Spherical Earth


The earliest documented mention of the spherical Earth concept dates from around the 5th century BC, when it was mentioned by ancient Greek philosophers. It remained a matter of speculation until the 3rd century BC, when Hellenistic astronomy established the spherical shape of the Earth as a physical fact and calculated the Earth's circumference. The paradigm was gradually adopted throughout the Old World during Late Antiquity and the Middle Ages. A practical demonstration of Earth's sphericity was achieved by Ferdinand Magellan and Juan Sebastián Elcano's circumnavigation.
The concept of a spherical Earth displaced earlier beliefs in a flat Earth: In early Mesopotamian mythology, the world was portrayed as a flat disk floating in the ocean with a hemispherical sky-dome above, and this forms the premise for early world 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.
The realization that the figure of the Earth is more accurately described as an ellipsoid dates to the 17th century, as described by Isaac Newton in Principia. In the early 19th century, the flattening of the earth ellipsoid was determined to be of the order of 1/300. The modern value as determined by the US DoD World Geodetic System since the 1960s is close to 1/298.25.

Cause

The Earth is massive enough that the pull of gravity maintains its roughly spherical shape. Most of its deviation from spherical stems from the centrifugal force caused by rotation around its north-south axis. This force deforms the sphere into an oblate ellipsoid.

Formation

The Solar System formed from a dust cloud that was at least partially the remnant of one or more supernovas that created heavy elements by nucleosynthesis. Grains of matter accreted through electrostatic interaction. As they grew in mass, gravity took over in gathering yet more mass, releasing the potential energy of their collisions and in-falling as heat. The protoplanetary disk also had a greater proportion of radioactive elements than the Earth today because, over time, those elements decayed. Their decay heated the early Earth even further, and continue to contribute to Earth's internal heat budget. The early Earth was thus mostly liquid.
A sphere is the only stable shape for a non-rotating, gravitationally self-attracting liquid. The outward acceleration caused by the Earth's rotation is greater at the equator than at the poles, so the sphere gets deformed into an ellipsoid, which represents the shape having the lowest potential energy for a rotating, fluid body. This ellipsoid is slightly fatter around the equator than a perfect sphere would be. Earth's shape is also slightly lumpy because it is composed of different materials of different densities that exert slightly different amounts of gravitational force per volume.
The liquidity of a hot, newly formed planet allows heavier elements to sink down to the middle and forces lighter elements closer to the surface, a process known as planetary differentiation. This event is known as the iron catastrophe; the most abundant heavier elements were iron and nickel, which now form the Earth's core.

Later shape changes and effects

Though the surface rocks of the Earth have cooled enough to solidify, the outer core of the planet is still hot enough to remain liquid. Energy is still being released; volcanic and tectonic activity has pushed rocks into hills and mountains and blown them out of calderas. Meteors also create impact craters and surrounding ridges. However, if the energy release from these processes halts, then they tend to erode away over time and return toward the lowest potential-energy curve of the ellipsoid. Weather powered by solar energy can also move water, rock, and soil to make the Earth slightly out of round.
Earth undulates as the shape of its lowest potential energy changes daily due to the gravity of the Sun and Moon as they move around with respect to the Earth. This is what causes tides in the oceans' water, which can flow freely along the changing potential.

Shapes of other bodies

The IAU definitions of planet and dwarf planet require that a Sun-orbiting body has undergone the rounding process to reach a roughly spherical shape, an achievement known as hydrostatic equilibrium. The same spheroidal shape can be seen from smaller rocky planets like Mars to gas giants like Jupiter.
Any natural Sun-orbiting body that has not reached hydrostatic equilibrium is classified by the IAU as a small Solar System body. These come in many non-spherical shapes which are lumpy masses accreted haphazardly by in-falling dust and rock; not enough mass falls in to generate the heat needed to complete the rounding. Some SSSBs are just collections of relatively small rocks that are weakly held next to each other by gravity but are not actually fused into a single big bedrock. Some larger SSSBs are nearly round but have not reached hydrostatic equilibrium. The small Solar System body 4 Vesta is large enough to have undergone at least partial planetary differentiation.
Stars like the Sun are also spheroidal due to gravity's effects on their plasma, which is a free-flowing fluid. Ongoing stellar fusion is a much greater source of heat for stars compared to the initial heat released during formation.

Effects and empirical evidence

The roughly spherical shape of the Earth can be confirmed by many different types of observation from ground level, aircraft, and spacecraft. The shape causes a number of phenomena that a flat Earth would not. Some of these phenomena and observations would be possible on other shapes, such as a curved disc or torus, but no other shape would explain all of them.

