The Greeks were outstanding mathematical geographers. Eratosthenes measured the circumference of the Earth with outstanding accuracy. Unfortunately the sources from this era are largely lost. Our main source is the much later work of Ptolemy, from a time when intellectual quality had deteriorated. Ptolemy’s geography is not great, but its errors are due to corruption of an earlier, excellent theory.

That is what I and others suspect. An article in the latest issue of Isis claims to disprove us:

> This essay seeks to explain the most glaring error in Ptolemy’s geography: the greatly exaggerated longitudinal extent of the known world as shown on his map. The main focus is on a recent hypothesis that attributes all responsibility for this error to Ptolemy’s adoption of the wrong value for the circumference of the Earth. This explanation has challenging implications for our understanding of ancient geography: it presupposes that before Ptolemy there had been a tradition of high-accuracy geodesy and cartography based on Eratosthenes’ measurement of the Earth. The essay argues that this hypothesis does not stand up to scrutiny. The story proves to be much more complex than can be accounted for by a single-factor explanation. A more careful analysis of the evidence allows us to assess the individual contribution to Ptolemy’s error made by each character in this story: Eratosthenes, Ptolemy, ancient surveyors, and others. As a result, a more balanced and well-founded assessment is offered: Ptolemy’s reputation is rehabilitated in part, and the delusion of high-accuracy ancient cartography is dispelled. (687)

As an aside, here’s a pro tip for academic novices: If you have little of substance to offer, make sure to lay it on thick with self-congratulatory posturing about how your work is supposedly based on “more careful analysis” and “more balanced and well-founded assessment,” showing everything to be “much more complex” than others think. After all, who would dare disagree with someone who is so careful and balanced and ever so sensitive to complexities?

Contrary to his smug proclamations, the author’s case is flimsy. For one thing he immediately admits that, indeed, Ptolemy’s error can be solved in a single stroke by recalculating his map with Eratosthenes’s excellent value for the circumference of the earth, yielding an “uncannily accurate” map (692). But he alleges that this is a mere “coincidence” (691). What is his evidence for this?

His main argument concerns the value of the length unit “stade.” A “‘short’ stade … is implied by the high accuracy of Eratosthenes’ value for the circumference” (694), for which there is no explicit evidence in the record, unlike a “long” or “common” value for a stade that is mentioned in some sources. Of course this is not strange since our hypothesis is based on precisely the claim that many excellent sources are lost. But there is in fact implicit evidence for the short stade, as the author himself in effect admits:

> The main argument for the “short” stade is based on comparison between ancient and modern distances: those measured on a modern map are divided by their ancient counterparts in stades, giving the length of an average stade. This comparison has been undertaken repeatedly, … and invariably the average stade comes out to be much closer to the “short” value of 157.5 m than to the “common” one of 185 m. On this basis, many researchers suggest that for practical purposes ancient surveyors used a special short stade, one never directly attested in extant sources. … This stade is often termed the “itinerary stade.” This result might have been regarded as a brilliant confirmation of the “short stade hypothesis” were it not for one “but”: strangely enough, in comparing ancient and modern distances, a crucial factor has been lost sight of——namely, “measurement error.” The proponents of the “itinerary stade” proceed from a tacit assumption that ancient distances were measured almost as accurately as modern ones. However, this cannot be true, for two main reasons. First, with rare exceptions, there is no indication that distances given by ancient sources were actual measurements on the ground, rather than rough estimates deduced, for example, from the duration and the average speed of travel. Second, and most important, even when ancient distances do derive from actual and quite accurate measurements, they were certainly measured not as a crow flies but including all the twists and turns of the route. (701-702)

The author’s counterargument is very weak. It merely asserts what he is trying to prove, namely that distance measurements in the time of Eratosthenes would have been poor and naive. The obvious reply, which the author does not consider, is that it is very possible that Eratosthenes and others used much more sophisticated mathematical methods such as triangulation. Instead the author expects us to believe that the generation that gave us the geometry of Archimedes was too stupid to account for “twists and turns of the route” when estimating distances for geographical purposes. There is also some reason to think that Eratosthenes defined a new stade based on his earth measurement——another possibility ignored by the author.

> We can apply a simple test: the same comparative approach may be used to determine the length of the Roman mile. Since it has been firmly established as equal to 1,480 m, such comparison will yield the average accuracy of Roman distance measurements. … For this purpose I have examined more than 160 distances given in Pliny’s Natural History. … If these distances are believed to be accurate, then, by the same logic that has led us to the “itinerary stade,” we have either to conclude that Pliny’s mile was equal to circa 1,190 m, which is impossible, or to assume that Roman measurements of distances were much less accurate than Greek ones, which is hard to believe. Otherwise, we have to conclude that Pliny’s distances were overestimated by 25 percent on average——that is, by approximately the same amount that Eratosthenes’ and Strabo’s distances must have been if they were expressed in the “common” stades of 185 m. (702-703)

Apparently it is “hard to believe” that the greatest generation of geometers who ever lived were better at measuring distances than a second-rate encyclopedist from a civilisation that never contributed an iota to mathematics during its entire lifespan.

The author also maintains that “the hypothesis … contains numerous logical fallacies” (693), namely:

> The match between the recalculated Ptolemy coordinates and the modern ones, however close, does not in itself mean that the longitudinal distortion in Ptolemy’s map was due entirely to a single cause—namely, the wrong value for the Earth’s circumference. Nor does it warrant discarding two other possible causes: the exaggeration of distances and the lengthening of the stade. Another crucial point to stress is that Ptolemy’s recalculated map turns out to be accurate only in terms of spherical coordinates. This does not mean that the actual distance measurements underlying it and Eratosthenes’ value for the Earth’s circumference were equally accurate. (693)

Calling these things “logical fallacies” is just ridiculous. Obviously no one ever claimed that these things were logical implications, only that they were the likeliest explanations. The author does however commit a logical fallacy himself when he draws the non sequitur that our hypothesis is a “delusion” from the fact that Ptolemy’s map errors can be explained by other means.

Altogether, the author’s preposterously overblown claims exaggerate his case by a mile and then some. I am surprised that such wording was allowed to stand in a respectable journal. The author has not taken our hypothesis seriously, let alone given a “careful” and “well-founded” demonstration that it “proves” to be a “delusion.”