Ancient World Mathematics

Written by Paul Dickson
(University of South Australia, 1996)

Thales of Miletus (640 - 546 BC)

Thales was born in Miletus in 640 BC and became a merchant as soon as his skills allowed, actual history concerning Thales is scarce but some stories about him have filtered down through the ages, whether they are true or not..... no one really knows.
Thales' major mathematical contribution is believed to be the theory of a triangle inscribed within a semi-circle being right angled at the corner touching the arc if one side is the diameter of the circle.


Figure 1: A Triangle inscribed in a Semi-circle makes a right angle.

Thales and the Salt Caravan

It is believed that while transporting salt which was loaded on mules, one of the animals slipped in a stream. The mule's load of salt was slightly dissolved by the water and it's load became lightened. This mule being smart at ways to get out of work rolled over at the next ford it came to and found it's load lighter again. Whether these mules were Thales or not is unclear bu the was consulted and came up with a plan to break the mule of this bad habit. The mule was loaded with sponges and rags, which when the mule rolled over, absorbed the water and made the load heavier. This eventually cured the mule of it's troublesome habit.

Thales and the Olive Oil Empire

In the ancient world of the mediterranean Olive Oil was an important commodity, as important as wheat or sugar is in todays. The Olive crop was a bumper havest one year and fearing that supply would outgrow demand for the coming Olive Oil production Thales quietly bought all the Olive presses he could afford to (no small task considering he was a very wealthy merchant by this time). Thus Thales controlled most of the Olive Oil production and 'cornered the market' of Olive Oil, a man much before his time Thales therefore became the first recorded man at about 600 BC to create a monopoly.

Thales meets Solon

Upon Solon's visit to Athens he met Thales and inquired of him why he had no wife or children, as he was a wealthy merchant who could afford to keep both in luxury. To this Thales made no immediate answer, instead he sent a messenger a couple of days later telling Solon of a great funeral for a youth. Upon questioning of the messenger Solon was able to figure out that the dead youth was his son, about this time Thales visited Solon and explained to him that his son was fine and the messenger was sent to prove a point. Thales had no wife or children because he did not want to ever feel as grief-stricken as Solon did upon hearing of his son's demise.

Eudoxus of Cnidus (408 - 355 BC)

Eudoxus was born in Cnidus where he spent his youth in poverty like many of his fellow mathematicians. It is said that Eudoxus 'inherited the mess that Zeno left the world and not much more' which is probably an apt summation considering the upheaval some of Zeno's ideas caused in the Greek mathematical community.
When he was old enough to leave Cnidus he travelled to Tarentum where he studied at the feet of Archytas (428 - 347 BC) who was himself a first-rate mathematician, scholar, and soldier, only twenty years Eudoxus' elder.
Upon completion of his study at Tarentum Eudoxus travelled to Athens to study at the Academy founded by Plato in 385 BC where he met the founder himself. Being poor, Eudoxus was forced to live in the nearby port of Piraeus which supplied Athens with its seafood. Initially Plato encouraged his friend to pursue mathematics but is thought that at some point this friendly encouragement turned to jealosy usually reserved for a rival.
Eudoxus travelled to Egypt with Plato where like Pythagoras before him Plato succumbed to the 'Eastern Number Mysticism' which was verging upon religion status. Eudoxus returned to Athens disappointed in the maths of Egypt and the Middle East.
After a time Eudoxus became unpopular in Athens and moved away to Cyzicus where he spent the remainder of his years in study and contemplation. He studied medicine, becoming an able physician, and philosophy following the works of his friend Plato. He also returned to Cnidus upon occasion to lecture in such fields as cosmology, theology, and meteorology.

Eudoxus' major contribution to mathematics was his 'Method of Exhaustion' which states that :

It is not necessary to worry about the validity of the assumption that there exist an infinite amount of small quantities, for mathematical purposes it is sufficient to be able to reach a magnitude as small as we please by the given division of other larger magnitudes.

Eratosthenes of Alexandria (230 - 194 BC)

Eratosthenes was born in Cyrene on the Northen coast of Africa. He grew up in Athens, where at the age of forty he was asked by Ptolemy III of Egypt to come to Alexandria to tutor his son and become the chief librarian at the University.
Eratosthenes was variously known as a poet, philosopher, geographer, astronomer, athlete, mathematician, as well as being a librarian.
In his lifetime Eratosthenes was resonsible for many different discoveries and advances for the Ancient world. He studied prime numbers, invented an instrument for duplicating a cube, measured the circumference of the earth, calculated the distance to the moon and produced a calendar that was used as the basis for todays Gregorian calendar.

The Sieve of Eratosthenes

Eratosthenes' major contribution to pure mathematics is named after him, Erathosthenes Sieve which is a logical systematic procedure for calculating the prime numbers. The steps are

A Mission to find the Earth's Circumference

When it came to using the Geometry of the Ancient Greeks not many were more practical than Eratosthenes, he wanted to know how big was the planet upon which he and his fellows lived.
To this end he arranged for a well in north africa to be observed, and when the noon-day sun was in such a position that there was no shadow in the well he marked the day. Then he travelled north along the same line of longitude until he was in northern Greece where he measured the shadow of a vertical stick marker at the same day of the year as the well had cast no shadow.
Using the geometry he was familiar with he calculated the circumference of the Earth to within 10% error, this estimate remained the nearest for many centuries.

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