Visibility of distant objects on Earth's surface

On a completely flat Earth with no visual interference the ground itself would never obscure distant objects; one would be able to see all the way to the edge of the surface. A spherical surface has a horizon which is closer when viewed from a lower altitude. In theory, a person standing on the surface with eyes above the ground can see the ground up to about away, but a person at the top of the Eiffel Tower at can see the ground up to about away.
This phenomenon would seem to present a method to verify that the Earth's surface is locally. If the degree of curvature was determined to be the same everywhere on the Earth's surface, and that surface was determined to be large enough, it would show that the Earth is spherical.
The shadow of the Earth on the Moon during a lunar eclipse is always a dark circle that moves from one side of the Moon to the other. The only shape that casts a round shadow no matter which direction it is pointed is a sphere, and the ancient Greeks deduced that this must mean the Earth is spherical.
The Moon's tidal lock to Earth results in the Moon always showing only one side to Earth. If Earth were flat, with the Moon hovering above it, then the portion of the Moon's surface visible to people on Earth would vary according to location on the Earth, rather than showing an identical face to everyone. If Earth were flat, with the Moon revolving around it tidally locked, then the Moon would be seen simultaneously at all places on Earth at once, but its apparent size would change for each viewer as its position moved across the sky over the course of the night.

Observation of the stars

On a perfectly spherical Earth, flat terrain or ocean, when viewed from the surface, blocks exactly half the sky without permanent distortion.
With the presumption of a spherical earth, an expedition commissioned by caliph al-Ma'mun used this phenomenon to calculate the Earth's circumference to within of the correct value of around, and possibly as accurately as.
The fixed stars can be demonstrated to be very far away, by diurnal parallax measurements. Unlike the Sun, Moon, and planets, they do not change position with respect to one another ; the shapes of the constellations are always the same. This makes them a convenient reference background for determining the shape of the Earth. Adding distance measurements on the ground allows calculation of the Earth's size.
The fact that the stars visible from the north and south poles do not overlap must mean that the two observation spots are on opposite sides of the Earth, which is not possible if the Earth is a single-sided disc, but is possible for other shapes.
The direction any intermediate spot on the Earth is facing can also be calculated by measuring the angles of the fixed stars and determining how much of the sky is visible. For example, New York City is about 40° north of the equator. The apparent motion of the Sun blots out slightly different parts of the sky from day to day, but over the course of the entire year it sees a dome of 280°. So for example, both Orion and the Big Dipper are visible during at least part of the year.
Making stellar observations from a representative set of points across the Earth, combined with knowing the shortest on-the-ground distance between any two given points makes an approximate sphere the only possible shape for the Earth.

Observing the Sun

On a flat Earth, an omnidirectional Sun would illuminate the entire surface at the same time, and all places would experience sunrise and sunset at the horizon at the same time. With a spherical Earth, half the planet is in daylight at any given time and the other half is experiencing nighttime. When a given location on the spherical Earth is in sunlight, its antipode - the location exactly on the opposite side of the Earth - is always experiencing nighttime. The spherical shape of the Earth causes the Sun to rise and set at different times in different places, and different locations get different amounts of sunlight each day.
In order to explain day and night, time zones, and the seasons, some flat Earth theorists propose that the Sun does not emit light in all directions, but acts more like a spotlight, only illuminating part of the flat Earth at a time. This theory is not consistent with observation; at sunrise and sunset, a spotlight Sun would be up in the sky at least a little bit, rather than at the horizon where it is always actually observed. A spotlight Sun would also appear at different angles in the sky with respect to a flat ground than it does with respect to a curved ground. Assuming light travels in straight lines, actual measurements of the Sun's angle in the sky from locations very distant from each other are only consistent with a geometry where the Sun is very far away and is being seen from a hemispherical surface. These two phenomena are related: a low-altitude spotlight Sun would spend most of the day near the horizon for most locations on Earth, but rise and set fairly close to the horizon. A high-altitude Sun would spend more of the day away from the horizon, but rise and set fairly far from the horizon.

