The Birth of Mathematical Mythologies
The late American mythologist Joseph Campbell, author of the book The Hero with a Thousand Faces, shares his knowledge about the influence of math on early man.
Settling sometime around 5000–4000 BC the Sumerians were the earliest people who settled in the region of Mesopotamia (the land between two rivers), which is now modern-day Iraq. During this time a mini-Ice Age cooled the Earth and made what is today a semiarid dessert into a lush fertile landscape that supported many important plants and animals and enabled these early settlers to create cities, invent agriculture, domesticate animals, establish religions, and most importantly, count things!
Map of the main cities of Lower Mesopotamia during the Early Dynastic period, with the approximate course of the rivers and the ancient shoreline of the Gulf. Near East_topographic_map-blank.svg: Sémhur
Cuneiform and the Sexagesimal Counting System
The Sumerians were an agriculturally based society and not nearly as warlike as their nomadic animal-herding neighbors to the north and the east. Despite the high walls they built and invention of both diplomacy and warfare, they were eventually conquered by various peoples; the most famous were the Babylonians. When we refer to the sciences and mathematics of Mesopotamia, the Babylonians take the credit, but their Sumerian predecessors should be credited.
Around 3100 BC the Sumerians invented a writing script called cuneiform, which were wedge-shaped marks made on clay tablets. Of the million or so tablets exhumed, only a small percentage has been translated. Unlike their paper counterparts, so many of these clay tablets survived because they were hardened and were effectively immortalized when burned by flames in kilns or during accidents or raids.
What do we know of the Sumerians based on these tablets? We know that they loved tales of adventure and danger, they wrote poetry about love and the appreciation of nature, they revered the gods, they traded with distant cultures, and they defended themselves against foreign invaders and warred between themselves.
The Sumerians also loved to calculate things. They invented a sexagesimal or base 60 counting system that has had far-reaching implications that affect us even today. The origin of this system is believed to derive from the counting of the individual segments of the fingers with the thumb of the right hand and then totaling 5 dozens with the fingers of the left hand. This counting method is still taught to children today in India and China.
Cuneiform counting system.
Sumerian mathematical exercise. University of Pennsylvania Museum of Archaeology and Anthropology, Philadelphia, Pennsylvania, USA
Babylonian Numbers: 1 to 9, presented by Dr. Eleanor Robson.
The Secret Success of the Sexagesimal System
The number 60 is a superior highly composite number with 12 factors, namely, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60; 2, 3, and 5 are prime numbers. This makes 60 a very easy number to divide and it affected every aspect of life in Mesopotamia. Not only was it used to calculate the more mundane events of urban life, such as tallying merchant ledgers, creating population census, or priests totaling up the number of offerings worshipers offered to the temple and hence to the gods, it was also applied to the number of days and months in a year (360 and 12), how days were divided into two parts of 12 totaling 24 hours, how many minutes were in each hour (60), and how many degrees in a circle (360).
There were no clocks then but they did use obelisks, which served as both important cultural city landmarks and sundials. As the sun moved across the sky, the obelisk shadow moved clockwise, also known as sun wise from left to right in a circle. Because they lived in the Northern Hemisphere, this is the naturally occurring direction a shadow moves around a gnomon. Had they lived in the Southern Hemisphere, clockwise would have gone the opposite direction.
Unlike the Egyptians, their sun-worshiping neighbors to the west, the cultures of Mesopotamia worshiped the moon and the constellations of the night sky. Each month was determined by the four phases of the moon with four weeks of 7 days. The seventh day was considered an evil or unlucky day when people were to rest because ill fortune may come to their endeavors. Although each month was determined by a complete cycle of the moon, there are actually a total of 13 full moons in a year. The thirteenth moon of the year was considered a magic one which is probably the origins the unlucky number thirteen and the expression, "A bad moon on the rise.". A year consisted of 360 days and every so many years they added an intercalary month to ensure that the seasons did not drift apart from one another.
The solar and lunar calendars each have their strengths. This segment from Dr. David Neiman's series, "Cradles of Civilization" discusses the method of reconciliation between these two systems and the adjustments made to the lunar system in the Jewish calendar.
How the Spread of Mesopotomian Culture Influenced the Ancient World
Trade during Sumerian rule was surprisingly far ranging with evidence of commerce as far west as Greece and east in the Indus Valley of Pakistan and India. The preceding rulers of Mesopotamia also traded with these cultures and beyond. There is no doubt that their culture, religious and mathematical beliefs traveled with them as they encountered other peoples. Although we don't know exactly how their system affected these other cultures the parallels they share are striking. Let's take a look at more applications of the base 60 system in other parts of the world.