Euclid (; Ancient Greek: Εὐκλείδης
– Eukleídēs, pronounced [eu̯.kleː.dɛːs];
fl. 300 BC), sometimes given the name Euclid
of Alexandria to distinguish him from Euclides
of Megara, was a Greek mathematician, often
referred to as the "founder of geometry" or
the "father of geometry". He was active in
Alexandria during the reign of Ptolemy I (323–283
BC). His Elements is one of the most influential
works in the history of mathematics, serving
as the main textbook for teaching mathematics
(especially geometry) from the time of its
publication until the late 19th or early 20th
century. In the Elements, Euclid deduced the
theorems of what is now called Euclidean geometry
from a small set of axioms. Euclid also wrote
works on perspective, conic sections, spherical
geometry, number theory, and rigor.
Euclid is the anglicized version of the Greek
name Εὐκλείδης, which means "renowned,
glorious".
== Life ==
Very few original references to Euclid survive,
so little is known about his life. He was
likely born c. 325 BC, although the place
and circumstances of both his birth and death
are unknown and may only be estimated roughly
relative to other people mentioned with him.
He is rarely mentioned by name by other Greek
mathematicians from Archimedes (c. 287 BC
– c. 212 BC) onward, and is usually referred
to as "ὁ στοιχειώτης" ("the author
of Elements"). The few historical references
to Euclid were written centuries after he
lived, namely by Proclus c. 450 AD.A detailed
biography of Euclid is given by Arabian authors,
mentioning, for example, a birth town of Tyre.
This biography is generally believed to be
fictitious. If he came from Alexandria, he
would have known the Serapeum of Alexandria,
and the Library of Alexandria, and may have
worked there during his time. Euclid's arrival
in Alexandria came about ten years after its
founding by Alexander the Great, which means
he arrived c. 322 BC.Proclus introduces Euclid
only briefly in his Commentary on the Elements.
According to Proclus, Euclid supposedly belonged
to Plato's "persuasion" and brought together
the Elements, drawing on prior work of Eudoxus
of Cnidus and of several pupils of Plato (particularly
Theaetetus and Philip of Opus.) Proclus believes
that Euclid is not much younger than these,
and that he must have lived during the time
of Ptolemy I (c. 367 BC – 282 BC) because
he was mentioned by Archimedes. Although the
apparent citation of Euclid by Archimedes
has been judged to be an interpolation by
later editors of his works, it is still believed
that Euclid wrote his works before Archimedes
wrote his. Proclus later retells a story that,
when Ptolemy I asked if there was a shorter
path to learning geometry than Euclid's Elements,
"Euclid replied there is no royal road to
geometry." This anecdote is questionable since
it is similar to a story told about Menaechmus
and Alexander the Great.
Euclid died c. 270 BC, presumably in Alexandria.
In the only other key reference to Euclid,
Pappus of Alexandria (c. 320 AD) briefly mentioned
that Apollonius "spent a very long time with
the pupils of Euclid at Alexandria, and it
was thus that he acquired such a scientific
habit of thought" c. 247–222 BC.Because
the lack of biographical information is unusual
for the period (extensive biographies being
available for most significant Greek mathematicians
several centuries before and after Euclid),
some researchers have proposed that Euclid
was not a historical personage, and that his
works were written by a team of mathematicians
who took the name Euclid from Euclid of Megara
(à la Bourbaki). However, this hypothesis
is not well accepted by scholars and there
is little evidence in its favor.
== Elements ==
Although many of the results in Elements originated
with earlier mathematicians, one of Euclid's
accomplishments was to present them in a single,
logically coherent framework, making it easy
to use and easy to reference, including a
system of rigorous mathematical proofs that
remains the basis of mathematics 23 centuries
later.There is no mention of Euclid in the
earliest remaining copies of the Elements,
and most of the copies say they are "from
the edition of Theon" or the "lectures of
Theon", while the text considered to be primary,
held by the Vatican, mentions no author. The
only reference that historians rely on of
Euclid having written the Elements was from
Proclus, who briefly in his Commentary on
the Elements ascribes Euclid as its author.
Although best known for its geometric results,
the Elements also includes number theory.
