Euclid's Elements
Euclid's
Elements is a
mathematical and
geometric treatise, consisting of 13 books, written by the
Hellenistic mathematician Euclid in
Egypt circa 300 BC. It comprises a collection of definitions, postulates , propositions , and proofs. 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 .
Encyclopedia
Euclid's Elements is a
mathematical and
geometric treatise, consisting of 13 books, written by the
Hellenistic mathematician Euclid in
Egypt circa 300 BC. It comprises a collection of definitions, postulates , propositions , and proofs. 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 . 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 used as a basic introduction to geometry today.
First principles
Euclid based his work in Book I on 23 definitions, such as point, line and
surface, five postulates and five "common notions" .
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.
- 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.
- 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 and philosophers 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.
Parallel postulate
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 , or none can .
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 .
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.
Problems with the Elements
In the construction of the first book, Euclid used a fact not postulated or proved . 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.
History
Euclid, an
Hellenistic mathematician who probably studied as a pupil under
Plato, wrote
Elements in
Egypt around 300 BC. 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 , 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 .
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 , 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.
Contents
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" .
- 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 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 solids inscribed in a sphere.
External links
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- - an unusual version using color rather than labels such as ABC
- - a course in how to read Euclid in the original Greek, with English translations and commentaries
Complete and fragmentary manuscripts of versions of Euclid's Elements :References
