Transcript Eratosthenes of Cyrene

Eratosthenes of Cyrene
(276-194 BC)
Finding Earth’s Circumference
January 21, 2013
Math 250
Eratosthenes’ Method
Results of Eratosthenes
• Knowing also that the arc of an angle this size
was 1/50 of a circle, and that the distance
between Syene and Alexandria was 5000
stadia, he multiplied 5000 by 50 to find the
earth's circumference.
His result, 250,000 stadia (about 46,250 km), is
quite close to modern measurements: The
circumference of the earth at the equator is
24,901.55 miles (40,075.16 kilometers). But, if
you measure the earth through the poles the
circumference is a bit shorter - 24,859.82 miles
(40,008 km).
Ecliptic
• Eratosthenes also determined the
obliquity of the Ecliptic, measured the
tilt of the earth's axis with great
accuracy obtaining the value of
23° 51' 15", prepared a star map
containing 675 stars, suggested that
a leap day be added every fourth year
and tried to construct an accuratelydated history.
Sieve of Eratosthenes
• He developed the “Sieve of
Eratosthenes” method of finding
prime numbers smaller than any
given number, which, in modified
form, is still an important tool in
number theory research.
Robert Fludd’s Celestial Monochord , 1618
Hipparchus-Ptolemy Models for Planetary Orbits
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Uniform-Circular-Motion Model
The following equation represents the circular
motion of a planet, P around the Earth, E.
z1 (t )  a1e 2it / T1 ,
  t  
a1  a1 ei1 ,
0  1  2
a1 = radius of orbit
T1 = period
1 = phase parameter
E
P at z1(t)
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Epicycle-on-Deferent Model
This equation demonstrates the observed retrograde motion
of a planet as seen from Earth. The planet, P, undergoes
uniform circular motion about a point that undergoes uniform
circular motion about the earth, E.
z2 (t )  z1 (t )  a2e2it / T2
0  2  2
a2  a2 ei2 ,
P at z2(t)
E
z1(t)
epicycle
deferent
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Three-Circle Model
zn (t )  a1e2it / T1  a2e2it / T2  a3e2it / T3
E
epicycle
deferent
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Connection to Fourier Series
zn (t )  a1e2it / T1  a2e2it / T2      ane2it / Tn
E
epicycle
deferent
•This motion is periodic only when T1, T2, … , Tn
are integral multiples of each other
•Hipparchus and Ptolemy found that if you shift
the position of Earth and keep the orbits where
they are, an even more accurate depiction of the
orbital motion can be obtained
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Hipparchus-Ptolemy Model Using Cartesian Coordinates
x1 (t )  R1 cos(1t  1 )  R2 cos( 2t  2 )  R3 cos(3t  3 )
y1 (t )  R1 sin(1t  1 )  R2 sin( 2t  2 )  R3 sin(3t  3 )
R1  2
R2  1
R3  .5
1  300
 2  60
 3  12
1  0
2   / 6
3   / 3
Note: Only a truncated Fourier Series if
 2 / 1 ,...., n / 1 are all rational numbers.
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Mathematica 5.1
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