Erathostenes Apparatus

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Introduction: Erathostenes Apparatus

About: I have a master's degree in physics and my hobbies are: 3D printing, CAD design, arduino, astronomy, astrophotography, cosmology and sci-fi :)

Recently I heard a lot of weird theories regarding Earth, so I have decided to make simple apparatus to repeat Erathostenes experiment conducted over 2200 years ago. I designed and 3D printed simple apparatus which I called 'Erathostenes apparatus' and you can use it to prove that Earth is a sphere and you can measure radius of Earth with pretty good accuracy.

Supplies

- Protractor

- PLA

- 30cm ruler

- Superglue

- M3 x 20mm bolt with nut

Tools:

- 3D printer

- Philips screwdriver

- Drill

Step 1: Experiment Principle

The experiment is simple. Erathostenes placed vertical rod in Syene at noon of the summer solstice so it didn't cast the shadow. The same size vertical rod was placed in Alexandria and it cast a shadow. The difference between those 2 locations was about 7 degrees. Knowing the distance between Syene and Alexandria, Erathostenes was able to calculate radius of Earth.

The experiment I designed is not super accurate, but in the age of internet and smartphones the results can be obtained immediately. You will need two exactly the same pieces of apparatus in two different locations.

On the same day and at the same time 2 people can move the rod so it won't cast a shadow and then measure the angle. Further the two measuring point will be, the greater the difference in the angles will be measured.

You can do it another way by placing the ruler and set up the rod the way, it will cast the same length shadow in both places and then read the angle. This is the method me and my friend used, as you can take multiple readings and calculate the average reducing the error significantly.

Step 2: Apparatus Assembly

- First you need to 3D print all the pieces

- Drill 3mm hole through the protractor centre and place it in the 3D printed cover

- Glue the top cover and then glue the case with protractor into the base

- Use M3 bolt and mount the rod (tighten the nut, but make sure you can still move it slightly)

- Glue the small 3D printed pointer into the rod (there is small slot for that)

Step 3: Setting Up the Apparatus

- Place the device on level ground and you can use spirit level to check if it's reasonably levelled

- At first you have to point the apparatus toward the Sun. The easiest way to do that is to keep rotating the plate until the shadow cast by protractor is parallel to shadow cast by the rod (see picture above)

- Move the rod so it won't cast any shadow. It's quite tricky and there is an error involved as the rod and the screw have certain thickness

- After that place the ruler or tape measure under the rod on the base

- Now you can start moving the rod so it will cast certain length shadow and measure the angle

- Record your readings in a table

NOTE: Remember to do that experiment with your friend or family member that lives quite far from you. The longer the distance the more accurate calculation will be.

Step 4: Taking Readings

My friend from Havant, UK made this apparatus and so did I. I'm based in Heywood, UK and we are about 320km apart. I know the distance because I drove to visit him few times and I tried to use most direct route as possible. Of course the distance I measured never will be super precise, but I took the average from 3 trips and then I compared it with Google maps. Based on that I estimated the distance between 320km to 330km.

As you can see on the pictures above we did video call and we took few readings from two different locations (Havant and Heywood), but on the same day and at the same time. The results are in table (see picture above).

Step 5: Radius Calculations

After taking reading, I calculated the average angle difference between the two locations. Having that you can use proportions to estimate the circumference (or radius L=2*Pi*R) of the Earth.

I also calculated the error and, although it's quite significant, the radius of Earth we calculated is within that error.

I'm very pleased with the results and the best thing about this experiment is that anyone can verify my calculations and find out radius of Earth themselves.

The final results: R = 6153m +/- 800km while the real value is R= 6371km

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    24 Comments

    0
    MuzicMaker
    MuzicMaker

    1 year ago

    This is a cool idea... I’m not a Flat Earther, but - if the earth WAS flat, and the sun orbited the flat earth, you’d still get no shadow at noon on solstice in one location and a shadow further away. Picture this experiment on a flat planet - the sun can only be directly above 1 point of the plane, any distance away and you’ll get a shadow. The tool will allow you to estimate a radius IF you assume that you’re on a ball. But I don’t see how it ‘proves’ the earth is round, to be fair to the doubters.

