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| {{Geodesy}}
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| <!-- yet at Geodesy template :''See also the main article on '''[[Geodesy]]'''.'' -->
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| [[File:A Brief History of Geodesy.ogv|thumb|350px|NASA/Goddard Space Flight Center's brief history of geodesy.<ref>{{cite AV media
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| | people = NASA/Goddard Space Flight Center
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| | year = 3 February 2012
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| | title = Looking Down a Well: A Brief History of Geodesy
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| | medium = digital animation
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| | language = English
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| | url = http://svs.gsfc.nasa.gov/vis/a010000/a010900/a010910/
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| | accessdate = 6 February 2014
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| | archiveurl = https://commons.wikimedia.org/wiki/File:A_Brief_History_of_Geodesy.ogv
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| | archivedate = 3 February 2012
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| | publisher = NASA/Goddard Space Flight Center
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| | id = Goddard Multimedia Animation Number: 10910
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| '''Geodesy''' (/dʒiːˈɒdɨsi/),[1] also named '''geodetics''', is the scientific discipline that deals with the measurement and representation of the Earth.
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| Early ideas about the figure of the Earth held the Earth to be flat (see [[flat earth]]), and the heavens a physical dome spanning over it. Two early arguments for a spherical Earth were that lunar eclipses were seen as circular shadows which could only be caused by a spherical Earth, and that [[Polaris]] is seen lower in the sky as one travels South.
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| ==Hellenic world==
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| The [[Ancient Greece|early Greeks]], in their speculation and theorizing, ranged from the flat disc advocated by [[Homer]] to the spherical body postulated by [[Pythagoras]] — an idea supported later by [[Aristotle]]. Pythagoras was a mathematician and to him the most perfect figure was a [[sphere]]. He reasoned that the gods would create a perfect figure and therefore the Earth was created to be spherical in shape. [[Anaximenes of Miletus|Anaximenes]], an early Greek scientist, believed strongly that the Earth was [[rectangular]] in shape. | |
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| Since the spherical shape was the most widely supported during the Greek Era, efforts to determine its size followed. [[Plato]] determined the circumference of the Earth (which is slightly over 40,000 km) to be 400,000 stadia (between 62,800 km/39,250 mi and 74,000 km/46,250 mi ) while [[Archimedes]] estimated 300,000 stadia ( 55,500 [[kilometre]]s/34,687 [[mile]]s ), using the Hellenic [[stadia (length)|stadion]] which scholars generally take to be 185 meters or 1/10 of a [[geographical mile]]. Plato's figure was a guess and Archimedes' a more conservative approximation.
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| ===Hellenistic world===
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| In Egypt, a Greek scholar and philosopher, [[Eratosthenes]] (276 BC– 195 BC), is said to have made more explicit measurements. He had heard that on the longest day of the summer [[solstice]], the midday sun shone to the bottom of a well in the town of Syene ([[Aswan]]). At the same time, he observed the sun was not directly overhead at [[Alexandria]]; instead, it cast a shadow with the vertical equal to 1/50th of a circle (7° 12'). To these observations, Eratosthenes applied certain "known" facts (1) that on the day of the summer solstice, the midday sun was directly over the [[Tropic of Cancer]]; (2) Syene was on this tropic; (3) Alexandria and Syene lay on a direct north-south line; (4) The sun was a relatively long way away ([[Astronomical unit]]). Legend has it that he had someone walk from Alexandria to Syene to measure the distance: that came out to be equal to 5000 stadia or (at the usual Hellenic 185 meters per stadion) about 925 kilometres.
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| [[File:Eratosthenes' method for determining the size of the Earth.svg|thumb|200px|right|Eratosthenes' method for determining the size of the Earth]]
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| From these observations, measurements, and/or "known" facts, Eratosthenes concluded that, since the angular deviation of the sun from the [[vertical direction]] at Alexandria was also the angle of the subtended arc (see illustration), the linear distance between Alexandria and Syene was 1/50 of the circumference of the Earth which thus must be 50×5000 = 250,000 stadia or probably 25,000 geographical miles. The circumference of the Earth is 24,902 miles (40,075.16 km). Over the poles it is more precisely 40,008 km or 24,860 statute miles. The actual unit of measure used by Eratosthenes was the stadion. No one knows for sure what his stadion equals in modern units, but some say that it was the Hellenic 185-meter stadion.
