# NavList:

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

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Re: Bygrave formula
From: Hanno Ix
Date: 2014 Jun 8, 10:47 -0700
Geoffry,

My question was: Why has Bygrave chosen his particular set of equations and his design?
Well, so far no answer. And I thought there would be a quick answer from those in the know.
So I started to investigate.

It appears to me the actual question is: What trig function F(phi)
1. is useful for CelNav math,
2. can be formed into a scale of log(F(phi)) of 20 to 30 meters length,
3. has a resolution of  at least 1/2 to 1 mm per arcmin for reading the results,
4. covers the range from phi  = 1 arc min to some angle close to 90 deg.
I hope this catches the problem of a scale on a cylindrical slide rule.

So far, my results show that sin(phi), cos(phi), sec(phi) and csec(phi) alone
are not good choices for F(phi). Reason: the log of these functions, which is actually identical
but for sign rules, has too large a range of step sizes: 4 above prohibits 3 and v.v.
for reasonable lengths of the scale. You can observe that effect on the cos() scale of the Bygrave.
You can compromise here and there and produce good results - for a limited range of angles.

Anyway, G. LaPook claims his cylindrical Bygrave implements a trig formulation with a
2 arcmin error or less. This might be so but I haven't convinced myself of this to the point where I would
start building one. However, there might indeed be an advantage in a clever choice of
2 trig functions working together as on the Bygrave.

The next part of my question aims at the physical implementation with simple materials and tools.
But it is not so easy according to the posts I found on this list. The biggest difficulty seems to be
the correct matching of the scales printed on drums of "slightly" different diameters.
BTW: the "slightly" has to be such that the drums telescope without sticking or slacking.

G. LaPook's "flat Bygrave" bypasses this problem. However, the user's fine-motoric needs to
be such that the scales line up well. I can do it with success working on a fixed surface -
but on a boat or even a light airplane? Also, there are no cursors as far as I know.
That complicates the use of the "flat Bygrave" quite a bit. One could substitute a pen with
erasable ink but that is messy.

For these latter reasons I am attracted to the Fuller / Riet concept. It seems much simpler.
In the Riet the scales are fixed in relation to each other and of the same diameter.
Cursors can be  implemented in form of fiducial marks on sliding transparent rolls.
It seems relatively easy to avoid sticking / slacking of those. The penalty might be a larger
size of the final product. But I don't know if that would be a problem.

Re: your surgery. I wish you the best results and a speedy recovery. This might help you:
I had one myself recently, and thanks to the highly advanced state of medicine in this country
I had almost no pain and started walking the next day. Only sleeping was a problem because of the
humming and beeping of the machines around me :)

Let's keep touch.

Hanno

On Sat, Jun 7, 2014 at 8:11 AM, Geoffrey Kolbe wrote:

```On 06/06/2014 07:11, Hanno Ix wrote:
> Why has Bygrave chosen his particular set of equations and his design?
>
> The formula :
>
> sin(hc) = sin(D)·sin(L) + cos(D)·cos(L)·cos(t)

And the formula for Zn.... how would you calculate that on a slide rule?

--
Dr Geoffrey Kolbe, Riccarton Farm, Newcastleton, TD9 0SN
Tel: 013873 76715 Mob: 07811 154621
```

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