Fundamental work of Euclid. Euclid's "Elements"

We invite you to meet such a great mathematician as Euclid. Biography, summary 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. Her main character- Euclid. 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 independent world 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 - these are golden ratio(division of a segment by the average and extreme respect) 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 theoretical arithmetic and geometry, were 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. The book of Euclid was the basic teaching aid in 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 to further activities. 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. Great scientist Newton built the "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 positions of Euclidean geometry presented in modern presentation, are usually divided into 5 groups of axioms: motion, order, continuity, combination and parallelism of Euclid.

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 (life years - 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 into scientific world for your brave 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. In mathematics he did important discoveries. This is confirmed by the fact that since 1482 the book “Principles” has gone through more than five hundred editions. various languages peace. 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.

For two thousand years, geometry was learned either from Euclid's Elements or from textbooks written based on this book. Only professional mathematicians turned to the works of other great Greek geometers: Archimedes, Apollonius - and geometers of later times. Classical geometry began to be called Euclidean, in contrast to those that appeared in the 19th century. "non-Euclidean geometries".

History has preserved so little information about this amazing man that doubts are often expressed about his very existence. What has reached us? Catalog of Greek geometers by Proclus Diadochos of Byzantium, who lived in the 5th century. AD, is the first serious source of information about Greek geometry. From the catalog it follows that Euclid was a contemporary of King Ptolemy I, who reigned from 306 to 283 BC.

Euclid must be older than Archimedes, who referred to the Elements. Information has reached our times that he taught in Alexandria, the capital of Ptolemy I, which was beginning to turn into one of the centers scientific life. Euclid was a follower of the ancient Greek philosopher Plato, and he probably taught four sciences, which, according to Plato, should precede the study of philosophy: arithmetic, geometry, theory of harmony, astronomy. In addition to the Elements, Euclid’s books on harmony and astronomy have reached us.

As for Euclid’s place in science, it is determined not so much by his own scientific research, how many pedagogical merits. Several theorems and new proofs are attributed to Euclid, but their significance cannot be compared with the achievements of the great Greek geometers: Thales and Pythagoras (VI century BC), Eudoxus and Theaetetus (IV century BC). Euclid’s greatest merit is that he summed up the construction of geometry and gave the presentation such a perfect form that for two thousand years the “Elements” became an encyclopedia of geometry.

Euclid with greatest art arranged the material across 13 books so that difficulties did not arise prematurely. Later, Greek mathematicians included two more books in the “Elements” - the XIVth and XVth, written by other authors.

The first book of Euclid begins with 23 “definitions”, among them the following: a point is that which has no parts; a line is length without width; the line is limited by points; a straight line is a line equally located relative to all its points; finally, two lines lying in the same plane are called parallel if they, however extended, do not meet. These are rather visual representations of the main objects, and the word “definition” in the modern sense does not accurately convey the meaning Greek word"horoi" used by Euclid.

Book I examines the basic properties of triangles, rectangles, and parallelograms, and compares their areas. This is where the theorem on the sum of the angles of a triangle appears. Then follow five geometric postulates: through two points one straight line can be drawn; each line can be extended as much as desired; with a given radius a circle can be drawn from a given point; all right angles are equal; if two straight lines are drawn to a third at angles that add up to less than two straight lines, then they meet on the same side of this straight line. All of these postulates, except one, were included in modern courses basic geometry. Following the postulates are general assumptions, or axioms - eight general mathematical statements about equalities and inequalities. The book ends with the Pythagorean theorem (see Pythagorean theorem).

Book II sets out geometric algebra, with the help of geometric drawings, solutions are given to problems that boil down to quadratic equations. Algebraic symbolism did not exist then.

Book III deals with the properties of the circle, properties of tangents and chords, in book IV - regular polygons, the foundations of the doctrine of similarity appear. Books VII-IX contain the beginnings of number theory (see Number theory), based on an algorithm for finding the largest common divisor, Euclid's algorithm is given (see Euclid's algorithm), this includes the theory of divisibility and the theorem on the infinity of the set of prime numbers.

