Eratosthenes of Cyrene(276-194 BC)

Finding Earth’s Circumference

January 21, 2013Math 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 accurately-dated 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

5

P at z1(t)

 

Uniform-Circular-Motion ModelThe following equation represents the

circular motion of a planet, P around the Earth, E.

1/211 )( Titeatz t

111

ieaa 20 1

,

,

1a = radius of orbit= period1T

1 = phase parameter

E

6

2/2212 )()( Titeatztz

222

ieaa 20 2 ,

Epicycle-on-Deferent ModelThis 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.

E

deferent

epicycle

z1(t)P at z2(t)

7

E

deferent

epicycle

Three-Circle Model

8

321 /23

/22

/21)( TitTitTit

n eaeaeatz

•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

nTitn

TitTitn eaeaeatz /2/2

2/2

121)(

Connection to Fourier Series

E

deferent

epicycle

9

Hipparchus-Ptolemy Model Using Cartesian Coordinates

)sin()sin()sin()(

)cos()cos()cos()(

3332221111

3332221111

tRtRtRty

tRtRtRtx

3/

6/

0

12

60

300

5.

1

2

3

2

1

3

2

1

3

2

1

R

R

R

Note: Only a truncated Fourier Series if are all rational numbers.

112 /,....,/ n

10

30

12090

60

150

0

11

Mathematica 5.1

240

270 300

210180

12

13