geodesy Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...

and

navigation Navigation is a field of study that focuses on the process of monitoring and controlling the motion, movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navig ...

, a meridian arc is the

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

between two points near the Earth's surface having the same

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 ...

. The term may refer either to a segment of the meridian, or to its

length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...

. Both the practical determination of meridian arcs (employing measuring instruments in field campaigns) as well as its theoretical calculation (based on geometry and abstract mathematics) have been pursued for many years.

Measurement

The purpose of measuring meridian arcs is to determine a

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 ...

. One or more measurements of meridian arcs can be used to infer the shape of the reference ellipsoid that best approximates the geoid in the region of the measurements. Measurements of meridian arcs at several latitudes along many meridians around the world can be combined in order to approximate a ''geocentric ellipsoid'' intended to fit the entire world. The earliest determinations of the size of a

spherical Earth Spherical Earth or Earth's curvature refers to the approximation of the figure of the Earth as a sphere. The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of Ancient Greek philos ...

required a single arc. Accurate survey work beginning in the 19th century required several arc measurements in the region the survey was to be conducted, leading to a proliferation of reference ellipsoids around the world. The latest determinations use astro-geodetic measurements and the methods of satellite geodesy to determine reference ellipsoids, especially the geocentric ellipsoids now used for global coordinate systems such as WGS 84 (see numerical expressions).

History of measurement

Early estimations of Earth's size are recorded from Greece in the 4th century BC, and from scholars at the

caliph A caliphate ( ) is an institution or public office under the leadership of an Islamic steward with Khalifa, the title of caliph (; , ), a person considered a political–religious successor to the Islamic prophet Muhammad and a leader of ...

House of Wisdom The House of Wisdom ( ), also known as the Grand Library of Baghdad, was believed to be a major Abbasid Caliphate, Abbasid-era public academy and intellectual center in Baghdad. In popular reference, it acted as one of the world's largest publ ...

Baghdad Baghdad ( or ; , ) is the capital and List of largest cities of Iraq, largest city of Iraq, located along the Tigris in the central part of the country. With a population exceeding 7 million, it ranks among the List of largest cities in the A ...

in the 9th century. The first realistic value was calculated by

Alexandria Alexandria ( ; ) is the List of cities and towns in Egypt#Largest cities, second largest city in Egypt and the List of coastal settlements of the Mediterranean Sea, largest city on the Mediterranean coast. It lies at the western edge of the Nile ...

n scientist

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 ...

about 240 BC. He estimated that the meridian has a length of 252,000 stadia, with an error on the real value between −2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres). Eratosthenes described his technique in a book entitled ''On the measure of the Earth'', which has not been preserved. A similar method was used by

Posidonius Posidonius (; , "of Poseidon") "of Apameia" (ὁ Ἀπαμεύς) or "of Rhodes" (ὁ Ῥόδιος) (), was a Greeks, Greek politician, astronomer, astrologer, geographer, historian, mathematician, and teacher native to Apamea (Syria), Apame ...

about 150 years later, and slightly better results were calculated in 827 by the

arc measurement Arc measurement, sometimes called degree measurement (), is the astrogeodetic technique of determining the radius of Earth and, by Circumference#Circle, extension, Earth's circumference, its circumference. More specifically, it seeks to determine ...

method, attributed to the Caliph

Al-Ma'mun Abū al-ʿAbbās Abd Allāh ibn Hārūn al-Maʾmūn (; 14 September 786 – 9 August 833), better known by his regnal name al-Ma'mun (), was the seventh Abbasid caliph, who reigned from 813 until his death in 833. His leadership was marked by t ...

Ellipsoidal Earth

Early literature uses the term ''oblate spheroid'' to describe a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

"squashed at the poles". Modern literature uses the term ''ellipsoid of revolution'' in place of

spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...

, although the qualifying words "of revolution" are usually dropped. An

ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a Surface (mathemat ...

that is not an ellipsoid of revolution is called a triaxial ellipsoid. ''Spheroid'' and ''ellipsoid'' are used interchangeably in this article, with oblate implied if not stated.

=17th and 18th centuries

= Although it had been known since

classical antiquity Classical antiquity, also known as the classical era, classical period, classical age, or simply antiquity, is the period of cultural History of Europe, European history between the 8th century BC and the 5th century AD comprising the inter ...

that the Earth was spherical, by the 17th century, evidence was accumulating that it was not a perfect sphere. In 1672,

Jean Richer Jean Richer (1630–1696) was a French astronomer and assistant (''élève astronome'') at the French Academy of Sciences, under the direction of Giovanni Domenico Cassini. Between 1671 and 1673 he performed experiments and carried out celestial ...

found the first evidence that

gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...

was not constant over the Earth (as it would be if the Earth were a sphere); he took a

pendulum clock A pendulum clock is a clock that uses a pendulum, a swinging weight, as its timekeeping element. The advantage of a pendulum for timekeeping is that it is an approximate harmonic oscillator: It swings back and forth in a precise time interval dep ...

