Astronomically determined localities, the core part of Ptolemy's Geography

ABSTRACT The paper deals with ancient methods of astronavigation and their potential use in finding geographical locations in Ptolemy’s Geographike Hyphegesis. Further, it describes the systematic errors in these methods and suggests how to correct them. The results include a new map which compares the locations of Ptolemy’s sites after removing the errors with their real locations. Subsequently, significant ancient locations according to Ptolemy’s list of noteworthy cities are identified on the map. In some cases, we presume that they were located on the map using astronavigation. The results of this Study imply that Ptolemy may have adopted a comparatively regular network of points from some older authors which was the basis of his extensive work.


Introduction
Claudius Ptolemy (Lat.Claudius Ptolemaeus) was an outstanding Greek scholar who lived in the second century B.C.E.In antiquity his work, Geographike Hyphegesis (abbr.Geography), was one of the best of its kind, although since then its quality has been questioned repeatedly, as the information contained therein seems somewhat distorted.Ptolemy's Geography has been the subject of numerous studies which can be classified into three groups depending on the methodology used: (1) Works that are a mere transcription or translation of Ptolemy's Geography, that do not deal with its interpretation and correction at all or only marginally; (2) Analytical works which aim to locate the geographical sites specified by Ptolemy more accurately on current maps, using traditional as well as modern methods and also suggest their etymological interpretation; (3) Studies focused on how Ptolemy's work was created and on identification of potential systematic errors which occurred during its completion.
Transcriptions and translations into world languages are used as sources for further works.The oldest extant manuscripts date from the thirteenth and fourteenth centuries, e.g. the Istanbul manuscript <Codex Seragliensis GI 57> or the Greek manuscript <Vatopedion codex Gr. 655>.Regarding the printed editions, works by German authors are often cited -F.W. Wilberg's (1838), C. F. A. Nobbe's (1843) or K. Müller's (1883); and more recently, for example, the Basel edition by Stückelberger and Graβhoff (2006).
Within the second group, some works deal with semantic issues and the search for similarities between Ptolemy's and contemporary toponyms using linguistic foundations (Alonso, 2005;Curchin, 1997;Parsons & Sims-Williams, 2002;Scheungraber & Grünzweig, 2014;Stempel, 2000).The study of spatial relations is also important.So far, a rather marginal section of the works tries to compare the distances between Ptolemy's locations derived from their coordinates to distances according to the itineraries of terrestrial routes (Cuntz, 1923) or waterways (Shcheglov, 2018).Modern GIS methods are more and more often being used to decipher the spatial data.The authors in this group use GIS methods when they try to correct any errors and transform Ptolemy's coordinates to current coordinate systems (Defaux, 2017;Isaksen, 2011;Livieratos et al., 2007;Marx, 2014;Mintz, 2011;Romera & Pérez-Acebo, 2019).
Our paper belongs to the third group, which deals with unravelling the possible process of creating Ptolemy's work and the identification of systematic errors.These errors are also discussed, for example, in: Knobloch et al., 2003;Stückelberger & Mittenhuber, 2009;Tsorlini, 2009;Marx, 2012;Tupikova & Geus, 2013;Russo, 2013;Shcheglov, 2016b;Graßhoff et al., 2017, which will also be discussed at the end of the paper.

Determination of locations in antiquity
Astronavigation, i.e. orientation on the basis of the observation of celestial bodies, was the most accurate navigation method in ancient times.While determining latitude was comparatively easy, the problem of a more accurate measurement of longitude continued until the beginning of the eighteenth century (Andrews, 1996;Čechurová, 2012;Graßhoff et al., 2016).

Latitude determination
Latitude is the angular distance of a location from the equator.Calculation from the angular distance of the Sun measured at noon at the equinox or during the summer or winter solstice was a standard method used in ancient history.An astronomical device called a gnomon was used for this purpose.This is a column erected vertically on a horizontal platform, where the angular altitude of the Sun is determined from the ratio of the height of the gnomon to the length of its shadow (Lelgemann et al., 2005).Measuring at the equinox, when it is not necessary to include the ecliptic inclination in the calculation, is the optimal time.Another innovation was Apollo's or Hipparchus' astrolabe.This is a portable device for measuring the position of celestial bodies.Latitude could also be derived from the length of the day, which was usually measured during the summer solstice.(Hadravová & Hadrava, 2001).