Length of the day

On a flat Earth with an omnidirectional Sun, all places would experience the same amount of daylight every day, and all places would get daylight at the same time. Actual day length varies considerably, with places closer to the poles getting very long days in the summer and very short days in the winter, with northerly summer happening at the same time as southerly winter. Places north of the Arctic Circle and south of the Antarctic Circle get no sunlight for at least one day a year, and get 24-hour sunlight for at least one day a year. Both the poles experience sunlight for 6 months and darkness for 6 months, at opposite times.
The movement of daylight between the northern and southern hemispheres happens because of the axial tilt of the Earth. The imaginary line around which the Earth spins, which goes between the North Pole and South Pole, is tilted about 23° from the oval that describes its orbit around the Sun. The Earth always points in the same direction as it moves around the Sun, so for half the year, the North Pole is pointed slightly toward the Sun, keeping it in daylight all the time because the Sun lights up the half of the Earth that is facing it. For the other half of the orbit, the South Pole is tilted slightly toward the Sun, and it is winter in the Northern Hemisphere. This means that at the equator, the Sun is not directly overhead at noon, except around the March and September equinoxes, when one spot on the equator is pointed directly at the Sun.
The length of the day varies because as the Earth rotates some places pass through only a short curve near the top or bottom of the sunlight half; other places travel along much longer curves through the middle.
The length of twilight would be very different on a flat Earth. On a round Earth, the atmosphere above the ground is lit for a while before sunrise and after sunset are observed at ground level, because the Sun is still visible from higher altitudes. Longer twilights are observed at higher latitudes due to a shallower angle of the Sun's apparent movement compared to the horizon. On a flat Earth, the Sun's shadow would reach the upper atmosphere very quickly, except near the closest edge of the Earth, and would always set at the same angle to the ground. The "spotlight Sun" theory is also not consistent with this observation, since the air cannot be lit without the ground below it also being lit.

Local solar time and time zones

Ancient timekeeping reckoned "noon" as the time of day when the Sun is highest in the sky, with the rest of the hours in the day measured against that. During the day, the apparent solar time can be measured directly with a sundial. In ancient Egypt, the first known sundials divided the day into 12 hours, though because the length of the day changed with the season, the length of the hours also changed. Sundials that defined hours as always being the same duration appeared in the Renaissance. In Western Europe, clock towers and striking clocks were used in the Middle Ages to keep people nearby appraised of the local time, though compared to modern times this was less important in a largely agrarian society.
Because the Sun reaches its highest point at different times for different longitudes, the local solar noon in each city is different except for those directly north or south of each other. This means that the clocks in different cities could be offset from each other by minutes or hours. As clocks became more precise and industrialization made timekeeping more important, cities switched to mean solar time, which ignores minor variations in the timing of local solar noon over the year, due to the elliptical nature of the Earth's orbit, and its tilt.
The differences in clock time between cities was not generally a problem until the advent of railroad travel in the 1800s, which both made travel between distant cities much faster than by walking or horse, and also required passengers to show up at specific times to meet their desired trains. In the United Kingdom, railroads gradually switched to Greenwich Mean Time, followed by public clocks across the country generally, forming a single time zone. In the United States, railroads published schedules based on local time, then later based on standard time for that railroad, and then finally based on four standard time zones shared across all railroads, where neighboring zones differed by exactly one hour. At first railroad time was synchronized by portable chronometers, and then later by telegraph and radio signals.
San Francisco is at 122.41°W longitude and Richmond, Virginia is at 77.46°W longitude. They are both at about 37.6°N latitude. The approximately 45° of longitude difference translates into about 180 minutes, or 3 hours, of time between sunsets in the two cities, for example. San Francisco is in the Pacific Time zone, and Richmond is in the Eastern Time zone, which are three hours apart, so the local clocks in each city show that the Sun sets at about the same time when using the local time zone. But a phone call from Richmond to San Francisco at sunset will reveal that there are still three hours of daylight left in California.

Determining the size of the Earth by [Eratosthenes]

Under the assumption that the Sun is very far away, the ancient Greek geographer Eratosthenes performed an experiment using the differences in the observed angle of the Sun from two different locations to calculate the circumference of the Earth. Though modern telecommunications and timekeeping were not available, he was able to make sure the measurements happened at the same time by having them taken when the Sun was highest in the sky at both locations. Using slightly inaccurate assumptions about the locations of two cities, he came to a result within 15% of the correct value.

Determining the shape of the Earth

On a given day, if many different cities measure the angle of the Sun at local noon, the resulting data, when combined with the known distances between cities, shows that the Earth has 180 degrees of north-south curvature. This is consistent with many rounded shapes, including a sphere, and is inconsistent with a flat shape.
Some claim that this experiment assumes a very distant Sun, such that the incoming rays are essentially parallel, and if a flat Earth is assumed, that the measured angles can allow one to calculate the distance to the Sun, which must be small enough that its incoming rays are not very parallel. However, if more than two relatively well-separated cities are included in the experiment, the calculation will make clear whether the Sun is distant or nearby. For example, on the equinox, the 0 degree angle from the North Pole and the 90 degree angle from the equator predict a Sun which would have to be located essentially next to the surface of a flat Earth, but the difference in angle between the equator and New York City would predict a Sun much further away if the Earth is flat. Because these results are contradictory, the surface of the Earth cannot be flat; the data is consistent with a nearly spherical Earth and a Sun which is very far away compared with the diameter of the Earth.

