XI Международная студенческая научная конференция Студенческий научный форум - 2019

Euclid is an ancient Greek mathematician (365-300 BC).

He was born in Athens (according to other sources, in Tire). The life of a scientist probably knows only that he was a pupil of Plato, and the flowering of his activity occurred at the time of the reign of Egypt in Ptolemy I Soter (IV century BC). Euclid is the author of the first theoretical treatise on mathematics that has come down to us. Ancient Greek thinker Euclid became the first mathematician of the Alexandrian school and the author of one of the most ancient theoretical mathematical treatises. The biography of this scientist is known much less than about his work. Thus, in the well-known work of "The Beginning" Euclid outlined the stereometry, planimetry, aspects of number theory, created the basis for the subsequent development of mathematics. Biography Euclid supposedly began in 325 BC (this is an approximate date, the exact year of birth is unknown) in Alexandria. Some researchers suggest that the future mathematician was born in Tire, and spent most of his adult life in Damascus. Probably, Euclid came from a rich family, since he studied at the Athenian school of Plato (at that time such education was available only to wealthy citizens). Euclid understood the philosophical concept of Plato and shared her basic theses. The above information about the personality and life path of Euclid is drawn by researchers from Proclus's comments, which he wrote to the first book of the "Beginning." Euclid is the first mathematician of the Alexandrian school. His main work is "The Beginnings". He sets out the fundamentals of planimetry, stereometry, number theory, algebra, describes methods for determining areas and volumes; besides, it contains a presentation of planimetry, stereometry and a number of questions in number theory; in it he summed up the preceding development of ancient Greek mathematics and created the foundation for the further development of mathematics. Of his other works in mathematics, it is necessary to note the "On the division of figures", preserved in the Arabic translation, 4 books "Conic sections", the material of which entered the work of the same name Apollonius of Perga, as well as "Porisms", the idea of ​​which can be obtained from " Mathematical Meeting »Papp of Alexandria. Euclid is the author of works on astronomy, optics, music, etc. Euclid is the author of a number of works on astronomy, optics, music, etc. Arabic authors ascribe to Euclid and various treatises on mechanics, including essays on weights and determination of specific gravity. The brilliant thinker formulated his knowledge in planimetry and stereometry in the form of axioms and postulates. The system of axioms concerned four concepts: a point, a straight line, a plane, a movement, and also the relationship of these concepts to each other. To construct concrete figures on a plane or in space, he developed a system of postulates that prescribe specific actions. A similar system of axioms and postulates in modern times was called "Euclidean geometry."

Information about the time and place of his birth has not reached us, but it is known that Euclid lived in Alexandria and the flowering of his work falls on the time of the reign of Egypt in Ptolemy I Soter. It is also known that Euclid was younger than the disciples of Plato (427-347 BC), but older than Archimedes (about 287-212 BC), since, on the one hand, he was a Platonist and knew the philosophy of Plato ( that is why he finished the "Beginning" with an exposition of so-called Platonic solids, that is, five regular polyhedra), and on the other hand his name is mentioned in the first of Archimedes' two letters to Dositheus "On the ball and cylinder". The name of Euclid is associated with the formation of Alexandrian mathematics (geometric algebra) as a science.

Proclus in the comments to the first book of "Beginnings" cites a well-known anecdote about the question that Ptolemy Euclid asked: "Is there a shorter path in geometry than (the one that is set forth) in the" Beginnings "? To which Euclid allegedly replied that "in geometry there is no royal road" (a similar anecdote is also told about Alexander and the disciple of Eudoxus Menehme, so that he belongs, apparently, to the number of "stray subjects").

Scientific achievements

The bulk of the scientist's work was written in mathematics:

"The Beginning";

"On the division of figures";

"Conical sections";

"Porisms" - about the curved lines and conditions that determine them;

"Pseudoria" - a treatise on errors that arise when geometric evidence.

"Data" - an introduction to geometric analysis.

There are known works of a scientist in related disciplines - music, astronomy, optics:

"Phenomena" - on the practical application of geometry to the study of astronomy;

"Optics" - about the light and the laws of its distribution;

"Catoptrika" - about mirrors and light refraction;

"Division of the Canon" - an elementary theory of music.

"The Beginning" of Euclid

The main work of Euclid - "Beginnings" (or "Elements", in the original "Stoichea"). The "beginnings" of Euclid consist of 13 books. Later two more books were added to them. The book consists of fifteen volumes:

In the book I the author tells about the properties of parallelograms and triangles, concluding the presentation using the Pythagorean theorem when calculating the parameters of rectangular triangles.

Book number II describes the principles and patterns of geometric algebra and goes back to the luggage of knowledge accumulated by the Pythagoreans.

In books III and IV, Euclid considers the geometry of circles described and inscribed in polygons. During the creation of these volumes, the author could turn to the use of the works of Hippocrates of Chios.

In the fifth book, the Greek mathematician examined the general theory of proportions developed by Eudoxus of Cnidus.

