Euclid (
Greek: ),
fl. 300 BC, also known as
Euclid of
Alexandria, was a
Greek
mathematician and is often referred to as the "Father of
Geometry."
He was active in Hellenistic Alexandria during the reign of Ptolemy
I (323–283 BC). His
Elements is the most successful
textbook and 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 it, the principles of what
is now called
Euclidean geometry
were deduced 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 , meaning "Good Glory".
Life
Little is known about Euclid's life, as there are only a handful of
references to him. In fact, the key references to Euclid were
written centuries after he lived, by
Proclus
and
Pappus of Alexandria.
Proclus introduces Euclid only briefly in his
Commentary on the
Elements, written in the fifth century, where he writes that
Euclid was the author of the
Elements, that he was
mentioned by
Archimedes, and that when
Ptolemy the First asked Euclid if
there was no shorter road to geometry than the
Elements,
he replied, "there is no royal road to geometry." Although the
purported 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 those of Archimedes. In
addition, the "royal road" anecdote is questionable since it is
similar to a story told about
Menaechmus
and
Alexander the Great. In the
only other key reference to Euclid, Pappus briefly mentioned in the
fourth century 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."
It is further believed
that Euclid may have studied at Plato's Academy in Greece.
The date and place of Euclid's birth and the date and circumstances
of his death are unknown, and only roughly estimated in proximity
to contemporary figures mentioned in references. No likeness or
description of Euclid's physical appearance made during his
lifetime survived antiquity. Therefore, Euclid's depiction in works
of art is the product of the artist's imagination.
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, 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.
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
third 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 to Euclid
is doubtful. Its author may have been Theon of Alexandria.
- 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
140 px
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.
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.
See also
Notes
References
- Artmann, Benno (1999). Euclid: The Creation of
Mathematics. New York: Springer. ISBN 0387984232.
- Heath, Thomas L. (1981). A History of Greek
Mathematics, 2 Vols. New York: Dover Publications. ISBN
0486240738 / ISBN 0486240746.
- Kline, Morris (1980).
Mathematics: The Loss of Certainty. Oxford: Oxford
University Press. ISBN 019502754X.
External links
- Euclid's elements, All thirteen books, with
interactive diagrams using Java. Clark University
- Euclid's elements, with the original Greek and an
English translation on facing pages (includes PDF version for
printing). University of Texas.
- Euclid's
elements, All thirteen books, in several languages as Spanish,
Catalan, English, German, Portuguese, Arabic, Italian, Russian and
Chinese .
- Elementa Geometriae 1482, Venice. From
Rare Book Room.
- Elementa 888 AD, Byzantine. From
Rare Book Room.
- Euclid biography by Charlene Douglass With extensive
bibliography.
- Texts on
Ancient Mathematics and Mathematical Astronomy PDF scans (Note:
many are very large files). Includes editions and translations of
Euclid's Elements, Data, and Optica,
Proclus's Commentary on Euclid, and other historical
sources.