Euclid's Elements


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Euclid’s Elements

 

a scientific work written by Euclid in the third century B.C. containing the fundamentals of ancient mathematics—elementary geometry, number theory, algebra, the general theory of proportions, and a method for determining areas and volumes, including elements of the theory of limits. In this work, Euclid summarized three centuries of Greek mathematics and created a reliable foundation for further mathematical research. His Elements is not, however, an encyclopedia of the mathematical knowledge of his time. Thus, the Elements does not deal with the then already extensive theory of conic sections or with computational methods.

The method of Euclid’s Elements is deductive. Euclid starts with definitions, postulates, and axioms. Then come statements and proofs of theorems. After defining the fundamental geometric concepts and objects (for example, points and lines), Euclid proves the existence of other geometric objects (for example, the equilateral triangle) by constructing them. These constructions rely on five postulates. The postulates assert the possibility of performing particular elementary constructions, for example, ”[it is possible to] draw a straight line from any point to any point” (Postulate I) and ”[it is possible to] describe a circle with any center and distance [radius]” (Postulate HI). A particularly important postulate is the fifth postulate (the parallel postulate), which states ”that if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.” The relative complexity of this postulate led to attempts by many mathematicians (over nearly 2, 000 years) to derive it from the other fundamental assumptions of geometry. Attempts to prove the fifth postulate continued right up to the time when N. I. Lobachevskii constructed the first system of non-Euclidian geometry, a system in which this postulate does not hold. Euclid’s postulates are followed by ”common notions,” or axioms. The axioms deal with properties of the relations of equality and inequality between magnitudes. The axioms include: ”Things which are equal to the same thing are also equal to one another.” (First Axiom) and ”The whole is greater than the part” (Fifth Axiom).

From the modern point of view, the system of axioms and postulates of Euclid’s Elements is not sufficient for a deductive development of geometry. The Elements has neither axioms of motion nor axioms of congruence (except for one). It also lacks axioms of order and continuity. Nevertheless, Euclid uses both motion and continuity in the proofs. The logical deficiencies in the structure of Euclid’s Elements were fully explained only at the end of the 19th century by D. Hubert. Until then, Euclid’s Elements had served for more than 2, 000 years as a model of scientific rigor. Geometry was studied using the Elements, either in its entirety or in abridged and revised form.

Euclid’s Elements consist of 13 books, or parts. Book I treats the fundamental properties of triangles, rectangles, and parallelograms and compares their areas. It concludes with the Pythagorean theorem. Book II contains some geometric algebra, that is, a geometric apparatus is constructed for solving problems that reduce to quadratic equations (Euclid’s Elements lacks algebraic notation). Properties of circles, including tangents and chords, are examined in Book III. (These problems were studied by Hippocratus of Chios in the second half of the fifth century B.C.) Regular polygons are examined in Book IV. The general theory of proportions, created by Eudoxus of Cnidus, is given in Book V. This theory may be considered as a prototype of the theory of real numbers, which was not developed until the second half of the 19th century. The general theory of proportions is the foundation for the theory of similarity (Book VI) and the method of exhaustion (Book VII), also due to Eudoxus. The beginnings of number theory, based on an algorithm for finding the greatest common divisor (Euclid’s algorithm), are presented in Books VII-IX. These books contain a theory of divisibility—including the theorem on the uniqueness of the factorization of an integer into primes and the theorem asserting the infinitude of primes—and a theory of proportions for integers, which is essentially equivalent to the theory of (positive) rational numbers. Book X gives a classification of quadratic and biquadratic irrationalities and proves various rules for transforming them. The results of Book X are used in Book XIII to find the lengths of edges of regular polyhedra. A major part of Books X and XIII (and probably Book VII) is due to Theaetetus (beginning of the fourth century B.C.).

The foundations of stereometry are set forth in Book XI. The ratio of the areas of two circles and the ratio of the volumes of a pyramid and a prism and of a cone and a cylinder are calculated in Book XII using the method of exhaustion. These theorems were first proved by Eudoxus. Finally, in Book XIII the ratio of the volumes of two spheres is calculated, five regular polyhedrons are constructed, and it is proved that no other regular solids exist. Books XIV and XV, which are not the work of Euclid, were added to the Elements by subsequent Greek mathematicians. They are often published with the basic text of Euclid’s Elements.

