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 project n 3 experiment

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مُساهمةموضوع: project n 3 experiment   الأربعاء مايو 25, 2011 8:51 am

Who Was Eratosthenes?

Eratosthenes (276 BC-194 BC) was a
Greek mathematician, geographer and astronomer. He was born in Cyrene
(now Libya) and died in Ptolemaic Alexandria. He is noted for devising a
map system based on latitude and longitude lines and computing the size
of the Earth.

Eratosthenes studied at Alexandria and for some
years in Athens. In 236 BC he was appointed by Ptolemy III Euergetes I
as librarian of the Alexandrian library. He made several important
contributions to mathematics and science, and was a good friend to
Archimedes. Around 255 BC he invented the armillary sphere (an
astronomical instrument for determining celestial positions), which was
widely used until the invention of the orrery in the 18th century.

Circa 200 BC Eratosthenes is thought to have coined or to have adopted the word geography, the descriptive study of the Earth.

Eratosthenes' other contributions include:

The Sieve of Eratosthenes as a way of finding prime numbers.

The
measurement of the Sun-Earth distance, now called the astronomical unit
(804,000,000 stadia, 1 stadion varies from 157 to 209 meter).

The measurement of the distance to the Moon (780,000 stadia).

The measurement of the inclination of the ecliptic with an angle error of 7'.

He compiled a star catalogue containing 675 stars, which was not preserved.

A map of the Nile's route as far as Khartoum.

A map of the entire known world, from the British Isles to Ceylon, and from the Caspian Sea to Ethiopia.


Eratosthenes' Experiment

Eratosthenes
will always be remembered for the calculation of the Earth's
circumference circa 240 BC, using trigonometry and knowledge of the
angle of elevation of the Sun at noon in Alexandria and Syene (now
Aswan, Egypt). The calculation is based on the assumption that the Earth
is spherical and that the Sun is so far away that its rays can be taken
as parallel.

Details of his method he published in a work On
the measurement of the Earth which unfortunately was lost. We know
indirectly about his method from other authors.

Before we begin a few definitions:

Tropic
of Cancer - is one of five major circles of latitude that mark maps of
the Earth. The Tropic of Cancer currently latitude is 23° 26′ 22″ north
of the Equator.

Local noon is when the sun is the highest in the sky and can be quite different from 12:00 noon on the clock.

Solstice
- is an astronomical event that happens twice each year, when the tilt
of the Earth's axis is most inclined toward or away from the Sun. In the
northern hemisphere, the maximum inclination toward the sun is around
21 June (the summer solstice) and with the maximum inclination away
around 21 December (the winter solstice). For the southern hemisphere
winter and summer solstices are exchanged.

What matters for our
experiment is the fact that on the summer solstice, local noon, the sun
rays are just overhead (at a right angle to the ground) on the Tropic of
Cancer.


Eratosthenes' Experiment

Eratosthenes knew
that on the summer solstice at local noon on the Tropic of Cancer, the
Sun would appear at the zenith, directly overhead (sun elavation of 90°)
- though Syene was in fact slightly north of the tropic. He also knew,
from using a vertical stick and measuring the cast shadow, that in his
hometown of Alexandria, the angle of elevation of the Sun would be 83°
or 7° south of the zenith at the same time. Assuming that Alexandria was
due north of Syene - Alexandria is in fact on a more westerly longitude
- he concluded, using geometry of parallel lines, that the distance
from Alexandria to Syene must be 7/360 of the total circumference of the
Earth. The distance between the cities was known from caravan
travellings to be about 5,000 stadia. He established a final value of
700 stadia per degree, which implies a circumference of 252,000 stadia.
The exact size of the stadion he used is no longer known (the common
Attic stadion was about 185 m), but it is generally believed that
Eratosthenes' value corresponds to between 39,690 km and 46,620 km. The
circumference of the Earth around the poles is now measured at around
40,008 km. Eratosthenes result is not bad at all.

Very
interesting is that the measurement of the distance between Alexandria
and Syene is based on the estimated average speed of a caravan of camels
that traveled this distance(!). Camels traveled the distance many times
to obtain an average estimate of the distance. Whether this is true is
not clear.


