The length of coastlines in Ptolemy ’ s Geography and in ancient periploi

The lengths of the coastlines in Ptolemy’s Geography are compared with the corresponding values transmitted by other ancient sources, presumably based on some lost periploi (literally “voyages around or circumnavigations”, a genre of ancient geographical literature describing coastal itineraries). The comparison reveals a remarkable agreement between them, suggesting that Ptolemy relied much more heavily on these or similar periploi than it used to be thought. Additionally, a possible impact of Ptolemy’s erroneous estimate of the circumference of the Earth is investigated. It is argued that this error resulted in two interrelated distortions of the coastal outlines in Ptolemy’s Geography. First, the north–south stretches of the coast that were tied to particular latitudes are shown compressed relative to the distances recorded in other sources in roughly the same proportion to which Ptolemy’s circumference of the Earth is underestimated relative to the true value. Second, in several cases this compression is compensated by a proportional stretching of the adjacent east–west coastal segments. In particular, these findings suggest a simple explanation for the strange shape of the Caspian Sea in Ptolemy’s Geography.


Introduction
Classics (or studies of Greco-Roman antiquity), which is the field that the history of ancient geography and cartography belongs to, is a very old discipline.So, as time goes by, the chance to find something really new in this field, or to do something that nobody has ever done before, tends to zero.Ptolemy's Geography (ca.AD 150) is a happy exception to this tendency.Although the Geography is one of the most famous and often-studied works in the history of cartography, its origins still remain an enigma.However, in the last decade, after the publication of its new edition by Stückelberger and Graßhoff (2006), Ptolemy's Geography has seen an upsurge of scholarly interest.Numerous researchers have developed a wide array of different methodological approaches to studying its nature and genesis (e.g.Isaksen, 2011Isaksen, , 2013;;Marx, 2011Marx, , 2016;;Heß, 2016;Shcheglov, 2017;Graßhoff et al., 2017;Arnaud, 2017;Defaux, 2017).One of the main approaches is a comparison with other ancient sources.Surprisingly, until this day nobody has ever tried to compare Ptolemy's Geography with the ample data on coastline length provided mostly by Agath-emerus in his Geographical Sketch (the 1st century AD) and by Pliny the Elder in the geographical books of his Natural History .The objective of the present study is to fill this gap.My contention is that this comparison can shed some new light on Ptolemy's modus operandi and the genesis of his Geography.

Periploi among the sources of Ptolemy's Geography
Ptolemy's Geographical Guide, or simply the Geography, was essentially a description of the world map in terms of spherical coordinates (latitude and longitude in degrees) assigned to each of the 6300 listed localities. 1 However, it is clear that only some of these coordinates could have been based on astronomical observations (e.g. the latitudes of the ever, deserve a more thorough examination elsewhere.Fortunately, thanks mainly to the Geographical Sketch by a certain Agathemerus (Diller, 1975) and to Books III-VI of Pliny's Natural History (Jan and Mayhoff, 1892), as well as to some other sources, we have a set of values for the total coastline lengths of the major regions as recorded by the most famous ancient geographers: Eratosthenes (the 3rd quarter of the 3rd century BC), 7 Artemidorus of Ephesus (ca.104-100 BC), 8 Marcus Vipsanius Agrippa (ca. 12 BC), 9 Isidore of Charax (around the turn of the 1st centuries BC and AD), and some others.These values evidently derive from some lost periploi.This data set covers the largest part of the then-known world (Figs. 2 and 3).First of all, Artemidorus, Isidore, and Agrippa give the lengths of the coastlines of the "Inner" Sea (Mediterranean, Black, and Azov seas) as a whole and as divided between the three continents: Europe from the Pillars of Hercules (or Calpe, i.e.Gibraltar) to the mouth of the Tanais (Don), Asia from the Tanais to the Canobic (westernmost) mouth of the Nile, and Libya (Africa) from this mouth to Tingis (Tangier).The same geographers and some other authors give the perimeters of the major peninsulas (the Pyrenean, the Italian, the Arabian), seas or gulfs (the Persian Gulf and the so-called "second" and "third" gulfs of Europe i.e. roughly speaking, the Adriatic, Ionic, and Aegean seas and the Red, Black, Azov, and Caspian seas), and the largest islands.The fact that different authors reproduce almost the same or similar set of values suggests that these data were regarded as a special category to be included in a geographical treatise.For the sake of completeness, we will add to this set several values of the total coastline lengths reported by the surviving "detailed" periploi: Pseudo-Arrian's Periplus and the Stadiasmus, as well as by Strabo's Geography (ca.AD 23), our main source on its subject matter (Radt, 2003(Radt, -2011)).
from Aradus to Miletus including the adjacent islands of Cyprus and Crete.The most reliable edition is still Müller (1855, p. 427-563); a new edition is being prepared by Pascal Arnaud.These periploi may be called "detailed" in the sense that, unlike other surviving periploi, they divide the route into hundreds of course legs with the median length of ca.90 stades or 16-17 km.On the ancient methods of maritime distance estimation, see Arnaud (1993). 7Eratosthenes was probably the most famous ancient geographer before Ptolemy.His geographical treatise has survived only in fragments; for the editions, see Berger (1880) and Roller (2010).
8 Schiano (2010). 9Marcus Vipsanius Agrippa (63-12 BC) was the right arm of the emperor Augustus.According to Pliny's testimony (Natural History 3.17), he left behind an unfinished geographical work which, as most scholars believe, served as the basis for a world map displayed in the Porticus Vipsania in the centre of Rome (before AD 14).The map was accompanied by explanatory notes with an account of dimensions and boundaries of 24 regions of the world.These notes have come down to us only in fragments; the standard edition is Klotz (1931).
The objective of this paper is to compare this data set with the lengths of the corresponding coastlines in Ptolemy's Geography.But before attempting the comparison there are several methodological issues we need to discuss.

