The western coast of Africa in Ptolemy ’ s Geography and the location of his prime meridian

A controversial question concerning ancient geography is the location of the prime meridian which underlies the position data in Ptolemy’s Geography and runs through the Fortunate Islands. An answer to this question is derived by means of a localisation of the places given by Ptolemy at the African western coast, i.e. in Mauritania Tingitana and Libya Interior, whose modern identifications are often uncertain or unknown. The origination of Ptolemy’s positions from the distance data of seafarings is considered. A comparison of his data with distances reported by Pliny yields a satisfactory agreement. The localisation of Ptolemy’s places is based on distances derived from Ptolemy’s coordinates and partly on further information by ancient authors. Through it previous identifications are confirmed and new identifications are found. It shows that the Fortunate Islands correspond to several eastern islands of the Canary Islands. Explanations are given for the errors of Ptolemy’s position data. A likely error by Ptolemy barely considered is his repetition of a part of Mauritania Tingitana in his description of Libya Interior. The existence of this repetition is confirmed by an adjustment of a transformation between the positions of the duplicated places and a statistical test of the arranged model. A comparison of longitudinal distances in different ancient sources reveals that the position of Ptolemy’s prime meridian is based on distances given by Marinos and Eratosthenes.


Introduction
The search for the origin of the concept of geographic coordinates (longitude and latitude) for the specification of a position on the earth's surface leads to , whose works, however, are lost.His geographical knowledge is mainly handed down by Strabo's (ca.63 BC-AD 23) Geography (G; see e.g.Jones, 1917Jones, -1932;;Aujac et al., 1969Aujac et al., -2015)), where longitudinal and latitudinal distances originating from Eratosthenes are reported (see Roller, 2010;Marx, 2015).applied spherical coordinates in astronomy and it seems likely that he used them also in his geographical works (cf.Dicks, 1960, 148-149).The first known discussion of the preparation of a graticule by means of meridians and parallels is given by Strabo (Berggren and Jones, 2000, p. 32).Probably the first comprehensive use of geographic coordinates is to be found in Ptolemy's (Klaudios Ptolemaios, ca.AD 100-170) Geography (Geographike Hyphegesis, GH).In a catalogue of locations in books 2-7 of the Geography, the positions of several thousand places are given by means of longitude and latitude stated in degrees.
The counting of the latitude is physically connected with the Equator, whereas the specification of the longitude necessitates the definition of a reference (prime) meridian.Eratosthenes used an assumed meridian through Alexandria, Rhodos, Byzantion and other places wrongly assigned to it (G 1.4.1, 2.5.7).A precursor of this meridian is to be found in the map created by Dicaearchus (ca.326-296 BC), which passed approximately through Rhodos (Aujac et al., 1987).Eratosthenes' prime meridian was adopted by Hipparchus (cf. G 1.4.1).Ptolemy also uses a prime meridian running through Alexandria in his Mathematike Syntaxis (MS; see Toomer (1984), e.g.MS 4.6) and in GH Book 8, but owing to his advanced geographical knowledge he does not locate the places at this meridian, which were wrongly assigned to it by Eratosthenes.For the catalogue of locations, Ptolemy applies a different prime meridian west of the African Atlantic coast which runs through the Insulae Fortunatae (Fortunate Published by Copernicus Publications. C. Marx: The western coast of Africa in Ptolemy's Geography and the location of his prime meridian Islands).These islands are usually equated with the Canary Islands (cf.Keyser, 1993;Stückelberger and Graßhoff, 2006, p. 14, fn. 38); yet another frequent identification is the archipelago of Cape Verde (e.g.Spaul, 1958, p. 8.10;Lacroix, 1998, p. 202).
An objective of the present contribution is to give an answer to the question of the location of Ptolemy's prime meridian.To this end, the Fortunate Islands are localised based on the places given by Ptolemy at the African western coast, which is described in connection with Africa in GH Book 4. This concerns the western coast of Mauritania Tingitana in GH 4.1 and of Libya Interior in GH 4.6.The modern counterparts of the majority of Ptolemy's places at the African western coast are not known for certain.For example, the Barrington Atlas (Talbert, 2000) only covers the northern part of Morocco and localises a few of Ptolemy's coastal places only.In the edition of the Geography by Stückelberger and Graßhoff (2006, referred to as S & G), modern identifications are not specified in many cases.Differing localisations are to be found in several treatises, e.g.Gosselin (1798Gosselin ( -1813, 120-64), 120-64), Mannert (1825, 467-632), Forbiger (1844, 862-92), Spaul (1958) and Lacroix (1998).
In the recent past, the places of GH books 2 and 3 have been identified, supported by the geodetic-statistical analysis method by Marx (2012); see e.g.Marx and Kleineberg (2012).This approach, however, is not suitable for the coastal regions of Mauritania Tingitana and Libya Interior, since the longitudinal differences of places are erroneous and the density of the locations is too low for the most part.Spaul (1958) locates Ptolemy's places of Mauritania Tingitana by means of distances derived from Ptolemy's coordinates.This approach is improved and applied in the present contribution.It is based on the coordinate values of the recension ( ) and the recension, represented by the manuscript Codex Vaticanus Graecus 191 (X), according to S & G.In addition to maps, the localisation was supported by internet-based mapping applications such as Google Earth (Google Inc., 2015).Gosselin (1798Gosselin ( -1813) ) assumes that places of the African western coast are repeatedly given by Ptolemy.This hypothesis is supported by e.g.Mauny (1978) but is not taken into account by the works mentioned above, and is reconsidered in the following.Pliny's Natural History (NH; see Bostock and Riley, 1855;Winkler and König, 1993;Brodersen, 1996) and the Periplus of Hanno (e.g.Lendering, 1998Lendering, -2014) ) are consulted as further ancient sources of information.
Initially, Ptolemy's coordinates and their origin are considered in Sect. 2. In Sect.3, the applied calculational methods for the localisation of places are described.In Sect.3.2, possible errors of the Ptolemaic distances are pointed out and the data are compared with distances given by Pliny.In Sect.4, the places of the African western coast are localised.The identity of places in GH 4.1 and GH 4.6 is investigated by means of a statistical test in Sect.4.4.In Sect.5, the errors of Ptolemy's data are analysed.In Sect.6, the Fortunate Islands are localised.In Sect.7, the origin of the position of Ptolemy's prime meridian is investigated.In addition, the findings reveal Ptolemy's determination of the longitude of Byzantion, which is given in Appendix D.
Appendix E lists the abbreviations used.Place names are based on S & G and Bostock and Riley (1855).Excerpts from the ancient sources are taken from the translations indicated.

Ptolemy's data
In the following, Ptolemy's locations dealt with by the present investigation are considered (Sect.2.1) as well as the origination of their position data (Sect.2.2).The places of GH 4.1 Mauritania Tingitana and GH 4.6 Libya Interior with given coordinates have been numbered consecutively according to their appearance in the manuscripts (see Tables 1-3).If necessary, the place numbers of GH 4.1 and 4.6 are distinguished by the prefixes "M" and "L", respectively.

Places
Figures 1 and 2 show the places of GH 4.1 Mauritania Tingitana and of the western part of GH 4.6 Libya Interior based on the Ptolemaic coordinates of .The Ptolemaic longitudes and latitudes of the considered places are given in Tables 1-3.Only the coordinate variant which showed to agree better with the actual position is specified; further values are given in Sect. 4.
Mauritania Tingitana was a Roman province which corresponds to the northern part of Morocco.Pliny mentions Volubilis (at Moulay Idriss) in the inland and Sala (Chellah) at the western coast as the southernmost places in his description of Mauritania Tingitana (NH 5.1, division according to Bostock and Riley, 1855).These places are also given by Ptolemy (nos.57 and 7).He describes the western side of Mauritania Tingitana in GH 4. 1.1-4 (nos. 1-24).( 59) Tokolosida (Bled Takourart, S & G) in the inland near the coast was the southernmost place of the Roman territory (Forbiger, 1844, p. 878, fn. 14).Ptolemy's description of the coast, however, contains several places which are situated further south.The southernmost (24) Bigger Atlas Mountain at = 26 • 30 is situated about 7 • more southerly than Volubilis and Sala.Ptolemy gives the south-eastern border point of Mauritania Tingitana at = 26 • (GH 4.1.8).Consequently, Ptolemy's Mauritania Tingitana ranges much further south than the Roman province (also Mannert, 1825, p. 473).
Ptolemy describes the coast of Libya Interior in GH 4. 6.5-7 (nos. 3-25).He states (4.6.4) that in the north its western side reaches up to the border point of Mauritania Tingitana (at = 26 • ; see above), and this is in agreement with his latitudes in Libya Interior, which are lower than those of Mauritania Tingitana (see Figs. 1 and 2).The southernmost location is the (2) Hesperian Gulf at = 4 • .

