Eratosthenes’ Contribution in Mathematics

EratosthenesEratosthenes was a man of many talents. He was a renowned mathematician, poet, geographer, astronomer, and music theorist. He was born in 271 BC in Cyrene, which is now known as Libya, in North Africa. He studied at Plato’s school in Athens. He was the chief librarian of the Great Library of Alexandria. While working in the library, he explored lots of books and synchronized all of his knowledge in three volumes of the book Geographika. In this book, he mapped the entire world. There are various other concepts associated with his name like Eratosthenes crater on the moon; Eratosthenes Seamount in the eastern Mediterranean Sea and Erosthenian period in the lunar geologic timescale. He is best known for calculating the circumference of the Earth. He was also a gifted poet with excellent imagination and sensibility. He was known to be the first person who coined the term Geography. He made a significant contribution to geography as he claimed that heavy rains sometimes fall in the regions near the source of a river. He also gave a correct explanation about the region “Eudaemon Arabia”, now Yemen as occupied by four distinct races. For his remarkable contributions in geography, he is also known as the ‘father of Geography.’ Despite all of his achievements, he never achieved the highest rank in any field. He was an all-rounder, who never came first but achieved the second rank in various contests, that’s why he was given the nickname Beta. He was also known to be close friends with Archimedes. He ended his life at the age of 82 by voluntary starvation in 194 BC because he feared the onset of blindness. His various other achievements still act as the foundation of modern scientific methods.

1. Measurement of Earth’s Circumference

He is mainly known for his discovery of the Earth’s circumference. He calculated the Earth’s circumference without leaving the boundaries of Egypt. To figure out this calculation, he made use of stadiums. Details about the calculations were given in his book ‘On the measurement of Earth,’ which is now lost. He used the lengths of the shadow to calculate, how high the sun was on a particular day and at a particular time. He was also aware of other places, where there was no shadow on the same day, which means the sun was overhead, further he estimated the distance between the two places. On the summer solstice, he measured the length of the tower’s shadow in the city of Alexandria and the angle of the sun from straight up. Then, in the city of Syene, there was a well, where he observed that there was no shadow, which means the sun was overhead. Finally, after calculating the sun’s angle and the distance between the two cities, he estimated that the circumference of the earth was 40,000 Km, which is very near to the actual circumference of the Earth. His percentage of accuracy varied between 0.5 to 17 per cent. Most of the scholars argued the accuracy of his result, and there were several papers published that discussed the same.

earth's circumference by eratosthenes

2. Estimation of Diameter and Distance to Sun and Moon

Given Macrobious, Eratosthenes measured the diameter of the sun to be 27 times that of the Earth. However, according to modern scientific results, the actual figure is about 109 times. He calculated the diameter with the help of a lunar eclipse. He noticed that at the time of the lunar eclipse, the shadow of the Earth on the Sun was twice its size, which helped him to determine that the Sun was twice the size of Earth. However, in the modern scientific world, results report that the Sun is four times the size of the Earth. He measured the distance of the Sun as 804,000,000 stadia and that of the Moon as 780,000 stadia from Earth.

diameter of sun


3. Measurement of Tilt of Earth’s Axis

He successfully measured the tilt of Earth’s axis to be 11/83 of 180°. The value 11/83 shocked mathematicians and they wrote several papers to find out the source of this value. Many scholars, along with Heath, believed that Eratosthenes used a value of 24° and the value 11/83 of 180° was due to Ptolemy. There were others also who believed that Eratosthenes used a value 2/15 of 180°.


4. Armillary Sphere

Hipparchus credited Eratosthenes for the discovery of the Armillary sphere. It is made up of a spherical framework of rings, centred on the Earth or the Sun. It is an instrument that is used to predict and project the motion of stars in the sky. When centred on the Earth, it is known as Ptolemaic, and when centred on the Sun, it is known as Copernican. It was later invented in China in the 4th century BC.

Armillary Sphere

5. Discovery of Calendar Having a Leap Year

When he was in the library at Alexandria, he proposed a calendar in which he calculated that there are 365 days in a year and 366 days in every fourth year, which was later termed as Leap year. He led the foundation of chronology by organizing the dates of literary and political events from the siege of Troy to his own time.

6. Doubling the Cube

It is believed that he wanted to construct catapults, and he made a machine (drawing device) to calculate the cube root of 2. However, there is a short story behind this problem of ‘doubling of the cube.’ This story involves the citizens of Delos, who were scared by the plague (a disease) sent by Apollo (God). They asked the oracle (advisor) of Delphi how to defeat the plague. The oracle advised that they should double the altar (table in front of god) of Apollo to get rid of the plague. The altar was a regular cube, and doubling the volume of the cube became a mathematical problem for the citizens of Delos. Then, they all went to Plato, and according to him, God wanted the citizens of Delos to stop neglecting mathematics and focus more on geometry.

Eutocius discovered a letter believed to have been written by Eratosthenes to Ptolemy III, Euergetes, in which he described the problem of doubling of cube. He writes,

One of the ancient tragic poets represented Minos having a tomb built for Glaucus, and that when Minos found that the tomb measured a hundred feet on every side, he said “Too small is the tomb you have marked out as the royal resting place. Let it be twice as large. Without spoiling the form, quickly double each side of the tomb”. This was clearly a mistake. For if the sides are doubled the surface is multiplied fourfold and the volume eightfold.’

The picture below gives some insight about the machine of Eratosthenes to find two mean proportional, i.e., for given lines a, b; find x, y such that a:x=x:y=y:b. Let the lengths of EA and HD be 2 and 1, respectively. Move the second and third triangles in such a way that points B and C lie on the straight line AD, giving the length of GC as 2 3.

Eratosthens machine to find cuberoot

7. Sieve of Eratosthenes

Eratosthenes provided an algorithm to find the prime numbers. This algorithm is known as the sieve of Eratosthenes. He started the process by marking the composite multiples of each prime, starting with 2. A sequence of multiples of prime, starting from that prime is generated, with a constant difference between them equal to that prime. When all the multiples of each prime have been marked as composites, then the remaining unmarked numbers are prime. For example, the process of calculating primes up to 15 is:-

Generate a list of integers from 2 to 15

2    3    4    5    6    7    8    9    10    11    12    13    14    15

The first number in the list is 2 so cross out every 2nd number in the list after 2 (in bold italic)

2    3    4    5    6    7    8    9    10    11    12    13    14    15

Now after 2, the number is 3, so cancel every 3rd number after 3

2    3    4    5    6    7    8    9    10    11    12    13    14    15  

The remaining unmarked numbers 2, 3, 5, 7, 11, 13 are primes up to 15

Algorithm GIF - Find on GIFER

8. Lost Works

a) Platonicus

Eratosthenes’s Platonicus mainly dealt with mathematics and was highly inspired by Plato’s philosophy. However, this book was lost, but Theon of Smyrna gave us an idea that Eratosthenes’s work revolved mainly around arithmetic and geometry. Also, Theon of Smyrna used his work, when he wrote Expositio rerum mathematicarum.

b) On means

This is one of the books written by Eratosthenes which is now lost. His mathematical work is majorly known from the writings of Greek mathematician, Pappus of Alexandria. He mentioned ‘On means’ to be one of the great books of geometry.















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