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The Cosmic Distance Ladder: Measuring Distances in Ancient
AstronomyASTR 250-B
Virginia Wahlig
Fall 2002
Foundations of Ancient Astronomy
Circumference of the Earth
Relative Distances to Sun Relative Sizes of Sun and and Moon
Moon
Distance to the Stars
References and Links
The Foundations of Ancient AstronomyAstronomy has sometimes been
called the world's first science. Since the beginnings of human
history, mankind has used astronomical objects both as tools for
improving daily life and as inspiration for systems of belief. Many
different cultures claim that the celestial realm is inhabited by
God or gods and that it has profound impact on their daily lives.
More practically, astronomical events help us to predict the
seasons, create calendars that keep track of past events, and
navigate on both land and sea. In the last millennia BC, astronomy
became a science of methodical records, changing paradigms, and the
first complex geometrical calculations. It is this scientific
approach that has led to the "Cosmic Distance Ladder," a series of
increasingly large measurements that have allowed us to get a
handle on the size, structure, beginnings, and possible endings of
our universe. Despite modern-day interest, measurements of distance
in the universe were not foremost in the minds of the first
astronomers. In ancient times, long before the development of
geometry, the telescope, or modern systems of measure, astronomy
was a tool used to predict food events like the ripening of local
plants or the migration patterns of animals. A group might starve
if they had no method of predicting the changes in the seasons.
Consider, especially, cultures like that of the ancient Egyptians,
whose entire economy depended on the annual flooding of the Nile
River. If they could not prepare for that event, many people would
probably die. Measuring the length of a year and the changes of
seasons, therefore, became essential to ancient peoples, and those
who had knowledge of such events were revered among their tribes.
(Weintraub, 08/29/01) Because of the necessity of such temporal
knowledge, it is likely that ancient cultures developed a way to
keep track of the seasons long before they began to consider
distances to objects in the universe. This idea of seasonal cycles
led to the development of calendars, as soon as the fear of
starvation was past; many cultures developed calendars early on in
their existence. Perhaps the most well-known of these ancient
cultures is the Babylonians. They are probably so well remembered
because their teachings went on to inspire the greatest of the
early mathematicians and astronomers in Greece. Their calendar,
like many others at that time, was "lunisolar," that is, based on
both the yearly motion of the sun and the monthly phases of the
moon. In fact, in 238 BC, King Ptolemy III ordered the introduction
of a leap year into the Egyptian calendar in order to maintain the
positions of the seasons within each year. The Babylonians also
invented the sundial and introduced the idea of the 360 circle.
(Motz, 6-7) During the same period, astronomy in China developed
strongly towards detailed star catalogs and records of astronomical
events like eclipses and a famous supernova. According to Motz,
they did not, however, develop a calendar that accounted for the
extra 6 hours tacked on to every 365 day year. The ancient Hindu
culture focused on the numerological significance of year lengths,
and so were repulsed by the idea that the year might not be an even
number of days. They had a thriving astronomical culture, but its
observations were mostly focused on creating astrological
significance. (Motz, 10) The ancient Mayans also developed an
incredibly accurate calendar, but did not then turn to a study of
other astronomical
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or events. (Motz, 12) Once appropriate calendars were developed,
the eyes of astronomers turned towards the philosophical
significance of astronomy. The ancient Greek
philosopher/astronomers tried to use logic to determine the
structure of the universe. Beginning with Thales around 600 BC, the
Pre-Socratic philosophers were determined to discover the principle
elements of the universe--what it was made of, what forces held it
together, and how those forces impacted life here on Earth. Around
550 BC, Anaximenes became the first to suggest that the celestial
realm must be subject to some kind of physical laws, rather than to
the whims of the ancient gods and goddesses. (Weintraub,
9/5/01-9/7/01) Pythagoras, famous for his theorem involving the
sides of right triangles, was influential in turning the minds of
astronomers towards mathematical models for celestial events.