Origination of coordinates
(M1) Cape Kotes/Cap Spartel and the (M3) Lix/Uad Lucus, the northernmost places at the western coast, have almost correct latitudes.They may be based on direct measurements at the place or in its vicinity (gnomon measurement).One kind of Ptolemy's data sources may have been itineraries, but this is only likely for the region of the Roman province Mauritania Tingitana, that is, for the places up to (M7) Sala.The majority of Ptolemy's positions at the coast is surely based on the information of the descriptions of sea routes along the coast.Known seafarings are the circumnavigation of Africa by Phoenicians under orders from Necho II (ca.600 BC) and the voyages of Hanno (5th century BC), Polybios and Eudoxos of Kyzikos (2nd century BC) along the African western coast (e.g.Mauny, 1978;Keyser, 1993).
Travel reports contained journey times (see e.g.Hanno's Periplus), which were converted into distances by means of assumptions about the speed.Either such estimated distances were available to Ptolemy or he himself converted journey times.Distance data may have been corrected for assumed errors.Ptolemy used to shorten distances in order to take into account bends of routes and anomalies of journeys, and in his examples a reduction of 1 3 is applied for each of these corrections (GH 1.2.4,1.13).Differences of coordinates were derived based on Ptolemy's setting (GH 1.7.1;st = stade) and further information or assumptions about the direction of the route.Berggren and Jones (2000, 16-17) give an overview of Ptolemy's procedures of the determination of coordinates.
According to GH 1.9.4 and 1.18.10,distances were specified as the number of day-and-night seafarings (DN) and day distances (D) (cf.S & G, p. 105, fn.117), and it applies 1 DN = 2 D. (2) Concerning the speed, Ptolemy says that one must trust the feasible daily shipping performance (GH 1.9.6).Forbiger (1842, p. 550) and Kroll (1921) give overviews of the specifications of ancient authors about the speed of seafarings.Ptolemy, referring to other sources, mentions the speed (GH 1.9.4) which also results from G 10.4.5 and which is a serviceable average value (similarly Forbiger, 1842, p. 551  of the stade underlying ancient data is often unknown, which also applies to Ptolemy's coordinates.Therefore two measures of the stade are applied in the following, by which the major range of the measures is covered.These are the 1 8 Rmi or Italian stade (Ist) according to G 7.7.4 and the Egyptian stade (Est) according to Hultsch (1882, p. 61): (Rmi = Roman mile, 1 Rmi = 1.4815 km).The former was probably used by Polybios (Pothecary, 1995), who may have been a source for Ptolemy's African coast, and both stades have been ascribed to Eratosthenes (see e.g.Pothecary, 1995).From Eq. ( 4) it follows that 1 DN = 185/158 km, 1 D = 93/79 km, 1 2 D = 46/39 km (Ist/Est).

Method of localisation
The unknown position of a place at the coast is determined uniquely by the distance from a known place and an indication of the direction along the coast.This information can be derived from the Ptolemaic coordinates so that they can be used for the localisation of the Ptolemaic places.In the following, the reconstruction of the ancient distances (Sect.3.1) and the accuracy of reconstructed distances (Sect.3.2) are considered.

Determination of distances
Seafarings along the African western coast were influenced by ocean currents and winds.The present current north of 10 • N is the Canary Current flowing in the southwesterly direction (see e.g.Gyory et al., 2013).Ancient ships were driven by oar and sail; the speed was mainly influenced by the direction of the wind (Casson, 1971, 270-82).The predominant wind at the northern African western coast is the northeastern trade wind.Its influence in this region can reach up to 35 • N (cf.Lockwood, 2005, p. 129, Fig. A79).A consideration of the current and the wind and of the speed of ancient journeys (Appendix A) shows that in ancient times the speed of a southward journey along the African western coast may have been about and the speed of a northward voyage For the localisation of Ptolemy's places, the distances between places are required.Assuming that the distance be-tween two Ptolemaic positions ( i , i ) and ( j , j ) is based on a journey time, the reconstruction of this distance should take into account the underlying ancient conversion.The procedure applied is the stepwise conversion (symbol x −→: step x of conversion).Since Ptolemy's procedure of the determination of geographic coordinates is unknown in individual cases, in step 1 the spherical distance (Eq.B1) is calculated between the two positions, which can be regarded as a sound approximation.In step 2, Eq. ( 1) is used.In step 3, the estimated ancient journey time is t = ŝ/v 3 .v 3 is the speed underlying the ancient conversion of a journey time into a distance.According to Sect.3.2.2, the common ancient speed v ∅ (Eq. 3) is an appropriate value, which is used.In step 4, the estimated actual distance is A special case of Eq. ( 9) is the equality of the ancient assumed speed v 3 and the actual ancient speed v 4 , e.g.v 3 = v 4 = v ∅ .Then, the conversion corresponds to the direct conversion i , i , j , j of Ptolemy's distance into a metric distance.Conversion Eq. ( 10) is applied to distances outside the area of the Canary Current and the trade wind and to inland distances.
If further coastal places are given by Ptolemy between the two coastal sites i and j being considered, the sum of the single distances ŝ of the intermediate places has to be determined (order of the indices = order of the places).The assumptions underlying the conversions according to Eqs. ( 9) and (10) may be wrong.Therefore, a correction factor a is applied to distances ŝ if they reveal an obvious distortion.Then, the calculated distance becomes Factor a is determined from the data.s i,j or s j i can be compared with the actual distance s.In the case of two inland places, the spherical distance is determined; in the case of coastal places, the path length along the coast is measured.The determination of actual distances was carried out by means of the Google Earth application (Google Inc., 2015).

Accuracy of Ptolemy's distances
The distances derived from Ptolemy's coordinates are possibly adulterated by diverse errors, such as rounding errors of the distances and coordinates and unsuitable conversions of journey times into distances.In the following, possible errors are considered (Sect.3.2.1),and Ptolemy's distances are compared with distances given by Pliny in order to gain an insight into the reliability of the data (Sect.3.2.2).

Errors of distances
The Ptolemaic coordinates and are rounded values.A few latitudes in GH 4.1 have a fraction of degree of 11  12 so that Ptolemy used a resolution of 1 12 • in these cases.The majority of the coordinate values surely has a rougher resolution (cf.Marx, 2011).An investigation of the propagation of the rounding errors of coordinates (Appendix B) shows that generally they may have caused distance errors up to 10 km (GH 4.1) or 30 km (GH 4.6), respectively.
Several large, partly systematic errors of the Ptolemaic distances (see Sect. 5) are not explicable by rounding errors.The following types of adulterations of distances may have occurred.
1. Distances were altered through a rough specification of journey times or the use of the measurement units D and DN, respectively (Sect.2).If the most precise reso- 2. The speed of a sea voyage was influenced by currents and winds (Sect.3.1), which was not taken into account in the conversion of a journey time into a distance.(a) If the assumed speed of a northward voyage was too large, the distance became too large.(b) If the assumed speed of a southward voyage was too small, the distance became too small.If a journey had the length s and if the actual speeds were v ns (Eq.7) and v sn (Eq.8), respectively, and provided that v ∅ (Eq. 3) was used for the conversion of the journey time into a distance s , then it follows that s = v ∅ /v ns s = 0.77 s and s = v ∅ /v sn s = 2.5 s.Consequently, the distances are significantly underestimated and overestimated.
3. The measurement units D and DN were confused because, for example, the distance was noted improperly as an x-day journey.(a) If a distance based on x D was mistaken for x DN, the distance became too large.(b) If a distance based on x DN was mistaken for x D, it became too small.

