Definitions

# History of geodesy

Humanity has always been interested in the Earth. During very early times this interest was limited, naturally, to the immediate vicinity of home and residency, and the fact that we live on a near spherical globe may or may not have been apparent. As humanity developed, so did its interest in understanding and mapping the size, shape, and composition of the Earth.

## Hellenic world

Early ideas about the figure of the Earth held the Earth to be flat, and the heavens a physical dome spanning over it. Two early arguments for a spherical earth were that lunar eclipses were seen as circular shadows which could only be caused by a spherical Earth, and that Polaris is seen lower in the sky as one travels South.

The early Greeks, in their speculation and theorizing, ranged from the flat disc advocated by Homer to the spherical body postulated by Pythagoras — an idea supported one hundred years later by Aristotle. Pythagoras was a mathematician and to him the most perfect figure was a sphere. He reasoned that the gods would create a perfect figure and therefore the earth was created to be spherical in shape. Anaximenes, an early Greek scientist, believed strongly that the earth was rectangular in shape.

Since the spherical shape was the most widely supported during the Greek Era, efforts to determine its size followed. Plato determined the circumference of the earth to be 400,000 stadia while Archimedes estimated 300,000 stadia, using the Hellenic stadion which scholars generally take to be 185 meters or 1/10 of a geographical mile. Plato's figure was a guess and Archimedes' a more conservative approximation. Meanwhile, in Egypt, a Greek scholar and philosopher, Eratosthenes, is said to have made more explicit measurements.

He had heard that on the longest day of the summer solstice, the midday sun shone to the bottom of a well in the town of Syene (Aswan). Figure 1. At the same time, he observed the sun was not directly overhead at Alexandria; instead, it cast a shadow with the vertical equal to 1/50th of a circle (7° 12'). To these observations, Eratosthenes applied certain "known" facts (1) that on the day of the summer solstice, the midday sun was directly over the Tropic of Cancer; (2) Syene was on this tropic; (3) Alexandria and Syene lay on a direct north-south line. Legend has it that he had someone walk from Alexandria to Syene to measure the distance: that came out to be equal to 5000 stadia or (at the usual Hellenic 185 meters per stadion) about 925 kilometres.

From these observations, measurements, and/or "known" facts, Eratosthenes concluded that, since the angular deviation of the sun from the vertical direction at Alexandria was also the angle of the subtended arc (see illustration), the linear distance between Alexandria and Syene was 1/50 of the circumference of the Earth which thus must be 50×5000 = 250,000 stadia or probably 25,000 geographical miles. The circumference of the Earth is 24,902 miles (40,075.16 km). Over the poles it is more precisely 40,008 km or 24,860 statute miles. The actual unit of measure used by Eratosthenes was the stadion. No one knows for sure what his stadion equals in today's units, but most current specialists in antiquities accept that it was the regular Hellenic 185 meter stadion, and few if any would incline to an obscure definition that happened to make Eratosthenes's result correct.

Had the experiment been carried out as described, it would not be remarkable if it agreed with actuality. What is remarkable is that the result was probably about one sixth too high. His measurements were subject to several inaccuracies: (1) though at the summer solstice the noon sun is overhead at the Tropic of Cancer, Syene was not exactly on the tropic (which was at 23° 43' latitude in that day) but about 22 geographical miles to the north; (2) Syene lies 3° east of the meridian of Alexandria; (3) the difference of latitude between Alexandria (31.2 degrees north latitude) and Syene (24.1 degrees) is really 7.1 degrees rather than the perhaps rounded (1/50 of a circle) value of 7° 12' that Eratosthenes used; (4) the actual solstice zenith distance of the noon sun at Alexandria was 31° 12' − 23° 43' = 7° 29' or about 1/48 of a circle not 1/50 = 7° 12', an error closely consistent with use of a vertical gnomon which fixes not the sun's center but the solar upper limb 16' higher; (5) the most importantly flawed element, whether he measured or adopted it, was the latitudinal distance from Alexandria to Syene (or the true Tropic somewhat further south) which he appears to have overestimated by a factor that relates to most of the error in his resulting circumference of the earth.

There is some cause to question the reality of the legendary "experiment". First, pacing the distance would be physically intimidating, across plenty of desert since the Nile isn't linear. Second, a traveller from Alexandria near the west extreme of the Nile delta would have had to veer on average over 20° east of due south to hit Syene, a nonsubtle conflict with Eratosthenes's reported experiment which put Syene directly south of Alexandria . Third, if the Hellenic stadion is assumed for Hellenic Eratosthenes, the resulting 250,000 stadia (later given as 252,000 for divisibility) is pretty close to the overlarge size of the earth one would find by simple mathematics and enormously less travel, through measuring a sea horizon's angular dip as seen from a known height, since the computational result will be about 6/5 of the correct result (1/5 too high) due to atmospheric refraction which for horizontal light is 1/6 of the curvature of the earth.