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| Had the experiment been carried out as described, it would not be remarkable if it agreed with actuality. What is remarkable is that the result was probably about one sixth too high. His measurements were subject to several inaccuracies: (1) though at the summer solstice the noon sun is overhead at the Tropic of Cancer, Syene was not exactly on the tropic (which was at 23° 43' latitude in that day) but about 22 geographical miles to the north; (2) the difference of latitude between Alexandria (31.2 degrees north latitude) and Syene (24.1 degrees) is really 7.1 degrees rather than the perhaps rounded (1/50 of a circle) value of 7° 12' that Eratosthenes used; (4) the actual solstice [[zenith distance]] of the noon sun at Alexandria was 31° 12' − 23° 43' = 7° 29' or about 1/48 of a circle not 1/50 = 7° 12', an error closely consistent with use of a vertical [[gnomon]] which fixes not the sun's center but the solar upper [[limb darkening|limb]] 16' higher; (5) the most importantly flawed element, whether he measured or adopted it, was the latitudinal distance from Alexandria to Syene (or the true Tropic somewhat further south) which he appears to have overestimated by a factor that relates to most of the error in his resulting circumference of the earth.
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| A parallel later ancient measurement of the size of the earth was made by another Greek scholar, [[Posidonius]]. He is said to have noted that the star [[Canopus]] was hidden from view in most parts of Greece but that it just grazed the horizon at Rhodes. Posidonius is supposed to have measured the elevation of Canopus at Alexandria and determined that the angle was 1/48th of circle. He assumed the distance from Alexandria to Rhodes to be 5000 stadia, and so he computed the Earth's circumference in stadia as 48 times 5000 = 240,000.<ref>Cleomedes 1.10</ref> Some scholars see these results as luckily semi-accurate due to cancellation of errors. But since the Canopus observations are both mistaken by over a degree, the "experiment" may be not much more than a recycling of Eratosthenes's numbers, while altering 1/50 to the correct 1/48 of a circle. Later either he or a follower appears to have altered the base distance to agree with Eratosthenes's Alexandria-to-Rhodes figure of 3750 stadia since Posidonius's final circumference was 180,000 stadia, which equals 48×3750 stadia.<ref>Strabo 2.2.2, 2.5.24; D.Rawlins, [http://www.dioi.org/cot.htm#gkpz Contributions]</ref> The 180,000 stadia circumference of Posidonius is suspiciously close to that which results from another method of measuring the earth, by timing ocean sunsets from different heights, a method which produces a size of the earth too low by a factor of 5/6, due to horizontal [[atmospheric refraction]].
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| The abovementioned larger and smaller sizes of the earth were those used by [[Claudius Ptolemy]] at different times, 252,000 stadia in the [[Almagest]] and 180,000 stadia in the later [[Geographia|Geographical Directory]]. His midcareer conversion resulted in the latter work's systematic exaggeration of degree longitudes in the Mediterranean by a factor close to the ratio of the two seriously differing sizes discussed here, which indicates <ref>D.Rawlins (2007). "[http://dioi.org/gad.htm#fftg Investigations of the ''Geographical Directory'' 1979–2007] "; [http://dioi.org/vols/w61.pdf DIO], volume 6, number 1, page 11, note 47, 1996.</ref> that the conventional size of the earth was what changed, not the stadion.
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| == Ancient India ==
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| The Indian mathematician [[Aryabhata]] (AD 476 - 550) was a pioneer of [[Indian astronomy|mathematical astronomy]]. He describes the earth as being spherical and that it rotates on its axis, among other things in his work [[Āryabhaṭīya]]. Aryabhatiya is divided into four sections. Gitika, Ganitha (mathematics), Kalakriya (reckoning of time) and Gola ([[celestial sphere]]). The discovery that the earth rotates on its own axis from west to east is described in Aryabhatiya ( Gitika 3,6; Kalakriya 5; Gola 9,10;).<ref name="autogenerated51">http://www.ias.ac.in/resonance/march2006/p51-68.pdf</ref> For example he explained the apparent motion of heavenly bodies is only an illusion (Gola 9), with the following simile;
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| :Just as a passenger in a boat moving downstream sees the stationary (trees on the river banks) as traversing upstream, so does an observer on earth see the fixed stars as moving towards the west at exactly the same speed (at which the earth moves from west to east.)
