In Geodesy, a
Meridian arc is a long measuring
line in northsouthern 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
astrogeodetic 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 AlexandriaSyene 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 alMa'mun.
The French expedition to Peru and Lappland
In the 18th century (17351740), 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.
This was carried out in Ecuador and Lappland by Pierre
Bouguer, Louis Godin, Charles Marie de La Condamine,
Pierre Louis Maupertuis and
Antonio de Ulloa.
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
 the exact distance between the two
end points of the arc
 the geographic latitudes of both points, φ_{s}
(standpoint) and φ_{f} (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 = φ
_{s}φ
_{f}.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 an
ellipse. 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.
See also