Around AD 150, in Alexandria, Claudius Ptolemy wrote down the known world as a list of coordinates: a longitude and a latitude for every town, cape and river mouth from Ireland to China. I matched 2,358 of his places to where they really are and measured his map against the real one.
Ptolemy's Geography contains no map. It is a set of instructions for drawing one, and a table. Book after book lists the places of the inhabited world, province by province, each with two numbers: degrees east of the Fortunate Isles at the western edge of the world, and degrees north of the equator. It is the first attempt we know of to put the whole known world on a grid, and for fourteen centuries, from Byzantine scribes to the printers of the Renaissance, maps of the world were drawn from these numbers.
Gerd Grasshoff and Oliver Defaux published the whole table as open data: 10,049 entries, 6,372 of them with coordinates, in the two main versions of the manuscripts, called Omega and Xi, and with the modern identification of each place where scholars know it. I matched those identifications to modern coordinates through the Pleiades gazetteer of ancient places, the Digital Atlas of the Roman Empire and my own work on coasts and capes, and kept only matches I could stand behind: 2,358 distinct real places. Click a place on either map to see it on the other.
Left: Ptolemy's longitudes and latitudes (Omega recension) on a plain grid of his degrees, with each degree of longitude drawn at the width it has on his parallel of Rhodes, 36 degrees north, as Marinus of Tyre drew his world map. Brown lines: his coasts, his coastal points joined in the order he lists them. Grey dots (Map layers): the 6,372 entries with coordinates that I could not, or did not try to, match. Right: the 2,358 places at their real positions. Arrows (Map layers): from each real place to where Ptolemy puts it once his longitudes are shrunk by 0.72. Sources: Grasshoff and Defaux (2019), CC BY 4.0; Pleiades (CC BY 3.0); DARE (CC BY-SA); Natural Earth.
The first thing to notice is that Ptolemy's world is far too wide. From the Sacred Cape in Portugal, Cape St Vincent, to Sera, the capital of the silk people, which scholars take to be Xi'an in China, he counts 174.8 degrees of longitude. The real figure is 117.9: he stretched the known world east to west by 1.48 times. Measured from Alexandria, where he worked, every 10 degrees of real longitude becomes 13.7 of his, to the west as well as to the east.
The usual explanation is the size of the Earth. Ptolemy followed Posidonius and Marinus in giving a degree of the great circle as 500 stadia, where Eratosthenes had reckoned 700. If his distances in stadia along the roads and sea routes were roughly right, turning them into degrees at 500 stadia a degree would make every gap too wide by 700 divided by 500, and to bring his degrees back to real ones you would have to multiply by 0.71. The best single correction I can find for his longitudes is 0.72, within 0.6% of that. Marinus's long reckoning of the road to the silk lands, which Ptolemy cut but did not cut enough, added to the overshoot in the far east.
How far from Alexandria: Ptolemy's longitude against the real one
Each dot is a matched place: its real longitude difference from Alexandria (across) and Ptolemy's (up). The dashed line is a perfect map; the solid line is his fitted slope of 1.37. Hover a dot.
Read exactly as written, with his longitudes lined up on Alexandria, Ptolemy's map is good near home and bad far away. Places within 1,000 km of Alexandria are a median 134 km out; places more than 4,000 km away, 1,445 km out. The rank correlation between distance from Alexandria and error is 0.91. Rome, 1,952 km away, is 532 km out as written; London 758 km; Taxila, in the Punjab, 2,017 km. Most of that is the stretch: once the longitudes are shrunk, Rome is only 79 km out and Samarkand 66.
Error against distance from Alexandria
Median error, as written (aligned on Alexandria) and after shrinking his longitudes by 0.72, for places in bands of distance from Alexandria. Hover a bar.
His latitudes are better than his longitudes, as you would expect: latitude can be measured from the height of the sun or the length of the longest day, while longitude needed a clock nobody had. Half his places are within 0.77 degrees of the right latitude. But they drift in a consistent way: Britain is a median 3.0 degrees too far north, Roman Africa -4.1 degrees too far south, which pushes Carthage -4.2 degrees off and squeezes the Mediterranean between them.
