Meridian arc

Both the practical determination of meridian arcs (employing measuring instruments in field campaigns) as well as its theoretical calculation (based on geometry and abstract mathematics) have been pursued for many years.

[citation needed] Early literature uses the term oblate spheroid to describe a sphere "squashed at the poles".

In 1687, Isaac Newton had published in the Principia as a proof that the Earth was an oblate spheroid of flattening equal to ⁠1/230⁠.

To resolve the issue, the French Academy of Sciences (1735) undertook expeditions to Peru (Bouguer, Louis Godin, de La Condamine, Antonio de Ulloa, Jorge Juan) and to Lapland (Maupertuis, Clairaut, Camus, Le Monnier, Abbe Outhier, Anders Celsius).

The resulting measurements at equatorial and polar latitudes confirmed that the Earth was best modelled by an oblate spheroid, supporting Newton.

[7]: 52 The question of measurement reform was placed in the hands of the French Academy of Sciences, who appointed a commission chaired by Jean-Charles de Borda.

[9] Apart from the obvious consideration of safe access for French surveyors, the Paris meridian was also a sound choice for scientific reasons: a portion of the quadrant from Dunkirk to Barcelona (about 1000 km, or one-tenth of the total) could be surveyed with start- and end-points at sea level,[10] and that portion was roughly in the middle of the quadrant, where the effects of the Earth's oblateness were expected not to have to be accounted for.

[12] [13][14][15][16][17] The task of surveying the meridian arc fell to Pierre Méchain and Jean-Baptiste Delambre, and took more than six years (1792–1798).

The technical difficulties were not the only problems the surveyors had to face in the convulsed period of the aftermath of the Revolution: Méchain and Delambre, and later François Arago, were imprisoned several times during their surveys, and Méchain died in 1804 of yellow fever, which he contracted while trying to improve his original results in northern Spain.

Although Méchain's sector was half the length of Delambre, it included the Pyrenees and hitherto unsurveyed parts of Spain.

[19] Delambre measured a baseline of about 10 km (6,075.90 toises) in length along a straight road between Melun and Lieusaint.

[19] Thereafter he used, where possible, the triangulation points used by Nicolas Louis de Lacaille in his 1739-1740 survey of French meridian arc from Dunkirk to Collioure.

[20] Méchain's baseline was of a similar length (6,006.25 toises), and also on a straight section of road between Vernet (in the Perpignan area) and Salces (now Salses-le-Chateau).

[21] To put into practice the decision taken by the National Convention, on 1 August 1793, to disseminate the new units of the decimal metric system,[24] it was decided to establish the length of the metre based on a fraction of the meridian in the process of being measured.

[25][26][27] In 1799, a commission including Johann Georg Tralles, Jean Henri van Swinden, Adrien-Marie Legendre, Pierre-Simon Laplace, Gabriel Císcar, Pierre Méchain and Jean-Baptiste Delambre calculated the distance from Dunkirk to Barcelona using the data of the triangulation between these two towns and determined the portion of the distance from the North Pole to the Equator it represented.

Pierre Méchain's and Jean-Baptiste Delambre's measurements were combined with the results of the French Geodetic Mission to the Equator and a value of ⁠1/334⁠ was found for the Earth's flattening.

Another flattening of the Earth was calculated by Delambre, who also excluded the results of the French Geodetic Mission to Lapland and found a value close to ⁠1/300⁠ combining the results of Delambre and Méchain arc measurement with those of the Spanish-French Geodetic Mission taking in account a correction of the astronomic arc.

[29][26][13][30] The distance from the North Pole to the Equator was then extrapolated from the measurement of the Paris meridian arc between Dunkirk and Barcelona and was determined as 5130740 toises.

[20] In the 19th century, many astronomers and geodesists were engaged in detailed studies of the Earth's curvature along different meridian arcs.

The analyses resulted in a great many model ellipsoids such as Plessis 1817, Airy 1830, Bessel 1841, Everest 1830, and Clarke 1866.

Historically a nautical mile was defined as the length of one minute of arc along a meridian of a spherical earth.

Even though latitude is normally confined to the range [−⁠π/2⁠,⁠π/2⁠], all the formulae given here apply to measuring distance around the complete meridian ellipse (including the anti-meridian).

[45] In 1825, Bessel[46] derived an expansion of the meridian distance in terms of the parametric latitude β in connection with his work on geodesics, with Because this series provides an expansion for the elliptic integral of the second kind, it can be used to write the arc length in terms of the geodetic latitude as The above series, to eighth order in eccentricity or fourth order in third flattening, provide millimetre accuracy.

With the aid of symbolic algebra systems, they can easily be extended to sixth order in the third flattening which provides full double precision accuracy for terrestrial applications.

For WGS84 an approximate expression for the distance Δm between the two parallels at ±0.5° from the circle at latitude φ is given by The distance from the equator to the pole, the quarter meridian (analogous to the quarter-circle), also known as the Earth quadrant, is It was part of the historical definition of the metre and of the nautical mile, and used in the definition of the hebdomometre.