- Measurements according to Eratosthenes -
&
Germany
South Africa
- Introduction -
Well, okay. I hope the majority
of you still remember me.
I’m Jana and I was at Sans
Souci for two weeks in August!
Some of my classmates and I decided with our physics teacher Mr. Ahrens to take part in a
competition – but you have already heard about that.
In our contribution we wanted to show that it is possible to measure the circumference of
the earth and the distance between earth and moon with the help of only simple means.
That does not work using only Lippstadt. We needed somebody who is able to observe the
sun and the moon from a place on the southern hemisphere with nearly the same degrees
of longitude – and this is what you are!
At first we all would like to say thank you for your cooperation! Without your help it would
have been impossible for us to take part in this competition.
- Eratosthenes
Part I -
First we would like to explain to you how we found out the circumference of the earth with your
help. I will start with explaining to you why you should make all these measurements in the
moment of the sun´s culmination.
This procedure, I mean measuring the shadow of a vertical standing object at the moment of
highest level of the sun in order to find out the circumferences of the earth, was discovered by the
Greek Eratosthenes who carried out these measurements not in Cape Town or Lippstadt but in
Alexandria and Kyene (Egypt) approximately in 200 BC.
He found out that, on places with
different degrees of latitude
(Alexandria and Kyene), the sun at the
same day and the same time
culminates in a different angle to the
horizon.
It is important that the locations have
nearly the same degrees of longitude.
- Eratosthenes
Part II -
Eratosthenes observed that in Kyene on the 21st June the sun is standing in the zenith on
midday, which means that objects do not cast shadows. But at the same time the objects in
Alexandria cast a shadow, because the sun does not stand in the zenith there.
The sun´s position in Alexandria
The sun´s position in
Kyene
- Eratosthenes
Part III zenith
zenith
sun
sun
From the ratio between shadow and
length of the vertical standing object he
calculated an angle of about 82.2°
between horizon and sun in Alexandria,
which means that the sun is standing
7.2° away from the zenith.
Kyene
Alexandria
- Eratosthenes
Part IV -
The genius idea of Eratosthenes was
the following: When the difference of
the angles at the moment of the sun´s
highest point (7,2°) is 1/50 of a full
circle (360°), then the distance
between Alexandria and Kyene is also
1/50 of the circumference of the earth.
Distance between Alexandria and Kyene
Kyene
If
7,2° / 360° = 1 / 50
then
Distance between
Kyene and Alexandria
Circumference of the earth
1
=
50
Take a look at this drawing and you will
understand what we mean.
- Eratosthenes
Part V -
He just had to measure the distance between Alexandria and Kyene, which was rather difficult with a
caravan of camels at that time. After that he multiplied this distance of about 800 kilometres with 50
and he got the approximate quantity of the circumference of the earth.
- Lippstadt – Cape Town
Part I -
With your help we were able to do these
measurements as well. Admittedly neither
in Lippstadt nor in Cape Town the sun is in
the zenith, nevertheless you can apply this
method also to these two towns, since they
are approximately on the same degree of
longitude.
We measured the shadow of a stick and calculated
from the ratio between shadow and length of the
vertical standing object the angle between horizon
and sun in Lippstadt on different days. Probably
you have found out the angle in a similar way.
date
Here you can see some of us measuring
the shadow.
angle Lippstadt
angle Cape Town
30-03-04
42,7°
53,8°
31-03-04
43,8°
53,0°
- Lippstadt – Cape Town Part II -
The difference between the two angles is about 90°. And that means that the distance
between Lippstadt and Cape Town is approximately ¼ of the circumference of the earth!
We wanted to know more about it in detail: Using an atlas you see that the distance between
Lippstadt and Cape Town is about 9500 km. The difference of the angles on 30.03.04 is about
83,5°.
Like Eratosthenes we had to multiply the distance between Lippstadt and Cape Town with
83,5° / 360° and we got the quantity of the circumference R of the earth.
R = 83,5° / 360° • 9500 km = 40958
km
With the readings on 31.03.04 we got R = 41106
km.
The exact value of the circumference of the earth is 40074 km and that means that
our measurements are not too bad!
With this contribution we wanted to show that it is possible to make these
calculations without any technical aid. But you can do this only with the help
of friends living on the other hemisphere of the earth!
- Distance
Part I -
Now we would like to present the second part of our project which is how to find out the distance between
earth and moon. As far as we know, we are the first to measure the distance between earth and moon this
way!
First, look at this
drawing:
Two locations on the
earth are facing the
moon. If you lengthen
the lines of the scale
drawing in direction to
the moon until they
cut each other. With
the help of the
drawing you can find
out the distance
between earth and
moon.
- Distance
Part II -
At first you have to measure the angle to the moon, when the moon is at its highest point.
At the beginning you have to look for a completely horizontal plain. At the plain you have to assemble a
vertical standing rod. We suggest putting another pipe over this. With this construction you are able to
measure the angle between the horizon and the moon by angling the pipe directly at the moon.
Here you can see
Nicolas measuring
the angle between
moon and horizon.
- Distance
Part III -
If you do this in Cape Town and in Lippstadt you can measure how far the moon is away from the
earth. We drew the earth in ratio 1cm =1000 km. Then we drew Lippstadt and Cape Town in a
correct distance of the angles of 84,7° .
direction to the moon
Now we just had to draw a line in the direction of the moon, which we both have measured at the same
day. By the way, in Lippstadt the moon was in the south, because on the northern hemisphere the sun
and the moon are reaching their highest level exactly in the south. We were very excited, whether there
would be a day without any clouds in Lippstadt as well as in Cape Town and whether we really would be
able to measure at the same day.
- Distance
Part IV -
In fact, we do have readings of both towns on 01.04.2004. The angle between horizon and moon in
Lippstadt was 56,2°, whereas in Cape Town it was 35,7°. We drew these two angles on our earth on the
paper. The moon is there, where the two lines are cutting each other. We had to extend the lines very
carefully and exactly. At our measurement the threads cut each other in a distance of 3,1m. If you convert
this in the scale of 1 cm = 1000 km, you´ll get a distance of 310000 km between earth and moon.
On this picture you can see
how Huy and Angela are
trying to find out the point
between the two angles of
vision with the help of long
threads. For this they are
using a globe; but this
method is too inaccurate.
- Distance
Part V -
The distance between the earth and the moon is not always the same, since the moon´s movement
around the earth is elliptical. On average the distance is 384000 km, on 01.04.2004 it was 388152 km.
So the difference between our measurement and the „right“ number is 20%. Considering the fact that
we used only the most simple means to draw our conclusions, we think that our results are quite
remarkable. We wouldn´t have managed to get so far all by ourselves – no matter how exact our
measurements might have been.
So even in astronomy co-operation of the northern and the southern part of the world is of great value –
in our case it even was a necessity!
Thank you ever so much for your cooperation.
d
Without your help this measuring wouldn‘t have been possible!!!