Spherical Earth

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

The concept of a spherical Earth was espoused by Pythagoras apparently on aesthetic grounds, as he also held all other celestial bodies to be spherical. It replaced 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. Other speculations as to the shape of Earth include a seven-layered ziggurat or cosmic mountain, alluded to in the Avesta and ancient Persian writings (see seven climes). In fact, the Earth is reasonably well-approximated by an oblate spheroid.

Early development

Yajnavalkya

Yajnavalkya (c. 9th–8th century BC) recognized that the Earth is spherical in his astronomical text Shatapatha Brahmana.

Pythagoras

Pythagoras (b. 570 BC) found harmony in the universe and sought to explain it. He reasoned that Earth and the other planets must be spheres, since the most harmonious geometric form was a circle.

Plato

Plato (427 BC - 347 BC) 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

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

Aristotle (384 BC - 322 BC) 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 were separated by a torrid zone near the equator, as well as two cold inhospitable regions, "one near our upper or northern pole and the other near the ... southern pole," both impenetrable and girdled with ice (Meteorologica, 362a31-35). Although no humans could survive in the frigid zones, inhabitants in the southern temperate regions could exist.

Eratosthenes

Eratosthenes (276 BC - 194 BC) estimated Earth's circumference around 240 BC. He had heard about a place in Egypt where the Sun was directly overhead at the summer solstice and used geometry to come up with a circumference of 250,000 stades. This estimate astonishes some modern writers, as it is within 2% of the modern value of the equatorial circumference, 40,075 kilometres. However, the length of a 'stade' is not precisely known; Eratosthenes' figure falls short if we do not use a fairly generous estimate for this length.

Claudius Ptolemy

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

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

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

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

Aryabhatta

The works of the classical Indian astronomer and mathematician Aryabhatta (CE 476 - 550) 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 24,835 miles, very close to the current value of 24,902 miles.. 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.

Geodesy

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

Geodesy is primarily concerned with positioning and the gravity field and geometrical aspects of their temporal variations, although it can also include the study of Earth's magnetic field. Especially in the German speaking world, geodesy is divided in 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.

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

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

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

Spherical models

The Earth as seen from the Apollo 17 mission.

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.

• 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: ${\sqrt[{3}]{a^{2}b}}$, or 6371.001 km.

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

See also

• Earth radius
• Figure of the Earth
• Flat Earth
• Celestial sphere
• Physical geodesy
• Terra Australis
• WGS 84

External links

• You say the earth is round? Prove it (from The Straight Dope)
• Oblate Spheroid