Eratosthenes

 

Eratosthenes of Cyrene Teaching in the Library of Alexandria

Eratosthenes was an Ancient Greek scientist born in the town of Cyrene in about 276 BC. Cyrene is now the town of Shahhat in Libya. He was educated in philosophy and mathematics in Athens. We do not really know what he looked like. This image above is from a painting by Bernardo Strozzi in the year 1635, 19 centuries after the era of Eratosthenes. It shows Eratosthenes teaching geography, the academic discipline he invented.

Quick Guide – Eratosthenes’ Greatest Achievements

• Assuming Earth was a sphere, in about 240 BC Eratosthenes calculated its size with good accuracy. This was a moment of triumph for the human intellect: first to realize our planet was a sphere, then to use the power of observation, deduction and mathematics to calculate its size.

• Eratosthenes knew that Earth was spinning once a day around an axis that formed an imaginary line from North Pole to South Pole through the center of the earth. He calculated the tilt of Earth’s axis with good accuracy.

• He produced the first map of the world which used meridian lines and parallel lines. These were similar to our modern lines of latitude and longitude. He marked the equator and its size, considered the size of the polar zones and how far these zones were from the tropics. (Evidently, the Ancient Greeks knew a lot about our planet!)

• He invented geography. Today we still use the word he invented for this new discipline. (‘Geo’ was Greek for ‘Earth’ and ‘graphy’ meant ‘field of study’.)

• He produced a reliable, logical method to discover prime numbers: The Sieve of Eratosthenes. In an updated form, this is still important in modern number theory.

• He produced a timeline recording all of the achievements of science since the time the Greeks had laid siege to Troy.

• He was the first person to explain why the River Nile flooded every year – i.e. that heavy, seasonal rains fall near the source of the river, resulting in an annual flood in Egypt.

• He rejected the commonly held view that people could be divided into ‘Greeks’ and ‘Barbarians.’ He held that people should be judged as individuals on their good and bad qualities.

A Rounded Character

Not restricting himself to science, Eratosthenes excelled at nearly all things intellectual. He wrote books on philosophy, geography, mathematics, astronomy, history, comedy, and also wrote poetry.

This all-round knowledge made him a shoe-in for one very special job. That job was the most prestigious role an academic could enjoy in Ancient Greece – Director of the Library of Alexandria, the greatest intellectual institution of the ancient world.

The Library of Alexandria is said to have contained over half a million books in scroll form. It was the place the greatest scientists, mathematicians, philosophers, poets, and dramatists gathered to talk about their intellectual quests. The Library had lecture halls and meeting rooms. Today, we would call it a great university.

Eratosthenes had his fans and his critics. Both seemed to have used the same nicknames for him. Eratosthenes’s had two nicknames we know about. One of these was ‘Beta’ the second letter of the Greek alphabet. This was because Eratosthenes, although he knew about everything, was not the absolute best at anything. ‘Beta’ could be used as an insult or a compliment.

The other nickname he was given was ‘the pentathlete’. The meaning is similar to Beta, in that a pentathlete has to be very good at five different sports, but will probably not be a champion in any of the individual sports.

Eratosthenes called himself ‘philologos’ – ‘lover of learning.’

The greatest minds of Greece would send their work to Eratosthenes as papyrus scrolls. Ever curious, Eratosthenes would take time to read many of these before his assistants cataloged them.

Library of Alexandria

Cataloging of scrolls at the Library of Alexandria, drawn in the 1800s based on the scholarship of the time.

His friend, the great Archimedes entrusted him with an enormously important treatise called “The Method.” In addition to containing the most advanced mathematics the world had ever seen, “The Method” gives us some clues to Eratosthenes. Archimedes writes (loosely translated):

“Since I know you care about your work, are an excellent teacher of philosophy, and greatly interested in mathematical investigations, I thought I’d let you know about my special method. The method will enable you to use mechanics to see the answers to mathematical questions…”

And so we learn that Eratosthenes is a fantastic teacher, as well as an intellectual. (Not all intellectuals are good teachers!)

Of course, Eratosthenes did not only read great work from other people. He did some great work himself.

We know him best for two important achievements; producing an accurate estimate of how big Earth is; and devising a method to find prime numbers.

Unfortunately, other than a few scraps, little remains of Eratosthenes’s original work. We usually have to rely on comments from people who were around at the time and in the following few centuries to get an idea of what Eratosthenes wrote.