In the materials of the VI book, the author applies the general theory of Eudoxus' proportions to the theory of similar figures.

Books numbered VII-IX describe the theory of numbers. When writing these volumes, the mathematician again turned to the materials created and assembled by the Pythagoreans - representatives of the teaching, in which the number plays a central role. In these works the author speaks about geometric progressions and proportions, proves the infinity of the set of prime numbers, studies even perfect numbers, introduces the concept of GCD (the greatest common divisor). The algorithm for finding such a divisor is now called the Euclidean algorithm. There is a suggestion that the Eighth book was written not by Euclid himself, but by Archite of Tarentum.

Volume number X is the most complex and voluminous work in the "Elements", which contains the classification of irrationalities. The authorship of this book is also not known for certain: it could be written both by Euclid himself and by the Teathet Athenian.

On pages XI of the book the mathematician tells about the fundamentals of stereometry.

Book XII contains proofs of theorems on the volumes of cones and pyramids, the ratio of the areas of circles. To construct these proofs, the exhaustion method is used. Most researchers agree that this book was not written by Euclid. The likely author is Eudoxus of Cnidus.

Materials in Book XIII contain information on the construction of five regular polyhedra ("Platonic solids"). Some part of the constructions given in the volume could be developed by the Teethet Athenian.

Books XIV and XV, according to a generally recognized opinion, also belong to other authors. So, the penultimate volume of the "Beginnings" was written by Gipsikl (also living in Alexandria, but later Euclid), and the last one was Isidore of Miletus (who built the temple of St. Sophia in Constantinople in the beginning of the sixth century BC).

Before the appearance of the "Elements" of Euclid, works of the same name, the essence of which consisted in the sequential presentation of the key facts of theoretical arithmetic and geometry, were compiled by Leont, Hippocrates of Chios, and Feb.. All of them have practically disappeared from use after the appearance of Euclid's work.

For two thousand years, fifteen volumes of "Elements" acted as a basic textbook on geometry. This work is translated in Arabiclanguage, then - into English. The "beginnings" were reprinted hundreds of times, and the basic mathematical calculations mentioned in them remain relevant to this day. A considerable part of the materials that the author has included in his work are not his own discoveries, but the theories known earlier. The essence of Euclid's work consisted in the processing of material, its systematization and the reduction of disparate data together. Some books Euclid began with a list of definitions, in the first book there is also a list of axioms and postulates.

In the writings of Euclid there is given a systematic exposition of the so-called. Euclidean geometry, the axiom system of which relies on the following basic concepts: point, line, plane, motion and the following relations: "the point lies on the line in the plane," "the point lies between the other two." In the present exposition, the system of axioms of Euclidean geometry is divided into the following five groups.

I. Axioms of combination.

Through every two points one can draw a straight line and moreover only one.

At least two points lie on each line. There are at least three points not lying on one line.

Through every three points not lying on one line, one can draw a plane and, moreover, only one.

There are at least three points on each plane and there are at least four points not lying in the same plane.

If two points of a given line lie on a given plane, then the line itself lies on this plane.

If two planes have a common point, then they have one more common point (and therefore a common straight line).

II. Axioms of order.

If the point B lies between A and C, then all three lie on the same line.

For every point A, B there is a point C such that B lies between A and C.

Of the three straight points, only one lies between the two others.

If a straight line intersects one side of a triangle, it intersects another side of it or passes through a vertex (the segment AB is defined as the set of points lying between A and B, respectively, the sides of the triangle are defined).

III. Axioms of motion.

The motion associates the points of a point, straight lines, planes of the plane, preserving the belonging of points to straight lines and planes.

Two successive movements give again motion, and for every motion there is the opposite.

If points A, A 'and half-planes a, a' are given, bounded by the extended half-lines a, a 'that start from the points A, A', then there is a movement, and the only one that transfers A, a, a to A ' , a ', a' (the half-line and half-plane are easily determined on the basis of the concepts of combination and order).

IV. Axioms of continuity.

Axiom of Archimedes: every segment can be covered by any segment, postponing it on the first sufficient number of times (the delay of the segment is carried out by movement).

Cantor's axiom: if a sequence of segments nested in one another is given, then they all have at least one common point.

V. The Euclidean parallelism axiom. Through the point A outside the straight line a in the plane passing through A and a, one can draw only one straight line that does not intersect a.

The appearance of Euclidean geometry is closely connected with visual representations about the world around us (straight lines - strained threads, rays of light, etc.). The long process of deepening our ideas led to a more abstract understanding of geometry. The discovery by NI Lobachevskii of a geometry other than Euclidean geometry has shown that our ideas about space are not a priori. In other words, Euclidean geometry can not claim to be the only geometry describing the properties of the surrounding space. The development of natural science (mainly physics and astronomy) has shown that Euclidean geometry describes the structure of the surrounding space only with a certain degree of accuracy and is not suitable for describing the properties of space associated with the displacements of bodies with velocities close to light. Thus, Euclidean geometry can be considered as the first approximation for describing the structure of a real physical space.

