Euclid's Elements
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The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570 |
Euclid's Elements (
Greek: ) is a
mathematical and
geometric treatise, consisting of 13 books, written by the
Hellenistic mathematician Euclid in
Egypt during the early
3rd century BC. It comprises a collection of definitions, postulates (
axioms), propositions (
theorems) and proofs thereof. Euclid's books are in the fields of
Euclidean geometry, as well as the ancient Greek version of
number theory. The
Elements is one of the oldest extant axiomatic deductive treatments of
geometry, and has proven instrumental in the development of
logic and modern
science.
It is considered one of the most successful textbooks ever written: the
Elements was one of the very first books to go to press, and is second only to the
Bible in number of editions published (well over 1000). 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 did it cease to be considered something all educated people had read. It is still (though rarely) used as a basic introduction to geometry today.
Euclid based his work in Book I on 23 definitions, such as
point,
line and
surface, five
postulates and five "common notions" (both of which are today called
axioms).
Postulates in Book I:
# A straight line segment can be drawn by joining any two points.# A straight line segment can be extended indefinitely in a straight line.# Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center.# All right angles are
congruent.# If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
Common notions in Book I:
# Things which equal the same thing are equal to one another. (
Transitive property of
equality)# If equals are added to equals, then the sums are equal. # If equals are subtracted from equals, then the remainders are equal. # Things which coincide with one another are equal to one another. (
Reflexive property of equality)# The whole is greater than the part.
These basic principles reflect the interest of Euclid, along with his contemporary Greek and Hellenistic mathematicians, in constructive geometry. The first three postulates basically describe the
constructions one can carry out with a
compass and an unmarked
straightedge. A marked
ruler, used in
neusis, is forbidden, probably because Euclid could not prove that verging lines meet.
The success of
Elements is due primarily to its logical presentation of much of the mathematical knowledge available to Euclid. Most of the material is not original to him, although a few of the proofs are his. Its systematic development from a small set of axioms to deep results encouraged its use as a textbook for hundreds of years, and still influences modern geometry books.
Throughout history there have been controversies surrounding many of Euclid's axioms and proofs. Nevertheless, the
Elements has withstood the test of time and is still considered a masterpiece in the application of
logic to
mathematics, and, historically, it has been enormously influential in many areas of
science. European scientists
Nicolaus Copernicus,
Johannes Kepler,
Galileo Galilei and especially Sir
Isaac Newton were all influenced by the
Elements, and applied their knowledge of it to their work. Mathematicians (
Bertrand Russell,
Alfred North Whitehead) and philosophers (
Baruch Spinoza) have also attempted to provide their own
Elements; that is, axiomatized deductive structures of their own respective disciplines. Even today, introductory mathematics textbooks often have the word elements in their title, e.g.
Elements of Information Theory.
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If the sum of the two interior angles equals 180°, the lines are parallel and will never intersect. |
Of the five postulates Euclid used, the last, so-called "
parallel postulate" seemed less obvious than the others. Many geometers suspected that it may be provable from the other postulates but all attempts to do this failed. By the mid-
19th century, it was shown that no such proof exists, because one can construct
non-Euclidean geometries where the parallel postulate is false, while the other postulates remain true.Mathematicians say that the parallel postulate is
independent of the other postulates.Two alternatives are possible: either an infinite number of parallel lines can be drawn through a point not on a straight line (
hyperbolic geometry, also called
Lobachevskian geometry), or none can (
elliptic geometry, also called
Riemannian geometry).That other geometries could be logically consistent was one of the most important discoveries in mathematics, with vast implications for science and philosophy.Indeed,
Albert Einstein's theory of
general relativity shows that the "real" space in which we live can be non-Euclidean (for example, around
black holes and
neutron stars).It is a testament to Euclid's dedication to a logical development from as few assumptions as possible that he recognized the independence of the parallel postulate. His statement of it as a fifth separate axiom predates by two millennia its acceptance as such by other mathematicians.