Method of comparison
How can we measure the lengths of the coastlines in Ptolemy's work?My contention is that the simplest and most natural way would be to take Ptolemy's Geography as what it essentially is, namely as a catalogue of spherical coordinates.The distance between two points specified by these coordinates is an arc of the great circle that can be calculated using the rules of spherical trigonometry. 10Accordingly, I propose to calculate the length of the coastlines in Ptolemy's Geography as the sum of individual arcs joining coastal points multiplied by the length of a degree of the great circle which he defines as 500 stades (Geogr. 1.7.1,11.2).
It is important to emphasize that actually next to nothing is known about how Ptolemy worked with his sources and how he converted their distance data into coordinates (Graßhoff et al., 2017, p. 2, 6-7;Defaux, 2017, p. 12, 255).It is hard to imagine that he would have calculated the coordinates for each of the 6000 points listed in his Geography.It would be more reasonable to assume that calculations may have been performed only for the most important points, whereas all others were localized by means of simpler methods, for example, Pythagoras' theorem and/or a simple ruler; cf.similarly Spaul (1958, p. 5-7) and Graßhoff et al. (2017, p. 1, 16, 25).
Ptolemy's coordinates are specified with precision up to 5 .This, however, must not mislead us about the accuracy of his data.The precision of Ptolemy's coordinates correlates clearly with the density of the points on his map: its "rarest" parts at the periphery (such as Asian Scythia or Inner Libya) have the highest percentage of coordinates in whole degrees, but as we approach the Mediterranean, the fractions of a degree become increasingly more frequent and smaller.In other words, Ptolemy's coordinates can be precise not so much because his sources were accurate, but rather in order to fit many points in a confined space.Statistical analysis of the frequency of different fractions of a degree used in Ptolemy's coordinates shows that he tended to round all values to the largest possible fractions. 11It means that, fig- 10 The distance between points A and B situated on a sphere can be calculated by the formula cos , where S AB is the distance between points A and B expressed in degrees of the great circle, λ AB is the longitudinal interval between them, ϕ A and ϕ B are their latitudes.Certainly, Ptolemy did not have trigonometric formulas in their modern form, but the theorems of Menelaus that he used to solve similar problems in the Almagest were an ancient equivalent of them (Neugebauer, 1975, p. 21-30). 11The frequency of different fractions of a degree in Ptolemy's catalogue of coordinates conforms to a normal distribution: the co-uratively speaking, Ptolemy's map had "low resolution": it could reproduce general outlines of large objects, but was quite inaccurate in details.For this reason, when comparing Ptolemy's data with the distances reported by other sources, it makes sense to consider only those applying to relatively long intervals.It also needs to be born in mind that, owing to the peculiarities of the map, close matches can hardly be expected from this comparison.
The text of Ptolemy's Geography has been handed down to us in two recensions: and (Fig. 1). 12Both of them descend from antiquity, but none of the extant manuscripts dates earlier than the late 13th century.The recension is considered to be earlier and more authentic, but it is represented by the sole manuscript Vaticanus Graecus 191 (ca.AD 1295) which omits all coordinates for the eastern half of the map (i.e. for the whole Asia, except for Asia Minor, Armenia, and Asiatic Sarmatia) and contains quite a few scribal errors.The recension is apparently secondary to , but it includes the majority of the manuscripts. 13In view of these circumstances, when comparing Ptolemy's data with the distances recorded in other sources, both recensions should be taken into account, but should be regarded as more reliable.
14 All Ptolemy's coordinates are taken from the electronic database attached to the newest edition of the Geography (Stückelberger and Graßhoff, 2006) and are appended to the present paper as Excel files in the Supplement.
www.hist-geo-space-sci.net/9/9/2018/ Hist.Geo Space Sci., 9, 9-24, 2018 Figure 1.The outlines of the Mediterranean Sea according to the and recensions of Ptolemy's Geography (Italy, the Balkans, and the Aegean Sea are taken as an example).This and all the other maps in the paper are drawn using the projection attributed by Ptolemy to Marinus of Tyre , whom he introduces as his immediate predecessor and the primary source of information (Geogr.1.6.1).In modern terms, it is an equidistant cylindrical projection, in which the ratio between equal latitudinal and longitudinal intervals is represented as 5 : 4. other sources, and how much deviates from , both in terms of stades and in terms of percentage.
What conclusions can be drawn from this comparison?First of all, our sample clearly falls into two parts: shorter and longer distances.Islands and other small features (such as Peloponnesus or no.18 in Table 1, the Great Syrtis or nos.7 and 17, etc.) or relatively short coastal stretches exhibit little similarity between Ptolemy's data and the periploi data (the only exceptions are Sicily nos.15 and 16, Crete no. 8, and Illyria nos. 1 and 9; Figs. 3 and 7) This result accords with the fact that Ptolemy's map had a low resolution and was in principle incapable of representing small features accurately.
However, for longer distances, with a threshold being approximately at 10 000 stades, the pattern changes radically.It is instructive to remember that, with only a few exceptions, these coastlines are very long and have very complex geometry.In these circumstances, one can hardly expect any similarity between Ptolemy's figures and the periploi figures at all.Nevertheless, the correspondences between them are so numerous and so close (Fig. 4) and their geographical coverage is so wide (Figs. 2 and 3) that this can reasonably be regarded as something more than just a series of coincidences.
Figure 4 shows that in most cases the differences between Ptolemy's data and the periploi-based sources fall between +4 and −1 %. 15 How can we interpret these values?Is it a lot or not?Previous researchers who compared Ptolemy's data with the distances recorded in other sources regarded 15 Of course, it is possible to find many contradictions between Ptolemy's data and distances reported by other sources, but there is no reason to expect them to match always and everywhere.For example, Pliny gives evidently incredible and fictional values for the coastlines of Germania (4.98 = Agrippa F 21; see Riese, 1878, p. 5) and Gaul (4.105 = Agrippa F 23 Riese = F 40 Klotz): 20 000 and 14 000 stades, respectively.
the 5-10 % difference between them as small enough to suggest a common origin (Cuntz, 1923, p. 120-122, 144-145;Urueña Alonso, 2014a, p. 164, 169-170;Marx, 2016, p. 33-34).But for a more robust assessment, it would be informative to compare the differences between Ptolemy and the other sources with the differences between the two recensions of Ptolemy's own work (see Table 1 and Figs. 5, 6).A convenient benchmark for this comparison is provided by the measurements of the "Inner" Sea (Mediterranean and Black seas without the Azov Sea). 16In the recension, its perimeter measures 133 792 stades, which is 2.8 % longer than according to Artemidorus (130 120, no.50 in Table 1), but in it is 1.8 % longer (136 231) than in .Remarkably, in a number of cases the differences between and are appreciably larger than those between and the other sources: Europe from Calpe to the Tanais (no.48 in Table 1), Hispania (33), Italy (29), the "second gulf" of Europe ( 26), and the Red Sea (28).If we still believe that and are, despite all the disagreements between them, two versions of the same work by the same author, the similarity between Ptolemy's values and those reported by the other sources cannot be dismissed as a coincidence.