Finally, Plato and his student Aristotle had the greatest impact on
ancient Greek astronomy. Plato suggested that the celestial realm
was the closest thing in the universe to perfection, and that it
could only be understood through logic and reasoning. (Motz, 27)
Aristotle was more of a scientist and argued that the universe
could be understood by observing it. He did, however, agree with
some of the teachings of Plato, and so claimed that the sun, moon,
stars, and planets could only move in perfect circles on crystal
spheres, and that the Earth had to be at the center of the
universe. Both his ideas of scientific study and of the structure
of the cosmos survived for a dozen centuries in the form of Ptolemy
and his geocentric view of the universe. (Weintraub,
9/10/01-9/24/01) Finally, with the coming of the last centuries BC,
Aristotle's philosophy, and the mathematics of Euclid (who was the
first to produce a definitive text on geometry around 300 BC), the
world of ancient Greece was ready to begin pondering the actual
size of the universe around them. About 240 BC, another disciple of
Plato, Aristarchus, was the first to create a scientific method of
determining the distance between the Earth, Moon, and Sun, relative
to the diameter of the Earth. He followed this calculation with
another to determine the relative sizes of the sun and moon. (Van
Helden, 6-7) A few years later in 235, Eratosthenes used a measure
of the lengths of shadows to determine the actual, numerical
circumference of the Earth to a startling degree of accuracy.
(Stern) Right before the beginning of the common era, Hipparchus,
the greatest of all ancient astronomers, lived only 33 years but
managed to catalog hundreds of stars, use a different method to
determine the distance to the moon, and make the first attempt to
determine the distance to that farthest crystal sphere: that which
contained the stars. (Measuring the Distances...) These three men
began the attempt of mankind to climb the cosmic distance ladder.
Below is a discussion of the major calculations of Aristarchus,
Eratosthenes, and Hipparchus. For a more details, please follow the
links in each section or refer to the titles mentioned in the
reference section.
Determining the Circumference of the Earth
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Assumptions:q
q
q
The Earth is approximately spherical. The Sun is approximately
infinitely far from the Earth, so all of the Sun's rays can be
assumed parallel. Alexandria is approximately due North of
Syene.
Figure 1. This figure shows the method Eratosthenes used to
determine the radius of the Earth. The pole at Alexandria casts a
shadow, while a pole at Syene would not. See description below. (As
Van Helden, 5) In approximately 235 BC, a pupil of Plato made the
first estimate of the size of the Earth that was based on empirical
observations. Eratosthenes, who was the third librarian at
Alexandria, knew that a certain stick (called a gnomon) cast a
short shadow in Alexandria at noon on the summer solstice. By
measuring the length of that shadow and conducting a lengthy series
of calculations (remember, there was no well-known trigonometry at
this time), Eratosthenes determined the noon Sun was approximately
7.2 from the zenith. This measurement corresponds to angle alpha in
Figure 1. Upon traveling one solstice to Syene (now Aswn), he
noticed that the same stick cast no shadow at all. This meant that
the Sun was directly overhead in Syene. (Stadel) Eratosthenes knew
that the Earth was approximately spherical, and so he correctly
assumed that the angle alpha corresponded to a percentage of the
Earth's circumference. All he had to do, then, was determine the
distance between Syene and Alexandria, and he could calculate the
circumference of the Earth! It is possible that Eratosthenes
measured this distance himself, or that he got it off a current
map, or even that he "ordered some soldiers to march off the
distance," as Stadel's page indicates.
The Calculations:
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(From O'Connor.) Obviously, the size of a "Stadium" has a great
influence over the accuracy of the calculations, but it is possible
that he came within 1000 km of the actual value (40,076 km). It
should be noted, however, that Alexandria and Syene do not exactly
share a meridian, nor is the Earth entirely spherical, so a certain
measure of error is introduced into the calculation due to
incorrect assumptions. No matter what the accuracy level, it is
clear that Eratosthenes had finally found a scientific way of
quantifying the size of the Earth. Combining his calculations with
those of Aristarchus below (which actually occurred before
Eratosthenes's work) would give quantitative values for the size of
the Moon, Sun, Earth, and the distances between the Earth and Moon
and the Earth and Sun.
Determining the Distance to the Sun and MoonAssumptions:q
q
The Earth is approximately spherical. The Sun is not infinitely
far from the Earth.
Figure 2. This figure shows Aristarchus's method of determining
the relative distances of the Sun
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Moon. In this diagram,