It considers the connection between perfect
numbers and Mersenne primes (known as the
Euclid–Euler theorem), the infinitude of
prime numbers, Euclid's lemma on factorization
(which leads to the fundamental theorem of
arithmetic on uniqueness of prime factorizations),
and the Euclidean algorithm for finding the
greatest common divisor of two numbers.
The geometrical system described in the Elements
was long known simply as geometry, and was
considered to be the only geometry possible.
Today, however, that system is often referred
to as Euclidean geometry to distinguish it
from other so-called non-Euclidean geometries
that mathematicians discovered in the 19th
century.
=== Fragments ===
The Papyrus Oxyrhynchus 29 (P. Oxy. 29) is
a fragment of the second book of the Elements
of Euclid, unearthed by Grenfell and Hunt
1897 in Oxyrhynchus. More recent scholarship
suggests a date of 75–125 AD.The classic
translation of T. L. Heath, reads:
If a straight line be cut into equal and unequal
segments, the rectangle contained by the unequal
segments of the whole together with the square
on the straight line between the points of
section is equal to the square on the half.
== Other works ==
In addition to the Elements, at least five
works of Euclid have survived to the present
day. They follow the same logical structure
as Elements, with definitions and proved propositions.
Data deals with the nature and implications
of "given" information in geometrical problems;
the subject matter is closely related to the
first four books of the Elements.
On Divisions of Figures, which survives only
partially in Arabic translation, concerns
the division of geometrical figures into two
or more equal parts or into parts in given
ratios. It is similar to a first-century AD
work by Heron of Alexandria.
Catoptrics, which concerns the mathematical
theory of mirrors, particularly the images
formed in plane and spherical concave mirrors.
The attribution is held to be anachronistic
however by J J O'Connor and E F Robertson
who name Theon of Alexandria as a more likely
author.
Phaenomena, a treatise on spherical astronomy,
survives in Greek; it is quite similar to
On the Moving Sphere by Autolycus of Pitane,
who flourished around 310 BC.
Optics is the earliest surviving Greek treatise
on perspective. In its definitions Euclid
follows the Platonic tradition that vision
is caused by discrete rays which emanate from
the eye. One important definition is the fourth:
"Things seen under a greater angle appear
greater, and those under a lesser angle less,
while those under equal angles appear equal."
In the 36 propositions that follow, Euclid
relates the apparent size of an object to
its distance from the eye and investigates
the apparent shapes of cylinders and cones
when viewed from different angles. Proposition
45 is interesting, proving that for any two
unequal magnitudes, there is a point from
which the two appear equal. Pappus believed
these results to be important in astronomy
and included Euclid's Optics, along with his
Phaenomena, in the Little Astronomy, a compendium
of smaller works to be studied before the
Syntaxis (Almagest) of Claudius Ptolemy.
=== Lost works ===
Other works are credibly attributed to Euclid,
but have been lost.
Conics was a work on conic sections that was
later extended by Apollonius of Perga into
his famous work on the subject. It is likely
that the first four books of Apollonius's
work come directly from Euclid. According
to Pappus, "Apollonius, having completed Euclid's
four books of conics and added four others,
handed down eight volumes of conics." The
Conics of Apollonius quickly supplanted the
former work, and by the time of Pappus, Euclid's
work was already lost.
Porisms might have been an outgrowth of Euclid's
work with conic sections, but the exact meaning
of the title is controversial.
Pseudaria, or Book of Fallacies, was an elementary
text about errors in reasoning.
Surface Loci concerned either loci (sets of
points) on surfaces or loci which were themselves
surfaces; under the latter interpretation,
it has been hypothesized that the work might
have dealt with quadric surfaces.
Several works on mechanics are attributed
to Euclid by Arabic sources. On the Heavy
and the Light contains, in nine definitions
and five propositions, Aristotelian notions
of moving bodies and the concept of specific
gravity. On the Balance treats the theory
of the lever in a similarly Euclidean manner,
containing one definition, two axioms, and
four propositions. A third fragment, on the
circles described by the ends of a moving
lever, contains four propositions. These three
works complement each other in such a way
that it has been suggested that they are remnants
of a single treatise on mechanics written
by Euclid.
== Legacy ==
The European Space Agency's (ESA) Euclid spacecraft
was named in his honor.
== See also ==
Axiomatic method
Euclid's orchard
Euclidean algorithm
Euclidean geometry
Euclidean relation
Extended Euclidean algorithm
List of topics named after Euclid
