Earth's circumference

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Eratosthenes' method for determining the circumference of the Earth, with sunbeams shown as two rays hitting the ground at two locations in Egypt – Syene (Aswan) and Alexandria. Eratosthenes' method for determining the size of the Earth.svg
Eratosthenes' method for determining the circumference of the Earth, with sunbeams shown as two rays hitting the ground at two locations in EgyptSyene (Aswan) and Alexandria.

Earth's circumference is the distance around the Earth, either around the equator (40,075.017 km [ 24,901.461 mi ] [1] ) or around the poles (40,007.86 km [ 24,859.73 mi ] [2] ).

In geometry, the circumference of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses. Informally, "circumference" may also refer to the edge itself rather than to the length of the edge.

Earth Third planet from the Sun in the Solar System

Earth is the third planet from the Sun and the only astronomical object known to harbor life. According to radiometric dating and other sources of evidence, Earth formed over 4.5 billion years ago. Earth's gravity interacts with other objects in space, especially the Sun and the Moon, Earth's only natural satellite. Earth revolves around the Sun in 365.26 days, a period known as an Earth year. During this time, Earth rotates about its axis about 366.26 times.

Equator Intersection of a spheres surface with the plane perpendicular to the spheres axis of rotation and midway between the poles

An equator of a rotating spheroid is its zeroth circle of latitude (parallel). It is the imaginary line on the spheroid's surface, equidistant from its poles, dividing it into northern and southern hemispheres. In other words, it is the intersection of the spheroid's surface with the plane perpendicular to its axis of rotation and midway between its geographical poles.

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Measurement of Earth's circumference has been important to navigation since ancient times. It was first calculated by Eratosthenes, which he did by comparing altitudes of the mid-day sun at two places a known North-South distance apart. [3] In the Middle Ages, Al Biruni calculated a more accurate version, becoming the first person to perform the calculation based on data from a single location.

Navigation The process of monitoring and controlling the movement of a craft or vehicle from one place to another

Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another. The field of navigation includes four general categories: land navigation, marine navigation, aeronautic navigation, and space navigation.

Eratosthenes ancient Greek scientist

Eratosthenes of Cyrene was a Greek mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria. He invented the discipline of geography, including the terminology used today.

In modern times, Earth's circumference has been used to define fundamental units of measurement of length: the nautical mile in the seventeenth century and the metre in the eighteenth. Earth's polar circumference is very near to 21,600 nautical miles because the nautical mile was intended to express 1/60TH of a degree of latitude, which is 21,600 partitions of the polar circumference. The polar circumference is even closer to 40,000 kilometres because the metre was originally defined to be one 10-millionth the distance from pole to equator. The physical length of each unit of measure has remained close to what it was determined to be at the time, but the precision of measuring the circumference has improved since then: the polar circumference is actually 21,566 nautical miles and 39,941 kilometres.

Nautical mile unit of distance (1852 m)

A nautical mile is a unit of measurement used in both air and marine navigation, and for the definition of territorial waters. Historically, it was defined as one minute of a degree of latitude. Today it is defined as 1,852 metres. The derived unit of speed is the knot, one nautical mile per hour.

History of the metre

The history of the metre starts with the scientific revolution that began with Nicolaus Copernicus's work in 1543. Increasingly accurate measurements were required, and scientists looked for measures that were universal and could be based on natural phenomena rather than royal decree or physical prototypes. Rather than the various complex systems of subdivision in use, they also preferred a decimal system to ease their calculations.

Treated as a sphere, determining Earth's circumference would be its single most important measurement. [4] (Earth actually deviates from spherical by about 0.3% as characterized by flattening.)

Flattening measure of compression between circle to ellipse or sphere to an ellipsoid of revolution

Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f and its definition in terms of the semi-axes of the resulting ellipse or ellipsoid is

History of calculation

Illustration showing a portion of the globe showing a part of the African continent. The sun beams shown as two rays hitting earth at Syene and Alexandria. Angle of sun beam and the gnomons (vertical sticks) is shown at Alexandria which allowed Eratosthenes' estimate of the circumference of Earth. Eratosthenes.png
Illustration showing a portion of the globe showing a part of the African continent. The sun beams shown as two rays hitting earth at Syene and Alexandria. Angle of sun beam and the gnomons (vertical sticks) is shown at Alexandria which allowed Eratosthenes' estimate of the circumference of Earth.