Surface circumnavigation

Since the 1500s, many people have sailed or flown completely around the world in all directions, and none have discovered an edge or impenetrable barrier.
Some flat Earth theories that propose the world is a north-pole-centered disc, conceive of Antarctica as an impenetrable ice wall that encircles the planet and hides any edges. This disc model explains east-west circumnavigation as simply moving around the disc in a circle. It is possible in this model to traverse the North Pole, but it would not be possible to perform a circumnavigation that includes the South Pole.
The Arctic Circle is roughly long, as is the Antarctic Cicle.. A "true circumnavigation" of the Earth is defined, in order to account for the shape of the Earth, to be about 2.5 times as long, including a crossing of the equator, at about.. On the flat Earth model, the ratios would require the Antarctic Circle to be 2.5 times the length of the circumnavigation, or 2.5x2.5=6.25 times the length of the Arctic Circle.
Explorers, government researchers, commercial pilots, and tourists have been to Antarctica and found that it is not a large ring that encircles the entire world, but actually a roughly disc-shaped continent smaller than South America but larger than Australia, with an interior that can in fact be traversed in order to take a shorter path from e.g. the tip of South America to Australia than would be possible on a disc.
The first land crossing of the entirety of Antarctica was the Commonwealth Trans-Antarctic Expedition in 1955-1958, and many exploratory airplanes have since passed over the continent in various directions.

Grids are distorted by spherical ground

A meridian of longitude is a line where local solar noon occurs at the same time each day. These lines define "north" and "south". These are perpendicular to lines of latitude that define "east" and "west", where the Sun is at the same angle at local noon on the same day. If the Sun were travelling from east to west over a flat Earth, meridian lines would always be the same distance apart - they would form a square grid when combined with lines of latitude. In reality, meridian lines get farther apart as one travels toward the equator, which is only possible on a round Earth. In places where land is plotted on a grid system, this causes discontinuities in the grid. For example, in areas of the Midwestern United States that use the Public Land Survey System, the northernmost and westernmost sections of a township deviate from what would otherwise be an exact square mile. The resulting discontinuities are sometimes reflected directly in local roads, which have kinks where the grid cannot follow completely straight lines.
Mercator projection has notable examples of size distortions.

Spherical vs. flat triangles

Because the Earth is spherical, long-distance travel sometimes requires heading in different directions than one would head on a flat Earth.
For example, consider an airplane that travels in a straight line, takes a 90-degree right turn, travels another, takes another 90-degree right turn, and travels a third time. On a flat Earth, the aircraft would have travelled along three sides of a square, and arrive at a spot about from where it started. But because the Earth is spherical, in reality it will have travelled along three sides of a triangle, and arrive back very close to its starting point. If the starting point is the North Pole, it would have travelled due south from the North Pole to the equator, then west for a quarter of the way around the Earth, and then due north back to the North Pole.
In spherical geometry, the sum of angles inside a triangle is greater than 180° unlike on a flat surface, where it is always exactly 180°.

Weather systems

Low-pressure weather systems with inward winds spin counterclockwise north of the equator, but clockwise south of the equator. This is due to the Coriolis force, and requires that the north and southern halves of the Earth are angled in opposite directions.

Gravity

The laws of gravity, chemistry, and physics that explain the formation and rounding of the Earth are well-tested through experiment, and applied successfully to many engineering tasks.
From these laws, we know the amount of mass the Earth contains, and that a non-spherical planet the size of the Earth would not be able to support itself against its own gravity. A flat disc the size of the Earth, for example, would likely crack, heat up, liquefy, and re-form into a roughly spherical shape. On a disc strong enough to maintain its shape, gravity would not pull downward with respect to the surface, but would pull toward the center of the disc, contrary to what is observed on level terrain.
Ignoring the other concerns, some flat Earth theorists explain the observed surface "gravity" by proposing that the flat Earth is constantly accelerating upwards. Such a theory would also leave open for explanation the tides seen in Earth's oceans, which are conventionally explained by the gravity exerted by the Sun and Moon.

Evidence based on modern technology

Observation of Foucault pendulums, popular in science museums around the world, demonstrate both that the world is spherical and that it rotates.
The mathematics of navigation by GPS assume that satellites are moving in known orbits around an approximately spherical surface. The accuracy of GPS navigation in determining latitude and longitude and the way these numbers map onto locations on the ground show that these assumptions are correct. The same is true for the operational GLONASS system run by Russia, and the in-development European Galileo, Chinese BeiDou, and Indian IRNSS.
Satellites, including communications satellites used for television, telephone, and Internet connections, would not stay in orbit unless the modern theory of gravitation were correct. The details of which satellites are visible from which places on the ground at which times prove an approximately spherical shape of the Earth.
Radio transmitters are mounted on tall towers because they generally rely on line-of-sight propagation. The distance to the horizon is further at higher altitude, so mounting them higher significantly increases the area they can serve. Some signals can be transmitted at much longer distances, but only if they are at frequencies where they can use groundwave propagation, tropospheric propagation, tropospheric scatter, or ionospheric propagation to reflect or refract signals around the curve of the Earth.