Longitude determination
Longitude is calculated as the angular distance of a place from the prime meridian, which is nowadays the Greenwich meridian.However, the prime meridian was also placed in other locations in the past, and Ptolemy himself located it in the Canary Islands, then called the Fortunate Islands.However, the prime meridian should not be confused with the reference meridian, from which the longitudes of other locations were calculated by local time difference.It has to be searched for rather in those locations where, according to contemporary reports, astronomical observations were mentioned more frequently.From this viewpoint, the Egyptian city of Alexandria is the most suitable point.
The first more accurate calculations of longitude were made in the second century B.C.E. by Hipparchus, who determined it according to the deviation of local time from time on the prime meridian, which at that time still passed through Greek Rhodes (Ptolemy, Almagest 4.11).Hipparchus suggested using the lunar eclipse as the reference time, which can be observed from various places for the same time interval of ca. 100 min (Figure 1).Since the lunar eclipse occurs only at night, it was not possible to measure the time using a sundial.At first, a clepsydra and later also an astrolabe were used.Besides Rhodes, similar measurements were also mentioned in Alexandria and in other locations as well (Graßhoff et al., 2016).

Astronomically determined locations as a basic pillar of Ptolemy's Geography
As confirmed by the results of this and other papers (Russo, 2013;Santoro, 2017;Shcheglov, 2018;Tupikova & Geus, 2013), the basic framework of Ptolemy's work was formed by the positions of more accurately localized sites.In this, Geography Ptolemy recommends starting the construction of a map by plotting the points of sites which have been localized on the basis of reliable observations and should serve as basic pillars of the map.Subsequently, data from other sources can be added according to them (Ptolemy, Geography 1.4.2).In the Geography, some of the 358 so-called 'Noteworthy Cities' that are mentioned in the eighth book of Geography, as well as in Ptolemy's 'Handy Tables', could serve as these pillars.Spatial information was likely taken from older documents, e.g. from Eratosthenes (Ptolemy, Almagest 1.12), or Hipparchus (Ptolemy, Geography 1.4.2.) and updated further by their successors.Ptolemy might have inherited some errors from Hipparchus, such as the erroneous determination of the locations of Carthage, Byzantium or Babylon.Similar conclusions are drawn, for example, by Ernst Honigmann, who assumes that Ptolemy's list of noteworthy cities might date as far back as the time of Hipparchus and was merely updated by Ptolemy (Honigmann, 1929).

Errors in latitude
In their calculations, ancient astronomers presumed that an imaginary line at the upper edge of the shadowed space between the gnomon top A and the end of the shadow cast on the ground B (cf. Figure 2) leads to the centre of the Sun's disc (Roger, 1742).However, in reality the line connecting points A and B leads to the margin of the Sun's disc.This error corresponds to the angular radius of the Sun disc of 16' (Faulkner, 2013;Roger, 1742).The measurement of the Sun's elevation was also distorted by atmospheric refraction which varies depending on the Solar altitude angle.For Southern and Eastern Europe, the value is, on average, 1'.As a result, the Sun's position was calculated to be higher than in reality.Since Ptolemy's Geography includes places located, in particular, to the north of the tropic of Cancer, when measuring using the gnomon the Sun is to the south of the location and the shadow points northwards of the gnomon.The apparently higher altitude of the Sun then results in a slight reduction of the latitude.Therefore, the total distortion is 16' + 1' = 17' = 0.2833°, which corresponds to a length of 31.5 km.

