Euclid
330 –260 BC
Euclid is one of the best known and most influential of
classical Greek mathematicians but almost nothing is known about
his life. He was a founder and member of the academy in
Alexandria, and may have been a pupil of Plato in Athens.
Despite his great fame Euclid was not one of the greatest of
Greek mathematicians and not of the same calibre as Archimedes.
Euclid's most celebrated work is the Elements, which is
primarily a treatise on geometry contained in 13 books. The
influence of this work not only on the future development of
geometry, mathematics, and science, but on the whole of Western
thought is hard to exaggerate. Some idea of the importance that
has been attached to the Elements is gained from the fact that
there have probably been more commentaries written on it than on
the Bible. The Elements systematized and organized the work of
many previous Greek geometers, such as Theaetetus and Eudoxus,
as well as containing many new discoveries that Euclid had made
himself. Although mainly concerned with geometry it also deals
with such topics as number theory and the theory of irrational
quantities. One of the most celebrated number theoretic results
is Euclid's proof that there are an infinite number of primes.
The Elements is in many ways a synthesis and culmination of
Greek mathematics. Euclid and Apollonius of Perga were the last
Greek mathematicians of any distinction, and after their time
Greek civilization as a whole soon became decadent and sterile.
Euclid's Elements owed its enormously high status to a number of
reasons. The most influential single feature was Euclid's use of
the axiomatic method whereby all the theorems were laid out as
deductions from certain selfevident basic propositions or
axioms in such a way that in each successive proof only
propositions already proved or axioms were used. This became
accepted as the paradigmatically rigorous way of setting out any
body of knowledge, and attempts were made to apply it not just
to mathematics, but to natural science, theology, and even
philosophy and ethics.
However, despite being revered as an almost perfect example of
rigorous thinking for almost 2000 years there are considerable
defects in Euclid's reasoning. A number of his proofs were found
to contain mistakes, the status of the initial axioms themselves
was increasingly considered to be problematic, and the
definitions of such basic terms as ‘line’ and ‘point’ were found
to be unsatisfactory. The most celebrated case is that of the
parallel axiom, which states that there is only one straight
line passing through a given point and parallel to a given
straight line. The status of this axiom was long recognized as
problematic, and many unsuccessful attempts were made to deduce
it from the remaining axioms. The question was only settled in
the 19th century when Janos Bolyai and Nicolai Lobachevski
showed that it was perfectly possible to construct a consistent
geometry in which Euclid's other axioms were true but in which
the parallel axiom was false. This epochmaking discovery
displaced Euclidean geometry from the privileged position it had
occupied. The question of the relation of Euclid's geometry to
the properties of physical space had to wait until the early
20th century for a full answer. Until then it was believed that
Euclid's geometry gave a fully accurate description of physical
space. No less a thinker than Immanuel Kant had thought that it
was logically impossible for space to obey any other geometry.
However when Albert Einstein developed his theory of relativity
he found that the appropriate geometry for space was not
Euclid's but that developed by Georg Riemann. It was
subsequently experimentally verified that the geometry of space
is indeed nonEuclidean.
In mathematical terms too, the discovery of nonEuclidean
geometries was of great importance, since it led to a broadening
of the conception of geometry and the development by such
mathematicians as Felix Klein of many new geometries very
different from Euclid's. It also made mathematicians scrutinize
the logical structure of Euclid's geometry far more closely and
in 1899 David Hilbert at last gave a definitively rigorous
axiomatic treatment of geometry and made an exhaustive
investigation of the relations of dependence and independence
between the axioms, and of the consistency of the various
possible geometries so produced.
Euclid wrote a number of other works besides the Elements,
although many of them are now lost and known only through
references to them by other classical authors. Those that do
survive include Data, containing 94 propositions, On Divisions,
and the Optics. One of his sayings has come down to us. When
asked by Ptolemy I Soter, the reigning king of Egypt, if there
was any quicker way to master geometry than by studying the
Elements Euclid replied “There is no royal road to geometry.”
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The Greek mathematician Euclid (active 300 B.C.) wrote the
"Elements", a collection of geometrical theorems. The oldest
extant major mathematical work in the Western world, it set a
standard for logical exposition for over 2,000 years.
Virtually nothing is known of Euclid personally. It is not even
known for certain whether he was a creative mathematician
himself or was simply good at compiling the work of others. Most
of the information about Euclid comes from Proclus, a
5thcenturyA.D. Greek scholar. Since Archimedes refers to
Euclid and Archimedes lived immediately after the time of
Ptolemy I, King of Egypt (ca. 306283 B.C.), Proclus concludes
they were contemporaries. Euclid's mathematical education may
well have been obtained from Plato's pupils in Athens, since it
was there that most of the earlier mathematicians upon whose
work the Elements is based had studied and taught.
No earlier writings comparable to the Elements of Euclid have
survived. One reason is that Euclid's Elements superseded all
previous writings of this type, making it unnecessary to
preserve them. This makes it difficult for the historian to
investigate those earlier mathematicians whose works were
probably more important in the development of Greek mathematics
than Euclid's. About 600 B.C. the Greek mathematician Thales is
said to have discovered a number of theorems that appear in the
Elements. It might be noted too that Eudoxus is also given
credit for the discovery of the method of exhaustion, whereby
the area of a circle and volume of a sphere and other figures
can be calculated. Book XII of the Elements makes use of this
method. Although mathematics may have been initiated by concrete
problems, such as determining areas and volumes, by the time of
Euclid mathematics had developed into an abstract construction,
an intellectual occupation for philosophers rather than
scientists.
