Science in School
The eratosthenes experiment: calculating the earth’s circumference teach article.
Author(s): Sevasti Malamou, Vasileios Kitsakis
On the shoulders of giants: follow in the footsteps of Eratosthenes and measure the circumference of the Earth like he did 2300 years ago.
The following learning scenario is made for secondary school students that are familiar with the concepts of equal angles, from geometry, and tangents, from trigonometry. Moreover, it can also be adjusted for primary school students not familiar with trigonometry: they can make measurements like Eratosthenes did and leave advanced calculations for the teacher.
Brief description
Students can measure the Earth’s circumference like Eratosthenes did approximately 2300 years ago using simple materials and a stick’s shadow made by the Sun. Even though there is a high probability that the measurement will not approximate the true value of the Earth’s circumference, like we know it today, the measuring itself provides a basis for simple mathematical reasoning and scientific thinking.
Ideally, the experiment should take place on the March or September equinoxes on a sunny, or almost sunny, day. Before starting to measure the Earth’s circumference, students should learn about Eratosthenes, his life, work, and the way he calculated the circumference of the Earth.
Learning objectives
- Describe the geometry of sunlight towards Earth (sunrays are parallel when falling to Earth)
- Understand that equinoxes and solstices are due to the Earth’s movement
- Realize the geographic coordinate system of Earth: latitude and longitude
- Describe how Eratosthenes measured the circumference of the Earth
- Measure angles and distances
- Compare angles and triangles
- Explain measurement errors and suggest ways to minimize them
- Collaborate with other schools on the same longitude
Optional introductory activity: Who was Eratosthenes? Why is his experiment so important nowadays?
Although it seems a simple and easy experiment, it takes time for students to really understand geometry, the direction of the Sun towards Earth on specific days, and the logical sequence of Eratosthenes’ thoughts.
The goal is not simply to measure the length of the stick and its shadow, but to understand Eratosthenes’ logic behind these simple measurements, and thus, highlight his ingenuity, since almost 2300 years ago he calculated the circumference of the Earth with relatively great accuracy.
We recommend carrying out the introductory activity , so that students understand the importance of Eratosthenes’ experiment, the era during which he did his experiment, what helped him to reach to his conclusions, and the way he managed to accomplish his experiment.
Activity 1: Identify the exact time of the measurement
Before implementing the experiment, which is actually to measure distances and make calculations, we must first determine when the Sun reaches its highest point in the sky on the day of the equinox; what we call local noon (or zenith). This time differs for each school and depends on the school’s position on the globe and should be determined with the best accuracy possible. At this specific time, sunrays fall perpendicularly on the equator, and they are parallel to the equatorial plane, so a vertical bar will have no shadow. On the other hand, in our place at this specific time, a vertical bar will have a shadow.
A calculator tool that can be used is SunCalc . The position and date have to be filled in for the Sun’s culmination time to be calculated. At this exact time on the day of the equinox, students have to measure the stick’s length and its shadow.
This activity only takes a few minutes, but it is also recommended to do the introductory activity to set the context. It should be possible to do the introductory activity and Activities 1 and 2 in one lesson; these should be completed a few days before the measurement lesson on the equinox (Activity 3).
- Internet connection and a suitable device (PC, laptop, tablet, smartphone)
At least 1–2 days before the equinox (when the measurements will be made), you should:
- Explain the experiment (ideally using the introductory activity ).
- Tell your students that many schools are doing this experiment on this specific day.
- Split your class into groups of two students and let them run the web tool SunCalc in the computer lab or use tablets.
- Find their city/village and school on the map
- Select the required date that the measurements are going to be made (equinoxes are preferred)
- Write down the culmination time
Alternatively, if you don’t have enough time or equipment, you can demonstrate the procedure.
Finding the zenith time for the Sun should only take a few minutes. You can also use the application as a learning object for students and ask them to investigate different sundial characteristics. The tool gives the chance to understand the concept of noon during the year. Students can change the date, and then see the culmination time for each date. Solstices and equinoxes are dates that should be investigated. Local noon is in the midpoint between sunrise and sunset times, and it depends on the latitude and date during the year.
Activity 2: Identify the school’s coordinates
At least one day before the experiment, students should identify the school’s coordinates using online tools. What is to be measured is the distance in kilometres from the schoolyard to the equator along the school’s meridian, which is going to be a curved line, following the Earth’s curvature. All points along this meridian have the same longitude.
Eratosthenes knew the distance between Alexandria and Syene (nowadays Aswan) in stadia (an ancient unit of length). Nowadays, we can measure the distance using electronic applications. You could also get a distance estimate using a real map and a ruler, just like students used to calculate real distances years ago. Especially for younger students, it would be a great chance to refresh their knowledge of map scales. The measurement won’t be as accurate as the one with online tools, but it is a simple estimation.
This activity should only take 10 min.
- Video on how to measure the distance from your school to the equator
- Alternatively, a map and a ruler
- Use a smartphone with a location function.
- Write down the latitude and longitude of the schoolyard.
- Measure the distance from your school to the equator. For this step, Google Maps or Google Earth can be used, as shown in the video .
- Write down the distance in kilometres. This information is what students need for their calculations.
Activity 3: Measure like Eratosthenes on the equinox
The day that everyone was expecting has arrived. The teacher has to organize all the required materials before the time that the Sun reaches its highest point. Everything should be prepared in advance because, once the Sun reaches its zenith, there is no time to lose. Students have to act quickly, and they have to know exactly what to do.
