Astronomy On-Line: How to measure the size of the EarthFrom: hpatel63
To: astronomyclubindia kutchastronomersclub
Sent: Friday, February 27, 2004 1:34 PM
Subject: FW : Astronomy On-Line How to measure the size of the Earth.htm


HOPE EVERY ONE FIND the SOLUTION
How to measure the size of the Earth
This project invites you to measure the circumference of the Earth, in a
collaboration with other Astronomy On-Line groups. To do so, you will have to
read carefully the instructions given here and then to contact other groups
which are interested in this type of project.

You may wish to contact actively those groups which are located more or less at
the same geographical longitude as your own. But this is not an absolute
condition.

You may also place a message about your interest in the Astronomy On-Line
Communications Archive. You may do so via the Marketplace (Group Communications'
Shop).

The measurement is not very difficult, and as long as the weather is not too bad
and you can see the Sun, you should be able to obtain quite accurate results.

The organisers shall be happy to hear about your experience and look forward to
your report(s). They will be brought in the Astronomy On-Line Newspaper.

Good luck!

From November 17, you may find the provisional report about this project here.

How big is the Earth?
Before Man started pondering over the question, it undoubtedly had been
necessary to realize first that Earth was spherical. This can easily be
understood during an eclipse of the Moon when one can see that the shadow cast
by Earth on the Moon is a portion of a disk.

Aristotle, the famous Greek natural philosopher, reports that mathematicians had
allegedly evaluated the dimension of Earth at 40.000 stadia, adding: `From their
supposition, it follows that the shape of Earth must be a sphere and also that
its size be small relative to the distance of other celestial bodies.'

It was generally agreed upon that measuring the size of Earth could be done by
measuring the altitude of a star from two cities situated on the same meridian.

Then, a difference expressed in degrees would be found. If the distance between
the two cities was known, from estimates by caravaneers for instance, it would
then be possible to find the value of a degree of meridian and hence derive the
value of the terrestrial circumference.

The stadium, Aristotle's unit length, apparently corresponds to 185 meters, so
the value of 74.000 km thus obtained is much too high. Archimedes, in his
treatise De Arenae Numero [On the number of sand grains] quotes a value of
300.000 stadia for the terrestrial circumference. This means that the
measurement must have been attempted several times.

Eratosthenes' measurement
Because he had been appointed Director to the Great Library at Alexandria by
Ptolemaeus III Evergetes, Eratosthenes had an access to innumerable sources of
knowledge.

He apparently made use of writings by Posidonius and reasoned thus:

a.. From his readings, he had learnt that once a year (on the day of the
Summer solstice), the bottom of a well situated at Syene in Upper Egypt was
illuminated by the Sun;
b.. However, at Alexandria, this never happened: obelisks always cast a
shadow;
c.. He believed that Earth was a sphere;
d.. He assumed that Alexandria and Syene were on the same meridian;
e.. He knew (or better, he assumed) that the distance between the two cities
was 5,000 stadia (as caravans covered the distance in 50 days at a rate of 100
stadia a day);
f.. He postulated that sunrays reached Earth as parallel beams (an idea that
was commonly held by the mathematicians of his time).
Information on ERATOSTHENES
a.. Born at Cyrene 275 BC
b.. Studied at Alexandria and Athens
c.. Appointed Director of the Alexandrian Library 236 BC
d.. Got blind 195 BC
e.. Starved himself to death 194 BC


So, on solstice day, he decided to measure the length of the meridian shadow
cast by a gnomon at Alexandria. He found a value of 1/50th of a circumference
(i.e. 7o 12') and derived the value of the terrestrial circumference: 50 x 5.000
= 250.000 stadia. Although our idea of the exact value of the stadium (which was
not the same at Athens, Alexandria or Rome) is fairly hazy, this puts the
terrestrial circumference at 40.000 km. The result is remarkable, although
several errors were introduced in the calculations:

a.. The distance between Alexandria and Syene is 729 km, not 800;
b.. The two cities are not on the same meridian (the difference in longitude
is 3o);
c.. Syene is not on the Tropic of Cancer (it is situated 55 km farther North);
d.. The angular difference is not 7o 12' but 7o 5'.
The most extraordinary thing is that the measurement rests on the estimated
average speed of a caravan of camels: one can certainly do better in the matter
of accuracy. Yet, in spite of all these flaws, it worked fine: around 250 BC,
Earth had at last a size.



Figure 1





Figure 2

Picard's measurement
The idea of measuring Earth kept running in the minds of scientists but there
was no improvement in the accuracy of the measurements until Galileo and the use
of the telescope for astronomical purposes. A few years later, a team of the
Royal Academy of Sciences in Paris decided to measure the value of the
terrestrial radius. Picard, who had been assigned the task, was to measure as
accurately as possible the linear distance between two points situated onthe
same meridian and whose latitudes differed by 1o. Then the distance that had
been measured would be multiplied by 360, thus yielding the value of the
terrestrial circumference.

