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Vladimiro: Thanks again for your comments. In short, I think you are right. I had discarded the stereographic from consideration because I do not expect meridional compression to arise out of its usual construction technique. But as I wrote to Waldo Tobler last night, the map as a conformal conic does not add up in too many ways: • The mathematics were not available; • The concept of conformality was not available; • The motivation to produce a conformal conic at this scale, for this map and no other, is obscure; • The standard parallel is inapt for the territory; something like 45° or 50° would be more reasonable. And, as I found last night: • While the parallel spacing and the cone constant fit a conformal conic, the parallels are shifted a constant 10-mm downward along the meridians. In the face of the contextual anomalies, the physical evidence for the conformal conic must be unassailable. It is not, given this last discovery. That led me to reexamine the stereographic. Even as a stereographic, the map suffers from constant parallel shift along the meridians, though the extent is harder to characterize because of the other major problem: the improbable compression of meridional spacing. While the fit to a stereographic is considerably poorer than to a conformal conic, the stereographic would not be an anachronism. As Vladimiro suggests, the map was likely stereographic in intent. It is just drafted particularly poorly, probably the result of taking some measurements from an existing map rather than starting from the basic geometry. We see plenty of examples of careless projection construction in maps of the period. Thanks, all, for the help. I am still interested to know when this map’s archetype appeared, and in what publication. Regards, — daan Strebe On Sep 30, 2011, at 1:53 AM, Vladimiro Valerio wrote: > This is a MapHist list message (when you hit 'reply' you're replying to the > whole list) > o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + > > Dear Daniel (Joel, Terry and all), > I fear that trying to find the right date of the "Carte Generale des Royaumes > Etats & Domaines, que les Rois & La Couronne de la Grande-Bretagne ont > Possédez ou Possédent en Europe en Afrique & en Amerique", is of no help for > the question posed by you, Daniel. > > Your question was: how is it possible we face a map "drafted on a Lambert > conformal conic with standard parallel of 75°" before its mathematization by > Lambert in 1772? (Daniel post of yesterday) > > The problem is that its projection is not at all a "Lambert conformal conic > with standard parallel of 75°" but simply and, I dare say, obviously, a polar > stereographic projection, known and used since long time. > Why you, Daniel, did not think about more obvious and simple solutions? And > why you did not check it, before trying a more sophisticated and > antihistorical solution? > > I wondered why the cartographer of the "Carte Generale . . ." had to choose a > "conformal conic with standard parallel of 75°" and not one of the many known > and used in his time!? In fact, he used a stereographic projection. > > The problem is solved whatever may be the date of compilation of that map. > > Ciao. > vladimiro > > > Il giorno 30/set/2011, alle ore 02.55, Daniel Strebe ha scritto: >> This is a MapHist list message (when you hit 'reply' you're replying to the >> whole list) >> o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + o + >> >> >> Vladimiro: >> >> Thanks for the comments. Normally I would not be (too) surprised to find a >> map projection used earlier than it was known to have been developed. Most >> map projections of the era were geometrical in conception; they were >> developed in order for their manual construction to be tractable. It would >> not be surprising for two people to come up with the same idea >> independently. Such ideas do not represent new mathematics; they represent >> new assemblages of known methods. >> >> This map is a another matter entirely. The mathematical principles upon >> which it was founded had not been even been described yet. For this map to >> exist from the era it appears to exist from, would mean that the history of >> mathematics needs to be revised. >> >> Only two conformal projections preceded this one. Neither was developed for >> its conformality. The fact that they were conformal was incidental and >> unrecognized in any modern sense. One of those is Mercator’s famed 1569 >> projection, developed for the virtue of straight rhumbs. The Mercator could >> not be constructed by purely geometric means; it required laborious >> calculations to set up the parallel spacing. The other early conformal >> projection is the stereographic, which is a perspective projection. It was >> widely used in that era because it could be constructed geometrically, and >> gave double hemispheric views with the meridians and parallels being >> orthogonal. >> >> By the known history of map projections (and mathematics), the conic >> projection of this map “Carte Generale des Royaumes Etats & Domaines…” ought >> to be a perspective conic or an equidistant conic or a conic with ad hoc >> parallel spacing. It is trivially not an equidistant conic because the >> parallel spacing varies blatantly. With more investigation, I have ruled out >> any perspective conic. There simply are no other candidates. This is either >> a Lambert conformal conic or an ad hoc contrivance that magically mimics a >> Lambert to within the margins of error inherent in the paper medium. The >> probability of that last happening is remote. >> >> The question then becomes, what motivated this projection? Unlike the >> Mercator or the stereographic, there is no rationale that could lead to the >> projection except the wish for a conformal projection. It has no other >> useful properties: Rhumbs are not portrayed favorably; distances are not >> correct across useful portions of the map; overall distortion is not >> minimized; directions in the large receive no favorable treatment. >> >> In other words, someone made it specifically to be a conformal conic. Yet >> the rigorous notion of conformality “should” not have existed. The >> mathematics needed to calculate the projection’s coordinates (the generating >> formulæ) are advanced for the time but not intractable. What was intractable >> was the mathematics needed to characterize the conformal condition and to >> thence construct those generating formulæ in order to fulfill the conformal >> conditions. >> >> I am more inclined to think that the map’s dating is wrong than to think >> someone developed (and then calculated!) such a projection in 1736. Hence my >> appeal to those with copies of the map in their collections. >> >> Best, >> — daan Strebe
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