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The concept of a spherical Earth displaced earlier beliefs in a [[flat Earth]]: In early [[Mesopotamia]]n thought, the world was portrayed as a flat disk floating in the ocean, and this forms the premise for early Greek maps like those of [[Anaximander]] and [[Hecataeus of Miletus]]. Other speculations on the shape of Earth include a seven-layered [[ziggurat]] or [[cosmic mountain]], alluded to in the [[Avesta]] and ancient [[Persian Empire|Persian]] writings (see [[seven climes]]).
The concept of a spherical Earth displaced earlier beliefs in a [[flat Earth]]: In early [[Mesopotamia]]n thought, the world was portrayed as a flat disk floating in the ocean, and this forms the premise for early Greek maps like those of [[Anaximander]] and [[Hecataeus of Miletus]]. Other speculations on the shape of Earth include a seven-layered [[ziggurat]] or [[cosmic mountain]], alluded to in the [[Avesta]] and ancient [[Persian Empire|Persian]] writings (see [[seven climes]]).


As determined by modern instruments, a sphere approximates the Earth's shape to within one part in 300. The actual [[figure of the Earth]] is not a sphere, but an [[oblate ellipsoid]] with a flattening of 1/300, which matches the Earth's more precisely.
As determined by modern instruments, a sphere approximates the Earth's shape to within one part in 300. The actual [[figure of the Earth]] is not a sphere, but an [[oblate ellipsoid]] with a flattening of 1/300, which matches the Earth's shape more precisely.


== Early Greek and Hellenistic development ==
== Early Greek and Hellenistic development ==

Revision as of 17:30, 2 March 2010

Medieval artistic representation of a spherical Earth - with compartments representing earth, air, and water (c.1400).

Spherical Earth refers to the idea that the Earth is shaped as a sphere. The concept appeared in ancient Greek philosophy around the 6th century BCE,[1] but remained a matter of philosophical speculation until the 3rd century BC when Hellenistic astronomy established the spherical shape of the earth as a physical given. The Greek paradigm was gradually adopted, either directly or via later European science, by all major civilizations and world regions, becoming the dominant concept of today.[2][3][4][5] The final, practical evidence of the spherical shape of the earth provided its circumnavigation by the Portuguese explorer Ferdinand Magellan (1519−1521).

The concept of a spherical Earth displaced earlier beliefs in a flat Earth: In early Mesopotamian thought, the world was portrayed as a flat disk floating in the ocean, and this forms the premise for early Greek maps like those of Anaximander and Hecataeus of Miletus. Other speculations on the shape of Earth include a seven-layered ziggurat or cosmic mountain, alluded to in the Avesta and ancient Persian writings (see seven climes).

As determined by modern instruments, a sphere approximates the Earth's shape to within one part in 300. The actual figure of the Earth is not a sphere, but an oblate ellipsoid with a flattening of 1/300, which matches the Earth's shape more precisely.

Early Greek and Hellenistic development

Pythagoras

Early Greek philosophers alluded to a spherical Earth, though with some ambiguity.[6] This idea influenced Pythagoras (b. 570 BCE), who saw harmony in the universe and sought to explain it. He reasoned that Earth and the other planets must be spheres, since the most harmonious geometric solid form is a sphere.[1] After the fifth century BCE, no Greek writer of repute thought the world was anything but round.[6]

Herodotus

In The Histories, written 431 BCE - 425 BCE, Herodotus dismisses a report of the sun observed shining from the north. This arises when discussing the circumnavigation of Africa undertaken c. 615-595 BCE. (The Histories, 4.43) His dismissive comment attests to a widespread ignorance of the ecliptic's inverted declination in a southern hemisphere.

Plato

Plato (427 BCE - 347 BCE) travelled to southern Italy to study Pythagorean mathematics. When he returned to Athens and established his school, Plato also taught his students that Earth was a sphere. If man could soar high above the clouds, Earth would resemble "one of those balls which have leather coverings in twelve pieces, and is decked with various colours, of which the colours used by painters on earth are in a manner samples." (Phaedo, 110b)

Aristotle
When a ship is at the horizon its lower part is invisible due to Earth's curvature. This was one of the first arguments favoring a round-Earth model.

