A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: finding your latitude through double altitudes (and elapsed time).
From: Joel Silverberg
Date: 2007 Apr 08, 14:30 -0700
From: Joel Silverberg
Date: 2007 Apr 08, 14:30 -0700
George, thank you for your comments in response to my inquiry. In particular your description of the diagram with its auxilliary constructions was very helpful. Your attempt to associate my variables with the diagrams was also helpful in helping me see the link between the diagram, the constructions and the equations. Frank's suggestion that I look at Benjamin Peirce's treatise provided me with just what I needed to make sense of all of these sources. I have written up a detailed summary of the method, including a complete illustrated derivation of all equations and will email it to Frank to see if he wishes to post it to the navlist. If you send me your email, I will mail you a copy as well. (It is an 8 page pdf file). Cotter does not name any angles or sides with variable names, only points on the sphere. That is part of what makes his explanation so difficult to follow. I have retained the variables that Bowditch and Peirce use for sides and angles, while changing the names of points on the sphere to match those that you use in your posting and that Cotter uses in his text. You are correct that my variable A is half the great circle distance between X and Y as you note below. Also, B is latitude (or declination) of point M. Variable C is not MR as you suggest, but the distance from the zenith to R. Variable Z (not to be confused with the zenith) is not the distance from the zenith to R, but the distance from M to R. Variable E is not PR, but it's complement... that is E is the declination (or latitude) of point R. The relationship between E, Z, and B with its complicated sign rules is now easier to understand. If E is the declination of R and B is the declination of M, then ... if R lies between M and P, then my variable z = declination of R - declination of M, that is z = E - B, which solving for E gives E = B + z. On the other hand, if M lies between R and P, then z = declination of M - declination of R . That is z = B - E. Solving for E, we have E = B - z. (The little z in these equations does not refer to the zenith, but the the distance between M and R. ) This clarifies Bowditch's explanation. R lies between M and P when the zenith and the pole lie on the same side of arc XY, while M lies between R and P when the zenith and pole lie on opposite sides of arc XY. > > The double altitude problem is also discussed at some length in Charles H Cotter's "A History of Nautical Astronomy", 1968, and Joel > might find it useful to refer to pages 152-155. I have a serious distrust of Cotter when he gets into anything mathematical, and a > long list of perceived errors in that book, collected by Jan Kalivoda and me, can be found atwww.huxtable.u-net.com/cotter01.htm . > However, he has delved into many obscure texts, and provided information which exists nowhere else. It's in interpreting them that > he seems to get out of his depth. It's a really useful book to have, but now hard to find and expensive. > > He looks into Ivory's method of 1821, and Ainsley's of 1867, both of which seem to correspond with the Bowditch procedure that Joel > describes so well. His diagram, figure 7, corresponds exactly in its lettering with Joel's descripion, but to understand what's > going on, some more construction is called for. > > The initial presumption is that changes in the Sun's declination during the observation are small enought to be neglected, as long > as a mid-value is used for declination. > > First, Joel needs to join points X and Y by a great circle (not by the path of the Sun, which is a great circle only at the dates of > the equinoxes). Because X and Y have the same declination, that great circle arc XY is symmetrical about its vertex, the mid point > where it comes closest to the pole P. That mid-point is to be marked M. The resulting two triangles PXM and PYM are both > right-angled (at M). The arc MX = MY is what Joel has taken as A. The latitude of M then corresponds to B. And his first two > equations then follow from the rules for right-angled spherical triangles. > > Next, some more construction. Join M by an arc to the pole P. Where that arc passes closest to the observer's zenith Z, mark another > point R, and join ZR. MRZ and PRZ are now two more right-angled triangles (at R). As I see it, Joel's C now corresponds to the arc > MR, his Z to the arc ZR, and E to the arc PR. > > Now, applying more spherical trig to the right-angled triangles ZMX, ZMY, and ZMR, brings him to his third equation (for C) and from > that to his fourth equation (for Z). I haven't even tried to follow the details of those manipulations, but they are all in Cotter. > > Then, if that's correct, I get a bit puzzled about his fifth equation, which he states to be E = Z +/- B, because as I see it, those > angles are not in the same plane. It seems more likely to me that the intention is to derive the lat (or perhaps colat) of R from > the lat (or perhaps colat) of M, using the difference angle C. So I think that fifth equation should be combining B and C to obtain > E, and suggest that Joel might take another look at it. > > Now we know E (= PR) and Z (= ZR) in the right-angled triangle PZR, so we can compute the third side PZ, which is the colat (and > from that the lat) of the observer. which was what was wanted. > > Note that I have completely ignored any subtleties in the complex sign-rules that Bowditch presents, in aiming just to give a > general picture of how that complex calculation was meant to work. Hope I've got it right, but not completely confident... Anyway, > it may give Joel a start-off in the right direction. > > George. > > contact George Huxtable at geo...---.u-net.com > or at +44 1865 820222 (from UK, 01865 820222) > or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavListemail@example.com -~----------~----~----~----~------~----~------~--~---