Errors in longitude
Ptolemy's exaggeration of longitude is a generally known fact.It is usually mentioned in connection with the different calculation of the Earth's circumference (Grijs, 2017;Shcheglov, 2016a).While Ptolemy and also Marinus of Tyr were of the opinion that the Earth's circumference was 180,000 stades (Ptolemy, Geography 7.5.12),their predecessor Eratosthenes calculated the circumference to be 252,000 stades (Strabo, Geography 2.5.7,2.5.34).During the research, it was ascertained that in order to correct the longitude, it is optimal to divide the value by 1.4.This on the one hand corresponds to the ratio of values 180,000/252,000 but, on the other hand, Ptolemy and Marinus used different lengths of stades, which should also be taken into consideration.Moreover, the question remains why this error did not impact the latitude as well, since the reduction of the circumference would logically result in its correction.The solution might lie in the following hypothesis: we assume that one part of Ptolemýs map is much older and was made at a time when Egyptian stades were in use.The older map might originally have been in the cylindrical rectangular projection, with a distance ratio of the meridians and parallels of 5/7.Subsequently, Marinus or one of his predecessors decided to redraw the map using the cylindrical equirectangular projection and this is where the mistake could have occurred.This error was subsequently taken over by Ptolemy.
Considering the technical possibilities of the time, it is evident that the original map included certain errors as well, which were caused by inaccurate determination of the times of the beginning and end of the lunar eclipse.The time interval between the eclipse beginning and the end is usually ca. 100 min, but the phenomenon proper is gradual, therefore discernible with difficulty.The Persian and Arabian astronomers al-Māhānī, al-Battānī, Banū Amājūr, ibn Yūnus and al-Bīrūnī were still facing the same problem in the Middle Ages.Regarding the above astronomers, Steele (1998) mentions 37 observations of the lunar eclipse in his paper, and the average error 11. 4 min.

Source data
The Basel edition of Ptolemy's Geography by Stückelberger and Graβhoff (Stückelberger & Graβhoff, 2006) and their colleagues from the Bern University Institute of Classical Philology has been used as the primary source.The book, titled <Klaudios Ptolemaios: Handbuch der Geographie>, was published as a bilingual edition in the original Greek and in parallel German translation.It is a revised edition of the Greek text based on the <Codex Seragliensis GI 57> found in 1927 in Istanbul at the Topkapi Palace Museum, which is, so far, the oldest discovered copy of Ptolemy's work (Figure 3).
Other complementary materials used were the publications by German authors from Berlin Technical University.The first of them, (Kleineberg et al., 2010) titled <Germania und die Insel Thule> is focused on the Magna Germania area and the second ( Kleineberg et al., 2012) <Europa in der Geographie des Ptolemaios> deals with some other European regions.In spite of the unquestioned assets of these publications, it is necessary to approach them with caution, as mentioned in Pápay's review (Pápay, 2011).Uncertain data were further reviewed using the Paris edition of Ptolemy's Geography according to Karl Müller (Müller, 1883) and, in some cases, also according to the Essen edition by Friedrich Wilhelm Wilberg (Wilberg, 1838).

The database of known ancient sites in Ptolemy's Geography
The database includes 632 localities from Ptolemy's Geography, whose positions are specified precisely on the basis of information found in other databases and in web map applications (cf.Table 1).Due to the considerable extent of Ptolemy's work and to the limited time available for processing our data, only a minor data sample was selected.The localities database properly includes a descriptive part as well as the original and newly corrected coordinates in the WGS84 (EPSG:4326) coordinate system.Within the sample of selected localities, 213 of them are also mentioned in the eighth book of Geography.Ptolemy rates them among the especially noteworthy cities.

Reconstruction map as an idealized image of Ptolemy's locations
A reconstruction map was created from the 632 localities in Ptolemy's Geography; it includes the correction of errors caused by incorrect interpretation of astronomical measurements and also a correction of erroneously set meridian intervals.The prime meridian shift has also been taken into consideration.It is an idealized map created on the basis of Ptolemy's coordinates, i.e. how it would look without systematic errors.The map also includes the locations of 213 of Ptolemy's noteworthy cities.The aim of the corrections was, in particular, to identify areas with higher concentrations of more accurately plotted locations where astronomically determined sites could be located, as well as to evaluate whether they coincided with the locations of the noteworthy cities.

Longitude correction
First of all, the original longitude values were divided by 1.4, thus adapting the meridian interval to the older original map, created according to Eratosthenes' model (cf.chapter 1.4).Next, 13.3°was subtracted so that the reference meridian position on Ptolemy's original map is equalized (in our case, it passes through Alexandria, cf.chapter 1.2) according to the relative meridian on the current map in WGS84.The value was calculated according to the formula λPt_Alexandria/1.4-λReal_Alexandria = 60.5/1.4-29.905.Subsequently, deviations were calculated for the entire data sample (cf.The method in chapter 2.6), where it was ascertained that on average, they deviate by 0.2°westward.In order to reduce the average deviation, the longitude value was additionally increased by +0.2°.Although this shifts Alexandria's position slightly, considering the technical possibilities of the time, this error is still acceptable.The resulting corrective formula is shown below:

