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FOLLOWING IN THE FOOTSTEPS OF ERATOSTHENES

MMeasureasuringing the circumference the circumference of Earthof Earth

Slavoljub Mitić, teacher

Primary School ‘’Bubanjski heroji’’

Niš, Srbija

ERATOSTHENES

Ερατοσθένης

276 BC - 194 BC

GreekGreek mathematicianmathematician, , geographer, geographer, astronomerastronomer

He lived in AlexandriaHe lived in Alexandria devised a simple way to measure devised a simple way to measure

the the circumference of circumference of the the EarthEarth

In Egypt, about 2200 years ago, a papyrus drew attention of a certain Eratosthenes, then Director of the Great Library of Alexandria (a town located on the side of the Mediterranean Sea): it was about a vertical stick which, on the first day of summer (that is to say on June the 21st) and at noon local solar time, did not cast any shadow on the ground (the Sun's rays reach the bottom of a well!). This happened very far from Alexandria, straight to the South, in a town called Syene (now Aswan). However, Eratosthenes noticed from his side that in Alexandria, on June the 21rst also and at the same time, a stick vertically driven in the ground did cast a shadow, even if such a shadow was relatively short.What the hell was this mystery?We invite you to discover it by yourselves. This will lead you pretty far since, as Eratosthenes showed, the key of this mystery will allow you to measure the circumference of the Earth, nothing less!

This text give to your studentsThis text give to your students

at noon Syene (now Aswan) - the Sun in vertical to the ground - in zenith, and the sun rays come to the bottom of the well, while the shadows of the vertical objects are only around them - the vertical objects do not cast the shadow Alexandria - the Sun is not in the vertical position and the objects cast a very short shadow

SyeneAlexandria

Why do the length of the shadows Why do the length of the shadows different, or why is it a shadow in different, or why is it a shadow in

one case and not in the other? one case and not in the other?

Eratosthenes used these starting Eratosthenes used these starting hypotheses hypotheses :: the Earth is flat the Sun should be close so the

objects of the same height have shadows of different length

1 2 3

The Earth is not flat, but has a curved

the Sun is far so the Sun rays are parallel while coming to the Earth

1

2

3

Eratosthenes used these starting Eratosthenes used these starting hypotheses hypotheses ::

Eratosthenes accepted the second Eratosthenes accepted the second hypothesishypothesis::

1

2

3

The Earth is not flat, but it is curved

the Sun is far a way so the Sun rays are parallel while coming to the Earth

Eratosthenes measured the length of the obelisk’s shadow, whose height he had known before.

According to the length of the shadow and height of the obelisk he calculated the angle which Sun rays form with the vertical.

The value of the angle is 7,20

Starting from the hypothesis that the Earth is spherical, he draw a

picture which can help him to calculate easily the circumference

of the Earth.

Alexandria

Syene

The extensions of the verticals in Alexandria (the obelisk) and in Syena (the well) intersect in the centre of the Earth .

The angle they form in the Earth’s centre is equal to the angle which Eratosthenes

measured with the shadow of the obelisk in Alexandria

7,20

Alexandria

Syene

angle 3600 – angle of the full circle

502,7

3600

0

the distance between Syena and Alexandria – 800km

the length of the circular curve corresponding to the angle of

7,20

kmkm 4000050*800 circumference of the Earth

shadow

= 7,20

d=800km

The Project Eratosthenes

2200 years later

The task

To measure the Earth meridian in the same way Eratosthenes did

that 2200 years ago

How to do it?

•Determine the local midday - at what time it is the noon in our town - at what time the Sun is in its zenith.•Measure the length of the shadow of vertical object at noon.•In cooperation with some other remote school calculate the circumference and the diameter of the Earth.

the noon is the moment when the Sun reaches the highest point in

the sky How to determine the real solar midday?

the shadow turns around and changes the length depending on the hour of a day

How to determine the real solar midday?• plant a stick into the ground and adjust it vertically by a plumbline or a level;

•late in the morning start measuring the length of the shadows;•being close to the noon, the shadow will be shorter and after the noon it will become longer and longer;•the shortest measured length will be the shadow at noon;

How to determine the real solar midday?

НИШ (Nish)latitude 43032’

longitude 22030’

the direction of the shortest shadow can be determined by a compass - in the relation to the bottom of the stick determine

the direction to the North

Constructing and application (usage) of the sundial - gnomon:

Material:•stick or a rod of 1 metre length;•a pedestal (a base);•a level, a protractor, a compass

at noon - measure the length of vertical object’s shadow

according to the length of the shadow and the height of the sundial (gnomon) determine the value of the angle

on a graph paper draw a minimized picture of gnomon and a shadow

connect the ends - you will get a right angle triangle

that measure the angle by a protractor;

Procedure:

to determine the value of the angle you can use these web sites

http://perbosc.eratosnoon.free.fr/spip.php?article191

http://isheyevo.ens-lyon.fr/eaae/groupspace/eratosthene/help-for-calculations/angle-of-the-

sun/

in cooperation with same other remote school calculate the

circumference and the diameter of the Earth

point A - the school is northwards (to the north)

point B - the school is southwards (to the south)

angle 1 - the angle measured at school which is in northwards

angle 2 - the angle measured at school which is in southwards

the angle which is necessary for calculation

= 1 - 2

1

2

1

А А

BB

if the school - partner in the project is situated in the southern hemisphere

the angles are added

= 1 + 2

12

2

1

d – distance between point А and В from north to south

that the results were more accurate distance should be as higher - at least 3 or 4 degrees of latitude

THE DISTANCE BETWEEN THE TOWNS

The schools are at the same meridian

The schools are not at the same meridian

the distance between the towns:

two schools are probably not at the same meridian,

you should determine the shortest distance between the parallels that go through the towns in which two schools are situated

according to the latitudes of the schools, determine the distance in a geographic map or write latitudes in the appropriate fields on these web sites and read the value of the distance

http://perbosc.eratosnoon.free.fr/spip.php?article187

http://isheyevo.ens-lyon.fr/eaae/groupspace/eratosthene/help-for-calculations/distance/