A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Date: 2014 Jun 11, 19:57 +0100
This debate is looking great!
Has anyone looked at the old Poor LOP circular slide rule from circa 1915? Even though 14 inch diameter, it had 5 scales (log Havs, natHavs, log Cos, log Sin, logs, to do the haversine method and no where near as accurate or quick as the slightly later Bygrave. Over to you maths guys to work it out.
I like the Haversine or versine (still not sure which is the better?) and the Bygrave first.. My new mini Fuller 2 is a close third (using the cosine formula, have just tried it with the haversine method and no real difference in terms of time.)
My gut feel is that Bygrave knew exactly what he was doing! Limited the whole thing to 3 equations and simple actions. (albeit with 2 intermediate angles y and X.) Reit writes up some of this I think? Gary LaPook knows all about it
Let’s see the maths?
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.
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 21°54'
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!