- Linear sticks (approximately 1 m long is ideal)
- Right-angle triangles, plumb bobs, carpenter’s levels, or an object that has a right angle to ensure verticality
- Metre sticks or tape measures
- Clock accurate to the minute (or a smartphone)
Student worksheet
- To measure the length of the sticks when the Sun is overhead, supply students with the necessary materials and worksheet a short time before the measurement time.
- Go with your students into the schoolyard at least 10–15 min before the zenith time for your latitude.
- Split the class into groups of four.
- Ask each member of the group to take on a specific role. Student 1 will be responsible for the time, student 2 will act as a scribe and record the measurements, and students 3 and 4 will do the measurements.
- Make sure that each group has a set of materials (a linear stick, a metre stick, a right-angle triangle, a pencil, a clock, and a worksheet).
- Ask students to measure the length of the stick before the zenith time and write their measurements on the worksheet.
- About 2–3 min before the zenith time, ask students to place and hold the sticks vertically.
- Use the right-angle triangle to make sure that the sticks are vertical. Check that all groups achieve verticality.
- When the Sun reaches its highest point in the sky, ask students to measure the stick’s shadow on the ground and write its value on the worksheet.
- If students don’t manage to measure the stick’s length before the zenith time, or if they want to be sure about it, they can measure it again after the local noon time.
After measuring the two lengths (stick length and shadow length) and writing the values on the worksheet, data processing begins. Calculations can be done in the schoolyard, with students working in groups and comparing their results with their classmates. Alternatively, if there is a lack of time, calculations can be done in the classroom at another time. Students can use scientific calculators (or the one on their smartphones) to calculate the angle θ .
Extension activities
There are several extension activities that can be done to make the learning experience more meaningful, such as calculating the radius of the Earth and collaborating with another school. These are described fully in the supporting material.
When doing the calculations and exporting the results, you could use the following questions as the basis for a discussion:
- What could be the measurement errors during the experiment?
- What can be done to minimize errors?
- What errors could have been made by Eratosthenes when he performed his own experiment 2300 years ago? To answer this, the globe could be used, with pins on Alexandria and Syene. Alternatively, students can observe the two cities on Google Earth.
- Angle θ , calculated during the experiment, also represents what?
- If you do the experiment during the summer/winter solstice, what would you change? Explain.
- Why did Eratosthenes make his measurements during the summer solstice? Could he do it on the spring or autumn equinoxes? Explain.
As a conclusion, you can emphasize to your students that science often develops from a simple idea and an inquisitive mind.
[1] Panhellenic Union of Heads of Laboratory Centers of Natural Sciences (Greek language): https://panekfe.gr/eratosthenes/
- Find more teaching resources relating to the Eratosthenes experiment on the eu website.
- Read more about the life and work of Eratosthenes of Cyrene .
- Watch a video about Eratosthenes by Carl Sagan.
- Explore data visualization by sketching graphs from story videos of everyday events: Reuterswärd E (2022) Graphing stories . Science in School 58 .
- Discover how physicists study very small and large objects that cannot be directly observed or measured: Akhobadze K (2021) Exploring the universe: from very small to very large . Science in School 55.
- Get your students to use their smartphones for some hands-on astronomy: Rath G, Jeanjacquot P, Hayes E (2016) Smart measurements of the heavens . Science in School 36: 37–42.
- Challenge your students to solve the mystery box puzzle while learning about the nature of science: Kranjc Horvat A (2022) The mystery box challenge: explore the nature of science . Science in School 59 .
- Measure distances to the stars like real astronomers with this classroom activity: Pössel M (2017) Finding the scale of space . Science in School 40 : 40–45.
- Measure the distance between Earth and the Moon with the help of radio signals: Middelkoop R (2017) To the Moon and back: reflecting a radio signal to calculate the distance . Science in School 41: 44–48.
Sevasti Malamou is a secondary school physics teacher at the Music School of Ioannina Nikolaos Doumpas, Greece. She has also taught chemistry, biology, and geography and enjoys using innovative methods to teach science. She holds master’s degrees in both electronics and telecommunications, and physics didactics.
Vasilios Kitsakis is a secondary school maths teacher and currently headmaster at the Music School of Ioannina Nikolaos Doumpas, Greece. He holds master’s degrees in studies in education and educational management. He has taken part in training programs and conferences as a speaker.
This article introduces ideas and activities that move teachers and students from a historical concept to its replication using modern technologies.
Marie Walsh, Science Lecturer, Ireland
Supporting materials
Introductory activity
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This measurement is more than 2,200 years old, and its accuracy is remarkable
Topic: Mathematics Education
Ancient mathematicians used their knowledge of geometry to take accurate measurements of earth ( Getty: Aris Messinis )
There are so many negative numbers and statistics flying around at the moment, it's a little overwhelming.
So let me tell you an inspiring story about numbers: it will be a relief to think about something else for a few minutes.
Travel back in time
About 2,250 years ago, there was a man called Eratosthenes.
He was one of those ancient Greeks who changed the world.
He was a polymath, someone with expert knowledge of a range of topics.
A mathematician, geographer, astronomer, philosopher, poet, and music theorist.
He's famous for being the first person known to have measured the earth's circumference.
How did he do it?
It's surprisingly simple. You just need some basic geometry.
Watch the short clip below of the great Carl Sagan to see how it was done.
I've set it to play from the 4 min 11 sec mark, because that's where Sagan explains the calculations. But the few minutes before that point are also wonderful.