It makes sense to take a closer look at the outliers.First of all, however obvious it may seem, it is worth noting that, when ancient sources give different estimates for the same coastline, Ptolemy's data cannot agree with all of them.For instance, when Ptolemy's data match one of the two known values for the perimeter of Arabia (no.45: 38 120 stades), they must naturally disagree with the other (no.44: 32 000).The same is true for the Great Syrtis (nos.7 and 17), the Maeotis (22,25), Hispania (33,38), the Pontus Euxinus (35-37, 39), Africa (40-43), and the Mediterranean without the Pontus (49,50).Taking this into account, only two outliers stand out: the Red Sea coast of Arabia (no.27) and the Mediterranean coast of .The estimated 14 000 stades for the Arabian coast, repeated by Eratosthenes, Artemidorus, and Agrippa, goes back as far as to Alexander's commander Anaxikrates and was regarded as "too much" ( ἐπὶ πλέον) already by Strabo (16.4.4 C768).
But it seems that the best way to explain a drastic reduction of this coast by 22 % in Ptolemy's Geography would be to link it to his adoption of an erroneous value for the circumference of the Earth (see below Sect.5).The case of Africa seems to be more complicated: Ptolemy's distance is 5 % shorter than the Stadiasmus' value for the coast between Alexandria and Utica (no.30), but it is 5.66 % longer than the largest value recorded by Pliny (no.43) for the coast between Canobus and Tingis.Yet, it is remarkable that Pliny calls this value "the average of all the various accounts" (6.208: ut media ex   for the mouth of the Cyrus.b Contrary to the general tendency, there are good reasons to think that for Sicily, recension is less reliable than , because in Cape Pelorus (one of the three principal capes of Sicily) is omitted, and Segesta (the westernmost point) is shifted considerably further to the east, whereas in the latter is quite reasonably placed on the same meridian with Ostia.c Since Ptolemy locates the island of Gades too far from the coast, it would be more correct to measure this distance from the temple of Hera on the coast (see the Supplement).d Codex Vaticanus Graecus 191 contains two clear errors.First, Paniardis is displaced far to the east (to 53 • 30 lat., 69 • 40 long.instead of 53 • 30 , 67 • 30 in ) relative to the rest of the coast.Second, Parthenion and Myrmekion on the European coast of the Cimmerian Bosporus (modern Kerch Strait) are placed south of Achilles' sanctuary on its Asian side (at 48 • 15 lat.and 48 • 10 lat., respectively).In , however, they are placed at the same latitude (at 48 • 30 lat., the latitude of the Borysthenes), which is more logical because Parthenion and Myrmekion are known to be situated opposite to Achilles' sanctuary in the narrowest part of the strait (Strabo 11.2.8 C494).Except for these two differences, matches with almost exactly.e Natural History 3.97: A Lacinio promunturio secundus Europae sinus incipit, magno ambitu flexus et Acroceraunio Epiri finitus promunturio, a quo abest LXXV; 3.150: universum autem sinum Italiae et Illyrici ambitu XVII.The latter quotation follows immediately after the fragment of Agrippa about the coast of Illyria (F 13 Riese = F 16 Klotz; no. 1 in this table), which allows Riese and Klotz to ascribe it also to Agrippa (F 13 Riese = F 47 Klotz).Detlefsen (1886, 243 p., 247-248) has reasonably noted that this quotation is also connected with Pliny's passages about the so-called "second" and "third" gulfs of Europe (3.97;4.1) which he ascribes to Varro, although they may well be attributed to Agrippa as well.f In a number of cases, the inconsistencies in the data have been corrected: the positions of Cape Circei, Tarracina, Privernum, and Hidruntum are given according to ; cf. also Cuntz (1923, inserted map), Polaschek (1965, cols. 726-727, 730-731, inserted map).The coordinates of Tarentum (41 • 30 long., 40 • lat.) are taken from the so-called Canon of the noteworthy cities (6.1; Stückelberger and Mittenhuber, 2009, p. 162); the outlines of the Gulf of Taranto are given according to the reconstruction proposed by Polaschek (1965, cols. 720-721, 730-731, map), even though it does not solve all contradictions in Ptolemy's data.g gives coordinates only for the north-western coast of the Caspian Sea, therefore all the rest are taken from .h "The provinces of Spain taken all together, measured from the two promontories of the Pyrenees along the sea line, are estimated to cover by the circumference of the whole coast 2924 miles, or by others 2600 miles" trans.by H. Rackham in the Loeb Classical Library (omnes autem Hispaniae a duobus Pyrenaei promunturiis per maria totius orae circuitu XXVIIII • XXIIII colligere existimantur, ab aliis XXVI).It means from Cape Oiason to the Temple of Aphrodite in Ptolemy's Geography.i For arguments, see Detlefsen (1877, p. 24) and Klotz (1931, p. 441-442).j omits the coordinates for the coast from Anthedon (64 • 50 long., 31 • 40 lat.) to the border of Cilicia (69 • long., 36 • 20 lat.), so they are taken from .k Pliny (4.77) lists other estimates of the circumference of the Pontus: 2150 m.p. = 17 200 stades according to Varro and "the old authorities generally" (fere veteres), 2500 m.p. = 20 000 stades in Cornelius Nepo, 2540 m.p. = 20 320 stades in Agrippa, and 2425 m.p. = 19 400 stades in Mucianus.Polybius (History 4.39.1)estimated it at 22 000 stades.Neither of these values agrees with Ptolemy's data.l Klotz (1931, p. 441-442) emends Pliny's figure to XXVIII XXIII (2823 m.p. = 22 584 stades).m However, the sum total stated in the Periplus (121 [92]) is 23,587.For an explanation of some of the numerical errors underlying this discrepancy, see Diller (1952, p. 104).n Remarkably, the sum of all distances given by Strabo for the coast of Libya amounts to a little more than 28 000 stades: 5000 from Cape Cotes to Cape Metagonium (17.3.6 C827), thence 6000 to Cape Tretum (17.3.9C829), 2500 to Carthage (17.3.13C832), more than 5000 to Cape Cephalae (in a straight line across the Lesser Syrtis) and thence 3930 along the Great Syrtis (17.3.18C835), 1000 from Berenice to Apollonia (17.3.20 C837), 2200 to Catabathmus (17.3.22C838), 900 to Paraetonium and 1300 to Alexandria (17.1.14C798), 120 to Canobus (17.1.17C801).With few exceptions, these distances disagree with Ptolemy's data, suggesting that Strabo and Ptolemy drew on different sources.o Müller (1855, p. 520).It is tempting to emend Marcianus' figure γ σ π (30 280) to γρπ (30 380) which would be in accord with Pliny 6.208.p circuitus Arabiae a Charace Laeana colligere proditur XLVII LXV p.Some secondary MSS read this figure as XLV LXV = 36 520 stades or XLVIII LXV = 38 920 stades.r omits the coordinates for the coast from Anthedon (64 • 50 long., 31 • 40 lat.) to the border of Cilicia (69 • long., 36 • 20 lat.), so they are taken from .s Artemidorus' length of the coast of Europe can be derived from the other values recorded by Pliny for the circumference of the Mediterranean Sea without the Maeotis (no.50 in Table 1), the perimeter of the Pontus (no.37), the coast of Africa (no.40), and the coast of Asia from Canobus to Chalcedon according to Timosthenes (no.34).t The deviation of this value from that of Artemidorus is most probably due to the fact that Agrippa's estimate of the circumference of the Pontus was about 6000 stades shorter than according to Artemidorus (Plin.4.77).Klotz (1931, p. 458-459), however, argues that this figure could not have come from Agrippa.