Recent books are devoted to stereometry. Book XI sets out the beginnings of stereometry; in Book XII, using the method of exhaustion, the ratio of the areas of two circles and the ratio of the volumes of a pyramid and a prism, a cone and a cylinder are determined. The pinnacle of stereometry in Euclid is theory regular polyhedra. One of them was not included in "Beginnings" greatest achievements Greek geometers - theory conic sections. Euclid wrote a separate book about them, “The Origins of Conic Sections,” which has not reached us, but was cited by Archimedes in his writings.

Euclid's "Elements" have not reached us in the original. Twelve centuries separate the oldest from Euclid famous lists, seven centuries - some detailed information about the “Principles”. In the medieval era, interest in mathematics was lost, some books of the Elements disappeared and were then difficult to recover from Latin and Arabic translations. And by that time, the texts had been overgrown with “improvements” by later commentators.

During the period of the revival of European mathematics (16th century), the “Principia” was studied and recreated anew. The logical construction of the “Principles” and the axiomatics of Euclid were perceived by mathematicians as something impeccable until the 19th century, when a period of critical attitude towards what had been achieved began, which ended with the new axiomatics of Euclidean geometry - the axiomatics of D. Hilbert. The presentation of geometry in the Elements was considered a model that scientists sought to follow outside of mathematics.

Essay

On the topic of:

Euclid and his “beginnings”

Completed: Gordienko Pavel.

Secondary school No. 31

2002.

Plan.

1. Euclid and his beginning.

2. Euclidean algorithm.

1. Euclid and his “Elements”

For two thousand years, geometry was learned either from Euclid's Elements or from textbooks written based on this book. Only professional mathematicians turned to the works of other great Greek geometers: Archimedes, Apollonius and geometers of later times. Classical geometry began to be called Euclidean, in contrast to the “non-Euclidean geometries” that appeared in the 19th century.

History has preserved so little information about this amazing man that doubts are often expressed about his very existence. What has reached us? The catalog of Greek geometers by Proclus Diadochos of Byzantium, who lived in the 5th century AD, is the first serious source of information about Greek geometry. From the catalog it follows that Euclid was a contemporary of King Ptolemy I, who reigned from 306-283 BC.

Euclid must be older than Archimedes, who referred to the Inception. Information has reached our times that he taught in Alexandria, the capital of Ptolemy I, which was beginning to turn into one of the centers of scientific life. Euclid was a follower of the ancient Greek philosopher Plato, and he probably taught four sciences, which, according to Plato, should precede the study of philosophy: arithmetic, geometry, theory of harmony, astronomy. In addition to the “Principles”, Euclid’s books on harmony and astronomy have reached us.

As for Euclid’s place in science, it is determined not so much by his own scientific research as by his pedagogical merits. Several theorems and new proofs are attributed to Euclid, but their significance cannot be compared with the achievements of the great Greek geometers: Thales and Pythagoras (VI century BC), Eudoxus and Theaetetus (IV century BC). Euclid’s greatest merit is that he summed up the construction of geometry and gave the presentation such a perfect form that for 2000 years the “Elements” became an encyclopedia of geometry.

Euclid, with the greatest skill, arranged the material across 13 books so that difficulties did not arise prematurely. Later, Greek mathematicians included two more books in the “Beginning” - XIV and XV, written by other authors.

The first book of Euclid begins with 23 “definitions,” among them the following: a point is something that has no parts; a line is length without width; the line is limited by points; a straight line is a line that is equally located relative to all its points; finally, two lines lying in the same plane are called parallel if they do not meet, no matter how extended they are. It's more likely visual representations about the main objects and the word “definition” in the modern understanding does not accurately convey the meaning of the Greek word “horoi”, which Euclid used.

Book I examines the basic properties of triangles, rectangles, and parallelograms, and compares their areas. This is where the theorem on the sum of the angles of a triangle appears. Then follow five geometric postulates: through two points one straight line can be drawn; each line can be extended as much as desired; with a given radius a circle can be drawn from a given point; all right angles are equal; if two straight lines are drawn to a third at angles that add up to less than two straight lines, then they meet on the same side of this straight line. All of these postulates, except one, are included in modern courses in basic geometry. Following the postulates are general assumptions, or axioms, - 8 general mathematical statements about equalities and inequalities. The book ends with the Pythagorean theorem.