Cayenne Cayenne (; ; ) is the Prefectures in France, prefecture and capital city of French Guiana, an overseas region and Overseas department, department of France located in South America. The city stands on a former island at the mouth of the Caye ...

French Guiana French Guiana, or Guyane in French, is an Overseas departments and regions of France, overseas department and region of France located on the northern coast of South America in the Guianas and the West Indies. Bordered by Suriname to the west ...

and found that it lost minutes per day compared to its rate at

Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, largest city of France. With an estimated population of 2,048,472 residents in January 2025 in an area of more than , Paris is the List of ci ...

. This indicated the

acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...

of gravity was less at Cayenne than at Paris. Pendulum gravimeters began to be taken on voyages to remote parts of the world, and it was slowly discovered that gravity increases smoothly with increasing

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 ...

gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag (physics), drag). This is the steady gain in speed caused exclusively by gravitational attraction. All bodi ...

being about 0.5% greater at the

geographical pole A geographical pole or geographic pole is either of the two points on Earth where its axis of rotation intersects its surface. The North Pole lies in the Arctic Ocean while the South Pole is in Antarctica. North and South poles are also defined ...

s than at the

Equator The equator is the circle of latitude that divides Earth into the Northern Hemisphere, Northern and Southern Hemisphere, Southern Hemispheres of Earth, hemispheres. It is an imaginary line located at 0 degrees latitude, about in circumferen ...

. In 1687,

Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...

had published in the '' Principia'' as a proof that the Earth was an oblate

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 ...

equal to .Isaac Newton
''Principia'', Book III, Proposition XIX, Problem III
translated into English by Andrew Motte. A searchable modern translation is available a
17centurymaths
Search the followin
pdf file
for 'spheroid'. This was disputed by some, but not all, French scientists. A meridian arc of Jean Picard was extended to a longer arc by

Giovanni Domenico Cassini Giovanni Domenico Cassini (8 June 1625 – 14 September 1712) was an Italian-French mathematician, astronomer, astrologer and engineer. Cassini was born in Perinaldo, near Imperia, at that time in the County of Nice, part of the Savoyard sta ...

and his son Jacques Cassini over the period 1684–1718.. Freely available online a
Archive.org
an
Forgotten Books
(). In addition the book has been reprinted b
Nabu Press
(), the first chapter covers the history of early surveys. The arc was measured with at least three latitude determinations, so they were able to deduce mean curvatures for the northern and southern halves of the arc, allowing a determination of the overall shape. The results indicated that the Earth was a ''prolate'' spheroid (with an equatorial radius less than the polar radius). To resolve the issue, 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 ...

(1735) undertook expeditions to Peru ( Bouguer, Louis Godin, de La Condamine,

Antonio de Ulloa Antonio de Ulloa y de la Torre-Guiral (12 January 1716 – 3 July 1795) was a Spanish Navy officer. He spent much of his career in the Spanish America, Americas, where he carried out important scientific work. As a scientist, Ulloa is re ...

, Jorge Juan) and to Lapland ( Maupertuis, Clairaut, Camus, Le Monnier, Abbe Outhier,

Anders Celsius Anders Celsius (; 27 November 170125 April 1744) was a Swedes, Swedish astronomer, physicist and mathematician. He was professor of astronomy at Uppsala University from 1730 to 1744, but traveled from 1732 to 1735 visiting notable observatories ...

). The resulting measurements at equatorial and polar latitudes confirmed that the Earth was best modelled by an oblate spheroid, supporting Newton. However, by 1743, Clairaut's theorem had completely supplanted Newton's approach. By the end of the century,

Jean Baptiste Joseph Delambre Jean Baptiste Joseph, chevalier Delambre (19 September 1749 – 19 August 1822) was a French mathematician, astronomer, historian of astronomy, and geodesist. He was also director of the Paris Observatory, and author of well-known books on the ...

had remeasured and extended the French arc from

Dunkirk Dunkirk ( ; ; ; Picard language, Picard: ''Dunkèke''; ; or ) is a major port city in the Departments of France, department of Nord (French department), Nord in northern France. It lies from the Belgium, Belgian border. It has the third-larg ...

to the

Mediterranean Sea The Mediterranean Sea ( ) is a sea connected to the Atlantic Ocean, surrounded by the Mediterranean basin and almost completely enclosed by land: on the east by the Levant in West Asia, on the north by Anatolia in West Asia and Southern Eur ...

