# Degree measurement

Memorial plaque on the Großer Feldberg (Taunus)

As Gradmessung one is astronomical - geodetic method called from the 16th to the 20th century to measure the earth's shape was used (the earth ellipsoid). The name comes from the precise determination of the distance (110.6–111.7 km) that lies between two latitudes that differ by 1 ° .

## Methodology and first measurements

The method is based on the measurement of the curvature of the earth between distant points by comparing their distance ( arc length  B) with the angle β between their astronomically determined plumb lines . The quotient B / β gives the mean radius of curvature of the earth between these points. It is best to choose these two locations for the vertical direction measurement in north-south direction, so that β corresponds to the difference in their geographical latitude .

Earth measurement of Eratosthenes

The principle of measuring degrees goes back to the Alexandrian mathematician and library director Eratosthenes ; he estimated the circumference of the earth to be around 240 BC. Chr. From different by 7.2 ° Sun between Alexandria and Syene (now Aswan). His result of 250,000 stadiums hit  the true value at around 10 percent , depending on the exact length of the stadium used .

The method was refined in the early Middle Ages by the Arabs under Al-Ma'mun to an accuracy of 1–2%. In France , in 1525 , Jean François Fernel (1497–1558) obtained a mean radius of the earth (approx. 6370 km) from an arc of meridian north of Paris, with an accuracy of a few kilometers, and determined the distance from the revolutions of the wheel of his carriage. The Dutchman Willebrord van Roijen Snell (Snellius) determined the distances for the first time in 1615 using the triangulation of large triangles. In 1670, Jean Picard was the first to determine the Paris - Amiens meridional arc by triangulation with quadrants that had telescopic sights with crosshair eyepieces to target the stars. This achieved a level of precision that was previously impossible.

The subsequent lengthening of this meridian arc to Dunkirk and Perpignan at the beginning of the 18th century suggested a locally varying curvature of the earth, i.e. deviations from the spherical shape. The controversial question of that time, whether the earth's curvature to the pole decreases or increases and the earth is flattened towards the pole or ovoid , was only clarified by the French earth measurements in Lapland and Peru.

In the 20th century there was a change from profile to area networks and the regional curvature of the earth was determined through various geoid studies and cross-border projects. Since the practicality of GNSS , many measurements no longer refer to the true shape of the earth ( geoid ), but to a mean earth ellipsoid - which of course results in problems with height measurement .

## French geography Lapland-Peru

Because of contradicting results, the Paris Académie des Sciences equipped two large expeditions to Peru ( Charles Marie de La Condamine ) and Lapland ( Pierre-Louis Moreau de Maupertuis ).

The results of these measurements (1735–1740) were intended to define a new international measure of length in addition to the earth's ellipsoid - with exactly 10,000,000 meters from the equator to the pole.

Various problems with rust and calibration of the scales used (see Toise ), however, led to ellipsoid radii being shortened by 1 km (today's data indicate the meridian quadrant at 10,002,249 meters).

The flattening of the earth ¹ resulted with f = 0.0046 (instead of 0.00335), with which the shortening of the earth's radius to the poles (6378 ⇒ 6357 km) or the increasing radius of curvature (6335 ⇒ 6400 km) was proven for the first time:

country observer latitude G
(arc / degree)
Ecuador Bouguer et al. −01 ° 31 ' 56,734  Toisen 6,335.5 km
France¹ Jean Picard + 49 ° 13 ' 57,060 Toisen 6,371.9 km
Lapland Maupertuis + 66 ° 20 ' 57,438 Toisen 6,414 km.

¹) Cassini's  follow-up measurement in 1740 resulted in flattening  f = 0.00329

## Other important meridian arcs in the 18th – 20th centuries century

management year
Bošković , Lemaine 1751-1753 2 ° Rimini - Rome , for the first time compensation calculation
Joseph Liesganig 1761-1765 3 ° Brno - Vienna - Varasdin
Delambre , Méchain 1792-1798 9 ° Dunkirk - Paris - Barcelona
Gauss for Hanover 1821-1823 2 ° Göttingen - Altona
English triangulation 1784-1858 9 ° Shetland - Isle of Wight
India , Lambton , Everest 1802-1841 23 ° Himalayas - Cape Comorin
Struve , Tenner 1821-1852 25 ° Struve arch : Hammerfest - mouth of the Danube
European degree measurement 1867 international coordination (see below)
eastern USA ~ 1900 20 ° Meridians + inclined chains
Peru meridian 1899-1906 6 ° Colombia - Ecuador - Peru
Parisian meridian 1906 27 ° Shetland - Algiers
Berliner Mer., F.Hopfner 1922 7 ° Großenhain-Kremsmünster-Pola (crossing the Alps!)
western north america 1922 50 ° Arctic Ocean - Mexico , later extended to the south
South Africa , R.Schumann ~ 1925 25 ° Tanganyika - Cape Land , equator axis ± 100 m!
Africa 30 °, D.Gill ~ 1940 65 ° Cairo - Tanganyika - Cape Town
Japan ~ 1940 20 ° Kuril Islands - southern Japan