Comparison with Pliny's distances
Pliny gives several distances between locations at the African western coast, see  1993;Brodersen, 1996), which can be compared with Ptolemy's distances.Müller (1902) and Spaul (1958, p. 6.2) make comparisons for the distances of NH 6.37 based on st and for the distances of NH 5.1 based on Rmi, respectively, revealing more or less significant differences from Ptolemy's distances.Müller (1902), however, remarks that the distances are rough estimates based on journey times.This is taken into account in the following comparison.Table 4 provides the sources of Pliny's information, Pliny's distances s N in Rmi and st (1 Rmi = 8 st) and the assumed original values in DN.The distances are grouped according to the assumed conversion between the units DN and st.
In group 1, the distances are multiples of 50 Rmi or of 400 st, so that they may be based on 1 DN = 800 st.The resulting distances are multiples of 1 2 DN and may be the original values.The distances of group 2 probably originate with Agrippa (Marcus Vipsanius Agrippa, ca. 64-12 BC;cf. translation Winkler and König, 1993).With one exception, these distances are multiples of 56 Rmi (Klotz, 1931) or of 448 st.Thus, the ancient conversion may be based on 1 DN = 900 st ≈ 112 Rmi.The distances of group 4 are multiples of 125 Rmi or 1000 st; they are probably based on 1 DN = 1000 st.
Pliny's and Ptolemy's distances can be used for an estimation of the relation between the units st and DN that underlies Ptolemy's distances.To do so, for each Ptolemaic distance ŝ (in st) of Table 4 the observation equation ŝi + v i = c s Ni is set up, where i = 1 . . .11 is the index of the distance, s Ni is Pliny's distance expressed in DN, c is an unknown factor in st DN −1 , and v i is a presumably random residual.c is determined by a least squares adjustment (see e.g.Böck, 1961;Baumann, 1993, 17-20); the result is 1008 st DN −1 ± 21 st DN −1 .Hence, the usual ancient relation 1 DN = 1000 st (Eq.4) can be assumed for Ptolemy's distances.Table 4 shows ŝ expressed in DN.It reveals that Pliny's and Ptolemy's distances are identical in most cases.A few distances are reviewed in the following.
Ptolemy's distance Fut-Atlas probably refers to the Anti-Atlas; see Sect.4.3; Pliny, however, may refer to the High Atlas, so that his distance is shorter.
Pliny's distance Lixos-Anatis cannot be derived from a journey time as the other distances.Spaul (1958, p. 6.2) assumes that 205 = CCV is a corruption of 250 = CCL.This is possible because the sum of Plinus' three distances Lixos-Sububus-Sala/Salat-Asana (= Anatis) also amounts to 250 Rmi (the River Salat is near the town of Sala).
The distance Hesperian Promontory-Theon Ochema is given directly by Pliny; it is also to be found in Peripl.16. (Pliny gives, referring to Agrippa, a differing value of 10 DN in NH 5.1.Since, however, Agrippa places the Atlas in the middle of this distance, the information seems to be unreliable and is not considered here.)Ptolemy's distance is calculated over the waypoints nos.L23-L25 and L29 and equals nearly exactly Pliny's distance.
Consequently, the investigation shows that the ancient distance data are consistent.

Ptolemy's places at the African western coast
In the following, the places of GH 4.1 Mauritania Tingitana (Sects.4.1-4.3)and GH 4.6 Libya Interior (Sects.4.5-4.7)at the African western coast as well as further places are localised.In Sect.4.4 the repetition of places of GH 4.1 in GH 4.6 is investigated.
Tables 1-3 list the modern counterparts of the Ptolemaic places considered and their geographic longitude λ (relative to Greenwich) and latitude φ; Fig. 3 shows the northern positions (no.M5 is omitted).Tables 5 and 6 give the localisations based on distances including these distances.Due to uncertainties and missing evidence, not all places can be localised.
The sequence of the Ptolemaic places in the following sections corresponds to that of their identification.Their modern counterparts are indicated by bold text.Unless otherwise stated, Pliny's information originates from NH 5.1.

Mauritania Tingitana part 1: up to the Asana
Firstly, the correction Eq. ( 12) is not used for distances ŝ.The (9) Smaller Atlas is not included in the sum Eq. ( 11) of distances; cf.Sect.3.2.2 and no. 9 below.
of X is in better agreement with of ( 7) Sala and of (51) Tamusida than of (6 • 10 ).
(5) Emporikos kolpos: Ptolemy places this bay between the (4) Subur and the (6) Sala, i.e. between the Oued Sebou and the Oued Bou Regreg.It has been assumed that it corresponds to the bay of Sagigi mentioned by Pliny (e.g.Spaul, 1958, p. 6.3;S &G , p. 383, fn. 9).According to Pliny (translation Winkler and König, 1993), the bay of Sagigi is situated between the (3) Lixos (Lix/Uad Lucus) and Mulelacha, which is further north than the Sububa (Subur) and is assumed to be Moulay Bou Selham (e.g.Winkler and König, 1993, p. 120).Thus, there is a distance of at least 80 km between the two bays so that they do not seem to be identical.Alternatively, Ptolemy's or Pliny's information may either be erroneous or both refer to a bay which reaches approximately from the Uad Lucus (Lix) to the Oued Bou Regreg (Sala).If Ptolemy refers to a coastal point, of X is to be preferred in agreement with no.6 ( : 6 • 10 ).
( From the ratio of ŝ12 7 and the corresponding s follows the correction factor which is the average of the results from Ist and Est. a 1 is applied in Eq. ( 12) up to no. 12. (8) River Duos: Spaul (1958, p. 6.11): Oued Cherrat; S & G: Oued Mellah.s 7,8 (a 1 ) of (no.8 = 33 • 20 ) is met by the Oued Mellah and the Oued Nefifikh (s .= 60 km), but then there would be no large river which satisfies the position of the (10) Kusa between the (8) Duos and the (12) Asana.
Therefore, the Oued Cherrat is chosen, which is in accordance with s 7,8 (a 1 ) of X.
(10) River Kusa: Spaul (1958, p. 6.11)  of Paina equals that of the (12) Asana/Oum Er-Rbia, and the island is positioned at a large distance to the coast.This is met by the Madeira archipelago.Ptolemy, however, often positions islands much too far from the coast (see e.g.England and Italy in Kleineberg et al., 2012;Marx and Kleineberg, 2012).Owing to its , Paina is probably situated near the (12) Asana/Oum Er-Rbia.There exists the island of Sidi Abderrahmane, which is only 10 more northern than the river mouth.

Mauritania Tingitana part 2: up to the Cape of Herakles
The sum ŝ24 12 from the Asana/Oued Oum er-Rbia to the Bigger Atlas amounts to 866/737 km (Ist/Est).The Bigger Atlas must correspond to the foothills of the High Atlas or of the Anti-Atlas near the coast.The resulting actual distances are significantly smaller than ŝ24 12 .Since the Anti-Atlas yields a smaller difference, it is preferred to the High Atlas.(Gosselin (1798-1813), Spaul (1958) and Lacroix (1998) also assume that the Bigger Atlas is further south than the High Atlas.)The foothills of the Anti-Atlas (at 29 (16) River Phthuth: Gosselin (1798Gosselin ( -1813, p. 125), p. 125), Mannert (1825, p. 476), Forbiger (1844, p. 869), Spaul (1958, p. 6.16): Oued Tensift.The Oued Tensift is situated in accordance with the lower values of s 16 14 (a 2 ), which is followed by only a few short rivers.
of fits better than of X (30 • 15 ).
Mountain Phokra: the Phokra reaches from the (9) Smaller Atlas/Moroccan Meseta west of the Middle Atlas to (19) Cape Usadion/Cap Sim (GH 4.1.12).In this region the Moroccan Meseta is situated at some distance from the coast so that the Phokra corresponds to the southern part of the Moroccan Meseta.