(the meridian arc of Delambre and Méchain). It was divided into five parts by four intermediate determinations of latitude. By combining the measurements together with those for the arc of Peru, ellipsoid shape parameters were determined and the distance between the Equator and pole along the

Paris Meridian 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 meri ...

was calculated as toises as specified by the standard toise bar in Paris. Defining this distance as exactly led to the construction of a new standard

metre The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...

bar as toises.

19th century

From the French revolution of 1789 came an effort to reform measurement standards, leading ultimately to an extravagant effort to measure the meridian passing through Paris in order to define the

. The question of measurement reform was placed in the hands of the

, who appointed a commission chaired by

Jean-Charles de Borda Jean-Charles, chevalier de Borda (4 May 1733 – 19 February 1799) was a French mathematician, physicist, and Navy officer. Biography Borda was born in the city of Dax to Jean‐Antoine de Borda and Jeanne‐Marie Thérèse de Lacroix. In 17 ...

. Instead of the seconds pendulum method, the commission of the French Academy of Sciences – whose members included Borda,

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLaplace Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...

Monge Gaspard Monge, Comte de Pelusium, Péluse (; 9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geom ...

and

Condorcet Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (; ; 17 September 1743 – 29 March 1794), known as Nicolas de Condorcet, was a French philosopher, political economist, politician, and mathematician. His ideas, including suppo ...

– decided that the new measure should be equal to one ten-millionth of the distance from the North Pole to the Equator (the quadrant of the Earth's circumference), measured along the meridian passing through Paris at the

of Paris pantheon, which became the central geodetic station in Paris.

otained the fundamental

co-ordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

of the Pantheon by triangulating all the geodetic stations around Paris from the Pantheon's dome. Apart from the obvious consideration of safe access for French surveyors, the

Paris meridian 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 meri ...

was also a sound choice for scientific reasons: a portion of the quadrant from

Barcelona Barcelona ( ; ; ) is a city on the northeastern coast of Spain. It is the capital and largest city of the autonomous community of Catalonia, as well as the second-most populous municipality of Spain. With a population of 1.6 million within c ...

(about 1000 km, or one-tenth of the total) could be surveyed with start- and end-points at sea level, and that portion was roughly in the middle of the quadrant, where the effects of the Earth's oblateness were expected not to have to be accounted for. The expedition would take place after the Anglo-French Survey, thus the French meridian arc, which would extend northwards across the

United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Northwestern Europe, off the coast of European mainland, the continental mainland. It comprises England, Scotlan ...

, would also extend southwards to

, later to

Balearic Islands The Balearic Islands are an archipelago in the western Mediterranean Sea, near the eastern coast of the Iberian Peninsula. The archipelago forms a Provinces of Spain, province and Autonomous communities of Spain, autonomous community of Spain, ...

Jean-Baptiste Biot Jean-Baptiste Biot (; ; 21 April 1774 – 3 February 1862) was a French people, French physicist, astronomer, and mathematician who co-discovered the Biot–Savart law of magnetostatics with Félix Savart, established the reality of meteorites, ma ...

and

François Arago Dominique François Jean Arago (), known simply as François Arago (; Catalan: , ; 26 February 17862 October 1853), was a French mathematician, physicist, astronomer, freemason, supporter of the Carbonari revolutionaries and politician. Early l ...

would publish in 1821 their observations completing those of Delambre and Mechain. It was an account of the length's variations of portions of one degree of amplitude of the meridian arc along the

as well as the account of the variation of the

seconds pendulum A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz. Principles A pendulum is a weight suspended from a pivot so tha ...

's length along the same meridian between

Shetland Shetland (until 1975 spelled Zetland), also called the Shetland Islands, is an archipelago in Scotland lying between Orkney, the Faroe Islands, and Norway, marking the northernmost region of the United Kingdom. The islands lie about to the ...

and the Balearc Islands. The task of surveying the meridian arc fell to

Pierre Méchain Pierre François André Méchain (; 16 August 1744 – 20 September 1804) was a French astronomer and surveyor who, with Charles Messier, was a major contributor to the early study of deep-sky objects and comets. Life Pierre Méchain was bo ...

and Jean-Baptiste Delambre, and took more than six years (1792–1798). The technical difficulties were not the only problems the surveyors had to face in the convulsed period of the aftermath of the Revolution: Méchain and Delambre, and later

, were imprisoned several times during their surveys, and Méchain died in 1804 of yellow fever, which he contracted while trying to improve his original results in northern Spain. Rodez-coquelicots480

