A more technical work on harmonic theory seems to lie behind schemes that P
attributes to Eratosthene ̄s (Harmonics 2.14) specifying numbers associated with the pitches
of three tuning systems: on Ptolemy’s understanding, these numbers represent the lengths
of a uniformly tensed string that would sound at the corresponding pitches. Two of
the schemes coincide perfectly with schemes that Ptolemy associates with A;
hence it appears that Eratosthene ̄s was attempting somehow to reconcile Aristoxenos’ the-
ory of tonal “distances” with the Pythagorean model of musical intervals as whole number
ratios.
- Geography. H (Dioptra 35) refers to Eratosthene ̄s’ work “on the measurement
of the Earth,” seemingly independent of his Geographica and in which he presented a
geometrical deduction of the length of the spherical Earth’s circumference from ostensibly
empirical data. K (1.7) reports a summary of Eratosthene ̄s’ approach, which
he characterizes as following a geo ̄metrike ̄ ephodos, a phrase that could mean a method involv-
ing surveying or, more likely, deductive argument expressed in the manner of the geometers.
The assumptions are (1) that the Sun is effectively at infinite distance from the Earth, so
that shadows cast in all localities are parallel, (2) that Alexandria is situated 5,000 stades
north of Sue ̄ne ̄ as measured along a meridian, (3) that for an observer at Sue ̄ne ̄ the Sun
passes through the zenith at noon on the summer solstice, and (4) that for an observer
at Alexandria the Sun is 1/50 of the meridian circle south of the zenith at the solstitial
noon. Of these data, (3) was probably derived from common report, and is accurate, and
the interval (2) between the two cities – not in fact on the same meridian – has the appear-
ance of a round estimate. Kleome ̄de ̄s, supported by M C’s (6.596–598)
dubious testimony, states that (4) was measured using a spherical sundial, though this
may be a didactic simplification. The resulting value of the Earth’s circumference, 250,000
stades, is often cited in ancient sources, but not always attributed to Eratosthene ̄s. Erato-
sthene ̄s himself is likely to be responsible for the well attested “rounding” of this number to
252,000, allowing a convenient equation of 700 stades with one degree of the meridian.
(Eratosthene ̄s apparently employed a division of the meridian into 60 units, however, rather
than into degrees.) In the same work, Eratosthene ̄s may have treated related questions of
mathematical geography, including estimates of the latitudes of Alexandria and other cities
derived from the ratio of a gnomon to its noon shadow on an equinox, and an estimate
of the obliquity of the ecliptic (or equivalently, the latitude of Sue ̄ne ̄), which Ptolemy
(Almagest 1.12) says was very near his own value, 11/83 of a semicircle.
The Geographica, in three books, was a treatise on the construction of a map of the
oikoumene ̄. Eratosthene ̄s may have coined the word geo ̄graphia (in the sense of “world-
cartography”) and terminology derived from it, reflecting a new emphasis on setting
map-making on a rational and quantitative scientific basis. Eratosthene ̄s thus initiated a
genre that was to lead, by way of M T, to Ptolemy’s Geography. Though
no longer extant, the Geographica is often mentioned in ancient authors, in particular
S, who reports many specific details. Strabo ̄n had direct access to Eratosthene ̄s’
work and also drew extensively from H’ lost polemic against it, and thus we
can recover from Strabo ̄n the general structure and character of the Geographica. Book 1
contained a critical review of earlier geographical authors and cartographers, a list from
which Eratosthene ̄s significantly excluded H. Book 2 appears to have addressed
methodology and the situation and dimensions of the oikoumene ̄. Book 3 provided the
detailed discussion of the dimensional and positional data necessary for drawing a map
of the oikoumene ̄, employing a division of the continents into large geometrically
ERATOSTHENE ̄S OF KURE ̄NE ̄