GEOG 862
GPS and GNSS for Geospatial Professionals

Development of the Ellipsoidal Model


Eratosthenes Calculation
Source: Jan Van Sickle

The first step in representing positions on the Earth has always been the building of a model of the Earth.

We have been trying to grasp that nature of the planet's size and shape for a long time. In 200 BC, Eratosthenes almost got the planet's circumference right, and, of course, the fundamental idea that it was in fact nearly a sphere.  He knew that Syene was 489 miles south of Alexandria, the rope stretchers had measured it. In Syene at noon on midsummer there were no shadows and the sun was directly overhead and visible in the water at the bottom of a deep well.  However, on the same day, a pillar did cast a shadow in Alexandria.  The shadow formed an angle of 7.2 degrees. Therefore, Eratosthenes reasoned the angle between Alexandria and Syene at the center of the Earth had to be 7.2 degrees. That is 1/50th of 360 degrees.  It followed that the circumference of the entire planet was 50 times 489 miles, or ~24,450 miles.  This was an extraordinary deduction for 200 BC, and not so terribly far from right (~24,874 miles). Another leap forward occurred when Newton's thesis that the Earth was an ellipsoid rather than a sphere came in the first edition of his Principia in 1687.

Newton’s idea that the actual shape of the Earth was slightly ellipsoidal was not entirely independent. There had already been some other suggestive observations. 15 years earlier, astronomer Jean Richer had found that to maintain the accuracy of the one-second clock he used in his observations in Cayenne, French Guiana, he had to shorten its pendulum significantly. The clock’s pendulum, regulated in Paris, tended to swing more slowly as it approached the equator. Newton reasoned that the phenomenon was attributable to a lessening of the force of gravity. Based on his own theoretical work, he explained the weaker gravity by the proposition, “the earth is higher under the equator than at the poles, and that by an excess of about 17 miles” (Philosophiae Naturalis Principia Mathematica, Book III, Proposition XX).

Obviously, if the Earth is to be more massive at the equator, bulging out at the equator, the Earth is ellipsoidal. It is like a squashed ball that bulges out in the middle. Although Newton’s model of the planet was supported by some of his contemporaries, notably Huygens, the inventor of Richer’s clock, it was attacked by others. The director of the Paris Observatory, Jean Dominique Cassini, for example, took exception to Newton’s concept. Even though the elder Cassini had himself observed the flattening of the poles of Jupiter in 1666, neither he nor his equally learned son Jacques were prepared to accept the same idea when it came to the shape of the Earth. It appeared they had some empirical evidence on their side.

two early drawings of people collecting baseline measurements
Early Baseline Measurements

For geometric verification of the Earth model, scientists had employed arc measurements at various latitudes. Establishing the latitude of their beginning and ending points astronomically, they measured a north-south cardinal line to discover the length of one degree of longitude along a meridian arc. Early attempts assumed a spherical earth, and the results were used to estimate its radius by simple multiplication. In fact, one of the most accurate of the measurements of this type, begun in 1669 by the French Abbé J. Picard, was actually used by Newton in formulating his own law of gravitation. However, Cassini noted that close analysis of Picard’s arc measurement, and others, seemed to show the length of one degree of longitude actually decreased as it proceeded northward. He concluded that the Earth was not flattened as proposed by Newton, but was prolate, elongated at the poles.

Prolate Ellipsoid (Incorrect)
Source: Jan Van Sickle

The argument was not resolved until two expeditions between about 1733 and 1744 were completed. They were sponsored by the Paris Académie Royale des Sciences and produced irrefutable proof. One group which included Clairaut and Maupertuis was sent to measure a meridian arc near the Arctic Circle, 66°20' Nφ, in Lapland. Another expedition with Bouguer and Godin, to what is now Ecuador, measured an arc near the equator, 01°31' Sφ. Newton’s conjecture was proved correct, and the contradictory evidence of Picard’s arc was charged to errors in the latter’s measurement of the astronomic latitudes.

Oblate Ellipsoid (Correct)
Source: Jan Van Sickle

Eventually, these measurements proved Newton correct, and, in fact, the Earth is flattened at the poles or is an oblate ellipsoid.

Parameters of a Biaxial Ellipsoid, see text below
Source: GPS for Land Surveyors

The ellipsoidal model bulging at the equator and flattened at the poles, has been used ever since as a representation of the general shape of the Earth’s surface. In fact, several reference ellipsoids have been established for various regions of the planet. They are precisely defined by their semi-major axis and flattening.  The semi-major axis you see there in blue. It is half of the major axis of an ellipse. The semi-minor axis is half of the minor axis of the ellipse.

Source: GPS for Land Surveyors

The ellipse can be rotated about the north-south semi-minor axis into the representation of a solid ellipsoid, in this case, a bi-axial ellipsoid. It bulges at the equator and is flattened at the poles.