Euclid or Euclid(ancient Greek Εὐκλείδης , from “good fame”, flourishing time - about 300 BC. BC) - ancient Greek mathematician, author of the first theoretical treatise on mathematics that has come down to us. Biographical information about Euclid is extremely scarce. The only thing that can be considered reliable is that his scientific activity took place in Alexandria in the 3rd century. BC e.

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  The most reliable information about the life of Euclid is considered to be the little that is given in the Commentaries of Proclus to the first book Started Euclid. Noting that “those who wrote on the history of mathematics” did not bring the development of this science to the time of Euclid, Proclus points out that Euclid was older than Plato’s circle, but younger than Archimedes and Eratosthenes and “lived in the time of Ptolemy I Soter,” “because Archimedes, who lived under Ptolemy the First, mentions Euclid and, in particular, says that Ptolemy asked him if there was a shorter way to study geometry than Beginnings; and he replied that there is no royal path to geometry.”

  Additional touches to Euclid's portrait can be gleaned from Pappus and Stobaeus. Pappus reports that Euclid was gentle and kind to everyone who could contribute even in the slightest degree to the development of mathematical sciences, and Stobaeus relates another anecdote about Euclid. Having begun to study geometry and having analyzed the first theorem, one young man asked Euclid: “What benefit will I get from this science?” Euclid called the slave and said: “Give him three obols, since he wants to make a profit from his studies.” The historicity of the story is questionable, since a similar one is told about Plato.

  Some modern authors interpret Proclus's statement - Euclid lived in the time of Ptolemy I Soter - in the sense that Euclid lived at the court of Ptolemy and was the founder of the Alexandrian Museion. It should be noted, however, that this idea was established in Europe in the 17th century, while medieval authors identified Euclid with the student of Socrates, the philosopher Euclid of Megara.

  Arab authors believed that Euclid lived in Damascus and published there " Beginnings» Apollonia. An anonymous 12th-century Arabic manuscript reports:

  Euclid, son of Naucrates, known as "Geometra", a scientist of old times, Greek by origin, Syrian by residence, originally from Tyre...

  In general, the amount of data about Euclid is so scarce that there is a version (though not widespread) that we are talking about the collective pseudonym of a group of Alexandrian scientists.

  « Beginnings» Euclid

  Euclid's main work is called Beginnings. Books with the same name, in which all the basic facts of geometry and theoretical arithmetic were consistently presented, were previously compiled by Hippocrates of Chios, Leontes and Feudius. However Beginnings Euclid pushed all these works out of use and remained the basic textbook of geometry for more than two millennia. When creating his textbook, Euclid included in it much of what was created by his predecessors, processing this material and bringing it together.

  Beginnings consist of thirteen books. The first and some other books are preceded by a list of definitions. The first book is also preceded by a list of postulates and axioms. As a rule, postulates define basic constructions (for example, “it is required that a straight line can be drawn through any two points”), and axioms - general rules of inference when operating with quantities (for example, “if two quantities are equal to a third, they are equal between yourself").

  In Book I the properties of triangles and parallelograms are studied; This book is crowned with the famous Pythagorean theorem for right triangles. Book II, going back to the Pythagoreans, is devoted to the so-called “geometric algebra”. Books III and IV describe the geometry of circles, as well as inscribed and circumscribed polygons; when working on these books, Euclid could have used the writings of Hippocrates of Chios. In Book V, the general theory of proportions, built by Eudoxus of Cnidus, is introduced, and in Book VI it is applied to the theory of similar figures. Books VII-IX are devoted to number theory and go back to the Pythagoreans; the author of Book VIII may have been Archytas of Tarentum. These books discuss theorems on proportions and geometric progressions, introduce a method for finding the greatest common divisor of two numbers (now known as the Euclid algorithm), construct even perfect numbers, and prove the infinity of the set of prime numbers. In the X book, which is the most voluminous and complex part Started, a classification of irrationalities is constructed; it is possible that its author is Theaetetus of Athens. Book XI contains the basics of stereometry. In the XII book, using the method of exhaustion, theorems on the ratios of the areas of circles, as well as the volumes of pyramids and cones are proved; The author of this book is generally acknowledged to be Eudoxus of Cnidus. Finally, Book XIII is devoted to the construction of five regular polyhedra; it is believed that some of the constructions were developed by Theaetetus of Athens.

  In the manuscripts that have reached us, two more books were added to these thirteen books. Book XIV belongs to the Alexandrian Hypsicles (c. 200 BC), and Book XV was created during the life of Isidore of Miletus, builder of the temple of St. Sophia in Constantinople (beginning of the 6th century AD).

  Beginnings provide a general basis for subsequent geometric treatises by Archimedes, Apollonius and other ancient authors; the propositions proven in them are considered generally known. Comments to Let's start in antiquity were Heron, Porphyry, Pappus, Proclus, Simplicius. A commentary by Proclus on Book I has been preserved, as well as a commentary by Pappus on Book X (in Arabic translation). From ancient authors, the commentary tradition passes to the Arabs, and then to Medieval Europe.

  In the creation and development of modern science Beginnings also played an important ideological role. They remained a model of a mathematical treatise, strictly and systematically presenting the main provisions of a particular mathematical science.

  Other works of Euclid

  Of the other works of Euclid, the following have survived:

  • Data (δεδομένα ) - about what is necessary to define a figure;
  • About division (περὶ διαιρέσεων ) - partially preserved and only in Arabic translation; gives the division of geometric figures into parts that are equal or consist of each other in a given ratio;
  • Phenomena (φαινόμενα ) - applications of spherical geometry to astronomy;
  • Optics (ὀπτικά ) - about the rectilinear propagation of light.