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| Aryabhatiya also estimates the circumference of Earth, with an accuracy of 1%, which is remarkable. Aryabhata gives the radii of the orbits of the planets in terms of the Earth-Sun distance as essentially their periods of rotation around the Sun. He also gave the correct explanation of lunar and solar eclipses and that the Moon shines by reflecting sunlight.<ref name="autogenerated51"/>
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| == Islamic world ==
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| {{Main|Geography and cartography in medieval Islam#Mathematical geography and geodesy|l1=Geography and cartography in medieval Islam: Mathematical geography and geodesy}}
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| The Muslim scholars, who held to the [[spherical Earth]] theory, used it to calculate the distance and direction from any given point on the earth to [[Mecca]]. This determined the [[Qibla]], or Muslim direction of prayer. [[Islamic mathematics|Muslim mathematicians]] developed [[spherical trigonometry]] which was used in these calculations.<ref>David A. King, ''Astronomy in the Service of Islam'', (Aldershot (U.K.): Variorum), 1993.</ref>
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| Around AD 830 Caliph [[al-Ma'mun]] commissioned a group of astronomers to measure the distance from Tadmur ([[Palmyra]]) to [[Ar Raqqah|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{{fraction|2|3}} miles and thus calculated the Earth's circumference to be 24,000 miles.<ref>''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)[http://www.bodley.ox.ac.uk/bookofcuriosities]</ref> Another estimate given was 56{{fraction|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.<ref>Edward S. Kennedy, ''Mathematical Geography'', pp. 187–8, in {{Harv|Rashed|Morelon|1996|pp=185–201}}</ref>
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| [[Islamic astronomy|Muslim astronomers]] and geographers were aware of [[magnetic declination]] by the 15th century, when the Egyptian Muslim astronomer [['Abd al-'Aziz al-Wafa'i]] (d. 1469/1471) measured it as 7 degrees from [[Cairo]].<ref>{{citation|doi=10.1086/373112|title=Turkish Mosque Orientation and the Secular Variation of the Magnetic Declination|first=Frank E.|last=Barmore|journal=Journal of Near Eastern Studies|volume=44|issue=2|date=April 1985|publisher=[[University of Chicago Press]]|pages=81–98 [98]}}</ref>
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| ===Biruni===
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| Of the medieval [[Persia]]n [[Abu Rayhan Biruni]] (973-1048) it is said:
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| {{quote|"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."<ref>John J. O'Connor, Edmund F. Robertson (1999). [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Biruni.html Abu Arrayhan Muhammad ibn Ahmad al-Biruni], ''MacTutor History of Mathematics archive''.</ref>}}
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| At the age of 17, Biruni calculated the [[latitude]] of Kath, [[Khwarazm]], using the maximum altitude of the Sun. Biruni also solved a complex geodesic equation in order to accurately compute the [[Earth]]'s [[circumference]], which were close to modern values of the Earth's circumference.<ref name=Khwarizm>{{cite web|url=http://muslimheritage.com/topics/default.cfm?ArticleID=482|title=Khwarizm|publisher=Foundation for Science Technology and Civilisation|accessdate=2008-01-22}}</ref><ref>James S. Aber (2003). Alberuni calculated the Earth's circumference at a small town of Pind Dadan Khan, District Jhelum, Punjab, Pakistan.[http://academic.emporia.edu/aberjame/histgeol/biruni/biruni.htm Abu Rayhan al-Biruni], [[Emporia State University]].</ref> 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.<ref>Lenn Evan Goodman (1992), ''Avicenna'', p. 31, [[Routledge]], ISBN 0-415-01929-X.</ref><ref name=Savizi>{{citation|title=Applicable Problems in History of Mathematics: Practical Examples for the Classroom|author=Behnaz Savizi|journal=Teaching Mathematics and Its Applications|volume=26|issue=1|year=2007|pages=45–50|publisher=[[Oxford University Press]]|doi=10.1093/teamat/hrl009}} ([[cf.]] {{cite web|title=Applicable Problems in History of Mathematics; Practical Examples for the Classroom|author=Behnaz Savizi|publisher=[[University of Exeter]]|url=http://people.exeter.ac.uk/PErnest/pome19/Savizi%20-%20Applicable%20Problems.doc|accessdate=2010-02-21}})</ref><ref name="Lumpkin"/> Abu Rayhan 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 (present day Pind Dadan Khan, Pakistan).