# Earthly Configurations

October 28, 2002
How scientists sought to determine if the Earth was flat.

Through the ages, scholars have attempted to determine the size and shape of the Earth. Many in the profession believe that Christopher Columbus first discovered that the Earth was round in 1492. However, centuries before Columbus, the early Greeks theorized the Earth’s surface to be something other than flat. The early Greek mathematician Pythagoras believed the sphere was the most perfect figure; therefore, the gods must have created the Earth in the shape of a sphere. Scholars such as Plato and Archimedes supported the spherical Earth theory and came up with surprisingly close estimates of the Earth’s circumference.

Another Greek scholar named Eratosthenes measured the sun directly overhead on the day of the summer solstice in the town of Syene. Further north in the town of Alexandria, it was also known on that day that the sun cast a shadow from the vertical that represented 1/50th of a circle (7?12'). The measured distance from Syene to Alexandria was a known value in stadia units, which worked out to be approximately 500 statute miles. From this, Eratosthenes extrapolated the Earth’s circumference by multiplying 50 (the 1/50th circle) by 500 miles=25,000 miles. The current accepted value of the Earth’s equatorial radius based on the World Geodetic System (WGS) is 24,901 miles.

Figure 1. Elements of an ellipse.

## The Earth is an Ellipsoid

A renaissance of astrometric ideas, methods and measurements occurred around the 17th century. The emergent field of geodesy, defined as the mathematical determination of the size and shape of the Earth, was the holy grail of astronomers and physicists. Improvements of telescopes, the introduction of logarithmic tables and triangulation methods enabled scientists to make remarkable discoveries of the shape of the Earth. One important discovery was the Earth taking on an elliptical cross sectional shape. In the early 18th century, Isaac Newton (1642-1727), by theory, predicted the Earth to be an oblate sphere, i.e., a sphere taking on a flattening aspect at the polar regions and a bulging mass at the equator.

In 1735-6, the Acad?e Royale De Sciences in Paris sent out two geodetic expeditions to measure the length of one degree of latitude along a meridian of longitude. The rationale behind this was to compare the arc length of a degree of latitude measured near the equator and near the north pole, whereby the size of the Earth and its flattening could be ascertained. The equatorial expedition led by Louis Godin, accompanied by Pierre Bouguer (1698-1758) went to Mitad del Mundo, Peru, to measure the meridional arc by astronomic observations. Bouguer was to measure the Earth’s density and therefore its gravitational variations by observing the deflection of the plumb bob from the vertical. In 1736, the polar expedition led by Pierre-Louis Moreau de Maupertuis (1698-1759) went to the Tornio valley in Lapland to measure its arc. Maupertuis used a special astronomic instrument having a removable telescope mounted on a trunnion axis, which later was the basic design for John Bird’s transit of equal altitude used by Mason and Dixon in their famous survey of the Maryland-Pennsylvania boundary.

The results of both expeditions were compared by the Acad?e Royale De Sciences, which declared that the Earth had an equatorial radius of 3,282,350 toises (6,996,172 yards), and a flattening ratio of 1/216.8. This proved that the length of an arc distance of one degree of latitude near the North Pole was longer than one near the equator (see Figure 1 below and Figure 2 on page 51). These values were used in North America for over 100 years until the British geodesist, Ross Clarke (1824-1914) developed his famous Clarke 1866 Spheroid for North America (1/294.98 flattening), which was used in the North American Horizontal Datum of 1927 (NAD27). In 1986, NAD83 was adopted based on the Geodetic Reference System (GRS), and is now the legal standard North American horizontal datum used in the United States.

Other countries around the world made great contributions to geodesy as well. In 1800, Great Britain sent surveyors to India to begin the longest measurement of the surface of the Earth at that time. Led by William Lambton, and later, George Everest, they undertook the impossible task of measuring the Great Indian Arc of the Meridian, or colloquially, “The Great Arc.” From the southern tip of India to the Himalayas, the Great Arc was the first accurate survey of the Indian subcontinent and the Himalayan mountain range. George Everest was honored for this great achievement, which led to the discovery of the world’s highest mountain by having it named after him. Taking nearly 50 years, the 1,600-mile survey used a precise theodolite weighing 1/2 ton, which had to be mounted on top of tall towers and mountain peaks for astronomical observations and triangulation. Through jungle and high-altitude blizzards, the Great Arc mapped the entire Indian subcontinent describing modern day India and producing ellipsoidal values of the Earth still used in that region today.