In case you couldn't watch or hear the video, I'll explain the story quickly.
Around 245 BC, when Eratosthenes was in his 30s, he was working as a librarian in the famous Library of Alexandria in Egypt.
It was there where he read about a water well in the city of Syene (modern-day Aswan in southern Egypt).
At midday every summer solstice, the sun would shine directly down into the well, illuminating the water at the bottom - but casting no shadow on the walls of the well.
It meant the sun sat directly above Syene at that exact moment.
So Eratosthenes wondered, if he stuck a pole in the ground in Alexandria at that same moment, would it cast a shadow?
And it turns out it did.
What did it prove?
His little experiment demonstrated that the surface of the earth was curved like a sphere.
Why? Because his pole in Alexandria was sticking straight into the air but the curvature of the earth made it face slightly away from the sun, causing the pole to throw a small shadow onto the ground.
And that allowed him to do something else.
Since he knew the height of the pole, and the length of the shadow it cast, it meant he knew the lengths of two sides of a right-angled triangle.
That meant he could figure out the length of the third side of the triangle, and he could also figure out the angle at the top of the pole, between the sunbeam and the pole itself.
It was 7.2 degrees.
Therefore, he knew the sun was hitting Alexandria at an angle of 7.2 degrees precisely at midday on the summer solstice.
When a fraction goes a long way
And that left him with one final measurement.
To figure out the circumference of the earth, he needed to somehow measure the distance between Alexandria and Syene.
So he asked someone (or a team of people) to walk it.
Those people were called "bematists", professional surveyors who were trained to measure vast distances extremely accurately by pacing the distance.
They estimated the distance between the two cities was roughly 5,000 stadia (or 800 kilometres).
And that was everything Eratosthenes needed.
He had all the ingredients to calculate the circumference of the earth.
A few assumptions help
Let's go.
Assume the earth is a perfect sphere (it's not, but it's not a problem for these calculations).
We know there are 360 degrees in a circle.
If you cut the earth in half, the earth's great circle will obviously have 360 degrees, and the circumference of that circle (i.e. the total length of its perimeter) could be divided up into equal bits of whatever length.
Eratosthenes knew that the distance between Syene and Alexandria was 7.2 degrees along the surface of the earth.
So how many of those distances would he need to stretch around the entire 360 degree circumference of the earth?
He divided 360 by 7.2, which gave a neat 50.
That meant, given the distance between Alexandria and Syene was 800 kilometres, all he had to do was multiply 800 by 50, which came to 40,000.
And that was it.
The circumference of the earth was 40,000 kilometres, according to Eratosthenes' calculations.
The bematists estimated that Syene was 5,000 stadia (or 800 kilometres) away from Alexandria, which gave Eratosthenes the final number he needed to work out the earth's circumference ( Source: Khan Academy, "Eratosthenes of Cyrene: Measuring the Circumference of the Earth." )
Was he correct?
He was incredibly close.
As it turns out, the meridional circumference of Earth (from pole to pole) is roughly 40,008 km, and the equatorial circumference is about 40,075 km (it's bigger at the equator because Earth slightly bulges in its middle).
Not bad for someone with such rudimentary tools.
Eratosthenes used his new knowledge to revolutionise map making.
He drew a map of the known world with parallels and meridians, making it possible to estimate real distances between objects, and plotted the names and locations of hundreds of cities over the grid.
It was the beginning of modern geography.
Anyway, I hope that's been a pleasant escape from reality.
When so much attention is focused on the maths of hospitalisations and vaccinations and contagion, it's easy to forget that maths can also be a source of innocent joy.
Take care this week.
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How did Eratosthenes calculate the size of the Earth?
Eratosthenes was born in the country we now call Lybia, but in those days was called Cyrene.
19th century reconstruction of Eratosthenes' map of the known world, c.194 BC. Source: Wikipedia.
Eratosthenes studied in Alexandria and claimed to have also studied for some years in Athens. In 236 BC he was appointed by Ptolemy III Euergetes I as librarian of the Alexandrian library, the center of science and learning in the ancient world, succeeding to Apollonius of Rhodes, in that post. He was the third chief librarian of the Great Library of Alexandria.
As chief librarian he read many documents and found out that at the Ancient Egyptian city of Swenet (known in Greek as Syene, and in the modern day as Aswan) that is located near the tropic of Cancer on June there is a well where on certain day of the year the sunlight goes down to the bottom of the well. He knew that in Alexandria there was no day that the great Obelisk did not produce shadow and he measured the shadow angle on the day the Sun was directly above the well in Aswan. He needed to know the distance from the well in Aswan to Alexandria and there several different versions of how he found out its value. The most popular one is that he send a slave to measure it in footsteps. The value that he used in his calculations was 8000 stadia (1 egiptian stadium is about 157.5 m, though the exact size of the stadium is often a theme of discussion).
With this information he measured the circumference of the Earth without leaving Egypt by assuming that Earth was a sphere and that the Sun rays are parallel when they arrive to Earth.
Eratosthenes assumed that Earth is a sphere and that the solar rays are parallel when they reach Earth.
If this was true then the angle (α) that the shadow made on the top of the obelisk in Alexandria would be the same as the diffrence in latitude between the two places.
Eratosthenes used a simple formula that relates the proportionality proportionality of distance on the meridian (d) and the difference in latitude (α) to the relation between the perimeter (P) and the angle of the circle (360º):
The shadow angle at the top of the obelisk measured by Eratosthenes was 7.2º, so he calculated that the Earth was about 252 000 stadia.