According to Cleomedes' On the Circular Motions of the Celestial Bodies, around 240 BC, Eratosthenes, the librarian of the Library of Alexandria, calculated the circumference of the Earth in Ptolemaic Egypt. [5] Using a scaphe, he knew that at local noon on the summer solstice in Syene (modern Aswan, Egypt), the Sun was directly overhead. (Syene is at latitude 24°05′ North, near to the Tropic of Cancer, which was 23°42′ North in 100 BC [6] ) He knew this because the shadow of someone looking down a deep well at that time in Syene blocked the reflection of the Sun on the water. He then measured the Sun's angle of elevation at noon in Alexandria by using a vertical rod, known as a gnomon, and measuring the length of its shadow on the ground. [7] Using the length of the rod, and the length of the shadow, as the legs of a triangle, he calculated the angle of the sun's rays. [8] This angle was about 7°, or 1/50th the circumference of a circle; taking the Earth as perfectly spherical, he concluded that the Earth's circumference was 50 times the known distance from Alexandra to Syene (5,000 stadia, a figure that was checked yearly), i.e. 250,000 stadia. [9] Depending on whether he used the "Olympic stade" (176.4m) or the Italian stade (184.8m), this would imply a circumference of 44,100 km (an error of 10%) or 46,100 km, an error of 15%. [9] In 2012, Anthony Abreu Mora repeated Eratosthenes's calculation with more accurate data; the result was 40,074 km, which is 66 km different (0.16%) from the currently accepted polar circumference. [8]

Cleomedes was a Greek astronomer who is known chiefly for his book On the Circular Motions of the Celestial Bodies.

Library of Alexandria one of the largest libraries in the ancient world, located in Alexandria, Egypt

The Great Library of Alexandria in Alexandria, Egypt, was one of the largest and most significant libraries of the ancient world. The Library was part of a larger research institution called the Mouseion, which was dedicated to the Muses, the nine goddesses of the arts. The idea of a universal library in Alexandria may have been proposed by Demetrius of Phalerum, an exiled Athenian statesman living in Alexandria, to Ptolemy I Soter, who may have established plans for the Library, but the Library itself was probably not built until the reign of his son Ptolemy II Philadelphus. The Library quickly acquired a large number of papyrus scrolls, due largely to the Ptolemaic kings' aggressive and well-funded policies for procuring texts. It is unknown precisely how many such scrolls were housed at any given time, but estimates range from 40,000 to 400,000 at its height.

Earth radius mean distance from the Earths center to its surface

Earth radius is the distance from a selected center of Earth to a point on its surface, which is often chosen to be sea level, or more commonly, the surface of an idealized ellipsoid representing the shape of Earth. Because Earth is not a perfect sphere, the determination of Earth's radius can have several values, depending on how it is measured; from its equatorial radius of about 6,378 kilometres to its polar radius of about 6,357 kilometres.

Posidonius calculated the Earth's circumference by reference to the position of the star Canopus. As explained by Cleomedes, Posidonius observed Canopus on but never above the horizon at Rhodes, while at Alexandria he saw it ascend as far as 7 12 degrees above the horizon (the meridian arc between the latitude of the two locales is actually 5 degrees 14 minutes). Since he thought Rhodes was 5,000 stadia due north of Alexandria, and the difference in the star's elevation indicated the distance between the two locales was 1/48 of the circle, he multiplied 5,000 by 48 to arrive at a figure of 240,000 stadia for the circumference of the earth. [10] It is generally thought that the stadion used by Posidonius was almost exactly 1/10 of a modern statute mile. Thus Posidonius's measure of 240,000 stadia translates to 24,000 mi (39,000 km), not much short of the actual circumference of 24,901 mi (40,074 km). [10] Strabo noted that the distance between Rhodes and Alexandria is 3,750 stadia, and reported Posidonius's estimate of the Earth's circumference to be 180,000 stadia or 18,000 mi (29,000 km). [11] Pliny the Elder mentions Posidonius among his sources and without naming him reported his method for estimating the Earth's circumference. He noted, however, that Hipparchus had added some 26,000 stadia to Eratosthenes's estimate. The smaller value offered by Strabo and the different lengths of Greek and Roman stadia have created a persistent confusion around Posidonius's result. Ptolemy used Posidonius's lower value of 180,000 stades (about 33% too low) for the earth's circumference in his Geography. This was the number used by Christopher Columbus in order to underestimate the distance to India as 70,000 stades. [12]

Alexandria Metropolis in Egypt

Alexandria is the second-largest city in Egypt and a major economic centre, extending about 32 km (20 mi) along the coast of the Mediterranean Sea in the north central part of the country. Its low elevation on the Nile delta makes it highly vulnerable to rising sea levels. Alexandria is an important industrial center because of its natural gas and oil pipelines from Suez. Alexandria is also a popular tourist destination.