  From brief descriptions we know:

  • Porisms (πορίσματα ) - about the conditions that determine the curves;
  • Conic sections (κωνικά );
  • Superficial places (τόποι πρὸς ἐπιφανείᾳ ) - about the properties of conic sections;
  • Pseudaria (ψευδαρία ) - about errors in geometric proofs;

  Euclid is also credited with:

  Euclid and ancient philosophy

  Already from the time of the Pythagoreans and Plato, arithmetic, music, geometry and astronomy (the so-called “mathematical” sciences; later called quadrivius by Boethius) were considered as a model of systematic thinking and a preliminary stage for the study of philosophy. It is no coincidence that a legend arose according to which the inscription “Let no one who does not know geometry enter here” was placed above the entrance to Plato’s Academy.

  Geometric drawings, in which by drawing auxiliary lines the implicit truth becomes obvious, serve as an illustration for the doctrine of recollection developed by Plato in Menone and other dialogues. Propositions of geometry are called theorems because to comprehend their truth it is necessary to perceive the drawing not with simple sensory vision, but with the “eyes of the mind.” Every drawing for a theorem represents an idea: we see this figure in front of us, and we reason and draw conclusions for all figures of the same type at once.

  Some “Platonism” of Euclid is also associated with the fact that in Timaeus Plato considers the doctrine of the four elements, which correspond to four regular polyhedra (tetrahedron - fire, octahedron - air, icosahedron - water, cube - earth), while the fifth polyhedron, the dodecahedron, “got to the lot of the figure of the universe.” Due to this Beginnings can be considered as a teaching on the construction of five regular polyhedra - the so-called “Platonic solids”, developed with all the necessary premises and connections, ending with a proof of the fact that there are no other regular solids besides these five.

  For the Aristotelian doctrine of evidence, developed in Second analytics, Beginnings also provide rich material. Geometry in Beginnings is constructed as an inferential knowledge system in which all propositions are sequentially deduced one after another along a chain based on a small set of initial statements accepted without proof. According to Aristotle, such initial statements must exist, since the chain of inference must begin somewhere in order not to be endless. Further, Euclid tries to prove statements of a general nature, which also corresponds to Aristotle’s favorite example: “if it is inherent in every isosceles triangle to have angles that add up to two right angles, then this is inherent in it not because it is isosceles, but because it is a triangle” (An. Post.85b12).

  Pseudo-Euclid

  Euclid is credited with two important treatises on ancient music theory: the Harmonic Introduction (Harmonics) and the Division of the Canon (

  Hello friends! The article “Euclid: short biography, discoveries, facts, video” is about the life of the ancient Greek mathematician and philosopher. "Euclid" - translated from ancient Greek means "good fame."

  Biography of Euclid

  According to some archival documents, he was born around 325 AD. e. The life of the thinker coincides in time with the reign of Ptolemy the First.

  The scientific activity of the great mathematician developed in Alexandria. He received his education from the followers of Plato, and from them he inherited a system of philosophical views. This allowed Euclid to open a mathematical school in Alexandria, where he became the first teacher.

  The scientist’s main work is “Principia” - the first treatise on theoretical mathematics in history. The treatise covered and systematized all the knowledge accumulated in Ancient Greece on planimetry, stereometry, and number theory.

  Euclid's algorithm, the currently used method for finding the greatest common divisor of two numbers, was formulated already in the Principia. The treatise laid the foundation not only for his subsequent scientific works, but also for the development of mathematics as a whole.

  What is "Euclidean geometry"?

  The brilliant thinker formulated his knowledge of planimetry and stereometry in the form of axioms and postulates. The system of axioms concerned four concepts: point, line, plane, motion, as well as the relationship of these concepts with each other.

  To construct specific figures on a plane or in space, he developed a system of postulates that prescribe specific actions. In modern times, such a system of axioms and postulates is called “Euclidean geometry.”

  Achievements of Euclid

  The bulk of the scientist’s works were written in mathematics:

  • "Beginnings";
  • “On the division of figures”;
  • "Conic sections";
  • “Porisms” are about curved lines and the conditions that determine them;
  • "Pseudarius" is a treatise on errors that arise in geometric proofs.

  The scientist’s works in related disciplines are known: music, astronomy, optics:

  • “Phenomena” - about the practical application of geometry to the study of astronomy;
  • “Optics” - about light and the laws of its propagation;
  • “Catoptrics” - and the refraction of light;
  • "Division of the canon" - elementary music theory.

  Arab scientists consider this mathematician to be the author of some works on mechanics and the determination of the specific gravity of bodies.

  This video contains additional and interesting information for the article “Euclid: short biography, discoveries, facts, video”

  Kupchinsky youth readings “Science. Creation. Search".
  Section "Mathematics"

  "Euclid and his contribution to science"

  The work was completed by a student of grade 6 "B"
  Surovegin Nikolay
  Head: Vasilyeva
  Daria Gennadievna

  St. Petersburg 2008

  I. Introduction………………………………….…3

  II. Mathematics in Ancient Greece……………..4

  III. Biography of Euclid……………………….….5

  IV. Euclidean algorithm……………………………8

  V. Axiomatics....…………………………….11

  VI. Euclidean geometry and V postulate………..12

  VII. Started………………………………………………………19

  VIII. Problems from Euclid's principles…………………...22

  IX. Solving problems……………………………..23

  X. Links to information sources......24

  XI. Conclusion…………………………………..25

  I. Introduction

  In this essay I will try to tell you everything I know about the great ancient Greek mathematician Euclid. The idea to write about him came to me after I learned about the Euclid algorithm. This scientist did a lot for algebra and geometry, and we use his discoveries constantly. The abstract also contains practical problems from the beginnings, the books of Euclid.