Book II presents geometric algebra, using geometric drawings to provide solutions to problems that reduce to quadratic equations. Algebraic symbolism did not exist then.

In book III the properties of the circle, the properties of tangents and chords are discussed, in book IV - regular polygons, the foundations of the doctrine of similarity appear. Books VII-IX set out the beginnings of number theory, and based on the algorithm for finding the greatest common divisor, Euclid's algorithm is given, this includes the theory of divisibility and the theorem on the infinity of the set of prime numbers.

Recent books are devoted to stereometry. Book XI sets out the beginnings of stereometry; in Book XII, using the method of exhaustion, the ratio of the areas of two circles and the ratio of the volumes of a pyramid and a prism, a cone and a cylinder are determined. The pinnacle of stereometry in Euclid is the theory of regular polyhedra. “Inception” did not include one of the greatest achievements of Greek geometers - the theory conic sections. Euclid wrote a separate book about them, “The Beginnings of Conic Sections,” which has not reached us, but was cited by Archimedes in his writings.

Euclid’s “Beginning” has not reached us in the original. Twelve centuries separate the oldest known copies from Euclid, seven centuries separate any detailed information about the Elements. In the medieval era, interest in mathematics was lost, some books of the Elements disappeared and were then difficult to recover from Latin and Arabic translations. And by that time, the texts had been overgrown with “improvements” by later commentators.

During the period of the revival of European mathematics (XVI century), the “Principia” was studied and recreated anew. The logical construction of the “Principia” and the axiomatics of Euclid were perceived by mathematicians as impeccable until the 19th century, when a period of critical attitude towards what had been achieved began, which ended with the new axiomatics of Euclidean geometry - the axiomatics D. Gilbert. The presentation of geometry in the Elements was considered a model that scientists sought to follow outside of mathematics.

2. Euclid Algorithm.

Euclid's algorithm is a method of finding the greatest common divisor of two integers, as well as the greatest common measure of two commensurate segments.

To find the greatest common divisor of two integers positive numbers, you need first larger number divide by the smaller number, then divide the second number by the remainder of the first division, then the first remainder by the second, etc. The last non-zero positive remainder in this process will be the greatest common divisor of these numbers.

Denoting the original numbers by A And b, positive remainders resulting from divisions, through r 1 ,r2

..., rn, and incomplete quotients through q1, q2, we can write the Euclidean algorithm in the form of a chain of equalities:

. . . . . . . . . .

Let's give an example. Let a=777, b=629. Then 777=629*1+148, 629=148*4+37, 148=37*4.

The last non-zero remainder 37 is the greatest common divisor of the numbers 777 and 629.

To find the greatest common measure of two segments proceed in a similar way. The division operation with a remainder is replaced by its geometric analogue: smaller segment postponed on the larger segment as many times as possible: the remaining part of the larger segment (taken as the remainder of the separation) is postponed on the smaller segment, etc. if the segments a and b are commensurable, then the last non-zero remainder will give the greatest common measure of these segments. In the case of incommensurable segments, the resulting sequence of non-zero remainders will be infinite.

Let's look at an example. Let us take as the initial segments the sides AB and AC of the isosceles triangle ABC, for which A=C = 72°, B= 36°. As the first remainder we will receive the segment AD (CD-bisector of angle C), and, as is easy to see, the sequence of zero remainders will be infinite. This means that segments AB and AC are not commensurable.

The Euclid algorithm has been known for a long time. It is already more than 2000 years old. This algorithm is formulated in Euclid’s Elements, where the properties of prime numbers, least common multiple, etc. are derived from it. As a method of finding the greatest common measure of two segments, Euclid's algorithm (sometimes called the method of alternating subtraction) was known to the Pythagoreans. TO mid-16th century V. Euclid's algorithm was extended to polynomials; from one variable, it was later possible to determine the Euclid algorithm for some other algebraic objects.

