Euclid's Elements
(Greek: ) is a mathematical and geometric treatise
consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It is a
collection of definitions, postulates (
axioms), propositions (
theorems
and
constructions), and
mathematical proofs of the
propositions. The thirteen books cover
Euclidean geometry and the ancient Greek
version of elementary
number theory.
With the exception of
Autolycus'
On the Moving Sphere, the
Elements is one of the
oldest extant Greek mathematical treatises and it is the oldest
extant axiomatic deductive treatment of
mathematics. It has proven instrumental in the
development of
logic and modern
science.
Euclid's
Elements is the most successful and influential
textbook ever written.
Being first set in type in Venice in 1482, it
is one of the very earliest mathematical works to be printed after
the invention of the printing press
and is estimated to be second only to the Bible in the number of editions published, with the
number reaching well over one thousand. It was used as the
basic text on geometry throughout the Western world for about 2,000
years. For centuries, when the
quadrivium
was included in the curriculum of all university students,
knowledge of at least part of Euclid's
Elements was
required of all students. Not until the 20th century, by which time
its content was universally taught through school books, did it
cease to be considered something all educated people had
read.
History
Basis in earlier work
Scholars believe that the
Elements is largely a collection
of theorems proved by other mathematicians supplemented by some
original work.
Proclus, a Greek
mathematician who lived several centuries after Euclid, wrote in
his commentary of the
Elements: "Euclid, who put together
the Elements, collecting many of
Eudoxus'
theorems, perfecting many of
Theaetetus', and also bringing to
irrefragable demonstration the things which were only somewhat
loosely proved by his predecessors".
Pythagoras was probably the source of most of
books I and II,
Hippocrates of book III,
and
Eudoxus book V, while books IV, VI, XI,
and XII probably came from other Pythagorean or Athenian
mathematicians. Euclid often replaced fallacious proofs with his
own, more rigorous versions. The use of definitions, postulates,
and axioms dated back to
Plato. The
Elements may have been based on an earlier textbook by
Hippocrates of Chios (not the
better known
Hippocrates of Kos), who
also may have originated the use of letters to refer to
figures.
Transmission of the text
In the fourth century C.E.
Theon of
Alexandria produced an edition of Euclid which was so widely
used that it became the only surviving source until François Peyrard's 1808 discovery at
the Vatican of a
manuscript not derived from Theon's.
Although known to, for instance,
Cicero,
there is no extant record of the text having been translated into
Latin prior to
Boethius in the fifth or
sixth century. The Arabs received the
Elements from the
Byzantines in approximately 760; this version, by a pupil of Euclid
called
Proclo, was translated into
Arabic under
Harun al
Rashid circa 800 AD. The Byzantine scholar
Arethas commissioned the copying of one of the
extant Greek manuscripts of Euclid in the late ninth century.
Although known in Byzantium, the
Elements was lost to
Western Europe until ca. 1120, when the English monk
Adelard of Bath translated it into Latin
from an Arabic translation.
The first printed edition appeared in 1482 (based on
Giovanni Campano's 1260 edition), and since
then it has been translated into many languages and published in
about a thousand different editions. Theon's Greek edition was
recovered in 1533. In 1570,
John Dee provided a widely
respected "Mathematical Preface", along with copious notes and
supplementary material, to the first English edition by
Henry Billingsley.
Copies of
the Greek text still exist, some of which can be found in the
Vatican
Library and the Bodleian Library in Oxford. The manuscripts available are of
variable quality, and invariably incomplete. By careful analysis of
the translations and originals, hypotheses have been drawn about
the contents of the original text (copies of which are no longer
available).
Ancient texts which refer to the
Elements itself and to
other mathematical theories that were current at the time it was
written are also important in this process. Such analyses are
conducted by
J.
L. Heiberg and Sir
Thomas Little Heath in their editions of
the text.
Also of importance are the
scholia, or
annotations to the text. These additions, which often distinguished
themselves from the main text (depending on the manuscript),
gradually accumulated over time as opinions varied upon what was
worthy of explanation or elucidation.
Influence
The
Elements is still considered a masterpiece in the
application of
logic to
mathematics. In historical context, it has
proven enormously influential in many areas of
science. Scientists
Nicolaus Copernicus,
Johannes Kepler,
Galileo Galilei, and Sir
Isaac Newton were all influenced by the
Elements, and applied their knowledge of it to their work.
Mathematicians and philosophers, such as
Bertrand Russell,
Alfred North Whitehead, and
Baruch Spinoza, have attempted to create
their own foundational "Elements" for their respective disciplines,
by adopting the axiomatized deductive structures that Euclid's work
introduced.
