Earth quadrant
A longitudinal earth quadrant is the spheroidally idealized distance at sea level from the North Pole to the equator .
Determination at the time of the meter definition (s)
The French National Convention of 1793 set the length of the meter as the ten millionth part of the route from the North Pole — Paris — Equator . A quadrant of the earth measured half the length of the meridian of Paris , or a quarter of the longitude of Paris. The French Geodesy rendered middle of the 18th century empirical evidence for the poleward flattening of the earth. Approximately simultaneous measurements by de Lacaille and Cassini III. confirmed these findings. The results of both measurements were used for a prototypical, physical realization of the meter definition of the Convention in the form of a brass rod.
Before the introduction of the meter, the Toise measure of length was used for work that had to determine the length of and on meridians .
Further degree measurements before the beginning of the 19th century redefined the earth quadrant , which, according to the original definition, also varied the length of the meter. The brass rod was too long. In order to avoid that redeterminations of the earth quadrant by means of improved measuring instruments and mathematical processes repeatedly change the length of the unit to be defined, the meter , the history of the definition of the meter changed fundamentally. The ten millionth part of the earth quadrant according to the calculation of 1799 was cast as a platinum rod and the meter was henceforth defined as the length of this object.
This platinum rod is also called the definitive standard meter . It marks a historical turning point because of the change in the reference system of the meter definition, from the previously targeted earth quadrant towards (or back) to the length of a certain object. Previously, the Paris line and Toise had been defined that way.
Determination in meters after 1800
The redefinition of the parameters of the earth's ellipsoid with the aim of greater accuracy continued in the 19th and 20th centuries . The meter was defined and could gradually replace the toise as a measure of length. The conversion of the length data turned out to be difficult, because reference objects of different lengths existed for the Toise, but the symbol of the unit of measurement did not distinguish them.
In 1837 Bessel determined the earth quadrant to be 10,000,565,278 m long, for which he had data from ten different degree measurements. Their results were noted in Toise, but sometimes did not state which Toise you (?) They specifically referred to.
Bessel corrected the value in 1841 to 10,000,855.76 m and gave a mean error as a measure of the inaccuracy of ± 498.23 m. The correction calculation is also based on the assumption that all measured values relate to one and the same Toise reference rod. The reference ellipsoid that resulted from his compensation calculation between the degree measurement data is known today as the Bessel ellipsoid .
For the ellipsoid of the World Geodetic System 1984 (WGS84) used with GPS , the length of the longitudinal earth quadrant is approx. 10,001,966 m.
Formula for determination
Given a spheroid, the length of the longitudinal quadrant can be determined as follows.
Using the example of the reference ellipsoid defined for WGS84, Sage is used to determine:
sage: a=6378137
sage: n=298.257223563
sage: f=1/n
sage: b=a*(1f)
sage: e=sqrt(1b^2/a^2)
sage: qlon=a*elliptic_ec(e^2)


Historical values
year  reference  Length in meters 
Length in toise 
Length in Paris lines 
rel. Deviation from WGS84 in ‰ 

1793 
de Lacaille Cassini III. 
10,003,248.394  5,132,407, 407  4,434,400,000  0.128 
1799 
Delambre Méchain 
10,000,000  5,130,740, 740  4,432,960,000  0.197 
1837  Bessel  10,000,565.278  5,131,030.77  4,433,210,585  −0.140 
1841  Bessel  10,000,855.762 ± 498.23 
5,131,179.81 ± 255.63 
4,433,339,356 ± 220,863 
−0.111 
~ 1980  WGS84  10,001,965.729  5,131,749,305  4,433,831,400  0 
See also
Individual evidence
 ↑ This designation is preferable to the term “earth meridian quadrant”, since the term “ meridian ” itself is used in contradictions. Logically, the etymology of the word meridian  simply meaning "noon"  suggests that it is only half a great circle . However, a wrong custom often equates the meridian and the full longitude.
 ↑ ^{a } ^{b } ^{c} Mr. Geh. Rat and Ritter Bessel : Determination of the axes of the elliptical spheroid of rotation which corresponds most closely to the existing measurements of meridian arcs of the earth . In: HC Schumacher (ed.): Astronomical news . tape 14 , no. 333 . Hammerich and Lesser, Altona 1837, p. 343–344 ( pages 343–344 in Google Book Search).
 ↑ ^{a } ^{b } ^{c} Mr. Geh. Rat and Ritter Bessel : About an error in the calculation of the French degree measurement and its influence on the determination of the figure of the earth . In: HC Schumacher (ed.): Astronomical news . tape 19 , no. 438 . Hammerich and Lesser, Altona 1842, p. 97–116 ( pages 115–116 in Google Book search).
 ↑ The Bessel ellipsoid as an idealized earth body approximates the previously empirically recorded measurement data used by FW Bessel as closely as possible using the method of least squares , so that the weighted average distance of all measurement points to the surface of the ellipsoid is minimal. The ellipsoid is the result of the adjustment calculation.
 ↑ The Earth according to WGS 84; calculated by Sigurd Humerfelt. ( Memento of the original from April 14, 2011 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.
 ↑ 1 Toise = ^{864000} ⁄ _{443296} m = ^{27000} ⁄ _{13853} m ≈ 1.949 036 meters (by definition from 1799, see Toise # Toise_du_Pérou )
 ^ 1 Toise = 864 Paris lines
 ↑ 10,003,248.3938 m according to today's definition of the meter; 10,000,000 m according to the definition of the provisional original meter from 1793