Warping History: Analytical Methods in Historical Geography

2012-01-24T14:15:00.000-05:00

Through Pollock's Eyes:

Reflections on the Fractal Nature of Geographic Curves and Abstract Cartographic Spaces

The idea that one gets a better and better approximation of the length of a shoreline by measuring it in finer and finer detail is false; the series of approximations does not coverge to an answer, it just gets bigger and bigger, to infinity.... ....................................................................---Tim Robinson, The Connemara Fractal

Many of my best mathematical reflections seem to come to me while wandering around in museums, especially the Museum of Modern Art in New York City. The abstraction of shapes and colors found in the paintings of many of the 20th century's greatest masters, all of which line the walls there, remind me of spatial and geographical forms that have been bled of reference, seemingly residual landscapes and spaces devoid of human action; the proverbial blank on the map or a vast bringing together of spatial silences. Recently, while standing in front of one of Jackson Pollock's great drip paintings, the connection that I have often felt between mathematical and geographic spaces and the planar surfaces created by the abstract expressionists suddenly crystalized in my mind as geometry. Fractal Geometry. I am of course not the first to think of this connection. J.R. Mureika for example, studied perceptual color space and related in to the fractal forms of Pollack's paintings ("Fractal dimensions in perceptual color space: A comparison study using Jackson Pollack's art," Chaos 15 (2005))

Both geographic spaces and most of Pollack's drip paintings are fractal in nature. In contrast to lines on a map, which are one-dimensional and are pure generalizations of reality, fractals consist of patterns that recur on finer and finer scales. Because of this 'scaling', fractals can build up natural shapes of immense complexity like coastlines, boundaries and even full landscapes. The Pollock drip paintings when looked at through finer and finer scales, such as those shown in the details below, are much more like geographical curves than cartographic maps actually are. His paintings, strangely enough, unlike maps, can be seen as scaled portions of the whole at larger and larger scales. Several scholars have looked at the fractal dimension of Pollock's paintings using one of the easiest, at least from a mathematical perspectuve, method to determine dimension which is called box-counting. The method is relatively easy to calculate and program and goes back to at least the 1930s when it was known as Kolmogorov entropy after its Russian inventor. The dimensions calculated for Pollack's paintings show them to be somewhere in the range of 1.3 to 1.7, and unlike those of cartographic expressions, are actually closer to that of real world geographic curves.

Real geographical curves are so complex in detail that their lengths are often infinite or to put it in a more accurate sense, undefinable. Many of these curves, such as those representing a coastline, are statistically 'self-similar', which means that each portion can be considered as a reduced-scale model of the whole much like a Pollock drip painting. Because of this one can immediately sense how geographical curves (boundaries for example and coastlines) are similar to the drip painting fragments shown below [1] . This peculiar feature of self-similarity, which is an artifact of scaling (something Galileo would have loved), can be described mathematically as a type of dimension, which unlike normal curves, is fractional.
This fractional dimensional property of coastlines was first posited by Benoit Mandelbrot in his classic article in Science from 1967, entitled, "How Long is the Coast of Britain ? Statistical Self-Similarity and Fractional Dimension."

Self Similarity of the von Koch Curve. It looks the same no matter what scale we look at it in.

In that article [2] Mandelbrot studied the form of geographic curves, and simpler examples like the von Koch curve shown above, for which the concept of 'length' has no apparent meaning. The von Koch curve is built up by an algorithmic procedure that at every stage in the operation the middle third of each interval or line segment is replaced by the other two sides of an equilateral triangle. These types of curves, to use Mandelbrot's language, can be considered as "superpositions of features of widely scattered characteristic sizes." As he puts it, "as even finer and finer features are taken into account, the total measured length increases, and there is usually no clear cut gap or crossover, between the realm of geography and details with which geography need not be concerned". [3]

Mandelbrot set of a quadratic function in the complex plane. Self-Similarity at every scale

One of the best reflections on this problem of crossover and its effect of cartographic thought comes from the writer and mapmaker Tim Robinson's essay, 'The Connemara Fractal' published in his Setting Foot on the Shores of Connemara and other essays. Robinson, who is one of the great writers on cartography and what for him is the solitary process of mapmaking, writes in the essay that he is "not very interested in maps from a technical point of view...so he will move on to the more interesting questions of what it is like to make a map...insofar as I can untangle my memories of the process." The process of mapmaking for Robinson is that of a long walk, "an intricate, knotted itinerary that visits every place within its territory." The idea of a long and very detailed walk that Robinson invokes was suggested to him by the extraordinary form of the southern coast of Connemara. Robinson tells us that, "it looks so complicated as to be unmapable; it is a challenge to be unraveled." ...a very Mandelbrotian series of images indeed.

Robinson crosses over to details that according to Mandelbrot "geography need not be concerned", when he starts looking at actual distances, what he finds to be possibly infinite distances. Robinson writes that the "distance from Ros a' Mhil to Roundstone is only about 20 miles, but the coastline in between is at least 250 miles long, even as estimated on a small-scale map." He continues, "when I wrote that I was ignorant of the work of Benoit Mandelbrot, who had proved that an outline as complex as a coastline does not have a definable length." Robinson first learned of Mandelbrot's work through a newspaper article sent to him by a reader of his essays on Connemara and he calls Mandelbrots work "a disturbing doctrine---disturbing to one who foundly imagined he had walked a coastline with due attention to its quiddity....for [Mandelbrots paper] was an annihilating critique of my essay's imagery."

In many respects Mandelbrot's paper was of course not really a critique of Robinson's essay, but rather a profound theoretical statement that opens up a clearing for us to concieve of a very different, and inherently less postivistic conceptual foundation for the cartographic spaces we create. For as Robinson says, the process of mapmaking, like that of discourse must stop somewhere. To be true of the world of fractals is to be infinitely and indefinately seeking a precise measurement of length or to possess the unexplainable talent of a Jackson Pollock. But to find our way in the real and empirical world it appears that you and I will have to be satisfied with our approximations and our finitude.

[1] For more on Pollack's paintings as fractals see, "The Visual Complexity of Pollack's Dripped Fractals by R.P. Taylor, et.al. at http://pages.uoregon.edu/msiuo/taylor/art/fractalexpr.html

[2] Benoit Mandelbrot, "How long is the coast of Britain? Science 156 (1967) 636-638. http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf

[3] For more on the concept of fractl dimension and measurement see M.C. Shellberg and Harold Moellering's classic paper, "Measuring the Fractal Dimension of Empirical Cartographic Curves." http://mapcontext.com/autocarto/proceedings/auto-carto-5/pdf/measuring-the-fractal-dimensions-of-empirical-cartographic-curves.pdf

2012-01-15T07:52:00.000-05:00

Modeling Roman Land Use and Environment:
Epigraphy, Servitudes, and Game Theory

...we have all too often lacked, or failed to consider, conceptual frameworks of theory in which to examine Man's relationship to his environment, the manner in which he weighs the alternatives presented, and the rationality of his choices once they have been made....
---Peter Gould

In the study of Roman agricultural patterns it is important to have a conceptual framework in which to place the fragmentary information and evidence that is available from epigraphy, Roman law, and landscape archaeology. For the past few months I have been experimenting with Game Theoretical Models and the concept of Nash Equilibrium trying to see what type of land use models would arise.

The basis of game theory was first laid down in the late 1940's by the mathematician John von Neumann and the economist Oskar Mogenstern in their now classic book

the Theory of Games and Economic Behavior. In the book von Neumann gives the proof of the Minimax Theorem, which is central to game theoretic reasoning and that he first approached in 1928. In the 1944 book, von Neumann placed the theorem within the context of linear inequalities and the theory of convexity, which was later updated with more formal proofs of equilibrium states by John Forbes Nash.

My current work on modeling land use and some of the environmental decisions made by Roman farmers takes its real start however, from a conversation that I had with Waldo Tobler, Emeritus Professor of Geography at California, Santa Barbara, about 8 years ago. I had just read Peter Gould’s paper on African farmers in General Systems Theory, a paper that would later lead me to his seminal work, Man against the Environment. I knew that Tobler was close to Gould and that he was also playing around with some game theory during these years, and so I asked Tobler about the paper. What was most impressive to me in all this was not really Gould’s mathematics, but rather his vision of what game theory might be able to do in geographical sciences, that even simple matrix games had a spatial component that few geographers had thought to utilize.

One of the things that Gould wrote and that struck me as profound was that , “we have all too often lacked, or failed to consider, conceptual frameworks of theory in which to examine Man's relationship to his environment, the manner in which he weighs the alternatives presented, and the rationality of his choices once they have been made.” The rationality part instantly jumped out at me. As you may or may not know, the idea of rationality is an area of hot debate when it comes to questions of the Roman economy. There are many scholars, especially after Finley’s seminal book called The Ancient Economy, who believe that to consider Roman farmers and landowners as ‘rational’, in the sense of their maximizing the yield from their farms and thinking about market forces, is to project too much of a modern conception of a market economy onto the past. More recently however, some scholars like Dennis Kehoe, Cynthia Jordan Bannon and D. W. Rathbone, using legal inscriptions and the everyday account books of farms that survive as papyrus fragments, have started to use economic models and things like the theory of the commons to talk about Roman markets and agricultural estate management. Each of them in their own way incorporates many of the terms and categories of game and decision theory in their analysis. Perhaps the best book that accepts and summarizes the presence of ‘rational’ actors in the Roman economy is a book by Paul Erdkamp, entitled, The Grain Market in the Roman Empire: a social, political and economic study. Erdkamp puts forward many different models in the book, and summarizes the economic theory in his historical examinations and reconstructions. His is the sort of book that makes you anxious when you read it, as it gives you a good idea of how much you do not know and how long it takes to make any real progress in this area.

My own models are simply extensions of this kind of work. One group of Gould’s papers, from which my research certainly takes its inspiration, was written by him in the 1960's. His papers, "Wheat on Kilimanjaro: the perception of choice in game and learning model frameworks," and "Man against His Environment: a game theoretic framework", were among the first attempts to use the concepts of game theory and Nash equilibrium to look into agricultural land use. These papers, and a few others, were also discussed in an early review article on these methods written by David Harvey, "Theoretical Concepts and the Analysis of Agricultural Land-Use Patterns in Geography." It is in fact from Harvey’s book, Models in Geography that my concept of geographic model derives.

Harvey asserts, in his review article on agricultural land use, that at the time he was writing, many geographers tended to ignore theoretical breakthroughs from other disciplines, mainly on the "grounds that they proved too abstract to help in the search for unique causes of specific events." To counter this he quotes from William Bunge, whose book Theoretical Geography transformed geography and opened up an analytical window for the field, suggesting a more theoretical and inherently mathematical approach to the study of geographical and spatial distributions. To me Bunge’s book is the most important work of geography in the 20th century and I still mine it for inspiration.

Most of Harvey's paper is dedicated to outlining the requirements for a set of theoretical and conceptual elements to constitute a model in geography. A model, according to Harvey, requires a set of relationships to be established that somehow link the input, status and output variables in a specific way. This linkage must quantify the model mathematically in order for it to be tested. For Harvey, the relationships of the variables in the model can be of three distinctive types:

1. Deterministic relationships which specify cause and effect sequences.
2. Probabilistic relationships which specify the likelihood of a particular cause leading to a particular effect.
3. Functional relationships which specify how two variables are related or correlated without necessarily having any causal connection at all.

For agricultural models Harvey makes a distinction between two types of frameworks, one in which the underlying structure is normative and therefore, describes what ought to be under certain assumptions. The second, is descriptive, and describes what it is that exists. These distinctions are extremely important when we try to interpret game theoretical models, especially in something as difficult to conceptualize as the Roman economy.

In his early research Gould, using a normative game theoretic model, studied a group of African farmers around Kilimanjaro and analyzed how they decided what to plant in varying environmental conditions. Gould understood the patterns of land-use and the choices made by farmers are the result of decisions made either individually or collectively and that it might be useful to try to model those decisions in a game theoretical framework. In Gould's models the environment is one player and the farmer is another. Each of the players is faced with a number of different strategies the solution of which is the game's equilibrium. Using simple matrix games he was able to construct cartographic representations of various equilibrium alternatives that could be compared to what was in the fields.In his early research, Gould studied a group of African farmers around Kilimanjaro using decision theory to analyze how they decided what to plant in varying environmental conditions. Gould understood the patterns of land-use and the choices made by farmers are the result of decisions made either individually or collectively and that it might be useful to try to model those decisions in a game theoretical framework. In Gould's models the environment is one player and the farmer is another. Each of the players is faced with a number of different strategies the solution of which is the game's equilibrium.

Importantly, Gould realized that the game theory of the time was still algorithmically primitive and that his results determined neither how the farmers actually behaved nor how they should have behaved in an absolute sense, but rather how they should behave if they want to achieve particular results. In strategic games, such as the one Gould proposed in his papers, Nash equilibria are a set of actions amongst the payers that lead to a steady state. It is a position in the game in which each player holds the correct expectation about the other player and behaves and acts rationally according to his choices. Gould uses the simple graphical solutions to the matrix games he creates which I found so attractive early on in Harold Kuhn’s lectures. For more on Kuhn and John Nash watch the video of a recent seminar they gave together at Princeton, here.

The concept of equilibrium is not so straightforward here as one might think, and it can be interpreted in several ways. For example, when we say that a physical system is in equilibrium we might mean that it is in a stable state, one in which all the causal forces internal to the system are in balance. This is the traditional economic meaning of equilibria. The variables are dynamic however, and the balance between them that makes up the equilibrium can be thought of as networks of mutually constraining relations. Equilibria can then be considered as endogenously stable states of the model. Some scholars however, interpret game theoretic equilibria as being explanatory of the process of strategic reasoning alone. For them a solution must be an outcome that a rational agent would predict using the mechanisms of rational computation alone. The meaning of equilibrium states is still a matter of discussion in the literature of game theory and has interesting philosophical implications to how we view and interpret what the models tell us outside of their mathematical formalism. (For more on the interpretation of game theoretical results see Ariel Rubinstein's seminal paper Comments on the Interpretation of Game Theory or the Philosophy of Game Theory by Grune-Yanoff.

The current models I am working with are of course much more complex than anything Gould could have considered, as he lacked both the mathematics and the computing power. New techniques like quantal response functions, which allow us to look at probable actions, are much more powerful and yield much more interesting results. They were first introduced by McKelvey and Palfrey in the late 1990s and considered mathematically for the possibility that the players will make mistakes and therefore they give more realistic results than anything Gould imagined, at least we hope they do.

The power of models in historical geography is that you can look at many different scenarios and compare them with the little actual historical data you have. I would never assert that what I am doing actually gives me any definitive answers on what decisions Roman farmers made or how they planted, rather they show me what possibilities there were and how to rank them. Most importantly however, they greatly inform my thinking about the Roman economy in its most empirical form, and since I do not have the disciplinary constraints on my ideas that an economic historian might, I can push the limits of the models for purely theoretical and curiosity reasons.
My hope is that these methods will yield an 'experimental' historical geography, an acceptance of simulation as a method in historical studies. These simulations have the potential to shed light on the decision alternatives that face farmers and estate owners acting within primitive or developing economies. They give us a glimpse into how historically farmers interacted with their environment on a mainly cognitive level, allowing us to consider the choices they made and how their decisions affected the landscape around them. This to me, and to other geographers before me, like Gould and Harvey, is certainly a central geographical question.

For those interested I am using a software package that can calculate the Nash equilibria for games with large numbers of players, or in this case environmental variables called GAMBIT.
http://www.gambit-project.org/doc/index.html
It is an open source program and you can have a great deal of fun experimenting with variables and how they change the equilibrium outcome.

2012-01-05T08:44:00.000-05:00

Random Walks Across the Atlantic:
Stochastic Processes and the Geometry of the Early Renaissance Portolan Chart

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

When one is considering trying to model the accuracy of the Medieval and Renaissance Portolan chart it is useful to reflect upon what types of data might have been used in order to construct these charts, such as the one from around 1300-1320, shown below and which is part of the collections of the Geography and Map Division at the Library of Congress. If we assume that these charts are simply graphic displays of information measured by sailors and navigators we can ask ourselves what type of information might have been used in the charts construction and what is the statistical error that might reside in such measurements?[1].

Calculated isolines of rotation that mimic lines of magnetic declination

see my presentation at the LOC's Conference on Portolans Charts at:

http://loc.academia.edu/JohnHessler/attachment/946350/full/Bi-dimensional-Regression-Revisited--Studying-the-Geometry-and-Form-of-the-Medieval-Portolan-Chart

Any model of measurement that includes measurement instruments, such as the compass or the hour-glass, can be thought of using classic measurment models which are composed of three parts.

1. a family of observables M (physical magnitudes like declination and longitude) each with a range of possible values.
2. a set states S...physical states of both the system measured and of the measuring system.
3. a stochastic response function P for each m in M and s in S, which is a probability measure of the range m with P to be interpreted as the probability that a measurement of m will give a value in E, if performed when the state is s [2].

