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 innacuracy 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 fondation 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 Δν_Cs.

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, it was later determined 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. This is why the polar circumference of the Earth is 40,008 km, instead of 40,000.^[2]

At that time, units of measurement were defined by primary standards, and unique artifacts made of different alloys with distinct coefficients of expansion were the legal basis of units of length. A wrought iron ruler, the Toise of Peru, also called Toise de l'Académie, was the French primary standard of the toise, and the metre was officially defined by an artifact made of platinum kept in the National Archives.^[57] Besides the latter, another platinum and twelve iron standards of the metre were made by Étienne Lenoir in 1799.^[58] One of them became known as the Committee Meter in the United States and served as standard of length in the United States Coast Survey until 1890.^[59] According to geodesists, these standards were secondary standards deduced from the Toise of Peru. In Europe, except Spain,^[60] surveyors continued to use measuring instruments calibrated on the Toise of Peru.^[61] Among these, the toise of Bessel and the apparatus of Borda were respectively the main references for geodesy in Prussia and in France. These measuring devices consisted of bimetallic rulers in platinum and brass or iron and zinc fixed together at one extremity to assess the variations in length produced by any change in temperature. The combination of two bars made of two different metals allowed to take thermal expansion into account without measuring the temperature.^[62][63] A French scientific instrument maker, Jean Nicolas Fortin, made three direct copies of the Toise of Peru, one for Friedrich Georg Wilhelm von Struve, a second for Heinrich Christian Schumacher in 1821 and a third for Friedrich Wilhelm Bessel in 1823. In 1831, Henri-Prudence Gambey also realised a copy of the Toise of Peru which was kept at Altona Observatory.^[58][64]

In 1816, Ferdinand Rudolph Hassler was appointed first Superintendent of the Survey of the Coast.^[65][66] Trained in geodesy in Switzerland, France and Germany, Hassler had brought a standard metre made in Paris to the United States in October 1805. He designed a baseline apparatus which instead of bringing different bars in actual contact during measurements,^[66] used only one bar calibrated on the Committee meter, an authenthic copy of the Mètre des Archives,^[59][24] and optical contact.^[66][67] Thus the metre became the unit of length for geodesy in the United States.^[68] This would result in the Metre Convention of 1875, when the metre was adopted as an international scientific unit of length for the convenience of continental European geodesists following forerunners such as Ferdinand Rudolph Hassler later Carlos Ibáñez e Ibáñez de Ibero.^{[69][60][70][71][68][12][46]}

In 1830, Hassler became head of the Office of Weights and Measures, which became a part of the Survey of the Coast. He compared various units of length used in the United States at that time and measured coefficients of expansion to assess temperature effects on the measurements.^[72]

In 1834, Hassler, measured at Fire Island the first baseline of the Survey of the Coast,^[73] shortly before Louis Puissant declared to the French Academy of Sciences in 1836 that Jean Baptiste Joseph Delambre and Pierre Méchain had made errors in the meridian arc measurement, which had been used to determine the length of the metre.^[9][74]

Egyptian astronomy has ancient roots which were revived in the 19th century by the modernist impetus of Muhammad Ali who founded in Sabtieh, Boulaq district, in Cairo an Observatory which he was keen to keep in harmony with the progress of this science still in progress.^[75][76] In 1858, a Technical Commission was set up to continue, by adopting the procedures instituted in Europe, the cadastre work inaugurated under Muhammad Ali. This Commission suggested to Viceroy Mohammed Sa'id Pasha the idea of buying geodetic devices which were ordered in France. While Mahmud Ahmad Hamdi al-Falaki was in charge, in Egypt, of the direction of the work of the general map, the viceroy entrusted to Ismail Mustafa al-Falaki the study, in Europe, of the precision apparatus calibrated against the metre intended to measure the geodesic bases and already built by Jean Brunner in Paris. Ismail Mustafa had the task to carry out the experiments necessary for determining the expansion coefficients of the two platinum and brass bars, and to compare the Egyptian standard with a known standard. The Spanish standard designed by Carlos Ibáñez e Ibáñez de Ibero and Frutos Saavedra Meneses was chosen for this purpose, as it had served as a model for the construction of the Egyptian standard.^[76][75] In addition, the Spanish standard had been compared with Borda's double-toise N° 1, which served as a comparison module for the measurement of all geodesic bases in France,^[77][78] and was also to be compared to the Ibáñez apparatus.^[79][77] In 1954, the connection of the southerly extension of the Struve Geodetic Arc with an arc running northwards from South Africa through Egypt would bring the course of a major meridian arc back to land where Eratosthenes had founded geodesy.^[80]

