Eratosthenes and his measurement
Eratosthenes as a Greek mathematician in the year 276 BC-194 BC, a remarkable astronomer and a geographer born in Cyrene and later died in the famed Ptolemaic Alexandria contributed significantly to evolution of science and mathematics. He is renowned for the significant work he did in devising the map system basing on the latitude and the longitude lines with computation of the earth’s size. Eratosthenes successfully studied at Alexandria for a few years then proceeds to Athens (Gow p. 27). In the late 236 BC, he had an appointment by the famed Ptolemy III Euergetes I as a librarian. He contributed significantly to science and mathematics and was the renowned friend to Archimedes. Around mid 255 BC, he made an invention of the armillary sphere recognized as the astronomical instrument used in the determination of celestial positions. It was applicable widely until the orrey invented in the late 18th century became active (Gow p. 27).
Eratosthenes was considered to have successfully adopted word geography as a description of the earth in the late 200 BC (Gow p. 27). Other contributions of Eratosthenes include the Sieve of the Eratosthenes applicable in the determination of prime numbers and making of the map containing the route of Nile towards Khartoum. The Sun-Earth distance measurement referred to as the astronomical unit of 804,000,000 stadia with one stadion varying from approximately 157 to 209 meter. The Earth-moon distance measurement which is approximately 780, 000 stadia was also one of his greatest contributions with the inclination of the ecliptic measurement having an angle error of seven. Eratosthenes managed to compile the star catalogue that contains 675 stars that was never preserved. Eratosthenes also made the amp that encompasses the entire world ranging from the Caspian Sea to the country of Ethiopia and British Isles to the Ceylon
The experiment of Eratosthenes
He is remembered for the remarkable contribution in the calculation of the circumference of the earth circa in the late 240 BC. This was through the application of the angle of elevation knowledge and the trigonometry of the sun at noon in Syene and Alexandria. The calculation bases on the relative assumption that the earth has a spherical nature with the Sun being far away to release parallel Rays (Nicastro p. 203). Some of the principal definitions used in his works as recorded in the measurement of the earth include the tropic of cancer is among the significant circles of the latitude marking the earth’s map. It is currently marked at 23° 26′ 22″ north of Equator. Local noon is the relative situation involving the sun being in the highest sky with a difference from the 12.00 noon.
Solstice is the astronomical event that considerably happens two times every year following the tilt of the earth’s axis inclined way or towards the sun. In the hemisphere lying to the north, the maximum inclination approximated towards the sun is 21 June referred to as the summer solstice. This is with a maximum inclination from the sun at around 21 December referred to as the winter solstice. For the hemisphere to the south, summer and winter solstices, are normally exchangeable (Nicastro p. 203). What is significant in the experiment of Eratosthenes is the significant fact with reference to the summer local noon, summer solstice, the sun Ray’s lay just overhead on the tropic of cancer.
Eratosthenes experiment
Eratosthenes had the relative knowledge that the summer solstice when at the local noon on the tropic of cancer would see the sun appear within the range of the zenith. This is directly overhead with an elevation of 90 degrees of the sun. He also considered the aspect from applying the vertical stick to measure the cast shadow in Alexandria with the Sun elevation being at 83 degrees south of zenith. Basing on the assumptions that Alexandria was directly due north of the Syene, Eratosthenes made a conclusion that, through the use of parallel lines in geometry, the relative distance from Syene to Alexandria had to be 7/360 of the earth’s total circumference. The distance amid the specified two cities was approximated from caravan traveling as of 5000 stadia. Eratosthenes later established the final value to be 700 stadia per every degree implying the circumference of about 252, 000 stadia (Eratosthenes p. 15).
The stadion exact size as used by Eratosthenes corresponds to an approximation 39,690 (km) to 46, 62 km. The earth’s circumference around the normal poles measures currently at around 40,008 (Eratosthenes p. 15). Through the calculation, Eratosthenes successfully calculated the difference in angles between the specified verticals at Alexandria and Syene. Eratosthenes formulated the experiment and realized that the angle value “A” is approximately 7.2 degrees. He also had the knowledge of the actual distance amid Syene and Alexandria to be 5040 stades. 7.2 degrees are approximately 7.2/360 around the globe. With the distance being 5040 stades, the total relative estimated distance around the earth is
The total distance = (360 degrees)/ (7.2 degrees) x 5040 stades
= 50 stades x 5040
= 250, 000 stades/ 2.5 x 105 stades
The argument of Eratosthenes
Eratosthenes bases majorly on assumptions as the relative hypotheses for the geometric approximation of the circumference of the earth. The method Eratosthenes devised forms the basis of the astrogeodetic complex method applicable in the measurement of earth’s circumference (Eratosthenes p. 15). The well-designed geometric argument of Eratosthenes illustration contributes to the justification of his contributions to the aspect of measurement. The claim of the argument states that the earth’s circumference approximation is about 250,000 stades (Fischer p. 1). This is proven through the aspect of Syene and Alexandria lying on approximately same meridian and the light rays straight from the sun striking the earth parallel. The relative distance between Syene and Alexandria being 5000 stades with the angle formation by shadow in Alexandria during the summer solstice equals the 1/50th the circle. These elements contribute to the evidence presented by Eratosthenes to prove that the earth has a spherical nature.
Through construction, the Alexandria staff is perpendicular to the corresponding ground. In the meridian plane, it has an orthogonal nature to the earth’s cross-sectional circle. Through the definition of the orthogonal aspect, the Alexandria staff has a perpendicular factor to the m line as the tangent aspect to the earth at the staff’s base (Fischer p. 1). Following the argument of Eratosthenes during the calculation, the Syene staff is perpendicular to the n line that is tangent to the earth relatively as the base of the staff. Through hypothesis, the shadow formed angle in Alexandria equals to the 1/50th of the circle. This results to a measure of 360°/50 = 7 1/5°. Following the alternating interior angle of Alexandria, the angle measure is approximately 360°/50 = 7 1/5° (Fischer p. 1).
Works cited
Eratosthenes, Duane W. Roller, and Strabo. Eratosthenes’ Geography. Princeton, N.J: Princeton University Press, 2010. Print.
Fischer, Irene. Another Look at Eratosthenes’ and Posidonius Determinations of the Earth’s Circumference.” Quarterly Journal of the Royal Astronomical Society 16 (2005): 152- 167.
Gow, Mary. Measuring the Earth: Eratosthenes and His Celestial Geometry. Berkeley Heights, NJ: Enslow, 2010. Print.
Nicastro, Nicholas. Circumference: Eratosthenes and the Ancient Quest to Measure the Globe. New York: St. Martin’s Press, 2008. Print.
