Eratosthenes, a Greek mathematician and astronomer from Alexandria in the 3rd century BC, devised a simple method to measure the circumference of the Earth. He noticed that on the summer solstice in Syene (now Aswan), Egypt, the sun was directly overhead at noon, while in Alexandria farther north, it cast a shadow. By measuring the angle of the shadow and knowing the distance between the two cities, he was able to calculate the circumference of the Earth. The document describes how students can follow in Eratosthenes' footsteps by measuring shadows and angles at solar noon to calculate the circumference and diameter of the Earth.

1. FOLLOWING IN THE FOOTSTEPSFOLLOWING IN THE FOOTSTEPS
OF ERATOSTHENESOF ERATOSTHENES
Measuring the circumference ofMeasuring the circumference of
EarthEarth
Grammar School ‘’Svetozar Marković’’Grammar School ‘’Svetozar Marković’’
Niš, SrbijaNiš, Srbija

2. ERATOSTHENES
Ερατοσθένης
276 BC - 194 BC
Greek mathematician, geographer,Greek mathematician, geographer,
astronomerastronomer
He lived in AlexandriaHe lived in Alexandria
devised a simple way to measure thedevised a simple way to measure the
circumference of the Earthcircumference of the Earth

3. 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

5. at noonat noon
Syene (now Aswan)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
AlexandriaAlexandria - the Sun is not in the vertical
position and the objects cast a very short shadow

6. SyeneAlexandria

7. Why do the length of the shadowsWhy do the length of the shadows
different, or why is it a shadow in onedifferent, or why is it a shadow in one
case and not in the other?case and not in the other?

8. Eratosthenes used these starting hypotheses :Eratosthenes used these starting hypotheses :
the Earth is flat
the Sun should be close so the objects
of the same height have shadows of
different length
α1 α2 α3

9. The Earth is not flat, but has a curved
the Sun is far so the Sun rays are
parallel while coming to the Earth
α1
α2
α3
Eratosthenes used these starting hypotheses :Eratosthenes used these starting hypotheses :

10. Eratosthenes accepted the second hypothesis:Eratosthenes accepted the second hypothesis:
α1
α2
α3
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

11. 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

12. 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.
α
Alexandria
Syeneα

13. 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
α
Alexandria
Syeneα

14. angle 3600
– angle of the full circle
50
2,7
360
0
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
α
αα
α= 7,20
d=800km

15. The Project EratosthenesThe Project Eratosthenes
2200 years later2200 years later

16. The taskThe task
To measure the Earth meridian in the
same way Eratosthenes did that 2200
years ago

17. How to do it?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.

18. the noon is the moment when the Sun
reaches the highest point in the sky
How to determine the real solar midday?How to determine the real solar midday?
the shadow turns around and changes the length
depending on the hour of a day

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

20. •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?How to determine the real solar midday?

21. НИШ (Nish)
latitude 430
18’N
longitude 210
53’E

22. 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

23. Constructing and application (usage)Constructing and application (usage)
of the sundial - gnomon:of the sundial - gnomon:
Material:
•stick or a rod of 1 metre
length;
•a pedestal (a base);
•a level, a protractor, a
compass

25. 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:

26. to determine the value of the angle youto determine the value of the angle you
can use these web sitescan 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/

27. in cooperation with same other remotein cooperation with same other remote
school calculate the circumference and theschool calculate the circumference and the
diameter of the Earthdiameter of the Earth
point A - the school is
northwards (to the north)
point B - the school is
southwards (to the south)
angle α1 - the angle measured
at school which is in
northwards
angle α2 - the angle measured
at school which is in
southwards
the angle which is necessary for calculation
α = α1 - α2
α1
α2
α
α2
α1
α
α
А А
BB

28. if the school - partner in
the project is situated in the
southern hemisphere the
angles are added
α = α1 + α2
α1
α2
α2
α1

29. 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 TOWNSTHE DISTANCE BETWEEN THE TOWNS
The schools are at the
same meridian
The schools are not at
the same meridian

30. the distance between the towns: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

31. 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/

32. http://perbosc.eratosnoon.free.fr/spip.php?article187
http://isheyevo.ens-
lyon.fr/eaae/groupspace/eratosthene/help-for-
calculations/distance/
write latitudes in the appropriate fields on thesewrite latitudes in the appropriate fields on these
web sites and read the value of the distanceweb sites and read the value of the distance

33. dO
O
d
α
α
0
0
360
360
=
=
π
π
2
2
O
R
RO
=
=
The calculation:The calculation:
d - the distance between two towns;
O - the circumference of the Earth;
α - the calculated angle;
R - the radius of the Earth.

