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The concept of a spherical Earth dates back to around the 6th century BC, in ancient Indian philosophy,[1] and in ancient Mediterranean thought.[2] It 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.[3][4][5][6] A practical demonstration of Earth's sphericity was achieved by Ferdinand Magellan and Juan Sebastián Elcano's expedition's circumnavigation (1519−1521).[7]

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).

The actual figure of the Earth, however, is not spherical.[8] As determined by modern instruments, a sphere approximates the Earth's shape to within one part in 300. An oblate ellipsoid shape with a flattening of 1/300 matches even more precisely. Recent measurements from satellites suggest that the Earth is, in fact, slightly pear-shaped.[9]

## HistoryEdit

### AntiquityEdit

#### Ancient Near EastEdit

Babylonians

In Mesopotamia, Babylonian cosmology depicted the Earth and the heavens as a "spatial whole, even one of round shape," with references to "the circumference of heaven and earth" and "the totality of heaven and earth." Their worldview was not exactly geocentric either. The idea of geocentrism, where the center of the Earth is the exact center of the universe, did not yet exist in Babylonian cosmology, but was established later by the Greek philosopher Aristotle's On the Heavens. In contrast, Babylonian cosmology suggested that the cosmos revolved around the "cult-place of the deity," who held ultimate authority as ruler of the cosmic polity.[10]

Phoenicians and Egyptians

The earliest evidence for a spherical Earth came from an ancient Phoenician expedition for ancient Egypt. The Egyptian pharaoh Necho II, during his reign from 610 BCE to 595 BCE, employed Phoenician sailors to circumnavigate around the entire African continent, then known as "Libya". In The Histories (written c. 431 BCE - 425 BCE), Herodotus described how the Phoenicians reported the sun being observed shining from the north. Herodotus wrote: [4]

Libya is washed on all sides by the sea except where it joins Asia, as was first demonstrated, so far as our knowledge goes, by the Egyptian king Necho, who, after calling off the construction of the canal between the Nile and the Arabian gulf, sent out a fleet manned by a Phoenician crew with orders to sail west about and return to Egypt and the Mediterranean by way of the Straits of Gibraltar. The Phoenicians sailed from the Arabian gulf into the southern ocean, and every autumn put in at some convenient spot on the Libyan coast, sowed a patch of ground, and waited for next year's harvest. Then, having got in their grain, they put to sea again, and after two full years rounded the Pillars of Heracles in the course of the third, and returned to Egypt. These men made a statement which I do not myself believe, though others may, to the effect that as they sailed on a westerly course round the southern end of Libya, they had the sun on their right - to northward of them. This is how Libya was first discovered by sea.

With this expedition, the Phoenicians and Egyptians were thus the first to discover evidence of the Earth being curved and therefore spherical.

There is also another possibility of how the sphericity of the Earth was discovered.[11] A plausible explanation is that it was "the experience of travellers that suggested such an explanation for the variation in the observable altitude and the change in the area of circumpolar stars,"[12] which would have been known to Phoenician sailors at the time.

#### IndiaEdit

Rigveda

The Rigveda is the oldest surviving Indian philosophical text, dating back to the 2nd millenium BC. According to K. V. Sarma: "One finds in the R.gveda intelligent speculations about the genesis of the universe from nonexistence, the configuration of the universe, the spherical self-supporting earth, and the year of 360 days divided into 12 equal parts of 30 days each with a periodical intercalary month."[13]

Aitareya Brahmana

Several writers such as Madame Blavatsky[14] and Subhash Kak[1] have noted that the concept of a spherical Earth is implicit in the Aitareya Brahmana, an ancient Indian philosophical text dating back to the early 1st millenium BC. Kak has interpreted a verse of the Aitareya Brahmana as suggesting that the Earth's rotation may be the cause of the apparent motion of the Sun rising and setting. He cites verse 4.18, which states:[1]

"The [sun] never really sets or rises. In that they think of him “He is setting,” having reached the end of the day, he inverts himself; thus he makes evening below, day above. Again in that they think of him “He is rising in the morning,” having reached the end of the night he inverts himself; thus he makes day below, night above. He never sets; indeed he never sets."

However, Shyam Singh Shashi interprets the verse as suggesting that the Sun has one bright and one dark side, its flipping around on itself being the cause of the apparent rising and setting.[15]

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.[16][17] 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.

