1.1e
describe how Eratosthenes made the first accurate calculation of the
circumference of the Earth
Eratosthenes lived from 276 – 194 BC. He found that in Aswan in southern Egypt
(then called Syene), on 21st June, when the Sun was at its highest point at noon, a
vertical drinking well reflected the light of the Sun straight back up the well. He
realised that to do this the Sun was directly overhead (at the ZENITH).
Eratosthenes lived in Alexandria and had noticed that the Sun cast a shadow at noon
on 21st June. He took note of the altitude of the Sun at noon on 21st June in another
year by setting up a vertical pole and measuring the angle of the shadow. He found
that Alexandria was just over 7˚ from the ZENITH (the angle between Alexandria and
Aswan represented 1/50 of the Earth).
Eratosthenes had measured the distance from Alexandria to Aswan and found it to
be 5,000 stadia (1 stadia = 185m).
Eratosthenes made 2 assumptions:
a). The rays of the Sun were parallel.
b). The Earth was spherical.
Eratosthenes used the following formula to work out the circumference of the
Earth:Circumference of the Earth
=
360
Distance between Alexandria and Aswan
Angle of shadow of vertical stick
Circumference of the Earth
5,000
=
Circumference of the Earth
=
5,000 x 50 = 250,000 stadia
=
250,000 x 0.185 = 46,250 km)
(In km, the circumference of the Earth
360
7.2
The actual circumference of the Earth from pole to pole is 40,008 km and so
Eratosthenes had achieved a very respectable result. One error in his measurement
was that Alexandria and Aswan were not on the same line of longitude.
ALEXANDRIA
ASWAN
1/50 of the Earth
in degrees
Centre of the
Earth’s core
Picture credit: NASA/JSC – Apollo 17 (Earth Observatory enhancement)
THE SUN IS 7.2˚ AWAY FROM THE ZENITH
(DIRECTLY OVERHEAD) AT ASWAN
ZENITH
ZENITH
ALEXANDRIA
ASWAN
7.2˚
ERATOSTHENES
REALISED THAT
THE 2 ANGLES
WERE THE SAME
DIRECTION OF THE SUN
(PARALLEL LIGHT RAYS)
7.2˚ = 1 EARTH
50
1 EARTH = 925 km
50
EARTH (A SPHERE)
The calculation for Eratosthenes becomes simply 50 x 925 = 46,250 km
The same calculation can be carried out from wherever you are observing. The
measurements need to be taken at the spring or autumn equinox – when the Sun is
known to be directly over the equator. You can use this fact to work out your own
value for the circumference of the Earth.
The process uses similar web pages as used in the production of a daylight length
graph through a year – as in section 1.4l. If a local newspaper publishes the sunrise
and sunset times for your area, these can be used instead of working through the
following two stages to find local sunrise and sunset times.
STAGE 1
You will need to know the latitude and longitude for your location. Use your mobile
phone, satnav, type into Google ‘your postcode lat long’, or click on the link below
to the website for ‘Satellite Signals’ after checking the details for using the website.
Using your postcode is best.

In the ‘Centre the map around box’, type in either your postcode or

the name of your city/town plus UK (if UK is missed out and there is
another place of the same name elsewhere in the world, you may
get the other option)

If your place-name is found elsewhere in the UK, add the county to
locate the correct place
Latitude and Longitude Finder
http://www.satsig.net/maps/lat-long-finder.htm
Website credit : Satellite Systems
(Make a note of the Latitude and Longitude)
STAGE 2
Data for a year can be obtained from the US Naval Observatory website using the
link below. You will arrive at the page:-
Sun or Moon Rise/Set Table
for One Year
US Naval Observatory
http://aa.usno.navy.mil/data/docs/RS_OneYear.html
Website credit : US Naval Observatory
Scroll down the page until you reach ‘Form B Locations Worldwide’.
I have included the information for Birmingham UK on the example following:Form B - Locations Worldwide
Specify year, type of table, and place:
Top of Form
Year:
Type of table:
Place Name: The place name you enter above is used only in the table header; you can enter any
identifier, or none (do not use punctuation characters).
Type in the latitude and longitude values for where your observations will take place:Longitude: Latitude: Time Zone: east west degrees minutes
north south degrees minutes
hours east of Greenwich west of Greenwich
Click the ‘Compute Table’ button
From the data that appears, select the equinox date that you require:-
YEAR
Spring or Vernal equinox
Autumn equinox
2012
20th March
22nd September
2013
20th March
22nd September
2014
20th March
23rd September
2015
20th March
23rd September
Read off the sunrise and sunset times and then take the midpoint of the two. This
will be the time for SOLAR NOON, when the Sun is at its highest point in the sky. As
the date is the equinox, the Sun will be directly above the equator. Check the angle
a shadow stick makes at SOLAR NOON and use the calculation that Eratosthenes
made to work out the circumference of the Earth.
To find the distance from your location to the equator, use an atlas or a globe to find
the value. Make sure that you measure along the same line of longitude down to the
equator. You can carry out a similar process and measure the angle of the shadow
stick at SOLAR NOON on the summer solstice. The Sun would be directly above the
Tropic of Cancer at 23.5˚N and so the distance to this point along the same line of
longitude would need to be found.