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I decided to work a sample problem, latitude 30� north; declination 30� 
north; hour angle 30�.
.
K&E solution using sine-cosine formulas: Hc 64.1�, Az 82.5�.

Flat Bygrave solution: Hc 64� 04', Az 82� 25'.

Cylindrical Bygrave: Hc 64� 05',Az 82� 23'.

H.O. 229: Hc 64� 05.7', Az 82.4�

Take your pick.

Even the K&E came within 0.3' of  the H.O. 229 Hc and the Az was within 
0.1�!

The cylindrical Bygrave within 0.7' and the flat Bygrave within 1.7' on 
the Hc and both had spot on Azs.

You can check the computation of the Bygrave by using a digital 
calculator and Bygrave's formulas.

Using the above test values and the Bygrave form:

Tan W = tan dec / cos H.
tan 30�/ cos 30� = .66666
W= inv tan .66666
W = 33.69006� = 33� 41' 24.2"

X= co-lat + W
X = 60� + 33.69007�
X = 93.69006�
Y = 180� - X
Y = 180�- 93.69007�
Y = 86.30993� = 86� 18' 35.7"

Tan az = cos W / Cos Y * tan H
Tan az = cos 33.69006�/cos 86.30993�  * tan 30�
Tan Az = 7.46410
Az= inv tan 7.46410
Az = 82.36925 = 82� 22' 09.3"

Tan Hc = cos az * tan Y
Tan Hc = cos 82.36925� * tan 86.30993�
Tan Hc = 2.05895
Hc = inv tan 2.05895
Hc = 64.09492� = 64� 05' 41.7"

Using the flat Bygrave the values were:
W = 33� 40'
X = 93� 40'
Y = 86� 20'
Az = 82� 25'
Hc = 64� 04'

gl





Gary LaPook wrote:
> Nothing in this world is perfect. You guys have identified a problem 
> that does not arise in practice with this model of the Bygrave. To the 
> level of accuracy expected from this device, the scale distortion 
> produced errors that you guys are concerned about just don't occur. The 
> errors you are concerned about "fall into the noise" of the two minute 
> expected accuracy. I have made 12 of these and worked more than a 
> hundred sample problems and checked the results against the results from 
> a digital calculator and all the results agreed within two minutes of 
> arc. Two minute accuracy is sufficient for practical off shore 
> navigation and is certainly good enough for a "backup" system. Even my 
> ten inch long Keuffel & Esser 4080-3 slide rule can produce calculated 
> altitudes that are accurate enough for off shore navigation but not at 
> all places on the scales as they become bunched near the ends. What 
> makes this flat version of the Bygrave very good for celestial 
> navigation is that the cotangent scale is not ten inches long it is 
> 351.5 inches long, 29.3 feet, 8.9 meters! Having logarithmic scales this 
> long allows for much greater accuracy than from a ten inch long slide 
> rule and they consistently produce results agreeing within two minutes 
> and often are in exact agreement.
>
> Many of the buildings you stand in, many of the airplanes that you fly 
> in and most of the bridges that you drive across today were designed 
> with the use of slide rules so they have for many years provided 
> calculations that we still rely upon for our daily safety.
>
> So make one and give it a try, just don't expect agreement within 
> one-tenth of a minute and you will see the usefulness of this for a 
> backup celnav system. As it says on the side of the medicine bottle, 
> "safe when used as directed."
>
> gl
>
>
> Brad Morris wrote:
>   
>> This problem does not exist on a �real� Bygrave because the cylinders 
>> are stiff and remain concentric with each other. I do recognize the 
>> difficulty in getting the scales mounted and keeping them referenced 
>> to each other when going through the zig-zag pattern of solution.
>>
>> I think that local distortions of the one scale to the other will 
>> clearly result in errors. Slide rules in general work when the one 
>> logarithmic scale is referenced to another logarithmic scale. 
>> Distorting one or the other cannot be permitted. NSG21 is absolutely 
>> correct.
>>
>> I remember being the last class to take slide rule instruction in High 
>> School. When you use a slide rule today, most people think of it as 
>> black magic and have no idea how it works. Further, the electronic 
>> calculator leads young engineers to give me as many decimal places as 
>> their calculator does, without judgment as to the meaning of those 
>> digits. Empty resolution without addition to accuracy.
>>
>> Best Regards
>>
>> Brad
>>
>> *From:* NavList@navlist.net [mailto:NavList@navlist.net] *On 
>> Behalf Of *Gary LaPook
>> *Sent:* Wednesday, February 25, 2009 4:39 PM
>> *To:* NavList@navlist.net
>> *Subject:* [NavList 7430] Re: New compact backup CELNAV system
>>
>> I haven't seen the problem you mentioned. I sealed the Cotangent scale 
>> in normal plastic protection sheets (about one buck each at Fryes) 
>> used for protecting documents which are quite rigid. I will experiment 
>> with bending the scale and working a sample problem and get back to you.
>>
>> gl
>>
>> --- On *Wed, 2/25/09, nsg21@hotmail.com //* wrote:
>>
>> From: nsg21@hotmail.com 
>> Subject: [NavList 7427] Re: New compact backup CELNAV system
>> To: NavList@navlist.net
>> Date: Wednesday, February 25, 2009, 9:47 AM
>>
>> I would like to share some experience in using this (transparency over printed)
>>
>> style of the slide rule. The surface it is placed for calculation MUST
>>  BE
>>
>> ABSOLUTELY flat. Even small warping of the surface (such as normally found on
>>
>> small plastic tables) leads to big errors in calculations.
>>
>>   
>>
>>   
>>
>>   
>>
>> From: glapook---NET
>>
>> Date: Tues, Feb 24 2009 12:06 pm
>>
>>     
>>> There are often posts on the Navlist regarding using celestial as a backup
>>>       
>> to
>>
>>     
>>> GPS and finding a simple way to do this. I think I have found a method
>>>       
>> that
>>
>>     
>>> is simple, self contained, takes up little space, needs no almanac or
>>>       
>> sight
>>
>>     
>>> reduction tables 
>>>       
>> ...
>>
>>   
>>
>>   
>>
>>   >
>>   
>>
>>   
>>
>>   
>>
>>
>>     
>
>
> >
>
>   


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