  Chapter II.
  Mathematics in Ancient Greece

  Mental development, and with it the development of science, has never progressed evenly throughout humanity. While some peoples stood at the head of the mental movement of mankind, others turned out to be barely out of their primitive state. When the latter, along with the improvement of their living conditions, appeared, under the influence of internal or external impulses, to acquire knowledge, then they had to first of all catch up with the advanced tribes. If at the same time the advanced tribes, having reached the highest level of development available to them by their abilities or by the living conditions created for them by history, degenerated and fell, stagnation or even a visible temporary decline occurred in the mental development of all mankind: the acquisition of new knowledge ceased and mental work humanity was reduced solely to the aforementioned assimilation by backward tribes of knowledge already acquired by humanity. Only after achieving this assimilation did the lagging tribes gain the opportunity to further pursue the acquisition of new knowledge and through this, in turn, become the head of the mental movement of mankind. Thus, in the history of the mental activity of every people that has ever taken a place among the leading figures of humanity and then completed its entire life cycle, the researcher must distinguish three periods: the period of assimilation of knowledge already acquired by humanity; a period of independent activity in the area of ​​acquiring new knowledge common to all humanity and, finally, a period of decline and mental degeneration. Turning from this general consideration of the course of mental development of mankind to that of its individual areas that appears to be the development of mathematics, we find that in the current state of historical and mathematical knowledge, we have access to the study of a completely completed cycle of activity of an individual people in the field of development of mathematics in only one nations, on the ancient Greeks.

  Chapter III Biography of Euclid

  EUCLID (Euclidc.356-300 VS)

  BIOGRAPHY

  Euclid is an ancient Greek mathematician, the author of the first theoretical treatises on mathematics that have reached us. Biographical information about the life and work of Euclid is extremely limited. It is known that he was from Athens and was a student of Plato. His scientific activity took place in Alexandria, where he created a mathematical school.

  ACHIEVEMENTS IN MATHEMATICS

  Euclid's main works "Elements" (Latinized title - "Elements") contain a presentation of planimetry, stereometry and a number of issues in number theory, algebra, the general theory of relations and the method of determining areas and volumes, including elements of limits (Method of exhaustion). In the Elements, Euclid summarized all the previous achievements of Greek mathematics and created the foundation for its further development. The historical significance of Euclid's Elements lies in the fact that they were the first to attempt the logical construction of geometry based on axiomatics. The main disadvantage of Euclid's axiomatics should be considered its incompleteness; there are no axioms of continuity, movement and order, so Euclid often had to appeal to intuition and trust the eye. Books XIV and XV are later additions, but whether the first thirteen books are the work of one man or of a school led by Euclid is not known. Since 1482 Euclid's Elements went through more than 500 editions. in all languages ​​of the world.

  "Beginnings"

  The first four books of the Elements are devoted to plane geometry, and they study the basic properties of rectilinear figures and circles.

  Book I is preceded by definitions of concepts used later. They are intuitive in nature, since they are defined in terms of physical reality: “A point is something that has no parts.” “A line is length without width.” "A straight line is one that is equally located in relation to the points on it." “The surface is that which has only length and width,” etc.

  These definitions are followed by five postulates: “Suppose:
  1) that a straight line can be drawn from any point to any point;
  2) and that a bounded line can be continuously extended along a straight line;
  3) and that a circle can be described from any center and by any solution;
  4) and that all right angles are equal to each other;
  5) and if a straight line falling on two straight lines forms interior angles on one side that are less than two right angles, then extended indefinitely these two straight lines will meet on the side where the angles are less than two right angles."

  The first three postulates ensure the existence of a straight line and a circle. The fifth, the so-called parallel postulate, is the most famous. It has always intrigued mathematicians, who tried to derive it from the four previous ones or discard it altogether, until in the 19th century. It was discovered that other, non-Euclidean geometries can be constructed and that the fifth postulate has a right to exist. Then Euclid formulated axioms that, in contrast to postulates that are valid only for geometry, are generally applicable to all sciences. Further, Euclid proves in Book I the elementary properties of triangles, among which are the conditions of equality. Then some geometric constructions are described, such as the construction of the bisector of an angle, the midpoint of a segment and the perpendicular to a line. Book I also includes the theory of parallels and the calculation of the areas of some plane figures (triangles, parallelograms and squares). Book II lays the foundations of the so-called geometric algebra, which dates back to the school of Pythagoras. All quantities in it are represented geometrically, and operations on numbers are performed geometrically. Numbers are replaced by line segments. Book III is entirely devoted to the geometry of the circle, and Book IV studies regular polygons inscribed in a circle, as well as circumscribed around it.

  The theory of proportions, developed in Book V, applied equally well to commensurate quantities and incommensurable quantities. Euclid included in the concept of “magnitude” lengths, areas, volumes, weights, angles, time intervals, etc. Refusing to use geometric evidence, but also avoiding resorting to arithmetic, he did not assign numerical values ​​to quantities. The first definitions of Book V of Euclid's Elements: 1. A part is a magnitude (of) a magnitude that is smaller (of) a larger one if it measures the larger one. 2. A multiple is the greater (from) the lesser, if it is measured by the lesser. 3. A ratio is a certain dependence of two homogeneous quantities in quantity. 4. Quantities are said to have a relationship with each other if they, taken as multiples, can exceed each other. 5. They say that quantities are in the same ratio: the first to the second and the third to the fourth, if equal multiples of the first and third are at the same time greater, or simultaneously equal, or at the same time less than equal multiples of the second and fourth each, for any multiplicity, if take them in the appropriate order. 6. Let quantities having the same ratio be called proportional. From the eighteen definitions placed at the beginning of the entire book, and the general concepts formulated in Book I, with admirable grace and almost without logical flaws, Euclid deduced (without resorting to postulates, the content of which was geometric) twenty theorems in which the properties of quantities and their relationships.

