## Eratosthenes Measurement of the Radius and Circumference of the Earth

The radius and circumference of the Earth were first measured in around 200 BCE by a man named Eratosthenes (276 BC – 194 BCE). A true polymath, Eratosthenes was an astronomer, mathematician, poet, and inventor, to name but a few of his arts. He also occupied the exalted post of Chief Librarian of the famed Library of Alexandria, and invented the discipline of geography.

His method for measuring the radius and the circumference of the Earth was truly ingenious, using nothing more than two sticks, basic trigonometry, and a single measurement within his actual experiment.

## Eratosthenes Experiment

Eratosthenes was born in Cyrene, Libya, and studied in Athens where he became recognized for his outstanding ability in the arts and disciplines touched on above, which is what ultimately brought him to Alexandria, Egypt, at the behest of Ptolemy III. The position of chief librarian of the Library at Alexandria no doubt was the best possible situation for this man of wide learning to continue his search for knowledge.

His education in Athens certainly was important, because it was one of the hubs of knowledge and learning at the time. However, even the hailed knowledge and intellectual advancement of Greece, thus Athens, at the time must ultimately be traced back to the advanced knowledge of Egypt, Mesopotamia, and even India, from whence it seems clear that much of Greek knowledge finds its roots.

As an intelligent astronomer, privy to the advanced astronomical knowledge that seems to have been handed down since the dawn of civilization in Sumer and of recorded history (indeed, astronomical texts where some of the first written down), Eratosthenes knew that on the summer solstice in Syene, Egypt (modern-day Aswan) the sun would be directly overhead. Perhaps this phenomenon was popularly known at the time, or perhaps he learned of it through his continued education during his time at the Library of Alexandria, or from the other learned individuals he would have come in contact with there. Frankly, we do not know.

What we do know is that this phenomenon occurs due to the Earths axial tilt (about 23.5 degrees from vertical, vertical relative to the plane of our orbit that is) and the orientation of the Earths axis towards and away from the Sun throughout our yearly orbit. It turns out that at the summer solstice in the northern hemisphere, when the Earths position in our orbit aligns the norther polar axis as directly towards the sun as it will get which it does at only this time throughout the year, it is from the latitude of about 23.5° N that the sun will be located precisely overhead. This latitude is known as the Tropic of Cancer. In other words, at precisely noon it will be about an equal 90 degrees in all directions from the true horizon.

This means that any vertical object on, or relatively close to, the Tropic of Cancer will not cast a shadow. Knowing this, and also being aware that in Alexandria at this time shadows where indeed cast, therefore that the sun was not directly overhead, and that the earth was a sphere, Eratosthenes devised an experiment that could take advantage of this phenomena and enable him to measure the radius and circumference of the Earth. He concluded that because the Earth is a sphere, if he erected a stick vertically in the ground in Alexandria, a shadow would be cast by the stick which, if measured at noon, would allow him to calculate the circumference of the Earth and then its radius.

## The Measurement

At noon on the summer solstice in Alexandria, which is about 843 km north of Syene/Aswan, Eratosthenes measured the length of the shadow of a stick that he had previously erected. He did not need to take any measurements in Syene because he already knew that at this time the sun would be directly overhead, thus no shadow would be cast.

By measuring the length of the shadow cast, and with the known length of the stick from the ground up, Eratosthenes was then able to use basic trigonometry to figure out the angle of the imaginary triangle drawn by the shadow and the length of the stick. The actual angle that he sought to measure was that created between the stick itself and the beam of light extending from the tip of the stick to the furthest length of the shadow. In the diagram to the right, this is the angle A.