The austere beauty of Euclidean geometry has been seen by many in
western culture as a glimpse of an otherworldly system of
perfection and certainty. Abraham Lincoln kept a copy of Euclid in
his saddlebag, and studied it late at night by lamplight; he
related that he said to himself, "You never can make a lawyer if
you do not understand what demonstrate means; and I left my
situation in Springfield, went home to my father's house, and
stayed there till I could give any proposition in the six books of
Euclid at sight".
Edna St.
Vincent Millay wrote in her sonnet
Euclid Alone Has Looked
on Beauty Bare, "O blinding hour, O holy, terrible day, When
first the shaft into his vision shone Of light anatomized!".
Einstein recalled a copy of the
Elements and a magnetic compass as two gifts that had a
great influence on him as a boy, referring to the Euclid as the
"holy little geometry book".
The success of the
Elements is due primarily to its
logical presentation of most of the mathematical knowledge
available to Euclid. Much of the material is not original to him,
although many of the proofs are his. However, Euclid's systematic
development of his subject, from a small set of axioms to deep
results, and the consistency of his approach throughout the
Elements, encouraged its use as a textbook for about 2,000
years. The
Elements still influences modern geometry
books. Further, its logical axiomatic approach and rigorous proofs
remain the cornerstone of mathematics.
Outline of Elements
Contents of the books
Books 1 through 4 deal with plane geometry:
- Book 1 contains Euclid's 10 axioms (5 named
postulates—including the parallel
postulate—and 5 named axioms) and the basic propositions of
geometry: the pons asinorum
(proposition 5) , the Pythagorean
theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a
triangle, and the three cases in which triangles are "equal" (have
the same area).
- Book 2 is commonly called the "book of geometrical algebra,"
because the most of the propositions are geometric interpretations
of algebraic identities, such as
a(b + c + ...) = ab + ac + ...
or
(2a + b)^{2} + b^{2} = 2(a^{2} + (a + b)^{2}).
- Book 3 deals with circles and their properties: inscribed angles, tangents,
the power of a point, Thales'
theorem.
- Book 4 constructs the incircle and
circumcircle of a triangle, and
constructs regular polygons with 4,
5, 6, and 15 sides.
Books 5 through 10 introduce
ratios and
proportions:
Books 11 through 13 deal with spatial geometry:
- Book 11 generalizes the results of Books 1–6 to space:
perpendicularity, parallelism, volumes of parallelepipeds.
- Book 12 studies volumes of cones, pyramids, and cylinders in detail, and shows for
example that the volume of a cone is a third of the volume of the
corresponding cylinder. It concludes by showing the volume of a
sphere is proportional to the cube of its
radius by approximating it by a union of many pyramids.
- Book 13 constructs the five regular Platonic solids inscribed in a sphere,
calculates the ratio of their edges to the radius of the sphere,
and proves that there are no further regular solids.
Euclid's method and style of presentation
Euclid's
axiomatic
approach and
constructive methods
were widely influential.
As was common in ancient mathematical texts, when a proposition
needed
proof in several different
cases, Euclid often proved only one of them (often the most
difficult), leaving the others to the reader. Later editors such as
Theon often interpolated their
own proofs of these cases.
Euclid's list of axioms was not exhaustive, but represented the
principles that were the most important. His proofs often invoke
axiomatic notions which were not originally presented in his list
of axioms.
Euclid's presentation was limited by the mathematical ideas and
notations in common currency in his era, and this causes the
treatment to seem awkward to the modern reader in some places. For
example, there was no notion of an angle greater than two right
angles, the number 1 was sometimes treated separately from other
positive integers, and as multiplication was treated geometrically
he did not use the product of more than 3 different numbers. The
geometrical treatment of number theory may have been because the
alternative would have been the extremely awkward
Alexandrian system of numerals.
The presentation of each result is given in a stylized form, which
originated with Euclid: enunciation, statement, construction,
proof, and conclusion. No indication is given of the method of
reasoning that led to the result, although the
Data does provide instruction about how
to approach the types of problems encountered in the first four
books of the
Elements.
Apocrypha
It was not uncommon in ancient time to attribute to celebrated
authors works that were not written by them. It is by these means
that the
apocryphal books XIV and XV of
the
Elements were sometimes included in the collection.
