The lost towns of Dacia, from Ptolemy's numbers

The list

Two numbers for every town#

Ptolemy records Porolissum, the Roman town near Moigrad in northern Romania, at 49 degrees of longitude and 48 of latitude. Every one of the 44 towns of Dacia is written the same way: a name and two numbers, usually rounded to five minutes of arc.

His degrees are not modern degrees. Longitude is measured from the Fortunate Isles in the Atlantic, and his Earth is smaller than the real one, so distances eastward come out too long. The latitudes are off too: Porolissum actually lies at 47°11′, not 48°.

So the numbers cannot be dropped straight onto a modern map. First the way his map is bent has to be worked out, and that can only be learned from the towns that have been found.

The fixed points

Nine towns that have been found#

Porolissum (Moigrad), Napoca (Cluj-Napoca), Potaissa (Turda), Salinae (Ocna Mureș), Apulum (Alba Iulia), Tibiscum (Jupa), Dierna (Orșova), Drobeta (Drobeta-Turnu Severin) and Cumidava (Râșnov) were identified from inscriptions or from the Roman road map, without using Ptolemy's numbers. Everything that follows rests on them.

Turned, shrunk and shifted as a whole, his map misses every one: half by more than 26 km, Porolissum by 53 km. The lines run from where he puts them to their ruins.

A tenth town, Germisara (Ptolemy's Zermizirga), near Geoagiu, also has an inscription naming it. It was kept aside, as a test.

The fit

His map stretches over the mountains#

A map can be bent, not just turned. Each point of Ptolemy's Dacia is moved according to the known towns around it, with near ones counting more than far ones. As you scroll, his red grid bends over the Carpathians and the nine towns come close to their ruins: half end up less than 12 km away.

Fitting his map to known towns is not a new idea. A team at the Technical University of Berlin (Andreas Kleineberg, Christian Marx, Eberhard Knobloch, Dieter Lelgemann) rebuilt Ptolemy's Germania that way (2010), and Marx and Kleineberg went on to the rest of Europe, Dacia included (2012). Șerban Drugaș computed positions for Dacia in 2020. None of their accessible work shows how well the method places a real town it was not fitted on; Sorin Forțiu (2013) checked some of their placements: Acmonia, for one, lands 88 km from its station on the Roman road map.

The test

Take a town out and see where the map puts it#

Pick one of the nine towns. The map is computed again without it, then Ptolemy's numbers put it somewhere: the blue patch shows where it should be sought, the square where it really is.

When each of the nine is taken out in turn, half land within 19 km of their ruins. Salinae comes nearest, at 14 km; Tibiscum farthest, at 47 km.

The hard test

On towns held out of the fit, the miss is three to four times larger#

68 kmthe typical miss on the towns held back

Taken out one at a time, the nine typically land 19 km from their ruins. On towns never used in the fit, the miss is three to four times larger. Seven towns were identified, with varying confidence, without Ptolemy's numbers (from inscriptions, the Roman road map or their names) and were left out of the fit entirely: Germisara, Aquae, Arcobadara, Aizisis, Acmonia, Amutrium and Angustia. On them the map typically misses by 68 km, from 31 km at Angustia to 95 km at Aquae. Three of them (Germisara, Arcobadara, Aquae) were added after the protocol was frozen; on the other four, fixed in advance, the typical miss is 60 km. Argidava, the fifth pre-registered test, is treated separately below.

The same test run on 82 towns with known sites in six neighbouring regions (Upper and Lower Pannonia, Upper Moesia, the Lower Moesian Danube, inland Dalmatia and inland Thrace) gives between 34 and 64 km. Ptolemy's Dacia is no better than they are.

The patches on the map are sized by the misses in those neighbouring regions, scaled so that the 50% zone holds about half of the towns hidden there (47%; the 90% zone, 91%). The Dacian towns played no part in the sizing, so they are the test: of the seven, three fell inside their 50% zone and six inside their 90% zone.

The lost towns

Where the others should be sought#

Each town without a sure place becomes a patch: the darker it is, the likelier the town stood there. The solid line encloses half the probability, the dotted one nine tenths. Tap a patch or a name to see the town.

The 50% zone typically covers 14,400 km², the area of a circle about 68 km in radius, from 5,000 km² for Marcodava to 57,000 km² for Carsidava. Ptolemy's numbers point to a district, not a hill.

Each zone holds dozens or hundreds of sites from Romania's national archaeological register. But the zones hold no larger share of Dacian or Roman sites than the same patches moved at random by 60–200 km: the difference passes the usual 5% threshold in one of 35 cases, about what chance alone would give. The map says where to look, not what will be found.

The copyists

One letter copied differently moves a town hundreds of kilometres#

The Geography survives only in handwritten copies. Its numbers were written in Greek letters, 49 as μθ and 45 as με, and one letter miscopied moves a town. Ziridava, long sought at Pecica near Arad, has longitude 49½ in most manuscripts. With it, the town's zone falls in southern Transylvania and Pecica is left 234 km away, outside even Ziridava's 90% zone. Some manuscripts, cited by Wilberg (1842) as E, Pal. 1 and M, have 45½, and Müller (1883) printed 45½ in his text: with that number Pecica falls inside the 12% zone.

Argidava is usually placed at Vărădia in the Banat, where the Roman road map has a station called Arcidava. The manuscripts give it longitude 49½, which puts it 192 km from Vărădia. The number that brings it closer to Vărădia, 46½, is a correction proposed by the editor F. G. Wilberg in 1842 and adopted by C. Müller in 1883. With it, Vărădia falls inside the 53% zone, but the number comes from no manuscript, so the fit is no test.

The buttons below show each reading on the map.

The capital

Which Sarmizegetusa?#

Ptolemy writes Sarmizegethusa basileion, "the royal one". The Barrington Atlas and Drugaș place it at Grădiștea de Munte, the citadel of the Dacian kings. Marx and Kleineberg, as reported by Sorin Forțiu, place it at Ulpia Traiana Sarmizegetusa, the Roman colony founded after the conquest, 45 km to the west.

His numbers put the town 26 km from the Roman colony and 60 km from the Dacian citadel. Allowing for the typical misses measured in the neighbouring regions, the colony is 3.3 times as likely as the Dacian citadel. That is a hint, not proof, and it fits the view, held by some historians, that Ptolemy's list describes Dacia after the conquest.

Beyond the mountains

In Moldavia the map has no fixed points#

Petrodava, Utidava, Carsidava, Patridava and Triphulum sit in the north-east corner of Ptolemy's Dacia, in today's Moldavia. No town there is securely identified, so the map is extended beyond its last fixed point, Cumidava, and the zones grow: the 50% zone covers 25,000 to 57,000 km².

Piatra Neamț, where Petrodava is usually sought, lies 124 km from the main estimate, outside the 50% zone and inside the 90% one. The numbers neither confirm it nor rule it out.