Sunday, October 15, 2006

What Can Waldseemüller's Projection Tell Us?

From what rests on the surface we are led into the depths
----Edmund Husserl

The 1507 World Map (click on figures to enlarge) shows the New World as a landmass detached from Asia, and was the subject of scholarly study and speculation even before the discovery of the only surviving copy by Joseph Fischer in the collections of the Wolfegg Castle in 1901. Scholars beginning with Alexander von Humboldt and Marie D’Avezac-Macaya in the early and mid-nineteenth century, through the more modern studies of Fischer himself, and later twentieth century investigators, have all concentrated on the map’s context and its place in cartographic history, showing little regard for its geometric accuracy, possible geographic sources, cartographic content, and structure.

The map, displays the continents of the New World with a shape, that when re-projected, is geometrically similar in form to the outlines of the continents as we recognize them today. The two aspects of shape and location of these landmasses, separated as they are from Asia, are chronologically and chronometrically problematic, in that in 1507, the map’s supposed creation date, neither Vasco Nunez de Balboa nor Ferdinand Magellan had reached the Pacific Ocean.

Waldseemüller discusses his portrayal of the New World in his Cosmographiae introductio, cum quibusdam geometricae ac astronomiae princpiis ad eam rem necessaries, printed in multiple editions in St. Dié under the patronage of Rene II, Duke of Lorraine, in 1507. According to Robert Karrow, “few books of its size have generated as much interest and speculation as the Cosmographiae Introductio”. The cause for this attention and speculation stems mostly from the mention on the title page of the two maps (descriptio tam in solido q[uam] plano) that constituted part of the book, one a flat map (plano) and the other a globe (solido). Waldseemüller and his collaborator Matthias Ringmann discuss these two maps in several places in the book that was printed to be a companion to the world maps. Neither of these maps appears to have actually accompanied the book when it was produced and they remained unknown until Lucien Louis Joseph Gallois’s discovery of the first copy of the globe gores in the Lichtenstein Collections in the late nineteenth century, followed by Fischer’s discovery of the Codex in 1901.

In the Cosmographiae Introductio Waldseemüller describes the New World by saying, “Hunc in midu terre iam quadripartite connscitiet; sunt tress prime partes cotinenentes Quarta est insula cu omni quaque mari circudata cinspiciat”. The semantics of his Latin are extremely important here. The passage translates, “the earth is now known to be divided into four parts. The first three parts are continents, while the fourth part is an island, because it has been found to be surrounded on all sides by sea.”

Waldseemüller uses highly suggestive phrases such as, “now known”, and “has been found”, both of which imply some form of empirical evidence rather than mere speculation. Other sources also testify to the form of this evidence. In a letter dated 12 August 1507, the humanist historian Johannes Trithemius wrote to his friend Veldicus Monapius that he had “a few days before purchased cheaply a handsome terrestrial globe of small size lately printed at Strasbourg, and at the same time a large map of the world…Containing the large islands and countries recently discovered by the Spaniard [sic] Americus Vespucius in the western sea, which extends south almost to the fiftieth parallel.”

This idea of empirical evidence is further expressed on the 1507 map itself where Waldseemüller tells us in the lower left hand text block that "All this we have carefully drawn on the map, to furnish true and precise geographical knowledge".

Among the many technical and theoretical problems that Waldseemüller faced in the construction of his map, one of the least trivial mathematically was the problem of projection. Dealing with a greatly enlarged earth, compared with the Ptolemaic models at his disposal, Waldseemüller modified Ptolemy’s second conic projection in a way that unfortunately distorted the shape of the new continents because they were forced to the far western portion of the map and hence greatly elongated.

During Waldseemüller’s time, new ideas were rapidly developing out of the theoretical discussions in Book I of Ptolemy’s Geographiae. Many commentators and cartographers realized that there was no reason to adhere to Ptolemy’s restriction of a correct representation of distances on three parallels, a restriction that was introduced in order to construct circular meridians. They found that by altering this arbitrary restriction on the form of the meridians and by applying Ptolemy’s methodology to any number of equidistant parallels, one could obtain a map correct on all parallels, with the meridians easily constructible as curves or polygons, connecting points of equal longitude.

This type of generalization was used on Ptolemy’s second conic projection by Waldseemüller to extend his world map, although not smoothly, as can be seen from the abruptness of the change in the meridians at the equator. A more continuous extension of the second conic projection was made in a less ad hoc way by Bernardus Sylvanus in a world map contained in his 1511 Claudii Ptholemaei Alexandrini liber geographiae cum tabulis universali fugura et cum additione locorum quae a recentioribus reperta sunt diligenti cura emendatus et impressus. Sylvanus’s generalization of Ptolemy’s mapping represented an extension of the area of the globe to between –40 and +80 degrees in latitude and between 70 degrees west and 290 degrees east in latitude using undistorted parallels.

In 1514, Johannes Werner produced his translation and commentary of Book I of Ptolemy’s Geographiae. Werner added to his translation a theoretical discussion of two generalizations of Ptolemy’s second conic projection in a section of his book entitled Libellus de quator terrarum orbis in plano figurationibus ab codem Ianne Verneo nouissime compertis et enarratis. In Werner’s Propositio IV (Figure 3) he modified Ptolemy’s methodology by requiring that lengths be preserved on all parallels, represented by concentric arcs, and on all radii. He further modified the projection in a way that made the North Pole the center of what in modern language would be called a system of polar coordinates. In Propositio V, he also required that a quadrant of the equator have the same length as the radius between a pole and the equator.
The modifications of Sylvanus and of Werner were the first solutions to the problem of representing the surface of a sphere within a finite area. Waldseemüller’s projection can be graphically approximated using the transformation equations that also can be used to represent an infinite series of projections that include Sylvanus’s, Werner’s and the later Bonne projection.

The value for the central parallel and an additive parameter can be changed in the equations for the Bonne Projection in such a way that an approximation to Waldseemüller’s projections results. The Sylvanus, Werner and Bonne projection in polar coordinates all contain an arbitrary parameter f > 0 such that r = + f. The image of the North Pole accordingly lies on the central meridian at a distance f below the center of the parallels. In the Bonne projection f is assigned in a way that the radii touch the meridian curves always on a given parallel. Sylvanus unknowingly uses a similar value to Bonne, f = 10, and if we assign f = 0 we arrive at Werner’s projection.

Waldseemüller’s map can be approximated in this same way using values of f between 7-8.5. The actual projection of the 1507 map differs from that represented in the above equations in that it has bends in the meridians at the equator, and the meridians are shown as segmented circular arcs rather than as continually changing curves. The modern coast of South America is projected in the figure above alongside the same region from the Waldseemüller map. It can be observed that on the Waldseemüller map that the western coast of South America is portrayed by a series of linear features and is labeled “terra ultra incognita”. These straight lines have been interpreted as Waldseemüller’s way of picturing regions for which he had no specific geographic information to make a more accurate representation. These same features, however, appear when the modern coast is projected on the approximate projection. Waldseemüller’s representation of the continent and the re-projected outline of modern South America are strikingly similar visually. Even though it is clear that Waldseemüller’s projection elongates the shape of the continent, it is very apparent that its width is close to that of the modern form.