Location, Trigonometry, and Measurement of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 33

trigonometric ideas. Eratosthenes sealed the gap between number theory
and the real world with the idea of using abstract tools to understand more
than what one could see, be that an infinity of whole numbers or an Earth
larger than a hometown—his brain was wider than his sky—his East and
West met in the abstract realm of pure mathematics. The transformational
idea of understanding a whole, which one could never see, applies to his
measurement of the circumference of the Earth (Arlinghaus and Arlinghaus,
2005).
Eratosthenes was the first known person to accurately measure the circumfer-
ence of the Earth, doing so by measuring Sun angles at two different points
along the same line of longitude (meridian) in North Africa. To perform his
astonishingly accurate measurement of the circumference of the Earth, he
used Euclidean geometry and simple measuring tools. Clearly, he understood
in some way, the theory underlying coordinate systems and seasons expressed
in Chapter 1. Spatial mathematics has deep roots in antiquity!
To understand how he might have had the intellectual combination of tools to
achieve this remarkable feat, consider a bit of his background. He was born
in Cyrene, now part of Libya, in northern Africa. After studying in Alexandria
and Athens, he became the director of the Great Library in Alexandria in 236
bc . The library housed a great deal of the learned and compiled knowledge of
the time. It was at the library that Eratosthenes read about a deep vertical well
near Syene (now Aswan) in southern Egypt. Once a year at noon at this well,
on the day of the summer solstice, the Sun lit the bottom of the well: The Sun
was directly overhead, its rays shining straight into the well.
Eratosthenes then placed a vertical post (obelisk) at Alexandria, which was
almost due north of Syene, and measured the angle of its shadow on the
same date and time. Making the assumptions that (a) the Earth is a sphere
and that (b) the Sun's rays are essentially parallel, Eratosthenes knew from
the geometry of Euclid that the size of the measured angle (7°12 ′ ) equaled
the size of the angle at the Earth's center between Syene and Alexandria. Also
knowing that the arc of an angle of this size was approximately 1/50 of a
circle, he then had to determine the distance between Syene and Alexandria.
This was a difficult task during that time, due to different strides of cam-
els and human error, and despite the best efforts of the King's surveyors,
required years of effort. It was finally determined to be 5000 stadia. Then it
became a straightforward matter to calculate the entire circumference: The
angular measure between the two locations was 1/50th of a circle. Thus,
Eratosthenes multiplied 5000 by 50 to find the Earth's circumference. His
result, 250,000 stadia (about 46,250 km), was amazingly close to the accepted
modern measurements (40,075 kilometers around the Equator and 40,008
kilometers around the poles).
Below, we show a more formal approach to the conceptual materials, and
their implications, associated with the measurement that he is said to have
made (different accounts give different details). See Figure 2.1 .