The Elements
The Elements consists of 13 books. Within each book is a
sequence of propositions or theorems, varying from about 10 to
100, preceded by definitions. In Book I, 23 definitions are
followed by five postulates. After the postulates, five common
notions or axioms are listed. The first is, "Things which are
equal to the same thing are also equal to each other." Next are
48 propositions which relate some of the objects that were
defined and which lead up to Pythagoras's theorem: in
rightangled triangles the square on the side subtending the
right angle is equal to the sum of the squares on the sides
containing the right angle. The usual elementary course in
Euclidean geometry is based on Book I.
The remaining books, although not so well known, are more
advanced mathematically. Book II is a continuation of Book I,
proving geometrically what today would be called algebraic
identities, such as (a + b)2 = a2 + b2 + 2ab, and generalizing
some propositions of Book I. Book III is on circles,
intersections of circles, and properties of tangents to circles.
Book IV continues with circles, emphasizing inscribed and
circumscribed rectilinear figures.
Book V of the Elements is one of the finest works in Greek
mathematics. The theory of proportions discovered by Eudoxus is
here expounded masterfully by Euclid. The theory of proportions
is concerned with the ratios of magnitudes (rational or
irrational numbers) and their integral multiples. Book VI
applies the propositions of Book V to the figures of plane
geometry. A basic proposition in this book is that a line
parallel to one side of a triangle will divide the other two
sides in the same ratio.
As in Book V, Books VII, VIII, and IX are concerned with
properties of (positive integral) numbers. In Book VII a prime
number is defined as that which is measured by a unit alone (a
prime number is divisible only by itself and 1). In Book IX
proposition 20 asserts that there are infinitely many prime
numbers, and Euclid's proof is essentially the one usually given
in modern algebra textbooks. Book X is an impressively
wellfinished treatment of irrational numbers or, more
precisely, straight lines whose lengths cannot be measured
exactly by a given line assumed as rational.
Books XIXIII are principally concerned with threedimensional
figures. In Book XII the method of exhaustion is used
extensively. The final book shows how to construct and
circumscribe by a sphere the five Platonic, or regular, solids:
the regular pyramid or tetrahedron, octahedron, cube,
icosahedron, and dodecahedron.
Manuscript translations of the Elements were made in Latin and
Arabic, but it was not until the first printed edition,
published in Venice in 1482, that geometry, which meant in
effect the Elements, became important in European education. The
first complete English translation was printed in 1570. It was
during the most active mathematical period in England, about
1700, that Greek mathematics was studied most intensively.
Euclid was admired, mastered, and utilized by all major
mathematicians, including Isaac Newton.
The growing predominance of the sciences and mathematics in the
18th and 19th centuries helped to keep Euclid in a prominent
place in the curriculum of schools and universities throughout
the Western world. But also the Elements was considered
educational as a primer in logic.
Euclid's Other Works
Some of Euclid's other works are known only through references
by other writers. The Data is on plane geometry. The word "data"
means "things given." The treatise contains 94 propositions
concerned with the kind of problem where certain data are given
about a figure and from which other data can be deduced, for
example: if a triangle has one angle given, the rectangle
contained by the sides including the angle has to the area of
the triangle a given ratio.
On Division (of figures), also on plane geometry, is known only
in the Arabic, from which English translations were made.
Proclus refers to it when speaking of dividing a figure into
other figures different in kind, for example, dividing a
triangle into a triangle and a quadrilateral. On Division is
concerned with more general problems of division. As an example,
one problem is to draw in a given circle two parallel chords
cutting off between them a given fraction of the area of the
circle.
The Conics appears to have been lost by the time of the Greek
astronomer Pappus (late 3d century A.D.). It is frequently
referred to by Archimedes. As the name suggests, it dealt with
the conic sections: the ellipse, parabola, and hyperbola, to use
the names given them later by Apollonius of Perga.
A work which has survived is Phaenomena. This is what today
would be called applied mathematics; it is about the geometry of
spheres applicable to astronomy. Another applied work which has
survived is the Optics. It was maintained by some that the sun
and other heavenly bodies are actually the size they appear to
be to the eye. This work refuted such a view by analyzing the
relationship between what the eye sees of an object and what the
object actually is. For example, the eye always sees less than
half of a sphere, and as the observer moves closer to the sphere
the part of it seen is decreased although it appears larger.
Another lost work is the Porisms, known only through Pappus. A
porism is intermediate between a theorem and a problem; that is,
rather than something to be proved or something to be
constructed, a porism is concerned with bringing out another
aspect of something that is already there. To find the center of
a circle or to find the greatest common divisor of two numbers
are examples of porisms. This work appears to have been more
advanced than the Elements and perhaps if known would give
Euclid a higher place in the history of mathematics.
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This web page was last updated on:
10 December, 2008