If we assume the Egyptian stadium this is about 39 817 km (252 808 stadium × 157.5 m/stadium) which has an error of less than 1% when compared to the accepted value of the meridional perimeter of Earth that is 40 007.86 km.
Eratosthenes' experiment was one of the most important experiments in antiquity and his estimate of the earth’s size was accepted for hundreds of years afterwards. It was, in fact, the most accurate estimate until Man was able to go to Space.
© EAAE - European Association for Astronomy Education 2024
September 7, 2017
Measure Earth's Circumference with a Shadow
A geometry science project from Science Buddies
By Science Buddies & Ben Finio
The earth is massive, but you don't need a massive ruler to measure its size. All you need are a few household items--and little bit of geometry!
George Retseck
Key concepts Mathematics Geometry Circumference Angles Earth's equator
Introduction If you wanted to measure the circumference of Earth, how long would your tape measure have to be? Would you need to walk the whole way around the world to find the answer? Do you think you can do it with just a meterstick in one location? Try this project to find out!
Before you begin, however, it is important to note this project will only work within about two weeks of either the spring or fall equinoxes (usually around March 20 and September 23, respectively).
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Background What is Earth’s circumference? In the age of modern technology this may seem like an easy question for scientists to answer with tools such as satellites and GPS—and it would be even easier for you to look up the answer online. It might seem like it would be impossible for you to measure the circumference of our planet using only a meterstick. The Greek mathematician Eratosthenes, however, was able to estimate Earth’s circumference more than 2,000 years ago, without the aid of any modern technology. How? He used a little knowledge about geometry!
At the time Eratosthenes was in the city of Alexandria in Egypt. He read that in a city named Syene south of Alexandria, on a particular day of the year at noon, the sun’s reflection was visible at the bottom of a deep well. This meant the sun had to be directly overhead. (Another way to think about this is that perfectly vertical objects would cast no shadow.) On that same day in Alexandria a vertical object did cast a shadow. Using geometry, he calculated the circumference of Earth based on a few things that he knew (and one he didn’t):
He knew there are 360 degrees in a circle.
He could measure the angle of the shadow cast by a tall object in Alexandria.
He knew the overland distance between Alexandria and Syene. (The two cities were close enough that the distance could be measured on foot.)
The only unknown in the equation is the circumference of Earth!
The resulting equation was:
Angle of shadow in Alexandria / 360 degrees = Distance between Alexandria and Syene / Circumference of Earth
In this project you will do this calculation yourself by measuring the angle formed by a meterstick’s shadow at your location. You will need to do the test near the fall or spring equinoxes, when the sun is directly overhead at Earth's equator. Then you can look up the distance between your city and the equator and use the same equation Eratosthenes used to calculate Earth’s circumference. How close do you think your result will be to the “real” value?
There is a geometric rule about the angles formed by a line that intersects two parallel lines. Eratosthenes assumed the sun was far enough away from our planet that its rays were effectively parallel when they arrived at Earth. This told him the angle of the shadow he measured in Alexandria was equal to the angle between Alexandria and Syene, measured at Earth’s center. If this sounds confusing, don’t worry! It is much easier to visualize with a picture. See the references in the “More to explore” section for some helpful diagrams and a more detailed explanation of the geometry involved.
Sunny day on or near the spring or fall equinoxes (about March 20 or September 23, respectively)
Flat, level ground that will be in direct sunlight around noon
Volunteer to help hold the meterstick while you take measurements (Or, if you are doing the test alone, you can use a bucket of sand or dirt to insert one end of the meter stick to hold it upright.)
Stick or rock to mark the location of the shadow
Long piece of string
Optional: plumb bob (you can make one by tying a small weight to the end of a string) or post level to make sure the meter stick is vertical
Preparation
Look at your local weather forecast a few days in advance and pick a day where it looks like it will be mostly sunny around noon. (You have a window of several weeks to do this project, so don’t get discouraged if it turns out to be cloudy! You can try again.)
Look up the sunrise and sunset times for that day in your local newspaper or on a calendar, weather or astronomy Web site. You will need to calculate “solar noon,” the time exactly halfway between sunrise and sunset, which is when the sun will be directly overhead. This will probably not be exactly 12 o’clock noon.
Go outside and set up for your materials about 10 minutes before solar noon so you have everything ready.
Set up your meter stick vertically, outside in a sunny spot just before solar noon.
If you have a volunteer to help, have them hold the meterstick. Otherwise, bury one end of the meterstick in a bucket of sand or dirt so it stays upright.
If you have a post level or plumb bob, use it to make sure the meterstick is perfectly vertical. Otherwise, do your best to eyeball it.
At solar noon, mark the end of the meterstick's shadow on the ground with a stick or a rock.
Draw an imaginary line between the top of the meterstick and the tip of its shadow. Your goal is to measure the angle between this line and the meterstick. Have your volunteer stretch a piece of string between the top of the meterstick and the end of its shadow.
Use a protractor to measure the angle between the string and the meterstick in degrees. Write this angle down.
Look up the distance between your city and the equator.
Calculate the circumference of the Earth using this equation:
Circumference = 360 x distance between your city and the equator / angle of shadow that you measured
What value do you get? How close is your answer to the true circumference of Earth (see “Observations and results” section)?
Extra: Try repeating your test on different days before, on and after the equinox; or at different times before, at and after solar noon. How much does the accuracy of your answer change?
