# The Eratosthenes-inator

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## Introduction: The Eratosthenes-inator

I am a high school astronomy teacher. Every year I have a few students tell me that they think the Earth is flat. I'm pretty sure they are joking, but I genuinely appreciate those students because they provide a great stepping off point for a discussion about how science works and the importance of evidence-based claims.

If I just explain to students how we know the Earth is round, it comes down to whether or not they believe me versus someone on the internet.

Also, the tricky part about astronomy is that the objects and phenomena we study are very big and/or very far away and/or unfold over very long stretches of time. Making astronomy a hands on class is challenging.

I wanted to come up with a direct and hands-on way for students to observe the evidence that supports the claim that the Earth is round. Enter...

... the Eratosthenes-inator!

A device that will...

1) demonstrate evidence used to support the claim that the Earth is a sphere

2) re-create the method used to measure the circumference of the Earth

3) demonstrate the way stars can be used to determine one's location on a spherical Earth

A quick note about the name ...

• Eratosthenes was a Greek scientist who first measured the circumference of the Earth using the method we will use. Here's a link to more information about him.
• the "-inator" is a nod to the great scientist and engineer Dr. Heinz Doofenshmirtz of Phineas and Ferb.

### Supplies:

• 1 of ... 2x6 8’
• 1 of ... 12” x 42” white tile board (.25” thick) ... comes in 4'x8' sheets
• 8 of ... 1.25” drywall screws
• 10 of ... #8x1” flat head wood screws
• 10 of ... #8 finish washers
• 2 of ... 4.5” hex head bolts
• 2 of ... neodymium disc magnets (5mm diameter)

Tools & building supplies

• butcher paper (for making cut templates)
• band saw or jigsaw
• drill

### Teacher Notes

Teachers! Did you use this instructable in your classroom?
Add a Teacher Note to share how you incorporated it into your lesson.

## Step 1: Making the Curved Sides Template

1. On a piece of butcher paper, draw a line 1 meter long
2. Draw two intersecting arcs 1 meter from the endpoints of the 1-m line.
3.  Draw an arc across the original 1 meters line, centered on the intersection in step 2
4. Cut out the enclosed arc
5. tape the arc to the 2x6 and use it to sketch the curve

The maximum height of the curved arc will be the same as the width of the 2x6 (which is 5.5" or 14 cm). If it is a little to tall or short, slide it so the top of the arc is at one edge of the board and the flat bottom is parallel to the other edge of the board. When you're using this in class, it's only the radius of the curve that really matters.

## Step 2: Cutting the Curved Sides and End Supports

1. Trace out the cut paths on the 2x6 ... If you arrange them as shown in the image above you will need only one 2x6.
2. Use a band saw or jigsaw to cut out the curved pieces.
3. Stack them and sand both curved edges to get a matching smooth curve.
4. Cut a 1-ft section from the remaining 2x6. Rip this in half to make two 1’x1-3/8”x2-5/8” pieces.

## Step 3: Assemble the Sides and Supports

Use wood glue and drywall screws to attach the supports to the ends of the curved sides.

Note: I recommend drilling pilot holes for the drywall screws. The tapered end of the curved side is susceptible to splitting. Glue and a nail gun could be used instead.

## Step 4: Prepping the Curved Top

1. Cut a 12" x 42" section of the tile board panel
2. Drill 1/8" holes at the locations shown in the image. The holes should be 3/4" in from the edges.

## Step 5: Attach the Curved Top and Hex Bolts

1. Use the #8x1” wood screws and #8 finish washers to attach the panel to the top of the curve sides.
2. The 4.5” bolts are held by the 1/2” neodymium magnets under the tile board. This enables them to be moved anywhere on the curve.

NOTE: I recommend not gluing the panel to the curved edges. Using whiteboard material enables students to mark measurements and make annotations. If the surface gets damaged over time it can be replaced if it is not glued down.

## Step 6: Lesson Plan Part 1 - the Earth Is Round!

One piece of evidence that was used to determine that the Earth is round was the observation of ships traveling over the horizon. Instead of graduating becoming smaller and smaller as they would on a flat Earth, ships disappear from the bottom up as they go over the horizon. Here's a photograph of this phenomenon in real life.

This can be demonstrated by observing the Eratosthenes-inator from one of the short ends and moving an image of a ship from that end to the other.  What the observer will see is shown in the images above.

## Step 7: Lesson Plan Part 2 - Measuring the Circumference of the Earth (BACKGROUND)

About 2000 years ago a Eratosthenes - a scientists living in Northern Egypt - developed a way to measure the circumference of the Earth.

There was a city in Southern Egypt called Syene at the time of Eratosthenes. From historical records he learned that on one day of they year at noon the sun is directly overhead when viewed from Syene. Today we know this occurs because Syene is located on the Tropic of Cancer and on June 22 the sun is directly over this line of latitude.