Architecture

The design of some large structures needs to take the shape of the Earth into account. For example, the towers of the Humber Bridge, although both vertical with respect to gravity, are farther apart at the top than the bottom due to the local curvature.

Aircraft, spacecraft

People in high-flying aircraft or skydiving from high-altitude balloons can plainly see the curvature of the Earth. Commercial aircraft do not necessarily fly high enough to make this obvious. Trying to measure the curvature of the horizon by taking a picture is complicated by the fact that camera lenses can produce distorted images depending on the angle used. An extreme version of this effect can be seen in the fisheye lens. Scientific measurements would require a carefully calibrated lens.
The fastest way for an airplane to travel between two distant points is a great circle route. This route shows as curved on any map except for one using a gnomonic projection.
Photos of the ground taken from airplanes over a large enough area also do not fit seamlessly together on a flat surface, but do fit on a roughly spherical surface. Aerial photographs of large areas must be corrected to account for curvature.
Many pictures have been taken of the entire Earth by satellites launched by a variety of governments and private organizations. From high orbits, where half the planet can be seen at once, it is plainly spherical. The only way to piece together all the pictures taken of the ground from lower orbits so that all the surface features line up seamlessly and without distortion is to put them on an approximately spherical surface.
Astronauts in low Earth orbit can personally see the curvature of the planet, and travel all the way around several times a day.
The astronauts who travelled to the Moon have seen the entire Moon-facing half at once, and can watch the sphere rotate once a day.

Watching the Sun set twice using an elevator

On level ground, the difference in the distance to the horizon between lying down and standing up is large enough to watch the Sun set twice by quickly standing up immediately after seeing it set for the first time while lying down. This also can be done with a cherry picker or a tall building with a fast elevator. On a flat Earth, one would not be able to see the Sun again due to a much faster-moving Sun shadow.
When the supersonic Concorde took off not long after sunset from London and flew westward to New York, the aircraft outran the sun's apparent motion westward, and therefore passengers aboard observed the sun rising in the west as they traveled. After landing in New York, passengers watched a second sunset in the west.
Because the speed of the Sun's shadow is slower in polar regions, even a subsonic aircraft can overtake the sunset when flying at high latitudes. One photographer used a roughly circular route around the North Pole to take pictures of 24 sunsets in the same 24-hour period, pausing westward progress in each time zone to let the shadow of the Sun catch up. The surface of the Earth rotates at at 80° north or south, and at the equator.

History

Antiquity

Though the earliest written mention of a spherical Earth comes from ancient Greek sources, there is no account of how the sphericity of the Earth was discovered. A plausible explanation given by the historian Otto E. Neugebauer is that it was "the experience of travellers that suggested such an explanation for the variation in the observable altitude of the pole and the change in the area of circumpolar stars, a change that was quite drastic between Greek settlements" around the eastern Mediterranean Sea, particularly those between the Nile Delta and Crimea.
Another possible explanation can be traced back to earlier Phoenician sailors. The first circumnavigation of Africa is described as being undertaken by Phoenician explorers employed by Egyptian pharaoh Necho II c. 610–595 BC. In The Histories, written 431–425 BC, Herodotus cast doubt on a report of the Sun observed shining from the north. He stated that the phenomenon was observed by Phoenician explorers during their circumnavigation of Africa who claimed to have had the Sun on their right when circumnavigating in a clockwise direction. To modern historians, these details confirm the truth of the Phoenicians' report. The historian Dmitri Panchenko theorizes that it was the Phoenician circumnavigation of Africa that inspired the theory of a spherical Earth, the earliest mention of which was made by the philosopher Parmenides in the 5th century BC. However, nothing certain about their knowledge of geography and navigation has survived, which means we have no evidence that they conceived of the Earth as spherical.

