Arc measurement, sometimes degree measurement (german: Gradmessung), is the astrogeodetic technique of determining of the

radius of Earth Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, deno ...

– more specifically, the local Earth radius of curvature of the

figure of the Earth Figure of the Earth is a Jargon, term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A Spherical Earth, sphere is a well-k ...

– by relating the

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

difference (sometimes also the

longitude Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...

difference) and the

geographic distance Geographical distance or geodetic distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. ...

(

arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...

)

surveyed Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Two-dimensional space#In geometry, two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of ...

between two locations on Earth's surface. The most common variant involves only

astronomical latitude In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...

s and the meridian arc length and is called ''meridian arc measurement''; other variants may involve only astronomical longitude (''

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...

arc measurement'') or both

geographic coordinates The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various ...

(''oblique arc measurement''). Arc measurement campaigns in Europe were the precursors to the

International Association of Geodesy ) , merged = , successor = , formation = , founder = , founding_location = , extinction = , merger = , type = scholarly society , tax_id ...

(IAG).

History

The first known arc measurement was performed by

Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...

(240 BC) between Alexandria and Syene in what is now Egypt, determining the radius of the Earth with remarkable correctness. In the early 8th century,

Yi Xing Yi Xing (, 683–727), born Zhang Sui (), was a Chinese astronomer, Buddhist monk, inventor, mathematician, mechanical engineer, and philosopher during the Tang dynasty. His astronomical celestial globe featured a liquid-driven escapement, the ...

performed a similar survey. The French physician

Jean Fernel Jean François Fernel ( Latinized as Ioannes Fernelius; 1497 – 26 April 1558) was a French physician who introduced the term "physiology" to describe the study of the body's function. He was the first person to describe the spinal canal. The l ...

measured the arc in 1528. The Dutch geodesist Snellius (~1620) repeated the experiment between

Alkmaar Alkmaar () is a city and municipality in the Netherlands, located in the province of North Holland, about 30 km north of Amsterdam. Alkmaar is well known for its traditional cheese market. For tourists, it is a popular cultural destination. The ...

and

Bergen op Zoom Bergen op Zoom (; called ''Berrege'' in the local dialect) is a municipality and a city located in the south of the Netherlands. Etymology The city was built on a place where two types of soil meet: sandy soil and marine clay. The sandy soil ...

using more modern geodetic instrumentation (''

Snellius' triangulation Willebrord Snellius (born Willebrord Snel van Royen) (13 June 158030 October 1626) was a Dutch astronomer and mathematician, Snell. His name is usually associated with the law of refraction of light known as Snell's law. The lunar crater Sn ...

''). Later arc measurements aimed at determining the

flattening 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 ...

of the Earth ellipsoid by measuring at different

geographic latitude In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pole ...

s. The first of these was the ''

French Geodesic Mission The French Geodesic Mission to the Equator (french: Expédition géodésique française en Équateur, also called the French Geodesic Mission to Peru and the Spanish-French Geodesic Mission) was an 18th-century expedition to what is now Ecuador c ...

'', commissioned by the

French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...

in 1735–1738, involving measurement expeditions to Lapland (

Maupertuis Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...

et al.) and Peru (

Pierre Bouguer Pierre Bouguer () (16 February 1698, Croisic – 15 August 1758, Paris) was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture". Career Bouguer's father, Jean Bouguer, one ...

et al.). Struve measured a

geodetic control network A geodetic control network (also geodetic network, reference network, control point network, or control network) is a network, often of triangles, which are measured precisely by techniques of terrestrial surveying or by satellite geodesy. ...

via

triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...

between the Arctic Sea and the

Black Sea The Black Sea is a marginal mediterranean sea of the Atlantic Ocean lying between Europe and Asia, east of the Balkans, south of the East European Plain, west of the Caucasus, and north of Anatolia. It is bounded by Bulgaria, Georgia, Roma ...

, the '' Struve Geodetic Arc''. Bessel compiled several meridian arcs, to compute the famous

Bessel ellipsoid The Bessel ellipsoid (or Bessel 1841) is an important reference ellipsoid of geodesy. It is currently used by several countries for their national geodetic surveys, but will be replaced in the next decades by modern ellipsoids of satellite geod ...