The Euclid algorithm has many applications. The equalities that define it make it possible to imagine greatest divisor d numbers a And b in the form d=ax+by (x;y are integers), and this allows you to find a solution to Diophantine equations of the 1st degree with two unknowns. The Euclidean algorithm is a means to represent rational number in the form of a continued fraction. It is often used in computer programs.

References.

Encyclopedic dictionary of a young mathematician.

(“Principles” of Euclid)

scientific work, written by Euclid in the 3rd century. BC e., containing the foundations of ancient mathematics: elementary geometry, number theory, algebra, general theory relations and a method for determining areas and volumes, which included elements of the theory of limits. Euclid summed up the three hundred years of development of Greek mathematics in this work and created a solid foundation for further mathematical research. "N." E. are not, however, an encyclopedia mathematical knowledge of his era. So, in "N." E. does not present the theory of conic sections, which was then quite developed, and there are no computational methods here.

"N." E. are constructed according to a deductive system: first, definitions, postulates and axioms are given, then the formulations of theorems and their proofs (see Deduction). Following the definition of the main geometric concepts and objects (for example, a point, a line) Euclid proves the existence of other objects of geometry (for example, equilateral triangle) by constructing them, which is carried out on the basis of five postulates. The postulates state the possibility of performing some elementary constructions, for example, “that from any point to any point (it is possible) to draw a straight line” (postulate 1); “And that from any center and from any solution a circle (can be) described” (III postulate). Special place Among the postulates, postulate V (the axiom of parallels) ranks: “And if a straight line falling on two straight lines forms internal angles on one side that are less than two right angles, then these straight lines extended without limit will meet the side where the angles are less than two right angles.” The relative complexity of the formulation led to the desire of many mathematicians (for almost 2 thousand years) to derive it as a theorem from other fundamental principles of geometry. Attempts to prove the V postulate continued until the works of N. I. Lobachevsky (See Lobachevsky) , who constructed the first system of non-Euclidean geometry in which this postulate is not satisfied (see Lobachevsky geometry). Behind the postulates in "N." E. axioms are given - propositions about the properties of relations of equality and inequality between quantities. For example: “Those who are equal to the same are equal to each other” (1st axiom); “And the whole is greater than the part” (8th axiom).

WITH modern point system of axioms and postulates "N." E. is not sufficient for the deductive construction of geometry. Thus, there are neither axioms of motion nor axioms of congruence (with the exception of one). The axioms of location and continuity are also missing. In fact, Euclid uses both motion and continuity in his proofs. Logical shortcomings of the “N.” construction E. were fully clarified only at the end of the 19th century. after the work of D. Hilbert (see Euclidean geometry) . Before that, for more than 2 thousand years, “N.” E. served as a model of scientific rigor; This book was used to study geometry in its entirety or in an abbreviated and revised form.

"N." E. consist of thirteen books (divisions, or parts). Book I examines the basic properties of triangles, rectangles, and parallelograms and compares their areas. The book of Pythagoras ends with a theorem (See Pythagorean theorem). Book II expounds the so-called geometric algebra, i.e., a geometric apparatus is constructed for solving problems that can be reduced to quadratic equations (there is no algebraic symbolism in “N.” E.). Book III discusses the properties of the circle, its tangents and chords (these problems were studied by Hippocrates of Chios (See Hippocrates of Chios) in the 2nd half of the 5th century BC), in book IV - regular polygons. Book V gives the general theory of relations of quantities, created by Eudoxus of Cnidus (See Eudoxus of Cnidus) ; it can be considered as a prototype of the theory real numbers, developed only in the 2nd half of the 19th century. The general theory of relations is the basis of the doctrine of similarity (Book VI) and the method of exhaustion (Book VII), also dating back to Eudoxus. Books VII-IX present the beginnings of number theory, based on the algorithm for finding the greatest common divisor (Euclidean algorithm). These books include the theory of divisibility, including theorems on the uniqueness of the decomposition of an integer into prime factors and about the infinity of the number of prime numbers; It also expounds the doctrine of the relation of integers, which is essentially equivalent to the theory of rational (positive) numbers. Book X gives a classification of quadratic and biquadratic irrationalities and substantiates some rules for their transformation. The results of Book X are used in Book XIII to find the edge lengths of regular polyhedra. A significant part of books X and XIII (probably also VII) belongs to Theaetetus (beginning of the 4th century BC). Book XI sets out the basics of stereometry. In Book XII, using the exhaustion method, the ratio of the areas of two circles and the ratio of the volumes of a pyramid and a prism, a cone and a cylinder are determined. These theorems were first proven by Eudoxus. Finally, in Book XIII, the ratio of the volumes of two balls is determined, five regular polyhedra are constructed and it is proved that there are no other regular bodies. Subsequent Greek mathematicians to "N." Books XIV and XV, which did not belong to Euclid, were added to E. They are often even now published together with the main text of “N.” E.

