Essay, Research Paper: Euclid

Mathematics

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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.
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