Hellenic and Hellenistic world

Pythagoras
Early Greek philosophers alluded to a spherical Earth, though with some ambiguity. Pythagoras was among those said to have originated the idea, but this might reflect the ancient Greek practice of ascribing every discovery to one or another of their ancient wise men. Some idea of the sphericity of the Earth seems to have been known to both Parmenides and Empedocles in the 5th century BC, and although the idea cannot reliably be ascribed to Pythagoras, it might nevertheless have been formulated in the Pythagorean school in the 5th century BC although some disagree. After the 5th century BC, no Greek writer of repute thought the world was anything but round.
Plato
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, though he offered no justifications. "My conviction is that the Earth is a round body in the centre of the heavens, and therefore has no need of air or of any similar force to be a support". 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."
In Timaeus, his one work that was available throughout the Middle Ages in Latin, we read that the Creator "made the world in the form of a globe, round as from a lathe, having its extremes in every direction equidistant from the centre, the most perfect and the most like itself of all figures", though the word "world" here refers to the heavens.
Aristotle
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."
Aristotle provided physical and observational arguments supporting the idea of a spherical Earth:
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. Although no humans could survive in the frigid zones, inhabitants in the southern temperate regions could exist.
Aristotle's theory of natural place relied on a spherical Earth to explain why heavy things go down, and things like air and fire go up. In this geocentric model, the structure of the universe was believed to be a series of perfect spheres. The Sun, Moon, planets and fixed stars were believed to move on celestial spheres around a stationary Earth.
Though Aristotle's theory of physics survived in the Christian world for many centuries, the heliocentric model was eventually shown to be a more correct explanation of the Solar System than the geocentric model, and atomic theory was shown to be a more correct explanation of the nature of matter than classical elements like earth, water, air, fire, and aether.
Archimedes
In proposition 2 of the First Book of his treatise "On floating bodies," Archimedes demonstrates that "The surface of any fluid at rest is the surface of a sphere whose centre is the same as that of the Earth". Subsequently, in propositions 8 and 9 of the same work, he assumes the result of proposition 2 that the Earth is a sphere and that the surface of a fluid on it is a sphere centered on the center of the Earth.
Eratosthenes
, a Hellenistic astronomer from Cyrenaica, estimated Earth's circumference around 240 BC, computing a value of 252,000 stades. The length that Eratosthenes intended for a 'stade' is not known, but his figure only has an error of around one to fifteen percent. Eratosthenes could only measure the circumference of the Earth by assuming that the distance to the Sun is so great that the rays of sunlight are practically parallel.
1,700 years after Eratosthenes, Christopher Columbus studied Eratosthenes's findings before sailing west for the Indies. However, ultimately he rejected Eratosthenes in favour of other maps and arguments that interpreted Earth's circumference to be a third smaller than it really is. If, instead, Columbus had accepted Eratosthenes' findings, he may have never gone west, since he didn't have the supplies or funding needed for the much longer eight-thousand-plus mile voyage.
Seleucus of Seleucia
, who lived in the city of Seleucia in Mesopotamia, wrote that the Earth is spherical.
Posidonius
put faith in Eratosthenes's 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 Eratosthenes. Posidonius furthermore expressed the distance of the Sun in Earth radii.

Roman Empire

The idea of a spherical Earth slowly spread across the globe, and ultimately became the adopted view in all major astronomical traditions.
In the West, the idea came 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. Pliny also considered the possibility of an imperfect sphere "shaped like a pinecone".
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 geographer Strabo, 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, citing a line from the Odyssey as indicating that the poet Homer knew 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.
Claudius Ptolemy
lived in Alexandria, the centre of scholarship in the 2nd century. In the Almagest, which remained the standard work of astronomy for 1,400 years, he advanced many arguments for the spherical nature of the Earth. Among them was the observation that when a ship is sailing towards mountains, observers note these seem to rise from the sea, indicating that they were hidden by the curved surface of the sea. He also gives separate arguments that the Earth is curved north-south and that it is curved east-west.
He compiled an eight-volume Geographia covering what was known about the Earth. 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. 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" at the extreme right, beyond the island of "Taprobane" and the "Aurea Chersonesus".
Ptolemy also devised and provided instructions on how to create maps both of the whole inhabited world 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. Calcidius's fourth-century Latin commentary on and translation of Plato's Timaeus, which was one of the few examples of Greek scientific thought that was known in the Early Middle Ages in Western Europe, discussed Hipparchus's use of the geometrical circumstances of eclipses to compute the relative diameters of the Sun, Earth, and Moon.
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. Learned Christian authors such as 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. 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. From the 8th century and the beginning medieval period, "no cosmographer worthy of note has called into question the sphericity of the Earth."

India

While the textual evidence has not survived, the precision of the constants used in pre-Greek Vedanga models, and the model's accuracy in predicting the Moon and Sun's motion for Vedic rituals, probably came from direct astronomical observations. The cosmographic theories and assumptions in ancient India likely developed independently and in parallel, but these were influenced by some unknown quantitative Greek astronomy text in the medieval era.
Greek ethnographer Megasthenes, c. 300 BC, has been interpreted as stating that the contemporary Brahmans believed in a spherical Earth as the center of the universe. With the spread of Hellenistic culture in the east, Hellenistic astronomy filtered eastwards to ancient India where its profound influence became apparent in the early centuries AD. The Greek concept of an Earth surrounded by the spheres of the planets and that of the fixed stars, vehemently supported by astronomers like Varāhamihira and Brahmagupta, strengthened the astronomical principles. Some ideas were found possible to preserve, although in altered form.
The works of the classical Indian astronomer and mathematician, Aryabhatta, 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 and the Gol, state that the Earth is spherical and that its circumference is 4,967 yojanas. In modern units this is, close to the current equatorial value of.