Figure 2.
Differences between the coastline lengths as derived from Ptolemy's coordinates and as given by Artemidorus (including those that he has inherited from Timosthenes and Eratosthenes: nos.20, 22-24, 27-28, 32, 40, 46-47 in Table 1).The percentages refer to the deviation of Ptolemy's values from those of Artemidorus.The outlines are drawn according to the recension supplemented with coordinates from for the areas of Asia that are missing in .(nos. 2, 5-6, 8, 12, 16-19, 21, 25, 28, 33, 39, 43-44 in Table 1).The percentages refer to the deviation of Ptolemy's values from those of other sources.The outlines are drawn according to the recension supplemented with coordinates from for the areas of Asia that are missing in .
Remarkably, Table 1 shows that the distances in are usually shorter than in , but longer than in the other sources.This ratio can easily be explained by the so-called coastline paradox: the well-known fact that a coastline length varies depending on the level of detail of its geometry.The periploi described, of course, the coastal routes.However, these routes did not have to replicate all the minor, but numerous curves and bends of the coast.It is unsurprising, therefore, that the coastline in Ptolemy's Geography, when measured with all its curves, proves to be a little longer than the distances recorded in periploi, and , as the second recen-sion, has usually a more detailed and, therefore, even longer coastline than (Fig. 1).

