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How two groups may measure the size of the Earth @
http://http.hq.eso.org/outreach/spec-
prog/aol/market/collaboration/erathostenes/

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:

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;
However, at Alexandria, this never happened: obelisks always cast a
shadow;
He believed that Earth was a sphere;
He assumed that Alexandria and Syene were on the same meridian;
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);
He postulated that sunrays reached Earth as parallel beams (an idea
that was commonly held by the mathematicians of his time).

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:

The distance between Alexandria and Syene is 729 km, not 800;
The two cities are not on the same meridian (the difference in
longitude is 3o);
Syene is not on the Tropic of Cancer (it is situated 55 km farther
North);
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.

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:

in Lapland, with Maupertuis, Clairaut, Camus and Lemonnier;
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:

semi-major axis of the reference ellipsoid: 6 378 245 m
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!

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.

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.

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.

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

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.