Aristotle (384 BCE - 322 BCE) was Plato's prize student and "the mind of the school." Aristotle observed "there are stars seen in Egypt and [...] Cyprus which are not seen in the northerly regions." Since this could only happen on a curved surface, he too believed Earth was a sphere "of no great size, for otherwise the effect of so slight a change of place would not be quickly apparent." (De caelo, 298a2-10)

Aristotle provided physical and observational arguments supporting the idea of a spherical Earth:

  • Every portion of the Earth tends toward the center until by compression and convergence they form a sphere. (De caelo, 297a9-21)
  • Travelers going south see southern constellations rise higher above the horizon; and
  • The shadow of Earth on the Moon during a lunar eclipse is round. (De caelo, 297b31-298a10)

The concepts of symmetry, equilibrium and cyclic repetition permeated Aristotle's work. In his Meteorology he divided the world into five climatic zones: two temperate areas separated by a torrid zone near the equator, and two cold inhospitable regions, "one near our upper or northern pole and the other near the ... southern pole," both impenetrable and girdled with ice (Meteorologica, 362a31-35). Although no humans could survive in the frigid zones, inhabitants in the southern temperate regions could exist.

Eratosthenes

Eratosthenes (276 BCE - 194 BCE) estimated Earth's circumference around 240 BCE. He had heard that in Syene the Sun was directly overhead at the summer solstice whereas in Alexandria it still cast a shadow. Using the differing angles the shadows made as the basis of his trigonometric calculations he estimated a circumference of around 250,000 stades. The length of a 'stade' is not precisely known, but Eratosthenes' figure only has an error of around five to ten percent.[7][8]

Seleucus of Seleucia

Seleucus of Seleucia (c. 190 BC), who lived in the Seleucia region of Mesopotamia, stated that the Earth is spherical (and actually orbits the Sun, influenced by the heliocentric theory of Aristarchus of Samos).

Claudius Ptolemy

Claudius Ptolemy (AD 90 - 168) lived in Alexandria, the centre of scholarship in the second century. Around 150, he produced his eight-volume Geographia.

The first part of the Geographia is a discussion of the data and of the methods he used. As with the model of the solar system in the Almagest, Ptolemy put all this information into a grand scheme. He assigned coordinates to all the places and geographic features he knew, in a grid that spanned the globe. Latitude was measured from the equator, as it is today, but Ptolemy preferred to express it as the length of the longest day rather than degrees of arc (the length of the midsummer day increases from 12h to 24h as you go from the equator to the polar circle). He put the meridian of 0 longitude at the most western land he knew, the Canary Islands.

Geographia indicated the countries of "Serica" and "Sinae" (China) at the extreme right, beyond the island of "Taprobane" (Sri Lanka, oversized) and the "Aurea Chersonesus" (Southeast Asian peninsula).

Ptolemy also devised and provided instructions on how to create maps both of the whole inhabited world (oikoumenè) and of the Roman provinces. In the second part of the Geographia he provided the necessary topographic lists, and captions for the maps. His oikoumenè spanned 180 degrees of longitude from the Canary Islands in the Atlantic Ocean to China, and about 81 degrees of latitude from the Arctic to the East Indies and deep into Africa. Ptolemy was well aware that he knew about only a quarter of the globe.

Global diffusion

From its Greek origins, the idea of a spherical earth, along with much of Greek astronomical thought, slowly spread across the globe and ultimately became the adopted view in all major astronomical traditions:[2][3][4][5]

Spread to the west

Spherical earth with the four seasons. Illustration in 12th century book Liber Divinorum Operum by Hildegard of Bingen

In the west, the idea came naturally to the Romans through the lengthy process of cross-fertilization with Greek civilization. Many Roman authors such as Cicero and Pliny refer in their works to the rotundity of the earth as a matter of course.[9]

Treated as self-evident by early Christian authorities like Aurelius Augustinus, the idea survived into the medieval corpus of knowledge and became increasingly traceable with the rise of scholasticism and medieval learning.[9] A non-exhaustive list of more than a hundred Latin and vernacular writers from late antiquity to the 15th century, who were aware that the earth was spherical, has been compiled by Reinhard Krüger, professor for Romance literature at the University of Stuttgart:[9]

Template:Seemore

  • Kings and politicians

Brunetto Latini, Visigoth king Sisebut, King Alfred of the Anglo-Saxons, Alfonso X of Castile