Extra: Ask a friend or family member in a different city to try the test on the same day and compare your results. Do you get the same answer?
Observations and results In 200 B.C. Eratosthenes estimated Earth’s circumference at about 46,250 kilometers (28,735 miles). Today we know our planet's circumference is roughly 40,000 kilometers (24,850 miles). Not bad for a more than 2,000-year-old estimate made with no modern technology! Depending on the error in your measurements—such as the exact day and time you did the test, how accurately you were able to measure the angle or length of the shadow and how accurately you measured the distance between your city and the equator—you should be able to calculate a value fairly close to 40,000 kilometers (within a few hundred or maybe a few thousand). All without leaving your own backyard!
More to explore Calculating the Circumference of the Earth , from Science Buddies Lesson: Measure the Earth's Circumference , from eGFI Angles, Parallel Lines and Transversals , from Math Planet Science Activities for All Ages! , from Science Buddies
This activity brought to you in partnership with Science Buddies
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Eratosthenes
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- APSNews - This Month in Physics History - June, ca. 240 B.C. Eratosthenes Measures the Earth
- Khan Academy - Eratosthenes of Cyrene
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- CORE - Eratosthenes on the “Measurement” of the Earth
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- Academia - Eratosthenes of Cyrene: Biography & Work as a Mathematician
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In addition to calculating Earth ’s circumference, Eratosthenes created the Sieve of Eratosthenes (a procedure for finding prime numbers), tried to fix the dates of literary and political events since the siege of Troy , and is thought to have created the armillary sphere (an early astronomical device for representing the great circles of the heavens).
Eratosthenes measured Earth’s circumference mathematically using two surface points to make the calculation. He noted that the Sun’s rays fell vertically at noon in Syene (now Aswān ), Egypt , at the summer solstice . In Alexandria , also in Egypt, at the same date and time, sunlight fell at an angle of about 7.2° from the vertical.
How did Eratosthenes die?
Eratosthenes died in his 80s in Alexandria, Egypt. He had become blind in his old age and could no longer work by 195 BCE. He reportedly fell into despair, and he is said to have committed suicide by voluntary starvation in 194 as a result.
Eratosthenes (born c. 276 bce , Cyrene , Libya—died c. 194 bce , Alexandria , Egypt) was a Greek scientific writer, astronomer, and poet, who made the first measurement of the size of Earth for which any details are known.
At Syene (now Aswān), some 800 km (500 miles) southeast of Alexandria in Egypt , the Sun’s rays fall vertically at noon at the summer solstice . Eratosthenes noted that at Alexandria, at the same date and time, sunlight fell at an angle of about 7.2° from the vertical. (Writing before the Greeks adopted the degree, a Babylonian unit of measure, he actually said “a fiftieth of a circle.”) He correctly assumed the Sun’s distance to be very great; its rays therefore are practically parallel when they reach Earth. Given an estimate of the distance between the two cities, he was able to calculate the circumference of Earth, obtaining 250,000 stadia. Earlier estimates of the circumference of Earth had been made (for example, Aristotle says that “some mathematicians” had obtained a value of 400,000 stadia), but no details of their methods have survived. An account of Eratosthenes’ method is preserved in the Greek astronomer Cleomedes’ Meteora . The exact length of the units (stadia) he used is doubtful, and the accuracy of his result is therefore uncertain. His measurement of Earth’s circumference may have varied by 0.5 to 17 percent from the value accepted by modern astronomers, but it was certainly in the right range. He also measured the degree of obliquity of the ecliptic (in effect, the tilt of Earth’s axis) and wrote a treatise on the octaëteris , an eight-year lunar-solar cycle. He is credited with devising an algorithm for finding prime numbers called the sieve of Eratosthenes , in which one arranges the natural numbers in numerical order and strikes out one, every second number following two, every third number following three, and so on, which just leaves the prime numbers.
Eratosthenes’ only surviving work is Catasterisms , a book about the constellations , which gives a description and story for each constellation , as well as a count of the number of stars contained in it, but the attribution of this work has been doubted by some scholars. His mathematical work is known principally from the writings of the Greek geometer Pappus of Alexandria , and his geographical work from the first two books of the Geography of the Greek geographer Strabo .
After study in Alexandria and Athens, Eratosthenes settled in Alexandria about 255 bce and became director of the great library there. He tried to fix the dates of literary and political events since the siege of Troy . His writings included a poem inspired by astronomy , as well as works on the theatre and on ethics . Eratosthenes was afflicted by blindness in his old age , and he is said to have committed suicide by voluntary starvation.
How Did Eratosthene Calculate The Circumference Of Earth In 240 BC?
The problem, the man for the job, the estimation, the advantages.
Eratosthene calculated the circumference of Earth by measuring the length of a stick’s shadow in Alexandria and the distance between Alexandria and Syrene on foot. He then took the inverse tangent of the ratio between the shadow’s length and the stick’s length to find the angle of inclination of the Sun. He calculated the total circumference of the Earth to be ((360/7.2) x D ) kilometers, with D being the distance between Alexandria and Syrene.
This might come as a surprise for our younger readers, but throughout history, not everything that people learnt could be found on the Internet. Before that, humans “wasted” time poring over books, and before books, our ancestors had to communicate knowledge verbally! With that in mind, how is it possible that someone could accurately predict the circumference of the Earth without ever using a computer?