In geodesy, a meridian arc measurement is the distance between two points with the same longitude, i.e., a segment of a meridian curve or its length. Two or more such determinations at different locations then specify the shape of the reference ellipsoid which best approximates the shape of the geoid. This process is called the determination of the figure of the Earth. The earliest determinations of the size of a spherical Earth required a single arc. The latest determinations use astro-geodetic measurements and the methods of satellite geodesy to determine the reference ellipsoids.

Strabo Greek geographer, philosopher and historian

Strabo was a Greek geographer, philosopher, and historian who lived in Asia Minor during the transitional period of the Roman Republic into the Roman Empire.

Diagram showing how Al Biruni was able to calculate the Earth’s circumference from a high point and a low point at the same location. Abu Reyhan Biruni-Earth Circumference.svg
Diagram showing how Al Biruni was able to calculate the Earth’s circumference from a high point and a low point at the same location.

A more accurate estimate was provided in Al-Biruni's Codex Masudicus (1037). [13] [14] In contrast to his predecessors, who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, al-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. [15]

1,700 years after Eratosthenes's death, while Christopher Columbus studied what Eratosthenes had written about the size of the Earth, he chose to believe, based on a map by Toscanelli, that the Earth's circumference was one-third smaller. Had Columbus set sail knowing that Eratosthenes's larger circumference value was more accurate, he would have known that the place that he made landfall was not Asia, but rather the New World. [16]

Units of measurement based on Earth’s circumference

Both the metre and the nautical mile were originally defined as a subdivision of the Earth's circumference; today the circumference around the poles is very nearly 40,000 km and 360 × 60 nautical miles long. [17]

In 1617 the Dutch scientist Willebrord Snellius assessed the circumference of the Earth at 24,630 Roman miles (24,024 statute miles). Around that time British mathematician Edmund Gunter improved navigational tools including a new quadrant to determine latitude at sea. He reasoned that the lines of latitude could be used as the basis for a unit of measurement for distance and proposed the nautical mile as one minute or one-sixtieth (1/60) of one degree of latitude. As one degree is 1/360 of a circle, one minute of arc is 1/21600 of a circle – such that the polar circumference of the Earth would be exactly 21,600 miles. Gunter used Snell's circumference to define a nautical mile as 6,080 feet, the length of one minute of arc at 48 degrees latitude. [18]

In 1791, the French Academy of Sciences selected the circumference definition over the alternative pendular definition because the force of gravity varies slightly over the surface of the Earth, which affects the period of a pendulum. [19] To establish a universally accepted foundation for the definition of the metre, more accurate measurements of this meridian were needed. The French Academy of Sciences commissioned an expedition led by Jean Baptiste Joseph Delambre and Pierre Méchain, lasting from 1792 to 1799, which attempted to accurately measure the distance between a belfry in Dunkerque and Montjuïc castle in Barcelona to estimate the length of the meridian arc through Dunkerque. This portion of the meridian, assumed to be the same length as the Paris meridian, was to serve as the basis for the length of the half meridian connecting the North Pole with the Equator. The problem with this approach is that the exact shape of the Earth is not a simple mathematical shape, such as a sphere or oblate spheroid, at the level of precision required for defining a standard of length. The irregular and particular shape of the Earth smoothed to sea level is represented by a mathematical model called a geoid, which literally means "Earth-shaped". Despite these issues, in 1793 France adopted this definition of the metre as its official unit of length based on provisional results from this expedition. However, it was later determined that the first prototype metre bar was short by about 200 micrometres because of miscalculation of the flattening of the Earth, making the prototype about 0.02% shorter than the original proposed definition of the metre. Regardless, this length became the French standard and was progressively adopted by other countries in Europe. [19]

Bibliography

See also

Related Research Articles

Geodesy The science of the geometric shape, orientation in space, and gravitational field of the Earth

Geodesy, is the earth science of accurately measuring and understanding the Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measurements for other planets. Geodynamical phenomena include crustal motion, tides, and polar motion, which can be studied by designing global and national control networks, applying space and terrestrial techniques, and relying on datums and coordinate systems.