  In Book VI, the theory of proportions of Book V is applied to rectilinear figures, to geometry on the plane and, in particular, to similar figures, and “similar rectilinear figures are those which have angles equal in order, and sides at equal angles proportional.” Books VII, VIII and IX constitute a treatise on the theory of numbers; the theory of proportions is applied to numbers in them. Book VII defines the equality of ratios of integers, or, from a modern point of view, builds the theory of rational numbers. Of the many properties of numbers studied by Euclid (parity, divisibility, etc.), we cite, for example, proposition 20 of Book IX, which establishes the existence of an infinite set of “firsts,” i.e., prime numbers: “There are more prime numbers than any proposed number first numbers." His proof by contradiction can still be found in algebra textbooks.

  Book X is difficult to read; it contains a classification of quadratic irrational quantities, which are represented there by geometric lines and rectangles. Here is how Proposition 1 is formulated in Book X of Euclid’s Elements: “If two unequal quantities are given and from the larger a part greater than half is subtracted, and from the remainder again a part greater than half, and this is repeated constantly, then someday a quantity remains which is less than the smaller of the given values." In modern language: If a and b are positive real numbers and a > b, then there always exists a natural number m such that mb > a. Euclid proved the validity of geometric transformations.

  Book XI is devoted to stereometry. In Book XII, which also probably dates back to Eudoxus, the areas of curvilinear figures are compared with the areas of polygons using the Method of Exhaustion. The subject of Book XIII is the construction of regular polyhedra. The construction of the Platonic solids, which apparently complete the Elements, gave reason to classify Euclid as a follower of Plato's philosophy.

  AREAS OF INTEREST

  In addition to the “Elements,” the following works of Euclid have reached us: a book under the Latin title “Data” (with a description of the conditions under which any mathematical image can be considered “data”); a book on optics (containing the doctrine of perspective), on catoptrics (outlining the theory of distortions in mirrors), a book “Division of Figures”. Euclid's pedagogical work "On False Conclusions" (in mathematics) has not survived. Euclid also wrote works on astronomy (“Phenomena”) and music.

  MERIT OF EUCLID

  EUCLID'S THEOREM about prime numbers: the set of prime numbers is infinite (Euclides' Elements, Book IX, Theorem 20). More accurate quantitative information about the set of prime numbers in the natural series is contained in Chebyshev’s theorem on prime numbers and the asymptotic formula. law of distribution of prime numbers.

  EUCLIDAN GEOMETRY - the geometry of space described by a system of axioms, the first systematic (but not sufficiently rigorous) presentation was given in Euclid's Elements. Usually the space of an electronic geometric system is described as a set of objects of three kinds, called “points,” “straight lines,” and “planes”; relations between them: belonging, order (“to lie between”), congruence (or the concept of movement); continuity. A special place in the axiomatics of E. is occupied by the axiom of parallels (fifth postulate). The first sufficiently strict axiomatics of J. g. was proposed by D. Hilbert (D. Hilbert, see Hilbert's system of axioms). There are modifications of the Hilbert axiom system and other variants of the axiomatics of E.G. For example, in the vector-point axiomatics the concept of a vector is taken as one of the basic concepts; The axiomatics of E. g. can be based on the symmetry relation (see).

  EUCLIDAN FIELD is an ordered field in which each positive element is a square. For example, the field R of real numbers is an E.p. The field Q of rational numbers is not an E.p. L. Popov.

  EUCLIDEAN SPACE is a space whose properties are described by the axioms of Euclidean geometry. In a more general sense, an E. space is a finite-dimensional real vector space Rn with the scalar product (x, y), x, which in appropriately chosen coordinates (Cartesian) is expressed by the formula

  Chapter IV Euclid's algorithm

  Euclidean algorithm- an algorithm for finding the greatest common divisor of two integers. This algorithm is also applicable to finding the greatest common divisor of polynomials, the rings in which the Euclidean algorithm is applicable are called Euclidean rings.

  Euclid described it in Book VII and Book X of the Elements. In both cases, he gave a geometric description of the algorithm for finding the “common measure” of two segments. Euclid's algorithm was known in ancient Greek mathematics at least a century before Euclid under the name "antifiresis" - "sequential mutual subtraction".

  Euclid's algorithm for integers

  Let a And b are integers that are not equal to zero at the same time, and a sequence of numbers

  determined by the fact that each rk this is the remainder of the division of the pre-previous number by the previous one, and the penultimate one is divided by the last one completely, i.e.

  a = bq 0 + r 1

  b = r 1q 1 + r 2

  r 1 = r 2q 2 + r 3

  https://pandia.ru/text/78/222/images/image004_176.gif" width="47" height="20">, is proven by induction on m.

  Correctness This algorithm follows from the following two statements:

  Let a = bq + r, Then ( a,b) = (b,r). (0,r) = r. for any non-zero r. Extended Euclidean algorithm and Bezout's relation

  Formulas for ri can be rewritten as follows:

  r 1 = a + b(- q 0)

  r 2 = b − r 1q 1 = a(− q 1) + b(1 + q 1q 0)

  margin-top:0cm" type="disc"> Ratio a / b can be represented as a continued fraction:

  .

  Attitude - t / s, in the extended Euclidean algorithm allows representation in the form of a continued fraction:

  .

  Variations and generalizations

  Rings in which the Euclidean algorithm is applicable are called Euclidean rings, which include, in particular, the ring of polynomials.

  Accelerated versions of the algorithm

  One method for accelerating the Euclidean integer algorithm is to choose symmetrical remainder:

  

  One of the most promising versions of the accelerated Euclidean algorithm for polynomials is based on the fact that the intermediate values ​​of the algorithm mainly depend on high powers. When applying the Divide & Conqurer strategy, a large acceleration in the asymptotic speed of the algorithm is observed.

  ChapterV.
  Axiomatics

  Axiom(Ancient Greek ἀξίωμα - statement, position) or postulate- a statement accepted without proof.

  Axiomatization theory - an explicit indication of a finite set of axioms. Statements that follow from axioms are called theorems.

