# Babylonian Mathematics

As merchants plied the Mediterranean Sea they wrote down the cargo they carried in their boats.  This was the beginning of the numerical system that eventually developed into the Roman, Indian, and European system we use today.  It was not the brainchild of scholars but of practical businessmen attempting to keep track of their commodities.  Clay tablets found show pictures of cows, denoted by horns, and sheep, long ovals.  Each animal was drawn, but after awhile they learned to save time by putting a dot or a line above an animal’s head to denote a quantity of more than one.

They inscribed cuneiform slashes and semi-circles with a reed that grew prevalently in the Euphrates-Tigris Rivers area (probably first used by the Sumerians who lived in the southwesterly region of what became Mesopotamia after the end of the Sumerian civilization).  It was cut at an oblique angle and easily made a wedge shaped slash or curve in a moist clay tablet.

Around 2000 BC the Babylonians invaded Sumer and took in most of their numerical system, eliminating the numbers between 1 (wedge) and 10 (hook) which they wrote as a wedge and a hook.  Two would be 2 wedges, etc. up to 10, then 11 would be a wedge and a hook up to 59 which was written as 5 hooks and 4 wedges.  They used a sexagesimal system with a base 60.  To represent 60, a wedge was placed in the 60’s place.  They invented the positional system and the zero to be used as a placeholder by using two oblique wedges.  Babylon is not credited with inventing the zero because they only used it as a placeholder (but only later, 200 years after the Persian invasion).  India is credited with inventing the zero concept because it was the first to use it as a number.

Tablets from the Old Babylonian period 1900-1600 BC reveal that they knew of the Pythagorean Theorem-finding the hypotenuse and legs of the right triangle-but did not prove it. It was not until 560-460 BC that the Greek mathematician Pythagoras of Samos provided the proof for it.

Babylonian mathematics was much more advanced than that of Egypt, whose main nongeometrical contribution was star clocks.  Babylon unfortunately, rather than having a great reputation for its mathematical advancement, is thought of as a superstitious culture because of its religious, mystical beliefs which limited its thinking, especially in the field of medicine (in which Egypt excelled), relying on spirits and other worldly powers to heal the body.

There was some use of symbols, but not much.  Like the Egyptians their algebra was essentially rhetorical (they discussed and solved problems without writing them down symbolically).  The procedures used to solve problems were taught through examples and no reasons or explanations were given.  Also, like the Egyptians, they recognized only positive rational numbers, although they did find approximate solutions to problems which had no exact rational solution.

“Their sexagesimal system facilitated their developing algebra: they had a general procedure equivalent to solving quadratic equations, although they recognized only one root and that had to be positive.  In effect, they had the quadratic formula.  They also dealt with the equivalent of systems of two equations in two unknowns and a few equivalent to solving equations of higher degree.” (History of Algebra, 2004).

Around 1700 BC the Babylonians found the approximation 1; 24; 51; 10 in their base 60 notation for the square root of 2, which works out to 1.414212963 in decimal notation.  However, they likely did not realize that 2 was a different kind of number-an irrational number.  Another problem they had was that there was no decimal point, sometimes resulting in ambiguous numbers.

Fragmentary, scattered texts from the Seleucid Era (310 BC-75 AD) and the Old Babylonian Period (1900-1600 BC) do not give enough information to verify that the Babylonian astronomers used the heliocentric model in their calculations, but they made some very precise synodic (luni-solar) observations and calculations which have led some to believe that they did actually use it in their calculations.  “One of the fundamental units of time and motion applied in Babylonian astronomy appears to have been the mean synodic month of 29;31,50,8,20 days.” (Harris, 2004).

John Harris writes in “Babylonian Planetary Theory and the Heliocentric Concept”: “The complete synodic arc and synodic period for Mars still provide the correct motion for Earth in degrees per day.  The standard unit of time in all cases is the Babylonian year of 12;22,8 mean synodic months treated as the time required for Earth to complete one sidereal revolution of 360 degrees.

“Given the undoubted awareness of accurate sidereal periods for the superior planets, implicit sidereal periods for the inferior planets, accurate sidereal, synodic, draconic, and anomalistic months, and varying velocity functions for the planets, sun, and moon-all readily understood in terms of a cohesive framework-it seems reasonable to conclude that the Babylonians almost certainly possessed a well-developed, fictive heliocentric planetary model by at least 250 BCE, and quite possibly much earlier.”

Harris also says that from the calculations of the Babylonians of mean sidereal revolutions and planetary mean synodic arcs, it seems that they might have used three different models to calculate the luni-solar, inferior and superior planets, and star motions, using both the geocentric model for the planetary and sidereal periods and the Earth as heliocentric for the superior planets.

The Babylonians worked with simple methodology, some likening it to counting; their tools were simple but they achieved a superiority in their time in mathematics which has left behind a legacy attesting to the greatness that their civilization once attained.

Britannica.com, 32 vols. 2002.

Harris, John.  “Babylonian Planetary Theory and the Heliocentric Concept.”

“Time and Tide.” 2003, 2004

http://www.spirasolaris.ca/sbb2c.html

“The History of Algebra.”  “I Only Have Eyes for Algebra.” 2004.

http://www.ucs.louisiana.edu/~sxw8045/history.htm

Yolkowski, James.  “Babylonian Mathematics.”  Math Lair. 2002, 2004.