## Connecting Datums of the World

By 1940, nations making great strides in the area of technology developed their own geodetic systems. These data were non-geocentric; i.e., the center of a local datum ellipsoid was not coincident with the center of the Earth. Using the Clarke 1866 ellipsoid in North America and orienting it by astro-geodetic methods, NAD27 based its non-geocentric center of mass at triangulation station Meade’s Ranch in Kansas to minimize the difference in the height of the ellipsoid and geoid. Other countries used their local ellipsoid to fit their portion of the Earth; consequently, each geodetic datum was incompatible with an adjoining countries’ datum. Significant errors resulted in connecting one continent to another.

United States national security demands for both intermediate and long-range defensive weapons systems and the increase of military distance requirements created the need for a global datum. During the late 1950s, the United States Department of Defense (DoD)—the leading proponent for intercontinental geodetic requirements developed the World Geodetic System of 1960 (WGS 60) to tie these relative data together allowing DoD mapping agencies global compatibility with widely separated sites of interest. The WGS 60 model used a combination of astrogeodetic and gravimetric data, LORAN (LOng RAnge Navigation) and incorporated the first use of satellite geodesy to yield a best-fit ellipsoid model based on satellite orbital geometry and an Earth-centered orientation for each individual datum.

In January 1966, the U.S. Army, Navy and Air Force headed up a World Geodetic System committee, which coordinated re-observations on the WGS 60 adjustment data, and used satellite geodesy in the form of Doppler and optical satellite data. In addition, the WGS 66 geoid model was produced. Implemented in 1967, WGS 66 served its purpose for five years.

In 1972, WGS 72 was developed from the first unified WGS solution, which was a large-scale least square adjustment. Both optical and electronic satellite data were used. The primary satellite data were derived from U.S. Navy and non-DoD agency Doppler observations. The U.S. Army also provided satellite data using the SECOR (Sequential Collation of Range) Equatorial Network. The Worldwide Geometric Satellite Triangulation Program provided optical data using BC-4 ballistic cameras. Some laser ranging was done as well.

As military and non-military geodetic communities were in constant demand for better accuracy, a new World Geodetic System was needed because WGS 72 couldn’t provide sufficient data coverage. The Geodetic Reference System of 1980 (GRS 80), consequently WGS 84, was developed, which coincides with the true center of the Earth whose ellipsoid surface yields an average fit of the geoid. Instead of the 100 geodetic stations used in WGS 72, WGS 84 used over 1,000 Doppler-determined station positions as well as incorporating Very Long Baseline Interferometry (VLBI), satellite radar altimetry, and dynamic Doppler for determining geoid heights, and an advanced least square adjustment system that takes into account gravity anomalies, such as deflection of the vertical. WGS 84 is the current Earth-centered Cartesian coordinate system used by the Global Positioning System (GPS) today.

Figure 2. One degree of latitude near the Equator and North Pole (approximate values).

## Applications Using the Ellipsoid

Geodetic latitude and longitude are derived from algorithms utilizing the parameters, a, b and f, which define the ellipsoid (see Figure 1 on page 49). This system can then be accurately transformed to Cartesian coordinates used in GPS, which is necessary for the computation of satellite orbits. There are some user problems associated with WGS 84 and national datums. NAD83, which is now based on WGS 84, is a metric system; using the wrong conversion factors from meters to feet can result in wrong state plane coordinate values.

With the recent explosion in the use of spatial data, the need for accurate mapping is critical. In this modern, global economy, users of GIS, GPS and other remote sensing platforms are realizing the importance of tying global data sets together. Many municipalities require plats tied to geodetic monuments. Surveys carried out in a “local” system are often no longer acceptable to users of spatial data except in small-scale projects. Now the surveyor is called on to reference his or her plat to a national datum or projection.

It is interesting that early geodesists and surveyors derived ellipsoid parameters that closely approximate modern values. With the advent of satellite geodesy, great strides continue to be made in the attempt to define a global, Earth-centered datum that links local and regional datums together.