This triangle was a right triangle, the right angle being that between the stick erected vertically, and the ground. With his two known values (the length of the shadow and the length of the stick), Eratosthenes then used the trigonometric relationship below to solve for θ, in this case angle A in the diagram, at the top of the stick towards the end of the shadow:

$tan\theta \quad =\frac { opposite }{ adjacent } =\frac { length\quad of\quad shadow }{ length\quad of\quad stick }$

Now, this angle represents a couple of different things. Firstly, since the sun would be directly overhead in Syene at this time, an imaginary line extending infinitely into space through the length of a stick planted vertically in Syene would pass through the Sun. Yet at this exact same time an imaginary line extending out of the top of the stick in Alexandria would not point directly at the sun, which would actually be off by about 7.2 degrees due south of this imaginary line in space, according to Eratosthenes measurement. So the difference in position of the sun viewed in Alexandria at this time, compared to Syene, was that it would have been about 7.2 degrees further south in the sky (from directly overhead).

This phenomenon occurs because of the curvature of the Earth, and Eratosthenes knew this. Alexandria is 5000 stadia away from Syene not in a straight line to the north, but along the curvature of the Earth’s circular circumference, making Alexandria further away from the sun at this time than Syene along this curvature of the Earth northwards (see the illustration above). So a perpendicular line drawn to the level of flat (horizontal ground) in both places would both be pointing in different directions, due to the curvature of the Earth, which is why a shadow would be cast in Alexandria and not in Syene. And it was this relationship between the difference in the angle of the sun and the distance between these two places which Eratosthenes was using to calculate the circumference of the Earth.

So if you drew a line from the center of the Earth to both Alexandria and Syene, as in the illustration above, the angle between these two lines (angle B) is equal to the calculated angle between the shadow and the stick, 7.2 degrees, which is also equal to the angle of perceived change in location of the Sun in the sky as viewed from these two locations. In Syene the sun would be directly overhead, but in Alexandria at the exact same time the sun would be about 7.2 degrees south of directly overhead. This means that the distance between these two locations (5000 stadia) also equates to 7.2 degrees of a full circle (360 degrees), which represents a 7.2/360 degree fraction of the full circumference of the Earth.

Interestingly, this method works with any two locations so long as the distance between them is known, and the angles created by the shadows are measured at the same time. The difference in this case is that if shadows are created in both places, the difference in angle is what equates to the distance between them. So if the measured difference in angle between two hypothetical places was 14.4 degrees, the distance between them would then be about 10,000 stadia, because according to Eratosthenes estimates, about 695 stadia is equal to 1 degree of arc along the circumference of the Earth (according of course to the estimations of Eratosthenes, and not precisely to modern measurements or calculations). In the case of Alexandria and Aswan, the difference in angles is 7.2 because there is no shadow created in Syene, so the value of the angle in Aswan (essentially that of the vertical declination of the sun) is zero.

In Eratosthenes case, it helps that Syene and Alexandria are close to the same longitude, because this means that the timing of the measurement in Alexandria was the same time as the phenomenon in Syene (noon), without having to account for factors such as time difference due to difference in longitude (which is where our modern time zones come from.

## Calculating the Circumference of the Earth

The distance between Aswan and Alexandria was estimated to be 5000 stadia. There are a few different theories as to where this value came from, but we do not actually know. The actual length of a stadia or stadion in modern units of distance measurement is debatable, as there were many different values for the stade that were used over time. (Refer to The Different Lengths of a Stadia for a table of these different values.)

Moreover, it is uncertain which length of stade that Eratosthenes actually used, which lends further uncertainty to the precision of his calculation. It is generally believed that he used the Italian/Ptolemaic stade (equal to about 185 meters), although this length apparently did not come into prominent use until roughly 300 years after Eratosthenes time. So we will do the calculations in the Olympic stades as well, measuring about 176 meters.

The distance between Alexandria and Syene was thought to be 5000 stadia, which equates with either 880 km (Olympic stade), or 925 km (Italian/Ptolemaic stade). The actual distance between Alexandria and modern Aswan is about 843 km. So either way that estimate is pretty good considering the level of technology at the time. However, if he did use the Olympic stade length, then this estimate was fairly accurate indeed, which will have made his estimate for the radius and circumference of the Earth correspondingly more accurate.

Eratosthenes knew that if he were to draw a sphere representing the Earth, and from the vertex in the center of the circle which represents the center of the Earth, draw one line to Alexandria, and another from the center to Syene, the resultant angle would be 7.2 degrees, as discussed above. And because he knew the distance between Syene and Alexandria, he knew that 5000 stadia of distance around the circumference of the Earth was equal to the 7.2 degree angle within the center of the Earth.