Identical places
If Ptolemy used different data sources for GH 4.1 Mauritania Tingitana and GH 4.6 Libya Interior, which contained different place names and distances, he might have not noticed identical places.Gosselin (1798Gosselin ( -1813, p. 129, p. 129) assumes that the following four coastal places of GH 4.1 also appear in GH 4.6 with similar names and in the same order: The first two pairs are inland towns.Nos.M14 and L11 are both equated with the promontory Soloeis of Peripl.3 (e.g.Forbiger, 1844, p. 867;Gosselin, 1798Gosselin, -1813, p. 130), p. 130).Consequently, identity is assumed for seven pairs of places here.This hypothesis is extended at the end of Sect.4.5.The similar positions of the presumably duplicated places are illustrated in Fig. 4 by means of a transformation of coordinates (see below), which positions the places of GH 4.6 in the vicinity of the places of GH 4.1.
A repetition of the places of GH 4.1 in GH 4.6 leads to the following assumptions.The positions in GH 4.6 are arranged south of the (M24) Bigger Atlas so that in GH 4.6 the coordinates are shifted.Additionally, differences in the scale may exist between GH 4.1 and GH 4.6, for example, because their data sources are based on different determinations and conversions of journey times and because Ptolemy scaled distances.Furthermore, the directions of the identical stretches of the coast may differ so that a small rotation is present.The described systematic differences of coordinates between GH 4.1 and GH 4.6 are modelled by means of a two-dimensional transformation of coordinates.Its parameters are estimated by means of a least squares adjustment.It is assumed that the size of the remaining differences after the transformation is explicable by the uncertainty of the coordinates.This hypothesis is tested by means of a statistical test (Appendix C).As a result, the hypothesis is accepted, and consequently the identity of the places considered can be assumed.Figure 4 shows remaining deviations between the positions of GH 4.1 and the transformed positions of GH 4.6.
(27) Sagapola Mountains, middle: the (3) Subos/Oued Sebou has its source in these mountains (GH 4.6.8); the source of the Oued Sebou is in the Middle Atlas.
(94) Island of Kerne: Spaul (1958, p. 8.10): one of the Canary Islands.According to Peripl.8, 10, Kerne has a circumference of 5 st ≈ 1 km and the journey time from the Pillars of Herakles (at the Strait of Gibraltar) to Kerne corresponds to that from Karchedon (Carthage) to the Pillars.It is often identified as Herne in the Bahia de Rio de Oro (e.g.Bunbury, 1879, p. 324, following Carl Müller), which roughly meets the information (the length of the sea route around Herne is ca.2.5 km).Pliny states (NH 6.36) that according to Polybios Kerne is situated 8 st from the coast opposite the mountain Atlas.The former is in accordance with Herne, whereas the latter is not.Ptolemy makes the same mistake by positioning Kerne and the (M24) Bigger Atlas at similar latitudes (25 • 40 and 26 • 30 ).
(48) Autolalai: it is the northernmost town at the coast of Libya Interior south of Ptolemy's Mauritania Tingitana, which can be related to the Autololes according to its name.Ptolemy seems to follow the information that the Autololes are situated south of Mauritania Tingitana (cf.NH 5.1).
The adjustment of a transformation of coordinates in Appendix C revealed a scaling factor of about 2 between the latitudes of GH 4.1 and 4.6 from the (3) Subos to the (11) Saluentia Capes.Since Ptolemy's coast runs almost southward, the factor 1 2 can be applied to the distances of GH 4.6 as a correction.Additionally, the distortion of the distances of GH 4.1 described by a 1 and a 2 (Eqs.13, 14) has to be considered.By means of their average of 0.621, the correction factor for Eq. ( 12) becomes a 4 = 0.5 • 0.621 = 0.311. (16) (7) Gannaria Capes: according to the name (Greek akra), Ptolemy refers to more than one cape, which is usually not taken into account (see e.g.Forbiger, 1844, p. 880;S & G).The most distinctive cape near s 6,7 (a 4 ) is the cape at Dar Bouazza.South and north of it, there are a small peninsula and a small headland, which may have been referred to.Alternatively, the cape at Casablanca, 20 km northeast of Dar Bouazza, is included additionally.
(8) River Ophiodes: the first large river with respect to s 7,8 (a 4 ) is the Oued Oum er-Rbia with acceptable s.
(11) Saluentia Capes: the name suggests more than one cape.One of them corresponds to the (M14) Helios Mountain/Cap Cantin (see Sects.4.2, 4.4, no.M14).Three kilometres south-south-east of it, there is a small headland with a height of 50 m, which is possibly referred to.Also, Cap de Safi 20 km south of Cap Cantin comes into consideration.
The localisation of the places reveals that there are two further places in GH 4.1 and GH 4.6, which refer to the same location or region: The Diur and the Nuios refer to the Sidi Moussa and Oualidia lagoon, which constitute a chain of lagoons (Hughes et al., 1992, p. 66).The inclusion of these places in the statistical hypothesis test of the identity of places results in an acceptance of this hypothesis (Appendix C).Owing to the large distance between nos.11 and 12, these places may originate from different data sources.Therefore, the correction factor a 4 (Eq.16) applied to the northern distances is not used in Eq. ( 12) for the following southern distances.
(  14) Daras/Oued Draa has its source in these mountains (GH 4.6.9).The confluences of the Draa rise in the middle part of the High Atlas; before it reaches lower areas, the Draa flows through a valley between the Anti-Atlas and Jbel Saghro.A part of these three mountains may correspond to the Kapha Mountains and may have been regarded as the location of the source of the Daras.Ptolemy's position far to the south is explicable by the southeasterly direction of the Draa from the coast, which may have been assumed for the entire course of the river. of X fits better than that of (10 • ).
(96-101) Fortunate Islands: Ptolemy's first and northernmost island (96) Aprositos has a somewhat smaller than the (12) Massa/Oued Massa.This is met by the Canary Islands.Fuerteventura is visible from the mainland (Keyser, 1993) so that this island must have been known.On the individual islands, see Sect. 6.
(17) Cape Arsinarion: since this cape is situated in the latitudes of the Fortunate Islands/Canary Islands, it is Cap Juby.
(15) Large Port: the port may have been at a river mouth; in the vicinity of ŝ14,15 , however, there are only very short rivers.The Khnifiss Lagoon (Sebkha Tazra) is situated in compliance with ŝ14,15 , which may have been suitable for a port.Accordingly, Arlett (1836, p. 298)  (18) Cape Rusaddir: according to Ptolemy, this cape is situated west of the (28) Rhussadion Mountains/highland southwest of Jebel Ouarkziz.This is in agreement with Ras Afkir Oum M'Bark.ŝ17,18 , however, does not support this identification; ŝ of fits better than that of X (X: no.17 = 13 • ).
(19) River Stacheir: according to X, the river mouth is east of (18) Cape Rusaddir/Ras Afkir Oum M'Bark, which is not given by the direction of the coast.According to , the river is further south than no.18 so that it may be the Wad As Saguia al Hamra.It is the largest and longest watercourse in Western Sahara (Hughes et al., 1992, p. 90).This identification, however, is not substantiated by ŝ18,19 .

Libya Interior part 3: up to the Ochema Theon
(29) Ochema Theon Mountain: according to Peripl.16, 17, flames and torrents of fire were observed on this mountain so that it is probably a volcano.Burton (1862) identifies it as Mount Cameroon, probably for the first time, which was the only volcano then active in the area under investigation (cf.Hennig, 1944, 86-95).Forbiger (1844, p. 880, fn. 18) assumes that the appearance of fire was not caused by a volcano but by people, which, however, does not explain the mentioned torrents of fire.Mount Cameroon at 4 • 13 latitude nearly fits = 5 • of the Ochema Theon.The source of the (24) Masitholos/Niger (see no. 24 below) is situated at the Ochema Theon (GH 4.6.9).The Niger has a tributary, the Benue, which has its sources in the Western High Plateau north-east of Mount Cameroon.Hence, Ptolemy refers to Mount Cameroon and the Western High Plateau, which explains why he does not place the Ochema Theon directly at the coast.
Ptolemy's distance from the (19) Stacheir/Wad As Saguia al Hamra to the (29) Ochema Theon/Mount Cameroon over nos.19-24 and 29 is a few thousands kilometres too short.Hence, some or all Ptolemaic distances south of no.19 are strongly shortened.The southern regions were surely less navigated so that only little and fragmentary information was available.A main reason for the shortening, however, is Ptolemy's repetition of a part of Mauritania Tingitana in GH 4.6, which diminished the available space.Additionally, the coast is shortened through its wrong direction from northwest to south-east.This direction corresponds to the ancient idea of the western side of Africa (e.g. G 2.5.15, Mela 1.20;cf. Berger, 1903, p. 400;Romer, 1998, p. 39, fn. 19), which Ptolemy probably makes use of.Consequently, the following localisations are mostly not based on distances.
Like the Nias, the Chretes in Peripl.9 and the Bambotus in NH 5.1 are also equated with the Senegal (e.g. S & G, p. 447, fn. 177;Winkler and König, 1993, p. 352).The equating of the Chretes with the Senegal is questionable.The text of Peripl.9,10 reads: "Sailing from there [Kerne/Herne], we crossed a river called Chretes, and reached a bay, which contained three islands [. . .] After a day's sail from here, we arrived at the end of the bay, which was overhung by some very great mountains.[. . .] Leaving from there, we arrived at another large, broad river teeming with crocodiles and hippopotamuses."The passage suggests that the three islands are not more than a few days' journey distant from Kerne.This is only met by the three islands at Cap d'Arguin, which are about 450 km from Kerne, or by Tidra and its surrounding islands.The Chretes is further north and cannot be the Senegal.The mentioned bay is probably the section between Cap d'Arguin ( 16• 32 W/20 • 33 N) and Cap Tafarit (16 • 16 W/20 • 08 N), and the islands are therefore at Cap d'Arguin.The actual distance of 55 km between both capes is in accordance with the given journey time if it is rounded up to 1 D. The Chretes may refer to the Bay of Lévrier ( 16• 54 W/20 • 53 N) and the salt marshes at its shores, which may have been taken for a river mouth.The mentioned mountains possibly correspond to Cap Tafarit, which has an extent of about 1 km and, in contrast to its vicinity at ground level, a height of about 40 m.The mentioned second river may be the Senegal (also Forbiger, 1844, p. 882;Lendering, 1998Lendering, -2014)), where there are crocodiles today (e.g.Hughes et al., 1992, p. 48).
(23) Hesperu Keras ("Horn of the West"): Mannert (1825, p. 525), Forbiger (1844, p. 881), Bunbury (1879, p. 325, fn. 6): Cap-Vert.A place of the same name is to be found in Peripl. 14 and NH 5.1,6.35,36.According to the Periplus, it is a bay, which is identified as the Bight of Benin (e.g.Winkler and König, 1993, p. 121).Later authors, including Pliny and Ptolemy, transformed it into a cape (cf.Bunbury, 1879).According to Peripl.14, 16, an island in the bay is a 4day journey from the (29) Ochema Theon/Mount Cameroon.This journey time is also specified in NH 6.35 for the distance Hesperu Keras-Ochema Theon so that Pliny and probably also Ptolemy actually refer to the bay.The journey time is consistent with Ptolemy's ŝ29 23 see Sect. 3.2.2,Table 4) and corresponds to about 740/630 km.The Bight of Benin is situated at this distance.
(2) Hesperian Gulf: Lacroix (1998, p. 263): Gulf of Guinea.The Hesperian Gulf lies south of the ( 24) Masitholos/Niger Delta.It may be the Gulf of Guinea.The places from the (19) Stacheir/Wad As Saguia al Hamra southwards, however, are also situated at the gulf (GH 4.6.7).This probably follows from a wrong idea of the direction of the coast; cf.Fig. 2. Thus, in the strict sense, Ptolemy's Hesperian Gulf refers to the Atlantic Ocean west and south of Western Africa.

