In this article we are going to delve into the exciting world of Arc measurement of Delambre and Méchain. From its origins to its influence today, we will explore all facets of Arc measurement of Delambre and Méchain and its impact in various fields. We will analyze its evolution over time, as well as the different perspectives that exist on Arc measurement of Delambre and Méchain. In addition, we will examine its relevance in today's society and its future projection. It doesn't matter if you are an expert in the field or if it is the first time you are hearing about Arc measurement of Delambre and Méchain, this article will offer you a complete and detailed insight into this exciting topic.
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.[2]
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.[3][4] 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, Pierre-Simon Laplace, 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 Panthéon, which would become the central geodetic station in Paris.[4][1]
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 Panthéon.[4][1] 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.[5] The Weights and Measures Commission would, in 1799, adopt a flattening of 1/334 based on analysis by Pierre-Simon Laplace who combined the French Geodesic Mission to the Equator and the data of the arc measurement of Delambre and Méchain.[6] Combining these two data sets Laplace succeeded to estimate the flattening of the Earth ellipsoid and was happy to find that it also fitted well with his estimate 1/336 based on 15 pendulum measurements.[6][5]
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 Toise de l'Académie 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: pieds) long ruler, part of the baseline measuring instrument devised for this survey.[7][4] When the final result was known, the Mètre des Archives 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.[4]
In 1834, Ferdinand Rudolph Hassler measured at Fire Island the first baseline of the Survey of the Coast,[8] shortly before Louis Puissant declared to the French Academy of Sciences in 1836 that there was an inaccuracy in the arc measurement of Delambre and Méchain.[9][10] Ferdinand Rudolph Hassler's use of the metre and the creation of the Office of Standard Weights and Measures as an office within the Coast Survey contributed to the introduction of the Metric Act of 1866 allowing the use of the metre in the United States,[11] and preceded the choice of the metre as international scientific unit of length and the proposal by the 1867 General Conference of the European Arc Measurement (German: Europäische Gradmessung) to establish the International Bureau of Weights and Measures.[12]
Ferdinand Rudolph Hassler was a Swiss-American surveyor who is considered the forefather of both the National Oceanic and Atmospheric Administration (NOAA) and the National Institute of Standards and Technology (NIST) for his achievements as the first Superintendent of the U.S. Survey of the Coast and the first U.S. Superintendent of Weights and Measures.[13][14] The foundation of the United States Coast and Geodetic Survey led to the actual definition of the metre, with Charles Sanders Peirce being the first to experimentally link the metre to the wave length of a spectral line.[15]
The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299792458 when expressed in the unit m⋅s−1, where the second is defined in terms of the caesium frequency ΔνCs.
Where older traditional length measures are still used, they are now defined in terms of the metre – for example the yard has since 1959 officially been defined as exactly 0.9144 metre.[16]
The French Academy of Sciences, responsible for the concept and definition of the metre, was established in 1666.[4] In the 18th 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.[17][18][19] 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.[4][5]
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.[20]
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 2+1⁄2 minutes per day compared to its rate at Paris.[21][22] 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 1/230.[23] 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.[24] 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.[24] However, by 1743, Clairaut's theorem had completely supplanted Newton's approach.
Clairaut confirmed that Newton's theory that the Earth was ellipsoidal was correct, but that his calculations were in error, and he wrote a letter to the Royal Society of London with his findings.[25] The society published an article in Philosophical Transactions the following year, 1737.[26] In it Clairaut pointed out (Section XVIII) that Newton's Proposition XX of Book 3 does not apply to the real earth. It stated that the weight of an object at some point in the earth depended only on the proportion of its distance from the centre of the earth to the distance from the centre to the surface at or above the object, so that the total weight of a column of water at the centre of the earth would be the same no matter in which direction the column went up to the surface. Newton had in fact said that this was on the assumption that the matter inside the earth was of a uniform density (in Proposition XIX). Newton realized that the density was probably not uniform, and proposed this as an explanation for why gravity measurements found a greater difference between polar regions and equatorial regions than what his theory predicted. However, he also thought this would mean the equator was further from the centre than what his theory predicted, and Clairaut points out that the opposite is true. Clairaut points out at the beginning of his article that Newton did not explain why he thought the earth was ellipsoid rather than like some other oval, but that Clairaut, and James Stirling almost simultaneously, had shown why the earth should be an ellipsoid in 1736.