Euclid’s Elements was widely known even in antiquity. Archimedes, Apollonius of Perga, and other scientists relied on the Elements in their studies in mathematics and mechanics. The original text of the Elements has not survived (the oldest copy that has been preserved dates from the second half of the ninth century A.D.). Translations of Euclid’s Elements into Arabic appeared at the end of the eighth or beginning of the ninth century. The first translation into Latin was made from Arabic by Adelard of Bath between 1100 and 1125. Ancient copies have substantially different texts. We have no definitive version of the text of the Elements. The first printed edition of Euclid’s Elements, a Latin translation by G. Campano of Novara, appeared in Venice in 1482 with figures in the margins. (The translation was made between 1250 and 1260, with Campano using both Arabic sources and the translation of Adelard of Bath.) The best modern edition is that of J. Heiberg (Euclidus Elementa, vols. 1–5, Leipzig, 1883–88), which has both the Greek text and its Latin translation. Euclid’s Elements have been published in Russian a number of times, beginning in the 18th century. The best Russian edition is a translation from the Greek with commentary by D. D. Mordukhai-Boltovskii (vols. 1–3, 1948–50).

REFERENCE

Istoriia matematiki s drevneishikh vremen do nachala novogo vremeni, vol. 1. Moscow, 1970.

I. G. BASHMAKOVA and A. I. MARKUSHEVICH

References in periodicals archive ?
Furthermore, while Viete made some major notational advances in algebra for solving equations prior to 1600, Galileo remained rooted in an older geometric form of ratio arithmetic that he learned from the recently recovered Book V of Euclid's Elements. In his earlier work, Hobart highlighted Viete's role in the new numeracy, but here Galileo is his protagonist.
Noting that the early Greeks were ignorant of algebra, Hahn sets aside the more familial- equation [a.sup.2] + [b.sup.2] = [c.sup.2] in favor of two theorems from Euclid's Elements: book 1, proposition 47 and theorems following it, and then the lesser-known book 6, proposition 31 and theorems following it.
(2.) Proclus Diadochus, Commentary on Euclid's Elements, Book I, Greek Mathematical Works, Volume I, The Loeb Classical Library, 1939, London, W.
He also helped translate Euclid's Elements of Geometry into Chinese and took care of the different gifts--such as mechanical clocks and a clavichord--he had given to the emperor.
Euclid's Elements was an influential work in which field of study?
Khayyam wrote a book entitled Explanations of the difficulties in the postulates in Euclid's Elements. The book consists of several sections on the parallel postulate (Book I), on the Euclidean definition of ratios and the Anthyphairetic ratio (modern continued fractions) (Book II), and on the multiplication of ratios (Book III).
'You do get collectors who like great things, and they collect across categories,' states Christie's London's specialist Margaret Ford--although categorising a book such as Darwin's Voyage of the Beagle or Euclid's Elements isn't straightforward.
Look at the oldest technical book (perhaps the only one from antiquity) that continues to be used unchanged: Euclid's Elements (ca.
While studying how Jesuits introduced Euclid's Elements into China during the early 17th century, Hart (history, U.
"Well," I said, "to give just one obvious example, '47' is the answer to several dozen questions, such as 'How many degrees of Latitude are there between the Tropics of Cancer and Capricorn?' or 'How many strings are there on a concert harp?' or 'What is the proposition number of Pythagoras' Theorem in Euclid's Elements?' or 'How many Ronin were there in the 18th century Japanese Samurai tale?' among countless others."
BUSARD, "The Translation of the Elements of Euclid from the Arabic into Latin by Hermann of Carinthia", in Ianus, 54, 1967, 1-140 e Ianus, 59, 1972, 125-187; Id., The Latin Translation of the Arabic Version of Euclid's Elements Commonly Ascribed to Gerard of Cremona, Leiden, 1983; Id., The Medieval Latin Translation of Euclid's Elements.