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Ancient astronomy was a puzzling field, with men using only visual observations, rather than the help of complex instruments. Even if they did have the intellectual prowess to tackle certain problems, most early scientists had to battle social and religious conventions. The idea of a spherical Earth appeared in Greek philosophy from Pythagoras in the 6th century BC and was sneered at because until then, Earth was considered completely flat! Aristotle provided evidence for the spherical shape of the Earth on empirical grounds by 330 BC, which is very slow progress when you think about it.
The problem with determining the Earth’s circumference held no importance if the Earth was flat, but gradually, once people started to acknowledge the actual shape, they needed to know exactly how big our world truly was. The only tool that astronomers had in 300 BC was their ability to recognize patterns.
Also Read: How Did People Figure Out That The Earth Was Round Without Any Technology?
Eratosthene was a Greek mathematician and the chief librarian at the Library of Alexandria. He is also credited for inventing the discipline of geography. His method for determining the size of the Earth was an elegant application of simple geometry to an otherwise very difficult problem. Although he took some small liberties with facts and made a number of assumptions, his calculations were pretty accurate. What’s even more amazing is the fact that this epic calculation took him a negligible amount of time, given that all his calculations could be done by hand using simple trigonometry.
Also Read: How Do We Know The Mass Of The Earth?
Eratosthenes knew that the Sun was directly overhead in the town of Syrene, in southern Egypt, on the Summer Solstice. Since the Sun was directly overhead, a well situated in that town didn’t cast a shadow at its bottom. He also knew that his hometown of Alexandria was further north of Syrene, and the exact distance on foot between the two cities was easily determined.
Next, he measured the length of a tall stick called a ‘gnomon’ and stuck it in the ground in Alexandria. When the sun’s rays struck the gnomon, it cast a shadow. Because the sun was directly overhead in Syrene, it had to be obliquely tilted if observed from Alexandria. Hence, a shadow of the stick could be observed in the latter town.
The length of the shadow and the actual length of the stick allowed Eratosthenes to calculate the angle of inclination of the Sun. This can be done by taking the inverse tangent of the ratio between the shadow’s length and the stick’s length.
The angle labelled “A” is shown in the illustration above. By simple rules of geometry, it can also be observed that A is the angle subtended from the center of the Earth by the distance between the two cities. This angle was found to be nearly 7.2 degrees.
Now that Eratosthenes knew the subtended angle and the actual distance between the two cities of Syrene and Alexandria, he could calculate the total circumference of the Earth. The concept he used was very simple.
Distance between Alexandria and Syrene = D kilometers
Angle subtended by them on the center of the Earth= 7.2 degrees
Total angle of a circle= 360 degrees
Total Circumference of the Earth= ((360/7.2) x D ) kilometers
The unit of measurement that was popular in Greece at the time was stadia. Historians have yet to figure out the true length of a single stadia, but popular estimates put it around 160 meters. Eratosthenes estimated the circumference to be 252,000 stades, which is approximately 40,074 kilometers.
Shockingly enough… the real polar circumference is only 66 kilometers greater than this estimate!
The few assumptions that Eratosthenes made were trivial, but they need to be mentioned anyway. He assumed that the Sun was sufficiently far away from the Earth, so that its rays were always falling in parallel at the two cities. However, in reality, the rays at Syrene and Alexandria are ever so slightly inclined. Another liberty that he took was assuming that the distance between Syrene and Alexandria was very small, as compared to the circumference of the Earth, so it could be treated as a flat surface, rather than the arc of a larger circle. Needless to say, the Earth isn’t a perfect sphere and the two cities do not lie on the same latitude. Although the calculation is slightly flawed, this primitive technique provided an answer with only a 0.16% error, which is rather phenomenal!
Firstly, it was of paramount importance for us to find out how big our home actually was! Geographical research improved by leaps and bound following this determination. Through the study of eclipses, the size of the moon and the Sun was also easier to estimate, since we knew how big of a shadow the Earth cast, relative to its size. We even had a better understanding of seasons and their predictability.
Christopher Columbus studied what Eratosthenes had written about the size of the earth, but instead chose to believe another geographer named Toscanelli, who calculated the Earth’s circumference and came up with a result roughly 33% smaller than the actual value. Had Columbus set sail believing that Eratosthenes’ larger circumference value was more accurate, he would have known that the place he made landfall was not Asia, but rather the real New World, now known as America!
- Eratosthenes - www.astro.cornell.edu:80
- Astronomy 101 Specials: Eratosthenes and the Size of the Earth. Bucknell University
- Eratosthenes Earth Measurement. umich.edu
Harsh Gupta graduated from IIT Bombay, India with a Bachelors degree in Chemical Engineering. His pedantic and ‘know-it-all’ nature made it impossible for him not to spread knowledge about (hopefully) interesting topics. He likes movies, music and does not shy away from talking and writing about that too.
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Some 1,700 years before the famous expedition of Magellan and Elcano, which took more than three years to circumnavigate the Earth to verify that it is not flat, but round, the Greek polymath Eratosthenes managed to make that same finding and also estimate its diameter with a straight-forward piece of mathematical reasoning, without leaving the city of Alexandria and with surprising accuracy. The power of the mathematics developed by the classical Greeks was the key to performing this remarkable feat and managing to measure the impossible.
Eratosthenes was born in Cyrene, a city located in modern-day Libya, around 276 B.C. and in the year 236 B.C. became Chief Librarian of the prestigious Library of Alexandria . He made contributions in fields as apparently disparate as poetry, philosophy, mathematics, astronomy, history and geography, among others. As a mathematician, he is well known for the so-called Sieve of Eratosthenes , which makes it possible to isolate and determine all prime numbers up to a given natural number and which is still used today.