So what does these mean for Portolan Charts? If, as I mentioned earlier, we think of a nautical chart as the graphic expression of empirical sailing measurements, we can ask ourselves how accurate is the data that went into the cartographic representation, and is there a way for us to compare the actual data, assuming it survives, with the chart in a way that is mathematically consistent and has significance tests. There are many surviving examples of log books from transatalantic voyages, but few if any statistcal studies of the actual data that was compiled in them. One important example of such a compilation was done by the cartographer Guillaime Delisle in 1705. He collected about 10,000 positional and declination measurements in a series of notebooks that still survive in the National Archives in Paris. This information has never been published, but contains a wealth of historical measurements taken at sea during the 16th and 17th centuries that might give us some idea of how accurate the data available to Portolan makers was.

Considering the date of Delisle's data, we must recognize that the positional measurements he cites are mostly based on dead-reckoning and astronomical navigation, coming as they do before the invention of the nautical chronometer. The fact that these measurements are based on dead-reckoning helps us to model the error involved because we can think of the error in positional measurement as serially correlated. Practically, this means that as a voyage proceeds the error in a particular positional measurement also incorporates the error found in the previous positional readings. The positional measurements were then corrected by the navigator when land was sighted, and hence the error forms a series of independent legs[3].

If we look at Delisle's data and a make a scatter plot of the individual errors in longitude accumulated as a function of time between points of land fixing, we can see by the figure below that the error forms a Gaussian distribution.

If we graph the error in positional fixation geographically using the difference in actual (modern) versus measured position we get a figure of the type shown below.

The fact that the scatter in the data is Gaussian and that it can be represented in figures like that shown above leads one to believe that the data on Portolan charts can be modeled using particular stochastic processes such as those known as the Random Walk and the Brownian Bridge[4].

According to the historical data found in most log books, two situations arise in the numbers that correspond to each of the models mentioned earlier:

1. a leg of a voyage starts from a known location and then a number of positional observations are made before the leg of the journey finishes with no geographical endpoint noted in the log book. This type of data fits the model of the classic Random walk.

2. a leg of a journey begins at a known starting point follwed by a number of positional observations and land-sightings, concluding at a known location. This type of data fits the pattern of a stochastic model known as a Brownian bridge[5]. In the case of the Random walk, the error pattern that emerges is the result of the accumulation of errors that are independent of each other, so-called independent increments. The idea of independent increments is applicable in the case of voyages where dead-reckoning was employed because the error contributes cumulatively to the positional uncertainty and the errors are not systematic but rather random with many causes.
We can therefore express the error accumulation by the above equation. Each time a positional measurement is made it increases the overall error by a small increment. Assuming a Gaussian distribution of the error leads to the summation,

The variance of this summation is the quantity that we are looking to model, as it will allow us to correlate the root mean square error for positional measurements taken from the log books, and that which would have been incorporated into the charts themselves. If we solve for the variance we can see that it is increasing with the length of the voyage, as we would of course expect from serially correlated errors.

When we actually graph the positional error from the log books against the models we find that the root mean square error for a voyage of, say, 50 days for example, is between 1 and 4.4 degrees with larger numbers for very long journeys. These two RMS minimums and maximums are shown in the log-log plot below. As it is very rare for a journey to last more than 5o days without a known positional fixation or land sighting, the Random walk model is probably representative of the error that might be found using the more mathematically complex Brownian bridge.

If we look at Portolan Charts that show the transatlantic regions, the Cantino Planisphere for example, we can compare our stochastic models with the calculations of scale error that we have accomplished using Huber tranformations.

Rotation and Scale Distortion on the Cantino Planisphere....for more on this see the Washington Post's Article on my research at:
http://www.washingtonpost.com/wp-dyn/content/article/2010/05/21/AR2010052104713.html

The RMS found in the data is comparable with the RMS found on the charts which, depending on local variations, is between 3 and 5.2 degrees. Although this shows us clearly that the data found in the log books matches the error found on the charts, it does not say anything about how the Portolan makers compiled the information they used...a problem that still awaits a real theory.

[1]. For more on Medieval Portolan Charts see Tony Campbell's seminal article and survey in Volume 1 of the History of Cartography, "Portolan Charts from the Late Thirteenth Century to 1500".
[2]. See A.R.T. Jonkers, Earth's Magnetism in the Age of Sail, John Hopkins University Press, 2003 for more on the surviving forms of positional and magnetic data and his models in "Four Centuries of Geomagnetic Secular Variation", Philosophical Transactions (2000) 957-990.
[3]. The philosophy and probabilty of measurement processes have been the subject of any number of articles. A good modern treatment can be found in Bass van Fraassen's, Scientific Representation: Paradoxes of Perspective, Oxford University Press, 2008. For a more formal treatment one should also consult the relevant sections on probability in his book, The Scientific Image, Oxford Library of Logic and Philosophy, 1980.
[4]. Rabi N. Bhattacharya and Edward C. Waymire, Stochastic Processes with Applications, Siam Classics in Applied Mathematics, 2009.
[5]. An interesting example of the type of data that we are concerned with here can be found in J.M. Vaquero's study "A note on some measurements of geomagnetic declination in 1776 and 1778", Physics of Earth and Planetary Interiors 152 (2005) 62-66.

2011-12-15T08:08:00.000-05:00

Written in Stone: Epigraphy, the Codex Iustinianus, and the Geography of Roman Petition and Response

Quacunque enim ingredimur in aquila historia vestigum imponimus.
[Wherever we step, we tread on one or another scene of history]
--Cicero, De Finibus, 5.5

This project centers around the epigraphy of Roman land ownership and environmental law, such as agrarian and water rights, and their relationship to the Codex of Justinian. Although the Codex records many of the imperial rescripts relating to these subjects, it does not contain most of the petitions that these recripts were written in response to. To look closely at this one must turn to legal records that have not been edited, shortened on interpreted by late-antique and medieval scribes and jurists. The only documents of this type are found in legal inscriptions, most of which survive from North Africa and the Middle East.

These inscriptions, when looked at through a more geographic lense, show regional variations in legal practice and shed light on how the Romans adapted themselves to differences in environment and the agricultural practises of the native populations in the provinces.

My research will consist in looking through the vast and very understudied collections of inscriptions from museums in Libya, Tunisia, Algeria, and other collections, along with writing geographical commentaries on some of the more famous inscriptions like Henchir Mettich, (pictured above in a photo I took in the storeroom of the Bardo in Tunis), Lamasba (Ain Merawa) and Aga Bey Koyu from the Usak Museum, along with many more.

2011-11-27T08:57:00.003-05:00

Finding the Antipodes: Mathematical Constructivism and the Changing Logic of Cartographic Objects, 1960-1975

Abstract of my AAG 2012 Paper

In mathematics everything is algorithm and nothing is meaning; even when it doesn't look like that because we seem to be using words to talk about mathematical things.
Even these words are used to construct an algorithm. ---Ludwig Wittgenstein

...a proof of the existence of a magnitude can only be seen as completely rigorous if it contains a method by which the magnitude whose existence is being claimed can really be found. ---Leopold Kronecker

We look upon maps not only as stores of spatially ordered information, but also as a means for the graphical solution of certain problems for which the mathematics proves to be intractable. --William Warntz

In the early years of computer cartography new levels of abstraction entered into the field of geographical analysis through the algorithmic development of theorems from pure mathematics. In an attempt to answer previously intractable geographical questions, concepts from pure mathematics, like existence theorems, whose basic logical structure contains statements that confirm or deny the existence of particular sets of mathematical objects, were employed in various computer mapping schemes. The development of these programs injected high levels of topological and algebraic abstraction into geographical analysis and changed the basic ontology of geographic objects. Existence theorems, although they provide logical proof for whatever mathematical entity they are claiming existence for, do not however, necessarily provide a way to find or calculate those objects. In the field of pure mathematics existence theorems had long been objects of controversy from both a practical and philosophical perspective and their use sparked debates among many mathematicians. Mathematicians and philosophers, like Leopold Kronecker and Ludwig Wittgenstein, questioned the utility of a mathematical proof that provided no algorithmic way to find the mathematical object whose existence is claimed, while others such as David Hilbert and Richard Dedekind, saw no conceptual or philosophical difficulties with their use. This debate among the so-called constructivists, like Wittgenstein, who believed that in mathematics “everything is algorithm”, and the formalists like Hilbert, has left a large body of philosophical literature that has deeply analyzed the ontology of mathematical objects. [1]

In the fields of geography and cartography, these theorems entered into early computer systems through the construction of practical algorithms that calculated particular sets of objects useful in geographic analysis. Two important papers that can be seen as case studies in the use of constructivist forms of existence theorems in early computer cartography were published in the series Harvard Papers in Theoretical Geography by William Warntz and his associates at the Harvard Lab for Computer Graphics and Spatial Analysis in the late 1960s and early 1970s. This series of papers developed algorithmic constructions of many existence theorems and two of the most interesting, because of the sheer complexity of the mathematics, the Borsuk-Ulam Theorem and the Ham Sandwich Theorem, were applied to real world geographic problems [2].
Besides using existence theorems, mathematical cartographers would also begin to re-conceptualize on a more general level questions about the use of pure mathematics and its role in defining the diagrammatic logic of maps. In an early lecture, later written as a discussion paper for the Michigan Inter-University Community of Mathematical Geographers, Warntz says that, "More than ever before geographers are using the tools of calculus, probability, topology, symbolic logic, the various algebras, geometries, for example, are being taken more literally than ever before." He elaborates on these comments by explaining to the reader that something as abstract and foreign to geography as Venn diagrams are being taken, "in a far more literal sense than they were originally intended and by substituting real space and attendent phenonema for ideal space and by insisting on the utilization of all geometric properties involved as well as just the topological ones, geographers can reinterpret, add to, and refine the conventional concepts in the methodology of uniform regional geography and provide it with a basis in logic." [3]

Many geographers at the time would push the concept logic form and notions from set theory further into geographic analysis and not just in the sense of a useful analogy. In a paper written for one of the classic compilations texts from early years mathematical geography called, The Philosophy of Maps, Warntz and others like Waldo Tobler, and William Bunge, would change not only the vocabulary used in analysis but would also alter the very form of its expression. In an article in the collection, called Some Elementary and Literal Notions About Geographical Analysis and Extended Venn Diagrams, Warntz would say that, "Maps showing regional classification can be regarded as logic diagrams. Mapping of sets is a general mathematical concept. Geographical mapping is merely a special case of this." [4] Warntz here sees almost a mereological or mereotopological relationship between the spatial extent of Venn diagrams and their isomorphic counterparts of geographic regions.

John Venn

It is quite remarkable that the two systems of logic that Warntz draws on in this paper, Venn diagrams and existential graphs, are both visual and not symbolic logical systems. Most of the work done in logic during the 20th century has focused on symbolic systems with little research, at least until quite recently, on the heterogeneous reasoning of the type Warntz is advocating. He says that, "It is part of our purpose here to extend the use of such diagrams to the mapping of geographical regions by making use of properties already inherent in Venn diagrams but as yet unutilized... We intend to apply spatial properties literally to real spatial distributions on the earth's surface..."

Venn diagrams can grow to extremely complex forms depending on the number of sets one is dealing with and recent research on the use of logical diagrams has shown that Warntz was ahead of his time in thinking that the spatial and geometrical component of logical diagrams would be useful analogs for spatial maps. [5]
As stated above, Warntz' paper calls on the work of Charles Sanders Peirce (above) and his existential logic diagrams, which he sees as mappings from non-spatial sets to geographical maps. Looking at the complexity of Peirce's systems, there is both an alpha and beta form depending on the required complexity, one wonders how deeply Warntz explored the subject of existential graphs. An important aspect of these graphs that Warntz thought useful for regional geographic analysis was the fact that a logic diagram can be drawn as a two-dimensional figure with spatial relations that are isomorphic with the structure of some logical statement. This is very important if one is going to try to apply set theory of the type Warntz is envisioning here, simply because these spatial relations are usually of a topographic nature.Logic diagrams, especially the type developed by Peirce (simple examples shown above with a page frm Peirce's notebook below), stand in the same relation to the various logical algebras as maps of areas stand in relation to their particular algebraic functions; they are simply other ways of symbolizing the same basic structure. [6]

In much of what Warntz has to say here we are reminded of the long way we have come when talking about set theory, topology and the formal properties of spatial structures and their relationship to cartography. One only has to look at books like Varzi and Casati's, Parts and Places: the Structures of Spatial Representations (MIT, 1999) [7] to get a feel for how our language and conceptual grasp of these topics has improved since Warntz and others involved in the early development of computer cartography were experimenting with what at the time were radically new ideas.
The current project envisioned here, which grew out of my research for the 20th century volume of the History of Cartography, will provide a mathematical and philosophical analysis of both of the Harvard papers mentioned above, along with others from this formative period that apply set theory and logical analysis, in an effort to show not only how constructivist methods migrated from mathematics to geography, but also to show how these new levels of abstraction changed the foundational ontology of geographic and cartographic objects. Using the philosophical debates that took place over things like existence theorems in the mathematical literature as a basis, this study will show that a foundational shift in the ontology of geographical objects opened the door to new conceptualizations of geographic space and formed the theoretical basis for the development of spatial logics and the current use of topological and abstract algebraic methods in geographical analysis.

[1] It is interesting to note that many early mathematical geographers had an interest in Wittgenstein. Waldo Tobler, in a private communication, told me recently that he was persuaded by Peter Gould (1932-2000) to take up the reading of Wittgenstein.
[2] The two papers are; Geography and an Existence Theorem: A Cartographic computer solution to the localization on a sphere of sets of equal-valued antipodal points for two-continuous distributions with practical applications to the real earth (1968) and The Sandwich Theorem: A basic one for geography (1971).
[3] A Note on Surfaces and Paths and Applications, William Warntz, Discussion Paper Number 6, Michigan Inter-University Community of Mathematical Geographers, 1965.
[4] The Philosophy of Maps, edited by John Nystuen, Michigan Inter-University Community of Mathematical Geographers Discussion Paper 12, 1968.
[5] For recent research on the logical status of Venn diagrams and the nature of spatial logic see, Eric Hammer (1995), Logic and Visual Information, Stanford CA: Center for the Study of Logic and Information; Nathaniel Miller (2007), Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry, Studies in the Theory and Applications of Diagrams, Stanford CA: Center for the Study of Logic and Information and Sun-Joo Shin (1994), The Logical Status of Diagrams, New York: Cambridge University Press.
[6] For more on Peirce's Existential Graphs see Sun-Jo Shin's seminal study, The Iconic Logic of Peirce's Graphs, MIT Press, 2002.
[7] Achille Varzi and Roberto Casati, Part and Places: The Structure of Spatial Representations, MIT Press. 1999.

2011-09-30T15:34:00.001-04:00

His aqueducts and his cartography: Frontinus, Roman law and the missing maps of the waters of Rome

The foundations of the science of land measurement lies in practical experience, since the truth about sites or area cannot be expressed without lines that can be geometrically measured.
--Frontinus, De arte mensoria

According to R.H.Rodgers, 'obscurity veils the early career of Julius Frontinus,' who in the year 97 was appointed curator aquarum of the city of Rome. Frontinus wrote two groups of texts that are important to us here in our study of Roman cartography; those being De Aquaeductu urbis Romae and a series of texts on Roman surveying. The work that remains extant on Roman surveying is found in the Corpus Agrimensorum and is very fragmentary. Karl Lachmann (shown below), who worked on the first edited edition of the text, believed that the full work comprised two books, the first consisting of De Agrorum Qualitate and De Controversiis, the second work containing De Limitibus, De arte Mensoria and some other more fragmentary texts from Urbicus, another writer on Roman surveying who may have copied his work from the now missing parts of Frontinus.

Besides his interest land surveying however, Frontinus is more well-known for his text on the aqueducts of Rome. (For more on this see the website www3.iath.virginia.edu/water/furst.html run by Katherine Rinne) In the text of De Aquaeductu urbis Romae he discusses the fact that he made maps used in the administration of the aqueducts. In the prologue to the book Frontinus refers to his work as a commentarius, and explains that it is a collection of data and other materials that he made primary for 'himself'. The contents of the book are quite technical and numerical, pertaining to sizes of individual aqueducts, the dates they were built, pipes and their sizes, the quantities of water delivered and legal matters relating to the right of private individuals to the use of public water. Although most of the material serves an adminstrative aim some of the text deals with methodological issues and it is these that are of interest for historians of cartography. In Chapter 17 of his book on aqueducts Frontinus writes:

Non alienum mihi visum est longitudines quoque rivorum cuiusque ductus etiam per species operum complecti. nam cum maxima huius officii pars in tutela eorum sit, scire praeposiutum oportet quae maiora impendia exigant. nostrae quidem sollicitudini non sufficit singula oculis subiecisse; formas quoque ductuum facere curavimus ex quibus adpareret ubi valles quantaeque, ubi flumina traicerentur, ubi montium lateribus specus adpliciti maiorem adsiduamque petendi ac muniendi vi exigerent curam[1].