Eratosthenes lived in Alexandria - a city on the Mediterranean Sea several hundred miles north of Syene. At noon on June 22 the sun is not directly overhead when observed from Alexandria.

Eratosthenes was able to use the length of shadows made by a tower in Alexandria to determine how many degrees around the curved Earth there are between the two cities. With this knowledge and a measurement of the distance between the two cities he was able to use the proportional relationship below to calculate the circumference of the Earth.

Here are a few videos that explain this:

## Step 8: Lesson Plan Part 2 - Measuring the Circumference of the Earth (PROCEDURE)

1. Lay the Eratosthenes-inator on its side on a table. Use the magnets to hold the two hex bolts to the curved surface and place them about 50 cm apart. One of these represents a tower in Syene, the other a tower in Alexandria.
2. Place a 60 W bulb light source on a table at least 10 meters away. Turn off the room lights so this bulb is the only source of light.

3. Arrange the Eratosthenes-inator and/or the bolts so that one casts no shadow and the other does. See the image above.

4. Use a whiteboard marker to trace the length of the shadow.

5. With the room lights back on, use the length of the shadow and height of the tower at Alexandria to determine the arc-angle between the two cities. Measure the distance between the cities. Then calculate the circumference and radius of the entire circle the arc is a part of.

6. The dimensions used to construct the Eratosthenes-inator result in an end-to-end arc of 60º. This means that if you build six of them they can be arranged in a way that creates the entire circle - as shown in the image above.

## Step 9: Lesson Plan Part 3 - Navigating With the Stars

Polaris (the North Star) has been used as a navigation tool by seafarers for centuries. It is used is not just to determine which way is North, but also to measure exactly how far North of the equator they are.

The Eratosthenes-inator can be used to demonstrate this. As shown in the image, protractors and drinking straws are attached to the hex bolts with rubber bands. The straws are arranged to point vertically upward.

The straw/protractor at the top represents the North Pole - Polaris is directly above an observer here.

The other straw/protractor represents an observer at some latitude North of the Equator. The angle this straw makes with the horizon (θ) can be measured with the protractor as shown.

The relationship between the altitude of Polaris and latitude can be demonstrated by sliding this hex nut up or down the arc and re-adjusting the straw to vertical. As the observer gets farther from the North Pole, Polaris gets closer to the horizon.

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## 31 Discussions

Surely Eratosthenes proved that the earth is round with a radius of 6300km and the sun is very far away. Or that the earth is flat and the sun is 6300km away. Or that the earth has a radius of 12700km and the sun is 12700km away. Or anything in between.

What proof did he have that the sun was a long way away?

He knew the earth cast a circular shadow on the moon so it's "round". But what if he'd done his experiment on Discworld?

There is indeed much other evidence that the world is a sphere of radius 6400km. I'm just not sure about the oft-repeated tale that Eratosthenes proved it or measured it.

I am always very suspicious of accounts of scientific history that appear to be hagiographies or what Stephen J Gould referred to, in many essays, as Whiggish History. In Time's Arrow, Time's Cycle Gould talked about Hutton and Lyell but the same applies to Newton or Darwin or even Eratosthenes. It's a "self-serving mythology."

The estimate that Eratosthenes arbitrarily chose for the distance to the sun was that of Aristarchus. Hipparchus had used parallax to calculate the distance to the moon (and was off by 7%). Aristarchus then tried to do the same for the sun but was off by a factor of 1000. Eratosthenes could have chosen other estimates that put the sun even closer or much further away. He seems to have simply assumed that the sun is a long way away and its rays could be treated as parallel.

He couldn't measure that the sun's rays are (almost) parallel. The difference between the rays of a sun 6400km away and 150000000km away are too small.

With hindsight, people cherry-pick the "facts" to say how amazingly accurate Eratosthenes was. In actuality, there were several other attempts to measure the earth at that time that gave very different results. 1700 years later it was still being argued about. Columbus used one of those results to claim that the earth was a half to 2/3 of its true size. Some of his contemporaries agreed with him but others thought it was bigger. They said he couln't reach India with just a month's food and they were right - he ran out before he even reached the Americas.

Ship observations don't help prove that the earth is spherical either. AFAIK, Flat-earthers claim that ships "disappear over the horizon" as a result of refraction caused by changing atmospheric pressure and density. Refraction effects due to temperature cause mirages so it would be a plausible explanation to Eratosthenes. IIRC, Hal Clement in A Mission Of Gravity pointed out that such refaction has the opposite effect; but Eratosthenes wouldn't know that.

This one specific observation (the angle of the sun relative to two points at the same time) does, as you point out, have multiple possible solutions. This one single observation is not sufficient to prove conclusively that the earth is in fact a sphere (as someone else pointed out, it could for instance be a cylinder).

