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The Problem of the Earth's Shape

Simulation Applet

In the late 17th and early 18th century there was a debate about the shape of the Earth: In his "Principia", appearing in 1687, Isaac Newton described the Earth as flattened at the poles (left) by the force caused by the rotation of the earth.
French astronomers (Cassini e. a.) affirmed to have
proved the opposite by arc measurements, i. e. the Earth is elongated at the poles (right).

In order to resolve the disagreement, the French Academy of Sciences, authorized by King Louis XV, equipped two expeditions to perform measurements along a meridian arc at places as distant as possible: one to the equator (Peru), the other to the high North (Lapland).

In 1735 Louis Godin, Charles-Marie de La Condamine, and Pierre Bouguer, members of the
French Academy of Sciences, accompanied by Joseph de Jussieu, Jean Séniergue, Jean Godin des Odonnais, Jean Verguin,  three assistants (Couplet, de Morainville, Hugot), and the Spanish officers Jorge Juan and Antonio de Ulloa, sailed to Peru (Spanish colony since 1542, now Ecuador). In the high Andes they triangulated 3 degrees south of Quito with high precision, completing the measurements as late as 1744.
There are some fascinating books describing the adventures of the expedition. Four of the members did not survive it:
Couplet, Séniergue, Morainville, Hugot.
The Lapponian expedition, led by Pierre-Louis Moreau de Maupertuis, lasting one year only, returned in 1737, having triangulated 1 degree north of Tornio.

Flattening f = (a-b)/a = 0.3 = 1/3.33

The length of 1 degree of meridian arc subtends a longer distance in polar regions than near equato, and the geocentric latitude phi is smaller than the local (true) latitude PHI. At sea level:
tan(phi) = (b/a)2 tan(PHI)

The Results for 1° meridian arc measurements:
using 1 Toise = 1.949.036 m

 Toises km (a-b)/a Peru (0° - 3° S) 56766 110.64 1/327 Lapland (66° - 67° N) 57438 111.95 1/92 France 57078 111.25 1/387

The figures in the last column are computed using the modern value of a= 6378.14 km. The flattening adopted by the IAU in 1976 is f = 298.257.

Formula for the ratio of radii from Menke and Abbott:

Select "Data Window" from the "Details" menu:

 Applet Newtons "Principia" Book III, Propos. XX
From Newton's table I computed a value of 1/(232 +/- 17) for the flattening of the shape.
Assuming the Earth to be a rotating fluid in hydrostatic equilibrum the shape is approximately an ellipsoid: f = 1/299

Newton: 1+0.006579*[1-cos(2*PHI)]
Modern: 1+0.005056*[1-cos(2*PHI)]

 Louis Godin 1704 - 1760 Pierre Bouguer 1698 - 1758 Charles Marie de La Condamine 1701 - 1774 Source: Wikipedia Commons

The length of 1° determined by Google Earth compared with the applet values:

 Northern Latitude Google Earth Applet 60° - 61° 111.40 km 111.42 km 50° - 51° 111.15 km 111.24 km 40° - 41° 111.04 km 111.04 km 30° - 31° 110.80 km 110.86 km 20° - 21° 110.72 km 110.71 km 10° - 11° 110.53 km 110.61 km 0° - 1° 110.68 km 110.57 km

The circumference of the Earth: Eratosthenes and GPS

 Web Links French Geodesic Mission (Wikipedia) J. R. Greenberg: The problem of the earth's shape from Newton to Clairau (Google Books) Sir Isaac Newton's Principia (Open Library) A Earth Shape Applet (TU Clausthal) The Shape of the Earth The shape of Planet Earth Quelle est la forme de la Terre : plate, oblongue ou aplatie aux pôles ? The Figure of the Earth P. Mohazzabi, Mark C. James: Plumb line and the shape of the earth (PDF) Finding g for a Rotating Ellipsoid Length of Latitude, Longitude Calculator Books William Menke, Dallas Abbott: Geophysical theory, Columbia University Press, New York/Oxford 1990. Jean Meeus: Astronomical Algorithms, Willmann-Bell, Chapter 10: The Earth's Globe. Paul Murdin: Die Kartenmacher - Der Wettstreit um die Vermessung der Welt, Artemis & Winkler, Mannheim 2010. Paul Murdin: Full Meridian of Glory - Perilous Adventures in the Competition to Measure the Earth, Springer Science, 2009. Robert Whitaker: Die Frau des Kartographen .. und das Rätsel um die Form der Erde, Karl Blessing, München 2005. Robert Whitaker: The Mapmaker's Wife. A True Tale of Love, Murder and Survival in the Amazon, Basic Books, New York 2004. Barbara Gretenkord (Hrsg.): Reise zur Mitte der Welt. Die Geschichte von der Suche nach der wahren Gestalt der Erde, Thorbecke, Ostfildern 2003. Florence Trystram: Le procès des étoiles - Récit de la prestigieuse expédition de trois savants français en Amérique du Sud et des aventure qui s'ensuivirent (1735-1771), Payot, Paris 2001. Florence Trystram: Der Prozess gegen die Sterne - Abenteuer einer Südamerika-Expedition, Brockhaus, Wiesbaden 1981. Michael Rand Hoare: The quest for the true figure of the Earth: Ideas And Expeditions In Four Centuries Of Geodesy, Ashgate Publishing, 2005.

Updated: 2016, Aug 04