A parallel later legendary ancient measurement of the size of the earth was made by another influential Greek scholar, Posidonius. He is said to have noted that the star Canopus was hidden from view in most parts of Greece but that it just grazed the horizon at Rhodes. Posidonius is supposed to have measured the elevation of Canopus at Alexandria and determined that the angle was 1/48th of circle. He assumed the distance from Alexandria to Rhodes to be 5000 stadia, and so he computed the earth's circumference in stadia as 48 times 5000 = 240,000 . Some scholars see these results as luckily semi-accurate due to cancellation of errors. But since the Canopus observations are both mistaken by over a degree, the "experiment" may be not much more than a recycling of Eratosthenes's numbers, while altering 1/50 to the correct 1/48 of a circle. Later either he or a follower appears to have altered the base distance to agree with Eratosthenes's Alexandria-to-Rhodes figure of 3750 stadia since Posidonius's final circumference was 180,000 stadia, which equals 48×3750 stadia . The 180,000 stadia circumference of Posidonius is suspiciously close to that which results from another unlaborious method of measuring the earth, by timing ocean sun-sets from different heights, a method which produces a size of the earth too low by a factor of 5/6, again due to horizontal refraction.

The abovementioned larger and smaller sizes of the earth were those used by Claudius Ptolemy at different times, 252,000 stadia in the Almagest and 180,000 stadia in the later Geographical Directory. His midcareer conversion resulted in the latter work's systematic exaggeration of degree longitudes in the Mediterranean by a factor close to the ratio of the two seriously differing sizes discussed here, which indicates that the conventional size of the earth was what changed, not the stadion.

## Ancient India

The great Indian mathematician Aryabhata (AD 476 - 550) was a pioneer of mathematical astronomy. He describes the earth as being spherical and that it rotates on its axis, among other things in his work Aryabhatia. Aryabhatiya is divided into four sections. Gitika,Ganitha (mathematics), Kalakriya (reckoning of time) and Gola (celestial sphere). The discovery that the earth rotates on its own axis from west to east is described in Aryabhatiya (Gitika 3,6; Kalakriya 5; Gola 9,10;) For example he explained the apparent motion of heavenly bodies is only an illusion (Gola 9), with the following simile;

Just as a passenger in a boat moving downstream sees the stationary (trees on the river banks) as traversing upstream, so does an observer on earth see the fixed stars as moving towards the west at exactly the same speed (at which the earth moves from west to east.

Aryabhatiya also estimates the circumference of Earth, accurate to 1% which is remarkable. Aryabhata gives the radius of planets in terms of the Earth-Sun distance as essentially their periods of rotation around the Sun. He also gave the correct explanation of lunar and solar eclipses and that the Moon shines by reflecting sunlight

## Medieval Persia

The medieval Persian geodesist Abu al-Rayhan al-Biruni (973-1048) is sometimes regarded as the "father of geodesy" for his significant contributions to the field. John J. O'Connor and Edmund F. Robertson write in the MacTutor History of Mathematics archive:

At the age of 17, al-Biruni calculated the latitude of Kath, Khwarazm, using the maximum altitude of the Sun. Al-Biruni also solved a complex geodesic equation in order to accurately compute the Earth's circumference, which were close to modern values of the Earth's circumference. His estimate of 6,339.9 km for the Earth radius was only 16.8 km less than the modern value of 6,356.7 km. In contrast to his predecessors who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, al-Biruni developed a new method of using trigonometric calculations based on the angle between a plain and mountain top which yielded more accurate measurements of the Earth's circumference and made it possible for it to be measured by a single person from a single location.

By the age of 22, al-Biruni had written several short works, including a study of map projections, Cartography, which included a method for projecting a hemisphere on a plane. Biruni's Kitab al-Jawahir (Book of Precious Stones) described minerals such as stones and metals in depth, and was regarded as the most complete book on mineralogy in his time. He conducted hundreds of experiments to gauge the accurate measurements of items he catalogued, and he often listed them by name in a number of different languages, including Arabic, Persian, Greek, Syriac, Hindi, Latin, and other languages. In the Book of Precious Stones, he catalogued each mineral by its color, odor, hardness, density and weight. The weights for many of these minerals he measured were correct to three decimal places of accuracy, and were almost as accurate as modern measurements for these minerals.

Around 1025, al-Biruni was the first to describe a polar equi-azimuthal equidistant projection of the celestial sphere. He was also regarded as the most skilled when it came to mapping cities and measuring the distances between them, which he did for many cities in the Middle East and western Indian subcontinent. He often combined astronomical readings and mathematical equations, in order to develop methods of pin-pointing locations by recording degrees of latitude and longitude. He also developed similar techniques when it came to measuring the heights of mountains, depths of valleys, and expanse of the horizon, in The Chronology of the Ancient Nations. He also discussed human geography and the planetary habitability of the Earth. He hypothesized that roughly a quarter of the Earth's surface is habitable by humans, and also argued that the shores of Asia and Europe were "separated by a vast sea, too dark and dense to navigate and too risky to try" in reference to the Atlantic Ocean and Pacific Ocean.

