(Eratosthenes assumed that the earth was perfectly circular). He accepted al-Ma'mūn's figure of \( 56 {2 \over 3} \) miles for a degree, which had come to Europe through the influential Latin translation of al-Farghānī (Latin, Alfraganus), available since the twelfth century. Earth's circumference was first accurately measured more than 2,000 years ago by the Greek astronomer Eratosthenes, who at the time lived in the Egyptian city of Alexandria.
Already have an account? If \( a, b, c \) are the three sides of a triangle, and \( A, B, C \) the corresponding angles, then the Law of Sines says that: Disclaimer, Eratosthenes - Measuring the Circumference of the Earth in 240 BC. One could look down the well and see his or her own shadow at the bottom, but no shadow from the sides of the well. Calculating the circumference of the Earth to within 100 miles is remarkably accurate given that it was done around 240 BC. ), Transaction Publishers (1995), ISBN 1-56000-210-7. Robert Helmert

Moreover, the 5000 stadia distance between Syene and Alexandria was probably the best measurement Eratosthenes had available at the time. He had heard that in the nearby town of Syene midday sunlight shines straight down to the bottom of deep wells on the same day each year, indicating that the Sun was directly overhead in Syene. But in Alexandria at the same hour pointers on sundials do cast a shadow, since this city is located further north than Syene. The foremost mathematical authorities in Europe for hundreds of years were Boethius (d. 524 / 525) and the Venerable Bede (d. 735); the first held forth on figurate numbers, the second on finger counting (Mathematics, by Michael S. Mahoney; Chapter 5 in Science in the Middle Ages, by David C. Lindberg (ed. Eratosthenes was born around 276 B.C., which is … \[ k^2 = a^2 + d^2 - 2ad \cos{\alpha}.\] \[ a = {{b \cdot \sin{A}} \over {\sin{B}}}, \hskip{20pt} c = {{b \cdot \sin{C}} \over {\sin{B}}}.\] ∠B \angle B∠B represents the angle subtended at the Earth’s center by the arc representing the distance between Alexandria and Syene. Since angle B is 1/501/501/50th of a circle and sweeps an arc at the earth’s surface of 5,000 stades, the Earth’s circumference would be 5,000×50=250,000 5,000\times 50 = 250,0005,000×50=250,000 stades. About 2000 years ago, an ancient Greek scholar, philosopher, poet, and mathematician named Eratosthenes used staggeringly simple geometry to calculate the size of Earth with very high accuracy. Therefore, simply multiplied by 50. He noted that in Alexandria at the same time, during the same day, sunlight fell at an angle of about 7.2 degrees from the vertical. The screw micrometer was another major innovation percolating as the 17th century advanced. Gratwick explains how averages were probably taken on the scaphe to account for the half degree or so of the sun's angular diameter. "There is no denying that a scientific dark age had descended on western Europe", p. 12. I will not participate in it. C = 360/7 x d (C-> circumference of Earth, d -> the distance between Syene and Alexandria) The distance between the cities was known from caravan travelings to be about 5,000 stadia (see notes 2).

\;\; 35' & {35 \over 60} \cdot {\pi \over 180} = 0.0101811 & 19,294 Would you like to be notified whenever we have a new post? That is the wonder of ancient astronomy, that they could do much of anything with their mostly eyeball techniques (more on this in Dicks, p. 9) That Eratosthenes could get the angle at Alexandria within a degree (and he did) is a small miracle, considering the tools at hand. Such handbooks were common in that day and for years to come in Christian Europe — diluted, confused, and plagiarized but sometimes extensive third-hand accounts for educated but non-expert readers as a barren dark age descended.[5]. Cleomedes' third assumption, that the rays of the sun all strike the earth in parallel, is also an approximation, the sun's disk extending over about 30 minutes of arc (half a degree) as viewed from the earth — this is called the sun's angular diameter. Log in here. Eratosthenes' method to calculate the Earth's circumference has been lost; what has been preserved is the simplified version described by Cleomedes to popularise the discovery. In the town of Syene (now Aswan) in Egypt, it was well known that at noon on the day of the Summer solstice, light from the sun would shine directly down a local well. Or take their aqueducts, including stretches falling at a grade of 1 in 5,000 at the time of Eratosthenes[7]. If he could determine the distance from Syene to Alexandria, all he would have to do to determine the circumference of the Earth was multiply this distance by 50. That is the required geometry, all of it. To see this, consider the simple two-triangle configuration pictured here — \( \bigtriangleup ABC \) above and \( \bigtriangleup ABC' \) below, sharing common side \( AB \). The science here is absolutely solid and elegantly simple. The point is rather to highlight Eratosthenes' (and al-Ma'mūn's) prescience, indeed their wisdom, in looking at the inclination of celestial bodies to measure the earth.
He determined that Alexandria is about 5,000 stadia (489 miles) North of Syene. He was born in Cyrene, Libya c. 276 BC.