In 1855, the Dufour map (French: Carte Dufour), the first topographic map of Switzerland for which the metre was adopted as the unit of length, won the gold medal at the Exposition Universelle.^[82][83] However, the baselines for this map were measured in 1834 with three toises long measuring rods calibrated on a toise made in 1821 by Jean Nicolas Fortin for Friedrich Georg Wilhelm von Struve.^[84][31] The Spanish standard, a geodetic measuring device calibrated on the metre devised by Carlos Ibáñez e Ibáñez de Ibero and Frutos Saavedra Meneses, was also displayed by Jean Brunner at the Exhibition.^[85][86] On the sidelines of the Exposition Universelle (1855) and the second Congress of Statistics held in Paris, an association with a view to obtaining a uniform decimal system of measures, weights and currencies was created in 1855.^[61] Under the impetus of this association, a Committee for Weights and Measures and Monies (French: Comité des poids, mesures et monnaies) would be created during the Exposition Universelle (1867) in Paris and would call for the international adoption of the metric system.^[87][61]

In 1866, an important concern was that the Toise of Peru, the standard of the toise constructed in 1735 for the French Geodesic Mission to the Equator, might be so much damaged that comparison with it would be worthless,^[88] while Bessel had questioned the accuracy of copies of this standard belonging to Altona and Koenigsberg Observatories, which he had compared to each other about 1840.^[89][58] This assertion was particularly worrying, because when the primary Imperial yard standard had partially been destroyed in 1834, a new standard of reference was constructed using copies of the "Standard Yard, 1760", instead of the pendulum's length as provided for in the Weights and Measures Act 1824,^[90] because the pendulum method proved unreliable.^[91][15] Nevertheless 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][92]

In 1867 at the second General Conference of the International Association of Geodesy held in Berlin, the question of an international standard unit of length was discussed in order to combine the measurements made in different countries to determine the size and shape of the Earth.^[93][94] According to a preliminary proposal made in Neuchâtel the precedent year,^[95][93] the General Conference recommended the adoption of the metre in replacement of the toise of Bessel,^[94][96] the creation of an International Metre Commission, and the foundation of a World institute for the comparison of geodetic standards, the first step towards the creation of the International Bureau of Weights and Measures.^[95][93]

Hassler's metrological and geodetic work also had a favourable response in Russia.^[72][66] In 1869, the Saint Petersburg Academy of Sciences sent to the French Academy of Sciences a report drafted by Otto Wilhelm von Struve, Heinrich von Wild, and Moritz von Jacobi, whose theorem has long supported the assumption of an ellipsoid with three unequal axes for the figure of the Earth, inviting his French counterpart to undertake joint action to ensure the universal use of the metric system in all scientific work.^[91][97] The French Academy of Sciences and the Bureau des Longitudes in Paris drew the attention of the French government to this issue. In November 1869, Napoleon III issued invitations to join the International Metre Commission.^[87]

The French government gave practical support to the creation of an International Metre Commission, which met in Paris in 1870 and again in 1872 with the participation of about thirty countries.^[98][87] There was much discussion within this Commission, considering the opportunity either to keep as definitive the units represented by the standards of the Archives, or to return to the primitive definitions, and to correct the units to bring them closer to them. Since its origin, the metre has kept a double definition; it is both the ten-millionth part of the quarter meridian and the length represented by the Mètre des Archives. The first is historical, the second is metrological. The first solution prevailed, in accordance with common sense and in accordance with the advice of the French Academy of Sciences. Abandoning the values represented by the standards, would have consecrated an extremely dangerous principle, that of the change of units to any progress of measurements; the Metric System would be perpetually threatened with change, that is to say with ruin. Thus the Commission called for the creation of a new international prototype metre which length would be as close as possible to that of the Mètre des Archives and the arrangement of a system where national standards could be compared with it.^[99]

