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Euclid's Elements

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Euclid's Elements (Greek: Στοιχεῖα) 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.

Contents

First principles

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:

  1. A straight line segment can be drawn by joining any two points.
  2. A straight line segment can be extended indefinitely in a straight line.
  3. Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center.
  4. All right angles are congruent.
  5. 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:

  1. Things which equal the same thing are equal to one another. (Transitive property of equality)
  2. If equals are added to equals, then the sums are equal.
  3. If equals are subtracted from equals, then the remainders are equal.
  4. Things which coincide with one another are equal to one another. (Reflexive property of equality)
  5. 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.

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 ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.
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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 ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.

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.

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 (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.

Problems with the Elements

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.

History

Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elementa.
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Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elementa.

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.

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:

Books 5 through 10 introduce ratios and proportions:

Books 11 through 13 deal with spatial geometry:

External links

Complete and fragmentary manuscripts of versions of Euclid's Elements :

References

 

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