  Examples of different but equivalent sets of axioms can be found in mathematical logic and Euclidean geometry.

  A set of axioms is called consistent if, using the rules of logic, it is impossible to arrive at a contradiction from the axioms of the set. Axioms are a kind of “reference points” for the construction of any science, while they themselves are not proven, but are derived directly from empirical observation (experience).

  The term “axiom” was first found in Aristo in 322 BC. BC) and moved to mathematics from the philosophers of Ancient Greece. Euclid distinguishes the concepts of “postulate” and “axiom” without explaining their differences. Since the time of Boethius, postulates have been translated as requirements (petitio), axioms - as general concepts. Originally, the word “axiom” meant “a truth that is obvious in itself.” In different manuscripts of Euclid's Elements, the division of statements into axioms and postulates is different, and their order does not coincide. Probably the scribes had different views on the difference between these concepts.

  ChapterVI. Euclidean geometry and the V postulate

  Euclidean geometry(old pronunciation - "Euclidean") - familiar geometry studied at school. Usually refers to two or three dimensions, although one can speak of multidimensional Euclidean space. Euclidean geometry is named after the ancient Greek mathematician Euclid. In his book "Principia", in particular, the geometry of the Euclidean plane is systematically described.

  Axiomatization

  The axioms given by Euclid in the Elements are as follows:

  Through every two points you can draw exactly one straight line. You can draw a straight line along any segment. Having a segment, you can draw a circle so that the segment is the radius, and one of its ends is the center of the circle. All right angles are equal. Euclid's axiom of parallelism: Through a point A outside the line a in the plane passing through A and a, only one straight line can be drawn that does not intersect a.

  To define three-dimensional Euclidean space, we need a few more axioms. There are other, modern axiomatizations.

  The problem of complete axiomatization of elementary geometry is one of the problems of geometry that arose in Ancient Greece in connection with criticism of this first attempt to construct a complete system of axioms so that all statements of Euclidean geometry followed from these axioms by purely logical conclusion without the clarity of drawings. The first such complete system of axioms was created by D. Hilbert in 1899; it already consists of 20 axioms divided into 5 groups.

  Euclid's Axiom of Parallelism or fifth postulate- one of the axioms underlying classical planimetry. First given in Euclid's Elements.

  And if a straight line falling onto two straight lines forms interior angles on one side that are less than two right angles, then if extended indefinitely, these straight lines will meet on the side where the angles are less than two right angles.

  Euclid distinguishes concepts postulate And axiom without explaining their differences; in different manuscripts of Euclid’s Elements, the division of statements into axioms and postulates is different, just as their order does not coincide. In the classic edition of Heyberg's Principia, the stated statement is the fifth postulate.

  In modern language, Euclid's text can be reformulated as follows:

  If the sum of interior angles with a common side formed by two straight lines when they intersect a third, on one side of the secant, is less than 180°, then these straight lines intersect, and, moreover, on the same side of the secant.

  School textbooks usually give another formulation, equivalent (equivalent) to postulate V and due to Proclus:

  margin-top:0cm" type="disc"> There is a rectangle ( at least one), that is, a quadrilateral with all right angles. There are triangles that are similar but not equal. Any figure can be proportionally increased. There is a triangle of any size. Through every point inside an acute angle it is always possible to draw a straight line intersecting both its sides. If two straight lines diverge in one direction, then in the other they come closer. Converging straight lines will intersect sooner or later. There are lines such that the distance from one point to another is constant. If two straight lines begin to approach each other, then it is impossible for them to then begin (in the same direction) to diverge. The sum of the angles is the same for all triangles. There is a triangle whose angles add up to two right angles. There are parallel lines, and two lines parallel to the third are parallel to each other. There are parallel lines, and a line that intersects one of the parallel lines will certainly intersect the other. For every triangle there is a circumscribed circle. The Pythagorean theorem is true.

  Their equivalence means that all of them can be proven if we accept the V postulate, and vice versa, replacing the V postulate with any of these statements, we can prove the original V postulate as a theorem.

  In non-Euclidean geometries, instead of the V postulate, a different axiom is used, which makes it possible to create an alternative, internally logically consistent system. For example, in Lobachevsky geometry the formulation is as follows: “ in a plane, through a point not lying on a given line, it is possible to draw at least two different lines that do not intersect with a given one" And in spherical geometry, where large circles act as analogues of straight lines, there are no parallel straight lines at all.

  It is clear that in non-Euclidean geometry all of the above equivalent statements are false.

  Attempts to prove

  The fifth postulate stands out sharply among others, which are quite obvious (see Euclid’s Elements). It looks more like a complex, non-obvious theorem. Euclid was probably aware of this, and therefore the first 28 sentences in the Elements are proved without his help.

  Since ancient times, mathematicians have tried to “improve Euclid” - either to exclude the fifth postulate from the number of initial statements, that is, to prove it based on the remaining postulates and axioms, or to replace it with another, as obvious as the other postulates. The hope for the achievability of this result was supported by the fact that Euclid’s IV postulate ( all right angles are equal) really turned out to be superfluous - it was strictly proven as a theorem and excluded from the list of axioms.

  Over two millennia, many proofs of the fifth postulate have been proposed, but in each of them, sooner or later, a vicious circle was discovered: it turned out that among the explicit or implicit premises there was a statement that could not be proven without using the same fifth postulate.

  The first mention of such an attempt that has reached us states that Claudius Ptolemy was involved in this, but the details of his proof are unknown. Proclus (5th century AD) gives his own proof, based on the assumption that the distance between two disjoint lines is a limited value; it later turned out that this assumption is equivalent to the fifth postulate.

  After the decline of ancient culture, mathematicians from Islamic countries took up postulate V. The proof of al-Abbas al-Jauhari, a student of al-Khwarizmi (9th century), implicitly implied: if when two straight lines intersect with any third, the cross-lying angles are equal, then the same occurs when the same two straight lines intersect with any other. And this assumption is equivalent to postulate V.