In addition, he knew how to apply basic mathematical knowledge, such as the calculation of the length of an arc of circumference—which is now studied in secondary school—in order to approximate the radius of the Earth very precisely, using only rudimentary instruments. In particular, Eratosthenes observed the shadow produced by the rays of the Sun during the summer solstice in two places far enough away from each other: Siena (now the Egyptian city of Aswan) and Alexandria, located north of Siena following the same meridian.
In the solar noon of that day, in a deep well of Siena, one could see for a very brief instant the reflection of the water it contained, which showed that the rays of the sun fell perpendicularly. This is true at the time of the summer solstice and on the Tropic of Cance (Eratosthenes placed Siena on that terrestrial parallel) However, at that same moment, in Alexandria (located about 7 degrees farther north) the rays fell at a slightly transversal angle, since obelisks or a simple cane stuck in the ground cast a small but perceptible shadow . This is already in itself a simple proof that the Earth cannot be flat, because if it were so, at that same moment in Alexandria the solar rays should also have fallen perpendicularly and not provided any shade.
A simple rule of three
Eratosthenes started from a model of a round Earth in the shape of a sphere, so he knew that the curvature of the Earth would cause this effect. He devised a method to calculate the diameter of the sphere from only two data points : the angle of incidence of the sun in Alexandria on the summer solstice (which is the same as the section of the circumference defined by the two cities) and the distance between them. In this way, with a simple rule of three he could calculate the length of the circumference of the Earth. If the angle of incidence gives rise to a length of an arc of circumference equal to the distance between Alexandria and Siena, then the total length will correspond to 360 degrees (the full circumference).
To calculate the angle of incidence of the sun’s rays in Alexandria on the summer solstice he had to use trigonometry concepts, which were already known to Greek mathematicians, although using methods very different from those used today. In current terminology, that angle of incidence is the value of the arctangent of the division between the shadow of an object and its height (see Figure 2). Eratosthenes obtained a value close to 7.2 degrees , or 1/50th the circumference of a circle.
To finish his calculation he needed a sufficiently accurate estimate of the distance between the two cities. Legend has it that Eratosthenes knew that a camel took fifty days to get from one city to another, traveling about a hundred stadia per day, so he estimated the distance at about five thousand stadia. The precision of his calculation is unknown, since the stadium is not a unit of measurement with a clear value. But if we consider as a measure of a stadium the one corresponding to the Egyptian stadium (157.5 metres), we would obtain an approximate distance of 787.5 km. Substituting these values in the rule of three above, we obtain a circumference length of 39,375 km . This is an excellent approximation of the actual value, which is about 40,075 km at the equator.
A model of the Earth that was quite successful
Eratosthenes had a model of the Earth and the solar system that was quite successful. Even though he made a series of assumptions that are not entirely accurate ( the Earth is not a sphere, the sun’s rays are not parallel, Siena is not directly on the Tropic of Cancer …), by combining modern capabilities with this same technique, a result extremely close to the real one can be obtained. Nowadays, this value is estimated using satellites and geolocation systems. These precise measurements allow us to detect even small modifications (of centimetres) on the surface of the Earth.
However, many centuries before, with hardly any technology, using the ingenuity and mathematics developed by their predecessors (Pythagoras, Archimedes, Euclid, Thales of Miletus…), other classical Greeks made amazing calculations, such as calculating the distance from the Earth to the Sun, predicting eclipses and the movement of known planets, and even proposing that the Sun was the centre of the Universe and not the Earth, as did Aristarchus of Samos. With these advances, they went beyond experimental knowledge, based only on direct measurements, to a much more ambitious conception of scientific knowledge, which allowed us to know things beyond our own immediate perception.
David Martín de Diego and Ágata Timón
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Eratosthenes (276 BC-194 BC) was a Greek mathematician, geographer and astronomer. He was born in Cyrene (now Libya) and died in Ptolemaic Alexandria. He is noted for devising a map system based on latitude and longitude lines and computing the size of the Earth. Eratosthenes studied at Alexandria and for some years in Athens. In 236 BC he was appointed by Ptolemy III Euergetes I as librarian of the Alexandrian library. He made several important contributions to mathematics and science, and was a good friend to . Around 255 BC he invented the (an astronomical instrument for determining celestial positions), which was widely used until the invention of the in the 18th century. Circa 200 BC Eratosthenes is thought to have coined or to have adopted the word geography, the descriptive study of the Earth. Eratosthenes' other contributions include: with an angle error of 7'. Before we begin a few definitions: Tropic of Cancer - is one of five major circles of latitude that mark maps of the Earth. The Tropic of Cancer currently latitude is 23° 26′ 22″ north of the Equator. Local noon is when the sun is the highest in the sky and can be quite different from 12:00 noon on the clock. Solstice - is an astronomical event that happens twice each year, when the tilt of the Earth's axis is most inclined toward or away from the Sun. In the northern hemisphere, the maximum inclination toward the sun is around 21 June (the summer solstice) and with the maximum inclination away around 21 December (the winter solstice). For the southern hemisphere winter and summer solstices are exchanged. What matters for our experiment is the fact that on the summer solstice, local noon, the sun rays are just overhead (at a right angle to the ground) on the Tropic of Cancer.
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Learn how the ancient Greek scholar Eratosthenes used geometry, trigonometry, and a well in Egypt to estimate the size of the Earth. Explore his achievements in geography, astronomy, and mathematics.
