Meridian arc

In geodesy and navigation, a meridian arc is the curve between two points near the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length. Both the practical determination of meridian arcs (employing measuring instruments in field campaigns) and their theoretical calculation (based on geometry and abstract mathematics) have been pursued for many years.

Measurement

The purpose of measuring meridian arcs is to determine a figure of the Earth. One or more measurements of meridian arcs can be used to infer the shape of the reference ellipsoid that best approximates the geoid in the region of the measurements. Measurements of meridian arcs at several latitudes along many meridians around the world can be combined in order to approximate a geocentric ellipsoid intended to fit the entire world. The earliest determinations of the size of a spherical Earth required a single arc. Accurate survey work beginning in the 19th century required several arc measurements in the region the survey was to be conducted, leading to a proliferation of reference ellipsoids around the world. The latest determinations use astro-geodetic measurements and the methods of satellite geodesy to determine reference ellipsoids, especially the geocentric ellipsoids now used for global coordinate systems such as WGS 84 (see numerical expressions).

History of measurement

Early estimations of Earth's size are recorded from Greece in the 4th century BC, and from scholars at the caliph's House of Wisdom in Baghdad in the 9th century. The first realistic value was calculated by Alexandrian scientist Eratosthenes about 240 BC. He estimated that the meridian has a length of 252,000 stadia, with an error on the real value between −2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres). Eratosthenes described his technique in a book entitled On the measure of the Earth, which has not been preserved. A similar method was used by Posidonius about 150 years later, and slightly better results were calculated in 827 by the arc measurement method, attributed to the Caliph Al-Ma'mun.

Ellipsoidal Earth

Early literature uses the term oblate spheroid to describe a sphere "squashed at the poles". Modern literature uses the term ellipsoid of revolution in place of spheroid, although the qualifying words "of revolution" are usually dropped. An ellipsoid that is not an ellipsoid of revolution is called a triaxial ellipsoid. Spheroid and ellipsoid are used interchangeably in this article, with oblate implied if not stated.

17th and 18th centuries Although it had been known since classical antiquity that the Earth was spherical, by the 17th century, evidence was accumulating that it was not a perfect sphere. In 1672, Jean Richer found the first evidence that gravity was not constant over the Earth (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, Isaac 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) 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. However, by 1743, Clairaut's theorem had completely supplanted Newton's approach. By the end of the century, Jean Baptiste Joseph Delambre had remeasured and extended the French arc from Dunkirk to the Mediterranean Sea (the meridian arc of Delambre and Méchain). 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.

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