## Medieval Europe

Revising the figures attributed to Posidonius, another Greek philosopher determined 18,000 miles as the earth's circumference. This last figure was promulgated by Ptolemy through his world maps. The maps of Ptolemy strongly influenced the cartographers of the Middle Ages. It is probable that Christopher Columbus, using such maps, was led to believe that Asia was only 3 or 4 thousand miles west of Europe. It was not until the 15th century that his concept of the earth's size was revised. During that period the Flemish cartographer, Mercator, made successive reductions in the size of the Mediterranean Sea and all of Europe which had the effect of increasing the size of the earth.

## Scientific revolution

The invention of the telescope and the theodolite and the development of logarithm tables allowed exact triangulation and grade measurement.

Jean Picard performed the first modern arc measurement. He measured a base line by the aid of wooden rods, used a telescope in his angle measurements, and computed with logarithms. Jacques Cassini later continued Picard's arc northward to Dunkirk and southward to the Spanish boundary. Cassini divided the measured arc into two parts, one northward from Paris, another southward. When he computed the length of a degree from both chains, he found that the length of one degree in the northern part of the chain was shorter than that in the southern part. Figure 2.

This result, if correct, meant that the earth was not a sphere, but an oblong (egg-shaped) ellipsoid -- which contradicted the computations by Isaac Newton and Christiaan Huygens. Newton's theory of gravitation predicted the Earth to be an oblate ellipsoid flattened at the poles to a ratio of 1:230.

The issue could be settled by measuring, for a number of points on earth, the relationship between their distance (in north-south direction) and the angles between their astronomical verticals (the projection of the vertical direction on the sky). On an oblate Earth the distance corresponding to one degree would grow toward the poles.

The French Academy of Sciences dispatched two expeditions. One expedition under Pierre Louis Maupertuis (1736-37) was sent to Lapland (as far North as possible). The second mission under Pierre Bouguer was sent to what is modern-day Ecuador, near the equator (1735-44).

The measurements conclusively showed that the earth was oblate, with a ratio of 1:210. Thus the next approximation to the true figure of the Earth after the sphere became the oblong ellipsoid of revolution.

In South America Bouguer noticed, as did George Everest in the 19th century Great Trigonometric Survey of India, that the astronomical vertical tended to be "pulled" in the direction of large mountain ranges, obviously due to the gravitational attraction of these huge piles of rock. As this vertical is everywhere perpendicular to the idealized surface of mean sea level, or the geoid, this means that the figure of the Earth is even more irregular than an ellipsoid of revolution. Thus the study of the "undulations of the geoid" became the next great undertaking in the science of studying the figure of the Earth.

## 19th century

In the late 19th century the Zentralbüro für die Internationale Erdmessung (that is, Central Bureau for International Geodesy) was established by Austria-Hungary and Germany. One of its most important goals was the derivation of an international ellipsoid and a gravity formula which should be optimal not only for Europe but also for the whole world. The Zentralbüro was an early predecessor of the International Association for Geodesy (IAG) and the International Union of Geodesy and Geophysics (IUGG) which was founded in 1919.

Most of the relevant theories were derived by the German geodesist F.R. Helmert in his famous books Die mathematischen und physikalischen Theorien der höheren Geodäsie (1880). Helmert also derived the first global ellipsoid in 1906 with an accuracy of 100 meters (0.002 percent of the Earth's radii). The US geodesist Hayford derived a global ellipsoid in ~1910, based on intercontinental isostasy and an accuracy of 200 m. It was adopted by the IUGG as "international ellipsoid 1924".

## References

• An early version of this article was taken from the public domain source at http://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003A.HTM#ZZ4.
• J.L. Greenberg: The problem of the Earth's shape from Newton to Clairaut: the rise of mathematical science in eighteenth-century Paris and the fall of "normal" science. Cambridge : Cambridge University Press, 1995 ISBN 0-521-38541-5
• M.R. Hoare: Quest for the true figure of the Earth: ideas and expeditions in four centuries of geodesy. Burlington, VT: Ashgate, 2004 ISBN 0-7546-5020-0
• D.Rawlins: "Ancient Geodesy: Achievement and Corruption" 1984 (Greenwich Meridian Centenary, published in Vistas in Astronomy, v.28, 255-268, 1985)
• D.Rawlins: "Methods for Measuring the Earth's Size by Determining the Curvature of the Sea" and "Racking the Stade for Eratosthenes", appendices to "The Eratosthenes-Strabo Nile Map. Is It the Earliest Surviving Instance of Spherical Cartography? Did It Supply the 5000 Stades Arc for Eratosthenes' Experiment?", Archive for History of Exact Sciences, v.26, 211-219, 1982
• C.Taisbak: "Posidonius vindicated at all costs? Modern scholarship versus the stoic earth measurer". Centaurus v.18, 253-269, 1974

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