    0
    poblocki1982
    poblocki1982

    Reply 1 year ago

    Please check picture below. In the experiment I'm also setting same length shadow at two different location and both location give different angles. It doesn't have to be at noon, actually I did my experiment at 2pm. As you can see on the picture above on flat Earth there would be no shadow at noon (you are right) but at any other time both location would cast exactly same shadow while on the sphere the shadow have different length at same angle or different angles at same length.

    You can actually do a simple experiment, use A3 paper and attach 2 cocktail stick with blutack and then do the same on beach ball. Then use torch and shine the light on the A3 paper and both cocktail stick will cast the same shadow. If you do that with ball the shadows ill have different length.

    I hope that helps :)

    FB_IMG_1589298403078.jpg
    0
    PavelP14
    PavelP14

    Reply 1 year ago

    That'd be true if the sun was "infinitely" far away, so the rays would be parallel. However, the flat-earthers usually assume the sun to be small and close casting non-parallel rays.
    Nonetheless, I think that if you took the measurement at three places at the same time, you couldn't explain it any more (I didn't do the maths though). Not that they would believe it ;)
    0
    poblocki1982
    poblocki1982

    Reply 1 year ago

    That's true, but then again flat-earthers don't do experiments and math calculations, they just say things without investigating it. You can easily calculate distance Earth-Sun yourself too using Aristarchus technique, then you find out that the Sun is so far that can be treated as it is in infinity. You just gave me an idea for another project :) Measuring distance to Sun using Aristarchus technique :)

    Once you calculate Radius of Earth and then distance to Sun you can work out orbits of other planets from Kepplers Law. You can use webcam to make a spetrometer and measure red shift and and calculate speed, then find a cepheid and measure period, then work out absolute magnitude and distance. Once you know that you can calculate Hubble constant and the age of Universe :D It's pretty cool and I've done it from my back garden.

    0
    weish
    weish

    Reply 1 year ago

    and sadly, no matter how many experiments you use to demonstrate a round earth, they'll just plug up their ears, close their eyes, and howl gibberish at you until you get tired of them and leave, at which point they declare victory over you. they don't believe in any science, or even math, that doesn't agree with their nonsense.

    0
    poblocki1982
    poblocki1982

    Reply 1 year ago

    I know and it's sad that they choose to belive in some pseudo science nonsense without any reasonable explanation rather to embrace the truth tgat they can experience with their own eyes. It's like them saying that water is dry and if you ask them to put hand in the water tank they will say its a trick or computer simulation and that's it feels wet 🤣

    0
    Godovrall
    Godovrall

    1 year ago

    Nicely done.... I think the planet is wider around the equator than the pole

    0
    poblocki1982
    poblocki1982

    Reply 1 year ago

    You're absolutely right, but the purpose of this experiment is to work out radius of Earth within 10% error using minimum equipment.
    To get precise radius you would need to measure the angle a lot more accurately, take into account that equator and pole radius are different, take into account elevation of two points and exact distance with GPS.

    It's definitely great idea for a college student. In my version of the experiment I wanted to make it as simple as possible and as close as possible to original experiment and that's why I called this apparatus as I did to honour Eratosthenes amazing experiment.

    0
    redbeardtrev
    redbeardtrev

    Reply 1 year ago

    You are quite correct, but the difference is small. Look up the original definition of a Nautical Mile, which is defined as 1/60th of a degree (or 1 Minute) of Latitude. At the poles it is approximately 1861m and only 1843m at the Equator. If those numbers appear to be the wrong way round, just draw an ellipse and subdivide it into equal angle segments.

    0
    scicior
    scicior

    1 year ago

    One thing I have never been able to figure out; how was Erathostenes able to make the measurement in two distant locations at the time time? He wasn't able to Facetime 2200 years ago...

    0
    frarugi87
    frarugi87

    Reply 1 year ago

    It was not a single measure, and from what Wikipedia says the actual method got lost.