Which translates as:

It has seemed to me not unfitting to include as well as description of the lengths and courses of each aqueduct, according to the classifications of construction. Because the greatest part of the duties of this position lies in the maintenance of the lines, the man in charge must know what thongs demand greater outlays. My sense of responsibility has not been satified with personal examination of particular items. I have also taken care to prepare maps of the lines, from which it is clear where there are valleys and how great they are, where rivers are crossed, and where channels attached to the sides of mountains demand greater and constant attention...for their repair.

Hence we learn here that Frontinus had detailed maps made of the aqueducts describing not only the lines themselves but also the topography of the surrounding countryside that they traversed. According to Harry B. Evans, in his Water Distributon in Ancient Rome, (Michigan, 1994), "Frontinus' mapmaking merits closer attention." Evans postulates that Frontinus' data, which he gives in the text on aqueducts, is in fact derived from the maps that he had made and that those sections of the text describing the actual lines are commentaries on the maps themselves. There are other indications in the text that Frontinus is using maps as he pinpoints some of the sources of the aqueducts by using exact spatial references to the existing road system outside of Rome.

...
Although none of Frontinus maps survive there are maps on inscriptions that show what aqueducts maps may have looked like. An example shown here (CIL 6.1261) displays in graphic form the lines of an aqueduct and the epigraphy gives indications of water flow and on what legal terms individuals may draw water from particular lines. The inscription contains the names of the people who shared the channel that came off the aqueduct, the volume of water that they where alloted, and the scheduled times that they could take that allotment.

The inscriptions translates as:

a. for Thyrsis, freedman of Augustus, two pipes from the second to the...hour, on the fourth day before...

b. for the freedman of C. Julius Caeser, C. Bicoleus Rufus Squaterianus, one pipe...

c. to the Aufidianum of Julius Hymetus, two pipes from the second to the sixth hour...

d. To Vibius...pipes, to C. Bicoleus, Freedman of C. Julius Caesar,... pipes from the sixth hour until sunset...

There are several known examples of this type of inscription and another is shown below (CIL 14.3676). This inscription describes a shared channel related to the supply of water at Tibur, a rural area outside Rome. The stone containing the inscription is found built into the side of the Church of Saint Peter at Tivoli and it preserves a fragment of a map showing two channels. The inscription itself lists the people to whom the water is to go to, the amount of water they have been alloted, and the time of day when it may be taken. (for more on water rights see Cynthia Jordan Bannon, Gardens and Neighbors: Private Water Rights in Roman Italy, Michigan, 2009). As Evans says, all of this deserves further work....and can help us understand some of the lesser known aspects of Roman cartography and its application.

[1] I have used the new edition of Frontinus by R.H. Rodgers, "De Aquaeductu urbis romae", Cambridge Classical Texts and Commentaries 42, 2004.

2011-09-29T09:42:00.001-04:00

A Cartographic Commentary on the Henchir Mettich Inscription from Central Tunisa: Roman Surveying in the Medjerda Valley

The foundations of the science of land measurement lies in practical experience, since the truth about sites or area cannot be expressed without lines that can be geometrically measured.
--Frontinus, De arte mensoria

As noted in my previous post, the inscription from Henchir Mettich in the Bagradas valley of Central Tunisia is important as a window on the landscape archaeology of the region and the history of agriculture during the Roman empire, but it is also important for the history of cartography as it relates to Roman law and the running of imperial estates in the second century AD. As one can see in the example of face one below, this coming from a series of photographs that I took last year in Tunisia, the inscription is badly damaged.

The inscription was published as CIL 25902 and very different transciptions of it can be found in Kehoe's, The Economics of Agriculture on Roman imperial Estates in North Africa and in van Nostrand's, The Imperial Domains of Africa Proconsularis: an epigraphical study.

The most important part of the inscription from a cartographic perspective is to be found on side one of the column shown below as published in the CIL and as a lithograph from Toutain's L'Inscription D'Henchir Mettich: un noveau document sur la propriete agricole dans L'Afrique Romaine.

The text on the first side of the column talks in some detail about the subject of subseciva or unallocated lands. The word is the subject of much discussion in the Corpus Agrimensorum and generally means lands unsuitable for allocation to settlers, either sirutated between the centuriae and the outer boundary of a communities territory or within centuriae. The word literally means "cut off" or "cut away below".

For example, Frontinus talks about the fact that he knows of fifteen different types of 'land dispute',
"...the position of boundary markers, a straight line boundary, boundary, site, area, ownership, possession, alluvial land, territorial juristdiction, subseciva, public places, places omitted and not enclosed, sacred and religious places, control of rain water and rights of way."

He continues later in his text on 'land disputes',
"A dispute over subseciva occurs when some or all of a centuria has not benn allocated and is possessed. Or if an adjacent landholder or someone else occupies any land from the edge of the allocated area, this also comes under disputes involving subseciva."

Hyginus also has much to say about subseciva. In his descritpion of categories of land he tells us that,
"Certain areas that protrude beyond the type of land which is curved or has angles, and are divided off by straight lines, are called subseciva, that is, pieces of land that remain when the boundary lines have cut them off and retain the character of peripheral areas."

The discussion of subseciva in the Henchir Mettich inscription begins after the dedication ends in line 6.

...qui eorum [i]ntra fundo Villae Mag-
[n](a)e Varian(a)e id est Mappalia Siga, eiseos agros qui su[b]-
[c]esiva sunt excolere permittitur lege Manciana
ita, ut eas qui excoluerit usum proprium habe-
at. [...]

The translation of this part of the column is not easy, but generally it says,

"To those coloni (who will have farmsteads) within the boundaries of the estate of Villae Magna or Mappalia Siga, who wish to cultivate more fields, permission is given to cultivate those fields which have not been alloted (subseciva) or have been classfied as unused, under the terms of the law of Mancia; namely that he who cultivates this lands shall have them for personal use."

One can infer from this that the land belonging to the Villa Magna was originally mapped and surveyed and then distributed to individuals, becoming some form of ager privatus. It certainly proves that this particluar area had boundaries drawn even though there are currently few physical remains of the Roman centuriation lines. Epigraphic evidence for Roman mapping has not been studied in a large scale fashion before and I hope to published my complete GIS of this information, at least from North Africa and Southern France, in the next year or so. For those who are interested I show the other four faces below...

Click on images to enlarge

Face II (above), Face III (below ), Face IV (below III)

2011-09-28T15:11:00.000-04:00

Epigraphic Evidence for Large-Scale Roman Mapping

What survives of their treatises [of the Roman surveyors] can appeal to few readers now, but so diverse are the manuscripts that preserve it, so many the names associated with its preservation, that no text opens the window wider on the transmission of Latin literature from Antiquity to print…
--L.D. Reynolds
Texts and Transmission

Besides the epigraphic cadastres from the colony of Orange in the South of France, a small fragment of which is shown in the figure below, there is other epigraphic evidence that the Romans actually made detailed maps of their territories. Although extremely rare, there are several examples of epigraphic inscriptions where explicit mention is made of the word "map'.

In the Corpus Agrimensorum, a compilation of Roman Surveying manuals from the 6th century, there are several words used for map. Writing in the text the surveyor Siculus Flaccus says,

“The maps are given various names: some are set up on wooden tablets, others on bronze, still others on skins, although ‘map’ is their generic term, they are sometimes called ‘territory’, ‘centuriation,’ ‘demarcation,’ ‘limitation,’ ‘grid-pattern,’ figures…”

Hence Latin words such as Forma, tabula, pertica, typon, and metatio all appear to mean map.

Epigraphic evidence from Tunisia shows other examples of the word Forma being used in this fashion. In the Corpus Inscriptorum Latinarum (CIL) we find two examples in which maps are mentioned as having been made or that are being referred to. The figure below shows CIL 22788, an inscription from Henchir Chenah, that is carved on four sides of stone.

CIL 22788

The part of the inscription that we are interested in reads:

sec]undu(m) [f]orma(m) missa(m) sibi ab posu[it]

and records a boundary settlement made "in accordance with the map".

A second inscription from Henchir ez Zoubia, CIL 23910, shown below records a longer inscription referring to a boundary stone set up between the land of two communities.

The inscription reads:

positum sic [secum] dum forman [um mar]

This refers to the fact that the settlement was set up once again "according to the map". From the remainder of the inscription we might imply that this was done by a soldier (perhaps a surveyor) attached to the XIIi Urban cohort based in Carthage.

CIL 23910

2011-09-07T14:02:00.002-04:00

Geometrical Problems from the Corpus Agrimensorum :
Mollweide meets Mommsen

A representation 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.
--Bas Van Fraassen

The group of Roman surveying texts known as the Corpus Agrimensorum have provided a great deal of information regarding the actual practices of Roman surveyors in the field and given scholars insight into how the Romans allocated and measured land. My current project in locating and mapping the surviving remains of Roman surveying in North Africa takes its starting point from this 6th century compilation of surveying manuals. The texts themselves and the illustrations attached to them have attracted the attention of many scholars in past including historians of Roman law and agrarian practices such as Theodor Mommsen and Max Weber. In this sence the historiography associated with the texts is almost as interesting as the texts themselves.

What is less known about this historiography is the attraction the texts have held for more mathematically inclined historians of cartography such as C.B. Mollweide. Mollweide is best known as map projectionist, but also did fundamental research into some of the unsolved geometrical problems found in the Corpus.

One of the most interesting of these problems concerns the method for finding south. In the Corpus there are several methods given for this, but the one that interests us here is that given by the writer, Hyginus.

There is also another method of obtaining South, by marking three shadows. On level ground we shall set up a gnomon AB, and note any three of its shadows, CDE. These shadows we shall mark with the set square, to see their distances from each other. If we set them up before noon, the first shadow will be the longest; if after noon, the last. We shall then draw these shadows in proportion by a footrule... Let AB be a gnomon, B the ground. Let us take the longest shadow and mark it [i.e. its end opposite to B] on the ground as C; the second likewise D, the third E... Let us project hypotenuses from C on to A and from D on to A. Now with centre A and radius E let us draw a circle. Then let us project lines parallel to the base, i.e. ground, on to the perpendicular [AB] from the intersections of the hypotenuses and the circumference, from F on to G and from I on to K. Then we shall apply the longest line, GF, to the largest shadow, and from B we shall mark out [the length of] GF ; the second line to the second shadow, and we shall mark out [the length of] KI. Then from F and I we shall project a straight line, and likewise from C and D, the shadow ends. These two lines will meet at T. Join TE; this will be east-west.

The above text by Hyginus is accompanied in the Codex Arcerianus A by the figure shown below.

The figure as drawn by the scribe is however totally inadequate to explain the complexities of the method outlined by Hyginus and one has to question its purpose in the manuscript and whether it was in fact added to the text by a later copiest with less understanding of the method. Mollweide analyzes the text in an article 'Erlduterung einer in der Scriptoribus rei agrariae. . . gegebenen Vorschrift..' published in Zach's Monatliche Correspon-denz zur Beforderung der Erd- und Himmelkunde (Volume 28, 1813. p. 396-425).

Click on Figures to enlarge

>>>>>>
The method that is being described in the text by Hyginus is extremely complicated and there are open questions concerning the level of mathematics and solid geometry involved.

Mollweide's solution is a complex affair and according to Dilke the method as described by Hyginus probably goes back to Alexandrian mathematical scholarship that has been lost but that must have been dependent on Apollonius' Conics.

The modern solution that Dilke adapted from Mollweide can be condensed to the following along with the figure below:

ABC, ABD, ABE are right-angled triangles. The lines CA, DA, EA go towards the centre of the sun. The arc EIF is part of a circle forming the base of a regular cone, parallel to the sun's daily round and so to the equator, and FI is a chord of this circle. Since GF is equal and parallel to BL, FL will be equal and parallel to GB; similarly IM to KB. As FL and IM are parallel to AB and so to each other, they lie in a plane in which FI and LM lie. But FI is also in the plane ACD, and LM is in the horizontal plane BCD; so FI is the intersection of the plane FIML with the plane ACD, and LM the intersection of the plane FIML with the horizontal plane BCD. As the plane ACD is cut by the horizontal plane at CD, which when produced meets LM produced at T, it follows that T is in the plane ACD and also in the plane FIML, and so is a point on the common intersection of both planes, i.e. of FI produced. Since the latter lies entirely on the plane of the circle through FIE, T is also in this plane, but likewise in the horizontal plane BCD, and so is a point on the common intersection of both planes. Since E is also such a point, it follows that ET is the intersection of a plane parallel to the equator plane with the horizontal plane, and so parallel to the east-west line.

One can see how different the figure which displays the actual construction as it described by Hyginus is from the original manuscript illustration.

2011-08-02T14:00:00.000-04:00

Fourier Finds Caesar: A Study in the Physical Evidence of Roman Agrarian Law and Land Usage Using Periodic Functions

Landscapes are dynamic constructions, with each community and each generation imposing its own cognitive map on an anthropogenic world of interconnected morphology, arrangement, and coherent meaning.

--Kurt Anschuetz, An Archaeology of Landscapes

Finding the physical and epigraphical remains of Roman surveying and centuriation throughout the Roman world remains an area of research that currently engages only a few historians of cartography. In the past the practice of Roman surveying was studied by many important Roman historians like Theodore Mommsen and Max Weber[1]. There remain however, many difficult and unanswered questions about Roman cartography, and the lack of actual extant maps has made me begin to look elsewhere for information that might shed light on its origins and methods. My current research on this problem employs GIS and image analysis to historical aerial photography and remote sensing imagery. It is my hope that in the near future it will produce the first complete map of North African sites that shows both the extent and orientation of Roman mapping. Several authors, such as Rita Compatangelo [2], J.W.M. Peterson [3], D.J. Bescoby [4] have pioneered the use of various mathematical transforms in the analysis of remote sensing imagery for the purpose of finding new sites and orientations. I have started to apply these methods in combination with edge detection algorithms in order to calculate the extent of Roman surveyng and the various types of orientations associated with the physical remains of limites.

The physical remains of Roman centuriation take on a number of sizes and orientations, but are typically discovered through the outlines of the limites that seperated the various regions from one another. Limites or Limes (singular) can be defined as a man-made boundary or balk, that is uncultivated and wide enough to form a road or pathway, which divided centuriae or other land division units from one another. These can take many forms from simple paths all the way to larger structures like the main roads of the decumanus and kardo maximus that were centrally located in a surveyed region. The feature that makes these remains discoverable through the use of transformational techniques is the fact that they appear on the landscape as periodic phenomena. This simply means that the pattern of centuration repeats itself over areas of the landscape, showing up as linear freatures that appear in remote sensing imagery as periodic pattern of grids over fixed distances. One of the most useful ways to study periodic phenomenon, at least from a mathematical perspective, is through the use of Fourier Series and transforms [5]. These transforms model any periodic phenomenon that we might be ineterested in as a infintie series of cosine and sine functions of varying frequencies.

This sum can be expressed more conviently through the use of complex exponentials which are easier to work with algebraically. Using a discrete and algorithmic version of the Fourier transform, known as the fast Fourier transform (FFT), Peterson and Compatangelo, in truely ground breaking papers, showed that one could calculate the most common period found in a group of periodic linear features found on more modern maps. I say the most common period, because many of the linear features found on the landscape today are subdivisions of modern and medieval origin, and it is sometimes extremely difficult to determine the date of the features whose period the transforms are measuring. As an illustrative example, one can think of the linear features found in the landscape as a more complex superposition of the images in the figure shown below. In the figure we see that there are linear features that repeat themselves and that in each of the figures they have different periods of repetition. One of the figures also has a different orientation than the other two. What we see in the landscape is typically a combination of all of these in the same region and on the same remotely sensed image. Peterson used the FFT to generate periodograms that produced the most common harmonics in a series of regions dislaying linear features that he took from 19th Century Ordnance Survey Maps of Scole-Dickleburgh area in South Norfolk. What the periodogram does is allow one to pick out the frequency of the linear features and the larger harmonics. An simple example of this is shown in the figure below. The periodograms not only show the most common distances between the linear features found on the map or on the satellite image, but they also yield a series of harmonics that might have the physical meaning. Larger harmonics beyond the most common one may show subdivisions in the original survey or different grid patterns from the type one is looking for. Because we are not only interested in the distances between linear features in the landscape buy also in their orientation, we have employed a second technique known as a Radon transform. This technique has been used by Bescoby to detect Roman boundaries in aerial photographs in Albania. The strength of this method is that it allows for the calculation of the angle and hence the orientation of the series of linear features. When combined with the two-dimensional version of the Fourier transform, this allows a complete characterization of the grid formed by the limites of Roman surveying. The Radon transform can be expressed by the equation below and its operation can be seen in the graph shown in the figure. Finding the size and orientation of linear features in a landscape lets us compare what we have calculated with known patterns of centriation found in literary and epigraphical evidence, such as that found in the 5th or 6th century Corpus Argimensorum. According to Hyginus, one of the authors found in this compilation of Roman surveying texts, the typical layout for an area of surveyed land is shown below. The letters and numbers define the parcel of land and very often appear as eppigraphic inscriptions on surviving boundary stones. The main intersection shown in the figure is that of the kardo and decumanus maximus which are the beginning points of any Roman survey. A typical kardo or decumanus can be seen in the photograph below that I took in a heavily centuriated area around Carthage just north of Tunis. The distances that the Romans typically used and their various names are shown in the schematic, with a century measuring 2400 Roman feet or about 705 meters. Many of these grids would however have been further subdivided in a variety of schemes that are not easily dated using physical evidence. The research that I have been doing has concentrated its efforts on the non-coastal regions of North Africa, taking in parts of Tunisia, Algeria and Libya. Below are two satellite images of the areas around Dougga and El Jem in Tunisia both of which are the sites of important Roman towns and ruins.