Errors of Ptolemy's data
Based on the localisations of the Ptolemaic places described in Sect.4, the errors of the Ptolemaic distances (Sect.5.1) and coordinates (Sect.5.2) are investigated.

Distances
In order to illustrate the errors of the Ptolemaic distances ŝ, the direct conversion of Eq. ( 10) is used for their derivation from the Ptolemaic coordinates.If identifications are missing between two places, the sum Eq. ( 11) is used; no.M9 is, however, excluded from it.Tables 7 and 8 give the deviation e = ŝ − s (17) between ŝ and the actual distance s along the coast.The strongly distorted distances south of the (L12) Stacheir are not considered; on their errors, see Sect.4.7.
A few ŝ have small deviations e of some kilometres.They are explicable by rounding errors of distances and coordinates, which may have caused distance errors of about 20 km (GH 4.1) or 30 km (GH 4.6); cf.Sect.3.2.1.Several deviations, however, are significantly larger.
The cause of the overly large distance (L11) Saluentia Capes-(L12) Massa may be insufficient information (Sect. 4.6,no. 12).Alternatively, it may have been altered by errors 2b and 3b.The actual s and a southward voyage with v ns (Eq.7) yield a journey time of 1.27/1.49days (Ist/Est).Possibly, this was specified as 1 1 2 DN and confused with 1 1 2 D. The determination of a distance by means of the smaller speed v ∅ results in 750 st, which is in accordance with Ptolemy's ŝ = 692 st.
The following errors of distances are possibly due to an intentional alteration by Ptolemy.The distance (L3) Subos-(L4) Salathos may have been enlarged by Ptolemy in order to integrate (L95) Hera/Autolala and (L48) Autolalai between the Subos and the Salathos (see Sect. 4.5,no. 48).The oversized distance (L15) Large Port-(L17) Cape Arsinarion is probably caused by the preset longitudinal distance of 60 • 30 between the Fortunate Islands and Alexandria (see Sect. 6).As a result, the distance of these islands from the African coast became too large.Ptolemy possibly shifted Cape Arsinarion to the west in order to compensate for this error and emphasise its position opposite the Fortunate Islands.
The adjustment of a transformation between the coordinates of GH 4.1 and 4.6 in the region from the Oued Sebou to Cap Cantin (Appendix C) reveals that the distances between the places concerned of GH 4.6 are significantly larger than those of GH 4.1.This is explicable by Ptolemy's enlargement of the distance L3-L4 (see above), by the increase in the distances L6-L7, L8-L10-L11 (e.g.error 2a) and by the decrease in the distances M11-M12-M13-M14.

Coordinates
Ptolemy's African western coast runs wrongly almost from north to south (Figs. 1 and 2).Longitudinal distances between places are often grossly erroneous and coastal stretches are wrongly oriented.The reason is surely fragmentary and erroneous information.For example, the Periplus deals with more than 30 places, but only five directions between them are indicated, two of them southward and two eastward.Such insufficient and erroneous information may have easily caused a wrong idea of the coastal direction.There are, however, some coastal stretches with approximately correct directions, for example (M7) Sala-(M8) Duos and (M13) Diur-(M14) Helios Mountain.
Ptolemy's latitudes of the northernmost places and of the southernmost location (L29) Ochema Theon are nearly correct.The northern latitudes in Libya Interior are shifted significantly southwards owing to Ptolemy's arrangement of Libya Interior south of Mauritania Tingitana and his repetition of a part of Mauritania Tingitana in Libya Interior (Sect.4.4).
between the (L3) Subos and (L17) Cape Arsinarion is 13 • ; the actual difference, however, is about 6 • .Thus, the latitudinal extent north of (L17) Cape Arsinarion is too large, which is explicable by the following.First, Ptolemy used 500 st per degree, which is too low and yields too large latitudinal differences (also Mannert, 1825, p. 480).Second, the wrong southward direction of the coast takes coastal places further south.Third, the lengths of some coastal stretches are enlarged.
of (L17) Cape Arsinarion and the (L29) Ochema Theon is 7 • , but the actual latitudinal difference amounts to about 21 • .This shortening is explicable by the southward shift of the places in Libya Interior and by a measured latitude in the region of the Ochema Theon, which constituted a southern limit.

The Fortunate Islands
The consideration of the repetition of a part of GH 4.1 Mauritania Tingitana in GH 4.6 Libya Interior reveals that Ptolemy's Fortunate Islands correspond to some of the Canary Islands (Sect.4.6).This is confirmed by the information of other ancient authors.In the following, this information is considered (Sect.6.1) and the single islands of the Fortunate Islands are identified (Sect.6.2).
According to Pliny's description of the Fortunate Islands (NH 6.37), they are beyond the Purple Islands/Mogador.In his description he refers to Sebosus and Juba; their information on the islands is compiled in Table 9 (based on Brodersen (1996); Bostock and Riley (1855) have some deviations).
Sebosus distinguished between the three islands of Junonia, Pluvialia and Capraria and the two Fortunate Islands of Invallis and Planasia.Junonia is 750 Rmi = 6 DN distant from Gades/Cádiz (Table 4).This distance is probably based on the journey time of a sea route from Cádiz along the African coast to Cap Sim and from there southwestwards to the Canary Islands.Its length is about 1150 km and corresponds to 5/6 DN (Ist/Est; reversion of Eq. 9, steps 5, 4, v 4 = v ns Eq. 7).Pluvialia and Capraria are 750 Rmi from Junonia in a westerly direction; the two Fortunate Islands are 250 Rmi from Pluvialia and Capraria in a southwesterly direction.If these distances refer to western islands of the Canary Islands, then they are much too large.
Juba mentioned six Fortunate Islands, of which five islands probably correspond to Sebosus' islands; see Table 9. Ninguaria received its name from perpetual snow; it may be Sebosus' Invallis (Latin vaulty), whose name is possibly a corruption of nivalis (Latin snowy ;Fischer, 1910 according to J. Partsch and Carl Müller;Brodersen, 1996, p. 248).Canaria may be Sebosus' Planasia (see below).Ombrios is the Greek name of Sebosus' Pluvialia (Müller, 1902).According to Juba, the Fortunate Islands are ". . .at a distance from the Purple Islands of six hundred and twenty-five miles [5 DN], the sailing being made for two hundred and fifty miles [2 DN] due west, and then three hundred and seventy-five [3 DN] towards the east."Müller (1902) points out that the distances correspond to 5000, 2000 and 3000 st and that they are rough estimates based on 1 DN = 1000 st (i.e.Eq. 4).The reported directions are contradictory.A westward seafaring from the Purple Islands/Mogador is diverted to the south due to the Canary Current and the trade winds.Possibly a southwestern journey over the sea to the Canary Islands (Alegranza) is meant, which has a length of about 425 km.This corresponds to 2 DN (calculation as above) in agreement with the distance of Juba's first, westward route.The second sea route of 375 Rmi may be a journey to further islands of the Canaries and along their coasts.Its eastward direction is wrong but explicable by a return journey from western to eastern islands.
According to the distance data of Sebosus and Juba, the Fortunate Islands may be some of the Canary Islands.This is substantiated by Mela, who states: "On the sandy part is Mt.Atlas [. . .] Opposite the sandy part, the Fortunate Isles abound . . ." (3.101, 102).Accordingly, the Fortunate Islands are situated in the latitudes of the Sahara and the Atlas mountains.This is met by the Canary Islands.