Middle Ages

In medieval Europe, knowledge of the sphericity of the Earth survived into the medieval corpus of knowledge by direct transmission of the texts of Greek antiquity, and via authors such as Isidore of Seville and Beda Venerabilis.
It became increasingly traceable with the rise of scholasticism and medieval learning.
Spread of this knowledge beyond the immediate sphere of Greco-Roman scholarship was necessarily gradual, associated with the pace of Christianisation of Europe. For example, the first evidence of knowledge of the spherical shape of the Earth in Scandinavia is a 12th-century Old Icelandic translation of Elucidarius.
A non-exhaustive list of more than a hundred Latin and vernacular writers from Late Antiquity and the Middle Ages who were aware that the earth was spherical has been compiled by Reinhard Krüger, professor for Romance literature at the University of Stuttgart.
;Late Antiquity
Ampelius, Chalcidius, Macrobius, Martianus Capella,
Basil of Caesarea, Ambrose of Milan, Aurelius Augustinus, Paulus Orosius, Jordanes, Cassiodorus, Boethius, Visigoth king Sisebut.
;Early Middle Ages
Isidore of Seville, Beda Venerabilis, Theodulf of Orléans, Vergilius of Salzburg,
Irish monk Dicuil, Rabanus Maurus, King Alfred of England, Remigius of Auxerre, Johannes Scotus Eriugena, :de:Leo von Neapel|Leo of Naples, Gerbert d’Aurillac.
;High Middle Ages
Notker the German of Sankt-Gallen, Hermann of Reichenau, Hildegard von Bingen, Petrus Abaelardus, Honorius Augustodunensis, Gautier de Metz, Adam of Bremen, Albertus Magnus, Thomas Aquinas, Berthold of Regensburg, Guillaume de Conches, :fr:Philippe de Thaon|Philippe de Thaon, Abu-Idrisi, Bernardus Silvestris, Petrus Comestor, Thierry de Chartres, Gautier de Châtillon, Alexander Neckam, Alain de Lille, Averroes, Snorri Sturluson, Moshe ben Maimon, Lambert of Saint-Omer, Gervasius of Tilbury, Robert Grosseteste, Johannes de Sacrobosco, Thomas de Cantimpré, Peire de Corbian, Vincent de Beauvais, Robertus Anglicus, :es:Juan Gil de Zamora|Juan Gil de Zámora, Ristoro d'Arezzo, Roger Bacon, Jean de Meung, Brunetto Latini, Alfonso X of Castile.
;Late Middle Ages
Marco Polo, Dante Alighieri, Meister Eckhart, Enea Silvio Piccolomini, Perot de Garbalei, Cecco d'Ascoli, :it:Fazio degli Uberti|Fazio degli Uberti, Levi ben Gershon, Konrad of Megenberg, Nicole Oresme, Petrus Aliacensis, :de:Alfonso de la Torre|Alfonso de la Torre, Toscanelli, :de:Brochard der Deutsche|Brochard the German, Jean de Mandeville, Christine de Pizan, Geoffrey Chaucer, William Caxton, Martin Behaim, Christopher Columbus.

Early Medieval Europe

;Isidore of Seville
Bishop Isidore of Seville taught in his widely read encyclopedia, The Etymologies, that the Earth was "round". The bishop's confusing exposition and choice of imprecise Latin terms have divided scholarly opinion on whether he meant a sphere or a disk or even whether he meant anything specific. Notable recent scholars claim that he taught a spherical Earth. Isidore did not admit the possibility of people dwelling at the antipodes, considering them as legendary and noting that there was no evidence for their existence.
;Bede the Venerable
The monk Bede wrote in his influential treatise on computus, The Reckoning of Time, that the Earth was round. He explained 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.". 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. 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."
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 centre 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 centre with perfect roundness on all sides."
;Anania Shirakatsi
The 7th-century Armenian scholar Anania Shirakatsi described the world as "being like an egg with a spherical yolk surrounded by a layer of white and covered with a hard shell."

