Arc Measurement Of Delambre And Méchain

The arc measurement of Delambre and Méchain was a geodetic survey carried out by Jean-Baptiste Delambre and Pierre Méchain in 1792–1798 to measure an arc section of the Paris meridian between Dunkirk and Barcelona. This arc measurement served as the basis for the original definition of the metre.

Until the French Revolution of 1789, France was particularly affected by the proliferation of length measures; the conflicts related to units helped precipitate the revolution. In addition to rejecting standards inherited from feudalism, linking determination of a decimal unit of length with the figure of the Earth was an explicit goal. This project culminated in an immense effort to measure a meridian passing through Paris in order to define the metre.

When question of measurement reform was placed in the hands of the French Academy of Sciences, a commission, whose members included Jean-Charles de Borda, Joseph-Louis Lagrange, Gaspard Monge and the Marquis de Condorcet, 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 longitude of Paris Observatory. Since this survey, the Panthéon became the central geodetic station in Paris.

In 1791, Jean Baptiste Joseph Delambre and Pierre Méchain were commissioned to lead an expedition to accurately measure the distance between a belfry in Dunkerque and Montjuïc castle in Barcelona in order to calculate the length of the meridian arc through the centre of Paris Observatory. The official length of the '' Mètre des Archives'' was based on these measurements, but the definitive length of the metre required a value for the non-spherical shape of the Earth, known as the flattening of the Earth. 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.

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 the length of the metre was established, in relation to the also called toise of Peru, which had been constructed in 1735 for the French Geodesic Mission to Peru, as well as to Borda's double-toise N°1, one of the four twelve feet (French: ) long ruler, part of the baseline measuring instrument devised for this survey. When the final result was known, the ''Mètre des Archives'' a platinum 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.

Scientific revolution in France and beginning of Greenwich arc measurement

The French Academy of Sciences, responsible for the concept and definition of the metre, was established in 1666. In the 17th century it had determined the first reasonably accurate distance to the Sun and organised important work in geodesy and cartography. In the 18th century, in addition to its significance for cartography, geodesy grew in importance as a means of empirically demonstrating Newton's law of universal gravitation, which Émilie du Châtelet promoted in France in combination with Leibniz's mathematical work and because the radius of the Earth was the unit to which all celestial distances were to be referred. Among the results that would impact the definition of the metre: Earth proved to be an oblate spheroid through geodetic surveys in Ecuador and Lapland.

Galileo had discovered gravitational acceleration explaining the fall of bodies at the surface of the Earth. He had also observed the regularity of the period of swing of the pendulum and that this period depended on the length of the pendulum. Marin Mersenne was the central figure in the dissemination of Galileo's ideas in France, thanks to his role as a translator and commentator of Galileo's work. In 1645 Giovanni Battista Riccioli had been the first to determine the length of a "seconds pendulum" (a pendulum with a half-period of one second). In 1656, Christiaan Huygens, inspired by Galileo, invented the first pendulum clock which greatly improved the accuracy of time measurement required for astronomical observations. In 1671, Jean Picard also measured the length of a seconds pendulum at Paris Observatory and proposed this unit of measurement to be called the astronomical radius (French: ''Rayon Astronomique''). He found the value of 36 ''pouces'' and 8 lignes of the Toise of Châtelet, which had been recently renewed. Proposals for decimal measurement systems from scientists and mathematicians also led to proposals to base units on reproducible natural phenomena, such as the motion of a pendulum or a fraction of the circle of latitude at the Equator.

The first reasonably accurate distance to the Sun was determined in 1684 by Giovanni Domenico Cassini. Knowing that directly measurements of the solar parallax were difficult he chose to measure the Martian parallax. Having sent Jean Richer to Cayenne, part of French Guiana, for simultaneous measurements, Cassini in Paris determined the parallax of Mars when Mars was at its closest to Earth in 1672. Using the circumference distance between the two observations, Cassini calculated the Earth-Mars distance, then used Kepler's laws to determine the Earth-Sun distance. His value, about 10% smaller than modern values, was much larger than all previous estimates.