The project was split into two parts – the northern section of 742.7 km from the belfry of the Church of Saint-Éloi, Dunkirk to Rodez Cathedral which was surveyed by Delambre and the southern section of 333.0 km from

Rodez Rodez (, , ; , ) is a small city and commune in the South of France, about 150 km northeast of Toulouse. It is the prefecture of the department of Aveyron, region of Occitania (formerly Midi-Pyrénées). Rodez is the seat of the communau ...

to the Montjuïc Fortress, Barcelona which was surveyed by Méchain. Although Méchain's sector was half the length of Delambre, it included the

Pyrenees The Pyrenees are a mountain range straddling the border of France and Spain. They extend nearly from their union with the Cantabrian Mountains to Cap de Creus on the Mediterranean coast, reaching a maximum elevation of at the peak of Aneto. ...

and hitherto unsurveyed parts of Spain. Delambre measured a baseline of about 10 km (6,075.90 ) in length along a straight road between

Melun Melun () is a commune in the Seine-et-Marne department in the Île-de-France region, north-central France. It is located on the southeastern outskirts of Paris, about from the centre of the capital. Melun is the prefecture of Seine-et-Marne, ...

and Lieusaint. In an operation taking six weeks, the baseline was accurately measured using four platinum rods, each of length two (a being about 1.949 m). Thereafter he used, where possible, the triangulation points used by

Nicolas Louis de Lacaille 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 ...

in his 1739–1740 survey of French meridian arc from

to Collioure. Méchain's baseline was of a similar length (6,006.25 ), and also on a straight section of road between Vernet (in the

Perpignan Perpignan (, , ; ; ) is the prefectures in France, prefecture of the Pyrénées-Orientales departments of France, department in Southern France, in the heart of the plain of Roussillon, at the foot of the Pyrenees a few kilometres from the Me ...

area) and Salces (now

Salses-le-Château Salses-le-Château (; ; ) or just Salses is a Communes of France, commune in the Pyrénées-Orientales Departments of France, department in southern France. It is located north of the city of Perpignan. Geography Salses-le-Château is located i ...

). To put into practice the decision taken by the

National Convention The National Convention () was the constituent assembly of the Kingdom of France for one day and the French First Republic for its first three years during the French Revolution, following the two-year National Constituent Assembly and the ...

, on 1 August 1793, to disseminate the new units of the decimal

metric system The metric system is a system of measurement that standardization, standardizes a set of base units and a nomenclature for describing relatively large and small quantities via decimal-based multiplicative unit prefixes. Though the rules gover ...

, it was decided to establish the length of the metre based on a fraction of the meridian in the process of being measured. The decision was taken to fix the length of a provisional metre (French: ''mètre provisoire'') determined by the measurement of the Meridian of France from

to Collioure, which, in 1740, had been carried out by

and Cesar-François Cassini de Thury. The length of the metre was established, in relation to the toise of the Academy also called toise of Peru, at 3 feet 11.44 lines, taken at 13 degrees of the temperature scale of René-Antoine Ferchault de Réaumur in use at the time. This value was set by legislation on 7 April 1795. It was therefore metal bars of 443.44 that were distributed in France in 1795-1796. This was the metre installed under the arcades of the

rue de Vaugirard ''Ruta graveolens'', commonly known as rue, common rue or herb-of-grace, is a species of the genus '' Ruta'' grown as an ornamental plant and herb. It is native to the Mediterranean. It is grown throughout the world in gardens, especially fo ...

, almost opposite the entrance to the

Senate A senate is a deliberative assembly, often the upper house or chamber of a bicameral legislature. The name comes from the ancient Roman Senate (Latin: ''Senatus''), so-called as an assembly of the senior (Latin: ''senex'' meaning "the el ...

. End of November 1798, Delambre and Méchain returned to Paris with their data, having completed the survey to meet a foreign commission composed of representatives of

Batavian Republic The Batavian Republic (; ) was the Succession of states, successor state to the Dutch Republic, Republic of the Seven United Netherlands. It was proclaimed on 19 January 1795 after the Batavian Revolution and ended on 5 June 1806, with the acce ...

: Henricus Aeneae and

Jean Henri van Swinden Jean Henri van Swinden (8 June 1746 – 9 March 1823) was a Dutch mathematician and physicist who taught at the University of Franeker and in the Athenaeum Illustre of Amsterdam. Biography His parents were the lawyer Phillippe van Swinden ...

Cisalpine Republic The Cisalpine Republic (; ) was a sister republic or a client state of France in Northern Italy that existed from 1797 to 1799, with a second version until 1802. Creation After the Battle of Lodi in May 1796, Napoleon Bonaparte organized two ...

