Transcript eratoshenes_earth_measurement

276-194 B.C.
© T Madas
276-194 B.C.
Eratosthenes was born in Cyrene which is now in
Libya in North Africa and was a friend of Archimedes.
His teachers included the scholar Lysanias of Cyrene
and the philosopher Ariston of Chios who had studied
under Zeno, the founder of the Stoic school of
philosophy. He also studied under the poet and scholar
Callimachus who had also been born in Cyrene. He
then spent some years studying in Athens.
Ptolemy III Euergetes succeeded his father in 245 BC and he persuaded
Eratosthenes to go to Alexandria as the tutor of his son Philopator. On
the death of Callimachus in about 240 BC, Eratosthenes became the
third librarian at Alexandria, in the library in a temple of the Muses called
the Mouseion. The library is said to have contained hundreds of
thousands of papyrus and vellum scrolls.
Eratosthenes is remembered for his prime number sieve, the 'Sieve of
Eratosthenes' which, in modified form, is still an important tool in
number theory research. A book written by Eratosthenes was On means
and, although it is now lost, it is mentioned by Pappus as one of the
great books of geometry.
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276-194 B.C.
In the field of geodesy, however, Eratosthenes will
always be remembered for his measurements of the
Earth. A firm believer of a spherical Earth, he made a
surprisingly accurate measurement of the circumference
of the Earth. Details were given in his treatise On the
measurement of the Earth which is now lost. However,
details of these calculations appear in works by other
authors such as Cleomedes and Theon of Smyrna.
In these calculations, he compared the noon shadow at midsummer
between Syene (now Aswan in Egypt) and Alexandria. He assumed that
the sun was so far away that its rays were essentially parallel, and then
with a knowledge of the distance between Syene and Alexandria, he
gave the length of the circumference of the Earth as 250,000 stadia.
(approx 39,200 km)
Eratosthenes also measured the distance to the sun as 804,000,000
stadia and the distance to the Moon as 780,000 stadia. He computed
these distances using data obtained during lunar eclipses. Ptolemy tells
us that Eratosthenes measured the tilt of the Earth's axis with great
accuracy obtaining the value of 11/83 of 180° , namely 23° 51' 15".
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Eratosthenes made many other major contributions to
the progress of science. He worked out a calendar
that included leap years. He is also said to have
compiled a star catalogue containing 675 stars.
276-194 B.C.
He also made major contributions to geography by
sketching, quite accurately, the route of the Nile to
Khartoum, showing the two Ethiopian tributaries.
Another contribution made to geography was his description of the
region "Eudaimon Arabia", now the Yemen.
Eratosthenes writings include the poem Hermes, inspired by astronomy,
as well as literary works on the theatre and on ethics which was a
favourite topic of the Greeks. Eratosthenes is said to have became blind
in old age and it has been claimed that he committed suicide.
Today, he is best known for his 2 most notable achievements:
•The use of his “Sieve” to isolate prime numbers
•His method for determining the distance around the Earth
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276-194 B.C.
© T Madas
Great Circle
“circumference”
of a sphere
Alexandria
Syene
Eratosthenes knew that at Syene,
at noon of the summer solstice
(21st of June each year) a vertical
stick casts no shadow on the
ground, as the sun is directly
overhead.
At the same time in Alexandria,
5200 stadia directly north of
Syenne a vertical stick on the
ground casts a shadow.
The assumption is made that rays
of light arriving from the distant
sun will be parallel to each other.
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Eratosthenes measured the angle
made by the top the stick and the
top of its shadow in Alexandria.
It was about 7½°.
Since the sun rays are parallel,
the alternate angle at the Earth’s
centre can be calculated.
Syene
not to scale
© T Madas
Eratosthenes measured the angle
made by the top the stick and the
top of its shadow in Alexandria.
It was about 7½°.
Since the sun rays are parallel,
the alternate angle at the Earth’s
centre can be calculated.
Thus 5200 stadia, between Syene
and Alexandria, correspond to a
7½° angle.
Syene
Therefore the circumference of
the Earth is estimated at 250,000
stadia, approximatelly.
degrees
stadia
7½°
5200
15°
10400
30°
20800
60°
41600
360°
249600
(1 stadium ≈ 157 metres)
This is around 39,000 km with
the modern accepted figure at
40,000 km!
© T Madas
Great Circle
“circumference”
of a sphere
Alexandria
degrees
stadia
7½°
5200
15°
10400
30°
20800
60°
41600
360°
249600
Eratosthenes measured the angle
made by the top the stick and the
top of its shadow in Alexandria.
It was about 7½°.
Since the sun rays are parallel,
the alternate angle at the Earth’s
centre can be calculated.
Thus 5200 stadia, between Syene
and Alexandria, correspond to a
7½° angle.
Therefore the circumference of
the Earth is estimated at 250,000
stadia, approximatelly.
(1 stadium ≈ 157 metres)
Syene
This is around 39,000 km with
the modern accepted figure at
40,000 km!
© T Madas
© T Madas