    In any case, since conditions are cyclic, you can just make a measurement today in one location and another in the other place next year...

    0
    poblocki1982
    poblocki1982

    Reply 1 year ago

    That's why it to took him over a year. He was traveling a lot between Syene and Alexandria so one year he did his measurements at one place and then next year at the other. That's why he choose specific date which was summer solstice and specific time, noon. Within one year he had readings for two points and he also knew the distance between two cities, so he estimated radius of Earth.

    When I was 10 I used that technique to calculate radius of Earth. At that time there was no mobile phones or Internet, so I measured the angle at my grandma when I was visiting her during summer vacation and next year I repeated the experiment at my house. The calculations where within 20% error, but I guess for 10 year old with a stick it's not that bad.

    0
    NigelMajor
    NigelMajor

    1 year ago on Step 3

    If you'd like to make it more accurate, add two sights (like the back sight of a rifle, just a tab with a hole in the middle), one at the base of the stick and the other at the top. When you align correctly the light from the top hole will pass through the bottom hole, when you're slightly off, the "dot" from the top sight will shine on the edge of the bottom sight, showing you where to adjust. The smaller the hole, the greater the accuracy and the greater patience needed to align.
    Thanks for a fun project.

    0
    poblocki1982
    poblocki1982

    Reply 1 year ago

    Hi, this i great idea, thank you :) I will make adjustments for v2.0. Thanks agin for your sugestions

    0
    NigelMajor
    NigelMajor

    Reply 1 year ago

    Looking forward to seeing it!

    0
    woodchipwilbur
    woodchipwilbur

    1 year ago

    I'm interested to note that there is no mention in the thesis to say that the base of the machine must be level! If the whole device is tilted, then the measurements would become very suspect... Add a spirit level?

    0
    poblocki1982
    poblocki1982

    Reply 1 year ago

    You're right. I must have forgot to mention that. That's why I added bottom plate so if place it on the even ground it should be reasonably levelled. I will add a sentence about placing it on level ground in a minute, thanks :)

    0
    Wayne in Granby CT
    Wayne in Granby CT

    1 year ago

    You wrote "Erathostenes placed vertical rod in Syene at noon of the summer solstice
    so it didn't cast the shadow. The same size vertical rod was placed in
    Alexandria and it cast a shadow too."
    Since the first rod in Syene DIDN'T cast a shadow, how can the rod at Alexandria "cast a shadow too."? Please remove the "too" to make it accurate.

    Could you place an arched vernier scale instead of a pointer to get a finer reading/calibration or is that asking too much?

    Am I missing something with the 83 and 90 degree angles on the first drawing? How can you get those angles measuring to an arc? The further away from the city center along the earth's radius would give you a different angle.

    Would the elevation difference between the two places to be measured make a big difference?

    Would the further away the two cities are from each other make for a more accurate reading?

    Things like this always invite more questions than I can answer.

    This will be a great project for me on my 48" x 52" engraving machine

    0
    poblocki1982
    poblocki1982

    Reply 1 year ago

    Hi, thanks for the correction, I already corrected the mistake. English isn't my first language and I'm always learning :)
    In the drawing where the angle is 90 degrees it's because there is no shadow for vertical rod, and at the other location there is a shadow and the angle between Earth and Sun rays is 83 degrees or between Sun rays and rod 7 degrees. It will also work it you tilt the rod at first location to cast a certain length shadow and you will measure for example 2 degrees so then at another location with same length shadow you will measure 9 degrees, so the difference will always be the same 7 degrees.

    As you said further away the two locations are the greater difference in angle, but there is limit to how far the two location can be as both location require the Sun. If one location is were there is a day and another where there is a night it won't work. Larger distance won't make the reading accurate, but taking readings between 3 locations and averaging them would improve accuracy.

    Elevation makes a difference, but for that you would need to add GPS which is easy to do really, but for this project I wanted to make it as simple as possible, just to estimate Radius of Earth within 10% error.

    0
    papa-ralph
    papa-ralph

    1 year ago

    Very nice but you will never be able to "sell" it to the Flat Earthers :-)