NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Another "emergency navigation" sight reduction method
From: Hanno Ix
Date: 2015 Jul 6, 08:42 -0700
Here, for your convenience, is a copy. Funny that D. Burch forgot how he did it :)
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David Burch's method is actually Ageton's.
N(x) = -1000* ln( sin(x) ) where ln() is log() taken to the basis e rather to the more common basis 10 which is written as log(). The rules for using it are the same in both. I bet your calculator has a ln() - key. ln() is frequently used in electronics.
H
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From: Hanno Ix
Date: 2015 Jul 6, 08:42 -0700
Brian,
I actually described the math of D. Burch's N(x) on the July 5 very much in detail. Did You miss that?H
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David Burch's method is actually Ageton's.
N(x) = -1000* ln( sin(x) ) where ln() is log() taken to the basis e rather to the more common basis 10 which is written as log(). The rules for using it are the same in both. I bet your calculator has a ln() - key. ln() is frequently used in electronics.
Example:
x =1; sin(1) = 0.01745; -1000*ln(sin(1)) = 4048 which is N(1) in D.
Burks table. I don't know why he chose ln(x) but that's what he did. So
N(90-x) must be -1000*ln(cos(x)).
Ageton described his method using sec(x) and cosec(x). Now, sec(x) = 1/cos(x) and cosec(x) = 1/sin(x).
According
to the rules of logarithms ln(cosec(x)) = - ln (sin(x)) and ln(sec(x)) =
- ln(cos(x)). The only thing this transformation did, then, was to
change the signs! That is not much of an advantage, and Ageton should
have stayed with sin(x) and cos(x) in my opinion because they are much
more common. The sign is in this case no issue at all.
Now
that you have -ln(sin(x)) and -ln(cos(x)), or N(x) and N(90-x) you can
calculate Ageton's partial triangles. I will not go into more detail
here.
D. Burch table N(x) lists -1000(ln(sin(x)) from
1 to 89 deg in steps of 1 deg. Astonishingly, with such a rough table
and with interpolation you can do Ageton's SR with an error of 10 sm or
less! And that makes his table a true Holy-Mary table that can save
your bun.
If you were to make a table N(x) in 1min
increments, which I did for the fun of it, you get what you get with
Ageton - warts and all - and can use it for standard SR.
So much for N(x).______________________________________________