#### Classical GreeceEdit

Pythagoras

Early Greek philosophers alluded to a spherical earth, though with some ambiguity.[18] Pythagoras (6th century BC) was among those said to have mentioned the idea, but this may reflect the ancient Greek practice of ascribing every discovery to one or another of their ancient wise men.[11] Some idea of the sphericity of the Earth seems to have been known to both Parmenides and Empedocles in the 5th century BC,[19] and although the idea cannot reliably be ascribed to Pythagoras,[20] it may, nevertheless have been accepted in the Pythagorean school in the 5th century BC.[11][19] After the 5th century BC, many Greek writers began to accept the world being round.[18]

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 "a ball made of twelve pieces of leather, variegated, a patchwork of colours."

Aristotle

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.

#### Hellenistic Near EastEdit

Eratosthenes

In Egypt, the Libyan mathematician 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 twenty percent.[21][22]

Seleucus of Seleucia

The Babylonian astronomer, Seleucus of Seleucia (c. 190 BC), who lived in the Seleucia region of Mesopotamia, stated that the Earth is spherical, and proposed a heliocentric model where the Earth actually orbits the Sun.

Posidonius

The Syrian geographer Posidonius (c. 135 – 51 BC) put faith in the Eratosthenes method, though by observing the star Canopus, rather than the sun in establishing the Earth's circumference. In Ptolemy's Geographia, his result was favoured over that of Erastosthenes. Posidonius furthermore expressed the distance of the sun in earth radii.

#### Roman Near EastEdit

The idea of a spherical earth slowly spread across the globe and ultimately became the adopted view in all major astronomical traditions.[3][4][5][6]

In the west, the idea came naturally to the Romans through the lengthy process of cross-fertilization with Hellenistic civilization. Many Roman authors such as Cicero and Pliny refer in their works to the rotundity of the earth as a matter of course.[23]

Strabo

It has been suggested that seafarers probably provided the first observational evidence that the Earth was not flat, based on observations of the horizon. This argument was put forward by the Anatolian geographer Strabo (c. 64 BC – 24 AD), who suggested that the spherical shape of the Earth was probably known to seafarers around the Mediterranean Sea since at least the time of Homer,[24] citing a line from the Odyssey[25] as indicating that the poet Homer was already aware of this as early as the 7th or 8th century BC. Strabo cited various phenomena observed at sea as suggesting that the Earth was spherical. He observed that elevated lights or areas of land were visible to sailors at greater distances than those less elevated, and stated that the curvature of the sea was obviously responsible for this.[26]

Claudius Ptolemy

The Egyptian astronomer, Claudius Ptolemy (CE 90 - 168), lived in Alexandria, Egypt, the centre of scholarship in the second century. Aroun 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.

Late Antiquity

Knowledge of the spherical shape of the Earth was received in scholarship of Late Antiquity as a matter of course, in both Neoplatonism and Early Christianity. Theological doubt informed by the flat Earth model implied in the Hebrew Bible inspired some early Christian scholars such as Lactantius, John Chrysostom and Athanasius of Alexandria, but this remained an eccentric current and learned Christian authors like Basil of Caesarea, Ambrose and Augustine of Hippo were clearly aware of the sphericity of the Earth. "Flat Earthism" lingered longest in Syriac Christianity, which tradition laid greater importance on a literalist interpretation of the Old Testament, and authors from that tradition such as Cosmas Indicopleustes presented the Earth as flat as late as in the 6th century. This last remnant of the ancient model of the cosmos disappeared during the 7th century, and from the 8th century and the beginning medieval period, "no cosmographer worthy of note has called into question the sphericity of the Earth."[27]

### Middle AgesEdit

#### ArmeniaEdit

Anania Shirakatsi (Template:Lang-hy), also known as Ananias of Sirak, (610685) 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)." [28]

Shirakatsi's work ‘Ashharatsuyts’ reports details and mapping of the ancient homeland of Bulgars in the Mount Imeon area of Central Asia.