Identification of the islands
The identification of the islands is dealt with by, e.g.Gosselin (1798Gosselin ( -1813, 146-159), 146-159), Buch (1819), Müller (1902); Krüss (1976) gives further works in this regard and discusses the etymology of the islands.Five of Ptolemy's six Fortunate Islands probably correspond to Sebosus' and Juba's islands; see Table 9.Four islands have identical or similar names; (101) Centuria (Pintuaria) may be Ninguaria (Fischer (1910) according to a conjecture by Carl Müller and Curt Müller).It was shown in Sect.3.2.2 that Sebosus' distances are consistent with Ptolemy's distances.Ptolemy's wrong arrangement of the islands from north to south is contradictory to Sebosus' directions, but a rotation of Ptolemy's positions brings them nearly in accordance with Sebosus' information.
The Canary Islands consist of seven large and further smaller islands.An assignment of the Fortunate Islands to the large islands leads to contradictions.If Canaria is Gran Canaria, then there are no three large islands in the east which could be the counterparts for the eastern islands Junonia, Pluvialia and Capraria.Buch (1819), for example, identifies Junonia with Fuerteventura and Pluvialia with Lanzarote, which contradicts the position of Pluvialia west of Junonia (Sebosus).Ptolemy and Juba mention only six islands so that (at least) one of the large Canary Islands does not occur.Therefore it is also possible that more than one large island is missing, and the small islands from Graciosa to Alegranza come into consideration.Through their inclusion, a solution can be found which is consistent with Sebosus' and Ptolemy's arrangement.The small islands have a largest extent of 2 to 9 km.Ptolemy lists islands with comparable sizes in other regions (e.g.Planasia/Pianosa in GH 3.1, Erikodes/Alicudi in GH 3.4; see Marx and Kleineberg, 2012, 26, 47).
(101) Centuria (Invallis, Ninguaria): owing to the perpetual snow (Juba), it may be Tenerife with the snowcovered Mount Teide (e.g.Buch, 1819;Müller, 1902).The fog (Juba) may refer to clouds being present around Mount Teide for a large part of the year (Müller, 1902).Sebosus gives a circumference of 300 Rmi.The islands with the largest circumference (sea route) are Tenerife and Lanzarote with about 250 km ≈ 170 Rmi.The circumference was possibly enlarged to 300 Rmi in order to take into account the jointed coast.
(96) Aprositos: this island is the northernmost island and may therefore be Alegranza, which is the northernmost of the Canary Islands.The name Aprositos means inaccessible, which applies to Alegranza; Arlett (1836) reports that there exists only one landing place on the southern side.
(97) Juno (Junonia): according to Sebosus, this island is the nearest to Gades, so that it is further north than Pluvialia/Lanzarote.Ptolemy positions it between Pluvialia/Lanzarote and Aprositos/Alegranza.Both conditions apply to Graciosa (also identified by Müller, 1902).
Smaller Junonia: it was suggested that Juba's smaller Junonia (Juba) corresponds to ( 96) Aprositos (see Fischer, 1910), which is not followed here.Pliny's text suggests that this Junonia is near to Junonia/Graciosa so that it is probably Isla de Montaña Clara.
Planasia: Sebosus' Planasia may be Ptolemy's and Juba's Canaria/Gran Canaria (also suggested by Müller (1902), who assumes a corruption of the name).Sebosus mentioned Fuerteventura (Capraria) and Tenerife (Invallis) so that the interjacent Gran Canaria should have been known to him.Sebosus gives the same relative position for Planasia and for Invallis/Tenerife with respect to Junonia so that Planasia and Invallis must be near to each other; the island nearest to Tenerife in the east is Gran Canaria.The name Planasia means plane (from Latin planus) and is probably chosen in view of the name of the island of Invallis/Tenerife; consistent with this, Gran Canaria is less high than Tenerife (likewise Gosselin, 1798Gosselin, -1813, p. 151), p. 151).
According to Juba, the Fortunate Islands are rich in fruits and birds of every kind, and Mela states: ". . . the Fortunate Isles abound in spontaneously generated plants; and with various ones always producing new fruit in rapid succession, the islands nourish people who want for nothing, whose islands are more blissfully productive than others are" (3.102).Hence, the good conditions for the vegetation growth gave the Fortunate Islands their name.This feature applies to Gran Canaria and Tenerife but not to Fuerteventura and Lanzarote, where the vegetation is sparse (cf.e.g.Chisholm, 1911, 174 pp.).It seems that this was taken into account by some ancient authors, since Sebosus did not assign the islands east of Gran Canaria to the Fortunate Islands.

Ptolemy's prime meridian
In his Mathematike Syntaxis and in GH Book 8, Ptolemy uses a prime meridian which passes trough Alexandria (see Sect. 1).For the catalogue of locations in GH books 2-7, he introduced a different prime meridian.The location of this prime meridian is considered in the following (Sect.7.1), and the origin of its position is investigated (Sect.7.2).

Location of the prime meridian
The geographic coordinate system which underlies Ptolemy's position data in GH Books 2-7 is described in GH 1.19.2.The longitude is counted with respect to a westernmost meridian which constitutes the western end of the known world (1.19.2, 1.22.4, 1.24.9).This western end or prime meridian, respectively, is at the Fortunate Islands (1.11.1,7.5.15,8.27.12).In the catalogue of locations, the Fortunate Islands have = 1 • in (4.6.34);deviating from this, the four islands Aprositos, Pluvialia, Capraria and Centuria (Pintuaria) have = 0 • in X.The value of 1 • is a substitute for 0 • due to the lack of a common sign for the value zero (S & G, p. 455, fn. 202).The longitude of Alexandria is = 60 • 30 (4.5.9).Furthermore, the longitudinal differences between Alexandria and the Fortunate Islands of as well as of 4 h are given (1 h = 15 • ; 7.5.14, 8.15.10, indirectly 8.27.12).The former yields 4 h 02 min; the latter corresponds to 60 • and is a rounded value.
The localisation of Ptolemy's Fortunate Islands (Sect.6) revealed that the westernmost of these islands is Tenerife.Therefore it can be regarded as the location of Ptolemy's prime meridian, which constitutes the western end of the known world.
For the definition of the new prime meridian at the Fortunate Islands, the longitudinal extent of the known world in the west had to be known.This, however, does not mean that Ptolemy had to determine the longitudinal distances of a multitude of places between the meridians of Alexandria and the Fortunate Island, since he adopted only a few primary distances from Marinos.

Origin of the position of the prime meridian
In ancient geography, the longitudinal extent of the known world was specified by means of the arc lengths r on the parallel through Rhodos.A comparison of such or comparable distances given by ancient authors is carried out by e.g.Blair (1784, 118-128) and Berggren and Jones (2000, 153-154).
From the Rhodos parallel it was assumed that it runs through the Pillars of Herakles at the Strait of Gibraltar.The origin of Ptolemy's longitudinal difference F,A (Eq.18) becomes evident by a consideration of the distances r P with respect to the Pillars of Herakles.In addition, arc lengths r M with respect to the Strait of Messina are considered.Table 10 compares some arc lengths r P and r M from the Atlantic to Issos (Dörtyol at the Gulf of Iskenderun, S & G) originating from Eratosthenes (G 1.4.5),Agrippa (NH 6.38), Polybios (NH 6.38),Strabo (G 2.1.40,2.4.3),Marinos (GH 1.12.11) and Ptolemy (GH 2.4.6,2.5.3,3.4.9,4.3.7,5.2.34,5.8.4); partly they are sums of further distances.Polybios' (ca.200-118 BC) and Agrippa's distances are given in Rmi by Pliny; they are converted by 1 Rmi = 8 st into st.Ptolemy assumes 400 st per 1 • for the Rhodos parallel (GH 1.11.2);his and Marinos' longitudinal differences are converted by means of this value.In the case of Ptolemy, Mountain Calpe (Rock of Gibraltar, = 7 • 30 ) is used for the Pillars, Messene

Marinos
In his discussion of the geographical works of Marinos, Ptolemy reports longitudinal distances from the Fortunate Islands to the Euphrat.Ptolemy complies closely with Marinos' distances (Berggren and Jones, 2000); the distances of Table 10

Eratosthenes
Strabo does not state explicitly that Eratosthenes' distances apply to the Rhodos parallel, but this is usually assumed (e.g.Thomson, 1948, 164-165).Eratosthenes' distances do not refer to Alexandria but to the Nile at Kanobos (Abukir, S & G); owing to the closeness of both locations, however, the distances also apply approximately to the Alexandria-Rhodos meridian.Eratosthenes arranged a distance of r P = −3000 st for the bulge of Europe.Furthermore, he added a second distance of 2000 st probably for promontories and islands and in order to obtain a longitudinal extent of the known world which is more than double the latitudinal extent (cf.G 1.4.5).(Eratosthenes applied both distances also to the eastern end of the known world.) For the later geographers Marinos and Ptolemy, the region of the 2000 st was obviously meaningless.First, the longitudinal and latitudinal extent was determined by considerations other than those mentioned by Strabo; see GH 1.7, 1.11.Second, Ptolemy's and probably also Marinos' westernmost place of Europe is the Sacred Cape, which is only = 5 • from the Pillars (Calpe), i.e. r P = −2000 st, so that Eratosthenes' first distance of −3000 st is not reached.This distance, however, was suitable for islands in the Atlantic whose location was not accurately known and which were assumed to be the westernmost ones.Marinos probably used Eratosthenes' distance and located the Fortunate Islands 3000 st ( = 7 • 30 ) from the Pillars.This corresponds to Ptolemy's longitude of Mountain Calpe.Ptolemy's = 53 • between Mountain Calpe and Alexandria yields r P = 21 200 st and is probably based on Eratosthenes' r P = 21 500 st, which corresponds to = 53 • 45 .This value may have been rounded to 53 • , possibly because Eratosthenes' distance refers to the Nile east of Alexandria.Consequently, Ptolemy's F,A or r = 24 200 st, respectively, from the Fortunate Islands at the western end of the known world to Alexandria is based on Eratosthenes' r = 24 500 st from an assumed western end of Europe to Alexandria.