In both of these photographs one can see a variety of linear features that may or may not be associated with Roman activity.

It would be interesting to know the extent to which the Romans actually produced maps of these areas considering their overall importance to the history of Roman colonization and occupation in the region. El Jem for example contains one of the best preserved Roman colesseums in all of Africa.

To apply these algorithms to remote sensgin imagery it was necessary to clean them up and enhance the linear features using edge detection algorithms. An example of this is shown in the figure below.

Once this is accomplished and the various transforms have been applied we can begin to compare the results with known grids based on our knowledge of Roman practice derived from the epigraphic and literary evidence.

Using ArcGIS I have generated maps with overlays showing the orientation and extent of the surveyed region under study. The map below shows a single division into centuriae around Enfida, Tunisia.

The map below shows both a division into centuriae and into a second subdivison which probably dates from a later time.

[1] There are many studies of Roman surveying. For a modern bibliography see Brian Campbell, The Writings of the Roman Land Surveyors, Society for the Promotion of Roman Studies, 2000.--
[2] Rita Compatangelo, Un Cadastre De Peirre Le Salento Romain, Annales Litteraires de l'Universite de Besancon, 1989.---

[3] John Peterson, Fourier Analysis of Field Boundaries, in G. Lock and J. Moffet, CAA91: Computer Applications and Quantitative Methods in Archaeology 1991. BAR International Series s577. Oxford, 149-156.

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[4] D.J. Bescoby, Detecting Roman land Boundaries in aerial photgraphs using Randon transforms, Journal of Archaeological Science (2006) 33, 735-743. See also, J. S. Bailly et'al "Agarian Landscapes linear features detection: application to artificial drainage networks" International Journal of Remote Sensing 29 (2008) 3489-3508 and E. Magli, et.al. Pattern Recognotion by means of the Radon transofrm and the continuous wavelet transform, Signal Processing 73 (9990 277-289.

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[5] See any of the recommended books on Fourier Analysis on this blog or for a good introduction to the subject see L. Solymar, Lectures on Fourier Series, Oxford University Press, 1988.

2011-05-20T11:25:00.006-04:00

The Morphology of the Early New England Landscape:
Searching for the Lost Maps of Henry David Thoreau

New York Times/Mattson Lecture
October 16th
Osher Map Library
University of Southern Maine

An article based on this lecture and intervening research will be pubished in the March 2011 issue of the Concord Saunterer: A Journal of Thoreau Studies.

The human organism has rarely been subjected to a severer test than the study of scientific problems, nor is there a truer hero than an investigator who never loses heart in a life-long grapple with the powers of the universe. It requires courage of the highest order to stand for years face to face with one of the enigmas of nature; to interrogate patiently, and hear no answers....
..................................................................................................................---Clarence King

Synopsis of the text of my lecture at the re-opening of the Osher Map Library and the "New Directions in the Study of Early American Cartographies" Conference...

In April of 1858, Ralph Waldo Emerson, in a letter to his friend H.S. Randall, wrote that, “Thoreau’s study seems at present to be equally shared between natural and civil history,” and that “he reads both with a keen and original eye.”

The civil history that Emerson refers to here is the history of the early exploration and discovery of the North American continent, especially the northeastern coast of New England and Canada. During the last 12 years of his life, from about 1850 thru 1862, Henry David Thoreau dedicated himself to historical and scientific studies that have either been ignored by or have puzzled generations of his commentators. What was the author of Walden and other works of transcendental literature doing out “in all weathers” as Emerson would say, counting tree rings, measuring the differences in the magnetic variation of compass needles, mapping the depths of streams or listing the blooming of plant species. Why was he borrowing the earliest exploration narratives of the New World from the Harvard Library, taking detailed notes on the names of places and the plants and animals mentioned, and making scaled copies of some of the earliest maps of North America?

During these last, but extremely productive, years of Thoreau’s life his interests turned sharply toward these types of more empirical and less transcendental studies—Thoreau being less influenced in his work at this time by Emerson than by the geographically oriented science of Humboldt and Darwin. Thoreau believed that the secrets of nature, and of humanities place within it, were ultimately revealed by identifying what was significant in the everyday world and that this in turn depended on meticulous attention to and an accounting of, the commonplace.

In this spirit Thoreau’s writings such as the Maine Woods, Cape Cod, Walden, A Yankee in Canada, his natural history essays, and of course his journals have occasionally been probed by humanist geographers and linked with the beginnings of modern environmental thought…this linkage stems mostly from Thoreau’s intense concern with the concept of place and his ability to see deep connections between historical process and environmental change.

Tonight I am going to talk to you about this linkage in a slightly different way than those geographers and historians who have probed Thoreau’s writings…for tonight rather than concentrate on his published works…I am going to speak with you about his cartography and how Thoreau’s cartographic explorations provided a link in his mind between natural and civil history… a link that led him to a very modern sense of man’s place in nature. To do this I am going to discuss, in more detail than perhaps has been done before, several important aspects of Thoreau’s mostly unknown and certainly understudied cartographic works
The first aspect, and perhaps the most important for his technical understanding of cartography and the process of mapmaking, was his work as a land surveyor. This work gave Thoreau the ability to look at maps critically and to understand not only their mathematical limits but also their broader cultural meaning. It also allowed him to wander the fields and woodlots of Concord and to observe nature closely in all seasons…in a way that his fellow transcendentalists certainly never would have.

The second aspect, and for the most part the driving reason for his engagement with early American cartography, was his interest in the history of the indigenous peoples of the north east. Thoreau would take nearly 4000 pages of notes on this subject…none of which has yet to be published and all of which are mostly unknown even to Thoreau specialists. In the course of this note taking Thoreau would copy and comment on many of the early maps of the northeastern United States and Canada by such seminal figures as Champlain, Ortelius, Smith, Verrazano and Wytfliet.

Finally, I am going to present to you two new maps that have recently been attributed by me to Thoreau and discuss my search for some of the other missing maps that he copied and annotated.

Some of what I am about to say might sound to some of you slightly academic, but really my talk tonight it is about something quite humble and speaks to the very heart and reason of why we are all here this evening. What am speaking about in the end is a story about a man in a Library, much like the one we celebrate this weekend, with a notebook, some old maps and his imagination on fire…

I am going to begin with one of those notebooks, namely Thoreau’s field notebook in which he kept the rough notes from his surveying activity. Even though we are talking tonight about Thoreau’s maps, it is with a pencil and a notebook that he truly found his way in the world…and we will have reason this evening to look at many of Thoreau’s notebooks.

....
The field notebook appears on the surface uninteresting as it is filled for the most part with measurements and locations…places around Concord, MA that Thoreau surveyed. But it provides a detailed record not only of how Thoreau worked but also how he approached the more technical aspects of cartography and we shall return to this notebook many times this evening. Thoreau surveyed many places around Concord and the list of his clients reads like a library of early American authors. Places like Bronson Alcott’s farm shown here.

....

Thoreau set himself up as a surveyor in late 1849 and by the end of 1851 was recognized as one of best and most accurate operating in the region. The question of exactly how Thoreau learned to survey is for the most part an open one but it does appear that he was mostly self taught. In early April of 1850 Thoreau would borrow from the Harvard Library a copy of Davies’ Elements of Surveying and navigation , [we will see a great deal of Library use by Thoreau this evening.] A copy of the 1847 edition of Davies was also owned by Thoreau. Davies was one of the most widely used books on surveying during the middle of the 19th century and with Galbraith’s mathematical tables, also borrowed from Harvard, represents a solid introduction to the principles of land measurement. Thoreau annotated his copy of Davies with notes from Galbraith on particular mathematical problems…this is Galbraith’s instructions on how to calculate the sine of an angle.

Thoreau took much more from Galbraith than mere mathematical instruction. In a journal entry dated June 9th of 1850 Thoreau lists nine books recommended by Galbraith’s text regarding the esoteric subject of the magnetic variation of compass needles. I am going to spend some time with this subject because I believe it shows how Thoreau thought through and imagined the complexities of mapmaking and his engagement with the subject can be used as a model for us to think through his transition from an Emersonian transcendentalist…to the more empirical view of the world that informed his readings and use of early American cartography

Thoreau’s interest in magnetic variation is first indicated on his advertising broadside where he explains the variation of the compass is noted so that the survey can be repeated.
Throughout the early 1850’s one finds references throughout his journals and field notebooks made to the books and articles Thoreau read on the subject and to the observations of compass needle variations that he made in and around Concord.

Magnetic declination for those of you who do not know is the variation that we see between magnetic north, or the north that a compass needle points to, and true north, the direction of the pole. The variation in the compass needle is caused by the earth’s magnetic field and was the subject of a great deal of scientific research in the mid-19th century. The exact direction that a compass needle points to is not constant even for a specific location, and although few people would notice these small changes, Thoreau, noted them quite explicitly…for example…in November of 1850, he made an entry in his journal that marks the beginning of what would be almost an obsession with the subject…

Thoreau writes,

“When I am considering the way which I walk, my needle is slow to settle, my compass varies by a few degrees and does not always point due southwest; and there is good authority for these variations in the heavens…”

Thoreau’s interest is more than just passing and he delves into the science of magnetic declination in a way that would become representative of his work not only in natural history but also in the way he approached cartography as well. In his field notebook he explains how he established the True meridian so he could continually check his surveys against the variation of his compass needle

True meridian slide

• Found the direction of the pole star at its western elongation (1,58-1/2) at 9h 26m PM (Feb 7th 1851).
• N coincides with a line drawn from the SE course of the stone post on the E side of our western small front gate, to the S side of the first door on the W side of the depot.

Thoreau has measured a reference line for the direction of true north… from the west gate of his home in Concord to the depot across the street.
By establishing a sight line for the True Meridian from his family’s house to the depot, Thoreau could easily check the declination of his compass before or after surveying. Thoreau would begin to include this information on his surveys even though it makes little difference to the purpose of the survey itself. For example we can see on his survey of Hosmer’s farm that Thoreau has added compass headings to each of the boundaries.

Thoreau would continue for some time to make detailed observations of magnetic declination and write them into in his field notebooks…
He would also go as far as to contact and correspond with William Cranch Bond, the director of the Harvard Observatory. Bond was conducting experiments in magnetic variation in Cambridge, and took thousands of measurements on magnetic declination in order to try to predict the changes that he and Thoreau saw in compass needles, a phenomenon that we now know to be chaotic. One can see just how chaotic by looking at one of the many graphs of Bond’s published measurements that Thoreau certainly read.

Thoreau soon found that Bond’s results were not applicable to Concord, even though it was just a few miles away.

One of the most amazing and important things about all this material is how it shows a change in Thoreau’s thought process and his turn away from the transcendentalist mode of thought that drove his early works. Among transcendentalists' core beliefs, at least as it was realized by Emerson, was an ideal spiritualstate that 'transcends' the physical and empirical and is only realized through the individual's intuition. In other words a real mind over matter philosophy. Emerson found little of higher worth in the empirical and downgraded most of science as “mere facts”. Thoreau would begin by the early 1850’s to leave these idealist tendencies behind and turn toward more realist studies of nature and history…to the point that by late 1852 he could write in his journal seemingly against Emerson, that “Mere facts & names & dates communicate to us much more than we suspect…”

This empirical turn toward a more scientific world view… would influence all of Thoreau’s thought from around 1852 onwards. Although we can never really call Thoreau a thinker who fully embraced the pure empiricism of late 19th century science, there is a change in his thought that effects all of his reading and observations even his interpretation of the early exploration narratives and the history of cartography… it is to his cartographic explorations that we will now turn…

In order to talk about Thoreau’s interactions with and interest in early American exploration and cartography we must look to what has become known as the “Canadian Notebook”. The notebook was begun shortly after Thoreau’s return from a trip to Canada and those few scholars who have actually read it….it remains unpublished… differ in their opinions of why it was composed. The notebook is divided into three parts… the second part is certainly the most intense interaction with cartography and cartographic literature that one will find in the entire Thoreau corpus.

This second part of the notebook contains notes from Thoreau’s reading of early exploration narratives and maps by such figures as Champlain, Lescarbot, John Smith, Ortelius and Wytfliet. Thoreau takes note of specific subjects like the changing of place names, the plants and animals that the explorers encountered, the size and flow of rivers, temperatures, snowfall, and the changing shape of the lakes and rivers shown on their maps. In other words facts, empirical data that describes the landscape and the conditions of place. This type of description is paralleled in his journal entries at the time where he is noting things like stream depths, tree ring counts, snowfall, and the blooming times of plants that he observed during his surveying of woodlots and farms around the Concord countryside.

Even as Thoreau was taking detailed notes on the maps and information found in these early exploration narratives he expressed his frustration with the study of human versus natural history. In his journal, on October 19, 1860, he writes,

“It is easier far to recover the history of the trees which stood here a century or more ago than it is to recover the history of the men who walked beneath them, How much do we know---how little more can we know—of these centuries of Concord life?”

It was to answer this question that Thoreau turned to early cartography and the texts that accompanied them. In the back of the Canadian notebook written in Thoreau’s hand, but in pencil, and in the wrong direction if one is reading from the front, Thoreau composes the following list of maps that he has copied…

On the back fly leaf written upside down in Thoreau’s hand:
“I have copied maps made ac. to…”
1.Verarzani’s plot in Hacklyts divers voyage 1582
2. Map made in forme sent from Seville in 1527 by Thorne
3. Map of Nova Francia etc. in Ramurio 3rd volume (1556) ac to discourse of a great sea captiane
4. Of America in Ortelius (1570 &e) who used Cabot and others
5. Of Norumbega and Virginia 1597 Wytfliet Lovanni
6. Nouvelle France Champlain 1612, 1632

The list, although there are 6 numbers, actually describes seven maps there being two by Champlain listed on the same line. I first encountered this list many years ago and became interested in determining if copies of the maps that Thoreau listed still survive, if he in fact drew them, and what they might have looked like.

Two of the maps on the list were quite easy to locate and are part of the Collection of Thoreau surveys and papers in the Concord Free Public Library. The one shown here was copied by Thoreau from Abraham Ortelius’ world atlas, the Theatrum Orbis Terrarum, published in 1570. Ortelius was the royal geographer to King Phillip II of Spain and a prolific atlas maker. Thoreau borrowed several different editions of the atlas from the Harvard Library over a period of years and I have not been able to determine when he made the sketch. Thoreau’s notes on Ortelius in the Canadian notebook are principally devoted to a commentary on this map…”a new description of America”…and they are very much concerned with place names…as is typical of these notebook entries that deal with maps Thoreau brings other sources in and makes comparisons…in the case of Ortelius Thoreau will ask about the antiquity of some of the place names the cartographer uses.

The second map found in the Concord Library is that of Cornelius Wytfleit. Wytfliet was a Flemish geographer who published an atlas called Descriptionis Ptolemaicae Augmentum in 1597. This sketch, which is cruder than the Ortelius, was made in 1855 again from a book borrowed many times from Harvard Library.

The other five maps have been more difficult to find and to identify. Three of them are very small and are found folded in the misc papers of Thoreau that are in the Morgan Library…they are unidentified and therefore do not appear in their catalog lists or finding aids. (an example shown below)

Of all of the notes in the Canadian notebook about cartography, by far the most extensive are associated with the narrative of Champlain’s voyages…time and time again Thoreau will borrow the 1613 and 1632 editions of this book from the Harvard Library…
Champlain made several voyages to the New World and explored the St. Lawerence River, along with most of the New England coast, at least as far south as Cape Cod. The narrative of his voyages is filled with maps and his reflections on the explorations.

Thoreau’s notes on Champlain’s texts are full of details on plants, animals, place names and especially the differences in the various editions of the maps Champlain made.…Thoreau writes praising Champlain’s accuracy as a geographer and he quotes in detail Champlain’s own commentary on the methods of his mapmaking.

But what about the two Champlain maps on Thoreau’s list of the maps he copied???…they appear in no inventory of any library, they are talked about in no scholarly articles and appeared to me to be lost.

Then one day, when I first came to the Library of Congress, almost ten years ago I happened to be doing some research on Thoreau and the depth of Walden Pond when Ron Grim…now of the Boston Public Library, mentioned to me that there were several maps in the Geography and Map Division thought to be by Thoreau but with no real attributions. Curious about this but not expecting much…for how could maps by Thoreau be sitting in the Library of Congress without firm attributions…I opened the folder and saw a manuscript…

Immediately I knew that I was looking at the missing Champlain maps. The manuscript that you see here is one of four pages that describes the two Champlain maps that Thoreau listed on the flyleaf of the Canadian notebook and copied, one from 1632,

This map was easy to identify as Thoreau’s because it is in his hand and fits the general form of his other map sketches. The second map however from 1612 was more difficult as it is a tracing and contains almost nothing that would directly link it with Thoreau. Thoreau does say in the manuscripts that accompany the maps that he annotated the 1612 map in red with later places names. This feature can be seen on the 1612 copy below.