The influence of Ptolemy's erroneous estimate of the circumference of the Earth
Distance data derived from periploi were bound to come into conflict with other Ptolemy sources, and particularly with his latitude data.In this case, contradictions must have been especially prominent because Ptolemy adopted an erroneous value for the Earth's circumference, namely 180 000 stades, which was about 17 % less than the actual value, if he used the stade of 185 m length (Shcheglov, 2016b).Owing to this www.hist-geo-space-sci.net/9/9/2018/ Hist.Geo Space Sci., 9, 9-24, 2018  error, all north-south distances which were tied to particular latitudes must also have been reduced by approximately the same amount.This effect may be illustrated by the example of the Red Sea coast of the Arabian Peninsula (no.27 in Table 1).In Ptolemy's Geography, this coast is squeezed between the latitude of its northern extremity (Elana, 29 • 15 ) and that of Ocelis ( 12• ) near the Strait of Deire (Bab-el-Mandeb), the latter being based on astronomical observations (Ptol.Geogr.1.7.4).It is unsurprising, therefore, that this coast exhibits a 22 % reduction in length.
However, I would like to draw attention to another possible and arguably more unusual side-effect of Ptolemy's underestimated circumference of the Earth.This effect is observed when a coastline in Ptolemy's Geography falls into two distinct parts: one oriented north-south, the other east-west.In a number of such cases the same pattern is seen: while the north-south part is shorter than the recorded periploi distance, which is understandable, the east-west part is proportionally longer, which requires explanation.It is reasonable to suppose that such a stretching of the east-west parts was intended to compensate for the shortening of the north-south part in order to maintain the total coastline length unchanged.If this explanation is valid, it can give us an important key to understanding Ptolemy's modus operandi.
The effect of proportional shortening/stretching of the neighbouring segments of the coast can be illustrated most clearly by three examples: Italy, the Caspian Sea, and the Red Sea.In the case of Italy, the southern part of the peninsula is squeezed between the latitudes of the Strait of Messina (at Rhegium) and Naples, whereas the part between Naples and Ostia is proportionally stretched from east to west (Fig. 7).The Caspian Sea is probably compressed from north to south together with the circumference of the Earth, but proportionally stretched from east to west (Figs. 2 and 3).In the case of the Red Sea, the largest part of its African coast is oriented from north to south and squeezed between the latitudes of its northern extremity and of Ptolemais Theron in the south, whereas its southern part, turning eastward to the Bab-el-Mandeb Strait, is proportionally stretched (Fig. 3).Below I consider these instances in detail.recension).The percentages refer to the deviation of Ptolemy's values for the coastline length from those given by Pliny (no.29 in Table 1) and Agrippa (nos. 1 and 9).

