**A Brief History of Greek Geometry**

__A Little Background:__

**The word geometry has its roots in the Greek work
geometrein, which means “earth
measuring”. Before the time of recorded history, geometry originated out
of practical necessity; it was the science of measuring land. Many ancient civilizations
(Babylonian, Hindu, Chinese, and Egyptian) possessed geometric information. The
first geometrical considerations “had their origin in simple observations
stemming from human ability to recognize physical form and to compare shapes
and sizes” (Historical Topics, 165). There were many circumstances
in which primitive people were forced to take on geometric topics, although
it
may not
have been recognized as such. For instance, man had to learn with situations
involving distance, bounding their land, and constructing walls and homes.
These types of situations were directly related to the geometric concepts
of vertical,
parallel, and perpendicular.
**

**The geometry of the ancient days was actually just a collection of rule-of-thumb
procedures, which were found through experimentation, observation of analogies,
guessing, and sometimes even intuition. Basically, geometry in the ancient days
allowed for approximate answers, which were usually sufficient for practical
purposes. For example, the Babylonians took p to be equal to 3. It is said that
the Babylonians were more advanced than the Egyptians in arithmetic and algebra.
They even knew the Pythagorean theorem long before Pythagoras was even born.
The Babylonians had an algebraic influence on Greek mathematics.
**

**Egyptian geometry was not a science in the way the Greeks viewed geometry. It
was more of a grab bag for rules for calculation without any motivation or justification.
Sometimes they guessed correctly, but other times they did not. One of their
greatest accomplishments was finding the correct formula for the volume of a
frustum of a square pyramid. However, they thought that the formula that they
had for the area of a rectangle could be applied to any quadrilateral.
**

**Primitive people could not escape geometry in the
same way that we cannot escape it today. The concept of the curve was
found in flowers and the
sun, a parabola
was represented by tossing an object, and spider webs posed an excellent
example of regular polygons. Symmetry could be seen in many living
objects, including
man, and the idea of volume had to be addressed when constructing a
device to hold water. Historical Topics for the Mathematics Classroom
calls
this type of
geometry “subconscious geometry”. This is the type of geometry
that very young children experience as they begin to play with objects.
This type
of geometry involves concrete objects.
**

**Still before the time of recorded history, man began
to consider situations that were more hypothetical. They were able to
take the knowledge they
had learned
from observation of concrete objects and come up with general algorithms
and procedures to be used in particular cases. This is what Historical
Topics for
the Mathematics Classroom refers to as “scientific geometry” (166-7).
Procedures such as trial and error, induction, and rule-of thumb were being used
to discover. This was mainly the geometry of the Babylonians and the Egyptians.
Although there is no evidence that they were able to deductively reason geometric
facts from basic principles, it is thought that they paved the way for Greek
geometry. Geometry remained this way (“scientific”) until
the Greek period.**

__A Bit of Greek Geometry (600 BC – 400
AD):__

**To see a chronological outline of the work of Greek geometers, click here. (An
entire mathematical chronology can be found by visiting: www-groups.dcs.st-and.ac.uk/~history/Chronology/full.html.)**

The Greeks worked to transform geometry into something much different
than the “scientific
geometry” of the people that worked before them. “The Greeks insisted
that geometric fact mush be established, … , by deductive reasoning; …” (Historical
Topics, 171). They believed that geometrical truth would be found by studying
rather than experimenting. They transformed the former “scientific geometry” into
a more “systematic geometry”.