On 6 May 1873 during the 6th session of the French section of the Metre Commission, Henri Étienne Sainte-Claire Deville cast a 20-kilogram platinum-iridium ingot from Matthey in his laboratory at the École normale supérieure (Paris). On 13 May 1874, 250 kilograms of platinum-iridium to be used for several national prototypes of the metre was cast at the Conservatoire national des arts et métiers.^[87] When a conflict broke out regarding the presence of impurities in the metre-alloy of 1874, a member of the Preparatory Committee since 1870 and president of the Permanent Committee of the International Metre Commission, Carlos Ibáñez e Ibáñez de Ibero intervened with the French Academy of Sciences to rally France to the project to create an International Bureau of Weights and Measures equipped with the scientific means necessary to redefine the units of the metric system according to the progress of sciences.^{[100][71][61][101]}

The Metre Convention was signed on 20 May 1875 in Paris and the International Bureau of Weights and Measures was created under the supervision of the International Committee for Weights and Measures. At the session on 12 October 1872 of the Permanent Committee of the International Metre Commission, which was to become the International Committee for Weights and Measures,^[46] Carlos Ibáñez e Ibáñez de Ibero had been elected president.^[102][103] His presidency was confirmed at the first meeting of the International Committee for Weights and Measures, on 19 April 1875.^[104] Three other members of the committee, the German astronomer, Wilhelm Julius Foerster, director of the Berlin Observatory and director of the German Weights and Measures Service,^[105] the Swiss meteorologist and physicist, Heinrich von Wild representing Russia,^[106] and the Swiss geodesist of German origin, Adolphe Hirsch were also among the main architects of the Metre Convention.^[46] In the 1870s, German Empire played a pivotal role in the unification of the metric system through the European Arc Measurement but its overwhelming influence was mitigated by that of neutral states. While the German astronomer Wilhelm Julius Foerster along with the Russian and Austrian representatives boycotted the Permanent Committee of the International Metre Commission in order to prompt the reunion of the Diplomatic Conference of the Metre and to promote the foundation of a permanent International Bureau of Weights and Measures,^[106] Adolphe Hirsch, delegate of Switzerland at this Diplomatic Conference in 1875, conformed to the opinion of Italy and Spain to create, in spite of French reluctance, the International Bureau of Weights and Measures in France as a permanent institution at the disadvantage of the Conservatoire national des arts et métiers.^[107]

In recognition of France's role in designing the metric system, the BIPM is based in Sèvres, just outside Paris. However, as an international organisation, the BIPM is under the ultimate control of a diplomatic conference, the Conférence générale des poids et mesures (CGPM) rather than the French government.^[16][108]

In 1889 the General Conference on Weights and Measures met at Sèvres, the seat of the International Bureau. It performed the first great deed dictated by the motto inscribed in the pediment of the splendid edifice that is the metric system: "A tous les temps, à tous les peuples" (For all times, to all peoples); and this deed consisted in the approval and distribution, among the governments of the states supporting the Metre Convention, of prototype standards of hitherto unknown precision intended to propagate the metric unit throughout the whole world.^[111]

For metrology the matter of expansibility was fundamental; as a matter of fact, the temperature measuring error related to the length measurement in proportion to the expansibility of the standard and the constantly renewed efforts of metrologists to protect their measuring instruments against the interfering influence of temperature revealed clearly the importance they attached to the expansion-induced errors. It was common knowledge, for instance, that effective measurements were possible only inside a building, the rooms of which were well protected against the changes in outside temperature, and the very presence of the observer created an interference against which it was often necessary to take strict precautions.^[111] Thus, the Contracting States also received a collection of thermometers whose accuracy made it possible to ensure that of length measurements.^[112] The international prototype would also be a "line standard"; that is, the metre was defined as the distance between two lines marked on the bar, so avoiding the wear problems of end standards.^[113][114]

