276 194 B C T Madas 276 194

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 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. © T Madas

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 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". © T Madas

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 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 © T Madas

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 (21 st 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. © T Madas

an ° dr 7½ Al ex Eratosthenes measured the angle made by the top of 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. ia Syene 7½ ° not to scale © T Madas

Eratosthenes measured the angle made by the top of 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. Al ex an dr ia 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. 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! ° stadia 7½ degrees © 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 of 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