The Rediscovery of Euclid's Elements

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The Rediscovery of Euclid's Elements

Overview

The principal Greek compendium of geometry, Euclid's Elements, was translated into Arabic in the ninth century. Muslim mathematicians were then able to combine geometry with the arithmetic and algebra they learned from the Hindus and develop new advances of their own. In the twelfth century the work was translated into Latin, making it more accessible to European scholars. The Elements was still regularly used as a textbook of geometry until about 1900.

Background

Euclid was a Greek mathematician who lived in Alexandria, Egypt, around 300 b.c., and founded the first school of mathematics there. He wrote many books, some of which have been lost. However his 13-volume treatise on geometry, called the Elements, was among the most important mathematical texts in history.

The Elements was a synopsis of the work of many mathematicians, including Hippocrates of Chios, Eudoxus, and Theaeteus. It was unique for its logical exposition of contemporary knowledge and techniques in geometry, with postulates that were self-evidently true and proofs of theorems derived from the postulates. It also included sections on algebra and number theory. For example, it includes a proof that the number of primes, which are divisible only by themselves and 1, are infinite.

Alexandria, along with the rest of Egypt, was conquered by the Romans in 31 b.c. The Roman Empire was heir to much of the Greek intellectual tradition. However, after the fall of the Roman Empire in the fifth century a.d., the task of preserving classical achievements in mathematics and science fell mainly to the Islamic world.

Impact

Islam expanded rapidly after the death of the prophet Muhammad in a.d. 632, eventually extending from Spain well into central Asia. Arabian warriors conquered Alexandria in 642. Muslim scholars who now had access to the ancient centers of learning took advantage of their opportunity to study and disseminate the classical texts. Many of these were received in a peace agreement with the Byzantine emperor.

Caliph al-Ma'mun, who ruled from Baghdad between 813 and 833, caused a translation and study center to be built there, along with an astronomical observatory. In this institution, called the House of Wisdom, both Muslim and Christian scholars were employed to translate texts from the Greek and Syriac languages into Arabic. Many of the translators of mathematical texts were themselves well known mathematicians. The enterprise was financed by the caliph and his successors, as well as other wealthy Muslim intellectuals.

Translations of the Elements were made by al-Hajjaj ibn Matar, Ishaq ibn Hunayn, and Tabit ibn Qurra. Euclid's Data, Optics, On Division, and his astronomical text Phaenomena were translated. Other important translations included Ptolemy's seminal work on astronomy, the Almagest, Archimedes's Sphere and Cylinder and Measurement of the Circle, Diophantus's Arithmetica, the Sphaerica of Menelaus, and almost all the works of Apollonius of Perga, known as the Great Geometer. In addition to the classical material, works written in Sanskrit by the great Hindu mathematicians were also translated. The translations disseminated from the Baghdad center stimulated independent mathematical research in the Islamic world for the next 600 years.

The Muslim mathematicians who studied at the House of Wisdom in the ninth century included Muhammad ibn Musa al-Khwarizmi, whose works brought Hindu arithmetic and algebra to the West. Al-Khwarizmi adopted the axiomatic (proof-oriented) presentation Euclid employed in his geometrical texts to elucidate these new mathematical disciplines.

With both the geometry of the Greeks and the advanced arithmetic and algebraic techniques of the Hindus, the Islamic world had a much more complete set of tools for dealing with mathematical problems. So, for example, solid geometry problems that could not be solved with a ruler and compass could be represented algebraically. Conversely, geometric curves could be used to represent algebraic equations. The brilliant Persian poet, astronomer and mathematician Omar Khayyam (c. 1050- c. 1123) found geometric solutions to cubic equations.

One of the lines of research that was vigorously pursued by Muslim mathematicians was the investigation of Euclid's parallel postulate. The Elements was based on theorems derived from five "common notions," or postulates. The first four are simple, stating that:

(1) Any two points can be joined with a straight line

(2) A finite straight line of any length can be drawn upon a straight line.

(3) A circle may be described with any center and radius.

(4) All right angles are equal.

The fifth, known as the parallel postulate, is considerably more complex. It states, "If a straight line meets two other straight lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles."

Later, a simpler wording, also known as Playfair's Axiom, was devised. It states, "Through a point on a given line, there passes not more than one parallel to the line."

Many mathematicians disagreed that this was a "common notion," and judged it insufficiently self-evident to be termed a postulate. Even Euclid proved his first 28 propositions without reference to the parallel postulate, as if reluctant to use it himself. The mathematicians who followed him attempted to prove the parallel postulate as a theorem, using only postulates 1 through 4, the 28 propositions proved using them exclusively, and the definitions given in Euclid's work. If they could do that, they would prove in effect that the entire geometry was based on the first four postulates alone.

A frequently encountered problem was petitio principii, or circular reasoning. This is the attempt to prove a statement by using another assertion which, when considered carefully, turns out to be equivalent to, or a consequence of, the one being proved. Of course, reputable scholars do not do this intentionally. Circular reasoning is generally a consequence of the fact that mathematical statements can be formulated in quite different ways, just as Euclid's formulation of the parallel postulate sounds nothing like Playfair's Axiom. Sometimes a proof would be accepted for many years before the circular reasoning was discovered.

Omar Khayyam approached the problem of proving the parallel postulate using a four-sided figure, or quadrilateral, with two equal sides perpendicular to the base. He realized that he could accomplish the proof by demonstrating that the other two angles in the figure were also right angles. Although he never succeeded in this effort, from then on the parallel postulate was generally discussed in terms of this quadrilateral.

The Elements was first translated into Latin in the 1100s, making it more readily available to European scholars. Advances in European understanding of geometry followed, paving the way for thirteenth-century advances in geometrical optics. The German monk Theodoric of Freiberg studied the reflection and refraction, or bending, of light in spherical droplets, and first understood how rainbows are formed. Euclid's work was also brought to India by the Muslims, returning the mathematical boost the Hindus had given them with their arithmetic and algebraic techniques.

The Elements was commonly used as a geometry textbook until the dawn of the twentieth century. Today mathematicians recognize two main classes of geometries, classical Euclidean geometry in which the parallel postulate is assumed to be true, and non-Euclidean geometries, such as spherical geometry, in which it does not exist.

SHERRI CHASIN CALVO