"N." E. became widely known already in ancient times. Archimedes, Apollonius of Perga and other scientists relied on them in their research in the field of mathematics and mechanics. Until our time, the ancient text “N.” E. did not reach (the oldest surviving copy dates back to the 2nd half of the 9th century). At the end of the 8th century. - early 9th century translations of “N.” appear E. on Arabic. First translation to Latin language was made from Arabic by Atelhard of Bath in the 1st quarter of the 12th century. Ancient lists differ in significant discrepancies; original text "N." E. definitely not restored. First printed edition"N." E., translated into Latin by G. Campano, appeared in Venice in 1482 with drawings in the margins of the book (the translation was completed around 1250-1260; Campano used both Arabic sources and the translation of Atelhard of Bath). The best edition is currently considered to be the edition of I. Heiberg (“Euclidis Elementa”, v. 1-5, Lipsiae, 1883-88), in which it is given as Greek. text and its Latin. translation. In Russian "N." E. have been published many times since the 18th century. Best edition- “Euclid’s Elements”, trans. from Greek and comments by D. D. Mordukhai-Boltovsky, vol. 1-3, 1948-50.

Lit.: History of mathematics from ancient times to the beginning of modern times, vol. 1, M., 1970.

I. G. Bashmakova, A. I. Markushevich.

• - a method of finding the greatest common divisor of two integers, two polynomials, or the common measure of two segments. Described in geometric terms. form in Euclid's Elements...

  Mathematical Encyclopedia

• - about prime numbers: the set of prime numbers is infinite. More accurate quantitative information about the set of prime numbers in the natural series is contained in Chebyshev's theorem on prime numbers and asymptotic...

  Mathematical Encyclopedia

• - essay on elementary mathematics ancient Greek scientist Euclid, the most widely distributed publication in the world, covering elementary geometry, number theory, algebra, the theory of measuring geometric quantities,...

  Beginnings modern Natural Science

• - in Christ. presented one of the nine ranks of angels. Mentioned in the New Testament. According to the classification of Pseudo-Dionysius the Areopagite, he is the seventh rank, forming, together with the archangels and angels, the third triad...

  Ancient world. encyclopedic Dictionary

• - a method of finding the greatest common divisor of two integers, two polynomials, or the common measure of two segments...
• - a scientific work written by Euclid in the 3rd century. BC e., containing the foundations of ancient mathematics: elementary geometry, number theory, algebra, general theory of relations and the method of determining areas and...

  Great Soviet Encyclopedia

• - a method of finding the greatest common divisor of two integers, two polynomials, or the common measure of two segments. Described in geometric shape Euclid...
• - "BEGINNINGS" OF EUCLID, a scientific work written by Euclid in the 3rd century. BC e., which sums up the 300-year development of Greek mathematics and creates the foundation for further mathematical research...

  Big encyclopedic Dictionary

• - for starters adv. circumstances decompression time 1. First, at first. 2. For the first time...

  Dictionary Efremova

• - began pl. 1. Basic provisions, principles of something. 2...

  Explanatory Dictionary by Efremova

• - ...

  Spelling dictionary-reference book

• - Razg. For a start; for the first time. he strictly organized labor and created teams. And he promised to “look after” something for her first...

