- See also the main article on '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. (276 BC– 195 BC)
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.
Eratosthenes' method for determining
the size of the Earth
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
Aryabhatiya. 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;)
[32381]. 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
[32382].
Islamic world
The Muslim
scholars who held to the spherical
Earth theory used it in an impeccably Islamic manner, to
calculate the distance and direction from any given point on the
earth to Mecca. This
determined the
Qibla, or Muslim direction of
prayer.
Muslim mathematicians
developed
spherical
trigonometry which was used in these calculations.
Around AD
830 Caliph al-Ma'mun commissioned a group
of astronomers to measure the distance from Tadmur (Palmyra) to al-Raqqah, in modern Syria. They
found the cities to be separated by one degree of latitude and the
distance between them to be 66 2/3 miles and thus calculated
the Earth's circumference to be 24,000 miles. Another estimate
given was 56 2/3 Arabic miles per degree, which corresponds to
111.8 km per degree and a circumference of 40,248 km, very close to
the currently modern values of 111.3 km per degree and 40,068 km
circumference, respectively.
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 a study of
map projections,
Cartography, which included a method for
projecting a
hemisphere on a
plane.
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".
Muslim astronomers and geographers
were aware of
magnetic
declination by the 15th century, when the Egyptian
Muslim astronomer 'Izz al-Din al-Wafa'i
(d.
1469/1471) measured it as 7 degrees from
Cairo.
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.
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 in 1699–70. 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. See the
illustration at right.
Cassini's ellipsoid; Huygens'
theoretical ellipsoid
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, 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".
See also
Notes
- Cleomedes 1.10. The Eratosthenes Nile map places the 1st
cataract (Syene) due south of the Nile delta. The delta's center is
actually more than 10° to the west of north of Syene.
- Cleomedes 1.10
- Strabo 2.2.2, 2.5.24; D.Rawlins, Contributions
- D.Rawlins (2007). " Investigations of the Geographical
Directory 1979-2007 "; DIO, volume 6,
number 1, page 11, note 47, 1996.
- David A. King, Astronomy in the Service of Islam,
(Aldershot (U.K.): Variorum), 1993.
- Gharā'ib al-funūn wa-mulah al-`uyūn (The Book of
Curiosities of the Sciences and Marvels for the Eyes), 2.1 "On the
mensuration of the Earth and its division into seven climes, as
related by Ptolemy and others," (ff. 22b-23a)[1]
- Edward S. Kennedy, Mathematical Geography, pp. 187-8,
in
- A. S. Ahmed (1984). "Al-Beruni: The First Anthropologist",
RAIN 60, p. 9-10.
- H. Mowlana (2001). "Information in the Arab World",
Cooperation South Journal 1.
- James S. Aber (2003). Alberuni calculated the Earth's
circumference at a small town of Pind Dadan Khan, District Jhelum,
Punjab, Pakistan. Abu Rayhan al-Biruni, Emporia
State University.
- Lenn Evan Goodman (1992), Avicenna, p. 31,
Routledge, ISBN
041501929X.
- David A. King (1996), "Astronomy and Islamic society: Qibla,
gnomics and timekeeping", in Roshdi Rashed, ed., Encyclopedia of
the History of Arabic Science, Vol. 1, p. 128-184 [153].
Routledge, London
and New York.
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
External links