  • Church fathers, popes, bishops, priests, members of religious orders

Basil of Caesarea, Ambrose of Milan, Aurelius Augustinus, Paulus Orosius, Jordanes, Cassiodorus, Isidore of Seville, Beda Venerabilis, Theodulf of Orléans, Vergilius of Salzburg, Irish monk Dicuil, Rabanus Maurus, Remigius of Auxerre, Johannes Scotus Eriugena, Leo of Naples Template:De icon, Gerbert d’Aurillac (Pope Sylvester II), Notker the German of Sankt-Gallen, Hermann the lame, Hildegard von Bingen, Petrus Abaelardus, Honorius Augustodunensis, Gautier de Metz, Adam of Bremen, Albertus Magnus, Thomas Aquinas, Berthold of Regensburg, Meister Eckhart, Enea Silvio Piccolomini (Pope Pius II)

  • Theologians, philosophers and encyclopedists

Ampelius, Chalcidius, Macrobius, Martianus Capella, Boethius, Guillaume de Conches, Philippe de Thaon Template:Fr icon, Abu-Idrisi, Bernardus Silvestris, Petrus Comestor, Thierry de Chartres, Gautier de Châtillon, Alexander Neckam, Alain de Lille, Averroes, Moshe ben Maimon, Lambert de Saint-Omer Template:De icon, Gervasius of Tilbury, Robert Grosseteste, Johannes de Sacrobosco, Thomas de Cantimpré, Peire de Corbian, Vincent de Beauvais, Robertus Anglicus, Juan Gil de Zámora Template:Es icon, Perot de Garbelei Template:De icon (divisiones mundi), Roger Bacon, Ristoro d'Arezzo, Cecco d'Ascoli, Fazio degli Uberti Template:It icon, Levi ben Gershon, Konrad of Megenberg, Nicole Oresme, Petrus Aliacensis, Alfonso de la Torre Template:De icon, Toscanelli

  • Poets, travellers, printers, seafarers, merchants

Snorri Sturluson, Marco Polo, Dante Alighieri, Brochard the German Template:De icon, Jean de Meung, Jean de Mandeville, Christine de Pizan, Geoffrey Chaucer, William Caxton, Martin Behaim, Christopher Columbus |}

Portuguese exploration of Africa and Asia, Columbus voyage to the Americas (1492) and finally Ferdinand Magellan's circumnavigation of the earth (1519-21) provided the final, practical proofs for the global shape of the earth, while European colonists planted the idea in the American colonies.

Spread to the east

With the rise of Greek culture in the east, Hellenistic astronomy filtered eastwards to ancient India where its profound influence became apparent in the early centuries AD.[10] The Greek concept of a spherical earth surrounded by the spheres of planets, vehemently supported by astronomers like Varahamihira and Brahmagupta, supplanted the long-standing Indian cosmological belief into a flat and circular earth disk.[10][11]

Islamic astronomy inherited the idea of a spherical earth from the Greek astronomical tradition.[12] The Islamic theoretical framework largely relied on the fundamental contributions of Aristotle (De caelo) and Ptolemy (Almagest), both of which worked with the premise that the earth was spherical and at the center of the universe (geocentric model).[12]

In the 17th century, the idea of a spherical earth, now considerably advanced by Western Astronomy, ultimately spread to Ming China, when Jesuit missionaries, who held high positions as astronomers at the imperial court, successfully challenged the Chinese belief that the earth was flat and square.[13][14][15]

Indian scholars

Rigveda

While Indian cosmology long maintained the belief in a disk-shaped earth, K. V. Sarma mentions in passing that the Rigveda contains speculations on the idea of a spherical self-supporting earth, but does not give more details nor references.[16] It is impossible to date when any particular piece of the Rigveda was composed. While the core of the text probably dates to around 1400 BCE, many revisions and redactions are thought to have been incorporated around 600 BCE. Nothing was written down until the 3rd century BCE at earliest, and possibly much later. See Dating and historical context of the Rigveda.

Aryabhata

The works of the classical Indian astronomer and mathematician, Aryabhata (476-550 AD), deal with the sphericity of the Earth and the motion of the planets. The final two parts of his Sanskrit magnum opus, the Aryabhatiya, which were named the Kalakriya ("reckoning of time") and the Gola ("sphere"), state that the Earth is spherical and that its circumference is 4,967 yojanas, which in modern units is 39,968 km, which is only 62 km less than the current value of 40,030 km.[17][18] He also stated that the apparent rotation of the celestial objects was due to the actual rotation of the Earth, calculating the length of the sidereal day to be 23 hours, 56 minutes and 4.1 seconds, which is also surprisingly accurate. It is likely that Aryabhata's results influenced European astronomy, because the 8th century Arabic version of the Aryabhatiya was translated into Latin in the 13th century.