Italy
Only a few ancient sources provide detailed information on individual distances along the coast.Pliny's description of Italy is among them (Natural History 3.49,51,56,62,70,73,95,97,99,100,111,115,127).Table 2 shows how Pliny's values for the separate coastal stretches constituting the perimeter of Italy relate to Ptolemy's data (see also Figs. 3  and 7).
Ptolemy's Italy may be divided into three parts (Fig. 7): (1) the northern part lying west of Ostia and Ancona, (2) the middle part lying east of Ostia and Ancona and north of Naples, and (3) the part lying to the south of Naples.Table 2 shows that for northern Italy Ptolemy's distances accord tolerably with Pliny's figures. 17Ptolemy's middle Italy 17 Ptolemy's coast from Ravenna to the mouth of the Formio matches Pliny's distances pretty well.The length of the coast from the Varus to Cape Circei is exaggerated by 8 % (or 42.5 m.p.) on is distinctly stretched in the east-west direction relative to Pliny's distances, whereas southern Italy is almost equally compressed.For example, Ptolemy's coast from Circei to Surrentum is stretched by 78 m.p., whereas that from Salernum to Rhegium is shortened by 83 m.p.Similar compressions and stretchings of these coastal sections are exhibited by Ptolemy's map relative to the modern map (Table 2, Fig. 8).
The compression of southern Italy in the north-south direction can be easily explained, first of all, by Ptolemy's underestimated value for the Earth's circumference.Ptolemy's latitudes for southern Italy are rather accurate: for example, the interval between his latitudes of Surrentum and Rhegium average as compared to Pliny's figures, which is quite a lot, but it can still be explained by the fact that Ptolemy's data almost always exhibit a similar overestimation of distances relative to the periploi data (Table 1).
www.hist-geo-space-sci.net/9/9/2018/ Hist.Geo Space Sci., 9, 9-24, 2018 across the open sea and the route along the coast, respectively.is 2 • 25 , which is pretty close to the true value of 2 • 30 .However, for Ptolemy the corresponding distance amounts to 150 m.p., whereas the correct value would have been 188 m.p.
Ptolemy's southern Italy is distinctly squeezed between the latitudes of Naples (40 • 55 in ; the true latitude is 40 • 50 ) in the north and that of the Strait of Messina (in particular Cape Pelorus, 38 • 35 in ; the true latitude is 38 • 16 ) in the south.These latitudes corresponded to the so-called klimata with the longest day of 15 and 14 3/4 h, respectively.Klimata (sing.klima) was a technical term for latitudes defined by the length of the longest day.These klimata were, in essence, the principal tool available to pre-Ptolemaic geographers for constructing a mathematically rigorous map of the world, at least, from Eratosthenes onwards. 18A set of these klimata constituted the basis of the geographical system of Marinus of Tyre, Ptolemy's immediate predecessor and the principal source (Honigmann, 1930;Wurm, 1931).Therefore, Ptolemy's latitudes of Naples and Pelorus go back, most probably, to Marinus or even further.
It is striking that various divergences between Ptolemy's and Pliny's data on the separate coastal stretches (Table 2) 19are in stark contrast with the close agreement between their values for the total length of the coast: Ptolemy's figures for the perimeter of Italy and for the southern side from the mouth of the Varus to Rhegium deviate from Pliny's values by only 2.4 % (no.29 in Table 1) and 3.7 % (no.19).Hence, it is tempting to explain this by assuming that the magnitude and direction of the disagreements may have been selected intentionally so that stretchings and compressions would have cancelled each other out.