(1841). Nowadays, the method is replaced by worldwide

geodetic network A geodetic control network (also geodetic network, reference network, control point network, or control network) is a network, often of triangles, which are measured precisely by techniques of terrestrial surveying or by satellite geodesy. A ...

s and by

satellite geodesy Satellite geodesy is geodesy by means of artificial satellites—the measurement of the form and dimensions of Earth, the location of objects on its surface and the figure of the Earth's gravity field by means of artificial satellite techniques ...

'The Adventures of Three Englishmen and Three Russians in South Africa' by Jules Férat 15

List of other instances

Al-Ma'mun's arc measurement The Arab, Arabic, or Arabian mile ( ar, الميل, ''al-mīl'') 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 ...

Posidonius' arc measurement Posidonius (; grc-gre, Ποσειδώνιος , "of Poseidon") "of Apameia" (ὁ Ἀπαμεύς) or "of Rhodes" (ὁ Ῥόδιος) (), was a Greek politician, astronomer, astrologer, geographer, historian, mathematician, and teacher nati ...

Swedish–Russian Arc-of-Meridian Expedition The Swedish–Russian Arc-of-Meridian expedition was a scientific expedition to Svalbard that took place from 1899 to 1902. The main purpose of the mission was to measure a meridian arc, in order to determine the earth flattening at the poles. Th ...

Picard's arc measurement Jean Picard (21 July 1620 – 12 July 1682) was a French astronomer and priest born in La Flèche, where he studied at the Jesuit Collège Royal Henry-Le-Grand. He is principally notable for his accurate measure of the size of the Earth, ba ...

* Dunkirk-Collioure arc measurement (Cassini, Cassini, and de La Hire) *

Dunkirk-Collioure arc measurement (Cassini de Thury and de Lacaille) The Paris meridian is a meridian line running through the Paris Observatory in Paris, France – now longitude 2°20′14.02500″ East. It was a long-standing rival to the Greenwich meridian as the prime meridian of the world. The "Paris meridi ...

* Meridian arc of Delambre and Méchain * West Europe-Africa Meridian-arc *

De Lacaille's arc measurement Abbé Nicolas-Louis de Lacaille (; 15 March 171321 March 1762), formerly sometimes spelled de la Caille, was a French astronomer and geodesist who named 14 out of the 88 constellations. From 1750 to 1754, he studied the sky at the Cape of Goo ...

* Fernel's arc measurement * Norwood's arc measurement * Boscovich and Maire's arc measurement * Maclear's arc measurement * Hopfner's arc measurement

Determination

Assume the

astronomic latitude In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

s of two endpoints,

\phi_s

(standpoint) and

\phi_f

(forepoint), are precisely determined by

astrogeodesy Geodetic astronomy or astronomical geodesy (astro-geodesy) is the application of astronomical methods into geodetic networks and other technical projects of geodesy. Applications The most important applications are: * Establishment of geodetic d ...

, observing the

zenith distance The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" ...

s of sufficient numbers of

star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

s (

meridian altitude Meridian altitude is a method of celestial navigation to calculate an observer's latitude. It notes the altitude angle of an astronomical object above the horizon at culmination. Principle Meridian altitude is the simplest calculation of celesti ...

method). The empirical

Earth's meridional radius of curvature Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, deno ...

at the midpoint of the meridian arc can then be determined as: :

R = \frac

where

\mathit'

is the

mean sea level There are several kinds of mean in mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...

(MSL). Historically, the distance between two places has been determined at low precision by pacing or

odometry Odometry is the use of data from motion sensors to estimate change in position over time. It is used in robotics by some legged or wheeled robots to estimate their position relative to a starting location. This method is sensitive to errors due t ...

. High precision land surveys can be used to determine the distance between two places at nearly the same longitude by measuring a baseline and a

triangulation network 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 by using trigonometry, rather than measuring distances to the point directly as ...

linking fixed points. The meridian distance

\mathit

from one end point to a fictitious point at the same latitude as the second end point is then calculated by trigonometry. The surface distance

\mathit

is reduced to the corresponding distance at MSL,

\mathit'

(see: Geographical distance#Altitude correction). Two arc measurements at different latitudinal bands serve to determine Earth's flattening.

References

{{DEFAULTSORT:Grade Measurement Geodetic surveys

History

List of other instances

Determination

See also

References