This paper presents a method for calculating ancient coordinates, known as Ptolemaic coordinates after the Alexandrian scholar Ptolemy. The starting point for the calculation is the value of the Hellenistic “great Ptolemaic foot,” defined as 356.838624 mm per foot. Based on this, the distance between two degrees of latitude in the coordinate system is calculated as *600 *6.3 *63 = approximately 84.97755 kilometers. The circumference of the Earth used is obtained by multiplying this value by 360. The calculation assumes that the ancient coordinates do not represent the definition of a point on the Earth’s surface. The ancient coordinates are points from which the measurement points on the Earth’s surface must first be calculated. This resolves the problem of the apparent inaccuracy of the Ptolemaic coordinates, which has been noted by all modern authors. The coordinates can be placed at the location intended for them in antiquity.
It is not clear from the explanations accompanying the coordinate descriptions that Ptolemy was familiar with the ancient calculation method. His explanatory texts are often of a technical nature. The coordinate system, on the other hand, contains numerological patterns, as the system itself is based on the numbers 600, 6.3, and 63. Divisions are possible, for example, using the numbers 7 and 3. Mathematics and numerology intertwine within the coordinate system.
The earliest literary mention of Ptolemy and the Geographica is found in the Res Gestae by Ammianus Marcellinus, in the second half of the 4th century AD. He cites the geographer Ptolemy as an authority for precise measurements of distances. Ammianus Marcellinus must also have had access to maps based on the Ptolemaic coordinate system, for he mentions in Sarmatia the far-northern location of two altars, roughly at the level of river mouths in the Baltic region, dedicated to Alexander the Great and the Emperor[1].
Ammianus Marcellinus regards the geographical location of the altars as absolute, which shows that he was no longer familiar with the ancient method of using the coordinate system. The latest possible date by which the ancient method of using the coordinates must still have been known is the second or third decade of the 2nd century AD. New entries of coordinates can still be found for this period. This is followed by a period of 250 years during which the understanding of the coordinate system must have changed, the ancient method of use was discontinued, and it was fundamentally revised.
In antiquity, the coordinates for the altars served as two axes from which distances to measurement points were calculated. In Germania, coordinates with a comparable axis function can be found in the form of the Nauaision coordinate and the Altar of the Flavians, which were required for the depiction of the Germanic campaigns and the campaigns of the Flavian emperors in the Neckar region. The far northern location of the altars of Alexander and the Emperor results from the great distances that Alexander the Great covered with his army from Macedonia to India and that the Emperor—presumably Caesar—covered.
The new interpretation of the ancient coordinates in this work also encompasses the textual corpus. RENATE BURRI is correct in noting that the manuscript tradition represents an extremely complex and still largely unresolved issue[2]. Limiting the analysis to two manuscripts and to the geographical area of Gallia Belgica and western Germania constitutes a selection that makes it possible to identify defining characteristics specific to this region, and, where possible, beyond it.
Overall, the manuscript tradition of the toponyms varies in quality, considering that changes occur in every language and that the presence of different spoken languages must be expected in the area under investigation. It becomes apparent that, in addition to Gallic and Germanic linguistic forms, a variety of the Greek language must also be expected to a significant extent as a living language in the study area, and that there were fluid transitions between these languages. Latin is also present. As a further language, the presence of Punic or, more generally, North African languages must be taken into account. However, due to a lack of knowledge about the Punic language itself, the connection can only be described in rudimentary terms. In addition, other linguistic elements appear that cannot be classified
When studying the transmission of toponyms, it is valuable to identify frequently occurring name components in order to gain insights that go beyond the individual toponym. This makes it possible to distinguish copyists’ errors from structural linguistic features. An example of the greater certainty that can be gained when dealing with the transmitted texts is the four-fold occurrence of the shift from Greek π to Greek φ, which, due to its frequency, does not represent a copyist’s error but rather attests to the linguistic development of Greek in the geographical area under investigation.
Part IV of this work compiles linguistic features that stand out during the analysis of the textual material and facilitate the understanding of individual toponyms and existing connections. The locative prefix, with an approximate meaning of “where … is located,” occupies a prominent position. It is found in several languages and had already ceased to serve its function of spatial orientation by the time of the Roman presence in Gallia Belgica and Germania. The Latin ubi still refers to forms of the locative prefix.
Overall, the linguistic conditions, the toponyms, and the mathematical principles of the coordinate system point to a time of origin for ancient coordinates that certainly dates back to the beginning of the first century BCE.
For the coordinate designations, I use the text corpus from Codex Vaticanus Graecus 177 and Codex Vaticanus Graecus 191, retrieved from the Vatican’s digital library[3] as well as the German translation by STÜCKELBERGER / GRASSHOFF.
To calculate the distances of the coordinates in the ancient coordinate system, I use the calculation method of the Polish-American geodesist Thaddeus Vincenty, which takes into account the Earth’s oblateness[4]. The Earth’s radius to be used for the ancient coordinates is derived from the construction of the coordinate system shown above, ranging from 4,868 to 4,869 kilometers. Once the distances between the ancient coordinates have been determined, they are projected into WGS 84 with identical distances. In this way, it is possible to convert the ancient coordinates into WGS 84 coordinates
This project includes the following datasets:
Ancient coordinates converted to WGS 84
The file contains the longitude and latitude values for approximately 210 ancient coordinates in WGS 84, in EPSG 4326.
Measurement points for the ancient coordinates in WGS 84
The file contains the longitude and latitude of the measurement points identified to date in WGS 84, in EPSG 4326. Approximately 350 data records.
