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

Using Edge Detection Algorithms to Search for the
Physical Remains of Roman Centuriation and Surveying

But the land surveyor is like a judge; the deserted fields become his forum,
crowded with eager spectators. You would fancy him a madman when you see
him walking along the most devious paths.
----Cassiodorus

It is well known that the remains of Roman Surveying throughout Northern Tunisia are the best preserved in the world, but the difficulties in geographically locating these areas has led to a serious lack of research and scholarship on these remains. The Romans, in the regions around Carthage, Dougga and Enfida, surveyed extensive areas, and left behind physical remains in the form of limites and field boundaries. I have recently begun studying these regions using both fieldwork and computer methods in an attempt to locate and map these important remnants of Roman colonization. In the figure below I have used an edge detection algorithm on a satellite photograph (Landsat) of the area around Carthage that allows for the enhancement of linear field boundary features and whose results can be statistically compared to the known forms of Roman cadastral surveys found in texts such as the Corpus Agrimensorum.





















[click on figures to enlarge]

Once the extent and size of these 'survey areas' has been determined a grid can be fitted and overlaid on a satellite photograph of the region in question. The figure below shows the area around the modern town of Enfida, Tunisia. The calculated grid lines up quite well with the current local path system around the fields and with the intersection known to have been the kardo and decumanus in Roman times.

Below one can see a GIS map that I produced showing several of the areas studied so far using these methods along with a grid over the approximate extent of the physical remains. More complicated algorithms using both Fourier and Radon transforms have also been used to locate and orient the regions shown. I have presented detailed results of this research at the International Conference on the History of Cartography held in Copenhagen in July of 2009, and will provide more on the mathematical details of this in a future post and publication.