Infinite Geometries:

Mathematical Notes on Werner's Commentary on Ptolemy's First Book and the Projection of the 1507 World Map by Martin Waldseemuller

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A representation is made with a purpose or a goal in mind, governed by criteria of adequacy pertaining to that goal, which guide its means, medium and selectivity. Hence there is even in those cases no general valid inference from what the representation is like to what the represented is like overall.

--Bas Van Fraassen, Scientific Representation

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The most naive view of representation might perhaps be put something like this: "A represents B if and only if A appreciably resembles B." Vestiges of this view, with assorted refinements, persist in most writing on representation. Yet more error could hardly be compressed into so short a formula.

--Nelson Goodman, Languages of Art: An Approach to the Theory of Symbols

Among the many technical and theoretical problems that Waldseemüller faced in the construction of his 1507 representation of the world, one of the least trivial mathematically and geometrically 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 (see figure below) 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. These of modifications result in the possibility of an infinte series of projections of the Waldseemuller type. This can be visualized by just a few examples from my models of Werner's projection below.

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. This difference is however trivial in the overall look of the projection and the distortions that is produces in the continents of the New World. Using these models the modern coast of South America has been projected in the figure below 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”. The straight lines that appear as the outlones of the west coast of South America 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, also appear on the modern coast when it is projected on the model projection that I am using to represent the geometry of the 1507 World Map. Waldseemüller’s representation of the continent and the re-projected outline of modern South America are strikingly similar visually.

Bi-dimensional Regression Revisited:
Studies in the geometry and form of the Medieval Portolan Chart

Introduction to my talk at the Library of Congress’ Conference
Re-Examining the Portolan Chart: History, Navigation and Science
May, 21st 2010

There are spaces in which the determination of position requires not a finite number, but either an endless series or a continuous manifold of determinations of quantity. Such manifolds are, for example, the possible determinations of a function for a given region, the possible shapes of a figure, and so on.
--Bernhard Riemann

What the historian of cartography should be concerned with is a systematic study of the factors effecting error, and seek to establish their cause and variability and the statistical parameters by which error is characterized...

--J.B. Harley

...secular magnetic variation is potentially as valuable in the history of cartography as the radiocarbon method in archaeology, though the calibrations have yet to be worked out.

--Tony Campbell, History of Cartography Volume 1

Click here to read the Washington Post Story on the LOC Portolan Conference

Those of you who know my more academic publications in the history of cartography realize that for the most part they tend to take an extremely phenomenological approach to cartographic objects. From my earliest publications on Fourier transforms and the Space Oblique Mercator projection through my current research on Topological existence theorems and mathematical constructivism in early computer cartography I have always been more interested in the conceptual and mathematical foundations of cartography than in any historical causalities or contingencies relating to maps themselves.

Calculated Distortion grid and vector displacements for the Library of Congress's 1320 Portolan Chart

Because of this phenomenological approach my paper this afternoon may be quite difficult for some of you (especially right after lunch) as it is extremely analytical and most of it is going to be concerned with very complex transformational geometry; discussions of things like Laplacian matrices and thin-plate splines. This being said, I promise you that if you keep your focus on the actual cartographic problems that I am trying to resolve much of the mathematics will dissolve into the background and in the end it is my hope that you will not only learn something about the geometric and mathematical structure of Portolan charts but also that this talk might serve as a methodological introduction to some of the computational techniques that I have helped develop and that I have been using in my cartometric research. These techniques have there beginnings with the work of Waldo Tobler whose paper and computer program called, Bi-dimensional Regression (see links section to read Tobler's paper) is where modern historical cartometry can be said to have started. Everytime I read this paper I am amazed at Tobler’s geometric insights and I find new inspiration in every re-reading. Using the integrated sums of the squares of the four partial derivatives was a real breakthrough and took incedible geometirc imagination.

Calculated rotation isolines for the Library of Congress' 1320 Portolan chart

My paper this afternoon will deal principally with three problems concerning the form of Portolan charts that have to date eluded solutions and whose logical structure goes directly to the heart of the geometric form that these early charts take. Borrowing a definition from the philosopher of science, Bas van Fraassen, “A representation [like a map] is made with a purpose or goal in mind, governed by criteria of adequacy pertaining to that goal, which guide its means, medium and selectivity”. In other words the form of a representation, in this case mathematical form, reflects the purpose for which the representation was created and hence my questions this afternoon are principally mathematical and not historical.

The first is the question of projection; Are Portolan charts purposely projected? Obviously, the fact they are the surface of a sphere drawn on the plane makes them projected, but the question is more specific; did their creators purposely project them in a consistent way and did these early mapmakers have any knowledge of the error they were introducing through this geometric transformation? This question is extremely difficult to answer because of the fact that any distortion that might be systematic from the projection is somewhat buried in the noise of the distortion caused by simple inaccuracy in the mapping of the coastlines. The most we can hope for is some statistical ruins that might be buried in the non-linear parts of the distortion...

The second has to do with the question of their apparent rotation; as has been pointed out this morning, the charts have various degrees of rotation; why do the parallels, at least in the area of Mediterranean Sea, appear to be rotated? Can we, by analyzing this rotation, gain some insight into the sources and measurement techniques used to construct the charts?