#### Islamic WorldEdit

Islamic astronomy may have inherited the idea of a spherical Earth from the earlier Indian and Hellenistic astronomical traditions.[29] While initially inspired by Aryabhata, the theoretical framework for classical Islamic astronomy was later influenced by the Greek Aristotle (De caelo) and Egyptian Ptolemy (Almagest), which worked with the premise that the Earth was spherical and at the center of the universe (geocentric model).[29]

Early Islamic scholars recognized Earth's sphericity,[30] leading Muslim mathematicians to develop spherical trigonometry[31] 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.

Applications

Muslim scholars who held to the round Earth theory used it for a quintessentially Islamic purpose: to calculate the distance and direction from any given point on the Earth to Makkah (Mecca).[32] This determined the Qibla, or Muslim direction of prayer.

A terrestrial globe (Kura-i-ard) was among the presents sent by the Persian Muslim astronomer Jamal-al-Din to Kubla Khan in 1267. It was made of wood on which "seven parts of water are represented in green, three parts of land in white, with rivers, lakes etc."[33] This was the earliest evidence of a spherical Earth in China.[34]

Al-Ma'mun

Around 830 AD, 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 meridian arc distance between them to be 66Template:Frac miles and thus calculated the Earth's circumference to be 24,000 miles.[35]

Another estimate given by his astronomers was 56Template:Frac Arabic miles (111.8 km) per degree, which corresponds to 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.[36]

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 (56Template:Frac Arabic miles) was much more accurate than the 60Template:Frac 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.[37]

Al-Biruni

Abu Rayhan al-Biruni (973-1048) used a new method, solving a complex geodesic equation, in order to accurately compute the Earth's circumference, by which he arrived at a value that was close to modern values for the Earth's circumference.[38] 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.[39][40] 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. From the top of the mountain, he sighted the angle to the horizon which, along with the mountain's height (which he calculated beforehand), allowed him to calculate the curvature of the Earth.[41][42] He also made use of algebra to formulate trigonometric equations and used the astrolabe to measure angles.[43]

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."[44]
Islamic scholars

Many Islamic scholars declared a mutual agreement (Ijma) that celestial bodies are round, among them Ibn Hazm (d. 1069), Abul-Faraj Ibn Al-Jawzi (d. 1200), and Ibn Taymiya (d. 1328).[45] Ibn Taymiya said, "Celestial bodies are round—as it is the statement of astronomers and mathematicians—it is likewise the statement of the scholars of Islam". Abul-Hasan ibn al-Manaadi, Abu Muhammad Ibn Hazm, and Abul-Faraj Ibn Al-Jawzi have said that the Muslim scholars are in agreement that all celestial bodies are round. Ibn Taymiyah also remarked that Allah has said, "And He (Allah) it is Who created the night and the day, the sun and the moon. They float, each in a Falak." Ibn Abbas says, "A Falaka like that of a spinning wheel." The word 'Falak' (in the Arabic language) means "that which is round."[46] [47]

The Muslim scholars who held to the round-earth theory used it in an impeccably Islamic manner, to calculate the distance and direction from any given point on the earth to Mecca. This determined the Qibla, or Muslim direction of prayer. Muslim mathematicians developed spherical trigonometry which was used in these calculations.[48] Ibn Khaldun (d. 1406), in his Muqaddimah, also identified the world as spherical.

#### Christian worldEdit

Isidore of Seville

Bishop Isidore of Seville (560–636) taught in his widely read encyclopedia, the Etymologies, that the Earth was round. While some writers have thought he referred to a spherical Earth;[49] this and other writings make it clear that he considered the Earth to be disk or wheel-shaped.[50] He didn't admit the possibility of people dwelling at the antipodes, considering them as legendary[51] and noting that there was no evidence for their existence.[52]

Bede the Venerable

The monk Bede (c. 672–735) wrote in his influential treatise on computus, The Reckoning of Time, that the Earth was round, explaining the unequal length of daylight from "the roundness of the Earth, for not without reason is it called 'the orb of the world' on the pages of Holy Scripture and of ordinary literature. It is, in fact, set like a sphere in the middle of the whole universe." (De temporum ratione, 32). The large number of surviving manuscripts of The Reckoning of Time, copied to meet the Carolingian requirement that all priests should study the computus, indicates that many, if not most, priests were exposed to the idea of the sphericity of the Earth.[53] Ælfric of Eynsham paraphrased Bede into Old English, saying "Now the Earth's roundness and the Sun's orbit constitute the obstacle to the day's being equally long in every land."[54]