: Lorenzo Mascheroni,

Kingdom of Denmark The Danish Realm, officially the Kingdom of Denmark, or simply Denmark, is a sovereign state consisting of a collection of constituent territories united by the Constitution of Denmark, Constitutional Act, which applies to the entire territor ...

: Thomas Bugge,

Kingdom of Spain Spain, or the Kingdom of Spain, is a country in Southern Europe, Southern and Western Europe with territories in North Africa. Featuring the Punta de Tarifa, southernmost point of continental Europe, it is the largest country in Southern Eur ...

: Gabriel Císcar and Agustín de Pedrayes,

Helvetic Republic The Helvetic Republic (; ; ) was a sister republic of France that existed between 1798 and 1803, during the French Revolutionary Wars. It was created following the French invasion and the consequent dissolution of the Old Swiss Confederacy, ma ...

: Johann Georg Tralles,

Ligurian Republic The Ligurian Republic (, , ) or Republic of Liguria was a French client republic formed by Napoleon on 14 June 1797. It consisted of the old Republic of Genoa, which covered most of the Ligurian region of Northwest Italy, and the small Imper ...

: Ambrogio Multedo,

Kingdom of Sardinia The Kingdom of Sardinia, also referred to as the Kingdom of Sardinia and Corsica among other names, was a State (polity), country in Southern Europe from the late 13th until the mid-19th century, and from 1297 to 1768 for the Corsican part of ...

: Prospero Balbo, Antonio Vassali Eandi,

Roman Republic The Roman Republic ( ) was the era of Ancient Rome, classical Roman civilisation beginning with Overthrow of the Roman monarchy, the overthrow of the Roman Kingdom (traditionally dated to 509 BC) and ending in 27 BC with the establis ...

: Pietro Franchini, Tuscan Republic: Giovanni Fabbroni who had been invited by Talleyrand. The French commission comprised

, Barnabé Brisson,

Charles-Augustin de Coulomb Charles-Augustin de Coulomb ( ; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of att ...

Jean Darcet Jean d'Arcet or Jean Darcet (7 September 1724 – 12 February 1801) was a French chemist, and director of the porcelain works at Sèvres. He was one of the first to manufacture porcelain in France. Darcet was probably born in Doazit, where his ...

René Just Haüy René Just Haüy () FRS MWS FRSE (28 February 1743 – 1 June 1822) was a French priest and mineralogist, commonly styled the Abbé Haüy after he was made an honorary canon of Notre-Dame de Paris, Notre Dame. Due to his innovative work on cryst ...

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaPierre- Simon Laplace, Louis Lefèvre-Ginneau, Pierre Méchain and Gaspar de Prony. In 1799, a commission including Johann Georg Tralles,

Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French people, French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transforma ...

Pierre-Simon Laplace Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...

, Gabriel Císcar, Pierre Méchain and Jean-Baptiste Delambre calculated the distance from Dunkirk to Barcelona using the data of the

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 these two towns and determined the portion of the distance from the North Pole to the Equator it represented. Pierre Méchain's and Jean-Baptiste Delambre's measurements were combined with the results of the French Geodetic Mission to the Equator and a value of was found for the Earth's flattening.

originally hoped to figure out the

Earth 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 ...

problem from the sole measurement of the arc from Dunkirk to Barcelona, but this portion of the meridian arc led for the flattening to the value of considered as unacceptable. This value was the result of a conjecture based on too limited data. Another flattening of the Earth was calculated by Delambre, who also excluded the results of the French Geodetic Mission to Lapland and found a value close to combining the results of Delambre and Méchain arc measurement with those of the Spanish-French Geodetic Mission taking in account a correction of the astronomic arc. The distance from the North Pole to the Equator was then extrapolated from the measurement of the

arc between Dunkirk and Barcelona and was determined as toises. As the metre had to be equal to one ten-millionth of this distance, it was defined as 0.513074 toise or 3 feet and 11.296 lines of the Toise of Peru, which had been constructed in 1735 for the French Geodesic Mission to Peru. When the final result was known, a bar whose length was closest to the meridional definition of the metre was selected and placed in the National Archives on 22 June 1799 (4 messidor An VII in the Republican calendar) as a permanent record of the result.

