' which is log cotan -1.92. You see the same thing in figure 27, two spirals to the left of 20' should be 31' but is, instead, 42'.
Can anybody explain how this can work?
--- On Thu, 1/31/13, Gary LaPook <garylapook---net> wrote:
From: Gary LaPook <garylapook---net>
Subject: [NavList 22190] Re: Re: Bygrave Scale Lengths
Date: Thursday, January 31, 2013, 1:07 AM
I just re-read you paper and I wonder if there is a slight error. You state that the inner tube of the HR2 has 27 spirals. Image 29 shows that the scale goes up to 89-40 and image 27 shows the scale goes all the way down to 20', (maybe even lower but I expect the photo was taken to the lowest end of the scale.) To cover that range requires 44 spirals like on the HR1 and Bygrave since the log cotan goes from 2.2 to -2.2 and each change of 0.1 requires a spiral, see:
--- On Thu, 1/31/13, Ronald van Riet <ronald---nl> wrote:
From: Ronald van Riet <ronald---nl>
Subject: [NavList 22188] Re: Bygrave Scale Lengths
Date: Thursday, January 31, 2013, 12:42 AM
First to answer Gary's question about the scale length of the HR2: as my paper shows, it is about the same as that of the HR1, even though the item itself is much larger. This is mainly due to the numbers being horizontal to a vertical scale, therefore the windings of the scale are farther apart than on the HR1.
But what I find expescially interesting in these pictures is the Werknummer of the HR2: 83989 which according to the black book should have been allocated to an HR1. I have just seen a picture of another HR2 with Werknummer 83995 which is in this same block of Werknummer.
I am attaching a photocopy of the relevant page in the little black book: there is no mistake about it, it clearly says HR1!
So this entire batch of 50 may have been HR2s, makes one wonder about the other entries in the same book.
Please keep those HR1/MHR1/HR2 Werknummer coming!!!
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