Before the Great War, there were quite a number of international associations active in geophysical sciences. Among them, the most powerful and oldest was the International Geodetic Association where German influence predominated and which had its central office at the Prussian Geodetic Institute in Potsdam.^[115]

In the second half of the 19th century, the creation of the International Geodetic Association would mark the adoption of new scientific methods which allowed to take into account observational errors in science.^[116][117] It then became possible to accurately measure parallel arcs, since the difference in longitude between their ends could be determined thanks to the invention of the electrical telegraph.^[88] Furthermore, advances in metrology combined with those of gravimetry have led to a new era of geodesy. If precision metrology had needed the help of geodesy, the latter could not continue to prosper without the help of metrology. It was then necessary to define a single unit to express all the measurements of terrestrial arcs and all determinations of the gravitational acceleration by means of pendulum.^[118]

The intimate relationships that necessarily existed between metrology and geodesy explain that the International Geodetic Association, founded to combine the geodetic operations of different countries, in order to reach a new and more exact determination of the shape and dimensions of the Globe, prompted the project of reforming the foundations of the metric system, while expanding it and making it international. Not, as it was mistakenly assumed for a certain time, that the Association had the unscientific thought of modifying the length of the metre, in order to conform exactly to its historical definition according to the new values that would be found for the terrestrial meridian. But, busy combining the arcs measured in the different countries and connecting the neighbouring triangulations, geodesists encountered, as one of the main difficulties, the unfortunate uncertainty which reigned over the equations of the units of length used.^[119] In 1867 at the second General Conference of the International Association of Geodesy held in Berlin, the question of an international standard unit of length was discussed in order to combine the measurements made in different countries to determine the size and shape of the Earth.^[93][94] According to a preliminary proposal made in Neuchâtel the precedent year,^[95][93] the General Conference recommended the adoption of the metre in replacement of the toise of Bessel,^[94][96] the creation of an International Metre Commission, and the foundation of a world institute for the comparison of geodetic standards, the first step towards the creation of the International Bureau of Weights and Measures.^[95][93]

When the metre was choosen as an international unit of length, it was well known that by measuring the latitude of two stations in Barcelona, Méchain had found that the difference between these latitudes was greater than predicted by direct measurement of distance by triangulation and that he did not dare to admit this inaccuracy.^[120][121] This was later explained by clearance in the central axis of the repeating circle causing wear and consequently the zenith measurements contained significant systematic errors.^[122] In 1889, the French Minister of Foreign Affairs, Eugène Spuller introduced the first General Conference on Weights and Measures with these words:

Your task, so useful, so beneficial to mankind, has been traversed by many vicissitudes for a hundred years. Like all the great things in this world, it has cost many pains, efforts, sacrifices, not to mention the difficulties, dangers, fatigue, tribulations of all kinds, which endured the two great French astronomers Delambre and Méchain, whose works are the basis of all yours. I am sure to be your interpreter, paying them supreme tribute on this day. Who does not remember with emotion the dangers to which Méchain so generously exposed his life? General Morin, who has been your worthy colleague for so long, wrote a few lines on this subject that you will be proud to hear: "To brave dangers similar to those which Méchain ran with the necessary calm, it is not enough to be devoted to science and to its duties; you must have an empire over your senses which will protect you from this kind of vertigo, in the shelter of which the most intrepid soldiers are not always. Someone who, without flinching, has faced the bullets a hundred times is, on the contrary, surprised by this insurmountable weakness in the presence of the emptiness that space offers him." It is a soldier speaking, Gentlemen; please listen to him again when he adds: "Science therefore also has its heroes who, happier than those of war, leave behind only works useful to humanity and not ruins and vengeful hatred".

