The Size and Shape of the Earth

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The Size and Shape of the Earth

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

Something to ponder before class How would you
measure the size of the Earth if alive in 200
BC? Assume that you have only sheep, some
primitive constructions (like a well and a hut)
and some free time.
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Activity How do we know that the Earth is Round?
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Falling off the edge
Imagine that you are living in the time of the
ancients. Your crazy uncle Enoed has vowed to
jump off the edge of the world. Though crazy,
Enoed is rather bright. List 3 arguments that
you could use to convince him that the Earth is
round. Your arguments must rely only on
observations available 2000 years ago.
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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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Earths Shadow
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The Stars Change with Latitude
Line of Horizon
The star can be seen from location A but not
location B
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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. Contributioned to Mathematics (The
Sieve of Eratosthenes), Astronomy, and Geography.
Determined the size tilt of the Earth
Constructed an accurate calendar including leap
years Mapped the Nile river and studied its
periodic flooding.
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Astronomy in the Tropics and the Overhead Sun
Chichen-Itza
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Longitude
Tucson latitude32 N longitude111
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Earth Coordinates Latitude
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Distance around a circle, the circumference,
is D 2 p R
R
R the radius p 3.14159
D
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Distance around a circle, the circumference,
is D 2 p R
L
R
360º
The angular distance around a circle, in degrees,
is by convention 360o
D
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Distance around a circle, the circumference,
is D 2 p R
L
?
R
Cut the circle in 4 sections ? 360/4 90 L
D/4 Or L D (?/360)
360º
D
L is the distance along a sector described by ?
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Distance around a circle, the circumference,
is 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
D
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Distance around a circle, the circumference,
is D 2 p R
L
?
R
Solve for D D L (360/ ?)
Distance along a section described by an angle ?
L D (?/360) That is the fraction of circle
defined by the angle
360º
D
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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. In Alexandria,
on the same day, an obelisk was seen to cast a
shadow.
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?
More Geometry
?
?
When a transversal cuts two parallel lines,
corresponding alternate angles are equal in
size.
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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.