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Learning to Think Spatially: GIS as a Support System in the K-12 Curriculum
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SPATIAL THINKING IN EVERYDAY LIFE, AT WORK, AND IN SCIENCE 57
the universe. This is the process by which generations of astronomers have attempted to infer the
structure and evolution of the universe from the basic observations, the spatial primitives of astronomy.
They did so by converting data about energy into representations, often graphic, that
allowed them to draw inferences about the physics of the stars and the cosmos. Spatial thinking is
so pervasive in astronomy that in our illustrative story, we will further restrict the discussion to
some of the key steps that built up a cosmic distance scale and enabled us to place objects in space,
and eventually in time, with increasing precision.
3.5.3 The First Step: The Shape and Size of Earth
The first step in the process involved inferences about the shape and size of Earth. Central to
this process were the attempts to measure the size of Earth and, outstanding among those attempts,
was the work of Eratosthenes.
Of the remarkable series of polymaths who were librarians of the Great Library in Alexandria,
Egypt, none was more remarkable than Eratosthenes of Cyrene (275–194 BC) (see Casson, 2001).
Eratosthenes was, among other things, a mathematician, a philosopher, and a geographer. In the last
of these avocations, he was a pioneer. He produced a remarkably accurate world map, centered on
the “known” world of the Mediterranean Sea. The map was the first to include parallel lines of
latitude. He suggested that Africa might be circumnavigated and that the major seas were connected.
He calculated the length of the year and proposed adding a leap year to accommodate for the
progressive discrepancy between Earth’s orbit and the calendar.
Of his geographic achievements, Eratosthenes is best remembered for his work in geodesy,
establishing the scientific grounding for that discipline through a brilliant exercise in spatial thinking.
He calculated the circumference of Earth:
He did this by employing a method that was perfectly sound in principle: first ascertaining by
astronomical observations the difference between the latitudes of two stations situated on the same
meridian, also by terrestrially measuring the distance between the same two stations, and finally, on
the assumption that the earth was spherical in shape, by computing its circumference. (Vrettos,
2002, p. 52)
The details of the actual procedure are even more remarkable. It combined knowledge, observation,
calculation, inference, and intuition in a way that captures the essence of spatial thinking.
Eratosthenes was aware of the idea, which had been proposed by earlier Greek natural philosophers,
that Earth is spherical in shape. When he learned from travelers that at noon on the summer
solstice in Syene (modern Aswan) the Sun shone directly into a deep well and its reflection was
visible on the surface of the water in the well, he realized that the Sun must be directly overhead at
that date and time, so that a gnomon (vertical stick) would cast no shadow. If Earth is indeed
spherical, on the same date and time, a gnomon where he lived, in Alexandria, would cast a shadow.
A smaller Earth, with a greater curvature, would produce a longer shadow than a larger Earth.
He realized that this general argument could be turned into a quantitative measurement by
envisioning a frame of reference with an origin at the center of Earth, and by assuming that the Sun
is so far away that its rays would be nearly parallel at Syene and Alexandria (Figure 3.2). In that
case, the angle formed by a gnomon and the tip of its shadow in Alexandria, would be the same
angular distance between Syene and Alexandria that would be observed from the center of Earth.
He measured this angle and found it to be 7 degrees and 12 minutes, or approximately 1/50th of a
circle. Since the overland distance between Alexandria and Syene had been measured by travelers
(5,000 stade), he multiplied that number by 50 to arrive at a distance for the circumference of Earth.
Starting from the generally accepted belief that Alexandria and Syene (modern Aswan) were
on the same meridian and his belief that Earth was spherical, Eratosthenes used a frame of reference
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