So in other words, 5000 stades along the circumference of the Earth was equal to 7.2 degrees of arc along the circumference of the Earth. And because 7.2 degrees is equal to 1/50th of a full sphere (360/7.2 = 50), he was able to do the simple calculation of 5000 stadia multiplied by 50 to estimate the value of the circumference of the Earth. As can be seen below:

$\frac { 1 }{ 50 } =\frac { 5000\quad stadia }{ Earth\quad Circumference }$

And so:

$c\quad =\quad 50x5000stadia\\ c\quad =\quad 250,000\quad stadia$

By this very calculation Eratosthenes arrived at his estimate for the circumference of the Earth being 250,000 stadions. This value has differing modern equivalents depending on the actual length of the stadia that Eratosthenes used. If we use the Italian/Ptolemaic length of the stade (=185 meters, or 0.185 km), from 250,000 stades we get a circumference of 46,250 km. The modern value, using all of our technology and advanced science is about 40,075 km. An error of only ~ 13%. If we use the Olympic stade, measuring around 176 m, we get a circumference of 44,000 km. An error of only 9%.

Though we are not certain of what the specific length of the stade was that Eratosthenes used (and the precision of his measurement depends directly on which length he used), the mathematical and geometrical principles, and the fantastic ingenuity of his method, still enable us today to acquire a relatively accurate measurement of the radius and circumference of the Earth using our far more accurate modern measurements of distance, as will be seen below. Thus his principles still apply, and the degree of error was due only in part to necessary generalizations and assumptions made in his method, but mostly due to the lack of precision in measurement of distance available during his time.

The modern measurement of the circumference of the Earth depends on how you measure it, because the Earth is not a perfect sphere. Moreover, it bulges at the equator due to the Earths rotation. However, derived from NASA’s value of the (presumably) average diameter of the Earth, 12,756 km, we can approximately calculate is circumference, using equations of the relationship between the diameter or radius of a circle, to its circumference, also incorporating π. Where c = circumference, d = diameter, and r = radius.

$c=\pi d\\ c=\pi \times 12,756km\\ c=40,074.16km$

Calculating the Earths circumference in this fashion is assuming that the Earth is a perfect sphere just as Eratosthenes did, which it is not. Nonetheless, this value is quite accurate. Depending on the length of the stadia that Eratosthenes used, the value could have been anywhere between 39,250 km – 52,250 km. But what if we do his same calculations, but with the modern measurement of distance between Alexandria and Syene? The modern distance between Alexandria and Syene is about 843 km, so:

$c\quad =\quad 50\times (distanceA-S)\\ c\quad =\quad 50\times 843km\\ c\quad =\quad 42,150km$

So using Eratosthenes methods and our modern measurement of distance we get a value of 42,150 km for the circumference of the Earth, an error of only 5%. Ingenious.

Once Eratosthenes had estimated the circumference of the Earth, the radius of the Earth was easy to figure out. So using Eratosthenes own estimate for the circumference of the Earth, and the equation above rearranged with basic algebra to solve for radius (r), we get:

$c=2\pi r\\ r=\frac { c }{ 2\pi } =\frac { 250,000 }{ 2\pi } \\ c\quad =\quad 39,789\quad stadions$

Once again, depending on the length of the stade used, this value of 39,789 stadions could translate to anywhere between 6,247 km – 8,316 km for the radius of the Earth. The modern value for the radius of the Earth being half the diameter used above, so about 6,378 km.

## Closing :: A Spherical Earth

It has been noted that Eratosthenes made two important assumptions:

1. That the Earth was a perfect sphere.
2. That Alexandria was 5000 stadia away from Syene/Aswan.

This is perfectly true. However, the assumption that the Earth was a perfect sphere or not is besides the point in my opinion. Even today we would assume the Earth to be perfectly spherical in some cases for mathematical purposes in order to get an approximation of the radius and circumference of the Earth, as I demonstrated above by deriving an accurate circumference measurement from NASA’s value for the Earth’s diameter. We have to assume the Earth is a sphere to use that equation at all. And the value is quite accurate too, for the modern measurement of the circumference of the Earth is actually 40,075 km. Moreover, the 5000 stadia distance between Syene and Alexandria was probably the best measurement Eratosthenes had available at the time.