The Caspian Sea
The coincidence between Ptolemy's and Artemidorus' values for the perimeter of the Caspian Sea gives us a key to explaining its strange configuration in Ptolemy's Geography, namely why it is shown as being longer from east to west (ca.8250 stades from Gangara to the mouth of the Polytimetos) than from north to south (ca.4400 stades between the mouths of the Rha and the Straton; see Fig. 3).
First of all, it is tempting to connect this anomaly with the fact that the whole of Ptolemy's map exhibits a similar stretching of all its outlines from east to west relative to the modern map. 20This stretching can largely be explained by Ptolemy's underestimated value for the Earth's circumference.This effect is due to the principal difference between the methods of measuring latitude and longitude (Shcheglov, 2016b;Graßhoff et al., 2016).Latitude can be determined by means of simple astronomical observations, which had been known to the Greeks since at least the 4th century BC, and starting at least with Hipparchus (2nd century BC) it was normally expressed in degrees.But it was not until the 18th century that an efficient and simple enough method of determining longitude was devised.For an ancient geographer, the main method to find longitude was to convert distance measurements from customary linear units (Greek stades, Roman miles, etc.) to angular units (degrees), for which the estimate of the Earth's circumference was essential.This is why an error in this estimate inevitably affected all longitude values based on distance measurements, but had no effect on latitude values when they had been originally expressed in degrees.A too low value for the circumference of the Earth results in that all distances projected on its surface (i.e.converted to degrees) become proportionally overestimated in angular terms, which made the outlines on the map stretch mostly in the east-west direction (e.g.Russo, 2013;Shcheglov, 2016a, b).However, Ptolemy's underestimation of the size of the Earth could have been responsible for a stretching by only 17 %, which is evidently insufficient to explain the configuration of the Caspian Sea in his Geography.
Furthermore, Ptolemy's Caspian Sea is not only stretched from east to west but also compressed from north to south.The only ancient estimate of the north-south extension of the Caspian Sea comes from Strabo (2.1.17C74): he defines the meridional distance between its southernmost end and the mouth entering the Ocean in the north as "about" ( περὶ ) 6000 stades.This value matches the actual straight-line length of the Caspian Sea pretty well (provided that 1 stade = 185 m).However, Ptolemy's latitude interval between the northern (the mouth of the Rha at 48 • 50 ) and the southern (the mouth of the Straton at 40 • ) extremities of the Caspian Sea corresponds to only 4417 stades (Fig. 3; of course, there is no reason to suppose that any of Ptolemy's latitudes for the Caspian Sea could have been based on actual astronomical observations).
The situation is complicated by the fact that there may have existed a hypothetical early version of Ptolemy's Geography that was based on a different estimate for the Earth's circumference, namely 252 000 stades, which was ca.16.5 % above the true value (252 000×185 m = 46 620 km).This estimate was first proposed by Eratosthenes and can be regarded as almost generally accepted in antiquity.As has been shown by Schnabel (1930, p. 218-219), Ptolemy probably used this value for calculating geographic longitudes in the Almagest, one of his early works.Even more importantly, Sarre and Herzfeld (1911, p. 143-153), Wurm (1937, 1940), and Shcheglov (2004, 2017) have shown, independently of one another, that a large area on Ptolemy's map was built on the basis of Eratosthenes' set of distances which were converted to spherical coordinates according to Eratosthenes' rate of 1 • = 700 stades.This area covers the whole of the Middle East, at least, between the Euphrates and the Indus including the Caspian Sea.More importantly, Ptolemy's breadth of the Caspian Sea, when expressed in Eratosthenes' 700-stade degrees, corresponds to 6183 stades which closely agrees with Strabo's distance, as was noted by Wurm (1937Wurm ( , p. 7, 13, 1940, p. 8, 11), p. 8, 11).
It is also remarkable that, on Ptolemy's map, the northern extremities of the Caspian Sea, as well as the northern coast of the Pontus Euxinus, are clearly tied to the latitude of the mouth of the Borysthenes (48 1/2 • ), one of the seven klimata or the key latitudes determining the structure of his map and going back, most probably, as far as Eratosthenes (Fig. 3). 21These correspondences make it possible to suppose that Ptolemy's meridional extension of the Caspian Sea had been originally defined as slightly more than 6000 stades and that this distance was expressed in degrees of latitude already in the early version of the map, which was based on Eratosthenes' value for the Earth's circumference.
Interestingly, 6183 stades is quite close to the quotient of

