# Elements-of-Euclid

2020 Mathematics Subject Classification: *Primary:* 01A20 *Secondary:* 51-03 [MSN][ZBL]

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A scientific work, written in the 3rd century B.C., containing the foundations of ancient mathematics: elementary geometry, number theory, algebra, the general theory of proportion, and a method for the determination of areas and volumes, including elements of the theory of limits. Euclid's Elements constitute a typical deductive system, containing the basic propositions of geometry and other branches of mathematics, on the basis of which all the theories are developed in a rigorously logical fashion.

The Elements follow a definite scheme that actually predates Euclid and is briefly expounded in Aristotle's works: first come definitions, postulates and axioms, then the statements of the theorems and their proofs. Besides theorems, the Elements also include problems solved by constructions or through the use of arithmetical algorithms. After defining the fundamental geometrical concepts and objects, Euclid proves the existence of other objects (such as the equilateral triangle) by constructing them; this he does on the basis of five postulates. The postulates state that the following elementary constructions are possible: 1) through two points one can draw a straight line; 2) a segment of a straight line can be extended indefinitely; 3) from a given point as centre one can describe a circle of given radius; 4) all right angles are equal to one another (this guarantees that the extension of a straight line is unique); and 5) if two straight lines lying in the same plane intersect a third, and if the sum of the interior angles on one side of the latter is less than the sum of two right angles, then the first two lines, if extended indefinitely, will intersect on that side. All the postulates (except the fourth, which is replaced by the condition that through two points passes a unique straight line) have been included as axioms in modern courses on the foundations of geometry. The fate of the fifth postulate is especially interesting. Attempts to prove it were already being made in antiquity. Such attempts continued until N.I. Lobachevskii constructed the first system of non-Euclidean geometry, in which this postulate is false (see Lobachevskii geometry). After the postulates, Euclid presents the axioms — propositions about the properties of the relations of equality and inequality between quantities: 1) things equal to the same thing are equal to one another; 2) if equals are added to equals, the results are also equal; 3) if equals are taken from equals, the remainders are also equal; 4) things that are superposable on one another are equal; and 5) the whole is greater than any of its parts (in some editions of the Elements there are four additional axioms).

Euclid's Elements consists of thirteen books (sections or parts). Book I treats the fundamental properties of triangles, rectangles and parallelograms, and compares their areas. The book ends with Pythagoras' theorem. Book II presents what might be called geometrical algebra, i.e. it constructs the geometrical tools for the solution of problems that reduce to quadratic equations. This is done with quantities represented by segments and products of two quantities — areas. The Elements contain no algebraic notation. Book III considers the properties of the circle, its tangents and chords (these problems were studied by Hippocrates of Chios in the second half of the 5th century B.C.), and Book IV studies regular polygons. In Book V, Euclid presents the general theory of proportion created by Eudoxus of Cnidus in the 4th century B.C.; his presentation, however, differs from the older one in its logical completeness and is basically equivalent to theory of Dedekind cuts, which is one of the rigorous approaches to the definition of real numbers. The general theory of proportion provides the basis for the theory of similarity (Book VI) and the method of exhaustion (Book XII), which also go back to Eudoxus. Books VII–IX present the elements of number theory, based on the algorithm for finding the greatest common divisor. In these books one finds the theory of divisibility, including the unique factorization of integers into prime factors and the fact that there are infinitely many primes (cf. Euclidean prime number theorem), and the construction of a theory of quotients of integers that is essentially equivalent to the theory of rational numbers. With the latter as basis, a classification is given in Book X of quadratic and biquadratic irrationalities and some rules for handling them are substantiated. The results of Book X are utilized in Book XIII to determine the edges of the five regular solids. A considerable part of Books X and XIII (and probably also Book VII) were written by Theaetetus (beginning of the 4th century B.C.). Book XI contains the elements of stereometry. Book XII uses the method of exhaustion (cf. Exhaustion, method of) to determine the ratio of the areas of two discs, and the ratio of the volumes of two pyramids and prisms, cones and cylinders. These theorems were first proved by Eudoxus. Finally in Book XIII he determines the ratio of the volumes of two spheres, constructs the five regular solids and proves that these are the only regular solids (cf. Regular polyhedra). Books XIV and XV were not written by Euclid but by later Greek mathematicians, although even today they are frequently printed together with the main text of the Elements. Their contents do not present much of scientific interest.

Euclid's Elements was widely known even in Antiquity. Archimedes, Apollonius of Perga and other scholars relied on the work for their own work in mathematics and mechanics. At some time around the end of the 8th century and the beginning of the ninth, Arabic translations of it appeared. The first translation into Latin was done (from the Arab) in the first quarter of the 12th century. The ancient editions of the work exhibit significant textual variations; they do not reproduce the original text exactly. The first printed edition of Euclid's Elements — in Latin translation — was published in 1482, with drawings in the margins. The definitive edition is that of J.L. Heiberg (5 vols., 1883–1888), in which both the Greek text and the Latin translation are reproduced. The following translations are available in Russian: I. Astarov — Euclid's Elements, abridged by Prof. A. Farkhvarson (8 books, 1739, transl. from the Latin); N. Kurganov — Euclid's Elements of Geometry (8 books, 1769, transl. from the French); P. Suvorov and V. Nikitin — Euclid's elements (8 books, 1-6, 11, 12; 1784, transl. from the Greek); F. Petrushevskii — Eight Books of Euclid's Elements, namely: the first six, the eleventh and the twelfth, containing the foundations of Geometry (1819, transl. from the Greek); F. Petrushevskii — Three Books of Euclid's Elements, namely: the seventh, eighth and ninth, containing the general theory of numbers of the Ancient geometers (1835, transl. from the Greek); M.E. Vashchenko-Zakharchenko — Euclid's Elements (1880); D.D. Mordukhai-Boltovskii — Euclid's Elements (3 vols., 1948-1950, transl. from the Greek).

#### Comments

See also Foundations of geometry.

By now people realize that the Elements are not really a deductive system, and indeed were not intended to be one.

Pythagoras' theorem is of course well-known: The square of the length of the hypothenuse of a plane triangle equals the sum of the squares of the lengths of the side of this triangle (cf. also Pythagorean numbers).

The algorithm for finding the greatest common divisor of two natural numbers is also called the Euclidean algorithm. For the history of Euclid's fifth postulate see also [a5].

#### References

[a1] | Sir Th.L. Heath, "The elements of Euclid" , Dent (1933) |

[a2] | Th.L. Heath, "The thirteen books of Euclid's elements" , Cambridge Univ. Press (1926) (Dover, reprint, 1956) |

[a3] | M. Greenberg, "Euclidean and non-Euclidean geometry" , Freeman (1974) |

[a4] | G. Choquet, "Geometry in a modern setting" , Kershaw (1969) |

[a5] | R. Bonola, "Non-Euclidean geometry" , Dover, reprint (1955) (Translated from Italian) |

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