Latitude correction
The latitude correction includes correction of errors which occurred in the measurement of the Sun's angular altitude (caused by refraction and the incorrect location of the centre of the sun's disc).Thus, 0.2833°was added to the original latitude values: w Cor = w Pt + 0.2833

Calculation of the deviations of current coordinates from those of Ptolemy after correction
Using the LinePlotter plug-in within the QGIS application, a line layer connecting precise points (which corresponds to the exact location of the site) and Ptolemaic points after correction was created.This means the line length captures the spatial expression of the deviation of the precise points from the corrected points.We computed distances using ellipsoidal geometry with WGS84 coordinate system.The values in the legend are divided into six intervals of 60 km each.The first three categories up to 180 km include deviations which could still be permissible for those locations where it is expected that their positions were located by astronomical observations.This limit was specified on the basis of the total evaluation of Ptolemy's 141 noteworthy cities (marked by magenta circles), for which Alexandria is considered to be a common reference point.First, 95% of the locations with the smallest deviations are selected from this total and a maximum deviation of 179.9 km is subsequently identified (Figure 4).

Subsequent shifting of locations
The size of deviations is checked in the next stage and for areas with major errors, correction by additional shifting is suggested.Major deviations are detected, for example, in the surroundings of Carthage, Byzantium, Babylon, Londinium, etc.The causes of these errors are explained well by Stückelberger and Mittenhuber (2009, pp. 245-252).

Conclusion
The results of this Study indicate that Ptolemy's work was based on several tens of astronomically localized locations, covering an area from the northwest corner of Africa as far as the eastern part of Asia to the city of Harappa, with a common reference point in Alexandria.The area includes 141 of Ptolemy's noteworthy cities, where, after the correction of systematic errors, an average error in longitude of 0.77°was found, which corresponds to a time deviation of up to a maximum 0.77 * (1440 min / 360°) = 3 min 5 s.
The results of our analysis further show that the locations of sites in the contiguous areas (the Black Sea coast, much of Asia, the British Isles, etc.) were calculated from incorrectly determined reference points taken probably from Hipparchus.As Mittenhuber states, for example (Stückelberger & Mittenhuber, 2009, p. 245), these are primarily errors in latitude for the locations of Carthage, Byzantium, and Babylon.According to our conclusions, the cities: Londinium, Iulia Caesareatoday's Cherchel on the northwest coast of Africa and Carthago Nova and Austrica Augusta on the Iberian Peninsula could also be erroneous reference points.
The coordinates of other localities have been calculated from the astronomically localized localities only by estimation, by staking road sections according to distance data, taken over from ships' logs (periploi) which also logged their cruise direction, or from travel itineraries.The relevant knowledge is also confirmed by other studies, e.g. in Graßhoff et al. (2017).
The research ascertained that minor deviations between the real and corrected positions occur only in those areas where repeated astronomical observations have been documented historically, or for locations from which some of the astronomers came.A concentration of localities with deviations less than 60 km was found, in particular, in the Nile delta including the city of Alexandria, and at the eastern coast of the Mediterranean, including the city of Antioch (the city of Apamea in the same region is just above the first category limit), on the island of Rhodes, in the Thessaly region with the city of Larissa, at the western coast of the Apennine Peninsula with the city of Rome, on the island of Sicily with the city of Syracuse, and at the foot of the Western Alps.Other localities with deviations up to 120 km are located here as well, including the city of Massalia.The second category also includes localities at the western coast of Asia Minor, including the city of Miletus, etc.

