Meridian arc

In geodesy, a meridian arc measurement is the distance between two points with the same longitude, i.e., a segment of a meridian curve or its length. Two or more such determinations at different locations then specify the shape of the reference ellipsoid which best approximates the shape of the geoid. This process is called the determination of the figure of the Earth. The earliest determinations of the size of a spherical Earth required a single arc. The latest determinations use astro-geodetic measurements and the methods of satellite geodesy to determine the reference ellipsoids. In geodesy, a meridian arc measurement is the distance between two points with the same longitude, i.e., a segment of a meridian curve or its length. Two or more such determinations at different locations then specify the shape of the reference ellipsoid which best approximates the shape of the geoid. This process is called the determination of the figure of the Earth. The earliest determinations of the size of a spherical Earth required a single arc. The latest determinations use astro-geodetic measurements and the methods of satellite geodesy to determine the reference ellipsoids. Those interested in accurate expressions of the meridian arc for the WGS84 ellipsoid should consult the subsection entitled numerical expressions. Early estimations of Earth's size are recorded from Greece in the 4th century BC, and from scholars at the caliph's House of Wisdom in the 9th century. The first realistic value was calculated by Alexandrian scientist Eratosthenes about 240 BC. He knew that on the summer solstice at local noon the sun goes through the zenith in the ancient Egyptian city of Syene (Assuan). He also knew from his own measurements that, at the same moment in his hometown of Alexandria, the zenith distance was 1/50 of a full circle (7.2°). Assuming that Alexandria was due north of Syene, Eratosthenes concluded that the distance between Alexandria and Syene must be 1/50 of Earth's circumference. Using data from caravan travels, he estimated the distance to be 5,000 stadia (about 500 nautical miles)—which implies a circumference of 252,000 stadia. Assuming the Attic stadion (185 m) this corresponds to 46,620 km, or 16% too great. However, if Eratosthenes used the Egyptian stadion (157.5 m) his measurement turns out to be 39,690 km, an error of only 1%. Syene is not precisely on the Tropic of Cancer and not directly south of Alexandria. The sun appears as a disk of 0.5°, and an estimate of the overland distance traveling along the Nile or through the desert couldn't be more accurate than about 10%. Eratosthenes' estimation of Earth's size was accepted for nearly two thousand years. A similar method was used by Posidonius about 150 years later, and slightly better results were calculated in 827 by the grade measurement of the Caliph Al-Ma'mun. Early literature uses the term oblate spheroid to describe a sphere 'squashed at the poles'. Modern literature uses the term ellipsoid of revolution in place of spheroid, although the qualifying words 'of revolution' are usually dropped. An ellipsoid that is not an ellipsoid of revolution is called a triaxial ellipsoid. Spheroid and ellipsoid are used interchangeably in this article, with oblate implied if not stated. 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. 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, Newton had published in the Principia as a proof that the Earth was an oblate spheroid of flattening equal to 1/230. This was disputed by some, but not all, French scientists. A meridian arc of Jean Picard was extended to a longer arc by Giovanni Domenico Cassini and his son Jacques Cassini over the period 1684–1718. The arc was measured with at least three latitude determinations, so they were able to deduce mean curvatures for the northern and southern halves of the arc, allowing a determination of the overall shape. The results indicated that the Earth was a prolate spheroid (with an equatorial radius less than the polar radius). To resolve the issue, the French Academy of Sciences (1735) proposed expeditions to Peru (Bouguer, Louis Godin, de La Condamine, Antonio de Ulloa, Jorge Juan) and Lapland (Maupertuis, Clairaut, Camus, Le Monnier, Abbe Outhier, Celsius). The expedition to Peru is described in the French Geodesic Mission article and that to Lapland is described in the Torne Valley article. The resulting measurements at equatorial and polar latitudes confirmed that the Earth was best modelled by an oblate spheroid, supporting Newton. By 1743, Clairaut's theorem however had completely supplanted Newton's approach. By the end of the century, Delambre had remeasured and extended the French arc from Dunkirk to the Mediterranean. It was divided into five parts by four intermediate determinations of latitude. By combining the measurements together with those for the arc of Peru,ellipsoid shape parameters were determined and the distance between the Equator and pole along the Paris Meridian was calculated as 5130762 toises as specified by the standard toise bar in Paris. Defining this distance as exactly 10000000 m led to the construction of a new standard metre bar as 0.5130762 toises.:22