I believe that the most crucial aspect of this entire calculation was that the entire experiment was literally founded on the understanding and awareness that the Earth was spherical. (Aside from the fact that Eratosthenes achieved the first ever estimates for the radius and circumference of the Earth of course, which where remarkably accurate.) For the understanding of a spherical Earth was not deeply understood again until the work of Nicolaus Copernicus (1473-1543 CE) about 1800 years later! And even then it wasn’t fully accepted and understood by the majority of people until around 100 years after Copernicus’ death, thanks to the work of Galileo, Tycho Brahe, and Kepler (and due also to the vicious resistance of the church). Even today there are people who believe the Earth is flat. To disprove this simply and conclusively, all one needs is a celestial understanding of the Earths season and the moon phases.

The fact is however, that Eratosthenes did not even set out to prove that the Earth was spherical. He knew this already. He was trying to determine the radius and circumference of the Earth, of our spherical Earth. Throughout his ingenious method he displayed his intelligence, creativity, and deep understanding of the advanced knowledge of mathematics, astronomy, and geometry which he was certainly well versed in. Knowledge that it seems obvious was far ahead of its time.

How is it that the knowledge in deep antiquity was superior to that of later times? The fact that Eratosthenes lived in Egypt and was the chief librarian of the Library of Alexandria is suggestive here, because it shows first that he had access to the most advanced – and ancient – texts and knowledge available at this time. A repository and storehouse of knowledge from antiquity, from the most advanced ancient cultures which Alexander the Great plundered from all of the great cities, libraries, centers of learning, and empires that he conquered on his way to India. But to his credit, he sought to preserve this advanced knowledge of antiquity, and sent it back to the library in Alexandria.

In fact, Eratosthenes was not even the first to say that the earth was round. Pythagoras from the 6th century BC taught that the Earth was round. Aristotle (385 – 323 BCE) also expounded on Pythagoras’ work, providing proofs such as that when one moves south, the southern constellations move higher in the sky. And when there is a lunar eclipse, the shadow of the Earth on the moon is spherical, and when a ship moves out to sea from the docks, its hull disappears first to an observer on land, eventually leaving only the tip of the mast visible as it moves further and further away. All of these being proof of the curvature of the Earth. Pythagoras’ students, the Pythagoreans, believed this also. That there where Pythagoreans who said that the Earth was a sphere is confirmed by no less a person than Nicolaus Copernicus in his landmark book derived from his own study of what was left of the work of the ancients:

“Some think the Earth is at rest; but Philolaus the Pythagorean says that it moves around the fire with on obliquely circular motion, like the sun and moon.” – Nicolaus Copernicus from On The Revolutions of the Heavenly Spheres, citing an excerpt from Plutarch.

It seems clear that in the above quote, Philolaus’ reference to the fire is a reference to the Sun, and that it was Plutarch who did not understand and then added “like the sun and moon”. Or neither of them understood and both where repeating an earlier, but fundamentally lost, tradition. Possibly that taught by Pythagoras himself. But he was known, or is at least believed to have studied in Egypt or the Near East, as most of the greatest thinkers of antiquity were known to have done. All where certainly the beneficiaries of this ancient advanced knowledge, yet their own contributions and development of this knowledge is not to be understated either.

There is other evidence and indications that there was high knowledge in antiquity, which we will cover in time. But beyond this we can speculate no further from the evidence at hand on the origins or extent of the advanced knowledge of antiquity. Nor where it came from. Nor the knowledge lost in the Library of Alexandria. For now let us leave the story there, and simply marvel at the work of an incredible man of antiquity, Eratosthenes, for giving us the first fairly accurate measurement of the radius and circumference of the Earth.

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