34. On march 2014, 25 classes of 13 countriesOn march 2014, 25 classes of 13 countries
have made measurements of shadows athave made measurements of shadows at
solar noon.solar noon.
http://www.eratosthenes.eu/spip/spip.php?http://www.eratosthenes.eu/spip/spip.php?
rubrique198rubrique198

35. MMarch 18arch 18thth
, 2014., 2014.

36. MMarch 18arch 18thth
, 2014., 2014.
Gimnazija ‘’Svetozar
Marković’’ Niš, Srbija
latitude longitude gnomon shadow angle (α1)
43.30
21.8830
100cm 96,5cm 440
Point St
Martin, Italy
Athens,
Greece
Ioannina,
Greece
Ghaziabad,
India
Kuantan,
Malaysia
latitude 45.60
N 38.0280
N 39.6670
N 28.6330
N 3.81°N
longitude 7.80
E 23.7260
E 20.850
E 77.4170
E 103.33°E
angle (α2) 46.30
38.50
40.50
29.80
3.40
angle
(α=α2- α1)
(α=α1- α2)
2.30
5.50
3.50
14.20
40.60
3600
/α 156.52 65.45 102.86 25.35 8.87
distance (d) 256km 586km 404km 1630km 4387km
O=d*(360/α) 40069.6km 38356.36km 41554.29km 41323.94km 38899.51km

37. MMarcharch 2020thth
, 2014., 2014.

38. Gimnazija ‘’Svetozar
Marković’’ Niš, Srbija
latitude longitude gnomon shadow angle (α1)
43.30
21.8830
100cm 93,5cm 43.10
MMarcharch 2020thth
, 2014., 2014.
Sault les
Rethel,
France
Gressonery
– St Jean,
Italy
Point St
Martin, Italy
Marina di
Carrara,
Italy
Kuantan,
Malaysia
Rosario,
Argentina
latitude 49.5°N 45.783°N 45.60
N 44.033°N 3.81°N 32.933°S
longitude 4.367°E 7.817°E 7.80
E 10.033°E 103.33°E 60.65°W
angle (α2) 49.20
45.70
45.40
43.80
3.70
32.90
angle
(α=α2-α1)
(α=α1-α2)
6.10
2.60
2.30
0.70
39.40
(α=α1+ α2)
760
3600
/α 59.02 138.46 156.52 514.29 9.14 4.74
distance (d) 689km 276km 256km 81km 4387km 8469km
O=d*(360/α) 40662.3km 38215.4km 40069.6km 41657.1km 40084.26km40116.32km

39. MMarch 21arch 21stst
, 2014., 2014.

40. MMarch 21arch 21stst
, 2014., 2014.

41. MMarch 21arch 21stst
, 2014., 2014.
Gimnazija ‘’Svetozar
Marković’’ Niš, Srbija
latitude longitude gnomon shadow angle (α1)
43.30
21.8830
100cm 94cm 43.20
Athens,
Greece
Ioannina,
Greece
HôChiMinhCit
y THPT,
Vietnam
Rio deJaneiro,
Brazil
latitude 38.0280
N 39.6670
N 10.7590
N 22.95°S
longitude 23.7260
E 20.850
E 106.6690
E 43.5°W
angle (α2) 37.70
39.40
10.90
20.10
angle
(α=α2- α1)
(α=α1- α2)
5.40
3.70
32.20
(α=α1+ α2)
63.20
3600
/α 65.45 102.86 25.35 8.87
distance (d) 586km 404km 3615km 7360km
O=d*(360/α) 38356.36km 38273.68km 40291.02km 41857.82km

42. http://www.fondation-lamap.org/en/node/9786
information and instructionsinformation and instructions
http://rukautestu.vin.bg.ac.rs/eratosten/
http://ciese.org/curriculum/noonday/
http://www.eratosthenes.eu/spip/
http://df.uba.ar/actividades-y-
servicios/difusion/proyecto-
eratostenes/informacion-conctacto-eratostenes

43. НИШ (Nish)
latitude 430
18’N
longitude 210
53’E