<ref name=Lumpkin/> 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. This was the earliest known use of dip angle and the earliest practical use of the law of sines.<ref name=Savizi/><ref name=Lumpkin>{{citation|title=Geometry Activities from Many Cultures|author=Beatrice Lumpkin|publisher=Walch Publishing|year=1997|isbn=0-8251-3285-1|pages=60 & 112–3}} [http://books.google.co.uk/books?id=Xpr_rBdY9PwC&pg=RA1-PA60&lpg=RA1-PA60&dq=biruni+mountain+earth&source=bl&ots=tLz9hhn-2h&sig=aWdOXExGAuxWabQYYLXyLZeUEc4&hl=en&ei=W6yAS7LLMNH__AawyrH8Bg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CBoQ6AEwBg#v=onepage&q=biruni%20mountain%20earth&f=false]</ref> He also made use of [[algebra]] to formulate trigonometric equations and used the [[astrolabe]] to measure angles.<ref name=Khalili>[[Jim Al-Khalili]], {{YouTube|id=nyWqx4qWr1s|title=The Empire of Reason 2/6 (Science and Islam - Episode 2 of 3)}}, [[BBC]]</ref> His method can be summarized as follows:
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| [[File:Abu Reyhan Biruni-Earth Circumference.svg|thumb|right|300px|Abu Rayhan Biruni accurately determined the Earth radius by formulating a [[Trigonometry|trigonometric]] equation relating the dip angle (between the true [[horizon]] and astronomical horizon) observed from the top of a mountain to the height of that mountain.]]
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| He first calculated the height of the mountain by going to two points at sea level with a known distance apart and then measuring the angle between the plain and the top of the mountain for both points. He made both the measurements using an astrolabe. He then used the following [[List of trigonometric identities|trigonometric formula]] relating the distance (''d'') between both points with the [[Tangent function|tangents]] of their angles (''θ'') to determine the height (''h'') of the mountain:<ref name=Khalili-3>Jim Al-Khalili, {{YouTube|id=WMXks8eib7s|title=The Empire of Reason 3/6 (Science and Islam - Episode 2 of 3)}}, BBC</ref>
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| <math>h = \frac{d \tan{\theta_1} \tan {\theta_2} }{\tan {\theta_2} - \tan{\theta_1} }</math>
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| He then stood at the highest point of the mountain, where he measured the dip angle using an astrolabe.<ref name=Khalili-3/> He applied the values he obtained for the dip angle and the mountain's height to the following trigonometric formula in order to calculate the Earth's radius:<ref name=Khalili-3/>
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| <math>R = \frac{h \cos{\theta} }{1 - \cos{\theta} }</math>
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| where<ref name=Khalili-3/>
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| *R = Earth radius
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| *h = height of mountain
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| *θ = [[dip angle]]
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| Biruni had also, by the age of 22, written a study of [[map projection]]s, ''[[Cartography]]'', which included a method for projecting a [[Sphere|hemisphere]] on a [[plane (mathematics)|plane]].{{Citation needed|date=June 2010}} Around 1025, Biruni was the first to describe a polar equi-[[azimuthal equidistant projection]] of the [[celestial sphere]].<ref>David A. King (1996), "Astronomy and Islamic society: Qibla, gnomics and timekeeping", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 1, p. 128-184 [153]. Routledge, London and New York.</ref> He was also regarded as the most skilled when it came to mapping cities and measuring the distances between them, which he did for many cities in the [[Middle East]] and western [[Indian subcontinent]]. He often combined astronomical readings and mathematical equations, in order to develop methods of pin-pointing locations by recording degrees of [[latitude]] and [[longitude]]. He also developed similar techniques when it came to measuring the heights of mountains, depths of [[valley]]s, and expanse of the horizon, in ''The Chronology of the Ancient Nations''. He also discussed [[human geography]] and the [[planetary habitability]] of the Earth. He hypothesized that roughly a quarter of the Earth's surface is habitable by humans, and also argued that the shores of Asia and Europe were "separated by a vast sea, too dark and dense to navigate and too risky to try".{{Citation needed|date=June 2010}}