The third question concerns their evolution as geodetic maps; do they get more accurate with time? Do they maintian their accuracy even though many charts have obviously been copied multiple times. Are there any correlations that can be found in looking not only at modern comparisons but also in intrasample variations? (it is here that thin-plate splines are useful, see link to Booksteins paper in links section) Are there any structural changes that we can perceive through their history as a cartographic form?

Principal warps of the LOC's 1320 Portolan Chart

I am going to begin with a bit of a theoretical and historical primer into both the mathematical and philosophical justification for the computational techniques that I am using…many of them have a long history in cartographic analysis and I think it will help you to understand the motivations for some of my research here and on other maps…

For the complete slides of this talk click on the academia.edu link on the right or on the small Portolan chart above it......

Schoner's Fragments:

Terrestrial and Celestial Globe Gore Fragments from the Schoner Sammelband

The discoverer of the Sammelband, Josef Fischer, removed the 1507 and 1516 world maps in order to produce a facsimile of them and in doing so recovered from the gutter of the binding fragments of a set of globe gores that belong to Schöner’s 1515 globe. There are only two other surviving examples of this globe, one owned by the Historisches Museum in Frankfurt am Main, and the other by the Herzogin Anna Amalia Bibliothek, Stiftung Weimar Klassik. The gore fragments were trimmed and glued onto gore outlines by Fischer and then rebound into the Sammelband when the 1507 and 1516 maps were replaced. The set of terrestrial fragments found in the Sammelband constitutes approximately 50 percent of the actual globe. Schöner’s 1515 globe depends heavily on Waldseemüller’s 1507 Universalis cosmographiae for much of its geographical information andmany of the legends that appear on the 1515 globe gores are small paraphrases from the larger 1507 map. The globe goes
much farther, however, in its description of the New World, in that it actually shows a complete passage around South America into the Pacific Ocean. A more complete description of the geography found on the gores can be found in the companion volume that Schöner wrote to accompany the globe, Luculentissima quaedam terrae totius descriptio. Besides the terrestrial fragments, a second set of vellum gore fragments was found in the Sammelband.

These are from Schöner’s celestial globes and represent a different edition of Schöner’s celestial gores than is found fully bound in the Sammelband. The fragments represent much less than half of the total globe. In contrast to the full paper gores described below, the celestial fragments show the equator of the earth projected onto the celestial sphere at an angle to the ecliptic. The gore fragments also show differences in the labeling of particular constellations such as
Delphini, and show signs of print stereotyping.

The celestial gores found in the Sammelband are printed on paper and form a complete set of Schöner’s gores from 1517. The gores are the first known set of printed celestial gores and are a great improvement over other star charts of the period. Although Schöner’s interest focused mostly on geography in the early period of his life, we still can see in his extant manuscripts interest in the accurate determinations of stellar positions for the purpose of casting horoscopes. This interest is further established by the annotations that he made to the 1515 Stabius star chart by Albrecht Dürer that originally constituted part of the Sammelband. The Dürer chart contains several well-known errors that Schöner corrected by annotating both the chart itself and his globe. One of the most remarkable features of Schöner’s celestial gores is the naming of several groups of stars in minor constellations that were unnamed on celestial charts. For example, the stars in the constellation Coma Berenicies are usually shown on star charts of the period but went unnamed until Schöner called them Trica (located just above Leo) on his globe gores. Schöner has annotated the gores in red ink mostly over the constellations of Andromeda, Perseus, and Orion.

The 1517 globe, called Solidi et sphaerici corporis sive globi astronomici canones usum et expeditam praxim ejusdem exprimentes, was dedicated to the Bishop of Bamberg, Georg Schenk von Limberg, as were many of Schöner’s works and letters. Several parts of the Schöner Sammelband have been removed over the course of its life, including the 1507 Universalis cosmographiae, now in the Library of Congress; an annotated Dürer star chart from 1515, still at Wolfegg Castle; and a manuscript drawing by Schöner of sheet six of the 1516
Carta Marina, privately held by Jay Kislak.

Some of the most interesting texts regarding Schoner's globes come from his manuscripts that are in the National Library in Vienna. Especially important is a compilation of texts that is listed in their catalog as MS. 3505. In that manuscript there is a treatise called Regionum sive civitatum distantiae, which is a short theoretical work that deals with the problem of locating place-names on a globe using a planar map as a source. In other words, Schoner is talking about the inverse projection problem. In the work Schoner lays out several methods for turning planar maps back into spheres and using them for sources when making globes. Many of the construction methods that he discusses are quite complex requiring mathematical skill and a fairly detailed knowledge of projections. More on this will be found in my forthcoming book, A Globemaker's Toolbox: the mathematical and geographical notebooks of Johannes Schoner, which will be published late next year.

For more information on Schoner's Globes see Chet van Duzer's forthcoming study from the American Philosophical Society and for more images and a complete description of the Sammelband see my articles in "The Jay Kislak Collection at the Library of Congress"

http://www.kislakfoundation.org/pdf/Catalog_05.pdf