The Red Sea
The general configuration of the Red Sea (or, in ancient terms, the Arabian Gulf) on Ptolemy's map is determined by the latitudes of four points: Berenice, Ptolemais Theron, Adulis, and Ocelis (Fig. 3). 25The length of the African coast from Heroonpolis (the northernmost point) to Deire (situated opposite to Ocelis) in Ptolemy's work is exactly the same as in Artemidorus (in it is only 0.1 % or 19 stades shorter, in it is 3.7 % or 574 stades longer; see Table 1).However, the lengths of the separate coastal stretches recorded by Eratosthenes and Artemidorus relate to Ptolemy's data as follows.
Table 3 shows that the north-south stretch from Heroonpolis to Ptolemais is ca.1425 stades shorter in Ptolemy than according to Artemidorus, while the next one turning to the east from Ptolemais to Deire is ca.1444 stades longer. 26Similar deformations of these stretches are exhibited by Ptolemy's of Bengal) and the Great Bay (the Tongking Gulf); see e.g.Stückelberger and Graßhoff (2006, p. 896-897, 902-903).

Conclusions and further perspectives
The main conclusion of our study is intuitively expectable and in this sense unsurprising, namely that Ptolemy's Geography was most likely based on some ancient periploi similar to those known from other sources.What seems unexpected and much more important is that our study has revealed a close numerical agreement between Ptolemy's Geography and the other periploi-based sources in the data on the length of relatively long coastlines (over 10 000 stades): in most cases the differences between Ptolemy's data and the other sources fall between +4 and −1 %.This observation can provide a new insight into the genesis of Ptolemy's Geography and the history of ancient geography in general.
A major challenge we face now is how to reconcile this striking agreement between Ptolemy's data and the other sources with the equally striking contradictions between them in the short coastal stretches of which the long ones consist (as was demonstrated by the example of Italy).In other words, with increase in the coastline lengths being compared, different discrepancies between Ptolemy and the other sources compensate one another until an almost complete agreement is reached.But what factors determined this transformation?Could it be intentional, or just accidental, and to what extent?This question compels us to revisit our understanding of Ptolemy's methods and the relevancy of our methods of examining his work.For the time being, we may only suggest a preliminary explanation.Most plausibly, Ptolemy's map was constructed not "bottom-up" -when its outlines would have been drawn successively point by point like beads are strung on a thread, but "top-down" -when the cartographer first drew the general outlines determined by long distances, and only after that he inserted minor details, fitted into the general picture.This assumption can be corroborated in part by the three cases (Italy, the Caspian Sea, and the Red Sea) where Ptolemy's Geography exhibits a proportional shortening of the north-south sections of coastline and stretching of the neighbouring east-west sections relative to the distances recorded in other sources.Whereas the shortening of the north-south sections was definitely caused by Ptolemy's underestimate of the circumference of the Earth, the stretching of the east-west sections is hard to explain, unless by assuming that the cartographer deliberately compensated the shortening.Of course, three cases are not enough for definitive conclusions, and the issue awaits further investigation.
Data availability.Ptolemy's coordinates are taken from the electronic database attached to the newest edition of the Geography (Stückelberger and Graßhoff, 2006).
Comparison of the coastline lengths in Ptolemy's Geography and in the other sources.

Figure 3 .
Figure 3. Differences between the coastline lengths as derived from Ptolemy's coordinates and as given by other periploi-based sources(nos.2, 5-6, 8, 12, 16-19, 21, 25, 28, 33, 39, 43-44  in Table1).The percentages refer to the deviation of Ptolemy's values from those of other sources.The outlines are drawn according to the recension supplemented with coordinates from for the areas of Asia that are missing in .

Figure 4 .
Figure 4. Deviation of Ptolemy's coastline length values from those given in the periploi-based sources for the distances over 10 000 stades (as expressed in the percentage terms).Ptolemy's data are mostly taken from and in a few cases from (nos.23-24, 27, 32, 44-45).Nos.31, 49-51 are omitted since they are the sums of the other values.

Figure 5 .
Figure 5. Differences between the coastline lengths in and in as expressed in terms of percentage of the values (for the same coastal stretches as shown on Fig. 2).

Figure 6 .
Figure 6.Differences between the coastline lengths in and in as expressed in terms of percentage of the values (for the same coastal stretches as shown on Fig. 3).

Figure 7 .
Figure 7. Pliny's points of the coast of Italy on Ptolemy's map (recension).The percentages refer to the deviation of Ptolemy's values for the coastline length from those given by Pliny (no.29 in Table1) and Agrippa (nos. 1 and 9).

Figure 8 .
Figure 8. Pliny's points of the coast of Italy on the modern map.

Figure A1 .
Figure A1.The outlines of Pliny's "third gulf" of Europe on Ptolemy's map ( recension) without the "minor gulfs" (shown with yellow).

Table 2 .
Comparison of the distances along the coast of Italy according to Pliny, Ptolemy, and measurements using Google Maps(Google  Inc., 2017).All distances are expressed in terms of Roman miles; 1 m.p. = 1480 m.The accuracy of this comparison should not be overestimated, because the routes on Google Maps have been measured rather arbitrarily.The lower and upper values in the table refer to the shortest route *

Table 3 .
Length of the Red Sea coast according to Artemidorus, Eratosthenes, Ptolemy, and measurements using Google Maps(Google Inc.,  2017).All distances are expressed in stades; 1 stade = 185 m.