### Video Transcript

In this video, we’re gonna take a look at how a Greek chap called Eratosthenes, who was a mathematician, astronomer, poet, music theorist, and chief librarian at The Great Library of Alexandria in Egypt, managed to make a remarkable calculation to estimate the circumference of the Earth over 200 years BC. He is believed to be the first person ever to come up with a relatively accurate way of doing this. But it’s actually quite tricky to say exactly how accurate he was. And we’ll explain why as we go through his method.

In one of the books in The Great Library, Eratosthenes read about a water well in the city of Syene — at fair distance south of Alexandria — in which the water remained in shadow all year round, apart from briefly at noon on the longest day of the year. A passage in the book described how the edge of the shadow passed down the side of the well until the water at the bottom was directly lit by the rays of the sun and no shadows existed.

And this set him thinking two things. Firstly, the Sun must have been directly overhead at Syene to make this happen as the Sun’s rays must have fallen at right angles to the Earth’s surface. And secondly, in Alexandria where he lived, he’d never seen that no-shadow situation. At noon on the longest day of the year there, the shadows were very short, but they did exist. Now, he figured that within a relatively local part of the Earth, the light rays from the Sun must all be roughly parallel because the Sun was so far away.

So the surface of the Earth must be curved to have different-length shadows in different places at the same time. If the Earth’s surface was curved all the way round, then maybe it was spherical. He decided to carry out an experiment to take some measurements and do some calculations to work out the circumference of the Earth.

Now, reports vary as to how he measured the angle of the Sun’s rays in Alexandria. Some say he used the shadow cast by a tall tower and other say that he used a special measuring rod called a gnomon. But either way, the method would basically be the same. The only real difference is that any slight measurement errors on a smaller rod would have a larger impact on the final accuracy of the estimate than similar errors on the tower. But we’ll come back to that a bit later.

To make it easier to talk about the method, let’s just say he used a rod. Although Eratosthenes would have done different calculations because of the mathematics and conventions that were available to him at that time, with our knowledge of high school trigonometry, we can calculate the measure of the angle of the Sun’s rays from the vertical.

Stand up a rod vertically from the ground, making sure it’s exactly 90 degrees to the ground, and then simply measure its height ℎ and the length of the shadow 𝑙. Then, the angle 𝜃 here at the top is equal to the inverse tan of the length of the shadow divided by the height of the rod. These days we can just tap that into our calculator to get an answer. But of course, that wasn’t an option for Eratosthenes. And when he did this measurement, he found that 𝜃 was about 7.2 degrees.

Then, let’s just draw a slightly exaggerated version of this diagram to make it easier to see what’s going on. Then from our diagram, we can see that if we extend the lines of the well and the rod down to the center of the Earth, the angle at the center of the Earth is alternate to the 7.2-degree angle. So it’s also 7.2 degrees.

That means that the arc on the surface of the Earth between the well and the rod is 7.2 over 360 of a full circle. And that simplifies to one fiftieth. So the distance between the rod and the well is one fiftieth of the Earth’s circumference. All he had to do was measure the distance between the well and the rod and multiply the answer by 50. Then, he had his estimate for the circumference of the Earth.

The typically used length for measuring the distance between towns of that time was the stadion, which was the length of a standard Greek sports stadium. One stadion was the same length as 600 Greek feet. Eratosthenes had been involved in a number of geographical surveys, in which people paced out the distances between towns in Egypt. So he was easily able to look up the distance between Alexandria and Syene. It was recorded as 5000 stadia. Then, it was just a matter of multiplying that by 50 to get an estimate of 250000 stadia for the Earth’s circumference.

So how accurate was he? Well, it depends. It’s widely reported that he was remarkably accurate. Well, certainly, he was much more accurate than the Greek astronomer called Posidonius, who used the angles of stars to estimate the circumference of the Earth over 100 years later and came up with a much smaller and much more inaccurate answer. It was Posidonius’s estimate that Christopher Columbus used many centuries later when he set off to find a different route from Europe to Asia and landed on the American continent instead.

