Measuring the Size of the Earth

With this activity, children will learn how the ancient Greek mathematician and geographer Eratosthenes determined the circumference of the Earth. As a result, they will realise that - in contrast to an often repeated urban legend - the notion of a spherical Earth has been established knowledge for over two millennia. In addition, The activities herein involve children in a journey that follows the fundamentals of the scientific method, from hypothesis to proving a theory, by recreating the state of knowledge in Eratosthenes time, the ideas that led to his experiment and finally the simulation of the experiment in miniature.

After this activity, children will be able to:

• name the Greek scientist who determined the circumference of the Earth around 300 BCE.
• explain the method of measuring the circumference of the Earth Eratosthenes employed.

During the activity, the teacher could ask the children:

• When did mankind discover that the Earth is a sphere?
• Who measured its circumference first?
• How did he do this?
• Why does this method work?

The teacher may ask the children to discuss why the Earth cannot be flat.

• Copy of the map of Egypt (see below)
• 2 cylindrical small objects (~3 cm high, ~5 mm wide) or game figurines
• Paper glue, adhesive tape, scissors
• Bendable cardboard or plastic placemat, size A4 (to keep the map straight)
• UNAWE Earth ball (plain inflated)
• A strong lamp or flashlight
• String or elastic band (aprox. 1.3 m)
• 2 pieces of stiff cardboard or carton (each ~9 x 9 cm to be cut into circles)
• 2 small pieces of cardboard (2 x 2 cm)
• 2 equally sized, small, cylindrical or conical pieces or game figurines
• Bowl, preferably >26 cm diameter (to hold the UNAWE Earthball)
• Small ruler, tape measure

By observing the shape of the Earth’s shadow projected on the Moon during a lunar eclipse, the philosopher Aristotle from Athens inferred the spherical shape of the Earth around 400 BCE. Some 100 years later, a multitalented Greek astronomer, mathematician and geographer named Eratosthenes living in Alexandria figured out a way to measure the size of the Earth by working with only the shadows cast by a simple, vertical stick. While dedicated to geography, Eratosthenes tried to develop maps of the known world. But by dividing the world in a grid, mapping distances and drawing these maps, the results turned out to be distorted and seemed wrong. Therefore the shape and size of the Earth needed to be known for not only accurate map making, but also to find shortest travelling directions, physical distances (flat versus curved earth), the planning of commercial routes, planning for distant warfare and troop supply and finally for the sake of knowledge itself!

Eratosthenes figured out that, on a curved Earth at local midday with the sun being highest in the sky, shadows at different latitudes on the same longitude casts shadows with different lengths. While on a flat Earth, at any latitude on the same meridian at local midday, the Sun will cause shadows of equal length. Earth is curved is thus the theory now to put to test, i.e. which shadows are actually cast in the real world.

Part 1: Observing shadows Procedure Glue the two cylindrical objects onto the map (Fig. 16a), one on Alexandria, and one on the city of Aswan (Syene) in the south. NB: You can see the river Nile starting at the northern edge of the great lakes.

Use adhesive tape to fix the map onto the bendable support so it stays flat when handled. Ensure the light source is slightly elevated over the map, shining down.

Position yourself and the map at least 2 m away from the lamp and expose the map to the light. Try to have the map perpendicular to the light beam and north up. This simulates local midday, with the sun highest in the south (or north in southern hemisphere). The shadows of the two pins should point in the same direction and have the same length even if the map is rotated on its N/S axes simulating morning and afternoon. Make sure the children gather around map, so all can see the shadows cast by the small objects (pins).

Orientate the map so both pins cast shadows of same length and point in the same direction (preferable northward). Orientate the map so both pins cast almost no shadow. That would be local midday on a flat Earth on summer solstice.

Fig. 16a Shadows on the flat map (Credits: Grom Matthies)

Bend the map or wrap it on the UNAWE Earthball with N/S proper alignment. Adjust the orientation in a way that the shadows point in the same direction toward the nearest pole (again, this would be local noon for both locations). Note how now the shadows differ only in size. Let or lead students to recognise:
(a) Shadows on a flat Earth are of same size and orientation. (b) Shadows on a curved Earth (on same longitude meridian) point in the same direction but will have different lengths.

_Fig. 16b Shadows on the curved map (Grom Matthies)

Finally, assuming northern hemisphere, orientate the map/ball so that pins cast shadows north in the same direction. Then rotate over N/S until the southern pin over Syene has no shadow. You will see that the pin at Alexandria will have a shadow. That would be local midday at summer solstice on a curved Earth – this is the starting idea and configuration for the Eratosthenes experiment.

Fig. 16c Map reproducing the Eratosthenes observation with no shadows on Syrene (Credits: Grom: Matthies)

Part 2: Using shadows to measure the size of the Earth Preparations

Inflate the UNAWE Earthball. Snuggly fit or tighten the string/elastic band on the ball passing over both poles. This string/elastic represents a longitude meridian. Note: A string tends to work a bit better than elastics if you intend to reposition the meridian several times.

