Wednesday, December 19, 2007

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)

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.

Tuesday, May 15, 2007

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 16th 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

  1. 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;
  2. the ratios of the equinoctial and solstical noon shadows to the gnomon
  3. the size of the difference of the longest and shortest day from the equinoctial day and all other phenomena which are studied concerning;
  4. the individual increases and decreases in length of day and night;
  5. the arcs of the equator which rise or set with arcs of the ecliptic;
  6. 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:

  1. The first parallel differing by ¼ hour, and distant from the equator by 4¼ degrees, as established approximately by geometrical demonstrations.
  2. The second, differing by ½ hour, and distant 8 5/12 degrees.
  3. 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.

Monday, February 12, 2007

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


Monday, January 29, 2007

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 20th 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.