Monday, October 30, 2006

Is the 1516 Carta Marina a Portolan Chart?, Continued.....

As we have discussed in a previous post the 1516 Carta Marina by Martin Waldseemuller has many of the characteristics of a Portolan Chart including the rotation of the axis of the Mediterrenean Sea that runs from Gibralter to Antioch.

The fact that our rotation studies have shown that the axis of the Mediterrenean Sea on the 1516 map is between 7.9 and 8.3 degrees leads us to believe that portolan charts where used as sources for the information it contains. This is further brought out through the comparison of our rotation calculations with the rotation on other charts, paleomagnetic declination studies, and a comparison with Waldseemüler's 1507 World Map. The figure above (double click on figure to enlarge) shows a graph of the calculated rotation of various portolan charts (red line) compared with the historical magnetic declination studies of Tanguy, Bucur and Thompson published in the journal Nature in 1985. This group studied lava flows from Mount Etna in Sicily from 1301 to 1901 to determine the change in magnetic declination. Their results are shown in the blue curve on our graph. It can be readily seen in the figure that the rotation values of the Portolan charts match very closely the paleomagnetic data. Further, the calculated rotation of the Carta Marina sits on the graph in the location one would expect if we were dealing with a Portolan Chart. The 1507 map on the other hand has a rotation value that is much less and does not reflect the use of a Portolan source. A good study on the history of secular geomagnetic variation is that of Jackson et. al. based on a large scale compilation of historical magnetic field data, The time-dependent ­ field model that they construct is based on a dataset that is parametrized spatially in terms of spherical harmonics and B-splines.

Because the paleomagnetic values that we use here are from Mount Etna, near the center of the region that we are interested in for our rotation models, the magnetic declination differences are comparable with the rotation of portolan charts for the Mediterranean Basin. Our results here make us fairly confident that the Carta Marina not only resembles a Portolan Chart but is in fact directly derived from one.

For further reading see:

Malin, S.R. 1985. "On the unpredictability of geomagnetic secular variation", Physics of the Earth and Planetary Interiors 39: 293-296

Tanguy, J.C., Bucur, I. and Thompson, J. 1985. "Geomagnetic Secular Variation in Sicily and revised ages of historical flows from Mt. Etna" Nature 318: 453-455

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.

Sunday, October 08, 2006

Is the 1516 Carta Marina a Portolan Chart?

The 1516 World Map by Martin Waldseemuller, known as the Carta Marina, looks very much like a Portolan Chart even though it is a printed map. Many scholars have surmised that the sources for the map were in fact Portolan Charts but no one has ever attempted any calculations that might allow conclusions beyond these speculations.

The above figure (click on figures to enlarge view) is the sheet from the 1516 Carta Marina that contains Europe and most of the Mediterranean. Portolan charts contain two analytic features that are very important in distingushing them from other maps of the period. First, the axis of the Mediterranean basin is deflected or rotated by between 5 and 11 degrees. This orientation shift most likely results from an orientation to magnetic and not true North. A.C. Mitchell's paper "Chapters in the history of terrestrial magnetism" (Terrestrial Magnetism and Atmospheric Electricity, XLII (1937) 241-280) is one of the earliest to suggest this and is worth reading.

In order to test the Carta Marina for rotation of the Mediterranean basin we used an affine transformation and a Hampel estimator to control the error distribution of the chosen landmark points. The figure below shows the sheet of the Carta Marina with a distortion grid calculated using M-estimators. M-estimators (see links) are part of a large family of statistical estimation functions (known as robust) that try to reduce the effect of outliers or points of large error on the overall tranformation calculations.

Hampel gives several answers to the question of when to apply Robust estimators:

There are two observations which when combined give an answer. Often in statistics one is using a parametric model implying a very limited set of probability distributions, such as the common model of normally distributed errors, or that of exponentially distributed observations. Classical (parametric) statistics derives results under the assumption that these models were strictly true (this is especially important in our cartographic applications where are chosen landmarks are rarely evenly distributed). However, apart from some simple discrete models perhaps, such models are never exactly true. We may try to distinguish three main reasons for the derivations: (i) rounding and grouping and other "local inaccuracies''; (ii) the occurrence of "gross errors'' such as blunders in measuring, wrong decimal points, errors in copying, inadvertent measurement of a member of a different population, or just "something went wrong''; (iii) the model may have been conceived only as an approximation anyway, e.g. by virtue of the central limit theorem.

One can imagine using these estimators as an application of Tobler's first law of geography, "everything is related to everything else, but near things are more related than distant things" (see Waldo Tobler's article in Economic Geography 46 234-40).

The application of the transformation to the European sheet yields a rotation of 7.6 degrees for the Mediterranean which is consistent with a Portolan source. Below is the bare grid displayed for clarity. The other feature that most Portolan's display is a lining up of the tip of Brittany with the location of Venice, showing both places on the same east-west line. The area of Brittany should be displaced by more than 3 degrees. The 1516 map shows a lining up of these two places consistent with most Portolans. The cause of this distortion most likely comes from an error in the interpretation of the data from the Atlantic coast, the prototype being measured in Catalan miles which are shorter than the Italian miles typically used in the Mediterranean.

Although none of this conclusively proves that Waldseemuller used Portolan charts as sources for the 1516 Carta Marina it a least implies it as a possibility and suggests a place to look for possible prototypes of this region..

More on this soon.

Thursday, October 05, 2006

The newest results on my studies of the Waldseemuller Map come from a technique known as a thin-plate spline. You can find more information on splines in the link section of the blog. The spline transformation technique allows for the isolation of scale and shape deformation at various scales and gives both local and global deformatiom information. This type of local information cannot be produced in more global techniques like Polynomial Warping. (see my paper in Cartographica, June 2006, Warping Waldseemüller: A Phenomenological and Computational Study of the 1507 World Map also see Coordinates article in links). A deformation grid can be generated for any area of the map. The regressions produced by these spline models show areas where different geographical sources may have been used.

The figure above shows the values of an eigenvector for the longitudinal displacement for the African sheet of the 1516 Carta Marina. These results have allowed us to compare the 1507 World map with the 1516 map.

The thin plate spline is the two-dimensional analog of the cubic spline in one dimension. It is the fundamental solution to the biharmonic equation.

Given a set of data points, a weighted combination of thin plate splines centered about each data point gives the interpolation function that passes through the points exactly while minimizing the so-called "bending energy." Bending energy is defined here as the integral over of the squares of the second derivatives. Regularization may be used to relax the requirement that the interpolant pass through the data points exactly. The name "thin plate spline" refers to a physical analogy involving the bending of a thin sheet of metal. In the physical setting, the deflection is in the direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the or coordinates within the plane. Thus, in general, two thin plate splines are needed to specify a two-dimensional coordinate transformation.

These thin-plate spline studies which will be highlighted in my November 29th talk at the Library of Congress show that both of these important maps are composites and that they come from very different geographical sources.