Postulates of Euclid

From Euclid's postulates it is clear that Euclid represented space as empty, boundless, isotropic and three-dimensional. The infinity and infinity of space is assumed by Euclid's postulates, such as the thesis that from any point to every point one can draw a straight line, that a bounded line can be continuously continued along a line such that a circle can be described from every center and every solution of the compass.

Especially famous is the fifth postulate of Euclid, which literally sounds like this (above we gave a retelling): "If a straight line that falls on two straight lines forms internal corners and on one side less than two straight angles, then these two straight lines continue unboundedly from the side where angles less than two straight lines. " Later, Proclus expressed this postulate as follows: "If a straight line crosses one of the two parallel lines, it will also cross the second parallel line." More familiar to us is the formula: "Through this point, you can draw only one parallel to a given line" - belongs to John Pleifer.

More than once attempts were made to prove the fifth postulate of Euclid (Ptolemy, Nasir al-Din, Lambert, Legendre). Finally, Karl Gauss expressed in 1816 a hypothesis that this postulate can be replaced by another. This conjecture was realized in parallel studies independently of each other by NI Lobachevsky (1792-1856) and Janos Bolja (1802-1866). However, both these researchers (both Russian and Hungarian) did not receive the recognition of other mathematicians, especially those who stood on the positions of Kantian a prioriism in the understanding of a space that allowed only one space - Euclidean. Only Bernhard Riemann (1826-1866) with his theory of varieties (1854) proved the possibility of the existence of many types of non-Euclidean geometry. B. Riemann himself replaced the fifth postulate of Euclid on the postulate, according to which there are no parallel lines at all, and the inner angles of the triangle are greater than two straight lines. Felix Klein (1849-1925) showed the ratio of non-Euclidean and Euclidean geometries. Euclidean geometry refers to surfaces with zero curvature, Lobachevsky's geometry to surfaces with positive curvature, and the Riemann geometry to a surface with negative curvature. Another significant work of Euclid - "Data" is an introduction to geometric analysis. The scientist also owns the "Phenomena" (devoted to elementary spherical astronomy), "Optics" (contains the doctrine of perspective) and "Catoptrika" (sets forth the theory of reflections in mirrors), a small treatise of the Section of the Canon (includes ten problems on musical intervals), a collection tasks on the division of the areas of the figures "On the divisions" (reached us in the Arabic translation).

In ancient times, philosophy was closely intertwined with many other branches of scientific knowledge. So, geometry, astronomy, arithmetic and music were considered mathematical sciences, understanding of which is necessary for a qualitative study of philosophy. Euclid developed Plato's doctrine of the four elements, which are associated with four regular polyhedra:

the elements of fire represent the tetrahedron;

the air element corresponds to an octahedron;

the elements of the earth are associated with the cube;

the water element contacts the icosahedron.

In this context, the "Elements" can be considered as a kind of teaching about the construction of "Platonic solids", that is, five regular polyhedra. The teaching contains all the necessary prerequisites, proofs and bundles. The proof of the possibility of constructing such bodies is completed by affirming the fact that no other regular bodies, with the exception of the given five, exist.

Practically every theorem of Euclid in the "Beginnings" corresponds also to the indicators of the doctrine of Aristotle's proof. So, the author consistently deduces the consequences from the causes, forming a chain of logical evidence. Moreover, he proves even statements of a general nature, which also corresponds to the teachings of Aristotle.

Personal life

We have received only some information about the work of Euclid in science, about his personal life, almost nothing is known. There is a legend that King Ptolemy, who decided to study geometry, was annoyed by its complexity. Then he turned to Euclid and asked him to point to an easier way to knowledge, to which the thinker replied: "There is no royal road to geometry." The expression subsequently became winged.

Euclid founded a mathematical school at the Alexandrinsky library. There is evidence that under the Alexandria library this ancient Greek scientist founded a private mathematical school. It was studied by the same enthusiasts of science as Euclid himself. Even at the end of his life, Euclid helped his students write works, create their own theories and develop relevant proofs. There is no precise data on the appearance of the scientist. His portraits and sculptures are the fruit of the imagination of their creators, a fictitious image passed down from generation to generation.

Presumably, Euclid died in the 260th years BC in Alexandria. The exact causes of death are not known. The scientist's legacy survived him for two thousand years and inspired many great people centuries after his death. There is an opinion that politician Abraham Lincoln liked to quote Euclid's statements in his speeches and had several volumes of "Elements" with him. The scientists of later years based the works on the works of Euclid. Thus, the Russian mathematician Nikolai Lobachevsky used the materials of an ancient Greek thinker to develop hyperbolic geometry, or Lobachevsky's geometry. The mathematics format that Euclid created was now known as "Euclidean geometry." The scientist also created a device for determining the pitch of the string and studied the interval relationships, contributing to the creation of keyboard musical instruments.

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