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.