Repeat Eratosthenes' Experiment

Eratosthenes
measured, at his local noon in Alexandria, the angle of elevation of
the sun on the summer solstice (21 June). Eratosthenes used the local
noon and no other time of the day since at local noon all relevant
places and sunrays are placed on the same imaginary plane enabling the
use of simple geometry for his calculations. In order to repeat
Eratosthenes’ experiment you’ll have to do the same.

First,
calculate your local noon because it may be quite different from 12:00
noon on the clock. There are several ways to compute its exact
occurrence. Basically, local noon is half-way between sunrise and
sunset. You can obtain sunrise and sunset times, for June 21, from your
local paper or from this link: http://aa.usno.navy.mil/data/docs/RS_OneDay.php
which also calculates local noon (sun transit). You can also obtain it
by yourself by using a sundial or find out when the shadow is the
shortest around noon time.

On June 21 erect a vertical straight
stick or pole of about 1 meter using a carpenter’s level and measure the
length of the shadow it casts at your local noon. With simple
trigonometry you can obtain the angle of the elevation of the sun. You
can also obtain the angle, without trigonometry, by drawing the stick
and shadow proportionally and measuring it with a protractor. You can
compare your results with a web based applet like this: http://www.jgiesen.de/azimuth
but be careful to use it correctly (insert your correct time zone,
local noon, coordinates, date and ensure that the dropdown menu points
to elevation).


After you get the angle of sun elevation, it’s
very easy to calculate the zenith angle by subtracting it from 90°,
like Eratosthenes did. Now you’ll have to measure the distance from your
location to the Tropic of Cancer latitude line - not by camel caravans
of course, the Eratosthenes way. You can use a relatively large scale
map, but take in account that maps tend to distort distance and the best
option is to use a globe. The distance from your location to the Tropic
of Cancer should be measured from north to south. In other words the
distance line has to cut the Tropic of Cancer at a right angle. There
are also web based calculators for this: http://facstaff.gpc.edu/~pgore/ISCI/earthcircumference.html.

Now it's easy to calculate the Earth circumference by using the following formula:


Likewise,
you can also perform this experiment on the winter solstice that takes
place around 21 December, but you’ll have to measure your distance from
the Tropic of Capricorn instead from the Tropic of Cancer because on
this date the sun reaches its highest degree of elevation on the Tropic
of Capricorn (23° 26′ 22″ south of the Equator).

It is also
possible to perform this experiment on the two Equinoxes which occur on
20 March and 23 September each year when the sun is crossing the equator
at the local noon on those dates and the sun rays are just overhead the
equator at a right angle to the ground. But instead to measure your
distance from the Tropic of Cancer or the Tropic of Capricorn you’ll
have to measure it from the equator.

There is another option and
you can perform this experiment on any other date of the year, at local
noon time, but you should have some partner located on your longitude
willing to measure sun elevation at the same time. Take in account that
you'll have to be a little careful treating correctly the sun angles
obtained in this case.

At any date the sun reaches its highest
position, at noon time, at some latitude. From here is clear that if the
two places involved are located on the same side of this latitude
(north or south) the shadows will be casted at the same direction and
the obtained angles should be subtracted from each other, whereas if the
places are located on different sides of this latitude the shadows will
be casted at different directions (southward or northward) and the
angles should be added up.

Further Reading

Links
Noon Day Project - Stevens Institute of Technology
The EarthDial Project: Eratosthenes experiment - The Planetary Society, Bill Nye
Eratosthenes Experiment: A Worldwide Science and Math Experiment - youth.net
How to Measure the Size of the Earth - Astronomy On-Line
Eratosthenes Finds Diameter of Earth! - Dennis P. Donovan, Rice University
Eratosthenes of Cyrene - Michael Lahanas
Measuring the Size of the Earth - D. Trapp, ie-Physics
Eratosthene's Diameter of Earth - John H. Lienhard
Eratosthenes of Cyrene - MacTutor
Ancient Measurements of the Circumference of the Earth - Livio C. Stecchini

Books
The Librarian Who Measured the Earth, Kathryn Lasky, Kevin Hawkes





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