19th century

In the 19th century, many astronomers and geodesists were engaged in detailed studies of the Earth's curvature along different meridian arcs. The analyses resulted in a great many model ellipsoids such as Plessis 1817, Airy 1830, Bessel 1841, Everest 1830, and Clarke 1866. A comprehensive list of ellipsoids is given under

The nautical mile

Historically a

nautical mile A nautical mile is a unit of length used in air, marine, and space navigation, and for the definition of territorial waters. Historically, it was defined as the meridian arc length corresponding to one minute ( of a degree) of latitude at t ...

was defined as the length of one minute of arc along a meridian of a spherical earth. An ellipsoid model leads to a variation of the nautical mile with latitude. This was resolved by defining the nautical mile to be exactly 1,852 metres. However, for all practical purposes, distances are measured from the latitude scale of charts. As the

Royal Yachting Association The Royal Yachting Association (RYA) is a United Kingdom national governing body for sailing, dinghy sailing, yacht and motor cruising, sail racing, RIBs and sportsboats, windsurfing and personal watercraft and a leading representative for i ...

says in its manual for day skippers: "1 (minute) of Latitude = 1 sea mile", followed by "For most practical purposes distance is measured from the latitude scale, assuming that one minute of latitude equals one nautical mile".

Calculation

On a sphere, the meridian arc length is simply the circular arc length. On an ellipsoid of revolution, for short meridian arcs, their length can be approximated using the Earth's meridional radius of curvature and the circular arc formulation. For longer arcs, the length follows from the subtraction of two ''meridian distances'', the distance from the equator to a point at a latitude . This is an important problem in the theory of map projections, particularly the

transverse Mercator projection The transverse Mercator map projection (TM, TMP) is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Merc ...

. The main ellipsoidal parameters are, , , , but in theoretical work it is useful to define extra parameters, particularly the

eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...

, , and the third

. Only two of these parameters are independent and there are many relations between them: :

\begin
 f&=\frac\,, \qquad  e^2=f(2-f)\,, \qquad n=\frac=\frac\,,\\
b&=a(1-f)=a\sqrt\,,\qquad  e^2=\frac\,.
\end

Definition

The meridian radius of curvature can be shown to be equal to: Section 5.6. This reference includes the derivation of curvature formulae from first principles and a proof of Meusnier's theorem. (Supplements
Maxima files
and
Latex code and figures
:

M(\varphi) = \frac,

The arc length of an infinitesimal element of the meridian is (with in radians). Therefore, the meridian distance from the equator to latitude is :

\begin
m(\varphi) &=\int_0^\varphi M(\varphi) \, d\varphi \\
&= a(1 - e^2)\int_0^\varphi \left(1 - e^2 \sin^2 \varphi \right)^ \, d\varphi\,.
\end

The distance formula is simpler when written in terms of the

parametric 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 ...

, :

m(\varphi) = b\int_0^\beta\sqrt\,d\beta\,,

where and . Even though latitude is normally confined to the range , all the formulae given here apply to measuring distance around the complete meridian ellipse (including the anti-meridian). Thus the ranges of , , and the rectifying latitude , are unrestricted.

Relation to elliptic integrals

The above integral is related to a special case of an incomplete elliptic integral of the third kind. In the notation of the online

NIST The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical s ...

handbook
Section 19.2(ii)
, :

m(\varphi)=a\left(1-e^2\right)\,\Pi(\varphi,e^2,e)\,.

It may also be written in terms of incomplete elliptic integrals of the second kind (See the NIST handboo
Section 19.6(iv)
, :

\begin
m(\varphi) &= a\left(E(\varphi,e)-\frac\right) \\
&= a\left(E(\varphi,e)+\fracE(\varphi,e)\right) \\
&= b E(\beta, ie')\,.
\end

The calculation (to arbitrary precision) of the elliptic integrals and approximations are also discussed in the NIST handbook. These functions are also implemented in computer algebra programs such as Mathematica and Maxima.

Series expansions

The above integral may be expressed as an infinite truncated series by expanding the integrand in a Taylor series, performing the resulting integrals term by term, and expressing the result as a trigonometric series. In 1755,

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

derived an expansion in the third eccentricity squared.

Expansions in the eccentricity ()

Delambre in 1799Delambre, J. B. J. (1799)
''Méthodes Analytiques pour la Détermination d'un Arc du Méridien''; précédées d'un mémoire sur le même sujet par A. M. Legendre
De L'Imprimerie de Crapelet, Paris, 72–73 derived a widely used expansion on , :

m(\varphi)=\fraca\left(D_0\varphi+D_2\sin 2\varphi+D_4\sin4\varphi+D_6\sin6\varphi+D_8\sin8\varphi+\cdots\right)\,,

where :

D_8 &= \tfrac e^8 + \cdots. \end

Richard Rapp gives a detailed derivation of this result.

Expansions in the third flattening ()

Series with considerably faster convergence can be obtained by expanding in terms of the third flattening instead of the eccentricity. They are related by :

e^2 = \frac\,.