Islamic astronomy

was developed on the basis of a spherical earth inherited from Hellenistic astronomy. The Islamic theoretical framework largely relied on the fundamental contributions of Aristotle and Ptolemy, both of whom worked from the premise that the Earth was spherical and at the centre of the universe.
Early Islamic scholars recognized Earth's sphericity, leading Muslim mathematicians to develop spherical trigonometry 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 geographers to measure the distance from Tadmur to Raqqa in modern Syria. They found the cities to be separated by one degree of latitude and the meridian arc distance between them to be 66 miles and thus calculated the Earth's circumference to be.
Another estimate given by his astronomers was 56 Arabic miles 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.
;Ibn Hazm
Andalusian polymath Ibn Hazm stated that the proof of the Earth's sphericity "is that the Sun is always vertical to a particular spot on Earth".
;Al-Farghānī
Al-Farghānī 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 was much more accurate than the 60 Roman miles 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.
;Biruni
Abu Rayhan Biruni used a new method 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. His estimate of 6,339.6 km for the Earth radius was only 31.4 km less than the modern mean value of 6,371.0 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. This yielded more accurate measurements of the Earth's circumference and made it possible for a single person to measure it from a single location.
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, allowed him to calculate the curvature of the Earth.
He also made use of algebra to formulate trigonometric equations and used the astrolabe to measure angles.
According to John J. O'Connor and Edmund F. Robertson,
;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 Mecca. This determined the Qibla, or Muslim direction of prayer.
A terrestrial globe was among the presents sent by the Persian Muslim astronomer Jamal-al-Din to Kublai Khan's Chinese court 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." Ho Peng Yoke remarks that "it did not seem to have any general appeal to the Chinese in those days".

High and late medieval Europe

During the High Middle Ages, the astronomical knowledge in Christian Europe was extended beyond what was transmitted directly from ancient authors by transmission of learning from Medieval Islamic astronomy. An early student of such learning was Gerbert d'Aurillac, the later Pope Sylvester II.
Saint Hildegard, depicted the spherical Earth several times in her work Liber Divinorum Operum.
Johannes de Sacrobosco wrote a famous work on Astronomy called Tractatus de Sphaera, based on Ptolemy, which primarily considers the sphere of the sky. However, it contains clear proofs of the Earth's sphericity in the first chapter.
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. Grant observes that no author who had studied at a medieval university thought that the Earth was flat.
The Elucidarium of Honorius Augustodunensis, 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 used the spherical Earth as an illustration in a sermon shows that he could assume this knowledge among his congregation. The sermon was preached in the vernacular German, and thus was not intended for a learned audience.
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 time zones of the Earth.
The Portuguese exploration of Africa and Asia, Columbus's voyage to the Americas and, finally, Ferdinand Magellan's circumnavigation of the Earth provided practical evidence of the global shape of the Earth.

Early Modern period

Circumnavigation of the globe

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. 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 what is now called 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.
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. The Transglobe Expedition was the first expedition to make a circumpolar circumnavigation, traveling the world "vertically" traversing both of the poles of rotation using only surface transport.

Ming China

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.
The Ge zhi cao treatise of Xiong Mingyu published in 1648 showed a printed picture of the Earth as a spherical globe, with the text stating that "the round Earth certainly has no square corners". The text also pointed out that sailing ships could return to their port of origin after circumnavigating the waters of the Earth.
The influence of the map is distinctly Western, as traditional maps of Chinese cartography held the graduation of the sphere at 365.25 degrees, while the Western graduation was of 360 degrees. Also of interest to note is on one side of the world, there is seen towering Chinese pagodas, while on the opposite side there were European cathedrals. The adoption of European astronomy, facilitated by the failure of indigenous astronomy to make progress, was accompanied by a sinocentric reinterpretation that declared the imported ideas Chinese in origin:
European astronomy was so much judged worth consideration that numerous Chinese authors developed the idea that the Chinese of antiquity had anticipated most of the novelties presented by the missionaries as European discoveries, for example, the rotundity of the Earth and the "heavenly spherical star carrier model." Making skillful use of philology, these authors cleverly reinterpreted the greatest technical and literary works of Chinese antiquity. From this sprang a new science wholly dedicated to the demonstration of the Chinese origin of astronomy and more generally of all European science and technology.

Although mainstream Chinese science until the 17th century held the view that the Earth was flat, square, and enveloped by the celestial sphere, this idea was criticized by the Jin-dynasty scholar Yu Xi, who suggested that the Earth could be either square or round, in accordance with the shape of the heavens. The Yuan-dynasty mathematician Li Ye firmly argued that the Earth was spherical, just like the shape of the heavens only smaller, since a square Earth would hinder the movement of the heavens and celestial bodies in his estimation. The 17th-century Ge zhi cao treatise also used the same terminology to describe the shape of the Earth that the Eastern-Han scholar Zhang Heng had used to describe the shape of the Sun and Moon.

Measurement and representation

, also called geodetics, is the scientific discipline that deals with the measurement and representation of the Earth, its gravitational field and geodynamic phenomena 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, which is concerned with measuring the Earth on a global scale, and surveying, which is concerned with measuring parts of the surface.
The Earth's shape can be thought of in at least two ways;
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.