The earth quadrant originally referred to for the definition of the meter between 1793 and 1799. On a spheroid, here specifically the earth ellipsoid, there are an infinite number of longitudinal quadrants, but they all have the same length

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.

 {\ displaystyle {\ begin {aligned} f & = {\ tfrac {1} {n}} \\ b & = a \, (1-f) \\\ varepsilon & = {\ sqrt {1 - {\ tfrac {b ^ {2}} {a ^ {2}}}}} \\ E (\ varepsilon) & = \ int \ limits _ {0} ^ {\ frac {\ pi} {2}} {\ sqrt {1- \ varepsilon ^ {2} (\ sin t) ^ {2}}} \ \ mathrm {d} t \\ Q_ {lon} & = a \, E (\ varepsilon) \ end {aligned}}} Using the example of the reference ellipsoid defined for WGS84, Sage is used to determine: ${\ displaystyle Q_ {lon}}$sage: a=6378137 sage: n=298.257223563 sage: f=1/n sage: b=a*(1-f) sage: e=sqrt(1-b^2/a^2) sage: qlon=a*elliptic_ec(e^2)  ${\ displaystyle Q_ {lon} = 10 \, 001 \, 965 {,} 729 \, 3127}$ ${\ displaystyle a \, ..}$ major semi-axis ${\ displaystyle b \, ..}$ small semi-axis ${\ displaystyle f \, ..}$ Flattening ${\ displaystyle n \, ..}$ inverse flattening ${\ displaystyle \ varepsilon \, ..}$ Numerical eccentricity ${\ displaystyle E (\ varepsilon) = E ({\ tfrac {\ pi} {2}}; k = \ varepsilon) \, ..}$Complete elliptical integral of the II. Type in Legendre form

## Historical values

Selection of length determinations of the longitudinal earth quadrant
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