Fig. 17 UNAWE ball with the string around it (Credits: Grom Matthies)

Cut both cardboards into a round shape of about 6-9 cm diameter. They represent the landscape up to the horizon. Glue a small figurine/cylinder in the middle of this circular shape (Fig. 18). The central piece represents the observer, the gnomon or the whole school at your location.

Use a strip of adhesive tape (~7 cm) and glue one of the small cardboards in the middle of the tape. This will acts as a glider. Then, fix one side of the tape onto the back of one of the round cardboards. Pass the string/elastic between the small glider piece and the round cardboard and glue the loose end of the adhesive tape (Fig. 19).

Fig. 18 Cardboard representing the landscape up to the horizon

Fig. 19 Passing the string between the glider piece and the round cardboard (Credits: Grom Matthies)

Repeat this procedure for the second round cardboard and attach that one also to the string/elastic. Both cardboard horizons should easily now moveable along the string (up/down) and, on the UNAWE Earthball. NB: It may be of advantage to have a 3rd cardboard horizon prepared and at hand (not on the string/elastic), which simplifies the explanation of the concept of the horizon for the children.

Procedure This part of the activity is best done outdoors using the Sun, but works also indoors with the lamp/flashlight, as long as it is placed distant enough.

1. Using a tape measure, determine the circumference over the poles of the inflated UNAWE Earthball. Do not share this result yet with the students, as this is the desired result of this miniature version of the Eratosthenes experiment.

2. Place the UNAWE Earthball on a bowl, which will serve as a support to avoid that the ball moves on its own. Ensure that the UNAWE Earthball is steady in place but can be rotated and aligned with the light source.

Fig. 20 UNAWE ball on a bowl (Credits: Grom Matthies)

1. Have both mini-horizons in the same hemisphere. Orientate the ball so that both central pins cast a midday shadow (i.e. aligned with the pole to pole string) toward the nearest pole.

2. Now rotate over the poles, until the southern pin casts no shadow (Fig. 21).

Fig. 21 Orientating the UNAWE ball until the southern pin shows no shadow (Credits: Grom Matthies)

1. Let some or all students measure and write down the length of the remaining shadow. Note: Measurements need to start in the center of the pins. Thus the radius of the pin-gnomon needs to be added to the shadow length.

2. Let students measure the distance between both gnomons centers using a tape measure.

3. Measure the height of the center pins (they should be the same height) and tell this to the students.

4. Finish this experiment indoors, by applying the required math. For students without trigonometry, either you do the math or show how to use a calculator. You may consider preparing a spreadsheet that records all measurements, does the math and provides mean values too. Otherwise, write the math sequence onto the board for all to copy and apply to their measurements.

Students need to calculate the angle of the incident sunrays. If there is no shadow at one of the locations, the angle of incidence for sun rays arriving in parallel at the other place is equal to the geocentric angular distance between those locations.

Angular distance = angle of incidence = arctan ( (length of shadow) / (height of gnomon) )

If students work with 2 shadows, one on each gnomon, they have to calculate the angle of the sunrays of incidence for each location. Subsequently they should subtract the smaller from the bigger angle to get the UNAWE Earthball-centric angular distance between the gnomons:

Angular distance = (bigger angle) - (smaller angle)

Calculate the number of times that the distance between Alexandria and Aswan (we will call this a 'slice' in this assignment) can fit around the UNAWE Earthball.

Number of slices = 360° / (angular distance)

You can now find the circumference of the UNAWE Earthball by multiplying this number of slices times the distance between the two cities on the Earthball:

Circumference of UNAWE Earthball = (number of slices) * (distance between gnomons) NB: Be wary that students consistently use the same units of measure!

1. Compare and discuss the results between the students, as they will most likely be quite diverging.

Calculate the number of times that the distance between Alexandria and Aswan (we will call this a 'slice' in this assignment) can fit around the UNAWE Earthball

Number of slices = 360° / (angular distance)

You can now find the circumference of the UNAWE Earthball by multiplying this number of slices times the distance between the two cities on the Earthball:

Note of Caution It is very likely that the final result from students will deviate quite a bit from the circumference you measured and will also diverge amongst students. One reason is that the UNAWE Earthball is smaller than the circumference described by the highest piece on the mini-horizons. The main discrepancy between the resulting values though is induced by imprecisions in measurements and the setup itself. This also applies to a real life experiment measuring Earth.

Remark Even if the results for the circumference of the UNAWE Earthball may not be as near to the desired value as wished, they still are reasonable measures. The same applies to Eratosthenes and his experiment done over 2000 years ago. Nowadays it is still an amazing feat and idea to use a simple stick, stuck in the ground for a shadow and measure the size of the world.

Space Awareness curricula topics (EU and South Africa)

The journey of ideas, Geography