Euclid
Euclid is one of the most influential and
best read mathematician of all time.
His prize work, Elements, was the
textbook of elementary geometry and logic up
to the early twentieth century.
For his work in the field, he is known as the
father of geometry and is
considered one of the great Greek mathematicians. Very
little is known about
the life of Euclid. Both the dates and places of his birth
and death are
unknown. It is believed that he was educated at Plato's academy in
Athens
and stayed there until he was invited by Ptolemy I to teach at his
newly
founded university in Alexandria. There, Euclid founded the school
of
mathematics and remained there for the rest of his life. As a teacher, he
was
probably one of the mentors to Archimedes. Personally, all accounts of
Euclid
describe him as a kind, fair, patient man who quickly helped and
praised the
works of others. However, this did not stop him from engaging in
sarcasm. One
story relates that one of his students complained that he had no
use for any of
the mathematics he was learning. Euclid quickly called to his
slave to give the
boy a coin because "he must make gain out of what he
learns." Another
story relates that Ptolemy asked the mathematician if there
was some easier way
to learn geometry than by learning all the theorems.
Euclid replied, "There
is no royal road to geometry" and sent the king to
study. Euclid's fame
comes from his writings, especially his masterpiece
Elements. This 13 volume
work is a compilation of Greek mathematics and
geometry. It is unknown how much
if any of the work included in Elements is
Euclid's original work; many of the
theorems found can be traced to previous
thinkers including Euxodus, Thales,
Hippocrates and Pythagoras. However,
the format of Elements belongs to him
alone. Each volume lists a number of
definitions and postulates followed by
theorems, which are followed by proofs
using those definitions and postulates.
Every statement was proven, no
matter how obvious. Euclid chose his postulates
carefully, picking only the
most basic and self-evident propositions as the
basis of his work. Before,
rival schools each had a different set of postulates,
some of which were very
questionable. This format helped standardize Greek
mathematics. As for the
subject matter, it ran the gamut of ancient thought. The
subjects include:
the transitive property, the Pythagorean theorem, algebraic
identities,
circles, tangents, plane geometry, the theory of proportions, prime
numbers,
perfect numbers, properties of positive integers, irrational numbers,
3-D
figures, inscribed and circumscribed figures, LCD, GCM and the
construction
of regular solids. Especially noteworthy subjects include the
method of
exhaustion, which would be used by Archimedes in the invention of
integral
calculus, and the proof that the set of all prime numbers is
infinite. Elements
was translated into both Latin and Arabic and is the
earliest similar work to
survive, basically because it is far superior to
anything previous. The first
printed copy came out in 1482 and was the
geometry textbook and logic primer by
the 1700s. During this period Euclid
was highly respected as a mathematician and
Elements was considered one
of the greatest mathematical works of all time. The
publication was used in
schools up to 1903. Euclid also wrote many other works
including Data, On
Division, Phaenomena, Optics and the lost books Conics and
Porisms.
Today, Euclid has lost much of the godlike status he once held. In his
time,
many of his peers attacked him for being too thorough and
including
self-evident proofs, such as one side of a triangle cannot be
longer than the
sum of the other two sides. Today, most mathematicians attack
Euclid for the
exact opposite reason that he was not thorough enough. In
Elements, there are
missing areas which were forced to be filled in by
following mathematicians. In
addition, several errors and questionable ideas
have been found. The most
glaring one deals with his fifth postulate, also
known as the parallel
postulate. The proposition states that for a straight
line and a point not on
the line, there is exactly one line that passes
through the point parallel to
the original line. Euclid was unable to prove
this statement and needing it for
his proofs, so he assumed it as true.
Future mathematicians could not accept
such a statement was unproveable and
spent centuries looking for an answer. Only
with the onset of non- Euclidean
geometry, that replaces the statement with
postulates that assume different
numbers of parallel lines, has the statement
been generally accepted as
necessary. However, despite these problems, Euclid
holds the distinction of
being one of the first persons to attempt to
standardize mathematics and set
it upon a foundation of proofs. His work acted
as a springboard for future
generations.