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    Re: New compact backup CELNAV system
    From: Gary LaPook
    Date: 2009 Mar 7, 14:39 -0800

    I gave an example of using an ordinary ten inch slide rule to compute
    Az and Hc and got a result that was within 0.3' of the correct Hc. Of
    course you can't expect this kind of accuracy with a normal slide
    rule, it only happened in this case because the sample values were
    exact and marked on the scale, 30�. Any visual interpolation would be
    less accurate but you can get accuracy fairly regularly of 6'
    depending where on the scales you find your values. The problem with
    the sine-cosine formula on a regular slide rule is when the sine value
    is found on the scale where the slide rule sine scale is spread out
    allowing precise readings the corresponding cosine scale is
    compressed.  You can also do this computation on the circular air
    navigation slide rule, MB-2a and the similar Felsenthal flight
    computers, since they have sine scales that are used to figure wind
    correction angles. You get the cosine value by finding the sine of 90�
    - value (but of course we all knew this.)
    
    gl
    
    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.
    >
    On Feb 26, 12:21�am, Gary LaPook  wrote:
    > 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@fer3.com [mailto:NavList@fer3.com] *On
    > >> Behalf Of *Gary LaPook
    > >> *Sent:* Wednesday, February 25, 2009 4:39 PM
    > >> *To:* NavList@fer3.com
    > >> *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, ns...---.com //* wrote:
    >
    > >> From: ns...---.com 
    > >> Subject: [NavList 7427] Re: New compact backup CELNAV system
    > >> To: NavList@fer3.com
    > >> 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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