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Figure 7.15. The geometry of Eratosthenes's method to determine the circumference of the Earth: The method needed a measurement of the noon Sun's zenith distance at Alexandria on the same date that the Sun was seen to appear at zenith at Syene. The method assumes that the Sun is at an infinite distance from the Earth and requires accurate knowledge of the physical distance between two sites involved in the measurement. Eratosthenes's method of the size of the Earth. Drawing by E.F. Milone.

Figure 7.15. The geometry of Eratosthenes's method to determine the circumference of the Earth: The method needed a measurement of the noon Sun's zenith distance at Alexandria on the same date that the Sun was seen to appear at zenith at Syene. The method assumes that the Sun is at an infinite distance from the Earth and requires accurate knowledge of the physical distance between two sites involved in the measurement. Eratosthenes's method of the size of the Earth. Drawing by E.F. Milone.

The determination rested on the measurement of the zenith distance of the Sun at Alexandria at the same date that the Sun was seen to appear at zenith at Syene. The measurement required the assumption that the Sun is at an infinite distance from the Earth, i.e., that the solar parallax is ignorable; it also required accurate knowledge of the physical distance between two sites involved in the measurement. The geometry is illustrated in Figure 7.15.

The relation between the zenith distance, z, and the difference in latitude, Df, between these two sites on approximately the same longitude circle is simply z = Df

so that the relation between the linear separation of these sites, s and z, is z = £_ 360 = C'

where C is the circumference of the Earth. Because C = 2pR, the radius of the Earth is thus found. Eratosthenes found z to be 7°2 or 1/50 of the circumference of a circle.36 His value for the distance between the two sites was 5000 stadia, and the resulting circumference, 250,000 stadia. This number cannot be directly and independently compared to modern units of length because the unit of stadium, whereas clearly refering to an athletic stadium is not uniquely defined in the ancient world; stadia were not of uniform dimensions. If there were 10 stadia to the mile, Eratosthenes's result is

36 Sarton (1970) states that the difference in latitude between Alexandria (27°31'N, 31°12E) and Syene (30°35'N, 24°05'E) is actually 7°07' and that they do not actually lie on the same longitude circle. The latter just means a slight difference in time between the two events, but the former indicates a slight error in angular measurement.

25,000 miles or 40,200 km), very close to the modern value, 24,900 miles (40,070 km). From Eratosthenes's data, the Earth's radius is then 39,800 stadia, and so 3980 miles (6400 km); the actual equatorial radius is 6378 km. If, however, there were only 8.5 or even 7.9 stadia to the statute mile, values found, for example, in the writing of the Roman historian Cassius Dio37 (Humphrey 1990), then the corresponding radius for the Earth is 4680 or 5040 miles (7530 or 8100 km), respectively. The distance between Syene and Alexandria is approximately 890 km, however, and if this represents the accurate distance between the two observing sites used by Eratosthenes, then 890 x 50 = 44,500 km, giving a relative error of about 11%. A similar but inherently more inaccurate technique was used by Poseidonios38 (140-150 b.c.), who used the difference in altitude of the star Canopus at two sites. At the island of Rhodes, it was just on the horizon, whereas at Alexandria, it was about 5° above the horizon.39 In principle, the method is fine. The refraction-corrected difference in altitude of a star observed at two sites differing in latitude by an amount Df would be

So, if the linear distance along the Earth's surface between the two sites is known, the circumference of the Earth is found. Due to refraction, the result, [email protected] = 4600 km, is less accurate than that seemingly obtained by Eratosthenes. Experiment §4 of Schlosser et al. (1991/1994) provides a modern challenge to carry out a slightly more accurate version of this ancient experiment (performed using zenith observations) and improve on the results.

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