Spuller, Eugène (1889), Compte rendus de la première Conférence générale des poids et mesures (PDF), p. 8

Scientific observations are marred by two distinct types of errors, systematic errors on the one hand, and random, on the other hand. The effects of random errors can be mitigated by the repeated measurements. Constant or systematic errors on the contrary must be carefully avoided, because they arise from one or more causes which constantly act in the same way, and have the effect of always altering the result of the experiment in the same direction. They therefore alter the value observed and repeated identical measurements do not reduce such errors.^[123][124]

The distinction between systematic and random errors is far from being as sharp as one might think at first glance. In reality, there are no or very few random errors. As science progresses, the causes of certain errors are sought out, studied, their laws discovered. These errors pass from the class of random errors into that of systematic errors. The ability of the observer consists in discovering the greatest possible number of systematic errors to be able, once he has become acquainted with their laws, to free his results from them using a method or appropriate corrections. It is the experimental study of a cause of error that has led to most of the great astronomical discoveries (precession, nutation, aberration).^[125] Polar motion predicted by Leonhard Euler and later discovered by Seth Carlo Chandler also had an impact on accuracy of latitudes' determinations.^{[126][127][128]}

Among all these sources of error, it was mainly an unfavourable vertical deflection that gave an inaccurate determination of Barcelona's latitude and a metre "too short" compared to a more general definition taken from the average of a large number of arcs.^[36] In 1841, errors in the method of calculating the length of the arc measurement of Delambre and Méchain were taken into account by Friedrich Wilhelm Bessel when he proposed his reference ellipsoid.^[43] Indeed, until the Hayford ellipsoid was calculated, vertical deflections were considered as random errors.^[129] Bessel, using the method of least squares calculated from several arc measurements, a new value for the flattening of the Earth, which he determined as ⁠1/299.15⁠.^[130][43] Bessel's reference ellipsoid would long be used by geodesists.

In 1901, Friedrich Robert Helmert made an accurate detemination of the Earth ellipsoid according to gravity measurements performed under the auspices of the International Geodetic Association.^[131][132]

Significant improvements in gravity measuring instruments must also be attributed to Bessel. He devised a gravimeter constructed by Adolf Repsold which was first used in Switzerland by Emile Plantamour,^[131] Charles Sanders Peirce and Isaac-Charles Élisée Cellérier (1818–1889), a Genevan mathematician soon independently discovered a mathematical formula to correct systematic errors of this device which had been noticed by Plantamour and Adolphe Hirsch.^[131][133] This would allow Friedrich Robert Helmert to determine a remarkably accurate value of ⁠1/298.3⁠ for the flattening of the Earth when he proposed his ellipsoid of reference.^[132] This was also the result of the Metre Convention of 1875, when the metre was adopted as an international scientific unit of length for the convenience of continental European geodesists following forerunners such as Ferdinand Rudolph Hassler later Carlos Ibáñez e Ibáñez de Ibero.^{[134][135][136]}

The 1875 Conference of the European Arc Measurement dealt with the best instrument to be used for the determination of gravity. The association decided in favor of the reversion pendulum and it was resolved to redo in Berlin, in the station where Friedrich Wilhelm Bessel made his famous measurements, the determination of gravity by means of devices of various kinds employed in different countries, in order to compare them and thus to have the equation of their scales, after an in-depth discussion in which an American scholar, Charles Sanders Peirce, took part.^[137] Indeed, as the figure of the Earth could be inferred from variations of the seconds pendulum length, the United States Coast Survey's direction instructed Charles Sanders Peirce in the spring of 1875 to proceed to Europe for the purpose of making pendulum experiments to chief initial stations for operations of this sort, in order to bring the determinations of the forces of gravity in America into communication with those of other parts of the world; and also for the purpose of making a careful study of the methods of pursuing these researches in the different countries of Europe.^[138]

The determination of gravity by the reversible pendulum was subject to two types of error. On the one hand the resistance of the air and on the other hand the movements that the oscillations of the pendulum imparted to its plane of suspension. These movements were particularly important with the apparatus designed by the Repsold brothers on the indications of Bessel, because the pendulum had a large mass in order to counteract the effect of the viscosity of the air. While Emile Plantamour was carrying out a series of experiments with this device, Adolph Hirsch found a way to demonstrate the movements of the pendulum's suspension plane by an ingenious process of optical amplification. Isaac-Charles Élisée Cellérier, a mathematician from Geneva and Charles Sanders Peirce would independently develop a correction formula that allowed the use of the observations made with this type of gravimeter.^[139][140]

