A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Hanno Ix
Date: 2014 Jun 11, 08:52 -0700
Given the choice Danioli vs Bygrave: What would Chichester have chosen?
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.
to survive a 1-week sailing trip in the Virgin Islands. ( Where and when can I sign up? )
for the full useful ranges of L,D,t. I am unsure, though, what "useful" means for our CelNav friends.
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);
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.
log vers LHA 9.9019
log cos Lat 9.9177
log cos Dec 9.9642
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!