Strabo
Strabo reports further distances from sources not indicated.

Agrippa
Pliny reports Agrippa's distance ". . . in a straight line from the Straits of Gades [Strait of Gibraltar] to the Gulf of Issus [Issos] . . ." (NH 6.38).Since this corresponds to the expected course of the Rhodos parallel, Agrippa's distances can be assumed to refer to it.Schnabel (1935, 418-420) supposes that already Agrippa placed Rhodos further west than Alexandria.According to Schnabel (1935), the assumed meridian through promontory Phycus (Cape Rasat in Libya, Bostock and Riley, 1855, 4.20, n. 20), Taenarum (Tainaron in Laconia, S & G) and promontory Criumetopon (Cape Krios on Crete, S & G) was adopted by Strabo (G 17.3.20,21)from Agrippa's world map, and a comparison of the distances Criumetopon-Rhodos and Phycus-Alexandria should have led Agrippa to the conclusion that Rhodos is further west than Alexandria (strictly speaking, Strabo states that Criumetopon is on the meridian of Apollonia, which is 170 st from Phycus).It is, however, questionable, whether Strabo used Agrippa's world map (cf.Jones, 1917Jones, -1932, 5.2, n. 124;, 5.2, n. 124;Dilke, 1985, 43-44), and it is unknown whether Agrippa applied the Phycus-Taenarum-Criumetopon meridian and which distances were available to him.Strabo would surely have mentioned such an important change of Eratosthenens' prime meridian through Alexandria and Rhodos, which is, however, not the case.
Pliny gives further longitudinal distances from Agrippa concerning the African northern coast; the two westernmost ones reach from the western end of Mauritania Tin-gitana to the Small Syrtis (Gulf of Gabès) and amount to 1038 and 580 Rmi (NH 5.1, 5.3;Klotz, 1931, frr. 36, 35).The sum is 12 944 st and corresponds approximately to Ptolemy's r = 13 133 st from Cape Kotes to Takape (Gabès, S & G) at the Small Syrtis.Hence, Ptolemy's distance may originate with Agrippa.
Agrippa's r A M = 10 800 st between the Strait of Messina and Alexandria is 2700 st smaller than Eratosthenes' value of 13 500 st and 1800 st larger than the common 9000 st to Rhodos reported by Strabo (see Sect. 7.2.3).Assuming that both distances were considered by Agrippa, the following cases can be distinguished.First, Agrippa used Eratosthenes' Alexandria-Rhodos meridian but located the Strait of Messina further east than Karchedon.This is less likely because then his r A M should be ≈ 9000 st.Second, Agrippa adopted Eratosthenes' Karchedon-Messene meridian and positioned Rhodos further west than Alexandria so that 9000 st < r A M ≤ 13 500 st.Third, Agrippa did not apply the two meridians of Eratosthenes and located the Strait of Messina and Rhodos as described in the first and second cases, respectively, so that 9000 st < r A M < 13 500 st.Fourth, Agrippa adopted the two meridians of Eratosthenes, but due to the contradiction between Eratosthenes' r M to Alexandria and the common 9000 st to Rhodos, he chose an intermediate distance.The conditions resulting from the second to fourth cases are fulfilled.According to the second and third cases, Agrippa may have located Rhodos further west than Alexandria.

Summary and conclusion
An answer to the disputed question of the location of Ptolemy's prime meridian at the Fortunate Islands can be found through a localisation of the places at the African western coast in GH 4.1 Mauritania Tingitana and GH 4.6 Libya Interior.The identifications of these places often differ in the literature or are missing.In the present contribution a localisation was carried out mainly based on the distances derived from the Ptolemaic coordinates.
The presumable origin of Ptolemy's coastal coordinates from the distance data of seafarings was considered including the applied measurement units and the cruising speed.The inaccuracy of the Ptolemaic distances was investigated which results from the rounding of the values of distances and coordinates and from errors concerning the ancient conversion of journey times into distances.The factor underlying the conversion between journey times and Ptolemy's distances was determined based on journey times derived from distances given by Pliny.As a result, Pliny's and Ptolemy's distances show a satisfactory match, which attests the reliability of the data.Gosselin (1798Gosselin ( -1813) ) points out that places at the African western coast are repeatedly given by Ptolemy, which is usually not recognised or not taken into account.In the present www.hist-geo-space-sci.net/7/27/2016/ Hist.Geo Space Sci., 7, 27-52, 2016 contribution, the repetition of a part of Mauritania Tingitana in Libya Interior is considered.Possibly duplicated places were selected based on their names, their relative positions and their localisations.The hypothesis of the identity of nine pairs of places was tested by means of a statistical test based on an adjustment of a transformation between the Ptolemaic coordinates.As a result, the hypothesis can be accepted.Further identical or similar places not attestable by the statistical test occur (Erythia/Hera, Atlas/Mandron Mountains), and possibly there exist more duplicates of places.
Port Rusibis (Rutubis) is usually taken for Mazagan.Ptolemy's position north of the Asana/Oued Oum er-Rbia and a consideration of the origination of Pliny's distances, however, lead to Casablanca or its vicinity.
Ptolemy's Mauritania Tingitana reaches much further south than the Roman province of the same name; the southernmost point at the coast is the Bigger Atlas which probably corresponds to foothills of the Anti-Atlas.
Ptolemy locates Libya Interior south of Mauritania Tingitana.Actually, the western coast of Ptolemy's Libya Interior begins in the north of Morocco at the Subos/Oued Sebou.
Cape Arsinarion is situated east of the Fortunate Islands and according to the Ptolemaic distances in the southwest of Morocco.Therefore, Cape Arsinarion is Cap Juby and the Fortunate Islands are some of the Canary Islands.The latter is substantiated by a distance by Sebosus.
It has been assumed that the Niger does not occur in ancient Greek and Latin sources (S & G, p. 449, fn. 190), but possibly it is the Masitholos because this river is situated between the Hesperu Keras/Bight of Benin and Ochema Theon/Mount Cameroon (and the Western High Plateau).
The single islands of the Fortunate Islands were identified based on their names, the position data of Ptolemy and Sebosus and further information from Sebosus and Juba.It shows that the ancient information is in accordance with the Canary Islands.Ptolemy's Fortunate Islands correspond to Alegranza, Graciosa, Lanzarote, Fuerteventura, Gran Canaria and Tenerife.From the ancient information follows that the smaller Junonia mentioned by Juba is Montaña Clara and that Sebosus' Planasia corresponds to Ptolemy's and Juba's Canaria/Gran Canaria.
Ptolemy's latitudes of the northernmost places in Morocco and of the southernmost location, the Ochema Theon, are nearly correct so that a determination of the latitude can be assumed in these cases.Some Ptolemaic distances between coastal places can be regarded as consistent with regard to the size of rounding errors.A few gross errors of distances were ascribed to alterations by Ptolemy.Mainly, the Ptolemaic positions show the following errors.(1) The African western coast runs in a wrong direction almost from north to south; only a few coastal stretches are oriented correctly.
Reasons for this are insufficient information.(2) The places of Libya Interior are strongly shifted to the south.This is due to the repetition of places of Mauritania Tingitana in Libya Interior and the arrangement of Libya Interior south of Mauritania Tingitana.(3) There are coastal stretches where the Ptolemaic distances are either systematically too large or too small.Their errors are explicable, for example, by wrong assumptions about the speed in a conversion of journey times into distances due to the ignorance of ocean currents and winds.(4) Coastal distances south of the Stacheir/Wad As Saguia al Hamra are strongly reduced.Reasons for this may be insufficient information about the southern regions and a reduction of the available space due to the shift of Libya Interior to the south.( 5) The distances of islands to the coast are significantly enlarged.This is a typical error of Ptolemy, which is also to be found in other regions.
In contrast to the common ancient prime meridian of Alexandria, Ptolemy's prime meridian is arranged at the western end of the known world, where Ptolemy located the Fortunate Islands.Hence, this prime meridian is situated at the Canary Islands or more precisely at Tenerife (Centuria), which is probably Ptolemy's westernmost island in reality.A comparison of arc lengths on the parallel of Rhodos given by ancient authors was carried out.Ptolemy's longitudinal distance of the Fortunate Islands from the westernmost continental point, the Sacred Cape, originates with Marinos.This also applies to the longitudinal distance of 7 • 30 between the Fortunate Islands and the Pillars of Herakles, which was probably derived from Eratosthenes' distance between an assumed western end of Europe and the Pillars.Ptolemy's longitudinal distance from the Pillars to Alexandria probably arose from Eratosthenes' value of this distance and a rounding to 53 • , which may have been given by Marinos.Consequently, Ptolemy's longitudinal distance of 60 • 30 between the Fortunate Islands and Alexandria is based on Marinos' and Eratosthenes' data.From Agrippa's distance between the Strait of Messina and Alexandria can be derived that he possibly located Rhodos west of the prime meridian of Alexandria for the first time.
In the test of the seven pairs of places given in Sect.4.4, the X variants of nos.M62 (X: Table 2, : = 32 • 50 ) and L4 (X: = 9 • , = 23 • , : Table 2) yield a better fit of the coordinates so that they are used for the parameter estimation (furthermore the X variant of no.M6; see Sect. 4.1,no. 6).The adjustment yields a = 0.30 ± 0.03 and a = 0.48 ± 0.06.Hence, the point cluster of GH 4.6 is strongly stretched in comparison to that of GH 4.1.This explains the longer distances and rougher resolution of the coordinate values in GH 4.6 (cf.Appendix B).It is to be expected that the random components of the coordinates are also enlarged so that σ and σ need to be increased.They are scaled by a factor of 2 resulting from a (it is determined geometrically more reliably than a ).A new adjustment leads to a = 0.30 ± 0.02, a = 0.49 ± 0.06, b = 4 • 36 ± 54 , b = 22 • 22 ± 1 • 20 , and = −4 • 19 ± 3 • 59 .Using a significance level of α = 5 %, the quantile χ 2 1−α,f is 16.9.Since T = 14.1 < χ 2 1−α,f , the hypothesis that the places considered are identical can be accepted.
The addition of the two pairs of places given in Sect.4.5 to the statistical test leads to χ 2 1−α,f = 22.4 and T = 19.9so that identity can be assumed for these places.