Bede was lucid about earth's sphericity, writing "We call the earth a globe, not as if the shape of a sphere were expressed in the diversity of plains and mountains, but because, if all things are included in the outline, the earth's circumference will represent the figure of a perfect globe... For truly it is an orb placed in the center of the universe; in its width it is like a circle, and not circular like a shield but rather like a ball, and it extends from its center with perfect roundness on all sides."[55]

Anania Shirakatsi

The 7th-century Armenian scholar Anania Shirakatsi 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)."[56]

High Middle Ages

During the High Middle Ages, the astronomical knowledge in Christian Europe is extended beyond what was transmitted directly from ancient authors by transmission of learning from Medieval Islamic astronomy. An early recipient of such learning was Gerbert d'Aurillac, the later Pope Sylvester II.

Saint Hildegard (Hildegard von Bingen, 1098–1179), depicts the spherical earth several times in her work Liber Divinorum Operum. [5]

Johannes de Sacrobosco (c. 1195 – c. 1256 AD) wrote a famous work on Astronomy called Tractatus de Sphaera, based on Ptolemy, which primarily considers the sphere of the sky but contains clear proofs of the earth's sphericity in the first chapter.[57][58]

Many scholastic commentators on Aristotle's On the Heavens and Sacrobosco's Treatise on the Sphere unanimously agreed that the earth is spherical or round.[59] Grant observes that no author who had studied at a medieval university thought that the earth was flat.[60]

Late Middle Ages

Dante's Divine Comedy, written in Italian in the early 14th century, portrays Earth as a sphere, discussing implications such as the different stars visible in the southern hemisphere, the altered position of the sun, and the various timezones of the Earth. Also, the Elucidarium of Honorius Augustodunensis (c. 1120), an important manual for the instruction of lesser clergy, which was translated into Middle English, Old French, Middle High German, Old Russian, Middle Dutch, Old Norse, Icelandic, Spanish, and several Italian dialects, explicitly refers to a spherical Earth. Likewise, the fact that Bertold von Regensburg (mid-13th century) used the spherical Earth as a sermonic illustration shows that he could assume this knowledge among his congregation. The sermon was held in the vernacular German, and thus was not intended for a learned audience.

Portuguese exploration of Africa and Asia, Columbus voyage to the Americas (1492) and finally Ferdinand Magellan's circumnavigation of the earth (1519–21) provided practical evidence of the global shape of the earth.

#### ChinaEdit

A terrestrial globe (Kura-i-ard) was among the presents sent by the Persian Muslim astronomer Jamal-al-Din to Kubla Khan in 1267. It was made of wood on which "seven parts of water are represented in green, three parts of land in white, with rivers, lakes etc."[61]  Islamic astronomy thus introduced the concept of a spherical Earth to Chinese astronomy, which previously believed the Earth to be flat. [6] However, it was not until the 17th century that the spherical Earth became widely accepted in China.[62]

The first direct demonstration of Earth's sphericity came in the form of the first circumnavigation in history, an expedition captained by Portuguese explorer Ferdinand Magellan.[63] The expedition was financed by the Spanish Crown. On August 10, 1519, the five ships under Magellan's command departed from Seville. They crossed the Atlantic Ocean, passed through the Strait of Magellan, crossed the Pacific, and arrived in Cebu, where Magellan was killed by Philippine natives in a battle. His second in command, the Spaniard Juan Sebastián Elcano, continued the expedition and, on September 6, 1522, arrived at Seville, completing the circumnavigation. Charles I of Spain, in recognition of his feat, gave Elcano a coat of arms with the motto Primus circumdedisti me (in Latin, "You went around me first").[64]

A circumnavigation alone does not prove that the earth is spherical. It could be cylindric or irregularly globular or one of many other shapes. Still, combined with trigonometric evidence of the form used by Eratosthenes 1,700 years prior, the Magellan expedition removed any reasonable doubt in educated circles in Europe.