While the finding of the two Champlain maps completes the list of the known cartographic copies that Thoreau made… it has opened up the question of how to understand Thoreau’s geographic explorations in relation to some of his larger projects and published works. For Thoreau’s relationship to cartography is a complicated one and has suffered from a lack of scholarly attention. In general Thoreau seems to have remained skeptical of maps even as he made constant use of them…in his journals he wrote,

How little there is on an ordinary map! ...between those lines indicating roads is a plain blank space in the form of a square or triangle or polygon or segment of a circle, and there is naught to distinguish this from another area…for on it are no moral lessons…

And in the Maine Woods he tells us that maps are “labyrinths of error.”

Thoreau’s true attitude toward cartography is not difficult to assess, if one takes the time to read through his extensive notes on the subject closely. The notes express an immediacy of experience that occurs when one is reading and observing directly…Thoreau did not think of historic maps from the past as obsolete, but rather as graphic and ideological documents that could help him understand what had been in a particular place before…In many ways we can consider Thoreau the first "modern" historian of cartography.

To conclude, I want to return to one of Thoreau’s surveys.

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It is a simple drawing of a woodlot but I think it sums up Thoreau’s relationship to cartography and its influence on his work. The surveying of woodlots was very much part of Thoreau’s daily routine and it took him into areas of Concord seldom seen by his fellow citizens…surveying a woodlot generally meant the lumber it contained was up for sale and it was going to be cut down. Thoreau would return to the lots after they had been cut and in his journals noted in detail the succession of plants and trees that would follow…these notes would result in Thoreau’s most important work of natural history, “The Dispersion of Seeds”, which he composed shortly after reading Darwin in 1860…

Unfortunately, Thoreau did not live long enough to complete the great work on geography and the indigenous peoples of North America that his extensive notes in the Canadian and Indian notebooks would lead us to believe he was working on…all we have are the notes…more than 4000 pages of them unpublished

I have said that in what Thoreau wrote in these notebooks, as dry and factual as they are, we get a sense of a transition, an empirical turn, that not only occurs in his thought, but that would begin to lay the foundations of modern geography and environmental history…

Besides this however, we may also sense something more important…something deeper that you can all take pride in tonight… Thoreau used many books, articles and maps in these notes, most of which were read in libraries just like the one we celebrate this weekend. In his life and work we can see how something as humble and as common, as book or a map in a library…can spark the imagination and inspire great thoughts…without a library there would be no Henry David Thoreau, there would be no notes and drawings for us to wonder and marvel at this evening… like Thoreau’s notes, libraries, through their collections of books, manuscripts and maps teach us that…

“Mere facts & names & dates communicate to us more than we suspect…”

Thank you all for listening….

2011-03-30T09:56:00.004-04:00

The Henchir Mettich Inscription from Central Tunisia:

Further Use of Epigraphy for the Study of Early Roman Property Law and Cadastral Cartography

...historians sometimes find much of what they are looking for when they step outside their rooms and look around in the landscape....

................................................................................................ --George Duby

The sociologist Max Weber, shown in the photograph below, wrote in his dissertation, Die Römische Agrargeschichte in ihrer Bedeutung für das Staats- und Privatrecht (Roman Agrarian History and its Significance for Public and Private Law), that Roman Surveyors distingushed three categories of land: ..

...

1. ager divisus et assignatus, land surveyed and assigned to owners which is further subdivided into two categories:

a. ager limitatus, that is land surveyed by centuriae and assigned to owners.

b. ager per scamna et striges land surveyed by scamna and striges and assigned to owners.

2. ager per extremitatem, land surveyed by boundaries.

3. ager arcifinius, land not surveyed. Weber completed the dissertation in 1891 and was greatly influenced by the work of great Roman historian, Theodor Mommsen.

...

Weber, in the dissertation, makes a case that the two broad categories of surveyed land, that of centuriae and that of scamna et striges are different systems of mapping based on land ownership laws. The centuriae surveys are more general and do not reflect the actual ownership boundaries of the land, while the scamna surveys follow actual property lines. Centuriation, the process of dividing the land into centuries, formed very regular patterns and, as is shown in the schematic below, had fixed sizes associated with it. Scamna and striges are different, in that the boundaries of the surveyed land form irregular rectangles of various sizes and extents.

....

Weber implies that this distinction has more than just structural meaning and cites the writings of the surveyor Hyginus from the Corpus Agrimensorum. Hyginus says that land surveyed per scamna et strigas was "subject to taxation" and for this reason "permanent property lines" had to be well defined. The question of the relationship of private property, taxation, and the role of surveying and cartography in the establishment of colonial tax and ownership rights in the Roman Empire is a very complex one and much more historical work needs to be accomplished in this area. One sourse of information regarding the interplay of these various bureacratic entities comes from epigraphic inscriptions. One important inscription regarding the role of ownership, rents and taxation that survives was found at Henchir Mettich near the village of Testour, Tunisa, west of Carthage. The inscription was originally part of the administrative documents associated with the estate or fundus of Villa Magna Variana. The section of the map below from the Barrington atlas shows the region to have been only lightly centuriated, yet the inscription, as I will show below implies that it was surveyed.

...

The inscription is currently part of the collections of the Bardo Museum in Tunis and is shown in the photograph below.

... ... The text found on the monument was first announced by M.R. Cagnat of the College du France and was published several times shortly after its discovery in 1896 by a Lieutenant Poulain, an officer in the French topographic corps.. The area around Henchir Mettich is currently a semi-arid district of mostly barren hills (approximately 400 mm-500mm rainfall per annum), some of which are currently cultivated with olive trees. The valley (modern Medjerda, Roman Babradas) is roughly sixty to eighty kilometers from the Mediterranean coast and is known as the Tell intérieur.

... ... While in Tunisia last year I took detailed photographs of the Henchir Mettich inscription at the Bardo (one of the series is shown in the above figure). The stone on which the inscription is written was in all probability moved from its original location to Henchir Mettich and is dedicated "to the safety of the Emperor Trajan" suggesting a date of around 115-117 AD. The test itself is quite long and is found on all four sides of the momument. Using edge dection algorithms I enhanced the inscription (figure to the left) so I could read it in its current condition, which appears degraded compared to some of the earlier publications of it. The inscription is addressed to the group of landowners and renters who are associated with the fundus. The actual boundaries of the fundus are not given in the inscription but it does tell us that the estate was large enough to have a wide variety of agricultural production. There are fields of wheat, barely and beans, along with orchards of olives and figs. This list is very much in line with the current agricultural output of the region. The inscription also mentions vineyards and pastures and home sites for workers. One of the most interesting parts of the inscription takes about the fact that some of the land is uncultivated and that a part of the land is leased to cultivators and also contains portions that are given over to the lessors for their own farms.

Those who worked the fields are described as coloni, of which the inscriptions defines two types: ...

1. those who lived on the fundus in their own homes (specifically homes they themselves built)

2. those who lived on the fundus but rented their buildings.

...

The inscription, whose entire contents I will translate in a future post, divides up in precise terms the rules associated with the various types of land associated with the fundus and talks about land renting and ownership rules and the terms under which things like unalloted farm lands may be occupied and cultivated. Much of the discussion of these rules implies a deep concern with boundary issues and surveying. For example the inscription states:

"To those coloni dwelling on the fundus of Villa Magna Variana, that is, of Mappalia Siga, who wish to cultivate the fields, permission is given to cultivate those fields which have not been alloted, under the terms of the Lex Manciana; namely that he who cultivates shall have personal use of the land. Of the crops which are raised on said land, the cultivators shall give to the owners, or lessors, or stewards of this fundus shares as fixed..."

The unalloted portions that are referred to resulted from the division of land for the purposes of distribution to private individuals. From this one can infer that the fundus was originally surveyed and distributed and therefore became ager privatus. If one believes the methodologies for such distribution as outlined in the Corpus Agrimensorum this implies the creation of a map at some time before 116-117 AD. There are other instances of this type of example. Mommsen in Volume X of CIL (pp 386) records an inscription from the ager Campanus, which settlers had begun to occupy without permission. The inscription says:

"He divided that territory into small farms (fundi) and leased them out at fixed rents. While in office he recovered more land than was expected, and he had the whole territory recorded on a surveyor's map (forma). This was then engraved on a bronze tablet..."

... The inscription's language is complex in a number of areas, especially in those referring to subseciva, or uncultivated, unalloted land. With the little evidence that survives however, regarding Roman surveying and cartography, epigraphy is just one more group of texts that need to be studied in a closer and more a critical way than historians of cartography have generally done...more on this inscription in later posts

...

The author practicing his epigraphy skills in the ruins of Carthage ...just outside of Tunis...

2011-03-08T12:44:00.000-05:00

In Nietzsche's Shadow:
Cartography, Geohistory and the Changing Nature of Landscape

...you would fancy him a madman when you see him walking along the most devious of paths...seeking for the traces of lost facts in rough woods and thickets...
--Cassiodorus on Roman Surveyors

My current work of searching for the remains of Roman mapping and cartography was born on the ancient paths and trails of Southern France. Hiking through these areas year after year continually brought me face to face with a type of historical research that I think has been missing from what has recently come to pass for the history of cartography. Wandering these paths and old Roman roads makes one consider the relationship that cartography has to landscape and with the kind of geohistory first imagined by members the French Annales School.

The village of Eze from Mont Bastide

The medieval historian Georges Duby said in his autobiography, History Continues, that,

"historians sometimes find much of what they are looking for when they step outside their rooms and look around."

I believe this to be especially true for the historian of cartography. Duby says that,

"What I was looking for in my wanderings through forests and fields was the reassurance of a concrete grasp on reality. The tattered, threadbare fabric that I was trying to mend stitch by stitch with the aid of my Latin texts needed solid support. I wanted to lay it down over a document of a very different kind: one just as rich...but without gaps and preserved not in the darkness of the archives but open to the sunlight and to life itself, namely, the landscape."

Duby talks at length about how,

"no technological revolution had yet radically transformed the agricultural system in my region, and forty years ago the network of paths was still much the same as it had been in the year 1000."

Today some things have changed but many of the old paths and field patterns survive and, as I have shown in my work on Tunisia, can be extremely useful when trying to reconstruct the Roman limits. This is especially true in places like Tunisia and Libya were a great deal survives in the way of field patterns and boundaries, but these same hints in the landscape can also be found in some areas on the continent. My own wanderings along the forgot paths of the moyenne and grande corniche, places like the Friedrich Nietzsche trail, which goes from the Mediterranean Sea up to the village of Eze (shown in the photo above) continuing to the summit of the Mount Bastide, have shown just how enlightening such an approach can be. Nietzsche himself used to walk this particular chemin daily while he was living in Eze, and composing parts of Thus Spake Zarathustra, but the history of the path goes back at least to the iron age. As Nietzsche said in Ecce Homo,

"The following winter [1883-84], under the halcyon skies of Nice, which glistened above me for the first time in my life, I discovered the third part of Zarathustra-and the book was finished. Scarcely a year for the composition of the whole. Many concealed spots and many heights in the landscape of Nice have become sacrosanct to me because of unforgettable moments there. That decisive part of the third book, 'Of Old and New Tablets,' was composed on the difficult and steep ascent from the railway station at Èze to the marvelous Moorish eagle's nest overhead.-My muscle tone was always greatest when my creative energies flowed most abundantly. The body is spirited-let us leave the 'soul' out of play. . . . One could often have spotted me dancing: at that time I could wander through the mountains for seven or eight hours at a time without tiring. I slept well. I laughed a lot-I was fit as I could be, and I was patient."

Along the trail one finds the remains of a large bastide or oppidum whose occupation dates back to neolithic times but that was also occupied and rebuilt by the Romans during the Julian-Claudian period. Many of the walls and some of the roads are still intact, as shown in the photo above. Ruins of this type, with parts of the road structure remaining, are useful in reconstructing how and where the Romans actually surveyed and mapped. Boundary stones and mile markers have also been found in this region and lend further help in producing accurate reconstructions of the type that I am engaged in.

Most of the theoretical foundation for these types of researches stem from the work of Fernand Braudel (1902-1985) and Marc Bloch (1866-1944), two of the founders of the Annales School. Braudelian landscape and geohistory can be reduced to a few basic assumptions.

1. geohistory has a specific concrete object that is 'tied to the soil', to elemental ecological conditions...to the landscape and its historical modifications.
2. geohistorical process, because it develops so slowly, represents a relatively immobile history, whose characteristic patterns last for long periods, things like field patterns, paths and roads.
3. geohistory is fundamental to other kinds of historical process and underlies other forms of historiocity.

Braudel wrote that within the bonds of his technological capacity man is free to do what he will with the landscape in which he dwells. A very interesting Heideggarian parrallel could be written here using Heidegger's essay Building, Dwelling, Thinking but I will not go into that now.

According to Braudel, the very formative capacity of the human endeavor, the ability to bend the landscape and change it in ways that last beyond single human lifetimes, creates constraints that become determinants of later human history because they are relatively 'fixed' or 'permanent'. Hence paths and field patterns last much longer and have their origins in many places in the remote past. These things become fixed and although we do not recognize them at first sight, old paths and field patterns can tell us a great deal about the history of an area and can open up new areas of physical research that help in our cartographic researches.

One look at Marc Bloch's French Rural History will give some insight into what I am getting at here. Bloch's essay is very important for a number of reasons that are methodologically interesting for this particular project. Bloch, in this short book (it is only 200 pages long), is concerned ironically with a long span of time, beginning in the 13th century and ending in the early 18th century. His conception of rural history is broad, taking into account not only farming techniques, but customs and the development of social norms. What was for the time most revolutionary however, was Bloch's systematic use of non-documentary sources like estate maps and the layout of the physical environment itself.

French Rural History came to my attention because of it's use of the so-called 'regressive method'. Bloch read history backwards on the grounds that we know more about later periods and that it makes logical sense to proceed from the known to the unknown. There were others before him who used this method like the English historian Frederick Seebohm who, in 1883, published The English Village Community. The book begins with an important chapter entitled, 'The English Open Field System Examined in its Modern Remains". Seebohm uses the surviving clues in the landscape to work backwards to the foundations of early English village life much in the same way as Bloch does with his maps of remaining field patterns in France and as I am trying to do with Roman Surveying. More modern studies, like Alan Baker's, Studies of Field systems in the British Isles give one some sense of what the historian can achieve if one simply gets outside. All of this of course requires a certain rethinking of how we write and conceptualize cartographic history as something that as George Duby said, "is open to the sunshine".

2010-12-15T14:56:00.000-05:00

Getting into Shape:
What Do Butterfly Wings and Renaissance Maps Have in Common?

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A Note on Method

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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

The comparison of the accuracy and geometric outline of an early map with a modern map is really just the calculation of the change in shape of two planar surfaces. It is a coordinate transformation that can be as complex or as simple as one likes. My interest in the mathematics of shape change came about while I was working in the Entomology Department of the Smithsonian Institution’s National Museum of Natural History.

I was first specifically interested in trying to quantify the variation in the shape of butterfly wing patterns. At first, my interest was centered on the genus Erebia from the Alps of France. Erebia are high mountain butterflies that speciated a great deal during the last large scale glaciations that drove them out of the valleys and isolated them on peaks throughout the Alps. The amazing complexity of their spatial distribution is not well understood. Some of the variation between species can be quanitifed by looking at the shape variation in their eyespots and the surrounding color bands on both pairs of wings.

Later, I worked, with Marc Epstein at the Smithsonian, on a group of moths where I became interested in one species known as Euclea Delphini. Within this species we find a great deal of variation in the wing patterns. In the case of Delphini there are five different forms that are not totally discrete but tend to make-up a pseudo-continuum. In order to study this with some meaningful statistics it is necessary to describe the shape of the features that are varying mathematically. These features take a variety of forms from lines and polygons through more complex curved concave and convex shapes.

There are no natural measures associated with the two-dimensional variation of these features in the wings of butterflies so what to measure becomes a problem. Nijhout’s(1) work in this area was very helpful in trying to define what the basic elements of wing patterns were and in helping to narrow down what parts of the patterns are varying in a meaningful way. (http://www.biology.duke.edu/nijhout/images/PatternElements.pdf) I began experimenting with some simple morphometric models and was naturally led to the work done on shape spaces by David Kendall and Fred Bookstein. The studies of shape spaces that they pioneered are very interesting and are themselves objects of mathematical research. It is not always useful to think of shapes as collections of points in Euclidean space. Shape is something that has a spatial structure that is quite peculiar and generally these spaces and subspaces do not occur in other contexts. Huiling Le and Kendall(2) wrote a beautiful paper on this entitled, “The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics” that shows that these spaces have a local Riemannian structure. I immediately became fascinated with these spaces and read more deeply than the application to the problem of butterfly wings suggested. The structure of these spaces is complex and unfortunately I still do not think that I have a good grasp of all of Kendall’s work. The key point here is that there exists a well defined mathematical machinery that lets us apply non-linear transformations to these spaces while maintaining some meaningful definitions of physical distance.

