Eratosthenes (276-195 BC). Mathematician, poet, philosopher, geographer.

One of the oldest and still most useful applications of mathematics is using geometric relationships to derive the dimensions of things without directly measuring them. About 2000 years ago, an Ancient Greek scholar, philosopher, poet, and mathematician named Eratosthenes used staggeringly simple geometry to calculate the size of the Earth; possibly with very high accuracy. The Ancient Greeks were remarkable for their ability to use geometry to describe the scale of things that they could not directly perceive. Their geometry  was always correct and clever, but their conclusions were often way off due to inaccuracies in the measurements that the calculations were based off of. The scale of the Earth is one measurement that they might have nailed pretty close.

Eratosthenes of Alexandria (276-195 BC) was born in Cyrene (modern Libya), studied in Athens, and became the head librarian at the Library of Alexandria. The Library of Alexandria was the center of knowledge in the Hellenic world. Learned men of the Ancient world were not fettered with specializations. Eratosthenes first gained fame as a poet, invented the first system of latitude and longitude, calculated the tilt of the earths axis, invented the discipline of geography, developed an algorithm for calculating prime numbers, and was the first person to attempt an accurate map of the world based off of physical data.  Supposedly, his colleagues nick-named him $\beta$, because he was reputed to be at least second best in the world at just about everything.

I am a big fan of his map of the world.

No original copies of Eratosthenes’s world map survive. This one is a 19th century reconstruction.

The map above was assembled from land surveys and celestial observations conducted during the time of the conquests of Alexander the Great. To me there are three impressive things about this map. Firstly, it covers an extremely large portion of the earth for having been made in a time when neither humans nor ideas traveled faster than a horse or boat. Secondly, much of the information in the map was probably passed from word of mouth and through the compilation of many observations, made by many people, over a long period of time. It also does a relatively impressive job of portraying the rough shapes and directions, of land masses, oceans, and rivers.

From his map, it is apparent that Eratosthenes had no first, second, or third hand knowledge about an entire hemisphere of the earth, and most of Asia and Africa. Despite this, using simple geometry taught to me as a child, he was able to deduce the size of the earth, and hence the amount of our planet that was unknown to him, or anyone he knew.

He observed that on noon on the summer solstice of the northern hemisphere, at two different places, the sun’s rays fell upon the earth at different angles. In the city of ancient Syene (modern Aswan, Egypt), at noon on the solstice, the sun was directly overhead and cast no shadows on the objects below it (Aswan is very near the Tropic of Cancer). This was proven by observing that the Sun completely illuminated the bottom of a very deep well in Syene at noon. The angle between the suns rays, and objects perpendicular to the earths surface was effectively zero.

In Alexandria, on that same day at noon, the sun cast slight shadows behind objects. By measuring the dimensions of the shadows and the objects that cast them. Eratosthenes was able to calculate that the Sun’s rays arrived in Alexandria at an angle “1/50th” of a circle( about 0.13 radians or 7.2 degrees). He knew the distance between Alexandria and Syene to be 5000 “stades”, from land surveys done between the two cities. A stade was an Ancient Greek unit of measurement (150-200 meters). He also assumed that Syene was due south of Alexandria and on the same meridian of longitude. This would mean that the distance between the two cities represented an arc of the earths circumference (In reality, Aswan is about three degrees of longitude east of Alexandria).

A cross-section of the Earth, at noon, on the northern hemisphere’s summer solstice. Figure not drawn to scale. The angles are exaggerated for illustration purposes.

Eratosthenes assumed that the Sun was far enough away from the Earth, such that its rays arrived at the Earth parallel to each other. In the diagram above, $\angle A$ is the angle that the Sun’s rays arrived to Earth at Alexandria on noon of  the solstice. $\angle B$ represents the angle subtended  at the Earth’s center by the arc representing the distance between Alexandria and Syene. From the equivalence of alternate interior angles across a line transverse to two parallel lines, $\angle A = \angle B = 1/50$ of a circle. Since angle B is 1/50th of a circle and sweeps an arc at the earth’s surface of 5,000 stades, the Earth’s circumference would be $5,000\times 50 = 250,000$ stades. I am awed by how trivially simple, yet brilliantly clever that is. The curvature of the earth is not the most obvious thing I notice when I stand on the Earth’s surface.