## Longitude measurements and subsequent networking

The degree measurement along meridians is easier to carry out because the astronomical work only requires latitude measurements . For precise continental projects, however, east-west profiles and measurements of geographical longitudes are also necessary - which, due to the need for precise time determinations, were only possible globally with radio time signals and precision chronometers :

continent year meridian
Europe 52 ° latitude 1895 69 ° Ireland - Germany - Poland - Urals
North America 39 ° latitude 1898 49 ° Atlantic - Pacific
USA / Mexico 19 °
Europe 48 ° Br., A.Galle 1923 19 ° Brest - Paris - Vienna - Astrakhan
India 24 °, R. Schumann ~ 1925 ~ 25 ° Panschab - Bengal
Australia south ~ 1930 40 ° including triangulation networks
Trans Siberia 51–52 ° ~ 1950 ~ 80 ° with Europe (52 °) 15,000 km profile
European network 40–60 ° 1951 ~ 90 ° Geoid determination to ± 1 m, H. Wolf

### International degree and earth measurements

Memorial stone on the Kiekeberg near Hamburg ( TP Vahrendorf 4/2525 or Vahrendorf station ), furnished by Heinrich Christian Schumacher .
Degree measurement pillars on the Trinity Mountain

The Central European Degree Measurement Commission was founded in 1862 on a German-Austrian initiative for the international coordination of the major projects mentioned . Its long-time leader was the Prussian general Johann Jacob Baeyer . It was expanded in 1867 to the European degree measurement and represents the forerunner of the international geodetic union IAG (1919), as well as today's geoscientific union IUGG .

Since around 1910 and 1940, the profiles in the north-south and east-west directions have no longer been observed or evaluated separately, but increasingly linked to form large surveying networks. The computational effort of such large-scale area networks and their compensation calculation increases enormously (with the 2nd to 3rd power of the number of points), but it is worthwhile due to higher accuracy and homogeneity. The first of these major projects involved the USA and Western Europe; The first networking of Eastern and Western European land surveys goes back to the “ Third Reich ”.

Since the 1970s and the development of IT , these area networks have also been combined with 3D measurements from satellite geodesy . As a result, the classic term of "degree measurement" is incorporated into that of " earth measurement ".

### Reference and earth ellipsoids

In national surveys , the individual states defined their own “geodetic date” ( reference system ) up to around 1850 . With the international extension and networking of the above-mentioned degree measurement profiles, the possibility and the desire to base the individual areas on large-scale valid data developed. The result was a series of so-called reference ellipsoids , which, with increasing expansion, approached the "mean earth ellipsoid ".

Of the approximately 200 national surveying networks around the world, over 90% are now based on data from a dozen large-area ellipsoids, which increases their quality and facilitates international cooperation. The older of these ellipsoids are based on the large meridian arcs of the 2nd section, the newer ones arose from intercontinental and satellite networks. The most important of these ellipsoids are:

Earth ellipsoid major axis a in meters minor axis b in meters 1 / flattening f
GB Airy 1830 6,377,563.4 m 6,356,256.91 299,3249646
Everest (India) 1830 6,377,276,345 300,8017
Bessel 1841 6,377,397.155 6,356,078.965 299.1528128
Clarke 1866 6,378,206,400 (partly Asia) 294.9786982
Clarke 1880 / IGN 6,378,249.15 293,465 (466)
Australian Nat. 6,378,160,000 298.25
Boarding school 1924 Hayford 6,378,388,000 (publ. 1909) 297.0
Krassowski 1940 6,378,245,000 298.3
Boarding school 1967 Lucerne 6,378,165,000 (for the first time + satellites) 298.25
SAD69 (South America) 6,378,160,000 (partly Astrogeoid / Mercury) 298.25
WGS72 (World Geodetic System 1972) 6,378,135,000 298.26
GRS 80 geographic reference system 6,378,137,000 (approx. = WGS 84) 298.257222¹
WGS84 (World Geodetic System 1984) 6,378,137,000 6,356,752,315 298.257223563

For many countries of Central Europe that is Bessel ellipsoid important also the ellipsoids of John Fillmore Hayford and Krasovsky and GPS survey the WGS 84 .

The pioneering work of Jean-Baptiste Joseph Delambre is based only on local measurements. On the other hand, the big difference between the ellipsoids of Everest (Asia) and Hayford (America) arises from the geologically determined geoid curvature of the two continents.