If you want to join our community I would advise you to refer to, What is the Orbital and Rotational Speed of Earth? Like so many ancient works, virtually nothing remains of Eratosthenes' The Measurement of the World[3]. Last time we looked at a couple questions about proving the earth is round, which led into questions about how Eratosthenes measured the earth (though that in itself did not prove the earth is not flat). By noting the angles of shadows in two cities on the Summer Solstice , and by performing the right calculations using his knowledge of geometry and the distance between the cities, Eratosthenes was able to make a remarkably accurate calculation of the circumference of Earth. Now since the two cities are located below a meridian (a great circle), if we draw an arc from the tip of the pointer's shadow on the sundial at Alexandria round to the base of the pointer, this arc will be a section of the great circle in the sundial's bowl, since the sundial's bowl is located beneath a great circle.If we next conceive of straight lines produced through the Earth from each of the pointers, they will coincide at the center of the Earth. It covers an extremely large portion of Earth, despite when it was created. Around 1020, al-Bīrūnī took an entirely different approach.

), University of Chicago Press (1978), ISBN 0-226-48232-4). ^ 1. Frost Fairs on the Thames River Eratosthenes Measures the Earth May 28, 2020 May 27, 2020 / Geometry / Real life / By Dave Peterson Last time we looked at a couple questions about proving the earth is round , which led into questions about how Eratosthenes measured the earth (though that in itself did not prove the earth is not flat). The Determination of the Coordinates of Positions for the Correction of Distances Between Cities: a Translation from the Arabic of al-Bīrūnī's Kitab Tahdid, by Jamil Ali, American University of Beirut (1967).

Fifth, assume that the arcs of circles standing on equal angles are similar, that is, have the same proportion (namely, the same ratio) to their own circles, as is also demonstrated by geometers. We have to assume the Earth is a sphere to use that equation at all. \;\; 34' & {34 \over 60} \cdot {\pi \over 180} = 0.0098902 & 20,446\\[4pt] (Kennedy Commentary, p 37) al-Bīrūnī's gives two values for the earth's radius in different places: \( 12,803,337;2,9 \) cubits and \( 12,851,369;50,42 \) cubits. Aryabhata was an Indian mathematician who lived from 476 to 550 AD and accurately wrote of the earth as being spherical. Eratosthenes - Measuring the Circumference of the Earth in 240 BC Eratosthenes was a Greek scientific writer, astronomer, and poet, who is credited with making the first approximation of the size of the Earth for which any details are know. Reverse-engineering the experiment, Eratosthenes was off by \( 7.59^{\circ} - 7.2^{\circ} = 0.39^{\circ} \), about \( 1/3^{\circ} \), so it seems his instruments were capable of accuracy on that order. The combined error was massive, Japan being closer to Spain to the east than to the west.

), Routledge (1996), ISBN 0-415-12410-7. But read Cleomedes for yourself, as translated by Bowen and Todd (pp. Mesure de la Terre (Measure of the Earth), by Jean Picard (1671). Cleomedes' fifth item, that arcs over equal angles are similar, is so fundamental that it is a definition in Euclid (Euclid III, definition 11). But the point is irrelevant, since the experiment will measure the great circle connecting Syene and Aswan, whether a meridian or not. (Kennedy Commentary, p 143) From the foregoing, this is nonsense, accuracy within \( 10\% \) being very much in doubt, not to say accuracy to within \( 1 / 3600 \) cubit, the length of this unit too being open to question as with the stadion and Arab mile. Because there is a scholarly debate as to what the definition of a “stade” is, there is no way of knowing exactly how close Eratosthenes was to today's currently accepted value for the circumference of the earth. was.

Snellius overtly retraced Eratosthenes, introducing triangulation as a good way to measure distances on earth. In 1906, the German geodesist Friedrich Robert Helmert determined that the earth was a global ellipsoid and calculated it to within 100 meters, 0.002% of modern measurements. The Origins of Summit Crosses ^ 11. \]

Much of the information in the map was probably passed from word of mouth and through the compilation of many observations made by many people over a long period of time. Eratosthenes records this angle to be "a fiftieth of a circle", as measuring angles in degrees had not yet been adopted from the Babylonians at this point.


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