EMAT 6700
by Jan White

The History of Ancient Mathematics

As long as humans were made up of nomadic tribes there was little need for them to perform

operations on numbers.  In fact if a hunter shot three arrows at three deer and had three women drag

them back to camp, the number three might be different in each case.  It took thousands of years for

humans to realize that these were all instances of the same abstract concept, the number 3.  When man

decided to set up permanent residences and cultivate the land was when mathematics finally took off.

The people of the Nile Valley, around 6,000 B.C., began settling along the Nile River, which

overflowed each year spreading fertile mud in the valley.  The name "Egypt" means "black earth".  The

people settled in communities built on mounds to avoid the floods.  The built irrigation systems, grain

storage, mastered metalworking and also developed writing.  By 3500 B.C. agriculture became the basis

for Egyptian life.  "There are two things on earth you can be sure of, death and taxes."  The Egyptian

people surely knew this.  Taxes were probably the reason for the development of geometry.  The

Pharaoh need money to support his armies, temples and his lifestyle.  What better way than by taxes.

Land taxes were based on the height of the year's flood and the area of the land.  The Egyptians had "a

collection of rules and rough measurements arrived at by trial and error", (Muir), for calculating the area

of squares, rectangles, triangles and trapezoids.  "To find the area of a circle, they approximated it by a

square with sides equal to eight ninths the diameter.  This is equivalent to using a value of 256/81, or

3.16, for pi, an overestimate, but off by only .6 percent." (Mlodinow).  Similar overestimates are used

even in today's taxes.  Borrowing was possible and the interest rate was simple, 100%.  Simple, you don't

pay, you lose your head.  Egyptian architects engineered many impressive structures.  How hard could it

be to build a pyramid?  First start with a square base and then build four equal triangular sides that meet at

a point say 450 feet high.  Of course if you are even a fraction off the pyramid will end up lopsided.

"The Pharaohs, worshipped as gods, with armies who cut the phalluses off enemy dead just to help them

keep count, were not the kind of of all-powerful deities you would want to present with a crooked

pyramid." (Mlodinow). A person called a haredonopta, "rope stretcher", would have three people stretch a

rope with knots at 30, 40, and 50 feet creating a triangle with the hypotenuse opposite the 90 degree

angle. Is this the Pythagorean Theorem?  The method was ingenious, but the theorem was never stated.

The Egyptian's collection of rules were used for practical reasons to build temples and tombs.  There was

never any generalization and even rarer that knowledge was passed on.

The Babylonians, between 2000 B.C. and 100 B.C., settled in the region between the Tigris and

Euphrates Rivers. They developed a system of numbers, sexagesimal (go here to see the Babylonian

number system), including place values which allowed them to develop a math much more sophisticated

than the Egyptians.  The babylonians did not write equations as we know them.  In stead they wrote them

as word problems such as, "four is the length and five is the diagonal.  What is the breadth?  Its size is not

known.  Four times four is sixteen. Five times five is twenty-five.  You take sixteen from twenty-five and

there remains nine.  What times what shall I take in order to get nine?  Three times three is nine.  Three is

the breadth." (Mlodinow)  Obviously this looks very much like the Pythagorean Theorem, a² + b² = c²,

but the Babylonians had written thousands of these Pythagorean triples without ever generalizing the

theorem. To find a Pythagorean triple such as 31,240,241, the Babylonians must have developed a fairly

sophisticated algorithm.  (To find out how the Babylonians might have come up with these Pythagorean

triples go here.)  They were able to approximate the answers to equations such as x2-3 = 0, an equation

the Greek were particularly bothered by ( go here for a demonstration of the Babylonian Square Root

method (pdf)),(mimio),(Streaming video).  They were know to calculate compound interest.  So, we have

the Babylonians to thank for the deceptive practice of quoting low interest rates versus the higher APR

rate.
The Golden Age of Greece began around 400 B.C.  Thales, a Greek philosopher and mathematician,

traveled to Egypt as well as Babylon learning their ideas about math and incorporating them into his own.

Phythagarus (c.580 B.C. - c.510 B.C.), a pupil of Thales, also traveled to Egypt and Babylon.

Phythagarus created a cult based on mathematics and his followers were forbidden to give away his

secrets on penalty of death.  "There is also a myth that Pythagorus returned from the dead, although,

according to the story, Pythagorus faked this by hiding in a secret underground chamber." (Mlodinow)

Unlike the Egyptians and Babylonians, who were only interested in "how", the Greeks' were interested

in "why".(Dunham)  They were able to express mathematics in a more abstract way. The famous

Pythagorean Theorem was now expressed in geometric terms not just in real life terms as the Egyptians

and Babylonians had.  The Golden Ratio also know by the Egyptians and Babylonians was used quite

often by the Greeks in architecture, sculpture and art.  It must have been particularly appealing to the

Greeks that many different geometric objects were quadrable, that is they were able to prove the

quadrature of a rectangle and triangle.  The symmetry and elegance shown by these proofs were

apparently known by many Greek geometers.  Hippocrates of Chios (not the medicine Hippocrates)

certainly knew them and was able to show that all rectilinear objects were quadrable.  He like many

thought that curvilinear objects must also be quadrable.  He was able to prove that three particular lunes

were quadrable. Hippocrates claimed that he could prove that a circle was quadrable. He apparently

forgot to write it down. (Mlodinow)  Two other lunes were found to be quadrable and those were found

using trigonometry by Euhler in the 1700's.  It was not until the 20th century that Tschebatorew and

Dorodnow proved that there were only five squareable lunes.  It was not until modern times that it was

proved that the quadrature of a circle was impossible. (Bold)  There were many great Greek

mathematicians during this period.  Euclid's, (c.300 B.C.), book the "Elements", one of the greatest

mathematical books of all time, was a compilation of all of what the Greeks new about geometry at that

time.  Archimedes, (c.287 B. C. - c.212 B.C.), works on finding areas would lead to the Integral in

Calculus.  More on the many Greek mathematicians to come.