Since the metre was originally defined, each time a new measurement is made, with more accurate instruments, methods or techniques, it is said that the metre is based on some error, from calculations or measurements.^[141] When Carlos Ibáñez e Ibáñez de Ibero first president of both the International Geodetic Association and the International Committee for Weigths and Measures took part to the remeasurement and extention of the arc measurement of Delambre and Méchain, mathematicians like Legendre and Gauss had developed new methods for processing data, including the least squares method which allowed to compare experimental data tainted with observational errors to a mathematical model.^[142][120] Moreover the International Bureau of Weights and Measures would have a central role for international geodetic measurements as Charles Édouard Guillaume's discovery of invar minimized the impact of measurement inaccuracies due to temperature systematic errors.^[143] The Earth measurements thus underscored the importance of the scientific method at a time when statistics were implemented in geodesy.^[120][19] As a leading scientist of his time, Carlos Ibáñez e Ibáñez de Ibero was one of the 81 initial members of the International Statistical Institute (ISI) and delegate of Spain to the first ISI session (now called World Statistic Congress) in Rome in 1887.^[135][144]

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.^[145] 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.^[146] 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.^[146] 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.^[147] 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.^[148]

The International Geodetic Association gained global importance with the accession of Chile, Mexico and Japan in 1888; Argentina and United-States in 1889; and British Empire in 1898. The convention of the International Geodetic Association expired at the end of 1916. It was not renewed due to the First World War. However, the activities of the International Latitude Service were continued through an Association Géodesique réduite entre États neutres thanks to the efforts of H.G. van de Sande Bakhuyzen and Raoul Gautier (1854–1931), respectively directors of Leiden Observatory and Geneva Observatory.^[4][32] After World War I, an essentially American and British idea was to group together the scientific unions relating to various disciplines under the authority of a Supreme Council. An international conference, which brought together in Brussels in July 1919 the scientists of the countries allied or associated in the fight against Germany and of a certain number of neutral states, created an International Science Council and various unions dependent on this Council; but, Geodesy, instead of being free and independent as before, was associated with the Geophysical Sciences in the International Union of Geodesy and Geophysics which first president was Charles Jean-Pierre Lallemand.^[115]

In 1920, Charles-Edouard Guillaume, was granted the Nobel Prize in Physics. Guillaume's Nobel Prize marked the end of an era in which metrology was leaving the field of geodesy to become a technological application of physics,^[149][150] as Charles Sanders Peirce's work promoted the advent of American science at the forefront of global metrology. Alongside his intercomparisons of artifacts of the metre and contributions to gravimetry through improvement of reversible pendulum, Peirce was the first to tie experimentally the metre to the wave length of a spectral line. According to him the standard length might be compared with that of a wave of light identified by a line in the solar spectrum. Albert Abraham Michelson soon took up the idea and improved it.^[15][151]

• Cartography of France
• Earth's circumference#Historical use in the definition of units of measurement
• Earth radius § History
• History of geodesy § Prime meridian and standard of length
• History of the metre § Meridional definition
• Meridian arc § 17th and 18th centuries
• Metre § Early adoption of the metre as a scientific unit of length: the forerunners
• Paris meridian#The West Europe-Africa Meridian-arc

• Hirsch, A.; von Oppolzer, Th., eds. (1884). "Rapport de la Commission chargée d'examiner les propositions du bureau de l'Association sur l'unification des longitudes et des heures" [Report of the Commission charged with examining the proposals of the Bureau of the Association on the unification of longitudes and times.]. Comptes-rendus des seances de la Septiéme Conférence Géodésique Internationale pour la mesure des degrés en Europe. Reunie a Rome du 15 au 24 Octobre 1863 [Proceedings of the Seventh International Geodesic Conference for the measurement of degrees in Europe. Held in Rome from 15 to 24 October 1863] (in French). Berlin: G. Reimer.