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    Re: Longhand Sight Reduction
    From: Hanno Ix
    Date: 2014 Jun 11, 08:52 -0700

    Given the choice Danioli vs Bygrave: What would Chichester have chosen?
    Assuming, of course, he had access to and enough experience with both.

    I celebrated Greg's RIC / Danioli yesterday by playing around with it.
    It indeed works and is fast. Comparison with the standard formula is attached below.

    Greg seems to think one can do that single multiplication with a 10" slide rule.
    I am skeptic. In praxis, ten inchers do not yield correct 4 digits consistently,
    and that's what I need for accuracy over the useful ranges of L,D,t.

    Now, there might be a challenge especially for you: a Fuller < = 10" that can

      - do 4 digit multiplication, i.e. yields vwxy = ABCD * abcd;  v, w, x, y being correct digits.
      - can be built with standard and garage tools plus a PC and printer.
      - in not more than, say, a week.
      - for about $50 or less,  $100 max.
    It need not look like an exhibit in a museum, although it should be sturdy enough
    to survive a 1-week sailing trip in the Virgin Islands.  ( Where and when can I sign up? )

    Re: formulas. One example, with good or flawed results, is not really sufficient to
    judge a formula or method. You need to show it yields accurate 4 digit results consistently
    for the full useful ranges of L,D,t. I am unsure, though, what "useful" means for our CelNav friends.
    Any ideas out there?

    I am studying the Bygrave in this respect right now. Stand by please.



    For the record, Danioli claims:

        sin(h) = n - ( n + m ) * a ;   n: cos(L-D);    m: cos(L+D);   a:  [1 -  cos(t) ] / 2 or hav(t);  

    Let's see. By inserting:

        sin(h) =  cos(L-D)   -  [ cos(L-D) + cos(L+D) ] *  [ 1 - cos(t) ] / 2;

    which is in more detail:

        sin(h) cos(L-D)  - cos(L-D) * [ 1 - cos(t) ] / 2  -  cos(L+D) * [1 - cos(t) ] / 2;

    and more detail yet:

        sin(h) =  cos(L-D)  -  cos(L-D) / 2  +   cos(L-D)*cos(t) / 2   -  cos(L+D) / 2 + cos(L+D)*cos(t) / 2;


         sin(h) =  cos(L-D)/2 - cos(L+D) / 2    +     [ cos(L-D) / 2 + cos(L+D) / 2 ] * cos(t);

                  =         sin(L) * sin(D)              +              cos(L) * cos(D) * cos(t);

    which is correct.

    On Tue, Jun 10, 2014 at 11:11 PM, Francis Upchurch <NoReply_Upchurch@fer3.com> wrote:

    Oh dear. Is it time to put my beloved Bygrave away? Cant wait to here more details of the Bygrave maths.Chichester said he preferred the Bygrave when flying single handed, because he made mistakes with log tables. (Perhaps he did not have Haversines?)  But, could someone explain the main difference/advantages/disadvantages of the versine method (Vers ZD=Vers LHAxCos Latx Cos Dec+Vers(Lat+/-Dec) and the Haversine method? My versine method (Reeds Astro Nav Tables) uses tables of natural and log versines and log cos (total 11 pages).Does not need sines.

    Versine method

    log vers LHA    9.9019

    log cos Lat      9.9177

    log cos Dec     9.9642

    add              29.7838

    Nat Vers of 9.7838=        0.6081

    Lat-Dec=11°13' Nat vers=0.0191. Add= 0.6272=68°6'. =ZD. 90°-68°6'= 21°54'

    Not a lot in it I would say? quicker for me than reduction tables and I understand what we are doing.

    Please correct me and explain the advantages of the Haversine over the versine. (I do not have haversines but do have versines! Where do I get haversines?)

    Bygrave. H=360°-LHA=78°21', co-lat=55°50', y(w)=64°31', X=colat+y(w)=120°21', Y=180°-X=59°39', > Az =76°24'> Hc 21054'

    No contest! Took a fraction of the time and no mistakes from looking up 4 figure logs etc. And I've got Az (OK done hundreds of Bygrave LOPs and only a couple of Versines!)

    I'll stick to my Bygrave!



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