The Man 2nd Best at Everything: Eratosthenes

Eratosthenes was a mathematician, born around 276 BC in a Greek colony in Cyrene (which is now Libya). He was educated by Lysanias of Cyrene, the philosopher Ariston of Chios, and the poet and scholar Callimachus, before spending years studying in Athens. In about 240 BC he was appointed the 3rd librarian at the Library of Alexandria. It was during this time that Eratosthenes was said to have invented geography through the writing of his books about the world, which he entitled Geography meaning "writing about the earth." Some call him the father of geography, as that was the first use of the word.

However, Eratosthenes is better known for his remarkably accurate experiment to determine the circumference of the earth. After hearing that sunlight only stuck the bottom of a deep well at Syene (southern Egypt) during the summer solstice, he figured out a way to make the calculation. At noon, the sun was directly over Syene proven by the fact that everything around showed no shadows. In Alexandria at the exact same time, the shadow of a wall was measured to be 7.2°. That value was divided into 360° to get 50. Knowing the distance between the two cities was 500 miles and that it must be 1/50th of the circumference, he multiplied it out to be 25,000 miles. Today we know the circumference at the equator to be 24,901 miles, making his calculation only 100 miles off.  

Another significant part of his life was the development of his method for finding prime numbers, called the Sieve of Eratosthenes. Prime numbers are numbers whose only two factors are 1 and itself; for example 2, 3, 5, and 7 are all prime numbers. Composite numbers have more than 2 possible factors; for example the factors of the number 4 include 1, 2, & 4, thus the number 4 is a composite number. The Sieve of Eratosthenes is a way to efficiently sort out all composites from a list of numbers, leaving only primes behind. Here’s how it works:

Make a list of all the integers less than or equal to n (and greater than one).

Strike out the multiples of all primes less than or equal to the square root of n.

Then the numbers that are left are the primes.

This method is easy as it doesn’t involve much more than counting, but it can become tedious. The Gale Encyclopedia of Science reports that, according to Ore, every table of primes has been constructed with the method described by Eratosthenes. This includes tables of all the primes up to one hundred million. If you would like to interact with a sieve, try this Seive of Eratothenes Applet by Albert Tremblay. 

Eratosthenes made many contributions to the world of science and mathematics, as well as to the general public, including his calculating the distance from the sun to the earth, determining the tilt of the earth’s axis, inventing the leap day, creating a map of the world (based on available knowledge of the time), and attempting to correct the dates of major political events since the conquest of Troy. He was nicknamed Beta because he was known to be the 2nd best at all subjects, but never seemed to come in first. Eratosthenes is said to have committed suicide through voluntary starvation as he was becoming deaf and blind around the age of 80, his life ended about 196 BC.

Sources: 

• Measuring the Circumference of the Earth
• Eratosthenes of Cyrene (by J J O'Connor and E F Robertson)
• Eratosthenes of Cyrene (by Nick Greene) 
• Eratosthenes (by Matt Rosenberg) 
• Eratosthenes (Wikipedia) 
• Sieve of Eratosthenes (by Jill Britton) 
• Sieve of Eratosthenes (Gale Encyclopedia of Science, pdf)

 

A Deeper Look into the Babylonian Number System

The Babylonian number system is one of the oldest number systems, dating back around 5,000 years ago. They used a type of cuneiform symbols to represent numbers, which were recorded on clay tablets. They used two different "wedge" type symbols, one to represent ten and the other to represent one. These symbols would be grouped together to represent various numbers.

 Unlike our decimal number system which is base 10, they used a sexagecimal number system which is base 60. To me the concept of a base 60 number system seems difficult to work with, although it's probably because I'm not used to it, but it's still an interesting concept. For example, the number 79,883 would have been laid out like this: 22 11 23, which is (22x60^2)+(11x60)+23 = 79883.

To increase my understanding, I found two good theories as to why a sexagecimal number system would have been chosen. The first theory is that it would have been a combination of two number systems from different tribes, such as base 5 and base 12 or base 6 and base 10. The resulting base 60 would have been useful for the tribes who trade between each other. Some evidence points to the combination of base 6 and base 10 to be more likely because of the way ones and tens were used to create unique symbols for numbers. Another common theory is that 60 was chosen for its prime factorization (2x2x3x5) and versatility, since it's the smallest positive integer whose factors include all numbers from 1-6, and it is also divisible by 10, 12, 15, 20, and 30.

Another part of their number system that I found to be rather interesting is the fact that they had no radix point and they had no symbol for zero for the first 1000 years. While this seems like this would make it very hard to distinguish numbers, they simply used the context to determine the exact number. An example of this issue is the number 1 21 ([1x60]+21), which equals 81 but can also mean the reciprocal of 81 or 1/81. You would have to understand the context in order to know which number it actually was. The lack of a zero symbol created the same issue. While it would make sense to write 3615 as 1 0 15 ([1x60^2]+[0x60]+15), it was written as 1 15 which also can equal 75 ([1x60^2]+15 or [1x60]+15). Again, context makes a huge difference. Even though these issues seem like they would result in a lot of errors, there is little evidence that the Babylonians struggled with them.

While their number system is very different from ours and may seem more difficult to use, I do understand some advantages it may have and see that it was fairly advanced for its time. 

Sources:

• Curiosities Regarding the Babylonian Number System
• The Babylonian Number System
• Babylonian Numerals