**Keep in mind that there exist virtually
no first-hand sources of early Greek geometry. Hence, the following is
based on manuscripts written
hundreds of years
after this early Greek geometry had been developed. According to these
manuscripts, Thales
of Miletus was the one who began early Greek geometry
in the sixth century
B.C. He is noted as one of the first known to indulge himself in deductive
methods in geometry. His credited elementary geometrical findings resulted
from logical
reasoning rather than intuition and experiment. He insisted that geometric
statements be established by deductive reasoning rather than by trial
and error. He was
familiar with the computations recorded from Egyptian and Babylonian
mathematics, and he developed his logical geometry by determining which
results were correct.
**

**The next mentioned great Greek geometer
is one who quite possibly studied under Thales of Miletus. This geometer
is Pythagoras, who founded the
Pythagorean school, which was “committed to the study of philosophy,
mathematics, and natural science” (Historical Topics, 172).
In the area of geometry, the members of this school developed the
properties of parallel to
prove
that the sum
of any angles of a triangle is equal to two right angles. They also
worked with
proportion to study similar figures. The deductive side of geometry
was further developed during this time. We all think of the Pythagorean
Theorem when we think of Pythagoras, however it is important to note
that this theorem was used (although it may not have been proved)
before his time.
**

**As an interesting side note, Pythagoras was regarded as a religious prophet
by his contemporaries. He preached the immortality of the soul and reincarnation,
and he even organized a brotherhood of believers. This brotherhood had
initiation
rites, they were vegetarian, and they shared all property. They did,
however, differ from other religious groups in one major way. They believed
that elevation
of the soul and union with God was achieved through the study of music
and mathematics.
**

**Hippocrates
of Chios was one of these
students at the Pythagorean school. It is suggested that he was the first
to attempt “a logical presentation of
geometry in the form of a single chain of propositions …” (Historical
Topics, 172). He is credited to writing the first “Elements of Geometry” where
he included geometric solutions to quadratic equations and some of the first
methods of integration. He studied the problem of squaring a circle and squaring
a lune. He also was the first to show that the ratio of the areas of two circles
equals the ratio of the squares of the circles’ radii (www.geometryalgorithms.com/history.htm).
**

**Although Plato did not make any major
mathematical discoveries himself, he did emphasize the idea of proof.
He insisted on accuracy,
which
helped pave
the way
for Euclid. It is correct to say that almost every significant
geometrical development can be traced back to three outstanding
Greek geometers:
Euclid, Archimedes,
and Apollonis. Euclid collected the theorems of Pythagoras,
Hippocrates, and others into a work called “The Elements”. (www.geometryalgorithms.com/history.htm).
Euclid is the most widely read author in the history of mankind. The teaching
of geometry has been dominated by Euclid’s approach to the subject. In
fact, Euclid’s axiomatic method is the prototype for all “pure mathematics”.
By “pure”, it is meant that all statements can
be verified through reasoning of demonstrations; no physical
experiments
are
necessary.
**

**Typically, the next mentioned Greek mathematician is regarded as the
greatest Greek mathematician by geometryalgorithms.com. His name was
Archimedes
of Syracuse.
He had many mathematical accomplishments as well as being the inventor
of the screw, the pulley, the lever, and other mechanical devices. He
perfected integration
using the method of exhaustion discovered my Eudoxus, and he was able
to find the areas and volumes of many objects. Inscribed on his tomb
was the result he
found that the volume of a sphere is two-thirds the volume of its circumscribed
cylinder. (http://geometryalgorithms.com/history.htm)
**

**Apollonius was an astronomer who had
his mathematical bid to fame in his work entitled Conic Sections. It
is this great
Greek geometer
who
provided
us with
the terms “ellipse,” “parabola,” and “hyperbola.” He
is also accredited with showing how to construct a circle which is tangent to
three objects. His approximation of π was even closer
than that of Archimedes. (http://geometryalgorithms.com/history.htm)
**

**Last, but certainly not least, Hypatia
of Alexandria was the first woman to substantially contribute to mathematics.
She
studied under
her father
and assisted
him in writing
a new version of Euclid’s Elements. She also wrote commentaries on other
great Greek geometer’s works. She was the “first woman in history
recognized as a professional geometer and mathematician” (http://geometryalgorithms.com/history.htm).**

References:

Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries:

Development and History. 3rd ed. New York: W.H. Freeman and Company,

1993. 6-19.

Historical Topics for the Mathematics Classroom. Washington D.C.:

National Council of Teachers of Mathematics, 1969.

http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Thales.html (March 2005)

http://geometryalgorithms.com/history.htm (March 2005)