In 1837,

Friedrich Bessel Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesy, geodesist. He was the first astronomer who determined reliable values for the distance from the Sun to another star by th ...

obtained one such series, which was put into a simpler form by Helmert, :

m(\varphi)=\frac2\left(H_0\varphi+H_2\sin 2\varphi+H_4\sin4\varphi+H_6\sin6\varphi+H_8\sin8\varphi+\cdots\right)\,,

with :

\begin
H_0 &= 1 + \tfrac n^2 + \tfrac n^4 + \cdots, \\
H_2 &= - \tfrac n + \tfrac n^3 + \cdots,&
H_6 &= - \tfrac n^3 + \cdots, \\
H_4 &= \tfrac n^2 - \tfrac n^4 - \cdots,\qquad&
H_8 &= \tfrac n^4 - \cdots.
\end

Because changes sign when and are interchanged, and because the initial factor is constant under this interchange, half the terms in the expansions of vanish. The series can be expressed with either or as the initial factor by writing, for example, :

\tfrac12(a+b) = \frac = a(1-n+n^2-n^3+n^4-\cdots)\,,

and expanding the result as a series in . Even though this results in more slowly converging series, such series are used in the specification for the

by the

National Geospatial-Intelligence Agency The National Geospatial-Intelligence Agency (NGA) is a combat support agency within the United States Department of Defense whose primary mission is collecting, analyzing, and distributing geospatial intelligence (GEOINT) to support national se ...

and the Ordnance Survey of Great Britain.A guide to coordinate systems in Great Britain
Ordnance Survey of Great Britain.

Series in terms of the parametric latitude

In 1825, Bessel English translation of Astron. Nachr. 4, 241–254 (1825), §5. derived an expansion of the meridian distance in terms of the

in connection with his work on

geodesics In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...

, :

m(\varphi) = \frac2\left(B_0\beta + B_2\sin2\beta + B_4\sin4\beta + B_6\sin6\beta + B_8\sin8\beta + \cdots\right),

with :

\begin
B_0 &= 1 + \tfrac n^2 + \tfrac n^4 + \cdots = H_0\,,\\
B_2 &= - \tfrac n + \tfrac n^3 + \cdots, &
B_6 &= - \tfrac n^3 + \cdots, \\
B_4 &= - \tfrac n^2 + \tfrac n^4 + \cdots, \qquad&
B_8 &= - \tfrac n^4 + \cdots.
\end

Because this series provides an expansion for the elliptic integral of the second kind, it can be used to write the arc length in terms of the

geodetic latitude Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a ''reference ellipsoid''. They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as geo ...

as :

&\quad -\frac\Biggr). \end

Generalized series

The above series, to eighth order in eccentricity or fourth order in third flattening, provide millimetre accuracy. With the aid of symbolic algebra systems, they can easily be extended to sixth order in the third flattening which provides full double precision accuracy for terrestrial applications. Delambre and Bessel both wrote their series in a form that allows them to be generalized to arbitrary order. The coefficients in Bessel's series can expressed particularly simply :

\dfrac\,, & \text k > 0\,, \end

where :

c_k = \sum_^\infty \frac n^

and is the

double factorial In mathematics, the double factorial of a number , denoted by , is the product of all the positive integers up to that have the same Parity (mathematics), parity (odd or even) as . That is, n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. Restated ...

, extended to negative values via the recursion relation: and . The coefficients in Helmert's series can similarly be expressed generally by :

H_ = (-1)^k (1-2k)(1+2k) B_\,.

This result was conjectured by Friedrich Helmert and proved by Kazushige Kawase. The extra factor originates from the additional expansion of

\frac

appearing in the above formula and results in poorer convergence of the series in terms of compared to the one in .

Numerical expressions

The trigonometric series given above can be conveniently evaluated using Clenshaw summation. This method avoids the calculation of most of the trigonometric functions and allows the series to be summed rapidly and accurately. The technique can also be used to evaluate the difference while maintaining high relative accuracy. Substituting the values for the semi-major axis and eccentricity of the

WGS84 The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descri ...

ellipsoid gives :

\begin
m(\varphi)&=\left(111\,132.952\,55\,\varphi^-16\,038.509\,\sin 2\varphi+16.833\,\sin4\varphi-0.022\,\sin6\varphi+0.000\,03\,\sin8\varphi\right)\mbox \\
&= \left(111\,132.952\,55\,\beta^-5\,346.170\,\sin 2\beta-1.122\,\sin4\beta-0.001\,\sin6\beta-0.5\times10^\,\sin8\beta\right)\mbox
\end

where is expressed in degrees (and similarly for ). On the ellipsoid the exact distance between parallels at and is . For WGS84 an approximate expression for the distance between the two parallels at ±0.5° from the circle at latitude is given by :

\Delta m=(111\,133 - 560\cos 2\varphi)\mbox

Quarter meridian

The distance from the equator to the pole, the quarter meridian (analogous to the quarter-circle), also known as the Earth quadrant, is :

m_\mathrm = m\left(\frac \pi 2\right)\,.