Latitude The angle between zenith at a point and the plane of the equator

In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications.

Longitude A geographic coordinate that specifies the east-west position of a point on the Earths surface

Longitude, is a geographic coordinate that specifies the east–west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians connect points with the same longitude. By convention, one of these, the Prime Meridian, which passes through the Royal Observatory, Greenwich, England, was allocated the position of 0° longitude. The longitude of other places is measured as the angle east or west from the Prime Meridian, ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. Specifically, it is the angle between a plane through the Prime Meridian and a plane through both poles and the location in question.

Mercator projection map projection

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for nautical navigation because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as straight segments that conserve the angles with the meridians. Although the linear scale is equal in all directions around any point, thus preserving the angles and the shapes of small objects, the Mercator projection distorts the size of objects as the latitude increases from the Equator to the poles, where the scale becomes infinite. So, for example, landmasses such as Greenland and Antarctica appear much larger than they actually are, relative to landmasses near the equator such as Central Africa.

Geographic coordinate system Coordinate system

A geographic coordinate system is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector. A common choice of coordinates is latitude, longitude and elevation. To specify a location on a plane requires a map projection.

Posidonius ancient greek philosopher

Posidonius "of Apameia" or "of Rhodes", was a Greek Stoic philosopher, politician, astronomer, geographer, historian and teacher native to Apamea, Syria.

Spherical Earth idea that the Earth is spherical

The earliest reliably documented mention of the spherical Earth concept dates from around the 6th century BC when it appeared in ancient Greek philosophy, but remained a matter of speculation until the 3rd century BC, when Hellenistic astronomy established the spherical shape of the Earth as a physical given and calculated 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 expedition's circumnavigation (1519–1522).

History of geodesy

Geodesy (/dʒiːˈɒdɨsi/), also named geodetics, is the scientific discipline that deals with the measurement and representation of the Earth. The history of geodesy began in pre-scientific antiquity and blossomed during the Age of Enlightenment.

Scaphe Sundial said to have been invented by Aristarchus

The Scaphe was a sundial said to have been invented by Aristarchus. There are no original works still in existence by Aristarchus, but the adjacent image is an accurate image of what it might have looked like, only his would have been made of stone. It consisted of a hemispherical bowl which had a vertical gnomon placed inside it, with the top of the gnomon level with the edge of the bowl. Twelve gradations inscribed perpendicular to the hemisphere indicated the hour of the day. Using this measuring instrument, Eratosthenes of Cyrene measured the length of Earth's meridian arc. The scaphe is also known as a skaphe, scaphion (diminutive) or Latin: scaphium.

Meridian (geography) line between the poles with the same longitude

A (geographical) meridian is the half of an imaginary great circle on the Earth's surface, terminated by the North Pole and the South Pole, connecting points of equal longitude, as measured in angular degrees east or west of the Prime Meridian. The position of a point along the meridian is given by that longitude and its latitude, measured in angular degrees north or south of the Equator. Each meridian is perpendicular to all circles of latitude. Each is also the same length, being half of a great circle on the Earth's surface and therefore measuring 20,003.93 km.

Grade measurement is the geodetic determination of the local radius of curvature of the figure of the Earth by determining the difference in astronomical latitude between two locations on the same meridian, the metric distance between which is known.

The stadion, formerly also anglicized as stade, was an ancient Greek unit of length, based on the length of a typical sports stadium of the time. According to Herodotus, one stadion was equal to 600 Greek feet (podes). However, the length of the foot varied in different parts of the Greek world, and the length of the stadion has been the subject of argument and hypothesis for hundreds of years. Various hypothetical equivalent lengths have been proposed, and some have been named. Among them are:

The Arab, Arabic, or Arabian mile was a historical Arabic unit of length. Its precise length is disputed, lying between 1.8 and 2.0 km. It was used by medieval Arab geographers and astronomers. The predecessor of the modern nautical mile, it extended the Roman mile to fit an astronomical approximation of 1 minute of an arc of latitude measured along a north-south meridian. The distance between two pillars whose latitudes differed by 1 degree in a north-south direction was measured using sighting pegs along a flat desert plane.