In the construction of the first book, Euclid used a fact not postulated or proved (i.e., two circles with centers at the distance of their radius will intersect in two points). Later, in the fourth construction, he used the movement of triangles to prove that if two sides and their angles are equal, then they are congruent. He didn't postulate or even define movement.
In the 19th century Euclid came under more criticism. The postulates were found to be both incomplete and superabundant. And at the same time, the non-Euclidean geometries attracted the attention of contemporary mathematicians. Attempts were made by leading mathematicians such as
Dedekind and
Hilbert to add axioms to the
Elements to make Euclidean geometry more complete, such as an axiom of continuity and an axiom of congruence.
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Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310) |
Elements was written in
Egypt during the early
3rd century BC by
Euclid, an ancient
Hellenistic mathematician who probably studied as a pupil under
Plato. Scholars believe that the
Elements is largely a collection of theorems proved by other mathematicians as well as containing some original work.
Proclus, a Greek mathematician who lived several centuries after Euclid, writes in his commentary of the
Elements: "Euclid, who put together the Elements, collecting many of Eudoxus's theorems, perfecting many of Theaetetus's, and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors".
A version by a pupil of Euclid called
Proclo was translated later into
Arabic after being obtained by the Arabs from
Byzantium and from those secondary translations into
Latin. 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. 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 also exist, e.g. in the
Vatican Library and the Bodlean library in Oxford. However, the manuscripts available are of very 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 footnotes 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. Some of these are useful and add to the text, but many are not.
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A proof from Euclid's Elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides. The proof is by construction: an equilateral triangle ΑÎ'Î" is made by drawing circles Î" and Ε centered on the points Α and Î', and taking one intersection of the circles as the third vertex of the triangle. |
Although
Elements is a geometric work, it also includes results that today would be classified as
number theory. Euclid probably chose to describe results in number theory in terms of geometry because he couldn't develop a constructible approach to arithmetic. A construction used in any of Euclid's proofs required a proof that it is actually possible. This avoids the problems the Pythagoreans encountered with irrationals, since their fallacious proofs usually required a statement such as "Find the greatest common measure of..."
The contents of the work are as follows:
Books 1 through 4 deal with plane geometry:
* Book 1 contains the basic properties of geometry: the
Pythagorean theorem, 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 material it contains may easily be interpreted as
algebra.
* Book 3 deals with circles and their properties:
inscribed angles,
tangents, the power of a point.
* Book 4 is concerned with inscribing and circumscribing triangles and
regular polygons.
Books 5 through 10 introduce
ratios and
proportions:
* Book 5 is a treatise on proportions of
magnitudes.
* Book 6 applies proportions to geometry:
Thales' theorem, similar figures.
* Book 7 deals strictly with number theory:
divisibility,
prime numbers,
greatest common divisor,
least common multiple.
* Book 8 deals with proportions in number theory and
geometric sequences.
* Book 9 applies the results of the preceding two books: the infinitude of prime numbers, the sum of a
geometric series,
perfect numbers.
* Book 10 attempts to classify
incommensurable (in modern language,
irrational) magnitudes by using the
method of exhaustion, a precursor to
integration.
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 calculates areas and volumes by using the method of exhaustion:
cones,
pyramids,
cylinders, and the
sphere.
* Book 13 generalizes Book 4 to space:
golden section, the five regular (or Platonic)
solids inscribed in a sphere.
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a bilingual edition (typset in PDF format, with the original Greek and an English translation on facing pages; free in PDF form, available in print)
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Heath's translation (HTML, without the figures, public domain)
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in ancient Greek (typeset in PDF format, public domain)
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Oliver Byrne's 1847 edition - an unusual version using color rather than labels such as ABC (scanned page images, public domain)
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Reading Euclid - a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)
Complete and fragmentary manuscripts of versions of Euclid's Elements :*
Sir Thomas More's
manuscript*
Latin translation by
Aethelhard of Bath