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| == Medieval Europe ==
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| Revising the figures attributed to Posidonius, another Greek philosopher determined 18,000 miles as the Earth's circumference. This last figure was promulgated by [[Ptolemy]] through his world maps. The maps of Ptolemy strongly influenced the cartographers of the [[Middle Ages]]. It is probable that [[Christopher Columbus]], using such maps, was led to believe that Asia was only 3 or 4 thousand miles west of Europe.{{Citation needed|date=November 2011}}
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| Ptolemy's view was not universal, however, and chapter 20 of ''Mandeville's Travels'' (c. 1357) supports Eratosthenes' calculation.
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| It was not until the 16th century that his concept of the Earth's size was revised. During that period the Flemish cartographer, [[Gerardus Mercator|Mercator]], made successive reductions in the size of the [[Mediterranean Sea]] and all of Europe which had the effect of increasing the size of the earth.
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| == Early modern period ==
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| {{merge section from|Meridian arc#The eighteenth century|date=December 2013}}
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| The invention of the [[telescope]] and the [[theodolite]] and the development of [[logarithm table]]s allowed exact [[triangulation]] and [[grade measurement]].
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| ===Europe===
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| [[Jean Picard]] performed the first modern [[meridian arc]] measurement in 1699–70. He measured a [[Baseline (surveying)|base line]] by the aid of wooden rods, used a telescope in his angle measurements, and computed with logarithms. [[Jacques Cassini]] later continued Picard's arc northward to [[Dunkirk, France|Dunkirk]] and southward to the Spanish boundary. Cassini divided the measured arc into two parts, one northward from [[Paris]], another southward. When he computed the length of a degree from both chains, he found that the length of one degree in the northern part of the chain was shorter than that in the southern part. See the illustration at right.
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| [[Image:CASSINIS' ELLIPSOID; HUYGEN'S THEORETICAL ELLIPSOID.GIF|thumb|200px|right|Cassini's ellipsoid; Huygens' theoretical ellipsoid]]
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| This result, if correct, meant that the earth was not a sphere, but an oblong (egg-shaped) [[ellipsoid]]—which contradicted the computations by [[Isaac Newton]] and [[Christiaan Huygens]]. <!-- what exactly was his role? --> Newton's [[Law of universal gravitation|theory of gravitation]] predicted the Earth to be an [[oblate spheroid]] with a [[flattening]] of 1:230.
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| The issue could be settled by measuring, for a number of points on earth, the relationship between their distance (in north-south direction) and the angles between their [[astronomical vertical]]s (the projection of the [[vertical direction]] on the sky). On an oblate Earth the meridional distance corresponding to one degree would grow toward the poles.
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| The [[French Academy of Sciences]] dispatched two expeditions – see [[French Geodesic Mission]]. One expedition under [[Pierre Louis Maupertuis]] (1736–37) was sent to [[Torne Valley]] (as far North as possible). The [[French Geodesic Mission|second mission]] under [[Pierre Bouguer]] was sent to what is modern-day [[Ecuador]], near the equator (1735–44).
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| The measurements conclusively showed that the earth was oblate, with a flattening of 1:210. Thus the next approximation to the true figure of the Earth after the sphere became the oblong [[ellipsoid of revolution]].