Wing Patterns or Coastlines?

Modeling the shape change on a modern and ancient map is not all that different from the butterfly wing problem, but it does vary in a number of ways. For example, when working with two maps, we do not have a large sample of objects that we are comparing. We are not looking for clusters or variations that seem to group themselves together in statistically meaningful ways. We really are comparing two discrete objects with the hope of quantifying an ill-defined concept that we call accuracy. The question of accuracy in historical cartometric studies is, at least according to the J.B Harley (3) , one of the least understood problems in the history of cartography and it is especially difficult to quantify.

Fortunately morphometrics does have a great deal to tell us about how to begin to study this problem of comparative accuracy. For instance in both cases we are still operating on sets of homologies. Homologies are points, locations or landmarks that are the same on the objects of interest. They could be the location of a particular structure on a butterfly’s wing or the location of Rome on two maps from different historical periods. In historical cartometry we select a set of these points and then transform one set of points into the other using any number of possible algebraic or differential transformations. We are performing a type of image warping which maps all the selected positions in one image plane (the modern map) to the positions in a second plane (the ancient map). The choice of the mathematical function that actually performs the “warp” is always a compromise between insisting that the distortion is smooth and achieving a good match between the two sets of points. There are and have been many approaches to this.

One of the earliest applications of this type of coordinate transform or “warp” to maps came from Waldo Tobler (4) who experimented with them back in the 1970’s. Any discussion of the applications of complex transformations to cartography must begin with his seminal paper and computer program, “Bi-dimensional Regression”. Bi-dimensional Regression was a statistical regression technique that allowed inferences to be made between two planes from point distributions. Tobler studied medieval maps and Portolan charts but was also interested in general questions of shape and was heavily influenced by D’Arcy Thompson’s early studies on deformations found in his book On Growth and Form. Although Tobler’s work was graphically somewhat primitive, the numerical results from his regressions gave the first look at what types of information it might be possible to add to historical studies of early maps through the use analytical comparisons. Tobler’s methods were formalized to shape spaces by Small in several important papers such as “Techniques of Shape Analysis on Sets of Points. (5) ”

A weakness in Tobler’s methods was the question of error distribution (6) and what have become known as outliers. Outliers are points formed from pairs of homologies that when transformed produce an error that is significantly larger than the other points in the sample. These points if not identified and somehow accounted for in the calculations can destabilize many transformations and can effect correlations and regressions in adverse ways. Hampel and Huber (7) have developed an entire field of study around this problem that has become known as Robust Estimation. The family of estimators that they developed and that are known as M-estimators have been widely used and have even found their way into specialty software programs used in historical cartometry such as MAPANALYST. Intuitively, these estimators allow the control of how much influence on the regressions distant points have on nearby points. One can imagine this 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”.

My current research derives from the work of Bookstein (8) and his groundbreaking paper, “ Principal Warps: Thin-Plate Splines and the Decomposition of Deformations” (see link section to view this paper). Bookstein uses a particular distance function that defines a convex surface between homologies and that is a solution to the biharmonic equation.
Through some very interesting algebra Bookstein derives functions that effectively separate the global and local error into affine and nonlinear components. The functions are vector valued and if the pairing of the points that are being transformed correspond to the homologies on our two maps we have effectively modeled the difference between the two point sets as a deformation.

At the foundation of Bookstein's method is the function shown above which corresponds to a portion of the surface,

---

where r is the distance from the origin. The U(r) function satisfies,

---

and is the solution to the so-called biharmonic equation. We can imagine this surface as a piece of metal that is subject to deformations resulting from the displacements of fixed points on a reference surface. In this case we are comparing points on a modern map and the same corresponding points on an older or different map of the same region. For a thin plate of this type subject to bending, the energy change at any point minimizes,

---

If we pick a group of points on the reference map we can place them on the spline grid as shown below.

---

The corresponding points on the map to be studied can then be transformed yielding a deformation grid that minimizes the energy necessary for the deformation. This deformation has both global and local components that allow us to look at the induced error from different scales and is equivalent to the well known measure of cartographic error, Tissot’s indicatrix. The information from these deformations can be used to generate scale isolines as below or vector displacements resembling the indicatrix.

Thus far I have utilized this thin-plate spline method on many maps and in a study that compares the longitudinal error in the Mediterranean basin on the 1507 and 1516 World Maps by Martin Waldseemüller. The method does have significant advantages over methods such as Polynomial Warping that I used in my study, “Warping Waldseemüller: A Phenomenological and Computational Study of the 1507 World Map” (9).

I have since utilized these methods on Roman and some Medieval cartography, like Portolan charts. Using these methods one can calculate distortion grids and scale and rotation isolines.

For the historian of cartography comparing the accuracy on a modern and early map has a number of formal and logical difficulties that must be considered before the application of any cartometric process.

1. The identification of tie points on the two maps to be compared is sometimes difficult and once selected are seldom distributed evenly across the surface of the two images. The problem is essentially one of homology. Finding homologous points on two maps may seem trivial but differences in landmark or coastline shape, an incorrect assignment of place names and changing scales require insight into the target image. The selection of points is especially important when using the simplest linear models without M-estimators where results can be error distribution sensitive.

2. The accuracy and error to be tested and compared may not be a relevant concept across the whole surface of the map. Accuracy is seldom evenly distributed especially in the case of maps of small scale that may be composed from a variety of sources. Discontinuities in error from scale changes are not only scalar but are vector quantities whose direction is important to determine. Rapid changes in accuracy across map surfaces may be difficult to handle with simple linear models and are better approached using local non-linear radial basis functions whose correlations may be statistically more relevant, but mathematically more complex to program. Decisions regarding the use of local or global methods cannot usually be made ahead of time and require experimentation in order to characterize the accuracy and decide on a methodology.

3. The substrate that the map is draw or printed on may have undergone distortion through shrinking, folding, or stretching. Distortion of this sort is especially important to consider in the case of environmentally sensitive materials such as vellum. The distortion of the medium not only effects the accuracy of statistical transformations that we are trying to perform but can also obscure the intent of the cartographer.

4. Spatial association does not necessarily imply causality. The warps and correlations that we calculate using cartometric techniques give real mathematical results but these may be extremely difficult to link to any historical meaning, event or cause. Care must be taken not to over interpret the results of these calculations and not to override historical and documentary context in the rush towards accuracy measures.

5. Historical cartometry is a problem in equifinality or process convergence. Similar types of distortion may arise from different causes making it difficult to derive exact causes for particular distortion patterns.

6. Experimentation is necessary and we must be prepared to use a variety of techniques to characterize the accuracy on a single map.

Considering the qualifications in the application of cartometric methods to historical materials it is obvious that we cannot hope to achieve exact “truth” using our methods or an absolute visualization and characterization of accuracy(10). Instead we assign meaningful notions of probability and statistical measures that are useful for historical comparisons. The analysis of accuracy is an important part of the historical characterization of early maps but the assignment of the adjectives, “accurate” or “inaccurate”, needs to be made more precise in our discourse. The addition of statements like “to this level of confidence by this method” to our use of these adjectives would go a long way to making the results of our calculations repeatable by others engaged in the same research. This more analytic understanding of accuracy requires a much more careful linguistic and conceptual use of these ideas than has been the norm in the historical literature.

Currently there are more algorithms being developed that will be useful for cartographic historians as the fields of medical imaging and shape analysis continue to provide fertile ground for the growth these mathematical techniques. There are many more ways to represent shapes on manifolds and to perform the types of similarity and coordinate transforms than we have discussed here and no doubt more applications shall be forthcoming. The areas of Stochastic Geometry(11) and Poisson Processes in Euclidean space are especially interesting (12).
In the end the researcher who would employ these methods needs to critically consider each map to be studied on a case-by-case basis and would be wise to consider the words of R.A. Skelton, which although written in a different context, can be used as model for the cautions we must recognize in historical cartometry.

---

“The content of the map, as a whole, cannot be assigned confidently to a single phase or horizon of geographical knowledge. Its outlines are in part transcribed from a map prototype or prototypes not precisely identifiable with any extant work; in part they illustrate texts, not all of which have come down to us. The information taken by the author of the map from these sources (graphical and textual) relates to events and concepts of various periods; most of it older by a least a century, and some of it much more, than the presumed date at which the existing map… was made. The delineations in the map before us are separated by long intervals of time not only from the original experience that they reflect, but also from the direct records of it. For the mapmaker was working always at one remove, sometimes (we cannot doubt) at two or more removes, from firsthand records; and it is evident that, to a degree and in senses which it is difficult for us to divine, he exercised his judgment in selection from and in adaptation of his sources, which are themselves partly unknown to us. (13) ”

---

(1)H.F. Nijhout, “Elements of Butterfly Wing Patterns,” Journal of Experimental Zoology 29 (2001): 213-225.

---

(2)Huiling Le and David Kendall, “The Riemann Structure of Euclidean Shape Spaces: A Novel Environment for Statistics,” Annals of Statistics 21 (1993): 1225-1271

(3) Brian Harley, “Concepts in the History of Cartography: A Review and Perspective,” Cartographica 17 (1980): 54.

---

(4)Waldo Tobler’s papers on the development, theory and applications of bi-dimensional regression include, “Computation of the Correspondence of Geographical Patterns”, Papers of the Regional Science Association 15 (1965): 131-39; “Medieval Distortions: The Projections of Ancient Maps”, Annals of Association of American Geographers 56 (1966): 351-61; “Bi-dimensional Regression”, reprinted in Geographical Analysis 26 (1994): 187-212.

(5) C.G. Small, “Techniques of Shape Analysis on Sets of Points,” International Statistical Review 56 (1988): 243-257.

(6) Tomoki Nakaya, “Statistical Inferences in Bi-dimensional Regression Models,” Geographical Analysis 29 (1997): 169-185.

(7)P.J. Huber, “Robust Estimation of a Location Parameter,” Annals of Mathematical Statistics 35 (1964): 73-101 and Robust Statistics (New York: John Wiley, 1981) see also F. Hampel, Contributions to the Theory of Robust Estimation, PhD Thesis (Berkeley: University of California, 1968).

(8) Fred Bookstein, “Principal Warps: Thin-Plate Splines and the Decomposition of Deformations”, IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (1989): 567-585 and Christopher G. Small, The Statistical Theory of Shape (Berlin: Springer-Verlag, 1996) 110

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(9)John Hessler, “Warping Waldseemüller: A Phenomenological and Computational Study of the 1507 World Map,” Cartographica 41 (2006): 101-113.

(10) Timothy R. Wallace and Charles van der Hanvel, “Truth and Accountability in Geographic and Historical Visualizations”, Cartographic Journal 42 (2005): 173-181.

(11) A. Braddeley, “Stochastic Geometry: an introduction and reading list,” International Statistical Review 50 (1982): 179-193.

---

(12) F. Morgan, Geometric Measure Theory: An Introduction (Boston: Academic Press, 1988).

---

(13)R.A. Skelton et. al., The Vinland Map and the Tartar Relation (New Haven, CT: Yale University Press, 1965) :228.

2010-07-15T10:43:00.000-04:00

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.

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.

2010-07-14T14:38:00.000-04:00

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

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......

2010-07-08T08:56:00.000-04:00

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

2009-09-03T14:03:00.000-04:00

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.

2009-06-21T08:02:00.000-04:00

Exploring Waldseemuller's World :

An International Symposium at the Library of Congress

For those of you who where unable to make it to the Library for our Waldseemuller Conference last month we have posted the complete conference on-line in four 2.5 hour parts... you can find the video webcasts at...

Part 1: Scholars and Scientists

http://www.loc.gov/today/cyberlc/feature_wdesc.php?rec=4568

Part 2: Exploring the Known and Unknown

http://www.loc.gov/today/cyberlc/feature_wdesc.php?rec=4567

Part 3: Sources and Texts

http://www.loc.gov/today/cyberlc/feature_wdesc.php?rec=4569

Part 4: Changes and Revolutions

http://www.loc.gov/today/cyberlc/feature_wdesc.php?rec=4566

Hope you all enjoy...

2009-02-04T08:08:00.001-05:00

Hidden Toponyms:
Corrections to Place Names on Waldseemüller's 1516 Carta Marina

“A man of genius makes no mistakes. His errors are volitional and portals of discovery.”
--James Joyce

On sheet number 9 of the 1516 Carta Marina by Martin Waldseemüller there are two ornamental shields, one of which is covered by a cutout piece of paper of the same shape (figure 1). It has been thought that beneath the shield occurs a series of printed errata that pertain to corrections that may have been made to the map. The full transcription and analysis of these errata have until now however proved illusive[1]. The hyperspectral imaging accomplished last year at the Library of Congress[2] has provided the means with which to finally transcribe these errata and analyze their relationship to the 1516 Carta Marina map.

Transmitted hyperspectral images of the map sheets were taken at a variety of wavelengths in the near infrared and have allowed the viewing of the printed text through the pastedown paper overlay (figure 2).

The inscription shown in the figure consists of a series of changes that needed to be made to the map and has provided evidence that one of the sheets (number 6) is an earlier proof copy. The changes consist mostly of misspelled and misplaced city and region names.

Sheet 6 of the Carta Marina shows the coast of West Africa and does not contain the red lines drawn on the rest of the map by its owner, the Nuremburg mathematician and astronomer, Johannes Schöner (1474-1547). This particular sheet was also not bound into the Schöner Sammelband in the same way as the other sheets of the 1507 and 1516 maps and displays a different watermark from the others.

On errata list above imaged using hyperspectral techniques there are two errors, “seu brachi” and “Teopardi”, that remain uncorrected on sheet 6 while all the other corrections listed in the errata shield have been made to the map. The fact that sheet 6 remains uncorrected in relation to the errata list has led us to conclude that it was most probably a proof sheet. This conclusion of course contrasts with the previously accepted view that sheet 6 was printed later.

Schöner reproduced sheet 6 as a manuscript on vellum and had it bound into the Sammelband in the same way as the 1507 and other sheets of the 1516 map. His drawing shows the changes that are suggested in the errata list suggesting that he either had access to the list or to a later non-proof copy of the map.

This technique has also been used to look at the pastedown on which the text block of the lower right-hand corner of the 1507 map is printed on. What one sees coming through is actually the text from a palm reading manual from 1515 indicating that at least this part of the Library of Congress' 1507 map was printed after 1515.

[1] E. Harris, “The Waldseemüller World Map: A Typographic Appraisal”, Imago Mundi 37 (1985), 30-53. In Harris’s article she reproduces the first videocon image of the errata shield using transmitted IR taken by Dianne van der Reyden at the Smithsonian’s National Museum of American History. Van der Reyden is currently the Director of the Preservation Division at the Library of Congress. Although Harris and van der Reyden were able to determine that the text was composed of errata, no transcription was possible at the time.

[2] Imaging Team for the Waldseemüller maps at the Library of Congress consisted of Michael Toth, R.B. Toth Associates; Dr. Keith Knox, Boeing Corp; Dr. Roger Easton, Rochester Institute of Technology; Dr. William Christen-Barry, Equipoise Imaging; Doug Emery, Emery IT; Dr. Fenella France, Library of Congress; Heather Wanser, Library of Congress; Kenneth Boydston, Megavision Imaging.

2009-02-03T13:56:00.000-05:00

Text Block from Sheet 9 of the 1516 Carta Marina

Below are the two halves of the text block form the 1516 Carta Marina that the discussion on MAPHIST has centered on. The figures overlap but I wanted you to be able to read them. There are several areas of intepretation that are very problematic. The first concerns the question of what map is actually being discussed when Waldseemuller is talking about the 1000 copies. Other problematic parts concern the sources that he is discussing, many of which seem to relate to Asia and the journeys of Marco Polo. I have placed my draft translation below the figures along with a few footnotes on some historical issues.

My translation contains much interpretation based on context, as the references in the text are at places very unclear. The text also has breaks in places that do not form complete sentences...these are made into complete sentences in the draft translation below. The red line annotations that you see in the text block were done by the map's original owner Johannes Schöner.
Good Luck on your attempts...

Click on Figures to enlarge

Draft Translation

Ilacomilus, Martin Waldseemuller, wishes to the reader good fortune.

We will seem to you reader, to have diligently presented and shown a representation of the world previously, which was filled with error, wonder and confusion. In this representation, we do believe that the reader disagrees with us in that we have represented irregular forms in our previous description of the land and sea (and these we certainly described with no deceiving rhetoric). Our previous representation pleased very few people, as we have lately come to understand. Therefore, since true seekers of knowledge rarely color their words in confusing rhetoric, and do not embellish facts with charm, but instead with a venerable abundance of simplicity, we must say that we cover ours heads with a humble hood. In the past we published an image of the whole world in 1000 copies, which was completed in a few years, not without hard work, and based on the tradition of Ptolemy, whose works are known to few because of his excessive antiquity. This representation took much effort to bring to light so that it would include the locations of the lands and the regions of peoples along with the manners and habits of men. We made it so that it would contain only the cities, the mountains, and the races of men along with their customs known to have flourished and to have been known by the people in the time of Ptolemy[1].

