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    Re: Another "emergency navigation" sight reduction method
    From: Hanno Ix
    Date: 2015 Jul 6, 08:42 -0700

    I actually described the math of D. Burch's N(x) on the July 5 very much in detail. Did You miss that?
    Here, for your convenience, is a copy. Funny that D. Burch forgot how he did it  :)



    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).



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