Author Climbing in the Queyras, Summer 2014

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.