# Meridian arc: Map

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In Geodesy, a Meridian arc is a long measuring line in north-southern direction along the Earth's surface or at the reference ellipsoid. In Astronomy, the term describes a method to determine the Earth's radius (through the circumference) by combining the length of the terrestrial arc with astronomic latitude observations at the two end points.

Early estimations of Earth's radius are recorded from Egypt 240 BC, and from Bagdad califes in the 9th century. In modern times, the Academy of Paris adopted the method 1735–1739 to determine the Earth ellipsoid. The results were used for the definition of the Meter.

In the 19th century, many astronomers and geodesists were engaged in detailed studies of the Earth's curvature along different meridians to derive the exact Figure of the Earth. The analyses resulted, e.g., in the Bessel ellipsoid and the Hayford ellipsoid, in the Indian meridian arc of Everest and in the International ellipsoids of the 20th century.

Nowadays, scientists no longer use meridians, but rather astro-geodetic measurements and the methods of Satellite geodesy.

## The Meridian arc of Eratosthenes

The Alexandrian scientist Eratosthenes was the first who calculated the circumference of the Earth: He knew that on the summer solstice at local noon the sun goes through the zenith in the Ancient Egyptian city of Syene (Assuan). On the other way, he knew from his own measurement, that in his hometown of Alexandria the zenith distance was 1/50 of a full circle (7.2°) at the same time. Assuming that Alexandria was due north of Syene he concluded that the distance Alexandria-Syene must be 1/50 of the Earth's circumference. Using data of caravan travels, he estimated the distance to be 5000 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, i.e. 16 per cent too large. However, if Eratosthenes used the Egyptian stadion (157.5 m) his measurement turns out to be 39,690 km, an error of only 1%. But considering geometry and the ancient conditions, a 16% error is more reliable:

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 overland distance from traveling along the Nile or in desert couldn't be more accurate than about 10%.

Eratosthenes' estimation of the Earth’s size was accepted for hundreds of years afterwards. A similar method was used by Posidonius about 150 years later, and slightly better results were calculated 827 by the Gradmessung of the Calife al-Ma'mun.

## The French expedition to Peru and Lappland

In the 18th century (1735-1740), the Academy of Paris applied the method to a pair of arcs and to 4 instead of 2 latitude measurements. The scientists had in mind to determine the Earth ellipsoid, comparing the length of a meridian degree in the neighbourhood of the equator and in an arctic region.

The data showed a significant difference in curvature, which is much greater near the equator than near the poles. The mathematical Figure of the Earth could be derived as an oblate ellipsoid, as proposed by Isaac Newton a few decades before.

## Meridian arc along the Earth ellipsoid

Nowadays the length of a meridional arc of the Earth's ellipsoid can be calculated exactly by means of elliptic integrals.

::M=M(\phi)=\frac{(ab)^2}{((a\cos(\phi))^2+(b\sin(\phi))^2)^{3/2}};\,\!

::M_r=\frac{2}{\pi}\int_{0}^{90^\circ}\!M(\phi)\,d\phi\;\approx\left[\frac{a^{1.5}+b^{1.5}}{2}\right]^{1/1.5};\,\!

These pure geometric methods need representative value of the two axes of the ellipsoid which can be derived by adjustment methods from measurements all over the world. Geometrically, this is the determination of the mean curvature of the geoid which mainly depends on geographic latitude, but also on the regional topography and geology.

To derive the Earth's curvature along a meridian arc, we have to measure
1. the exact distance between the two end points of the arc
2. the geographic latitudes of both points, φs (standpoint) and φf (forepoint).

The latitude determinations are done by Astrogeodesy, observing the zenith distances of adequate stars. The surface distance Δ is reduced to mean sea level (Δ') and compared with the latitude differenceβ = |φsf|.This results in the mean radius of curvature R = Δ'/β.

If we know the radius R of a second meridian arc, we can derive the relation between R and the geographic latitude. This leads to the Earth's oblateness, resp. the two axes of the Earth ellipsoid. A meridian arc on an ideal surface of the Earth has the exact form of anellipse. If the arc starts at the equator, its length B to a point of latitude φ can be calculated by elliptic integrals or by a power series,

B = C · φ + D · sin(2φ) + E · sin(4φ) + F . sin (6φ) + ...

The first coefficient C depends on the mean Earth radius. At the Bessel ellipsoid (1856), C = 111,120 km per degree.The second coefficient D depends on the Earth's oblateness (the relative difference of the equatorial axis a and the polar axis b). At Bessel's ellipsoid it is D = 15,988 km. The values of other reference ellipsoids differ just at the 4th digit.

Since the 20th century, Geodesy does not use simple meridian arcs, but complex network with hundreds of fixed points.