-Figure 4 .
Figure 4. Presumably identical places of GH 4.1 Mauritania Tingitana (upright) and GH 4.6 Libya Interior (italic).The positions of GH 4.6 are geometrically transformed by means of an adjustment of a transformation of coordinates.

Table 1 .
Places of Ptolemy's Geography at the African western coast as well as further inland places and their identifications, part 1. S.: source of and .

Table 2 .
Places of Ptolemy's Geography at the African western coast as well as further inland places and their identifications, part 2. S.: source of and .

Table 3 .
Places of Ptolemy's Geography at the African western coast as well as further inland places and their identifications, part 3. S.: source of and .
Table 4 (based on Winkler and König,

Table 5 .
Localisation of the places of GH 4.1 Mauritania Tingitana.S.: source of the Ptolemaic position; a: correction factor; s(a): corrected Ptolemaic distance Eq. (12); s: actual distance; Ist, Est: based on the Italian/Egyptian stade.

Table 6 .
Localisation view of that of the other ancient data.Pliny's two distances may originate from rough specifications as 2 DN; see Sect.3.2.2.Third, 205 Rmi may be a corruption of 250; three other distances given by Pliny yield the sum of 250 Rmi for this distance; see Sect.3.2.2.In contrast to the usual identification Mazagan, Casablanca is consistent with Ptolemy's positioning and also with his distance s 10,11 (a 1 ).

River Agna:
Müller (1902) the location of his prime meridian Soloeis and north of the River Masathat, which probably correspond to Cap Cantin and the Oued Massa (e.g.Winkler  and König, 1993, p. 120)as well as to Ptolemy's (14) Helios Mountain and (L12) Massa (seeSect.4.2, no.14; Sect.4.6,  no.12).In conclusion, the Autololes were situated between Cap Cantin and the High Atlas.The only islands in this region are at Essaouira (Mogador).Thus, these are the Purple Islands, and Erytheia refers to Mogador, the largest of them.It has been suggested that the Purple Islands are the Canary Islands (e.g.Hennig, 1944, p. 45), where orchil lichen may have been used for dyeing.Müller (1902), however, points out that Pomponius Mela and Pliny mention seashells as the origin of the Gaetulian Purple (Mela: "Those coasts [of the Nigritae and the Gaetuli] are very famous for purple and murex -the most effective dyeing materials", Chorography 3.104, Romer, 1998; Pliny: ". . .all the rocks of Gaetulia are searched for the murex and the purple").
Forbiger (1844, p. 882), S & G: Oued Massa.The name of the river suggests the Oued Massa.In accordance with Ptolemy's information, it is situated further south than the (11) Saluentia Capes/Cap Cantin and has its source in the (26) Mandron Mountains/Atlas (see no. 26 below).ŝ11,12 , however, is much too small.Probably, Ptolemy had no accurate information about this very long distance.

range of the Middle, High and Anti-Atlas, whose
assumes that the Portuguese port Porto Consado shown on old maps of this region was at the lagoon.The name Large Port may have arisen from the extent of the lagoon.The five rivers nos.4,6, 8, 10 and 12 have their sources in the Mandron Mountains(GH 4.6.8).The locations of the actual sources are the following: (4) Salathos/Oued Bou Regreg: Middle Atlas; (6) Chusarios/Oued Mellah: at Khouribga, nearly halfway between the coast and Middle Atlas; (8) Ophiodes/Oued Oum er-Rbia: Middle Atlas; and (12) Massa/Oued Massa: Anti-Atlas.Thus, it can be assumed that the Mandron Mountains are the entire mountain middle is in the High Atlas.A part of the Mandron Mountains is already given in GH 4.1 by the (M24) Bigger Atlas.The (10) Nuius refers to the Oualidia lagoon; the source of this supposed river was assumed to be in easterly mountains, i.e. in the (26) Mandron Mountains, middle: S & G: High Atlas.

Table 7 .
Deviations e = ŝ − s between the Ptolemaic distances ŝ Eq. (10) and the actual distances s in GH 4.1 Mauritania Tingitana.S.: source of the Ptolemaic position; Ist, Est: based on the Italian/Egyptian stade.

Table 8 .
Deviations e = ŝ − s between the Ptolemaic distances ŝ Eq. (10) and the actual distances s in GH 4.6 Libya Interior.S.: source of the Ptolemaic position; Ist, Est: based on the Italian/Egyptian stade.

Table 10 .
Arc lengths on the parallel of Rhodos from different ancient sources expressed in st.r P , r M : arc length with respect to the meridian of the Pillars of Herakles/the Strait of Messina.=39 • 30 , S & G) for the Strait of Messina and the westernmost and easternmost places on Rhodos ( = 58 • and 58 • 40 ) for Rhodos.Polybios' r P to the Gulf of Issos and r M to Rhodos apply only approximately because Polybios refers to Seleucia Pieria (near the Gulf of Issos) and to Sicily, respectively.His distances are not consistent with the other ancient sources and are not further considered.
are in accordance.Apart from islands, the westernmost place of the entire Geography is the Sacred Cape (Cabo de San Vicente, S & G) in Hispania Lusitania.Ptolemy's = 2 • 30 from the Fortunate Islands to the Sacred Cape was already specified by Marinos; this also applies to Ptolemy's = 7 • 30 from the Fortunate Islands to the Pillars of Herakles.Marinos' positioning of Alexandria is unknown, but it is conceivable that also Ptolemy's Table 10 apply only approximately, since Strabo specifies some distances imprecisely.His r P to Rhodos corresponds approximately to Marinos' and Ptolemy's distances.Therefore, Marinos' and Strabo's distances may have been derived from an identical source.Strabo states: ". . .Eratosthenes says [. . .] that the distance from Alexandria to Carthage [Karchedon] is more than thirteen thousand stadia, though it is not more than nine thousand -if Caria [region in the southwest of Turkey] and Rhodes lies, as Eratosthenes says, on the same meridian as Alexandria, and the Strait of Sicily [Strait of Messina] on the same meridian as Carthage.In fact, all agree that the voyage from Caria to the Strait of Sicily [Messina] is not more than nine thousand stadia . . ." (G 2.1.40).Strabo recognises the inconsistency caused by the wrong courses of Eratosthenes' two meridians through Karchedon and the Strait of Messina and through Alexandria and Rhodos.He, however, accepts these meridians and refuses Eratosthenes' distance between Karchedon and Alexandria.In agreement with the mentioned distance of ≤ 9000 st, the sum of Strabo's distances in G 2.4.3 from the Strait of Messina to Rhodos is approximately 8500 st.