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 (as it would be if the Earth were a sphere); 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.. 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 (1735) undertook expeditions to Peru ( Bouguer, Louis Godin, de La Condamine, Antonio de Ulloa, Jorge Juan) and to Lapland ( Maupertuis, Clairaut, Camus, Le Monnier, Abbe Outhier, Anders Celsius). The resulting measurements at equatorial and polar latitudes confirmed that the Earth was best modelled by an oblate spheroid, supporting Newton. In the 1740s an account was published in the Paris ''Mémoires'', by Cassini de Thury, of a remeasurement by himself and Nicolas Louis de Lacaille of the meridian of Paris. With a view to determine more accurately the variation of the degree along the meridian, they divided the distance from Dunkirk to Collioure into four partial arcs of about two degrees each, by observing the latitude at five stations. The results previously obtained by Giovanni Domenico and Jacques Cassini were not confirmed, but, on the contrary, the length of the degree derived from these partial arcs showed on the whole an increase with increasing latitude.

Geodetic surveys found practical applications in French cartography and in the Anglo-French Survey, which aimed to connect Paris and Greenwich Observatories and led to the Principal Triangulation of Great Britain. The unit of length used by the French was the , while the English one was the yard, which became the geodetic unit used in the British Empire.

In 1783 the director of the Paris Observatory, César-François Cassini de Thury, addressed a memoir to the Royal Society in London, in which he expressed grave reservations about the latitude and longitude measurements undertaken at the Royal Greenwich Observatory. He suggested that the correct values might be found by combining the Paris Observatory figures with a precise trigonometric survey between the two observatories. This criticism was roundly rejected by Nevil Maskelyne who was convinced of the accuracy of the Greenwich measurements but, at the same time, he realised that Cassini's memoir provided a means of promoting government funding for a survey which would be valuable in its own right.

For the triangulation of the Anglo-French Survey, César-François Cassini de Thury was assisted by Pierre Méchain. They used the repeating circle, an instrument for geodetic surveying, developed from the reflecting circle by Étienne Lenoir in 1784. He invented it while an assistant of Jean-Charles de Borda, who later improved the instrument. It was notable as being the equal of the great theodolite created by the renowned instrument maker, Jesse Ramsden. It would later be used to measure the meridian arc from Dunkirk to Barcelona by Jean Baptiste Delambre and Pierre Méchain as improvements in the measuring device designed by Borda and used for this survey also raised hopes for a more accurate determination of the length of the French meridian arc.

Invention and international adoption of the metre as scientific unit of length

From the French revolution of 1789 came an effort to reform measurement standards, leading ultimately to remeasure 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, 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 (the quadrant of the Earth's circumference), 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 Panthéon by triangulating all the geodetic stations around Paris from the Panthéon'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 (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, 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 (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 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 (6,075.90 ) 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 (a being about 1.949 m). These measuring devices consisted of bimetallic rulers in platinum and brass fixed together at one extremity to assess the variations in length produced by any change in temperature. Borda's double-toise N°1 became the main reference for measuring all geodetic bases in France. Intercomparisons of baseline measuring devices were essential, because of thermal expansion. Indeed, geodesists tried to accurately assess temperature of standards in the field in order to avoid temperature systematic errors. Thereafter he used, where possible, the triangulation points used by Nicolas Louis de Lacaille in his 1739-1740 arc measurement. Méchain's baseline was of a similar length (6,006.25 ), and also on a straight section of road between Vernet (in the Perpignan area) and Salces (now Salses-le-Chateau).

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 (French: ''mètre provisoire'') determined by the French meridian arc measurement, which had been carried out from Dunkirk to Perpignan by Nicolas Louis de Lacaille and Cesar-François Cassini de Thury and published by the latter in 1744. 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, almost opposite the entrance to the Senate.

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

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