Triangulation (surveying)

In surveying, triangulation is the process of determining the location of a point by measuring only angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly as in trilateration. The point can then be fixed as the third point of a triangle with one known side and two known angles.

References

  1. World Geodetic System (WGS-84). Available online from National Geospatial-Intelligence Agency.
  2. Humerfelt, Sigurd (26 October 2010). "How WGS 84 defines Earth". Archived from the original on 24 April 2011. Retrieved 29 April 2011.
  3. Ridpath, Ian (2001). The Illustrated Encyclopedia of the Universe. New York, NY: Watson-Guptill. p. 31. ISBN   978-0-8230-2512-1.
  4. Shashi Shekhar; Hui Xiong (12 December 2007). Encyclopedia of GIS. Springer Science & Business Media. pp. 638–640. ISBN   978-0-387-30858-6.
  5. Van Helden, Albert (1985). Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley. University of Chicago Press. pp. 4–5. ISBN   978-0-226-84882-2.
  6. Balasubramaniam, R. (10 August 2017). Story of the Delhi Iron Pillar. Foundation Books. ISBN   9788175962781 via Google Books.
  7. "Astronomy 101 Specials: Eratosthenes and the Size of the Earth". www.eg.bucknell.edu. Retrieved 19 December 2017.
  8. 1 2 "How did Eratosthenes measure the circumference of the earth?". 3 July 2012.
  9. 1 2 "Eratosthenes and the Mystery of the Stades – How Long Is a Stade? – Mathematical Association of America". www.maa.org.
  10. 1 2 Posidonius, fragment 202
  11. Cleomedes (in Fragment 202) stated that if the distance is measured by some other number the result will be different, and using 3,750 instead of 5,000 produces this estimation: 3,750 x 48 = 180,000; see Fischer I., (1975), Another Look at Eratosthenes' and Posidonius' Determinations of the Earth's Circumference, Ql. J. of the Royal Astron. Soc., Vol. 16, p.152.
  12. John Freely, Before Galileo: The Birth of Modern Science in Medieval Europe (2012)
  13. King, David A. (1996). Rashed, Roshdi, ed. Astronomy and Islamic society: Qibla, gnomics and timekeeping (PDF). Encyclopedia of the History of Arabic Science. 1. pp. 128–184. ISBN   978-0203711842 . Retrieved 10 November 2016.
  14. Aber, James Sandusky (2003). "Abu Rayhan al-Biruni". academic.emporia.edu. Emporia State University . Retrieved 10 November 2016.
  15. Goodman, Lenn Evan (1992). Avicenna. Great Britain: Routledge. p. 31. ISBN   978-0415019293 . Retrieved 10 November 2016. It was Biruni, not Avicenna, who found a way for a single man, at a single moment, to measure the earth's circumference, by trigonometric calculations based on angles measured from a mountaintop and the plain beneath it – thus improving on Eratosthenes' method of sighting the sun simultaneously from two different sites, applied in the ninth century by astronomers of the Khalif al-Ma'mun.
  16. Gow, Mary. "Measuring the Earth: Eratosthenes and His Celestial Geometry, p. 6 (Berkeley Heights, NJ: Enslow, 2010).
  17. Garrison, Peter (April 2008). "Old Wings". Flying Magazine: 90. ISSN   0015-4806.: "The kilometer, which is the foundation of the entire SI or “metric” system, was originally intended to be 1/10,000th of a quadrant of a meridian — that is, 1/40,000th of the earth's polar circumference — and was established as such in 1793... The nautical mile, like the kilometer, is a unit based on the dimensions of the earth. It is the length of one minute of arc along a meridian. (The meridians are the lines that run from pole to pole on the globe; the other ones are called parallels, and minutes of arc on them shrink toward the poles.) One minute of arc is 1/21,600th of a full circumference, and so the polar circumference of the earth... is 21,600 nautical miles."
  18. Marine Insight, Why Nautical Mile and Knot Are The Units Used at Sea?
  19. 1 2 Alder, Ken (October 2003). The Measure of All Things: The Seven-Year Odyssey and Hidden Error That Transformed the World. Simon and Schuster. ISBN   978-0-7432-1676-0.