That's why we combine the observations made at many points on the globe at different times, all of which combine to demonstrate beyond reasonable doubt that the earth is in fact a sphere. This experiment can be conducted with any pair of points in the world and the results will be consistent; ships disappear below the horizon in all directions; the heavens rotate about a point in the sky with an elevation equal to your latitude; different latitudes have different climates, daylight throughout a day and the seasons throughout a year change in ways consistent with a spherical earth; every other planet in the solar system is a sphere, because a mass of rock thousands of miles across couldn't remain in a stable non-spherical* shape because of gravity, and so on.

* yes slightly oblate spheroidal if we're being pedantic

This is just one piece of evidence that supports the spherical earth model. An additional part of that would be the relative distance to the sun. They were a bit off on that but using eclipses and the relative size of the moon and sun, they were able to calculate that the sun was about 93 million miles away giving approximately parallel light at the surface of the earth and the moon was 89000 miles away. This coupled with his shadow experiment, and the ship observations, were plenty to come to the conclusion that the earth is spherical.

The wise Eratosthenes already believed the earth was a sphere, his experiment was to measure its radius. To show it’s a sphere just use a third tower in line with the other two and you will find only one mathematical solution.

Indeed. Eratosthes's experiment did nothing to prove the earth was a sphere. It made two assumptions - that the earth was a sphere and that the sun was a long way away. Either of those assumptions could have been wrong.

The two points he used were, lets say, 800km apart (Alexandria and Syene). That gives an angle of 7.2 degrees. If you had a tower half way between them (in Asyut), then the angle of its shadow would be 3.6 degrees.

Let's assume that the sun wasn't a long way away. Let's say the radius of the earth is about equal to the distance from the surface to the sun.

Radius of the earth = 12804km
Surface to sun = 12779km

The angle at Alexandria (800km from Syene) would again be 7.2 degrees but at Asyut, the angle would be 3.6026 degrees. He didn't have the tools to tell 3.6 degrees from 3.6026 degrees.

Let's take the worst case: the earth is flat. The angle at Asyut would be 3.614 degrees. Even that would be difficult for him.

I can find no evidence that either Eratosthenes or Aristarchus were able to calculate the distance to the sun with any accuracy at all.

"But what if he'd done his experiment on Discworld?"

Then he'd have been fictional, too. (You do know Discworld is fiction, right?) ;-)

I mean... that's patently not true. Any "evidence" which I've ever seen claimed for a flat earth has glaring scientific flaws visible to anyone with reasonably science literacy or rely on misunderstandings of simple principles. Every observation in astronomy, including very simple ones anyone can do like watching the stars rotate at night indicates a spherical earth, as does any map of "where is it daylight now", as does the fact that you can fly from South Africa to Australia, and every photo of the earth taken from space, and in fact every map not sold at a flat earth conference. The "evidence" I see held up for a flat earth is things like claiming the moon emits cold laser light which lowers in temperature the more you concentrate it, questioning whether the force of gravity actually exists (easily demonstrated with simple lab equipment) and so on.

Excellent work. I just wished you kept using metric units all along.

I never really thought about it before but I think I switch from metric to imperial system when I go from my science classroom to the school's wood shop.

You have a point. It is just that as science teachers, we need to promote the SI units and stay away from the confusing US Customary units. I invite you to join our growing association at usma.org
I promise to have my students to try making it.

Great demonstration! The only error I found was in your explanation of navigating by the stars, in which you referred to the angle between the horizon and the North Star Polaris as the "altitude" of Polaris. I think you meant the "elevation" of Polaris, which is the angle of any specific direction above the horizon, usually in an outdoor setting. The "altitude" is the height of an object, such as an airplane or mountaintop, above sea level.

I was unaware of the use of the term "altitude" as an angle in astronomy terminology. It appears that the terms "altitude angle" and "elevation angle" are synonymous in the discipline of astronomy, as seen in the article https://solarsystem.nasa.gov/basics/chapter2-2/ under the topic HA-DEC versus AZ-EL Radio Telescopes where it states "In an AZ-EL system anywhere on Earth, east is 90 degrees AZ, and halfway up in EL or altitude (ALT) would be 45 degrees. AZ-EL and ALT-AZ are simply different names for the same reference system, ALTitude being the same measurement as ELevation."

Thanks for the follow up ... and for helping me learn something new!

Great tool. Thanks.

I'd consider fixing only one end of the whiteboard, (or just the middle of the whiteboard) to the arcs and use something temporary to bend the ends against the arcs. Then you can demonstrate the same setup in the counterfactual flat-earth configuration and see the differences.

Great idea! Maybe I'll play around with using velcro to hold down one end.
Or attach flat piece of whiteboard to the bottom of the device (again with velcro so they can still attach and move the bolts). thank you!

When I saw the title of your Instructable in my email I wondered about the "inator" part.