How Big is Planet Earth?

As Director of the Library of Alexandria, Eratosthenes had the closest thing in the ancient world to a modern internet search engine. Everything that had been learned about the world by the Greeks – and that was a lot – was just about at his fingertips.

Eratosthenes had learned that at midday, on the longest day of the year, walls in the city of Syene cast no shadows, because the sun was directly overhead. He could see with his own eyes in Alexandria that there were small shadows at midday on the longest day.

Syene was more-or-less due south of Alexandria, therefore the angle of the shadow must mean something. What did it mean?

Eratosthenes reasoned that if:

1. you make the assumption that planet Earth is a sphere
2. the sun’s rays are parallel to one another when they reach Earth
3. you measure the angle of the shadow in Alexandria when there is no shadow in Syene
4. you know the distance between Alexandria and Syene

then, you could calculate how big Earth is.

Some Key Parts of Eratosthenes’ Reasoning

How Eratosthenes Calculated Earth's Size

The Calculation

On the diagram above, the angle z is the angle of shadow Eratosthenes found in Alexandria. He found it was one-fiftieth of a whole circle.

Using a little simple Euclidean geometry, he knew that by drawing a line downward from each wall to the center of the earth, they would form the same angle z.

This meant that the distance from Syene to Alexandria was one-fiftieth of the distance all the way around planet Earth.

His maps told Eratosthenes that the distance from Alexandria to Syene was 5000 stades. Eratosthenes multiplied 50 x 5000 to get an answer of 250,000 stades for Earth’s circumference. He then added a correction of 2000 stades (lacking his original work, we don’t know why he did this) and concluded that:

Earth’s circumference is equal to 252,000 stades

All we need to do now is convert stades to modern units. (Some things are easier said than done!)

How Good Was Eratosthenes’ Estimate of Earth’s Size?

We can’t say exactly how good his estimate was, because the ‘stade’ unit of length meant different things to different people.

Most likely, Eratosthenes would have said which type of stade he was using when he wrote his book, ‘On the Measurement of the Earth,’ but the book is lost in the mists of time.

Whichever stade was used, Eratosthenes overestimated the size of Earth. Depending on which stade he used, we can say that his estimate was, at best, within 1% and, at worst, within about 30% of the value we use now. (Our current value for Earth’s polar circumference is 40,075.16 km or 24,901.55 miles.) Whichever way you look at it, this was an enormous advance in an era when most people in the world had no idea that our planet is approximately spherical.

The Sieve of Eratosthenes

Using his ‘seive’ Eratosthenes solved the problem of how to find prime numbers logically and systematically.

Prime numbers are those numbers with no factors except for themselves and 1. Mathematicians look on them in the same way as chemists look on the chemical elements. Prime numbers are the building blocks of all other numbers.

The first eight prime numbers are 2, 3, 5, 7, 11, 13, 17 and 19. The prime numbers go on forever. There are an infinite number of them. This had already been proven by the Greeks, and the proof was written in Euclid’s Elements.

To use the Sieve of Eratosthenes, first decide the highest number you wish to check. Then, starting at 2, write down all of the numbers up to the highest number. Say you wanted to check all the numbers up to 110, you would write:

Sieve of Eratosthenes - 1

You would then remove 1, since it’s not prime, and leaving 2, which is prime, you would remove every second number from 2 onwards, to get:

Sieve of Eratosthenes 2

And now, leaving 3 alone, you remove every multiple of 3, to get:

Sieve of Eratosthenes 3

The next number to leave is 5. Remove every multiple of 5 to get:

Sieve of Eratosthenes 5

The next number to leave is 7. Remove every multiple of 7 to get:

Sieve of  Eratosthenes 7

Repeat this process until all the primes have been found. In the case of primes up to 110, they are all shown in the final table above. All of the numbers that are not prime have been sieved out using Eratosthenes’ method.

The procedure can be used to find primes to as high a limit as you like.

The End of Eratosthenes

Legend has it that Eratosthenes went blind and died by starving himself when he was 80 to 82 years old.

Our Cast of Characters

Eratosthenes lived in Ancient Greece. He was born in about 276 BC and died in about 194 BC.
Archimedes lived in Ancient Greece. He was born in about 287 BC and died in 212 BC.
Euclid lived in Ancient Greece. He was born in about 325 BC and died in about 265 BC.

Author of this page: The Doc
© All rights reserved.

 

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>