The spurious Book XIV was likely written by
Hypsicles on the basis of a treatise by
Apollonius. The book continues Euclid's
comparison of regular solids inscribed in spheres, with the chief
result being that the ratio of the surfaces of the
dodecahedron and
icosahedron inscribed in the same sphere is the
same as the ratio of their volumes, the ratio being
- \sqrt{\tfrac{10}{3(5-\sqrt{5})}} =
\sqrt{\tfrac{5+\sqrt{5}}{6}}.\
The spurious Book XV was likely written, at least in part, by
Isidore of Miletus. This book
covers topics such as counting the number of edges and solid angles
in the regular solids, and finding the measure of dihedral angles
of faces that meet at an edge.{{cite
book|last=Boyer|authorlink=Carl Benjamin
Boyer|title=|year=1991|chapter=Euclid of
Alexandria|pages=118-119|quote=In ancient times it was not uncommon
to attribute to a celebrated author works that were not by him;
thus, some versions of Euclid's
Elements include a
fourteenth and even a fifteenth book, both shown by later scholars
to be apocryphal. The so-called Book XIV continues Euclid's
comparison of the regular solids inscribed in a sphere, the chief
results being that the ratio of the surfaces of the dodecahedron
and icosahedron inscribed in the same sphere is the same as the
ratio of their volumes, the ratio being that of the edge of the
cube to the edge of the icosahedron, that is,
\sqrt{10/[3(5-\sqrt{5})]}. It is thought that this book may have
been composed by Hypsicles on the basis of a treatise (now lost) by
Apollonius comparing the dodecahedron and icosahedron. [...] The
spurious Book XV, which is inferior, is thought to have been (at
least in part) the work of Isidore of Miletus (fl. ca. A.D. 532),
architect of the cathedral of Holy Wisdom (Hagia Sophia) at
Constantinople. This book also deals with the regular solids,
counting the number o edges and solid angles in the solids, and
finding the measures of the dihedral angles of faces meeting at an
edge.}}
Editions
Translations
- 1505, Bartolomeo Zamberti (Latin)
- 1543, Venturino Ruffinelli (Italian)
- 1555, Johann Scheubel (German)
- 1562, Jacob Kündig (German)
- 1564, Pierre Forcadel de Beziers (French)
- 1570, Henry Billingsley
(English)
- 1576, Rodrigo de Zamorano
(Spanish)
- 1594, Typografia Medicea (edition of the Arabic translation of
Nasir al-Din al-Tusi)
- 1607, Matteo Ricci, Xu Guangqi (Chinese)
- 1660, Isaac Barrow (English)
- early 1700s Jagannatha Samrat (Sanskrit)
Currently in print
"Euclid's Elements - All thirteen books in one volume" Green Lion
Press. ISBN 1-888009-18-7Based on Heath's translation.
Notes
- Ball (1960).
- Encyclopedia of Ancient Greece (2006) by Nigel Guy Wilson, page
278. Published by Routledge Taylor and Francis Group.
Quote:"Euclid's Elements subsequently became the basis of all
mathematical education, not only in the Romand and Byzantine
periods, but right down to the mid-20th century, and it could be
argued that it is the most successful textbook ever written."
- The Historical Roots of Elementary Mathematics by Lucas
Nicolaas Hendrik Bunt, Phillip S. Jones, Jack D. Bedient (1988),
page 142. Dover publications. Quote:"the Elements became
known to Western Europe via the Arabs and the Moors. There the
Elements became the foundation of mathematical education.
More than 1000 editions of the Elements are known. In all
probability it is, next to the Bible, the most widely
spread book in the civilization of the Western world."
- W.W. Rouse Ball, A Short Account of the History of Mathematics,
4th ed., 1908, p. 54
- Ball, p. 43
- Ball, p. 38
- Russell, Bertrand. A History of Western Philosophy. p.
212.
- L.D. Reynolds and Nigel G. Wilson, Scribes and
Scholars 2nd. ed. (Oxford, 1974) p. 57
- One older work claims Adelard disguised himself as a Muslim
student in order to obtain a copy in Muslim Córdoba (Rouse Ball, p.
165). However, more recent biographical work has turned up no clear
documentation that Adelard ever went to Muslim-ruled Spain,
although he spent time in Norman-ruled Sicily and Crusader-ruled
Antioch, both of which had Arabic-speaking populations. Charles
Burnett, Adelard of Bath: Conversations with his Nephew
(Cambridge, 1999); Charles Burnett, Adelard of Bath
(University of London, 1987).
- Henry Ketcham, The Life of Abraham Lincoln, at Project
Gutenberg, http://www.gutenberg.org/ebooks/6811
- Dudley Herschbach, "Einstein as a Student," Department of
Chemistry and Chemical Biology, Harvard University, Cambridge, MA,
USA, page 3, web: HarvardChem-Einstein-PDF: about Max Talmud
visited on Thursdays for six years.
- Heath, p. 62
- Ball, p. 55
- Ball, pp. 58, 127
- Ball, p. 54
References
- Heath's authoritative translation plus extensive historical
research and detailed commentary throughout the text.
External links