But the Earth isn’t actually exactly spherical. It’s what we call an oblate spheroid. So it’s a bit like a sphere, but squashed slightly between the poles — although not as exaggerated as my diagram suggests. With all our modern technology, we now know that depending on the way you measure it, the circumference is between about 40008 and 40075 kilometers. And because Eratosthenes was attempting to measure the circumference through the poles, the actual circumference he was trying to measure was close to 40008 kilometers.

Now, it’s quite tricky to interpret Eratosthenes’s answer of 250000 stadia and convert it into kilometers for a comparison. Archaeological digs have revealed that Greek sport stadia actually varied in size. The ones we know about range from about 157 meters to 209 meters in length. So that would mean that his circumference could have been as low as 39250 kilometers or as high as 52250 kilometers. Now lots of people suggest that he would have used the Olympic Stadium length of 176.4 meters for his unit, which would have made the Earth’s circumference 44100 kilometers, about 10 percent larger than the actual circumference.

But more importantly, as with all good school math, it’s not just the answer we need to think about. It’s the method. You might come up with a near perfect numerical answer to a problem. But maybe it’s only accurate by coincidence if you used an incorrect method. The impressive thing is to get the right answer and show that you used a proper valid method to get it.

When we look at Eratosthenes’s method and assumptions, we discover a few minor issues. And it’s important to understand these before we talk about how accurate or remarkable his estimate of the Earth’s circumference was. When you make approximations and small mistakes in your assumptions, sometimes, you get lucky and their effects cancel each other out. And sometimes, you get unlucky and their effects compound each other to make your answer even more incorrect.

Firstly, Eratosthenes assumed that the Earth is a perfect sphere. And as we said, this isn’t quite true. Let’s look at this exaggerated comparison of a spherical Earth with an oblate spheroid like Earth actually is. You can see that this circumference around the poles is in the shape of an ellipse. And the same angle at the center will mark out a different-length part of the circumference depending on how close to the equator or pole you are.

If that 5000-stadia distance between the two towns was near the equator and you multiplied it by 50, then you’d be overestimating the circumference. But if it was near the poles and you multiplied it by 50, then you’ll be underestimating it. In fact, the two towns are about 24.1 degrees and 31.2 degrees north of the equator. So it looks like we’re in the region where it will lead to a slight overestimate of the answer we’re looking for.

Secondly, he figured that the Sun was directly overhead in Syene at midday on the longest day. But, from our more accurate modern knowledge, we know that the town was just under half a degree further north than the point at which the Sun would have been directly overhead at the Tropic of Cancer. This means that he should have subtracted a very small amount from the length of the shadow in Alexandria to get the difference in shadow length between the two towns. And that would’ve led him to a slightly smaller angle between the vertical and the Sun’s rays. So he should have multiplied the 5000 stadia by slightly more than 50. And this led to an underestimate of the circumference.

Thirdly, he assumed that the Sun’s rays were exactly parallel, but they would actually be very slightly splayed. So the shadow he measured in Alexandria was slightly longer than it would’ve been if they were parallel. And this would have led to a small overestimate of the angle of the center of the Earth and hence a small underestimate of the circumference of the Earth.

Fourthly, this rounded figure for the distance between the two towns was very likely to be inaccurate although we don’t know for sure whether it’s an underestimate or an overestimate. So it’s difficult to know what impact this would have had on the final calculations.

Lastly, he assumed that Alexandria was directly north of Syene so that they were on the same circumference. But there’s actually about a three-degree difference between their longitudes. And a little bit of work with the Pythagorean theorem enables us to find out that the 5000-stadia distance between the two towns was up to eight percent larger than the distance between the latitudes that he actually needed. And this led to an overestimate of the circumference of the Earth.

So overall, Eratosthenes’s method was basically right although some of his measurements led to an answer that was too big, while some of the other measurements cancelled out some of these errors in the other direction, leaving him with an answer that was probably about 10 percent too large overall in the end.

Now, thinking about how accurate each piece of the puzzle is and using that to analyze how far out your final answer might be is an incredibly useful thing in real-world mathematics and engineering. So Eratosthenes was certainly a remarkable scholar and his endeavor to estimate the circumference of the Earth was brilliant. But his work also serves to highlight why we shouldn’t concentrate too much on just a single answer. We can also try to evaluate how confident we are in that answer, and what are reasonable upper and lower bounds given the source of measurement errors and rounding errors we’re likely to have made along the way.