  Thabit ibn Qurra (9th century) gave 2 proofs; in the first, he relies on the assumption that if two straight lines move away from each other on one side, they are necessarily approaching on the other side. In the second, it proceeds from the existence of equidistant straight lines, and Ibn Kurra tries to deduce this fact from the idea of ​​“simple motion,” i.e., uniform motion at a fixed distance from a straight line (it seems obvious to him that the trajectory of such motion is also a straight line). Each of the two mentioned statements of Ibn Qurra is equivalent to postulate V.

  https://pandia.ru/text/78/222/images/image011_109.gif" width="180" height="229">

  Essay by Saccheri

  An in-depth study of the V postulate, based on a completely original principle, was carried out in 1733 by the Italian Jesuit monk and mathematics teacher Girolamo Saccheri. He published a work entitled " Euclid, cleared of all stains, or a geometric attempt to establish the very first principles of all geometry". Saccheri's idea was to replace the V postulate with the opposite statement, derive as many consequences as possible from the new system of axioms, thereby constructing a “false geometry”, and find contradictions or obviously unacceptable provisions in this geometry. Then the validity of the V postulate will be proven from the opposite.

  Saccheri considers the same three hypotheses about the 4th angle of the Lambert quadrilateral. He immediately rejected the obtuse angle hypothesis for formal reasons. It is easy to show that in this case, in general, all the lines intersect, and then we can conclude that Euclid’s V postulate is valid - after all, it precisely states that under certain conditions the lines intersect. From this it is concluded that “ the obtuse angle hypothesis is always completely false, since it destroys itself» .

  After this, Saccheri moves on to refuting the “acute angle hypothesis,” and here his research is much more interesting. He admits that it is true, and, one after another, proves a whole series of consequences. Without suspecting it, he moves quite far in constructing Lobachevsky's geometry. Many of the theorems proved by Saccheri seem intuitively unacceptable, but he continues the chain of theorems. Finally, Saccheri proves that in “false geometry” any two lines either intersect or have a common perpendicular, according to both sides from which they move away from each other, or move away from each other on one side and indefinitely approach each other on the other. At this point Saccheri makes an unexpected conclusion: “ the acute angle hypothesis is completely false, since it contradicts the nature of a straight line» .

  Apparently, Saccheri felt that this “evidence” was unfounded, because the research continues. He considers the equidistant - the geometric locus of points on the plane equidistant from the straight line; Unlike his predecessors, Saccheri knows that in this case it is not a straight line at all. However, when calculating the length of its arc, Saccheri makes a mistake and comes to a real contradiction, after which he finishes the study and with relief declares that he “ tore out this evil hypothesis by the roots».

  In the second half of the 18th century, more than 50 works on the theory of parallels were published. The review of those years () examines more than 30 attempts to prove the V postulate and proves their fallacy. A famous German mathematician and physicist, with whom Klügel corresponded, also became interested in the problem; his Theory of Parallel Lines was published posthumously in 1786.

  Spherical geometry: all lines intersect

  Lambert was the first to discover that “obtuse angle geometry” is realized on a sphere, if by straight lines we mean great circles. He, like Saccheri, deduced many consequences from the “acute angle hypothesis”, and went much further than Saccheri; in particular, he discovered that the addition of the sum of the angles of a triangle to 180° is proportional to the area of ​​the triangle.

  In his book, Lambert astutely noted:

  It seems to me very remarkable that the second hypothesis [of the obtuse angle] is justified if instead of flat triangles we take spherical ones. I would almost have to draw a conclusion from this - the conclusion that the third hypothesis takes place on some imaginary sphere. In any case, there must be a reason why it is not as easy to refute on a plane as could be done in relation to the second hypothesis.

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  Lobachevsky and Bolyai showed greater courage than Gauss, and almost simultaneously (around 1830), independently of each other, published an exposition of what is now called Lobachevsky's geometry. As a high-class professional, Lobachevsky advanced the furthest in the study of new geometry, and it rightfully bears his name. But his main merit is not this, but the fact that he believed in the new geometry and had the courage to defend his conviction (he even proposed to experimentally test the V postulate by measuring the sum of the angles of a triangle).

  The tragic fate of Lobachevsky, ostracized in the scientific world and official circles for his too bold thoughts, showed that Gauss’s fears were not in vain. But his struggle was not in vain. Several decades later, mathematicians (Bernhard Riemann), and then physicists (General Relativity, Einstein), finally put an end to the dogma of the Euclidean geometry of physical space.

  Neither Lobachevsky nor Bolyai were able to prove the consistency of the new geometry - then mathematics did not yet have the necessary means for this. Only 40 years later, the Klein model and other models appeared that implemented the axiomatics of Lobachevsky geometry on the basis of Euclidean geometry. These models convincingly prove that the negation of postulate V does not contradict the other axioms of geometry; from here it follows that postulate V is independent of the other axioms, and it is impossible to prove it. The centuries-old drama of ideas has ended.

  Chapter VII. Euclid's beginnings.

  Greek text Began.

  During excavations of ancient cities, several papyri were found containing small fragments of Euclid's Elements. The best known was found in the ruins of the ancient city of Oxyrhynchus, near the modern village of Behnesa (about 110 miles up the Nile from Cairo and 10 miles west of it) and contains the wording II prop. 5 with a picture.

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  We invite you to meet such a great mathematician as Euclid. A biography, a summary of his main work and some interesting facts about this scientist are presented in our article. Euclid (life years - 365-300 BC) - mathematician dating back to the Hellenic era. He worked in Alexandria under Ptolemy I Soter. There are two main versions of where he was born. According to the first - in Athens, according to the second - in Tire (Syria).