Anania Shirakatsi

Anania Shirakatsi (Armenian: Անանիա Շիրակացի), also known as Ananias of Sirak, (610–685) was an Armenian scholar, mathematician, and geographer. His most famous works are Geography Guide (‘Ashharatsuyts’ in Armenian), and Cosmography (Tiezeragitutiun). He described the world as "being like an egg with a spherical yolk (the globe) surrounded by a layer of white (the atmosphere) and covered with a hard shell (the sky)." [19]

Islamic scholars

Early Islamic scholars recognized earth's sphericity[20], leading Muslim mathematicians to develop spherical trigonometry[21] in order to further mensuration and to calculate the distance and direction from any given point on the Earth to Mecca. This determined the Qibla, or Muslim direction of prayer.

Al-Ma'mun

Around 830 CE, Caliph Al-Ma'mun commissioned a group of Muslim astronomers and Muslim geographers to measure the distance from Tadmur (Palmyra) to al-Raqqah, in modern Syria. They found the cities to be separated by one degree of latitude and the distance between them to be 66 2/3 miles and thus calculated the Earth's circumference to be 24,000 miles.[22]

Another estimate given by his astronomers was 56 2/3 Arabic miles per degree, which corresponds to 111.8 km per degree and a circumference of 40,248 km, very close to the currently modern values of 111.3 km per degree and 40,068 km circumference, respectively.[23]

Al-Farghānī

Al-Farghānī (Latinized as Alfraganus) was a Persian astronomer of the 9th century involved in measuring the diameter of the Earth, and commissioned by Al-Ma'mun. His estimate given above for a degree (56 2/3 Arabic miles) was much more accurate than the 60 2/3 Roman miles (89.7 km) given by Ptolemy. Christopher Columbus uncritically used Alfraganus's figure as if it were in Roman miles instead of in Arabic miles, in order to prove a smaller size of the Earth than that propounded by Ptolemy.[24]

Biruni

Abu Rayhan Biruni (973-1048) solved a complex geodesic equation in order to accurately compute the Earth's circumference, which was close to modern values of the Earth's circumference.[25][26] His estimate of 6,339.9 km for the Earth radius was only 16.8 km less than the modern value of 6,356.7 km. In contrast to his predecessors who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, Biruni developed a new method of using trigonometric calculations based on the angle between a plain and mountain top which yielded more accurate measurements of the Earth's circumference and made it possible for it to be measured by a single person from a single location.[27][28][29] Biruni's method was intended to avoid "walking across hot, dusty deserts" and the idea came to him when he was on top of a tall mountain in India.[30] From the top of the mountain, he sighted the dip angle which, along with the mountain's height (which he calculated beforehand), he applied to the law of sines formula.[31][30] He also made use of algebra to formulate trigonometric equations and used the astrolabe to measure angles.[32]

John J. O'Connor and Edmund F. Robertson write in the MacTutor History of Mathematics archive:

"Important contributions to geodesy and geography were also made by Biruni. He introduced techniques to measure the earth and distances on it using triangulation. He found the radius of the earth to be 6339.6 km, a value not obtained in the West until the 16th century. His Masudic canon contains a table giving the coordinates of six hundred places, almost all of which he had direct knowledge."[33]

Geodesy

Geodesy, also called geodetics, is the scientific discipline that deals with the measurement and representation of the Earth, its gravitational field and geodynamic phenomena (polar motion, Earth tides, and crustal motion) in three-dimensional time-varying space.

Geodesy is primarily concerned with positioning and the gravity field and geometrical aspects of their temporal variations, although it can also include the study of Earth's magnetic field. Especially in the German speaking world, geodesy is divided into geomensuration ("Erdmessung" or "höhere Geodäsie"), which is concerned with measuring the Earth on a global scale, and surveying ("Ingenieurgeodäsie"), which is concerned with measuring parts of the surface.

The Earth's shape can be thought of in at least two ways;

  • as the shape of the geoid, the mean sea level of the world ocean; or
  • as the shape of Earth's land surface as it rises above and falls below the sea.