Discussion
As mentioned in the Introduction, many other authors have also dealt with the identification of systematic errors in Ptolemy's Geography, namely Knobloch et al. (2003), Stückelberger and Mittenhuber (2009), Tsorlini (2009), Marx (2012), Russo (2013), Tupikova and Geus (2013), Shcheglov (2016b), Graßhoff et al. (2017).Therefore, their methods are evaluated and compared with our new conclusions.Marx (2012) and Tsorlini (2009) discuss modern coordinate transformation methods.Using a major data sample, they try to compare Ptolemy's coordinates to the real ones and to subsequently identify similar errors which are repeated and which could be identified as systematic errors.However, this is not an identification of a single universal error, but rather a set of several errors related to specific areas or isolines.
Most of the other works mentioned above focused on a systematic error in longitude, which is explained as a consequence of the difference in Ptolemy's estimate of the Earth's circumference from that of Eratosthenes.The correction of a systematic error in latitude is the subject of a small number of the cited papers.
The results presented by Russo (2013) are probably the closest to our outputs.He compared Ptolemy's values to the real ones on a sample of ca.80 localities which he then applied in the relation: y = 1.428x + 17.05, where y is Ptolemy's value and x is the real value.
The work by Knobloch et al. (2003) deals with the correction of both longitude and latitude.In this case, various lengths of stades are also taken into consideration.The latitude and longitude correction is calculated using the same correction factor 1.4.
The factor of 1.4 for longitude correction is also mentioned by Graßhoff et al. (2017) together with an explanation of the errors in latitude and an illustrative map of deviations in the form of displacement vectors, which partly resembles our results.However, a more accurate corrective equation is missing in the article.Stückelberger and Mittenhuber (2009, s. 243) present the opinion that the distortion of latitude increases with growing distance eastward.Therefore, it is impossible to obtain a uniform 'equalization factor' to solve the problem.
In the article (Shcheglov, 2016b), the author documents that the sizes of deviations between Ptolemy's and real latitudes differ quite significantly for various areas.Hence, applying only one corrective equation for Ptolemy's work as a whole is impossible.To a certain extent, the conclusions comply with our research results, where it was necessary to conduct a subsequent shift of localities in selected areas in addition to the basic correction.However, the selection of localities differs.The document does not deal with latitude correction either.
The authors (Tupikova & Geus, 2013) are of the opinion that Ptolemy himself had, if any, just a few reliable observations of lunar eclipses, therefore, in order to determine the latitude, he had to use terrestrial measurements recorded in travelogues and itineraries.

Software
Data was processed in QGIS 2.18.28,where the positions of individual localities were specified by entering the coordinates from a CSV text file loaded into the application using the function 'add a layer with separate text'.The separate text file included Ptolemy's original and corrected coordinates as well as the present-day coordinates in WGS 84.All the recalculations of coordinates were done separately using Microsoft Excel, which is a part of Microsoft 365.The line layer of deviations was created in QGIS, using the LinePlotter plug-in module.The open source relief map and river network map available at https://maps-for-free.com/ and https://mapy.cz/ in the form of the map service Tile Server (XYZ) was used for the background.

Figure 1 .
Figure 1.Longitude determination from the local time difference at the moment of the lunar eclipse using the example of Rome and Alexandria (the times mentioned here are for information only).Notes: The method for determining the local time at Alexandria and Rome at the moment of the lunar eclipse using a clepsydra is shown schematically on the left.Calculation of the longitude of Rome derived from the known longitude of Alexandria on the basis of the local time difference converted to degrees is explained in the section on the right.

Figure 2 .
Figure 2. Latitude determination from the Sun's angular altitude using a gnomon on the day of the equinox and the most frequent measuring errors.Notes: Latitude determination from the Sun's angular altitude using the gnomon is described in the left part of the text.Measuring errors caused by incorrect localization of the centre of the Sun's disc and by refraction are shown here.The right part includes calculation of latitude from the Sun's altitude which also takes into account the error correction mentioned above.

Figure 3 .
Figure 3. Manuscript Codex Seragliensis GI 57, found in 1927 in the Topkapi Palace Museum, Istanbul.A facsimile(Şengör, 2017) is also available.Notes: Damaged Greek map of the inhabited world according to Ptolemy's second projection.

Figure 4 .
Figure 4. Deviations between the real position and the location specified by Ptolemy after correction.Places and areas mentioned in the text: (A) Alexandria and surroundings, (B) eastern coast of the Mediterranean, including Antioch and Apamea, (C) Rhodes, (D) western coast of Asia Minor including Miletus, (E) Thessaly, (F) western coast of the Apennine peninsula including Rome, (G) the West Alps foothills, including Massalia, (H) Sicily including Syracuse.Notes: The actual map of Europe and of its close surroundings, complemented by point plotting of selected locations according to Ptolemy's coordinates after correction and by point plotting of real positions of the same locations.Deviations of the real positions from those by Ptolemy after correction are plotted in the map in the form of connecting lines with six categories differing by colour at 60 km intervals.The locations mentioned in the text are highlighted on the map.

Table 1 .
Map sources used, where settlements, roads and other objects of the Roman Empire are located, based on contemporary maps (OpenStreetMap). https://omnesviae.org/