## Summary of evidence for a spherical earthEdit

These are given in an order which approximates how they were observed historically:

1. The Phoenician sailors, on the c. 600 BC Egyptian expedition, reached the southern hemisphere where they observed the Sun being observed shining from the north, in contrast to the northern hemisphere where the Sun is observed shining from the south.
2. When at sea it is possible to see high mountains or elevated lights in the distance before lower lying ground and the masts of boats before the hull. It is also possible to see further by climbing higher in the ship, or, when on land, on high cliffs.
3. The sun is lower in the sky as you travel away from the tropics. For example, when traveling northward, stars such as Polaris, the north star, are higher in the sky, whereas other bright stars such as Canopus, visible in Egypt, disappear from the sky.
4. The length of daylight varies more between summer and winter the farther you are from the equator.
5. The earth throws a circular shadow on the moon during a lunar eclipse.
6. The times reported for lunar eclipses (which are seen simultaneously) are many hours later in the east (e.g. India) than in the west (e.g. Europe). Local times are confirmed later by travel using chronometers and telegraphic communication.
7. When you travel far south, to Ethiopia or India, the sun throws a shadow south at certain times of the year. Even farther (e.g. Argentina) and the shadow is always in the south.
8. It is possible to circumnavigate the world; that is, to travel around the world and return to where you started.
9. Travelers who circumnavigate the earth observe the gain or loss of a day relative to those who did not. See also International Date Line.
10. An artificial satellite can circle the earth continuously and even be geostationary.
11. The earth appears as a disc on photographs taken from space, regardless of the vantage point.

Several of these arguments have alternative explanations by themselves. e.g. the shadow thrown by a lunar eclipse could be caused by a disk-shaped earth. Similarly the north-south movement of stars in the sky with travel could mean they are much closer to earth. However, the arguments strengthen each together.

## GeodesyEdit

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 modelsEdit

There are several reasonable ways to approximate Earth's shape as a sphere. Each preserves a different feature of the true 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 real Earth. 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: $\frac{2a}{\pi}\int_{0}^{\frac{\pi}{2}}\sqrt{\cos^2\phi + \frac{b^2}{a^2}\sin^2 \phi}\,d\phi$. 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: $\sqrt{\frac{a^2+b^2}{2}}\,\!$, about 6,367.454 km.
• Preserve the average circumference. As there are different ways to define an ellipsoid's average circumference (radius vs. arcradius/radius of curvature; elliptically fixed vs. ellipsoidally "fluid"; different integration intervals for quadrant-based geodetic circumferences), there is no definitive, "absolute average circumference". The ellipsoidal quadratic mean is one simple model: $\sqrt{\frac{3a^2+b^2}{4}}\,\!$, giving a spherical radius of 6,372.798 km.
• Preserve the surface area of the real Earth. This gives the authalic radius: $\sqrt{\frac{a^2+\frac{ab^2}{\sqrt{a^2-b^2}}\ln{(\frac{a+\sqrt{a^2-b^2}}b)}}{2}}\,\!$, or 6,371.007 km.
• Preserve the volume of the real Earth. This volumetric radius is computed as: $\sqrt[3]{a^2b}$, or 6,371.001 km.

Note that the authalic and volumetric spheres have radii that differ by less than 7 meters, yet both preserve important properties. Hence both, and occasionally an average of the two, are used.

## ReferencesEdit

1. 1.0 1.1 1.2 Subhash Kak (2000), "Birth and Early Development of Indian Astronomy", in Helaine Selin (2000), Astronomy Across Cultures: The History of Non-Western Astronomy, pp. 303-40, Boston: Kluwer, ISBN 0-7923-6363-9
2. Dicks, D.R. (1970). Early Greek Astronomy to Aristotle. pp. 72-198. ISBN 9780801405617.
3. 3.0 3.1 Continuation into Roman and medieval thought: Reinhard Krüger: "Materialien und Dokumente zur mittelalterlichen Erdkugeltheorie von der Spätantike bis zur Kolumbusfahrt (1492)"
4. 4.0 4.1 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
5. 5.0 5.1 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, ISBN 0-415-96930-1, p. 463
6. 6.0 6.1 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)
7. Pigafetta, Antonio (1906). Magellan's Voyage around the World. Arthur A. Clark. [1]
8. Michael Hoskin (1999), The Cambridge concise history of astronomy, Cambridge University Press, pp. 4, 174, 183, 336-7, ISBN 0521576008
9. Hugh Thurston, Early Astronomy, (New York: Springer-Verlag), p. 119. ISBN 0-387-94107-X.
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