Clairaut's article did not provide a valid equation to back up his argument as well. This created much controversy in the scientific community. It was not until Clairaut wrote Théorie de la figure de la terre in 1743 that a proper answer was provided. In it, he promulgated what is more formally known today as Clairaut's theorem.
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.[27][28] The unit of length used by the French was the Toise de Paris, while the English one was the yard, which became the geodetic unit used in the British Empire.[29][30][31]
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.[32]
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.[32]
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.[33]: 52 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 (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.[34][35] Jean Baptiste Joseph Delambre otained the fundamental co-ordinates of the Pantheon by triangulating all the geodetic stations around Paris from the Pantheon's dome.[35][36]
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,[37] 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.[38]
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.[39][40]
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.[41]
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.[42]
Delambre measured a baseline of about 10 km (6,075.90 toises) 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 (a toise being about 1.949 m).[42][7] 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.[43][44] Borda's double-toise N°1 became the main reference for measuring all geodetic bases in France.[4] 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.[45] 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.[46] Méchain's baseline was of a similar length (6,006.25 toises), and also on a straight section of road between Vernet (in the Perpignan area) and Salces (now Salses-le-Chateau).[47]
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,[50] 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 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.[50] It was therefore metal bars of 443.44 lignes that were distributed in France in 1795-1796.[41] This was the metre installed under the arcades of the rue de Vaugirard, almost opposite the entrance to the Senate.[46]
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.[51][52][53]
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 1/334 was found for the Earth's flattening.[52][37] 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 1/150 considered as unacceptable.[49][52][54] 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 1/300 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.[52][55][56] 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 5130740 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.[51][37] 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.[46]
However, Louis Puissant declared in 1836 to the French Academy of Sciences that Jean Baptiste Joseph Delambre and Pierre Méchain had made errors in the triangulation of the meridian arc, which had been used for determining the length of the metre.[57][9] This is why Antoine Yvon Villarceau verified the geodetic operations at eight points of the Paris meridian arc from 1861 to 1866. Some of the errors in the operations of Delambre and Méchain were then corrected.[9]
Moreover it was later asserted that the Mètre des Archives was short by about 200 micrometres because of miscalculation of the flattening of the Earth ellipsoid, making the prototype about 0.02% shorter than the original proposed definition of the metre. Regardless, this length became the French standard and was progressively adopted by other countries in Europe.[2][58] Most of the difference was due to the failure to take vertical deflections into account; which was beyond the reach of Delambre and Méchain because the Earth's gravitational field had not yet been studied.[59][36]
In 1870 Ibáñez founded the Spanish National Geographic Institute which he then directed until 1889.[61][62] At the time it was the world's biggest geographic institute.[63] It encompassed geodesy, general topography, leveling, cartography, statistics and the general service of weights and measures.[63] Spain had adopted the metric system in 1849. The Government was urged by the Spanish Royal Academy of Sciences to approve the creation of a large-scale map of Spain in 1852.[64]
The following year Carlos Ibáñez e Ibáñez de Ibero was appointed to undertake this task.[63] As all the scientific and technical equipment for a vast undertaking of this kind had to be created, Ibáñez, in collaboration with Frutos Saavedra Meneses drew up the project of a new apparatus for measuring bases. He recognized that the end standards with which the most perfect devices of the eighteenth century and those of the first half of the nineteenth century were still equipped, that Jean-Charles de Borda or Friedrich Wilhelm Bessel simply joined measuring the intervals by means of screw tabs or glass wedges, would be replaced advantageously for accuracy by the system, designed by Ferdinand Rudolph Hassler for the United States Coast Survey, and which consisted of using a single standard with lines marked on the bar and microscopic measurements. Regarding the two methods by which the effect of temperature was taken into account, Ibáñez used both the bimetallic rulers, in platinum and brass, which he first employed for the central base of Spain, and the simple iron ruler with inlaid mercury thermometers which was used in Switzerland.[63][65]