Join the global event on March 21, 2024 to calculate the circumference of the Earth using simple tools and eLearning. See the results, photos and winners of the previous Eratosthenes Experiment and the photo contest.
Learn how to measure the Earth's circumference like Eratosthenes did 2300 years ago using a stick's shadow and simple calculations. Find out the geometry, trigonometry, and geography behind this ancient feat of scientific thinking.
His little experiment demonstrated that the surface of the earth was curved like a sphere. ... Eratosthenes knew that the distance between Syene and Alexandria was 7.2 degrees along the surface of ...
Eratosthenes' experiment was one of the most important experiments in antiquity and his estimate of the earth's size was accepted for hundreds of years afterwards. It was, in fact, the most accurate estimate until Man was able to go to Space. Details. Last Updated: 14 June 2019.
Observations and results In 200 B.C. Eratosthenes estimated Earth's circumference at about 46,250 kilometers (28,735 miles). Today we know our planet's circumference is roughly 40,000 kilometers ...
Learn how Eratosthenes calculated the circumference of Earth using the angle of the sun at two locations. Find out the equipment, methods, and examples for this ancient experiment.
Eratosthenes reasoned that the ratio of the angular difference in the shadows to the number of degrees in a circle (360°) must equal the ratio of the distance to the circumference of the Earth. The resulting estimate, about 25,000 miles (40,234 km), is astonishingly accurate. In making his calculations Eratosthenes measured distance in stadia ...
So Eratosthenes made this measurement and found that the value for angle ``A'' is 7.2 degrees. He also knew that the actual distance between Alexandria and Syene was 5040 stades (1 stade = about 160 m) because somebody had measured it out by foot. Well 7.2 degrees is only 7.2/360ths of the way around the globe (since all the way around is 360 ...
Eratosthenes (born c. 276 bce, Cyrene, Libya—died c. 194 bce, Alexandria, Egypt) was a Greek scientific writer, astronomer, and poet, who made the first measurement of the size of Earth for which any details are known.. At Syene (now Aswān), some 800 km (500 miles) southeast of Alexandria in Egypt, the Sun's rays fall vertically at noon at the summer solstice.
Eratosthene calculated the circumference of Earth by measuring the length of a stick's shadow in Alexandria and the distance between Alexandria and Syrene on foot. He then took the inverse tangent of the ratio between the shadow's length and the stick's length to find the angle of inclination of the Sun. He calculated the total ...
Learn how the Greek mathematician Eratosthenes calculated the circumference of the Earth using only a stick, a well and trigonometry. Discover his ingenious method and the accuracy of his result, despite the limitations of his time.
Repeat Eratosthenes' Experiment. A simplified explanation of Eratosthenes' Experiment Eratosthenes and the Circumference of the Earth from Rogue Robot on Vimeo. Eratosthenes measured, at his local noon in Alexandria, the angle of elevation of the sun on the summer solstice (21 June). Eratosthenes used the local noon and no other time of the day ...
Eratosthenes' ingenious experiment involved observing the sun's angle at noon on the summer solstice in two different cities - Syene and Alexandria - which were located at the same longitude. He noticed that at noon on the summer solstice, the sun was directly overhead of a well in Syene, without casting any shadows, while in Alexandria ...
Eratosthenes of Cyrene (/ ɛr ə ˈ t ɒ s θ ə n iː z /; Greek: Ἐρατοσθένης [eratostʰénɛːs]; c. 276 BC - c. 195/194 BC) was an Ancient Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist.He was a man of learning, becoming the chief librarian at the Library of Alexandria.His work is comparable to what is now known as the study of geography, and ...
In the mid-20th century we began launching satellites into space that would help us determine the exact circumference of the Earth: 40,030 km. But over 2000 ...
He could then use simple proportions to find the Earth's circumference—7.2 degrees is 1/50 of 360 degrees, so 800 times 50 equals 40,000 kilometers. And just like that, an ancient Greek calculated precisely the circumference of our entire planet with just a stick and his brain over two thousand years ago. Eratosthenes accomplished many ...
Modern scholars disagree about the length of the stadium used by Eratosthenes. Values between 500 and about 600 feet have been suggested, putting Eratosthenes' calculated circumference between about 24,000 miles and about 29,000 miles. The Earth is now known to measure about 24,900 miles around the equator, slightly less around the poles.
Eratosthenes found a way, using none of the modern tools that we have, to measure the circumference of the Earth. And in this video, we're going to see how he did this. So the heart of Eratosthenes's measurement is a simple geometry problem. So consider the circle shown here, which has points A and B.
Eratosthenes (l. c. 276-195 BCE) was a Greek astronomer, geographer, mathematician, and poet best known for being the first to calculate the circumference of the earth and its axial tilt. He is also recognized for his mathematical innovation, the Sieve of Eratosthenes, which identified prime numbers, and his position as head of the Library at Alexandria.
A versatile scholar, Eratosthenes of Cyrene lived approximately 275-195 BC. He was the first to estimate accurately the diameter of the earth. For several decades, he served as the director of the famous library in Alexandria. He was highly regarded in the ancient world, but unfortunately only fragments of his writing have survived.
Eratosthenes was a talented mathematician and geographer as well as an astronomer. He made several other important contributions to science. Eratosthenes devised a system of latitude and longitude, and a calendar that included leap years. He invented the armillary sphere, a mechanical device used by early astronomers to demonstrate and predict ...