Meridian arc
In geodesy and navigation, a meridian arc is the curve between two points near the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. Both the practical determination of meridian arcs and their theoretical calculation have been pursued for many years.
Measurement
The purpose of measuring meridian arcs is to determine a figure of the Earth. 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 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.
History of measurement
Early estimations of Earth's size are recorded from Greece in the 4th century BC, and from scholars at the caliph's House of Wisdom in Baghdad in the 9th century. The first realistic value was calculated by Alexandrian scientist Eratosthenes 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%. 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 about 150 years later, and slightly better results were calculated in 827 by the arc measurement method, attributed to the Caliph Al-Ma'mun.Ellipsoidal Earth
Early literature uses the term oblate spheroid to describe a sphere "squashed at the poles". Modern literature uses the term ellipsoid of revolution in place of spheroid, although the qualifying words "of revolution" are usually dropped. An ellipsoid 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 that the Earth was spherical, by the 17th century, evidence was accumulating that it was not a perfect sphere. In 1672, Jean Richer found the first evidence that gravity was not constant over the Earth ; he took a pendulum clock to Cayenne, French Guiana and found that it lost minutes per day compared to its rate at Paris. This indicated the acceleration 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, gravitational acceleration being about 0.5% greater at the geographical poles than at the Equator.In 1687, Isaac Newton had published in the Principia as a proof that the Earth was an oblate spheroid of flattening equal to. 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 and his son Jacques Cassini over the period 1684–1718. 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. To resolve the issue, the French Academy of Sciences undertook expeditions to Peru and to Lapland. 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 had remeasured and extended the French arc from Dunkirk to the Mediterranean Sea. 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 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 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 metre.The question of measurement reform was placed in the hands of the French Academy of Sciences, who appointed a commission chaired by Jean-Charles de Borda. Instead of the seconds pendulum method, the commission of the French Academy of Sciences – whose members included Borda, Lagrange, Laplace, Monge and Condorcet – decided that the new measure should be equal to one ten-millionth of the distance from the North Pole to the Equator, measured along the meridian passing through Paris at the longitude of Paris pantheon, which became the central geodetic station in Paris. Jean Baptiste Joseph Delambre obtained the fundamental co-ordinates 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 was also a sound choice for scientific reasons: a portion of the quadrant from Dunkirk to Barcelona 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, would also extend southwards to Barcelona, later to Balearic Islands. Jean-Baptiste Biot and François Arago 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 Paris meridian as well as the account of the variation of the seconds pendulum's length along the same meridian between Shetland and the Balearc Islands.
The task of surveying the meridian arc fell to Pierre Méchain and Jean-Baptiste Delambre, and took more than six years. 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 François Arago, 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.
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 to the Montjuïc Fortress, Barcelona which was surveyed by Méchain. Although Méchain's sector was half the length of Delambre, it included the Pyrenees and hitherto unsurveyed parts of Spain.
Delambre measured a baseline of about 10 km in length along a straight road between Melun and Lieusaint. In an operation taking six weeks, the baseline was accurately measured using four platinum rods, each of length two toises. Thereafter he used, where possible, the triangulation points used by Nicolas Louis de Lacaille in his 1739–1740 survey of French meridian arc from Dunkirk to Collioure. Méchain's baseline was of a similar length, and also on a straight section of road between Vernet and Salces.
To put into practice the decision taken by the National Convention, on 1 August 1793, to disseminate the new units of the decimal metric system, 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 determined by the measurement of the Meridian of France from Dunkirk to Collioure, which, in 1740, had been carried out by Nicolas Louis de Lacaille 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 lignes that were distributed in France in 1795-1796. This was the metre installed under the arcades of the rue de Vaugirard, almost opposite the entrance to the Senate.
File:Castell de Montjuic - Fossat entrada - Barcelona.jpg|thumb|Montjuïc Castle in Barcelona, Spain – the southern end of the meridian arc|185x185px|left
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: Henricus Aeneae and Jean Henri van Swinden, Cisalpine Republic: Lorenzo Mascheroni, Kingdom of Denmark: Thomas Bugge, Kingdom of Spain: Gabriel Císcar and Agustín de Pedrayes, Helvetic Republic: Johann Georg Tralles, Ligurian Republic: Ambrogio Multedo, Kingdom of Sardinia: Prospero Balbo, Antonio Vassali Eandi, Roman Republic: Pietro Franchini, Tuscan Republic: Giovanni Fabbroni who had been invited by Talleyrand. The French commission comprised Jean-Charles de Borda, Barnabé Brisson, Charles-Augustin de Coulomb, Jean Darcet, René Just Haüy, Joseph-Louis Lagrange, Pierre- Simon Laplace, Louis Lefèvre-Ginneau, Pierre Méchain and Gaspar de Prony.
In 1799, a commission including Johann Georg Tralles, Jean Henri van Swinden, Adrien-Marie Legendre, Pierre-Simon Laplace, Gabriel Císcar, Pierre Méchain and Jean-Baptiste Delambre calculated the distance from Dunkirk to Barcelona using the data of the triangulation 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. Pierre-Simon Laplace originally hoped to figure out the Earth ellipsoid 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 Paris meridian 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 as a permanent record of the result.