# 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