After the bold citizens of Venice[2], the great pontiffs Clement IV[3] and Gregory X[4], and after both Christopher Columbus and Americo Vespucci, captains of Portugal, published the accounts of their discoveries many things were added to our knowledge. Although it is well known that the machinery of the world has not varied since the time of Ptolemy, it is indeed a fact that the passing of time inverts and changes things so that it is difficult to find one city or region in twenty which has kept its ancient name or that has not been newly developed after his time. Because of this and because nothing in these matters is clear in hindsight, difficulties may arise in our understanding of very distant regions and cities. Where are now located Augusta, Rauricum, Elcebu, Berbetomagus, or, among the foreign maritime powers, Byzantium, Aphrodisium, Carthage, Ninive, whose names and locations have been transferred to us which great accuracy by Ptolemy? This is of course a difficult question. Are they close by, next to the Rhine River, or far away and concealed? Who has knowledge of, who can tell apart and who can make known to us the Sequani people, the Hedui, the Helvetians, the Leuci, the Vangioni, the Hagoni, the Mediomatrices, all of whom where so well known at one time. I acknowledge that it is possible that no one could now know the manners of the ancients and could come upon knowledge of Celtic Gaul and Belgium, Austrasia, Noricam, Sarmatia, Synthia, Thaurica and the golden Chersonses, the bay of Caticolphi, the bay of Ganges and the very well known island of Taprobane. Time is expansive; it renews, and brings change into the affairs of men.

Many years ago a traveler set upon a long and laborious journey[5] and, as in more recent times explored the lands of men because even the lands whose names have not changed may have been carelessly reported as things in other zones and at the equator have been. It is obvious that the boundaries of Ethiopia and indeed the fortunate islands, now called the Canaries, could be more north, and the boundaries of India, by the persuasion of its leaders, could be more south than the locations passed on by Ptolemy. Is it not possible that Ptolemy did not judge the accounts of travelers so critically and that information from travelers who believed in some absurdity was transferred to him so that his work now persuades people that the new cosmographers rather than the ancient are to be imitated, lest some important change or alteration remain unknown or uncertain. Moved by these considerations, I have prepared this second image of the whole world for the benefit of the learned, so that as the representation of the whole of the land and sea by the ancient authors stood together, not only would the new and present image of the world shine through, but also the natural change that has taken place in the intervening time would be so evident that you would have a unique view of what sort of things become perishable. These things whatever they may have been in the past and whatever they may become in the future are presented so this change may in no way be doubted as time goes on.

Therefore it has pleased us to create an image and description of the whole world as a marine chart after the manner of modern cartographers to the point that we copied their style in the descriptions of the sea from the most accepted nautical records. In consequence we have generally copied the accounts of journeys, chorographies and the reports of recent travelers in the description of the Mediterranean, of Asia and of Africa. We used accounts of the brother Ascelius[6], who took care of many business affairs under the Supreme Pontiff Innocence, of brother Odoricus de Foro[7], of Julius de Parca Leonis, of Peter de Alaicus, of brother of John de Plano Carpio[8], of Massius and Marcus, Venetian citizens, of Casper, the Jewish informer, whose book of travels was copied and dedicated to the King of Portugal, of Francis de Albiecheta, Joseph of India, of Aloysius de Cadamosco, of Peter Aliaris, of Christopher Columbus, of Ianuensus Ludoicus, and of Vatomanus Bononien. All of these travels, experiences and descriptions of places found on the globe, communicated to us by the patrons and admirers of this affair, we have rendered into this single Marine chart. We took great care in making sure that not a single word of our description be embellished in some sweet style or adorned with some kind of festive arrangement. For it is always better to speak in a humble and truthful style. For this reason we ask you to look upon us with a benevolent spirit.
Farewell.

[1] In this paragraph Waldseemuller appears to be describing his 1507 World Map. Several scholars have put forth other suggestions for the identity of the map discussed here such as the maps contained in the 1513 Ptolemy (See Peter Dickson’s, The Magellan Myth, 2007, for more on this). The maps found in the 1513 Ptolemy however do not have any reference to particular customs and peoples found in the various locations as are found on the 1507 map and seemingly described here by Waldseemuller in this paragraph. The description is also problematic in relation to the 1507 world map as it of course it contains, as Dickson points out, much more than simply the lands known to Ptolemy.
[2] Probably a reference to Marco Polo’s travels.
[3] (d. 1268) Both Clement IV Gregory X expanded the known world from the Ptolemaic view by sending ambassadors to the Mongols.
[4] (1210 – 1276)
[5] Polo?
[6] Dominican priest who led an envoy from Pope Innocent IV to the Mongol kings in around 1245.
[7] Marco Polo speaks of Unc-Khan as the great prince who is called Prester John, the whole world speaking of his great power". In 1229 the celebrated missionary John of Monte Corvino converted a Nestorin prince belonging to this tribe, who afterwards served Mass for him (Rex Gregorius de illustri genere Magni Regis qui dictus fuit Presbyter Johannes)). Many of the actual missionaries, who at this time were trying to convert the Mongolian princes of Upper Asia, paid much attention to the extravagant embellishments of the legend. One of these missionaries, Odoricus de Foro Julii, wrote "that not a hundredth part of the things related of Prester John were true".
[8] Franciscan who visited the Mongols also in around 1245.

2009-01-08T13:09:00.000-05:00

Annotations in the 1482 Ulm Edition
of Ptolemy's Geographia

...once owned by Johannes Schöner

Although it is well known that Johannes Schöner owned the only surviving copy of the famous 1507 and 1516 World Maps by Martin Waldseemüller, few, if any studies have been done on the other cartographic materials found in his library. The remains of that library currently reside in the National Library of Austria and are filled with discoveries waiting to happen (there is a disseration here). One of the most interesting groups of texts that Schöner owned is a series of editions of the Geographia by Ptolemy. Schöner's copy of the 1482 Ulm edition in particular is heavily annotated and has never been adequately studied by historians of cartography.

Most of Schöner's annotations are technical and many are corrections to the text that we know, based on his inscription, he purchased in 1507. Schöner's notes are very much in line with that of Regiomontanus, who attempted to correct the early Latin translation of Ptolemy, but sadly died before finishing them. They were first published in the 1525 Pirkheimer edition.

(click on image to enlarge)

Regiomontanus says in relation to the Latin Geographia, in his Dialogus adverus Gerardum Creminensem in planetarum theoricas deliramenta,

–What will happen if the first copy has been rendered obscure by a careless translator, or transformed by the first starving copyist who happens along? Both of these things can be seen in the work that today is passed off as being Ptolemy’s Geographia, in which the literal structure intended by the Greek author does not correspond to the phrases written by Jacobus Angelus….
–…who mistakes the meaning of words and in which the appearance of the maps do not preserve the appearance intended by Ptolemy.

(click on image to enlarge)

Schöner would also try to correct Ptolemy and made annotations in all of the editions of the book that he could find. He owned the 1482, 1507, 1513 editions and a manuscript edition from 1509.

(click on images to enlarge)

2008-12-31T15:29:00.000-05:00

Sketching the Unknown:
A Phenomenological and Computational Study
of the Rossi “Map With Ship”

The actual maps that exist are but a tiny subset
of the theoretical maps that could exist. These
real maps are products of a very small number
of trajectories through cartographic space...each
with its own unique place in this mathematical
construction. Every real map is surrounded by a
tiny cluster of real or unreal neighbors
who are its ancestors and descendents...

The motivation for this investigation stems from the problem of trying to identify the possible sources and the overall chronological similarity of the Rossi “Map With Ship” to other maps that may be contemporary with it. The history and origin of “Map With Ship” has been a problem for historians of cartography almost since its discovery and it has never been adequately studied from an analytic perspective. A recent C-14 dating of the vellum yielded two age distributions both after 1475.The map purportedly shows the coast of Asia, including Japan, and the coast of North America in the area of Alaska and the Aleutian Islands. Scholars have drawn attention to the fact that the strait shown on the map is depicted in a form that recalls several European maps of the region from the late sixteenth century and later. In particular the maps of Zaltieri of 1566 and that of Forlani of 1574 have been discussed in recent studies[1]. The date of the map has been variously claimed to be from the time of Marco Polo’s journeys, to be a fifteenth or sixteenth century copy of a map from the time of Polo, or to be an outright fake or forgery. Although there have been several studies of the Rossi materials in the past no real conclusions have been reached as to the veracity of any of the above claims[2]. The “Map With Ship” is a strange mixture of elements that graphically do not seem to fit the style of any particular cartographic time frame. It is closely related to other maps in the Rossi corpus such as the “Map of the New World” and the “Sirdomap Map” and it appears to be a palimpsest in several areas on both the verso and recto. The map also contains script in three languages: Venician, Chinese and Arabic[3].
Marcian Rossi, the original owner of the collection of maps and documents, corresponded in the early 1930’s with Leo Bagrow who wrote the first study of the corpus[4]. In this study Bagrow reproduces part of Rossi’s letter:

“Marco Polo entrusted the maps to Admiral Rujerus Sanseverinus who had graduated the Nautical School at Amalfi. A number of centuries later his descendent Ruberth Sanseverinus married Elizabeth Feltro Dell Rovere, Duchess of Urbino. In the year 1539 Julius Cesare de Rossi, count of Bergeto married Maddalena Feltro Rovere Sanseverinus…”

Rossi continues this genealogical tree down to his great-grandfather Marciano de Rossi from whom he received the group of map and documents. None of this has however been verified by scholarship.
The following paper will attempt to shed some light on the date of “Map with Ship” by taking a more phenomenological approach to the information actually shown on the map through the use of several mathematical models. By modeling the projection of the Map with Ship, assuming that there is one, and by looking closely at the scale error displayed in the coastal outlines, this study shows that the information on the map is consistent with the geometric methods that would have been available to cartographers in the late medieval period or to a fifteenth or sixteenth century copyist. This is not to say in any way that the map dates from this early period but rather to more conservatively conclude that there is nothing in its geometrical makeup that would exclude such a dating. The calculations for the Rossi map contained several sources of error most especially in the selection of homologies. The final selected points required experimentation and trail and error as the map is somewhat devoid of specific geographic details. The calculated transformational grid overlaid on the Rossi map is shown in the Figure.

The transformational grid shows that we have induced a scale error change from north to south and east to west that is consistent with the family of pseudo-conic projections. Almost all of the error in the Rossi Map is located in angular scale deviations in the Alaskan region, the very region whose existence on the map is the most suspect. The variation in the transformational grid is extremely interesting in that it implies that the information on the map was probably not copied verbatim from a Portolan chart (there is scale deviation and there is little rotation) but rather from a projected map. The pseudo-conic projection scale variation shown on the Rossi map is however nearly indistinguishable at this large level of error from the only other small scale map projections that would have been available to someone before 1550, that of Ptolemy's first and second projections. Hence although the map is projected, and could not have been derived from nautical charts, it is consistent with the geometrical tools that were available to cartographers before the discovery of the information that the map shows. Based on the results of the spline and warping calculations we also calculated the two-dimensional the similarity coefficients of the Rossi map compared to other possible projections[5]. http://www.geog.ucsb.edu/~tobler/publications/pdf_docs/cartography/ScalProb.pdf

These coefficients yield a measure of how similar or dissimilar a map is to different projections using distance functions and calculating least-square norms. We attempted this calculation in order to show that the Rossi map was not similar to any other commonly used modern projections and to determine how far it was geometrically from the family of conic projections that characterize those of Ptolemy. This calculation, the results of which are shown in the graph, confirm the results of the spline calculations and show that the Rossi map is most similar to the conic projections of Ptolemy and has little in common with more modern forms.

One can easily see in the diagram below that the Rossi map’s coefficients place it in the location of the two conic Ptolemaic models and far from the cylindrical Sinusoidal and Mercator projections.

(Click on image to enlarge)

Conclusion

-Bernhard Riemann

In conclusion we have shown that the Rossi “Map With Ship” has no internal or geometric inconsistencies that would lead us to believe that it was definitively copied from a modern map. However, the models employed in coming to this conclusion unfortunately do not totally rule out this possibility. For although we have suggested that the map was not copied verbatim from a Portolan chart we have no way of narrowing down the possible geographic sources more precisely at the present time. All we can say based on this study is that it is still possible that the Rossi map was copied from or based on geographic sources that are consistent in their construction, geometry and scale error with those that COULD have been produced or copied from late medieval and early modern sources. The question of the date of the map and its authenticity must however await further studies.

[1] Benjamin B Olshin, “The Mystery of the ‘Marco Polo’ Maps: An Introduction to a Privately-Held Collection of Cartographic Materials Relating to the Polo Family”, Terrae Incognitae, 39 (2007): 1-23

[2] Leo Bagrow, “The Maps from the Home Archives of the Descendants of a friend of Marco Polo,” Imago Mundi 5 (1948): 3 –13.
[3] The Chinese symbols on the map are unreadable and appear to be either nonsensical or copied by someone who did not know the language as is some of the Arabic script.

[4] Bagrow, (1948)

[5] Waldo Tobler, “Measuring the Similarity of Map Projections,” American Cartographer 13 (1986) 135-39.

2008-11-27T10:29:00.000-05:00

Sententiae in quatuor IV libris distinctae:
Manuscript Fragments from Library of Congress’ 1513 Edition of Ptolemy’s Geographia by Martin Waldseemüller

Structure of Manuscript

The copy of Martin Waldseemüller’s 1513 edition of Ptolemy’s Geographia housed in the Geography and Map Division of the Library Congress contains a group of interesting vellum fragments used as guards to attach the maps found in the book to the binding. The fragments are manuscripts in two different hands, the identity of which, until now, has been unknown since the book’s acquisition by the Library in 1867.

The manuscript found in the Ptolemy is made up of a series of heavily annotated fragments with glossae volatiles from the Sententiae in quatuor IV libris distinctae (Four Books of Sentences) by Peter Lombard (ca.1100-1160). The fragments, which total 46 in number, each consist of from four to seven lines of text in four columns and all come from the first book, “On the Mystery of the Trinity”. The text is written in Northern Textualis Formata[1] indicating a possible date range of 1275-1350.

(click on figure to enlarge)
The first book of the Sententiae discusses, from a purely cosmological viewpoint, the evidence for the existence of God. The doctrine of the Trinity is also discussed using a series of analogies that come from Late Antiquity, mostly taken from the works of Augustine. Lombard denies that any real knowledge of the doctrine can be obtained from these analogies without positive revelation and faith, and emphasizes, again with Augustine, the fact that human speech cannot give a satisfactory account of the nature of God.

The glossae volatiles (flying glosses) on many of the fragments are all in the same hand and are found throughout the text. These glosses consist of three types. The first are corrections made to the Latin orthography using cross outs and replacement text. These corrections are simply grammatical changes made to the manuscript text. These glosses may have been used in preparing of a printed edition. The second type of gloss consists in simple annotations regarding the location of portions of the text that are quotes from other works. The third type of annotation is much more extensive and deals with the subject of the written text itself. An example of this is where the annotator is troubled by the phrase “from the altar the coal by which the mouths of the faithful”. The annotator alludes to a similarity of this passage to Isaiah 6:6: “And there flew toward me one of the Seraphim, and in his hand a coal, which he had taken with forceps from the altar.”

The manuscript contains rubrics (red characters) that fulfill several functions in this particular manuscript.
1 .They mark subdivisions within the chapters.
2. They identify sources of quotations or the beginnings of quotations.
3. They point out important or difficult issues. For example there are several cases of “be careful” or “pay attention here” found in the fragments.

The 1513 Ptolemy is the only one of Waldessumüller’s works for which a printer and firm date are known. The text and the maps of the Geographia were printed by Johannes Schott of Strasbourg and many of the pages of the 1513 edition have the same watermarks as those found on Waldseemüller’s 1507 and 1513 World Maps[2]. The fragments may have been discarded after the preparation of a printed edition as several editions of the Sententiae in quatuor IV libris distinctae were made in Strasbourg in the late 15th and early sixteenth centuries. The manuscript shows commonalities with two of these editions and may have been used for the Duns Scotus Commentary on the Sententiae in quatuor IV libris distinctae, printed in Strasbourg in 1474.

[1] Albert Derolez, The Paleography of Gothic Manuscript Books: from the twelfth to the early sixteenth century. (Cambridge : Cambridge University Press), 2003.
[2] John Hessler, The Naming of America: Martin Waldseemüller’s 1507 World Map and the Cosmographiae Introductio, (London: Giles and the Library of Congress), 2008.

I want to thank Lage Carlsen, Senior Book Conservator at the Library of Congress, for providing the photographs of the manuscript fragments and Chet van Duzer for the intial attribution of the Lombard fragments.

2007-12-19T09:53:00.000-05:00

Translation of Large Text Block on Sheet 9 of 1516 Carta Marina
by Martin Waldseemuller

The text block on sheet 9 of the 1516 Carta Marina is the only source for information regarding the number of copies of the 1507 World Map by Waldseemüller that may have been printed. A full translation of the text block has never been published and is found below.