The limits of the arc to be measured were 6 km from LaFert-Alais, a small city
North of Paris on one side and 20 km south of Amiens on the other side. The
problem was to use a unit length that would be accepted by everybody. Picard's
idea was quite clever: using the length of a pendulum oscillating seconds (mean
solar time). Unfortunately, he did not know that the length of such a pendulum
varies with latitude, which ruined all his efforts.

Anyhow, with a rigorous method and a concern for accuracy that remain exemplary,
he set to work finally publishing in 1671 a treatise of about 30 pages, entitled
Mesure de la Terre [The measurement of Earth]. The length of a degree of
meridian was set at 57.057 toises, i.e. between 111 and 112 km, corresponding to
a terrestrial radius of 6372 km.(1)

Earth had at last been measured with some more accuracy but there remained a lot
to discover.

A "revolutionary" Earth
The French Revolution burst out at the end of the XVIIIth century and it
established a new system of scientific education. Important decisions were made
regarding the units of measurements. To define the meter, the new universal
standard, it was decided to measure a part of a terrestrial meridian. The
adventurous task was led by Jean-Baptiste Delambre and Pierre Mechain, from 1792
to 1799, between Dunkirk (northern end) to Perpignan (southern end).

Similar missions had previously taken place:

a.. in Lapland, with Maupertuis, Clairaut, Camus and Lemonnier;
b.. in Peru, with Godin, Bouguer, la Condamine and one of the Jussieu
brothers.
Much later, on September 3, 1957, the Toronto Colloquium of the International
Association of Geodesy and Geophysics assessed the results of three centuries of
measurements:

a.. semi-major axis of the reference ellipsoid: 6 378 245 m
b.. polar flattening: 1/298.3
Since then, the problem has been dealt with the help of satellites, and as the
measurements became more and more accurate, it became more and more complex. But
this is another story!

Determinations of the polar flattening through the centuries
Newton: 1/230
Huygens: 1/578
1733 Cassini: -1/284
1737 Maupertuis: 1/178
1810 Delambre: 1/308
1841 Bessel: 1/299
1880 Clarke: 1/293
1924 Hayford: 1/297
1957 I.A.G. 1/298.257



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Figure 3: During an eclipse of the Moon, it is possible to evaluate the ratio
between the size of Earth's shadow and the size of the Moon (LUN207).


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Figure 4: Seen from space, Earth seems to be an ideal sphere, but an improvement
in the accuracy of the measurements has shown that this ideal shape was a mere
appearance. The polar flattening can hardly be noticed on the image, but
remote-sensing satellites can measure it (TER207).


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Figure 5: Thanks to remote-sensing satellites as ERS1, a new vision of Earth is
now available. They can monitor variations over more restricted zones as on this
view showing the Atlantic Ocean (ATER0439).


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The codes in the figure captions above correspond to references in the Geospace
Picture Library (address below). These images have been digitized for inserting
in a text file. Click on the images to see larger JPG-versions (but beware of
the sizes of those files (14k, 144k, and 70k, respectively)!


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Annotation:
(1) These values can be compared with present measurements, i.e.

- mean equatorial radius: 6378 km
- mean polar radius: 6357 km


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How two groups may measure the size of the Earth
Here is then how your group, together with another group, will be able to repeat
this fundamental measurement.

a.. Requisites
2 vertical poles of the same height (think of sports equipment's), one for
each site. 1 telephone set or any computer allowing an Internet login.

b.. Method
In two separate sites distant of at least a few hundreds of kilometers and
situated on the same meridian (say, for example: Lille and Montpellier in
France), on any given day and at the same time, the shadows cast by the Sun are
measured and the results are shared through a telephone or Internet link. If the
two cites are on a different meridian, the mathematics involved will be a little
more complicated.

c.. Calculation
As in the case of Eratosthene's experiment, there is very little math
involved. Angle A = angle (l) measured on one site - angle (m) measured on the
other site
= l - m





Figure 6

A represents a part of the terrestrial circumference. The only thing to do is
then to extrapolate, knowing the distance ML between the two sites.





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Please direct all enquiries and also send your results to:

Contact: Dr. Bernard Pellequer
National Representative of EAAE in France
Geospace, Observatoire d'Aniane
Tel: +33 (0)4 67 04 02 22 Fax: 33 (0)4 67 54 26 75
e-mail: bernard.pellequer@...
Website 1: www.eaae-france.org
Website 2: www.geospace-online.com/index_en.htm


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