NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Certain Errors in "Certaine Errors".
From: Herbert Prinz
Date: 2007 Dec 10, 10:35 -0500
From: Herbert Prinz
Date: 2007 Dec 10, 10:35 -0500
Garry and George have been puzzling over the inner workings of Wright's geometrical method for finding GCD. It's based on an analemma, whereby various orthogonal projections are sitting on top of each other. To add a little confusion, some lines are shifted around, other ones are reused for more than one purpose, all in an effort to minimize the effort of plotting. E is London, N is Jerusalem, F and O are the projections of these places onto the equatorial plane. FO can be constructed from the known latitudes and longitudes by means of projections into the respective meridional planes. In trapezoid FONE, EN is the chord of the required distance. EF and NO are parallel to each other and perpendicular to FO. FE = sin(Lat(London)) and NO = sin(Lat(Jerusalem)). The trapezoid FONE can thus be broken down into a rectangle FONN* and a right triangle NN*E with the right angle at N*. The rectangle is of no interest, only the triangle gets drawn as PFQ, with right angle at F. The recipe is actually explained for "him who desireth a demonstration", but there are two small errors in the example for London and Jerusalem, which may confuse the issue: 1. Read "[make] EQ equal to NO" instead of "PQ equal to NO". (This is mentioned in the errata.) 2. In the corresponding diagram, the point Q is supposed to be a member of the line EF; not of FO, as it is drawn. The line from P to Q should thus meet the line EF in Q. (This is not mentioned in the errata. It may be just a flaw in the printing of the copy at hand.) I hope everything falls into place now. If not, I may try to come up with a 3-dimensional diagram of what's going on. Analemmata have been employed since antiquity to solve problems in spherical astronomy. They are orthographic projections. Their purpose was to convert problems of spherical trigonometry into ones of plane trig. The latter could be solved numerically by means of proportions. For a long time, this was the only way of expressing trigonometric equations. The analemma also found direct use in geometrical construction, e.g. of sun dials. Lacaille proposed a graphical method applying an analemma for the the reduction of time sights and lunar distances as late as 1761. At that time the moon predictions were still the limiting factor of accuracy in solving the longitude problem. After the introduction of Mayer's tables, graphical reduction was no longer adequate. I would not say that it disappeared. It pops up now and then in lifeboat navigation and the like. Other projections (e.g. stereographic) have been suggested since for graphical or mechanical methods with little success. Herbert Prinz --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To , send email to NavList-@fer3.com -~----------~----~----~----~------~----~------~--~---