  Biography of Euclid: interesting facts

  There's not much of that about life. There is a message belonging to Pappus of Alexandria. This man was a mathematician who lived in the 2nd half of the 3rd century AD. He noted that the scientist we were interested in was kind and gentle with all those who could somehow contribute to the development of certain mathematical sciences.

  There is also a legend reported by Archimedes. Its main character is Euclid. A short biography for children usually includes this legend, since it is very interesting and can arouse interest in this mathematician among young readers. It says that King Ptolemy wanted to study geometry. However, it turned out that this is not easy to do. Then the king called the scientist Euclid and asked him if there was any easy way to comprehend this science. But Euclid replied that there was no royal road to geometry. So this expression, which became popular, came to us in the form of a legend.

  

  At the beginning of the 3rd century BC. e. founded the Alexandria Museum and Euclid. A short biography and his discoveries are associated with these two institutions, which were also educational centers.

  Euclid - Plato's student

  This scientist went through the Academy founded by Plato (his portrait is presented below). He learned the main philosophical idea of ​​this thinker, which was that there is an independent world of ideas. It is safe to say that Euclid, whose biography is sparse in details, was a Platonist in philosophy. This attitude strengthened the scientist in the understanding that everything that was created and outlined by him in his “Principles” has eternal existence.

  

  The thinker we are interested in was born 205 years later than Pythagoras, 63 years later than Plato, 33 years later than Eudoxus, 19 years later than Aristotle. He became acquainted with their philosophical and mathematical works either independently or through intermediaries.

  The connection between Euclid's Elements and the works of other scientists

  Proclus Diadochus, a Neoplatonist philosopher (years of life - 412-485), author of comments to the "Elements", expressed the idea that this work reflects Plato's cosmology and the "Pythagorean doctrine ...". In his work, Euclid outlined the theory of the golden section (books 2, 6 and 13) and (book 13). Being an adherent of Platonism, the scientist understood that his “Principles” contributed to Plato’s cosmology and to the ideas developed by his predecessors about the numerical harmony that characterizes the universe.

  Proclus Diadochos was not the only one who appreciated the Platonic solids and (years of his life - 1571-1630) was also interested in them. This German astronomer noted that there are 2 treasures in geometry - the golden ratio (division of a segment in the average and extreme ratio) and the Pythagorean theorem. He compared the value of the last of them to gold, and the first to a precious stone. Johannes Kepler used the Platonic solids in creating his cosmological hypothesis.

  Meaning "Started"

  

  The book "Elements" is the main work that Euclid created. The biography of this scientist, of course, is marked by other works, which we will discuss at the end of the article. It should be noted that works with the title “Principles”, which set out all the most important facts of theoretical arithmetic and geometry, were also compiled by his predecessors. One of them is Hippocrates of Chios, a mathematician who lived in the 5th century BC. e. Theudius (2nd half of the 4th century BC) and Leontes (4th century BC) also wrote books with this title. However, with the advent of Euclidean "Principles" all these works were forced out of use. Euclid's book was the basic textbook on geometry for more than 2 thousand years. The scientist, creating his work, used many of the achievements of his predecessors. Euclid processed the available information and brought the material together.

  In his book, the author summed up the development of mathematics in Ancient Greece and created a solid foundation for further discoveries. This is the significance of Euclid’s main work for world philosophy, mathematics and all science in general. It would be wrong to believe that it consists in strengthening the mysticism of Plato and Pythagoras in their pseudo-universe.

  Many scientists appreciated Euclid's Elements, including Albert Einstein. He noted that this is an amazing work that gave the human mind the self-confidence necessary for further activity. Einstein said that the person who did not admire this creation in his youth was not born for theoretical research.

  Axiomatic method

  It should be noted separately the significance of the work of the scientist we are interested in in the brilliant demonstration in his “Principles”. This method in modern mathematics is the most serious of those used to substantiate theories. It also finds wide application in mechanics. The great scientist Newton built his "Principles of Natural Philosophy" on the model of the work created by Euclid.

  Basic provisions of "Beginnings"

  

  The book "Principia" systematically expounds Euclidean geometry. Its coordinate system is based on concepts such as plane, straight line, point, motion. The relations that are used in it are the following: “a point is located on a line lying on a plane” and “a point is located between two other points.”

  The system of provisions of Euclidean geometry, presented in a modern presentation, is usually divided into 5 groups of axioms: motion, order, continuity, combination and parallelism of Euclidean.

  

  In the thirteen books of “Principles,” the scientist presented arithmetic, stereometry, planimetry, and relations according to Eudoxus. It should be noted that the presentation in this work is strictly deductive. Every book of Euclid begins with definitions, and in the first of them they are followed by axioms and postulates. Next come sentences, divided into problems (where you need to build something) and theorems (where you need to prove something).

  Disadvantage of Euclid's Mathematics

  The main drawback is that the axiomatics of this scientist are not complete. The axioms of motion, continuity and order are missing. Therefore, the scientist often had to trust his eye and resort to intuition. Books 14 and 15 are later additions to the work authored by Euclid. There is only a very brief biography of him, so it is impossible to say for sure whether the first 13 books were created by one person or are the fruit of the collective work of a school led by a scientist.

  Further development of science

  The emergence of Euclidean geometry is associated with the emergence of visual representations of the world around us (rays of light, stretched threads as an illustration of straight lines, etc.). Then they deepened, thanks to which a more abstract understanding of such a science as geometry arose. N.I. Lobachevsky (years of life - 1792-1856) - Russian mathematician who made an important discovery. He noted that there is a geometry that differs from Euclidean. This changed scientists' ideas about space. It turned out that they are by no means a priori. In other words, the geometry set out in Euclid’s Elements cannot be considered the only one describing the properties of the space surrounding us. The development of natural science (primarily astronomy and physics) has shown that it describes its structure only with a certain accuracy. In addition, it cannot be applied to the entire space as a whole. Euclidean geometry is the first approximation to understanding and describing its structure.

