## Juan de la Cosa’s Projection: A Fresh Analysis of the Earliest Preserved Map of the Americas

Fig. 7: Axes of reference X and Y defined along the equinocial and meridional lines, respectively.

The fact that the equinocial line represents the equator is expressed mathematically as Y LAT=0 = 0 . Similarly, assuming that liña meridional is equivalent to meridian 26ÂºW (which has been called LON0) results in the constraint X LON=LON0 = 0.

Regarding the ecliptic, if we make the assumption that the earth is a perfect sphere, it can be described by the following equation:

where parameter E is a constant equal to tan (23.4Âº) and LON0 is the longitude of the meridian at which the ecliptic is tangent to the Tropic of Cancer, equal to 26ÂºW in this case.

The projection we are looking for has to transform the ecliptic into a straight line of equation Y = constant. This will be verified if the Y component of the candidate projection is an exclusive function of the ratio

Table 1 summarizes the mathematical conditions deduced from the hypotheses on the three main lines of La Cosa map.
Liña meridional = meridian 26ÂºW
Cancro = ecliptic

Equinocial = equator

Table 1: Mathematical conditions imposed by the three hypotheses on the nature of the main lines of Juan de la Cosa’s map

In theory there exist infinite mathematical projections that satisfy the three conditions listed in Table 1, but in fact few of them have ever been described in the literature. The best-known one is the gnomonic projection, which will be discussed in next section. It will be shown, however, that a map of the Atlantic plotted in the gnomonic projection does not fit well with the coastlines drawn by La Cosa. This will lead to testing a second projection that is far less well known, but provides a substantially better match.

4. Gnomonic Projection

The gnomonic projection (GP), also known as gnomic or central, is a perspective projection obtained by plotting the sphere from its center onto a plane tangent to its surface. Its main property is that all great circles of the sphere are represented as straight lines on the plane. While it was probably known at least in theory since antiquity, no map in gnomonic projection dated before 1600 has been preserved.[29]

The mathematical equations of the gnomonic projection in the equatorial case are:

where LON stands for longitude, LAT for latitude and X and Y represent the horizontal and vertical coordinates, respectively, of the flat projected map. Longitude is measured with respect to a certain meridian of reference, LON0. Both LAT and (LON – LON0) are restricted to the range (-90Âº, +90Âº). These equations verify the conditions of Table 1.

Fig. 8: Star map of 1674 in gnomonic projection. The ecliptic and the celestial equator are parallel horizontal lines in this particular case of the projection.[30]

A real image of the Atlantic basin has been plotted in the gnomonic projection, using the point LON0 = 0Âº, LON0 = 26ÂºW as the centre of projection. The image was built with Microsoft Excel and, in order not to overload the graphs, only selected coastal features and islands were plotted. The result has been superimposed over La Cosa map. In addition, the discrepancies in latitude and longitude of a set of control points have been computed numerically and reported in Annex 1.

• 21-7-2010

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