The Size of Earth

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The Size of Earth

• All men by nature desire to know.
• - Aristotle

Interests 1) Astronomy (Planets, Stars,
Cosmology 2) Current Issues (e.g.
Greenhouse warming) 3) Quantum Mechanics
4) String and Unified Field Theory 5)
Others (Biology Genetics, Cultural Astronomy
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The News

• BRAZIL Amazon is deforested at a rate of 20, 000
  km2 (an area the size of New Jersey) each year.
  New plan to bring logging to the heart of the
  Brazilian rain forest for the first time.
• BIOMEDICINE 100 trillion bacteria live in the
  human intestinal track. Bowel diseases result in
  part from abnormal response to this flora.
  Scientists are trying to sort this out.

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Aristotle (384-322 BC)
Used physical arguments to explain nature.
Regarding the spherical shape of Earth Ships
disappear over the horizon, Travelers see new
stars above the horizon, The shadow of Earth
on the moon is round. Earth is at the center of
the Universe No parallax of the stars
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Distance to Horizon
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Earth Coordinates Latitude
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Longitude
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Earths Shadow
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Parallax
1 light year 9.4x1015 m
The nearest star (Proxima Centauri) exhibits a
parallax of 0.77233 arc seconds. That is the
angle subtended by a quarter 5.3 km away. (Alpha
Centauri is 4.22 light yrs or 3.8x1013 km away.)
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Eratosthenes of Alexandria (276-195 B.C.)
The second librarian at the Library of
Alexandria. Made many contributions to
Mathematics (The Sieve of Eratosthenes),
Astronomy, and Geography. He determined the
size of the Earth and the tilt of the Earths
axis He constructed an accurate calendar
including leap years He made a good map of the
Nile river and deduced the correct explanations
for its periodic flooding.
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A Well with no Shadow
Eratosthenes, was told of a well in the city of
Syene (now Aswan) where on a certain day in
summer the wells cast no shadow, i.e. the sun
shines directly down the well. Being well
educated, he knew that the Earth was a sphere and
realized that he could use this well to measure
the size of the Earth. He had a student measure
the distance from Alexandria to Syene by counting
the steps required to walk there. The name of
the student so privileged has been lost to
history. Fortunately, the other details of
Eratosthenes method were carefully recorded.
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Measurement of ? Tan(?) shadow/height
?
Obelix in Alexandria
The angle measured from the obelisk, ?, equals
the angle that defines the section of the Earth
from Alexandria to Syene.
?
Well in Syene
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Distance around a circle, the circumference D
2 p R
L
?
R
Distance along a section described by an angle ?
L D (?/360) That is the fraction of circle
defined by the angle
360º
Solve for D D L (360/ ?)
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Solution
Distance around the Earth is D L (360/
?) The slave student measured values of ?
7.2 L5000 stades Thus, D 250,000 stades
39,250 kilometers Modern Value is 40,070
kilometers !!!!
1 stade 600 Greek feet 157 meters
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SIZE OF THE EARTH Summary
L Distance from Alexandria to Syene D
Circumference of Earth L (7.2/360)D L 5000
stades D(360/7.2)L D250,000 stades 1
stade 157 meters D157250000
meters D39,250,000 meters D 39,250
kilometers Modern Value 40,070 kilometers
Size of a Greek stadium at, for example, Olympia
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The Greek Universe

• Were sitting on a ball of rock of radius 6,000
  km.
• The Sun is a red-hot stone 1.3x108 km away.
• The moon is a smaller ball of rock 1.2x105 km
  away.
• The Sun and the moon rotate about the Earth at
  the center of the Universe.

Actual Earth-Moon distance is 3.9x105 km
Earth-Sun distance is 1.5x108 km
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In summary

• Aristotle tried to explain nature from
  observation.
• His explanations are testable, and indeed he got
  some things wrong.
• Nonetheless he others pioneered the scientific
  method.
• Eratosthenes used mathematics (geometry) and
  observation to reveal the size of Earth, and the
  Earth-Sun and Earth-Moon distances. In short he
  explained the world around him.

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Summary of Technical Info
Geometry Distance around a circle D 2 p
R Length of section described by angle ? L D
(?/360) Units 1 light year 9.4x1015
m Note m is meters, R is radius, D is
circumference
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We see in detail how the Greeks knew not only
that the Earth was a sphere, but indeed how they
calculated its immense size, using measurements
geometry. Humans had not yet figured out how
the world worked. First they had to measure the
basic characteristics of the world around them,
e.g. the sizes of the Earth, Moon and Sun, and
their distances from one another.
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Had we never seen the stars, and the sun, and
the heaven, none of the words which we have
spoken about the Universe would ever have been
uttered. But now the sight of day and night, and
the months and the revolutions of the years, have
created numbers, and have given us a conception
of time, and the power of enquiring about the
nature of the Universe and from this source we
have derived philosophy, that which no greater
good ever was or will be given by the gods to
mortal man. Plato
January 18, 2006
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Mayan Observations
900 AD Chichen Itza, Mexico