Linkages between the ancient coordinates and the corresponding measurement points
The file contains data on the links (“mappings”) between coordinates and survey points, including the respective distance intervals. Approximately 1,650 data records.
Since the ancient coordinates represent not only a geography but also a historiography, a region-based representation faces particular challenges. This challenge is further compounded by the fact that Ptolemy’s regional division can only be reconstructed with considerable difficulty today. The study area is therefore divided into 16 modern regions, to which measurement points are assigned, thereby establishing a connection to contemporary geographical areas.
The data contained in the coordinates as measurement points, along with depictions of Roman military campaigns, is presented in a separate context outside the regional representation. The rivers Seine, Saône, Doubs, and Rhône are also depicted in context. Fundamentally, the separation of the representation into regions and cross-regional features remains a challenge that requires further processing at a later stage.
[1] AMMIANUS MARCELLINUS, Das römische Weltreich vor dem Untergang, übersetzt ins Deutsche von Otto Veh, Amsterdam 1997:22,8,40. Die Koordinatenverzeichnis nennt nur ‚Kaiser‘.
[2] BURRI, RENATE, Die Geographie des Ptolemaios im Spiegel der griechischen Handschriften, Berlin und Boston 2013: 88.
[3] https://digi.vatlib.it/view/MSS_Vat.gr.177 ; https://digi.vatlib.it/view/MSS_Vat.gr.191
[4] Zum Rechenweg siehe Anlage.
Coordinates Today and in Antiquity
What are coordinates for? There are simple ways to get from one point on the Earth’s surface to another and to determine the distance and travel time in advance. A route planner on a smartphone displays the route and the estimated travel time. For trips by train, ship, or plane, departure and arrival times are readily available. What is less well known, however, is the ability to mathematically calculate the shortest distance between two points by specifying their coordinates in degrees and minutes. For example, the distance between the city halls of London, England, and Tokyo, Japan, is approximately 9.553,5 kilometers when the great circle distance is calculated using the Haversine formula—in this case, by an AI.
Smartphones, train and flight schedules, and AI did not exist 2.000 years ago, but coordinates did exist, and with their help and the application of mathematics, distances could be calculated theoretically. We have no information from antiquity regarding the application of mathematical knowledge for this purpose. However, it is also possible that information was preserved in manuscripts but has not been included in analyses over the past 700 years because the scientific understanding of Roman antiquity was no longer compatible with the natural-scientific understanding of science that has developed since the European modern era.
This work will demonstrate that in geography, the significance of coordinates in antiquity—of which between 6.500 and 10.500 have been preserved, depending on the method of counting—fundamentally differs from that of modern coordinates. Modern coordinates describe a point on the Earth’s surface. The ancient coordinates themselves do not specify the location of a point on the Earth’s surface, but they do make it possible to calculate points on the Earth’s surface based on them. As will be seen, the calculation is performed using an interval system that must be referred to as the Pythagorean interval system.
This paper does not provide a comprehensive examination of the scientific understanding of the late Roman Republic and early Roman Empire, but focuses only on the few points necessary for a basic understanding of ancient coordinates. To this end, it is important to demonstrate the ability of the ancient world to calculate distances based on the few sources available today.
The mathematician Heron of Alexandria is known to have summarized a method for the astronomical calculation of the distance between the cities of Alexandria in Egypt and Rome, based on the observation of the same lunar eclipse at both locations. Heron apparently also presents the result of the calculation as part of a great circle.[1] For the great circle, the circumference of the Earth must always be specified for the calculation. Heron of Alexandria, who lived and worked in the first century AD, cites Eratosthenes’ figure of 252.000 stadia for the circumference of the Earth at . Ptolemy later gives 180.000 stadia, a value for the Earth’s circumference that had already been corrected by Eratosthenes.[2]
Ptolemy, author of the Geography, provides guidance on measuring distances. He describes a method of calculation in which points lie on meridians and parallels (longitude and latitude) and thus represent coordinate points. The distance between these is calculated as a segment of a great circle[3] .
An often overlooked benefit of coordinates lies in the easy availability of distance information. With 10.500 measurement points, this results in over 55 million individual distance values between the points. This volume of information is unmanageable; adding new data or removing data that is no longer needed is simply not feasible under any circumstances. Specifying only a single datum for each measurement point minimizes the effort and ensures that the data can be managed and used as needed. This argument remains valid even when working with only a small portion of the data. The effort involved in working with coordinates and calculating distances as needed is always much less than storing pre-calculated distances between points.
Herodotus provides an indication in the Histories that longitudes were used in the 6th century BCE. He writes there of the sources of the Nile River, which he assumes are on the same longitude as the sources of the Ister River (the Danube). The fact that his assumption is incorrect is irrelevant in our context. What is crucial is that Herodotus offers no further explanation of the technical term, leading to the conclusion that his readers, for whom he wrote, were already familiar with the concept; otherwise, he would have included some explanation.
[1] SIDOLI, NATHAN, Heron’s Dioptra 35 and Annalemma Methods: An Astronomical Determination of the Distance between Two Cities, in: CENTAURUS 2005, Vol. 47, pp. 236–258.
[2] STÜCKELBERGER, ALFRED and GRASSHOFF, GERD. Ptolemaios Handbuch der Geographie, Bd. 1, Einleitung und Buch 1-4, 2nd edition, Basel 2017: 69.
[3] Information based on the German translation from the Greek in STÜCKELBERGER / GRASSHOFF 2017 1: 63.