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| ===Asia and Americas===
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| In [[South America]] Bouguer noticed, as did [[George Everest]] in the 19th century [[Great Trigonometric Survey]] of India, that the astronomical vertical tended to be pulled in the direction of large mountain ranges, due to the [[gravitation]]al attraction of these huge piles of rock. As this vertical is everywhere perpendicular to the idealized surface of mean sea level, or the [[geoid]], this means that the figure of the Earth is even more irregular than an ellipsoid of revolution. Thus the study of the "[[undulation of the geoid]]" became the next great undertaking in the science of studying the figure of the Earth.
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| == 19th century ==
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| [[Image:Litography archive of the Bayerisches Vermessungsamt.jpg|thumb|250px|Archive with [[lithography]] plates for maps of [[Bavaria]] in the ''Landesamt für Vermessung und Geoinformation'' in [[Munich]]]]
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| [[Image:Litography negative stone and positive paper.jpg|thumb|250px|Negative litography stone and positive print of a historic map of Munich]]
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| In the late 19th century the '''Zentralbüro für die Internationale Erdmessung''' (that is, Central Bureau for International Geodesy) was established by [[Austria-Hungary]] and [[Germany]]. One of its most important goals was the derivation of an international [[ellipsoid]] and a [[gravity]] formula which should be optimal not only for [[Europe]] but also for the whole world. The Zentralbüro was an early predecessor of the [[International Association of Geodesy]] (IAG) and the [[International Union of Geodesy and Geophysics]] (IUGG) which was founded in 1919.
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| Most of the relevant theories were derived by the German geodesist [[Friedrich Robert Helmert]] in his famous books [http://books.google.com/books?id=qt2CAAAAIAAJ ''Die mathematischen und physikalischen Theorieen der höheren Geodäsie'', Einleitung und 1. Teil] (1880) and [http://books.google.com/books?id=p9-CAAAAIAAJ 2. Teil] (1884); English translation: [http://geographiclib.sourceforge.net/geodesic-papers/helmert80-en.html Mathematical and Physical Theories of Higher Geodesy, Vol. 1] and [http://geographiclib.sourceforge.net/geodesic-papers/helmert84-en.html Vol. 2]. Helmert also derived the first global ellipsoid in 1906 with an accuracy of 100 meters (0.002 percent of the Earth's radii). The [[United States|US]] geodesist [[John Fillmore Hayford|Hayford]] derived a global ellipsoid in ~1910, based on intercontinental [[isostasy]] and an accuracy of 200 m. It was adopted by the IUGG as "international ellipsoid 1924".
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| ==See also==
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| *[[Figure of the Earth]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *An early version of this article was taken from the public domain source at http://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003A.HTM#ZZ4.
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| *J.L. Greenberg: ''The problem of the Earth's shape from Newton to Clairaut: the rise of mathematical science in eighteenth-century Paris and the fall of "normal" science.'' Cambridge : Cambridge University Press, 1995 ISBN 0-521-38541-5
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| *M.R. Hoare: ''Quest for the true figure of the Earth: ideas and expeditions in four centuries of geodesy''. Burlington, VT: Ashgate, 2004 ISBN 0-7546-5020-0
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| *D.Rawlins: "Ancient Geodesy: Achievement and Corruption" 1984 (Greenwich Meridian Centenary, published in ''Vistas in Astronomy'', v.28, 255-268, 1985)
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| *D.Rawlins: "Methods for Measuring the Earth's Size by Determining the Curvature of the Sea" and "Racking the Stade for Eratosthenes", appendices to "The Eratosthenes-Strabo Nile Map. Is It the Earliest Surviving Instance of Spherical Cartography? Did It Supply the 5000 Stades Arc for Eratosthenes' Experiment?", ''Archive for History of Exact Sciences'', v.26, 211-219, 1982
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| *C.Taisbak: "Posidonius vindicated at all costs? Modern scholarship versus the stoic earth measurer". ''Centaurus'' v.18, 253-269, 1974
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| ==External links==
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| *[http://www.juliantrubin.com/bigten/eratosthenes.html Eratosthenes: The Measurement of the Earth's Circumference]
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| {{DEFAULTSORT:History Of Geodesy}}
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| [[Category:Geodesy]]
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