# Meridian arc

As a meridian arc, a north- south running measuring distance on the earth's surface or its mathematical equivalent is on the Erdellipsoid referred (see Meridian ).

Former test section can be used in the "method of measurement of a degree " to determine the mean curvature of the earth and thus the radius of the earth. This also includes the latitudes of the two line endpoints ( φ1, φ2 ) must be measured. This width determinations are carried out astronomical, by the elevation of stars observed.

The system is then reduced to sea level and compared their length with the difference of latitudes. If the meridian arc length B and the width difference β the amount = | φ1 - φ2 |, we obtain the local radius of curvature R = B / β. May Together with a second arc of the meridian from the shape of the earth ellipsoid are derived - such as 1735-1740 when the famous expeditions of the Paris Academy to Lapland and Peru.

Since about 1900, however, extensive surveying networks are used in geodesy instead of the meridian method.

## Significant meridian arcs to 1900

The first meridian arcs of the history of science were demonstrated spherical earth's shape and size. After Geodesy of Eratosthenes here is the degree measure of the Caliph al-Ma'mun to mention a 2 ° long bow in Baghdad, for the meridian degree 122 km gave ( value today according to the latitude 111-112 km). Reported is the measurement of the meridian of Paris Fernel 1525 between Paris and Amiens by solar observations and cartwheel rotation counts. 1615 surveyed Willibrord van Roijen Snell a meridian arc between Alkmaar and Bergen- op-Zoom.

As a noticeable deviation from the spherical shape - ie the ellipsoidal figure of the earth - was to be expected, followed in the 18th and 19th centuries, several significant degree measurements, especially

- The French Geodesy in Lapland and Peru
- The meridian arc of Kremsmuenster in Upper Austria
- Further degree bends in Germany, Italy and Austria - Hungary, v. a Rome - Rimini ( Roger Boscovich 1751 ) and Brno - Varasdin (Wiener Meridian ) by Father Liesganig 1761
- The Delambre - Méchain arc from Dunkirk to Barcelona ( for the first time over 1000 miles), 1792 to 1798
- The Gaussian sheet Göttingen - Altona (1821-1823), the first time by means of strictly tutoring. adjustment
- The 2,500 km long Great Arc of India ( 1802 to 1841 as part of the Great Trigonometric Survey )
- The 3,000 km long Struve Geodetic Arc in Eastern Europe ( 1816-1852 ) and
- Its integration into other forms for the European Bessel ellipsoid
- The meridian arc Großenhain - Kremsmuenster - Pola (~ 1880-1910 ) of Saxony to the Adriatic
- And first degree measurements in the U.S. for the Hayford ellipsoid.

## Significant Meridian arches of the 20th century

From about 1900 besides astronomical width measurements or meridionally aligned surveying networks and length determinations were (ie east-west profiles) measured that differed from around 1910 gradually extended area networks - see Astro- geodetic network adjustment.

Of the modern meridian arcs are of particular importance:

- The arc of the meridian of Spitzbergen (measured from Russia & Sweden from 1898 to 1902 ). At 4.2 ° amplitude it is relatively short, but the northernmost usable for Geodesy arc
- The South American meridian arc of Colombia throughout Ecuador to northern Peru - 1899-1906 was the final measurement of the famous French arch of 17.
- The West European-African arc ( Paris meridian ) from the Shetland Islands to Laghouat in Algiers. He has 27 ° amplitude ( width difference ) and 38 stations
- The North American Latitude measurement in the 98 ° meridian of Mexico to the Arctic Ocean (1922 )
- And other Meridionalketten the U.S. and the 23 ° long, crooked bow along the east coast
- The African arc measurement in longitude 30 ° East, already initiated by Sir D.Gill, but only in 1953 completed. Course from Cairo to Cape Town, with 65 ° amplitude, the longest measuring profile of classical geodesy.

## Mathematical Description

A meridian arc on a spheroid has the exact shape of an ellipse. One can therefore its length - counted from the equator - calculated as the elliptic integral and represent φ as a series for functions of latitude:

The first coefficient C is related to the mean radius of the Earth and is the Bessel ellipsoid 111.120 km / degree. The second coefficient D is related to the Earth flattening and is 15.988 km. The values for other ellipsoids differ from the fourth position.

The development of means of eccentricity e2 are already Jean -Baptiste Joseph Delambre in 1799:

Friedrich Robert Helmert used in 1880:

General formulas were Kazushige Kawase 2009:

Said.