Ibáñez and Saavedra went to Paris to supervise the production by Jean Brunner of a measuring instrument calibrated against the metre which they had devised and which they later compared with Borda's double-toise N°1 which was the main reference for measuring all geodetic bases in France and whose length was by definition 3.8980732 metres at a specified temperature.[66][67][68][69][70] The four-metre-long Spanish measuring instrument, which became known as the Spanish Standard (French: Règle espagnole), was replicated in order to be used in Egypt.[66][71][72][73] In 1863, Ibáñez and Ismail Effendi Mustafa compared the Spanish Standard with the Egyptian Standard in Madrid.[74][75][76] These comparisons 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.[72]
Jean Brunner displayed the Ibáñez-Brunner apparatus at the Exposition Universelle of 1855.[77][78] Copies of the Spanish standard[77] were also made for France[79][80] and Germany.[81] These standards would be used for the most important operations of European geodesy.[72] Indeed, the southward extension of Paris meridian's triangulation by Pierre Méchain (1803–1804), then François Arago and Jean-Baptiste Biot (1806–1809) had not been secured by any baseline measurement in Spain.[82][57]
In 1858 Spain's central geodetic base of triangulation was measured in Madridejos (Toledo) with exceptional precision for the time thanks to the Spanish Standard.[63][71] Ibáñez and his colleagues wrote a monograph which was translated into French by Aimé Laussedat.[83] The experiment, in which the results of two methods were compared, was a landmark in the controversy between French and German geodesists about the length of geodesic triangulation bases, and empirically validated the method of General Johann Jacob Bayer, founder of the International Association of Geodesy.[84]
From 1865 to 1868 Ibáñez added the survey of the Balearic Islands with that of the Iberian Peninsula.[71][85] For this work, he devised a new instrument, which allowed much faster measurements.[71] In 1869, Ibáñez brought it along to Southampton where Alexander Ross Clarke was making the necessary measurements to compare the Standards of length used in the World.[63][68][86] Finally, this second version of the appliance, called the Ibáñez apparatus, was used in Switzerland to measure the geodetic bases of Aarberg, Weinfelden and Bellinzona.[63][87]
In 1865 the triangulation of Spain was connected with that of Portugal and France.[83][76] In 1866 at the conference of the Association of Geodesy in Neuchâtel, Ibáñez announced that Spain would collaborate in remeasuring and extending the French meridian arc.[63][88] From 1870 to 1894, François Perrier, then Jean-Antonin-Léon Bassot proceeded to a new survey.[9][79] In 1879 Ibáñez and François Perrier completed the junction between the geodetic networks of Spain and Algeria and thus completed the measurement of a meridian arc which extended from Shetland to the Sahara.[89] This connection was a remarkable enterprise where triangles with a maximum length of 270 km were observed from mountain stations (Mulhacén, Tetica, Filahoussen, M'Sabiha) over the Mediterranean Sea.[90][89][91][79]
This meridian arc was named West Europe-Africa Meridian-arc by Alexander Ross Clarke and Friedrich Robert Helmert. It yielded a value for the equatorial radius of the earth a = 6 377 935 metres, the ellipticity being assumed as 1/299.15 according to Bessel ellipsoid.[92][93] The radius of curvature of this arc is not uniform, being, in the mean, about 600 metres greater in the northern than in the southern part.[60]
According to the calculations made at the central bureau of the International Geodetic Association, the net does not follow the meridian exactly, but deviates both to the west and to the east; actually, the meridian of Greenwich is nearer the mean than that of Paris.[60]
In the 19th century, astronomers and geodesists were concerned with questions of longitude and time, because they were responsible for determining them scientifically and used them continually in their studies. The International Geodetic Association, which had covered Europe with a network of fundamental longitudes, took an interest in the question of an internationally-accepted prime meridian at its seventh general conference in Rome in 1883.[94] Indeed, the Association was already providing administrations with the bases for topographical surveys, and engineers with the fundamental benchmarks for their levelling. It seemed natural that it should contribute to the achievement of significant progress in navigation, cartography and geography, as well as in the service of major communications institutions, railways and telegraphs.[95] From a scientific point of view, to be a candidate for the status of international prime meridian, the proponent needed to satisfy three important criteria. According to the report by Carlos Ibáñez e Ibáñez de Ibero, it must have a first-rate astronomical observatory, be directly linked by astronomical observations to other nearby observatories, and be attached to a network of first-rate triangles in the surrounding country.[95] Four major observatories could satisfy these requirements: Greenwich, Paris, Berlin and Washington. The conference concluded that Greenwich Observatory best corresponded to the geographical, nautical, astronomical and cartographic conditions that guided the choice of an international prime meridian, and recommended the governments should adopt it as the world standard.[96] The Conference further hoped that, if the whole world agreed on the unification of longitudes and times by the Association's choosing the Greenwich meridian, Great Britain might respond in favour of the unification of weights and measures, by adhering to the Metre Convention.[97]
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