Demonstration

An experiment carried out by Eratosthenes over 2000 years ago.

Apparatus and Materials

• Pole, 2–3 m long

Health & Safety and Technical Notes

Read our standard health & safety guidance

Procedure

1. Two distant schools or colleges must co-operate. Preferably they should lie on a north-south line.
2. Set up a pole of known height at each site. Use a plumb line to ensure that the poles are vertical. The ground around them must be horizontal.
3. At noon on an agreed day, measure the length of the pole’s shadow at each site. Exchange details of the apparatus and measurements by phone, email or internet.

Teaching Notes

• To estimate the size of the Earth, the distance between the two sites must be known. This can be found from maps, or by using the co-ordinates of each location (which can be obtained from, for example, Google Maps) and a tool to calculate the distance between co-ordinates on the Earth’s surface, such as the one available on:

Ed Williams's Aviation website

• On a sketch of the Earth, continue the line of each pole down into the centre of the Earth as a radius. The angles produced at the poles, A and B, also give their latitude north of a place where the Sun’s rays are directly overhead (since a straight line intersecting two parallel lines has opposite angles equal).
• This means that the radius from each position meets at an angle C where C = B − A.


• The slanting lines of sunlight are parallel, assuming that the Sun is infinitely distant. Angles A and B can be calculated from the height of the pole and the length of the shadow.
tan angle = shadow lengthpole length
• Or a scale drawing could be made.
• The radius of the Earth is calculated from
sin C = distance between sitesradius of Earth
• An experiment of this type was first done by Eratosthenes (~235 BC). He used observations at Alexandria and at Syene, 500 miles further south. He knew that at noon on midsummer day, sun beams falling on a deep well there were reflected back from the water surface. This showed the Sun was directly overhead at Syene on that day. On that same day, he found the shadow cast by a tall obelisk in Alexandra's made 7 ½ ° with the vertical. His calculation of the Earth's circumference was within 5% of today's accepted value.

This experiment was safety-tested in July 2007

Class Practical

An estimate of the diameter of the Moon using a photograph of a lunar eclipse.

Apparatus and Materials

• Photograph of a partial eclipse of the Moon, as below.

Health & Safety and Technical Notes

Read our standard health & safety guidance

Procedure

1. Enlarge a photograph of a partial eclipse of the Moon. Preferably get students to take this photograph themselves, firmly securing the camera during the exposure to avoid camera shake.
2. On the photograph, estimate the radius of the shadow-bite on the Moon and the radius of the moon itself. The diameter of the moon can be calculated if the diameter of the Earth is known (as from the experiment

   The Moon’s distance from Earth

Teaching Notes

• The Greek method of estimating the distance of the Moon is to know the Moon’s diameter and then calculate that it is 110 Moon diameters away. See the experiment

  The Moon’s distance from Earth

• An early Greek method was to measure the time that the Moon spends in a total eclipse and work out from that the size of the Earth’s shadow in Moon diameters.

• Diagrams to illustrate eclipses are often drawn with incorrect proportions as in figure 1 which shows an eclipse of the sun and in figure 2 which shows an eclipse of the Moon. However that figure does show how the Moon’s shadow just tapers to a point by the time it reaches the Earth. We know that must be so because the Sun and Moon look the same size. Also, in an eclipse of the Sun, the Moon just manages to blot the Sun out.

• The true proportions are shown in figure 3. There you can imagine that you see the moon’s shadow tapering to nothing at the Earth and the Earth’s shadow also growing smaller with the same angle of taper.
• By watching a total eclipse of the Moon, the Greeks found that the Earth’s shadow is 2.5 Moon diameters wide out at the Moon (found by rough timing methods). The shadow has had the whole radius of the Moon’s orbit in which to taper. Knowing that such tapering makes a shadow lose one Moon diameter in that distance, you expect the Earth’s shadow to be 3.5 Moon diameters just behind the Earth. Therefore the Earth’s diameter is 3.5 Moon diameters.
• The Greeks already knew the diameter of the Earth, about 13,000 km and so they could then calculate the Moon’s diameter (3700 km). As the Moon is 110 Moon diameters away then the Moon’s distance from Earth is more than 400 000 km away.
• When an eclipse happens today then the length of time for an eclipse is given and could be used in the measurements. (For the purist, these times are arrived at by using information which we are trying to calculate.)

• This diagram illustrates why the true radius of the Earth is one moon diameter larger than its shadow of the Moon.
• From the value obtained from the Moon’s diameter from this experiment and knowing that the Moon is 110 Moon diameters away from the Earth then the distance of the Moon can be calculated.

This experiment was safety-tested in July 2007.

Class practical

An estimate of the ratio of the Moon's distance from the Earth compared to its diameter.

Apparatus and Materials

• Coin, small (e.g. penny)

Health & Safety and Technical Notes

Great care must be taken not to look directly at the Sun. Wear sunglasses or use a filter to reduce the Sun's glare.

Read our standard health & safety guidance

Procedure

1. Hold a small coin just beyond arm's length and adjust its distance from the eye until the disc of the coin just obscures the disc of the moon. A partner can then measure the distance from the eye to the coin.

Teaching Notes

• A British penny has a diameter of 2 cm so a clamp stand will be needed to support it. The coin will be approximately 110 coin-diameters away. So the Moon's distance is about 110 Moon-diameters away.
• A similar technique can be applied to the Sun, preferably when it is somewhat veiled by mist or cloud.
• By chance, the Sun looks about the same size as the Moon so the Sun is also about 110 Sun-diameters away. The Sun and the Moon both subtend an angle of about ½ ° at the Earth.
• Nowadays scientists measure how far the Moon is from the Earth with radar, by timing a pulse of radio waves to the Moon and back or by placing a reflector on the moon and timing a flash of laser light there and back. Greek astronomers were able to make measurements of the radius of the Earth and the distance of the Moon and Sun with reasonable accuracy 22 centuries ago with no radar, no radio time signals, and no telescopes and with only a small part of the world explored.

This experiment was safety-tested in July 2007}.