As the science of geodesy measured Earth more accurately, the shape of the geoid was first found not to be a perfect sphere but to approximate an oblate spheroid, a specific type of ellipsoid. More recent measurements have measured the geoid to unprecedented accuracy, revealing mass concentrations beneath Earth's surface.

Spherical models

Main article: Earth radius
The Earth as seen from the Apollo 17 mission.

There are several reasonable ways to approximate Earth's shape as a sphere. Most preserve a different feature of an ellipsoid that closely models the real Earth in order to compute the radius of the spherical model. All examples in this section assume the WGS 84 datum, with an equatorial radius a of 6,378.137 km and a polar radius b of 6,356.752 km. A sphere being a gross approximation of the spheroid, which itself is an approximation of the geoid, units are given here in kilometers rather than the millimeter resolution appropriate for geodesy.

  • Preserve the equatorial circumference. This is simplest, being a sphere with circumference identical to the equatorial circumference of the ellipsoidal model. Since the circumference is the same, so is the radius, at 6,378.137 km.
  • Preserve the lengths of meridians. This requires an elliptic integral to find, given the polar and equatorial radii: . A sphere preserving the lengths of meridians has a rectifying radius of 6,367.449 km. This can be approximated using the elliptical quadratic mean: , about 6,367.454 km; or even just the mean of the two axes: , about 6,367.445 km.
  • Preserve the surface area of the ellipsoidal model. This gives the authalic radius (denoted by the International Union of Geodesy and Geophysics): , or 6,371.007 km.
  • Preserve the volume of the ellipsoidal model. This volumetric radius (denoted by the IUGG) is computed as: , or 6,371.001 km.
  • Synthesize some mean radius. The IUGG defines the mean radius (denoted ) to be , giving 6,371.009 km. There are other ways to define the mean.