  Phrasebook Russian literary language

• - Without beginning, without end, and not God...

  IN AND. Dahl. Proverbs of the Russian people

• - first of all, first of all, in the first place, first of all, first of all, in the first place, first of all, first of all, before, initially, first thing, at first, first of all, first of all, first of all, first of all,...

  Synonym dictionary

• - adverb, number of synonyms: 1 per factory...

  Synonym dictionary

• - again, again, great, again, again, twenty-five again, ice from the eggs, anew, from...

  Synonym dictionary

"Euclid's Elements" in books

Appendix 2. Euclid's proof of the irrationality of the number √2

by Singh Simon

Appendix 2. Euclid's proof of the irrationality of the number?2 Euclid's goal was to prove that the number?2 cannot be represented as a fraction. Since Euclid used proof by contradiction, the first step was to assume that the opposite was true

Appendix 5. Euclid's proof of the existence of an infinite number of Pythagorean triples

From book Great Theorem Farm by Singh Simon

Appendix 5. Euclid's proof of existence infinite number Pythagorean triples A Pythagorean triple is a set of three integers such that the sum of the squares of two of them is equal to the square of the third number. Euclid was able to prove that he exists infinitely

Euclid's 5th postulate

From the book Hypotheses and Misconceptions You Should Know About modern man author Tribis Elena Evgenevna

5th postulate of Euclid Knowledge of the basics of geometry became necessary for humanity as economic relations developed, accompanied by the division of land and the construction of various structures. Born as pure applied Science, geometry gradually

Criticism of Newtonian mechanics and Euclidian geometry

From the book Course in the History of Physics author Stepanovich Kudryavtsev Pavel

Criticism of Newtonian mechanics and Euclidian geometry Electrodynamics of moving media in the theory of electrons led to many radical conclusions, primarily to the collapse of the idea of ​​unchanging solid particles. Solids And unchanging particles not in nature, shape and size of bodies

Eudoxus of Cnidus, predecessor of Euclid

From the book History Persian Empire author Olmsted Albert

Eudoxus of Cnidus, predecessor of Euclid Eudoxus of Cnidus began his career as a physician. He visited Athens and there found himself a teacher in Plato, who upon his return from Egypt became a philosopher. Eudox received letter of recommendation from Agesilaus to King Nekht-har-khebi and

What is the fundamental difference between the geometric optics of the Greek Euclid and the Arab Alhazen?

From book Newest book facts. Volume 3 [Physics, chemistry and technology. History and archaeology. Miscellaneous] author Kondrashov Anatoly Pavlovich

What is the fundamental difference between geometric optics the Greek Euclid and the Arab Algazen? Trying to explain the phenomenon of vision, the ancient Greek thinkers of the Pythagorean school put forward a hypothesis about a special fluid that is emitted by the eyes and “feels” (like

Euclid's "Elements"

From the book Big Soviet Encyclopedia(ON) the author TSB

Euclidean algorithm

From the book Great Soviet Encyclopedia (EB) by the author TSB

Chapter 3 Euclid's Competitors

From the book When Straight Lines Curve [Non-Euclidean Geometries] by Gomez Juan

Chapter 3 Euclid's Competitors Over the centuries, the fifth postulate has been the subject of much commentary and criticism in the writings of the most famous geometers. Many of them were convinced that this postulate could be proven using other postulates, and they concentrated their efforts on searching

Chapter 1 Before Euclid - Prehistoric Times

From the book In Pursuit of Beauty author Smilga Voldemar Petrovich

Chapter 1 Before Euclid - prehistoric times The true beginning of this story is lost in the mists of apocryphal times. Where, how and when did geometry begin... Where, how and when did it acquire its completed form and earn the right to be called science... Who was that unknown, first,

§ 3. Euclid's algorithm

From the book Invitation to Number Theory by Ore Oistin

§ 3. Euclid's algorithm Let's return to our fractions a/b. If a > b, then the fraction is a number greater than 1, and we often divide it into the whole part and correct fraction, less than one. Examples. We write 32/5 = 6 + 2/5 = 6 2/5, 63/7 = 9 + 0/7 = 9.V general case we use division with remainder numbers a and