The most famous error in the Geography is Scotland. In Ptolemy's coordinates, north of the Solway and the Tyne, Britain does not run north but turns sharply east, so that the northern tip of Scotland points towards Norway. On the real map the Solway Firth to Duncansby Head is a bearing of 3 degrees, almost due north. On his it is 79 degrees, almost due east. Fitting the best rotation to each part separately, his Scotland (22 places) is turned 78 degrees further clockwise than his England and Wales (60 places).
Britain in Ptolemy's coordinates and in reality
The matched places of Albion, coloured by real latitude from the south coast (light) to the far north of Scotland (dark), drawn at the same scale. Hover a dot; click to see it on the maps.
Nobody knows exactly why. One long-standing idea is that Ptolemy, believing nothing habitable lay much further north than his Thule at 63 degrees, folded Scotland over to keep it inside the limit; another is that he simply joined two separate sources at the wrong angle. Whatever the cause, his Britain is still recognisably Britain coast by coast: the Clyde, the Forth, the Tay, the Wash and the Thames are all in order, just rotated.
In the east the errors stop being a matter of scale. Real India is a great triangle hanging south from the Tropic: from Barygaza, modern Bharuch, to Cape Komaria, Cape Comorin, is 13.6 degrees of latitude. In Ptolemy's coordinates it is 3.8. The peninsula is flattened against the equator, and the west coast that should run south south east, at 161 degrees, runs east south east, at 115. His Cape Komaria lies at 13.5 degrees north, 5.4 degrees too far north.
Off its tip, his Taprobane, Sri Lanka, swells to fill the space. Its coast, as he lists it, runs from 12.5 degrees north to -2.5 degrees south, across the equator: 15.0 degrees of latitude against the real island's 3.9. Measured at the same overall scale as the rest of his map, his island covers about 14.6 times the area of the real one.
India and Sri Lanka in Ptolemy's coordinates and in reality
Left: his matched places in India and Taprobane, and the coast of Taprobane as he lists it (brown). Right: the same places and the real coastline, at the same scale. Hover a dot.
Ptolemy got one thing right that many before him had wrong: the Caspian is a lake. Eratosthenes and Strabo had made it a gulf of the northern ocean; Ptolemy closes it on every side. But his Caspian has the wrong shape. Taking the 8 river mouths on its shore that can be matched, from the Volga and the Ural in the north to the Kura and the Sefid Rud in the south, the real ones span 1,054 km north to south and 387 km east to west; his span 815 km and 790 km. A sea 2.7 times as long as it is wide becomes one as wide as it is long. He also has the Oxus and the Jaxartes, the Amu Darya and the Syr Darya, flowing into it, when their real mouths are in the Aral Sea, 470 and 620 km from the Caspian shore.
The Indian Ocean he closed by mistake. In his text, unknown land runs south of the Indian Sea from beyond Cape Prason in East Africa to beyond Kattigara, the last port of the Sinae. Prason, which the editors identify with Cape Delgado in Mozambique, he puts at -15 degrees south, 25.7 degrees too far south; Kattigara, if it is near Hanoi as they suggest, at -8.5 degrees south, 29.5 degrees too far south and 3,403 km from where the rest of his map would place it. The eastern end of his world bends down to meet the southern continent and the ocean becomes an inland sea.
The Caspian in Ptolemy's coordinates and in reality
River mouths on the Caspian shore, and the mouths of the Oxus and Jaxartes, which he puts on the Caspian and which really reach the Aral Sea. Same scale on both sides. Hover a dot.
To see which parts of the map are well drawn, I took each of Ptolemy's 26 regional maps separately and found the best way to lay his coordinates on the real ones, allowing any shift, rotation and scale. What is left over is the error in the shape of the region itself. For the 18 regional maps with at least 15 matched places, the best are Sardinia and Sicily (a median 42 km), Dacia, Moesia and Thrace (52 km) and Greece (52 km); the worst are India (528 km), Inner Libya and Ethiopia (448 km) and the British Isles (128 km). Placed on the world map as a whole, after the shrinking of longitudes, Cyprus, Syria and Mesopotamia come out best (71 km) and Egypt and Cyrenaica next (93 km): the eastern Mediterranean, the world Ptolemy knew best.