See also

References

  1. ^ a b Dicks, D.R. (1970). Early Greek Astronomy to Aristotle. Ithaca, N.Y.: Cornell University Press. pp. 72–198. ISBN 9780801405617.
  2. ^ a b Continuation into Roman and medieval thought: Reinhard Krüger: "Materialien und Dokumente zur mittelalterlichen Erdkugeltheorie von der Spätantike bis zur Kolumbusfahrt (1492)"
  3. ^ a b Direct adoption of the Greek concept by Islam: Ragep, F. Jamil: "Astronomy", in: Krämer, Gudrun (ed.) et al.: Encyclopaedia of Islam, THREE, Brill 2010, without page numbers
  4. ^ a b Direct adoption by India: D. Pingree: "History of Mathematical Astronomy in India", Dictionary of Scientific Biography, Vol. 15 (1978), pp. 533−633 (554f.); Glick, Thomas F., Livesey, Steven John, Wallis, Faith (eds.): "Medieval Science, Technology, and Medicine: An Encyclopedia", Routledge, New York 2005, ISBN0-415-96930-1, p. 463
  5. ^ a b Adoption by China via European science: Jean-Claude Martzloff, “Space and Time in Chinese Texts of Astronomy and of Mathematical Astronomy in the Seventeenth and Eighteenth Centuries”, Chinese Science 11 (1993-94): 66-92 (69) and Christopher Cullen, "A Chinese Eratosthenes of the Flat Earth: A Study of a Fragment of Cosmology in Huai Nan tzu 淮 南 子", Bulletin of the School of Oriental and African Studies, Vol. 39, No. 1 (1976), pp. 106-127 (107)
  6. ^ a b Dicks, D.R. (1970). Early Greek Astronomy to Aristotle. Ithaca, N.Y.: Cornell University Press. p. 68. ISBN 9780801405617.
  7. ^ "JSC NES School Measures Up". NASA. 11 April 2006. Retrieved 24 January 2008.
  8. ^ "The Round Earth". NASA. 12 December 2004. Retrieved 24 January 2008.
  9. ^ a b c Reinhard Krüger: "Materialien und Dokumente zur mittelalterlichen Erdkugeltheorie von der Spätantike bis zur Kolumbusfahrt (1492)"
  10. ^ a b D. Pingree: "History of Mathematical Astronomy in India", Dictionary of Scientific Biography, Vol. 15 (1978), pp. 533−633 (533, 554f.)
  11. ^ Glick, Thomas F., Livesey, Steven John, Wallis, Faith (eds.): "Medieval Science, Technology, and Medicine: An Encyclopedia", Routledge, New York 2005, ISBN0-415-96930-1, p. 463
  12. ^ a b Ragep, F. Jamil: "Astronomy", in: Krämer, Gudrun (ed.) et al.: Encyclopaedia of Islam, THREE, Brill 2010, without page numbers
  13. ^ "Jean-Claude Martzloff, "Space and Time in Chinese Texts of Astronomy and of Mathematical Astronomy in the Seventeenth and Eighteenth Centuries", Chinese Science 11 (1993-94): 66-92 (69)" (PDF).
  14. ^ Christopher Cullen, “Joseph Needham on Chinese Astronomy”, Past and Present, No. 87. (May, 1980), pp. 39-53 (42 & 49)
  15. ^ Christopher Cullen, "A Chinese Eratosthenes of the Flat Earth: A Study of a Fragment of Cosmology in Huai Nan tzu 淮 南 子", Bulletin of the School of Oriental and African Studies, Vol. 39, No. 1 (1976), pp. 106-127 (107-109)
  16. ^ Sarma, K.V. (2008), "Astronomy in India", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition) edited by Helaine Selin, pp. 317–321, Springer, ISBN 9781402045592.
  17. ^ Aryabhata_I biography
  18. ^ http://www.gongol.com/research/math/aryabhatiya The Aryabhatiya: Foundations of Indian Mathematics
  19. ^ Hewson, Robert H. "Science in Seventh-Century Armenia: Ananias of Sirak, Isis, Vol. 59, No. 1, (Spring, 1968), pp. 32-45
  20. ^ Muhammad Hamidullah. L'Islam et son impulsion scientifique originelle, Tiers-Monde, 1982, vol. 23, n° 92, p. 789.
  21. ^ David A. King, Astronomy in the Service of Islam, (Aldershot (U.K.): Variorum), 1993.
  22. ^ Gharā'ib al-funūn wa-mulah al-`uyūn (The Book of Curiosities of the Sciences and Marvels for the Eyes), 2.1 "On the mensuration of the Earth and its division into seven climes, as related by Ptolemy and others," (ff. 22b-23a)[1]
  23. ^ Edward S. Kennedy, Mathematical Geography, pp=187-8, in (Rashed & Morelon 1996, pp. 185–201)
  24. ^ Felipe Fernández-Armesto, Columbus and the conquest of the impossible, pp. 20-1, Phoenix Press, 1974.
  25. ^ Khwarizm, Foundation for Science Technology and Civilisation.
  26. ^ James S. Aber (2003). Alberuni calculated the Earth's circumference at a small town of Pind Dadan Khan, District Jhelum, Punjab, Pakistan.Abu Rayhan al-Biruni, Emporia State University.
  27. ^ Lenn Evan Goodman (1992), Avicenna, p. 31, Routledge, ISBN 041501929X.
  28. ^ Behnaz Savizi (2007), "Applicable Problems in History of Mathematics: Practical Examples for the Classroom", Teaching Mathematics And Its Applications, 26 (1), Oxford University Press: 45–50, doi:10.1093/teamat/hrl009 (cf. Behnaz Savizi. "Applicable Problems in History of Mathematics; Practical Examples for the Classroom". University of Exeter. Retrieved 2010-02-21.)
  29. ^ Beatrice Lumpkin (1997), Geometry Activities from Many Cultures, Walch Publishing, pp. 60 & 112-3, ISBN 0825132851 [2]
  30. ^ a b Beatrice Lumpkin (1997), Geometry Activities from Many Cultures, Walch Publishing, pp. 60 & 112-3, ISBN 0825132851 [3]
  31. ^ Behnaz Savizi (2007), "Applicable Problems in History of Mathematics: Practical Examples for the Classroom", Teaching Mathematics And Its Applications, 26 (1), Oxford University Press: 45–50, doi:10.1093/teamat/hrl009 (cf. Behnaz Savizi. "Applicable Problems in History of Mathematics; Practical Examples for the Classroom". University of Exeter. Retrieved 2010-02-21.)
  32. ^ Jim Al-Khalili, The Empire of Reason 2/6 (Science and Islam - Episode 2 of 3) on YouTube, BBC
  33. ^ O'Connor, John J.; Robertson, Edmund F., "Al-Biruni", MacTutor History of Mathematics Archive, University of St Andrews
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