Euclid's law

From the book Agents of Russia author Udovenko Yuri Alexandrovich

Euclid's Law Back in the third century BC, the ancient Greek mathematician Euclid said: angle of incidence equal to angle reflections! Isaac Newton's third law echoes: the force of action is equal to the force of reaction... These laws are relevant in everything, including in the agent field

Don Quixote from the Rue Euclid

From the book From the Potomac to the Mississippi: An Unsentimental Journey Through America author Sturua Melor Georgievich

Don Quixote from the Rue Euclid (Chronicle of a Strike)

Chapter 10. From Thales to Euclid

author Turchin Valentin Fedorovich

Chapter 10. From Thales to Euclid 10.1. Proof Neither in the Egyptian nor in the Babylonian texts do we find anything that was even remotely similar to mathematical proof. The concept of proof was introduced by the Greeks, and this is their greatest merit. Somehow

Chapter 11. From Euclid to Descartes

From the book The Phenomenon of Science. Cybernetic approach to evolution author Turchin Valentin Fedorovich

Chapter 11. From Euclid to Descartes 11.1. Number and Magnitude During the time of Pythagoras and the early Pythagoreans, the concept of number occupied a leading position in Greek mathematics. The Pythagoreans believed: God put numbers at the basis of the world order. God is unity and the world is plurality.

Euclid was born around 330 BC, presumably in Alexandria. Some Arab authors believe that he came from rich family from Knocrate. There is a version that Euclid could have been born in Tyre, and all his life later life to be held in Damascus. According to some documents, Euclid studied at the ancient school of Plato in Athens, which was only possible wealthy people. After this, he moved to Alexandria in Egypt, where he laid the foundation for the branch of mathematics now known as “geometry”.

The life of Euclid of Alexandria is often confused with the life of Euclid of Meguro, making it difficult to discover any reliable sources biography of a mathematician. What is known for certain is that it was he who attracted public attention to mathematics and brought this science to a completely new level. new level, making revolutionary discoveries in this area and proving many theorems. In those days, Alexandria was not only largest city in the western part of the world, but also the center of a large, thriving papyrus industry. It was in this city that Euclid developed, recorded and presented to the world his works on mathematics and geometry.

Scientific activity

Euclid is rightly considered the “father of geometry.” It was he who laid the foundations of this field of knowledge and elevated it to proper level, revealing to society the laws of one of the most complex branches of mathematics at that time. After moving to Alexandria, Euclid, like many scientists of that time, wisely spent most time in Library of Alexandria. This museum, dedicated to literature, art and sciences, was founded by Ptolemy. Here Euclid begins to unify geometric principles, arithmetic theories And irrational numbers V unified science geometry. He continues to prove his theorems and compiles them into the colossal work “Principia.”

For all the time of its little-explored scientific activity, the scientist completed 13 editions of the Elements, covering wide range questions ranging from axioms and statements to stereometry and the theory of algorithms. Along with the nomination various theories, he begins to develop a method of proof and logical justification for these ideas that will prove the statements proposed by Euclid.

His work contains more than 467 statements regarding planimetry and stereometry, as well as hypotheses and theses that put forward and prove his theories regarding geometric concepts. It is known for certain that as one of the examples in his Elements, Euclid used the Pythagorean theorem, which established the relationship between the sides right triangle. Euclid stated that "the theorem is true for all cases of right triangles."

It is known that during the existence of “Principles”, right up to the 20th century, more copies of this book were sold than the Bible. The Principia, published and republished countless times, was used by various mathematicians and authors in their work. scientific works. Euclidean geometry knew no boundaries, and the scientist continued to prove new theorems in completely different areas, such as, for example, in the field of “prime numbers”, as well as in the field of fundamentals arithmetic knowledge. Through a chain of logical reasoning, Euclid sought to discover secret knowledge to humanity. The system that the scientist continued to develop in his “Principles” would become the only geometry that the world would know until the 19th century. However modern mathematicians discovered new theorems and hypotheses of geometry, and divided the subject into “Euclidean geometry” and “non-Euclidean geometry”.

