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Gary writes:


Did you mean "planisphere" and not "analemma"? All I can find about
analemmas is that they relate to sundials.


gl

On Dec 10, 7:35 am, Herbert Prinz  wrote:
> 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
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