Because there is a scholarly debate as to what the definition of a “stade” is, there is no way of knowing exactly how close Eratosthenes was to our currently accepted value for the circumference of the earth. It is thought that a “stade” is defined as the length of an Ancient Greek stadium used for athletic competitions. Most of them are about 1/10th of a mile, but some are longer and some are shorter. There is evidence that the word stade meant different things in different contexts, at different times, and in different places. It is not certain which definition of stade Eratosthenes used. The actual circumference of the Earth is 40,075 km at the equator.

Though his accuracy cannot be known for certain. It is safe to say, that he calculated the size of the earth to within an order of magnitude, and possibly got very close. If he was really accurate (2% error), then he was also probably lucky. Error’s could have arisen from the fact that Syene and Alexandria are not exactly on the same meridian of longitude. Measurement error would have been significant in establishing the distance overland between Alexandria and Syene. There would have been measurement error in measuring the angles cast by the shadows at Alexandria.

It is a common modern misconception to assume that past civilizations believed in a flat earth of unknown dimensions. Eratosthenes’s calculation proliferated far and wide and was preserved by successive generations, and geographically disparate civilizations. This estimate of the size of the earth and it’s method, would remain canonical for the educated elite of the future civilizations of Europe, North Africa,  the Middle East, and South Asia till the 17th century, when more precise measurements could be made to constrain the size of the earth..

From → General

its more useful and interesting method..

It’s beutiful

that is very intersting i like it.

thoughtful and interesting

Its amazing and wonderful. I really like it.

7. brilliant way to start a trig lesson! Thanks

A very interesting topic and well written.

why would the 3 degree error make any difference?if the distance between the cities is the same,then it is still an arc of the earth’s circumference just not on any meridian.stating that the 3 degrees caused the difference is like stating a clock is incorrect if it is slanted.of course this is assuming the distance was 5000 stades all the time.

Good question! You are right that the distance (the way a crow flies) between the two cities is an arc of the earth’s circumference whether that arc is in the north-south plane or not. The error arises because if the arc deviates from the north south plane, then the angular distance of that arc cannot be equated to the difference in the the angles at which the sun arrives at the two cities. The difference in arrival angle of the sun effectively measures differences in latitude. As it is currently written that is not made explicitly clear. Thanks for noticing.

thanks for the explanation

very beautiful and the deliberation of the topic can really catch the interests of the readers

Thank you to everyone for the feedback.

may i know why the discussion page is not displayed??????????plzzzz answer………

There is a sample of the calculation in the blog post. Here is a link to a more general discussion of measuring the arc of a meridian.

may i know why the discussion page is not displayed??

Measuring Earth’s Radius à la al-Biruni.
Recently I got interested in the story of a great Central Asian polymath Abu Rayhan al-Biruni (known as Albironius in the west). Biruni lived in the 10th and 11th centuries AD and contributed to numerous scientific disciplines that I am not going to list here, but which can be easily found on the web.

The way I approached the topic was via the book Brilliant Biruni by M. Kamiar. Except for a few supremacist statements embedded, the book is very easy to read and provides an interesting account of Biruni’s life.

One thing I particularly enjoyed was the way Biruni applied abstract geometry developed by the Greeks to the real world problems. In particular, Biruni provided a very elegant way of calculating the earth’s radius.

Biruni’s method proceeded in two steps. First, Biruni chose a mountain and measured its height by choosing two points at the sea level with known distances between them (Fig 1.)

Fig 1: Measuring the height of a mountain

Suppose that we know the angles and the distance d in the Fig 1, then we can easily determine the height with the following formula

Eq 1: Computing the height

In practice, Biruni used an astrolab to measure the angles. Then, Biruni climbed the mountain and used the astrolab again to measure the angle to the horizon (Fig 2.)

Fig 2: Measuring the earth’s radius

Then finally he could compute the radius via the following formula

In 10th century Beruni obtained the number 6339.6 km, which is remarkably close to the equatorial radius of 6378 km.

Note that an astrolab is just another name for an inclinometer, which can be easily assembled at home by combining a protractor, a string, a straw, and some weight [click here].

So, here it is a brilliant application of trigonometry by a famous mathematician whose works played a significant role in our development. To conclude I quote a well known joke for mathematicians:
Mathematicians never die – they only lose some of their functions.

radius of earth according to al biruni method =h*cos α/1-cosα

h is the height of the mountain

cosα is the inclintion of sun rays at a top of a mountain beside a sea

Al Biruni is a fascinating man, and another example of an impressive polymath thinker. Here is a link to a wikipedia figure illustrating the method that “Anonymous” has described above

18. The article is great! Here’s something from me on the same theme:

http://m4t3m4t1k4.wordpress.com/page/4/