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.

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.

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.