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: Στοιχεῖα) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises 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^[1] and it is the oldest extant axiomatic deductive treatment of mathematics.^[2] It has proven instrumental in the development of logic and modern science.

Euclid's Elements is the most successful^[3][4] and influential^[5] 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,^[5] with the number reaching well over one thousand.^[6] 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.^[2]

[edit] History

Euclid was a Greek mathematician who wrote Elements at the University at Alexandria during the Hellenistic period (around 300 BC).

[edit] Basis in earlier work

The frontispiece of an Adelard of Bath Latin translation of Euclid's Elements, c. 1309–1316; the oldest surviving Latin translation of the Elements is a 12th century work by Adelard, which translates to Latin from the Arabic.^[7]

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.^[8] Euclid often replaced fallacious proofs with his own, more rigorous versions.^[9] The use of definitions, postulates, and axioms dated back to Plato.^[10] 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.^[11]

[edit] 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.^[7] 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.^[7] The Elements was lost to Christian Europe until ca. 1120, when the English monk Adelard of Bath disguised himself as a Muslim student in order to obtain a copy in Muslim Córdoba.^[12]

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.

[edit] 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.^[13] 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".^[14]

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.

[edit] Outline of Elements

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.

[edit] Contents of the books

A fragment of Euclid's elements found at Oxyrhynchus, which is dated to circa 100 AD. The diagram accompanies Proposition 5 of Book II of the Elements.

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 +b²= 2(a² + (a+b)²).
• Book 3 deals with circles and their properties: inscribed angles, tangents, the power of a point.
• 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:

• Book 5 is a treatise on proportions of magnitudes. Proposition 25 has as a special case the inequality of arithmetic and geometric means.
• Book 6 applies proportions to geometry: Thales' theorem, similar figures.
• Book 7 deals strictly with elementary number theory: divisibility, prime numbers, Euclid's algorithm for finding the greatest common divisor, least common multiple. Propositions 30 and 32 together are essentially equivalent to the fundamental theorem of arithmetic stating that every positive integer can be written as a product of primes in an essentially unique way, though Euclid would have had trouble stating it in this modern form as he did not use the product of more than 3 numbers.
• Book 8 deals with proportions in number theory and geometric sequences.
• Book 9 applies the results of the preceding two books and gives the infinitude of prime numbers (proposition 20), the sum of a geometric series (proposition 35), and the construction of even perfect numbers (proposition 36).
• 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 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.

[edit] Euclid's method and style of presentation

Euclid's axiomatic approach was widely influential.

Euclidean geometry is constructive.

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.^[15]

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,^[16] 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.^[17]

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.^[18]

[edit] 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.^[19] 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 .

The spurious Book XV was likely written, at least in part, by Isidore of Miletus. This inferior 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.^[19]

[edit] Editions

The Italian Jesuit Matteo Ricci (left) and the Chinese mathematician Xu Guangqi (right) published the Chinese edition of Euclid's Elements (幾何原本) in 1607.

• 1460s, Regiomontanus (incomplete)
• 1533, editio princeps by Simon Grynäus
• 1572, Commandinus
• 1574, Christoph Clavius

[edit] Translations

• 1505, Bartolomeo Zamberti (Latin)
• 1543, Venturino Ruffinelli (Italian)
• 1555, Johann Scheubel (German)
• 1562, Jacob Kündig (German)
• 1564, Pierre Forcadel de Beziers (French)
• 1570, John Day (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)

[edit] Currently in print

"Euclid's Elements - All thirteen books in one volume" Green Lion Press. ISBN 1-888009-18-7 Based on Heath's translation.

[edit] Notes

[edit] References

• Ball, W.W. Rouse (1960). A Short Account of the History of Mathematics (4th ed. [Reprint. Original publication: London: Macmillan & Co., 1908] ed.). New York: Dover Publications. pp. 50–62. ISBN 0-486-20630-0. 
• Heath, Thomas L. (1956) (3 vols.). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3).  Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.
• Boyer, Carl B. (1991). A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. ISBN 0471543977. 

[edit] External links

Wikisource has original text related to this article:

The Elements of Euclid

Wikimedia Commons has media related to: Elements of Euclid

• Euclid (David E. Joyce, ed. 1997) [c. 300 BC]. Elements. http://aleph0.clarku.edu/~djoyce/java/elements/toc.html. Retrieved on 2006-08-30.  In HTML with Java-based interactive figures.
• Richard Fitzpatrick 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) ISBN-13 978-0615179841
• Heath's English translation (HTML, without the figures) (accessed May 6, 2007)
  • Heath's English translation and commentary, with the figures (Google Books): vol. 1, vol. 2, vol. 3, vol. 3 c. 2
• Euclid's Elements in ancient Greek (typeset in PDF format, public domain. available in print--free download)
• Oliver Byrne's 1847 edition - an unusual version using color rather than labels such as ABC (scanned page images, public domain)
• The First Six Books of the Elements by John Casey and Euclid scanned by Project Gutenberg.
• Heath's translation (HTML, without the figures, public domain)
• In ancient Greek (typeset in PDF format, public domain)
• Reading Euclid - a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)
• Mind Map of The Elements Interactive.
• Sir Thomas More's manuscript
• Latin translation by Aethelhard of Bath
• Euclid Elements - The original Greek text Greek HTML