  By the way, Lobachevsky’s fate turned out to be tragic. He was not accepted in the scientific world for his bold thoughts. However, this scientist’s struggle was not in vain. The triumph of Lobachevsky's ideas was ensured by Gauss, whose correspondence was published in the 1860s. Among the letters were the scientist’s enthusiastic reviews of Lobachevsky’s geometry.

  Other works of Euclid

  

  The biography of Euclid as a scientist is of great interest in our time. He made important discoveries in mathematics. This is confirmed by the fact that since 1482 the book “Principles” has gone through more than five hundred editions in various languages ​​of the world. However, the biography of the mathematician Euclid is marked by the creation of not only this book. He owns a number of works on optics, astronomy, logic, and music. One of them is the book “Data,” which describes the conditions that make it possible to consider one or another mathematical maximum image as “data.” Another work of Euclid is a book on optics, which contains information about perspective. The scientist we are interested in also wrote an essay on catoptrics (in this work he outlined the theory of distortions that occur in mirrors). Euclid's book entitled "Division of Figures" is also known. The work on mathematics “Unfortunately, it has not survived.

  So, you met such a great scientist as Euclid. We hope you found his brief biography useful.

  The ancient mathematician and philosopher Euclid lived in the 3rd century BC. And he was truly an outstanding mathematician - not only for his time, but also for our time. After all, the very geometry that schoolchildren around the world study today is called Euclidean. It is based on five axioms derived by him. Without exaggeration, this scientist laid the foundation of modern geometry and, in many ways, mathematics as a science.

  And many will probably be interested in learning some interesting facts from the life of Euclid.
  
  

  Where and when

  It is noteworthy that it is not known for certain when exactly and in what place Euclid was born. From scant records from Arabic books of the 12th century, it can be judged that his father’s name was Naukrates, and the future great mathematician himself was born in Greece.

  It is assumed that he began to receive his education at Plato's Academy, at the entrance to which, by the way, there was an inscription: “No one who does not know geometry will ever enter here.”

  However, the circumstances and even the exact date of Euclid’s death are also shrouded in mystery: it is assumed that this sad event occurred no later than 265 BC.

  Royal ways

  One of the most famous legends about Euclid came to us from the words of Archimedes himself. He said that one day King Ptolemy himself decided to start studying geometry according to Euclid’s Elements. However, science seemed very difficult to the royal person and was not given at all. And then Ptolemy asked if there was any easier and faster way to master everything... To which Euclid uttered the now catchphrase: “There are no royal paths in geometry.”
  

  Profitable Science

  There is also a known case when one student asked the famous mathematician how geometry could be beneficial to him in life. To which Euclid called the servant and ordered to give the student three obols (a monetary unit), saying:

  - Give him money, since he only wants profit from science.

  Many Beginnings

  It is interesting that Euclid’s “Elements” were not the only “Elements” before him. Previously, many scientists wrote scientific works, and they were called “Principles”. However, only the Euclidean ones became famous throughout the centuries.

  But the great geometer did not build his works from absolutely nothing. In fairness, it is worth noting that many of his theorems were built on the basis of knowledge already available at that time. But Euclid put them together, classified them and was able to justify them from a scientific point of view.

  According to a strict logical chain

  It was in his Elements that Euclid did what today seems self-evident: he began to base all his conclusions on a chain of strict logical deductions. At the same time, he considered it important that the chain should begin somewhere, and not grow from empty space, since in this case it may never end. The very name of his scientific work must be connected with this. But, since it was very difficult to get to the very initial judgment, Euclid himself formulated his famous axioms - statements that do not require proof. And only on these axioms did he manage to derive all the other proofs and theorems.
  
  

  Plato is my friend

  As already mentioned, Euclid studied at school with Plato himself. It is not surprising that in his philosophical judgments he belonged to the so-called Platonist. In particular, he believed that everything was based on four elements - water, air, earth and fire.

  Unproven works of Euclid

  The Arabs - and not only them - often attribute other works to Euclid in many fields of knowledge, from music to medicine. For example, the fundamental work on music theory “Harmonics”, as well as “Division of Canons”. However, already in our time it has been proven that the mathematician has nothing to do with these works. Most likely, their author was the Pythagorean Kleonidas. Although this is not known for certain.

  Good math

  Another ancient mathematician, Pappus, reports that Euclid was unusually gentle and kind towards those who, firstly, could help in the spread of mathematics as a science, and secondly, if he saw that a person really had a craving for geometry . He was even able to change his opinion about this or that person if he suddenly found out that he was interested or, on the contrary, not interested, in mathematics.

  Both a museum and a library

  It is also known that Euclid, at the turn of the third century BC, organized the opening of a museum and library in the city of Alexandria. Here he subsequently made many of his discoveries. In addition, both the museum and the library under Euclid played the role of ancient scientific centers.
  

  "Eternal" book

  Submitting to the school of Plato, Euclid believed that everything that he sets out in his “Principles” not only is not questioned, but will also exist forever. Be that as it may, for more than 2 thousand years, it is from the works of Euclid that students master the wisdom of geometry.

  Non-Euclidean geometry

  And only after more than 2 thousand years, the Russian mathematician Lobachevsky doubted the absolute validity of Euclid’s geometry. He developed “his own” geometry, which was based not on a plane, but on a pseudosphere. It is interesting that all the Axioms derived by Euclid were preserved. Except one - about parallel lines.

  In addition to Lobachevsky, the German mathematician Riemann also developed “his” geometry. Currently, three geometries strangely coexist in the world - Euclidean, Riemann and Lobachevsky.

  Whether it was so, as some stories about Euclid describe, or maybe nothing like that happened at all, is not so important. The author of "Mathematical Principles" forever inscribed his name in the annals of science, where he will remain - along with such geniuses as Newton, Galileo, Socrates or Pythagoras.