Arc measurement, sometimes called degree measurement (), is the astrogeodetic technique of determining the radius of Earth and, by extension, its circumference. More specifically, it seeks to determine the local Earth radius of curvature of the

figure of the Earth In geodesy, the figure of the Earth is the size and shape used to model planet Earth. The kind of figure depends on application, including the precision needed for the model. A spherical Earth is a well-known historical approximation that is ...

, by relating the

latitude In geography, latitude is a geographic coordinate system, geographic 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 t ...

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 lett ...

difference) and the

geographic distance Geographical distance or geodetic distance is the distance measured along the surface of the Earth, or the shortest arch length. The formulae in this article calculate distances between points which are defined by geographical coordinates in ...

(

arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...

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

astronomical latitude In geography, latitude is a geographic 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 ...

s and the

meridian arc In geodesy and navigation, a meridian arc is the curve (geometry), curve between two points near the Earth's surface having the same longitude. The term may refer either to a arc (geometry), segment of the meridian (geography), meridian, or to its ...

length and is called ''meridian arc measurement''; other variants may involve only astronomical longitude ('' parallel arc measurement'') or both

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

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

International Association of Geodesy The International Association of Geodesy (IAG) is a constituent association of the International Union of Geodesy and Geophysics focusing on the science which measures and describes the Figure of the Earth, Earth's shape, its rotation and gravity ...

(IAG). Nowadays, the method is replaced by worldwide geodetic networks 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 ...

History

The first known arc measurement was performed by

Eratosthenes Eratosthenes of Cyrene (; ; – ) was an Ancient Greek polymath: a Greek mathematics, mathematician, geographer, poet, astronomer, and music theory, music theorist. He was a man of learning, becoming the chief librarian at the Library of A ...

(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 Yixing (, 683–727) was a Buddhist monk of the Tang dynasty, recognized for his accomplishments as an astronomer, a reformer of the calendar system, a specialist in the ''I Ching, Yijing'' (易經), and a distinguished Buddhist figure with exp ...

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 ...

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

Alkmaar Alkmaar () is a List of cities in the Netherlands by province, city and List of municipalities of the Netherlands, municipality in the Netherlands, located in the Provinces of the Netherlands, province of North Holland. Alkmaar is well known fo ...

and

Bergen op Zoom Bergen op Zoom (; called ''Berrege'' in the Brabantian dialect, local dialect) is a List of cities in the Netherlands by province, city and Municipalities of the Netherlands, municipality in southwestern Netherlands. It is located in the Province ...

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, commonly known as Snell. His name is usually associated with the law of refraction of light known as Snell's law. The ...

''). 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 f ...

of the Earth ellipsoid by measuring at different

geographic latitude In geography, latitude is a geographic 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 ...

s. The first of these was the ''

French Geodesic Mission The French Geodesic Mission to the Equator (), also called the French Geodesic Mission to Peru and the Spanish-French Geodesic Mission, was an 18th-century expedition to what is now Ecuador carried out for the purpose of performing an arc measu ...

'', commissioned by the

French Academy of Sciences The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...

in 1735–1738, involving measurement expeditions to Lapland ( Maupertuis et al.) and Peru (

Pierre Bouguer Pierre Bouguer () (16 February 1698, Le 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, ...

et al.). Struve measured a

geodetic control network A geodetic control network is a network, often of triangles, that are measured precisely by techniques of ''control surveying'', such as terrestrial surveying or satellite geodesy. It is also known as a geodetic network, reference network, contro ...

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 m ...

between the

Arctic Sea The Arctic Ocean is the smallest and shallowest of the world's five oceanic divisions. It spans an area of approximately and is the coldest of the world's oceans. The International Hydrographic Organization (IHO) recognizes it as an ocean, a ...

and the

Black Sea The Black Sea is a marginal sea, marginal Mediterranean sea (oceanography), mediterranean sea 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 bound ...

, the ''

Struve Geodetic Arc The Struve Geodetic Arc is a chain of survey triangulations stretching from Hammerfest in Norway to the Black Sea, through ten countries and over , which yielded the first accurate measurement of a meridian arc. The chain was established ...

''.

Chronological list

This is a partial chronological list of arc measurements: * 230 B.C.:

Eratosthenes' arc measurement Eratosthenes of Cyrene (; ; – ) was an Ancient 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. His work is comparable ...

Principal Triangulation of Great Britain The Principal Triangulation of Britain was the first high-precision triangulation survey of the whole of Great Britain and Ireland, carried out between 1791 and 1853 under the auspices of the Board of Ordnance. The aim of the survey was to estab ...

* 1792-1798: meridian arc of Delambre and Méchain * 1802–1841: the Great Trigonometric Survey of India * 1806-1809: Arago and Biot's arc measurement * 1816-1855:

* 1821-1825: Gauss' geodetic survey * 1841-1848: Maclear's arc measurement * 1879: West Europe-Africa Meridian-arc * 1899-1902: Swedish–Russian Arc-of-Meridian Expedition * 1921: Hopfner's arc measurement

Determination

Assume the astronomic latitudes of two endpoints,

\phi_s

(standpoint) and

\phi_f

(forepoint) are known; these can be determined by astrogeodesy, observing the zenith distances of sufficient numbers of

star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...

s (

meridian altitude {{inline, date=June 2024 Meridian altitude is a method of celestial navigation to determine the latitude of an observer. It notes the altitude angle of an astronomical object above the horizon at culmination. Principle Meridian altitude is the si ...

method). Then, the empirical Earth's meridional radius of curvature at the midpoint of the meridian arc can then be determined inverting the

great-circle distance The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path between the two points on the surface of the ...

(or circular arc length) formula: :

R = \frac

where the latitudes are in radians and

\mathit'

is the

mean sea level A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...

(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 ...

. 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 linking

fixed points Fixed may refer to: * ''Fixed'' (EP), EP by Nine Inch Nails * ''Fixed'' (film), an upcoming animated film directed by Genndy Tartakovsky * Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System * Fi ...

. 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).

Extensions

Additional arc measurements, at different latitudinal bands (each delimited by a new pair of standpoint and forepoint), serve to determine Earth's flattening. Bessel compiled several

s, to compute the famous Bessel ellipsoid (1841). Clarke (1858) combined most of the arc measurements then available to define a new

reference ellipsoid An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximation ...

References

{{DEFAULTSORT:Grade Measurement Geodetic surveys History of measurement

History

Chronological list

Determination

Extensions

See also

References