It was part of the historical definition of the metre and of the

, and used in the definition of the hebdomometre. The quarter meridian can be expressed in terms of the complete elliptic integral of the second kind, :

m_\mathrm=aE(e)=bE(ie').

where

e, e'

are the first and second eccentricities. The quarter meridian is also given by the following generalized series: :

m_\mathrm = \frac4 c_0 = \frac4 \sum_^\infty\left(\frac\right)^2 n^\,,

(For the formula of ''c''₀, see section #Generalized series above.) This result was first obtained by

James Ivory James Francis Ivory (born Richard Jerome Hazen June 7, 1928) is an American film director, producer, and screenwriter. He was a principal in Merchant Ivory Productions along with Indian film producer Ismail Merchant (his domestic and professio ...

. The numerical expression for the quarter meridian on the WGS84 ellipsoid is :

\begin
m_\mathrm &= 0.9983242984312529\ \frac\ a\\
&= 10\,001\,965.729\mbox
\end

Full meridian (polar perimeter)

The polar

Earth's circumference Earth's circumference is the distance around Earth. Measured around the equator, it is . Measured passing through the poles, the circumference is . Treating the Earth as a sphere, its circumference would be its single most important measuremen ...

is simply four times quarter meridian: :

C_p=4m_p

The

perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...

of a meridian ellipse can also be rewritten in the form of a rectifying circle perimeter, . Therefore, the rectifying Earth radius is: :

M_r=0.5(a+b)/c_0

It can be evaluated as .

The inverse meridian problem for the ellipsoid

In some problems, we need to be able to solve the inverse problem: given the arc length , find the latitude . This may be solved by

Newton's method In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a ...

, iterating :

\varphi_ = \varphi_i - \frac\,,

until convergence. A suitable starting guess is given by where :

\mu = \frac2 \frac m

is the rectifying latitude. Note that it there is no need to differentiate the series for , since the formula for the meridian radius of curvature can be used instead. Alternatively, Helmert's series for the meridian distance can be reverted to give Adams, Oscar S (1921)
''Latitude Developments Connected With Geodesy and Cartography''
US Coast and Geodetic Survey Special Publication No. 67. p. 127. :

\varphi = \mu + H'_2\sin2\mu + H'_4\sin4\mu + H'_6\sin6\mu + H'_8\sin8\mu + \cdots

where :

\begin
H'_2 &= \tfrac n - \tfrac n^3 + \cdots,&
H'_6 &= \tfrac n^3 + \cdots, \\
H'_4 &= \tfrac n^2 - \tfrac n^4 + \cdots,\qquad&
H'_8 &= \tfrac n^4 + \cdots.
\end

Similarly, Bessel's series for in terms of can be reverted to give :

\beta = \mu + B'_2\sin2\mu + B'_4\sin4\mu + B'_6\sin6\mu + B'_8\sin8\mu + \cdots,

where :

\begin
B'_2 &= \tfrac n - \tfrac n^3 + \cdots,&
B'_6 &= \tfrac n^3 - \cdots, \\
B'_4 &= \tfrac n^2 - \tfrac n^4 + \cdots,\qquad&
B'_8 &= \tfrac n^4 - \cdots.
\end

showed that the distance along a geodesic on a spheroid is the same as the distance along the perimeter of an ellipse. For this reason, the expression for in terms of and its inverse given above play a key role in the solution of the geodesic problem with replaced by , the distance along the geodesic, and replaced by , the arc length on the auxiliary sphere. The requisite series extended to sixth order are given by Charles Karney,
Addenda
Eqs. (17) & (21), with playing the role of and playing the role of .

References

External links

Online computation of meridian arcs on different geodetic reference ellipsoids
{{DEFAULTSORT:Meridian Arc Geodesy Meridians (geography) History of measurement

Measurement

History of measurement

Ellipsoidal Earth

=17th and 18th centuries

19th century

19th century

The nautical mile

Calculation

Definition

Relation to elliptic integrals

Series expansions

Expansions in the eccentricity ()

Expansions in the third flattening ()

Series in terms of the parametric latitude

Generalized series

Numerical expressions

Quarter meridian

Full meridian (polar perimeter)

The inverse meridian problem for the ellipsoid

See also

References

External links