(click on figure to enlarge)

2007-05-15T10:53:00.000-04:00

The Longest Day:

A Latitudinal Converter from Johannes Schoner's Copy of the 1482 Ulm Edition Ptolemy

Schöner's Drawing of Ptolemy's First Projection

(click on figures to enlarge)

One of the most remarkable features, at least from a cartographic perspective, of the extant manuscripts of the Nuremburg astronomer and globe maker Johannes Schöner (1477-1547) comes from his annotations in the various editions of Ptolemy’s Geography that he owned. This literature, which is today owned by the Osterreichisches Nationalbibliothek, in Vienna, is characterized by a great number of handwritten corrections and complex annotations that show his thinking about theoretical cartography and the state of the art in the early sixteenth century.

After Schöner’s death the contents of his library passed into the hands of Georg Fugger (d. 1569) and from him it was handed down to his son Philipp Eduard (1546-1618) and then to his great-grandson Albert III (1624-1682). The entire contents of Fugger’s library, containing mostly books and manuscripts on mathematics and astronomy, was purchased for the Hofbibliothek in Vienna, by the Emperor Ferdinand III in 1656. Schöner’s library contained some of the most important books on cartography and geography that were available to him at the time, including copies of the 1482, 1509 and 1513 editions of Ptolemy’s Geography, the Cosmographiae Introductio, and of course the only surviving copies of the 1507 and 1516 World Maps by Martin Waldseemüller.

While we are not certain which one of Schöner’s Ptolemaic atlases he may have purchased first, we are sure that his copy of the 1482 Ulm Ptolemy came into his possession in 1507. According to an annotation in the text that is in Schöner’s hand he purchased the book on October 16^th of that year. The book, which is bound between heavy wooden covers connected with leather-backs that show blind imprinting all in the same manner as the codex containing the 1507 world map, also contains a number of manuscripts in Schöner’s hand. The manuscripts found in Schöner’s Ulm Ptolemy are the description De locis ac mirabilibus mundi et primo de tribus orbis partibus, together with the Registrum super tractatum de tribus partibus, the Registrum alphabeticum super octo libros Prolomei and the De mutatione nominorum.

The texts of the manuscripts are followed by three drawings that are glued into the atlas on sheets of paper smaller than the other pages of the atlas. Two of the drawings are studies of Ptolemy’s first two map projections and are shown in Figures 1 and 2. The third, and most interesting of the three drawings is shown in Figure 3. This diagram, entitled Lineares demonstrations Parallelorum Ptholemei is a type of computational device that allows the continuous conversion of the length of the longest day on most parts of the globe to the latitude of the that location and the corresponding parallel of Ptolemy. The problem of understanding Ptolemy’s conception of latitude and operation and uniqueness of this calculator are main the subjects of this brief paper.

Schöner's Drawing of Ptolemy's Second Projection

(click on figures to enlarge)

Schöner's Latitudinal Calculator

The concept of latitude in Ptolemaic astronomical and cartographic theory is a complex one and is quite different from the modern notion of a group of equally spaced parallel lines on the surface of the globe or a map. For Ptolemy, latitude was an angle of inclination, which varied with the location of an observer and determined which stars were capable of being seen in that location. In Ptolemy’s writings, especially in the Almagest, he assumes that the observer is at intermediate latitude, somewhere in the northern hemisphere, and therefore, that the stars in the possible universe fall into three groups. The groupings are based on observability and include the stars that never set but are always above the horizon, the stars that both rise and set, and the stars that never rise in that location and are therefore always invisible. Using two parallel lines of equal size Ptolemy separates these three groups of stars on the celestial sphere.

The two circles used to separate the groups of stars were also used by Ptolemy, and all classical geographers, to define what we now know as compass directions. For example, as one proceeds northward from the equator the circle of always-visible stars will be seen to increase until one reaches the North Pole at which time it will coincide with the line of the horizon, while at the same time the circle of invisible stars also increases. Ptolemy demonstrated that a locality X is north of some locality Y just in the cases where some star that is always visible at X, rises and sets at Y, or if some star that cannot be viewed at X, rises and sets in Y. Because of the fact all these phenomena were seen not to change if we move from east to west on the earth’s surface they were used to define a parallel of latitude. Hence latitude is in general defined astronomically, rather than terrestrially for Ptolemy.

In Book II of the Almagest Ptolemy explains that there are many types of phenomena that are characteristic of latitude:

“The individual points [concerning the sphaera obliqua] which might be considered most appropriate to study for the subject we have undertaken are the more important phenomena which are particular to each of the northern parallels to the equator and to the region of the earth directly beneath each. These are

the distance of the poles of the first motion from the horizon, or the distance form the zenith from the equator, measured along the meridianfor those regions where the sun reaches the zenith, when and how this often occurs;
the ratios of the equinoctial and solstical noon shadows to the gnomon
the size of the difference of the longest and shortest day from the equinoctial day and all other phenomena which are studied concerning;
the individual increases and decreases in length of day and night;
the arcs of the equator which rise or set with arcs of the ecliptic;
and the particulars and quantities of angles between the more important great circles.”

In section 6 of Book II Ptolemy describes particular characteristics of the various parallels and defines their exact locations on the surface of the earth using constant increments of the longest day at various locations. The latitudes corresponding to this regular series of daylight increments are not equally spaced but become more crowded the farther one moves from the equator. This way of defining latitude on a map produces a much different form of graticule than is found on modern maps and it is the attempt to understand this relationship that caused Schöner produce the diagram shown in figure 3. Schöner’s diagram reproduces Ptolemy’s relationship between length of day and latitude in a unique geometrical way that allows one to quickly convert from one to the other and I shall explain its operation in what follows.

Ptolemy reproduces the values of the length of day and latitude that he gave in the Almagest in Book 1 chapter 23 of the Geography as follows: (I do not produce the entire list, just some examples for use in discussing Schöner’s calculator)

“These limiting meridians will enclose twelve hour intervals according to what has been demonstrated above. However we have decided it is appropriate to draw the meridians at intervals of a third of an equinoctial hour, that is, at intervals of five of the chosen units of the equator, and to draw the parallels north of the equator as follows:

The first parallel differing by ¼ hour, and distant from the equator by 4¼ degrees, as established approximately by geometrical demonstrations.
The second, differing by ½ hour, and distant 8 5/12 degrees.
The third, differing by ¾ hour, and distant 12½ degrees.

….

10. The tenth, differing by 2 ½ hours, and distant 36 degrees, which is drawn through Rhodes

…

20. The twentieth, differing by 7 hours, and distant 61 degrees

21. The twenty-first, differing by 8 hours, and distant 63 degrees, which is drawn through Thule.”

Schöner’s calculator reproduces this list graphically, allowing for the conversion of length of day to latitude and of latitude to length of day in a way that would have been very convenient for a map and globe maker of the early sixteenth century. The bottom of figure 3 shows a curved line marked with numbers from 12 to 20 and that continues unmarked for three more divisions making a total of 24 intervals. These divisions represent the hours in a day past the twelve-hour period that makes up the equatorial day. In Ptolemy’s list of hours that define the parallels he expresses the duration of the length of the longest day at various latitudes as an additive difference from this twelve-hour day. Schöner displays the difference geometrically, allowing for the relationship of the latitude to the time of the longest day to be calculated in a continuous and not simply discrete way. Above the curved lower line in figure 3 we there is a series of additional numbers marked from 0 through 21 representing the numerical sequence of parallels defined by Ptolemy in the above list. For example, if we observe the number 21 we can see that it corresponds to the number 20 on the curved line below it. The parallel 21 of Ptolemy is the line that runs through Thule and has a difference of 8 hours in the duration of its longest day from the equatorial day. In this case because Schöner’s beginning point on the curved line is 12 hours representing the equatorial day, the addition of 8 hours gives the number 20 shown on the curved line. To use the computer to calculate the number of degrees corresponding to any particular length of the longest day one follows the vertical line on the figure, continuing with the twenty-hour line as an example, until it intersects the diagonal line that bisects the center of the drawing. One can then follow the horizontal line at the intersection over to the right to determine the latitude. In the case of the twenty-hour line we find, using Schöner’s diagram, the latitude of 63 degrees, corresponding with Ptolemy values.

The construction of the calculator is more complicated than its use. The curvature of the lower line is based on the variation in great circle distances of latitudes and the change in the length of day. One can see this variation by simply looking at the distances between the twelve and thirteen hour points and comparing it to the distance between the nineteen and twenty-hour points. Schöner began the construction of the diagram by laying out the large quarter circle and drawing the various angles for the latitude values from the origin located at the intersection of the diagonal line and the x-y axis of the quarter circle. The angles then provide him with precise measure of the variable spacing between the degree markings that can be seen on the left hand (y-axis) of the diagram. While it is impossible to determine how he actually constructed the figure, he found many sources for the numerical information that it contains in the many Ptolemaic atlases in his possession.

We can only speculate on what Schöner may have used the calculator for but it would have been an extremely useful tool for any globe maker or cartographer in the early sixteenth century. The fact that Ptolemy still held a prominent place in the geographical imagination of the period meant that in order to keep current on the latest information one would have had to go back and forth between new information provided by cartographers like Waldseemüller and the traditional forms of latitude found in the Geography. Schöner’s diagram represents just one of many innovations found in the annotations of his Library and shows quite clearly a solution to one of the many conceptual problems that beset early sixteenth century cartographers as they wrestled to use and improve upon the concepts of theoretical cartography and the mathematical techniques bequeathed to them from Ptolemy.

2007-02-12T11:31:00.000-05:00

New Spline Interpolation Function

Click on figures to enlarge

ESRI has finally undated their rubber-sheeting functions in ArcMap (9.2) to allow for Spline georectification and full rubber-sheeting. The new function also allows for the use of raster data making it much easier to compare the features of old and new maps and to display them in a pleasing and informative way. Below is a georectified sheet of the Waldseemüller 1516 Carta Marina that shows how it must be rotated in order to get the best spline fit.

For more information on splines see http://mathworld.wolfram.com/Spline.html

2007-01-29T09:30:00.000-05:00

How to Map a Sandwich:
Potential Theory,Topological Existence Theorems, and the Changing History of the Ontology of Cartographic Objects

Abstract (full article coming soon)

In the 1960s and 1970s the most important work being accomplished in mathematical cartography had to do with the topological properties of surfaces and their relationship to geographical and spatial analysis. The Harvard Laboratory for Computer Graphics and Spatial Analysis was a hotbed of such work and was led into new areas by the ideas of the theoretician William Warntz. While most other researchers in the field where looking at the numerical properties of surfaces Warntz’s approach centered on understanding their topology. He recognized that the most important properties of surfaces from a mathematical point of view had nothing to do with numbers but rather their invariance under transformations. Warntz described the relationship of the topological properties of a surface to cartography in a number of important papers that adopted a terminology and methodology built on the work of the mathematician Arthur Cayley (1859). Warntz was particularly interested in mapping thematic surfaces and adopted a macrogeographical theoretical perspective that led not only to fundamental mathematical breakthroughs but also yielded philosophical insight into the nature of the objects described by the “science” of cartography. This paper focuses on one particular aspect of the work of Warntz and one of his students at the Harvard Laboratory; existence theorems. Existence theorems contain a statement of existential quantification such as “there is” and prove the existence of a particular set of mathematical objects. They do not however contain any directions of how such objects might actually be constructed algorithmically or numerically.

The researchers at the Lab published two very important works on existence theorems in the influential and now largely forgotten series the Harvard Papers in Theoretical Geography. We will provide a close reading of two of these papers, “The Sandwich Theorem: A Basic One for Geography”, and “Geography and an Existence Theorem: A Cartographic Solution to the Localization of Sets of Equal-Valued Antipodal Points”, in order to show how the Lab used a mathematical approach that was underexploited in cartography and in doing so changed the accepted notions of the nature of cartographic objects.

This shift in the nature of what constituted geographical and cartographic objects is discussed in this study within the framework of Thomas Kuhn's Structure of Scientific Revolutions. Kuhn provides an example in his analysis of the development of the theory of relativity in the beginning of the 20^th century of the type of profound conceptual shifts that took place in cartography in the 1960s and 70s. These shifts were not simply dramatic changes in beliefs about the world or even in scientific and geographic methodology, but rather in the very concepts that define the structure and formal properties (topological and transformationally invariant) of the objects of inquiry. In this way Kuhn’s framework and lexicon provides us with a solid philosophical and historical framework in which to discuss the same type of radical shifts that took place at the foundations of mathematical cartography. These changes in the conceptual framework of cartographic science redefined the nature of geographical objects (what is it that is mapped) and laid the foundations for the development of topological data structures and modern GIS.

2006-12-27T09:39:00.000-05:00

Schöner's Cosmographical Miscellany and His Annotations in the Waldseemüller Sammelband

In 1656 the Emperor Ferdinand III of Austria purchased the Library of Georg Fugger for the Hofbibliothek in Vienna. The collection included the Library of Johannes Schöner and was handed down by Fugger to his son (Phillip Eduard, 1546-1618) and his great-grandson (Albert III, 1624-1682). Besides Schöner's Library Fugger's collection also contained the best mathematical and scientific literature then available.

Portrait of Schöner

The codex containing the 1507 and 1516 World Maps by Martin Waldseemüller was once part of this collection but how and when it became separated from Schöner's original Library and made its way to the Wolfegg Castle in Wurttemberg where it was discovered by Joseph Fischer in 1901 remains an historical mystery. The Waldseemüller Sammelband contained additional items besides the two famous world maps and originally included a set of Celestial Gores by Schöner and the star-chart of Stabius as rendered by Albrecht Dürer. Only one of the star-charts was bound into the codex and it shows the stars visible in the southern hemisphere. The figure below shows Schöner's annotations on the chart.

Contained in Schöner's Library that now resides in the Austrian National Library in Vienna are several volumes that resemble the Waldseemüller Sammelband in that they are bound in the same manner with heavy wooden covers connected with leather backs and also display Schöner's bookplate. These volumes include Schöner's copy of the 1482 Ulm edition of Ptolemy and his copy of Waldseemüller's 1513 edition of the same book. Both books are annotated with the same red-lines found on the 1507 and 1516 World maps and are held into the volumes with slices of printed vellum globe gores and pieces of the Elinger Map.

Schöner's Vellum Manuscript Drawing of a Sheet from the 1516 Carta Marina

Also included in of Schöner's Library is an interesting cosmographical miscellany that is unpublished but holds a great deal of interest for historians of cartography. The miscellany bares the title "Cosmographia" and contains the following items:

1. A short treatise with title "Regionum sive civitatum distantae."
2. Tables of latitudes and longitudes that are similar in content and structure to the so-called University Tables.
3. Notes on various units used to measure distance.
4. A table that displays the number of miles in a degree of longitude for each parallel similar to that found in Waldseemüller and Ringmann's Cosmographiae Introductio.
5. Instructions for measuring the distances between two cities on a map that has coordinates.
6. The University Coordinate Tables.
7. The Tabula Regionum of Regiomontanus. This is of course not the only Regiomontanus in Schöner's Library. Schöner inherited Regiomonanus' manuscripts and published his very important work On Triangles.
8. An outline for the chapter headings of an incomplete work on Cosmography.
9. Excerpts from the Geographiae.

The most interesting of all these works is the Regionum sive Civitatum. The treatise begins by describing at set of instructions for constructing a terrestrial globe. The initial part of the text describes the process by which one inscribes on a globe the locations of the cities and regions on the earth. The first set divides the earth into four equal areas by means of two arcs that intersect each other at ninety degree angles. Once these circles have been inscribed on the globe another great circle is drawn that bisects the other two and forms the equator. The next step divides that part of the equator that lies along the "habitable" part of the earth into 180 degrees of longitude numbering them in units of five. For marking the globe with latitude lines a strip of heavy vellum is used, equal in length to the distance from the pole to the equator. This type of construction continues in the various regions until the whole surface of the globe has coordinates. In order to transpose the points and locations of cities and regions from the globe to a plane the method is essentially that of an azimuthal projection from any point on the surface of the earth. At first the globe-maker selects the city or point that he wishes to make the center of the projected map. Then with a compass he inscribes a circle on the surface of the globe that is large enough in diameter to include the area to be reproduced. The smaller the circle, the larger the scale of the resulting map and the greater ease involved in measuring distances. The second method described in the book outlines what appears to be a conic projection. To do this two new vellum strips are used equal in length to the diameter of the circle drawn on the surface of the globe. The strips are divided into the same number of degrees as the strips used in the first method. They are them placed tangentially along the circle, running north to south. The text says that this method can be used either on a square (quadratam) or a circler (rotundam) map. All this is simply to suggest that Schöner was experimenting a great deal with different methods for measuring distances and for transferring coordinates from maps to globes and vise versa and obviously drew his annotations on the 1507 and 1516 World maps by Waldseemüller for this purpose.

2006-10-30T09:28:00.000-05:00

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, http://earth.leeds.ac.uk/~earccf/animations/Jacksonetal2000.pdf. 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

2006-10-15T11:03:00.000-04:00

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.

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.

2006-10-08T15:16:00.000-04:00

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.

2006-10-05T13:48:00.000-04:00

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.