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No one really understood music unless he was a scientist, her father had declared, and not just a scientist, either, oh, no, only the real ones, the theoreticians, whose language is mathematics. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships. "And relationships," he had told her, "contained the essential meaning of life."
The Goddess Abides, Pt. I, 1972.
Eratosthenes and the Mystery of the Stades
The Basic Problem
The simplicity and elegance of Eratosthenes’ measurement of the circumference of the Earth is an excellent example of ancient Greek ingenuity. While working at the library, he learned that on the first day of summer the Egyptian town of Syene cast no shadows [8, p.339]. This happens because at noon on the day of the summer solstice the Sun is positioned directly above the town of Syene, near the modern city of Aswan, Egypt [3, p.115 ].
In contrast, on that same day in Alexandria a staff, or gnomon, did cast a shadow [8, p.339]. With a few measurements, some assumptions, and a little geometry, Eratosthenes was ready to approximate the circumference of the Earth [8, p.339]. Eratosthenes’ original account of this measurement does not survive, but his argument has been preserved in the writings of many other ancient scholars such as Cleomedes, Strabo, and Ptolemy [4, p.1 ].
Eratosthenes makes five assumptions which he will use as hypotheses in his argument [11, p. 109 ].
1. That Alexandria and Syene lie on the same meridian.
2. That light rays from the Sun which strike the Earth are parallel.
3. That the distance between Alexandria and Syene is 5000 stades.
4. That the angle formed by the shadow and the staff in Alexandria at the
summer solstice is equal to 1/50 th of a circle.
5. That the Earth is a sphere.
Let us examine each of Eratosthenes’ assumptions along with their respective justifications.