The scientist himself called this a “generalized approach”, based not on trial and error, but on the presentation of indisputable facts of theories. At a time when access to knowledge was limited, Euclid began to study issues completely different areas, including “arithmetic and numbers”. He concluded that the discovery of "the largest prime number"It's physically impossible. He justified this statement by the fact that if one is added to the largest known prime number, this will inevitably lead to the formation of a new prime number. This classic example is proof of the clarity and accuracy of the scientist’s thoughts, despite his venerable age and the times in which he lived.

Axioms

Euclid said that axioms are statements that do not require proof, but at the same time he understood that blind acceptance of these statements on faith cannot be used in the construction of mathematical theories and formulas. He realized that even axioms must be supported by indisputable evidence. Therefore, the scientist began to draw logical conclusions that confirmed his geometric axioms and theorems. To better understand these axioms, he divided them into two groups, which he called “postulates.” The first group is known as " general concepts”consisting of accepted scientific statements. The second group of postulates is synonymous with geometry itself. The first group includes such concepts as “whole more than the amount parts" and "if two quantities are separately equal to the same third, then they are equal to each other." These are just two of the five postulates written down by Euclid. The five postulates of the second group relate directly to geometry, stating that “all right angles are equal to each other” and that “a straight line can be drawn from any point to any point.”

The scientific activity of the mathematician Euclid flourished, and in the early 1570s. his "Principles" were translated from Greek language into Arabic, and then into English language John Dee. Since its writing, "Principia" has been reprinted 1,000 times and, in the end, took pride of place in classrooms XX century. There are many known cases when mathematicians tried to challenge and refute geometric and mathematical theories Euclid, but all attempts invariably ended in failure. The Italian mathematician Girolamo Saccheri sought to improve the works of Euclid, but abandoned his attempts, unable to find the slightest flaw in them. And only a century later a new group mathematicians will be able to present innovative theories in the field of geometry.

Other jobs

Without ceasing to work on changing the theory of mathematics, Euclid managed to write a number of works on other topics, which are used and referred to to this day. These works were pure assumptions, based on irrefutable evidence, running like a red thread through all the “Principles”. The scientist continued his study and discovered new area optics - catoptrics, which largely asserted mathematical function mirrors His work in the field of optics, mathematical relationships, data systematization and the study of conic sections was lost in the mists of time. It is known that Euclid successfully completed eight editions, or books, on theorems concerning conic sections, but none of them has survived to this day. He also formulated hypotheses and assumptions based on the laws of mechanics and the trajectory of bodies. Apparently, all these works were interconnected, and the theories expressed in them grew from a single root - his famous “Principles”. He also developed a number of Euclidean "constructions" - the basic tools needed to perform geometric constructions.

Personal life

There is evidence that Euclid discovered at the Library of Alexandria private school, to be able to teach mathematics to enthusiasts like himself. There is also an opinion that in late period Throughout his life, he continued to help his students develop their own theories and write works. We don’t even have a clear idea of ​​the scientist’s appearance, and all the sculptures and portraits of Euclid that we see today are only a figment of the imagination of their creators.

Death and legacy

The year and causes of Euclid's death remain a mystery to humanity. There are vague hints in the literature that he may have died around 260 BC. The legacy left by the scientist is much more significant than the impression he made during his lifetime. His books and works were sold all over the world until the 19th century. Euclid's legacy survived the scientist for as many as 200 centuries, and served as a source of inspiration for such personalities as, for example, Abraham Lincoln. According to rumors, Lincoln always superstitiously carried the “Principia” with him, and in all his speeches he quoted the works of Euclid. Even after the death of a scientist, mathematician different countries continued to prove theorems and publish works under his name. In general, at a time when knowledge was closed to the general public, Euclid, in a logical and scientific way, created a format for the mathematics of antiquity, which today is known to the world under the name “Euclidean geometry”.

Biography score

New feature! average rating, which this biography received. Show rating