Error by regional map
Median error in the shape of each region (after its own best shift, rotation and scale) and in its placement on the whole map (after shrinking longitudes by 0.72). Regional maps with at least 15 matched places. Hover a bar.
All 26 regional maps
Matched places, median error in shape and in placement, and the median latitude error (positive: too far north). Click a row to see the region on both maps.
The Geography survives in two families of manuscripts. Most follow the Omega recension; one important manuscript, the Vaticanus Graecus 191, follows the Xi recension, which covers Books 2 to 5 and the start of Book 6. Of the 4,406 entries that have coordinates in both, 69% agree exactly. Of the 1,373 that differ, 681 differ only in latitude and 448 only in longitude; the median difference is 20 minutes of arc, about 31 km, but 245 differ by a degree or more. Book 2, on the far west of Europe, differs most (42% of entries), Book 5, on Asia Minor and the Levant, least (21%).
Neither version is clearly better. Fitted the same way on the 2,049 matched places that have both, both have a median error of 144 km and almost identical stretch factors (0.71 and 0.71). Where they differ, Xi is closer to the real place 364 times and Omega 266 times. Turn on the Xi positions in Map layers to see the two side by side.
How far the two recensions differ
Entries with coordinates in both Omega and Xi, by the larger of the two differences in longitude and latitude. Hover a bar.
The catalogue. Grasshoff and Defaux's catalogue of the main books of the Geography (Books 2 to 7) gives each entry an identifier by book, chapter and section, the regional map it belongs to, its name (in German), the editors' modern identification (also in German, blank where the place is unknown) and its coordinates in the Omega and Xi recensions. I sorted the entries by the kind of feature their names describe, and used only points: towns, capes, river mouths, harbours and islands. Of 5,667 such points with coordinates, the editors identify 2,871 (51%). I never matched an entry they leave unidentified.
Matching. I matched 2,439 of those 2,871 identified points, 85%, giving 2,358 distinct real places once repeats are removed. 1,436 are places in Pleiades or the Digital Atlas of the Roman Empire whose ancient names include Ptolemy's name and whose modern names include the editors' identification (high confidence); 261 match on the ancient name alone and 202 on the modern name alone, each the only such place in that part of the world (medium); and 459, mostly capes, river mouths and the cities of the far east, are my own coordinates for the editors' identification. In all 1,742 places (74%) are high confidence. Spellings were compared on a skeleton that ignores Greek and Latin spelling and German transliteration. Because Ptolemy's errors are smooth, a wrong match stands out as a place displaced unlike its neighbours: I dropped 21 single-name matches that departed from their neighbours by more than 300 km, inspected every other outlier by hand and rejected 2; 56 more were left out as ambiguous.
Fitting. Errors are great-circle distances between the real place and where a transform of Ptolemy's coordinates puts it. As written, aligned on Alexandria, the median error is 425 km. Shrinking longitudes by a fitted factor and shifting both axes (three numbers) brings it to 144 km; a general affine transform (six numbers) to 142 km; a rotation and one scale (a Helmert transform) to 144 km; fitting the shrink and shift separately for each regional map, to 105 km. That the simple three-number fit does as well as the affine and Helmert transforms says the main error really is a stretch in longitude. On that fit Ptolemy's zero of longitude falls at 12.9 degrees west, about where the eastern Canary Islands are.
What this is not. The matched places are not a random sample: they are the places scholars have identified, so they favour the Roman Mediterranean (1,365 in Europe, 297 in Africa, 696 in Asia, only 67 east of 60 degrees east). Figures for India, Arabia, Central Asia and China rest on few places and on identifications that are themselves debated, Kattigara and Sera above all. Some coastlines, river mouths and harbours have moved since antiquity; the mouths of the Nile delta, the Meander and the Tigris are the obvious cases, and I located few of them. The Ptolemaic coasts on the left map are his coastal points joined in the order he lists them, not his own drawing. My hand coordinates are good to a few kilometres, far finer than the errors measured, but they follow the editors' identifications, which are scholarship, not certainty.